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+{"text":"\\section{Introduction}\n\nArbitrary-spin massless particles are expected to play a crucial role in the understanding of Quantum Gravity. Lower-spin theories may be realized as low-energy limits of\nspontaneously-broken higher-spin gauge theories since lower-spin symmetries are subgroups of higher-spin ones. It is believed that the tensionless limit of string theory\nis a theory of higher-spin gauge fields. The study of fermionic fields is interesting in this regard because they are required by supersymmetry.\n\nHigher-spin gauge fields can be described in the framework of two different formulations: frame-like and metric-like. The frame-like formulation generalizes the Cartan formulation\nof gravity where the gauge fields are described in terms of differential forms carrying irreducible representations of the fiber Lorentz group. This is available\nin Minkowski~\\cite{V-flat,hygra,AD} as well as in Anti-de Sitter (AdS)~\\cite{Vasiliev:1986td,Lopatin:1987hz,V-AdS,Vasiliev:2001wa} spaces.\nThe metric-like formulation, on the other hand, is a generalization of the metric formulation of linearized gravity~\\cite{deWit}. Originally developed by\nFronsdal~\\cite{Fronsdal:1978rb,Fronsdal:1978vb} and Fang--Fronsdal~\\cite{FF,Fang:1979hq}, it encodes the degrees of freedom of higher-spin particles in symmetric tensors and tensor-spinors.\nIn this approach, the construction of a gauge-invariant action for a higher-spin field requires that the field and the gauge parameter obey some off-shell algebraic constraints\n(see~\\cite{Rahman:2015pzl,Campoleoni:2017vds} for a recent review). Note that the latter requirement can be avoided by recourse to other formulations~\\cite{Francia:2002aa,Bekaert:2003az,Buchbinder:2004gp,Francia:2005bu,Bekaert:2006ix,Francia:2007qt,Buchbinder:2007ak,Buchbinder:2007vq,Francia:2007ee,Campoleoni:2009gs} (see\nAppendix~\\ref{sec:A}).\n\nBoth these approaches are geometric, albeit in different manners, in that the frame-like formulation extends Cartan geometry whereas the metric-like formulation extends Riemannian geometry.\nThe latter is however a particular gauge of the former just like in the case of gravity. The construction of interacting theories for higher-spin fields, fermions in particular,\nappears to be in dire need of the frame-like formulation. The metric-like formulation, in contrast, seems rather clumsy in managing the non-linearities required by gauge-theoretic consistency.\nYet it has the advantage of having a simplified field content that may make some features of the interactions more transparent. Understanding the connections\nbetween the two \nmay therefore provide valuable information~\\cite{Campoleoni:2012hp,Fredenhagen:2014oua,Campoleoni:2014tfa,Boulanger:2015ova}.\n\nIn this article, we will focus exclusively on higher-spin gauge fermions. These fields appear naturally in the supersymmetric versions of Vasiliev\ntheory~\\cite{Konstein:1989ij,Vasiliev:1990vu,Vasiliev:1992av,Sezgin:1998gg,Sezgin:1998eh,Sezgin:2002ru,Engquist:2002vr} (see~\\cite{Sezgin:2012ag} for a recent review) and also\nin the tensionless limit of superstring theory compactified on AdS$_5\\times S^5$. The frame-like formulation of gauge fermions~\\cite{V-flat,AD,hygra,V-AdS} has been discussed more\nrecently by various authors~\\cite{Alkalaev:2001mx,Alkalaev:2006hq,Sorokin:2008tf,Z,Z2,Skvortsov:2010nh}.\nThe Fang-Fronsdal metric-like approach for higher-spin fermions, on the other hand, has been studied in\narbitrary dimensions in Ref.~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}. We will consider the free theory of a spin $s=n+\\tfrac{1}{2}$ massless fermionic field\nin flat and AdS spaces. Although we consider Majorana fermions for simplicity, our main results are valid almost verbatim for Dirac fermions in arbitrary spacetime dimensions.\nA crucial property of frame-like fermions in flat space is their shift symmetry w.r.t.~a gauge parameter which is an irreducible tensor-spinor in the fiber space with the symmetry\nproperty of the Young diagram $\\mathbb{Y}(n-1,1)$. This symmetry makes it almost manifest that the free Lagrangian is equivalent to that of the metric-like formulation~\\cite{V-flat}.\nIn AdS space, however, the constraints on this parameter may receive nontrivial corrections which vanish in the flat limit~\\cite{Sorokin:2008tf,Z}. This is tantamount to having no such\ncorrections provided that some appropriate mass-like terms appear in the gauge transformation. In other words, one can have a gauge-invariant Lagrangian description for frame-like fermions\nin AdS space that does not deform of the flat-space constraints on the field and the gauge parameters.\n\nThe organization of this article is as follows. In the remaining of this section we spell out our notations and conventions. A review of frame-like higher-spin massless fermions in flat space\nappears in Section~\\ref{sec:FFF}, where we write down the free Lagrangian~\\cite{Z,Skvortsov:2010nh} and discuss its gauge symmetries along with the constraints on the field and the gauge parameters.\nWe also show how this theory simplifies in $D=3, 4$. Section~\\ref{sec:FFA} formulates the free theory in AdS space with a trivial but convenient modification of the well-known mass-like\nterm~\\cite{Sorokin:2008tf,Z}. By virtue of judiciously-chosen terms in the gauge transformation, we ensure that the constraints on the field and the gauge parameters mimic their flat-space counterparts.\nThe value of the mass parameter, determined uniquely by gauge invariance, is in complete agreement with the known results~\\cite{Metsaev:2013wza,Metsaev:2003cu}.\nIn Section~\\ref{sec:Eqv}, we demonstrate explicitly the equivalence of the frame-like Lagrangian to the metric-like one at the free level. We conclude in Section~\\ref{sec:remarks} with some remarks,\nespecially on the subtleties\nthat may arise in an interacting theory. An appendix summarizes the essentials of the metric-like formulation of higher-spin gauge fermions.\n\n\\subsubsection*{Conventions \\& Notations}\\label{subsec:convnot}\nWe adopt the conventions of Ref.~\\cite{Freedman:2012zz}, with mostly positive metric signature $(-+\\cdots+)$.\nThe expression $(i_1\\cdots i_n)$  denotes a totally symmetric one in all the indices $i_1,\\cdots,i_n$ with no normalization factor, e.g., $(i_1i_2)=i_1i_2+i_2i_1$ etc.\nThe totally antisymmetric expression $[i_1\\cdots i_n]$ has the same normalization.\nThe number of terms appearing in the (anti-)symmetrization is assumed to be the possible minimum.\nA prime will denote a trace w.r.t. the background metric, e.g., $A^{\\prime}=\\bar{g}^{\\mu\\nu}A_{\\mu\\nu}=A^{\\mu}_{~\\mu}$.\nThe Levi-Civita symbol is normalized as $\\varepsilon_{01\\ldots D-1}= +1$, where $D$ is the spacetime dimension.\n\nFiber indices and world indices will respectively be denoted with lower case Roman letters and Greek letters.\nRepeated indices with the same name (appearing all as either covariant or contravariant ones)\nare (anti-)symmetrized with the minimum number of terms. This results in the following rules: $a(k)a=aa(k)=(k+1)a(k+1)$,~\n$a(k)a(2)=a(2)a(k)=\\binom{k+2}{2}\\,a(k+2)$,~ $a(k)a(k')=a(k')a(k)=\\binom{k+k'}{k}\\,a(k+k')$ etc,\nwhere $a(k)$ has a unit weight by convention, and so the proportionality coefficient gives the weight of the right hand side.\n\nThe $\\gamma$-matrices satisfy the Clifford algebra: $\\{\\gamma^a,\\gamma^b\\}=+2\\eta^{ab}$, and $\\gamma^{a\\,\\dagger}=\\eta^{aa}\\gamma^a$.\nTotally antisymmetric products of $\\gamma$-matrices, $\\gamma^{a_1\\ldots a_r}=\\tfrac{1}{r!}\\gamma^{[a_1}\\gamma^{a_2}\\cdots\\gamma^{a_r]}$, have unit weight.\nA ``slash'' will denote a contraction with $\\gamma$-matrix, e.g., $\\displaystyle{\\not{\\!\\!A\\!\\,}}=\\gamma^a A_a$.\n\nA Majorana spinor $\\chi$ obeys the reality condition: $\\chi^C=\\chi$. Two Majorana spinors $\\chi_{1,\\,2}$ follow the bilinear identity:\n$\\bar\\chi_1\\gamma^{a_1\\ldots a_r}\\chi_2=t_r\\,\\bar\\chi_2\\gamma^{a_1\\ldots a_r}\\chi_1$, where a ``bar'' denotes Majorana conjugation, and $t_r=\\pm 1$,\ndepending on the value of $r$ and spacetime dimensionality~\\cite{Freedman:2012zz}.\n\n\\section{Frame-like Fermions in Flat Space}\\label{sec:FFF}\n\nIn the frame-like formulation, a fermion of spin $s=n+\\tfrac{1}{2}$ is described by a vielbein-like 1-form $\\Psi^{a(n-1)}$, which is\na symmetric rank-($n-1$) irreducible tensor-spinor in the fiber space:\n\\begin{equation}\\label{fr1}\n\\Psi^{a(n-1)}=\\Psi_\\mu{}^{a(n-1)}dx^\\mu,\\qquad \\gamma_a\\Psi^{ab(n-2)}=0.\n\\end{equation}\nThe Minkowski background is described by the vielbein $\\bar{e}^{\\,a}=\\bar{e}_\\mu^{\\,a}dx^\\mu$ that satisfies $\\eta_{ab}\\bar{e}_\\mu^{\\,a}\\bar{e}_\\nu^{\\,b}=\\eta_{\\mu\\nu}$, and the spin-connection\n$\\bar{\\omega}^{ab}=\\bar{\\omega}_\\mu{}^{ab}dx^\\mu=-\\bar{\\omega}_\\mu{}^{ba}dx^\\mu$, which fulfill the following equations:\n\\begin{equation}\\label{fr2}\nT^a\\equiv d\\bar{e}^{\\,a}+\\bar{\\omega}^a{}_b\\bar{e}^{\\,b}=0,\\qquad \\rho^{ab}\\equiv d\\bar{\\omega}^{ab}+\\bar{\\omega}^a{}_c\\bar{\\omega}^{cb}=0.\n\\end{equation}\nIn the Cartesian coordinates, in particular, the solution of Eqs.~(\\ref{fr2}) is given by $\\bar{e}_\\mu^{\\,a}=\\delta_\\mu^a$ and $\\bar{\\omega}_\\mu{}^{ab}=0$. We will however work\nwith a generic coordinate system in order to facilitate the transition to AdS space. The following quantities will be useful in the subsequent discussion:\n\\begin{eqnarray}\n  {}^*\\bar{e}_{a_1}\\ldots\\bar{e}_{a_p}&\\equiv&\\tfrac{1}{(D-p)!}\\,\\epsilon_{a_1\\ldots a_pa_{p+1}\\ldots a_D}\\bar{e}^{\\,a_{p+1}}\\ldots\\bar{e}^{\\,a_D},\\label{fr2a}\\\\\n  \\eta^{a_1a_2|b_1b_2}&\\equiv&\\tfrac{1}{2}\\left(\\eta^{a_1b_1}\\eta^{a_2b_2}-\\eta^{a_1b_2}\\eta^{a_2b_1}\\right).\\label{fr2b}\n\\end{eqnarray}\nThe frame-like free action for a Majorana gauge fermion, in arbitrary dimensions\\footnote{Majorana fermions exist in $D=3,4,8,9,10$ and $11$. In dealing with such objects it is\nimportant to assume the anti-commuting nature of fermions already at the classical level (before quantization).}, reads~\\cite{Z,Skvortsov:2010nh}:\n\\begin{equation}\\label{fr3}\nS=-\\tfrac{1}{2}\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3},\n\\end{equation}\nwhere $\\hat{D}$ denotes the Lorentz covariant derivative, and\n\\begin{equation}\\label{fr4}\n\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\equiv\\tfrac{1}{6n}\\left(\\gamma^{a_1a_2a_3}\\eta^{b_1b_2}+2(n-1)\\eta^{b_1b_2|[a_1a_2}\\gamma^{a_3]}\\right).\n\\end{equation}\nThe action~(\\ref{fr3}) enjoys the following gauge invariance:\n\\begin{equation}\\label{gauge-fr0}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)}+\\bar{e}_b\\lambda^{b,\\,a(n-1)},\n\\end{equation}\nwhere the 0-form gauge parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$ are irreducible tensor-spinors of rank ($n-1$) and rank $n$ respectively with the symmetry of the Young diagrams\n$\\mathbb{Y}(n-1)$ and $\\mathbb{Y}(n-1,1)$, i.e.,\n\\begin{equation}\\label{YD-1}\n\\zeta^{a(n-1)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_{n-1}\n\\end{aligned}~,\n\\qquad\\qquad\n\\lambda^{b,\\,a(n-1)}\\sim\\overbrace{\\begin{aligned}\n&\\begin{tabular}{|c|c|c|c|}\\hline\n   $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}\\\\[-4pt]\n&\\begin{tabular}{|c|}\n   $\\phantom{a}$\\\\\\hline\n\\end{tabular}\n\\end{aligned}}^{n-1}~.\n\\end{equation}\nThese irreducible tensor-spinors are subject to the following constraints:\n\\begin{equation}\\label{identity-fr1}\n\\gamma_b\\zeta^{ba(n-2)}=0,\\qquad \\gamma_b\\lambda^{b,\\,a(n-1)}=0,\\qquad \\gamma_c\\lambda^{b,\\,ca(n-2)}=0,\\qquad \\lambda^{a,\\,a(n-1)}=0.\n\\end{equation}\n\nIt is obvious that the action~(\\ref{fr3}) is invariant, up to a total derivative term, under the gauge transformation of the parameter $\\zeta^{a(n-1)}$, since $\\hat{D}^2=0$ in flat space.\nTo prove the shift symmetry w.r.t. the parameter $\\lambda^{b,\\,a(n-1)}$, let us make use of the identity: $\\bar{e}^c{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}={}^*\\bar{e}_{[a_1}\\bar{e}_{a_2}\\delta_{a_3]}^c$,\nso that the variation of the action can be written as\n\\begin{equation}\\label{gauge-fr2}\n\\delta_\\lambda S=-3\\int\\left[\\bar{\\Psi}_{b_1}{}^{c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\lambda_{a_3,\\,b_2c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}.\n\\end{equation}\nNow, let us take a careful look at the identity:\n\\begin{eqnarray}\n  6n\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}&=&\\left(\\gamma^{a_1a_2}\\eta^{b_1b_2}+2(n-1)\\eta^{a_1a_2|b_1b_2}\\right)\\gamma^{a_3}+(n-1)\\gamma^{[a_1}\\eta^{a_2]b_1}\\eta^{a_3b_2}\\nonumber\\\\\n  &&~~~~~~~~~~~~~~~~~~~~~~~~-\\gamma^{[a_1}\\eta^{a_2]a_3}\\eta^{b_1b_2}-(n-1)\\gamma^{[a_1}\\eta^{a_2]b_2}\\eta^{a_3b_1}.\\label{fr5c}\n\\end{eqnarray}\nWhen plugged into the gauge variation~(\\ref{gauge-fr2}), the first line on the right hand side of this identity gives vanishing contribution on account of the\n$\\gamma$-trace constraints~(\\ref{identity-fr1}) on the gauge parameter $\\lambda^{b,\\,a(n-1)}$. The two terms in the second line, on the other hand, cancel each other, thanks\nto the property $\\lambda^{a,\\,a(n-1)}=0$. This proves the shift symmetry since $\\delta_\\lambda S=0$.\n\nLet us count the number independent of components of the parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$. Because the frame indices are $\\gamma$-traceless,\nthe number of possible values each index can take is essentially ($D-1$). Then it is easy to compute the number of components of the corresponding Young diagrams~(\\ref{YD-1});\nthey respectively turn out to be $\\binom{D+n-3}{n-1}f_D$ and $(n-1)\\binom{D+n-3}{n}f_D$, where\n\\begin{equation}\\label{fD-defined} f_D\\equiv2^{D\/2+((-)^D-5)\/4},\\end{equation}\nfor a Majorana fermion in $D$ dimensions. On the other hand, one needs to take into account the vanishing of the trace when one contracts two indices from different rows of\n$\\lambda^{b,\\,a(n-1)}$, which removes $\\binom{D+n-4}{n-2}f_D$ components. Therefore, the total numbers are given by\n\\begin{equation}\\label{z-dof} \\Delta_\\zeta=\\binom{D+n-3}{n-1}f_D,\\qquad \\Delta_\\lambda=(n-1)\\binom{D+n-3}{n}f_D-\\binom{D+n-4}{n-2}f_D.\\end{equation}\nThis counting will be useful later on.\n\n\\subsubsection*{\\underline{Special Case: $D=3$}}\n\nThe case of $D=3$ is important in the context of hypergravity theories~\\cite{hygra} (see also~\\cite{Troncoso} for a recent discussion). In this case,\nnote that the quantity ${}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}$ reduces to the Levi-Civita tensor $\\epsilon_{a_1a_2a_3}$. Furthermore, one has at one's disposal the useful\n$D$-dimensional identity:\n\\begin{equation}\\label{fr5b}\n  \\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}=\\tfrac{1}{6}\\gamma^{a_1a_2a_3}\\eta^{b_1b_2}+\\left(\\tfrac{n-1}{6n}\\right)\\gamma^{a_1a_2a_3b_1b_2}-\\left(\\tfrac{n-1}{12n}\\right)\\left(\\gamma^{b_1}\\gamma^{b_2}\\gamma^{a_1a_2a_3}+\\gamma^{a_1a_2a_3}\\gamma^{b_1}\\gamma^{b_2}\\right).\n\\end{equation}\nThe second term on the right hand side in the above identity is zero in $D=3$, whereas the last term gives vanishing contribution because of the $\\gamma$-trace condition on the field.\nOn account of the relation: $\\gamma^{a_1a_2a_3}\\epsilon_{a_1a_2a_3}=(3!)\\mathbb{I}$, therefore, the action~(\\ref{fr3}) reduces to the well-known Aragone-Deser form~\\cite{hygra}:\n\\begin{equation}\\label{fr6}\nS_{D=3}=-\\tfrac{1}{2}\\int\\bar{\\Psi}_{a(n-1)}\\hat{D}\\Psi^{a(n-1)}.\n\\end{equation}\nOn the other hand, the gauge symmetry~(\\ref{gauge-fr0})--(\\ref{identity-fr1}) reduces to\n\\begin{equation}\\label{gauge3D}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)},\\qquad \\gamma_b\\zeta^{ba(n-2)}=0.\n\\end{equation}\nThis is because in $D=3$ the shift parameter $\\lambda^{b,\\,a(n-1)}$ is trivial but $\\zeta^{a(n-1)}$ is not,\n\\begin{equation}\\label{bagh} \\Delta_\\lambda=0,\\qquad \\Delta_\\zeta=n,\\end{equation}\nas one can easily see from Eq.~(\\ref{z-dof}).\n\n\\subsubsection*{\\underline{Special Case: $D=4$}}\n\nIn this case, the quantity ${}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}$ reduces to the 1-form $\\epsilon_{a_1a_2a_3b}\\bar{e}^{\\,b}$, while only the first piece on the right hand side of the identity~(\\ref{fr5b})\ncontributes. Then the dimension-dependent identity: $\\gamma^{a_1a_2a_3}=-i\\epsilon^{a_1a_2a_3b}\\gamma_5\\gamma_{b}$, reduces the action~(\\ref{fr3}) to\n\\begin{equation}\\label{fr7}\nS_{D=4}=-\\tfrac{i}{2}\\int\\bar{\\Psi}_{a(n-1)}\\gamma_5\\gamma_b\\bar{e}^{\\,b}\\hat{D}\\Psi^{a(n-1)}.\n\\end{equation}\nBecause $\\Delta_\\zeta=n(n+1)\\neq0,~\\Delta_\\lambda=(n-1)(n+2)\\neq0$, both the parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$ are nontrivial, and so the gauge symmetry has the full general form of~(\\ref{gauge-fr0}).\nThe Lagrangian~(\\ref{fr7}) appeared in both Ref.~\\cite{V-flat} and~\\cite{AD}, but only the former reference could correctly identify the gauge symmetries.\n\n\\section{Frame-like Fermions in AdS Space}\\label{sec:FFA}\n\nThe AdS background is described by the vielbein $\\bar{e}^{\\,a}=\\bar{e}_\\mu^{\\,a}dx^\\mu$ that satisfies $\\eta_{ab}\\bar{e}_\\mu^a\\bar{e}_\\nu^b=\\bar{g}_{\\mu\\nu}$, and the spin-connection\n$\\bar{\\omega}^{ab}=\\bar{\\omega}_\\mu{}^{ab}dx^\\mu=-\\bar{\\omega}_\\mu{}^{ba}dx^\\mu$, which fulfill the following equations:\n\\begin{equation}\\label{fr2AdS}\nT^a\\equiv d\\bar{e}^{\\,a}+\\bar{\\omega}^a{}_b\\bar{e}^{\\,b}=0,\\qquad \\rho^{ab}\\equiv d\\bar{\\omega}^{ab}+\\bar{\\omega}^a{}_c\\bar{\\omega}^{cb}=-\\frac{1}{l^2}\\bar{e}^{\\,a}\\bar{e}^{\\,b},\n\\end{equation}\nwhere $l$ is the AdS radius.\nLet us write the free action for a Majorana gauge fermion in AdS space by augmenting the kinetic term, already studied in the context of flat space, by a mass term:\n\\begin{eqnarray}\nS&=&-\\tfrac{1}{2}\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}\\nonumber\\\\\n&&-\\tfrac{1}{2}\\mu\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2},\\label{fr3Ad}\n\\end{eqnarray}\nwhere $\\mu$ is some parameter with the dimensions of mass, to be specified later, and\n\\begin{equation}\\label{fr4Ad}\n\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\equiv\\tfrac{1}{2n}\\left[\\gamma^{a_1a_2}\\eta^{b_1b_2}+2(n-1)\\eta^{a_1a_2|b_1b_2}-\\tfrac{1}{2}\\left(\\tfrac{n-1}{D+2n-4}\\right)\\left(\\gamma^{b_1}\\gamma^{b_2}\\gamma^{a_1a_2}+\\gamma^{a_1a_2}\\gamma^{b_1}\\gamma^{b_2}\\right)\\right].\n\\end{equation}\nNote that our choice of $\\mathcal{B}^{a_1a_2,\\,b_1b_2}$ differs from that of Ref.~\\cite{Sorokin:2008tf,Z} by a trivial term which vanishes upon implementing\nthe constraint on the field. Yet this term will be useful for our purpose.\n\nIt suffices to consider, invoking another mass parameter $\\tilde{\\mu}$, the gauge transformation:\n\\begin{equation}\\label{gauge-fr0Ad}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)}+\\tilde{\\mu}\\bar{e}_b\\left[\\gamma^b\\zeta^{a(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma^a\\zeta^{a(n-2)b}\\right]+\\bar{e}_b\\lambda^{b,\\,a(n-1)},\n\\end{equation}\nwhich is compatible with the $\\gamma$-trace constraint, $\\gamma_a\\Psi^{ab(n-2)}=0$, on the field without requiring any modification of the\nproperties~(\\ref{YD-1}) and~(\\ref{identity-fr1}) of the gauge parameters. In other words, the choice of this gauge transformation~(\\ref{gauge-fr0Ad}) is such that the field and the gauge\nparameters mimic their flat-space properties. This point is implicit in the choice made in Ref.~\\cite{Sorokin:2008tf,Z}.\n\nTo see that the shift transformation w.r.t.~the parameter $\\lambda^{b,\\,a(n-1)}$ is a symmetry of the Lagrangian~(\\ref{fr3Ad}), let us first note that the invariance of the kinetic term\nfollows exactly the flat-space logic. Then, from the variation of the mass term, we have\n\\begin{equation}\\label{mass-var}\n\\delta_\\lambda S=-2\\mu\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\lambda_{a_2,\\,b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}.\n\\end{equation}\nOn account of the identity:\n\\begin{eqnarray}\n  2n\\mathcal{B}^{a_1a_2,\\,b_1b_2}&=&\\eta^{b_1b_2}\\gamma^{a_1}\\gamma^{a_2}+(n-1)\\eta^{a_1b_1}\\eta^{a_2b_2}-\\tfrac{1}{2}\\left(\\tfrac{n-1}{D+2n-4}\\right)\\left(\\gamma^{a_1a_2b_1}\\gamma^{b_2}+\\eta^{b_2[a_1}\\gamma^{a_2]}\\right)\\nonumber\\\\\n  &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\\eta^{a_1a_2}\\eta^{b_1b_2}-(n-1)\\eta^{a_1b_2}\\eta^{a_2b_1},\\label{fr4Ad1}\n\\end{eqnarray}\nwe then see that $\\delta_\\lambda S=0$. The cancellations happen in much the same way as the identity~(\\ref{fr5c}) eliminates contributions from the kinetic term.\n\nThe symmetry requirement of the Lagrangian~(\\ref{fr3Ad}) w.r.t.~the $\\zeta$-transformation in~(\\ref{gauge-fr0Ad}) would relate the mass parameters $\\mu$ and $\\tilde{\\mu}$ to each other\nand with the inverse AdS radius. There are a priori three kinds on contributions resulting from the $\\zeta$-transformation: 2-derivative, 1-derivative and 0-derivative ones. Not surprisingly,\nby virtue of the commutator formula:\n\\begin{equation}\\label{dsq-Ad}\n\\hat{D}^2\\zeta_{a(n-1)}=-\\frac{1}{l^2}\\bar{e}^{\\,b}\\bar{e}^{\\,c}\\left[\\eta_{ab}\\zeta_{ca(n-2)}+\\tfrac{1}{4}\\gamma_{bc}\\zeta_{a(n-1)}\\right],\n\\end{equation}\nthe 2-derivative piece actually reduces to a 0-derivative piece. The explicit computation makes use of the identities:\n$\\bar{e}^{\\,b}\\bar{e}^{\\,c}{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}={}^*\\bar{e}_{[a_1}\\delta^b_{a_2}\\delta^c_{a_3]}$ and\n$\\bar{e}^{\\,b}{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}={}^*\\bar{e}_{[a_1}\\delta^b_{a_2]}$, and leads straightforwardly to\n\\begin{equation}\\label{AdS-last0} -\\tfrac{(D+2n-3)(D+2n-4)}{4\\,n}\\tfrac{1}{l^2}-\\tfrac{(D-2)(D+2n-3)}{n(D+2n-4)}\\mu\\tilde{\\mu}=0,\\end{equation}\nin order that the even-derivative terms cancel each other. Cancellation of the 1-derivative terms,\non the other hand, requires that the following condition be met:\n\\begin{equation}\\label{mm-rel}-(D-2)\\tilde{\\mu}-\\mu=0.\\end{equation}\nConditions~(\\ref{AdS-last0}) and~(\\ref{mm-rel}) can be combined into the relation:\n\\begin{equation}\\label{AdS-last} \\mu^2 l^2=\\left(n+\\tfrac{D-4}{2}\\right)^2>0,\\end{equation}\nwhich gives, up to a sign, the real mass parameter $\\mu$ in terms of the inverse AdS radius. The parameter $\\tilde{\\mu}$ is then also determined from Eq.~(\\ref{mm-rel}).\nThis uniquely fixes the Lagrangian~(\\ref{fr3Ad}) as well as the gauge transformation~(\\ref{gauge-fr0Ad}) while the field and gauge parameters mimic their respective\nflat-space properties.\n\nThe physical significance of the mass parameter $\\mu$ will be made clear in the next section as we work out the gauge fixed equations of motion. To proceed,\nlet us forgo the language of differential forms and rewrite the action~(\\ref{fr3Ad}) as:\n\\begin{equation}\\label{fr3Ad2}\nS=-\\tfrac{1}{2}\\int d^Dx\\,\\bar{e}\\,\\bar{\\Psi}_{\\mu,\\,ac(n-2)}\\left(6\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}\\hat{D}_\\rho+2\\mu \\mathcal{B}^{\\mu\\nu,\\,ab}\\right)\\Psi_{\\nu,\\,b}{}^{c(n-2)},\n\\end{equation}\nwhere $\\bar{e}\\equiv\\det\\bar{e}_\\mu^{\\,a}$ is the determinant of the background AdS vielbein. The resulting Lagrangian equations of motion for the frame-like fermion field\n$\\Psi_{\\mu,\\,a(n-1)}$ take the form:\n\\begin{equation}\\label{eom1}\n\\mathcal{R}^{\\mu,\\,a(n-1)}\\equiv\\left(\\tfrac{6}{n-1}\\right)\\left(\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}\\hat{D}_\\rho+\\tfrac{1}{3}\\mu\\,\\mathcal{B}^{\\mu\\nu,\\,ab}\\right)\\Psi_{\\nu,\\,b}{}^{a(n-2)}=0.\n\\end{equation}\nHere, the normalization factor keeps the equations of motion well defined also for $n=1$, as we will see.\nWe emphasize that the equations of motion~(\\ref{eom1}) are $\\gamma$-traceless in the fiber indices, i.e.,\n\\begin{equation}\\label{xxx} \\gamma_b\\mathcal{R}^{\\mu,\\,ba(n-2)}=0,\\end{equation}\nas they should be. Actually, the very choices of $\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}$ and $\\mathcal{B}^{\\mu\\nu,\\,ab}$ made respectively in Eqs~(\\ref{fr4}) and~(\\ref{fr4Ad})\nwere such that the action~(\\ref{fr3Ad2}) manifestly has the following form:\n\\begin{equation}\\label{fr3Ad2.5}\nS=-\\tfrac{1}{2}\\int d^Dx\\,\\bar{e}\\,\\bar{\\Psi}_{\\mu,\\,a(n-1)}\\mathcal{R}^{\\mu,\\,a(n-1)}.\n\\end{equation}\nClearly, the equations of motion~(\\ref{eom1}) share the gauge symmetries~(\\ref{gauge-fr0Ad}) of the action:\n\\begin{equation}\\label{gauge-xxx}\n\\delta \\Psi_{\\mu,\\,a(n-1)}=\\hat{D}_\\mu\\zeta_{a(n-1)}+\\tilde{\\mu}\\bar{e}_\\mu^{\\,b}\\left[\\gamma_b\\zeta_{a(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma_a\\zeta_{a(n-2)b}\\right]+\\bar{e}_\\mu^{\\,b}\\lambda_{b,\\,a(n-1)}\\,.\n\\end{equation}\nIn the next section we will fix these gauge symmetries to find among other things the number of physical degrees of freedom, which should match with that of a Majorana fermion\nof spin $s=n+\\tfrac{1}{2}$\\,.\n\n\\section{Equivalence of Frame- \\& Metric-like Formulations}\\label{sec:Eqv}\n\nThe first step to establish the equivalence of the frame- and metric-like descriptions of a gauge fermion is to find a match in the respective number of local degrees of freedom.\nTo count this for a frame-like fermion~\\cite{V-AdS}, we rewrite the equations of motion~(\\ref{eom1}) exclusively in terms of world indices:\n\\begin{equation}\\label{eom2}\n\\mathcal{R}^{\\mu,\\,\\alpha(n-1)}\\equiv\\left(\\gamma^{\\mu\\rho\\nu}\\nabla_\\rho+\\mu\\gamma^{\\mu\\nu}\\right)\\Psi_{\\nu,}{}^{\\alpha(n-1)}+\\tfrac{1}{2n}\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\Psi_{\\nu,\\,\\beta}{}^{\\alpha(n-2)}=0,\n\\end{equation}\nwhere $\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}$ is an operator antisymmetric in the $\\mu,\\nu,\\beta$ indices, given by\n\\begin{equation}\\label{eom3}\n\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\equiv\\left[\\gamma^\\alpha,\\gamma^{\\mu\\rho\\nu\\beta}\\right]\\nabla_\\rho-\\mu\\left\\{\\gamma^\\alpha,\\gamma^{\\mu\\nu\\beta}\\right\\}-\\left(\\tfrac{2}{D+2n-4}\\right)\\mu\\gamma^\\alpha\\gamma^{\\mu\\nu\\beta}.\n\\end{equation}\nSome of the dynamical modes however are not physical because of gauge invariance. In order to exclude the correct number of pure gauge modes, let us rewrite the gauge\ntransformations~(\\ref{gauge-xxx}) as:\n\\begin{equation}\\label{gauge-xw}\n\\delta \\Psi_{\\mu,\\,\\alpha(n-1)}=\\nabla_\\mu\\zeta_{\\alpha(n-1)}+\\tilde{\\mu}\\left[\\gamma_\\mu\\zeta_{\\alpha(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma_{\\alpha}\\zeta_{\\alpha(n-2)\\mu}\\right]+\\lambda_{\\mu,\\,\\alpha(n-1)}\\,.\n\\end{equation}\nNow one can use this freedom to choose the following covariant gauge:\n\\begin{equation}\\label{gf1}\n\\displaystyle{\\not{\\!\\Psi\\!\\,}}_{\\alpha(n-1)}\\equiv\\gamma^\\mu\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad\\Longrightarrow\\qquad \\Psi'_{\\alpha(n-2)}\\equiv\\bar{g}^{\\mu\\nu}\\Psi_{\\mu,\\,\\nu\\alpha(n-2)}=0.\n\\end{equation}\nAs a consequence, the equations of motion~(\\ref{eom2}) reduce to the following form:\n\\begin{equation}\\label{eom-gf}\n\\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\mu\\right)\\Psi^{\\mu,}{}_{\\alpha(n-1)}-\\gamma^\\mu\\nabla^\\nu\\Psi_{\\nu,\\,\\alpha(n-1)}+\\tfrac{1}{2n}\\mathcal{C}^{\\mu\\nu\\rho,}{}_\\alpha\\,\\chi_{\\nu,\\,\\rho\\alpha(n-2)}=0,\n\\end{equation}\nwhere $\\chi_{\\mu,\\,\\alpha(n-1)}$ is the irreducible part of the field $\\Psi_{\\mu,\\,\\alpha(n-1)}$ with the symmetry of the Young diagram $\\mathbb{Y}(n-1,1)$, i.e., it has exactly the same properties as\nthe gauge parameter $\\lambda_{\\mu,\\,\\alpha(n-1)}$. Its appearance in the last term of Eq.~(\\ref{eom-gf}) is easy to understand. The antisymmetry property of $\\mathcal{C}^{\\mu\\nu\\rho,\\,\\alpha}$\nremoves the completely symmetric part of $\\Psi_{\\mu,\\,\\alpha(n-1)}$, while the $\\gamma$-trace parts are trivial  by the gauge choice~(\\ref{gf1}).\n\nThe condition~(\\ref{gf1}) is however not a complete gauge fixing. This can be seen by taking its gauge variation, which results in the Dirac equation for $\\zeta_{\\alpha(n-1)}$:\n\\begin{equation}\\label{gf2}\n\\delta\\displaystyle{\\not{\\!\\Psi\\!\\,}}_{\\alpha(n-1)}\\equiv\\left[\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\left(\\tfrac{D+2n-2}{D+2n-4}\\right)\\mu\\right]\\zeta_{\\alpha(n-1)}=0.\n\\end{equation}\nNot only does this allow for nontrivial solutions for $\\zeta_{\\alpha(n-1)}$ but it also leaves $\\lambda_{\\mu,\\,\\alpha(n-1)}$ completely unaffected. Therefore, one can use to freedom of the shift parameter\n$\\lambda_{\\mu,\\,\\alpha(n-1)}$ to further gauge fix:\n\\begin{equation}\\label{gf3} \\chi_{\\mu,\\,\\alpha(n-1)}=0.\\end{equation}\nThis finally reduces the equations of motion~(\\ref{eom-gf}) to the Dirac form plus the divergence constraint:\n\\begin{equation}\\label{eom7}\n  \\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\mu\\right)\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad \\nabla^\\mu\\Psi_{\\mu,\\,\\alpha(n-1)}=0.\n\\end{equation}\nTo exhaust the residual freedom of $\\zeta_{\\alpha(n-1)}$ let us choose the gauge:\n\\begin{equation}\\label{gf9}\\Psi_{0,\\,\\alpha(n-1)}=0.\\end{equation}\nIts is easy to see that no residual freedom of $\\zeta_{\\alpha(n-1)}$ is left. A would-be residual parameter must obey some screened Poisson equation with no source term,\nwhich has no nontrivial solutions.\n\nThe count of local physical degrees of freedom is now immediate. The system~(\\ref{eom7}) describes $(D-1)\\Delta_\\zeta$ many dynamical variables, where $\\Delta_\\zeta$ is given in Eq.~(\\ref{z-dof}).\nBut the gauge choices~(\\ref{gf1}), (\\ref{gf3}) and~(\\ref{gf9}) respectively remove $\\Delta_\\zeta$, $\\Delta_\\lambda$ and $\\Delta_\\zeta$ degrees of freedom. Therefore, the total number of physical degrees\nof freedom is $(D-3)\\Delta_\\zeta-\\Delta_\\lambda$, which is the same as\n\\begin{equation}\\label{dof-adv} \\Delta_{\\text{Frame}}=\\binom{D+n-4}{n}f_D\\,.\\end{equation}\nThis confirms, in view of Eq.~(\\ref{dofF}), that the count matches in the two formulations: $\\Delta_\\text{Frame}=\\Delta_{\\text{Metric}}$\\,.\n\nThe physical significance of the mass parameter $\\mu$ is now clear from the Dirac equation in~(\\ref{eom7}). While Eq.~(\\ref{AdS-last}) says that $\\mu$ must be real, one may choose $\\mu>0$\nwithout any loss of generality. Then,\n\\begin{equation}\\label{AdS-last1} \\mu=\\frac{1}{l}\\left(n+\\tfrac{D-4}{2}\\right)>0.\\end{equation}\nOur $\\mu$ corresponds to the lowest value of the mass parameter $m$ for a fermion carrying a unitary irreducible representation of the AdS isometry algebra:\n\\begin{equation}\\label{kutta}  \\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-m\\right)\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad m\\geq\\mu>0.\\end{equation}\nThe bound saturates for the massless representation~\\cite{Metsaev:2006zy,Metsaev:2013wza,Metsaev:2003cu}, as we see.\n\nNext we will show that the two formulations are equivalent at the level of the free Lagrangian. With this end in view, let us decompose the fermion field\n$\\Psi_{\\mu,\\,\\alpha(n-1)}$ into totally symmetric, $\\gamma$-traceless mixed-symmetric and $\\gamma$-trace parts:\n\\begin{equation}\\label{decomp} \\Psi_{\\mu,\\,\\alpha(n-1)}=\\psi_{\\mu\\alpha(n-1)}+\\chi_{\\mu,\\,\\alpha(n-1)}+\\gamma_{[\\mu}\\theta_{\\alpha]\\alpha(n-2)},\\end{equation}\nwhere the fields appearing on the right hand side have the symmetry of the following Young diagrams:\n\\begin{equation}\\label{bilai1}\n\\psi_{\\alpha(n)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_n\\,,\n\\end{aligned}\n\\quad\n\\chi_{\\mu,\\,\\alpha(n-1)}\\sim\\overbrace{\\begin{aligned}\n&\\begin{tabular}{|c|c|c|c|}\\hline\n   $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}\\\\[-4pt]\n&\\begin{tabular}{|c|}\n   $\\phantom{a}$\\\\\\hline\n\\end{tabular}\n\\end{aligned}}^{n-1}\\,,\n\\quad\n\\theta_{\\alpha(n-1)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_{n-1}\\,.\n\\end{aligned}\n\\end{equation}\nWe have imposed irreducibility conditions on $\\chi_{\\mu,\\,\\alpha(n-1)}$, so that it is subject to the following constraints:\n\\begin{equation}\\label{bilai2}\n\\gamma^\\mu\\chi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad \\gamma^\\beta\\chi_{\\mu,\\,\\alpha(n-2)\\beta}=0,\\qquad \\chi_{\\alpha,\\,\\alpha(n-1)}=0.\n\\end{equation}\nOf course there will be additional constraints on the fields  $\\psi_{\\alpha(n)}$ and $\\theta_{\\alpha(n-1)}$ coming from the $\\gamma$-trace condition on the\nparent field $\\Psi_{\\mu,\\,\\alpha(n-1)}$ in the $\\alpha$-indices. To find them, let us first take a $\\gamma$-trace of Eq.~(\\ref{decomp}) in an $\\alpha$-index.\nThis results in\n\\begin{equation}\\label{bilai3} \\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\mu\\alpha(n-2)}-(D-2)\\theta_{\\mu\\alpha(n-2)}-(n-1)\\gamma_\\mu\\displaystyle{\\not{\\!\\theta}}_{\\alpha(n-2)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}_{\\mu\\alpha(n-3)}=0.\\end{equation}\nAnother $\\gamma$-trace w.r.t. the $\\mu$-index gives\n\\begin{equation}\\label{bilai4}\n\\psi'_{\\alpha(n-2)}-(Dn-2n+2)\\displaystyle{\\not{\\!\\theta}}_{\\alpha(n-2)}-\\gamma_\\alpha\\theta'_{\\alpha(n-3)}=0.\n\\end{equation}\nNow a third $\\gamma$-trace in an $\\alpha$-index yields:\n\\begin{equation}\\label{bilai5} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}-(Dn+D-4)\\theta^{\\prime}_{\\alpha(n-3)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}^{\\,\\prime}_{\\alpha(n-4)}=0.\\end{equation}\nOn the other hand, one could also have obtained a triple $\\gamma$-trace by first contracting the $\\mu$ index with an $\\alpha$ index in Eq.~(\\ref{decomp}) and then taking a $\\gamma$ trace.\nThis however produces a different result:\n\\begin{equation}\\label{bilai6} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}-(D+n-4)\\theta^{\\prime}_{\\alpha(n-3)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}^{\\,\\prime}_{\\alpha(n-4)}=0.\\end{equation}\nEqs.~(\\ref{bilai5}) and~(\\ref{bilai6}) impose the following constraints:\n\\begin{equation}\\label{bilai7} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}=0,\\qquad \\theta^{\\prime}_{\\alpha(n-3)}=0,\\end{equation}\ni.e., the symmetric rank-$n$ field $\\psi_{\\alpha(n)}$ must be triply $\\gamma$-traceless, whereas the symmetric rank-$(n-1)$ field $\\theta_{\\alpha(n-1)}$ must be traceless. This in turn results,\nfrom Eqs.~(\\ref{bilai3}) and~(\\ref{bilai4}), in the following relation:\n\\begin{equation}\\label{bilai8}\n\\theta_{\\alpha(n-1)}=\\left(\\tfrac{1}{D-2}\\right)\\left[\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha(n-1)}-\\left(\\tfrac{1}{nD-2n+2}\\right)\\gamma_\\alpha\\psi'_{\\alpha(n-2)}\\right].\n\\end{equation}\nFinally, plugging the above expression into the decomposition~(\\ref{decomp}), we obtain:\n\\begin{eqnarray}\n\\Psi_{\\mu,\\,\\alpha(n-1)}&=&\\psi_{\\mu\\alpha(n-1)}+\\chi_{\\mu,\\,\\alpha(n-1)}+\\left(\\tfrac{1}{D-2}\\right)\\left[\\gamma_{[\\mu}\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha]\\alpha(n-2)}-\\left(\\tfrac{2}{Dn-2n+2}\\right)\\gamma_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}\\right]\\nonumber\\\\\n&&~~~~~~~~~~~+\\tfrac{1}{(D-2)(Dn-2n+2)}\\left[(n-2)\\gamma_\\alpha\\gamma_\\mu\\psi'_{\\alpha(n-2)}-2\\bar{g}_{\\alpha(2)}\\psi'_{\\mu\\alpha(n-3)}\\right].\\label{bilai9}\n\\end{eqnarray}\nThis decomposition generalizes that of Ref.~\\cite{V-flat} to arbitrary dimensions.\n\nIt will be convenient to write the covariant equations of motion~(\\ref{eom2}) in the following form:\n\\begin{equation}\\label{ge1} \\mathcal{R}^{\\mu,\\,\\alpha(n-1)}\\equiv\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Psi_{\\nu,\\,\\beta(n-1)}=0,\\end{equation}\nwhere we have defined the operator $\\mathcal O$ as:\n\\begin{equation}\\label{ge2}\n\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\equiv\\left(\\gamma^{\\mu\\rho\\nu}\\nabla_\\rho+\\mu\\gamma^{\\mu\\nu}\\right)\\bar{g}^{\\,\\alpha(n-1),\\,\\beta(n-1)}+\\tfrac{1}{2n(n-1)}\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\bar{g}^{\\,\\alpha(n-2),\\,\\beta(n-2)},\n\\end{equation}\nwith $\\bar{g}^{\\,\\alpha(k),\\,\\beta(k)}\\equiv\\tfrac{1}{k^2}\\bar{g}^{\\,\\alpha\\beta}\\bar{g}^{\\,\\alpha\\beta}\\ldots\\bar{g}^{\\,\\alpha\\beta}$ (multiplicity $k$) denoting the unit-strength symmetric tensor product of $k$ background\nmetric tensors. This enables us to present the corresponding Lagrangian as:\n\\begin{equation}\\label{ge3} \\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}=-\\tfrac{1}{2}\\bar{\\Psi}_{\\mu,\\,\\alpha(n-1)}\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Psi_{\\nu,\\,\\beta(n-1)}\\,.\\end{equation}\n\nWhen the decomposition~(\\ref{bilai9}) is plugged into the above Lagrangian, the irreducible mixed-symmetric part $\\chi_{\\mu,\\,\\alpha(n-1)}$ completely drops out, thanks to the shift symmetry.\nThe fact that the parameter $\\lambda_{\\mu,\\,\\alpha(n-1)}$ enjoys exactly the same properties as $\\chi_{\\mu,\\,\\alpha(n-1)}$ plays a crucial role in this regard. The resulting Lagrangian contains only\nthe completely symmetric part $\\psi_{\\alpha(n)}$ and can be viewed as a gauge-fixed version of the original Lagrangian~(\\ref{ge3}) with the gauge fixing: $\\chi_{\\mu,\\,\\alpha(n-1)}=0$. The explicit\nderivation of this Lagrangian is tedious but straightforward. The calculations can however be simplified by noting that, on account of the $\\gamma$-tracelessness of the equations of\nmotion~(\\ref{ge1}) in the $\\alpha$-indices, the Lagrangian splits into the sum of two pieces:\n\\begin{equation}\\label{ge4} \\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}=-\\tfrac{1}{2}\\bar{\\Xi}_{\\mu,\\,\\alpha(n-1)}\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Xi_{\\nu,\\,\\beta(n-1)}\n+\\tfrac{1}{2}\\bar{\\xi}_{\\mu,\\,\\alpha(n-2)}\\gamma_\\alpha\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\gamma_\\beta\\xi_{\\nu,\\,\\beta(n-2)}\\,,\\end{equation}\nwhere the tensor-spinors $\\Xi_{\\mu,\\,\\alpha(n-1)}$ and $\\xi_{\\mu,\\,\\alpha(n-2)}$ are given by:\n\\begin{eqnarray}\n  \\Xi_{\\mu,\\,\\alpha(n-1)} &=& \\psi_{\\mu\\alpha(n-1)}+\\left(\\tfrac{1}{D-2}\\right)\\left[(n-1)\\gamma_\\mu\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha(n-1)}-\\left(\\tfrac{2}{Dn-2n+2}\\right)\\bar{g}_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}\\right],\\nonumber\\\\\n  \\xi_{\\mu,\\,\\alpha(n-2)} &=& \\left(\\tfrac{1}{D-2}\\right)\\left[-\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\mu\\alpha(n-2)}+\\left(\\tfrac{1}{Dn-2n+2}\\right)\\left(n\\gamma_\\mu\\psi'_{\\alpha(n-2)}-\\gamma_\\alpha\\psi'_{\\mu\\alpha(n-3)}\\right)\\right].\\label{ge5}\n\\end{eqnarray}\nOne can explicitly carry out the calculations to get to the following result:\n\\begin{eqnarray}\n-\\tfrac{2}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}&=&\\bar{\\psi}_{\\alpha(n)}\\left(\\not{\\!\\nabla\\!}-\\mu\\right)\\psi^{\\alpha(n)}\n+n\\bar{\\displaystyle{\\not{\\!\\psi\\!\\,}}}_{\\alpha(n-1)}\\left(\\not{\\!\\nabla\\!}+\\mu\\right)\\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\alpha(n-1)}\n-2n\\bar{\\displaystyle{\\not{\\!\\psi\\!\\,}}}_{\\alpha(n-1)}\\!\\nabla_\\mu\\psi^{\\mu\\alpha(n-1)}\\nonumber\\\\\n&&-\\tfrac{1}{4}n(n-1)\\bar{\\psi}'_{\\alpha(n-2)}\\left(\\not{\\!\\nabla\\!}-\\mu\\right)\\psi'^{\\,\\alpha(n-2)}\n-n(n-1)\\bar{\\psi}'_{\\alpha(n-2)}\\nabla_\\mu\\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\mu\\alpha(n-2)}.\\label{ge6}\n\\end{eqnarray}\nThis indeed coincides with the Lagrangian~(\\ref{f00}) for a metric-like gauge fermion in AdS space. Because only the symmetric part of the parent field $\\Psi_{\\mu,\\,\\alpha(n-1)}$\nappears in this Lagrangian, the corresponding gauge symmetry is obtained simply by a total symmetrization of the indices in Eq.~(\\ref{gauge-xw}). The result is:\n\\begin{equation}\\label{gauge-xwm}\n\\delta \\psi_{\\alpha(n)}=\\tfrac{1}{n}\\left(\\nabla_\\alpha\\zeta_{\\alpha(n-1)}-\\tfrac{1}{2l}\\gamma_{\\alpha}\\zeta_{\\alpha(n-1)}\\right),\n\\end{equation}\nwhich also matches perfectly with the metric-like gauge symmetry~(\\ref{tamm2}).\n\nThis hardly comes as a surprise. The symmetric part of $\\Psi_{\\mu,\\,\\alpha(n-1)}$ has all the characteristics of a metric-like gauge fermion; in particular it is\ntriple $\\gamma$-traceless as we have shown in Eq.~(\\ref{bilai7}). Moreover, it transforms w.r.t.~a symmetric $\\gamma$-traceless gauge parameter $\\zeta_\\alpha(n-1)$.\nThe gauge-invariant Lagrangian description for such a system is unique~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}.\nSo, $\\psi_{\\alpha(n)}$ is a metric-like gauge fermion in every sense.\n\n\\section{Remarks}\\label{sec:remarks}\n\nIn this article, we have elaborated on some key features of higher-spin gauge fermions and the connections between their frame- and metric-like formulations\nat the free level. A gauge-invariant frame-like Lagrangian description in AdS space, with the constraints on the\nfields and the gauge parameters resembling their flat-space cousins, facilitates the explicit derivation of the corresponding metric-like Lagrangian as a gauge\nfixing. This derivation generalizes that of Ref.~\\cite{V-flat} to AdS space and arbitrary dimensions. Although the equivalence of the frame-\nand metric-like formulations at the free level may not come as a surprise, our work fills a gap in the literature.\n\nAs is well-known, the frame-like formulation packages the non-linearities in an interacting theory in a very efficient way. For higher-spin fermions this can be seen\nin a very simple setup: the Aragone-Deser hypergravity~\\cite{hygra}$-$a consistent gauge theory of a spin $s=n+\\tfrac{1}{2}$ massless Majorana fermion coupled to Einstein\ngravity in 3D flat space. While only fermion bilinears appear in the frame-like formulation~\\cite{hygra}, the metric-like formulation will also include four-fermion couplings\nthat originate from integrating out the spin-connection, just like in supergravity~\\cite{Freedman:2012zz}. Moreover, the fermion-bilinear terms will look more complicated\nin the metric-like variables. To see this, note that with frame-like fermions the cubic cross-coupling in the covariant language has the simple form~\\cite{ours}:\n\\begin{equation}\\label{last0}\n\\mathcal{L}_3\\sim\\bar\\Psi_{\\mu,\\,\\alpha(n-1)}\\gamma^{\\mu\\nu\\rho}\\gamma^{\\sigma\\lambda}\\Psi_{\\nu,}{}^{\\alpha(n-1)}\\partial_\\sigma h_{\\rho\\lambda},\n\\end{equation}\nwhere $h_{\\mu\\nu}$ is the metric perturbation. Because the irreducible hook part $\\chi_{\\mu,\\,\\alpha(n-1)}$ of the frame-like fermion is trivial in $D=3$, the\ndecomposition~(\\ref{bilai9}) amounts to a complicated field redefinition:\n\\begin{equation}\\label{last1}\n\\Psi_{\\mu,\\,\\alpha(n-1)}=\\psi_{\\mu\\alpha(n-1)}+\\gamma_{[\\mu}\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha]\\alpha(n-2)}+\\left(\\tfrac{1}{n+2}\\right)\\left[n\\gamma_\\alpha\\gamma_\\mu\\psi'_{\\alpha(n-2)}-2\\eta_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}+2\\eta_{\\alpha(2)}\\psi'_{\\mu\\alpha(n-3)}\\right],\n\\end{equation}\nwhere $\\psi_{\\alpha(n)}$ is the metric-like fermion. After this redefinition is performed, the cubic coupling~(\\ref{last0}) will look cumbersome in terms of the metric-like fermion.\nWithin the metric-like formulation, it would be more difficult to construct or to prove the consistency of this cubic coupling, say using the techniques of\nRef.~\\cite{Henneaux:2012wg,Henneaux:2013gba}. The fermion-bilinear cross-couplings do not stop at any finite order in the graviton fluctuations and the situation gets only worse\nat higher orders, while the frame-like formulation captures all the non-linearities in a very neat way~\\cite{hygra}.\n\nIn higher dimensions the difference between the two formulations becomes more drastic. The hook part of the frame-like fermion never shows up in the interacting Lagrangian because\nof the deformed shift symmetry. However, there appear the so-called ``extra'' fields: a set of additional fields that arises when one tries to construct a complete set of gauge-invariant\nobjects (curvatures)\\footnote{The extra fields are generalizations of the spin-connection. The number of extra fields depends on the spin; the higher the spin, the more are the\nextra fields needed for constructing curvatures. The extra fields however do not enter the free action, and so they are not expressed in terms of physical fields via equations of motion.\n}~\\cite{Fradkin:1986ka}.\nTo understand the role of these extra fields that are absent in the free Lagrangian, one may express them in terms of the physical fields by means of appropriate constraints\nimplemented via Lagrange multipliers~\\cite{Vasiliev:1986td,V-AdS,Fradkin:1986ka,Fradkin:1987ks,Fradkin:1986qy}. Then, up to pure gauge parts, the extra fields are given by\nderivatives of the physical fields. The extra fields therefore induce higher-derivative terms in the interactions, while their absence in the free Lagrangian merely reflects\nthe absence of higher-derivative kinetic terms.\nExplicit solution of the aforementioned constraints are difficult, and actually not needed. The main idea of the so-called Fradkin-Vasiliev\nformalism~\\cite{Fradkin:1986ka,Fradkin:1987ks,Fradkin:1986qy} is that one can treat the extra fields as independent variables since most of the gauge-invariant curvatures\nvanish on shell.\n\n\\subsection*{Acknowledgments}\n\nThe author is grateful to N.~Boulanger, A.~Campoleoni, G.~Lucena G\\'omez, M.~Henneaux, and especially to  E.~D.~Skvortsov for valuable inputs and useful comments.\nHe would like to thank the organizers of the 4th Mons Workshop on Higher Spin Gauge Theories (2017), during which this study was initiated.\n\n\\begin{appendix}\n\\numberwithin{equation}{section}\n\\section{Metric-like Formulation}\\label{sec:A}\n\nThe metric-like formulation of gauge fermions originated in the work of Fang and Fronsdal~\\cite{FF,Fang:1979hq}, who studied the massless limit of the Lagrangian for massive\nhigher-spin fermions. The Fang-Fronsdal Lagrangian can be derived uniquely by considering gauge invariance and supersymmetry transformations for a massless system involving\nthe pair of spins $\\left(s, s+\\tfrac{1}{2}\\right)$~\\cite{Curtright:1979uz}. The construction was later generalized for maximally symmetric spaces with arbitrary dimension in Ref.~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}. In the metric-like formulation, a spin $s=n+\\tfrac{1}{2}$ gauge fermion is described by a completely symmetric rank-$n$\ntensor-spinor $\\psi_{\\mu(n)}$ in the world indices. It satisfies the triple $\\gamma$-trace condition:\n\\begin{equation}\\label{tg1}\\displaystyle{\\not{\\!\\psi\\!\\,}}'_{\\mu(n-3)}=0.\\end{equation}\nIt is convenient to describe metric-like theories in the operator formalism, where contraction and symmetrization of indices are realized through auxiliary variables\nand tensor operations are simplified in terms of operator calculus. Symmetric tensor-spinor fields are represented by:\n\\begin{equation}\\label{field}\\psi(x,u)=\\tfrac{1}{n!}\\,\\psi_{\\mu_1\\ldots\\mu_n}(x)\\,\\bar{e}^{\\,\\mu_1}_{a_1}(x)u^{a_1}\\,\\ldots\\,\\bar{e}^{\\,\\mu_n}_{a_n}(x)u^{a_n},\\end{equation}\nwhere $\\bar{e}^{\\,\\mu}_a(x)$ is the background vielbein and $u^a$ is an auxiliary tangent variable. The action of the covariant derivative is defined as\na differential operation involving both $x$ and $u$:\n\\begin{equation}\\label{covD}\\nabla_\\mu=\\bar{\\nabla}_\\mu+\\bar{\\omega}_\\mu{}^{ab}u_a\\tfrac{\\partial}{\\partial u^b},\\end{equation}\nwhere $\\bar{\\nabla}_\\mu$ is the standard covariant derivative acting on naked tensorial indices, and $\\bar{\\omega}_\\mu{}^{ab}$ the background spin connection.\nIn what follows we work only with the contracted auxiliary variable and the associated derivative:\n\\begin{equation}\\label{u-du} u^\\mu\\equiv \\bar{e}^{\\,\\mu}_{a}(x)u^{a},\\quad \\partial_u^\\mu\\equiv \\bar{e}^{\\,\\mu a}(x)\\tfrac{\\partial}{\\partial u^a}.\\end{equation}\nThe vielbein postulate then implies that $[\\nabla_\\mu,u^\\nu]=0$~as well as $[\\nabla_\\mu,\\partial_u^\\nu]=0$. The commutator of covariant derivatives on a spinor function of $u$ and $\\partial_u$\nwill be given by:\n\\begin{equation}\\label{commutator}[\\nabla_\\mu,\\nabla_\\nu]=R_{\\mu\\nu\\rho\\sigma}(x)u^\\rho\\partial_u^\\sigma+\\tfrac{1}{4}R_{\\mu\\nu\\rho\\sigma}(x)\\gamma^{\\rho\\sigma}.\\end{equation}\nOne would have to use the following set of operators~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}:\n\\begin{equation}\\label{sett} \\mathbb{G}=\\left\\{\\displaystyle{\\not{\\!\\nabla\\!\\,}},\\,\\partial_u\\!\\cdot\\!\\nabla,\\,u\\!\\cdot\\!\\nabla,\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}},\\,\\displaystyle{\\not{\\!u\\!\\,}},\\,\\partial_u^2,\\,u^2,\\,u\\!\\cdot\\!\\partial_u\\right\\}.\\end{equation}\nThe set comprises eight operators: the Dirac operator $\\displaystyle{\\not{\\!\\nabla\\!\\,}}$, divergence $\\partial_u\\!\\cdot\\!\\nabla$, symmetrized-gradient $u\\!\\cdot\\!\\nabla$, $\\gamma$-trace $\\displaystyle{\\not{\\!\\partial_u\\!\\,}}$, symmetrized-$\\gamma$ $\\displaystyle{\\not{\\!u\\!\\,}}$, trace $\\partial_u^2$,\nsymmetrized-metric $u^2$ and rank $u\\!\\cdot\\!\\partial_u$.\nThese operators have nontrivial commutation relations because of $[\\partial_u^\\mu,u^\\nu]=\\bar{g}^{\\,\\mu\\nu}$ and the non-commutativity~(\\ref{commutator}) of the\ncovariant derivatives if the background is non-flat.\n\nThen, the Lagrangian for a massless fermionic field in AdS space can be written as (for a Majorana fermion, certain terms in the Lagrangian are equivalent up to total\nderivatives)~\\cite{Metsaev:2013wza}:\n\\begin{eqnarray}\n\\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}&=&-\\tfrac{1}{2}\\bar{\\psi}(\\ast_n)\\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-u\\!\\cdot\\!\\nabla\\displaystyle{\\not{\\!\\partial_u\\!\\,}}-\\displaystyle{\\not{\\!u\\!\\,}}\\,\\partial_u\\!\\cdot\\!\\nabla+\\displaystyle{\\not{\\!u\\!\\,}}\\,\\displaystyle{\\not{\\!\\nabla\\!\\,}}\\displaystyle{\\not{\\!\\partial_u\\!\\,}}+\\tfrac{1}{2}\\displaystyle{\\not{\\!u\\!\\,}}\\,u\\!\\cdot\\!\\nabla\\,\\partial_u^2+\\tfrac{1}{2}u^2\\,\\partial_u\\!\\cdot\\!\\nabla\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\right)\n\\psi\\nonumber\\\\&&-\\tfrac{1}{2}\\bar{\\psi}(\\ast_n)\\left(-\\tfrac{1}{4}u^2\\displaystyle{\\not{\\!\\nabla\\!\\,}}\\,\\partial_u^2\\right)\\psi+\\tfrac{1}{2}\\mu\\,\\bar{\\psi}(\\ast_n)\\left(1-\\displaystyle{\\not{\\!u\\!\\,}}\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}}-\\tfrac{1}{4}u^2\\,\\partial_u^2\\right)\\psi,\\label{f00}\n\\end{eqnarray}\nwhere the operation: $(\\ast_k)\\equiv\\left(\\overleftarrow{\\partial_u}\\cdot\\overrightarrow{\\partial_u}\\right)^k$ enables contraction between two rank-$k$ tensor-spinors, and has the properties:\n$(\\ast_k)u^\\mu=k\\overleftarrow{\\partial_u}^\\mu(\\ast_{k-1})$ and $(\\ast_k)\\overrightarrow{\\partial_u}^\\mu=(k+1)^{-1}u^\\mu(\\ast_{k+1})$\\,. The mass parameter:\n\\begin{equation}\\label{tamm0}\\mu=\\frac{1}{l}\\left(n+\\tfrac{D-4}{2}\\right),\\end{equation}\nis uniquely fixed by gauge invariance~\\cite{Metsaev:2006zy,Metsaev:2013wza}, where $l$ is the AdS radius.\nThe gauge symmetry of the Lagrangian~(\\ref{f00}) is w.r.t.~a symmetric $\\gamma$-traceless rank-$(n-1)$ tensor-spinor parameter:\n\\begin{equation}\\label{tamm0.5}\n\\varepsilon=\\tfrac{1}{(n-1)!}\\,\\varepsilon_{\\mu_1\\ldots\\mu_{n-1}}u^{\\mu_1}\\ldots u^{\\mu_{n-1}}, \\qquad \\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\varepsilon=0,\n\\end{equation}\nwhile the triple $\\gamma$-tracelessness condition~(\\ref{tg1}) on the field translates in the operator formalism to:\n\\begin{equation}\\label{tamm1} \\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\partial_u^2\\psi=\\partial_u^2\\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\psi=0.\\end{equation}\nExplicitly, the gauge transformations are given by:\n\\begin{equation}\\label{tamm2}\\delta\\psi=u\\!\\cdot\\!\\nabla\\varepsilon-\\frac{1}{2l}\\displaystyle{\\not{\\!u\\!\\,}}\\,\\varepsilon.\\end{equation}\nThis can be verified by using the commutator~(\\ref{commutator}), which reduces in AdS space to:\n\\begin{equation}\\label{commutator-AdS}[\\nabla_\\mu,\\nabla_\\nu]=-\\frac{1}{l^2}\\left(u_{[\\mu} d_{\\nu]}+\\tfrac{1}{2}\\gamma_{\\mu\\nu}\\right),\\end{equation}\nand the various commutators of the operators in $\\mathbb{G}$ given the properties~(\\ref{tamm0.5}) and~(\\ref{tamm1}).\n\nThe metric-like description of higher-spin gauge fermions in flat-space is easily obtained by taking the limit $l\\rightarrow\\infty$ of the\ngauge invariant system~(\\ref{f00})--(\\ref{commutator-AdS}). The degrees of freedom count in flat~\\cite{Rahman:2015pzl} and AdS~\\cite{Campoleoni:2017vds}\nspaces are of course the same, and is given by:\n\\begin{equation}\\label{dofF} \\Delta_{\\text{Metric}}=\\binom{D+n-4}{n}f_D,\\end{equation}\nwhere $f_D$ for a Majorana fermion is given in Eq.~(\\ref{fD-defined}), while for a Dirac fermion the value is twice as much.\nNote that Eq.~(\\ref{dofF}) counts the number of physical dynamical fields plus their conjugate momenta.\nIn AdS space, one of course gets the same number since the counting of dynamical equations, constraints and gauge freedom works in the same way.\n\nAs already mentioned in the Introduction, the $\\gamma$-trace constraints~(\\ref{tamm0.5})--(\\ref{tamm1}) on the gauge parameter and the higher-spin fermionic field can be avoided by\nrecourse to other formulations. These include the non-local formulation~\\cite{Francia:2002aa}, the BRST\nformulation~\\cite{Buchbinder:2004gp,Buchbinder:2007vq}, the higher-derivative compensator formulation~\\cite{Francia:2007qt}, the quartet formulation~\\cite{Buchbinder:2007ak}\nand the non-minimal formulation with no higher derivatives~\\cite{Campoleoni:2009gs}.\n\n\\end{appendix}\n\n\\bibliographystyle{ws-rv-van}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nIn the theory of planet formation, planets are thought to have formed in\nprotoplanetary disks through mutual collisions and coalescence of\nplanetesimals. The formation process of planetesimals, on the other\nhand, still has a large uncertainty. Before the planetesimal formation,\ndust grains grow through their collisional coalescence in a\nprotoplanetary disk and settle to the disk mid-plane,  which forms a\ndense dust layer at the mid-plane \\citep[e.g.,][]{saf69,nak81,tan05,dul05}.\nPlanetesimals would be formed in the dust layer through gravitational\ninstability \\citep[e.g.,][]{gol73,sek98,you02},\nstreaming instability \\citep[e.g.,][]{you05,you07}\nor simple coalescence \\citep[e.g.,][]{wei93,bra08a,bra08b}.\nIn these models of planetesimal formation, motion of dust\ngrains is an important factor because it determines the spatial\ndistribution of dust grains and the collision speed between\nthem. Furthermore, their motion is governed by the drag forces from the\ndisk gas.\n\nGas drag forces on dust grains strongly depend \non their internal\nstructure (or their bulk densities). Most studies of dust growth in\nprotoplanetary disks have assumed compact structure of dust\ngrains. However, dust grains \ngrowing\nthrough mutual\ncollisions would actually be aggregates of (sub-micron) primitive grains\nand the aggregates have a fluffy structure with an extremely low bulk\ndensity, as reported by experimental and theoretical studies\n\\citep[e.g.,][]{blu04b,orm07,suy08,oku09,zsom11}.\nSuch fluffy aggregates\nhave large ratios of their \ngeometrical \ncross sections to masses, which\nsignificantly enhance \ngas drag forces\non them compared with compact\ndust grains. \nHence, in order to clarify dust growth and planetesimal formation \nin protoplanetary disks, we have to examine the internal structure and \nthe geometrical \ncross sections of dust aggregates \n(we use a term `cross section' in referring to `geometrical cross\nsection', hereafter). \n\n\\citet{suy08} (hereafter S08) performed $N$-body numerical simulations of\nsequential aggregate collisions to examine the compression process of\ngrowing aggregates. \nThe sequential collisions mean that we repeat collisions of\naggregates obtained at the previous collisions.\nWith such a simulation, we can observe a natural evolution of \nthe aggregate structure.\nTheir numerical results showed that large aggregates have an\nextremely low bulk density \nin spite of\ncompression at aggregate collisions. In the early stage of dust growth,\naggregates just stick without any restructuring because of their low \nimpact energy and they have a fluffy structure with an extremely \nlow bulk density as they grow. \nIn the later stage in which the impact energy exceeds a\ncritical energy, aggregates are gradually compressed. \nEven in this compression stage, their density remains very low.\nIt is found that the compressed aggregates have a low fractal\ndimension of 2.5. This structural feature causes the low density\nof the compressed aggregates.\nS08 also\nderived a formula describing the density evolution of growing\naggregates. To estimate their bulk densities, S08 \nused\nthe\nso-called gyration radii of the aggregates but did not examine their\ncross sections. \nHowever, cross sections are directly related to the gas drag \nforces\nrather than gyration radii. \nIt is necessary to clarify the evolution of\ncross sections of dust aggregates during their growth. \n\nCross sections of aggregates depend on their internal\nstructure. There are two simple aggregate models. \nOne is Ballistic Cluster-Cluster Aggregation (BCCA). \nA BCCA cluster is formed through\ncollisions between \ntwo equal-sized clusters. \nSecond is Ballistic Particle-Cluster Aggregation (BPCA). \nA BPCA cluster is formed through\ndeposition of small monomer particles on a large cluster. \nFor both BCCA and BPCA, restructuring is assumed to be\nnegligible at each collision.\nThe BCCA clusters have very fluffy and open structures and \nthe BPCA clusters have relatively\ncompact structures. \nFigure~\\ref{fig:spmintro} shows the ratios of cross\nsections to masses of aggregates. \nIt is shown that the cross section per mass strongly\ndepend on aggregate types. \nCross sections of dust aggregates are expected to be between those of\nBCCA and BPCA clusters. \nOne may consider that a cross section is approximately given by the\nsquare of a gyration radius.\nCross sections of aggregates are, however, generally\nindependent of their gyration radii, especially for highly fluffy\naggregates.\nThe non-dimensional ratio of the cross section to the square\nof the gyration radius gradually decreases with their growth for BCCA\nclusters (Minato et al.~2006; see also Fig.~6).\nOn the other hand, this ratio is almost constant in the growth\nof BPCA clusters. \nOkuzumi et al. (2009) proposed a useful relation between \nthe cross section and the gyration radius\nfor various aggregates formed through \nhit-and-stick growth (as well as BCCA and BPCA). \nPaszun and Dominik (2009) also derived another relation.\nNevertheless, it is not clear whether these relations are\nalso valid for aggregates compressed at collisions.\nWe check the validity of these relations,\nusing the resultant aggregates obtained by S08.\n\nOnce we find a valid relation between the cross sections and the \ngyration radii, it would be very helpful to describe the evolution \nof the cross sections because the compression model by S08\ncan describe gyration radii of growing aggregates.\nThe compression model by S08, however, has some limitations.\nThis model is not directly applicable to low-energy collisions\n(i.e., hit-and-stick collisions) or to non-equal-mass collisions.\nIn order to describe gyration radii and cross sections of aggregates\nfor all growth stages seamlessly, we further refine the compression model, \nby removing these limitations.\n\n\nIn laboratory experiments, \\citet{weidl09}\nexamined compression of aggregates consisting of \n1.5$\\mu$m-diameter SiO$_2$ spheres at their multiple \nrebounds and also developed an empirical compression model. \nInitial aggregates in their experiments\npossess a volume filling factor of $\\sim 0.1$, \nwhich is approximately equal to \nthat of BPCA clusters. On the other hand, S08 and the \npresent paper focus on the compression of fluffier \naggregates of which filling factor is between BCCA and \nBPCA clusters during their collisional growth. \nHence our compression model and theirs are\ncomplementary to each other. As mentioned above, dust \naggregates are expected to have much smaller bulk densities\nthan BPCA clusters at the early stage of their \ngrowth in protoplanetary disks. Our compression model is \nuseful as long as bulk densities of aggregates is lower \nthan that of BPCA clusters.\n\nIt should be noticed that S08 only considered head-on collisions of\naggregates in their numerical simulations of sequential\ncollisions. The oblique collisions are expected to hinder the\ncompression (Wada et al. 2007; Paszun and Dominik 2009).\nIn the present study, however, we use the results in S08 as the \nfirst step. We will examine the effects of oblique collisions \nin future work.\n\nIn the next section, we briefly summarize the results of S08. In Section\n3, we numerically \ncalculate\ncross sections for the aggregates obtained by S08. \nWe find that Okuzumi et al's relation between\nthe cross section and the gyration radius is valid\nfor compressed aggregates, too.\nIn Section 4, we refine the compression model by S08,\nby removing its limitations in a reasonable way.\nWe find that the refined compression model reproduces well \nboth of gyration radii and cross sections of aggregates\nobtained by the numerical simulation,\nwith the help of Okuzumi et al's relation.\nWe also check the validity of the refined model for non-equal-mass \ncollisions with additional numerical simulations of aggregate collisions.\nA summary is given in the last section.\n\\section{RESULTS OF AGGREGATE COMPRESSION IN $N$-BODY SIMULATIONS BY S08}\n\nSuyama et al.~(2008) performed $N$-body numerical simulations of head-on\naggregate collisions and examined the density evolution of aggregates\ngrowing through the collisions. We examine the cross section\nof the resultant aggregates obtained by S08. Before\nthat, we briefly describe the numerical results of S08.\n\nIn the simulations, aggregates consist of a large number of icy spherical\nparticles with a radius of $r_1=0.1\\mu$m.\nS08 adopted the particle-interaction\nmodel by Wada et al.~(2007). In the interaction model, \nrepulsive and adhesive forces in the normal direction between particles\nin contact\nare given by the JKR theory (Johnson et al. 1971).\nA\ntangential force and a torque also arise to\nresist the slide, roll, and twist motions between them. Aggregate\ncompression is regulated mainly by inelastic rolling motions of the constituent\nparticles (e.g., Dominik \\& Tielens 1997;\nWada et al. 2007, 2008 [hereafter W07,W08]~; G\\\"{u}ttler et al. 2010).\nThe rolling energy $E_\\mathrm{roll}$\nis the energy required for rolling \na particle on its contact neighbor by an angle of $\\pi\/2$.\nThe rolling\nenergy is given by (W07, S08)\n\\begin{equation}\n E_\\mathrm{roll}=6\\pi^2\\gamma r_1\\xi_\\mathrm{crit},\\label{eroll} \n\\end{equation}\nwhere $\\xi_\\mathrm{crit}$ is the critical displacement for inelastic\nrolling motion.\nThe parameter range of $\\xi_\\mathrm{crit}$ is set to be from 2 to 16\n$\\mathrm{\\mbox{\\AA}}$ in S08. \nA large\nrolling energy $E_\\mathrm{roll}$\nsuppresses the restructuring of\naggregates. \nTo examine the structure evolution of\ngrowing aggregates, S08 performed $N$-body simulations of sequential\ncollisions. Each\nsimulation starts from a collision of aggregates composed of two\nparticles (i.e., dimers) and ends with a collision of aggregates\ncomposed of 16,384 particles. \nThe resultant aggregate obtained in the previous\ncollision is used as initial aggregates at each collision in the\nsimulation of sequential collisions.\nThe impact velocity is constant in sequential collisions. \nFor various (constant) impact velocities and critical rolling displacements,\nthey performed a large number of runs of sequential collisions.\n\nAs an index of the \nsize of an aggregate,\nS08 adopted the\nradius of gyration, $r_g$, defined by\n\\begin{equation}\n r_g\\equiv\n  \\sqrt{\\sum_{i=1}^N\n  \\frac{|\\mbox{\\boldmath{$x$}}_{i}-\\mbox{\\boldmath{$x$}}_\\mathrm{M}|^2}\n  {N}\n  },\n\\end{equation}\nwhere $\\mbox{\\boldmath{$x$}}_i$ is the position of particle $i$,\n$\\mbox{\\boldmath{$x$}}_\\mathrm{M}$ is\nthe position of the center of mass of the aggregate, and $N$ is the\nnumber of particles composing the aggregate.\nUsing the radius of gyration, \nthe volume $V$ and \nthe bulk density \n$\\rho$ \nof the aggregate are \nevaluated to be (Mukai et al. 1992; W08)\n\\begin{equation}\n V(r_g)=\\frac{4\\pi}{3}\\left(\\sqrt{\\frac{5}{3}}r_g\\right)^3,\n\\label{vdef}\n\\end{equation}\n\\begin{equation}\n \\rho(r_g)=\\frac{m_1N}{V(r_g)},\n\\label{rhodef}\n\\end{equation}\nrespectively,\nwhere \n$\\sqrt{5\/3}r_{g}$ is the so-called characteristic radius of an aggregate\nand \n$m_1$ is the mass of a constituent particle.\n\nAggregates are expected to have a BCCA structure for collisions at\nsufficiently low velocity because of sticking together of equal-mass\naggregates without any restructuring. If the compression is effective at\ncollisions, the gyration radii of aggregates would become smaller than\nthose of BCCA clusters. It is meaningful to compare the\nobtained aggregates with BCCA clusters. \nSince BCCA clusters have a fractal dimension of $\\sim 2$, the radius of\ngyration of  the BCCA cluster is given for large $N$ by (e.g., Mukai et\nal. 1992; W08\n)\n\\footnote{Exactly speaking,\nequation (\\ref{bccarg}) is satisfied for BCCA clusters formed through\nhead-on (hit-and-stick) collisions, which have the fractal dimension\nof 2.0. When offset collisions are also included at the formation\nof BCCA, their fractal dimension is 1.9 and the gyration radii\nproportional to $N^{0.52}$ (Okuzumi et al. 2009). \nAlthough the later BCCA is more realistic,\nthe former BCCA is used in S08 and the present study \nsince S08 consider only \nhead-on collisions in their simulations.}\n\n\\begin{equation}\n r_{g,\\mathrm{BCCA}}\\simeq N^{0.50}r_1.\\label{bccarg}\n\\end{equation}\n\nFigure~\\ref{fig:rgrho}a shows the gyration radius of the aggregates in the\nsimulations of sequential collisions performed by S08 for various values of\nparameters, $\\xi_\\mathrm{crit}$ and the impact velocity\n$v_\\mathrm{imp}$.\nThe density of monomer particles is given by \n$\\rho_m (\\equiv 3m_1\/[4\\pi r_1^3])$. \nIn the simulation of sequential collisions, the size of growing\naggregates is dependent on \nthe direction of each collision.\nS08 did 30 runs of the simulation of sequential\ncollisions and obtained the averaged value of $r_g$ from 30 runs for each\n$\\xi_\\mathrm{crit}$ and $v_\\mathrm{imp}$.\nIn Figure~\\ref{fig:rgrho}a,  the horizontal axis is the number of the\nconstituent particles, $N$, in the growing aggregates and the vertical\naxis is the gyration radii divided by $N^{1\/2}r_1$ for comparison\nwith BCCA clusters.\nThe dashed line represents the radius of the BCCA cluster\nand it is \nalmost flat for\nlarge $N$\nas expected from equation (\\ref{bccarg}).\nThe size of small aggregates produced in our simulation is almost the same as\nthat of BCCA clusters. \nThis is because\nthe impact energy is small enough at the early stage of the aggregate\ngrowth and the compression is ineffective at\neach collision. As the aggregates grow,\nthe impact energy increases.\nWhen the impact energy attains to $E_\\mathrm{roll}$, the compression of\nthe aggregate starts: aggregates become smaller than BCCA clusters. \nThe critical number of\nparticles, $N_\\mathrm{crit}$, in the aggregate for compression is given\nby (W08, S08)\n\\begin{equation}\n N_\\mathrm{crit} = \\beta\\frac{8E_\\mathrm{roll}}{m_1v_\\mathrm{imp}^2},\n\\label{ncrit}\n\\end{equation}\nwhere $\\beta$ is a non-dimensional coefficient. In Figure~\\ref{fig:rgrho}a,\nwe also plot the critical number $N_\\mathrm{crit}$ with filled circles on\neach curve, by setting $\\beta=0.5$.\nFigure~\\ref{fig:rgrho}b shows bulk densities of growing aggregates in\nthese simulations, which evaluated with $r_g$ by\nequation~(\\ref{rhodef}). \nWe also plot the density of BCCA clusters.\nIt decreases as $N^{-0.50}$ for large $N$.\nAfter the onset of compression (i.e., $N > N_\\mathrm{crit}$), the\nbulk densities of the aggregates obtained by the numerical simulation \nare larger than that of the BCCA but still keep on decreasing gradually\nin all cases,\nindicative of the inefficiency of collisional compression.\nS08 also developed the compression model,\nwhich reproduces \nthe density evolution of growing aggregates in\nthe compression stage. We will describe the compression model \nin Section 4.\n\\section{GEOMETRICAL CROSS SECTIONS OF AGGREGATES PRODUCED IN THE\n SEQUENTIAL COLLISIONS}\n\nWe numerically \ncalculate\ncross sections of resultant aggregates \nobtained by\nS08. The cross section of a dust aggregate is given by\nthe area of the shadow of the aggregate projected onto a plane. The area\nof the shadow is calculated by counting the number of square meshes\nin the shadow (Fig.~\\ref{fig:ssmosikizu}). The width of the square\nmeshes is set to be\n$0.0055 r_1$. This width is much smaller than the radius of the\nmonomer particle, $r_1$, \nthough meshes with a much wider width are\ndrawn to emphasize them in Figure~\\ref{fig:ssmosikizu}.\nThe area of the shadow is dependent on the\nplane onto which the shadow is projected.\nWe calculate the areas of the shadows for 30 orientations randomly\nchosen and define the cross section of the aggregate\nby the mean values of the areas. Figure~\\ref{fig:ss1headon} shows the\ncross section\ncalculated in this way for  aggregates produced in the simulations of\nsequential collisions. \nThe vertical axis is the cross\nsection divided by $N \\pi r_1^2$, \nwhich corresponds to the (non-dimensional) cross section per mass. \nIf the overlapping of the monomer particles in the shadow is negligibly \nsmall, the value of the vertical axis \napproaches\nunity. A filled circle in Figure~\\ref{fig:ss1headon}a \nindicates the shadow area\nfor each orientation and the line shows the mean of them. \nThe cross section per mass\ndecreases as an aggregate grows in the simulation of sequential\ncollisions due to the overlapping of the constituent particles. \nSince S08 did 30 independent runs of sequential collisions, the mean cross\nsections are calculated and plotted as thin lines in\nFigure~\\ref{fig:ss1headon}b. \nThen we obtain the averaged value of the\n30 mean cross sections as shown by the thick line in\nFigure~\\ref{fig:ss1headon}b.\nThe dispersion of the mean geometrical cross section can be evaluated\nin Figure~\\ref{fig:ss1headon}b.\nThe standard deviation of the mean geometrical\ncross section is equal to or less than 11\\% of its averaged value\nduring the aggregate growth.\nIn this way, we did two kinds of averaging to calculate the cross section\nof growing aggregates in the simulation.\nWe show and discuss the cross sections of aggregates by using these\nfinally-obtained cross sections hereafter.\n\nFigure~\\ref{fig:ssheadon} shows the cross section of resultant\naggregates for various values\nof the parameter set ($\\xi_\\mathrm{crit}, v_\\mathrm{imp}$).\nThe averaged cross section of the BCCA cluster is also calculated and\nplotted. \nThe cross section of BCCA we obtained agrees \nwith the result of\n\\citet{min06b}.\nEven for BCCA, the ratio $S\/(N \\pi r_1^2)$\ngradually decreases with an increase in $N$ due to the overlapping of\nconstituent particles.\nIn the early stage of the aggregate growth (i.e. for small $N$), the cross\nsections of the resultant aggregates change almost along the line of\nBCCA. As the aggregates grow, however, their cross sections deviate from\nthe line of BCCA and become much smaller than that of BCCA, which\nis due to compression at collisions. This qualitative tendency \nis consistent\nwith the evolution of the radius of gyration (Fig.~\\ref{fig:rgrho}a).\nAlthough the change in the cross sections is gradual and the starting\npoints of compression in the cross sections are not clear compared with\nthose in the gyration radii, the starting points are also described by\nequation~(\\ref{ncrit}), by setting the parameter\n$\\beta$ to be 2.0.\nThe larger value of $\\beta$ than Figure~\\ref{fig:rgrho} indicates that\nthe onset of compression in the cross section is later than that in the\ngyration radius.\n\nIn Figure~\\ref{fig:ssrgheadon} we plot the ratio of the cross section $S$ \nto $\\pi r_g^2$ for all resultant aggregates with solid lines. \nThe ratio of  $S$ to $\\pi r_g^2$ decreases for small aggregates, \nwhich is consistent with the BCCA case (dotted lines). \nFor sufficiently large aggregates, the ratio increases as a result of\ntheir compression.\nOkuzumi et al. (2009) proposed a useful expression of \nthe cross section $S$ for aggregates formed through hit-and-sticks.\nThe expression is given by\n\\begin{equation}\nS(r_g, N) = \\left(\n\\frac{1}{S\\sub{BCCA}(N)} + \\frac{1}{ \\pi (5\/3) r_g^2}\n-\\frac{1}{ \\pi (5\/3) r\\sub{$g$, BCCA}(N)^2}\n\\right)^{-1},\n\\label{s-O09}\n\\end{equation}\nwhere the cross section of the BCCA cluster is given by \n(Minato et al. 2006)\n\\begin{eqnarray}\n\\frac{S_\\mathrm{BCCA}}{\\pi r_1^2}=\\left\\{\n \\begin{array}{cc}\n  12.5N^{0.685}\\exp(-2.53\/N^{0.0920}) &(N < 16),\\\\\n  0.352N+0.566N^{0.862} &(N \\geq 16).\\label{sbcca}\n \\end{array}\n  \\right.\n\\end{eqnarray}\nEquation~(\\ref{bccarg}) is used as the expression of\n$r\\sub{$g$,BCCA}(N)$.\nWe also plotted the cross sections obtained from equation (\\ref{s-O09})\nwith dashed lines in Figure~\\ref{fig:ssrgheadon}.\nIt is found that the expression by Okuzumi et al. (2009) \nreproduces the cross section surprisingly well for compressed \naggregates as well as hit-and-stick aggregates, by using the gyration \nradius $r_g$. In this expression, the information of compression\nis correctly included through the gyration radius.\nPaszun and Dominik (2009) also derived another relation\nbetween $S$ and the aggregate size (i.e., eq.[11] of their paper).\nFigure~7 is the same as the left-bottom panel of Figure~6\nbut the prediction by Paszun and Dominik is also plotted.\nAlthough the prediction by Paszun and Dominik is consistent with\nthe numerical results,\nit overestimates $S$ when the ratio $S\/(\\pi r_g^2)$ is larger than unity\n(i.e., for relatively compact aggregates)\nand underestimates for $S\/(\\pi r_g^2)<0.7$.\nThe underestimation was also reported by Okuzumi et al.\nThey found that the underestimation in Paszun and Dominik's model\nis severe especially for large and fluffy aggregates.\nFor other $\\xi_\\mathrm{c}$,\nwe also find the same trend as in the case of \n$\\xi_\\mathrm{c}=8\\mathrm{\\mbox{\\AA}}$ shown in \nFigure~\\ref{fig:ssrgheadon2}.\nHence it is concluded that the model of Okuzumi et al.~is \nmore accurate than that of Paszun and Dominik.\nOnce an accurate compression model describing $r_g$ is obtained,\nit enables us to calculate the evolution of the cross section with the\nhelp of Okuzumi et al's model.\n\n\n\\section{COMPRESSION MODEL}\nA compression model describing $r_g$ was developed by S08\nbut it has two limitations.\nThe model is not directly applicable to low-energy collisions\n(i.e., hit-and-stick collisions at the early growth stage) or \nto non-equal-mass collisions.\nIn order to describe gyration radii and cross sections of aggregates\nfor both the early hit-and-stick stage and the compression stage seamlessly,\nwe refine our compression model,\nby removing these limitations in a natural way.\nBefore that, we briefly describe the compression model by S08.\n\n\\subsection{Compression Models of W08 and S08}\nW08 developed a compression model by introducing the pressure\n(or the strength) of aggregates to explain their numerical results on\ncollisions between BCCA clusters.\nThe compression model of S08 is based on that of W08.\nAt a collision of two aggregates with the impact energy \n$E_\\mathrm{imp}$, the compression of the merged aggregate from the \ninitial volume, $V_\\mathrm{initial}$, to the final\nvolume, $V_\\mathrm{final}$, is described in the model of W08 by\n\\begin{equation}\n E_\\mathrm{imp} = -\\int_{V_\\mathrm{initial}}^{V_\\mathrm{final}} P dV.\n\\label{eimpp}\n\\end{equation}\nThe initial volume $V_\\mathrm{initial}$ is defined by the volume of the merged\naggregate at the moment that the two aggregates just stick. After the\nmoment of the sticking, the compression proceeds. The volumes before\nand after the compression are evaluated with the radius of gyration,\n$r_g$, as in equation~(\\ref{vdef}). The pressure $P$ of the aggregates\nis given by\n\\begin{equation}\n P = 2 \\left(\\frac{5}{3}\\right)^6  \n\\frac{b E_\\mathrm{roll}\\rho_m}{m_1}\n \\left(\\frac{\\rho}{\\rho_m }\\right)^{13\/3} N^{2\/3},\\label{pform}\n\\end{equation}\nwhere the fitting parameter $b$ is set to be 0.15. Note that the\npressure $P$ of the aggregates is dependent on the total number of\nconstituent particles (or the total mass) as well as the density.\nThat is, $P$ is not an intensive variable.\nThis strange property in the pressure comes from the fractal\nstructure of the aggregates. W08 showed with their simulation of\ncollisions between BCCA clusters that the compressed aggregates \nhave internal structures with a fractal dimension of\n2.5. The simulation of sequential collisions done by S08 showed that\ntheir resultant aggregates also have the same fractal dimension of 2.5.\n\nIn order to describe the compression of such fractal aggregates,\nW08 also introduced the fractal volume defined by\n\\begin{equation}\n V_f(r_g)\\equiv ar_g^{2.5},\n\\end{equation}\nwhere \nthe coefficient $a$ is given by \n$(9\\pi\/5)^{5\/4}\\Gamma(9\/4) \\simeq 7.7$.\nUsing the fractal volume, the fractal density is defined by\n\\begin{equation}\n \\rho_f(r_g)\\equiv\\frac{m_1N}{V_f(r_g)}=\\frac{m_1N}{a}r_g^{-2.5}\n  \\label{rhofdef}.\n\\end{equation}\nThe dimensions of the fractal volume and the fractal density differ\nfrom those of the ordinary volume and density. These fractal\nquantities are related with the ordinary quantities $V$ and $\\rho$ as\n\\begin{eqnarray}\n\\frac{V(r_g)}{v_m}&=&\\left(\\frac{5}{3}\\right)^{3\/2}\n \\left( \\frac{V_f(r_g)}{v_{f,1}}\\right)^{6\/5},\\label{ord_fra_v}\\\\\n \\frac{\\rho(r_g)}{\\rho_m}&=&\\left(\\frac{3}{5}\\right)^{3\/2} \n  \\left(\\frac{\\rho_f(r_g)}{\\rho_{f,1}}\\right)^{6\/5}\n N^{-1\/5},\\label{ord_fra}\n\\end{eqnarray}\nwhere $v_m$ ($=4\/3\\pi r_1^3$) is the volume of a monomer and $v_{f,1}$ is\ngiven by $ar_1^{2.5}$.\nUsing the fractal volume $V_f$ (and the fractal density $\\rho_f$),\nequation~(\\ref{eimpp}) is rewritten as\n\\begin{equation}\n E_\\mathrm{imp} = -\\int_{V_{f,\\mathrm{initial}}}^{V_{f,\\mathrm{final}}} P_f\n  dV_f,\n  \\label{eimppf}\n\\end{equation}\nwhere the fractal pressure $P_f$ is given by\n\\begin{equation}\n P_f \\equiv P \\frac{dV}{dV_f}\n     =  4 \\frac{bE_\\mathrm{roll}\\rho_{f,1}}{m_1}\n     \\left(\\frac{\\rho_f}{\\rho_{f,1}}\\right)^5.\\label{pf} \n\\end{equation}\nIt should be noticed that the fractal pressure is dependent on\n$\\rho_f$ but not on the total mass. \nThat is, $P_{f}$ is an intensive variable.\nEquation (\\ref{eimppf}) (or\n[\\ref{eimpp}]) reproduces the numerical results on the compression\nat collisions between BCCAs.\n\nS08 pointed out that W08's compression model\nneeds a minor modification to describe the compression of\npartially compressed aggregates at their collisions, which occurs\nin their simulation of sequential collisions. At the moment of\nsticking at each collision, large voids are produced in the merged\naggregate. The volume of the new voids is included in the initial\nvolume of the merged aggregate, $V_{f,\\mathrm{initial}}$ in\nequation~(\\ref{eimppf}).\nThe energy required for compression of the new voids is\n$\\sim E_\\mathrm{roll}$ and it is much smaller that that predicted by\nequation~(\\ref{eimppf})\nat collisions between partially compressed aggregates. To describe\nthe compression at such collisions, S08 modified W08's\nmodel. Since the energy required for the crush of the new voids is\nnegligible, the initial fractal volume of the merged aggregate in\nequation~(\\ref{eimppf}) is set to be the sum of the fractal volumes of two\ncolliding aggregates, by removing the volume of the new voids.\nThat is,\n\\begin{equation}\n V_{f,\\mathrm{initial}} = V_{f,1} + V_{f,2}\\label{vfinit}. \n\\end{equation}\nUsing equations~(\\ref{eimppf})-(\\ref{vfinit}), we have the (final)\nfractal density of the merged aggregate, $\\rho_{f,\\mathrm{final}}$,\nproduced at collisions of two equal-mass aggregates with the fractal\ndensity $\\rho_{f,0}~(=Nm_1\/[V_{f,1}+V_{f,2}])$\n\\begin{equation}\n \\left(\\frac{\\rho_{f,\\mathrm {final}}}{\\rho_{f,1}}\\right)^4=\n  \\left(\\frac{\\rho_{f,0}}{\\rho_{f,1}}\\right)^4\n   +\\frac{E_\\mathrm{imp}}{bNE_\\mathrm{roll}}.\\label{oldmodel}\n\\end{equation}\nwhere $N$ is the number of constituent particles in the merged one.\nEquation~(\\ref{oldmodel}) describes the density evolution of partially\ncompressed aggregates growing through mutual collisions.\nEquation~(\\ref{oldmodel}) with $b=0.15$ reproduces \nthe density evolution of growing aggregates in Figure~\\ref{fig:rgrho}\nfor $N>N_\\mathrm{crit}$, as seen in Figure~8 of S08.\n\n\n\\subsection{Refinement of the Compression Model}\n\nThe compression model by S08 is not applicable to the early growth stage \n($N<N_\\mathrm{crit}$) where the compression is ineffective. \nThis is because the energy\nrequired for the crush of the new voids is neglected in the model.\nThis limitation is removed, by taking into account the compression process \nof the new voids produced at sticking of two aggregates.\nFurthermore, the model of S08 assumes the collisions of equal-mass aggregates.\nWe also remove this limitation in a reasonable way.\nAt the extension of our compression model to non-equal-mass collisions, \nwe assume that the compressed aggregates have the fractal dimensions \nof 2.5 as well as in the case of equal-mass collisions. \nThe validity of the assumption will be discussed in Section 5.\n\nDensity evolution of aggregates at each collisions\nis divided into the following three steps: \n\\begin{enumerate}\n \\item Creation of new voids at the sticking of two aggregates.\n \\item Compression of the new voids in the merged aggregate. \n \\item Further compression of the merged aggregate after the crush of\n       the new voids.\n\\end{enumerate}\nThese steps are schematically explained by Figure~\\ref{fig:refinedmodel}.\nWe describe the change in the volume or the \ndensity of the aggregates at each step in detail below.\n\nIn Step~1, the density decreases because of the new voids in the merged\naggregate. The change in the density\nis described by Okuzumi et al. (2009).\nWhen the two aggregates with the masses $M_1$, $M_2$, and the\nvolumes $V_1$, $V_2$ ($\\le V_1$) collide with each other, the volumes\nof the merged aggregate $V'_{1+2}$ (before the compression)\nis given by\n\\begin{equation}\n V'_{1+2} = V_1 + V_2 + V_\\mathrm{void},\\label{vdash}\n\\end{equation}\nwhere the volume of the voids $V_\\mathrm{void}$ is obtained from\nthe empirical formula\n\\begin{equation}\n V_\\mathrm{void} =\\min\\left[\n                       \\chi\\sub{BCCA}-1.03\\ln\\left(\\frac{V_1+V_2}{2V_2}\\right),\n                       6.94\\right]V_2\n\\end{equation}\n(see Okuzumi et al. [2009] for the derivation).\nThe collision of equal-mass\naggregates is not assumed in Okuzumi et al. (2009).\nNote that $\\chi\\sub{BCCA} (\\equiv 2^{3\/d_f} -2)$ is 0.83 \nfor BCCA clusters formed through head-on collisions having the\nfractal dimension $d_f=2.0$.\nThe density of the merged aggregates before the compression\nis given by ($M_1+M_2)\/V'_{1+2}$. The fractal volume $V'_{f,1+2}$\nand the fractal density $\\rho'_{f,1+2}$ are obtained from\nequations~(\\ref{ord_fra_v}) and (\\ref{ord_fra}), respectively.\n\n\n\nIn Step~2 and 3, the merged aggregate is compressed\nand the density increases. Step~2 is the compression of the\nnew voids. We put the energy required for the crush of the\nnew voids to be $b'E_\\mathrm{roll}$, where $b'$ is the non-dimensional\nparameter. By fitting with results of the simulation,\nthis parameter is fixed to be $b'=3b$, as will be shown in the next\nsubsection. This relation would be reasonable because both parameters \nare related to the beginning of compression.\n\nWhen the impact energy $E_\\mathrm{imp}$ is smaller than\n$b'E_\\mathrm{roll}$,\nthe new voids are only partially compressed at Step~2.\nIn this case, the compression at Step 3 does not occur \nbecause of the small impact energy.\nThe impact energy $E_\\mathrm{imp}$ is \nevaluated using the reduced mass, $(1\/M_1+1\/M_2)^{-1}$.\nIn this case, the decrease in the fractal volume of the merged\naggregates, $\\Delta V_{f,\\mathrm{1+2}}$, is evaluated to be\n\\begin{equation}\n \\Delta V_{f,1+2} = \\frac{E_\\mathrm{imp}}{b'E_\\mathrm{roll}}\n  V_{f,\\mathrm{void}},\n\\end{equation}\nwhere\n\\begin{equation}\n V_{f,\\mathrm{void}} = V'_{f, 1+2} - V_{f,1} - V_{f,2}.\n\\label{vfvoid}\n\\end{equation}\nIt should be noticed that the fractal volume $V_{f,\\mathrm{void}}$\ndefined by equation~(\\ref{vfvoid}) is negative in collisions where\nthe volume ratio $V_1\/V_2$ is larger than $8 \\times 10^4$.\nA prescription for such high-volume-ratio collisions will be described \nat the end of this subsection.\nHere we consider the case where $V_{f,\\mathrm{void}}$ is positive.\nThen the final fractal volume $V_{f, 1+2}$ after the compression\nis given by\n\\begin{equation}\n V_{f, 1+2} = V'_{f, 1+2} - \\Delta V_{f,1+2}.\\label{modellow}\n\\end{equation}\nand the final fractal density $\\rho_{f,\\mathrm{final}}$ of the merged\naggregate is given by $(M_1+M_2)\/V_{f, 1+2}$.\n\nWhen the impact energy is larger than $b'E_\\mathrm{roll}$, \nthe new voids are crushed completely and the merged\naggregate is further compressed at Step~3. Since the fractal\nvolume of the merged aggregate is $V_{f,1}+V_{f,2}$ at the end\nof Step~2, the final fractal volume $V_{f, 1+2}$ after Step 3\nis obtained from the equation\n\\begin{equation}\nE_\\mathrm{imp} - b'E_\\mathrm{roll} =\n - \\int_{ V_{f,1}+V_{f,2} }^{ V_{f, 1+2} } P_f(\\rho_f) dV_f.\n\\label{modelhigh}\n\\end{equation}\nIntegrating the RHS of equation~(\\ref{modelhigh}), we obtain the final\nfractal density for $E_\\mathrm{imp} > b'E_\\mathrm{roll}$ as\n\\begin{equation}\n \\left(\\frac{\\rho_{f,\\mathrm {final}}}{\\rho_{f,1}}\\right)^4=\n  \\left(\\frac{\\rho_{f,0}}{\\rho_{f,1}}\\right)^4\n   +\\frac{E_\\mathrm{imp}-b'E_\\mathrm{roll}}{bNE_\\mathrm{roll}},\n   \\label{rhoevol_up1}\n\\end{equation}\nwhere we used \n\\begin{equation}\n\\rho_{f,0}= \\frac{M_1+M_2}{V_{f,1}+V_{f,2}}.\n\\label{rhof0}\n\\end{equation}\nIn the limit of\n$E_\\mathrm{imp}\\gg b'E_\\mathrm{roll}$,\nequation~(\\ref{rhoevol_up1}) is identical to\nequation~(\\ref{oldmodel}) (or the compression model by S08).\nThe final fractal volume $V_{f,1+2}$ is given by\n$(M_1+M_2)\/\\rho_{f,\\mathrm{final}}$.\n>From the fractal density, we obtain the gyration radius $r_g$,\nusing equation~(\\ref{rhofdef}). The cross section $S$\nis also obtained from Okuzumi et al's expression (eq.[\\ref{s-O09}]).\nIn this way, we can calculate the density evolution \n(i.e., the evolution of $r_g$ and $S$) at both low- and high-energy \ncollisions, using equation~(\\ref{modellow}) for \n$E_\\mathrm{imp} < b'E_\\mathrm{roll}$ and equation~(\\ref{rhoevol_up1}) for \n$E_\\mathrm{imp} > b'E_\\mathrm{roll}$.\n\nIn high-volume-ratio collisions where $V_1\/V_2 > 8\\times10^4$,\nas noticed above, $V_{f,\\mathrm{void}}$ is negative and\na special prescription is necessary. \nSince a negative $V_{f,\\mathrm{void}}$ means no voids,\nStep 2 should be omitted and \nthe merged aggregate is compressed only with Step 3.\nThat is, equations~(\\ref{modelhigh})-(\\ref{rhof0}) are used\nfor all impact energies in this case.\nIn equations~ (\\ref{modelhigh})-(\\ref{rhof0}), the terms of \n$b'E_\\mathrm{roll}$ is omitted and $V_{f,1}+V_{f,2}$ is replaced by \n$V'_{f,1+2}$ because Step 2 does not occur.\n\n\\subsection{Test of the Refined Compression Model}\n\nLet us test the refined compression model with the numerical results.\nUsing the refined compression model, we calculate the evolution of\ngyration radius for the same condition as the numerical simulations by\nS08 and also obtain the cross sections of the aggregates with \nequation~(\\ref{s-O09}). The results are shown in Figures~\\ref{fig:rgfit}\nand \\ref{fig:sfitoku}. In Figure~\\ref{fig:rgfit}, we plot the evolution\nof the gyration radius calculated with the refined compression model\nand compared it with the numerical results by S08. The parameters $b$ and \n$b'$ are set to be $b=0.15$ and $b'=3b~(=0.45)$, respectively.\nWith this setting of the parameters, the refined model reproduces well \nthe numerical results at both the early growth stage and the compression \nstage. Figure~\\ref{fig:sfitoku} shows the evolution of the cross sections\nand indicates that the refined model also succeeds in describing the \ncross sections with the help of equation~(\\ref{s-O09}). \n\nIn the above, the evolution of gyration radius of growing aggregates\nare calculated with the refined compression model\nand their cross sections are indirectly calculated, by using\n$r_g$ and equation~(\\ref{s-O09}).\nWe also propose another way to describe the cross sections of aggregates.\nWe define alternative characteristic sizes of aggregates $r_S$ by \n\\begin{equation}\nr_S=\\sqrt{S\/\\pi}.\n\\label{eq:rs}\n\\end{equation}\nIt would be possible to describe the evolution of $r_S$\ndirectly (instead of the gyration radius)\nwith the refined compression model in the following way.\nUsing this characteristic size $r_S$ instead of $\\sqrt{5\/3}r_g$,\nwe can define the volume and the bulk density of the aggregate\nby the similar equations to (\\ref{vdef}) and (\\ref{rhodef}).\nThe fractal volume and the fractal density are also defined in the same way.\nThen, applying the refined compression model to the fractal density\ndefined with $r_S$, we can describe the evolution of $r_S$\nas well as in the case of $r_g$. \nThe evolution of the cross section $S$ is calculated with \nequation~(\\ref{eq:rs}).\nThis is a direct way to describe the\ncross section rather than the above.\nIn this calculation of $S$,\nwe have to be cautious with the following two points.\nOne is the modification in equation~(\\ref{vdash}).\nAt a collision of sufficiently fluffy aggregates, \nequation~(\\ref{vdash}) can give the volume $V'_{1+2}$ larger\nthan that of the BCCA cluster with the same mass, $V_\\mathrm{BCCA}$\nwhen the size $r_S$ is used instead of $r_g$.\nSuch a large $V'_{1+2}$ is not realistic. In this case, we set\nthe volume $V'_{1+2} = V_\\mathrm{BCCA}$ instead of equation~(\\ref{vdash}).\nThe other point is the parameter $b$.\nAlthough $b$ is set to be 0.15 in the case of $r_g$,\nwe have to calibrate the parameter $b$ again in the case of $r_S$\nas a result of the fitting with numerical results.\nIn Figure~\\ref{fig:sfit}, we plot the evolution\nof $S$ calculated with this direct way.\nIn this calculation, the parameters are set to be $b=0.6$ and $b'=3b$.\nWe see that the refined model works well for the evolution of the\ncross sections with this direct way, too.\\footnote{\nIn the early growth stage of Figure~\\ref{fig:sfit},\nwe used $V'_{1+2} = V_\\mathrm{BCCA}$\ninstead of equation~(\\ref{vdash}) \nwhen the volume of equation~(\\ref{vdash})\nis larger than $V_\\mathrm{BCCA}$, as mentioned above.\n}\n\nIt is found that the refined compression model enables us to describe \nthe whole evolution of the radius of gyration $r_g$ and the cross \nsection $S$ of growing aggregates.\nNote that the refined model is applicable to non-equal-mass\ncollisions though the tests in Figure~\\ref{fig:rgfit}-\\ref{fig:sfit} are \ndone only in the equal-mass case. \nAt the extension of the refined model to non-equal-mass\ncollisions, the fractal dimension of compressed aggregates\nis assumed to be 2.5 even in the case of non-equal-mass collisions\nthough it is not verified with $N$-body simulations in non-equal-mass cases.\nIt is possible that a very large mass ratio increases the fractal dimension,\nas seen in BPCA clusters.\nOkuzumi et al. (2009) examined the effect of the mass ratio\non the fractal dimension for aggregates growing with hit-and-stick\ncollisions with $N$-body simulations. They showed that\nthe collisions with the mass ratio of 10 increases\nthe fractal dimension $d_f$ only by 0.1 (see their figure 6). Furthermore, \ncollisions with such a mass ratio have a major contribution at dust \ngrowth in protoplanetary disks, as shown by Okuzumi et al.~(2009,2012).\nHence the effect of non-equal-mass collisions would not \nchange largely the fractal dimension of compressed aggregates\nin the realistic growth process.\n\nTo confirm the validity of the refined model in non-equal-mass\ncollisions, we have further performed additional $N$-body simulations of \naggregate collisions. Similar to W08, we consider collisions of two BCCA \nclusters but their masses are not equal in the present case.\nThe projectile BCCA cluster consists of 1024 particles (or 4096 particles)\nwhile the number of constituent particles of the target BCCA cluster \nis 16384. Their mass ratio is $1\/16$ (or $1\/4$).\nThe constituent particles are icy ones with the radius with 0.1$\\mu$ m.\nThe impact velocity $v\\sub{imp}$ is a parameter.\nWe set $v\\sub{imp} \\le 4.4$ms$^{-1}$ since we focus on the compression \nprocess rather than fragmentation (Wada et al. 2007,2008).\nThe numerical results of compression at the non-equal-mass collisions\nare shown in Figure~\\ref{fig:additional}.\nThe predictions by the refined model with $b=0.15$ are plotted by solid \nlines and the dashed line indicates the formula by W08 (their equation~[45]).\nThe numerical results in the non-equal-mass collisions approximately \nagree with the predictions by the refined model though upper shifts by \n$\\sim$20\\% are observed in the case of $M_p\/M_t=1\/16$.\nAs well as in the case of equal-mass collisions,\nthe slope in $r_g$-$E\\sub{imp}$ relation is approximately given by -0.1,\nwhich indicates that compressed aggregates have the fractal dimension of \n2.5. The upper shifts in the numerical results indicate that the \ncompression requires larger impact energy in non-equal-mass collisions \nthan the equal-mass case. This effect in non-equal-mass collisions would \nbe included by adopting a larger parameter $b$ in the refined model.\nFurthermore, for high-mass-ratio collisions with $M_t\/M_p \\gg 10$,\nthe refined model is not verified through $N$-body simulations yet\nalthough such collisions have only a minor contribution in dust \ngrowth (Okuzumi et al.~2009,2012).\nIn the future work, the effect of non-equal-mass collisions\nshould be further examined in the numerical simulation of sequential \ncollisions as done by S08 in order to calibrate the parameter\n$b$ more accurately.\n\\section{SUMMARY}\nWe examined the evolution of the geometrical cross section of the\ngrowing (icy) aggregates obtained by $N$-body simulations of\nsequential head-on collisions (S08)\nand constructed to construct a refined compression model,\nwhich is applicable to the description of the evolution of both\ngeometrical cross sections and gyration radii of growing aggregates.\nThe results are summarized as follows:\n\\begin{enumerate}\n \\item We examined geometrical cross sections of the aggregates\n       produced in the simulation of sequential collisions done in our\n       previous paper. \n\tAs aggregates grow, compression becomes effective and makes their \n\tcross sections smaller than those of the BCCA clusters. The \n\tbeginning of the compression is given by equation~(\\ref{ncrit}), \n\tas seen in the evolution of the gyration radius.\n \\item The relation between the cross section and the gyration radius\n       seen in aggregates obtained by S08\n\tis well described by Okuzumi et al's expression.\n       This indicates that Okuzumi et al's expression is valid\n        for compressed aggregates as well as hit-and-stick aggregates.\n       If the evolution of the gyration radius is well described by\n       a compression model, Okuzumi et al's expression enables us to\n       calculate the cross section, too.\n \\item We further refined the compression model of S08, by including \n       the compression energy for the voids produced at the sticking \n       of two aggregates. The refined model is also extended to\n       non-equal-mass collisions in a reasonable way.\n\tWith the refined model, we can accurately reproduce the evolution \n        of both the gyration radius and the cross section of aggregates\n        obtained by S08 from their early growth stage.\n       The validity of the refined compression model\n       for non-equal-mass collisions is also checked by\n       additional numerical simulations of BCCA collisions.\n       Although S08 considered only icy aggregates in the numerical \n       simulation, our compression model would be also applicable to \n       silicate aggregates by using a suitable value of $E_\\mathrm{roll}$. \n\\end{enumerate}\n\nOur $N$-body simulations of aggregate collisions and the refined \ncompression model indicate that collisional compression is not so \neffective. As a result, dust aggregates (or initial planetesimal material) \nwould have extremely low bulk densities, as suggested by S08. Okuzumi \net al. (2012) showed that such extremely low bulk densities of aggregates \naccelerate their growth in protoplanetary disks and help their overcoming \nof the radial drift barrier against the planetesimal formation. However, \nsolar-system bodies do not have such low densities at present. Dust \naggregates (or planetesimals) should be compressed by other processes. \nA steady ram pressure due to the gas drag on aggregates and a \nself-gravity of sufficiently large aggregates would be candidates for \naggregate compression, as indicated by S08 and Okuzumi et al. (2012). \nFor relatively compact dust cakes \n($\\rho \\sim0.1$g$\/$cm$^3$) made of micron-sized silicate particles, \ncompression is observed at a pressure $> 100$Pa (Blum and \nSchr\\\"{a}pler 2004). However, for icy aggregates with very low bulk \ndensities ($\\rho \\ll 0.1$g$\/$cm$^3$), compressive strength \nhas not yet been measured. In future work, compression strength of \nvery fluffy aggregates should be measured in numerical simulations \nand laboratory experiments.\n\nIn the present study,\nwe focus on the aggregates obtained at head-on collisions. \nAt oblique collisions, the merged aggregates are elongated\n(W07; Paszun and Dominik 2009). Although our model\nindicates inefficient compression at aggregate collisions, the effect of \noblique collisions would further hinder compression.\nIn future work, we should clarify the validity of our compression model \nin the case where oblique collisions are included.\n\\acknowledgments\nThe authors would like to thank Hiroshi Kimura, \nTetsuo Yamamoto and Hiroshi Kobayashi for\ntheir valuable comments. We would also like to thank Takeshi Chigai for\ntechnical support with respect to the computer setup. This study was\nsupported by a Grant-in-Aid from JSPS(22540242, 22740299).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter[Aging rate: exploring the growth of the coherence length]{Aging rate: exploring the growth \\\\ of the coherence length} \\labch{aging_rate}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\nExperiments in \\gls{SG}s are developed in out-of-equilibrium conditions most of the times\\footnote{As we have mentioned before, recent experiments in thin-film geometry~\\cite{guchhait:14,guchhait:15b} stands as honorable exceptions.}. Typically, the experimental setup consists of a system that it is rapidly cooled from $T_1>\\ensuremath{T_\\mathrm{c}}\\xspace$ to $T_2<\\ensuremath{T_\\mathrm{c}}\\xspace$, and its off-equilibrium evolution, which we have already termed as \\textit{aging\\index{aging}} (see~\\refsubsec{aging_memory_rejuvenation}), is studied.\n\nUnder these non-equilibrium conditions, it was originally predicted in the context of the droplet\\index{droplet!picture} theory (see~\\refsubsec{theoretical_pictures}) that domains\\index{magnetic domain} of correlated spins start to grow at the microscopic level~\\cite{fisher:88}. Although with some differences in the nature of the domains\\index{magnetic domain}, \\gls{RSB} also expects a similar behavior. The linear size of those domains\\index{magnetic domain} is known as \\textit{coherence length\\index{coherence length}}.\n\nThis coherence length\\index{coherence length} have been measured in numerical simulations~\\cite{huse:91,marinari:96,komori:99,komori:00} and also in experiments~\\cite{joh:99,bernardi:01} long time ago. The initial expectation~\\cite{fisher:88} for the growth of the coherence length\\index{coherence length} with the time was $\\xi \\sim \\left(\\log \\ensuremath{t_\\mathrm{w}}\\xspace\\right)^{1\/\\psi}$ and some numerical simulations found that ansatz to be compatible with their results (see, e.g.~\\cite{kisker:96}). However, the mainstream usually accepts the alternative growth functional form described by $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$ which better describes the results.\n\nThe aging rate\\index{aging!rate} $z(T) = {\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace \/ {\\mathrm{d}} \\log \\xi$ was difficult to measure in traditional experiments based on the study of the shift of the peak in the relaxation\\index{relaxation} rate $S(\\ensuremath{t_\\mathrm{w}}\\xspace)$~\\cite{joh:99}. However, it can be now experimentally measured with excellent accuracy through the study of activation energies in \\gls{SG}s with thin-film geometry~\\cite{zhai:17}.\n\nA strong discrepancy has been found between numerical and experimental measurements of $z(T)$. We solve that discrepancy in Ref.~\\cite{janus:18} and this chapter is devoted to exposing the results of the cited reference that have been obtained in this thesis. In this chapter, we first motivate the work and describe the state of the art in~\\refsec{why_study_aging} and~\\refsec{how_can_study_aging}. Then, we describe the numerical simulation that we have performed in~\\refsec{numerical_simulation_aging}. We describe the problem in~\\refsec{controversy_aging} and finally we show the results in~\\refsec{large_xi_limit}.\n\n\\section{Why should we care about off-equilibrium dynamics?} \\labsec{why_study_aging}\nWe have stated several times that experiments and theory focus on different regimes, off-equilibrium, and equilibrium respectively. Moreover, we have also stated that, traditionally, simulations have been a powerful tool in theoretical research.\n\nThe increase of computational power has recently allowed us to promote the numerical simulations to a higher ``responsibility'' role in the development of the \\gls{SG} field. Now, our simulations are in the border of the experimental regime~\\cite{janus:08b,janus:09b,janus:17b,janus:18} and therefore, numerical work is in a privileged situation. On the one hand, it still has the classical advantages of the simulations: we are able to access microscopic configurations\\index{configuration} that are difficult to access from the experiments, and we have total control of our system which is desirable in many senses (for example, our protocols are totally reproducible with no source of error). On the other hand, our numerical data can be now confronted with the experimental one through mild extrapolations (see e.g. ~\\cite{janus:17b,janus:18,zhai-janus:20,zhai-janus:21}). This is extremely useful, not only because it allows us to test the numerical models but also because we can compute theoretical quantities, not accessible from experiments, in our experiment-compatible system at relevant time-scales.\n \nMoreover, in the last years, the development of a statics-dynamics dictionary~\\cite{barrat:01,janus:08b,janus:10,janus:10b,janus:17} has been a milestone in the development of the numerical research. According to the statics-dynamics equivalence\\index{statics-dynamics equivalence}, the off-equilibrium properties of an (effective) infinite system that ages for a finite-time $\\ensuremath{t_\\mathrm{w}}\\xspace$ with a coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, are tightly bounded with the equilibrium properties of a finite-size system with linear length $L \\sim \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. The study of out-equilibrium systems may be helpful in order to extend the statics-dynamics dictionary and establish new relations.\n\n\\section{Coherence length, a fundamental quantity to study off-equilibrium dynamics}\\labsec{how_can_study_aging}\nThe experimental study of out-equilibrium \\gls{SG}s is usually focused on the characterization of magnetic responses when an external magnetic field is applied. As we have already discussed in~\\refsec{experimental_spinglass}, both time scales, the aging\\index{aging} before turn on (off) the magnetic field and the subsequent magnetic evolution of the system, turned out to be essential to describe the off-equilibrium phenomenon.\n\nHowever, different aging\\index{aging!rate} rates $z(T)$ for different temperatures make the coherence length\\index{coherence length} $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$ a much more convenient quantity to describe the ``aging\\index{aging} state'' of the system. The meaning of the aging\\index{aging!rate} rate $z(T)$ comes directly from the Arrhenius law. As we have mentioned many times along this thesis, the \\gls{SG}s exhibit extremely slow dynamics. If we assume that the slow dynamics are due to the presence of many valleys in a rugged free-energy\\index{free energy!landscape} landscape, it is natural to propose the Arrhenius law to characterize the typical time-scale that the system needs to overcome these free-energy barriers\\index{free energy!barrier} and explore different valleys\\index{free energy!valley}.\n\\begin{equation}\n\\ensuremath{t_\\mathrm{w}}\\xspace = t(\\xi) = \\tau_0 \\exp \\left[ \\beta \\Delta(\\xi) \\right] \\, , \\labeq{arrhenius_law}\n\\end{equation}\nbeing $\\tau_0 \\propto \\beta_\\mathrm{c}$ a microscopic time-scale and the exponent $\\Delta(\\xi)$ is the size of the energetic barriers\\index{free energy!barrier} in units of $1\/\\beta=k_{\\mathrm{B}}T$. Therefore, the aging\\index{aging!rate} rate\n\\begin{equation}\nz(T,\\xi) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi} = \\dfrac{{\\mathrm{d}} \\left[ \\beta \\Delta(\\xi)\\right]}{{\\mathrm{d}} \\log \\xi} \\, , \\labeq{def_aging_rate}\n\\end{equation}\ngive us the information of the evolution of the free-energy\\index{free energy!barrier} barriers with $\\log \\xi$. Actually, as we will discuss in \\refsec{large_xi_limit}, different hypothesis about the specific form of $\\beta \\Delta(\\xi)$ will lead to very different behavior in the large-$\\xi$ limit.\n\nMoreover, as mentioned in the introduction of the chapter, experimental studies on thin-film geometry \\gls{SG}s have achieved accurate measurements of this, once elusive, quantity~\\cite{zhai:17}. Besides, it has been found that the experimental estimation of $\\xi$ and the numerical one matches~\\cite{janus:17b}.\n\nThe recent advances in the experimental determination of the coherence length\\index{coherence length} $\\xi$ have also brought a discrepancy in the estimation of the aging\\index{aging!rate} rate $z(T)$ from numerical simulations and experiments~\\cite{zhai:17}. To solve this discrepancy, it is fundamental to estimate $\\xi$ with unprecedented accuracy. Two main factors have allowed us to compute the data needed to perform this research.\n\nFirst, the dedicated hardware Janus\\index{Janus} II~\\cite{janus:14} has a central role in this work. The simulation of very large systems to very long times has been the result of thousands of computational hours with the largest special-purpose\\index{special-purpose computer} machine focused on \\gls{SG}s.\n\nSecond, our particular choice of the simulation parameters has turned out to be fortunate. The numerical effort is usually focused on increasing the number of samples\\index{sample} $\\ensuremath{N_{\\text{S}}}\\xspace$ as much as possible and simulating the minimum number of replicas\\index{replica} needed to compute the observables, typically $\\ensuremath{N_{\\text{Rep}}}\\xspace=2$ or $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$. However, we had in mind to study the \\textit{temperature chaos}\\index{temperature chaos} phenomenon (see~\\refch{out-eq_chaos}), where the determination of the error of the observables of interest is greatly benefited by a maximization number of overlaps\\index{overlap} $\\ensuremath{N_{\\text{ov}}}\\xspace = \\ensuremath{N_{\\text{Rep}}}\\xspace (\\ensuremath{N_{\\text{Rep}}}\\xspace-1)\/2$. Unexpectedly, this has led to a dramatic increase in precision. We analyze this reduction of the error in~\\refsec{Nr_aging}.\n\nThis study is a clear demonstration of the importance of the high-precision results for the investigation of glassiness. Indeed, without the dramatic reduction of the error bars\\index{error bars}, we would not be able to solve the discrepancy between numerical simulations and experiments.\n\n\\section{Numerical simulation} \\labsec{numerical_simulation_aging}\nIn this work, we simulate in the FPGA-based\\index{FPGA} [\\gls{FPGA}] computer Janus\\index{Janus} II an \\gls{EA}\\index{Edwards-Anderson!order parameter} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. This assumption is sound, provided that $L\\gg\\xi$ (see~\\refsec{finite_size_effects}). Note that this condition limits the maximum time at which we can safely ignore finite-size effects\\index{finite-size effects}. The temperature remains constant throughout the whole simulation.\n\nWe shall perform direct quenches from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds with the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample}, we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ replicas\\index{replica}. As we have already said, this simulation had the original aim to study the temperature chaos\\index{temperature chaos} phenomenon under non-equilibrium conditions (see~\\refch{out-eq_chaos}), however, the reader may notice that in that study we use a total number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$. Indeed, this study about the aging\\index{aging!rate} rate was performed much earlier and we had at our disposal ``only'' $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$.\n\nThe main observable of this study is the coherence length\\index{coherence length} $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, estimated by $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, computed from integral estimators of the correlation\\index{correlation function!four point} function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. Both observables have been described with great detail in \\refsubsec{observables_introduction}. The large number of replicas\\index{replica} simulated has allowed us to follow the decay of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ over six decades (see inset of~\\reffig{growth_xi}). The $\\xi$ estimation used in our work is plotted in~\\reffig{growth_xi}.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.8\\linewidth]{aging\/xi12_and_C4}\n\\caption[\\textbf{Growth of the coherence length \\boldmath $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Growth of the coherence length\\index{coherence length} \\boldmath $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Growth of the coherence length\\index{coherence length} $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with the waiting time $\\ensuremath{t_\\mathrm{w}}\\xspace$ after a quench to temperature $T$ in a log-log  scale [the critical\\index{critical temperature} temperature is $\\ensuremath{T_\\mathrm{c}}\\xspace=1.102(3)$]. Given the smallness of the statistical errors, instead of error bars\\index{error bars} we have plotted two lines for each $T$, which enclose the error estimate.  At this scale, the curves seem linear for long times, indicating a power-law growth but, see~\\reffig{a2}, there is actually a measurable curvature. \\textbf{Inset:} Spatial four-point correlation\\index{correlation function!four point} function of the overlap\\index{overlap!field} field $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$, plotted as a function of distance at the last simulated time for several temperatures. Note the six orders of magnitude in the vertical axis.}\n\\labfig{growth_xi}\n\\end{figure}\n\n\\section{The controversy of the aging rate}\\labsec{controversy_aging}\nThe growth of the coherence length\\index{coherence length} has been a debated issue in the \\gls{SG} literature (see, e.g. \\cite{fisher:88,huse:91,marinari:96,joh:99,bernardi:01,bouchaud:01,berthier:02}). However, despite the existence of different proposals, the simplest functional form that was able to fit the data was the power law\n\\begin{equation}\n\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace,T) = A(T) \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)} \\quad , \\quad z(T) \\approx z(\\ensuremath{T_\\mathrm{c}}\\xspace)\\dfrac{\\ensuremath{T_\\mathrm{c}}\\xspace}{T} \\, , \\labeq{xi_powerlaw}\n\\end{equation}\nbeing $z(T)={\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace \/ {\\mathrm{d}} \\log \\xi$ the so-called aging\\index{aging!rate} rate. Indeed, the experimental measurements concerns to the renormalized aging\\index{aging!rate} rate\n\\begin{equation}\nz_c = z(T) \\dfrac{T}{\\ensuremath{T_\\mathrm{c}}\\xspace} \\, . \\labeq{renormalized_aging_rate}\n\\end{equation}\nExperiments performed in thin-film geometry systems~\\cite{zhai:17} has measured $z_c \\approx 9.62$, which is very far from the value predicted by numerical simulations $z_c = 6.86(16)$~\\cite{janus:08b} and $z_c=6.80(15)$~\\cite{lulli:16}. \n\nThose experiments are performed in \\gls{CuMn} films with $20$ nm of thickness which translates to a distance of 38 lattice spacings (the typical Mn-Mn distance is 5.3 \\r{A}). Therefore, we will need to extrapolate our results to $\\xi_{12} \\approx 38$ in order to confront the numerical simulations and the experiments.\n\n\\subsection[The growth of $\\xi$ does not follow a power law]{The growth of \\boldmath $\\xi$ does not follow a power law} \\labsubsec{no-powerlaw}\nThe increase of the precision of the data shows that the pure power law of~\\refeq{xi_powerlaw} is not a faithful description anymore. Indeed, in order to discern if our data of~\\reffig{growth_xi} presents a deviation from a power law, we propose a very naive ansatz\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace(T,\\xi_{12}) = a_0(T) + a_1(T)\\log \\xi_{12} + a_2(T) \\log^2 \\xi_{12}  \\, , \\labeq{naive_divergent}\n\\end{equation}\nwhere $a_0(T)$, $a_1(T)$ and $a_2(T)$ are meaningless coefficients, only useful to interpolate our data. An absence of curvature [$a_2(T)=0$] would reduce~\\refeq{naive_divergent} to~\\refeq{xi_powerlaw}. On the contrary, $a_2(T)>0$ indicates a slowing down in the dynamics for increasing $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\n\\reffig{a2} is telling us that $a_2 \\geq 0$ and only vanishes for $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ with $z_c=6.69(6)$ which improves the accuracy of previous estimations. Therefore, the solution of the discrepancy of the results of $z_c$ seems to be the introduction of a (very mild) scale dependence in the \\textit{effective} dynamical exponent which is defined as\n\\begin{equation}\nz(T,\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} \\, . \\labeq{definition_aging_rate}\n\\end{equation}\n\nThe reader may think about the possibility that our deviation from the power-law behavior might be due to the existence of finite-size effects\\index{finite-size effects}, however, two main reasons against this argument can be defended. First, the curvature decreases when increasing the temperature (see~\\reffig{a2}) and one would expect the opposite behavior in presence of finite-size effects\\index{finite-size effects}\\footnote{Actually, finite-size effects\\index{finite-size effects} would be controlled by $\\xi\/L$ which is smaller for the lower temperatures.}. Second, exhaustive checks have been done in order to establish our system size $L=160$ as a safe choice, see~\\refsec{finite_size_effects}.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/a2_vs_T}\n\\caption[\\textbf{Deviation of \\boldmath $\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ from a simple power-law growth.}]{\\textbf{Deviation of \\boldmath $\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ from a simple power-law growth.} We plot the quadratic parameter $a_2$ in a fit to~\\refeq{naive_divergent}. This quantity is zero at the critical point, but  has a positive value at low temperatures, indicating that the growth  of $\\xi_{12}$ slows down over the simulated time range.}\n\\labfig{a2}\n\\end{figure}\n\nOf course, our naive ansatz of~\\refeq{naive_divergent} is only useful for interpolations. If we want to explore the growth of $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ in the large-$\\xi_{12}$ limit, we need some insight from theory.\n\n\\section[The large-$\\xi$ limit]{The \\boldmath large-$\\xi$ limit} \\labsec{large_xi_limit}\nIn this section, we explore the growth of $\\xi$ in the large-$\\xi$ limit. For that purpose, we need to propose extrapolations from our numerical data. We introduce and study two functional forms for the function $\\log \\ensuremath{t_\\mathrm{w}}\\xspace$ that will allow us to extrapolate $z(T,\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace))$ in~\\refsubsec{ansatzs_aging}. However, before extrapolating our data, we need to compute the exponent $\\vartheta$ appearing in the long-distance decay of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ [see \\refeq{long_distance_C4}] because it is required by one of our ans\\\"atze. We recall the relation between $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ and $\\vartheta$ for the reader's convenience and compute the exponent $\\vartheta$ in \\refsubsec{vartheta_aging}. Finally, we extrapolate our results and confront them with the experimental ones in~\\refsubsec{extrapolations_aging}.\n\\subsection[Computing $\\vartheta$]{Computing \\boldmath $\\vartheta$} \\labsubsec{vartheta_aging}\nThe correlation\\index{correlation function!four point} function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ presents the following long-distance decay\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) = r^{-\\vartheta} f(r\/\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)) \\, , \\labeq{recall_long_distance_C4}\n\\end{equation}\nwhere $f(x)$ is an unknown function which vanishes faster than exponentially. As we have already introduced in~\\refsubsec{observables_introduction}, the exponent $\\vartheta$ at $\\ensuremath{T_\\mathrm{c}}\\xspace$ is given by $\\vartheta = 1 + \\eta$ where $\\eta=-0.390(4)$~\\cite{janus:13} is the anomalous dimension. For $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ the two main pictures, namely droplets\\index{droplet!picture} and \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, have differing expectations: coarsening\\index{coarsening} domains\\index{magnetic domain!compact} with $\\vartheta=0$ is the \\index{droplet!picture}droplets' prediction while in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory, $\\vartheta$ is given by the replicon\\index{replicon} (see~\\cite{janus:10b} for a detailed discussion). The best previous numerical study of $\\vartheta$ found $\\vartheta=0.38(2)$~\\cite{janus:09b}.\n\nIf we recall the integral $I_k= \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r$ [see~\\refeq{integral_estimator_xi}] it is easy to prove that $I_2(T,\\xi_{12}) \\propto \\xi_{12}^{3-\\vartheta}$ and, therefore, we can estimate the value of $\\vartheta$ from the numerical derivative of $I_2$\n\\begin{equation}\n\\vartheta=3-\\dfrac{{\\mathrm{d}} \\log I_2}{{\\mathrm{d}} \\log \\xi_{12}} \\, .\n\\end{equation}\n\nOur estimations of $\\vartheta$ show that, for $T=\\ensuremath{T_\\mathrm{c}}\\xspace$, its value is compatible with $1+\\eta$, as expected. However, for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ we found a slow decrease as $\\xi$ increases or $T$ decreases, i.e. $\\vartheta \\to \\vartheta(T,\\xi_{12})$.\n\nThis result is unsatisfactory because we expect $\\vartheta$ to be constant (possibly 0) in the large-$\\xi$ limit. For the sake of clarity, we will call that theoretically expected value $\\vartheta_{\\infty}$. The simplest explanation is that the values of $\\vartheta(T,\\xi_{12})$ are affected by critical effects of the fixed point at $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ with $\\vartheta(\\ensuremath{T_\\mathrm{c}}\\xspace,\\xi_{12}) \\approx 0.61$. For large $\\xi$, we expect those critical effects to vanish and the value of $\\vartheta(T,\\xi_{12}\\to \\infty)$ should be dominated by the $T=0$ fixed point, i.e. $\\vartheta(T,\\xi_{12}\\to \\infty) = \\vartheta_{\\infty}$.\n\nIn analogy with the ferromagnetic phase\\index{phase!ferromagnetic} of the $O(N)$ model, we study this crossover from $\\vartheta(T=\\ensuremath{T_\\mathrm{c}}\\xspace)$ to the unknown $\\vartheta(T=0)$ in terms of the Josephson length $\\ell_J$~\\cite{josephson:66}. The Josephson length is expected to behave as $\\ell_J \\sim (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu}$ with $\\nu=2.56(4)$~\\cite{janus:13} for temperatures $T \\to \\ensuremath{T_\\mathrm{c}}\\xspace$. The scaling corrections for $\\ell_J(T)$ at lower temperatures are explained in~\\refsec{Josephson_length}.\n\nWe can test the crossover hypothesis by considering the ratio of two different estimations of $\\xi$: $\\xi_{23}\/\\xi_{12}$. We plot this ratio against the scaling variable $x=\\ell_J\/\\xi_{12}$. This ratio should be scale-invariant in the large-$\\xi_{12}$ limit because different determinations of $\\xi$ should grow with the same rate but with a different prefactor (see~\\cite{janus:09b} and~\\refsubsec{observables_introduction}).\n\n\\reffig{testing_crossover} shows us that the ratio grows towards the critical value (represented with a gray line) for the curves with the largest temperatures when $x$ is large, i.e. for a given curve, which is $T$-constant, when $\\xi_{12}$ is small. Then, it relaxes to the value corresponding to the $T=0$ fixed point (small $x$). The lowest temperatures, namely $T=0.55$, $T=0.625$ and $T=0.7$, seem to be free of critical effects.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/josephson_crossover_inset}\n\\caption[\\textbf{Testing the crossover hypothesis with \\boldmath $\\xi_{23}\/\\xi_{12}$.}]{\\textbf{Testing the crossover hypothesis with \\boldmath $\\xi_{23}\/\\xi_{12}$.} We consider the ratio $\\xi_{23}\/\\xi_{12}$ between two definitions of the coherence length\\index{coherence length}, which should be constant in the large-$\\xi_{12}$ (or $x\\to0$) limit. For $T$ close to $\\ensuremath{T_\\mathrm{c}}\\xspace$, this ratio initially grows, approaching the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ value (represented by the thick gray line) and eventually relaxes towards the $T=0$ fixed point.}\n\\labfig{testing_crossover}\n\\end{figure}\n\nThis positive result encourages us to perform a similar analysis for $\\vartheta(T,\\xi_{12})$. If the hypothesis of the crossover is correct, we should observe a collapse of the $\\vartheta(T,\\xi_{12})$ values when we plot them in terms of the scaling variable $x=\\ell_J\/\\xi_{12}$. \n\nThe functional form of $\\vartheta(T,\\xi_{12})$ should be, in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture\n\\begin{equation}\n\\vartheta(x) = \\vartheta_{\\infty} + b_2 x^{2-\\vartheta_{\\infty}} + b_3 x^{3-\\vartheta_{\\infty}} + \\cdots \\, . \\labeq{vartheta_RSB}\n\\end{equation}\nThe reader may find a derivation of this expression in~\\refsec{Josephson_length}. On the contrary, for the droplets\\index{droplet!picture} picture, it should be \n\\begin{equation}\n\\vartheta(x)=C x^\\zeta \\, , \\labeq{vartheta_droplet}\n\\end{equation}\nwhere $\\zeta\\approx 0.24$~\\cite{boettcher:04} is the stiffness coefficient\\index{stiffness!coefficient}.\n\n\\reffig{vartheta_collapse} shows us a nice collapse of the $\\vartheta(x)$ values. Moreover, we have to keep in mind that our goal is to extrapolate the values of the aging\\index{aging!rate} rate to the experimental $\\xi$ length-scale which roughly corresponds to $\\xi_{\\mathrm{films}}\\approx 38$. \n\nFor the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, a fit of the $\\vartheta(x)$ values to~\\refeq{vartheta_RSB} gives us the value $\\vartheta_{\\infty} \\approx 0.30$, although we take $\\vartheta^{\\mathrm{upper}}=0.35$ as our upper bound for $\\vartheta_{\\infty}$. For the droplets\\index{droplet!picture} picture, a fit to~\\refeq{vartheta_droplet} can be performed only by considering\\footnote{Similar results were found in~\\cite{janus:10}} $\\zeta \\approx 0.15$. It is worthy to note that we find this exponent very sensitive to the fitting range. We extrapolate the droplet\\index{droplet!picture} $\\vartheta(x)$ to $\\xi = \\xi_{\\mathrm{films}}=38$ and we obtain $\\vartheta(\\xi_{\\mathrm{films}}=38) \\approx 0.28$.\n\nHowever, because our determination of $\\xi$ through the estimator $\\xi_{12}$ may differ from the experimental estimation of $\\xi$ by a small constant factor, we consider also $\\xi_{\\mathrm{films}}=76$ and we obtain $\\vartheta(\\xi_{\\mathrm{films}}=76) \\approx 0.25$.\n\nThe same conclusions found in~\\cite{janus:10,janus:10b} stands here: for the experimental relevant scales, the physics is well described by a non-coarsening\\index{coarsening} picture with $0.25<\\vartheta(\\xi_{\\mathrm{films}})<0.35$ depending on the theory we use to extrapolate the data and the exact value chosen for the experimental scale $\\xi_{\\mathrm{films}}$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/josephson_crossover_main}\n\\caption[\\textbf{Crossover between the \\boldmath $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the $T=0$ fixed points controlled by a Josephson length $\\ell_\\text{J}(T)$.}]{\\textbf{Crossover between the \\boldmath $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the $T=0$ fixed points controlled by a Josephson length $\\ell_\\text{J}(T)$.} We plot the evolution of the replicon\\index{replicon} exponent $\\vartheta$ for several temperatures against the relevant scaling variable $x=\\ell_\\text{J}(T)\/\\xi_{12}$. We show two possible extrapolations (dashed lines) to infinite $\\xi_{12}$: one with finite $\\vartheta$, as expected in the RSB picture, and one with $\\vartheta=0$, as expected in the droplet\\index{droplet!picture} picture. For the latter, we also show the extrapolated value for the experimental scale corresponding to experiments in \\gls{CuMn} films~\\cite{zhai:17}, which we estimate between $\\xi_{12}=38$ and $\\xi_{12}=76$.}\n\\labfig{vartheta_collapse}\n\\end{figure}\n\n\\subsection{The convergent and the divergent ansatz} \\labsubsec{ansatzs_aging}\nNow, we discuss the possible extrapolations of $z(T,\\xi_{12})$ to the large-$\\xi_{12}$ limit. The most natural assumption is to consider that $z(T,\\xi_{12} \\to \\infty) = z_{\\infty}(T)$ with a convergence $z(T,\\xi_{12}) - z_{\\infty}(T) \\propto \\xi_{12}^{-\\omega}$. Taking into account the~\\refeq{definition_aging_rate}, the expresion of $\\ensuremath{t_\\mathrm{w}}\\xspace$ should be\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace = C_1(T) + z_{\\infty}(T) \\log \\xi_{12} + C_2(T)\\xi_{12}^{-\\omega} \\, , \\labeq{convergent_ansatz}\n\\end{equation}\nwhere $\\omega$ is the exponent that controls finite-$\\xi_{12}$ corrections. The value of $\\omega$ for the critical\\index{critical temperature} temperature $T_c$ is $\\omega=1.12(10)$~\\cite{janus:13,fernandez:15,lulli:16}. In the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, the leading behavior is given by $\\omega=\\vartheta$, see \\cite{janus:10b} for a detailed discussion.\n\nThe effective exponent in the \\textit{convergent} ansatz would be, therefore\n\\begin{equation}\nz_{\\mathrm{conv}}(T,\\xi_{12}) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} = z_{\\infty}(T) - \\omega C_2(T) \\xi_{12}^{-\\omega} \\, . \n\\end{equation}\n\nThe fits to~\\refeq{convergent_ansatz} have two main sources of error. First, the value of $\\vartheta$ has associated some uncertainty. We choose $\\vartheta=0.35$, as explained above. Second, we consider possible systematic effects due to the fitting range. We follow objective criteria to select a minimum $\\xi_{12}^{\\min}$ for the fitting range. Further details can be found in~\\refsec{parameters_aging}.\n\nAn alternative approach was proposed in~\\cite{bouchaud:01,berthier:02}. In those works, the authors proposed a crossover to activated dynamics. This approach is a refinement of the droplet\\index{droplet!picture} proposal that we expose at the beginning of the chapter [$\\xi \\sim (\\log \\ensuremath{t_\\mathrm{w}}\\xspace)^{1\/\\psi}$]. In this case, the authors propose\n\\begin{equation}\n\\ensuremath{t_\\mathrm{w}}\\xspace = \\tau_0 \\xi^{z_c} \\exp \\left(\\dfrac{\\Upsilon(T) \\xi^{\\psi}}{k_{B}T} \\right) \\, , \\labeq{original_divergent}\n\\end{equation}\nbeing $\\Upsilon(T) = \\Upsilon_0(1-T\/T_c)^{\\psi\\nu}$ and $k_{B}$ the Boltzmann\\index{Boltzmann!constant} constant. Here, we express the original ansatz in logarithmic form with generic coefficients\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace = D_1(T) + z_c \\log \\xi_{12} + D_2(T) \\xi_{12}^{\\psi} \\, . \\labeq{divergent_ansatz}\n\\end{equation}\nThe exponent $\\psi$ has been used before in experiments~\\cite{schins:93} and numerical simulations~\\cite{rieger:93} with values $\\psi \\approx 1$. Moreover, the reader may noticed that $D_2(T) \\sim (T_c-T)^{\\psi\\nu}$ [see~\\refeq{original_divergent}]. This can be regarded as another way to present the crossover between the $T=T_c$ fixed point and the $T=0$ fixed point. We need $\\xi_{12}(T_c-T)^{\\nu} \\gg 1$ in order to perceive deviations from the pure power-law with an aging\\index{aging!rate} rate $z_c$ equal to the critical one.\n\nThe effective exponent in the \\textit{divergent} ansatz would be, therfore\n\\begin{equation}\nz_{\\mathrm{div}}(T,\\xi_{12})=\\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} =z_c + D_2(T) \\psi \\xi_{12}^{\\psi} \\, .\n\\end{equation}\n\nWe find fair fits to our simulated data for both ans\\\"atze,~\\refeq{convergent_ansatz} and~\\refeq{divergent_ansatz}. Indeed, for~\\refeq{divergent_ansatz} we find $\\psi \\approx 0.4$. The next step is clear, we need to test both proposals at the experimental length-scales.\n\n\\subsection{Extrapolation to experimental regime} \\labsubsec{extrapolations_aging}\nWe perform extrapolations of the renormalized aging\\index{aging!rate} rate $z_c(T,\\xi) = z(T,\\xi) T\/T_c$ at the experimental length-scale $\\xi_{\\mathrm{films}}$. The value of $z(T,\\xi)$ is computed for both $z_{\\mathrm{conv}}(T,\\xi_{\\mathrm{films}})$ and $z_{\\mathrm{div}}(T,\\xi_{\\mathrm{films}})$.\n\nIn a similar way as we did in the previous analysis of~\\refsubsec{vartheta_aging}, we use safe estimations of $\\xi_{\\mathrm{films}}$ and we consider $\\xi_{\\mathrm{films}}=38$ and $\\xi_{\\mathrm{films}}=76$. All the relevant data from the fits can be found in~\\refsec{parameters_aging}, however, we plot in~\\reffig{zc} the relevant information.\n\nThe main plot shows the renormalized aging\\index{aging!rate} rate $z_c(T,\\xi_{\\mathrm{films}})$ plotted against the reduced temperature $T\/\\ensuremath{T_\\mathrm{c}}\\xspace$. We see that the convergent ansatz of~\\refeq{convergent_ansatz} is very successful in reproducing the experimental behavior for both $\\xi_{12}=38$ and $\\xi_{12}=76$.  Compatible results with experiments are found for a wide range of temperatures.\n\nThe inset shows the divergent ansatz of~\\refeq{divergent_ansatz}. In this case this proposal for $z_c(T,\\xi_{\\mathrm{films}})$ is not able to reproduce the constant value found in experiments for $T\/T_c < 0.8$.\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/zc_vs_T_inset}\n\\caption[\\textbf{Numerical and experimental aging rate.}]{\\textbf{Numerical and experimental aging rate.} Value of the experimental aging\\index{aging!rate} rate for \\gls{SG}s $z_c(T,\\xi_{\\mathrm{films}}) = z(T,\\xi_\\text{films}) T\/\\ensuremath{T_\\mathrm{c}}\\xspace$, extrapolated from our data for values of the coherence length\\index{coherence length} corresponding to thin  \\gls{CuMn} films. The main plot considers an ansatz~\\refeq{convergent_ansatz} with a finite $z_{\\infty}(T)T\/\\ensuremath{T_\\mathrm{c}}\\xspace$, which agrees very well with the experimental value of $z_c(T)\\approx 9.62$~\\cite{zhai:17}, indicated by the straight line, whose width represents the experimental temperature range. Notice that critical effects are only visible for $T>0.7$. \\textbf{Inset:} Same plot but now considering a crossover to activated dynamics~\\refeq{divergent_ansatz}, as in~\\cite{bouchaud:01}. This is less successful at reproducing the roughly constant $z_c(T)$ observed in experiments.}\n\\labfig{zc}\n\\end{figure}\n\n\n\\section*{Agradecimientos}\n\n\n\\section*{Agradecimientos}\n\nEsta tesis ha sido posible gracias a la ayuda de muchas personas que han puesto su granito de arena para que yo pudiera avanzar en la consecuci\u00f3n de los objetivos (no s\u00f3lo acad\u00e9micos) que me he impuesto a lo largo de los cuatro \u00faltimos a\u00f1os.\n\nA pesar de que mi tesis se ha desarrollado principalmente en la Universidad de Zaragoza, he tenido la suerte de poder colaborar estrechamente con la Universidad Complutense de Madrid. Quiero agradecer en primer lugar a mis directores de tesis, David \u00cd\u00f1iguez Dieste de la Universidad de Zaragoza y V\u00edctor Mart\u00edn Mayor de la Universidad Complutense de Madrid, su ayuda y dedicaci\u00f3n a lo largo de estos cuatro a\u00f1os.\n\nDavid \u00cd\u00f1iguez me recibi\u00f3 con los brazos abiertos en mi primera toma de contacto con Zaragoza y siempre me ha dado todas las facilidades del mundo, tanto al principio para establecerme en mi puesto de trabajo y en la ciudad (nueva para m\u00ed) como a lo largo de toda la tesis donde sab\u00eda que cruzando el pasillo y llamando a su puerta siempre lo encontrar\u00eda dispuesto a echar una mano.\n\nA V\u00edctor Mart\u00edn Mayor lo conoc\u00ed en la realizaci\u00f3n del m\u00e1ster de F\u00edsica Te\u00f3rica en la Universidad Complutense de Madrid. Con V\u00edctor ha sido con la persona que m\u00e1s he aprendido a lo largo de esta tesis, ya fuera en sesudas sesiones de trabajo en su despacho, por videoconferencias o escap\u00e1ndonos en alg\u00fan hueco entre charlas en una conferencia en Argentina para hacer una reuni\u00f3n improvisada. Su labor como director ha ido mucho m\u00e1s all\u00e1 de sus obligaciones, y su dedicaci\u00f3n y esfuerzo ha sido todav\u00eda mayor teniendo en cuenta que la mayor\u00eda de veces tuvimos que trabajar a distancia.\n\nQuiero agradecer tambi\u00e9n toda la ayuda que me han prestado a Alfonso Taranc\u00f3n y a Luis Antonio Fern\u00e1ndez. Alfonso Taranc\u00f3n oficialmente figura como mi ``Tutor de Tesis'', un t\u00edtulo que no hace justicia a toda la ayuda que he recibido de su parte. Al igual que me ocurr\u00eda con David, sab\u00eda que cruzando el pasillo y llamando a su puerta encontrar\u00eda su ayuda para cualquier problema que se me presentase. Fue la persona que me introdujo a la docencia y creo que eso no se olvida. Podr\u00eda decirse que ha sido un director no oficial de mi tesis. Y si Alfonso Taranc\u00f3n ha sido mi director de tesis no oficial en Zaragoza, sin duda, Luis Antonio ha sido mi director de tesis no oficial en Madrid. Desde luego, Luis Antonio ha tenido la dedicaci\u00f3n de un director tesis sin tener ninguna obligaci\u00f3n a ello. Siempre ha estado dispuesto a ayudarme con cualquier cosa y recuerdo con especial cari\u00f1o las largas tardes delante de su ordenador, explic\u00e1ndome las tripas de alg\u00fan programa que me servir\u00eda de base para desarrollar uno nuevo, en las que yo muchas veces sal\u00eda asustado con un mont\u00f3n de notas y con las cosas no del todo claras, pero que m\u00e1s tarde comenzaban a cobrar sentido cuando me met\u00eda en harina y descubr\u00eda que esa extra\u00f1a l\u00ednea de c\u00f3digo era uno de los checks exhaustivos que ahora son tambi\u00e9n imprescindibles en todos mis programas.\n\nA mis directores de tesis y tambi\u00e9n a Alfonso y a Luis Antonio a quienes considero mis ``directores no oficiales'', muchas gracias. Sin vuestra ayuda esta tesis no habr\u00eda sido posible.\n\nTambi\u00e9n me gustar\u00eda mencionar en estos agradecimientos a la gente con la que realic\u00e9 mis estancias de investigaci\u00f3n fuera de Espa\u00f1a. Mi primera estancia de investigaci\u00f3n la realic\u00e9 en Roma, en la Universidad La Sapienza, donde fui recibido por Enzo Marinari y Andrea Maiorano. En el trabajo del d\u00eda a d\u00eda con quien m\u00e1s tiempo pas\u00e9 fue con Andrea, que en aquella \u00e9poca estaba prepar\u00e1ndose unas oposiciones y a\u00fan as\u00ed sacaba tiempo de donde no lo hab\u00eda para echarme una mano e interesarse por mi trabajo. A Enzo, Federico, Giorgio, y especialmente a Andrea, me gustar\u00eda agradecerles su c\u00e1lida acogida durante mi estancia en Roma. \n\nMi segunda estancia de investigaci\u00f3n la realic\u00e9 en Paris, en la Universidad de Paris Sud. Debido a la pandemia las condiciones fueron mucho peores por razones obvias, sin embargo agradezco enormemente a Cyril y a Aur\u00e9lien tanto su ayuda (y m\u00e1s teniendo en cuenta la situaci\u00f3n tan delicada que hab\u00eda) como la posibilidad que me brindaron de comenzar a investigar en algo tan apasionante como el Machine Learning.\n\nTambi\u00e9n quiero agradecer a todos los investigadores de los cuales he aprendido mucho en estos cuatro a\u00f1os. Especialmente me gustar\u00eda mencionar a Juan Jes\u00fas Ruiz que me introdujo en el mundo de los espines en mi TFG, que siempre me prest\u00f3 su ayuda desde Badajoz y que incluso organiz\u00f3 un curso intensivo sobre el grupo de renormalizaci\u00f3n con el \u00fanico objetivo de ayudarnos en nuestra tesis, sacrificando\nmuchas horas sin tener ninguna obligaci\u00f3n a ello. Tambi\u00e9n a David Yllanes, que siempre est\u00e1 al pie del ca\u00f1\u00f3n, y al que le tengo en gran estima. A Sergio P\u00e9rez Gaviro, que me ayud\u00f3 much\u00edsimo en mi primera toma de contacto en Zaragoza ense\u00f1\u00e1ndome todos los detalles t\u00e9cnicos, d\u00e1ndome programas para ayudarme en mi tarea hasta abrumarme y bajando conmigo al CPD para verle las tripas a los Janus. Y tambi\u00e9n a Ilaria e Isidoro, con los que he compartido director de tesis y a los que espero que les vaya realmente bien en su carrera investigadora y en su vida.\n\nPor supuesto, no puedo olvidarme de toda la gente de Zaragoza con la que he trabajado estos \u00faltimos cuatro a\u00f1os. Isabel Vidal me ayud\u00f3 con todo el tema administrativo. Nunca he visto a una persona hacer su trabajo tan bien como lo hac\u00eda ella, con tanto cuidado, cari\u00f1o y siempre con una sonrisa. El grupo de Yamir Moreno, que me acogi\u00f3 desde el primer d\u00eda con los brazos abiertos y con los que he vivido grandes momentos trabajando juntos. Toda la gente del BIFI, especialmente Pedro, Daniel, Sergio y Rub\u00e9n que me ayudaron cada vez que tuve alg\u00fan problema con \\textit{Cierzo}.\n\nDejo para la parte final a aquellos a los que conoc\u00ed mucho antes de iniciar el doctorado y que s\u00e9 que seguir\u00e1n formando parte de mi vida mucho despu\u00e9s de completarlo. A mis amigos Kike, \u00c1lvaro, Pilar, Carlos, Helena, Cienfu, Gordillo, Kubicki, Luiso y V\u00e1zquez, muchas gracias por estar ah\u00ed estos a\u00f1os y los que vendr\u00e1n. Por supuesto, muchas gracias a mi familia, a Tito y a Ariadna. Tambi\u00e9n a Toni, Candela, Lolo y Ana. Sin vuestra ayuda, que va mucho m\u00e1s all\u00e1 de la tesis, nada de esto hubiera salido adelante.\n\nNo me olvido de Sarita, con la que llevo media vida y con la que espero que me queden muchos a\u00f1os por delante. La persona que ha sido mi apoyo emocional durante toda esta tesis y a la que le debo tanto que me sabe a poco dedicarle solo estas pocas l\u00edneas. Sin ti, no s\u00f3lo no hubiera sido posible esta tesis, tampoco muchas otras cosas en mi vida. Te quiero. A la Juani y al Ram\u00f3n, los perros m\u00e1s fant\u00e1sticos del mundo. Que comparten habitaci\u00f3n conmigo (son perros investigadores) y cuya alegr\u00eda al verme siempre me levanta el \u00e1nimo (te echar\u00e9 much\u00edsimo de menos Juanita). Y por \u00faltimo a mam\u00e1, que lo ha hecho todo por m\u00ed. Probablemente la persona con la que m\u00e1s haya discutido, y tambi\u00e9n la \u00fanica persona que me ha acompa\u00f1ado en cada paso que he dado y que no me ha dejado caerme nunca. Sin ella no es que no hubiese sido posible esta tesis, es que no hubiera podido hacer nada en esta vida. Much\u00edsimas gracias mam\u00e1, te quiero.\n\nEstos agradecimientos pretend\u00edan ser unas pocas l\u00edneas pero sois muchos los que me hab\u00e9is ayudado y me siento afortunado por teneros. A todas las personas que he nombrado, a todas aquellas que mi terrible memoria no me haya permitido incluir y a mi yaya Paula, a la que le dedico esta tesis, muchas gracias. Hab\u00e9is dejado huella en m\u00ed y hab\u00e9is contribuido a que esta tesis llegase a buen puerto.\n\nMuchas gracias a todos.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Multispin Coding} \\labch{AP_multispin_coding}\n\\setlength\\epigraphwidth{.7\\textwidth}\n\\epigraph{\\textit{Premature optimization is the root of all evil.}}{-- Donald Knuth, \\textit{The Art of Computer Programming} }\n\nParallel computing\\index{parallel!computing} has turned out to be essential in scientific computing to achieve the computational power that we enjoy nowadays. The tasks that usually tackle the computational hardware do not need a completely sequential work-flow. Instead, many computational tasks can be cut up into independent smaller tasks that can be performed at the same time.\n\nThe evolution of the hardware through the years has reflected this fact. From the most general-purpose hardware, the \\gls{CPU}, to the (originally) game-oriented hardware, the \\gls{GPU}, the mainstream-design tends to increase the independent cores with great benefits for the parallel\\index{parallel!computation} computation. Of course, there exists a lot of hardware capable of performing parallel tasks. In addition to the above-mentioned \\gls{CPU} and \\gls{GPU}, the \\gls{FPGA}\\index{FPGA} stands as a great example of parallel hardware.\n\nIn this appendix, we focus on the \\gls{CPU} parallel\\index{parallel!computation} computation. Specifically, we will focus on the streaming extensions proposed for the first time by Intel with the MultiMedia eXtension (MMX)~\\cite{yu:97} and that was subsequently improved until the most recent iteration of this technology, the Advanced Vector Extensions (AVX, AVX-2, and AVX-512)~\\cite{intel_avx}. This technology allows one-clock-cycle boolean\\index{boolean} operations of registers of 128 or 256 bits, i.e. we can perform 128 or 256 boolean\\index{boolean} operations at the same time in one cycle of the \\gls{CPU}'s clock.\n\nWe first introduce the Multispin Coding\\index{Multispin Coding} as a general concept in~\\refsec{what_is_multispin_coding} and then we explain with great detail one implementation that takes advantage of the \\gls{CPU} streaming extensions AVX in \\refsec{overlap_spheres_multispin_coding}. Last, we explain how we use Multispin Coding\\index{Multispin Coding} for more general and complex simulation programs in \\refsec{musa_musi_multispin_coding}.\n\n\\section{What is the Multispin Coding?} \\labsec{what_is_multispin_coding}\nThe \\gls{MSC}\\index{Multispin Coding}~\\cite{friedberg:70,jacobs:81} is a method that is born out of the necessity of performing simulations with limited computational resources. The basic idea is that, for Ising\\index{Ising} spins, we are wasting a lot of memory and computational resources if we encode each spin as an integer number. Indeed, in a 32-bit processor\\footnote{Actually, nowadays the standard of 64-bit processors is imposing.} we can encode a variable with $2^{32}$ possible values, and an Ising\\index{Ising} spin needs only $2$ values to be encoded.\n\nFor binary variables, the solution is clear. By using an integer, we can store $32$ spins at the same time. However, our problem is not the memory but the performance. Our main task is not to be more efficient by storing data, but by running our algorithm. Here is where the streaming extensions (specifically we will focus on the family AVX) take center stage.\n\nThe new Intel and AMD processors are able to execute the AVX set of instructions, which allows us to perform one-clock-cycle instructions for registers of $128$ or $256$ bits. We are interested in perform boolean\\index{boolean} operations like the AND boolean\\index{boolean} operation \\\\\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n0 & 0 & 1 & $\\cdots$ & 1 \\\\\n\\hline\n\\end{tabular} $\\&$ \\begin{tabular}{|c|c|c|c|c|}\n\\hline\n1 & 0 & 1 & $\\cdots$ & 0 \\\\\n\\hline\n\\end{tabular} $=$ \\begin{tabular}{|c|c|c|c|c|}\n\\hline\n0 & 0 & 1 & $\\cdots$ & 0\\\\\n\\hline\n\\end{tabular} ,\n\\end{center}\nbut also in performing rotations of the bits in our registers\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \n\\hline\n$s_0$ & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$ & $s_6$ & $s_7$ \\\\\n\\hline\n\\end{tabular} $\\to$ \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$s_7$ & $s_0$ & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$ & $s_6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nAll the operations available for each one of the instructions sets can be found on the official page of Intel (\\href{https:\/\/software.intel.com\/sites\/landingpage\/IntrinsicsGuide\/}{Intel Intrinsics Guide}).\n\nNow, we know how to efficiently compute a large number of boolean\\index{boolean} operations but, how can we use this in a real situation? The next two sections are devoted to explain specific implementations in programs developed during this thesis. All the programs showed here have been coded in the programming language C.\n\n\\section{An easy example: computing overlaps inside spheres} \\labsec{overlap_spheres_multispin_coding}\nWe start with a simple implementation of \\gls{MSC}\\index{Multispin Coding}. The program that we describe here was coded to compute the chaotic parameter of a given sphere (see~\\refch{out-eq_chaos}). We will briefly describe the program and focus on the implementation of the \\gls{MSC}.\n\nIn the program, we receive as an input two $L=160$ three-dimensional cubic lattices for each of the $512$ replicas\\index{replica}, where the nodes correspond to the spins and the edges to the couplings\\index{couplings}. One of the lattices has been simulated with a thermal reservoir at temperature $T_1$ and the other one with $T_2$, both at a time $\\ensuremath{t_\\mathrm{w}}\\xspace$ such that the coherence length\\index{coherence length} of both systems $\\xi_1=\\xi_2=\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. \n\nOnce we have stored the lattices, we select (randomly) 8000 centers in the dual lattice\\footnote{The dual lattice of a cubic lattice with \\gls{PBC}\\index{boundary conditions!periodic} is another cubic lattice of the same size, and with \\gls{PBC}\\index{boundary conditions!periodic} as well. The nodes of the dual lattice are the centers of the elementary cells of the original lattice. See \\refch{out-eq_chaos}.} and we build for each center a sphere of radius $r$. The number of spins inside the sphere is $N_r$, for example, the smallest sphere $r=1$ has $N_r=8$ spins inside.\n\nWe want to compute the chaotic parameter of the sphere, defined in \\refeq{def_chaotic_parameter} and repeated here for the reader's convenience\n\\begin{equation} \nX^{s,r}_{T_1,T_2}(\\xi) = \\dfrac{\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T}{\\sqrt{\\langle[q_{T_1,T_1}^{s,r}(\\xi)]^2\\rangle_T \\,\\langle[q_{T_2,T_2}^{s,r}(\\xi)]^2\\rangle_T}} \\, . \n\\end{equation} \n\nThe square of the overlaps\\index{overlap} $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ has to be averaged over the thermal noise. As long as we are in an out-of-equilibrium simulation, our estimation of that thermal average would be an average over the replicas\\index{replica}. Therefore, if we focus now on a given sphere, we have a total of $N_r \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ number of spins at each temperature. Furthermore, we have to repeat this procedure for the 8000 spheres, for different sizes of spheres $r$, for different coherence lengths\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ and for different pairs of temperatures $T_1$ and $T_2$. \n\nJust to put the reader in context, if we fix $T_1$, $T_2$ and $\\xi$, we have to perform $2.5526747136 \\cdot 10^{14}$ overlaps\\index{overlap}. This is, indeed, a huge amount of computations that are completely independent of each other. Hence, we can greatly benefit from the \\gls{MSC} in this situation. \n\nFirst, to each sphere, we associate a vector of pairs of $256$-bit registers. To this purpose, we coded an specific function \\textit{void fill\\_sphere(int, int*, int, int)}\n\\begin{lstlisting}[language=C,style=mystyle]\n\n#define NR_PIECES 2 \/\/The number of words of 256 bits\ntypedef __m256i MY_WORD; \/\/MY_WORD is a register of 256 bits (see Intel Intrinsics for further information)\n\nMY_WORD replicas[NR_PIECES][V],replicas2[NR_PIECES][V]; \/\/Full lattice for temperature T1 and T2 respectively\nMY_WORD sphere_replicas[NR_PIECES][V], sphere_replicas2[NR_PIECES][V]; \/\/Spins inside the sphere for the replica 1 and replica 2.\n\nvoid fill_sphere(int size,int* sphere_index, int size_word, int temperature_flag){\n  int is; \/\/loop variable to run over the spins of the sphere\n  int i512; \/\/loop variables to run over the two words of 256-bits\n  \n  \/\/Auxiliar variables  \n  MY_WORD* aux1[NR_PIECES];\n  MY_WORD* aux2[NR_PIECES];\n  \n  \/\/The value of the flag is arbitrary, but this specific set of values\n  \/\/allows us to iterate over it in a loop\n  \n  for(i512=0;i512<NR_PIECES;i512++){\n    if(temperature_flag==0){ \/\/Means T1T1\n      aux1[i512]=replicas[i512];\n      aux2[i512]=replicas[i512];\n    }else if(temperature_flag==1){ \/\/Means T2T2\n      aux1[i512]=replicas2[i512];\n      aux2[i512]=replicas2[i512];\n    }else if(temperature_flag==2){ \/\/Means T1T2\n      aux1[i512]=replicas[i512];\n      aux2[i512]=replicas2[i512];\n    }else{ \/\/There are no more options\n      print_and_exit(\"Temperature flag cannot be differente from 0,1 or 2 -> temperature_flag =\n    }\n  }\n\n  if(size_word==512){\n    for(is=0;is<size;is++){\n      for(i512=0;i512<NR_PIECES;i512++){\n\t\tsphere_replicas[i512][is]=aux1[i512][sphere_index[is]];\n\t\tsphere_replicas2[i512][is]=aux2[i512][sphere_index[is]];\n      }\n    }\n  }\n}\n\\end{lstlisting}\nThis function is really easy, it only associates each spin of the sphere (\\textit{sphere\\_replicas} and \\textit{sphere\\_replicas2}) with its corresponding value of the full lattice. Now, we have built our spheres. The variable \\textit{temperature\\_flag} is selecting the proper temperature of the replicas\\index{replica} to construct $[q_{T_1,T_1}^{s,r}(\\xi)]^2$, $[q_{T_2,T_2}^{s,r}(\\xi)]^2$ or $[q_{T_1,T_2}^{s,r}(\\xi)]^2$.\n\nFrom now on, we focus on the $T_1 \\neq T_2$ case for the sake of simplicity, but for the $T_1=T_2$ case, the situation is almost the same (below, the reader can find a minor discrepancy). The next step is to compute the overlap\\index{overlap} between the $512$ replicas\\index{replica}. The code for this operation, using the AVX instructions, is really easy:\n\\begin{lstlisting}[language=C,style=mystyle]\nMY_WORD q[NR_PIECES][V]; \/\/ Matrix (2 x V) of registers of size 256-bits\nint i512,j; \/\/Loop variables\n\nfor (j=0;j<sphere_size;j++){\n   for(i512=0;i512<NR_PIECES;i512++){\n      q[i512][j]=_mm256_xor_si256(sphere_replicas[i512][j],sphere_replicas2[i512][j]);\n   }\n}\n\n\\end{lstlisting}\n\nNow, the variable \\textit{q[NR\\_PIECES][V]} contains, for each site inside the sphere, a set of $512$ overlaps\\index{overlap}. We can imagine it as a $(512 \\times n_{\\mathrm{sph}})$ matrix, where $n_{\\mathrm{sph}}$ is the number of spins in the sphere. To compute $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ we fix one of the $512$ bits of the register and compute the square of the boolean\\index{boolean} sum over the $n_{\\mathrm{sph}}$ elements. At the end of this process, we will have computed $512$ values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$.\n\nHowever, we can compute much more values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$. Indeed, we can rotate the 512 replica-spins\\index{replica} in one of the spheres (as done in the example of~\\refsec{what_is_multispin_coding}) and compute again the overlap\\index{overlap}. This code makes the rotations of the bits\n\n\\begin{lstlisting}[language=C,style=mystyle]\ntypedef union {\n    __m256i sse;\n    uint64_t vec[4];\n} VectorUnion64;\n\nvoid rota(int size, MY_WORD *sphere1, MY_WORD *sphere2)\n{\n\n  VectorUnion64 w[2];\n  uint64_t resto[8];\n\n  int i,iw; \/\/Loop variables\n  \n  for (i=0;i<size;i++){\n     w[0].sse=sphere1[i];\n     w[1].sse=sphere2[i];\n     for (iw=0;iw<8;iw++){\n        resto[iw]=w[iw\/4].vec[i\n\t    w[iw\/4].vec[i\n     }\n    \n     for (iw=0;iw<8;iw++){\n\t    w[iw\/4].vec[i\n     }\n     sphere1[i]=w[0].sse;\n     sphere2[i]=w[1].sse;\n  }\n}\n\\end{lstlisting}\n\nThis rotation process can be repeated $511$ times until we reach again the initial disposition of spins. Therefore, we compute a total of $512^2$ values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ that will allow us to estimate the thermal average $\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T$. Here is where the small discrepancy between the $T_1 \\neq T_2$ and the $T_1 = T_2$ cases appears. For the $T_1=T_2$ case, we can compute only $512(512-1)\/2$ overlaps\\index{overlap}.\n\nWith this simple code, we are able to reduce by a factor $512$ the total number of operations we have to compute. We have considered explaining with detail this simple case because it is very illustrative with a relatively small amount of code and technical details. Nonetheless, the same idea sketched here can be extended to more complex simulations as we will discuss in the next section.\n\n\\section{Spin glass simulations by using Multispin Coding} \\labsec{musa_musi_multispin_coding}\n\nThe equilibrium simulations performed for the works appearing in \\refch{metastate} and \\refch{equilibrium_chaos} required a huge numerical effort. Because those simulations were performed in \\gls{CPU}s, we could take advantage of the \\gls{MSC}\\index{Multispin Coding} method to save a great amount of time. Here, we explain two different optimizations that are based on the \\gls{MSC}\\index{Multispin Coding} technique.\n\n\\subsection{Multisample Multispin Coding} \\labsubsec{multisample_multspin_coding_appendix}\nAs said in~\\refch{equilibrium_chaos}, is known for around 20 years that we can perform the Metropolis update of a single spin by using a sequence of boolean\\index{boolean} operations \\cite{newman:99}. Thus, we can take advantage of the \\gls{MSC}\\index{Multispin Coding} technique to simulate several independent \\gls{EA}\\index{Edwards-Anderson!model} models with Ising\\index{Ising} spins and $\\pm 1$ couplings\\index{couplings}. This method is widely used in computational physics \\cite{newman:99,leuzzi:08,fernandez:09f,banos:12,janus:13,manssen:15,lulli:15,billoire:17,billoire:18,gonzalez-adalid-pemartin:19,fernandez:19} and it is known as \\gls{MUSA}\\index{Multispin Coding!Multisample}. Because \\gls{MUSA} is well known (see e.g. Appendix B.1 in~\\cite{seoane:13}) we will only briefly describe it here.\n\nThe idea is to take a set of $N$ independent systems (in our particular case, $N$ different samples\\index{sample}) that can be simulated in a parallel\\index{parallel!computation} way and encode, for each site of the lattice, the spins of the $N$ systems in a bit-register of size $N$. Then, all the computations can be done by performing boolean\\index{boolean} operations and rotations of spins, just like we have exemplified in \\refsec{overlap_spheres_multispin_coding}. In our particular case, with the available hardware (Intel Xeon E5-2680 and AMD Opteron Processor 6272 processors), we found that the $128$ bits version was the most efficient one.\n\nHowever, as introduced in~\\refch{equilibrium_chaos}, this method is not efficient for all the samples\\index{sample}. Indeed, if only a few of the 128 samples\\index{sample} coded in a computer word are not yet thermalized, continuing the simulation of the already equilibrated samples\\index{sample} is a waste of computer time. In \\reffig{all_L_prob_tau} one can understand that the width of the autocorrelation time distribution increases with $L$, which makes this problem more important for large sizes. For those samples\\index{sample}, we need an extra level of optimization.\n\n\\subsection{Multisite Multispin Coding} \\labsubsec{musi}\nThe \\gls{MUSI}\\index{Multispin Coding!Multisite} method provides a solution for the problem of slow equilibration mentioned at the end of \\refsubsec{multisample_multspin_coding_appendix}. In this case, we found that the $256$ version was more efficient. The idea is to use the 256 bits in a computer word to code 256 distinct spins of a single replica\\index{replica} of a single sample\\index{sample}~\\cite{fernandez:15}. In this way, we execute the Metropolis algorithm in $L^3\/256$ steps. The major problem of the \\gls{MUSI}\\index{Multispin Coding!Multisite} method is the generation of random numbers. Because we are performing $256$ updates of spins at the same time, we need $256$ independent random numbers. In our realization of the \\gls{MUSI}\\index{Multispin Coding!Multisite} we circumvent this problem as explained in~\\cite{fernandez:15}.\n\nThe geometry used to encode the lattice with $L=16$ is the same that appears in~\\cite{fernandez:15}. Here, we briefly recall it.\n\nThe physical lattice of Cartesian coordinates $0\\leq x,y,z<L$ is mapped to a \\emph{super-spin} lattice. Each super-spin is coded in a 256-bits computer word (of course, the 256 bits correspond to 256 physical spins which are updated in parallel). The crucial requirement is that spins that are nearest-neighbors in the physical lattice are coded into nearest-neighbors super-spins. In particular, our super-spins are placed at the nodes of a cubic lattice with the geometry of a parallelepiped of dimensions $L_x=L_y=L\/8$, and $L_z=L\/4$. The relation between physical coordinates $(x,y,z)$ and the coordinates in the super-spin lattice $(i_x,i_y,i_z)$ is\n\\begin{eqnarray}\nx&=& b_x L_x + i_x\\,,\\ 0\\leq i_x<L_x\\,,\\ 0\\leq b_x < 8\\,,\\nonumber\\\\\\labeq{MSC-lattice}\ny&=& b_y L_y + i_y\\,,\\ 0\\leq i_y<L_y\\,,\\ 0\\leq b_y < 8\\,,\\\\\nz&=& b_z L_z + i_z\\,,\\ 0\\leq i_z<L_z\\,,\\ 0\\leq b_z < 4\\,.\\nonumber\n\\end{eqnarray}\nIn this way, exactly 256 sites in the physical lattice are given the same super-spin coordinates $(i_x,i_y,i_z)$. We distinguish between them by means of the bit index:\n\\begin{equation}\ni_b=64 b_z+8b_y+b_x\\,,\\ 0\\leq i_b\\leq 255\\,.\n\\end{equation}\nSince we have to simulate $N_T$ independent system copies in our \\gls{PT} simulation (see~\\refch{equilibrium_chaos} and \\refsec{thermalizing_PT}), we simply carry out successively the simulation of the $N_T$ systems. \n\nThe alert reader will note that the above geometric construction is very anisotropic (we start with a cube, but end-up with a parallelepiped). Fortunately, this unsightly feature can be easily fixed by noticing that the single-cubic lattice is bipartite. Indeed the lattice splits into the \\emph{even} and \\emph{odd} sub-lattices according to the parity of $x+y+z$. The two sub-lattices contain $L^3\/2$ sites. Furthermore, odd spins interact only with even spins and vice versa. It follows that the update ordering is irrelevant, provided that our full-lattice sweep first updates all the (say) odd sites and next to all the even sites. Now, provided that $L_x$, $L_y$ and $L_z$ are all \\emph{even}, the parity of $x+y+z$ and $i_x+i_y+i_z$ coincide.  This implies that all the spins coded in a single super-spin share the same parity, making irrelevant the super-spin lattice asymmetry. For $L=16$ one finds that $L_x=L_y=2$ and $L_z=4$, the three of them even numbers, and hence the above geometric construction works smoothly. \n\nDespite the success for $L=16$, we found that changes in the geometry were needed in order to correctly simulate the $L=24$ system because one has $L_x=L_y=3$ and $L_z=6$ which implies that the super-spin lattice cannot be split into even and odd sub-lattices.\n\nOur solution consisted in introducing \\emph{logical} super-spins of $512$ physical spins, that were later on coded into two computer words of 256 bits each. The geometrical correspondence was ($L_x=L_y=L_z=L\/8$)\n\\begin{eqnarray}\nx&=& \\tilde b_x L_x + j_x\\,,\\ 0\\leq j_x<L_x\\,,\\ 0\\leq \\tilde b_x < 8\\,,\\nonumber\\\\\\labeq{MSC-lattice-L24}\ny&=& \\tilde b_y L_y + j_y\\,,\\ 0\\leq j_y<L_y\\,,\\ 0\\leq \\tilde b_y < 8\\,,\\\\\nz&=& \\tilde b_z L_z + j_z\\,,\\ 0\\leq j_z<L_z\\,,\\ 0\\leq \\tilde b_z < 8\\,.\\nonumber\n\\end{eqnarray}\nIn this way, exactly 512 sites in the physical lattice are given the same super-spin coordinates $(j_x,j_y,j_z)$. We distinguish between them by means of the bit index:\n\\begin{equation}\nj_b=64 \\tilde b_z+8\\tilde b_y+\\tilde b_x\\,,\\ 0\\leq i_b\\leq 511\\,.\n\\end{equation}\nNow, the crucial observation is that (because $L_x=L_y=L_z=3$ for $L=24$) the parity of $x+y+z$ coincides with that of $j_x+j_y+j_z$ if (and only if) the parity of $\\tilde b_x+\\tilde b_y+\\tilde b_z$ is even. In other words, given super-spin coordinates $(j_x,j_y,j_z)$, the 512 spins coded in the super-spin split into 256 even spins and 256 odd spins. Because same-parity spins are guaranteed to be mutually non-interacting, we decided to code the 256 bits with the same parity in the same computer word, with the corresponding bit index being the integer part of $j_b\/2$.\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\\chapter{Parallel Tempering. Technical details.} \\labch{AP_PT}\n\n\\section{On the selection of relevant parameters of the simulation} \\labsec{selection_parameters}\nIn this section, we try to clarify some of the choices made for the parameters of the simulation that can be obscure from the reader's point of view.\n\n\\subsection{Setup-independence of the results}\nAt the beginning of the chapter, we assured that our results do not depend on the particular setup of our simulations, motivated by the metastate\\index{metastate} study exposed in~\\refch{metastate}. Now, it is time to quantitatively justify our statement.\n\nIf we focus on the relevant observable from the dynamical point of view, $\\tintf=\\tau$ we can check in~\\reffig{selection_samples_dinamica} that the internal disorder\\index{disorder} is affecting almost nothing the probability distribution of our observable. The average over the outer disorder\\index{disorder} (which we can call the metastate\\index{metastate} average, as introduced in~\\refch{metastate}) dramatically reduces the fluctuations due to the internal disorder\\index{disorder}.\n\nThe same discussion is straightforwardly applicable to the chaos integral in~\\reffig{selection_samples_estatica}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_interiores_g}\n\\caption[\\textbf{The integrated time is independent on the simulation setup.}]{\\textbf{The integrated time is independent on the simulation setup.} Empirical probability distribution function of $\\tau$ represented for the $10$ inner samples\\index{sample} separately.  $L=16$ case (top) and $L=24$ case (bottom). Averaging over the metastate\\index{metastate} (i.e. the outer samples\\index{sample}) with fixed inner couplings\\index{couplings} strongly reduces the fluctuations between the inner samples\\index{sample}.}\n\\labfig{selection_samples_dinamica}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_I_interiores_g}\n\\caption[\\textbf{The chaos integral is independent on the simulation setup.}]{\\textbf{The chaos integral is independent on the simulation setup.} Empirical probability distribution function of the integrated chaotic parameter. \\textbf{Top} We compare the distribution (labeled as \"Metastate\"\\index{metastate}) obtained with our particular choice of samples\\index{sample} with the distribution obtained from $4000$ fully independent samples\\index{sample} (data from Janus\\index{Janus}). \\textbf{Bottom:} Distributions obtained for the $10$ inner samples\\index{sample} plotted separately. Averaging over the metastate\\index{metastate} (over the outer couplings\\index{couplings}) strongly reduces the fluctuations between the inner samples\\index{sample}.}\n\\labfig{selection_samples_estatica}\n\\end{figure}\n\n\\subsection{Temperature-mesh's choice}\nOur main results are mainly concerned about the $L=16$ and $L=24$ simulations. The reader may find it surprising that we have two simulations for the $L=16$ case, one with $\\Tmin=0.479$ and the other one with $\\Tmin=0.698$. The latter is obvious since is the $\\Tmin$ needed to compare with the other lattice sizes in our scaling result of~\\refsec{scaling_eq_chaos}. However, the choice of $\\Tmin=0.479$ is not capricious. We carefully choose this $\\Tmin$ with the aim of having similar $\\tint$ for the most chaotic samples\\index{sample} in both simulations ($L=24$, $N=24$, $\\Tmin=0.698$) and ($L=16$, $N=16$, $\\Tmin=0.479$). This is shown in~\\reffig{comparison_dynamics_L24_L16}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_comparative_g}\n\\caption[\\textbf{Empirical probability distribution function of $\\mathbf{\\tau}$.}]{\\textbf{Empirical probability distribution function of $\\mathbf{\\tau}$.} Comparison of results for the simulations ($L=24$,$T_\\mathrm{min}=0.698$) and ($L=16$,$T_\\mathrm{min}=0.479$). Note that at the high-end of very difficult samples\\index{sample}, these two simulations are similarly challenging.}\n\\labfig{comparison_dynamics_L24_L16}\n\\end{figure}\n\n\\section{Thermalizing with Parallel Tempering} \\labsec{thermalizing_PT}\nThe reader used to deal with thermalization\\index{thermalization} protocols might find surprising that we focus on the temperature-random-walk\\index{random walk!temperature} to establish our particular thermalization\\index{thermalization} protocol. This appendix is devoted to justify our decision (which is not pioneer and can be find in other works, see for example~\\cite{janus:10}).\n\nWe have introduced in~\\refsubsec{Monte_Carlo} several properties and theorems of the Markov\\index{Markov chain} chains. Specifically, there exists one theorem~\\cite{sokal:97} that assures, for an aperiodic\\index{aperiodicity} Markov\\index{Markov chain} chain that fulfills the balance\\index{balance condition} condition, that the time evolution of the Markov\\index{Markov chain} chain eventually reaches a stationary distribution (which is unique).\n\nIn the case of \\gls{PT}, that stationary distribution appears in~\\refeq{PT_eq_distribution} but we recall it here for the reader's convenience.\n\\begin{equation}\nP_{\\mathrm{eq}}(\\{ \\pi, \\{s^{(\\alpha)}\\}_{\\alpha=1}^N\\}) = \\dfrac{1}{N!}\\prod_{\\alpha=1}^N \\dfrac{\\exp \\left[-\\beta_{\\alpha} \\mathcal{H}(\\{s^{\\pi^{-1}(\\alpha)}\\})\\right]}{Z_{\\beta_{\\alpha}}} \\, , \n\\end{equation}\nIn the previous expression, $N$ is the number of temperatures in the temperature-mesh of our \\gls{PT} method, the index $\\alpha$ labels a given clone and, here, $\\pi(\\alpha)$ is the permutation of the clone $\\alpha$ at a given time and $\\pi^{-1}$ is its inverse.\n\nThis equilibrium configuration\\index{configuration} for the \\gls{PT} is telling us two main things. On the one hand, the marginal probability for the clones permutation is uniform as the factor $1\/N!$ implies. On the other hand if we focus on the marginal probability for the spins of one clone at temperature $T_{\\alpha'}$, namely $\\pi(\\alpha) = \\alpha'$, we find that the equilibrium probability at a given temperature is\n\\begin{equation}\nP_{\\mathrm{eq}}(\\{\\{s^{(\\alpha)}\\} | \\pi(\\alpha)=\\alpha' ) = \\dfrac{\\exp \\left[-\\beta_{\\alpha'} \\mathcal{H}(\\{s^{\\alpha}\\})\\right]}{Z_{\\beta_{\\alpha'}}} \\, , \n\\end{equation}\nwhich is none but the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution.\n\nThe message is clear, when the random-walks\\index{random walk} equilibrate, the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution is reached by all the $N$ clones simultaneously.\n\nThe theorem ensuring the equilibration in the Markov\\index{Markov chain} chains is fundamental because it legitimates its use to study equilibrated systems. However, that theorem only assures that we will reach, at some point, our desired equilibrium distribution, but it says nothing about the convergence to that limit. We have succinctly explained in~\\refsec{time_scales_eq_chaos} the usual way to study this convergence through the computation of time autocorrelation functions\\index{correlation function!time auto-} $\\hat{C}_f(t)$. \n\nNevertheless, the reader familiarized with those methods may be used to build that time autocorrelation functions\\index{correlation function!time auto-} from spin-dependent functions instead of focusing on the temperature random-walk\\index{random walk!temperature} of \\gls{PT}. Actually, this assumption is customary rather than necessary. All the statements done concerning the time autocorrelation function\\index{correlation function!time auto-} $\\hat{C}_f(t)$ are valid from any function $f$ related to the Markovian\\index{Markovian} dynamics. The reader can check that no special condition, restricting the function $f$, was imposed in~\\refsec{time_scales_eq_chaos}. In our particular case, it is really convenient to study the random-walk\\index{random walk!temperature} in the temperature. We expose here an example to illustrate our statements.\n\nWe shall consider an $L=24$ sample\\index{sample} chosen from the simulated samples\\index{sample} with $N=24$ and $T_{\\min}=0.698$ (see~\\refsec{numerical_simulations_eq_chaos}). Specifically, the chosen sample\\index{sample} will be more difficult (i.e. will have a larger autocorrelation time) than the $90\\%$ of the simulated samples\\index{sample}.\n\nNow, we consider two different \\gls{PT} setups. The first one corresponds to the setup used in the simulations, namely a temperature mesh with $N=24$ temperatures and $T_{\\min}=0.698$. The second one will consist only of the four lowest temperatures: $T_1=T_{\\min}=0.698$, $T_2 = 0.735$, $T_3=0.771$ and $T_4=0.808$.\n\nThe $N=24$ setup is found to need $2\\cdot10^9$ Metropolis sweeps to meet our thermalization\\index{thermalization} criteria and we decided to run both \\gls{PT} setups for that number of Metropolis sweeps. We know that the $N=24$ system will thermalize and the natural expectation for the $N=4$ system is that it will not reach our thermalization\\index{thermalization} criteria. The reason for that expectation is that, while the standard simulation will spend around $1\/2$ of the total time above the critical\\index{critical temperature} temperature, the truncated simulation ($N=4$) will not. \n\nActually, we know that, for the highest temperature in the (standard) mesh, $T=1.6$, the exponential autocorrelation time\\index{autocorrelation time!exponential} for Metropolis dynamics is about $10^4$ lattice sweeps~\\cite{ogielski:85}, which corresponds to $1000$ times more lattice sweeps that each clone will spend at the highest temperature ($2\\cdot 10^9\/24 \\approx 8 \\cdot 10^7$). On the contrary, for the truncated system with $N=4$ the highest temperature is still in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} and the time that each clone spends at the highest temperature is not enough to decorrelate the spins.\n\nNow, we need a function $f$ depending on the spin configurations\\index{configuration} in order to compare with our function $f$ depending on the temperature random-walk\\index{random walk!temperature} of the \\gls{PT}. Using the fact that we have already equilibrated this sample\\index{sample}, we have selected randomly four equilibrium spin configurations\\index{configuration} at our lowest temperature $T_{\\min}=0.698$. We denote those configurations\\index{configuration} as $\\{ \\tau_{x,a}\\}$ with $a=$1, 2, 3, 4. Our four proposed functions (for each clone $\\alpha$) $f_{a,\\alpha}$ will be the time-dependent overlap\\index{overlap} between the spin configurations\\index{configuration} of the clone $\\alpha$, and each of the four selected equilibrium configurations\\index{configuration} $\\{ \\tau_{x,a} \\}$\n\\begin{equation}\nf_{a,\\alpha} = q_{a,\\alpha}(t) = \\dfrac{1}{L^3} \\sum_x \\tau_{x,a} s_x^{(\\alpha)}(t) \\, . \\labeq{spin-dependent_observable}\n\\end{equation}\n\nWe compute those overlaps\\index{overlap} for a set of $10$ new standard simulations (i.e. $N=24$) and for the truncated simulations $N=4$. Because we are particularly interested in the thermalization\\index{thermalization} process, we measure $q_{a,\\alpha}(t)$ very often (every $5\\cdot 10^4$ Metropolis sweeps\\footnote{This suppose a total of $40000$ measure from $t=0$ to $t=2\\cdot 10^9$ Metropolis sweeps.}). \n\nThe global spin-flip symmetry of the \\gls{EA}\\index{Edwards-Anderson!Hamiltonian} Hamiltonian\\index{Hamiltonian} implies that the equilibrium distribution for $q_{a,\\alpha}$ is symmetric under overlap\\index{overlap} inversions $q_{a,\\alpha} \\leftrightarrow -q_{a,\\alpha}$. It is important to check this symmetry, since it is believed that the largest dynamical barriers are related to global spin-flips~\\cite{billoire:01}.\n\nOnce we simulate both setups, we notice that the truncated simulation is not able to thermalize within the time span of our simulations. To illustrate our findings, we show~\\reffig{historia_q}.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/historia_q}\n\\caption[\\textbf{Overlap history. Exploring the thermalization process.}]{\\textbf{Overlap\\index{overlap} history. Exploring the thermalization\\index{thermalization} process.}\\textbf{ Top:} Monte\\index{Monte Carlo} Carlo history for the overlap\\index{overlap} $q_4(t)$. Note that our simulation time is too short to expose the symmetry $q_4 \\leftrightarrow -q_4$. As a consequence, we know for sure that thermal equilibrium has not been reached for the truncated simulation. \\textbf{Bottom:} as in the top panel, for the first four clones in one of our standard simulations with $N = 24$ temperatures (there were 10, completely independent, standard simulations). For each clone, the overlap\\index{overlap} $q_4(t)$ changes sign many times along the simulation (as it is to be expected for a well-equilibrated simulation). Note that, with small probability, each clone reaches a state where $|q_4| \\sim 0.8$. These events, which are not observed for the other three overlaps\\index{overlap} $q_a$ with $a=$1, 2, 3, make it particularly interesting to study the dynamics of $q_4$.}\n\\labfig{historia_q}\n\\end{figure}\n\nIn the top panel of this figure, we represent the time-dependent overlap\\index{overlap} $q_4(t)$ for each of the four clones in the truncated simulation. Furthermore, in the bottom panel, we represent the same quantity for the clones corresponding to the four lowest temperatures in the standard simulation. We clearly observed that nor the overlap\\index{overlap} symmetry neither the clone equivalence is respected in the truncated simulation. On the contrary, the standard simulation with $N=24$ displays the expected overlap\\index{overlap} symmetry. The Monte\\index{Monte Carlo} Carlo histories (in the standard simulation) for $q_{a,\\alpha}$ with $a =$ 1, 2, 3 (not shown) are symmetric as well. However, only $q_4$ uncovers a state that arises with small probability, characterized by $|q_4| \\sim 0.8$. This feature suggests that $q_4$ is the most interesting overlap\\index{overlap} to look at.\n\nIt is clear, from the above discussion, that thermalization\\index{thermalization} is not reached. But we have focused on the overlaps\\index{overlap} $q_a$ with $a=$ 1, 2, 3, 4. One could wonder if looking at the functions based on the temperature random-walk\\index{random walk!temperature} of the \\gls{PT} we would infer a different result, after all, the truncated simulation is composed only of $4$ copies of the system and one could think that it should not be so difficult to equilibrate the clone permutation. The answer is negative as can be seen in~\\reffig{histogramas}. The fact that the spins are out of equilibrium makes it also impossible to equilibrate the clone permutations.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/histogramas}\n\\caption[\\textbf{Time spent in each temperature by the four clones of the truncated Parallel Tempering simulation.}]{\\textbf{Time spent in each temperature by the four clones of the truncated  Parallel Tempering simulation.} For each of the four clones in the truncated simulation, we show the histogram of temperature (i.e. the number of times that each clone can be found at $T_1$, $T_2$, \\dots). The temperature state was sampled every $5\\times 10^4$ Metropolis sweeps (per clone). Had the simulation equilibrated, we would have expected the occupation histograms to be uniform.}\n\\labfig{histogramas}\n\\end{figure}\n\nNow, we have exemplified that if the equilibrium is not reached by spin-related quantities it would neither be reached by clone-permutations-related quantities but, are the autocorrelation timescales of the spin-related quantities and the clone-permutations-related quantities equivalent?\n\nTo answer this question we focus on the standard setup with $N=24$, which is the only one that equilibrates and, therefore, the only one in which we can compute the equilibrium autocorrelation functions\\index{correlation function!time auto-}. In~\\reffig{correlaciones} we plot the time autocorrelation functions\\index{correlation function!time auto-} for the four quantities $q_a$ with $a=$ 1, 2, 3, 4, and also the time autocorrelation function\\index{correlation function!time auto-} for the temperature random-walk\\index{random walk!temperature} (labeled as $T$-correlation), specifically the piecewise linear function with $T^* = T_3$ (see~\\refch{equilibrium_chaos}).\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/correlaciones}\n\\caption[\\textbf{Equilibrium time dependent correlation functions, as computed from the standard simulation with \\boldmath $N_T=24$.} ]{\\textbf{Equilibrium time dependent correlation functions, as computed from the standard simulation with \\boldmath $N_T=24$.}  We consider five observables, one related to temperature (computed from the piece-wise linear function with $T^*=T_3$, see \\refch{equilibrium_chaos}), and the overlaps\\index{overlap} $q_a$ with $a=1,2,3,4$. The fact that the $T$ and $q_4$ correlations become parallel in this semi-logarithmic scale indicates that we are safely computing the exponential auto-correlation time (which is independent of the observable). Instead, the $q_{a=1,2,3}$ correlations do not become parallel to the other curves, at least not within the range we can measure, which probably indicates that the amplitudes $A_{n=1,q_{a=1,2,3}}$, see~\\refeq{autocorrelation_function_decomposition}, are much smaller for these observables.}\n\\labfig{correlaciones}\n\\end{figure}\n\nWe observe that the long-time behavior of both the $q_4$-correlation function and the $T$-correlation function are equivalent. Recall that this long-time behavior is giving us the exponential autocorrelation time\\index{autocorrelation time!exponential} (which is independent of the particular quantity we are studying). This is indicating us that we are safely computing the exponential autocorrelation time\\index{autocorrelation time!exponential} and, furthermore, that both timescales are equivalent. Specifically, and measuring time in Metropolis sweeps, we find $\\texp = 3.0(4)\\cdot 10^{7}$ from $q_4$, while we find the fully compatible value $\\texp = 3.1(6) \\cdot 10^7$ from the $T$ random-walk\\index{random walk!temperature}. \n\nA final warning should be made in this respect. The reader might wonder, since we have shown the equivalence of both methods, if it is not simpler to study $q_4$ rather than the clone permutations. The answer is negative. In order to discover the $q_4$-clone-permutation equivalence we had to thermalize the system and find the equilibrium configuration\\index{configuration} $\\{\\tau_{x,4}\\}$. Moreover, we randomly pick four different configurations\\index{configuration} and only one of them was useful. It is not guaranteed that one can identify an interesting overlap\\index{overlap} by randomly picking a small number of equilibrated configurations\\index{configuration}.\n\n\\chapter{Statistical tools} \\labch{AP_statistics}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Most people use statistics like a drunk man uses a lamppost; more for support than illumination.}}{-- Andrew Lang}\n\nIn this appendix, we expose the main statistical tools that we have used throughout this thesis. The huge amount of data (often correlated) and the non-linear nature of many quantities in our analysis are the main reasons for our interest in non-traditional statistics. We will discuss each of the methods used in this thesis and we provide examples.\n\n\\section{Estimating error bars} \\labsec{estimating_errorbars}\nLet us consider a random variable $x$ which follows a \\gls{pdf} with mean $\\mu_x$ and variance $\\sigma_x$. The common situation in both, numerical and real experiments, is that the \\gls{pdf} of $x$ is completely unknown for us. Thus, we can only \\textit{sample} the \\gls{pdf} with a finite number $N$ of measurements of the variable $x$: $\\{x_1,x_2,\\dots,x_N\\}$. The usual estimators for $\\mu_x$ and $\\sigma_x$ are the arithmetic mean of the data $m(x)$ and the sample variance $s^2(x)$\n\\begin{equation}\nm(x) = \\dfrac{1}{N} \\sum_{i=1}^N x_i \\quad , \\quad s^2(x) = \\dfrac{1}{N-1} \\sum_{i=1}^N \\left[x_i - m(x)\\right]^2 \\, .\n\\end{equation}\n\nThese estimators are perfectly suitable for random variables and a linear combination of them. However, it is usual to be interested in a non-linear combination of random variables where the previous simple method does not work. In this case, we need a more advanced analysis in order to estimate the mean and the variance.\n\nThe traditional method is the well-known \\textit{error propagation}. This method is based on Taylor expansions and is unbiased (up to arbitrary precision, determined by the number of terms we retain in the Taylor expansion). Nonetheless, it is dependent on the specific expression of the non-linear quantity studied and, therefore, a different computation (involving derivatives) has to be performed for each particular case.\n\nHere, we introduce two alternatives that consist of different strategies to resample the data and compute the error bars\\index{error bars}: the Jackknife\\index{Jackknife} method and the Bootstrap\\index{Bootstrap} method.\n\n\\subsection{The Jackknife method} \\labsubsec{jk_method}\nJust as before, suppose we have $N$ independent measures $\\{x_1,x_2,\\dots,x_N\\}$. We define the Jackknife\\index{Jackknife} variables as\n\\begin{equation}\nx^{\\mathrm{JK}}_i = m(x) + \\dfrac{1}{N-1} \\left[ m(x) - x_i \\right] = \\dfrac{1}{N-1} \\sum_{j \\neq i} x_j \\, . \\labeq{jackknife_variable_definition}\n\\end{equation}\nNow, suppose we have a set of random variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$ (each one with $N$ independent measures $x_{\\alpha,i}$, $x_{\\beta,i}$, $\\dots$ with $i=1,\\dots,N$) and a non-linear combination of them $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$. We define the Jackknife\\index{Jackknife} variable (or Jackknife\\index{Jackknife} block) of that non-linear function as\n\\begin{equation}\nf^{\\mathrm{JK}}_i = f(x^{\\mathrm{JK}}_{\\alpha,i},x^{\\mathrm{JK}}_{\\beta,i},x^{\\mathrm{JK}}_{\\gamma,i},\\dots) \\, .\n\\end{equation}\nOur aim is to estimate $\\mu_f=f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ and the standard deviation of the quantity $f$, $\\sigma_f$. To estimate that quantity, we introduce the mean of the Jackknife\\index{Jackknife} blocks\n\\begin{equation}\nm(f^{\\mathrm{JK}}) = \\dfrac{1}{N} \\sum_{i=1}^N f^{\\mathrm{JK}}_i \\, .\n\\end{equation}\nFor non-linear quantities is easy to show~\\cite{young:12} that there is a bias in the estimation of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ with the quantity $m(f^{\\mathrm{JK}})$. Indeed, the elimination of the leading bias leads to the following estimator $f_{\\mathrm{JK,estimator}}$ of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$:\n\\begin{equation}\nf_{\\mathrm{JK,estimator}} = N f\\left(m(x_{\\alpha}),m(x_{\\beta}),\\dots\\right) - (N-1) m(f^{\\mathrm{JK}}) \\, .\n\\end{equation}\n\nIn this case, the bias is of order $1\/N^2$. If we just take the quantity $m(f^{\\mathrm{JK}})$ as an estimator of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ the bias is of order $1\/N$. As we will discuss below, this is not a big problem since the bias in the statistical error is of order $1\/\\sqrt{N}$ which is much bigger than $1\/N$ for large $N$.\n\nWe can also define the Jackknife\\index{Jackknife} variance as\n\\begin{equation}\ns^2(f^{\\mathrm{JK}}) = m\\left((f^{\\mathrm{JK}})^2\\right) - \\left[m(f^{\\mathrm{JK}})\\right]^2 \\, .\n\\end{equation}\n\nAfter some easy mathematical steps~\\cite{young:12}, we can reach an expression for the estimator $\\sigma_{\\mathrm{JK,estimator}}$ of the variance $\\sigma_f$\n\\begin{equation}\n\\sigma^2_{\\mathrm{JK,estimator}} = (N-1) s^2(f^{\\mathrm{JK}}) \\, .\n\\end{equation}\nThis is the great success of the Jackknife\\index{Jackknife} method, the estimator of the variance can be obtained from the raw data, and no derivatives are needed to be computed. Hence, the error bars\\index{error bars} of our desired quantity, that usually corresponds to the standard deviation of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$, would be \n\\begin{equation}\n\\sigma_{\\mathrm{JK,estimator}} = \\sqrt{N-1} \\, s(f^{\\mathrm{JK}}) = \\sqrt{N-1} \\left[  m\\left((f^{\\mathrm{JK}} \\right)^2) - \\left[m(f^{\\mathrm{JK}})\\right]^2 \\right]^{1\/2} \\, . \\labeq{errorbar_estimator}\n\\end{equation}\n\nThe reader can found an example of a specific application of this method in~\\refsec{procedure}. \n\nAnother advantage of this method is that the Jackknife\\index{Jackknife} blocks can be easily propagated to perform further computations. This propagation usually involves non-linear operations, fortunately, we have just discussed that the Jackknife\\index{Jackknife} method can deal with them. We illustrate our statement with one example.\n\n\\subsubsection{Computing the error of the coherence length}\nAs we have said, the coherence length\\index{coherence length} is fundamental in order to characterize the off-equilibrium dynamics. Let us sketch the procedure to compute its error bars\\index{error bars}.\n\nFirst, we recall that our off-equilibrium simulation consists of an \\gls{EA}\\index{Edwards-Anderson!model} model with Ising\\index{Ising} spins for which we have simulated $\\ensuremath{N_{\\text{S}}}\\xspace=16$ different samples\\index{sample} and, for each sample\\index{sample}, $512$ replicas.\n\nBefore explaining the method, we briefly recall the definition of the estimator of the coherence length\\index{coherence length} $\\xi_{12}(t)$ from the four-point\\index{correlation function!four point} correlation function $C_4(T,r,t)$\n\\begin{align}\nC_4(T,r,t) & =  \\overline{\\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}} \\, , \\\\\nI_k(t) & = \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r \\, , \\\\\n\\xi_{12}(t) & = \\dfrac{I_{2}(t)}{I_1(t)} \\, .\n\\end{align}\n\nWe start by computing $C_4(T,r,t)$. The Jackknife\\index{Jackknife} method will allow us to estimate the disorder\\index{disorder!average} average $\\overline{(\\cdots)}$. For each sample\\index{sample} $\\mathcal{J}_i$, we compute\n\\begin{equation}\nC_4^{\\mathcal{J}_i}(T,r,t) = \\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}_{\\mathcal{J}_i} \\, ,\n\\end{equation}\nwhere the label $\\mathcal{J}_i$ stresses the fact that the quantity is sample-dependent. Then we can build the Jackknife\\index{Jackknife} blocks for this quantity\\footnote{The details on the computation of $ C_{4,i}^{\\mathrm{JK}}(T,r,t)$ can be found in~\\refsec{finite_size_effects}.}\n\\begin{equation}\nC_{4,i}^{\\mathrm{JK}}(T,r,t) = \\dfrac{1}{\\ensuremath{N_{\\text{S}}}\\xspace-1} \\sum_{i \\neq j} C_4^{\\mathcal{J}_j}(T,r,t)\n\\end{equation}\n\nIf we were interested only in computing the four-point\\index{correlation function!four point} correlation function, we would simply estimate the errors of $C_4(T,r,t)$ by using~\\refeq{errorbar_estimator}. However, we are interested in the quantity $\\xi_{12}(t)$. The \\textit{propagation} of the Jackknife\\index{Jackknife} method to derived quantities is quite simple. We now compute $I_k(t)$ [in our case we are interested in $I_1(t)$ and $I_2(t)$] for each Jackknife\\index{Jackknife} block of the four-point\\index{correlation function!four point} correlation function. Therefore, we will obtain\n\\begin{equation}\nI^{\\mathrm{JK}}_{k,i}(t) = \\int_0^{\\infty} r^k C_{4,i}^{\\mathrm{JK}}(T,r,t) {\\mathrm{d}} r \\, .\n\\end{equation}\nFinally we can, again, propagate the Jackknife\\index{Jackknife} block by computing\n\\begin{equation}\n\\xi^{\\mathrm{JK}}_{12,i}(t) = \\dfrac{I^{\\mathrm{JK}}_{2,i}(t)}{I^{\\mathrm{JK}}_{1,i}(t)} \\, .\n\\end{equation}\n\nIn the end, we will have $\\ensuremath{N_{\\text{S}}}\\xspace$ Jackknife\\index{Jackknife} blocks and we can estimate the error bars\\index{error bars} of $\\xi_{12}(t)$ by using~\\refeq{errorbar_estimator}. \n\n\n\\subsection{The Bootstrap method} \\labsubsec{bootstrap_method}\nAlthough most of the time we use the Jackknife\\index{Jackknife} method, there exists an alternative that also takes advantage of resampling the data: the Bootstrap\\index{Bootstrap} method. \n\nLet us consider again that we have a set of $N$ points $\\{x_1,x_2,\\dots,x_N\\}$. In the Jackknife\\index{Jackknife} method, we build $N$ blocks from all the points of the original data except one. On the contrary, in the Bootstrap\\index{Bootstrap} method, we build $N_{\\mathrm{boots}} \\neq N$ data sets, each one containing $N$ points picked at random from the original data. The probability of selecting one point of the original data is $1\/N$ and does not depend on whether it has been selected before.\n\nFor a given data set $k$, with $k$ running from 1 to $N_{\\mathrm{boots}}$, we can define the mean of the points inside this data set as\n\\begin{equation}\nx^{B}_k = \\dfrac{1}{N} \\sum_{i=1}^N n^{B}_{i,k} x_i \\, ,\n\\end{equation}\nwhere $n^{B}_{i,k}$ is the number of times that the point $x_i$ have been selected for the data set $k$. The mean of all the $x^{B}_k$ over the $N_{\\mathrm{boots}}$ data sets is\n\\begin{equation}\nm\\left(x^{B}\\right) = \\dfrac{1}{N_{\\mathrm{boots}}} \\sum_{k=1}^{N_{\\mathrm{boots}}} x_k^{B} \\, . \\labeq{def_mean_boots}\n\\end{equation}\n\nIt can be shown~\\cite{young:12}, that for $N_{\\mathrm{boots}} \\to \\infty$ the value of $m\\left(x^{B}\\right)$ corresponds to the value of $m(x)$, which is an unbiased estimator of $\\mu_x$. The variance of $x^{B}$ is\n\\begin{equation}\ns^2\\left(x^{B}\\right) = m \\left( (x^{B})^2\\right) - \\left[ m\\left( x^{B} \\right) \\right]^2 \\, .\n\\end{equation}\n\nTo address the more interesting scenario, suppose again that we have a set of random variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$ (each one with $N$ independent measures $x_{\\alpha,i}$, $x_{\\beta,i}$, $\\dots$ with $i=1,\\dots,N$) and a non-linear combination of them $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$.  We are again interested in estimating $\\mu_f= f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ and its standard deviation $\\sigma_f$. The Bootstrap\\index{Bootstrap} variable will be\n\\begin{equation}\nf^{B}_k = f(x^{B}_{\\alpha,k},x^{B}_{\\beta,k},x^{B}_{\\gamma,k},\\dots) \\, . \\labeq{bootstrap_variable}\n\\end{equation}\n\nSimilarly to the Jackknife\\index{Jackknife} case, the mean and the variance of $f^{B}_k$ would be\n\\begin{equation}\nm\\left( f^{B} \\right) = \\dfrac{1}{N_{\\mathrm{boots}}} \\sum_{k=1}^{N_{\\mathrm{boots}}} f^{B}_{k} \\, ,\n\\end{equation}\n\\begin{equation}\ns^2 \\left( f^{B} \\right) = m\\left( \\left(f^{B} \\right)^2 \\right)- \\left[ m\\left(f^{B}\\right) \\right]^2 \\, .\n\\end{equation}\n\nAgain, the bias in $m\\left( f^{B} \\right)$ is of order $1\/N$, much smaller than the statistical error. Indeed, the Bootstrap\\index{Bootstrap} estimator $\\sigma_{\\mathrm{B,estimator}}$ of the standard deviation $\\sigma_f$ is\n\\begin{equation}\n\\sigma_{\\mathrm{B,estimator}} = \\sqrt{\\dfrac{N}{N-1}} \\left[ m\\left( \\left(f^{B} \\right)^2 \\right)- \\left[ m\\left(f^{B}\\right) \\right]^2 \\right]^{1\/2} \\, . \\labeq{standard_deviation_bootstrap}\n\\end{equation}\nIn the $N\\to \\infty $ limit, the prefactor tends to $1$.\n\nIt is worthy to note that a high enough number of data sets $N_{\\mathrm{boots}}$ is needed to ``achieve'' the $N_{\\mathrm{boots}} \\to \\infty$ limit in~\\refeq{def_mean_boots}. How large should be $N_{\\mathrm{boots}}$ to achieve that limit? We have not performed a systematic study but the relation $N_{\\mathrm{boots}} = 10N$ seems to be safe. Indeed, our use of the Bootstrap\\index{Bootstrap} method is limited and the computations involved do not require large computational times, but a much more detailed study should be done for a different situation.\n\nAs a final note, the reader may wonder why we would use the Bootstrap method if we can use the Jackknife method that involves less computations and provide similar results. The Jackknife method assumes a Gaussian behavior of the variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$, on the contrary, the Bootstrap method allows us to access to the probability distribution of $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$ in those cases in which $x_{\\alpha}$ does not follow a Gaussian distribution. \n\n\\subsubsection{Computing the Pearson coefficient}\nLet us consider a set of $N$ pairs of independent measures $\\{(x_1,y_1),(x_2,y_2),\\dots,(x_N,y_N)\\}$. In some situations, we want to test the linear dependence between the pairs of measures. To test that dependence, the proper tool is the Pearson coefficient $r$ which is between $-1$ and $1$. The extreme values are clear, $r=-1$ corresponds to a complete linear-anticorrelation between $x$ and $y$ and the value $r=1$ corresponds to a perfect correlation between $x$ and $y$.\n\nThe Pearson coefficient is defined as\n\\begin{equation}\nr = \\dfrac{\\sum_{i=1}^N x_i y_i - N m(x) m(y)}{(n-1) \\sqrt{m(x^2) - [m(x)]^2} \\sqrt{m(y^2) - [m(y)]^2}} \\, , \\labeq{pearson_coefficient}\n\\end{equation}\nwhere $m(x)=(1\/N)\\sum_i x_i$.\n\nThe reader may find one practical example of this computation in \\refch{equilibrium_chaos}. Indeed, the Bootstrap\\index{Bootstrap} estimation of the error for this quantity is very simple. From the complete cloud of $N$ points, we randomly (with uniform probability) select $N$ pairs with repetition (i.e. picking more than once the same pair is allowed). We repeat this procedure $N_{\\mathrm{boots}}$ times and, for each of the $N_{\\mathrm{boots}}$ clouds of $N$ points, we compute the Pearson coefficient $r$. \n\nThe correspondence between each of the $r_k$ computed values (with $k$ running from 1 to $N_{\\mathrm{boots}}$) and the values of $f^{B}_k$ appearing in~\\refeq{bootstrap_variable} is evident. Then, the estimation of the error bars\\index{error bars} is straightforward, we have just to apply~\\refeq{standard_deviation_bootstrap}.\n\n\n\\section{Improving the statistics} \\labsec{improving_statistics}\nHere, we explain how the correlation between variables (in a controlled situation) can reduce the error bars\\index{error bars} of the quantities we are interested in. We take advantage of the \\textit{control\\index{control variate} variates} for that purpose. The reader may find further information in~\\cite{fernandez:09c,ross:14}.\n\n\\subsection{Control Variates}\nThe starting point is the same of~\\refsec{estimating_errorbars}, consider a random variable $x$ which follows a \\gls{pdf} with mean $\\mu_x$ and standard deviation $\\sigma_x$ and a set $N$ measurements of the variable $x$: $\\{x_1,x_2,\\dots,x_N\\}$. Since we only have a finite number of measurements, we can only aspire to estimate the mean and the standard deviation through the arithmetic mean $m(x)$ and the sample standard deviation $s(x)$ (or equivalently, the sample variance $s^2(x)$)\n\\begin{equation}\nm(x) = \\dfrac{1}{N} \\sum_{i=1}^N x_i \\quad , \\quad s^2(x) = \\dfrac{1}{N-1} \\sum_{i=1}^N \\left[x_i - m(x)\\right]^2 \\, .\n\\end{equation}\n\nIn order to apply the control\\index{control variate} variates method, we need another variable $y$ (the control\\index{control variate} variate) from which we know exactly its mean $\\mu_y$. Hence, we define a new variable $z$\n\\begin{equation}\nz= x + c(y-\\mu_y) \\, , \\labeq{cv_new_variable}\n\\end{equation}\nwhere $c \\in \\mathbb{R}$ is a constant. It is clear that \n\\begin{equation}\nm(z) = m\\left(x + c(y-\\mu_y) \\right) = m(x) + c(m(y) - \\mu_y) = m(x) \\, .\n\\end{equation}\nTherefore, $m(z)$ is an unbiased estimator of $\\mu_x$. The situation is more interesting for the variance\n\\begin{equation}\ns^2(z) = s^2 \\left( x + c(y -\\mu_y)\\right) = s^2(x) + c^2 s^2(y) + 2c \\mathrm{Cov}(x,y) \\, ,\n\\end{equation}\nbeing $\\mathrm{Cov}(x,y)=m(xy) - m(x)m(y)$ the covariance between the random variables $x$ and $y$. As long as we are interested in reduce the variance of $z$ we can look for the constant value $c$ minimizing $s^2(z)$\n\\begin{equation}\n\\dfrac{\\partial s^2(z)}{\\partial c} = 0 \\implies c^* = - \\dfrac{\\mathrm{Cov}(x,y)}{s^2(y)} \\, ,\n\\end{equation}\nwhere $c^*$ is the value of the constant $c$ that minimizes the variance of $s^2(z)$. Then, the variance of $z$ will be\n\\begin{equation}\ns^2(z) = s^2\\left(x+c^*(y-\\mu_y)\\right) = s^2(x) - \\dfrac{\\left[ \\mathrm{Cov}(x,y) \\right]^2}{s^2(y)} \\, .\n\\end{equation}\nThe plan for reducing the error bars\\index{error bars} and improve the statistics is clear: we have to find a quantity $y$ with the smallest possible variance and the biggest possible covariance with $x$.\n\n\\subsubsection{Improving the energy statistics}\nOne practical example of the control\\index{control variate} variates has been already mentioned in \\refsec{numerical_simulation_mpemba}, here, we provide details of our particular implementation. To put the reader in context, we recall here that we are interested in computing the energy-density\\index{energy!density} $e(t,\\mathcal{J})$ to study the Mpemba\\index{Mpemba effect} effect.\n\nThe definition of the energy\\index{energy!density} density in our three-dimensional lattice of linear size $L$ is\n\\begin{equation}\ne(t,\\mathcal{J}) = \\dfrac{1}{L^3} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\quad , \\quad e(t) = \\overline{e(t,\\mathcal{J})} \\, .\n\\end{equation}\n\nIn order to apply the above-exposed method, we need to introduce a control\\index{control variate} variate with a large covariance with the energy-density\\index{energy!density}. Our proposal for the control\\index{control variate} variate is the following. For each of the $\\ensuremath{N_{\\text{S}}}\\xspace=16$ samples\\index{sample} simulated, we took a fully random initial configuration\\index{configuration}, placed suddenly at $T=0.7$. We perform an isothermal\\index{isothermal} simulation for $t=2^{21}$ Monte\\index{Monte Carlo} Carlo steps, and we store in our disk those configurations\\index{configuration} that fulfills $t = \\Floor{2^{n\/8}}$ with $n$ integer. Then, we choose all the configurations\\index{configuration} with times $\\mathcal{C} \\, \\in \\, [2^{19},2^{21}]$, the number of selected times is $N_{\\mathcal{C}}$. Then, we compute\n\\begin{equation}\n{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} = \\dfrac{1}{L^3 N_{\\mathcal{C}}}  \\sum_{t \\in \\mathcal{C}} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\, . \\labeq{control_variate}\n\\end{equation}\n\nThe alert reader may wonder why we choose this quantity ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ if we do not know the real value of its mean $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$. At the first sight, it seems that we have made no improvements. However, although we do not know $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$ we can estimate it with great accuracy. Indeed, we perform a separated simulation with $1.6 \\cdot 10^5$ samples\\index{sample} and two replicas for each sample\\index{sample} (i.e. the energy\\index{energy!density} was averaged over $3.2 \\cdot 10^5$ systems). Thus, the estimation of $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$ would have negligible error compared with $e(t)$.\n\nThen, the final estimates of the internal energy\\index{energy!density} is\n\\begin{equation}\n\\tilde{e}(t,\\mathcal{J}) = e(t,{\\mathcal{J}}) - \\left[ {\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} - \\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}} \\right] \\quad \\, \\quad \\tilde{e}(t) = \\overline{\\tilde{e}(t,\\mathcal{J})} \\, . \n\\end{equation}\n\nThe benefits of using this control\\index{control variate} variate are obvious from~\\reffig{control_variate}. Moreover, the importance of the high covariance between the studied quantity [in our case, the energy-density $e(t,\\mathcal{J})$] and the control\\index{control variate} variate ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ is evident. In the three simulations showed in~\\reffig{control_variate}, the larger improvement in the error estimation is achieved for $T=0.7$ for a coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ with $\\ensuremath{t_\\mathrm{w}}\\xspace$ roughly corresponding to $\\ensuremath{t_\\mathrm{w}}\\xspace \\in [2^{19},2^{21}]$. Because our control\\index{control variate} variate has been computed with great accuracy for those conditions, it is logical that we achieve the best performance in the method for that case, although, improvements are also achieved for different temperatures (see for example $T=0.9$).\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/control_variate.pdf}\n\\caption[\\textbf{Improving the accuracy with control variates.}]{\\textbf{Improving the accuracy with control\\index{control variate} variates.} The figure shows the ratio of statistical errors, as a function of $\\xi(t)$, for the naive $e(t)$ and improved $\\tilde{e}(t)$ estimates of the energy density. The data shown correspond to three different relaxations\\index{relaxation}. Two of them are isothermal\\index{isothermal} relaxations\\index{relaxation} starting from $\\xi=0$ at $t=0$. The third relaxation\\index{relaxation} corresponds to the preparation starting at $(T=0.9,\\xi=5)$ which is quenched to $T=0.7$ at $t=0$. The error reduction is largest for the isothermal\\index{isothermal} relaxation\\index{relaxation} at $T=0.7$ and $2^{19}\\leq t\\leq 2^{21}$, of course (after all, this is the temperature and time region defining the control\\index{control variate} variate), but the error reduction is also very significant at other times and temperatures.}\n\\labfig{control_variate}\n\\end{figure}\n\n\n\\chapter{The aging rate. Technical details.} \\labch{AP_technical_details_aging}\nIn this appendix, we expose the technical details that are fundamental to understand the procedure followed in~\\refch{aging_rate} but that would blur the mean message of the chapter and would make reading difficult.\n\nIn~\\refsec{Nr_aging} we explore the error reduction for the four-point\\index{correlation function!four point} correlation function when a high number of replicas\\index{replica} is simulated. We perform exhaustive checks to our $L=160$ system to be sure that our results are not affected by finite-size effects\\index{finite-size effects} in~\\refsec{finite_size_effects}. Then, in~\\refsec{Josephson_length}, we give additional details of the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ fixed point and the $T=0$ one, which is controlled by the Josephson length. Finally, we discuss the choices of the parameters in our fits in~\\refsec{parameters_aging}.\n\n\\section{Error reduction for high number of replicas}\\labsec{Nr_aging}\nAs mentioned in~\\refch{aging_rate}, the choice of the number of replicas\\index{replica} ($\\ensuremath{N_{\\text{Rep}}}\\xspace$) and samples\\index{sample} ($\\ensuremath{N_{\\text{S}}}\\xspace$) was taken with the aim of improving the estimation of observables related to \\gls{TC}\\index{temperature chaos} in future work, where it is important to maximize the number of possible overlaps\\index{overlap} (pairs of replicas\\index{replica}) $N_\\text{ov}=\\ensuremath{N_{\\text{Rep}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace - 1)\/2$.\n\nUnexpectedly, this has led to a dramatic increase in precision.~\\reffig{error-NR} shows the reduction of the statistical error in the correlation function $C_4$ as a function of $1\/\\sqrt{\\ensuremath{N_{\\text{ov}}}\\xspace}$. Moreover, this effect is enhanced as $r$ increases, which leads to a qualitative improvement in the computation of the $I_k(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ integrals (\\reffig{error-r2Cr}).\n\nThe reader should be alert that, in view of the above results, a previous study can be done for any quantity which involves disorder\\index{disorder!average} averages and thermal averages. The general form of the variance of such a quantity would be\n\\begin{equation}\n\\sigma^2 (\\ensuremath{N_{\\text{Rep}}}\\xspace,\\ensuremath{N_{\\text{S}}}\\xspace) = \\left[ \\sigma_{\\mathrm{S}}^2 + \\sigma_{\\mathrm{R}}^2 \\left( \\dfrac{2}{\\ensuremath{N_{\\text{Rep}}}\\xspace (\\ensuremath{N_{\\text{Rep}}}\\xspace-1)}\\right)^x \\right] \\dfrac{1}{\\ensuremath{N_{\\text{S}}}\\xspace} \\, ,\n\\end{equation}\nwhere $\\sigma_{\\mathrm{S}}^2$ and $\\sigma_{\\mathrm{R}}^2$ are the sample\\index{sample} and thermal contributions to the variance respectively and $x$ is the exponent controlling the error reduction due to the number of overlaps\\index{overlap} $\\ensuremath{N_{\\text{ov}}}\\xspace$ which should be between $0.5$ and $1$.\n\nA careful analysis in the context of thin-film \\gls{SG}s can be found in \\cite{fernandez:19b}. The great advantage of this analysis is that we can choose, for a given computational effort $E=\\ensuremath{N_{\\text{Rep}}}\\xspace \\ensuremath{N_{\\text{S}}}\\xspace$, the number of replicas\\index{replica} (or equivalently, of samples\\index{sample}) that minimizes the error instead of assuming that the maximum possible number of samples\\index{sample} should be simulated (as it was usual).\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/error-NR_2w}\n\\caption[\\textbf{Reduction of statistical error with the number of overlaps.}]{\\textbf{Reduction of statistical error with the number of overlaps.\\index{overlap}} Rapid reduction in the statistical error for the overlap\\index{overlap} autocorrelation function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with increasing number of replicas\\index{replica} $N_\\text{R}$. We plot $\\Delta C(r)$ for $T=0.7$ and $\\ensuremath{t_\\mathrm{w}}\\xspace=2^{32}$ and several values of $r$ as a function of the number of possible overlaps\\index{overlap} [$N_\\text{ov} = N_\\text{R}(N_\\text{R}-1)\/2$]. For large values of $r$ the error is essentially linear in $1\/N_\\text{R}$. The inset shows the whole range from $N_\\text{R}=2$ (the minimum to define $C_4$), while the main plot is a close up of the large-$N_\\text{R}$ sector. The simulations reported in this paper have $N_\\text{R}=256$.}\n\\labfig{error-NR}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.7\\linewidth]{aging\/error-r2Cr}\n\\caption[\\textbf{Improving the determination of \\boldmath $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Improving the determination of \\boldmath $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Comparison of $r^2 C_4(r)$, function whose integral is used to estimate the coherence length\\index{coherence length}, as computed with $4$ and $256$ replicas\\index{replica} and the same number $N_\\text{S}=16$ of samples\\index{sample} ($T=0.7, \\ensuremath{t_\\mathrm{w}}\\xspace=2^{34}$). We have marked on the $x$ axis the resulting self-consistent cutoffs (see \\refsec{finite_size_effects}) used to estimate the $I_k$ integrals in both cases.}\n\\labfig{error-r2Cr}\n\\end{figure}\n\n\\section{Controlling finite-size effects} \\labsec{finite_size_effects}\nIn order to obtain an estimate for $\\xi_{k,k+1}(T,\\ensuremath{t_\\mathrm{w}}\\xspace) = I_{k+1}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)\/I_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ we need to compute the integrals \\begin{equation}\nI_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace) = \\int_0^\\infty r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace){\\mathrm{d}} r \\, . \\labeq{I_SI} \n\\end{equation}\nAs discussed in~\\cite{janus:09b}, the main difficulty  in this computation is handling the large-$r$ tail where relative errors [$\\Delta C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) \/ C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$] are big. Our main goal in this part is to minimize the statistical errors and to check for finite-size effects\\index{finite-size effects} (which will appear when $\\xi\/L$ becomes relatively large).\n\nTo compute those $I_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, we proceed in a similar way as in~\\cite{janus:09b,fernandez:18b}. We separate the integration range in two parts. From $0$ to $r_{\\mathrm{cutoff}}$, we compute the numerical integral of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. From $r_{\\mathrm{cutoff}}$ to infinity, we estimate the contribution of the tail with a smooth extrapolating function $F(r) \\sim r^{-\\vartheta} f\\bigl(r\/\\xi)$.\n\nIn short, the procedure is\n\\begin{enumerate}\n\\item Obtain $F(r)$ with fits of $C_4$ in a self-consistent region $[r_\\text{min},r_\\text{max}]$  where the signal-to-noise ratio is still good. \n\\item Integrate $C_4$ numerically up to some cutoff and add the analytical integral of $F(r)$  beyond the\ncutoff to estimate the tail contribution.\n\\end{enumerate}\nThe selection of the function $F(r)$ can be done in several ways, here, we propose two particular functions. First, we consider\n\\begin{equation}\nF_1(r) = A_1 r^{-\\vartheta} {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_1}} \\, . \\labeq{F1}\n\\end{equation}\nWhere $\\vartheta$ is the replicon\\index{replicon} exponent and we fit for $A_1,\\beta_1$ and $\\xi$. This analytical form is motivated by the fact that this is the simplest choice that avoids a pole singularity in the Fourier transform of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ at finite $\\ensuremath{t_\\mathrm{w}}\\xspace$ and has been used in several previous works, for example, see~\\cite{janus:09b}. In order to check for finite-size effects\\index{finite-size effects}, we also consider a second function $F_2(r)$ resulting from fits that include the first-image term:\n\\begin{equation}\nF_2^*(r) = A_2 \\left[ \\frac{ {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_2}}}{r^\\vartheta} + \\frac{ {\\mathrm{e}}^{-((L-r)\/\\xi)^{\\beta_2}}}{(L-r)^\\vartheta}\\right],\n\\end{equation}\nso we have a second extrapolating function\n\\begin{equation}\\labeq{F2}\nF_2(r) = A_2 r^{-\\vartheta} {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_2}}.\n\\end{equation}\nFor these fits we used $\\vartheta=0.35$. However, this value has very little effect on the final computation of $\\xi_{k,k+1}$. We have checked this by recomputing the integrals with $\\vartheta = 0 $ and $\\vartheta = 1+\\eta\\approx 0.61$ \n(prediction of the droplet\\index{droplet!picture} theory and influence of the $T=T_\\mathrm{c}$ fixed point respectively). The different choices of $\\vartheta$  led to a systematic effect smaller than $20 \\%$ of the error bars\\index{error bars} in the worst case. \n\nOnce we have our two extrapolating functions $F_1$ and $F_2$, we can combine them with the $C_4$ data in several ways:\n\\begin{eqnarray}\nI^1_k &=& \\int_0^{r_{\\mathrm{max}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{max}}}^\\infty {\\mathrm{d}} r~r^k F_1(r) \\, , \\\\\nI^2_k &=& \\int_0^{r_{\\mathrm{max}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{max}}}^\\infty {\\mathrm{d}} r~r^k F_2(r) \\, , \\\\\nI^3_k &=& \\int_0^{r_{\\mathrm{min}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{min}}}^\\infty {\\mathrm{d}} r~r^k F_2(r) \\, . \\labeq{I3}\n\\end{eqnarray}\n\nThe difference between $I^2_k$ and $I^1_k$ is always under $1\\%$ in the error, so choosing between them has no effect in any computation. In contrast, $I^1_k$ and $I^3_k$ present measurable differences for long $\\ensuremath{t_\\mathrm{w}}\\xspace$, at least for our highest temperatures, where the faster dynamics allows us to reach higher values of $\\xi_{12}\/L$. As a (very conservative) cutoff we have discarded all the $\\ensuremath{t_\\mathrm{w}}\\xspace$ where $|I^1-I^3|$ is larger than $20\\%$ of the error bar\\index{error bars}, thus obtaining a $\\xi_{12}^\\text{max}(T)$ below which we are sure we do not have any finite-size effects\\index{finite-size effects} in our $L=160$ systems. The reader can find the values in~\\reftab{selected_ximax}.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{cccccccc}\n\\toprule\n\\toprule\n$T$ & $0.55$ & $0.625$ & $0.7$ & $0.8$ & $0.9$ & $1.0$ & $1.1$  \\\\\n$\\xi_\\mathrm{max}$ & - & - & - & - & $18.1$ & $17.3$ & $17$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Cutoff values of \\boldmath $\\xi_{12}^\\mathrm{max}(T)$.}]{\\textbf{Cutoff values of \\boldmath $\\xi_{12}^\\mathrm{max}(T)$.} Cutoff values of $\\xi_{12}^\\mathrm{max}(T)$ below which we are sure that no finite-size effects\\index{finite-size effects} are present in our $L=160$ lattices. For $T<0.9$ the growth of $\\xi_{12}$ is very slow and we never reach the cutoff value.}\n\\labtab{selected_ximax}\n\\end{table}\n\nAs a final check that our data is not affected by finite-size effects\\index{finite-size effects}, we have compared our $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with that of~\\cite{fernandez:15}. This reference considers shorter simulations but with $L=256$ and $50$ samples\\index{sample}. As shown in~\\reffig{xi256}, the $L=160$ and $L=256$ data coincide even  beyond our cutoff $\\xi_{12}^\\text{max}$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.7\\linewidth]{aging\/xi12_conL256}\n\\caption[\\textbf{Comparision of \\boldmath $\\xi_{12}$.}]{\\textbf{Comparision of \\boldmath $\\xi_{12}$.} Comparison between the $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ computed in $L=160$ lattices with $N_\\text{S}=16$ samples\\index{sample} and $N_\\text{R}=256$ replicas\\index{replica}, and that of~\\cite{fernandez:15} ($L=256$; $N_\\text{S}=50$; $N_\\text{R}=4$  for $T=0.9,1.0$ and $N_\\text{R}=8$ for $T=1.1$). For the $L=160$ simulations, we plot with two parallel lines the error interval for $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$. Only the \\ensuremath{t_\\mathrm{w}}\\xspace range depicted with  continuous lines is used in the paper, the  extension with dashed lines represents the discarded times with $\\xi_{12}>\\xi_{12}^\\text{max}$ (see~\\reftab{selected_ximax}). These curves were generated with the $I^3_k$ estimator for the  integrals~\\refeq{I3}. The values from the $L=256$ simulations are plotted with conventional error bars\\index{error bars}. Notice that both curves are compatible even beyond this cutoff.}\n\\labfig{xi256}\n\\end{figure}\n\n\\section{Josephson length} \\labsec{Josephson_length}\nThe Josephson length $\\ell_J$ is expected to grow as $\\ell_J(T) \\sim (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu}$ with $\\nu = 2.56(4)$~\\cite{janus:13} for temperatures close to $\\ensuremath{T_\\mathrm{c}}\\xspace$. Scaling corrections are expected for lower temperatures\n\\begin{equation}\n\\ell_J(T) = (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu} \\left[ a_0 + a_1(\\ensuremath{T_\\mathrm{c}}\\xspace-T)^{\\nu} + a_2(\\ensuremath{T_\\mathrm{c}}\\xspace-T)^{\\omega \\nu}   \\right] \\, , \\labeq{josephson_length}\n\\end{equation}\nwhere $a_0$, $a_1$ and $a_2$ are coefficients chosen to perform the best collapse in \\reffig{vartheta_collapse} aging\\index{aging} and $\\omega=1.12(10)$~\\cite{janus:13}.\n\nAssuming $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace) \\gg \\ell_J(T)$, the crossover from the fixed point at $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the fixed point at $T=0$ is affecting our basic quantity $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ in the following way\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) \\sim\n\\begin{cases}\n  \\displaystyle \\dfrac{1}{r^{D-2+\\eta}}\\,,  & r\\ll \\ell_\\text{J}(T)\\,, \\\\[3mm]\n  \\displaystyle \\dfrac{\\ell_\\text{J}^\\vartheta}{\\ell_\\text{J}^{D-2+\\eta}} \\dfrac{1}{r^\\vartheta} f(r\/\\xi)\\,, & r \\gg \\ell_\\text{J}(T) \\,.\n\\end{cases} \\labeq{C4_josephson}\n\\end{equation}\nThe prefactor $\\ell_J^{\\vartheta}\/\\ell_\\text{J}^{D-2+\\eta}$ is fixed by the condition that the two asymptotic expansions in $r$ connect smoothly at $r=\\ell_J$.\n\nFrom this expression, we arrive at an asymptotic expansion for the $I_k$ integrals\n\\begin{equation}\nI_k = \\int_0^{\\infty} r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) {\\mathrm{d}} r = \\dfrac{F_k}{\\ell_\\text{J}^{D-2+\\eta}} \\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{k+1-\\vartheta} \\left[ 1 + a_k\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{k+1-\\vartheta}+\\ldots\\right]\\,, \\labeq{Ik_expansion}\n\\end{equation}\nwhere $F_k$ and $a_k$ are amplitudes.\n\nFinally, we need to eliminate the unknown $\\xi$ in favor of the computable $\\xi_{12}$,\n\\begin{equation}\n\\xi_{12}(T,\\xi) = \\dfrac{F_2}{F_1} \\xi \\left[ 1+ a_1'\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{2-\\vartheta} + a_2\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{3-\\vartheta}+\\ldots\\right]\\,,\n\\end{equation}\nwhere $a_1'$ considers contributions both from the numerator ($-a_1$) and  from the denominator.  The easiest way to obtain $\\vartheta$ is to study the evolution of $\\log I_2$ as a function of $\\log \\xi$. However, we have to settle for using $\\log\\xi_{12}$ as independent variable (see~\\reffig{I2xi}).\n\n\nWe can define an effective $\\vartheta(T,\\xi_{12})$ as\n\\begin{equation}\n\\vartheta(T,\\xi_{12}) = 3 - \\dfrac{{\\mathrm{d}} \\log I_2(T,\\xi_{12})}{{\\mathrm{d}} \\log \\xi_{12}} = \\vartheta + b_2 \\left(\\dfrac{\\xi_{12}}{\\ell_\\text{J}}\\right)^{\\vartheta-2}+b_3 \\left(\\dfrac{\\xi_{12}}{\\ell_\\text{J}}\\right)^{\\vartheta-3}+\\ldots\\, . \\labeq{fit}\n\\end{equation}\nTo estimate this derivative for a given $\\xi_{12}^*$, we  fit $\\log I_2$ to a quadratic polynomial in $\\log \\xi_{12}$  in a $[0.75\\xi_{12}^*,1.25\\xi_{12}^*]$ window. We then take the derivative of this polynomial at $\\xi^*$. The procedure, as well as the wiggles in the resulting values of $\\vartheta$ due to the extreme data correlation (see~\\reffig{replicon_sin_reescalar}), may remind the reader of Fig.~1 in~\\cite{janus:08b}.\n\nWe have computed a fit to the first two terms in~\\refeq{fit} in the range $0\\leq \\ell_\\text{J}\/\\xi_{12}\\leq 0.33$, resulting in the value of $\\vartheta\\approx 0.30$ reported in the main text.\n\nThe previous analysis solves the problem of the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and $T=0$ fixed points. However, in the framework of the droplet\\index{droplet!picture} picture, one would also need to consider corrections to scaling at the $T=0$ fixed point. This is precisely what the droplet\\index{droplet!picture} fit in the main text to $\\vartheta(x) \\simeq Cx^\\zeta$ does.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\linewidth]{aging\/I2xi}\n\\caption[\\textbf{Integral \\boldmath $I_2$ as a function of $\\xi_{12}$ in a logarithmic scale, for all our $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ temperatures.}]{\\textbf{Integral \\boldmath $I_2$ as a function of $\\xi_{12}$ in a logarithmic scale, for all our $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ temperatures.} We use the numerical derivative of this curve to compute the replicon\\index{replicon} exponent $\\vartheta$.}\n\\labfig{I2xi}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\linewidth]{aging\/theta-xi12}\n\\caption[\\textbf{The replicon \\boldmath $\\vartheta(T,\\xi_{12})$.}]{\\textbf{The replicon\\index{replicon} \\boldmath $\\vartheta(T,\\xi_{12})$.} Value of the replicon\\index{replicon} exponent $\\vartheta(T,\\xi_{12})$ computed from a numerical derivative of $\\log I_2$ as a function of $\\log \\xi_{12}$, nicely illustrating the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and  $T=0$ fixed points.}\n\\labfig{replicon_sin_reescalar}\n\\end{figure}\n\n\n\n\\section{Parameter choices in our fits}\\labsec{parameters_aging}\nWe will discuss separately the choice of $\\xi_{\\mathrm{min}}$ for different\ntemperatures and the choice of the value of $\\omega$.\n\n\\subsection[Selection of $\\xi_{12}^{\\mathrm{min}}$ for each temperature]{Selection of \\boldmath $\\xi_{12}^{\\mathrm{min}}$ for each temperature}\nWe have reported fits of our data to three different functional forms\n\n\\begin{align}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace&= a_0(T) + a_1(T) \\log \\xi_{12} + a_2(T) \\log^2 \\xi_{12},\\\\\n\\log t_\\mathrm{w} &= C_1(T) + z_\\infty(T) \\log \\xi_{12} + C_2(T) \\xi_{12}^{-\\omega} \\, , \\labeq{RSB}\\\\\n\\log t_\\mathrm{w} &= D_1(T) + z_\\mathrm{c} \\log \\xi_{12} + D_2(T) \\xi_{12}^{\\psi} \\, .\n\\labeq{Bouchaud}\n\\end{align}\n\nIn these fits we have used $z_\\text{c}=6.69$ and $\\omega=0.35$ ($T<\\ensuremath{T_\\mathrm{c}}\\xspace$), $\\omega=1.12$ ($T=\\ensuremath{T_\\mathrm{c}}\\xspace$), as discussed in the~\\refch{aging_rate}. Full results for the fits to~\\refeq{RSB} and~\\refeq{Bouchaud} can be seen in~\\reftab{RSB_omega_0.35} and~\\reftab{Saclay_6.69}, for different fitting ranges. We include for both cases the extrapolated values of $z(T,\\xi)$ for the experimental scale (as explained in the~\\refch{aging_rate} we use both $\\xi_{12}=38$ and $\\xi_{12}=76$) and for~\\refeq{RSB} also the value of the $\\xi\\to\\infty$ aging\\index{aging!rate} rate $z_\\infty$.\n\nIn order to make the choice of the fitting range, we have followed two criteria. Firstly we require the parameters of the fit to be stable inside the error when we increase $\\xi_{12}^\\mathrm{min}$. Secondly, we impose $\\xi_\\mathrm{min}$ to be monotonically increasing in $T$ (with the exception of \\ensuremath{T_\\mathrm{c}}\\xspace, which has different behavior).~\\reftab{selected_ximin} shows our final choices for $\\xi_{12}^\\text{min}(T)$, which is the same for all three fits.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{cccccccc}\n\\toprule\n\\toprule\n$T$ & $0.55$ & $0.625$ & $0.7$ & $0.8$ & $0.9$ & $1.0$ & $1.1$  \\\\\n$\\xi_{12}^\\mathrm{min}$ & $4$ & $5$ & $6$ & $8$ & $8$ & $9$ & $5$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Values of $\\xi_{12}^\\text{min}(T)$ determining the common fitting range $\\xi_{12}\\geq\\xi_{12}^\\text{min}$ for our three different fits of $\\log\\ensuremath{t_\\mathrm{w}}\\xspace$ as a function of $\\log \\xi_{12}$.}\n\\labtab{selected_ximin}\n\\end{table}\n\n\\subsection[Selection of $\\omega$]{Selection of \\boldmath $\\omega$}\nFor our most important result, namely the extrapolation of the aging\\index{aging!rate} rate to the experimental scale of $\\xi_{12}=38,76$, we have repeated our fits with our upper and lower bounds for $\\omega=\\vartheta(\\xi_\\text{films})$ (\\gls{RSB}\\index{replica!symmetry breaking (RSB)} and droplet\\index{droplet!picture} extrapolations, respectively).  The results are completely compatible, as we can see in~\\reftab{omega_xi38}.\n\n\\begin{table}[h!]\n\\begin{tabular}{lccccc}\n\\toprule\n\\toprule\n& \\multicolumn{2}{c}{$z(T,\\xi_{12}=38)$} & & \\multicolumn{2}{c}{$z(T,\\xi_{12}=76)$} \\\\ \n       & $\\omega = 0.35$    & \\multicolumn{1}{c}{$\\omega = 0.28$} & \n       & $\\omega = 0.35$    & \\multicolumn{1}{c}{$\\omega = 0.25$} \\\\ \n\\hline\n$T=0.55$  & 19.80(20) & 20.08(22) &  & 20.75(24) & 21.41(27)                    \\\\\n$T=0.625$ & 16.90(19)  & 17.07(20)& & 17.69(24)  & 18.13(27)                   \\\\\n$T=0.7$   & 14.81(15) & 14.93(16)&  & 15.54(19) & 15.87(21)                   \\\\\n$T=0.8$   & 12.73(22) & 12.81(23)&  & 13.47(30) & 13.71(32)                    \\\\\n$T=0.9$   & 10.55(25) & 10.61(26)&  & 11.11(34) & 11.28(37)                   \\\\\n$T=1.0$   & 8.63(32) & 8.68(33)  &  & 8.98(44) & 9.02(42)                  \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Estimations of the aging rate.}]{\\textbf{Estimations of the aging rate.} Comparison of our estimates of the experimental aging\\index{aging!rate} rate $z(T,\\xi_{12}=\\xi_\\text{films})$ for $\\xi_\\text{films}=38$ and $\\xi_\\text{films}=76$ using our lower and upper bounds for $\\omega=\\vartheta(\\xi_\\text{films})$. The choice of $\\omega$ is immaterial, since even in the worst case (lowest temperatures for $\\xi_\\text{films}=76$) there is  only a two-sigma difference.}\n\\labtab{omega_xi38}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{lcccccccc}\n\\toprule\n\\toprule\n& & $\\xi_{\\mathrm{min}}$= 3.5 & $\\xi_{\\mathrm{min}}$= 4 & $\\xi_{\\mathrm{min}}$= 5 & $\\xi_{\\mathrm{min}}$= 6 & $\\xi_{\\mathrm{min}}$= 7 & $\\xi_{\\mathrm{min}}$= 8 & $\\xi_{\\mathrm{min}}$= 9 \\\\ \\hline\n\\multirow{4}{*}{{$T=0.55$}}&\n$z_\\infty$ & 23.61(28)   & \\bfseries 24.22(40)   & 25.30(86)   & 22.9(31)   & & &   \\\\ \n&$z (\\xi\\!=\\!38)$ & 19.49(15)   &  \\bfseries 19.80(20)   & 20.32(41)  & 19.2(14)   & & &   \\\\\n&$z (\\xi\\!=\\!76)$ & 20.38(18)   &  \\bfseries 20.75(24)   & 21.39(51)   & 20.0(18)   & & &   \\\\\n&$\\chi^2$\/dof &  $40(17)\/133$ &  \\bfseries 10.2(54)\/111 & 3.0(12)\/73 & 1.71(76)\/40  &  &  & \\\\ \n\\hline\n\\multirow{4}{*}{{$T=0.625$}}&\n$z_\\infty$ & 19.85(17)   & 20.26(23)   &  \\bfseries 20.60(41)   & 20.16(84)   &  &  &  \\\\ \n&$z (\\xi\\!=\\!38)$ & 16.538(91)   & 16.74(12)   &  \\bfseries 16.90(19)   & 16.71(37)   &  &  &  \\\\\n&$z (\\xi\\!=\\!76)$ & 17.25(11)   & 17.50(14)   &  \\bfseries 17.69(24)   & 17.45(47)   &  &  &  \\\\\n&$\\chi^2$\/dof & 81(34)\/167\n &  18(10)\/147\n &  \\bfseries  8.3(21)\/114\n&  5.1(19)\/86\n&  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.7$}}&\n$z_\\infty$ & 17.04(18)   & 17.23(21)   & 17.61(27)   &  \\bfseries 18.23(35)   & 18.63(62)   &  &  \\\\ \n&$z (\\xi\\!=\\!38)$ & 14.295(88)   & 14.38(11)   & 14.55(13)   &  \\bfseries 14.81(15)   & 14.96(25)&  &  \\\\\n&$z (\\xi\\!=\\!76)$ & 14.89(11)   & 15.00(13)   & 15.21(16)   &  \\bfseries 15.54(19)   & 15.75(32)   &  &  \\\\ \n&$\\chi^2$\/dof & 116(40)\/190 & 66(36)\/173 & 33(24)\/144 &  \\bfseries 9.3(84)\/119 & 4.9(21)\/98\n &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.8$}}&\n$z_\\infty$ & 13.76(15)   & 14.06(19) & 14.53(26)   & 15.19(35)   & 15.68(42)   &  \\bfseries 16.18(58)   & 16.55(78)   \\\\ \n& $z (\\xi\\!=\\!38)$ & 11.787(73)   & 11.921(89)   & 12.11(12)   & 12.37(15)   & 12.55(17)   & \\bfseries  12.73(22)   & 12.85(28)   \\\\\n& $z (\\xi\\!=\\!76)$ & 12.211(93)   & 12.38(11)   & 12.63(15)   & 12.98(19)   & 13.23(23)   & \\bfseries  13.47(30)   & 13.65(39)   \\\\ \n& $\\chi^2$\/dof & 351(104)\/185 & 188(72)\/170& 93(41)\/146& 27(16)\/125& 12.4(82)\/107\n& \\bfseries  6.2(31)\/91\n& 4.9(21)\/77\n\\\\ \\hline\n\\multirow{4}{*}{{$T=0.9$}}&\n$z_\\infty$ & 11.00(13)  & 11.29(18)   & 11.54(24)   & 11.80(31)   & 12.55(41)   & \\bfseries  13.16(68)   & 12.3(13)   \\\\ \n& $z (\\xi\\!=\\!38)$ & 9.748(65)   & 9.883(93)   & 9.98(11)   & 10.08(13)   & 10.34(16)   & \\bfseries  10.55(25)   & 10.33(41)   \\\\\n& $z (\\xi\\!=\\!76)$ & 10.017(82)   & 10.18(11)   & 10.32(14)   & 10.45(17)   & 10.82(21)   &  \\bfseries 11.11(34)   & 10.80(60)   \\\\ \n& $\\chi^2$\/dof & 310(150)\/165 & 129(64)\/152 & 79(44)\/131 & 63(35)\/113 & 22(13)\/98\n & \\bfseries  5.9(21)\/84\n & 6.4(79)\/72\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.0$}}&\n$z_\\infty$ & 8.60(11)   & 8.69(15)   & 8.83(20)   & 8.97(53)   & 9.29(45)   & 9.36(46)   &  \\bfseries 10.28(89)   \\\\\n& $z (\\xi\\!=\\!38)$ & 8.041(59)   & 8.080(73)   & 8.132(93)   & 8.21(22)   & 8.27(18)   & 8.34(17)   &  \\bfseries 8.63(32)   \\\\ \n& $z (\\xi\\!=\\!76)$ & 8.162(74) & 8.210(86) & 8.28(11) & 8.38(29) & 8.46(24) & 8.57(24) & \\bfseries  8.98(44)   \\\\ \n& $\\chi^2$\/dof & 43(30)\/137\n & 27(18)\/126\n & 16(13)\/107\n & 12(25)\/91\n & 10.4(91)\/78\n & 8.4(60)\/66\n &  \\bfseries 2.9(21)\/55\n\\\\ \\hline\n\\multirow{4}{*}{{$T=1.1$}}&\n$z_\\infty$ & 6.672(44) & 6.671(41) & \\bfseries  6.689(63) & 6.751(84) & 6.80(12) & 7.00(18) & 7.02(21)  \\\\\n& $z (\\xi\\!=\\!38)$ & 6.682(32) & 6.673(41) & \\bfseries  6.694(50) & 6.732(68) & 6.77(10) & 6.92(14) & 6.94(16)   \\\\\n& $z (\\xi\\!=\\!76)$ & 6.677(33) & 6.671(41) & \\bfseries  6.691(54) & 6.742(72) & 6.79(11) & 6.96(16) & 6.99(16)   \\\\\n& $\\chi^2$\/dof & 32(19)\/119\n & 31(20)\/109\n &  \\bfseries 26(16)\/92\n & 19(10)\/78\n & 16.8(74)\/66\n & 5.9(20)\/55\n & 6.3(27)\/46 \\\\ \n \\bottomrule\n\\end{tabular}}\n\\caption[\\textbf{Parameters for the convergent ansatz.}]{\\textbf{Parameters for the convergent ansatz.}Parameters of the fits to~\\refeq{RSB} for different fitting ranges $\\xi_{12}\\geq\\xi_{12}^\\text{min}$.  We use $\\omega = 0.35$ ($\\omega = 1.12$ for $T = T_\\mathrm{c}$). The fitting range that we choose for our final values is highlighted in boldface.}\n\\labtab{RSB_omega_0.35}\n\\end{table}\n\n\\begin{table}[h!]\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{lcccccccc}\n\\toprule\n\\toprule\n& & $\\xi_{\\mathrm{min}}$= 3.5 & $\\xi_{\\mathrm{min}}$= 4 & $\\xi_{\\mathrm{min}}$= 5 & $\\xi_{\\mathrm{min}}$= 6 & $\\xi_{\\mathrm{min}}$= 7 & $\\xi_{\\mathrm{min}}$= 8 & $\\xi_{\\mathrm{min}}$= 9 \\\\ \\hline\n\n\\multirow{4}{*}{{$T=0.55$}}& \n$z(\\xi\\!=\\!38)$ & 24.07(41)   & \\bfseries24.25(55)  & 24.6(11)& 24.7(81)  &  &  & \\\\\n&$z(\\xi\\!=\\!76)$ & 28.86(69)   & 2\\bfseries9.18(95)  & 29.9(19)   & 30(15)  &  &  & \\\\\n& $G(T)$ & 13.78(65)   & \\bfseries13.45(92) & 12.8(18)  & 18(13) &  &  &  \\\\ \n& $\\psi$ & 0.3512(92)   & \\bfseries0.355(21)   & 0.372(33)   & 0.29(24)  &  &  & \\\\\n& $\\chi^2$\/dof & 13.3(47)\/133\n & \\bfseries6.8(20)\/111\n & 3.2(15)\/73\n & 1.7(27)\/40\n &  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.625$}}& \n $z(\\xi\\!=\\!38)$ & 19.73(22)   & 19.72(28)   & \\bfseries19.36(45)   & 18.53(77)  &  &  & \\\\\n& $z(\\xi\\!=\\!76)$ & 23.33(38)   & 23.31(49)   & \\bfseries22.66(79)   & 21.1(13)  &  &  & \\\\\n& $G(T)$ & 10.36(37)   & 10.39(52)   & \\bfseries11.3(11)   & 14.0(30)  &  &  & \\\\\n& $\\psi$ & 0.354(14)   & 0.352(12)   & \\bfseries0.334(21)   & 0.290(39)  &  &  &  \\\\\n& $\\chi^2$\/dof & 19(10)\/167\n & 15(10)\/147\n & \\bfseries8.5(33)\/114\n & 4.5(14)\/86\n &  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.7$}}& \n$z(\\xi\\!=\\!38)$ & 16.58(22)   & 16.44(23)   & 16.35(27)   &\\bfseries 16.51(32)   & 16.55(52) &  &   \\\\ \n& $z(\\xi\\!=\\!76)$ & 19.40(37)   & 19.14(40)   & 18.98(47)   &\\bfseries 19.29(58)   & 19.4(10)  &  &  \\\\\n& $G(T)$ & 7.32(34)   & 7.63(43)   & 7.84(59)   &\\bfseries 7.41(75)   & 7.3(14)   &  & \\\\ \n& $\\psi$ & 0.364(13)   & 0.354(12)   & 0.354(24)   &\\bfseries 0.358(23)   & 0.360(39)   &  & \\\\\n& $\\chi^2$\/dof & 49(38)\/190\n & 28(20)\/173\n & 10.5(83)\/144\n & \\bfseries6.3(31)\/119\n & 5.7(29)\/98\n &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.8$}}& \n$z(\\xi\\!=\\!38)$ & 13.37(18)   & 13.39(21) & 13.45(25) & 13.68(31) & 13.80(35) & \\bfseries13.94(45) & 14.1(17)   \\\\\n& $z(\\xi\\!=\\!76)$ & 15.44(33) & 15.48(37) & 15.60(46) & 16.06(59)   & 16.31(68)   & \\bfseries16.59(93)   & 17.1(38)   \\\\ \n& $G(T)$ & 4.16(25)   & 4.13(29)   & 4.01(38)   & 3.57(43)   & 3.36(48)   & \\bfseries3.13(66)   & 3.0(19)   \\\\\n& $\\psi$ & 0.392(13) & 0.390(19) & 0.395(18)   & 0.421(27) & 0.443(31) & \\bfseries0.447(50) & 0.46(18) \\\\ \n& $\\chi^2$\/dof & 31(19)\/185\n & 29(19)\/170\n & 22(17)\/146\n & 10.0(60)\/125\n & 7.5(34)\/107\n & \\bfseries5.5(21)\/91\n & 5(11)\/77\n \\\\ \\hline\n\\multirow{4}{*}{{$T=0.9$}}& \n$z(\\xi\\!=\\!38)$ & 10.76(17) & 10.86(21) & 10.82(24) & 10.86(27) & 11.23(35) & \\bfseries11.49(54) & 11.13(56) \\\\\n& $z(\\xi\\!=\\!76)$ & 12.12(30) & 12.31(39) & 12.24(45) & 12.31(53) & 13.12(73) & \\bfseries13.7(12)   & 12.9(12)  \\\\\n& $G(T)$ & 2.15(19) & 2.01(23) & 2.07(31) & 2.00(39) & 1.52(31)   & \\bfseries1.18(45)   & 1.67(77)  \\\\\n& $\\psi$ & 0.417(23)  & 0.430(34) & 0.431(28) & 0.427(41) & 0.490(51) & \\bfseries0.546(88) & 0.47(10) \\\\\n& $\\chi^2$\/dof & 68(44)\/165\n & 46(25)\/152\n & 41(25)\/131\n & 38(24)\/113\n & 17(10)\/98\n & \\bfseries8.7(45)\/84\n & 4.8(25)\/72\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.0$}}& \n$z(\\xi\\!=\\!38)$ & 8.53(15) & 8.54(18) & 8.55(20) & 8.56(30) & 8.59(60) & 8.74(72) &\\bfseries 9.22(18)  \\\\\n& $z(\\xi\\!=\\!76)$ & 9.19(27) & 9.20(33) & 9.22(39) & 9.25(58) & 9.3(12) & 9.7(16) &\\bfseries 10.9(45)  \\\\\n& $G(T)$ & 0.85(14) & 0.84(18) & 0.83(22) & 0.79(40) & 0.7(10)   & 0.5(10) &\\bfseries 1.4(19)  \\\\\n& $\\psi$ & 0.440(36)   & 0.441(51)   & 0.444(64) & 0.45(10) & 0.49(25) & 0.55(30)   & \\bfseries0.34(71) \\\\ \n& $\\chi^2$\/dof & 12.6(95)\/137\n & 12.1(90)\/126\n & 10.0(87)\/107\n & 9.3(83)\/91\n & 8.2(97)\/78\n & 07(11)\/66\n & \\bfseries11(11)\/55\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.1$}}& \n$z(\\xi\\!=\\!38)$ & 6.684(12)   & 6.682(14)   &\\bfseries 6.684(11)   & 6.672(31)   & 6.683(32)& 6.694(31)& 6.712(41)   \\\\ \n& $z(\\xi\\!=\\!76)$ & 6.684(13)   & 6.681(11)   & \\bfseries 6.682(14)   & 6.674(41)   & 6.684(41)& 6.692(31)& 6.721(42)  \\\\ \n& $G(T)$ & 1.9(10)   & 1.71(91) & \\bfseries0.4(27) & 0.02(64)  & 0.0(26)   & 1.5(26)   & 1.2(10)   \\\\ \n& $\\psi$ & -0.0030(49)   & 0.0037(68)   &\\bfseries 0.03(15)   & 0.29(42)   & 0.37(55)   & 0.002(16)   & 0.023(51)   \\\\\n& $\\chi^2$\/dof & 34(20)\/119\n & 33(19)\/109\n &\\bfseries 27(18)\/92\n & 25(18)\/78\n & 23(15)\/66\n & 21(14)\/55\n & 11.0(26)\/46\n \\\\ \n \\bottomrule\n\\end{tabular}}\n\\caption[\\textbf{Parameters for the divergent ansatz.}]{\\textbf{Parameters for the divergent ansatz.}Parameters of the fits to ~\\refeq{Bouchaud} for different fitting ranges $\\xi_{12}\\geq\\xi_{12}^\\text{min}$.  We use $z_\\text{c} = 6.69$. The fitting range that we choose for our final values is highlighted in boldface.}\n\\labtab{Saclay_6.69}\n\\end{table}\n\n\n\n\n\\chapter{Temperature Chaos. Technical details.} \\labch{AP_technical_details_out-eq_chaos}\n\n\\section{Our procedure to obtain the distribution functions} \\labsec{procedure}\nHere, we explain the details in our computation of the distribution functions $F(X,T_1,T_2,\\xi,r)$ or, rather, the quantity we really compute, namely its inverse $X(F,T_1,T_2,\\xi,r)$. First, in~\\refsubsec{parameters} we provide the relevant parameters for the construction of the distribution functions. Next, in~\\refsubsec{construction} we explain how we compute the chaotic parameter for a given sphere and a given number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace$. In~\\refsubsec{sample_to_sample_fluctuations} we explain our computation of $X(F,T_1,T_2,\\xi,r)$ for a given $\\ensuremath{N_{\\text{Rep}}}\\xspace$, including our procedure for the computation of the error bars\\index{error bars}. Finally, in~\\refsubsec{extrapolation} we explain the process of the extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$.\n\n\\subsection{The parameters in our computation} \\labsubsec{parameters}\nThe computation of $X^{s,r}_{T_1,T_2}(\\xi)$, namely the chaotic parameter for a given sphere $s$ of radius $r$, recall~\\refsubsec{observables-locales}, is specified by five parameters: two temperatures $T_1$ and $T_2$, the radius $r$, the coherence length\\index{coherence length} $\\xi$ and the number of replicas\\index{replica} used to estimate the thermal noise $\\ensuremath{N_{\\text{Rep}}}\\xspace$. Our choice of the parameters has been the following:\n\n\\begin{itemize}\n\\item \\textbf{Temperatures:} we impose $T_1<T_2$ with $T_1 \\in \\{0.625,0.7,0.8\\}$ and $T_2 \\in \\{ 0.7,0.8,0.9,1.0\\}$.\n\\item \\textbf{Radius:} integer values of $r$ from $r=1$ to $r=15$ are simulated.\n\\item \\textbf{Coherence lengths\\index{coherence length}:}  from $\\xi_{\\min} = 3$ to $\\xi_{\\max,T_1}$ with intervals $\\Delta_\\xi=0.25$ where $\\xi_{\\max,T_1}$ is the maximum $\\xi$ simulated for temperature $T_1$\\footnote{This condition is imposed by~\\refeq{el_reloj_doble}. Indeed, we need $\\xi$ to be reached at both temperatures $T_1$ and $T_2$. So, as long as $\\xi_{\\max}$ increases with temperature (recall~\\reftab{xi_max}), the maximum $\\xi$ has to be $\\xi_{\\max,T_1}$}.\n\\item \\textbf{Number of replicas\\index{replica}:} in order to quantify the thermal noise, we have computed $X^{s,r}_{T_1,T_2}(\\xi)$ by using different number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace \\in \\{ 16,32,64,128,256,512\\}$ (see below).\n\\end{itemize}\n\n\\subsection{On the computation of $X^{s,r}_{T_1,T_2}(\\xi)$}\\labsubsec{construction}\nThe reader will recall from~\\refeq{def_chaotic_parameter} that the computation of $X^{s,r}_{T_1,T_2}(\\xi)$ requires an infinite number of replicas\\index{replica}. Unfortunately we only have $512$ replicas\\index{replica} at our disposal. Our choice has been to produce different estimates of $X^{s,r}_{T_1,T_2}(\\xi)$ by varying the number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace$. Specifically, our procedure has been the following:\n\n\\begin{enumerate}\n\\item For each $\\ensuremath{N_{\\text{Rep}}}\\xspace < \\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ we order randomly the $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ replicas\\index{replica} and divide them in $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}\/\\ensuremath{N_{\\text{Rep}}}\\xspace$ groups of $\\ensuremath{N_{\\text{Rep}}}\\xspace$ replicas\\index{replica}.\n\\item In this way, we get $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}\/\\ensuremath{N_{\\text{Rep}}}\\xspace$ independent estimates of $X^{s,r}_{T_1,T_2}(\\xi)$.\n\\item In order to eliminate the effect of the initial permutation of the $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ replicas\\index{replica}, we repeat this procedure $10$ times for all $\\ensuremath{N_{\\text{Rep}}}\\xspace < \\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$.\n\\end{enumerate}\n\nIn a nutshell, for every sphere of radius $r$ we obtain $\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$ where\n\\begin{equation}\n\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace<\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}) = 10 \\times \\dfrac{\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}}{\\ensuremath{N_{\\text{Rep}}}\\xspace} \\, ,\n\\end{equation}\nor\n\\begin{equation}\n\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace=\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}) = 1 \\, .\n\\end{equation}\n\n\\subsection{The computation of $X(F,T_1,T_2,\\xi,r)$}\\labsubsec{sample_to_sample_fluctuations}\nGiven that, for every radius $r$, we have computed the chaotic parameter for $8000$ different spheres in each sample\\index{sample}, we shall fix the probability level $F$ as\n\\begin{equation}\nF_i=\\dfrac{i}{8000} \\, , \\quad i=1,2,\\ldots,7999 \\, .\n\\end{equation}\nErrors have been computed with the Jackknife method (see~\\refsubsec{jk_method}). Let us briefly describe our procedure:\n\n\\begin{enumerate}\n\\item \\textbf{Our estimate of $X(F,T_1,T_2,\\xi,r)$ for each $\\ensuremath{N_{\\text{Rep}}}\\xspace$.}\n\t\\begin{enumerate}\n\t\\item We group together the $\\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$.\n\t\\item We sort them in increasing order.\n\t\\item Our estimate of the central values of $X(F=F_i,T_1,T_2,\\xi,r)$ is the $X^{s,r}$ that occupies the position $F_i \\times \\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ in the sorted list. Note that our choice of $F_i$ ensures that $F_i \\times \\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ is an integer.\n\t\\end{enumerate}\n\n\\item \\textbf{Computation of the fluctuations.}\\\\\n\tFor the $j$-th Jackknife block, $j=1,2,\\ldots,\\ensuremath{N_{\\text{S}}}\\xspace$, we proceed as follows:\n\t\\begin{enumerate}\n\t\\item We group together the $(\\ensuremath{N_{\\text{S}}}\\xspace-1) \\times 8000 \\times \\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$, obtained excluding the sample\\index{sample} $j$.\n\t\\item We sort them in increasing order.\n\t\\item Our estimate of $X(F=F_i,T_1,T_2,\\xi,r)$ for the $j$ Jackknife block, is the $X^{s,r}$ that occupies the position $F_i \\times (\\ensuremath{N_{\\text{S}}}\\xspace-1) \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ in the sorted list.\n\t\\end{enumerate}\n\tThe errors $X(F,T_1,T_2,\\xi,r)$ are computed through the Jackknife blocks using the standard formula (see e.g.~\\cite{amit:05} and \\refch{AP_statistics}).\n\\end{enumerate}\nExamples of these computations can be seen in~\\reffig{linear_quadratic} and~\\reffig{exponente_libre}.\n\n\\subsection{Extrapolation to infinite number of replicas}\\labsubsec{extrapolation}\n\nThe computation of the thermal expectation values necessary to calculate the chaotic parameter, see~\\refeq{def_chaotic_parameter}, requires an infinite number of replicas\\index{replica}. As the maximum number of replicas\\index{replica} in our simulation is $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}=512$, we need an extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$. \n\nFixing $T_1$, $T_2$, $\\xi$, $r$ and $F$ we find a smooth behavior of $X(F,T_1,T_2,\\xi,r)$ with $\\ensuremath{N_{\\text{Rep}}}\\xspace$ (see~\\reffig{linear_quadratic} and~\\reffig{exponente_libre}). This smooth evolution with $\\ensuremath{N_{\\text{Rep}}}\\xspace$ makes feasible the extrapolation to the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ limit. Our main ansatz for the extrapolation is\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{A}{\\ensuremath{N_{\\text{Rep}}}\\xspace} \\>\\>\\> , \\labeq{extrapolacion_lineal}\n\\end{equation}\nwhere $A$ is an amplitude, $X_{\\ensuremath{N_{\\text{Rep}}}\\xspace}$ is a short hand for $X(F,T_1,T_2,\\xi,r;\\ensuremath{N_{\\text{Rep}}}\\xspace)$ and also a short hand for $X_\\infty=X(F,T_1,T_2,\\xi,r; \\ensuremath{N_{\\text{Rep}}}\\xspace=\\infty)$. As a check for the linear ansatz in~\\refeq{extrapolacion_lineal}, we consider two alternative functional forms for the extrapolation:\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{B}{\\ensuremath{N_{\\text{Rep}}}\\xspace} + \\dfrac{C}{\\ensuremath{N_{\\text{Rep}}}\\xspace^2} \\>\\>\\> , \\labeq{extrapolacion_cuadratica}\n\\end{equation}\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{D}{\\ensuremath{N_{\\text{Rep}}}\\xspace^\\gamma} \\>\\>\\> , \\labeq{extrapolacion_libre}\n\\end{equation}\nwhere $B$, $C$ and $D$ are amplitudes and $\\gamma$ is a free exponent. We perform independent fits to~\\refeq{extrapolacion_lineal},~\\refeq{extrapolacion_cuadratica} and~\\refeq{extrapolacion_libre} for every value of the parameters $(F,T_1,T_2,\\xi,r)$. We reject fits with a diagonal $\\chi^2\/\\text{d.o.f.}\\geq 1.1$.\\index{degree of freedom} Errors in $X_{\\infty}$ are computed from the fluctuations of the Jackknife blocks. Indeed, we perform separated fits for each Jackknife block (the fitting procedure consists in minimizing the diagonal $\\chi^2$, see~\\cite{yllanes:11}).\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{off-eq_chaos\/lineal_cuadratica_T0708_xi11_r08_nb000000_X.pdf}\n  \\caption[\\textbf{Equivalence of linear and quadratic extrapolations.}]{\\textbf{Linear and quadratic extrapolations,~\\refeq{extrapolacion_lineal} and~\\refeq{extrapolacion_cuadratica}, turn out to be equivalent for the tail of the distribution function.} The continuous lines are the linear (golden curves) and quadratic (blue curves) extrapolations to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ for $F(X,T_1,T_2,\\xi,r)$ as a function of $X$. The data shown correspond to the case $T_1=0.7$, $T_2=0.8$, $\\xi=11$ and $r=8$. The two curves shown for each extrapolation correspond to the central value plus or minus the standard error. We show horizontal error bars\\index{error bars} because we are computing the inverse distribution function $X(F,T_1,T_2,\\xi,r)$. We only show extrapolated data when $\\chi^2\/\\text{d.o.f.} < 1.1$\\index{degree of freedom} in the fits to~\\refeq{extrapolacion_lineal} or to~\\refeq{extrapolacion_cuadratica}. For comparison, we also plot the data corresponding to $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ and $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ (yellow and blue dots respectively) \\textbf{Inset:} As in the main plot, but with the vertical axis in log-scale.}\n\\labfig{linear_quadratic}\n\\end{figure} \n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{off-eq_chaos\/extrapola_free_expo_T0708.pdf}\n  \\caption[\\textbf{Free-exponent extrapolation.}]{\\textbf{The exponent $\\gamma$ in~\\refeq{extrapolacion_libre}, remains close to one when it becomes a fit parameter.} The distribution function $F(X,T_1,T_2,\\xi,r)$ is plotted as a function of $X$ for $\\ensuremath{N_{\\text{Rep}}}\\xspace=\\{512, 256, 128 ,64, 32, 16\\}$ together with their extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ as obtained from a fit to~\\refeq{extrapolacion_libre}. The data shown correspond to $T_1=0.7$, $T_2=0.8$, $\\xi=11$ and $r=8$. In order not to clutter the figure, we do not show error bars\\index{error bars} in $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ extrapolation. \\textbf{Left inset:} exponent $\\gamma$, which is plotted against the probability $F$, remains close to $\\gamma=1$ for all $F$, with the exception of the unstable behavior at $F \\approx 0.35$, where curves for different $\\ensuremath{N_{\\text{Rep}}}\\xspace$ cross (see also top right inset). \\textbf{Bottom right inset:} $\\chi^2$ per degree of freedom\\index{degree of freedom} is plotted against $F$. The blue line corresponds to $\\chi^2\/\\text{d.o.f.} = 1$.\\index{degree of freedom} \\textbf{Top right inset:} Zoom of the main plot, emphasizing the crossing region at $F \\approx 0.35$. Note that at that particle value of $F$ data show almost no dependence with $\\ensuremath{N_{\\text{Rep}}}\\xspace$, which makes unstable the fit to~\\refeq{extrapolacion_libre}.}\n \\labfig{exponente_libre}\n\\end{figure}\n\nAs a first check, we compare the linear and the quadratic extrapolations (see~\\reffig{linear_quadratic} for an illustrative example). The figure shows that even for our largest $\\ensuremath{N_{\\text{Rep}}}\\xspace$, namely $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$, and $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$, we are still far from the extrapolation to the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ limit. Fortunately, the linear and the quadratic extrapolations provide compatible results in our region of interest, i.e. the tail of the distribution function. We remark that the consistency condition $\\chi^2\/\\text{d.o.f.} < 1.1$\\index{degree of freedom} is met in a larger range for the quadratic extrapolation ($F<0.9$) than in the linear extrapolation ($F<0.7$). However, because both coincide in the low-$F$ range we are interested in, we have kept the simpler linear extrapolation.\n\nOur second check in~\\refeq{extrapolacion_libre} seeks the natural exponent $\\gamma$ for the extrapolation, which is a fitting parameter. We have found, see~\\reffig{exponente_libre}, that the consistency condition $\\chi^2 \/ \\mathrm{d.o.f.}<1.1$\\index{degree of freedom} is met for $F<0.85$. Fortunately, $\\gamma$ turns out to be very close to the value $\\gamma=1$, with the exception of the instability in the crossing region around $F\\approx 0.35$. \n\nIn summary, the quadratic and the free-exponent extrapolations support our choice of~\\refeq{extrapolacion_lineal} as the preferred form for the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ extrapolation.\n\n\\section{On the most convenient variable to characterize the sphere size}\\labsec{cambio_de_r}\n\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/Nr_vs_r_example.pdf}\n  \\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/Nr_vs_r_theoretical.pdf}\t\n  \\caption[\\textbf{$N_r^{1\/3}$ postulates as a better variable to describe short length scales.}]{\\textbf{$N_r^{1\/3}$ postulates as a better variable to describe short length scales.} \\textbf{Left:} complementary of \\gls{TC}\\index{temperature chaos} $1-X^{s,r}_{T_1,T_2}(\\xi)$ against the region size for the two discussed independent variables, namely $N_r^{1\/3}$ and the radius $r$. The continuous lines are fits to~\\refeq{functional_form} taking as variables $z=r$ (golden curve) and $z=N_r^{1\/3}$ (blue curve). The shown data correspond to $T_1=0.7$, $T_2=0.9$, $F=0.01$ and $\\xi=7$. We zoom the region of small spheres, where both independent variables most differ. \\textbf{Right:} the cubic root of the volume of a sphere (blue curve) is plotted as a function of the radius of the sphere $r$. The golden curve is $N_r^{1\/3}$, namely the cubic root of the number of lattice points contained in a sphere of radius $r$, centered at a node of the dual lattice corresponding to our cubic lattice. Values of $N_r^{1\/3}$ corresponding to integer $r$ are highlighted as black dots.}\n \\labfig{Nr_vs_r}\n\\end{figure}\n\nHere we explain our rationale for choosing the cubic root of the number of spins contained in the sphere $N_r^{1\/3}$, rather than its radius $r$, to characterize the size of the spheres considered in our analysis.\n\nWe asked ourselves this question because our first attempt to fit the peaks of $1-X$ to~\\refeq{functional_form} by using as an independent variable the radius of the spheres $r$ failed. Indeed, see~\\reffig{Nr_vs_r} left, $1-X$ is not a smooth function of $r$. After some reflection, we realized that the number of lattice points in our spheres is not a smooth function of $r$ either (see~\\reffig{Nr_vs_r} right). At that point, it was only natural to trade $r$ with $N_r^{1\/3}$ as the independent variable. In fact, see~\\reffig{Nr_vs_r} left, the new independent variable $N_r^{1\/3}$ solved our problem of fitting to~\\refeq{functional_form}. Of course, the difference between both independent variables becomes immaterial for very large spheres.\n\n\\section{Difficulties in peak characterization for the weak temperature chaos regime}\\labsec{peak_characterization}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/T062507_F01.pdf}\n  \\caption[\\textbf{Too weak a temperature chaos makes it difficult to characterize the peak.}]{\\textbf{Too weak a temperature chaos makes it difficult to characterize the peak.} The complementary of the chaotic parameter $1-X^{s,r}_{T_1,T_2}(\\xi)$ is represented against $N_r^{1\/3}$ for the temperatures $T_1=0.625$ and $T_2=0.7$, the probability $F=0.01$ and different values of $\\xi$. A quick growth for small $N_r^{1\/(3}$ followed by a plateau is observed in all the plots.}\n \\labfig{T062507_F01}\n\\end{figure}\n\nHere we illustrate the difficulty of characterizing the peak of the function $f(z)$ defined in~\\refeq{functional_form} when \\gls{TC}\\index{temperature chaos} is extremely weak. Indeed, as~\\reffig{T062507_F01} shows, the size of the error bars makes\\index{error bars} data almost compatible with a plateau (rather than a peak). Moreover, mind the vertical scale in~\\reffig{T062507_F01}, \\gls{TC}\\index{temperature chaos} is almost nil, which suggests that this set of parameters ($T_1=0.625,T_2=0.7,F=0.01$) is not suitable to study \\gls{TC}\\index{temperature chaos}. Consequently, we have decided to exclude from the analysis the data obtained with the temperatures $T_1=0.625$ and $T_2=0.7$ and the probability $F=0.01$.\n\n\\chapter{Conclusions} \\labch{conclusions}\n\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Si el pasado y el presente\\\\\nSe reflejan y no mienten\\\\\nTengo que hacer algo \\\\\npor mi porvenir.}}{-- Evaristo P\u00e1ramos, \\textit{Ya no quiero ser yo}}\n\nThe traditional picture of a solitary theoretical physicist, working alone with paper and pencil as her only tools, has evolved through the years. Instead, computational research has been an important part of the development of physics for the last decades and the discoveries are often performed by groups of physicists working together. However, although the increase of computational power is not new, it has been only recently that we can collect, process, and analyze a huge amount of data at affordable times.\n\nThe increase of computational power has particularly benefited theoretical physics. Specifically for spin glasses, the development of special-purpose\\index{special-purpose computer} hardware has allowed simulating systems up to the experimental time-scale. This thesis is another iteration in the development of a field that takes the general path of science and embraces the interplay between experiments, theory, and computing as a fruitful relation to make relevant advances.\n\nThroughout this thesis, we have studied the spin glasses from a numerical point of view. The study of the metastate\\index{metastate} in~\\refch{metastate} has been focused on addressing a theoretical problem through the first construction of the equilibrium metastate\\index{metastate} in numerical simulations. In~\\refch{aging_rate} and \\refch{mpemba} the off-equilibrium dynamics of spin glasses has been studied. In the former, we introduce the coherence length\\index{coherence length} as the most relevant quantity characterizing the aging\\index{aging} state of a spin glass, and we solve a numerical discrepancy in the aging\\index{aging!rate} rate between experiments and previous numerical simulations. In the latter, we discuss the Mpemba\\index{Mpemba effect} effect, which is a memory\\index{memory effects} effect that takes place in the off-equilibrium dynamics, providing a good example of how the coherence length\\index{coherence length} governs many out-of-equilibrium phenomena. The final part of the thesis, \\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos}, is devoted to introduce and analyze the Temperature Chaos\\index{temperature chaos} phenomenon in spin glasses. Actually, we tackle this problem from the equilibrium point of view (\\refch{equilibrium_chaos}) and we also observe and characterize the phenomenon in the off-equilibrium dynamics (\\refch{out-eq_chaos}).\n\nIn this chapter we outline the main results of this thesis, revisiting all the parts and summarizing the relevant messages.\n\n\\section{A brief thought about the importance of the data}\nThis thesis is mainly focused on the numerical study of spin glasses. One idea that is central throughout this thesis is the fundamental importance of high-quality data. The reader may wonder what could we say in~\\refch{aging_rate} without our precise estimation of the coherence length\\index{coherence length}, or which conclusions could we extract from~\\refch{out-eq_chaos} without our accurate estimations of the chaotic parameter. None of these works would be possible without our high-quality data.\n\nThis section aims to emphasize the role of the special-purpose\\index{special-purpose computer} FPGA-based\\index{FPGA} hardware, Janus\\index{Janus} II, in this thesis. It is usual in the physicist work to spend a lot of time by using (or developing) sophisticated statistical methods to improve the quality of the data, and, it is actually, a very important task. Nevertheless, if we can combine the statistical methods with powerful hardware, we can, indeed, obtain data at the forefront of the field.\n\nSpecifically, in this thesis, we have taken advantage (in some works) of the largest spin-glass simulation in off-equilibrium dynamics. We have simulated an Edwards-Anderson\\index{Edwards-Anderson!model} model with Ising\\index{Ising} spins, for a lattice size $L=160$ for a modest number of samples\\index{sample} $\\ensuremath{N_{\\text{S}}}\\xspace=16$, but a huge number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ (a choice that turned to be fundamental). The simulations have been performed up to temperatures well deep in the spin-glass phase\\index{phase!low-temperature\/spin-glass} (for instance, $T=0.625$) to unprecedented long times.\n\nIt is worthy to mention also the fundamental role of other computers such as the Madrid's Cluster in the UCM and the supercomputer Cierzo in BIFI. They have made possible the analysis of the data in this thesis through thousands of hours of computational time.\n\n\n\\section{Conclusions on the metastate} \\labsec{conclusion_metastate}\nIn its origins, the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} theory aimed to explain the nature of the low-temperature phase\\index{phase!low-temperature\/spin-glass} in spin glasses assuming infinite-size systems. However, some mathematical procedures in that development were ill-defined. In particular, for disordered\\index{disorder!systems} systems, the thermodynamic limit\\index{thermodynamic limit} $L \\to \\infty$ for a Gibbs state may not exist. This problem is originated from a phenomenon known as \\textit{chaotic size dependence}. Mathematical physics offered a new approach in the context of disordered\\index{disorder!systems} systems and brought a solution to this problem: \\textit{the metastate}\\index{metastate}. This concept is a generalization of the concept of Gibbs states. Nonetheless, this discussion used to be limited to the theoretical work, without any numerical or experimental counterpart.\n\nIn~\\refch{metastate} we have shown that the state of the art in numerical simulations allows the construction of the Aizenman-Wehr metastate\\index{metastate!Aizenman-Wehr}. Indeed, our numerical data suggest that the $1 \\ll W \\ll R \\ll L$ limit required from the construction of this metastate\\index{metastate!Aizenman-Wehr} (see~\\refsubsec{aw_metastate}) can be relaxed to $W\/R \\approx 0.75$ and $R\/L \\approx 0.75$ without changing substantially the thermodynamic physical behavior.\n\nThe main quantitative result of our work is the numerical computation of the exponent $\\zeta$. According to Read~\\cite{read:14}, it is possible to partially discriminate between the competing pictures trying to describe the nature of the spin-glass phase\\index{phase!low-temperature\/spin-glass} in\\footnote{Recall that $d_{\\mathrm{L}}$ and $d_{\\mathrm{U}}$ are the lower and the upper critical dimension\\index{critical dimension!lower}\\index{critical dimension!upper} respectively.} $d_{\\mathrm{L}} < d < d_{\\mathrm{U}}$ by computing this exponent $\\zeta$. Indeed, $\\zeta$ is related to the number of different states that can be measured in a system of size $W$. This number behaves as $\\log n_{\\mathrm{states}} \\sim W^{d -\\zeta}$. Therefore, $\\zeta < d$ would lead, for $W \\to \\infty$, to a metastate\\index{metastate} of infinitely many states.\n\nHere, we find $\\zeta = 2.3 (3)$. We compare our result to previous estimations of this exponent, computed in different contexts, and we summarize our knowledge about the behavior of this exponent as a function of the dimension $d$ in~\\refsec{relating_numerical_theory_metastate}.\n\nAll the numerical evidence strongly supports the existence of a spin-glass metastate\\index{metastate!dispersed} dispersed over infinitely many states for $d=3$. This result probably holds for $d>d_{\\mathrm{L}}$. Our findings are incompatible with the droplet\\index{droplet!picture} model, while they are compatible with both the chaotic pair picture and the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} scenario.\n\nThe results concerning this work are published in~\\cite{billoire:17}.\n\n\\section{Conclusions on the study of the aging rate}\nThe off-equilibrium dynamics in spin glasses is of glaring importance since the experiments are always conducted out-of-equilibrium. Under these conditions, domains\\index{magnetic domain} of correlated spins start to grow at the microscopical level. The linear size of these domains\\index{magnetic domain} is the so-called \\textit{coherence length\\index{coherence length}} $\\xi$. The \\textit{aging\\index{aging!rate} rate} $z(T)$, which is none but the variation rate of the free-energy\\index{free energy!barrier} barriers with the logarithm of the coherence length\\index{coherence length} $\\xi$ (see \\refsec{how_can_study_aging}), is directly related to the growth of the coherence length\\index{coherence length} with the time. Previous numerical evidences pointed to a dependency of the form $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$. However, numerical discrepancies in the determination of $z(T)$ were recently found between experiments~\\cite{zhai:17} and numerical simulations~\\cite{janus:08,lulli:16}.\n\nIn~\\refch{aging_rate} we have taken advantage of the previously mentioned simulations performed in Janus\\index{Janus} II to study the growth of the coherence length\\index{coherence length} in the glassy phase\\index{phase!low-temperature\/spin-glass} and solved this discrepancy. Specifically, we found that the growth of the coherence length\\index{coherence length} is controlled by a time-dependent aging\\index{aging!rate} rate $z(T,\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace))$. \n\nIn this work, we have described the dynamics as governed by a crossover between a critical and a low-temperature fixed point. The characterization of that crossover has allowed us to quantitatively model the growth of the aging\\index{aging!rate} rate. In particular, we have considered two different ans\\\"atze for that growth, and we have found that, for the convergent one [\\refeq{convergent_ansatz}], the computation of the aging\\index{aging!rate} rate is consistent with the most recent experimental measures~\\cite{zhai:17}.\n\nBesides, we find clear evidence of non-coarsening dynamics at the experimental scale and find that temperatures $T \\lesssim 0.7$ are free of critical effects and therefore safe for numerical studies of the spin-glass phase\\index{phase!low-temperature\/spin-glass}.\n\nThe results concerning this work are published in~\\cite{janus:18}.\n\n\\section{Conclusions on the Mpemba effect}\nConsider two beakers of water that are identical to each other except for the fact that one is hotter than the other. If we put both of them in contact with a thermal reservoir (for example, a freezer) at some temperature lower than the freezing point of the water, under some circumstances, it can be observed that the initially hotter water freezes faster than the colder one. This phenomenon is known as the Mpemba\\index{Mpemba effect} effect~\\cite{mpemba:69}.\n\nIn~\\refch{mpemba} we have shown that the Mpemba\\index{Mpemba effect} effect is present in spin glasses, where it appears as an intrinsically non-equilibrium process, ruled by the spin-glass coherence length\\index{coherence length} $\\xi$.\n\nFirst, we have identified the relevant quantities to mimic the effect in spin glasses, namely the energy-density\\index{energy!density} (as the role of the temperature in the classical Mpemba effect), and the Monte\\index{Monte Carlo} Carlo time. However, the introduction of the coherence length\\index{coherence length} as a hidden quantity ruling the process turned out to be fundamental.\n\nIndeed, we have provided the first explanation of this phenomenon (in the spin-glass context), by using the relation between the energy density\\index{energy!density} and the coherence length\\index{coherence length} [\\refeq{energy_coherence_length_relation}] to characterize the effect. Although this description is approximate, it is accurate enough to describe the Mpemba\\index{Mpemba effect} effect.\n\nOur results explain how the most natural experimental setup (prepare two identical systems at $T_1,T_2>T_\\text{c}$ with an identical protocol, then quench them) can fail to observe the effect. Indeed, for spin glasses at least, a different starting $\\xi$ is required. \n\nFinally, we have investigated the inverse Mpemba\\index{Mpemba effect} effect (\\refsec{inverse_mpemba}). In the spin-glass phase\\index{phase!low-temperature\/spin-glass}, the inverse Mpemba\\index{Mpemba effect} effect is completely symmetrical to the classical Mpemba\\index{Mpemba effect} effect. However, we found that above the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ the inverse Mpemba\\index{Mpemba effect} effect is strongly suppressed because our description of \\refeq{energy_coherence_length_relation} is not valid for $T> \\ensuremath{T_\\mathrm{c}}\\xspace$.\n\nThe Mpemba\\index{Mpemba effect} effect is peculiar among the many memory\\index{memory effects} effects present in spin glasses. Indeed, this phenomenon can be studied through quantities, such as the energy density\\index{energy!density}, which are just measured at one-time scale (rather than the usual two times~\\cite{young:98,jonason:98,janus:17b}). However, our setup poses an experimental challenge, because we are not aware of any measurement of the non-equilibrium temperature associated with the magnetic degrees of freedom\\index{degree of freedom}. Perhaps one could adapt the strategy of Ref.~\\cite{grigera:99}, connecting dielectric susceptibility\\index{susceptibility} and polarization noise in glycerol, to measurements of high-frequency electrical noise in spin glasses~\\cite{israeloff:89}.\n\nOur investigation of the Mpemba\\index{Mpemba effect} effect offers as well a new perspective into an important problem, namely the study of the glassy coherence length\\index{coherence length} in supercooled liquids and other glass formers~\\cite{cavagna:09}. Indeed, the identification of the right correlation function to study experimentally (or numerically) is still an open problem. Spin glasses are unique in the\ngeneral context of the glass transition\\index{phase transition}, in both senses: we know which correlation functions should be computed microscopically~\\cite{edwards:75,edwards:76}, while accurate experimental determinations of the coherence length\\index{coherence length} have been obtained~\\cite{guchhait:17}.\n\nThe results concerning this work are published in~\\cite{janus:19}.\n\n\\section{Conclusions on the equilibrium Temperature Chaos}\nIn a spin glass, the Temperature\\index{temperature chaos} Chaos phenomenon refers to the complete reorganization of the Boltzmann\\index{Boltzmann!weight} weights that determines the frequency with which each configuration\\index{configuration} of spins will appear, upon an arbitrary small change in the temperature $T$.\n\nIn~\\refch{equilibrium_chaos} we have studied the Temperature Chaos\\index{temperature chaos} phenomenon in equilibrium simulations and have proposed an efficient variational method\\index{variational method} to estimate the elusive exponential autocorrelation time\\index{autocorrelation time!exponential} of a Monte\\index{Monte Carlo} Carlo Markov\\index{Markov chain} chain, specific to the case of a Parallel Tempering\\index{parallel!tempering} simulation.\n\nThis variational method\\index{variational method} takes into account three parameters and performs a maximization of the estimation of the integrated autocorrelation time\\index{autocorrelation time!integrated} in the phase-space\\index{phase space} of these parameters. Since the exponential autocorrelation time\\index{autocorrelation time!exponential} is an upper bound of the parameter-dependent integrated autocorrelation time\\index{autocorrelation time!integrated}, our procedure leads to robust estimations.\n\nIn addition, we have studied the scaling properties of the probability distribution of the autocorrelation time, obtained using the proposed variational\\index{variational method} approach. In particular, we have shown that scaling holds for lattices of sizes $L \\geq 24$, consistently with previous studies using effective potentials.\n\nThe presence of Temperature Chaos\\index{temperature chaos} is related to the poor performance of the Parallel Tempering simulations in the spin-glass phase\\index{phase!low-temperature\/spin-glass}. Then, the exponential autocorrelation time\\index{autocorrelation time!exponential} provides us a \\textit{dynamic} characterization of this phenomenon.\n\nIn this work, we have also characterized the Temperature Chaos\\index{temperature chaos} from a \\textit{static} point of view by studying the equilibrium configurations\\index{configuration} of the system. The observable that allows us to quantitatively study the Temperature Chaos\\index{temperature chaos} is the so-called \\textit{chaotic parameter}. The empirical observation of the most chaotic samples\\index{sample} (from the dynamical point of view), led to the construction of observables derived from the chaotic parameter to characterize the Temperature Chaos\\index{temperature chaos} (see \\cite{fernandez:13,fernandez:16}). In our work, we introduce a new observable to characterize the phenomenon.\n\nFinally, we have checked the statistical correlations between these \\textit{static} chaotic indicators and the dynamical correlation times. The introduction of the new \\textit{static} indicator has improved previous results in the correlations.\n\nThe results concerning this work are published in~\\cite{billoire:18}.\n\n\\section{Conclusions on the Temperature Chaos in off-equilibrium dynamics}\nIn~\\refch{out-eq_chaos} we have studied an interesting phenomenon in off-equilibrium dynamics that closely mimics Temperature Chaos\\index{temperature chaos}. Indeed, as we have just discussed in the previous section, Temperature Chaos\\index{temperature chaos} is defined as an equilibrium phenomenon. Therefore, studying it in a non-equilibrium system is an open challenge that we have addressed here. \n\nFirst, we tried a naive approach to non-equilibrium Temperature Chaos\\index{temperature chaos} (see \\refsec{average_killed_chaos_signal}), which found only an exceedingly small chaotic signal. Fortunately, the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17} combined with the rare events analysis performed in equilibrium simulations, see~\\cite{fernandez:13}, provide the crucial insight to approach the problem. Specifically, the statics-dynamics equivalence allows us to relate the non-equilibrium dynamics of a spin glass (of infinite size) with a finite coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, with small samples\\index{sample} of size $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ which can be equilibrated.\n\nOur numerical protocol considers spherical-like regions of radius $r$. We focus on the probability distribution function of the chaotic parameter as computed over the spheres. We find that only the spheres in the tail of the distribution exhibit a strong Temperature Chaos\\index{temperature chaos}.\n\nChoosing a suitable length scale $r$ for the spherical-like regions turns out to be instrumental in the study of dynamic Temperature Chaos\\index{temperature chaos}. This optimal length scale is proportional to the coherence length\\index{coherence length}. However, our data mildly suggests that the importance of choosing exactly the correct $r$ becomes less critical in the $\\xi \\to \\infty$ limit.\n\nA striking link emerges between the dynamic and the static faces of the Temperature Chaos\\index{temperature chaos} phenomenon. Indeed, we find a characteristic length scale $\\xi^*(T_2-T_1,F)$ at which the crossover between the weak chaos and the strong chaos regimes occurs. The physical meaning of the characteristic length $\\xi^*$ suggests that the equilibrium chaotic length $\\ell_c$~\\cite{fisher:86,bray:87b} is its equilibrium counterpart. In fact, both quantities depend on $T_2-T_1$ in the same way and the exponent\\footnote{The reader should be warned that this exponent $\\zeta$ is completely different from the exponent $\\zeta$ in \\refsec{conclusion_metastate}. We follow here the usual notation in the literature and denote both of them with the same letter despite their completely different meanings.} $\\zeta$, controlling the temperature-dependence of $\\ell_c$ turns out to be equal to $\\zeta_{\\mathrm{NE}}$, obtained from the off-equilibrium estimation, at the two-$\\sigma$ level. We regard this coincidence as new and important evidence for the statics-dynamics equivalence\\index{statics-dynamics equivalence}.\n\nIn the second part of the work we have performed temperature-varying\\index{temperature-varying protocol} simulations to address the cumulative aging\\index{aging!cumulative} problem~\\cite{jonsson:02,bert:04,komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. We have found small but clear violations of cumulative aging\\index{aging!cumulative}, which are stronger upon \\textit{cooling} than upon \\textit{heating}. Although both protocols display a memory-erasing process, the \\textit{cooling} process shows better memory, i.e. larger coherence-lengths are needed to lose the memory of the previous state at the higher temperature. \n\nThe results concerning this work are published in~\\cite{janus:21}.\n\n\n\\chapter{Conclusiones} \\labch{conclusiones_castellano}\nLa visi\u00f3n tradicional de un f\u00edsico te\u00f3rico solitario, trabajando s\u00f3lo con papel y l\u00e1piz como sus \u00fanicas herramientas, ha ido evolucionando a lo largo de los a\u00f1os. En su lugar, la investigaci\u00f3n computacional ha sido una parte importante del desarrollo de la f\u00edsica en las \u00faltimas d\u00e9cadas y los descubrimientos son normalmente hechos por grupos de f\u00edsicos trabajando juntos. Sin embargo, aunque el incremento de la capacidad computacional no es nuevo, ha sido recientemente cuando hemos conseguido recolectar, procesar y analizar una gran cantidad de datos en tiempos razonables.\n\nLa f\u00edsica te\u00f3rica ha sido una de las grandes beneficiadas de este incremento de la capacidad computacional. Si nos centramos en los vidrios de esp\u00edn, el desarrollo de hardware dedicado ha permitido simular sistemas hasta las escalas de tiempo experimentales. Esta tesis es otra iteraci\u00f3n en el desarrollo de un campo que ha tomado el camino general de la ciencia y ha abrazado la interacci\u00f3n entre experimentos, teor\u00eda y computaci\u00f3n como una relaci\u00f3n fruct\u00edfera para hacer avances relevantes.\n\nA lo largo de esta tesis, hemos estudiado los vidrios de esp\u00edn desde un punto de vista num\u00e9rico. El estudio del metaestado en el Cap\u00edtulo 2 ha estado enfocado a abordar un problema te\u00f3rico a trav\u00e9s de la primera construcci\u00f3n del metaestado en equilibrio en simulaciones num\u00e9ricas. En los cap\u00edtulos 3 y 4, hemos estudiado la din\u00e1mica fuera del equilibrio de los vidrios de esp\u00edn. En el Cap\u00edtulo 3, hemos introducido la longitud de coherencia como la cantidad m\u00e1s relevante para caracterizar el estado de envejecimiento de un vidrio de esp\u00edn y hemos resuelto una discrepancia num\u00e9rica en el ratio de envejecimiento entre experimentos y simulaciones num\u00e9ricas. En el cap\u00edtulo 4, hemos discutido el efecto Mpemba, que es un efecto de memoria que tiene lugar en la din\u00e1mica fuera del equilibrio, d\u00e1ndonos un buen ejemplo de c\u00f3mo la longitud de coherencia gobierna una multitud de fen\u00f3menos en este r\u00e9gimen. La parte final de esta tesis, los cap\u00edtulos 5, 6 y 7, est\u00e1n dedicados a introducir y analizar el fen\u00f3meno del Caos en Temperatura en los vidrios de esp\u00edn. De hecho, abordamos este problema desde un punto de vista de equilibrio (Cap\u00edtulo 6) y tambi\u00e9n caracterizamos este fen\u00f3meno en la din\u00e1mica de fuera del equilibrio (Cap\u00edtulo 7).\n\nEn este cap\u00edtulo, resaltamos los principales resultados de esta tesis, revisitando todas las partes de la misma y resumiendo los mensajes m\u00e1s relevantes.\n\n\\section{Una breve reflexi\u00f3n sobre la importancia de los datos}\nEsta tesis est\u00e1 principalmente enfocada al estudio num\u00e9rico de los vidrios de esp\u00edn. Una idea que es central a lo largo de esta tesis es la importancia fundamental de los datos de alta calidad. El lector puede preguntarse qu\u00e9 podr\u00edamos haber dicho en el Cap\u00edtulo 3 sin una estimaci\u00f3n precisa de la longitud de coherencia, o qu\u00e9 conclusiones podr\u00edamos sacar del Cap\u00edtulo 7 sin una estimaci\u00f3n precisa del par\u00e1metro ca\u00f3tico. Ninguno de estos trabajos ser\u00eda posible sin nuestros datos de alta calidad.\n\nEsta secci\u00f3n tiene por objetivo enfatizar el rol del hardware dedicado basado en FPGA Janus II en esta tesis.  Es usual en el trabajo de un f\u00edsico dedicar mucho tiempo a usar (o desarrollar) m\u00e9todos estad\u00edsticos sofisticados para mejorar la calidad de los datos y esta es una tarea muy importante. No obstante, si podemos combinar \u00e9stos m\u00e9todos estad\u00edsticos con un hardware potente, podremos obtener datos en la vanguardia del campo.\n\nEn concreto, en esta tesis hemos tenido acceso (para varios trabajos) a la simulaci\u00f3n m\u00e1s grande en vidrios de esp\u00edn fuera del equilibrio. Hemos simulado un modelo de Edwards-Anderson con espines de Ising para una red de tama\u00f1o $L=160$, para un modesto n\u00famero de samples $\\ensuremath{N_{\\text{S}}}\\xspace=16$, pero para un gran n\u00famero de r\u00e9plicas $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ (una elecci\u00f3n que ha resultado ser fundamental). Las simulaciones han sido llevadas a cabo a temperaturas bastante bajas, en la fase v\u00edtrea (por ejemplo $T=0.625$) hasta tiempos muy largos, sin precedentes.\n\nTambi\u00e9n cabe destacar el papel fundamental de otros ordenadores como el Cluster de Madrid de la UCM y el superordenador Cierzo en el BIFI. Estos ordenadores han hecho posible el an\u00e1lisis de los datos de esta tesis a trav\u00e9s de miles de horas de tiempo computacional.\n\n\\section{Conclusiones sobre el metaestado}\nEn sus or\u00edgenes, la teor\u00eda de \\textit{Replica Symmetry Breaking} ten\u00eda como objetivo explicar la fase de bajas temperaturas en los vidrios de esp\u00edn asumiendo sistemas de tama\u00f1o infinito. Sin embargo, algunos procedimientos matem\u00e1ticos involucrados en la teor\u00eda estaban mal definidos. En concreto, para sistemas desordenados, el l\u00edmite termodin\u00e1mico $L\\to\\infty$ para un estado de Gibbs puede no existir. Este problema est\u00e1 originado por un fen\u00f3meno conocido como la \\textit{dependencia ca\u00f3tica con el tama\u00f1o}. La irrupci\u00f3n de la f\u00edsica matem\u00e1tica en el contexto de los sistemas desordenados trajo la soluci\u00f3n a este problema: el metaestado. Este concepto es una generalizaci\u00f3n del concepto de estado de Gibbs. No obstante, esta discusi\u00f3n sol\u00eda estar limitada al trabajo te\u00f3rico, sin una contraparte num\u00e9rica o experimental.\n\nEn el cap\u00edtulo 2 hemos demostrado que el estado del arte en simulaciones num\u00e9ricas permite la construcci\u00f3n del metaestado de Aizenman-Wehr~\\cite{aizenman:90}. De hecho, nuestros datos num\u00e9ricos sugieren que el l\u00edmite $1 \\ll W \\ll R \\ll L$ requerido para la construcci\u00f3n del metaestado puede relajarse a $W\/R \\approx 0.75$ y $R\/L \\approx 0.75$ sin que haya cambios sustanciales en el comportamiento f\u00edsico del mismo.\n\nEl principal resultado cuantitativo de nuestro trabajo es la computaci\u00f3n num\u00e9rica del exponente $\\zeta$. Seg\u00fan Read \\cite{read:14}, es posible discriminar parcialmente entre las teor\u00edas contrapuestas que tratan de describir la naturaleza de la fase v\u00edtrea entre la dimensi\u00f3n cr\u00edtica inferior $d_{\\mathrm{L}}$ y la dimensi\u00f3n cr\u00edtica superior $d_{\\mathrm{U}}$, calculando este exponente $\\zeta$. De hecho, $\\zeta$ est\u00e1 relacionado con el n\u00famero de estados distintos que pueden ser medidos en un sistema de tama\u00f1o $W$. Este n\u00famero se comporta como $\\log n_{\\mathrm{states}} \\sim W^{d -\\zeta}$. Por lo tanto, si $\\zeta < d$ tendr\u00edamos que, para el l\u00edmite $W \\to \\infty$, el metaestado tendr\u00eda un n\u00famero infinito de estados.\n\nNosotros encontramos $\\zeta = 2.3(3)$. Comparamos nuestro resultado num\u00e9rico con estimaciones previas de este exponente, calculadas en contextos distintos, y resumimos todo nuestro conocimiento sobre el comportamiento de dicho exponente como funci\u00f3n de la dimensi\u00f3n $d$ en la secci\u00f3n 2.8.\n\nTodas las evidencias num\u00e9ricas apoyan firmemente la existencia de un metaestado disperso en vidrios de esp\u00edn, formado por infinitos estados, en $d=3$. Este resultado, probablemente es v\u00e1lido para todo $d>d_{\\mathrm{L}}$. Nuestros resultados son incompatibles con el modelo \\textit{droplet}, mientras que son compatibles tanto con el modelo \\textit{chaotic pair} como con el modelo \\textit{replica symmetry breaking}.\n\nLos resultados de este trabajo est\u00e1n publicados en~\\cite{billoire:17}.\n\n\\section{Conclusiones sobre el estudio del ratio de envejecimiento}\nLa din\u00e1mica fuera del equilibrio tiene una importancia palmaria en los vidrios de esp\u00edn puesto que los experimentos siempre son llevados a cabo en estas condiciones. En estos experimentos, dominios de espines correlacionados empiezan a crecer a nivel microsc\u00f3pico. La longitud de estos dominios es conocida como \\textit{longitud de coherencia} $\\xi$. La \\textit{tasa de envejecimiento} $z(T)$, que no es m\u00e1s que la tasa de variaci\u00f3n de las barreras de energ\u00eda libre con el logaritmo de la longitud de coherencia $\\xi$ (v\u00e9ase la Secci\u00f3n 3.2), est\u00e1 directamente relacionada con el crecimiento de la longitud de coherencia con el tiempo. La evidencia num\u00e9rica se\u00f1ala que el comportamiento de este crecimiento con el tiempo es, esencialmente, $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$. Sin embargo, discrepancias num\u00e9ricas en la determinaci\u00f3n de $z(T)$ han sido encontradas recientemente entre experimentos~\\cite{zhai:17} y simulaciones num\u00e9ricas~\\cite{janus:08,lulli:16}.\n\nEn el cap\u00edtulo 3 hemos aprovechado las simulaciones previamente mencionadas, llevadas a cabo por Janus II, para estudiar el crecimiento de la longitud de coherencia en la fase v\u00edtrea. En concreto, hemos descubierto que el crecimiento de la longitud de coherencia est\u00e1 controlado por un ratio de envejecimiento que depende del tiempo $z(T, \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace))$.\n\nEn este trabajo hemos descrito la din\u00e1mica mediante el cruce entre la influencia del punto fijo a temperatura cr\u00edtica y el punto fijo a temperatura 0. La caracterizaci\u00f3n de ese cruce nos ha permitido modelizar cuantitativamente el crecimiento del ratio de envejecimiento. En concreto, hemos considerado dos hip\u00f3tesis para ese crecimiento y hemos encontrado que la hip\u00f3tesis convergente [\\refeq{convergent_ansatz}] es consistente con el ratio de envejecimiento obtenido por las medidas experimentales m\u00e1s recientes~\\cite{zhai:17}.\n\nAdem\u00e1s, hemos encontrado claras evidencias de din\u00e1mica \\textit{non-coarsening} a la escala de tiempo experimental y hemos encontrado que las temperaturas $T \\lesssim 0.7$ est\u00e1n libres de efectos cr\u00edticos y, por lo tanto, son seguras para el estudio num\u00e9rico.\n\nLos resultados correspondientes a este cap\u00edtulo est\u00e1n publicados en~\\cite{janus:18}.\n\n\\section{Conclusiones sobre el efecto Mpemba}\nConsidere dos recipientes de agua que son id\u00e9nticos entre ellos con la \u00fanica excepci\u00f3n de que el agua contenida en uno de ellos est\u00e1 m\u00e1s caliente que la del otro recipiente. Si ponemos ambos recipientes en contacto con un ba\u00f1o t\u00e9rmico (por ejemplo, un congelador) a una temperatura por debajo del punto de congelaci\u00f3n del agua, bajo ciertas circunstancias, puede observarse que el agua inicialmente m\u00e1s caliente se congela antes que la fr\u00eda. Este fen\u00f3meno es conocido como el efecto Mpemba \\cite{mpemba:69}.\n\nEn el Cap\u00edtulo 4 hemos demostrado que el efecto Mpemba est\u00e1 presente en los vidrios de esp\u00edn, donde es un proceso que ocurre fuera del equilibrio y que est\u00e1 gobernado por la longitud de coherencia $\\xi$.\n\nEn primer lugar, hemos identificado los observables relevantes para imitar el efecto Mpemba cl\u00e1sico en los vidrios de esp\u00edn. Esos observables han sido la densidad de energ\u00eda, cumpliendo el rol de la temperatura en el efecto Mpemba cl\u00e1sico, y el tiempo de Monte Carlo. Sin embargo, la introducci\u00f3n de la longitud de coherencia como el observable oculto que gobierna el proceso ha resultado ser fundamental.\n\nHemos dado una primera explicaci\u00f3n de este fen\u00f3meno en los vidrios de esp\u00edn usando la relaci\u00f3n entre la densidad de energ\u00eda y la longitud de coherencia [\\refeq{energy_coherence_length_relation}] para caracterizar dicho fen\u00f3meno. Aunque la descripci\u00f3n dada es aproximada, es suficientemente precisa para caracterizar el efecto Mpemba.\n\nNuestros resultados explican c\u00f3mo la configuraci\u00f3n experimental m\u00e1s habitual (preparar dos sistemas id\u00e9nticos a $T_1,T_2 > \\ensuremath{T_\\mathrm{c}}\\xspace$ con un protocolo id\u00e9ntico y despu\u00e9s enfriarlos) puede fallar para ver este efecto. De hecho, al menos para los vidrios de esp\u00edn, diferentes longitudes de coherencia $\\xi$ iniciales son necesarias.\n\nFinalmente, hemos investigado el efecto Mpemba inverso. En la fase v\u00edtrea, el efecto Mpemba inverso es completamente sim\u00e9trico respecto del efecto Mpemba cl\u00e1sico. Sin embargo, hemos encontrado que por encima de la temperatura cr\u00edtica $\\ensuremath{T_\\mathrm{c}}\\xspace$, el efecto Mpemba inverso se ve fuertemente menguado debido a que la descripci\u00f3n de \\refeq{energy_coherence_length_relation} no es v\u00e1lida para $T>\\ensuremath{T_\\mathrm{c}}\\xspace$.\n\nEl efecto Mpemba es peculiar dentro de los muchos efectos de memoria presentes en los vidrios de esp\u00edn. De hecho, este fen\u00f3meno puede estudiarse a trav\u00e9s de cantidades como la densidad de energ\u00eda que requieren una sola escala temporal para ser medidas (al contrario que las dos escalas temporales que suelen requerir estos fen\u00f3menos para ser estudiados y caracterizados~\\cite{young:98,jonason:98,janus:17b}). Sin embargo, nuestra configuraci\u00f3n para el experimento num\u00e9rico plantea un desaf\u00edo experimental, dado que no tenemos noticia de ninguna medida de temperatura fuera del equilibrio asociada con los grados de libertad magn\u00e9ticos. Quiz\u00e1s podr\u00eda adaptarse la estrategia de~\\cite{grigera:99}, que conecta la susceptibilidad diel\u00e9ctrica y el ruido de polarizaci\u00f3n en glicerol, con las mediciones de ruido el\u00e9ctrico de alta frecuencia en vidrios de esp\u00edn~\\cite{israeloff:89}.\n\nNuestro estudio del efecto Mpemba inverso sugiere una v\u00eda experimental m\u00e1s f\u00e1cil, donde los sistemas son calentados, en lugar de enfriados. En este caso, aunque la respuesta de la energ\u00eda es muy peque\u00f1a, el proceso va acompa\u00f1ado de un efecto de memoria dram\u00e1tico en la longitud de coherencia. Esta cantidad tiene una evoluci\u00f3n temporal no mon\u00f3tona al calentarse desde la fase v\u00edtrea a la fase paramagn\u00e9tica, antes de converger a la curva maestra (isoterma). Adem\u00e1s puede medirse con las t\u00e9cnicas experimentales actuales.\n\nNuestra investigaci\u00f3n del efecto Mpemba ofrece tambi\u00e9n una nueva perspectiva sobre un problema importante, a saber, el estudio de la longitud de coherencia v\u00edtrea en l\u00edquidos sobreenfriados y otros formadores de vidrio~\\cite{cavagna:09}. De hecho, la identificaci\u00f3n de la funci\u00f3n de correlaci\u00f3n correcta para el estudio experimental (o num\u00e9rico) sigue siendo un problema abierto. Los vidrios de esp\u00edn son \u00fanicos en el contexto general de la transici\u00f3n v\u00edtrea: sabemos qu\u00e9 funciones de correlaci\u00f3n deben calcularse microsc\u00f3picamente~\\cite{edwards:75,edwards:76} y se han obtenido determinaciones experimentales precisas de la longitud de coherencia~\\cite{guchhait:17}.\n\nLos resultados relacionados con este trabajo se han publicado en~\\cite{janus:19}.\n\n\\section{Conclusiones sobre el Caos en Temperatura en equilibrio}\nEn un vidrio de esp\u00edn, el efecto del \\textit{caos en temperatura} es la completa reorganizaci\u00f3n de los pesos de Boltzmann que determinan la frecuencia con la que cada configuraci\u00f3n de espines aparece, debido a un cambio arbitrariamente peque\u00f1o en la temperatura $T$.\n\nEn el cap\u00edtulo 6 hemos estudiado el fen\u00f3meno del caos en temperatura en simulaciones de equilibrio y hemos propuesto un eficiente m\u00e9todo variacional para estimar la elusiva cantidad del tiempo de autocorrelaci\u00f3n exponencial en una cadena de Markov, espec\u00edficamente para el caso del \\textit{Parallel Tempering}.\n\nEste m\u00e9todo variacional toma en cuenta tres par\u00e1metros y realiza una maximizaci\u00f3n de la estimaci\u00f3n del tiempo de autocorrelaci\u00f3n integrado en el espacio de fases de dichos par\u00e1metros. Puesto que el tiempo de autocorrelaci\u00f3n exponencial es una cota superior del tiempo de autocorrelaci\u00f3n integrado (que depende de los par\u00e1metros anteriormente mencionados), nuestro procedimiento es capaz de dar estimaciones robustas.\n\nAdem\u00e1s, hemos estudiado propiedades de escalado de la distribuci\u00f3n de probabilidad del tiempo de autocorrelaci\u00f3n obtenido a trav\u00e9s del m\u00e9todo variacional. En concreto, hemos demostrado que el escalado se mantiene para redes de tama\u00f1os $L \\geq 24$. Este resultado es consistente con resultados previos que usaban potenciales efectivos.\n\nLa presencia del caos en temperatura est\u00e1 relacionada con el pobre desempe\u00f1o del algoritmo de \\textit{Parallel Tempering} en la fase v\u00edtrea. Por lo tanto, el tiempo de autocorrelaci\u00f3n exponencial constituye una caracterizaci\u00f3n din\u00e1mica de este fen\u00f3meno.\n\nEn este trabajo, tambi\u00e9n hemos caracterizado el caos en temperatura desde un punto de vista est\u00e1tico mediante el estudio de configuraciones de equilibrio del sistema. El observable que nos permite estudiar cuantitativamente el caos en temperatura es el par\u00e1metro ca\u00f3tico. La observaci\u00f3n experimental de las \\textit{samples} m\u00e1s ca\u00f3ticas (desde un punto de vista din\u00e1mico) llev\u00f3 a la construcci\u00f3n de cantidades derivadas del par\u00e1metro ca\u00f3tico para estudiar el caos en temperatura (v\u00e9ase \\cite{fernandez:13,fernandez:16}). En nuestro trabajo, hemos introducido un nuevo observable para caracterizar este fen\u00f3meno.\n\nFinalmente, hemos comprobado las correlaciones estad\u00edsticas entre los estimadores est\u00e1ticos del caos y los estimadores din\u00e1micos. La introducci\u00f3n del nuevo estimador est\u00e1tico del caos ha mejorado considerablemente la correlaci\u00f3n con los estimadores din\u00e1micos con respecto a los estudios previos.\n\nLos resultados relacionados con este trabajo est\u00e1n publicados en~\\cite{billoire:18}.\n\n\n\\section{Conclusiones sobre el Caos en Temperatura fuera del equilibrio}\nEn el Cap\u00edtulo 7 hemos estudiado el interesante fen\u00f3meno del caos en temperatura en din\u00e1mica fuera del equilibrio que imita de cerca al caos en temperatura en equilibrio. De hecho, como hemos visto en la secci\u00f3n previa, el caos en temperatura est\u00e1 definido como un fen\u00f3meno de equilibrio y, por lo tanto, el estudio de este fen\u00f3meno fuera del equilibrio ha sido hist\u00f3ricamente un desaf\u00edo abierto que nosotros trataremos aqu\u00ed.\n\nEn primer lugar, intentamos una metodolog\u00eda \\textit{naive} para explicar el fen\u00f3meno (v\u00e9ase secci\u00f3n 7.2), con la cual encontramos un caos extremadamente d\u00e9bil. Afortunadamente, la correspondencia est\u00e1tica-din\u00e1mica ~\\cite{barrat:01,janus:08b,janus:10b,janus:17} junto con el an\u00e1lisis de eventos raros llevado a cabo en simulaciones de equilibrio~\\cite{fernandez:13}, nos proporcionaron el enfoque correcto para abordar el problema. En concreto, la equivalencia est\u00e1tica-din\u00e1mica nos permite relacionar la din\u00e1mica fuera del equilibrio de un vidrio de esp\u00edn (de tama\u00f1o infinito) con una longitud de coherencia $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, con peque\u00f1as \\textit{samples} de tama\u00f1o $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ que pueden ser llevadas al equilibrio.\n\nNuestro protocolo num\u00e9rico considera regiones quasi-esf\u00e9ricas de radio $r$. En este trabajo, nos centramos en la funci\u00f3n densidad de probabilidad del par\u00e1metro ca\u00f3tico calculado para dichas esferas. Encontramos que s\u00f3lo para las esferas en la cola de la distribuci\u00f3n puede observarse un caos en temperatura fuerte.\n\nEscoger una escala de longitud $r$ adecuada para las regiones esf\u00e9ricas ha resultado ser fundamental para estudiar el caos en temperatura. Esta escala de longitud \u00f3ptima es proporcional a la longitud de coherencia. Merece la pena resaltar que nuestros datos sugieren que la importancia de escoger la escala de longitud $r$ correcta es menos cr\u00edtica para el l\u00edmite $\\xi \\to \\infty$.\n\nUna conexi\u00f3n llamativa surge entre la din\u00e1mica y la est\u00e1tica en lo que al caos en temperatura se refiere. Encontramos una longitud caracter\u00edstica $\\xi^*(T_2-T_1,F)$ que marca el l\u00edmite entre un r\u00e9gimen de caos d\u00e9bil y un r\u00e9gimen de caos fuerte. El significado f\u00edsico de esta longitud caracter\u00edstica $\\xi^*$ sugiere que la llamada \\textit{longitud ca\u00f3tica de equilibrio} $\\ell_c$~\\cite{fisher:86,bray:87b} podr\u00eda ser su contrapartida en los sistemas en equilibrio. De hecho, ambas cantidades dependen de $T_2-T_1$ de la misma forma, a trav\u00e9s del exponente $\\zeta$\\footnote{Advertimos al lector que este exponente $\\zeta$ es completamente diferente al exponente $\\zeta$ de la Secci\u00f3n 8.2. Nosotros seguimos la notaci\u00f3n habitual en la literatura, denotando ambas cantidades con la misma letra a pesar de sus significados completamente distintos.}, que ha resultado ser el mismo para el equilibrio, y para fuera del equilibrio $\\zeta_{\\mathrm{NE}}$ dentro del nivel de dos desviaciones est\u00e1ndar. Contemplamos esta coincidencia como una nueva e importante evidencia de la equivalencia est\u00e1tica-din\u00e1mica.\n\nEn la segunda parte de este trabajo hemos llevado a cabo simulaciones con protocolos de cambios de temperatura para referir el problema del \\textit{cummulative aging}~\\cite{jonsson:02,bert:04,komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. Hemos encontrado peque\u00f1as pero claras violaciones al \\textit{cummulative aging} que son m\u00e1s fuertes cuando se enfr\u00eda el sistema que cuando se calienta. Aunque ambos protocolos (calentar y enfriar), muestran procesos de borrado de memoria, el protocolo de enfriamiento muestra una mejor memoria, es decir, se necesitan tiempos de coherencia m\u00e1s largos para borrar la memoria del estado previo a mayor temperatura.\n\nLos resultados relativos a este trabajo est\u00e1n publicados en~\\cite{janus:21}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Dynamic variational study of Temperature Chaos} \\labch{equilibrium_chaos}\n\nThe process of taking a \\gls{SG} sample to equilibrium in numerical simulations requires the use of dynamic Monte\\index{Monte Carlo} Carlo methods. Unfortunately, as we have already commented in \\refsubsec{Monte_Carlo}, the sluggish dynamics exhibited by \\gls{SG}s impedes the use of simple methods like the Metropolis-Hasting algorithm. One solution comes from the use of the \\gls{PT} method, which equilibrates at once a set of $N$ copies of the system running at different temperatures.\n\nHowever, the \\gls{TC}\\index{temperature chaos} phenomenon represents a major obstacle in the performance of \\gls{PT}~\\cite{fernandez:13}. In this chapter, we follow the ideas proposed in previous studies~\\cite{fernandez:13,martin-mayor:15,fernandez:16} and we take advantage of that fact by quantitatively characterizing the \\gls{TC}\\index{temperature chaos} through a careful study of the process of thermalization\\index{thermalization} of the system when using the \\gls{PT} method. This work also extends the study of \\gls{TC}\\index{temperature chaos} performed in previous papers~\\cite{fernandez:13,fernandez:16} by the development of a variational method\\index{variational method}.\n\nMoreover, we also focus on the very definition of \\gls{TC}\\index{temperature chaos} and we study it by comparing equilibrium configurations\\index{configuration} at different temperatures. Both characterizations, namely dynamic and static, are found to correlate very well~\\cite{fernandez:13,fernandez:16}. Here, we propose new observables to study the \\textit{static} chaos and we found large correlations between the main observables of both characterizations, static and dynamic.\n\nAll the results exposed in this chapter came from the original work~\\cite{billoire:18} which has been developed during this thesis.\n\n\\section{Numerical simulations} \\labsec{numerical_simulations_eq_chaos}\nIn order to keep clean the rest of the chapter of technical details and to focus on the physical results, we explain here the simulations performed.\n\nFirst of all, it is fundamental to mention that the data used here come from the study of the metastate\\index{metastate} (see~\\refch{metastate}) and, therefore, the structure of the couplings\\index{couplings} is not conventional. We briefly recall here, for the reader's convenience, the particularities of this simulation.\n\nThe system, composed of $L^3$ spins, is divided into an inner region of $(L\/2)^3$ spins and an outer region surrounding it. For each of the $10$ realizations of the inner disorder\\index{disorder}, we have a set of $1280$ realizations of the outer disorder\\index{disorder}. Hence, we have a total of $12800$ samples\\index{sample} and for each one, we have simulated $\\Nrep=4$ different replicas\\index{replica}. \n\nA natural question is whether this particular setup's choice is affecting the results. One could imagine that those samples\\index{sample} sharing the same inner disorder\\index{disorder} would be strongly correlated and, hence, the statistics coming from only 10 different inner realizations could be not enough to deal with the sample-to-sample fluctuations\\index{sample-to-sample fluctuations}. However, this choice is irrelevant for the studied observables in this work. The interested reader can find a detailed discussion in \\refsec{selection_parameters}.\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{cccccccc}           \n\\toprule\n\\toprule\n\\multicolumn{7}{c}{MUSA-MSC} \\\\\n\\hline\n$L$ & $L_{\\text{int}}$ & $N_T$ & $T_{\\mathrm{min}}$ & $T_{\\max}$ & $N_\\text{Met}$ ($ \\times 10^6$) & $\\text{ps\/s}$  \\\\\n\\hline\n24 & 12 & 24 & 0.698 & 1.538 & 500 & 104 \\\\ \n16 & 8 & 16 & 0.479 & 1.575 & 250 & 107  \\\\ \n16 & 8 & 13 & 0.698 & 1.575 & 250 & 119  \\\\ \n16 & 12 & 13 & 0.698 & 1.575 & 250 & 119 \\\\ \n14 & 12 & 13 & 0.698 & 1.575 & 500 & 120 \\\\\n12 & 6 & 13 & 0.698 & 1.575 & 250 & 119  \\\\ \n8 & 4 & 13 & 0.698 & 1.575 & 250 & 126 \\\\\n\\bottomrule\n\\end{tabular}\n\\\\[5mm]\n\n\\begin{tabular}{ccccccccc}  \n\\toprule\n\\toprule         \n\\multicolumn{8}{c}{MUSI-MSC} \\\\\n\\hline\n$L$ & $L_{\\text{int}}$ & $N_T$ & $N_\\text{samp}$ & $N_\\text{Met,min}$ & $N_\\text{Met,mean}$ & $N_\\text{Met,max}$ & $\\text{ps\/s}$ \\\\\n\\multicolumn{4}{c}{} & $\\times 10^6$ & $\\times 10^6$ & $\\times 10^6$ & \\\\\n\\hline\n24 & 12 & 24 & 2441 & 1000 & 4262 & 326000 & 57 \\\\ \n16 & 8 & 16  & 2898 & 500 & 5096 & 355500 & 304 \\\\ \n16 & 8 & 13 & 338 & 500 & 543 & 4000 & 306 \\\\ \n16 & 12 & 13 & 314  & 500 & 578 & 8000 & 306 \\\\ \n\\bottomrule\n\\end{tabular}\n\n\\caption[\\textbf{Parameters of the simulations MUSA-MSC and MUSI-MSC.}]{\\textbf{Parameters of the simulations MUSA-MSC and MUSI-MSC.} $L$ is the lattice size; $L_\\text{int}$ the size of the inner part of the lattice; $N_T$, $T_{\\mathrm{min}}$ and $T_{\\max}$ are the number of temperatures, the minimum and the maximum temperatures used in the \\gls{PT} method; $N_\\text{Met}$ is the number of Metropolis sweeps (at each temperature); $\\text{ps\/spin}$ is the average CPU time per spin-flip in MUSI-MSC, using an Intel Xeon CPU E5-2680 processors; $N_\\text{samp}$ denotes the number of bad samples\\index{sample} whose simulations had to be extended in order to thermalize and finally $N_\\text{Met,min}$, $N_\\text{Met,mean}$ and $N_\\text{Met,max}$ are the minimum, mean and maximum number of Metropolis sweeps per temperature needed to reach thermalization\\index{thermalization} (bad samples\\index{sample}). The set of temperatures used is clearly the same in the MUSI-MSC and MUSA-MSC parts of this Table. The number of Metropolis sweeps between two consecutive \\gls{PT} updates is always $N_\\text{MpPT} = 10$.  For the MUSI-MSC simulation of $L=24$ we parallelized\\index{parallel!computation}, using \\emph{Pthreads}, by distributing the $N_T=24$ system copies among 12 CPU cores in the Intel Xeon CPU E5-2680.}\n\\labtab{parameters_simulation_MUSA_MUSI}\n\\end{table}\n\nThe samples\\index{sample} have been equilibrated by using the \\gls{PT} method with Metropolis updates between two consecutive \\gls{PT} exchanges. We increase the performance of the Metropolis update via multispin coding and we apply two methods widely used in numerical simulations in statistical physics, namely the \\gls{MUSA}\\index{Multispin Coding!Multisample}~\\cite{newman:99} and the \\gls{MUSI}\\index{Multispin Coding!Multisite}~\\cite{fernandez:15}. The basic idea of these methods is the parallelization\\index{parallel!computation} of operations that are, indeed, independent from each other by taken advantage of the streaming extensions of the current computer processors. Our simulations were carried out using either Intel Xeon E5-2680 or AMD Opteron Processor 6272. Further details can be found in \\refch{AP_multispin_coding}.\n\nThe selection of the parameters of the simulation are in \\reftab{parameters_simulation_MUSA_MUSI} and the reason for the choice of some of them is explained in~\\refsec{selection_parameters}. Although all the simulations are included for completeness, some of them were only used in the metastate\\index{metastate} study (see~\\refch{metastate}). We focus here only in those simulations with $L_{\\mathrm{int}} = L\/2$. It is worthy to note that in most of this chapter, for the $L=16$ system, we are using the simulation with $N=16$ temperatures, barring the discussion on the impact of the minimum temperature of the \\gls{PT} mesh in the \\gls{TC}\\index{temperature chaos} (see \\refsec{correlation_dynamics_static}), where we will use the simulation with $N=13$.\n\n\n\n\n\\section{Monte Carlo, why have you forsaken me?}\\labsec{Monte_Carlo_forsaken}\nWe have already sketched the main idea motivating this work. Traditional Monte\\index{Monte Carlo} Carlo methods like the Metropolis-Hasting algorithm are not useful to study (at equilibrium) the low-temperature phase\\index{phase!low-temperature\/spin-glass} of a \\gls{SG} because the presence of many free-energy\\index{free energy!valley} local minima often causes the numerical simulation to get trapped and, as a consequence, the correct sampling of the phase space\\index{phase space} gets severely harmed.\n\nThe \\gls{PT} method solves this problem. The introduction of $N$ copies at different temperatures, and the possibility for each copy to exchange its temperature with another different copy, allows the copies to visit the high-temperature phase\\index{phase!high-temperature\/paramagnetic}, where it decorrelates very quickly from its previous state. When the copy \\textit{comes back} to the low-temperature phase\\index{phase!low-temperature\/spin-glass}, it visits another different free-energy\\index{free energy!valley} local minima and the performance of the thermalization\\index{thermalization} process boosts.\n\nThe \\gls{TC}\\index{temperature chaos} phenomenon dramatically decreases that performance. Intuitively one can understand why the \\gls{TC}\\index{temperature chaos} represents a major obstacle in the \\gls{PT} temperature flow~\\cite{janus:10,fernandez:13,martin-mayor:15,fernandez:16}. Imagine we have two sets of configurations\\index{configuration} (states) of two equilibrated systems at different temperatures $T_1<T_2$. If \\gls{TC}\\index{temperature chaos} occurs, then the typical configurations\\index{configuration} will be very different for both temperatures. In the simpler scenario, we have a level crossing in which at a given temperature the free-energy\\index{free energy} for both sets of configurations\\index{configuration} are almost equal but their specific heats\\index{specific heat} are rather different. Actually, this discrepancy in the value of the specific heat\\index{specific heat} is very harmful to the \\gls{PT} method (see, for example, \\cite{rathore:05,katzgraber:06b,sabo:08,malakis:13}). As a consequence, to place a $T_1$\\textit{-like} set of configurations\\index{configuration} at $T_2$ is possible, but, due to the specific heat\\index{specific heat} difference, it will have a hard traveling to higher temperatures (the inverse example is also true). In addition, equilibrium in \\gls{PT} implies equilibrium at all the temperatures, hence, even a single \\textit{dynamic chaotic event} as described above can ruin our expectations.\n\nBut these \\textit{a priori} inconveniences are not a real reason to be sad. Actually, generally in Markov\\index{Markov chain} chains, the exponential time $\\tau_{\\exp}$~\\cite{sokal:97} tells us how long we have to simulate the system in order to reach the equilibrium. Those samples\\index{sample} which suffer stronger \\gls{TC}\\index{temperature chaos} would also need longer times to thermalize and the study of the exponential time could help us to point those chaotic samples\\index{sample}~\\cite{fernandez:13,fernandez:16}. \n\nUnfortunately, $\\tau_{\\exp}$ has proven to be a very elusive quantity, and its accurate computation is not an easy task. It has been suggested that $\\tau_{\\exp}$ is best computed by studying the temperature-flow of the system copies of the \\gls{PT} simulation~\\cite{janus:10}. In the next section, we revisit the problem of the computation of $\\tau_{\\exp}$ and present a variational method\\index{variational method} that can potentially save a large amount of time.\n\n\\section{Time scales in a Markov chain}\\labsec{time_scales_eq_chaos}\nWe have stated in~\\refsubsec{Monte_Carlo} that a well-behaved Monte\\index{Monte Carlo} Carlo method, based on a Markov\\index{Markov chain} chain, assures the system to reach the equilibrium (i.e. the configurations\\index{configuration} of the system are distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution), no matter which particular initial condition we choose [recall~\\refeq{thermalization_condition}]. However, knowing how long we need to simulate the system to reach that equilibrium is an important question.\n\nLet us consider an equilibrated system, for a given quantity $f$ its \\textit{autocorrelation function}\\index{correlation function!time auto-} would be\n\\begin{equation}\nC_f(t) = \\mcav{f(t_1)f(t_2)} - \\mcav{f(t_1)}^2 \\, , \\quad t=t_1 - t_2 \\, , \\labeq{unnormalized_autocorrelation_function}\n\\end{equation}\nwhere $\\mcav{\\cdots}$ stands for the expectation value\\footnote{It is worthy to note that $\\mcav{\\cdots}$ is different for the average $\\braket{\\cdots}$ defined in~\\refeq{average_o}. It is true that, for an equilibrated system, the Markov\\index{Markov chain} chain samples\\index{sample} the configurational\\index{configuration!space} space according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution and, therefore, $\\mcav{\\cdots}$ can estimate $\\braket{\\cdots}$, but it is not true in general for a non-equilibrated system. We stress the difference between them by adopting different notations.} of the quantity $f$. The required equilibrium condition implies that $\\mcav{f(t_1)} = \\mcav{f(t_2)}$. \n\nThe normalized autocorrelation function\\index{correlation function!time auto-} $\\hat{C}(f)$ is\n\\begin{equation}\n\\hat{C}_f(t) = \\dfrac{C_f(t)}{C_f(0)} \\, . \\labeq{normalized_autocorrelation_function}\n\\end{equation}\n\nThis quantity decays exponentially for large $t$ as $\\hat{C}_f(t) \\sim \\exp\\left(-\\abs{t}\/\\texpf\\right)$. For each quantity we can extract the value of $\\texpf$\n\\begin{equation}\n\\texpf = \\limsup_{t \\to \\infty} \\dfrac{t}{- \\log \\abs{\\hat{C}_f(t)}} \n\\end{equation}\nand the supreme of $\\texpf$ for every quantity $f$ \n\\begin{equation}\n\\texp = \\sup_f \\texpf \\, , \\labeq{exponential_autocorrelation_time}\n\\end{equation}\nis what we call \\textit{exponential autocorrelation time}\\index{autocorrelation time!exponential} which is the time scale ruling the transition of the system from an arbitrary configuration\\index{configuration} to the equilibrium, that is, this is the quantity we are looking for.\n\nNonetheless, there are two main difficulties in the computation of $\\texp$ from a practical point of view. The first one concerns the selection of the quantity $f$: we do not know which is the quantity $f$ that maximizes $\\texpf$. The second one is a technical problem: the fit of a numeric autocorrelation function\\index{correlation function!time auto-} to an arbitrary function from which we only know that its long-term behavior is an exponential decay is not an easy task. Procedures followed in previous works~\\cite{janus:10} suggest considering the autocorrelation function\\index{correlation function!time auto-} as a sum of two decaying exponentials from which we can extract the value of $\\texp$. This procedure is very delicate because a large number of simulated samples\\index{sample} requires an automatic way to obtain $\\texp$ and this automatic protocol can fail, requiring human intervention and, most of the time, demanding to extend the simulations. The full protocol is carefully detailed in Appendix A of \\cite{janus:10}.\n\nIt would be desirable to introduce a different time scale with better properties. At this point, we define the \\textit{integrated autocorrelation time}\\index{autocorrelation time!integrated} $\\tintf$\n\\begin{equation}\n\\tintf = \\dfrac{1}{2}+ \\sum_{t=1}^\\infty \\hat{C}_f(t) \\, . \\labeq{integrated_autocorrelation_time}\n\\end{equation}\n\nIt is not hard to prove~\\cite{sokal:97} that $\\tintf$ controls the statistical errors in measuring the quantity $f$. Specifically, if an equilibrated system generates one configuration\\index{configuration} at the Monte\\index{Monte Carlo} Carlo time $t$, we need to wait until $t + 2\\tintf$ to obtain a statistically independent configuration\\index{configuration}, but only as far as the quantity $f$ is concerned.\n\nThe normalized autocorrelation function\\index{correlation function!time auto-} can be expressed in terms of the eigenvalues $\\lambda_n$ of the transition probability matrix\\index{transition matrix} projected on the subspace orthogonal to its eigenvalue 1 ($1>|\\lambda_1|\\geq |\\lambda_2|\\geq\\ldots$), see Ref.~\\cite{sokal:97},\n\\begin{equation}\n\\hat C_f(t)=\\sum_n A_{n,f} \\lambda_n^{|t|}\\,,\\quad \\sum_n A_{n,f}=1\\,, \\labeq{autocorrelation_function_decomposition}\n\\end{equation}\nwhere the index $n$ runs from 1 to the size of the transition matrix\\index{transition matrix}, in our case $N_T!2^{N_TL^D} - 1$.\n\nThe amplitudes $A_{n,f}$  depend on $f$, while the $\\lambda_n$ are $f$-independent. We can plug~\\refeq{autocorrelation_function_decomposition} into \\refeq{integrated_autocorrelation_time} and, by computing the sum of a geometric series, we have\n\\begin{equation}\n\\tintf = \\dfrac{1}{2} + \\sum_n A_{n,f} \\dfrac{\\lambda_n}{1-\\lambda_n} \\, .\n\\end{equation}\n\nNow, in practical applications the (leading) $A_{n,f}$'s and $\\lambda_n$'s are real positive. Hence, $\\lambda_n=\\mathrm{e}^{-1\/\\tau_n}$ defines the characteristic time $\\tau_n$. The exponential autocorrelation time\\index{autocorrelation time!exponential} of the Markov\\index{Markov chain} chain $\\texp$ is just $\\tau_1$, the largest of the $\\tau_n$ (see \\cite{sokal:97}). Now, for $\\tau_n\\gg 1$ we can perform a Taylor expansion and we obtain $\\lambda_n\/(1-\\lambda_n)=\\tau_n - \\dfrac{1}{2} + \\mathcal{O}(1\/\\tau_n)$. \\refeq{normalized_autocorrelation_function} and \\refeq{integrated_autocorrelation_time} become\n\\begin{equation}\n\\hat{C}_f(t) = \\sum_n A_{n,f} e^{-\\abs{t}\/\\tau_n} \\>\\> , \\>\\> \\tintf = \\sum_n A_{n,f} \\tau_n \\, . \\labeq{autocorrelation_decomposition}\n\\end{equation}\nThe integrated autocorrelation\\index{autocorrelation time!integrated} time $\\tintf$ for the quantity $f$ is just an average of the decay modes of the correlation function, being the slower mode the exponential autocorrelation time\\index{autocorrelation time!exponential} $\\texp$ and the weights of that average the coefficients $A_{n,f}$. It is straightforward to prove that \n\\begin{equation}\n\\tintf \\leq \\texp \\, , \\labeq{inequality_autocorrelation_time}\n\\end{equation}\nand the equality is reached when $A_{1,f} = 1$.\n\n$\\tintf$ is the quantity we were looking for because, although it has the same disadvantage that $\\texpf$ with respect to the chosen $f$, it is much simpler to compute. In addition, in this work we overcome the problem of the quantity $f$ by using a variational method\\index{variational method} very similar to the Rayleigh-Ritz variational principle in Quantum Mechanics.\n\nThe last consideration has to be done before explaining our variational method\\index{variational method}. In this work, although we have proposed a new thermalization\\index{thermalization} protocol, we needed to compare our results with the reliable results provided by the computation of the exponential autocorrelation time\\index{autocorrelation time!exponential}. The thermal protocol followed here is the same described in appendix A of~\\cite{janus:10}.\n\n\\section{The variational method: dynamic Temperature Chaos} \\labsec{variational_method_eq_chaos}\nThis sections aims to describe the variational method\\index{variational method} used to compute our estimation of $\\texp$. The idea is very simple, from~\\refeq{autocorrelation_decomposition} and~\\refeq{inequality_autocorrelation_time} we can deduce that a very good estimation of $\\texp$ from $\\tintf$ would imply to choose a quantity $f$ with $A_{1,f} \\approx 1$ and $A_{n>1,f} \\approx 0$. As we have already introduce above, it has been suggested that we should focus on the temperature flow along the \\gls{PT} dynamics in order to compute the autocorrelation function\\index{correlation function!time auto-}~\\cite{janus:10,fernandez:13,martin-mayor:15}.\n\nThe computations of time autocorrelation functions\\index{correlation function!time auto-} are usually performed with spin-configuration\\index{configuration} functions $f$. In our protocol, we focus on the temperature random-walk\\index{random walk!temperature} that each copy of the system performs over the temperature mesh in the \\gls{PT} dynamics. At the first sight, it might be surprising that this temperature random-walk\\index{random walk!temperature} can provide information about the thermalization\\index{thermalization} of the system. The reader may find in \\refsec{thermalizing_PT} a detailed discussion about this fact.\n\nLet us consider, for a given sample\\index{sample}, the $N$ system copies in the \\gls{PT} dynamics. Our Markov\\index{Markov chain} chain will be the temperature random-walk\\index{random walk!temperature} through the index $i_t$ that indicates that, at time $t$, our system copy is at temperature $T_{i_t}$. At equilibrium, all the copies spend the same time in each temperature and, therefore, the probability for $i_t$ is just the uniform probability over the set $\\{1,2,\\dots,N \\}$. In consequence, the expectation value of the quantity $f$ is just the arithmetic mean\n\\begin{equation}\n\\mcav{f} = \\dfrac{1}{N} \\sum_{i=0}^N f(i) \\, . \\labeq{expectation_value_f}\n\\end{equation} \nWe shall consider, as well, functions that depends on pairs of system copies. For a given time $t$, these pairs will be described by two indices $i_t \\neq j_t$. The equilibrium value for an arbitrary function of a pair of system copies is\n\\begin{equation}\n\\mcav{f_{\\mathrm{pairs}}} = \\dfrac{1}{N(N-1)}\\sum_{i=0}^N \\sum_{j\\neq i}^N f_{\\mathrm{pairs}}(i,j) \\, .\n\\end{equation}\nBy looking at the definition of the time autocorrelation function\\index{correlation function!time auto-} in~\\refeq{unnormalized_autocorrelation_function} it is clear that, for computational purposes, it is convenient to define a function $f$ with $\\mcav{f}=0$. Therefore, for each function $f$ we define\n\\begin{equation}\n\\tilde{f} = f - \\mcav{f} \\, . \\labeq{f_expectation_0}\n\\end{equation}\nWe can define now our (not normalized) time autocorrelation function\\index{correlation function!time auto-} as\n\\begin{equation}\nC_f(t) = \\dfrac{1}{N_s-t_0-t} \\sum_{t'=t_0}^{N_s-t} \\tilde{f}(i_{t'})\\tilde{f}(i_{t'+t}) \\, ,\n\\end{equation}\nwhere $N_s$ is the total number of times we have \\textit{measured} the state of the \\gls{PT} indices $i_t$. Here, of course, we are considering the equilibrium autocorrelation function\\index{correlation function!time auto-} and, therefore, $t_0 \\gg \\tintf$\\footnote{The reader may find this statement a little bit contradictory since we are trying to estimate $\\tintf$, however a self-consistent procedure is followed, similarly to that one explained in~\\cite{janus:10}.}. Note that $C_f(t)$ is independent of the system copy as well as of the replica\\index{replica} and, hence, we can improve our statistics by averaging over the $N \\times \\Nrep$ numerical estimations of $C_f(t)$. All the statements for $f$ depending on a single copy are straightforwardly transferable to functions depending of a pair of system copies. \n\nOnce we have an estimation of $C_f(t)$ we can estimate the integrated autocorrelation\\index{autocorrelation time!integrated} time\n\\begin{equation}\n\\tintf \\approx \\nmet \\left( \\dfrac{1}{2} + \\sum_{t=0}^W \\hat{C}_f(t) \\right) \\, , \\labeq{window_integrated_autocorrelation_time}\n\\end{equation}\nwhere $\\nmet$ is the periodicity with which we record the time indices $i_t$ of our random walker\\index{random walk} and $\\hat{C}_f(t) = C_f(t)\/C_f(0)$ is the normalized autocorrelation function. In our simulations $\\nmet=25000$ Metropolis sweeps most of the times. The  original definition of $\\tintf$ [see \\refeq{integrated_autocorrelation_time}] involves an infinite sum, here we restrict the sum only to the first $W$ values, being $W$ a self-consistent window (see \\cite{sokal:97}) that avoids the divergence of the variance of $\\tintf$. We impose $\\tintf < 10W$.\n\nIn order to compute $\\tintf$ as closely as possible to the $\\texp$ value, we consider three different parameters to optimize: the type of function $f$, the temperature $T^*$ at which $f$ is zero, and a Wilson-Kadanoff\\index{Wilson-Kadanoff} renormalization\\index{renormalization group} block length $\\lblo$. We describe here the three parameters.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.6\\columnwidth}{@{\\extracolsep{\\fill}}cc}\n\\toprule\n\\toprule\n\\textbf{Identifier} & \\textbf{Function} \\\\\n\\toprule\n$0$ & piecewise constant \\\\\n$1$ & piecewise linear\\\\\n$2$ & piecewise quadratic\\\\\n$3$ & piecewise cubic\\\\\n$|$ & OR in couples\\\\\n$\\&$ & AND in couples\\\\\n$\\wedge$ &  XOR in couples\\\\\n$*$ &  Multiplication in couples \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Functions of the variational method.}]{\\textbf{Functions of the variational method\\index{variational method}.} Different choices of the function $f$ used in the variational method\\index{variational method}.}\n\\labtab{functions_variational_method}\n\\end{table}\n\n\\begin{itemize}\n\\item \\textbf{The type of function \\boldmath $f$.} Similarly to the Rayleigh-Ritz variational principle in Quantum Mechanics we consider test-functions $f$, belonging to eight different classes (see~\\reftab{functions_variational_method}). The four first functions depend only on a single system copy. Specifically, the function labeled as $0$ is just the complementary of the Heaviside function $1-\\Theta(T^*)$. The function labeled as $1$ is a piecewise linear function that has already been used before in~\\cite{janus:10}. As is quite evident by the description, the functions labeled as $2$ and $3$ are quadratic and cubic piecewise functions respectively. We will specify their specific functional form below. \n\nThe last four functions depend on two system copies. For the functions labeled as $\\lvert$, $\\&$ and $\\wedge$, each system copy of the pair has associated the value of the function labeled as $0$. Then, the value of the function is the corresponding binary operation of the pair of values obtained for each system copy. Finally, the $*$ function is just the multiplication of the piecewise linear function for each system copy (with the corresponding normalization).\n\n\\item \\textbf{The temperature $\\mathbf{T^*}$.} We require for the temperature $T^* \\in \\{T_1,T_2, \\dots, T_{N\/2}\\}$ that $f(T^*) = 0$. The value of $T^*$ is our second variational parameter. In addition, the condition $\\mcav{f_{T^*}}=0$ together with $f(T^*) = 0$ define our test-functions. Specifically, the linear piecewise function is\n\\begin{equation}\n\\begin{aligned}\nT > T^*\\,&: \\quad &f_{T^*}(T) = a_+ (T-T^*) \\, ,\\\\\nT < T^*\\,&: \\quad &f_{T^*}(T) = a_- (T-T^*) \\, .\n\\end{aligned}\n\\end{equation}\nWe require $a_+$ and $a_-$ to be positive and their ratio is fixed by the condition $\\mcav{f_{T^*}}=0$. Indeed, we only need to fix the ratio, because the overall scale of the test function $f_{T^*}$ is irrelevant.\n\nWith an analogous procedure we can define the quadratic ($p=1$) and the cubic ($p=2$) functions\n\\begin{equation}\n\\begin{aligned}\nT>T^* \\, : \\quad f_{T^*}(T) = a_+ (T-T^*)^p (2T_N - T^* -T) \\, ,\\\\\nT<T^* \\, : \\quad f_{T^*}(T) = a_- (T-T^*)^p (2T_1 - T^* -T) \\, .\\\\\n\\end{aligned}\n\\end{equation}\nAgain, we choose $a_+>0$ and $a_->0$ and the ratio is fixed by $\\mcav{f_{T^*}}=0$. We try all the possible values of $T^*$ in the lower half part of the set of temperatures in our \\gls{PT} simulation.\n\n\\item \\textbf{The renormalization\\index{renormalization group} time-block \\boldmath $\\lblo$.} We can modify the value of the time autocorrelation function\\index{correlation function!time auto-} by changing the function $f$ itself (as it has been introduced in the two previous parameters of the variational method\\index{variational method}) but we can also modify it by changing the temporal series from which we compute that autocorrelation function\\index{correlation function!time auto-}. We build Wilson-Kadanoff\\index{Wilson-Kadanoff} blocks: the Monte\\index{Monte Carlo} Carlo sequence $f_{T^*}(i_1),f_{T^*}(i_2),\\dots,f_{T^*}(i_{N_s})$ is divided into blocks of $\\lblo$ consecutive data (see e.g. \\cite{amit:05}). For each block, we compute the average and we build a new sequence $f'_{T^*}(j_1),f'_{T^*}(j_2),\\dots,f'_{T^*}(j_{N_s\/\\lblo})$ from which we compute the integrated autocorrelation time\\index{autocorrelation time!integrated} just as we did for $\\lblo=1$. Of course, after computing the integrated autocorrelation time\\index{autocorrelation time!integrated}, we need to rescale it to recover the original time units. The idea of this parameter is to reduce the high-frequency fluctuations of the time autocorrelation function\\index{correlation function!time auto-}.\n\nHowever, the parameter $\\lblo$ needs to be controlled, otherwise, it can become larger than $\\tintfTlblo$ erasing all the information of the autocorrelation function\\index{correlation function!time auto-} and giving us spurious results. If that happens, each block would be just the expectation value (well, a finite estimation) of the function $f$, and the autocorrelation function\\index{correlation function!time auto-} would vanish for times $t\\neq 0$. From~\\refeq{window_integrated_autocorrelation_time} we deduce that, after converting $\\tintf$ to the correct time units, we would have $\\tintfTlblo = \\lblo \\nmet\/2$, which diverges with $\\lblo$. With the aim to control that spurious effect, we impose\n\\begin{equation}\n\\tintfTlblo > \\dfrac{5}{2} \\nmet \\lblo \\, . \\labeq{tintfTlblo_condition}\n\\end{equation}\nThe values of $\\lblo$ are taken from the list $\\lblo =\\{1,2,5,10,20,50,100,200,500,1000$\n$,2000\\}$.\n\\end{itemize}\n\nOur estimation of the integrated autocorrelation time\\index{autocorrelation time!integrated}, namely $\\tintvar$, is just the highest value of all the $\\tintfTlblo$. This estimation has great advantages from its predecessor (which corresponds with the linear piecewise function at the critical\\index{critical temperature} temperature and $\\lblo=1$). Firstly, our estimation is robust in the sense that it does not produce spurious values. Moreover, the process can be easily implemented in an automatic way which is a \\textit{sine qua non} condition given the huge number of possible combinations of parameters.\n\nThe worse scenario would be to found that our effort has been in vain and the automatic process always chooses the piecewise linear function with $T^* = \\ensuremath{T_\\mathrm{c}}\\xspace$ and $\\lblo=1$. Fortunately, this is not the case, in~\\reftab{frequency_functions_variational_method} we can see the numbers of times that our method chooses each function. Indeed, it is notorious that almost all of the time, the method chooses a single-copy function. The same happens with $T^*$, the chosen value is not always $\\ensuremath{T_\\mathrm{c}}\\xspace$, actually, we called the chosen temperature the \\textit{dynamic chaotic temperature} $T_d$ that will be useful in the following analysis. The effects of $\\lblo$ or, more accurately, the effects of the discretization of $\\lblo$ can also be observed in the results (for example \\reffig{log_tau_I}). Specifically, the low density in the $\\lblo$ mesh leads to small gaps in the determination of the autocorrelation time\\index{autocorrelation time!integrated} $\\tintvar$.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}cccccccccc}\n\\toprule\n\\toprule\n$L$ & $0$ & $1$ & $2$ & $3$ & $|$ & $\\&$ & $\\wedge$ & $*$ & Total  \\\\\n\\toprule\n$16$ & $2032$ & $5320$ &  $3875$ & $1374$  & $4$ & $115$ & $74$ & $6$ & $12800$ \\\\\n$24$ & $1556$ & $7196$ &  $3089$ & $820$  & $0$ & $127$ & $11$ & $1$ & $12800$ \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Frequency of choice of the variational method.}]{\\textbf{Frequency of choice of the variational method\\index{variational method}.} Number of times the variational method\\index{variational method} has picked one of the eight choices among the functions $f$ described in the text. $L$ denotes the lattice size.}\n\\labtab{frequency_functions_variational_method}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/func_corr}\n\\caption[\\textbf{Improvement of the estimation of the time autocorrelation functions.}]{\\textbf{Improvement of the estimation of the time autocorrelation functions.}\\index{correlation function!time auto-} Auto-correlation function for the most chaotic sample\\index{sample} for $L=16$ (left) and $L=24$ (right): (Top) Auto-correlation function computed using the method of \\cite{janus:10} and (Bottom) using the variational method\\index{variational method} presented here. Note that the improvement of the new method is notorious is we focus on the y-intercept (further details can be found in the text). \\textbf{Inset:} Linear-log plot showing the small $t$ behavior of the autocorrelation function\\index{correlation function!time auto-}.}\n\\labfig{example_autocorrelation_function}\n\\end{figure}\n\nAnother question has to be answered. It is true that our method chooses a variety of parameters, to the detriment of the classical choice but, is there a significant improvement in the estimation of $\\texp$ or are all the estimations just small fluctuations of the previous estimation $\\tint$? An example of the improvement obtained in the computation of the autocorrelation function\\index{correlation function!time auto-} is shown in~\\reffig{example_autocorrelation_function}. The main problem of the previous estimation becomes obvious from the figure: the value of $A_{1,f}$ [see~\\refeq{autocorrelation_decomposition}] could be, indeed, fairly small $A_{1,f} \\approx 0.1$. In the figure, this amplitude roughly corresponds to the abscissa of the initial point of the linear decreasing that we are able to see in the log-log scale (i.e. the beginning of the domination of the large time-scale corresponding to $\\texp$). Our new estimations (bottom panels) are rather better.\n\nThis hand-waving argument can be made quantitative. Let us denominate $\\tintold$ to the methodology of estimation of $\\tint$ for previous works~\\cite{janus:10} and $\\tintvar$ to our variational-method\\index{variational method} estimation. In~\\reffig{histograma_taus_multiplot_g} we separate our samples\\index{sample} in deciles according to its $\\tintvar$ value so that the first decile corresponds to the $1280$ samples\\index{sample} with smaller $\\tintvar$. We have argued that the most chaotic samples\\index{sample} will have larger $\\texp$, so those deciles are our proposal for separating the \\textit{most chaotic} and \\textit{less chaotic} samples\\index{sample}.\n\nThen, we build up the histogram of the ratio $\\tintold\/\\tintvar$ for the samples\\index{sample} on a given decile. Top panels in~\\reffig{histograma_taus_multiplot_g} show that the gain of considering $\\tintvar$ is sizable but, if we focus on the most chaotic samples\\index{sample} (i.e. the tenth decile, in the bottom panels) the benefits of our variational method\\index{variational method} are more than evident with a significant fraction of the samples\\index{sample} with $\\tintold\/\\tintvar<0.1$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/histograma_taus_multiplot_g}\n\\caption[\\textbf{A quantitative argument for the variational method.}]{\\textbf{A quantitative argument for the variational method\\index{variational method}.} Conditional probability density function of the ratio $\\tintold\/\\tintvar$, given that $\\tintvar$ belongs to a given decile is plotted. We show the data for the first decile (left) and the tenth decile (right) for $L=16$ (top) and $L=24$ (bottom).}\n\\labfig{histograma_taus_multiplot_g}\n\\end{figure}\n\n\n\n\\section{Static Temperature Chaos}\nAt this point, we momentarily forget about previous disquisitions on \\gls{TC}\\index{temperature chaos} from the dynamical point of view and we come back to the classical static definition: \\gls{TC}\\index{temperature chaos} is the complete rearrangement of the equilibrium configurations\\index{configuration} upon any change of temperature. This phenomenon has been traditionally studied~\\cite{billoire:00,billoire:02,katzgraber:07} through the probability density function of the overlap\\index{overlap!distribution} between the spin configurations\\index{configuration} at temperatures $T_1$ and $T_2$,\n\\begin{equation}\nq_{T_1,T_2} = \\dfrac{1}{V} \\sum_i s_i^{T_1} s_i^{T_2} \\, . \\labeq{overlap_2T}\n\\end{equation}\nDue to the limitation of the maximum size that can be simulated, this quantity is strongly affected by finite-size effects\\index{finite-size effects}. We focus therefore on other quantity, introduced in~\\cite{ritort:94}: the \\textit{chaotic parameter}\n\\begin{equation}\n\\xchaos = \\dfrac{\\braket{q^2_{T_1,T_2}}_J}{\\sqrt{\\braket{q^2_{T_1,T_1}}_J\\braket{q^2_{T_2,T_2}}_J}} \\, , \\labeq{chaotic_parameter}\n\\end{equation}\nwhere $\\braket{\\cdots}_J$ stands for the usual thermal average but we stress the sample\\index{sample} dependency with the sub-index $J$. It has been proposed that the \\gls{TC}\\index{temperature chaos} phenomenon should be studied through a detailed analysis of the distribution of this sample-dependent chaotic parameter~\\cite{fernandez:13,billoire:14}.\n\nThe reader should notice that $0 < \\xchaos \\lesssim 1$. The extreme values are clear; $\\xchaos = 1$ means that both configurations\\index{configuration} at temperatures $T_1$ and $T_2$ are indistinguishable i.e. absence of chaos. On the contrary, $\\xchaos = 0$ means that both configurations\\index{configuration} are completely different which would indicate strong chaos.\n\nWe select the most chaotic samples\\index{sample} and the less chaotic ones accordingly to the estimation $\\tintvar$ and we plot in~\\reffig{X_TminT} their chaotic parameter as a function of temperature by keeping $T_1$ to the lower simulated temperature $T_{\\min}$ and varying $T_2$. It is clear that qualitative different behaviors on the quantity $X_{T_{\\min},T}^J$ are present in both sets. The less chaotic ones tend to decrease smoothly as $T_2$ increases while the more chaotic ones suffer sharp drops at well-defined temperatures, namely \\textit{chaotic events}. In addition, it was empirically observed~\\cite{fernandez:13} that chaotic events occurring at low temperatures are more harmful to the performance of \\gls{PT}. With this information in mind, we are looking for a single number that could quantify the \\textit{chaoticity} of a given sample\\index{sample}. The introduced observable~\\cite{fernandez:13} was the chaotic integral\n\\begin{equation}\nI = \\int_{T_{\\min}}^{T_{\\max}} \\xchaosmin dT_2 \\, . \\labeq{chaotic_integral}\n\\end{equation}\nThis quantity will be smaller if the sample\\index{sample} suffers a chaotic event that ``cuts'' the integral. Moreover, for the chaotic samples\\index{sample} is usual that, once the chaotic event takes place at temperature $T^*$, the chaotic signal for temperatures $T>T^*$ is low but the fluctuations of the value are still present. To minimize this effect, we propose the parameter $I_2$ that reduces the integration range to the first half of the simulated temperatures.\n\nFinally, looking at~\\reffig{X_TminT}, we noticed that some samples\\index{sample} presented strong decays but then, they maintain a relatively high value of the chaotic parameter for higher temperatures (for example, look at the purple curves in the top panels). To take into account that effect we define the quantity\n\\begin{equation}\nK_i = 1-X_{T_{i},T_{i+1}} \\, , \\labeq{finite_differences}\n\\end{equation} \nwhich is essentially the finite difference of two consecutive points in the curve. After some trials based on heuristic arguments and after seeing a lot of $\\xchaosmin$ vs $T$ curves, we define the quantity\n\\begin{equation}\nI_X = aI_2 - b \\min_i \\left( -\\log K_i^2\\right) - c \\sum_i \\left(-\\log K_i^2 \\right) \\, ,\n\\end{equation}\nwhere the coefficients $a$, $b$ and $c$, that depend on the lattice size $L$, are obtained through a minimization of the correlation Pearson coefficient $r$ between $I_X$ and $\\log(\\tintvar)$ (as we will discuss in the following section).\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/X_TminT}\n\\caption[\\textbf{Temperature dependence of the chaotic parameter.}]{\\textbf{Temperature dependence of the chaotic parameter.} Plot of $X_{T_{\\min},T}^J$ versus $T$ for the five most chaotic samples\\index{sample} (top) and the five less chaotic ones (bottom): $L=16$ case (left) and $L=24$ case (right).}\n\\labfig{X_TminT}\n\\end{figure}\n\n\\section{Correlation dynamics-statics} \\labsec{correlation_dynamics_static}\nWe study here the correlation between the static characterization of chaos through the previously defined observables and the dynamic one, through the autocorrelation time\\index{autocorrelation time!integrated} $\\tintvar$ that we will call from now on, simply $\\tint$. \n\nWe also find it useful to address the failures in the process of finding quantities relating to both perspectives. Usually, space requirements in publications or the clarity of the message make the rules of choosing the appropriate results to show, and it is perfectly reasonable. However, this privileged format allows us to extend a little bit more and show the path of the research which often contains failures. In addition, addressing those quantities not related to chaos would be also helpful for future works.\n\\subsection{The failures}\nHere, we address some \\textit{a priori} reasonable quantities that turned out to be not related with \\gls{TC}\\index{temperature chaos}.\n\nFirst, we recall the previously defined $T_d$ which is the temperature $T^*$ chosen by the variational method\\index{variational method}. We compute also, from the static characterization, the temperature $T_s$ at which the \\textit{bigger} chaotic event occurs\\footnote{Note that, for the less chaotic samples\\index{sample} with a smooth decay of the function $\\xchaosmin$ against $T_2$, this chaotic event can be fairly small.} i.e. the temperature for which $\\xchaosmin$ presents the maximum (negative) slope.\n\nThe correlation between both quantities is almost absent as can be seen in~\\reffig{Td_Ts}. We can see an over-density, however, the points out of the principal density are too dispersed. For $L=16$ (top) the number of points within the lines is $8017$ ($62.63 \\%$ of the total) while for $L=24$ (bottom) the number of points within the lines is $7539$ ($58.90 \\%$ of the total). If we compute the correlation coefficients we obtain~\\reftab{coef_correla_TdTs}. The errors in the determination of the correlation coefficients are computed from the Bootstrap\\index{Bootstrap} method (see \\refsec{estimating_errorbars}).\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{eq_chaos\/Td_Ts_g}\n\\caption[\\textbf{\\boldmath Scatter plot of $T_d$ versus $T_s$.}]{\\textbf{Scatter plot of $\\mathbf{T_d}$ versus $\\mathbf{T_s}$.} We present the $L=16$-data (top) and the $L=24$-ones (bottom). Points are calculated with a special procedure. First, samples\\index{sample} are classified on deciles according to $\\log(\\tau_{\\mathrm{int}})$. The points coordinates were obtained by computing the median $T_d$ and the median $T_s$ within each decile (errors from Bootstrap\\index{Bootstrap}). The golden parallel lines enclose the area of over-density that presents a higher correlation for later recount.}\n\\labfig{Td_Ts} \n\\end{figure}\n\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.5\\columnwidth}{@{\\extracolsep{\\fill}} ccccc}\n\\toprule\n\\toprule\n&$L$  & & $r$ & \\\\\n\\toprule\n&$16$ & & $0.348 \\pm 0.008$ &\\\\\n&$24$ & & $ 0.342 \\pm 0.007$ & \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{\\boldmath $T_d$ vs $T_s$.}]{\\textbf{$\\mathbf{T_d}$ vs $\\mathbf{T_s}$.} Correlation coefficients of the scatter plot of $T_d$ against $T_s$ for the simulated two lattice sizes.}\n\\labtab{coef_correla_TdTs}\n\\end{table}\n\n\nWe can try to relate $T_s$ to other dynamic estimation of chaos, for example, the integrated autocorrelation time\\index{autocorrelation time!integrated} $\\tint$. Unfortunately, although slightly better than our previous attempt, we observe a weak correlation between both estimators, $\\tint$ and $T_s$, (see \\reffig{log_tau_Ts}) and we can check it quantitatively through \\reftab{coef_correla_Ts}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{eq_chaos\/log_tau_Ts_g}\n\\caption[\\textbf{\\boldmath Scatter plot of $\\log(\\tint)$ against $T_s$.}]{\\textbf{Scatter plot of $\\mathbf{\\log(\\tint)}$ against $\\mathbf{T_s}$.} We show $L = 16$ (top) and $L=24$ (bottom).}\n\\labfig{log_tau_Ts} \n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.5\\columnwidth}{@{\\extracolsep{\\fill}}ccccc}\n\\toprule\n\\toprule\n&$L$   && $r$ & \\\\\n\\toprule\n&$16$  && $-0.621 \\pm 0.006$ &\\\\\n&$24$  && $ -0.621 \\pm 0.006$ &\\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Correlation coefficients for the scatter plot of $\\mathbf{\\log(\\tau_{\\mathrm{int}})}$ versus $\\mathbf{T_s}$ for the two simulated lattice sizes.}]{\\textbf{Correlation coefficients for the scatter plot of $\\mathbf{\\log(\\tau_{\\mathrm{int}})}$ versus $\\mathbf{T_s}$ for the two simulated lattice sizes.}}\n\\labtab{coef_correla_Ts}\n\\end{table}\n\n\n\\subsection{The success}\nHere, we present our most successful attempts to relate both dynamic and static characterization of chaos. In \\reffig{log_tau_I}, we confront the most representative\nestimator for the dynamical chaos, namely the largest integrated autocorrelation time\\index{autocorrelation time!integrated} $\\tint$ found in our variational\\index{variational method} study, with the static chaotic integrals $I$, $I_2$ and $I_X$. We can observe how spurious values of the original parameter $I$ (i.e. large values of $I$ associated with large $\\tau_\\mathrm{int}$) are displaced towards lower values when we use the improved parameters $I_2$ and $I_X$.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/log_tau_I_g}\n\\caption[\\textbf{Scatter plot of $\\mathbf{\\log(\\tintvar)}$ versus integrated chaotic parameters.}]{\\textbf{Scatter plot of $\\mathbf{\\log(\\tintvar)}$ versus integrated chaotic parameters.} We present data for two lattice sizes and for the three definitions of the integrated chaotic parameter defined in the text ($I, I_2$ and $I_X$). The pattern of depleted horizontal bands is due to our choice of a few $l_\\mathrm{blo}$.}\n\\labfig{log_tau_I}\n\\end{figure}\n\nThe value of the correlation coefficients is reported in~\\reftab{coef_correla} (as before, the errors are computed by using a Bootstrap\\index{Bootstrap} method, see \\refsec{estimating_errorbars}). We observe a strong anti-correlation in  $I_X$, which improves over the previous indicator of correlation $I$. \\cite{fernandez:13} The improvement is less clear for $I_2$.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.6\\columnwidth}{@{\\extracolsep{\\fill}}ccccc}\n\\toprule\n\\toprule\n&$L$ & Integral & $r$  \\\\\n\\toprule\n&$16$ & $I$   & $-0.714 \\pm 0.005$ \\\\\n&$16$ & $I_2$ & $-0.751 \\pm 0.005$ \\\\\n&$16$ & $I_X$ & $-0.795 \\pm 0.004$ \\\\\n\\toprule\n&$24$ & $I$   & $-0.725 \\pm 0.005$ \\\\\n&$24$ & $I_2$ & $-0.746 \\pm 0.005$ \\\\\n&$24$ & $I_X$ & $-0.786 \\pm 0.004$ \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Correlation coefficients for $\\mathbf{\\log(\\tint)}$ versus the integrated chaotic parameters.}]{\\textbf{Correlation coefficients for $\\mathbf{\\log(\\tint)}$ versus the integrated chaotic parameters.} Correlation coefficients are shown for each two lattice sizes and for the three definitions of the parameter ($I, I_2$ and $I_X$).}\n\\labtab{coef_correla}\n\\end{table}\n\n\\section{Finite size scaling} \\labsec{scaling_eq_chaos}\n\\index{finite size scaling}\nThis section is devoted to study the size-scaling behavior of the dynamic characterization of \\gls{TC}\\index{temperature chaos}. It has been observed \\cite{fernandez:13} that chaotic events are less common in small systems. This suggests a large $L$ limit for the chaotic behavior that we investigate here.\n\nAn implicit assumption of our study is that the scaling behavior of $\\tint$ is mostly decided by the value $\\Tmin$. Other details, such as the number of temperatures in the \\gls{PT} mesh, are expected to play a minor role (if kept in a reasonable range). Our choice of simulated parameters (see, \\reftab{parameters_simulation_MUSA_MUSI}) does not allow us to check the impact of the number of temperatures in the \\gls{PT} mesh, but we can justify that $\\Tmin$ has a deep impact on the determination of $\\tint$.\n\n\\subsection{Temperature chaos depends on \\boldmath $\\Tmin$}\nIn order to study how the range of temperatures in the \\gls{PT} affects the dynamics, we have confronted both simulations for $L=16$, one with $N=13$ and $\\Tmin=0.698$, the other one with $N=16$ and $\\Tmin=0.479$. We need to increase the number of temperatures $N$ in the mesh to keep the interval between adjacent temperatures fixed.\n\nSince the simulation with $N = 16$ reaches a lower minimum temperature than the simulation with $N = 13$, we expect to find chaos events (i.e a jam in the \\gls{PT} temperature flow) that the simulation with $N=13$ cannot ``see''. In~\\reffig{cociente_taus} we show a scatter plot of $\\log(\\tau_{\\mathrm{int},16}\/\\tau_{\\mathrm{int},13})$ versus $T_d$ for the 12800 samples\\index{sample} ($\\tau_{\\mathrm{int},16}$ and $\\tau_{\\mathrm{int},13}$ are the autocorrelation times for $N=16$ and $N=13$ respectively). \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/cociente_taus_g}\n\\caption[\\textbf{Scatter plot of $\\mathbf{\\log({\\tau_{\\mathrm{int,}16}}\/{\\tau_{\\mathrm{int,}13}})}$ versus $\\mathbf{T_d}$.}]{\\textbf{Scatter plot of $\\mathbf{\\log({\\tau_{\\mathrm{int,}16}}\/{\\tau_{\\mathrm{int,}13}})}$ versus $\\mathbf{T_d}$.} The lattice size is $L=16$, $\\tau_{\\mathrm{int},16}$ is the relaxation\\index{relaxation} time for $N=16$ ($T_\\mathrm{min}=0.479$), $\\tau_{\\mathrm{int},13}$ is the relaxation\\index{relaxation} time for $N=13$ ($T_\\mathrm{min}=0.698$), $T_d$ is the temperature of chaos from a dynamical point of view (defined in the variational method\\index{variational method}) of the simulation with $N = 16$. Both simulations have the same number of disorder\\index{disorder} samples\\index{sample}. The vertical black line represents the minimum temperature simulated in the $N= 13$ simulation. (We added a small Gaussian\\index{Gaussian!noise} white noise to $T_d$, which is a discrete variable, to avoid the cluttering of data in vertical lines). }\n\\labfig{cociente_taus}\n\\end{figure}\n\nFor $T_d>0.698$ the ratio takes values of order one for most samples\\index{sample}, while for $T_d < 0.698$ there is a huge number of samples\\index{sample} with $\\tau_{\\mathrm{int},16} \\gg \\tau_{\\mathrm{int},13}$,\ni.e. there are a lot of samples\\index{sample} with a chaotic behavior in a temperature-range  below $T_\\mathrm{min}=0.698$.\n\nThe same idea can be analyzed from a different point of view. Imagine that we have studied with great care a given sample\\index{sample} down to some temperature $T_\\mathrm{min}$. Can we say something about possible chaotic effects at lower temperatures? The question is answered negatively in \\reffig{no-prediction-from-small-Tmin}: the probability that a sample\\index{sample} has a large $\\tau_{\\mathrm{int}}$ for the simulation with a lower $T_\\mathrm{min}$ is not correlated to the value of $\\tau_{\\mathrm{int}}$ for the first simulation.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_quintiles_g}\n\\caption[\\textbf{Conditional probability distribution of $\\mathbf{\\tint}$.}]{\\textbf{Conditional probability distribution of $\\mathbf{\\tint}$.} The empirical probability distribution as a function of $\\tint$ for the $N=16$ simulation, conditional to the $\\tint$ obtained from $N=13$ simulation belonging to a given quintile. The non-conditional probability distribution function is also shown ($L_{16}$ curve).  \\textbf{Inset.} Blowup of the top right part of the main figure. For the hard samples\\index{sample}, the simulation with $T_\\mathrm{min}=0.698$ conveys little or no information on the difficulty of the $T_\\mathrm{min}=0.479$ simulation.}\n\\labfig{no-prediction-from-small-Tmin}\n\\end{figure}\n\n\\subsection{The scaling}\nWe have discussed that $\\Tmin$ has a great impact on the $\\tint$ value, therefore, we need to fix the same $\\Tmin$ for all the simulations in order to establish fair comparisons. We study the \\gls{PT} dynamics for $L=8,12,16,24$ and $32$ with $\\Tmin\\approx 0.7$. An important advantage of $T_\\mathrm{min}\\approx 0.7$ is that \\gls{TC}\\index{temperature chaos} has been already characterized at such temperatures, in the equilibrium setting~\\cite{fernandez:13}. Lowering $T_\\mathrm{min}$ would increase chaos effects, which would have been good in principle, but it would have been also extremely difficult to reach thermal equilibrium. Instead, increasing $T_\\mathrm{min}$ to approach the critical point would make the results irrelevant, because samples\\index{sample} displaying \\gls{TC}\\index{temperature chaos} would be too scarce (besides, we want to study the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, rather than critical effects).\n\nThe $L=32$ data are from Ref.~\\cite{janus:10} and have been obtained with the dedicated Janus\\index{Janus} computer~\\cite{janus:09}. The Janus\\index{Janus} simulation used heat bath dynamics, rather than Metropolis, and the \\gls{PT} there had $N_T=34$ and $T_\\mathrm{min}=0.703$. In order to be sure that heat bath autocorrelation times are consistent with Metropolis times (as we would expect) we simulated with Janus\\index{Janus} ten randomly selected samples\\index{sample} with both algorithms, finding that $\\tau_\\mathrm{Metropolis}\\approx \\tau_\\mathrm{heat-bath}\/3$.\n\nWe show in~\\reffig{all_L_prob_tau} the cumulative distribution function of $\\tau = \\tint$, $F(\\tau)$. It can be seen qualitatively from the figure that the maximum slope of $F$ decreases with $L$ for the small systems, and it stabilizes between $L=24$ and $L=32$; indeed these two distributions can be approximately superposed by a simple translation. This is reminiscent of a critical slowing-down~\\cite{zinn-justin:05}\n\\begin{equation}\\label{eq:zPT-def}\n\\tau\\sim L^{z^\\mathrm{PT}(T_\\mathrm{min})}\\,.\n\\end{equation}\n\nIt is not obvious \\emph{a priori} that such a simple scaling should hold in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. As a working, simplifying hypothesis we assume that the exponent $z^\\mathrm{PT}$ only depends on the value of the lowest temperature in the \\gls{PT} grid, $T_\\mathrm{min}$ (and not on the number of temperatures).\n\nThe reader may warn that in~\\reffig{all_L_prob_tau} the distribution functions are not drawn for small values of $F(\\tau)$ in the $L=8$ and $L=16$ cases. The reason is that we could not find with our variational method\\index{variational method} a $\\tint$ that fulfills the condition of~\\refeq{tintfTlblo_condition} and therefore we can not provide a safe computation of $\\tau$. As long as we are concerned about the top part of the curve, we ignore this problem that does not appear in the simulation of $L=16$ with $N=16$, which is the simulation used in the rest of the study.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/all_L_prob_tau_g}\n\\caption[\\textbf{Empirical probability distribution of $\\mathbf{\\tau}$ for $\\mathbf{L=8,12,16,24}$ and 32.}]{\\textbf{Empirical probability distribution of $\\mathbf{\\tau}$ for $\\mathbf{L=8,12,16,24}$ and 32.} For $L=8$ and $L=16$ some of the samples\\index{sample} have $\\tau$ smaller than our minimal resolution (if $\\tau<n_{\\mathrm{Met}}$ we cannot compute it safely). We show only the part of the distribution function that can be safely computed.}\n\\labfig{all_L_prob_tau}\n\\end{figure}\n\nIt is clear from~\\reffig{all_L_prob_tau} that a simple translation would not attains the collapse of all the curves. Therefore, we study the exponent $z^{\\mathrm{PT}}$ for different parts of the distribution function, i.e. for different \\textit{percentiles}. We use the different simulated sizes to compute an effective $z^{\\mathrm{PT}}$ exponent for each pair of lattices ($L_1,L_2$) by means of the definition\n\\begin{equation}\nz^\\mathrm{PT}(L_1,L_2,p)=\\frac{\\log(\\tint(L_1,p)\/\\tint(L_2,p))}{\\log(L_1\/L_2)}\\,,\n\\end{equation}\nwhere $\\tau(L_i,p)$ is determined by the implicit equation $F(\\tau(L_i,p))=p\/100$ where $p=1,\\ldots,100$ is the so called percentile rank (i.e. $\\tau(L_i,p)$ is the $p$-th percentile of the distribution for the size $L_i$).  \n\nWe have computed $z^\\mathrm{PT}$ for three pairs of lattice sizes, (12,24), (16,24), and (24,32) and in~\\reffig{z_percentile} we show the results as a function of the rank. As expected, we see that the scaling does not hold for the whole curve in general. However, we observe that for higher ranks, i.e. the most chaotic samples\\index{sample}, the scaling hold for all the pairs of lattices. In addition, for the largest pair of lattices (24,32), the exponent $z^{\\mathrm{PT}}$ is independent of the percentile, within the statistical errors. This result is consistent with the previously drawn picture in~\\cite{fernandez:13}: for small system sizes, it is almost impossible to find samples\\index{sample} displaying strong \\gls{TC}\\index{temperature chaos} and one needs to go to larger lattices, like $L=32$ to find chaotic samples\\index{sample} with a significant probability.\n\nAn interesting coincidence with the results of non-equilibrium simulations~\\cite{janus:08b,janus:09b,fernandez:15,janus:17} could have a deep meaning. Indeed in non-equilibrium conditions one finds that the \\gls{SG} coherence length\\index{coherence length} $\\xi$, in a lattice of size $L\\gg\\xi$, at temperature $T=0.7$ grows with the simulation time $t_\\mathrm{w}$ as~\\cite{janus:09b}\n\\begin{equation}\n\\xi(t_\\mathrm{w})\\propto t_w^{1\/z(T)}\\,,\\quad z(T=0.7)=11.64(15)\\,, \\labeq{zeta_janus}\n\\end{equation}\nwhere $z(T)$ is the so-called dynamic critical exponent\\index{dynamic critical exponent}\\index{dynamic critical exponent!zzzzz@\\Also{aging rate}|gobbleone}, that turns out to be strongly temperature-dependent in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} $z(T)\\propto T_\\mathrm{c}\/T$, see \\refch{aging_rate}.\n\nOur results for the lattice pair $(24,32)$ suggest that\n\\begin{equation}\nz(T=0.7)\\approx z^\\mathrm{PT}(T_\\mathrm{min}=0.7)\\,. \\labeq{z-coincide}\n\\end{equation}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/z_percentil_g}\n\\caption[\\textbf{The effective exponent $\\mathbf{z^{\\mathrm{PT}}}$.}]{\\textbf{The effective exponente $\\mathbf{z^{\\mathrm{PT}}}$.} The effective exponent $z^\\mathrm{PT}(L_1,L_2,p)$ for three different pairs of lattice sizes (12,24), (16,24) and (24,32) as a function of the percentile rank $p$.  The two horizontal lines are the bounds for the off-equilibrium value $z=11.64(15)$ [see \\refeq{zeta_janus}]. The numerical values of $z^\\mathrm{PT}$ for the largest pair are compatible with the off-equilibrium value.}\n\\labfig{z_percentile}\n\\end{figure}\n\nThe surprising result of~\\refeq{z-coincide} suggests the rescaling of the whole probability distribution as a test. This is done in figure \\reffig{all_L_prob_tau_reescaled} (main) that shows $F(\\tau)$ as a function of $y = \\tau\/L^z$. \n\nAs expected, the data for $L=24$ and $L=32$ present a nice collapse. The curve corresponding to $L=16$ collapses into them for percentile ranks higher than $80$ only and the curve corresponding to $L=12$ collapses for percentile ranks higher than $90$. This is pointing to us that the chaotic behavior of the large $L$ limit is harder to find as we go to smaller $L$.\n\nIn figure~\\ref{fig:all_L_prob_tau_reescaled} (inset), we show a log-log plot of $1\\!-\\!F(\\tau)$ as a function of $\\tau\/L^z$, that emphasizes the large $\\tau$ tail of the distribution. The fit presented shows that the probability density function of $\\tau$ behaves, asymptotically for large $y$, like a fat tailed distribution:\n\\begin{equation}\n\\rho(y\\equiv \\tau\/L^z) \\sim y^{-1-a_1}\\,,\\quad a_1\\approx 1.38 \\,.\n\\end{equation}\n\nThe distribution seems to reach its asymptotic form for $L\\geq 24$. Perhaps unsurprisingly, the thermodynamic (i.e. equilibrium) effective potential that characterizes temperature chaos turns out to be also asymptotic for $L\\geq 24$~\\cite{fernandez:13}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/all_L_prob_tau_reescaled_g}\n\\caption[\\textbf{Probability distribution function of the rescaled variable \\boldmath $y = \\tau\/L^{z}$}.]{\\textbf{Probability distribution function of the rescaled variable \\boldmath $y = \\tau\/L^{z}$}. $z$ is the dynamic exponent corresponding to $T_\\mathrm{min}=0.7$, namely $z(T=0.7)=11.64(15)$. \\textbf{(Inset)} Plot of $\\log(1-F(\\tau))$ versus $\\log\\left(\\tau\/L^{z}\\right)$; the straight black line is a fit to the form $a_0 - a_1 \\log(\\tau\/L^z)$ yielding $a_0 = -29.33$ and $a_1=-1.38$.}\n\\labfig{all_L_prob_tau_reescaled}\n\\end{figure}\n\n\n\n\n\\chapter{General Introduction}\n\\labch{general_introduction}\nAs usually happens in science, we have just used the light of our predecessors to open the door in front of us and we expect that our work provides enough light to those who are to come.\n\nThis thesis pretends to be another step in the development of numerical research in disordered\\index{disorder!systems} systems. Specifically, we will focus on spin glasses which have demonstrated to be a fertile field from both, experimental and theoretical approaches. Throughout this thesis, we will discuss a variety of interesting phenomenons and we will also open new avenues to previously unexplored effects in the context of spin glasses. However, without a doubt, the leitmotiv conducting this thesis is the role of numerical simulations as a valuable tool to explore spin-glass physics.\n\nIndeed, the development of this thesis has been possible due to the high-quality data obtained from a huge numerical effort. In particular, the role of the FPGA\\index{FPGA}-based supercomputer Janus\\index{Janus} II has been determinant in this thesis. Unprecedented large off-equilibrium simulations have allowed obtaining high-accuracy data that are the basis of several chapters in the present document. The Janus\\index{Janus} II success has been possible due to the Janus\\index{Janus} Collaboration, an international project involving five different universities from Italy and Spain that have been joining in their efforts to design and program the spin-glass dedicated computer which is at the forefront of numerical research. In addition to Janus\\index{Janus} II, thousands of hours of computational time in CPUs have been also fundamental in the analysis of data. Specifically I would like to mention Cierzo supercomputer in BIFI and the Madrid's cluster in the UCM.\n\nThis thesis is organized into five different parts. The first part, containing the~\\refch{def_spinglass}, is focused on introducing the spin glasses to the reader. It is impossible to account for all the work developed in the spin-glass field. Nonetheless, we try to properly introduce the reader into the spin-glass context, starting from the very definition of the studied system. We also discuss relevant experimental results, trying to focus on those that will come up throughout the thesis. The most relevant theoretical results are also provided in this introduction. Finally, numerical simulations, the main research-resource in this thesis, are introduced.\n\nThe second part, containing the~\\refch{metastate}, is dedicated to discussing the metastate\\index{metastate}. The theoretical development of the spin glasses in the thermodynamic limit\\index{thermodynamic limit} has historically suffered mathematical inconsistencies. The irruption of the physical-mathematics in the context of spin glasses brought a solution to this problem introducing the concept of metastate\\index{metastate}. However, for finite dimensions, there exist different metastate\\index{metastate} pictures with different predictions. Since the invention of the metastate\\index{metastate} in 1990, this concept has been in the theoretical \\textit{world}, separated from experiments and numerical simulations, nonetheless, the current state of the art in numerical simulations have allowed us to numerically construct the metastate\\index{metastate} and to (partially) elucidate between the competing metastate\\index{metastate} pictures.\n\nThe third part, shaped by the~\\refch{aging_rate} and \\refch{mpemba}, is devoted to studying the off-equilibrium dynamics in spin glasses. This topic is of glaring importance in the spin glasses since experiments are (almost) always conducted in off-equilibrium conditions. Moreover recently, quantitative relevant relations have been found between equilibrium and out-equilibrium spin glasses. Learning about off-equilibrium dynamics might be useful to unveil the equilibrium properties of spin glasses concerning their very nature. Specifically, \\refch{aging_rate} is focused on discussing the growth of the coherence length\\index{coherence length} in spin glasses, a key quantity that characterizes the off-equilibrium evolution of those systems. In that chapter, we will witness a \\textit{tour de force} of Janus\\index{Janus} II, and we will be able to solve a discrepancy between numerical simulations and experiments by extrapolating our results to the relevant experimental time-scales. In~\\refch{mpemba} we will discuss an interesting phenomenon: the Mpemba effect. It has been observed that, under some circumstances, if two beakers of water, one hotter than the other, are put in contact with a thermal reservoir at low temperatures, the hot water freezes faster than the cold water. We translate this off-equilibrium phenomenon to the spin-glass context for the first time and provide a satisfactory explanation.\n\nThe fourth part, containing the~\\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos} is devoted to study the Temperature Chaos phenomenon in spin glasses. In~\\refch{Introduction_chaos} we introduce the main historical results on Temperature Chaos, from its origins to the last steps. In~\\refch{equilibrium_chaos} we study equilibrated spin glasses and we characterize the Temperature Chaos from a static and a dynamical point of view. We also reproduce previous results by relating both approaches through an exploration of the correlations between them and proposing new observables to improve those correlations. Moreover, in this chapter, we develop a numerical method to improve the dynamical estimation of Temperature Chaos. In~\\refch{out-eq_chaos} we tackle the problem of characterizing Temperature Chaos in off-equilibrium dynamics. The very definition of Temperature Chaos refers to the reorganization of the \\textit{equilibrium} configurations\\index{configuration} of a spin glass upon small changes in the temperature. Then, the existence of the phenomenon in out-equilibrium systems needs to be established. In that chapter, we find a phenomenon that closely mimics the equilibrium Temperature Chaos. Indeed, we observe a strong relationship between the equilibrium and the off-equilibrium Temperature Chaos.\n\nThe fifth and last part of the main body of the thesis corresponds to the conclusions (\\refch{conclusions}). There, we recall the relevant results of the thesis and we also discuss possible future works.\n\nFinally, some appendixes can be found in the last part of the document. In~\\refch{AP_statistics} we explain the relevant statistical tools that we have used in the analysis of this thesis. In~\\refch{AP_technical_details_aging} and~\\refch{AP_technical_details_out-eq_chaos} we provide technical details on the~\\refch{aging_rate} and~\\refch{out-eq_chaos} respectively. \\refch{AP_PT} is devoted to explaining technical details on the use of the parallel\\index{parallel!tempering} tempering method to compute the characteristic time-scales in Markov\\index{Markov chain} chains. \\refch{AP_multispin_coding} provides technical details on general methods of parallelization\\index{parallel!computation} that are useful in our simulations and analysis.\n\nThe original results of this thesis are published in the following articles\n\\begin{itemize}\n\\item A. Billoire, L. A. Fernandez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo. 'Numerical Construction of the Aizenman-Wehr Metastate'. In: Phys. Rev. Lett. 119 (3 July 2017), p. 037203. doi: 10.1103\/PhysRevLett.119.037203 \\cite{billoire:17}\n\n\\item A Billoire, L A Fernandez, A Maiorano, E Marinari, V Martin-Mayor, J Moreno-Gordo, G Parisi, F Ricci-Tersenghi, and J J Ruiz-Lorenzo. 'Dynamic variational study of chaos: spin glasses in three dimensions'. In: Journal of Statistical Mechanics: Theory and Experiment 2018.3 (2018), p. 033302. doi: 10.1088\/1742-5468\/aaa387 \\cite{billoire:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Aging Rate of Spin Glasses from Simulations Matches Experiments'. In: Phys. Rev. Lett. 120 (26 June 2018), p. 267203. doi: 10.1103\/PhysRevLett.120.267203 \\cite{janus:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'The Mpemba effect in spin glasses is a persistent memory effect'. In: Proceedings of the National Academy of Sciences 116.31 (2019), pp. 15350\u201315355 \\cite{janus:19}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, I. Gonzalez-Adalid Pemartin, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, I. Paga, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Temperature chaos is present in off-equilibrium spin-glass dynamics'. In: Communications Physics 4.1 (Apr. 2021), p. 74 \\cite{janus:21}\n\\end{itemize}\n\n\n\\setcounter{chapter}{-1}\n\\chapter{Introducci\u00f3n General}\n\\labch{introduccion_general}\n\nComo habitualmente ocurre en ciencias, nuestro cometido ha sido usar la luz de nuestros predecesores para poder abrir la puerta que se encontraba delante de nosotros. Esperamos que nuestro trabajo d\u00e9 la suficiente luz a aquellos que est\u00e1n por venir.\n\nEsta tesis pretende ser otro paso m\u00e1s en el desarrollo de la investigaci\u00f3n num\u00e9rica en los sistemas desordenados. En concreto, nos centraremos en los vidrios de esp\u00edn, que han demostrado ser un campo de conocimiento f\u00e9rtil tanto para los experimentos como para el desarrollo te\u00f3rico. A lo largo de esta tesis discutiremos una gran variedad de fen\u00f3menos interesantes y tambi\u00e9n abriremos nuevos caminos hacia efectos que no hab\u00edan sido explorados previamente en el contexto de los vidrios de esp\u00edn. Sin embargo, sin g\u00e9nero de duda, el leitmotiv de esta tesis es el papel de las simulaciones num\u00e9ricas como una valiosa herramienta para explorar la f\u00edsica de los vidrios de esp\u00edn.\n\nCiertamente, el desarrollo de esta tesis ha sido posible gracias a la alta calidad de los datos obtenidos con un inmenso esfuerzo num\u00e9rico. En concreto, el rol de la supercomputadora, basada en FPGA, Janus II ha sido determinante en esta tesis. Simulaciones masivas fuera del equilibrio sin precedentes han permitido obtener datos altamente precisos que son la base de varios cap\u00edtulos en este texto. El \u00e9xito de Janus II ha sido posible gracias a la colaboraci\u00f3n Janus, un projecto internacional que involucra cinco universidades distintas de Italia y Espa\u00f1a que han unido sus fuerzas para dise\u00f1ar y programar un hardware dedicado a la simulaci\u00f3n de vidrios de esp\u00edn en la vanguardia de la investigaci\u00f3n num\u00e9rica. Adem\u00e1s de Janus II, miles de hora de tiempo computacional en CPU han tenido tambi\u00e9n un papel fundamental en el an\u00e1lisis de datos. En concreto, me gustar\u00eda mencionar la supercomputadora Cierzo, en el BIFI y el cluster de Madrid, en la UCM.\n\n\nEsta tesis est\u00e1 organizada en cinco partes diferentes. La primera parte, formada por el Cap\u00edtulo 1, est\u00e1 centrada en introducir los vidrios de esp\u00edn al lector. Es imposible dar cuenta de todo el trabajo desarrollado en el campo de los vidrios de esp\u00edn. No obstante, intentamos introducir apropiadamente al lector al contexto de los vidrios de esp\u00edn, empezando por la definici\u00f3n del sistema en cuesti\u00f3n. En este cap\u00edtulo tambi\u00e9n discutimos los resultados experimentales m\u00e1s importantes, tratando de dar m\u00e1s relevancia a aquellos que guardan relaci\u00f3n directa con los temas tratados en esta tesis. Los resultados te\u00f3ricos m\u00e1s relevantes tambi\u00e9n se plasman en este cap\u00edtulo introductorio. Finalmente, las simulaciones num\u00e9ricas, el principal recurso para el desarrollo de la investigaci\u00f3n en esta tesis, son introducidas en este cap\u00edtulo.\n\nLa segunda parte, que incluye el Cap\u00edtulo 2, est\u00e1 dedicada a discutir el metaestado. El desarrollo te\u00f3rico de los vidrios de esp\u00edn en el l\u00edmite termodin\u00e1mico ha sufrido hist\u00f3ricamente de inconsistencias matem\u00e1ticas. La irrupci\u00f3n de la f\u00edsica-matem\u00e1tica en el contexto de los vidrios de esp\u00edn trajo la soluci\u00f3n a este problema introduciendo el concepto del metaestado. Sin embargo, para dimensi\u00f3n finita, existen diferentes teor\u00edas del metaestado con diferentes predicciones. Desde la invenci\u00f3n del metaestado en 1990, el concepto ha estado restringido al mundo te\u00f3rico, separado de experimentos y de simulaciones num\u00e9ricas, no obstante, el actual estado del arte en las simulaciones num\u00e9ricas ha permitido construir num\u00e9ricamente el metaestado y (al menos parcialmente) dilucidar entre las distintas teor\u00edas que lo describen.\n\nLa tercera parte, formada por los Cap\u00edtulos 3 y 4, est\u00e1 dedicada al estudio de la din\u00e1mica de no equilibrio de los vidrios de esp\u00edn. Este tema es de palmaria importancia en estos sistemas puesto que los experimentos (casi) siempre son llevados a cabo fuera del equilibrio. Adem\u00e1s, relaciones cuantitativas relevantes han sido halladas entre los vidrios de esp\u00edn en equilibrio y fuera del equilibrio. Aprender sobre la din\u00e1mica fuera del equilibrio puede ser \u00fatil para desvelas las propiedades de equilibrio de los vidrios de esp\u00edn, relacionadas con la misma naturaleza de estos sistemas. Espec\u00edficamente, el Cap\u00edtulo 3 est\u00e1 centrado en discutir el crecimiento de la longitud de coherencia en los vidrios de esp\u00edn, una cantidad clave que caracteriza la evoluci\u00f3n fuera del equilibrio de estos sistemas. En este cap\u00edtulo, presenciaremos un aut\u00e9ntico \\textit{tour de force} de Janus II y seremos capaces de resolver una discrepacia entre experimentos y simulaciones num\u00e9ricas extrapolando nuestros resultados hasta las escalas de tiempo relevantes para los experimentos. En el Cap\u00edtulo 4 discutiremos un fen\u00f3meno interesante: el efecto Mpemba. Se ha observado que, bajo ciertas circunstancias, si dos recipientes de agua, uno caliente y otro fr\u00edo, son puestos en contacto con un ba\u00f1o t\u00e9rmico a baja temperatura, el agua caliente se congela antes que el agua fr\u00eda. Nosotros traducimos este fen\u00f3meno que se da fuera del equilibrio al contexto de los vidrios de esp\u00edn por primera vez y damos una primera explicaci\u00f3n del mismo.\n\nLa cuarta parte, formada por los Cap\u00edtulos 5, 6 y 7, est\u00e1 dedicada a estudiar el fen\u00f3meno del Caos en Temperatura en vidrios de esp\u00edn. En el Cap\u00edtulo 5 introducimos los principales resultados hist\u00f3ricos sobre el Caos en Temperatura, desde sus or\u00edgenes hasta los \u00faltimos pasos. En el Cap\u00edtulo 6 estudiamos vidrios de esp\u00edn en equilibrio y caracterizamos el Caos en Temperatura desde un punto de vista est\u00e1tico y din\u00e1mico. Adem\u00e1s, relacionamos ambos enfoques explorando las correlaciones entre ambos. Adem\u00e1s, en este cap\u00edtulo desarrollamos un m\u00e9todo num\u00e9rico para mejorar la estimaci\u00f3n din\u00e1mica del Caos en Temperatura. En el Cap\u00edtulo 7, atacamos el problema de caracterizar el Caos en Temperatura en una din\u00e1mica fuera del equilibrio. La propia definici\u00f3n del Caos en Temperatura se refiere a las configuraciones de \\textit{equilibrio} del vidrio de esp\u00edn bajo un cambio arbitrariamente peque\u00f1o de temperatura. Por lo tanto, la existencia del fen\u00f3meno fuera del equilibrio debe ser establecida. En este cap\u00edtulo, encontramos un fen\u00f3meno que imita a la perfecci\u00f3n al fen\u00f3meno del Caos en Temperatura en equilibrio. De hecho, observamos una fuerte relaci\u00f3n entre el Caos en Temperatura en equilibrio y fuera del equilibrio.\n\nLa quinta y \u00faltima parte del cuerpo principal de la tesis corresponde a las conclusiones. En el Cap\u00edtulo 8 revisitamos los resultados m\u00e1s relevantes de esta tesis y tambi\u00e9n discutimos sobre posibles futuros trabajos.\n\nFinalmente, el lector puede encontrar algunos ap\u00e9ndices en la parte final de este texto. En el ap\u00e9ndice A explicamos cu\u00e1les son las herramientas estad\u00edsticas m\u00e1s relevantes que hemos usado a lo largo de esta tesis. En los ap\u00e9ndices B y D se dan detalles t\u00e9cnicos de los cap\u00edtulos 3 y 7 respectivamente. En el ap\u00e9ndice C se explican los detalles t\u00e9cnicos del uso del m\u00e9todo de \\textit{Parallel Tempering} para calcular las escala de tiempo caracter\u00edsticas en las cadenas de Markov. Por \u00faltimo, en el ap\u00e9ndice E se dan los detalles t\u00e9cnicos de m\u00e9todos generales de paralelizaci\u00f3n que han sido \u00fatiles en nuestras simulaciones y an\u00e1lisis.\n\nLos resultados originales de esta tesis est\u00e1n publicados en los siguientes art\u00edculos\n\\begin{itemize}\n\\item A. Billoire, L. A. Fernandez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo. 'Numerical Construction of the Aizenman-Wehr Metastate'. In: Phys. Rev. Lett. 119 (3 July 2017), p. 037203. doi: 10.1103\/PhysRevLett.119.037203 \\cite{billoire:17}\n\n\\item A Billoire, L A Fernandez, A Maiorano, E Marinari, V Martin-Mayor, J Moreno-Gordo, G Parisi, F Ricci-Tersenghi, and J J Ruiz-Lorenzo. 'Dynamic variational study of chaos: spin glasses in three dimensions'. In: Journal of Statistical Mechanics: Theory and Experiment 2018.3 (2018), p. 033302. doi: 10.1088\/1742-5468\/aaa387 \\cite{billoire:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Aging Rate of Spin Glasses from Simulations Matches Experiments'. In: Phys. Rev. Lett. 120 (26 June 2018), p. 267203. doi: 10.1103\/PhysRevLett.120.267203 \\cite{janus:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'The Mpemba effect in spin glasses is a persistent memory effect'. In: Proceedings of the National Academy of Sciences 116.31 (2019), pp. 15350\u201315355 \\cite{janus:19}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, I. Gonzalez-Adalid Pemartin, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, I. Paga, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Temperature chaos is present in off-equilibrium spin-glass dynamics'. In: Communications Physics 4.1 (Apr. 2021), p. 74 \\cite{janus:21}\n\\end{itemize}\n\n\n\n\n\n\n\n\\chapter[A brief introduction to spin glasses]{A brief introduction \\\\ to spin glasses}\n\\labch{def_spinglass}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\n\\epigraph{\\textit{Todo empez\u00f3 aquel fat\u00eddico d\u00eda \\dots}}{-- Jonathan Stroud, \\textit{El amuleto de Samarkanda} }\n\nThe first signs of unusual properties of alloys between noble metals and transition metals such as \\gls{AuFe} or \\gls{CuMn}, which later become the paradigmatic examples of spin glasses, were discovered in the second half of the 50's and the 60's through measurement of the specific heat\\index{specific heat} at low temperatures \\cite{denobel:59,zimmerman:60}, observation of time-dependent remanent magnetization\\index{magnetization} \\cite{kouvel:60,kouvel:61} and measurements of the ESR spectra \\cite{owen:56}. In parallel, the first interpretations of those experimental results from a theoretical point of view came out \\cite{marshall:60,klein:63} through \\gls{RKKY}\\index{RKKY interaction} interactions \\cite{ruderman:54,kasuya:56,yosida:57}.\n\nThe experiments and the first developed theories suggested that those systems present a low-temperature magnetic order\\index{magnetic order} which exhibits different properties from the known condensed-matter phases. Therefore, a natural question arose: there exists a sharp phase transition\\index{phase transition} from a high-temperature phase\\index{phase!high-temperature\/paramagnetic} to a low-temperature phase\\index{phase!low-temperature\/spin-glass} capable of describing the observed phenomena? \n\nThe very existence of this sharp phase transition\\index{phase transition} was discussed during the 1960s and the first half of the 1970s \\cite{violet:66,violet:67,beck:71,tholence:74}. However, the experiments of Canella, Mydosh, and Budnick \\cite{cannella:71,cannella:72} unveiled a sharp cusp in the ac susceptibility\\index{susceptibility!ac} of spin glasses such as \\gls{CuMn}. Those experiments were strong support for the sharp transition\\index{phase transition} hypothesis between a high-temperature paramagnetic phase\\index{phase!high-temperature\/paramagnetic} and a, at that moment new, low-temperature phase\\index{phase!low-temperature\/spin-glass}. Indeed, subsequent works in the following years brought evidence against this sharp transition\\index{phase transition}~\\cite{mulder:81,mulder:82} unveiling the rounded nature of the susceptibility\\index{susceptibility} curve. It was in 1991 when the community got convinced about the existence of the phase transition\\index{phase transition} in experimental spin-glasses \\cite{gunnarsson:91}. Nonetheless, at that time (the early 1970s), although still debated, the experimental results were supporting the phase transition\\index{phase transition} to occur.\n\nMoreover, previous experiments of neutron scattering\\index{neutron scattering}, indicated that there was not an order of the constitutive elements responsible for the magnetization\\index{magnetization}, the spins, in the low-temperature phase\\index{phase!low-temperature\/spin-glass} \\cite{arrott:65}. \n\nThe evidence of no order in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, together with the results that suggested a sharp phase transition\\index{phase transition}, led Edwards and Anderson to propose their theory \\cite{edwards:75}. This theory was a breakthrough in the study of the spin glasses because, although it was a mean-field\\index{Mean-Field!theory} theory, proposed a model that put aside the \\gls{RKKY}\\index{RKKY interaction} interactions in favor of random interactions following probability distribution. This simple conceptual change opened the door to a large class of systems to be studied in the same terms that spin glasses did.\n\nAfter the Edwards-Anderson theory\\index{Edwards-Anderson!theory} not only the spin glasses became a hot topic in the condensed matter\\index{condensed matter} research, but also the divergence between the theoretical and the experimental researches was considerable. Some effort was indeed done to establish some interplay, for example, scaling theories of domain\\index{magnetic domain} growth \\cite{fisher:88,koper:88,ocio:90,bouchaud:01}, however, the mainstream of both, theory and experiments, were quite apart from each other, as we briefly discuss below.\n\nDue to the glassy nature of the spin glasses in low temperatures, experiments always were developed in off-equilibrium conditions while the theoretical work put its efforts on capture the nature of spin glasses at low temperatures, with results that require access to the spin configurations\\index{configuration} for equilibrated systems in the thermodynamic limit\\index{thermodynamic limit}.\n\nTraditionally, numerical work has been a powerful tool of theoretical research. Numerical studies were, almost always, developed in small systems at thermal equilibrium, very far from the experiments, most of them performed out of equilibrium in very large systems. Moreover, the numerical simulations performed in off-equilibrium conditions could only reach time-scales much smaller than the experimental ones. However, the continuous increase of the computational power and the development of special-purpose computers\\index{special-purpose computer} have allowed the study of the spin glasses at low temperatures out of equilibrium in the relevant experimental scales.\n\nAn exciting new venue for numerical research is bridging the experimental approach to spin glasses and the theoretical one. This thesis pretends to be a step forward in this sense.\n\nIn this chapter, the reader may find a general introduction to spin glasses. First of all, a first definition of the concept of spin glass will be introduced (see \\refsec{definition_spinglass}). Then, some experimental background will be provided in \\refsec{experimental_spinglass}. The main theoretical pictures, with their different quantitative predictions, will be exposed in \\refsec{theoretical_spinglass}. Finally, a brief introduction to the numerical work performed in this thesis together with some useful definitions can be found in \\refsec{numerical_spinglass}.\n\nAll the results presented in this chapter are not original from this thesis. In addition to the cited papers that will appear in this chapter, the reader may find useful general references about spin glasses in \\cite{binder:86,mezard:87,mydosh:93,fischer:93,young:98,dotsenko:01,dedominicis:06}.\n\n\\section{A first definition of spin glass} \\labsec{definition_spinglass}\nSpin glasses are magnetic systems whose low-temperature phase\\index{phase!low-temperature\/spin-glass} is frozen and disordered\\index{disorder}. From a practical point of view, we consider the spins (i.e. the magnetic moments) as the constitutive elements of the spin glass. We denote $\\vec{s}_{\\vec{r}}$ as the spin at the position $\\vec{r}$, and $\\braket{\\dots}_t$ as the mean over the experimental time.\n\nA direct consequence of the low-temperature phase\\index{phase!low-temperature\/spin-glass} to be frozen is $\\braket{\\vec{s}_{\\vec{r}}}_t \\neq 0$. Moreover, the direction in which the spins freeze are random, which has a direct implication in the magnetization\\index{magnetization} $\\sum_{\\vec{r}} \\braket{\\vec{s}_{\\vec{r}}}_t = 0$. Actually, no long-range magnetic order\\index{magnetic order} will appear in the low-temperature phase\\index{phase!low-temperature\/spin-glass}\n\\begin{equation}\n \\vec{M}_{\\vec{k}} = \\dfrac{1}{N} \\sum_i e^{-i\\vec{k}\\cdot\\vec{r}}\\braket{\\vec{s}_{\\vec{r}}}_t = 0 \\quad \\,\\, \\forall \\, \\vec{k} .  \\labeq{no_magnetic_order}\n\\end{equation} \nThis general expression includes the ferromagnetic order parameter\\index{order parameter} $\\vec{k} = \\vec{0}$ and the antiferromagnetic one $\\vec{k} = (\\pi,\\pi,\\pi)$.\n\nThese two characteristics, freezing and disorder\\index{disorder}, give us the key to understand the name \\gls{SG} which was proposed for the first time by Anderson \\cite{anderson:70}. The sluggish evolution of the system together with the disorder\\index{disorder} reminds the structural glass\\index{structural glass} main properties trading the magnetic moments by the position of the particles. In structural glasses\\index{structural glass}, particles occupy random positions with no time-evolution (disorder\\index{disorder} and freezing).\n\nThere exists a far well establish consensus about the principal ingredients that one model which aspires to exhibit the \\gls{SG}'s phenomenology must include: \\textit{randomness}\\index{randomness} in the interactions and \\textit{frustration}\\index{frustration} \\cite{toulouse:77,blandin:78}. \n\n\\subsection{A key property: frustration}\nSuppose that a pair of spins $\\vec{s}_{\\vec{r}_i}$ and $\\vec{s}_{\\vec{r}_j}$ interact with each other through the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{ij} = -J_{ij}\\vec{s}_{\\vec{r}_i}\\cdot \\vec{s}_{\\vec{r}_j}$, being $J_{ij}$ the \\textit{coupling}\\index{couplings} interaction between the spins at the positions $\\vec{r}_i$ and $\\vec{r}_j$. For $J_{ij}>0$ the more energetically favorable values for the pair of spins $\\vec{s}$ i.e. the more energetically favorable \\textit{configurations},\\index{configuration} are those in which the spins are parallel to each other. In the same way, for $J_{ij}<0$ the spins tends to align in an anti-parallel way. For a collection of spins, the Hamiltonian\\index{Hamiltonian} can be easily generalized\n\\begin{equation}\n\\mathcal{H} = -\\sum_{i,j}J_{ij}\\vec{s}_{\\vec{r}_i}\\cdot \\vec{s}_{\\vec{r}_j} \\labeq{first_Hamiltonian} \\, .\n\\end{equation}\nWe will say that a system is frustrated\\index{frustration} when it is not possible to satisfy simultaneously all the pairwise interactions i.e. there is no way to maximize simultaneously all the summands of \\refeq{first_Hamiltonian}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/frustration}\n\\caption[\\textbf{A graphical example of frustration.}]{\\textbf{A graphical example of frustration\\index{frustration}.} Plaquette\\index{plaquette} 1 is said to be frustrated\\index{frustration} because the spins, that tend to align in a parallel or anti-parallel way depending on the couplings\\index{couplings} (+ or - respectively), can not satisfy all the interactions simultaneously. However, in the unfrustrated plaquette\\index{plaquette} 2, we observe that neither the size of the plaquette\\index{plaquette} nor the mixture of positive and negative interactions is responsible for the frustration\\index{frustration}. The key is the number of negative couplings\\index{couplings} (respectively positive): if the number of, say, negative couplings\\index{couplings} is odd, then we will have a frustrated\\index{frustration} plaquette\\index{plaquette}, otherwise, we will have an unfrustrated plaquette\\index{plaquette}.}\n\\labfig{frustration}\n\\end{figure}\n\nThe canonical example of a frustrated\\index{frustration} system can be found in \\reffig{frustration} where the spins lie in the nodes of a square-regular\\index{regular lattice!square} lattice\\footnote{As far as we are just dealing with the sign of the couplings\\index{couplings}, that particular disposition of the spins only try to simplify the visualization without any generality loss.} in which the edges represent the coupling\\index{couplings} interactions between the spins. We firstly focus our attention in the closed-loop, also called \\textit{plaquette}\\index{plaquette}, labeled with the number 1. As long as we are discussing here just if the spins are parallel or anti-parallel, let us take for the sake of simplicity spins with values $\\uparrow$ (up) and $\\downarrow$ (down). Suppose that the spin (1) is $\\uparrow$, following to spin (2) through a positive coupling\\index{couplings} ($+$) which favors the parallel interactions, the value of spin (2) also should be $\\uparrow$. The same reasoning can be applied to the spin (3). Finally, the spin (4) is at the end of the loop and, therefore, has to satisfy two couplings\\index{couplings}: the positive ($+$) coupling\\index{couplings} with the spin (3) and the negative ($-$) coupling\\index{couplings} with the spin (1). Any value that the spin (4) takes will lead to one unsatisfied interaction. We say that the plaquette\\index{plaquette} 1 is frustrated\\index{frustration}. \n\nHowever, if we focus now on plaquette\\index{plaquette} 2 and we apply the same mental exercise, we can easily find a configuration\\index{configuration} of spins that satisfies all the interactions, for example, the sequence (1)$\\uparrow$, (2)$\\uparrow$, (3)$\\downarrow$, (4) $\\uparrow$, (5) $\\uparrow$, (6) $\\uparrow$, (7) $\\uparrow$, (8) $\\uparrow$, (9) $\\downarrow$ and (10) $\\downarrow$. The length of the plaquette\\index{plaquette} is irrelevant and one can infer from these examples that the key is the number of negatives (or equivalently positives) interactions; if the number of negative (positive) interactions that favor anti-parallel (respectively parallel) spin-alignment is odd, then the plaquette\\index{plaquette} will be frustrated\\index{frustration}, otherwise, we will say that the plaquette\\index{plaquette} is unfrustrated.\n\nIn frustrated\\index{frustration} systems there exist many configurations\\index{configuration} with several unsatisfied interactions in which any local change would lead to an increase of the energy\\index{energy}, thus, frustration\\index{frustration} draws a rugged free-energy\\index{free energy!landscape} landscape with many metastable\\index{metastability} states and high-energy barriers. Indeed, the rugged free-energy\\index{free energy!landscape} landscape is directly related to the frozen nature of \\gls{SG}s and other characteristic properties as the slow time-evolution.\n\nTwo clarifications should be made at this point. First, the mixture of positive and negative interactions do not guarantee the system to be frustrated\\index{frustration},\\index{frustration} see \\reffig{frustration}, plaquette\\index{plaquette} 2, or Mattis model\\index{Mattis model} \\cite{mattis:76} which can be mapped to a uniform ferromagnet\\index{ferromagnet}. Lastly, frustration\\index{frustration} without randomness\\index{randomness} (or vice versa) does not lead to \\gls{SG} behavior, the most simple example is the regular\\index{regular lattice!triangular} triangular lattice with antiferromagnetic\\index{antiferromagnetic} interactions\\footnote{Exactly solvable, see \\cite{wannier:50}}, which is a fully frustrated\\index{frustration} system with a large ground-state\\index{ground-state} degeneracy but without phase transition\\index{phase transition} at finite temperature\\footnote{More examples of frustrated\\index{frustration} systems without randomness\\index{randomness} can be found in \\cite{villain:77,villain:77b,wolff:82,wolff:83,wolff:83b,mackenzie:81}}.\n\nWe conclude that frustration\\index{frustration} and randomness\\index{frustration}\\index{randomness} are necessary conditions to have a \\gls{SG} but not sufficient ones. An illustrative example with this respect can be found in the Ising\\index{Ising} ferromagnet in a small random magnetic field, where frustration\\index{frustration} and randomness\\index{randomness} appear in a very weak way and it is possible to find long-range magnetic order\\index{magnetic order}.\n\n\\subsection{Why spin glasses?}\nA natural question is, if the \\gls{SG}s reproduce the glassy behavior at low temperatures, why should we focus on them instead on the structural\\index{structural glass} glasses? The main reason to study the glassy behavior through \\gls{SG}s is their simplicity. This simplicity allows the development of theoretical tools in \\gls{SG}s that can be later applied to other fields of the complex systems \\cite{mezard:85c,mezard:86b,amit:85,amit:85b,goldstein:92} (paradoxically, including the structural\\index{structural glass} glasses \\cite{charbonneau:14}).\n\nMoreover, \\gls{SG}s exhibit a wide set of characteristic phenomenons of glassy behavior with several advantages from the theoretical and experimental points of view. \n\nFirst of all, it is worthy to note that, unlike the structural\\index{structural glass} glasses, the phase transition\\index{phase transition} is well-known in \\gls{SG}s, both in experiments through the study of the susceptibility\\index{susceptibility} \\cite{gunnarsson:91} and in the theoretical models \\cite{palassini:99,ballesteros:00}. The physicists can greatly benefit from this fact by establishing quantitative criteria to determine whether or not they are studying the glassy phase\\index{phase!low-temperature\/spin-glass}.\n\nBesides, from the experimental point of view, \\gls{SQUID}\\index{Superconducting Quantum Interference Device (SQUID)} allows for very precise measures, much more difficult to perform in structural\\index{structural glass} glasses.\n\nFinally, a technical reason is that \\gls{SG}s are much simpler to simulate than structural\\index{structural glass} glasses. The lattice models are very easy to simulate numerically.\n\n\\subsubsection{Further motivations in the study of spin glasses}\nIn addition to the above-exposed advantages of studying \\gls{SG}s in order to explore the glassy behavior, there exist other fields in which the study of these systems is interesting and prolific. The study of complexity in optimization problems constitutes a paradigmatic example in this regard~\\cite{barahona:82b}. \n\nThe Turing machine\\index{Turing machine} is a theoretical machine widely used in computation theory introduced by Turing~\\cite{turing:37}. The deterministic version of the machine is able to give at most one result for every situation while the nondeterministic one is able to provide more than one result in each situation.\n\nThe set of problems that can be solved by a deterministic Turing machine\\index{Turing machine} in polynomial time belongs to the set P\\index{complexity!P} of problems. In the same way, the set of problems that can be solved by a non-deterministic Turing machine in polynomial time belong to the set of NP problems. It is trivial to see that P\\index{complexity!P} problems are a subset of NP\\index{complexity!NP} problems.\n\nSpecifically, there exists a subset of NP\\index{complexity!NP} which is called NP-complete\\index{complexity!NP-complete}\\footnote{The concept of NP-completeness was introduced in~\\cite{cook:71}.}, which is of special interest. We say that a problem is NP-complete if it belongs to the complexity class NP and all the NP problems are reducible (in polynomial time) to that problem. From a computational point of view, the NP-complete\\index{complexity!NP-complete} problems are the hardest in the set NP\\index{complexity!NP} and are equivalent to each other (see Cook-Levin theorem~\\cite[p.38]{garey:79}), in the sense that founding a polynomial-bounded algorithm for any one of them would effectively yield a polynomial-bounded algorithm for all.\n\nThere exist many problems in the NP-complete\\index{complexity!NP-complete} set (see, for instance ~\\cite{karp:72}) and, specifically, the problem of finding the ground-state\\index{ground-state} for a three-dimensional Ising \\gls{SG} \\footnote{See~\\refsubsec{source_randomness} for the concept of Ising \\gls{SG}.} is NP-complete\\index{complexity!NP-complete}~\\cite{barahona:82}. \n\nAlthough the three-dimensional case is of special interest in this thesis, there exist several models of \\gls{SG}s that have been studied with great detail from the complexity point of view. In particular, the question of the \\textit{planarity}\\index{planar graph} of the \\gls{SG} lattice has proven to be central in the complexity discussion~\\cite{istrail:00}.\n\nThe concept of planar graph\\index{planar graph} is rather intuitive. A graph is said to be planar\\index{planar graph} if it can be drawn in a plane without edge-crossing\\footnote{Although the concept is rather intuitive, in general, prove that a graph is planar\\index{planar graph} requires non-trivial criteria like Kuratowski's theorem.}. On the contrary, the fact that the problem of finding the ground-state\\index{ground-state} in a non-planar graph\\index{non-planar graph} is NP-complete\\index{complexity!NP-complete} is certainly not intuitive. The reader may find an interesting study of this problem for several graphs in~\\cite{istrail:00}.\n\nAn interesting fact in the case of \\gls{SG}s is the two-dimensional case. The typical case in the study of spin systems is to consider the spins placed in the vertex of a lattice and only take into account nearest-neighbors interactions. In this case, for the two-dimensional case, finding the ground-state is a P problem. However, if one considers next-nearest-neighbors interactions, the problem of finding the ground-state becomes NP-complete. This case might be surprising at first sight since the basic elements building the lattice are complete graphs of fourth order\\footnote{A complete graph of $n^{\\mathrm{th}}$ order, also knows as $K_n$ is a graph with $n$ nodes where every node is connected to the rest of them. In other contexts are also known as \\textit{fully-connected networks}.}, that fulfill Kuratowski's criteria and, therefore, are planar. The reader may find a deep discussion in this respect in~\\cite{istrail:00}.\n\nTherefore, the study of \\gls{SG}s is not only interesting from the statistical physics or the solid-state physics point of view but also from the complexity point of view.\n\n\\subsection{Beyond spin glasses. Weakly disordered versus strongly disordered systems}\nThroughout this first approach to \\gls{SG}s, we have set that the disorder is one of the essential characteristics that a system must have in order to exhibit the \\gls{SG} behavior. However, there exist a variety of systems exhibiting disorder that cannot be identified as \\gls{SG}s.\n\nWe have already introduced the Hamiltonian for spin systems in~\\refeq{first_Hamiltonian}. The simplest case, with $J_{ij}=J$ constant, corresponds to the Ising ferromagnet. Therefore, the introduction of disorder can be regarded as an additional random term\n\\begin{equation}\nJ_{ij} = J + \\delta J_{ij} \\, .\n\\end{equation}\n\nThe limiting cases $\\delta J_{ij} \\ll J$ and $\\delta J_{ij} \\gg J$ correspond to weak and strong disorder respectively. Specifically, in this thesis we will focus on systems with a strong disorder\\index{disorder!systems}, which is the case of \\gls{SG}s. However, there exist a variety of systems exhibiting weak disorder with a very different but rich phenomenology.\n\nThe study of weak-disorder systems\\index{disorder!systems} is a natural generalization of the study of pure systems\\footnote{In this context, pure systems refer to the absence of impurities in their composition.} by the introduction of impurities that are unavoidable in real systems. In these systems, the ground-state\\index{ground-state} and the equilibrium properties keep a close relationship with the pure system obtained by removing its impurities. However, the presence of these impurities may affect the behavior of the system in the neighborhood of the critical temperature\\index{critical temperature} \\cite{mccoy:70,balagurov:73,harris:74,harris:74b,khmelnitskii:75,grinstein:76}. \n\nMoreover, for systems undergoing a first-order phase transition, the effects of the weak disorder have been widely studied. Particularly, in two-dimensional systems, any arbitrarily small amount of disorder makes the first-order transition to become a second-order one~\\cite{hui:89,aizenman:90,cardy:97,jacobsen:98,chatelain:98}. The three-dimensional case is more difficult to study but there exist results suggesting that a change of the order of the transition from first- to second-order occurs when the disorder increases~\\cite{fernandez:12}. This is known as the Cardy-Jacobsen conjecture\\index{Cardy-Jacobsen conjecture}~\\cite{cardy:97,jacobsen:98}.\n\nOn the contrary, in strong-disorder systems, like \\gls{SG}s, the ground-state, the equilibrium properties of the system, and the phase transition completely differs from the pure system, as we will illustrate throughout this thesis.\n\nThe disorder can also be present in the form of an external random field $h_i$. This type of disorder, with the absence of the coupling disorder, leads to the well-known and widely-studied \\gls{RFIM}\\index{Random Field Ising Model}~\\cite{imry:75,nattermann:98,belanger:98}. This model is paradigmatic in the study of disordered systems and can be regarded as an intermediate case between weak and strong disorder.\n\nIn the two-dimensional case, the ordered phase is destroyed by the random field~\\cite{binder:84,aizenman:90}, i.e. the effects in the transition are strong and the properties of the pure system give no clue to understand the disordered system. However, in the three-dimensional case, the critical behavior of the system is changed by the presence of the random fields but still exhibits an ordered phase~\\cite{imbrie:84,bricmont:87}, similarly to what occur in the weak-disorder systems.\n\nThe importance of the \\gls{RFIM} in the statistical physics literature yields over several reasons. Standing as one of the simplest disordered systems with a rich phenomenology is one. Moreover, this model has numerous representatives in nature, for example, the diluted antiferromagnets in a homogeneous external field~\\cite{fishman:79} and it has been intensively studied, from both the experimental~\\cite{belanger:98} and the theoretical~\\cite{nattermann:98} point of view. Besides, contrary to the \\gls{SG} case (as we will discuss in subsequent sections), the theoretical and experimental \\gls{RFIM} have been developed in parallel for many years.\n\nThe number of weak-disorder systems and the results characterizing them is large and of central importance in the statistical physics field, as we have illustrated above. Nonetheless, the presence of weak disorder is not always affecting the physics of the pure system. We discuss this issue next.\n\n\\subsubsection{The Harris criterion}\nWe have stated above that the presence of impurities may affect the critical behavior of a system. There exists a quantitative criterion to know if the presence of the disorder is going to be relevant: the Harris criterion\\index{Harris criterion}~\\cite{harris:74}.\n\nThe argument behind the original formulation of the Harris criterion is rather simple and a useful discussion can be found in~\\cite{brooks:16}. The main idea is that the sign of the critical exponent associated with the specific heat\\index{specific heat} ($\\alpha$) for the pure system, is the stability condition itself.\n\nIf $\\alpha>0$, the disorder will change the critical behavior. On the contrary, the disorder will be irrelevant and the critical behavior of the system will not change.\n\n\n\\section{Experimental spin glasses} \\labsec{experimental_spinglass}\nAs we said in the introduction of the chapter, the paradigmatic examples of \\gls{SG}s are transition metal impurities in noble metal hosts like \\gls{CuMn}, \\gls{AuFe}, \\gls{AgMn}, \\gls{CuFe}, etc. However, there exist other types of materials that exhibit the \\gls{SG} phenomenology. Is the case of rare earth constituents in metallic host \\gls{YDy}, \\gls{YEr}, \\gls{ScTb}, etc. Also the same holds for ternary systems, e.g \\gls{LaGdAl}. We will discuss the interactions that are the source of the randomness\\index{randomness} and frustration\\index{frustration} necessary to have a \\gls{SG} low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\refsubsec{source_randomness} and we will expose relevant experiments that had a historical impact in the development of the \\gls{SG}s field and that are somehow related to the results presented in this thesis in \\refsubsec{aging_memory_rejuvenation}\n\n\\subsection{Internal structure and magnetic interactions: the source of randomness}\n\\labsubsec{source_randomness}\nWe have set that one of the main ingredients to have a \\gls{SG} is the randomness\\index{randomness}. How the real systems achieve that randomness\\index{randomness} in the interactions? There exist two main ways to obtain it: \\textit{bond randomness}\\index{randomness!bond} and \\textit{site randomness}\\index{randomness!site}.\n\nBond randomness\\index{randomness!bond} is a type of disorder\\index{disorder} present in real systems like \\gls{RbCuCoF} and \\gls{FeMnTiO}. These systems present regular\\index{regular lattice} lattices where the dominant magnetic interactions are short-ranged, then the impurities (cobalt and manganese respectively) are introduced. This procedure mimics what is called \\textit{ideal spin-glass} i.e. a regular\\index{regular lattice} lattice of spins interacting with their nearest-neighbors in a ferromagnetic or antiferromagnetic random way.\n\nSite randomness\\index{randomness!site} is the type of disorder\\index{disorder} that is present in the most commonly studied \\gls{SG}s such as \\gls{CuMn}. In this system, the substitution of the magnetic solute for the non-magnetic solvent should occur completely randomly\\footnote{There are other procedures used to create this type of disorder\\index{disorder} by destroying the crystal structure of the materials, making them amorphous.}. However, this type of disorder\\index{disorder} needs something else to generate randomness\\index{randomness} in the interactions. We need a kind of magnetic interaction that depending on the distance between the magnetic impurities generates antiferromagnetic or ferromagnetic couplings\\index{couplings}.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nThe dominant interaction in those systems is the so-called \\gls{RKKY}\\index{RKKY interaction} interaction \\cite{ruderman:54,kasuya:56,yosida:57}. This interaction is long-ranged and the underlying mechanism is the conduction electrons of the host metal acting as intermediaries between the magnetic moments of the magnetic solute. \n\nA magnetic impurity placed at $\\vec{r}_i$ changes the susceptibility\\index{susceptibility} of the conduction electrons surrounding it through hyperfine interaction. A second magnetic impurity placed at $\\vec{r}_j$ will behave in the same way, thus the two \\gls{RKKY}\\index{RKKY interaction} polarization will overlap, establishing an effective interaction between the two spins of the magnetic impurities. This interaction is given by\n\\begin{equation}\nJ(r_{ij}) = 6\\pi Z J^2 N(E_F) \\left[ \\dfrac{\\sin (2k_F r_{ij})}{(2k_F r_{ij})^4} - \\dfrac{\\cos (2k_F r_{ij})}{(2k_Fr_{ij})^3}\\right] \\quad , \\labeq{RKKY}\n\\end{equation}\nwhere $Z$ is the number of conduction electrons per atom, $J$ is the s-d exchange constant, $N(E_F)$ is the density of states at the Fermi level, $k_F$ is the Fermi momentum and $r_{ij} = \\lvert \\vec{r}_i - \\vec{r}_j \\rvert$. At large distance the quartic term of the distance becomes irrelevant with respect to the cubic one, therefore\n\\begin{equation}\nJ(r_{ij}) \\approx \\dfrac{J_0 \\cos (2k_F r_{ij} + \\phi)}{(2k_F r_{ij})^3} \\quad ,\\labeq{RKKY_reduced}\n\\end{equation}\nbeing $J_0$ a constant which agglutinates all the constant terms of \\refeq{RKKY} and $\\phi$ a phase that takes into account the charge difference between impurity and host. The reader may wonder whether the decaying-sinusoidal behavior of \\refeq{RKKY} or \\refeq{RKKY_reduced} would be enough to generate random interactions. We would like to stress the fact that the Fermi moment $k_F$ is, actually, quite large (of the order of the inverse of the interatomic spacing). That makes the sinusoidal oscillations to be very sensitive to any change of the distance $r_{ij}$.\n\nOf course, there exist other types of interactions between spins capable to generate randomness\\index{randomness}\\footnote{For example, superexchange interaction is relevant in insulating and semiconducting materials due to the lack of conduction electrons. Moreover, there exist weaker interactions like dipolar\\index{dipolar} interaction that play an important role because they introduce anisotropies\\index{anisotropy}.}, however, we only stop to explain \\gls{RKKY}\\index{RKKY interaction} for historical and practical reasons. From the historical point of view, \\gls{RKKY}\\index{RKKY interaction} interactions are bounded to the birth of the \\gls{SG} research. Moreover, along this thesis experiments with \\gls{CuMn} will have an important role and the dominant interaction in \\gls{CuMn} turn out to be the \\gls{RKKY}\\index{RKKY interaction} interaction.\n\nLast, we have to keep in mind that the computation of \\refeq{RKKY} involves several approximations. The assumption of the free electron and the random impurities are the stronger ones. On the one hand, the consideration of an electronic band structure leads to considerable modifications of the \\gls{RKKY}\\index{RKKY interaction} interaction, see for example \\cite{narita:84} or, for computations in specific materials like graphene \\cite{annica:10,sherafati:11}. On the other hand, the positions of the impurities are not truly random as we assumed before. Experimentally it is possible to find significant correlations in the position of the impurities through neutron-scattering\\index{neutron scattering} techniques. Actually, the knowledge of those correlations allows the experimental computation for the couplings\\index{couplings} in different \\gls{SG}s, see \\reffig{RKKY} extracted from \\cite{morgownik:83}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{intro\/RKKY.png}\n\\caption[\\textbf{RKKY interaction for real Spin Glasses.}]{\\textbf{RKKY interaction for real Spin Glasses.} Dotted line represents the original computation for the \\gls{RKKY}\\index{RKKY interaction} coupling\\index{couplings} as a function of the interatomic distance. Computations over real \\gls{SG}s shows significant differences with the theoretical results due to the correlations between the position of the impurities in the host metal. Figure from \\cite{morgownik:83}.}\n\\labfig{RKKY}\n\\end{figure}\n\nThe naive computation may not be quantitatively accurate but this, \\textit{a priori}, unfortunate fact turns out to be a hope for the study of \\gls{SG}s. The \\gls{RKKY}\\index{RKKY interaction} model still captures the fundamental requirements to find glassy behavior and thus, open the door to theoretical models which we will discuss in future sections that do not reproduce the couplings\\index{couplings} distribution of the real systems but which still contains the two main ingredients needed for finding \\gls{SG} behavior: randomness\\index{randomness} and frustration\\index{frustration}.\n\n\\subsubsection{Subtle but fundamental: anisotropies}\nUp to now, the magnetic interaction that we have attributed to \\gls{CuMn} real systems, the \\gls{RKKY}\\index{RKKY interaction} interaction, presents isotropic behavior, thus, there is no reason to restrict the value of the spin $\\vec{s}$ of the impurities in any dimension. This three-dimensional spin leads to the so-called \\textit{Heisenberg\\index{Heisenberg} spin glass}.\n\nHowever, even the purest real system presents some anisotropies\\index{anisotropy}. The role of those anisotropies\\index{anisotropy} is fundamental because they can restrict the degrees of freedom\\index{degree of freedom} of the spins to a plane (resulting in the $XY$ \\textit{spin glass}) or to a single dimension, leading to the known as \\textit{Ising\\index{Ising} spin glass}.\n\nThroughout this thesis, several results will be compared with real \\gls{CuMn} systems which are, essentially, Heisenberg\\index{Heisenberg}-like \\gls{SG}. A lot has been said about the effect of the anisotropies\\index{anisotropy} in the \\gls{CuMn} \\gls{SG}s \\cite{prejean:80,fert:80,levy:81,bray:82,mendels:87,chu:94,petit:02,bert:04,zhai-janus:21} where the main ones that we can found are the dipolar\\index{dipolar} anisotropies\\index{anisotropy}, weak but present in every spin system, and the \\gls{DM} anisotropies\\index{anisotropy}\\index{Dzyaloshinksii-Moriya}, whose origin is a large spin-orbit coupling\\index{couplings} of the conduction electron with the impurities, acting the conduction electron as an intermediary (similar to \\gls{RKKY}\\index{RKKY interaction}). \n\nFrom the computational point of view, Ising\\index{Ising} \\gls{SG}s are very convenient since they are much easier to simulate than Heisenberg\\index{Heisenberg} \\gls{SG}s and the research developed in the context of this thesis is focused on the former. Differences between Heisenberg\\index{Heisenberg} and Ising\\index{Ising} \\gls{SG}s are numerous\\footnote{For example, the very existence of a phase transition\\index{phase transition} in 3D systems is not clear in the Heisenberg\\index{Heisenberg} models while is commonly accepted in 3D Ising\\index{Ising} \\gls{SG}s.}, therefore, a natural question is whether or not the results obtained in this thesis are general.\n\nThis question is positively answered in \\cite{baityjesi:14}, actually, we know now that the ruling universality\\index{universality} class in presence of coupling\\index{couplings} anisotropies\\index{anisotropy} is Ising\\index{Ising} and even the purest real \\gls{SG} will present some anisotropies\\index{anisotropy}.\n\n\\subsection{Aging, memory and rejuvenation}\\labsubsec{experiments_introduction}\n\\labsubsec{aging_memory_rejuvenation}\nHere, we present emblematic experiments showing that \\gls{SG}s are out of equilibrium in the experimental time-scales. We also take the opportunity to expose those experiments that will be, at least conceptually, related in some way to the original results presented throughout this thesis.\n\n\\subsubsection{Aging}\\index{aging}\nWe have said that \\gls{SG}s, in absence of an external magnetic field, have null magnetization\\index{magnetization!zero}\\footnote{Not only, but also no magnetic order\\index{magnetic order} can be found, see \\refeq{no_magnetic_order}} $M=\\sum_{\\vec{r}} \\braket{\\vec{s}_{\\vec{r}}}=0$, however, when a magnetic field is applied, a magnetization\\index{magnetization} $M \\neq 0$ can be measured. When the external magnetic field is turned off, the system evolves from $M=M(t_0)\\neq 0 $ to $M(t_f)=0$. This process is called \\textit{relaxation}\\index{relaxation}.\n\n\\gls{SG}s exhibit in the low-temperature phase\\index{phase!low-temperature\/spin-glass} extremely-large relaxation\\index{relaxation} times, remaining out of equilibrium during the whole experiments. Still, the most striking feature is the emergence of a second time-scale: the relaxation\\index{relaxation} process strongly depends on the time that the system spent in the low-temperature phase\\index{phase!low-temperature\/spin-glass} before turning off the external field. We say that the system \\textit{ages}.\n\nThere exist two mirror experimental setups that are proven to be equivalent \\cite{nordblad:86}: the relaxation\\index{relaxation} of the \\gls{TRM} and the relaxation\\index{relaxation} of the \\gls{ZFC} magnetization\\index{magnetization!thermo-remanent}.\n\nThe typical protocol to study the \\gls{TRM}\\index{magnetization!thermo-remanent} is the following. First, we set a small external magnetic field and the system is cooled in its presence from a temperature $T_0$ above the critical\\index{critical temperature} temperature\\footnote{The temperature that separates the low-temperature phase\\index{phase!low-temperature\/spin-glass} and the high-temperature one\\index{phase!high-temperature\/paramagnetic}.} $\\ensuremath{T_\\mathrm{c}}\\xspace$ to a temperature $T_1<\\ensuremath{T_\\mathrm{c}}\\xspace$. We let the system age at temperature $T_1$ for a waiting time $t_w$ and then, the external magnetic field is switched off. At that moment, the decreasing magnetization\\index{magnetization} is recorded as a function of time $t$ where $t=0$ corresponds to the moment in which we turned off the field. One part of the total magnetization\\index{magnetization} falls immediately, the so-called reversible magnetization\\index{magnetization}. The other part is the so-called remanent magnetization\\index{magnetization!thermo-remanent} and falls slowly with the time $t$. The slow fall of the remanent magnetization\\index{magnetization!thermo-remanent} is shown in \\reffig{TRM_relaxation} from \\cite{vincent:97}. The reader may find similar experiments of \\gls{TRM}\\index{magnetization!thermo-remanent} relaxation\\index{relaxation} processes in \\cite{chamberlin:84,ocio:85,nordblad:86,alba:86}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/TRM_relaxation.png}\n\\caption[\\textbf{Thermo-remanent magnetization in spin glass.}]{\\textbf{Thermo-remanent magnetization in spin glass.} Thermo-remanent magnetization\\index{magnetization!thermo-remanent} $M$ normalized by the field-cooled value $M_{fc}$ is plotted against the time $t$ in a semi-log scale. \\gls{AgMn} with the $2.6 \\%$ of impurities is cooled from $T_0 >\\ensuremath{T_\\mathrm{c}}\\xspace=10.4$ K to $T_1=9$ K$ = 0.87 \\ensuremath{T_\\mathrm{c}}\\xspace$ in the presence of an external field of $0.1$ Oe. The system stays at $T_1$ with the field for a time $t_w$, then, the field is cut and the decaying magnetization\\index{magnetization} is recorded as a function of time $t$ where $t=0$ corresponds to the moment of turning off the field. Different curves corresponds to different $t_w$. Figure from \\cite{vincent:97}.}\n\\labfig{TRM_relaxation}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/TRM_relaxation_collapse.png}\n\\caption[\\textbf{The relevant time-scale of the thermo-remanent magnetization relaxation.}]{\\textbf{The relevant time-scale of the thermo-remanent magnetization relaxation\\index{relaxation}.} Thermo-remanent magnetization\\index{magnetization!thermo-remanent} $M$ normalized by the field-cooled value $M_{fc}$ is plotted against $t\/t_w$ in a semi-log scale in the same experimental conditions that in \\reffig{TRM_relaxation}. An almost perfect collapse is observed. Figure from \\cite{vincent:97}.}\n\\labfig{TRM_relaxation_collapse}\n\\end{figure}\n\nIn fact, one can observe in \\reffig{TRM_relaxation} for every curve an inflection point which roughly corresponds to $t=t_w$. The natural representation is, therefore, the one showed in \\reffig{TRM_relaxation_collapse} where the abscissa axis corresponds to the time normalized as $t\/t_w$. A perfect collapse of the curves under this representation is known as \\textit{full aging\\index{aging}}. However, the collapse is only approximated. The scaling variable\\footnote{The original symbol associated with this quantity, and the most used is $\\xi$, however, we use here $\\tau$ to avoid confusion with the coherence length\\index{coherence length} $\\xi$.} $\\tau$, firstly used in structural\\index{structural glass} glasses \\cite[p.~129]{struik:80} and lately introduced in \\gls{SG} by \\cite{ocio:85} as quoted by \\cite{rodriguez:03}, solves the problem and achieves a much better collapse\n\\begin{equation}\n\\tau=\\dfrac{t_w^{1-\\mu}}{1-\\mu}\\left[ \\left( 1 + \\dfrac{t}{t_w} \\right)^{1-\\mu} -1 \\right] \\quad \\mu < 1 \\,\\, . \\labeq{not_full_aging}\n\\end{equation}\n\nPutting the subtleties aside, we now know that the relevant time-scale of the aging\\index{aging} processes is $t_w$ and this is fundamental, as we will discuss in the following chapters because it has a deep relation with the coherence length\\index{coherence length} $\\xi$ acting as the key quantity governing the non-equilibrium phenomena, see~\\refch{aging_rate}.\n\nThe mirror protocol is the \\gls{ZFC}. In that protocol, the sample\\index{sample} is cooled from a temperature $T_0 > \\ensuremath{T_\\mathrm{c}}\\xspace$ to a temperature $T_1<\\ensuremath{T_\\mathrm{c}}\\xspace$ in zero-field. After a time $t_w$ a small field is turned on and the magnetization\\index{magnetization} is recorded as a function of time $t$, analogously to the previous experiment, $t=0$ corresponds to the moment in which the field is applied. This protocol is equivalent to the \\gls{TRM}\\index{magnetization!thermo-remanent} but with an increasing magnetization\\index{magnetization}. Furthermore, the sum of the \\gls{ZFC}-magnetization\\index{magnetization!zero-field-cooled} plus the \\gls{TRM}\\index{magnetization!thermo-remanent} is equal to the field-cooled magnetization\\index{magnetization}\\footnote{This is not true in general, the reader should note that we are in the small field limit where the only relevant term is the linear one. The equality is not guaranteed for larger fields where the non-linear responses are sizable, see for example recent relevant works~\\cite{zhai-janus:20,zhai-janus:21}}. For experimental results of this protocol, see \\cite{lundgren:83,nordblad:86}.\n\nAn experiment that can be regarded as a generalization (if that word can be used in the context of experiments) was performed, actually, earlier by Nagata \\textit{et al.}, see \\cite{nagata:79}. The main results of their research can be summarized in \\reffig{nagata_dcsuscept}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/nagata_dcsuscept.png}\n\\caption[\\textbf{DC-Susceptibility in CuMn spin glass.}]{\\textbf{DC-Susceptibility\\index{susceptibility!dc} in CuMn spin glass.}  Susceptibility\\index{susceptibility!dc} of a DC applied-field $h=5.9$ G is measure in a $1.08 \\%$ and a $2.02 \\%$ CuMn spin-glass and plotted against the temperature. Curves (a) and (c) corresponds to a field cooled protocol while curves (b) and (d) corresponds to a zero-field cooled protocol. Arrows indicate the reversibility ($\\leftrightarrow$) or irreversibility ($\\rightarrow$) of the protocol. Figure from \\cite{nagata:79}.}\n\\labfig{nagata_dcsuscept}\n\\end{figure}\n\nIn this experiment, the authors measure the magnetic response to an applied magnetic field i.e. \\textit{the susceptibility}\\index{susceptibility} $\\chi = \\lvert \\vec{M} \\lvert \/ \\lvert \\vec{H} \\lvert$.\n\nTwo different protocols are performed in this experiment. In the protocol corresponding to the curves (a) and (c), the sample\\index{sample} is cooled in the presence of a constant field $h = 5.9$ G. Below the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ the susceptibility\\index{susceptibility} remains almost constant and the process is reversible.\n\nOn the contrary, in the protocol corresponding to the curves (b) and (d), the sample\\index{sample} is cooled in absence of any field. Then, the sample\\index{sample} is heated and the susceptibility\\index{susceptibility} increases monotonically until it reaches the critical\\index{critical temperature} temperature. Moreover, cooling again the system in presence of the field leads to an irreversible behavior. Above $\\ensuremath{T_\\mathrm{c}}\\xspace$ both protocols are identical.\n\nMoreover, if we switch on the field after \\gls{ZFC} and let the system evolve at a fixed temperature, we observe that the susceptibility\\index{susceptibility} grows towards the field-cooled value but without reaching it in the experimental time-scales. This is the connection to the aging\\index{aging}-experiments showed above, where the temperature cycle\\index{temperature cycle} was performed only between two temperatures, but the aging\\index{aging} and the non-equilibrium behavior were captured the same.\n\nThe experiments exposed above are stressing us two main things:\n\\begin{enumerate}\n\\item Experimental \\gls{SG}s are out of equilibrium in the experimental time-scales.\n\\item The system ages, i.e. the time that the system expends below $\\ensuremath{T_\\mathrm{c}}\\xspace$ is a key quantity to understand its non-equilibrium behavior in the low-temperature phase\\index{phase!low-temperature\/spin-glass}.\n\\end{enumerate}\n\n\\subsubsection{Memory and rejuvenation}\n\n\\begin{figure}[h]\n\\centering\n\\hspace{-3cm}\n\\includegraphics[width=0.8\\textwidth]{intro\/memory_rejuvenation.png}\n\\caption[\\textbf{Memory and rejuvenation experiment in spin glasses.}]{\\textbf{Memory\\index{memory effects} and rejuvenation\\index{rejuvenation} experiment in spin glasses.} Out-of-phase susceptibility\\index{susceptibility!ac} is recorded when temperature vary in the presence of a sinusoidal field $h_\\mathrm{ac}$ of frequency $f=0.04$ Hz and peak amplitude of $0.3$ Oe. Main plot corresponds to CdCr$_{1.7}$In$_{0.3}$S$_4$ and the inset corresponds to the same plot for CuMn. Figure from \\cite{jonason:98}.}\n\\labfig{memory_rejuvenation}\n\\end{figure}\n \n \nIn previous experiments, when a constant field was applied, we defined the so-called \\textit{dc-susceptibility}\\index{susceptibility!dc}, therefore, a straightforward generalization is the \\textit{ac-susceptibility}\\index{susceptibility!ac}, which is none but the magnetic response of the system when a sinusoidal field is applied to it. \n\nIn \\gls{SG}s, the susceptibility\\index{susceptibility!ac} $\\chi_\\mathrm{ac}$ measured from a sinusoidal applied field $h_{\\mathrm{ac}}$ is also a sinusoidal quantity that presents a phase-delay $\\varphi$ with respect to the field $h_{\\mathrm{ac}}$. This delay lead to the definition of two quantities: the in-phase susceptibility\\index{susceptibility!ac} $\\chi'$ and the out-of-phase susceptibility\\index{susceptibility!ac} $\\chi''$\n\\begin{equation}\n\\begin{split}\n\\chi' & = \\chi \\cos \\varphi \\\\\n\\chi'' & = \\chi \\sin \\varphi\n\\end{split}\n\\end{equation}\n\nIn \\cite{jonason:98} \\footnote{Experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation} in temperature cycles\\index{temperature cycle} of two temperatures were performed earlier, see for example \\cite{lefloch:92} however, the richer phenomenology of \\cite{jonason:98} make us to focus on this experiment for the sake of clarity.}, the authors measured the out-of-phase susceptibility\\index{susceptibility!ac} when a $h_\\mathrm{ac}$ field of frequency $f=0.04$ Hz and peak amplitude of $0.3$ Oe was applied to a \\gls{CdCrInS} \\gls{SG}. The main result of this research is summarized in \\reffig{memory_rejuvenation}\n\nThe system was cooled from $T>\\ensuremath{T_\\mathrm{c}}\\xspace=16.7$ K down to $T_f=5$ K at a constant cooling-rate of $0.1$ K\/min. Then, the system was heated back continuously at the same rate.\n\n$\\chi''$ was recorded during the cooling and heating protocol, resulting in two close curves (still slightly different) where the heated one was used as a reference curve (continuous line in \\reffig{memory_rejuvenation}). Next, the experiment was repeated but the system was let age at temperatures $T_1=12$ K and $T_2=9$ K for a waiting time $t_w=7$ h for $T_1$ and $t_w=40$ h for $T_2$. After the age, the cooling was restarted at the same rate and the out-of-phase susceptibility\\index{susceptibility!ac} merged back with the reference curve, it is said that the system \\textit{rejuvenates}. The measurements of this protocol correspond to the white squares in \\reffig{memory_rejuvenation}. \n\nLast, the system was reheated at a constant heating rate. When the temperature approached the age temperatures $T_1$ and $T_2$, the system, somehow, ``remembered'' the age and followed the susceptibility\\index{susceptibility} curve, reproducing the data of the cooling protocol. In the inset, the authors included the same experiment appearing in \\cite{djurberg:99} in a \\gls{CuMn} \\gls{SG}, where the behavior was completely similar.\n\nThis experiment sends a very clear message: the aging\\index{aging} at a temperature $T$ does not affect the susceptibility\\index{susceptibility} value at lower temperatures. This temperature independence has been commonly related to the \\textit{temperature chaos}\\index{temperature chaos} phenomenon that we extensively treat in \\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos} and, in particular, the \\refch{out-eq_chaos} sums up the spirit of this thesis: separated researches from the experimental and the theoretical point of view that can be related by numerical simulations.\n\n\\section{Theoretical spin glasses} \\labsec{theoretical_spinglass}\nIn this section, we will briefly expose the theoretical development of the \\gls{SG}'s theory. The aim of this section is not to deeply review all the important results on \\gls{SG} but to provide some context of its history and to highlight the main theoretical predictions which we will deal with by means of numerical simulations.\n\nThe tools of statistical mechanics that we use in this section will be introduced in \\refsubsec{tools_statistical_mechanics}. Then, the most popular theoretical models will be shown in \\refsubsec{theoretical_models}. Finally, we will present the main theoretical pictures which predict different results in the \\gls{SG} low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\refsubsec{theoretical_pictures}. One of the tasks of our numerical simulations will be partially to discriminate between them.\n\n\\subsection{The tools of statistical mechanics}\n\\labsubsec{tools_statistical_mechanics}\nWe will present some basic results on statistical mechanics that will be useful in \\gls{SG} theory. The reader may find general references in \\cite{landau:80,lebellac:91,amit:05,greiner:12}.\n\nAlthough most of the things we are saying in this part are general for statistical mechanics, the notation and the language will be always focused on spin systems.\n\nWe consider a system described by $N$ microscopic variables $\\{s_i\\}$. The most basic quantity describing the system is the \\textit{energy}\\index{energy}\n\\begin{equation}\nE(\\{s_i\\}) \\equiv \\mathcal{H}(s_0,s_1,\\dots,s_n) \\, , \\labeq{Hamiltonian}\n\\end{equation}\na function (usually called \\textit{Hamiltonian}\\index{Hamiltonian}) of the microscopic variables $\\{s_i\\}$ that tends to be minimized when the system approach the thermal equilibrium.\n\nHowever, the energy\\index{energy} is not the only quantity describing the system, nor the only quantity we are interested in. A general quantity $\\mathcal{O}$ depending on the concrete values of the microscopic variables is called \\textit{observable}: $\\mathcal{O}(\\{s_i\\})$. \n\nIn general, the fluctuation of the observables due to the fluctuation of the microscopic variables makes desirable to consider averaged quantities\n\\begin{equation}\n\\braket{\\mathcal{O}} = \\lim_{t\\to \\infty} \\dfrac{1}{t} \\int_0^t \\mathcal{O}(\\{s_i\\}(\\tau)) d\\tau \\, . \\labeq{time_average}\n\\end{equation}\n\nThe problem now is clear. To compute these averages we need to know the time-evolution of a macroscopic number of variables, usually interacting with each other. Here emerges the fundamental principle of the statistical mechanics: the system, when it is at equilibrium, will eventually reach every possible set of microscopical variables, namely \\textit{configuration}\\index{configuration}. If we could know which is the probability of each configuration\\index{configuration} to appear, we could trade the integral over infinitely large periods with the weighted sum over all the possible configurations\\index{configuration} of the system.\n\nIn the canonical ensemble\\index{canonical ensemble}, the probability distribution of the configurations\\index{configuration} $\\{s_i\\}$ is\n\\begin{equation}\nP(\\{s_i\\}) = \\dfrac{e^{-\\beta \\mathcal{H}(\\{s_i\\})}}{Z} \\, , \\labeq{prob_configuration}\n\\end{equation}\nwhere $\\beta$ is the inverse of the temperature $T$ of the heat bath\\footnote{Which coincides with the temperature of the system in thermal equilibrium.} in units such that the \\textit{Boltzman constant} $k_B=1$ and $Z$ is a normalization factor called \\textit{Zustandssumme} or \\textit{partition function}\\index{partition function}\\footnote{For convenience, we are assuming a discrete number of possible configurations\\index{configuration} in the phase-space\\index{phase space}. In general, the expression of the partition function\\index{partition function} involves an integral instead of a sum.}\n\\begin{equation}\nZ = \\sum_{\\{s_i\\}} e^{-\\beta \\mathcal{H}(\\{s_i\\})} \\, . \\labeq{partition_function}\n\\end{equation}\n\nTherefore, the computation of the averages is\n\\begin{equation}\n\\braket{\\mathcal{O}} = \\dfrac{ \\sum_{\\{s_i\\}} \\mathcal{O}(\\{s_i\\}) e^{-\\beta \\mathcal{H}(\\{si\\})}}{Z} \\, . \\labeq{average_o}\n\\end{equation}\n\nIt is worthy to note that we have fully characterized the system at equilibrium through the partition function\\index{partition function} however, the very related quantity $F$, the \\textit{free energy\\index{free energy}}\\footnote{Note that we use $\\log$ as the symbol for the natural logarithm.}\n\\begin{equation}\nF = - \\dfrac{1}{\\beta} \\log Z(\\{s_i\\}) \\, ,\n\\end{equation}\nturns out to be much more practical because it is directly related to measurable quantities, fundamental in the thermodynamics of the system e.g. the \\textit{energy}\\index{energy} $U$ or the \\textit{magnetization}\\index{magnetization} $M$\n\\begin{equation}\nU = - \\left. \\dfrac{\\partial \\left( \\beta F\\right)}{\\partial T}\\right|_H \\quad , \\quad M = - \\left. \\dfrac{\\partial \\left( \\beta F\\right)}{\\partial H}\\right|_T \\labeq{thermodynamic_quantities} \n\\end{equation}\n\n\\subsubsection{Disorder and self-averaging}\nThe systems appearing along this thesis present a particularity: the interaction between any pair of particles is random. This randomness\\index{randomness}, extended in the whole system is what we call \\textit{disorder}\\index{disorder} and needs to be characterized by additional variables, taking into account the interaction between particles. We denote those variables $\\{J\\}$.\n\nThe disorder\\index{disorder} variables $\\{J\\}$ can exhibit a dynamical evolution and the time-scale of its evolution will determine if we are treating with \\textit{annealed} disorder or \\textit{quenched} disorder.\n\nOn the one side, the annealed disorder\\index{disorder!annealed} occurs when the time-scale of the dynamical evolution of $\\{J\\}$ is shorter than the observation time. Therefore, the interactions can be regarded as a sort of dynamic variables and we can average over them the same that we average over the configuration\\index{configuration} of the system\n\\begin{equation}\n\\overline{\\mathcal{O}_J} = \\int \\mathcal{O}(J) P(J) dJ \\, , \\labeq{average_disorder}\n\\end{equation}\nwhere $P(J)$ is the probability distribution that follows the disorder variables $J$. Therefore, in this situation, the free energy\\index{free energy} of the system is\n\\begin{equation}\nF = - \\dfrac{1}{\\beta} \\log \\overline{Z_J} \\, . \\labeq{free_energy_annealed}\n\\end{equation}\n\nThe computations associated with the annealed disorder have no additional difficulty, we have just added a fashion hat to our quantities, but the treatment is essentially the same. \n\nOn the other side, we have the quenched disorder\\index{disorder!quenched}, in which the time-scale of the dynamical evolution of $\\{J\\}$ is much larger than the observation time. In this case, each system is different due to the disorder\\index{disorder!quenched} \n\\begin{equation}\nF_J = - \\dfrac{1}{\\beta} \\log Z_J \\, .\n\\end{equation}\n\n\\textit{A priori}, there is no hope of universality\\index{universality} in those systems and we are forced to study them individually. However, the \\textit{self-averaging}\\index{self-averaging} property emerges to rescue us. In the thermodynamic limit\\index{thermodynamic limit}, for a system with $N$ degrees of freedom\\index{degree of freedom}, we have\n\\begin{equation} \n\\lim_{N \\to \\infty}\\dfrac{F_J}{N} = f_\\infty \\, , \\labeq{self_average}\n\\end{equation}\nbeing $f_\\infty$ the free-energy\\index{free energy!density} density in the thermodynamic limit\\index{thermodynamic limit}, which is independent of the disorder\\index{disorder} variables $\\{J\\}$.\n\nAn argument supporting this property can be found in \\cite{brout:59}. The reasoning is the following: any macroscopic system can be divided into a statistically large number $n$ of macroscopic systems. Due to the quenched disorder\\index{disorder!quenched}, every subsystem will have a different free energy\\index{free energy} $F_{J_j}$ with $j=(0,1,\\dots,n)$. Can we relate those free energies\\index{free energy} with the free energy\\index{free energy} of the whole system? \n\nThe free energy\\index{free energy} is just the logarithm of the partition function\\index{partition function} whose dependence of the interactions is codified in the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_J(\\{s_i\\})$. If we assume short-range interactions, which is a physically reasonable assumption, the total free energy\\index{free energy} will be the sum of the free energies\\index{free energy} of the subsystem plus the contribution to the free energy\\index{free energy} of the interface between the subsystems. As long as $N \\to \\infty$, the interface between subsystems will be negligible against the volume of those subsystems and, therefore \\refeq{self_average} holds. The implicit corollary is that computing the free energy\\index{free energy} of a large enough system is equivalent to compute the sum of the free energy\\index{free energy} for smaller systems.\n\nIf the distribution probability of the couplings\\index{couplings} $J$ is not pathological, we expect\n\\begin{equation}\n\\overline{F_J^2} - \\overline{F_J}^2 \\propto \\dfrac{1}{N} \\, .\n\\end{equation}\n\nThe computation of the disorder\\index{disorder!average} average of the logarithm \n\\begin{equation}\nF = \\overline{F_J} = -\\dfrac{1}{\\beta} \\overline{\\log Z_J} \\, ,\n\\end{equation}\nis one of the biggest difficulties on studying statistical mechanics on disordered\\index{disorder!systems} systems and we deal with it by using the so-called \\textit{replica method}\\index{replica!method\/trick}.\n\nIt is worthy to note that, the crucial hypothesis for Brout's argument is that the contribution of the boundaries of those subsystems is negligible. There exist some situations in which this assumption is no longer valid (see for instance \\cite{binder:86} for a more detailed explanation). For example, when a phase transition\\index{phase transition} occurs, the boundary conditions\\index{boundary conditions} become crucial and the system is no longer self-averaging\\index{self-averaging}. \n\nIt is well known that, for a spin glass below the critical\\index{critical temperature} temperature, some quantities are non-self-averaging\\index{self-averaging!non-}. This feature is closely related to the concept of \\textit{dispersed metastate\\index{metastate!dispersed}} (see \\refch{metastate}).\n\n\\subsubsection{The replica method}\nIn order to compute $\\overline{F_J}$ we use the so-called \\textit{replica method}\\index{replica!method\/trick} or \\textit{replica trick}, that was firstly introduced in the context of \\gls{SG}s in \\cite{edwards:75}.\n\nThe method is based on the expression\n\\begin{equation}\n\\log Z = \\lim_{n\\to 0} \\dfrac{Z^n -1}{n} \\, , \\labeq{replica_expression}\n\\end{equation}\nwhich is direct if we use the Taylor expansion of the right-side expression\n\\begin{equation}\n\\dfrac{Z^n -1}{n} = \\dfrac{e^{n \\log Z}-1}{n} = \\dfrac{1 + n \\log Z + O(n^2)-1}{n} = \\log Z + O(n) \\, . \\labeq{replica_expression_explained}\n\\end{equation}\n\nAt this point, we consider $n$ identical systems $Z_J^{(a)}$ $a=(0,1,\\dots,n)$ also called \\textit{replicas}\\index{replica} i.e. systems with the same realization of the disorder\\index{disorder} $J$, and we define\n\\begin{equation}\nZ_n \\equiv \\overline{Z_J^n} = \\overline{\\prod_{a=1}^n Z_J^{(a)}} \\, ,\n\\end{equation}\nand \n\\begin{equation}\nF_n = -\\lim_{n\\to 0} \\dfrac{1}{n \\beta}  \\log Z_n \\, .\n\\end{equation}\n\nBy using \\refeq{replica_expression}, is easy to see now that $F_n = \\overline{F_J}$\n\\begin{equation}\nF_n = -\\lim_{n\\to 0} \\dfrac{1}{n \\beta}  \\log \\overline{Z_J^n} = -\\dfrac{1}{\\beta} \\lim_{n \\to 0} \\dfrac{\\log \\left(1+n\\overline{\\log Z_J}+ O(n^2) \\right)}{n} = - \\dfrac{1}{\\beta} \\overline{\\log Z_J} = \\overline{F_J} \\, . \\labeq{replica_trick}\n\\end{equation}\n\nThe computation of $F_n$ is easier than the computation of $\\overline{F_J}$ if we first compute $\\log \\overline{Z_J^n}$ with $n$ integer and then we take the limit $n \\to 0$. This is, indeed, a very doubtful step in the mathematical sense. We consider $n$ to be an integer in order to define $Z_n$, then, we take the analytical extension of $Z_n$ for $n \\in \\mathbb{R}$ and finally we take the limit of $n\\to 0$. In \\refsubsec{theoretical_models} we will introduce a practical use for the replica method\\index{replica!method\/trick}.\n\nIn the cases in which the free energy\\index{free energy} is an analytic function of the temperature\\footnote{which usually happens in the high-temperature phase\\index{phase!high-temperature\/paramagnetic} of magnetic systems.} $\\beta$, the replica method\\index{replica!method\/trick} is exact. Moreover, when other methods are available to compute the free-energy\\index{free energy}, the results coincide with the predictions of the replica method\\index{replica!method\/trick}.\n\nThere exist also an alternative approximation to the replica method\\index{replica!method\/trick} which not requires the \\textit{trick} of the duality nature of $n$ integer-real, see \\cite{dotsenko:93,coolen:93,dotsenko:94}.\n\n\\subsection{Theoretical models} \\labsubsec{theoretical_models}\nHere, we discuss the main models that capture the \\gls{SG} physics and that are relevant in the development of the field. The trade-game is clear, on the one hand, the model should be detailed enough to exhibit the main \\gls{SG} phenomenology. We have discussed which are the basic ingredients for that: randomness\\index{randomness} and frustration\\index{frustration}. On the other hand, the model should be also simple enough to be analytically tractable. The \\gls{EA}\\index{Edwards-Anderson!model} model is the first we are going to introduce here, for historical reasons, but also due to its relevance in the actual context of \\gls{SG} and the present thesis. We are also going to define the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, which represents the mean-field\\index{Mean-Field!model} approximation, which is analytically solvable, and which characterizes the \\gls{SG}s at infinite dimension.\n\nNonetheless, the aim of this part is not to deeply review all the theoretical results in \\gls{SG}s, that can be found in several places \\cite{mezard:87,dotsenko:01,dedominicis:06}, but to give some historical context and present results affecting the very nature of the \\gls{SG}s, an issue still debated nowadays at finite dimensions.\n\n\\subsubsection{Edwards-Anderson model}\nThis model was proposed by Edwards and Anderson\\index{Edwards-Anderson!model} in \\cite{edwards:75}. Now, the general degrees of freedom\\index{degree of freedom} we defined in \\refsubsec{tools_statistical_mechanics} are, indeed, spins which lie in a regular lattice\\index{regular lattice} and that interact with each other through the couplings\\index{couplings} $\\{J\\}$. The Hamiltonian\\index{Hamiltonian} of this model is\n\\begin{equation}\n\\mathcal{H} = - \\sum_{\\braket{i,j}} J_{ij} \\vec{s}_i \\vec{s}_j - \\sum_i \\vec{h}_i \\vec{s}_i \\, , \\labeq{ea_Hamiltonian_theory}\n\\end{equation}\nwhere $\\vec{s}_i$ is a unitary vector, $\\vec{h}$ is an external magnetic field and the sum over $\\braket{i,j}$ denotes the sum over the pairs of spins $s_i$, $s_j$ that are bounded by a coupling\\index{couplings} $J_{i,j}$ and that actually depends on the concrete form of the considered lattice. Along this thesis we will focus on the case $\\vec{h}_i = \\vec{0}$, therefore, from now on we address the particular case of non external magnetic field for the sake of simplicity.\n\nIf the spin vector is 3-dimensional, the system is called \\textit{Heisenberg\\index{Heisenberg} spin glass}, if the vector is 2-dimensional it is called $XY$ \\textit{spin glass} and, if the spin only can take values $s_i = \\pm 1$ we say that it is an \\textit{Ising\\index{Ising} spin glass}. From now on, we will focus on the Ising\\index{Ising} \\gls{SG}, which is actually a reasonable assumption in many systems (see \\refsubsec{source_randomness}) and is the particular case of our numerical simulations in the research developed throughout this thesis. \n\nMoreover, the most popular choice is to consider only nearest-neighbor interactions between the spins\\footnote{The rationale of this approach is the short-ranged nature of the interactions between spins.}, with the quenched variables $J_{ij}$ following a Gaussian\\index{Gaussian!distribution} probability distribution, however, the particular shape of the distribution is not very important and a very popular choice, apart from the Gaussian\\index{Gaussian!distribution}, is the bimodal one ($J_{ij} = \\pm J$) that we will use in the numerical simulations.\n\nThe \\gls{EA}\\index{Edwards-Anderson!model} model also brought a proposal for the order parameter\\index{order parameter!Edwards-Anderson} controlling the phase transition\\index{phase transition}: the \\textit{overlap\\index{overlap}}. The traditional order-parameters displayed in \\refeq{no_magnetic_order} are not valid in \\gls{SG} because, by definition, \\gls{SG} exhibit no long-range order. Nonetheless, the frozen and disorder\\index{disorder} nature of the glassy phase\\index{phase!low-temperature\/spin-glass} suggests a different order parameter\\index{order parameter!Edwards-Anderson} based on time correlations on the same site.\n\\begin{equation}\nq_{EA} = \\dfrac{1}{N} \\lim_{t \\to \\infty}  \\sum_i \\braket{s_i(t=0)s_i(t)}_t \\, . \\labeq{order_parameter_SG}\n\\end{equation}\n\nThe question that is answered by \\refeq{order_parameter_SG} is, therefore, how similar is the configuration\\index{configuration} of the system at a time $t$ compared to the configuration\\index{configuration} at time $t=0$? This time average does not seem very useful, but fortunately, by the same reasoning made in \\refsubsec{tools_statistical_mechanics} we can trade the time average by the weighted sum over all the possible configurations\\index{configuration}\n\\begin{equation}\nq_{EA} = \\dfrac{1}{N} \\sum_i \\braket{s_i}^2 \\, . \\labeq{order_parameter_SG_2}\n\\end{equation}\n\nWe expect $q_{EA} = 0$ for $T > \\ensuremath{T_\\mathrm{c}}\\xspace$ i.e. in the paramagnetic phase\\index{phase!high-temperature\/paramagnetic} $\\braket{s_i}=0$. The expectation for $T \\to 0$ is $q_{EA} \\to 1$.\n\nAs a final remark, let us note that the very existence of a phase transition\\index{phase transition} in the Ising\\index{Ising} \\gls{EA}\\index{Edwards-Anderson!model} model was not completely accepted (even with the existence of an earlier consensus~\\cite{kawashima:96,iniguez:96,iniguez:97,berg:98,janke:98,marinari:98}) until the beginning of the XXI century~\\cite{palassini:99,ballesteros:00}.\n\n\\subsubsection{Mean Field: the Sherrington-Kirkpatrick model}\n\nEven with the aim of simplicity in mind, the \\gls{EA}\\index{Edwards-Anderson!model} model is not simple to solve, nor its mean-field\\index{Mean-Field!model} version, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model:\n\\begin{equation}\n\\mathcal{H} = - \\sum_{i<j} J_{ij} s_i s_j \\, .\n\\end{equation}\n\nThe reader may have noticed one main difference between this Hamiltonian\\index{Hamiltonian} and that appearing in \\refeq{ea_Hamiltonian_theory}, in addition to the absence of an external field $\\vec{h}$. Now, the sum is performed over every pair of sites with the restriction of $i<j$ to avoid double counting. In consequence, no useful concept of distance between spins exists, the interactions are infinite-ranged.\n\nMoreover, for convenience, the couplings\\index{couplings} $J_{ij}$ follow a Gaussian\\index{Gaussian!distribution} distribution probability\n\\begin{equation}\nP(J_{ij}) = \\dfrac{1}{\\sqrt{2\\pi \\sigma^2}} \\exp \\left[-\\dfrac{(J_{ij}-\\mu)^2}{2\\sigma^2} \\right]  \\, ,\n\\end{equation}\nwhere $\\mu$ and $\\sigma^2$ stand for the mean and the variance of the Gaussian\\index{Gaussian!distribution} distribution and take the values\n\\begin{equation}\n\\mu = \\overline{J_{ij}} = 0 \\quad , \\quad \\sigma^2 = \\overline{J_{ij}^2} = \\dfrac{1}{N} \\, .\n\\end{equation}\n\nThis particular choice of $\\sigma^2$ makes the free-energy\\index{free energy!density} density $f=F\/N$ independent of the total number of spins $N$.\n\n\\subsubsection{The replica symmetric solution}\n\nWith this information, we are able to compute the key quantity, the free energy\\index{free energy}, or equivalently, $Z_n$\n\\begin{align}\n Z_n & = \\int_{\\mydots} \\int P(J) \\prod_{a=1}^n \\sum_{\\{s^a\\}} \\exp \\left( \\beta \\sum_{i<j}^N J_{ij} s^a_i s^a_j \\right) {\\mathrm{d}} \\{J\\} = \\nonumber \\\\ \n & = \\left(\\dfrac{N}{2\\pi}\\right)^{N(N-1)\/4} \\sum_{\\{s^a\\}} \\int_{\\mydots} \\int \\exp \\left(\\beta \\sum_{a=1}^n \\sum_{i<j}^N J_{ij} s_i^a s_j^a - \\dfrac{1}{2}N \\sum_{i<j}^N J^2_{ij} \\right) {\\mathrm{d}} \\{J\\} \\, ,\n\\end{align} \nwhere the notation $\\int_{\\mydots} \\int$ and ${\\mathrm{d}} \\{J\\} = \\prod_{i<j} {\\mathrm{d}} J_{ij}$ denotes that we are dealing with multiple integrals, one for each pair $i<j$ and the sum over the configurations\\index{configuration} now runs over all the possible configurations\\index{configuration} for each replica\\index{replica} with superindex $a$.\n\nAfter some computations for the exchange in the sum order, we get the expression\n\\begin{equation}\nZ_n = \\sum_{\\{s^a\\}} \\exp \\left[ \\dfrac{\\beta^2 N n}{4} + \\dfrac{\\beta^2 N}{2} \\sum_{a<b}^n \\left( \\dfrac{1}{N}\\sum_i s_i^a s_i^b \\right)^2 \\right] \\, .\n\\end{equation}\n\nWe can now step back by introducing an integral variable depending on the pair of replicas\\index{replica} $a,b$ instead of the pair of sites $i,j$ with the help of the Hubbard-Stratonovich\\index{Hubbard-Stratonovich} transformation\n\\begin{equation}\nZ_n = \\beta \\sqrt{\\dfrac{N}{2\\pi}} \\int_{\\mydots}\\int \\sum_{\\{s^a\\}} \\exp \\left( \\dfrac{\\beta^2 N n}{4}-\\dfrac{\\beta^2 N}{2} \\sum_{a<b}^n Q_{ab}^2 + \\beta^2 \\sum_{a<b}^n \\sum_i^N Q_{ab}s_i^a s_i^b \\right) {\\mathrm{d}} \\{Q\\} \\, , \\labeq{partition_function_mean_field_1}\n\\end{equation}\nwhere now, $\\int_{\\mydots}\\int$ and $d\\{Q\\} = \\prod_{a<b} {\\mathrm{d}} Q_{ab}$ represent multiple integrals, one for each pair $a<b$.\n\nAfter some basic algebra, we can write \\refeq{partition_function_mean_field_1} as\n\\begin{equation}\nZ_n = \\int_{\\mydots}\\int e^{-\\beta N n f_n(\\{Q\\})} {\\mathrm{d}} \\{Q\\} \\, , \\labeq{partition_function_mean_field_2}\n\\end{equation}\nwhere we can see that $f_n(\\{Q\\})$ has a free-energy-like\\index{free energy} structure\n\\begin{equation}\nf_n(\\{Q\\})= -\\dfrac{\\beta}{4} + \\dfrac{\\beta}{2n}\\sum_{a<b}^n Q_{ab}^2 - \\dfrac{1}{\\beta N n} \\log \\left[ \\sum_{\\{s^a\\}} \\exp \\left( \\beta^2 \\sum_{i}^N \\sum_{a<b}^n Q_{ab} s^a_i s^b_i \\right) \\right] \\, ,\n\\end{equation}\nwith the effective Hamiltonian\\index{Hamiltonian}\n\\begin{equation}\n\\mathcal{H}_{\\mathrm{eff}} (\\{Q\\}) = -\\beta \\sum_i^N \\sum_{a<b}^n Q_{ab}s_i^a s_i^b \\, . \\labeq{effective_Hamiltonian_mean_field}\n\\end{equation}\n\nIn the thermodynamic limit\\index{thermodynamic limit} we can compute the integral of \\refeq{partition_function_mean_field_2} through the saddle-point\\index{saddle point} approximation. The minimization of $Z_n$ in \\refeq{partition_function_mean_field_2} is given by the equations $\\partial f_n(\\{Q\\})\/ \\partial Q_{ab}=0$ for all the pairs $a<b$\n\\begin{equation}\n\\dfrac{\\partial f(\\{Q\\})}{\\partial Q_{ab}} = \\dfrac{\\beta}{n} Q_{ab} - \\dfrac{\\beta}{nN} \\dfrac{\\sum_{\\{s^a\\}} \\sum_i^N s_i^a s_i^b e^{-\\beta \\mathcal{H}_{\\mathrm{eff}}(\\{Q\\})}}{\\sum_{\\{s^a\\}} e^{-\\beta \\mathcal{H}_{\\mathrm{eff}}(\\{Q\\})}} = 0 \\, ,\n\\end{equation}\nwhich leads to the solution\n\\begin{equation}\nQ_{ab} = \\dfrac{1}{N} \\sum_i^N \\braket{s_i^a s_i^b}_Q \\, , \\labeq{mean_field_solution_general}\n\\end{equation}\nwhere $\\braket{\\cdot}_Q$ represents the usual average defined in \\refeq{average_o} but with the effective Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathrm{eff}}$.\n\nUnfortunately, this is all that we can do with an arbitrary matrix $Q$. The first and most natural ansatz for the form of $Q$, as all the replicas\\index{replica!equivalence} were supposed to be equivalent, is the \\textit{replica symmetric} ansatz\\index{replica!symmetric ansatz}. In that case, we have the symmetric form\n\\begin{equation}\nQ_{ab} = (1-\\delta_{ab})q \\, , \\labeq{replica_ansatz}\n\\end{equation}\nwhere $\\delta_{ab}$ is the Kronecker delta.\n\nThis particular form of $Q$ leads, with standard methods of Gaussian\\index{Gaussian!integration} integration\\footnote{see for example~\\cite{dotsenko:01} for a detailed computation.}, to the saddle-point\\index{saddle point} equation\n\\begin{equation}\nq = \\int_{-\\infty}^\\infty \\dfrac{1}{\\sqrt{2\\pi}} e^{-z^2\/2} \\tanh^2 \\left( \\beta z \\sqrt{q} \\right) {\\mathrm{d}} z \\, .\n\\end{equation}\n\nThis equation can be numerically solved and it is straightforward to prove that, for $\\beta<1$ i.e. $T>\\ensuremath{T_\\mathrm{c}}\\xspace=1$, the only solution is the trivial one $q=0$ corresponding to the paramagnetic phase\\index{phase!high-temperature\/paramagnetic}. On the contrary, for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ not only $q \\neq 0$, but also $\\lim_{T \\to 0} q(T) = 1$.\n\nA deeper connection can be established by computing the disorder\\index{disorder!average} average of the \\gls{EA} order parameter\\index{order parameter!Edwards-Anderson} introduced in \\refeq{order_parameter_SG_2}\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\overline{\\braket{s_i}^2} = \\dfrac{1}{N} \\sum_i \\overline{\\left(\\dfrac{\\sum_{\\{s\\}} s_i \\exp \\left[-\\beta \\mathcal{H}_J(\\{s\\})\\right] }{Z_J}\\right)^2}\n\\end{equation}\nNow, we multiply the numerator and the denominator by $Z_J^{n-2}$ in order to use the replica\\index{replica!method\/trick} trick\n\\begin{equation}\n\\overline{q_{EA}}  =  \\dfrac{1}{N} \\sum_i^N \\overline{\\left( \\dfrac{\\sum_{\\{s^a\\}}s^r_i \\exp \\left[ -\\beta \\mathcal{H}_J^a(\\{s^a\\}) \\right]}{Z_J^n}\\right)^2} \\, ,\n\\end{equation}\nwhere $s^r$ represents an arbitrary replica\\index{replica} $r$. When the limit $n\\to 0$ is taken, the denominator $Z_J^n$ tends to $1$ and we finally get\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\overline{\\braket{s_i}^2} = \\dfrac{1}{N} \\sum_i^N \\lim_{n \\to 0} \\overline{\\braket{s_i^\\alpha s_i^\\beta}} \\, ,\n\\end{equation}\nhowever, performing the disorder\\index{disorder!average} average in the \\gls{SK}\\index{Sherrington-Kirkpatrick} formalism, as we have just done above, leads to\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\lim_{n \\to 0} \\braket{s_i^\\alpha s_i^\\beta}_Q \\, ,\n\\end{equation}\nand finally, from \\refeq{mean_field_solution_general} and \\refeq{replica_ansatz} we have that $q = \\overline{q_{EA}}$. $q$ is nothing but the order parameter\\index{order parameter!Edwards-Anderson} in the \\gls{EA}\\index{Edwards-Anderson!model} model.\n\nThis solution is, unfortunately, wrong. The first sign of a deep error in the replica symmetric\\index{replica!symmetric solution} solution was the computation of low-temperature entropy\\index{entropy} that turned out to be negative $S(T=0) = -1\/2\\pi <0$. Moreover, a detailed analysis of the solution \\cite{dealmeida:78} showed that it is unstable in the low-temperature phase\\index{phase!low-temperature\/spin-glass}.\n\n\\subsubsection{Parisi's solution: the Replica Symmetry Breaking}\n\n\\captionsetup{justification=centering}\n\nThe unsatisfactory previous results suggested that the replica symmetry \\index{replica!symmetric ansatz}should be broken and some attempts can be found in \\cite{bray:78} and \\cite{blandin:78} who actually proposed the first step of the general iterative solution. That solution came from Parisi \\cite{parisi:79,parisi:80,parisi:80b} and it is known as the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} solution. The starting point is the replica symmetry\\index{replica!symmetric matrix} matrix $Q_{ab}^{\\text{RS}}$, see \\reffig{rs_qab}\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{RS}} = \\left(\\begin{array}{cccccccc}\n0 & & & & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\n & 0 & &  & \\\\\n &  & 0 &  & \\\\\n & &  & 0 & \\\\\n\\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 &  &  &  \\\\\n & & & &   & 0 &  &   \\\\\n& & &  &  &  & 0 &   \\\\\n & & & &  &  &  & 0\n\\end{array}\\right) \n\\end{equation*}\n\\caption[\\textbf{Replica symmetry ansatz for the matrix $Q_{ab}$.}]{\\textbf{Replica symmetry ansatz\\index{replica!symmetric ansatz} for the matrix $Q_{ab}$.}}\n\\labfig{rs_qab}\n\\end{figure}\n\nFrom here the proposed matrix $Q_{ab}^{\\text{RSB}}$ is constructed through successive iterations. The first step, called the one-step \\gls{RSB}\\index{replica!symmetry breaking (RSB)} consists of dividing the $n$ replicas\\index{replica} into $n\/m_1$ groups, where $n$ and $n\/m_1$ are supposed to be integers at this point. The $n\/m_1$ groups are $m_1 \\times m_1$ squares placed in the diagonal of the matrix $Q_{ab}^{\\text{1-step}}$. All the elements of the matrix where $a=b$ remain equal to $0$, the elements in the $m_1 \\times m_1$ squares with $a \\neq b$ are equal to $q_1$ and the rest of the elements are equal to $q_0$ . The compact form to represent the values of $Q_{ab}^{\\text{1-step}}$ is\n\\begin{equation}\nQ^{\\text{1-step}}_{ab} =\n\\begin{cases}\n0 & \\text{if $a=b$}\\\\\nq_0 & \\text{if $\\Ceil{a\/m_1} \\neq \\Ceil{b\/m_1}$} \\\\\nq_1 & \\text{if $\\Ceil{a\/m_1} = \\Ceil{b\/m_1}$}\n\\end{cases}\n\\, , \\labeq{1step_RSB_compact} \n\\end{equation}\nbeing $\\Ceil{x}$ the ceiling function. The schematic representation of \\refeq{1step_RSB_compact} is in \\reffig{1step_qab}.\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{1-step}} = \\left(\\begin{array}{cccc|cccc}\n0 &  &\\multicolumn{2}{r|}{\\multirow{2}{*}{\\Large $\\ \\ q_1$}} & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\n & 0 & &  & \\\\\n\\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 & & \\\\\n &  &  & 0 & \\\\\n\\hline\n\\multicolumn{4}{c|}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 & & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}\\\\\n & & & &  & 0 &   &  \\\\\n& & & & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 &   \\\\\n & & & &  & &  & 0\n\\end{array}\\right)\n\\end{equation*}\n\\caption[\\textbf{First step of Replica Symmetry Breaking.}]{\\textbf{First step of Replica Symmetry Breaking.\\index{replica!symmetry breaking (RSB)}}}\n\\labfig{1step_qab}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{2-steps}} = \\left(\\begin{array}{cc|cc|cc|cc}\n0 & q_2 &\\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $\\ \\ q_1$}} & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\nq_2 & 0 & &  & \\\\\n\\cline{1-4}\n\\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 &q_2 & \\\\\n &  & q_2 & 0 & \\\\\n\\hline\n\\multicolumn{4}{c|}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 &q_2 & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}\\\\\n\\multicolumn{4}{c|}{} &q_2  & 0 &   &  \\\\\n\\cline{5-8}\n\\multicolumn{4}{c|}{} & \\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 & q_2  \\\\\n\\multicolumn{4}{c|}{} &  & &q_2  & 0\n\\end{array}\\right) \n\\end{equation*}\n\\caption[\\textbf{Second step of Replica Symmetry Breaking.}]{\\textbf{Second step of Replica Symmetry Breaking.\\index{replica!symmetry breaking (RSB)}}}\n\\labfig{2step_qab}\n\\end{figure}\n\\captionsetup{justification=raggedright}\n\nWe can repeat the computation of the free energy\\index{free energy}\\footnote{It is worthy to note that, contrary as usual, in the framework of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} formalism, the free-energy\\index{free energy} should be \\textit{maximized}. The formal reason is the number of components of the matrix $Q_{ab}$ becomes negative in $n \\to 0$ limit, see \\cite{mezard:87,dotsenko:01}.} with this matrix (see e.g. \\cite{dotsenko:01}) and the related thermodynamic quantities. The zero-temperature entropy\\index{entropy} is $S^{\\text{1-step}}(T=0) \\approx -0.01$ i.e. $|S^{\\text{1-step}}(T=0)| < |S^{RS}(T=0)|$ and the instability of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} is also reduced\\footnote{Actually, what is reduced is the most negative eigenvalue of the free-energy\\index{free energy} Hessian matrix near the critical\\index{critical temperature} temperature.}.\n\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} procedure can be generalized to an infinite number of steps. To obtain the two-steps \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, we proceed in the same way as we did in the one-step \\gls{RSB}\\index{replica!symmetry breaking (RSB)} for each of the diagonal blocks of size $m_1 \\times m_1$, now, dividing them in blocks of size $m_2 \\times m_2$. An schematic view of the $Q_{ab}^{\\text{2-steps}}$ is in \\reffig{2step_qab}.\n\nSuccessive steps of \\gls{RSB}\\index{replica!symmetry breaking (RSB)} lead to $S(T=0) \\to 0$ and a less unstable solution in the low-temperature phase\\index{phase!low-temperature\/spin-glass}. It took a while, but finally, it was rigorously proved that the infinite-steps \\gls{RSB}\\index{replica!symmetry breaking (RSB)} produces the correct solution for the free energy\\index{free energy} in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model \\cite{guerra:02,guerra:03,talagrand:06}.\n\nThe infinite-step solution, therefore, depends on an infinite number of parameters $q_i$. Each of those $q_i$ appear with a different weight in the \\gls{pdf} of the overlap\\index{overlap} $q$, that takes the form\n\\begin{align}\np(q) & = \\dfrac{1}{n(n-1)} \\sum_{a \\neq b} \\delta(Q^{\\infty\\text{-steps}}_{ab}-q) = \\nonumber \\\\\n& = \\dfrac{n}{n(n-1)}\\left[ (n-m_1)\\delta(q-q_0) + (m_1-m_2)\\delta(q-q_1) + \\dots \\right] \\, . \\labeq{pdf_q}\n\\end{align}\nThe $n \\to 0$ limit here is a delicate procedure in which we move the $n\\times n$ matrix $Q^{\\text{RSB}}_{ab}=Q^{\\infty\\text{-steps}}_{ab}$ to a $0 \\times 0$ matrix-space. Moreover, the construction of the matrix suggests the assimilation of $m_0 = n$ and, with the restriction of $n$ to be an integer, $m_\\infty=1$. Obviously, $m_k > m_{k+1}$, so $n=m_0 > m_1 > \\dots > m_{\\infty} = 1$. The analytical extension of $n$ and the limit $n \\to 0$ implies that there is no reason to still considering $m_k$ with $k=0,1,\\dots$ to be integers and, therefore, $m_k \\in [0,1]$. The direct implication is the reversing of the order of the coefficients $m_k$, the \\gls{pdf} now is\n\\begin{equation}\np(q) = m_1 \\delta(q-q_0) + (m_2-m_1)\\delta(q-q_1) + \\dots \\, ,\n\\end{equation} \nand the \\gls{SG} order parameter\\index{order parameter} is no longer a discrete set of parameters but a function $q(x)$ with $x \\in [0,1]$ defined as\n\\begin{equation}\nq(x) = q_k \\quad, \\quad 0 \\leq m_k < x < m_{k+1} \\leq 1 \\, .\n\\end{equation}\n\nWe stop here our brief recall of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} results in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, nonetheless, there exists a huge number of interesting results, for example in the rich physical interpretation (see e.g. \\cite{parisi:83,rammal:86}) or numerical results that agree with the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} predictions \\cite{vannimenus:81,sommers:84,crisanti:02,aspelmeier:08}.\n\n\\subsection{Theoretical pictures in finite-dimension spin glasses} \\labsubsec{theoretical_pictures}\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} computation gives us the solution to the mean-field\\index{Mean-Field!model} model, but the behavior of the \\gls{SG}s in the low-temperature phase\\index{phase!low-temperature\/spin-glass} at finite dimensions is still a widely debated issue. Here, we briefly review the differences between the diverse pictures explaining the equilibrium \\gls{SG}-phase\\index{phase!low-temperature\/spin-glass} and we show the main predictions that will be crucial to elucidate their validity through experiments, analytical results, or in the case of this thesis, numerical simulations.\n\n\\subsubsection{The Droplet picture}\nAfter Parisi's solution for the mean-field\\index{Mean-Field!model} model, numerical studies of domain\\index{magnetic domain!walls} walls in \\gls{SG}s and their scaling properties \\cite{mcmillan:84,mcmillan:85,bray:84,bray:87}, based on Migdal-Kadanoff\\index{Midgal-Kadanoff!zzzzz@\\Also{Wilson-Kadanoff}|gobbleone} renormalization\\index{renormalization group} computations (that are exact for dimension $d=1$), were the seed for a completely different approach to explain the low-temperature phase\\index{phase!low-temperature\/spin-glass} in short-ranged Ising\\index{Ising} \\gls{SG}s. This picture, introduced along seminal works by Fisher, Huse, Bray and Moore \\cite{fisher:86,bray:87,fisher:88} is known as the \\textit{droplets}\\index{droplet!picture} picture.\n\nThe droplet\\index{droplet!picture} picture understands the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} from its ground-state\\index{ground-state}. The basic object, the \\textit{droplet}\\index{droplet}, consists of a compact domain\\index{magnetic domain!compact} of linear size $L$ of coherently flipped spins with respect to the ground-state\\index{ground-state}, which constitutes the lowest-energy\\index{energy} excitation at this length-scale $L$. Actually, the droplets\\index{droplet} are expected to have fractal boundaries with a surface area of order $L^{d_s}$, $d-1 \\leq d_s < d$ \\cite{fisher:86}.\n\nThe droplets\\index{droplet} with zero energy\\index{energy} occurs with a probability $P \\propto L^{-\\theta}$ being $\\theta< (d-1)\/2$ the so-called \\textit{stiffness exponent\\index{stiffness!exponent}} and the free-energy\\index{free energy} cost of generating a droplet\\index{droplet} of linear size $L$ is $F_L \\sim \\varUpsilon L^\\theta$ where $\\varUpsilon$ is the \\textit{stiffness modulus\\index{stiffness!modulus}}. The computation of $\\theta$ have been performed numerically for $d=3$ resulting in $\\theta=0.27$ \\cite{boettcher:04,boettcher:05}, $\\theta=0.26$ \\cite{monthus:14}. For $d=2$ the exponent $\\theta$ is negative ($\\theta \\sim -0.28$ \\cite{boettcher:04}), thus, in the thermodynamic limit\\index{thermodynamic limit} the free-energy\\index{free energy} cost for generating a droplet\\index{droplet} tends to zero and the \\gls{SG} transition\\index{phase transition} disappears.\n\nThe most relevant results of the droplet\\index{droplet!picture} pictures are the following:\n\\begin{itemize}\n\\setlength\\itemsep{0.3cm}\n\n\\item The spatial correlation decays with the exponent $\\theta$ as \n\\begin{equation}\nC(r_{ij}) = \\overline{\\braket{s_i s_j}^2} - \\overline{\\braket{s_i}^2} \\overline{\\braket{s_j}^2} \\propto r_{ij}^{-\\theta} \\, .\n\\end{equation}\n\n\\item As a direct consequence of the long-distance vanishing limit of the correlation\\index{correlation function!four point} functions \\cite{fisher:86} the overlap\\index{overlap!distribution} distribution is trivial i.e. corresponds to a delta function $p(q) = \\delta(q^2 - q^2_{EA})$. The many-states nature of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} displayed by the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} is no longer valid in the droplet\\index{droplet!picture} picture where only a pair of equilibrium states, related by spin-flip, appears\\footnote{It is commonly said that, according to the droplet\\index{droplet!picture} picture, the \\gls{SG} is a ``ferromagnet in disguise``, that is, a system with a complicated spin configuration\\index{configuration} for the ground-state\\index{ground-state} due to the randomness\\index{randomness!bond} of the couplings\\index{couplings} but that can be mapped to a ferromagnet by performing gauge transformations, similar to the Mattis model\\index{Mattis model} \\cite{mattis:76}.}. \n\n\\item Related to the dynamics, the aging\\index{aging} in the droplet\\index{droplet!picture} picture is explained through the growth of coherent domains\\index{magnetic domain} of spins. Moreover, the coarsening\\index{coarsening} domains\\index{magnetic domain!compact} would be compact objects where the overlap takes one of the two possible values associated with the two pure states allowed $q = \\pm q_{EA}$ \\cite{fisher:88}.\n\n\\item The presence of an external magnetic field suppresses the transition\\index{phase transition} to the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. The argument underlying this prediction is a generalization of the Imry-Ma \\cite{imry:75} argument. The energy cost of reversing the spins inside a droplet\\index{droplet} is, by an assumption of the droplets\\index{droplet!picture} model, proportional to $L^\\theta$, and the magnetization\\index{magnetization} of the droplet\\index{droplet} scales as $L^{d\/2}$. By introducing the Zeeman\\index{energy!Zeeman} energy, we can write the free-energy\\index{free energy} cost for flipping the droplet\\index{droplet} in the presence of a small external magnetic field $h$ as $L^\\theta - hL^{d\/2}$. Since $\\theta < (d-1)\/2 < d\/2$, the \\gls{SG} becomes unstable under the presence of any field $h$.\n\n\\item The \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} exhibits a chaotic behavior under small changes of the temperature \\cite{banavar:87,bray:87}. This feature is a direct consequence of free-energy\\index{free energy} scaling ansatz $F_L \\propto L^\\theta$. The \\textit{naive} expectation for the free-energy\\index{free energy} of the droplet\\index{droplet} with surface $L^{d_s}$ would be $F_L \\propto L^{d_s}$, and since $d_s \\geq d - 1 > (d-1)\/2 > \\theta$, the difference between the \\textit{naive} expectation and the scaling ansatz is the presence of large cancellations of the contribution to the free-energy\\index{free energy} in different parts of the boundaries. This precarious equilibrium would be sensitive to small changes in the temperature due to changes in the sign of the free-energy\\index{free energy} at large scales (see e.g. \\cite{katzgraber:07} for further details). Thus, one would expect a complete reorganization of the spin equilibrium configurations\\index{configuration} upon small changes of the temperature. This sensitivity of the system is known as \\textit{temperature chaos}\\index{temperature chaos}. In~\\refch{Introduction_chaos} the reader may find a deeper discussion about this issue.\n\\end{itemize}\n\n\\subsubsection{The RSB picture}\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} solution for the mean-field\\index{Mean-Field!model} model is expected to be correct in short-ranged models like the \\gls{EA}\\index{Edwards-Anderson!model} model for dimensions higher than the \\textit{upper critical dimension}\\index{critical dimension!upper} $d>d_u = 6$. However, its validity in lower dimensions (in particular we are interested in dimension $d=3$) is still a debated issue.\n\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory in short-ranged finite dimensions is obtained from perturbative computations from the original mean-field\\index{Mean-Field!solution} solution but the physical behavior drawn is very similar to the mean-field\\index{Mean-Field} predictions. The most remarkable results are:\n\\begin{itemize}\n\\setlength\\itemsep{0.3cm}\n\\item In the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, the order parameter\\index{order parameter} is a function $q(x): [0,1] \\longrightarrow [-q_{EA},+q_{EA}]$. In particular, in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, each pair of states will have an overlap\\index{overlap} $q \\in [-q_{EA},+q_{EA}]$ which follows the \\gls{pdf} of \\refeq{pdf_q} and that can be written as\n\\begin{equation}\np(q) = \\dfrac{{\\mathrm{d}} x(q)}{{\\mathrm{d}} q} \\, ,\n\\end{equation}\nby the introduction of the inverse function of $q(x)$\n\\begin{equation}\nx(q) = \\int_0^q P(q') {\\mathrm{d}} q' \\, .\n\\end{equation}\nThis non-trivial \\gls{pdf} is the sign of one of the most distinctive features of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}: in the low-temperature phase\\index{phase!low-temperature\/spin-glass} there exist infinitely many states.\n\\item The organization of those infinitely many states is studied through the \\gls{pdf} of three pure states, see e.g. \\cite{rammal:86,dotsenko:01}. The main result is that, for any arbitrary tern of states $\\alpha$, $\\beta$ and $\\gamma$, the overlap\\index{overlap} between them must fulfill the condition\n\\begin{equation}\nq_{ab} = q_{bc} \\leq q_{ac} \\quad \\forall \\,\\, (a,b,c) \\in \\mathrm{Sym}(\\{\\alpha,\\beta,\\gamma\\}) \\, \\labeq{ultrametric}\n\\end{equation}\nbeing Sym$(\\{\\alpha,\\beta,\\gamma\\})$ the set of all permutations of the three states. Equivalently, \n\\begin{equation}\nq_{ab} \\geq \\min(q_{bc},q_{ac}) \\quad \\forall \\,\\, (a,b,c) \\in \\mathrm{Sym}(\\{\\alpha,\\beta,\\gamma\\}) \\, .\n\\end{equation}\nThis property defines a measure over the space of states and those spaces that present this particular metric are known as \\textit{ultrametric} spaces. Therefore, in the space of \\gls{SG} states, there exist no triangles with all three sides being different.\n\nThe usual way to visualize the ultrametricity\\index{ultrametricity} in \\gls{SG} is displayed in \\reffig{ultrametric} from \\cite{mydosh:93}. For each pair of states $\\alpha$ and $\\beta$, the overlap\\index{overlap} $q_{\\alpha \\beta}$ is obtained by going back in the tree until reaching the first common level. The ultrametric property of \\refeq{ultrametric} can be easily checked if one picks any three states (labeled with a number) in the referred figure.\n\\begin{figure}[h]\n\\includegraphics[width=0.5\\textwidth]{intro\/ultrametric.png}\n\\caption[\\textbf{Ultrametric organization of Replica Symmetry Breaking states.}]{\\textbf{Ultrametric organization of Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} states.} The tree representation of the Replica Symmetry Breaking states\\index{replica!symmetry breaking (RSB)}. For any pair of states $\\alpha$ and $\\beta$ their corresponding overlap\\index{overlap} is obtained by downing the tree until reaching the encounter point. Figure from \\cite{mydosh:93}.}\n\\labfig{ultrametric}\n\\end{figure}\n\n\\item The ultrametricity\\index{ultrametricity} is argued to be related to the origin of the temperature chaos\\index{temperature chaos} phenomenon in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, see for instance \\cite{vincent:97}. The ultrametric hierarchical structure of states is temperature-dependent, that is, the free-energy\\index{free energy!landscape} landscape changes with the temperature as sketched in \\reffig{ultrametric_temperature}. \n\nIn the thermodynamic limit\\index{thermodynamic limit}, any small change of the temperature will relocate the state to a different local minimum, leading to a complete reorganization of its equilibrium configuration\\index{configuration}. Furthermore, this explanation of the temperature chaos\\index{temperature chaos} phenomenon would also explain the experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation}, commonly associated with it \\cite{picco:01,takayama:02,maiorano:05,jimenez:05} but not unanimously \\cite{berthier:02}.\n\nIn this picture, the system at a temperature $T$ would explore the metastable\\index{metastability} states\\footnote{In the thermodynamic limit\\index{thermodynamic limit}, the system would need infinite time to ``jump'' from one state to another.}. When the temperature is lowered by an amount $\\delta T$, the system would move in the branch corresponding to its actual state and restart the aging\\index{aging} process, this is the \\textit{rejuvenation}\\index{rejuvenation} effect. When the temperature is back to its previous value $T$, the system comes back to the initial state by moving in the same branch of the tree, this is the \\textit{memory}\\index{memory effects} effect.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.7\\textwidth]{intro\/ultrametric_temperature.png}\n\\caption[\\textbf{Sketch of ultrametric structure as a function of the temperature.}]{\\textbf{Sketch of ultrametric structure as a function of the temperature.} The hierarchical structure of states as a function of temperature is commonly argued to be related to the temperature chaos\\index{temperature chaos} phenomenon in the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} picture. Figure from \\cite{vincent:97}.}\n\\labfig{ultrametric_temperature}\n\\end{figure}\n\n\\item In the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, contrarily to the expected in the droplet\\index{droplet!picture} picture, the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} is not destroyed by a small magnetic field. The temperature-magnetic field plane is separated by the so-called \\textit{de Almeida-Thouless} line \\cite{dealmeida:78}. The part of the plane with a large magnetic field $h$ and a large temperature is paramagnetic-like while the opposite limit presents a \\gls{SG} behavior.\n\n\\item Similarly to the droplet\\index{droplet!picture} picture, the aging\\index{aging} in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture is explained through the growth of coherent domains\\index{magnetic domain} of spins. However, the predictions of both pictures split when trying to explain the nature of those domains\\index{magnetic domain}. The \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory expects space-filling domains\\index{magnetic domain!space-filling} i.e. the fractal dimension of the surface is $d_s=d$. \n\\end{itemize}\n\n\n\\subsubsection{Problems with early interpretation: the concept of metastate}\nHowever, the classic interpretation of \\gls{RSB}\\index{replica!symmetry breaking (RSB)} described above presents some issues. The properties of the theory were thought to be present in infinite systems but the procedure to obtain them was to average over the disorder\\index{disorder!average} and, only after that, the infinite-size limit was taken. The problem in disordered\\index{disorder!systems} systems is that, even if that limit exists for averaged quantities or distributions of quantities, it does not imply that an infinite system from which we obtain these quantities or distributions exists.\n\nIn order to solve this problem, mathematical approaches irrupted the physical debate through the concept of metastate\\index{metastate}, firstly proposed in a general context of disordered\\index{disorder!systems} systems by Aizenman and Wehr~\\cite{aizenman:90} and later introduced in the specific context of \\gls{SG}s to deal with this problem, by Newman and Stein~\\cite{newman:92,newman:96,newman:98,newman:03}.\n\nBy using the metastate\\index{metastate} formalism, two main pictures were introduced that rigorously solve the problem of taking the infinite-size limit: the metastate\\index{metastate} interpretation of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, and the chaotic-pairs picture. The reader may find a detailed discussion in~\\refch{metastate}.\n\n\n\\section{Numerical simulations in spin glasses} \\labsec{numerical_spinglass}\nThe previous sections showed us that, in general, the main theoretical results were far apart from the main experimental results. One role of numerical simulations is to fill that gap. On the one side, experiments are restricted to off-equilibrium conditions, and access to microscopical configurations\\index{configuration} is prohibited. On the other side, the theoretical works have been focused to understand the nature of the low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\gls{SG}. Furthermore, the analytical results are only exact in unrealistic models: droplets\\index{droplet!picture}, exact in one dimensional \\gls{SG}s, and \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, exact in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model and, with almost total consensus, in \\gls{EA}\\index{Edwards-Anderson!model} model for dimensions $d>6$.\n\nNumerical simulations, mostly focused on Monte\\index{Monte Carlo} Carlo simulations \\cite{landau:05}, allow us to study off-equilibrium and equilibrium \\gls{SG}s. Moreover, from numerical data we can access the microscopical configurations\\index{configuration} and we have total control of the system. However, there are also obstacles in the path of numerical work. The equilibrium simulations are restricted to small system-sizes $L$ and temperatures $T$ not very far from the critical\\index{critical temperature} temperature due to the sluggish dynamics exhibited in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, thus, the suspect of finite-size and critical effects hovers over the results. In the off-equilibrium case, again due to the extremely slow dynamics of \\gls{SG}s, the time-scale of the numerical work was traditionally very far from the time-scale of experiments.\n\nFortunately, this situation has improved significantly during the last years. The year-to-year increase of the computational power, the emergence of special-purpose\\index{special-purpose computer} computer like Janus\\index{Janus} \\cite{janus:06,janus:09} and Janus\\index{Janus} II \\cite{janus:14}, and the implementation of algorithms like Parallel\\index{parallel!tempering} Tempering have allowed to simulate larger systems with unprecedented precision and to achieve time-scales comparable with the experimental ones \\cite{janus:08b,janus:09b,janus:18}.\n\nThis thesis aims to be a step forward in the conversion of the numerical simulations from an extremely useful tool for theoretical studies to a bridge between theory and experiments. Along this document we will present several original works with at least one of the following tasks on mind:\n\\begin{enumerate}\n\\item \\textbf{Simulated systems must capture the observed phenomenology of real \\gls{SG}.} Physics is an experimental science and, therefore, every model that we take into consideration to explain the \\gls{SG}s physics must exhibit the same phenomenology observed in real systems. The state of the art for numerical simulations allows extrapolations to experimental scales (see~\\refch{aging_rate}) and the comparison between numerical and experimental results. \n\\item \\textbf{Relate theoretical results with experiments.} A few years ago, the idea of the statics-dynamics equivalence\\index{statics-dynamics equivalence} was proposed in numerical simulations \\cite{janus:08b,janus:10,janus:17}. Numerical evidence is suggesting that off-equilibrium systems of coherence length\\index{coherence length} $\\xi$ could be regarded as a set of equilibrated systems of linear size $L \\sim \\xi$. This concept allows us to relate theoretical predictions in equilibrated \\gls{SG}s with off-equilibrium measures that can be compared with experimental results (see \\refch{aging_rate}).\n\\item \\textbf{Discern between proposed theoretical pictures.} Although the new avenues open by the improvement of the computational power are exciting, we still can take advantage of the traditional purposes of the numerical works. We have discussed how different pictures provide different predictions in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} (\\refsubsec{theoretical_pictures}). Throughout this thesis we will face those predictions and we will compare them with numerical results (see~\\refch{metastate}, \\refch{aging_rate}).\n\\item \\textbf{Open new paths for experimental work.} The level of control that we have over simulated systems makes the numerical work an ideal field to find new phenomena that can be later addressed by experiments (see~\\refch{mpemba} and \\refch{out-eq_chaos}).\n\\item \\textbf{Develop new tools for numerical simulations.} Of course, the numerical simulations are not perfect, and not only the development of the hardware is capable of improving their performance. The numerical research to find new methods have been fundamental, from a historical point of view. Here we also focus on improve the numerical simulations, for example, in the study of the Temperature Chaos\\index{temperature chaos} (see~\\refch{equilibrium_chaos}), but also by implementing well-established methods in our works (\\refch{AP_statistics}, \\refch{AP_technical_details_aging}, \\refch{AP_PT}, \\refch{AP_technical_details_out-eq_chaos}, \\refch{AP_multispin_coding}).\n\\end{enumerate}\n\nThroughout this thesis we will be focus on the numerical study of the 3D \\gls{EA}\\index{Edwards-Anderson!model} model by using Monte\\index{Monte Carlo} Carlo methods, in the rest of this section, we will introduce the model and the observables computed with the goal to avoid repetitions in the subsequent chapters.\n\n\\subsection{3D Edwards-Anderson model} \\labsubsec{3D_EA_model}\nAll the numerical simulations carried out in the original works of this thesis are performed in the three-dimensional Ising\\index{Ising} \\gls{EA}\\index{Edwards-Anderson!model} model. In our simulations, the spins are disposed in a cubic lattice $\\Lambda_L$  with \\gls{PBC}\\index{boundary conditions!periodic} where the vertices corresponds to the location of the Ising\\index{Ising} spins $s_i = \\pm 1$. The edges of the cubic lattice correspond to the quenched couplings\\index{couplings} $J_{ij}$ and the energy\\index{energy} of the system is defined by the Hamiltonian\\index{Hamiltonian}\n\\begin{equation}\n\\mathcal{H}_{\\{J\\}} (\\{s\\})= - \\sum_{\\braket{i,j}} J_{ij}s_is_j \\, . \\labeq{EA_Hamiltonian}\n\\end{equation}\nIn our simulations, the couplings\\index{couplings} $J_{ij}$ are independent and identically distributed random variables drawn from a bimodal distribution ($J_{ij}= \\pm 1$ with a $50 \\%$ probability). This model exhibits a spin-glass transition\\index{phase transition} at temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.1019(29)$ \\cite{janus:13}.\n\nEach realization of the couplings\\index{couplings} $\\{J\\}$ is called a \\textit{sample}\\index{sample} and allows us to estimate the average over the disorder\\index{disorder!average}. For each sample\\index{sample}, we simulate statistically independent system copies, each of them evolving under the same couplings\\index{couplings} $\\{J\\}$ but with different thermal noise. Each of these copies is called a \\textit{replica}\\index{replica}. The need of simulating different replicas\\index{replica} will be exposed below.\n\nThe parameters of the simulation (number of samples\\index{sample}, number of replicas\\index{replica}, simulated temperatures, the size of the lattice, \\dots) and the corresponding Monte\\index{Monte Carlo} Carlo method used will be specified in the following chapters, depending on the simulation carried out. \n\n\\subsection{Monte Carlo simulations} \\labsubsec{Monte_Carlo}\nAs we have already anticipated, the studies carried out along this thesis are performed through Monte\\index{Monte Carlo} Carlo simulations \\cite{landau:05,amit:05}. Here, we briefly introduce this method for the reader unfamiliar with the Monte\\index{Monte Carlo} Carlo methods. We will quickly explain the basics of Markov\\index{Markov chain} chains, we will introduce the Metropolis\\index{Metropolis-Hastings} algorithm and the Parallel\\index{parallel!tempering} Tempering. Nonetheless, advanced applications of Markov\\index{Markov chain} chains and, specifically Parallel\\index{parallel!tempering} Tempering, related to the thermalization\\index{thermalization} process will be described in~\\refch{equilibrium_chaos}.\n\\subsubsection{Markov chains}\nWe are interested in the study of a very specific spin system in a lattice, as we have just discussed. The problem is that even for fairly small systems, the integrals involved in the computation of typical quantities have a very high dimensionality, which makes the numerical methods of integration very inefficient. We use, instead, a well-established method to obtain a sample\\index{sample} of configurations\\index{configuration} with the appropriate \\gls{pdf}: a dynamic Monte\\index{Monte Carlo} Carlo method.\n \nTo that purpose, we consider a random walk\\index{random walk} in the configuration\\index{configuration!space} space which hopefully allows us, for each temperature $T=1\/\\beta$, to \\textit{move} from an arbitrary point in the configuration\\index{configuration!space} space\\footnote{In general, we have no \\textit{a priori} information about how the typical configuration\\index{configuration} should be at the desired temperature.} to the relevant configurations\\index{configuration} at that temperature (i.e. the \\textit{thermalization}\\index{thermalization}) and to sample\\index{sample} configurations\\index{configuration} according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution [see \\refeq{prob_configuration}].\n\nBesides, our random walk\\index{random walk} should be Markovian\\index{Markovian}\\footnote{The next state depends only on the actual state and not on the history of the random walker\\index{random walk}.} and will be represented by a transition matrix\\index{transition matrix} $\\pi$. The transition matrix\\index{transition matrix} is 2-dimensional and the rows (equivalently the columns) correspond to every possible configuration\\index{configuration} of the system. The elements $\\pi_{XY}$ denote the probability of change from a configuration\\index{configuration} $X$ to another configuration\\index{configuration} $Y$ in the next step of the random walk\\index{random walk}. Therefore $\\pi_{XY} \\geq 0$ $\\forall X,Y$ and $\\sum_{Y} \\pi_{XY} = 1$.\n\nAs long as the system is Markovian\\index{Markovian}, the probability for going from one configuration\\index{configuration} $X_0$ to one configuration\\index{configuration} $X_n$ in $n>1$ steps is just the sum of the probabilities of the system running over all the possible paths from $X_0$ to $X_n$ in $n$ steps. In those paths, due to the \\textit{no-memory} characteristic of the Markovian\\index{Markovian}\\index{Markov chain} chains, the probability is just the product of the probabilities $\\pi^{(n)}_{X_0X_n} = \\sum_{ \\{ X_1,X_2,\\dots,X_{n-1} \\} } \\pi_{X_0X_1} \\pi_{X_1X_2} \\cdots \\pi_{X_{n-1}X_n}$. \n\nIn addition to those properties, which are inherent to all the Markov\\index{Markov chain} chains, we require other properties that are fundamental for the thermalization\\index{thermalization} process.\n\nFirst, the so-called \\textit{balance\\index{balance condition}} condition\n\\begin{equation}\n\\dfrac{\\exp\\left(-\\beta \\mathcal{H}(Y) \\right)}{Z} = \\sum_{X} \\pi_{XY} \\dfrac{\\exp\\left((-\\beta \\mathcal{H}(X) \\right)}{Z} \\, . \\labeq{balance_condition}\n\\end{equation}\nThis condition express the fact that, if we have a set of configurations\\index{configuration} distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} probability distribution at one step $t$, the set of configurations\\index{configuration} after one step ($t+1$) of the random walk\\index{random walk} for each element of the set will be also distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} probability distribution.\n\nMoreover, the \\textit{irreducibility}\\index{irreducibility}\\index{irreducibility} condition assures that all the configurations\\index{configuration} $Y$ are accessible from any configuration\\index{configuration} $X$, mathematically this is expressed as \n\\begin{equation}\n\\forall \\, X,Y \\, \\exists \\, n>0 \\, : \\, \\sum_{ \\{ X_1,X_2,\\dots,X_{n-1} \\} } \\pi_{X_0X_1} \\cdots \\pi_{X_{n-1}X_n} > 0 \\quad \\mathrm{with} \\quad X_n=Y \\, . \\labeq{irreducibility}\n\\end{equation}\nThere exists a more restricting condition, the \\textit{aperiodicity}\\index{aperiodicity} which needs the introduction of the concept of \\textit{period}. A period $d_X$ (being $X$ a configuration\\index{configuration}), is the greatest common divisor of the length of all the paths starting at configuration\\index{configuration} $X$ and finishing at the same configuration\\index{configuration} $X$. If that period is $d_X=1$ for all the states, we say that the Markov\\index{Markov chain} chain is aperiodic\\index{aperiodicity}.\n\nThe above properties allow us to introduce a specially useful theorem~\\cite{sokal:97}: if one Markov\\index{Markov chain} chain satisfies both, the aperiodic\\index{aperiodicity} and the balance\\index{balance condition} condition, then\n\\begin{equation}\n\\lim_{n\\to \\infty} \\pi^{(n)}_{XY} = \\dfrac{\\exp\\left( -\\beta \\mathcal{H}(Y)\\right)}{Z} \\, . \\labeq{thermalization_condition}\n\\end{equation}\nThis theorem implies that the starting configuration\\index{configuration} is immaterial, the random walk\\index{random walk} will eventually sample the desired Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution. This theorem is not but a mathematical warranty of the fact that our system will thermalize.\n\n\\subsubsection{The Metropolis-Hastings algorithm}\nIn our numerical simulations, most of the time we are using the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm\\footnote{Often called just Metropolis\\index{Metropolis-Hastings}, for short.}, which is nothing but a dynamic Monte\\index{Monte Carlo} Carlo method that fulfills the previously described conditions. This method is generally applicable to a multitude of contexts, but it is not our goal to provide general results on Monte\\index{Monte Carlo} Carlo methods\\footnote{To that purpose, the reader may consult \\cite{sokal:97,amit:05,landau:05}.}. Hence, we focus here on our particular context and identify a configuration\\index{configuration} $X$ with the set of all the spins in our lattice. \n\nThere exist several possible ways to propose a change from the configuration\\index{configuration} $X$ to the configuration\\index{configuration} $Y$. One very common choice is to attempt the change of individual spins in the lattice in a sequential way. As far as we are focus on the Ising\\index{Ising} model, for each spin there is only a possible change to perform: to flip the spin.\n\nIn the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm, for one temperature $T=1\/\\beta$, we proceed in the following way:\n\\begin{enumerate}\n\\item Compute the energy\\index{energy} of the configuration\\index{configuration} $X$, namely $\\mathcal{H}(X)$\n\\item Flip the spin $i$ and compute the energy\\index{energy} of the new configuration\\index{configuration} $Y$: $\\mathcal{H}(Y)$.\n\\item Draw a random number $r \\in [0,1)$. If $r<\\exp\\left[ -\\beta(\\mathcal{H}(Y)-\\mathcal{H}(X))\\right]$ then we accept the change $X \\to Y$, otherwise, we remain in the configuration\\index{configuration} $X$.\n\\item Repeat the previous steps for all the spins $i$ in the lattice. We denote the realization of the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm to all the lattice a \\textit{lattice sweep} or simply, a \\textit{sweep}.\n\\end{enumerate}\nThis algorithm makes the system evolve whenever the energy\\index{energy} is diminished, but also allows, with probability $e^{-\\beta \\Delta\\mathcal{H}}$, local moves which increase the energy\\index{energy} by an amount $\\Delta \\mathcal{H}$.\n\n\\subsubsection{Parallel Tempering}\nThe sluggish dynamics exhibited by the \\gls{SG}s is a major obstacle in the classical dynamic Monte\\index{Monte Carlo} Carlo methods, such as the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm, because the necessary time to thermalize, even small systems, would be prohibitive. Several algorithms try to palliate this problem, here we introduce the \\gls{PT} algorithm \\cite{hukushima:96,marinari:98b}.\n\nThe general idea of the \\gls{PT} is to thermalize at the same time a set of $N$ identical copies which are at different temperatures $T_1 < T_2 < \\cdots < T_N$ (or equivalently $\\beta_N > \\cdots > \\beta_2 > \\beta_1$). For those samples\\index{sample} above the critical\\index{critical temperature} temperature $T>\\ensuremath{T_\\mathrm{c}}\\xspace$, the evolution of the system will be fast. On the contrary, for those systems that lie at temperatures $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, the configurations\\index{configuration} will be almost frozen. The \\gls{PT} algorithm consists of two alternating sets of steps.\n\nFirst, each system copy independently undergoes standard Monte\\index{Monte Carlo} Carlo dynamics (for example Metropolis\\index{Metropolis-Hastings}) at its own temperature; one can use one or more Monte\\index{Monte Carlo} Carlo steps each time. Second, pairs of spin configurations\\index{configuration} attempt to exchange their temperatures by permuting the $N$ copies of configurations\\index{configuration} in the temperature mesh. The exchange rule between two copies labeled as $\\alpha$ and $\\alpha'$ with configurations\\index{configuration} $\\{s^{(\\alpha)}\\}$ and $\\{s^{(\\alpha')}\\}$ of the system follows the Metropolis\\index{Metropolis-Hastings} scheme but we trade the accepting probability to\n\\begin{equation}\nr < \\exp \\left\\lbrace \\left[\\beta_{\\alpha} - \\beta_{\\alpha'}\\right] \\left[\\mathcal{H}(\\{s^{(\\alpha)}\\}) - \\mathcal{H}(\\{s^{(\\alpha')}\\}) \\right] \\right\\rbrace \\, . \\labeq{PT_exchange_rule}\n\\end{equation}\n\nThe goal of the \\gls{PT} algorithm is to sample sets of $N$ configurations\\index{configuration} (one for each copy of the system $\\{s^{(\\alpha)}\\}_{\\alpha=1}^N$), each one at its given temperature. Of course, those copies are not ordered in the temperature mesh with the $\\alpha$ index because of the permutations introduced by the algorithm. In order to fully characterize the state of the system at a given time, we need to introduce $\\pi(\\alpha)$: the permutation of the $\\alpha=1,2,\\dots,N$ copies of systems in the temperature mesh. Now, the state of the \\gls{PT} can be described by $X=\\{ \\pi, \\{s^{(\\alpha)}\\}_{\\alpha=1}^N\\}$ and the stationary distribution of the \\gls{PT} algorithm would be\n\\begin{equation}\nP_{\\mathrm{eq}}(X) = \\dfrac{1}{N!}\\prod_{\\alpha=1}^N \\dfrac{\\exp \\left[-\\beta_{\\alpha} \\mathcal{H}(\\{s^{\\pi^{-1}(\\alpha)}\\})\\right]}{Z_{\\beta_{\\alpha}}} \\, , \\labeq{PT_eq_distribution}\n\\end{equation}\nbeing $Z_{\\beta_{\\alpha}}$ the partition function\\index{partition function} at the temperature $\\beta_{\\alpha}$ and $\\pi^{-1}(\\alpha)$ the inverse permutation of $\\pi$ that fulfills the condition $\\pi^{-1}\\left( \\pi(\\alpha)\\right) = \\alpha$ ; $\\forall$ $\\alpha$.\n\nThe rationale behind the \\gls{PT} method is simple. Each system copy undergoes a random walk\\index{random walk} in temperature space. When a system copy is at a low temperature, it only explores the nearby free-energy\\index{free energy} local minima. When its temperature is high, however, free-energy\\index{free energy!barrier} barriers disappear: the copy can freely wander in phase space\\index{phase space}, and when it cools again it will typically fall in a different free-energy\\index{free energy!valley} valley, with different local minima. For \\gls{PT} to effectively thermalize, it is crucial that any copy of the system spends its time roughly evenly at every temperature: high temperatures are needed to ensure visiting all the phase space\\index{phase space}; low temperatures are needed to visit its low free-energy\\index{free energy} regions. \n\nIn fact, \\gls{PT} is currently used in a very large number of very different applications (for example in physics, biology, chemistry, engineering, statistics), and considerable efforts have been devoted to improving it from various communities. Various temperature-exchange rules have been developed and tested \\cite{sugita:99,calvo:05,earl:05,brenner:07,bittner:08,malakis:13}. Furthermore, it has been suggested that a significant gain can be achieved by optimizing the choice of the $N$ temperatures \\cite{katzgraber:06,sabo:08}. Further details on the \\gls{PT} method can be found in~\\refch{equilibrium_chaos} and \\refsec{thermalizing_PT}.\n\n\n\\subsection{Observables}\\labsubsec{observables_introduction}\nHere, we present the observables that we will measure in most of the performed simulations. Nonetheless, in the subsequent chapters, we will present some observables that have to be announced in their context in order to be fully understood.\n\nOne more consideration needs to be done before defining the observables. The Hamiltonian\\index{Hamiltonian} \\refeq{EA_Hamiltonian} presents a gauge symmetry, specifically, the transformation \n\\begin{equation}\ns_i \\longrightarrow \\epsilon_i s_i \\quad \\quad J_{ij} \\longrightarrow \\epsilon_i \\epsilon_j J_{ij} \\, ,\n\\end{equation}\nleaves it unchanged, being $\\epsilon_i$ a random sign $\\pm 1$ for each site of the lattice $i$. When several samples\\index{sample} are simulated and averages over the disorder\\index{disorder!average} are taken, this gauge symmetry is just the expression of redundant degrees of freedom\\index{degree of freedom} of the system. If the measured observables are insensible to this symmetry, it would be possible to find pairs of $\\{ \\{J\\}, \\{s\\}\\}$ related by a gauge transformation and providing different \\textit{weights} to the disorder\\index{disorder!average} average i.e. totally equivalent configurations\\index{configuration} from the energetic point of view, provide different results for some observables in different samples\\index{sample}. Therefore, it is desirable to define gauge-invariant observables. The usual way to do that is by introducing \\textit{replicas}\\index{replica} (see \\refsubsec{3D_EA_model}). \n\nThe overlap\\index{overlap!field} field between two replicas\\index{replica} is defined as\n\\begin{equation}\nq^{\\sigma,\\tau}_{\\vec{x}}(t) = s_{\\vec{x}}^\\sigma(t) s_{\\vec{x}}^\\tau(t) \\, , \\labeq{def_overlap}\n\\end{equation}\nwhere $\\sigma$ and $\\tau$ are labels to denote two different replicas\\index{replica} and subscripts $\\vec{x}$ represent the position in the lattice. The four-point spatial correlation\\index{correlation function!four point} function is\n\\begin{equation}\nC_4(T,\\vec{r},t) = \\overline{\\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}} \\, , \\labeq{def_C4}\n\\end{equation}\nwhere $\\overline{\\left( \\cdots \\right)}$ is the disorder\\index{disorder!average} average defined in \\refeq{average_disorder} and $\\braket{\\cdots}$ the average over the thermal noise\\footnote{This correlation\\index{correlation function!four point} function will be measured in off-equilibrium systems along this thesis, therefore, the mean given by \\refeq{average_o} that holds for equilibrium systems do not apply. We estimate the thermal noise by averaging over all the pairs of replicas\\index{replica} $\\sigma$ and $\\tau$.}. In numerical simulations we can only estimate these means, since we only have finite number of samples\\index{sample} (much smaller than the possible set of couplings\\index{couplings} $\\{J\\}$) and a finite number of replicas\\index{replica}. The long distance decay of $C_4(T,\\vec{r},t)$ defines the coherence length\\index{coherence length} $\\xi(t)$, an observable of central importance as we will discuss below\n\\begin{equation}\nC_4(T,\\vec{r},t) \\sim r^{-\\vartheta} f(r\/\\xi(t)) \\, . \\labeq{long_distance_C4}\n\\end{equation}\nThe function $f(x)$ decreases faster than exponentially for large $x$, $f(x) \\sim e^{-x^{\\beta}}$ with $\\beta \\approx 1.7$ (see \\cite{jimenez:05}). The exponent $\\vartheta$ at the critical\\index{critical temperature} temperature $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ is related to the anomalous dimension $\\eta$ (see for example \\cite{amit:05} for a definition), being $\\vartheta(\\ensuremath{T_\\mathrm{c}}\\xspace) = 1 + \\eta$ with $\\eta = -0.390(4)$ \\cite{janus:13}. For $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, droplets\\index{droplet!picture} picture predicts compact domains\\index{magnetic domain!compact} and, therefore, $\\vartheta=0$. On the contrary, \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture expects space-filling domains\\index{magnetic domain!space-filling} where $C_4(T,\\vec{r},t)$ vanishes at fixed $r\/\\xi$ as $t$ grows. The value of $\\vartheta$ in this picture is given by the replicon\\index{replicon}, a critical mode analogous to magnons in Heisenberg\\index{Heisenberg} ferromagnets (see \\cite{janus:10b} for a detailed discussion and \\cite{dedominicis:06} for theoretical introduction of the replicon\\index{replicon}). Previous numerical studies give us the value $\\vartheta=0.38(2)$ \\cite{janus:09} with a small dependence on the temperature $T$ that was vaguely attributed to the effect of the critical point. We will discuss widely about this exponent in~\\refch{aging_rate}.\n\n\n\\subsubsection{Coherence length}\\labsubsubsec{coherence_length}\nThe coherence length\\index{coherence length} $\\xi(t)$ is a fundamental observable in the dynamics of the \\gls{SG}s because it rules the off-equilibrium phenomena a multitude of times as we will discuss along this thesis (see~\\refch{aging_rate},~\\refch{mpemba}, and \\refch{out-eq_chaos}). Conceptually, the coherence length\\index{coherence length} is a characteristic length-scale of the off-equilibrium systems that measure the linear size of domains\\index{magnetic domain} of correlated spins. Although, the name ``coherence length\\index{coherence length}''\\footnote{This coherence length\\index{coherence length} should not be confused with the so-called coherence length\\index{coherence length} in optics.} is not universal and some authors denominate it \\textit{dynamical correlation length} (see for example \\cite{parisi:99c}) or, by abuse of language, simply \\textit{correlation length} (see \\cite{joh:99}).\n\nThe relation of the coherence length\\index{coherence length} with the length of correlated spins deserves a real example which will (hopefully) help us to visualize the concept. Consider one three dimensional lattice of linear size $L=160$ in which we have simulated the \\gls{EA}\\index{Edwards-Anderson!model} model (see~\\refsubsec{3D_EA_model}) with a Metropolis-Hastings\\index{Metropolis-Hastings} algorithm for $t=2^{36}$ number of Monte\\index{Monte Carlo} Carlo steps at temperature $T=0.7$. In the left panels of~\\reffig{coherence_length_snapshot} two projections of configurations\\index{configuration} at this time over the $xy$ plane are showed. The \\textit{up} spins are plotted in yellow and the \\textit{down} spins are plotted in blue. We can no see any pattern in those configurations\\index{configuration} that appear to be random.\n\nHowever, if we consider the overlap\\index{overlap} between them (right panel of~\\reffig{coherence_length_snapshot}) by using the same \\textit{color code}, a pattern emerges. Islands of correlated spins appear in the plot. This, of course, is only a visual sketch of the concept but it encodes the idea that the four-point correlation\\index{correlation function!four point} function (which is just a correlation\\index{correlation function!four point} function of the overlaps\\index{overlap}) has encrypted the correlation length. Yet, the quantitative obtaining of such observable have required a long time in both, numerical simulations and experiments.\n\nThe problem of finding characteristic lengths in off-equilibrium systems have been widely discussed. The integral estimators have been used since 1982 \\cite{cooper:82}. Detailed numerical studies have concluded that a well-behaved estimator of $\\xi(t)$, that is very convenient from the numerical point of view (see \\cite{janus:09b}), should be computed through the integrals\\footnote{The alert reader may notice that in this expression, the correlation\\index{correlation function!four point} function $C_4(T,\\vec{r},t)$ has changed its vectorial dependence of the distance $\\vec{r}$ to a simple scalar dependence $r$. This rotational invariance is assumed and justified numerically with a careful study in \\cite{janus:09b}.}\n\\begin{equation}\nI_k(t) = \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r \\, . \\labeq{integral_estimator_xi}\n\\end{equation}\nLet us identify the correlation\\index{correlation function!four point} function $C_4(T,r,t)$ with its long range behavior displayed in \\refeq{long_distance_C4}. In that case, taking $x=r\/\\xi$ would lead to\n\\begin{equation}\nI_k(t) = \\int_0^{\\infty} \\xi^{k-\\vartheta} \\left( r\/\\xi \\right)^{k-\\vartheta} f(r\/\\xi) \\xi \\dfrac{{\\mathrm{d}} r}{\\xi} = \\xi^{k+1-\\vartheta} \\int_0^{\\infty} x^{k-\\vartheta} f(x) {\\mathrm{d}} x \\, .\n\\end{equation}\n\nTherefore, the knowledge about the concrete form of the function $f(x)$ is no longer needed and one finds an estimator of $\\xi$\n\\begin{equation}\n\\xi_{k,k+1}(t) = \\dfrac{I_{k+1}(t)}{I_k(t)} \\propto \\xi(t) \\, . \\labeq{def_xi}\n\\end{equation}\n\nHowever, we should not forget the previous assumption made concerning the correlation\\index{correlation function!four point} function. The estimation of \\refeq{def_xi} involves systematic errors and that result would be only valid in the $r \\to \\infty$ limit. The larger the value of $k$, the smaller is the deviation from the asymptotic behavior due to the term $r^k$, which rules the short distances. However,  increasing the exponent $k$ moves the peak of $r^k C_4(T,r,k)$ to larger values of $r$ where the relative error of the correlation\\index{correlation function!four point} function is larger. A compromise solution of the value $k$ is needed, here, we use $k=1$. This decision is numerically justified in \\cite{janus:09b} and used in several works \\cite{janus:10b,janus:14b,fernandez:15,manssen:15,janus:17,fernandez:18b,fernandez:19}. Further details of computation in numerical simulations are provided in~\\refsec{finite_size_effects}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/coherence_length}\n\\caption[\\textbf{Spin-glass coherence length.}]{\\textbf{Spin-glass coherence length\\index{coherence length}.} \\textbf{Top left:} A snapshot of a configuration\\index{configuration} $\\{s_{\\boldsymbol{x}}^{(a)}\\}$, which has evolved for $t=2^{36}$ Monte\\index{Monte Carlo} Carlo steps at $T=0.7\\approx 0.64T_\\text{c}$.  We show the average magnetization\\index{magnetization} on the $xy$ plane, averaging over $z$. \\textbf{Bottom left:} Another configuration\\index{configuration} $\\{s_{\\boldsymbol{x}}^{(b)}\\}$ of the same sample\\index{sample}, prepared in the same way as $\\{s_{\\boldsymbol{x}}^{(a)}\\}$.  No visible ordering is present in either configuration\\index{configuration} because the preferred pattern of the magnetic domains\\index{magnetic domain} cannot be seen by eye ($s=1$ is plotted in yellow, and $-1$ in blue). \\textbf{Right:} If one measures the overlap\\index{overlap} between the two configurations\\index{configuration}, and with the same color code used for the spins, the preferred pattern of the magnetic domains\\index{magnetic domain}, of size $\\xi$, becomes visible. Figure from~\\cite{janus:19}.}\n\\labfig{coherence_length_snapshot}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\chapter[Sweet introduction to Temperature Chaos]{Sweet introduction to \\\\ Temperature Chaos} \\labch{Introduction_chaos}\n\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Una racha de viento nos visit\u00f3 \\\\\nPero nuestra veleta ni se inmut\u00f3.\\\\\nLa canci\u00f3n de que el viento se parara \\\\\nDonde nunca pasa nada.}}{-- Extremoduro, \\textit{Dulce introducci\u00f3n al caos}}\n\nBefore the second half of the XIX century, it was commonly accepted that the \\textit{predictability} of a physical system was only constrained for technical reasons such as the limited knowledge of the position and speed of the particles. In the last part of the XIX century, however, Henri Poincar\u00e9, in his geometrical study of the stability of the Solar System, introduced the idea of the extreme sensibility of a system to small changes on its initial conditions.\n\nThat idea was relatively forgotten in the mainstream physics literature until $1963$ when Lorenz \\cite{lorenz:63} realized that this extreme sensitivity was exhibited by a system of coupled differential equations. He was simulating a simplified model of convection rolls and he noticed that starting his numerical simulations from two slightly different initial conditions led to completely different results, even in relatively short times. This evidence about the impossibility of long-term predictions in certain systems was, indeed, very attractive for the physics community and, the interest in that research topic notoriously increased.\n\nAlthough the concept of \\textit{chaos} have considerably evolved through the years and it is well-defined in the mathematical context~\\cite{hasselblatt:03}, it has been historically associated with the extreme sensitivity to small perturbations~\\cite{strogatz:18}. \\gls{SG}s borrow the term to describe the fragility of the glassy phase\\index{phase!low-temperature\/spin-glass} in response to perturbations.\n\nThe sensitivity of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} upon changes in the couplings\\index{couplings}, namely \\textit{disorder chaos}\\index{disorder!chaos}~\\cite{ney-nifle:97,ney-nifle:98,sasaki:05,katzgraber:07}, or in the external magnetic field~\\cite{kondor:89,ritort:94,billoire:03}, have been widely studied and satisfactorily described. \n\nThe temperature counterpart of this fragility is known as \\gls{TC}\\index{temperature chaos}, which means that the spin configurations\\index{configuration} which are typical from the Boltzmann\\index{Boltzmann!weight} weight at temperature $T_1$ are very atypical at temperature $T_2$ (no matter how close the two temperatures $T_1$ and $T_2$ are). This phenomenon has proved to be very elusive~\\cite{bray:87b,banavar:87,kondor:89,kondor:93,ney-nifle:97,ney-nifle:98,billoire:00,mulet:01,billoire:02,krzakala:02,rizzo:03,sasaki:05,katzgraber:07,parisi:10,fernandez:13,billoire:14,wang:15,billoire:18,janus:21} and remains to be fully understood.\n\nThis chapter\\footnote{The name of chapter is a small tribute to rock band Extremoduro and their song \\textit{Dulce introducci\u00f3n al caos} (Sweet introduction to chaos).} pretends to be a very brief introduction to the \\gls{TC}\\index{temperature chaos} phenomenon by wandering through the main historical results in this field. The aim is to understand the starting points of the two following chapters (\\refch{equilibrium_chaos} and \\refch{out-eq_chaos}) that will be devoted to exposing the original results of this thesis in \\gls{TC}\\index{temperature chaos}.\n\n\\section{The origin of Temperature Chaos} \\labsec{origin_tc}\nThe \\gls{TC}\\index{temperature chaos} phenomenon was originally predicted in finite-dimension \\gls{SG}s by Bray, Moore, and Banavar\\footnote{Although it was originally predicted in \\gls{SG}s, other systems like polymers~\\cite{sales:02,dasilveira:04} also exhibit it.}~\\cite{bray:87,bray:87b,banavar:87} in the context of renormalization\\index{renormalization group} studies and scaling arguments (the germ of the later-called \\textit{droplets picture}\\index{droplet!picture}, see~\\refsubsec{theoretical_pictures}). For the sake of clarity, we briefly recall the main characteristics of the droplets\\index{droplet!picture} to understand the emerging chaos feature in this theory (see for example \\cite{katzgraber:07} for further details).\n\nIn the droplet\\index{droplet!picture} picture, the low-temperature phase\\index{phase!low-temperature\/spin-glass} is understood in terms of excitation of the ground-state\\index{ground-state}. The energy\\index{energy} excitation occur through the so-called \\textit{droplets}\\index{droplet!picture}, which are compact domains\\index{magnetic domain!compact} of spins with linear size $L$ that have been coherently flipped and whose boundaries are expected to be fractal, with a surface area of the order $L ^{d_s}$, $d-1 \\leq d_s < d$. The free-energy\\index{free energy} cost of generating such a droplet\\index{droplet} is $F_L(T) \\sim \\Upsilon(T) L^{\\theta}$ with $0<\\theta<(d-1)\/2$, being $\\Upsilon$ and $\\theta$ the stiffness modulus\\index{stiffness!modulus} and the stiffness exponent\\index{stiffness!exponent} respectively. The entropy\\index{entropy} in the droplets\\index{droplet!picture} picture scales with the size of the droplet\\index{droplet} as $S = \\sigma(T) L^{d_s\/2}$, being $\\sigma(T)$ the \\textit{entropy\\index{entropy!stiffness} stiffness}. \n\nThe key to understanding the \\gls{TC}\\index{temperature chaos} in the droplets\\index{droplet!picture} picture is to focus on the scaling behavior of the free energy\\index{free energy}. One would \\textit{naively} expect, for domains\\index{magnetic domain!compact} of spin of volume $\\sim L^{d_s}$, that the free-energy\\index{free energy} would also scales as $\\sim L^{d_s}$. However, as we have mentioned before, the free-energy\\index{free energy} scales as $F_L(T) \\sim L^{\\theta}$ with $\\theta < d_s$. This happens due to large cancellations of the contribution to the free\\index{free energy} energy from different parts of the boundary of the droplet\\index{droplet}, and this delicate equilibrium is the key to understand the \\gls{TC}\\index{temperature chaos} phenomenon. In this picture, \\gls{TC}\\index{temperature chaos} appears if the free energy\\index{free energy} of a droplet\\index{droplet} changes its sign upon a small change in the temperature. The length scale in which this happens is the so-called \\textit{chaotic length} $\\ell_c$.\n\nThe computation of $\\ell_c$ is performed through thermodynamic arguments. The free\\index{free energy} energy is $F(T) = U(T) - TS(T)$ being $U(T)$ the internal energy\\index{energy} of the droplet\\index{droplet} and $S(T)$ the entropy\\index{entropy} of the same droplet\\index{droplet}, both at temperature $T$. However, when the changes of temperature are small enough, the internal energy\\index{energy} can be considered as an independent quantity with respect to the temperature, and therefore\n\\begin{equation}\nF(T_2) = U(T_2) - T_2S(T_2) \\approx U(T_1) - T_2S(T_2) = F(T_1) + T_1S(T_1) - T_2S(T_2) \\, .\n\\end{equation}\nTaking into account the scaling behavior of the free\\index{free energy} energy and the entropy\\index{entropy} in a droplet\\index{droplet}, we can compute the length scale $\\ell_c$ at which $F(T_2)$ inverts its sign\n\\begin{equation}\n\\ell_c = \\left(\\dfrac{\\Upsilon(T_1)}{T_2 \\sigma(T_2) - T_1 \\sigma(T_1)}\\right)^{1\/\\zeta} \\quad \\mathrm{being} \\quad \\zeta = d_s\/2 - \\theta \\, . \\labeq{def_chaotic_length}\n\\end{equation}\nThe usual approach\\footnote{In~\\cite{katzgraber:07} the authors avoid this simplification with small changes in the final result.} is to take $\\sigma(T_2) \\approx \\sigma(T_1)$ when $\\lvert T_2 - T_1 \\rvert \\ll 1$ and, therefore\n\\begin{equation}\n\\ell_c \\sim \\lvert T_2 - T_1 \\rvert^{-1\/\\zeta} \\, . \\labeq{scaling_chaotic_length}\n\\end{equation}\nThe meaning of $\\ell_c$ is clear: small changes in the temperature make domains\\index{magnetic domain} of spins of length scales greater than $\\ell_c$ to flip, leading to two separate regimes. On the one side, in the short-length regime, the chaos is absent or it is rather weak. On the other side, in the large-length regime, two equilibrium configurations\\index{configuration} at temperatures $T_1$ and $T_2$ are completely uncorrelated, leading to a strong chaos phenomenon.\n\nIn this framework, a multitude of numerical work~\\cite{ney-nifle:97,ney-nifle:98,aspelmeier:02,krzakala:04,sasaki:05,katzgraber:07,monthus:14} has been performed. Indeed, the scaling of the chaotic length showed in~\\refeq{scaling_chaotic_length} was numerically found and the exponent $\\zeta$ was computed. However, still this approach presents major problems. The equilibrium simulations performed at that time were limited to $L \\sim 10$ which made the system to be in the $L \\ll \\ell_c$ regime, where the chaos is almost absent. Moreover, the scaling of~\\refeq{scaling_chaotic_length} extends beyond the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ where \\gls{TC}\\index{temperature chaos} should not occur. Thus, the numerical evidence supporting this picture is quite weak.\n\n\\section{Temperature Chaos in Mean Field}\nIn Mean-Field\\index{Mean-Field!model} models\\footnote{Those models that can be exactly solved through Mean-Field approximations.}, the \\gls{TC}\\index{temperature chaos} has proved to be particularly elusive. Specifically, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model stoically resisted numerical attempts to characterize the \\gls{TC}\\index{temperature chaos} phenomenon~\\cite{billoire:00,billoire:02}, later solved by~\\cite{billoire:14} as we discuss below. The lack of numerical evidence of \\gls{TC}\\index{temperature chaos} and the publication of studies which, indeed, presented evidence against it~\\cite{mulet:01,rizzo:01} led to the conclusion that \\gls{TC}\\index{temperature chaos} did not take place in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model.\n\nHowever, in 2003, a \\textit{tour de force}~\\cite{rizzo:03} showed that the \\gls{SK}\\index{Sherrington-Kirkpatrick} model presented an exceedingly small \\gls{TC}\\index{temperature chaos}, and it was necessary to compute up to the ninth order in a perturbative expansion in the replica\\index{replica!symmetry breaking (RSB)} framework\\footnote{Actually,~\\cite{rizzo:01} found no \\gls{TC}\\index{temperature chaos} because the computations in this paper were performed ``only'' until the fifth order in the perturbation expansion.} to find it.\n\nThis study is based on the use of a large-deviation functional. The idea is that, under the \\gls{TC}\\index{temperature chaos} hypothesis, the overlap\\index{overlap} between any pair of equilibrium states at temperatures $T_1$ and $T_2$ ($T_1 \\neq T_2$) should be zero. Therefore, the shape of the probability distribution of overlaps\\index{overlap!distribution} $q$ between equilibrium configurations\\index{configuration} at different temperatures should tend to a Dirac's delta function peaked on $q=0$ as the size of the system grows. Moreover, the scaling of this probability distribution is given by the large-deviations formula\n\\begin{equation}\nP(q) \\sim \\exp \\left[-N\\Delta F(q)\\right] \\, ,\n\\end{equation}\nwhere $N$ is the size of the system and $\\Delta F(q)$ takes account of the free-energy\\index{free energy} cost of constraining two replicas\\index{replica} to have a given mutual overlap\\index{overlap} at equilibrium. This functional $\\Delta F(q)$ is computed in~\\cite{rizzo:03} through a perturbative approach. It is necessary to reach the ninth order to find a non-vanished term, hence, it was demonstrated that \\gls{SK}\\index{Sherrington-Kirkpatrick} model presents, though pathologically, the \\gls{TC}\\index{temperature chaos} phenomenon.\n\nNonetheless, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model has not been the only Mean-Field\\index{Mean-Field!model} model in which \\gls{TC}\\index{temperature chaos} has been studied. For example, it has been found that diluted Mean-Field\\index{Mean-Field!model} \\gls{SG}s present much stronger \\gls{TC}\\index{temperature chaos}~\\cite{parisi:10}. Besides, in $p$-spin models, which are just a generalization of the \\gls{SK}\\index{Sherrington-Kirkpatrick} model in which interactions occur between $p \\geq 3$ spins, different behaviors have been found. On the one hand, we have a recent mathematical proof of the absence of \\gls{TC}\\index{temperature chaos} in the homogeneous spherical $p$-spin model~\\cite{subag:17}, in agreement with a previous claim based on physical arguments~\\cite{kurchan:93}. On the other hand, \\gls{TC}\\index{temperature chaos} should be expected when one mixes several values of $p$~\\cite{barrat:97}, as confirmed by a quite recent mathematical analysis~\\cite{chen:14,panchenko:16,chen:17,arous:20}.\n\n\\section{Memory and rejuvenation}\\labsec{memory_rejuvenation_introduction_chaos}\nThe reader may note that all the previous discussion about \\gls{TC}\\index{temperature chaos} assumes that it is an equilibrium phenomenon (since its very definition), however, most of the experimental work in spin-glasses is carried out under non-equilibrium conditions as we have already discussed in~\\refsubsec{aging_memory_rejuvenation}\\footnote{With the notable exception of\nexperiments in a thin-film geometry, see~\\cite{guchhait:14}. In fact, the experimental study of \\gls{TC}\\index{temperature chaos} in thin films has been initiated~\\cite{guchhait:15b}.}.\n\nThe spectacular rejuvenation\\index{rejuvenation} and memory\\index{memory effects} effects~\\cite{jonason:98,lundgren:83,jonsson:00,hammann:00} (described in~\\refsubsec{aging_memory_rejuvenation}) have been commonly related to the phenomenon of \\gls{TC}\\index{temperature chaos}~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. Yet, the situation is far from clear.\n\nThe idea is that, due to the \\gls{TC}\\index{temperature chaos} phenomenon, the equilibrium configurations\\index{configuration} at two slightly different temperatures $T_1$ and $T_2$ would be completely different, thus, the aging\\index{aging} performed at temperature $T_1$ would not be useful at the temperature $T_2$, and the aging\\index{aging} process restarts, leading to the so-called \\textit{rejuvenation}\\index{rejuvenation} phenomenon. The opposite behavior, the \\textit{cumulative aging\\index{aging!cumulative}}, means that the relaxation\\index{relaxation} work carried out at temperature $T_1$ is still useful (partly useful at least) when the temperature is varied to $T_2$.\n\nIndeed, some experiments~\\cite{jonsson:02,bert:04} and most of the numerical work~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05} trying to simulate temperature-varying\\index{temperature-varying protocol} experimental protocols can be interpreted as cumulative aging\\index{aging!cumulative}. At this point, opinions are split. Some authors find only cumulative aging\\index{aging!cumulative} in their simulations~\\cite{picco:01,takayama:02,maiorano:05}, while others find some traces of aging\\index{aging} restart~\\cite{komori:00,berthier:02}. However, this restarting of aging\\index{aging} occurs on exceedingly short times~\\cite{jimenez:05}. Perhaps more worryingly, it has been found numerically that a site-diluted ferromagnet (where no \\gls{TC}\\index{temperature chaos} is expected) behaves analogously to the spin glass~\\cite{jimenez:05}.\n\nIf a strong connection between \\gls{TC}\\index{temperature chaos} and the experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation} exists, some work is needed in order to establish it.\n\n\\section{Last steps}\nThis historical tour about \\gls{TC}\\index{temperature chaos} allows us to understand the general feeling in the field at the beginning of the 2010s. The \\gls{TC}\\index{temperature chaos} phenomenon seemed to be extremely weak, with gradual increasing effects that we were not able to perceive due to the difficulty of equilibrating large systems in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. However, an alternative weak-\\gls{TC}\\index{temperature chaos} scenario~\\cite{sales:02} could be compatible with the results. In this scenario, almost all the samples\\index{sample} exhibit no \\gls{TC}\\index{temperature chaos} at all but a few of them suffer dramatic effects upon temperature changes. Actually, this scenario was not completely unknown and some numerical studies mentioned that situation~\\cite{katzgraber:07}, but a quantitative study was lacking.\n\nThe main idea is that very few samples\\index{sample} undergo \\textit{chaotic events} i.e. at well-defined temperatures $T^*$, the samples\\index{sample} suffer first-order\\index{phase transition!first order} like transitions (rounded in finite-systems) such that the typical spin configurations\\index{configuration} below and above $T^*$ differ. While the majority of samples\\index{sample} do not have any chaotic event, some of them display one (or more) with a $T^*$ which seems to be located randomly within the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. Yet, the fraction of samples\\index{sample} lacking chaotic events decreases upon increasing the system size. Indeed, one expects~\\cite{rizzo:03,parisi:10} that the fraction\nof samples\\index{sample} lacking TC will decrease exponentially (in the system size).\n\nIt was in 2013 when the \\gls{TC}\\index{temperature chaos} was quantitatively studied as a rare-event-driven phenomenon~\\cite{fernandez:13}. This numerical study was based on the study of large-deviations functional which was fundamental in order to deal with the wild sample-to-sample fluctuations\\index{sample-to-sample fluctuations}. The mean problem in the numerical study of \\gls{TC}\\index{temperature chaos} in short-ranged \\gls{SG} was, indeed, the statistical methods used to deal with the chaotic observables: the majority of non-chaotic samples\\index{sample} killed any chaos signal when taking the disorder\\index{disorder!average} average\\footnote{As quoted by ``The Buggles'': \\textit{Average Killed The Chaos Signal}.}.\n\nLater, further works~\\cite{billoire:14,martin-mayor:15,fernandez:16,billoire:18,janus:21} showed that the rare-event analysis was the appropriate protocol in order to study the \\gls{TC}\\index{temperature chaos} phenomenon. In the subsequent chapters (\\refch{equilibrium_chaos} and \\refch{out-eq_chaos}), we will develop two different rare-event analysis in 3D Ising\\index{Ising} \\gls{SG}s in order to study the \\gls{TC}\\index{temperature chaos} phenomenon.\n \n\n\n\n\n\\chapter{Metastate} \\labch{metastate}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\n\\epigraph{\\textit{Si paso por Florida te recuerdo\\\\\nSi paso por la Valle me es igual,\\\\\nQue si estoy en Corrientes, que si estoy en Palermo\\\\\nPor todo Buenos Aires conmigo siempre est\u00e1s.}}{-- Julio Jaramillo, \\textit{No me toquen ese vals} }\n\n\nThis chapter is dedicated to discussing the metastate\\index{metastate}. We start by introducing the concepts of mixed and pure states in lattice systems, see \\refsec{mixed_pure_states_metastate}, which would be needed in order to understand further discussions. Then, we describe the problem of taking the thermodynamic limit\\index{thermodynamic limit} in disordered\\index{disorder!systems} systems in \\refsec{the_problem_metastate} and we introduce the proposed solution in \\refsec{the_solution_metastate}. The different theoretical pictures described in \\refsubsec{theoretical_pictures} provide different predictions for some observables in the metastate\\index{metastate} formalism, we introduce those observables and discuss the different scenarios in \\refsec{observables_predictions_metastate}. \n\nAt this point, we present an original contribution to the metastate\\index{metastate} problem developed during this thesis~\\cite{billoire:17} by explaining the numerical setup \\refsec{simulation_parameters_metastate} and by exploring the metastate\\index{metastate} from the numerical point of view in \\refsec{results_metastate}. Finally, we relate our numerical results to theory in~\\refsec{relating_numerical_theory_metastate}.\n\n\\section{Mixed and pure states} \\labsec{mixed_pure_states_metastate}\n\nConsider a spin system in a lattice $\\Lambda \\subset \\mathbb{Z}^d$ where the vertices correspond to the spins $s_i$ and the edges correspond to the couplings\\index{couplings} between spins $J_{ij}$, leading to nearest-neighbor interactions defined through a Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}} = \\sum_{\\braket{i,j}} J_{ij} s_i s_j$. As far as we are dealing with general definitions, we are neither restricting the values of the spins, nor the values of the couplings\\index{couplings}. \n\nOne \\textit{configuration}\\index{configuration} of the system is determined by the value of the set of all the spins $\\mathcal{S} = \\{s_i\\}$ as $i$ runs over all the lattice sites. In the same way, a concrete \\textit{sample}\\index{sample} is determined by the value of the set of all the couplings\\index{couplings} $\\mathcal{J} = \\{J_{ij}\\}$ as the pair $(i,j)$ runs over all the pairs of spins. \n\nThe restriction of the lattice to finite size $L$ is simply done by considering a cubic lattice $\\Lambda_L$ composed of $L^d$ spins. However, in the case of finite systems, an additional problem arises with the choice of the boundary conditions\\index{boundary conditions}. It is possible to define several boundary conditions\\index{boundary conditions} but we will consider a very common choice, the \\gls{PBC}\\index{boundary conditions!periodic}.\n\nAt a given temperature $T=1\/\\beta$, a Gibbs state $\\Gamma_{L,\\mathcal{J}}$ for a finite system $\\Lambda_L$ is a probability distribution over the configurations\\index{configuration} $\\mathcal{S}$ where each configuration\\index{configuration} has a probability to appear equal to \n\\begin{equation}\n\\Gamma_{L,\\mathcal{J}}(\\mathcal{S}_{\\Lambda_L}) = \\dfrac{\\exp\\left(-\\beta \\mathcal{H}_{L,\\mathcal{J}}(\\mathcal{S})\\right)}{Z_L} \\, , \\labeq{Gibbs_probability_state}\n\\end{equation}\nbeing $\\mathcal{H}_{L,\\mathcal{J}}$ the Hamiltonian\\index{Hamiltonian} of the system restricted to the lattice $\\Lambda_L \\subset \\mathbb{Z}^d$ and $Z_L$ the partition function\\index{partition function}\\footnote{For the sake of simplicity, we tacitly assume that each spin can only take a finite set of possible values and, therefore, the set of configurations\\index{configuration} $\\mathcal{S}$ is countable and we can perform the sum. In the case of infinite-uncountable possible values for each spin, the trade between the sum and an integral is needed.} $Z_L=\\sum_{\\mathcal{S}} \\exp \\left( -\\beta \\mathcal{H}_{L,\\mathcal{J}}\\right)$.\n\nWhen considering an infinite lattice, this definition of state is not useful anymore because the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}}$ involves sums of infinite terms that do not converge. Nonetheless, there exist a well-established definition for state in infinite lattice: the \\gls{DLR} states \\cite{ruelle:04,sinai:14,friedli:17}. A probability distribution of states in an infinite-size lattice is a Gibbs state $\\Gamma_{\\mathcal{J}}$ if, for any finite subset $\\Lambda_W \\subset \\mathbb{Z}^d$, two conditions are fulfilled:\n\\begin{enumerate}\n\\item The Hamiltonian\\index{Hamiltonian} for a spin configuration\\index{configuration} inside the subset $\\Lambda_W$ conditioned to the values of the rest of spins in the infinite lattice $\\mathbb{Z}^d \\setminus \\Lambda_W$, namely $\\mathcal{H}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})$, and the partition function\\index{partition function} restricted to that subset $Z_{W}$ are finite for almost every $\\mathcal{S}$.\\\\\n\nIn the case of the \\gls{EA}\\index{Edwards-Anderson!model} model the expression for $\\mathcal{H}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})$ is very easy due to the short-ranged nature of the Hamiltonian\\index{Hamiltonian}. We have to take care only with the frontier $\\partial \\Lambda_W$ where the Hamiltonian\\index{Hamiltonian} includes terms with spins out of $\\Lambda_W$ and spins inside. Therefore, we have\n\\begin{equation}\n\\mathcal{H}^{\\mathrm{EA}}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}) = \\sum_{J_{ij} \\in \\Lambda_W \\setminus \\partial \\Lambda_W} J_{ij} s^{\\mathrm{int}}_i s^{\\mathrm{int}}_j + \\sum_{J_{ij} \\in \\partial \\Lambda_W} J_{ij} s^{\\mathrm{int}}_i s^{\\mathrm{out}}_j \\, , \\labeq{hamiltonian_conditioned}\n\\end{equation}\nwhere the superindex \\textit{int} stands for spins belonging to $\\Lambda_W$ and the superindex \\textit{out} stands for spins outside of $\\Lambda_W$. The reader should notice that, if $\\Lambda_W$ contains a finite number of spins, the Hamiltonian\\index{Hamiltonian} of~\\refeq{hamiltonian_conditioned} only involves finite sums.\n\n\\item The conditional probability $\\Gamma_{W,\\mathcal{J}}$ of a configuration\\index{configuration} $\\mathcal{S}$ in $\\Lambda_W$ given the rest of the spins $\\mathbb{Z}^d \\setminus \\Lambda_W$ is absolutely continuous\\footnote{The reader may be confuse about the term ``absolutely continuous'' in this context. This continuity is defined over a measure of the random variables, the spins in our particular case. The reader should consult~\\cite{sinai:14} for a much more detailed discussion with rigorous proofs.} and it is defined by the expression\n\\begin{equation}\n\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})  = \\dfrac{\\exp \\left[-\\beta \\left( \\mathcal{H}_{W,\\mathcal{J}}(S_{\\Lambda_W}) + \\mathcal{H}_{W,\\mathcal{J}}(S_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}) \\right)  \\right]}{Z_{W}} \\, . \\labeq{def_DLR}\n\\end{equation}\n\\end{enumerate}\n\nFrom \\refeq{def_DLR} one can conclude that, for any two spin configurations\\index{configuration} $\\mathcal{S}_{1,\\Lambda_W}$ and $\\mathcal{S}_{2,\\Lambda_W}$ with the rest of the spins fixed $\\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}$, the ratio of their conditional probabilities would be, simply\n\\begin{equation}\n\\dfrac{\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})}{\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})} = \\exp \\left[ -\\beta \\left( \\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W})-\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W}) \\right) \\right] \\, . \\labeq{ratio_DLR}\n\\end{equation}\n\nDoes this definition hold for our particular case? It is easy to prove that the \\gls{EA}\\index{Edwards-Anderson!Hamiltonian} Hamiltonian\\index{Hamiltonian} defined in \\refeq{EA_Hamiltonian} is finite for any finite lattice $\\Lambda_W$ and, therefore, the partition function\\index{partition function}, which involves a finite number of summands since we are considering Ising\\index{Ising} spins, is simply a finite sum of finite terms. The second condition is also fulfilled because $\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W})-\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W})$ is finite, since the lattice $\\Lambda_W$ is finite by definition. The reader may find rigorous proof of the existence of the Gibbs states in the \\gls{EA}\\index{Edwards-Anderson!model} model in~\\cite{ruelle:04}.\n\nActually, given a Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}}$ and given a temperature $T=1\/\\beta$, the previous definition of infinite state allows the existence of many different Gibbs states $\\Gamma_\\mathcal{J}$. The set of all of the possible Gibbs states is convex and compact, so there exist \\textit{extremal} Gibbs states that are not convex combinations of any other Gibbs states. The extremal Gibbs states are called \\textit{pure} states while the non-extremal Gibbs states are called \\textit{mixed} states and can be decomposed uniquely into a convex combination of pure states in the following way\\footnote{We assume, for the sake of simplicity, that the decomposition is discrete.}\n\\begin{equation}\n\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}} = \\sum_{\\alpha} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\braket{\\cdots}_{\\alpha} \\, , \\labeq{decomposition_pure_states}\n\\end{equation}\nbeing $\\alpha$ a pure state and $\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}}$ an appropriate weight for that pure state that fulfills the condition $\\sum_\\alpha \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} = 1$, with $\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\geq 0$ $\\forall \\alpha$.\n\nMoreover, the overlap\\index{overlap} between two pure states labeled as $\\alpha$ and $\\beta$ can be defined in the lattice $\\Lambda_W$ as\n\\begin{equation}\nq_{\\alpha \\beta} = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} \\braket{s_x}_{\\alpha}\\braket{s_x}_{\\beta} \\, , \\labeq{overlap_pure_states}\n\\end{equation}\nbeing $s_x$ the spin at the position $x$. The \\gls{pdf} of the overlap\\index{overlap!distribution} can, therefore, be defined as\n\\begin{equation}\nP_{\\Gamma_{\\mathcal{J}}}(q) = \\sum_{\\alpha,\\beta}\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\omega_{\\beta,\\Gamma_{\\mathcal{J}}} \\delta(q-q_{\\alpha \\beta}) \\, . \\labeq{pq_pure_states}\n\\end{equation}\n\n\n\n\\section{The problem: Chaotic Size Dependence} \\labsec{the_problem_metastate}\n\nIn the previous section, we have properly defined the infinite-size states through the \\gls{DLR} states, however, that definition is not physically relevant because experiments are always conducted in large but finite systems. It would be desirable from the physical point of view, to connect the \\gls{DLR} states with a sequence of growing systems of linear size $L$ as $L \\to \\infty$.\n\nThis connection can be made for transitional-invariant Hamiltonians\\index{Hamiltonian} like the ferromagnet one. However, in systems with quenched disorder\\index{disorder!quenched}, such as the \\gls{EA}\\index{Edwards-Anderson!order parameter} model, that connection is still much of a mystery \\cite{aizenman:90}. The problem of taking the $L \\to \\infty$ limit have been already introduced in \\refsubsec{theoretical_pictures}: the \\gls{CSD}. \n\nLet us consider a system of linear size $L$ in a lattice $\\Lambda_L$ and fix an internal region of linear size $W$, $\\Lambda_W \\subset \\Lambda_L$. If we take the limit $L \\to \\infty$, remaining constant the couplings\\index{couplings} of the lattice $\\Lambda_W$, the state $\\Gamma_{W,\\mathcal{J}}$ changes chaotically as $L$ grows and also the observables measured in $\\Lambda_W$. \n\nThis extreme sensitivity of the system to the addition of couplings\\index{couplings} at the boundaries as long as $L$ tends to infinity is called \\gls{CSD}. This problem remained apparently oblivious to the \\gls{SG} literature and was pointed out for the first time by \\cite{newman:92}.\n\n\\section{The solution: the Metastate} \\labsec{the_solution_metastate}\nThe solution to that problem is the concept of Metastate\\index{metastate!Aizenman-Wehr} which was introduced by Aizenman and Wehr in~\\cite{aizenman:90} when studying first-order transitions\\index{phase transition!first order} in general disordered\\index{disorder!systems} systems. Two years later, Newman and Stein introduced this concept of metastate\\index{metastate!Newman-Stein} to solve the \\gls{CSD} problem in the particular case of \\gls{SG}s~\\cite{newman:92}. The concept of the metastate\\index{metastate} \\cite{aizenman:90,newman:92,newman:96,newman:98,newman:03} is just a generalization of the concept of Gibbs state that we have exposed in \\refsec{mixed_pure_states_metastate}. A Gibbs state in a finite system can be regarded as a probability distribution of configurations\\index{configuration} $\\mathcal{S}$, each one with an associated probability given by \\refeq{Gibbs_probability_state}. In the same way, the metastate\\index{metastate}, which we denote as $\\kappa_{\\mathcal{J}}(\\Gamma_\\mathcal{J})$, is a probability distribution over the states $\\Gamma_{\\mathcal{J}}$. \n\nThe reader may notice that the description of the metastate\\index{metastate} concept considers infinite-size states $\\Gamma_{\\mathcal{J}}$, this poses the problem of the construction of that metastate\\index{metastate} from finite-size systems. There exist two definition of metastates\\index{metastate} in the literature that propose the solution to that problem: the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} \\cite{aizenman:90} and the \\gls{NS} metastate\\index{metastate!Newman-Stein} \\cite{newman:92}. Although, it has been argued that both proposals should be equivalent (see \\cite{read:14} for a detailed discussion).\n\n\\subsection{Newman-Stein Metastate}\nWe first introduce the \\gls{NS} metastate\\index{metastate!Newman-Stein}, so-named in \\cite{newman:96}. Consider a growing sequence of $n$ systems of size $L_0<L_1<L_2< \\dots L_n$. For the smallest system of size $L_0$, we fix the couplings\\index{couplings} $\\mathcal{J}$ randomly in the lattice $\\Lambda_{L_0}$ and we define a hypercube of linear size $W < L_0$, usually called \\textit{window}, in the lattice $\\Lambda_W \\subset \\Lambda_L$. \n\nIn this system, we are able to define the Gibbs state $\\Gamma_{L_0,\\mathcal{J}}$ and to restrict this state to the window $W$, obtaining a probability distribution on the spins in $\\Lambda_W$ consisting in $2^{W^d}$ real numbers. This probability distribution is a Gibbs state $\\Gamma_{W,\\mathcal{J}}$. If we consider now the space of finite states $\\Gamma_{W,\\mathcal{J}}$, each time that a specific state $\\Gamma_{W,\\mathcal{J}}$ appears, we can ``record'' it as a Dirac delta $\\delta$ of integral $1\/n$. Repeating this procedure for the $n$ systems would lead to an empirical probability distribution over the space of finite states $\\Gamma_{W,\\mathcal{J}}$.\n\nThen, the limit $n \\to \\infty$ is taken and that empirical distribution may tend to a limit. If this limit exists for every window $W$ as $W \\to \\infty$, then, the resulting distribution of states $\\Gamma_\\mathcal{J}$ is the so-called \\gls{NS} metastate\\index{metastate!Newman-Stein}.\n\n\\subsection{Aizenman-Wehr Metastate} \\labsubsec{aw_metastate}\nWe introduce now the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} that was indeed introduced earlier \\cite{aizenman:90} in the context of first-order transitions\\index{phase transition!first order} but was later related with this concept of metastate\\index{metastate} as a limiting object for quenched-disordered\\index{disorder!quenched} systems \\cite{newman:92}.\n\nThe definition of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} requires the introduction of three length scales $W \\ll R \\ll L$ which will be the linear size of three lattices $\\Lambda_W \\subset \\Lambda_R \\subset \\Lambda_L$ as sketched in \\reffig{sketch_AW_metastate}.\n\n\\captionsetup{justification=centering}\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{metastate\/sketch_AW_metastate.png}\n\\caption[\\textbf{Sketch of the Aizenman-Wehr metastate construction.}]{\\textbf{Sketch of the Aizenman-Wehr metastate\\index{metastate!Aizenman-Wehr} construction.}}\n\\labfig{sketch_AW_metastate}\n\\end{figure}\n\\captionsetup{justification=raggedright}\n\nConsider the lattice $\\Lambda_L$ (with a given boundary conditions\\index{boundary conditions}, usually \\gls{PBC}\\index{boundary conditions!periodic}). This lattice $\\Lambda_L$ is divided into an inner region $\\Lambda_R$ and an outer region $\\Lambda_L \\setminus \\Lambda_R$. Now, consider the state $\\Gamma_{L,\\mathcal{J}}$ for a set of couplings\\index{couplings} $\\mathcal{J}$ in the lattice $\\Lambda_L$. \n\nThe first step in the construction of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is to take an average over the states $\\Gamma_{L,\\mathcal{J}}$. We keep constant all the couplings\\index{couplings} that have both extremes inside $\\Lambda_R$ while changing the rest of them, obtaining a set of states $\\{\\Gamma_{L,\\mathcal{J}_1},\\Gamma_{L,\\mathcal{J}_2},\\dots\\}$. Then, we take the average of this set of states.\n\nFinally, we take the limit of the three-length scales to infinity while respecting the relation $W \\ll R \\ll L$. The obtained distribution of states should not depend on the arbitrary choice of the fixed internal couplings\\index{couplings} and will, hopefully, have a smooth limit: the metastate\\index{metastate}.\n\nThe lattice $\\Lambda_W$ plays a fundamental role in the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} as a \\textit{measuring window}, we are restricting our attention to that hypercube when measuring the relevant quantities in order to avoid boundary effects that may appear as long as $R$ is finite.\n\n\\section{Theoretical pictures in the metastate formalism} \\labsec{pictures_metastate}\nAs we have introduced in~\\refsubsec{theoretical_pictures}, the mathematical concept of metastate\\index{metastate} irrupted in the physics debate of the nature of a \\gls{SG} at finite dimensions. The metastate\\index{metastate} came to the \\gls{SG} field to solve some mathematical ambiguities in the early formalism of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}.\n\nThe introduction of the concept of metastate\\index{metastate}, although, brought also a deep debate about the validity of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} picture. According to \\cite{newman:03}, the existence of states composed of a mixture of infinitely many pure states would not be allowed. The Newman-Stein statement was not fully accepted in mathematical physics and it was debated in subsequent works. For example, see the work of Talagrand~\\cite{talagrand:06}, who demonstrated that Parisi's formula reproduced exactly the free-energy\\index{free energy} for the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, and the work of Read, who questioned the arguments of Newman and Stein to rule out the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture (part IV of \\cite{read:14}).\n\nThis debate leads to the proposal of an alternative picture: the \\textit{chaotic pairs}\\footnote{The term ``chaotic pair'' is intimately related with the droplet\\index{droplet!picture} picture, but, as quoted by Newman and Stein \\cite{newman:03} \\textit{The term 'chaotic pairs' was chosen in reference to spin-symmetric boundary conditions\\index{boundary conditions}, such as periodic. If one considers a fixed boundary condition metastate\\index{metastate}, then it would be more appropriate to refer to this picture as 'chaotic pure states' because the Gibbs state in a typical large volume $\\Lambda_L$ with fixed boundary conditions\\index{boundary conditions} will be (approximately) a single pure state that varies chaotically with L.}}.\n\nHere, we (very) briefly introduce both pictures that together with the classical Droplet\\index{droplet!picture} picture stands as the mathematical consistent formalism to describe the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}.\n\n\\subsection{The RSB metastate}\n \nThe adequacy of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} formalism to the metastate\\index{metastate!RSB} concept lead to the so-baptized as \\textit{nonstandard \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture} \\cite{newman:03}. However, as quoted by \\cite{read:14} we find this name unfortunate because this picture should be the standard in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}. Thus, we adopt the terminology proposed by Read and we call this picture the \\textit{\\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate}\\index{metastate!RSB}. \n\nIn this picture, each of the states that form the metastate\\index{metastate!RSB} is \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}-like, that is, each state is composed of a hierarchical structure of pure states and presents all the properties exposed in~\\refsubsec{theoretical_pictures}. Moreover, the metastate\\index{metastate!dispersed} is \\textit{disperse} i.e. the metastate\\index{metastate!dispersed} is composed of infinitely many states. This dispersion has, indeed, deep consequences. For instance, the disperse metastate\\index{metastate!dispersed} is responsible for the lack of the self-averaging\\index{self-averaging!non-} property in certain quantities for the \\gls{SG}s in the low-temperature phase\\index{phase!low-temperature\/spin-glass} (see, for example \\cite{binder:86}, for the concept of non-self-average\\index{self-averaging}). The reader may notice that, despite the deep meaning of these results, the study of a finite fixed-size system would not change the predictions of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} because, for that system, a single state is present and we have already stated that, in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} picture, each state is \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}-like.\n\n\\subsection{Chaotic pairs picture}\nChaotic pairs can be regarded as a generalization of the Droplet\\index{droplet!picture} picture in the metastate\\index{metastate!chaotic pairs} formalism, therefore, each of the states composing the metastate\\index{metastate!chaotic pairs} would be trivial i.e. each state will be composed of a single pair of spin-flip related pure states. However, in this picture, the pair of pure states will change chaotically with $L$ due to the chaotic size dependence.\n\n\\section{Observables and predictions} \\labsec{observables_predictions_metastate}\nThe formalism of the metastate\\index{metastate} allows the computation of several quantities that would behave differently in the different theoretical pictures described above. Should we obtain these quantities with enough accuracy, it would be possible to discriminate (at least partially) between those competing pictures. This section is devoted to introducing the observables that we have measured in our numerical simulations and to recalling the predictions made for those observables.\n\n\\subsection{The observables} \\labsubsec{observables_metastate}\nThe overlap\\index{overlap} in the measure window $\\Lambda_W$ is defined taking advantage of the metastate\\index{metastate} setup. Consider one realization of the internal couplings\\index{couplings} $i$ and a pair of external realizations of the outer couplings\\index{couplings} $o$ and $o'$. One possible definition of the overlap\\index{overlap} (that will be useful later) is\n\\begin{equation}\nq^{i;o,o'} = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} s_x^{\\sigma;i,o}s_x^{\\tau;i,o'}  = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} q^{i,o,o'}_x \\, , \\labeq{metastate overlap}\n\\end{equation}\nwhere $\\sigma$ and $\\tau$ denotes replicas\\index{replica} of the system as in \\refeq{def_overlap}.\n\nAt this point, we should introduce a new type of average that will be useful in the study of the metastate\\index{metastate}. We have introduced the notion of metastate\\index{metastate} $\\kappa_{\\mathcal{J}}(\\Gamma_\\mathcal{J})$, which is a distribution over states, and therefore, we have a new kind of average. In addition to the usual thermal average\\footnote{In this chapter, we add the subscript $\\Gamma_{\\mathcal{J}}$ in order to avoid confusion with other averages.} $\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}}$ we can now average over the metastate\\index{metastate!averaged state} $\\left[ \\cdots \\right]_{\\kappa}$. \n\nThose averages can be combined in several ways but one particular combination is of great interest here. The \\gls{MAS} $\\rho$ is defined via the average $\\braket{\\cdots}_{\\rho} = \\left[ \\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}} \\right]_{\\kappa}$. \n\nThe \\gls{MAS} is a mixed Gibbs state that can be decomposed in terms of pure states\\footnote{As stated in \\cite{read:14}, the organization of the pure states in the \\gls{MAS} is not ultrametric, on the contrary to each one of the states that compounds the metastate\\index{metastate} according to the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture.}, as explained in \\refsec {mixed_pure_states_metastate}.\n\nThe main objects in our numerical study will be the \\gls{pdf} of the overlaps\\index{overlap!distribution}\n\\begin{equation}\nP(q) = \\overline{\\left[\\braket{\\delta(q-q^{i;o,o})}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}}\\, , \\labeq{usual_pq}\n\\end{equation}\nwhere we remark that the overlap\\index{overlap} is defined with the same outer part $o$ for both replicas\\index{replica} and $\\overline{\\left(\\cdots\\right)}$ denotes the disorder average\\index{disorder!average}, as usual. This probability density function corresponds to the usual $P(q)$ in \\gls{SG}s because we are forcing the outer couplings\\index{couplings} to be the same, but we can also define a \\index{metastate!averaged state}metastate-averaged $P(q)$ as\n\\begin{equation}\nP_{\\rho}(q) = \\overline{\\left[ \\braket{\\delta(q-q^{i;o,o'})}_{\\Gamma_\\mathcal{J}} \\right]_{\\kappa}} \\, . \\labeq{metastate_pq}\n\\end{equation}\n\nThe \\gls{MAS} spin correlation\\index{correlation function!four point} function is\n\\begin{equation}\nC_{\\rho}(x) = \\overline{\\left[ \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\Gamma_\\mathcal{J}}\\right]^2_{\\kappa}} = \\overline{\\left[ \\braket{q^{i,o,o'}_0 q^{i,o,o'}_x}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}} \\, .\n\\end{equation}\nThis correlation\\index{correlation function!four point} function has been studied in other contexts \\cite{dedominicis:98} (see also \\cite{dedominicis:06,marinari:00}) and it has been found that $C_{\\rho}(x) \\sim \\lvert x \\rvert^{-(d-\\zeta)}$ for large $\\lvert x \\rvert$. The exponent $\\zeta$ has a central importance in the Read's discussion of \\cite{read:14} as we discuss below.\n\nFinally, from this correlation\\index{correlation function!four point} function, we can define\n\\begin{equation}\n\\chi_{\\rho} = \\sum_{x \\in \\Lambda_W} C_{\\rho} = \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\Gamma_\\mathcal{J}}\\right]^2_{\\kappa}} =  \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\braket{q^{i,o,o'}_0 q^{i,o,o'}_x}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}}\\, .\n\\end{equation}\nWe can make some computations with this quantity, by decomposing it into pure states by using \\refeq{decomposition_pure_states}.\n\\begin{equation}\n\\chi_{\\rho} = \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\sum_{\\alpha} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\alpha}\\right]^2_{\\kappa}}  \\, . \\labeq{susc_intermediate}\n\\end{equation}\nIt is easy to prove that, because the pure states exhibit the clustering property $\\braket{s_0 s_x}_\\alpha \\sim \\braket{s_0}_{\\alpha}\\braket{s_x}_{\\alpha}$ as $\\lvert x \\rvert \\to \\infty$ and because as long as $W \\to \\infty$ the dominant terms in the sum are those with large $x$, the expression \\refeq{susc_intermediate} can be written as:\n\\begin{equation}\n\\chi_{\\rho} \\approx \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\sum_{\\alpha,\\beta} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\omega_{\\beta,\\Gamma_{\\mathcal{J}}} \\braket{s^{\\sigma;i,o}_0}_{\\alpha} \\braket{s^{\\tau;i,o'}_0}_{\\beta}  \\braket{s^{\\sigma;i,o}_x}_{\\alpha} \\braket{s^{\\tau;i,o'}_x}_{\\beta}\\right]_{\\kappa}} \\, .\n\\end{equation}\nNow, using the relation $\\int_{-\\infty}^{\\infty} \\delta(x-a) dx = 1$ $\\forall a \\in \\mathbb{R}$ and the definitions of \\refeq{overlap_pure_states} and \\refeq{pq_pure_states} we can rewrite the previous expression \n\\begin{equation}\n\\chi_{\\rho} \\approx W \\int_{-\\infty}^\\infty q^2 P_{\\rho}(q) {\\mathrm{d}} q \\, .\n\\end{equation}\n\nThe quantity $\\chi_{\\rho}$ turns out to be the \\gls{MAS} susceptibility\\index{susceptibility}, and hence, it is related with the variance of the distribution function of the metastate\\index{metastate}. Recalling the long term behavior of the correlation\\index{correlation function!four point} function, the exponent $\\zeta$ emerges again\n\\begin{equation}\n\\chi_{\\rho} \\sim W^\\zeta \\, . \\labeq{susceptibility_scaling}\n\\end{equation}\n\n\\subsection{Theoretical predictions of the metastate} \\labsubsec{theoretical_predictions_metastate}\nAs introduced in \\refsubsec{theoretical_pictures} and in \\refsec{pictures_metastate} we have three mathematical consistent pictures for the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. First, the droplet\\index{droplet!picture} model whose metastate\\index{metastate} is concentrated on a single trivial state\\footnote{Recall that we name \\textit{trivial state} to a mixture of two pure states related by global spin-flip symmetry.}. Second, the chaotic pictures \\cite{newman:92,newman:96,newman:98,newman:03}, predicting a metastate\\index{metastate!chaotic pairs} composed of infinitely many states i.e. disperse metastate\\index{metastate!dispersed}, yet each state is trivial. Finally, the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} \\cite{read:14}, which is disperse and every state drawn from it contains the Parisi hierarchical tree of pure states. There might exist alternatives to these three pictures but are much limited by recent rigorous results \\cite{arguin:15}.\n\nThe Read's proposal \\cite{read:14} to partially discriminate between them is to compute the exponent $\\zeta$ because the logarithm of the number of pure states that one can discriminate in a window $\\Lambda_W$ of linear size $W$ scales as $\\sim W^{d-\\zeta}$. Therefore, $\\zeta<d$ implies an infinitely large number of states as $W \\to \\infty$, which in turn implies a disperse metastate\\index{metastate!dispersed}.\n\n\n\\section{Numerical construction of the metastate} \\labsec{simulation_parameters_metastate}\nWe simulate the \\gls{EA}\\index{Edwards-Anderson!order parameter} model introduced in \\refsubsec{3D_EA_model} for the sizes $L=\\{8,12,16,24\\}$ using Monte\\index{Monte Carlo} Carlo simulations consisting in Metropolis single spin-flip updates and \\gls{PT} exchanges. Due to the \\gls{PT} method, we have a temperature mesh of $n$ different values, however, in this work, we use only the lowest one $T=0.698 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$, well below the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.102(3)$~\\cite{janus:13}.\n\nThe thermalization\\index{thermalization} protocol is explained with great detail in \\refsec{time_scales_eq_chaos} and \\refsec{variational_method_eq_chaos}, and for the largest systems, it required the use of multispin coding (see, \\refch{AP_multispin_coding}). Moreover, deeper explanation of the simulation can be found in~\\refsec{numerical_simulations_eq_chaos}.\n\nOur computation is performed for $N_i = 10$ different realizations of the internal couplings\\index{couplings} in the lattice $\\Lambda_R$ (see \\reffig{sketch_AW_metastate}) and, for each internal realization, we simulate a total of $N_o=128$ different outer-coupling\\index{couplings} realizations. Therefore, we have a total of $N_\\mathcal{J}=1280$ different samples\\index{sample} and for each one, we simulate $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$ different replicas\\index{replica}.\n\nWe take $N_i \\ll N_o$ because we expect all the internal disorder\\index{disorder} to be ``typical''~\\cite{read:14} when computing metastate\\index{metastate!averaged state} averages at $R\\gg 1$. However, we find sizable sample-to-sample fluctuations\\index{sample-to-sample fluctuations} for the system sizes we consider. \n\nNumerically, the averages used in the computation of observables (see \\refsubsec{observables_metastate}) are approached in the following way:\n\\begin{itemize}\n\\item Thermal averages over the Gibbs state, that we denote as $\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}}$ are estimated via Monte\\index{Monte Carlo} Carlo thermal averages $\\braket{\\cdots}_{\\text{MC}}$ in a fixed sample\\index{sample}.\n\\item The metastate\\index{metastate!averaged state} average $\\left[ \\cdots \\right]_{\\kappa}$ is estimated by averaging over the outer disorder\\index{disorder!average} $\\sum_o \\sum_{o'} (\\cdots)\/N^2_o$.\n\\item The disorder\\index{disorder!average} average $\\overline{\\left(\\cdots\\right)}$ is estimated by averaging over the internal disorder\\index{disorder!average} $\\sum_i (\\cdots)\/N_i$.\n\\end{itemize}\nWith these estimations, the observables would be\n\\begin{equation}\nP(q) = \\dfrac{\\sum_i \\sum_o \\braket{\\delta(q-q_{i;o,o})}_{\\text{MC}}}{N_i N_o} \\, ,\n\\end{equation}\n\\begin{equation}\nP_{i,\\rho}(q) = \\dfrac{\\sum_{o,o'} \\braket{\\delta(q-q_{i;o,o'})}_{\\text{MC}}}{N^2_o} \\quad , \\quad P_{\\rho}(q) = \\dfrac{\\sum_i P_{i,\\rho}(q)}{N_i} \\, ,\n\\end{equation}\n\\begin{equation}\nC_{\\rho}(x) = \\dfrac{1}{N_i}\\sum_i \\dfrac{1}{N_o^2}\\sum_{o,o'} \\braket{s_0^{\\sigma;i,o}s_x^{\\sigma;i,o}s_0^{\\tau;i,o'}s_x^{\\tau;i,o'}}_{\\text{MC}} \\, .\n\\end{equation}\nFinally, due to the number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$, it is worthy to note that, for each set of $\\{i,o,o'\\}$ we have $\\ensuremath{N_{\\text{Rep}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace-1)\/2=6$ different estimations of the observables if $o=o'$ and $\\ensuremath{N_{\\text{Rep}}}\\xspace^2=16$ estimations otherwise.\n\n\n\\section{Exploring the numerical metastate} \\labsec{results_metastate}\nIn this section, we will focus on the main results obtained from our work in the construction of the numerical metastate\\index{metastate}. First of all, the exponent $\\zeta$ is computed numerically in the metastate\\index{metastate} setup. Moreover, one of the requisites for the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is the relation of the three-length scales $W \\ll R \\ll L$. We investigate which are the ratios $W\/R$ and $R\/L$ that allow safe computations (given an accuracy level). \n\nFinally, we explore the scaling of the functions $P_{\\rho}(q)$ and $P(q)$ as $W$ grows.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{metastate\/RL_ratio.pdf}\n\\caption[\\textbf{\\boldmath $\\mathbf{R\/L}$ ratio in the metastate.}]{\\textbf{\\boldmath $\\mathbf{R\/L}$ ratio in the metastate\\index{metastate}.} Main plot: The \\gls{MAS} overlap\\index{overlap!distribution} distribution $P_{\\rho}(q)$ with $R=12$ at $T=0.698 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$ shows no statistically significant dependence on the lattice size $L$ (the error bars\\index{error bars}, computed from fluctuations on the inner disorder\\index{disorder}, are shown for only one value of $q$ for the sake of clarity). \\textbf{Insets:} $P_{i,\\rho}(q)$ for two specific configurations\\index{configuration} of the inner disorder\\index{disorder} (the error bars\\index{error bars}, computed from fluctuations on the outer disorder\\index{disorder}, are smaller than the data points).}\n\\labfig{RL_ratio}\n\\end{figure}\n\n\\subsection[The $R\/L$ ratio]{The \\boldmath $R\/L$ ratio}\nIn the main plot of~\\reffig{RL_ratio} we see that the \\gls{MAS} $P_{\\rho}(q)$ measured with $R=12$ and both $L=24$ and $L=16$ are statistically compatible, suggesting that the $R\/L = 3\/4$ is already a safe choice. As we have mentioned before, the error bars\\index{error bars} are quite large due to an unexpected high dependence of $P_{i,\\rho}(q)$ on the internal disorder\\index{disorder} for the values of $W$ and $R$ that we simulate. The reader can check in the inset of~\\reffig{RL_ratio} two examples of $P_{i,\\rho}(q)$, where the error bars\\index{error bars} are smaller than the data point and can appreciate the sample-to-sample\\index{sample-to-sample fluctuations} fluctuation. \n\nAn additional check can be performed by representing the susceptibility\\index{susceptibility} $\\chi_{\\rho}$ against the size of the measuring window $W$ for different values of $L$ by keeping constant $R$, see~\\reffig{RL_WR}. We see that, for $R\/L \\leq 0.75$ the data are statistically compatible, while data with $R\/L = 6\/7$ show significant deviations.\n\nIn view of the previous results, we establish the ratio $R\/L=0.5$ as a safe choice for the following analysis.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{metastate\/RL_WR.pdf}\n\\caption[\\textbf{Susceptibility $\\mathbf{\\chi_{\\rho}}$ for different length scales $\\mathbf{\\{W,R,L\\}}$.}]{\\textbf{Susceptibility\\index{susceptibility} $\\mathbf{\\chi_{\\rho}}$ for different length scales $\\mathbf{\\{W,R,L\\}}$.} The susceptibility\\index{susceptibility} $\\chi_{\\rho}$ is plotted against $W$ for a fixed $R=12$ in three different curves $L=14,16,24$. Deviation from the asymptotic behavior $R\/L \\ll 1$ are evident only for $R\/L>0.75$.} \n\\labfig{RL_WR}\n\\end{figure}\n\n\\subsection[The $W\/R$ ratio]{The \\boldmath $W\/R$ ratio}\nThe susceptibility\\index{susceptibility} scaling with $W$ in the $W\/R \\ll 1$ limit has been already expressed in~\\refeq{susceptibility_scaling}. Therefore, the~\\reffig{susceptibility_scaling} gives us relevant information about the validity range for $W\/R$. We note that the expected power-law behavior in the $W\\ll R$ limit actually extends up to $W\/R \\approx 0.75$, where corrections to the asymptotic power-law appear. Therefore, similarly to the $R\/L$ case, the $W\/R \\approx 0.75$ stands as a safe choice.\n\n\\subsection{The exponent \\boldmath $\\zeta$}\nIn this section, we fix $R=L\/2$ (which is in the safe side, given our bound $R< 3L\/4$) and $W\/R \\approx 0.75$. We can now compute the exponent $\\zeta$ by taking advantage of the relation~\\refeq{susceptibility_scaling}. Fitting the data with $W\/R \\leq 0.75$ we found $\\zeta = 2.3 \\pm 0.3$. Moreover, we are concerned about the finite-size effects\\index{finite-size effects} that our small-lattices simulations could suffer.\n\nA finite-size scaling~\\cite{cardy:12}\\index{finite size scaling} is needed to study the size effects\\index{finite-size effects}. Indeed, we expect for finite $R$ and $W$ a scaling behavior\n\\begin{equation}\n\\chi_\\rho(W,R) = R^\\zeta f(W\/R)\\, , \\labeq{R-finite}\n\\end{equation}\nwhich is compatible to~\\refeq{susceptibility_scaling} for $f(x) \\propto x^\\zeta$ in the $x\\to 0$ limit. \\refeq{R-finite} is expected to be exact only in the limit of large $W$ and $R$~\\cite{cardy:12}, hence one needs to check for size corrections. We do so with the quotients method\\index{quotients method}~\\cite{nightingale:76,ballesteros:96,amit:05} which produces effective $\\zeta$ estimates at a well defined length scale. The size dependence can be assessed later on. Specifically, we take two sizes-pairs $(W_1,R_1)$, $(W_2,R_2)$ with the same value of $W\/R$, which ensures the cancellation of scaling functions in the quotient\n\\begin{equation}\n\\frac{\\chi_\\rho(W_2=x R_2,R_2)}{\\chi_\\rho(W_1=x R_1,R_1)} = \\frac{\\left(W_2\/x\\right)^\\zeta f(x) }{\\left(W_1\/x\\right)^\\zeta f(x)}=\\left(\\frac{W_2}{W_1}\\right)^\\zeta\\, . \\labeq{quotients}\n\\end{equation}\n\nThe resulting determination of $\\zeta$, see \\reftab{zeta_exponent_metastate}, is fully compatible with the computed result $\\zeta=2.3 \\pm 0.3$. Furthermore, no significant size-dependence emerges\nfrom \\reftab{zeta_exponent_metastate}. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{cclc}\n \\toprule \n \\toprule\n$W\/R$ & $L\/R$ & $(W_1,W_2)$ & $\\zeta^\\mathrm{eff}$\\\\\\hline\\hline\n1\/2 & 2 & (4,6) & 2.18(40)\\\\\\hline\n2\/3 & 2 & (4,8) & 2.59(22)\\\\\\hline\n1   & 2& (8,12) & 2.37(26)\\\\\n    &  &(6,8)  & 2.14(37)\\\\\n    &  &(6,12) & 2.28(18)\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Effective $\\mathbf{\\zeta}$ exponent.}]{\\textbf{Effective $\\mathbf{\\zeta}$ exponent.} The effective $\\zeta$ exponent depends on the two lengths $W_1$ and $W_2$ and\non the ratio $W_1\/R_1=W_2\/R_2$. }\n\\labtab{zeta_exponent_metastate}\n\\end{table}\n\nBesides, in~\\reffig{susceptibility_scaling}, the \\gls{MAS} susceptibility\\index{susceptibility} has been rescaled by using the previously defined scaling relations. A power-law behavior is exhibited for $W\/R <0.75$ as expected and our $\\zeta$ estimation interpolates the data nicely in that region.\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\columnwidth]{metastate\/susceptibility_scaling}\n\\caption[\\textbf{Scaling behavior of the \\gls{MAS} susceptibility.}]{\\textbf{Scaling behavior of the \\gls{MAS} susceptibility\\index{susceptibility}.} \\gls{MAS} susceptibility\\index{susceptibility} data measured with fixed $R\/L=1\/2$ at $T=0.698 \\approx 0.64 \\ensuremath{T_\\mathrm{c}}\\xspace$ as a function of the $W\/R$ ratio.}\n\\labfig{susceptibility_scaling}\n\\end{figure}\n\n\\subsection{Size dependence of \\boldmath $P(q)$ and \\boldmath $P_{\\rho}(q)$}\nWe show in~\\reffig{pq_metastate}, for $L=24$ and $R\/L=1\/2$ the dependence of the functions $P_{\\rho}(q)$ and $P(q)$ on $W$. The expectation for a dispersed metastate\\index{metastate!dispersed}~\\cite{read:14} is that both distributions are different in the thermodynamic limit\\index{thermodynamic limit}. We found here that they are distinct objects even for moderate sizes of $W$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\columnwidth]{metastate\/pq.pdf}\n\\caption[\\textbf{Size dependence of $\\mathbf{P(q)}$ and $\\mathbf{P_{\\rho}(q)}.$}]{\\textbf{Size dependence of $\\mathbf{P(q)}$ and $\\mathbf{P_{\\rho}(q)}.$} Functions $P_{\\rho}(q)$ and $P(q)$ for $L=24$, $R=L\/2$ and $T=0.698 \\approx 0.64 \\ensuremath{T_\\mathrm{c}}\\xspace$. Different panels corresponds to different measuring window size $W=4,8,12$.}\n\\labfig{pq_metastate}\n\\end{figure}\n\n\\section{Relating numerical results and theory} \\labsec{relating_numerical_theory_metastate}\nTo the best of our knowledge, this is the first numerical construction of the metastate\\index{metastate} carried out in equilibrium simulations. Nonetheless, there exists one previous study in the non-equilibrium regime~\\cite{manssen:15}, however, a deep relation between this \\textit{dynamic metastate}\\index{metastate!dynamic} and the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is still to be explored. \n\nWe have shown that the actual state of the art in the numerical \\gls{SG}s allows the simulation of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} in the \\gls{EA}\\index{Edwards-Anderson!order parameter} for $d=3$. We cannot extrapolate safely to the thermodynamic limit\\index{thermodynamic limit} and the unexpected dependence of the \\gls{MAS} on the inner disorder\\index{disorder} is still important at the accessible system sizes. Nevertheless, we have found a convincing scaling law for the \\gls{MAS} susceptibility\\index{susceptibility} and we have estimated the exponent $\\zeta(d=3)=2.3 \\pm 0.3$, which strongly suggest $\\zeta < d$, in addition, we have found no substantial size-dependence for this exponent.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{metastate\/zeta_exponent.pdf}\n\\caption[\\textbf{The exponent \\boldmath $\\zeta$ as a function of  $d$.}]{\\textbf{The exponent \\boldmath$\\zeta$ as a function of $d$.} Different predictions of the exponent $\\zeta$ for $d=3$ and $d=4$ are plotted. Above $d=6$, the mean-field\\index{Mean-Field!solution} solution $\\zeta=4$ is correct. The line $\\zeta=d$ separates the disperse metastate\\index{metastate!dispersed} for the trivial one.}\n\\labfig{zeta_exponent_metastate}\n\\end{figure}\n\nIn~\\reffig{zeta_exponent_metastate} we have summarized our knowledge about the $\\zeta$ exponent. Below the lower critical dimension\\index{critical dimension!lower}\\footnote{The lower critical dimension\\index{critical dimension!lower} is the dimension below which there is no phase transition\\index{phase transition} at finite temperature $T$.} $d_\\mathrm{L}$ (at zero magnetic-field), the droplets\\index{droplet!picture} picture is expected to be valid, and, therefore, the exponent $\\zeta$ should be $\\zeta \\geq d$. In this work we have found $\\zeta = 2.3 \\pm 0.3$ (green-squared point in~\\reffig{zeta_exponent_metastate}). Moreover, alternative estimations of the $\\zeta$ exponent come from the decay of the four-spins spatial correlation\\index{correlation function!four point} function at equilibrium.\n\nIn the equilibrium systems, the spins configurations\\index{configuration} of different replicas\\index{replica} follow a distribution function $P(q)$ (see~\\refsubsec{theoretical_models}) and, therefore, the $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ has contributions of pair of replicas\\index{replica} with all the possible values of $q$, weighted with the $P(q)$. However, the metastate\\index{metastate} is in the $q=0$ by definition, and therefore, in order to compare both determinations, we have to restrict the computation of the correlation\\index{correlation function!four point} functions to the zero overlap\\index{overlap!zero sector} sector in the equilibrium results.\n\nThe four-point correlation\\index{correlation function!four point} function conditioned to the $q=0$ sector is\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace | q=0) \\sim \\lvert x \\rvert^{-\\vartheta} \\, ,\n\\end{equation}\nwith $\\vartheta=d-\\zeta_{q=0}$, see~\\refeq{long_distance_C4}. Previous studies found $\\zeta_{q=0}(d=3) = 2.62 \\pm 0.02$~\\cite{janus:09b,janus:10} and $\\zeta_{q=0}(d=4) = 2.62 \\pm 0.02$~\\cite{nicolao:14} (blue circles in~\\reffig{zeta_exponent_metastate}). \n\nFinally, in $d\\geq6$, where the mean-field\\index{Mean-Field} computations are correct, we found $\\zeta=4$~\\cite{dedominicis:98,dedominicis:99}. A gentle extrapolation with the values of $\\zeta_{q=0}(d=3,4)$ and the value of $z(d=6)=4$ (dashed line in~\\reffig{zeta_exponent_metastate}) seems to meet, as expected, the yellow line corresponding to $d=\\zeta$ around $d=2.5$, which is a general accepted estimation for the lower critical dimension\\index{critical dimension!lower} from numerical and experimental results (see e.g.~\\cite{franz:94,boettcher:04,boettcher:04b,boettcher:05,guchhait:14,maiorano:18}).\n\nAs we have previously discussed in~\\refsubsec{theoretical_predictions_metastate}, the exponent $\\zeta$ is related with the number of states that can be discriminated in a measuring window of size $W$, scaling that number with $\\sim W^{d-\\zeta}$~\\cite{read:14}. This numerical estimation of $\\zeta$ for $d=3$ supports the pictures of the metastate\\index{metastate} with infinitely many states, namely \\gls{RSB}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} and chaotic pairs\\index{metastate!chaotic pairs}.\n\n\n\n\n\n\n\n\n\\chapter[The Mpemba effect]{The Mpemba effect} \\labch{mpemba}\n\n\\epigraph{\\textit{Aqu\u00ed el que no carneguea, borreguea.}}{-- Acervo popular}\n\nConsider two beakers of water that are identical to each other except for the fact that one is hotter than the other. If we put both of them in contact with a thermal reservoir (for example, a freezer) at some temperature lower than the freezing point of the water, under some circumstances, it can be observed that the, initially, hotter water freezes faster than the colder one. This phenomenon is known as the Mpemba\\index{Mpemba effect} effect~\\cite{mpemba:69}.\n\nThe history of this phenomenon is, indeed, very curious and constitutes one paradigmatic example of the importance of the scientific method in the development of science. Although the first written record comes from Aristotle, it would be probably a well-known fact for most of the people~\\cite{aristotle-lee:89}. This effect was sporadically mentioned through the ages~\\cite{bacon-burke:62,bacon:11,descartes:65} but it received little attention from the scientific community until the second half of the XX century.\n\nIn 1969 this phenomenon was brought back to the scientific debate by Erasto Mpemba, a young student in Tanzania, and Denis Osborne, a teacher of the University College Dar es Salaam, Tanzania~\\cite{mpemba:69}. The same year, Dr. Kell reported the same phenomenon in an independent publication~\\cite{kell:69}. \n\nDifferent arguments were given to explain this phenomenon~\\cite{kell:69,deeson:71,firth:71,walker:77}, but there is no consensus neither in the explanations~\\cite{osborne:79,freeman:79,wojciechowski:88} nor in the very existence of the effect.~\\cite{burridge:16}. We will briefly discuss the situation later in \\refsec{water_complicated_mpemba}.\n\nThis phenomenon is not specific to water and has been reported in other systems like nanotube resonators~\\cite{greaney:11}, clathrate hydrates~\\cite{ahn:16}, granular fluids~\\cite{lasanta:17} and colloidal systems~\\cite{kumar:20}. This chapter is devoted to discussing the Mpemba\\index{Mpemba effect} effect in \\gls{SG}s. Besides, the Mpemba\\index{Mpemba effect} effect constitutes a great example to stress the importance of the coherence length\\index{coherence length} as a fundamental quantity to describe the off-equilibrium phenomena in \\gls{SG}s.\n\nWe begin with a brief historical introduction\\footnote{The reader may consult also a fantastic historical review in~\\cite{jeng:06}.} to the phenomenon in \\refsec{historical_introduction_mpemba} and with some of the proposed explanations in~\\refsec{water_complicated_mpemba}. Then, we explain the numerical simulation, performed in the Janus\\index{Janus} II custom-built computer, that has allowed the study of the Mpemba\\index{Mpemba effect} effect in \\gls{SG} (see~\\refsec{numerical_simulation_mpemba}). At this point, we are ready to discuss the results. We first identify in \\refsec{identifying_mpemba} the Mpemba\\index{Mpemba effect} effect in \\gls{SG} by choosing the adequate quantity to represent the \\textit{temperature} of our system. We found in \\refsec{coherence_length_mpemba} that the quantity controlling the phenomenon is, indeed, the coherence length\\index{coherence length} $\\xi$. Finally, we study the Inverse Mpemba\\index{Mpemba effect} effect in \\refsec{inverse_mpemba}.\n\nThe results described in this chapter were published in~\\cite{janus:19}.\n\n\\section{A historical introduction}\\labsec{historical_introduction_mpemba}\n\nThe first record of the Mpemba\\index{Mpemba effect} effect is attributed to Aristotle in his \\textit{Metereologica} around 350 B.C.~\\cite{aristotle-lee:89}. His discussion there, suggests that the phenomenon was a well-known fact and he used it as an example to illustrate his theory of \\textit{antiperistasis}\\footnote{The concept of antiperistasis refers to the reaction between two opposite \\textit{forces}, when one increases, the other have to do it.}:\n\n\\textit{``If the water has been previously heated, this contributes to the rapidity with which it freezes [sic] for it cools more quickly (Thus so many people when they want to cool water quickly first stand it in the sun and the inhabitants of Pontus when they encamp on the ice to fish --they catch fish through a hole which they make in the ice-- pour hot water on their rods because it freezes quicker, using the ice like solder to fix their rods.) \\dots ''}\n\\\\[5pt]\n\\rightline{{ --- Aristotle, Metereologica Book I, Chapter XII}}\n\nThe Mpemba\\index{Mpemba effect} effect probably remained in the popular heritage through the centuries, but actually, did not receive too much attention from academics. Yet, it was mentioned in some important texts, for example in the \\textit{Opus Majus} of Roger Bacon~\\cite{bacon-burke:62} or in the \\textit{Novum Organum} of Francis Bacon~\\cite{bacon:11}. \n\nSome years after Francis Bacon mentioned the phenomenon, Descartes wrote about it in \\textit{Les Meteores}~\\cite{descartes:65}. Indeed, he stressed the importance of the experiments, and actually, he proposed a specific experiment that is not exactly the standard Mpemba effect (which is the most commonly studied). \n\nDescartes proposed to fill a beaker (and he also specified that should have a long straight neck) with hot water that has been kept over the fire for a long time, then the water should be let to reach room temperature. He proposed to do the same with another beaker of water but now, without boiling it. Then, both beakers should be put in contact with the [sic] ``freezing cold air'' and one observes that the beaker which has held for a long time over the fire, freezes first.\n\nHe stated in a letter to Mersenne (1638) that he performed that experiment and he defended that there was nothing incorrect in his methods. \n\nHowever, the Mpemba\\index{Mpemba effect} effect was relegated to oblivion in mainstream physics by the emergence of thermodynamics, which was supported by an unprecedented success, both in describing reality and in the creation of modern machines. Apparently, the Mpemba\\index{Mpemba effect} effect contradicts the knowledge provided by thermodynamics. It has been suggested~\\cite{kuhn:12} that the theoretical views of scientists may condition the experiments that they decide to perform, and this could be the answer to the small numbers of experiments researching this interesting effect before the XX century. Actually, this points also to one of the weaknesses of the human being, that affects the application of the scientific method: the confirmation bias.\n\nIn 1963, one young student of a secondary school in Tanzania, Erasto Mpemba, put this phenomenon under the scrutiny of the scientific community~\\cite{mpemba:69}. In his school, he used to make ice-cream by boiling milk, mixing it with sugar, and by putting the mix into the freezer. One day, some students were doing ice-cream but the space in the freezer was scarce. Despite the warnings for not introducing the hot mix directly in the freezer because that could damage it, Mpemba decided to do it anyway in order to do not lose his space in it.\n\nHe observed that his mix had frozen before that of other boys who had followed the \\textit{standard protocol} and had let the mix cool at the room's temperature before introducing it in the freezer. Persistent questions of Mpemba to different physics teachers about this fact led to the same answer ``That is impossible'', one of the teachers said that ``That is Mpemba's physics, not universal physics''.\n\nNonetheless, Mpemba did not surrender and he took the opportunity to ask this question to a university professor, Denis Osborne, that went to Mpemba's High School to give a talk. From the first time that Mpemba observed the phenomenon, he did great advances on building up a specific protocol to develop the experiment in a reproducible way and he asked a very concrete question:\n\n\\textit{``If you take two beakers with equal volumes of water, one at 35\u00baC and the other at 100\u00baC, and put them into a freezer, the one that started at 100\u00baC freezes first. Why?''}~\\cite{mpemba:69}.\n\nFortunately, Osborne did not dismiss the Mpemba's claim although he confessed that he thought at the first moment that Mpemba was wrong. Indeed, he asks a technician to make the experiment. The result was the following~\\cite{mpemba:69}:\n\n\\textit{``The technician reported that the water that started hot did indeed freeze first and added in a moment of unscientific enthusiasm: But we'll keep on repeating the experiment until we get the right result.''}\n\nOf course, further tests led to the same results and they started to think about an explanation that we will briefly sketch in the next section. The same year of the publication of the article of Mpemba and Osborne, Dr. Kell in Canada reported the same experiment~\\cite{kell:69}.\n\nA curious fact is that this phenomenon, though very far from academic physics, was conserved in the popular heritage through the years. Indeed, Mpemba remembers that the ice-cream makers in his town were aware of this effect and they used it commonly to obtain their ice-cream faster~\\cite{mpemba:69}.\n\nSeveral explanations came up in the following years (see~\\refsec{water_complicated_mpemba}) and the phenomenon was found to be not exclusive from water. Indeed, it was found in nanotube resonators~\\cite{greaney:11}, clathrate hydrates~\\cite{ahn:16}, granular fluids~\\cite{lasanta:17}, colloidal systems~\\cite{kumar:20} and, as we will expose in this chapter, in spin glasses~\\cite{janus:19}.\n\n\\section{Water is too complex} \\labsec{water_complicated_mpemba}\nAfter the work of Mpemba and Osborne~\\cite{mpemba:69}, several explanations were proposed, however, one of the main difficulties in the study of the Mpemba\\index{Mpemba effect} effect is the high number of parameters that may affect the phenomenon. The shape of the beaker, the composition of the water, its temperature distribution \\dots All of these parameters might play an important role in the explanation of the effect and should be taken into account.\n\nThe mass of the water may be one of these parameters that could (at least partially) explain the effect. The hotter system shall lose more mass due to evaporation processes than the colder one and it has been suggested~\\cite{kell:69} that this would be the reason for the Mpemba\\index{Mpemba effect} effect to occur. However, other experiments claimed that this mass loss would be insufficient to explain Mpemba\\index{Mpemba effect} effect~\\cite{osborne:79,freeman:79,wojciechowski:88}.\n\nAnother parameter that may play a role in the phenomenon is the temperature distribution of the water. Cold water is denser than hot water\\footnote{Above 4\u00baC.} and, when preparing the hot beaker, if the heating process is not uniform that may induce convection currents in it. Due to those convection currents and the different densities of the water as a function of the temperature, the top part of the hot beaker would be at a lower temperature than the bottom part. This could favor the creation of a layer of ice on the top of the hot beaker before than in the colder one. Besides, the convection currents may work together with other factors, like the above-mentioned evaporation, to provoke the Mpemba\\index{Mpemba effect} effect in the water. These convection currents would in turn be affected by other parameters like the shape of the beaker. Actually, experiments in which the hotter beaker was stirred in order to make the temperature gradient disappear, showed a sizable raise of the time of freezing~\\cite{deeson:71}.\n\nThe above examples are only two of the variety of explanations proposed for the Mpemba\\index{Mpemba effect} effect. However, the situation is far from clear and there exist experiments claiming the nonexistence of the phenomenon, see for example~\\cite{burridge:16}.\n\nIt is clear that the complexity of water makes it too difficult to study the Mpemba\\index{Mpemba effect} effect and it would be desirable to have a much better-controlled system to study the phenomenon. Here, we take advantage of the numerical simulations in \\gls{SG}s to address the Mpemba\\index{Mpemba effect} effect and studying it with total control of the system.\n\n\\section{Numerical simulation}\\labsec{numerical_simulation_mpemba}\nIn this work, we use the same simulations performed for the study of the aging\\index{aging!rate} rate (\\refch{aging_rate}) and we add some simulations with temperature-varying\\index{temperature-varying protocol} protocols. We briefly remind here the parameters of the simulation for the reader's convenience.\n\nWe simulate in the \\gls{FPGA}-based\\index{FPGA} computer Janus\\index{Janus} II an \\gls{EA}\\index{Edwards-Anderson!model} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. \n\nWe shall perform two different protocols. The isothermal\\index{isothermal} protocol consists of a direct quench from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds to the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. To uniquely identify this protocol, it is enough to label it with its temperature $T$.\n\nThe temperature-varying\\index{temperature-varying protocol} protocol begins in the same way that the isothermal\\index{isothermal} one, by quenching the system from configurations\\index{configuration} of randomly initialized spins to the working temperature $T_1$. When the system reaches a certain coherence length\\index{coherence length} $\\xi'(\\ensuremath{t_\\mathrm{w}}\\xspace)$ we change the temperature of the thermal reservoir to a temperature $T_2$. Hence, this protocol should be labeled with a pair of temperatures (the initial one $T_1$ and the final one $T_2$), and with the coherence length\\index{coherence length} at which the temperature-change was produced $\\xi'(\\ensuremath{t_\\mathrm{w}}\\xspace)$. We use the following notation $T_1,\\xi' \\to T_2$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample}, we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ replicas\\index{replica}. As said in~\\refch{aging_rate}, this simulation had the original aim to study the temperature chaos\\index{temperature chaos} phenomenon under non-equilibrium conditions (see~\\refch{out-eq_chaos}), however, the reader may notice that in that study we use a total number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$. Indeed, this study about the Mpemba\\index{Mpemba effect} effect was performed much earlier and we had at our disposal ``only'' $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$.\n\nThe main observables of this study are the coherence length\\index{coherence length} $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ and the energy\\index{energy!density} density $e(t)$. The coherence length\\index{coherence length} is estimated by $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, computed from integral estimators of the correlation function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. These two observables have been described with great detail in \\refsubsec{observables_introduction}. The energy\\index{energy!density} density is defined as\n\\begin{equation}\ne(t,{\\mathcal{J}}) = \\dfrac{1}{L^3} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\quad  \\, \\quad e(t) = \\overline{e(t,\\mathcal{J})} \\, , \\labeq{energy-density_definition}\n\\end{equation}\nwhere $\\overline{(\\cdots)}$ is the exact mean over the disorder\\index{disorder!average}. Although this estimation is perfectly correct, we decided to increase the accuracy of our estimations by using a control variate\\index{control variate}~\\cite{fernandez:09,ross:14} ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ depending on the sample\\index{sample} $\\mathcal{J}$ and with an exact disorder\\index{disorder!average} average $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$. \n\nThe studied quantity would be now\n\\begin{equation}\n\\tilde{e}(t,\\mathcal{J}) = e(t,{\\mathcal{J}}) - \\left[ {\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} - \\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}} \\right] \\quad \\, \\quad \\tilde{e}(t) = \\overline{\\tilde{e}(t,\\mathcal{J})} \\, . \\labeq{energy-density_control_variate}\n\\end{equation}\n\nThis new quantity $\\tilde{e}(t,\\mathcal{J})$ has the same disorder\\index{disorder!average} mean that the usual energy-density\\index{energy!density} $e(t,\\mathcal{J})$ but has a significantly lower variance. The reader may find further details of the implementation of the control variate\\index{control variate} in \\refsec{improving_statistics}.\n\n\\section{Identifying the Mpemba Effect}\\labsec{identifying_mpemba}\nThe aim of identifying the Mpemba\\index{Mpemba effect} effect in \\gls{SG} is composed of two main tasks. Firstly, we have to identify what ``temperature'' means in an out-of-equilibrium \\gls{SG}. Secondly, we have to establish a protocol to mimic the traditional protocol of the classic Mpemba\\index{Mpemba effect} effect. \n\nThe natural candidate to take the place of the temperature, which is telling us if a system is hotter than another, is the energy-density\\index{energy!density} $e(t)$ [or equivalently, $\\tilde{e}(t)$] because it is the observable conjugated with (the inverse of the) temperature. Furthermore, at equilibrium, where the temperature $T$ of the thermal reservoir corresponds to the temperature of the system by definition, there exists a monotonically increasing correspondence between the energy-density\\index{energy!density} and the temperature.\n\nThe protocol followed in our numerical experiment strongly resembles the original Mpemba's protocol~\\cite{mpemba:69}. We study the evolution of three different off-equilibrium systems. The first one is quenched from infinite temperature (random configuration\\index{configuration}) to a temperature (of the thermal reservoir) $T_1=1.3$, which is above the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.102(3)$~\\cite{janus:13}. This system is labeled with the number $1$. We let it evolve until it reaches an energy\\index{energy!density} $\\tilde{e}_1(t=0)\\approx -1.6428$ and we set this time as our starting point of the (numerical) experiment. \n\nThe second system (labeled with the number $2$) is prepared in a similar way but the temperature of the reservoir is now $T_2=1.2$ and we let the system reach a much lower energy\\index{energy!density} $\\tilde{e}_2(t=0)\\approx -1.6714$. \n\nIn the last one, labeled with the number $3$, the temperature of the reservoir is again $T_3=1.2$ but the starting point is at even lower energy\\index{energy!density} $\\tilde{e}_3(t=0) \\approx -1.6738$. \n\nAt that point, we quenched the three systems to a temperature $T_{\\mathrm{f}} =0.7 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$, we let them evolve and we record their energies. The results are shown in \\reffig{first_mpemba} where we can appreciate the classical Mpemba\\index{Mpemba effect} effect. The \\textit{hot start} system (system 1) crosses the other two curves in a way indicating a faster cooling process.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/first_mpemba.pdf}\n\\caption[\\textbf{Classical Mpemba protocol}]{\\textbf{Classical Mpemba protocol}. We show the time evolution of the energy\\index{energy!density} of spin-glass systems initially prepared at a higher temperature ($T=1.3$, yellow line) or a lower temperature ($T=1.2$, blue and green lines), but always in the paramagnetic (high-temperature) phase\\index{phase!high-temperature\/paramagnetic} ($T_\\text{c}\\approx1.102$). In all three cases, the systems are initially left to evolve out of equilibrium until they reach the internal energies shown in the figure key. At $t=0$ all preparations are quenched, that is, put in contact with a thermal reservoir at temperature $T=0.7\\approx0.64 T_\\text{c}$.  As discussed in the text, the instantaneous energy-density\\index{energy!density} is a measure of the (off-equilibrium) sample\\index{sample} temperature. In agreement with the original Mpemba experiment~\\cite{mpemba:69}, the system originally at the higher energy\\index{energy!density} cools faster. \\textbf{Bottom left inset:} Closeup of the first crossing between energy\\index{energy!density} curves, showing the very small error bars\\index{error bars}, equal to the thickness of the lines. \\textbf{Top right inset:} Closeup of the second crossing between energy\\index{energy!density} curves.}\n\\labfig{first_mpemba}\n\\end{figure}\n\nThe reader should notice that the crossing of $\\tilde{e}_1(t)$ and $\\tilde{e}_2(t)$ takes place at much longer times that the crossing of $\\tilde{e}_1(t)$ and $\\tilde{e}_3(t)$, even if the initial energies of $\\tilde{e}_2(t)$ and $\\tilde{e}_3(t)$ differ only by a $0.15\\%$. We need a control parameter that helps us to quantitatively characterize the Mpemba\\index{Mpemba effect} effect.\n\n\\section{Coherence length controls the Mpemba Effect in spin glasses}\\labsec{coherence_length_mpemba}\nThe natural candidate, that characterizes the dynamical state of an off-equilibrium \\gls{SG}, is the coherence length\\index{coherence length} $\\xi(t)$. Indeed, in terms of the coherence length\\index{coherence length} $\\xi(t)$ our three systems are very different. The \\textit{hot start} system [$T_1=1.3$ and $\\tilde{e}_1(t=0)\\approx -1.6428$] has $\\xi_1(t=0)=12$, the \\textit{cold start} system [$T_2=1.2$ and $\\tilde{e}_2(t=0)\\approx -1.6714$] has $\\xi_2(t=0)=5$ and the \\textit{colder start} system [$T_3=1.2$ and $\\tilde{e}_3(t)\\approx -1.6738$] has $\\xi_3(t=0)=8$.\n\nThis perspective is pointing us that the out-equilibrium \\gls{SG}s for the study of the Mpemba\\index{Mpemba effect} effect should not be labeled only with the temperature of the thermal reservoir, not even with the temperature of the thermal reservoir plus the energy density\\index{energy!density} at some time $t$. We need the coherence length\\index{coherence length} to fully characterize the state of the system and understand the Mpemba\\index{Mpemba effect} effect in \\gls{SG}s. Next, we test this hypothesis.\n\n\\subsection{A first test} \\labsubsec{first_test_mpemba}\nIf our hypothesis is correct, crossing the critical\\index{critical temperature} temperature should not matter in order to observe the Mpemba\\index{Mpemba effect} effect. We only require that both starting points fulfill the next conditions: $T_{\\mathrm{A}} > T_{\\mathrm{B}}$ and $\\xi_{\\mathrm{A}} > \\xi_{\\mathrm{B}}$.\n\nTo test this hypothesis we focus on the low-temperature phase\\index{phase!low-temperature\/spin-glass}. We set the final temperature $T_\\mathrm{f}=0.7$ and we simulate $4$ different systems, $3$ of them with the temperature-varying\\index{temperature-varying protocol} protocol described in \\refsec{numerical_simulation_mpemba} and the other with the isothermal\\index{isothermal} protocol. We use the temperature-varying\\index{temperature-varying protocol} notation also for the isothermal\\index{isothermal} protocol in this case to stress that we set the time $t=0$ at a given coherence length\\index{coherence length} $\\xi$. The protocols are\n\\begin{itemize}\n\\item \\textbf{Preparation A:} $T=0.7, \\xi=6 \\to T=0.7$ (this is the isothermal\\index{isothermal} protocol).\n\\item \\textbf{Preparation B:} $T=0.9, \\xi=5 \\to T=0.7$.\n\\item \\textbf{Preparation C:} $T=0.9, \\xi=8 \\to T=0.7$.\n\\item \\textbf{Preparation D:} $T=0.9, \\xi=15 \\to T=0.7$.\n\\end{itemize}\n\nThe time at which we quench the four systems to $T=0.7$ (in the case of the isothermal\\index{isothermal} protocol simply corresponds to the time at which the system reaches $\\xi=6$) will be $t=0$. Then, we plot $\\tilde{e}(t)$ against time for the four preparations and we show the results in \\reffig{test_mpemba}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/test_mpemba.pdf}\n\\caption[\\textbf{Mpemba effect in the spin-glass phase.}]{\\textbf{Mpemba\\index{Mpemba effect} effect in the spin-glass phase\\index{phase!low-temperature\/spin-glass}.} As in \\reffig{first_mpemba}, but all four initial preparations are now carried out in the spin-glass phase\\index{phase!low-temperature\/spin-glass} ($T<T_\\text{c}$). The preparation that cools faster is not the initially coldest one, but the one with the largest initial coherence length\\index{coherence length}. \\textbf{Inset:} A zoom of the \\emph{second} crossing point between the curves for preparations $(T=0.9,\\xi=8$; green curve) and $(T=0.7,\\xi=6$; yellow curve). This second crossing is not the Mpemba\\index{Mpemba effect} effect. Rather, the Mpemba\\index{Mpemba effect} effect arises at the \\emph{first} crossing at $t\\approx 5\\times 10^4$. The second crossing disappears if one plots parametrically $\\tilde{e}(t)$ as a function of $\\xi(t)$.}\n\\labfig{test_mpemba}\n\\end{figure}\n\n\nBefore initiating our discussion, we want to clarify the figure. The reader may wonder about the peculiar behavior of the curve belonging to the isothermal\\index{isothermal} protocol. This peculiar behavior is just an optical artifact. Indeed, the times at which we record the configuration\\index{configuration} of our system and measure the relevant quantities are exponential ($t=\\Floor{2^{n\/8}}$, with $n$ integer). Thus, for $T=0.7,\\xi=6$ the measuring times are quite far from each other and the first point roughly corresponds to $t\\approx 7 \\cdot 10^5$. However, between two consecutive measurements, the change of the energy-density\\index{energy!density} is very small and that explains the almost horizontal line.\n\nWe observe that preparations at initial temperature $T=0.9$ with coherence length\\index{coherence length} $\\xi>6$, namely preparation C and preparation D, cool faster than preparation A. Indeed, we observe the Mpemba\\index{Mpemba effect} effect for preparation D at time $t\\approx 10^3$, when it crosses the isothermal\\index{isothermal} preparation. For preparation C we observe the same effect at time $t \\approx 5 \\cdot 10^4$. Nevertheless, for preparation B we observe no Mpemba\\index{Mpemba effect} effect.\n\nIt is worthy to mention that a second crossing can be observed for higher times $t\\approx 5\\cdot 10^8$. This crossing does not correspond to the Mpemba\\index{Mpemba effect} effect. Actually, we will observe that this crossing disappears under an appropriate representation.\n\n\n\\subsection[The $\\tilde{e}-\\xi$ phase-diagram]{The \\boldmath $\\tilde{e}-\\xi$ phase-diagram}\\labsubsec{e_xi_phase_diagram_mpemba}\n\nAlthough we have identify the coherence length\\index{coherence length} as the hidden parameter controlling the Mpemba\\index{Mpemba effect} effect, we need to explore the relation between them to make our interpretation quantitative. Numerical and heuristic arguments \\cite{marinari:96,parisi:97,janus:09b} suggest\n\\begin{equation}\n\\tilde{e}(t) = \\tilde{e}_{\\infty}(T) + \\dfrac{e_1}{\\xi^{d_\\mathrm{L}}(t)} + \\cdots \\, , \\labeq{energy_coherence_length_relation}\n\\end{equation}\nwhere $d_{\\mathrm{L}}\\approx 2.5$~\\cite{boettcher:05,maiorano:18} is the lower critical dimension\\index{critical dimension!lower} at zero magnetic field, and the dots stand for scaling corrections, subdominant for large $\\xi$. This relation makes sense only for the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}~\\cite{parisi:88}.\n\n\n\n\\subsubsection{The isothermal protocols}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/phase_diagram_isothermal.pdf}\n\\caption[\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the  $\\xi$ for isothermal\\index{isothermal} protocols.}]{\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for isothermal\\index{isothermal} protocols.} As suggested by~\\refeq{energy_coherence_length_relation} for isothermal\\index{isothermal} relaxations\\index{relaxation} $\\tilde{e}$ is an essentially linear function of $1\/\\xi^{2.5}$, (at least for the plotted range of  $\\xi>4.8$). Furthermore, the dependence of the slope on temperature is marginal.}\n\\labfig{phase_diagram_isothermal}\n\\end{figure}\n\nWe test the relation defined by \\refeq{energy_coherence_length_relation} in \\reffig{phase_diagram_isothermal} by plotting the energy-density\\index{energy!density} $\\tilde{e}$ against $1\/\\xi^{d_{\\mathrm{L}}}$, arrows indicate the direction of the points for increasing $t$. First, we observe that the isothermal\\index{isothermal} protocols ($T=0.7$ and $T=0.9$) are (almost) straight lines in our representation. In addition, both isothermal\\index{isothermal} protocols are (almost) parallel to each other. Of course, we need to make these observations quantitative. To that purpose we fit our data to\n\\begin{equation}\n\\tilde{e}(t) = \\tilde{e}_{\\infty}(T) + \\dfrac{e_1}{\\xi^{d_\\mathrm{L}}(t)} + \\dfrac{e_2}{\\xi^{2d_\\mathrm{L}}(t)} \\, , \\labeq{quadratic_fit_mpemba}\n\\end{equation}\nwhich is just~\\refeq{energy_coherence_length_relation} with a simple quadratic correction in $1\/\\xi^{d_\\mathrm{L}}(t)$ that would be negligible for large $\\xi$. \n\nBecause this subdominant term (the quadratic one) becomes less important for the interesting limit (the large-$\\xi$ limit), we decide to establish an objective criterion to select the fitting range. We perform the fit for the range $[\\xi_{\\min},\\xi_{\\max}]$ by setting $\\xi_{\\max}$ to the maximum $\\xi$ simulated and by varying $\\xi_{\\min}$. We set $\\xi_{\\min}$ to be the lowest value of $\\xi$ that stabilizes the values of $\\tilde{e}_{\\infty}$, $e_1$ and $e_2$ (within the error bars\\index{error bars}), and, for the desired temperatures $T=0.7$ and $T=0.9$ we found $\\xi_{\\min}=6$. In addition, to describe the quality of the fit, we report the figure of merit $\\chi^2$\/d.o.f.\\index{degree of freedom} The results can be consulted in \\reftab{fit_results_mpemba}\n\n\\begin{table}[b!]\n\\begin{tabular}{ccccc}\n\\toprule\n\\toprule\n$T$ & $\\tilde{e}_{\\infty}$ & $e_1$ & $e_2$ & $\\chi^2\/$d.o.f.\\index{degree of freedom} \\\\\n\\hline\n0.7 & $-1.7708070(7)$ & $0.2217(3)$ & $1.17(2)$ & $20.9(1)\/119$ \\\\\n0.9 & $-1.7443347(6)$ & $0.2251(3)$ & $1.08(2)$ & $11.0(4)\/118$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Mpemba parameters of the quadratic fit.}]{\\textbf{Mpemba parameters of the quadratic fit.} We report the results of the fits to \\refeq{quadratic_fit_mpemba}. For each fit, the figure of merit $\\chi^2$\/d.o.f.\\index{degree of freedom} is also reported. Errors are computed by using the Jackknife\\index{Jackknife} method.}\n\\labtab{fit_results_mpemba}\n\\end{table}\n\nAs we said, both curves are almost parallel ($e^{T=0.7}_1\/e^{T=0.9}_1 \\approx 0.9849$) and are also straight lines, because the effect of the curvature is around 1\\% of the effect of the linear term $e_1\/\\xi^{d_{\\mathrm{L}}}$ for $\\xi \\approx8$ that is a typical coherence length\\index{coherence length} in our data.\n\n\\subsubsection{The temperature-varying protocols}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/phase_diagram.pdf}\n\\caption[\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for temperature-varying\\index{temperature-varying protocol} protocols.}]{\\textbf{\\boldmath Relationship between the energy\\index{energy!density} density $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for temperature-varying\\index{temperature-varying protocol} protocols.} Temperature-varying\\index{temperature-varying protocol} protocols are seen to be essentially vertical moves between the straight lines corresponding to isothermal\\index{isothermal} relaxations\\index{relaxation} at the initial and final temperatures. These vertical moves are very quick initial transients, in which (in moves to higher temperatures only), $\\xi$ slightly decreases and then increases again.}\n\\labfig{phase_diagram}\n\\end{figure}\n\nWe add the temperature-varying\\index{temperature-varying protocol} protocols to the analysis, see \\reffig{phase_diagram}. Those protocols where the temperature of the thermal reservoir decreases correspond to preparations B, C, and D in \\reffig{test_mpemba}. We can see that the time-scales of the energy-density\\index{energy!density} and the coherence length\\index{coherence length} are totally decoupled. The energy-density\\index{energy!density} $\\tilde{e}$ is a \\textit{fast variable} and, as a first approximation, when a quick temperature change takes place, $\\tilde{e}$ instantaneously takes the value of the energy-density\\index{energy!density} corresponding to its new thermal reservoir. However, the coherence length\\index{coherence length} $\\xi$ is a \\textit{slow variable} that basically remains unchanged when a temperature change takes place. The combination of both effects is translated into almost vertical movements between isothermal\\index{isothermal} protocols in \\reffig{phase_diagram}.\n\nIn this representation, the crossing points in \\reffig{test_mpemba} are not so evident. Now, the temperature-varying\\index{temperature-varying protocol} protocols experiment a very fast decrease of the energy-density\\index{energy!density}, while the isothermal\\index{isothermal} protocols need longer times (equivalently, longer coherence length\\index{coherence length}) to reach those values of the energy-density\\index{energy!density} and, therefore, the temperature-varying\\index{temperature-varying protocol} protocol ``cools'' faster. Of course, the previous analysis is a simplification, and measurable (still small) transient effects can be seen in \\reffig{phase_diagram}, however, it provides a very simple explanation of the Mpemba\\index{Mpemba effect} effect.\n\nThe curves corresponding to an increase in the temperature of the thermal reservoir are analyzed next.\n\n\\section{The Inverse Mpemba Effect} \\labsec{inverse_mpemba}\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/inverse_me_ener.pdf}\n\\caption[\\textbf{A tiny inverse Mpemba effect.}]{\\textbf{A tiny inverse Mpemba\\index{Mpemba effect} effect.} Time evolution of the energy\\index{energy!density}, for the three different preparations (namely 1,2 and 3), compared with an isothermal\\index{isothermal} protocol with $T=1.4$ (top curve). In the three preparations, the initial temperature is in the spin-glass phase\\index{phase!low-temperature\/spin-glass}, and the final temperature is $T=1.4>T_\\mathrm{c}$. A very small Mpemba\\index{Mpemba effect} effect is found at the time pointed by the arrow, only when warming up samples\\index{sample} with similar starting energy\\index{energy!density}.}\n\\labfig{inverse_me_ener}\n\\end{figure}\n\nWe focus now on the inverse Mpemba\\index{Mpemba effect} effect protocol that was first suggested in~\\cite{lu:17,lasanta:17}. Now, the final temperature of the thermal reservoir is chosen to be higher than the starting one. We see in \\reffig{phase_diagram} that both curves corresponding to that protocol behave in a symmetrical way concerning to the classical protocol. In \\reffig{phase_diagram} all the temperatures are below the critical one, and the natural question is, does the inverse Mpemba\\index{Mpemba effect} effect survive for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$? The question is not trivial because \\refeq{energy_coherence_length_relation} is not expected to hold for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/inverse_me_xi.pdf}\n\\caption[\\textbf{Coherence length: Undershooting and convergence to a master curve.}]{\\textbf{coherence length\\index{coherence length}: Undershooting and convergence to a master curve.} coherence length\\index{coherence length}s $\\xi$ of the experiments described in \\reffig{inverse_me_ener}. The time evolution of $\\xi$ tends to converge towards the curve corresponding to isothermal\\index{isothermal} protocol with $T=1.4$ (bottom curve), giving rise to an undershoot of $\\xi$ when its initial value is higher than the equilibrium $\\xi$ at $T=1.4$.}\n\\labfig{inverse_me_xi}\n\\end{figure}\n\nTo answer this question, we use our temperature-varying\\index{temperature-varying protocol} protocol, but this time the final temperature will be at $T>\\ensuremath{T_\\mathrm{c}}\\xspace$. We propose three starting conditions:\n\\begin{itemize}\n\\item \\textbf{Preparation 1:} $T=0.7, \\xi=2.5 \\to T=1.4$.\n\\item \\textbf{Preparation 2:} $T=0.7, \\xi=11.7 \\to T=1.4$.\n\\item \\textbf{Preparation 3:} $T=0.8, \\xi=15.8 \\to T=1.4$.\n\\end{itemize}\nThe reader should be aware that, although for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ the coherence length\\index{coherence length} grows without bonds\\footnote{In an infinite system.}, this is not the case for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$. Specifically, for $T=1.4$ the asymptotic equilibrium value for the coherence length\\index{coherence length} is $\\xi=8.95(5)$. We also compare these temperature-varying\\index{temperature-varying protocol} protocols with the isothermal\\index{isothermal} protocol at $T=1.4$.\n\nIf we study the relaxation\\index{relaxation} of the energy-density\\index{energy!density} we can observe a small Mpemba\\index{Mpemba effect} effect between protocols 2 and 3 for $t\\approx 20$ (see \\reffig{inverse_me_ener}). However, between protocols 1 and 3 or 1 and 2, the Mpemba\\index{Mpemba effect} effect is clearly absent.\n\nWe can study also the relaxation\\index{relaxation} of the coherence length\\index{coherence length} $\\xi$, see \\reffig{inverse_me_xi}. In the paramagnetic phase\\index{phase!high-temperature\/paramagnetic}, the growth of the magnetic domains\\index{magnetic domain} does not follow~\\refeq{xi_powerlaw} as it becomes evident in the figure. In addition, we observe that all the protocols tend to the isothermal\\index{isothermal} one very fast (for $t \\leq 10^5$).\n\nWe can see here again that both time scales, the $\\xi$ one and the $\\tilde{e}$ one, are clearly decoupled. In \\reffig{inverse_me_ener} and \\reffig{inverse_me_xi} we can see that both quantities tend to their equilibrium values at very different time scales. Furthermore, if we focus on protocol 2 we can see that the undershoot present for the coherence length\\index{coherence length} does not correspond to a similar behavior for the energy-density\\index{energy!density}. Although for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$ the Mpemba\\index{Mpemba effect} effect is strongly suppressed, this decoupling between both time scales seems to be necessary for the Mpemba\\index{Mpemba effect} effect to take place.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter[Temperature Chaos phenomenon in off-equilibrium Spin Glasses]{Temperature Chaos phenomenon in \\\\ off-equilibrium spin glasses} \\labch{out-eq_chaos}\n\n\\gls{TC} phenomenon is described in terms of equilibrium configurations\\index{configuration}, and therefore, all the previous work studying it is focused on equilibrated spin glasses. However, experiments are almost always carried out in non-equilibrium conditions. Moreover, the concept of \\gls{TC}\\index{temperature chaos} is not totally alien to the non-equilibrium regime in \\gls{SG}. Actually, it has been related to the memory\\index{memory effects} and rejuvenation\\index{rejuvenation} effects~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}.\n\nHowever, at this point, the relation of \\gls{TC}\\index{temperature chaos} with memory\\index{memory effects} and rejuvenation\\index{rejuvenation} effects is far from clear and no quantitative description of \\gls{TC}\\index{temperature chaos} under off-equilibrium conditions has been provided. In this chapter, we present a numerical work in off-equilibrium conditions that try to be the first step to fill that gap. All the simulations have been performed in the dedicated computer Janus\\index{Janus} II~\\cite{janus:14} and the high-accuracy achieved could not be possible without its computational power.\n\nIn the course of the chapter, we will explain why traditional approaches to study \\gls{TC}\\index{temperature chaos} do not work and we need, again, a rare-event analysis in order to fully understand the phenomenon. Moreover, the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17} shows us the path to quantitatively study an equilibrium phenomenon in a non-equilibrium system.\n\nThe results of this work show us how, again, the coherence length\\index{coherence length} $\\xi$ rules the off-equilibrium phenomena in \\gls{SG}s. In particular, a crossover behavior between a weak chaos regime and a strong chaos regime is found when $\\xi$ grows. The characteristic length scale where this occurs, $\\xi^*$, is related to its equilibrium counterpart, the chaotic length $\\ell_c$ defined in~\\refeq{def_chaotic_length}.\n\nIn \\refsec{numerical_simulations_tc}, we give all the information about the performed numerical simulations. We explore the first \\textit{naive} attempt to characterize \\gls{TC}\\index{temperature chaos} in off-equilibrium dynamics in~\\refsec{average_killed_chaos_signal}. The computed observables to perform our rare-event analysis are introduced in~\\refsec{observables_tc}. The rare-event analysis can be found in~\\refsec{char} and its results in~\\refsec{results}. In~\\refsec{scaling_fixed_r} we explore the scaling behavior of our chaos-related quantities and, in \\refsec{T-changes}, we focus on temperature changing protocols in order to make contact with the cumulative-aging\\index{aging!cumulative} controversy (described in \\refsec{memory_rejuvenation_introduction_chaos}) and lay the groundwork to future numerical works trying to relate simulations and experiments.\n\nAll the results provided in this chapter were originally published (in a reduced form) in~\\cite{janus:21}.\n\n\\section{Numerical parameters of the simulation} \\labsec{numerical_simulations_tc}\nIn this work, we simulate the \\gls{EA}\\index{Edwards-Anderson!model} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. This assumption is sound, provided that $L\\gg\\xi$ (see~\\reftab{xi_max}). Note that this condition limits the maximum time at which we can safely ignore finite-size effects\\index{finite-size effects}. The temperature remains constant through the whole simulation, with the only exception of the runs reported and discussed in~\\refsec{T-changes}.\n\nWe shall perform direct quenches from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds with the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample} we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ \\emph{replicas}\\index{replica} \\footnote{It has been noted in~\\cite{janus:18} (see also \\refsec{Nr_aging}) that, for global observables (see~\\refsubsec{observables-globales}), it is advantageous to have $\\ensuremath{N_{\\text{Rep}}}\\xspace\\gg \\ensuremath{N_{\\text{S}}}\\xspace$. However, working with $\\ensuremath{N_{\\text{Rep}}}\\xspace\\gg \\ensuremath{N_{\\text{S}}}\\xspace$ is not only a matter of numerical convenience for us. In fact, the local observables in~\\refsubsec{observables-locales} are well defined only in the limit of $\\ensuremath{N_{\\text{Rep}}}\\xspace\\to\\infty$.}.\n\nThe simulation has been performed in the dedicated \\gls{FPGA}-based\\index{FPGA} computer Janus\\index{Janus} II~\\cite{janus:14} by using Metropolis dynamics. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{l c r c c c r}\n\\toprule\n\\toprule\n$T$ &  \\hspace{1cm} & $\\ensuremath{t_\\mathrm{w}}\\xspace$ (MCs) & \\hspace{1cm} & $\\log_2(\\ensuremath{t_\\mathrm{w}}\\xspace)$ & \\hspace{1cm} & $\\xi_{\\max}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ \\\\\n\\toprule\n$0.625$ & \\hspace{1cm} & 42669909513 & \\hspace{1cm} & 35.3 & \\hspace{1cm} & 9.52(1) \\\\\n\\hline\n$0.7$ & \\hspace{1cm} & 48592007999 & \\hspace{1cm} & 35.5 & \\hspace{1cm} & 12.02(2) \\\\\n\\hline\n$0.8$ & \\hspace{1cm} & 34359738368 & \\hspace{1cm} & 35\\phantom{.5} & \\hspace{1cm} & 15.84(5) \\\\\n\\hline \n$0.9$ & \\hspace{1cm} & 17179869184 & \\hspace{1cm} & 34\\phantom{.5} & \\hspace{1cm} & 20.34(6)  \\\\\n\\hline\n$1.0$ & \\hspace{1cm} & 4294967296 & \\hspace{1cm} & 32\\phantom{.5} & \\hspace{1cm} & 24.4(1)\\phantom{0} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Parameters of the simulations.}]{\\textbf{Parameters of the simulations.} Maximum value of $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ simulated for each temperature. The central columns show the $\\ensuremath{t_\\mathrm{w}}\\xspace$ corresponding value for the computed $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}\n\\labtab{xi_max}\n\\end{center}\n\n\\end{table}\n\n\n\\section{Taking spatial averages kills the chaotic signal} \\labsec{average_killed_chaos_signal}\n\nThe first naive attempt to study \\gls{TC}\\index{temperature chaos} phenomenon in out-equilibrium \\gls{SG}s consisted of studying global quantities affecting the whole system. However, in close analogy with equilibrium studies~\\cite{fernandez:13}, we find that \\gls{TC}\\index{temperature chaos} is extremely weak when the full system is considered on average, see \\reffig{xi_T1T2}, although the effect increases when the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ grows (just as \\gls{TC}\\index{temperature chaos} becomes more visible in equilibrium when the system size increases).\n\nIn view of the above negative result, we have followed Ref.~\\cite{fernandez:13} and performed a rare-event analysis that provides a satisfactory quantification of the \\gls{TC}\\index{temperature chaos} phenomenon. The\nrationale for this approach is the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17}: we expect to learn about the non-equilibrium dynamics of a spin glass (of infinite size), with a finite coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, by studying small samples\\index{sample} of size $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ which can be equilibrated. \n\n\\begin{figure}[t!] \n\\centering \n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/xi_T1T2.pdf}\n\\caption[\\textbf{Non-equilibrium \\gls{TC}\\index{temperature chaos} is weak when averaging over the whole system.}]{\\textbf{Non-equilibrium \\gls{TC}\\index{temperature chaos} is weak when averaging over the whole system.} We compare typical spin configurations\\index{configuration} at temperature $T_1$ and time $t_{\\mathrm{w},1}$ with configurations\\index{configuration} at $T_2$ and time $t_{\\mathrm{w},2}$. The comparison is carried through a global estimator of the coherence length\\index{coherence length} of their overlap\\index{overlap} $\\xi^{T_1T_2}_{1,2}$, see \\refeq{def_correlation_length_2T} (physically, $\\xi^{T_1T_2}_{1,2}$ is the maximum length scale at which configurations\\index{configuration} at temperatures $T_1$ and $T_2$ still look similar). The two times $t_{\\mathrm{w},1}$ and $t_{\\mathrm{w},2}$ are chosen in such a way that the configurations\\index{configuration} at both temperatures have glassy-domains\\index{magnetic domain} of the same size, namely $\\xi_{1,2}(t_{\\mathrm{w},1},T_1)=\\xi_{1,2}(t_{\\mathrm{w},2},T_2)=\\xi$. The figure shows the ratio $\\xi^{T_1T_2}_{1,2}\/\\xi$ as a function of $\\xi$ for two pairs of temperatures $(T_1,T_2)$, recall that $\\ensuremath{T_\\mathrm{c}}\\xspace\\approx 1.1$, see~\\refsubsec{3D_EA_model}. Under the hypothesis of fully developed \\gls{TC}\\index{temperature chaos}, we would expect $\\xi^{T_1T_2}_{1,2}$ to be negligible as compared to $\\xi$. Instead, our data show only a small decrease of $\\xi^{T_1T_2}_{1,2}\/\\xi$ upon growing $\\xi$ (the larger the difference $T_2-T_1$ the more pronounced the decrease).}\n \\labfig{xi_T1T2}\n\\end{figure}\n\nIn our case, we shall be considering spatial regions (spheres) of linear size $\\sim \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, chosen randomly within a very large spin glass. Just as found with the small samples\\index{sample} in equilibrium~\\cite{fernandez:13}, we expect that a small fraction of our spheres will display strong \\gls{TC}\\index{temperature chaos}. The important question will be how this rare-event phenomenon evolves as $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ grows. In fact, we expect that our description of non-equilibrium \\gls{TC}\\index{temperature chaos} will allow us to perform sensible extrapolations to values of $\\xi$ of experimental interest (for comparison, a typical experimental value is $\\xi\\sim 100$ lattice spacings, while in our simulations $\\xi\\sim 10$ lattice spacings).\n\n\n\\section{Observables} \\labsec{observables_tc}\nIn this section, we briefly introduce the quantities that will help us to characterize \\gls{TC}\\index{temperature chaos} in off-equilibrium systems. The global observables (\\refsubsec{observables-globales}) will be fundamental in order to characterize the relevant length (equivalently time) scales of the system. On the contrary, local observables (\\refsubsec{observables-locales}) will be necessary in order to perform a rare-event analysis of the \\gls{TC}\\index{temperature chaos}.\n\n\\subsection{Global observables}\\labsubsec{observables-globales}\nThe out-equilibrium time evolution is usually characterized by the growth of the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ at temperature $T$, see~\\refsubsec{observables_introduction}. In order to compute it, two basic observables are needed: the overlap\\index{overlap!field} field and the four-point\\index{correlation function!four point} spatial correlation function [see~\\refeq{def_overlap} and~\\refeq{def_C4}]. We repeat here the definitions for the reader's convenience\n\\begin{equation}\nq^{\\sigma,\\tau}(x,\\ensuremath{t_\\mathrm{w}}\\xspace) = s_{x}^\\sigma(\\ensuremath{t_\\mathrm{w}}\\xspace) s^\\tau_{x}(\\ensuremath{t_\\mathrm{w}}\\xspace) \\, ,\n\\end{equation}\n\\begin{equation} \nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) = \\overline{\\langle q^{\\sigma,\\tau}(x,\\ensuremath{t_\\mathrm{w}}\\xspace) q^{\\sigma,\\tau}(x+r,\\ensuremath{t_\\mathrm{w}}\\xspace)\\rangle_T} \\, . \\labeq{def_corr_func}\n\\end{equation} \nIn the previous definitions $ s_{x}^\\sigma(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is the spin of the replica\\index{replica} $\\sigma$ in the lattice position $x$ at time $\\ensuremath{t_\\mathrm{w}}\\xspace$, $\\langle\\dots\\rangle_T$ is the average over thermal noise at temperature $T$, and $\\overline{(\\cdots)}$ is the average over the disorder\\index{disorder!average}. Of course, the two replica\\index{replica} indices $\\sigma$ and $\\tau$ should be different.\n\nThe correlation function in~\\refeq{def_corr_func} can be extended for a pair of temperatures $T_1$ and $T_2$ in the following\nway\n\\begin{equation}\nC_4^{T_1T_2}(T_1,T_2,t_{\\mathrm{w}1},t_{\\mathrm{w}2},r) = \\overline{\\langle q^{\\sigma(T_1),\\tau(T_2)} (x,t_{\\mathrm{w}1},t_{\\mathrm{w}2}) q^{\\sigma(T_1),\\tau(T_2)}(x+r,t_{\\mathrm{w}1},t_{\\mathrm{w}2})\\rangle_{T}} \\, , \\labeq{def_corr_func_2T}\n\\end{equation}\nwhere now the thermal averages are taken at temperature $T_1$ for the replica\\index{replica} $\\sigma$, and at temperature $T_2$ for the replica\\index{replica} $\\tau$. From the four-point\\index{correlation function!four point} correlation function we can compute the coherence length\\index{coherence length} as have been explained in~\\refsubsec{observables_introduction} and \\refsec{finite_size_effects}.\n\nOf course, the coherence length\\index{coherence length} can be straightforwardly extended to a pair of temperatures $T_1$ and $T_2$ by using  $C_4^{T_1T_2}$ instead of $C_4$:\n\\begin{equation} \nI^{T_1T_2}_k( t_{\\mathrm{w}1},t_{\\mathrm{w}2}) = \\int_{0}^{\\infty}r^k\\,C^{T_1T_2}_4(r,t_{\\mathrm{w}1},t_{\\mathrm{w}2})\\,\\mathrm{d} r \\, , \\labeq{def_integral_2T}\n\\end{equation}\nand\n\\begin{equation}\n\\xi^{T_1T_2}_{k,k+1}(t_{\\mathrm{w}1},t_{\\mathrm{w}2}) = \\dfrac{I^{T_1T_2}_{k+1}(t_{\\mathrm{w}1},t_{\\mathrm{w}2})}{I^{T_1T_2}_k(t_{\\mathrm{w}1},t_{\\mathrm{w}2})} \\, . \\labeq{def_correlation_length_2T}\n\\end{equation}\nAs a rule, we shall fix the two times $t_{\\mathrm{w}1}$ and $t_{\\mathrm{w}2}$ through the condition\\footnote{Because our $\\ensuremath{t_\\mathrm{w}}\\xspace$ are in a discrete grid, we solve~\\refeq{el_reloj_doble} for the \\emph{global} overlaps\\index{overlap} defined in~\\refsubsec{observables-globales} through a (bi)linear interpolation.}:\n\\begin{equation}\n\\xi(t_{\\mathrm{w}1},T_1)=\\xi(t_{\\mathrm{w}2},T_2)=\\xi\\,, \\labeq{el_reloj_doble}\n\\end{equation}\nthat ensures that we are comparing spin-configurations\\index{configuration} which are ordered on the same length scale.\n\n\\subsection{Local observables}\\labsubsec{observables-locales}\n\nIn order to explore the heterogeneity of the system, we construct here local observables that will allow us to generalize the rare-event analysis in~\\cite{fernandez:13}. Specifically, we shall be studying the properties of spherical regions.\n\nWe start by choosing $N_{\\mathrm{sph}}=8000$ centers for the spheres, on each sample\\index{sample}. The sphere centers are chosen randomly, with uniform probability, on the dual lattice~\\footnote{The dual lattice of a cubic lattice with \\gls{PBC}\\index{boundary conditions!periodic} is another cubic lattice of the same size, and with \\gls{PBC}\\index{boundary conditions!periodic} as well. The nodes of the dual lattice are the centers of the elementary cells of the original lattice.}. The radius of the spheres is varied, but their centers are held fixed. Let $B_{s,r}$ be the $s$-th ball of radius $r$. \n\nSimilarly as in the previous chapter (\\refch{equilibrium_chaos}), our basic observable will be the overlap\\index{overlap} between temperatures $T_1$ and $T_2$\n\\begin{equation}\nq_{T_1,T_2}^{s,r}(\\xi) = \\dfrac{1}{N_r} \\sum_{x\\in B_{s,r}} s_{x}^{\\sigma,T_1}(t_{\\mathrm{w}1}) s_{x}^{\\tau,T_2}(t_{\\mathrm{w}2}) \\>\\> , \\labeq{sphere_overlap}\n\\end{equation}\nwhere $N_r$ is the number of spins within the ball of radius $r$, and the two times $t_{\\mathrm{w}1}$ and $t_{\\mathrm{w}2}$ are chosen according to~\\refeq{el_reloj_doble}\\footnote{Because the local overlaps\\index{overlap} in~\\refeq{sphere_overlap} have much larger fluctuations than the global overlaps\\index{overlap} in~\\refsubsec{observables-globales}, in this case we solve~\\refeq{el_reloj_doble} in a cruder way. We just select the value of $t_{\\mathrm{w}1}$ that yields the $\\xi(t_{\\mathrm{w}1},T_1)$ nearest to our target $\\xi$ value. The same procedure is followed with $t_{\\mathrm{w}2}$.}.\nNext, again as in the previous chapter, we introduce the so called \\textit{chaotic parameter}~\\cite{ritort:94,ney-nifle:97,fernandez:13,billoire:14} which now is restricted to the balls $B_{s,r}$\n\\begin{equation} \nX^{s,r}_{T_1,T_2}(\\xi) = \\dfrac{\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T}{\\sqrt{\\langle[q_{T_1,T_1}^{s,r}(\\xi)]^2\\rangle_T \\,\\langle[q_{T_2,T_2}^{s,r}(\\xi)]^2\\rangle_T}} \\, . \\labeq{def_chaotic_parameter}\n\\end{equation} \nThe extreme values of the chaotic parameter, just in close analogy with the equilibrium case, have a very clear interpretation: $X^{s,r}_{T_1,T_2}=1$ corresponds to a situation in which spin configurations\\index{configuration} in the ball $B_{s,r}$, at temperatures $T_1$ and $T_2$, are completely indistinguishable (absence of chaos) while $X^{s,r}_{T_1,T_2}=0$ corresponds to completely different configurations\\index{configuration} (which means strong \\gls{TC}\\index{temperature chaos}). A representative example of our results is shown in \\reffig{hetereogeneity_chaos}.\n\n\\begin{figure}[h!] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/paraview.png}\n\\caption[\\textbf{Dynamic temperature chaos is spatially heterogeneous.}]{\\textbf{Dynamic temperature chaos is spatially heterogeneous.} The 8000 randomly chosen spheres in a sample\\index{sample} of size $L=160$ are depicted with a color code depending on $1-X$ [$X$ is the chaotic parameter, \\refeq{def_chaotic_parameter}, as computed for spheres of radius $r=12$, $\\xi=12$ and temperatures $T_1=0.7$ and $T_2=1.0$]. For visualization purposes, spheres are represented with a radius $12(1-X)$, so that only fully chaotic spheres (i.e., $X=0$) have their real size.}\n\\labfig{hetereogeneity_chaos}\n\\end{figure}\n\nWe shall focus our attention on the distribution function\n\\begin{equation}\nF(X,T_1,T_2,\\xi,r)=\\text{Probability}[X^{s,r}_{T_1,T_2}(\\xi)<X] \\, ,\\labeq{F-def}\n\\end{equation}\nand its inverse $X(F,T_1,T_2,\\xi,r)$.\n\nThe alert reader will note that the computation of $X^{s,r}_{T_1,T_2}(\\xi)$ requires the \\emph{exact} computation of thermal expectation values, which would require an infinite number of replicas\\index{replica}.  Fortunately, we have been able of extrapolating $X(F,T_1,T_2,\\xi)$ to the limit $\\ensuremath{N_{\\text{Rep}}}\\xspace\\to\\infty$. Details about this extrapolation are provided in~\\refsubsec{extrapolation}.\n\nA final note: as explained in~\\refsec{cambio_de_r}, we have found it advantageous to trade the sphere radius $r$ by $N_r^{1\/3}$. Of course, $N_r$ equals $4\\pi r^3\/3$ for large $r$, but $N_r^{1\/3}$ provides smoother interpolations at small $r$.\n\n\\section{The temperature-chaos distribution functions} \\labsec{char}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/distribution_function_xi.pdf}\n\\caption[\\textbf{Temperature chaos increases with \\boldmath $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Temperature chaos increases with \\boldmath $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} The figure shows the distribution function $F(X,T_1,T_2,\\xi,r)$, see~\\refeq{F-def}, for $T_1=0.625$ and $T_2=0.9$, for spheres of radius $r=4$ (left) and $r=8$ (right), as computed for various values of $\\xi$. The distributions have been extrapolated to $\\ensuremath{N_{\\text{Rep}}}\\xspace=\\infty$. Error bars\\index{error bars} are horizontal, because we have actually extrapolated the inverse function $X(F,T_1,T_2,\\xi,r)$ (the extrapolation is not always safe for $F>0.3$, see~\\refsubsec{extrapolation}; the same caveat applies to all the distribution functions shown in~\\refsec{char} and~\\refsec{results}). Most of the spheres have a chaotic parameter very close to $X=1$.}\n\\labfig{distribution_function_xi}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/distribution_function_r.pdf}\n\\caption[\\textbf{Dependency of Temperature Chaos on the size of the observation region.}]{\\textbf{Dependency of Temperature Chaos on the size of the observation region.} The figure shows the distribution function $F(X,T_1,T_2,\\xi,r)$ for $T_1=0.625$ and $T_2=0.9$, for coherence length\\index{coherence length} $\\xi=5$ (left) and $\\xi=9$ (right), as computed for spheres of various radius $r$. If we focus on some low probability ($F=0.01$, for instance), we find that there is an optimal size for the observation of chaos (in the sense of a smallest chaotic parameter $X$).}\n \\labfig{distribution_function_r}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/plot_Xvsr_examples.pdf}\n\\caption[\\textbf{The complementary chaotic parameter $\\mathbf{1-X(F,T_1,T_2,\\xi,r)}$ for fixed $F$ as a function of $\\mathbf{r}$.}]{\\textbf{The complementary chaotic parameter $\\mathbf{1-X(F,T_1,T_2,\\xi,r)}$ for fixed $F$ as a function of $\\mathbf{r}$.} The difference $1-X(F,T_1,T_2,\\xi,r)$ [recall that $X(F,T_1,T_2,\\xi,r)$ is the inverse of the distribution function, see~\\refeq{F-def}] as a function of the cubic root of the number of points in the spheres $N_r^{1\/3}$, as computed for different values of $F$, $T_1$, $T_2$ and $\\xi$.  Our rationale for choosing $N_r^{1\/3}$ as independent variable, rather than the radius of the spheres $r$, is explained in~\\refsec{cambio_de_r}. In this representation, the size of the spheres which are optimal for the observation of chaos (for given parameters $F$, $T_1$, $T_2$ and $\\xi$) appears as the maximum of these curves. Continuous lines are fits to~\\refeq{functional_form}.}\n\\labfig{Xvsr_examples}\n\\end{figure}\n\nSome examples of the distribution functions $F(X,T_1,T_2,\\xi,r)$ can be found in~\\reffig{distribution_function_xi} and~\\reffig{distribution_function_r}, for typical (fixed) values of $T_1$ and $T_2$. Although most spheres are clearly non-chaotic ($X>0.9$), the situation is far more interesting for low probabilities (say $F=0.01$). For the sake of simplicity, consider first spheres of a fixed size (\\reffig{distribution_function_xi}). For small $F$, we find that $X$ decreases significantly (and monotonically) upon growing $\\xi$. The situation is more complex if we consider spheres of different sizes, for given $F$ and $\\xi$.  As \\reffig{distribution_function_r} shows, when the size of the spheres grows the chaotic parameter is non-monotonic.\n\nThe situation clarifies when we fix both the probability $F$ and the coherence length\\index{coherence length} $\\xi$, see~\\reffig{Xvsr_examples}. Rather than the chaotic parameter, let us consider the difference $1-X$ (which grows when \\gls{TC}\\index{temperature chaos} becomes stronger). We find that $1-X$ peaks for one size of the spheres which indicates the optimal length scale for the study of \\gls{TC}\\index{temperature chaos} (however, see~\\reffig{Xvsr_examples}, this peak is asymmetric and becomes broader when $\\xi$ increases). Our main analysis in~\\refsec{results} will correspond to the scaling with $\\xi$ of these peaks.\n\nLet us remark that, at least close to a maximum, any smooth curve is characterized by the position, height and width of the peak. In order to meaningfully compute these three parameters from our data (see e.g.~\\reffig{Xvsr_examples}), we fit $1-X$ to\n\\begin{equation}\nf(z) = \\dfrac{az^b}{1+cz^d} \\quad ,\\quad z=N_r^{1\/3}\\,, \\labeq{functional_form}\n\\end{equation}\n($a$, $b$, $c$, and $d$ are the parameters of the fit). We extract the position, width and height from the fitted function $f(z)$. In order to compute errors in (say) the peak position $N_{r,\\max}^{1\/3}$ we use a Jackknife\\index{Jackknife} method (see \\refsec{estimating_errorbars} for further details): we perform a separated fit for each Jackknife\\index{Jackknife} block, extract $N_{r,\\max}^{1\/3}$ from the fit for that block, and compute errors from the block fluctuations. Of course, Jackknife\\index{Jackknife} blocks are formed from our $\\ensuremath{N_{\\text{S}}}\\xspace=16$ samples\\index{sample}. Let us stress that~\\refeq{functional_form} is meant to be only a convenient way of characterizing the peak, without any deep meaning attached to it.\n\nHowever, the reader may question whether or not the local peak description (i.e. position, height, and width) is sensible for the full curve. We provide some positive evidence in this respect in~\\refsubsec{taylor}.\n\n\\subsection{Global versus local description of the peaks}\\labsubsec{taylor} \n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/taylor_0710_algunos.pdf}\\\\\n\n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/taylor_examples.pdf}\n\\caption[\\textbf{Universality in $1-X$ extends beyond the trivial Taylor's Universality.}]{\\textbf{Universality in $1-X$ extends beyond the trivial Taylor's Universality.} The upper part shows $1-X$ in units of its peak value, for the temperatures $T_1=0.7$, $T_2=1.0$ and $F=0.01$. Taylor's theorem implies that, using the independent variable $y$ [see~\\refeq{Taylor-universality}], the different curves should coincide close to $y=0$. However, we see that the coincidence holds beyond the quadratic approximation (as evinced by the strong asymmetry of the master curve). The lower panel shows the same set of temperatures $T_1$ and $T_2$ and probabilities $F$ shown in~\\reffig{Xvsr_examples} (we have added data for several coherence length\\index{coherence length}). Mixing different values of $F$, $T_1$ and $T_2$ leads to significant discrepancies for large values of $|y|$. Nevertheless, the collapse of the curves is still present in the range $y \\in (-0.3,0.5)$ where the asymmetry is not negligible.}\n\\labfig{taylor}\n\\end{figure}\n\nConsider any smooth, positive function $H(z)$, with a local maximum at $z=z_{\\max}$. Close to this peak, Taylor's theorem implies some (trivial) Universality\n\\begin{equation}\n\\frac{H(z)}{H(z_{\\max})}=1-\\frac{1}{2} y^2+{\\cal O}(y^3)\\,,\\text{ where } \\,\ny=\\sqrt{\\frac{|H''(z_{\\max})|}{H(z_{\\max})}}(z-z_{\\max})\\,. \\labeq{Taylor-universality}\n\\end{equation}\nNote that, in the language of the previous paragraph, the peak position is $z_{\\text{max}}$, its heigth is $H(z_{\\max})$ and its (inverse) width is $\\sqrt{|H''(z_{\\max})|\/H(z_{\\max})}$. Of course, in principle, there is no reason for~\\refeq{Taylor-universality} to be accurate away from the peak. However, \\refeq{Taylor-universality} suggests yet another representation for our $1-X$ curves, see~\\reffig{taylor}. We note that, in this new representation, the $1-X$ curves are invariant under changes of coherence length\\index{coherence length} $\\xi$ (\\reffig{taylor} upper panel). However, when considering changes in the temperatures $T_1$ and $T_2$ and the probability $F$, the curves mildly differ away from the peak (see~\\reffig{taylor} lower panel). This (approximate) independence in $(T_1,T_2,F,\\xi)$ is a fortunate fact because the complexity of the problem gets reduced to the study of the scaling with $\\xi$ of the three peak parameters while keeping constant $(T_1,T_2,F)$.\n\n\n\n\n\\section{The off-equilibrium characterization of Temperature Chaos}\\labsec{results}\nIn this section we present the scaling of the peak position (\\refsubsec{peaks-position}), the peak height (\\refsubsec{peaks-height}) and the peak (inverse) width (\\refsubsec{peaks-width}) with the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nDue to the difficulty of characterizing peaks which exhibit weak \\gls{TC}\\index{temperature chaos}, in the following analysis we exclude the data corresponding to the pair of temperatures ($T_1=0.625,T_2=0.7$) at the probability level $F=0.01$ (see~\\refsec{peak_characterization} for further details).\n\n\\subsection{The peak position}\\labsubsec{peaks-position}\n\\begin{figure}[b!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{off-eq_chaos\/Nr_xi.pdf}\n  \\caption[\\textbf{Peak position increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Peak position increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Position of the peak against $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is plotted for all the simulated pair of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel).}\n \\labfig{Nr_xi}\n\\end{figure}\n\nLet us recall that the peak position indicates the most convenient length-scale for studying \\gls{TC}\\index{temperature chaos} (for a given coherence length\\index{coherence length} $\\xi$, probability $F$ and temperatures $T_1$ and $T_2$). Dimensional analysis suggests the linear fit as the natural ansatz to study the scaling of the peak position $N_{r,\\max}^{1\/3}$ with the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ (indeed, both quantities are lengths):\n\\begin{equation}\nN_{r,\\max}^{1\/3} = a \\> \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace) + b \\, \\, . \\labeq{Nr_xi}\n\\end{equation}\n\\reffig{Nr_xi} and~\\reftab{parametros_Nmax} show the fits to~\\refeq{Nr_xi}. In all cases, values of the parameter $b$ are compatible with $0$ (at the two-$\\sigma$ level). In addition, the proportional parameter $a$ shows a monotone increasing behavior with $T_2-T_1$ and with the probability $F$. Hence, our naive expectation $N_{r,\\max}^{1\/3} \\propto \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is confirmed.\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $a$ & \\hspace{1cm} &$b$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 0.60(12) & \\hspace{1cm} & 0.9(9) & \\hspace{1cm} & 22.12\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.81(7) & \\hspace{1cm} & 0.0(5) & \\hspace{1cm} & 11.52\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.93(10) & \\hspace{1cm} & 0.1(6) & \\hspace{1cm} & 5.35\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.13(13) & \\hspace{1cm} & -0.6(8) & \\hspace{1cm} & 3.99\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.86(8) & \\hspace{1cm} & -0.5(6) & \\hspace{1cm} & 43.40\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.98(8) & \\hspace{1cm} & -0.1(6) & \\hspace{1cm} & 14.90\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.08(7) & \\hspace{1cm} & -0.2(6) & \\hspace{1cm} & 22.32\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.29(5) & \\hspace{1cm} & -0.2(3) & \\hspace{1cm} & 22.30\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.47(6) & \\hspace{1cm} & -0.5(4) & \\hspace{1cm} & 7.32\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.65(6) & \\hspace{1cm} & -0.8(4) & \\hspace{1cm} & 4.83\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.19(6) & \\hspace{1cm} & 0.1(4) & \\hspace{1cm} & 53.23\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.48(9) & \\hspace{1cm} & -0.7(6) & \\hspace{1cm} & 17.19\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.63(9) & \\hspace{1cm} & -0.8(6) & \\hspace{1cm} & 10.81\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak position characterization.}]{\\textbf{Peak position characterization.} Parameters obtained in the fits of our data for $N_{r,\\max}^{1\/3}$ to \\refeq{Nr_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_Nmax}\n\\end{center}\n\\end{table}\n\n\n\\subsection{The peak height}\\labsubsec{peaks-height}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{off-eq_chaos\/fmax_xi.pdf}\n  \\caption[\\textbf{Peak height increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Peak height increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Height of the peak against $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is plotted for all the simulated pair of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel). Curves display a monotone trend with the difference of temperatures $T_2-T_1$.}\n \\labfig{fmax_xi}\n\\end{figure}\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{off-eq_chaos\/zeta_exponent_fixed.pdf}\n\t\\caption[\\textbf{The exponent $\\mathbf{\\zeta_{\\text{NE}}}$ turns out to be independent of $\\mathbf{F}$ and $\\mathbf{T_1}$.}]{\\textbf{The exponent $\\mathbf{\\zeta_{\\text{NE}}}$ turns out to be independent of $\\mathbf{F}$ and $\\mathbf{T_1}$.} The characteristic length $\\xi^*$ is plotted against the temperature difference $T_2-T_1$ in a log-log scale. Each curve is uniquely identified by the probability level $F$ and the smallest temperature of each pair $T_1$. Fits to~\\eqref{eq:def_zeta}, enforcing a common exponent, are shown with continuous lines and result in a chaotic exponent $\\zeta_\\text{NE}=1.19(2)$.}\n \\labfig{zeta_exponent}\n\\end{figure}\n\nThe peak height $H_{\\max} \\equiv H(z_{\\max})$ is an indication of the strength of \\gls{TC}\\index{temperature chaos} (for a given coherence length\\index{coherence length} $\\xi$, probability $F$ and temperatures $T_1$ and $T_2$). In order to study the scaling with $\\xi$, we have considered the following ansatz:\n\\begin{equation}\n  H_{\\max}(\\xi) = \\dfrac{\\varepsilon(\\xi)}{1+ \\varepsilon(\\xi)} \\, , \\text{ with } \\varepsilon(\\xi)=(\\xi\/\\xi^*)^\\alpha\\,. \\labeq{fmax_xi}\n\\end{equation}\nThe fit parameters are the characteristic length scale $\\xi^*$ and the exponent $\\alpha$. The rationale behind~\\refeq{fmax_xi} is that, although in cases of extremely weak chaos $1-X$ may grow with $\\xi$ as a power law, $1-X$ should eventually approach its upper bound $1-X=1$ (when chaos becomes strong).\nNevertheless, a consistency check necessary to give some physical meaning to~\\refeq{fmax_xi}, is that exponent $\\alpha$ should not depend neither on temperatures $T_1$ and $T_2$ nor on the chosen probability $F$.\n\nWe find fair fits to~\\refeq{fmax_xi}, see~\\reffig{fmax_xi} and~\\reftab{parametros_fmax}. Fortunately, in all cases exponent $\\alpha$ turns out to be very close to $\\alpha \\approx 2.1$ (actually, all the $\\alpha$ obtained in the fits turn out to be compatible with $2.1$ at the two-$\\sigma$ level). Under these conditions, we can interpret $\\xi^*$ as a characteristic length indicating the crossover from weak to strong \\gls{TC}\\index{temperature chaos}, at the probability level indicated by $F$ (the relatively large value of exponent $\\alpha$ indicates that this crossover is sharp). The trends for the crossover-length $\\xi^*(F,T_1,T_2)$ are very clear: it grows upon increasing $F$ or upon decreasing $T_2-T_1$. At this point, we can try to be more quantitative. \n\nIndeed, because $\\xi^*$ indicates the crossover between weak and strong chaos, it must be the non-equilibrium analogue of the equilibrium chaotic length $\\ell_\\text{c}(T_1,T_2)$~\\cite{fisher:86,bray:87b} (see~\\refsec{origin_tc}). Now, the equilibrium $\\ell_\\text{c}(T_1,T_2)$ has been found to scale for the 3D Ising\\index{Ising} spin glass as \n\\begin{equation}\n\\ell_\\text{c}(T_1,T_2) \\propto (T_2-T_1)^{-1\/\\zeta} \\, \\, , \\labeq{def_zeta_equilibrium}\n\\end{equation}\nwith $\\zeta \\approx 1.07$~\\cite{katzgraber:07} and $\\zeta \\approx 1.07(5)$~\\cite{fernandez:13}. These considerations suggest the following ansatz for the non-equilibrium crossover length\n\\begin{equation}\n\\xi^*(T_1,T_2,F) = B(F) \\, (T_2-T_1)^{-1\/\\zeta_{\\text{NE}}} \\, \\, , \\labeq{def_zeta}\n\\end{equation}\nwhere $B(F)$ is an amplitude.\n\nWe have tested~\\refeq{def_zeta} by computing a joint fit for four $(T_1, F)$ pairs as functions of $T_2-T_1$, allowing each curve to have its own amplitude but enforcing a common $\\zeta_\\text{NE}$ (see~\\reffig{zeta_exponent}). The resulting $\\chi^2\/\\text{d.o.f.} = 7.55\/7$\\index{degree of freedom} validates our ansatz, with an exponent $\\zeta_\\text{NE}=1.19(2)$ fairly close  to the equilibrium result $\\zeta=1.07(5)$~\\cite{fernandez:13}. This agreement strongly supports our physical interpretation of the crossover length. We, furthermore, find that $B$ is only weakly dependent on $T_1$.\nNevertheless, the reader should be warned that it has been suggested~\\cite{fernandez:13} that the equilibrium exponent $\\zeta$ may be different in the weak- and strong-chaos regimes.\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $\\xi^*$ & \\hspace{1cm} &$\\alpha$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 55(4) & \\hspace{1cm} & 2.10(7) & \\hspace{1cm} & 14.10\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 23.5(7) & \\hspace{1cm} & 2.22(5) & \\hspace{1cm} & 38.00\/19\\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 16.8(3) & \\hspace{1cm} & 2.09(4) & \\hspace{1cm} & 28.88\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 13.24(15) & \\hspace{1cm} & 2.04(3) & \\hspace{1cm} & 8.77\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 43.5(15) & \\hspace{1cm} & 2.12(5) & \\hspace{1cm} & 41.05\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 22.9(5) & \\hspace{1cm} & 2.09(4) & \\hspace{1cm} & 33.32\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 16.3(2) & \\hspace{1cm} & 2.04(4) & \\hspace{1cm} & 22.32\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 28.4(4) & \\hspace{1cm} & 2.26(2) & \\hspace{1cm} & 49.15\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 20.1(2) & \\hspace{1cm} & 2.16(2) & \\hspace{1cm} & 48.07\/19\\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 15.87(16) & \\hspace{1cm} & 2.08(2) & \\hspace{1cm} & 23.93\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 51.4(12) & \\hspace{1cm} & 2.17(3) & \\hspace{1cm} & 8.06\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 27.7(4) & \\hspace{1cm} & 2.13(2) & \\hspace{1cm} & 65.66\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 19.9(2) & \\hspace{1cm} & 2.05(2) & \\hspace{1cm} & 31.78\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak height characterization.}]{\\textbf{Peak height characterization.} Parameters obtained in the fits of our data for $H_{\\max}$ to~\\refeq{fmax_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_fmax}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{0.25cm} & $T_1$ &  \\hspace{0.25cm} & $B(F)$ & \\hspace{0.25cm} & $\\zeta_{\\text{NE}}$ & \\hspace{0.25cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & & 0.625 & & 5.77(11) & & 1.19(2) & & 2.14\/2\\\\\n\\hline\n0.01 & & 0.625 & & 6.94(14) & & 1.19(2) & & 1.57\/2\\\\\n\\hline\n0.001 & & 0.7 & & 5.99(13) & & 1.19(2) & & 2.46\/2\\\\\n\\hline\n0.01 & & 0.7 & & 7.28(16) & & 1.19(2) & & 1.38\/2\\\\\n\\hline\n\\hline\n0.001 & & 0.625, 0.7 & & 5.85(11) & & 1.19(2) & & 11.54\/5\\\\\\hline\n0.01 & & 0.625, 0.7 & & 7.08(14) & & 1.19(2) & & 21.19\/5\\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{The chaotic exponent $\\zeta$.}]{\\textbf{The chaotic exponent $\\zeta$.} The smallest temperature $T_1$ is fixed in each fit in the upper part of the table. The two last rows in the table correspond to the fit including all the available pairs of temperatures (i.e. in these fits we mix data with $T_1=0.625$ and $T_1=0.7$). Points with $(T_1=0.625,T_2=1.0)$ for both $F=0.001$ and $F=0.01$ are not considered in these fits.}\n\\labtab{zeta}\n\\end{center}\n\\end{table}\n\n\\subsection{The (inverse) peak width}\\labsubsec{peaks-width}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{off-eq_chaos\/width_xi_power_law.pdf}\n  \\caption[\\textbf{Curvature $\\kappa$ decays as a power law when increasing $\\xi$.}]{\\textbf{Curvature $\\kappa$ decays as a power law when increasing $\\xi$.} The inverse peak width $\\kappa(\\xi)$ is plotted against the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ for all the simulated pairs of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel). Similar decaying exponent $\\beta$, actually compatible at the two-$\\sigma$ level (see~\\reftab{parametros_width}), is displayed for all the pairs of temperatures in both probability levels.}\n \\labfig{width_xi}\n\\end{figure}\n\nThe peak width provides the answer to the following question: how critical is it to select the right length-scale to study \\gls{TC}\\index{temperature chaos}? Obviously, if the peak width becomes larger than its position (see~\\refsubsec{peaks-position}), this choice is no longer critical.\n\nWe study the inverse peak width (i.e. the curvature)\n\\begin{equation}\n\\kappa(\\xi)=\\sqrt{\\dfrac{|H''(z_{\\max})|}{H(z_{\\max})}} \\, , \\labeq{def_curvature}\n\\end{equation}\nand propose a power law decaying with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ characterized by the ansatz\n\\begin{equation}\n\\kappa(\\xi) = A(F) \\, \\xi^{-\\beta} \\, , \\labeq{width_xi}\n\\end{equation}\nwhere $A(F)$ is an amplitude while $\\beta$ is the power law exponent. Results are shown in~\\reffig{width_xi} and~\\reftab{parametros_width}.\n\nThe value of $A(F)$ turns out to be compatible for all the pairs of temperatures $(T_1,T_2)$ at fixed probability $F$. Furthermore, at the current precision of the data, exponent $\\beta$ does not exhibit any significant dependency on the temperature pair ($T_1,T_2$) or the probability $F$.\n\nLet us now recall the linear relation between the peak position and the coherence length\\index{coherence length}, see~\\refeq{Nr_xi}. Consider the ratio between the position of the maximum and its width, $N_{r,\\max}\\kappa(\\xi) \\sim \\xi^{1-\\beta}$. The~\\reftab{parametros_width} mildly suggests that $\\beta$ is slightly greater than 1, which implies that the ratio goes to zero (very slowly) in the limit of large $\\xi$. The parameter $\\beta$ would have, indeed, a critical meaning for the large $\\xi$ limit. If greater than one, as mildly suggested by the results, the chaotic behavior would be present at any scale $\\xi_c$. On the contrary, if further works find $\\beta < 1$, in the $\\xi \\to \\infty$ limit, the chaos would only be visible at the coherence-length scale.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $A$ & \\hspace{1cm} &$\\beta$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 0.8(3) & \\hspace{1cm} & 0.9(2) & \\hspace{1cm} & 18.72\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.6(4) & \\hspace{1cm} & 1.27(14) & \\hspace{1cm} & 8.07\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.4(3) & \\hspace{1cm} & 1.32(12) & \\hspace{1cm} & 10.05\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.3(2) & \\hspace{1cm} & 1.37(9) & \\hspace{1cm} & 5.60\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.1(3) & \\hspace{1cm} & 1.10(12) & \\hspace{1cm} & 35.26\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.26(16) & \\hspace{1cm} & 1.25(7) & \\hspace{1cm} & 25.90\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.19(17) & \\hspace{1cm} & 1.29(7) & \\hspace{1cm} & 23.01\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.63(9) & \\hspace{1cm} & 1.11(7) & \\hspace{1cm} & 20.44\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.59(10) & \\hspace{1cm} & 1.14(8) & \\hspace{1cm} & 6.08\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 0.58(15) & \\hspace{1cm} & 1.21(12) & \\hspace{1cm} & 9.05\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.59(11) & \\hspace{1cm} & 1.05(11) & \\hspace{1cm} & 21.26\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.63(8) & \\hspace{1cm} & 1.15(7) & \\hspace{1cm} & 18.46\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 0.59(12) & \\hspace{1cm} & 1.18(9) & \\hspace{1cm} & 17.93\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak width characterization.}]{\\textbf{Peak width characterization.} Parameters obtained in the fits of our data for $\\kappa(\\xi)$ to~\\refeq{width_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_width}\n\\end{center}\n\\end{table}\n\n\\subsection{On the relation between experimental and numerical results} \\labsubsec{relation_experiment_numerical_simulations}\nThis characterization of \\gls{TC}\\index{temperature chaos} in the off-equilibrium dynamics of a large \\gls{SG} paves the way to a major interplay between numerical simulations and experiments.\n\nAlthough we have considered in this work fairly small values of the chaotic system fraction $F$, a simple extrapolation, linear in $\\log F$, predicts $\\xi^* \\approx 60$ for $F=0.1$ at $T_1=0.7$ and $T_2=0.8$ (our closest pair of temperatures in~\\reftab{parametros_fmax}). A spin-glass coherence length\\index{coherence length} well above $60 a_0$ is experimentally reachable\nnowadays~\\cite{zhai:19,zhai:20b,zhai-janus:20,zhai-janus:21} ($a_0$ is the typical spacing between spins), which makes our dynamic \\gls{TC}\\index{temperature chaos} significant.\n\nIndeed the \\gls{TC}\\index{temperature chaos}-closely related experimental study~\\cite{zhai:20b} reported a value for exponent\\footnote{The authors propose several schemes and different computations of the exponent are provided.} $\\zeta_{\\text{NE}}$ in fairly good agreement with our result of $\\zeta_\\text{NE}=1.19(2)$ in~\\reffig{zeta_exponent}.\n\nA deeper relation between the \\gls{TC}\\index{temperature chaos} phenomenon in experiments and in numerical simulations would be desirable. Actually, simple temperature-varying protocols\\index{temperature-varying protocol} (in which temperature sharply drops from $T_2$ to $T_1$, see, e.g.~\\cite{zhai:20b}) seems more accessible to a first analysis than memory\\index{memory effects} and rejuvenation\\index{rejuvenation} experiments~\\cite{jonason:98,lundgren:83,jonsson:00,hammann:00}.\n\nThe rupture point between both approaches is the difference in the measured observables. An important problem is that the correlation functions that are studied theoretically are not easily probed experimentally. Instead, experimentalists privilege the magnetization\\index{magnetization!density} density (which is a spatial average over the whole sample\\index{sample}). The study of the magnetization\\index{magnetization!density} density from a numerical point of view, on the other side, is clearly bounded by the computational power available nowadays since its global nature makes the chaos signal almost disappear for our achievable coherence lengths\\index{coherence length}. Therefore an important theoretical goal is to predict the behavior of the non-equilibrium time-dependent magnetization\\index{magnetization} upon a temperature drop.\n\n\\section{Scaling at fixed \\boldmath $r$}\\labsec{scaling_fixed_r}\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/extrapola_q2_T1T1_v2.pdf}\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/log_extrapola_q2_T1T1_v2.pdf}\t\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/extrapola_q2_T1T2_v2.pdf}\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/log_extrapola_q2_T1T2_v2.pdf}\t\n  \\caption[\\textbf{Extrapolation of $\\mathbf{F(q^2)}$ to the $\\mathbf{\\xi \\to \\infty}$ limit.}]{\\textbf{Extrapolation of $F(q^2_{T_1T_1})$ (up) and  $F(q^2_{T_1T_2})$ (down) to the $\\xi \\to \\infty$ limit, both in linear scale (left) and logarithmic scale (right).} \\textbf{Main plots:} We extrapolate the distribution functions for spheres of radius $r=1$ to the $\\xi \\to \\infty$ limit. Both ans\\\"atze, the one in~\\refeq{extrapolacion_estatica_lineal}, golden curves, and the one in~\\refeq{extrapolacion_estatica_cuadratica}, blue curves, produce equivalent extrapolations. For the sake of clarity, we plot two curves for each extrapolation (with the same color) which corresponds to the computed central value plus (minus) the standard deviation. We compare the results with the equilibrium data from an \\gls{EA}\\index{Edwards-Anderson!model} model in a cubic lattice of linear size $L=32$ (red curve, see main text for further details). The equilibrium distribution and the extrapolated one are compatible for the $q^2_{T_1T_1}$ curves. Instead, $q^2_{T_1T_2}$ is compatible for percentiles of order one but not for probabilities smaller than $F=0.1$. \\textbf{Insets:} As in the main plots, for spheres of radius $r=2$.}\n\\labfig{extrapola_estatica_q2}\n\\end{figure}\n\nUntil this point, the analysis of the chaotic parameter (its characterization through the size of the sphere and the dependence of the peak with the coherence length\\index{coherence length}) have been inspired by the representation of the data given by the~\\reffig{distribution_function_r} and the idea of an optimal scale to observe \\gls{TC}\\index{temperature chaos}. However, a different approach can be performed by regarding the representation of the data in~\\reffig{distribution_function_xi}.\n\nFixing the two temperatures $T_1$ and $T_2$ and the size of the sphere, the distribution function $F(X,T_1,T_2,\\xi,r)$ exhibit a monotonic behavior with the coherence length\\index{coherence length} $\\xi$ (see~\\reffig{distribution_function_xi}). Specifically, regarding at the lowest possible size of the sphere $r=1$, we found there was no convergence (apparently) to a limit curve when increasing the coherence length\\index{coherence length} $\\xi$.\n\nThe absence of a limit curve led us to consider slow convergence as a hypothesis. In this section, we propose algebraic extrapolations to explain the convergence to the $\\xi \\to \\infty$ limit. Moreover, we compare the extrapolations with the equilibrium data obtained from an \\gls{EA}\\index{Edwards-Anderson!model} Ising\\index{Ising} model simulated with a \\gls{PT} algorithm in a cubic lattice of linear size $L=32$ (see~\\cite{janus:10,janus:10b}). We expect that, for the smallest sphere radius at least, $L=32$ will be representative of the thermodynamic limit\\index{thermodynamic limit}.\n\nLet us fix the probability level $F$ and the radius $r$ of the spheres, we propose two different extrapolations to the $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace) \\to \\infty$ limit\n\\begin{equation}\n\\Omega(T_1,T_2,\\xi,r,F) = \\Omega(T_1,T_2,\\xi= \\infty,r,F) +  \\Phi(F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{\\delta(F)} \\, , \\labeq{extrapolacion_estatica_lineal}\n\\end{equation}\n\\begin{equation}\n\\Omega(T_1,T_2,\\xi,r,F) = \\Omega(T_1,T_2,\\xi= \\infty,r,F) + \\Psi_1 (F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{\\epsilon(F)} + \\Psi_2(F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{2\\epsilon(F)} \\, , \\labeq{extrapolacion_estatica_cuadratica}\n\\end{equation}\nwhere $\\Omega(T_1,T_2,\\xi,r)$ is the extrapolated quantity $\\Omega \\in \\lbrace q^2_{T_1T_1},q^2_{T_2T_2},q^2_{T_1T_2},X_{T_1T_2} \\rbrace$ at the fixed $F$ probability level, $\\Omega(T_1,T_2,\\xi= \\infty,r)$ is the value of that quantity in the $\\xi \\to \\infty$ limit, $\\Phi$, $\\Psi_1$ and $\\Psi_2$ are the coefficients of the fit, and $\\delta$ and $\\epsilon$ are the exponents of the fits.\n\nFrom now on, we focus on the $r=1$ and $r=2$ cases. The extrapolations for $q^2_{T_1T_1}$ and $q^2_{T_2T_2}$ show the agreement between the extrapolations to the $\\xi \\to \\infty$ limit and the equilibrium data, see~\\reffig{extrapola_estatica_q2} (up panels), for the whole range. Besides, the quadratic and the linear extrapolations are statistically equivalent.\n\nThe extrapolation for $q^2_{T_1T_2}$ keeps the agreement between the extrapolations to the $\\xi \\to \\infty$ limit and the equilibrium data for high values of the distribution function, but differs for short values of $F(q^2_{T_1T_2})$, see~\\reffig{extrapola_estatica_q2} (down panels). Again, linear and quadratic extrapolation show no difference within the statistical error.\n\nFor the chaotic parameter $X_{T_1T_2}$ we find no convergence at all at the current precision of the data and the simulated scales of the coherence length\\index{coherence length}.\n\nWe also explore the $r>2$ case, however, the greater the sphere size $r$, the bigger the simulated coherence length\\index{coherence length} $\\xi$ needed for a reliable fit to~\\refeq{extrapolacion_estatica_lineal} and~\\refeq{extrapolacion_estatica_cuadratica}. Extreme events stop providing reliable fits when increasing the sphere size $r$.\n\nIndeed, this approach provides reasonable results in the extrapolation to the $\\xi \\to \\infty$ limit for $q^2$ distribution functions when compared with the equilibrium data. However, the convergence is slow and the fits are difficult at our current precision.\n\nFinally, we examine our results in search of scale invariance. Due to the compatibility of the linear and quadratic extrapolations, we will focus on the former for the sake of simplicity, see~\\refeq{extrapolacion_estatica_lineal}. We study the $r=1$ and $r=2$ cases which more likely hold the limit $r\/\\xi \\ll 1$. \n\nIf the scale invariance is present in our results, we would expect the exponent $\\delta$ to be independent of the radius $r$. We find a similar behavior for the exponent in the extrapolations for spheres of radius $r=1$ and $r=2$ (\\reffig{factor_exponente} main plots), nevertheless the results are not compatible for all the percentiles $F(q^2_{T_1T_1})$ and $F(q^2_{T_1T_2})$.\n\nWe also expect the ratio of the coefficients $\\Phi(r=2)\/\\Phi(r=1)$ to be constant if the scale invariance holds. We find small changes in the ratio between the amplitudes $\\Phi(F,r)$, see~\\reffig{factor_exponente} insets. The ratio remains around $1$ for all the percentiles, however, the results are not compatible for the whole range with $1$.\n\nOur results mildly suggest the existence of scale invariance for the limit $r\/\\xi \\ll 1$, in our cases represented by the simulations with sphere radius $r=1$ and $r=2$. However, the difficulty of the extrapolations and the noise in the amplitude and the exponent, prevent us from making robust statements.\n\n\n\\begin{figure}[h!]\n  \\centering\t\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/factor_exponente.pdf}\n  \\caption[\\textbf{Scale invariance test.}]{\\textbf{Scale invariance test for parameters in~\\refeq{extrapolacion_estatica_lineal}.} Indeed, scale invariance demands that both the exponent $\\delta(F)$ and the amplitude $\\Phi(F)$ should be independent of the sphere radius. The figure compares both quantities for the radius $r=1$ and $r=2$. \\textbf{Main panels:} exponent $\\delta(F)$ as computed for spheres of radius $r=1$ (golden curves) and $r=2$ (blue curves), as a function of $F(q^2_{T_1T_1})$ (left panel) or $F(q^2_{T_1T_2})$ (right panel). \\textbf{Insets: }Ratio of amplitudes $\\Phi(F,r=2)\/\\Phi(F,r=1)$ for the same fits to~\\refeq{extrapolacion_estatica_lineal} reported in the main panels. The horizontal black line is set at $1$ as a reference.}\n\\labfig{factor_exponente}\n\\end{figure}\n\n\n\\section{Temperature changing protocols}\\labsec{T-changes}\n\nIn this section, we explore the impact of temperature-varying\\index{temperature-varying protocol} simulations in the \\gls{TC}\\index{temperature chaos} results and we also address the cumulative aging\\index{aging!cumulative} controversy: Is the aging\\index{aging} performed at temperature $T_a$ useful at different temperature $T_b$?\n\nSpecifically, the cumulative aging\\index{aging!cumulative} hypothesis~\\cite{jonsson:02,bert:04} considers a protocol in which the system is first suddenly quenched from high temperature to temperature $T_a < \\ensuremath{T_\\mathrm{c}}\\xspace$, where it ages for time $t_a$. Then, the temperature is changed to $T_b < \\ensuremath{T_\\mathrm{c}}\\xspace$ and the system evolves for a time $t_b$. The hypothesis is made that this two-steps protocol is equivalent to isothermal\\index{isothermal} aging\\index{aging} at temperature $T_b$ for an effective time $t^{\\mathrm{eff}}_b$ defined through the equation \n\\begin{equation}\n\\xi (t_a ; t_b ; T_a \\rightarrow T_b) = \\xi(t^{\\mathrm{eff}}_b,T_b) \\> . \\labeq{cumulative_hypotesis}\n\\end{equation}\n\nFortunately, we now know how to compute the coherence length\\index{coherence length} $\\xi$~\\cite{janus:09b,fernandez:18b} accurately (see also \\refsec{finite_size_effects}). Therefore, we are able to compare the evolution of systems that have undergone either the two-steps protocol or the isothermal\\index{isothermal} one, choosing the time scales to precisely match~\\refeq{cumulative_hypotesis}.\n\nHere, we will consider two symmetric two-steps proposals. On the one hand, we let age the system at temperature $T_a=0.8$ (by \\textit{the system} we mean the $\\ensuremath{N_{\\text{S}}}\\xspace$ samples\\index{sample} and the $\\ensuremath{N_{\\text{Rep}}}\\xspace$ replicas\\index{replica}) until it reaches coherence length\\index{coherence length} $\\xi(t_a)=6$. At this point, we suddenly quench to the temperature $T_b=1.0$ and let the system evolve until $\\xi(t_a;t_b; T_a=0.8 \\rightarrow T_b=1.0)=9$. On the other hand we perform the symmetric two-steps protocol, starting at temperature $T_a=1.0$ and cooling the system to temperature $T_b=0.8$. On the technical side, let us mention that we compute the distribution function as explained in~\\refsec{procedure}.\n\nAs everywhere else in this chapter, we will compute the overlaps\\index{overlap} between configurations\\index{configuration} at two temperatures $T_1<T_2$. However, we shall need to introduce notation in order to specify the protocol. \n\nWe will put the superscript \\textit{iso} on a temperature to indicate that the system has always been at that temperature (\\textit{iso} is a short for isothermal\\index{isothermal}). For instance, the notation $\\ensuremath{T^{\\text{iso}}}\\xspace_1 = 0.8$ will mean that we are computing the overlap\\index{overlap} between a system that has always been at temperature $0.8$ and a system that currently is at some higher temperature. Instead, $\\ensuremath{T^{\\text{iso}}}\\xspace_2 = 0.8$ means that the overlap\\index{overlap} is computed between our isothermal\\index{isothermal} system and a system which currently is at some lower temperature.\n\nWe will put the superscript $T_a \\to T_b$ on a temperature to indicate that the system has followed the two-steps protocol from temperature $T_a$ to temperature $T_b$. Notice that the current temperature of the system is $T_b$. For instance, the notation $X(\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8,T^{0.8 \\to 1.0}_2=1.0)$ indicates that we are computing overlaps\\index{overlap} between the configurations\\index{configuration} of a system that has always been at temperature $0.8$ and those of a system that has followed the two-steps protocol from temperature $T_a=0.8$ to $T_b=1.0$.\n\nWe organize the obtained results into two subsections. The first one (see~\\refsubsec{no-equivalence}) captures the asymmetric behavior when the system is cooled or heated. The second one (see~\\refsubsec{auxiliary}) takes advantage of an auxiliary temperature to study the effect of the two-steps protocol on the \\gls{TC}\\index{temperature chaos} phenomenon.\n\n\\subsection{On the asymmetry between cooling and heating}\\labsubsec{no-equivalence}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/no_auxiliar.pdf}\n  \\caption[\\textbf{Comparison between isothermal and two-step protocol.}]{\\textbf{Comparison of the main parameters in the \\boldmath $1-X$ curves for the two-steps protocol and the isothermal\\index{isothermal} protocol. The two protocols approach to each other for large $\\xi$.} Position of the peak (up panels), height of the peak (middle panels) and curvature (down panels) against $\\xi(t_w)$ are plotted for the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ and for the temperature-varying\\index{temperature-varying protocol} simulations at two different probability levels $F=0.001$ (left) and $F=0.01$ (right). Heating and cooling two-steps protocols are not equivalent.}\n \\labfig{xi_noaux}\n\\end{figure}\n\nWe analyze the \\gls{TC}\\index{temperature chaos} phenomenon through the same three observables used in~\\refsec{results}: the peak position, the peak height, and the (inverse) peak width (i.e. the curvature). Each of those quantities are extracted from one of the following overlap\\index{overlap} combination. First we have the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$. Next we have the so called \\textit{cooling} curve $T_1^{1.0 \\to 0.8} = 0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$. Finally we have the so called \\textit{heating} curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $T_2^{0.8 \\to 1.0} = 1.0$.\n\nThe prediction of the cumulative aging\\index{aging!cumulative}, see~\\refeq{cumulative_hypotesis}, is that the three curves, namely the reference, the \\textit{heating} and the \\textit{cooling} curves, should coincide. Indeed, we choose the times in such a way that the coherence length\\index{coherence length} is the same for all the four systems involved: $\\ensuremath{T^{\\text{iso}}}\\xspace=0.8$, $\\ensuremath{T^{\\text{iso}}}\\xspace=1.0$, $T^{0.8 \\to 1.0} = 1.0$ and $T^{1.0 \\to 0.8} = 0.8$. Instead, our data in~\\reffig{xi_noaux} show that cumulative aging\\index{aging!cumulative} holds only in an approximated way. In fact, both the \\textit{heating} and the \\textit{cooling} curves significantly differ from the reference curve at coherence length\\index{coherence length} $\\xi \\approx 6$, although the three curves tend to merge as $\\xi$ grows. Nevertheless, we notice that the \\textit{cooling} curve does not converge to the other two curves, at least in the simulated range of $\\xi$. In this sense, there exist an asymmetry between the \\textit{cooling} curve and the \\textit{heating} curve. Furthermore, by comparing the left and right panels in~\\reffig{xi_noaux}, we conclude that the larger the probability $F$, the stronger the asymmetry.\n\nFinally, the reader may wonder about the possibility of computing overlaps\\index{overlap} between configurations\\index{configuration} at temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $T_2^{1.0 \\to 0.8} = 0.8$ or configurations\\index{configuration} at temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=1.0$ and $T_2^{0.8 \\to 1.0} = 1.0$. Indeed, we have attempted to compute those overlaps\\index{overlap} but we found very large values of the chaotic parameter that made impossible the characterization of the peak.\n\n\\subsection{Auxiliary-temperature analysis}\\labsubsec{auxiliary}\nIn this subsection, we study the behavior of the two-steps protocol simulations by comparing them with the most ordered configurations\\index{configuration} we have, namely, the simulations of temperature $\\ensuremath{T^{\\text{iso}}}\\xspace=0.625$. We establish two reference curves $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ and we compare them with the curves $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{1.0 \\to 0.8}=0.8$, and $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{0.8 \\to 1.0}=1.0$.\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/Nr_width_xi_changeT08.pdf}\t\n  \\caption[\\textbf{Temperature-varying\\index{temperature-varying protocol} simulations of the peak position and curvature exhibit non-symmetric behavior when symmetric cooling and heating protocols are applied.}]{\\textbf{Temperature-varying\\index{temperature-varying protocol} simulations of the peak position and curvature exhibit non-symmetric behavior when symmetric cooling and heating protocols are applied.} The peak position (upper panels) and the curvature (lower panels) are plotted against the coherence length\\index{coherence length} $\\xi(t_w)$. We use a constant auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ combined with two fixed-temperature simulations as the reference curves ($\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$) and the constant auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ combined with two temperature-varying\\index{temperature-varying protocol} simulations as the studied curves ($T^{0.8 \\to 1.0}_2=1.0$ and $T^{1.0 \\to 0.8}_2=0.8$). We compute all the curves at the probability levels $F=0.001$ (left panels) and $F=0.01$ (right panels).}\n \\labfig{Nr_width_xi_changeT}\n\\end{figure}\n\nWe can observe the migration, as the coherence length\\index{coherence length} grows, of the curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{0.8 \\to 1.0}_2=1.0$ from the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ to the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ by increasing the position of the peak (\\reffig{Nr_width_xi_changeT} upper panels) and the curvature (\\reffig{Nr_width_xi_changeT} lower panels). However, the opposite temperature-change, namely the curve corresponding to $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $T^{1.0 \\to 0.8}_2=0.8$, does not lead to the symmetric behavior. The position of the peak (respectively, the curvature) does not decrease (respectively, increase) when cooling the temperature to merge with the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$. Thus, the optimal scale to observe \\gls{TC}\\index{temperature chaos} is a monotonically increasing function of the coherence length\\index{coherence length} while the curvature seems a monotonically decreasing function of the coherence length\\index{coherence length}.\n\nAlthough we do not know the behavior of the peak position and the peak curvature for values of the coherence length\\index{coherence length} higher than those simulated, one could expect, as a naive ansatz, that the curve relative to the \\textit{cooling} protocol would merge with its corresponding reference curve ($\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$) when the coherence length\\index{coherence length} is increased and then it would behave as the isothermal\\index{isothermal} simulation.\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/fmax_xi_changeT08_twopanels.pdf}\t\n  \\caption[\\textbf{Peak height of cooled two-steps protocol simulations with an auxiliary temperature converges slower than the symmetric heating protocol.}]{\\textbf{Peak height of cooled two-steps protocol simulations with an auxiliary temperature converges slower than the symmetric heating protocol.} The height of the peak is plotted against the coherence length\\index{coherence length} $\\xi(t_w)$ for fixed-temperature simulations and for two-steps protocol simulations with an auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ at the probability levels $F=0.001$ (left panel) and $F=0.01$ (right panel).}\n \\labfig{fmax_xi_changeT}\n\\end{figure}\n\nAs for the peak height (\\reffig{fmax_xi_changeT}), we may appreciate both studied curves approaching the reference curves. However, the same information provided in the previous subsection (\\refsubsec{no-equivalence}) emerges here: the curve with the \\textit{cooling} protocol needs a higher coherence length\\index{coherence length} to converge to the reference curve. \n\nRecalling the fact that the height of the peak is related to the strength of \\gls{TC}\\index{temperature chaos}, the system tends to erase the memory of its previous relaxation\\index{relaxation} state and exhibits the same amount of \\gls{TC}\\index{temperature chaos} as it could be expected from its current pair of temperatures. Nonetheless, the erasing process is not complete in the cooling protocol and a small amount of information of the previous state lasts in the system, at least for the simulated coherence lengths\\index{coherence length}.\n\nLet us conclude this section with a small summary of the results. The reader might imagine a \\gls{TC}\\index{temperature chaos} movie in which each frame corresponds to a function that represents $1-X(F,T_1,T_2,\\xi,r)$ against the size of the studied region $N_r^{1\/3}$, similar to those appearing in~\\reffig{Xvsr_examples}. When both temperatures $T_1$ and $T_2$ follow an isothermal\\index{isothermal} protocol, the function will, monotonically, increase the height of its peak, will move to the right (i.e. higher values of the sphere size), and will become \\textit{flatter} (i.e. the curvature decreases). \n\nThe same situation is expected to happen (qualitatively) when $T_2$ is suddenly heated following the two-steps protocol. However, in this situation, the increase of the peak position, the peak height, and the decrease of the curvature \\textit{accelerates} (i.e. requires shorter values of coherence length\\index{coherence length}) compared with the $T_2$-isothermal\\index{isothermal} movie.\n\nFinally, when $T_2$ is suddenly cooled following the two-steps protocol, the height of the peak will decrease, while the position of the peak and the curvature will \\textit{freeze}. The naive expectation is that the \\textit{freezing} of the position of the curve and the curvature will end at higher values of the coherence length\\index{coherence length} than the simulated ones and then, the curve will behave as the isothermal\\index{isothermal} one. \n\n\n\n\\section*{Preface}\n\n\\input{chapters\/aknowledgment.tex}\n\\checkoddpage\n\\ifoddpage{\n\t\\afterpage{\\blankpage}\n}\\fi\n\n\\oldmainmatter\n\n\\setlength{\\overflowingheadlen}{\\linewidth}\n\\addtolength{\\overflowingheadlen}{\\marginparsep}\n\\addtolength{\\overflowingheadlen}{\\marginparwidth}\n\n\\pagestyle{scrheadings}\n\\input{chapters\/general_introduction.tex}\n\\addpart{A brief introduction to Spin Glasses}\n\\input{chapters\/Introduction.tex}\n\\addpart{Metastate}\n\\input{chapters\/Metastate.tex}\n\\addpart{Off-equilibrium phenomena}\n\\input{chapters\/Aging}\n\\input{chapters\/Mpemba}\n\\addpart{Temperature Chaos}\n\\input{chapters\/Introduction_chaos}\n\\input{chapters\/Equilibrium_chaos}\n\\input{chapters\/Out_equilibrium_chaos}\n\\addpart{Conclusions}\n\\input{chapters\/Conclusions}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary information}\n\n\\begin{table}[hbtp]\n\\begin{center}\n\\caption{The complete listing of the reflections used to index the Fast Fourier transforms of region (A) of the HRTEM image in Fig~\\ref{hrtem-23-a}. Reflections are assigned to  \\ce{Mg}, with beam direction, $\\boldsymbol{B}=[2\\overline{4}2\\overline{3}]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$. Interplanar angles are measured with respect to the $10\\overline{1}0$ reflection (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided.\n\\label{tab-fft-hrtem-23a}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t& \t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){5-7}\\cmidrule(l){8-10} \n\t&\t\t&&\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$) &\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\t$\\cdot$\t&l\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t\t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n1\t&\t0\t&\t$\\overline{1}$\t&\t0*\t&\t0.279\t&\t0.277\t&\t0.01\t&\t0.0\t&\t0.0\t&\t0.0\t\\\\\n$\\overline{1}$\t&\t0\t&\t1\t&\t0\t&\t0.271\t&\t0.277\t&\t0.02\t&\t179.7\t&\t180.0\t&\t0.3\t\\\\\n1\t&\t$\\overline{1}$\t&\t0\t&\t2\t&\t0.188\t&\t0.190\t&\t0.01\t&\t70.3\t&\t70.0\t&\t0.3\t\\\\\n0\t&\t$\\overline{1}$\t&\t1\t&\t2\t&\t0.188\t&\t0.190\t&\t0.01\t&\t110.9\t&\t110.0\t&\t0.9\t\\\\\n$\\overline{1}$\t&\t1\t&\t0\t&\t$\\overline{2}$\t&\t0.189\t&\t0.190\t&\t0.00\t&\t110.1\t&\t110.0\t&\t0.1\t\\\\\n0\t&\t1\t&\t$\\overline{1}$\t&\t$\\overline{2}$\t&\t0.189\t&\t0.190\t&\t0.00\t&\t70.4\t&\t70.0\t&\t0.4\t\\\\\n1\t&\t1\t&\t$\\overline{2}$\t&\t$\\overline{2}$\t&\t0.135\t&\t0.136\t&\t0.01\t&\t137.8\t&\t137.6\t&\t0.3\t\\\\\n$\\overline{1}$\t&\t$\\overline{1}$\t&\t2\t&\t2\t&\t0.136\t&\t0.136\t&\t0.00\t&\t42.8\t&\t42.4\t&\t0.4\t\\\\\n2\t&\t0\t&\t$\\overline{2}$\t&\t0\t&\t0.140\t&\t0.139\t&\t0.01\t&\t0.0\t&\t0.0\t&\t0.0\t\\\\\n$\\overline{2}$\t&\t0\t&\t2\t&\t0\t&\t0.138\t&\t0.139\t&\t0.01\t&\t180.0\t&\t180.0\t&\t0.0\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{A complete listing of the reflections used to index the Fast Fourier transforms of region (B) of the HRTEM image in Fig~\\ref{hrtem-23-a}. Reflections are assigned to the  \\ce{Mg4Zn7} phase, with beam direction, $\\boldsymbol{B}=[010]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$. Interplanar angles are measured with respect to the 206 reflection (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided.\\label{tab-fft-hrtem-23b}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){4-6}\\cmidrule(l){7-9} \n\t&\t\t&\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$\t)&\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\tl\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t \t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n\n2\t&\t0\t&\t6*\t&\t0.222\t&\t0.228\t&\t0.00\t&\t0\t&\t0.0\t&\t0\t\\\\\n$\\overline{4}$\t&\t0\t&\t6\t&\t0.222\t&\t0.244\t&\t0.06\t&\t33.2\t&\t32.2\t&\t1.0\t\\\\\n4\t&\t0\t&\t$\\overline{6}$\t&\t0.220\t&\t0.244\t&\t0.07\t&\t148.1\t&\t147.8\t&\t0.3\t\\\\\n$\\overline{2}$\t&\t0\t&\t$\\overline{6}$\t&\t0.218\t&\t0.228\t&\t0.02\t&\t179.5\t&\t180.0\t&\t0.5\t\\\\\n12\t&\t0\t&\t$\\overline{3}$\t&\t0.213\t&\t0.211\t&\t0.04\t&\t97.8\t&\t92.6\t&\t5.2\t\\\\\n$\\overline{12}$\t&\t0\t&\t3\t&\t0.210\t&\t0.211\t&\t0.02\t&\t83.0\t&\t87.4\t&\t4.4\t\\\\\n8\t&\t0\t&\t5\t&\t0.178\t&\t0.194\t&\t0.05\t&\t23.4\t&\t26.6\t&\t3.2\t\\\\\n$\\overline{8}$\t&\t0\t&\t$\\overline{5}$\t&\t0.176\t&\t0.194\t&\t0.06\t&\t156.1\t&\t153.4\t&\t2.7\t\\\\\n$\\overline{8}$\t&\t0\t&\t7\t&\t0.167\t&\t0.194\t&\t0.11\t&\t45.4\t&\t46.8\t&\t1.4\t\\\\\n8\t&\t0\t&\t$\\overline{7}$\t&\t0.164\t&\t0.194\t&\t0.12\t&\t135.5\t&\t133.2\t&\t2.2\t\\\\\n12\t&\t0\t&\t0\t&\t0.192\t&\t0.211\t&\t0.06\t&\t63.7\t&\t67.4\t&\t3.6\t\\\\\n$\\overline{12}$\t&\t0\t&\t0\t&\t0.191\t&\t0.211\t&\t0.07\t&\t116.6\t&\t112.6\t&\t3.9\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{A complete listing of the reflections used to index the Fast Fourier transforms of region (B) of the HRTEM image in Fig~\\ref{hrtem-n20}. The reflections are assigned to the \\ce{MgZn2} phase, with beam direction, $\\boldsymbol{B}=[0001]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$.     Interplanar angles are measured with respect to the $\\overline{1}2\\overline{1}0$ reflection  (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided. \\label{tab-fft-_N20_6}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t& \t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){5-7}\\cmidrule(l){8-10} \n\t&\t\t& &\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$) &\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\t$\\cdot$\t&l\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t\t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n$\\overline{1}$\t&\t2\t&\t$\\overline{1}$\t&\t0*\t&\t0.259\t&\t0.260\t&\t0.00\t&\t0.3\t&\t0.0\t&\t0.3\t\\\\\n1\t&\t1\t&\t$\\overline{2}$\t&\t0\t&\t0.226\t&\t0.260\t&\t0.13\t&\t63.7\t&\t60.0\t&\t3.7\t\\\\\n2\t&\t$\\overline{1}$\t&\t$\\overline{1}$\t&\t0\t&\t0.227\t&\t0.260\t&\t0.13\t&\t116.5\t&\t120.0\t&\t3.5\t\\\\\n1\t&\t$\\overline{2}$\t&\t1\t&\t0\t&\t0.252\t&\t0.260\t&\t0.03\t&\t177.1\t&\t180.0\t&\t2.9\t\\\\\n$\\overline{1}$\t&\t$\\overline{1}$\t&\t2\t&\t0\t&\t0.222\t&\t0.260\t&\t0.14\t&\t118.3\t&\t120.0\t&\t1.8\t\\\\\n$\\overline{2}$\t&\t1\t&\t1\t&\t0\t&\t0.237\t&\t0.260\t&\t0.09\t&\t64.5\t&\t60.0\t&\t4.5\t\\\\\n0\t&\t3\t&\t$\\overline{3}$\t&\t0\t&\t0.147\t&\t0.150\t&\t0.02\t&\t32.6\t&\t30.0\t&\t2.6\t\\\\\n3\t&\t0\t&\t$\\overline{3}$\t&\t0\t&\t0.133\t&\t0.150\t&\t0.11\t&\t89.7\t&\t90.0\t&\t0.3\t\\\\\n3\t&\t$\\overline{3}$\t&\t0\t&\t0\t&\t0.143\t&\t0.150\t&\t0.05\t&\t146.0\t&\t150.0\t&\t4.0\t\\\\\n0\t&\t$\\overline{3}$\t&\t3\t&\t0\t&\t0.144\t&\t0.150\t&\t0.04\t&\t147.9\t&\t150.0\t&\t2.1\t\\\\\n$\\overline{3}$\t&\t0\t&\t3\t&\t0\t&\t0.132\t&\t0.150\t&\t0.12\t&\t91.7\t&\t90.0\t&\t1.7\t\\\\\n$\\overline{3}$\t&\t3\t&\t0\t&\t0\t&\t0.149\t&\t0.150\t&\t0.01\t&\t34.9\t&\t30.0\t&\t4.9\t\\\\\n$\\overline{2}$\t&\t4\t&\t$\\overline{2}$\t&\t0\t&\t0.130\t&\t0.130\t&\t0.00\t&\t5.5\t&\t0.0\t&\t5.5\t\\\\\n2\t&\t$\\overline{4}$\t&\t2\t&\t0\t&\t0.130\t&\t0.130\t&\t0.00\t&\t178.5\t&\t180.0\t&\t1.5\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\section{Introduction}\n\nSevere plastic deformation (SPD) comprises a suite of techniques which impose high plastic strain on a material in order to reduce the grain size to the micron or nanometer regime. Such grain refinement substantially increases the yield stress via the Hall-Petch effect, while simultaneously improving the ductility \\cite{Yamashita2001,DingChang2009,LiWei2011}, and can lead to high-strain rate superplasticity \\cite{Komura1998}. This desirable combination of properties has generated great interest in SPD techniques, and particularly in equal-channel angular extrusion (ECAE) and high-pressure torsion (HPT) which impose high strains with no net shape change. \n\nThe ultrafine grain sizes that develop during HPT are inherently thermodynamically unstable \\cite{WangIwahashi1996} and restriction of grain growth during and after processing is important in order to retain the improved mechanical properties. Grain growth occurs more rapidly and at lower temperatures than usual due to the high defect densities and the resulting ultra-fast atomic diffusion that occurs during SPD \\cite{Amouyal2007,Divinski2010,Ribbe2009a}. During HPT, dynamic grain growth occurs even without external heating and results in ``saturation'' where the rate of dynamic grain growth balances that of strain-induced recrystallisation and a further increase in strain will not lead to a reduction of grain size.\n\nOne approach used to restrict grain growth is the use of fine precipitates to pin grain boundaries. For example, fine-scale precipitation of Al-Sc particles has been shown to reduce the rate of grain growth of ECAE Al-Mg-Sc \\cite{Komura1998} leading to high-strain rate superplasticity. More recently it was demonstrated that grain sizes of approximately 140\\,nm could be achieved in HPT-deformed Mg-Zn \\cite{MengRosalie2014}. This is a much more refined grain size than in pure Mg, which reaches saturation at a grain size of  $\\sim1\\,\\mu$m \\cite{EdalatiHpt2011}. The difference in grain size was attributed to the presence of fine-scale zinc-rich grain boundary precipitates that were observed in high angle annular dark field (HAADF) STEM images after 20 revolutions of HPT deformation.\n\nIn this work we identify the precipitates formed during HPT of a Mg-Zn alloy using high resolution transmission electron microscopy (HRTEM), and quantify the precipitate size distribution using small angle X-ray scattering (SAXS). The complementary TEM and SAXS methodology has recently been shown to be a valuable tool for quantifying particle size and volume fraction in similar alloys \\cite{RosaliePauw2014} and here we demonstrate its applicability to studies of  precipitation in SPD materials.\n\n\\section{Experimental details}\n\nThe experiments were conducted on disc-shaped samples (10\\,mm diameter, $\\sim$0.85\\,mm height) cut in cross-section from an extruded rod of a Mg-3.4at.\\%Zn alloy (The alloy casting and prior heat treatment is described in detail by Rosalie and Pauw \\cite{RosaliePauw2014}). Discs were solution-treated at 300$^\\circ$C for 1\\,h in an Ar atmosphere and quenched in ambient temperature water. They were subsequently deformed at room temperature by HPT. Each disc was compressed with an applied pressure of 5\\,GPa. This load was maintained for 30\\,s before and during the application of rotational strain at a rotation speed of 1\\,RPM. Previous studies on a \\ce{Zr50(Cu , Al)50} alloy under similar deformation conditions found that frictional heating only  increased the anvil temperature adjacent to the sample to $\\sim$306\\,K \\cite{MengTsuchiya2013}. The temperature increase for the present material is expected to be significantly less and insufficient to facilitate precipitation during HPT. The height of the discs after HPT treatment was $\\sim$0.55\\,mm.\n\nTEM foils were prepared from two HPT discs, one of which was subjected to 3\\,HPT rotations (N=3), and the second subjected to 20 rotations (N=20). These conditions were selected as N$>$1 was required to give a homogeneous hardness distribution and microstructure \\cite{MengRosalie2014}, thus ensuring that the TEM foils are representative of the disc as a whole. The N=3 condition corresponds to the peak hardness condition, whereas the N=20 sample was in the strain-saturated condition \\cite{MengRosalie2014}. The samples were prepared for TEM by mechanical grinding and polishing, followed by dimple grinding and thinning to perforation by precision ion polishing. Foils were examined using JEOL 2100 and 2100F microscopes, operating at 200\\,kV. The precipitate phases were identified from Fast Fourier transforms of high-resolution images. A Hann filter was used to remove high frequency noise from the region of interest before performing the FFT.\n\nSAXS analysis was performed on samples in four different states: one sample preceding the HPT treatment (i.e. solution-treated, ``ST''), one where the sample was compressed but not rotated (``Compressed'', or N=0), one sample after 1\\,HPT rotation (N=1), and one at N=20. The samples were thinned to approximately 0.1\\,mm, and measured for 21600\\,s each. SAXS measurements were performed on a Bruker Nanostar instrument with a chromium target producing 5.4\\,keV photons, and a sample-to-detector distance of 1.05m, resulting in a Q-range coverage of $0.05\\leq$Q$\\leq 1.3\\, \\mbox{nm}^{-1}$ (with $Q=4\\pi\/\\lambda \\sin(\\theta)$, where $2\\theta$ denotes the scattering angle, and $\\lambda$ the X-ray photon wavelength). A photon-counting wire array (delay line) detector was used to collect the scattered photons.\n\nCollected images were corrected for image distortion using Bruker's ``2D SAXS'' software supplied with the instrument. The following corrections were subsequently performed using an in-house developed data correction procedure (see \\cite{Pauw2013a} for the details of each correction step): data read-in corrections (DS), darkcurrent (DC), pixel masking (MK), flatfield (FF), time (TI), flux (FL), transmission (TR), sample self-absorption (SA), spherical distortion (SP), background (BG), and thickness (TH), followed by a data integration in logarithmically-spaced Q-bins. Uncertainty estimates, used as weights in the fitting procedure, have been determined as the largest of either the propagated Poisson uncertainties, 1\\% of the intensity in the bin, or the standard error in the bin. \n\nAfter correction, the data was subjected to a Monte Carlo-based size distribution determination procedure assuming spherical scatterers \\cite{Pauw2013}, whose size (radii) range is defined by the Q-range to between 2.5 and 60\\,nm. On average, the scattering patterns fit to within the uncertainty of the data over the entire Q-range (i.e. to $\\chi^2_r \\leq 1$). \n\n\\section{Results and discussion}\n\n\\subsection{TEM}\n\nTEM observations on samples deformed to N=3 show narrow regions of stronger atomic contrast at the grain boundaries. These darker regions were typically 5--20\\,nm in calliper diameter (see Fig.~\\ref{hrtem-15}, in agreement with SAXS results) with widths of 5-10\\,nm, no clearly defined facets, and extended along the grain boundary as a film. Figure~\\ref{hrtem-23-a} shows a HRTEM image of the grain boundary region in this condition.  The two Mg grains present ($A$, left) and  ($C$, right)  are separated by a darker region  ($B$) at the grain boundary. A FFT of region $B$ is shown in Fig~\\ref{hrtem-23-fft-Mg4Zn7} with reflections assigned to the  \\ce{Mg4Zn7} phase. This phase generally forms in the shape of high aspect ratio rods in isothermally aged Mg-Zn alloys \\cite{Gao2007,Singh2007}. \\ce{Mg4Zn7} has been reported at grain boundaries in conventionally cast Mg-Zn alloys  \\cite{GaoInter2007} where it is present in Mg--\\ce{Mg4Zn7} lamellae. The beam direction, $\\boldsymbol{B}$, is  parallel to the ${[}2\\overline{4}2\\overline{3}{]}$ zone axis in ($A$) and is close to the [010]  zone axis of \\ce{Mg4Zn7} in ($B$). This orientation relationship is not among those previously reported  for this phase in Mg \\cite{Singh2007} and lacks the usual alignment of the $[010]_\\ce{Mg4Zn7}$ ($d=0.52$\\,nm) parallel to $[002]_\\ce{Mg}$ ($d=0.26$\\,nm). The adjacent grain, (C), is not aligned along a rational, low-index direction. The insets show an enlargement of the matrix (left) and a simulated HRTEM image (right) for defocus=70\\,nm, thickness=70\\,nm.\nThe absence of  i) a favourable orientation relationship, ii) clear crystallographic facets and and iii) the usual rod-like morphology suggest that the precipitate nucleated at the grain boundary rather than intragranularly.  A full listing of the reflections used to index the phases in regions $(A, B)$ is set out in Tables~\\ref{tab-fft-hrtem-23a} and \\ref{tab-fft-hrtem-23b} in the supplementary information (SI).   \n\nMore extensive deformation results  in the grain-boundary precipitates coalescing into roughly equiaxed particles. A typical micrograph showing this condition is shown in Figure~\\ref{N20_10}. The larger, dark regions are Mg grains in a strongly diffracting condition. Precipitates are concentrated at the grain boundaries and triple points, with two examples indicated by arrows. \nIt was possible to obtain atomic resolution images of some of the precipitates, as shown in Figure~\\ref{hrtem-n20}.\nThe Mg matrix (lower right) is close to a two beam condition with $\\boldsymbol{g}=0002$. The precipitate (centre and left region)  has stronger atomic contrast.\nFigure~\\ref{hrtem-n20-fft} shows the FFT of the precipitate region which was assigned to the \\ce{MgZn2} phase, with beam direction $[0001]$. A full listing of the reflections used to index the phase is set out in Table~\\ref{tab-fft-_N20_6} in the SI.   \n\n\\begin{figure}\n\\begin{center}\n\t\\begin{center}\n\t\\makebox[2ex]{}\n\t\\hfill\n\tTEM\n\t\\hfill\\\n\t\\hfill\n\tHRTEM\n\t\\hfill\\\n\t\\hfill\n\tFFT\n\t\\hfill\\\n\t\\hfill\\\n\t\\end{center}\n\t\n\t\\raisebox{0.15\\textwidth}[0pt][0pt]{\n\t\t\\begin{rotate}{90}\n\t\t\tN=3\n\t\t\\end{rotate}}\n\t\\hfill\n       \\subfigure[\\label{hrtem-15}]{\\includegraphics[width=0.3\\textwidth]{Tem-2014-04-11_15}}\\hfill\n       \\subfigure[ \\label{hrtem-23-a}]{\\includegraphics[width=0.3\\textwidth]{Tem-2014-04-11_23b}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-23-fft-Mg4Zn7}]{\\includegraphics[width=0.3\\textwidth]{FFT_Tem-2014-04-11_23-center-assign}}\n\t\\hfill\\\n\n\t\\raisebox{0.15\\textwidth}[0pt][0pt]{\n\t\t\\begin{rotate}{90}\n\t\t\tN=20\n\t\t\\end{rotate}}\n\t\\hfill\n\\subfigure[\\label{N20_10}]{\\includegraphics[width=0.3\\textwidth]{MgZnN20_10b}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-n20}]{\\includegraphics[width=5cm]{N20_6a}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-n20-fft}]{\\includegraphics[width=5cm]{FFT_N20_6-prec}}\n\t\\hfill\\\n\n\t\\caption{Electron micrographs of grain boundary precipitates in HPT Mg-Zn. Figures (a)--(c) are from samples deformed to N=3 and (d)--(f) are for N=20. (a,d) show typical diffraction contrast images of grain boundary precipitation. High resolution images are presented in Figs.~(b,e) with FFTs of the precipitate-containing regions presented in (c) and (f), respectively. For N=3 (c) the FFT is indexed to the \\ce{Mg4Zn7} phase  (circles) and Mg matrix (squares). For N=20 (f) the FFT is indexed to the \\ce{MgZn2} phase. A Hann filter was used to remove high frequency noise from the regions of interest in (b,e) before performing the FFT.\n \\label{hrtem-23}}\n\\end{center}\n\\end{figure}\n\n\nThe TEM observations of N=3 samples suggest that the initial grain boundary precipitates are comprised of the \\ce{Mg4Zn7} phase. Further solute diffusion of Zn to the grain boundaries continues during HPT deformation, resulting in the formation  of the \\ce{MgZn2} phase. Although first principles calculations show \\ce{MgZn2}  has a lower formation enthalpy for Zn $>$66at.\\% \\cite{Xie2013} the difference is minimal and microdomains of the \\ce{Mg4Zn7} and \\ce{MgZn2} phases can co-exist within individual precipitates  \\cite{RosalieSingh2011,SinghRosalie2010,RosalieSomekawa2010} giving rise to a continuous spectrum of compositions. \nIt is probable that such mixed-phase precipitates are present in the HPT material but extensive analysis would be required to confirm this.\n\n\\subsection{SAXS}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{SAXSyStuff}\n\t\t\\caption{Left: Small-angle scattering data and fits (points with uncertainties, and solid lines, respectively). Right: Size distributions of scatterers determined from fitting of the scattering data. Error bars indicate $\\pm$1\\,standard deviation (SD). \\label{fig-saxs}}\n\t\\end{center}\n\\end{figure}\n\nThe scattering behaviour of the ST alloy shows little evidence for any structure in the measurable size range, and only shows the $I\\propto Q^{-4}$ scattering from larger scattering structures (Fig.~\\ref{fig-saxs}). After compression (N=0), there are slight but measurable changes in the scattering. Major changes occur upon HPT rotation, with strong, almost identical shoulders appearing in the scattering patterns for the N=1 and N=20 samples. \n\nThe Monte Carlo method can retrieve a size distribution from the scattering patterns provided that a general scatterer shape is defined \\cite{Pauw2013}. Given the diversity of grain boundary precipitate morphologies observed in TEM, a generally applicable spherical shape was chosen for the characterisation of the precipitate dimensions. Application of this method shows that the shoulders in the scattering patterns for both HPT samples can be interpreted as scattering originating from a broad distribution of objects, with radii ranging from about 2.5--20\\,nm, and maximum volume-weighted radii of around 3--5\\,nm. This size is consistent with the particle sizes observed via TEM. Due to technical limitations, the volume fractions shown in Fig.~\\ref{fig-saxs} are not on absolute scale. They are, however, internally consistent and therefore allow for comparisons between samples  \\footnote{Analysis of simulated data indicates that the determined volume fraction remains largely unchanged provided the aspect ratio of globular scatterers is less than 1:5.}. The volume fraction and size of detected precipitates in the N=1 and N=20 samples exhibit striking similarity. \n\n\\section{Conclusions}\nIn conclusion, HRTEM and SAXS provided direct evidence of precipitation in an HPT-deformed Mg-3.4at.\\%Zn alloy without post-deformation heat treatment as well as the characterisation of the precipitates. Both techniques show fine-scale precipitates after deformation with dimensions between 2.5--20\\,nm, with the precipitate radii centred at around 3--5\\,nm. HRTEM observations indicate that precipitation initially takes the form of a grain boundary film consisting  of the \\ce{Mg4Zn7} phase.  \n\n\n\nSAXS shows little difference in the precipitate size and volume fraction in materials subjected to 20 rotations, where the alloy is in the the strain-saturated condition. However solute segregation to the grain boundaries continues during deformation, as evidenced by HRTEM showing a transition from a microstructure with a grain boundary \\ce{Mg4Zn7} film to one with more equiaxed particles of the \\ce{MgZn2} phase.\n\n\\section*{Author contributions}\nThe HPT samples were prepared by FM. JMR and FM performed the SEM and TEM investigations, with the analysis of the HRTEM micrographs being done by JMR. BRP performed the SAXS measurements, corrections and analysis. JMR and BRP wrote the majority of the manuscript, with contributions from FM and KT and with feedback and comments from all authors. HM and HK are responsible for the SAXS instrument used in the studies. KT supervised the investigations presented herein.\n\n\\section*{Acknowledgements}\nThe authors would like to thank Dr. H. Somekawa for supplying the extruded material used in the investigations. This work was supported in part by a Grant-in-Aid for Scientific Research on Innovative Area, \"Bulk Nanostructured Metals\", through MEXT, Japan (contract no. 22102004). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMarkov Chain Monte Carlo algorithms (MCMC) enjoy great success in simulating \nmany theories from the Ising model up to Lattice QCD. \nAlbeit the potential, MCMC has a hard time whenever the action \nbecomes complex-valued due to the associated Boltzmann weight loosing its \ninterpretability as probability distribution. \n\nUsing MCMC, we focus on the Hubbard model capturing electronic properties \nof systems with strongly interacting electrons propagating on a fixed \nspatial lattice of ions.\nExamples for such systems are carbon nano structures like Graphene and \nFullerene $C_n$.\nIn the Hubbard model the sign problem is observed at finite chemical potential as well as on \nnon-bipartite lattices\\footnote{ \n    Bipartite describes lattice geometries at which the sites can be two \n    coloured such that no neighbouring sites have the same colour.\n    For example, the square is bipartite while the triangle is non-bipartite.\n}.\nReweighting can treat the complex-valued Boltzmann weight though, at the \nsame time, introducing an exponentially hard to estimate normalization.\n\nDeforming the region of integration onto Manifolds with an almost constant imaginary \naction showed great promise in reducing the sign problem substantially~\\cite{\nKashiwa:2018vxr, alexandru2020complex, DetmoldPathIntegral, Detmold:2021ulb}.\nPractically, this deformation requires numerical integration of differential \nequations which becomes infeasible for larger systems. \nWe aim to identify efficient Neural Network architectures to learn such  \nbeneficial deformations.\nThis removes the cost of numerically integrating configurations and enables \nsimulations of large systems with a sign problem beyond the standard reweighting\napproach.\n\nIn this proceedings, we collect material from our earlier publications~\\cite{leveragingML,rodekampMitigating} and a master thesis~\\cite{christophThesis}.\nThe manuscript is organized in the following way.\nIn section~\\ref{sec:Formalism} a brief introduction to the Hubbard model and \nthe tested system is presented along a short discussion of the sign problem.\nThis discussion is then followed by the definition of the neural network \narchitectures as published in~\\cite{leveragingML,rodekampMitigating}.\nFurther, in section~\\ref{sec:NumericalResults} correlator results are presented \nand we discuss the obtained charge density of one of the larger systems.\n %\n\\label{sec:Introduction}\n\n\\section{Formalism}\\label{sec:Formalism}\nThe Hubbard model~\\cite{Hubbard1963} describes a fixed spatial lattice $X$ of ions on which electrons can move and interact.\nIts Hamiltonian, in particle-hole basis, is\n\\begin{equation}\n    \\nonumber\n    \\mathcal{H}\\left[K, V, \\mu\\right]\n    =\n           - \\sum_{x,y\\in X} \\left( p_x^\\dagger K^{xy} p_y - h_x^\\dagger K^{xy} h_y \\right)\n         + \\frac{1}{2} \\sum_{x,y\\in X} \\rho_x V^{xy} \\rho_y\n            + \\mu \\sum_{x\\in X} \\rho_x,\n    \\label{eq:hubbard-hamiltonian}\n\\end{equation}\nwhere the amplitudes in $K$ encode the hopping of fermionic particles $p$ \nand holes $h$, the potential $V$ describes the interactions between charges\n\\begin{equation}\n    \\rho_x = p^{\\dagger}_x p_x - h^{\\dagger}_x h_x\n    \\label{eq:net-charge}\n\\end{equation}\nand the chemical potential $\\mu$ incentivizes charge.\nBy adjusting the hopping and lattice symmetry $K$ as well as the interaction $V$ \nthis model can describe a wide variety of physical systems.\nIn the following investigation, five systems are considered as displayed in figure~\\ref{fig:systems}.\nThe 2 site system describes one unit cell of the honeycomb structure used \nfor Graphene type models which we successively built up with the 8 and 18 site ones.\nSecondly, we present preliminary results for fullerenes $C_{20}$ and $C_{60}$ at zero chemical potential.\nIn all cases $K$ encodes nearest-neighbor hopping\nand we assume an on-site interaction,\n\\begin{align}\n    K &= \\kappa \\delta_{\\langle xy \\rangle}\n    &\n    V &= U \\delta_{xy}.\n\\end{align}\nIn figure~\\ref{fig:systems} the sites, i.e.\\@ ions and their nearest neighbor connections are depicted. \nLines stretching out display periodic boundary of the spatial lattice (suppress in the 18 site case).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1] {figure\/systemPlots.pdf}\n    \\caption{\n        Showing the different geometries considered in this proceedings. \n        Each vertex represents an ion and each (dashed) line depicts the \n        nearest-neighbor hopping that is allowed by the Hubbard model. \n        Dashed lines indicate periodic boundary condition where possible.\n    }\n    \\label{fig:systems}\n\\end{figure}\n %\n\\subsection{Simulation Setup}\nCalculating observables follows the standard procedure~\\cite{browerHybridMonteCarlo2012} \nof evaluating the thermal trace.\nAfter trotterizing into $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}$ time slices, inserting Grassmannian resolutions of\nthe identity and linearizing the interaction via the Hubbard-Stratonovich \ntransformation~\\cite{Hubbard1959} the Hamiltonian is transformed into the action\n\\begin{equation}\n    S\\left[\\Phi \\,\\vert \\, K, V, \\mu \\right]\n       =\n            -   \\log\\det{ M\\left[\\Phi\\,\\vert\\, K,\\mu\\right] \\cdot M\\left[-\\Phi\\,\\vert\\, -K,-\\mu\\right] }\n         +   \\frac{1}{2} \\sum_{t=0}^{\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}-1}\\sum_{x,y\\in X} \\Phi_{tx} \\delta V^{-1\\,xy} \\Phi_{ty},\n    \\label{eq:hubbard-action}\n\\end{equation}\nwhere $\\Phi \\in \\ensuremath{\\mathbb{R}}^{\\abs{\\Lambda}}$ is an auxiliary field on the \nspacetime lattice $\\Lambda = [0, N_t-1]\\otimes X$ and $\\delta=\\beta\/N_t$ \nis the (temporal) lattice spacing controlled by the inverse temperature $\\beta$.\nThe fermion matrix is not uniquely defined on the lattice, we choose the exponential discretization~\\cite{Wynen:2018ryx}\n\\begin{equation}\n    M\\left[\\Phi\\,\\vert\\, K,\\mu\\right]_{x't';xt} \n    =   \n        \\delta_{x'x}\\delta_{t't}\n    -  \\left( e^{\\delta(K + \\mu)} \\right)_{x'x} e^{+ i \\Phi_{xt}} \\mathcal{B}_{t'}\\delta_{t'(t+1)}\n\\end{equation}\nwhere $\\mathcal{B}$ encodes the anti-periodic boundary conditions in time.\nFor bipartite systems we may replace $-K$ with $+K$ in the holes' fermion \nmatrix~\\cite{browerHybridMonteCarlo2012}.\nThe thermal trace for this is expressed as path integral\n\\begin{equation}\n    \\expval{\\ensuremath{\\mathcal{O}}}\n    =\n    \\frac{1}{\\ensuremath{\\mathcal{Z}}} \\int  \\DD{\\Phi} e^{- S\\left[\\Phi\\right]} \\ensuremath{\\mathcal{O}}\\left[\\Phi\\right]\n    =\n    \\int  \\DD{\\Phi} p_S\\left[\\Phi\\right] \\ensuremath{\\mathcal{O}}\\left[\\Phi\\right]\n    \\label{eq:true-expectation-value}\n\\end{equation}\nFor cases of real action we can apply MCMC to generate $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}$ configurations according to the Boltzmann distribution $p_S\\left[\\Phi\\right] $\nand estimate observables \\eqref{eq:true-expectation-value} by an unweighted expectation value.\nIf the action is complex valued we apply reweighting\n\\begin{align}\n        \\expval{\\ensuremath{\\mathcal{O}}}\n        &   = \\frac{\\expval{e^{-i\\Im{S}}\\ensuremath{\\mathcal{O}}}_{\\Re{S}}}{\\expval{e^{-i \\Im{S}}}_{\\Re{S}}}\n            = \\frac{1}{\\Sigma} \\expval{ e^{-i \\Im{S}} \\ensuremath{\\mathcal{O}}}_{\\Re{S}}.\n        \\label{eq:reweighting}\n\\end{align}\nand sample configurations according to the Boltzmann distribution under the real part of the action.\nIt has been shown~\\cite{berger2021complex} that an effective number of configurations\n\\begin{equation}\n    \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\textrm{eff}} = \\abs{\\Sigma}^2 \\cdot \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} \\label{eq:effective-Nconf}\n\\end{equation}\ncontrols the scaling of statistical errors $\\sim \\left(\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\textrm{eff}}\\right)^{-1\/2}$.\nThis translates the sign problem to the ability of calculating the denominator $\\Sigma$, \ni.e. the statistical power, reliably~\\cite{berger2021complex,leveragingML,PhysRevD.93.014504,mori2018lefschetz}.\n\\\\\n \nA promising approach to mitigate, or even remove, the sign problem is to \ndeform the region of integration $\\Phi \\in \\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}} = \\ensuremath{\\mathbb{R}}^{\\abs{\\Lambda}}$ onto a manifold \non which the imaginary part of the action is (nearly\\footnote{\n    If $\\Im{S}\\approx const$, the statistical power $\\abs{\\Sigma} = \n    \\abs{\\expval{e^{i\\Im{\\ensuremath{S_{\\mathrm{eff}}}}}}} \\approx \\abs{\\expval{e^{i \\cdot const}}} \\approx 1$\n    yielding nearly no reduction in effective number of configurations \n    $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\mathrm{eff}} \\approx \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}$.\n}) constant~\\cite{PhysRevD.86.074506,alexandru2020complex}\n, $\\ensuremath{\\mathcal{M}} = \\left\\{ \\Phi \\in \\ensuremath{\\mathbb{C}}^\\abs{\\Lambda} \\, \\vert \\, \\Im {S\\left[\\Phi\\right]} = const. \\right\\}$.\nIf $\\ensuremath{\\mathcal{M}}$ is in the same homology class as $\\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}}$\nan analogue of the Cauchy integral theorem ensures that the observables are unchanged.\nParametrizing fields $\\Phi(\\phi)\\in\\ensuremath{\\mathcal{M}}$ then adds a Jacobian defining the \nused effective action\n\\begin{equation}\n    \\ensuremath{S_{\\mathrm{eff}}}\\left[\\phi\\right] = S[\\Phi\\left(\\phi\\right)] - \\log\\det{J\\left[\\phi\\right]}, \\quad J_{ij} = \\pdv{\\Phi(\\phi)_i}{\\phi_j}\n    \\label{eq:effective-action}\n\\end{equation}\n\nThere are many strategies for picking target manifolds $\\ensuremath{\\mathcal{M}}$~\\cite{Tanizaki:2017yow}.\nOne choice is to try to approximate the Lefschetz thimbles~\\cite{PhysRevD.86.074506}.\nEach thimble contains a critical point $\\Phi_{cr}$ that is a fixed point \nof the holomorphic flow\n\\begin{equation}\n    \\frac{d\\Phi(\\tau)}{d\\tau} = \\pm\\left(\\frac{\\partial S\\left[ \\Phi(\\tau) \\right]}{\\partial \\Phi(\\tau)}\\right)^*, \\quad \\Phi(0) = \\phi\n    \\label{eq:holomorphic-flow}\n\\end{equation}\nintroducing the fictitious flow time $\\tau$.\nA thimble is the set of complexified configurations that flow to a critical \npoint under downward flow, i.e. $-$ in~\\eqref{eq:holomorphic-flow}.\nAn integrator for~\\eqref{eq:holomorphic-flow} will always be computationally \nexpensive~\\cite{alexandru2020complex,rodekampMitigating,leveragingML}.\nHowever, the non-interacting solution $\\phi = 0$ for~\\eqref{eq:holomorphic-flow}\nassuming a constant field $\\Phi_{t,x} = \\Phi_{t',x'}$ defines \na (hyper-) plane parallel to the real plane $\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ that goes through the main critical point $i \\Phi_c^0$.\nThis so called tangent plane $\\ensuremath{\\mathcal{M}}_{T} = \\{\\phi + i \\Phi_c^0 \\, \\vert \\, \\forall \\phi \\in \\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}}\\}$\nshowed promise in sufficiently mitigating the sign problem, at least for smaller systems~\\cite{PhysRevD.93.014504,Alexandru:2018ddf, Warrington:2019kzf, leveragingML, rodekampMitigating}.\n\n \n\\subsubsection{Neural Network Architectures}\nTo improve beyond the tangent plane it seems plausible to \nidentify a transformation that transforms a given configuration $\\phi\\in\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ \nto a target manifold $\\tilde{\\ensuremath{\\mathcal{M}}}$ that may be closer to $\\ensuremath{\\mathcal{M}}$ than the tangent plane.\nSuch a transformation may be parametrized by a neural network \n\\begin{equation}\n    \\ensuremath{\\mathrm{SHIFT}}: \\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}} \\to \\tilde{\\ensuremath{\\mathcal{M}}}, \\, \\phi \\mapsto \\phi + i\\left(\\Phi_c^0 + NN\\left( \\phi \\right)\\right).\n    \\label{eq:SHIFT}\n\\end{equation}\nFor the neural network part $NN$ we pick two linear layers $\\omega \\phi + b$ with real trainable weights $\\omega$ and biases $b$\nwhich are separated by a leacky-ReLU. \nAs the effective action~\\eqref{eq:effective-action} suggests the defining \ntransformations Jacobian needs to be computed \n\\begin{equation}\n    \\log\\det{J_{\\ensuremath{\\mathrm{SHIFT}}}[\\phi]} = \\log\\det{\\mathds{1} + i \\pdv{NN\\left(\\phi\\right)}{\\phi} } \n\\end{equation}\nwhich requires an $\\order{V^3}$ algorithm for the determinant calculation.\nThis scaling is not feasible for large scale systems but still much cheaper \nthan applying a Runge-Kutta --- or similar algorithms --- to integrate the \nholomorphic flow equations.\n\nTo improve the scaling, we identify a neural network that has a cheaper determinant. \nOne such architectures is given by Affine Coupling Layers~\\cite{albergo2021introduction,foreman2021hmc}\nthat approximate the integrator $\\Phi(\\phi) \\approx \\ensuremath{\\mathcal{NN}}(\\phi) $\n\\begin{equation}\n    \\ensuremath{\\mathcal{NN}}_\\ell(\\Phi) =\n    \\begin{cases}\n        c_\\ell\\left[ \\Phi_A, \\, \\Phi_B \\right] & A_\\ell \\text{ components} \\\\\n        \\Phi_B & B_\\ell \\text{ components}\n    \\end{cases}\n    \\label{eq:ACL-def}\n\\end{equation}\nHere $A$ and $B$ are layer-specific partitions of the input vector $\\Phi$ of equal cardinality $\\nicefrac{1}{2}\\abs{\\Lambda}$. $\\Phi_{A,B}$ are the components of the input belonging to the indicated partition.\nWe apply the affine coupling~\\cite{albergo2021introduction}\n\\begin{equation}\n    c_\\ell\\left[\\Phi_A, \\Phi_B \\right] = e^{m_\\ell\\left(\\Phi_B\\right)} \\odot \\Phi_A + a_\\ell\\left(\\Phi_B\\right).\n    \\label{eq:affine-coupling}\n\\end{equation}\nThe coupling functions \n$m_\\ell,a_\\ell: \\mathbb{C}^{\\nicefrac{\\abs{\\Lambda}}{2}} \\to \\mathbb{C}^{\\nicefrac{\\abs{\\Lambda}}{2}}$ \nare again two linear layers separated by the non-linear Softsign function.\nTo ensure that the neural network produces a complex configuration as is required by the holomorphic flow, \nthe weights and biases need to be complex valued which is discussed in more detail in~\\cite{rodekampMitigating}.\nA single layer just transforms half of the configuration, we thus pair it up with a second layer that is set up in the same way but with the roles of $A$ and $B$ interchanged.\nThis setup allows to express the Jacobian, with $L\/2$ of these pairs, in the form\n\\begin{equation}\n    \\log{\\det{J_{\\ensuremath{\\mathcal{NN}}}(\\phi)}} = \\sum_{\\ell = 1}^{L} \\sum_{i =0}^{\\abs{A}-1} m_\\ell\\left( \\Phi_{\\ell-1}(\\phi)_B\\right)_i.\n    \\label{eq:logDetJ-NN}\n\\end{equation}\nwhich adds only an $\\order{V}$ cost to the application of the transformation.\n\nFor any of these architectures we perform Molecular Dynamics on $\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ using \na standard leapfrog algorithm and then apply the network to move onto $\\tilde{\\ensuremath{\\mathcal{M}}}$.\nAccept\/Reject is then performed according to the effective action~\\eqref{eq:effective-action}\nusing the transformed configuration from the Network and the Jacobian defined by the \nnetwork.\nThis machine learning enhanced Hybrid Monte Carlo is referred to as ML HMC.\n\n \n\\subsection{Observables}\nWe are interested in the electronic properties of a given system.\nEuclidean correlation functions of a single particle or a single hole %\n\\begin{equation}\n    C_{xy}^{\\tiny\\substack{p\\\\h}}(t) = \\expval{ p_x^\\dagger(t) p_y(0) } = \\expval{ M[\\pm\\Phi | \\pm K, \\pm\\mu ]^{-1}_{xt;y0} },\n\\end{equation}\nmomentum projected and averaged we obtain $C_{sp}(t)$~\\cite{rodekampMitigating}.\nIn the future we aim to match the parameters $\\nicefrac{U}{\\kappa},\\mu$ to real-world systems\nand extract the band-gap $C_{xy}(t) \\sim e^{-t\\cdot \\Delta E} $.\nFurther, the charge density is defined by\n\\begin{equation}\n    \\rho(\\mu) = \\frac{1}{\\abs{X}} \\sum_{x\\in X} \\expval{\\rho_x} \n            = \\frac{1}{\\abs{X}} \\sum_{x\\in X} C_{x,x}^p(0) - C_{x,x}^h(0).\n            \\label{eq:charge-density}\n\\end{equation}\nIt is point symmetric around the electric neutral half-filling point, $\\mu = 0$, due to \nparticle-hole symmetry.\nFor very large $\\mu\\to \\pm \\infty$ the charges (+) or holes (-) are favoured\nyielding a charge density of $\\pm 1$.\nQualitatively, it is expected that the system's charge \nequals integer multiples of the electric charge $n\\cdot e^{-}$ with \n$n \\in [-\\ensuremath{\\mathrm{N}_{\\mathrm{x}}},\\ensuremath{\\mathrm{N}_{\\mathrm{x}}}]$, i.e.\\@ $\\rho(\\mu) = \\nicefrac{n}{\\ensuremath{\\mathrm{N}_{\\mathrm{x}}}}$.\n\n \n\\section{Numerical Results}\\label{sec:NumericalResults}\nWe experimented with different training setups. \nForemost, supervised training using ADAM to minimize the $L1-$Loss.\nThe training data consists out of $\\order{\\num{10000}}$ pairs $(\\phi,\\Phi(\\phi))$\nobtained by a Runge-Kutta of \\nth{4} order.\nFor further details consider~\\cite{leveragingML,rodekampMitigating}.\n\n\n %\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth,height = 0.4\\textheight]{figure\/Correlators}\n\\caption{\n    Each row in this figure shows the correlators\n    of a system with 2 (upper row) and 8 (lower row) sites~\\cite{rodekampMitigating}.\n    The different columns correspond to a number of configurations, \n    $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}\\in\\{\\num{1000},\\num{100000}\\}$, used to estimate the correlators.\n    Three methods --- ML HMC with coupling layer $\\ensuremath{\\mathcal{NN}}$ (blue), HMC on the Tangent Plane (red) and exact diagonalization (black) --- \n    are compared to show the effectiveness and correctness of the introduced \n    machine learning enhanced method. \n    The sign problem manifests as a loss of signal, \n    i.e. small number of effective configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\mathrm{eff}}$~\\eqref{eq:effective-Nconf}, \n    and greatly increases as the number of sites expands.\n    It can be seen that the ML HMC has a substantially reduced sign problem.\n}\n\\label{fig:SmallSystemCorrelators}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth,height = 0.3\\textheight]{figure\/18Site_Correlators.pdf}\n\\caption{The correlators of Graphene with 18 ions are shown~\\cite{rodekampMitigating}.\n    $\\num{100000}$ measurements are taken.\n    With this larger lattice direct diagonalization as in figure~\\ref{fig:SmallSystemCorrelators} \n    is not tractable any more hence only the two statistical methods ML HMC using the coupling network $\\ensuremath{\\mathcal{NN}}$ (blue)\n    and HMC on the tangent plane (red) are compared. \n    As expected the ML HMC improves the signal drastically.\n}\n\\label{fig:18SiteCorrelator}\n\\end{figure}\n\\begin{figure}\n    \\centering\n\\includegraphics[width = 0.9\\linewidth,height = 0.4\\textheight]{figure\/ChargeDensity.pdf}\n\\caption{\n    We computed the charge density for several values of the chemical potential $\\mu\\in \\left[0,\\,5.2\\right]$\n    for an 18 site Graphene sheet~\\cite{christophThesis}.\n    For most smaller and larger values of $\\mu$ the sign problem is small enough\n    that estimation with HMC on the tangent plane (red) is sufficient. \n    However, in the region $\\mu \\in \\left[2,3\\right]$ an ML HMC (blue) is used \n    to narrow particular values for which the sign problem becomes untraceable.\n    The features at $\\mu = 0$ and $\\mu \\to \\infty$ are captured as expected.\n    Finding the charge plateaus at $\\rho(\\mu)\\sim n$, however, is yet unavailable \n    due to the small $\\beta$.\n    The dashed line at $\\rho(\\mu)=\\nicefrac{4}{18}$ \n    indicate an expected plateau which may be surmised around $\\mu \\approx 1$.\n}\n\\label{fig:ChargeDensityScan}\n\\end{figure}\n\\begin{figure}\n    \\centering\n\\includegraphics[width = 0.9\\linewidth,height = 0.3\\textheight]{figure\/Correlators_buckyballs.pdf}\n\\caption{\n    The correlators of Fullerene $C_{20}$ (upper row) and $C_{60}$ (lower row) are shown.\n    A real plane (red) and a tangent plane (blue) HMC at \n    inverse temperature $\\beta = 8$ and $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$ time slices are compared.\n    We consider an on-site interaction $U = 3$ and zero chemical potential.\n    The, here negligible, sign problem solely stems from the non-bipartiteness \n    of the system due to the pentagonal rings.  \n    Already, at small number of configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} \\leq \\num{10000}$ the \n    signals are very good.\n}\\label{fig:buckyball-correlators}\n\\end{figure}\n %\nIn figure~\\ref{fig:SmallSystemCorrelators} correlators for systems with \n2 and 8 sites are compared using the different algorithms HMC (red) --- on the tangent plane --- \nML HMC (blue), applying the coupling network $\\ensuremath{\\mathcal{NN}}$, and exact diagonalization (black)~\\cite{rodekampMitigating}.\nWe use $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}\\in\\{\\num{1000},\\num{100000}\\}$ to portray the effect of the \nstatistical power on the effective number of configurations. \nCorresponding statistical powers $\\abs{\\Sigma}$ can be found in~\\cite{rodekampMitigating}.\nThe system parameters --- $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 4$, $U = 4$ and $\\mu = 3$ --- are kept fix.\nThe ML HMC outperforms the HMC in terms of signal.\nThe 8 site system has a stronger sign problem to an extend that HMC retrieves no signal.\nIf a signal is obtained, both algorithms agree with the exact diagonalization.\nIn figure~\\ref{fig:18SiteCorrelator} the correlators for the system with 18 sites are displayed~\\cite{rodekampMitigating}.\nThe system is computed at $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 4$, $U=3$ and $\\mu = 3$.\nThe sign problem is much stronger than in the previous cases due to the larger volume.\nNevertheless the ML HMC extracts a good signal for the correlators. \nSimilar to the 8 site case HMC can't keep up.\nDue to the number of sites exact diagonalization is not feasible. \n\n %\nContinuing the 18 site model --- with $U = 2$, $\\beta = 5$, $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}=32$ ---\nwe want to study the charge density~\\eqref{eq:charge-density} subjected to the chemical potential. \nThis can be seen in figure~\\ref{fig:ChargeDensityScan}~\\cite{christophThesis}.\nWe compare tangent plane HMC (red) and ML HMC (blue) using \nthe SHIFT network.\nVarying the chemical potential has shown that for small and large values \nthe sign problem is mild (on the tangent plane).\nHowever, in the intermediate range of $\\mu \\in [2,3]$ the tangent plane is \nnot sufficient for a reasonable estimate, where we apply ML HMC with the SHIFT network.\nThe point at $\\mu = 2.5$ requires more attention and we plan to address it\nwith the coupling network in the future expecting much better results.\nThe expected behaviour of the charge density, $\\rho(\\mu=0) = 0$ and $\\rho(\\mu\\to\\infty) \\to 1$, \nis found numerically.\nThe dashed line exemplarily indicates an expected plateau at $\\expval{\\rho(\\mu)} = \\nicefrac{4}{18}$.\nAs it can be seen, this plateau is not fully deducible but may be surmised around $\\mu \\approx \\num{1}$.\nStudies of smaller systems, see~\\cite{christophThesis}, indicate increasing $\\beta$\nmakes these plateaus more pronounced. %\n\n \n\nTo probe our method in physically more relevant systems than the 18 Site Graphene sheet, \nwe started an investigation of Fullerene $C_{20}$ and $C_{60}$.\nThe correlators at $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 8$, $U = 3$ and zero chemical potential \nare displayed in figure~\\ref{fig:buckyball-correlators}.\nThe mild sign problem is solely due to the non-bipartiteness of the lattice structure.\nWe compare standard HMC (red) with tangent plane HMC (blue) to show \nthat the tangent plane obtains a good signal already at small \nnumber of configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} = 1000$ in both systems.\nFor $C_{60}$ the sign problem is negligible and the real plane HMC gives a\ngood signal too.\nFor finite chemical potential the sign problem\naggravates as it imposes a second source. \nWe are currently working on this particular lattice geometry. \n\n \n\\section{Conclusions}\nSimulating systems with strongly correlated electrons is a rather challenging \ntask due to the innate sign problem for doped systems.\nCurrent methods, like deformation onto Lefschetz thimbles, suffer from a very difficult \nscaling in computational cost.\nWe overcome this issue by identifying efficient Neural Network architectures and incorporating them\nin a HMC algorithm.\nWe present first studies of doped Graphene sheets using this enhanced HMC and \ndemonstrate a substantial improvement of the signal, effectively mitigating the sign problem.\nConsidering systems with increasing volume illustrates some stability of this \nmethod for larger volumes.\nFurther, preliminary simulation of Fullerene $C_{20}$ and $C_{60}$ at \nvanishing chemical potential are shown.\nIn the near future we will apply the neural network enhanced HMC also\nat finite chemical potential.\n\n \n\\begin{acknowledgments}\nWe thank the original authors of our recent papers for many helpful discussions and hard work.\nThis work was funded in part by the NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) through the funds provided to the Sino-German Collaborative\nResearch Center ``Symmetries and the Emergence of Structure in QCD''\n(NSFC Grant No.~12070131001, DFG Project-ID 196253076 -- TRR110)\nas well as the STFC Consolidated Grant ST\/T000988\/1.  \nMR was supported under the RWTH Exploratory Research Space (ERS) grant PF-JARA-SDS005.\nWe gratefully acknowledge the computing time granted by the JARA Vergabegremium and provided on the JARA Partition part of the supercomputer JURECA at Forschungszentrum J\u00fclich.\n\\end{acknowledgments}\n \n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nA lot of work has been done in the field of Twitter sentiment analysis till date. Sentiment analysis has been handled as a Natural Language Processing task at many levels of granularity. Most of these techniques use Machine Learning algorithms with features such as unigrams, n-grams, Part-Of-Speech (POS) tags. However, the training datasets are often very large, and hence with such a large number of features, this process requires a lot of computation power and time. The following question arises: What to do if we do not have resources that provide such a great amount of computation power? The existing solution to this problem is to use a smaller sample of the dataset. For sentiment analysis, if we train the model using a smaller randomly chosen sample, then we get low accuracy [16, 17]. In this paper, we propose a novel technique to sample tweets for building a sentiment classification model so that we get higher accuracy than the state-of-the-art baseline method, namely Distant Supervision, using a smaller set of tweets. Our model has lower computation time and higher accuracy compared to baseline model. \n\nUsers often express sentiment using subjective expression. Although objective expressions can also have sentiment, it is much rare. Determining subjectivity is quite efficient compared to determining sentiment. Subjectivity can be determined for individual tweets. But to do sentiment classification, we need to build a classification model with positive and negative sentiment tweets. The time to train a sentiment classification model increases with the increase in the number of training tweets. In this paper, we use tweet subjectivity to select the best training tweets. This not only lowers the computation time but also increases the accuracy because we have training data with less noise. Even the created features will be more relevant to the classification task. The computation cost will reduce due to small training data size and better set of features. Thus if users do not have enough computational resources, they can filter the training dataset using a high value of subjectivf\nity threshold. This ensures reliable prediction on a smaller training dataset, and eventually requires less computational time. The above approach, and some of the intricacies that invariably seep in, need to be considered, and are described in the later sections of the paper. In this paper we also integrate a lot of meticulous preprocessing steps. This makes our model more robust, and hence leads to higher accuracy.\n\nAlong with the machine learning algorithms being used, we use a heuristic-based classification of tweets. This is based on the EFWS of a tweet, which is described in later sections. This heuristic basically takes into account the polarity scores of frequently used words in tweets, and is able to achieve around 85\\% accuracy on our dataset, hence boosting the overall accuracy by a considerable amount.\n\nOur training data consists of generic (not topic-specific) Twitter messages with emoticons, which are used as noisy labels. We show that the accuracy obtained on a training dataset comprising 100K tweets, and a test dataset of 5000 tweets gives an accuracy of around 80\\% on the following classifiers: Naive Bayes, RBF-kernel Support Vector Machine, and Logistic Regression. Our model takes roughly half the time to train and achieves higher accuracy (than the baseline model) on all the classifiers. Because the amount of training time is expected to increase exponentially as the training data increases, we expect our model to outperform (in terms of higher accuracy) the baseline model at a speed which is at least twofold the speed of the baseline model on larger datasets.  \n\n\\section{Related Work}\nThere has been a large amount of prior research in sentiment analysis of tweets. Read [10] shows that using emoticons as labels for positive and sentiment is effective for reducing dependencies in machine learning techniques. Alec Go [1] used Naive Bayes, SVM, and MaxEnt classifiers to train their model. This, as mentioned earlier, is our baseline model. Our model builds on this and achieves higher accuracy on a much smaller training dataset.\\\\\nAyushi Dalmia [6] proposed a model with a more involved preprocessing stage, and used features like scores from Bing Liu's Opinion Lexicon, and number of positive, negative POS tags. This model achieved considerably high accuracies considering the fact that their features were the not the conventional bag-of-words, or any n-grams. The thought of using the polarity scores of frequently used tweet words (as described in our EFWS heuristic) was inspired from this work. [14] created prior probabilities using the datasets for the average sentiment of tweets in different spatial, temporal and authorial contexts. They then used a Bayesian approach to combine these priors with standard bigram language models. \\\\ \nAnother significant effort in sentiment analysis on Twitter data is by Barbosa [16]. They use polarity predictions from three websites as noisy labels to train a model and use 1000 manually labelled tweets for tuning and another 1000 for testing. They propose the use of syntax features of tweets like punctuation, retweet, hashtags, link, and exclamation marks in addition with features like prior polarity of words and POS of words. \\\\\nSome works leveraged the use of existing hashtags in the Twitter data for building the training data. (Davidov, Tsur, and Rappoport 2010) also use hashtags for creating training data, but they limit their experiments to sentiment\/non-sentiment classification, rather than 3-way polarity classification, as [15] does. Our model integrates some of the preprocessing techniques this work used. Hassan Saif [9] introduced a novel approach of adding semantics as additional features into the training set for sentiment analysis. This approach works well for topic specific data. Hence, we thought of taking a different approach for a generic tweet dataset like ours.   \n\n\\section{Subjectivity}\nSubjectivity refers to how someone's judgment is shaped by personal opinions and feelings instead of outside influences. An objective perspective is one that is not influenced by emotions, opinions, or personal feelings - it is a perspective based in fact, in things quantifiable and measurable. A subjective perspective is one open to greater interpretation based on personal feeling, emotion, aesthetics, etc. \\\\\nSubjectivity classification is another topic in the domain of text classification which is garnering more and more interest in the field of sentiment analysis. Since a single sentence may contain multiple opinions and subjective and factual clauses, this problem is not as straightforward as it seems. Below are some examples of subjective and objective sentences. \\\\\n\\newline \nObjective sentence with no sentiment: So, the Earth revolves around the Sun. \\\\\nObjective sentence with sentiment: The drug relieved my pain.\\\\\nSubjective sentence with no sentiment: I believe he went home yesterday.\\\\\nSubjective sentence with sentiment: I am so happy you got the scholarship.\\\\ \n\\newline \nClassifying a sentence as subjective or objective provides certain conclusions. Purely objective sentences do not usually convey any sentiment, while most of the purely subjective sentences have a clear inclination towards either the positive or negative sentiment. Sentences which are not completely subjective or objective may or may not convey a sentiment. Libraries like TextBlob, and tools like Opinion Finder can be used to find the extent to which a sentence can be considered subjective.\\\\\nSince tweets are usually person-specific, or subjective, we use this intuition to reduce the size of the training set by filtering the sentences with a subjectivity level below a certain threshold (fairly objective tweets).      \n\n\\section{Implementation}\nIn this section, we explain the various preprocessing techniques used for feature reduction, and also the additional step of filtering the training dataset using the subjectivity score of tweets. We further describe our approach of using different machine learning classifiers and feature extractors. We also propose an additional heuristic for sentiment classification which can be used as a tag-along with the learning heuristics. \n\\subsection{Corpus}\nOur training dataset\\footnote{The URL is http:\/\/twittersentiment.appspot.com\/. This page has a link to our training data and test data. It is also a public tool that other researchers can use to build their own data sets.} has 1.6 million tweets, and 5000 tweets in the test dataset. Since the test dataset provided comprised only 500 tweets, we have taken part of the training data (exactly 5000 tweets, distinct from the training dataset) as the test dataset. We remove emoticons from our training and test data. The table below shows some sample tweets.\\\\ \n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n    \\small Tweet & \\small Sentiment \\\\\n\\hline\n    \\small @MrZeroo00 Yeah! tks man & \\small Positive \\\\\n   \n  \n    \\small oh so bored...stuck at home & \\small Negative\\\\\n    \\small pizza night and i feel too sick & \\small Negative\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\subsection{Subjectivity Filtering}\nThis is a new step we propose to achieve higher accuracy on a smaller training dataset. We use TextBlob to classify each tweet as subjective or objective. We then remove all tweets which have a subjectivity level\/score (score lies between 0 and 1) below a specified threshold. The remaining tweets are used for training purposes. We observe that a considerable number of tweets are removed as the subjectivity threshold increases. We show the effect of doing this procedure on the overall accuracy in the evaluation section of the paper.\n\\subsection{Preprocessing}\nThe Twitter language model has many unique properties. We take advantage of the following properties to reduce the feature space. Most of the preprocessing steps are common to most of the previous works in the field. However, we have added some more steps to this stage of our model. \n\\subsubsection{Basic steps}\nWe first strip off the emoticons from the data. Users often include twitter usernames in their tweets in order to direct their messages. We also strip off usernames (e.g. @Chinmay) and URLs present in tweets because they do not help us in sentiment classification. Apart from full stops, which are dealt in the next point, other punctuations and special symbols are also removed. Repeated whitespaces are replaced with a single space. We also perform stemming to reduce the size of the feature space.\n\\subsubsection{Full Stops}\nIn the previous works, full stops are just usually replaced by a space. However, we have observed that casual language in tweets is often seen in form of repeated punctuations. For example, ``this is so cool...wow\". We take into consideration this format, and replace two or more occurrences of ``.\" and ``-\" with a space. Also, full stops are also quite different in usage. Sometimes, there isn't any space in between sentences. For example, ``It's raining.Feeling awesome\". We replace a single occurrence of a full stop with a space to ensure correct feature incorporation.  \n\\subsubsection{Parsing Hashtags}\nIn the case of hashtags, most of the previous works just consider the case of hashtags followed by a single word; they just remove the hashtag and add the word to the feature vector. However, sometimes, there are multiple words after a hashtag, and more often than not, these words form an important, conclusive part of the Tweet. For example, \\#ThisSucks, or \\#BestMomentEver. These hashtags need to be dealt with in a proper fashion. We split the text after hashtags after before each capital letter, and add these as tokens to the feature vector. For hashtags followed by a single word, we just replace the pattern \\#word with the word, as conventional models do. The intuition behind this step is that quite often, the sentiment of a tweet is expressed in form of a hashtag. For example, \\#happy or \\#disappointed are frequently used hashtags, and we don't want to lose this information during sentiment classification. \n\\subsubsection{Repeated letters}\nTweets contain very casual language as mentioned earlier. For example, if we search ``wow\" with an arbitrary number of o's in the middle (e.g. wooow, woooow) on Twitter, there will most likely be a non-empty result set. We use preprocessing so that any letter occurring more than two times in a row is replaced with two occurrences. In the samples above, these words would be converted into the token ``woow\". After all the above modifications, tweets are converted into lowercase to avoid confusion between features having same content, but are different in capitalization. \n\\subsubsection{Stopwords, Acronyms and Negations}\nWe gather a list of 400 stopwords. These words, if present in the tweets, are not considered in the feature vector.  \\\\\nWe store an acronym dictionary which has over 5000, frequently-used acronyms and their abbreviations. We replace such acronyms in tweets with their abbreviation, since these can be of great use while sentiment classification.\\\\\nAll negative words like 'cannot', 'can't', 'won't', 'don't' are replaced by 'not', which effectively keeps the sentiment stable. It is observed that doing this makes the training faster, since the model has to deal with a smaller feature vector.  \n\\subsection{Baseline model}\nThe baseline model for our experiments is explained in the paper by Alec Go [1]. The model uses the Naive Bayes, SVM, and the Maximum Entropy classifiers for their experiment. Their feature vector is either composed of Unigrams, Bigrams, Unigrams + Bigrams, or Unigrams + POS tags. \\\\\nThis work achieved the following maximum accuracies:\\\\\na) 82.2 for the Unigram feature vector, using the SVM classifier,\\\\\nb) 83.0 for the Unigram + Bigram feature vector, using the MaxEnt classifier, and 82.7 using the Naive Bayes classifier. \\\\\nc) 81.9 for the Unigram + POS feature vector, using the SVM classifier.\\\\\nThese baseline accuracies were on a training dataset of 1.6 million tweets, and a test dataset of 500 tweets. We are using the same training dataset for our experiments. We later present the baseline accuracies on a training set of 200K tweets, and a test dataset of 5000 tweets; we compare our model's accuracy with these baseline accuracy values on the same test data of 5000 tweets.  \n\\subsection{Effective Word Score (EFWS) Heuristic}\nWe have described our baseline model above. So the feature vectors we collate results for, are Unigram, Unigram + Bigram, and Unigram + POS. We have already made two major changes before the training starts on our dataset as compared to our baseline model. Firstly, our training dataset will be filtered according to the subjectivity threshold. And secondly, our preprocessing is much more robust as compared to their work.\\\\\nNow let us look at an additional heuristic we use to obtain labels for our test data. Along with dictionaries for stop words and acronyms, we also maintain a dictionary of a list of frequently used words and their polarity scores. This dictionary has around 2500 words and their polarity score ranging from -5 to 5. At runtime, we also use all synonyms of a word (from WordNet) present in a tweet and also the dictionary, and assign them the same score as the dictionary word. There is a reasonable assumption here, that the synonyms aren't very extremal in nature, that is, a word with a polarity score of 2 cannot have a synonym which has a polarity score of 5. Now, we calculate the Effective Word Scores of a tweet.\\\\ \n\\newline\nWe define the Effective Word Score of score x as \\\\\n~\\\\\n\\textit{EFWS(x) = N(+x) - N(-x)},\\\\\n~\\\\\nwhere N(x) is the number of words in the tweet with polarity score x.\\\\ \n\\newline\nFor example, if  a tweet has one word with score 5, three words with score 4, two with score 2, three with with score -2, one with score -3, and finally two with score -4, then the effective word scores are:\\\\\n\\newline\nEFWS(5) = N(5) - N(-5) = 1 - 0 = 1\\\\\nEFWS(4) = N(4) - N(-4) = 3 - 2 = 1\\\\\nEFWS(3) = N(3) - N(-3) = 0 - 1 = -1\\\\\nEFWS(2) = N(2) - N(-2) = 2 - 3 = -1\\\\\nEFWS(1) = N(1) - N(-1) = 2 - 0 = 2\\\\\n\\newline \nWe now define the heuristic for obtaining the label of a Tweet. \\\\\n\\begin{algorithmic}\n\\IF {(EFWS(5) $\\geq$ 1 or EFWS(4) $\\geq$ 1) and (EFWS(2) $\\geq$ 1)}\n\t\\STATE Label = positive \n\\ENDIF\n\\end{algorithmic}\n~\\\\\nSimilarly, \\\\\n\\begin{algorithmic}\n\\IF {(EFWS(5) $\\leq$ -1 or EFWS(4) $\\leq$ -1) and (EFWS(2) $\\leq$ -1)}\n\t\\STATE Label = negative\n\\ENDIF\n\\end{algorithmic}\n~\\\\\nThe basic intuition behind such a heuristic is that we found tweets having one strongly positive and one moderately positive word more than the number of strongly negative and the moderately negative words respectively, usually conveyed a positive sentiment. Similar was the case for negative sentiments. The tweets getting a label from this heuristic are not sent into the training phase. After considerable amount of experimenting, and analyzing the nature of our dataset, which is not domain specific, we have reached the conclusion that the heuristic mentioned above is optimal for obtaining labels. We found that the heuristic accuracy was around 85\\% for a training dataset of 100K and a test dataset of 5K, where the total number of test tweets labelled by the heuristic were around 500. This means that around 425 out of the 500 tweets received a correct prediction of sentiment using this heuristic. \\\\\nThus, using this heuristic improves the overall accuracy, as well as saves time by reducing the number of tweets to be tested by the ML algorithms. \n\\subsection{Training Model}\nWe use the following classifiers for our model. \n\\subsubsection{Naive Bayes}\nNaive Bayes is a simple model which works well on text categorization. We use a Naive Bayes model. Class c* is assigned to tweet d, where\tc* = argmax P(c$|$d). \\[P_{NB}(c|d) = P(c) * \\sum_{i=1}^{m} P(f|c)^{n_i(d)}\\] And $P_{NB}(c|d)$ is calculated using Bayes Rule. In this formula, f represents a feature and $n_i(d)$ represents the count of feature $f_i$ found in tweet d. There are a total of m features. Parameters P(c) and $P(f|c)$ are obtained through maximum likelihood estimates.\n\\subsubsection{Support Vector Machines}\nSupport vector machines are based on the Structural Risk Minimization principle from computational learning theory. SVM classification algorithms for binary classification is based on finding a separation between hyperplanes defined by classes of data. One remarkable property of SVMs is that their ability to learn can be independent of the dimensionality of the feature space. SVMs can generalize even in the presence of many features as in the case of text data classification. We use a non-linear Support Vector Machine with an RBF kernel.\n\\subsubsection{Maximum Entropy Model}\nMaximum Entropy Model belongs to the family of discriminative classifiers also\nknown as the exponential or log-linear classifiers.. In the naive Bayes classifier, Bayes rule is used to estimate this best y indirectly from the likelihood $P(x|y)$ (and the prior $P(y)$) but a discriminative model takes this direct approach, computing $P(y|x)$ by discriminating among the different possible values of the class y rather than first computing a likelihood. \\[ \\hat{y} = \\underset{y}{\\operatorname{argmax}} P(y|x) \\]\nLogistic regression estimates $P(y|x)$ by combining the feature set  linearly (multiplying each feature by a weight and adding them up), and then applying a function to this combination.\n\n\\section{Evaluation}\nIn this section, we present the collated results of our experiments. To show that our model achieves higher accuracy than the baseline model and on a smaller training dataset, we first fix the test dataset. Our test dataset, as mentioned before, consists of 5000 tweets. We conducted our experiments on an Intel Core i5 machine (4 cores), with 8 GB RAM. The following are the accuracies of the baseline model on a training set of 200K tweets:\n\n\\begin{center}\n\\begin{tabular}{|p{2.2cm}|p{1.5cm}|p{1cm}|p{2.5cm}|}\n\\hline\n     & \\footnotesize Naive Bayes & \\footnotesize SVM & \\footnotesize Logistic Regression\\\\\n\\hline\n    \\footnotesize Unigram & 78.23\\% & 74.10\\% & 79.03\\% \\\\\n    \\footnotesize Unigram + Bigram & 77.5\\% & 71.3\\% & 80.2\\% \\\\\n    \\footnotesize Unigram + POS & 76.7\\% & 71.8\\% & 79.7\\% \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\nWe filtered the training set with a subjectivity threshold of 0.5. By doing this, we saw that the number of tweets reduced to approximately 0.6 million tweets from an earlier total of 1.6 million. We then trained our model described in earlier sections on a 100K tweets randomly picked from this filtered training dataset, and observed the following accuracies:\n\\begin{center}\n\\begin{tabular}{|p{2.2cm}|p{1.5cm}|p{1cm}|p{2.5cm}|}\n\\hline\n     & \\footnotesize Naive Bayes & \\footnotesize SVM & \\footnotesize Logistic Regression\\\\\n\\hline\n    \\footnotesize Unigram & 79.2\\% & 77.8\\% & 80.5\\% \\\\\n    \\footnotesize Unigram + Bigram & 77.9\\% & 71.7\\% & 81.7\\% \\\\\n    \\footnotesize Unigram + POS & 77.5\\% & 73.6\\% & 79.9\\% \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\nNote that all the accuracies in the tables above have been recorded as the average of 3 iterations of our experiment. We achieve higher accuracy for all feature vectors, on all classifiers, and that too from a training dataset half the size of the baseline one. \\\\\n\\newline \nWe now see the intricacies of the subjectivity threshold parameter. It is clear that more and more tweets get filtered as the subjectivity threshold parameter increases. This can be seen in the Figure 1 shown below. We have plotted the number of tweets that remain after filtering from two sources: TextBlob, Opinion Finder Tool\\footnote{\\*This tool can be found at: http:\/\/mpqa.cs.pitt.edu\/opinionfinder\/}. TextBlob has an inbuilt function that provides us the subjectivity level of a tweet. On the other hand, Opinion Finder only provides the information of which parts of the text are subjective, and which are objective. From that, we define the subjectivity level of that text as: \\\\\n\\newline\nSubjectivity level = $\\dfrac{\\sum{\\text{Length of subjective clauses}}}{\\text{Total length of the text}}$ \n\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[\n    xlabel={Subjectivity Threshold},\n    ylabel={Tweets (in millions)},\n    xmin=0, xmax=1,\n    ymin=0, ymax=2000000,\n    xtick={0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},\n    ytick={0,200000,400000,600000,800000,1000000,1200000,1400000,1600000,1800000},\n    legend pos=north east,\n]\n\\addplot[color=red]\n    coordinates{\n      (0, 1600000)\n      (0.1, 939785)\n      (0.2, 873054)\n      (0.3, 804820)\n      (0.4, 712485)\n      (0.5, 571864)\n      (0.6, 449286)\n      (0.7, 304874)\n      (0.8, 211217)\n      (0.9, 135788)\n    };\n    \n \\addplot[color=blue]\n     coordinates{\n       (0, 1600000)\n       (0.1, 602313)\n       (0.2, 499173)\n       (0.3, 392223)\n       (0.4, 262109)\n       (0.5, 169477)\n       (0.6, 154667)\n       (0.7, 139613)\n       (0.8, 126148)\n       (0.9, 116842)\n     };\n     \\legend{Textblob, Opinion Finder}    \n\n\\end{axis}    \n\\end{tikzpicture}\nFigure 1: Number of tweets with subjectivity greater than the subjectivity threshold\n\\end{center}\n\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[\n    xlabel={Subjectivity Threshold},\n    ylabel={Accuracy (from 0 to 1)},\n    xmin=0, xmax=1,\n    ymin=0.7, ymax=1,\n    xtick={0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},\n    ytick={0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},\n    legend pos=north east,\n]\n\\addplot[color=red]\n  coordinates{\n    (0.1, 0.753871866) \n    (0.2, 0.779442897)\n    (0.3, 0.763421155) \n    (0.4, 0.783231198)\n    (0.5,0.805132645)\n    (0.6,0.807373259)\n    (0.7,0.808587744)\n    (0.8,0.817799443)\n    (0.9,0.823872989)\n};\n\\end{axis}\n\\end{tikzpicture}\nFigure 2: Variation of accuracy (*Training data of 100K, Test data of 5K) with subjectivity threshold. *TextBlob is used to filter the tweets to form the training dataset. \n\\end{center}\n\nWe now focus on the issue of choosing the optimum threshold value. As the subjectivity threshold parameter increases, our model trains on tweets with a higher subjectivity level, and the overall accuracy increases. We observed the following accuracies on subjectivity level 0.8 (Unigrams as features):\\\\\nNaive Bayes: 80.32\\% \\\\\nNon-linear SVM: 80.15 \\% \\\\\nLogistic Regression: 81.77\\% \\\\\n\\newline \nWe should consider the fact that a lot of useful tweets are also lost in the process of gradually increasing the parameter, and this could cause a problem in cases when the test data is very large, because the model will not train on a generic dataset. Researchers may use a higher subjectivity threshold for their experiments if they are confident that most of the important information would be retained. This is most likely to happen in case of topic-specific or domain-specific data.\\\\\n\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[\n    ybar,\n    enlargelimits=0.15,\n    legend style={anchor=north},\n    legend pos= north east,\n    ylabel={Training time (in minutes)},\n   \n    symbolic x coords={baseline,subjectivity=0.5,subjectivity=0.8},\n    xtick=data,\n   \n   \n    ]\n\\addplot coordinates {(baseline,17.4) (subjectivity=0.5,12.55) (subjectivity=0.8,10.68)};\n\\addplot coordinates {(baseline,16.23) (subjectivity=0.5,12.31) (subjectivity=0.8,10.34)};\n\\addplot coordinates {(baseline,31.9) (subjectivity=0.5,18.24) (subjectivity=0.8,16.3)};\n\\legend{Logistic Regression,Naive Bayes,SVM}\n\\end{axis}\n\\end{tikzpicture}\n\nFigure 3: Comparison of training times for Unigrams\n\\end{center}\n\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[\n    ybar,\n    enlargelimits=0.15,\n    legend style={anchor=north},\n    legend pos= north east,\n    ylabel={Training time (in minutes)},\n   \n    symbolic x coords={baseline,subjectivity=0.5,subjectivity=0.8},\n    xtick=data,\n   \n   \n    ]\n\\addplot coordinates {(baseline,28.41) (subjectivity=0.5,14.09) (subjectivity=0.8,11.3)};\n\\addplot coordinates {(baseline,16.6) (subjectivity=0.5,13.51) (subjectivity=0.8,12.66)};\n\\addplot coordinates {(baseline,35.2) (subjectivity=0.5,20.6) (subjectivity=0.8,19.2)};\n\\legend{Logistic Regression,Naive Bayes,SVM}\n\\end{axis}\n\\end{tikzpicture}\nFigure 4: Comparison of training times for Unigrams + Bigrams\n\\end{center}\n\nWe use Logistic regression for classification and unigrams as the feature vector with K-fold cross validation for determining the accuracy. We choose an optimal threshold value of 0.5 for our experiment, considering the fact that the model should train on a more generic dataset. Figure 2 shows the variation of accuracy with the subjectivity threshold. The training size is fixed at 100K and the test dataset (5K tweets) is also same for all the experiments.\\\\ \n\\newline \nWe also measure the time taken to train our model, and compare it to the baseline model. Our observation was that our model took roughly half the amount of time in some cases and yet obtained a higher accuracy. Figures 3 and 4 show the difference in training time of the baseline model, our model on a 0.5 subjectivity-filtered dataset, and our model on a 0.8 subjectivity-filtered dataset on unigrams and unigrams + bigrams respectively. The times recorded are on a training dataset of 100K for our model and 200K for the baseline model, and a test dataset of 5K was fixed in all the recordings. The winning point, which can be seen from the plots, is that our model is considerably faster, and even has twofold speed in some cases. And alongside saving computation time, it achieves higher accuracy. This can be attributed to the fact that as the subjectivity threshold increases, only the tweets with highly polar words are retained in the training set and this makes the whole process faster.      \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe show that a higher accuracy can be obtained in sentiment classification of Twitter messages training on a smaller dataset and with a much faster computation time, and hence the issue of constraint on computation power is resolved to a certain extent. This can be achieved using a subjectivity threshold to selectively filter the training data, incorporating a more complex preprocessing stage, and using an additional heuristic for sentiment classification, along with the conventional machine learning techniques. As Twitter data is abundant, our subjectivity filtering process can achieve a better generalised model for sentiment classification.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the most remarkable structures in the energy spectrum of cosmic rays is\na change of the spectral index $\\gamma$ of the power law $dN\/dE\\propto\nE^\\gamma$ at an energy of about 4~PeV, the so called {\\sl knee}\\xspace\n\\cite{naganowatson,pg,haungsrebelroth}.  The origin of the {\\sl knee}\\xspace has not been\nresolved yet and a convincing explanation of the {\\sl knee}\\xspace structure is thought to\nbe a corner stone in understanding the origin of galactic cosmic rays.  In the\nliterature various reasons for the {\\sl knee}\\xspace are discussed, being related to the\nacceleration and propagation processes of cosmic rays as well as to\ninteractions in interstellar space or the Earth's atmosphere\n\\cite{origin,ecrsreview}.\n\nThe steeply falling energy spectrum requires large detection areas at high\nenergies.  The largest balloon-borne experiment with single-element resolution\n(TRACER, 5~m$^2$\\,sr) reaches energies of a few $10^{14}$~eV \\cite{tracer05}.\nTo study higher energies, experiments covering several $10^4$~m$^2$ and exposure\ntimes exceeding several years are necessary, which, at present, can only be\nrealized in ground-based installations. They measure the secondary products\ngenerated by high-energy cosmic-ray particles in the atmosphere -- the\nextensive air showers.  The challenge of these investigations is to reveal the\nproperties of the shower inducing primary particle behind an absorber -- the\natmosphere -- with a total thickness at sea level corresponding to 11 hadronic\ninteraction lengths or 30 radiation lengths.\n\nOne of the most advanced experiments in the energy range from \n$10^{13}$~eV to $10^{17}$~eV is the experiment KASCADE (\"KArlsruhe\nShower Core and Array DEtector\") \\cite{kascadenim}.  It is continuously\noperating since 1996, detecting simultaneously the three main components of air\nshowers.  A $200\\times 200$~m$^2$ scintillator array measures the\nelectromagnetic and muonic components ($E_\\mu>0.23$~GeV).  The central detector\nsystem combines a large hadron calorimeter, measuring the energy, as well as\npoint and angle of incidence for hadrons with energies $E_h>50$~GeV\n\\cite{kalonim}, with several muon detection systems ($E_\\mu>0.49$, 2.4~GeV)\n\\cite{mwpcnim}.  In addition, high-energy muons are measured by an underground\nmuon tracking detector equipped with limited streamer tubes ($E_\\mu>0.8$~GeV)\n\\cite{mtdnim}.\n\nResults of the KASCADE experiment are summarized in\nSects.\\,\\ref{wwsect} - \\ref{especsect}. \nIn several astrophysical models a transition from galactic to extragalactic\ncosmic radiation is expected at energies from $10^{17}$ to $10^{18}$~eV.  A\nparticularity in the energy spectrum in this range is the second {\\sl knee}\\xspace at about\n400~PeV \\cite{naganowatson,pg}.\nThe KASCADE-Grande experiment operates in this energy region, recent results\nwill be reported in Sect.\\,\\ref{grandesect}.\nAn alternative method to investigate high-energy cosmic rays is the detection\nof radio signals from air showers, the status of the LOPES experiment will be\ndiscussed in Sect.\\,\\ref{radiosect}.\n\n\\section{High-energy interactions and air showers}\\label{wwsect}\n\n\n\n\n\n         \n\n\n\nAddressing astrophysical questions with air-shower data necessitates the\nunderstanding of high-energy interactions in the atmosphere. \nThe interpretation of properties of primary radiation derived from air-shower\nmeasurements depends on the understanding of the complex processes during the\ncascade development.  Recent investigations indicate inconsistencies in the\ninterpretation of air shower data \\cite{rothnn,chicagoknee,pg}.  Thus, one of\nthe goals of KASCADE is to investigate high-energy interactions and to improve\ncontemporary models to describe such processes.\nThe program CORSIKA \\cite{corsika} is applied to calculate the development of\nextensive air showers. It contains several models to describe hadronic\ninteractions at low and high energies.  Their predictions are compared with\nexperimental results in order to check their correctness.\n\nStudies of the shower development in the atmosphere have been performed with\nthe multi-detector set-up and interaction models have been improved\n\\cite{wwtestjpg,kascadelateral,rissejpg,kascadeabslength,hadrisvhecricern,annaprd,mayerapp}.\nA valuable tool to test high-energy interaction models are correlations between\ndifferent shower components \\cite{jenskrakow,jenspune}.  \nA couple of years ago some models like SIBYLL\\,1.6, DPMJET\\,2.5, or NEXUS\\,2\nfailed to describe the measurements of particular correlations. On the other\nhand, for contemporary models like QGSJET\\,01, SIBYLL\\,2.1, or DPMJET~2.55, the\nKASCADE measurements are compatible with predictions for various correlations\nbetween the electromagnetic, muonic, and hadronic components, i.e.\\ the\nmeasurements are bracketed by the extreme assumptions of primary protons and\niron nuclei \\cite{jenskrakow,jenspune}.\nWhile in previous analyses  pure proton or iron compositions have been assumed\nas extreme cases, at present, more detailed analyses are performed\n\\cite{jenspune,jrhtorun}. They take into account the spectra for elemental\ngroups as obtained from investigations of the electromagnetic and muonic\ncomponents (see below) and reveal deviations between measurements and\nsimulations for the hadronic component of the order of 10\\% to 20\\%.\n\n\nIn conclusion, the models QGSJET\\,01 \\cite{qgsjet}, SIBYLL\\,2.1 \\cite{sibyll21},\nand DPMJET\\,2.55 \\cite{dpmjet} seem to be the most reliable models to describe\nhigh-energy hadronic interactions.  However, also they are not able to describe\nthe data fully consistently \\cite{ulrichapp}.  This illustrates the possibility\nof experiments like KASCADE, which are able to study details of high-energy\ninteractions in the atmosphere and indicates the progress made in this field\nduring the last decade.  At the same time, this stimulates new efforts, with\nthe objective to improve the present interaction models. Examples are new\ntheoretical concepts included in the QGSJET model \\cite{qgsjet2}, or the\ninvestigations of dedicated changes of physical parameters like the inelastic\nproton-proton cross-section or the inelasticity of hadronic interactions on\nair-shower observables \\cite{wq,isvhecri04kascadewq}.  The latter indicate that\nvariations of interaction parameters within the error bounds of accelerator\nmeasurements yield significant and measurable changes in the air shower\ndevelopment \\cite{kascadewqpune}.\n\n\\section{Cosmic-ray anisotropy}\n\\begin{figure}[t]\n \\includegraphics[width=0.49\\textwidth]{gmaier-aniso.eps}\\hspace*{\\fill}\n        \n\t\n\t\n\t\n \\begin{minipage}[b]{0.47\\textwidth}\t \n  \\caption{Rayleigh amplitudes as function of energy for various\n\t   experiments, for references see \\cite{kascade-aniso}.  Additionally,\n\t   model predictions for Leaky Box models \\cite{ptuskinaniso} and a\n\t   diffusion model \\cite{candiaaniso} are shown. For the latter, the\n\t   lines indicate the expected anisotropy for primary protons, iron\n\t   nuclei, and all particles.}\n  \\label{aniso}\t  \n \\end{minipage}\t  \n\\end{figure}\n\nSupernova remnants, such as Cassiopeia A, have been observed in electromagnetic\nradiation in a wide energy range up to TeV-energies.  Calculations indicate\nthat the observed multi-wavelength spectra are consistent with the acceleration\nof cosmic-ray electrons and hadrons in supernova remnants \\cite{berezhko-casa}.\nRecent observations by the H.E.S.S. experiment reveal a shell structure of the\nsupernova remnant RXJ-1713 and an energy spectrum of $\\gamma$-rays $\\propto\nE^{-2.2}$ in agreement with the idea of particle acceleration in a shock front\n\\cite{hesssnr}.\n\nAlso, of great interest is to study the arrival direction of charged cosmic\nrays to search for potential point sources.  The arrival directions of showers\nwith energies above 0.3~PeV covering a region from $10^\\circ$ to $80^\\circ$\ndeclination have been investigated with KASCADE \\cite{kascade-points}.  No\nsignificant excess has been observed neither for all showers, nor for muon-poor\nevents.  The analysis has been deepened by investigating a narrow band\n($\\pm1.5^\\circ$) around the Galactic plane. Also circular regions around 52\nsupernova remnants and 10 TeV-$\\gamma$-ray sources have been studied.  None of\nthe searches provided a hint for a point source, neither by taking into account\nall events, nor selecting muon-poor showers only.  Upper limits for the fluxes\nfrom point like sources are determined to be around\n$10^{-10}$~m$^{-2}$s$^{-1}$. In addition, no clustering of the arrival\ndirection for showers with primary energies above 80~PeV is visible.\n\nWhile the search for point sources is related to the investigation of\ncosmic-ray acceleration sites, the large scale anisotropy is expected to reveal\nproperties of the cosmic-ray propagation.  The Rayleigh formalism is applied to\nthe right ascension distribution of extensive air showers measured by KASCADE\n\\cite{kascade-aniso}.  No hints of anisotropy are visible in the energy range\nfrom 0.7 to 6~PeV. This accounts for all showers, as well as for subsets\ncontaining showers induced by predominantly light or heavy primary particles.\nUpper limits for Rayleigh amplitudes are shown in \\fref{aniso}.\nThe increase of the amplitudes as function of energy is predicted by\ncalculations using a diffusion model to describe the cosmic-ray propagation in\nthe Galaxy \\cite{candiaaniso}.  This indicates that leakage from the Galaxy\nplays an important part during cosmic-ray propagation and most likely, the\nleakage is also (partly) responsible for the origin of the {\\sl knee}\\xspace. On the other\nhand, simple Leaky-Box models seem to be ruled out by the measurements\n\\cite{kascade-aniso,maierflorenz,aspenphen}.\n\n\n\\section{Energy spectra and mass composition of cosmic rays}\\label{especsect}\n\\begin{figure}[t]\n \\includegraphics[width=0.49\\textwidth]{ulrich-qgsjet.eps}\\hspace*{\\fill}\n \\includegraphics[width=0.49\\textwidth]{ulrich-sibyll.eps} \\vspace*{-0mm}\n \\caption{Energy spectra for five cosmic-ray mass groups for measurements\n\t  interpreted with two different models to describe high-energy\n\t  hadronic processes in the atmosphere: QGSJET (left) and SIBYLL\n\t  (right) \\cite{ulrichapp}. The bands represent systematic errors.}\n \\label{holger}\t  \n\\end{figure}\n\nThe main objective of KASCADE is to determine the energy spectra and mass\ncomposition of cosmic rays. The problem has been approached from various points\nof view.  It could be shown that a {\\sl knee}\\xspace exists in all three main shower\ncomponents, i.e.\\ electrons, muons, and hadrons at energies $\\approx 4-5$~PeV\n\\cite{allknee}.  The primary energy spectrum could be established based on the\nelectromagnetic and muonic \\cite{glasstetterslc} as well as the hadronic and\nmuonic components \\cite{hknie}.  An analysis of muon densities showed that the\n{\\sl knee}\\xspace in the all-particle spectrum is caused by a suppression of light elements\n\\cite{muden}.  Analyses of the electromagnetic and muonic shower components\n\\cite{weber}, the hadronic and muonic components \\cite{kascadehm}, as well as\nvarious combinations of them \\cite{rothnn} indicate an increase of the mean\nlogarithmic mass of cosmic rays as function of energy in the {\\sl knee}\\xspace region.  The\nlongitudinal development of the muonic shower component is studied with the\nmuon tracking detector of the KASCADE-Grande experiment \\cite{buettner}.  The\nmeasured flux of unaccompanied hadrons at ground level has been used to derive\nthe spectrum of primary protons \\cite{kascadesh}. The resulting flux follows a\nsingle power law in the energy range from 100~GeV to 1~PeV and is compatible\nwith direct measurements. \n\nAn advanced analysis is founded on the measurement of the electromagnetic and\nmuonic shower components \\cite{ulrichapp}.  It is based on the\ndeconvolution of a two-dimensional electron muon number distribution.\nUnfolding is performed using two hadronic interaction models (QGSJET\\,01 and\nSIBYLL\\,2.1) to interpret the data. The spectra obtained for five elemental\ngroups are displayed in \\fref{holger}. They exhibit sequential cut-offs in the\nflux for the light elements.  For both models a depression is visible for\nprotons around 3 to 4~PeV and at higher energies for helium nuclei.  The\nsystematic differences in flux for the spectra derived with QGSJET and SIBYLL\namount to a factor of about two to three.  The silicon and iron groups show a\nrather unexpected behavior for both models.  The increase of the flux for both\ngroups (QGSJET) and the early cut-off for the silicon group (SIBYLL) is not\ncompatible with contemporary astrophysical models.  The discrepancies are\nattributed to the fact that none of the models is able to describe the observed\ndata set in the whole energy range consistently \\cite{ulrichapp}. \n\nDespite of the discrepancies, the spectra compare well to the results obtained\nby the EAS-Top experiment \\cite{eastopspec} and extend the results of direct measurements to\nhigh energies \\cite{aspenreview}.  Considering the energy range above 10~GeV,\nat least a qualitative picture of the energy spectra for individual mass groups\nemerges: the spectra seem to be compatible with power laws with a cut-off at\nhigh energies.  The cut-off behavior indicated by the measurements is reflected\nby theoretical considerations taking into account the maximum energy attained\nduring acceleration in supernova remnants \\cite{sveshnikova} or diffusive\npropagation of cosmic rays in the Galaxy \\cite{kalmykov}. \n\n\\section{Towards the second knee and the transition to extragalactic cosmic\n         rays}\\label{grandesect}\n\nEnergy spectra have been reconstructed with KASCADE data up to energies of\n100~PeV. At these energies statistical errors start to dominate the overall\nerror. To improve this situation, the experiment has been enlarged.  Covering\nan area of 0.5~km$^2$, 37 detector stations, containing 10~m$^2$ of plastic\nscintillators each, have been installed to extend the original KASCADE set-up\n\\cite{grande}.  Regular measurements with this new array and the original\nKASCADE detectors, forming the KASCADE-Grande experiment, are performed since\nsummer 2003 \\cite{chiavassapune}.  In parallel,  a flash ADC system is being\ndeveloped to measure the time structure of air showers \\cite{brueggemannpune}.\nThe objective is to reconstruct energy spectra for groups of elements up to\n$10^{18}$~eV \\cite{haungsaspen}, covering the energy region of the second\n{\\sl knee}\\xspace, where the galactic cosmic ray spectrum is expected to end\n\\cite{aspenphen}.\n\n\\begin{figure}[t]\n \\includegraphics[width=0.49\\textwidth]{grandenenm.eps} \\hspace*{\\fill}\n \\begin{minipage}[b]{0.47\\textwidth}\t \n \\caption\n         \n\t \n\t  Reconstructed number of electrons as function of the number of muons.\n\t  The dashed lines indicate estimates for the primary energy for\n\t  showers with zenith angles of 0$^\\circ$ and 18$^\\circ$.\n\t  Parallel to the lines light elements are at the top and heavy \n\t  elements at the bottom of the distribution\n\t  \\cite{glasstetterpune}.}\n \\label{grande}\t  \n \\end{minipage}\n\\end{figure}\n\nFirst analyses extend the lateral distributions of electrons and muons up to\n600~m \\cite{glasstetterpune,vanburenpune}.  \nA measured two-dimensional shower size spectrum is shown in \\fref{grande},\nthe number of electrons is plotted as function of the number of muons.\nTo guide the reader, the dashed lines indicate an estimated primary energy.\nOne recognizes that already now with this data set, based on one year of\nmeasurements, energies close to $10^{18}$~eV are reached. It is planned to\nconduct an unfolding analysis, similar to the one described above, and reveal\nthe energy spectra for groups of elements up to $10^{18}$~eV.\n\n\\section{Radio emission from air showers}\\label{radiosect}\n\\begin{figure}[t]\n \\includegraphics[width=0.49\\textwidth]{radiogeom.eps}\\hspace*{\\fill}\n \\includegraphics[width=0.49\\textwidth]{radionm.eps}\\vspace*{-0mm}\n \\caption{Dependence of the measured radio signal yield on the angle between\n\t  the geomagnetic field and the shower axis ({\\sl left}) and the number\n\t  of muons in the shower ({\\sl right}).}\n \\label{lopes}\t  \n\\end{figure}\n\nRadio emission from air showers is known since the 1960ies \\cite{jelleynature}.\nMost likely its origin are electrons deflected in the geomagnetic field and\nemitting synchrotron radiation in the radio frequency range \\cite{allanrev}.\nCalculations show, that on ground level signals in the range of\n$\\sim\\mu$V\/(m\\,MHz) are expected at frequencies of a few tens of MHz and\ndistances to the shower core smaller than 250~m \\cite{huegefalcke}.\nThe LOPES experiment registers radio signals in the frequency range from 40 to\n80~MHz \\cite{lopesspie}. In this band are few strong man made radio\ntransmitters only, the emission from air showers is still strong (it decreases\nwith frequency), and background emission from the Galactic plane is still low.  \nAn active short dipole has been chosen as antenna. \nAn inverted V-shaped dipole is positioned about 1\/4 of the\nshortest wavelength above an aluminum ground plate. In this way a broad\ndirectional beam pattern is obtained.\nThirty antennas have been installed at the site of the KASCADE-Grande\nexperiment \\cite{nehlspune}. LOPES is triggered on large air showers detected\nwith KASCADE-Grande.  All antennas, including the complete analog electronic\nchain, have been individually calibrated with a reference radio source\n\\cite{nehlsarena}.\n\nOne of the most important results is the dependence of the radio signal pulse\nheight on the angle between the shower axis and the geomagnetic field\n\\cite{radionature,hornefferarena}. The measured radio pulse height normalized\nto the number of muons in the respective shower is presented as function of the\ncosine of the angle with respect to the geomagnetic field in \\fref{lopes} ({\\sl\nleft}). The signal yield clearly depends on the orientation with respect to the\nmagnetic field. This can be interpreted as confirmation for the proposed origin\nof the radio frequency radiation, i.e.  synchrotron radiation in the\ngeomagnetic field.  It is planned to measure the polarization of the radio\nsignal as well, this will clarify the situation.  The measured radio pulse\nheight is shown as function of the registered number of muons in \\fref{lopes}\n({\\sl right}).  The radio signal yield increases as function of muon number.\nThe latter is strongly correlated to the shower energy (nearly independent of\nthe mass of the primary), the range depicted corresponds roughly to about\n$10^{17}$ to $6\\cdot10^{17}$~eV.\n\nIn addition, alternative antenna designs are investigated \\cite{gemmekearena}.\nThe experimental work is accompanied by efforts to include models for the radio\nsignal generation into the standard air shower simulation program CORSIKA\n\\cite{huegepune}.\n\n\n\\section*{References}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Optimality and Effectiveness} \\label{sec:optimality}\n\n\nAlg.~\\ref{alg:ict} computes an optimum flow $\\vect F^*$, whose components are determined by the quantities $r$ in step 4. Namely, the components of the $i$-th row of $\\vect F^*$, are given recursively as $F^*_{i,\\vect s[1]} = \\min(p_i, q_{\\vect s[1]})$ and $F^*_{i,\\vect s[l]} = \\min(p_i - \\sum_{u=1}^{l-1} F^*_{i,\\vect s[u]}, q_{\\vect s[l]})$ for $l=2, \\ldots, h_q$.\n\n\\begin{lemma} \\label{lemma:ict}\n  Each row $i$ of the flow $\\vect F^*$ of Algorithm~\\ref{alg:ict} has a certain number $k_i$, $1\\leq k_i\\leq h_q$ of nonzero components, which are given by\n  $F^*_{i,\\vect s[l]} = q_{\\vect s[l]}$ for $l=1,\\ldots, k_i-1$ and\n  $F^*_{i,\\vect s[k_i]} = p_i - \\sum_{l=1}^{k_i-1}q_{\\vect s[l]}$. \n\\end{lemma}\nThe Lemma follows by keeping track of the values of the term $r$ in step~4 in Alg.~\\ref{alg:ict}.\nAn immediate implication is that the flow $F^*$ satisfies the constraints (\\ref{eq:outflow}) and  (\\ref{eq:relaxed}). \nOne can also show that $F^*$ is a minimal solution of (\\ref{eq:objective}) under the constraints (\\ref{eq:outflow}) and  (\\ref{eq:relaxed}),\nand this leads to the following theorem.\n\n\\begin{theorem}\n  (i) The flow $F^*$ of Algorithm~\\ref{alg:ict} is an optimal solution of the relaxed minimization problem given by (\\ref{eq:objective}), (\\ref{eq:outflow}) and (\\ref{eq:relaxed}).\n  (ii) ICT provides a lower bound on EMD.\n\n\\end{theorem}\n\n\\begin{proof}\n\nProof of part (i): It has already been shown that the flow $\\vect F^*$ satisfies constraints (\\ref{eq:outflow}) and (\\ref{eq:relaxed}), and it remains to show that $\\vect F^*$ achieves the minimum in (\\ref{eq:objective}). To this end, let  $\\vect F$ be any nonnegative flow, which satisfies (\\ref{eq:outflow}) and (\\ref{eq:relaxed}). To show that  $\\vect F^*$ achieves the minimum in (\\ref{eq:relaxed}), it is enough to show that for every row $i$, one has\n$\\sum_{j} F_{i,j} C_{i,j} \\geq \\sum_{j} F^*_{i,j} C_{i,j}$, which then implies $\\sum_{i,j} F_{i,j} C_{i,j} \\geq \\sum_{i,j} F^*_{i,j} C_{i,j}$.\n\nBy Alg.~\\ref{alg:ict}, there is a reordering given by the list $\\vect s$ such that\n\\begin{equation} \\label{eq:proof1}\nC_{i,\\vect s[1]}\\leq C_{i,\\vect s[2]}\\leq \\ldots \\leq C_{i,\\vect s[n_q]}.\n\\end{equation}\nBy Lemma~\\ref{lemma:ict}, there is a $k_i\\leq n_q$ such that $\\sum_{l=1}^{k_i} F^*_{i,\\vect s[l]} = p_i$ and  $F^*_{i,\\vect s[l]} = 0$ for $l>k_i$. Furthermore by Lemma~\\ref{lemma:ict} and by constraint (\\ref{eq:relaxed}) on $\\vect F$ , it follows that\n\\begin{equation} \\label{eq:proof2}\nF_{i,\\vect s[l]} \\leq q_{\\vect s[l]} = F^*_{i,\\vect s[l]} \\quad \\text{for $l=1, \\ldots, k_i-1$}.\n\\end{equation}\nThe outflow-constraint (\\ref{eq:outflow}) implies\n$\\sum_{j} F_{i,j} = p_i = \\sum_{j} F^*_{i,j}$ or, equivalently,\n\\begin{equation} \\label{eq:proof3}\n\\sum_{l=k_i}^{n_q} F_{i,\\vect s[l]} =\n                   F^*_{i,\\vect s[k_i]} + \\sum_{l=1}^{k_i-1}(F^*_{i,\\vect s[l]} - F_{i,\\vect s[l]}).\n\\end{equation}\n\nIn the following chain of inequalities, the first inequality follows from (\\ref{eq:proof1}), and  (\\ref{eq:proof3}) implies the equality in the second step.\n\\begin{eqnarray*}\n\\sum_{l=k_i}^{n_q} C_{i,\\vect s[l]}F_{i,\\vect s[l]}\n&{\\geq}&C_{i,\\vect s[k_i]}\\sum_{l=k_i}^{n_q} F_{i,\\vect s[l]} \\\\\n  &{=}&C_{i,\\vect s[k_i]}(F^*_{i,\\vect s[k_i]} + \\sum_{l=1}^{k_i-1}(F^*_{i,\\vect s[l]} - F_{i,\\vect s[l]})) \\\\\n  &{=}&C_{i,\\vect s[k_i]}F^*_{i,\\vect s[k_i]} + \\sum_{l=1}^{k_i-1}C_{i,\\vect s[k_i]}(F^*_{i,\\vect s[l]} - F_{i,\\vect s[l]}) \\\\\n  &{\\geq}&C_{i,\\vect s[k_i]}F^*_{i,\\vect s[k_i]} + \\sum_{l=1}^{k_i-1}C_{i,\\vect s[l]}(F^*_{i,\\vect s[l]} - F_{i,\\vect s[l]}).\n\\end{eqnarray*}\nThe inequality in the last step follows from (\\ref{eq:proof1}) and the fact that the terms\n$F^*_{i,\\vect s[l]} - F_{i,\\vect s[l]}$ are nonnegative by (\\ref{eq:proof2}). By rewriting the last inequality, one obtains the desired inequality\n\\begin{eqnarray*}\n\\sum_{j} F_{i,j} C_{i,j} &=& \\sum_{l=1}^{n_q} F_{i,\\vect s[l]} C_{i,\\vect s[l]} \\\\\n&\\geq& \\sum_{l=1}^{k_i} F^*_{i,\\vect s[l]} C_{i,\\vect s[l]} \\\\\n&=& \\sum_{j} F^*_{i,j} C_{i,j} , \n\\end{eqnarray*}\nwhere in the last equation  $F^*_{i,\\vect s[l]} = 0$ for $l>k_i$ is used.\n\nProof of part (ii): Since ICT is a relaxation of the constrained minimization problem of the EMD, ICT provides a lower bound on EMD given by the output of Alg.~\\ref{alg:ict}, namely, $\\sum_{i,j} F^*_{i,j} C_{i,j} = \\text{ICT}(\\vect p, \\vect q) \\leq \\text{EMD}(\\vect p, \\vect q)$.\n\n\\end{proof}\n\nSimilar to Alg.~\\ref{alg:ict}, Alg.~\\ref{alg:aict} also determines an optimum flow $F^*$, which now depends on the number of iterations $k$. \n\\begin{lemma} \\label{lemma:aict}\n  Each row $i$ of the flow $\\vect F^*$ of Algorithm~\\ref{alg:aict} has a certain number $k_i$, $1\\leq k_i\\leq k$ of nonzero components, which are given by\n  $F^*_{i,\\vect s[l]} = q_{\\vect s[l]}$ for $l=1,\\ldots, k_i-1$ and\n  $F^*_{i,\\vect s[k_i]} = p_i - \\sum_{l=1}^{k_i-1}q_{\\vect s[l]}$. \n\\end{lemma}\nBased on this Lemma, one can show that the flow $F^*$ from Algorithm~\\ref{alg:aict} is an optimum solution to the minimization problem given by (\\ref{eq:objective}), (\\ref{eq:outflow}) and (\\ref{eq:relaxed}), in which the constraint (\\ref{eq:relaxed}) is further relaxed in function of the predetermined parameter $k$. Since the constrained minimization problems for ICT, ACT, OMR, RWMD form a chain of increased relaxations of EMD, one obtains the following result. \n\n\\begin{theorem} \n  For two normalized histograms $\\vect p$ and $\\vect q$:\n  $\\text{RWMD}(\\vect{p},\\vect{q}) \\leq  \\text{OMR}(\\vect{p},\\vect{q}) \\leq \\text{ACT}(\\vect{p},\\vect{q}) \\leq \\text{ICT}(\\vect{p},\\vect{q}) \\leq \\text{EMD}(\\vect{p},\\vect{q})$.\n\\end{theorem}\n\n\nWe call a nonnegative cost function $\\vect C$ {\\it effective}, if for any indices\n$i, j$, the equality $C_{i,j}= 0$ implies $i = j$. For a topological\nspace, this condition is related to the Hausdorff property. For\nan effective cost function $\\vect C$, one has $C_{i,j} > 0$ for all $i \\neq j$,\nand, in this case, $\\text{OMR}(\\vect p, \\vect q) = \\sum_{i,j}C_{i,j}F^*_{i,j}=0$ implies $F^*_{i,j}=0$ for $i \\neq j$ and, thus, $k_i=1$ in\nLemma~\\ref{lemma:aict} and, thus, $\\vect F^*$ is diagonal with $F^*_{i,i}=p_i$. This implies $p_i\\leq q_i$ for all $i$ and, since both histograms are normalized, one must have $\\vect p = \\vect q$.\n\\begin{theorem}\n  If the cost function $\\vect C$ is effective, then  $\\text{OMR}(\\vect p, \\vect q) = 0$ implies $\\vect p = \\vect q$, i.e., OMR is effective.\n\\end{theorem}\n\n\\begin{remark}\nIf  $\\text{OMR}$ is effective, then, a fortiori, ACT and ICT are also effective. However, RWMD does not share this property.\n\\end{remark}  \n\n\n\\section{Complexity Analysis} \\label{sec:complexity}\n\n\nThe algorithms presented in Section 3 assume that the cost matrix $\\vect{C}$ is given, \nyet they still have a quadratic time complexity in the size of the histograms. \nAssume that the histograms size is $h$. Then, the size of $\\vect{C}$ is $h^2$.\nThe complexity is determined by the row-wise reduction operations on $\\vect{C}$. \nIn case of the OMR method, the top-$2$ smallest values are computed in each row of $\\vect{C}$ and a \nmaximum of two updates are performed on each bin of $\\vect{p}$. Therefore, the complexity is $O(h^2)$. \nIn case of the ACT method, the top-$k$ smallest values are computed in each row, and  \nup to $k$ updates are performed on each histogram bin. Therefore, the complexity is $O(h^2\\log{k}+hk)$.\nThe ICT method is the most expensive one because 1) it fully sorts the rows of $\\vect{C}$, and 2) it \nrequires $O(h)$ iterations in the worst case. Its complexity is given by $O(h^2\\log{h})$.\n\n\nIn Section~\\ref{sec:linear-complexity}, the complexity of Phase 1 of the LC-ACT algorithm is $O(vhm + nh\\log{k})$ \nbecause the complexity of the matrix multiplication that computes $\\vect{D}$ is $O(vhm)$, and the complexity of \ncomputing top-$k$ smallest distances in each row of $\\vect{D}$ is $O(nh\\log{k})$. The complexity of performing (\\ref{eq:derivey}), \n(\\ref{eq:derivex}), (\\ref{eq:derivet1}), and (\\ref{eq:derivet2}) are $O(nh)$ each. When $k-1$ \niterations of Phase 2 is applied, the overall time complexity of the LC-ACT algorithm is $O(vhm + knh)$.\nNote that when the number of iterations $k$ performed by LC-ACT is a constant, LC-ACT and LC-RWMD \nhave the same time complexity. When the number of iterations are in the order of the dimensionality \nof the coordinates (i.e., $O(k)=O(m)$) and the database is sufficiently large (i.e., $O(n)=O(v)$), LC-ACT \nand LC-RWMD again have the same time complexity, which increases linearly in the size of the histograms $h$. \nIn addition, the sizes of the matrices $\\vect{X}$, $\\vect{V}$, $\\vect{D}$, and $\\vect{Z}$ are $nh$, $vm$, $vh$, and $vk$, respectively. \nTherefore, the overall space complexity of the LC-ACT algorithm is $O(nh+vm+vh+vk)$.\n\n\\section{Background} \\label{sec:background}\n\nEMD can be considered as the discrete version of the Wasserstein distance, and can be used to quantify\nthe affinity between discrete probability distributions. Each probability distribution is modelled as a histogram,\nwherein each bin is associated with a weight and a coordinate in a multi-dimensional vector \nspace. For instance, when measuring the distance between greyscale images, the histogram weights are given by\nthe pixel values and the coordinates are defined by the respective pixel positions (see Fig.~\\ref{fig:illustration} (a)).  \n\n\nThe distance between two histograms is calculated as the cost of transforming one into the other. \nTransforming a first histogram into a second one involves moving weights from the bins of the first\nhistogram into the bins of the second, thereby constructing the second histogram from the first. \nThe goal is to minimize the total distance travelled, wherein the pairwise distances between different \nhistogram bins are computed based on their respective coordinates. This optimization problem is \nwell studied in transportation theory and is the discrete formulation of the Wasserstein distance. \n\n\\begin{figure}[tb!]\n  \\centering%\n  \\includegraphics*[width=.8\\linewidth]{flow}\n \n  \\caption{(a) Converting a 28x28 image into a histogram with $h$=28x28=784 bins. The weights are the pixel values and the embedding vectors are the pixel coordinates. \n  (b) Computing the EMD between two flattened histograms for an $h$x$h$ cost matrix $\\vect{C}$.}\n  \\vspace{-0.05in}\n  \\label{fig:illustration}\n\\end{figure}\n\nAssume that histograms $\\vect{p}$ and $\\vect{q}$ are being compared, where $\\vect{p}$ has $h_p$ entries and $\\vect{q}$ has $h_q$ entries. \nAssume also that an $h_p \\times h_q$ nonnegative cost matrix $\\vect{C}$ is available.\nNote that ${p}_i$ indicates the weight stored in the $i$th bin \nof histogram $\\vect{p}$, ${q}_j$ the weight stored in the $j$th bin of histogram $\\vect{q}$, and ${C}_{i,j}$ the distance between the coordinates of the $i$th bin of $\\vect{p}$ and the $j$th bin of $\\vect{q}$ (see Fig.~\\ref{fig:illustration} (b)). \nSuppose that the histograms are $L_1$-normalized: $\\sum_i {p}_i = \\sum_j {q}_j = 1$. \n\nWe would like to discover a non-negative flow matrix $\\vect{F}$, where ${F}_{i,j}$ indicates how much of the bin $i$ of $\\vect{p}$ has to \\textit{flow} to the bin $j$ of $\\vect{q}$, such that the cost of moving $\\vect{p}$ into $\\vect{q}$ is minimized. Formally, the objective of EMD is as follows:\n\\begin{equation}\n\\text{EMD}(\\vect{p},\\vect{q}) = \\min_{{F}_{i,j} \\geq 0} \\sum_{i,j} {F}_{i,j}\\cdot{C}_{i,j} . \n\\label{eq:objective}\n\\end{equation}\n\n\nA valid solution to EMD has to satisfy the so-called \\textit{out-flow} (\\ref{eq:outflow}) and \\textit{in-flow} (\\ref{eq:inflow}) constraints.\nThe \\textit{out-flow} constraints ensure that, for each $i$ of $\\vect{p}$, the sum of all the flows exiting $i$ is equal to ${p}_i$. \nThe \\textit{in-flow} constraints ensure that, for each $j$ of $\\vect{q}$, the sum of all the flows entering $j$ is equal to ${q}_j$. \nThese constraints guarantee that all the mass stored in $\\vect{p}$ is transferred and $\\vect{q}$ is reconstructed as a result. \n\\begin{equation}\n\\sum_j {F}_{i,j} = {p}_i\n\\label{eq:outflow}\n\\end{equation}\n\\begin{equation}\n\\sum_i {F}_{i,j} = {q}_j\n\\label{eq:inflow}\n\\end{equation}\n\nComputation of EMD requires solution of a minimum-cost-flow problem on a bi-partite graph, wherein \nthe bins of histogram $\\vect{p}$ are the source nodes, the bins of histogram $\\vect{q}$ are the sink nodes, and\nthe edges between the source and sink nodes indicate the pairwise transportation costs. Solving \nthis problem optimally takes supercubical time complexity in the size of the input histograms~\\cite{Ahuja1993}.\n\n\\subsection{Relaxed Word Mover's Distance}~\\label{sec:rwmd}\n\n\nTo reduce the complexity, an approximation algorithm called Relaxed Word Mover's Distance (RWMD) was proposed~\\cite{kusnerskw15}. \nRWMD computation involves derivation of two asymmetric distances. \nFirst, the \\textit{in-flow} constraints are relaxed and the relaxed problem is solved using only \\textit{out-flow} constraints.\nThe solution to the first relaxed problem is a lower bound of EMD.\nAfter that, the \\textit{out-flow} constraints are relaxed and a second relaxed optimization problem is solved using only \\textit{in-flow} constraints, which computes a second lower bound of EMD.\nRWMD is the maximum of these two lower bounds. Therefore, it is at least as tight as each one. In addition, it is symmetric. \n\nFinding an optimal solution to RWMD involves mapping the coordinates of one histogram to the closest coordinates of the other. \nJust like EMD, RWMD requires a cost matrix $\\vect{C}$ that stores the pairwise distances between coordinates of $\\vect{p}$ and $\\vect{q}$. \nFinding the closest coordinates corresponds to row-wise and column-wise minimum operations in the cost matrix (see Fig.~\\ref{fig:rwmd}).\nTo compute the first lower bound, it is sufficient to find the column-wise minimums in the cost matrix, and then \nperform a dot product with the weights stored in $\\vect{p}$. Similarly, to compute the second lower bound, it is sufficient \nto find the row-wise minimums and then perform a dot product with the weights stored in $\\vect{q}$.\nThe complexity of RWMD is given by the cost of constructing the cost matrix $\\vect{C}$: it requires quadratic \ntime and space in the size of the input histograms. Computing the row-wise and column-wise minimums of $\\vect{C}$ also has quadratic time complexity.\n\n\\begin{figure}[tb!]\n  \\centering%\n  \\includegraphics*[width=.5\\linewidth]{rwmd}\n  \\vspace{-0.1in}\n  \\caption{Quadratic-complexity RWMD computation}\n \\label{fig:rwmd}\n  \\vspace{-0.2in}\n\\end{figure}\n\n\n\n\n\\subsection{Linear-Complexity RWMD (LC-RWMD)}\n\nWhen computing RWMD between only two histograms, it is not possible to avoid a quadratic time complexity. \nHowever, in a typical information retrieval system, a query histogram is compared with a large database \nof histograms to identify the top-$\\ell$ most similar histograms in the database. \nIt is shown in~\\cite{AtasuBigData17} that the RWMD computation involves redundant and repetitive\noperations in such cases, and that eliminating this redundancy reduces the average time complexity from quadratic to linear. \n\nAssume that a query histogram is being compared with two database histograms. Assume also\nthat the two database histograms have common coordinates. A simple replication of the RWMD computation\nwould involve creation of two cost matrices with identical rows for the common coordinates. \nAfterwards, it would be necessary to perform reduction operations on these identical rows to compute the row-wise minimums. \nIt is shown in~\\cite{AtasuBigData17} that both of these redundant operations can be eliminated by \n1) constructing a vocabulary that stores the union of the coordinates that occur in the database histograms, \nand 2) computing the minimum distances between the coordinates of the vocabulary and the coordinates of the query only once.\n\nTable~\\ref{tab:parameters} lists the algorithmic parameters that influence the \ncomplexity. Table~\\ref{tab:complexity} shows the complexity of RWMD and LC-RWMD \nalgorithms when comparing one query histogram with $n$ database histograms. \nWhen the number of database histograms ($n$) is in the order of the size of the vocabulary \n($v$), the LC-RWMD algorithm reduces the complexity by a factor of the average histogram \nsize ($h$). Therefore, whereas the time complexity of a brute-force RWMD implementation \nscales quadratically in the histogram size, the time complexity of LC-RWMD scales only linearly. \n\n\\begin{table}[t!]\n\\caption{Algorithmic Parameters}\n\\label{tab:parameters}\n\\begin{center}\n\\begin{tabular}{lc}\n  \\hline\n  $n$           & Number of database histograms\\\\\n  $v$           & Size of the vocabulary\\\\ \n  $m$           & Dimensionality of the vectors \\\\ \n  $h$           & Average histogram size \\\\ \n  \\hline\n\\vspace{-0.4in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t!]\n\\caption{Complexity of Computing $n$ RWMD Distances}\n\\label{tab:complexity}\n\\begin{center}\n\\begin{tabular}{ccc}\n  \\hline\n              & Time Complexity & Space Complexity \\\\ \n  \\hline\n  LC-RWMD     & $O(vhm+nh)$        & $O(nh+vm+vh)$ \\\\\n  RWMD        & $O(nh^2m)$         & $O(nhm)$ \\\\  \n  \\hline\n  \\vspace{-0.4in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{Conclusions}\n\nThis paper offers new theoretical and practical results for improving \nthe efficiency and the accuracy of approximate EMD computation in both\nhigh and low dimensions. \nWe identify the shortcomings of the RWMD measure and propose improved lower \nbounds that result in a higher nearest-neighbors-search accuracy and robustness without \nincreasing the computational complexity significantly. Under realistic assumptions, the \ncomplexity of our methods scale linearly in the size of the input probability distributions. \nIn addition, our methods are data-parallel and well-suited for GPU acceleration. \nThe experiments demonstrate a four orders of magnitude improvement \nof the performance without any loss of nearest-neighbors-search accuracy \nwith respect to WMD on high-dimensional text datasets. \nSimilar improvements have been achieved with \nrespect to Sinkhorn's algorithm on two-dimensional image datasets.\n\n\\section{Evaluation}\n\nWe performed experiments on two public datasets: \\emph{20 Newsgroups} \nis a text database of newsgroup documents, partitioned evenly across 20 different classes\\footnote{http:\/\/qwone.com\/~jason\/20Newsgroups\/}, \nand \\emph{MNIST} is an image database of \ngreyscale hand-written digits that are partitioned evenly across 10 classes\\footnote{http:\/\/yann.lecun.com\/exdb\/mnist\/}. \nThe MNIST images are mapped to histograms as illustrated in Fig.~\\ref{fig:illustration}, \nwherein the weights are normalized pixel values and the embedding vectors indicate the coordinates of the pixels.\nThe words in 20 Newsgroups documents are mapped to a 300-dimensional real-valued embedding \nspace using Word2Vec vectors that are pre-trained on Google News\\footnote{https:\/\/code.google.com\/archive\/p\/word2vec\/}, \nfor which the size of the vocabulary ($v$) is 3M words and phrases. \nIn our setup, each histogram bin is associated with a word from this vocabulary. \nThe first 100 words of the vocabulary are treated as stop words. These stop words \nand the Word2Vec phrases are not mapped to histogram bins.\nThe histogram weights indicate normalized frequencies of words found in each document. \nIn addition, 20 Newsgroups histograms are truncated to store only the most-frequent 500 words found in each document.\nTab.~\\ref{tab:benchmarks} shows some properties of the datasets and the results of our preprocessing.\nNote that the size of the vocabulary used has an impact on the complexity of our methods (see Tab.~\\ref{tab:complexity2}).\n\nThe results we provide in the remainder of the text are associated \nwith linear complexity implementations of RWMD, OMR, and ACT methods.\nTo improve the robustness of these methods, we compute two asymmetric \nlower bounds and take the maximum of the two as discussed in Section~\\ref{sec:rwmd}. \nWord2Vec embedding vectors are $L_2$-normalized, but MNIST embedding \nvectors are not normalized in our setup. In addition, when computing \nour approximations, the histogram weights are always \n$L_1$-normalized. Lastly, the transportation cost between two words or \npixels is the Euclidean ($L_2$) distance between their embedding vectors.\n\n\\begin{table}[t!]\n\\vspace{-0.1in}\n\\caption{Dataset properties: no. docs ($n$), average size of histograms ($h$), original vocabulary size ($v$), size of used vocabulary}\n\\label{tab:benchmarks}\n\\begin{center}\n\\begin{tabular}{lcccc}\n  \\hline\n                 & Size $n$ & Average $h$ & Original $v$ & Used $v$ \\\\ \n  \\hline\n  20 News        & 18828  & 78.8  & 3M  & 69682  \\\\\n  MNIST          & 60000  & 149.9 & 784 & 717 \\\\\n  \\hline\n  \\vspace{-0.3in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nIn our experiments, we treated each document of the database as a query and compared it with \nevery other document in the database. Based on the distance measure used in the \ncomparison, for each query document, we identified the top-$\\ell$ nearest neighbors in the \ndatabase. After that, for each query document, we computed the percentage of documents \nin its nearest-neighbors list that have the same label.\nWe averaged this metric over all the query documents and computed it as a function \nof $\\ell$. The result is the average \\emph{precision @ top-$\\ell$} for the query documents,\nand indicates the expected accuracy of nearest neighbors search queries.\n\nWe compared our new distance measures with simple baselines, such as, Bag-of-Words (BoW) \ncosine similarity and the Word Centroid Distance (WCD)~\\cite{kusnerskw15} measures, both \nof which exhibit a lower algorithmic complexity than our methods. The BoW approach does not \nuse the proximity information provided by the embedding vectors. It simply computes a dot \nproduct between two sparse histograms after an $L_2$ normalization of the weights. \nThe complexity of computing BoW cosine similarity between one query histogram and $n$ \ndatabase histograms is $O(nh)$. The WCD measure, on the other hand, is closely related \nto document embedding techniques. For each document, it first computes a centroid vector of \nsize $m$, which is a weighted average of the embedding vectors, and then determines  \nthe Euclidean distances between these centroid vectors. The complexity of \ncomputing WCD between one query document and $n$ database documents is $O(nm)$.  \n\n\nWe compared our distance measures with state-of-the-art EMD approximations as well, \nsuch as WMD and Sinkhorn distance. WMD uses the FastEMD library\\footnote{https:\/\/github.com\/LeeKamentsky\/pyemd} to approximate the EMD,\nwhich uses a thresholding technique~\\cite{Pele2009} to reduce the time to compute EMD. \nIn addition, WMD uses an RWMD-based pruning technique to reduce the number of calls to FastEMD~\\cite{kusnerskw15}.\nTo improve the WMD performance further, we developed a multi-threaded CPU implementation of the pruning technique.\nWe also compared our methods with Cuturi's open-source implementation\nof Sinkhorn's algorithm\\footnote{http:\/\/marcocuturi.net\/SI.html}, which can be executed on both CPUs and GPUs. \nWe used $\\lambda=20$ as the entropic regularization parameter because it offered \na good trade-off between the accuracy and the speed of Sinkhorn's algorithm. \n\n\nWe have developed GPU-accelerated implementations of the WCD, RWMD, OMR, ACT, and BoW cosine similarity \nmethods and evaluated their performance on an NVIDIA\\textsuperscript{\\textregistered} GTX 1080Ti GPU. \nCuturi's Sinkhorn implementation has also been executed on the same GPU.  \nOur multithreaded WMD implementation has been deployed on an 8-core Intel\\textsuperscript{\\textregistered} i7-6900K CPU. \nTop-$\\ell$ calculations have been performed on the same Intel\\textsuperscript{\\textregistered} i7-6900K CPU in all cases.\n\nTheorem~\\ref{theorem:ordering} states that the more complex the considered algorithms, \nthe smaller the gap to the EMD and, hence, the better the accuracy. The least complex \nACT algorithm is the RWMD, which corresponds to the ACT-0 with zero iterations \nin Phase 2 (see Fig.~\\ref{fig:lc-ict}). The second most complex algorithm is the OMR. \nThe third most complex ACT algorithm is ACT-1 with a single iteration in Phase 2.\nThe most complex ACT algorithm we considered was ACT-15 with 15 iterations in Phase 2. \nOur experiments show that the search accuracy improves with the complexity and, \nthus, illustrate the accuracy vs complexity trade-off. Typically, most of the \nimprovement in the search accuracy is achieved by the first iteration of Phase 2,\nand subsequent iterations result in a limited improvement only. As a result, \nACT-1 offers very favorable accuracy and runtime combinations. \n\n\nFigure~\\ref{fig:combined} (a) shows the accuracy and runtime trade-offs between \ndifferent methods on the 20 Newsgroups dataset. Note that even though ACT-1 is \napproximately 20-fold slower than BoW cosine similarity, it offers a 4.5\\% to 7.5\\% higher search accuracy.\nTypically, the accuracy improvement with respect to BoW becomes larger as we increase $\\ell$. \nFig.~\\ref{fig:combined} (a) also shows that ACT-1 is approximately 20000-fold faster than WMD, \nbut offers a similar search accuracy. It is only 30\\% slower than RWMD, but results in a 2\\% to 3.5\\% \nhigher search accuracy. OMR is somewhere between RWMD and ACT-1 in terms of both runtime and search accuracy.\nFinally, ACT-7 is approximately 10000-fold faster than WMD, and offers a slightly higher search accuracy! \n\n\\begin{figure}[tb!]\n  \\centering%\n  \\vspace{-0.03in}\n  \\includegraphics*[width=1.0\\linewidth]{Results_combined}\n  \\vspace{-0.3in}\n  \\caption{Runtime vs accuracy for 20News and the MNIST subset}\n  \\vspace{-0.17in}\n  \\label{fig:combined}\n\\end{figure}\n\nIn case of MNIST, because the number of dimensions is small \n($m=2$), RWMD is almost as fast as BoW cosine similarity. However, \nthe runtime of the Phase 2 of the ACT-1 method is much more significant \nthan that of its Phase 1. Therefore, the runtime increase with respect \nto BoW is around ten fold for ACT-1. Nevertheless, when using ACT-1, computing\nall pairwise distances between 60000 MNIST training images (i.e., 3.6 billion \ndistance computations) takes only 3.3 minutes. The accuracy comparisons between\nBoW, RWMD, and ACT methods for the complete MNIST database are given in Tab.~\\ref{tab:mnist_wo_bg}.\nThe accuracy is already very high when using BoW because the images are normalized and centered. \nOur methods are comparable to BoW for small $\\ell$, but outperform it for large enough $\\ell$.\n\nComputing all pairwise distances for the complete set of MNIST training images \nwould take months when using WMD and Sinkhorn's algorithm. To enable comparisons \nwith these two methods, we have set up a simpler experiment. We used only \nthe first 6000 MNIST training images as our query documents and compared them \nwith the complete set of 60000 MNIST training images. The results are given \nin Fig.~\\ref{fig:combined} (b). We observe that ACT-1 is four orders of magnitude \nfaster than Sinkhorn's algorithm when running on the same GPU, yet it achieves a higher search accuracy! \nSimilarly, ACT-1 is five orders of magnitude faster than WMD while achieving a higher search accuracy! \nThese results show that the OMR and the ACT measures we proposed are meaningful on their own, and using more \ncomplex measures, such as WMD or Sinkhorn does not necessarily improve the accuracy of nearest-neighbors search.\n\n\n\n\nTable~\\ref{tab:mnist} illustrates the sensitivity of RWMD to a minor \nchange in the data representation. Here, we simply explore the impact \nof including the background (i.e. the black pixels) in the MNIST \nhistograms. The most immediate result is that when comparing two \nhistograms, all their coordinates overlap. As a result, the distance \ncomputed between the histograms by RWMD is always equal to zero, and \nthe top-$\\ell$ nearest neighbors are randomly selected, resulting in a \nprecision of 10\\% for RWMD. The OMR technique solves this problem \nimmediately even though its accuracy is lower than that of BoW cosine \nsimilarity. In fact, several iterations of ACT are required to \noutperform BoW. However, these results demonstrate\nthe improved robustness and effectiveness of our methods in comparison to RWMD.\n\n\\begin{table}[t!]\n\\caption{Precision @ top-$\\ell$ for MNIST (without background)}\n\\label{tab:mnist_wo_bg}\n\\begin{center}\n\\begin{tabular}{lccccc}\n  \\hline\n  $\\ell$      & BoW & RWMD & ACT-1 & ACT-3 & ACT-7 \\\\ \n  \\hline\n  1           & 0.9771 & 0.9752 & 0.9776 & 0.9780 & 0.9781 \\\\\n  16          & 0.9480 & 0.9481 & 0.9510 & 0.9520 & 0.9521 \\\\\n  128         & 0.8874 & 0.8963 & 0.8997 & 0.9014 & 0.9016 \\\\\n  \\hline\n  \\vspace{-0.4in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t!]\n\\caption{Precision @ top-$\\ell$ for MNIST (with background)}\n\\label{tab:mnist}\n\\begin{center}\n\\begin{tabular}{lccccc}\n  \\hline\n  $\\ell$      & BoW & RWMD & OMR & ACT-7 & ACT-15 \\\\ \n  \\hline\n  1           & 0.9771 & 0.1123 & 0.9707 & 0.9756 & 0.9783\\\\\n  16          & 0.9480 & 0.1002 & 0.9368 & 0.9470 & 0.9520\\\\\n  128         & 0.8874 & 0.1002 & 0.8692 & 0.8872 & 0.8999\\\\\n  \\hline\n  \\vspace{-0.4in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Introduction}\n\nEarth Mover's Distance (EMD) was initially proposed in the image retrieval field to quantify the similarity between images~\\cite{rubnertg98}. In the optimization theory, a more general formulation of EMD, called Wasserstein distance, has been used extensively to measure the distance between probability distributions~\\cite{villani2003}. In statistics, an equivalent measure is known as Mallows distance~\\cite{emd_mallows}. \nThis paper uses the EMD measure for similary search in image and text databases. \n\nIn the text retrieval domain, an adaptation of EMD, called Word Mover's Distance (WMD), has emerged as a state-of-the-art semantic similarity metric~\\cite{kusnerskw15}. \nWMD captures semantic similarity by using the concept of word embeddings in the computation of EMD. Word embeddings map words into a high-dimensional vector space such that the words that are semantically similar are close to each other. These vectors can be pre-trained in an unsupervised way, e.g., by running the Word2vec algorithm~\\cite{mikolovcorr2013} on publicly available data sets. The net effect is that, given two sentences that cover the same topic, but have no words in common, traditional methods, such as cosine similarity, fail to detect the similarity. \nHowever, WMD detects and quantifies the similarity by taking the proximity between different words into account.\n\nWhat makes EMD-based approaches attractive is their high search and classification accuracy. However, such an accuracy does not come for free. \nIn general, the time complexity of computing these measures grows cubically in the size of the input probability distributions. \nSuch a high complexity renders their use impractical for large datasets. Thus, there is a need for low-complexity approximation methods.\n\nEMD can be computed in quadratic time complexity when an $L_1$ ground distance is used~\\cite{Ling2007,Gudmundsson07}. \nIn addition, approximations of EMD can be computed in linear time by embedding EMD into Euclidean space~\\cite{Indyk2003}. \nHowever, such embeddings result in high distortions in high-dimensional spaces~\\cite{Naor2006}. \nAn algorithm for computing EMD in the wavelet domain has also been proposed~\\cite{ShirdhonkarJ08}, which achieves linear time complexity \nin the size of the input distributions. However, the complexity grows exponentially in the dimensionality of the underlying vector space. \nThus, both linear-complexity approaches are impractical when the number of dimensions is more than three or four. \nFor instance, they are not applicable to WMD because the word vectors typically have several hundred dimensions.\n\nA linear-complexity algorithm for computing approximate EMD distances over high-dimensional vector spaces has also been proposed~\\cite{AtasuBigData17}. \nThe algorithm, called Linear-Complexity Relaxed Word Mover's Distance (LC-RWMD), \nachieves four orders of magnitude improvement in speed with respect to WMD. \nIn addition, on compact and curated text documents, it computes high-quality search results that are comparable to those found by WMD.\nDespite its scalability, the limitations of LC-RWMD are not well understood. Our analysis shows that 1) it is not applicable to dense histograms, and 2) its accuracy decreases when comparing probability distributions with many overlapping coordinates. Our main contributions are as follows: \n\\begin{itemize}\n\\itemsep0em \n\\item We propose new distance measures that are more robust and provably more accurate than LC-RWMD.\n\\item We show that the new measures effectively quantify the similarity between dense as well as overlapping probability distributions, e.g., greyscale images.\n\\item We show that the new measures can be computed in linear time complexity in the size of the input probability distributions: the same as for LC-RWMD.\n\\item We propose data-parallel algorithms that achieve four orders of magnitude speed-up with respect to state of the art without giving up any search accuracy.\n\\end{itemize}\n\n\\section{Linear-Complexity Implementations} \\label{sec:linear-complexity}\n\nIn this section, we focus on the ACT method because 1) it is a generalization of all the other methods presented, \nand 2) its complexity and accuracy can be controlled by setting the number $k$ of iterations  performed. \nWe describe a data-parallel implementation of ACT, which achieves a linear time complexity when \n$k$ is a constant. Unlike the previous section, we do not assume that the cost matrix is given. We \ncompute the transportation costs on the fly, and take into account the complexity of computing these costs as well.\n\nA high-level view of the linear-complexity ACT algorithm (LC-ACT) is given in Figure~\\ref{fig:lc-ict}. \nLC-ACT is strongly inspired by LC-RWMD. Just like LC-RWMD, it assumes that \n1) a query histogram is compared with a large number of database histograms, and \n2) the coordinate space is populated by the members of a fixed-size vocabulary. \nThe complexity is reduced by eliminating the redundant and repetitive operations that \narise when comparing one query histogram with a large number of database histograms. \n\n\\begin{figure}[tb!]\n  \\centering%\n \n  \\includegraphics*[width=0.95\\linewidth]{LC-ICT}\n  \\vspace{-0.20in}\n  \\caption{Linear Complexity ACT}\n  \\vspace{-0.05in}\n  \\label{fig:lc-ict}\n\\end{figure}\n\n\n\nSuppose that the dimension of the coordinates is $m$ and the size of the vocabulary is $v$. \nLet $\\vect{V}$ be an $v \\times m$ matrix that stores this information. Given a query histogram $\\vect{q}$ of size $h$,\nwe construct a matrix $\\vect{Q}$ of size $h \\times m$ that stores the coordinates of the histogram\nentries. Phase 1 of LC-ACT (see Fig.~\\ref{fig:phase1}) performs a matrix-matrix multiplication \nbetween $\\vect{V}$ and the transpose of $\\vect{Q}$ to compute all pairwise distances between the coordinates of \nthe vocabulary and the coordinates of the query. The result is a $v \\times h$ distance matrix, denoted by $\\vect{D}$.\nAs a next step, the top-$k$ smallest distances are computed in each row of $\\vect{D}$. The result\nis stored in a $v \\times k$ matrix $\\vect{Z}$. Furthermore, we store the indices of $\\vect{q}$ that are associated\nwith the top-$k$\nsmallest distances in a $v \\times k$ matrix $\\vect{S}$. We can then construct another $v \\times k$ \nmatrix $\\vect{W}$, which stores the corresponding weights of $\\vect{q}$ by defining $\\vect{W}_{i,l}=\\vect{q}_{{S}_{i,l}}$ for $i=1, \\ldots, v$ and $l=1, \\ldots, k$.\nThe matrices $\\vect{Z}$ and $\\vect{W}$ are then used in Phase 2 to transport the largest possible\nmass, which are constrained by $\\vect{W}$, to the smallest possible distances, which are given by $\\vect{Z}$.\n\nThe database histograms are stored in a matrix $\\vect{X}$ (see Fig.~\\ref{fig:phase2}), wherein each row \nstores one histogram. These histograms are typically sparse. Thus, the matrix $\\vect{X}$ is \nstored using a sparse representation, e.g., in compressed sparse rows (csr) format. For simplicity, \nassume that $\\vect{X}$ is stored in a dense format and $\\vect{X}_{u,i}$ stores the weight of the $i$-th coordinate \nof the vocabulary in the $u$-th database histogram. Note that if the histograms have $h$ entries on average, \nthe number of nonzeros of the matrix $\\vect{X}$ would be equal to $nh$. \n\n\\begin{figure}[tb!]\n  \\centering%\n  \\includegraphics*[width=0.9\\linewidth]{Phase1}\n  \\vspace{-0.2in}\n  \\caption{Phase 1 of LC-ACT}\n  \\vspace{-0.1in}\n  \\label{fig:phase1}\n\\end{figure}\n\n\\begin{figure}[tb!]\n  \\centering%\n \n  \\includegraphics*[width=0.9\\linewidth]{Phase2}\n  \\vspace{-0.2in}\n  \\caption{Phase 2 of LC-ACT}\n  \\vspace{-0.1in}\n  \\label{fig:phase2}\n\\end{figure}\n\nPhase 2 of ACT iterates the columns of $\\vect{Z}$ and $\\vect{W}$ and iteratively transfers weights from the database histograms $\\vect{X}$ to the query histogram $q$. \nLet $\\vect{X}^{(l)}$ represent the residual mass remaining in $\\vect{X}$ after $l$ iterations, where $\\vect{X}^{(0)}=\\vect{X}$. \nLet $\\vect{Y}^{(l)}$ store the amount of mass that is transferred from $\\vect{X}^{(l-1)}$ in iteration $l$, which is  the difference between $\\vect{X}^{(l-1)}$ and $\\vect{X}^{(l)}$. \nLet $\\vect{z}^{(l)}$ and $\\vect{w}^{(l)}$ be the $l$-th columns of $\\vect{Z}$ and $\\vect{W}$, respectively; thus, \n$\\vect{z}^{(l)}_{u}$ is the $l$-th smallest distance between the coordinate $u$ of the vocabulary and the coordinates of the query, \nand $\\vect{w}^{(l)}_{u}$ is the respective weight of the query coordinate that produces the $l$-th smallest distance. \nThe iteration $l$ of Phase 2 computes $\\vect{Y}^{(l)}$ and $\\vect{X}^{(l)}$:  \n\\vspace{-0.1in}\n\\begin{equation}\n  \\vect{Y}^{(l)}_{u,i} = \\min_{\n\t\\scriptsize \n        \\begin{array}{c}\n        u{\\in}\\{1 \\ldots v\\} \\\\\n        i{\\in}\\{1 \\ldots v\\}\n        \\end{array}}  (\\vect{X}^{(l-1)}_{u,i}, \\vect{w}^{(l)}_{u}) . \n\\label{eq:derivey}\n\\end{equation}\n\\vspace{-0.1in}\n\\begin{equation}\n\\vect{X}^{(l)} = \\vect{X}^{(l-1)} - \\vect{Y}^{(l)}. \n\\label{eq:derivex}\n\\end{equation}\nThe cost of transporting $\\vect{Y}^{(l)}$ to $q$ is given by $\\vect{Y}^{(l)} \\cdot \\vect{z}^{(l)}$.\nLet $\\vect{t}^{(l)}$ be a vector of size $n$ that accumulates all the transportation \ncosts incurred between iteration $1$ and iteration $l$:\n\\begin{equation}\n\\vect{t}^{(l)} = \\vect{t}^{(l-1)} + \\vect{Y}^{(l)} \\cdot \\vect{z}^{(l)}.\n\\label{eq:derivet1}\n\\end{equation}\nAfter $k-1$ iterations of Phase 2, there might still be some mass remaining in $\\vect{X}^{(l-1)}$.\nPhase 3 approximates the cost of transporting the remaining mass to $q$ by multiplying $\\vect{X}^{(l-1)}$ \nwith $\\vect{z}^{(k)}$. The overall transportation cost $\\vect{t}^{(k)}$ is:\n\\begin{equation}\n\\vect{t}^{(k)} = \\vect{t}^{(k-1)} + \\vect{X}^{(l-1)} \\cdot \\vect{z}^{(k)}.\n\\label{eq:derivet2}\n\\end{equation}\n\nThe main building blocks of LC-ACT are matrix-matrix and matrix-vector multiplications, \nrow-wise top-$k$ calculations, and parallel element-wise updates, all of which are data-parallel \noperations. Table~\\ref{tab:complexity2} shows the complexity of computing LC-ACT between one \nquery histogram and $n$ database histograms. Note that when $k$ is a constant, LC-ACT and LC-RWMD methods have the same complexity. \n\n\\begin{table}[t!]\n\\caption{Complexity of LC-ACT ($n$ distances, $k$ iterations)}\n  \\vspace{-0.05in}\n\\label{tab:complexity2}\n\\begin{center}\n\\begin{tabular}{ll}\n  \\hline\n  Time & Space \\\\ \n  \\hline\n  $O(vhm+nhk)$    & $O(nh+vm+vh+vk)$ \\\\\n  \\hline\n  \\vspace{-0.35in}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe would like to thank Celestine D\\\"{u}nner, Haralampos Pozidis, Ahmet Solak, and Slavisa Sarafijanovic from IBM Research -- Zurich, Marco Cuturi from Google Brain and Institut Polytechnique de Paris, and the anonymous reviewers of ICML 2019 Conference for their valuable comments.\n\n\\input{Bibliography}\n\\bibliographystyle{icml2019}\n\n\\pagebreak\n\\input{Appendix}\n\n\\end{document}\n\n\n\n\\section{New Relaxation Algorithms}\n\n\nIn this section, we describe improved relaxation algorithms that address the weaknesses of the RWMD measure and its linear-complexity implementation (LC-RWMD). \nAssume that we are measuring the distance between two histograms $\\vect{p}$ and $\\vect{q}$. \nAssume also that the coordinates of the two histograms fully overlap but the respective weights are different (see Fig.~\\ref{fig:same_coordinates}). \nIn other words, for each coordinate $i$ of $\\vect{p}$, there is an identical coordinate $j$ of $q$, for which $\\vect{C}_{i,j}=0$. \nTherefore, RWMD estimates the total cost of moving $\\vect{p}$ into $\\vect{q}$ and vice versa as zero even though $p$ and $q$ are \nnot the same. This condition arises, for instance, when we are dealing with dense histograms. In other cases, the data of interest \nmight actually be sparse, but some background noise might also be present, which results in denser histograms. \n\n\\begin{figure}[tb!]\n  \\centering%\n  \\includegraphics*[width=0.8\\linewidth]{same_coordinates}\n  \\vspace{-0.15in}\n  \\caption{Different histograms with identical coordinates}\n  \\vspace{-0.15in}\n  \\label{fig:same_coordinates}\n\\end{figure}\n\nIn general, the more overlaps there are between the coordinates of $\\vect{p}$ and $\\vect{q}$, the higher \nthe approximation error of RWMD is. The main reason for the error is that when coordinate \n$i$ of $\\vect{p}$ overlaps with coordinate $j$ of $\\vect{q}$, RWMD does not take into account the fact that the\nrespective weights $p_i$ and $q_j$ can be different. In an optimal solution, we would not be moving\na mass larger than the minimum of $p_i$ and $q_j$ between these two coordinates. This is a fundamental \ninsight that we use in the improved solutions we propose.\n\nGiven $\\vect{p}$, $\\vect{q}$ and $\\vect{C}$, our goal is to define new distance measures that relax fewer EMD constraints than RWMD,\nand therefore, produce tighter lower bounds on EMD. Two asymmetric distances can be computed by deriving 1) the cost of moving \n$\\vect{p}$ into $\\vect{q}$ and 2) the cost of moving $\\vect{q}$ into $\\vect{p}$. If both are lower bounds on $\\text{EMD}(\\vect{p},\\vect{q})$, \na symmetric lower bound can be derived, e.g., by using the maximum of the two. Thus, we consider only the computation of the cost of moving $\\vect{p}$ into $\\vect{q}$ without loss of essential generality.\n\nWhen computing the cost of moving $\\vect{p}$ to $\\vect{q}$ using RWMD, the \\textit{in-flow} constraints of (\\ref{eq:inflow}) are removed. \nIn other words, all the mass is transferred from $\\vect{p}$ to the coordinates of $\\vect{q}$, but the resulting distribution is not the same as $\\vect{q}$.\nTherefore, the cost of transforming $\\vect{p}$ to $\\vect{q}$ is underestimated by RWMD. To achieve better approximations of $\\text{EMD}(\\vect{p},\\vect{q})$, instead of \nremoving the \\textit{in-flow} constraints completely, we propose the use of a relaxed version of these constraints: \n\\begin{equation}\nF_{i,j} \\leq {q}_j \\quad \\text{for all $i, j$} .\n\\label{eq:relaxed}\n\\end{equation} \nThe new constraint ensures that the amount of weight that can be moved from a coordinate $i$ of $\\vect{p}$ to a coordinate $j$ of $\\vect{q}$ cannot \nexceed the weight ${q}_j$ at coordinate $j$. However, even if (\\ref{eq:relaxed}) is satisfied, the total weight moved to coordinate $j$ of $\\vect{q}$ from all the coordinates of $\\vect{p}$ can exceed ${q}_j$, potentially violating (\\ref{eq:inflow}). Namely,  (\\ref{eq:inflow}) implies (\\ref{eq:relaxed}), but not vice versa.\n\nWhen (\\ref{eq:relaxed}) is used in combination with (\\ref{eq:outflow}), we have:\n\\begin{equation}\nF_{i,j} \\leq \\min{(p_i,q_j)} \\quad \\text{for all $i, j$} .\n\\label{eq:relaxed2}\n\\end{equation} \nNote that we are essentially imposing capacity constraints on the edges of the flow network (see Fig.\\ref{fig:example}) based on (\\ref{eq:relaxed2}).\n\n\\begin{figure}[tb!]\n  \\centering%\n  \\includegraphics*[width=0.80\\linewidth]{example}\n  \\vspace{-0.15in}\n  \\caption{Imposing capacity constraints on the edges}\n  \\vspace{-0.15in}\n  \\label{fig:example}\n\\end{figure}\n\n\n\nWe would like to stress that in the framework of this work, which considers the discrete and not the \ncontinuous case of Wasserstein distances, the only requirement on the cost matrix is that it is nonnegative. \nSince any nonnegative cost $c$ between two locations can be written as the $p$-th power of the $p$-th \nroot of $c$ for $p \\geq 1$, one can assume that we are dealing with a $p$-th Wasserstein distance~\\cite{Villani}.\n\nIn the following subsections, we describe three new approximation methods. \nThe Overlapping Mass Reduction (OMR) method imposes the relaxed constraint (\\ref{eq:relaxed}) only between overlapping coordinates, and is the lowest-complexity and the least accurate approximation method.\nThe Iterative Constrained Transfers (ICT) method imposes constraint (\\ref{eq:relaxed}) between all coordinates of $\\vect{p}$ and $\\vect{q}$, and is the most complex and most accurate approximation method.\nThe Approximate Iterative Constrained Transfers (ACT) method imposes constraint (\\ref{eq:relaxed}) incrementally between coordinates of $\\vect{p}$ and $\\vect{q}$, and is an approximation of the ICT method.\nTherefore, both its complexity and its accuracy are higher than those of OMR, but lower than those of ICT.\n\n\\subsection{Overlapping Mass Reduction}\n\nThe OMR method imposes (\\ref{eq:relaxed}) only between overlapping coordinates.\nThe main intuition behind OMR method is that if the coordinate $i$ of $\\vect{p}$ and the coordinate $j$ of $\\vect{q}$ overlap (i.e., ${C}_{i,j}=0$), \na transfer of $\\min({p}_i,{q}_j)$ can take place free of cost between $\\vect{p}$ of $\\vect{q}$. After that, the remaining weight in $p_i$ is transferred simply to the second closest coordinate in $\\vect{q}$ \nas this is the next least costly move. Therefore, the method computes only the top-2 smallest values in each row of $\\vect{C}$.\nA detailed description is given in Algorithm~\\ref{alg:omr}.\n\n\n\\begin{algorithm}\n\\begin{algorithmic}[1]\n\t\\Function{OMR}{$\\vect{p}, \\vect{q}, \\vect{C}$}\n\t\\State $t = 0$\\ \\Comment{initialize transportation cost $t$} \n\t\\For{$i=1\\ldots,h_p$}\\ \\Comment{iterate the indices of $\\vect{p}$}\n                \\State $\\vect{s} =\\text{argmin}_{2}({C}_{i,[1 \\ldots h_q]})$ \\Comment{find top-$2$ smallest}\n\t\t\\If{${C}_{i,\\vect{s}[1]}==0$}\\ \\Comment{if the smallest value is 0}\n\t\t\t\\State{$r = \\min({p}_i, {q}_{\\vect{s}[1]})$}\\ \\Comment{size of max. transfer} \n\t\t\t\\State{${p}_i= {p}_i - r$}\\ \\Comment{move $r$ units of ${p}_i$ to ${q}_{\\vect{s}[1]}$} \n\t\t\t\\State{$t = t + {p}_i \\cdot {C}_{i,\\vect{s}[2]}$}\\ \\Comment{move the rest to ${q}_{\\vect{s}[2]}$}  \n\t\t\\Else\n\t\t\t\\State{$t = t + {p}_i \\cdot {C}_{i,\\vect{s}[1]}$}\\ \\Comment{move all of ${p}_i$ to ${q}_{\\vect{s}[1]}$}  \n\t\t\\EndIf\n\t\\EndFor\n\t\\State \\Return $t$\\ \\Comment{return transportation cost $t$}\n\t\\EndFunction\n\\end{algorithmic}\n\\caption{Optimal Computation of OMR}\n\\label{alg:omr}\n\\end{algorithm}\n\n\n\\subsection{Iterative Constrained Transfers}\n\nThe ICT method imposes the constraint (\\ref{eq:relaxed}) between all coordinates of $\\vect{p}$ and $\\vect{q}$. \nThe main intuition behind the ICT method is that because the inflow constraint (\\ref{eq:inflow}) is relaxed, the optimal flow exiting each source node can be determined independently.\nFor each source node, finding the optimal flow involves sorting the destination nodes in the ascending order of transportation costs, and then performing iterative mass transfers \nbetween the source node and the sorted destination nodes under the capacity constraints (\\ref{eq:relaxed}). Algorithm~\\ref{alg:ict} describes the ICT method in full detail.\n\n\n\\begin{algorithm}\n\\begin{algorithmic}[1]\n\t\\Function{ICT}{$\\vect{p}, \\vect{q}, \\vect{C}$}\n\t\\State $t = 0$\\ \\Comment{initialize transportation cost $t$} \n\t\\For{$i=1\\ldots,h_p$}\\ \\Comment{iterate the indices of $\\vect{p}$}\n\t\t\\State $\\vect{s}=\\text{argsort}({C}_{i,[1 \\ldots h_q]})$ \\Comment{sort indices by value}\n\t\t\\State $l=1$\\ \\Comment{initialize $l$}\n\t\t\\While{${p}_i > 0$}\\ \\Comment{while there is mass in ${p}_i$}\n\t\t\t\\State{$r = \\min({p}_i,{q}_{\\vect{s}[l]})$}\\ \\Comment{size of max. transfer} \n\t\t\t\\State{${p}_i= {p}_i - r$}\\ \\Comment{move $r$ units of ${p}_i$ to ${q}_{\\vect{s}[l]}$} \n\t\t\t\\State{$t = t + r \\cdot {C}_{i,\\vect{s}[l]}$}\\ \\Comment{update cost}  \n\t\t\t\\State{$l=l+1$}\\ \\Comment{increment $l$} \n\t\t\\EndWhile\n\t\\EndFor\n\t\\State \\Return $t$\\ \\Comment{return transportation cost $t$}\n\t\\EndFunction\n\\end{algorithmic}\n\\caption{Optimal Computation of ICT}\n\\label{alg:ict}\n\\end{algorithm}\n\\vspace{-0.05in}\n\nAlgorithm~\\ref{alg:aict} describes an approximate solution to ICT (ACT), which offers the possibility to terminate the ICT iterations before all the mass is transferred from $\\vect{p}$ to $\\vect{q}$.\nAfter performing a predefined number $k-1$ of ICT iterations, the mass remaining in $\\vect{p}$ is transferred to the $k$-th closest coordinates of $\\vect{q}$, making the solution approximate. \n\n\nTheorem~\\ref{thm:ict} establishes the optimality of Algorithm~\\ref{alg:ict}. Theorem~\\ref{theorem:ordering} establishes the relationship between different distance measures. \nThe proofs and the derivation of the complexity of the algorithms are omitted for brevity.\n\n\\begin{theorem} \\label{thm:ict}\n  (i) The flow $F^*$ of Algorithm~\\ref{alg:ict} is an optimal solution of the relaxed minimization problem given by (\\ref{eq:objective}), (\\ref{eq:outflow}) and (\\ref{eq:relaxed}).\n  (ii) ICT provides a lower bound on EMD.\n\n\\end{theorem}\n\n\\begin{theorem} \\label{theorem:ordering}\n  For two normalized histograms $\\vect p$ and $\\vect q$:\n  $\\text{RWMD}(\\vect{p},\\vect{q}) \\leq  \\text{OMR}(\\vect{p},\\vect{q}) \\leq \\text{ACT}(\\vect{p},\\vect{q}) \\leq \\text{ICT}(\\vect{p},\\vect{q}) \\leq \\text{EMD}(\\vect{p},\\vect{q})$.\n\\end{theorem}\n\n\\begin{algorithm}\n\\begin{algorithmic}[1]\n\t\\Function{ACT}{$\\vect{p}, \\vect{q}, \\vect{C}, k$}\n\t\\State $t = 0$\\ \\Comment{initialize transportation cost $t$} \n\t\\For{$i=1\\ldots,h_p$}\\ \\Comment{iterate the indices of $\\vect{p}$}\n\t\t\\State $\\vect{s}=\\text{argmin}_{k}({C}_{i,[1 \\ldots h_q]})$ \\Comment{find top-$k$ smallest}\n\t\t\\State $l=1$\\ \\Comment{initialize $l$}\n\t\t\\While{$l < k$}\\\n\t\t\t\\State{$r = \\min({p}_i, {q}_{\\vect{s}[l]})$}\\ \\Comment{size of max. transfer} \n\t\t\t\\State{${p}_i= {p}_i - r$}\\ \\Comment{move $r$ units of ${p}_i$ to ${q}_{\\vect{s}[l]}$} \n\t\t\t\\State{$t = t + r \\cdot {C}_{i,j}$}\\ \\Comment{update cost}  \n\t\t\t\\State{$l=l+1$}\\ \\Comment{increment $l$} \n\t\t\\EndWhile\n\t\t\\If{${p}_{i} \\neq 0$}\\ \\Comment{if ${p}_i$ still has some mass}\n\t\t\t\\State{$t = t + {p}_i \\cdot{C}_{i,\\vect{s}[k]}$}\\ \\Comment{move the rest to ${q}_{\\vect{s}[k]}$} \n\t\t\\EndIf\n\t\\EndFor\n\t\\State \\Return $t$\\ \\Comment{return transportation cost $t$}\n\t\\EndFunction\n\\end{algorithmic}\n\\caption{Approximate Computation of ICT}\n\\label{alg:aict}\n\\end{algorithm}\n\\vspace{-0.05in}\n\n\n\\section{Related Work}\n\n\nA regularized version of the optimal transport problem can be solved more efficiently than network-flow-based approaches~\\cite{Cuturi13}. \nThe solution algorithm is based on Sinkhorn's matrix scaling technique~\\cite{Sinkhorn}, and thus, it is called Sinkhorn's algorithm. \nConvolutional implementations can be used to reduce the time complexity of Sinkhorn's algorithm~\\cite{Solomon2015}, e.g., when operating on images.\nGiven an error term $\\epsilon$, the time complexity of Sinkhorn's algorithm is $O((h^2\\log{h})\/\\epsilon^3)$ when computing the distance between histograms of size $O(h)$~\\cite{Altschuler2017}.\nIn addition, a cost matrix has to be constructed, which incurs an additional complexity of $O(h^2m)$ when using $m$-dimensional coordinates.\n\nSeveral other lower bounds of EMD have been proposed~\\cite{assent2008efficient,xu2010efficient,ruttenberg2011indexing,wichterich2008efficient,xu2016emd,huang2016heads,huang2014melody}.\nThese lower bounds are typically used to speed-up the EMD computation based on pruning techniques.\nAlternatively, EMD can be computed approximately using a compressed representation~\\cite{Uysal2016approximation,Pele2009}. \n\nA greedy network-flow-based approximation algorithm has also been proposed~\\cite{Gottschlich2014}, \nwhich does not relax the \\textit{in-flow} or \\textit{out-flow} constraints. Therefore, it is not a data-parallel algorithm \nand its complexity is quadratic in the histogram size. In addition, it produces an upper bound rather than a lower bound of EMD. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec-intro}}\nThe total cross section for\nelectron-positron annihilation into hadrons is a very important\nphysical observable weighting significantly on the theory error\nof the muon anomalous magnetic moment and the running electromagnetic\n fine structure constant used in the tests of the Standard Model and its extensions.\nFor recent reviews see e.g.~\\cite{Actis:2010gg,Hagiwara:2011af,Davier:2010nc,Jegerlehner:2009ry,Harlander:2002ur}.\n At high energies, the cross section can be calculated using perturbative QCD,\n however at low energies one has to rely on the experimental measurements.\n\nOne of the methods used to extract the hadronic cross section\nis the\n''radiative return'', exploiting the fact that the cross section\nof the process with initial state photon radiation can be factorised\ninto a known perturbative factor and the hadronic cross section\nwithout initial state radiation at the energy lowered by the emitted\n  photons.  Due to the complexity of the experimental setup, the extraction of the hadronic cross\n section within a realistic\n experimental framework can be achieved only by means of an event generator. \n\n The most important contribution to the hadronic cross section\n is the pion pair production channel. Its accuracy is an issue \n as it provides the main source of error in the evaluation of the muon anomalous \n magnetic moment~\\cite{Hagiwara:2011af}. In view of the planned improvement of the direct measurement\n of the muon anomalous magnetic moment~\\cite{Venanzoni:2012vha}, with the expected error four times smaller than\n the present one, it is crucial to pull down the theoretical error as much as possible. \n The pion production cross section in $e^+e^-$\n scattering was measured, using the radiative return method, by BaBar~\\cite{Lees:2012cj,Aubert:2009ad} and\nKLOE~\\cite{Babusci:2012rp,Ambrosino:2010bv,Ambrosino:2008aa}.\n Both experiments quote individual errors at the level of a fraction of a percent, while the discrepancy\n between them is up to 2\\% at the $\\rho$ peak and 5\\% when approaching the energy of 1~GeV. \nThe origin of the discrepancy remains unclear.\n Since both experiments use the PHOKHARA~\\cite{Czyz:2002np,Czyz:2005as}\n event generator to extract the hadronic cross section, it is necessary to\n check carefully its physical content. The PHOKHARA event generator was used to generate the \n reactions $e^+e^-\\to \\pi^+\\pi^- + \\ {\\rm photons}$ and $e^+e^-\\to \\mu^+\\mu^- + \\ {\\rm photons}$. \n The\n latter process is used for monitoring the luminosity. So far, the version of PHOKHARA used\n by BaBar and KLOE included the dominant next-to-leading order (NLO) radiative corrections. \nIn view of the above mentioned  discrepancy between BaBar and KLOE, it is essential to make a full NLO\ncalculation and to establish the importance of the missing contributions.\n\nIn this article, the \ncomplete radiative NLO Quantum Electrodynamics~(QED) corrections to the reaction\n  $e^+e^-\\to \\mu^+\\mu^- \\gamma$ are calculated, tested and implemented into the event generator \n PHOKHARA.\nFrom a technical point of view, the pentagon diagrams are the most challenging.\nBecause there is no scale available which might lead to logarithmically enhanced contributions, they are\nexpected to be small. However, it is known that logarithmic enhancements\n  can be\n generated in some regions of the phase space within\n complicated experimental setups. \nThe fact that we can not neglect the small electron mass for the same reason poses an additional\n challenge in the calculation of the virtual amplitudes since ratios of the order of $s\/m_e^2$ can appear, where $s$ is the\n energy of the collider. This demands a good control on the numerical\n accuracy of our amplitudes~\\footnote{Technically, the accuracy problems in the calculation\nof the radiative corrections are similar to the ones in\n$e^+e^-\\to \\bar t t \\gamma$,  solved in Ref.~\\cite{Khiem:2012bp}\nusing the GRACE system.  The electroweak radiative corrections were calculated there, but with \n the photon emitted at large angles only.}.\nConsequently, detailed Monte Carlo studies are mandatory. \nFirst studies were presented some time ago by the Ka\\-to\\-wi\\-ce\/Zeuthen group~\\cite{Kajda:2009aa}, in PhD\ntheses~\\cite{kajphd,yundin-phd-2012--oai:export} and also by an independent\nalternative approach~\\cite{Actis:2009zz,Actis:2009uq}. \nThis enabled us to compare specific phase space points with~\\cite{Actis:2009zz,Actis:2009uq} with high\nprecision as a first numerical test; see\nRef.~\\cite{yundin-phd-2012--oai:export} for details.\nHere, we perform a complete calculation in the frame of a realistic Monte Carlo environment, PHOKHARA9.0.\nBecause there are several known sources of numerical instabilities, and because we have no external\ncross-check available, we organized for two independent implementations of the QED virtual corrections. \nHaving implemented the complete radiative corrections, detailed physics studies\n became possible, and their results are presented here.\n\n The article is organised as follows: In Section~\\ref{sec:class} and Appendices~\\ref{appex:twophoton} and~\\ref{sec-asoft} we give a detailed description\n of the calculation of the radiative corrections. Section~\\ref{sec-impl}\n sketches the implementation of the radiative corrections into \n PHOKHARA9.0. \nIn Section~\\ref{sec-tests}, \n the main tests of the correctness and the numerical stability  of the code are presented.\nFurther, the relevance of the NLO\nradiative corrections, which were missing \n  in PHOKHARA7.0, is investigated.\nIn Section~\\ref{sec-impact}, the \npossible impact of the radiative corrections, analogous to those studied here, on the  pion\nform factor measurements of BaBar and KLOE is discussed. \n Section~\\ref{sec-conc} contains the conclusions.\n\n\n\n \\section{Radiative corrections to $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$ \\label{sec:class} }\n \nThe tree level diagrams contributing to the leading order (LO) amplitude \nare shown in Fig.~\\ref{fig:eemmg0}.  \nThere are two types of contributions, those  with initial state photon emission (ISR)\nand final state photon emission (FSR). \nThe ISR and FSR pairs of diagrams are separately gauge invariant.  \n\n\\begin{figure}[htb]%\n    \\centering\n     \\includegraphics[width=1.\\textwidth]{muons_fig_tree.pdf} %\n    \\caption{The tree diagrams for $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$.} %\n    \\label{fig:eemmg0}\n\\end{figure}\n\n\nAt NLO QED,   \nthere are the virtual and the real corrections resulting in three\ntypes of contributions to the cross section, the ISR and the FSR contributions, and their ISR-FSR interference terms.\n\nWe use dimensional regularization~\\cite{'tHooft:1972fi} to regularize the ultraviolet (UV) and \ninfrared (IR) divergences. \nThe UV divergences of the virtual amplitude are removed by the renormalization\ncounter-terms.\nBoth the virtual and the real corrections are infrared divergent. These divergences cancel\nin the sum for infrared-safe observables. The IR divergences are canceled \nand both the virtual+real~(soft) and the real~(hard) corrections\n become separately numerically integrable.  Details of\nthe real emission calculation are given in\nSection~\\ref{sec-nlo-real}. In the following, we describe the method used to\ncompute the virtual amplitudes.\n\n \n \\subsection{Virtual corrections \\label{sec:class1}  }\n\n\nBesides photonic self-energy corrections, there are 32 diagrams contributing to $e^+e^-\\to\\mu^+ \\mu^-\\gamma$\nat NLO QED. They can be classified into several independent gauge invariant\nsubsets, which we will call Penta-Box, Box-Triangle-Bubble and Triangle\ncontributions. The first class involves loop corrections with the two\nlepton lines attached to the loop. The most challenging diagrams are the four pentagon diagrams,\nshown in  Fig.~\\ref{fig:pentagons}, \nwhere a real photon is emitted from an  internal line.\n\n\\begin{figure}[htb]%\n    \\centering\n     \\includegraphics[width=1.\\textwidth]{muons_fig_pen.pdf} %\n    \\caption{The four diagrams with  pentagon topology for $e^+ e^- \\to \\mu^+\n      \\mu^- \\gamma$} %\n    \\label{fig:pentagons}\n\\end{figure}\n\n\nThey do not constitute a class of gauge independent diagrams by themselves.\nGauge invariant groups are formed when a pentagon is associated with\ntwo box diagrams where a photon is radiated from the same external (electron or muon) line. This is shown\nschematically in Fig.~\\ref{fig:pentabox}.\nThe contribution of these twelve Penta-Box diagram combinations, interfering with the tree level\ndiagrams of Fig.~\\ref{fig:eemmg0}, will be discussed in detail in Section~\\ref{sec-tests}. \n\n\n\n\\begin{figure}[h!]%\n    \\centering\n   \\includegraphics[width=1.\\textwidth]{muons_fig_penbox.pdf} %\n    \\caption{One of the four gauge invariant combinations of a pentagon with two boxes with external photon\nemission for $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$.  }%\n    \\label{fig:pentabox}\n\\end{figure}\n\nBox-Triangle-Bubble and Triangle\ncontributions contain corrections to one lepton line (electron or muon) and are further\nclassified depending whether the loop and the real photon are attached to the\nsame lepton line.\nThe Box-Triangle-Bubble class contains all the loop corrections to a lepton \nline with a real (on-shell) photon and a second off-shell photon, connecting to\nthe other lepton line. The contributing  boxes, vertices and bubbles can be found in\nFig.~\\ref{fig:wholeBT}. There are two independent gauge invariant subsets, for\nFSR (two upper lines of Fig.~\\ref{fig:wholeBT}) and for ISR~(two lower lines of Fig.~\\ref{fig:wholeBT}).\nThe triangle\ncontributions are given in Fig.~\\ref{fig:triangle}. There, a real photon is emitted from one fermion line, \nand the other photon (off-shell) entering a 3-point function is connected to the\nother fermion line.  \n\n\\begin{figure}[h!]%\n    \\centering\n   \\includegraphics[width=1.\\textwidth]{muons_fig_boxp.pdf} %\n    \\caption{The  set of sixteen one-loop Box, triangle  and self-energy diagrams with internal photon emission  in $e^+ e^-\n\\to \\mu^+ \\mu^- \\gamma$. \n}%\n    \\label{fig:wholeBT}\n\\end{figure}\n\n\n \n\n\\begin{figure}[h!]%\n   \\centering\n   \\includegraphics[width=1.\\textwidth]{muons_fig_tri.pdf} %\n    \\caption{The four triangle diagrams with external photon emission  in $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$.}%\n    \\label{fig:triangle}\n\\end{figure}\n \nFinally, we mention the diagrams with external photon emission and self-energy insertions to the photon\npropagator~\\footnote{The sum of contributions\nfrom diagrams with real emission from fermionic self-energy insertions to the photon propagator vanishes due\nto the Furry theorem.}.\nThey constitute a gauge-invariant universal correction which can be accounted for in any QED calculation by\nsimply\nrunning the fine structure constant to the appropriate scale~\\cite{Bohm:1986rj,Denner:1991kt}.\nThese self-energies  are treated separately and have been omitted  from our fixed-order loop amplitude\ndefinition in Fig.~\\ref{fig:wholeBT}.\nThe treatment of vacuum polarisation in the PHOKHARA event generator, together \n with narrow resonance contributions is described in detail \nin Ref.~\\cite{Czyz:2010hj} and will not be discussed here. \n\n\nIn the present article, two independent programs using two different  methods are used.  One is based on\nthe trace method and the calculations are done using double precision\nnumerical routines, including the \\texttt{PJFry} libraries~\\cite{pjfry-project}. \nWe refer to it as ``Double precision - Trace method'' (DT-method). \nThe other one is based on the helicity formalism as  \n  described in Ref.~\\cite{Campanario:2011cs}, and we refer to it as ``Quadruple precision - Helicity method''\n(QH-method) because numerical calculations are done partially using\nquadruple  precision. Such independent implementations are necessary to gain\nsufficient numerical reliability.\n\n\n\n\\subsubsection{The DT-method \\label{subsubsec-DT}}\n\nWith the DT-method, topologies are generated by QGRAF~\\cite{Nogueira:1991ex} and then dressed with\nparticles and\nmomenta\nby the DIANA program~\\cite{Tentyukov:1999is} according to the QED model description file.\nThe resulting output contains a list of Feynman diagrams in the textual representation,\nwhich is defined by the TML markup language script~\\cite{Tentyukov:1999yq}.\nNext, the diagrams are passed through the FORM~\\cite{Vermaseren:2000nd} script,\nwhich substitutes Feynman rules according to the selected model.\nFurther manipulations are done with FORM.\nIn addition, some general simplifications can be enabled by setting configuration parameters.\nThis includes gamma algebra identities like $\\gamma^\\mu\\gamma^\\nu\\gamma_\\mu=(2-d)\\gamma^\\nu$,\nthe transversality condition, e.g.  $p_1 \\cdot \\epsilon(p_1)=0$, usage of Dirac equation and momentum\nconservation.\n The resulting expressions are written in the FORM tablebase.\nWe use it as an input in the squaring program which sums the diagrams and multiplies them\nby the complex conjugated set of Born diagrams. \nThe fermion lines are connected by the completeness relation. Then, Dirac traces are taken. \n \n\nFor the calculation of the newly added one-loop pentagon contributions, \none has to calculate 5-point tensor Feynman integrals up to rank $R=3$. \nWe reduce the tensor integrals in $d=4-2\\epsilon$ dimensions to scalar 1-\nto 4-point functions.\nThey depend on the reduction basis chosen.\nOften one uses as basis momenta, the external momenta of the diagram as in Refs.~\\cite{Passarino:1978jh,Campanario:2011cs}.\nOur choice (with a one-to-one correspondence) are the so-called chords, the shifts of internal momenta\nwith respect to the loop momentum~\\cite{pjfry-project}. \n \nAdvanced tensor integral calculations became a standard task in recent years,\nmainly\ntriggered by LHC physics.\nNevertheless, ensuring sufficient numerical stability is  demanding for several reasons.\nAn often discussed issue is the treatment (or avoidance) of small or vanishing inverse Gram determinants.\nAnother one is just the extreme spread of scales met in our physical process, because we cannot \nneglect\nthe electron mass $m_e \\approx 1\/2000$ GeV as an\nindependent parameter.\nWith $\\sqrt{s} = 1 - 10$ GeV, one faces e.g. a ratio $m_e^2\/s \\sim 10^{-7}-10^{-9}$.  \nThe DT implementation of tensor integral calculation\nrelies on the approach developed in Refs.\n\\cite{Davydychev:1991va,Tarasov:1996br,Fleischer:1999hq,Diakonidis:2008ij,Diakonidis:2009fx,\nFleischer:2010sq,Almasy:2013uwa}\nand uses the PJFry tensor reduction package~\\cite{yundin-phd-2012--oai:export,Fleischer:2011zz},\ncombined with QCDLoop\/FF~\\cite{Ellis:2007qk,vanOldenborgh:1990yc} or OneLOop~\\cite{vanHameren:2010cp} for\nscalar integrals.\nMore technical details can be found in Ref.~\\cite{yundin-phd-2012--oai:export}~\\footnote{A new approach to the treatment of pentagon diagrams is under development in the OLEC project~\\cite{olec-project-2013,Fleischer:2012ad,Almasy:2013uwa}.\nIt is alternative to tensor\nreduction and relies instead on the direct calculation of tensor contractions~\\cite{Fleischer:2010sq,Fleischer:2011nt}.\nIt will be interesting to see whether this improves speed or stability of the numerics.}.\n \n\n\\subsubsection{The QH-method \\label{subsubsec-QH}}\n\nThe second implementation (QH-method) uses the  helicity formalism as  \n  described in Ref.~\\cite{Campanario:2011cs}. To build the virtual amplitude, four\n  building blocks are used. Corrections to a lepton line with two real\n  (on-shell or off-shell) photons \n  attached in a fixed order of external\n  bosons, Fig.~\\ref{fig:boxtriangle}, constitute the first building block, which we call\n  Boxline and also include the corresponding counter-terms which are not shown in Fig.~\\ref{fig:boxtriangle}. We used the effective current approach, thus, $V_1$ and $V_2$ should be understood as generic off-shell\ncurrents which can be, in this case, an on-shell photon or an off-shell\nphoton, which forms the second lepton line.  The  physical amplitude is built by considering\n  all physical permutations and contractions with external\n  currents yielding the Box-Triangle-Bubble gauge invariant\n  subsets of Fig.~\\ref{fig:wholeBT}. In addition, we use the\n  vertex corrections to a lepton line with one real photon attached to it. All\n  possible contributions result in the triangle contributions of\n  Fig.~\\ref{fig:triangle}. %\n\\begin{figure}[h!]%\n    \\centering\n   \\includegraphics[width=1\\textwidth]{muons_fig_boxline.pdf} %\n    \\caption{Boxline contributions for $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$. }%\n    \\label{fig:boxtriangle}\n\\end{figure}\nThe third\nbuilding group is formed by the Penta-Box diagrams depicted in\nFig.~\\ref{fig:pentabox}, which involve the\npentagon diagrams. The last building block is obtained by crossing the two\ninitial fermion lines in Fig.~\\ref{fig:pentabox} and constitute an\nindependent gauge group.\nTo compute them, we generalized the software developed\nin Ref.~\\cite{Campanario:2011cs} to be able to compute diagrams with two\nfermion lines. This includes the use of  Chisholm identities~\\cite{Sirlin:1981pi} which reduces\nthe CPU time required to evaluate the Penta-Box contributions by a factor\nten. At the same time, this improves the stability of the code since it makes\nexplicit terms proportional to the small electron mass.\nThe calculation of tensor \nintegrals is done by using Passarino-Veltman reduction~\\cite{Passarino:1978jh} \nfor up to $4$-point diagrams and the method of~\\cite{Denner:2005nn,Binoth:2005ff}, following\nthe convention of Ref.~\\cite{Campanario:2011cs} for higher-point tensor integrals. \nThe scalar integrals are calculated as in\nRefs.~\\cite{'tHooft:1978xw,Denner:1991kt}. %\n\nWe use a cache system in all the building blocks such that the\ninformation of the loop-dependent parts are stored. This is particularly important for this process since up to 32\ndifferent helicity amplitudes exist corresponding to the different helicity\nand polarization combinations of the external particles. After the first\nhelicity is computed, which include the evaluation of the loop-dependent parts, any additional helicity amplitude is computed with less than\n10$\\%$ of CPU time, reducing the CPU time of the code by a factor 10. \n\n\nThe building blocks do not use special properties like \ntransversality or being on-shell for the real photons attached to the lepton\nlines, instead, we assume external effective current attached to them. This\nallows us to use Ward identities, by replacing\nan effective current with the corresponding momentum, to check the accuracy of the computed amplitudes. We classified our\ncontributions in gauge invariant subsets so that the Ward\nidentities are fulfilled.  Those identities are called gauge tests and are\nchecked with a small additional computing cost, using the\ncache system. They are checked for every phase space point and each gauge invariant subset distinguishing between\nFSR and ISR contributions.\nThis is important because the phase space integration \nof the virtual contribution shows numerical instabilities \nin the calculation of the one-loop tensor integrals~\\cite{Campanario:2011cs}.\n\nWe have implemented a rescue system for\nphase space points where the Ward identities are not satisfied with an\naccuracy of at least three digits.  First, we calculate the amplitudes applying quadruple precision only\nto the scalar integrals and tensor reduction routines. This requires reconstructing the external momenta\nin quadruple precision, so that global energy-momentum conservation is still fulfilled at the higher numerical accuracy retaining the external particles on their mass-shell.\nIf the Ward identities are not satisfied, the rescue system evaluates the \namplitude using quadruple precision in all parts of the code.\nWith this system, we find that the proportion of phase space points that do not pass \nthe Ward identities for a requested accuracy of\n$\\epsilon=10^{-3}$ is well below one in ten thousand. The rescue system\nadds an additional $10\\%$ to the CPU time. \n\n\nDespite the cache system and the use of Chisholm identities, this implementation is\nstill seven times slower that the DT-method which can be traced back to the\nevaluation of the 32 helicity amplitudes and the parts evaluated in quadruple precision.\n\n We have tried to evaluate the\namplitudes only in double precision to improve the speed of the code by a\nfactor two using dedicated\nsubroutines for small Gram determinants. These involve the evaluation of\nthree- and four-point functions up to Rank 11 and 9, respectively,\nfollowing the notation of Ref.~\\cite{Campanario:2011cs}.  The high rank of the\nrescue system allows to obtain full double precision for mild cancellations\nin the Gram determinants.  However, two problems arise. First, for the failing phase space points, there was\nalways some internal combination for which the expansion breaks down due to the\npresence of additional cancellations in sub-Cayley determinants. Thus, \nfull double precision is not achieved for all tensor\nintegrals coefficients. Second, within the\nhelicity method formalism, there exist extremely large cancellations between the different helicity\namplitudes resulting into an additional loss of precision which can be larger\nthan the one due to the presence of small Gram determinants. These numerical cancellations are related to \nthe fact that the mass of the electron has to be retained, and many numerical\ncancellation occur. For example, in the numerical calculation of \n$\\slash \\kern-6pt {p}_e u(p_e)= m_e u(p_e) $ cancellations of the order of\n $s\/m_e^2$ would appear, where s is the energy\nof the collider, if the Dirac equation is not applied or is not treated\ncarefully.\n\nThese two problems are reflected in a bad accuracy of the gauge test and,\ntherefore, in the large number of identified unstable points using\nonly double precision. The second problem is naturally solved using the DT-method since the\nsummation and averaging over spins are done analytically and\nmany of these numerical cancellations are avoided. \nWe decided to implement the DT-method in PHOKHARA9.0 and compare it with the code implemented in full quadruple precision where the gauge test ensures the numerical\naccuracy of the code.\n\n\\subsection{Real photon emission \\label{sec-nlo-real}}\n\n The real two-photon emission, which contributes to the $e^+e^-\\to\\mu^+\\mu^-\\gamma$\ncross section at NLO is now included in the PHOKHARA code\n completely in contrast to the implementation of Refs.~\\cite{Rodrigo:2001kf,Czyz:2004rj}, where\n subleading contributions were neglected.  We distinguish between soft and 2-hard photon\n emission. \n\\subsubsection{Soft photon emission \\label{sec-soft}}\n  \n  In the soft photon contribution, the phase space of one of the photons ($k_1)$\n  is integrated out analytically. The integrals to be performed, which factorise\n in front of the square of the full amplitude describing the\n $e^+e^-\\to\\mu^+\\mu^-\\gamma$~ reaction, read\n\\begin{eqnarray}\nF(p_1,p_2,q_1,q_2,r) = \\frac{-\\alpha}{4 \\pi} \\int \\frac{d^3k_{1}}{E_{k_1}} \\left[ \\left( \\frac{p_1}{p_1 \\cdotp  k_1} - \\frac{p_2}{k_1 \\cdotp p_2} \\right) +  \\left( \\frac{q_2 }{q_2 \\cdotp  k_1} - \\frac{q_1}{k_1 \\cdotp q_1} \\right) \\right]^2,\\label{ffdef}\n\\end{eqnarray}\nwhere $p_1$, $p_2$, $q_1$ and  $q_2$  are the momenta of the positron,\nelectron, antimuon and muon, respectively.\n The infrared regulator --- the photon mass $\\lambda$ and the photon energy\n cut-off\n$E_{max}$ dependence can be cast into a single parameter\n\\begin{eqnarray}\nr=\\frac{2E_{max}}{\\lambda}. \n\\end{eqnarray}\n\n In principle, this is a well known formula, which can be found in\nthe literature~\\cite{Berends:1987ab,Rodrigo:1999qg}. However, as the \nintegral over the photon energy is performed only\n up to a given cut-off $E_{max}$,\n its form depends on the frame\n in which this cut-off is applied. We found it suitable to give the formulae,\n which are valid in any frame in which the cut-off is defined. They are a bit\n longer than the usual ones as we express everything through the four momenta\n $p_1,p_2,q_1,q_2$ given in the frame, where the cut-off is defined, in contrast\n  to the usual expressions which use invariants, but are less universal.\n The explicit expression for $F(p_1,p_2,q_1,q_2,r)$ is given  in\n Appendix~\\ref{sec-asoft}.\n\n\\subsubsection{Two-hard photon emission \\label{sec-2hard}}\n\nFor the two-hard photon emission, the helicity method for the calculation of the\namplitudes \nwas used and cross checked with a dedicated code based on the trace method of spin summation. \n The convention for the helicity amplitude method \n introduced in Ref.~\\cite{Rodrigo:2001kf} was adopted. \n\nThe only amplitude,\n  which was missing in earlier versions of PHOKHARA\n was the two-photon FSR.\n Interferences between the coded amplitudes, with infrared divergences\n matching the ones from Penta-Box diagrams,\n were also not included. \n After some algebra similar to Refs.~\\cite{Rodrigo:2001kf,Czyz:2004rj},\n a very compact form for the two-photon FSR amplitude is obtained\n\\begin{eqnarray}\n \\kern-15pt M\\left(\\lambda_{\\mu^+},\\lambda_{\\mu^-},\\lambda_1,\\lambda_2\\right) =\n v_I^{\\dagger}(\\lambda_{\\mu^+}) A\\left(\\lambda_1,\\lambda_2\\right)\n u_I(\\lambda_{\\mu^-}) \n   + v_{II}^{\\dagger}(\\lambda_{\\mu^+}) B\\left(\\lambda_1,\\lambda_2\\right)\n u_{II}(\\lambda_{\\mu^-})\n \\ , \n\\label{eq:a1}\n\\end{eqnarray} \nwhere the matrices A and B and the convention used to define\nthe spinors are given in Appendix~\\ref{appex:twophoton}. \nThe energy of one of the photons has to be bigger than the cut-off  $E_{max}$\n and the sum of the soft and hard contributions should not depend on \nthis cut-off\n up to terms $\\sim E_{max}$, which are neglected in the analytic calculation.\n\n\\section{Implementation of the radiative corrections in the event generator PHOKHARA9.0\n\\label{sec-impl}}\nPHOKHARA9.0 is available from the webpage \\url{http:\/\/ific.uv.es\/~rodrigo\/phokhara\/}.\n As stated already, all new parts of the released computer code\n  were calculated independently by two methods and\/or groups\n of the authors of this article.\nTo ensure the stability of the virtual corrections, \n we use the two independent codes\ndescribed in Section~\\ref{sec:class}.   \n The faster routine in the released version of PHOKHARA9.0 is\n used, which is the code based on the DT-method.\n The other one can be obtained on request.\n We sketch here shortly the new ingredients of the released code.\n The listed changes concern only the $e^+e^-\\to \\mu^+\\mu^- \\gamma$ mode,\n when it is running with the complete NLO radiative corrections:\n \\begin{itemize}\n \\item{The\n  virtual corrections are calculated in \n double precision and the sum over polarisations is done with \n the trace method. The software PJFry~\\cite{pjfry-project} is used\n for this purpose and the relevant parts of the libraries developed there\n are distributed with PHOKHARA9.0. }\n \\item{The soft photon emission is calculated using the formulae discussed in \n  Section~\\ref{sec-soft} and in Appendix~\\ref{sec-asoft}.}\n \\item{The two-hard-photon emission part uses the helicity amplitudes\n  as defined in Ref.~\\cite{Czyz:2004rj}. The newly added part --- the two photon FSR is described in Section~\\ref{sec-2hard} and coded accordingly. }\n\\end{itemize}\n\n\\section{Tests of the code \\label{sec-tests}}\n The released code was tested very extensively both for the real and the virtual\n contributions to assure the technical\n accuracy of the code to be much better than the one required for experimental\n measurements. The necessity of retaining a finite electron mass possesses a\n potential threat of numerical instabilities both for the real and the\n virtual contributions since cancellations of the order\n of $s\/m_e^2$ can appear.\n\nThe virtual corrections constitute the most challenging part.  \nThe presence of Gram determinants can constitute an\nadditional source of instabilities which can be more challenging for some\nrealistic experimental selection cuts where forward photon emissions are favoured, resulting\nin collinear photon emissions. \n %\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multicolumn{5}{|c|}{$\\sqrt{s}$=1.02 GeV}\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{QH}$ = 6.332(1) [nb] }\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{DT}$ = 6.332(1) [nb] }\\\\\n\\hline\n$|\\Delta| >$ & $\\sigma_{\\Delta_{DT}}$ [nb]& $\\sigma_{\\Delta_{QH}}$ [nb]& $\\sigma_{\\Delta_{DT}}\/\\sigma_{DT}$& N$_{event}$\\\\\n\\hline\n0.1    & 0& 0             &0&0\\\\\n\\hline\n0.01   & $4(4) \\cdotp 10^{-8}$& $4(4) \\cdotp 10^{-8}$&$6(6) \\cdotp 10^{-9}$&1\\\\\n\\hline\n0.001  & $1.4(3) \\cdotp 10^{-6} $& $1.4(3) \\cdotp 10^{-6} $&$2.2(4) \\cdotp 10^{-7}$&32\\\\\n\\hline\n0.0001 & $2.1(1) \\cdotp 10^{-4}$& $2.1(1) \\cdotp 10^{-4}$&$3.4(2) \\cdotp 10^{-5}$&521\\\\\n\\hline\n0.00001& $2.7(1) \\cdotp 10^{-4}$& $2.7(1) \\cdotp 10^{-4}$&$4.2(2) \\cdotp 10^{-5}$&787\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison between the codes based on the DT-method  and the \n  QH-method at $\\sqrt{s}$=1.02 GeV. No selection cuts are applied. See text\n  for details on the definition of the Table entries. } \\label{test1}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multicolumn{5}{|c|}{$\\sqrt{s}$=10.56 GeV}\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{QH}$ = 0.07004(4) [nb] }\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{DT}$ = 0.07004(4) [nb] }\\\\\n\\hline\n$|\\Delta| >$ & $\\sigma_{\\Delta_{DT}}$ [nb]& $\\sigma_{\\Delta_{QH}}$ [nb]& $\\sigma_{\\Delta_{DT}}\/\\sigma_{DT}$& N$_{event}$\\\\\n\\hline\n0.1    & $6(6) \\cdotp 10^{-7}$&$2(1) \\cdotp 10^{-8}$  &$9(9) \\cdotp 10^{-6}$&125\\\\\n\\hline\n0.01   & $7(5) \\cdotp 10^{-7}$&$1.4(4) \\cdotp 10^{-7}$ &$1.1(9) \\cdotp 10^{-5}$&1044\\\\\n\\hline\n0.001  & $7.7(6) \\cdotp 10^{-6} $&$7.1(2) \\cdotp 10^{-6} $ &$1.10(9) \\cdotp 10^{-4}$&9599\\\\\n\\hline\n0.0001 & $8.3(1) \\cdotp 10^{-5}$&$8.3(1) \\cdotp 10^{-5}$ &$1.18(2) \\cdotp 10^{-3}$&42621\\\\\n\\hline\n0.00001& $2.24(2) \\cdotp 10^{-4}$& $2.24(2) \\cdotp 10^{-4}$&$3.21(4) \\cdotp 10^{-3}$&115091\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison between the codes based on the DT-method  and\n  QH-method at $\\sqrt{s}$=10.56 GeV. No selection cuts are applied. See text\n  for details on the definition of the Table entries.} \\label{test2}\n\\end{table}\n\nWe performed very detailed tests for the different gauge invariant blocks\nseparately~\\cite{Gunia:PhD}.  \nHere, we mainly show the results of the tests concerning the sum \nof all contributions. \nThey are summarized  in Tables~\\ref{test1}-\\ref{test4}. The tests were performed for two\n different energies  1.02~GeV and 10.56~GeV without any event selection\n (Tables~\\ref{test1} and~\\ref{test2}) and with event selections close to\n the ones used in the experiments KLOE (Table~\\ref{test3}) and BaBar\n (Table~\\ref{test4}) -- the specific cuts applied are found in Appendix~\\ref{appex:KLOBA}.\n The integrated cross sections for both codes, DT-method ($\\sigma_{DT}$)\n and QH-method ($\\sigma_{QH}$), are in perfect agreement in all cases and the\n statistical errors are well below the per mille level.\n\n\nFor ten million of examined events with one photon in the final state,\n we count for how many events, N$_{event}$,\n the predictions for the cross section\ndisagree at the relative accuracy $|\\Delta|$.  \nTo\n check whether these events might have an impact on the differential cross\n section, we also calculate the cross section corresponding to these events for both codes\n $\\sigma_{\\Delta_{DT}}$ (DT-method) and $\\sigma_{\\Delta_{QH}}$ (QH-method). \nAt 1.02~GeV, Tabs.~\\ref{test1} and~\\ref{test3},  one retains at least 1 digit of accuracy for the\nmatrix element squared and the cross section of these events is irrelevant.\n At 10.56~GeV,  one observes in Tab.~\\ref{test2} that one can lose completely\n the accuracy (the order of magnitude\n  of the results is however always the same),  but that does\n not happen for the BaBar event selection cuts, Tab.~\\ref{test4}, where at\n least two digits  are correct.\n  Even if Table~\\ref{test2} shows that the cross\n section from events for which one loses the precision is small (below 0.3\\%), the\n released program based on the DT-method should be used with care at high\n energies if no event selection is applied and a cross check with the \n QH-method is recommended. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multicolumn{5}{|c|}{KLOE event selection}\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{QH}$ = 1.575(2) [nb] }\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{DT}$ = 1.575(2) [nb] }\\\\\n\\hline\n$|\\Delta| >$ & $\\sigma_{\\Delta_{DT}}$ [nb]& $\\sigma_{\\Delta_{QH}}$ [nb]& $\\sigma_{\\Delta_{DT}}\/\\sigma_{DT}$& N$_{event}$\\\\\n\\hline\n0.01  & 0&0&0&0\\\\\n\\hline\n0.001  & $2(2) \\cdotp 10^{-9}$&$2(2) \\cdotp 10^{-9}$&$2(1) \\cdotp 10^{-9}$&2\\\\\n\\hline\n0.0001 & $7.7(3) \\cdotp 10^{-5}$& $7.7(3) \\cdotp 10^{-5}$&$4.9(2) \\cdotp 10^{-5}$&713\\\\\n\\hline\n0.00001& $1.02(4) \\cdotp 10^{-4}$&$1.02(4) \\cdotp 10^{-4}$&$6.5(2) \\cdotp 10^{-5}$&1852\\\\\n\\hline\n0.000001&$1.17(4) \\cdotp 10^{-4}$&$1.17(4) \\cdotp 10^{-4}$&$7.4(2) \\cdotp 10^{-5}$&5068\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison between the codes based on the DT-method  and the \n  QH-method for KLOE event selection cuts~(see Appendix~\\ref{appex:KLOBA}). \nThe contribution $\\sigma_{\\Delta}$ to the cross section $\\sigma$  for a \nchosen $|\\Delta|$. Subscripts $QH$ and $DT$ denote\n QT-method and DT-method respectively. $q^2 \\in (0.34,0.96)$ GeV$^2$. See text\n  for details on the definition of the Table entries.} \\label{test3}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multicolumn{5}{|c|}{BaBar event selection}\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{QH}$ = 0.0005655(7) [nb] }\\\\\n\\hline\n\\multicolumn{5}{|c|}{$\\sigma_{DT}$ =  0.0005655(7)  [nb] }\\\\\n\\hline\n$|\\Delta| >$ & $\\sigma_{\\Delta_{DT}}$ [nb]& $\\sigma_{\\Delta_{QH}}$ [nb]& $\\sigma_{\\Delta_{DT}}\/\\sigma_{DT}$& N$_{event}$\\\\\n\\hline\n0.0001 & 0&0&0&0\\\\\n\\hline\n0.00001& $3(1) \\cdotp 10^{-10}$& $3(1) \\cdotp 10^{-10}$&$5(2) \\cdotp 10^{-7}$&6\\\\\n\\hline\n0.000001&$1.2(2) \\cdotp 10^{-9}$&$1.2(2) \\cdotp 10^{-9}$&$2.1(4) \\cdotp 10^{-6}$&26\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison between the codes based on the DT-method and\n  QH-method for BaBar event selection cuts~(see Appendix~\\ref{appex:KLOBA}). \nThe contribution $\\sigma_{\\Delta}$ to the cross section $\\sigma$  for a \nchosen $|\\Delta|$. Subscripts $QH$ and $DT$ denote\n QT-method and DT-method respectively. $q^2 \\in (0.34,0.96)$ GeV$^2$. See text\n  for details on the definition of the Table entries. \\label{test4}}\n\\end{table}\n\nThe checks clearly show the control on the numerical accuracy of the\nvirtual corrections.  Additionally, using the DT-method  and realistic cuts (see Appendix~\\ref{appex:KLOBA} for their definition) for KLOE and\nBabar energies, we have studied the relevance of the most challenging\ncontribution in this article. For this purpose, we compare the contributions from one-loop Penta-Box diagrams\ndefined in Section~\\ref{sec:class} with the Born contributions in\nFig.~\\ref{fig:kloebabarcos} for muon angular distributions and in Fig.~\\ref{fig:kloebabarqq} for\nthe $\\mu^+ \\mu^-$ invariant mass distribution. As we can clearly see from the\nmuon and antimuon angular distributions, the size of the Penta-Box contributions\ncan reach the percent level and they cannot be neglected for the charge\n odd observables. We confirm here the expectations that\n the neglected corrections~\\cite{Czyz:2004rj} for the charge even distributions\n are indeed small.\n For a classification of the charge odd and even contributions, we refer the reader\n to Ref.~\\cite{Czyz:2003ue}, where it was done for charged pions in the final state.\n Replacing pions with muons does not change the classification presented there.\n\n\n\n  \n\\begin{figure}[ht]%\n      \\centering\n    \\includegraphics[scale=0.6,angle=0]{102cos1} %\n  \\includegraphics[scale=0.6,angle=0]{102cos_average} %\n\\\\\n   \\includegraphics[scale=0.6,angle=0]{1056cos1} %\n  \\includegraphics[scale=0.6,angle=0]{1056cos_average} %\n\\caption{Relevance of NLO Penta-Box contributions at KLOE  (above) and Babar (below) energies for muon angular\ndistributions: $\\theta_{\\mu^+}$ is the $\\mu^+$ polar angle, while $\\theta$ is \n the angle of $\\mu^+$ or $\\mu^-$ for the charge 'blind' observable. \nThese definitions are the same in all the figures and will not be repeated\n in captions.   }\n\\label{fig:kloebabarcos}\n  \\end{figure}\n\n \n\\begin{figure}[ht]%\n      \\centering\n       \\includegraphics[scale=0.6,angle=0]{102_qq} %\n\t  \\includegraphics[scale=0.6,angle=0]{1056_qq} %\n\\caption{Relevance of NLO Penta-Box contributions at KLOE  (left) and Babar (right) energies for $q^2$ $\\mu^+\n\\mu^-$ distributions.}\n\\label{fig:kloebabarqq}\n  \\end{figure}\n\n\\begin{figure}[ht]%\n  \\hskip 0.5cm  \\vskip 0.55cm \\includegraphics[scale=0.35,angle=0]{PJFRYa} \\vskip -5.75cm  \\hskip 7.5 cm\n\\includegraphics[scale=0.31,angle=270]{KLOE-loop-7-mum_LT90loop-9-mum_GT90}\n\\caption{On left: Muon pair distributions including 5-point functions at KLOE\n  calculated with \\texttt{PJFry} (bottom: absolute error estimate). On right:\n  the same calculated without decicated routines to avoid small Gram determinants.\nApproximately $4\\cdot 10^{10}$ ($10^9$) events have been generated\n }\n\\label{fig:acc}\n  \\end{figure}\n\n\n\n To appreciate these results,   \nin Fig.~\\ref{fig:acc}, we plot the Penta-Box contributions using the\n\\texttt{PJFry} package with (left) and without (right) using expansion for small Gram\ndeterminants. The right panel reveals discrepancies only after\nincreasing the number of Monte Carlo events to $10^9$~\\cite{kajphd,Gluza:2012yz}.\n\\texttt{PJFry} treats properly \n small Gram determinants, as discussed in details in~\\cite{yundin-phd-2012--oai:export}.  \n  With the \\texttt{PJFry} package, the leading  inverse Gram determinants\n  $|G^{(5)}|$ are eliminated in the tensor reduction and \nsmall inverse Gram determinants $|G^{(4)}|$ are avoided using asymptotic expansions and Pade approximants. \nFor more details concerning the numerical stability of the tensor reductions, see\nRefs.~\\cite{yundin-phd-2012--oai:export,Fleischer:2011bi}. The results are\ncompletely stable and well under control. \n\n \nThe soft real emission analytic formulae were also checked. Firstly, by\ncomparing to the integral obtained by means of a Monte Carlo methods. A good agreement was found even if the\nmethod is limited in some cases to an\naccuracy of  $2\\cdot 10^{-4}$. Secondly, the numerical stability of the code\n was tested comparing the quadruple and the double precision versions of the same\n code. The relative accuracy of the double precision version used \n in the released code\n of the generator is at the level of $10^{-7}$ at 1~GeV, while at 10~GeV , it\n was only about $10^{-3}$ in some corners of the phase space. However, since\n those phase space regions did not\n affect the relevant observables (invariant mass and angular \n distributions), the code was not changed to cure this behaviour by means of appropriate expansions. \n\n The new contributions of the real two-photon emission\n were tested comparing two completely independent codes.\n  In one, the trace method (T) and FORM~\\cite{Kuipers:2012rf}\n  was used to obtain an analytic result,\n  in the second one, the helicity amplitude method (H),\n  described in~\\cite{Rodrigo:2001kf} and in Appendix~\\ref{appex:twophoton},\n was applied.\n\n \n \n   The biggest observed relative difference of the codes was at the level of $10^{-4}$\n  even if both codes were using double precision only. Additionally, in both\n  cases gauge invariance was checked. For the T-method\n  analytically, and for the H-method numerically, obtaining a relative accuracy of $10^{-15}$.\n\n Both the soft and the real parts were tested checking the independence \n  of the cross section and differential distributions of the separation parameter between the soft part,\nwhere the integral\n  over the one photon phase space is performed analytically, and the hard\n part, where the integral is obtained using the Monte Carlo generation.  The accuracy of this test was\n $2\\cdot 10^{-4}$. A perfect agreement at this level of accuracy \n was observed.\n \n  \n \n\\section{Impact of the radiative corrections added\nto the event generator on the  pion form factor measurements at BaBar and KLOE\\label{sec-impact}}\n\n PHOKHARA7.0 has been used by BaBar and KLOE until quite recently. In fact, from version 4.0 to 7.0 the\nmuon production channel was not changed. \nComparing numerics with PHOKHARA9.0, which includes  the\n complete NLO corrections, one has to distinguish between the charge average distributions \n for which the bulk of the NLO corrections was already included in PHOKHARA7.0 and the charge sensitive observables \nfor which version 7.0 was limited to the leading order only. In the experimental framework for the extraction\n of the hadronic cross section,  charge averaged observables were used. \nThe most important invariant muon pair mass distribution (i.e. the  $q^2$ distribution), from\nwhich the hadronic cross section is extracted, is shown in Fig.~\\ref{q2reldif}.\n As one can see, the radiative corrections missing in  PHOKHARA7.0 are small. They reach up to 0.1~\\%\n for the KLOE event selection and up to 0.25\\% for the BaBar event selection. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\vspace*{-4.0cm}\n\\hspace*{-2.0cm}\\subfloat{\\includegraphics[width=.8\\textwidth]{kloe_all}}\n\\hspace*{-4.0cm}\\subfloat{\\includegraphics[width=.8\\textwidth]{babar_all}}\n\\end{center}\n\\vspace*{-9.0cm}\n\\caption[]{The relative difference between differential cross sections of PHOKHARA7.0 \n and PHOKHARA9.0 (with subscript `new').\n }\n\\label{q2reldif}\n\\end{figure}\n\nThe charge averaged angular \n distributions are also not very much different as shown in Fig.~\\ref{ang1reldif} and Fig.~\\ref{ang1reldif1}\n for different $q^2$ bins. For other muon pair invariant mass ranges, the results are similar to the ones\n shown. We can conclude at this point that in the experiments using the charge averaged observables, \n the missing radiative corrections are very small and should not have affected the extraction\n of the hadronic cross section.\n  \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{kl5455angav}\n\\includegraphics[width=.49\\textwidth]{bb5455angav}\n\\end{center}\n\\caption[]{The relative difference between differential cross sections of PHOKHARA7.0 \n and PHOKHARA9.0 (with subscript `new'). $q^2\\in (0.54,0.55)$.\n }\n\\label{ang1reldif}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{kl9495angav}\n\\includegraphics[width=.49\\textwidth]{bb7475angav}\n\\end{center}\n\\caption[]{The relative difference between differential cross sections of PHOKHARA7.0 \n and PHOKHARA9.0 (with subscript `new'). $q^2\\in (0.94,0.95)$ for KLOE cuts;\n$q^2\\in (0.74,0.75)$ for BaBar cuts.\n }\n\\label{ang1reldif1}\n\\end{figure}\n\nFor the charge sensitive observables, which were available at PHOKHARA7.0 only at LO \n (the ISR-FSR interference was present at LO only) the new corrections are relatively bigger \n and reach typically a few percent as expected from NLO corrections. The KLOE event selection\n  was designed to diminish the FSR radiative corrections and as such it was also mostly killing \n  the asymmetry coming from one photon emission. The asymmetry coming from the two photon emission\n is however surviving the cuts as shown in Fig.~\\ref{asym}. For BaBar, the asymmetry\n is naturally suppressed by tagging the photon at large angles. At low invariant masses, as\n compared to the energy available at the experiment, it forces the muons to fly in the opposite\n direction to the photon and thus the suppression. The asymmetry is at the level of  few percent and it\n is dominated by the LO contributions as shown in Fig.~\\ref{asym1}. \n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{kl5455asym}\n\\includegraphics[width=.49\\textwidth]{kl9495asym}\n\\end{center}\n\\caption[]{The asymmetries given by\n  PHOKHARA7.0 (denoted as $PH$)\n and PHOKHARA9.0 (denoted as $PH_{new}$). $q^2\\in (0.54,0.55)$ - left plot;\n $q^2\\in (0.94,0.95)$ - right plot.\n }\n\\label{asym}\n\\end{figure}\n\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{bb5455asym}\n\\includegraphics[width=.49\\textwidth]{bb8485asym}\n\\end{center}\n\\caption[]{The asymmetries given by\n  PHOKHARA7.0 (denoted as $PH$)\n and PHOKHARA9.0 (denoted as $PH_{new}$). $q^2\\in (0.54,0.55)$ - left plot;\n $q^2\\in (0.74,0.75)$ - right plot.\n }\n\\label{asym1}\n\\end{figure}\n\n\n\\section{Conclusions  \\label{sec-conc}}\n     The presented studies allow for the development of a numerically\n  stable Monte Carlo event generator PHOKHARA9.0 simulating the reaction\n  $e^+e^- \\to \\mu^+\\mu^-\\gamma$ with full NLO QED accuracy. \nThe radiative corrections which were missing in the previous\n  versions of the generator can reach a few percent.\nThough, it was shown that the charge blind observables used\n by the BaBar and KLOE collaborations are affected only at the level of \n 0.1\\% for KLOE and 0.3\\% for BaBar. \nWe conclude that the observed discrepancies\n  between these experiments cannot be attributed to the missing corrections for  the reaction  $e^+e^- \\to\n\\mu^+\\mu^-\\gamma$ in PHOKHARA4.0~\\cite{Czyz:2004rj,Czyz:2004ua} to PHOKHARA8.0~\\cite{Czyz:2013xga} .\n\n\n \\acknowledgments{This work has been supported by the\nResearch Executive Agency (REA) of the European Union under\nthe Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet),\nby the Polish Ministry of Science and High Education\nunder grant number N N202 102638,\nby the Spanish Government and EU ERDF funds\n(grants FPA2011-23778, FPA2011-23596 and CSD2007-00042\nConsolider Project CPAN), by GV (PROMETEUII\/2013\/007) and by the Deutsche Forschungsgemeinschaft via the\nSonderforschungsbereich\/Transregio SFB\/TR-9 Computational Particle Physics. This work is  part of the activity of the ``Working Group on Radiative\n Corrections and Monte Carlo Generators for Low Energies\"  \n [\\url{http:\/\/www.lnf.infn.it\/wg\/sighad\/}].\n FC is funded by a Marie Curie fellowship (PIEF-GA-2011-298960).\nMG was supported by ``\\'Swider'' PhD program.\n}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nLet $F$ be an infinite field and let $F[y_1,\\dots,y_n]^{S_n}$ be the\nring of \\emph{symmetric polynomials} in $n$ variables. The general\nlinear group $GL(n,F)$ acts by conjugation on the full ring\n$Mat(n,F)$ of $n\\times n$ matrices over $F$. Denote by\n$F[Mat(n,F)]^{GL(n,F)}$ the ring of the polynomial invariants for\nthis actions. It is well known that\n\\begin{equation}F[Mat(n,F)]^{GL(n,F)}\\cong\nF[y_1,\\dots,y_n]^{S_n}.\\label{cla}\\end{equation} The above result\ncan be restated by saying that the scheme parameterizing\n$n-$dimensional linear representations of $F[x]$ up to basis change\nis isomorphic to the symmetric product $(\\mathbb{A}^1)^n\/S_n$ which\nis defined as $\\mathrm{Spec}\\,F[y_1,\\dots,y_n]^{S_n}$.\n\nThe main result of this paper is to show that the same thing happens\nfor any commutative algebra $A$ over a characteristic zero field.\n\n\\noindent Namely, when $F$ is a characteristic zero field, we prove\nthat the scheme $\\mathcal{R}_n(A)\/\/Gl(n,F)$ that parameterizes as a coarse\nmoduli space $n-$dimensional linear representations of a commutative\n$F-$algebra $A$ up to basis change is isomorphic to the symmetric\nproduct of its prime spectrum, i.e.\n\\[\\mathcal{R}_n(A)\/\/Gl(n,F)\\cong\nX^n\/S_n=\\mathrm{Spec}\\,(A^{\\otimes n})^{S_n},\\] where\n$X=\\mathrm{Spec}\\,A$.\n\nSuppose $A$ is generated by say $m$ elements, then on geometric\npoints one can identify the symmetric product with a subscheme of\nthe $m-$tuples of diagonal $n\\times n$ matrices and the above result\ncan be read as theorem of simultaneous diagonalization.\n\nIn positive characteristic we prove an analogous result for the associated varieties.\n\nThe proof is based on the carefully analysis of the morphism induced\nby the composition $det\\cdot \\rho$ of the determinant with a\nrepresentation. This will be developed for flat algebra over a\ncommutative ring.\n\nAs a complement we give a presentation by generators and relations\nof $(A^{\\otimes n})^{S_n}$ that holds for $A$ flat and over any\ncommutative base ring $F$ giving then the equations of $X^{(n)}$ in\nthe flat case.\n\nThank to this we describe $V_n(A)^G\\cong (A^{\\otimes n})^{S_n}$ in\nthe characteristic zero case in term of traces and polynomial\nidentities and in positive characteristic in term of the coefficients of\nthe characteristic polynomial of polynomial of generic matrices.\n\nThese results generalizes the one given in {\\cite{vf,v3}}.\n\n\\centerline{\\textbf{Acknowledgements}}\nI would like to thank M.Brion, C. De Concini and last but not least C.Procesi for useful discussions.\n\n\\section{Notation} Unless otherwise stated we adopt the following\nnotations:\n\\begin{itemize}\n\\item $F$ is a fixed base ring\n\\item we write {\\it{algebra}} to mean {\\it {commutative $F-$algebra}}\n\\item we denote by $\\mathcal{C}_F$ the category of commutative $F-$algebras\n\\item $Sets$ the category of sets\n\\item we write $\\mathcal{A}(B,C):=Hom_{\\mathcal{A}}(B,C)$ in a category $\\mathcal{A}$ with\n$B,C\\in Ob(\\mathcal{A})$ objects in $\\mathcal{A}$.\n\\item for $A$ a set and any\nadditive monoid $M$, we denote by $M^{(A)}$ the set of functions\n$f:A\\rightarrow M$ with finite support.\n\\item let $\\alpha\\in M^{(A)}$,\nwe denote by $\\mid \\alpha \\mid$ the (finite) sum $\\sum_{a\\in A}\n\\alpha(a)$,\n\\item given a set $I$ we denote by $\\sharp I$ its cardinality.\n\\end{itemize}\n\n\\section{Representations}\\label{rep}\nGiven an algebra $B$ write $Mat(n,B)$ for the ring of $n\\times n$\nmatrices with entries in $B$.\n\\begin{definition}\\label{rep}\n For a $n-$dimensional representation\nof $A$ over $B$ one means an algebra homomorphism $A\\to Mat(n,B)$.\nWe denote by $\\mathcal{R}_n(A,B)$ the set of these representations. Given any\nalgebras $A,B$ and a homomorphism $\\rho:A\\to B$ we write $(\\rho)_n$\nfor the induced map $Mat(n,A)\\to Mat(n,B)$.\n\\end{definition}\nThe assignment $B\\to \\mathcal{R}_n(A,B)$ gives a covariant functor\n$\\mathcal{R}_n^A:\\mathcal{C}_R\\to Sets$ as can be easily checked.\n\\begin{proposition}({\\cite{dp}} Sec.1\\,)\nFor all $n\\in\\mathbb{N}$ and $A\\in\\mathcal{C}_F$ there exist a unique algebra\n$V_n(A)$ and a unique representation $\\pi_n^A\\in\\mathcal{R}_n(A,V_n(A))$ such\nthat the map $\\rho\\mapsto(\\rho)_n\\cdot \\pi_n^A$ is an isomorphism\n\\[\\mathcal{C}_F(V_n(A),B)\\xrightarrow{\\cong}\\mathcal{R}_n(A,B),\\]\nfor all algebra $B$, i.e. the functor from schemes to sets\nassociated to $\\mathcal{R}_n^A$ is represented by the affine scheme\n$\\mathcal{R}_n(A):=\\mathrm{Spec}\\,V_n(A)$\n\\end{proposition}\n\\begin{proof} Let $\\Omega$ be a set and consider\n$F[\\xi_{ij,\\omega}]$, a polynomial ring where $i,j=1,\\dots,n$ and\n$\\omega\\in\\Omega$. Note that $F[\\xi_{ij,\\omega}]$ is isomorphic to\nthe symmetric algebra on the dual of $Mat(n,F)^{\\Omega}$.\n\nLet $A_{\\Omega}=F\\{x_{\\omega}\\}_{\\omega\\in\\Omega}$ be the free\nassociative algebra on $\\Omega$ then\n\\[\\mathcal{R}_n(A_{\\Omega},S)\\cong Mat(n,S)^{\\Omega} \\cong\n\\mathcal{C}_F(F[\\xi_{ij,\\omega}],S)\\] for any $S\\in\\mathcal{C}_F$. More precisely write\n$D:=Mat(n,F[\\xi_{ij,\\omega}])$ and let $\\xi_{\\omega}\\in D$ be given\nby $(\\xi_{\\omega})_{ij}=\\xi_{ij,\\omega}$\\,, $\\forall\\, i,j,\\omega$,\nthese are called the ($n\\times n$) {\\em generic matrices} and were\nintroduced by C.\\,Procesi (see {\\cite{prolin}}).\n\nLet $\\pi:A_{\\Omega}\\to D$ be the $n-$dimensional representation\ngiven by $x_{\\omega}\\mapsto \\xi_{\\omega}$. We have then that given\nany $\\rho\\in \\mathcal{R}_n(A_{\\Omega},S)$, with $S\\in\\mathcal{C}_F$ there is a unique\n$\\bar{\\rho}\\in \\mathcal{C}_F(F[\\xi_{ij,\\omega}],S)$ given by\n$\\xi_{ij,h}\\mapsto (\\rho(x_{\\omega}))_{ij}$ and it is such that the\nfollowing diagram commutes\n\\begin{equation}\n\\xymatrix{\n  A_{\\Omega} \\ar[dr]_{\\rho} \\ar[r]^{\\pi}\n                & B \\ar[d]^{(\\bar{\\rho})_n}  \\\\\n                & Mat(n,S)             }\n\\end{equation}\nWe now substitute $A_{\\Omega}$ with an algebra $A\\in\\mathcal{C}_F$ and we let\n\\[0\\longrightarrow J \\longrightarrow A_{\\Omega} \\longrightarrow A\n\\longrightarrow 0\\] be a presentation by generators and relations.\n\nLet $I$ be the unique ideal in $F[\\xi_{ij,\\omega}]$ such that\n\\begin{equation}\\label{ideal}\nMat(n,I)=D\\,\\pi(J)\\,D\n\\end{equation}\nthen as one can easily check that, for all $S\\in\\mathcal{C}_F$\n\\begin{equation}\n\\mathcal{R}_n(R,S)\\cong \\mathcal{C}_F(F[\\xi_{ij,\\omega}]\/I,S)\n\\end{equation}\nvia the lifting to a representation of $F$. Set now\n$V_n(A):=F[\\xi_{ij,\\omega}]\/I$. Let\n$\\{a_{\\omega}\\}_{\\omega\\in\\Omega}$ be a set of generators of $A$, in\nthe same way as above we have a universal representation\n\\begin{equation}\n\\pi_n^A:\\begin{cases}\n        A\\longrightarrow Mat(n,V_n(A)) \\\\\n        r_{\\omega}\\mapsto \\xi_{\\omega}^A\n      \\end{cases}\n\\end{equation}\nwhere $\\xi_{\\omega}^A$ is the image of $\\xi_{\\omega}$ via the\nsurjection $Mat(n,F[\\xi_{ij,\\omega}])\\to Mat(n,V_n(A))$.\n\\end{proof}\n\\begin{remark} Note that $\\mathcal{R}_n(A)$ could be quite complicated, as an example,\nwhen $A=\\mathbb{C}[x,y]$ we obtain that $\\mathcal{R}_n(A)$ is the\n\\textit{commuting scheme} and it is not even known (but\nconjecturally true) if it is reduced or not, see \\cite{v3}.\n\\end{remark}\n\\begin{definition}\nWe set $\\mathcal{G}_n(A):=\\pi_n^A(A)\\subset Mat(n,V_n(A))$.\n\\end{definition}\n\\begin{proposition}\nThe universal representation $\\pi_n^A$ gives an isomorphism\n$A\\cong\\mathcal{G}_n(A)$.\n\\end{proposition}\n\\begin{proof}\nThe representation $A\\to Mat(n,A)$ given by $a\\mapsto\ndiag(a,\\dots,a)$ is injective so is $\\pi_n^A$ by universality.\n\\end{proof}\n\\begin{definition}\\label{coef}\nThe subalgebra of $V_n(A)$ generated by the coefficients of the\ncharacteristic polynomial of the elements of $\\mathcal{G}_n(A)$ will\nbe denoted by $C_n(A)$.\n\\end{definition}\n\\section{Symmetric Products}\\label{sym}\nLet $A$ be an algebra and $X=\\mathrm{Spec}\\,A$ its prime spectrum.\nThe symmetric group $S_n$ acts on the $n-$th tensor power\n$A^{\\otimes n}$ and as usual we write $TS^n_F(A)$ or simply\n$TS^n(A)$ to denote the subalgebra of the invariants for this\nactions, i.e. the symmetric tensors of order $n$ over $A$. The\n$n-$th symmetric product of the affine scheme $X$ is the quotient\nscheme of $X^n$ with respect to the above action and is usually\ndenoted by $X^{(n)}$. By definition\n$X^{(n)}:=X^n\/S_n:=\\mathrm{Spec}\\, TS^n(A)$.\n\\subsection{Polynomial\nLaws}\\label{poly} To link symmetric tensors to linear\nrepresentations we shall use the determinant so that we are lead to\nthe topic of polynomial laws: we recall the definition of this kind\nof map between $F-$modules that generalizes the usual polynomial\nmapping between free $F-$modules (see {\\cite{bo,rl,rs}}).\n\\begin{definition} Let $A$ and $B$ be two\n$F$-modules. A \\emph{polynomial law} $\\varphi$ from $A$ to $B$ is a\nfamily of mappings $\\varphi_{_{L}}:L\\otimes_{F} A \\longrightarrow\nL\\otimes_{F} B$, with $L\\in\\mathcal{C}_{F}$ such that the following diagram\ncommutes\n\\begin{equation}\n\\xymatrix{\n  L\\otimes_{F}A \\ar[d]_{f\\otimes id_A} \\ar[r]^{\\varphi_L}\n                & L\\otimes_{F} B \\ar[d]^{f\\otimes id_B}  \\\\\n  M\\otimes_{F} A \\ar[r]_{\\varphi_M}\n                & M\\otimes_{F} B             }\n\\end{equation}\nfor all $L,\\,M\\in\\mathcal{C}(F)$ and all $f\\in \\mathcal{C}_{F}(L,M)$.\n\\end{definition}\n\\begin{definition}\nLet $n\\in \\mathbb{N}$, if $\\varphi_L(au)=a^n\\varphi_L(u)$, for all $a\\in\nL$, $u\\in L\\otimes_{F} A$ and all $L\\in\\mathcal{C}_{F}$, then $\\varphi$ will\nbe said \\emph{homogeneous of degree} $n$.\n\\end{definition}\n\\begin{definition}\nIf $A$ and $B$ are two $F$-algebras  and\n\\[\n\\begin{cases} \\varphi_L(xy)&=\\varphi_L(x)\\varphi_L(y)\\\\\n         \\varphi_L(1_{L\\otimes A})&=1_{L\\otimes B}\n         \\end{cases}\n         \\]\nfor $L\\in\\mathcal{C}_{F}$ and for all $x,y\\in L\\otimes_{F} A$, then $\\varphi$\nis called \\emph{multiplicative}.\n\\end{definition}\n\\begin{remark} A polynomial law $\\varphi:A\\rightarrow B$  is a\nnatural transformation $-\\otimes_{F}A\\rightarrow -\\otimes_{F}B$.\n\\end{remark}\nLet $A$ and $B$ be two $F$-modules and $\\varphi:A\\rightarrow B$ be a\npolynomial law. The following result on polynomial laws is a\nrestatement of Th\\'eor\\`eme I.1 of {\\cite{rl}}.\n\\begin{theorem}\\label{roby} Let $S$ be a set.\n\\begin{enumerate}\n\\item Let $L=F[x_s]_{s\\in S}$ and let $\\{a_{s}\\, :\\,s\\in S\\}\\subset A$\nbe such that $a_{s}=0$ except for a finite number of $s\\in S$, then\nthere exist $\\varphi_{\\xi}((a_{s}))\\in B$, with $\\xi \\in \\mathbb{N}^{(S)}$,\nsuch that:\n\\[\\varphi_{_{L}}(\\sum_{s\\in S} x_s\\otimes\na_{s})=\\sum_{\\xi \\in \\mathbb{N}^{(S)}}  x^{\\xi}\\otimes\n\\varphi_{\\xi}((a_{s})),\\] where $x^{\\xi}:=\\prod_{s\\in S}\nx_s^{\\xi_s}$.\n\\item Let $R$ be any commutative $F$-algebra and let\n$(r_s)_{s\\in S}\\subset R$, then: \\[\\varphi_{_{R}}(\\sum_{s\\in S}\nr_s\\otimes a_{s})=\\sum_{\\xi \\in \\mathbb{N}^{(S)}}  r^{\\xi}\\otimes\n\\varphi_{\\xi}((a_{s})),\\] where $r^{\\xi}:=\\prod_{s\\in S}\nr_s^{\\xi_s}$.\n\\item If $\\varphi$ is homogeneous of degree $n$, then one has\n$\\varphi_{\\xi}((a_{s}))=0$ if $\\mid \\xi \\mid$ is\ndifferent from $n$. That is: \\[\\varphi_{_{R}}(\\sum_{a\\in A}\nr_a\\otimes a)=\\sum_{\\xi \\in \\mathbb{N}^{(A)},\\,\\mid \\xi \\mid=n}\nr^{\\xi}\\otimes \\varphi_{\\xi}((a)).\\] In particular, if $\\varphi$ is\nhomogeneous of degree $0$ or $1$, then it is constant or linear,\nrespectively.\n\\end{enumerate}\n\\end{theorem}\n\\begin{remark}\\label{coef}\nThe above theorem means that a polynomial law $\\varphi:A\\rightarrow\nB$ is completely determined by its coefficients\n$\\varphi_{\\xi}((a_{s}))$, with $(a_s)_{s\\in S} \\in S^{(\\mathbb{N})}$.\n\\end{remark}\n\\begin{remark} If $A$ is a free $F$-module and $\\{a_{t}\\,\n:\\, t\\in T\\}$ is a basis of $A$, then $\\varphi$ is completely\ndetermined by its coefficients $\\varphi_{\\xi}((a_{t}))$, with $\\xi\n\\in \\mathbb{N}^{(T)}$. If also $B$ is a free $F$-module with basis\n$\\{b_{u}\\, :\\, u\\in U\\}$, then $\\varphi_{\\xi}((a_{t}))=\\sum_{u\\in\nU}\\lambda_{u}(\\xi)b_{u}$. Let $a=\\sum_{t\\in T}\\mu_{t}a_{t}\\in A$.\nSince only a finite number of $\\mu_{t}$ and $\\lambda_{u}(\\xi)$ are\ndifferent from zero, the following makes sense:\n\\begin{eqnarray*}\\varphi(a)=\\varphi(\\sum_{t\\in T}\\mu_{t}a_{t}) =  \\sum_{\\xi\\in\n\\mathbb{N}^{(T)}} \\mu^{\\xi}\\varphi_{\\xi}((a_{t}))& = & \\sum_{\\xi\\in\n\\mathbb{N}^{(T)}} \\mu^{\\xi}(\\sum_{u\\in U}\\lambda_{u}(\\xi)b_{u})\\\\ & = &\n\\sum_{u\\in U}(\\sum_{\\xi\\in \\mathbb{N}^{(T)}}\\lambda_{u}(\\xi)\n\\mu^{\\xi})b_{u}.\\end{eqnarray*} Hence, if both $A$ and $B$ are free\n$F$-modules, a polynomial law $\\varphi:A\\rightarrow B$ is simply a\npolynomial map.\n\\end{remark}\n\\begin{definition}\\label{polset}\nLet $A,B\\in\\mathcal{C}_F$ be two algebras we write $P_n(A,B)$ for the\nmultiplicative homogeneous polynomial mapping $A\\to B$ of degree\n$n$.\n\\end{definition}\nThe assignment $B\\to P_n(A,B)$ determines a functor $P_n^A:\\mathcal{C}_F\\to\nSets$ as one can easily check.\n\n\\subsection{Symmetric products as representing schemes}\\label{sym1}\nFrom now on $A$ will be a flat algebra. There is an element\n$\\gamma_n\\in P_n(A,TS^n(A))$ given by $\\gamma_n(a)= a^{\\otimes n}$\nsuch that the composition $\\mathcal{C}_F(TS^n(A),B)\\ni\\rho\\mapsto\\rho\\cdot\\gamma_n\\in P_n(A,B)$ gives an isomorphism\n\\begin{equation}\\mathcal{C}_F(TS^n(A),B)\\xrightarrow{\\cong}P_n(A,B),\\end{equation}\n i.e. the\nfunctor $P_n^A$ is represented by the symmetric product $X^{(n)}$,\n(see {\\cite{bo}} chap.IV). We call $\\gamma_n$ {\\textit{the universal\nmapping of degree n}}.\n\nThe proof goes as follows: by Rem.~\\ref{coef} a polynomial law is\ndetermined by its coefficients. If $A$ is free then one can extract\nfrom the set of coefficients of $\\gamma_n$ a linear basis of\n$TS^n(A)$. Then any polynomial law homogeneous of degree $n$\ncorrespond to a specialization of these coefficients of $\\gamma_n$.\nThe requirement to be multiplicative corresponds to the requirement\nthe above specialization to be an algebra homomorphism, since\n$\\gamma_n$ is obviously multiplicative. Then one applies the good\nproperties of $TS^n$ with respect to inverse limits and get the\ndesired result for flat algebras, see \\cite{laz}.\n\n\\subsection{Generators}\nLet $a\\in A$, there is an algebra homomorphism\n$\\eta_a:F[x_1,\\dots,x_n]^{S_n}\\cong TS^n(F[x])\\to TS^n(A)$ induced\nby the evaluation of $x$ at $a$. We write $e_i^n(a)=\\eta_a(e_i^n)$\nwhere $e_i^n$ is the $i-$th elementary symmetric polynomial in $n$\nvariables. Given and independent variable $t$ we have an induced map\n$\\overline{\\eta}_a:F[t][x_1,\\dots,x_n]^{S_n}\\cong TS^n(F[t])\\to\nTS^n(A[t])$ such that\n\\begin{equation}\nt^n+\\sum_ie_i^n(a)=\\overline{\\eta}_a(t^n+\\sum_ie_i^n)=\\overline{\\eta}_a(\\prod_i(t+x_i))=(t+a)^{\\otimes\nn}\n\\end{equation}\nso that $e_i^n(a)$ is the orbit sum of $a^{\\otimes i}\\otimes\n1^{\\otimes n-i}$. Note that $e_n^n(a)=a^{\\otimes n}=\\gamma_n(a)$.\n\n\\begin{proposition}[Generators]\nLet $A$ be a commutative flat algebra generated by $\\{a_i\\}_{i\\in\nI}$ then $TS^n_F(A)$ is generated by $e_i^n(a)$ where $a=\\prod_i\nr_i^{\\alpha_i}$ is such that $\\sum_i\\alpha_i\\leq max(n,\\sharp I(n-1))$ and\nthe $\\alpha_i$ are coprime.\n\\end{proposition}\\label{gen}\n\\begin{proof} In \\cite{vf} Th.1 we prove the above statement for\n$A=P$ a free polynomial $F-$algebra. Given $A=P\/I$ flat then\n$TS^n_F(P)\\to TS^n_{F}(A)$ is onto and we are done.\n\\end{proof}\n\n\\begin{proposition}\\label{newt}\nLet $F\\supset \\mathbb{Q}$ with $\\mathbb{Q}$ the rational integers\nand let $A$ be generated by $\\{a_i\\}_{i\\in I}$ then $TS^n_F(A)$ is\ngenerated by $e_1^n(a)$ where $a=\\prod_i r_i^{\\alpha_i}$ is such that\n$\\sum_i\\alpha_i\\leq n$.\n\\end{proposition}\n\\begin{proof}\nIt follows from Prop.\\ref{gen}, Newton formulas and Noether's bound.\n\\end{proof}\n\n\n\n\\section{Determinant and isomorphism}\nWe prove the key result of this paper.\n\\begin{theorem}\\label{main}\nThe composition $det\\cdot \\pi_n^A$ induces an isomorphism\\\\\n$$\\delta_n^A:TS^n_F(A)\\to C_n(A)$$ for all flat $A\\in \\mathcal{C}_F$.\n\\end{theorem}\n\\begin{proof}\nBy Sect.~\\ref{sym1} there is a unique algebra homomorphism\n$\\delta_n^A:TS^n_F(A)\\to C_n(A)\\subset A_n(A)$ such that\n$det\\cdot\\pi_n^A=\\delta_n^A\\cdot\\gamma_n$. We have an identification\n\\[1_{F[t]}\\otimes \\delta_n^A(t\\otimes 1_A+1\\otimes a)^{\\otimes n}=det(t\\otimes\n1_A+1\\otimes\\pi_n^A(a))\\]\nfor all $a\\in A$ with $t$ and independent\nvariable. By definition of $C_n(A)$ and Prop.\\ref{gen} one has that $\\delta_A^n$ is surjective\nBy Sect.~\\ref{rep} the representation $\\partial_n:A\\to\nMat(n,A^{\\otimes n})$ given by $\\partial(a)= diag(a\\otimes\n1^{\\otimes n-1},\\,1\\otimes a\\otimes 1^{\\otimes\nn-2},\\,\\dots\\,,\\,1^{\\otimes n-1}\\otimes a)$ corresponds uniquely to\na homomorphism $\\overline{\\partial_n}:V_n(A)\\to A^{\\otimes n}$ such\nthat $\\partial_n=(\\overline{\\partial_n})_n\\cdot \\pi_n^A$. Observe\nthat\n$\\overline{\\partial_n}\\cdot\\det\\cdot\\pi_n^A=det\\cdot\\partial_n(a)=a^{\\otimes\nn}=\\gamma_n(a)$ hence the restriction of $\\partial_n$ to $C_n(A)$ is surjective by Prop.\\ref{gen} and gives an inverse to $\\delta_n^A$ and we are done.\n\\end{proof}\n\n\n\n\n\\section{Invariants}\\label{inv}\nWe want to study in a GIT fashion the equivalence classes of the\nrepresentations of $A$ under basis changes, i.e. under the action of\nthe general linear group $G:=Gl(n,F)$. The right object is the\ncategorical quotient\n$\\mathcal{R}_n(A)\/\/G:=\\mathrm{Spec}\\,(V_n(A)^G)$, where as usual\n$V_n(A)^G$ denotes the invariants for the $G-$action induced on\n$V_n(A)$ by basis change on $F^n$ .\n\\begin{theorem}\nIf $F$ is an algebraically closed field the $F-$points of the variety associated to $X^{(n)}$ are in one to one correspondence with the equivalence classes of the semisimple $n-$dimensional linear representations of $A$.\n\\end{theorem}\n\\begin{proof}\nIt follows from Th.\\ref{main} plus Sec.4\\cite{pis}.\n\\end{proof}\n\\begin{theorem}\\label{ch0}\nLet $F\\supset \\mathbb{Q}$ then\n$$\\delta_n^A:TS^n_F(A)\\cong V_n(A)^G$$ i.e.\n$$X^{(n)}\\cong \\mathcal{R}_n(A)\/\/G$$\n\\end{theorem}\n\\begin{proof}\nIn \\cite{v3} we proved that the statement is true when $A$ is a free\npolynomial ring. Then the characteristic zero case follows by\nTh.\\ref{main} and the reductivity of $G$.\n\n\\noindent {\\it{Another proof}.} The statement also follows observing\nthat, by C.\\,Procesi \\cite{p1} one has\n$F[\\xi_{ij,\\omega}]^G=C_n(A_{\\Omega})$ and again the result follows\nby Th.\\ref{main} and the reductivity of $G$.\n\\end{proof}\n\\begin{remark}\nLet $A=F[x]$ then Th.\\ref{ch0} implies $$F[x_1,\\dots,x_m]^{S_n}\\cong\nF[Mat(n,F)]^G$$ a well know result.\n\\end{remark}\n\\begin{remark}\nIf $A$ is reduced then Th.\\ref{ch0} implies that $V_n(A)^G$ is\nreduced too. This gives some support to the conjecture that\n$V_n(\\mathbb{C}[x,y])$ is reduced.\n\\end{remark}\n\n\\subsection{Equations}\nLet now $F$ be again an arbitrary commutative ring. Suppose you have\nan homomorphism $f:A\\to B$ of algebras then on can easily check that\nthe kernel of $TS^nf:TS^n(A)\\to TS^n(B)$ is linearly generated by\nthe orbit sums (under $S_n$) of elements $a_1\\otimes\\cdots\\otimes\na_n$ such that $\\exists k\\in \\{1,\\dots,n\\}$ with $a_k\\in \\ker f$.\nNow any such element can be expressed as a polynomial in the\n$e_i^n(a)$ with $a$ varying into the set of monomials in the $a_j$\n(see \\cite{vf} Lemma 1.2 and Cor.2.3). Thus we have the following\n\\begin{proposition}\\label{ker1}\nLet $f:A\\to B$ be an algebra homomorphism then the kernel of\n$TS^nf:TS^n(A)\\to TS^n(B)$ is generated as an ideal by the elements\n$e_i^n(a)$ with $i=1,\\dots,n$ and $a\\in \\ker f$.\n\\end{proposition}\n\\begin{corollary}\nSuppose $F\\supset \\mathbb{Q}$ then $\\ker TS^nf$ is generated by\n$e_1^n(a)$ with $a\\in \\ker f$.\n\\end{corollary}\n\\begin{proof}\nIt follows from the above Proposition by Newton's formulas.\n\\end{proof}\n\nLet $P_{\\Omega}=F[x_{\\omega}]_{\\omega\\in\\Omega}$ be the free\npolynomial algebra on $\\Omega$. We set $P^+_{\\Omega}$ for the\naugmentation ideal i.e. the kernel of the evaluation\n$x_{\\omega}\\mapsto 0$ for all $\\omega$.\n\nConsider the polynomial ring $F[e_{i,\\mu}]$ freely generated by the\nsymbols $e_{i,\\mu}$ with $1\\geq i$ and $\\mu$ that varies in the set\nof monomials $\\prod_{\\omega}x_{\\omega}^{\\alpha_{\\omega}}$ having\ncoprime exponents. By Prop.\\ref{gen} for all $n$ there is a\nsurjective homomorphism $\\kappa_n:F[e_{i,\\mu}]\\to TS^n(P_{\\Omega})$\ngiven by $e_{i,\\mu}\\mapsto e_i^n(\\mu)$ if $i\\leq n$ and\n$e_{i,\\mu}\\mapsto 0$ for $i>n$.\n\nGiven $f\\in P_{\\Omega}^+$ we can compute $e_m^n(f)=$ orbit sum under\n$S_n$ of $f^{\\otimes m}\\otimes 1^{\\otimes n-m}$ for all $n\\geq m$\n(for $n<m$ it is zero) and express it as a polynomial in the\n$e_i^n(\\mu)$ with $\\mu$ monomial with coprime exponents. It has been\nshown that this expression stabilizes as $n>>m$ (see \\cite{vf}\nProp.3.4 or \\cite{vz} 9.1). Thus we have a well defined polynomial\nlaw $\\overline{e}_m:P_{\\omega}^+\\to F[e_{i,\\mu}]$ homogeneous of\ndegree $m$. In \\cite{vf} we prove that $\\ker\\kappa_n$ is linearly\ngenerated by the coefficients of $\\overline{e}_m$ with $m>n$.\n\nLet us give an improvement of that result.\n\\begin{theorem}[Relations]\\label{rel}\n\\begin{enumerate}\n\\item Let $A=P_{\\omega}\/I$ be a flat algebra, then the kernel of the surjection\n$F[e_{i,\\mu}]\\xrightarrow{\\kappa_n}TS^n(P_{\\Omega})\\to TS^n(A)$ is\nlinearly generated by the coefficients of $\\overline{e}_m$ with\n$m>n$ plus the lifting to $F[e_{i,\\mu}]$ of $e_k^n(g)$ where $k\\leq\nn$ and $g\\in I\\subset P_{\\Omega}$.\n\\item Suppose $F$ is an infinite field: the kernel of $\\kappa_n$ is\ngenerated as an ideal by $e_m(f)$ with $m>n$ and $f\\in\nP_{\\Omega}^+$.\n\\item Suppose $F$ is an infinite field and let $A=P_{\\omega}\/I$.\nThe kernel of the surjection\n$F[e_{i,\\mu}]\\xrightarrow{\\kappa_n}TS^n(P_{\\Omega})\\to TS^n(A)$ is\ngenerated as an ideal by $e_m(f)$ with $m>n$ and $f\\in P_{\\Omega}^+$\nplus the lifting to $F[e_{i,\\mu}]$ of the $e_k^n(g)$ where $k\\leq n$\nand $g\\in I\\subset P_{\\Omega}$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\\,\n\n\\begin{enumerate}\n\\item It follows from the above discussion and Prop.\\ref{ker1}\n\\item A linear form that is zero on the linear subspace\ngenerated by $e_k(f)$ with and $f\\in P^+_{\\Omega}$ is zero also on\nthe subspace generated by the coefficients as $f$ varies.\n\\item By \\textbf{1.} and \\textbf{2.}\n\\end{enumerate}\n\\end{proof}\n\\begin{remark}\nThe above Theorem give the equations of the symmetric product of any\naffine scheme.\n\\end{remark}\n\n\\subsection{Traces} Along this paragraph we suppose\nagain $F\\supset \\mathbb{Q}$. When we have a polynomial law $\\varphi$\nhomogeneous of degree $n$ we can consider its full polarization\n$\\varphi_{\\mathbf{1}_n}$ that is the coefficient of $t_1t_2\\cdots\nt_n$ in $\\varphi(t_1 x_1+\\dots +t_1 x_n)$ where $t_i$ are commuting\nindependent variables and the $x_j$ are generic elements in the\ndomain of $\\varphi$. It is well known that the process of\npolarization is effective via restitution\n$n!\\varphi(x)=\\varphi_{\\mathbf{1}_n}(x,x,\\dots,x)$.\n\n\\noindent What happens for $e_n^n$? We write \\[e_n^n(t_1 x_1+\\dots\n+t_n x_n)=t_1t_2\\cdots t_n e_{\\mathbf{1}_n}^n(x_1,\\dots,x_n) + Z\\]\nand if one observe that $e_n^n(t_1 x_1+\\dots +t_n x_n)=(t_1\nx_1+\\dots +t_n x_n)^{\\otimes n}$ then it follows easily that\n$e_{\\mathbf{1}_n}^n(x_1,\\dots,x_n)=\\sum_{\\sigma\\in\nS_n}x_{\\sigma(1)}\\otimes\\cdots\\otimes x_{\\sigma(n)}$\\,. We would\nlike now to express $e_{\\mathbf{1}_n}^n(x_1,\\dots,x_n)$ in terms of\n$e_1^n(\\mu)$ with $\\mu$ a monomial in the $x_i$.\n\nIt is clear that $\\delta_n^A(e_i^n(a))=\\psi_i^n(\\pi_n^A(a))$ where\n\\begin{equation}det(t+\\pi_n^A(a))=t^n+\\sum_i\\psi_i^n(\\pi_n^A)t^{n-i}\\end{equation}\nin particular $e_1^n(a)$ is identified with the trace of\n$\\pi_n^A(a)$ and $e_n^n(a)$ with the determinant. It is well known\nthat (see \\cite{p1}) the full polarization $\\chi_{\\mathbf{1}_n}$ of\nthe determinant can be expressed as a special polynomial in\ntraces of monomials, namely consider the cycle decomposition\n$\\sigma=\\sigma_1\\cdots\\sigma_k\\in S_n$ and let correspond to it the\nproduct $T_{\\sigma}=tr(\\mu_1)\\cdots tr(\\mu_k)$ where\n$\\mu_h=x_{h_1}\\cdots x_{h_l}$ being $\\sigma_h=(h_1\\,h_2\\,\\dots\nh_l)$, then\n\\begin{equation}\n\\Psi_{\\mathbf{1}_n}=\\sum_{\\sigma\\in S_n}\\epsilon_{\\sigma}T_{\\sigma}\n\\end{equation}\nand it is a well celebrated theorem due to Procesi \\cite{p1} and\nRazmyslov\\cite{ra} that all the relations (in characteristic zero)\nbetween the invariants of matrices, i.e. between traces of monomial\nof generic $n\\times n$ matrices are consequences, in the sense of\n$T-$ideals, of $\\Psi_{\\mathbf{1}_{n+1}}$.\n\nLet us summarize all we are able say in the characteristic zero\ncase.\n\\begin{theorem}\nLet $A=P_{\\Omega}\/I$ be a commutative $F-$algebra. The ring of the\ninvariants $V_n(A)^G$ is generated by traces of monomial of generic\nmatrices $\\xi^A$ and the ideal of relations is generated by the\nevaluation of $\\Psi_{\\mathbf{1}_{n+1}}$ at the elements of\n$\\mathcal{G}_n(A)$ and by the traces $tr(\\pi_n^A(f))$ with $f\\in I$.\nThe same obviously holds \\textit{mutatis mutandis} in $TS^n_F(A)$\n\\end{theorem}\n\\begin{proof}\nIt follows for the above discussion, Th.\\ref{rel}.3 using Newton's\nformulas.\n\\end{proof}\n\n\\section{Positive characteristic}\nIn this subsection $F$ will be an infinite field of arbitrary\ncharacteristic. Set $N_n(R)$ for the nilradical of $V_n(A)$ and\n$\\mathcal{R}_{n,\\,red}(A):= \\mathrm{Spec}\\,V_n(R)\/N_n(R)$ for the reduced\nscheme associates to $\\mathcal{R}_n(A)$. Since the nilradical it is\n$G$-stable then the action of $G$ on $\\mathcal{R}_n(A)$ can be restricted to\nthe variety $\\mathcal{R}_{n,\\,red}(A)$.\n\\begin{theorem}\nThe above isomorphism gives\n$$TS^n_R(A)_{red}\\cong V_n(A)_{red}^G$$\ni.e. $$X^{(n)}_{red}\\cong \\mathcal{R}_{n,red}(A)\/\/G.$$\n\\end{theorem}\n\\begin{proof}\nThe injectivity follows passing to an algebraically closed field and\nthen observing that a tuple of commuting matrices can be put\nsimultaneously in upper triangular form. One can then reach a tuple\nof diagonal matrices via a one parameter subgroup of $G$. Hence if\nan invariant regular function is zero on a tuple of diagonal\nmatrices then $f=0$.\nSurjectivity comes from $TS^n_F F[x_1,\\dots,x_m]\\twoheadrightarrow TS^n_FA$ and Cor.4.1 in {\\cite{v3}}\nthat states that the morphism $F[Mat(n,F)]^G\\to  TS^n_F F[x_1,\\dots,x_m]$ induced by restriction to diagonal matrices is onto for any commutative ring $F$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:Intro}\n    \n        Quantum computer hardware is quickly developing~\\cite{de2021materials} together with the theory of quantum computing algorithms~\\cite{scaling}. Problems historically solved classically have been translated into quantum computing formulations~\\cite{bhaskar2015quantum, ANDERSSON2022100754, novel,container}. Problems formulated for a particular quantum computing hardware have a completely different formulation for different hardware. In this work we attempt to understand how well constraint optimization can be implemented between two well known quantum computing approaches, (i) quantum annealing, and (ii) the gate-based model, that reflect on two problem formulations: Quadratic Unconstrained Binary Optimization (QUBO) and Quantum Approximate Optimization Algorithm (QAOA). The goal of this paper is to assess the performance of the algorithm when solving a particular problem related to Mission Covering Optimization (MCO).\n        \n        MCO is a variant of the Generalized Assignment Problem\\cite{gap}. The goal is to perform a set of missions given constraints on the available resources. Missions are organized efforts to achieve an objective. Examples of mission types include industrial, logistic, and militaristic, where the problem can be anything from getting packages to destinations, to efficiently producing and maintaining equipment devices~\\cite{Venturelli2016JobSS, Asset_Sustainemnt}. To assure that a mission is successful, a set of resources performing different tasks are used. The global solutions of such problem is computationally challenging. It is a complex optimization problem~\\cite{Ng2022}, often non-convex. Heuristic approaches are used to find feasible solutions, but they depend on domain knowledge and problem decomposition, leading to inaccurate solutions. So this category of problems is a good candidate for quantum acceleration. In this effort, we will address an abstraction of a real MCO problem.\n        \n        We formulate the problem for a quantum annealer, specifically the D-Wave quantum computer, as well as investigate formulation for a gate-based model approach, specifically the IBM quantum computer. Quantum annealer is a type of quantum discrete optimization algorithm that exploits the power of tunneling to reach a global minimum \\cite{qa_optimization}. Gate-based quantum computer solve problems applying a sequence of unitary operators to qubits in superposition.\n       \n       \n        From using results from the QPU and through simulations, we present strengths and weaknesses found with both models, and outline fundamentals of this new type of optimization problem. \n        \n        This paper is divided into 4 main sections. \\Sec{sec:Formalism} describes MCO in greater formal detail. \n        \\Sec{sec:MCO:Scenarios} outlines two different scenarios of MCO that are tested in this study. \n        \\Sec{sec:Alg:Impl}  details how each scenario is implemented using three different algorithmic techniques. \n        \\Sec{sec:Compare:Results} describes the results of each scenario and compares the different algorithmic techniques by cost, timing, and constraint-holding metrics.\n        In \\Sec{sec:Conclusion}, we summarize and conclude our findings of this work.\n        Lastly, \\Sec{sec:Future:Work} discusses research directions for future work.\n\n   \n   \n   \n\n   \n   \n   \n   \n   \n   \n                    \n                    \n                    \n                    \n   \n   \n\n    \\section{Formalism for Mission Covering Optimization}\\label{sec:Formalism}\n                \n        The objective of an MCO problem is to allocate a set of resources to missions such that each resource is assigned to at most one mission, and each mission's requirements are satisfied. \n        \n        A MCO problem is described by a 7-tuple{}: \n        \n        $$(\\MSet, \\RSet, \\QSet, \\capability, \\textsf{REQ}_{M}, \\textsf{REQ}_R, \\comment{\\textsf{PR},} f_{obj})$$\n          \n          \n        \\begin{enumerate}\n            \\item $\\mathbf{M}$ is the set of missions\n            \\item $\\mathbf{R}$ is the set of resources\n            \\item $\\mathbf{Q}$ is the set of qualifications\n            \\item $\\capability$ is the capability function\n            \\item $\\textsf{REQ}_{M}$ is the mission's requirements function\n            \\item $\\textsf{REQ}_R$ is the resource's requirements function\n            \\item $f_{obj}$ is the objective function that scores problem solutions      \n        \\end{enumerate}\n    \n        In the rest of this section these terms will be explained in detail.\n        \\subsection{Missions}\n    \n            Missions are operations to achieve specific results. They require resources for specific tasks. For example, a mission could be transporting children to school. The resources are the buses and the bus drivers. In manufacturing, a mission could represent the process of building an asset from the design to delivery. It includes design, engineering, sourcing, suppliers, production, control, and packaging. $N_M$ denotes the number of missions. In an MCO, missions need to be completed at the same time; we denote the mission set as $\\mathbf{M} =\\{m_1, m_2, \\ldots, m_{N_M}\\}$.\n    \n        \\subsection{Resources}\n            $\\mathbf{R}$ represents the set of all resources. Resources are all the assets needed for mission completion. Resources can be objects, machines or people. For example, a given MCO may define its resources as four planes, four pilots and two engineers. $\\mathbf{R} = \\{r_1, r_2, \\ldots, r_{N_R}\\}$ is the set of $N_R$ resources within the MCO.\n               \n        \\subsection{Qualifications}\n            $\\mathbf{Q}$ represents the set of all qualifications. Resources are specialized in the sense that they have qualifications and they can be assigned to the mission that requests a resource with such qualification. For example `Can transport', `Can pilot', and `Can troubleshoot' are qualifications a mission may require.  $\\mathbf{Q} = \\{q_1, \\ldots, q_{N_Q}\\}$ is the number of $N_Q$ qualifications in the MCO problem.           \n               \n        \\subsection{Capabilities}\n           $\\capability$ represents the capability function. Each resource is scored for its qualifications. Capabilities are integer number that represent how well a resource can perform a certain qualification. The higher the number, the more qualified the resource is. For example, consider the resource set composed by a senior engineer, intern engineer, and an HR Manager and with capabilities of 2, 1, 0 respectively to a certain qualification titled 'Troubleshoot carburetor'. This reflects the fact that the senior engineer has a higher capability than the intern engineer  to troubleshoot carburetor because he\/she is more experienced in the field. Conversely, the HR manager has no qualification capability to 'Troubleshoot carburetor', therefore has a score of zero. Capability is the function\n            $$\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\mathbb{Z}_{\\ge 0}},$$\n            which returns a zero if and only if the resource is not qualified for the specified qualification. When this happens, we say that this resource has no capability for that qualification.\n      \n            Missions require resources with specific qualifications. For example, mission $m_1$ requires $n$ resources with qualification $q_1$ where $m_1\\in\\mathbf{M}$,  $n\\in\\mathbb{N}$ and $q_1\\in\\mathbf{Q}$, which represents `mission 1 requires 10 resources that can fly a plane'. We assume that resources with higher capability for certain qualifications are chosen.\n     \n     \n        \\subsection{Mission's Requirements}\n            The mission's requirement function is described as     \n            $$\\fncType{\\textsf{REQ}_{M}}{\\mathbf{M}\\times\\mathbf{Q}}{\\mathbb{N}},$$\n            and represents the qualification requirement for each mission. This function returns the number of required resources that satisfy the qualification needed for that mission.\n        \\subsection{Resource's Requirements}\n            The resource's requirement function is described as     \n            $$\\fncType{\\textsf{REQ}_R}{\\mathbf{R}\\times\\mathbf{Q}}{\\mathbb{N}},$$\n            and represents the qualification requirement for each mission. This function returns the number of required resources that satisfy the qualification needed for that resource. This will be used on scenario 2.\n    \n        \\subsection{Solutions and Solution Score}\n            A solution of the MCO is defined as a function that associates resources to missions \n            $$\\fncType{f_s}{\\mathbf{R}}{\\mathbf{M}},$$\n            If $N_F$ is the number of functions that are the solutions of the MCO, then $\\mathbf{F}= \\{f_1, f_2, \\ldots, f_{N_F}\\}$ is the set of all possible solutions. The goal is to find the best solution in which all missions meet requirements as well as possible. The objective function maps solutions to real numbers, which measures how far off each solution is from meeting requirements. The objective function represents a cost function:\n            $$\\fncType{f_{obj}}{F_{s}}{\\mathbb{R}}$$ s.t. $$F_{s} = \\{\\{ (m,r) \\in \\mathbf{M} \\times \\mathbf{R} \\mid m = f_s(r)\\} \\mid \\fncType{f_s}{\\mathbf{R}}{\\mathbf{M}}\\}.$$\n            Therefore, the best solution to an MCO is achieved by minimizing the objective function and retrieving the minimizer.\n    \n    \\section{MCO Scenarios}\\label{sec:MCO:Scenarios} \n        The MCO optimization problem involves the parameters (missions, resources, mission require, resource require). From this, there are different metrics of cost that can be used to represent the objective function. Two specific MCO scenarios are formalized and implemented on different quantum computing hardware machines. These scenarios are not computationally difficult to compute classically. However, these scenario's design is intentional for comparing results in this study, as brute force methods for finding the absolute best cost-minimizers can be used to without being throttled by the exponential time complexity since the scenario's are designed to permutationally symmetric. The primary focus of each of these readily solvable MCO scenarios is to identity how well each algorithm implementation performs when constraints are introduced.  Results are compared in terms of cost, timing, and constraints met. In the general MCO formulation, resources can be assigned to at most one mission, so that resources could be unused. In this paper, in both scenarios, it is assumed that each resource is assigned to one and only one mission. This is done by introducing an additional mission that includes the unused resources. The following subsections describe each scenario.\n    \n        \\subsection{Scenario 1: Primary and Secondary Resources}\n            The first scenario involves two categories of resources: primary and secondary. Primary resources are ready to be used for mission completion. Secondary resources are allocated when primary resources are not able to perform their duty. Primary resources that cannot be allocated to missions are removed from the set of primary resources. For example, the mission covering optimization solution shown in figure~\\Fig{Figure_0} is composed by three missions requiring three, two and two pilots respectively for mission completion. Suppose in the planning phase that seven pilots were allocated to cover the three missions; they are on-duty (in the primary resource set), and three other pilots are on-call (in the secondary resource set). Suppose at a certain point in time, two of the pilots got sick and they were removed from the resource group. This is an emergency, unforeseen situation that requires to re-run an optimization algorithm to cover the missions including secondary resources. The pilots are pulled from the set of five pilots on-duty, ready to cover the missions and two pilots from the secondary resource set should be also included. The challenge is to find the best allocation of pilots to missions using all the pilots on-duty and two pilots on-call. The goal of our effort is to use quantum computing to solve this problem and analyse results from different quantum computing hardware implementations. \n            \n            The following outlines all the rules in this specific scenario:\n    \n            \\begin{enumerate}\n                \\item The set of missions in the problem is $\\mathbf{M} =\\{m_1, \\ldots, m_{N_m-1}, U\\}$.\n                \\item The set of resources is $\\mathbf{R} = \\{r_1, \\ldots, r_{N_r}\\}$.\n                \\item There only exists one qualification, which is the ability to fly a plane, $\\mathbf{Q} = \\{q_1\\}$. \n                \\item Resources can have a capability of 1 or 2. So the capability function is then $\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\{1, 2\\}}$. This is a way to represent how ready the resource is to be allocated to a mission. It can be though of as an ordering for the allocation of resources; resources with capability 2 should be allocated before ones with 1. Resources that have capability 2 are referred to as primary resources, which are assigned to missions first. Resources that have capability 1 are referred to as secondary resources, which are assigned to missions only if primary resources become unavailable.\n                \\item $\\textsf{REQ}_{M}$ is the mission's requirement.\n                For example:\n                \\begin{itemize}\n                    \\item $\\textsf{REQ}_{M}(m_1, q_1)=3$.\n                    \\item $\\textsf{REQ}_{M}(m_2, q_1)=2$.\n                    \\item $\\textsf{REQ}_{M}(m_3, q_1)=2$.\n                \\end{itemize}\n                \\item There are no resource requirements. Therefore, the resource requirement function is $$\\fncType{\\textsf{REQ}_R}{\\mathbf{R}\\times\\mathbf{Q}}{\\{0\\}}.$$\n                \\item $f_{obj}$ is the function used to score the different problem solutions in terms of cost\n            \\end{enumerate}\n    \n            The mission cost includes two parts:\n            \\begin{itemize}\n                \\item The mission cost represents how many of the mission's requirements are not met. It is a penalty introduced every time a mission is not able to accomplish its goal due to a lack of resources.\n                \\item The precedence cost measures how well each solution allocates resources with higher capabilities before others to missions. In the specific example, it means that it is desirable that primary pilots are allocated to missions before secondary pilots.\n            \\end{itemize}\n                \n    \n            As discussed previously, MCO covers a broad spectrum of problems distinguishable only by it's objective function. In the scenario described here, the objective function reflects the cost which is the sum of the mission cost and precedence cost. All solutions are measured in terms of the cost in order to find the optimal solution. This scenario can be described by using a matrix arrangement of Boolean variables, shown in \\Fig{Figure_0}. Each row of the table in the figure represents a mission, and each column represents a resource.\n    \n            \\fig{Figure_0}{\\textbf{Matrix view for Solution}: This represents a table representing the solution of a problem with three missions, five primary resources and three secondary resources. A purple circle symbolize a Boolean variable $x_{m, r}\\in\\{0,1\\}$ where $m$ represents the mission (row) while $r$ represents the resource (column). A circle filled in with orange represents $x_{m, r}=1$, and the fact that resource $r$ is allocated to mission $m$.}\n            \n            The final row of the table represents an artificial mission created for the purpose of having all the resources utilized. When resources are not allocated to any of the other missions, it has to be allocated this special mission. It will be referred to as the unallocated mission (U mission for short) and has no mission cost associated.\n      \n                \n            A separate column specifies the required amount of qualifying resources for each mission. Since there is only one qualification in the scenario, this number just represents how many resources should be allocated to the mission. The example in \\Fig{Figure_0} shows a solution where each mission satisfies its requirements, but only after using all primary resources before it uses any of the secondary ones. Since each resource can only be assigned to one mission, no more than one Boolean variable may be true in a single column of the matrix representation of a MCO. In general, the column constraint assures that a resource can be assigned at most to one mission. This is a hard constraint. \n        \n        \\subsection{Scenario 2: The Buddy System}\n            \n            This scenario was designed to see how different algorithms deal with additional constraints. There are two groups of resources of different types. If a resource from one group is assigned to a mission, it is required that a resource from the other group joins the same mission. Consider the previous example where a set of planes and pilots must be allocated to a set of aerial missions. A pilot and plane are required to complete a mission. And every time a plane is chosen, a pilot needs to be chosen. From an application standpoint, this scenario highlights the modeling of resource dependencies (planes depend on pilots for allocation). This scenario contains an additional row constraint. The list of constraints is as follows:\n            \\begin{itemize}\n                \\item Column constraint: a resource can be assigned at most to one mission. This is the hard constraint of the problem.\n                \\item Row constraint: if a type of resource is chosen then a resource of the second type must also be chosen. We call it the {\\it buddy} constraint. This is the additional constraint added in this scenario.\n            \\end{itemize}\n    \n    \n            The entire set of rules for this scenario is as follows:\n            \\begin{enumerate}\n                \\item The set of missions in the problem is $\\mathbf{M} =\\{m_1, \\ldots, m_{N_m-1}, U\\}$.\n                \\item The set of resources is $\\mathbf{R} = \\{r_1, \\ldots, r_{N_r}\\}$.\n                \\item There are two qualifications $\\mathbf{Q} = \\{q_1, q_2\\}$, which means that resources are divided in two groups: $\\mathbf{R}_1$ and $\\mathbf{R}_2$.\n                \\item Resource's capability is $\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\{0, 1\\}}$, which means that all resources that are qualified have the same capability. Resources have only one qualification:\n                \\begin{itemize}\n                    \\item The set of all resources that have capability 1 for qualification $q_1$ is notated as $\\mathbf{R}_1$.\n                    \\item The set of all resources that have capability 1 for qualification $q_2$ is notated as $\\mathbf{R}_2$.\n                    \\item The sets $\\mathbf{R}_1$ and $\\mathbf{R}_2$ do not contain the same resource: $\\mathbf{R}_1\\cap\\mathbf{R}_2 = \\{\\}$.\n                    \\item The sets $\\mathbf{R}_1$ and $\\mathbf{R}_2$ contain all resources: $\\mathbf{R}_1\\cup\\mathbf{R}_2 = \\mathbf{R}$.\n                    \\item The number of resources in $\\mathbf{R}_1$ and $\\mathbf{R}_2$ are the same $|\\mathbf{R}_1| = |\\mathbf{R}_2|$.\n                \\end{itemize}\n                \\item Missions require resources with qualification $q_1$ and not $q_2$. Therefore, the mission require function can be formally described as:  $\\fncType{\\textsf{REQ}_{M}}{\\mathbf{M}\\times\\{q_1\\}}{\\mathbb{Z}_{\\ge 0}}$.\n                \\item The resource's requirement is called the \\textit{buddy requirement}. Every resource when allocated requires another resource with the opposite qualification.\n                \\begin{itemize}\n                    \\item $\\textsf{REQ}_R(r, 1) = 0 \\And \\textsf{REQ}_R(r, 2) = 1 \\quad \\forall r\\in\\mathbf{R}_1$\n                    \\item $\\textsf{REQ}_R(r, 1) = 1 \\And \\textsf{REQ}_R(r, 2) = 0 \\quad \\forall r\\in\\mathbf{R}_2$\n                \\end{itemize}\n                \\item As in the previous scenario, the objective function measures the total mission cost.\n            \\end{enumerate}\n                \n            The mission cost formulation is exactly the same as the secondary resources scenario. However, the way we view the problem in its matrix representation is slightly different, as shown in \\Fig{Figure_7}. \n            \n            \\fig{Figure_7}{\\textbf{Matrix view for Solution}. This solution is shown for a problem with 3 missions, four resources of type $\\mathbf{R}_1$ and four resources of type $\\mathbf{R}_2$. $\\textsf{REQ}_{M}$ describes how many $\\mathbf{R}_1$ resources must be assigned to mission $m$. Each $\\mathbf{R}_1$ resource is paired with exactly one $\\mathbf{R}_2$ resource for each mission.}\n            \n            The primary difference between this solution and the prior scenario's solution is that now there are \\textit{buddy} constraints present along the rows of the matrix (for this reason it's also called the \\textit{row} constraint). For example, in \\Fig{Figure_7}, the first row related to the first mission has two resources allocated from $\\mathbf{R}_1$. In order for the buddy constraint to not be violated, two resources from $\\mathbf{R}_2$ are allocated to that mission. The solution shown is valid since all no rows or columns violate any of the constraints. The solution also has the lowest mission cost since the mission requirements were all exactly met.\n            \n            Finding optimal solutions for this scenario is trivial, since all resources within the same group are indistinguishable from each other in terms of allocation cost. However, when adding different capabilities to resources within this MCO, the complexity of finding the optimal solution increases. The case where resources have different and multiple capabilities has not been studied and is part of future work. While this does not change substantially how the problem is formulated and implemented on the quantum device, it is more complex to check how well the quantum algorithms solutions are compared to optimal solution. To do this, the optimal solution needs to be known, so brute force methods are used to find it. If resources are indistinguishable from one another, then it takes far less time to brute force to the optimal solution due to the permutation symmetry across resources.\n\n    \\section{Algorithm Implementation}\\label{sec:Alg:Impl}\n        In this work we tested three techniques to solve the MCO problem with the two scenarios described in the previous section, quantum annealing and two types of QAOA algorithms. \n        \\begin{itemize}\n            \\item Quantum Annealing (QA)(\\cite{qa}\n            \\item Quantum Alternating Operator Ansatz (QAOA) \\cite{qaoa}. It implements constraints by means of Lagrange multiplier embedded into a cost function Hamiltonian.\n            \\item QAOAH which is a version of QAOA with a constrained mixer. It is denoted as QAOAH since it was developed by Hadfield \\textit{et al.} \\cite{qaoa_had}. It engineers the constraints to remain within a constraint space during the entire solution process.\n        \\end{itemize}\n        All three approaches use similar means for encoding the objective function but differ in the way they implement their constraints and what machines support them.\n\n        \\subsubsection{Quadratic Unconstrained Binary Optimization}\n\n            Quantum Annealing is an quantum optimization methodology performed on a specific implementation of quantum computers such as D-Wave quantum annealing machines. Quantum Annealing is an adiabatic quantum computing technique that capitalizes on a unique feature of quantum mechanics, quantum tunneling, which is the capability to surf and dive though a energy landscape until it hits a minimum energy level\\cite{qa_optimization}. This property of quantum mechanics is engineered to solve Quadratic Unconstrained Binary Optimization problems\\cite{QUBO}. QUBO, is a type of optimization where the solution is a binary vector that minimizes an objective function described with terms up to a quadratic degree. These problems are unconstrained, but there are methods to incorporate the behavior of constraints\\cite{QUBO_CONST} into the cost function. To encode an MCO onto a quantum annealing machine, we must translate the objective function into a QUBO.\n        \n        \\subsubsection{Quantum Alternating Operator Ansatz}       \n            Quantum Alternating Operator Ansatz QAOA is method for solving combinatorial optimization problems on NISQ devices\\cite{qaoa}. The algorithm is supported by gate-based quantum computers such as the IBM, Rigetti, IonQ or Xanadu machines. QAOA In this effort we focus on implementing the problem on an IBM quantum computer~\\cite{QAOA_QISKIT}. At the current time, IBM's Qiskit implementation of QAOA uses Variational Quantum Eigensolver (VQE) to find the expectance of a parameterized ansatz eigenstate. This quantity is then used to calculate the minimum of the cost function which is embedded in a cost Hamiltonian $H_C$. Like QA, it is limited to solving binary quadratic problems\\cite{PhysRevApplied.5.034007}.\n        \\subsubsection{Quantum Alternating Operator Ansatz with Constraints (QAOAH)}        \n            In QAOA, the starting state is in an equal superposition of all computational basis states, and the mixing Hamiltonian is a sum of Pauli-X operators acting on each qubit. In this configuration, every possible classical-state solution can be traversed to making it ideal for unconstrained problems where there are no states to be filtered out. However, it is possible to incorporate constraints by altering the default mixing Hamiltonian $H_M$ and the initial quantum state $\\ket{\\psi}$\\cite{qaoa_had}. Since the mixing operator (the exponentiated mixing Hamiltonian) specifies how to explore the solution space, it is possible to specify a mixing operator that determines how to move to another solution within the constraints of the problem.\n        \n            Consider an initial quantum state $\\ket{\\psi}$ which represents a solution that does not violate the constraints of the optimization problem, but is not necessarily the minimizer. Consider the mixing operator $H_M$ that is constructed such that after an application on a state within the constrained space, the output state is also guaranteed to be within the constrained space. The idea is to construct $H_M$ and $\\ket{\\psi}$ so that when stopping QAOA at any iteration, the final encoded solution is always within the constrained solution space. If this mixing operator can mix such that it allows the algorithm to reach every possible classical-state in the constrained solution space, it is then possible to obtain the minimizer in that same space \\cite{qaoa_had}.\n        \n            Since QA and QAOA are both derived from the same formulation, the implementation details are described together. In QAOAH, constraints are embedded into the Mixing Hamiltonian which brings to a different formulation. The following sub-sections outline the implementation for both scenarios using the three approaches described above.\n\n        \n        \\subsection{Scenario 1 (One Constraint)}\n            \\subsubsection{QA and QAOA}\n                The goal of a MCO problem is to minimize the total cost. Various ways are used to define mission cost. The following formulation takes into account limitations of QUBO. In the rest of the paper $QA$ denotes QUBO problems performed on quantum annealing machines while QAOA and QAOAH denote QUBO problems executed on gate-based architectures.\n                \n                The optimal mission cost occurs when all the resources required are allocated. Or, formally, consider mission $m$; the optimal mission cost occurs when:\n                \\be{zi3GOCbzYwWHuuutPmGK}\n                    \\sum\\nolimits_{r \\in \\mathbf{R}} x_{m,r} = \\textsf{REQ}_{M}(m, 1).\n                \\end{equation}\n                \n                The $x$ values represent Booleans (an alternative representation of the solution $f_s$), such that when indexed by $m\\in\\mathbf{M}$ and $r\\in\\mathbf{R}$, e.g. $x_{m, r}$, it represents whether or not the solution mapped resource $r$ to mission $m$.\n                \n                As an example, suppose there are three resources, $\\mathbf{R} = \\{1, 2, 3\\}$ and the first mission requires two of them, then the optimal mission cost for the mission occurs when:\n                \\be{dFx-f3K0sdSgXNW7_ilp}\n                    x_{1,1} + x_{1,2} + x_{1,3} = 2.\n                \\end{equation}\n                \\Eq{dFx-f3K0sdSgXNW7_ilp} is true when 2 out of the 3 variables are true. The mission cost represents the penalty added any time a mission lacks of one or more needed resources. The penalty is higher when more resources are missing. A squared error is used to represent the mission cost:\n                \\be{5onSFKdjQxjrP-eN0BfE}\n                    \\textsf{MC}(x, m) = \\left(\\sum\\nolimits_{r \\in \\mathbf{R}} x_{m,r} - \\textsf{REQ}_{M}(m, 1)\\right)^2,\n                \\end{equation}\n                where $\\textsf{MC}(x, m)$ is the cost associated to mission $m$. It can be seen from \\Eq{5onSFKdjQxjrP-eN0BfE} that the minimum mission cost, when $\\textsf{MC}(x, m) = 0$, yields the optimal case as described in \\Eq{zi3GOCbzYwWHuuutPmGK}. The mission cost is quadratic. The example that yielded the optimal case in \\Eq{dFx-f3K0sdSgXNW7_ilp} in terms of the mission cost function is:\n                \\be{EeWWAXtMbm09rvIbOAEE}\n                    \\textsf{MC}(x, 1) = \\left( x_{1,1} + x_{1,2} + x_{1,3} - 2\\right)^2.\n                \\end{equation}\n                \n                For this optimization problem, secondary resources should be allocated only after primary resources are allocated first. Therefore, we introduce precedence cost for resources. \\Eq{lPsZgJ1LWm9RbsTBps8B} shows the ideal condition regarding the allocation of secondary resources.\n                \\be{lPsZgJ1LWm9RbsTBps8B}\n                    \\sum\\nolimits_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}} x_{m, r} = \\capability(r, 1) - 1 \\quad\\forall r\\in\\mathbf{R}\n                \\end{equation}\n                The precedence cost is dependant on the capability of a resource, and therefore dependant on whether or not this resource is primary or secondary. When the resource is primary, the ideal condition means that the it must be allocated to one of the active missions (apart from the unallocated\/psuedo mission). For the other secondary resources, ideally none are used.  \n                \n                The precedence cost may or may not be met when optimizing, so just like mission cost, precedence cost is the squared error of this equality:\n                \\be{n_F7J1oYAc7WIuRr5Nme}\n                    \\textsf{PC}(x, r) = \\left(\\sum\\nolimits_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}} x_{m, r} - \\capability(r, 1) + 1\\right)^2.\n                \\end{equation}\n                \n                Finally, the objective function is then the total mission cost and the total precedence cost in the MCO:\n                \n                \\be{vB-eQejEL5xyYQUiOqbr}\n                    f_{obj}(x) = \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\frac{1}{|\\mathbf{R}|}\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{PC}(x, r).\n                \\end{equation}\n                \n                We weight the precedence cost by $\\frac{1}{|\\mathbf{R}|}$ to ensure that the mission cost is minimized before precedence cost. The total cost is the sum of the mission cost and the precedence cost and it is reflected in the objective function.\n                \n                \n                \n                The constraint that a resource must be paired to exactly one mission is formulated:\n                \\be{_0nqf1oq3S9d62ZOk3Pg}\n                    \\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} = 1 \\quad\\forall r\\in\\mathbf{R}.\n                \\end{equation}\n                The constraint function used for this scenario is defined as \n                \n                \\be{0FakCSC5k-YoFIaG_KAN}\n                    \\textsf{CONSTR}(x, r) = \\left(\\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} - 1\\right)^2.\n                \\end{equation}               \n               \n                Since the QUBO is used to solve problems without constraints, we must add it to the objective function so that when it minimizes, the constraints will be met. The method of Lagrange multipliers is a strategy for finding the local maxima and minima of functions subject to equality constraints. \n                \n                If $f_{obj}(x)$ is the objective function to be minimized, the lagrangian function is\n                \\be{bHljve4SovW75-5yEWxd}\n                    \\mathcal{L}(x, \\lambda) = f_{obj}(x) + \\lambda\\cdot\\textsf{CONSTR}(x).\n                \\end{equation}\n                And the solution to the original constrained problem is always a saddle point of this function. Setting a large value for $\\lambda$, the term related to the constraint, will have the greatest impact on the optimization problem. And the solution will minimize the constraint first and then the cost.\n                \n                The new objective function that includes the constraints is:\n                \\be{8Mu4WZnVmok5H4uWjbmn}\n                    \\begin{split}\n                        f_{obj}(x) &= \\mathcal{L}(x, \\lambda) \\\\\n                                          &= \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\frac{1}{|\\mathbf{R}|}\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{PC}(x, r) \\\\\n                                          &\\quad\\quad+ \\lambda\\cdot\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{CONSTR}(x, r).        \n                    \\end{split}\n                \\end{equation}\n                \n                QAOA for this method uses an equal superposition for the starting state $\\ket{\\Psi}$ over $N$ states:\n                \n                \\be{R36C6d1UqdqSZIrDXMcw}\n                    \\ket{\\Psi} = \\frac{1}{\\sqrt{N}}\\sum_{i=0}^{N-1}\\ket{i}.\n                \\end{equation}\n                \n                The value $N$ is equal to $2^n$ where $n$ is the number of qubits used. For this problem, the number of qubits used is $N_{M}\\cdot M_{R}$, the number of missions times the number of resources.\n                \n                The mixing operator is constructed using a Hamiltonian, which is the sum of Pauli-X as follows:\n                \\be{vZTjtWDfeNFtO0FXr-a2}\n                    \\begin{matrix}\n                        \\\\\n                        X_i = & I & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & I \\\\\n                            &  1  &  &  i & & n\n                    \\end{matrix},\n                \\end{equation}\n                \\be{Iyl7tZpP2CXZiW4FHcok}\n                    H_M = \\sum_{i=1}^{n} X_i.\n                \\end{equation}\n                \n            \\subsubsection{QAOAH}\n                In the last section, Lagrange multipliers are used to encode constraints into the QUBO problem. Alternatively, by choosing the appropriate mixing Hamiltonian and initial state, we can constrain the solution space outside of the QUBO formulation in QAOA\\cite{qaoa_had}.\n                \n                The initial state must be within the constrained solution space. A trivial starting configuration that is known not to violate the constraint is when all resources are set to the unallocated mission as shown in \\Fig{Figure_9}.\n                \\fig{Figure_9}{\\textbf{Example Initial State:} All resources are initialized to be unallocated, or allocate to mission $U$, the last row.}\n                \n                This initial state used is:\n\n                \\be{lpOGOEbsbF-qt7v0fJUy}\n                    \\ket{\\psi} = \\bigotimes_{(m, r)\\in\\mathbf{M}\\times\\mathbf{R}} \n                    \\begin{cases} \n                        \\ket{1} & m = U \\\\\n                        \\ket{0} & m \\neq U\n                    \\end{cases}\\quad.\n                \\end{equation}\n\n                The mixing Hamiltonian describes how to move from the starting state, as well as all subsequent states, such that they are also in the constrained space. To be in the constrained space only one qubit per column must be active. The Hamiltonian should describe how to cycle a qubit in active state throughout the column so that it can reach every possible combination of configurations that still satisfy the constraint. The identity operator and the $\\textsf{SWAP}$ gates are used for this cycling action. For the 3 mission example, we can confirm that an individual column can have each of it's possible states reached using 3 swap gates (see \\Fig{Figure_8}).\n                \\comment{TODO: Talk about the possibility of using all permutation of the SWAP gate in a column.}\n                \\fig{Figure_8}{\\textbf{Resource 1's Mixing Operators:} The beginning scenario, (leftmost column) details the starting state before applying the mixing operator. The second scenario shows a identity mixing operator 2nd column from left which allowed the current state to remain unchanged. The 3 right-most scenarios each use one swap to cycle qubit's \\textbf{on} state to missions 1,2, and 3}\n                \n                Thus, a single resource mixing $\\textsf{MIX}(r)$ is:\n                \n                \\be{x5oRpxylQTEhdYB0w4Xw}\n                    \\textsf{MIX}(r) = \\sum_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}}{\\textsf{SWAP}_{(U,r),(m, r)}},\n                \\end{equation}\n                \n                where $(m, r)$ encodes the index of the qubit representing a mapping of resource $r$ and mission $m$. The $U$ in $(U, r)$ represents the unallocated mission. The total mixing operator $H_m$ is the sum of $\\textsf{MIX}(r)$ on each resource with the identity operation: \n                \n                \\be{smK9UzTkSju8UiVCq-bx}\n                    H_m = I^{\\otimes n} + \\sum_{r\\in\\mathbf{R}}{\\textsf{MIX}(r)}.\n                \\end{equation}\n                \n                In order to embed the mixing operation as a Hamiltonian to run on the IBM machines, it must be described as a composition of tensored Pauli gates. Each $\\textsf{SWAP}$ gate can be decomposed in terms of Pauli gates:\n                \n                \\be{Ba0eQI2c0iOaj6eSiUhJ}\n                    \\textsf{SWAP} = \\frac{1}{2}\\left(I\\otimes I + X\\otimes X + Y\\otimes Y + Z\\otimes Z\\right).\n                \\end{equation}\n                \n                When $\\textsf{SWAP}$ is indexed by $i$ and $j$, the Pauli gates fall on the $i^{th}$ and $j^{th}$ qubits respectively:\n                \n                \\be{NIl6X5cf18D9nG3spUZy}\n                    \\textsf{SWAP}_{i, j} = \\frac{1}{2}\\left(I^{\\otimes n} + X_{i, j} + Y_{i, j} + Z_{i, j}\\right),\n                \\end{equation}\n                \n                \\be{sIsNHYzXiF_r_TB4fqeK}\n                    \\begin{matrix}\n                        \\\\\n                        X_{i, j} = & I & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & I \\\\\n                            & 1 & & i & & j & & n\n                    \\end{matrix}\\quad.\n                \\end{equation}\n                \n                Each Pauli gate used in \\Eq{NIl6X5cf18D9nG3spUZy} ($X_{i, j}$, $Y_{i, j}$, $Z_{i, j}$) is defined similarly to what is defined in \\Eq{sIsNHYzXiF_r_TB4fqeK}. For clarity, \\Eq{sIsNHYzXiF_r_TB4fqeK} places the corresponding Pauli gate only at the specified indicies $i$ and $j$ in a tensor product of identities $I$.\n        \n        \\subsection{Scenario 2 (Two Constraints)}\n            \\subsubsection{QA and QAOA}\n                In this scenario, the objective function measures just mission cost, as opposed to the previous scenario that also measures precedence cost. Therefore, the objective function is the sum of mission cost and the constraint function:\n                \n                \\be{NpmMn5bAksngEsR1UKBE}\n                    f_{obj}(x) = \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\lambda\\cdot\\textsf{CONSTR}(x).\n                \\end{equation}\n                \n                Two different constraints are embedded into the objective function using Lagrange multipliers. The first one was discussed in the previous scenario assures that resources are allocated to no more than one mission. It is formulated in a similar way as before:\n                \n                \\be{LUEPhowwvgU8g5_4BDtn}\n                    \\textsf{CONSTR}_1(x, r) = \\left(\\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} - 1\\right)^2.\n                \\end{equation}\n                \n                \n                The addional constraint (buddy constraint) requires that the amount of resources allocated from set $\\mathbf{R}_1$ must be the same as the number of resources allocated from set $\\mathbf{R}_2$. This hard equality is formulated as:\n\n                \n               \n               \n               \n                \n               \n                \n                \\be{WlIgzOycMu3ESpVQt26I}\n                    \\textsf{CONSTR}_2(x, m) = \\left(\\sum_{r\\in\\mathbf{R}_1} x_{m, r} - \\sum_{r\\in\\mathbf{R}_2} x_{m, r}\\right)^2.\n                \\end{equation}\n                \n                The total constraint function expressed in \\Eq{NpmMn5bAksngEsR1UKBE} can be expressed as the sum of both of these constraints:\n                \n                \\be{gm0YwnJ1TSQjD78B5eVR}\n                    \\begin{split}\n                        \\textsf{CONSTR}(x) = \\sum_{r\\in\\mathbf{R}}\\textsf{CONSTR}_1(x, r) \\\\ + \\sum_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}}\\textsf{CONSTR}_2(x, m)\n                    \\end{split}\\quad.\n                \\end{equation}\n                \n                For QAOA, the starting state and the Mixing Hamiltonian are the same as defined in \\Eq{R36C6d1UqdqSZIrDXMcw} and \\Eq{Iyl7tZpP2CXZiW4FHcok}.\n                \n            \\subsubsection{QAOAH}\n                For this scenario, it is more challenging to construct the mixing Hamiltonian to describe how to move in the constrained space in QAOA. Unlike the Lagrange multiplier's case, it is not easy to linearly combine two constraint encodings to get the final constraint Hamiltonian. In other words, one cannot add $H_{m1} + H_{m2}$ to get $H_{m1+m2}$, where $H_{m1}$ $H_{m2}$, $H_{m1 \\& m2}$ are mixing Hamiltonian's representing $1^{\\text{st}}$, $2^{\\text{nd}}$ and $1^{\\text{st}}$ \\& $2^{\\text{nd}}$ constraints respectively. This scenario presents two challenges:\n                \\begin{enumerate}\n                    \\item If a mixing operator allowed a resource in $\\mathbf{R}_1$ to move from an unallocated state to an allocated state by pairing it with a mission, then it must also move a resource from $\\mathbf{R}_2$ to the same mission\n                    \\item the mixing operator must operate such that it is possible after multiple applications to visit every classical state from the starting quantum state. These two obstacles require a slightly different mixing operator and more qubits.\n                \\end{enumerate}\n                \n                The strategy for creating a two-constraint mixing operator is to reallocate resources in pairs - one from $\\mathbf{R}_1$ and one from $\\mathbf{R}_2$. These pairs will always move together. However, the problem is that not every possible classical state can be produced from the mixing. For example, let's say that there are resources $\\mathbf{R}_1=\\{1, 2, 3, 4\\}$ and $\\mathbf{R}_2=\\{5, 6, 7, 8\\}$. If resource 1 and resource 5 move together, then it's never possible to see resource 2 just be paired with resource 5 without also being paired with 1. A way to resolve this problem is to introduce an additional mixing operator to swap entire columns within just $\\mathbf{R}_1$  or $\\mathbf{R}_2$ resources. However, every time columns are swapped, the classical state must remember what mappings are paired with which others so that if another reallocation is done, the buddy constraint won't be violated. \n                \n                For this reason new qubits are introduced to each column. These qubits represent the pair ID that is present in the columns for resources in $\\mathbf{R}_1$ and $\\mathbf{R}_2$. Resources with the same ID are reallocated together. When columns are swapped, the pair IDs of the columns are also swapped. \n                \n                Consider a MCO with 3 missions (plus the unallocated mission) and 8 resources evenly split between $\\mathbf{R}_1$ and $\\mathbf{R}_2$. Our initialized state is shown in \\Fig{Figure_10}. Each column has 2 extra ID qubits (lowest two rows) with a unique bit-encoding that matches another column from the opposite qualification type. This means that these columns are paired together.\n\n                \\begin{figure*}[ht]\n                    \\subfloat[\\label{fig:Figure_10}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_10.PNG}\n                    }\n                    \\hspace*{\\fill}\n                    \\subfloat[\\label{fig:Figure_11}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_11.PNG}\n                    }\n                   \n                    \\vspace{1mm}\n                    \\subfloat[\\label{fig:Figure_12}]{\n                      \\includegraphics[width=1\\columnwidth]{Figure_12.PNG}\n                    }\n                    \\hspace*{\\fill}\n                    \\subfloat[\\label{fig:Figure_13}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_13.PNG}\n                    }\n                        \n                    \\caption{\n                        (a) Initial State Layout. The initial state is similar to the QAOA case except with additional ID qubits to be used to support the constraint mixer. Each column of ID qubits form a unique binary number for that resource within its resource set.\n                        (b) First Control Dual-Swap Operation. Through mixing, resource 2 is allocated to mission 2. In order to not violate the row constraint, resource 6 from $\\mathbf{R}_2$ is allocated using a Dual-Swap gate with controls on the ID qubits.\n                        (c) Column Swap. The mixing operator can never produce the valid solution where just resource 4 and 6 are paired to mission 2 using just Controlled Dual-Swap Operations. Therefore, column-swap operations are permitted, which swap any two columns in the table within its resource set. Here, resource 2 and 4 in resource set $\\mathbf{R}_1$ have there entire columns swapped.\n                        (d) Second Control Dual-Swap Operation. This operation is exactly the same as (b), but using the Control Dual-Swap operator to unallocate resource 4 from mission 2 to mission $U$. Notice that the Control Dual-Swap operator has its control configuration identical on both resource 4 and 6 ID-qubit columns; this is to ensure that the row constraint cannot be violated. Violation occurs when leaving resource 6 with no pair on mission 2.\n                    }\n                    \\label{fig:29udh23923nd}\n                \\end{figure*}\n                \n                For example, consider the case where the mixer operation reallocates resource 2 from the unallocated mission to mission 2. Since this column has an ID of `$10_{\\textsf{b}}$` (top ID qubit true, bottom one false) it is paired with resource 6 because it has the same ID. In order to respect this pairing, both resource 2 and 6 are swapped together using a dual swap gate $\\textsf{C-DSWAP}$ as shown in \\Fig{Figure_11}.\n                \n               Consider the situation when the mixing operator performs a column swap between columns 2 and 4 (see \\Fig{Figure_12}). Note that the IDs of these columns are also swapped.\n                %\n                If the mixer chooses to move resource 6 back to the unallocated state, it also would move resource 4 into the unallocated state since they have the same ID. This can be ensured if the mixer uses the dual swap operation once again (see \\Fig{Figure_13}).\n                \n                \n                The added qubits to represent the IDs of each of the columns are notated as $\\ket{\\textsf{ID}}$ and defined as:\n                 \n                \\be{L3eP8k09CpNEzp75j3vG}\n                    \\ket{\\textsf{ID}} = \\bigotimes_{j_1=0}^{\\textsf{ID}_{max}}\\ket{j_1} \\otimes \\bigotimes_{j_2=0}^{\\textsf{ID}_{max}}\\ket{j_2}.\n                \\end{equation}\n                \n                                \n                The starting state $\\ket{\\psi}$ is:\n                \n                \\be{OjfQ5En0Z3rMRB-BC7c6}\n                    \\ket{\\psi} = \\ket{\\textsf{ID}} \\otimes \\bigotimes_{(m, r)\\in\\mathbf{M}\\times\\mathbf{R}} \n                    \\begin{cases} \n                        \\ket{1} & m = U \\\\\n                        \\ket{0} & m \\neq U\n                    \\end{cases}\\quad,\n                \\end{equation}\n                \n                and the mixing Hamiltonian $H_m$ becomes\n                \n                \\be{eTDqpOkD5RxRJqJRkGde}\n                    \\begin{split}\n                        \\begin{aligned}\n                            H_m = & \\sum_{p\\in\\mathbf{R}_1\\times\\mathbf{R}_2\\times\\mathbf{M}}\\sum_{j=0}^{\\textsf{ID}_{max}}\\textsf{C-DSWAP}(p, j) \\\\ \n                                  & + \\frac{1}{2} \\sum_{r_1\\in\\mathbf{R}_1}\\sum_{\\substack{r_1'\\in\\mathbf{R}_1 \\\\ r_1\\neq r_1'}}\\textsf{COL-SWAP}(r_1, r_1') \\\\ \n                                  & + \\frac{1}{2} \\sum_{r_2\\in\\mathbf{R}_2}\\sum_{\\substack{r_2'\\in\\mathbf{R}_2 \\\\ r_2\\neq r_2'}}\\textsf{COL-SWAP}(r_2, r_2')\n                        \\end{aligned}\n                    \\end{split}\\quad.\n                \\end{equation}                \n                \n \n                The constant $\\textsf{ID}_{max}$ represents the maximum required binary states to represent all columns. This is the number of resources in either $\\mathbf{R}_1$ or $\\mathbf{R}_2$  and it is represented by $\\textsf{ID}_{max}$:\n                \n                \\be{objg8PD2X1Euux-XNdH8}\n                    \\textsf{ID}_{max} = |\\mathbf{R}_1| = |\\mathbf{R}_2|.\n                \\end{equation}\n\n                The column-swap gate, notated as $\\textsf{COL-SWAP}(r, r')$, swaps the columns represented by resources $r$ and $r'$. Its decomposition is trivial as it employs many swaps tensored together.\n                \n                The control dual-swap, notated as $\\textsf{C-DSWAP}(p,j)$, has parameters $p$ and $j$. The parameter $p=(r_1, r_2, m)$ is a tuple composing of a resource from $\\mathbf{R}_1$, a resource from $\\mathbf{R}_2$, and a mission $m$ in $\\mathbf{M}$. This gate applies a control \n                %\n                %\n                %\n                swap gate to resource $r_1$ and $r_2$ between the mission $m$ and the unallocated mission U. The parameter $j$ is an ID which represents how to control the swap gate. For example, $10_\\textsf{b}$ is applying a control-true, control-false gate to both IDs in the columns represented by $r_1$ and $r_2$. \n\n\n                \n                To implement the algorithm on the IBM machine, we must decompose $H_m$ such that it is a sum of tensored Pauli gates. The column-swap gates $\\textsf{COL-SWAP}(r, r')$ can be decomposed, knowing that they are made up of swap gates, as from \\Eq{Ba0eQI2c0iOaj6eSiUhJ}.\n                \n                Decomposing a generalized version of the control dual-swap gate is tedious, so we provided an example decomposition for $\\textsf{C-DSWAP}(*, 10_\\textsf{b})$, which is the gate used in \\Fig{Figure_11} and \\Fig{Figure_13}. First, we present the Pauli-decomposition of the control-true and control-false unitary operations shown in \\Eq{ftKEkEa3NI2i0cp516l0} and \\Eq{PiyQA_TRJ5ErW6-h-hTo}, respectively. Unitary $A$ is arbitrarily acting on $m$ qubits.\n                \n                \n                \\be{ftKEkEa3NI2i0cp516l0}\n                    \\textsf{C-T}(A) = \\frac{1}{2}\\left(\\left(I+Z\\right)\\otimes I^{\\otimes m} + \\left(I-Z\\right)\\otimes A\\right).\n                \\end{equation}\n                \n                \\be{PiyQA_TRJ5ErW6-h-hTo}\n                    \\textsf{C-F}(A) = \\frac{1}{2}\\left(\\left(I-Z\\right)\\otimes I^{\\otimes m} + \\left(I+Z\\right)\\otimes A\\right).\n                \\end{equation}\n                \n                Now, $\\textsf{C-DSWAP}(*, 10_\\textsf{b})$ can be represented in terms of control-true and control-false unitaries and the dual-swap gate $\\textsf{DSWAP}$:\n                \n                \\be{qB_sgrFogjI-OAM0F7nF}\n                    \\begin{split}\n                        \\begin{aligned}\n                            \\textsf{C-DSWAP}&(*, 10_\\textsf{b}) = \\\\\n                            & \\textsf{C-T}\\left(\\textsf{C-F}\\left( \\textsf{C-T}\\left({\\textsf{C-F}\\left({\\textsf{DSWAP}}\\right)}\\right)\\right)\\right).\n                        \\end{aligned}    \n                    \\end{split}\n                \\end{equation}\n                \n                Following this, the dual-swap gate $\\textsf{DSWAP}$ is two swap gates tensored together:\n                \\be{4qKC5CJNtD0ez2KQLJ8e}\n                    \\textsf{DSWAP} = \\textsf{SWAP}\\otimes\\textsf{SWAP}.\n                \\end{equation}\n                \n                The $\\textsf{DSWAP}$ can further be decomposed using \\Eq{Ba0eQI2c0iOaj6eSiUhJ}.\n                \n                As mentioned in the previous section, our two-constraint MCO problem has permutation symmetry between the resources in the same set\/group. So the column swap terms can be effectively removed, and $H_m$ becomes\n                \n                 \\be{12345CJNtD0ez2KQ6789}\n                    \\begin{split}\n                        \\begin{aligned}\n                            H_m = & \\sum_{p\\in\\mathbf{R}_1\\times\\mathbf{R}_2\\times\\mathbf{M}}\\sum_{j=0}^{\\textsf{ID}_{max}}\\textsf{C-DSWAP}(p, j).\n                        \\end{aligned}\n                    \\end{split}\n                \\end{equation}       \n                \n                This effectively shrinks the search space from the total constraint space. However, because of the symmetry, it is known that the optimal solution still lays inside the smaller subspace. When different capabilities are introduced to each resource, this optimization technique cannot be done, since the permutation symmetry is not guaranteed.\n\n                \n    \\section{Analyses of Results}\\label{sec:Compare:Results}\n        In this section we compare the results of the different MCO implementations. Employing the implementation methods discussed above, the MCO problem was run on the D-Wave and on IBM machines, capturing several key metrics:\n\n        \\begin{itemize}\n            \\item Number of qubits \n            \\item Quantum processor time\n            \\item Cost\n            \\item Number of constraints violated\n        \\end{itemize} \n        \n        \\Tbl{tbl:results_table}(top) indicates the execution status (quantum hardware or simulation) of the\n       \n       \n        \\begin{table}[ht]\n            \\centering\n            \\includegraphics[width=3.0in,height=2.25in]{table_paul_done.png}\n            \\caption{\n                S1 and S2 stands for Scenario 1 and 2, respectively.\n                MCO implementations: \n                (top) Hardware\/simulation execution environment.\n                (bottom) number of constraints violated.\n            }\n            \\label{tbl:results_table}\n        \\end{table}\n        MCO algorithms, while \\Tbl{tbl:results_table}(bottom) shows the average number of constraints violated per implementation.\n    \n\n        \n        For both scenarios, 100 random MCO configurations were generated using up to 27-qubits (200 different MCO configurations in total). Quantum Annealing, QAOA, QAOAH, and Brute Force (BF) methods were run for each generated configuration. For Quantum Annealing, the DW\\_2000Q\\_6 machine was used, while ibmq\\_toronto, ibm\\_hanoi, ibm\\_cairo, ibmq\\_mumbai, and ibmq\\_montreal machines were used for running QAOA and QAOAH. The Lagrange multipliers were set to 5 for both scenarios, and a $p$-value of 2 is used for QAOA (this parameter is discussed in the original paper\\cite{qaoa}). The Quantum annealing runs each sampled the anneal 50 times, while each IBM job sampled the state-vector 1000 times. These parameters were chosen based off of a good balance of timing, cost, and constraints satisfied found by preliminary results not discussed in this paper.\n        \n        \\subsection{Scenario 1}\n           \n            \\Fig{Figure_15} shows the timing averages for QA, QAOA, and QAOAH, respectively, versus the problem qubit size.\n           \n            \\fig[.4]{Figure_15}{\\textbf{QPU Results} QPU Times for Scenario 1 Runs. QA timing includes the anneal time of each 50 samples, each 20 microseconds per sample. QAOA and QAOAH varied in the amount of jobs that ran, each of which ran the parameterized circuit 1000 times.}\n           \n           \n            \\begin{figure*}[ht]\n                \\subfloat[\\label{fig:Figure_14}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_14.PNG}\n                }\n                \\hspace*{\\fill}\n                \\subfloat[\\label{fig:Figure_16}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_16.PNG}\n                }\n                \\caption{\n                    \\textbf{QPU Results}\n                    (a) Average Relative Cost versus the number of qubits used. The relative cost is the mission plus the precedence cost obtained minus the lowest possible mission plus precedence cost that can be achieved (while keeping constraints) for the problem; The lowest possible cost is found by brute force: Relative Cost = Algorithm's Solution Cost - Brute Force Solution's Cost. Negative relative costs imply that the solution violated constraints.\n                    (b) Average number of constraints violated versus the number of qubits used. The number of constraints counts up how many extras qubit are on within a column in the matrix view representation. This count includes the amount of columns that do not have any qubits on at all.\n                }\n                \\label{fig:results_1}\n            \\end{figure*}\n           \n            Because the timing differed by different levels of magnitude, three y-axes with different scales are displayed. QA run time outperforms the other methods by having an overall constant run-time of 0.012 seconds, while QAOAH can use the QPU for 5 hours across all jobs in certain worse-case instances (this excludes queue-time and creation time on QPU for IBMQ devices). For all methods, timing was calculated based on qpu time (not wall-clock time). For the QA method, 'qpu\\_sampling\\_time' was used to calculate the total qpu time, while 'running time' is used for QAOA and QAOAH.\n            \n            \\Fig{results_1}(left) shows the average costs for all MCO configurations versus the number of qubits it took to encode each scenario 1 run. These costs do not include the cost accumulated from the embedded constraint functions \\Eq{0FakCSC5k-YoFIaG_KAN} and \\Eq{gm0YwnJ1TSQjD78B5eVR} as these are used to make the constraints hold. Also, all the costs plotted are actually the cost for that particular run minus the best possible cost it can receive. This best possible cost is computed by using brute force search methods in simulation. This difference is referred to as the relative cost. Therefore, the best possible relative cost a run could have is zero. For most runs, it can be seen that QA and QAOA runs have a relative cost around zero, but as the number of qubits increase, QAOA becomes less optimal compared to QA. The QAOAH approach, when using large amount of qubits, actually had a negative relative cost indicating that it must have violated some constraints in order to achieve a cost below the solution found with brute-force.\n\n            In \\Fig{Figure_16}, the average number of constraints violated is plotted against qubit size. The number of constraints is calculated by counting the amount of resources that were assigned to more than one mission. For example, if resource 1 was assigned to two extra missions, and resource 2 was assigned to three extra missions, then the number of constraints violated is 5. QA mostly had no constraints violated at any sized qubit problem, while QAOA and QAOAH suffered constraint violations when the problem size increased.\n\n        \\subsection{Scenario 2}\n            For scenario 2, the QAOAH method was calculated via IBM's state-vector numerical simulation tool due to the long running-time of runs in scenario 1. Because of this shift from running on actual hardware (scenario 1), to simulation (scenario 2), the QAOAH method is plotted using its wall-clock times against QA's \\& QAOA's QPU time in \\Fig{Figure_18}. Even with this change, the magnitudes of running-time are very diverse, so a third y-axis is added, as before. Furthermore, for the QAOAH approach, the mixing operator in \\Eq{12345CJNtD0ez2KQ6789} is used instead of \\Eq{eTDqpOkD5RxRJqJRkGde} because removing the column-swap terms reduces the total gate count of the overall computation, making simulation times faster.\n            \n            In scenario 2, QA times are quite faster than QAOA methods. It can be seen that the QAOAH method has far less flexibility in terms of qubit-range. This is because in our MCO algorithm implementation, extra qubits are required to represent each resource's ID. To keep consistency with the QA and QAOA methods, the x-axis in each plot in this section represents the number of qubits used minus the amount used to represent the IDs.\n            \n            \n            \\fig[.5]{Figure_18}{\\textbf{QPU \\& Simulated Results} QA \\& QAOA QPU Times and Simulated QAOAH Wall-Clock times for Scenario 2. QA timing includes the anneal time of each 50 samples, each 20 microseconds per sample. QAOA ran with 1000 shots. QAOAH ran in undermined amount of jobs using statevector-simulator to get results. Unlike Scenario 1, the times recorded are total wall-clock times for QAOAH.}\n            \n            As in scenario 1, the relative costs for each method is plotted in \\Fig{Figure_17}. Both QAOAH and QA methods have zero relative costs. However, QAOA by itself did not exhibit a positive relative cost. For both cases in this subplot, the data is insufficient to deduce whether or not these violated constraints.\n            \n            In \\Fig{Figure_19}, the average number of constraints violated is plotted against the number of qubits the problem encodes. The number of constraints violated is calculated similarly to scenario 1, but now including all violations of the second constraint. For example, if four resources of type-1 and 2 of type-2 were allocated to a mission, then the number of constraints violated is $4-2=2$. \\Fig{Figure_19} shows that QA and QAOAH did not violate constraints at any qubit size, while QAOA on average did. QAOA violated constraints mostly likely because it found a solution where the mission cost exceeded the cost incurred by the constraint function. QAOAH in simulation, however, did not violate constraints because the mixing operator that was used transforms solutions without leaving the constrained space. These results contrast with the QPU runs for QAOAH in Scenario 1 where it did violate constraints. Since the simulation ran without any noise profiles, it is expected that QAOAH shouldn't violate constraints in theory, but this is not the case when running on the actual quantum machine. The source of noise on actual hardware is most likely due to the gate noise and noise from measurement.\n            \n            \\begin{figure}\n                \\subfloat[\\label{fig:Figure_17}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_17.PNG}\n                }\n                \\vspace{1mm}\n               \n                \\subfloat[\\label{fig:Figure_19}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_19.PNG}\n                }\n                \\caption{\n                    \\textbf{QPU \\& Simulated Results}\n                    (a) Average relative cost versus the number of qubits used. The relative cost is the mission cost obtained minus the lowest possible mission cost that can be achieved (while keeping constraints) for the problem. The lowest possible cost is found by brute force. Negative relative costs must mean that the solution violated constraints.\n                    (b) Average number of constraints of each method versus the number of qubits used. The number of constraints counts up how many extras qubit are on within a column in the matrix view representation plus the absolute difference of qubits on within a row between resource sets ($\\mathbf{R}_1$ and $\\mathbf{R}_2$). This count includes the amount of columns that do not have any qubits on at all.\n                }\n                \\label{fig:results_2}\n            \\end{figure}\n            \n    \\section{Summary and Conclusion}\\label{sec:Conclusion}\n        In this paper, we introduced Mission Covering Optimization (MCO), implemented three different constraint optimization techniques (QA, QAOA, and QAOAH) to find solutions of two scenarios of MCO, and discussed results after running implementations on the IBM machine, D-Wave Machine, and on a state vector simulator. Results were compared based on timing, relative cost, qubits used, and constraints violated. From the 200 tests performed on each scenario, QA  achieved the quickest results while using the least number of qubits and violating the least number of constraints. We conjecture that QAOA and QAOAH approaches may have taken longer for the gradient descent algorithm to be convinced that an optimal solution was found because of the abundance of noise in current hardware. It was found that it is nontrivial to engineer in multiple constraints by embedding into the Mixing-Hamiltonian, especially when compared to the ease of using Lagrange multipliers, in simulation, where adding constraints entails simply adding terms together. The study conducted here suggests that the additional complexity in the QAOAH approach poses potential scalability challenges (due to the additional qubits required to ID the constraints) for problems similar to MCO with multiple types of unique constraints.\n        \n    \\section{Future Work}\\label{sec:Future:Work}\n        Adding further capabilities to different resources in Scenario 2 would make for a more interesting\/realistic optimization problems for quantum computers to solve. This is just one of the many alterations that can be done to MCO to increase the complexity of the optimization problem. \n        \n        In this paper, resources dependencies are modeled within MCO, however there may be missions with different priorities along with mission dependencies. Also, in this work resources were only shown to possess only one type of qualification. However, there can be cases where a resource may have many types of qualifications. At the heart of MCO, it is an optimization problem concerning allocation of resources invariant to time. An interesting direction for future study is how well this type of optimization problem can be ported to an extension of a Job shop problem.\n        \n        Although error mitigation is not a focus of this study, research for mitigating error for QAOA are being studied by others\\cite{PhysRevA.103.042412}. Employing error mitigation techniques for MCO is another interesting direction of study.\n\n\n        Lastly, another interesting solution method might entail the use of a ``bang-bang\" strategy~\\cite{bangbang} for multiple constraints. Here, for each constraint $C_i$ associated with a constraint Hamiltonian $H_i$, one might randomly cycle through applications of individual $H_i$ for each time step, as opposed to the application of the joint Hamiltonian $H = \\sum_i H_i$. While each $H_i$ only preserves constraints $C_i$, the supposition is that the application of $H_i$ might only partially violated constraints $C_{j\\ne i}$, if applied for a short time, and randomly. This solution approach to MCO-like problems will be investigated in future work.\n\n        \n    \\section{Acknowledgments}\n        The authors would like to thank David Vernooy and the exponential campaign at GE Research for supporting this effort.\n        %\n        The views expressed are those of the authors and do\n        not reflect the official guidance or position of the United\n        States Government, the Department of Defense, the\n        United States Air Force or General Electric. \n        %\n        The appearance of external hyperlinks does not constitute endorsement by the United States Department of Defense or General Electric of the linked websites, or the information, products, or services contained therein.  The Department of Defense and General Electric do not exercise any editorial, security, or other control over the information you may find at these locations.\n    \n   \n    \n   \n   \n   \n   \n   \n   \n    \n    \\nocite{*}\n\n    \n\\section{Introduction}\\label{sec:Intro}\n    \n        Quantum computer hardware is quickly developing~\\cite{de2021materials} together with the theory of quantum computing algorithms~\\cite{scaling}. Problems historically solved classically have been translated into quantum computing formulations~\\cite{bhaskar2015quantum, ANDERSSON2022100754, novel,container}. Problems formulated for a particular quantum computing hardware have a completely different formulation for different hardware. In this work we attempt to understand how well constraint optimization can be implemented between two well known quantum computing approaches, (i) quantum annealing, and (ii) the gate-based model, that reflect on two problem formulations: Quadratic Unconstrained Binary Optimization (QUBO) and Quantum Approximate Optimization Algorithm (QAOA). The goal of this paper is to assess the performance of the algorithm when solving a particular problem related to Mission Covering Optimization (MCO).\n        \n        MCO is a variant of the Generalized Assignment Problem\\cite{gap}. The goal is to perform a set of missions given constraints on the available resources. Missions are organized efforts to achieve an objective. Examples of mission types include industrial, logistic, and militaristic, where the problem can be anything from getting packages to destinations, to efficiently producing and maintaining equipment devices~\\cite{Venturelli2016JobSS, Asset_Sustainemnt}. To assure that a mission is successful, a set of resources performing different tasks are used. The global solutions of such problem is computationally challenging. It is a complex optimization problem~\\cite{Ng2022}, often non-convex. Heuristic approaches are used to find feasible solutions, but they depend on domain knowledge and problem decomposition, leading to inaccurate solutions. So this category of problems is a good candidate for quantum acceleration. In this effort, we will address an abstraction of a real MCO problem.\n        \n        We formulate the problem for a quantum annealer, specifically the D-Wave quantum computer, as well as investigate formulation for a gate-based model approach, specifically the IBM quantum computer. Quantum annealer is a type of quantum discrete optimization algorithm that exploits the power of tunneling to reach a global minimum \\cite{qa_optimization}. Gate-based quantum computer solve problems applying a sequence of unitary operators to qubits in superposition.\n       \n       \n        From using results from the QPU and through simulations, we present strengths and weaknesses found with both models, and outline fundamentals of this new type of optimization problem. \n        \n        This paper is divided into 4 main sections. \\Sec{sec:Formalism} describes MCO in greater formal detail. \n        \\Sec{sec:MCO:Scenarios} outlines two different scenarios of MCO that are tested in this study. \n        \\Sec{sec:Alg:Impl}  details how each scenario is implemented using three different algorithmic techniques. \n        \\Sec{sec:Compare:Results} describes the results of each scenario and compares the different algorithmic techniques by cost, timing, and constraint-holding metrics.\n        In \\Sec{sec:Conclusion}, we summarize and conclude our findings of this work.\n        Lastly, \\Sec{sec:Future:Work} discusses research directions for future work.\n\n   \n   \n   \n\n   \n   \n   \n   \n   \n   \n                    \n                    \n                    \n                    \n   \n   \n\n    \\section{Formalism for Mission Covering Optimization}\\label{sec:Formalism}\n                \n        The objective of an MCO problem is to allocate a set of resources to missions such that each resource is assigned to at most one mission, and each mission's requirements are satisfied. \n        \n        A MCO problem is described by a 7-tuple{}: \n        \n        $$(\\MSet, \\RSet, \\QSet, \\capability, \\textsf{REQ}_{M}, \\textsf{REQ}_R, \\comment{\\textsf{PR},} f_{obj})$$\n          \n          \n        \\begin{enumerate}\n            \\item $\\mathbf{M}$ is the set of missions\n            \\item $\\mathbf{R}$ is the set of resources\n            \\item $\\mathbf{Q}$ is the set of qualifications\n            \\item $\\capability$ is the capability function\n            \\item $\\textsf{REQ}_{M}$ is the mission's requirements function\n            \\item $\\textsf{REQ}_R$ is the resource's requirements function\n            \\item $f_{obj}$ is the objective function that scores problem solutions      \n        \\end{enumerate}\n    \n        In the rest of this section these terms will be explained in detail.\n        \\subsection{Missions}\n    \n            Missions are operations to achieve specific results. They require resources for specific tasks. For example, a mission could be transporting children to school. The resources are the buses and the bus drivers. In manufacturing, a mission could represent the process of building an asset from the design to delivery. It includes design, engineering, sourcing, suppliers, production, control, and packaging. $N_M$ denotes the number of missions. In an MCO, missions need to be completed at the same time; we denote the mission set as $\\mathbf{M} =\\{m_1, m_2, \\ldots, m_{N_M}\\}$.\n    \n        \\subsection{Resources}\n            $\\mathbf{R}$ represents the set of all resources. Resources are all the assets needed for mission completion. Resources can be objects, machines or people. For example, a given MCO may define its resources as four planes, four pilots and two engineers. $\\mathbf{R} = \\{r_1, r_2, \\ldots, r_{N_R}\\}$ is the set of $N_R$ resources within the MCO.\n               \n        \\subsection{Qualifications}\n            $\\mathbf{Q}$ represents the set of all qualifications. Resources are specialized in the sense that they have qualifications and they can be assigned to the mission that requests a resource with such qualification. For example `Can transport', `Can pilot', and `Can troubleshoot' are qualifications a mission may require.  $\\mathbf{Q} = \\{q_1, \\ldots, q_{N_Q}\\}$ is the number of $N_Q$ qualifications in the MCO problem.           \n               \n        \\subsection{Capabilities}\n           $\\capability$ represents the capability function. Each resource is scored for its qualifications. Capabilities are integer number that represent how well a resource can perform a certain qualification. The higher the number, the more qualified the resource is. For example, consider the resource set composed by a senior engineer, intern engineer, and an HR Manager and with capabilities of 2, 1, 0 respectively to a certain qualification titled 'Troubleshoot carburetor'. This reflects the fact that the senior engineer has a higher capability than the intern engineer  to troubleshoot carburetor because he\/she is more experienced in the field. Conversely, the HR manager has no qualification capability to 'Troubleshoot carburetor', therefore has a score of zero. Capability is the function\n            $$\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\mathbb{Z}_{\\ge 0}},$$\n            which returns a zero if and only if the resource is not qualified for the specified qualification. When this happens, we say that this resource has no capability for that qualification.\n      \n            Missions require resources with specific qualifications. For example, mission $m_1$ requires $n$ resources with qualification $q_1$ where $m_1\\in\\mathbf{M}$,  $n\\in\\mathbb{N}$ and $q_1\\in\\mathbf{Q}$, which represents `mission 1 requires 10 resources that can fly a plane'. We assume that resources with higher capability for certain qualifications are chosen.\n     \n     \n        \\subsection{Mission's Requirements}\n            The mission's requirement function is described as     \n            $$\\fncType{\\textsf{REQ}_{M}}{\\mathbf{M}\\times\\mathbf{Q}}{\\mathbb{N}},$$\n            and represents the qualification requirement for each mission. This function returns the number of required resources that satisfy the qualification needed for that mission.\n        \\subsection{Resource's Requirements}\n            The resource's requirement function is described as     \n            $$\\fncType{\\textsf{REQ}_R}{\\mathbf{R}\\times\\mathbf{Q}}{\\mathbb{N}},$$\n            and represents the qualification requirement for each mission. This function returns the number of required resources that satisfy the qualification needed for that resource. This will be used on scenario 2.\n    \n        \\subsection{Solutions and Solution Score}\n            A solution of the MCO is defined as a function that associates resources to missions \n            $$\\fncType{f_s}{\\mathbf{R}}{\\mathbf{M}},$$\n            If $N_F$ is the number of functions that are the solutions of the MCO, then $\\mathbf{F}= \\{f_1, f_2, \\ldots, f_{N_F}\\}$ is the set of all possible solutions. The goal is to find the best solution in which all missions meet requirements as well as possible. The objective function maps solutions to real numbers, which measures how far off each solution is from meeting requirements. The objective function represents a cost function:\n            $$\\fncType{f_{obj}}{F_{s}}{\\mathbb{R}}$$ s.t. $$F_{s} = \\{\\{ (m,r) \\in \\mathbf{M} \\times \\mathbf{R} \\mid m = f_s(r)\\} \\mid \\fncType{f_s}{\\mathbf{R}}{\\mathbf{M}}\\}.$$\n            Therefore, the best solution to an MCO is achieved by minimizing the objective function and retrieving the minimizer.\n    \n    \\section{MCO Scenarios}\\label{sec:MCO:Scenarios} \n        The MCO optimization problem involves the parameters (missions, resources, mission require, resource require). From this, there are different metrics of cost that can be used to represent the objective function. Two specific MCO scenarios are formalized and implemented on different quantum computing hardware machines. These scenarios are not computationally difficult to compute classically. However, these scenario's design is intentional for comparing results in this study, as brute force methods for finding the absolute best cost-minimizers can be used to without being throttled by the exponential time complexity since the scenario's are designed to permutationally symmetric. The primary focus of each of these readily solvable MCO scenarios is to identity how well each algorithm implementation performs when constraints are introduced.  Results are compared in terms of cost, timing, and constraints met. In the general MCO formulation, resources can be assigned to at most one mission, so that resources could be unused. In this paper, in both scenarios, it is assumed that each resource is assigned to one and only one mission. This is done by introducing an additional mission that includes the unused resources. The following subsections describe each scenario.\n    \n        \\subsection{Scenario 1: Primary and Secondary Resources}\n            The first scenario involves two categories of resources: primary and secondary. Primary resources are ready to be used for mission completion. Secondary resources are allocated when primary resources are not able to perform their duty. Primary resources that cannot be allocated to missions are removed from the set of primary resources. For example, the mission covering optimization solution shown in figure~\\Fig{Figure_0} is composed by three missions requiring three, two and two pilots respectively for mission completion. Suppose in the planning phase that seven pilots were allocated to cover the three missions; they are on-duty (in the primary resource set), and three other pilots are on-call (in the secondary resource set). Suppose at a certain point in time, two of the pilots got sick and they were removed from the resource group. This is an emergency, unforeseen situation that requires to re-run an optimization algorithm to cover the missions including secondary resources. The pilots are pulled from the set of five pilots on-duty, ready to cover the missions and two pilots from the secondary resource set should be also included. The challenge is to find the best allocation of pilots to missions using all the pilots on-duty and two pilots on-call. The goal of our effort is to use quantum computing to solve this problem and analyse results from different quantum computing hardware implementations. \n            \n            The following outlines all the rules in this specific scenario:\n    \n            \\begin{enumerate}\n                \\item The set of missions in the problem is $\\mathbf{M} =\\{m_1, \\ldots, m_{N_m-1}, U\\}$.\n                \\item The set of resources is $\\mathbf{R} = \\{r_1, \\ldots, r_{N_r}\\}$.\n                \\item There only exists one qualification, which is the ability to fly a plane, $\\mathbf{Q} = \\{q_1\\}$. \n                \\item Resources can have a capability of 1 or 2. So the capability function is then $\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\{1, 2\\}}$. This is a way to represent how ready the resource is to be allocated to a mission. It can be though of as an ordering for the allocation of resources; resources with capability 2 should be allocated before ones with 1. Resources that have capability 2 are referred to as primary resources, which are assigned to missions first. Resources that have capability 1 are referred to as secondary resources, which are assigned to missions only if primary resources become unavailable.\n                \\item $\\textsf{REQ}_{M}$ is the mission's requirement.\n                For example:\n                \\begin{itemize}\n                    \\item $\\textsf{REQ}_{M}(m_1, q_1)=3$.\n                    \\item $\\textsf{REQ}_{M}(m_2, q_1)=2$.\n                    \\item $\\textsf{REQ}_{M}(m_3, q_1)=2$.\n                \\end{itemize}\n                \\item There are no resource requirements. Therefore, the resource requirement function is $$\\fncType{\\textsf{REQ}_R}{\\mathbf{R}\\times\\mathbf{Q}}{\\{0\\}}.$$\n                \\item $f_{obj}$ is the function used to score the different problem solutions in terms of cost\n            \\end{enumerate}\n    \n            The mission cost includes two parts:\n            \\begin{itemize}\n                \\item The mission cost represents how many of the mission's requirements are not met. It is a penalty introduced every time a mission is not able to accomplish its goal due to a lack of resources.\n                \\item The precedence cost measures how well each solution allocates resources with higher capabilities before others to missions. In the specific example, it means that it is desirable that primary pilots are allocated to missions before secondary pilots.\n            \\end{itemize}\n                \n    \n            As discussed previously, MCO covers a broad spectrum of problems distinguishable only by it's objective function. In the scenario described here, the objective function reflects the cost which is the sum of the mission cost and precedence cost. All solutions are measured in terms of the cost in order to find the optimal solution. This scenario can be described by using a matrix arrangement of Boolean variables, shown in \\Fig{Figure_0}. Each row of the table in the figure represents a mission, and each column represents a resource.\n    \n            \\fig{Figure_0}{\\textbf{Matrix view for Solution}: This represents a table representing the solution of a problem with three missions, five primary resources and three secondary resources. A purple circle symbolize a Boolean variable $x_{m, r}\\in\\{0,1\\}$ where $m$ represents the mission (row) while $r$ represents the resource (column). A circle filled in with orange represents $x_{m, r}=1$, and the fact that resource $r$ is allocated to mission $m$.}\n            \n            The final row of the table represents an artificial mission created for the purpose of having all the resources utilized. When resources are not allocated to any of the other missions, it has to be allocated this special mission. It will be referred to as the unallocated mission (U mission for short) and has no mission cost associated.\n      \n                \n            A separate column specifies the required amount of qualifying resources for each mission. Since there is only one qualification in the scenario, this number just represents how many resources should be allocated to the mission. The example in \\Fig{Figure_0} shows a solution where each mission satisfies its requirements, but only after using all primary resources before it uses any of the secondary ones. Since each resource can only be assigned to one mission, no more than one Boolean variable may be true in a single column of the matrix representation of a MCO. In general, the column constraint assures that a resource can be assigned at most to one mission. This is a hard constraint. \n        \n        \\subsection{Scenario 2: The Buddy System}\n            \n            This scenario was designed to see how different algorithms deal with additional constraints. There are two groups of resources of different types. If a resource from one group is assigned to a mission, it is required that a resource from the other group joins the same mission. Consider the previous example where a set of planes and pilots must be allocated to a set of aerial missions. A pilot and plane are required to complete a mission. And every time a plane is chosen, a pilot needs to be chosen. From an application standpoint, this scenario highlights the modeling of resource dependencies (planes depend on pilots for allocation). This scenario contains an additional row constraint. The list of constraints is as follows:\n            \\begin{itemize}\n                \\item Column constraint: a resource can be assigned at most to one mission. This is the hard constraint of the problem.\n                \\item Row constraint: if a type of resource is chosen then a resource of the second type must also be chosen. We call it the {\\it buddy} constraint. This is the additional constraint added in this scenario.\n            \\end{itemize}\n    \n    \n            The entire set of rules for this scenario is as follows:\n            \\begin{enumerate}\n                \\item The set of missions in the problem is $\\mathbf{M} =\\{m_1, \\ldots, m_{N_m-1}, U\\}$.\n                \\item The set of resources is $\\mathbf{R} = \\{r_1, \\ldots, r_{N_r}\\}$.\n                \\item There are two qualifications $\\mathbf{Q} = \\{q_1, q_2\\}$, which means that resources are divided in two groups: $\\mathbf{R}_1$ and $\\mathbf{R}_2$.\n                \\item Resource's capability is $\\fncType{\\capability}{\\mathbf{R}\\times\\mathbf{Q}}{\\{0, 1\\}}$, which means that all resources that are qualified have the same capability. Resources have only one qualification:\n                \\begin{itemize}\n                    \\item The set of all resources that have capability 1 for qualification $q_1$ is notated as $\\mathbf{R}_1$.\n                    \\item The set of all resources that have capability 1 for qualification $q_2$ is notated as $\\mathbf{R}_2$.\n                    \\item The sets $\\mathbf{R}_1$ and $\\mathbf{R}_2$ do not contain the same resource: $\\mathbf{R}_1\\cap\\mathbf{R}_2 = \\{\\}$.\n                    \\item The sets $\\mathbf{R}_1$ and $\\mathbf{R}_2$ contain all resources: $\\mathbf{R}_1\\cup\\mathbf{R}_2 = \\mathbf{R}$.\n                    \\item The number of resources in $\\mathbf{R}_1$ and $\\mathbf{R}_2$ are the same $|\\mathbf{R}_1| = |\\mathbf{R}_2|$.\n                \\end{itemize}\n                \\item Missions require resources with qualification $q_1$ and not $q_2$. Therefore, the mission require function can be formally described as:  $\\fncType{\\textsf{REQ}_{M}}{\\mathbf{M}\\times\\{q_1\\}}{\\mathbb{Z}_{\\ge 0}}$.\n                \\item The resource's requirement is called the \\textit{buddy requirement}. Every resource when allocated requires another resource with the opposite qualification.\n                \\begin{itemize}\n                    \\item $\\textsf{REQ}_R(r, 1) = 0 \\And \\textsf{REQ}_R(r, 2) = 1 \\quad \\forall r\\in\\mathbf{R}_1$\n                    \\item $\\textsf{REQ}_R(r, 1) = 1 \\And \\textsf{REQ}_R(r, 2) = 0 \\quad \\forall r\\in\\mathbf{R}_2$\n                \\end{itemize}\n                \\item As in the previous scenario, the objective function measures the total mission cost.\n            \\end{enumerate}\n                \n            The mission cost formulation is exactly the same as the secondary resources scenario. However, the way we view the problem in its matrix representation is slightly different, as shown in \\Fig{Figure_7}. \n            \n            \\fig{Figure_7}{\\textbf{Matrix view for Solution}. This solution is shown for a problem with 3 missions, four resources of type $\\mathbf{R}_1$ and four resources of type $\\mathbf{R}_2$. $\\textsf{REQ}_{M}$ describes how many $\\mathbf{R}_1$ resources must be assigned to mission $m$. Each $\\mathbf{R}_1$ resource is paired with exactly one $\\mathbf{R}_2$ resource for each mission.}\n            \n            The primary difference between this solution and the prior scenario's solution is that now there are \\textit{buddy} constraints present along the rows of the matrix (for this reason it's also called the \\textit{row} constraint). For example, in \\Fig{Figure_7}, the first row related to the first mission has two resources allocated from $\\mathbf{R}_1$. In order for the buddy constraint to not be violated, two resources from $\\mathbf{R}_2$ are allocated to that mission. The solution shown is valid since all no rows or columns violate any of the constraints. The solution also has the lowest mission cost since the mission requirements were all exactly met.\n            \n            Finding optimal solutions for this scenario is trivial, since all resources within the same group are indistinguishable from each other in terms of allocation cost. However, when adding different capabilities to resources within this MCO, the complexity of finding the optimal solution increases. The case where resources have different and multiple capabilities has not been studied and is part of future work. While this does not change substantially how the problem is formulated and implemented on the quantum device, it is more complex to check how well the quantum algorithms solutions are compared to optimal solution. To do this, the optimal solution needs to be known, so brute force methods are used to find it. If resources are indistinguishable from one another, then it takes far less time to brute force to the optimal solution due to the permutation symmetry across resources.\n\n    \\section{Algorithm Implementation}\\label{sec:Alg:Impl}\n        In this work we tested three techniques to solve the MCO problem with the two scenarios described in the previous section, quantum annealing and two types of QAOA algorithms. \n        \\begin{itemize}\n            \\item Quantum Annealing (QA)(\\cite{qa}\n            \\item Quantum Alternating Operator Ansatz (QAOA) \\cite{qaoa}. It implements constraints by means of Lagrange multiplier embedded into a cost function Hamiltonian.\n            \\item QAOAH which is a version of QAOA with a constrained mixer. It is denoted as QAOAH since it was developed by Hadfield \\textit{et al.} \\cite{qaoa_had}. It engineers the constraints to remain within a constraint space during the entire solution process.\n        \\end{itemize}\n        All three approaches use similar means for encoding the objective function but differ in the way they implement their constraints and what machines support them.\n\n        \\subsubsection{Quadratic Unconstrained Binary Optimization}\n\n            Quantum Annealing is an quantum optimization methodology performed on a specific implementation of quantum computers such as D-Wave quantum annealing machines. Quantum Annealing is an adiabatic quantum computing technique that capitalizes on a unique feature of quantum mechanics, quantum tunneling, which is the capability to surf and dive though a energy landscape until it hits a minimum energy level\\cite{qa_optimization}. This property of quantum mechanics is engineered to solve Quadratic Unconstrained Binary Optimization problems\\cite{QUBO}. QUBO, is a type of optimization where the solution is a binary vector that minimizes an objective function described with terms up to a quadratic degree. These problems are unconstrained, but there are methods to incorporate the behavior of constraints\\cite{QUBO_CONST} into the cost function. To encode an MCO onto a quantum annealing machine, we must translate the objective function into a QUBO.\n        \n        \\subsubsection{Quantum Alternating Operator Ansatz}       \n            Quantum Alternating Operator Ansatz QAOA is method for solving combinatorial optimization problems on NISQ devices\\cite{qaoa}. The algorithm is supported by gate-based quantum computers such as the IBM, Rigetti, IonQ or Xanadu machines. QAOA In this effort we focus on implementing the problem on an IBM quantum computer~\\cite{QAOA_QISKIT}. At the current time, IBM's Qiskit implementation of QAOA uses Variational Quantum Eigensolver (VQE) to find the expectance of a parameterized ansatz eigenstate. This quantity is then used to calculate the minimum of the cost function which is embedded in a cost Hamiltonian $H_C$. Like QA, it is limited to solving binary quadratic problems\\cite{PhysRevApplied.5.034007}.\n        \\subsubsection{Quantum Alternating Operator Ansatz with Constraints (QAOAH)}        \n            In QAOA, the starting state is in an equal superposition of all computational basis states, and the mixing Hamiltonian is a sum of Pauli-X operators acting on each qubit. In this configuration, every possible classical-state solution can be traversed to making it ideal for unconstrained problems where there are no states to be filtered out. However, it is possible to incorporate constraints by altering the default mixing Hamiltonian $H_M$ and the initial quantum state $\\ket{\\psi}$\\cite{qaoa_had}. Since the mixing operator (the exponentiated mixing Hamiltonian) specifies how to explore the solution space, it is possible to specify a mixing operator that determines how to move to another solution within the constraints of the problem.\n        \n            Consider an initial quantum state $\\ket{\\psi}$ which represents a solution that does not violate the constraints of the optimization problem, but is not necessarily the minimizer. Consider the mixing operator $H_M$ that is constructed such that after an application on a state within the constrained space, the output state is also guaranteed to be within the constrained space. The idea is to construct $H_M$ and $\\ket{\\psi}$ so that when stopping QAOA at any iteration, the final encoded solution is always within the constrained solution space. If this mixing operator can mix such that it allows the algorithm to reach every possible classical-state in the constrained solution space, it is then possible to obtain the minimizer in that same space \\cite{qaoa_had}.\n        \n            Since QA and QAOA are both derived from the same formulation, the implementation details are described together. In QAOAH, constraints are embedded into the Mixing Hamiltonian which brings to a different formulation. The following sub-sections outline the implementation for both scenarios using the three approaches described above.\n\n        \n        \\subsection{Scenario 1 (One Constraint)}\n            \\subsubsection{QA and QAOA}\n                The goal of a MCO problem is to minimize the total cost. Various ways are used to define mission cost. The following formulation takes into account limitations of QUBO. In the rest of the paper $QA$ denotes QUBO problems performed on quantum annealing machines while QAOA and QAOAH denote QUBO problems executed on gate-based architectures.\n                \n                The optimal mission cost occurs when all the resources required are allocated. Or, formally, consider mission $m$; the optimal mission cost occurs when:\n                \\be{zi3GOCbzYwWHuuutPmGK}\n                    \\sum\\nolimits_{r \\in \\mathbf{R}} x_{m,r} = \\textsf{REQ}_{M}(m, 1).\n                \\end{equation}\n                \n                The $x$ values represent Booleans (an alternative representation of the solution $f_s$), such that when indexed by $m\\in\\mathbf{M}$ and $r\\in\\mathbf{R}$, e.g. $x_{m, r}$, it represents whether or not the solution mapped resource $r$ to mission $m$.\n                \n                As an example, suppose there are three resources, $\\mathbf{R} = \\{1, 2, 3\\}$ and the first mission requires two of them, then the optimal mission cost for the mission occurs when:\n                \\be{dFx-f3K0sdSgXNW7_ilp}\n                    x_{1,1} + x_{1,2} + x_{1,3} = 2.\n                \\end{equation}\n                \\Eq{dFx-f3K0sdSgXNW7_ilp} is true when 2 out of the 3 variables are true. The mission cost represents the penalty added any time a mission lacks of one or more needed resources. The penalty is higher when more resources are missing. A squared error is used to represent the mission cost:\n                \\be{5onSFKdjQxjrP-eN0BfE}\n                    \\textsf{MC}(x, m) = \\left(\\sum\\nolimits_{r \\in \\mathbf{R}} x_{m,r} - \\textsf{REQ}_{M}(m, 1)\\right)^2,\n                \\end{equation}\n                where $\\textsf{MC}(x, m)$ is the cost associated to mission $m$. It can be seen from \\Eq{5onSFKdjQxjrP-eN0BfE} that the minimum mission cost, when $\\textsf{MC}(x, m) = 0$, yields the optimal case as described in \\Eq{zi3GOCbzYwWHuuutPmGK}. The mission cost is quadratic. The example that yielded the optimal case in \\Eq{dFx-f3K0sdSgXNW7_ilp} in terms of the mission cost function is:\n                \\be{EeWWAXtMbm09rvIbOAEE}\n                    \\textsf{MC}(x, 1) = \\left( x_{1,1} + x_{1,2} + x_{1,3} - 2\\right)^2.\n                \\end{equation}\n                \n                For this optimization problem, secondary resources should be allocated only after primary resources are allocated first. Therefore, we introduce precedence cost for resources. \\Eq{lPsZgJ1LWm9RbsTBps8B} shows the ideal condition regarding the allocation of secondary resources.\n                \\be{lPsZgJ1LWm9RbsTBps8B}\n                    \\sum\\nolimits_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}} x_{m, r} = \\capability(r, 1) - 1 \\quad\\forall r\\in\\mathbf{R}\n                \\end{equation}\n                The precedence cost is dependant on the capability of a resource, and therefore dependant on whether or not this resource is primary or secondary. When the resource is primary, the ideal condition means that the it must be allocated to one of the active missions (apart from the unallocated\/psuedo mission). For the other secondary resources, ideally none are used.  \n                \n                The precedence cost may or may not be met when optimizing, so just like mission cost, precedence cost is the squared error of this equality:\n                \\be{n_F7J1oYAc7WIuRr5Nme}\n                    \\textsf{PC}(x, r) = \\left(\\sum\\nolimits_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}} x_{m, r} - \\capability(r, 1) + 1\\right)^2.\n                \\end{equation}\n                \n                Finally, the objective function is then the total mission cost and the total precedence cost in the MCO:\n                \n                \\be{vB-eQejEL5xyYQUiOqbr}\n                    f_{obj}(x) = \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\frac{1}{|\\mathbf{R}|}\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{PC}(x, r).\n                \\end{equation}\n                \n                We weight the precedence cost by $\\frac{1}{|\\mathbf{R}|}$ to ensure that the mission cost is minimized before precedence cost. The total cost is the sum of the mission cost and the precedence cost and it is reflected in the objective function.\n                \n                \n                \n                The constraint that a resource must be paired to exactly one mission is formulated:\n                \\be{_0nqf1oq3S9d62ZOk3Pg}\n                    \\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} = 1 \\quad\\forall r\\in\\mathbf{R}.\n                \\end{equation}\n                The constraint function used for this scenario is defined as \n                \n                \\be{0FakCSC5k-YoFIaG_KAN}\n                    \\textsf{CONSTR}(x, r) = \\left(\\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} - 1\\right)^2.\n                \\end{equation}               \n               \n                Since the QUBO is used to solve problems without constraints, we must add it to the objective function so that when it minimizes, the constraints will be met. The method of Lagrange multipliers is a strategy for finding the local maxima and minima of functions subject to equality constraints. \n                \n                If $f_{obj}(x)$ is the objective function to be minimized, the lagrangian function is\n                \\be{bHljve4SovW75-5yEWxd}\n                    \\mathcal{L}(x, \\lambda) = f_{obj}(x) + \\lambda\\cdot\\textsf{CONSTR}(x).\n                \\end{equation}\n                And the solution to the original constrained problem is always a saddle point of this function. Setting a large value for $\\lambda$, the term related to the constraint, will have the greatest impact on the optimization problem. And the solution will minimize the constraint first and then the cost.\n                \n                The new objective function that includes the constraints is:\n                \\be{8Mu4WZnVmok5H4uWjbmn}\n                    \\begin{split}\n                        f_{obj}(x) &= \\mathcal{L}(x, \\lambda) \\\\\n                                          &= \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\frac{1}{|\\mathbf{R}|}\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{PC}(x, r) \\\\\n                                          &\\quad\\quad+ \\lambda\\cdot\\sum\\nolimits_{r \\in \\mathbf{R}}\\textsf{CONSTR}(x, r).        \n                    \\end{split}\n                \\end{equation}\n                \n                QAOA for this method uses an equal superposition for the starting state $\\ket{\\Psi}$ over $N$ states:\n                \n                \\be{R36C6d1UqdqSZIrDXMcw}\n                    \\ket{\\Psi} = \\frac{1}{\\sqrt{N}}\\sum_{i=0}^{N-1}\\ket{i}.\n                \\end{equation}\n                \n                The value $N$ is equal to $2^n$ where $n$ is the number of qubits used. For this problem, the number of qubits used is $N_{M}\\cdot M_{R}$, the number of missions times the number of resources.\n                \n                The mixing operator is constructed using a Hamiltonian, which is the sum of Pauli-X as follows:\n                \\be{vZTjtWDfeNFtO0FXr-a2}\n                    \\begin{matrix}\n                        \\\\\n                        X_i = & I & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & I \\\\\n                            &  1  &  &  i & & n\n                    \\end{matrix},\n                \\end{equation}\n                \\be{Iyl7tZpP2CXZiW4FHcok}\n                    H_M = \\sum_{i=1}^{n} X_i.\n                \\end{equation}\n                \n            \\subsubsection{QAOAH}\n                In the last section, Lagrange multipliers are used to encode constraints into the QUBO problem. Alternatively, by choosing the appropriate mixing Hamiltonian and initial state, we can constrain the solution space outside of the QUBO formulation in QAOA\\cite{qaoa_had}.\n                \n                The initial state must be within the constrained solution space. A trivial starting configuration that is known not to violate the constraint is when all resources are set to the unallocated mission as shown in \\Fig{Figure_9}.\n                \\fig{Figure_9}{\\textbf{Example Initial State:} All resources are initialized to be unallocated, or allocate to mission $U$, the last row.}\n                \n                This initial state used is:\n\n                \\be{lpOGOEbsbF-qt7v0fJUy}\n                    \\ket{\\psi} = \\bigotimes_{(m, r)\\in\\mathbf{M}\\times\\mathbf{R}} \n                    \\begin{cases} \n                        \\ket{1} & m = U \\\\\n                        \\ket{0} & m \\neq U\n                    \\end{cases}\\quad.\n                \\end{equation}\n\n                The mixing Hamiltonian describes how to move from the starting state, as well as all subsequent states, such that they are also in the constrained space. To be in the constrained space only one qubit per column must be active. The Hamiltonian should describe how to cycle a qubit in active state throughout the column so that it can reach every possible combination of configurations that still satisfy the constraint. The identity operator and the $\\textsf{SWAP}$ gates are used for this cycling action. For the 3 mission example, we can confirm that an individual column can have each of it's possible states reached using 3 swap gates (see \\Fig{Figure_8}).\n                \\comment{TODO: Talk about the possibility of using all permutation of the SWAP gate in a column.}\n                \\fig{Figure_8}{\\textbf{Resource 1's Mixing Operators:} The beginning scenario, (leftmost column) details the starting state before applying the mixing operator. The second scenario shows a identity mixing operator 2nd column from left which allowed the current state to remain unchanged. The 3 right-most scenarios each use one swap to cycle qubit's \\textbf{on} state to missions 1,2, and 3}\n                \n                Thus, a single resource mixing $\\textsf{MIX}(r)$ is:\n                \n                \\be{x5oRpxylQTEhdYB0w4Xw}\n                    \\textsf{MIX}(r) = \\sum_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}}{\\textsf{SWAP}_{(U,r),(m, r)}},\n                \\end{equation}\n                \n                where $(m, r)$ encodes the index of the qubit representing a mapping of resource $r$ and mission $m$. The $U$ in $(U, r)$ represents the unallocated mission. The total mixing operator $H_m$ is the sum of $\\textsf{MIX}(r)$ on each resource with the identity operation: \n                \n                \\be{smK9UzTkSju8UiVCq-bx}\n                    H_m = I^{\\otimes n} + \\sum_{r\\in\\mathbf{R}}{\\textsf{MIX}(r)}.\n                \\end{equation}\n                \n                In order to embed the mixing operation as a Hamiltonian to run on the IBM machines, it must be described as a composition of tensored Pauli gates. Each $\\textsf{SWAP}$ gate can be decomposed in terms of Pauli gates:\n                \n                \\be{Ba0eQI2c0iOaj6eSiUhJ}\n                    \\textsf{SWAP} = \\frac{1}{2}\\left(I\\otimes I + X\\otimes X + Y\\otimes Y + Z\\otimes Z\\right).\n                \\end{equation}\n                \n                When $\\textsf{SWAP}$ is indexed by $i$ and $j$, the Pauli gates fall on the $i^{th}$ and $j^{th}$ qubits respectively:\n                \n                \\be{NIl6X5cf18D9nG3spUZy}\n                    \\textsf{SWAP}_{i, j} = \\frac{1}{2}\\left(I^{\\otimes n} + X_{i, j} + Y_{i, j} + Z_{i, j}\\right),\n                \\end{equation}\n                \n                \\be{sIsNHYzXiF_r_TB4fqeK}\n                    \\begin{matrix}\n                        \\\\\n                        X_{i, j} = & I & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & X & \\otimes\\cdots\\otimes & I \\\\\n                            & 1 & & i & & j & & n\n                    \\end{matrix}\\quad.\n                \\end{equation}\n                \n                Each Pauli gate used in \\Eq{NIl6X5cf18D9nG3spUZy} ($X_{i, j}$, $Y_{i, j}$, $Z_{i, j}$) is defined similarly to what is defined in \\Eq{sIsNHYzXiF_r_TB4fqeK}. For clarity, \\Eq{sIsNHYzXiF_r_TB4fqeK} places the corresponding Pauli gate only at the specified indicies $i$ and $j$ in a tensor product of identities $I$.\n        \n        \\subsection{Scenario 2 (Two Constraints)}\n            \\subsubsection{QA and QAOA}\n                In this scenario, the objective function measures just mission cost, as opposed to the previous scenario that also measures precedence cost. Therefore, the objective function is the sum of mission cost and the constraint function:\n                \n                \\be{NpmMn5bAksngEsR1UKBE}\n                    f_{obj}(x) = \\sum\\nolimits_{m \\in \\mathbf{M}} \\textsf{MC}(x, m) + \\lambda\\cdot\\textsf{CONSTR}(x).\n                \\end{equation}\n                \n                Two different constraints are embedded into the objective function using Lagrange multipliers. The first one was discussed in the previous scenario assures that resources are allocated to no more than one mission. It is formulated in a similar way as before:\n                \n                \\be{LUEPhowwvgU8g5_4BDtn}\n                    \\textsf{CONSTR}_1(x, r) = \\left(\\sum\\nolimits_{m \\in \\mathbf{M}} x_{m, r} - 1\\right)^2.\n                \\end{equation}\n                \n                \n                The addional constraint (buddy constraint) requires that the amount of resources allocated from set $\\mathbf{R}_1$ must be the same as the number of resources allocated from set $\\mathbf{R}_2$. This hard equality is formulated as:\n\n                \n               \n               \n               \n                \n               \n                \n                \\be{WlIgzOycMu3ESpVQt26I}\n                    \\textsf{CONSTR}_2(x, m) = \\left(\\sum_{r\\in\\mathbf{R}_1} x_{m, r} - \\sum_{r\\in\\mathbf{R}_2} x_{m, r}\\right)^2.\n                \\end{equation}\n                \n                The total constraint function expressed in \\Eq{NpmMn5bAksngEsR1UKBE} can be expressed as the sum of both of these constraints:\n                \n                \\be{gm0YwnJ1TSQjD78B5eVR}\n                    \\begin{split}\n                        \\textsf{CONSTR}(x) = \\sum_{r\\in\\mathbf{R}}\\textsf{CONSTR}_1(x, r) \\\\ + \\sum_{\\substack{m\\in\\mathbf{M} \\\\ m\\neq U}}\\textsf{CONSTR}_2(x, m)\n                    \\end{split}\\quad.\n                \\end{equation}\n                \n                For QAOA, the starting state and the Mixing Hamiltonian are the same as defined in \\Eq{R36C6d1UqdqSZIrDXMcw} and \\Eq{Iyl7tZpP2CXZiW4FHcok}.\n                \n            \\subsubsection{QAOAH}\n                For this scenario, it is more challenging to construct the mixing Hamiltonian to describe how to move in the constrained space in QAOA. Unlike the Lagrange multiplier's case, it is not easy to linearly combine two constraint encodings to get the final constraint Hamiltonian. In other words, one cannot add $H_{m1} + H_{m2}$ to get $H_{m1+m2}$, where $H_{m1}$ $H_{m2}$, $H_{m1 \\& m2}$ are mixing Hamiltonian's representing $1^{\\text{st}}$, $2^{\\text{nd}}$ and $1^{\\text{st}}$ \\& $2^{\\text{nd}}$ constraints respectively. This scenario presents two challenges:\n                \\begin{enumerate}\n                    \\item If a mixing operator allowed a resource in $\\mathbf{R}_1$ to move from an unallocated state to an allocated state by pairing it with a mission, then it must also move a resource from $\\mathbf{R}_2$ to the same mission\n                    \\item the mixing operator must operate such that it is possible after multiple applications to visit every classical state from the starting quantum state. These two obstacles require a slightly different mixing operator and more qubits.\n                \\end{enumerate}\n                \n                The strategy for creating a two-constraint mixing operator is to reallocate resources in pairs - one from $\\mathbf{R}_1$ and one from $\\mathbf{R}_2$. These pairs will always move together. However, the problem is that not every possible classical state can be produced from the mixing. For example, let's say that there are resources $\\mathbf{R}_1=\\{1, 2, 3, 4\\}$ and $\\mathbf{R}_2=\\{5, 6, 7, 8\\}$. If resource 1 and resource 5 move together, then it's never possible to see resource 2 just be paired with resource 5 without also being paired with 1. A way to resolve this problem is to introduce an additional mixing operator to swap entire columns within just $\\mathbf{R}_1$  or $\\mathbf{R}_2$ resources. However, every time columns are swapped, the classical state must remember what mappings are paired with which others so that if another reallocation is done, the buddy constraint won't be violated. \n                \n                For this reason new qubits are introduced to each column. These qubits represent the pair ID that is present in the columns for resources in $\\mathbf{R}_1$ and $\\mathbf{R}_2$. Resources with the same ID are reallocated together. When columns are swapped, the pair IDs of the columns are also swapped. \n                \n                Consider a MCO with 3 missions (plus the unallocated mission) and 8 resources evenly split between $\\mathbf{R}_1$ and $\\mathbf{R}_2$. Our initialized state is shown in \\Fig{Figure_10}. Each column has 2 extra ID qubits (lowest two rows) with a unique bit-encoding that matches another column from the opposite qualification type. This means that these columns are paired together.\n\n                \\begin{figure*}[ht]\n                    \\subfloat[\\label{fig:Figure_10}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_10.PNG}\n                    }\n                    \\hspace*{\\fill}\n                    \\subfloat[\\label{fig:Figure_11}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_11.PNG}\n                    }\n                   \n                    \\vspace{1mm}\n                    \\subfloat[\\label{fig:Figure_12}]{\n                      \\includegraphics[width=1\\columnwidth]{Figure_12.PNG}\n                    }\n                    \\hspace*{\\fill}\n                    \\subfloat[\\label{fig:Figure_13}]{\n                        \\includegraphics[width=1\\columnwidth]{Figure_13.PNG}\n                    }\n                        \n                    \\caption{\n                        (a) Initial State Layout. The initial state is similar to the QAOA case except with additional ID qubits to be used to support the constraint mixer. Each column of ID qubits form a unique binary number for that resource within its resource set.\n                        (b) First Control Dual-Swap Operation. Through mixing, resource 2 is allocated to mission 2. In order to not violate the row constraint, resource 6 from $\\mathbf{R}_2$ is allocated using a Dual-Swap gate with controls on the ID qubits.\n                        (c) Column Swap. The mixing operator can never produce the valid solution where just resource 4 and 6 are paired to mission 2 using just Controlled Dual-Swap Operations. Therefore, column-swap operations are permitted, which swap any two columns in the table within its resource set. Here, resource 2 and 4 in resource set $\\mathbf{R}_1$ have there entire columns swapped.\n                        (d) Second Control Dual-Swap Operation. This operation is exactly the same as (b), but using the Control Dual-Swap operator to unallocate resource 4 from mission 2 to mission $U$. Notice that the Control Dual-Swap operator has its control configuration identical on both resource 4 and 6 ID-qubit columns; this is to ensure that the row constraint cannot be violated. Violation occurs when leaving resource 6 with no pair on mission 2.\n                    }\n                    \\label{fig:29udh23923nd}\n                \\end{figure*}\n                \n                For example, consider the case where the mixer operation reallocates resource 2 from the unallocated mission to mission 2. Since this column has an ID of `$10_{\\textsf{b}}$` (top ID qubit true, bottom one false) it is paired with resource 6 because it has the same ID. In order to respect this pairing, both resource 2 and 6 are swapped together using a dual swap gate $\\textsf{C-DSWAP}$ as shown in \\Fig{Figure_11}.\n                \n               Consider the situation when the mixing operator performs a column swap between columns 2 and 4 (see \\Fig{Figure_12}). Note that the IDs of these columns are also swapped.\n                %\n                If the mixer chooses to move resource 6 back to the unallocated state, it also would move resource 4 into the unallocated state since they have the same ID. This can be ensured if the mixer uses the dual swap operation once again (see \\Fig{Figure_13}).\n                \n                \n                The added qubits to represent the IDs of each of the columns are notated as $\\ket{\\textsf{ID}}$ and defined as:\n                 \n                \\be{L3eP8k09CpNEzp75j3vG}\n                    \\ket{\\textsf{ID}} = \\bigotimes_{j_1=0}^{\\textsf{ID}_{max}}\\ket{j_1} \\otimes \\bigotimes_{j_2=0}^{\\textsf{ID}_{max}}\\ket{j_2}.\n                \\end{equation}\n                \n                                \n                The starting state $\\ket{\\psi}$ is:\n                \n                \\be{OjfQ5En0Z3rMRB-BC7c6}\n                    \\ket{\\psi} = \\ket{\\textsf{ID}} \\otimes \\bigotimes_{(m, r)\\in\\mathbf{M}\\times\\mathbf{R}} \n                    \\begin{cases} \n                        \\ket{1} & m = U \\\\\n                        \\ket{0} & m \\neq U\n                    \\end{cases}\\quad,\n                \\end{equation}\n                \n                and the mixing Hamiltonian $H_m$ becomes\n                \n                \\be{eTDqpOkD5RxRJqJRkGde}\n                    \\begin{split}\n                        \\begin{aligned}\n                            H_m = & \\sum_{p\\in\\mathbf{R}_1\\times\\mathbf{R}_2\\times\\mathbf{M}}\\sum_{j=0}^{\\textsf{ID}_{max}}\\textsf{C-DSWAP}(p, j) \\\\ \n                                  & + \\frac{1}{2} \\sum_{r_1\\in\\mathbf{R}_1}\\sum_{\\substack{r_1'\\in\\mathbf{R}_1 \\\\ r_1\\neq r_1'}}\\textsf{COL-SWAP}(r_1, r_1') \\\\ \n                                  & + \\frac{1}{2} \\sum_{r_2\\in\\mathbf{R}_2}\\sum_{\\substack{r_2'\\in\\mathbf{R}_2 \\\\ r_2\\neq r_2'}}\\textsf{COL-SWAP}(r_2, r_2')\n                        \\end{aligned}\n                    \\end{split}\\quad.\n                \\end{equation}                \n                \n \n                The constant $\\textsf{ID}_{max}$ represents the maximum required binary states to represent all columns. This is the number of resources in either $\\mathbf{R}_1$ or $\\mathbf{R}_2$  and it is represented by $\\textsf{ID}_{max}$:\n                \n                \\be{objg8PD2X1Euux-XNdH8}\n                    \\textsf{ID}_{max} = |\\mathbf{R}_1| = |\\mathbf{R}_2|.\n                \\end{equation}\n\n                The column-swap gate, notated as $\\textsf{COL-SWAP}(r, r')$, swaps the columns represented by resources $r$ and $r'$. Its decomposition is trivial as it employs many swaps tensored together.\n                \n                The control dual-swap, notated as $\\textsf{C-DSWAP}(p,j)$, has parameters $p$ and $j$. The parameter $p=(r_1, r_2, m)$ is a tuple composing of a resource from $\\mathbf{R}_1$, a resource from $\\mathbf{R}_2$, and a mission $m$ in $\\mathbf{M}$. This gate applies a control \n                %\n                %\n                %\n                swap gate to resource $r_1$ and $r_2$ between the mission $m$ and the unallocated mission U. The parameter $j$ is an ID which represents how to control the swap gate. For example, $10_\\textsf{b}$ is applying a control-true, control-false gate to both IDs in the columns represented by $r_1$ and $r_2$. \n\n\n                \n                To implement the algorithm on the IBM machine, we must decompose $H_m$ such that it is a sum of tensored Pauli gates. The column-swap gates $\\textsf{COL-SWAP}(r, r')$ can be decomposed, knowing that they are made up of swap gates, as from \\Eq{Ba0eQI2c0iOaj6eSiUhJ}.\n                \n                Decomposing a generalized version of the control dual-swap gate is tedious, so we provided an example decomposition for $\\textsf{C-DSWAP}(*, 10_\\textsf{b})$, which is the gate used in \\Fig{Figure_11} and \\Fig{Figure_13}. First, we present the Pauli-decomposition of the control-true and control-false unitary operations shown in \\Eq{ftKEkEa3NI2i0cp516l0} and \\Eq{PiyQA_TRJ5ErW6-h-hTo}, respectively. Unitary $A$ is arbitrarily acting on $m$ qubits.\n                \n                \n                \\be{ftKEkEa3NI2i0cp516l0}\n                    \\textsf{C-T}(A) = \\frac{1}{2}\\left(\\left(I+Z\\right)\\otimes I^{\\otimes m} + \\left(I-Z\\right)\\otimes A\\right).\n                \\end{equation}\n                \n                \\be{PiyQA_TRJ5ErW6-h-hTo}\n                    \\textsf{C-F}(A) = \\frac{1}{2}\\left(\\left(I-Z\\right)\\otimes I^{\\otimes m} + \\left(I+Z\\right)\\otimes A\\right).\n                \\end{equation}\n                \n                Now, $\\textsf{C-DSWAP}(*, 10_\\textsf{b})$ can be represented in terms of control-true and control-false unitaries and the dual-swap gate $\\textsf{DSWAP}$:\n                \n                \\be{qB_sgrFogjI-OAM0F7nF}\n                    \\begin{split}\n                        \\begin{aligned}\n                            \\textsf{C-DSWAP}&(*, 10_\\textsf{b}) = \\\\\n                            & \\textsf{C-T}\\left(\\textsf{C-F}\\left( \\textsf{C-T}\\left({\\textsf{C-F}\\left({\\textsf{DSWAP}}\\right)}\\right)\\right)\\right).\n                        \\end{aligned}    \n                    \\end{split}\n                \\end{equation}\n                \n                Following this, the dual-swap gate $\\textsf{DSWAP}$ is two swap gates tensored together:\n                \\be{4qKC5CJNtD0ez2KQLJ8e}\n                    \\textsf{DSWAP} = \\textsf{SWAP}\\otimes\\textsf{SWAP}.\n                \\end{equation}\n                \n                The $\\textsf{DSWAP}$ can further be decomposed using \\Eq{Ba0eQI2c0iOaj6eSiUhJ}.\n                \n                As mentioned in the previous section, our two-constraint MCO problem has permutation symmetry between the resources in the same set\/group. So the column swap terms can be effectively removed, and $H_m$ becomes\n                \n                 \\be{12345CJNtD0ez2KQ6789}\n                    \\begin{split}\n                        \\begin{aligned}\n                            H_m = & \\sum_{p\\in\\mathbf{R}_1\\times\\mathbf{R}_2\\times\\mathbf{M}}\\sum_{j=0}^{\\textsf{ID}_{max}}\\textsf{C-DSWAP}(p, j).\n                        \\end{aligned}\n                    \\end{split}\n                \\end{equation}       \n                \n                This effectively shrinks the search space from the total constraint space. However, because of the symmetry, it is known that the optimal solution still lays inside the smaller subspace. When different capabilities are introduced to each resource, this optimization technique cannot be done, since the permutation symmetry is not guaranteed.\n\n                \n    \\section{Analyses of Results}\\label{sec:Compare:Results}\n        In this section we compare the results of the different MCO implementations. Employing the implementation methods discussed above, the MCO problem was run on the D-Wave and on IBM machines, capturing several key metrics:\n\n        \\begin{itemize}\n            \\item Number of qubits \n            \\item Quantum processor time\n            \\item Cost\n            \\item Number of constraints violated\n        \\end{itemize} \n        \n        \\Tbl{tbl:results_table}(top) indicates the execution status (quantum hardware or simulation) of the\n       \n       \n        \\begin{table}[ht]\n            \\centering\n            \\includegraphics[width=3.0in,height=2.25in]{table_paul_done.png}\n            \\caption{\n                S1 and S2 stands for Scenario 1 and 2, respectively.\n                MCO implementations: \n                (top) Hardware\/simulation execution environment.\n                (bottom) number of constraints violated.\n            }\n            \\label{tbl:results_table}\n        \\end{table}\n        MCO algorithms, while \\Tbl{tbl:results_table}(bottom) shows the average number of constraints violated per implementation.\n    \n\n        \n        For both scenarios, 100 random MCO configurations were generated using up to 27-qubits (200 different MCO configurations in total). Quantum Annealing, QAOA, QAOAH, and Brute Force (BF) methods were run for each generated configuration. For Quantum Annealing, the DW\\_2000Q\\_6 machine was used, while ibmq\\_toronto, ibm\\_hanoi, ibm\\_cairo, ibmq\\_mumbai, and ibmq\\_montreal machines were used for running QAOA and QAOAH. The Lagrange multipliers were set to 5 for both scenarios, and a $p$-value of 2 is used for QAOA (this parameter is discussed in the original paper\\cite{qaoa}). The Quantum annealing runs each sampled the anneal 50 times, while each IBM job sampled the state-vector 1000 times. These parameters were chosen based off of a good balance of timing, cost, and constraints satisfied found by preliminary results not discussed in this paper.\n        \n        \\subsection{Scenario 1}\n           \n            \\Fig{Figure_15} shows the timing averages for QA, QAOA, and QAOAH, respectively, versus the problem qubit size.\n           \n            \\fig[.4]{Figure_15}{\\textbf{QPU Results} QPU Times for Scenario 1 Runs. QA timing includes the anneal time of each 50 samples, each 20 microseconds per sample. QAOA and QAOAH varied in the amount of jobs that ran, each of which ran the parameterized circuit 1000 times.}\n           \n           \n            \\begin{figure*}[ht]\n                \\subfloat[\\label{fig:Figure_14}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_14.PNG}\n                }\n                \\hspace*{\\fill}\n                \\subfloat[\\label{fig:Figure_16}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_16.PNG}\n                }\n                \\caption{\n                    \\textbf{QPU Results}\n                    (a) Average Relative Cost versus the number of qubits used. The relative cost is the mission plus the precedence cost obtained minus the lowest possible mission plus precedence cost that can be achieved (while keeping constraints) for the problem; The lowest possible cost is found by brute force: Relative Cost = Algorithm's Solution Cost - Brute Force Solution's Cost. Negative relative costs imply that the solution violated constraints.\n                    (b) Average number of constraints violated versus the number of qubits used. The number of constraints counts up how many extras qubit are on within a column in the matrix view representation. This count includes the amount of columns that do not have any qubits on at all.\n                }\n                \\label{fig:results_1}\n            \\end{figure*}\n           \n            Because the timing differed by different levels of magnitude, three y-axes with different scales are displayed. QA run time outperforms the other methods by having an overall constant run-time of 0.012 seconds, while QAOAH can use the QPU for 5 hours across all jobs in certain worse-case instances (this excludes queue-time and creation time on QPU for IBMQ devices). For all methods, timing was calculated based on qpu time (not wall-clock time). For the QA method, 'qpu\\_sampling\\_time' was used to calculate the total qpu time, while 'running time' is used for QAOA and QAOAH.\n            \n            \\Fig{results_1}(left) shows the average costs for all MCO configurations versus the number of qubits it took to encode each scenario 1 run. These costs do not include the cost accumulated from the embedded constraint functions \\Eq{0FakCSC5k-YoFIaG_KAN} and \\Eq{gm0YwnJ1TSQjD78B5eVR} as these are used to make the constraints hold. Also, all the costs plotted are actually the cost for that particular run minus the best possible cost it can receive. This best possible cost is computed by using brute force search methods in simulation. This difference is referred to as the relative cost. Therefore, the best possible relative cost a run could have is zero. For most runs, it can be seen that QA and QAOA runs have a relative cost around zero, but as the number of qubits increase, QAOA becomes less optimal compared to QA. The QAOAH approach, when using large amount of qubits, actually had a negative relative cost indicating that it must have violated some constraints in order to achieve a cost below the solution found with brute-force.\n\n            In \\Fig{Figure_16}, the average number of constraints violated is plotted against qubit size. The number of constraints is calculated by counting the amount of resources that were assigned to more than one mission. For example, if resource 1 was assigned to two extra missions, and resource 2 was assigned to three extra missions, then the number of constraints violated is 5. QA mostly had no constraints violated at any sized qubit problem, while QAOA and QAOAH suffered constraint violations when the problem size increased.\n\n        \\subsection{Scenario 2}\n            For scenario 2, the QAOAH method was calculated via IBM's state-vector numerical simulation tool due to the long running-time of runs in scenario 1. Because of this shift from running on actual hardware (scenario 1), to simulation (scenario 2), the QAOAH method is plotted using its wall-clock times against QA's \\& QAOA's QPU time in \\Fig{Figure_18}. Even with this change, the magnitudes of running-time are very diverse, so a third y-axis is added, as before. Furthermore, for the QAOAH approach, the mixing operator in \\Eq{12345CJNtD0ez2KQ6789} is used instead of \\Eq{eTDqpOkD5RxRJqJRkGde} because removing the column-swap terms reduces the total gate count of the overall computation, making simulation times faster.\n            \n            In scenario 2, QA times are quite faster than QAOA methods. It can be seen that the QAOAH method has far less flexibility in terms of qubit-range. This is because in our MCO algorithm implementation, extra qubits are required to represent each resource's ID. To keep consistency with the QA and QAOA methods, the x-axis in each plot in this section represents the number of qubits used minus the amount used to represent the IDs.\n            \n            \n            \\fig[.5]{Figure_18}{\\textbf{QPU \\& Simulated Results} QA \\& QAOA QPU Times and Simulated QAOAH Wall-Clock times for Scenario 2. QA timing includes the anneal time of each 50 samples, each 20 microseconds per sample. QAOA ran with 1000 shots. QAOAH ran in undermined amount of jobs using statevector-simulator to get results. Unlike Scenario 1, the times recorded are total wall-clock times for QAOAH.}\n            \n            As in scenario 1, the relative costs for each method is plotted in \\Fig{Figure_17}. Both QAOAH and QA methods have zero relative costs. However, QAOA by itself did not exhibit a positive relative cost. For both cases in this subplot, the data is insufficient to deduce whether or not these violated constraints.\n            \n            In \\Fig{Figure_19}, the average number of constraints violated is plotted against the number of qubits the problem encodes. The number of constraints violated is calculated similarly to scenario 1, but now including all violations of the second constraint. For example, if four resources of type-1 and 2 of type-2 were allocated to a mission, then the number of constraints violated is $4-2=2$. \\Fig{Figure_19} shows that QA and QAOAH did not violate constraints at any qubit size, while QAOA on average did. QAOA violated constraints mostly likely because it found a solution where the mission cost exceeded the cost incurred by the constraint function. QAOAH in simulation, however, did not violate constraints because the mixing operator that was used transforms solutions without leaving the constrained space. These results contrast with the QPU runs for QAOAH in Scenario 1 where it did violate constraints. Since the simulation ran without any noise profiles, it is expected that QAOAH shouldn't violate constraints in theory, but this is not the case when running on the actual quantum machine. The source of noise on actual hardware is most likely due to the gate noise and noise from measurement.\n            \n            \\begin{figure}\n                \\subfloat[\\label{fig:Figure_17}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_17.PNG}\n                }\n                \\vspace{1mm}\n               \n                \\subfloat[\\label{fig:Figure_19}]{\n                    \\includegraphics[width=1\\columnwidth]{Figure_19.PNG}\n                }\n                \\caption{\n                    \\textbf{QPU \\& Simulated Results}\n                    (a) Average relative cost versus the number of qubits used. The relative cost is the mission cost obtained minus the lowest possible mission cost that can be achieved (while keeping constraints) for the problem. The lowest possible cost is found by brute force. Negative relative costs must mean that the solution violated constraints.\n                    (b) Average number of constraints of each method versus the number of qubits used. The number of constraints counts up how many extras qubit are on within a column in the matrix view representation plus the absolute difference of qubits on within a row between resource sets ($\\mathbf{R}_1$ and $\\mathbf{R}_2$). This count includes the amount of columns that do not have any qubits on at all.\n                }\n                \\label{fig:results_2}\n            \\end{figure}\n            \n    \\section{Summary and Conclusion}\\label{sec:Conclusion}\n        In this paper, we introduced Mission Covering Optimization (MCO), implemented three different constraint optimization techniques (QA, QAOA, and QAOAH) to find solutions of two scenarios of MCO, and discussed results after running implementations on the IBM machine, D-Wave Machine, and on a state vector simulator. Results were compared based on timing, relative cost, qubits used, and constraints violated. From the 200 tests performed on each scenario, QA  achieved the quickest results while using the least number of qubits and violating the least number of constraints. We conjecture that QAOA and QAOAH approaches may have taken longer for the gradient descent algorithm to be convinced that an optimal solution was found because of the abundance of noise in current hardware. It was found that it is nontrivial to engineer in multiple constraints by embedding into the Mixing-Hamiltonian, especially when compared to the ease of using Lagrange multipliers, in simulation, where adding constraints entails simply adding terms together. The study conducted here suggests that the additional complexity in the QAOAH approach poses potential scalability challenges (due to the additional qubits required to ID the constraints) for problems similar to MCO with multiple types of unique constraints.\n        \n    \\section{Future Work}\\label{sec:Future:Work}\n        Adding further capabilities to different resources in Scenario 2 would make for a more interesting\/realistic optimization problems for quantum computers to solve. This is just one of the many alterations that can be done to MCO to increase the complexity of the optimization problem. \n        \n        In this paper, resources dependencies are modeled within MCO, however there may be missions with different priorities along with mission dependencies. Also, in this work resources were only shown to possess only one type of qualification. However, there can be cases where a resource may have many types of qualifications. At the heart of MCO, it is an optimization problem concerning allocation of resources invariant to time. An interesting direction for future study is how well this type of optimization problem can be ported to an extension of a Job shop problem.\n        \n        Although error mitigation is not a focus of this study, research for mitigating error for QAOA are being studied by others\\cite{PhysRevA.103.042412}. Employing error mitigation techniques for MCO is another interesting direction of study.\n\n\n        Lastly, another interesting solution method might entail the use of a ``bang-bang\" strategy~\\cite{bangbang} for multiple constraints. Here, for each constraint $C_i$ associated with a constraint Hamiltonian $H_i$, one might randomly cycle through applications of individual $H_i$ for each time step, as opposed to the application of the joint Hamiltonian $H = \\sum_i H_i$. While each $H_i$ only preserves constraints $C_i$, the supposition is that the application of $H_i$ might only partially violated constraints $C_{j\\ne i}$, if applied for a short time, and randomly. This solution approach to MCO-like problems will be investigated in future work.\n\n        \n    \\section{Acknowledgments}\n        The authors would like to thank David Vernooy and the exponential campaign at GE Research for supporting this effort.\n        %\n        The views expressed are those of the authors and do\n        not reflect the official guidance or position of the United\n        States Government, the Department of Defense, the\n        United States Air Force or General Electric. \n        %\n        The appearance of external hyperlinks does not constitute endorsement by the United States Department of Defense or General Electric of the linked websites, or the information, products, or services contained therein.  The Department of Defense and General Electric do not exercise any editorial, security, or other control over the information you may find at these locations.\n    \n   \n    \n   \n   \n   \n   \n   \n   \n    \n    \\nocite{*}\n\n    ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\\setcounter{equation}{0}\nOne of the main goals of modern experiments of heavy-ion collision (HIC) at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) is to study the quark matter under extreme conditions. These include extremely high temperatures ($>10^{12}$ K \\cite{rajagopal2018}), very large densities (up to $5\\rho_{0}$ with $\\rho_0=2.7\\times 10^{14}$ gr\/cm$^{3}$ \\cite{lacey2002}), extremely large electromagnetic fields ($10^{18}-10^{20}$ G \\cite{warringa2007}), and, in particular, large angular frequencies ($10^{22}$ Hz \\cite{becattini2016}). The aim is to imitate the circumstances of the early Universe, which is believed to be made of a hot plasma of free quarks and gluons. It is known that the plasma of quarks and gluons undergoes certain Quantum Chromodynamics (QCD) phase transitions upon cooling, and this leads to hadronization. These transitions include a deconfinement\/confinement and, in particular, a chiral phase transition. Theoretically, the QCD phase transitions can be studied using various effective QCD-like models, e.g. the Nambu-Jona--Lasinio (NJL) model \\cite{klevansky1992} and its extensions. Numerically, it is the merit of lattice QCD simulations at zero density \\cite{bazavov2013}, that show, \\textit{inter alia}, that these transitions occur at the same critical temperature, and they are nothing other than a smooth crossover. As concerns the QCD matter under high density\/high baryon chemical potential, it is shown, via model building, that it undergoes a certain spontaneous color symmetry breaking, that leads to the formation of diquarks in a color superconductive medium \\cite{fayazbakhsh2011}. Another important feature of noncentral HICs is the generation of very strong magnetic fields, which has many exciting effects on the Quark matter created in these collisions \\cite{warringa2007, skokov2009, huang2015}. These effects, including the (inverse) magnetic catalysis (see \\cite{shovkovy2015} and the references therein)\nand the chiral magnetic effect \\cite{fukushima2008}, are the subject of intensive studies in recent years. In particular, the impact of a constant magnetic field on the QCD phase diagram is studied intensively in the literature \\cite{fayazbakhsh2011, cao2021}. The main focus here is on the catalytic effect of constant magnetic fields. This enhances the formation of chiral condensates and thus leads, in comparison to the field-free case, to an increase of the critical temperature of the chiral phase transition $T_c$. There are, however, pieces of evidence from lattice QCD simulations that in the absence of baryonic chemical potential, $T_c$ decreases with increasing the strength of the magnetic field \\cite{bali2012-1, bali2012-2, delia2013,bruckmann2013}. This effect, which is previously dubbed \"inverse magnetic catalysis\" \\cite{rebhan2011}, is shown to be present in dense quark matter \\cite{fayazbakhsh2011}, or once the anomalous magnetic moment of the quark matter is nonzero \\cite{fayazbakhsh2014}, or an axial vector interaction is present\n\\cite{huang2014}, or when the scalar coupling constant of effective models depends on the magnetic field \\cite{farias2014, ferrer2014}, or for nonlocal chiral quark models, \\cite{scoccola2017}. The true reason for the inverse magnetic catalysis is still under debate (see \\cite{cao2021} and the references therein).\n\\par\nApart from extreme temperatures, densities, and external electromagnetic fields, the plasma of quarks and gluons created at RHIC and LHC possesses extremely large vorticity. This is the purpose of the present paper to focus on the interplay between rotation, magnetic field, and temperature on the chiral symmetry breaking (see below for more explanation). \nA simple estimate of the nonrelativistic vorticity $\\boldsymbol{\\Omega}=\\frac{1}{2}\\boldsymbol{\\nabla}\\times\\boldsymbol{v}$ is made in  \\cite{becattini2016}. Assuming that the difference between the $z$ component of the collective velocity in a HIC close to the target and projectile spectators is  about 0.1 (in the units of the speed of light), and that the transverse size of the system is about $5$ fm, the vorticity $\\Omega$ turns out to be of the order 0.02 fm$^{-1}\\sim 10^{22}$ Hz \\cite{becattini2016}. Many interesting transport phenomena are related to a rotating quark matter, whose macroscopic description is mainly made by relativistic hydrodynamics. Some of them are the chiral vortical effect and wave, in analogy to chiral magnetic effect and wave (see \\cite{kharzeev2015} and references therein). Similar to the case of magnetic fields, there are several attempts to study the phase structure of QCD under rotation. In \\cite{liao2016}, the effect of rotation on the formation of two different condensates in a hot and dense QCD matter, the chiral condensate and the color superconductivity, are studied, and the $T$-$\\Omega$ phase portrait is presented. It is found that a generic rotational suppression effect occurs, in particular, on the scalar pairing states. This effect is supposed to be caused by a rotational polarization effect induced by the global rotation. In order to check whether pairing states with nonzero angular momentum are favorable, the effect of rotation on the chiral phase transition in an NJL model with a vector interaction is studied in \\cite{huang2018}.  It is shown that whereas the phase structure in the $T$-$\\mu$ plane is sensitive to the coupling strength in the vector channel, the phase structure in $T$-$\\Omega$ plane is not. \nThe aforementioned suppression of the chiral condensate is originally found in \\cite{fukushima2015}. Here, the Dirac equation of a single flavor fermionic system is solved in the presence of rotation and magnetic field, and the corresponding energy dispersion relation is found. The latter indicates a close analogy between the rotation and the chemical potential because the energy spectrum is shifted similarly by a term proportional to the angular frequency $\\Omega$ of the fermionic system.\nIn \\cite{fukushima2015}, after solving the Dirac equation in the presence of rotation and magnetic field, and after determining the energy dispersion relation, the authors introduce the temperature and magnetic field in a system without boundary conditions. The zero temperature case is then derived by taking the limit $T\\to 0$. In this way, the dynamical mass exhibits $\\Omega$ dependence, and decreases with increasing $\\Omega$. At a certain critical $\\Omega$ the dynamical mass vanishes, and the chiral symmetry is restored. \nThe fact that the chiral condensate is suppressed in the presence of finite rotation is interpreted as the inverse magnetic catalysis, a phenomenon which occurs, in general, in low energy effective models at finite densities \\cite{fayazbakhsh2011}.  \nIt is referred to as \"rotational magnetic inhibition\".\\footnote{In this paper, we use the term \"inverse magneto-rotational effect\".} In \\cite{ebihara2017}, it is, however, shown that in an explicit computation at zero temperature, the dynamical mass does not depend on $\\Omega$. A fact that is also confirmed in the present paper. In the absence of magnetic fields, the authors in \\cite{ebihara2017} also introduce a global boundary condition to avoid causality-violating problems. This is also systematically done in a series of papers by Chernodub et al. \\cite{chernodub2016-1, chernodub2016-2, chernodub2016-3} in the absence and presence of magnetic fields. Here, another MIT boundary condition is imposed on the fermions on the surface of the cylinder, and its effect of the phase diagram of a QCD-like model in the presence of rotation is studied. The spectral and MIT boundary conditions are originally introduced in \\cite{kdrothe1980} and \\cite{chodos1974, lutken1984}. Various effects of these boundary conditions on the thermal expectation values of the fermion condensate, neutrino charge, and stress-energy tensor are studied intensively in \\cite{ambrus2016}. Other recent studies of the effect of rotation on the confinement\/deconfinement phase transition and mesonic condensation are studied in \\cite{chernodub2020, fukushima2021, braguta2021} and \\cite{zhang2020, cao2019}. \n\\par\nIn the present paper, we continue studying the interplay between rotation and magnetic field at zero and finite temperatures using a global boundary condition, and gain additional insights into IMRC. To do this, we use a one flavor NJL model, and solve numerically the corresponding gap equation for different fixed parameters $T,eB,\\Omega$, and $r$. The aim is, in particular, to find pieces of evidence for this effect in the phase diagrams of our model. The organization of the paper is as follows: In Sec. \\ref{sec2}, we solve the  Dirac equation within a cylinder using the Ritus eigenfunction method \\cite{ritus1972}. In Sec. \\ref{sec2a}, the solution is presented for a system with no boundary condition, and in  Sec. \\ref{sec2b}, it is given for a system with a global boundary condition. In Secs. \\ref{sec2a3} and \\ref{sec2b3}, the quantization of fermionic fields in a system without and with boundary conditions is demonstrated. It is then used in Sec. \\ref{sec2b4} to derive the fermion propagator of fermions in a bounded, rotating, and magnetized system. \nIn this context, the Ritus eigenfunction formalism is introduced as a methodical novelty in the present paper, though the same notations as previously introduced and utilized in \\cite{fayazbakhsh2011, fayazbakhsh-ritus, sadooghi2016, tabatabaee2020} are used. \n\\par\nIn Sec. \\ref{sec3}, the numerical solutions of the gap equation at zero and nonzero temperatures are presented (see Secs. \\ref{sec3a} and \\ref{sec3b}). At zero temperature, we mainly focus on the $r$ and $eB$ dependence of the dynamical mass for different values of NJL couplings. We show, in particular, that the $eB$ dependence of the dynamical mass at some fixed distance relative to the rotation axis and for a relatively large coupling exhibits certain oscillations. These are due to successive filling of the Landau levels. We then focus on the $T,eB,\\Omega$ and $r$ dependence of the dynamical mass at finite temperature.  We show, in particular, that the $eB$ dependence of $\\bar{m}$  decreases with increasing $eB$. Moreover, $\\bar{m}$ decreases with increasing $\\Omega$. These are clear pieces of evidence of IMRC. We study the $G,eB,\\Omega$, and $r$ dependence of the critical temperature $T_c$, and show that it decreases with $eB$ and $\\Omega$. We finally examine the $G,eB, T$, and $r$ dependence of the critical angular frequency $\\Omega_c$, and show that it decreases with $eB$ and $T$. The latter results can be viewed as a new piece of evidence of the IMRC. We devote Sec. \\ref{sec4} to a number of concluding remarks.       \n\\section{Ritus Eigenfunction formalism and rotating fermions in a constant magnetic field}\\label{sec2}\n\\setcounter{equation}{0}\nIn this section, we use the Ritus eigenfunction method \\cite{ritus1972} to solve the Dirac equation of a charged and massive fermion in the presence of a constant magnetic in a system that uniformly rotates with a constant angular velocity $\\Omega$ about a fixed axis. Being interested on the boundary effects, we set the system under certain global boundary condition, and explore its consequences for the solution of the corresponding Dirac equation. We assume that the magnetic field is aligned in the $z$-direction, and that all spatial regions of the system have the same angular velocity about the same axis (rigid rotation). This system is thus cylindrical symmetric around this axis, and is naturally described by the cylindrical coordinate system $x^{\\mu}=(t,x,y,z)=(t,r\\cos\\varphi,r\\sin\\varphi,z)$. The corresponding line element reads \\cite{chernodub2016-1}\n\\begin{eqnarray}\\label{N1}\nds^2&=&g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}=\\left(1-r^{2}\\Omega^{2}\\right)dt^2-dx^2\\nonumber\\\\\n&&+2\\Omega ydtdx-dy^2-2\\Omega xdtdy-dz^{2}.\n\\end{eqnarray}\nThis is equivalent to the metric\n\\begin{eqnarray}\\label{N2}\n\\hspace{-0.5cm}g_{\\mu\\nu}=\\left(\n\\begin{array}{cccc}\n1-\\Omega^2(x^2+y^2)&+\\Omega y&-\\Omega x&0\\\\\n+\\Omega y&-1&0&0\\\\\n-\\Omega x&0&-1&0\\\\\n0&0&0&-1\n\\end{array}\n\\right).\n\\end{eqnarray}\nAdopting the conventional notations in the curved space, we use the vierbein $e^{\\mu}_{~a}$ to connect the general coordinate with the Cartesian coordinate in the local rest frame (tangent space), $x^{\\mu}=e^{\\mu}_{~a}x^{a}$. Here, the Greek indices $\\mu=t,x,y,z$ refer to the general coordinate in the rotating frame, while the Latin indices $a=0,1,2,3$ to the Cartesian coordinate in the local rest frame. We choose the nonvanishing components of $e^{\\mu}_{~a}$ as \\cite{chernodub2016-1, fukushima2015}\n\\begin{eqnarray}\\label{N3}\ne^{\\mu}_{~a}:&\\quad& e^{t}_{~0}=e^{x}_{~1}=e^{y}_{~2}=e^{z}_{~3}=1,\\nonumber\\\\\n&& e^{x}_{~0}=+y\\Omega,\\quad e^{y}_{~0}=-x\\Omega.\n\\end{eqnarray}\nThey lead together with $g_{\\mu\\nu}$ from \\eqref{N2} to the metric $\\eta_{ab}=g_{\\mu\\nu}e^{\\mu}_{~a}e^{\\nu}_{~b}=\\mbox{diag}\\left(1,-1,-1,-1\\right)$.\n\\par\nIn a curved spacetime, the Dirac equation of a charged massive fermion in a constant background magnetic field is given by\n\\begin{eqnarray}\\label{N4}\n\\bigg[i\\gamma^{\\mu}\\left(D_{\\mu}^{(q)}+\\Gamma_{\\mu}\\right)-m_{q}\\bigg]\\psi(x)=0,\n\\end{eqnarray}\nwith $D_{\\mu}^{(q)}\\equiv \\partial_{\\mu}-iqeA_{\\mu}$. Here, $m_{q}$ is the mass of the fermion with charge $eq, e>0$. The gauge field $A_{\\mu}$ in the rotating frame is defined by $A_{\\mu}=e^{a}_{~\\mu}A_{a}$. Here, $e^{a}_{~\\mu}$s satisfy $e^{a}_{~\\mu}e^{\\mu}_{~b}=\\delta^{a}_{~b}$, and are given by\n\\begin{eqnarray*}\ne^{a}_{~\\mu}:&\\quad& e^{0}_{~t}=e^{1}_{~x}=e^{2}_{~y}=e^{3}_{~z}=1, \\nonumber\\\\\n&& e^{t}_{~1}=-y\\Omega,\\quad e^{t}_{~2}=+x\\Omega.\n\\end{eqnarray*}\nChoosing $A_{a}=\\left(0,-\\boldsymbol{A}\\right)=\\left(0,By\/2,-Bx\/2,0\\right)$, we arrive at a magnetic field aligned in the $z$-direction $\\boldsymbol{B}=B\\boldsymbol{\\hat{z}}$ with $B>0$. In \\eqref{N4}, the affine connection $\\Gamma_{\\mu}$ is defined in terms of the spin connection $\\omega_{\\mu ab}$ and vierbeins $e^{\\mu}_{~a}$ as\n\\begin{eqnarray}\\label{N5}\n\\Gamma_{\\mu}&\\equiv&-\\frac{i}{4}\\omega_{\\mu ab}\\sigma^{ab},\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{N6}\n\\omega_{\\mu ab}\\equiv g_{\\alpha\\beta}e^{\\alpha}_{~a}\\left(\\partial_{\\mu}e^{\\beta}_{~b}+\\Gamma^{\\beta}_{\\mu\\nu}e^{\\nu}_{~b}\\right),\n\\end{eqnarray}\nand $\\sigma^{ab}\\equiv\\frac{i}{2}[\\gamma^{a},\\gamma^{b}]$. In \\eqref{N6}, the Christoffel connection $\\Gamma^{\\beta}_{\\mu\\nu}\\equiv \\frac{1}{2}g^{\\beta\\sigma}\\left(\\partial_{\\mu}g_{\\sigma\\nu}+\\partial_{\\nu}g_{\\mu\\sigma}-\\partial_{\\sigma}g_{\\mu\\nu}\\right)$. As it turns out, for the metric \\eqref{N2}, the nonvanishing components of $\\Gamma^{\\beta}_{\\mu\\nu}$ are given by\n\\begin{eqnarray}\\label{N7}\n\\Gamma_{tt}^{x}=-\\Omega^{2}x,&\\quad&\\Gamma_{tt}^{y}=-\\Omega^{2}y,\\nonumber\\\\\n\\Gamma_{tx}^{y}=\\Gamma_{xt}^{y}=\\Omega,&\\quad& \\Gamma_{ty}^{x}=\\Gamma_{yt}^{x}=-\\Omega.\n\\end{eqnarray}\nThe affine connection $\\Gamma_{\\mu}$ is then given by\n\\begin{eqnarray}\\label{N8}\n\\Gamma_{t}=-\\frac{i}{2}\\Omega\\sigma^{12},\\quad\\Gamma_{x}=\\Gamma_{y}=\\Gamma_{z}=0.\n\\end{eqnarray}\nMoreover, the $\\gamma$-matrices in \\eqref{N4} are defined by $\\gamma^{\\mu}=e^{\\mu}_{~a}\\gamma^{a}$. For $e^{\\mu}_{~a}$ given in \\eqref{N3}, they read \\cite{chernodub2016-1}\n\\begin{eqnarray}\\label{N9}\n\\begin{array}{rclcrcl}\n\\gamma^{t}&=&\\gamma^{0},&\\quad&\\gamma^{x}&=&y\\Omega\\gamma^{0}+\\gamma^{1},\\\\\n\\gamma^{y}&=&-x\\Omega \\gamma^{0}+\\gamma^{2},&\\quad& \\gamma^{z}&=&\\gamma^{3}.\n\\end{array}\n\\end{eqnarray}\nPlugging $\\Gamma_{\\mu}$ from \\eqref{N8} and $\\gamma^{\\mu}$ from \\eqref{N9} into \\eqref{N4}, the explicit form of the Dirac equation of a rotating fermionic system in  a constant magnetic field reads\n\\begin{eqnarray}\\label{N10}\n\\left(\\gamma\\cdot \\Pi^{(q)}-m_{q}\\right)\\psi^{(q)}=0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{N11}\n\\gamma\\cdot\\Pi^{(q)}&\\equiv&\ni\\gamma^{0}\\left(\\partial_{t}-i\\Omega \\hat{J}_{z}\\right)+i\\gamma^{1}\\left(\\partial_x+iqeBy\/2\\right)\n\\nonumber\\\\\n&&+i\\gamma^{2}\\left(\\partial_y-iqeBx\/2\\right)+i\\gamma^{3}\\partial_{z},\n\\end{eqnarray}\nand $\\hat{J}_{z}\\equiv\\hat{L}_{z}+\\Sigma_{z}\/2$ with $\\hat{L}_{z}\\equiv-i\\left(x\\partial_{y}-y\\partial_{x}\\right)$, the total angular momentum in the $z$-direction, and $\\Sigma_{z}\\equiv \\mathbb{I}_{2\\times 2}\\otimes \\sigma^{3}$. Here, we used the Weyl representation of the $\\gamma$-matrices\n\\begin{eqnarray}\\label{N12}\n\\gamma^{0}=\\left(\\begin{array}{cc}\n0&1\\\\\n1&0\n\\end{array}\n\\right),\\quad\\boldsymbol{\\gamma}=\\left(\n\\begin{array}{cc}\n0&\\boldsymbol{\\sigma}\\\\\n-\\boldsymbol{\\sigma}&0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwith $\\boldsymbol{\\sigma}=\\left(\\sigma^{1},\\sigma^{2},\\sigma^{3}\\right)$ are the Pauli matrices, and $[\\sigma^{i},\\sigma^{j}]=2i\\epsilon^{ijk}\\sigma^{k}$ to get $\\sigma^{12}=\\frac{i}{2}[\\gamma^{1},\\gamma^{2}]=\\Sigma_{z}$.   Moreover, $\\mathbb{I}_{2\\times 2}\\equiv \\text{diag}(1,1)$.\n\\par\nSimilar to the description presented in \\cite{tabatabaee2020}, in the Ritus eigenfunction method, we start solving \\eqref{N10} by making use of the Ansatz $\\psi_{+}^{(q)}=\\mathbb{E}_{\\lambda,\\ell,+}^{(q)}u\\left(\\tilde{p}_{\\ell,+}\\right)$ for the positive frequency solution and $\\psi_{-}^{(q)}=\\mathbb{E}_{\\lambda,\\ell,-}^{(q)}v\\left(\\tilde{p}_{\\ell,-}\\right)$ for the negative frequency solution. Here, $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ with $\\kappa=\\pm 1$ satisfies the Ritus eigenfunction relation\n\\begin{eqnarray}\\label{N13}\n\\left(\\gamma\\cdot \\Pi^{(q)}\\right)\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\kappa \\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\left(\\gamma\\cdot\\tilde{p}^{(q)}_{\\lambda,\\ell,\\kappa}\\right),\n\\end{eqnarray}\nwhere $\\Pi^{(q)}$ is defined in \\eqref{N11}. The aim is to determine the Ritus function $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ and the Ritus momentum $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$ in terms of $\\lambda$. The latter plays the role of Landau levels in a rotating system (see below). Using the Weyl basis \\eqref{N12} for the $\\gamma$-matrices, the operator $\\gamma\\cdot \\Pi^{(q)}$ turns out to be\n\\begin{eqnarray}\\label{N14}\n\\gamma\\cdot \\Pi^{(q)}=\\left(\n\\begin{array}{cc}\n0&\\Pi^{(q)}_{R}\\\\\n\\Pi^{(q)}_{L}&0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwith\n\\begin{widetext}\n\\begin{eqnarray}\\label{N15}\n\\Pi^{(q)}_{R}&=&\\left(\n\\begin{array}{ccc}\ni\\partial_t+\\Omega\\left(\\hat{L}_{z}+1\/2\\right)+i\\partial_z&&+i\\left(\\partial_{x}+iqeBy\/2\\right)+\\left(\\partial_{y}-iqeBx\/2\\right)\\\\\n+i\\left(\\partial_{x}+iqeBy\/2\\right)-\\left(\\partial_{y}-iqeBx\/2\\right)&&i\\partial_t+\\Omega\\left(\\hat{L}_{z}-1\/2\\right)-i\\partial_z\n\\end{array}\n\\right),\\nonumber\\\\\n\\Pi^{(q)}_{L}&=&\\left(\n\\begin{array}{ccc}\ni\\partial_t+\\Omega\\left(\\hat{L}_{z}+1\/2\\right)-i\\partial_z&~~&-i\\left(\\partial_{x}+iqeBy\/2\\right)-\\left(\\partial_{y}-iqeBx\/2\\right)\\\\\n-i\\left(\\partial_{x}+iqeBy\/2\\right)+\\left(\\partial_{y}-iqeBx\/2\\right)&~~&i\\partial_t+\\Omega\\left(\\hat{L}_{z}-1\/2\\right)+i\\partial_z\n\\end{array}\n\\right).\n\\end{eqnarray}\nIn a cylinder coordinate system $(r,\\varphi,z)$ with $\\left(x=r\\cos\\varphi, y=r\\sin\\varphi, z\\right)$, \\eqref{N15} is equivalently given by\n\\begin{eqnarray}\\label{N16}\n\\Pi^{(q)}_{R}&=&\\left(\n\\begin{array}{ccc}\ni\\partial_{t}+\\Omega\\left(-i\\partial_{\\varphi}+\\frac{1}{2}\\right)+i\\partial_z&~~&+ie^{-i\\varphi}\\left(\\partial_{r}-\\frac{i}{r}\\partial_{\\varphi}-\\frac{qeB}{2}r\\right)\\\\\n+ie^{+i\\varphi}\\left(\\partial_{r}+\\frac{i}{r}\\partial_{\\varphi}+\\frac{qeB}{2}r\\right)&~~&\ni\\partial_{t}+\\Omega\\left(-i\\partial_{\\varphi}-\\frac{1}{2}\\right)-i\\partial_{z}\n\\end{array}\n\\right),\\nonumber\\\\\n\\Pi^{(q)}_{L}&=&\\left(\n\\begin{array}{ccc}\ni\\partial_{t}+\\Omega\\left(-i\\partial_{\\varphi}+\\frac{1}{2}\\right)-i\\partial_z&~~&-ie^{-i\\varphi}\\left(\\partial_{r}-\\frac{i}{r}\\partial_{\\varphi}-\\frac{qeB}{2}r\\right)\\\\\n-ie^{+i\\varphi}\\left(\\partial_{r}+\\frac{i}{r}\\partial_{\\varphi}+\\frac{qeB}{2}r\\right)&~~&\ni\\partial_{t}+\\Omega\\left(-i\\partial_{\\varphi}-\\frac{1}{2}\\right)+i\\partial_{z}\n\\end{array}\n\\right).\n\\end{eqnarray}\n\\end{widetext}\nTo arrive at \\eqref{N16}, we used\n\\begin{eqnarray}\\label{N17}\ni\\partial_{x}\\pm\\partial_{y}=ie^{\\mp i\\varphi}\\left(\\partial_{r}\\mp \\frac{i}{r}\\partial_{\\varphi}\\right),\n\\end{eqnarray}\nand replaced $\\hat{L}_{z}$ with $\\hat{L}_{z}=-i\\partial_{\\varphi}$. Plugging $\\psi_{+}^{(q)}=\\mathbb{E}_{\\lambda,\\ell,+}^{(q)}u\\left(\\tilde{p}_{\\ell,+}\\right)$ and $\\psi_{-}^{(q)}=\\mathbb{E}_{\\lambda,\\ell,-}^{(q)}v\\left(\\tilde{p}_{\\ell,-}\\right)$ into \\eqref{N10}, and using \\eqref{N13}, we arrive at\n\\begin{eqnarray}\\label{N18}\n\\left(\\gamma\\cdot \\tilde{p}_{\\lambda,\\ell,+}^{(q)}-m_{q}\\right)u\\left(\\tilde{p}_{\\ell,+}\\right)&=&0,\\nonumber\\\\\n\\left(\\gamma\\cdot \\tilde{p}_{\\lambda,\\ell,-}^{(q)}+m_{q}\\right)v\\left(\\tilde{p}_{\\ell,-}\\right)&=&0.\n\\end{eqnarray}\nThe solutions are the standard Dirac spinors of free electrons with $p^{\\mu}=(p_0,\\boldsymbol{p})$ replaced with $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$, where $\\kappa=+1$ ($\\kappa=-1$) denotes the positive (negative) frequency solution of the Dirac equation. \n\\par\nIn what follows, we first determine $\\mathbb{E}_{\\lambda,\\ell, \\kappa}^{(q)}$ and  $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$ in a system with no boundary condition. We then consider a certain global boundary condition, and determine $\\mathbb{E}_{\\lambda,\\ell, \\kappa}^{(q)}$ and  $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$. In both cases, we present the quantization relations for fermionic field operators $\\bar{\\psi}^{(q)}$ and $\\psi^{(q)}$.\n\\subsection{Rotating magnetized fermions in a system with no boundary condition}\\label{sec2a}\n\\subsubsection{Determination of $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$}\\label{sec2a1}\nTo determine $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ in this case, we use, similar to the nonrotating case \\cite{tabatabaee2020}, the Ansatz\n\\begin{eqnarray}\\label{N19}\n\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=e^{-i\\kappa\\left(E_{\\lambda,\\ell,\\kappa} t-p_{z}z\\right)}\\mathbb{P}_{\\lambda,\\ell}^{(q)},\n\\end{eqnarray}\nwith the projector defined by\n\\begin{eqnarray}\\label{N20}\n\\mathbb{P}_{\\lambda,\\ell}^{(q)}\\equiv P_{+}f_{\\lambda,\\ell,s_q}^{+}+P_{-}f_{\\lambda,\\ell,s_q}^{-},\n\\end{eqnarray}\n$s_{q}\\equiv \\text{sgn}\\left(qeB\\right)$ and the spin projector\n\\begin{eqnarray}\\label{N21}\nP_{\\pm}\\equiv \\frac{1\\pm i\\gamma^{1}\\gamma^{2}}{2}.\n\\end{eqnarray}\nIn \\eqref{N19}, $E_{\\lambda,\\ell,\\kappa}$ and $p_{z}$ are the zeroth and fourth components of the Ritus momentum $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$. Because of the specific structure of the $\\gamma$-matrices in the Weyl representation, $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ reduces to a block diagonal matrix in the form\n\\begin{eqnarray}\\label{N22}\n\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\left(\n\\begin{array}{cc}\n\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}&0\\\\\n0&\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}\n\\end{array}\n\\right),\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{N23}\n\\hspace{-1cm}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}=e^{-i\\kappa\\left(E_{\\lambda,\\ell,\\kappa}^{(q)}t-p_z z\\right)}\\left(\n\\begin{array}{cc}\nf_{\\lambda,\\ell,s_q}^{+}&0\\\\\n0&f_{\\lambda,\\ell,s_q}^{-}\n\\end{array}\n\\right).\n\\end{eqnarray}\nPlugging this Ansatz into\n\\begin{eqnarray}\\label{N24}\n\\hat{J}_{z}\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\left(\\ell+\\frac{1}{2}\\right)\\mathbb{E}_{\\kappa,\\lambda}^{(q)},\n\\end{eqnarray}\nwith $\\hat{J}_{z}=\\hat{L}_{z}+\\Sigma_{z}\/2$, we arrive at\n\\begin{eqnarray}\\label{N25}\n\\hat{L}_{z}f_{\\lambda,\\ell,s_q}^{+}&=&\\ell f_{\\lambda,\\ell,s_q}^{+},\\nonumber\\\\\n\\hat{L}_{z}f_{\\lambda,\\ell,s_q}^{-}&=&\\left(\\ell+1\\right) f_{\\lambda,\\ell,s_q}^{-}.\n\\end{eqnarray}\nPlugging $\\hat{L}_{z}=-i\\partial_{\\varphi}$ into \\eqref{N25}, we arrive immediately at\n\\begin{eqnarray}\\label{N26}\nf_{\\lambda,\\ell,s_q}^{+}&=&e^{i\\ell\\varphi}\\chi_{\\lambda,\\ell,s_q}^{+},\\nonumber\\\\\nf_{\\lambda,\\ell,s_q}^{-}&=&e^{i\\left(\\ell+1\\right)\\varphi}\\chi_{\\lambda,\\ell,s_q}^{-},\n\\end{eqnarray}\nwith unknown functions $\\chi_{\\lambda,\\ell,s_q}^{\\pm}$. To determine these functions, we consider first the quadratic equation\n\\begin{eqnarray}\\label{N27}\n\\left(\\gamma\\cdot \\Pi^{(q)}\\right)^2\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}= \\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\left(\\gamma\\cdot\\tilde{p}^{(q)}_{\\lambda,\\ell,\\kappa}\\right)^2.\n\\end{eqnarray}\nPlugging $\\gamma\\cdot \\Pi^{(q)}$ from \\eqref{N14} and $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ from \\eqref{N22} into \\eqref{N27}, and using $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)2}=m_{q}^{2}$ as well as $\\Pi_{L}^{(q)}\\Pi_{R}^{(q)}=\\Pi_{R}^{(q)}\\Pi_{L}^{(q)}$, we arrive at\n\\begin{eqnarray}\\label{N28}\n\\Pi_{L}^{(q)}\\Pi_{R}^{(q)}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}=m_{q}^{2}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)},\n\\end{eqnarray}\nwith  $\\Pi_{L}^{(q)}\\Pi_{R}^{(q)}$ given by\n\\begin{eqnarray}\\label{N29}\n\\Pi_{L}^{(q)}\\Pi_{R}^{(q)}=\\left(\\begin{array}{cc}\n\\mathscr{O}_{+}&0\\\\\n0&\\mathscr{O}_{-}\n\\end{array}\n\\right).\n\\end{eqnarray}\nHere,\n\\begin{eqnarray}\\label{N30}\n\\mathscr{O}_{\\pm}&\\equiv&\\left(i\\partial_{t}-\\Omega\\left(i\\partial_{\\varphi}\\mp 1\/2\\right)\\right)^{2}+\\partial_{r}^{2}+\\frac{1}{r}\\partial_{r}+\\frac{\\partial_{\\varphi}^{2}}{r^{2}}\\nonumber\\\\\n&&-qeB\\left(i\\partial_{\\varphi}\\mp 1\\right)-\\left(\\frac{qeB}{2}\\right)^{2}r^{2}+\\partial_{z}^{2}.\n\\end{eqnarray}\nThe differential equation for $\\chi_{\\lambda,\\ell,s_q}^{\\pm}$ arise by plugging $\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ from \\eqref{N23} with $f_{\\lambda,\\ell,s_q}^{\\pm}$ from \\eqref{N26} into \\eqref{N28}. We thus arrive at\n\\begin{eqnarray}\\label{N31}\n\\bigg[x\\partial_{x}^{2}+\\partial_{x}+\\lambda-\\frac{\\ell^{2}}{4x}+\\frac{s_{q}(\\ell+1)}{2}-\\frac{x}{4}\\bigg]\\chi_{\\lambda,\\ell,s_q}^{+}=0,\\nonumber\\\\\n\\bigg[x\\partial_{x}^{2}+\\partial_{x}+\\lambda-\\frac{(\\ell+1)^{2}}{4x}+\\frac{s_{q}\\ell}{2}-\\frac{x}{4}\\bigg]\\chi_{\\lambda,\\ell,s_q}^{-}=0,\\nonumber\\\\\n\\end{eqnarray}\nwhere $x\\equiv \\frac{|qeB|r^{2}}{2}$ and\n\\begin{eqnarray}\\label{N32}\n\\lambda\\equiv \\frac{\\left(E_{\\lambda,\\ell,\\kappa}^{(q)}+\\kappa\\Omega j\\right)^{2}-p_z^2-m_q^2}{2|qeB|},\n\\end{eqnarray}\nwith $j\\equiv \\ell+1\/2$. Hence, according to \\eqref{N32}, the energy dispersion relation for a rotating and magnetized fermionic system reads\n\\begin{eqnarray}\\label{N33}\n\\hspace{-1cm}E_{\\lambda,\\ell,\\kappa}^{(q)}=-\\kappa \\Omega j\\pm\\sqrt{2\\lambda|qeB|+p_{z}^{2}+m_{q}^{2}}.\n\\end{eqnarray}\nTo solve the differential equations \\eqref{N31}, we use the Ansatz\n\\begin{eqnarray}\\label{N34}\n\\chi_{\\lambda,\\ell,s_q}^{+}&=&e^{-x\/2}x^{|\\ell|\/2}g_{\\lambda,\\ell,s_q}^{+},\\nonumber\\\\\n\\chi_{\\lambda,\\ell,s_q}^{-}&=&e^{-x\/2}x^{|\\ell+1|\/2}g_{\\lambda,\\ell,s_q}^{-},\n\\end{eqnarray}\nwhich leads to\n\\begin{eqnarray}\\label{N35}\n\\hspace{-1cm}&&\\big[x\\partial_{x}^{2}+\\left(|\\ell|+1-x\\right)\\partial_{x}+\\mathscr{N}_{\\lambda,s_q}^{+}\\big]g_{\\lambda,\\ell,s_q}^{+}=0,\\nonumber\\\\\n\\hspace{-1cm}&&\\big[x\\partial_{x}^{2}+\\left(|\\ell+1|+1-x\\right)\\partial_{x}+\\mathscr{N}_{\\lambda,s_q}^{-}\\big]g_{\\lambda,\\ell,s_q}^{-}=0,\n\\end{eqnarray}\nupon plugging \\eqref{N34} into \\eqref{N31}. Here,\n\\begin{eqnarray}\\label{N36}\n\\mathscr{N}_{\\lambda,s_q}^{+}&\\equiv&\\lambda+\\frac{s_{q}\\left(\\ell+1\\right)-|\\ell|-1}{2},\\nonumber\\\\\n\\mathscr{N}_{\\lambda,s_q}^{-}&\\equiv&\\lambda+\\frac{s_{q}\\ell-|\\ell+1|-1}{2}.\n\\end{eqnarray}\nComparing the differential equations \\eqref{N35} with Kummer's differential equation\n$$\n\\left(z\\partial_{z}^{2}+\\left(b-z\\right)\\partial_z-a\\right)g(z)=0,\n$$\nwhose solution\n$$\ng(z)=A~{}_{1}F_{1}\\left(a;b;z\\right)+B~ U\\left(a;b;z\\right),\n$$\nis a linear combination of a hypergeometric function of the first and second kind ${}_{1}F_{1}\\left(a;b;x\\right)$ and $U\\left(a;b;x\\right)$, and requiring that $g^{\\pm}_{\\lambda,\\ell,s_q}$ are regular at $x\\to 0$,\\footnote{This is equivalent to $r\\to 0$.} we arrive at\n\\begin{eqnarray}\\label{N37}\ng_{\\lambda,\\ell,s_q}^{+}&=&\\mathscr{A}^{+}~{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{+};|\\ell|+1;x\\right),\\nonumber\\\\\ng_{\\lambda,\\ell,s_q}^{-}&=&\\mathscr{A}^{-}~{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1;x\\right).\n\\end{eqnarray}\nHere, $\\mathscr{A}^{\\pm}$ are appropriate normalization factors, which are determined by using the orthonormality relation\n\\begin{eqnarray}\\label{N38}\n\\lefteqn{\\hspace{-0.8cm}\n\\int d^{4}r~\\bar{\\mathbb{E}}_{\\lambda^{\\prime},\\ell^{\\prime},\\kappa}^{(q)}(r)\n\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}(r)}\\nonumber\\\\\n&&\\hspace{-0.5cm}=\\delta\\left(E_{\\lambda^{\\prime},\\ell^{\\prime},\\kappa}^{(q)}-E_{\\lambda,\\ell,\\kappa}^{(q)}\\right)\\delta\\left(k_{z}-k_{z}^{\\prime}\\right)\n\\delta_{\\lambda,\\lambda^{\\prime}}\\delta_{\\ell,\\ell^{\\prime}}.\n\\end{eqnarray}\nIn cylinder coordinate system, we have $d^{4}r=dt r dr d\\varphi dz$. Moreover, $\\bar{\\mathbb{E}}_{\\lambda^{\\prime},\\ell^{\\prime},\\kappa}^{(q)}(r)=\\gamma^{0}\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)\\dagger}\\gamma^{0}$. Here, $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ is given in \\eqref{N22} with $\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}$\nfrom \\eqref{N23}, and $f_{\\lambda,\\ell,s_q}^{\\pm}$ read\n\\begin{eqnarray}\\label{N39}\nf_{\\lambda,\\ell,s_q}^{+}&=&\\mathscr{A}^{+}e^{i\\ell\\varphi}e^{-x\/2}x^{|\\ell|\/2}~{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{+};|\\ell|+1;x\\right),\\nonumber\\\\\nf_{\\lambda,\\ell,s_q}^{-}&=&\\mathscr{A}^{-}e^{i\n\\left(\\ell+1\\right)\\varphi}e^{-x\/2}x^{|\\ell+1|\/2}\\nonumber\\\\\n&&\\times{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1;x\\right).\n\\end{eqnarray}\nThese solutions are general and valid for both cases of a rotating fermionic system without and with a boundary condition. Let us now assume, that the rotating magnetized fermions are in a system with no spatial boundary condition. As it turns out, in this case, the parameter $\\lambda$ in \\eqref{N32} is a positive integer, i.e. $\\lambda\\in\\mathbb{N}_{0}$. Thus $\\lambda$ plays the role of Landau levels similar to the case of nonrotating fermions in a magnetic field. On the other hand, if the first argument $-\\mathscr{N}_{\\lambda,s_q}^{\\pm}$ in ${}_{1}F_{1}$ appearing in \\eqref{N39} is a nonpositive integer, the hypergeometric function can be replaced by the associated Laguerre polynomials,\n \\begin{eqnarray*}\n{}_{1}F_{1}\\left(-n;m+1;z\\right)=\\frac{m! n!}{(m+n)!}L_{n}^{m}(z).\n\\end{eqnarray*}\nThe solutions \\eqref{N39} thus read\n\\begin{eqnarray}\\label{N40}\nf_{\\lambda,\\ell,s_q}^{+}(x)&=&\\frac{\\mathscr{A}^{+}\\mathscr{N}_{\\lambda,s_q}^{+}!|\\ell|!}{\\left(\\mathscr{N}_{\\lambda,s_q}^{+}+|\\ell|\\right)!}e^{i\\ell\\varphi}e^{-x\/2}x^{|\\ell|\/2}L_{\\mathscr{N}_{\\lambda,s_q}^{+}}^{|\\ell|}\\left(x\\right),\\nonumber\\\\\nf_{\\lambda,\\ell,s_q}^{-}(x)&=&\\frac{\\mathscr{A}^{-}\\mathscr{N}_{\\lambda,s_q}^{-}!|\\ell+1|!}{\\left(\\mathscr{N}_{\\lambda,s_q}^{-}+|\\ell+1|\\right)!}e^{i\n\\left(\\ell+1\\right)\\varphi}e^{-x\/2}x^{|\\ell+1|\/2}\\nonumber\\\\\n&&\\times L_{\\mathscr{N}_{\\lambda,s_q}^{-}}^{|\\ell+1|}\\left(x\\right).\n\\end{eqnarray}\nUsing then \\eqref{N38} for $r\\in[0,\\infty[$ and the orthonormality relations of the Laguerre polynomial\n\\begin{eqnarray*}\n\\int_{0}^{\\infty}dz z^{\\alpha} e^{-z}L_{n}^{\\alpha}(z)L_{m}^{\\alpha}(z)=\\frac{(n+\\alpha)!}{n!}~\\delta_{m,n},\n\\end{eqnarray*}\nfor $\\mbox{Re}(\\alpha)>-1$, $\\mathscr{A}^{\\pm}$ are determined. We finally arrive at\n\\begin{eqnarray}\\label{N41}\nf_{\\lambda,\\ell,s_q}^{+}&=&\\left(\\frac{|qeB|}{2\\pi}\\frac{\\mathscr{N}_{\\lambda,s_q}^{+}!}{\\left(\\mathscr{N}_{\\lambda,s_q}^{+}+|\\ell|\\right)!}\\right)^{1\/2}e^{i\\ell\\varphi}e^{-x\/2}x^{|\\ell|\/2}\\nonumber\\\\\n&&\\times L_{\\mathscr{N}_{\\lambda,s_q}^{+}}^{|\\ell|}(x),\\nonumber\\\\\nf_{\\lambda,\\ell,s_q}^{-}&=&\n\\left(\\frac{|qeB|}{2\\pi}\\frac{\\mathscr{N}_{\\lambda,s_q}^{-}!}{\\left(\\mathscr{N}_{\\lambda,s_q}^{-}+|\\ell+1|\\right)!}\\right)^{1\/2}\ne^{i\n\\left(\\ell+1\\right)\\varphi}e^{-x\/2}\\nonumber\\\\\n&&\\times x^{|\\ell+1|\/2}L_{\\mathscr{N}_{\\lambda,s_q}^{-}}^{|\\ell+1|}\\left(x\\right).\n\\end{eqnarray}\nHere,  $P_{\\pm}^{\\dagger}=\\gamma_{0}P_{\\pm}\\gamma_{0}$, $P_{\\pm}^{2}=P_{\\pm}$, and $P_{\\pm}P_{\\mp}=0$ are also used. In Table \\ref{table1}, $\\mathscr{N}_{\\lambda,s_q}^{\\pm}$ for $s_q=+1$ and $s_q=-1$, corresponding to $q>0$ and $q<0$, are listed. Let us notice that the Laguerre polynomials $L_{\\mathscr{N}_{\\lambda,s_q}^{\\pm}}^{\\cdots}$ appearing in \\eqref{N41} are defined only for $\\mathscr{N}_{\\lambda,s_q}^{\\pm}\\geq 0$. This constraints the choice for $\\ell$ for positively and negatively charged particles with positive and negative spins, $s=+1$ and $s=-1$, respectively.\\footnote{Let us remind that the positive $+$ and negative $-$ upper indices on $\\mathscr{N}^{\\pm}_{s_q}$ denote the up ($s=+1$) and down ($s=-1$) spin orientations.} In Table \\ref{table2}, the allowed values of $\\ell$ for different choices of $\\lambda$ are demonstrated. According to this table, the lowest energy level (LEL) with $\\lambda=0$ is only occupied either with positively charged particles with $s=+1$ and $\\ell\\geq 0$ or with negatively charged particles with $s=-1$ and $\\ell\\leq -1$. As concerns the higher energy levels with $\\lambda \\geq 1$, they can be occupied with positively and negatively charged particles with both spin orientations $s=+1$ (spin up) and $s=-1$ (spin down). For positively charged particles the allowed values for $\\ell$ are $\\ell=-\\lambda,-\\lambda+1,\\cdots,-2,-1,0,1,2,\\cdots$ and for negatively charged particles $\\ell=\\cdots,-2,-1,0,1,2,\\cdots,\\lambda-2,\\lambda-1$.\n\\begin{center}\n\\begin{table}[ht]\n    \\centering\n\\caption{The values for $\\mathscr{N}_{\\lambda,s_q}^{\\pm}$ appearing in the first argument of the hypergeometric functions ${}_{1}F_{1}(a;b;x)$ in \\eqref{N39}. Assuming $eB>0$, $s_{q}=+1$ and $s_{q}=-1$ correspond to $q>0$  and $q<0$, respectively.}\\label{table1}    \\vspace{1ex}\n    \\vspace{1ex}\n        \\begin{tabular}{c|rclcrcl}\n            \\hline\\hline\n&$\\ell$&$\\leq$&-1&\\qquad\\qquad&$\\ell$&$\\geq$&$0$\\\\\n\\hline\n  $q>0,s=+1$&$\\mathscr{N}_{\\lambda,+}^{+}$&$=$&$\\lambda+\\ell$&\\qquad\\qquad&$\\mathscr{N}_{\\lambda,+}^{+}$&=&$\\lambda$ \\\\\n\n  $q>0,s=-1$&$\\mathscr{N}_{\\lambda,+}^{-}$&$=$&$\\lambda+\\ell$&\\qquad\\qquad&$\\mathscr{N}_{\\lambda,+}^{-}$&=&$\\lambda-1$ \\\\\n\n  $q<0,s=+1$&$\\mathscr{N}_{\\lambda,-}^{+}$&$=$&$\\lambda-1$&\\qquad\\qquad&$\\mathscr{N}_{\\lambda,-}^{+}$&=&$\\lambda-\\ell-1$ \\\\\n\n  $q<0,s=-1$&$\\mathscr{N}_{\\lambda,-}^{-}$&$=$&$\\lambda$&\\qquad\\qquad&$\\mathscr{N}_{\\lambda,-}^{-}$&=&$\\lambda-\\ell-1$ \\\\\n            \\hline\\hline\n        \\end{tabular}\n\\end{table}\n\\end{center}\n\\begin{table*}[ht]\n    \\centering\n    \\caption{The allowed values of $\\ell$ for the LEL, $\\lambda=0$, and higher energy levels, $\\lambda\\geq 1$, and for different combinations of particles' charge $Q$ and spin $s$. In an infinitely extended rotating fermionic system in a constant magnetic field, we have $\\lambda\\in \\mathbb{N}_{0}$.}\n    \\label{table2}\n    \\vspace{1ex}\n        \\begin{tabular}{c|rclcl}\n            \\hline\\hline\n            \\multirow{2}{*}{$q>0,s=+1$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&$\\ell=0,1,2,\\cdots$\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=-\\lambda,-\\lambda+1,\\cdots,-2,-1,0,1,2,\\cdots$ \\\\\n            \\hline\n            \\multirow{2}{*}{$q>0,s=-1$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&---\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=-\\lambda,-\\lambda+1,\\cdots,-2,-1,0,1,2,\\cdots$ \\\\\n            \\hline\\hline\n            \\multirow{2}{*}{$q<0,s=+1$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&---\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1,0,1,2,\\cdots,\\lambda-2,\\lambda-1$ \\\\\n            \\hline\n            \\multirow{2}{*}{$q<0,s=-1$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1$\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1,0,1,2,\\cdots,\\lambda-2,\\lambda-1$ \\\\\n            \\hline\\hline\n        \\end{tabular}\n\\end{table*}\nPlugging $f_{\\lambda,\\ell,s_q}^{\\pm}$ from \\eqref{N40} into \\eqref{N23} and the resulting expression into \\eqref{N22}, \nthe Ritus function $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ for an infinitely large fermionic system in a constant magnetic field is determined. \nLet us notice, however, that since the Ansatz \\eqref{N20} for $\\mathbb{P}_{\\lambda,\\ell}^{(q)}$ does not take the boundaries for $\\ell$ demonstrated\n in Table \\ref{table2} into account, it has to be accordingly modified. This is done in Sec. \\ref{sec2a3}, where the final expression for the quantization of the \n fermionic fields in a multiflavor system under rotation and constant magnetic field with no boundary condition is presented.\n\\subsubsection{Determination of $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$}\\label{sec2a2}\nTo determine $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)}$, let us first consider the Ritus relation \\eqref{N13} with $\\gamma\\cdot \\Pi^{(q)}$ from \\eqref{N14}, and $\\Pi_{R\/L}^{(q)}$ from \\eqref{N16}. Plugging $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ from \\eqref{N22} into the left hand side of \\eqref{N13}, we obtain\n\\begin{eqnarray}\\label{N42}\n\\hspace{-1cm}\\left(\\gamma\\cdot \\Pi^{(q)}\\right)\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\n\\left(\\begin{array}{cc}\n0&\\Pi_{R}^{(q)}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\\\\n\\Pi_{L}^{(q)}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}&0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwith $\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ given in terms of $f_{\\lambda,\\ell,s_q}^{\\pm}$ [see \\eqref{N23}]. For the sake of generality, we use $f_{\\lambda,\\ell,s_q}^{\\pm}$, from \\eqref{N39} in terms of the hypergeometric function ${}_{1}F_{1}(a;b;z)$. To determine the resulting differential equations $-ie^{\\pm i\\varphi}\\left(\\partial_{r}\\pm \\frac{i}{r}\\partial_{\\varphi}\\pm\\frac{qeB}{2}r\\right)$ which appear on the right hand side (r.h.s.) of \\eqref{N42}, we use following relations\n\\begin{eqnarray}\\label{N43}\n&&\nb{}_{1}F_{1}\\left(a;b;z\\right)-b{}_{1}F_{1}\\left(a-1;b;z\\right)-z{}_{1}F_{1}\\left(a;b+1;z\\right)=0,\\nonumber\\\\\n&&\n{}_{1}F_{1}\\left(a;b;z\\right)-\\frac{(b+z)}{b}{}_{1}F_{1}\\left(a;b+1;z\\right)-\\frac{\\left(a-b-1\\right)}{b(b+1)}\\nonumber\\\\\n&&\\qquad \\times{}_{1}F_{1}\\left(a;b+2;z\\right)=0, \\nonumber\\\\\n&&\\left(a-b+1\\right){}_{1}F_{1}\\left(a;b;z\\right)-a~{}_{1}F_{1}\\left(a+1;b;z\\right)\\nonumber\\\\\n&&\\qquad-\\left(1-b\\right){}_{1}F_{1}\\left(a;b-1;z\\right)=0,\n\\end{eqnarray}\nand arrive after some work at\n\\begin{eqnarray}\\label{N44}\n\\lefteqn{\\hspace{-2cm}-ie^{\\pm i\\varphi}\\left(\\partial_{r}\\pm \\frac{i}{r}\\partial_{\\varphi}\\pm\\frac{qeB}{2}r\\right)f_{\\lambda,\\ell,s_q}^{\\pm}\n}\\nonumber\\\\\n&&=\\pm is_{\\ell}\\sqrt{2\\lambda|qeB|}f_{\\lambda,\\ell,s_q}^{\\mp},\n\\end{eqnarray}\nwith $s_{\\ell}\\equiv \\mbox{sgn}(\\ell)=1$ for $\\ell\\geq 0$ and $=-1$ for $\\ell\\leq -1$. Plugging these results into the r.h.s. of \\eqref{N42}, we obtain\n\\begin{eqnarray}\\label{N45}\n\\Pi_{R}^{(q)}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\kappa\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\Xi_{R}^{(q)},\\qquad\n\\Pi_{L}^{(q)}\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\kappa\\mathscr{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\Xi_{L}^{(q)},\\nonumber\\\\\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{N46}\n\\Xi_{R}^{(q)}&\\equiv& \\left(\n\\begin{array}{cc}\n\\epsilon_{\\lambda}^{(q)}-p_{z}&+i\\kappa s_\\ell\\sqrt{2\\lambda|qeB|}\\\\\n+i\\kappa s_\\ell\\sqrt{2\\lambda|qeB|}&\\epsilon_{\\lambda}^{(q)}+p_{z}\n\\end{array}\n\\right),\\nonumber\\\\\n\\Xi_{L}^{(q)}&\\equiv&\\left(\n\\begin{array}{cc}\n\\epsilon_{\\lambda}^{(q)}+p_{z}&-i\\kappa s_\\ell\\sqrt{2\\lambda|qeB|}\\\\\n+i\\kappa s_\\ell\\sqrt{2\\lambda|qeB|}&\\epsilon_{\\lambda}^{(q)}-p_{z}\n\\end{array}\n\\right),\\nonumber\\\\\n\\end{eqnarray}\nwith $\\epsilon_{\\lambda}^{(q)}\\equiv E_{\\lambda,\\ell,\\kappa}^{(q)}+\\kappa\\Omega j$.\nThese lead eventually to\n\\begin{eqnarray}\\label{N47}\n\\left(\\gamma\\cdot \\Pi^{(q)}\\right)\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}=\\kappa\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}\\left(\n\\begin{array}{cc}\n0&\\Xi_{R}^{(q)}\\\\\n\\Xi_{L}^{(q)}&0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwith $\\Xi_{R\/L}^{(q)}$ from \\eqref{N46}. Comparing, at this stage, \\eqref{N47} with the r.h.s. of \\eqref{N13}, we arrive immediately at the Ritus momentum in a rotating fermionic system\n\\begin{eqnarray}\\label{N48}\n\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)\\mu}=\\left(\\epsilon_{\\lambda}^{(q)}, 0, \\kappa s_\\ell\\sqrt{2\\lambda|qeB|},p_{z}\\right),\n\\end{eqnarray}\nwhere $\\epsilon_{\\lambda}^{(q)}=E_{\\lambda,\\ell,\\kappa}^{(q)}+\\kappa\\Omega j$ with $j\\equiv \\ell+\\frac{1}{2}$. Let us notice that since $\\tilde{p}_{\\lambda,\\ell,\\kappa}^{(q)2}=m_{q}^{2}$,  \\eqref{N48} leads to\n\\begin{eqnarray}\\label{N49}\n\\epsilon_{\\lambda}^{(q)}=\\pm\\left(m_{q}^{2}+2\\lambda|qeB|+p_{z}^{2}\\right)^{1\/2}.\n\\end{eqnarray}\nThus, $\\epsilon_{\\lambda}^{(q)}$ depends only on $\\lambda$,\\footnote{Here, $\\kappa^{2}=1$ and $\\mbox{sgn}^{2}(\\ell)=1$ are used.} which, for a rotating fermionic system with no boundary condition, is a positive integer, i.e. $\\lambda\\in \\mathbb{N}_{0}$. Hence, the expression under the squared root in \\eqref{N49} turns out to be always positive. Let us also notice that according to our construction, $E_{\\lambda,\\ell,\\kappa}^{(q)}$ is always positive, while $\\epsilon_{\\lambda}$ is allowed to be positive and negative.\n\\subsubsection{Quantization of fermionic fields in an infinitely extended rotating system}\\label{sec2a3}\nCombining the above results, the quantization relations for a magnetized fermion in a rotating system without boundary read\n\\begin{eqnarray}\\label{N50}\n\\psi_{\\alpha}^{(q)}(x)&=&\\sum_{\\ell,\\lambda,s}\\int \\frac{dp_z}{2\\pi}\\frac{1}{\\sqrt{2|\\epsilon_{\\lambda}^{(q)}}|}\\left\\{\ne^{-i\\left(E_{\\lambda,\\ell,+}^{(q)}t-p_{z}z\\right)}a_{p_{z}}^{\\lambda,\\ell,s}\\right.\\nonumber\\\\\n&&\\left.\\times \\big[\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}\\left(x\\right)\\big]_{\\alpha\\rho}u_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right)\\Theta\\left(E_{\\lambda,\\ell,+}^{(q)}\\right)\\right.\\nonumber\\\\\n&&\\left.+\ne^{+i\\left(E_{\\lambda,\\ell,-}^{(q)}t-p_{z}z\\right)}b_{p_{z}}^{\\lambda,\\ell,s\\dagger}\\big[\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}\\left(x\\right)\\big]_{\\alpha\\rho}^{\\dagger}\\right.\\nonumber\\\\\n&&\\left.\\times v_{s,\\rho}\\left(\\tilde{p}_{\\ell,-}\\right)\\Theta\\left(E_{\\lambda,\\ell,-}^{(q)}\\right)\\right\\}. \\nonumber\\\\\n\\bar{\\psi}_{\\alpha}^{(q)}(x)&=&\\sum_{\\ell,\\lambda,s}\\int \\frac{dp_z}{2\\pi}\\frac{1}{\\sqrt{2|\\epsilon_{\\lambda}^{(q)}}|}\\left\\{\ne^{+i\\left(E_{\\lambda,\\ell,+}^{(q)}t-p_{z}z\\right)}a_{p_{z}}^{\\lambda,\\ell,s\\dagger}\\right.\\nonumber\\\\\n&&\\left.\\times \\bar{u}_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right) \\big[\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}\\left(x\\right)\\big]_{\\rho\\alpha}^{\\dagger}\\Theta\\left(E_{\\lambda,\\ell,+}^{(q)}\\right)\\right.\\nonumber\\\\\n&&\\left.+\ne^{-i\\left(E_{\\lambda,\\ell,-}^{(q)}t-p_{z}z\\right)}b_{p_{z}}^{\\lambda,\\ell,s}\n\\bar{v}_{s,\\rho}\\left(\\tilde{p}_{\\ell,-}\\right)\\right. \\nonumber\\\\\n&&\\left.\\times\n\\big[\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}\\left(x\\right)\\big]_{\\rho\\alpha}\n\\Theta\\left(E_{\\lambda,\\ell,-}^{(q)}\\right)\\right\\}.\n\\end{eqnarray}\nHere, $a_{p_{z}}^{\\lambda,\\ell,s\\dagger}$ and $a_{p_{z}}^{\\lambda,\\ell,s}$, as well as\n$b_{p_{z}}^{\\lambda,\\ell,s\\dagger}$ and $b_{p_{z}}^{\\lambda,\\ell,s}$ are the\ncreation and annihilation operators of particles and antiparticles. They satisfy the commutation relations\n\\begin{eqnarray}\\label{N51}\n\\hspace{-0.9cm}\n\\{a_{p_{z}}^{\\lambda,\\ell,s},a_{p_{z}^{\\prime}}^{\\lambda^{\\prime},\\ell^{\\prime},s^{\\prime}\\dagger}\\}&=&2\\pi\\delta\\left({p_{z}}-p_{z}^{\\prime}\\right)\\delta_{\\lambda,\\lambda^{\\prime}}\\delta_{\\ell,\\ell^{\\prime}}\\delta_{s,s^{\\prime}},\\nonumber\\\\\n\\hspace{-0.9cm}\n\\{b_{p_{z}}^{\\lambda,\\ell,s},b_{p_{z}^{\\prime}}^{\\lambda^{\\prime},\\ell^{\\prime},s^{\\prime}\\dagger}\\}&=&2\\pi\\delta\\left({p_{z}}-p_{z}^{\\prime}\\right)\\delta_{\\lambda,\\lambda^{\\prime}}\\delta_{\\ell,\\ell^{\\prime}}\\delta_{s,s^{\\prime}}.\n\\end{eqnarray}\nIn \\eqref{N50}, $\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}$, the modified version of $\\mathbb{P}_{\\lambda,\\ell}^{(q)}$ from \\eqref{N20}, reads\n\\begin{eqnarray}\\label{N52}\n\\hspace{-0.5cm}\\widetilde{\\mathbb{P}}_{\\lambda,\\ell}^{(q)}=\\left(\\mathscr{P}_{+}^{(q)}f_{\\lambda,\\ell,s_q}^{+s_q}+\\Pi_{\\lambda}\\mathscr{P}_{-}^{(q)}f_{\\lambda,\\ell,s_q}^{-s_{q}}\\right)\\Gamma_{\\lambda,\\ell,q},\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{N53}\n\\mathscr{P}^{(q)}_{\\pm}\\equiv \\frac{1\\pm is_{q}\\gamma_1\\gamma_{2}}{2},\n\\end{eqnarray}\nwhich leads to $\\mathscr{P}_{\\pm}^{(+)}=\\mathscr{P}_{\\pm}$ and $\\mathscr{P}_{\\pm}^{(-)}=\\mathscr{P}_{\\mp}$, and\n\\begin{eqnarray}\\label{N54}\n\\Pi_{\\lambda}&\\equiv& 1-\\delta_{\\lambda,0},\\nonumber\\\\\n\\Gamma_{\\lambda,\\ell,q}&\\equiv&\\Theta\\left(q\\right)\\Theta\\left(\\ell+\\lambda\\right)+\\Theta\\left(-q\\right)\\Theta\\left(-\\ell+\\lambda-1\\right). \\nonumber\\\\\n\\end{eqnarray}\nHere, $\\Pi_{\\lambda}$ considers the degeneracy of Landau levels. The fact that positively charged particles with negative spins, and negatively charged particles with positive spins do not occupy the LEL (see Table \\ref{table2}) is considered by $\\Gamma_{\\lambda,\\ell,q}$.\nThe above definitions are written in the same language as previously presented in \\cite{tabatabaee2020} for the solutions of the Dirac equation of nonrotating fermions in a constant magnetic field. \n\\par\nIn \\eqref{N52}, $f_{\\lambda,\\ell,s_q}^{\\pm}$ are given in \\eqref{N41}. Moreover, in \\eqref{N50} and \\eqref{N54}, the Heaviside function $\\Theta(z)=+1$  and $=0$ for $z\\geq 0$ and  $z<0$, respectively.\nAs aforementioned, the modification of $\\mathbb{P}_{\\lambda,\\ell}^{(q)}$ according to Table \\ref{table2} is necessary because in this way, the allowed values of $\\ell$ for positively and negatively charged fermions with up or down spins are considered directly in the solutions of the Dirac equation as well as the quantization of the Dirac fields for a  system without boundary conditions. In what follows, we introduce the global boundary condition for the fermionic field to avoid the system having a velocity that exceeds the speed of light \\cite{fukushima2017}.\n\\subsection{Rotating magnetized fermions in a system with a global boundary condition}\\label{sec2b}\n\\subsubsection{Imposing a global boundary condition}\\label{sec2b1}\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig1a}\n\\includegraphics[width=8cm,height=6cm]{fig1b}\n\\caption{color online. (a) The $\\ell$ dependence of the first, second, and third roots $\\lambda_k, k=1,2,3$ of the hypergeometric functions appearing in \\eqref{N61} ($\\ell\\geq 0$) and \\eqref{N62} ($\\ell\\leq -1$). (b) The $\\ell$ dependence of the energies $RE_{\\lambda_1,\\ell,+}^{(+)}$ of the first root $\\lambda_1$ for different $R\\Omega=0, 0.3,0.6$. The energy $E_{\\lambda_1,\\ell,+}^{(+)}$ corresponds to a positively charged particle ($s_q=+1, \\kappa=+1$).}\\label{fig1}\n\\end{figure*}\n\\begin{figure*}\n\t\\includegraphics[width=8cm,height=6cm]{fig2a}\n\t\\includegraphics[width=8cm,height=6cm]{fig2b}\n\t\\caption{color online. The $R\\Omega$ dependence of $E_{\\lambda_1,\\ell,+}^{(+)}$ (panel a) and $E_{\\lambda_1,\\ell,-}^{(-)}$ (panel b). }\\label{fig2}\n\\end{figure*}\nThe solutions of the Dirac equation for magnetized fermions in a nonrotating system with the global boundary condition are already presented in \\cite{fukushima2017}. In what follows, we use the solutions from Sec. \\ref{sec2a} for a rotating quark matter with no boundary, and impose the same global boundary condition at $r=R$ as in \\cite{fukushima2017},\n\\begin{eqnarray}\\label{N55}\nI\\equiv \\int_{-\\infty}^{+\\infty}dz\\int_{0}^{2\\pi} d\\varphi \\bar{\\psi}^{(q)}\\gamma^{r}\\psi^{(q)}\\bigg|_{r=R}=0,\n\\end{eqnarray}\nwith $R$  the cylinder radius  and $\\gamma^{r}=\\gamma^{1}\\cos\\varphi+\\gamma^{2}\\sin\\varphi$. In contrast to \\cite{fukushima2017}, the solutions for $\\psi^{(q)}$ and $\\bar{\\psi}^{(q)}$ are derived using the Ritus eigenfunction method (see Sec. \\ref{sec2a}).\nPlugging  $\\psi^{(q)}=\\mathbb{E}_{\\lambda,\\ell,+}^{(q)}u(\\tilde{p}_{\\ell,+})$ and\n$\\bar{\\psi}^{(q)}=\\bar{u}(\\tilde{p}_{\\ell,+})\\mathbb{E}_{\\lambda,\\ell,+}^{(q)}$\nfor positive frequency solution ($\\kappa=+1$) and $\\psi^{(q)}=\\mathbb{E}_{\\lambda,\\ell,-}^{(q)}v(\\tilde{p}_{\\ell,-})$ as well as  $\\bar{\\psi}^{(q)}=\\bar{v}(\\tilde{p}_{\\ell,-})\\mathbb{E}_{\\lambda,\\ell,-}^{(q)}$ for negative frequency solution ($\\kappa=-1$), we arrive first at\n\\begin{eqnarray}\\label{N56}\n\\text{I}_{+}\n&\\equiv&\\int_{-\\infty}^{+\\infty}dz\\int_{0}^{2\\pi} d\\varphi \\bar{u}\\left(\\tilde{p}^{\\prime}_{\\ell^{\\prime},+}\\right)\\nonumber\\\\\n&&\\times\\mathbb{E}_{\\lambda^{\\prime},\\ell^{\\prime},+}^{(q)}\\gamma^{r}\\mathbb{E}_{\\lambda,\\ell,+}^{(q)}u\\left(\\tilde{p}_{\\ell,+}\\right)\\bigg|_{r=R}=0, \\nonumber\\\\\n\\text{I}_{-}&\\equiv&\\int_{-\\infty}^{+\\infty}dz\\int_{0}^{2\\pi} d\\varphi \\bar{v}\\left(\\tilde{p}^{\\prime}_{\\ell^{\\prime},-}\\right)\\nonumber\\\\\n&&\\times\\mathbb{E}_{\\lambda^{\\prime},\\ell^{\\prime},-}^{(q)}\\gamma^{r}\\mathbb{E}_{\\lambda,\\ell,-}^{(q)}v\\left(\\tilde{p}_{\\ell,-}\\right)\\bigg|_{r=R}=0.\n\\end{eqnarray}\nUsing then $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(q)}$ from \\eqref{N19} with $\\mathbb{P}_{\\lambda,\\ell}^{(q)}$ from \\eqref{N20}, and $f_{\\lambda,\\ell,s_{q}}^{\\pm}$ from \\eqref{N40} as well as $\\gamma^{r}P_{\\pm}=P_{\\mp}\\gamma^{r}$, $I_{\\pm}$ become proportional to\n\\begin{eqnarray}\\label{N57}\n0=\\text{I}_{+}&\\propto& \\bar{u}\\left(\\tilde{p}_{\\lambda,\\ell,+}^{(q)}\\right)\\mathscr{H}_{\\lambda^{\\prime},\\lambda,\\ell}u\\left(\\tilde{p}_{\\lambda^{\\prime},\\ell,+}^{(q)}\\right),\\nonumber\\\\\n0=\\text{I}_{-}&\\propto& \\bar{v}\\left(\\tilde{p}_{\\lambda,\\ell,+}^{(q)}\\right)\\mathscr{H}_{\\lambda^{\\prime},\\lambda,\\ell}v\\left(\\tilde{p}_{\\lambda^{\\prime},\\ell,+}^{(q)}\\right),\n\\end{eqnarray}\nwhere $\\alpha_b\\equiv x(r={R})=|qeB|R^{2}\/2$, and\n\\begin{eqnarray}\\label{N58}\n\\mathscr{H}_{\\lambda^{\\prime},\\lambda,\\ell}\\equiv\n\\left(\n\\begin{array}{cccc}\n0&0&0&+h_{\\lambda^{\\prime},\\lambda,\\ell}^{(1)}\\\\\n0&0&+h_{\\lambda^{\\prime},\\lambda,\\ell}^{(2)}&0\\\\\n0&-h_{\\lambda^{\\prime},\\lambda,\\ell}^{(1)}&0&0\\\\\n-h_{\\lambda^{\\prime},\\lambda,\\ell}^{(2)}&0&0&0\n\\end{array}\n\\right),\\nonumber\\\\\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{N59}\nh_{\\lambda^{\\prime},\\lambda,\\ell}^{(1)}&\\equiv& {}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda^{\\prime},s_q}^{+};|\\ell|+1;\\alpha_{b}\\right)\\nonumber\\\\\n&&\\times{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1;\\alpha_{b}\\right),\n\\end{eqnarray}\nand $h_{\\lambda^{\\prime},\\lambda,\\ell}^{(2)}\\equiv h_{\\lambda,\\lambda^{\\prime},\\ell}^{(1)}$. In order to fulfill the boundary condition $I=0$ with $I$ from \\eqref{N55}, we have to find the solution of\n\\begin{eqnarray}\\label{N60}\n{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{+};|\\ell|+1;\\alpha_{b}\\right)&=&0,\\nonumber\\\\\n{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1;\\alpha_{b}\\right)&=&0.\n\\end{eqnarray}\nBut, before doing this let us notice that the hypergeometric function ${}_{1}F_{1}(a;b;z)$, being defined as\n\\begin{eqnarray*}\n{}_{1}F_{1}(a;b;z)=\\sum_{k=0}^{\\infty}\\frac{\\left(a\\right)_{k}}{\\left(b\\right)_{k}}\\frac{z^{k}}{k!},\n\\end{eqnarray*}\nyields only a polynomial with a finite number of terms, when $a<0$, and either $b>0$ or $b<a$ \\cite{mathworld}. Here, $(x)_{n}\\equiv\\Gamma(x+n)\/\\Gamma(x)$ is the Pochhammer symbol. In Table \\ref{table3}, we summarize the intervals for $-\\mathscr{N}_{\\lambda,s_q}^{+}<0$ and $|\\ell|+1>0$ or $|\\ell|+1<-\\mathscr{N}_{\\lambda,s_q}^{+}$  corresponding to\n${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{+};|\\ell|+1;\\alpha_{b}\\right)$ as well as\n$-\\mathscr{N}_{\\lambda,s_q}^{-}<0$ and $|\\ell+1|+1>0$ or $|\\ell+1|+1<-\\mathscr{N}_{\\lambda,s_q}^{-}$ corresponding to\n${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1;\\alpha_{b}\\right)$.\n\\begin{table*}[ht]\n    \\centering\n    \\caption{ The allowed intervals of $\\ell$ for which the hypergeometric functions ${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{+};|\\ell|+1;\\alpha_{b}\\right)$ and ${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,s_q}^{-};|\\ell+1|+1\\right)$ yield polynomials with a finite number of terms. Here, $\\lambda>0$ is assumed.}\n    \\label{table3}\n    \\vspace{1ex}\n        \\begin{tabular}{c|rclcl}\n            \\hline\\hline\n            \\multirow{2}{*}{${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,+}^{+};|\\ell|+1;\\alpha_{b}\\right)$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&$\\ell=0,1,2,\\cdots$\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=-\\lambda,-\\lambda+1,\\cdots,-2,-1,0,1,2,\\cdots$ \\\\\n            \\hline\n            \\multirow{2}{*}{${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,+}^{-};|\\ell+1|+1;\\alpha_{b}\\right)$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&---\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=-\\lambda,-\\lambda+1,\\cdots,-2,-1,0,1,2,\\cdots$ \\\\\n            \\hline\\hline\n            \\multirow{2}{*}{${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,-}^{+};|\\ell|+1;\\alpha_{b}\\right)$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&---\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1,0,1,2,\\cdots,\\lambda-2,\\lambda-1$ \\\\\n            \\hline\n            \\multirow{2}{*}{${}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,-}^{-};|\\ell+1|+1;\\alpha_{b}\\right)$}\n            &$\\lambda$&$=$&$0$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1$\\\\\n            &$\\lambda$&$\\geq$&$1$&\\qquad\\qquad&$\\ell=\\cdots,-2,-1,0,1,2,\\cdots,\\lambda-2,\\lambda-1$ \\\\\n            \\hline\\hline\n        \\end{tabular}\n\\end{table*}\n\\par\nAs concerns the solutions of the hypergeometric functions appearing in \\eqref{N60}, we first consider $s_q=+1$, and choose $\\alpha_{b}=7$,\\footnote{The numerical results presented in Sec. \\ref{sec3} correspond to $q>0$ leading to $s_q=1$.}. Then, setting\n\\begin{eqnarray}\\label{N61}\n\\begin{array}{rclcccc}\n{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,+}^{+};|\\ell|+1;\\alpha_{b}\\right)&=&0,&~~&\\mbox{for}&~~&\\ell\\geq 0,\n\\end{array}\\nonumber\\\\\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{N62}\n\\begin{array}{rclcccc}\n{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda,+}^{-};|\\ell+1|+1;\\alpha_{b}\\right)&=&0,&~~&\\mbox{for}&~~&\\ell\\leq -1,\n\\end{array}\\nonumber\\\\\n\\end{eqnarray}\nwe determine numerically the roots of these two functions.\nIn Fig. \\ref{fig1}(a), the results of the first, second, and third roots of the hypergeometric functions in \\eqref{N61} and \\eqref{N62} are plotted. The roots are not symmetrically distributed around $j=0$, i.e., there is a certain asymmetry with respect to $j\\to -j$ or equivalently $\\ell \\to -\\ell-1$, which is also observed in \\cite{fukushima2017}. Here, it is argued that this is because of broken $\\mathcal{C}$ and $\\mathcal{CP}$ symmetry in the case of nonvanishing magnetic field. Let us denote these roots with $\\lambda_{k}, k=1,2,\\cdots$. In contrast to the previous case of no boundary condition, $\\lambda_{k}\\in \\mathbb{R}$ (i.e. they are not necessarily integers). Plugging $\\lambda_{k}$ into the energy dispersion equation [see also \\eqref{N40}],\n\\begin{eqnarray}\\label{N63}\n\\hspace{-0.4cm}E_{\\lambda_{k},\\ell,\\kappa}^{(q)}=-\\kappa \\Omega j+\\sqrt{\\frac{4\\lambda_{k}\\alpha_b}{R^2}+p_{z}^{2}+m_{q}^{2}},\n\\end{eqnarray}\nwhere $j=\\ell+1\/2$ and $\\alpha_b=|qeB|R^2\/2$, yields the value of each energy level. At this stage, it is important to check whether $E_{\\lambda_{k},\\ell,\\kappa}^{(q)}$ for given values of $\\kappa, \\Omega, m_q,p_z,R,\\alpha_b$ and for $\\ell\\in(-\\infty,+\\infty)$ remains positive. To do this, let us consider $\\kappa=+1, s_q=+1 (\\text{or equivalently}~q>0), m_{q}=0, p_{z}=0$, and $\\alpha_{b}=7$. Plugging $\\lambda_{1}$ ($k=1$) from Fig. \\ref{fig1}(a) into \\eqref{N63}, and assuming that the transverse size of the QGP in the early stage of the collision to be $R=6$ fm \\cite{becattini2016}, we arrive at $E_{\\lambda_{1},\\ell,+}^{(+)}$ as a function of $\\ell$ and $\\Omega$. In Fig. \\ref{fig1}(b), $RE_{\\lambda_{1},\\ell,+}^{(+)}$ is plotted for $R\\Omega=0, 0.3, 0.6$ and $-19\\leq\\ell\\leq19$. Since according to \\cite{becattini2016}, $\\Omega=0.02\\text{~fm}^{-1}\\sim 10^{22}\\text{~s}^{-1}$, $R\\Omega=0.3$ and $R\\Omega=0.6$ with $R=6$ fm correspond to $2.5\\times 10^{22}\\text{~s}^{-1}$ and $5\\times 10^{22}\\text{~s}^{-1}$, respectively. The energy  $E_{\\lambda_{1},\\ell,+}^{(+)} $ turns out to be positive in the whole interval of $\\ell$.\n\\par\nIn Fig. \\ref{fig2}, we have shown that the LEL is affected by the choice of the angular frequency $\\Omega$. In Fig. \\ref{fig2}(a), the $R\\Omega$ dependence of $RE_{\\lambda_1,\\ell,+}^{(+)}$ is demonstrated for $\\ell=0$ and $\\ell=1$. The data corresponding to up (down) spin $s=+1$ ($s=-1$) arise by plugging $m_{q}=p_z=0, R=6~\\text{fm},\\kappa=+$ ($\\kappa=-$) into $\\eqref{N63}$ with $\\lambda_1$ determined from \\eqref{N61} for $s=+1$ as well as \\eqref{N62} for $s=-1$ with $s_q=+1$ (or equivalently a positively charged particle) and $\\ell=0$ and $\\ell=1$.\\footnote{Let us remind that the superscripts $\\pm$ in $\\mathscr{N}^{+}_{\\lambda,s_q}$ and $\\mathscr{N}^{-}_{\\lambda,s_q}$ appearing in \\eqref{N61} and \\eqref{N62} correspond to fermions with spin up ($+$) and down ($-$).} In Fig. \\ref{fig2}(b), the same is done for $\\kappa=-1$ and $s_q=-1$~(or equivalently a negatively charged antiparticle).  As it turns out, for positively charged particles in the regime $R\\Omega<0.6$, the energy level corresponding to $\\ell=0$ and $s=+1$ is lower than $\\ell=0, s=+1$ (green squares) and $\\ell=1, s=+1$  (yellow triangles). For $R\\Omega= 0.6$, however, the energy level for $\\ell=+1$ becomes lower than that corresponding to $\\ell=0$, and for $R\\Omega>0.7$ negative $E_{\\lambda_1,\\ell,+}^{(+)}$ appear, which are unacceptable. The same effect\nis observed in Fig. \\ref{fig2}, for $\\ell=0, s=-1$ (gray circles) and $\\ell=1, s=-1$ (red stars). The same plot shows that for $s_q=+1$ (or equivalently $q>1$), in general, the energy levels for $s=+1$ is lower than the energy levels for $s=-1$.\n\\par\nAs concerns the results for a negatively charged antiparticle in Fig. \\ref{fig2}(b), it turns out that, in contrast to the positively charged particle, the energies corresponding to $\\ell=0$ and spin orientations $s=+1$ and $s=-1$ are lower than the energies corresponding to $\\ell=1$ with $s=\\pm 1$. We thus conclude that in general, the spin degeneracy in the LEL for magnetized and rotating Dirac fermions with a global boundary condition is to be determined numerically.\n\\subsubsection{Normalization of the wave functions with the global boundary condition}\\label{sec2b2}\nIn Sec. \\ref{sec2a}, we used the Ritus eigenfunction method, and derived the solutions to the Dirac equation in a rotating system of fermions in a constant background magnetic field. When the system is infinitely extended, i.e. when no boundary conditions are imposed, the Ritus function $\\mathbb{E}_{\\lambda,\\ell,\\kappa}^{(\\kappa)}$ is given by \\eqref{N19}-\\eqref{N20}, with $f^{\\pm}_{\\lambda,\\ell,s_q}$ given in \\eqref{N41}. Here, the normalization factors $\\mathscr{A}^{\\pm}$ from \\eqref{N40} are determined by using the orthonormality of the Laguerre polynomials. In what follows, we determine $\\mathscr{A}^{\\pm}$ for a fermionic system under the global boundary condition \\eqref{N55}.\n\\par\nTo do this, we follow the method introduced in \\cite{fukushima2017}. Here, $f^{\\pm}_{\\lambda_{k},\\ell,\\kappa}$ from \\eqref{N39} are given by\n\\begin{eqnarray}\\label{N64}\nf_{\\lambda_k,\\ell,s_q}^{+}&=&\\mathscr{C}_{k,\\ell,s_q}^{+}e^{i\\ell\\varphi}\\Phi_{\\lambda_k,\\ell,s_q}^{+},\\nonumber\\\\\nf_{\\lambda_k,\\ell,s_q}^{-}&=&\\mathscr{C}^{-}e^{i(\\ell+1)\\varphi}\\Phi_{\\lambda_k,\\ell,s_q}^{-},\n\\end{eqnarray}\nwhere two functions $\\Phi_{\\lambda_k,\\ell,s_q}^{\\pm}$ are defined as\\footnote{Let us remind that in this case ($R\\to \\infty$) the hypergeometric functions appearing in \\eqref{N65} are to be replaced with the Laguerre function $L_{n}^{m}$, as described in Sec. \\ref{sec2a}.}\n\\begin{eqnarray}\\label{N65}\n\\Phi_{\\lambda_{k},\\ell,s_q}^{+}&\\equiv&\\frac{1}{|\\ell|!}\\left(\\frac{|qeB|}{2\\pi}\\frac{\\left(\\mathscr{N}_{\\lambda_k,s_q}^{+}+|\\ell|\\right)!}{\\mathscr{N}_{\\lambda_k,s_q}^{+}!}\\right)^{1\/2}e^{-x\/2}\\nonumber\\\\\n&&\\times x^{|\\ell|\/2}{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda_k,s_q}^{+};|\\ell|+1;x\\right),\\nonumber\\\\\n\\Phi_{\\lambda_{k},\\ell,s_q}^{-}&\\equiv&\\frac{1}{|\\ell+1|!}\\left(\\frac{|qeB|}{2\\pi}\\frac{\\left(\\mathscr{N}_{\\lambda_k,s_q}^{-}+|\\ell+1|\\right)!}{\\mathscr{N}_{\\lambda_k,s_q}^{-}!}\\right)^{1\/2}\\nonumber\\\\\n&&\\times e^{-x\/2}x^{|\\ell+1|\/2}{}_{1}F_{1}\\left(-\\mathscr{N}_{\\lambda_k,s_q}^{-};|\\ell+1|+1;x\\right). \\nonumber\\\\\n\\end{eqnarray}\nThe normalization factors  $\\mathscr{C}_{k,\\ell,s_q}^{\\pm}$ in a cylinder with a finite radius $R$ are defined so that by taking the limit $R\\to \\infty$, the results from \\eqref{N41} of an unbounded rotating system are reproduced \\cite{fukushima2017},\n \\begin{eqnarray}\\label{N66}\n\\mathscr{C}_{k,\\ell,s_q}^{+}&=&\\left(\\frac{|qeB|}{2\\pi\\int\\limits_{0}^{\\alpha_b}dx\\left(\\Phi_{\\lambda_k,\\ell,s_q}^{+}(x)\\right)^{2}}\\right)^{1\/2},\\nonumber\\\\\n\\mathscr{C}_{k,\\ell,s_q}^{-}&=&\\left(\\frac{|qeB|}{2\\pi\\int\\limits_{0}^{\\alpha_b}dx\\left(\\Phi_{\\lambda_k,\\ell,s_q}^{-}(x)\\right)^{2}}\\right)^{1\/2}.\n\\end{eqnarray}\nFor $\\lambda_{k}$s that satisfy two conditions in \\eqref{N61} and \\eqref{N62}, it turns out that $\\mathscr{C}_{k,\\ell,s_q}^{+}=\\mathscr{C}_{k,\\ell,s_q}^{-}\\equiv \\mathscr{C}_{k,\\ell,s_q}$. This can be shown numerically. This is the same result as previously reported in \\cite{fukushima2017} for a nonrotating quark matter in the presence of a constant magnetic field. \n\\subsubsection{Quantization of fermionic fields in a system with global boundary conditions}\\label{sec2b3}\nIn Sec. \\ref{sec2a3}, we presented the quantization relations of fermions in a rotating system without boundary condition [see \\eqref{N50}-\\eqref{N54}]. Imposing a global boundary condition does not change this quantization too much. The exact quantization relations for fermionic fields $\\psi$ and $\\bar{\\psi}$ are given by\n\\begin{eqnarray}\\label{N67}\n\\psi_{\\alpha}^{(q)}(x)&=&\\sum_{k,\\ell,s}\\int \\frac{dp_z}{2\\pi}\\frac{\\mathscr{C}_{k,\\ell,s_q}}{\\sqrt{2|\\epsilon_{\\lambda_{k}}^{(q)}}|}\\left\\{\ne^{-i\\left(E_{\\lambda_{k},\\ell,+}^{(q)}t-p_{z}z\\right)}a_{p_{z}}^{\\lambda_{k},\\ell,s}\\right.\\nonumber\\\\\n&&\\left.\\times \\big[\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}\\left(x\\right)\\big]_{\\alpha\\rho}u_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right)\\Theta\\left(E_{\\lambda_{k},\\ell,+}^{(q)}\\right)\\right.\\nonumber\\\\\n&&\\left.+\ne^{+i\\left(E_{\\lambda_{k},\\ell,-}^{(q)}t-p_{z}z\\right)}b_{p_{z}}^{\\lambda_{k},\\ell,s\\dagger}\\big[\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}\\left(x\\right)\\big]_{\\alpha\\rho}^{\\dagger}\\right.\\nonumber\\\\\n&&\\left.\\times v_{s,\\rho}\\left(\\tilde{p}_{\\ell,-}\\right)\\Theta\\left(E_{\\lambda_{k},\\ell,-}^{(q)}\\right)\\right\\}. \\nonumber\\\\\n\\bar{\\psi}_{\\alpha}^{(q)}(x)&=&\\sum_{k,\\ell,s}\\int \\frac{dp_z}{2\\pi}\\frac{\\mathscr{C}_{k,\\ell,s_q}}{\\sqrt{2|\\epsilon_{\\lambda_{k}}^{(q)}}|}\\left\\{\ne^{+i\\left(E_{\\lambda_{k},\\ell,+}^{(q)}t-p_{z}z\\right)}a_{p_{z}}^{\\lambda_{k},\\ell,s\\dagger}\\right.\\nonumber\\\\\n&&\\left.\\times \\bar{u}_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right) \\big[\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}\\left(x\\right)\\big]_{\\rho\\alpha}^{\\dagger}\\Theta\\left(E_{\\lambda_{k},\\ell,+}^{(q)}\\right)\\right.\\nonumber\\\\\n&&\\left.+\ne^{-i\\left(E_{\\lambda_{k},\\ell,-}^{(q)}t-p_{z}z\\right)}b_{p_{z}}^{\\lambda_{k},\\ell,s}\n\\bar{v}_{s,\\rho}\\left(\\tilde{p}_{\\ell,-}\\right)\\right. \\nonumber\\\\\n&&\\left.\\times\n\\big[\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}\\left(x\\right)\\big]_{\\rho\\alpha}\n\\Theta\\left(E_{\\lambda_{k},\\ell,-}^{(q)}\\right)\\right\\}.\n\\end{eqnarray}\nHere, $a_{p_{z}}^{\\lambda_{k},\\ell,s\\dagger}$ and $a_{p_{z}}^{\\lambda_{k},\\ell,s}$ as well as\n$b_{p_{z}}^{\\lambda_{k},\\ell,s\\dagger}$ and $b_{p_{z}}^{\\lambda_{k},\\ell,s}$ are the\ncreation and annihilation operators of particles and antiparticles, and satisfy the commutation relations\n\\begin{eqnarray}\\label{N68}\n\\hspace{-0.9cm}\n\\{a_{p_{z}}^{\\lambda_{k},\\ell,s},a_{p_{z}^{\\prime}}^{\\lambda_{k}^{\\prime},\\ell^{\\prime},s^{\\prime}\\dagger}\\}&=&2\\pi\\delta\\left({p_{z}}-\np_{z}^{\\prime}\\right)\\delta_{\\lambda_{k},\\lambda_{k}^{\\prime}}\\delta_{\\ell,\\ell^{\\prime}}\\delta_{s,s^{\\prime}},\\nonumber\\\\\n\\hspace{-0.9cm}\n\\{b_{p_{z}}^{\\lambda_{k},\\ell,s},b_{p_{z}^{\\prime}}^{\\lambda_{k}^{\\prime},\\ell^{\\prime},s^{\\prime}\\dagger}\\}&=&2\\pi\\delta\\left({p_{z}}-\np_{z}^{\\prime}\\right)\\delta_{\\lambda_{k},\\lambda_{k}^{\\prime}}\\delta_{\\ell,\\ell^{\\prime}}\\delta_{s,s^{\\prime}}.\n\\end{eqnarray}\nIn \\eqref{N50}, $\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}$, the modified version of $\\mathbb{P}_{\\lambda_{k},\\ell}^{(q)}$ from \\eqref{N20}, reads\n\\begin{eqnarray}\\label{N69}\n\\hspace{-0.5cm}\\widetilde{\\mathbb{P}}_{\\lambda_{k},\\ell}^{(q)}=\\left(\\mathscr{P}_{+}^{(q)}f_{\\lambda_{k},\\ell,s_q}^{+s_q}+\\Pi_{\\lambda_{k}}\\mathscr{P}_{-}^{(q)}f_{\\lambda_{k},\\ell,s_q}^{-s_{q}}\\right)\\Gamma_{\\lambda_{k},\\ell,q},\\nonumber\\\\\n\\end{eqnarray}\nwith $\\mathscr{P}^{(q)}_{\\pm}$ given in \\eqref{N53} with $f_{\\lambda_{k},\\ell,s_q}^{\\pm}$ from \\eqref{N64}. Here, $\\epsilon_{\\lambda_k}^{(q)}=E_{\\lambda_k,\\ell,\\kappa}^{(q)}+\\kappa\\Omega j$ with $j=\\ell+1\/2$, as in the previous section.\nHence, according to these results, \\eqref{N50}-\\eqref{N53} are still valid, except that in \\eqref{N52}, two factors $\\Pi_{\\lambda}$ and $\\Gamma_{\\lambda_{k},\\ell,q}$ are to determined numerically [see our descriptions in \\ref{sec2b2}].\\footnote{In what follows, we use instead of $\\widetilde{\\mathbb{P}}_{\\lambda_k,\\ell}^{(q)}$ from \\eqref{N69}, $\\mathbb{P}_{\\lambda_k,\\ell}^{(q)}$ from \\eqref{N20}, keeping in mind that the restrictions for $\\ell$ are automatically dictated by the properties of the hypergeometric functions appearing in $f_{\\lambda_k,\\ell,s_q}^{\\pm}$  from \\eqref{N64}.} Moreover, the summation over $k$ and $\\ell$ shall be performed according to the description in this section. Let us also remind that $\\Pi_{\\lambda}$ was introduced to consider the degeneracy of the energy levels, and $\\Gamma_{\\lambda,\\ell,q}$ to consider the lower and upper bounds of $\\lambda$ for positive and negative charges according to Table \\ref{table2}. We also notice that index $k$ in the above expressions counts the number of the roots of the hypergeometric functions \\eqref{N61} and \\eqref{N62}, $\\lambda_{k}, k=1,2,\\cdots$.\n\\subsubsection{The fermion propagator in a magnetized rotating fermionic system with boundary condition}\\label{sec2b4}\nThe main purpose of this paper is to compute the chiral condensate at zero and finite temperature $T$ and zero chemical potential, and to study the effect of rotation on its $T$ dependence for a fixed magnetic field. To do this, we use in Sec. \\ref{sec3}, the mass gap relation\n\\begin{eqnarray}\\label{N70}\n\\bar{m}_{q}=G\\lim_{r\\to r^{\\prime}} \\mbox{Tr}\\left(S^{(q)}\\left(r,r^{\\prime}\\right)\\right),\n\\end{eqnarray}\nwhere $G$ is a dimensionful coupling constant, and $S\\left(x,x^{\\prime}\\right)$ is the fermion propagator of magnetized Dirac fermions in a rotating system with a global boundary condition. In what follows, we show that the fermion propagator is given by\n\\begin{widetext}\n\\begin{eqnarray}\\label{N71}\n\\hspace{-1cm}\nS_{\\alpha\\beta}^{(q)}(r,r^{\\prime})=i\\sum_{k,\\ell}\\int\\frac{dp_{0}dp_z}{\\left(2\\pi\\right)^{2}}\\mathscr{C}_{k,\\ell,s_q}^{2} e^{-ip_0\\left(t-t^{\\prime}\\right)+ip_z\\left(z-z^{\\prime}\\right)}[\\mathbb{P}_{\\lambda_{k},\\ell}^{(q)}(x)]_{\\alpha\\rho}\\left(\\frac{\\gamma\\cdot \\tilde{p}_{\\lambda_{k},\\ell,+}^{(q)}+m_{q}}{\\left(p_{0}+\\Omega j\\right)^{2}-\\epsilon_{\\lambda_{k}}^{(q)2}}\\right)_{\\rho\\sigma}[\\mathbb{P}_{\\lambda_{k,\\ell}}^{(q)}(x^{\\prime})]^{\\dagger}_{\\sigma\\beta},\n\\end{eqnarray}\n\\end{widetext}\nwith $\\epsilon_{\\lambda_{k}}^{(q)2}=m_{q}^{2}+2\\lambda_{k}|qeB|+p_{z}^{2}$ from \\eqref{N49}. The functions $\\mathbb{P}_{\\lambda_{k,\\ell}}^{(q)}$ are given in \\eqref{N20} with $f^{\\pm}_{\\lambda_{k,\\ell,s_q}}$ from \\eqref{N64}. To show this, let us start with the definition of the fermion propagator\n\\begin{eqnarray}\\label{N72}\nS_{\\alpha\\beta}^{(q)}=\\theta(t-t')\\langle \\psi_{\\alpha}(x)\\bar{\\psi}_{\\beta}(x')\\rangle-\\theta(t-t')\\langle\\bar{\\psi}_{\\beta}(x')\\psi_{\\alpha}(x)\\rangle.\\nonumber\\\\\n\\end{eqnarray}\nUsing the quantization relation \\eqref{N67}, we arrive first at\n\\begin{widetext}\n\\begin{eqnarray}\\label{N73}\n\\langle \\psi_{\\alpha}(x)\\bar{\\psi}_{\\beta}(x')\\rangle&=&\\sum_{k,\\ell,s}\\int\\frac{dp_z}{2\\pi}\\frac{\\mathscr{C}_{k,\\ell,s_q}^{2}}{2|\\epsilon_{\\lambda_k}^{(q)}|}\n\\left\\{e^{-iE_{\\lambda_k,+}^{(q)}(t-t')+ip_{z}(z-z')}\n[\\mathbb{P}_{\\lambda_k,\\ell}^{(q)}(x)]_{\\alpha\\rho}u_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right)\\bar{u}_{\\sigma,s}\\left(\\tilde{p}_{\\ell,+}\\right)[\\mathbb{P}_{\\lambda,k}^{(q)}]^{\\dagger}_{\\sigma\\beta}\\right\\},\\nonumber\\\\\n\\langle \\bar{\\psi}_{\\beta}(x')\\psi_{\\alpha}(x)\\rangle&=&\\sum_{k,\\ell,s}\\int\\frac{dp_z}{2\\pi}\\frac{\\mathscr{C}_{k,\\ell,s_q}^{2}}{2|\\epsilon_{\\lambda_k}^{(q)}|}\n\\left\\{e^{+iE_{\\lambda_k,-}^{(q)}(t-t')-ip_{z}(z-z')}\n[\\mathbb{P}_{\\lambda_k,\\ell}^{(q)}(x)]_{\\alpha\\rho} u_{s,\\rho}\\left(\\tilde{p}_{\\ell,+}\\right)\\bar{v}_{\\rho,s}\\left(\\tilde{p}_{\\ell,-}\\right)[\\mathbb{P}_{\\lambda,k}^{(q)}]^{\\dagger}_{\\sigma\\beta}\\right\\}.\n\\end{eqnarray}\nPlugging then these expressions into \\eqref{N72}, and using $E_{\\lambda_k,\\ell,\\kappa}^{(q)}=\\epsilon_{\\lambda_k}^{(q)}-\\kappa\\Omega j$,\n\\begin{eqnarray}\\label{N74}\n\\theta\\left(\\pm z\\right)=\\lim\\limits_{\\varepsilon\\to 0^{+}}\\mp\\int\\frac{dp_0}{2\\pi}\\frac{e^{izt}}{p_0\\mp i\\varepsilon},\n\\end{eqnarray}\nas well as \\eqref{N18}, we obtain\n\\begin{eqnarray}\\label{N75}\nS_{\\alpha\\beta}^{(q)}\\left(x,x'\\right)&=&-i\\sum\\limits_{k,\\ell}\\int\\frac{dp_0 dp_z}{(2\\pi)^{2}}\\frac{\\mathscr{C}_{k,\\ell,s_q}^{2}e^{i\\Omega j\\left(t-t'\\right)}}{2\\epsilon_{\\lambda_{k}}^{(q)}}[\\mathbb{P}_{\\lambda_k,\\ell}^{(q)}(x)]_{\\alpha\\rho}\\bigg\\{\\frac{\\gamma\\cdot\\tilde{p}_{\\lambda_k,\\ell,+}^{(q)}+m_q}{p_0-i\\varepsilon}e^{i\\left(p_0-\\epsilon_{\\lambda_k}^{(q)}\\right)(t-t')+ip_z\\left(z-z'\\right)}\\nonumber\\\\\n&& +\\frac{\\gamma\\cdot\\tilde{p}_{\\lambda_k,\\ell,-}^{(q)}-m_q}{p_0+i\\varepsilon}e^{i\\left(p_0+\\epsilon_{\\lambda_k}^{(q)}\\right)(t-t')-ip_z\\left(z-z'\\right)}\\bigg\\}_{\\rho\\sigma}\n[\\mathbb{P}_{\\lambda_k,\\ell}^{(q)}(x')]_{\\sigma\\beta}.\n\\end{eqnarray}\n\\end{widetext}\nPerforming a shift of variables $p_0\\to  -p_0+\\epsilon_{\\lambda_k}^{(q)}$ and $p_{0}\\to -p_{0}-\\epsilon_{\\lambda_k}^{(q)}$, and eventually $p_{0}\\to p_{0}+\\Omega j$, we arrive at \\eqref{N71}, as claimed. In the next section, \\eqref{N71} is used to determine the chiral condensate at zero and finite temperature.\n\\section{Inverse magneto-rotational Catalysis at zero and finite temperature; Numerical results}\\label{sec3}\n\\setcounter{equation}{0}\nOne of the aims of this paper is to elaborate on the interplay between the rotation and the presence of a constant magnetic field, in particular, on the formation of bound states. It is known that external magnetic fields enhance chiral symmetry breaking. This is the well-established magnetic catalysis. There are a number of attempts exploring the effect of the rigid rotation of a system of quark matter on magnetic catalysis. In this section, after reviewing the results for zero temperature by shedding light on some new aspects, which are not discussed before in the literature, we introduce the temperature $T$, and explore the $T,eB,\\Omega$, and $r$ dependence of the dynamical mass. We then present numerical results for the $G,eB,\\Omega$, and $r$ dependence of the critical temperature $T_c$, and $G,eB,T$, and $r$ dependence of certain critical frequency $\\Omega_c$.      \n\\subsection{Zero temperature}\\label{sec3a}\n\\setcounter{equation}{0}\nIn this section, after presenting the relations which are used to study the effect of rotation on the magnetic catalysis in a fermionic system with boundary at zero temperature, we explore, in particular, the $r$ dependence of the dynamical mass $\\bar{m}$. Here, it is shown that the angular frequency plays no role in the behavior of $\\bar{m}$.  \n\\par\nFirst, we focus on the mass gap relation \\eqref{N70}. Plugging the propagator $S^{(q)}$ from \\eqref{N71} into \\eqref{N70}, and performing the trace over $\\gamma$-matrices, we arrive at\n\\begin{eqnarray}\\label{E1}\n\\frac{\\bar{m}_{q}}{G}=\\frac{i\\bar{m}_{q}}{2\\pi^{2}}\\sum\\limits_{k,\\ell}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\\int\\frac{ dp_0 dp_{z}}{\\big[\\left(p_0+\\Omega j\\right)^{2}-\\epsilon_{\\lambda_{k}}^{(q)2}\\big]}, \\nonumber\\\\\n\\end{eqnarray}\nwith\n$$\n\\Phi_{\\lambda_{k,\\ell,s_q}}^{2}\\equiv \\Phi^{+2}_{\\lambda_{k},\\ell,s_q}+\\Phi^{-2}_{\\lambda_{k},\\ell,s_q}.\n$$\nPerforming then a shift of variable $p_{0}\\to p_{0}-\\Omega j$, the integration over $p_0$ can be immediately carried out. The resulting expression reads\n\\begin{eqnarray}\\label{E2}\n\\frac{\\bar{m}_{q}}{G}=\\frac{i\\bar{m}_{q}}{2\\pi^{2}}\\sum\\limits_{k,\\ell}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\\int dp_{z}\\frac{1}{\\epsilon_{\\lambda_{k}}^{(q)}}.\n\\end{eqnarray}\nAs it turns out, the angular frequency $\\Omega$ is canceled from the computation, and has indeed no effect on the mass $m_q$, arising from \\eqref{E2}. This is in contrast to the results presented in \\cite{fukushima2015}, where the zero temperature case is considered as a limit of the finite temperature case. In this case, the $p_{0}$ integration appearing in \\eqref{E1} is replaced with a sum over Matsubara frequencies, and the $\\Omega j$ dependence thus appears in a Heaviside $\\theta$-function, arising from \n$$\n\\lim\\limits_{T\\to 0}T\\ln\\left(1+e^{-\\frac{x}{T}}\\right)=-x\\theta(-x). \n$$  \nThe integration over $p_z$ is carried out by introducing the ultraviolet smooth cutoff \\cite{fukushima2015,fukushima2017,shovkovy2011}\n\\begin{eqnarray}\\label{E3}\n\\hspace{-0.5cm}f\\left(p;\\Lambda,\\delta\\Lambda\\right)=\\frac{\\sinh\\left(\\Lambda\/\\delta\\Lambda\\right)}{[\\cosh\\left(p\/\\delta\\Lambda\\right)+\\cosh\\left(\\Lambda\/\\delta\\Lambda\\right)]},\n\\end{eqnarray}\nwith $p=\\sqrt{2\\lambda_{k,\\ell,s_q}|qeB|+p_{z}^{2}}$. For the limit $\\delta\\Lambda\/\\Lambda\\to 0$, the function $f(p;\\Lambda,\\delta\\Lambda)$ approaches the Heaviside $\\Theta$-function\n\\begin{eqnarray}\\label{E4}\n\\hspace{-1cm}\\lim\\limits_{\\delta\\Lambda\/\\Lambda\\to 0}f\\left(p;\\Lambda,\\delta\\Lambda\\right)\\to \\Theta\\left(\\Lambda^2-2\\lambda_{k,\\ell,s_q}|qeB|-p_z^2\\right).\\nonumber\\\\\n\\end{eqnarray}\nPlugging the $\\Theta$-function into the remaining $p_{z}$ integral in \\eqref{E1}, and integrating $p_z$ from $-\\left(\\Lambda^2-2\\lambda_{k,\\ell,s_q}|qeB|\\right)^{1\/2}$ to $+\\left(\\Lambda^2-2\\lambda_{k,\\ell,s_q}|qeB|\\right)^{1\/2}$, dictated by the $\\Theta$-function, we arrive at\n\\begin{eqnarray}\\label{E5}\n\\frac{\\bar{m}_q}{G}&=&\\frac{\\bar{m}_{q}}{\\pi}\\sum\\limits_{k,\\ell}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\n\\nonumber\\\\\n&&\\times \\mbox{tanh}^{-1}\\left(\\frac{\\sqrt{\\Lambda^{2}-2\\lambda_{k,\\ell,s_q}|qeB|}}{\\Lambda^{2}+m_{q}^{2}}\\right)\\nonumber\\\\\n&&\\times\\Theta\\left(\\Lambda^{2}-2\\lambda_{k,\\ell,s_q}|qeB|\\right).\n\\end{eqnarray}\nAssuming $\\bar{m}_{q}\\neq 0$, the nontrivial solutions to \\eqref{E5} can be determined numerically by fixing $q, \\Lambda,R, eB$ and $G$.\nIn what follows we choose\n\\begin{eqnarray}\\label{E6}\nq=+1,\\qquad\\Lambda=1~\\mbox{GeV},\\qquad R=6~\\mbox{fm}.\n\\end{eqnarray}\nInstead of $eB$, it is more appropriate to work with the dimensionless quantity $\\alpha_b= eB R^2\/2$ which is introduced in the previous section. To generate our data, we use $\\alpha_b=1,\\cdots,10$ that correspond to $eB\/m_{\\pi}^{2}$ given in Table \\ref{tab4} for $R=6$ fm.\\footnote{Here, we use $1$ fm$^{-1}\\sim m_{\\pi}$ in MeV, where $m_{\\pi}\\sim 200$ MeV is the pion mass.}\n\\begin{figure}[h]\n\\includegraphics[width=8cm,height=6cm]{fig3}\n\\caption{color online. The minimum value of the coupling constant for which the mass gap possesses nonvanishing solution in the interval $x\\in[1,\\alpha_{b}]$, with $x=eBr^2\/2$ and $\\alpha_{b}=eBR^2\/2$. }\\label{fig3x}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=8cm,height=6cm]{fig4}\n\\caption{color online. The $r^{2}\/R^{2}$ dependence of the mass gap $\\bar{m}$ is plotted for $\\alpha_{b}=7$ ($eB\\sim 0.4 m_{\\pi}^{2}$) and three different values of $G$. As it turns out, for a fixed $eB$, increasing $G$ enhances the chiral symmetry breaking.}\\label{fig4}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig5a}\n\\includegraphics[width=8cm,height=6cm]{fig5b}\n\\caption{color online. The $x$ dependence of $\\bar{m}$ is plotted for $\\alpha_{b}=2,4,6,8,10$ and $G\\Lambda^2$ equal to $G_{m}\\Lambda^2$ from Table \\ref{tab5} (panel a) and a constant $G\\Lambda^2=32$ (panel b). }\\label{fig5}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig6a}\n\\includegraphics[width=8cm,height=6cm]{fig6b}\n\\caption{color online. (a) The $r^2\/R^2$ dependence of $\\bar{m}$ is plotted for $\\alpha_b=2,4,6,8,10$, and $G\\Lambda^2=G_m\\Lambda^2$. The coupling are chosen so that $\\bar{m}$ remains almost constant in the range $r^2<0.8 R^2$. (b) Using same couplings $G_m\\Lambda^2$ from Table \\ref{tab4}, the $eB\/m_{\\pi}^2$ dependence of $\\bar{m}$ is plotted for fixed $r^2=0.2R^2$ and $r^2=0.8 R^2$. The behavior reflects the dependence of $G_m\\Lambda^2$ on $eB\/m_{\\pi}^2$ (see Fig. \\ref{fig3x}).    }\\label{fig6}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig7a}\n\\includegraphics[width=8cm,height=6cm]{fig7b}\n\\caption{color online. (a) The $r^2\/R^2$ dependence of $\\bar{m}$ is plotted for $\\alpha_b=2,4,6,8,10$, and $G\\Lambda^2=32$. In contrast to the results from Fig. \\ref{fig6}(a), weak magnetic fields do not affect the dependence of $\\bar{m}$ on $r$. Only small perturbations occur, which are explicitly shown in panel (b), where the $eB\/m_{\\pi}^2$ dependence of $\\bar{m}$ is plotted for fixed $r^2=0.2R^2$ and $r^2=0.8 R^2$. The oscillations are due to successive filling of the Landau levels (de Haas-van Alfven effect). }\\label{fig7}\n\\end{figure*}\n\\begin{table}[hbt]\n\\begin{tabular}{ccccccc}\n\\hline\n$\\alpha_{b}$&$\\quad$&$eB\/m_{\\pi}^{2}$&$\\qquad$&$\\alpha_{b}$&$\\quad$&$eB\/m_{\\pi}^{2}$\\\\\n\\hline\\hline\n$1$&&$0.05$&$\\qquad\\quad$&$6$&&$0.33$\\\\\n$2$&&$0.10$&$\\qquad\\quad$&$7$&&$0.39$\\\\\n$3$&&$0.17$&$\\qquad\\quad$&$8$&&$0.44$\\\\\n$4$&&$0.22$&$\\qquad\\quad$&$9$&&$0.50$\\\\\n$5$&&$0.28$&$\\qquad\\quad$&$10$&&$0.55$\\\\\n\\hline\n\\end{tabular}\n\\caption{The values of $eB\/m_{\\pi}^{2}$ for $R=6$ fm and given $\\alpha_{b}$s. }\\label{tab4}\n\\end{table}\nAs concerns G, let us notice that the mass gap $\\bar{m}_{q}$ arising in \\eqref{N70} is related to the chiral condensate which is created as a result of a spontaneous chiral symmetry breaking in a QCD-like model. An appropriate example is the Lagrangian density of one flavor NJL model\n\\begin{eqnarray}\\label{E7}\n\\mathscr{L}=\\bar{\\psi}\\left(\\gamma\\cdot \\Pi-m\\right)\\psi+\\frac{G}{2}[\\left(\\bar{\\psi}\\psi\\right)^{2}+\\left(\\bar{\\psi}i\\gamma^{5}\\psi\\right)^{2}].\\nonumber\\\\\n\\end{eqnarray}\nHere, $\\Pi\\equiv \\Pi^{(q=1)}$ is given in \\eqref{N11}, $m\\equiv m_{q=1}$ is the current mass of a particle with $q=1$. For the sake of simplicity, we assume the fermion to be massless ($m=0$). The solution of the mass gap $\\bar{m}$ is related to the value of chiral condensate $\\langle\\bar{\\psi}\\psi\\rangle$ through\n\\begin{eqnarray}\\label{E8}\n\\bar{m}=-G\\langle \\bar{\\psi}\\psi\\rangle.\n\\end{eqnarray}\nFor vanishing magnetic fields and in a nonrotating fermionic system the condensate is built only when $G\\Lambda^2$ is large enough \\cite{klevansky1992,miransky1995}. However, it is known that external magnetic fields enhance the condensation so that even moderate values of $G\\Lambda^2$ would be enough for the formation of the condensate \\cite{miransky1995}. It is thus interesting to determine the minimum value of $G\\Lambda^2$ for which the gap equation \\eqref{E5} possesses a nontrivial solution in a rotating system for a given $\\alpha_b$ and for a relatively large interval $x\\in[0,\\alpha_{b}]$ with $x=eB r^2\/2$. Denoting these kinds of $G$'s by $G_m$, we plotted them as a function of $eB\/m_{\\pi}^{2}$ in Fig. \\ref{fig3x}. Their values are listed in Table \\ref{tab5}. As it turns out, for small values of $\\alpha_b\\leq 4$ ($eB\\leq 0.22 m_{\\pi}^{2}$) the NJL coupling $G_m$ increases with increasing $\\alpha_{b}$. This means that the magnetic field is not yet strong enough to hold the constituent mass nonvanishing in the interval $x\\leq 10$ fm. However, once $\\alpha_{b}$ increases, it becomes strong enough and enhances the production of the dynamical mass even when the coupling is not very large. This is why $G_{m}$ decreases with increasing $\\alpha_{b}\\geq 4$.\n \\begin{table}[hbt]\n\\begin{tabular}{ccccccccccc}\n\\hline\n$\\alpha_{b}$&&$eB\/m_{\\pi}^{2}$&&$G_m$&$\\qquad\\quad$\n&$\\alpha_{b}$&&$eB\/m_{\\pi}^{2}$&&$G_{m}$\\\\\n\\hline\\hline\n$1$&&$0.05$&&$22.60$&$\\qquad\\quad$&$6$&&$0.33$&&$30.22$\\\\\n$2$&&$0.10$&&$28.71$&$\\qquad\\quad$&$7$&&$0.39$&&$28.37$\\\\\n$3$&&$0.17$&&$31.30$&$\\qquad\\quad$&$8$&&$0.44$&&$26.55$\\\\\n$4$&&$0.22$&&$31.89$&$\\qquad\\quad$&$9$&&$0.50$&&$24.11$\\\\\n$5$&&$0.28$&&$31.37$&$\\qquad\\quad$&$10$&&$0.55$&&$21.57$\\\\\n\\hline\n\\end{tabular}\n\\caption{The $eB\/m_{\\pi}^{2}$ dependence of $G_{m}$ as the minimum value of G for which a nonvanishing constituent mass $\\bar{m}$ arises in the interval $x\\in[1,\\alpha_b]$.}\\label{tab5}\n\\end{table}\n\\par\nIn Fig. \\ref{fig4}, the mass gap $\\bar{m}$ is plotted as a function of $x\/\\alpha_{b}=r^{2}\/R^{2}$ for $\\alpha_{b}=7$ ($eB \\sim 0.4 m_{\\pi}^{2}$), and three different $G\\Lambda^{2}$, $G_{1}\\Lambda^{2}=24, G_{2}\\Lambda^{2}=32$, and $G_{m}\\Lambda^{2}=28.37$. It is demonstrated how larger values of $G$ enhances the chiral symmetry breaking. The qualitative dependence of the constituent mass $\\bar{m}$ on the position relative to $R$ does not change dramatically by increasing $G$.\n\\par\nIn Fig. \\ref{fig5}, the $x$ dependence of $\\bar{m}$ is plotted for $\\alpha_{b}=2,4,6,8,10$. In Fig. \\ref{fig5}(a), we used the corresponding $G_{m}\\Lambda^2$ to each $\\alpha_b$ (see Table \\ref{tab4}), while in Fig. \\ref{fig5}(b), $G\\Lambda^{2}=32$ is used. The color code in Fig. \\ref{fig5}(b) is the same as in Fig. \\ref{fig5}(a). As it is shown, the quantitative dependence of $\\bar{m}$ does not change by increasing $\\alpha_{b}$, but the position where $\\bar{m}$ starts to decrease depends on $\\alpha_{b}$, because, according to its definition, the maximum value of $x=eB r^{2}\/2$ is equal to $\\alpha_{b}=eBR^2\/2$. In Fig. \\ref{fig5}(b), we consider only the interval $\\bar{m}\\in [700,820]$ in the vertical axis for fixed $G\\Lambda^{2}=32$ and the same values of $\\alpha_{b}$ as demonstrated in Fig. \\ref{fig5}(a). It is shown that independent of $\\alpha_{b}$, $\\bar{m}$ exhibits small oscillations as a function of $x$. The amplitudes of the oscillations become large in the vicinity of the boundary. At boundary $R$ ($x=\\alpha_{b}$), $\\bar{m}$ decreases rapidly.\n\\par\nBeing a function of $eB$, the parameter $x$ is not a natural quantity to demonstrate the dependence of $\\bar{m}$ on the position $r$ relative to the boundary $R$. This is why, we plotted $\\bar{m}$ as a function of $r^2\/R^2$ in Fig. \\ref{fig6}(a) for $\\alpha_{b}=2,4,6,8,10$. Here, $G\\Lambda^2$ is chosen to be $G_{m}\\Lambda^2$ which are different for different $\\alpha_{b}$ (see Table \\ref{tab4}). As expected from the previous results in Fig. \\ref{fig5}(a), independent of $\\alpha_b$, $\\bar{m}$ remains relatively constant for a large interval $r^2 \\in [0,0.8R^2]$ before it starts decreasing at the boundary $r\\sim R$. However, for a fixed $r\/R$, it has a non-monotonic dependence on $\\alpha_b$. First, it increases and then decreases with $\\alpha_{b}$. To scrutinize this dependence, we plotted in Fig. \\ref{fig6}(b) the $eB$ dependence of $\\bar{m}$ for fixed $r^2\/R^2=0.2,0.8$ and for $G\\Lambda^2=G_{m}\\Lambda^2$. As it is demonstrated here, it increases first as a function of $eB$ and then decreases with increasing $eB$. This specific behavior is mainly related to the $\\alpha_{b}$-dependence of the NJL coupling $G_m$, demonstrated in Fig. \\ref{fig3x}.\n\\par\nThe $r^2\/R^2$ dependence of $\\bar{m}$ for $\\alpha_{b}=2,4,6,8,10$ and a fixed $G\\Lambda^2=32$ is plotted in Fig. \\ref{fig7}. As it is shown, $\\bar{m}$ remains almost constant for $r^{2}<0.8 R^2$, and rapidly decreases for $r\\to R$. The values of $\\bar{m}$ corresponding to $\\alpha_{b}$ are slightly different. In order to see the difference between $\\bar{m}$'s in the interval $r^2 \\in [0,0.8R^2]$, the $eB$ dependence of $\\bar{m}$ is plotted for $r^2=0.2R^2, 0.8R^2$ and relatively large $G\\Lambda^{2}=32$ in Fig. \\ref{fig7}(b). In the interval $\\bar{m}\\in[780,810]$, the constituent mass oscillates with $eB$. This is in contrast to the behavior of $\\bar{m}$ demonstrated in Fig. \\ref{fig6}. The positions of the maxima and minima appearing in Fig. \\ref{fig7} do not change by increasing $r^{2}$ from $r^{2}=0.2 R^2$ to $r^2=0.8 R^2$. The oscillations are related to the de Haas-Alfven effect, and are because of the successive filling of Landau levels.  \n\\subsection{Finite temperature}\\label{sec3b}\nIn this section, we generalize our previous results to the case of finite temperature. We demonstrate the $T,R\\Omega,eB$ and $r^{2}\/R^{2}$ dependence of $\\bar{m}$ for a fixed set of parameters $\\{\\alpha_{b}, x,R\\Omega,G\\Lambda^2,T\\}$. We also determine the phase diagram $T_{c}$ ($\\Omega_{c}$) versus $R\\Omega, eB$ ($T,eB$) and $r^2\/R^2$ for a fixed set of parameters\n$\\{\\alpha_{b}, x,R\\Omega,G\\Lambda^2\\}$ ($\\{\\alpha_{b}, x,T,G\\Lambda^2\\}$). We demonstrate, in particular, the IMRC, in which a finite rotation neutralizes the magnetic catalysis induced by a constant magnetic field.  As a consequence $\\bar{,m}$ decreases with increasing $eB$ for relatively large $R\\Omega$ and small coupling $G\\Lambda^2$. Moreover, in exploring the phase diagram of $T_{c}$ versus $R\\Omega$, this effect is reflected in reducing $T_{c}$ as a function of $eB$ for large value of $R\\Omega$. The same effect is also demonstrated in the phase diagram $R\\Omega_{c}$ versus $eB$ and $T$. The dependence of $T_c$ and $\\Omega_c$ on the coupling $G\\Lambda^2$ and $r^2\/R^2$ is also explored. \n\\par\nTo introduce the temperature, let us consider \\eqref{E1}, and use\n\\begin{eqnarray}\\label{E9}\np_0\\to i\\omega_{n}=i\\pi T(2n+1),\\qquad \\int\\frac{dp_0}{2\\pi}\\to iT\\sum_{n=-\\infty}^{+\\infty}, \\nonumber\\\\\n\\end{eqnarray}\nwhere $\\omega_{n}$ is the corresponding Matsubara frequencies for fermions. We arrive first at\n\\begin{eqnarray}\\label{E10}\n\\frac{\\bar{m}_{q}}{G}=\\frac{\\bar{m}_{q}}{2\\pi^{2}}\\sum\\limits_{k,\\ell,n}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\\int\\frac{dp_{z}}{\\big[\\left(p_0-i\\Omega j\\right)^{2}+\\epsilon_{\\lambda_{k}}^{(q)2}\\big]}, \\nonumber\\\\\n\\end{eqnarray}\nwith $\\epsilon_{\\lambda_{k}}^{(q)}$ from \\eqref{N49}. Using then\n\\begin{eqnarray}\\label{E11}\nT\\sum_{n=-\\infty}^{+\\infty}\\frac{1}{\\left(\\omega_{n}-i\\mu\\right)^{2}+\\epsilon^2}=\\frac{1-f\\left(\\epsilon+\\mu\\right)-f(\\epsilon-\\mu)}{2\\epsilon},\\nonumber\\\\\n\\end{eqnarray}\nwhere $f(\\epsilon\\pm\\mu)\\equiv\\left(e^{\\left(\\epsilon\\pm\\mu\\right)\/T}+1\\right)^{-1}$\nis the Fermi-Dirac distribution function, the gap equation \\eqref{E10} is separated into a  $T$ independent and a $T$ dependent part,\n\\begin{eqnarray}\\label{E12}\n\\lefteqn{\n\\frac{\\bar{m}_q}{G}=\\frac{\\bar{m}_{q}}{\\pi}\\sum\\limits_{k,\\ell}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\n}\n\\nonumber\\\\\n&&\\times \\mbox{tanh}^{-1}\\left(\\frac{\\sqrt{\\Lambda^{2}-2\\lambda_{k,\\ell,s_q}|qeB|}}{\\Lambda^{2}+m_{q}^{2}}\\right)\\nonumber\\\\\n&&\\times\\Theta\\left(\\Lambda^{2}-2\\lambda_{k,\\ell,s_q}|qeB|\\right)\\nonumber\\\\\n&&-\\frac{\\bar{m}_{q}}{\\pi}\\sum\\limits_{k,\\ell,n}\\mathscr{C}_{k,\\ell,s_q}^{2}\\Phi^{2}_{\\lambda_{k},\\ell,s_q}\\nonumber\\\\\n&&\\times \\int_{0}^{\\infty} dp_{z}\\frac{f(\\epsilon_{\\lambda_{k}}^{(q)}+\\Omega j)+f(\\epsilon_{\\lambda_{k}}^{(q)}-\\Omega j)}{\\epsilon_{\\lambda_{k}}^{(q)}}.\n\\end{eqnarray}\nThe $T$ independent part of the gap equation is regularized in the same manner as in \\eqref{E2}. This yields \\eqref{E5}, which appears again in the first term on the r.h.s.  of \\eqref{E12}. Here, $\\Lambda$ is the corresponding cutoff, as appears also in \\eqref{E5}. Concerning the $T$ dependent part of the gap equation, the distribution functions prevent the corresponding integrals to be divergent. In what follows, we present first the numerical results for the gap equation \\eqref{E12} for fixed parameters $\\{q,\\Lambda,R\\}$ from \\eqref{E6}. Then, focusing on the critical temperature as well as angular frequency, we study, in particular, their $eB$ dependence. \n\\subsubsection{The constituent mass as a function of $T, eB,  R\\Omega$,  and $r^2\/R^2$}\\label{sec3b1}\n\\begin{figure*}\n\t\\includegraphics[width=8cm,height=6cm]{fig8a}\n\t\\includegraphics[width=8cm,height=6cm]{fig8b}\\vspace{.5cm}\n\t\\includegraphics[width=8cm,height=6cm]{fig8c}\n\t\\includegraphics[width=8cm,height=6cm]{fig8d}\n\t\\caption{color online. a) The $T$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=24,26$, $\\alpha_b=10, x=5$ and $R\\Omega=0.5$. The results indicate a second-order phase transition at certain critical temperature $T_{c}$.\n\t\tb) The $eB\/m_{\\pi}^{2}$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=26,27$, $x=\\alpha_{b}\/2,T=200 $ MeV, and $R\\Omega=0.8$. In the regime of weak magnetic field $eB\\leq 0.5 m_{\\pi}^{2}$, $\\bar{m}$ decreases with increasing $eB$. This is an indication for the IMRC.\n\t\tc) The $R\\Omega$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=25,27,30$, $\\alpha_b=10, x=5$, and $T=200$ MeV.  While for large couplings, $\\bar{m}$ turns out to be almost constant, it decreases with increasing $R\\Omega$ for small $G\\Lambda^2=25,27$. This is another indication of the IMRC, in particular for small couplings. This effect can also be observed in the $R\\Omega$ dependence of the critical temperature $T_c$ in Fig. \\ref{fig9}. The critical $R\\Omega$, for which the dynamical mass vanishes, increases with increasing coupling $G\\Lambda^2$ (see also Fig. \\ref{fig11}).\n\t\td) The $r^2\/R^2$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=24,26$, $\\alpha_b=10, T=200$ MeV, and $R\\Omega=0.5$. As it turns out, for small couplings, $\\bar{m}$ decreases with increasing $r$.\n\t\tThe larger the velocity, and consequently the kinetic energy as well as the centrifugal force of a rotating system is, it is most probably in the chirally restored phase, where the dynamical mass vanishes.}\\label{fig8}\n\\end{figure*}\nIn Fig. \\ref{fig8}(a), the $T$ dependence of the constituent mass $\\bar{m}$ is plotted for fixed $\\alpha_{b}=10,x=5$ and $R\\Omega=0.5$, and two different choices of $G\\Lambda^{2}=24$ and $G\\Lambda^{2}=26$.\\footnote{By combining the definitions of $x$ and $\\alpha_b$, we arrive at $x=\\alpha_{b}r^2\/R^2$. Hence, $\\{\\alpha_b=10,x=5\\}$  corresponds to $r^2=0.5 R^2$. \nMoreover, the choice $x=\\alpha_b\/2$ in Figs. \\ref{fig8}, \\ref{fig9}, and \\ref{fig11} corresponds to $r^{2}=0.5R^{2}$ in the whole range of $eB$.} \nAs expected, for fixed $\\alpha_b,R\\Omega$ and $T$, $\\bar{m}$ increases with increasing $G\\Lambda^2$.  The same is also true for the critical temperature $T_c$. As it is demonstrated in Fig. \\ref{fig8}(a), the corresponding critical temperatures for $G\\Lambda^{2}=24$ and $26$ are $T_c\\sim 220$ MeV and $\\sim 250$ MeV, respectively. As expected, $T_c$ increases with increasing coupling.\nThe results presented in Fig. \\ref{fig8}(a) indicate also a second-order chiral phase transition.\nThis is in contrast to the results presented in \\cite{fayazbakhsh2011}, where it is shown that the presence of external magnetic fields leads principally to a first-order chiral phase transition. \n\\par\nIn Fig. \\ref{fig8}(b), the $eB$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=26,27$, $x=x_{\\text{max}}\/2=\\alpha_{b}\/2,T=200 $ MeV, and $R\\Omega=0.8$. As it turns out, in the regime of weak magnetic fields $eB\\leq 0.5 m_{\\pi}^{2}$, the dynamical mass $\\bar{m}$ decreases with increasing $eB$. Let us remind that in a nonrotating system, because of the magnetic catalysis effect, the dynamical mass increases with increasing $eB$ \\cite{fayazbakhsh2011}. In contrast, the results presented in Fig. \\ref{fig8}(b) show that the rotation of a bounded system neutralizes this effect, and leads to IMRC.  A similar effect is introduced in \\cite{fukushima2015} for an unbounded system. It has been dubbed \"the rotational magnetic inhibition\". Let us notice that the IMRC is best demonstrated for large $R\\Omega$ and small $G\\Lambda^2$. This is because it is mainly an effect of rotation in combination with the magnetic field.  \n\\par\nThe results presented in Fig. \\ref{fig8}(c) and, in particular, Fig. \\ref{fig9}(c) are another demonstration for this effect. In Fig. \\ref{fig8}(c), the $R\\Omega$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=25,27,30$, $\\alpha_b=10, x=5$, and $T=200$ MeV. For small couplings $G\\Lambda^2=25,27$, $\\bar{m}$ decreases significantly with $R\\Omega$. It even vanishes at some critical $R\\Omega_{c}$. The value of $R\\Omega_{c}$ increases with increasing $G\\Lambda^2$. This is because larger coupling enhances the production of the condensate, whereas rotation has a\ncounter-effect. There is thus a competition between rotation\/coupling to destroy\/produce chiral condensates. For a larger value of $G\\Lambda^2\\geq 30$, $\\bar{m}$ decreases with increasing $R\\Omega$, but it does not vanish, at least in the allowed regime of $ 0\\leq R\\Omega\\leq 1$. According to the above results, for a constant magnetic field, the IMRC occurs in a bounded system of quark matter.\n\\par\nIn Fig. \\ref{fig8}(d) the $r^2\/R^2$ dependence of $\\bar{m}$ is plotted for $G\\Lambda^2=24,26$, $\\alpha_b=10, T=200$ MeV, and $R\\Omega=0.5$. As it turns out, for intermediate value of $R\\Omega$, and fixed $\\alpha_{b}$ and $T$, the dynamical mass decreases with increasing $r^2\/R^2$. Moreover, the results show that in order to keep $\\bar{m}$ almost constant in the whole range of $0<r\\leq R$, the coupling $G\\Lambda^2$ has to be large enough.\nThe $r$ dependence of $\\bar{m}$, demonstrated in Fig. \\ref{fig8}(d) indicates that the IMRC is in fact induced by the linear velocity $v=r\\Omega$. For a given $\\Omega$, the farther the condensate is from the axis of rotation, i.e. the larger $r$ is, the larger is the rotational kinetic energy of the condensate as well as the centrifugal force it feels, and the smaller is the value of the dynamical mass $\\bar{m}$. This has, by itself, positive consequences for the chiral symmetry restoration in rotating systems (see Fig. \\ref{fig11-x} for a sketch of a system of quark matter in a rotating cylinder). \n\\par\nLet us notice at this stage that the above results are only valid for intermediate values of the magnetic fields. For a larger value of $\\alpha_b$, the dynamical mass increases near the boundary. This is known as \"the surface magnetic catalysis\", and is elaborated, \\textit{inter alia}, in \\cite{fukushima2017}. \n\\begin{figure}\n\\includegraphics[width=6.5cm,height=6.5cm]{cylinder}\n\\caption{color online. A sketch of a system of quark matter in a rotating cylinder. It can be imagined to consist of infinite number of rotating cylinders with radii $r_{i}\\leq R, i=1,2,3,\\cdots$, made of chiral condensate. According to the results from Fig. \\ref{fig8}(c), for a given angular velocity $\\Omega$, the larger $r$ is, the smaller is $\\bar{m}$, thus the most probable is the chiral symmetry restoration.\t\nMoreover, the larger $r$ is, the larger are the kinetic energy and the centrifugal force applied on each layer. The fact that $\\bar{m}$ decreases with increasing $r$ indicates that larger kinetic energy and centrifugal force have a positive impact on destroying the chiral condensate, and thus restoring the chiral symmetry in a rotating system.}\\label{fig11-x}\n\\end{figure}\n\\subsubsection{Critical temperature as a function of $G\\Lambda^2, eB, R\\Omega$, and $r^2\/R^2$}\\label{sec3b2}\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig9a}\n\\includegraphics[width=8cm,height=6cm]{fig9b}\\vspace{.3cm}\n\\includegraphics[width=8cm,height=6cm]{fig9c}\n\\includegraphics[width=8cm,height=6cm]{fig9d}\n\\caption{color online. a) The $G\\Lambda^2$ dependence of $T_c$ is plotted for $\\alpha_{b}=10,R\\Omega=0.5$ and $x=1,5,10$. As expected, $T_c$ increases with increasing $G\\Lambda^2$. This confirms the fact that larger couplings enhance the formation of the dynamical mass.\nb) The $eB\/m_{\\pi}^{2}$ dependence of $T_{c}$ is plotted for $x=\\alpha_b\/2, R\\Omega=0.8$, and $G\\Lambda_{2}=24,26$. Because of the IMRC, $T_{c}$ decreases with increasing $eB$. The effect is mainly induced by the rotation so that for a smaller value of $R\\Omega$, the slope of the curves becomes smaller. c)  The  $R\\Omega$ dependence of $T_c$ is plotted for $x=\\alpha_{b}\/2,G\\Lambda^2=30$ and\n$\\alpha_{b}=1,3,5,10$. As it turns out, $T_{c}$ decreases with increasing $R\\Omega$. This indicates an IMRC induced apparently by the interplay between rotation and the presence of a magnetic field in a bounded system of quark matter.\nd) The $r^{2}\/R^{2}$ dependence of $T_{c}$ is plotted for $\\alpha_b=10, R\\Omega=0.5$, and $G\\Lambda^2=24,30$. Only at the boundary $0.8 R^2<r^2<R^2$, the critical temperature $T_c$ decreases with $r\/R$, otherwise it remains almost constant.}\\label{fig9}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=5.5cm,height=5cm]{fig10a}\n\\includegraphics[width=5.5cm,height=5cm]{fig10b}\n\\includegraphics[width=5.5cm,height=5cm]{fig10c}\n\\caption{color online. The $R\\Omega$ dependence of $\\Delta_{c}$ is plotted for $x=\\alpha_{b}\/2$ as well as $G\\Lambda^2=25$ (panel a), $G\\Lambda^2=30$ (panel b), and $G\\Lambda^2=40$ (panel c). As it turns out $\\Delta_{c}$ increases with increasing $R\\Omega$. A comparison between the results in the panels a, b, and c shows that the maximum value of $\\Delta_{c}$ decreases with increasing $G\\Lambda^2$.}\\label{fig10}\n\\end{figure*}\nTo explore the interplay between the background magnetic field and the rotation of a bounded system of quark matter, and in particular, to scrutinize their possible effects on the critical temperature $T_c$ of a chiral phase transition, the $G\\Lambda^2, eB, R\\Omega$, and $r$ dependence of the critical temperature $T_{c}$ are plotted for various fixed parameters (see Fig. \\ref{fig9}). In Fig. \\ref{fig9}(a), the $G\\Lambda^2$ dependence of $T_c$ is plotted for $\\alpha_{b}=10,R\\Omega=0.5$ and $x=1,5,10$. The latter correspond to $r^2\/R^2=0.1, 0.5, 1$, respectively. As expected, $T_c$ increases with increasing coupling. This is because of large values of $G\\Lambda^2$ the formation of a chiral condensate is enhanced, and a phase transition to a chiral symmetry restored phase becomes only possible at higher temperatures. Moreover, the transition temperature depends on the distance (velocity\/angular kinetic energy\/centrifugal force) of the system with respect to the rotation axis. The farther bound states are from the origin (rotation axis), the lower is the critical temperature of a chiral phase transition.\n\\par\nIn Fig. \\ref{fig9}(b), the $eB$ dependence of $T_{c}$ is plotted for $x=\\alpha_{b}\/2, R\\Omega=0.8$, and $G\\Lambda^{2}=24,26$. We choose an appropriate large value of $R\\Omega$, and small values of $G\\Lambda^2$, to best demonstrate the IMRC. As it turns out, $T_{c}$ decreases with increasing $eB$. This is definitely a sign for an IMRC, induced solely by the rotation of a bounded system of quark matter. Let us remind that the results presented in \\cite{fayazbakhsh2011} for a nonrotating two-flavor NJL model at zero chemical potential $\\mu$ show an increase of $T_c$ as a function of $eB$. This is because of the catalytic effect of the magnetic field. In contrast, it is also shown that for nonvanishing chemical potential, $T_{c}$ decreases with increasing $eB$, and thus inverse magnetic catalysis occurs for $\\mu\\neq 0$ \\cite{fayazbakhsh2011}. There are, however, pieces of evidence for inverse magnetic catalysis arising from an \\textit{ab initio} lattice QCD computations \\cite{bali2013}.  Here, it is shown that $T_{c}$ decreases with $eB$ even in a system with zero chemical potential.\n\\par\nThe IMRC is best demonstrated in the $T_{c}$ versus $R\\Omega$ phase portrait in Fig. \\ref{fig9}(c). Here, the  $R\\Omega$ dependence of $T_c$ is plotted for $x=\\alpha_{b}\/2,G\\Lambda^2=30$ and $\\alpha_{b}=1,3,5,10$. The critical temperature decreases with increasing $R\\Omega$, but the slope of the corresponding curves increases with increasing $\\alpha_{b}$. Moreover, for a constant $R\\Omega$, $T_{c}$ decreases with increasing $\\alpha_{b}$.  These effects become more significant in the regime of large $R\\Omega$, e.g. $0.4<R\\Omega\\leq 1$  (see also Fig. \\ref{fig10}).\n\\par\nIn Fig. \\ref{fig9}(d), the $r^{2}\/R^{2}$ dependence of $T_{c}$ is plotted for $\\alpha_b=10, R\\Omega=0.5$, and $G\\Lambda^2=24,30$. In the regime $0<r^2<0.8 R^2$, $T_c$ decreases\nslightly with $r\/R$. Only near the boundary, for $0.8 R^2<r^2<R^2$ the critical temperature $T_c$ decreases drastically with $r\/R$.\nThese results indicate that as long as the ratio $r^2\/R^2$ is not larger than $r^2\/R^2=0.8$, the above conclusions concerning the appearance of the IMRC do not depend on $r$. The reason for the slight decrease of $T_c$ as a function of $r$ is the fact that for small values of coupling, the dynamical mass decreases with increasing $r$ [see Fig. \\ref{fig8}(d)], and thus smaller temperatures are necessary to destroy the bound states.\n\\par\nTo have a measure for the effect of large $R\\Omega$ on IMRC, let us define a quantity $\\Delta_c$ as\n\\begin{eqnarray}\n\\Delta_{c}\\equiv\\frac{T_{c}(\\alpha_{b,1})-T_{c}(\\alpha_{b,2})}{T_{c}(\\alpha_{b,1})}\\time 100\\qquad\\mbox{in \\%},\n\\end{eqnarray}\nfor fixed $x$ and $G\\Lambda^2$. In Fig. \\ref{fig10}, the $R\\Omega$ dependence of $\\Delta_{c}$ is plotted for $\\alpha_{b,1}=1,5$ and $\\alpha_{b,2}=10$, fixed $x=\\alpha_{b}\/2$, and $G\\Lambda^2=25,30,40$ [See Figs. \\ref{fig10}(a)-\\ref{fig10}(c)].\\footnote{Note that $T_{c}(\\alpha_{b}=10)\\leq T_{c}(\\alpha_{b}=1),\\mbox{and},~T_{c}(\\alpha_{b}=3)$. Hence, $\\Delta_{c}\\geq 0$ in these two cases.} The results reveal that independent of the value of the coupling $G\\Lambda^2$, $\\Delta_{c}$ increases with increasing $R\\Omega$.\n\\subsubsection{Critical $R\\Omega_{c}$ as a function of $G\\Lambda^2, eB,T$, and $r^2\/R^2$}\\label{sec3b3}\nFocusing on the behavior of the dynamical mass and the critical temperature of the chiral phase transition with respect to $eB$ and $R\\Omega$, the IMRC is demonstrated in Figs. \\ref{fig8}(b), \\ref{fig8}(c), as well as \\ref{fig9}(b) and \\ref{fig9}(c). As it is shown in Fig. \\ref{fig8}(c), for fixed values of $\\alpha_b,x,T$, and, in particular, for small values of $G\\Lambda^{2}$, the dynamical mass $\\bar{m}$ vanishes for certain critical velocity $R\\Omega_c\\leq 1$. For $G\\Lambda^2=25$ and $G\\Lambda^{2}=27$, $R\\Omega_{c}\\sim 0.78$ and $R\\Omega_c=0.93$, respectively. In what follows, the dependence of $R\\Omega_{c}$ on $G\\Lambda^2, eB,T$, and $r^2\/R^2$ is explored.\n\\par\nIn Fig. \\ref{fig11}(a), the $G\\Lambda^2$ dependence of $R\\Omega_{c}$ is plotted for $\\alpha_b=10, T=200$, and $x=1,3,5$. Although for all values of $x$, $R\\Omega_c$ increases with $G\\Lambda^2$, but the slope of the curves decreases with increasing $x$. For a given value of $G\\Lambda^2$, the smaller the value of $x$ is, the larger $\\Omega$ is necessary to break the condensate, and to restore the chiral symmetry. This is because, for a given $\\Omega$, the larger $x$ is, the larger is the kinetic energy of a bound state in the rotating system of quark matter and the centrifugal force applied on it. Apparently, larger kinetic energy\/centrifugal force applied on constituents helps to destroy the corresponding bound state.\nThe same effect is also observed in Fig. \\ref{fig11}(d), where the $r^2\/R^2$ dependence of $R\\Omega_c$ is plotted for $\\alpha_b=10, T=200$ MeV and $G\\Lambda^2=23, 25$. Here, for a fixed $\\Omega$, the larger $r$ is, the larger is the rotational kinetic energy as well as the centrifugal force, thus the smaller values of $\\Omega$ are necessary to destroy the chiral condensate and restore the chiral symmetry.\n\\par\nIn Fig. \\ref{fig11}(b), the $eB$ dependence of $R\\Omega_c$ is plotted for fixed $x=\\alpha_{b}\/2, T=200$ MeV, and $G\\Lambda^2=24,25$. Because of the IMRC, in which the dynamical mass decreases with increasing $eB$ [see Fig. \\ref{fig8}(b)], once the larger $eB$ is, the smaller values of $R\\Omega$ are necessary to restore the chiral symmetry. The same is also true for the temperature dependence of $\\Omega_{c}$. Since higher temperatures help to destroy the chiral condensate, the larger $T$ is, the smaller values of $\\Omega$ are necessary to restore the chiral symmetry.\n\\begin{figure*}\n\\includegraphics[width=8cm,height=6cm]{fig11a}\n\\includegraphics[width=8cm,height=6cm]{fig11b}\\vspace{0.3cm}\n\\includegraphics[width=8cm,height=6cm]{fig11c}\n\\includegraphics[width=8cm,height=6cm]{fig11d}\n\\caption{\ncolor online. a) The $G\\Lambda^2$ dependence of $R\\Omega_c$ is plotted for $\\alpha_{b}=10,T=200$ MeV, and $x=1,3,5$. As expected, $R\\Omega_c$ increases with increasing $G\\Lambda^2$. For a given value of $G\\Lambda^2$, the smaller $x$ is, the larger values of $\\Omega$ is necessary to restore the chiral symmetry.\nb) The $eB\/m_{\\pi}^{2}$ dependence of $R\\Omega_{c}$ is plotted for $x=\\alpha_{b}\/2, T=200$ MeV, and $G\\Lambda_{2}=24,25$.  Because of the IMRC,  $R\\Omega_{c}$ decreases with increasing $eB$.\nc)  The  $T$ dependence of $R\\Omega_c$ is plotted for $\\alpha_b=10, x=5$, and $G\\Lambda^2=24,26$.  As expected, $R\\Omega_{c}$ decreases with increasing temperature, which, by itself, trigger the chiral symmetry restoration.\nd) The $r^{2}\/R^{2}$ dependence of $R\\Omega_{c}$ is plotted for $\\alpha_b=10, T=200$ MeV, and $G\\Lambda^2=23,25$. For a given value of $eB,\\Omega,T$ and $G$, the farther the distance from the rotational axis is, the larger is the kinetic energy of a bound state as well as the centrifugal force applied on it, thus the smaller values of $\\Omega$ are necessary to destroy the bound state, and restore the chiral symmetry.\n}\\label{fig11}\n\\end{figure*}\n\n\\section{Concluding remarks}\\label{sec4}\n\\setcounter{equation}{0}\nOne of the key ingredients in studying the possible effects of the simultaneous presence of the rotation and constant magnetic field on fermionic systems is the solution of the corresponding Dirac equation. In the first part of the present work, we presented a systematic derivation of the solutions of the Dirac equation in a rotating and magnetized system, using the Ritus eigenfunction method. Using the Ritus eigenfunctions and energy spectrum, we derived the corresponding quantization relations for fermions in an infinitely extended system with no boundary condition. We then imposed a global boundary condition on this fermionic system to avoid causality-violating effects, and scrutinized the effect of boundary conditions on the energy spectrum of fermions once they are bounded in a cylinder with radius $R$. This derivation, which did not appear in the literature before, gives us new insight, in particular, into the way how fermions occupy the lowest energy levels. We showed, in particular, that for a bounded system the lowest energy level is to be determined numerically. Using the aforementioned quantization relation of a bounded magnetized and rotating system, we derived the corresponding fermion propagator, from which appropriate expressions for the gap equations of a QCD-like model at zero and nonzero temperatures were found. \n\\par\nWe showed that at zero temperature, the dynamical mass $\\bar{m}$ has no dependence on the angular frequency $\\Omega$. This confirms the statement in \\cite{chernodub2016-1} that \"cold vacuum does not rotate\". Its dependence on the distance $r$ from the rotation axis could be regulated by choosing appropriate coupling $G_m$ for each fixed $eB$. The values of $G_{m}$ are determined so that the dynamical mass remains almost constant in a relatively large interval of $r$ (see Table \\ref{tab5}). We then studied the $r$ and $eB$ dependence $\\bar{m}$ for different values of $G_m$ and fixed values of $eB$ and $r$, respectively, and showed how the $eB$ dependence of $G_m$ is reflected in the $eB$ dependence of $\\bar{m}$ [compare the results demonstrated in Figs. \\ref{fig3x} and \\ref{fig6}(b)].    \n\\par\nIn the finite temperature case, we numerically determined the $T, eB,\\Omega$, and $r$ dependence of $\\bar{m}$, and plotted the complete phase portraits of $T_c$ versus $G\\Lambda^2$, $eB\/m_{\\pi}^{2}$, $R\\Omega$, and $r^2\/R^2$ as well as $R\\Omega_c$ versus $G\\Lambda^2$, $eB\/m_{\\pi}^{2}$, $R\\Omega$, and $r^2\/R^2$. Our results show that there are, at least, three signatures for the fact that rigid rotation initiates the IMRC. The first one is the decrease of $\\bar{m}$ with $eB$, once $G\\Lambda^2, T, r$, and $\\Omega$ are kept fixed [see Fig. \\ref{fig8}(b)]. As it turns out, the slope is larger for smaller values of $G\\Lambda^2$ and $R\\Omega$. Hence, an appropriate choice of these two parameters negatively affects the production of the dynamical mass and leads to IMRC. Apart from this specific behavior of $\\bar{m}$, the fact that $T_{c}$ decreases with increasing $eB$ is a result of the IMRC [see Fig. \\ref{fig9}(b)]. This is similar to ordinary inverse magnetic catalysis in a nonrotating quark matter, whose evidence from lattice QCD is exactly the same phenomenon. But, there is also a third and novel evidence of IMRC. This is the decrease of $\\Omega_c$ with $eB$ [see Fig. \\ref{fig11}(b)]. Recent studies show similar results for a two-flavor quark matter \\cite{tabatabaee2021-2}.          \n\\par\nApart from the IMRC, the $r$ dependence of $\\bar{m}$ is striking. It turns out that, for small enough coupling $G\\Lambda^2$ and even for large $eB$, the dynamical mass decreases with increasing $r\/R$ [see Fig. \\ref{fig8}(d)]. A similar conclusion is also made in \\cite{chernodub2016-1}. Here, we must be cautious. Actually, we solved the gap equation with the assumption of $\\partial_r\\bar{m}\\ll \\bar{m}(r)^2$ \\cite{fukushima2017}. Hence,  $\\bar{m}(r)$  curves with very large slopes in Fig. \\ref{fig8}(d) are unacceptable. As it turns out, for fixed values of $eB,T$, and $R\\Omega$,  an appropriate choice of $G\\Lambda^2$ affects the slope of the $\\bar{m}(r)$ curves. Without this assumption, we are facing an integral equation, whose solution leads to an inhomogeneous mass gap. In \\cite{wang2019-1, wang2019-2}, the nonlocal chiral condensate in $2+1$ dimensions is treated using a Bogoliubov-de Gennes like method \\cite{buballa2015}. It is shown that for sufficiently large angular frequency, chiral vortices are built in the ground state. These are topological defects in analogy to the ones appearing in superfluids and superconductors \\cite{wang2019-2}. It would be interesting to extend this work to the case of magnetized quark matter, which is relevant in the physics of neutron stars. These compact stars are subject to extremely large angular frequencies and very large magnetic fields. Other methods to treat inhomogeneous chiral condensates are presented in \\cite{ferrer2015}, where the Magnetic Dual Chiral Density Wave phase is studied. A possible extension of this work is to introduce the rotation, and to study its effect on the phase diagram of this system \\cite{sadooghi2021-3}. Implications of rotation on the phenomenological observables of HIC, like dilepton and photon production rates, are also most relevant in HIC's physics. We postpone these interesting subjects to our future publications.  \n\\section{Acknowledgments}\nN. S. thanks M. H. Gholami for useful discussions. This work is supported by Sharif University of Technology's Office of Vice President for Research under Grant No: G960212\/Sadooghi. \n\n\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe investigation of exactly solvable models over the last many years has been\nproven very fruitful in extracting physical information regarding phenomena\nlying in strongly coupled regimes \\cite{1dimHubbard}.  For integrable models\nin one spatial dimension, the framework of quantum inverse scattering method\n(QISM) \\cite{Faddeev:1979gh, Korepin} provides an efficient way of computing\nthe various physical quantities. In the present article we study the small\npolaron model, which can be also regarded as a graded version of the XXZ\nquantum spin chain. The $R$-matrix satisfies the graded Yang-Baxter equation\n\\cite{Yang_Baxter_eq, Kulish_Sklyanin, Kulish:1985bj} \n\\begin{equation}\n\\label{eqYBE}\nR_{12}(\\lambda) \\,\nR_{13}(\\lambda+\\mu) \\, R_{23}(\\mu) = R_{23}(\\mu) \\, R_{13}(\\lambda+\\mu) \\, R_{12}(\\lambda) \\, ,\n\\end{equation}\nand the resulting bulk Hamiltonian coincides with the bulk Hamiltonian of the\nXXZ spin chain, after employing a Jordan-Wigner transformation. The effects of\nsupersymmetry become visible when periodicity of the boundary conditions is\nrelaxed, hence more general boundaries are considered, corresponding to open\nboundaries in the spin chain picture.\n\nWithin a graded version of Sklyanin's reflection algebra\n\\cite{Sklyanin:1988yz} these boundary conditions can be implemented while\nkeeping the integrability of the model.  For purely diagonal boundary fields,\ni.e.\\ respecting the $U(1)$ symmetry of the bulk, the spectrum of the model\ncan be obtained using algebraic Bethe Ansatz (ABA) methods.\nFor non diagonal boundary conditions, however, this symmetry is broken and a\nsimple reference state does not exist anymore, rendering the ABA framework\ninsufficient.  \n\n\nFor the respective ungraded model various methods have been successfully\nemployed in the past, such as the fusion hierarchy of the transfer matrices\nand its truncation \\cite{Nepomechie:2001uz, Nepomechie:2003vv}, the\nconstruction of a vacuum state by using gauge transformations \\cite{Cao_2002},\nand generalized TQ equations \\cite{Murgan:2006cx}.  All of these approaches,\nhowever, rely on the boundary parameters satisfying certain algebraic\nrelations or are restricted to particular values of the bulk anisotropy\nparameter. No such constraints on the system parameters appear in solutions\nusing the representation theory of the $q$-Onsager algebra \\cite{BaKo07}, in\nterms of functional relations derived directly from the Yang-Baxter algebra\n\\cite{Galleas:2008zz}, or by separation of variables\n\\cite{FrSW08,FGSW11,Nicc12}.  So far, an actual solution of the eigenvalue\nproblem in these formulations is possible for relatively small systems only\nand it is unclear how the thermodynamic limit can be approached.\n\nMotivated by a recent analysis of the free fermion case\n\\cite{Grab_Frahm_free_fermi} where the Grassmannian nature of the nondiagonal\nboundary parameters was proven sufficient to solve the model without imposing\nany constraints, we studied the small polaron model with nondiagonal boundary\nconditions, expecting that a similar situation may also hold. As it will\nbecome transparent in the next sections, it turns out that this is the case\nfor the interacting fermions as well: supersymmetry lifts the need of imposing\nconstraints on the boundary parameters.  Furthermore, the structure of the\neigenvectors is greatly restricted, so that certain eigenstates can be\ncomputed exactly for an arbitrary number of chain sites.\n\nThe paper is organized as follows: first, we describe the basic facts of the\nsmall polaron model and setup our conventions.\nIn Section 3 we first recall the functional equations relating the commuting\ntransfer matrices obtained within the fusion approach obtained in\nRef.~\\cite{Grab_Frahm}.  Based on these findings we proceed to derive higher\norder functional equations for the transfer matrix of the small polaron model\nfor particular values of the anisotropy parameter which can be formulated as a vanishing\ndeterminant condition.  The latter allows to formulate the spectral problem in\nterms of a TQ equation.  Based on the comparison with a particular limit of\nthe fusion hierarchy we conjecture that this equation actually describes the\nspectrum of the model without any restrictions on the system parameters, i.e.\\\nanisotropy and boundary fields.  This is supported by the fact that the\nsolution for diagonal boundary conditions can be shown to be equivalent to\nwhat has been found previously using algebraic or coordinate Bethe ansatz\nmethods.\nIn Section 4 we proceed with solving the full nondiagonal model, by deforming\nthe corresponding problem of the diagonal case. The transfer matrix eigenvalues \nare found to depend on two distinct sets of\nBethe roots, which satisfy two coupled sets of Bethe Ansatz Equations (BAE).\nIn Section 5 we compute the 'vacuum eigenstate' of the model, i.e.\\ the unique\nstate which reduces to the Fock vacuum in the limit of diagonal boundary\nfields.  Based on this result we propose an expression for the generic\nstructure of the eigenvectors of the model which allows for a complete\nsolution of the spectral problem in principle.  We conclude with discussing\nour results and future directions.\n\n\n\n\n\n\n\\section{The small polaron with open boundary conditions}\nThe small polaron model \\cite{Fedyanin_1, Fedyanin_2} describes the motion of\nan additional electron in a polar crystal.  The physics of these interactions\nis captured by the following bulk Hamiltonian\n\\begin{equation}\nH_\\textrm{bulk} = \\sum_{j=1}^{N-1} - t\\, \\big(c^\\dagger_{j+1}\\, c_j + c^\\dagger_j\\,  c_{j+1}\\big) \n+V \\, \\big(n_{j+1} \\, n_j + \\bar{n}_{j+1}\\, \\bar{n}_j\\big) \\, ,\n\\label{Hbulk}\n\\end{equation}\nwhere $c^\\dagger_k$ and $c_k$ denote the creation and annihilation operators\nof spinless fermions at site $k$ respectively, obeying anticommutation\nrelations $\\{c_k^\\dagger, c_l\\} = \\delta_{kl}$. We have also introduced the\noccupation number operators $n_k = c_k^\\dagger c_k$ and $\\bar{n}_k = 1-n_k$,\nso that the parameters $t$ and $V$ may be interpreted as hopping amplitude and\ndensity-density interaction strength respectively. It is possible to derive\nthe above Hamiltonian through the QISM framework, thus rendering the system\nintegrable \\cite{Pu_Zhao}. In the Hamiltonian (\\ref{Hbulk}), periodic boundary\nconditions are to be assumed. However, periodicity can be relaxed and one may\nconsider integrable open boundary conditions, by using the framework of\nquantum integrability for models with open boundaries\n\\cite{Sklyanin:1988yz}. The full Hamiltonian of the system contains the\nadditional boundary terms, which in general may be non-diagonal ones\n\\begin{equation}\nH = H_\\textrm{bulk} + H_\\textrm{diag}+ H_\\textrm{nondiag} \\, .\n\\label{Hfull}\n\\end{equation}\nSince we deal with a fermionic lattice model, the local space of states is\n$\\mathbb{Z}_2$-graded \\cite{Gohmann_Murakami}.  The tensor product is graded\naccording to the rule\n\\begin{equation}\n(A \\otimes_s B)^{ac}_{~~bd} = (-1)^{[p(a)+p(b)]p(c)} A^a_{~b}\\,  B^c_{~d} \\, ,\n\\end{equation}\nwhere the parity $p(a)$ is equal to zero (one) for bosonic (fermionic)\nindices. All matrix operations below, such as the super trace of a matrix, are\nthen to be understood as operations on super matrices. We omit the definitions\nof these matrix operations and refer the interested reader to the references\n\\cite{Cornwell, Gohmann_Murakami}, whose conventions we follow.\n\nThe fundamental super transfer matrix of the model is given by the super trace\n\\begin{equation}\n t(u) = \\textrm{str}_0 \\Big\\{ K^+(u)\\, T(u)\\, K^-(u)\\, \\hat{T}(u) \\Big\\} \\, ,\n \\label{super_transfer_matrix}\n\\end{equation}\nwhere the monodromy matrices are defined as\n\\begin{equation}\nT(u) = R_{N0}(u) \\cdots R_{20}(u) \\, R_{10}(u) \\, , \n\\qquad \\hat{T}(u) = R_{01}(u)R_{02}(u) \\cdots R_{0N}(u) \\, .\n\\label{monodromies_defs}\n\\end{equation}\nThe $R$-matrix is given by\n\\begin{equation}\n R_{ij}(u)=\\frac{1}{\\sin(2\\eta)}\n      \\begin{pmatrix}\n        \\sin(u+2\\eta)&0&0&0\\\\\n        0&\\sin(u)&\\sin(2\\eta)&0\\\\\n        0&\\sin(2\\eta)&\\sin(u)&0\\\\\n        0&0&0&-\\sin(u+2\\eta)\n      \\end{pmatrix}\\,\n \\label{eq:RMatrix}\n\\end{equation}\nacting on the tensor product $V_i\\otimes V_j$ of two linear spaces $\\sim\n\\mathbb{C}^2$.  It satisfies the graded Yang-Baxter equation (\\ref{eqYBE}) and\nenjoys several useful properties, such as unitarity\n\\begin{equation}\nR_{12}(u)R_{21}(-u) = \\zeta(u) \\, ,\n\\end{equation}\ncrossing symmetry \n\\begin{equation}\nR_{21}^{st_2}(-u-4\\eta) R_{21}^{st_1}(u)=\\zeta(u+2\\eta) \\, ,\n\\end{equation}\nand periodicity\n\\begin{equation}\nR_{12}(u+\\pi) = -\\sigma_1^z R_{12}(u) \\sigma_1^z = -\\sigma_2^z R_{12}(u) \\sigma_2^z \\, .\n\\end{equation}\nIn the above we have also defined\n\\begin{equation} \n\\zeta(u) \\equiv g(u)g(-u), \\qquad \\textrm{and} \\qquad g(u) \\equiv \n\\frac{\\sin(u-2\\eta)}{\\sin(2\\eta)}\\, .\n\\end{equation}\n\nThe $K$-matrices, which contain the boundary information, satisfy the graded\nreflection algebra \\cite{Sklyanin:1988yz, Bracken} and have the following\ngeneric expressions (see also \\cite{Zhou, Zhou2, GuanGrimmRoemer})\n\\begin{equation}\n \\begin{aligned}\n   K^\\sss{-}(u)&=\\omega^\\sss{-}\n          \\begin{pmatrix}\n            \\sin(u+\\psi_\\sss{-})&\\alpha^\\sss{-}\\sin(2u)\\\\\n            \\beta^\\sss{-}\\sin(2u)&-\\sin(u-\\psi_\\sss{-})\n          \\end{pmatrix} \\,,\n\\\\[1em]\n   K^\\sss{+}(u)&=\\omega^\\sss{+}\n          \\begin{pmatrix}\n            \\sin(u+2\\eta+\\psi_\\sss{+} ) & \\alpha^\\sss{+}\\sin(2u+4\\eta)\\\\\n            \\beta^\\sss{+}\\sin(2u+4\\eta) & \\sin(u+2\\eta-\\psi_\\sss{+})\n          \\end{pmatrix} \\, ,\n \\end{aligned}\n \\label{eq:generalKs}\n\\end{equation}\nwith normalizations $\\omega^\\sss{\\pm}$ defined by\n\\begin{equation}\n  \\omega^\\sss{-} \\equiv \\frac{1}{\\sin\\psi_\\sss{-}} \\qquad \\text{and} \\qquad\n  \\omega^\\sss{+} \\equiv \\frac{1}{2\\cos2\\eta \\, \\sin\\psi_\\sss{+}} \\, .\n\\end{equation}\nThe boundary parameters $\\psi_\\sss{\\pm}$ are commuting numbers with a\nnon-vanishing complex part, while the parameters $\\alpha^\\sss{\\pm}, \\beta^\\sss{\\pm}$\nare odd Grassmann numbers and anticommute:\n\\begin{equation}\n[\\psi_\\sss{+}, \\, \\psi_\\sss{-}] = 0 = \\{\\alpha^\\sss{\\pm}, \\, \\alpha^\\sss{\\pm} \\} \n= \\{\\alpha^\\sss{\\pm}, \\,\\beta^\\sss{\\pm} \\} = \\{\\beta^\\sss{\\pm}, \\, \\beta^\\sss{\\pm} \\} \\, . \n\\end{equation}\nMoreover, the reflection algebra is only satisfied provided that the odd\nGrassmann numbers are subject to the condition $\\alpha^\\sss{+} \\cdot \\beta^\\sss{+} =0\n= \\alpha^\\sss{-}\\cdot \\beta^\\sss{-}$.\n\nThrough the framework of QISM, the super transfer matrix\n(\\ref{super_transfer_matrix}) gives rise to a commutative family of conserved\ncharges, among them the Hamiltonian of the small polaron with open\nboundaries, which can be derived as\n\\begin{equation}\n\\left.\\frac{d}{du}t(u) \\right|_{u=0} = 2H +\\textrm{const}\\, .\n\\end{equation}\nThe bulk part of this Hamiltonian coincides with the expression (\\ref{Hbulk})\nafter identifying $t=-\\csc 2\\eta$ and $V=\\cot2\\eta$.  The boundary terms\nobtained with (\\ref{eq:generalKs}) read as\n\\begin{equation}\n\\begin{split}\n& H_{\\textrm{diag}} = \\mathcal{N}_+ \\, \\bar{n}_N - \\mathcal{N}_-\\, n_N +\n\\frac{1}{2}\\cot\\psi_\\sss{-} \\, (\\bar{n}_1 -n_1) \\cr \n& H_{\\textrm{nondiag}} =\\csc \\psi_\\sss{-} \\,  ( \\alpha^\\sss{-} c_1 - \\beta^\\sss{-}\nc^\\dagger_1)  \n+ \\csc\\psi_\\sss{+} \\, ( \\alpha^\\sss{+} c_N - \\beta^\\sss{+} c^\\dagger_N) \\, ,\n\\end{split}\n\\label{H_boundary}\n\\end{equation}\nwhere the following shorthands have been also introduced:\n$\\mathcal{N}_\\pm\\equiv\\frac{1}{2}\\csc(2\\eta) \\csc(\\psi_\\sss{+})\n\\sin(2\\eta\\pm\\psi_\\sss{+})$.\nThe diagonal boundary terms can be identified with static boundary chemical\npotential at the first and last site of the lattice, respectively.  By means\nof a Jordan-Wigner transformation the bulk and diagonal boundary terms can be\nmapped to spin-1\/2 XXZ Heisenberg chain with boundary magnetic fields, see\nalso Appendix A.  This is not possible for the off-diagonal terms breaking the\n$U(1)$ symmetry of the system.  In the small polaron formulation the terms in\n$H_{\\textrm{nondiag}}$ can be interpreted as sources and sinks for injection\nof additional particles into the system.  Their amplitudes are odd Grassman\nnumbers, reflecting the fermionic nature of the corresponding reservoir.  The\nJordan-Wigner transformation of these terms yields a non-local expression in\nthe spin chain formulation.\n\n\nThe super transfer matrix enjoys crossing symmetry\n\\begin{equation} \nt(-u-2\\eta) = t(u) \\, ,\n\\label{trns_crs_symm}\n\\end{equation}\nand periodicity\n\\begin{equation}\nt(u+\\pi) = t(u)\\, . \n\\end{equation}\nFor future use, we also note that the open transfer matrix is normalized as\n$t(0) = \\mathds{1} $ and becomes diagonal in the semi-classical limit\n$\\eta\\to0$\n\\begin{eqnarray}\n&& t(u)\\Big|_{\\eta=0} = \\frac{(\\sin u)^{2N}}{\\sin\\psi_\\sss{+}\\sin\\psi_\\sss{-}} \\Big[ 2\\sin^2u  \\cos^2u \\, (\\beta^\\sss{+}\\alpha^\\sss{-} - \n\\alpha^\\sss{+}\\beta^\\sss{-}) \\, \\prod_{k=1}^N \\sigma^z_k \\cr \n&& \\hspace{3cm} - \\big(\\cos^2u \\, \\sin\\psi_\\sss{-}\\sin \\psi_\\sss{+}  +\\sin^2u \\cos\\psi_\\sss{-} \\cos\\psi_\\sss{+}\\big) \\,  \n\\mathds{1} \\Big]\\, .\n\\label{transfer_semiclassical_limit}\n\\end{eqnarray}\nFinally, taking the limit $z\\equiv e^{iu} \\to \\infty$, the asymptotic behavior\nof the super transfer matrix is obtained, which is needed for later\ncomparison. The leading term contains only odd Grassmann boundary parameters\nand has the expression\n\\begin{equation}\nt(z)  =  \\left(\\frac{z}{2i\\sin 2\\eta}\\right)^{2N} \\frac{\\omega^\\sss{+} \\omega^\\sss{-}}{4}\\,   z^4 \\, e^{4i\\eta} \\, \n (\\beta^\\sss{+}\\alpha^\\sss{-} - \\alpha^\\sss{+}\\beta^\\sss{-})  \\, \\prod_{j=1}^N(\\bar{n}_j - e^{2i\\eta}n_j)(n_j+e^{2i\\eta}\n \\bar{n}_j) + \\mathcal{O}(z^2)\\, .\n \\label{asymptotic_transfer}\n\\end{equation}\nThis particular combination of the odd Grassmann boundary parameters emerges\nin many different relations, and as will become transparent below it appears\nuniquely in all eigenvalues of the transfer matrix, thus henceforth it will be\ndenoted as\n\\begin{equation}\n\\mathcal{G} \\equiv \\beta^\\sss{+}\\alpha^\\sss{-} - \\alpha^\\sss{+}\\beta^\\sss{-} \\, .\n\\label{mathcalg}\n\\end{equation}\n\n\n\n\\section{Functional relations}\nAs stated in the introduction our analysis of the spectral problem for the\ntransfer matrix of the small polaron model relies on its reformulation in\nterms of functional equations.  To keep the presentation of our results\nself-contained we begin by recalling the results of Ref.~\\cite{Grab_Frahm},\nnamely the infinite hierarchy of fused transfer matrices and its truncation\nfor particular values of the anisotropy parameter $\\eta$.  Then we use the\ntruncation identity to rewrite the fusion hierarchy as a higher order\nfunctional relation for the transfer matrix (\\ref{super_transfer_matrix})\nalone.  This relation is then shown to be equivalent to a vanishing\ndeterminant condition which allows for the formulation of the spectral problem\nin terms of a TQ equation.  We then argue that this TQ equation holds for\narbitrary values of the anisotropy and generic non-diagonal boundary\nconditions.\n\n\n\\subsection{Fusion hierarchy and truncation identities}\nHaving a transfer matrix at hand, one may construct a family of commuting\ntransfer matrices derived from auxiliary spaces of higher dimension, through a\nfusion procedure \\cite{Kulish:1981gi, Mezincescu:1991ke}.  Choosing a suitable\nnormalization the fused transfer matrices form a hierarchy related by the\nrecursion relations \n\\begin{equation}\n  t^{(n)}(u) \\cdot t^{(1)}(u+n\\cdot 2\\eta) =\n  t^{(n+1)}(u) -  \n          \\tilde{\\Delta}(u+[n-1]\\cdot 2\\eta) \\cdot t^{(n-1)}(u) \\, .\n  \\label{eq:OBCFusionHierarchyNoTilde}\n\\end{equation}\nwith $t^{(0)} \\equiv \\mathds{1}$ and $t^{(1)} \\equiv -t(u)$.\nThe function $\\tilde\\Delta(u)$ is up to a scaling factor  the\nsuper quantum determinant of the small polaron model \\cite{Grab_Frahm}\n\\begin{equation}\n\\begin{aligned} \n  \\tilde\\Delta(u)\\,\\zeta(2u+4\\eta) &\\equiv \\Delta(u)\\\\ & =\n  \\zeta^{2N}(u+2\\eta) \\, g(-2u-6\\eta) \\, g(2u+2\\eta) \\det\n  K^+(u) \\det K^-(u+2\\eta) \\, . \n\\end{aligned}\n\\end{equation}\nNote that the quantum determinant does not depend on the off-diagonal elements\nof the boundary matrices $K^\\pm(u)$ as a consequence of the nilpotency of the\nodd Grassmann parameters.\nSince the fused transfer matrices commute with each other, the fusion\nhierarchy can be also read as a set of relations between their eigenvalues,\nwhich after shifting the spectral parameter $u\\to u-n\\cdot 2\\eta$ gives\n\\begin{equation}\n\\label{FUS-n}\n -\\Lambda(u) = \\frac{\\Lambda^{(n+1)}(u-n\\cdot\n  2\\eta)}{\\Lambda^{(n)}(u-n\\cdot 2\\eta)} - \\tilde{\\Delta}(u-2\\eta) \n\\frac{\\Lambda^{(n-1)}(u-n\\cdot 2\\eta)}{\\Lambda^{(n)}(u-n\\cdot 2\\eta)}\\, .\n\\end{equation}\nFor later use we rewrite these equations in terms of the functions\n\\begin{equation}\n  \\Lambda^{(n)}(u-n\\cdot2\\eta) \n  \\equiv Q^{(n)}(u) \\prod_{k=0}^n \\kappa(u-k\\cdot 2\\eta) \\, ,\n\\end{equation}\ngiving\n\\begin{equation}\n\\label{TQeq-n}\n\\begin{aligned}\n  -\\Lambda(u) &= h^+(u) \\frac{Q^{(n+1)}(u+2\\eta)}{Q^{(n)}(u)}\n                      - h^-(u) \\frac{Q^{(n-1)}(u-2\\eta)}{Q^{(n)}(u)}\\,,\\\\\n  & h^+(u) = \\kappa(u+2\\eta)\\,,\\qquad \n    h^-(u) = \\tilde{\\Delta}(u-2\\eta)\\frac{1}{\\kappa(u)}\\,.\n\\end{aligned}\n\\end{equation}\nNote that while the coefficient functions $h^{\\pm}(u)$ can be modified by\nchoosing different factors $\\kappa(u)$ in the definition of $Q^{(n)}(u)$ they\nalways factorize the rescaled quantum determinant, i.e.\\ $h^+(u) h^-(u+2\\eta)\n= \\tilde{\\Delta}(u)$.  \n\n\n\nThe functional equations (\\ref{FUS-n}) constitute a set of relations for an\ninfinite set of unknown functions $\\tilde{\\Lambda}^{(n)}(u)$.  Restricting the\nanisotropy parameter to 'roots of unity', $\\eta_n=\\frac{\\pi\/2}{n+1}$, is has\nbeen shown that the fused transfer matrices at level $n+1$ and $n-1$ are related\nby the truncation identity \\cite{Grab_Frahm}\n\\begin{equation}\nt^{(n+1)}(u,\\eta_{n}) =  \\phi^{\\text{id}}_{n}(u) \\cdot \\mathbb{I} -\n\\phi^{\\tau}_{n}(u)  \n\\cdot t^{(n-1)}(u+2\\eta_{n},\\eta_{n}) \\, ,\n\\label{eq:obcTauTruncation}\n\\end{equation}\nwhere the functions $\\phi^{\\text{id}}_{n}(u), \\phi^{\\tau}_{n}(u)$ are given by\nthe following expressions \n\\begin{eqnarray} \n&& \\phi^{\\text{id}}_{n}(u)  =  \\mathcal{M}_{n}^{2N}(u)\\ \\mu^{\\sss\n  +}_{n}(u)\\mu^{\\sss -}_{n}(u) \n[ \\nu^{\\sss +}_{n}(-u)\\nu^{\\sss -}_{n}(u) + \\nu^{\\sss +}_{n}(u)\\nu^{\\sss\n  -}_{n}(-u) ] \\nonumber \\\\ \n&& \\phi^{\\tau}_{n}(u) = \\zeta^{2N}(u) \\mu_n^+(u) \\mu_n^-(u) \\, ,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\mathcal{M}_n(u) & \\equiv & \\left(\\frac{1\/2}{\\sin2\\eta_n}\\right)^n \n\\frac{\\sin([n+1]u)}{\\sin2\\eta_n} \\cr\n\\mu^\\sss{\\pm}_n(u) &\\equiv& \\pm \\delta\\{K^\\sss{\\pm}(\\mp\nu-2\\eta_n,\\eta_n)\\}\\frac{\\sin(2\\eta_n)}{\\sin(2u-2\\cdot 2\\eta_n)} \n\\prod_{k=2}^{2n}\\frac{\\sin(2u+k\\cdot 2\\eta_n)}{\\sin(2\\eta_n)}\\cr\n\\nu^\\sss{\\pm}_n(u) &\\equiv& \\mp\\frac{\\omega_n^\\sss{\\pm}}{\\mu^\\sss{\\pm}_n(u)}\n\\left(\\frac{\\omega_n^\\sss{\\pm}}{2}\\right)^n \n\\sin([n+1][u\\mp\\psi_\\sss{\\pm}])\n\\prod_{i=1}^n \\prod_{j=1}^i \\frac{\\sin(2u+[i+j]\\cdot 2\\eta_n)}{\\sin(2\\eta_n)}\\ .\n\\end{eqnarray}\n\n\n\n\\subsection{Higher order functional equation and determinant representation}\nCombining the fusion hierarchy with the truncation identity for a particular\nvalue $n$, yields an $(n+1)$-order functional equation for the transfer matrix\nof the model with the corresponding anisotropy $\\eta=\\eta_n$. Unfortunately,\nit does not seem possible to write the functional relations in a closed form.\nStarting from low values of $n$, one observes that at each level new terms\nemerge and their number is given by the Fibonacci numbers $F(n)$. In\nparticular, at a given value $n$, there are\n\\[\nF(n+1) + F(n-1) +1 ~, \\qquad n=2,3, \\cdots \\, \n\\]\nterms in total. Nevertheless, after carefully examining the structure of the\nfunctional relations, it is seen that the following simple schematic structure\nappears\n\\begin{equation}\n \\label{funct_rel}\n \\begin{aligned}\n   & t(u) \\,  t(u+2\\eta) \\cdots t(u+(2n-2)\\eta) + \\mho_1\n   + \\phi^\\tau_{n-1}(u) \\cdot \\mho_0 \\\\\n   & \\qquad = \\phi_{n-1}^{id}(u) + \\prod_{k=0}^{\\frac{n}{2}-1}\n   \\tilde{\\Delta}(u+4k\\eta) \n   + \\, \\phi^\\tau_{n-1}(u) \\, \\prod_{k=1}^{\\frac{n}{2}-1}\n   \\tilde{\\Delta}(u+(4k-2)\\eta)\\,. \n \\end{aligned}\n\\end{equation}\nFirst, we should note that the last two terms in the RHS are present only for\neven values of $n$. Next, the symbol $\\mho$ stands for the following sequence:\n\\begin{eqnarray}\n && \\mho_1 :=   \\sum_{q_0 = 0}^{n-2} \\, \\tilde{\\Delta}(u+2q_0 \\eta) ~ \\mathcal{T}_{q_0}(u) \\nonumber \\\\ \n && \\qquad +  \\sum_{q_1=0}^{n-4} \\, \\tilde{\\Delta}(u+ (2q_1+4) \\eta) \n \\sum_{q_0=0}^{q_1} \\, \\tilde{\\Delta}(u+2q_0 \\eta) ~ \\mathcal{T}_{q_0q_1}(u) \\nonumber \\\\ \n && \\qquad +  \\sum_{q_2=0}^{n-6}\\, \\tilde{\\Delta}(u+ (2q_2+8) \\eta)  \\sum_{q_1=0}^{q_2} \\, \n \\tilde{\\Delta}(u+ (2q_1+4) \\eta)\n \\sum_{q_0=0}^{q_1} \\tilde{\\Delta}(u+2q_0 \\eta) ~ \\mathcal{T}_{q_0q_1q_2}(u) \\nonumber \\\\ \n & & \\qquad + \\cdots \\, ,\n\\label{om_seq}\n\\end{eqnarray}\nwith \n\\[\n \\mathcal{T}_{q_0q_1\\cdots q_{\\ell}}(u) \\equiv \\prod_{\\substack{m=0 \\\\ m\\notin\n     \\mathcal{Q}}}^{n-1} t (u+2m\\eta),  \n \\qquad \\mathcal{Q}= \\bigcup_{k=1}^\\ell  \\{q_k +2k, q_k+ 2k +1\\}\\,.\n\\]\nA similar structure holds for $\\mho_0$ as well, although some lower\/upper\nlimits of the sums and the product are different. Nevertheless, the structure\nabove is reminiscent of a path-ordered exponential, and appears as some\ndeformed discrete version of the latter one. It would be interesting to see if\nthis observation possesses some physical meaning, i.e., if the corresponding\noperator in the path-ordered exponential plays some physical role here.\n\nDespite the fact that the functional relations cannot be written in a closed\nform, they can be cast into a vanishing determinant representation as a\nconsequence of the\nrestriction $m \\notin \\mathcal{Q}$ in Eq.~(\\ref{om_seq}) above.\nThis\nleads to the derivation of a TQ equation for the eigenvalues of the transfer\nmatrix (\\ref{super_transfer_matrix}) \\cite{Bazhanov:1987zu}. For the small\npolaron model, it turns out that the vanishing determinant representation has\nthe same structure with the corresponding one of the XXZ model\n\\cite{Nepomechie:2003vv}. In particular, the functional relations\n(\\ref{funct_rel}) can be also written in the following form\n\\begin{equation}\n\\det \n\\begin{pmatrix}\n  \\Lambda_0 & -\\tilde{h}_{-1} & 0 & \\cdots &  0& -h_0 \\cr\n -h_1 & \\Lambda_1 & -\\tilde{h}_0 & 0  & \\cdots   & 0 \\cr\n  0 & -h_2 & \\Lambda_2 & -\\tilde{h}_1 & \\ddots & \\vdots \\cr\n  \\vdots & 0 & \\ddots  & \\ddots & \\ddots & 0\\cr \n  0 & \\vdots  & \\ddots & -h_{n-1} & \\Lambda_{n-1} & -\\tilde{h}_{n-2} \\cr\n  -\\tilde{h}_{n-1}  & 0 & \\cdots & 0 & -h_n & \\Lambda_n\n\\end{pmatrix} = 0~,\n\\label{det_rep}\n\\end{equation}\nwhere $\\Lambda_k \\equiv \\Lambda(u+2k\\eta)$ is an eigenvalue of\n$t(u+2k\\eta)$, $~h_{k} \\equiv h(u+2k\\eta)$ and $\\tilde{h}_k \\equiv\n\\tilde{h}(u+2k\\eta)$.  At this point, the functions $h_k, \\tilde{h}_k$ are\nunknown which have to be determined by requiring relations (\\ref{funct_rel})\nand (\\ref{det_rep}) to be identical.  This gives\n\\begin{equation}\n   \\begin{split}\n    h(u+2\\eta) \\, \\tilde{h}(u-2\\eta) & =   -\\tilde{\\Delta}(u)\\,, \\\\\n    h(u) \\, \\tilde{h}(u+ 2(n-1) \\eta) & =   -\\phi^\\tau_n(u)\\,, \\\\\n    \\prod_{k=0}^n h(u+2k\\eta) + \\prod_{k=-1}^{n-1} \\tilde{h}(u+2k\\eta) & =\n    \\phi^{id}_{n}(u)\\,. \n  \\end{split}\n  \\label{condtosolve}\n\\end{equation}\nInspired by the structure of the $h$-functions in the XXZ case\n\\cite{Nepomechie:2003vv}, as well as the functions found in \\cite{Grab_Frahm},\nwe consider the following expressions\n\\begin{equation}\n \\tilde{h}(u) = h(-u-4\\eta), \\qquad h(u) = \\left(\\frac{\\sin(u+2\\eta)}{\\sin 2\\eta}\\right)^{2N} \n \\frac{\\sin(2u+4\\eta)}{\\sin(2u+2\\eta)} g_-(u)g_+(u) \\, ,\n \\label{h_trnc_interm}\n\\end{equation}\nwhere the boundary information is contained in the functions $g_\\pm(u)$. Substituting\nthe above expressions into (\\ref{condtosolve}), the following condition arises\nregarding the functions $g_{\\pm}(u)$:\n\\begin{equation}\n g_-(u +2\\eta )g_-(-u-2\\eta )g_+(u+2\\eta)g_+(- u - 2\\eta) = - \\det K^+(u) \\det K^-(u+2\\eta) \\, .\n\\end{equation}\nAssuming that the functions $g_{\\pm}(u)$ factorize the determinants of the \nreflection matrices and recalling the explicit expressions for the determinants of the \nboundary matrices, one obtains\n\\begin{equation}\n  \\begin{aligned}\n    & g_-(u )g_-(-u) = \\det K^-(u) = -(\\omega^\\sss{-})^2 \\sin(u  +\n    \\psi_\\sss{-})\\sin(u - \\psi_\\sss{-}) \\\\ \n    & g_+(u)g_+(- u ) = - \\det K^+(u-2\\eta) = -(\\omega^\\sss{+})^2 \\sin(u +\n    \\psi_\\sss{+})\\sin(u - \\psi_\\sss{+}) \\, , \n  \\end{aligned}\n\\end{equation}\npinpointing to the natural solutions\n\\begin{equation}\ng_-(u) = \\omega^\\sss{-} \\sin(u+\\psi_\\sss{-})\\, , \\qquad \ng_+(u) = \\omega^+ \\sin(u+\\psi_\\sss{+}) \\, .\n\\label{gsol}\n\\end{equation}\nUsing the expressions (\\ref{h_trnc_interm}) and (\\ref{gsol}) then, it is\nstraightforward to check that all three relations in (\\ref{condtosolve}) are\nautomatically satisfied, so that the expressions for $h(u), \\tilde{h}(u)$ are\nnow proven. It is interesting to point out that the structure we have found\nhere is identical with the corresponding structure of the purely diagonal XXZ\ncase \\cite{Nepomechie:2003vv}.\n\nThe vanishing determinant guarantees the existence of a non-trivial null\nvector, $(Q_0, Q_1, \\cdots Q_n)$, so that the following relations hold\n\\begin{eqnarray}\n\\Lambda_0 Q_0 - \\tilde{h}_{-1} Q_1 - h_0 Q_n & = & 0 \\, , \\nonumber \\\\\n-h_k Q_{k-1} + \\Lambda_k Q_k - \\tilde{h}_{k-1} Q_{k+1} & = & 0 \\, , \\qquad\nk=1, \\cdots, n-1 \\, ,\\\\ \n-\\tilde{h}_{n-1} Q_0 - h_{n} Q_{n-1} + \\Lambda_n Q_n & = & 0 \\, . \\nonumber\n\\end{eqnarray}\nFor the present choice of the quasi-classical parameter, i.e.\\\n$\\eta=\\eta_n=\\frac{\\pi\/2}{n+1}$, these equations can be recast as TQ equations\nfor a periodic function $Q(u)=Q(u+\\pi)$ by identifying $Q_k \\equiv Q(u+2\\eta\nk)$:\n\\begin{equation}\n \\Lambda(u) = h(u) \\frac{Q(u-2\\eta)}{Q(u)} + h(-u-2\\eta)\n \\frac{Q(u+2\\eta)}{Q(u)}\\,, \n \\quad \\mathrm{for~}\\eta=\\eta_n. \n\\label{fnctrel_tq_eq}\n\\end{equation}\nHere the degree of the Fourier polynomial for the function $Q(u)$ depends on\nthe choice of the anisotropy $\\eta=\\eta_n$ (or, equivalently, the level of the\nfusion hierarchy where the truncation identity (\\ref{eq:obcTauTruncation})\nappears).  We note that as a consequence of Eqs.\\ (\\ref{trns_crs_symm}) and\n(\\ref{fnctrel_tq_eq}) the function $Q(u)$ is crossing symmetric.\n\nWe emphasize that up to this point we have made no assumptions concerning the\nform of the reflection matrices $K^\\pm(u)$.  In particular the functional\nequation (\\ref{fnctrel_tq_eq}) holds for both diagonal and non-diagonal\nboundary conditions and coincide with the limit $n\\to\\infty$ of\nEq.~(\\ref{TQeq-n}) of the fusion hierarchy provided that the limit\n$\\lim_{n\\to\\infty} Q^{(n)}(u) = Q(u)$ exists \\cite{Yang:2005ce}.  The first of the conditions\n(\\ref{condtosolve}) used to derive the determinant representation of the\nfunctional equation guarantees the factorization $h^+(u) h^-(u+2\\eta) =\n-h(u+2\\eta) h(-u-2\\eta) = \\tilde\\Delta(u)$ of the quantum determinant with\n\\begin{equation}\n  h^-(u) = -h^+(-u-2\\eta) = -\\omega^+\\omega^- \\left(\\frac{\\sin(u+2\\eta)}{\\sin2\\eta}\\right)^{2N}\n  \\frac{\\sin(2u+4\\eta)}{\\sin(2u+2\\eta)} \\sin(u+\\psi_+)\\sin(u+\\psi_-) \\, ,\n  \\label{hfunctions}\n\\end{equation}\ncorresponding to \n\\begin{equation}\n  \\label{kappaTQ}\n  \\kappa(u) = \\omega^- \\omega^+ \\left(-\\frac{\\sin(u-2\\eta)}{\\sin2\\eta}\\right)^{2N} \\frac{\\sin(2u-4\\eta)}{\\sin(2u-2\\eta)}\n  \\sin(u-\\psi_-)\\sin(u-\\psi_+) \\, ,\n\\end{equation}\nin the definition of the $Q^{(n)}(u)$.  \n\nFor the limit $\\lim_{n\\to\\infty} Q^{(n)}(u)$ to exist, however, this choice of\nthe function $h(u)$ is not sufficient for general boundary conditions: in the\nfollowing Section we shall introduce deformations of the functions $h$ and $Q$\nwhich allow for an algebraic solution of the TQ equation (\\ref{fnctrel_tq_eq})\nin the case of non-diagonal boundary conditions.\n\nIn the case of particle number conserving \\emph{diagonal} boundary matrices\nthe TQ equation can be solved in terms of Fourier polynomials of fixed degree\nfor the $Q$-functions.  Comparing to what is obtained using the algebraic or\ncoordinate Bethe ansatz they are given by \\cite{Umeno_Fan_Wadati, Fan_Guan, GuanFanYang}\n\\begin{equation}\n  Q(u) = \\prod_{\\ell=1}^M \\sin(u-v_\\ell^{(0)}) \\sin(u+v_\\ell^{(0)} +2\\eta) \\, ,\n  \\label{Q_fnc_diag}\n\\end{equation}\nfor all anisotropies $0\\le\\eta\\le \\pi\/2$.\nFor small system sizes $N$ we have checked by explicit computation of the\nfunctions $Q^{(n)}(u)$ from the recursion relations (\\ref{TQeq-n}) for\ndiagonal boundary fields that they do in fact converge to the expression\n(\\ref{Q_fnc_diag}) for $n\\to\\infty$ for all anisotropy parameters.\nThe zeros $v_\\ell^{(0)}$ of the $Q$-functions are determined by requiring\nanalyticity of the eigenvalues $\\Lambda(u)$ of the transfer matrix from\nthe TQ equation (\\ref{fnctrel_tq_eq}).  This yields the BAE for the diagonal\nmodel\n\\begin{equation}\n\\frac{h^-(v_\\ell^{(0)})}{h^+(v_\\ell^{(0)})} = \\frac{Q(v_\\ell^{(0)} +\n  2\\eta)}{Q(v_\\ell^{(0)} - 2\\eta)} \\, , \n\\end{equation}\nor upon substitution \n\\begin{eqnarray} &&\n\\left(\\frac{\\sin(v_\\ell^{(0)} + 2\\eta)}{\\sin v_\\ell^{(0)}}\\right)^{2N}\n\\frac{\\sin(v_\\ell^{(0)}+\\psi_\\sss{-})\\sin(v_\\ell^{(0)}+\\psi_\\sss{+})}{\\sin(v_\\ell^{(0)}+2\\eta-\\psi_\\sss{+})\n  \\sin(v_\\ell^{(0)}+2\\eta - \\psi_\\sss{-})} = \\nonumber \\\\\n&&\\hspace{4cm} = \\prod_{j\\neq \\ell}^M \\frac{\\sin(v_\\ell^{(0)} - v_j^{(0)}\n  +2\\eta)\\sin(v_\\ell^{(0)} + v_j^{(0)} + 4\\eta)} {\\sin(v_\\ell^{(0)} -v_j^{(0)}\n  - 2\\eta)\\sin(v_\\ell^{(0)} + v_j^{(0)})} \\, .\n\\label{BAE_diag}\n\\end{eqnarray}\n\n\n\n\n\n\n\n\\section{Non-diagonal boundaries}\nHaving reporduced the spectrum of the diagonal model, we now turn our\nattention to the case where nondiagonal boundary conditions are considered.\nBased on our findings above we assume that the spectral problem is given by\nthe TQ equation (\\ref{fnctrel_tq_eq}) for arbitrary values of the anisotropy\n$\\eta$.\nIt turns out that the incorporation of the nondiagonal contributions can be\ndone efficiently by appropriately deforming the $h$ and $Q$-functions of the\ndiagonal model.  Thus our starting point would be a deformation of the TQ\nequations themselves\n\\begin{equation}\n\\label{TQeq}\n  \\Lambda(u) = H^-(u) \\frac{\\tilde{Q}(u-2\\eta)}{\\tilde{Q}(u)} \n  - H^+(u) \\frac{\\tilde{Q}(u+2\\eta)}{\\tilde{Q}(u)} \\, .\n\\end{equation}\nwhere $\\Lambda(u)$ denote the eigenvalues of the full, nondiagonal problem. \nRegarding the $H$-functions first, since the diagonal case should be contained \nwithin the construction as a\nspecial limit, we propose the following deformations \n\\begin{equation}\nH^{\\pm}(u) = h^{\\pm}(u) \\big(1 + \\mathcal{G} \\, f^{\\pm}(u)\\big) \\, ,\n\\label{H_deform}\n\\end{equation}\nwhere $h^{\\pm}(u)$ are given in (\\ref{hfunctions}), $\\mathcal{G}$ is the\ncombination of the odd Grassmann parameters defined in (\\ref{mathcalg}) and\n$f^{\\pm}(u)$ are generic functions to be determined. There exist two\nconstraints on the expressions of these unknown functions. First, their\ncontribution in the factorization of the quantum determinant should vanish,\nyielding\n\\begin{equation}\nf^+(u-2\\eta) = - f^-(u) \\, .\n\\end{equation}\nFurthermore, the crossing symmetry of the einvalues $\\Lambda(u)$ entailed\nby (\\ref{trns_crs_symm}) should be preserved.  We therefore assume that the \n$\\tilde{Q}$-functions enjoy crossing symmetry as well, similarly to the\nfunctional equation (\\ref{fnctrel_tq_eq}) derived from the truncation\nidentities.   This implies that the $H$-functions are related\nthrough $H^-(u) = - H^+(-u-2\\eta)$. This\nrelation holds automatically for the $h$-functions of the diagonal case,\nwhereas it also provides a second constraint on $f^\\pm(u)$:\n\\begin{equation}\nf^+(-u-2\\eta) = f^-(u) \\, .\n\\label{2nd_cnstrnt_f}\n\\end{equation}\nA set of solutions to these functional relations is given by\n\\begin{eqnarray}\nf^+(u) & = & \\mathcal{W} \\, \\sin(2u+4\\eta) \\cr\nf^-(u) & = & -\\mathcal{W} \\, \\sin(2u) \\, , \n\\label{ffunct}\n\\end{eqnarray}\nwith the coefficient $\\mathcal{W}$ to be determined from the asymptotic behavior (\\ref{asymptotic_transfer}) \nof the transfer matrix eigenvalues. It should be stressed\nout that the above choice is not unique, but it was found to be the only \none consistent with the various limits and with the constraints arising from the\nfunctional relations. \n\nA similar deformation should be considered for the $\\tilde{Q}$-functions as\nwell, since they are assumed to originate from the higher spin eigenvalues\nthrough a limiting procedure. The most general ansatz extending the functions\n$Q(u)$ of the diagonal case (\\ref{Q_fnc_diag}) and containing the special\ncombination of Grassmann numbers, $\\mathcal{G}$, reads as \\begin{equation} \\tilde{Q}(u) =\nQ(u) + \\mathcal{G} \\, b(u) \\, ,\n\\label{Q_deform}\n\\end{equation} \nwith $b(u)$ being a function to be determined. Substituting the deformations\n(\\ref{H_deform}) and (\\ref{Q_deform}) into the TQ equation (\\ref{TQeq}), the latter \none becomes\n\\begin{eqnarray}\n&& \\Lambda(u) =  \\Lambda^{\\textrm{diag}}(u)\\left(1 - \\mathcal{G} \\, \\frac{b(u)}{Q(u)}\\right)  \n + \\mathcal{G}\\left[h^-(u)\\left( \\frac{b(u - 2\\eta)}{Q(u)} + f^-(u)\\frac{Q(u-2\\eta)}{Q(u)}\\right) \\right.  \\nonumber \\\\\n&& \\hspace{3cm} \\left. -h^+(u)\\left( \\frac{b(u+2\\eta)}{Q(u)} + f^+(u)\\frac{Q(u+2\\eta)}{Q(u)} \\right)\\right] \\, .\n\\label{TQnd}\n\\end{eqnarray}\n %\nAdditional requirements are needed to\ndetermine the function $b(u)$ in (\\ref{Q_deform}).  Again the choice of this\nfunction has to guarantee the analyticity of the additional terms appearing in\nthe functional equation (\\ref{TQnd}) for the eigenvalues. In addition, $b(u)$\nshould enjoy crossing symmetry, which was assumed in order to derive the \nconstraint (\\ref{2nd_cnstrnt_f}).\nIn this spirit, we propose that the nondiagonal correction to the\n$\\tilde{Q}$-functions is given by\n\\begin{equation}\nb(u) = \\prod_{\\ell=1}^{M'} \\sin(u-v_\\ell^{(1)})\\sin(u+v_\\ell^{(1)}+2\\eta) \\, ,\n\\label{b_funct}\n\\end{equation}\nwith $\\{v_\\ell^{(1)} \\} \\neq \\{v_\\ell^{(0)}\\}$ in general.  Matching of the\nasymptotic behavior dictates that the upper limits of the products in\neqs. (\\ref{Q_deform}) and (\\ref{b_funct}) should be equal, $M'=M$. Furthermore, the \nasymptotics (\\ref{asymptotic_transfer}) provide the explicit expression for the\ncoefficient $\\mathcal{W}$ as\n\\begin{equation}\n \\mathcal{W} = \\frac{1}{\\sin[\\psi_\\sss{+} + \\psi_\\sss{-} + (N-2M-1)2\\eta]} \\, .\n \\label{wcoef}\n\\end{equation}\nApart from the asymptotics, one should also consider additional limits of the\nTQ equation in order to check the consistency of the procedure. The limit\n$u\\to0$ gives $\\Lambda(0) = 1$, which stems from the normalization of the\ntransfer matrix. Setting $u=0$ into the TQ equation, after a quick inspection\none is lead to the following constraint for $b(u)$\n\\begin{equation}\n b(-2\\eta) = b(0)\\, ,\n\\end{equation}\nwhich is a special case of a crossing symmetry requirement and automatically\nsatisfied by the choice (\\ref{b_funct}).  Finally, it is interesting to\nconsider the semi-classical limit $\\eta\\to 0$. In this limit the $b(u)$\nfunctions drop out from the TQ equation and the eigenvalues become\n\\begin{equation}\n\\Lambda(u)\\Big|_{\\eta\\to0} = \\big(h^-(u) - h^+(u)\\big) + \\mathcal{G}\\,  f^-(u) \\big(h^-(u) + h^+(u)\\big) \\, ,\n\\end{equation}\nfinding perfect agreement with the semi-classical limit obtained from the\ntransfer matrix (\\ref{transfer_semiclassical_limit}), after recalling the\nexpressions for $h^\\pm(u)$ (\\ref{hfunctions}) and $f^\\pm(u)$ (\\ref{ffunct}) \nwith the coefficient $\\mathcal{W}$ (\\ref{wcoef}) for $\\eta\\to0$.\n\nThe parameters $v_\\ell^{(1)}$ in (\\ref{b_funct}) are determined by requiring\nanalyticity of the eigenvalues of the full transfer matrix.  The purely complex part of \nthe TQ equation (\\ref{TQnd}) yields the\nalready derived BAE (\\ref{BAE_diag}), while the one containing Grassmann\nnumbers provides a second set of relations, involving both sets of roots\n$v^{(0)}$, $v^{(1)}$, which reads as\n\\begin{equation}\n \\frac{h^-(v_\\ell^{(0)})}{h^+(v_\\ell^{(0)})} = \\frac{\\Lambda^{\\textrm{diag}}(v_\\ell^{(0)}) \\, b(v_\\ell^{(0)})}\n{h^+(v_\\ell^{(0)})\\left(b(v_\\ell^{(0)}-2\\eta)+f^-(v_\\ell^{(0)})Q(v_\\ell^{(0)}-2\\eta)\\right)}\n+ \\frac{b(v_\\ell^{(0)}+2\\eta) + f^+(v_\\ell^{(0)}) Q(v_\\ell^{(0)}+2\\eta)}\n{b(v_\\ell^{(0)}-2\\eta)+f^-(v_\\ell^{(0)})Q(v_\\ell^{(0)}-2\\eta)} \\, .\n\\label{BAE_nondiag}\n\\end{equation}\nNote that the roots $v^{(0)}$ and $v^{(1)}$ also enter this set of equations\nthrough the functions $Q(u)$ and $b(u)$.  \n\nIn summary, we have two sets of algebraic equations fixing the parameters\nappearing in the ansatz for $\\tilde{Q}(u)$.  These equations are of nested\nBethe Ansatz type, similar to those appearing in integrable models based on\nhigher rank symmetries. It should be stressed out that the \nexpression for the eigenvalues of the super transfer matrix (\\ref{TQnd}) and the \ncorresponding BAE (\\ref{BAE_nondiag}) rely on the conjectural deformations of \nthe $h$ and $Q$-functions in equations (\\ref{H_deform}) and (\\ref{Q_deform}) \nrespectively. However, we have explicitly verified that our proposed scheme is compatible \nwith the various limits and symmetry requirements, namely the truncation to the \ndiagonal case, the crossing symmetry, the limits $u\\to0$ and $\\eta\\to 0$ and \nthe asymptotics of the transfer matrix. Moreover, the deformations of the $h$-\nfunctions are compatible with the constraints arising from the functional \nrelations, namely eqs. (\\ref{condtosolve}), which provides an additional, nontrivial\ncheck of our expressions. In order to further enforce the validity of  \nour proposed scheme for the eigenvalues as computed from the TQ equation, we \nhave  analytically diagonalized the transfer matrix for a small number of chain \nlengths. The comparison of the exact eigenvalues with the ones obtained from \n(\\ref{TQnd}) exhibits perfect agreement.\n\n\n\n\n\n\n\\section{The eigenstates of the model} \nIn the previous sections we have obtained the eigenvalues of the transfer\nmatrix from the the fusion hierarchy and the resulting TQ equation.  Similar\nas in the case of diagonal boundary matrices the eigenvalues can be associated\nto sectors labelled by the integer $M$ of parameters appearing in the\nsolution.  This is remarkable since the related $U(1)$ symmetry is broken when\nthe nondiagonal boundary conditions are applied.\n\nAs is common to functional Bethe Ansatz approaches our solution of the\nspectral problem thus far does not provide information regarding the\neigenstates of the model.\nHowever, exploiting the existence of the odd Grassmann numbers and their\nnilpotency, it is possible to exactly compute the $M=0$ state of the model for\nan arbitrary number $N$ of chain sites, solely by using the derived\nexpressions of the eigenvalues.  To this end we choose $k$-particle\nstates, $k=0,\\ldots,N$,\n\\begin{equation}\n|i_1i_2\\cdots i_k\\rangle \\equiv c^\\dagger_{i_1}c^\\dagger_{i_2} \\cdots\nc^\\dagger_{i_k } | \\Omega\\rangle   \\, ,  \\qquad\n1\\le i_1<i_2<\\ldots<i_k\\le N\n\\end{equation}\nas basis of the Hilbert space of the system, $| \\Omega\\rangle$ is the Fock\nvacuum of the system (and the $M=0$ eigenvector of the diagonal problem).\n\n\nThe key observation used for the construction of the unique $M=0$ eigenvector\nof the model with nondiagonal boundary conditions is that for generic chain\nlength $N$, only the Fock vacuum and basis states containing one or two\nparticles contribute with a non-vanishing amplitude, all other sectors\ndecouple.  This allows to express the $M=0$ eigenvector of the nondiagonal\nmodel as \n\\begin{equation} |\\Psi \\rangle_{M=0} = |\\Omega \\rangle +\n\\beta^\\sss{+} \\sum_{i=1}^N b^\\sss{+}_i |i\\rangle + \\beta^\\sss{-} \\sum_{i=1}^N\nb^\\sss{-}_i |i\\rangle +\\beta^\\sss{+}\\beta^\\sss{-} \\sum_{i<j}^N B_{ij} |ij \\rangle \\,\n.\n\\label{ground_state_ansatz}\n\\end{equation}\nThe eigenvalue problem for the complete Hamiltonian reads as\n\\begin{equation}\nH |\\Psi\\rangle = (\\lambda_\\textrm{diag} + \\mathcal{G}\\, \\lambda_{\\textrm{nondiag}})\n|\\Psi\\rangle\\, . \n\\end{equation}\nSplitting the terms with respect to the order of $\\beta^\\pm$ we obtain\n\\begin{eqnarray}\n\\mathcal{O}(\\beta): && (H_\\textrm{bulk} + H_\\textrm{diag}) \\, \\beta^\\kappa \\sum_i b^\\kappa_i |i\\rangle + \nH_\\textrm{nondiag} |\\Omega\\rangle = \\lambda_\\textrm{diag} \\, \\beta^\\kappa \\sum_i b^\\kappa_i |i\\rangle\\cr\n\\mathcal{O}(\\beta^2): &&  (H_\\textrm{bulk}+H_\\textrm{diag}) \\, \\beta^+ \\beta^- \\sum_{i<j} B_{ij} |ij\\rangle + \nH_\\textrm{nondiag} \\, \\beta^\\kappa \\sum_i b^\\kappa_i |i\\rangle = \\cr\n&& \\hspace{4cm} = \\mathcal{G} \\, \\lambda_\\textrm{nondiag}|\\Omega\\rangle + \\lambda_\\textrm{diag} \\, \n\\beta^\\sss{+} \\beta^\\sss{-} \\sum_{i<j} B_{ij}|ij\\rangle\\, .\n\\end{eqnarray}\nRegarding the terms linear in $\\beta^\\sss{\\pm}$ first, after the relevant computations and splitting the resulting\nrelations with respect to the appropriate excited states, one ends up with the following six \nrelations that determine the coefficients $b_i^\\sss{\\pm}$\n\\begin{eqnarray}\n&& \\beta^\\sss{+}|1\\rangle: \\qquad \n-t\\, b^\\sss{+}_2 + b_1^\\sss{+} \\big[V(N-2) - \\tfrac{1}{2}\\cot\\psi_\\sss{-} + \\mathcal{N}_\\sss{+}  -\\lambda_\\textrm{diag}\\big]  = 0 \\cr\n&& \\beta^\\sss{+}|N\\rangle: \\qquad -t\\, b^\\sss{+}_{N-1} - \\csc \\psi_\\sss{+} \n+ b_N^\\sss{+} \\big[V(N-2) + \\tfrac{1}{2}\\cot\\psi_\\sss{-} -\\mathcal{N}_\\sss{-} - \\lambda_\\textrm{diag}\\big]  = 0 \\cr\n&& \\beta^\\sss{-}|1\\rangle: \\qquad \n-t\\, b^\\sss{-}_2 - \\csc\\psi_\\sss{-} + b_1^\\sss{-} \\big[V(N-2) - \\tfrac{1}{2}\\cot\\psi_\\sss{-} \n+ \\mathcal{N}_\\sss{+} - \\lambda_\\textrm{diag}\\big]  = 0 \\cr\n&& \\beta^\\sss{-}|N\\rangle: \\qquad \n-t\\, b^\\sss{-}_{N-1} + b_N^\\sss{-} \\big[V(N-2) + \\tfrac{1}{2}\\cot\\psi_\\sss{-} \n-\\mathcal{N}_\\sss{-} - \\lambda_\\textrm{diag}\\big]  = 0 \\cr\n&& \\beta^\\sss{\\pm} |\\ell\\rangle: \\qquad -t \\, ( b_{\\ell-1}^\\sss{\\pm} +  b_{\\ell+1}^\\sss{\\pm} ) \n+ b_\\ell^\\sss{\\pm} \\big[ V(N-3) + \\tfrac{1}{2}\\cot\\psi_\\sss{-} \n+\\mathcal{N}_\\sss{+} - \\lambda_\\textrm{diag} \\big] =0 \\, , \n\\label{linear_b} \n\\end{eqnarray}\nwhere the recursion relations are valid for $2 \\leq \\ell \\leq N-1$ and can be solved analytically, giving\n\\begin{equation}\nb_\\ell^\\sss{\\pm} = \\frac{1}{2^\\ell}\\left[ \\left(\\frac{\\mathcal{C}_0 - \n\\sqrt{\\mathcal{C}_0^2 -4t^2}}{t}\\right)^\\ell \\mathcal{C}_1^\\sss{\\pm}\n+\\left(\\frac{\\mathcal{C}_0 + \\sqrt{\\mathcal{C}_0^2 -4t^2}}{t}\\right)^\\ell \\mathcal{C}_2^\\sss{\\pm}\n\\right] \\, , \n\\end{equation}\nwith $\\mathcal{C}_0 \\equiv V(N-3) + \\tfrac{1}{2}\\cot\\psi_\\sss{-} \n+\\mathcal{N}_\\sss{+} - \\lambda_\\textrm{diag}$ and $\\mathcal{C}_{1,2}^\\sss{\\pm}$ constants to be determined. Proceeding to \nthe quadratic terms, the ones proportional to $\\beta^\\sss{+} \\alpha^\\sss{-}$ and $\\beta^\\sss{-}\\alpha^\\sss{+}$ first give the constraints\n\\begin{equation}\n\\csc (\\psi_\\sss{-}) \\, b_1^\\sss{+} = \\lambda_{\\textrm{nondiag}} = \\csc(\\psi_\\sss{+}) \\, b_N^\\sss{-} \\, .\n\\end{equation}\nThese two constraints, combined with the set of relations (\\ref{linear_b}) are sufficient to completely \ndetermine the constants $\\mathcal{C}_{1,2}^\\sss{\\pm}$ and therefore all coefficients $b_\\ell^\\sss{\\pm}$. \nWe conclude that the coefficients $b_\\ell^\\sss{\\pm}$ are given by\n\\begin{eqnarray}\n&& b_\\ell^\\sss{+} = -\\frac{\\sin(\\psi_\\sss{-} + (\\ell-1)2\\eta)}{\\sin((N-1)2\\eta + \\psi_\\sss{-} +\\psi_\\sss{+})} \\nonumber \\\\\n&& b_\\ell^\\sss{-} = -\\frac{\\sin(\\psi_\\sss{+} + (N-\\ell)2\\eta)}{\\sin((N-1)2\\eta + \\psi_\\sss{-} +\\psi_\\sss{+})} \\, .\n\\label{coeffs_final}\n\\end{eqnarray}\nConcerning the rest of the \nquadratic terms, after some algebra and splitting the terms with respect to various states, we conclude that the \ncoefficients $B_{k\\ell}$ satisfy the following relations\n\\begin{eqnarray}\n&&\n\\big(\\Xi^\\sss{+}_3 +\\mathcal{N}_\\sss{-} \\big) B_{1N} \n+ t (B_{1N-1} + B_{2N}) + b^\\sss{+}_N \\csc\\psi_\\sss{-} + b_1^\\sss{-} \\csc\\psi_\\sss{+} =0 \\cr\n&&  \\big(\\Xi^\\sss{+}_2 - \\mathcal{N}_\\sss{+}  \\big) B_{12} + t \\, B_{13} + b^\\sss{+}_2 \\csc\\psi_\\sss{-} =0 \\cr\n&& \\big(\\Xi^\\sss{-}_2 + \\mathcal{N}_\\sss{-}\\big) B_{N-1N} \n+ t B_{N-2N} +  b_{N-1}^\\sss{-} \\csc\\psi_\\sss{+} = 0 \\cr \n&& \\big(\\Xi^\\sss{-}_4  + \\mathcal{N}_\\sss{-} \\big) B_{\\ell N} \n+ t(B_{\\ell-1 N}+ B_{\\ell+1N} + B_{\\ell N-1}) + b_\\ell^\\sss{-} \\csc\\psi_\\sss{+} = 0, \\qquad 1<\\ell<N-1 \\cr\n&& \\big(\\Xi^\\sss{+}_4 - \\mathcal{N}_\\sss{+} \\big) B_{1\\ell} + \nt (B_{1\\ell-1} + B_{1\\ell+1} + B_{2\\ell}) +  b_\\ell^\\sss{+} \\csc\\psi_\\sss{-} =0, \\qquad 2<\\ell<N \\cr\n&&  \\big(\\Xi^\\sss{-}_3 - \\mathcal{N}_\\sss{+} \\big)  B_{\\ell\\ell+1} \n+t (B_{\\ell-1\\ell+1} + B_{\\ell\\ell+2}) = 0, \\qquad\\qquad\\qquad\\qquad 1<\\ell<N-1 \\cr\n&& \\big(\\Xi^\\sss{-}_5  - \\mathcal{N}_\\sss{+} \\big)B_{k\\ell} \n+ t(B_{k-1\\ell} + B_{k+1\\ell} + B_{k\\ell-1} +B_{k\\ell+1}) =0, \\qquad1<k<N-2,~~ \\ell>k+1\n\\nonumber\n\\end{eqnarray}\nwhere for the sake of readability we have defined\n\\begin{equation}\n\\Xi_q^{\\sss{\\pm}} \\equiv \\lambda_{\\textrm{diag}} \\pm \\tfrac{1}{2}\\cot\\psi_\\sss{-} - V(N-q) \\, .\n\\end{equation}\nIn principle, having acquired the exact \nexpressions for $b_\\ell^\\sss{\\pm}$, the above set of relations provides the expressions of $B_{k\\ell}$ as well. \nSince it is hard to solve these relations analytically, one may resort to a numerical analysis for \na given number of chain sites. However, solving these equations for small numbers of chain length, we were able \nto observe the emerging pattern which governs the coefficients $B_{k\\ell}$. In short, we have found that they are \neventually given by the very simple expressions\n\\begin{equation}\nB_{k\\ell} = \\frac{\\sin((N-1)2\\eta + \\psi_\\sss{-} + \\psi_\\sss{+}) }{\\sin((N-2)2\\eta + \\psi_\\sss{-} + \\psi_\\sss{+})} \n\\big ( b_{k+1}^\\sss{+} \\, b_\\ell^\\sss{-} + b_k^\\sss{-} \\, b_{\\ell-1}^\\sss{+} \\big ) \\, .\n\\label{Bkl_coefs}\n\\end{equation}\nWe have explicitly confirmed that the expressions (\\ref{Bkl_coefs}) satisfy all the generic constraints and\nrelations derived above. In conclusion, the $M=0$ eigenvector of the model is completely determined for \nan arbitrary number of chain sites. \n\nThe decoupling of higher\/lower sectors, due to the nilpotency of the Grassmannian parameters, further constraints \nthe expressions of the eigenvectors which correspond to excited states. For generic values of $M$ then, we \npropose that the corresponding eigenvector will be given by the following schematic expression\n\\begin{eqnarray}\n|\\Psi\\rangle_{M} & = &  (c_1 + c_2 \\, \\alpha^\\sss{+} \\beta^\\sss{-} + c_3 \\, \\beta^\\sss{+}\\alpha^\\sss{-}) |M \\rangle \n+ c_4 \\, \\beta^\\sss{+} |M+1 \\rangle + c_5 \\, \\beta^\\sss{-} |M+1 \\rangle \\cr\n &+ &  c_6 \\, \\beta^\\sss{+}\\beta^\\sss{-} |M+2 \\rangle + c_7 \\, \\alpha^\\sss{+} |M-1 \\rangle + c_8 \\, \\alpha^\\sss{-} |M-1 \\rangle +\n c_9 \\, \\alpha^\\sss{+} \\alpha^\\sss{-} |M-2 \\rangle \\, ,\n\\label{generic_eigenvector}\n\\end{eqnarray}\nwhere with $|M\\rangle$ we denote the states with $M$ particles present, or equivalently the states with $M$ \nspins down in the spin picture, i.e. $M$ excitations from the ground state. The number of states with the \nsame $M$ is given by the binomial coefficient \n\\begin{equation}\n\\begin{pmatrix}\n N \\cr M\n\\end{pmatrix} \n= \\frac{N!}{M!(N-M)!} \\, ,\n\\end{equation}\nso that the states $|M\\rangle$ in (\\ref{generic_eigenvector}) are to be understood as collections of states \nspanning the degeneracy space for a particular $M$. In the same spirit, the coefficients $c_i \\in \\mathbb{C}$ \nappearing also in (\\ref{generic_eigenvector}) are to be interpreted as sets of coefficients of the degenerate\nstates.\n\n\n\n\n\n\n\n\\section{Discussion}\nIn the present work, we have solved the small polaron model with nondiagonal\nboundary conditions. \nThe eigenvalues of the transfer matrix have been\nextracted by using the fusion hierarchy of the transfer matrices and also by\nconsidering the functional relations for particular values of the anisotropy\nparameter. \nStarting from the fusion hierarchy of transfer matrices together with its\ntruncation at particular values of the anisotropy parameter we have formulated\nthe spectral problem as a functional TQ equation.  The latter has been solved\nby means of appropriate deformations needed to account for the nondiagonal\nnature of the model.  The eigenvalues have been found to depend on two sets of\nBethe roots, for which coupled Bethe ansatz equations have been presented.\n\nAn interesting aspect of the model's solution is that, unlike in the case of\nthe XXZ model with nondiagonal boundary conditions, no restrictions emerge for\nthe boundary parameters.  This extra freedom, as well as the remnants of\nparticle number conservation leading to sectors of the Hilbert space labelled\nby the integer $M$ appear to be inherited from the supersymmetric nature of\nthe model.  Furthermore, supersymmetry heavily restricts the structure of the\neigenvectors and has rendered possible to exactly compute the $M=0$ eigenstate\nof the model. A more detailed analysis should provide the complete expressions\nof all eigenvectors.\n\nSince supersymmetry lifted any possible constraints between the boundary\nparameters for the small polaron, it is interesting to consider other\nsupersymmetric models with nondiagonal boundaries.  Particularly interesting\nwould be an extension of the analysis of the supersymmetric t-J model with\nopen boundaries \\cite{Gonz94,Essl96} to this case.  In this case nondiagonal\nterms breaking either the $U(1)$ charge symmetry or the $SU(2)$ spin symmetry\nof the model can be added.  \nThe latter problem has been partially solved by Galleas \\cite{Gall07} and\nleads to problems similar to those encountered in the XXX Heisenberg chain\nwith non-diagonal boundary fields.  As to boundary terms breaking the charge\nsymmetry of the model we expect that they can be dealt with in a similar\nmanner as in the small polaron model.\n\nAnother route to follow is to consider operator valued representations of the\nreflection algebras \\cite{Sklyanin:1988yz, FrSl99, FrPa06} instead of\n$c$-number solutions, and attempt to solve the model for boundary conditions\nbreaking the bulk symmetries.  The construction of suitable, generalized\nJordan-Wigner transformations for the nondiagonal boundary terms would provide\nvaluable information in this spirit.\n\n\n\n\n\\subsection*{Acknowledgments}\nWe would like to thank A. Doikou for useful discussions on the subject. \nThis work has been supported by the Deutsche Forschungsgemeinschaft under\ngrant no. Fr 737\/6.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and motivation}\nSince the discovery of the accelerated expansion of our universe, there has been intensified interest in the study of spacetime physics with a positive cosmological constant, $\\Lambda$. A tiny positive $\\Lambda$ can very satisfactorily explain the data corresponding to the current cosmological evolution of our universe. Thus, black hole spacetimes endowed with a positive cosmological constant are expected to provide us reasonable and physically well motivated models to study the global properties of the black holes living in our current universe. \n\nOne of the most exotic features of spacetimes endowed with a positive cosmological constant perhaps is the existence of a cosmological event horizon, when the parameters of that solution obey certain conditions~\\cite{Hawking}. It can be regarded as a complementary part of the black hole event horizon, arising due to the repulsive effect due to positive $\\Lambda$, at large length scales. The cosmological event horizon acts as a causal boundary surrounding us, as no communication along a future directed path is possible beyond it~\\cite{Hawking}. When we have a black hole located inside the cosmological horizon, we call the entire spacetime a de Sitter black hole. Thus for such black holes the natural region of interest is the region between the black hole and the cosmological event horizon.  \n\nJust like the black hole event horizon, the cosmological event horizon also creates particles~\\cite{Hawking}, and possesses thermodynamical properties, see. e.g.~\\cite{Kastor:1992nn}-\\cite{Bhattacharya:2013tq} and references therein\nfor some recent developments in this direction.\n\nGiven that a black hole has thermodynamical properties and it creates particles, it has been an exciting topic to understand this from \nthe symmetry and the microscopic point of view. In particular, for black hole spacetimes, the Bekenstein-Hawking entropy can solely be derived utilizing the near horizon conformal properties~\\cite{Carlip}-\\cite{Kang:2004js}. Precisely, these formalisms use some suitable fall-off conditions near a horizon and investigates under action of which vector fields these near horizon structures are `preserved'. The algebra of charges corresponding to these symmetry generating vector fields gives a Virasoro algebra with a central extension. Then one uses the Cardy formula~\\cite{Cardy1}-\\cite{Cardy2} to determine the entropy of the spacetime, which, under suitable choice of the mode functions, coincides with the Bekenstein-Hawking entropy. In other words, the entropy of a black hole spacetime is solely determined by the local symmetry at its boundary, i.e the event horizon.  \n\nWe further refer our reader to~\\cite{Carlip:2014pma} and references therein for a recent review on various approaches to understand black hole thermodynamics, the derivation of the Bekenstein-Hawking entropy and its quantum corrections.\n\n\nA recent approach for deriving the thermal properties of a Killing horizon can be found in~\\cite{Majhi:2011ws}-\\cite{Majhi:2014lka}.\nThis novel formalism uses the the Noether current associated with the variation of the Gibbons-Hawking-York boundary term to obtain the conserved charges associated with the diffeomorphism generating vector fields. It turns out that the variation of the surface term gives a boundary integral of the Noether current located on the event horizon. Then the requirement of the near horizon symmetry gives the Bekenstein-Hawking entropy of the spacetime. This method has also recently been applied to time dependent cosmological black hole spacetimes~\\cite{Majhi:2014hpa}, and to conformal scalar hairy black holes~\\cite{Meng:2014dfa}.\n\nLet us return to our focus -- black holes in the de Sitter spacetime. Since such a spacetime has two Killing horizons, it is natural to expect that its entropy will be the sum of the two horizon areas, e.g.~\\cite{Kastor:1992nn}. Our precise goal in this work is to derive this total entropy using the formalism of~\\cite{Majhi:2011ws}-\\cite{Majhi:2014lka}. Since such spacetimes has two natural boundaries, i.e. the two horizons, we get two surface integrals located at the two horizons. Then we shall find out the modes of the vector fields that preserve both the horizons' structure to find a Virasoro algebra `effectively' representing the whole spacetime, giving us the total entropy (Sec.~3). To the best our knowledge, this has not been done before. We refer our reader to~\\cite{Davies:2003me} for a verification of the second law of thermodynamics with this definition of entropy for some specific cases. See also~\\cite{Maeda:1997fh} for some further discussions motivating this definition of the total entropy, from the point of view of area theorem.\n\nLet us now emphasize a rather peculiar feature of this total entropy. We consider the Schwarzschild-de Sitter spacetime, \n\\begin{eqnarray}\nds^2=-\\left(1-\\frac{2MG}{r}-\\frac{\\Lambda r^2}{3}\\right)dt^2 +\\left(1-\\frac{2MG}{r}-\\frac{\\Lambda r^2}{3}\\right)^{-1}dr^2+r^2\\left(d\\theta^2+\\sin^2\\theta d\\phi^2\\right),\n\\label{e1}\n\\end{eqnarray}\nwhere $M$ is the mass parameter. For $3MG\\sqrt{\\Lambda}<1$, we have two Killing horizons located at,\n\\begin{eqnarray}\nr_{ H}=  \\frac{2}{\\sqrt{\\Lambda}}\\cos\\left[\\frac13 \\cos^{-1}\\left(3MG\\sqrt{\\Lambda}\\right)+\\frac{\\pi}{3} \\right],~~ r_{C}=  \\frac{2}{\\sqrt{\\Lambda}}\\cos\\left[\\frac13 \\cos^{-1}\\left(3MG\\sqrt{\\Lambda}\\right)-\\frac{\\pi}{3} \\right],\n\\label{e2}\n\\end{eqnarray}\nwhere the smaller root $r_{H}$ is the black hole and the larger $r_{\\rm C}$ is the cosmological horizon. For $3MG\\sqrt{\\Lambda}=1$,  $r_{H}$ and $r_{C}$ merge to $1\/\\sqrt{\\Lambda}$, known as the Nariai limit. \nThe entropy and the temperature of the black hole and the cosmological horizons are respectively given by $(A_{H}\/4G, \\kappa_{H}\/2\\pi)$, and $(A_{C}\/4G, \\kappa_{C}\/2\\pi)$, where $A$ denotes the horizon area, and $\\kappa_H$ and $-\\kappa_C$ are respectively the two horizon's surface gravity (with $\\kappa_C>0$, and $\\kappa_H\\geq \\kappa_C$)~\\cite{Hawking}.    \n\n\nNow, if we consider the variation of the total entropy of the spacetime, $S=(A_H+A_C)\/4G$, we obtain a Smarr formula with an effective equilibrium temperature, $T_{\\rm eff}=\\frac{\\kappa_H \\kappa_C}{2\\pi(\\kappa_H+\\kappa_C)}$, e.g.~\\cite{Urano:2009xn, Bhattacharya:2013tq}.  This implies that even though the two horizons have different characteristic temperatures, there can be an effective thermal equilibrium state \nwhen their entropies are combined. This has been demonstrated earlier in~\\cite{Shankaranarayanan:2003ya} via semiclassical tunneling method, but to the best of our knowledge, any field theoretic derivation of this is yet unknown.  \n\n\nThus, it is highly motivating to understand this effective thermal equilibrium state, and quite naturally, a first step would be to actually derive the total entropy of such spacetimes. \n\nThe derivation of the entropy of the cosmological event horizon in any dimension can be found in~\\cite{Lin:1999gf}, where suitable \nfall-off condition on an asymptotic de Sitter spacetime and  the near horizon symmetry has been used. A discussion on the relation  \nbetween the Friedman equation and the Cardy formula can be seen in~\\cite{Wang:2001bf}. We also refer our reader to~\\cite{Zhang:2014jfa} for a discussion on phase transition of de Sitter black holes using the aforementioned effective thermal\nequilibrium state. \n\nThe paper is organized as follows. In the next section we outline the general geometric scheme in which we work in. This will help \nus to deal with general stationary axisymmetric spacetimes with arbitrary number of commuting Killing vector fields in arbitrary spacetime dimensions. Precisely, this will provide us a timelike (non-Killing) vector field that foliates the spacetime between the two horizons. This vector field becomes null and Killing on both the horizons. This will enable us to identify Rindler geometries in the vicinity of both the horizons, and hence to treat them in an equal footing. Apart from the existence of the cosmological event horizon, we shall not assume any further explicit form or fall-off for the metric there. In other words, our method works well for the Nariai class de Sitter black holes as well, where the two horizons have comparable length scales.\n\nIn Sec.~3, we use this general framework to derive the total entropy of such spacetimes extending the formalism of~\\cite{Majhi:2011ws}-\\cite{Majhi:2014lka}. We point out there that the mode functions corresponding to the symmetry generating vector fields near the horizons have to be restricted in a certain manner, compared to the single horizon spacetimes, in order to obtain the total entropy. Sec.~4 is devoted to address known\nnon-trivial explicit examples from four and higher dimensions, to demonstrate they all fall under the general framework we built. We also address two cases of non-minimal couplings here and show in particular, some qualitative difference for the Brans-Dicke field for our case, when compared to the asymptotically flat spacetimes. Finally, we discuss our results in Sec.~5.\n\n\nWe shall work with mostly positive signature of the metric and set $c=k_{B}=\\hbar=1$ throughout, but will retain Newton's constant, $G$. In different spacetime dimensions, different values of it will be understood.   \n\n    \n\\section{The general near horizon geometry}\n\nWe shall derive below the general geometric framework we will be working in. \nThe first part of which essentially deals with the construction of Killing horizons in stationary axisymmetric spacetimes of general dimensions with two or more commuting Killing vector fields. This will help us to deal with both the black hole and the cosmological horizon in an equal footing and in a much convenient manner than dealing with exact solutions case by case. The essential details of this can be found in e.g.~\\cite{Bhattacharya:2013caa, Wald:1984rg} and references therein. For the sake of self consistency and convenience of the reader, we shall briefly outline them here. \n\nThe next part consists of identifying an $(1+1)$-dimensional geometry in a general way, such that it becomes the Rindler on any of the two Killing horizons.  \n\n\nWe assume that the spacetime is an $n$-dimensional torsion-free manifold and satisfies Einstein's equations. \nWe assume that the spacetime is stationary, axisymmetric and is endowed with two or more than two commuting Killing vector fields,\n\\begin{eqnarray}\n\\nabla_{(a}\\xi_{b)} &=&0= \\nabla_{(a}\\phi^{(i)}_{b)}, \\nonumber\\\\\n\\label{g0}\n\\pounds_{\\xi}\\phi^{(i)b}&=&0= \\pounds_{\\phi^{(i)}}\\phi^{(j)b},~(i, j=1,2 \\dots m,~{\\rm with}~m<n)\n\\label{g1}\n\\end{eqnarray}\nwhere $\\xi^a$ and $\\phi^{(i)}$ respectively generates stationarity and axisymmetries and the vanishing commutators are represented by the vanishing Lie derivatives. Since the spacetime is stationary, not static, we take $\\xi^a\\phi^{(i)}_a\\neq 0$ for all $i=1,2,\\dots m$. Also, to allow sufficient generality in our method, we further assume that the axisymmetric Killing vector fields $\\phi^{(i)}$'s are not mutually orthogonal as well. There can be additional spatial isometries orthogonal to the stationary Killing vector field $\\xi^a$, but for our present purpose we need to worry only about isometries non-orthogonal to $\\xi^a$. \n\nFor convenience, we shall first discuss the case of three commuting Killing vector fields ($\\xi,~\\phi^{(1)},~\\phi^{(2)}$). Generalization to higher numbers or specialization to two such commuting fields will be clear from this, as we shall see below. \n\nWe assume that the ({\\it n}$-$3)-dimensional\nspacelike surfaces orthogonal to these three commuting Killing vector fields form integral submanifolds, which essentially means the vector fields spanning the subspace form a Lie algebra between themselves, which in turn implies Frobenius-like conditions~\\cite{Wald:1984rg},\n\\begin{eqnarray}\n\\phi^{(1)}_{[a}\\phi^{(2)}_b\\xi_c\\nabla_d \\xi_{e]}=\\phi^{(1)}_{[a}\\xi_b\\phi^{(2)}_c\\nabla_d \\phi^{(2)}_{e]}=\\phi^{(2)}_{[a}\\xi_b\\phi^{(1)}_c\\nabla_d \\phi^{(1)}_{e]}=0.\n\\label{g9}\n\\end{eqnarray}\nClearly, the chief difference between the static and stationary axisymmetric spacetime is that, for the later the timelike Killing\nvector field is not hypersurface orthogonal. For our convenience, we shall now construct a foliation of the spacetime by constructing a hypersurface orthogonal (non-Killing) vector field. To do this, we define a 1-form $\\chi_a$ as\n\\begin{eqnarray}\n\\chi_a=\\xi_a+\\alpha_1 (x)\\phi^{(1)}_a+\\alpha_2(x)\\phi^{(2)}_a,\n\\label{g10}\n\\end{eqnarray}\nso that $\\chi_a\\phi^{(1)a}=0=\\chi_a\\phi^{(2)a}$ identically everywhere, giving\n\\begin{eqnarray}\n\\alpha_1(x)=\\frac{(\\xi\\cdot\\phi^{(1)})f_2^2 -(\\xi\\cdot\\phi^{(2)})(\\phi^{(1)}\\cdot\\phi^{(2)})  }{(\\phi^{(1)}\\cdot\\phi^{(2)})^2-f_1^2f_2^2},\\quad \\alpha_2(x)=\\frac{(\\xi\\cdot\\phi^{(2)})f_1^2 -(\\xi\\cdot\\phi^{(1)})(\\phi^{(1)}\\cdot\\phi^{(2)})  }{(\\phi^{(1)}\\cdot\\phi^{(2)})^2-f_1^2f_2^2},\n\\label{g11}\n\\end{eqnarray}\nwhere we have written for the norms, $\\phi^{(1)}\\cdot\\phi^{(1)}=+f_1^2$ and $\\phi^{(2)}\\cdot\\phi^{(2)}=+f_2^2$. Let $\\xi^a\\xi_a=-\\lambda^2$. Then the norm of $\\chi_a$ is given by\n\\begin{eqnarray}\n\\chi_a\\chi^a=-\\beta^2=\\left(-\\lambda^2+ \\frac{\\left(f_2(\\xi\\cdot \\phi^{(1)})-f_1(\\xi\\cdot\\phi^{(2)})\\right)^2 }{(\\phi^{(1)}\\cdot\\phi^{(2)})^2-f_1^2f_2^2}\\right).\n\\label{g12}\n\\end{eqnarray}\nSince for any two spacelike vector fields $A$ and $B$, we always have $A\\cdot B\\leq |A| |B|$, the denominator of the second term on the right hand side is negative. This shows that $\\chi_a$ is timelike when $\\beta^2>0$.\nThe price we have paid doing this orthogonalization is that, $\\chi_a$ is not a Killing field in general,\n\\begin{eqnarray}\n\\nabla_{(a}\\chi_{b)}=\\phi^{(1)}_{(a}\\nabla_{b)}\\alpha_1(x)+\\phi^{(2)}_{(a}\\nabla_{b)}\\alpha_2(x).\n\\label{g13}\n\\end{eqnarray}\nIn terms of $\\chi_a$, the first of Eq.~(\\ref{g9}) can be written as $\\phi^{(1)}_{[a}\\phi^{(2)}_b\\chi_c\\nabla_d \\chi_{e]}=0$, which admits general solution\n\\begin{eqnarray}\n\\nabla_{[a} \\chi_{b]}=\\mu_{1[a}\\chi_{b]}+\\mu_{2[a}\\phi^{(1)}_{b]}+\\mu_{3[a}\\phi^{(2)}_{b]}+\\nu_1\\chi_{[a}\\phi^{(1)}_{b]}\n+\\nu_2\\chi_{[a}\\phi^{(2)}_{b]}+\\nu_3\\phi^{(1)}_{[a}\\phi^{(2)}_{b]}\n\\label{g15}\n\\end{eqnarray}\nwhere $\\mu_{ia}$ ($i=1,2,3$) are 1-forms orthogonal to $\\chi^a$, $\\phi^{(1)a}$ and $\\phi^{(2)a}$, and $\\nu_i(x)$ ($i=1,2,3$) are functions. These functions and 1-forms can be determined exactly, chiefly using the commutativity of the Killing vector fields~\\cite{Bhattacharya:2013caa},  \n\\begin{eqnarray}\n\\nabla_{[a}\\chi_{b]}=\\beta^{-2}\\left(\\chi_b\\nabla_a\\beta^2-\\chi_a\\nabla_b\\beta^2\\right),\n\\label{g17}\n\\end{eqnarray}\nwhich shows $\\chi_a$ satisfies the Frobenius condition, $\\chi_{[a}\\nabla_{b}\\chi_{c]}=0$, and hence is orthogonal to the family of\n($n-1$)-dimensional spacelike hypersurfaces containing $\\phi^{(i)}_a$'s. This is a crucial result for proceeding further.\n\nHaving obtained the foliation of the spacetime, we now proceed to define the Killing horizons. Eq.~(\\ref{g17}) shows by the torsion-free condition that, for any $\\beta^2=0$ hypersurface (say ${\\cal H}$),  \n\\begin{eqnarray}\n\\chi_{[b}\\nabla_{a]}\\beta^2\\big\\vert_{\\beta^2\\to 0}=\\beta^2\\partial_{[a}\\chi_{b]}\\big\\vert_{\\beta^2\\to 0}\\to 0,\n\\label{g22'}\n\\end{eqnarray}\nso that on any such hypersurface ${\\cal{H}}$, we may write\n\\begin{eqnarray}\n\\nabla_a\\beta^2=2\\kappa(x)\\chi_a,\n\\label{g22}\n\\end{eqnarray}\nwhere $\\kappa(x)$ is a smooth function defined on ${\\cal H}$. \n\nThe next step is to prove that any such compact surface ${\\cal H}$ is a Killing horizon, in the sense that the functions \n$\\alpha_i(x)$ (Eq.s~(\\ref{g10}), (\\ref{g11})) becomes constant on ${\\cal H}$. \nThis involves constructing a null geodesic congruence for $k_a=e^{-\\kappa(x)\\tau}\\chi_a$ (with $\\chi^a\\nabla_a\\tau:=1$), and then solving for the Raychaudhuri equation on ${\\cal H}$. We shall not go into the details of this proof here referring our reader to~\\cite{Bhattacharya:2013caa} and references therein for this.\n\nThen following similar steps as in four spacetime dimensions~\\cite{Wald:1984rg}, we can show that $\\kappa$ is a constant on ${\\cal H}$, and is given by\n\\begin{eqnarray}\n\\kappa^2=\\frac{\\left(\\nabla_a\\beta^2\\right)\\left(\\nabla^a\\beta^2\\right)}{4\\beta^2}\\Big\\vert_{\\cal H},\n\\label{g21'}\n\\end{eqnarray}\nknown as the surface gravity of the Killing horizon. We shall assume in the following $\\kappa\\neq 0$, always.\n\nThus we have seen that the foliation timelike vector field $\\chi^a$, smoothly becomes the horizon Killing vector field (say, $\\chi^a_{\\cal H}$). For the de Sitter black hole spacetimes we wish to deal with, we have two such compact $\\beta^2=0$ surfaces. The smaller one is the black hole event horizon and the larger one is the cosmological event horizon, and the vector field $\\chi^a$ smoothly becomes  null and Killing on both of them. \n\nThe next step is to show using this general framework that, we can select a part of the near horizon geometry, which is Rindler-like. In order to see this, we define a 1-form    \n\\begin{eqnarray}\n\\widetilde{X}_a:=\\frac{1}{\\kappa}\\nabla_a\\beta,\n\\label{g21}\n\\end{eqnarray}\nwhich is orthogonal to $\\chi^a$ and $\\phi^{(i)a}$, as can be seen by using (\\ref{g1}). Also, Eq.~(\\ref{g21'}) shows, $\\widetilde{X}_a\\widetilde{X}^a=+1$. Let $\\widetilde{X}$ be the parameter along $\\widetilde{X}^a$, such that $\\widetilde{X}^a\\nabla_a\\widetilde{X}:=1$. We have from the action of a vector field on functions~\\cite{Wald:1984rg},  \n\\begin{eqnarray}\n\\widetilde{X}_a\\widetilde{X}^a=1=\\frac{1}{\\kappa}\\widetilde{X}^a\\nabla_a\\beta=\\frac{1}{\\kappa}\\frac{d\\beta}{d\\widetilde{X}}, \n\\label{g22}\n\\end{eqnarray}\nwhich gives, $\\widetilde{X}=\\frac{\\beta}{\\kappa}$. If we choose $\\widetilde{X}^a$ to be one of the basis vectors orthogonal to $\\chi^a$, it is clear that the metric infinitesimally close to ${\\cal H}$ takes the form\n\\begin{eqnarray}\ng_{ab}=-\\frac{1}{\\kappa^2 \\widetilde{X}^2} \\chi_{{\\cal H} a}\\chi_{{\\cal H} b}+\\widetilde{X}_a\\widetilde{X}_b+\\gamma_{ab},\n\\label{g23}\n\\end{eqnarray}\nwhere the spacelike compact $(n-2)$-section $\\gamma_{ab}$ is orthogonal to both $\\chi^a$ and $\\widetilde{X}^a$. It is clear that while $\\chi_{\\cal H}$\nis tangent to ${\\cal H}$, the vector field $\\widetilde{X}^a$ defines orthogonality or `away from' ${\\cal H}$.  The `$\\chi-\\widetilde{X}$' part of the near horizon coincides with the Rindler metric.\n\nWe assume that the basis vectors spanning $ \\gamma_{ab}$ have neither vanishing nor diverging norms. This is just because otherwise\nwe will have either vanishing or diverging horizon `area' ($:=\\int (\\det {\\gamma_{ab}})$).\n\n\nWe note that the function $\\kappa^2(x)=\\frac{(\\nabla_a\\beta^2)(\\nabla^a\\beta^2)}{4\\beta^2}$, where $\\beta^2$ is not necessarily vanishing this time, smoothly coincides with $\\kappa^2$ (Eq.~(\\ref{g21'})) on ${\\cal H}$. Then it is clear that with the vector field $\\widetilde{X}_a=\\frac{\\nabla_a \\beta}{\\kappa(x)}$, we may write the general spacetime metric as  \n\\begin{eqnarray}\ng_{ab}=-\\beta^{-2}\\chi_{a}\\chi_{ b}+\\widetilde{X}_a\\widetilde{X}_b+\\gamma_{ab},\n\\label{g24}\n\\end{eqnarray}\nwhich smoothly coincides with (\\ref{g23}) on any of the two Killing horizons. This helps us to deal with the black hole and the cosmological horizon in an equal footing. \n \n\nTo summarize, we have found a foliation of an $n$-dimensional stationary axisymmetric spacetime with three commuting Killing vector fields along a timelike vector field. Whenever that vector field becomes null on a compact surface, it becomes Killing as well, making the null surface a Killing horizon. For our concern, we have two such Killing horizons.  We have also shown that the general spacetime metric (\\ref{g24}) coincides with (\\ref{g23}), in an infinitesimal neighborhood of any of the horizons.\n\nFor two commuting Killing vector fields, we set any one of $\\phi^{(1)}$ and $\\phi^{(2)}$ (and hence $f_{12}$, Eq.s~(\\ref{g11})) to zero. Also, there is only two conditions analogous to (\\ref{g9}) now. They include $\\xi$ and any one of the axisymmetric Killing vector fields.\n\nFor more than three commuting Killing vector fields, say four, we assume that the $(n-4)$-subspace orthogonal to those Killing vector fields form integral submanifolds. Accordingly, we have four Frobenius-like conditions (\\ref{g11}). We proceed then as earlier to define $\\chi_a$ as $\\chi_a=\\xi_a+\\alpha_i\\phi^{(i)}_a$ with $i=1,2,3$. We find $\\alpha_i$ by imposing the orthogonality\nbetween $\\chi_a$ and the axisymmetric Killing vector fields and proceed as earlier for the rest of the construction. This process may go on and clearly can accommodate arbitrary number of commuting Killing vector fields.  \n\nFor our convenience we further define a new coordinate $X=\\sqrt{\\kappa \\widetilde{X}\/2}$ in (\\ref{g23}) to get\n\\begin{eqnarray}\ng_{ab}=-\\frac{1}{2\\kappa X} \\chi_{{\\cal H} a}\\chi_{{\\cal H} b}+ (2\\kappa X) X_aX_b+\\gamma_{ab},\n\\label{g25}\n\\end{eqnarray}\nwhere $X^a$ is the tangent vector field associated with the new coordinate $X$, and $X_a X^a=(2\\kappa X)^{-1}$. \n\nFinally, we note that since the Killing vector fields commute, we may specify coordinates along them, at least locally.\nOn the other hand, since the horizon Killing vector field $\\chi_{{\\cal H}}^a$ is a linear combination of those Killing fields with constant coefficients, it is clear that we can treat $\\chi_{{\\cal H}}^a$ as a coordinate Killing vector field.\nWe shall denote the surface gravities ($\\kappa$) of the black hole and the cosmological event horizon by $\\kappa_H$ and $-\\kappa_C$\n(with $\\kappa_C>0$) respectively and will always work with the absolute value of the cosmological event horizon's surface gravity, in order to maintain the correct signature of the metric (\\ref{g25}).\n\nWith all these necessary geometric ingredients, we are now ready to go into the derivation of entropy.\n\n\n\n\n\n\n\n\\section{General derivation of the entropy}\nWe shall use below the formalism developed in~\\cite{Majhi:2011ws}-\\cite{Majhi:2014lka} using the Gibbons-Hawking-York \nsurface-counterterm in order to calculate the total entropy of a stationary axisymmetric de Sitter black hole spacetime.     \n\nLet us briefly review the formalism first. The Gibbons-Hawking-York surface term subject to the variation at the boundary is\ngiven by,\n\\begin{eqnarray}\nA_{\\rm sur}=\\frac{1}{8\\pi G}\\int_{\\partial {\\cal M}} [d^{n-1}x] K=\\frac{1}{8\\pi G}\\int_{{\\cal M}} [d^nx]\\nabla_a\\left(K(x) N^a(x)\\right),\n\\label{en1}\n\\end{eqnarray}\nwhere $G$ is the Newton constant in dimension $n$, and $[d^{n-1}x]$, $[d^{n}x]$ stand for the invariant volume measures in respective dimensions. $K$ is the trace of the extrinsic curvature of an $(n-1)$-dimensional boundary hypersurface, $\\partial {\\cal M}$. In the second integral, which is valid on the entire spacetime manifold ${\\cal M}$, $K(x)$ and $N^a(x)$ are respectively a function and vector field that smoothly coincide with $K$ and $N^a$ on ${\\partial {\\cal M}}$, where $N^a$ is the unit normal to $\\partial {\\cal M}$. \n\nThe conserved Noether charge corresponding to the variation of the integrand of the second integral of (\\ref{en1}) under infinitesimal diffeomorphism generated by a vector field $\\zeta^a$ is given by \n\\begin{eqnarray}\nQ[\\zeta]=\\frac12\\int d\\Sigma_a J^a=\\frac12 \\int \\sqrt{\\gamma} d\\gamma_{ab} J^{ab},\n\\label{en2}\n\\end{eqnarray}\nwhere $J^a$ is the conserved Noether current, $J^{ab}$ is an antisymmetric tensor field given by, $J^{ab}=\\frac{K}{8\\pi G}(\\zeta^a N^b-\\zeta^b N^a)$ and is interpreted as the Noether potential, as $J^a[\\zeta]=\\nabla_b J^{ab}[\\zeta]$. $\\Sigma$ is a suitable hypersurface, and the second integral is the\n $(n-2)$-dimensional boundary of $\\Sigma$. The choice of $\\Sigma$ is made in such a way that its boundary coincides with the $(n-2)$-section of $\\partial{\\cal M}$. $\\sqrt {\\gamma}$ is the determinant of the induced metric on that boundary, and $\\sqrt{\\gamma}d\\gamma_{ab}=-\\sqrt{\\gamma}d^{n-2}x (N_a M_b-N_b M_a)$ is the area element. The vector fields \n$N^a$ and $M^a$ are chosen to be unit spacelike and timelike, respectively. \n\nFor a black hole spacetime, the natural choice of the $(n-1)$-dimensional hypersurface in~(\\ref{en1}) is clearly the event horizon.\nThe hypersurface $\\Sigma$ in~(\\ref{en2}) is a spatial hypersurface, and the $(n-2)$-dimensional subspace in~(\\ref{en2}) is the compact spatial section of the event horizon, spanned by angular coordinates. \n\nThe bracket algebra of charges~(\\ref{en2}) generated by different vector fields is given by\n\\begin{eqnarray}\n[Q[\\zeta_m], Q[\\zeta_n]]:=\\frac12(\\delta_{\\zeta_m} Q[\\zeta_n]-\\delta_{\\zeta_n} Q[\\zeta_m]) = \\frac12 \\int \\sqrt{\\gamma} d\\gamma_{ab} (\\zeta_n^a J^b[\\zeta_m]-\\zeta_m^a J^b[\\zeta_n]),\n\\label{en3}\n\\end{eqnarray}\nwhere $\\delta_{\\zeta_m} Q[\\zeta_n]:=\\int_{\\Sigma} d\\Sigma_a \\pounds_{\\zeta_m}(\\sqrt{\\det g}J^a[\\zeta_n])$. \n\nThe next step is to identify an infinite discrete set of diffeomorphism generating vector fields $\\{\\zeta^a_m\\}$ which leave the near horizon geometry invariant. It can then be shown that for such vector fields,~(\\ref{en3}) can be identified with the Virasoro algebra with a central extension. One then uses the Cardy formula~\\cite{Cardy1, Cardy2} in order to compute the entropy of the spacetime.\n\nFor the de Sitter black holes, we have two natural boundaries -- the black hole horizon along with the cosmological event horizon. In other words, the so called `bulk' of the de Sitter black hole spacetimes is the region between these two horizons.     \nFor such two natural boundaries, the first integral in Eq.~(\\ref{en1}) splits into two pieces -- on the two hypersurfaces located at the two horizons. \n\nAccordingly, the Noether charge in (\\ref{en2}) will consist of two integrals at the two Killing horizons, similarly for the \nalgebra satisfied by the charges, Eq.~(\\ref{en3}). The hypersurface $\\Sigma$ is orthogonal to the foliation vector field $\\chi^a$, derived in the previous section. Since $\\chi^a$ smoothly becomes null and Killing ($\\chi_{{\\cal H}}^a$) on both the horizons, we have obtained the two boundary integrals in a quite natural manner.  \n    \nLet us first evaluate the charge corresponding to the horizon Killing vector field $\\chi_{{\\cal H}}^a$ in Eq.~(\\ref{en2}).\nWe have seen in the previous section that the spacetime metric formally looks the same in the neighborhood of both the Killing horizons, (\\ref{g25}). We choose for the black hole horizon, $N^a=\\sqrt{2\\kappa_H X} X^a$, and $M^a=\\frac{\\chi_{{\\cal H}}^a}{\\sqrt{2\\kappa_H X}}$. The trace of the extrinsic curvature at the black hole event horizon is given by $K_H=-\\sqrt{\\kappa_H\/2X}$. Likewise, we get the unit vectors and the trace of the extrinsic curvature on the cosmological horizon, by replacing $\\kappa_H$ with $\\kappa_C$. We find  \n\\begin{eqnarray}\nQ=\\frac12\\int d\\Sigma_a J^a= \\frac12 \\int_H \\sqrt{\\gamma} d\\gamma_{ab} J^{ab}+\\frac12 \\int_C \\sqrt{\\gamma} d\\gamma_{ab} J^{ab}=\\frac{\\kappa_H A_{H}+\\kappa_C A_C}{8\\pi G},\n\\label{en4}\n\\end{eqnarray}\nwhere we have defined the $(n-2)$-dimensional `area' as $A=\\int \\sqrt{\\gamma} d^{n-2} x$, which correspond to the compact spatial sections at the two horizons.\n\nWe shall next obtain a Virasoro algebra for the charges generated by vector fields which preserves the near horizon structures. Eq.~(\\ref{en3}) for our case becomes, \n\\begin{eqnarray}\n[Q[\\zeta_m], Q[\\zeta_n]]=\\frac12 \\int_{H} \\sqrt{\\gamma} d\\gamma_{ab} (\\zeta_n^a J^b[\\zeta_m]-\\zeta_m^a J^b[\\zeta_n])+\\frac12 \\int_{C} \\sqrt{\\gamma} d\\gamma_{ab} (\\zeta_n^a J^b[\\zeta_m]-\\zeta_m^a J^b[\\zeta_n]).\n\\label{en5}\n\\end{eqnarray}\nWe shall look for the set of vector fields $\\{\\zeta^a\\}$, which has only non-vanishing `time' and $X$ components, with respect the spacetime metric (\\ref{g25}). Let us collectively denote the spatial coordinates and basis vectors tangent to the horizon by $\\{\\Theta^i\\}$ and $\\{\\Theta^a\\}$ respectively,\nspanning $\\gamma_{ab}$ in Eq.~(\\ref{g25}). Let $\\Phi (x)$ be the norm of any of the basis vectors $\\{\\Theta^a\\}$. Clearly, in order to have the area of the horizon to be finite and non-vanishing, $\\Phi$ has to finite and non-vanishing, too.\nThus, in the neighborhood of a Killing horizon, we may expand $\\Phi= \\Phi_1(\\Theta^i)+ {\\cal O}(\\beta)+~{\\rm higher~order~terms~in}~\\beta$. Also, since $\\Theta^a$'s are tangent to the horizon, where $\\beta=0$, we must have $\\Theta^a\\nabla_a\\beta\\to 0$ in the infinitesimal neighborhood of any Killing horizon. Putting these all in together, using Eq.s~(\\ref{g22}), (\\ref{g23}), (\\ref{g25}) and the chain rule for the partial derivatives, we get  \n\\begin{eqnarray}\n\\partial_X \\Phi= (\\partial_{\\beta}\\Phi) (\\partial_{\\widetilde{X}}\\beta)( \\partial_{X}\\widetilde{X}),\n\\label{en6}\n\\end{eqnarray}\nto be vanishing in the neighborhood of any Killing horizon. This is analogous to the static and spherically symmetric case : the metric functions spanning the 2-sphere (at $r=r_H$) do not depend upon the spacelike Rindler coordinate, $X$, on or in the infinitesimal neighborhood of the horizon.\n\nIn order to preserve the near horizon geometry, we must impose $\\pounds_{\\zeta} g_{\\tau\\tau}=0=\\pounds_{\\zeta} g_{XX}$, where $\\tau$ represents the parameter or coordinate along the horizon Killing vector field $\\chi^a_{\\rm H}$. Solving these two equations involves only the Rindler part ($\\tau-X$) of the metric, and one obtains~\\cite{Majhi:2011ws, Majhi:2012tf},\n\\begin{eqnarray}\n\\zeta^{\\tau}=T(\\tau,X, \\Theta^i)-\\frac{1}{2\\kappa}\\partial_{\\tau} T(\\tau,X, \\Theta^i), \\quad \\zeta^X=-X\\partial_{\\tau} T(\\tau,X, \\Theta^i),\n\\label{en7}\n\\end{eqnarray}\nwhere $T$ is some smooth but otherwise arbitrary function. In terms of this vector field, Eq.~(\\ref{en4}) reads\n\\begin{eqnarray}\nQ=\\frac{1}{8\\pi G} \\int_H \\sqrt{\\gamma} d^{n-2}x \\left(\\kappa_H T-\\frac12\\partial_{\\tau}T \\right)+\\frac{1}{8\\pi G} \\int_C \\sqrt{\\gamma} d^{n-2}x \\left(\\kappa_C T-\\frac12\\partial_{\\tau}T \\right).\n\\label{en8}\n\\end{eqnarray}\nWe now expand the function $T$ in terms of infinite number of discrete eigenmodes as, $T=\\sum\\limits_{m}A_m T_m$ with $m$\ninteger, so that for each  $m$, we call the corresponding vector field as $\\zeta_m^a$. It is usual to choose $T_m=\\frac{1}{l_0}e^{im(l_0\\tau+ l_i\\phi^i+g(x))}$, where $\\phi^i$'s are the parameters along the axisymmetric Killing vector fields, $l_0$ and $l_i$'s are constants, $g(x)=-l_0\\int \\frac{dX}{2\\kappa X}$. With this choice of the modes, $\\zeta_m$'s satisfy an infinite dimensional discrete Lie algebra over a circle,   \n\\begin{eqnarray}\n[\\zeta_m,\\zeta_n]^a_{LB}=-i(m-n)\\zeta_{m+n},\n\\label{en9}\n\\end{eqnarray}\nwhere the subscript `LB' denotes the Lie bracket.\n\nThus the modes formally look the same on both the horizons, but the eigenvalue $l_0$ may be different. We shall call them as $l_{0H}$ and $l_{0C}$ respectively, for the black hole and the cosmological event horizon. Moreover, we have to fix them uniquely as well in order to derive the entropy of the whole system, as we shall see below.  This is qualitatively different from the single horizon system discussed in~\\cite{Majhi:2011ws, Majhi:2012tf}, where one can leave $l_0$ completely arbitrary.\n\nNow, due to the axisymmetric geometry, the mode functions must be periodic in the Killing parameters $\\phi^i$ of the axisymmetric Killing vector fields. Then, since we have assumed the horizons to be compact, $\\phi^i$'s are tangent to them and (\\ref{en8}) becomes \n\\begin{eqnarray}\nQ_m=\\frac{(\\kappa_H\/l_{0H}) A_H +(\\kappa_C\/l_{0C}) A_C}{8\\pi G}\\delta_{m,0}.\n\\label{en10}\n\\end{eqnarray}\nLikewise, the algebra of the charges, Eq.~(\\ref{en3}) gives for our two horizon spacetimes,\n\\begin{eqnarray}\n[Q_m, Q_n]= -{\\frac{i(m-n)}{8\\pi G}}\\left((\\kappa_H\/l_{0H}) A_H+(\\kappa_C\/l_{0C}) A_C\\right)\\delta_{m+n,0}-\\frac{im^3}{16\\pi G}\\left(A_H (l_{0H}\/\\kappa_H) +A_C (l_{0C}\/\\kappa_C) \\right)\\delta_{m+n,0},\\nonumber\\\\\n\\label{en11}\n\\end{eqnarray}\nwhich is a Virasoro algebra, effectively encompassing both the boundaries, and hence the bulk.\nWe can identify the zero mode energy or the Hamiltonian and the central charge, $C$ from Eq.s~(\\ref{en10}), (\\ref{en11}),\n\\begin{eqnarray}\nQ_0=\\frac{(\\kappa_H\/l_{0H}) A_H +(\\kappa_C\/l_{0C}) A_C}{8\\pi G},\\quad \\frac{C}{12}=\\frac{A_H (l_{0H}\/\\kappa_H) +A_C (l_{0C}\/\\kappa_C)}{16\\pi G}.\n\\label{en12}\n\\end{eqnarray}\nAccording to the Cardy formula~\\cite{Cardy1, Cardy2}, the entropy of the system is given by $S=2\\pi\\,\\sqrt{\\frac{CQ_0}{6}}$. Then it is clear that in order to get the entropy, we must set $l_{0H}=\\kappa_H$ and $l_{0C}=\\kappa_C$ in (\\ref{en12}). This choice gives the same modes as one obtains via the method of `asymptotic' fall-off near the horizon, e.g.~\\cite{Dreyer:2013noa}.\n\nWith this choice, we get via the Cardy formula  \n\\begin{eqnarray}\nS=2\\pi\\,\\sqrt{\\frac{CQ_0}{6}}=\\frac{A_H+A_C}{4G}\n\\label{en13}\n\\end{eqnarray}\ni.e. the total Bekenstein-Hawking entropy of the de Sitter black hole spacetimes.  \n\nBefore proceeding further, let us summarize what we have done so far. Since the de Sitter black holes have two\nKilling horizons, the Gibbons-Hawking-York surface counter-term splits into two pieces,  corresponding to the two horizons. The hypersurface $\\Sigma$ appearing in Eq.~(\\ref{en2}) is the one which is orthogonal \nto the timelike vector field $\\chi^a$, derived in Sec.~2. In terms of the two boundary integrals and choice of appropriate \nmode functions on the horizons, we have actually derived the total entropy of such spacetimes. \n\nWe have also seen that\nas long as the derivation of the total entropy is concerned, the choice of the mode functions are much more restricted \nthan the single horizon spacetimes.  \nClearly, doing so is absolutely justified. We start with considering vector fields generating diffeomorphism in the entire spacetime, and due to the existence of boundaries, we only consider their explicit forms on the boundaries themselves. Since our spacetime has two horizons, the diffeomorphism generating vector fields $\\zeta_m$'s assume different forms on them. In other words, this analysis can be regarded as `doubly local' instead of `local'~\\cite{Majhi:2011ws, Majhi:2012tf}, as the single horizon systems. Most importantly, both the surface gravities can be formally expressed by~(\\ref{g21'}), so that $\\kappa_H$ and $\\kappa_C$ are nothing but the $\\beta^2\\to 0$ limits of the smooth function\n$\\kappa(x)=\\sqrt{(\\nabla_a\\beta^2)(\\nabla^a\\beta^2)\/4\\beta^2}$. Then it is clear that even though $l_{0H}$ and $l_{0C}$ have different values numerically, they are formally exactly the {\\it same}.\n\n\nWe shall consider below some explicit and non-trivial exact solutions in order to demonstrate that they indeed fall under the scope of the general analysis we have done so far.   \n\n\n\\section{Explicit examples}\n\\subsection{The Kerr-Newman- and the Plebanski-Demianski-de Sitter families}\nLet us begin with the Kerr-Newman-de Sitter spacetime in four spacetime dimensions, whose metric in the Boyer-Lindquist coordinate reads\n\\begin{eqnarray}\nds^2&=&-\\frac{\\Delta_r-a^2\\sin^2\\theta \\Delta_{\\theta}}{\\rho^2}dt^2 - \\frac{2a\\sin^2\\theta}{\\rho^2 \\Xi} \\left(\\left(r^2+a^2\\right)\\Delta_{\\theta}-\\Delta_r\\right)dtd\\phi \\nonumber\\\\&+&\n \\frac{\\sin^2\\theta}{\\rho^2 \\Xi^2} \\left(\\left(r^2+a^2\\right)^2\\Delta_{\\theta}-\\Delta_ra^2\\sin^2\\theta \\right)d\\phi^2+\\frac{\\rho^2}{\\Delta_r}dr^2+\\frac{\\rho^2}{\\Delta_{\\theta}}d\\theta^2,\n\\label{ex1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Delta_r&=&\\left(r^2+a^2\\right)\\left(1-\\Lambda r^2\/3\\right)-2MGr+q^2,\\quad \\Delta_{\\theta}=1 +\\Lambda a^2 \\cos^2\\theta\/3\\nonumber\\\\\n\\Xi&=&1+\\Lambda a^2\/3,\\quad \\rho^2=r^2+a^2\\cos^2\\theta,\n\\label{ex2} \n\\end{eqnarray}\nwith $M$, $a$ and $q$ are respectively the mass, rotation parameter and charge. For a Dyonic black hole, $q^2$ is understood as the sum of the square of the electric and magnetic charges.   \n\nFirstly, since the timelike and axisymmetric Killing vector fields are coordinate fields ($\\xi^a=(\\partial_t)^a,\\,\\phi^a=(\\partial_{\\phi})^a$), they trivially commute\\footnote{Note that in our general analysis, we did not need to assume that the Killing fields are global coordinate fields everywhere inside the bulk.}.\nThe 2-planes orthogonal to these Killing vectors are spanned by the coordinate vector fields, $(\\partial_r)^a$ and $(\\partial_{\\theta})^a$. Hence they also commute to give a trivial Lie algebra and thus form integral submanifolds~\\cite{Wald:1984rg}. This was a crucial assumption made in Sec.~2. Also, as we have seen in Sec.~2, this guarantees the existence of the hypersurface orthogonal timelike vector field $\\chi^a=\\xi^a-(\\xi\\cdot\\phi\/\\phi\\cdot\\phi)\\phi^a=\\partial_t^a-(g_{t\\phi}\/g_{\\phi\\phi})\\partial_{\\phi}^a$.\n\nThe black hole and the cosmological event horizon of (\\ref{ex1}) correspond to the largest and the next to largest positive roots of $\\Delta_r=0$. The inner or the Cauchy horizon will not concern us for our present purpose. \n\nThe norm $-\\beta^2$ of $\\chi^a$, close to any of the horizons $(\\Delta_r\\to 0)$ is given by\n\\begin{eqnarray}\n-\\beta^2=g_{tt}-\\frac{g_{t\\phi}^2}{g_{\\phi\\phi}}=-\\frac{\\Delta_r \\rho^2}{\\left(r^2+a^2\\right)^2}+{\\cal O}(\\Delta_r^2),\n\\label{ex3}\n\\end{eqnarray}\nwhich is null on both the horizons. Also, it is eassy to check from the metric functions (\\ref{ex1}) that the function\n$\\alpha=\\frac{\\xi\\cdot\\phi}{\\phi\\cdot\\phi}$ becomes a constant whenever $\\Delta_r=0$.\n\nThus the Kerr-Newman-de Sitter spacetime falls into the general geometric category we discussed in Sec.~2.\n\nLet us now discuss the derivation of the Rindler coordinate following Sec.~2. According to Eq.~(\\ref{g21}), infinitesimally close to any horizon, we choose \n\\begin{eqnarray}\n\\widetilde{X}_a^{H,C}=\\frac{\\nabla_a\\beta}{\\kappa_{H,C}},\n\\label{exa3}\n\\end{eqnarray}\nwhere the subscripts or superscripts `$H,C$' denote black hole an the cosmological horizon respectively. Using Eq.s~(\\ref{ex1}), (\\ref{ex3}), we have $(\\partial_{\\theta})^a\\widetilde{X}_a\\sim {\\cal O} (\\sqrt{\\Delta_r})$ in the infinitesimal vicinity of any of the horizons.  Likewise, for the Rindler coordinate $X^a$ (Eq.~(\\ref{g25})), it is easy to check that\n$(\\partial_{\\theta})^a\\widetilde{X}_a\\sim {\\cal O} (\\Delta_r^{1\/4})$.\n Thus infinitesimally close to the horizon, the vector fields $\\partial_{\\theta}^a$ becomes orthogonal to $\\widetilde{X}_a$  or $X^{a}$.\n\nSo, the Kerr-Newman-de Sitter spacetime takes the form given in Eq.~(\\ref{g25}) infinitesimally close to the horizons with $\\gamma_{ab}$ spaanned by $(\\partial_{\\theta})^a$ and $(\\partial_{\\phi})^a$. \nThus, it falls into the general scheme discussed in Sec.s 2 and 3.  \n\n\nAn exact asymptotically anti-de Sitter black hole solution with two independent rotation parameters in five dimensional minimal supergravity can be seen in~\\cite{Chong}. From this solution we can obatain a de Sitter black hole spacetime via analytic continuation~\\cite{dolan2},\n\\begin{eqnarray}\nds^2 = &-&\\left[\\frac{\\Delta_{\\theta}\\left(1-g^2r^2\\right)}{\\Xi_a\\Xi_b}-\n\\frac{\\Delta_{\\theta}^2\\left(2M\\rho^2-q^2-2abqg^2\\rho^2\n\\right)}{\\rho^4\\Xi_a^2\\Xi_b^2}\n\\right]dt^2+\\frac{\\rho^2}{\\Delta_r}dr^2+\\frac{\\rho^2}{\\Delta_\\theta}\nd\\theta^2 \\nonumber\\\\\n&+& \\left[\\frac{\\left(r^2+a^2\\right)\\sin^2\\theta}{\\Xi_a}+\n\\frac{a^2 \\left(2MG\\rho^2-q^2\\right)\\sin^4\\theta +2abq\\rho^2\\sin^4\\theta }{\\rho^4 \\Xi_a^2}\\right] d\\phi^2 \\nonumber\\\\\n&+&\\left[\\frac{\\left(r^2+b^2\\right)\\cos^2\\theta}{\\Xi_b}+\n\\frac{b^2 \\left(2MG\\rho^2-q^2\\right)\\cos^4\\theta+2abq\\rho^2\\cos^4\\theta }\n{\\rho^4 \\Xi_b^2}\\right] d\\psi^2\\nonumber\\\\\n&-&\\frac{2\\Delta_{\\theta}\\sin^2\\theta\\left[a\\left(2MG\\rho^2-q^2\\right)\n+ bq\\rho^2\\left(1-a^2g^2\\right) \\right]}\n{\\rho^4\\Xi_a^2\\Xi_b}dtd\\phi \\nonumber\\\\\n&-&\\frac{2\\Delta_{\\theta}\\cos^2\\theta\\left[b\\left(2MG\\rho^2-q^2\\right)\n+ aq\\rho^2\\left(1-b^2g^2\\right) \\right]}\n{\\rho^4\\Xi_a\\Xi_b^2}\ndtd\\psi\\nonumber\\\\\n&+&\\frac{2\\sin^2\\theta\\cos^2\\theta\\left[ab\\left(2MG\\rho^2-q^2\\right)\n+ q\\rho^2\\left(a^2+b^2\\right) \\right]}\n{\\rho^4\\Xi_a\\Xi_b}\nd\\phi d\\psi\n\\label{ex6}\n\\end{eqnarray} \nwhere \n\\begin{eqnarray}\n\\rho^2&=&\\left(r^2+a^2\\cos^2\\theta+b^2\\sin^2\\theta\\right), \\quad \\Delta_{\\theta}=\\left(1+a^2g^2\\cos^2\\theta+b^2g^2\\sin^2\\theta\\right), \\quad \\Xi_a=(1+a^2g^2),\\nonumber\\\\\\quad \\Xi_b&=&(1+b^2g^2), \\quad \\Delta_r=\\left[\\frac{(r^2+a^2)(r^2+b^2)(1-g^2r^2)+q^2+2abq}{r^2}-2MG\\right].\n\\label{ex7}\n\\end{eqnarray} \nThe parameters $M,~a,~b,~q$ specify respectively the mass, two independent rotations and the charge of the black hole and $g^2$ is the positive cosmological constant.\n\nClearly, being coordinate fields, all the three Killing vector fields $(\\partial_t)^a,\\,(\\partial_{\\phi})^a, (\\partial_{\\psi})^a$\ncommute. The 2-planes orthogonal to them are spanned by coordinate fields along $r$ and $\\theta$, which commute and trivially form a Lie algebra. Thus those two planes are integral submanifolds. This gurantees the existence of the timelike \nvector field $\\chi^a$ orthogonal to the $(r,\\theta, \\phi,\\psi)$ family of hypersurfaces, which from the discussions of Sec.~2 is written as\n\\begin{eqnarray}\n\\chi^a =(\\partial_{t})^a-\\frac{\\left(g_{t\\phi}g_{\\psi\\psi}-g_{t\\psi}g_{\\phi\\psi}\\right)}\n{\\left(g_{\\phi\\phi}g_{\\psi\\psi}-(g_{\\psi\\phi})^2\\right)}\n(\\partial_{\\phi})^a\n-\\frac{\\left(g_{t\\psi}g_{\\phi\\phi}-g_{t\\phi}g_{\\phi\\psi}\\right)}\n{\\left(g_{\\phi\\phi}g_{\\psi\\psi}-(g_{\\psi\\phi})^2\\right)}(\\partial_{\\psi})^a.\n\\label{5d1}\n\\end{eqnarray} \nThe horizons of this spacetime are the positive roots of $\\Delta_r=0$.  A discussion on thermodynamics of this spacetime can be seen in~\\cite{dolan2}. Infinitesimaly close to any of the horizons, the norm of $\\chi^a$ takes the form, \n\\begin{eqnarray}\n\\chi^a\\chi_a\n=-\\beta^2=-\\frac{\\rho^2r^4\\Delta_r} {\\left[(r^2+a^2)(r^2+b^2)+abq\n\\right]^2}+{\\cal{O}}({\\Delta_r^2}).\n\\end{eqnarray}\nThus $\\chi^a$ becomes null on the horizons. It is easy to check from the metric functions that infinitesimally close to any of the horizons, coefficient functions in (\\ref{5d1}) become constants\n\\begin{eqnarray}\n\\frac{\\left(g_{t\\phi}g_{\\psi\\psi}-g_{t\\psi}g_{\\phi\\psi}\\right)}\n{\\left(g_{\\phi\\phi}g_{\\psi\\psi}-(g_{\\psi\\phi})^2\\right)}\n\\Bigg\\vert_{r=r_H,\\,r_C}\n=-\\frac{a (r^2+b^2)(1-g^2r^2)+bq}\n{(r^2+a^2)(r^2+b^2)+abq}\\Bigg\\vert_{r=r_H,\\,r_C}, \\nonumber\\\\\n\\frac{\\left(g_{t\\psi}g_{\\phi\\phi}-g_{t\\phi}g_{\\phi\\psi}\\right)}\n{\\left(g_{\\phi\\phi}g_{\\psi\\psi}-(g_{\\psi\\phi})^2\\right)}\n\\Bigg\\vert_{r=r_H,\\,r_C}=-\n\\frac{b(r^2+a^2)(1-g^2r^2)+aq}\n{(r^2+a^2)(r^2+b^2)+abq}\\Bigg\\vert_{r=r_H,\\,r_C},\n\\label{e13}\n\\end{eqnarray} \nwhere $r_H$ and $r_C$ denote respectively, the balck hole and the cosmological horizon radii.\n\nAs earlier, it can also be checked easily that the near horizon Rindler coordinates can be defined and $\\gamma_{ab}$ in Eq.~(\\ref{g25}) for this case is spanned by $(\\theta,\\phi, \\psi)$. Thus the general analysis of Sec.s 2 and 3 holds perfectly for~(\\ref{ex6}).\n\n\n\n\n\n\nWe shall next consider the Plebansky-Demianski-de Sitter black hole spacetimes. This class is a geneailazation of the Kerr-Newman family, in the sense that apart from mass, charge, rotation and the cosmological constant, it contains additional parameters.\nThe complete family of the Plebanski-Demianski-de Sitter class spacetimes which might represent de Sitter black holes has the metric~\\cite{Griffiths:2005qp},\n\\begin{eqnarray}\nds^2=\\frac{1}{\\Omega^2}\\left[-\\frac{\\Delta_r}{\\rho^2}\\left(dt-  \\left(a\\sin^2\\theta+4l\\sin^2\\frac{\\theta}{2}\\right)d\\phi \\right)^2+\\frac{\\rho^2}{\\Delta_r}dr^2  + \\frac{P}{\\rho^2} \\left(adt-\\left(r^2 +(a+l)^2 \\right) d\\phi \\right)^2      +\\frac{\\rho^2}{P}\\sin^2\\theta d\\theta^2 \\right]\n\\label{ex8}\n\\end{eqnarray} \nwhere\n\\begin{eqnarray}\n\\Omega&=&1-\\frac{\\alpha}{\\omega}\\left( l+a\\cos\\theta\\right)r, \\quad \\rho^2=r^2+\\left( l+a\\cos\\theta\\right)^2, \\qua\nP=\\sin^2\\theta \\left(1-a_3\\cos\\theta-a_4\\cos^2\\theta\\right)\\nonumber\\\\\n\\Delta_r&=&\\left(\\omega^2 k+q^2+q_m^2\\right)-2MGr+\\epsilon r^2 -\\frac{2\\alpha n}{\\omega}r^3-\\left(\\alpha^2 k+\\Lambda\/3\\right)r^4,\n\\label{ex9}\n\\end{eqnarray} \nThe parameters $\\alpha$, $\\omega$, $q$, $q_m$, $\\epsilon$ and \n$k$ are arbitrary, and $a_3$ and $a_4$ are determined via them. These parameters have their physical meaning\nin certain special sub-classes only. \n\n\n\n\nFor example, for $\\alpha=0$, the above metric becomes the Kerr-Newman-NUT-de Sitter solution~\\cite{Griffiths:2005qp},\n\\begin{eqnarray}\nds^2=-\\frac{Q}{\\rho^2}\\left[dt-(a\\sin^2\\theta+4l \\sin^2\\theta\/2)d\\phi\\right]^2+\\frac{\\rho^2}{Q}dr^2\n+\\frac{P}{\\rho^2}\\left[adt-(r^2+(a+l)^2)d\\phi\\right]^2+\\frac{\\rho^2}{P}\\sin^2\\theta d\\theta^2,\n\\label{ex11}\n\\end{eqnarray} \nwhere\n\\begin{eqnarray}\n\\rho^2&=&r^2+\\left( l+a\\cos\\theta\\right)^2, \\qua\nP=\\sin^2\\theta \\left(1+ \\frac{4\\Lambda a l}{3} \\cos\\theta+ \\frac{\\Lambda a^2 \\cos^2\\theta}{3}\\cos^2\\theta\\right)\\nonumber\\\\\n\\Delta_r&=&\\left(a^2-l^2+q^2+q_m^2\\right)-2MGr+r^2 -\\Lambda\\left((a^2-l^2)l^2+(a^2\/3+2l^2)r^2+ r^4\/3\\right),\n\\label{ex10}\n\\end{eqnarray} \nwhere $q$ and $q_m$ are electric and magnetic charges and $l$ is the NUT parameter.\n\n\nWe shall consider the most general class given by~(\\ref{ex8}), implicitly assuming it represents de Sitter black holes. The black hole and the cosmological horizons are the two largest roots \nof $\\Delta_r=0$. It is easy to argue the existence of the hypersurface orthogonal timelike vector field $\\chi^a$ as earlier. The norm of $\\chi^a$ behaves as $\\Delta_r\\to 0$ as,\n\\begin{eqnarray}\n\\beta^2=\\frac{P \\Delta_r}{\\Omega^2 \\rho^2}  \\left[r^2+(a+l)^2-a^2(a\\sin^2\\theta+4l \\sin^2\\theta\/2)^2\\right]+{\\cal O}(\\Delta_r^2),\n\\label{ex12}\n\\end{eqnarray} \nwhich is null. It can also be verified from the metric functions that the function $(\\xi\\cdot \\phi)\/(\\phi\\cdot \\phi)$ becomes \na constant on $\\Delta_r=0$. Thus the most general class of Plebanski-Demianski metrics (\\ref{ex8}) falls into our geometrical \nassumptions, and clearly, when it represents a de Sitter black hole spacetime (such as~(\\ref{ex11})),  we can derive its total entropy.\n\nFinally, we shall briefly discuss the Kerr-de Sitter spacetime in generic spacetime dimensions $n$~\\cite{Gibbons1, Gibbons2}.  The metric of which reads in the Boyer-Lindquist like coordinates~\\cite{dolan2, Gibbons1, Gibbons2}, \n\\begin{eqnarray}\nds^2=-W\\left(1-g^2r^2\\right)dt^2+\\frac{2MG}{U}\\left(Wdt-\\sum\\limits_{i=1}^{N} \\frac{a_i\\mu_i^2 d\\phi_i}{\\Xi_i} \\right)^2+\\sum\\limits_{i=1}^{N}\\frac{(r^2+a_i)^2}{\\Xi_i}\\left(\\mu_i^2d\\phi_i^2 +d\\mu_i^2\\right)+\\frac{U dr^2}{X-2MG}\\nonumber\\\\+ \\epsilon r^2 d\\nu^2\n+\\frac{g^2}{W(1-g^2r^2)} \\left(\\sum\\limits_{i=1}^{N}\\frac{r^2 +a_i^2}{\\Xi_i}\\mu_id\\mu_i+\\epsilon r^2 \\nu d\\nu \\right)^2, \n\\label{ex4}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n2\\Lambda&=& (n-1)(n-2)g^2,\\quad W= \\sum\\limits_{i=1}^{N} \\mu_i^2\/\\Xi_i+\\epsilon \\nu^2,\\quad X=r^{\\epsilon-2}\\left(1-g^2r^2\\right)\\prod_{i=1}^N\\left(r^2+a_i^2\\right)\\nonumber\\\\\nU&=&\\frac{Z}{1-g^2 r^2}\\left(1-\\sum\\limits_{i=1}^{N} \\frac{a_i^2\\mu_i^2}{r^2+a_i^2}\\right),\\quad \\Xi_i=1+g^2a_i^2,\n\\label{ex5}\n\\end{eqnarray}\nwhere $N$ is the integer part of $(n-1)\/2$. The constant $\\epsilon$ is $+1\\,(0)$ for even (odd) spacetime dimensions.\n$\\phi_i$'s are the coordinates along the axisymmetric Killing vector fields, and $a_i$ are the corresponding independent\nrotation parameters and $g^2$ is the cosmological constant. The coordinates $\\mu_i$ and $\\nu$ are not independent, but are related via the constraint,\n\\begin{eqnarray}\n\\sum\\limits_{i=1}^{N} \\mu_i^2+\\epsilon \\,\\nu^2=1,\n\\label{ex5a1}\n\\end{eqnarray}\nensuring the correct dimensionality of the spacetime.\n\nDiscussion of thermodynamic properties of this spacetime including that of the variation of the cosmological constant can be found in~\\cite{dolan2}. \n\n\nThe spacetime is endowed with total $(N+1)$ Killing vector fields. Since all of them are coordinate Killing vector fields, they commute. The $(n-N-1)$-dimensional subspace is spanned by coordinate vector fields $\\{(\\partial_{\\mu_i})^a,\\, (\\partial_{\\nu})^a\\}$.\nSince the spatial coordinates along these vector fields satisfy the constraint~(\\ref{ex5a1}), we may replace by using the chain rule of the partial derivatives, any specific $\\partial_{\\mu_i}$\nas a linear combination of partial derivatives of the set $\\{\\partial_{\\mu_j},\\, \\partial_{\\nu}\\},~j\\neq i$. Then it is clear that the Lie brackets like $[\\partial_{\\mu_i},\\,\\partial_{\\mu_k}]_{LB}^a$ or $[\\partial_{\\mu_i},\\,\\partial_{\\nu}]_{LB}^a$, will always be a linear combination of the coordinate vector fields spanning this subspace. This is a Lie algebra, but not trivial as the earlier examples. Nevertheless, this ensures that the subspace orthogonal to the Killing vector fields form an integral submanifolds~\\cite{Wald:1984rg}. \n \nThen from this, the existence of the hypersurface orthogonal vector field $\\chi^a$, and its Killing property when it becomes null follows as earlier. \n\n\n\n\n\n\n\\subsection{Non-minimal couplings}\nBefore we end, we shall address two cases of non-minimal couplings, the hairy black hole with conformal scalar field and the Brans-Dicke theory in the Jordan frame. Since for any non-minimal coupling the Ricci scalar term in the action gets modified  as,  $f(\\varphi)\\, R$, where $\\varphi$ is the scalar field, the presence of such coupling modifies the surface counterterm in Eq.~(\\ref{en1}), as well.  \n\nFor static and spherically symmetric de Sitter black hole spacetimes, a derivation of Bekenstein-Hawking-Wald entropy using the surface counterterm has recently been done in~\\cite{Meng:2014dfa}, for the black hole event horizon. The surface term reads in this case, \n\\begin{eqnarray}\nA_{\\rm sur}=\\frac{1}{8\\pi G}\\int_{\\partial {\\cal M}} [d^{n-1}x]\\left(1-\\frac{2\\pi G(n-2)\\varphi^2}{(n-1)} \\right)K.\n\\label{nm1}\n\\end{eqnarray}\nWe shall briefly discuss this case for stationary axisymmetric spacetimes. It was shown in~\\cite{Bhattacharya:2013hvm} that a black hole with conformal scalar hair and a positive cosmological constant cannot have a slow rotation in four spacetime dimensions. As was also argued there, it may be  possible though, that a solution with generic rotation exists -- at least one cannot rule out the possibility. In the following, we shall assume that such a solution indeed exists, and it falls into the geometric category described in Sec.~2. \n\n\nFollowing~\\cite{Meng:2014dfa}, the conserved Noether charge $Q$ corresponding to the diffeomorphism, instead of Eq.~(\\ref{en8}), in this case becomes\n\\begin{eqnarray}\nQ=\\frac{1}{8\\pi G} \\int_H \\sqrt{\\gamma} d^{n-2}x \\left(1-\\frac{2\\pi G(n-2)\\varphi^2}{(n-1)}\\right)\\left(\\kappa_H T-\\frac12\\partial_{\\tau}T \\right)\\nonumber\\\\+\\frac{1}{8\\pi G} \\int_C \\sqrt{\\gamma} d^{n-2}x \\left(1-\\frac{2\\pi G(n-2)\\varphi^2}{(n-1)} \\right)\\left(\\kappa_C T-\\frac12\\partial_{\\tau}T \\right)\n\\label{nm2}\n\\end{eqnarray}\nWe note that for axisymmetric spacetimes, the field $\\varphi$ may depend on the non-Killing coordinates tangent to the horizons (such as the polar angle $\\theta$). Thus unlike the spherically symmetric spacetimes~\\cite{Meng:2014dfa}, we cannot pull out the scalar field out of the integration in Eq.~(\\ref{nm2}).\n\nUsing the suitable mode decomposition as described in Sec.~3, we find\n\\begin{eqnarray}\nQ_m=\\frac{1}{8\\pi G}\\left[ \\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_H}{(n-1)}\\right)(\\kappa_H A_H)\/l_{0H}+ \\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_C}{(n-1)}\\right)(\\kappa_C A_C)\/l_{0C}\\right]\\, \\delta_{m,0},\n\\label{nm3}\n\\end{eqnarray}\nwhere on any of the horizons we have defined,\n\\begin{eqnarray}\n\\langle\\varphi^2\\rangle_{H, C}:=  \\frac{1}{A_{H, C}}\\int_{H, C} \\sqrt{\\gamma} d^{n-2}x \\varphi^2.\n\\label{nm4}\n\\end{eqnarray}\nLikewise, we find the algebra of charges  \n\\begin{eqnarray}\n[Q_m, Q_n]= -{\\frac{i(m-n)}{8\\pi G}}\\left[ \\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_H}{(n-1)}\\right) (\\kappa_H\/l_{0H}) A_H+\\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_C}{(n-1)}\\right)(\\kappa_C\/l_{0C}) A_C\\right]\\delta_{m+n,0}\\nonumber\\\\-\\frac{im^3}{16\\pi G}\\left[\\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_H}{(n-1)}\\right)  A_H (l_{0H}\/\\kappa_H) + \\left(1-\\frac{2\\pi G(n-2)\\langle\\varphi^2\\rangle_C}{(n-1)}\\right)A_C (l_{0C}\/\\kappa_C) \\right]\\delta_{m+n,0}.\\nonumber\\\\\n\\label{nm5}\n\\end{eqnarray}\nSetting $l_{0H}=\\kappa_H$ and $l_{0C}=\\kappa_C$, we obtain the entropy.\n\nFinally, we shall discuss thermodynamics of black holes in the Brans-Dicke theory (see, e.g.~\\cite{Faraoni} and references therein, for discussions on asymptotically flat spacetimes). The action of the Einstein-Brans-Dicke theory with a cosmological constant in the Jordan frame reads (see e.g.~\\cite{Bhattacharya:2015iha} and references therein) \n\\begin{eqnarray}\nS=\\int[d^nx]\\left[\\varphi R-2\\Lambda-\\frac{\\omega}{\\varphi}\\left(\\nabla \\varphi\\right)^2\\right],\n\\label{nm6}\n\\end{eqnarray}\nwhere $\\varphi$ is the Brans-Dicke scalar field, and $\\omega$ is called the Brans-Dicke parameter. The inverse of the Brans-Dicke field, $\\varphi^{-1}$ acts as a spacetime dependent or dynamical Newton's `constant'. Also, for $\\omega=\\infty$, the field becomes a constant and the theory\ncoincides with the Einstein gravity, $\\varphi=\\varphi^{(0)}=\\frac{1}{16\\pi G}$.\n\nThe derivation of the surface couterterm for the Brans-Dicke theory is similar to that of~(\\ref{nm1}), giving  \n\\begin{eqnarray}\nA_{\\rm sur}=2\\int_{\\partial {\\cal M}} [d^{n-1}x] \\varphi K.\n\\label{nm7}\n\\end{eqnarray}\nNow, for stationary asymptotically flat black hole spacetimes in four spacetime dimensions, the Brans-Dicke theory obeys a no hair theorem~\\cite{Hawking:1972qk},\nwhich states that $\\varphi$ is necessarily a constant in the exterior of such black hole spacetimes. However, it does not constrain\nthe parameter $\\omega$ anyway.\n\nThus, it was argued in e.g.~\\cite{Faraoni} that the entropy of such black holes should be $4\\pi\\varphi_0\\, A $, where $A$ is the horizon\narea and $\\varphi_0$ is constant (corresponding to some finite value of $\\omega$), i.e not necessarily it equals $\\varphi^{(0)}=\\frac{1}{16\\pi G}$.\n\nHowever, when we consider the de Sitter black hole spacetimes, it turns out that $\\varphi$ is not only a constant between the black hole and the cosmological event horizon, but also we must have $\\omega=\\infty$~\\cite{Bhattacharya:2015iha}. In other words, for such spacetimes, we must have $\\varphi_0=\\varphi^{(0)}=\\frac{1}{16\\pi G}$. It is then clear that the total entropy of the de Sitter black holes in Brans-Dicke theory equals~(\\ref{en13}). Clearly, this is qualitatively different from the asymptotically flat black hole spacetimes.  \n\n\n\\section{Discussions}\nIn this work we have considered the thermodynamics of stationary axisymmetric de Sitter black hole spacetimes. We have utilized the\nformalism developed using the Gibbons-Hawking-York surface couterterm and near horizon \nsymmetries~\\cite{Majhi:2011ws}-\\cite{Majhi:2014lka} to derive the total entropy of such two horizon spacetimes. To the best of our knowledge, this has not been done before.  Since the spacetimes we have considered are endowed with two Killing horizons, the surface counterterm method provides us a natural convenience to deal with them. We have used a very general geometric framework to perform the derivation of the entropy in Sec.s 2 and 3. We have also \nseen in Sec.~3 that in order to do that, we have to choose uniquely, constants in the mode functions\npreserving the near horizons' geometries. Such restriction is not present for single horizon spacetimes.\n\nThis entire analysis can be thought of as doubly-local, instead of local as the single horizon spacetimes. Since nowhere in our analysis we assumed any precise asymptotic behaviour of the metric, our analysis is absolutely valid for the Nariai class de Sitter black holes, where the black hole and the cosmological event horizons have comparable sizes. \n\nWe also emphasize here that considering two boundary integrals in such two horizon spacetimes is the most natural choice. \n\nAfter performing this general analysis, we have considered known non-trivial stationary axisymmetric solutions belonging to the Kerr-Newmann- or more general Plebanski-Demianski-de Sitter classes, and have demonstrated that for all of them the general derivation of the total entropy holds. We have also considered two cases of non-minimal couplings. In particular, we have pointed out a qualitative difference of the de Sitter black hole entropy in the presence of a Brans-Dicke scalar field, compared to the asymptotically flat spacetimes.   \n\n\nThe analysis we have done above holds well as long as the spacetime is not exactly extremal, as the very use of Eq.~(\\ref{g25}) requires, even though $\\kappa$ could be `small', but could never be vanishing. This is just because of the fact that the extremal black holes are qualitatively different objects from the usual non-extremal or even near-extremal ones. The vanishing of the surface gravity indicates vanishing temperature and this leads to a debate about the entropy of such black holes : should the entropy of such black holes vanish?  We refer our reader to~\\cite{Stotyn:2015qva}\nfor a recent discussion and a novel proposal about the calculation of entropy of extremal Schwarzschild-de Sitter spacetime,\nwhere the black hole and cosmological event horizon coincide. In that spacetime, there exists no timelike Killing vector field. It was shown in~\\cite{Stotyn:2015qva} that there exists a vector field, which is not Killing on the bulk, but becomes so and also null on the coinciding horizons, thereby enhancing the symmetry of the spacetime near the horizons. This is in accordance with the usual requirement that the near horizon structure must contain the conformal group. Rather curiously, this particular vector field gives bifurcate Killing horizons, spatially coincident with the original horizons we started with, which might give rise to a nonvanishing entropy. \nIt should be interesting to attempt to relate the current formalism we have used, with that of~\\cite{Stotyn:2015qva}. Precisely, for non-extremal spacetimes, where the black hole and the cosmological horizon are separated, one might look for such a vector field and check its near horizon properties. The enhanced algebra containing this vector field\nmight yield meaningful result for all values of $\\kappa$, and hence might give a meaningful notion of the yet not-well understood  $\\kappa \\to 0$ limit.  This procedure seems to have some qualitative similarity with what we did for the stationary axisymmetric spacetimes -- we found a specific timelike non-Killing vector field $\\chi^a$ in the bulk, becoming Killing and null on both the horizons. \n  \n\nAs we also have emphasized in the beginning, the current work is a step towards understanding aspects of de Sitter black hole thermodynamics\nwhen we treat the entire two horizon spacetime as a thermodynamic system as a whole. The variation of the total entropy of such spacetimes gives a Smarr formula predicting an effective equilibrium temperature. Thus the next step is to understand how or under what circumstances or with what choices of the field mode functions one can actually  derive such thermal states using field theory. We hope to return to this issue in future works.  \n\n\n\n\n\n\n\\section*{Acknowledgement}\nThis research was implemented under the ``ARISTEIA II\" Action of the Operational Program ``Education and Lifelong Learning\" and is co-funded by the European Social Fund (ESF) and Greek National Resources, when I was a post doctoral researcher at ITCP and Dept. of Physics of University of Crete, Heraklion, Greece. My current research is supported by IUCAA, Pune, India.  I thank Amit Ghosh, Avirup Ghosh and Amitabha Lahiri for useful discussions and encouargement. \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nAdvertisement is the primary way to get revenue or gathering interest for a product or service. Most of the current solutions use implicit placement of ads without regard of context or content, which has been shown to be an effective way of raising brand awareness. In order to well target the user, there is a need to analyze the content and relevant advertisements that exist nowadays.\nIn order to captivate their customers and make better decisions, companies have the need to analyze the presence of their brand in images and other types of content.\nBrand logos help to assess the identity of something or someone. Most solutions use graphic logos as the main target for brand detection since they often present distinct shapes and appear in high contrast regions. It is a challenge since they are often subject to multiple angles and sizes, varying lighting conditions and noise. \nTherefore, although they give perceptual distinctiveness, it is a challenge due to all the conditions they can appear in. \nMost previous works in this context have been based considerably on SIFT \\cite{Lowe2004}. This method provides representations and transformations to image gradients that are invariant to affine transformations and robust when facing lighting conditions and clutter. They can also detect stable salient points in the image across multiple scales, usually called key-points.\nThese previous works build models upon these representations to better capture specific patterns present in graphic logos. For instance, in~\\cite{RombergICMR2011}, a shape representation built with found key-points and their respective SIFT representation is proposed for scalable logo recognition in images. Similarly,  in~\\cite{romberg2013bundle}  bundles of SIFT features are built from local regions around each key-point to index specific graphic logo patterns.  Although there are many successful works covering a broad range of methods that have been applied in the past, recently, we assisted to the blossom of Convolution Neural Networks (CNNs) in the area of computer vision. \nCNNs have been producing an impressive impact on image recognition. CNNs are composed by many layers which resemble the simple and complex cells in the primary visual cortex~\\cite{Wiesel59}. Their structure is hierarchical and are specifically designed to recognize visual patterns directly from image pixels. The convolutional layers alternate with subsampling layers which may vary in number and size. The works in ~\\cite{Fukushima80}~\\cite{Fukushima2003},~\\cite{Schmidhuber96} and~\\cite{lecun1998gradient} have been pioneers  for the current CNNs that are researched today. Recently CNNs have been in the center of object recognition research. The rekindled interest in CNNs is largely attributed to \\cite{krizhevsky2012imagenet} CNN model, that showed significantly higher image classification accuracy on the 2012 ImageNet Large Scale Visual Recognition Challenge (ILSVRC)~\\cite{ILSVRC15}. Their success resulted from a model inspired by LeCun previous work and a few twists that enabled training with 1.2 million labeled images (\\textit{e.g.} GPU programming, max($x$,0) rectifying non-linearities and \\textit{dropout} regularization).\nOur work is focused on providing a way towards graphic logo detection by utilizing general region proposal algorithms and state-of-the-art object recognition systems. However, due to the lack of a large scale dataset with such graphic logos, training a modern system with CNNs from the scratch is mostly unfeasible, therefore, we use transfer learning and data augmentation to ameliorate this problem.\n\nThe remaining of this paper is organized as follows. Section~\\ref{sec:approach} emphasizes the contributions for automatic logo recognition with deep convolutional neural networks.  Section \\ref{sec:transfer} focuses on the transfer learning methodology which apart from fine tuning of parameters enables less costly architectures. \nSection~\\ref{sec:Fast} describes the architecture for object detection dubbed as Fast Region-based Convolutional Networks (FRCN). Section~\\ref{sec:experimental} presents the research design, the metrics and the discusses the performance results.\nFinally, in Section~\\ref{sec:conclusion} we address the main contributions of this paper and propose improvements that can be further explored. \n\\section{Our approach and contributions}\n\\label{sec:approach}\nWe extend and build upon general concepts and models of object recognition in images. Thus, we propose a solution that takes advantage of specific characteristics of graphic logos based on current cutting-edge research.\nWe focus on company graphic logos since they give a distinctive way of assessing a particular advertising campaign or brand.\nThe system we propose uses transfer learning to leverage image representations learned with CNNs on large-scale annotated datasets. The transferred representation leads to significantly improved results for brand detection. This is  very important in this study and we empirically show that it performs well and can be as well applied in other contexts and problems.\n\\section{Transfer learning with CNNs}\n\\label{sec:transfer}\nThe convolutional layer of a CNN consists of a set of learnable filters that activate with specific image features.\nEarlier layers of these networks learn to detect generic features like edges, and as we move into further layers, the learned features begin to model high level concepts and get more and more specific towards the original dataset. It is then possible to continue training and re-purpose, or transfer the weights so that they adapt to other datasets. Several authors have shown that fine-tuning these networks and transferring features, even from tasks that are not closely similar, can be advantageous when compared to training from the scratch \\cite{chatfield2014return}. The process takes advantage of the more generic learned features, given that both datasets `live' in a similar domain, in this case, object detection in images. This method enables training a large network with a small dataset without overfitting. \n\\section{Fast Region-based Convolutional Networks (FRCN)}\n\\label{sec:Fast}\nGraphic logos are not usually the main focus of an image, so they are often present in small sizes and partially occluded. Performing classification using the full image would introduce high amounts of background noise. Therefore, performing a more exhaustive search is required in this context. To prevent such an exhaustive search recent methods  employ region `proposals' to reduce the initial search space and perform detection using powerful CNN models~\\cite{frcn_girshick2015fast}.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.485\\textwidth]{frcn}\n\\end{center}\n   \\caption{FRCN model. Adapted from \\cite{frcn_girshick2015fast}.}\n\\label{fig:frcn}\n\\end{figure}\nRoss Girshick successfully proposed in~\\cite{frcn_girshick2015fast} a single-stage object detection model by using a region proposal algorithm and CNN features. This method performs object detection by classifying multiple regions in the image. This method has shown state-of-the-art performance in PASCAL VOC 2012~\\cite{everingham2010pascal}. Its high level representation is shown in Figure \\ref{fig:frcn} slightly adapted from~\\cite{frcn_girshick2015fast}.\nUsing category-independent region `proposals', the model performs classification in a reduced set of regions that are likely to have present an object; this helps to reduce the potential false-positives and the number of regions to analyze. In particular, it uses the approach in~\\cite{uijlings2013selective} based on a selective search for object recognition. This method generates regions using clustering of similar regions and several diversification methods, by capturing possible regions where an object might be, simply without the use of sliding windows. \nFeature extraction is performed using a CNN for each one of the proposed regions. These regions were then classified using a softmax classifier with a Fully Connected (FC) layer and post-processed using non-maxima suppression. After classification priority is given to regions with a high confidence value (local maxima). Duplicate region are iteratively removed with non-maximum suppression. Moreover, localization of the object is further refined using bounding-box regressors. \n\\section{Experimental Results}\n\\label{sec:experimental}\nIn order to assess our approach and the validity of our statements we conducted a set of experiments on a logo benchmark dataset. We provide further details of the dataset, indicate the performance measures, describe the augmentation pre-processing procedure, present  and discuss the results.\n\\subsection{Experimental Setup}\nWe evaluate the FRCN model for brand detection with the FlickrLogos-32 dataset~\\cite{dataset_website} which contains 32 graphic logo class images. The training set contains 320 images, the validation set 960 and the test set 960 images, each of which showing at least one single logo. Additionally, the test set contains 3000 images that do not contain any logo from the 32 classes.  A detailed characterization of the dataset is given in Table~\\ref{table:flickr32_dist}. \n\\begin{table}[htb]\n\\caption{Flickr32 Logo Dataset\\cite{RombergICMR2011} partitions\/subsets. Slightly adapted from \\cite{dataset_website}.}\n\\centering\n\\begin{tabular}{p{4em}|p{13em}|l|l}\nPartition & Description & Images\/class & \\#Images \\\\\n\\hline\nTraining set & Hand-picked images & 10 per class & 320 \\\\\n\\hline\nValidation set & Images showing at least a single logo under various views & 30 per class & 960  \\\\\n& Non-logo images & 3000 & 3000 \\\\\n\\hline\nTest set & Images showing at least a single logo under various views & 30 per class & 960\\\\\n& Non-logo images & 3000 & 3000 \\\\\\hline\n\\end{tabular}\n\\label{table:flickr32_dist}\n\\end{table}\nThe dataset is annotated with image binary masks. Testing is performed in images that do not only contain graphic logos, but also some amount of background noise.\n\\begin{figure}[htb]\n\\includegraphics[width=1.0\\linewidth]{visualsummary3-test}\\\\\n\\caption{Graphic Logos, FlickrLogos-32 dataset \\protect\\cite{RombergICMR2011}}\n\\label{fig:dataset}\n\\end{figure}\n\\subsection{Model Evaluation}\n\n\n\\paragraph{Intersection Over Union}\n\n\nAs mentioned in Section~\\ref{sec:Fast}, the method from\\cite{uijlings2013selective} is used to generate region `proposals' for each image during training. The \\acf{IoU} metric is used to evaluate which regions should be considered as a positive sample given the region bounding box overlap with the ground-truth object.\n\nThis metric is often used to judge correct object detections in images. Detections usually are considered correct when the area of overlap $a_0$ between the predicted bounding box and ground-truth exceeds 50\\% \\cite{everingham2010pascal} using Equation \\ref{eq:iou}.\n\n\\begin{equation}\n\ta_0 = \\frac{area \\left( B_p \\cap B_{gt} \\right)}{ area \\left( B_p \\cup B_{gt} \\right)}\n\\label{eq:iou}\n\\end{equation}\n\n\n\\paragraph{Non-maximum Suppression}\n\nSince we are classifying multiple region `proposals' in each testing image, many of the detected regions will be heavily overlapped, not only with the ground-truth object, but also with each other. Non-maximum Suppression is a technique that has been used in the edge thinning in the Canny edge detection algorithm \\cite{rcnn_GirshickDDM13}. In this context it is used to remove redundant regions that have a low score, but also merging multiple regions that indicate the same object that are given an high score by the classifier.\n\n\\paragraph{Mean Average Precision}\nWhile precision and recall are single values based on list of unranked predictions, \\acf{mAP} is often used in information retrieval. In image context, we look at a specific detection of a logo in an image. Since the retrieved list is ranked, it is desirable to consider the order that the returned images are presented and give more relevance to correct items at the top.\nTo obtain the value of \\ac{mAP}, P (precision) and R (recall) are then calculated at every position in the ranked list of images for each query. Average Precision is defined as:\n\\begin{equation}\n\tAverage\\,\\, Precision = \\sum_{k=1}^{N} P(k) \\cdot \\Delta R(k) \\nonumber\n\t\\label{eq:avep}\n\t\\end{equation}\nwhere $k$ is the rank in the list of retrieved documents, $N$ the total number of retrieved documents for this query, $P(k)$ the precision value at the $k$ point of the list and $\\Delta R(k)$ the change in recall value since the last point in the list.\nMean Average Precision (\\ac{mAP}) is the mean value of average precision values obtained across all queries or classes.\nThe Average Precision also summarizes the shape of the precision\/recall curve built from the method's ranked output. Most of the recent computer vision works use this metric for classification and detection.\n\\subsection{Pre-Processing}\nWe performed data augmentation by horizontally flipping the training images. Since the training set only contains $10$ images per class we also used the original validation set for training. In addition, as a separate experiment we randomly distort each image with a shear operation ([-5,5] degrees) and apply a slight shift in color values (3 percent) on each image.\nThe method from \\cite{uijlings2013selective} is used to generate region `proposals' for each image in the dataset. Selective Search in~\\cite{uijlings2013selective} first applies image segmentation and then performs bottom-up hierarchical grouping of the neighboring segments to generate regions. This step produces a varying number or regions depending on the image characteristics and the diversification methods when grouping segments. Defining different sizes for the initial segments is a way to diversify the produced regions. It is possible to generate regions from multiple initial sizes of segments.\nWe generate regions using Selective Search in two distinct modes, which we call Fast and Quality mode. These two modes differ on the number of initial segment sizes used for grouping. Quality mode uses four different segment sizes while Fast mode uses only two sizes during the initial grouping. For that reason, quality mode generates a significantly larger and more diversified set of regions than Fast mode.\nAs mentioned above, using this method will help to reduce background noise and guide the model toward regions where an object is probable to appear in. They are used not only during the testing phase but also during training, where these regions are sampled to fine tune the model.\n\\subsection{Tools}\nWe used the Caffe deep learning framework \\cite{jia2014caffe,caffe} implementation of the FRCN model~\\cite{edison} and present the main results yielded by using two pre-trained CNNs. The first is the Caffe implementation of the CNN described by \\cite{krizhevsky2012imagenet} called Caffenet, the second, is a larger model, which has the same depth but wider convolutional layers, VGG{\\_}CNN{\\_}M{\\_}1024, described in \\cite{chatfield2014return}. Both were pre-trained with the ILSVRC ImageNet dataset.\nWe refer to \\cite{krizhevsky2012imagenet,chatfield2014return} for more network architecture details.\n\\subsection{ Setup Summary}\n An overall summary of the experimental setup is given below.\n\\begin{itemize}\n\t\\item FlickrLogos-32 dataset\n\t\\begin{itemize}\n\t\t\\item Real-world images showing brand logos\n\t\t\\item $32$ different brand logos\n\t\t\\item $70$ images per class\/brand logo\n\t\t\\item $6000$ non-logo images\n\t\\end{itemize}\n\t\\item CNN models pre-trained with the ILSVRC ImageNet dataset (1000 categories and 1.2 million images)\n\t\\begin{itemize}\n\t\t\\item Caffenet [2] (BD-FRCN-M$_1$)\n\t\t\\item VGG{\\_}CNN{\\_}M{\\_}1024 [3] (BD-FRCN-M$_2$)\n\t\\end{itemize}\n\t\\item Caffe Deep Learning Framework\n\\end{itemize}\n\\subsection{Results and Discussion}\nIn this section we first present and discuss the results of the proposed model using \\ac{mAP} values. We analyse the detection and recognition F1-scores not only in images with graphic logos, but also in images with no logos.\nConvolutional Neural Networks are now considered a strong candidate for visual recognition tasks \\cite {razavian2014cnn}. The results we present herein and which will be described in this section, seem to corroborate that fact, moreover if we compare them to our previous efforts~\\cite{Oliveira2015}.\nA major concern from the beginning was the amount of data available. If we join the training set and validation set of the dataset, $40$ images of each class could be obtained for training. \n\nAs previously mentioned, pre-training a deep network with a similar domain dataset is advantageous. In this case, by using transfer learning we avoid having to build a new large scale dataset for our domain to train a deep network from the scratch.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{montage5s}\n\\caption{Top-5 confidence scores for several example images}\n\\label{fig:top-5}\n\\end{figure*}\nThe flexibility given by the original \\ac{FRCN} model is greatly suitable for the brand detection problem. Graphic logos appear in small regions in the image and since the first step is to generate object `proposals', the model gives an intuitive and modern way to solve this issue. Additionally, the model deals directly with background noise by using a special background class and also object `proposals' to reduce the number of regions that are processed in the image.\nMoreover, the model is able to detect and localize multiple instances of a graphic logo. As already mentioned, that functionality is helpful in situations where contextual advertisement insertion is needed.\nThe training process of the FRCN model jointly optimizes a softmax classifier and bounding-box regressors while adapting the CNN weights to our task of graphic logo detection. We start Stochastic Gradient Descent at a small learning rate of $0.001$, which allows fine-tuning of the CNN network to occur without losing the capabilities gained during pre-training. We fine-tune Caffenet and VGG{\\_}CNN{\\_}M{\\_}1024 during $80000$ iterations while saving a snapshot of the models each $10000$ iterations. We use these snapshots to analyze each model performance across the whole fine-tuning process. We alternatively designate the Caffenet model as M$_1$, and the larger model, VGG{\\_}CNN{\\_}M{\\_}1024 as M$_2$. We will abbreviate the name of our final Brand Detection model to BD-FRCN. The specification of the models BD-FRCN-M$_1$ and BD-FRCN-M$_2$ have been above indicated in the Setup Summary.\n\\begin{table*}\n\\small\\centering\n\\caption{Comparison of obtained \\ac{mAP} values with those of the work in \\cite{romberg2013bundle}. Results obtained at 60000 iterations.}\n\\begin{tabular}{l|l|l|l|l}\nMethod    & Learning  rate & Selective Search & \\ac{mAP} \\\\  \\hline\n\\ac{BoW} (tf-idf-sqrt, vocabulary-1000)\\cite{romberg2013bundle}     &            &                                     & 0.545      \\\\\nBundle Min Hashing (1p-wgc-ransac) \\cite{romberg2013bundle}                               &            &                                & 0.568      \\\\\nBD-FRCN-M$_2$                           & 0.001              & Fast             & 0.7347     \\\\\nBD-FRCN-M$_1$                                  & 0.001              & Fast             & 0.7314     \\\\\nBD-FRCN-M$_2$                        & 0.005              & Fast             & 0.7347     \\\\\nBD-FRCN-M$_1$                               & 0.005              & Fast             & 0.7314     \\\\\nBD-FRCN-M$_2$                        & 0.001              & Quality          & 0.6972     \\\\\nBD-FRCN-M$_1$                                & 0.001              & Quality          & 0.6898     \\\\\nBD-FRCN-M$_2$ - (Shear \\& Color)     & 0.001              & Fast             & 0.6941     \\\\\nBD-FRCN-M$_1$ - (Shear \\& Color)              & 0.001              & Fast             & 0.6912    \n\\end{tabular}\n\\label{table:map_comparison}\n\\end{table*}\n\n\n\nIn Figure \\ref{fig:top-5} we can observe that regions with a graphic logo are often assigned with a higher value of confidence than background regions. Most graphic logos tend to be simplistic and therefore contain less information than more complex ones. For example, the HP logo will blend more easily into background noise than the Starbucks logo since it has less characteristic features. To reduce this problem and subsequently the number of false positives, more training data is required. Given the fact that graphic logos are mostly presented as a flat object or imposed over other objects, this is a scenario where it would be possible to perform data-augmentation or procedurally generating new images for this purpose.\nThe softmax classifier will give a confidence value for each class, throughout all proposed regions. After reducing the set of classified regions using non-maximum suppression we will use the region with top confidence value to classify each image. As an example, Figure \\ref{fig:esso} shows the top-5 confidence scores produced by this system for a test image.\n\nTable \\ref{table:map_comparison} compares our results with other referenced methods, more specifically in~\\cite{romberg2013bundle}. It does also illustrate our results obtained with the approach from two sides: learning rate and selective search. In the former (i) learning rates are changed in the training and in the latter (ii) the idea is to generate window `proposals' in the pre-processing stage.  We can verify that having a higher base learning rate of $0.005$ is not beneficial for learning, since both models are overfitting at $10000$ iterations, whereas starting with a learning rate of $0.001$ achieves higher testing performance from the beginning.\nFrom the results, it is counter-intuitive that the Quality mode achieves worst performance as compared with the Fast mode. This could be explained by the difference of regions analyzed between the two modes. Although the Quality mode generates a larger and more diverse set of regions, they tend to be misclassified more often than in the Fast mode. \nIn summary, by using the quality version variant of Selective Search, it does not lead necessarily to better performance. This might be explained due to a higher number of windows (that do not contain a graphic logo) that have to be evaluated.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.375\\textwidth]{adidastest-rd}\n\\end{center}\n   \\caption{Top-5 confidence scores for an example image belonging to the Adidas class.}\n\\label{fig:esso}\n\\end{figure}\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{map_iter2}\n  \\caption{\\ac{mAP} values across iterations with each network and several configurations.}\n    \\label{fig:map}\n\\end{figure*}\nIn Figure \\ref{fig:map} we compare the test set \\ac{mAP} values during the fine-tuning process for the two networks and several configurations. This means that the values across iterations with each network and configuration are indicated. The choice to measure test performance across iterations is mostly due to the fact that we lost validation data by joining the training and validation set for training. It is possible to achieve high \\ac{mAP} if both precision and recall are high across detections.  As it can be seen, one of the configurations is performed with data augmentation.\n\\begin{figure}[h]\n  \\centering\n \t\\subfigure[Original]{\\includegraphics[width=0.175\\textwidth]{aug_orig} \n    \\subfigure[Variation]{\\includegraphics[width=0.175\\textwidth]{aug_var} }\n  \\caption{Example of image variations produced.}\n      \\label{fig:variations}\n\\end{figure}\nAdditionally to the data augmentation performed by randomly flipping each region, we performed a small experiment by randomly distorting each image with a shear and slightly shifting color values on the image. We can observe a higher average precision for some classes, such as the Google, Guinness and BMW, but this fact is not reflected across all classes. In practice we doubled the number of images on our training set, by saving each variation. Further study on this aspect is required. Figure \\ref{fig:variations} shows and example of such  variation.\n\nSince our model classifies each image based on regions, and we do not have explicit confidence scores for the no-logo class, we use the region on the image with top confidence value and a threshold value to infer if only background regions are present. If the top confidence value is below a certain threshold value, we classify that image as a no-logo image, instead of the predicted logo class.\n\\begin{figure}[!]\n\\begin{center}\n\\includegraphics[width=0.375\\textwidth]{thresholdsplot4}\n\\end{center}\n   \\caption{F1-scores for both BD-FRCN-M$_1$ and BD-FRCN-M$_2$.}\n\\label{fig:thresholds}\n\\end{figure}\n \\begin{figure}[!]\n  \\centering\n\\subfigure[Class: no-logo]{{\\includegraphics[width=0.20\\textwidth]{04780378432779_no-logo_test} }}  ~ \n\\subfigure[Class: no-logo]{{\\includegraphics[width=0.20\\textwidth]{047803444260953_no-logo_test} }}  ~            \n  \\caption{Examples of incorrect detections in the no-logo class.}\n      \\label{fig:no_logo_examples}\n\\end{figure}\nIntroducing the $3000$ no-logo images from the test set, we computed recognition F1-scores for both networks across learning rates iterations and configurations, for each threshold value on a range 0-1.\nSo far we have assumed that all images contain a graphic logo. We introduce a threshold value to deal with images without graphic logos (non-logo). Images with top confidence values below the threshold will be considered as having no logo present. Figure \\ref{fig:thresholds} shows consolidated F1-scores for both detection and recognition metrics and the two tested CNN models. Using a threshold value of $0.3$ we achieve the top recognition F1-score, although by setting an adequate threshold value still allows the model to maintain high recognition scores without losing the ability to perform detection between images with logos and no graphic logos (\\textit{e.g.} threshold $0.8$). \nWe can observe in Figure \\ref{fig:thresholds}, both metrics interacting. As we set higher threshold values the model starts to correctly identify more no-logo images and the detection score rises, at the same time, recognition scores start lowering since we start to incorrectly classify logo images as having no-logo. At high threshold values both scores start to go on a downward trend, since few top confidence values reside on that range.\n \nOur model is able to distinguish most of the background regions from logos. However under specific conditions ends up giving high confidence values for regions that do not contain a logo from the list of 32 classes.\nFigure \\ref{fig:no_logo_examples} show some of those images, for example, the Apple and Milka logo are often confused for white backgrounds and purple backgrounds, respectively. Graphic logos, that are present in the images and do not belong to the $32$ classes are also sometimes wrongly detected.\n \n Table \\ref{table:values} also shows the state-of-the-art results achieved by other authors on this specific dataset that we know of, and as we can see, our results with this model are better, specifically if we look at the F1 score. \nThe top recognition F1-score of $0.931$ was found using the BD-FRCN-M$_2$ model, with a base learning rate of $0.001$ at $40000$ iterations and a threshold value of $0.32$.\nThe BD-FRCN-M$_1$ model achieved a top recognition F1-score of $0.909$ with a base learning rate of $0.001$ at $30000$ iterations and with a threshold of $0.4$.\n\\begin{table}[thb]\n\\centering\n\\caption{Recognition scores}\n\\begin{tabular}{l|l|l|l}\n\\hline\nMethod                         & Precision & Recall    & F1        \\\\\n\\hline \\hline\nBrugman AlexNet-logos-3000~\\cite{brugman}     & 0.713     & 0.569     & 0.633     \\\\\nBrugman NIN-logos-3000~\\cite{brugman}  & 0.705     & 0.604     & 0.651     \\\\\nBrugman Caffenet-logos-3000~\\cite{brugman}    & 0.729     & 0.565     & 0.637     \\\\\nRomberg et al.~\\cite{RombergICMR2011} & 0.98      & 0.61      & 0.752     \\\\\nRomberg and Lienhart~\\cite{romberg2013bundle} & 0.99      & 0.832     & 0.904     \\\\\nBD-FRCN-M$_1$ (thresh 0.4)   & \\textbf{0.928} & \\textbf{0.891} & \t\\textbf{0.909} \\\\\nBD-FRCN-M$_2$ (thresh 0.81)  & \\textbf{0.987} & \\textbf{0.846} & \t\\textbf{0.911} \\\\\nBD-FRCN-M$_2$ (thresh 0.32)  & \\textbf{0.955} & \\textbf{0.908} & \t\\textbf{0.931} \\\\\n\\end{tabular}\n      \\label{table:values}\n\\end{table}\n\nAs a side note, the model was able to find some inconsistencies in the dataset. It is stated that no two graphic logos are present in the same image. However, Figure \\ref{fig:incongruences} shows two examples where the two top windows are correctly assigned to the class, but the top detection does not correspond to dataset ground truth.\n \\begin{figure}[h]\n  \\centering\n \t\\subfigure[Class: Tsingtao]{{\\includegraphics[width=0.175\\textwidth]{tsingtao} }}  ~\n    \\qquad\n    \\subfigure[Class: Heineken]{{\\includegraphics[width=0.175\\textwidth]{heineken} }}  ~ \n  \\caption{Examples of two different graphic logos in the image}\n      \\label{fig:incongruences}\n\\end{figure}\n\\section{Conclusion}\n\\label{sec:conclusion}\nA key contribution of this work has been the introduction of graphic logo detection using regions `proposals' and CNNs, by taking advantage of transfer learning.  \nWe experimented with a modern detection model and two CNNs pre-trained for a general object recognition task with abundant data and fine-tuned these networks for a task where the data is scarce - graphic logo detection -. \nBoth  proposed models perform better than those of the state-of-the-art performance almost out of the box, with a wide margin for improvement that we intend to explore further.\nIn the future, we will  extend this model by exploring specific ways to generate regions where the presence of a graphic logo is probable, instead of using a class agnostic method like Selective Search. We are also planning on exploring data augmentation techniques to generate realistic transformations of graphic logos. Saving each image variation is not feasible due to disk space constraints, so implementing a module that records this data during training is further required. Another future issue will be the identification of brand logos from video data.\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Acknowledgement}\nWe would like to express our gratitude to all members of\n the J-PARC Accelerator and Hadron Experimental Facility groups for their support.\n We also thank the KEK Computing Research Center for KEKCC\n and the National Institute of Information for SINET4.\n This work was supported by the Ministry of Education,\n Culture, Sports, Science, and Technology (MEXT) of\n Japan and the Japan Society for the Promotion of\n Science (JSPS) under the MEXT KAKENHI Grant No.~JP18071006,\n the JSPS KAKENHI Grants No.~JP23224007,\n and No.~JP16H06343, by the research\n fellowship program\n for postdoctoral scientists No.~17J02178, and\n through the Japan-U.S. Cooperative Research\n Program in High Energy Physics; the U.S. Department of Energy,\n Office of Science, Office of High Energy Physics,\n under Awards No.~DE-SC0006497, No.~DE-SC0007859, and\n No.~DE-SC0009798; the Ministry of Education and the Ministry\n of Science and Technology in Taiwan under Grants\n No.~104-2112-M-002-021, 105-2112-M-002-013 and\n 106-2112-M-002-016; and the National Research Foundation of Korea\n (2017R1A2B2011334 and 2019R1A2C1084552).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\nThere are four kilometre-scale interferometric gravitational-wave\n(GW) detectors currently in operation: the 4\\,km and 2\\,km\ninterferometers (IFOs) at the LIGO Hanford Observatory (LHO) (known\nrespectively as H1 and H2), the 4\\,km IFO at the LIGO Livingston\nObservatory (LLO) (known as L1), and the 3\\,km Virgo IFO (known as\nV1).  The LLO and LHO IFOs are currently conducting operations at\ntheir design sensitivity, and operate together with the 600\\,m GEO600\nIFO (known as G1) under the auspices of the LIGO Scientific\nCollaboration (LSC).  Virgo's first full science run commences in May\n2007.\n\nOne of the signals targeted by ground-based GW IFOs is a stochastic GW\nbackground (SGWB), which can be either of cosmological or\nastrophysical origin, in the latter case being produced by a\nsuperposition of unresolved sources.  The standard technique to search\nfor a SGWB looks for correlations in the outputs of multiple\ndetectors.  We describe in this paper how the inclusion of correlation\nmeasurements involving Virgo could improve the sensitivity of the\ncurrent LLO-LHO network.\n\n\\section{All-sky sensitivity at design}\n\n\\subsection{Observing Geometry}\n\nThe effect of a SGWB is to generate correlations in the outputs of a\npair of GW detectors, which can be described for an isotropic\nbackground in the Fourier domain by\n\\begin{equation}\n  \\label{eq:h1h2corr}\n  \\langle \\widetilde{h}_1^*(f)\\,\\widetilde{h}_2(f')\\rangle\n  = \\frac{1}{2}\\delta(f-f')\\,\\gamma_{12}(f)\\,S_{\\text{gw}}(f)\n  \\ .\n\\end{equation}\n\\begin{table}\n  \\centering\n  \\input{gammaDC}\n  \\caption{Limiting behaviour of overlap reduction\n    functions of detector pairs.\n    H refers to either of the IFOs at the LHO site, L to LLO, V to Virgo,\n    and G to GEO600.\n    At $f=0$, the ORF is determined by the alignment of the detectors.\n    The reduced inverse light travel time $(2\\pi T_{12})^{-1}$ gives\n    a characteristic frequency for the onset of ``high-frequency''\n    behaviour, which includes a sinc function of the ratio of $f$ to\n    that frequency.  However, the limiting form in \\eqref{e:gammalim}\n    also includes a geometric projection factor, leading to an overall\n    envelope $\\gamma^{\\text{env}}_{12}$ which is shown in the third row.\n    In particular, while the light-travel time $T_{HL}$ is less than\n    $T_{HV}$ or $T_{LV}$, the projection factor more than makes up\n    for this, which makes the mean amplitudes of $\\gamma_{\\text{HV}}(f)$\n    and $\\gamma_{\\text{LV}}(f)$\n    at high frequencies larger than that for $\\gamma_{\\text{HL}}(f)$.}\n  \\label{tab:overlap}\n\\end{table}\nThe raw correlation depends on the (one-sided) power spectral density\n$S_{\\text{gw}}(f)$ the SGWB would generate in an IFO with\nperpendicular arms, as well as the observing geometry.  The\ngeometrical dependence manifests itself via the overlap reduction\nfunction (ORF)\\cite{Flanagan:1993}, which can be written\nas\\cite{Whelan:2006}\n\\begin{equation}\n  \\label{e:gammadef}\n  \\gamma_{12}(f)={d_{1ab}}\\, {d_2^{cd}}\\,\n  \\frac{5}{4\\pi} \\iint d^2\\Omega_{\\hat{n}}\\,\n  P^{\\text{TT}\\hat{n}}{}^{ab}_{cd}\\,\n  e^{i2\\pi f\\hat{n}\\cdot(\\vec{r}_2-\\vec{r}_1)\/c}\n\\end{equation}\nwhere each IFO's geometry is described by a response tensor\nconstructed from unit vectors $\\hat{x}$ and $\\hat{y}$ down the two\narms\n\\begin{equation}\n  d^{ab} = \\frac{1}{2}(\\hat{x}^a \\hat{x}^b - \\hat{y}^a \\hat{y}^b)\n  \\ ,\n\\end{equation}\n$\\vec{r}_{1,2}$ is the respective interferometer's location and\n$P^{\\text{TT}\\hat{n}}{}^{ab}_{cd}$ is a projector onto traceless\nsymmetric tensors transverse to the unit vector $\\hat{n}$.  At zero\nfrequency, the ORF is determined entirely by detector orientation.\nThe LHO and LLO sites are aligned as nearly as possible given their\nseparation on the globe, so that $\\gamma_{HL}(0)=-0.89$.\nIn contrast, the Virgo and GEO600 sites are poorly oriented with\nrespect to one another, so $\\gamma_{GV}(0)=-0.08$.  However, the\nfrequency-dependence of the ORFs means that the situation is quite\ndifferent at frequencies above 40\\,Hz, where the IFOs are sensitive.\nIn particular, the amplitude of $\\gamma_{GV}(f)$ does not drop\nappreciably for $f$ below about $350\\un{Hz}$.\nFor the other pairs, the behaviour is determined by the\nhigh-frequency limiting form of the ORF,\n\\begin{equation}\n  \\label{e:gammalim}\n  \\gamma_{12}(f)\\longrightarrow\n  5\n  {d_{1ab}}\\,P^{\\text{TT}\\hat{s}_{12}}{}^{ab}_{cd}\\,{d_2^{cd}}\\,\n  \\mathop{\\rm sinc}\\nolimits(2\\pi f T_{12})\n  = \\frac{\\gamma^{\\text{env}}_{12}}{f} \\sin(2\\pi f T_{12})\n\\end{equation}\nwhere $T_{12}$ is the light travel time between the detector sites and\n$\\hat{s}_{12}$ is a unit vector pointing from one site to the other.\nWhile trans-Atlantic light travel times like $T_{\\text{HV}}$ and\n$T_{\\text{LV}}$ are greater than $T_{\\text{HL}}$, leading to more\noscillations in the ORF, the overall envelope\n$\n\\gamma_{12}^{\\text{env}} =\n(5{d_{1ab}}P^{\\text{TT}\\hat{s}_{12}}{}^{ab}_{cd}{d_2^{cd}}) \/ (2\\pi T_{12})\n$\nincludes geometric projection factors, which more than make up for\nthis discrepancy, as summarised in Table~\\ref{tab:overlap}.\nThe result is that in the full overlap reduction function\n(Fig.~\\ref{fig:overlap}) the typical amplitudes\n$\\abs{\\gamma_{\\text{HV}}(f)}$ and $\\abs{\\gamma_{\\text{LV}}(f)}$ are\nlarger than the typical $\\abs{\\gamma_{\\text{HL}}(f)}$ for\n$f\\gtrsim 200\\un{Hz}$.\n\\begin{figure}\n  \\begin{center}\n     \\includegraphics[width=3in,angle=0]{overlap}\n  \\end{center}  \n  \\caption{\n    The overlap reduction functions for pairs of detector sites.\n    Note that the ORF for the two LIGO sites goes off the scale of\n    this plot at 40\\,Hz, which is the ``seismic wall'' below which\n    LIGO data are too noisy to be of any use.  The proximity and\n    alignment of LLO and LHO makes HL the most favourable pair of\n    detector sites for observations below 150\\,Hz or so.  However,\n    the proximity of the Virgo and GEO600 sites means the GV ORF is\n    substantial out to higher frequencies, overcoming the\n    low-frequency suppression due to their poor alignment.  On the\n    other hand, as shown in Table~\\protect\\ref{tab:overlap}\n    the HV and LV ORFs, while they oscillate\n    rapidly with increasing frequency, do not decay as precipitously\n    as the HL ORF, making them more favourable\n    than HL (but less than GV)\n    for $f\\gtrsim200\\un{Hz}$.\n  }\n  \\label{fig:overlap}\n\\end{figure}\n\n\\subsection{Definition of Sensitivity}\n\nThe strength of an isotropic stochastic background can be written in\nterms of the one-sided power spectral density $S_{\\text{gw}}(f)$ it\nwould generate in an IFO with perpendicular arms.\n\n\nThe standard cross-correlation method seeks to measure the amplitude\n$S_R=S_{\\text{gw}}(f)\/\\mc{S}(f)$ of a background whose\n$S_{\\text{gw}}(f)$ is assumed to have a specified shape $\\mc{S}(f)$.\nGiven co\\\"{\\i}ncident data between times $t_1$ and $t_2$ from\ndetectors with one-sided noise power spectral densities (PSDs)\n$P_{1,2}(f)$, we can make an optimally-filtered cross-correlation\nstatistic\n\\begin{equation}\n  \\label{e:CC}\n  Y = \\int_{t_1}^{t_2} \\int_{t_1}^{t_2} s_1(t)\\,Q(t-t')\\,s_2(t)\\,dt\\,dt'\n\\end{equation}\nwith the optimal filter defined by its Fourier transform\n\\begin{equation}\n  \\widetilde{Q}(f) = \\mc{N}\\frac{\\gamma_{12}(f)\\mc{S}(f)}{P_1(f)P_2(f)}\n\\end{equation}\nand $\\mc{N}$ chosen so that $\\langle Y\\rangle=S_R$.  If the geometric\nmean of the noise PSDs is large compared to $S_{\\text{gw}}(f)$, the\nexpected variance of the statistic will be\n\\begin{equation}\n  \\sigma^2 = \\frac{1}{2T} \n  \\left(\n    \\int_{0}^{\\infty}\n    \\frac{\\left[\\gamma_{12}(f)\\,\\mc{S}(f)\\right]^2}{P_1(f)\\,P_2(f)}\n  \\right)^{-1}\n\\end{equation}\nwhere $T=t_2-t_1$ is the duration of the data analysed.\nThe squared signal-to-noise ratio of the standard\ncross-correlation statistic will thus be\n\\begin{equation}\n  \\mathrm{SNR}^2 := \\frac{\\langle Y\\rangle^2}{\\sigma^2}\n  = 2T\\,S_R^2\n  \\int_{0}^{\\infty}\n  [\\mc{S}(f)]^2\n  \\mc{I}_{12}(f)\n  \\ df\n  \\ ,\n\\end{equation}\nwhere we have defined a ``sensitivity integrand'' which illustrates\nthe contribution to the sensitivity of different frequencies:\n\\begin{equation}\n  \\mc{I}_{12}(f) =\n  \\frac{[\\gamma_{12}(f)]^2}{P_{1}(f)\\,P_{2}(f)}\n\\end{equation}\nWe plot $\\mc{I}(f)$ for several pairs of detectors in\nFigure~\\ref{fig:sensints}, using the design sensitivities\nin\\cite{ligonoise,virgonoise,geonoise}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=2.5in,angle=0]{sensints}\n  \\includegraphics[width=2.5in,angle=0]{sensints_zoom}\n  \\caption{Sensitivity integrands $\\mc{I}(f)$ for pairs of detector sites,\n    where pairs like HL include e.g., the combined sensitivity of\n    H1-L1 and H2-L1.  To get an overall sensitivity, these\n    need to be multiplied by the square of the shape\n    of the SGWB spectrum $S_{\\text{gw}}(f)$.    \n    The HL pair is the\n    most sensitive for flat and low-frequency spectra, but as shown in\n    the closeup at the right, above 200\\,Hz, pairs involving Virgo are\n    more sensitive when all detectors are operating at their design\n    sensitivities.    \n    The GH and GL pairs are not shown, since the GEO600 noise spectrum\n    \\cite{geonoise} means that only the GV pair contributes\n    significantly to the overall sensitivity at these frequencies.}\n  \\label{fig:sensints}\n\\end{figure}\nAs shown in \\cite{Allen:1999}, the optimal method for combining\ncorrelation measurements from different detector pairs is the same as\nthat for combining measurements from different times: average the\npoint estimates $Y$ with a relative weighting of $\\sigma^{-2}$, and\nthe resulting variance will be the inverse of the sum of the\n$\\sigma^{-2}$ values.  This produces a\nsensitivity integrand which is the sum of the integrands for\nindividual pairs:\n\\begin{equation}\n  \\mc{I}(f) = \\sum_{\\text{pair}} \\mc{I}_{\\text{pair}}(f)\n  \\ .\n\\end{equation}\nAn immediate application of this is to define sensitivity integrands\nthat combine pairs involving H1 and H2, e.g.,\n$\\mc{I}_{\\text{HL}}=\\mc{I}_{\\text{H1,L1}}+\\mc{I}_{\\text{H2,L1}}$.\nThis is the same as using the spectrum of an optimally combined H\npseudo-detector as described in \\cite{Lazzarini:2004}.\n\nFigure~\\ref{fig:net_sensints} shows the combined sensitivity for four\nnetworks of detectors operating at design sensitivity: the existing\nH-L network, an H-L-G network in which GEO600 is also operating at\ndesign sensitivity, and H-L-V and H-L-V-G networks which also include\na design-sensitivity Virgo.  The H1-H2 pair is not included in\nthese networks, because the presence of correlated environmental noise\nnecessitates special treatment of this pair \\cite{Fotopoulos:2006}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=2.5in,angle=0]{net_sensints}\n  \\includegraphics[width=2.5in,angle=0]{net_sensints_zoom}\n  \\caption{Combined sensitivity integrands for networks of detectors.\n    In each case a network including ``H'' includes correlations between\n    both H1 and H2 and the detectors at other sites.  As\n    the closeup on the right shows, addition of the LV, HV and GV pairs\n    to the HL network\n    increases sensitivity to backgrounds with significant power above\n    200\\,Hz.}\n  \\label{fig:net_sensints}\n\\end{figure}\n\nSince the power in the faintest detectable ``white'' stochastic\nbackground is proportional to the square root of the area under the\nsensitivity integrand, we see that the addition of Virgo to the\nLHO-LLO network would be most useful in improving sensitivity to a\nnarrow-band background peaked above 200\\,Hz, or to one whose spectrum\nrises with increasing frequency.  As an illustration,\nTable~\\ref{tab:network_sens} shows for several detector networks the\nfaintest detectable background with constant $S_{\\text{gw}}(f)$ in a\n100\\,Hz band, assuming one year of observation time and an SNR\nthreshold of 3.29, associated with 5\\% false alarm and false dismissal\nrates.\n\\begin{table}\n  \\centering\n  \\caption{Smallest detectable band-limited background using each of\n    the detector networks defined in Fig.~\\ref{fig:net_sensints}.  In\n    each case, this is the strain power spectrum, in units of\n    $10^{-48}\\un{Hz}^{-1}$, that could be detected with 5\\% false alarm\n    and 5\\% false dismissal rates, using one year of co\\\"{\\i}ncident\n    data at design sensitivity.}\n  \\input{network_sens}\n  \\label{tab:network_sens}\n\\end{table}\n\n\\section{Simulations}\n\n\\label{s:simulations}\n\nTo test cross-correlation analyses of LIGO and Virgo data, we injected\na simulated SGWB signal into simulated LIGO and Virgo noise.  The\nsimulated noise data were the 24 hours of H1, H2, L1, and V1 data, all\nat nominal design sensitivity, initially generated for simulated\nsearches for GW bursts and inspiralling compact object binaries\n\\cite{LVBurst,LVInspiral}, known as ``project 1b''.  We chose a\nspectral shape designed to highlight the performance of the LV and\nHV pairs at $f\\gtrsim200\\un{Hz}$, but which corresponds to a model of\nan astrophysical SGWB.  The spectrum we used\nis associated with the superposition of the tri-axial\nemission from the extra-galactic population of spinning magnetars with\ntype I superconducting interior, as described in model B of\n\\cite{Regimbau:2006}, but updated using the star formation history of\n\\cite{Hopkins:2006}.\n\nIn Figure~\\ref{fig:magnetar} we show the\nspectrum and the associated sensitivity integrand in the corresponding\ndetectors.\nSince this spectrum rises with increasing frequency up to about\n400\\,Hz, it is useful for illustrating the utility of a network\ninvolving Virgo to search for a broad-band astrophysical source.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=2.5in,angle=0]{magnetar_spectrum}\n  \\includegraphics[width=2.5in,angle=0]{magnetar_sensints}\n  \\caption{The magnetar spectrum used to generate simulated signals,\n    and the associated integrand for squared signal-to-noise-ratio.\n    Note that since $S_{\\text{gw}}(f)$ increases approximately\n    linearly with frequency up to about 400\\,Hz, this spectrum, while\n    broad-band, tends to favour the higher frequencies where LIGO-Virgo\n    detector pairs are more sensitive.  Beyond 400\\,Hz, the spectrum\n    is no longer linear, but we show the SNR that would result from\n    attempting to detect it with a $\\mc{S}(f)\\propto f$ filter; the\n    corrections are negligible below about 500\\,Hz, and still small\n    throughout the frequency range displayed.  Note that integrating\n    the area under the H-L-V curve on the right still gives an SNR\n    below $10^{-6}$ from 24 hours of data, so we scale up the injected\n    strain signal by a factor of several thousand in the simulations\n    described in Section~\\protect\\ref{s:simresults}.}\n  \\label{fig:magnetar}\n\\end{figure}\nSince the model signal would be too weak to detect with\nfirst-generation interferometers, we scale up the signal strength,\ninjecting a signal with the same spectral shape, but a much\nlarger amplitude, in our simulations.\n\n\\subsection{Simulation algorithm}\n\nThe problem of simulation of the signal in a pair of detectors due to\nan isotropic and Gaussian SGWB has been considered previously in e.g.,\n\\cite{Allen:1999,Bose:2003,LLOALLEGRO}.  For this work we generalise\nthat to a network of $N$ GW detectors.\\footnote{In our case, to\n  simulate signals in H1, H2, L1, and V1, $N=3$, because H1 and H2\n  have the same response tensor and therefore the same simulated GW\n  signal can be used for both of them.}  We need to satisfy\n\\eqref{eq:h1h2corr} for each pair of detectors; treating $\\{h_A(f)\\}$\nas the elements of a column vector $\\mathbf{\\widetilde{h}}(f)$ and\n$\\{\\gamma_{AB}(f)\\}$ as the elements of a real, symmetric\nmatrix\\footnote{For non-isotropic backgrounds, the ORF is complex\n  rather than real, and more care must be taken with the definition of\n  the Hermitian matrix $\\boldsymbol{\\gamma}(f)$.}\n$\\boldsymbol{\\gamma}(f)$, we can write this as a matrix equation\n\\begin{equation}\n  \\label{eq:covariance}\n  \\langle \\mathbf{\\widetilde{h}}(f)\\,\\mathbf{\\widetilde{h}}(f')^\\dagger \\rangle\n  = \\frac{1}{2}S_{\\text{gw}}(f) \\boldsymbol{\\gamma}(f) \\delta(f-f')\n\\end{equation}\nIf we can define a matrix $\\boldsymbol{\\beta}(f)$ which factors\n$\\boldsymbol{\\gamma}(f)$:\n\\begin{equation}\n  \\label{eq:factorise}\n  \\boldsymbol{\\gamma}(f)=\\boldsymbol{\\beta}(f)\\boldsymbol{\\beta}(f)^\\dagger\n  \\ ,\n\\end{equation}\nthen we can generate $N$ independent white noise data streams\n$\\{\\widetilde{\\eta}_A(f)\\}$ which satisfy\n\\begin{equation}\n  \\langle \\boldsymbol{\\widetilde{\\eta}}(f)\n  \\,\\boldsymbol{\\widetilde{\\eta}}(f')^\\dagger \\rangle\n  =  \\boldsymbol{1} \\delta(f-f')\n\\end{equation}\nand then convert them into the desired coloured correlated data streams\nvia\n\\begin{equation}\n  \\label{eq:fdsim}\n  \\mathbf{\\widetilde{h}}(f) = \\sqrt{\\frac{S_{\\text{gw}}(f)}{2}}\n  \\boldsymbol{\\beta}(f)\\boldsymbol{\\widetilde{\\eta}}(f)\n\\end{equation}\nFor a given $\\boldsymbol{\\gamma}(f)$, there are different choices of\n$\\boldsymbol{\\widetilde{\\eta}}(f)$ which achieve the factorisation\n\\eqref{eq:factorise}.\n\nSince \\eqref{eq:covariance} is a covariance matrix, it is positive\nsemi-definite, from which it follows (since $S_{\\text{gw}}(f)>0$) that\n$\\boldsymbol{\\gamma}(f)$ is positive semidefinite as well.\n$\\boldsymbol{\\gamma}(f)$ could have one or more zero eigenvalues in\nthe presence of linear dependence between detector outputs.  A\npractical example of this is two detectors sharing the same geometry\nand location, such as H1 and H2, are included in the\nnetwork.\\footnote{Less trivial examples can be constructed, for\n  example three detectors in the same location and in the same plane.}\nWe avoid this problem by generating a simulated signal for H1 and\ninjecting it into both H1 and H2.\n\nIn the generic case where $\\boldsymbol{\\gamma}(f)$ is positive\ndefinite, we can make the straightforward choice of the Cholesky\ndecomposition\\cite{Golub:1989}, in which $\\boldsymbol{\\beta}(f)$ is a\nlower diagonal matrix.  In the case $N=2$, this reduces to the form\nused in e.g.,\\cite{Allen:1999}.  For $N=3$, if the diagonal elements\nof $\\boldsymbol{\\gamma}(f)$ are unity\\footnote{This is the case for\n  interferometers with perpendicular arms, but \\emph{not} for GEO600\n  or for resonant bar detectors; see \\cite{Whelan:2006}}, the explicit\nform is\n\\begin{equation}\n  \\boldsymbol{\\beta}(f) =\n  \\left(\\begin{array}{ccc}\n      1 & 0 & 0\\\\\n      \\gamma_{12} & \\sqrt{1-\\gamma_{12}^{2}} & 0 \\\\\n      \\gamma_{13} \n      & \\frac{\\gamma_{23}-\\gamma_{12}\\gamma_{13}}{\\sqrt{1-\\gamma_{12}^{2}}}\n      & \\sqrt{\\frac{1+2\\gamma_{12}\\gamma_{13}\\gamma_{23}-\\gamma_{12}^{2}-\\gamma_{13}^{2}-\\gamma_{23}^{2}}{1-\\gamma_{12}^{2}}}\\end{array}\\right)\n\\end{equation}\nHowever in practice we can simply use a fast iterative algorithm for\nthe Cholesky decomposition.\n\nOther factorisation strategies which treat the different detectors\nmore symmetrically (e.g., defining\n$\\boldsymbol{\\beta}(f)=\\boldsymbol{\\Lambda}(f)^{1\/2}\\mathbf{U}(f)$\nwhere $\\boldsymbol{\\Lambda}(f)$ is the diagonal matrix of eigenvalues\nof $\\boldsymbol{\\gamma}(f)$ and $\\mathbf{U}(f)$ is the matrix\nconstructed from the corresponding eigenvectors) may be more\ndemanding in terms of computational power, but more numerically stable\nwhen correlations between the detectors are large and off-diagonal\nelements of $\\boldsymbol{\\gamma}(f)$ are comparable to unity. In\nparticular, this strategy can deal directly with the case when\n$\\boldsymbol{\\gamma}(f)$ has one or more zero eigenvalues.\n\n\\subsection{Filtering strategy}\n\nThe continuous frequency-domain idealisation \\eqref{eq:fdsim} needs to\nbe applied with care to finite stretches of real detector data.  In\nthe time domain, the multiplication \\eqref{eq:fdsim} amounts to a\nconvolution\n\\begin{equation}\n  \\mathbf{h}(t) = \\int_{-\\infty}^{\\infty}\n  \\mathbf{K}(t-t')\\,\\boldsymbol{\\eta}(t')\\,dt'\n\\end{equation}\nwith a kernel which is the inverse Fourier transform of\n\\begin{equation}\n  \\mathbf{\\widetilde{K}}(f) =\n  \\sqrt{\\frac{S_{\\text{gw}}(f)}{2}} \\boldsymbol{\\beta}(f)\n  \\ .\n\\end{equation}\nIf the time-domain kernel $\\mathbf{K}(\\tau)$ is negligible outside the\ninterval $-\\tau_{-} < \\tau < \\tau_{+}$, we can use the standard\noverlap-and-add strategy to generate a continuous stream of\ntime-series data, as illustrated in Figure~\\ref{fig:OverlapAdd}:\n\\begin{figure}\n  \\includegraphics[width=1\\columnwidth]{OA}\n  \\caption{\\label{fig:OverlapAdd}The overlap-and-add algorithm.}\n\\end{figure}\n\\begin{enumerate}\n\\item Generate a sequence of ``buffers'' of white noise data for each\n  of the $N$ detectors, each of length $T$.\n\\item Convolve each buffer with the kernel $\\mathbf{K}(\\tau)$ to\n  obtain a time series of length $\\tau_{-}+T+\\tau_{+}$, with an\n  associated start time $\\tau_{-}$ before the start and end time\n  $\\tau_{+}$ after the end of the original buffer.  (This is most\n  naturally done in the frequency domain, zero-padding the white noise\n  by $\\tau_{-}$ at the beginning and $\\tau_{+}$ at the end, then\n  Fourier transforming and multiplying by $\\mathbf{\\widetilde{K}}(f)$\n  before inverse-Fourier-transforming.)\n\\item Add together the processed data buffers, overlapping by\n  $\\tau_{+}$ on one end and $\\tau_{-}$ on the next, producing\n  correlated coloured time-series data of duration $T$ times the number\n  of buffers, plus transients of $\\tau_{-}$ at the beginning and\n  $\\tau_{+}$ at the end, which are discarded.\n\\end{enumerate}\n\nThis strategy was implemented in code based on Virgo's ``Noise\nAnalysis Package'' ({\\sc nap})~\\cite{NAP}.\n\n\\section{Analysis of simulated data}\n\\label{s:simresults}\n\nThe continuous signals described in Section~\\ref{s:simulations} were injected\ninto the ``project 1b'' simulated noise and the resulting time series\nanalysed using the {\\sc matapps} stochastic analysis code developed by the\nLSC. \\cite{matapps}.\nIn particular, the cross-correlation \\eqref{e:CC} was performed in the\nfrequency domain without any need to resample the time-series data,\nusing a variation of the method described in \\cite{Whelan:2005} and\napplied in \\cite{LLOALLEGRO}.\nSeveral simulation runs were performed in which\nsame set of simulated signals were injected into the four data\nstreams, scaled up by a different factor for each run.  The results of\ntwo of those simulations are shown here.  In Figure~\\ref{fig:p1b} we\nplot the individual point estimates and error bars in each of the five\ndetector pairs (every combination except for H1-H2).  The amplitude\nmeasure $S_R$ quoted is the one-sided strain PSD at 200\\,Hz.\n\\begin{figure}\n  \\begin{center}\n     \\includegraphics[width=3in,angle=0]{project1b_results_pairs}\n  \\end{center}  \n  \\caption{The individual point estimates and one-sigma error bars\n    with signals injected at two different levels.  The strength of\n    the injection is shown on the vertical axis, as a strain power at\n    200\\,Hz, while the estimate of this quantity from each detector\n    pair is on the horizontal axis.  The error bars are drawn\n    vertically for ease of reading, and because they are frequentist\n    error bars as quoted.  Note that the signals are seen at\n    comparable strength in all five pairs of detectors.}\n  \\label{fig:p1b}\n\\end{figure}\nThe optimal filter used assumed the shape $\\mc{S}(f)\\propto f$,\nalthough the injected signals only had that form below about 400\\,Hz.\nThe analysis was done over a frequency band 50--500\\,Hz to avoid\ndifficulties arising from this mismatch, as illustrated in\nFigure~\\ref{fig:magnetar}.\n\nThe optimal combination of these results is shown in\nTable~\\ref{tab:p1b}, both for the full network and for the network\nconsisting only of the two LHO-LLO pairs.  Our error bars are reduced\nby 15\\% via the inclusion of the LIGO-Virgo pairs.  Note that although\nthis source has more power at the intermediate frequencies favoured by\nthe LIGO-Virgo pairs, it is still broad-band.  A narrow-banded source\nwith most of its power above 200\\,Hz would favour LIGO-Virgo pairs to\nan even greater extent.\n\\begin{table}\n  \\input{project1b_results}\n  \\centering\n  \\caption{The values of $S_{\\text{gw}}(200\\un{Hz})$ calculated from\n    the simulated project 1b data, with associated one-sigma error bars,\n    for the H-L network consisting of the H1-L1 and H2-L1 pairs, and\n    for the H-L-V network consisting of those plus the H1-V1, H2-V1,\n    and L1-V1 pairs.  We see that including Virgo in the network reduces\n    our error bars for this broad-band astrophysical spectrum by 15\\%.}\n  \\label{tab:p1b}\n\\end{table}\n\n\\section{Conclusions and Outlook}\n\nWe have demonstrated how the inclusion of LIGO-Virgo and possibly\nGEO600-Virgo detector pairs can enhance the sensitivity of the global\nGW detector network to an isotropic background of gravitational waves,\nparticularly at frequencies above 200\\,Hz.  As a practical\nillustration, we have adapted and applied pipelines for generating\ncorrelated simulated signals in the LSC and Virgo detectors, and for\nanalysing co\\\"{\\i}ncident data via the standard cross-correlation\ntechnique.  The specific astrophysical model we used\n(which was chosen because its frequency spectrum was peaked at\nfrequencies where LIGO-Virgo pairs at design will be more sensitive\nthan LLO-LHO pairs)\nhad to have its\namplitude increased to be detectable by any pair of first-generation\nIFOs.  Nonetheless, the exercise illustrates\nhow multiple detector pairs can be\nused to discover an ``unexpected'' background.\n\nVirgo is not yet at its nominal design sensitivity, but has improved\nits sensitivity markedly over the past year, and its\nfirst full science run starts in May 2007, to be analysed\nin conjunction with the end of LIGO's S5 run.\n\n\n\\ack\n\nThe authors would like to thank their colleagues in the LIGO\nScientific Collaboration and the Virgo project.  JTW gratefully\nacknowledges Loyola University New Orleans and the University of Texas\nat Brownsville.\nThis work was supported by the National Science Foundation under grant\nPHY-0300609 and by the Max-Planck-Society.\nThis paper has been assigned LIGO Document Number\n{LIGO-P070028-03-Z} and AEI document number AEI-2007-017.\n\n\\section*{References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nType Ia supernovae (SNe Ia) result from the explosions of carbon-oxygen (CO) white dwarfs \n\\citep{Fow60}. Their extremely high and very similar peak luminosities\nhave made SNe Ia important standard candles for measuring distances in the Universe, and therefore \nestimating cosmological parameters\ncritical for our understanding of its evolution \\citep{Per98,Sch98,Rie01}.\nThey also play a critical role in nucleosynthesis, as contributors to the abundances of both \niron-peak (titanium to zinc) nuclei and intermediate mass (silicon to calcium) elements. \n\nDespite the widespread use of and interest in SNe Ia, their explosion mechanism is not yet fully \nunderstood, and their progenitor systems have not been unambiguously identified. We \nhave chosen to explore the single-degenerate scenario of SNe Ia, which involves the growth of a \nCarbon-Oxygen white dwarf toward the Chandrasekhar limit in a binary system and its ultimate \nthermonuclear ignition \nand disruption. The initial phases of the explosion involve a subsonic thermonuclear flame \n(deflagration) which gives rise to the expansion of the white dwarf prior to the initiation of a \nsupersonic burning front (detonation wave) \\citep{Kho91, Hof95, Nie97}. The density of the white \ndwarf decreases as a consequence of its expansion. Thus, the detonation burns material at \nlower densities, producing higher abundances of intermediate mass elements \\citep{Hil00}. \nAccording to observations, approximately 0.4 solar masses of intermediate mass \nelements are synthesized during the explosion \\citep{Whe90}. A variety of models\ninvolve the scenario of a deflagration phase followed by a detonation phase, such as the \ndeflagration-to-detonation transition (DDT)\\citep{Kho91,Gam04}, the pulsating reverse detonation \n(PRD) \\citep{Bra06,Bar09,Bra09}, or the gravitationally confined detonation (GCD) \\citep{Ple04, Tow07, Jor08, Mea09, Jor12}. These models differ primarily in the method by which the deflagration \nphase leads to a detonation wave. A common feature of the models, however, is that all of them \ninvolve the \npropagation of the detonation wave through a white dwarf that is either expanding or contracting, \nwhere the stellar velocity profile depends on both time and space. The detonation wave thus \npropagates into material that is not at rest. \n\nIn this paper, we explore the post-detonation expansion of the white dwarf and its consequences for \nthe production of $\\alpha$-particle nuclei, including intermediate-mass nuclei and $^{56}$Ni, using \ncentral and off-center detonation models with different velocity profiles at \ndetonation\\footnote{A related work is the study by \\citet{Gar07} of the nucleosynthetic yield produced by a subsonic deflagration wave in the low-density outer layers of an expanding white dwarf.}. \nWe describe the details of the numerical methods used in our simulations  \nin $\\S  \\ref{Numerical Methods}$. We performed a total of nine simulations; we give the \nsetups for these simulations in $\\S \\ref{Model Setup}$. We present our results in $\\S \\ref{Results}$,  \nfocusing primarily on the central explosion models.  Our results include the detonation structure for \nnon-static stars ($\\S \\ref{Detonation Structure for Non-Steady \nStars}$), the expansion of the flow at different time scales behind the detonation \nand the nucleosynthesis using the thermal properties of the flow ($\\S  \\ref{The Effect of Expansion \nTime \nScales on Nucleosynthesis}$), and an empirical relation between certain properties of the detonation \nwave and the final nucleosynthetic yields of $\\alpha$-particle nuclei, including $^{56}$Ni, in the \ndensity range of  \n$5 \\times 10^{6}  \\sim 2.5 \\times 10^{7} g \\ cm^{-3}$ ($\\S \\ref{An Empirical Relation}\n$). We present the results for the off-center detonation models \nin $\\S \\ref{Off-Center Detonation}$. We conclude with a \nsummary of our findings and of their implications for our understanding of the consequences of the \ndetonation under the conditions appropriate for a pulsating white dwarf environment ($\\S  \n\\ref{Conclusion}$).\n    \n\\section{Numerical Methods}\n\\label{Numerical Methods}\n\\subsection{FLASH}\n\nWe performed one-dimensional spherically-symmetric simulations using the FLASH code \n\\citep{Fry00}. FLASH is  a modular, block-structured Eulerian code with adaptive mesh refinement \ncapabilities that solves Euler's equations for compressible gas dynamics with a directionally split \npiecewise parabolic method \\citep{Col84}. PPM is particularly well-suited to flows \ninvolving discontinuities, such as detonation waves, and allows handling of non-ideal \nequations of state (EOS) \\citep{Col85}. The implementation of the EOS uses a tabular \nHelmholtz free energy EOS appropriate for stellar conditions encountered here including \ncontributions from blackbody radiation, ions, and electrons of an arbitrary degree of degeneracy\n \\citep{Tim00}. \n \nFor the energetics required to treat detonation waves, we track three distinct burning stages: (1) the \nburning of carbon to O, Ne, Na, Mg, and $\\alpha$-particles; (2) the subsequent burning of oxygen to a \nQuasi-Static Equilibrium (QSE) state comprised predominantly of Si group \nelements; and (3) if the temperature is high enough, the evolution of the system to Nuclear Statistical \nEquilibrium (NSE), which at the low entropies found in Chandrasekhar mass white dwarf, is primarily of comprised iron-group elements. The energy \nreleased due to carbon burning is explicitly calculated using the carbon fusion rate of \\citet{Cau88}, \nwhile the energy released during Oxygen and Si-group burning are \ncontrolled by the timescales for reaching the QSE state and NSE respectively. \nAdditional details concerning the energetics are presented in \\citet{Cal07}, \\citet{Tow07}, \n\\citet{Mea09}, and Seitenzahl et al. (2009). \n\nFor the hydrodynamics, we use the the conservation equations for the reacting gas flow together with \ntwo source terms: gravity and nuclear reactions. A detonation wave is a form of combustion \nin which a shock wave propagates into the fluid, heating it to high temperatures and triggering nuclear \nreactions. The energy that is released supports the detonation wave. \nSince the detonation wave propagates supersonically, molecular diffusion, thermal \nconduction, and viscous effects are negligible. Therefore, the simplified set of the Euler equations \nfor the reacting gas flow is:\n\n\\begin{equation}\n\\frac{\\partial \\rho }{\\partial t} \\ + \\ \\mathbf{\\triangledown} \\cdot (\\rho \\mathbf{v}) \\ = \\ 0\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\rho \\mathbf{v}}{\\partial t} \\ + \\ \\mathbf{\\triangledown} \\cdot (\\rho \\mathbf{vv}) \\ +\n\\mathbf{\\triangledown} P= \\ \\rho \\mathbf{g}\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\rho E}{\\partial t} \\ + \\ \\mathbf{\\triangledown} \\cdot [(\\rho E \\ + P)\\mathbf{v}] = \\ \\rho \n\\mathbf{v\\cdot g} + \\sum_{l}\\rho q_{l}\\dot{\\Phi _{l}}\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\rho \\Phi_{l}}{\\partial t}\\ +\\ \\mathbf{\\triangledown} \\cdot (\\rho\\Phi_{l}\\mathbf{v})\\ = \\ \\rho\n\\dot{\\Phi _{l}},\n\\end{equation}\nwhere $\\rho$ is the fluid density, $\\mathbf{v}$ is the fluid velocity, $P$ is the pressure, $E$ is the \nsum of the internal energy $\\epsilon $ and kinetic energy per unit mass with $E \\  = \\ \\epsilon \\ + \\ \n\\frac{\\mathbf{v}^2}{2}$, $\\mathbf{g}$ is the gravitational acceleration, $\\Phi_{l}\\ (l = 1, 2, 3)$ is a \nreaction progress variable that tracks the three burning stages (C-, O-, and Si-burning), \n$\\dot{\\Phi _{l}}$ is the reaction rate, and \\textit{$q_{l}$} is the corresponding \nmass-specific energy release, (see \\citet{Cal07}; \\citet{Tow07}; \\citet{Sei09} for further details).\nNuclear burning is suppressed within the shock to prevent instabilities. We performed the simulations \nusing a one-dimensional, spherically symmetric computational\ndomain with radius 22,016 km.  We employed twelve levels of refinement, which\ngave a maximum spatial resolution of 0.5 km during the simulations of the detonation \nphase.  \n\n\\subsection{Libnucnet}\n\\label{Libnucnet}\nThe network code used for the post processing is a version of the network code based on \nlibnucnet, a library of C codes for storing and managing nuclear reaction networks \\citep{Mey07}. \nThe network contains 230 nuclear species ranging from neutrons, protons, and $\\alpha$-particles, \ngiving reliable results for the evolving abundances of the most important species, including the \nintermediate-mass elements (IME)  and iron-peak elements. The nuclear and reaction data for \nthe calculations were taken from the JINA reaclib database \\citep{Cyb10}. \nIn this network, electron-capture rates were not included since at densities $\\sim 10^{7} \\ g\\ cm^{-3}$, \nthese reactions occur much more \nslowly than the detonation-induced explosion of the white dwarf. \n\n\\section{Model Setup}\n\\label{Model Setup}\nWe have computed nine one-dimensional detonation models. Although supernovae are multi-dimensional phenomena, the one-dimensional treatment greatly simplifies the problem. We \nneglect the multi-dimensional effects such as a multicellular structure \\citep{Gam99,Tim00}. However, \nthe simplicity of the one-dimensional models allows us to draw general conclusions regarding the \nsubject of primary interest to us: the nucleosynthetic yields produced by a detonation wave \npropagating across a star that is not static, but is instead either expanding or \ncontracting. While our main focus is central-detonation models, we also explore the \ncase in which the detonation occurs off-center.\n\n\\begin{deluxetable*}{c c c c c c}\n\\tablewidth{0pt}\n\\tablecaption{Physical Characteristics of the White Dwarf at the Time of Detonation.}\n\\tablehead{\n\\colhead{Physical Characteristics} & \\colhead{$\\mathbf{LCV}$\\footnotemark[1]} & \\colhead{$\\mathbf{HCV}$\\footnotemark[1]} & \\colhead{$\\mathbf{NOV}$} & \\colhead{$\\mathbf{HEV}$\\footnotemark[1]} & \n\\colhead{$\\mathbf{LEV}$\\footnotemark[1]}}\n\\startdata\nVelocity-Radius Ratio [$sec^{-1}$] & -1\/8 & -1\/4 & 0 & 1\/4 & 1\/8\\\\\nCentral Density [$g\/cm^{3}$] & $2.7 \\times 10^7$ & $3.0 \\times 10^7$ & $2.5 \\times 10^7$ & $3.0 \n\\times 10^7$ & $2.7 \\times 10^7$\\\\\n\\enddata\n\\label{init_table}\n\\footnotetext[1]{We investigate both central and off-center detonations for these initial profiles.}\n\\end{deluxetable*}\n\nEach initial stellar model is a cold ($T = 3 \\times 10^{7}K$), isothermal white dwarf (WD) in hydrostatic \nequilibrium with a central density of $2.2\\times 10^{9}\\ g\/cm^{^{3}}$ and a radius of approximately \n2000 km. Its composition consists of equal parts of carbon and oxygen by mass \nthroughout the star. \n\nIn the whole-star simulations of SNe Ia models that involve both deflagration and detonation phases, \nthe deflagration phase leads to expansion of the WD prior to \ndetonation.  In the present one-dimensional models, we artificially expand the star out of hydrostatic \nequilibrium by exciting its lowest vibrational mode with a velocity profile that varies linearly with the \nstellar radius. The \nslope of the velocity profile is such that the resulting kinetic energy is 30\\% of the WD binding \nenergy. This causes the star to first expand and then contract. Figure~\\ref{tem_evo} shows the \ntemporal evolution of the density and velocity profiles over the time during which the star expands and \ncontracts. We initiated a detonation by setting the \ntemperature in selected computational cells to $5.0\\times 10^{9}\\ K$. We detonated five WD models at \nthe center of the star, while we detonated four models at $2.0\\times 10^{8}cm$ slightly off-center. The \nset of central and off-center ignition models share the same initial stellar profiles and differ only by the \nignition location. In total, we studied five initial density and velocity profiles at the time of the \nignition; these are shown in Figure~\\ref{init_cond}. The velocity profiles are nearly linear \nwith radius for densities higher than $5.0\\times 10^{6}\\ g\/cm^{^{3}}$. We were therefore able to \nchoose two expanding white dwarf models in which the ratio between the\ninitial velocity and radius profiles is approximately 1\/4 ($\\mathbf{HEV}$) and 1\/8 ($\\mathbf{LEV}$) in \nunits of sec$^{-1}$. For the contracting models, we choose two models in which the ratio between \nthe initial velocity and radius profiles is approximately -1\/4 ($\\mathbf{HCV}$) and -1\/8 ($\\mathbf{LCV}\n$) in units of sec$^{-1}$. Finally, we choose an initial profile ($\\mathbf{NOV}$) whose velocity profile is \napproximately 0.0 sec$^{-1}$ to represent the static case. For simplicity, we denote these models by \nthe ratio of the initial velocity and radius profiles throughout this paper. Unless explicitly stated \notherwise, the data used are from the central detonation models. Table~\\ref{init_table} summarizes \nthe physical characteristics of the models we study.\nWe adopt the following nomenclature for the models we study: the first letter indicates whether the \nvelocity is relatively $\\mathbf{L}$ow or $\\mathbf{H}$igh, the second letter indicates whether the star is  \n$\\mathbf{C}$ontracting or $\\mathbf{E}$xpanding mode, while the third letter stands for $\\mathbf{V}\n$elocity.  The letters ``NOV\" stand for $\\mathbf{NO\\ V}$elocity. \n\nEvery simulation included Lagrangian tracer particles distributed by mass that were \npassively advected with the fluid during the course of the simulation. The temperature \nand density histories of the tracer particles were calculated by interpolating the corresponding \nquantities using the underlying Eulerian grid. \nEach tracer particle represents a fluid parcel, all of which have the same mass. We used the \ntemperature and density histories of the tracer particles to calculate the nucleosynthetic yields in a \npost-processing step using libnucnet, and to \nprovide additional diagnostics for the complex flows. Each simulation contained approximately \n$10^{4}$ tracer particles. \n\n\\section{Results}\n\\label{Results}\nThis section is largely composed of two parts: the detailed analysis of the central-detonation models \nin $\\S \\ref{Detonation Structure for Non-Steady Stars}$, $\\S  \\ref{The Effect of Expansion Time Scales \non Nucleosynthesis}$, and $\\S \\ref{An Empirical Relation}$, and the exploration of the off-center \ndetonation models in $\\S \\ref{Off-Center Detonation}$.   \n\n\\subsection{Detonation Structure for Non-Steady Stars}\n\\label{Detonation Structure for Non-Steady Stars}\nIn this section, we discuss the properties of a one-dimensional spherical detonation as it \npropagates through a WD that is either expanding or contracting. In what follows, we distinguish \namong three types of hydrodynamic motion that interact nontrivially at different phases of the \nevolution of the explosion: the expanding or contracting flow belonging to the initial pre-expanded \nmodel; the expanding rarefaction that trails the detonation wave; and the bulk expansion experienced \nby the stellar material downstream of the detonation wave due to the energy release by the wave. We \nshow that the combined effect of the \nfirst two flows is responsible for creating the specific cooling conditions behind the shock that \ndetermine the time available for nuclear burning, and hence the nucleosynthetic yields.\n\n\n\\subsubsection{Hydrodynamics of a Detonation Wave}\n\\label{Hydrodynamics of a Detonation Wave}\nThe initial high-temperature disturbance in the center of the WD forms a shock wave that \nquickly transitions to a detonation whose wave speed is roughly the Chapman-Jouguet (CJ) speed \n$\\sim \\ 10^{9} \\ cm\/s$. The CJ speed corresponds to the smallest possible detonation speed, and \nmany detonations fall into this category \\citep{Sei09}. In our study, the detonation wave speed in \nthe frame of the upstream fuel remains almost constant (varying by only $\\pm 5\\%$) as it propagates \nthrough \nthe entire star, and this behavior is true in all models (see Figure~\\ref{sh_str}). This implies that the \nshock strength is independent of the upstream velocities and is insensitive to the upstream density \nof the fuel.  \n\nThe propagation of the detonation wave is powered by the exothermic nuclear reactions that occur \nimmediately behind it. The detonation wave\npropagates outward from the center of the WD, consuming the carbon and oxygen in the stellar core. \nCarbon burning occurs almost instantly in the high-density regions close to the \ncenter of the WD, while its burning length scale becomes comparable to the maximum resolution of \nthe simulation (500m) at densities lower than $\\sim \\ 1.0\\times 10^{6} \\ g~cm^{-3} $. The length scale \nfor Oxygen burning becomes resolved at densities $< \\ \\sim \\ 5\\times 10^{6}$\\ g~cm$^{-3}$, \nwhile the length scale for Silicon burning is larger than the maximum resolution at all densities \n\\citep{Dom11}.  As the detonation wave approaches the WD surface, it is still burning almost all of the \ncarbon but leaves unburned Oxygen \nin the low-density outer layers of the star. Not enough energy is released by nuclear burning behind \nthe front in these low-density outers to continue to power the detonation wave; as a result, \nit transitions to a shock wave and propagates out of the star.   \n\n\\subsubsection{Rarefaction and Bulk Expansion}\n\\label{Rarefaction and Bulk Expansion}\nThe temperature and density of the stellar material increase sharply due to the detonation wave; \nimmediately afterward, the shocked material experiences a strong rarefaction. As an example, we \nshow the behavior of four models at an upstream density of $7.5\\times 10^{6} \\ g\\ cm^{-3}$. Figure~\n\\ref{prof_d_t} shows the density and temperature structure of the four models. Since the different \nmodels have different \ninitial velocity profiles, the time it takes for the detonation wave to reach approximately the same upstream \ndensity differs. Therefore, we aligned the profiles so that their \nupstream densities overlap.\n\nFor the above upstream density, the matter in all models enter the detonation wave and is strongly \ncompressed and heated to similar post-shock densities and temperatures ($2.1 \\times 10^{7} \\ g\\ \ncm^{-3}$ and $5.3 \\times 10^{9} \\ K$). However, the strength of \nthe rarefaction wave behind the front differs greatly, depending on the model. \nIn general, the speed of the rarefaction wave behind the front is higher in expanding cores than in \ncontracting ones. This trend is consistent with the thermal history of the tracer particles, which we will \ndiscuss in $\\S \\ref{The Effect of Expansion Time Scales on Nucleosynthesis}$. \n\nIn view of the fact that\n the strength of the detonation wave is similar in all models, as illustrated in \nFigure~\\ref{sh_str}, the difference in the expansion rates behind the detonation wave shows that the \nphysics in the rarefaction zone is not dominated by the structure of the detonation wave; rather, \nit is a result of the superposition of the flow due to the detonation wave and that due to the pre-existing \nvelocity profile in the star. This is the reason the nuclear yields are different in the different models;\n had the detonation structure dominated the preexisting flow, all of the models would have had similar \n yields. Besides the rarefaction wave, there is another expanding phase due to the bulk expansion of\n  the white dwarf.\n\n Figures~\\ref{tracer} and~\\ref{hydro} \nshow Lagrangian and Eulerian views of the typical flow properties behind the outwardly moving \ndetonation wave for the \\textbf{LCV} model. The yellow dot marks the position of a tracer \nparticle in time in Figure~\\ref{tracer} and in space in Figure~\\ref{hydro}. The time histories of the tracer \nparticle in velocity \nand acceleration are shown in Figure~\\ref{tracer}. The tracer particle which is originally located at a radius of $4.5 \\times 10^{7} cm$ at t = 0 sec travels towards the center of the WD while the detonation wave front propagates through the contracting star. As the tracer particle enters the detonation wave, it \nmomentarily experiences a strong acceleration outward due to compression in the detonation wave. It is then accelerated toward the center (see the bottom panel of Figure~\\ref{tracer}), as the pressure \ndecays due to the expansion of the wave front. At t = 2.57 sec, the tracer particle experiences outward acceleration due to the bulk expansion of the star. The turnover in the acceleration can be seen \nin the time history of the tracer particle in velocity. This behavior is explained by \nFigure~\\ref{hydro}, which shows the velocity profile and the acceleration  ($\\alpha$) profile of \nthe star t = 2.57 sec when the propagating \ndetonation wave is $\\sim 3.4 \\times 10^8 cm$ from the center of the star. \nThe pressure soon dominates over the inward gravitational force as the \nmatter is heated by the energy that is released in the nuclear burning, leading to bulk \nexpansion of the star (the location of the tracer particle is shown by the yellow dot).    \n\n\\subsection{The Effect of Expansion Time Scales on Nucleosynthesis}\n\\label{The Effect of Expansion Time Scales on Nucleosynthesis}\nWe calculated the final nucleosynthetic yields for each tracer particle by integrating the reaction \nnetwork in libnucnet, starting when the tracer particle experienced the peak temperature $T_{peak}$ \nproduced by the detonation wave. We continued the \nintegration until the temperature decreased to $1.0 \\times 10^{9} \\ K$, below which \nthe nuclear abundances do not change significantly except by beta decay. \nWe found that, in all of the models, freeze-out to $^{56}$Ni occurs during the strong rarefaction \nphase and before the bulk expansion phase during which the pressure force drives the incinerated matter into \nexpansion.  \n\n\\subsubsection{Nuclear Abundances}\n\nAt the center of all of the models, the detonation wave converts the material \nchiefly into $^{56}$Ni by complete silicon burning. As the wave propagates through the \nlow-density, outer layers, the nucleosynthesis is characterized by carbon and oxygen \nburning, then only carbon \nburning, and finally the burning ceases. Thus, the amount of $^{56}$Ni is, in general, a function of \nthe initial density of the fuel, and increases with the fuel density. Figure~\\ref{ni56_upD} shows the final \nmass fraction of $^{56}$Ni as a function of the upstream density $\\rho_{up}$ of fuel, where \n``upstream'' denotes the \nundisturbed matter entering the detonation wave. While there is a tight relation between the final\nabundance of $^{56}$Ni and $\\rho_{up}$ in each simulation, we find that the final abundance of \n$^{56}$Ni for a given \n$\\rho_{up}$ varies across the five simulations, with the $\\mathbf{HEV}$ model producing the least \n$^{56}$Ni and the $\\mathbf{HCV}$ model producing the most.\n\nThe differences in the final abundances of $^{56}$Ni across the five models are due to the differences \nin the bulk expansion time scales, as we demonstrate in the next section. \n\n\\subsubsection{Time Evolution of Thermodynamic Properties}\n\nIn general,  the detailed behavior of the density and temperature is coupled to the \nnuclear burning and hydrodynamics. In the case of a simple one-dimensional detonation model, \nthe detonation wave propagates outward from the point of ignition by heating  \nthe upstream fuel to a temperature above the ignition temperature. Once nuclear burning begins, the\nenergy released by it influences the hydrodynamic behavior of the ash, and \nthe subsequent hydrodynamic behavior controls the time evolution of temperature and density of the ash. \nBecause nuclear reactions are highly temperature-sensitive, the interplay of the \ntime scales between burning and hydrodynamic expansion determines the final nucleosynthetic \nyields. \n\nFigure~\\ref{time_his} shows for all five models the density and temperature histories of tracer particles \nwhose upstream densities are $4.4 \\times \\ 10^{6} \\ g\\ cm^{-3}$, $7.0 \\times 10^{6} \\ g\\ cm^{-3}$, and \n$1.2 \\times 10^{7} \\ g\\ cm^{-3}$, which spans the density range for which the final abundance of \n$^{56}$Ni \nlies between 0.1 and 1.0. The tracer particle histories were aligned such that their \npeak temperatures begin at time = 0.0 second.  \n\nThe peak density and the peak temperature are the same to within 3\\% across the five models, \nbut decrease on different time scales. We find that the flows with higher incoming \nspeeds relative to the detonation wave (simulations: $\\mathbf{HCV}$ and $\\mathbf{LCV}$) expand \nmore slowly compared to those with lower incoming speeds (simulations: $\\mathbf{HEV}$ and $\n\\mathbf{LEV}$). Most burning to $^{56}$Ni occurs within $\\sim$ 0.4 seconds after the temperature \nreaches its peak.\n\n\\subsubsection{The Effect of Expansion Time Scale on the Nuclear Yields}\n\nWithin the stellar material that has a final $^{56}$Ni mass fraction above 0.1, the peak\ntemperature is high enough to ensure silicon burning. The final $^{56}$Ni yield depends on the \npeak temperature and the expansion time scale. Freeze out occurs for a nuclide when the \ntemperature drops low enough that reactions become too slow to alter its abundance. \nBecause this condition occurs at different temperatures for different nuclides, the final abundances \ndepend on the rate at which the material cools.    \n\nThe thermodynamic trajectories  of the tracer particles (i.e. individual Lagrangian mass elements) \nprocessed by the detonation wave are well characterized by an exponential temperature dependence \n\\citep{Arn71, Woo73, THW00, Mea09}  \n\\begin{equation}\nT(t) = T_{0}\\; \\mathrm{exp}^{-t\/\\tau},\n\\end{equation}\nwhere $T_{0}$ is the initial temperature at which the nuclear burning begins and $\\tau$ is the time \nscale for the temperature to decrease to 1\/e of its initial value. This time scale characterizes the \nexpansion of the fluid after it is compressed by the leading shock. \nThe density is related to the temperature by the fitted formula \n\\begin{equation}\n\\rho(t) = \\rho_{0}\\; \\left [\\mathrm{exp}^{-t\/\\tau} \\right ]^n,\n\\end{equation}\nwhere n is a function of upstream density and varies between 3.2 and 3.6 over the range of densities \nfor which the final $^{56}$Ni mass fraction lies above 0.1, as shown in Figure~\\ref{exponent}. In this \nwork, we adopt a central value of $n = 3.4$.  \n \nIn order to study the sensitivity of the final $^{56}$Ni abundance to variations in $\\tau$, we \nused libnucnet to perform nucleosynthesis using thermal profiles that correspond to an \nexponential expansion. We chose peak density and temperature values that are characteristic of \nour tracer particle data. The initial composition consisted of equal masses of C and O, and the \nexpansion timescale $\\tau$ varied between 0.1 and 0.6, reflecting the estimated range seen in the \ntracer particle time histories in our models. The time evolution of selected isotopes is shown in \nFigure~\\ref{tau_dep} for different expansion timescales and two different initial thermal conditions: an \ninitial peak temperature of $5.0 \\times\n10^{9}$\\ K and a density of $1.2\\times 10^{7}$\\ g\\ cm$^{-3}$ (left panel), and an initial peak \ntemperature of $4.4 \\times 10^{9}$ K and density of \n$9.4\\times 10^{6}$\\ g\\ cm$^{-3}$ (right panel).   \n\nStellar material with a similar initial density and temperature ends up with different abundances of $^{56}$Ni, \ndepending on the history of the hydrodynamic expansion during the nuclear burning. Even in the case \nof stellar material that reaches the same peak temperature, the thermal profile with the longer \n(0.6 sec) expansion timescale produces more $^{56}$Ni while burning more $^{28}$Si than does \nwith the shorter (0.1 sec) expansion timescale \\citep{Cha12}.\n\nTo demonstrate this point, we compare the nucleosynthetic yield for tracer particles in different models \nwith that from the exponential thermal profiles in (5) and (6). For a given upstream \ndensity of $\\rho = 7.0 \\times 10^6\\ g\\ cm^{-3}$, the temperature and density history of a tracer particle \nwas analyzed and its expansion time scale estimated in three models: 0.4s in $\\mathbf{HEV}$, \n0.45s in $\\mathbf{LCV}$, and 0.51s in $\\mathbf{HCV}$. Nucleosynthesis was \nperformed using both the temperature history of the tracer particles and the exponential temperature \nprofiles. We compare the results in Figure~\\ref{tau_nuc}. \nThe tracer particle from the model $\\mathbf{HCV}$ model (shown in blue) has the longest expansion \ntime scale, while the one from the $\\mathbf{HEV}$ model (shown in red) has the shortest. Importantly, \nthe nucleosynthesis produced by each time history is consistent with that produced by an exponential \ntemperature profile with a similar  time scale. We conclude that the final $^{56}$Ni abundance is \nsensitive to the expansion time scale, and that the expansion time scale experienced by the tracer \nparticles depends on the model.  It does so because the models differ in the velocity of the upstream \nstellar material relative to the detonation wave in the laboratory frame.   \n\n\\subsection{Identification of a Physical Parameter That Can Be Used to Estimate Nucleosynthetic Yields}\n\\label{An Empirical Relation}\n\nIn $\\S\\ref{The Effect of Expansion Time Scales on Nucleosynthesis}$, we showed that, even for similar\nfuel densities upstream of the detonation wave, the final nucleosynthetic yields vary depending on the \nthermal expansion history behind the detonation wave. Since the expansion time scale depends on \nthe incoming velocity, we have demonstrated that the upstream velocity affects the final \nnucleosynthetic yields. In this section, we identify a physical parameter that \nis correlated with the expansion time scale, and that can therefore be used to estimate the \nfinal nucleosynthetic yields.\n\nThe detonation wave connects two different flows: upstream and \ndownstream. Here we define upstream and downstream as the physical states immediately before \nand after the detonation wave. For the same $\\rho_{up}$, \nwe found that the flow with a higher downstream velocity ($v_{down}$)\nexpands faster than the flow with a low downstream velocity. This is illustrated in \nFigure~\\ref{etau_peakv}, and indicates that the downstream velocity is a second important parameter \naffecting the thermal expansion behind the shock front and therefore the nucleosynthetic yield.  In \nother words, flows with a higher downstream velocity yield lower $^{56}$Ni abundances. This \nsuggests that a physical parameter formed by the ratio of $\\rho_{up}$ and $v_{down}$, i.e. $\\rho_{up}\/\nv_{down}$, might be an even better estimator of the final abundance of $^{56}$Ni than $\\rho_{up}$ \nalone.  Figure~\\ref{scale} shows that this is indeed the case: the physical parameter formed by the \nratio of upstream density and downstream velocity is able to predict the final abundance of $^{56}$Ni to within 10$\\%$.    \n\n\\subsubsection{Intermediate Mass Elements}\nThe timescale to burn to $^{56}$Ni is much longer than those to burn to the primary intermediate mass \nelements such as $^{20}$Ne, $^{24}$Mg, $^{28}$Si, $^{32}$S, $^{36}$Ar, and $^{40}$Ca. We \ntherefore hypothesize that the physical parameter $\\rho_{up}\/v_{down}$ should be a good predictor \nfor the final abundances of intermediate mass elements as well. Figure~\\ref{int_scale}, which shows \nthe final mass fraction of the intermediate mass elements as a function of the ratio of $\\rho_{up}\/\nv_{down}$, shows that it is. \n\nWe also note two distinctive features in Figure~\\ref{int_scale}. First, on the high-density branch (where \n$\\rho_{up}\/v_{down} > 0.025$), the \nintermediate mass elements form at the expense of $^{56}$Ni. This is most likely due to the\nincreased in entropy this mixture of elements represents. Second, on the low density branch (where $\n\\rho_{up}\/v_{down} \\sim 0.0$), \nthe light $\\alpha$-nuclei capture $^{4}$He and burn to heavier $\\alpha$-nuclei, thus reducing the \nmass fractions of the lighter nuclei. \nThe mass fractions of the most abundant elements in the $\\rho_{up}\/v_{down}$ range of interest are \nshown in the Figure~\\ref{scale_lin}. While $^{56}$Ni is the dominant product in the high density \nregion, at lower densities, relaxation to NSE is incomplete and the final product is a mixture of \nintermediate mass elements.\n\n\\subsection{Off-Center Detonation Models}\n\\label{Off-Center Detonation}\nWe have also investigated off-center detonation \nscenarios using one-dimensional models detonated at a finite radius. These models enable us to determine \nwhether the empirical relation we found in $\\S \\ref{An Empirical Relation}$ also holds in off-center (but \nstill spherically symmetric) detonation models. To investigate this question, we used the same four \ninitial density and velocity profiles as before: $\\mathbf{HCV}$, $\\mathbf{LCV}$, $\\mathbf{LEV}$, \nand $\\mathbf{HEV}$. We initiated an off-centered detonation in each at a distance of $2.0 \\times \n10^8$ cm from the stellar center.  \n\nFigure~\\ref{off_scale} shows the final mass fraction of $^{56}Ni$ \nas a function of the empirical physical parameter $\\rho_{up}\/v_{down}$ for the four off-center detonation \nmodels. In the off-center \ndetonation models, detonation produces an ingoing detonation wave as well as an outgoing detonation wave.  The latter converges toward the center of the star, triggering a second explosion. In \nthe region \nof the star where nuclear burning takes place as a result of the ingoing detonation \nwave, matter is driven inward by the detonation wave and the star is contracting. Consequently, v$_{down}$ is negative in this region, as is the ratio of \n$\\rho_{up}$ and v$_{down}$. An empirical relation still holds, but it is  different for this contracting (high-density) branch\nwhere $\\rho_{up}\/v_{down}$ $<0$ than for the expanding (low-density) branch. \n\nIn order to better compare the relation between the physical parameter $\\rho_{up}\/v_{down}$ and the final mass fraction of  \n$^{56}$Ni for the two regions, we replace v$_{down}$ with its absolute value; the parameter then becomes $\\rho_{up}\/|v_{down}|$. The resulting relation is \nshown in the left panel of Figure~\\ref{scale_comp}. The \ndata in the vicinity of the detonation point are numerically noisy, and we have omitted them from the \nfigure. The two branches join when plotted in this way.  Although \nthe empirical relation is not as tight for the contracting (high-density) branch as it is for the expanding (low-density) branch, the \nphysical variable $\\rho_{up}\/|v_{down}|$ is able to predict the final abundance of  $^{56}$Ni to \nwithin 15$\\%$. This suggests that \nthis physical parameter is a fairly robust predictor of the final nucleosynthetic yields. \n\nThe right panel in the Figure~\\ref{scale_comp} compares the empirical relation between $\\rho_{up}\/|\nv_{down}|$ and the final abundance of  $^{56}$Ni for the center and off-center detonation models for \nthe $\\mathbf{HCV}$ initial stellar velocity profile. The results demonstrate again that the final nucleosynthetic \nyields cannot be characterized by the local values of density and velocity of the matter alone. Rather, \nthey are affected by \nboth the pre-existing velocity flow in the star and the post-detonation expansion of the star. \n\n\\section{Conclusion}\n\\label{Conclusion}\nThe explosion mechanism of Type Ia SNe is still an active topic of research, and among the leading paradigms for the explosion mechanism are the delayed detonation models. While many different delayed detonation scenarios have been published, we direct our attention to investigating the key common feature of the explosion that involves the propagation of the detonation wave in a non-static velocity field of a white dwarf. Although the most satisfactory approach to this investigation would be to simulate a multidimensional delayed detonation model on scales of an exploding white dwarf, this is a numerically challenging task, requiring a considerable amount of data storage and computation time. Furthermore, our primary interest lies in the role played by the velocity profile settled during the previous deflagrative phase in the evolution of the detonation. Therefore, we create simplified one-dimensional models that test mainly the effects of the pre-detonation stellar internal velocity profile and the post-detonation velocity of expansion on the production of $\\alpha$-particle nuclei, including $^{56}$Ni.   \n\nWe have studied the flow structure behind the traveling detonation wave front and observed two \ndistinct expansion phases: rarefaction just behind the detonation wave and bulk expansion of the \nexpanding white dwarf. The rarefaction timescale, or the \nexpansion time scale studied in this work, was found to be not a function of detonation strength \nonly, but is also influenced by pre-existing flow structure, depending on whether the initial \nconfiguration of the star is expanding or contracting. In our models, almost all the burning to $^{56}$Ni \noccurred only in rarefaction phase.\n\nThe relationship between the flow properties and the \nresulting $^{56}$Ni yields showed that the final yields of burned matter emerging from \nthe detonation are highly sensitive to the expansion time scales of the flow behind the detonation, \nwhich is strongly conditioned by the pre-existing flow in the expanding or contracting progenitor \nstar. The expansion time is longer for contracting stars and shorter for expanding stars, as the \nrarefaction proper to the detonation is combined with the pre-existing flow. With the greater \nexpansion time scale, both temperature and density evolve relatively slowly, providing more time \nfor $^{56}$Ni production.   \n \nWe have also found an empirical relationship between the ratio of the upstream density to the \npost-shock velocity and the final $^{56}$Ni yield for central detonations. It is not surprising to find that \n$^{56}$Ni yield increases with increasing upstream density of fuel. However, \nwe also find that the final post-detonation velocity is another parameter that influences $^{56}$Ni\nyield. The quantity $\\rho_{up}$\/$v_{down}$ is tightly related with $^{56}$Ni yield, and produces \nrelatively small scatter ($< 10\\%$) about the relationship; however, the picture is more complicated for \noff-centered detonations. Currently this relationship is limited to one-dimensional models and we caution that the above results are preliminary. In more realistic SNe Ia models, \ndetonations in white dwarfs not only show a rich multilevel cellular structure \\citep{Gam99,Tim00} but also travel through \nnon-spherically symmetric stellar density distributions \\citep{Jor08, Mea09}. In addition, some explosion scenarios involve \ndetonations propagating through partially burnt material as well as unburnt fuel. Thus, further analysis in \nhigher dimensions will need to be carried out. \n\n\\newpage\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm, scale=1.0, angle=90]{figure1-eps-converted-to.pdf}\n\\caption{Density profile (left panel) and velocity profile (right panel) at five different times ($t$ = 0.4, \n0.8, 1.6, 2.4, and 3.0 seconds) during the expansion phase (red), at maximum expansion (black), and \nthe contraction phase (blue).  }\n\\label{tem_evo}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm, scale=1.0, angle=90]{figure2-eps-converted-to.pdf}\n\\caption{Density profile (left panel) and velocity profile (right panel) of five one-dimensional models at \nthe time of detonation. The \nrelative ratio (sec$^{-1}$) between velocity and radius is approximately 1\/8 ($\\mathbf{LEV}$), 1\/4 ($\n\\mathbf{HEV}$), -1\/4 ($\\mathbf{HCV}$), and -1\/8 ($\\mathbf{LCV}$). The velocity in the model $\n\\mathbf{NOV}$ is close to zero, and was chosen to represent the static case. In the left panel, the $\n\\mathbf{HEV}$ and $\\mathbf{HCV}$ models have similar density profiles, while the $\n\\mathbf{LEV}$ and $\\mathbf{LCV}$ models have similar density profiles.      }\n\\label{init_cond}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm, scale=1.0, angle=0]{figure3-eps-converted-to.pdf}\n\\caption{The speed of the shock wave in the frame of the upstream fuels as a function of upstream \ndensity for four different models: HCV (blue), LCV\n(magenta), HEV (red), and LEV (green). }\n\\label{sh_str}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=0]{figure4-eps-converted-to.pdf}\n\\caption{Stellar density profiles (left panel) and temperature profiles (right panel) for \nfour different models: HCV (blue), LCV \n(magenta), HEV (red), and LEV (green). The profiles are aligned such that their upstream densities \noverlap. The sharp discontinuity in both density and temperature profiles represents the detonation \nwave, which is traveling outward in radius. The structure of the thermodynamic properties behind the \ndetonation wave front differs for the different models. }\n\\label{prof_d_t}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm,  angle=90]{figure5-eps-converted-to.pdf}\n\\caption{Velocity and acceleration of a typical tracer particle as a function of time. Shown are the \nvelocity (black curve) and the total acceleration (red curve) of the fluid as a function of time as given by\na typical tracer particle. At time t = 2.57 sec (indicated by the yellow dot), \nthe fluid experiences positive acceleration and its outward velocity begins to increase with time.}\n\\label{tracer}\n\\end{figure*}\n\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=90]{figure6-eps-converted-to.pdf}\n\\caption{Velocity and acceleration profile of the star. Shown are the velocity (black curve), the \nacceleration due \nto the pressure gradient (blue curve), the acceleration due to gravitational force (green curve), and the \nsum of the two accelerations (red curve). The yellow dot marks the location at time = 2.57 sec after the \nonset of the detonation of a tracer particle that was originally at a radius of $4.5 \\times 10^{7} cm$ at $t$ = 0 sec.}\n\\label{hydro}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=0]{figure7-eps-converted-to.pdf}\n\\caption{Final mass fraction of $^{56}$Ni as a function of upstream density derived from tracer \nparticles. }\n\\label{ni56_upD}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=0]{figure8-eps-converted-to.pdf}\n\\caption{Density and temperature histories of tracer particles whose upstream \npre-detonation wave densities are \n$4.4 \\times 10^{6} g\\ cm^{-3}$, $7.0\\times 10^{6} g\\ cm^{-3}$, and $1.2 \\times 10^{7} g\\ cm^{-3}$ \nrespectively from top to bottom. Shown are the results for five different models: $\\mathbf{HCV}$ \n(blue), $\\mathbf{LCV}$ (magenta), $\\mathbf{NOV}$ (yellow), $\\mathbf{LEV}$ (green), and $\n\\mathbf{HEV}$ (red).}\n\\label{time_his}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm,angle=0]{figure9-eps-converted-to.pdf}\n\\caption{Values of the exponent $n$ in equation (6) for the density, which were estimated from fits to a \nset of approximately \n7000 tracer particle time histories over the range in upstream density that yields a final mass fraction \nof $^{56}$Ni\nbetween 0.1 and 1.0. The central value $n$ = 3.4 was adopted for calculations that investigated   \nthe sensitivity of the final yields to variations in the expansion time scale $\\tau$. }\n\\label{exponent}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm,angle=90]{figure10-eps-converted-to.pdf}\n\\caption{Time evolution of selected light elements, intermediate mass elements, and iron group \nisotopes during the expansion, which is parameterized by temperature. Left panel: initial peak \ntemperature of $5.0 \\times 10^{9}$\\ K and density of $1.2\\times 10^{7}$\\ g\\ cm$^{-3}$.  Right panel: Initial peak \ntemperature of $4.4 \\times 10^{9}$ K and density of $9.4\\times 10^{6}$\\ g\\ cm$^{-3}$. For each \nspecies, the abundances have been calculated with a nuclear network (see $\\S\\ref{Results}$) using an \nexponential temperature evolution for three different expansion time scales: 0.1 s (dashed), 0.3 s \n(dotted), and 0.6 s (solid). These figures illustrate the dependence of the nuclear \nabundances on the expansion time scale, especially for the heavy nuclei.}\n\\label{tau_dep}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm,angle=0]{figure11-eps-converted-to.pdf}\n\\caption{Time evolution of $^{56}$Ni parameterized by the temperature. The solid lines represent \nthe final $^{56}$Ni abundances derived from the density and temperature histories of tracer particles \nwhose upstream densities \nare all $\\rho = 7.0 \\times 10^6\\ g\\ cm^{-3}$ for three different models: $\\mathbf{HCV}$ (blue),\n$\\mathbf{LCV}$ (magenta), and $\\mathbf{HEV}$ (red). The final $^{56}$Ni abundances derived from \nfitting equation (6) to tracer particle data are \ncalculated for expansion time scales of 0.40 (dashed), 0.45 (dot-dashed), and 0.51 (dotted).  }\n\\label{tau_nuc}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm, angle=90]{figure12-eps-converted-to.pdf}\n\\caption{Left panel: Downstream flow velocities derived from tracer particles as a function of \nupstream density. Right panel:  \nExpansion time scales of the flow downstream of the detonation wave front as a function of upstream \ndensity. Shown are the results four different models: \nHCV (blue), LCV (magenta), HEV (red), and LEV (green).   }\n\\label{etau_peakv}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm, angle=0]{figure13-eps-converted-to.pdf}\n\\caption{Mass fraction of $^{56}$Ni as a function of the ratio of mass density immediately upstream \nof  and the velocity immediately downstream of the detonation wave front. Shown are the results for\nfive different models: HCV (blue), LCV (magenta), NOV (yellow), LEV (green), and HEV (red).   }\n\\label{scale}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=0]{figure14-eps-converted-to.pdf}\n\\caption{Mass fraction of the intermediate-mass elements:$^{20}$Ne, $^{24}$Mg, \n$^{28}$Si, $^{32}$S, $^{36}$Ar, and $^{40}$Ca as a function of the ratio of $\\rho_{up}\/v_{down}$. \nShown are the results for four different models: HCV (blue), LCV (magenta), LEV (green), and HEV \n(red).   }\n\\label{int_scale}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=16cm, angle=0]{figure15-eps-converted-to.pdf}\n\\caption{Mass fraction of the dominant nuclei in mass fraction after $^{56}$Ni: $^{4}$He, $^{28}$Si, \n$^{32}$S, $^{36}$Ar, $^{40}$Ca, and $^{57}$Ni as a function of the ratio of $\\rho_{up}\/v_{down}$ in \nthe region where the empirical relation holds. Shown are the results for four different models: HCV \n(blue), LCV (magenta), LEV (green), and HEV (red).    }\n\\label{scale_lin}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm, angle=0]{figure16-eps-converted-to.pdf}\n\\caption{Mass fraction of $^{56}$Ni as a function of the ratio of mass density immediately upstream \nof  and the velocity immediately downstream of the detonation wave front. Shown are the results for \nfour different\noff-center detonation models: $\\mathbf{HCV}$ (blue), $\\mathbf{LCV}$ (magenta), $\n\\mathbf{LEV}$ (green), and $\\mathbf{HEV}$ (red).   }\n\\label{off_scale}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=8cm, angle=90]{figure17-eps-converted-to.pdf}\n\\caption{Left panel: Mass fraction of $^{56}$Ni as a function of $\\rho_{up}\/|v_{down}|$. \nShown are the results of four different off-center models: HCV (blue), LCV (magenta), LEV (green), \nand \nHEV (red). Right panel: the relation between $^{56}$Ni and $\\rho_{up}\/|v_{down}|$ for the off-center \ndetonation model (red) is compared to the empirical relation in the central detonation model (black). \nBoth models had an HCV initial profile.  }\n\\label{scale_comp}\n\\end{figure*}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\nMany prominent recent successes in deep learning have come from training models on large datasets~\\citep{alexnet, bert, gpt3}.\nBy contrast, the reinforcement learning (RL,~\\citet{sutton_book_2018}) paradigm is typically characterized by agents learning from \\emph{online interaction}, which is often expensive or potentially even impractical to collect in abundant quantities.\n\\emph{Offline} (or batch) reinforcement learning~\\citep{batchmodeRL, offlinerl_survey} aims to address these issues by leveraging pre-collected datasets to train and deploy autonomous agents without requiring any online interaction with an environment.\n\nWhile offline reinforcement learning algorithms, both model-based~\\citep{mopo, morel, mbop} and model-free~\\citep{cql, kostrikov2021iql, td3bc, brac, bcq, kumar19bear}, have mastered challenging continuous control tasks, most prior works have relied on access to proprioceptive states~\\citep{d4rl}.\nBy contrast, there exists only a single study of offline reinforcement learning from \\emph{visual observations}~\\citep{lompo} for continuous control tasks, a problem setting for which there exist neither well-designed benchmarking tasks, nor any carefully evaluated baselines.\n\nTraining agents offline from visual observations provides an opportunity to make reinforcement learning more widely applicable to real-world settings, where we have access to vast quantities of visual observations of desirable behaviors.\nFor example, in autonomous driving~\\citep{kendall2018learning}, large quantities of visual offline data already exist but have not been fully utilized~\\citep{RobotCarDatasetIJRR, bdd100k}.\nSimilarly, in robotics, data collection is expensive due to costs associated with set-up and human supervision. \nEffective, transferable offline reinforcement learning could allow us to reuse datasets gathered previously for different tasks or settings for unseen new problems~\\citep{chebotaractionablemodels21}.\nUnlocking this potential would represent significant progress towards learning general-purpose agents for realistic problems.\n\nTo enable the development of effective, robust, and adaptive algorithms for offline RL from visual observations, we present a suite of carefully designed datasets and benchmarking tasks for this burgeoning domain.\nWe use these tasks to establish simple performance baselines, to study how the composition of vision-based offline datasets affects performance of different types of RL algorithms, and to evaluate the extent to which algorithms for offline RL from visual observations satisfy a set of \\emph{desiderata}, including robustness to visual distractions, generalization across environment dynamics, and improved performance at scale.\nOur evaluation identifies clear failure modes of the baseline methods and highlights opportunities and open problems for future work which can be tackled with our benchmark.\n\nRecent progress in offline reinforcement learning from proprioceptive observations has been driven by well-designed and easy-to-use evaluation testbeds and baselines.\nWe hope that \\textsc{v-d4rl} and the analysis in this paper will help facilitate the development of robust RL agents that leverage large, diverse, and often imperfect offline datasets of visual observations across tasks and deployment settings. We open-source our code and data at \\url{https:\/\/github.com\/conglu1997\/v-d4rl}.\n\nThe core contributions of this paper are as follows:\\vspace*{-3mm}\n\\begin{enumerate}[leftmargin=12pt]\n\\setlength\\itemsep{0.1em}\n\\item We present a benchmark for offline RL from visual observations of \\textsc{DMControl} \\textsc{Suite} (DMC) tasks~\\citep{tassa2020dmcontrol}.\nThis benchmark, \\mbox{\\textbf{Vision Datasets for Deep Data-Driven RL} (\\textsc{v-d4rl})}, follows the design principles of the popular \\textsc{d4rl} benchmark~\\citep{d4rl}.\n\\item\nWe identify \\textbf{three key desiderata} for realistic offline RL from visual observations: robustness to distractions~\\citep{distractingcontrolsuite}, generalization across dynamics~\\citep{zhang2021learning}, and improved performance for offline reinforcement learning at scale.\nWe present a suite of evaluation protocols designed to test whether offline RL algorithms satisfy these desiderata.\n\\item\nWe establish model-based and model-free baselines for offline RL from visual observations.\nWe do so by modifying state-of-the-art online RL algorithms, \\textbf{DreamerV2}~\\citep{dreamerv2} and \\textbf{DrQ-v2}~\\citep{drqv2}, which showcase the relative strengths of model-based and model-free algorithms for offline RL from visual observations.\nWe use these algorithms to provide simple baselines for the aforementioned desiderata to serve as a measure of progress for future advances in this domain.\n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n\\vspace{-2mm}\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/general\/grid_envs.png}\n\\vspace{-1mm}\n\\caption{\n    \\small{Sampled images from \\textsc{v-d4rl}.}\n}\n\\label{fig:vd4rl_picture}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/general\/mbmf-bidirectional.pdf}\n\\vspace{-1mm}\n\\caption{\\small{Model-based vs. Model-free on \\textsc{v-d4rl}.}}\n\\label{fig:intro_comparison}\n\\end{subfigure}\n\\vspace{-1mm}\n\\caption{\\small{We introduce \\textsc{v-d4rl}, a benchmarking suite for offline reinforcement learning from visual observations, which includes a comprehensive set of \\textsc{d4rl}-style datasets and modalities unique to learning from visual observations.}}\n\\vspace{-5mm}\n\\end{figure}\n\\section{Preliminaries}\n\\label{sec:prelims}\n\\vspace*{-5pt}\n\nWe model the environment as a Markov Decision Process (MDP), defined as a tuple $M = (\\mathcal{S}, \\mathcal{A}, P, R, \\rho_0, \\gamma)$, where $\\mathcal{S}$ and $\\mathcal{A}$ denote the state and action spaces respectively, $P(s' | s, a)$  the transition dynamics, $R(s, a)$  the reward function, $ \\rho_0$  the initial state distribution, and $\\gamma \\in (0, 1)$ the discount factor.\nThe standard goal in online reinforcement learning is to optimize a policy $\\pi (a | s)$ that maximizes the expected discounted return $\\mathbb{E}_{\\pi, P, \\rho_0}\\left[\\sum_{t=0}^\\infty \\gamma^t R(s_t, a_t)\\right]$ through interactions with the environment.\n\nIn \\textit{offline reinforcement learning}, the policy is not deployed in the environment until test time.\nInstead, the algorithm has access to a fixed dataset $\\mathcal{D}_{\\text{env}} = \\{(s_i, a_i, r_i, s_{i+1})\\}_{i=1}^{N}$, collected by one or more behavioral policies $\\pi_b$.\nFollowing \\citet{mopo}, we refer to the distribution from which $\\mathcal{D}_{\\text{env}}$ was sampled as the \\textit{behavioral distribution}.\n\nWe first describe recent advancements in offline RL and RL from visual observations through the lens of model-based and model-free methods.\n\n\\subsection{Offline Reinforcement Learning Paradigms}\n\\textbf{Model-based.} A central problem in offline reinforcement learning is over-estimation of the value function~\\cite{sutton_book_2018} due to incomplete data~\\citep{kumar19bear}.\nModel-based methods in offline RL provide a natural solution to this problem by penalizing the reward from model-rollouts by a suitable measure of uncertainty.\n\\citet{mopo} provide theoretical justification for this approach by constructing a pessimistic MDP (P-MDP) by lower-bounding the expected true return, $\\eta_M(\\pi)$, using the error between the estimated and true model dynamics.\n\nHowever, this quantity is usually not available without access to the true environment dynamics, so algorithms such as MOPO and MOReL in practice~\\citep{mopo, morel} penalize reward with a surrogate measure of uncertainty.\nThose algorithms train an ensemble of $K$ probabilistic dynamics models~\\citep{nixweigend}, and define a heuristic based on the ensemble predictions.\nRecent work~\\citep{lu2021revisiting} has shown that a more optimal choice to approximate true dynamics error is the standard deviation of the ensemble's mixture distribution, as proposed by \\citet{deep_ensembles}.\n\n\\textbf{Model-free.} In the model-free paradigm, we lose the natural measure of uncertainty provided by the model.\nIn lieu of this, algorithms such as CQL~\\citep{cql} attempt to avoid catastrophic over-estimation by penalizing actions outside the support of the offline dataset with a wide sampling distribution over the action bounds.\nRecently, \\citet{td3bc} have shown that a minimal approach to offline reinforcement learning works in proprioceptive settings, where offline policy learning with TD3~\\citep{td3} can be stabilized by augmenting the loss with a behavioral cloning term.\n\n\\subsection{Reinforcement Learning from Visual Observations}\n\\label{subsec:visionbasedrl}\nRecent advances in reinforcement learning from visual observations have been driven by use of data augmentation, contrastive learning and learning powerful recurrent world models of the environment.\nWe describe the current state-of-the-art in model-based and model-free methods.\n\n\\textbf{Model-based (DreamerV2, \\citet{dreamerv2}).} \nDreamerV2 learns a powerful model of the environment using a Recurrent State Space Model (RSSM,~\\citet{planet, dreamer}, and predicts ahead using compact model latent states.\nThe particular instantiation used in DreamerV2 uses model states $s_t$ containing a deterministic component $h_t$, implemented as the recurrent state of a Gated Recurrent Unit (GRU,~\\citep{gru}), and a stochastic component $z_t$ with categorical distribution.\nThe actor and critic are trained from imagined trajectories of latent states, starting at encoded states of previously encountered sequences.\n\n\\textbf{Model-free (DrQ-v2, \\citet{yarats2021mastering}).} DrQ-v2 is an off-policy algorithm for vision-based continuous control, which uses data-augmentation~\\citep{rad, drq} of the state and next state observations.\nThe base policy optimizer is DDPG~\\citep{ddpg}, and the algorithm uses a convolutional neural network (CNN) encoder to learn a low-dimensional feature representation.\n\n\\section{Baselines for Offline Reinforcement Learning from Visual Observations}\n\\vspace*{-4pt}\n\\label{sec:baselines}\n\nIn this section, we begin by motivating our creation of a new benchmark (\\textsc{v-d4rl}), introduce our new simple baselines combining recent advances in offline RL and vision-based online RL, and present a comparative evaluation of current methods on \\textsc{v-d4rl}. Comprehensive and rigorous benchmarks are crucial to progress in nascent fields.\nTo our knowledge, the only prior work that trains vision-based offline RL agents on continuous control tasks is LOMPO~\\citep{lompo}.\nWe analyze their datasets in~\\Cref{subsubsec:lompo_eval} and find they do not conform to standard \\textsc{d4rl} convention.\n\n\\subsection{Adopting \\textsc{d4rl} Design Principles}\n\\label{subsec:d4rl_eval}\n\\vspace*{-2pt}\n\nIn this section, we outline how to generate \\textsc{d4rl}-like vision-based datasets for \\textsc{v-d4rl}.\n\nTo generate offline datasets of visual observations, we consider the following three \\textsc{DMControl Suite} (DMC) environments:\\vspace*{-3pt}\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=12pt]\n\\setlength\\itemsep{0.1em}\n\\item\n\\textbf{walker-walk}: a planar walker is rewarded for being upright and staying close to a target velocity.\n\\item\n\\textbf{cheetah-run}: a planar biped agent is rewarded linearly proportional to its forward velocity.\n\\item\n\\textbf{humanoid-walk}: a 21-jointed humanoid is rewarded for staying close to a target velocity. Due to the huge range of motion styles possible, this environment is \\emph{extremely challenging} with many local minima and is included as a stretch goal.\n\\end{itemize}\n\nFrom these environments, we follow a \\textsc{d4rl}-style procedure in considering five different behavioral policies for gathering the data.\nAs in \\textsc{d4rl}, the base policy used to gather the data is Soft Actor--Critic (SAC, \\citet{sac-v2}) on the proprioceptive states.\n\\vspace*{-3pt}\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=12pt]\n\\setlength\\itemsep{0.1em}\n\\item\n\\textbf{random}: Uniform samples from the action space.\n\\item\n\\textbf{medium-replay (mixed)}: The initial segment of the replay buffer until the SAC agent reaches medium-level performance.\n\\item\n\\textbf{medium}: Rollouts of a fixed medium-performance policy.\n\\item\n\\textbf{expert}: Rollouts of a fixed expert-level policy.\n\\item\n\\textbf{medium-expert (medexp)}: Concatenation of medium and expert datasets above.\n\\end{itemize}\n\nBy default, each dataset consists of 100,000 total transitions (often 10$\\times$ less than in \\textsc{d4rl}) in order to respect the memory demands of vision-based tasks.\nThe cheetah and humanoid medium-replay datasets consist of 200,000 and 600,000 transitions respectively due to the increased number of samples required to train policies on these environments.\nFull statistics of each dataset are given in~\\Cref{app:data_characteristics}. Further details on data generation are given in~\\Cref{app:data_generation}.\n\n\\subsection{Baselines}\n\\label{subsec:baselines}\n\n\\begin{figure*}[t]\n\\centering\n\\small\n\\vspace{-1mm}\n\\includegraphics[width=\\textwidth]{figures\/exp1\/all_combined_100k.pdf}\n\\includegraphics[width=0.5\\textwidth, trim={0 100 0 100}, clip]{figures\/exp1\/combined_legend_100k.pdf}\n\\vspace{-2mm}\n\\caption{\n    \\small{Rigorous comparison on datasets from the \\textsc{v-d4rl} benchmark, each setting is averaged over 6 seeds with error bar showing one standard deviation. Total gradient steps are normalized under epochs, and we plot the un-normalized evaluated return. We note that the model-free and BC baselines are far more stable than the model-based.}\n}\n\\label{fig:d4rlp_comparison}\n\\end{figure*}\n\nWe show that for the two state-of-the-art online vision-based RL algorithms described in~\\Cref{subsec:visionbasedrl}, simple adjustments from the proprioceptive literature suffice to transfer them to the offline setting. In~\\Cref{subsec:comp_eval}, we demonstrate that these baselines provide a new frontier on our benchmark and on prior datasets. Additional details and hyperparameters for our algorithms are given in~\\Cref{app:algodetails}.\n\n\\textbf{Model-based.} For DreamerV2, the latent stochastic states are sampled from an ensemble, which allows us to naturally define a penalty for the reward based on the dynamics uncertainty as in~\\citep{mopo}.\nWe adopt a similar approach to that studied in \\citep{lu2021revisiting} and use the mean-disagreement of the ensemble.\nThus, the reward at each step becomes:\n\\begin{align}\n\\SwapAboveDisplaySkip\n    \\tilde{r}(s, a) = r(s, a) \\textcolor{violet}{- \\lambda \\sum_{k=1}^{K} (\\mu_{\\phi}^{k}(s, a) - \\mu^{*}(s, a))^2 } ,\n\\end{align}\nwhere $\\lambda$ is a normalization constant and $\\mu^{*}(s, a)=\\frac{1}{K} \\sum_{k=1}^{K} \\mu_{\\phi}^{k}(s, a)$ is the mean over the dynamics ensemble.\nInstead of interleaving model-training steps and policy optimization steps, we simply perform one phase of each.\nWe refer to this algorithm as \\textbf{Offline DV2}.\n\n\\textbf{Model-free.} For DrQ-v2, we note that the base policy optimizer shares similarities with TD3~\\cite{td3}, which has recently been applied effectively in offline settings from proprioceptive states by simply adding a regularizing behavioral-cloning term to the policy loss, resulting in the algorithm TD3+BC~\\citep{td3bc}.\nConcretely, the policy objective becomes:\n\\begin{align}\n    \\pi=\\underset{\\pi}{\\operatorname{argmax}} \\mathbb{E}_{(s, a) \\sim \\mathcal{D}_{\\text{env}}}\\left[\\textcolor{violet}{\\lambda} Q(s, \\pi(s))\\textcolor{violet}{-(\\pi(s)-a)^{2}}\\right] ,\n\\end{align}\n\nwhere $\\lambda$ is a normalization term, $Q$ is the learned value function and $\\pi$ is the learned policy.\nWe apply the same regularization to DrQ-v2, and call this algorithm: \\textbf{DrQ+BC}.\n\n\\textbf{Prior work.} \nSince DrQ-v2 is an actor-critic algorithm, we may also use it to readily implement the \\textbf{CQL}~\\cite{cql} algorithm by adding the CQL regularizers to the Q-function update. We additionally compare against \\textbf{LOMPO}~\\citep{lompo}, and behavioral cloning (\\textbf{BC}, \\citep{bain1995framework, bratko1995behavioural}), where we apply supervised learning to mimic the behavioral policy.\nOffline DV2 is closely related to LOMPO as both use an RSSM~\\citep{planet, dreamer} as the fundamental model, however Offline DV2 is based on the newer discrete RSSM with an uncertainty penalty more closely resembling the ensemble penalties in supervised learning~\\citep{deep_ensembles} and uses KL balancing during training~\\cite{dreamerv2}.\n\n\\setlength{\\tabcolsep}{12.0pt}\n\\begin{table}[t]\n\\centering\n\\caption{\\small{Final performance on \\textsc{v-d4rl} averaged over 6 seeds, with one standard deviation given as error for full transparency. Evaluated return is mapped from $[0, 1000]$ to $[0, 100]$. Our model-based method does best on the diverse low-reward datasets, model-free on the diverse high-reward datasets and behavioral cloning on the narrow expert data. We list the `average rank' for each algorithm computed over each individual dataset.}}\n\\label{tab:all_comparison}\n\\scalebox{0.85}{\n\\begin{tabular}{@{}llccccc@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Environment}}                  & \\textbf{Offline DV2}              & \\textbf{DrQ+BC}   & \\textbf{CQL}                & \\textbf{BC}                      & \\textbf{LOMPO}           \\\\ \\toprule\n\\multirow{5}{*}{walker-walk} & random                     & \\cellcolor[gray]{0.9}\\textbf{28.7 \\addtiny{$\\pm$13.0}} & 5.5 \\addtiny{$\\pm$0.9}         & 14.4 \\addtiny{$\\pm$12.4}  & 2.0 \\addtiny{$\\pm$0.2}           & \\cellcolor[gray]{0.9}\\textbf{21.9 \\addtiny{$\\pm$8.1\\hphantom{0}}}  \\\\\n                             & mixed                      & \\cellcolor[gray]{0.9}\\textbf{56.5 \\addtiny{$\\pm$18.1}} & 28.7 \\addtiny{$\\pm$6.9\\hphantom{0}}          & 11.4 \\addtiny{$\\pm$12.4}   & 16.5 \\addtiny{$\\pm$4.3\\hphantom{0}}           & 34.7 \\addtiny{$\\pm$19.7} \\\\\n                             & medium                     & 34.1 \\addtiny{$\\pm$19.7}          & \\cellcolor[gray]{0.9}\\textbf{46.8 \\addtiny{$\\pm$2.3\\hphantom{0}}}          & 14.8 \\addtiny{$\\pm$16.1}   & 40.9 \\addtiny{$\\pm$3.1\\hphantom{0}}           & 43.4 \\addtiny{$\\pm$11.1} \\\\\n                             & medexp                     & 43.9 \\addtiny{$\\pm$34.4}          & \\cellcolor[gray]{0.9}\\textbf{86.4 \\addtiny{$\\pm$5.6\\hphantom{0}}}           & 56.4 \\addtiny{$\\pm$38.4}  & 47.7 \\addtiny{$\\pm$3.9\\hphantom{0}}           & 39.2 \\addtiny{$\\pm$19.5} \\\\\n                             & expert                     & 4.8 \\addtiny{$\\pm$0.6}            & 68.4 \\addtiny{$\\pm$7.5\\hphantom{0}}          & \\cellcolor[gray]{0.9}\\textbf{89.6 \\addtiny{$\\pm$6.0\\hphantom{0}}}   & \\cellcolor[gray]{0.9}\\textbf{91.5 \\addtiny{$\\pm$3.9\\hphantom{0}}}           & 5.3 \\addtiny{$\\pm$7.7}   \\\\ \\midrule\n\\multirow{5}{*}{cheetah-run} & random                     & \\cellcolor[gray]{0.9}\\textbf{31.7 \\addtiny{$\\pm$2.7\\hphantom{0}}}  & 5.8 \\addtiny{$\\pm$0.6}          & 5.9 \\addtiny{$\\pm$8.4}   & 0.0 \\addtiny{$\\pm$0.0}           & 11.4 \\addtiny{$\\pm$5.1\\hphantom{0}}  \\\\\n                             & mixed                      & \\cellcolor[gray]{0.9}\\textbf{61.6 \\addtiny{$\\pm$1.0\\hphantom{0}}}  & 44.8 \\addtiny{$\\pm$3.6\\hphantom{0}}          & 10.7 \\addtiny{$\\pm$12.8}   & 25.0 \\addtiny{$\\pm$3.6\\hphantom{0}}           & 36.3 \\addtiny{$\\pm$13.6} \\\\\n                             & medium                     & 17.2 \\addtiny{$\\pm$3.5\\hphantom{0}}           & \\cellcolor[gray]{0.9}\\textbf{53.0 \\addtiny{$\\pm$3.0\\hphantom{0}}}          & 40.9 \\addtiny{$\\pm$5.1}   & \\cellcolor[gray]{0.9}\\textbf{51.6 \\addtiny{$\\pm$1.4\\hphantom{0}}}           & 16.4 \\addtiny{$\\pm$8.3\\hphantom{0}}  \\\\\n                             & medexp                     & 10.4 \\addtiny{$\\pm$3.5\\hphantom{0}}           & 50.6 \\addtiny{$\\pm$8.2\\hphantom{0}}          & 20.9 \\addtiny{$\\pm$5.5}   & \\cellcolor[gray]{0.9}\\textbf{57.5 \\addtiny{$\\pm$6.3\\hphantom{0}}}           & 11.9 \\addtiny{$\\pm$1.9\\hphantom{0}}  \\\\\n                             & expert                     & 10.9 \\addtiny{$\\pm$3.2\\hphantom{0}}           & 34.5 \\addtiny{$\\pm$8.3\\hphantom{0}}          & \\cellcolor[gray]{0.9}\\textbf{61.5 \\addtiny{$\\pm$4.3}}   & \\cellcolor[gray]{0.9}\\textbf{67.4 \\addtiny{$\\pm$6.8\\hphantom{0}}}           & 14.0 \\addtiny{$\\pm$3.8\\hphantom{0}}  \\\\ \\midrule\n\\multirow{5}{*}{humanoid-walk} & random & 0.1 \\addtiny{$\\pm$0.0} & 0.1 \\addtiny{$\\pm$0.0} & 0.2 \\addtiny{$\\pm$0.1} & \\hphantom{0}0.1 \\addtiny{$\\pm$0.0} & 0.1 \\addtiny{$\\pm$0.0} \\\\\n                               & mixed  & 0.2 \\addtiny{$\\pm$0.1} &\\cellcolor[gray]{0.9}\\textbf{ 15.9 \\addtiny{$\\pm$3.8}}\\hphantom{0}& 0.1 \\addtiny{$\\pm$0.0} & \\cellcolor[gray]{0.9}\\textbf{18.8 \\addtiny{$\\pm$4.2}} & 0.2 \\addtiny{$\\pm$0.0} \\\\\n                               & medium & 0.2 \\addtiny{$\\pm$0.1} & 6.2 \\addtiny{$\\pm$2.4} & 0.1 \\addtiny{$\\pm$0.0} & \\cellcolor[gray]{0.9}\\textbf{13.5 \\addtiny{$\\pm$4.1}} & 0.1 \\addtiny{$\\pm$0.0} \\\\\n                               & medexp & 0.1 \\addtiny{$\\pm$0.0} & 7.0 \\addtiny{$\\pm$2.3} & 0.1 \\addtiny{$\\pm$0.0} & \\cellcolor[gray]{0.9}\\textbf{17.2 \\addtiny{$\\pm$4.7}} & 0.2 \\addtiny{$\\pm$0.0} \\\\\n                               & expert & 0.2 \\addtiny{$\\pm$0.1} & 2.7 \\addtiny{$\\pm$0.9} & 1.6 \\addtiny{$\\pm$0.5} & \\cellcolor[gray]{0.9}\\textbf{\\hphantom{0}6.1 \\addtiny{$\\pm$3.7}} & 0.1 \\addtiny{$\\pm$0.0} \\\\ \\bottomrule\n\\end{tabular}}\n\\vspace*{-5pt}\n\\end{table}\n\n\\subsection{Comparative Evaluation}\n\\label{subsec:comp_eval}\nWe now evaluate the five algorithms described in~\\Cref{subsec:baselines} on a total of fifteen datasets.\nTo provide a fair evaluation, we provide full training curves for each algorithm in~\\Cref{fig:d4rlp_comparison} and summarize final performance with error in~\\Cref{tab:all_comparison}.\nSince no online data collection is required, we measure progress through training via an ``offline epochs'' metric which we define in~\\Cref{app:hyperparams}.\n\n\\Cref{tab:all_comparison} shows a clear trend: Offline DV2 is the strongest on the random and mixed datasets, consisting of lower-quality but diverse data, DrQ+BC is the best on datasets with higher-quality but still widely-distributed data and pure BC outperforms on the high-quality narrowly-distributed expert data. We see from~\\Cref{tab:all_comparison} and~\\Cref{fig:d4rlp_comparison} that DrQ+BC is extremely stable across seeds and training steps and has the highest overall performance.\nCQL is also a strong baseline, especially on expert data, but requires significant hyperparameter tuning per dataset, often has high variance across seeds, and is also far slower than DrQ+BC to train.\nFinally, no algorithm achieves strong performance on the challenging humanoid datasets, mirroring the online RL challenges~\\citep{dreamerv2, drqv2}, with only the supervised BC and, by extension, DrQ+BC showing marginal positive returns (for more graphs, see \\cref{app:humanoid}).\n\nPerhaps surprisingly, Offline DV2 learns mid-level policies from random data on DMC environments.\nFurthermore, the random data are more challenging than their \\textsc{d4rl} equivalents because there is no early termination, and thus mostly consists of uninformative failed states; this shows the strength of model-based methods in extracting signal from large quantities of suboptimal data.\nOn the other hand, Offline DV2 is considerably weaker on the expert datasets that have narrow data distributions.\nFor these environments, we find the uncertainty penalty is uninformative, as discussed in \\cref{app:modeltrain}.\n\nTaking all these findings into consideration leads us to our first open problem, which we believe continued research using our benchmark can help to answer:\n\n\\hypothesis{Open Problem 1}{Can a single algorithm outperform both the model-free and model-based baselines, and produce strong performance across \\emph{all} offline datasets?}\n\\vspace*{5pt}\n\n\\subsubsection{Comparison to the LOMPO Benchmark}\n\\label{subsubsec:lompo_eval}\nFor a fair comparison to LOMPO, we also benchmark on the data used in \\citet{lompo} on the DMC Walker-Walk task. In the LOMPO benchmark, the datasets are limited to three types: \\{medium-replay, medium-expert and expert\\}.\nWe provide final scores in~\\Cref{tab:lompo_data}.\nWhile LOMPO struggles on \\textsc{v-d4rl}, it performs reasonably on its own benchmark.\nHowever, LOMPO is still outperformed by Offline DV2 on all datasets, whereas DrQ+BC is the best on two datasets.\n\nWe may explain the relative strength of LOMPO on this benchmark by noting that the medium-expert dataset used by \\citet{lompo} is described as consisting of the second half of the replay buffer after the agent reaches medium-level performance, thus containing far more diverse data than a bimodal \\textsc{d4rl}-style concatenation of two datasets.\nFurthermore, the expert data is far more widely distributed than that of a standard SAC expert, as we confirm in the statistics in~\\Cref{tab:lompo-stats} of~\\Cref{app:data_characteristics}.\n\n\n\\setlength{\\tabcolsep}{18.5pt}\n\\begin{table}[t]\n\\centering\n\\small\n\\vspace{-1mm}\n\\caption{\n    \\small{We confirm that our simple baselines outperform LOMPO on the original walker-walk data provided by \\citet{lompo}.\n    We report final performance mapped from $[0, 1000]$ to $[0, 100]$ averaged over 6 seeds.\\textsuperscript{1}\n    We show our baselines are more performant than LOMPO on their benchmark. CQL numbers taken from \\citep{lompo}, we were unable to verify these results using \\textsc{v-d4rl}-CQL.}\n}\n\\label{tab:lompo_data}\n\\begin{tabular}{ccccc}\n\\toprule\n\\textbf{LOMPO Dataset} & \\textbf{LOMPO} & \\textbf{Offline DV2} & \\textbf{DrQ+BC} & \\textbf{CQL} \\\\ \\toprule\nmedium-replay       & 61.3 \\addtiny{$\\pm$9.1\\hphantom{0}} & \\cellcolor[gray]{0.9}\\textbf{76.3 \\addtiny{$\\pm$3.1\\hphantom{0}}} & 31.1 \\addtiny{$\\pm$3.7\\hphantom{0}} & 14.7 \\\\\nmedium-expert       & 69.0 \\addtiny{$\\pm$24.1} & 72.3 \\addtiny{$\\pm$20.1} & \\cellcolor[gray]{0.9}\\textbf{73.3 \\addtiny{$\\pm$3.5\\hphantom{0}}} & 45.1 \\\\\nexpert              & 52.4 \\addtiny{$\\pm$35.7} & 59.4 \\addtiny{$\\pm$26.6} & \\cellcolor[gray]{0.9}\\textbf{90.8 \\addtiny{$\\pm$2.2\\hphantom{0}}} & 40.3 \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4mm}\n\\end{table}\n\\footnotetext[1]{ It is unclear how the original scores in \\citet{lompo} were normalized.}\n\\section{Desiderata for Offline Reinforcement Learning from Visual Observations}\n\\vspace{-5pt}\nA key ingredient in recent advances in deep learning is the use of large and diverse datasets, which are particularly prevalent in vision, to train models.\nSuch datasets enable the learning of \\emph{general} features that can be successfully transferred to downstream tasks, even when the original task bears little immediate similarity with the transferred task~\\citep{pretrainMedical, BertPretrain}.\nThis is a clear rationale for adopting visual observations in offline RL; by leveraging large quantities of diverse high-dimensional inputs, we should be able to learn generalizable features and agents for control.\nHowever, combining rich visual datasets with RL presents its own unique challenges.\nIn this section, we present important desiderata that highlight this, and conclude each with an open problem that requires further research.\\footnote{As CQL is quite sensitive to hyperparameters per environment, in the following sections we use the more robust Offline DV2 and DrQ+BC algorithms.}\n\n\\begin{figure*}[ht]\n\\centering\n\\vspace{-2mm}\n\\includegraphics[width=0.80\\textwidth]{figures\/exp_distracted\/distraction_all.pdf}\n\\begin{subfigure}[c]{0.14\\textwidth}\n\\vspace*{-145pt}\n\\centering\n\\includegraphics[width=\\textwidth, trim={0 0 0 0}, clip]{figures\/exp_distracted\/distraction.png}\n\\end{subfigure}\n\\vspace{-3mm}\n\\caption{\\small{Both DrQ+BC and Offline DV2 readily support training datasets with different distractions (mixture of original and shifted train). Offline DV2 additionally shows the ability to generalize to unseen distractions (shifted test) whereas DrQ+BC is more brittle. Return is normalized against unshifted performance without distractions from \\cref{tab:all_comparison} and averaged over 6 seeds. Unnormalized returns are provided in Tables~\\ref{tab:dv2-random-distractions} and~\\ref{tab:drq-medexp-distractions}.\n}}\n\\vspace{-6mm}\n\\label{fig:distraction_comp}\n\\end{figure*}\n\n\\subsection{Robustness to Visual Distractions}\n\\label{subsec:distractions}\n\\vspace{-4pt}\n\nOne desideratum for offline RL from visual observations is the ability to learn a policy from data collected under multiple different settings.\nFor example, quadrupedal robots deployed at different times of day (e.g., morning and night) will likely gather data having significantly different visual properties, such as brightness and range of visibility. Although the robot may produce similar behaviors in both deployments, superficial differences in visual outputs present a host of opportunities for the agent to learn spurious correlations that prevent generalization~\\citep{Song2020Observational, idaac}. \n\nA key opportunity that arises is the potential to \\emph{disentangle} sources of distractions through training on multiple settings, facilitating the learning of general features. By contrast, proprioceptive observations do not generally have distractions, as they are typically low-dimensional and engineered for the task of interest~\\citep{stateEngineered}. This also limits their ability to transfer, as it is unclear how to incorporate features learned under one set of agent geometries to another.\n\nTo test a simplified version of this challenge, we train our baseline agents using newly created datasets featuring varying visual augmentations from the Distracting Control Suite~\\citep{distractingcontrolsuite}.\nThis suite provides three levels of distractions (i.e., low, moderate, high), and each distraction represents a shift in the data distribution. We subsequently refer to the level of distraction as the ``shift severity''~\\citep{schneider2020improving}.\nThe offline datasets are then constructed as mixtures of samples from the original environment \\emph{without distractions} and samples from an environment with a \\emph{fixed distraction} level. Further details of how we generate this data are given in~\\Cref{app:data_generation}. The learned policies are then evaluated on test environments featuring unseen distractions of the same shift severity.\n\nWe compare the baseline algorithms, Offline DV2 and DrQ+BC on datasets that they excel on in \\Cref{sec:baselines} and \\Cref{subsec:large_datasets}: walker-walk random with 100,000 datapoints and cheetah-run medium-expert with 1 million respectively.\nThe returns normalized by unshifted performance are shown in~\\Cref{fig:distraction_comp}.\nOffline DV2 is able to accommodate datasets with mixed distractions and generalizes reasonably well to unseen test distractions, especially when trained with `low' and `moderate' levels of shift severity.\nSimilarly, DrQ+BC is robust to multiple training distractions, with little to no degradation in performance.\nHowever, the policy learned is brittle with respect to unseen distractions, and performs significantly worse on the test environments.\n\nOverall, both Offline DV2 and DrQ+BC represent strong baselines for mixed datasets.\nInterestingly, Offline DV2 demonstrates strong generalization to unseen distractions. This can be explained by generalization that occurs in the trained world-model, which uses a self-supervised loss; we discuss reasons behind this in~\\Cref{app:modelbasedextrap}.\nThis could be improved even further with recent reconstruction-free variants of DreamerV2~\\citep{dreaming, tpc} which have shown robustness to distractions.\nOn the other hand, we observe DrQ+BC generalizes poorly to unseen distractions, presenting a direction for future work using our datasets to learn robust \\emph{model-free} agents.\nThis directly leads us to our next open problem:\n\n\\vspace{-2pt}\n\\hypothesis{Open Problem 2}{How can we improve the generalization performance of offline model-free methods to unseen visual distractions?}\n\n\\subsection{Generalization Across Environment Dynamics}\n\\vspace{-3pt}\n\n\\label{subsec:dynamics_gen}\n\nAnother desideratum of offline RL from visual observations is learning policies that can generalize across multiple dynamics. This challenge manifests in three clear ways. Firstly, we will likely collect data from multiple agents that each have different dynamics, and must therefore learn a policy that can perform well when deployed on any robot that gathered the data (i.e., train time robustness). Secondly, we may be provided with asymmetric data, featuring scarce coverage of particular dynamics, and therefore require the ability to leverage data from more abundant sources (i.e., transferability). Thirdly, we may be presented with \\emph{unseen} dynamics at deployment time, and must therefore learn a policy that is robust to these changes (i.e., test time robustness).\n\nA key opportunity that arises in visual observations is the improved richness of the underlying dataset compared to proprioceptive data. For instance, \\emph{some dynamics changes may be visually obvious} (e.g., changed limb sizes, broken actuators), whereas in the proprioceptive setting such information may not be available.\nWithout this information, we must turn to meta-RL~\\cite{pearl, varibad} or HiPMDP~\\cite{hipmdp, zhang2021learning} approaches that try to infer the missing information from gathered trajectories, adding complexity to the RL process.\nIn contrast, this information can be contained explicitly in visual observations, and should allow adaption to a range of downstream tasks without complex inference methods.\n\nTo test this hypothesis, we consider two settings from the MTEnv benchmark~\\citep{Sodhani2021MTEnv} which adapts DMC: cheetah-run with modified torso length and walker-walk with modified leg length.\nWe follow an analogous approach to \\citep{zhang2021learning} where we consider eight different settings \\{A - H\\} ordered in terms of limb length, and construct new offline datasets using \\{B, C, F, G\\} in equal proportions as our training data.\nThe settings \\{A, H\\} are considered the \\emph{extrapolation} generalization environments and  \\{D, E\\} \\emph{interpolation} generalization. Data generation details are provided in~\\Cref{app:data_generation}.\n\n\\setlength{\\tabcolsep}{7pt}\n\\begin{wraptable}[13]{r}{5.5cm}\n\\centering\n\\small\n\\caption{\n    \\small{Evaluation on the DMC-Multitask benchmark using \\emph{medexp} data for Offline DV2, DrQ+BC and BC. Normalized performance from {[}0, 1000{]} to {[}0, 100{]} over 6 seeds. Our algorithms learn multitask policies from visual observations, with a slight generalization gap for extrapolation tasks.}}\n\\label{tab:all_multitask}\n\\vspace{-2mm}\n\\scalebox{0.66}{\n\\begin{tabular}{@{}lllccc@{}}\n\\toprule\n  \\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{Algorithm}}} &\n  \\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{Environment}}} &\n  \\multicolumn{3}{c}{\\textbf{Eval. Return}} \\\\ \\cmidrule(l){3-5} \n\\multicolumn{1}{c}{} &\n  \\multicolumn{1}{c}{} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Train\\\\ Tasks\\end{tabular}} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Test\\\\ Interp.\\end{tabular}} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Test\\\\ Extrap.\\end{tabular}} \\\\ \\midrule\n\\multirow{2}{*}{DrQ+BC}      & walker  & 90.8 & 91.4 & 65.1 \\\\\n                             & cheetah & 71.6 & 65.1 & 43.2 \\\\\n\\multirow{2}{*}{BC}          & walker  & 61.2 & 61.4 & 47.2 \\\\\n                             & cheetah & 69.7 & 61.3 & 39.6 \\\\\n\\multirow{2}{*}{Offline DV2} & walker  & 23.2 & 16.5 & 19.8 \\\\\n                             & cheetah &  8.2 &  7.2 &  9.6 \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4mm}\n}\n\\end{wraptable}\n\nWe evaluate Offline DV2 and DrQ+BC on medium-expert datasets of size 100,000 and 1 million respectively and show the results in~\\Cref{tab:all_multitask}.\nWe see that DrQ+BC learns policies that are suitable for transfer across multiple tasks in both walker and cheetah, and maintains that performance on the interpolation test environments.\nFor the extrapolation environments, we see an average drop of around 30\\%.\nWhile this may represent adequate performance, especially compared to a medium policy, it is a striking drop when compared to performance on in-distribution dynamics.\nThis suggests there is a dynamics generalization gap that remains for model-free methods when extrapolating, and represents clear opportunities for further research.\n\nOffline DV2 displays similar trends (results on random datasets are in~\\cref{app:model_random_multi}).\nOn the random data, Offline DV2 learns a similar quality policy to that on the base environment, but experiences no deterioration in performance on the test environments in walker or cheetah.\nThus, we demonstrate the sufficiency of both Offline DV2 and DrQ+BC as baselines in multitask offline RL \\emph{without any modification};\nmodel-based approaches further admit opportunities for zero-shot generalization~\\citep{ball2021augwm}.\n\nWe now contrast our work to that of~\\citep{zhang2021learning}, where a multitask policy was trained using a total of 3.2 million timesteps of online data collection.\nWhilst it is hard to compare offline and online results, our DrQ+BC algorithm uses less data, with 1 million total timesteps, and obtains similar extrapolation return on the walker environments.\nThis supports a similar conclusion reached by~\\citep{kurin2022defense} who show that our approach, simply minimizing the sum of the task losses, is drastically underestimated in the literature.\nAs noted before, we suffer a comparatively larger drop in performance, lending further evidence that closing this generalization gap should be prioritized.\n\nIn conclusion, we believe there are many further avenues for future research using these benchmarks; an immediate open problem we have identified is as follows:\n\\vspace*{-5pt}\n\\hypothesis{Open Problem 3}{How can we improve generalization to new dynamics that are not contained in the offline dataset?}\n\n\\vspace{-5pt}\n\\subsection{Improved Performance with Scale}\n\\label{subsec:large_datasets}\n\\vspace{-3pt}\n\n\\begin{wrapfigure}[11]{r}{6.65cm}\n\\vspace{-13.2mm}\n\\centering\n\\small\n\\includegraphics[width=0.5\\textwidth, trim={0 5 0 0}, clip]{figures\/exp2\/size_ablations.pdf}\n\\vspace{-1mm}\n\\includegraphics[width=0.5\\textwidth, trim={5 100 5 100}, clip]{figures\/exp2\/size_abl_legend.pdf}\n\\caption{\\small{Sensitivity analysis on dataset size for both Offline DV2 (walker-walk random) and DrQ+BC (cheetah-run medexp). Both methods scale with more data, but receive diminishing returns past 2$\\times$ the original size. Performance averaged over 4 seeds.}}\n\\label{fig:size_comparison}\n\\end{wrapfigure}\n\nLearning from large datasets presents huge opportunities for learning general agents for control.\nTo make use of them, we need to understand how our baselines scale with dataset size.\nWe analyze our base choice of 100,000 observations for \\textsc{v-d4rl} in~\\Cref{fig:size_comparison}, where we vary the size of the walker-walk random dataset for Offline DV2 and the cheetah-run medexp dataset for DrQ+BC in the range of $\\{ 0.25\\times,\\dots , 4\\times\\}$ the size of the original dataset.\nWe observe a monotonic increase in the performance of both Offline DV2 and DrQ+BC with increasing dataset size but hit diminishing returns past 2$\\times$ the original size. \nWe note, perhaps surprisingly, that Offline DV2 can reach $\\approx$500 total return from random data that average 10$\\times$ less.\n\n\\setlength{\\tabcolsep}{10.2pt}\n\\begin{table}[t]\n\\centering\n\\small\n\\caption{\n    \\small{The reinforcement learning algorithms readily scale to higher dataset sizes, compared to supervised behavioral cloning, \n    with a particular benefit to the \\emph{medexp} and \\emph{expert} datasets for DrQ+BC and the \\emph{random} and \\emph{medium} datasets for Offline DV2.\n    Results are averaged over 6 seeds, with one standard deviation given as error.\n    The evaluated return is mapped from $[0, 1000]$ to $[0, 100]$, and the \\emph{mixed} dataset is excluded. The \\emph{medexp} dataset is a concatenation of the \\emph{medium} and \\emph{expert} datasets.}\n}\n\\label{tab:500k_exps}\n\\scalebox{0.85}{\n\\begin{tabular}{llcccccc}\n\\toprule\n\\multicolumn{2}{c}{\\multirow{2}{*}{\\textbf{Environment}}} & \\multicolumn{2}{c}{\\textbf{Offline DV2}} & \\multicolumn{2}{c}{\\textbf{DrQ+BC}}              & \\multicolumn{2}{c}{\\textbf{BC}}                    \\\\ \\cline{3-8} \n\\multicolumn{2}{c}{}                                      & \\textbf{100K}           & \\textbf{500K}  & \\textbf{100K}           & \\textbf{500K}           & \\textbf{100K}           & \\textbf{500K}            \\\\ \\midrule\n\\multirow{4}{*}{walker}           & random           & 28.7 \\addtiny{$\\pm$13.0} & 49.9 \\addtiny{$\\pm$1.6\\hphantom{0}} & 5.5 \\addtiny{$\\pm$0.9}  & 3.5 \\addtiny{$\\pm$0.6}  & 2.0 \\addtiny{$\\pm$0.2}  & 2.1 \\addtiny{$\\pm$0.3}   \\\\\n                                       & medium           & 34.1 \\addtiny{$\\pm$19.7} & 61.3 \\addtiny{$\\pm$10.9} & 46.8 \\addtiny{$\\pm$2.3\\hphantom{0}} & 51.0 \\addtiny{$\\pm$1.1\\hphantom{0}} & 40.9 \\addtiny{$\\pm$3.1\\hphantom{0}} & 40.9 \\addtiny{$\\pm$3.0\\hphantom{0}}  \\\\\n                                       & medexp           & 43.9 \\addtiny{$\\pm$34.4} & 38.9 \\addtiny{$\\pm$28.1} & 86.4 \\addtiny{$\\pm$5.6\\hphantom{0}} & 94.1 \\addtiny{$\\pm$2.0\\hphantom{0}} & 47.7 \\addtiny{$\\pm$3.9\\hphantom{0}} & 48.8 \\addtiny{$\\pm$5.3\\hphantom{0}}  \\\\\n                                       & expert           & 4.8 \\addtiny{$\\pm$0.6}   & 7.1 \\addtiny{$\\pm$5.3} & 68.4 \\addtiny{$\\pm$7.5\\hphantom{0}} & 94.2 \\addtiny{$\\pm$2.3\\hphantom{0}} & 91.5 \\addtiny{$\\pm$3.9\\hphantom{0}} & 95.1 \\addtiny{$\\pm$2.5\\hphantom{0}}  \\\\ \\midrule\n\\multirow{4}{*}{cheetah}           & random           & 31.7 \\addtiny{$\\pm$2.7\\hphantom{0}}  & 40.8 \\addtiny{$\\pm$4.2\\hphantom{0}} & 5.8 \\addtiny{$\\pm$0.6}  & 10.6 \\addtiny{$\\pm$0.7\\hphantom{0}} & 0.0 \\addtiny{$\\pm$0.0}  & 0.0 \\addtiny{$\\pm$0.0}   \\\\\n                                       & medium           & 17.2 \\addtiny{$\\pm$3.5\\hphantom{0}}  & 39.2 \\addtiny{$\\pm$14.4} & 53.0 \\addtiny{$\\pm$3.0\\hphantom{0}} & 57.3 \\addtiny{$\\pm$1.2\\hphantom{0}} & 51.6 \\addtiny{$\\pm$1.4\\hphantom{0}} & 52.9 \\addtiny{$\\pm$1.3\\hphantom{0}}  \\\\\n                                       & medexp           & 10.4 \\addtiny{$\\pm$3.5\\hphantom{0}}  & 9.7 \\addtiny{$\\pm$5.0} & 50.6 \\addtiny{$\\pm$8.2\\hphantom{0}} & 79.1 \\addtiny{$\\pm$5.6\\hphantom{0}} & 57.5 \\addtiny{$\\pm$6.3\\hphantom{0}} & 69.6 \\addtiny{$\\pm$10.6} \\\\\n                                       & expert           & 10.9 \\addtiny{$\\pm$3.2\\hphantom{0}}  & 11.3 \\addtiny{$\\pm$4.7\\hphantom{0}} & 34.5 \\addtiny{$\\pm$8.3\\hphantom{0}} & 75.3 \\addtiny{$\\pm$7.5\\hphantom{0}} & 67.4 \\addtiny{$\\pm$6.8\\hphantom{0}} & 87.8 \\addtiny{$\\pm$1.9\\hphantom{0}}  \\\\ \\midrule\n\\multicolumn{2}{c}{Average Overall}                       & 22.7 \\addtiny{$\\pm$10.1} & 32.3 \\addtiny{$\\pm$9.3\\hphantom{0}} & 43.9 \\addtiny{$\\pm$4.6\\hphantom{0}} & 58.1 \\addtiny{$\\pm$2.6\\hphantom{0}} & 44.8 \\addtiny{$\\pm$3.2\\hphantom{0}} & 49.7 \\addtiny{$\\pm$3.1\\hphantom{0}}   \\\\ \\midrule\n\\multicolumn{2}{c}{Percentage Gain}                       & \\multicolumn{2}{c}{+42.1\\%} & \\multicolumn{2}{c}{+32.5\\%}                        & \\multicolumn{2}{c}{+10.8\\%}                         \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table}\n\nFor the walker and cheetah datasets with fixed distributions---random, medium, medium-expert and expert---\\Cref{tab:500k_exps} shows an average increase of 42.1\\% for Offline DV2 and 32.5\\% for DrQ+BC compared to 10.8\\% for BC when we scale the dataset size to 500,000, showing that the reinforcement learning algorithms make far better use of increased data than supervised behavioral cloning.\nHowever, a crucial difference between Offline DV2 and DrQ+BC is that DrQ+BC handles larger offline datasets far more readily.\nDrQ+BC policy training for the same number of epochs on 500,000 and 100,000 observations takes 8 and 1.6 hours respectively on a V100 GPU.\nThis is significantly quicker than Offline DV2, which takes 10 hours to train on 100,000 observations; we discuss this further in~\\Cref{app:modeltrain}.\nThis significant computational discrepancy leads to a clear open problem:\n\\vspace*{-4pt}\n\\hypothesis{Open Problem 4}{How can we scale model-based methods to larger datasets?}\n\n\\section{Related Work}\n\\label{sec:related_work}\nThere has been significant recent progress in offline RL, giving rise to many benchmarks. We list several here; to our knowledge, no contemporary works feature tasks related to distractions or changed dynamics.\n\n\\textbf{Benchmarks for continuous control on states.} \\textsc{d4rl}~\\citep{d4rl} is the most prominent benchmark for continuous control with proprioceptive states. The large variety of data-distributions have allowed for comprehensive benchmarking~\\citep{cql, morel, mopo, yu2021combo, kostrikov2021offline} and understanding the strengths and weaknesses of offline algorithms. Our work aims to establish a similar benchmark for vision-based tasks. RL Unplugged~\\cite{gulcehre2020rl} also provides data on the DMControl Suite, but focuses on proprioceptive states.\n\n\\textbf{Analysis on characteristics of offline datasets.} Recent work ~\\citep{florence2021implicit} has sought to understand when offline RL algorithms outperform behavioral cloning in the proprioceptive setting. \\citet{kumar2021should} recommend BC for many settings but showed theoretically that offline RL was preferable in settings combining expert and suboptimal data, which we confirm in Tables~\\ref{tab:all_comparison}~and~\\ref{tab:500k_exps}.\n\n\\textbf{Vision-based discrete control datasets.} Whilst there has been a lack of suitable benchmarks for vision-based offline continuous control, vision-based datasets for discrete control have been created for Atari~\\citep{agarwal2020optimistic}.\nHowever, at 50M samples per environment, this poses \\href{https:\/\/github.com\/google-research\/batch_rl\/issues\/10}{significant challenges}; we believe that \\textsc{v-d4rl}'s 100K benchmark represents a far more approachable challenge for practitioners.\n\n\\textbf{Offline vision-based robotics.} Offline vision-based learning is an active area of robotics research.\nIn~\\citep{kalashnikovQTOPT}, policies are fine-tuned on real interactions, after offline pre-training on visual data of robot interactions.\n\\citep{chebotaractionablemodels21} follows a similar setup, and explores the representation learning for controllers.\nOur work complements this, is not focused on goal-conditioned learning, and is open source.\n\n\\section{Conclusion}\nIn this paper, we took the first steps towards establishing benchmarking tasks and competitive baselines for offline reinforcement learning from visual observations.\nUntil now, work in this space has been nascent, with ad-hoc analyses leading to unclear comparisons.\nTo address the lack of meaningful evaluations and comparative analyses in this space, we provided a set of straightforward and standardized benchmarking tasks that follow popular low-dimensional equivalent experiment setups and presented competitive model-based and model-free baselines.\nWe analyzed key factors that help explain the performance of these approaches, while also demonstrating their ability to generalize in more challenging settings that are unique to visual observations.\nWe hope this work will be useful to practitioners and researchers alike and that it will provide a springboard for developing offline reinforcement learning methods for real-world continuous-control problems and spark further progress in this space.\n\n\n\\section*{\\LARGE \\centering Supplementary Material}\n\\label{sec:appendix}\n\n\\vspace{8pt}\n{\\hrule height 0.3mm}\n\\vspace{24pt}\n\n\\crefalias{section}{appsec}\n\\crefalias{subsection}{appsec}\n\\crefalias{subsubsection}{appsec}\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\section*{Table of Contents}\n\\vspace*{-10pt}\n\\startcontents[sections]\n\\printcontents[sections]{l}{1}{\\setcounter{tocdepth}{2}}\n\n\\clearpage\n\n\\section{Offline Dataset Characteristics}\n\\label{app:data_characteristics}\nWe provide explicit statistics on the returns of each episode for the datasets used in our main evaluation. This provides a reasonable proxy to how diverse each dataset is.\n\n\\setlength{\\tabcolsep}{5.0pt}\n\\begin{table}[ht]\n\\centering\n\\caption{Full summary statistics of per-episode return in the \\textsc{v-d4rl} benchmark.}\n\\label{tab:d4rlp-stats}\n\\begin{tabular}{@{}llcccccccc@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Dataset}} &\n  \\textbf{Timesteps} &\n  \\textbf{Mean} &\n  \\textbf{Std. Dev.} &\n  \\textbf{Min.} &\n  \\textbf{P25} &\n  \\textbf{Median} &\n  \\textbf{P75} &\n  \\textbf{Max.} \\\\ \\midrule\n\\multirow{5}{*}{walker}      & random & 100K & 42.3  & 8.7   & 30.0  & 34.9  & 41.3  & 46.8  & 74.6  \\\\\n                             & mixed  & 100K & 144.5 & 155.9 & 10.9  & 44.3  & 69.4  & 162.4 & 604.9 \\\\\n                             & medium & 100K & 439.6 & 48.4  & 176.2 & 423.1 & 445.5 & 466.7 & 538.0 \\\\\n                             & medexp & 200K & 704.1 & 267.7 & 176.2 & 445.5 & 538.0 & 969.1 & 990.6 \\\\\n                             & expert & 100K & 969.8 & 12.4  & 909.2 & 963.9 & 969.1 & 979.5 & 990.6 \\\\ \\midrule\n\\multirow{5}{*}{cheetah}     & random & 100K & 6.6   & 2.6   & 1.1   & 4.7   & 6.3   & 8.4   & 16.3  \\\\\n                             & mixed  & 200K & 191.2 & 144.6 & 2.5   & 48.3  & 191.9 & 303.4 & 473.8 \\\\\n                             & medium & 100K & 523.8 & 25.5  & 325.3 & 509.1 & 524.2 & 538.3 & 578.3 \\\\\n                             & medexp & 200K & 707.0 & 184.9 & 325.3 & 524.2 & 578.3 & 894.1 & 905.7 \\\\\n                             & expert & 100K & 891.1 & 11.2  & 843.0 & 886.9 & 894.2 & 898.5 & 905.7 \\\\ \\midrule\n\\multirow{5}{*}{humanoid}    & random & 100K &   1.1 &   0.8 &   0.0 &   0.5 &   1.0 &   1.5 &   5.7 \\\\\n                             & mixed  & 600K &   275.7 &  176.1 &  0.0 & 93.7 &  341.7 & 423.1 & 529.0 \\\\\n                             & medium & 100K & 573.0   & 16.7  &  526.5 & 560.6  & 572.9  & 584.9  & 609.4  \\\\\n                             & medexp & 200K &  715.9  &  146.1 & 526.5  & 572.9  &  620.6 & 877.4  & 889.8  \\\\\n                             & expert & 100K &  858.1  &  42.4 &  631.8 &  846.4 & 877.6  & 885.5  & 889.8  \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nWe compare this to the LOMPO datasets and find that they are more widely distributed, due to the differing data-collection method. \n\n\\setlength{\\tabcolsep}{6.0pt}\n\\begin{table}[ht]\n\\centering\n\\caption{Summary statistics of per-episode return in the LOMPO DMC walker-walk datasets.}\n\\label{tab:lompo-stats}\n\\begin{tabular}{@{}llcccccccc@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Dataset}} &\n  \\textbf{Timesteps} &\n  \\textbf{Mean} &\n  \\textbf{Std. Dev.} &\n  \\textbf{Min.} &\n  \\textbf{P25} &\n  \\textbf{Median} &\n  \\textbf{P75} &\n  \\textbf{Max.} \\\\ \\midrule\n\\multirow{3}{*}{walker} & mixed  & 100K & 208.9 & 144.1 & 33.3  & 75.1  & 172.4 & 340.5 & 496.7 \\\\\n                             & medexp & 100K & 674.4 & 92.9  & 501.3 & 596.8 & 679.8 & 752.5 & 869.1 \\\\\n                             & expert & 100K & 920.6 & 81.6  & 17.9  & 905.9 & 950.3 & 957.9 & 987.0 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nAs we see in \\Cref{tab:lompo-stats} and \\Cref{fig:lompo_histogram}, the LOMPO walker-walk expert dataset has a standard deviation roughly 8x higher than our expert dataset and has an extremely wide [min, max] range. Furthermore, whilst our medexp dataset is bimodal, the LOMPO medexp dataset's returns are a continuous progression. This reflects that the LOMPO data is sampled from the second half of a replay buffer after medium-level performance is attained, akin to the medium-replay (mixed) datasets.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/histogram\/lompo_histogram_with_legend.pdf}\n\\caption{\\small{Comparison of the episodic returns from the LOMPO and \\textsc{v-d4rl}. We see LOMPO has significantly more diversity in the medium-expert and expert datasets. Note that there is a single episode in the LOMPO expert dataset which has a low return of $17.9$.}}\n\\label{fig:lompo_histogram}\n\\vspace{-2mm}\n\\end{figure*}\n\n\n\\section{Offline Data Generation Details}\n\\label{app:data_generation}\n\n\\subsection{Standard Datasets}\n\\label{subsec:standard_gen}\n\nFor the standard offline medium and expert datasets, we first train SAC~\\cite{sac-v2} policies on the proprioceptive states until convergence, taking checkpoints every 10,000 frames of interaction. We use a frame skip of 2, the default in other state-of-the-art vision RL algorithms~\\citep{dreamerv2,drqv2}.\nWe define expert policies as those that have converged in the limit. For DMControl tasks, this typically means near 1,000 reward on the environment they were trained on.\nWe define medium policies as the first saved agent during training that is able to consistently achieve above 500 reward in the environment.\nWe confirm these thresholds are reasonable, as we observe a noticeable gap between the behavior of the medium and expert level policies.\n\nIn order to generate the offline visual observations, we deploy the proprioceptive agents in the environment, and save the visual observation rendered from the simulator instead of the proprioceptive state.\nThis provides us with the flexibility to \\emph{generate observations of any size} without having to retrain for that resolution (e.g., $84\\times84$ or $64\\times64$).\nAs was done in D4RL, we generate data using a stochastic actor, which involves sampling actions from the Gaussian posterior distribution of the SAC policy, featuring a parameterized variance head that determines the amount of action stochasticity at each state.\n\nFor the mixed datasets, we simply store the replay buffer of the medium agent when it finishes training, and convert the proprioceptive observations into visual observations. The cheetah-run mixed dataset is larger because the SAC agent takes longer to reach medium-level performance.\n\n\\subsection{Visually Distracted Datasets}\nThe visual distractions from the Distracting Control Suite do not manifest themselves in the proprioceptive states, so we can naturally generate the shifted visual observations by simply rendering the distractions on top of the existing visual observations.\nThus, we can simply use the same ``medium'' and ``expert'' level policies as before, trained on proprioceptive states.\n\n\\subsection{Multitask Datasets}\nFor the multitask dataset, we follow the same procedure described in~\\Cref{subsec:standard_gen} but instead train a SAC on the proprioceptive states from the modified tasks (e.g., \\{A-H\\}).\nThe medium and expert policies are defined in the same way.\n\n\\subsection{Choice of Behavioral Policy for Offline Data}\n\nInterestingly, we found that using online DrQ-v2 as the offline behavioral policy made the tasks significantly easier to learn for all agents. This suggests that the proprioceptive agent may learn behavior modes that are less biased towards being easy under vision-based methods; for instance, DrQ-v2 may be biased towards behavior modes that induce fewer visual occlusions compared to proprioceptive SAC.\n\n\\subsection{Broader Issues with Visual Data}\n\\label{appsec:societal}\n\nLarge datasets consisting of images often contain systematic biases, which can damage generalization.\nThe datasets constructed in this paper are all synthetic from simulated reinforcement learning environments.\nHowever, as we move towards applying offline RL from visual observations to real-world tasks, it is important to take these potential dangers into account and extend existing work in algorithmic fairness from computer vision to our setting.\n\n\n\\section{Algorithmic Details}\n\\label{app:algodetails}\n\nWe provide additional details for both algorithms here and indicate where our modifications have been made.\n\n\\subsection{Offline DV2}\n\\label{appsubsec:offline_dv2}\nIn the offline setting, it suffices to simply perform one phase of model training and one phase of policy training for the DreamerV2~\\citep{dreamerv2} algorithm.\nEach episode in the offline dataset is ordered sequentially to facilitate sequence learning.\nTo this instantiation of DreamerV2, we simply add a reward penalty corresponding to the mean disagreement of the dynamics ensemble.\nDuring standard DreamerV2 policy training, imagined latent trajectories $\\left\\{\\left(s_{\\tau}, a_{\\tau}\\right)\\right\\}_{\\tau=t}^{t+H}$ are assigned reward $r_{\\tau} = \\mathrm{E}\\left[q_{\\theta}\\left( \\cdot \\mid s_{\\tau}\\right)\\right]$ according to the mean output of the reward predictor.\nThe imagined latent states $s_t$ consist of a deterministic component $h_t$, implemented as the recurrent state of a GRU, and a stochastic component $z_t$ with categorical distribution.\nThe logits of the categorical distribution are computed from an ensemble (with input $h_t$) over which we compute the mean disagreement.\n\n\\subsection{DrQ+BC}\n\\label{appsubsec:drqbc}\nHere, we simply modify the policy loss term in DrQ-v2~\\citep{drqv2} to match the loss given in~\\citet{td3bc}. Following the notation from~\\citet{drqv2}, the DrQ-v2 actor $\\pi_\\phi$ is trained with the following loss: \n$$\n\\mathcal{L}_{\\phi}(\\mathcal{D})=-\\mathbb{E}_{\\boldsymbol{s}_{t} \\sim \\mathcal{D}}\\left[Q_{\\theta}\\left(\\boldsymbol{h}_{t}, \\boldsymbol{a}_{t}\\right)\\right]\n$$\nwhere $\\boldsymbol{h}_{t}=f_{\\xi}\\left(\\operatorname{aug}\\left(\\boldsymbol{s}_{t}\\right)\\right)$ is the encoded augmented visual observation, $\\boldsymbol{a}_{t}=\\pi_{\\phi}\\left(\\boldsymbol{h}_{t}\\right)+\\epsilon$ is the action with clipped noise to smooth the targets $\\epsilon \\sim \\operatorname{clip}\\left(\\mathcal{N}\\left(0, \\sigma^{2}\\right),-c, c\\right)$. Note that we also do not update encoder weights with the policy gradient. We also train an ensemble of two fully connected Q-networks, which both use features from a single encoder, and take their minimum when calculating target values and actor losses. The resultant algorithm can be viewed as TD3 \\citep{td3}, but with decaying smoothing noise parameters $c$.\n\nIn DrQ+BC, this loss becomes:\n$$\n\\mathcal{L}_{\\phi}(\\mathcal{D})=-\\mathbb{E}_{\\boldsymbol{s}_{t}, \\boldsymbol{a}_{t} \\sim \\mathcal{D}}\\left[\\textcolor{violet}{\\lambda} Q_{\\theta}\\left(\\boldsymbol{h}_{t}, \\boldsymbol{a}_{t}\\right) \\textcolor{violet}{-(\\pi_{\\phi}\\left(\\boldsymbol{h}_{t}\\right)-\\boldsymbol{a}_{t})^{2}}\\right]\n$$\nwhere $\\lambda=\\frac{\\alpha}{\\frac{1}{N} \\sum_{\\left(h_{i}, a_{i}\\right)}\\left|Q\\left(h_{i}, a_{i}\\right)\\right|}$ is an adaptive normalization term computed over minibatches. $\\alpha$ is a behavioral cloning weight, always set to 2.5 in~\\citet{td3bc}, which we also adopt. We experimented performing an extensive grid search over $\\alpha$, but did not observe any noticeable benefit in our offline datasets by deviating away from the default value.\n\n\\subsection{CQL}\n\\label{appsubsec:cql}\nOur CQL implementation was also built on top of the DrQv2 codebase for comparability.\nWe use the CQL($\\mathcal{H}$) objective with fixed weight, as we found this to be the most performant.\nThis corresponds to choosing the KL-divergence to a uniform prior as the regularizer $\\mathcal{R}(\\mu)$ in~\\citet{cql}. \nConcretely, the Q-function objective becomes\n$$\n\\min _{Q} \\alpha_{\\textrm{CQL}} \\mathbb{E}_{\\mathbf{s} \\sim \\mathcal{D}}\\left[\\log \\sum_{\\mathbf{a}} \\exp (Q(\\mathbf{s}, \\mathbf{a}))-\\mathbb{E}_{\\mathbf{a} \\sim \\hat{\\pi}_{\\beta}(\\mathbf{a} \\mid \\mathbf{s})}[Q(\\mathbf{s}, \\mathbf{a})]\\right]+\\frac{1}{2} \\mathbb{E}_{\\mathbf{s}, \\mathbf{a}, \\mathbf{s}^{\\prime} \\sim \\mathcal{D}}\\left[\\left(Q-\\hat{\\mathcal{B}}^{\\pi_{k}} \\hat{Q}^{k}\\right)^{2}\\right]\n$$\nwhere $\\alpha_{\\textrm{CQL}}$ is a trade-off factor, $\\hat{\\pi}_{\\beta}$ refers to the empirical behavioral policy and $\\hat{\\mathcal{B}}^{\\pi_{k}}$ to the empirical Bellman operator that backs up a single sample.\nThis is approximated by taking gradient steps and sampling actions from the given bounds.\n\n\\section{Hyperparameter and Experiment Setup}\n\\label{app:hyperparams}\n\\subsection{Offline DV2}\nOur Offline DV2 implementation was built on top of the official DreamerV2 repository at: \\url{https:\/\/github.com\/danijar\/dreamerv2} with minor modifications.\nThe code was released under the MIT License.\n\\Cref{tab:dv2_hyperparams} lists the hyperparameters used for Offline DV2.\nFor other hyperparameter values, we used the default values in the DreamerV2 repository.\n\n\\setlength{\\tabcolsep}{25pt}\n\\begin{table}[ht!]\n\\centering\n\\caption{Offline DV2 hyperparameters.}\n\\label{tab:dv2_hyperparams}\n\\begin{tabular}{l|c}\n\\toprule\n\\textbf{Parameter}                    & \\textbf{Value(s)}   \\\\ \\hline\nensemble member count (K)             & 7                   \\\\\nimagination horizon (H)               & 5                   \\\\\nbatch size                            & 64                   \\\\\nsequence length  (L)                  & 50                   \\\\\naction repeat                         & 2                   \\\\\nobservation size                      & [64, 64]              \\\\\ndiscount ($\\gamma$)                   & 0.99              \\\\\noptimizer                             & Adam              \\\\\nlearning rate                         & \\{model = $3\\times 10^{-4}$, actor-critic = $8\\times 10^{-5}$\\}              \\\\\nmodel training epochs                 & 800                   \\\\\nagent training epochs                 & 2,400                   \\\\\nuncertainty penalty                   & mean disagreement  \\\\\nuncertainty weight ($\\lambda$)        & in [3, 10]  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nWe found a default value of $\\lambda=10$ works for most settings. The only settings where this changes are $\\lambda=3$ for both random datasets and $\\lambda = 8$ for the walker-walk mixed dataset.\n\nFor the penalty choice, we chose mean disagreement of the ensemble because it comprises one half of the ensemble variance, which was shown to be an optimal choice for offline model-based reinforcement learning in~\\citet{lu2021revisiting}. We found that the other component of the ensemble variance, the average variance over the ensemble, was uninformative and so discarded it.\n\n\\subsection{DrQ+BC}\nOur DrQ+BC implementation was built on top of the official DrQ-v2 repository at: \\url{https:\/\/github.com\/facebookresearch\/drqv2}.\nThe code was released under the MIT License.\n\\Cref{tab:drq_hyperparams} lists the hyperparameters used for DrQ+BC.\nFor other hyperparameter values, we used the default values in the DrQ-v2 repository. \nDue to the size of some of our offline datasets, we found the default replay buffer would not scale to the offline datasets.\nThus, we used a NumPy \\cite{harris2020array} array implementation instead.\n\n\\setlength{\\tabcolsep}{46.0pt}\n\\begin{table}[ht!]\n\\centering\n\\vspace{-3mm}\n\\caption{DrQ+BC hyperparameters.}\n\\label{tab:drq_hyperparams}\n\\begin{tabular}{l|c}\n\\toprule\n\\textbf{Parameter}                    & \\textbf{Value}   \\\\ \\hline\nbatch size                            & 256                   \\\\\naction repeat                         & 2                   \\\\\nobservation size                      & [84, 84]              \\\\\ndiscount ($\\gamma$)                             & 0.99              \\\\\noptimizer                             & Adam              \\\\\nlearning rate                             & $1\\times 10^{-4}$              \\\\\nagent training epochs                 & 256                   \\\\\n$n$-step returns.                     & 3                    \\\\\nExploration stddev. clip              & 0.3                   \\\\\nExploration stddev. schedule.         & linear(1.0, 0.1, 500000) \\\\\nBC Weight ($\\alpha$)                  & 2.5  \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-2mm}\n\\end{table}\n\nWe tuned $\\alpha$ within \\{1.5, 2.5, 3.5\\} but as~\\citet{td3bc} found, we did not observe any noticeable benefit from deviating away from the default value for $\\alpha$.\n\n\\subsection{Behavioral Cloning}\nOur BC implementation shares the exact same policy network and hyperparameters in DrQ+BC but just minimizes MSE on the offline data. Consequently, we must also optimize the learned encoder using the supervised learning loss (unlike in DrQ+BC, where the TD-loss only contributes to the encoder representation learning, and not the policy loss).\n\n\\subsection{CQL}\nSimilarly to BC, our CQL implementation is also based on the same networks and hyperparameters in DrQ+BC.\nCQL introduces one extra hyperparameter, the trade-off factor $\\alpha_{\\textrm{CQL}}$.\nWe perform a hyperparameter sweep for this over the range: $\\{ 0.5, 1, 2, 5, 10, 20\\}$.\nWe chose the following values per environment:\n\n\\setlength{\\tabcolsep}{58.0pt}\n\\begin{table}[ht]\n\\centering\n\\caption{CQL trade-off factor per environment for Walker and Cheetah. Humanoid omitted as all choices performed equally.}\n\\label{tab:cql-alpha}\n\\begin{tabular}{@{}llc@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Dataset}} &\n  \\textbf{Trade-off Factor ($\\alpha_{\\textrm{CQL}}$)} \\\\ \\midrule\n\\multirow{5}{*}{walker}      & random  & 0.5 \\\\\n                             & mixed   & 0.5 \\\\\n                             & medium  & 2 \\\\\n                             & medexp  & 2 \\\\\n                             & expert  & 5 \\\\ \\midrule\n\\multirow{5}{*}{cheetah}     & random  & 0.5 \\\\\n                             & mixed   & 0.5 \\\\\n                             & medium  & 10 \\\\\n                             & medexp  & 1 \\\\\n                             & expert  & 20 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{LOMPO}\nFor our LOMPO evaluation, we used the official repository at: \\url{https:\/\/github.com\/rmrafailov\/LOMPO}.\nThe code was open-sourced without license.\nWe perform a hyperparameter search over the uncertainty weight $\\lambda$ in the range $\\{ 1, 5 \\}$. The default value used in~\\citet{lompo} is $\\lambda=5$, but we found that $\\lambda=1$ worked better for random datasets.\nThe accompanying data for the DMC Walker-Walk data from \\citet{lompo} was very kindly provided by the authors.\n\n\\subsection{Computational Cost}\n\\label{appsubsec:computation}\nThe experiments in this paper were run on NVIDIA V100 GPUs. On the standard \\textsc{v-d4rl} 100K datasets, DrQ+BC took 1.6 hours, Offline DV2 took 10 hours, and CQL took 12 hours.\n\nSince we wish to compare several offline algorithms using the same dataset, we define a notion of ``offline epoch'' for all algorithms to show training performance over time. We simply normalize the total number of gradient steps, so training progress falls within $[0, 1000]$.\n\n\\section{Further Tabular Results and Training Curves}\n\n\\subsection{Training Curves for the Humanoid Environment}\n\\label{app:humanoid}\nWe include additional training curves for the Humanoid environment, as in \\cref{fig:d4rlp_comparison} for all our evaluated algorithms in \\cref{fig:humanoid_comparison}.\nAs we noted previously, only the supervised BC and by extension DrQ+BC achieve meaningful return on this environment.  \n\n\\begin{figure*}[ht]\n\\centering\n\\small\n\\vspace{-1mm}\n\\includegraphics[width=\\textwidth, trim={0 0 0 0}, clip]{figures\/exp1\/humanoid_100k.pdf}\n\\includegraphics[width=0.5\\textwidth, trim={0 100 0 100}, clip]{figures\/exp1\/combined_legend_100k.pdf}\n\\vspace{-2mm}\n\\caption{\n    \\small{Rigorous comparison on the humanoid datasets from the \\textsc{v-d4rl} benchmark, each setting is averaged over 6 seeds with error bar showing one standard deviation. Total gradient steps are normalized under epochs, and we plot the un-normalized evaluated return.}\n}\n\\label{fig:humanoid_comparison}\n\\end{figure*}\n\n\\subsection{Full Tabular Results for Experiments with Distractions}\nWe give the full un-normalized results for \\Cref{subsec:distractions} in \\Cref{tab:dv2-random-distractions} and \\Cref{tab:drq-medexp-distractions}. The highlighted base results are the same as in \\Cref{tab:all_comparison} for Offline DV2 and \\Cref{tab:500k_exps} for DrQ+BC. The Offline DV2 results use 100K datapoints, and the DrQ+BC results use 1 million, in light of the discussion in \\Cref{subsec:large_datasets}.\n\\setlength{\\tabcolsep}{23.0pt}\n\\begin{table}[ht]\n\\centering\n\\caption{Offline DV2 shows surprisingly good generalization to unseen distractions and can handle multitask datasets. Final mean performance is averaged over 6 seeds and base undistracted performance is highlighted. The environment used is walker-walk random.}\n\\label{tab:dv2-random-distractions}\n\\scalebox{0.9}{\n\\begin{tabular}{lcccc}\n\\toprule\n\\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Shift Severity\\end{tabular}}}} &\n  \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\% Shifted\\end{tabular}}} &\n  \\multicolumn{3}{c}{\\textbf{Eval. Return}} \\\\ \\cline{3-5} \n\\multicolumn{1}{c}{} &\n   &\n  \\textbf{Original} &\n  \\textbf{Dis. Train} &\n  \\textbf{Dis. Test} \\\\ \\midrule\n\\multirow{5}{*}{low}   & 0\\%   & \\cellcolor[gray]{0.9}28.7 & 25.9 & 20.0 \\\\\n                        & 25\\%  & 28.1 & 22.0 & 14.0 \\\\\n                        & 50\\%  & 22.5 & 16.0 & 17.7 \\\\\n                        & 75\\%  & 16.3 & 21.9 & 20.1 \\\\\n                        & 100\\% & 29.4 & 29.7 & 18.6 \\\\ \\midrule\n\\multirow{5}{*}{moderate} & 0\\%   & \\cellcolor[gray]{0.9}28.7 & 21.0 & 15.1 \\\\\n                        & 25\\%  & 28.6 & 19.1 & 14.9 \\\\\n                        & 50\\%  & 19.8 & 20.5 & 11.2  \\\\\n                        & 75\\%  & 24.4 & 20.0 & 11.9 \\\\\n                        & 100\\% & 20.1 & 22.9 & 15.5 \\\\ \\midrule\n\\multirow{5}{*}{high}   & 0\\%   & \\cellcolor[gray]{0.9}28.7 & 14.1 & 10.7 \\\\\n                        & 25\\%  & 24.6 & 10.1 & 6.4  \\\\\n                        & 50\\%  & 10.7 & 19.1 & 6.0  \\\\\n                        & 75\\%  & 20.3 & 18.9 & 5.54  \\\\\n                        & 100\\% & 3.2 & 25.4 & 5.2  \\\\ \\bottomrule\n\\end{tabular}}\n\\vspace{-6mm}\n\\end{table}\n~\n\\begin{table}[ht]\n\\centering\n\\caption{DrQ+BC can adapt to multiple different distractions but is extremely brittle to settings it has not seen and struggles to generalize. Final mean performance is averaged over 6 seeds and base undistracted performance is highlighted. The environment used is cheetah-run medexp.}\n\\label{tab:drq-medexp-distractions}\n\\scalebox{0.9}{\n\\begin{tabular}{lcccc}\n\\toprule\n\\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Shift Severity\\end{tabular}}}} &\n  \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\% Shifted\\end{tabular}}} &\n  \\multicolumn{3}{c}{\\textbf{Eval. Return}} \\\\ \\cline{3-5} \n\\multicolumn{1}{c}{} &\n   &\n  \\textbf{Original} &\n  \\textbf{Dis. Train} &\n  \\textbf{Dis. Test} \\\\ \\midrule\n\\multirow{5}{*}{low}   & 0\\%   & \\cellcolor[gray]{0.9}79.1 & 0.5 & 1.1 \\\\\n                        & 25\\%  & 73.9 & 77.5 & 8.7  \\\\\n                        & 50\\%  & 72.3 & 82.2 & 8.5 \\\\\n                        & 75\\%  & 67.6 & 85.5 & 7.6  \\\\\n                        & 100\\% & 4.2 & 86.6 & 4.5  \\\\ \\midrule\n\\multirow{5}{*}{moderate} & 0\\%   & \\cellcolor[gray]{0.9}79.1 & 0.2 & 0.9 \\\\\n                        & 25\\%  & 74.6 & 77.9 & 3.1  \\\\\n                        & 50\\%  & 68.0 & 83.6 & 3.4  \\\\\n                        & 75\\%  & 55.4 & 86.1 & 3.7  \\\\\n                        & 100\\% & 0.9 & 86.7 & 2.0  \\\\ \\midrule\n\\multirow{5}{*}{high}   & 0\\%   & \\cellcolor[gray]{0.9}79.1 & 0.1 & 0.5 \\\\\n                        & 25\\%  & 76.7 & 76.8 & 1.3  \\\\\n                        & 50\\%  & 66.7 & 82.5 & 1.1  \\\\\n                        & 75\\%  & 48.4 & 84.8 & 1.2  \\\\\n                        & 100\\% & 0.4 & 86.1 & 0.9  \\\\ \\bottomrule\n\\end{tabular}}\n\\end{table}\n\n\\clearpage\n\n\\subsection{Further Offline DV2 Results on Multitask Datasets}\n\\label{app:model_random_multi}\nOn the random multitask data, Offline DV2 learns a similar quality policy to that on the base environment, but experiences no deterioration in performance on the test environments in walker or cheetah.\n\n\\setlength{\\tabcolsep}{23.0pt}\n\\begin{table}[ht]\n\\centering\n\\small\n\\caption{\n    \\small{Evaluation on the DMControl-Multitask benchmark using \\emph{random} data for Offline DV2. Normalized performance from {[}0, 1000{]} to {[}0, 100{]} is averaged over 6 seeds. Offline DV2 shows a strong ability to generalize to the extrapolation test environments.}}\n\\label{tab:random_multitask}\n\\begin{tabular}{@{}lllccc@{}}\n\\toprule\n  \\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{Algorithm}}} &\n  \\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{Environment}}} &\n  \\multicolumn{3}{c}{\\textbf{Eval. Return}} \\\\ \\cmidrule(l){3-5} \n\\multicolumn{1}{c}{} &\n  \\multicolumn{1}{c}{} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Train\\\\ Tasks\\end{tabular}} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Test\\\\ Interp.\\end{tabular}} &\n  \\textbf{\\begin{tabular}[c]{@{}c@{}}Test\\\\ Extrap.\\end{tabular}} \\\\ \\midrule\n\\multirow{2}{*}{Offline DV2} & walker  & 24.4 & 25.3 & 24.9 \\\\\n                             & cheetah & 31.6 & 31.1 & 31.1 \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4mm}\n\\end{table}\n\n\\section{Further Ablation Studies}\n\n\\subsection{DrQ+BC Random-Expert Ablation Studies}\n\\label{app:randexp}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figures\/appendix\/randexp.pdf}\n\\caption{\\small{Comparison of DrQ+BC and BC on random-expert datasets with 500K of each datatype, results averaged over 6 seeds.}}\n\\label{fig:randexp}\n\\end{figure}\n\nAnalogously to \\citet{td3bc}, we continue to see that DrQ+BC does well on mixed datasets with high reward with experiments on the concatenated random-expert datasets.\nSurprisingly, we see behavioral cloning also does reasonably well on cheetah-run with random-expert data.\nThis is likely to be because there is significant distribution shift between the states of random and expert trajectories for that environment.\nThis would lead to minimal destructive interference between similar states which contain completely different actions, as these state distributions largely do not overlap.\nThis is in contrast to the medium-expert dataset, which experiences higher state overlap between its two modes (i.e., states generated by a medium and an expert policy respectively), resulting in a marginal ``average\" action being learned which is likely suboptimal, as can be seen by the poor performance of the BC agent in Tables~\\ref{tab:all_comparison} and \\ref{tab:500k_exps}.\n\n\\subsection{Offline DV2 Training Time and Uncertainty Penalty}\n\\label{app:modeltrain}\nOne of the major factors preventing algorithms which use a RSSM like Offline DV2 and LOMPO from scaling to larger datasets is the time required for model training. The standard number of epochs of model training for our 100K datasets in \\Cref{sec:baselines} is 800 epochs, which takes around 6 hours on a V100 GPU. This scales linearly with number of training points if we maintain the same batch size. As we show in \\Cref{fig:modeltrainabl}, this is mandatory for performance and is a fundamental limitation of model-based methods. We can see that evaluated return increases and becomes more tightly distributed as the number of training epochs increases until 800.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{figures\/appendix\/dv2_model_train.pdf}\n\\caption{\\small{Evaluated return against model train epochs for Offline DV2 on a random dataset of size 400K. We train a single model up to 800 epochs and evaluate the model periodically on 3 seeds. We see that full model training, which scales linearly with training points, is mandatory for good performance.}}\n\\vspace{-2mm}\n\\label{fig:modeltrainabl}\n\\end{figure}\n\nIn \\cref{tab:all_comparison}, we see that Offline DV2 is considerably weaker on the expert datasets that have narrow data distributions.\nOn these expert datasets, we found that the model uncertainty often had lower variance across a trajectory compared to models trained on other datasets, and was thus uninformative.\nWe give summary statistics for the uncertainty penalty on states generated by a random policy in the walker-walk environment in \\cref{tab:penalty-over-traj}.\nSince we followed the author's DreamerV2 implementation~\\cite{dreamerv2}, only the stochastic portion of the latent state is predicted by the ensemble; this may mean crucial calibration is lost when ignoring the impact of the deterministic latent.\nFuture work could involve investigating SVSG~\\citep{jain2021learning}, an extension to DreamerV2 with a purely stochastic latent state.\n\n\\setlength{\\tabcolsep}{74.0pt}\n\\begin{table}[ht]\n\\centering\n\\caption{Mean and standard deviation of the uncertainty penalty computed over 1,024 states sampled from the `random' dataset on the walker-walk environment. Note that the model trained on expert data reports considerably smaller and tighter uncertainty values (compared to `medium' and `medexp'), despite the large distribution shift that exists from `random' to `expert' data. Instead, we'd expect the `expert' trained model to exhibit the largest mean uncertainty when tested on the `random' data.}\n\\label{tab:penalty-over-traj}\n\\begin{tabular}{@{}lcc@{}}\n\\toprule\n\\textbf{Dataset Type} & \\textbf{Mean} & \\textbf{Std.} \\\\ \\midrule\nrandom  & 0.223 & 0.040 \\\\\nmixed   & 0.226 & 0.035 \\\\\nmedium  & 0.341 & 0.034 \\\\\nmedexp  & 0.338 & 0.034 \\\\\nexpert  & 0.262 & 0.020 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Understanding Model-Based Extrapolation}\n\\label{app:modelbasedextrap}\nWe further investigate the reasons why our model-based baseline, Offline DV2, appears to generalize to unseen distractions.\nThe RSSM includes a latent decoder for the visual observations, which is primarily used during training to provide a self-supervised reconstruction loss. During deployment, the policy solely relies on the latent states from the encoder, and the decoder is effectively discarded.\nHowever, we can still use the decoder at test-time in order to understand the information contained in the latent states. To do this, we take the latent states generated during rollouts in the extrapolation experiments, and `translate' them into natural images by passing them through the decoder.\n\nWe first investigate the setting where each RSSM is trained only on data from a single distraction and then transferred. This setting is the most successful, as can be seen in~\\cref{fig:distraction_comp}. Thus, in \\cref{fig:distracted_recon} we first show the ground truth observation provided to the agent, then below this we show the decoder output reconstructed from the latent. In many cases, we can recover the original pose despite ending up in different visual surroundings. This is an indication that despite the decoder overfitting to the exogenous factors in the visual input (e.g., background, colors), the latent captures the salient \\emph{state} information of the agent (e.g., joint positions and angles), explaining the strong test-time transfer performance.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/appendix\/distracted_easy_recon.png}\\\\\n\\vspace{5mm}\n\\includegraphics[width=0.8\\textwidth]{figures\/appendix\/distracted_medium_recon.png}\\\\\n\\vspace{5mm}\n\\includegraphics[width=0.8\\textwidth]{figures\/appendix\/distracted_hard_recon.png}\n\\caption{\\small{Reconstruction of random episodes shifted by a fixed distractor by models trained on data shifted by a different distractor. The top row shows the original ground truth, and the bottom two rows the model reconstruction for a different RSSM. We can see that in many cases, the original pose of the walker is still able to be recovered.}}\n\\label{fig:distracted_recon}\n\\end{figure}\n\nIn~\\cref{fig:distraction_comp}, we note that settings with a mixture of distractors often had worse test-time transfer performance than those with a single distractor. To explain this, we consider an RSSM trained on images with a combination of two fixed distractors, and examine the reconstruction of an episode under a third distraction. We see in \\cref{fig:mixed_recon}, that the RSSM latents are split between the two modes of the data and the reconstruction switches robot color and background midway. This significantly confuses the recovered pose of the walker and likely causes the degradation in performance. Disentangling the factors of variation~\\citep{DBLP:journals\/corr\/abs-1804-03599, pmlr-v97-mathieu19a} represents important future work for offline RL from visual observations.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/appendix\/mixed_recon.png}\n\\caption{\\small{Reconstruction of a distracted random episode using an RSSM trained on a mixture of two differently distracted environments. We can see that the RSSM latents are split between the two modes of the data and the reconstruction switches robot color and background midway.}}\n\\label{fig:mixed_recon}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAmong all the gravitational waves events\nannounced\\cite{Abbott:2016blz,TheLIGOScientific:2016wfe,\n\tAbbott:2016nmj,Abbott:2017vtc,\n\tAbbott:2017oio,TheLIGOScientific:2017qsa,\n\tGBM:2017lvd,Abbott:2017gyy,LIGOScientific:2018mvr}\n by the  \nLIGO\/Virgo Collaboration, still the GW150914 event remains \nthe one showing the strongest signal to noise ratio.\nIt is because of this that many \nstudies\\cite{Liu:2016kib,Naselsky:2016lay,Creswell:2017rbh,\n\tGreen:2017voq,Creswell:2018tsr,Liu:2018dgm}\non gravitational waves data\ndevote special attention to it.\nWe are here concerned with the first manipulation on the data\nprevious to the full study of the nature of the physical signal\ncontained in it\n\nIn order to study the physical content of the Handford and Livingstone LIGO data\nfor the GW150914 event, the LIGO\/Virgo Collaboration has applied pre-processing\nfiltering techniques that include whitening filter,  35\u2013350 Hz bandpass filter\nand band-reject filters to remove the strong instrumental spectral lines,\nin \\cite{Abbott:2016blz},\nand in \\cite{TheLIGOScientific:2016wfe}\nthey only state to have whitened the data by the noise power spectral\ndensity.\nIn \\cite{TheLIGOScientific:2016qqj} \nthe LIGO\/Virgo Collaboration carried out a\nmatched-filter search for GW150914 using relativistic models of compact-object binaries\nusing a couple of techniques; but in both they\nuse a low-frequency cutoff of 30 Hz for the search,\nand in one of them they also used whitening filter.\nAlso in the analysis of reference \\cite{TheLIGOScientific:2016uux} they\nreport to have used whitening filters.\n\nThis article deals with the strategy for the pre-processing filtering\ntechniques; in particular we present arguments against the use\nof whitening filters, and we present a set of finite impulse filters (FIR)\nthat will show to be more convenient, for the pre-processing stage.\nAfter this then one can more safely process the data\nwith for example matched filter searches\\cite{TheLIGOScientific:2016qqj}.\n\n\nThe organization of this article is as follows.\nIn section \\ref{sec:preliminar} we present a preliminary study of the nature\nof the data and a review of the LIGO Collaboration filtering techniques.\nOur suggestion for the pre-processing filtering is presented in\nsection \\ref{sec:newfiltering}; and in\nsection \\ref{sec:final} we include some final comments\non the work and suggest further study.\n\n\n\\section{Preliminary view of the raw data}\\label{sec:preliminar}\n\nBefore embarking in the presentation of the new filtering approach,\nlet us review the main characteristics of the LIGO data around the event\nGW150914.\n\n\\subsection{Spectrograms near the event}\n\nOne of the main indications that something interesting happens around\nthe time of the GW150914 event that we take it to be:\n$t_e = 1126259462.422$GPS, equivalent to: {\\tt Mon Sep 14 09:50:45 GMT 2015},\nas suggested by LIGO at first,\nis the study of the spectrograms that we show in\nfigure \\ref{fig:spectrogrmas-raw}.\nOne can see that a faint\nalmost vertical line on both spectrograms at around the time\nassigned to the event.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_raw_spectrogram_fs10-65_32-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_raw_spectrogram_fs10-65_32-p1.png}\n\\caption{Spectrograms of raw data from Hanford on the left,\n\t and Livingston on the right; $\\pm$4 seconds around the time of event GW150914,\n\t in the range from 0 to 750Hz.}\n\\label{fig:spectrogrmas-raw}\n\\end{figure}\nSince the detectors are separated by about 3000km, the fact that both detectors\nshow the same appearance of an increasing in frequency signal, with a time\nseparation of less than 10ms indicates that this event is probably\ncaused by an astrophysical system far from Earth.\n\nIt is worthwhile to emphasize that this preliminary study does not need\nfor any a priori knowledge of calculated templates; since it is\nnoticed in the raw data.\n\n\\subsection{The amplitude spectral density of 256s around the event}\n\nWe are concerned with the details of the intrinsic noises of the detectors;\nfor this reason we do almost all studies with \na fairly wide extension of data of length 256 seconds\naround the time of the event GW150914. \nThis choice allows us to work with broader windows when performing studies\nand constructing filters.\nFor example in figure \\ref{fig:ASD-raw}, where we show the amplitude\nspectral densities (ASD) of both detectors, we also present the details\nof ASD in the range 34-37Hz, where calibration lines appear as\nvery narrow peaks.\nThese have been of concern in \\cite{Creswell:2017rbh}, where they have\nworked with a 32 seconds time lapse of data; and it can be seen that\nour figure \\ref{fig:ASD-raw} which compares with their figure 2,\nshows that these peaks are very narrow when studied with 90 seconds windows.\nThis is important for our work since this allows for constructing\nnarrow stopband filters to eliminate these unwanted contributions to the strain.\nThe three graphs of figure \\ref{fig:ASD-raw} have been constructed with\n90seconds windows on the 256 seconds strain.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_H_ASDs_256s_90-21-4s-p1.png}\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_L_ASDs_256s_90-21-4s-p1.png}\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_ASDs_256s_90s_33,9_37,1Hz-p1.png}\n\\caption{Amplitude spectral density of raw data from Hanford on the left,\n\tand Livingston at the center, in the range from 4 to 9000Hz,\n\tfor the interval of 256s around the time of the event with 90s windows; \n\twhere one can observe the cutoff at the Nyquist frequency.\n\tOn the right, a detail from 34 to 37Hz from both detectors; where due to the fact\n\tthat we are using a 90 seconds windows, one can see that the calibrating \n\tsignals are very narrow. The vertical lines mark the frequencies 34.7Hz and 35.3Hz\n\tfor Livingston, and 35.9Hz  and 36.7Hz for Hanford.\n}\n\\label{fig:ASD-raw}\n\\end{figure}\nIn the left and center graphs we have also superimposed the amplitude spectral density\nof each detector using also windows of 21s and 4s.\nIt can be seen that the 90s windows show the widest ASD;\nthen the 21s windowsASD(in light blue) are narrower and \nfinally the 4s windows ASD(in black) are the thinnest.\nThis is motivating for our work since it means that when one\nincreases the statistics, the ASD show wider variations than\nthose calculated with short windows, since they have increasing frequency\nresolution.\n\n\n\\subsection{The usual whitening procedure performed by LIGO}\n\nDetails of the GW150914 data have been presented by LIGO Collaboration through\nthe study of various filtering techniques. In figure 6 of reference\n\\cite{TheLIGOScientific:2016wfe} one can see graphs of the data,\nsampled at 2048Hz,\nwhere whitening filters were applied.\nIn figure \\ref{fig:whiten} we present the effects on the \namplitude spectral density of 256s intervals\nof data, using 4s windows, and the full sampling rate of 16384Hz;\nwhere each whitening filter is normalized in order to preserve the\nstrain amplitude at the respective minimum of the amplitude spectral density\nfor each detector.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_whitenosv_ASDs_I=256s_W=04s-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_whitenosv_ASDs_I=256s_W=04s-p1.png}\n\\caption{Amplitude spectral density of the whitened strains for 256s intervals,\n\tand windows of 4s, as suggested in LIGO public python scripts.\n\tHanford data on the left and Livingston's on the right.\n}\n\\label{fig:whiten}\n\\end{figure}\n\nIn LIGO reference \\cite{Abbott:2016blz} they have used a 35\u2013350Hz bandpass \nButterworth filter,\nwhich we also apply to the data we are considering,\naccording to the indications in the public LIGO python scripts. \nIn figure \\ref{fig:whiten-bandpass} we show the amplitude spectral density\ngraphs of the 256s intervals, after applying the bandbass filter, \nwhere we have used the same axis limits,\nin order to emphasize the effects of this filter.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_whitenosvbp_ASDs_I=256s_W=04s-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_whitenosvbp_ASDs_I=256s_W=04s-p1.png}\n\\caption{Amplitude spectral density of the whitened and bandpass strains \n\tfor 256s intervals,\n\tand windows of 4s, as suggested in LIGO public python scripts.\n\tHanford data on the left and Livingston's on the right.\n}\n\\label{fig:whiten-bandpass}\n\\end{figure}\n\n\n\n\n\\subsection{Phase diagrams of the raw data and after filtering}\n\nIn figure \\ref{fig:phase-raw} we show the phase diagrams of the raw data\nof both detectors for 256s centered at the time\nof the GW150914 event, up to 500Hz.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_phase_raw_256s-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_phase_raw_256s-p1.png}\n\\caption{Phase diagrams as a function of frequency for the raw data of 256s length\n\taround the time of the event GW150914, showing Hanford on the left and Livingston\n\ton the right.\n}\n\\label{fig:phase-raw}\n\\end{figure}\nIt can be noticed that they \nshow strong correlations and since we are dealing with the full sampling rate\nof the data at $f_s = 16384$Hz, and with a 256s time interval,\nour graphs are also different from those\nshown in \\cite{Creswell:2017rbh}.\nSince these studies depend very much\non the length of the time interval, and the sampling rate,\nthis reinforces the interpretation that at this raw stage the noise is not Gaussian.\n\n\nWe show the phases as function of frequencies, after applying the LIGO filtering\nprocedures, in figure \\ref{fig:phase-filtered}, up to 350Hz.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_strain_H_phase_whitenosvbp-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_strain_L_phase_whitenosvbp-p1.png}\n\\caption{Phase diagrams as a function of frequency after applying the LIGO \n\twhitening and bandpass filter\n\tto the data of 256s length\n\taround the time of the event GW150914, showing Hanford on the left and Livingston\n\ton the right.\n}\n\\label{fig:phase-filtered}\n\\end{figure}\nIt can be seen in figure \\ref{fig:phase-filtered}\nthat the 256s statistics results in narrower bands, than those\nfound in the studies shown in figure 3 of reference  \\cite{Creswell:2017rbh};\ntherefore, the concerns expressed there, on the phase behavior, are augmented here.\n\nWe show these figures not to participate in a debate on\nthe meaning of the signal at both detectors,\ninstead, we just want to illustrate that the application of whitening\nand bandpass filters are to be handled with most care; since otherwise\nunwanted side effects involving phase behavior\nand group delay\\cite{Chatterji2005} might appear.\nBut our main concern is related to the severe relative attenuation\nof low frequencies in the whitening procedure with \nthe characteristics of LIGO amplitude spectral density of\neach detector, as it can be noticed below in our study\nof the effects of LIGO filtering on the templates.\n\n\\subsection{Effects of the whitening and bandpass LIGO filtering on templates}\n\nThe LIGO Scientific Collaboration has also made public the components of \nthe waveform templates that where used in matched filtering.\nWe follow the indications of the public LIGO python scripts to find,\nfrom both components of the calculated template,\nthose that match for Hanford and Livingstone detectors;\nthat below will appear as \\emph{template\\_H} and \\emph{template\\_L}\nin the graphs.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.48\\textwidth]{graficos\/GW150914_templates_-20,0_20,0s-p1.png}\n\\includegraphics[clip,width=0.48\\textwidth]{graficos\/GW150914_templates_-0,8_0,2s-p1.png}\n\\includegraphics[clip,width=0.48\\textwidth]{graficos\/GW150914_templates_whitenbp_-20,0_20,0s-p1.png}\n\\includegraphics[clip,width=0.48\\textwidth]{graficos\/GW150914_templates_whitenbp_-0,8_0,2s-p1.png}\n\\caption{\n\tMatched templates for Hanford and Livingston GW150914 event.\n\tIn top left graph, the full time extend of the original templates; \n\twhile on the\n\ttop right, the detail of 1s length around the time of the event.\n\tIn the bottom row, the corresponding graphs after applying the\n\twhitening and bandpass of 35-350Hz filter. \n}\n\\label{fig:templates-1}\n\\end{figure}\nOne can see in the graphs of figure \\ref{fig:templates-1} that the filtering\nprocedure suggested by LIGO, severely changes the shape of the matched templates;\nwhich are supposed to give a close representation of the expected physical\nsignal hidden in the observed data.\nIn particular, in the lower right graph, it is noted that by applying\nthese filters, one can only use a very limited lapse of time in the\nsignal, on the order of 0.1second.\nWe will argue below against this limitation imposed by the filtering approach\nsuggested by the LIGO Collaboration.\n\n\n\n\\section{The new filtering scheme without whitening}\\label{sec:newfiltering}\n\nA comparison of the U shape shown in figure \\ref{fig:ASD-raw} \nand the flat shape shown in \\ref{fig:whiten} indicates that\nif there were a physical signal with frequencies below that one\nat minimum of the ASD; then they will be severely attenuated\nby the whitening procedure. \nWe present here a new approach to the initial filtering in order\nto circumvent this effect.\n\n\n\\subsection{Initial bandpass filter}\n\nIt is sensible to apply an initial bandpass filter in order to concentrate\non the main frequency band that might contain physically interesting signals.\nFor this reason we choose a very wide band that goes from 22Hz to 1024Hz.\nHowever, instead of the infinite impulse response (IIR) filters\nused in the public LIGO python scripts,\nwe choose to apply finite impulse response (FIR) filters;\nwhich are supposed to have safer phase behavior.\n\nSince we are interested in an interval of 256s and the FIR bandpass filters\nmight include boundary effects, we start with a 288s interval centered\nat the time of the event, and after applying the FIR bandpass filter,\nwe crop it to an interval of 256s centered at the event time,\nwhose resulting ASD are shown in figure \\ref{fig:osvfilt-hipa-lopa}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_ASDs_hipa22Hz_lopa1024Hz_256s_21s-p1.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_ASDs_hipa22Hz_lopa1024Hz_256s_21s-p1.png}\n\\caption{Amplitude spectral density of the strains after bandpass FIR filter\n\tat frequencies 22-1024Hz; Hanford on the left and Livingston on the right.\n}\n\\label{fig:osvfilt-hipa-lopa}\n\\end{figure}\nComparing with the graphs in figure \\ref{fig:ASD-raw} one can see the sharp\nattenuations at the chosen boundary frequencies.\nOne can still observed the narrow bands of intrinsic detector noise\nat well defined frequencies; that we handle next.\n\n\n\n\\subsection{Filtering intrinsic detector frequencies}\nA common way to deal with signals contained in complex noisy data is to\napply the whitening filters; that had been using the LIGO Collaboration.\nInstead of this, we use stopband FIR filters for each of the\nwell defined narrow frequencies generated by each of the detectors.\n\nThe first thing to do is to identify precisely the value of the frequency\nand width of every intrinsic instrumental excitation introduced by each detector in the strain.\nAfter this we apply the stopband FIR filter to each strain.\nAgain, to avoid boundary effects, we apply the filter to an interval of length 272s\nwhich was obtained from the bandpass strain of 288s by clipping the extremes.\nThen, we trim again the interval to the desired length of 256s.\nThe result is shown in figure \\ref{fig:osvfilt-hipa-lopa+stopbands}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_ASDs_H_filt11_256s_21s-p1-b.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_ASDs_L_filt11_256s_21s-p1-b.png}\n\\caption{Amplitude spectral density of the strains after bandpass and stopband FIR filter;\n Hanford on the left and Livingston on the right.\n We use the same axis limits as in previous graphs in order to facilitate the comparison.\n}\n\\label{fig:osvfilt-hipa-lopa+stopbands}\n\\end{figure}\n\nIt can be seen that the resulting ASD behavior is fairly flat\nexcept for the initial curved behavior at low frequencies;\nwhich it could be related to colored noise.\nThe idea behind the decision of not filtering the low frequencies at this stage\nis that we would like to allow for possible interesting\nphysical signals at this initial portion of the spectrum.\nBelow, we do find such low frequency signals.\n\nThe strategy is then to avoid making changes to the astrophysical signal\nin the pre-processing filtering stage.\nThis is completely different from the approach used up to now \nin LIGO\/Virgo Collaboration \narticles\\cite{Abbott:2016blz,TheLIGOScientific:2016wfe,\n\tTheLIGOScientific:2016qqj,TheLIGOScientific:2016uux}.\n\n\n\n\\subsection{Phase behavior after filtering}\nIn figure \\ref{fig:fase-osvfilt} we show how the phases are distributed\nacross the frequency range of 22 to 500Hz, \non the 256s strain of both detectors,\nafter the first set of FIR filters, bandpass and stopband, have been\napplied. \n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_H_filt_phase_256s-p1-b.png}\n\\includegraphics[clip,width=0.49\\textwidth]{graficos\/GW150914_L_filt_phase_256s-p1-b.png}\n\\caption{Phase graphs of strains after new first FIR filtering round;\n\tHanford on the left and Livingston on the right.\n}\n\\label{fig:fase-osvfilt}\n\\end{figure}\nIt can be seen that phases are evenly distributed in the frequency range of interest;\nwhich it must be compared with the corresponding phase study after applying\nthe first LIGO filtering procedures to the strains, shown in figure \\ref{fig:phase-filtered}.\n\nThis is a strong indication that we were able to suppress the initial phase\ncorrelation in the raw data; and also that this correlation was due to the intrinsic\nexcitations of the detectors.\n\nOf course an evenly distribution of phases is a good sign, since indicates\nan almost Gaussian noise behavior.\n\n\\subsection{Time domain graphs after filtering}\n\nSince the initial data is obtained in the time domain, we should\nsee what is the result of our filtering approach in it,\nand also if we are able to obtain some new insight.\n\nFigure \\ref{fig:strain-H-time-domain+LIGOtemplates} and \\ref{fig:strain-L-time-domain+LIGOtemplates}\npresents respectively the graph\nof the H and L strains along with the corresponding LIGO matched templates.\nTo avoid very high frequency contribution from the strain, that is not contained\nin the template, we have applied for these graphs a lowpass FIR 200Hz filter.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.75\\textwidth]{graficos\/GW150914_-H_data_fir200Hz_templates-roll0015_shifted74_-0,6_0,02s-b.png}\n\\caption{\n\tStrain of Hanford detector after a lowpass 200Hz FIR-filter, along with\n\tits adjusted by LIGO script template,\n\tclose to the time of the event. \n}\n\\label{fig:strain-H-time-domain+LIGOtemplates}\n\\end{figure}\nIt can be seen in the figure \\ref{fig:strain-H-time-domain+LIGOtemplates} that\nthere is a remarkable coincidence in frequency, phase and amplitude of\nthe strain and the template in the time interval that goes from about -0.5s to -0.1s\nbefore the time of the event; that we have marked with a light green color rectangle.\nThe light yellow band that includes the time of the event, and extends for about\n0.1s, is to indicate approximately the lapse of time that is allowed by the\nwhitening LIGO procedure.\nThis was shown in the lower right graph of figure \\ref{fig:templates-1} above.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.75\\textwidth]{graficos\/GW150914_L_data_fir200Hz_templates-roll0015_-0,6_0,02s-b.png}\n\\caption{\n\tStrain of Livingston detector after a lowpass 200Hz FIR-filter, along with\n\tits adjusted by LIGO script template,\n\tclose to the time of the event.\n}\n\\label{fig:strain-L-time-domain+LIGOtemplates}\n\\end{figure}\nObserving the Livingston graph of figure \\ref{fig:strain-L-time-domain+LIGOtemplates}\none can see that there is a very good agreement in frequency, phase and amplitude of\nthe strain and the template in the time lapse that goes from about -0.5s to -0.2s\nbefore the time of the event; that we have marked with a light green color rectangle.\nThere is a short period of time from about -0.2 to -0.1s where the detector\ncould not record properly the signal.\n\nThe fact that the same type of indication appears in both detectors\ninvites to present the two sets of data in the same graph,\nwhich appears in figure \\ref{fig:strain-H+L-time-domain+LIGOtemplates}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.75\\textwidth]{graficos\/GW150914_All_-H_data_fir200Hz_templates-roll0015_shifted74_-0,6_0,02s-a.png}\n\\caption{\n\tStrain of both detectors after a lowpass 200Hz FIR-filter, along with\n\ttheir adjusted by LIGO script templates,\n\tclose to the time of the event.\n\t}\n\\label{fig:strain-H+L-time-domain+LIGOtemplates}\n\\end{figure}\nIn figure \\ref{fig:strain-H+L-time-domain+LIGOtemplates} the H strain has been inverted\nan appropriately shifted, in order to agree in phase at the high amplitude part of the signal\nwith the L strain.\n\nIt can be noticed that even if one considers only the light green region in which\nboth detectors show independently a match of the strain with the\ncorresponding templates, \nwe find several coincidences:\nthe shown interval is very close to the event time,\nat each detector the template matches the strain up to about -0.5s,\nin phase, amplitude and frequency\nand very importantly both detectors synchronize in the shown interval.\nThis is compelling evidence that the lapse of time\nup to about -0.5 seconds before the time of the event\ncontains physically interesting signals.\nOur findings can be contrasted with the claims in LIGO publications\\cite{TheLIGOScientific:2016zmo}\nin which they recognize up to 0.2s of signal.\n\nThis discovery shows precisely the benefit to use filtering techniques\nthat avoid to change the expected nature of the astrophysical signal.\n\n\n\n\\subsection{Spectrograms of the filtered signals}\n\nIn figure \\ref{fig:spectrogrmas-filtered}\nwe present the graphs corresponding to the spectrograms after applying our filters\nto the data, where one can see that the rising in frequency signals\nreaches more than 250Hz in the Hanford data, and more than 200Hz in the\nLivingston one.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_H_filtered11_spectrogram_d2_fs4-23_20-p1.png}\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_L_filtered11_spectrogram_d2_fs4-23_20-p1.png}\n\\includegraphics[clip,width=0.32\\textwidth]{graficos\/GW150914_median_filtered11_spectrogram_d2_fs4-23_20-p1.png}\n\\caption{Spectrograms of filtered data from Hanford on the left,\n\tand Livingston at the center; $\\pm$2 seconds around the time of event GW150914,\n\tin the range from 22 to 300Hz.\nWe have also included on the right the corresponding spectrogram for the median;\nwhere the H strain has been inverted and shifted.\n}\n\\label{fig:spectrogrmas-filtered}\n\\end{figure}\nDue to the chosen directions of the arms of the LIGO detectors,\nHanford and Livingston observatories will be sensible to almost the same component\nof the polarization of the signal. This is the reason why both\nmatched templates are so similar.\nBecause of this one is tempted also to look at the median of\nboth strains, after the corresponding inversion and time shift\nhas been applied to one of the data; since in particular\nby doing so one expects to obtain better reduction of the noise.\nFor this reason we have also included in the last graph of figure \\ref{fig:spectrogrmas-filtered}\nthe corresponding spectrogram for the median.\nThere one can see that we have gain some noise reduction and also that\nthere seems to be some coincidence on the data\nat low frequencies, within the last second before the time of the event;\nwhich reinforces our claims observed in the time domain graphs.\n\n\n\\section{Final comments}\\label{sec:final}\n\nLet us summarize what we have presented so far.\nIn section \\ref{sec:preliminar} we have reviewed\nthe basic characteristics of the data of the GW150914 event,\nand showed the limitations in the pre-processing filtering techniques used\nin the LIGO Collaboration studies; based on the whitening procedure,\nand occasionally IIR bandpass filtering.\n\nIn order to circumvent the shortcoming of the whitening procedure\nof too much attenuation for low frequencies, \nwe have presented a new straight forward filtering approach based\non FIR filters, in section \\ref{sec:newfiltering},\nthat performs and initial bandpass, and secondly\na careful stopband filter; that yields a minimum touched\nstrain which respects the possibility of low frequency\nsignals.\n\nWhen observing the effect of our filter on the phase diagrams in figure \\ref{fig:fase-osvfilt},\nwe deduce that our approach handles successfully the initial correlations\nof phases shown in the raw data graphs, presented in section \\ref{sec:preliminar}.\nIn this way we also avoid the awkward phase behavior after applying the\nLIGO filtering techniques, shown in figure \\ref{fig:phase-filtered}.\n\nWe have given evidence that, very close to the time\nof the GW150914 event,\nin a lapse of time of few tenths of a second:\nthere is coincidence of frequency, amplitude and phase,\nbetween the detectors and with their respective templates used by the LIGO\nCollaboration.\nIt is highly improbable that this concomitance should be attributed\njust to chance. \nTherefore, there is more signal to be studied which is\nencoded in the data of the GW150914 event.\n\nThis article is devoted to the presentation of a new approach\nfor the pre-processing of the data;\nso that we intend to study in detail in separate works\nthe strain we have obtained in this way.\nFor this reason we have not quantified the coincidence\nof strains at both detectors near the time of the event;\nwe do plan to present this with a new technique to\nlook for similar signals in a pair of detectors,\nin a separate work.\nWe also intend to apply the filtering approach presented here\nto all the available gravitational wave data of other events.\n\nWe suggest that many of previous analysis of the data,\nshould be carried out after applying the type of filtering we have\npresented here; instead of the usual whitening pre-processing approach.\n\n\n\n\\subsection*{Acknowledgments}\n\nWe are very grateful to the LIGO\/Virgo Collaboration for making available the\ndata and the python scripts on data analysis\nat \\href{https:\/\/www.gw-openscience.org\/}{https:\/\/www.gw-openscience.org\/}.\n\nWe thank Emanuel Gallo for a careful reading of the manuscript and for suggestions.\n\nWe acknowledge support from CONICET, SeCyT-UNC and Foncyt.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nAny programmable response of  a material to external stimulation can be interpreted as computation. \nTo implement a logical function in a material one must map space-time dynamics of an internal structure of a material onto  a space of logical values. \nThis is how experimental laboratory prototypes of unconventional computing devices are made: logical gates, circuits and binary adders employing interaction of wave-fragments in light-sensitive Belousov-Zhabotinsky media \\cite{ref1}, swarms of soldier crabs \\cite{ref2}, growing lamellipodia of slime mould  Physarum polycephalum~\\cite{adamatzky2016logical}, crystallisation patterns in ``hot ice'' \\cite{adamatzky2009hot},  peristaltic waves in protoplasmic tubes \\cite{adamatzky2014slime}.  In many cases logical circuits are `built'  or evolved from a previously disordered material \\cite{miller2014evolution}, e.g. networks of slime mould \\emph{Physarum polycephalum} \\cite{whiting2016towards}, bulks of nanotubes \\cite{broersma2012nascence}, nano particle ensembles \\cite{bose2015evolution, broersma2017computational}. In these works the computing structures could be seen as growing on demand, and logical gates develop in a continuum where an optimal distribution of material minimised internal energy.  A continuum exhibiting such properties can be coined as a ``self-optimising continuum''. Slime mould of \\emph{Physarum polycephalum} well exemplifies such a continuum: the slime mould is capable of solving many computational problems, including mazes and adaptive networks  \\cite{adamatzky2016advances}. Other examples of the material behaviour include bone remodelling \\cite{christen2014bone}, \nroots elongation \\cite{mazzolai2010plant}, sandstone erosion \\cite{bruthans2014sandstone}, \ncrack and lightning propagation \\cite{achtziger2000optimization}, growth of neurons and blood vessels etc. \nSome other physical systems suitable for computations were also proposed in \\cite{miller2014evolution, turner2014neuroevolution, banzhaf2006guidelines,  miller2002evolution}. In all these cases, a phenomenon of the formation of an optimum layout of material is related to non-linear laws of material behaviour, \nresulting in the evolution of material structure governed by algorithms similar to those used in a topology optimisation of structures~\\cite{klarbring2010dynamical}. We develop the ideas of material optimisation further and show, in numerical models, how logical circuits can be build in a conductive material self-optimise its structure governed by configuration of inputs and outputs. \n\nThe paper is structured as follows. In Sect.~\\ref{topologyoptimisation} we introduce  topology optimisation aimed to solve a problem of a stationary heat conduction.\nGates {\\sc and} and {\\sc xor} are designed and simulated in Sects.~\\ref{andgate} and \\ref{xorgate}. We design one-bit half-adder in Sect.~\\ref{onebithalfadder}.\nDirections of further research are outlined in Sect.~\\ref{discussion}.\n\n\n\n\\section{Topology optimisation}\n\\label{topologyoptimisation}\n\n\nA topology optimisation in continuum mechanics aims to find a layout of a material within a given \ndesign space that meets specific optimum performance targets \\cite{bendsoe2013topology, hassani2012homogenization, huang2010evolutionary}.  \nThe topology optimisation is applied to solve a wide range of problems \\cite{bendsoe2005topology}, \ne.g. \nmaximisation of heat removal for a given amount of  heat conducting material \\cite{bejan1997constructal}, \nmaximisation of fluid flow within channels \\cite{borrvall2003topology},\nmaximisation of structure stiffness and strength \\cite{bendsoe2005topology}, \ndevelopment of meta-materials satisfying specified  mechanical and thermal physical properties \\cite{bendsoe2005topology}, \noptimum layout of plies in composite laminates \\cite{stegmann2005discrete}, \nthe design of an inverse acoustic horn \\cite{bendsoe2005topology}, modelling of amoeboid organism growing towards food sources \\cite{Safonov20161},\noptimisation of photonics-crystal band-gap structures \\cite{men2014robust}. \n\n\nA standard method of the topology optimisation employs a modelling material layout that uses a density of material, $\\rho$, \nvarying from 0 (absence of a material) to 1 (presence of a material),  where a dependence of structural properties on the density \nof material is described by a power law. This method is known as Solid Isotropic Material with Penalisation (SIMP) \\cite{zhou1991coc}. \nAn optimisation of the objective function consists in finding an optimum distribution of $\\rho$:  $\\min_\\rho f(\\rho)$.\n\nThe problem can be solved in various numerical schemes, including the sequential quadratic programming (SQP) \\cite{wilson1963simplicial}, \nthe method of moving asymptotes (MMA) \\cite{svanberg1987method}, and the optimality criterion (OC) method \\cite{bendsoe2005topology}. \nThe topology optimisation problem can be replaced with a problem of finding a stationary point of an Ordinary Differential Equation (ODE) \\cite{klarbring2010dynamical}. Considering density constraints\non $\\rho$, the right term of ODE is equal to a projection of the negative gradient of the objective function. Such\noptimisation approach is widely used in the theory of projected dynamical systems \\cite{nagurney2012projected}. Numerical schemes of topology\noptimisation solution can be found using simple explicit Euler algorithm. As shown in \\cite{klarbring2012dynamical} iterative schemes match the\nalgorithms used in bone remodelling literature \\cite{harrigan1994bone}.\n\n\nIn this work the topology optimisation problem as applied to heat conduction problems \\cite{gersborg2006topology}. \nConsider a region in the space $\\Omega$ with a boundary  \n$\\Gamma=\\Gamma _D \\cup \\Gamma _N$,  $\\Gamma _D \\cap \\Gamma_N= \\emptyset$,  \nseparated for setting the Dirichlet (D) and the Neumann (N) boundary conditions. \nFor the region $\\Omega $  we consider the steady-state heat equation given in: \n\n\\begin{equation}\n\\nabla \\cdot k \\nabla T +f=0  \\text{ in } \\Omega \n\\end{equation}\n\n\\begin{equation}\nT = T_0  \\text{ on } \\Gamma_D \n\\end{equation}\n\n\n\\begin{equation}\n(k \\nabla T) \\cdot n = Q_0 \\text{ on } \\Gamma_N \n\\end{equation}\n\n\n\n\nwhere  $T$ is a temperature, \n$k$ is a heat conduction coefficient, \n$f$ is a volumetric heat source, and \n$n$ is an outward unit normal vector. \nAt the boundary $\\Gamma_D$  a temperature  $T=T_0$ is specified in the form of Dirichlet\nboundary conditions, and at the boundary  $\\Gamma _N$ of  the heat flux $(k \\nabla T) \\cdot n$ is specified using Neumann boundary conditions. \nThe condition $(k \\nabla T) \\cdot n = 0$   specified at the part of  $\\Gamma_N$  means a thermal insulation  (adiabatic conditions).\n\nWhen stating topology optimisation problem for a solution of the heat conduction problems it is necessary to find an optimal distribution for a limited volume of conductive material in order to minimise heat release, which corresponds to designing a thermal conductive device. It is necessary to find an optimum distribution of material density $\\rho$  within a given area  $\\Omega$ in order to minimise the cost function:\n\n\\begin{equation}\n\\text{Minimize } C(\\rho) = \\int_\\Omega \\nabla T \\cdot (k (\\rho) \\nabla T)\n\\end{equation}\n\n\\begin{equation}\n\\text{Subject to } \\int_\\Omega \\rho <M \n\\end{equation}\n\n\n\n\nIn accordance with the SIMP method the region being studied can be divided into finite elements\nwith varying material density $\\rho_i$ assigned to each finite element $i$. A relationship\nbetween the heat conduction coefficient and the density of material is described by a power law as follows:\n\n\\begin{equation}\nk_i = k_{\\min} + (k_{\\max} - k_{\\min}) \\rho^p_i ,  \\hspace{5mm}  \\rho_i \\in \\lfloor 0, 1 \\rfloor\n\\end{equation}\nwhere  \n$k_i$ is  a value of heat conduction coefficient at the $i$-th finite element,\n$\\rho_i$ is a  density value at the $i$-th element,\n$k_{\\max}$ is a heat conduction coefficient at $\\rho_i=1$, \n$k_{\\min}$ is a heat conduction coefficient at $\\rho_i=0$,\n$p$ is a penalisation power ($p>1$).\n\nIn order to solve the problem (1)--(6) we apply the following techniques used in the dynamic systems modelling. \nAssume that  $\\rho$  depends on a time-like variable $t$.  \nLet us consider the following differential equation to determine density in $i$-th finite element,  $\\rho_i$, \nwhen solving the problem stated in (1)--(6):\n\n\\begin{equation}\n\\acute{\\rho_i}=\\lambda (\\frac{C_i(\\rho_i)}{\\rho_i V_i} - \\mu), \\hspace{5mm}  C_i(\\rho_i) = \\int_{\\Omega_i} \\nabla T \\cdot (k_i(\\rho) \\nabla T) d\\Omega\n\\end{equation}\nwhere dot above denotes the derivative with respect to $t$, \n$\\Omega_i$ is a  domain of  $i$-th finite element, \n$V_i$ is a volume of $i$-th element, \n$\\lambda$ and $\\mu$ are positive constants characterising behaviour of the model. \nThis equation can be obtained by applying methods of the projected dynamical systems \\cite{klarbring2012dynamical} \n or bone remodelling methods \\cite{harrigan1994bone, mullender1994physiological, payten1998optimal}.\n\n\nFor numerical solution of equation (8) a projected Euler method is used \\cite{nagurney2012projected}. \nThis gives an iterative formulation for  the solution finding $\\rho_i$ \\cite{klarbring2010dynamical}:\n\\begin{equation}\n\\rho^{n+1}_i = \\rho^n_i + q[\\frac{C_i(\\rho^n_i)}{\\rho^n_i V_i} - \\mu^n]\n\\end{equation}\nwhere $q = \\lambda \\Delta t$, \n$\\rho^{n+1}_i$ and $\\rho^n_i$ are the numerical approximations of  $\\rho_i(t+\\Delta t)$ and $\\rho_i(t)$,   \n$\\mu^n =\\frac{\\sum_i C_i(\\rho^n_i)}{\\sum_i \\int_{\\Omega_i} \\rho_{ev} d\\Omega}$, $\\rho_{ev}$ is a specified mean value of density.\n\nWe consider a modification of equation (8):\n\\begin{equation}\n\\rho^{n+1}_i = \n\\begin{cases}\n\\rho^n_i + \\theta \\text{ if } \\frac{C_i(\\rho^n_i)}{\\rho^n_i V_i} - \\mu^n \\geq 0, \\\\\n \\rho^n_i - \\theta \\text{ if } \\frac{C_i(\\rho^n_i)}{\\rho^n_i V_i} - \\mu^n < 0,\n\\end{cases}\n\\end{equation}\nwhere $\\theta$ is a positive constant.\n\nThen we calculate a value of $\\rho _i^{n+1}$  using equation (9) and project $\\rho _i$ onto a set of constraints:\n\n\\begin{equation}\n\\rho^{n+1}_i = \n\\begin{cases}\n\\rho_{\\max} \\text{ if } \\rho^{n+1}_i > \\rho_{\\max}, \\\\\n\\rho^{n+1}_1 \\text{ if } \\rho_{\\min} \\leq \\rho^{n+1}_i \\leq \\rho_{\\max},\\\\\n\\rho_{\\min} \\text{ if } \\rho^{n+1}_i < \\rho_{\\min} \n\\end{cases}\n\\end{equation}\nwhere  $\\rho_{\\min}$ is a specified minimum value of $\\rho_i$ and $\\rho_{\\max}$ is a specified maximum value of $\\rho_i$. \nA minimum value is taken as the initial value of density for all finite elements:  $\\rho_i^0=\\rho_{\\min}$.\n\n\\section{Specific parameters}\n\\label{Specificparameters}\n\n\nThe algorithm above is implemented in ABAQUS \\cite{Abaqus2014} using the modification of the structural topology optimisation plug-in, \nUOPTI, developed previously \\cite{Safonov2015}. Calculations were performed using topology optimisation methods for the finite element model\nof $200 \\times 200 \\times 1$ elements. Cube-shaped linear hexahedral elements of DC3D8 type with a unit\nlength edges were used in calculations. The elements used have eight integration points. The cost function value is\nupdated for each finite element as a mean value of integration points for an element under consideration  \\cite{Abaqus2014}. \n\n\nThe model can be described by the following parameters: \n$\\rho_{\\min}$ and  $\\rho_{\\max}$ are minimum and maximum values of $\\rho_i$, \n$M=\\sum_i \\int_{\\Omega_i} \\rho_{ev} d\\Omega$ is a mass of the conductive material, \n$\\theta$ is an increment of $\\rho_i$ at each time step, \n$p$ is a penalisation power, \n$k_{\\max}$ is a heat conduction coefficient at $\\rho_i = 1$, \n$k_{\\min}$ is a heat conduction coefficient at $\\rho_i = 0$. \nAll parameters but $M$ are the same for all six (three devices with two type of boundary conditions) implementations:\n$\\rho_{\\max}=1$,\n$\\rho_{\\min}=0.01$,\n$\\theta=0.03$,\n$p=2$,\n$K_{\\max}=1$,\n$K_{\\min}=0.009$. \n\nThe parameter $M$ is specified as follows: \n$M=2000$ for {\\sc and}, {\\sc xor} in Dirichlet boundary conditions on inputs, and one-bit half-adder for both types of boundary conditions;\n$M=800$ for {\\sc and} gate and $M=400$ for {\\sc xor} gate in Neumann boundary conditions. \n\nWe use the following notations. Input logical variables are $x$ and $y$, output logical variable is $z$. \nThey takes values 0 ({\\sc False}) and 1 ({\\sc True}).\nSites in input stimuli the simulated material are $I_x$ and $I_y$ (inputs),  $O$, $O_1$, $O_2$ (outputs). \nSites of outlets are $V$, $V_1$ and $V_2$ (temperature are set to 0 in the outlet, \nso we use symbol $V$  by analogy with vents in fluidic devices).  Temperature at the sites is shown as \n$T_{I_x}$, $T_{I_y}$, $T_{O}$, $T_{O_1}$ etc. We show distances between as $l(I_x, I_y)$, $l(I_x, O)$ etc.\n\n\n\nLogical values are represented by temperature: $x=1$ is $T_{I_x}=100$ and $x=0$ is  $T_{I_x}=0$, the same for $y$. \nWe input data in the gates by setting up thermal boundary conditions are set at the input sites and  adiabatic boundary conditions \nfor other nodes. The temperature at each point is specified by setting equal values in 4 neighbour nodes belonging to the same finite \nelement. Temperature at outputs and outlets is set to zero of all experiments: $T_O=T_{O_1}=T_{O_2}=0$,\n $T_V=T_{V_1}=T_{V_2}=0$. To maintain specified boundary conditions we setup a thermal flow  through the boundary points. Intensity of the flows is determined via solution of the thermal conductivity equation at each iteration. Therefore  intensity of the thermal streams via input, output and outlet sites changes during the simulation  depending on a density distribution of the conductive material. Namely, if we define zero temperature at a site the intensity of the stream though the site will be negative if a density of the conductive material is maximal; the intensity will be zero if the material density is minimal. In case when we do not define a temperature at a site the intensity is non-zero if the density is maximal and zero if the density is minimal.\n Therefore, instead of talking about temperature at the output we talk about thickness of the conductive material. \n Namely, if the material  density value at the output site $O$ is minimal, $\\rho_O = \\rho_{\\min}$, we assume logical output 0 ({\\sc False}). \n If the density $\\rho_O = \\rho_{\\max}$ we assume logical output 1 ({\\sc True}).  The material density for all finite elements is set to a\nminimum value $\\rho _i^0=\\rho _{\\min}$ at the beginning of computation. \n\n In case of Dirichlet boundary conditions in inputs, in $x=0$ and $y=0$  the temperature is constant and equal to zero at all points, therefore the temperature gradient is also zero, $\\nabla T=0$. The cost function is also equals to zero at all points: $C_i(\\rho _i)=0$. \nAs the initial density for all finite elements is set to a minimum value $\\rho_i^0=\\rho_{\\min}$ then \n from equations (9) and (10) follows that the density stays constant and equal to its minimum value  \n $\\rho_i^n=\\rho_{\\min}$.  Therefore, the density value at  $O$ point is minimal,  $\\rho_O=\\rho_{\\min}$ which indicates \n logical output 0. Further we consider only situations when one of the inputs is non-zero.\n \n In case of Neumann boundary conditions in inputs a flux in each site is specified by setting the flux through the face of the finite element to which the site under consideration belongs.  Adiabatic boundary conditions are set for other nodes. The logical value of $x$ is represented by the value of given flux in $I_x$, $Q_{I_x}$. The logical value of $y$ is represented by the value of given flux in $I_y$, $Q_{I_y}$. Flux $Q_{I_x}=0$ represents $x=0$ and flux $Q_{I_x}=1$ represents $x=1$. \n \n Figures in the paper show density distribution of the conductive material. The maximum values of $\\rho$ are shown by red colour, \n the minimum values by blue colour. \n \n\n \n\n\n\\section{{\\sc and} gate}\n\\label{andgate}\n\n\\subsection{Dirichlet boundary conditions}\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.25]{fig1a} }\n\\subfigure[]{\\includegraphics[scale=0.25]{fig1b}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig1c}}\n\\caption{{\\sc and} gate implementation with Dirichlet boundary conditions. \n(a) Scheme of inputs and outputs. (bc) Density distribution $\\rho$ for inputs (b) $x=1$ and $y=0$ and \n(c ) $x=1$ and $y=1$. }\n\\label{fig1}\n\\end{figure}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[$t=10$]{ \\includegraphics[scale=0.85]{fig2a} }\n\\subfigure[$t=20$]{ \\includegraphics[scale=0.85]{fig2b} }\n\\subfigure[$t=30$]{ \\includegraphics[scale=0.85]{fig2c} }\n\\subfigure[$t=40$]{ \\includegraphics[scale=0.85]{fig2d} }\n\\subfigure[$t=50$]{ \\includegraphics[scale=0.85]{fig2e} }\n\\subfigure[$t=100$]{ \\includegraphics[scale=0.85]{fig2f} } \n\\caption{Density distribution,  $\\rho$, in the implementation of {\\sc and} gate for inputs  $x=1$  and  $y=0$, \nDirichlet boundary conditions for input points. The snapshots are taken at t=10, 20, 30, 40, 50, and 100 steps.}\n\\label{fig2}\n\\end{figure}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[$t=10$]{ \\includegraphics[scale=0.85]{fig3a} }\n\\subfigure[$t=20$]{ \\includegraphics[scale=0.85]{fig3b} }\n\\subfigure[$t=30$]{ \\includegraphics[scale=0.85]{fig3c} }\n\\subfigure[$t=40$]{ \\includegraphics[scale=0.85]{fig3d} }\n\\subfigure[$t=50$]{ \\includegraphics[scale=0.85]{fig3e} }\n\\subfigure[$t=100$]{ \\includegraphics[scale=0.85]{fig3f} }  \n\\caption{Density distribution,  $\\rho$, in the implementation of {\\sc and} gate for inputs  $x=1$  and  $y=1$, \nDirichlet boundary conditions for input points. The snapshots are taken at t=10, 20, 30, 40, 50, and 100 steps.}\n\\label{fig3}\n\\end{figure}\n\nLet us consider implementation of a {\\sc and} gate in case of the Dirichlet boundary conditions in the input sites. \nThe input $I_x$ and $I_y$ and output $O$ sites are arranged at the vertices of an isosceles triangle (Fig.~\\ref{fig1}a):\n$l(I_x, I_y)=102$, $l(I_x, O)=127$, $l(I_y, O)=127$. The Dirichlet boundary conditions are set to $I_x$, $I_y$ and $O$.\nThe material density distribution for inputs $x=1$ and $y=0$ is shown in Fig.~\\ref{fig1}b. The maximum density region connects \n$I_x$ with $I_y$ and no material is formed at site $O$, thus output is 0. The space-time dynamics of the gate is shown in Fig.~\\ref{fig2}. \nWhen both inputs are {\\sc True}, $x=1$ and $y=1$, domains with maximum density of the material span input sites with output site, \n($I_x, O$) and ($I_y, O$) (Fig.~\\ref{fig1}c). Therefore the density value at the output is maximal, $\\rho_O = \\rho_{\\max}$ which indicated \nlogical output 1 ({\\sc True}). Figure~\\ref{fig3} shows intermediate results of density distribution in the gate for $x=1$ and $y=1$. Supplementary videos can be found here \\cite{Safonov2016}. \n\n\n\n\\subsection{Neumann boundary conditions.}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.25]{fig5a}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig5b}}\\\\\n\\subfigure[]{\\includegraphics[scale=0.25]{fig5c}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig5d}}\n\\caption{{\\sc and} gate implementation in case of Neumann boundary conditions.\n(a) Scheme of the gate. Density distribution, $\\rho$, for inputs\n(b)  $x=1$, $y=0$,\n(c ) $x=0$, $y=1$,\n(d)  $x=1$, $y=1$.}\n\\label{fig5}\n\\end{figure}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[$t=10$]{ \\includegraphics[scale=0.85]{fig6a} }\n\\subfigure[$t=20$]{ \\includegraphics[scale=0.85]{fig6b} }\n\\subfigure[$t=30$]{ \\includegraphics[scale=0.85]{fig6c} }\n\\subfigure[$t=40$]{ \\includegraphics[scale=0.85]{fig6d} }\n\\subfigure[$t=50$]{ \\includegraphics[scale=0.85]{fig6e} }\n\\subfigure[$t=200$]{ \\includegraphics[scale=0.85]{fig6f} } \n\\caption{Density distribution,  $\\rho$, in the implementation of {\\sc and} gate for inputs  $x=1$  and  $y=1$, \nNeumann boundary conditions for input points. The snapshots are taken at t=10, 20, 30, 40, 50, and 200 steps.}\n\\label{fig6}\n\\end{figure}\n\nLet us consider the implementation of the {\\sc and}  in case of Neumann boundary conditions for input points. \nScheme of the gate is shown in Fig.~\\ref{fig5}a.  The distance between $I_x$ and $I_y$ is 40 points, the distance between $I_x$ \nand $V$ is 70 points and between $I_y$ and outlet $V$ 90 points. The output site $O$ is positioned in the middle of the segment\n$(I_x, I_y)$. Boundary conditions in $I_x$, $I_y$ and $V$ are set as fluxes, i.e. Neumann boundary conditions.\n\nFigure~\\ref{fig5}b shows density distribution, $\\rho$ for inputs $x=1$ and $y=0$. The maximum density region\ndevelops along the shortest path $(I_x, V)$. Therefore, the density value at $O$ is minimal,  $\\rho_O=\\rho_{\\min}$, which represents\nlogical output {\\sc False}.  For inputs $x=0$ and $y=1$ (Fig.\\ref{fig5}c) the maximal density region is formed along the path \n$(I_y, V)$, i.e. $\\rho_O=\\rho_{\\min}$ and the logical output is {\\sc False}. The material density distribution for inputs $x=1$ and $y=1$ is shown in \nFig.\\ref{fig5}d.  The maximum density region develops along the path $(I_y, I_x, V)$. Thus $\\rho_O=\\rho_{\\max}$ and logical output is {\\sc True}.\n\n\nFigure \\ref{fig6} shows intermediate results of simulating density distribution, $\\rho$, for inputs $x=1$ and $y=1$. At beginning of computation \nthe material develops in proximity of $I_x$, $I_y$ and $V$ (Fig.~ \\ref{fig6}a). Then $I_x$ and $V$ are connected by a domain with highest density of the material (Fig.~ \\ref{fig6}b). The thinner region of high-density material is further develops between $I_x$ and $I_y$ (Fig.~ \\ref{fig6}c--f).\n\n\n\n\n\\section{{\\sc xor} gate}\n\\label{xorgate}\n\n\\subsection{Dirichlet boundary conditions}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.24]{fig4a} }\n\\subfigure[]{\\includegraphics[scale=0.24]{fig4b} }\n\\subfigure[]{\\includegraphics[scale=0.24]{fig4c} }\n\\caption{{\\sc xor} gate implementation with Dirichlet boundary conditions. \n(a) Scheme of inputs and outputs. (bc) Density distribution $\\rho$ for inputs (b) $x=1$ and $y=0$ and \n(c ) $x=1$ and $y=1$. }\n\\label{fig4}\n\\end{figure}\n\n\nLet us consider the implementation of the {\\sc xor} gate in case of Dirichlet boundary conditions for input points. We use similar design as in {\\sc and} gate (Fig.~\\ref{fig1}) but  use two inputs $I_x$ and $I_y$, output $O$  and outlet $V$. The site of output $O$ in {\\sc and} gate is assigned outlet $V$ function\nand the output site $O$ is positioned in the middle of the segment connecting sites $I_x$ and $I_y$ (Fig~\\ref{fig4}a). \nThe temperature at $V$ point is set to 0, $T_V=0$, no temperature boundary conditions are set at $O$. If only one input is {\\sc True} a region of maximum density material is formed along a shortest path between $I_x$ and $I_y$. Therefore, the density value $\\rho_O=\\rho_{\\max}$ thus indicated output {\\sc True} (Fig.~\\ref{fig4}b, $x=1$, $y=0$). When both inputs variables  are {\\sc True}, $x=1$ and $y=1$, maximum density regions are formed along the path \n$(I_x, V)$ and $(I_y, V)$ not along $(I_x, I_y)$. Thus $\\rho_0 = \\rho_{\\min}$, i.e. logical output {\\sc False} (Fig.~\\ref{fig4}c, $x=1$, $y=1$). \n\n\n\n\\subsection{Neumann boundary conditions.}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.25]{fig7a}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig7b}}\\\\\n\\subfigure[]{\\includegraphics[scale=0.25]{fig7c}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig7d}}\n\\caption{{\\sc xor} gate implementation in case of Neumann boundary conditions.\n(a) Scheme of the gate. Density distribution, $\\rho$, for inputs\n(b)  $x=1$, $y=0$,\n(c )  $x=0$, $y=1$,\n(d)  $x=1$, $y=1$.}\n\\label{fig7}\n\\end{figure}\n\n\nLet us consider the implementation of {\\sc xor} gate  in case of Neumann boundary conditions for input points. \nThe gate has five sites: inputs $I_x$ and $I_y$, output $O$, outlets $V_1$ and $V_2$ (Fig.~\\ref{fig7}a).  \nSites $I_x$, $I_y$, $V_1$ and $V_2$ are vertices of the square with the side length 42 points. The output site $O$\nis positioned at the intersection of diagonals of the square. Boundary conditions in $I_x$, $I_y$, $V_1$ and $V_2$ \nare set as fluxes, i.e. Neumann boundary conditions. To ensure convergence of solutions for the stationary problem of \nheat conduction (1) the fluxes at $V_1$ and $V_2$  are set equal to the negative half-sum of fluxes in $I_x$ and $I_y$:\n$Q_{V_1} = Q_{V_2} = - \\frac{Q_{I_x}+Q_{I_y}}{2}$. \n\n\nFor $x=1$ and $y=0$ the maximum density domain is formed between $I_x$ and $V_1$  and \nbetween $I_x$ and $V_2$ (Fig~\\ref{fig7}b). The output site $O$ sits at the $(I_x, V_2)$ diagonal, \ntherefore $\\rho_O = \\rho_{\\max}$, and thus the logical  output is {\\sc True}. When inputs are $x=0$ and $y=1$\nthe maximum density domain is formed between $I_y$ and $V_2$  and  between $I_y$ and $V_1$ (Fig~\\ref{fig7}c).\nThe output site $O$ sits at the $(I_y, V_1)$ diagonal, therefore $\\rho_O = \\rho_{\\max}$, and thus the logical output is {\\sc True}.\nWhen goths inputs are {\\sc True}, $x=1$ and $y=1$, domains of high-density material develop along shortest paths \n$(I_x, V_1)$ and $(I_y, V_2)$ (Fig~\\ref{fig7}d). These domain do not cover the site $O$, therefore logical output is {\\sc False}.\n\n\n\\section{One-bit half-adder}\n\\label{onebithalfadder}\n\n\\subsection{Dirichlet boundary conditions}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.25]{fig8a}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig8b}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig8c}}\n\\caption{One-bit half-adder implementation in case of Dirichlet boundary conditions.\n(a) Scheme of the adder. Density distribution, $\\rho$, for inputs\n(b)  $x=1$, $y=0$,\n(c )  $x=1$, $y=1$.\n}\n\\label{fig8}\n\\end{figure}\n\n\n\n\nTo implement the one-bit  half-adder in case of Dirichlet boundary conditions for input points we combine designs of {\\sc and} and {\\sc xor} gates\n(Figs.~\\ref{fig1}a and \\ref{fig4}a). We introduce the following changes to the scheme shown in Fig~\\ref{fig4}a:  the former outlet $V$\nis designated as output $O_1$, the former output $O$ is designated as output $O_2$ (Fig.~\\ref{fig8}a).  Temperature value at $O_1$ is set zero, \n$T_{O_1}=0$. No temperature boundary conditions are set at $O_2$. The output $O_1$ indicated logical value $xy$ and the output $O_2$ logical \nvalue $x \\oplus y$. When only one of the inputs is {\\sc True} and other {\\sc False}, e.g. $x=1$ and $y=0$ as shown in Fig.~\\ref{fig8}b, the density\nvalue at $O_1$ is minimal, $\\rho_{O_1}=\\rho_{\\min}$, and the density value at $O_2$ is maximal, $\\rho_{O_2}=\\rho_{\\max}$.  Thus $O_1$ indicated {\\sc False} and $O_2$ {\\sc True}. For inputs $x=1$ and $y=1$ we have $\\rho_{O_1}=\\rho_{\\max}$ and $\\rho_{O_2}=\\rho_{\\min}$, i.e. logical outputs \n{\\sc True} and {\\sc False}, respectively. \n\n\n\\subsection{Neumann boundary conditions}\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.25]{fig9a}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig9b}}\\\\\n\\subfigure[]{\\includegraphics[scale=0.25]{fig9c}}\n\\subfigure[]{\\includegraphics[scale=0.25]{fig9d}}\n\\caption{One-bit half-adder implementation in case of Neumann boundary conditions.\n(a) Scheme of the adder. Density distribution, $\\rho$, for inputs\n(b)  $x=1$, $y=0$,\n(c ) $x=0$, $y=1$,\n(b)  $x=1$, $y=1$.\n}\n\\label{fig9}\n\\end{figure}\n\n\n\n\\begin{figure}[!tbp]\n\\centering\n\\subfigure[$t=10$]{ \\includegraphics[scale=0.85]{fig10a} }\n\\subfigure[$t=20$]{ \\includegraphics[scale=0.85]{fig10b} }\n\\subfigure[$t=30$]{ \\includegraphics[scale=0.85]{fig10c} }\n\\subfigure[$t=40$]{ \\includegraphics[scale=0.85]{fig10d} }\n\\subfigure[$t=50$]{ \\includegraphics[scale=0.85]{fig10e} }\n\\subfigure[$t=69$]{ \\includegraphics[scale=0.85]{fig10f} } \n\\caption{Density distribution,  $\\rho$, in the implementation of one-bit half-adder for inputs  $x=1$  and  $y=1$, \nNeumann boundary conditions for input points. The snapshots are taken at t=10, 20, 30, 40, 50, and 69 steps.}\n\\label{fig10}\n\\end{figure}\n\n\nLet us consider the implementation of one-bit half-adder in case of Neumann boundary conditions for input points. \nThe devices consists of seven sites: two inputs $I_x$ and $I_y$, two outputs $O_1$ and $O_2$, \nthree outlets $V_1$, $V_2$ and $V_3$ (Fig.~\\ref{fig9}a). Sites $I_x$, $I_y$, $O_1$ and $O_2$ are vertices of a square with \nthe side length 40. The output $O_2$ is positioned at the intersection of diagonals of this square. The output $O_1$ is positioned \nat the middle of the segment connecting $V_2$ and $V_3$. The distance between $V_1$ and $V_3$ is 36 points, the distance between \n$V_3$ and $V_2$ is 51 points. The output $O_1$ represent logical function $xy$ and the output $O_2$ function $x \\oplus y$. \nBoundary conditions in $I_1$, $I_2$, $V_1$, $V_2$ and $V_3$ are set as fluxes,  thus corresponding to Neumann boundary \nconditions. To ensure convergence of solutions for the stationary problem of heat conduction (1) the flux values at\n$V_1$, $V_2$ and $V_3$ are set equal to one third of the negative sum of fluxes in  $I_x$ and $I_2$: \n$Q_{V_1} = Q_{V_2} = Q_{V_3} = - \\frac{Q_{I_x}+Q_{I_y}}{3}$.\n\nFigure~\\ref{fig9}b shows results of calculating density distribution $\\rho$ for inputs $x=1$ and $y=0$. There the \nmaximum density region connects $I_x$ with $V_1$, $V_2$ and $V_3$. The density domain $(I_x, V_3)$ is not a straight line because\nthe system benefits most when a of the segment $(I_x, V_3)$ coincide with the segment $(I_x, V_1)$. The site $O_2$ is covered by maximum \ndensity domain $(I_x, V_1)$, $\\rho_{O_2} = \\rho_{\\max}$, thus representing logical value {\\sc True}; the output $O_1$ is {\\sc False} because\n$\\rho_{O_2} = \\rho_{\\min}$. \n\nThe density distribution $\\rho$ calculated for inputs $x=0$ and $y=1$ is shown in Fig.~\\ref{fig9}c. The maximum density region \nconnects $I_y$ with $V_1$, $V_2$ and $V_3$ via paths $(I_y, V_1)$, $(I_y, V_2)$, $(I_y, V_3)$. The output $O_2$ belongs to $(I_y, V_2)$ therefore it \nindicates logical output {\\sc True}. The output $O_1$ indicated {\\sc False} because it is not covered by a high density domain. \n\nThe density distribution $\\rho$ calculated for inputs $x=1$ and $y=1$ is shown in Fig.~\\ref{fig9}d. The maximum density regions are developed along paths\n$(I_1, V_2)$, $(I_2, V_1)$, $(I_2, V_3)$ and $(V_2, V_3)$. There is heat flux between $V_1$ and $V_2$ which forms a segment of high density material.\nThe high density material covers $O_1$, therefore the output $O_1$ indicate logical value {\\sc True}. The output $O_2$ is not covered by a high density material, \nthus {\\sc False}. \n\n\nFigure \\ref{fig10} shows intermediate results of simulating density distribution $\\rho$ for inputs $x=1$ and $y=1$.\n\n\n\\section{Discussion}\n\\label{discussion}\n\nWe implemented logical gates and circuits using optimisation of conductive material when solving stationary problems of heat conduction. \nIn the simplest case of two sites with given heat fluxes the conductive material is distributed between the sites in a straight line. \nThe implementations of gates presented employ several sites, exact configuration of topologically optimal structures of the conductive \nmaterial is determined by value of input variables. The algorithm of optimal layout of the conductive material is similar to a biological process of \nbone remodelling. The algorithm proposed can be applied to a wide range of biological networks, including  neural networks, \nvascular networks, slime mould, plant routes, fungi mycelium. These networks will be the subject of further studies. In future we can also \nconsider an experimental laboratory testing of the numerical implementations of logical gates, e.g. via dielectric breakdown tests, because\nthe phenomenon is also described by Laplace's stationary heat conduction equation which takes into account the evolution of conductivity of \na medium determined by the electric current. The approach to developing logical circuits, proposed by us, could be used in fast-prototyping of \nexperimental laboratory unconventional computing devices. Such devices will do computation by changing properties of their material substrates. \nFirst steps in this direction have been in designing Belousov-Zhabotinsky medium based computing devices for pattern recognition~\\cite{fang2016pattern} and \nconfigurable logical gates~\\cite{wang2016configurable}, learning slime mould chip~\\cite{whiting2016towards}, \nelectric current based computing~\\cite{ayrinhac2014electric},\nprogrammable excitation wave propagation in living bioengineered tissues~\\cite{mcnamara2016optically}, heterotic computing~\\cite{kendon2015heterotic}, memory devices in digital collides~\\cite{phillips2014digital}.\n\n\\section*{Supplemetary materials}\n\n\\subsection*{{\\sc xor} gate, Neumann boundary conditions}\n\\begin{itemize}\n\\item inputs $x=0$, $y=1$: \\url{https:\/\/www.youtube.com\/watch?v=osB12UqM3-w}\n\\item inputs $x=1$, $y=0$: \\url{https:\/\/www.youtube.com\/watch?v=lKMeu1nFuak}\n\\item inputs $x=1$, $y=1$: \\url{https:\/\/www.youtube.com\/watch?v=AxdCVVtIqgk}\n\\end{itemize}\n\n\\subsection*{One-bit half-adder, Neumann boundary conditions} \n\\begin{itemize}\n\\item inputs $x=0$, $y=1$: \\url{https:\/\/www.youtube.com\/watch?v=i81WTCrg8Lg}\n\\item inputs $x=1$, $y=0$: \\url{https:\/\/www.youtube.com\/watch?v=impbwJXjCAM}\n\\item inputs $x=1$, $y=1$: \\url{https:\/\/www.youtube.com\/watch?v=ubrgfzlAQQE}\n\\end{itemize}\n\n\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{}}\n\n\n\\section{Introduction}\n\n\n\nDespite rapid progress in computer technology, there are still computational problems that are\ndifficult to solve for any known algorithm that uses modern computers. However, the theory that\ndescribes physics on atomic length-scales, quantum mechanics, suggests a new way to attack\nhard computational problems in a more efficient way. A computer using operations involving\nthe entanglement of quantum states is known as a quantum computer. \\cite{Deutsch1985} A number of algorithms proposed for\nquantum computers are expected to solve many classically hard problems. \\cite{nielsen2010quantum} For example,\nShor's algorithm \\cite{Shor1997} can efficiently solve the prime factorization problem, which is difficult to\nsolve even with state-of-the-art classical algorithms and computers.\nThe quantum analog of the bit -- the fundamental information storage unit in classical\ncomputers -- is called the qubit. Many physical realizations of  qubits for quantum computers\nare being developed, including semiconductor, superconductor, nuclear, and optical qubit\nsystems. The mature semiconductor manufacturing industry today offers several advantages in\nthe construction of semiconductor qubits. In particular, the nanoscale semiconductor structure\nknown as the quantum dot has been proposed as a possible realization of the semiconductor\nqubit. \\cite{LossDiVincenzo1998}\n\nSemiconductor qubits constructed from typical semiconductor (e.g. GaAs) quantum dots have limited usefulness, because\nquantum information stored in the device can be lost due to spin-decoherence. Sources of\nspin-decoherence in semiconductors include the spin-orbit, hyperfine, and electron-phonon interactions.\n{Graphene, \\cite{RevModPhys.81.109,Kotov2012,DasSarma2011} a two-dimensional lattice structure formed by carbon, is a promising material for\navoiding spin-decoherence. \\cite{trauzettel2007spin} Graphene has not only weak spin-orbit coupling, but also a\nnegligible hyperfine interaction, since carbon-12 has zero nuclear spin. These advantages of\ngraphene have driven researchers towards developing a graphene-based qubit. One proposed\nmodel for the graphene qubit is the graphene nanoribbon (GNB) quantum dot, \\cite{trauzettel2007spin,BreyFertig2006,Gucclu2014} which has been\nexperimentally realized by several groups. \\cite{Han2007,Li2008,Stampfer2009,Guttinger2012,Liu2010,Wei2013}}\n\nPrevious theoretical efforts to model GNB quantum dots \\cite{trauzettel2007spin} offer only\nestimates of the electron-electron interaction and many-particle effects in graphene.\nConsequently, it is not clear if the predicted long-range Heisenberg exchange coupling between\nthe dots---which is necessary for qubit operations in universal quantum\ncomputing---can be achieved in practice. Nor are the effects of gate voltage changes on the\nelectronic structure and the exchange coupling in the multi-electron regime well understood.\nRealistic models of graphene qubit operation require a better understanding of these effects. \\cite{LossDiVincenzo1998}\nTo understand how the GNB quantum dot qubit functions in real\napplications, we study a more complete model of the graphene qubit using a numerical\napproach. {  To study the exchange coupling between two graphene qubits, we use an unrestricted Hartree-Fock method with a generalized-valence-bond (GVB) wave function. \\cite{Ostlund1996,Hu2000,Yannouleas2001,Fang2002} This is a double-Slater-determinant approach, which takes into account the correlation effect due to charge separation and is suitable for describing diatomic molecules. In contrast, the conventional Hartree-Fock method, which uses factorized single-particle product states (i.e. a single determinant) cannot describe the correlation effect. The GVB method allows us to capture the physics of entangled states (crucial for quantum computing), which cannot be described by the single-determinant Hartree-Fock method. \\cite{Horodecki2009} The computational cost of the GVB method is much lower than that of the configuration interaction calculations. \\cite{Ostlund1996,Dutta2008} Thus, it is easier to analyze the exchange coupling between two qubits in various gate configurations with the GVB method.} In this work, we employ this numerical scheme and the Dirac equation to provide a realistic simulation of a GNB quantum dot qubit.\n\n\n\n\\section{Method}\n\n\\subsection{One-particle problem}\n\nConsider a graphene nanoribbon with width $W$ and length $L$ and armchair boundary conditions. Let the x-axis be the direction across the width of the nanoribbon and the y-axis be the direction parallel to the length of the ribbon. { The behavior of an electron with energy close to the Fermi level in this system can be described by the Dirac equation \\cite{Semenoff1984,DiVincenzo1984,trauzettel2007spin,BreyFertig2006,RevModPhys.81.109}\n\n\\begin{eqnarray}\n&& H_1 |\\psi \\rangle = \\epsilon   | \\psi \\rangle \\\\\n&& H_1 =  -i \\hbar v\n\\begin{pmatrix}\n\\sigma_x  \\partial_x + \\sigma_y  \\partial_y & 0 \\\\\n0 & -\\sigma_x  \\partial_x + \\sigma_y  \\partial_y  \\\\\n\\end{pmatrix}\n +V(y) ,\\notag\n\\end{eqnarray}\nwhere $\\hbar$ is Planck's constant, $v$ is the Fermi velocity of graphene, $\\sigma_x,\\sigma_y$ are Pauli matrices for the pseudospin describing two sublattices of graphene, $\\partial_x,\\partial_y$ are partial derivatives, and $V(y)$ is the electrical confining potential along the y-axis.} $|\\psi \\rangle$ is a 4-component spinor in the form\n\\begin{eqnarray}\n|\\psi \\rangle =\n\\begin{pmatrix}\n\\psi (K,A)\\\\\n\\psi (K,B)\\\\\n-\\psi(K',A)\\\\\n-\\psi(K',B)\\\\\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $K,K'$ label the two valleys in the Brillouin zone of graphene, $A,B$ label the two sublattices of graphene, and $\\psi$ is the envelope function.\n\n\nWe expand the electron envelope function within a basis set as follows\n\\begin{eqnarray}\n |\\psi \\rangle &=& \\sum_{m,s,n} \\phi^{m,n}_s   |\\psi^{m,n}_s \\rangle, \\\\\n \\langle x,y  |\\psi^{m,n}_s \\rangle &=&\n\\frac{1}{\\sqrt{2WL}}  \\begin{pmatrix}\n\\chi_s e^{iq_nx} \\\\\n\\chi_s e^{-iq_nx}  \\\\\n\\end{pmatrix}\n f_m (y) ,\n\\end{eqnarray}\n where $\\{ f_m(y) \\}$ is a set of basis functions of variable $y$, $s=A,B$ labels the two components of the pseudospin describing two sublattices. The basis vectors for the two-component pseudospinors are\n\\begin{eqnarray}\n&\\chi_A &=\n\\begin{pmatrix}\n1 \\\\\n0 \\\\\n\\end{pmatrix} \\\\\n&\\chi_B &=\n\\begin{pmatrix}\n0 \\\\\n1 \\\\\n\\end{pmatrix}.\n\\end{eqnarray}\nThe boundary conditions are\n\\begin{eqnarray}\n&\\langle x,y|\\psi \\rangle |_{x=0} &=\n\\begin{pmatrix}\n 0 & {\\bf I}_{2} \\\\\n {\\bf I}_{2} & 0 \\\\\n\\end{pmatrix}\n\\langle x,y |\\psi \\rangle |_{x=0} \\label{bc1} \\\\\n&\\langle x,y|\\psi \\rangle |_{x=W} &=\n\\begin{pmatrix}\n 0 & e^{+ i\\frac{2\\pi\\mu}{3} }{\\bf I}_{2} \\\\\n e^{ -i\\frac{2\\pi\\mu}{3} }{\\bf I}_{2} & 0 \\\\\n\\end{pmatrix}\n\\langle x,y |\\psi  \\rangle |_{x=W} , \\label{bc2}\n\\end{eqnarray}\nwhere $\\mu =\\pm 1, 0$ is a constant determined by the termination of the ribbon edge. $\\mu =\\pm 1$ defines the semiconducting boundary condition. The symbol ${\\bf I}_{2}$ in Eqs. (\\ref{bc1}) and (\\ref{bc2}) denotes a $2\\times2$ identity matrix in pseudospin space. {Imposing the  semiconducting boundary conditions on the  basis functions leads to quantization of the electronic states in the x-direction\n\\begin{eqnarray}\n &q_n &=\\frac{\\pi}{W}(n + \\frac{\\mu}{3}) , n \\in \\mathbb{Z} \\\\\n &&= (3n + \\mu )q_0 ,\n\\end{eqnarray}\nwhere the characteristic momentum scale $q_0 =  \\frac{\\pi}{3W}$ is defined by the width of the ribbon $W$. In this work, $1\/q_0$ is the characteristic length scale and $\\hbar v q_0$ is a characteristic energy scale. Throughout this paper, we consider only the condition with $n=0$ and $\\mu=+1$, since the ribbon is narrow enough such that the energies of higher confined states are outside the range of interest.}\n\n\nThe basis functions $f_m(y)$ are chosen to be sinusoidal functions confined within the interval $[0,L]$,\n\\begin{eqnarray}\nf_m(y) &=& \\sqrt{2}\\sin (\\frac{\\pi m y}{L}) .\n\\end{eqnarray}\nThe matrix elements of the one-particle Hamiltonian and overlap can then be written down analytically. {The Dirac equation is cast into the form of a generalized eigenvalue problem\n\\begin{eqnarray}\n\\sum_{ms} \\langle \\psi^{m',n}_{s'} | H_1 | \\psi^{m,n}_{s} \\rangle  \\phi^{m,n}_s = \\epsilon \\sum_{ms} \\langle \\psi^{m',n}_{s'}  | \\psi^{m,n}_s \\rangle \\phi^{m,n}_s , \\notag \\\\\n\\end{eqnarray}\nwhich is solved numerically.}\n\n\n\n\n\\subsection{Two-particle problem}\n\nTo evaluate the exchange coupling between two electrons separately located in two neighboring GNB quantum dots, we need to consider the mutual Coulomb interaction and exchange term between them.  The Coulomb interaction in two-dimension is given by {\n\\begin{equation}\n v_{ee} = (\\frac{e^2}{4\\pi\\epsilon \\hbar v }) \\hbar v\n  \\frac{1}{\\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}}, \\notag\n\\end{equation}}\nwhere $(\\frac{e^2 }{4\\pi\\epsilon \\hbar v })$ is a dimensionless Coulomb parameter, which can be viewed as the fine-structure constant of graphene. {One expects $\\frac{e^2 }{4\\pi\\epsilon \\hbar v }=2.2$ or smaller for a suspended graphene.\\cite{Hwang2012,Reed2010,Kotov2012} In this work we use $\\frac{e^2 }{4\\pi\\epsilon \\hbar v }=1.43$ for graphene on quartz substrate. \\cite{Hwang2012}}\n\nWe adopt the unrestricted Hartree-Fock method with generalized-valence-bond wave function \\cite{Fang2002} to solve the two-particle problem. Because the spin-orbit interaction is omitted here, the total wavefunction can be written as the product of the spatial wavefunction, denoted by $\\Psi_+ (\\Psi_-)$, and the corresponding two-particle spinor for the singlet (triplet) state. The spatial wave function $\\Psi_+ (\\Psi_-)$ of the spin singlet (triplet) must be symmetric (antisymmetric) with respect to the exchange of two particles to make the total wavefunction antisymmetric. The spatial wavefunctions take the form\n\\begin{eqnarray}\n | \\Psi_\\pm \\rangle =\n\\frac{1}{\\sqrt{2(1\\pm S^2)}}( | \\psi_L , \\psi_R \\rangle \\pm | \\psi_R , \\psi_L \\rangle),\n\\end{eqnarray}\nwhere $\\psi_L$ ($\\psi_R$) denotes a four-component single-particle wavefunction (for the Dirac particle with two bands and two valleys at K and K') localized at the left (right) quantum dot, and $S=\\langle \\psi_L | \\phi_R \\rangle$, which is enhanced when the Klein tunneling condition is met.  The two-particle Hamiltonian can be written as\n\\begin{eqnarray}\nH = H_1 \\otimes 1 + 1\\otimes H_1 + v_{ee},\n\\end{eqnarray}\nwhere $H_1$ is the single-particle Hamiltonian for a Dirac particle as defined in Eq.~(1).\nThe two-particle Schrodinger equation is\n\\begin{eqnarray}\n& H | \\Psi_\\pm \\rangle  &= E | \\Psi_\\pm \\rangle .\n\\end{eqnarray}\n$\\psi_L$ ($\\psi_R$) appearing in $| \\Psi_\\pm \\rangle$ is expanded in a non-orthonormal basis set $\\{|\\nu \\rangle\\}$ as defined in Eq.~(4) with $\\nu$ being a composite index for $(m,n,s)$.\nIn each iteration with a given $\\psi_R$, we write $| \\psi_L \\rangle=\\sum_{\\nu}C^L_{\\nu}|\\nu \\rangle$ and solve for the expansion coefficients of $C^L_{\\nu}$ according to the following projected Scr\\\"{o}dinger equation for $| \\Psi_\\pm \\rangle$.\n\\begin{eqnarray}\n& \\langle \\nu',\\psi_R|H| \\Psi_\\pm \\rangle  &= E \\langle \\nu',\\psi_R | \\Psi_\\pm \\rangle ,\n\\end{eqnarray}\nwhere $\\langle\\nu' , \\psi_R|$ denotes a product state of basis state $\\langle \\nu'|$ and $\\langle \\psi_R|$.\n\nThe generalized eigenvalue problem to be solved becomes\n\\begin{eqnarray}\n\\sum_{\\nu} \\langle \\nu' | H_{GVB} |\\nu \\rangle C^L_{\\nu} = E \\sum_{\\nu} \\langle \\nu' | S_{GVB} |\\nu \\rangle C^L_{\\nu},\n\\end{eqnarray}\nwhere the Hamiltonian and overlap matrices elements are\n\\begin{eqnarray}\n\\langle \\nu' | H_{GVB} |\\nu \\rangle &=&  \\langle \\nu' |H_1| \\nu \\rangle  + \\langle \\nu' | \\nu \\rangle   \\langle \\psi_R |H_1| \\psi_R \\rangle \\\\\n& \\pm & \\langle \\nu' |H_1| \\psi_R \\rangle \\langle \\psi_R |\\nu \\rangle \\pm \\langle \\nu' | \\psi_R \\rangle \\langle \\psi_R |H_1| \\nu \\rangle   \\notag \\\\\n&+& \\langle \\nu',\\psi_R | v_{ee} | \\nu, \\psi_R \\rangle  \\pm  \\langle \\nu',\\psi_R | v_{ee} |\\psi_R , \\nu \\rangle \\notag \\\\\n \\langle \\nu' | S_{GVB} |\\nu \\rangle &=& \\langle \\nu' | \\nu\\rangle   \\pm \\langle \\nu' | \\psi_R \\rangle  \\langle \\psi_R |\\nu\\rangle .\n\\end{eqnarray}\nThe iteration continues until self-consistency is reached. For the triplet state, we carry out the reorthonormalization and projection procedure described in \\textcite{Fang2002} to resolve the linear-dependence problem in the generalized eigenvalue problem.\n\n\\subsection{The double well model}\n\n\\begin{figure}\n\\includegraphics[trim = 0mm 0mm 0mm 0mm,width=0.4\\textwidth]{fig1.eps}\n\\caption{Double well potential profile along the graphene ribbon length. The width of each well is $w$, the well-to-well separation is $d$,  the { barrier  height outside the double wells is $\\mu_{out}$, the height of the middle potential barrier is $\\mu_b$, and the potential at the bottom of the well is set to zero.}\\label{fig1}}\n\\end{figure}\n\n\nWe model a GNB double-dot system by a double square well potential in the y-direction, as shown in Fig.~\\ref{fig1}. Unless specified, we use the following parameters throughout this work. The physical parameters used in the model are those reported in the experimental study by \\textcite{Liu2010} The width of the ribbon is {$W=20 (nm)\\approx 1q_0^{-1}$}, and hence our characteristic energy is {$\\hbar v q_0\\approx 32.9 (meV)$}. The length of the ribbon is {$L=800 (nm)\\approx 40q_0^{-1}$}. The width of each dot (i.e., the width of each confining well) is $w$. The separation between the dots (i.e., the width of the potential barrier between two dots) is $d$. {The potential heights of the barrier and outside region are given by $\\mu_b$ and $\\mu_{out}$, respectively. The potential at the bottom of the well is set to zero.} We use $\\mu_{out}=1.5 \\hbar vq_0 =49.4 (meV)$ as suggested by previous theoretical work. \\cite{trauzettel2007spin} The Fermi energy is fixed at $E_F=1 \\hbar vq_0$. The number of sinusoidal basis functions used is 50.\n\n\n\n\n\n\\section{Results and Discussion}\n\n\\subsection{{Single-particle solutions}}\n\n\\begin{figure}\n\\includegraphics[trim = 20mm 0mm 80mm 0mm,width=0.45\\textwidth]{fig2.eps}\n\\caption{ (a) Single particle energy levels as functions of confining potential height for a single well. The well width is $w=2q_0^{-1}$. The conducting states (states forming the left upper triangle) and the Klein tunneling states (states forming the right lower triangle) are also shown in our calculation. (b) Single particle energy levels as functions of inter-well distance for a double well. The width of the wells is $w=4q_0^{-1}$. The ground state (solid blue) is generally above but very close to the top of the barrier valence band (dotted black), and hence is a {Klein} tunneling assisted state. (c) Energy splittings ($\\Delta E_m= E_{m+1}-E_m$) as functions of inter-well distance for a double well. \\label{fig2}}\n\\end{figure}\n\n\nWe first examine the single-particle behavior of single and double potential wells. Figure \\ref{fig2}(a) shows the single-particle energy levels as functions of confining barrier height $\\mu_{out}$ for a single potential well, where the width of the well is $w=2q_0^{-1}$ and the length of the ribbon is $L=16 q_0^{-1}$. { The left upper triangle region contains many slanted lines which describe the discretized states of the conduction bands derived from the GNB regions outside the well. Similarly, the right lower triangle region also contains many slanted lines which describe the discretized states of the GNB valence bands. The discretization is a result of quantum confinement due to the finite length ($L$) of GNB considered. The two regions are separated by a constant $2 \\hbar v q_0$, which corresponds to the band gap of GNB. In the gap, there are 3 quantized levels, which are the quantum confined states derived from the middle well. The dependence of these energy levels on the barrier height $\\mu_{out}$ matches the results obtained by solving the transcendental equation as described in \\textcite{trauzettel2007spin}. The good agreement validates our numerical procedure. It should be noted that our numerical method based on the expansion within a nearly complete basis set can handle not only the simple case of a GNB quantum dot defined by a square well but also arbitrary potential profile along the $y$ axis. This makes it easy to extend to double wells and the two-particle problem for finding the exchange coupling between two neighboring qubits.}\n\n\nFigure~\\ref{fig2}(b) shows energy levels as functions of inter-well separation, $d$ for a double square well. The width of the wells is $w=4q_0^{-1}$. This figure involves some high-energy excited states, so the number of sinusoidal basis functions used in this calculation is enlarged to 100 for higher accuracy. These energy levels come in pairs (indicated by the same color but different curve types), corresponding to two split levels associated with the inter-well coupling of one energy level in the left dot and the corresponding one in the right. The amount of energy splitting reflects the coupling strength of the two states located in the two dots, which would be degenerate in energy in the absence of the inter-well coupling. {Above the energy of the valence band maximum (VBM) of the barrier (black dotted line), there are three pairs of bound states indicated by blue, green, and red colors.  The ground state (blue) pair  is very close to the VBM of the barrier, and one expects to see enhanced inter-well coupling due to the Klein-tunneling effect.}\n\n\nFigure~\\ref{fig2}(c) shows the energy splittings due to inter-well coupling ($\\Delta E_m=E_{m+1}-E_m$) as functions of the well-to-well separation, $d$ for the same double square well potential in semi-log scale. $E_0$ is the ground state energy, $E_1$ is the 1st-excited state energy ...etc. The curves are almost linear in semi-log scale, indicating that the energy splittings decay exponentially with the well-to-well separation. For small separation the ground state splitting, $\\Delta E_0$ is smaller than the excited state splittings $\\Delta E_m, m=2,4$. However, for large separation ($q_0d>4$) the ground state splitting becomes larger than the excited state splittings, indicating the ground state splitting has a smaller decay rate comparing to the excited state splittings. This suggests that the inter-well coupling is enhanced by the Klein tunneling for the ground state pair, in qualitative agreement with the prediction in Ref.~\\onlinecite{trauzettel2007spin} based on simple estimation.\n\n\n\\subsection{{Effects of barrier height on two-particle solutions}}\n\n\n\\begin{figure}\n\\includegraphics[trim = 35mm 0mm 45mm 0mm,width=0.5\\textwidth]{fig3.eps}\n\\caption{Negative exchange coupling $-J_{ex}$ as a function of the inter-well barrier potential height $\\mu_b$ for $q_0d=2$ (blue solid) and $q_0d=4$ (red dashed). The width of the wells is $w=4q_0^{-1}$. (a) Linear plot. (b) Semi-log plot. \\label{fig3}}\n\\end{figure}\n\n\n{ Here, we study the effects of barrier height on the exchange coupling ($J_{ex}=E_{triplet}-E_{singlet}$) between two electrons in the GNB double well. Figure~\\ref{fig3}(a) shows  $-J_{ex}$ as a function of barrier potential height $\\mu_b$. For small barrier heights, $-J_{ex}$ can be either negative (for $q_0d=2$, blue solid) or positive (for $q_0d=4$, red dashed) depending on the well-to-well separation.} For small barrier heights, the potential profile behaves like a single confining well instead of a double well. A singlet-triplet ground state transition is expected for barrier heights lower than a critical value, which shall be discussed further in Section \\ref{distance}. In this section we focus on the regime in which the barrier height is larger than or equal to the critical value. In our model, the critical value can be estimated by comparing the barrier height with both the single-particle ground state energy and the bottom of the conduction bands associated with the wells. In Fig.~\\ref{fig3}(a), the green dashed line marks the point when $\\mu_b$ crosses $\\langle H_1 \\rangle$, the expectation value of $H_1$ in the triplet solution for $q_0d=2$, which has a weak dependence on $\\mu_b$. We label this point by $\\bar E_1$. $\\bar E_1$ happens to be almost the same as the average of $\\langle H_1 \\rangle$ over $\\mu_b$ for $\\mu_b$ from 0 to 2$\\hbar vq_0$.\nThe black dashed line indicates where the barrier height equals to the bottom of the conduction bands of the wells, which is at $1\\hbar vq_0$. {The critical value $\\approx 1.1\\hbar vq_0$, which lies between $1\\hbar vq_0$ and $\\bar E_1$.}\nFor $\\mu_b$ larger than the critical value {($\\approx 1.1\\hbar vq_0$) $-J_{ex}$} is always positive, and it increases monotonically up to $\\mu_b= 1.9\\hbar vq_0$.\n\n\nFigure~\\ref{fig3}(b) shows $-J_{ex}$ for $\\mu_b > 1.1\\hbar vq_0$ on a semi-log scale. The magnitude of  $-J_{ex}$ grows exponentially for $q_0d=2$, and super-exponentially for $q_0d=4$. The coupling for $q_0d=4$ can be almost as large as the case for $q_0d=2$ as $\\mu_b$ reaches $ 1.9\\hbar vq_0 $. The exchange coupling for $q_0d=2$ (blue solid) is linear in the semi-log plot, which indicates an exponential growth. For a longer well-to-well separation with $q_0d=4$, one can see super-exponential growth of the exchange coupling. For $q_0d=4$ and $\\mu_b >1.92 \\hbar vq_0$, the electrons in the singlet state are in the first-excited single-particle state of both dots, while the electrons in the triplet state are still in the single-particle ground state. The exchange coupling can not be defined in this case.\n\n\n\n \\begin{table\n \\caption{Singlet total energy $E_{singlet}$, triplet total energy $E_{triplet}$, and triplet single particle energy $E_1$ for some selected inter-dot distance $d$ and barrier height $\\mu_b$.\\label{table1}}\n \\begin{ruledtabular}\n \\begin{tabular}{c c c c c}\n $q_0d$ & $\\mu_b\/\\hbar vq_0$ & $E_{singlet}\/\\hbar vq_0$ & $E_{triplet}\/\\hbar vq_0$ & $E_1\/\\hbar vq_0$\\\\ \\hline\n 2 & 1.5 & 2.63897  & 2.63869 & 1.20982 \\\\\n 2 & 1.9 & 2.65891 &   2.65711 & 1.21937 \\\\\n 4 & 1.5 & 2.57493 &  2.57489  & 1.20796 \\\\\n 4 & 1.9 &  2.59815 &   2.59668 &   1.21833   \\\\\n 8 & 1.5 & 2.51309 &  2.51309  & 1.20725 \\\\\n 8 & 1.9 &  2.53504  &  2.53501 & 1.21845 \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n\n\nThe super-exponential growth of $-J_{ex}$ as $\\mu_b$ increases for larger well-to-well separation as shown in Fig.~\\ref{fig3}(b) is a special characteristic of the GNB quantum dot qubit. The overlap between the wave functions of the electrons in the left dot and the right dot is expected to be enhanced by the Klein tunnelling of Dirac particles when the valence band maximum of the barrier is close to the energies of conduction band states in the wells. \\cite{trauzettel2007spin} This long-distance coupling of Dirac particles is suggested as a possible advantage of the GNB quantum dot qubit over qubits in conventional systems. For the GNB qubit, the exchange coupling for the long distance case ($q_0d=4$) is almost as large as that in the short distance case ($q_0d=2$) as the barrier height approaches $\\mu_b=2\\hbar vq_0$. This implies that the valence band maximum in the barrier region is approaching the bottom of conduction band in the well. This result lends support to the proposal in \\textcite{trauzettel2007spin}.\n\n\n\n\\begin{figure}\n\\includegraphics[trim = 35mm 0mm 35mm 0mm,width=0.33\\textwidth]{fig4.eps}\n\\caption{Projected charge densities $\\rho_\\pm(y)$ along the ribbon for the case $q_0d=4$ for (a) singlet state and (b) triplet state for various barrier potential strengths. $\\mu_b=0.3\\hbar vq_0$ (red dotted), $\\mu_b=0.6\\hbar vq_0$ (green dash-dot), $\\mu_b=1.2\\hbar vq_0$ (blue dashed), $\\mu_b=1.9\\hbar vq_0$ (black solid). The density in the barrier region decreases as $\\mu_b$ increasing from zero to $0.6$ and $1.2 \\hbar vq_0$, but increases as $\\mu_b$ further increases to a higher value $1.9\\hbar vq_0$. The increment of density in the barrier region is significantly larger for the singlet state than that of the triplet state. The width of the wells is $w=4q_0^{-1}$.\\label{fig4}}\n\\end{figure}\n\n\n\n{This overlap enhancement can be illustrated by examining the projected charge density, defined as\n\\begin{equation} \\rho_\\pm (y_1)=\\int {\\Psi}_\\pm(x_1,y_1;x_2,y_2)\\cdot \\Psi_\\pm(x_1,y_1;x_2,y_2) dx_1dx_2dy_2, \\end{equation}\n where $\\Psi_\\pm$ is a four-dimensional vector defined in Eq.~(13). { The projected charge density can be written explicitly as\n\\begin{eqnarray} \\rho_\\pm (y_1)&=&\\int dx_1 [|\\psi_L(x_1,y_1)|^2\/2+|\\psi_R(x_1,y_1)|^2\/2 \\nonumber \\\\\n&\\pm & S \\psi_L(x_1,y_1)\\psi_R(x_1,y_1)]\/(1\\pm S^2). \\end{eqnarray} \nFor all cases, the $\\rho_\\pm (y)$ is dominated by the sum of the first two terms, since the overlap $S$ is small. We plot $\\rho_\\pm (y)$ as a function of $y$ in Fig.~\\ref{fig4} for singlet and triplet states for various barrier heights. The electron density is mainly localized in the two potential wells. The charge density between two wells decreases as the barrier potential is raised from $\\mu_b=0.3\\hbar vq_0$ (red dotted) to $0.6\\hbar vq_0$ (green dash-dot) and $1.2\\hbar vq_0$ (blue dashed). However, as the barrier height is raised to a higher value $\\mu_b=1.9\\hbar vq_0$ (black solid), the charge density between the wells becomes significantly larger for the singlet state as seen in Fig.~4(a). In contrast, there is much smaller change in charge density for the triplet state for the corresponding change in barrier height. The enhanced charge density at the middle for the singlet is caused by the long tails of $|\\psi_L(x_1,y_1)|^2$ and $|\\psi_R(x_1,y_1)|^2$ extending into the opposite well, which result from the Klein tunneling effect. The long tail is enhanced (reduced) for the singlet (triplet) state, which is driven by the positive (negative) exchange term.}\n\nThis counter-intuitive behavior due to Klein tunneling is one of the special characteristics of Dirac particles. As the barrier height approaches $\\mu_b=2\\hbar vq_0$, the VBM of the barrier is aligned with the conduction band minimum in the well. This leads to an enhancement of the overlap between the two electrons. This result is consistent with the result shown in Fig.~\\ref{fig3}.\n\n\n\n\n\\subsection{{Effects of inter-dot distance on two-particle solutions}\\label{distance}}\n\n\n\\begin{figure}\n\\includegraphics[trim = 30mm 0mm 40mm 0mm,width=0.45\\textwidth]{fig5.eps}\n\\caption{Negative exchange coupling ($-J_{ex}$) as a function of well-to-well distance $d$ for various barrier heights in (a) linear scale and (b) semi-log scale. The width of the wells is $w=4q_0^{-1}$. $\\mu_b=0$ (red dash-dot), $\\mu_b=1.5\\hbar vq_0$ (blue solid), $\\mu_b=1.9\\hbar vq_0$ (green dashed). For zero barrier, the singlet-triplet ground state transition occurs roughly at critical distance {with $q_0d_c\\approx 3$} (black vertical dashed). For finite barrier,  $-J_{ex}$ decays exponentially for $q_0d>3$.\\label{fig5}}\n\\end{figure}\n\n\nFigure~\\ref{fig5}(a) shows  $-J_{ex}$  as a function of inter-dot distance for different barrier heights. In the absence of a barrier (red dash-dot),  $-J_{ex}$  starts with negative values and increases to positive values for $q_0d>3$. For $\\mu_b=1.5\\hbar vq_0$ and $\\mu_b=1.9\\hbar vq_0$,  $-J_{ex}$  starts with a positive value, and decays exponentially for $q_0d>3$, as shown in Fig.~\\ref{fig5}(b).\n\n\nFor $\\mu_b=0$, there is no barrier and we only have one confining potential well. For $\\mu_b=0$, increasing $d$ is the same as increasing the width of a single potential well (which equals to $d+2w$, as shown in Fig.~1). {This situation has been studied by using various first-principle calculations, as summarized in Ref.~\\onlinecite{Rayne2011}. The ground state is a singlet for a short ribbon, and a triplet for a long ribbon. Our result is consistent with the previous studies for this limiting case. In particular, the red dash-dot line is similar to Fig. 2(a) in Ref.~\\onlinecite{Rayne2011}. The change of sign of $J_{ex}$ as $q_0d$ varies has a simple physical explanation. Whether the ground state is the singlet or triplet state depends the relative strength of the kinetic energy and the potential energy due to mutual Coulomb interaction. Similar to interacting electrons in a jellium model, the kinetic energy of the system scales like $1\/r_s^2$ and potential energy scales like $1\/r_s$, where $r_s$ denotes the average distance between electrons in the system. For small inter-dot separation, the kinetic energy dominates and the singlet state has lower total energy since its wavefunction has less spatial variation (due to symmetric behavior) as compared to the the triplet state. A the separation gets larger, the potential energy dominates and the total energy of the singlet state becomes higher, since it has more charge piling toward the center as compared to the the triplet state as illustrated in Fig.~4.} In our calculation, the singlet-triplet ground state transition occurs roughly at the critical distance {$d_c\\approx 3q_0^{-1}$}, which is labeled by the black vertical dashed line. For larger barrier heights, the singlet state has a larger density at the central barrier region and hence has higher Coulomb energy. This is why the triplet state is the ground state and  $-J_{ex}$ is always positive for $\\mu_b=1.5\\hbar vq_0$ and $\\mu_b=1.9\\hbar vq_0$.\n\nFor medium barrier height $\\mu_b=1.5\\hbar vq_0$, the exchange coupling decreases exponentially when the inter-dot distance increases, as shown in Fig.~\\ref{fig5}. For higher barrier height $\\mu_b=1.9\\hbar vq_0$, where Klein paradox assisted tunneling occurs,  $-J_{ex}$  is generally larger than that for $\\mu_b=1.5\\hbar vq_0$. For small values of $q_0d$,  $-J_{ex}$  increases with increasing separation before reaching the singlet-triplet ground state transition point {$q_0d_c\\approx 3$}. For inter-dot distances longer than the critical distance {$d_c\\approx 3q_0^{-1}$}, we see the expected exponential decay. Hence in the Klein tunneling regime, the location of the maximum of  $-J_{ex}$ can be roughly predicted by looking at the zero barrier height solution.\n\n\n\\subsection{Effects of well width on two-particle solutions}\n\n\n\\begin{figure}\n\\includegraphics[trim = 35mm 0mm 35mm 0mm,width=0.36\\textwidth]{fig6.eps}\n\\caption{Projected charge density $\\rho_\\pm(y)$ along the ribbon with $q_0d=4$ and $\\mu_b=1.9\\hbar vq_0$ for (a) singlet state and (b) triplet state for various well widths. $q_0w=4$ (red dotted), $q_0w=6$ (green dash-dot), $q_0w=8$ (blue dashed).   \\label{fig6}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[trim = 30mm 0mm 40mm 0mm,width=0.45\\textwidth]{fig7.eps}\n\\caption{Negative exchange coupling ($-J_{ex}$) as a function of well width $w$ for various barrier heights in (a) linear scale and (b) semi-log scale. \\label{fig7}}\n\\end{figure}\n\n\nFigure~\\ref{fig6} shows the charge density along the y-axis for various well widths for (a) the singlet states and (b) the triplet states. The inter-well separation is fixed at $d=4q_0^{-1}$ and the barrier height is $\\mu_b=1.9\\hbar vq_0$. For small well width with $q_0w=4$ (black solid), the density in the barrier region for the singlet state is much higher than that of the triplet state. For larger well widths, the charge densities spread out from the center, and the difference of the charge densities between the singlet state and the triplet state in the barrier region become less significant. {The absolute value of exchange coupling is hence expected to be very small for large well width.} This is shown in Fig.~\\ref{fig7}, where $-J_{ex}$  is plotted as functions of well width. For barrier height larger than the critical value $\\mu_b=1.5\\hbar vq_0$ (blue solid) and $\\mu_b=1.9\\hbar vq_0$ (green dashed), the exchange splitting decays exponentially, as shown in the semi-log scale  in Fig.~\\ref{fig7}(b). {The exchange splitting of $\\mu_b=1.9\\hbar vq_0$ is much larger than that of $\\mu_b=1.5\\hbar vq_0$, since Klein tunneling emerges as the VBM of the barrier  gets close to the conduction band states of the wells.} For zero barrier height (red dash-dot), there is only a single confining well, and the curve is just the long-distance extension of the same curve (red dash-dot) in Fig.~\\ref{fig5}(a) discussed in the previous section.\n\n\n\\subsection{Qubit operation}\n\nOne figure of merit for qubit operation is the ratio of coherence time to switching time ${T_c}\/{\\tau_s}$, where $T_c$ is the coherence time and $\\tau_s$ is the switching time required for a $\\mbox{SWAP}^{\\frac{1}{2}}$ gate operation. \\cite{LossDiVincenzo1998,Shi2014}Within our model and range of parameters, the maximum exchange splitting for graphene nanoribbon quantum dot qubit is $|J_{ex}|_{graphene}\\approx 0.002 \\hbar v q_0=66 \\mu eV$, which is of the same order as the typical value for GaAs quantum dot qubit $|J_{ex}|_{GaAs}\\approx 100 \\mu eV$. \\cite{Burkard2000,Schliemann2001,Engel2004} Various sources of spin decoherence in graphene quantum dot are investigated, including spin-orbit coupling, \\cite{Kane2005,Min2006,Struck2010,Hachiya2014} electron-phonon interaction,  \\cite{Mariani2008,Struck2010,Tikhonov2014} and hyperfine interaction. \\cite{Fischer2009,Recher2010,Fuchs2012}. The coherence time for graphene is expected to be $T_c \\approx 80 \\mu s$, three orders of magnitude longer than that of GaAs. \\cite{Recher2010,Kloeffel2013} Since the coherence time is much longer and the switching time is similar, we expect the figure of merit of graphene is much better than that of GaAs. In the original proposal, the exchange coupling is estimated to be $J_{ex}\\approx 0.1 \\sim 1.5 meV$ using single-particle picture and empirical value for Coulomb interaction, \\cite{trauzettel2007spin} which is an overestimation comparing to our calculation. Our calculation provides details of the exchange splitting versus various parameters associated with gate operation, and confirms that the magnitude of $J_{ex}$ required for qubit operation is achievable in the presence of electron-electron interaction, even though it is somewhat smaller than the previous estimation.\n\n{\nThe barrier height of the outside region $\\mu_{out}$ should be carefully chosen to avoid the electron leakage caused by Klein tunneling. $\\mu_{out}=1.5\\hbar vq_0$ in the current model, so an electron state will be confined in the double well if its single-particle energy lies in the band gap of outside region ($0.5\\hbar vq_0 < E <2.5\\hbar vq_0$). However, if the single-particle energy is too close to the VBM ($0.5\\hbar vq_0$) or the bottom of conduction band ($2.5\\hbar vq_0$) of the outside region, the electrons could also tunnel out from the double well. In our numerical tests, it is safe to set $\\mu_{out}=1.5\\hbar vq_0$  and the Fermi energy  $E_F$ is set to be slightly above the conduction band minimum ($1\\hbar vq_0$). This region of operation could be identified by measuring the charge stability diagram of a graphene double quantum dot device.}\n\n\n\n\\section{Conclusion}\n\nWe have performed theoretical studies of the electronic structures of GNB quantum dot qubits using the Dirac equation and a double square well potential. The two electron wave functions and exchange splitting are calculated for various potential configurations by using a GVB wave function within the unrestricted Hartree-Fock approximation. As the barrier height approaches $2\\hbar v q_0$ (the band gap of the nanoribbon), the magnitude of the exchange coupling is enhanced by the Klein tunneling. This enhancement can make the long distance coupling almost as large as the short distance coupling. {We found that the exchange coupling between two GNB quantum dot qubits can switch sign as the average inter-particle distance varies. This behavior is consistent with previous first-principle studies for two electrons in a finite-length GNB (which corresponds to the zero barrier limit of our GNB double dot system).\\cite{Rayne2011} For higher barriers, the magnitude of the exchange coupling decays, but it can be magnified by the Klein tunneling when the valence band maximum of the barrier is close to the conduction band states of the wells. We found that the magnitude for the exchange splitting required for qubit operation is achievable, and the figure of merit of the GNB qubit is expected to be significantly better than that of the GaAs quantum dot qubit.}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank Lance Cooper and Jason Chang for valuable discussions. This work was supported in part by Ministry of Science and Technology, Taiwan under Contract No. NSC 101-2112-M-001-024-MY3.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nData-driven optimal control of a nonlinear system is a problem that has significant interest with applications ranging from vehicle autonomy, robotics to manufacturing and power systems.\nThe traditional approach to optimal control problem (OCP) relies on solving the Hamilton Jacobi Bellman (HJB) equation \\cite{fleming2012deterministic}. The HJB equation is a nonlinear partial differential equation and challenging to solve. Existing algorithms for solving HJB equations rely on iterative scheme \\cite{beard1997galerkin,bertsekas2011approximate}. This iterative scheme is also at the heart of the variety of reinforcement learning (RL) algorithms for the data-driven optimal control \\cite{sutton2018reinforcement}. In this paper, we present an alternate approach based on the dual formulation of the OCP. This dual approach leads to a convex optimization formulation of the OCP, which can be solved using a single-shot algorithm. This is in contrast to the iterative scheme used for solving the HJB equation.\nFurthermore, the iterative algorithm required for solving the HJB equation requires an initial stabilizing controller. Finding stabilizing controller for a nonlinear system is, in general, a nontrivial problem. However, the computational framework for solving the OCP problem in the dual space does not require an initial stabilizing controller. \n\nThe dual formulation to the OCP we present is based on the theory of linear operators, namely the P-F and Koopman operators \\cite{Lasota} and is developed in \\cite{vaidyaocpconvex}. However, there are differences between the results presented in this paper and \\cite{vaidyaocpconvex} as discussed in our contributions. The convex formulation to the OCP in the dual space of densities and occupation measure has been extensively studied in \\cite{henrion2013convex,korda2017convergence,lasserre2008nonlinear}. The computational framework in these works relies on moment-based relaxation of the infinite-dimensional optimization problem. In contrast, our proposed computational framework uses data and depends on the linear operator theory for the finite-dimensional approximation of the infinite-dimensional convex optimization problem. The convex formulation for optimal control is also extended to study stochastic OCP, control design with safety constraints, and data-driven stabilization problems \\cite{safetyPF,choi2020convex1}. There is a growing body of literature on the use of the Koopman operator for data-driven control, where the control dynamical system is lifted in the space of functions or observables using the Koopman operator \\cite{kaiser2021data,huang2018feedback,arbabi2018data,ma2019optimal,korda2020optimal1,mauroy2013spectral,huang2020data}. However, in this paper, we lift the control system using the P-F operator, which is dual to the Koopman operator. Unlike Koopman-based lifting, P-F lifting of the control dynamical system leads to a convex formulation of the OCP \\cite{raghunathan2013optimal,vaidya2010nonlinear}. \n\nThe main contributions of this paper are stated as follows. First, we provide a convex formulation to the infinite horizon OCP with discounted cost involving continuous-time dynamics. We consider OCP problems with both positive and negative discount. For the continuous-time OCP, the negative (positive) discount corresponds to the case where the cost function is exponentially decreasing (increasing) with time. There is extensive literature on the OCP with negative discount factor \\cite{modares2014linear,modares2016optimal,ghosh1993optimal}. One of the main contributions of this paper is to provide condition for the existence of optimal control problem with a positive discount. The condition arises in the form of a stronger notion of almost everywhere exponential stability \\cite{Vaidya_TAC}. Unlike \\cite{vaidyaocpconvex}, the computation framework relies on the use of polynomial basis for the approximation of linear Koopman operator using generator Extended Dynamic Mode Decomposition (gEDMD) algorithm \\cite{klus2020data}. Hence, we employ sum-of-square (SOS) optimization methods for solving a finite-dimensional optimization problem. The finite-dimensional approximation of the infinite-dimensional optimization problem is written as a semi-definite program (SDP). SOS-based optimization toolbox is then used to solve the SDP in a numerically efficient manner. Existing rigorous results for the convergence analysis of Koopman operator in the limit of data and number of basis functions goes to infinity are leveraged to provide convergence analysis of data-driven optimization problem. Simulation results are presented to verify the efficacy of the developed framework. The results presented in this paper are an extension of our conference paper \\cite{moyalan2021sum}. In particular, the data-driven framework for optimal control design and theorems involving OCP with discounted cost function are new to this paper. \n\nThe rest of the paper is structured as follows. In Section \\ref{sec:background}, we introduce preliminaries and notations used throughout the paper. The main results of the paper on the convex formulation to OCP are presented in Section \\ref{section_main}. The SOS and Koopman-based computation framework for the data-driven approximation of the convex optimization problem is discussed in Section \\ref{sec:SOS Control}. Simulation results are presented in Section \\ref{sec:examples}. Conclusions are presented in Section \\ref{sec:conclusion}. \n\n\n\n\n\n\n\n\n\n\\section{Preliminaries and Notations}\\label{sec:background}\n\\noindent{\\bf Notation:} ${\\mathbb R}^n$ denotes the $n$ dimensional Euclidean space  and ${\\mathbb R}^n_{\\geq 0}$ is the positive orthant. Given ${\\mathbf X}\\subseteq {\\mathbb R}^n$ and ${\\mathbf Y}\\subseteq {\\mathbb R}^m$, let ${\\cal L}_1({\\mathbf X},{\\mathbf Y}), {\\cal L}_\\infty({\\mathbf X},{\\mathbf Y})$, and ${\\cal C}^k({\\mathbf X},{\\mathbf Y})$ denote the space of all real valued integrable functions, essentially bounded functions, and space of $k$ times continuously  differentiable functions mapping from ${\\mathbf X}$ to ${\\mathbf Y}$ respectively. If the space ${\\mathbf Y}$ is not specified then it is understood that the underlying space is ${\\mathbb R}$. ${\\cal B}({\\mathbf X})$ denotes the Borel $\\sigma$-algebra on ${\\mathbf X}$ and ${\\cal M}({\\mathbf X})$ is the vector space of real-valued measure on ${\\cal B}({\\mathbf X})$. ${\\mathbf s}_t({\\mathbf x})$ denotes the solution of dynamical system $\\dot {\\mathbf x}={\\bf f}({\\mathbf x})$ starting from initial condition ${\\mathbf x}$.  We will use the notation ${\\cal N}_\\delta$ to denote the $\\delta$ neighborhood of the equilibrium point at the origin for some fixed $\\delta>0$ and ${\\mathbf X}_1:={\\mathbf X}\\setminus {\\cal N}_\\delta$. \n\\subsection{Koopman and Perron-Frobenius Operators}\nConsider a dynamical system\n\\begin{align}\\dot {\\mathbf x}={\\mathbf f}({\\mathbf x}),\\;\\;{\\mathbf x}\\in {\\mathbf X}\\subseteq \\mathbb{R}^n \\label{dyn_sys}\\end{align} where the vector field is assumed to be ${\\bf f}({\\mathbf x})\\in {\\cal C}^1({\\mathbf X},{\\mathbb R}^n)$. There are two different ways of linearly lifting the finite dimensional nonlinear dynamics from state space to infinite dimension space of functions, Koopman and Perron-Frobenius operators.   \n\n\\noindent{\\bf Koopman Operator:} $\\mathbb{K}_t :{\\cal L}_\\infty({\\mathbf X})\\to {\\cal L}_\\infty({\\mathbf X})$ for dynamical system~\\eqref{dyn_sys} is defined as \n\\[[\\mathbb{K}_t \\varphi]({\\mathbf x})=\\varphi({\\mathbf s}_t({\\mathbf x})),\\;\\;\\varphi\\in {\\cal L}_\\infty. \n\\]\nThe infinitesimal generator for the Koopman operator is\n\\begin{equation}\n\\lim_{t\\to 0}\\frac{\\mathbb{K}_t\\varphi-\\varphi}{t}={\\mathbf f}({\\mathbf x})\\cdot \\nabla \\varphi({\\mathbf x})=:{\\cal K}_{{\\mathbf f}} \\varphi. \\label{K_generator}\n\\end{equation}\n\\noindent{\\bf Perron-Frobenius Operator:} $\\mathbb{P}_t:{\\cal L}_1({\\mathbf X})\\to {\\cal L}_1({\\mathbf X})$ for  system~\\eqref{dyn_sys} is defined as\n\\begin{equation}[\\mathbb{P}_t \\psi]({\\mathbf x})=\\psi({\\mathbf s}_{-t}({\\mathbf x}))\\left|\\frac{\\partial {\\mathbf s}_{-t}({\\mathbf x}) }{\\partial {\\mathbf x}}\\right|,\\;\\;\\psi\\in {\\cal L}_1,\\label{pf-operator} \n\\end{equation}\nwhere $\\left|\\cdot \\right|$ stands for the determinant. The infinitesimal generator for the P-F operator is given by \n\\begin{align}\n\\lim_{t\\to 0}\\frac{\\mathbb{P}_t\\psi-\\psi}{t}=-\\nabla \\cdot ({\\mathbf f}({\\mathbf x}) \\psi({\\mathbf x}))=: {\\cal P}_{{\\mathbf f}}\\psi. \\label{PF_generator}\n\\end{align}\nThese two operators are dual to each other where the duality is expressed as\n\\begin{align*}\n\\int_{\\mathbb{R}^n}[\\mathbb{K}_t \\varphi]({\\mathbf x})\\psi({\\mathbf x})d{\\mathbf x}=\n\\int_{\\mathbb{R}^n}[\\mathbb{P}_t \\psi]({\\mathbf x})\\varphi({\\mathbf x})d{\\mathbf x}.\n\\end{align*}\n\n\n\n\n\n\n\n\n\\subsection{Sum of squares} \\label{sec:SOS}\nSum of squares (SOS) optimization \\cite{Topcu_10,Pablo_03_SDP,parrilo2003minimizing,Pablo_2000} is a relaxation of positive polynomial constraints appearing in polynomial optimization problems. SOS polynomials are in a set of polynomials that can be described as a finite linear combination of monomials, i.e., $p = \\sum_{i=1}^\\ell d_i p_i^2$ where $p$ is a SOS polynomial, $p_i$ are polynomials, and $d_i$ are nonnegative coefficients. Hence, SOS is a sufficient condition for the nonnegativity of a polynomial. Thus SOS relaxation provides a lower bound on the minimization problems of polynomial optimizations. Using the SOS relaxation, a large class of polynomial optimization problems with positive constraints can be formulated as SOS optimization as\n\\begin{align} \\label{eq:SOSOPT}\n\\begin{split}\n    \\min_{{\\mathbf d}} \\, \\mathbf{w}^\\top \\mathbf{d} \\,\\,\\, \\mathrm{s.t.} \\,\\,\\, p_s({\\mathbf x},{\\mathbf d}) \\in \\Sigma[{\\mathbf x}], \\, p_e({\\mathbf x};{\\mathbf d}) = 0,\n\\end{split}\n\\end{align}\nwhere $\\Sigma[{\\mathbf x}]$ denotes SOS set, $\\mathbf{w}$ is weighting coefficient, $p_s$ and $p_e$ are polynomials parametrized by coefficients ${\\mathbf d}$. The problem in~\\eqref{eq:SOSOPT} can be translated into a Semidefinite Programming (SDP)~\\cite{Pablo_03_SDP, Laurent_2009}. There are readily available SOS optimization packages such as SOSTOOLS~\\cite{sostools_Parrilo} and SOSOPT~\\cite{Seiler_2013_SOSOPT} for solving~\\eqref{eq:SOSOPT}.\n\n\n\\subsection{Almost everywhere  uniform stability and Stabilization}\n\nThis section derives results on a stronger notion of stability used in formulating optimal control problem with discounted cost. We first present the notion of, a.e., uniform stability as introduced in \\cite{rajaram2010stability}. In the rest of the paper, we will use the following notation.\n\n\n\\begin{definition}\\label{definitiona.euniform}[a.e. uniform stability]\nThe equilibrium point of \\eqref{dyn_sys} is said to be a.e. uniform  stable w.r.t. measure $\\mu\\in {\\cal M}({\\mathbf X})$ if, for every given $\\epsilon$, there exists a time $T(\\epsilon)$ such that\n\\begin{align}\n\\int_{T(\\epsilon)}^\\infty \\mu (B_t)dt<\\epsilon\\label{eq_aeunifrom}\n\\end{align}\nwhere $B_t:=\\{{\\mathbf x} \\in {\\mathbf X}: {\\mathbf s}_t({\\mathbf x})\\in B\\}$ for any set $B\\in {\\cal B}({\\mathbf X}_1)$. \n\\end{definition}\nThe above stability definition essentially means that given any arbitrary set $B$ not containing the origin, the measure of the set of all initial conditions that stay inside $B$ can be made arbitrarily small after a sufficiently long time. Note that the above definition of a.e. uniform stability is stronger than the almost everywhere stability notion as introduced in \\cite{Rantzer01} (refer to \\cite{rajaram2010stability} for the proof). The following definition of a.e. exponential stability is introduced here and is stronger than the above Definition \\ref{definitiona.euniform}.  The following exponential stability definition is a continuous-time counterpart of the discrete-time definition studied in \\cite{Vaidya_TAC}.\n\n\\begin{definition}\n\\label{definition_aeexponential}[a.e. exponential stability]\nThe equilibrium point is said to be almost everywhere  exponential stable w.r.t. measure $\\mu$\nwith rate $\\gamma>0$ if there exists a constant $M$ such that \n\\begin{align}\n\\mu(B_t)\\leq M e^{-\\gamma t}\n\\label{a.eexponential}\n\\end{align}\nwhere $B_t:=\\{{\\mathbf x}\\in {\\mathbb R}^n: {\\mathbf s}_t({\\mathbf x})\\in B\\}$ for any set $B\\in {\\cal B}({\\mathbf X}_1)$.\n\\end{definition}\n\n\n\n\n\n\n In the following, we state theorems providing necessary and sufficient condition for a.e. uniform and a.e uniform exponential stability. These results are proved under the following assumption on the equilibrium point of  (\\ref{dyn_sys}). \n\\begin{assumption}\\label{assume_localstability} We assume that ${\\mathbf x}=0$ is locally stable equilibrium point for the system (\\ref{dyn_sys}) with local domain of attraction denoted by $\\mathcal{D}$ and let  $0\\in {\\cal N}_\\delta\\subset \\mathcal{D}$.\n\\end{assumption}\n\\begin{theorem}\\label{theorem_necc_suffuniform}\nThe equilibrium point ${\\mathbf x}=0$ for the system (\\ref{dyn_sys})  satisfying Assumption \\ref{assume_localstability} is almost everywhere uniform stable w.r.t. measure $\\mu$ if and only if there exists a density function $\\rho({\\mathbf x})\\in{\\cal C}^1({\\mathbf X}\\setminus \\{0\\},{\\mathbb R}^+)$ which is integrable on ${\\mathbf X}_1$ and satisfies \n\\begin{align}\n\\nabla\\cdot ({\\bf f}\\rho )=h_0\\label{steady_pde_uniform}\n\\end{align}\nwhere $h_0\\in {\\cal C}^1({\\mathbf X})$ is the density function corresponding to the measure $\\mu$.\n\\end{theorem}\nRefer to \\cite[Theorem 13]{rajaram2010stability} for the proof.\n\n\n\n\n\n\n\n\n\\section{Convex Formulation of Optimal Control Problem}\\label{section_main}\nIn this section we briefly summarize the main results from \\cite{vaidyaocpconvex} on the convex formulation of the optimal control problem. Consider a control affine system of the form \n\\begin{align} \\label{cont_syst1}\n\\dot {\\mathbf x}=\\bar {\\bf f}({\\mathbf x})+{\\bf g}({\\mathbf x})\\bar {\\mathbf u},\\;\\;{\\mathbf x} \\in {\\mathbf X}\\subseteq{\\mathbb R}^n\n\\end{align}\nwhere, ${\\mathbf x}$ is the state, $\\bar {\\mathbf u}=[\\bar u_1,\\ldots,\\bar u_m]^\\top\\in \\mathbb{R}^m$ is the control input and ${\\mathbf g}({\\mathbf x})=({\\mathbf g}_1({\\mathbf x}),\\ldots,{\\mathbf g}_m({\\mathbf x}))$. All the vector fields are assumed to belong to ${\\cal C}^1({\\mathbf X},{\\mathbb R}^n)$.\n\n\\begin{remark}\nThe affine control assumption for a dynamical control system is not restrictive as any non-affine dynamical control system can be converted to control-affine by extending the state space. In particular, consider the control dynamical system of the form\n\\[\\dot {\\mathbf x}={\\mathbf f}({\\mathbf x},{\\mathbf u})\\]\nthen we can define ${\\mathbf u}$ as a new state and introduce $\\tilde {\\mathbf u}$ as another control input to write the above system as the following affine in the input control system\n\\begin{align}\n&\\dot {\\mathbf x}={\\mathbf f}({\\mathbf x},{\\mathbf u})\\nonumber,\\;\\;\\;\\dot {\\mathbf u}=\\tilde {\\mathbf u}\n\\end{align}\n\\end{remark}\nThe following assumption is made on (\\ref{cont_syst1}).\n\\begin{assumption}\\label{assume_localstable}\nWe assume that the linearization of the nonlinear system dynamics (\\ref{cont_syst1}) at the origin is stabilizable i.e., the pair $(\\frac{\\partial \\bar {\\bf f}}{\\partial {\\mathbf x}}(0),{\\mathbf g}(0))$ is stabilizable. \n\\end{assumption}\nUsing the above stabilizability assumption, we can design a local stable controller using data. The detailed procedure for the design of such controller is given in Section \\ref{sec:SOS Control}. Let ${\\mathbf u}_\\ell$ be the locally stable controller. Defining ${\\bf f}({\\mathbf x}):=\\bar {\\bf f}({\\mathbf x})+{\\mathbf g}({\\mathbf x}) {\\mathbf u}_\\ell$ and ${\\mathbf u}=\\bar {\\mathbf u}-{\\mathbf u}_\\ell$, we can rewrite control system (\\ref{cont_syst1}) as\n\\begin{align}\n\\dot {\\mathbf x}={\\bf f}({\\mathbf x})+{\\bf g}({\\mathbf x}) {\\mathbf u}.\\label{cont_syst}\n\\end{align}\nThe following is valid for the above dynamical system.  With the control input ${\\mathbf u}=0$, the origin of system (\\ref{cont_syst}) is  almost sure asymptotically stable locally in small neighborhood $\\mathcal{D}$ of the origin such that ${ B}_\\delta \\subset {\\cal D}$.\n\n\n\n\n\n\n\nConsider the discounted cost OCP of the form\n\\begin{align}\nJ^\\star(\\mu_0)\\!\\!=\\!&\\inf_{\\mathbf u}\\!\\!\\int_{{\\mathbf X}_1}\\!\\!\\left[\\!\\int_0^\\infty\\!\\!\\! e^{\\gamma t}(q({\\mathbf x}(t))+ \\beta {\\mathbf u}(t)^\\top {\\mathbf R} {\\mathbf u}(t)) \\;dt\\right] d\\mu_0\\nonumber\\\\\n&{\\rm subject\\;to\\;(\\ref{cont_syst})}\\label{cost_function}\n\\end{align}\nwhere $\\gamma\\in {\\mathbb R}$. The existing literature on OCP with discounted cost address the case where $\\gamma$ is negative, i.e., negative discount factor. In this paper, with the stronger notion of a.e. uniform stability with geometric decay, we can address the case of cost with a positive discount. Note that the cost function is a function of initial measure $\\mu_0$, and this dependency on $\\mu_0$ can be explained as follows.\nThe cost function can be written as \n\\begin{align}J(\\mu_0)=\\int_{{\\mathbf X}_1}V({\\mathbf x})d\\mu_0({\\mathbf x})\\label{costmu}\n\\end{align}\nwhere, $V({\\mathbf x})$ can be written as \n\\[V({\\mathbf x})=\\int_0^\\infty e^{\\gamma t}(q({\\mathbf x}(t))+\\beta {\\mathbf u}^\\top (t){\\mathbf R} {\\mathbf u}(t))dt\\]\nwith ${\\mathbf x}(\\cdot)$ being a trajectory with initial condition ${\\mathbf x}(0)={\\mathbf x}$.While $V({\\mathbf x})$ can be recognized with the familiar cost function used in the formulation of OCP in primal domain, the cost function $J(\\mu_0)$ is minimized w.r.t. set of initial condition distributed with initial measure $\\mu_0$. In the rest of the paper we assume that the initial measure $\\mu_0$ is equivalent to Lebesgue with density function $0<h_0({\\mathbf x})\\in  {\\cal L}_1({\\mathbb R}^n,{\\mathbb R}_{> 0})\\cap {\\cal C}^1({\\mathbb R}^n)$. \nWe make the following assumption on the OCP. \n\\begin{assumption}\\label{assumption_onocp}\n We assume that the state cost function $q: {\\mathbb R}^n\\to {\\mathbb R}_{\\geq 0}$ is zero at the origin and uniformly bounded away from zero outside the neighborhood ${\\cal N}_\\delta$ and ${\\mathbf R}>0$. Furthermore, there exists a feedback control for which the cost function in (\\ref{cost_function}) is finite and  that the optimal control is feedback in nature, i.e., ${\\mathbf u}^\\star={\\mathbf k}^\\star({\\mathbf x})$ with the function ${\\mathbf k}^\\star$\n being in ${\\cal C}^1({\\mathbf X},{\\mathbb R}^m)$. \n\\end{assumption}\nWith the assumed feedback form of the optimal control input, the OCP can be written as \n{\\small\n\\begin{eqnarray}\n \\inf\\limits_{{\\mathbf k}\\in {\\cal C}^1({\\mathbf X})} &\\int_{{\\mathbf X}_1}\\left[\\int_0^\\infty e^{\\gamma t}(q({\\mathbf x}(t))+ \\beta {\\mathbf k}({\\mathbf x}(t))^\\top {\\mathbf R} {\\mathbf k}({\\mathbf x}(t)))\\;dt\\right] d\\mu_0\\nonumber\\\\\n {\\rm s.t.}&\\dot {\\mathbf x}={\\bf f}({\\mathbf x})+{\\mathbf g}({\\mathbf x}){\\mathbf k}({\\mathbf x})\n\\label{ocp_main_discounted}\n\\end{eqnarray}\n}\nThe following is the main theorem on the OCP with discounted cost function.\n\n\\begin{theorem}\\label{theorem_maingeometric}\nConsider the OCP (\\ref{ocp_main_discounted}) with discount factor $\\gamma\\leq 0$ and assume that the cost function and optimal control satisfy Assumption \\ref{assumption_onocp}. Then the OCP (\\ref{ocp_main_discounted}) can be written as the following infinite dimensional convex optimization problem \n\\begin{eqnarray}\nJ^\\star(\\mu_0)&=&\\inf_{\\rho\\in {\\cal S},\\bar {\\boldsymbol \\rho}\\in {\\cal C}^1({\\mathbf X}_1)} \\;\\;\\; \\int_{{\\mathbf X}_1} q({\\mathbf x})\\rho({\\mathbf x})+\\beta\\frac{\\bar {\\boldsymbol \\rho}({\\mathbf x})^\\top {\\mathbf R}\\bar {\\boldsymbol \\rho}({\\mathbf x})}{\\rho} d{\\mathbf x}\\nonumber\\\\\n{\\rm s.t}.&&\\nabla\\cdot ({\\bf f}\\rho +{\\mathbf g}\\bar {\\boldsymbol \\rho})=\\gamma \\rho+ h_0\n\\label{eqn_ocpdiscount_L2}\n\\end{eqnarray}\nwhere $\\bar {\\boldsymbol \\rho}=(\\bar \\rho_1,\\ldots, \\bar\\rho_m)$ and ${\\cal S}:={\\cal L}_1({\\mathbf X}_1)\\cap {\\cal C}^1({\\mathbf X}_1,{\\mathbb R}_{\\geq 0})$. The optimal feedback control input is recovered from the solution of the above optimization problem as  \n\\begin{align}\n{\\mathbf k}^\\star({\\mathbf x})=\\frac{\\bar {\\boldsymbol \\rho}^\\star({\\mathbf x})}{\\rho^\\star({\\mathbf x})}\\label{feedback_input}.\n\\end{align}\nFurthermore, if $\\gamma=0$, then optimal control ${\\mathbf k}^\\star({\\mathbf x})$ is a.e. uniformly stabilizing   w.r.t. measure $\\mu_0$.\n\\end{theorem}\n\nProof of theorem \\eqref{theorem_maingeometric} is given in Appendix.\n\n\n\nNext, we consider discounted cost OCP with ${\\cal L}_1$ norm on control term \n\\begin{eqnarray}\n \\inf\\limits_{{\\mathbf k}\\in {\\cal C}^1({\\mathbf X})} &\\int_{{\\mathbf X}_1}\\left[\\int_0^\\infty e^{\\gamma t}(q({\\mathbf x}(t))+ \\beta \\|{\\mathbf k}({\\mathbf x}(t))\\|_1)\\;dt\\right] d\\mu_0({\\mathbf x})\\nonumber\\\\\n {\\rm s.t.}&\\dot {\\mathbf x}={\\bf f}({\\mathbf x})+{\\mathbf g}({\\mathbf x}){\\mathbf k}({\\mathbf x}).\n\\label{ocp_main_discounted1}\n\\end{eqnarray}\nWe make the following assumption on the nature of optimal control for the ${\\cal L}_1$-norm OCP (\\ref{ocp_main_discounted1}). \n\\begin{assumption}\\label{assume_OCP1}\nWe assume that the state cost function $q: {\\mathbb R}^n\\to {\\mathbb R}_{\\geq 0}$ is zero at the origin and uniformly bounded away from zero outside the neighborhood ${\\cal N}_\\delta$ and ${\\mathbf R}>0$. Furthermore, there exists a feedback control input for which the cost function in \\eqref{ocp_main_discounted1} is finite. Furthermore,  the optimal control is feedback in nature, i.e., ${\\mathbf u}^\\star={\\mathbf k}^\\star({\\mathbf x})$ with the function ${\\mathbf k}^\\star$ is assumed to be ${\\cal C}^1({\\mathbf X},{\\mathbb R}^m)$. \n\\end{assumption}\n\\begin{theorem}\\label{theorem_maingeometric1}\nConsider the OCP (\\ref{ocp_main_discounted1}) with discount factor $\\gamma\\leq 0$ and assume that the cost function and optimal control satisfy Assumption \\ref{assume_OCP1} respectively. Then the OCP (\\ref{ocp_main_discounted1}) can be written as following infinite dimensional convex optimization problem \n\\begin{eqnarray}\nJ^\\star(\\mu_0)&=&\\inf_{\\rho\\in {\\cal S},\\bar {\\boldsymbol \\rho}\\in {\\cal C}^1({\\mathbf X}_1)} \\;\\;\\; \\int_{{\\mathbf X}_1} q({\\mathbf x})\\rho({\\mathbf x})+\\beta\\|\\bar {\\boldsymbol \\rho}({\\mathbf x})\\|_1 d{\\mathbf x}\\nonumber\\\\\n{\\rm s.t}.&&\\nabla\\cdot ({\\bf f}\\rho +{\\mathbf g}\\bar {\\boldsymbol \\rho})=\\gamma \\rho+ h_0\n\\label{eqn_ocpdiscount}\n\\end{eqnarray}\nwhere $\\bar {\\boldsymbol \\rho}=(\\bar\\rho_1,\\ldots, \\bar\\rho_m)$ and ${\\cal S}:={\\cal L}_1({\\mathbf X}_1)\\cap {\\cal C}^1({\\mathbf X}_1,{\\mathbb R}_{\\geq 0})$. The optimal feedback control input is recovered from the solution of the above optimization problem as  \n\\begin{align}\n{\\mathbf k}^\\star({\\mathbf x})=\\frac{\\bar {\\boldsymbol \\rho}^\\star({\\mathbf x})}{\\rho^\\star({\\mathbf x})}.\n\\end{align}\nFurthermore, if $\\gamma=0$, then optimal control ${\\mathbf k}({\\mathbf x})$ is a.e. uniformly stabilizing w.r.t. measure $\\mu_0$.\n\\end{theorem}\n\n\n\\begin{proof}\nThe proof of Theorem \\ref{theorem_maingeometric1} follows along similar lines to the proof of Theorem \\ref{theorem_maingeometric}. \n\\end{proof}\n\n\\begin{remark}\\label{remark_singularity}\nIt is important to emphasize that the optimal feedback controller with discount factor $\\gamma= 0$ is stabilizing in almost everywhere sense. This is analogous to the optimal control design in the primal formulation. The optimal cost function also serves as a Lyapunov function, thereby ensuring the stability of the feedback control system. In our proposed dual setting, the optimal density function serves as a.e. stability certificate for the case of discount factor $\\gamma= 0$. However, due to the dual nature of the Lyapunov function and density function \\cite{Vaidya_TAC,Rantzer01}, the density function has a singularity at the origin. Because of this singularity at the origin, the cost function is evaluated in ${\\mathbf X}_1$ excluding the small region around the origin. Hence it may become necessary to design a local stabilizing or local optimal controller. The existence of such a local stabilizing controller is ensured following Assumption \\ref{assume_localstable}. The local controller, say ${\\mathbf k}_\\ell$, can be blended with global control ${\\mathbf k}^\\star$ using the following  formula  \\cite{rantzer2001smooth}. \n\n\\begin{equation}\n    u({\\mathbf x}) = \\frac{\\rho_L}{\\rho_L+\\rho_N}{\\mathbf k}_\\ell({\\mathbf x}) + \\frac{\\rho_N}{\\rho_L+\\rho_N}{\\mathbf k}^*({\\mathbf x}) \\nonumber\n\\end{equation}\n \n\\begin{equation}\n    \\rho_L({\\mathbf x}) = max\\{(({\\mathbf x}^T P {\\mathbf x}))^{-3}-\\Delta,0\\} \\nonumber\n\\end{equation}\nwhere matrix $P > 0$ define a control Lyapunov function. The parameter $\\Delta$ determines the region of operation for the local controller. \n\\end{remark}\n\n\n\nThe optimal control results involving ${\\cal L}_2$ and ${\\cal L}_1$ norm on the control input with positive discount factor, i.e., $\\gamma>0$, are proved under the following assumption. \n\n\\begin{assumption}\\label{assumption_ocppositivediscount}\nWe assume that the state cost function $q: {\\mathbb R}^n\\to {\\mathbb R}_{\\geq 0}$ is zero at the origin and uniformly bounded away from zero outside the neighborhood ${\\cal N}_\\delta$ and ${\\mathbf R}>0$. Furthermore, there exists a feedback control for which the cost function in (\\ref{cost_function}) is finite and  that the optimal control is feedback in nature, i.e., ${\\mathbf u}^\\star={\\mathbf k}^\\star({\\mathbf x})$ with the function ${\\mathbf k}^\\star$\n being in ${\\cal C}^1({\\mathbf X},{\\mathbb R}^m)$. Furthermore, the feedback controller is assumed to be almost everywhere exponentially stabilizing  (Definition \\ref{definition_aeexponential}) with decay rate $\\gamma'>\\gamma>0$.\n\\end{assumption}\nNote that Assumption \\ref{assumption_ocppositivediscount} is same as Assumption \\ref{assumption_onocp} except for the additional requirment that the feedback controller is a.e. exponentially stabilizing with decay rate strictly large than $\\gamma$.\n\n\\begin{theorem}\\label{theorem_maingeometric_positivediscount}\nConsider the OCP (\\ref{ocp_main_discounted}) with discount factor $\\gamma> 0$ and assume that the cost function and optimal control satisfy Assumption \\ref{assumption_ocppositivediscount}. Then the OCP (\\ref{ocp_main_discounted}) can be written as the following infinite dimensional convex optimization problem \n\\begin{eqnarray}\nJ^\\star(\\mu_0)&=&\\inf_{\\rho\\in {\\cal S},\\bar {\\boldsymbol \\rho}\\in {\\cal C}^1({\\mathbf X}_1)}\\int_{{\\mathbf X}_1} q({\\mathbf x})\\rho({\\mathbf x})+\\beta\\frac{\\bar {\\boldsymbol \\rho}({\\mathbf x})^\\top {\\mathbf R}\\bar {\\boldsymbol \\rho}({\\mathbf x})}{\\rho} d{\\mathbf x}\\nonumber\\\\\n{\\rm s.t}.&&\\nabla\\cdot ({\\bf f}\\rho +{\\mathbf g}\\bar {\\boldsymbol \\rho})=\\gamma \\rho+ h_0\n\\label{eqn_ocpdiscount_L2}\n\\end{eqnarray}\nwhere $\\bar {\\boldsymbol \\rho}=(\\bar \\rho_1,\\ldots, \\bar\\rho_m)$ and ${\\cal S}:={\\cal L}_1({\\mathbf X}_1)\\cap {\\cal C}^1({\\mathbf X}_1,{\\mathbb R}_{\\geq 0})$. The optimal feedback control input is recovered from the solution of the above optimization problem as  \n\\begin{align}\n{\\mathbf k}^\\star({\\mathbf x})=\\frac{\\bar {\\boldsymbol \\rho}^\\star({\\mathbf x})}{\\rho^\\star({\\mathbf x})}\\label{feedback_input}.\n\\end{align}\n\\end{theorem}\nThe proof of this theorem is provided in the Appendix. Theorem analogous to Theorem \\ref{theorem_maingeometric1} can be stated and proved for the case involving ${\\cal L}_1$ control norm and with positive discount factor $\\gamma>0$. \\\\\n\n\\begin{remark}\nIn the above formulations of the OCPs, we did not explicitly impose constraints on the control input. Explicit constraints on the magnitude of the control input can be imposed in a convex manner as follows:\n\\begin{align}\n   \\|{\\mathbf u}\\|_1\\leq M\\iff   |\\bar \\rho_1({\\mathbf x})|^2+\\ldots+|\\bar \\rho_m({\\mathbf x})|^2\\leq M \\rho({\\mathbf x}),\\label{control_constraints}\n\\end{align}\nfor some positive constant $M$. To arrive at (\\ref{control_constraints}) we have used the formula for the optimal feedback control, i.e., (\\ref{feedback_input}) and the fact that $\\rho({\\mathbf x})>0$.  \nThe above constraints are linear in the optimization variables $\\bar {\\boldsymbol \\rho}$ and $\\rho$ and hence can be implemented convexily. So the OCP involving explicit norm constraints on the control input can be implemented convexily by augmenting the optimization problem  with linear constraints in (\\ref{control_constraints}). \n\\end{remark}\n\n\\begin{remark}\nIn the above formulations of the OCP problem, we have assumed that density function $h_0>0$, implying that the initial measure $\\mu_0$ is equivalent to Lebesgue. This assumption is necessary as $h_0>0$ guarantees that $\\rho>0$ (refer to Eq.  (\\ref{definingrho})) and hence the  the feedback control input ${\\mathbf k}=\\frac{\\bar {\\boldsymbol \\rho}}{\\rho}$ is well defined. However, it is possible to relax this assumption and work with density function $h_0\\geq 0$. This will correspond to the case where the initial measure $\\mu_0$ is continuous w.r.t. Lebesgue measure and not equivalent to Lebesgue. In order to ensure that the feedback control input is well defined when $\\mu_0$ is continuous w.r.t. Lebesgue measure, we need to impose the following constraints on the control input\n\\[  |\\bar \\rho_k({\\mathbf x})|^2\\leq M \\rho({\\mathbf x}),\\;\\;k=1,\\ldots,m\\]\nfor some large constant $M$. The above constraints will ensure that the for a.e. ${\\mathbf x}$ if $\\rho({\\mathbf x})=0\\implies \\bar {\\boldsymbol \\rho}_k({\\mathbf x})=0$ for $k=1,\\ldots, m$ thereby the feedback control input is well defined. Working with absolutely continuous initial measure $\\mu_0$ or equivalently $h_0\\geq 0$ will correspond to the case where optimality is guaranteed only from set of initial condition with support on $\\mu_0$.      \n\\end{remark}\nIn the following, we demonstrate how the results involving dual formulation to the OCP problem works out for the special case of scalar linear system. The main conclusion is that the optimal control obtained using dual formulation matches with the control obtained using a primal formulation of OCP, namely the linear quadratic regulator problem for a particular choice of $h_0({\\mathbf x})$. Note that the initial measure or the density function $h_0({\\mathbf x})$ is unique to our dual formulation with no parallel in the primal formulation. \n\\begin{comment}\n\\begin{example}\nConsider a linear scalar control system \n\\[\\dot x=ax+u,\\] with $q(x)=q x^2$ and $h_0(x)=\\frac{1}{x^2}$,  we then have \n\\begin{align*}\n\\inf_{\\rho,\\bar \\rho}&\\int_{X_1} qx^2\\rho(x)+\\frac{\\bar \\rho^2(x)}{\\rho(x)}dx\n\\\\\n&{\\rm s.t.}\\;\\;\\frac{d}{dx}\\left(ax \\rho+\\bar \\rho\\right)=\\frac{1}{x^2}\n\\end{align*}\nAssuming following parameterization for $\\rho(x)=\\frac{1}{px^{2\\beta}}$ and $\\bar \\rho(x)=\\frac{1}{\\lambda x^\\alpha}$, we get\n\\[\\frac{d}{dx}\\left(ax \\rho+\\bar \\rho\\right)=\\frac{1}{x^2}\\implies \\beta=1, \\alpha=1, \\lambda=\\frac{-p}{a+p}\\]\nSubstituting for $\\alpha, \\beta$, and $\\lambda$ in the minimization, we get\n\\[\n\\inf_{\\rho,\\bar \\rho}\\int_{X_1} qx^2\\rho(x)+\\frac{\\bar \\rho^2(x)}{\\rho(x)}dx\\propto\\min_{p} \\frac{q}{p}+\\frac{(a+p)^2}{p}.\n\\]\n\nSolving for $p$, we obtain, $p=\\sqrt{q+a^2}$, and hence\n\n\\[\\bar\\rho(x)=-\\frac{(a+p)}{p x},\\;\\;\\rho(x)=\\frac{1}{px^2}\\]\nwith feedback control\n\\begin{align}u=\\frac{\\bar \\rho(x)}{\\rho(x)}=kx=-(a+p)x=-(a+\\sqrt{a^2+q})x.\\label{control_linearscalar}\\end{align}\nThe above formula for feedback control matches the solution for OCP obtained using the primal formulation. In particular, the Riccati solution for the linear system in  variable $d$ and feedback control can be written as \n\\[ d=a\\pm\\sqrt{a^2+q},\\;\\;u=-(a+\\sqrt{a^2+q})x\\]\nwhich matches with (\\ref{control_linearscalar})\n\\end{example}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\\section{Koopman and SOS-based Computation Framework for Optimal Control} \\label{sec:SOS Control}\nIn this section, we provide Koopman and SOS-based computational framework for the finite dimensional approximation of optimal control formulation involving ${\\cal L}_1$\/${\\cal L}_2$ costs. \n\\chen{switch the order of the $L_1, L_2$ problems in the following presentation}\nConsider the parameterization of $\\rho({\\mathbf x})$ and $\\bar{{\\boldsymbol \\rho}}({\\mathbf x})$ in~\\eqref{steady_pde_gdecay} as follows:\n\\begin{align}\n    \\rho({\\mathbf x}) = \\frac{a({\\mathbf x})}{b({\\mathbf x})^{\\alpha}},\\;\\; \\bar{{\\boldsymbol \\rho}}({\\mathbf x}) = \\frac{{\\mathbf c}({\\mathbf x})}{b({\\mathbf x})^{\\alpha}}, \\label{rational_parametrization}\n\\end{align}\nwhere $a({\\mathbf x})$ and $b({\\mathbf x})$ are positive polynomials (positive at ${\\mathbf x} \\neq 0$) and ${\\mathbf c}({\\mathbf x}) = \\begin{bmatrix} c_1({\\mathbf x}),\\, \\ldots,\\, c_m({\\mathbf x})\\end{bmatrix}^\\top$. Here, $b({\\mathbf x})$ is a fixed positive polynomial and $\\alpha$ is a positive constant which is sufficiently large for integrability condition (\\cite{Prajna04}, Theorem~1). Using~\\eqref{rational_parametrization}, we can restate~\\eqref{steady_pde_gdecay} as \\chen{use $h$ or $h_0$?}\n\\begin{align} \\label{eq:parameterized stability}\n&\\nabla  \\cdot ({\\mathbf f}\\rho  + {\\mathbf g}\\bar{{\\boldsymbol \\rho}}) = \\gamma \\rho + h_0 \\nonumber\\\\\n\\Rightarrow &\\nabla  \\cdot \\left[\\frac{{\\mathbf f} a + {\\mathbf g}{\\mathbf c}}{b^{\\alpha}} \\right] = \\frac{\\gamma a}{b^\\alpha} + h_0 \\nonumber\\\\\n\\Rightarrow &(1 + \\alpha) b \\nabla  \\cdot ({\\mathbf f} a + {\\mathbf g}{\\mathbf c}) - \\alpha \\nabla \\cdot (b{\\mathbf f} a + b{\\mathbf g}{\\mathbf c}) - \\gamma a b \\geq 0.\n\\end{align}\nThe inequality in~\\eqref{eq:parameterized stability} is from the fact that $h$ is a positive function. Using~\\eqref{rational_parametrization} and~\\eqref{eq:parameterized stability}, the $\\mathcal{L}_1$ OCP problem in~\\eqref{eqn_ocpdiscount} can be rewritten as follows:\n\\begin{align}\\label{eqn_ocp3}\n\\begin{split}\n\\inf_{a({\\mathbf x}), {\\mathbf c}({\\mathbf x})} &\\;\\;\\; \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x}) a({\\mathbf x})}{b({\\mathbf x})^\\alpha}+\\frac{\\beta||{\\mathbf c}({\\mathbf x})||_1}{b({\\mathbf x})^\\alpha} d{\\mathbf x}\\\\\n        {\\rm s.t}.& \\;\\;\\;\\mathrm{LHS~of}~\\eqref{eq:parameterized stability} \\geq 0,\\,\\, a({\\mathbf x}) \\geq 0.\n\\end{split}\n\\end{align}\nRecall that we exclude a small neighborhood of the origin $\\mathcal{B}_\\delta$ due to singularity of the density $\\rho$ at the origin. To guarantee the optimality in $\\mathcal{B}_\\delta$, we design an optimal linear quadratic regulator (LQR) controller for $\\mathcal{B}_\\delta$ using the identified linear dynamics. \n\nTaking one step further, we introduce dummy polynomials ${\\mathbf s}({\\mathbf x}) = [s_1({\\mathbf x}),\\ldots,s_m({\\mathbf x})]^\\top$ such that $s_j({\\mathbf x}) \\ge |c_j({\\mathbf x})|$, which leads to the following additional constraints to~\\eqref{eqn_ocp3}:\n\\begin{align} \\label{eq:const on sx}\n    {\\mathbf s}({\\mathbf x}) - {\\mathbf c}({\\mathbf x}) \\geq 0,\\, {\\mathbf s}({\\mathbf x}) + {\\mathbf c}({\\mathbf x}) \\geq 0.\n\\end{align}\nThe objective function of the OCP in~\\eqref{eqn_ocp3} is rational polynomial functions. To make~\\eqref{eqn_ocp3} a polynomial optimization, we define a vector of polynomial basis functions\n\\begin{align} \\label{eq:basis functions}\n    {\\boldsymbol \\Psi}({\\mathbf x}) = [\\psi_1({\\mathbf x}), \\ldots, \\psi_Q({\\mathbf x})]^\\top,\n\\end{align}\nand coefficient vectors ${\\boldsymbol{\\mathcal C}}_a$, ${\\boldsymbol{\\mathcal C}}_{c_j}$, and ${\\boldsymbol{\\mathcal C}}_{s_j}$ such that\n\\begin{align} \\label{eq:a&cj}\n    a({\\mathbf x}) \\hspace{-0.03in}=\\hspace{-0.03in} {\\boldsymbol{\\mathcal C}}_a^\\top {\\boldsymbol \\Psi}({\\mathbf x}), c_j({\\mathbf x})\\hspace{-0.03in}=\\hspace{-0.03in}{\\boldsymbol{\\mathcal C}}_{c_j}^\\top {\\boldsymbol \\Psi}({\\mathbf x}), s_j({\\mathbf x})\\hspace{-0.03in}=\\hspace{-0.03in}{\\boldsymbol{\\mathcal C}}_{s_j}^\\top {\\boldsymbol \\Psi}({\\mathbf x}),\n\\end{align}\nfor $j=1,\\ldots,m$. Furthermore, we introduce ${\\boldsymbol{\\mathcal C}}_{ab}$, ${\\boldsymbol{\\mathcal C}}_{bc_1}$, \\ldots, ${\\boldsymbol{\\mathcal C}}_{bc_m}$ such that \n\\begin{align} \\label{eq:ab&bc}\n    a({\\mathbf x})b({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{ab}^\\top {\\boldsymbol \\Psi} ({\\mathbf x}), b({\\mathbf x}) c_j({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{bc_j}^\\top {\\boldsymbol \\Psi} ({\\mathbf x}).\n\\end{align}\nUsing the parameterization in~\\eqref{eq:const on sx}--\\eqref{eq:ab&bc} and also treating positive constraints as sum of squares constraints denoted by $\\Sigma[{\\mathbf x}]$, \\eqref{eqn_ocp3} is expressed as a SOS problem given below:\n\\begin{align} \\label{eqn_ocp5}\n        \\min_{\\underset{j=1,\\ldots,m}{{\\boldsymbol{\\mathcal C}}_a, {\\boldsymbol{\\mathcal C}}_{c_j}, {\\boldsymbol{\\mathcal C}}_{s_j}}} &\\,\\,\\, {\\mathbf d}_1^\\top {\\boldsymbol{\\mathcal C}}_a + \\beta \\textstyle{\\sum}_{j=1}^{m} {\\mathbf d}_2^\\top {\\boldsymbol{\\mathcal C}}_{s_j} \\nonumber\\\\\n        {\\rm s.t}. &\\,\\,\\, \\mathrm{LHS~of}~\\eqref{eq:parameterized stability} \\in \\Sigma[{\\mathbf x}],\\,\\, a({\\mathbf x}) \\in \\Sigma[{\\mathbf x}],\\\\\n        &\\,\\,\\, ({\\mathbf s}({\\mathbf x}) - {\\mathbf c}({\\mathbf x})) \\in \\Sigma[{\\mathbf x}],\\, ({\\mathbf s}({\\mathbf x}) + {\\mathbf c}({\\mathbf x})) \\in \\Sigma[{\\mathbf x}].\\nonumber\n\\end{align}\nwhere\n\\begin{align} \\label{eq:integral}\n        {\\mathbf d}_1 = \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x}){\\boldsymbol \\Psi}({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x},\\,\\, {\\mathbf d}_2 = \\int_{{\\mathbf X}_1} \\frac{{\\boldsymbol \\Psi}({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x}.\n\\end{align}\n\n\n\n\n\n\nSimilarly, using the same parameterization in~\\eqref{rational_parametrization} and~\\eqref{eq:parameterized stability}, $\\mathcal{L}_2$ OCP in~\\eqref{eqn_ocpdiscount_L2} is restated as below:\n\\begin{align}\\label{eq:eqn_ocp6}\n\\begin{split}\n    \\min_{a,{\\mathbf c}} &\\quad \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x})a({\\mathbf x})}{b({\\mathbf x})^\\alpha} + \\frac{{\\mathbf c}({\\mathbf x})^\\top {\\mathbf R} {\\mathbf c}({\\mathbf x})}{a({\\mathbf x})b({\\mathbf x})^\\alpha} dx\\\\\n    \\mathrm{s.t.} &\\quad \\mathrm{LHS~of}~\\eqref{eq:parameterized stability} \\geq 0,\\,\\, a({\\mathbf x}) \\geq 0.\n\\end{split}\n\\end{align}\nSubsequently, we introduce a dummy variable $w({\\mathbf x})$ such that $\\frac{{\\mathbf c}({\\mathbf x})^\\top {\\mathbf R} {\\mathbf c}({\\mathbf x})}{a({\\mathbf x})} \\leq w({\\mathbf x})$ and apply Schur complement lemma on the new inequality constraint. Then, \\eqref{eq:eqn_ocp6} is reformulated as follows:\n\\begin{align}\\label{eq:eqn_ocp7}\n\\begin{split}\n    \\min_{a,{\\mathbf c},w} &\\quad \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x})a({\\mathbf x})}{b({\\mathbf x})^\\alpha} + \\frac{w({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x}\\\\\n    \\mathrm{s.t.} &\\quad \\mathrm{LHS~of}~\\eqref{eq:parameterized stability} \\geq 0,\\,\\, a({\\mathbf x}) \\geq 0,\\\\\n    &\\quad {\\boldsymbol{\\mathcal M}}({\\mathbf x}) = \\begin{bmatrix} w({\\mathbf x}) & {\\mathbf c}({\\mathbf x})^\\top \\\\ {\\mathbf c}({\\mathbf x}) & a({\\mathbf x}){\\mathbf R}^{-1} \\end{bmatrix} \\succcurlyeq 0.\n\\end{split}\n\\end{align}\nNow we algebraically express the positive-definite polynomial matrix ${\\boldsymbol{\\mathcal M}}({\\mathbf x})\\succcurlyeq 0$ by using the following lemma: \n\\begin{lemma}[Positive semidefinite polynomial] \\label{lem:polyPSD}\nA $p \\times p$ matrix ${\\mathbf H}({\\mathbf x})$ whose entries are polynomials is positive semidefinite with respect to the monomial vector ${\\mathbf z}({\\mathbf x})$, \\textit{if and only if}, there exist ${\\mathbf D} \\succcurlyeq 0$ such that\n\\begin{align*}\n    {\\mathbf H}({\\mathbf x}) = \\left({\\mathbf z}({\\mathbf x}) \\otimes {\\mathbf I}_p \\right)^\\top {\\mathbf D} \\left({\\mathbf z}({\\mathbf x}) \\otimes {\\mathbf I}_p\\right),\n\\end{align*}\nwhere $\\otimes$ denotes a Kronecker product (tensor product) and ${\\mathbf I}_p$ is an identity matrix with dimension $p$ \\cite{Scherer_2006}.\n\\end{lemma}\nFollowing Lemma~\\ref{lem:polyPSD}, ${\\boldsymbol{\\mathcal M}}({\\mathbf x})$ is PSD when there exists ${\\mathbf D} \\succcurlyeq 0$ such that ${\\boldsymbol{\\mathcal M}}({\\mathbf x})={\\mathbf H}({\\mathbf x})$ with ${\\mathbf z}({\\mathbf x})$ being a monomial vector with the maximum degree equal to $\\mathrm{floor}(\\mathrm{deg}({\\boldsymbol \\Psi}({\\mathbf x}))\/2)+1$, then ${\\boldsymbol{\\mathcal M}}({\\mathbf x})$. As a result, a SOS problem equivalent to~\\eqref{eq:eqn_ocp7} can be formulated as follows:\n\\begin{align} \\label{ocp_9}\n        \\min_{\\underset{j=1,\\ldots,m}{{\\boldsymbol{\\mathcal C}}_a, {\\boldsymbol{\\mathcal C}}_w, {\\boldsymbol{\\mathcal C}}_{c_j}}} &\\,\\,\\, {\\mathbf d}_1^\\top {\\boldsymbol{\\mathcal C}}_a + {\\mathbf d}_2^\\top {\\boldsymbol{\\mathcal C}}_{w}\\nonumber\\\\\n        \\mathrm{s.t.} &\\quad \\mathrm{LHS~of}~\\eqref{eq:parameterized stability} \\in \\Sigma[{\\mathbf x}],\\,\\, a({\\mathbf x}) \\in \\Sigma[{\\mathbf x}],\\\\\n        &\\quad w({\\mathbf x}) - {\\mathbf H}_{11}({\\mathbf x}) = 0,\\, {\\mathbf c}({\\mathbf x}) - {\\mathbf H}_{12}({\\mathbf x}) = 0,\\nonumber\\\\\n        &\\quad a({\\mathbf x}) {\\mathbf R}^{-1} - {\\mathbf H}_{22}({\\mathbf x}) = 0,\\, {\\mathbf D} \\succcurlyeq 0\\nonumber\n\\end{align}\nwhere ${\\mathbf H}_{ij}({\\mathbf x})$ denotes the ${ij}$th entry of~${\\mathbf H}({\\mathbf x})$; and $\\boldsymbol{\\mathcal{C}}_w$ is a coefficient vector of~$w({\\mathbf x})$ with respect to ${\\boldsymbol \\Psi}({\\mathbf x})$, i.e., $w({\\mathbf x})=\\boldsymbol{\\mathcal{C}}_w^\\top {\\boldsymbol \\Psi}({\\mathbf x})$.\n\n\n\\subsection{Data-driven OCPs using linear operator approximations} \\label{sec:data-driven}\nIn this section, we solve the OCPs in~\\eqref{eqn_ocp5} and~\\eqref{ocp_9} using time-series data without known explicit models ${\\mathbf f}$ and ${\\mathbf g}$. Especially, we leverage infinitesimal P-F generator approximation using time-series data given as below:\n\\begin{align} \\label{PF_gen_approx}\n-{\\cal P}_{\\mathbf f} \\psi \\hspace{-0.02in} = \\hspace{-0.02in} \\nabla\\cdot({\\mathbf f} \\psi) \\hspace{-0.02in} = \\hspace{-0.02in} {\\mathbf f} \\cdot \\nabla \\psi+\\nabla\\cdot {\\mathbf f} \\psi \\hspace{-0.02in} = \\hspace{-0.02in} {\\cal K}_{\\mathbf f} \\psi+\\nabla \\cdot {\\mathbf f} \\psi\n\\end{align}\nwhere $\\cal P_{\\mathbf f}$ and $\\cal K_{\\mathbf f}$ denote P-F and Koopman generators corresponding to a function ${\\mathbf f}$. We approximate the Koopman generator $\\cal K_{\\mathbf f}$ using the algorithm shown in~\\cite{Klus_2019}.\nThe brief sketch of the procedure is given as follows. First, we collect finite-time-series data by injecting different control inputs to the system ${\\mathbf f}$ and ${\\mathbf g}$: i) zero control inputs, ${\\mathbf u}=0$, and ii) unit step control inputs, ${\\mathbf u}=\\mathbf{e}_j$, where ${\\mathbf e}_j \\in \\mathbb{R}^m$ denotes unit vectors, i.e., $j$th entry of $\\mathbf{e}_j$ is 1, otherwise 0, into the data matrices:\n\\begin{align} \\label{eq:data_matrix}\n    \\mathbf{X}_i = \\left [ \\mathbf{x}_1, \\ldots, \\mathbf{x}_{T_i} \\right ], \\,\\, \\dot{\\mathbf{X}}_i = \\left [ \\dot{\\mathbf{x}}_1, \\ldots, \\dot{\\mathbf{x}}_{T_i} \\right ],\n\\end{align}\nwith $i=0,1,\\ldots,m$ for zero and step control inputs where $T_i$ are the number of data points for $i$th input case. The samples in $\\mathbf{X}_i$ and $\\mathbf{\\dot{X}}_i$ can be collected from a concatenation of multiple experiment\/simulation trajectories without imposing any specific orders. Also, state derivatives $\\dot{{\\mathbf x}}$ can be numerically identified using algorithms in~\\cite{Xiao_2011,Na_1979}. Next, we construct \\textit{dictionary functions} ${\\boldsymbol \\Psi}({\\mathbf x})$ which constitutes polynomial basis vector as shown in~\\eqref{eq:basis functions}. The types of polynomial basis include monomials, or Legendre\/Hermite polynomials. Based on the choice of the dictionary functions, we calculate the derivative of ${\\boldsymbol \\Psi}({\\mathbf x})$ as below:\n\\begin{align} \\label{eq:basis functions time derivatives}\n\\dot{{\\boldsymbol \\Psi}}({\\mathbf x},\\dot{{\\mathbf x}}) = [\\dot{\\psi}_1({\\mathbf x},\\dot{{\\mathbf x}}), \\ldots, \\dot{\\psi}_Q({\\mathbf x},\\dot{{\\mathbf x}})]^\\top,\n\\end{align}\n\nwhere $\\dot{\\psi}_k({\\mathbf x},\\dot{{\\mathbf x}}) = (\\nabla_{\\mathbf x} \\psi_k)^\\top \\dot{{\\mathbf x}} = \\textstyle{\\sum}_{j=1}^n \\frac{\\partial \\psi_k}{\\partial x_j} \\frac{dx_j}{dt}$.\n{\\color{red} How is $\\dot x$ computed from data is not clear here..as this will depend on $\\Delta t$ the sampling time and the dependence is not shown here explicitly..??}\nThen, the Koopman generator approximates ${\\mathbf L}_i$ for each input case are approximated as follows:\n\\begin{align} \\label{eq:finite Koopman generator}\n\\begin{split}\n    {\\mathbf L}_i &= \\underset{{\\mathbf L}_i}{\\mathrm{argmin}} \\, ||{\\mathbf B}_i - {\\mathbf A}_i {\\mathbf L}_i||_F,\\\\\n      \\mathbf{A}_i &= \\frac{1}{T_i} \\textstyle{\\sum}_{\\ell=1}^{T_i} \\, \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell}) \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell})^\\top,\\\\\n        \\mathbf{B}_i &= \\frac{1}{T_i} \\textstyle{\\sum}_{\\ell=1}^{T_i} \\, \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell}) \\dot{\\boldsymbol{\\Psi}}({\\mathbf X}_{i,\\ell},\\dot{{\\mathbf X}}_{i,\\ell})^\\top,\n\\end{split}\n\\end{align}\nand ${\\mathbf X}_{i,\\ell}$ and $\\dot{{\\mathbf X}}_{i,\\ell}$ denote $\\ell$th column of ${\\mathbf X}_i$ and $\\dot{{\\mathbf X}}$, respectively. \\eqref{eq:finite Koopman generator} admits explicit solution $\\mathbf{K}_i = \\mathbf{A}_i^\\dagger \\mathbf{B}_i$, where $\\dagger$ stands for pseudo-inverse. Since the Koopman generator has linear properties, i.e., ${\\cal K}_{{\\mathbf f} + {\\mathbf g}_j} = \\cal{K}_{\\mathbf f} +\\cal{K}_{{\\mathbf g}_j}$, Koopman generators corresponding to ${\\mathbf g}_j$ are found by:\n\\begin{align}\n{\\cal K}_{{\\mathbf g}_j}\\approx {{\\mathbf L}}_j-{\\mathbf L}_0,\\;\\;\\;j=1,\\ldots, m. \\label{K_gapprox}\n\\end{align}\nNote that ${\\cal K}_{{\\mathbf f}}$ and ${\\cal K}_{{\\mathbf g}_j}$ can be identified using different algorithms, e.g., standard \\textit{extended dynamic mode decomposition (EDMD)}, and also, can be approximated jointly by using trajectories subject to arbitrary inputs.\nLastly, the divergence of ${\\mathbf f}$ shown in~\\eqref{PF_gen_approx} can be found:\n\\begin{align}\n\\nabla \\cdot {{\\mathbf f}} =\\nabla \\cdot [{\\cal K}_0 x_1,\\ldots, {\\cal K}_0 x_n]^\\top \\approx \\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_0 {\\boldsymbol \\Psi}) \\label{approx_divergence}\n\\end{align}\nwhere ${\\boldsymbol{\\mathcal C}}_x$ is a coefficient vector for ${\\mathbf x}$ with respect to ${\\boldsymbol \\Psi}({\\mathbf x})$, i.e., ${\\mathbf x} = {\\boldsymbol{\\mathcal C}}_x^\\top {\\boldsymbol \\Psi} ({\\mathbf x})$, which can be found easily if ${\\boldsymbol \\Psi}({\\mathbf x})$ includes 1st-order monomials (i.e., ${\\mathbf x}$ itself). \nSimilarly, the divergence of vector fields ${\\mathbf g}_j$ are approximated as \n\\begin{align}\n    \\nabla\\cdot ({\\mathbf g}_j)\\approx \\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_j {\\boldsymbol \\Psi}),\\;\\;j=1,\\ldots,m.\\label{divg_approx}\n\\end{align}\nUsing the results in~\\eqref{K_gapprox}--\\eqref{divg_approx}, P-F generators are approximated by:\n\\begin{align} \\label{eq:PF generators}\n\\begin{split}\n    {{\\mathbf P}}_i &={\\mathbf L}_i+\\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_i {\\boldsymbol \\Psi}) {\\bf I},\n\\end{split}\n\\end{align}\nfor $i=0,1,\\ldots,m$. Now, using approximated infinitesimal PF generators in~\\eqref{eq:PF generators}, the OCPs in~\\eqref{eqn_ocp5} and~\\eqref{ocp_9} using data by restating the LHS of~\\eqref{eq:parameterized stability} as follows:\n\\begin{align} \\label{eq:stability_approximation}\n\\begin{split}\n&(1+\\alpha) b({\\mathbf x}) \\left ( {\\boldsymbol{\\mathcal C}}_a^\\top {\\mathbf P}_0 {\\boldsymbol \\Psi}({\\mathbf x}) + \\textstyle{\\sum}_{j=1}^m {\\boldsymbol{\\mathcal C}}_c^\\top {\\mathbf P}_j {\\boldsymbol \\Psi}({\\mathbf x}) \\right )\\\\\n&- \\alpha \\left( {\\boldsymbol{\\mathcal C}}_{ab} {\\mathbf P}_0 {\\boldsymbol \\Psi}({\\mathbf x}) + \\textstyle{\\sum}_{j=1}^m {\\boldsymbol{\\mathcal C}}_{bc_j}^\\top {\\mathbf P}_j {\\boldsymbol \\Psi}({\\mathbf x}) \\right).\n\\end{split}\n\\end{align}\n\\end{comment}\n\n\\section{Koopman and SOS-based Computation Framework for Optimal Control} \\label{sec:SOS Control}\nThis section provides Koopman and SOS-based computational framework for the finite-dimensional approximation of OCP involving ${\\cal L}_1$\/${\\cal L}_2$ costs. We begin with the following parameterization for the optimization variables $\\rho$ and $\\bar{\\boldsymbol \\rho}$. \n\\begin{equation}\n    \\rho({\\mathbf x}) = {a({\\mathbf x})}\/b({\\mathbf x})^{\\alpha},\\;\\; \\bar{{\\boldsymbol \\rho}}({\\mathbf x}) = {\\mathbf c}({\\mathbf x})\/b({\\mathbf x})^{\\alpha}, \\label{rational_parametrization}\n\\end{equation}\nwhere $a({\\mathbf x})\\geq 0$  and ${\\mathbf c}({\\mathbf x}) = \\begin{bmatrix} c_1({\\mathbf x}),\\, \\ldots,\\, c_m({\\mathbf x})\\end{bmatrix}^\\top$. Here, $b({\\mathbf x})$ is a positive polynomial (positive at ${\\mathbf x} \\ne 0$), and $\\alpha$ is a positive constant which is sufficiently large for integrability condition. In fact $b({\\mathbf x})$ is chosen to be control Lyapunov function based on the  linearized control dynamics at the origin. The data-driven procedure for the identification of the linear dynamics used in the construction of $b({\\mathbf x})$ is explained in Remark \\ref{remark_bconstrucntion}. The particular form for the parameterization of the optimization variable in (\\ref{rational_parametrization}) is chosen because of the fact that $\\rho$ has singularity at the origin (Remark \\ref{remark_singularity}). Using~\\eqref{rational_parametrization}, we can write the constraints for the optimization problem as  in~\\eqref{ocp_main_discounted1} and~\\eqref{eqn_ocpdiscount} as\n\\begin{align*} \nh &=\\nabla  \\cdot ({\\mathbf f}\\rho  + {\\mathbf g}\\bar{{\\boldsymbol \\rho}}) - \\gamma \\rho = \\nabla  \\cdot \\left[({\\mathbf f} a + {\\mathbf g}{\\mathbf c})\/b^{\\alpha} \\right] - \\gamma \\frac{a}{b^{\\alpha}}\\\\\n&=\\frac{1}{b^{\\alpha +1}} [(1 + \\alpha) b \\nabla  \\cdot ({\\mathbf f} a + {\\mathbf g}{\\mathbf c}) - \\alpha \\nabla \\cdot (b{\\mathbf f} a + b{\\mathbf g}{\\mathbf c})\\\\\n&\\;\\;\\;\\;-\\gamma a b].\n\\end{align*}\nWith the above form of the constraints, we assume the following parameterization for $h=\\frac{d}{b^{\\alpha+1}}$, \nwhere $d$ is an arbitrary positive polynomial. With the assumed form of $h$, we write the constraints in the optimization variable, $a$ and ${\\mathbf c}$ \n as\n\\begin{align} \\label{rational_parametrization_1}\n(1 + \\alpha) b \\nabla  \\cdot ({\\mathbf f} a + {\\mathbf g}{\\mathbf c}) - \\alpha \\nabla \\cdot (b{\\mathbf f} a + b{\\mathbf g}{\\mathbf c}) - \\gamma a b = d\n\\end{align}\nThe above constraint in the optimization problem can be written in terms of the P-F generator as follows:\n\\begin{align}\n    &(1+\\alpha)b \\left({\\cal P}_{\\bf f}a+\\sum_{i=1}^m{\\cal P}_{\\bf g_i}c_i\\right)-\\alpha \\left({\\cal P}_{\\bf f}(ba)+\\sum_{i=1}^m{\\cal P}_{\\bf g_i}(bc_i)\\right)\\nonumber\\\\\n    &-\\gamma a=d\\label{ss12}\n\\end{align}\n\\subsection{Data-driven Approximation of the Generators}\\label{section_datagenerator}\nFrom (\\ref{ss12}), it follows that the data-driven approximation of the constraints in the optimization problems (\\ref{eqn_ocpdiscount}) and (\\ref{eqn_ocpdiscount_L2}) involves approximation of the P-F generators, ${\\cal P}_{\\bf f}$ and ${\\cal P}_{{\\mathbf g}_i}$. Furthermore, the P-F generator can be expressed in terms of the Koopman generator as \n\\begin{eqnarray}\n-{\\cal P}_{\\mathbf f} \\psi \\hspace{-0.02in} = \\hspace{-0.02in} \\nabla\\cdot({\\mathbf f} \\psi) \\hspace{-0.02in} = \\hspace{-0.02in} {\\mathbf f} \\cdot \\nabla \\psi+\\nabla\\cdot {\\mathbf f} \\psi \\hspace{-0.02in} = \\hspace{-0.02in} {\\cal K}_{\\mathbf f} \\psi+\\nabla \\cdot {\\mathbf f} \\psi.\\label{PF_gen_approx}\n\\end{eqnarray}\n\nExpressing the P-F generator in terms of the Koopman generator allows us to use data-driven methods used to approximate the Koopman generator for the approximation of the P-F generator. In particular, we use generator Extended Dynamic Mode Decomposition (gEDMD) algorithm from~\\cite{klus2020data} for the approximation.\nTo approximate Koopman generators, we first collect time-series data from the dynamical system in~\\eqref{cont_syst1} by injecting different control inputs: i) zero control inputs, ${\\mathbf u}=0$, and ii) unit step control inputs, ${\\mathbf u}=\\mathbf{e}_j$\\footnote{${\\mathbf e}_j \\in \\mathbb{R}^m$ denotes unit vectors, i.e., $j$th entry of $\\mathbf{e}_j$ is 1, otherwise 0.} for $j=1,\\ldots,m$ for a finite time horizon with sampling step $\\delta t$. Let, \n\\begin{align} \\label{eq:data_matrix}\n    \\mathbf{X}_i = \\left [ \\mathbf{x}_1, \\ldots, \\mathbf{x}_{T_i} \\right ], \\,\\, \\dot{\\mathbf{X}}_i = \\left [ \\dot{\\mathbf{x}}_1, \\ldots, \\dot{\\mathbf{x}}_{T_i} \\right ],\n\\end{align}\nwith $i=0,1,\\ldots,m$ for zero and step control inputs where $T_i$ are the number of data points for $i$th input case. The samples in $\\mathbf{X}_i$ do not have to be from a single trajectory; it can be a concatenation of multiple experiment\/simulation trajectories. Also, time derivatives of the states $\\dot{{\\mathbf x}}$ can be accurately estimated using numerical algorithms such as finite differences. Next, we construct a polynomial basis vector:\n\\begin{align} \\label{eq:basis functions}\n{\\boldsymbol \\Psi}({\\mathbf x}) = [\\psi_1({\\mathbf x}), \\ldots, \\psi_Q({\\mathbf x})]^\\top,\n\\end{align}\nwhich can include monomials or Legendre\/Hermite polynomials. The data-driven approximation of the generator will essentially involve the projection of the infinite-dimensional Koopman and P-F generators on the finite-dimensional space spanned by the basis function (\\ref{eq:basis functions}). Following \\cite{klus2020data}, we define:\n\\begin{align} \\label{eq:basis functions time derivatives}\n\\dot{{\\boldsymbol \\Psi}}({\\mathbf x},\\dot{{\\mathbf x}}) = [\\dot{\\psi}_1({\\mathbf x},\\dot{{\\mathbf x}}), \\ldots, \\dot{\\psi}_Q({\\mathbf x},\\dot{{\\mathbf x}})]^\\top,\n\\end{align}\n\\begin{align}\n    \\dot{\\psi}_k({\\mathbf x},\\dot{{\\mathbf x}}) = (\\nabla_{\\mathbf x} \\psi_k)^\\top \\dot{{\\mathbf x}} = \\textstyle{\\sum}_{j=1}^n \\frac{\\partial \\psi_k}{\\partial x_j} \\frac{dx_j}{dt}\n\\end{align}\nThe partial derivatives of the basis function are computed analytically which is required for $\\dot{\\psi}_k({\\mathbf x},\\dot{{\\mathbf x}})$. Note that we also need $\\frac{dx_j}{dt}$ which is simply denoted by $\\dot{x}_j$. The value of $\\dot{x}_j$ is  approximated using finite differences:\n\\begin{align}\n    \\dot{x}_j \\approx \\frac{x_j - x_{j-1}}{\\Delta t}\n\\end{align}\nwhere $x_{j-1}$ and $x_j$ are the $j-1^{th}$ and $j^{th}$ data point in the system trajectory and $\\Delta t$ is the time difference between two consecutive data points. A more sophisticated finite-difference method can be used, e.g., total variation regularization~\\cite{Chartrand2011-ni} for noisy data and discontinuous derivatives, and also second-order central difference for better accuracy. Then, the Koopman generator approximate $\\mathbf{L}_i$ for each input case can be approximated as:\n\\begin{align} \\label{eq:finite Koopman generator}\n\\begin{split}\n    \\mathbf{L}_i &= \\underset{{\\mathbf L}_i}{\\mathrm{argmin}} \\, ||{\\mathbf B}_i - {\\mathbf A}_i {\\mathbf L}_i||_F,\\\\\n      \\mathbf{A}_i &= \\frac{1}{T_i} \\textstyle{\\sum}_{\\ell=1}^{T_i} \\, \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell}) \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell})^\\top,\\\\\n        \\mathbf{B}_i &= \\frac{1}{T_i} \\textstyle{\\sum}_{\\ell=1}^{T_i} \\, \\boldsymbol{\\Psi}({\\mathbf X}_{i,\\ell}) \\dot{\\boldsymbol{\\Psi}}({\\mathbf X}_{i,\\ell},\\dot{{\\mathbf X}}_{i,\\ell})^\\top,\n\\end{split}\n\\end{align}\nand ${\\mathbf X}_{i,\\ell}$ and $\\dot{{\\mathbf X}}_{i,\\ell}$ denote $\\ell$th column of ${\\mathbf X}_i$ and $\\dot{{\\mathbf X}}$, respectively.  The solution of~\\eqref{eq:finite Koopman generator} is explicitly known, $\\mathbf{K}_i = \\mathbf{A}_i^\\dagger \\mathbf{B}_i$, where $\\dagger$ stands for pseudo-inverse. Given the Koopman generator approximates for ${\\mathbf f}$, ${\\cal K}_{{\\mathbf f}}\\approx {\\mathbf L}_0$, using the linearity of the generators,\n\\begin{equation}\\label{eq:L0}\n{\\cal K}_{{\\mathbf g}_j}\\approx {{\\mathbf L}}_j-{\\mathbf L}_0,\\;\\;\\;j=1,\\ldots, m.\n\\end{equation}\n\nThe above is one method to estimate ${\\cal K}_{{\\mathbf f}}$ and ${\\cal K}_{{\\mathbf g}_j}$. They can also be approximated jointly by using trajectories subject to arbitrary inputs.\nNext, we approximate the divergence of vector field ${\\mathbf f}$ as\n\\begin{equation}\n\\nabla \\cdot {{\\mathbf f}} =\\nabla \\cdot [{\\cal K}_0 x_1,\\ldots, {\\cal K}_0 x_n]^\\top \\approx \\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_0 {\\boldsymbol \\Psi}) \\label{approx_divergence}\n\\end{equation}\nwhere ${\\boldsymbol{\\mathcal C}}_x$ is a coefficient vector for ${\\mathbf x}$, i.e., ${\\mathbf x} = {\\boldsymbol{\\mathcal C}}_x^\\top {\\boldsymbol \\Psi}$, which can be found easily if ${\\boldsymbol \\Psi}$ includes 1st-order monomials (i.e., ${\\mathbf x}$). \nSimilarly, the divergence of vector fields ${\\mathbf g}_j$ are approximated as \n\\begin{equation}\n\\nabla\\cdot ({\\mathbf g}_j)\\approx \\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_j {\\boldsymbol \\Psi}),\\;\\;j=1,\\ldots,m.\\label{divg_approx}\n\\end{equation}\nfrom~\\eqref{PF_gen_approx},~\\eqref{eq:L0}--\\eqref{divg_approx}, P-F generators are approximated by\n\\begin{equation} \\label{eq:PF generators}\n{{\\mathbf P}}_j\\hspace{-0.03in}=\\hspace{-0.03in}{\\mathbf L}_j+\\nabla \\cdot({\\boldsymbol{\\mathcal C}}_x^\\top {\\mathbf L}_j {\\boldsymbol \\Psi}) {\\bf I}\n\\end{equation}\nfor $j=0,1,\\ldots,m$. \n\\begin{remark}\nWhile the above procedure describes an approach for the approximation of the Koopman generators corresponding to the drift vector field, ${\\bf f}({\\mathbf x})$, and control vector fields, ${\\mathbf g}_j({\\mathbf x})$, for $j=1,\\ldots,m$ using zero input and step input, it is also possible to identify these vector field using random inputs. The problem of data-driven identification of the system dynamics using random or arbitrary control input will involve identifying bilinear vector fields. It can again be reduced to a least-square optimization problem similar to (\\ref{eq:finite Koopman generator}).\n\\end{remark}\n\nTo parameterize the optimization variables, we express polynomial functions $a({\\mathbf x})$, $b({\\mathbf x})$, and $c_j({\\mathbf x})$ with respect to the basis ${\\boldsymbol \\Psi}({\\mathbf x})$. Let ${\\boldsymbol{\\mathcal C}}_a$, ${\\boldsymbol{\\mathcal C}}_b$, and ${\\boldsymbol{\\mathcal C}}_{c_j}$ be the coefficient vectors used in the expansion of $a({\\mathbf x})$, $b({\\mathbf x})$, and $c_j({\\mathbf x})$:\n\\begin{align} \\label{eq:a&cj}\n    a({\\mathbf x}) \\hspace{-0.03in}=\\hspace{-0.03in} {\\boldsymbol{\\mathcal C}}_a^\\top {\\boldsymbol \\Psi}({\\mathbf x}), b({\\mathbf x}) \\hspace{-0.03in}=\\hspace{-0.03in} {\\boldsymbol{\\mathcal C}}_b^\\top {\\boldsymbol \\Psi}({\\mathbf x}), c_j({\\mathbf x})\\hspace{-0.03in}=\\hspace{-0.03in}{\\boldsymbol{\\mathcal C}}_{c_j}^\\top {\\boldsymbol \\Psi}({\\mathbf x}).\n\\end{align}\nNote that ${\\boldsymbol{\\mathcal C}}_b$ contains constant coefficients of $b({\\mathbf x})$ since $b({\\mathbf x})$ is a known polynomial function. Similarly, let ${\\boldsymbol{\\mathcal C}}_{ab}$, ${\\boldsymbol{\\mathcal C}}_{bc_1}$, \\ldots, ${\\boldsymbol{\\mathcal C}}_{bc_m}$ denote coefficient vectors of polynomials, $a({\\mathbf x}) b({\\mathbf x})$, and $b({\\mathbf x}) c_j({\\mathbf x})$, for $j=1,\\ldots,m$, namely,\n\\begin{align} \\label{eq:ab&bc}\n    a({\\mathbf x})b({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{ab}^\\top {\\boldsymbol \\Psi} ({\\mathbf x}), b({\\mathbf x}) c_j({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{bc_j}^\\top {\\boldsymbol \\Psi} ({\\mathbf x}).\n\\end{align}\nConstructing the coefficient vectors ${\\boldsymbol{\\mathcal C}}_{ab}$, and ${\\boldsymbol{\\mathcal C}}_{bc_j}$ from the coefficient vectors ${\\boldsymbol{\\mathcal C}}_{a}$, ${\\boldsymbol{\\mathcal C}}_b$, and ${\\boldsymbol{\\mathcal C}}_{c_j}$ requires trivial numerical procedures. In case that ${\\boldsymbol \\Psi}({\\mathbf x})$ is a monomial, finding these coefficient vectors are straightforward. If ${\\boldsymbol \\Psi}({\\mathbf x})$ is not a monomial vector, it involves more complicated numerical steps. One approach is to express the basis with respect to a common monomial vector $\\mathcal{M}({\\mathbf x})$ such that ${\\boldsymbol \\Psi}({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_\\Psi^\\top \\mathcal{M}({\\mathbf x})$ where ${\\boldsymbol{\\mathcal C}}_\\Psi$ is a coefficient matrix that connects ${\\boldsymbol \\Psi}({\\mathbf x})$ and $\\mathcal{M}({\\mathbf x})$. Then, we can find coefficient vectors in terms of the monomial vector $\\mathcal{M}({\\mathbf x})$, i.e., $a({\\mathbf x}) b({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{ab}^{'} \\mathcal{M}({\\mathbf x})$ and $b({\\mathbf x}) c_j({\\mathbf x}) = {\\boldsymbol{\\mathcal C}}_{bc_j}^{'} \\mathcal{M}({\\mathbf x})$, and convert these coefficient vectors back to the original basis ${\\boldsymbol \\Psi}({\\mathbf x})$ by multiplying pseudo-inverse of ${\\boldsymbol{\\mathcal C}}_\\Psi$, i.e., ${\\boldsymbol{\\mathcal C}}_{ab} = {\\boldsymbol{\\mathcal C}}_\\Psi^\\dagger {\\boldsymbol{\\mathcal C}}_{ab}^{'}$ and ${\\boldsymbol{\\mathcal C}}_{bc_j} = {\\boldsymbol{\\mathcal C}}_\\Psi^\\dagger {\\boldsymbol{\\mathcal C}}_{bc_j}^{'}$. The implementation of this procedure can be done easily using polynomial toolbox provided in SOSOPT in Matlab~\\cite{Seiler_2013_SOSOPT}.\n\n\n\n\nNow, using approximated infinitesimal PF generators in~\\eqref{eq:PF generators}, we restate the LHS of~\\eqref{rational_parametrization_1} in~\\eqref{eqn_ocpdiscount_L2} and~\\eqref{eqn_ocpdiscount} as:\n\\begin{align} \\label{eq:stability_approximation}\n\\begin{split}\n&(1+\\alpha) b({\\mathbf x}) \\left ( {\\boldsymbol{\\mathcal C}}_a^\\top {\\mathbf P}_0 {\\boldsymbol \\Psi}({\\mathbf x}) + \\textstyle{\\sum}_{j=1}^m {\\boldsymbol{\\mathcal C}}_c^\\top {\\mathbf P}_j {\\boldsymbol \\Psi}({\\mathbf x}) \\right )\\\\\n&- \\alpha \\left( {\\boldsymbol{\\mathcal C}}_{ab}^\\top {\\mathbf P}_0 {\\boldsymbol \\Psi}({\\mathbf x}) + \\textstyle{\\sum}_{j=1}^m {\\boldsymbol{\\mathcal C}}_{bc_j}^\\top {\\mathbf P}_j {\\boldsymbol \\Psi}({\\mathbf x}) \\right) - \\gamma a({\\mathbf x}) b({\\mathbf x}).\n\\end{split}\n\\end{align}\n\n\n\\begin{remark}\nIn \\cite{korda2018convergence,klus2020data}, convergence results for the finite-dimensional approximation of the Koopman operator and generators in the limit as the number of basis functions and data points goes to infinity is studied. These convergence results combined with the finite-dimensional approximation of the cost function for the optimization problem can be used to provide theoretical justification for the solution obtained using the finite-dimensional approximation of the infinite-dimensional optimization problem. \n\\end{remark}\nIn the following sections, we discuss the finite dimensional approximation of the cost function leading up to the finite dimensional approximation of the infinite dimensional optimization problems (\\ref{eqn_ocpdiscount}) and (\\ref{eqn_ocpdiscount_L2}) involving ${\\cal L}_1$ and ${\\cal L}_2$ norms on control input respectively.\n\n\\subsection{Optimal Control with $\\mathcal{L}_1$ norm of feedback control}\n\n\n\n\nUsing the assumed paramaterization for the \n$\\rho$ and $\\bar{\\boldsymbol \\rho}$ from (\\ref{rational_parametrization}) the cost function for the ${\\cal L}_1$ OCP problem in~\\eqref{eqn_ocpdiscount} can be written as \n\\begin{align}\\label{eqn_ocp3}\n\\begin{split}\n\\inf_{a\\geq 0,{\\mathbf c}} &\\;\\;\\; \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x}) a({\\mathbf x})}{b({\\mathbf x})^\\alpha}+\\frac{\\beta||{\\mathbf c}({\\mathbf x})||_1}{b({\\mathbf x})^\\alpha} d{\\mathbf x}\n\\end{split}\n\\end{align}\nwhere a small neighborhood of the origin, ${\\cal N} = \\{{\\mathbf x} \\in {\\mathbf X}: |{\\mathbf x}| \\le \\epsilon,\\, \\epsilon > 0\\}$, \nis chosen as a polytope and excluded from the integration of the cost function to remove singularity of the density at the origin (refer to Remark \\ref{remark_singularity}).\n\n\nTo make~\\eqref{eqn_ocp3} solvable, we introduce dummy polynomials ${\\mathbf s}({\\mathbf x}) = [s_1({\\mathbf x}),\\ldots,s_m({\\mathbf x})]^\\top$,  adding constraints:\n\\begin{align} \\label{eq:const on sx}\n    {\\mathbf s}({\\mathbf x}) - {\\mathbf c}({\\mathbf x}) \\geq 0,\\, {\\mathbf s}({\\mathbf x}) + {\\mathbf c}({\\mathbf x}) \\geq 0.\n\\end{align}\nThe polynomial ${\\mathbf s}$ is expressed in terms of the basis function using the coefficient vector ${\\boldsymbol{\\mathcal C}}_{s_j}$ as:\n\\begin{align} \\label{eq:a&cj}\n   \n    s_j({\\mathbf x})\\hspace{-0.03in}=\\hspace{-0.03in}{\\boldsymbol{\\mathcal C}}_{s_j}^\\top {\\boldsymbol \\Psi}({\\mathbf x}),\n\\end{align}\nfor $j=1,\\ldots,m$.\nSubstituting (\\ref{eq:ab&bc}) in the integral cost (\\ref{eqn_ocp3}), we obtain following finite dimensional approximation of the cost function.\n\\begin{align} \\label{eq:L1 cost precomp}\n    {\\mathbf d}_1^\\top {\\boldsymbol{\\mathcal C}}_a + \\beta \\textstyle{\\sum}_{j=1}^{m} {\\mathbf d}_2^\\top {\\boldsymbol{\\mathcal C}}_{s_j}\n\\end{align}\nwhere ${\\mathbf d}_1$ and ${\\mathbf d}_2$ are the coefficient vectors given by\n\\begin{align} \\label{eq:integral}\n        {\\mathbf d}_1 = \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x}){\\boldsymbol \\Psi}({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x},\\,\\, {\\mathbf d}_2 = \\int_{{\\mathbf X}_1} \\frac{{\\boldsymbol \\Psi}({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x}.\n\\end{align}\nUsing~\\eqref{eq:const on sx}--\\eqref{eq:integral} and SOS positivity constraints denoted by $\\Sigma[{\\mathbf x}]$, \\eqref{eqn_ocp3} can be expressed as a SOS problem as:\n\\begin{align} \\label{eqn_ocp5}\n\\begin{split}\n        \\min_{\\underset{j=1,\\ldots,m}{{\\boldsymbol{\\mathcal C}}_a, {\\boldsymbol{\\mathcal C}}_{c_j}, {\\boldsymbol{\\mathcal C}}_{s_j}}} &\\,\\,\\, {\\mathbf d}_1^\\top {\\boldsymbol{\\mathcal C}}_a + \\beta \\textstyle{\\sum}_{j=1}^{m} {\\mathbf d}_2^\\top {\\boldsymbol{\\mathcal C}}_{s_j}\\\\\n        &{\\rm s.t}. \\,\\; \\eqref{eq:stability_approximation} \\in \\Sigma[{\\mathbf x}],\\,\\, a({\\mathbf x}) \\in \\Sigma[{\\mathbf x}],\\\\\n        &\\, ({\\mathbf s}({\\mathbf x}) - {\\mathbf c}({\\mathbf x})) \\in \\Sigma[{\\mathbf x}],\\, ({\\mathbf s}({\\mathbf x}) + {\\mathbf c}({\\mathbf x})) \\in \\Sigma[{\\mathbf x}].\n\\end{split}\n\\end{align}\n\n\n\n\n\n\n\\subsection{Optimal Control with $\\mathcal{L}_2$ norm of feedback control}\n$\\mathcal{L}_2$ OCP in~\\eqref{eqn_ocpdiscount_L2} is restated as:\n\\vspace{-0.0in}\n\\begin{align}\\label{eq:eqn_ocp6}\n\\begin{split}\n    \\min_{a,{\\mathbf c}} &\\quad \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x})a({\\mathbf x})}{b({\\mathbf x})^\\alpha} + \\beta \\frac{{\\mathbf c}({\\mathbf x})^\\top {\\mathbf R} {\\mathbf c}({\\mathbf x})}{a({\\mathbf x})b({\\mathbf x})^\\alpha} dx\\\\\n    &\\mathrm{s.t.} \\quad \\eqref{eq:stability_approximation} \\geq 0,\\,\\, a({\\mathbf x}) \\geq 0.\n\\end{split}\n\\end{align}\nby following the same parameterization in~\\eqref{rational_parametrization}. Subsequently, we reformulate \\eqref{eq:eqn_ocp6} as follows:\n\\vspace{-0.0in}\n\\begin{align}\\label{eq:eqn_ocp7}\n\\begin{split}\n    \\min_{a,{\\mathbf c},w} &\\quad \\int_{{\\mathbf X}_1} \\frac{q({\\mathbf x})a({\\mathbf x})}{b({\\mathbf x})^\\alpha} + \\beta \\frac{w({\\mathbf x})}{b({\\mathbf x})^\\alpha} d{\\mathbf x}\\\\\n    \\mathrm{s.t.} &\\quad \\eqref{eq:stability_approximation} \\geq 0,\\,\\, a({\\mathbf x}) \\geq 0,\\\\\n    &\\quad {\\boldsymbol{\\mathcal M}}({\\mathbf x}) = \\begin{bmatrix} w({\\mathbf x}) & {\\mathbf c}({\\mathbf x})^\\top \\\\ {\\mathbf c}({\\mathbf x}) & a({\\mathbf x}){\\mathbf R}^{-1} \\end{bmatrix} \\succcurlyeq 0,\n\\end{split}\n\\end{align}\nwhere the positive semidefinite (PSD) of ${\\boldsymbol{\\mathcal M}}({\\mathbf x})$ is a result of applying the Schur complement lemma on $\\mathcal{L}_2$ cost bounded by $w({\\mathbf x})$, i.e., $\\frac{{\\mathbf c}({\\mathbf x})^\\top {\\mathbf R} {\\mathbf c}({\\mathbf x})}{a({\\mathbf x})} \\leq w({\\mathbf x})$. Now, to algebraically express ${\\boldsymbol{\\mathcal M}}({\\mathbf x})\\succcurlyeq 0$, we introduce the lemma:\n\\begin{lemma}[\\scriptsize Positive semidefinite polynomial matrix] ~\\cite{Scherer_2006} \\label{lem:polyPSD}\nA $p \\times p$ matrix ${\\mathbf H}({\\mathbf x})$ whose entries are polynomials is positive semidefinite with respect to the monomial vector ${\\mathbf z}({\\mathbf x})$, \\textit{if and only if}, there exist ${\\mathbf D} \\succcurlyeq 0$ such that\n\\begin{align*}\n    {\\mathbf H}({\\mathbf x}) = \\left({\\mathbf z}({\\mathbf x}) \\otimes {\\mathbf I}_p \\right)^\\top {\\mathbf D} \\left({\\mathbf z}({\\mathbf x}) \\otimes {\\mathbf I}_p\\right),\n\\end{align*}\nwhere $\\otimes$ denotes a Kronecker product (tensor product) and ${\\mathbf I}_p$ is an identity matrix with dimension $p$.\n\\end{lemma}\n\nFollowing Lemma~\\ref{lem:polyPSD}, let ${\\mathbf z}({\\mathbf x})$ be a monomial vector with the maximum degree equal to $\\mathrm{floor}(\\mathrm{deg}({\\boldsymbol \\Psi}({\\mathbf x}))\/2)+1$, then ${\\boldsymbol{\\mathcal M}}({\\mathbf x})$ in~\\eqref{eq:eqn_ocp7} is PSD when there exists ${\\mathbf D} \\succcurlyeq 0$ such that ${\\boldsymbol{\\mathcal M}}({\\mathbf x})={\\mathbf H}({\\mathbf x})$. Using this result and~\\eqref{eq:integral}, a SOS problem equivalent to~\\eqref{eq:eqn_ocp7} can be formulated as follows:\n\\begin{align} \\label{ocp_9}\n        \\min_{\\underset{j=1,\\ldots,m}{{\\boldsymbol{\\mathcal C}}_a, {\\boldsymbol{\\mathcal C}}_w, {\\boldsymbol{\\mathcal C}}_{c_j}}} &\\,\\,\\, {\\mathbf d}_1^\\top {\\boldsymbol{\\mathcal C}}_a + \\beta {\\mathbf d}_2^\\top {\\boldsymbol{\\mathcal C}}_{w}\\nonumber\\\\\n        \\mathrm{s.t.} &\\quad \\eqref{eq:stability_approximation} \\in \\Sigma[{\\mathbf x}],\\,\\, a({\\mathbf x}) \\in \\Sigma[{\\mathbf x}],\\\\\n        &\\quad w({\\mathbf x}) - {\\mathbf H}_{11}({\\mathbf x}) = 0,\\, {\\mathbf c}({\\mathbf x}) - {\\mathbf H}_{12}({\\mathbf x}) = 0,\\nonumber\\\\\n        &\\quad a({\\mathbf x}) {\\mathbf R}^{-1} - {\\mathbf H}_{22}({\\mathbf x}) = 0,\\, {\\mathbf D} \\succcurlyeq 0\\nonumber\n\\end{align}\nwhere ${\\mathbf H}_{ij}({\\mathbf x})$ denotes the ${ij}$th entry of~${\\mathbf H}({\\mathbf x})$; and $\\boldsymbol{\\mathcal{C}}_w$ is a coefficient vector of~$w({\\mathbf x})$, i.e., $w({\\mathbf x})=\\boldsymbol{\\mathcal{C}}_w^\\top {\\boldsymbol \\Psi}({\\mathbf x})$.\n\n\n\n\n\n\n\\begin{remark}\\label{remark_bconstrucntion}\nTo obtain $b({\\mathbf x})$, the control Lyapunov function for the linearized dynamics, we first identify the linearized control system dynamics from time-series data collected near the origin. to identify the linear dynamics, we use the gEDMD algorithm discussed  in Section \\ref{section_datagenerator} for the special case of identity basis functions i.e., ${\\boldsymbol \\Psi}({\\mathbf x})={\\mathbf x}$. Once linearized system dynamics is identified we use linear quadratic regulator based optimal control for the construction of $b({\\mathbf x})$, namely $b({\\mathbf x})={\\mathbf x}^\\top P{\\mathbf x}$, where $P$ is the solution of algebraic Riccati equation (ARE). Following Assumption \\ref{assume_localstable}, we know that there exists a positive definite solution, $P$, to the ARE, which serves as control Lyapunov function for the linearized control system. \n\\end{remark}\n\n\\section{Simulation Results}\\label{sec:examples}\nIn this section, we present simulation results to illustrate the proposed data-driven control framework. All the simulation results are performed using MATLAB on a desktop computer with 64GB RAM. We have taken the value of $\\alpha = 4 $ and $\\beta = 1$ for all our examples. The cost function and control matrix for each example is $q({\\mathbf x}) = {\\mathbf x}^T{\\mathbf x}$ and $R=1$ respectively. Furthermore, the cost function is computed outside the region $\\cal N$ of the neighborhood. However, we did not implement a local stabilizing controller around the origin and hence no blending controller (Remark \\ref{remark_singularity}). Also, the maximum degree of $a({\\mathbf x})$  is taken to be 1. We take the simulation time step $\\Delta t = 0.01 \\, \\mathrm{[s]}$ for sampling time-series data, and also, Legendre polynomials are used as dictionary functions for all examples. We use SOSOPT~\\cite{Seiler_2013_SOSOPT} toolbox to solve the formulated SOS optimization problems for the OCPs in~\\eqref{eqn_ocp5} and~\\eqref{ocp_9}. All other parameters used for each example are listed in Table \\ref{table:OCP param}.\n\n\\begin{table}[b]\n\\small\n\\centering\n\\caption{Parameters for different examples}\n\\begin{tabular} {*5c}\n\\toprule\n& \\textbf {Ex~1} &  \\textbf {Ex~2}&  \\textbf{Ex~3} &\\textbf{Ex~4}\\\\\n\\midrule\n$\\mathbf{X}$ &$[-5,5]^2$ & $[-5,5]^2$ & $[-5,5]^2$ & $[-5,5]^3$ \\\\\n$\\mathcal{N}$ & $[-0.1,0.1]^2$ & $[-0.1,0.1]^2$ & $[-0.1,0.1]^2$ & $[-0.1,0.1]^3$ \\\\\ndeg($c({\\mathbf x})$) & $2$ & $6$ & $3$ & $6$  \\\\ \ndeg($s({\\mathbf x})$) & 2& $7$ & $7$ & $6$ \\\\\n${\\boldsymbol \\Psi}({\\mathbf x})$ & ${4^{th}}$ order & ${9^{th}}$ order & ${7^{th}}$ order & ${8^{th}}$ order \\\\\n\\toprule\n\\end{tabular}\n\\label{table:OCP param}\n\\end{table}\n\n\n\n\n\n\n\\noindent\\textbf{Example 1:} Consider the dynamics of controlled simple nonlinear numerical system:\n\\begin{align*}\n\\dot x_1=-x_1+x_2,\\;\\;\n\\dot x_2= -0.5(x_1+x_2) + 0.5x_1^2x_2 + x_1 u.\n\\end{align*}\nWith this example we use ${\\cal L}_2$ cost on the control input. For this example, optimal control and optimal cost can be found by solving the HJB equation~\\cite{primbs1996optimality} analytically and are given as below:\n\\[u^\\star({\\mathbf x})=-x_1x_2,\\;\\;V^\\star({\\mathbf x})=0.5 x_1^2+x_2^2.\\]\nNext, using the proposed method, we get an optimal control with discount factor $\\gamma=0$ as follows.\n\\[k^\\star({\\mathbf x}) = -1.38x_1x_2 + 0.00005x_1 - 0.00004x_2 - 0.0063\\]\nBy rounding off the coefficients of $k^\\star({\\mathbf x})$, we see that $u^\\star({\\mathbf x}) = k^\\star({\\mathbf x})$. The small mismatch in decimal is due to the choice of $h_0({\\mathbf x})$, which is unique to our formulation but is absent from the primal formulation of OCP. The simulation results are obtained by solving inequality in the constraints corresponding to the case where $h_0\\geq 0$. In this example, we collected $2 \\times 10^4$ time-series data points by simulating the system to estimate the Koopman generator. Figure~\\ref{E7L2_1} shows the comparison of trajectories simulated from the closed-loop system using the optimal control solutions obtained by the HJB approach (dotted red) and the proposed data-driven convex approach (solid black).  \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{E7L2_1.eps}\n\\caption{$\\mathcal{L}_2$ OCP results of system in Example 1, showing converging state trajectories for $\\gamma=0$.}\n\\label{E7L2_1}\n\\end{figure}\n\n\\noindent{\\textbf{Example 2.}} Consider the dynamics of controlled Van der Pol oscillator as follows:\n\\begin{equation*}\n\\dot{x}_1 = x_2,\\,\\, \\dot{x}_2 = (1-x_1^2)x_2 - x_1 + u.\n\\end{equation*}\nIn this example, we solve the ${\\cal L}_2$ OCP for different values of the discount factor. Total $2 \\times 10^4$ time-series data points are collected from repeated simulations to estimate the Koopman generator. Fig.\\ref{E2L2_1} and~\\ref{E2L2_6} show the trajectories of the closed-loop system starting from arbitrary initial points obtained from discount factor values, $\\gamma=0$ and $\\gamma=1$, respectively. We notice that the controller becomes more aggressive for larger $\\gamma$  and trajectories converge to the origin faster. This is expected as the OCPs achieve optimal control solutions at an exponential rate for $\\gamma > 0$ for which the closed-loop system converges faster than uniform stability.\nOn the other hand, we observe that, for negative discount factor $\\gamma < 0$, the control solution is not guaranteed to stabilize the system to the origin, and the closed-loop dynamics converge to a limit cycle as shown in Fig.~\\eqref{E2L2_6} resulting from $\\gamma = -5$. This is again expected as the cost function is decreasing exponentially, and even without the stabilizing feedback controller, the optimal cost function is finite.\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{tract_0.eps}\n\\caption{Trajectories of closed-loop system with $\\mathcal{L}_2$ OCP solution for $\\gamma=0$ for Van der pol Oscillator.}\n\\label{E2L2_1}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{tract_1.eps}\n\\caption{Trajectories of closed-loop system with $\\mathcal{L}_2$ OCP solution for $\\gamma=1$ for Van der pol Oscillator.}\n\\label{E2L2_2}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{tract_m_5.eps}\n\\caption{Trajectories of closed-loop system with $\\mathcal{L}_2$ OCP solution for $\\gamma=-5$ for Van der pol Oscillator.}\n\\label{E2L2_6}\n\\end{figure}\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\\noindent\\textbf{Example 3:} Consider the dynamics of controlled simple inverted pendulum involing nonpolynomial dynamics:\n\\begin{equation*}\n\\dot x_1=x_2,\\;\\;\\dot x_2=-\\sin x_1 - 0.2 x_2 + u\n\\end{equation*}\nThe number of data points used in the estimation of the Koopman operator equals $2 \\times 10^4$. \nThe simulation results for the ${\\cal L}_2$ optimal control for different discount factor $\\gamma$ are shown in Fig.\\ref{E5L2_1} - Fig.\\ref{E5L2_5}. Similar to Example 2, we notice that, for zero and positive discount factor $\\gamma$, the controller obtained by positive discount factor can stabilize the origin at a faster rate than the case of $\\gamma = 0$ whereas, for negative discount factor, the origin is not stabilized for the closed-loop system.\n\n\n\n\n\n\\begin{comment}\n\\begin{figure} [ht]\n  \\centering\n  \\includegraphics[scale=0.17]{Lorentz_L2_2D.eps}\n  \\caption{Trajectories in states vs. time of Lorentz attractor simulated from open-loop\nas well as optimal control with $\\mathcal{L}_2$ control norm, starting from some disturbed\ninitial points converge to the origin while open-loop dynamics\nshows chaotic behavior.}\n  \\label{Lorentz2}\n\\end{figure}\n\\begin{figure} [ht]\n  \\centering\n  \\includegraphics[scale=0.17]{lorentzL2_trajectory.eps}\n  \\caption{Dynamics of Lorentz attractor, for both closed-loop system using optimal control with $\\mathcal{L}_2$ norm for control input (blue) converging to the origin (denoted by black dot) as well as the open-loop dynamics (red), starting from initial points (denoted by $\\times$).}\n  \\label{lorentzL2_trajectory}\n\\end{figure}\n\\end{comment}\n\n\\noindent\\textbf{Example 4:} Consider the  controlled Lorentz attractor:\n\\begin{equation*}\n\\dot x_1=\\sigma(x_2-x_1),\\dot x_2=x_1(\\rho-x_3) - x_2+u,\\dot x_3=x_1x_2 - \\eta x_3,\n\\end{equation*}\nwhere $\\sigma=10$, $\\rho = 28$, and $\\eta = \\frac{8}{3}$. The open-loop dynamics of the Lorentz system with the above parameter values are chaotic. For this example, we provide simulation results with ${\\cal L}_1$ optimal control with $\\gamma=0$. We notice that the optimal control can stabilize the system.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{pend_tract_0.eps}\n\\caption{$\\mathcal{L}_2$ OCP results of simple inverted pendulum, showing converging state trajectories for $\\gamma=0$.}\n\\label{E5L2_1}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{pend_tract_2.eps}\n\\caption{$\\mathcal{L}_2$ OCP results of simple inverted pendulum, showing converging state trajectories for $\\gamma=2$.}\n\\label{E5L2_2}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{pend_tract_m_5.eps}\n\\caption{$\\mathcal{L}_2$ OCP results of simple inverted pendulum, showing converging state trajectories for $\\gamma=-5$.}\n\\label{E5L2_5}\n\\end{figure}\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=1\\linewidth]{Lor_tract1_0.eps}\n\\caption{$\\mathcal{L}_1$ OCP results of Lorentz attractor, showing converging state trajectories for $\\gamma=0$.}\n\\label{E4L1_1}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nA systematic convex optimization-based framework is provided for optimal control of nonlinear systems with a discounted cost function. In contrast to the existing literature of OCP with discounted cost, we consider the OCP problem with both positive and negative discount factor and provide a condition for the existence of optimal control. The OCP is formulated in the dual space of density function as an infinite-dimensional convex optimization problem. A new data-driven algorithm is provided for the computation of optimal control combining methods from Sum-of-Squares optimization and data-driven approximation of linear transfer operators. Simulation results are presented to verify the developed framework on data-driven optimal control. \n\n\n\n\n\n\\section*{Appendix}\n\n\n\\textbf{ Proof of Theorem \\ref{theorem_maingeometric}}\\\\\nConsider the feedback control system\n\\[\\dot {\\mathbf x}={\\bf f}({\\mathbf x})+{\\mathbf g}({\\mathbf x}){\\mathbf k}({\\mathbf x})\\]\nand let ${\\mathbb P}^c_t$ and ${\\mathbb U}^c_t$ be the P-F and Koopman operator for the feedback control system. Using the definition of the Koopman operator, the cost in (\\ref{ocp_main_discounted}) can be written as \n\\[\nJ(\\mu_0)=\\int_{{\\mathbf X}_1}\\int_0^\\infty e^{\\gamma t}[{\\mathbb U}_t^c (q+{\\mathbf k}^\\top {\\mathbf R}{\\mathbf k})]({\\mathbf x})dt h_0({\\mathbf x})d{\\mathbf x}.\n\\]\nUsing the duality and linearity of the Koopman and P-F operators, we obtain\n\\[\nJ(\\mu_0)=\\int_{{\\mathbf X}_1}\\int_0^\\infty (q+{\\mathbf k}^\\top {\\mathbf R}{\\mathbf k})({\\mathbf x})e^{\\gamma t}[{\\mathbb P}_t^c h_0]({\\mathbf x})dt d{\\mathbf x}.\n\\]\nDefining \n\\begin{equation}\\rho({\\mathbf x}):=\\int_0^\\infty e^{\\gamma t}[{\\mathbb P}_t^c h_0]({\\mathbf x}) dt, \\label{definingrho}\n\\end{equation} \nthe $J(\\mu_0)$ can be written as \n\\begin{equation}\nJ(\\mu_0)=\\int_{{\\mathbf X}_1} (q({\\mathbf x})+{\\mathbf k}({\\mathbf x})^\\top {\\mathbf R}{\\mathbf k}({\\mathbf x})) \\rho({\\mathbf x})d{\\mathbf x}.\\label{ocp_costdiscountproof}\n\\end{equation}\nDefining $\\bar {\\boldsymbol \\rho}({\\mathbf x}):={\\mathbf k}({\\mathbf x})\\rho({\\mathbf x})$, the cost function can be written in the form given in \\eqref{eqn_ocpdiscount_L2}.\nWe next show that $\\rho({\\mathbf x})$ and $\\bar {\\boldsymbol \\rho}$ satisfies the constraints in \\eqref{eqn_ocpdiscount_L2}. Following Assumptions \\ref{assumption_onocp}, we know that the state cost $q$ is uniformly bounded away from zero in ${\\mathbf X}_1$ and optimal cost function is finite and hence we have\n\\begin{equation}\\infty>J(\\mu_0)\\geq \\int_{{\\mathbf X}_1}q({\\mathbf x})\\rho({\\mathbf x})d{\\mathbf x}\\geq c\\int_{{\\mathbf X}_1}\\rho({\\mathbf x})d{\\mathbf x}\\label{ss}\n\\end{equation}\nwhere $c$ is the lower bound for the state cost function $q({\\mathbf x})$ on ${\\mathbf X}_1$. The above proves that there exists a constant $M$ such that \n\\begin{eqnarray}\n\\int_{{\\mathbf X}_1}\\rho({\\mathbf x})d{\\mathbf x}=\\int_{{\\mathbf X}_1}\\int_0^\\infty e^{\\gamma t}[{\\mathbb P}_t^c h_0]({\\mathbf x})dt d{\\mathbf x}\\leq M.\\label{before_claim}\n\\end{eqnarray}\nWe next claim that\n\\begin{eqnarray}\n\\lim_{t\\to \\infty}e^{\\gamma t}[{\\mathbb P}_t h]({\\mathbf x})=0\\label{claim}\n\\end{eqnarray}\nfor $\\mu_0$ almost all ${\\mathbf x}\\in {\\mathbf X}_1$. Using Barbalat Lemma, we know that for $f(t)\\in {\\cal C}^1$, and $\\lim_{t\\to \\infty} f(t)=\\alpha$. If $f'(t)$ is uniformly continuous, then $\\lim_{t\\to \\infty} f'(t)=0$.\\\\\nLetting $f(t)=\\int_0^t e^{\\gamma \\tau}\\int_{{\\mathbf X}_1}[{\\mathbb P}^c_\\tau h_0]({\\mathbf x})d{\\mathbf x} d\\tau$ and hence $f'(t)=e^{\\gamma t} \\int_{{\\mathbf X}_1}[{\\mathbb P}_t h_0]({\\mathbf x})d{\\mathbf x}$. By Barbalat Lemma, since $e^{\\gamma t}\\int_{{\\mathbf X}_1}[{\\mathbb P}^c_t h_0]({\\mathbf x})d{\\mathbf x}$  is uniformly continuous w.r.t. time for  $\\gamma\\leq 0$. The uniformly continuity follows from the definition of the P-F semi-group and the fact that solution of dynamical system is uniformly continuous w.r.t. time. We have \n\\[0=\\lim_{t\\to \\infty}e^{\\gamma t}\\int_{{\\mathbf X}_1}[{\\mathbb P}^c_t h_0]({\\mathbf x})d{\\mathbf x}=\\lim_{t\\to \\infty}\\int_{{\\mathbf X}_1}e^{\\gamma t}[{\\mathbb P}_t^c h_0]({\\mathbf x})d{\\mathbf x}\\]\nwhich implies \n\\begin{equation}\\lim_{t\\to \\infty}e^{\\gamma t}[{\\mathbb P}_t^c h_0]({\\mathbf x})=0\n\\label{convergence}\n\\end{equation}\nfor a.e. ${\\mathbf x}$ w.r.t. $\\mu_0$ in ${\\mathbf X}_1$.\nNext, we claim that $\\rho({\\mathbf x})$ as defined in (\\ref{definingrho}) can be obtained as a solution of the following equation\n\\begin{equation}\n\\nabla\\cdot (({\\bf f}({\\mathbf x})+{\\mathbf g}({\\mathbf x}){\\mathbf k}({\\mathbf x}))\\rho({\\mathbf x}))=\\gamma \\rho({\\mathbf x})+h({\\mathbf x}),\\label{steady_pdegeometric}\n\\end{equation}\nfor $x\\in {\\mathbf X}_1$.\nSubstituting (\\ref{definingrho}) in (\\ref{steady_pdegeometric}), we obtain\n\\begin{eqnarray}\n\\nabla\\cdot ({\\bf f}_c \\rho)=\\int_0^\\infty \\nabla\\cdot({\\bf f}_c ({\\mathbf x})e^{\\gamma t}[{\\mathbb P}^c_t h_0]({\\mathbf x}))dt\\nonumber\\\\\n=\\int_0^\\infty -e^{\\gamma t}\\frac{d}{dt}[{\\mathbb P}^c_t h_0]({\\mathbf x})dt\\nonumber\\\\\n-e^{\\gamma t} [{\\mathbb P}_t^c h_0]({\\mathbf x})|_{t=0}^\\infty+\\int_0^\\infty \\gamma e^{\\gamma t} [{\\mathbb P}_t^c h_0]({\\mathbf x}) dt\\nonumber\\\\\n=h_0({\\mathbf x})+\\gamma \\rho({\\mathbf x})\\label{eq22}\n\\end{eqnarray}\nIn deriving (\\ref{eq22}) we have used the infinitesimal generator property of P-F operator Eq. (\\ref{PF_generator}) and the fact that $\\lim_{t\\to \\infty} e^{\\gamma t}[\\mathbb{P}_t^c h_0]({\\mathbf x})=0$  following (\\ref{convergence}).\nFurthermore, since $h_0({\\mathbf x})> 0$, it follows that $\\rho({\\mathbf x})> 0$ from the positivity property of the P-F operator. Combining (\\ref{ocp_costdiscountproof}) and (\\ref{eq22}) along with the definition of $\\bar \\rho$, it follows that the OCP problem can be written as  convex optimization problem (\\ref{eqn_ocpdiscount}). The optimal control ${\\mathbf k}^\\star({\\mathbf x})$ obtained as the solution of optimization problem (\\ref{eqn_ocpdiscount}) is  a.e. uniform stable (for $\\gamma=0$) follows from the results of Theorem \\ref{theorem_necc_suffuniform} using the fact that closed loop system satisfies (\\ref{steady_pdegeometric}) with $\\rho$ that is integrable. The optimal solution  $\\rho^\\star({\\mathbf x})\\in {\\cal L}_1({\\mathbf X}_1)\\cap {\\cal C}^1({\\mathbf X}_1,{\\mathbb R}_{\\geq 0})$ follows from the fact that $h_0\\in {\\cal L}_1({\\mathbb R}^n,{\\mathbb R}_{> 0})\\cap {\\cal C}^1({\\mathbb R}^n)$  and the definitions of $\\rho$ (\\ref{definingrho}) and P-F operator (\\ref{pf-operator}). \\\\\n\n\n\\textbf{ Proof of Theorem \\ref{theorem_maingeometric_positivediscount}} The proof of this theorem follows exactly along the lines of proof of Theorem \\ref{theorem_maingeometric} until equation (\\ref{before_claim}). Unlike the proof of Theorem \\ref{theorem_maingeometric}, where the claim (\\ref{claim}) is proved using Barbalat Lemma, in this proof $\\lim_{t\\to \\infty}[{\\mathbb P}_t h_0]({\\mathbf x})=0$ for $\\mu_0$ almost all ${\\mathbf x} \\in {\\mathbf X}_1$ follows from Assumption \\ref{assumption_ocppositivediscount}. As $\\mu_0(B_t)\\leq M e^{-\\gamma' t}$ implies $e^{\\gamma t}[{\\mathbb P}_t h_0]({\\mathbf x})\\to 0$ for $\\mu_0$ a.e. ${\\mathbf x}\\in {\\mathbf X}_1$. The rest of the proof then follows again along the lines of proof of Theorem \\ref{theorem_maingeometric}. \n\n\n\n\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSecure communication based on message encryption with controlled noise (pseudo-noise or PN)\nstarted with the work of the actress-engineer Hedy Lamarr and her husband-pianist \nGeorges Antheil in 1941 who were interested in military communications during World War II.\n\nLamarr invented frequency hopping to prevent an intruder from jamming a signal sent to \ncontrol torpedoes remotely, since using a single frequency might be easily detected and blocked.\nFrequency hopping is used presently in Bluetooth and other types of wireless communication \nand is called FHSS~\\cite{Carlson} (Frequency Hopping Spread Spectrum). \n\nGiven a set of frequency values $[f_1, f_2, f_3...]$, one selects a well-defined \nsequence of frequencies following an apparently random pattern (picked from PN values) \nshared solely between transmit and receive ends. \nFor an eavesdropper unaware of the sequence used, the signal appears as white noise containing no\nvaluable information and that is the reason why it is termed spread-spectrum communication\ngiven that noise has broader bandwidth than the signal.\n\nFHSS needs another important ingredient to be completely operational: sender-receiver \nperfect synchronization in order to be able to modulate-demodulate with the right frequency. \nIt is Antheil, exploiting his musician skills, who developed the \nsynchronization~\\cite{Carlson} method between sender and \nreceiver enabling them to encrypt-decrypt ongoing transmitted information.\n\nThe analog to digital conversion of frequency hopping gave birth to DSSS\n(Direct Sequence Spread Spectrum) methods based on Galois polynomials, the generators of\nPN sequences that we describe below and that are used presently in Wi-Fi and other \ntypes of digital communications (Smartphones...).\n\n\nConsequently, it is important to relate noise to communications, how it might \nbe used to alter the nature of the signal and ultimately transmit hidden information\nin a way such that it is  properly retrieved by the target receiver.\n\nAn ordinary resistor has a fluctuating\nvoltage across it whether standing free or belonging to an electronic \ncircuit. The resistor embodies free electrons that are thermally agitated, \ninducing random voltage fluctuations. \n\nMost physicists\/engineers refer to this thermal voltage fluctuation as \nJohnson-Nyquist noise, after J.B. Johnson~\\cite{Nyquist}, who was first \nto observe the effect at Bell laboratories in 1928 and H. Nyquist~\\cite{Nyquist} \nwho first explained it. \n\nCircuit noise studies in Bell laboratories have a very peculiar history since\nin the 1950's, H. E. D. Scovil and his associates  built the world's lowest-noise\nmicrowave amplifiers cooled by liquid helium to reduce noise  \nand incorporated in extremely sensitive radiometers used in radio-astronomy.\n\nRadiometers usually contain calibration noise sources consisting of a resistor \nat a known temperature. During the 1960's Penzias and Wilson~\\cite{Penzias} \nwhile improving these radiometers discovered  serendipitously Big-Bang cosmic \nbackground radiation in 1965.\n\n\nB. Yurke~\\cite{Yurke} and his collaborators embarked, in the 1980's, \non a pioneering study of Quantum noise through the quantization of $LC$ networks \ndrawing from an analogy between an $LC$ circuit and the harmonic oscillator. \nQuantum effects in circuits occur when we deal with low temperature \n(as in superconductors) or at very high frequency. Usual telecommunication and \nsignal processing frequencies are in the kHz-GHz range whereas \nTera-Hz ($10^{12}$ Hz) devices encountered in medical imaging and \noptical devices operate at $10^{14}$ Hz. Consequently, kHz-GHz frequencies\nare classical whereas Tera-Hz and optical devices should be considered as quantum.\nWith the progress of integrated circuits toward the nanometer scale (presently \nthe minimal feature used in the semiconductor industry is 14 nm) and \nsingle electron as well as quantum dot (akin to synthetic atom) devices, \nwe expect large quantum effects implying quantum noise becoming more \nimportant than thermal. \n\n\nThe equivalence between an impedance\nand an oscillator is a very important idea that will trigger and sustain steady\nprogress in several areas of Quantum information and communication.\n\nNyquist derived an expression for White Noise based on the interaction between \nelectrons and electromagnetic waves propagating along a transmission line using\narguments based on black-body radiation. This means that Nyquist is in fact\na true pioneer in Quantum noise. \n\nHe based his work on Johnson measurements who \nfound that thermal agitation of electricity in conductors produces a random \nvoltage variation between the ends of the conductor $R$ of the form:\n\n\\begin{equation}\n\\langle(V - \\langle V \\rangle)^2 \\rangle= \\mean{\\delta V^2}= 4R k_B T \\Delta f\n\\end{equation}\n\n$\\langle ...\\rangle$ is the average value, voltage\nfluctuation is $\\delta V = V-\\mean{V}$ and $V$ is the instantaneous voltage measured at\nthe ends of the resistance $R$. \n$k_B$ is Boltzmann constant and $T$ is absolute temperature. $ \\Delta f$ is the\nbandwidth of voltage fluctuations (see Appendix A).\nThis frequency interval spans the range of a few Hz to several tens of GHz.\n\nThe voltage fluctuation developed across the\nends of the conductor due to Thermal noise is unaffected by the presence or \nabsence of direct current. This can be explained by the fact that electron thermal \nvelocities in a conductor are much greater ($\\sim 10^3$ times) than electron drift velocities. \n\nSince electromagnetic waves are equivalent to photons through Quantum Mechanics \nDuality principle~\\cite{Duality}, Nyquist derivation is based on blackbody radiation that was\nexplained earlier by Planck.\n\nIn Quantum Mechanics language, a (zero rest mass) photon is a special case of \na harmonic oscillator since the energy levels are separated by the same energy $\\hbar \\omega$ \ni.e. the $n$-th level $E_n=n\\hbar \\omega$ (ignoring zero-point energy $\\hbar \\omega\/2$) \ncorresponds to an integer number $n$ of photons. \nMoreover $\\omega=2\\pi f$ is the electromagnetic pulsation\nand not the mechanical one $\\sqrt{k\/m}$ where $k,m$ are the respective \nspring constant and  mass of the mechanical oscillator.\n\n\n\nWhile classical pseudo-noise used in spread-spectrum communications hides the signal\nfrom intrusion by an eavesdropper through a crypting operation (FHSS frequencies follow\na PN sequence whereas in DSSS, the signal is directly multiplied by the PN sequence) using\na set of keys (corresponding to a given PN sequence) that are shared solely between \nthe transmitter and the receiver, Quantum mechanics can be used to encrypt the \nsignal in a completely different fashion.\n\n\nQuantum Mechanics can be used to generate naturally random instead of \ndeterministic pseudo-random numbers. In the early days of computing\ncosmic rays or radioactive sources~\\cite{Schmidt} were used for generating \nnon-deterministic random numbers.\nQuantum phenomena being essentially non-deterministic, \nwould be able to produce truly \nrandom numbers and the corresponding devices are called Quantum Random Number\nGenerators~\\cite{Schmidt} (QRNG).  \n\nObviously, this is not the only advantage of Quantum Mechanics since\nat the Garching Max Planck Institute for Quantum Optics (MPQ) in Germany \nand the Technical University of Vienna, \ncommunication experiments showed that Quantum Mechanics provides entanglement \nas an alternative concept to secure information transfer between two remote sites. \n\nEntanglement, first introduced by Einstein (who called it \"spooky \naction at a distance\"), Podolsky, and Rosen~\\cite{EPR}, and \nSchr\\\"odinger~\\cite{Schrodinger} in 1935, can arise when two \nquantum systems are produced from \na common source, e.g. when a spinless particle decays into two particles\ncarrying opposite spins. Such states violate a set of \ninequalities~\\cite{Bell} established by J.S. Bell in 1964, implying \nthat quantum theory embodies non-locality (see section IV for the mathematical\nimplication). Bell inequalities are the statistical measure of entanglement\nand their violation can be demonstrated by measuring correlations between\nquantum states.\n\nEntangled quantum systems behave as if they can \naffect each other instantaneously, even when they are extremely far from \neach other, due to the essential non-local~\\cite{Buscemi} character of \nentanglement. \n\nThe strongest advantage of noise-based communication is that by hiding a\nsignal in noise, it is extremely difficult or even impossible to detect it\nif the eavesdropper does not know the keys or the algorithm used between \nthe transmitter and the receiver. In Quantum communication (QC), entanglement\nties together in a very stringent fashion both parties and any intrusion\nattempted by an eavesdropper, when detected, triggers immediately disruption \nof communication.\n\nThis work can be taught as an application chapter \nin a general Statistical Physics course at the Graduate  \nor in a specialized Graduate course \nrelated to applications of Quantum Mechanics and Statistical Physics\nsince physicists generally interested in the applications of\nQuantum Mechanics and Statistical Physics are keen to expand their \nknowledge to areas of Quantum Information Processing and Communications (QIPC).\n\nThis paper is organized as follows: after reviewing several derivations\nof White noise by Nyquist and others in section II, we discuss in section III\nthe Fluctuation Dissipation theorem and its quantum version in order \nto derive in a rigorous way, Nyquist result with modern quantum noise approach\nand lay the foundations of secure communication from the classical and quantum\npoints of view. In section IV we apply the analysis to secure communications\nwith classical noise (spread-spectrum) and entanglement based Quantum information \nprocessing and transfer. Discussions and Conclusions are in section V.\n\n\\section{Derivations of Thermal Noise}\n\nNyquist work is based on phenomenological thermodynamic considerations and electric\ncircuit theory, including the classical equipartition theorem.\nThe latter is based on the physical system number of degrees of freedom.\nThis number~\\cite{freedom} is well defined when the different contributions to system energy \n(translational, rotational, vibrational, electromagnetic...) \nare quadratic and decoupled with presence of weak interactions.\nIn the general case (non-quadratic energy or strong interactions), \none evaluates the partition function in order to derive thermodynamical properties.\n\n\\subsection{Nyquist derivation of Thermal Noise}\nNyquist based his derivation on Einstein remark that many physical systems would \nexhibit Brownian motion and that Thermal noise in circuits is nothing more \nthan Brownian motion of electrons due to ambient temperature.\nDespite the fact one might find several strange assumptions and even flaws in \nNyquist derivation, it remains a pioneering interesting approach since it paves \nthe way to quantization of electrical circuits and noise in circuits.\nNyquist considered thermal noise in a resistor $R$ as stemming from \nelectrons interacting with electromagnetic waves represented by \na one-dimensional black-body thermal radiator. Electromagnetic waves travel \nthrough an ideal (lossless) one-dimensional\ntransmission line of length $\\ell$ joining two resistances $R$.\nHence the transmission line characteristic impedance being equal to $R$ amounts \nto considering that its impedance is matched at both ends and \nthat any voltage wave propagating along the line is completely absorbed by the \nend resistor $R$ without any reflection, exactly like a black-body. \n\nEach resistor $R$ has a thermally-fluctuating voltage at\ntemperature $T$ which will be transmitted down the wires \nwith a current and voltage wave appearing across the other resistor.  \n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[angle=0,width=50mm,clip=]{nyquist.pdf}  \n\\vspace*{-3mm} \n\\caption{Transmission line  of length $\\ell$ matched by two resistors $R$ at its ends.}\n    \\label{fig1}\n\\end{figure} \n\nA voltage wave propagating along the transmission line is expressed at any point $x$\nand any time $t$ as:\n$V(x,t) = V_0 \\exp[i(\\kappa x - \\omega t)]$  \nwhere $\\kappa$ is the wavenumber and $\\omega$ the angular frequency of the wave.\n\n\nThe  velocity $v = \\omega \/\\kappa$ in the line is typically $c\/10$ where \n$c$ is light velocity in vacuum.\nConsidering the transmission line of length $\\ell$ as a domain between $x=0$ and $x = \\ell$ and \nimposing the boundary condition $V (\\ell,t) = V (0,t), \\hspace{0.5cm} \\forall t$,\nwe infer that possible propagating wavenumbers are \ngiven by $\\kappa \\ell = 2n\\pi $, where $n$ is any integer, \nand there are $\\Delta n = (1\/2\\pi)d\\kappa=d\\omega\/(2\\pi v)$  such modes per unit \nlength of the line in the frequency range between $\\omega $ and $\\omega  + d\\omega $. \n\n\nIn the Canonical ensemble, the Bose-Einstein distribution for photons gives the mean number\nof photons  $\\langle n \\rangle$  per mode at energy $\\hbar \\omega$ and temperature $T$ as:\n\\begin{equation}\n\\langle n \\rangle= \\frac{1}{e^{\\beta \\hbar \\omega} -1}\n\\label{Bose}\n\\end{equation}\n\n$\\beta = \\frac{1}{k_B T}$ is inverse temperature~\\cite{Reif} coefficient. \nThus the energy of the photon gas (ignoring zero-point energy) \nis $\\langle n \\rangle \\hbar \\omega$.\n\n\nDetailed balance allows to equate the power absorbed by a resistor\n(in any angular frequency range $\\omega$  and $\\omega  + d\\omega$) \nto the power emitted by it. The energy in the interval $\\omega$  and $\\omega  + d\\omega$\nis proportional (see Appendix C) to the number of  propagating modes per \nunit length in this frequency range. The mean energy per unit time incident\nupon a resistor in this frequency range is:\n\n\\begin{equation}\nP_{in}=v \\left(\\frac{d\\omega}{2\\pi v}\\right) E(\\omega)= \\frac{1}{2  \\pi} E(\\omega) d\\omega \n\\end{equation}   \n\nwhere $E(\\omega)$ is the electromagnetic energy at $\\omega$.\nNyquist considers that the total resistance making the circulating\ncurrent $I$ is $2R$ and thus $I = V \/2R$ as if the line  \nwhose characteristic impedance $R$ did not contribute at all to\nthe total resistance. Perfect matching at both ends implies that no resistance is \ncontributed by the line since current\/voltage waves are not subjected to scattering. \nThe mean power emitted down the line and absorbed by the resistor at the other end is \n\\begin{equation}\nR\\langle I^2\\rangle= R \\left<\\frac{V^2}{4R^2}\\right>= \\frac{1}{4R} \n\\int\\limits_0^{\\infty} S_V(\\omega) d\\omega\n\\end{equation}\n\nwhere $S_V$ is the voltage Power Spectral Density (PSD) (see Appendix A).\n\nHence we have:\n\n\\begin{equation}\n\\frac{1}{2  \\pi} E(\\omega) d\\omega= \\frac{1}{4R} S_V(\\omega) d\\omega\n\\end{equation}\n\nwith:\n\n\\begin{equation}\nS_V(\\omega) = \\frac{2R}{\\pi} \\frac{\\hbar \\omega}{e^{\\beta \\hbar \\omega} -1}\n\\end{equation}\n\n\nMoving from angular to linear frequency, we get: \n\n\\begin{equation}\nS_V(f) = 4R\\frac{hf}{e^{\\beta hf} -1}\n\\label{frequency_dependent}\n\\end{equation}\n\nVoltage fluctuations are given by (see Appendix A):\n\n\\begin{equation}\n\\sigma_V^2=\\mean{\\delta V^2}=\\int\\limits_{0}^{\\infty}4R\\frac{hf}{e^{\\beta hf} -1} df\n\\end{equation}\n\nPerforming a change of variable $x=\\beta hf$, we get:\n\n\\begin{equation}\n\\sigma_V^2=\\frac{4R (k_B T)^2}{h} \\int\\limits_{0}^{\\infty}\\frac{x}{e^{x} -1} dx\n\\label{integral}\n\\end{equation}\n\nUsing the integral~\\cite{Landau,Gradstein}:\n\n\\begin{equation}\n\\int\\limits_{0}^{\\infty}\\frac{x^{2n-1}}{e^{x} -1} dx= \\frac{(2\\pi)^{2n}B_n}{4n}\n\\end{equation}\n\nwith $B_n$ the Bernoulli polynomial coefficients, we select $n=1$ and $B_1=\\frac{1}{6}$.\n\nThe value of the integral in eq.~\\ref{integral} is thus $\\frac{\\pi^2}{6}$. \n\nThe fluctuations are then given by:\n\n\\begin{equation}\n\\sigma_V^2=\\frac{2R (\\pi k_B T)^2}{3h}\n\\end{equation}\n\nThis surprising result implies that, in the classical case, \n($ h \\rightarrow 0$), fluctuations become extremely large. In fact, \nordinary frequencies $f \\sim $kHz-GHz are low with respect to \n6.25 $\\times 10^{12}$ Hz frequencies that correspond to \nthermal room temperature energy $k_B T$. \nThus $hf \\ll k_B T$ and the number of photons per mode is large since \n$\\langle n \\rangle \\approx k_B T\/hf$.  In the classical \nlimit, the number of photons being very large, we get wave-like behaviour\nwhereas in the quantum limit a small $\\langle n \\rangle$ produces particle-like \n(photon) behaviour. \n\n\nExpanding the PSD eq.\\ref{frequency_dependent} at low frequency $hf \\ll k_B T$:\n\n\\begin{equation}\nS_V(f) \\approx 4R k_B T (1 -\\frac{hf}{2 k_B T})\n\\label{low_frequency}\n\\end{equation}\n\nThus quantum effects no longer intervene in the low frequency limit \n$ f \\rightarrow 0$, yielding Nyquist result: \n\n\\begin{equation}\nS_V(0) = 4R k_B T\n\\label{frequency_independent}\n\\end{equation}\n\nAnother divergence is encountered when we ignore the frequency dependence of \n$S_V(f)$ in eq.~\\ref{frequency_dependent} and consider\n$S_V(0)$ to be valid for all frequencies as usually considered for \"White Noise\"\n(flat spectrum for all frequencies):\n\n\\begin{equation}\n\\sigma_V^2= \\int\\limits_{0}^{\\infty} 4R k_B T df= 4R k_B T \\int\\limits_{0}^{\\infty} df= \\infty\n\\end{equation}\n\n\nThis divergence is similar to the Ultra-Violet catastrophe encountered in black-body\nradiation since $hf \\ll k_B T$ corresponds to Rayleigh-Jeans regime and its \nsolution is that voltages are filtered and we never encounter \nin practice an infinite frequency domain. \n\nTherefore let us assume we have a finite bandwidth $\\Delta f$ for voltage \nfluctuations, then:\n\n\\begin{equation}\n\\sigma_V^2= \\int\\limits_{0}^{\\Delta f} 4R k_B T df= 4R k_B T \\Delta f\n\\end{equation}\n\nTo sum up, in order to recover the Johnson-Nyquist result we have to respect two\nconditions: finite band $\\Delta f < \\infty$ and low frequencies (kHz-GHz range) \n$\\Delta f \\ll k_B T\/h$.\n\nAdditionally, it is surprising to note that Nyquist considered a 1D photon gas \nwith a single polarization\ndespite the fact the photon had two polarizations (circular left and right) and in sharp \ncontrast with the evaluation of the blackbody radiation by Planck who considered \na 3D gas with two polarizations (see Appendix C). Moreover, Nyquist ignored zero-point \nenergy in spite of its importance in quantum circuits \nand the fact Planck introduced it in his second paper on black-body radiation \n(see further below).\n\n\n\\subsection{RC circuit classical derivation of Thermal Noise}\n\nNyquist's theorem can be proven with the help of a parallel $RC$ circuit containing\na random source representing interactions with a thermal reservoir.\nThe resistor $R$ is parallel to the capacitor $C$ and the result of \nrandom thermal agitation of the electrons in the resistor will charge\nand discharge  the capacitor in a random fashion. \n\nStarting from the time dependent equation of motion of the $RC$ circuit, we have:\n\n\\begin{equation}\nR\\frac{dq(t)}{dt}= -\\frac{q(t)}{C} + \\xi(t)\n\\label{langevin}\n\\end{equation}\n\n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[angle=0,width=40mm,clip=]{RC.pdf}  \n\\caption{RC circuit with the noise source $\\xi(t)$ originating from thermal contact with a reservoir.}\n\\vspace*{-3mm} \n    \\label{RC}\n\\end{figure} \n\n$q(t)$ is the capacitor charge and $\\xi(t)$ is a stochastic voltage \n(see Appendix A)  stemming from interactions with\na reservoir at temperature $T$ with the following statistical properties: \n\n\\begin{equation}\n\\mean{\\xi(t)}=0 \\mbox{  and  } \\mean{\\xi(t)\\xi(t')} = \\lambda \\delta(t-t') \n\\label{white2}\n\\end{equation}\n\n$\\lambda$ is a constant that will be determined later and \neq.~\\ref{langevin} is called a Langevin  equation~\\cite{Gardiner} (see Appendix B) due to the\npresence of the time-dependent random term $\\xi(t)$.\n\nAssuming the capacitor is uncharged at time $t=-\\infty$, direct integration of the \nfirst-order differential equation yields:\n\n\\begin{equation}\nq(t)= \\frac{1}{R} \\exp(-t\/RC) \\int\\limits_{-\\infty}^{t}   \\exp(t'\/RC) \\xi(t') dt'\n\\label{langevin2}\n\\end{equation}\n\nThe voltage $V(t)$ across the capacitor is related to charge through: $q(t)=C V(t)$, \ntherefore evaluating the charge PSD (see Appendix A) is equivalent to voltage PSD.\n\nUsing properties of $\\xi(t)$ given in eq.~\\ref{white2}, we obtain:\n\n\\begin{equation}\n\\mean{q(t)q(t')}= C^2 \\mean{V(t) V(t')}= \\frac{\\lambda C}{R} \\exp \\left(-\\frac{|t-t'|}{RC}\\right)\n\\label{spectrum}\n\\end{equation}\n\nThis result is expected as discussed in Appendix A since the auto-correlation\n$\\mean{q(t)q(t')}$ must be a decreasing function of the argument $|t-t'|$\ncontrolled by the relaxation time $RC$. \n\nSetting $t=t'$ we have the equality: $C^2 \\mean{V^2(t)}= \\frac{\\lambda C}{R}$, hence\n$\\lambda$ is determined as: $\\lambda= RC  \\mean{V^2(t)}$.\n\nThe average $\\mean{V^2(t)}$ can be determined from the energy  \n$\\frac{1}{2}C \\mean{V^2}$  stored in the capacitor\nthrough the classical equipartition theorem: \n$\\frac{1}{2}C \\mean{V^2}=\\frac{1}{2} k_B T$ as if a capacitor is equivalent\nto a single degree of freedom (see next section).\n\nThe equipartition theorem can be proven as follows.\nIf a system is at temperature $T$, the probability that it is in a state of energy $E$ \nis proportional to the Boltzmann factor $\\exp(-E\/k_B T )$.\n\n\nIn the $RC$ circuit the probability element $dp$ of finding\na voltage between $V$ and $(V + dV)$ is $dp= A \\exp(-E\/k_B T ) dV$ corresponding \nto an energy $E=\\frac{1}{2}C V^2$ stored in the capacitor $C$. \n\n\nThe prefactor $A$ normalizes the probability density:\n\\begin{equation}\n\\int\\limits_{-\\infty}^{\\infty} A  e^{(-C V^2\/2 k_B T )} dV =1\n\\end{equation}\nUsing the result $\\int\\limits_{-\\infty}^{\\infty} e^{-x^2} dx= \\sqrt{\\pi}$ we get \n$A=\\sqrt{\\frac{C}{2 \\pi k_B T}}$.\nThe mean square value of the voltage is obtained from the probability density \nas:\n\\begin{equation}\n\\mean{V^2}= A \\int\\limits_{-\\infty}^{\\infty}  V^2 e^{(-C V^2\/2 k_B T )} dV\n\\end{equation}\n\nThus $\\mean{V^2}= \\frac{k_B T}{C}$.\n\n\nThe voltage fluctuation PSD may be evaluated with the equipartition theorem as:\n\n\\begin{equation}\nS_V (\\omega)= \\mean{V^2} \\frac{2RC}{1+(RC \\omega)^2}= \\frac{2R k_B T}{1+(RC \\omega)^2}\n\\label{spectrum2}\n\\end{equation}\n\nAt frequencies such that $\\omega \\ll 1\/RC$ we get $S_V (\\omega)= 2R k_B T$\nrecovering the Johnson-Nyquist result (see note~\\cite{factor2}).\n\n\n\\subsection{Derivation of Thermal noise from Einstein thermodynamic fluctuation theory}\nA macroscopic system at thermodynamic equilibrium is an ensemble of subsystems  which are in \nthermodynamic equilibrium with each other. The actual values of the variables, \nhowever, may differ from mean equilibrium values. \n\nThe departure from equilibrium is due to fluctuations in the subsystems.\n\nThe probability, $p$, for entropy fluctuation $\\delta S$ is obtained by reverting \nBoltzmann principle $S=k_B \\ln W$ as $p(\\delta S)=p(S-\\langle S\\rangle) \\propto e^{\\delta S\/k_B}$.\nThe probability $p$ is proportional to $W$ the number of microscopic available states \nand we assume the validity of applying Boltzmann principle to the entropy fluctuation $\\delta S$.\n\nGenerally, entropy is a function of state variables $X_i$, \ni.e. $S = S(X_1, X_2, . . ., X_i, . . .)$. It \ncan be expanded as $dS$ around equilibrium since it is an analytical \nfunction for most thermodynamic systems when\nsmall fluctuations of the $X_i$ are considered.\n\nAt equilibrium $S$ is maximum and all the first order derivatives \n$\\frac{\\partial S}{\\partial X_i}=0, \\forall i$\nimplying that the first non-vanishing terms  are quadratic.\n\n\nThus one has around equilibrium:\n\n\\begin{equation}\n\\delta S \\approx \\frac{1}{2} \\sum_{i,j} \\frac{\\partial^2 S}{\\partial X_i \\partial X_j} \\delta X_i \\delta X_j\n\\end{equation}\n\nThe series expansion is performed at the equilibrium value of $X_i$, hence  \n$\\delta X_i =  X_i -\\langle X_i\\rangle$. \n\nThe combination of Boltzmann principle and second-order expansion of entropy about equilibrium\nresults in a Gaussian probability density function (PDF) for finding subsystems with the non-equilibrium value of the variable $X_i$ (akin to the central limit-theorem),\n\n\\begin{equation}\np(\\delta X_i)= \\frac{1}{\\sqrt{2\\pi \\sigma_i^2}} e^{-\\frac{\\delta X_i^2}{2\\sigma_i^2}}\n\\end{equation}\n\nmeaning that the average value (or macroscopic equilibrium) of $X_i$ is $\\langle X_i\\rangle$\nand that the standard deviation away from equilibrium is \n$\\sigma_i^2=\\mean{\\delta X_i^2}=-k_B\/(\\frac{\\partial^2 S}{\\partial X_i^2})$.\n\n\nThe entropy of a single phase, one component system is given in terms of the energy $U$, volume \n$\\Phi$ and pressure $P$ as $dS=\\frac{dU}{T}+\\frac{P}{T}d\\Phi$.\n\n\nIn the presence of a voltage $V$, the entropy expression becomes \n$dS=\\frac{dU}{T}+\\frac{P}{T}d\\Phi -\\frac{q}{T}dV$ since charge $q$ couples to\nvoltage $V$. This yields the value \nof the entropy derivative $\\left(\\frac{\\partial S}{\\partial V}\\right)_{T,\\Phi}=-\\frac{q}{T}$.\nThe second derivative is thus obtained as:\n$\\left(\\frac{\\partial^2 S}{\\partial V^2}\\right)_{T,\\Phi}= -\\frac{1}{T} \\left(\\frac{\\partial q}{\\partial V}\\right)_{T,\\Phi}$.\n\nThe voltage fluctuation is expressed as:\n\n\\begin{equation}\n\\sigma_V^2=\\mean{\\delta V^2}=-k_B\/\\left(\\frac{\\partial^2 S}{\\partial V^2}\\right)_{T,\\Phi}=k_B T \\left(\\frac{\\partial V}{\\partial q}\\right)_{T,\\Phi}\n\\end{equation} \n\nAssuming the validity of Ohm's law: $I=\\frac{dq}{dt}=\\frac{V-V_0}{R}$ where \n$V_0$ is some reference voltage (e.g. $V_0= \\mean{V}$), \nwe infer that voltage fluctuations are given by:\n\n\n\\begin{equation}\n\\sigma_V^2=k_B T R \\frac{\\left[d(V-V_0)\/dt\\right]}{(V-V_0)}\n\\end{equation}\n\nEstimating the time derivative: $\\frac{d(V-V_0)}{dt} \\sim \\frac{(V-V_0)}{\\tau}$ with\n$\\tau$ as a typical time variation of the voltage and given \nthat for a band-limited signal of bandwidth~\\cite{Carlson} $\\Delta f$ with \n$\\Delta f \\sim \\frac{1}{2 \\tau}$, we finally obtain for\nthe voltage fluctuation expression :\n\n\\begin{equation}\n\\sigma_V^2= 2R k_B T\\Delta f\n\\end{equation}\n\nthat agrees with Johnson-Nyquist result (see note~\\cite{factor2}).\n\n\\section{The quantum fluctuation-dissipation theorem (QFDT)}\n\\label{QFDT}\nThe classical fluctuation-dissipation theorem derived in Appendix B provides a\nrelation between equilibrium fluctuations and dissipative transport\ncoefficients. Besides, it is an interesting route to quantize classical noise.\n\nCallen and Welton~\\cite{Callen} proved the QFDT with the correspondence theorem \nallowing to transpose classical results to quantum ones such that\na classical physical quantity is transformed into its quantum counterpart\nwith an observable operator.\n\nFor a single degree of freedom, linear response theory~\\cite{Hanggi} yields for the\nchange of the expectation value of an operator-valued observable $B$ due to\nthe action of a (classical) force $F(t)$ that couples to the conjugate\ndynamical operator $A$:\n\n\\begin{equation}\n\\langle\\delta B(t)\\rangle = \\int\\limits_{-\\infty}^{t}ds \\chi_{BA}(t-s) F(s)\\,.\n\\end{equation}\n\n$\\delta B(t)=B(t)-\\langle B\\rangle_0$ denotes the difference with respect to\nthe thermal equilibrium average $\\langle B\\rangle_0$ in force absence.\nThe dissipative part of the response function  $\\chi_{BA}(t)$ is given by:\n\n\\begin{equation}\n\\chi_{BA}^D(t)= \\frac{1}{2i}[\\chi_{BA}(t) - \\chi_{AB}(-t)]\\,.\n\\end{equation}\n\nThe fluctuations are described by the equilibrium correlation function\n\\begin{equation}\nC_{BA}(t) = \\langle\\delta B(t)\\delta A(0)\\rangle_\\beta\n\\end{equation}\n\n\nThe thermal average is taken at an inverse temperature $\\beta$ (see Appendix A).\n\nThe correlation function is\ncomplex-valued because the operators $B(t)$ and $A(0)$ in general do not\ncommute. While the antisymmetric part of $C_{BA}(t)$ is directly related \nto the response function by linear response theory, the symmetrized correlation function PSD:\n\n\\begin{equation}\nS_{BA}(t)=\\frac{1}{2}\\langle\\delta B(t)\\delta A(0)+\\delta A(0)\\delta B(t)\\rangle\n\\end{equation}\n\ndepends on the Fourier transform of the dissipative part of the response function:\n\n\\begin{equation}\nS_{BA}(\\omega) = \\hbar\\coth\\left(\\frac{\\beta \\hbar\\omega}{2}\\right) X_{BA}^D(\\omega)\\,.\n\\end{equation}\n\nwhere $X_{BA}^D(\\omega)$ is the Fourier transform of $\\chi_{BA}^D(t)$. \nThis is the quantum version of the fluctuation-dissipation theorem\nas it links the fluctuations $S_{BA}(\\omega)$ to dissipation as in the\nclassical case (Appendix B).\n\nNote that $S_{BA}$ is a two variable extension of the PSD\npreviously used and defined in Appendix A with a single variable.\nConsequently $S_V$ should be written in fact as  $S_{VV}$.\n\nAnalyzing the response of a current $\\delta I$ through an \nelectric circuit subject to a voltage change $\\delta V$, \nimplies $B=I$ and $A=Q$, since voltage couples to charge $Q$.\n\nA circuit response is determined by $\\delta I(\\omega)=Y(\\omega) \\delta V(\\omega)$ \nwhere $Y(\\omega)$ is the admittance. \nGiven $I=\\dot Q$, the symmetrized current PSD is $S_{II}(\\omega)=i\\omega S_{IQ}(\\omega)$ yielding:\n\n\\bea\nS_{II} (\\omega) &= \\hbar\\omega\\coth\\left(\\frac{\\beta \\hbar\\omega}{2}\\right)\n\\Re Y(\\omega)  \\nonumber \\\\\n&=2 (\\mean{n}+\\frac{1}{2})\\hbar\\omega \\Re Y(\\omega)\\,.\n\\eea\n\nwhere $\\mean{n}$ is the Bose-Einstein factor (given by eq.~\\ref{Bose}) and $\\Re$ is real part symbol.\nIn the high temperature limit $k_B T \\gg \\hbar\\omega$, we recover the Johnson-Nyquist result \n$ S_{II}(\\omega)= 2\\Re Y(\\omega) k_B T$ (see note~\\cite{factor2}). \n\nNyquist, in the last paragraph of his 1928 paper~\\cite{Nyquist},\nhad already anticipated the quantum case. However, he made use of\nPlanck first paper on black-body radiation which does not contain \nzero-point energy term $\\frac{1}{2} \\hbar \\omega$. By missing this term, Nyquist\nignored the 1912 second~\\cite{Planck} paper  on black-body radiation by Planck\nwho aimed at correcting his previous work by introducing zero-point energy\nin order to recover the right classical~\\cite{classical} limit of an oscillator \nmean energy per mode. Let us add that zero-point energy can also be shown\nto originate from Heisenberg uncertainty as in the 1D harmonic oscillator \ncase~\\cite{Landauq}.\n\nIf the oscillator mean energy per mode is taken as $\\mean{n} \\hbar \\omega$, \nwe obtain to order $O(\\frac{1}{k_B T})$ the classical (high temperature) limit $k_B T \\gg \\hbar\\omega$:\n\n\\bea\n\\mean{n} \\hbar \\omega & \\approx & \\frac{\\hbar \\omega}{1+ \\frac{\\hbar \\omega}{k_B T} + \\frac{1}{2} {(\\frac{\\hbar \\omega}{k_B T})}^2... -1} \\nonumber \\\\\n & \\approx & \\frac{k_B T}{1+ \\frac{1}{2} (\\frac{\\hbar \\omega}{k_B T})...} \\approx k_B T - \\frac{1}{2} \\hbar \\omega.\n\\eea\n\nThus one should rather write the mean energy per mode as  $(\\mean{n}+ \\frac{1}{2}) \\hbar \\omega$ \nin order to retrieve the right classical limit $ k_B T$ originating\nfrom the comparison between Planck photon spectral density expression \n$\\frac{2 \\omega^2}{\\pi^2 c^3}  \\left[\\frac{\\hbar \\omega}{e^{\\beta \\hbar \\omega} -1} \\right]$\nand Rayleigh-Jeans~\\cite{electromag} expression $ \\frac{2 \\omega^2}{\\pi^2 c^3} [k_B T]$\n($c$ is the velocity of light).\n\n\\subsection{$LC$ circuit quantum derivation of Thermal Noise}\nWe start with an analogy~\\cite{Devoret} between the harmonic oscillator and the $LC$ resonator. \nMoving from an $RC$ to an $LC$ circuit\nstems from the fact a resistance may be defined from $ R = \\sqrt{\\frac{L}{C}} $ \nand that an oscillator underlying a resistor allows a ready route to quantization.\n\nLater on when we consider a semi-infinite transmission line with $L$ inductance \nand $C$ capacitance per unit length, the line resistance\n$R = \\sqrt{\\frac{L}{C}} $ is same as the single $LC$ resonator.\nThus the transmission line  might be viewed  simply as a large collection \nof harmonic oscillators (normal modes) and hence can be readily quantized. \nThe resistance picture that links the resonator to the transmission line \nis very appealing and has has been introduced for the first time by \nCaldeira and Leggett~\\cite{Devoret} to describe a continuum as sets of harmonic oscillators \nas described below.\n\n\nLet us write the Hamiltonian of a single $LC$ resonator circuit in the form:\n\n\\begin{equation}\n\\mathscr{H}_0= \\frac{q^2}{2C_0} + \\frac{\\phi^2}{2L_0}\n\\end{equation}\n\n\nwhere variables $q$ and $\\phi$ are capacitor charge and flux in the inductor.\nDrawing from complete analogy with the harmonic oscillator, we quantize variables with:\n\n\\begin{equation}\nq=\\sqrt{\\frac{\\hbar}{2R}}(a+a^{\\dagger}), \\phi=-i\\sqrt{\\frac{\\hbar R}{2}}(a-a^{\\dagger}) \n\\end{equation}\n\nwhere $ R = \\sqrt{\\frac{L_0}{C_0}} $ and $a^{\\dagger}$, $a$  are ladder operators  \ncharacterized by commutation property: $[a,a^{\\dagger}]=1$.\n\nThe Hamiltonian is transformed into standard harmonic oscillator form\n$\\mathscr{H}_0= \\hbar \\omega_0 (a^{\\dagger}a + \\frac{1}{2})$\nwith $\\omega_0= 1\/{\\sqrt{L_0 C_0}}$, the classical resonance frequency.\n \nMoving from a single oscillator to a continuum, \nwe follow Goldstein~\\cite{Goldstein} treatment by considering a transmission\nline  of length $\\ell$ characterized by an inductance $L$ and capacitance  $C$ per unit length.  \n\nThe Hamiltonian of the system is \n\\begin{equation} \\mathscr{H}(t) = \\int\\limits_0^\\ell dx\\,\\left[ \\frac{q^2(x,t)}{2C} + \\frac{\\phi^2(x,t)}{2L}\\right], \n\\end{equation}\n\nwhere $\\phi(x,t)$ is the local flux density and $q(x,t)$ is the local charge density.\n\n\nWe  define a new variable \n\\begin{equation}\n\\theta(x,t) = \\int\\limits_0^x dx'\\, q(x',t) \n\\end{equation} \n\nto express current density $ j(x,t) = -\\frac{\\partial \\theta(x,t)}{\\partial t}$\nand charge density $ q(x,t) = \\frac{\\partial \\theta(x,t)}{\\partial x}$\nsuch that charge conservation rule:\n\n\\begin{equation}\n\\frac{\\partial}{\\partial x} j(x,t) + \\frac{\\partial}{\\partial t} q(x,t) = 0.\n\\end{equation} \n\nis obeyed.\n\nThe Hamiltonian is written as:\n\\begin{equation} \\mathscr{H}(t) = \\int\\limits_0^\\ell dx\\,\n\\left[ \\frac{1}{2C}\\left(\\frac{\\partial \\theta}{\\partial x}\\right)^2 \n+\\frac{L}{2}\\left(\\frac{\\partial \\theta}{\\partial t}\\right)^2 \\right] \n\\end{equation} \n\n\nFrom Hamilton equations of motion, we get the wave equation \n$\\frac{\\partial^2 \\theta}{\\partial x^2} - \\frac{1}{v^2} \\frac{\\partial^2 \\theta}{\\partial t^2}= 0 $ \nwith velocity $v=1\/{\\sqrt{LC}}$.\n\nThe normal mode expansion when the transmission line is considered with stationary\nboundary conditions at both ends $\\theta(0,t) = \\theta(\\ell,t)=0$ is given by:\n\n\n\\begin{equation} \n\\theta(x,t) = \\sqrt{\\frac{2}{\\ell}}\\sum_{n=1}^\\infty b_n(t) \\sin k_n x, \n\\end{equation} \n\nwhere $b_n(t)$ is the time-dependent mode amplitude and quantized wavevectors $k_n = \\frac{n\\pi}{\\ell}$.\nAfter substitution of this form into the Hamiltonian and integrating over $x$ exploiting orthogonality\nof the basis functions $[\\cos k_n x, \\sin k_n x]$ over the interval $\\ell$ gives: \n\n\\begin{equation} \n\\mathscr{H}(t) =\\sum_{n=1}^\\infty  \\frac{k_n^2}{2C} {[b_n(t)]}^2 + \\frac{L}{2} \\left[\\frac{d b_n(t)}{dt}\\right]^2\n\\end{equation} \n\nQuantizing the system in terms of harmonic oscillator ladder operator sets\nusing the correspondence: \n\\begin{equation}\nb_n(t) \\rightarrow \\sqrt{\\frac{\\hbar C}{2}} \\frac{\\sqrt{\\omega_n}}{k_n}[a^\\dagger_n(t) + a_n(t)]\n\\end{equation} \n\nwhere $\\omega_n= \\frac{n v \\pi}{\\ell}$ controls Heisenberg time dependence of ladder operators through\n$a^\\dagger_n(t)=\\exp(i \\omega_n t) a^\\dagger_n(0)$  and $a_n(t)~=~\\exp(-i\\omega_n t)~a_n(0)$,\nyields, from charge density $ q(x,t) = \\frac{\\partial \\theta(x,t)}{\\partial x}$, the voltage at $x=0$:\n\n\\bea \nV(t) & =&\\frac{1}{C}\\left[\\frac{\\partial \\theta(x,t)}{\\partial x}\\right]_{x=0} \\nonumber \\\\\n    & =& \\sqrt{\\frac{\\hbar}{\\ell C}}\\sum_{n=1}^\\infty \n\\sqrt{ \\omega_n} [e^{i \\omega_n t} a^\\dagger_n(0) + e^{-i\\omega_n t}~a_n(0)] \\nonumber \\\\\n\\eea \n\n\n\nThe voltage PSD is obtained after quantum averaging the voltage time correlation (see Appendix A): \n\\begin{equation}\nS_{V}(\\omega) = \\frac{2 \\pi}{\\ell C} \\sum_{n=1}^\\infty \\hbar\\omega_n [ n (\\omega_n)\\delta(\\omega+\\omega_n) \n+ [n (\\omega_n)+1]  \\delta(\\omega-\\omega_n) ], \n\\end{equation}\n\n\nwhere $n (\\omega)=\\mean{a^\\dagger_n(0)a_n(0)}$ is the photon Bose-Einstein distribution \nwith energy $\\hbar\\omega$ defined previously as $\\mean{n}$ in eq.~\\ref{Bose}. \n\nTaking the limit $\\ell \\rightarrow\\infty$ and converting summation to integration through the replacement\n\n\n\\begin{equation} \n \\sum_{n=1}^\\infty f(\\omega_n) \\approx \\frac{\\ell}{v\\pi} \\int_0^\\infty f(\\omega) d\\omega\n\\end{equation}\n\nyields \n\n\\begin{equation} \nS_{V}(\\omega) = 2R \\hbar\\omega \\{-n (-\\omega) \\Upsilon(-\\omega) +[n (\\omega)+1] \\Upsilon(\\omega) \\}, \n\\end{equation} \n\nwhere $\\Upsilon(\\omega)=\\int_0^\\infty \\delta(\\omega-x) dx$ is the Heaviside step function.\nPhysically the negative $\\omega$ term corresponds to energy absorption whereas in the\npositive $\\omega$ case,  $n(\\omega)$ represents stimulated emission and +1 represents\nspontaneous emission leading to $S_{V}(\\omega)$ being asymmetric with respect \nto $\\omega$ in contrast to the classical oscillator case (see Appendix A).\n\nIn the $\\omega > 0$ case ($\\Upsilon(\\omega)$ term retained), the spectral density: \n\\begin{equation} \nS_{V}(\\omega) = \\frac{2R \\hbar\\omega}{1-e^{-{\\hbar \\omega \/ k_{B}T}} },\n\\end{equation} \n\nreduces, in the classical limit~\\cite{classical}  $k_{B}T \\gg \\hbar\\omega$,\nto Johnson-Nyquist noise result $S_{V}(\\omega) = 2R k_{B}T$ (see note~\\cite{factor2}).\n\nIn order to retrieve the quantum fluctuation-dissipation theorem\\cite{Callen},\nwe take the symmetric part of $S_{V}(\\omega)$ by adding positive and negative \nspectral contributions:\n\n\\begin{equation} \nS_{V}(\\omega) + S_{V}(-\\omega) = 2R \\hbar\\omega \\coth \\left(\\frac{\\hbar\\omega}{2k_{B}T} \\right)\n\\end{equation} \n\n\n\\section{Application to secure communications}\nThe main question in this section deals with the possible way to communicate\nsecurely with a classical approach based upon acting on communication bits with \ncontrolled noise (shift-register generated pseudo-random bits) or through\nQuantum Communications based on entanglement.\n\n\\subsection{Spread spectrum communications}\nThe principle of spread-spectrum communications such as DSSS used in Wi-Fi and \ncordless telephony is based on multiplying the message (made of 0's and 1's) \nby a sequence of pseudo-random bits. \nPseudo-Random Binary Sequences (PRBS), the digital version of PN\nsequences are produced in a controlled fashion\nwith a deterministic algorithm akin to pseudo-random numbers used in \na Monte-Carlo algorithm or some other type of simulation~\\cite{Recipes}.\n\nThe main goal of PRBS generation, is to draw 0 or 1 in an equally probable fashion\nin order to have highly efficient crypting of the message (largest bandwidth or spreading). \nA particularly efficient method for producing PRBS is based on primitive polynomials modulo 2 \nor Galois polynomials~\\cite{Knuth} with the following arithmetic: \\\\\n$0\\oplus 0=0, 0\\oplus 1=1, 1\\oplus 0=1, 1\\oplus 1=0$. \\\\\n$\\oplus$  is the usual symbol for modulo 2 arithmetic corresponding\nto the logical XOR operation.\n \n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{70mm}{!}{\\includegraphics[angle=0,clip=]{shift9.pdf}} \n\\vspace*{-3mm}  \n\\caption{Shift register connections with feedback set up for PRBS~\\cite{Recipes} generation \non the basis of a (modulo 2) primitive polynomial given by $1+x^4+x^9$. It is of order 9 and  \ntap connections (9,4,0) are shown.}\n\\label{XOR}\n\\end{figure} \n\nThe coefficients of primitive polynomials modulo 2 are zero or one e.g. \n$x^4+x^3+1$, moreover they cannot be decomposed into a product of simpler modulo 2 polynomials.\nAn illustrative example is $x^2+1$ that cannot be decomposed into simpler\npolynomials with real coefficients but can be decomposed into polynomials  with \ncomplex coefficients $x^2+1=(x+i)(x-i)$ with $i=\\sqrt{-1}$. When this polynomial\nis viewed as a Galois polynomial, it is not primitive since it can be decomposed into a product \nof simpler polynomials $x^2+1 \\equiv x^2+2 x +1=(x+1)(x+1)$ since in modulo 2 arithmetic the term\n$2 x$ is equivalent to 0 according to the above arithmetic rule ($1\\oplus 1=0$).\n\n\nThe method for producing PRBS illustrated in fig.~\\ref{XOR} requires only a single shift \nregister $n$ bits long and a few XOR or mod 2 bit addition operations ($\\oplus$ gates).\n\nThe terms that are allowed to be XOR summed together are indicated by shift register taps.\nThere is precisely one term for each nonzero coefficient in the primitive polynomial \nexcept the constant (zero bit) term. Table~\\ref{Galois} contains a list of polynomials for $n \\le 15$,\nshowing that for a primitive polynomial of degree $n$, the first and last term are 1. \n \n\n\\begin{table}[htbp]\n\\begin{tabular}{l|r}\n\\hline\nConnection Nodes & Equivalent Polynomial \\\\\n\\hline\n   (1,\\hspace{0.3cm}0)  &  $1+x$ \\\\ \n    (2,\\hspace{0.3cm}1,\\hspace{0.3cm}0) &  $1+x+x^2$  \\\\  \n    (3,\\hspace{0.3cm}1,\\hspace{0.3cm}0)  & $1+x+x^3$ \\\\\n    (4,\\hspace{0.3cm}1,\\hspace{0.3cm}0)  & $1+x+x^4$\\\\ \n    (5,\\hspace{0.3cm}2,\\hspace{0.3cm}0)  & $1+x^2+x^5$\\\\    \n    (6,\\hspace{0.3cm}1,\\hspace{0.3cm}0)  & $1+x+x^6$\\\\ \n    (7,\\hspace{0.3cm}1,\\hspace{0.3cm}0)  & $1+x+x^7$\\\\ \n    (8,\\hspace{0.3cm}4,\\hspace{0.3cm}3,\\hspace{0.3cm} 2,\\hspace{0.3cm} 0) & $1+x^2+x^3+x^4+x^8$ \\\\\n    (9,\\hspace{0.3cm}4,\\hspace{0.3cm}0)  & $1+x^4+x^9$\\\\ \n    (10,\\hspace{0.3cm}3,\\hspace{0.3cm}0)  & $1+x^3+x^{10}$\\\\ \n    (11,\\hspace{0.3cm}2,\\hspace{0.3cm}0)  & $1+x^2+x^{11}$\\\\   \n    (12,\\hspace{0.3cm}6,\\hspace{0.3cm}4,\\hspace{0.3cm}1,\\hspace{0.3cm} 0) & $1+x+x^4+x^6+x^{12}$ \\\\\n    (13,\\hspace{0.3cm}4,\\hspace{0.3cm}3,\\hspace{0.3cm}1,\\hspace{0.3cm} 0) & $1+x+x^3+x^4+x^{13}$ \\\\ \n    (14,\\hspace{0.3cm}5,\\hspace{0.3cm}3,\\hspace{0.3cm}1,\\hspace{0.3cm} 0)  & $1+x+x^3+x^5+x^{14}$\\\\ \n    (15,\\hspace{0.3cm}1,\\hspace{0.3cm}0) &  $1+x+x^{15}$\\\\  \n\\hline\n\\end{tabular}\n\\caption{List of the first 15 Galois polynomials.}\n\\label{Galois}\n\\end{table}  \n\n\nA Maximum-Length Sequence (MLS) $x[n]$ is a balanced sequence made from \nequally probable symbols with values +1 and -1 such that the MLS averages to zero. \nChoosing $x[n] = (-1)^{a[n]}$ with $a[n]\u00a0=$\u00a00 or 1 originating from PRBS yields the \ndesired values $x[n]$\u00a0=\u00a0+1 or -1 with +1\/-1 equally probable.\nThe PRBS sequence $a[n]$ is produced with a shift register XOR operation \nas discussed previously and illustrated in fig.~\\ref{XOR}.\nThe MLS has many attractive features in addition to the balanced character: \nits standard deviation and peak values are both equal to 1 making its crest factor \n(peak\/standard deviation) equal to 1, the lowest value it can get~\\cite{Recipes}. \nThat is why MLS has noise-immune property~\\cite{Recipes} required in communication\nelectronics. MLS are used not only in secure communications but also in synchronization \nof digital sequences. \n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{60mm}{!}{\\includegraphics[angle=0,clip=]{MLS.pdf}} \n\\vspace*{-3mm}  \n\\caption{Property of MLS auto-correlation $R_{xx}[n]$ showing peaks that enables decoding\nthe message and displaying maximum length $N=2^{n_c} -1$ with $n_c$ the number of coding bits.}\n\\label{MLS}\n\\end{figure} \n\nA message $x(t)$ transmitted through a linear time-invariant medium is    \nconvoluted with the channel impulse response $h(t)$ resulting in an output message:\n\n\\begin{equation}\ny(t)=h(t)*x(t)=\\int_{-\\infty}^{\\infty}h(t-t') x(t') dt'\n\\label{reception}\n\\end{equation}\n\n\nThe decoding process of the message is based on a correlation operation\nbased on the $x[n]$ auto-correlation given by: \n                         \n\\bea\nR_{xx}[n] &=& \\frac{1}{N-1}  \\sum_{i=0}^{N-2}{ x[i] x[n+i] }   \\nonumber \\\\\n         & =& \\frac{1}{N-1}  \\sum_{i=0}^{N-2}{ (-1)^{(a[i] \\oplus a[n+i])} }                         \n\\eea\n\nwith $N=2^{n_c} -1$ where $n_c$ is the number of coding bits or MLS order. $N$ is the period or the\nlength of the MLS.\n\nAs an example, the auto-correlation  $R_{xx}[n]$ of order $n_c=9$ shown in fig.~\\ref{Auto} displays a \n$\\delta$ function-like behaviour required for message decoding or synchronization (shown in fig.~\\ref{MLS}).\n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{60mm}{!}{\\includegraphics[angle=0,clip=]{auto9.pdf}} \n\\vspace*{-3mm}  \n\\caption{Auto-correlation $R_{xx}[n]$ of order 9 MLS displaying the peak value of 2$^9$-1=511 \nover the period interval [0-511].}\n\\label{Auto}\n\\end{figure} \n\nAnother application  of the MLS  is the determination of the impulse response $h(t)$ of any communication channel \nby sending through the channel a PRBS signal $x(t)$ whose auto-correlation is a delta function that will be used\nto identify $h(t)$ at the receiver (see eq.~\\ref{reception}) since:\n\n\\begin{equation}\nh(t)=\\int_{-\\infty}^{\\infty}h(t-t') \\delta(t') dt'\n\\label{identification}\n\\end{equation}\n\n\nThe impulse response determined with MLS is known to be immune to distortion. This is why \ndespite the fact, many other methods~\\cite{Stan} exist to measure it with various success, \nMLS is still preferred when distortion is an issue. \n\n\n\\subsection{Quantum communication (QC)}\nQuantum Mechanics provides several important ingredients to information communication\nnot present in its classical counterpart~\\cite{Ekert}.\nFirstly the information itself sent across a communication channel can be either \nclassical or quantum. The same applies to the channel that might be classical or quantum.\nInformation transmission is measured with input-output correlations performed \nacross the channel that can also be classical or quantum, the signature of entangled states. \nCopying a bit in classical communication is a trivial voltage replication operation\nwhereas in QC the no-cloning theorem~\\cite{Scarani} forbids copying quantum information\nwithout leaving a trace. Crypting information can be made with classical keys (as in PRBS) or\nquantum keys. The generation of random numbers through quantum means\n(QRNG) are superior to PRBS despite their many interesting properties.\n\nQuantum networks across which quantum information is carried is also different from its\nclassical counterpart and finally classical noise as well as quantum noise should be\nproperly described in order to evaluate information error rates.\n\nWe describe every element of quantum communication below.\n\n\\subsubsection{Quantum unit of information: the qubit}\n\nThe discrete~\\cite{continuous} unit quantum information in 2D Hilbert space is the qubit, \nthe two-state quantum counterpart of the classical bit (see fig.~\\ref{bloch}). \n\nIt is represented by a \ntwo-component wavefunction (or spinor~\\cite{spinor}) $\\ket{\\psi(\\theta,\\phi)}$.\n\nComputationally, a qubit is representable with \n128 classical bits considering that it is made of two complex numbers that \nare equivalent themselves to four 32 bit (single precision) float numbers.\n\nIn the case of photons, quantum states $\\ket{0}$ and  $\\ket{1}$ are equivalent \nto orthogonal polarization states (see fig.~\\ref{bloch}).\n  \nThe photon is the logical choice as the basic information carrier in quantum\ncommunications proceeding between nodes that make quantum networks. \nInformation can be encoded in photon polarization, orbital momentum, spatial mode \nor time and any manipulation targeting processing\nor information transfer can be made with optical operations, \nsuch as using birefringent waveplates to encode polarization...\n\nOn the other hand, atoms are the natural choice to make quantum memories since some of \ntheir electronic states can retain quantum information for a very long time.\n\nQuantum networks convey quantum information with nodes that allow for its \nreversible exchange. The latter may be done with two coupled single-atom nodes \nthat communicate via coherent exchange of single photons. In comparison, classical\nfiber-optic networks use pulses containing typically 10$^7$ photons each.\n\nIn order to prevent change in information or even its loss, it is necessary \nto have tight control over all quantum network components. Considering the \nsmallest memory for quantum information as a single atom with single photons \nas message carriers, efficient information transfer between an atom and a \nphoton requires strong interaction between the two components not achievable \nwith atoms in free space but in special optical cavities.\n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{60mm}{!}{\\includegraphics[angle=0,clip=]{bloch.pdf}} \n\\vspace*{-3mm}  \n\\caption{Bloch sphere (Poincar\\'e sphere for photons) representing the possible \nvalues of a quantum information unit in 2D Hilbert space or qubit shown as the\nquantum wavefunction $\\ket{\\psi(\\theta,\\phi)}$. \nThe classical bits (0,1) are the poles of the sphere. A qubit is\nany two-component wavefunction given by $(\\alpha \\ket{0} + \\beta \\ket{1})$ \nwith complex coefficients $\\alpha,\\beta$. A pure state exists over the Bloch sphere with\n$|\\alpha|^2 +|\\beta|^2 =1$ whereas a mixed state lies inside the sphere with\n$|\\alpha|^2 +|\\beta|^2 <1$. A classical\nanalog~\\cite{continuous} state lies anywhere on the vertical axis linking the poles.}\n\\label{bloch}\n\\end{figure} \n\nA low-loss cavity made with a set of strongly reflective mirrors  \nalters the distribution of modes with which the atom interacts modifying \nthe density of vacuum fluctuations that it experiences at a given frequency\nenhancing or reducing atomic radiative properties. As a consequence, \nspontaneous emission from the atom excited state being a \nmajor source of decoherence can be inhibited in a cavity. \n\nA low-loss optical cavity possesses a high quality factor ($Q > 10^3$) allowing\na photon entering the cavity to be reflected between mirrors making the cavity several \nthousand times per second strongly enhancing its coupling with the atom leading\nto its absorption by the atom in a highly efficient coherent fashion. \n\nOn the other hand, photon emission by an atom inside a cavity is highly directional\nand can be sent to other network nodes in a precisely controlled fashion. \n\n\nControlling qubit states means that an operator is required to allow switching from one qubit state to another. \nA rotation matrix $\\hat{R}_z(\\theta,\\phi)$ represents such an operator in the $\\ket{0},\\ket{1}$\nbasis:\n\n\\begin{equation}\n   \\hat{R}_z(\\theta,\\phi)= \\left [\n   \\begin{array}{cc}\n   \\cos\\frac{\\theta}{2}&-ie^{i\\phi}\\sin\\frac{\\theta}{2}\\\\\n   -ie^{-i\\phi}\\sin\\frac{\\theta}{2}&\\cos\\frac{\\theta}{2}\n   \\end{array}\\right ] \\label{rotation matrix}\n\\end{equation} \n\nApplying $ \\hat{R}_z(\\theta,\\phi)$ on the state $\\ket{0}$ allows us \nto produce an arbitrary state $(\\theta \\ne 0,\\phi \\ne 0)$  on the Bloch sphere: \n$\\ket{\\psi(\\theta,\\phi)}=  \\hat{R}_z(\\theta,\\phi) \\ket{0}= \n\\cos\\frac{\\theta}{2}\\ket{0} -ie^{-i\\phi}\\sin\\frac{\\theta}{2}\\ket{1} \\equiv (\\alpha \\ket{0} + \\beta \\ket{1})$. \nThis applies only for pure states that lie on the\nBloch sphere surface ($|\\alpha|^2 +|\\beta|^2 =1$). In the case of mixed states \n($|\\alpha|^2 +|\\beta|^2 <1$) we need additional operators that alter also the \nwavefunction modulus. Experimentally, photon polarization can be rotated\nwith a half-wavelength plate (called also a Hadamard gate), moreover it can be\nseparated into individual components with a polarizing beam-splitter \n(see ref.~\\cite{Obrien}).\n\n\\subsubsection{Entanglement}\n\nQuantum networks possess special features that are not found in their classical counterparts. \nThis is due to the intrinsic nature of the information processed: \nwhile a classical bit is either 0 or 1 (see fig.~\\ref{bloch}), a qubit \n(quantum bit wavefunction) can take both values at the same time due to coherent \nsuperposition inherent to Quantum Mechanics linearity.\n \nQuantum Mechanics embodies the notion of entanglement \ndetected with violation of Bell inequalities~\\cite{Bell} and\nthat brings a paradigm shift into information processing.\n\nGiven two qubits $\\ket{Q_1}=~\\frac{1}{\\sqrt{2}}(c_0 \\ket{0} + c_1 \\ket{1})$ \nand $\\ket{Q_2}~=~\\frac{1}{\\sqrt{2}}(d_0 \\ket{0} + d_1 \\ket{1})$, \nit is possible to build a state $\\ket{Q_1,Q_2}=\\ket{Q_1}\\otimes \\ket{Q_2}$ such that:\n\\bea\n\\ket{Q_1,Q_2}=\\frac{1}{\\sqrt{4}}(c_0 \\ket{0} + c_1 \\ket{1})\\otimes (d_0 \\ket{0} + d_1 \\ket{1}) \\nonumber \\\\\n =\\frac{1}{\\sqrt{4}}(c_0 d_0 \\ket{0,0}+c_0 d_1 \\ket{0,1}+c_1 d_0 \\ket{1,0}+c_1 d_1 \\ket{1,1}) \\nonumber \\\\\n\\eea\n\nQuantum Mechanics, however, allows for building other states such as:\n$\\ket{Q}=\\frac{1}{\\sqrt{2}}(c_0 d_1 \\ket{0,1}+c_1 d_0 \\ket{1,0})$\nwhich are not decomposable into products of constituent states. \n\nThese states are called entangled and can be mapped onto the polarization of single photons \nwhich can be transferred through an optical fiber between two nodes consisting respectively \nof atoms in state $\\ket{A}$ and state $\\ket{B}$. \n\nQuantum mechanical entanglement ought to be achieved between the two nodes in order to\nhave successful QC maintained over the coherence time preserving\nintegrity of quantum information transfer.\n\nIn order to achieve entanglement between two remote network nodes, polarization of the \nsingle photon emitted by atom in state $\\ket{A}$ is entangled with the atomic quantum state. \n\nOnce the photon gets absorbed, the entanglement is transferred onto atom in state $\\ket{B}$\nand reversible exchange of quantum information is performed between the two nodes. \n\n\nExperimental production of entanglement can be made between two particles \n(bipartite) or between several particles (multipartite) and for each number of particle \n(two, three, four ...) case or  particle type (atoms, electrons, photons...) \nseveral experimental procedures readily exist. It is not limited to microscopic\nparticles since it can be induced by a light pulse between two macroscopic \nobjects~\\cite{Julsgaard} consisting each of a gas containing \nabout 10$^{12}$ Cesium atoms.\n\nEntanglement can occur when particles interact and kept in contact or \nwhen they emerge from a common ancestor as in the EPR~\\cite{EPR} case where \na spinless particle decays into two particles carrying opposite spins... \nAnother example is the case of a photon interacting with a non-linear crystal. \nIt can be destroyed and replaced with a lower energy entangled photon pair \n(the process is called SPDC~\\cite{Kwiat} or Spontaneous Parametric Down-Conversion).\n\nHeralded~\\cite{Monroe} entanglement may occur between non-interacting remote particles\n(Yb ions held in two ion traps, 1 meter apart) not possessing a common ancestor, \nhowever the entanglement probability $p_E$ is very low ($p_E \\approx 10^{-9}$) since\nentanglement results from the interaction of decay photons emitted by each ion after \ntheir excitation by picosecond laser pulses. Thus $p_E$ needs to be increased substantially \nin order to make it applicable to mass QC. \n\n\n\n\\subsubsection{Quantum Random Number Generation}\nClassical Random Number Generators (RNG) are based on Uniform\nRNG and the standard statistical quality tests target \nthe uniformity~\\cite{Recipes} of the numbers generated.\nQuantum Mechanics introduce a predictability test to further improve\nquality of RNG.\n\nThis means that even if the RN is perfectly uniformly distributed, \nit may contain hidden deterministic information and is therefore \nprone to be predictable. For instance, PN and PRBS generate uniformly \ndistributed numbers but since they are produced with a deterministic \nalgorithm, an eavesdropper might, by drawing values and performing\nstatistical analysis~\\cite{Recipes}, be able to make an educated guess \nand access the cipher password, key...\n\nThus, statistical uniformity tests are necessary but not sufficient to guarantee that any given \nRNG is not prone to attack and guess by an intruder. Quantum RNG (QRNG) offers \"true\nRN\" generation that is very difficult to predict. Using a special program called \n\"randomness extractor\"~\\cite{Colbeck} one might eliminate all bit strings originating from \nan implicit deterministic algorithm and keep only truly random bit strings. For this reason,\nthe method is also called, amplification of weak randomness~\\cite{Colbeck}.\n\nRandomness extraction procedure exploits entropy hierarchy (see Appendix D) that\nattributes a number of bits depending on the entropy estimation used. \nR\\'enyi min-entropy is very efficient computationally wise\nand a string of perfectly random bits has unit min-entropy per bit as derived in Appendix D.\n\nStarting from $l$ input bits $X_i$ of low-entropy per bit ($s < 1$), the extractor \ncomputes a number $k < l $ of higher-entropy ($s' \\approx 1$) output bits $Y_j$ with\na linear transformation via multiplication by a matrix $m$:\n\\begin{equation}\nY_j = \\sum^l_{i=1}m_{ji}X_i, \\hspace{1cm} j=1...k\n\\label{extractor}\n\\end{equation}\n\n$m$ is built from $l\\times k$ random bits that can be generated with Galois\npolynomials and all arithmetic operations are done modulo 2 with AND and XOR logic.\n\nThis \"Whitening\" procedure can be viewed as the quantum counterpart of the Maximum Entropy\nMethod that is widely used in Image Processing for deblurring images~\\cite{Recipes}.\n\nAs a direct application of this concept, Sanguinetti  {\\it et al.}~\\cite{Sanguinetti} used \nSmartphone cameras to produce Quantum Random Numbers. \nAfter uniform illumination of the camera image sensor \nby a LED and estimation of the number of photons generated per pixel, a randomness \nextractor algorithm such the above (eq.~\\ref{extractor}) is used to \ncompute truly random numbers.\n\nFor $X_i$ input bits with low entropy per bit ($s < 1$), the probability that \nthe output $Y_j$ deviates from a perfectly random bit string (with high entropy\nper bit $s' \\approx 1$) is bounded~\\cite{Cover} by:\n\\begin{equation}\n\\epsilon=2^{-(l s - k s')\/2},\n\\end{equation}\n\nPicking a CCD image sensor with 16 bits per pixel (detection capability) \nand a photon flux producing $2\\times 10^4$ electrons per \npixel gives $R_\\infty=8.469$ bits\/pixel (from eq.~\\ref{Renyi}) \nyielding a min-entropy per bit $s=0.529$  (in comparison, Shannon Entropy \nis  9.191 bits\/pixel or 0.574 per bit). Selecting input $l=2000$, \noutput $k=400$ and $s'=1$, we get $\\epsilon =2.57 \\times 10^{-197}$.\n\nAs a result, an eavesdropper would have to generate \nan extremely large~\\cite{Cover} amount of random numbers (about $1.97 \\times 10^{99}$)\nbefore noticing any departure from a perfectly random sequence, indicating\nthe superior performance of QRNG with respect to any classical RNG.\n\n\n\n\\subsubsection{Quantum Keys}\nClassical cryptography is based on two types of keys that are used to\nencode and decode messages: secret or symmetric keys and public or asymmetric keys.\nSymmetric keys are same for encoding and decoding messages whereas in \npublic cryptography systems, one needs a public key and a private key.\nIn the PGP (Pretty Good Privacy) secure mailing system over the Internet,\nthe sender encodes the message with receiver public key and the receiver\ndecodes the message with his private key. \nIn quantum cryptography, the simplest example of secret key sharing among\nsender and receiver (Alice and Bob) in QKD is the BB84~\\cite{Scarani} protocol. \nAlice and Bob communicate through two channels: one quantum to send\npolarized single photons and one classical to send ordinary messages.\nAlice selects two bases in 2D Hilbert space consisting each of two\northogonal states: $\\bigoplus$ basis with $(0,\\pi\/2)$\nlinearly polarized photons,   \nand $\\bigotimes$ basis with $(\\pi\/4, -\\pi\/4)$ linearly polarized photons.\n\nFour symbols: $\\ket{\\rightarrow}, \\ket{\\uparrow},\\ket{\\nearrow},\\ket{\\searrow}$\nrepresenting polarized single photons are used to transmit quantum data with\n$\\ket{\\nearrow}=\\frac{1}{\\sqrt{2}}(\\ket{\\rightarrow}+ \\ket{\\uparrow})$\nand $\\ket{\\searrow}=\\frac{1}{\\sqrt{2}}(\\ket{\\rightarrow}- \\ket{\\uparrow})$.\n\nIn the $(basis,data)$ representation, the symbols are given by \n$\\ket{\\rightarrow}=(\\bigoplus,0)$, $\\ket{\\uparrow}=(\\bigoplus,1)$ in the $\\bigoplus$\nbasis whereas $\\ket{\\searrow}=(\\bigotimes,0)$, $\\ket{\\nearrow}=(\\bigotimes,1)$ in the\n$\\bigotimes$ basis.\n\nA message transmitted by Alice to Bob over the Quantum channel is a stream of \nsymbols selected randomly among the four described above. \n\nBob performs polarization measurements over the received symbols selecting \nrandomly bases $\\bigoplus$ or $\\bigotimes$.\n\nAfterwards Bob and Alice exchange via the classical channel their mutual\nchoice of bases without revealing the measurement results.\n\nIn the ideal case (no transmission errors, no eavesdropping) \nAlice and Bob should discard results pertaining to \nmeasurements done in different bases (or when Bob failed to detect \nany photon).  This process is called \"key sifting\" after which the raw key is determined.\n\nAfter key sifting, another process called key distillation~\\cite{Scarani} must be performed.\nThis process entails three steps~\\cite{Scarani}: error correction, privacy amplification and\nauthentication in order to reveal classical or quantum errors of transmission,\ndetect eavesdropping (with the no-cloning theorem~\\cite{Scarani}) and act against it. \n\nIgnoring, for simplicity, key distillation, the raw key size is typically \nabout one quarter of the data sent since both Alice and Bob are selecting \ntheir bases at random (total probability is roughly  \n$\\frac{1}{2} \\times \\frac{1}{2}=\\frac{1}{4}$).\n\n\nA random number generator (RNG) or rather\na random bit generator can be used to select $\\bigoplus$ or $\\bigotimes$ bases. \nUsing PRBS or, even better, QRNG to select measurement bases, we infer that\nby comparison with the classical FHSS crypting method, Quantum Mechanics \nprovides extra flexibility through basis selection. \nSuch option is simply not available in classical communication.\n\n\n\nOn the negative side, there are several problems that may come up with the \nBB84 scheme. One major obstacle is that\npresently, it is difficult, on a large scale level, to produce single photons.  \nOne approximate method for doing this, is to \nuse attenuated laser pulses containing several photons that might be intercepted\nin the quantum channel by an eavesdropper with a PNS (Photon Number Splitting) attack. \n\nQuantum Communications can be made more secure when QKD is implemented with\nentanglement~\\cite{Scarani} providing a secure way to distribute secret keys between remote \nusers such that when some eavesdropper is detected, the transmission is halted and the data discarded.\n\n\nThe BBM92~\\cite{Scarani} scheme is an entanglement based version of the BB84 protocol. \nPolarization entangled photon pairs (called EPR pairs or Bell states) are sequentially \ngenerated with one photon polarization measured by Alice and the other \nmeasured by Bob. EPR pairs are produced after emerging from SPDC~\\cite{EPR} by using \na birefringent phase shifter or slightly rotating the non-linear crystal \nitself since the state produced by SPDC is:\n\n\\begin{equation}\n\\ket{\\psi}=\\frac{1}{\\sqrt{2}}(\\ket{\\rightarrow \\uparrow} + e^{i \\varphi} \\ket{\\uparrow \\rightarrow})\n\\end{equation}\n\nThus it suffices to modify $\\varphi$ to 0 or $\\pi$ or place a quarter wave-plate \ngiving a 90\\deg shift in one photon path to generate all \nBell states~\\cite{Kwiat}.\nThese states are polarization entangled~\\cite{Scarani} photons :\n\n\\begin{equation}\n\\ket{\\psi^\\pm}=\\frac{1}{\\sqrt{2}}(\\ket{\\rightarrow \\uparrow} \\pm \\ket{\\uparrow \\rightarrow}),\n\\ket{\\phi^\\pm}=\\frac{1}{\\sqrt{2}}(\\ket{\\rightarrow \\rightarrow} \\pm \\ket{\\uparrow \\uparrow})\n\\end{equation}\n\nThe set forms a complete orthonormal basis in 4D Hilbert space for all polarization states\nof a two-photon system.\n\nAlice and Bob  choose randomly one of the two bases $\\bigoplus$ or $\\bigotimes$\nto perform photon polarization measurement.\n\nAfterwards Alice and Bob communicate over the classical channel \nwhich basis they used for each photon successfully received by Bob.\n\nThe raw key is obtained by retaining the results obtained when the bases used are same.\nNeither RNG nor QRNG are used in this case since randomness is inherent \nto the EPR pair polarization measurement~\\cite{Scarani}.\nMoreover, no Bell inequality tests are needed since all measurements \nmust be perfectly correlated or anti-correlated.\n\nFor instance in the $\\ket{\\psi^+}$ state, if one photon is measured to be in the \n$\\ket{\\rightarrow}$ state, the other must be in the $\\ket{\\uparrow}$ since \nthe probabilities of measuring ${\\rightarrow \\rightarrow}$ or ${\\uparrow \\uparrow}$ are given by \n$|\\bra{\\rightarrow \\rightarrow}\\ket{\\psi^+}|^2=|\\bra{\\uparrow \\uparrow}\\ket{\\psi^+}|^2=0$, \nwhereas the probabilities of measuring ${\\rightarrow \\uparrow}$ and ${\\uparrow \\rightarrow}$ are \n$|\\bra{\\rightarrow \\uparrow}\\ket{\\psi^+}|^2=|\\bra{\\uparrow \\rightarrow}\\ket{\\psi^+}|^2=\\frac{1}{2}$.\nThis is termed perfect anti-correlation.\n\nWhen the polarization measurements are performed in the $\\bigotimes$ basis, we get\nrather, perfect correlation. That means \nthe probabilities of measuring ${\\nearrow \\searrow}$ or ${\\searrow \\nearrow}$ are \n$|\\bra{\\nearrow \\searrow}\\ket{\\psi^+}|^2=|\\bra{\\searrow \\nearrow}\\ket{\\psi^+}|^2=0$,\nwhereas the probabilities of measuring ${\\nearrow \\nearrow}$ or ${\\searrow \\searrow}$ are given by \n$|\\bra{\\nearrow \\nearrow}\\ket{\\psi^+}|^2=|\\bra{\\searrow \\searrow}\\ket{\\psi^+}|^2=\\frac{1}{2}$. \n\nNote that if we rather consider the  $\\ket{\\psi^-}$ state, we get perfect \nanti-correlation in both bases $\\bigoplus$ and $\\bigotimes$.\n\n\n\\subsubsection{Quantum Networks}\n\nIn classical communications, channel transfer function, the Fourier transform\nof its impulse response $h(t)$ is a function of\nfrequency and distance. Channel bandwidth and signal attenuation are functions\nof distance. When a pulse (representing a communication symbol made of several\nbits depending on the modulation method used) is sent through an optical fiber, \nit undergoes broadening leading to inter-symbol interference, attenuation\nleading to signal loss and alteration due to noise. \nThus it is required to evaluate the largest distance\nthat could be covered at the end of which a repeater is placed in order to filter\nout noise and restore pulse shape to its original form.\n\nIn QKD, Alice and Bob should be able to determine efficiently their shared secret key \nas a function of distance $L$ separating them. Since, the secure key is determined after\nsifting and distillation, secure key rate is expressed in bps (bits per symbol) given\nthat Alice sends symbols to Bob to sift and distill with the remaining bits making the secret key.\n\nThe simplest phenomenological way to estimate secure key rate versus distance $K(L)$ is to consider \na point-to-point scenario with $K(L) \\propto [A(L)]^n$  where $A(L)= 10^{-\\alpha_0 L\/10}$\nis signal attenuation versus distance. $\\alpha_0$ is the attenuation \ncoefficient per fiber length and $n=1,2...$. $\\alpha_0$ depends strongly on the wavelength $\\lambda$\nused to transmit information through the fiber. For the standard Telecom wavelength~\\cite{Carlson}\n$\\lambda=1.55 \\mu$m, $\\alpha_0$=0.2 dB\/km.\n\nThe optimal distance~\\cite{Scarani} $L_{opt}$ is determined by the maximum of the objective\nfunction $L K(L)$. Taking  the derivative and solving, we get \n${L_{opt}=\\frac{10}{n \\alpha_0 \\ln(10)}}$.\n\nThis yields $L_{opt}=$21.7 kms for $n=1$, $L_{opt}=$10.86 kms for $n=2$ and \n$L_{opt}=$5.43 kms for $n=4$. \n\nErrors produced by noise, interference and damping are represented by a $BER$\n(Bit Error Rate), the ratio of wrong bits over total number of transmitted bits.\n$BER$ versus distance is an important indicator of communication quality as much as\ncommunication speed is represented by bit rate versus distance. \n\nIn the quantum case, the $QBER$ (Quantum $BER$) $Q_e(L)$ versus distance \nis the quantity of interest. Regarding the BB84 protocol case, a simple model~\\cite{GYS} \ndelivers the expression:\n\n\\begin{equation}\nQ_e(L)=\\frac{P_e}{ A(L) \\mu  \\eta_{Bob}  +2P_e}\n\\end{equation}\n\nwhere $P_e$ is the probability of error per clock cycle (measured to be\n8.5 $\\times 10^{-7}$). $\\mu=0.1$ is the  average photon flux used by Alice\nto transmit symbols and $\\eta_{Bob}=0.045$ is Bob apparatus detection efficiency. \nThe results are displayed versus distance $L$ in fig.\\ref{QBER}.\n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{80mm}{!}{\\includegraphics[angle=0,clip=]{QBER.pdf}} \n\\vspace*{-3mm}  \n\\caption{(Color on-line) Classical $BER$ and quantum $QBER$  versus distance\n$L$ along an optical fiber using the BB84 protocol considering they start from\nthe same value at $L=0$.}\n\\label{QBER}\n\\end{figure}\n\n\nFig.~\\ref{QBER} shows that the $QBER$ increases faster and takes larger values \nthan the optical fiber classical $BER$. For many digital lightwave systems\nusing ON-OFF modulation~\\cite{Carlson} (1 for light pulse, 0 for no pulse), the classical $BER$ \nis typically about $10^{-9}$ and may reach values in the $[10^{-16}-10^{-15}]$ range.\n\n\nMoving on to estimate the secure key generation rate in bits per symbol (bps) emitted \nby Alice, a simple model for the BB84 protocol~\\cite{GLLP} gives:\n\n\\begin{equation}\nK(L)=G_\\mu \\{ -h_2(Q_e(L)) + \\Omega [1-h_2(e_1)]  \\}\n\\end{equation}\n\nwhere $G_\\mu$ is the gain for an average photon flux $\\mu$. \n$\\Omega$ is the fraction of events detected by Bob and produced\nby single-photon signals emitted by Alice. $e_1$ is the corresponding $QBER$ and\n$h_2$ is the binary Shannon entropy~\\cite{Carlson} given by \n$h_2(x)=-x\\log_2(x)-(1-x)\\log_2(1-x)$. Using the same parameters as in Ref.~\\cite{GYS} and \nbounds for $\\Omega$ and $e_1$ estimated in Ref.~\\cite{GLLP}, we are able to plot \nthe secure key rate versus distance for several\nvalues of the detector error rate $e_D$ as displayed in fig.~\\ref{Rate}.\n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{80mm}{!}{\\includegraphics[angle=0,clip=]{Rate.pdf}}  \n\\vspace*{-3mm} \n\\caption{(Color on-line) Key rate $K(L)$ in bps versus distance $L$ using the same parameters \nas in Ref.~\\cite{GLLP} for several detector error rates $e_D$=0.01, 0.1, 0.2 and 0.4.}\n\\label{Rate}\n\\end{figure} \n\n\n\nFig.~\\ref{Rate} shows that the key rate is small and given that security increases \nwith key length, a major improvement with respect to this simple approach \nshould be undertaken in order to increase substantially the bps rate.\n\nRecently, a joint team from Cambridge Science Park and University of Cambridge~\\cite{Comandar} \nsucceeded to increase substantially the secure key rate using detectors operating at room temperature.\nThe secure key rates obtained were between 1.79 Mbit\/s and 1.2 kbit\/s for \nfiber lengths between 40 km and 100 km, respectively.\n\n\n  \n\nRegarding network building developments, the first elementary quantum network based on interfaces \nbetween single atoms and photons located at two network nodes installed in two distant \nlaboratories connected by an optical fiber link was made in 2012 by a team of \nscientists~\\cite{Ritter} at the Garching MPQ.\n\nUsing the above procedures, Ritter  {\\it et al.}~\\cite{Ritter} were able to generate \nentanglement between two remote nodes in two different laboratories separated by a \ndistance of 21 meters and linked by an optical fiber. \nThey were able to maintain entanglement for about 100 microseconds while entanglement \ngeneration itself took about a single microsecond.\n\nLater, a team from Technical University of Vienna~\\cite{Vienna} succeeded in coupling Cesium \natoms to an optical fiber and storing quantum information over a period of time that is\nlong enough to sustain entanglement over distances (hundreds of kilometers) that are large enough\nto achieve reliable long distance communication.\n\nThe Vienna team extended coherence time to several milliseconds  \nand given that speed of light in an optical fiber is about 200 kilometers per \nmillisecond, a substantial separation increase is henceforth achievable potentially\nreaching several hundred kilometers between nodes over which entanglement and \ncoherence are maintained, paving the way to long-distance QC.\n\n\\subsubsection{Quantum noise}\n\n\nAt low temperature, very high frequency $hf >  k_B T$, mesoscopic scale or when considering\nsingle carrier, quantum dot devices... quantum noise becomes larger than thermal \nimplying a full reconsideration of traditional electronics that has long been\ndescribed by White (thermal noise with no relaxation time), Shot noise based on \na single relaxation time (such as generation-recombination noise in semiconductors),\nPink noise ($1\/f$) originating from a distribution of relaxation times...\n \nRecently, entanglement has been shown to appear spontaneously in photon-assisted \nelectrical noise occurring in quantum conductors consisting of an ac-biased tunnel \njunction  cooled at low temperature~\\cite{Sherbrooke}. \n\nThe experiments were performed in Sherbrooke~\\cite{Sherbrooke} at 18 mK~\\cite{field} on \na Al\/Al$_2$O$_3$\/Al tunnel junction with resistance of 70 $\\Omega$, \nthe signal being emitted by the junction analyzed at two frequencies $f_1$=7 GHz and $f_2$=7.5 GHz.\n\nThe total voltage applied on the junction is given by $V_{dc}+V_{ac} \\cos 2 \\pi f_0 t$\nwith frequency $f_0=f_1+f_2$ chosen to produce optimal junction response as explained below.\n\nFirstly, junction noise becomes photon-assisted because ac-biasing injects photons in the junction.\n\nSecondly, statistical correlations between currents at $f_1$ and $f_2$ as a function of dc voltage\nshowed that photons generated in pairs in the junction are entangled since \ntheir correlations violate Bell inequalities~\\cite{Bell} as discussed below.\nDefining \"position\" $X_1,X_2$ and \"momentum\" operators $P_1, P_2$ from frequency dependent\ncurrent operators $I(\\pm f_{1}), I(\\pm f_{2})$ as:\n\n\\begin{equation}\nX_{1,2}=\\frac{I(f_{1,2})+I(-f_{1,2})}{\\sqrt{2}}, \\, P_{1,2}=\\frac{I(f_{1,2})-I(-f_{1,2})}{i\\sqrt{2}}\n\\end{equation}\n\nwe use the QFDT (see section~\\ref{QFDT}) to evaluate the various quantum correlations versus\ndc voltage $V_{dc}$ applied to  the Al\/Al$_2$O$_3$\/Al tunnel junction for a fixed ac \nvoltage $V_{ac}$= 37 $\\mu$V in fig.~\\ref{correlations}. \nViolation of Bell inequalities displayed by $\\langle X_1X_2\\rangle$ and $\\langle P_1P_2\\rangle$  \nfor non-zero $V_{ac}$ indicate entanglement in contrast with the other correlators that\ndo not display any variation with $V_{dc}$. $\\langle X_1X_2\\rangle_0$ and $\\langle P_1P_2\\rangle_0$ that \nare evaluated for $V_{ac}=0$ do not show violation\nof Bell inequalities indicating that it is $V_{ac}$ that induces quantum correlations thus\nyielding a simple electrical control parameter for entanglement.\n\n\n\\begin{figure}[htbp]\n  \\centering\n    \\resizebox{60mm}{!}{\\includegraphics[angle=0,clip=]{correlations.pdf}}  \n\\vspace*{-3mm} \n\\caption{(Color on-line) Correlations in Kelvin units between currents at frequencies \n$f_1$ and $f_2$ versus $V_{dc}$ voltage applied to  the Al\/Al$_2$O$_3$\/Al tunnel junction for a fixed \nac voltage $V_{ac}$= 37 $\\mu$V. Violation of Bell inequalities is displayed only \nby $\\langle X_1X_2\\rangle$ and $\\langle P_1P_2\\rangle$ quantum correlations for non-zero \napplied ac voltage whereas all other correlators are null including $\\langle X_1X_2\\rangle_0$ \nand $\\langle P_1P_2\\rangle_0$ that are evaluated when the ac voltage is zero.}\n\\label{correlations}\n\\end{figure} \n  \n\n\\section{Discussions and conclusions}\nIn this work, the main unifying thread for the description of fluctuations, noise\nand noise-based communication is the ubiquitous presence of harmonic oscillators \nrepresented mostly by photons.\n\nWhile secure classical noise-based communication uses spread-spectrum sequences,\nsecure QC based on QKD implemented with entanglement\nties communicating parties in a way such that any attempt by some\neavesdropper to intercept or interfere in the communication process is\nimmediately sensed and treated appropriately.\n\nEntanglement may be done between quantum objects such as atoms, electrons, photons etc...\nhowever the preferred information carrier is the photon and the entanglement \nthat can be based on polarization, momentum, spatial mode or time can be sustained over \nvery large distances as demonstrated by the Vienna experiment.\n\nHeralded entanglement not necessitating a common ancestor has even been applied by the\nsame Garching~\\cite{Ritter2} (MPQ) group to transfer a polarization qubit from a photon to \na single atom with 39\\% efficiency and perform the reverse process, that is from the atom \nto a given photon with an efficiency of 69\\%, proving once again that a long-distance\nQC network based on entangled photons is a serious contender for secure communication.\n\nOn the other hand, the Sherbrooke experiment shows that a major component \nof noise-based QC is built within quantum noise since entanglement is produced \nin quantum conductors by a simple electrical (ac voltage) control.\n\nEven if presently such entanglement occurs at very low temperature (18 mK), \nthe result is still important since that particular type of entanglement could be \nexploited after appropriate conditioning with quantum cryptography techniques in order\nto  secure information transfer and communication. \n\n\nPresently several secure QC schemes not based on entanglement exist, moreover\nsome other protocols not relying on key generation and distribution have \nalso been developed.\n\nFor instance QSDC (Quantum Secure Direct Communication) is a branch of QC \nin which the message is sent directly between remote users without \ngenerating a key to encrypt it.\n\nPracticality and robustness of schemes used in QC for securing transmission\nof information will finally decide which of the different methods and \nprotocols will be ultimately adopted as reliable for secure mass communication.    \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNew models of infinitely divisible random matrices have emerged in recent\nyears from both applications and theory. On the one hand, they have been\nimportant in multivariate financial L\\'{e}vy modelling where stochastic\nvolatility models have been proposed using L\\'{e}vy and Ornstein-Uhlenbeck\nmatrix valued processes; see \\cite{BNSe07}, \\cite{BNSe09}, \\cite{BNS11} and\n\\cite{PiSe09a}. A key role in these models is played by the positive-definite\nmatrix processes and more general matrix covariation processes.\n\nOn the other hand, in the context of free probability, Bercovici and Pata\n\\cite{BP} introduced a bijection $\\Lambda$ from the set of classical\ninfinitely divisible distributions to the set of free infinitely divisible\ndistributions. This bijection was explained in terms of random matrix\nensembles by Benaych-Georges \\cite{BG} and Cabanal-Duvillard \\cite{CD},\nproviding in a more palpable way the bijection $\\Lambda$ and producing a new\nkind of infinitely divisible random matrix ensembles. Moreover, the results in\n\\cite{BG} and \\cite{CD} constitute a generalization of Wigner's result for the\nGaussian Unitary Ensemble and give an alternative simple infinitely divisible\nrandom matrix model for the Marchenko-Pastur distribution, for which the\nWishart and other empirical covariance matrix ensembles are not infinitely divisible.\n\nMore specifically, it is shown in \\cite{BG} and \\cite{CD} that for any\none-dimensional infinitely divisible distribution $\\mu$ there is an ensemble\nof Hermitian random matrices $(M_{d})_{d\\geq1}$, whose empirical spectral\ndistribution converges weakly almost surely to $\\Lambda(\\mu)$ as $d$ goes to\ninfinity. Moreover, for each $d\\geq1$, $M_{d}$ has a unitary invariant matrix\ndistribution which is also infinitely divisible in the matrix sense. From now\non we call these models BGCD matrix ensembles. We consider additional facts of\nBGCD models in Section 3.\n\nA problem of further interest is to understand the matrix L\\'{e}vy processes\n$\\left \\{  M_{d}(t)\\right \\}  _{t\\geq0}$ associated to the BGCD matrix\nensembles. It was pointed out in \\cite{DRA}, \\cite{PAS} that the L\\'{e}vy\nmeasures of these models are concentrated on rank one matrices. This means\nthat the random matrix $M_{d}$ is a realization, at time one, of a matrix\nvalued L\\'{e}vy process $\\left \\{  M_{d}(t)\\right \\}  _{t\\geq0}$ with rank one\njumps $\\Delta M_{d}(t)=M_{d}(t)-M_{d}(t-).$\n\nThe purpose of this paper is to study the structure of a $d\\times d$ Hermitian\nL\\'{e}vy process $\\left \\{  L_{d}(t)\\right \\}  _{t\\geq0}$ with rank one jumps.\nIt is shown in Section 4 that if $L_{d}$ is a $d\\times d$ complex matrix\nsubordinator, it is the quadratic variation of an $\\mathbb{C}^{d}$-valued\nL\\'{e}vy process $X_{d}$, being the converse and extension of a known result\nin dimension one, see \\cite[Example 8.5]{CT}. The process $X_{d}$ is\nconstructed via its L\\'{e}vy-It\\^{o} decomposition. In\\ Section 5 we consider\nnew realizations in terms of covariation of $\\mathbb{C}^{d}$-valued L\\'{e}vy\nprocess for matrix compound Poisson process as well as sample path\napproximations for L\\'{e}vy processes associated to general BGCD ensembles. A\nnew insight on Marchenko-Pastur's type results for empirical covariance matrix\nensembles was recently given in \\cite{BGCD} by considering compound Poisson\nmodels (then infinitely divisible). In this direction our results show the\nrole of covariation of $d$-dimensional L\\'{e}vy processes as an alternative to\nempirical covariance processes.\n\nFor convenience of the reader, and since the material and notation in the\nliterature is disperse and incomplete, we include Section 2 with a review on\npreliminaries on complex matrix semimartingales and matrix valued L\\'{e}vy\nprocesses that are used later on in this paper.\n\n\\section{Preliminaries on matrix semimartingales and matrix L\\'{e}vy \\newline\nprocesses}\n\nLet $\\mathbb{M}_{d\\times q}=\\mathbb{M}_{d\\times q}\\left(  \\mathbb{C}\\right)  $\ndenote the linear space of $d\\times q$ matrices with complex (respectively\nreal) entries with scalar product $\\left \\langle A,B\\right \\rangle\n=~\\mathrm{tr}\\left(  AB^{\\ast}\\right)  $ and the Frobenius norm $\\left \\Vert\nA\\right \\Vert =\\left[  \\mathrm{tr}\\left(  AA^{\\ast}\\right)  \\right]  ^{1\/2}$\nwhere $\\mathrm{tr}$ denotes the (non normalized) trace. If $q=d,$ we write\n$\\mathbb{M}_{d}=\\mathbb{M}_{d\\times d}$. The set of Hermitian random matrices\nin $\\mathbb{M}_{d}$ is denoted by $\\mathbb{H}_{d}$. Likewise, let\n$\\mathbb{U}_{d\\times q}=\\mathbb{U}_{d\\times q}\\left(  \\mathbb{C}\\right)\n=\\left \\{  U\\in \\mathbb{M}_{d\\times q}:U^{\\ast}U=\\mathrm{I}_{q}\\right \\}  .$ If\n$q=d,$ $\\mathbb{U}_{d}=\\mathbb{U}_{d\\times d}$.\n\nWe denote by $\\mathbb{H}_{d(1)}$ the set of matrices in $\\mathbb{H}_{d}$ of\nrank one and by $\\mathbb{H}_{d}^{+}$ ($\\overline{\\mathbb{H}}_{d}^{+}$) the set\nof positive (nonnegative) definite matrices in $\\mathbb{H}_{d}$. Likewise\n$\\mathbb{H}_{d(1)}^{+}=\\mathbb{H}_{d(1)}\\cap \\overline{\\mathbb{H}}_{d}^{+}$ is\nthe closed cone of $d\\times d$ nonnegative definite matrices of rank one. Let\n$\\mathbb{S}(\\mathbb{H}_{d(1)})$ denote the unit sphere of $\\mathbb{H}_{d(1)}$.\n\n\\begin{remark}\n\\label{Decomp}(a) Every $V\\in \\mathbb{H}_{d(1)}^{+}$ can be written as\n$V=xx^{\\ast}$ where $x\\in \\mathbb{C}^{d}$. One can see that $x$ is unique if we\nrestrict $x$ to the set $C_{+}^{d}=\\{x=\\left(  x_{1},x_{2},\\ldots\n,x_{d}\\right)  \\allowbreak:\\allowbreak x_{1}\\allowbreak \\geq \\allowbreak\n0,\\allowbreak$ $x_{j}\\allowbreak \\in \\allowbreak \\mathbb{C},\\allowbreak$\n$j\\allowbreak=2,\\allowbreak...,\\allowbreak d\\}$.\n\n(b) Every $V\\in \\mathbb{H}_{d\\left(  1\\right)  }$ can be written as $V=\\lambda\nuu^{\\ast}$ where $\\lambda$ the eigenvalue of $V$ and $u$ is a unitary vector\nin $\\mathbb{C}^{d}$. In this representation the $d\\times d$ matrix $uu^{\\ast}$\nis unique.\n\\end{remark}\n\n\\paragraph{Covariation of complex matrix semimartingales}\n\nAn $\\mathbb{M}_{d\\times q}$-valued process $X=\\left \\{  (x_{ij})(t)\\right \\}\n_{t\\geq0}$ is a matrix semimartingale if $x_{ij}(t)$ is a complex\nsemimartingale for each $i=1,...,d,j=1,...,q.$ Let $X=\\left \\{  (x_{ij\n)(t)\\right \\}  _{t\\geq0}$ $\\in \\mathbb{M}_{d\\times q}$ and $Y=\\left \\{\n(y_{ij})(t)\\right \\}  _{t\\geq0}\\in \\mathbb{M}_{q\\times r}$ be semimartingales.\nSimilar to the case of matrices with real entries in \\cite{BNSe07}, we define\nthe matrix covariation of $X$ and $Y$ as the $\\mathbb{M}_{d\\times r}$-valued\nprocess $\\left[  X,Y\\right]  :=\\left \\{  \\left[  X,Y\\right]  (t):t\\geq\n0\\right \\}  $ with entrie\n\\begin{equation}\n\\left[  X,Y\\right]  _{ij}(t)=\\sum \\limits_{k=1}^{q}\\left[  x_{ik\n,y_{kj}\\right]  (t)\\text{,} \\label{DefCov\n\\end{equation}\nwhere $\\left[  x_{ik},y_{kj}\\right]  (t)$ is the covariation of the\n$\\mathbb{C}$-valued semimartingales $\\left \\{  x_{ik}(t)\\right \\}  _{t\\geq0}$\nand $\\left \\{  x_{kj}(t)\\right \\}  _{t\\geq0}$; see \\cite[pp 83]{Pr04}. One has\nthe decomposition into a continuous part and a pure jump part as follow\n\\begin{equation}\n\\left[  X,Y\\right]  (t)=\\left[  X^{c},Y^{c}\\right]  (t)+\\sum_{s\\leq t}\\left(\n\\Delta X(s)\\right)  \\left(  \\Delta Y(s)\\right)  \\text{,} \\label{ForCov\n\\end{equation}\nwhere $\\left[  X^{c},Y^{c}\\right]  _{ij}(t):=\\sum \\nolimits_{k=1}^{q}\\left[\nx_{ik}^{c},y_{kj}^{c}\\right]  (t).$ We recall that for any semimartingale $x$,\nthe process $x^{c}$ is the $a.s.$ unique continuous local martingale $m$ such\nthat $\\left[  x-m\\right]  $ is purely discontinuous.\n\nWe will use the facts that $\\left[  X\\right]  =\\left[  X,X^{\\ast}\\right]  $ is\na nonnegative definite $d\\times d$ matrix, that $\\left[  X,Y\\right]  ^{\\top\n}=\\left[  Y^{\\top},X^{\\top}\\right]  $ and that for any nonrandom matrices\n$A\\in \\mathbb{M}_{m\\times d},C\\in \\mathbb{M}_{r\\times n}$ and semimartingales\n$X\\in \\mathbb{M}_{d\\times q},Y\\in \\mathbb{M}_{q\\times r}$\n\\begin{equation}\n\\left[  AX,YC\\right]  =A\\left[  X,Y\\right]  C\\text{.} \\label{CovBil\n\\end{equation}\n\n\nThe natural example of a continuous semimartingale is the standard complex\n$d\\times q$ matrix Brownian motion $B=\\left \\{  B(t)\\right \\}  _{t\\geq\n0}=\\left \\{  b_{jl}(t)\\right \\}  _{t\\geq0}$ consisting of independent\n$\\mathbb{C}$-valued Brownian motions $b_{jl}(t)=\\operatorname{Re\n(b_{jl}(t))+\\mathrm{i}\\operatorname{Im}(b_{jl}(t))$ where $\\operatorname{Re\n(b_{jl}(t)),\\operatorname{Im}(b_{jl}(t))$ are independent one-dimensional\nBrownian motions with common variance $t\/2$. Then we have $\\left[  B,B^{\\ast\n}\\right]  ^{ij}(t)=\\sum \\nolimits_{k=1}^{q}\\left[  b_{ik},\\overline{b\n_{jk}\\right]  (t)=qt\\delta_{ij}$ and hence the matrix quadratic variation of\n$B$ is given by the $d\\times d$ matrix process\n\\begin{equation}\n\\left[  B,B^{\\ast}\\right]  (t)=qt\\mathrm{I}_{d}\\text{.} \\label{CovCB\n\\end{equation}\nThe case $q=1$ corresponds to the $\\mathbb{C}^{d}$-valued standard Brownian\nmotion $B$.$\\ $We observe this corresponds to $\\left[  B,B^{\\ast}\\right]\n_{t}=t\\mathrm{I}_{d}$ instead of the common $2t\\mathrm{I}_{d}$ used in the literature.\n\nOther examples of complex matrix semimartingales are L\\'{e}vy processes\nconsidered next.\n\n\\paragraph{Complex matrix L\\'{e}vy processes}\n\nAn infinitely divisible random matrix $M$ in $\\mathbb{M}_{d\\times q}$ is\ncharacterized by the L\\'{e}vy-Khintchine representation of its Fourier\ntransform $\\mathbb{E}\\mathrm{e}^{\\mathrm{itr}(\\Theta^{\\ast}M)}\\allowbreak\n\\ =\\  \\allowbreak \\exp(\\psi(\\Theta))$ with Laplace exponent\n\\begin{equation}\n\\psi(\\Theta)={}\\mathrm{itr}(\\Theta^{\\ast}\\Psi \\text{ }){}-{}\\frac{1\n{2}\\mathrm{tr}\\left(  \\Theta^{\\ast}\\mathcal{A}\\Theta^{\\ast}\\right)  {}+{\n\\int_{\\mathbb{M}_{d\\times q}}\\left(  \\mathrm{e}^{\\mathrm{itr}(\\Theta^{\\ast\n\\xi)}{}-1{}-\\mathrm{i}\\frac{\\mathrm{tr}(\\Theta^{\\ast}\\xi)}{1+\\left \\Vert\n\\xi \\right \\Vert ^{2}}{}\\right)  \\nu(\\mathrm{d}\\xi),\\  \\Theta \\in \\mathbb{M\n_{d\\times q}, \\label{LKRgen\n\\end{equation}\nwhere $\\mathcal{A}:\\mathbb{M}_{q\\times d}\\rightarrow \\mathbb{M}_{d\\times q}$ is\na positive symmetric linear operator $($i.e. $\\mathrm{tr}\\left(  \\Phi^{\\ast\n}\\mathcal{A}\\Phi^{\\ast}\\right)  \\geq0$ for $\\Phi \\in \\mathbb{M}_{d\\times q}$ and\n$\\mathrm{tr}\\left(  \\Theta_{2}^{\\ast}\\mathcal{A}\\Theta_{1}^{\\ast}\\right)\n=\\mathrm{tr}\\left(  \\Theta_{1}^{\\ast}\\mathcal{A}\\Theta_{2}^{\\ast}\\right)  $\nfor $\\Theta_{1},\\Theta_{2}\\in \\mathbb{M}_{d\\times q})$, $\\nu$ is a measure on\n$\\mathbb{M}_{d\\times q}$ (the L\\'{e}vy measure) satisfying $\\nu(\\{0\\})=0$ and\n$\\int_{\\mathbb{M}_{d\\times q}}(1\\wedge \\left \\Vert x\\right \\Vert ^{2\n)\\nu(\\mathrm{d}x)<\\infty$, and $\\Psi \\in \\mathbb{M}_{d\\times q}$. The triplet\n$(\\mathcal{A},\\nu,\\Psi)$ uniquely determines the distribution of $M$.\n\n\\begin{remark}\n\\label{ObsGaPart}The notation $\\mathcal{A}\\Theta^{\\ast}$ means the linear\noperator $\\mathcal{A}$ from $\\mathbb{M}_{q\\times d}$ to $\\mathbb{M}_{d\\times\nq}$ acting on $\\Theta^{\\ast}\\in \\mathbb{M}_{q\\times d}$. Some interesting\nexamples of $\\mathcal{A}$ and its corresponding matrix Gaussian distributions are:\n\n(a) $\\mathcal{A}\\Theta^{\\ast}=\\Theta$. This corresponds to a Gaussian matrix\ndistribution invariant under left and right unitary transformations in\n$\\mathbb{U}_{d}$ and $\\mathbb{U}_{q}$, respectively.\n\n(b) $\\mathcal{A}\\Theta^{\\ast}=\\Sigma_{1}\\Theta \\Sigma_{2}$ for $\\Sigma_{1}\\in$\n$\\overline{\\mathbb{H}}_{d}^{+}$ and $\\Sigma_{2}\\in \\overline{\\mathbb{H}\n_{q}^{+}$. In this case the corresponding matrix Gaussian distribution is\ndenoted by $\\mathrm{N}_{d\\times q}(0,\\Sigma_{1}\\otimes \\Sigma_{2})$ \\ and\n$\\Sigma_{1}\\otimes \\Sigma_{2}$ is called a Kronecker covariance. It holds that\nif $N$ has the distribution $\\mathrm{N}_{d\\times q}(0,\\mathrm{I}_{d}{\n\\otimes \\mathrm{I}_{q})$, then $\\Sigma_{1}^{1\/2}N\\Sigma_{2}^{1\/2}$ has\ndistribution $\\mathrm{N}_{d\\times q}(0,\\Sigma_{1}\\otimes \\Sigma_{2})$.\n\n(c) When $q=d$, $\\mathcal{A}\\Theta^{\\ast}=$ $\\mathrm{tr}(\\Theta)\\mathrm{I\n_{d}$ is the covariance operator of the Gaussian random matrix $g\\mathrm{I\n_{d}$ where $g$ is a one-dimensional random variable with a standard Gaussian distribution.\n\\end{remark}\n\nLet $\\mathbb{S}_{d\\times q}$ be the unit sphere of $\\mathbb{M}_{d\\times q}$\nand let $\\mathbb{M}_{d\\times q}^{0}=\\mathbb{M}_{d\\times q}\\backslash \\{0\\}$. If\n$\\nu$ is a L\\'{e}vy measure on $\\mathbb{M}_{d\\times q}$, then there are a\nmeasure $\\lambda$ on $\\mathbb{S}_{d\\times q}$ with $\\lambda(\\mathbb{S\n_{d\\times q})\\geq0$ and a measure $\\nu_{\\xi}$ for each $\\xi \\in \\mathbb{S\n_{d\\times q}$ with $\\nu_{\\xi}((0,\\infty))>0$ such that\n\\[\n\\nu(E)=\\int_{\\mathbb{S}_{d\\times q}}\\lambda(\\mathrm{d}\\xi)\\int_{(0,\\infty\n)}1_{E}(u\\xi)\\nu_{\\xi}(\\mathrm{d}u),\\qquad E\\in \\mathcal{B}(\\mathbb{M}_{d\\times\nq}^{0}).\n\\]\nWe call $(\\lambda,\\nu_{\\xi})$ a polar decomposition of $\\nu$. When $d=q=1$,\n$\\nu$ is a L\\'{e}vy measure on $\\mathbb{R}$ and $\\lambda$ is a measure in the\nunit sphere $\\mathbb{S}_{1\\times1}=\\left \\{  -1,1\\right \\}  $ of $\\mathbb{R}$.\n\nAny $\\mathbb{M}_{d\\times q}$-valued L\\'{e}vy process $L=\\left \\{  L(t)\\right \\}\n_{t\\geq0}$ with triplet $(\\mathcal{A},\\nu,\\Psi)$ is a semimartingale with the\nL\\'{e}vy-It\\^{o} decomposition\n\\begin{equation}\nL(t)=t\\Psi+B_{\\mathcal{A}}(t)+\\int_{[0,t]}\\int_{\\left \\Vert V\\right \\Vert \\leq\n1}V\\widetilde{J}_{L}(\\mathrm{d}s\\mathrm{,d}V)+\\int_{[0,t]}\\int_{\\left \\Vert\nV\\right \\Vert >1}VJ_{L}(\\mathrm{d}s,\\mathrm{d}V)\\text{, }t\\geq0, \\label{LID\n\\end{equation}\nwhere:\n\n(a) $\\left \\{  B_{\\mathcal{A}}(t)\\right \\}  _{t\\geq0}$ is a $\\mathbb{M}_{d\\times\nq}$-valued Brownian motion with covariance $\\mathcal{A}$, i.e. it is a\nL\\'{e}vy process with continuous sample paths (a.s.) and each $B_{\\mathcal{A\n}(t)$ is centered Gaussian with\n\\[\n\\mathbb{E}\\left \\{  \\mathrm{tr(}\\Theta_{1}^{\\ast}B_{\\mathcal{A}}(t){\n)\\mathrm{tr}\\left(  \\Theta_{2}^{\\ast}B_{\\mathcal{A}}(s){}\\right)  {}\\right \\}\n=\\min(s,t)\\mathrm{tr}\\left(  \\Theta_{1}^{\\ast}\\mathcal{A}\\Theta_{2}^{\\ast\n}\\right)  {}\\text{for each }\\Theta_{1},\\Theta_{2}\\in \\mathbb{M}_{d\\times q},\n\\]\n\n\n(b) $J_{L}(\\cdot,\\cdot)$ is the Poisson random measure of jumps on\n$[0,\\infty)\\times \\mathbb{M}_{d\\times q}^{0}$. That is, $J_{L}(t,E)=\\# \\{(0\\leq\ns\\leq t:\\allowbreak \\Delta L_{s}\\in E\\},$ $E$ $\\in \\mathbb{M}_{d\\times q}^{0},$\nwith intensity measure $Leb\\otimes \\nu$, and independent of $\\left \\{\nB_{\\mathcal{A}}(t)\\right \\}  _{t\\geq0}$,\n\n(c) $\\widetilde{J}_{L}$ is the compensator measure of $J_{L}$, i.e.\n\\[\n\\widetilde{J}_{L}(\\mathrm{d}t,\\mathrm{d}V)=J_{L}(\\mathrm{d}t,\\mathrm{d\nV)-\\mathrm{d}t\\nu(\\mathrm{d}V);\n\\]\nsee for example \\cite{Ap07} for the most general case of L\\'{e}vy processes\nwith values in infinite dimensional Banach spaces.\n\nAn $\\mathbb{M}_{d\\times q}$-valued L\\'{e}vy process $L=\\left \\{  L(t)\\right \\}\n_{t\\geq0}$ has bounded variation if and only if its L\\'{e}vy-It\\^{o}\ndecomposition takes the form\n\\begin{equation}\nL(t)=t\\Psi_{0}+\\int_{[0,t]}\\int_{\\mathbb{M}_{d\\times q}^{0}}VJ_{L\n(\\mathrm{d}s,\\mathrm{d}V)=t\\Psi_{0}+\\sum_{s\\leq t}\\Delta L(s)\\text{, }t\\geq0,\n\\label{LIFV\n\\end{equation}\nwhere $\\Psi_{0}=$ $\\Psi-\\int_{\\left \\Vert V\\right \\Vert \\leq1}V\\nu\n(\\mathrm{d}V).$\n\nThe matrix quadratic variation (\\ref{ForCov}) of $L$ is given by the\n$\\overline{\\mathbb{H}}_{d}^{+}$-valued process\n\\begin{equation}\n\\lbrack L](t)=\\left[  B_{\\mathcal{A}},B_{\\mathcal{A}}^{\\ast}\\right]\n(t)+\\int_{[0,t]}\\int_{\\mathbb{M}_{d\\times q}^{0}}VV^{\\ast}J_{L}(\\mathrm{d\ns,\\mathrm{d}V)=\\left[  B_{\\mathcal{A}},B_{\\mathcal{A}}^{\\ast}\\right]\n(t)\\mathcal{+}\\sum_{s\\leq t}\\Delta L(s)\\Delta L(s)^{\\ast}. \\label{QVLP\n\\end{equation}\n\n\nIn\\ Section 3 we prove a partial converse of the last result in the case\n$q=1.$\n\n\\begin{remark}\n\\label{ObsGaPartqv} On the lines of Remark \\ref{ObsGaPart} we have the\nfollowing observations for the quadratic variation of the continuous part in\n(\\ref{QVLP}):\n\n(a) When $\\mathcal{A}\\Theta^{\\ast}=\\Theta,$ $\\left[  B_{\\mathcal{A\n},B_{\\mathcal{A}}^{\\ast}\\right]  (t)=qt\\mathrm{I}_{d}$. This follows from\n(\\ref{CovCB}) since $B_{\\mathcal{A}}(t)$ is a standard complex $d\\times q$\nmatrix Brownian motion.\n\n(b) When $\\mathcal{A}\\Theta^{\\ast}=\\Sigma_{1}\\Theta \\Sigma_{2}$ for $\\Sigma\n_{1}\\in$ $\\overline{\\mathbb{H}}_{d}^{+}$ and $\\Sigma_{2}\\in \\overline\n{\\mathbb{H}}_{q}^{+}$, we have $B_{\\mathcal{A}}(t)=\\Sigma_{1}^{1\/2\nB(t)\\Sigma_{2}^{1\/2}$ where $B=\\left \\{  B(t)\\right \\}  _{t\\geq0}$ is a standard\ncomplex $d\\times q$ matrix Brownian motion. Then, using (\\ref{CovBil}) we\nhave\n\\[\n\\left[  B_{\\mathcal{A}},B_{\\mathcal{A}}^{\\ast}\\right]  (t)=\\left[  \\Sigma\n_{1}^{1\/2}B\\Sigma_{2}^{1\/2},\\Sigma_{2}^{1\/2}B^{\\ast}\\Sigma_{1}^{1\/2}\\right]\n(t)=\\Sigma_{1}^{1\/2}\\left[  B\\Sigma_{2}^{1\/2},\\Sigma_{2}^{1\/2}B^{\\ast}\\right]\n(t)\\Sigma_{1}^{1\/2}=t\\mathrm{tr}(\\Sigma_{2})\\Sigma_{1},\n\\]\nwhere we have also used the easily checked fact $\\left[  B\\Sigma_{2\n^{1\/2},\\Sigma_{2}^{1\/2}B^{\\ast}\\right]  (t)=t\\mathrm{tr}(\\Sigma_{2})I_{d}$.\n\n(c) When $q=d$ and $\\mathcal{A}\\Theta^{\\ast}=$ $\\mathrm{tr}(\\Theta\n)\\mathrm{I}_{d}$, we have $\\left[  B_{\\mathcal{A}},B_{\\mathcal{A}}^{\\ast\n}\\right]  (t)=t\\mathrm{I}_{d}$ since $B_{\\mathcal{A}}(t)=b(t)\\mathrm{I}_{d}$\nwhere $b=\\left \\{  b(t)\\right \\}  _{t\\geq0}$ is a one-dimensional Brownian motion.\n\\end{remark}\n\nThe extension of the notion of a real subordinator to the matrix case relies\non cones. A cone $K$ is a nonempty, closed, convex subset of $\\mathbb{M\n_{d\\times q}$ such that if $A\\in K$ and $\\alpha \\geq0$ imply $\\alpha A\\in K$. A\ncone $K$ determines a partial order in $\\mathbb{M}_{d\\times q}$ by defining\n$V_{1}\\leq_{K}V_{2}$ for $V_{1},V_{2}\\in \\mathbb{M}_{d\\times q}$ whenever\n$V_{2}-V_{1}\\in K$. A $\\mathbb{M}_{d\\times q}$-valued L\\'{e}vy process\n$L=\\left \\{  L(t)\\right \\}  _{t\\geq0}$ is $K$- increasing if $L(t_{1})\\leq\n_{K}L(t_{2})$ for every $t_{1}<t_{2}$ almost surely. A $K$-increasing L\\'{e}vy\nprocess with values in $\\mathbb{M}_{d\\times q}$ is called a matrix\nsubordinator. It is easy to see that if $L=\\left \\{  L(t)\\right \\}  _{t\\geq0}$\nis a L\\'{e}vy process in $\\mathbb{M}_{d\\times q}$ then $L$ is a subordinator\nif and only if $L$ takes values in $K$. In this sense the matrix quadratic\nvariation L\\'{e}vy process in (\\ref{QVLP}) with values in the cone\n$\\overline{\\mathbb{H}}_{d}^{+}$ is a matrix subordinator$.$\n\n\\paragraph{Approximation of L\\'{e}vy processes}\n\nThe following are useful results on the sample path approximation of complex\nmatrix L\\'{e}vy processes; see \\cite[Th 15.17]{Ka} and \\cite[Th. 8.7]{Sato1}.\nThey follow from their corresponding real vector case by the usual\nidentification of $\\mathbb{M}_{d\\times q}\\rightarrow$ $\\mathbb{R}^{2dq}$ via\n$A\\rightarrow \\mathrm{vec}(A),$ $A\\in$ $\\mathbb{M}_{d\\times q}$ and the fact\nthat $\\mathrm{tr}\\left(  A^{\\ast}B\\right)  =\\mathrm{vec}(A)^{\\ast\n\\mathrm{vec}(B)$, where $\\mathrm{vec}(A)$ is the $dq$ column complex vector\nobtained by stacking the columns of $A$ one down each other.\n\n\\begin{proposition}\n\\label{convproc}Let $L$ and $L^{n}$ $n=1,2,...$ be complex matrix L\\'{e}vy\nprocesses in $\\mathbb{M}_{d\\times q}$ with $L^{n}(1)\\allowbreak \\overset\n{\\mathcal{L}}{\\rightarrow}L(1)$. Then there exist processes $\\tilde{L}^{n}$\nwith the same distribution that $L^{n}$ such tha\n\\[\n\\sup_{0\\leq s\\leq t}\\left \\vert \\tilde{L}^{n}(s)-L(s)\\right \\vert \\overset{\\Pr\n}{\\longrightarrow}0,\\quad \\forall t\\geq0.\n\\]\n\n\\end{proposition}\n\n\\begin{proposition}\n\\label{convternas}Let $M^{n},n=1,2,...$ be infinitely divisible random\nmatrices in $\\mathbb{M}_{d\\times q}$ with triplet $(\\mathcal{A}^{n},\\nu\n^{n},\\Psi^{n})$. Let $M$ be a random matrix in $\\mathbb{M}_{d\\times q}$. Then\n$M^{n}\\overset{\\mathcal{L}}{\\rightarrow}M$ if and only if $M$ is infinitely\ndivisible whose triplet $(\\mathcal{A},\\nu,\\Psi)$ satisfies the following three\nconditions:\\newline a) If $f:\\mathbb{M}_{d\\times q}\\rightarrow \\mathbb{M\n_{d\\times q}$ is bounded and continuous function vanishing in a neighborhood\nof $0$ the\n\\[\n\\lim_{n\\rightarrow \\infty}\\int \\nolimits_{\\mathbb{M}_{d\\times q}}f(\\xi)\\nu\n^{n}(\\mathrm{d}\\xi)=\\int \\nolimits_{\\mathbb{M}_{d\\times q}}f(\\xi)\\nu\n(\\mathrm{d}\\xi)\\text{.\n\\]\nb) Define the positive symmetric operator $\\mathcal{A}^{n,\\epsilon\n:\\mathbb{M}_{q\\times d}\\rightarrow \\mathbb{M}_{d\\times q}$ b\n\\[\n\\mathrm{tr}\\left(  \\Theta^{\\ast}\\mathcal{A}^{n,\\epsilon}\\Theta^{\\ast}\\right)\n=\\mathrm{tr}\\left(  \\Theta^{\\ast}\\mathcal{A}^{n}\\Theta^{\\ast}\\right)\n+\\int \\nolimits_{\\left \\Vert \\xi \\right \\Vert \\leq \\varepsilon}\\left \\vert\n\\mathrm{tr}\\left(  \\Theta^{\\ast}\\xi \\right)  \\right \\vert ^{2}\\nu_{n\n(\\mathrm{d}\\xi)\\quad \\text{for }\\Theta \\in \\mathbb{M}_{d\\times q}\\text{.\n\\]\nThen\n\\[\n\\lim_{\\varepsilon \\downarrow0}\\limsup_{n\\rightarrow \\infty}\\left \\vert\n\\mathrm{tr}\\left(  \\Theta^{\\ast}\\mathcal{A}^{n,\\epsilon}\\Theta^{\\ast}\\right)\n-\\mathrm{tr}\\left(  \\Theta^{\\ast}\\mathcal{A}\\Theta^{\\ast}\\right)  \\right \\vert\n=0,\\quad \\text{for }\\Theta \\in \\mathbb{M}_{d\\times q}\\text{.\n\\]\nc) $\\Psi^{n}\\rightarrow \\Psi$.\n\\end{proposition}\n\n\\section{BGCD random matrix ensembles}\n\nWe now consider the matrix L\\'{e}vy processes associated to the BGCD matrix\nensembles $(M_{d})_{d\\geq1}$ mentioned in the introduction.\n\nWhen $\\mu$ is the standard Gaussian distribution, $M_{d}$ is a Gaussian\nunitary invariant random matrix, $\\Lambda(\\mu)$ is the semicircle distribution\nand $\\left \\{  M_{d}(t)\\right \\}  _{t\\geq0}$ is the Hermitian matrix valued\nprocess given by $M_{d}(t)=\\left(  1\/\\sqrt{d+1}\\right)  (B(t)+dg(t)\\mathrm{I\n_{d})$ where $B\\left(  t\\right)  $ is a $d\\times d$ Hermitian matrix Brownian\nmotion independent of the one-dimensional Brownian motion $g(t);$ see\n\\cite[Remark 3.5]{BG}.\n\nLikewise, if $\\mu$ is the Poisson distribution with parameter $\\lambda>0,$\n$\\left \\{  M_{d}(t)\\right \\}  _{t\\geq0}$ is the $d\\times d$ matrix compound\nPoisson process $M_{d}(t)=\\sum_{k=1}^{N(t)}u_{k}^{d}u_{k}^{d\\ast}$ where\n$\\left \\{  u_{k}^{d}\\right \\}  _{k\\geq1}$ is a sequence of independent uniformly\ndistributed random vectors on the unit sphere of $\\mathbb{C}^{d}$ independent\nof the Poisson process $\\left \\{  N(t)\\right \\}  _{t\\geq0}$, and $\\Lambda(\\mu)$\nis the Marchenko-Pastur distribution of parameter $\\lambda>0$; see\n\\cite[Remark 3.2]{BG}. We observe that in this case $\\left \\{  M_{d\n(t)\\right \\}  _{t\\geq0}$ is a matrix covariation (quadratic) process rather\nthan a covariance matrix process as in the Wishart or other empirical\ncovariance processes.\n\nProposition \\ref{polar} below collects computations in \\cite{BG}, \\cite{CD}\nand \\cite{DRA} to summarize the L\\'{e}vy triplet of a general BGCD matrix\nensemble in an explicit manner. Let $\\nu|_{(0,\\infty)}$ and $\\nu\n|_{(-\\infty,0)}$ denote the corresponding restrictions to $\\left(\n0,+\\infty \\right)  $ and $\\left(  -\\infty,0\\right)  $ for any L\\'{e}vy measure\n$\\nu$, respectively.\n\n\\begin{proposition}\n\\label{polar}Let $\\mu \\ $be an infinitely divisible distribution in\n$\\mathbb{R}$ with L\\'{e}vy triplet $(a^{2}$,$\\mathcal{\\psi},\\nu)$ and let\n$(M_{d})_{d\\geq1}$ be a BGCD matrix ensemble for $\\Lambda(\\mu)$. Then, for\neach $d\\geq1$ $M_{d}$ has the L\\'{e}vy-Khintchine representation\n(\\ref{LKRgen}) with L\\'{e}vy triplet $(\\mathcal{A}_{d},\\Psi_{d},\\nu_{d})$ where\n\na) $\\Psi_{d}=\\mathcal{\\psi}\\mathrm{I}_{d}$\n\nb)\n\\begin{equation}\n\\mathcal{A}_{d}\\Theta=a^{2}\\frac{1}{d+1}(\\Theta+\\mathrm{tr}(\\Theta\n)\\mathrm{I}_{d}),\\quad \\Theta \\in \\mathbb{H}_{d}. \\label{GPBGCD\n\\end{equation}\n\n\nc)\n\\begin{equation}\n\\nu_{d}\\left(  E\\right)  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})}\\int\n_{0}^{\\infty}1_{E}\\left(  rV\\right)  \\nu_{V}\\left(  \\mathrm{d}r\\right)\n\\Pi \\left(  \\mathrm{d}V\\right)  \\text{,\\quad}E\\in \\mathcal{B}\\left(\n\\mathbb{H}_{d}\\backslash \\left \\{  0\\right \\}  \\right)  \\text{,} \\label{PDBGCD\n\\end{equation}\nwhere $\\nu_{V}=\\nu|_{(0,\\infty)}$ or $\\nu|_{(-\\infty,0)}$ according to\n$V\\geq0$ or\\ $V\\leq0$ and $\\Pi$ is a measure on $\\mathbb{S}(\\mathbb{H\n_{d(1)})$ such that\n\\begin{equation}\n\\Pi \\left(  D\\right)  =\\int \\limits_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap\n\\overline{\\mathbb{H}}_{d}^{+}}\\int \\limits_{\\left \\{  -1,1\\right \\}  \n1_{D}\\left(  tV\\right)  \\lambda \\left(  \\mathrm{d}t\\right)  \\omega_{d}\\left(\n\\mathrm{d}V\\right)  \\text{,\\quad}D\\in \\mathcal{B}\\left(  \\mathbb{S\n(\\mathbb{H}_{d(1)})\\right)  \\text{,} \\label{pi\n\\end{equation}\nwhere $\\lambda$ is the spherical measure of $\\nu$ and $\\omega_{d}$ is the\nprobability measure on $\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H\n}_{d}^{+}$ induced by the transformation $u\\rightarrow V=uu^{\\ast}$, where $u$\nis a uniformly distributed column random vector in the unit sphere of\n$\\mathbb{C}^{d}$.\n\\end{proposition}\n\n\\begin{proof}\n(a) It follows from the first term in the L\\'{e}vy exponent of $M_{d}$ in page\n$635$ of \\cite{CD}, where the notation $\\Lambda_{d}(\\mu)$ is used for the\ndistribution of $M_{d}$. For (b), the form of the covariance operator\n$\\mathcal{A}_{d}$ was implicitly computed in the first example in Section II.C\nof \\cite{CD}. Finally, the polar decomposition of the L\\'{e}vy measure\n(\\ref{PDBGCD}) was found in \\cite{DRA}.\n\\end{proof}\n\nThe L\\'{e}vy-It\\^{o} decomposition of the L\\'{e}vy process associated to the\nBGCD model $M_{d}$ is given by\n\\begin{equation}\nM_{d}(t)=\\mathcal{\\psi}td\\mathrm{I}_{d}+B_{\\mathcal{A}_{d}}(t)+\\int\n_{[0,t]}\\int_{\\left \\{  \\left \\Vert V\\right \\Vert \\leq1\\right \\}  \\cap\n\\mathbb{H}_{d(1)}}V\\widetilde{J}_{d}(\\mathrm{d}s\\mathrm{,d}V)+\\int_{[0,t]\n\\int_{\\left \\{  \\left \\Vert V\\right \\Vert >1\\right \\}  \\cap \\mathbb{H}_{d(1)\n}VJ_{d}(\\mathrm{d}s,\\mathrm{d}V)\\text{,} \\label{LIDbgcd\n\\end{equation}\nwhere $t\\geq0$, $\\mathcal{A}_{d}\\Theta=a^{2}\\frac{1}{d+1}(\\Theta\n+\\mathrm{tr}(\\Theta)\\mathrm{I}_{d})$, $J_{d}(t,E)=\\# \\left \\{  0\\leq s\\leq\nt:\\Delta M_{d}(s)\\in E\\right \\}  =J_{d}(t,E\\cap \\mathbb{H}_{d(1)})$ for any\nmeasurable $E$ $\\in \\mathbb{H}_{d}\\backslash \\{0\\}$. Its quadratic variation is\nobtained by (\\ref{QVLP}) as the matrix subordinato\n\\begin{subequations}\n\\begin{equation}\n\\left[  M_{d}\\right]  (t)=a^{2}t\\mathrm{I}_{d}+\\int_{[0,t]}\\int_{\\mathbb{H\n_{d(1)}\\backslash \\{0\\}}VV^{\\ast}J_{d}(\\mathrm{d}s,\\mathrm{d}V)=a^{2\nt\\mathrm{I}_{d}+\\sum_{s\\leq t}\\Delta M_{d}(s)\\left(  \\Delta M_{d}(s)\\right)\n^{\\ast}.\\nonumber\n\\end{equation}\n\n\\end{subequations}\n\\begin{remark}\nIt is possible to obtain BGCD models of symmetric random matrices rather than\nHermitian. Indeed, slight changes in the proof of \\cite[Theorem 3.1]{BG} give\nfor each $d\\geq1$, a $d\\times d$ real symmetric random matrix $M_{d}$ with\northogonal invariant infinitely divisible matrix distribution. The asymptotic\nspectral distribution of the corresponding Hermitian and symmetric ensembles\nis the same, similarly as the semicircle distribution is the asymptotic\nspectral distribution for the Gaussian Unitary Ensemble and Gaussian\nOrthogonal Ensemble.\n\\end{remark}\n\n\\section{Bounded variation case}\n\nIt is well known that the quadratic variation of a one-dimensional L\\'{e}vy\nprocess is a subordinator, see \\cite[Example 8.5]{CT}. The following result\ngives a converse and a generalization to matrix subordinators with rank one\njumps. The one dimensional case is given in \\cite[Lemma 6.5]{Se10}.\n\n\\begin{theorem}\n\\label{Sub} Let $L_{d}=\\left \\{  L_{d}(t):t\\geq0\\right \\}  $ be a L\\'{e}vy\nprocess in $\\overline{\\mathbb{H}}_{d}^{+}$ whose jumps are of rank one almost\nsurely. Then there exists a L\\'{e}vy process $X=\\left \\{  X(t):t\\geq0\\right \\}\n$ in $\\mathbb{C}^{d}$ such that $L_{d}(t)=\\left[  X\\right]  (t)$.\n\\end{theorem}\n\n\\begin{proof}\nWe construct $X$ as a L\\'{e}vy-It\\^{o} decomposition realization. Using\n(\\ref{LIFV}), for each $d\\geq1$, $L_{d}$ is an $\\overline{\\mathbb{H}}_{d}^{+\n$-process of bounded variation with L\\'{e}vy-It\\^{o} decompositio\n\\[\nL_{d}(t)=\\Psi_{0}t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{H\n_{d\\left(  1\\right)  }^{+}\\backslash \\{0\\}}VJ_{L_{d}}(\\mathrm{d}s,\\mathrm{d\nV)\\text{, }t\\geq0\\text{,\n\\]\nwhere $\\Psi_{0}\\in \\mathbb{H}_{d}^{+}$ and $J_{L_{d}}$ is the Poisson random\nmeasure of $L_{d}$. Let $Leb\\otimes \\nu_{L_{d}}$ denote the intensity measure\nof $J_{L_{d}}$.\n\nConsider the cone $C_{+}^{d}=\\left \\{  x=\\left(  x_{1},x_{2},\\ldots\n,x_{d}\\right)  :x_{1}\\geq0,\\text{ }x_{j}\\in \\mathbb{C},\\text{ \nj=2,...,d\\right \\}  $ and let $\\varphi_{+}:\\mathbb{R}_{+}\\times \\mathbb{H\n_{d(1)}^{+}\\rightarrow \\mathbb{R}_{+}\\times C_{+}^{d}$ be defined as\n$\\varphi_{+}\\left(  t,V\\right)  =(t,x)$ where $V=xx^{\\ast}$ and $x\\in\nC_{+}^{d}$. Let $\\overline{\\varphi}_{+}:\\mathbb{H}_{d(1)}^{+}\\rightarrow\nC_{+}^{d}$ be defined by $\\overline{\\varphi}_{+}\\left(  V\\right)  =x$ for\n$V=xx^{\\ast}$ and $x\\in C_{+}^{d}$. By Remark \\ref{Decomp} (a) the functions\n$\\varphi_{+}$ and $\\overline{\\varphi}_{+}$ are well defined.\n\nLet us define $J(\\mathrm{d}s,\\mathrm{d}x)=\\left(  J_{L_{d}}\\circ \\varphi\n_{+}^{-1}\\right)  \\left(  \\mathrm{d}s,\\mathrm{d}x\\right)  $ the random measure\ninduced by the transformation $\\varphi_{+}$ which is a Poisson random measure\non $\\mathbb{R}_{+}\\times C_{+}^{d}$. Observe that $\\mathbb{E}\\left[\nJ(t,F)\\right]  =\\mathbb{E}\\left[  J_{L_{d}}\\circ \\varphi_{+}^{-1}\\left(\n\\left \\{  t\\right \\}  \\times F\\right)  \\right]  =t\\nu_{L_{d}}\\left(\n\\overline{\\varphi}_{+}\\left(  F\\right)  \\right)  =t\\left(  \\nu_{L_{d}\n\\circ \\overline{\\varphi}_{+}^{-1}\\right)  \\left(  F\\right)  $ for\n$F\\allowbreak \\in \\allowbreak \\mathcal{B(}\\allowbreak C_{+}^{d}\\allowbreak\n\\backslash \\left \\{  0\\right \\}  )$. Let us denote $\\nu=\\nu_{L_{d}}\\circ\n\\overline{\\varphi}_{+}^{-1}$ which is a L\\'{e}vy measure on $C_{+}^{d}$ sinc\n\\[\n\\int_{C_{+}^{d}\\backslash \\left \\{  0\\right \\}  }\\left(  1\\wedge \\left \\vert\nx\\right \\vert ^{2}\\right)  \\nu(\\mathrm{d}x)=\\int_{C_{+}^{d}\\backslash \\left \\{\n0\\right \\}  }\\left(  1\\wedge \\left \\vert x\\right \\vert ^{2}\\right)  \\nu_{L_{d\n}\\circ \\overline{\\varphi}_{+}^{-1}(\\mathrm{d}x)\n\\\n\\[\n=\\int_{C_{+}^{d}\\backslash \\left \\{  0\\right \\}  }\\left(  1\\wedge \\mathrm{tr\n\\left(  xx^{\\ast}\\right)  \\right)  \\nu_{L_{d}}\\circ \\overline{\\varphi}_{+\n^{-1}(\\mathrm{d}x)=\\int_{\\mathbb{H}_{d(1)}^{+}\\backslash \\left \\{  0\\right \\}\n}\\left(  1\\wedge \\mathrm{tr}\\left(  V\\right)  \\right)  \\left(  \\nu_{L_{d}\n\\circ \\overline{\\varphi}_{+}^{-1}\\right)  \\circ f^{-1}(\\mathrm{d}V)\n\\\n\\[\n=\\int_{\\mathbb{H}_{d(1)}^{+}\\backslash \\left \\{  0\\right \\}  }\\left(\n1\\wedge \\mathrm{tr}\\left(  V\\right)  \\right)  \\nu_{L_{d}}(\\mathrm{d\nV)<\\infty \\text{,\n\\]\nwhere $\\left(  \\nu_{L_{d}}\\circ \\overline{\\varphi}_{+}^{-1}\\right)  \\circ\nf^{-1}=\\nu_{L_{d}},$ with $f\\left(  x\\right)  =xx^{\\ast}$ and we have used\nthat $\\mathrm{tr}\\left(  V\\right)  \\leq \\alpha \\left \\Vert V\\right \\Vert $ for\nsome constant $\\alpha>0$. Thus $Leb\\otimes \\nu$ is the intensity measure of the\nPoisson random measure $J$.\n\nLet us take the L\\'{e}vy process in $\\mathbb{C}^{d}\n\\begin{equation}\nX(t)=\\left \\vert \\Psi_{0}\\right \\vert ^{1\/2}B_{I}(t)+\\int \\nolimits_{\\left[\n0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert \\leq\n1\\}}x\\widetilde{J}(\\mathrm{d}s,\\mathrm{d}x)+\\int \\nolimits_{\\left[  0,t\\right]\n}\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert >1\\}}xJ(\\mathrm{d\ns,\\mathrm{d}x)\\text{, }t\\geq0\\text{,} \\label{LIX\n\\end{equation}\nwhere $B_{I}$ is a $\\mathbb{C}^{d}$-valued standard Brownian motion with\nquadratic variation $tI_{d}$, (i.e. (\\ref{CovCB}) with $q=1$). Thus the\nquadratic variation of $X$ is given b\n\\[\n\\left[  X\\right]  (t)=\\left[  \\left \\vert \\Psi_{0}\\right \\vert ^{1\/2}B_{I\n,B_{I}^{\\ast}\\left \\vert \\Psi_{0}\\right \\vert ^{1\/2}\\right]  (t)+\\int\n\\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{C}^{d}\\backslash \\{0\\}}xx^{\\ast\n}J(\\mathrm{d}s,\\mathrm{d}x)\n\\\n\\[\n=\\Psi_{0}t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{C}^{d\n\\backslash \\{0\\}}xx^{\\ast}J_{L_{d}}\\circ \\varphi_{+}^{-1}(\\mathrm{d\ns,\\mathrm{d}x)=\\Psi_{0}t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int\n_{\\mathbb{H}_{d(1)}^{+}\\backslash \\left \\{  0\\right \\}  }VJ_{L_{d}}\\circ\n\\varphi_{+}^{-1}\\circ h^{-1}(\\mathrm{d}s,\\mathrm{d}V)\n\\\n\\[\n=\\Psi_{0}t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{H}_{d(1)\n^{+}\\backslash \\left \\{  0\\right \\}  }VJ_{L_{d}}(\\mathrm{d}s,\\mathrm{d\nV)=L_{d}(t),\n\\]\nwhere $J_{L_{d}}\\circ \\varphi_{+}^{-1}\\circ h^{-1}=J_{L_{d}},$ with $h\\left(\nt,x\\right)  =\\left(  t,xx^{\\ast}\\right)  .$\n\\end{proof}\n\n\\smallskip\n\nFor the general bounded variation case we have the following Wiener-Hopf type decomposition.\n\n\\begin{theorem}\n\\label{Bdv}Let $L_{d}=\\left \\{  L_{d}(t):t\\geq0\\right \\}  $ be a L\\'{e}vy\nprocess in $\\mathbb{H}_{d}$ of bounded variation whose jumps are of rank one\nalmost surely. Then there exist L\\'{e}vy processes $X=\\left \\{  X(t):t\\geq\n0\\right \\}  $ and $Y=\\left \\{  Y(t):t\\geq0\\right \\}  $ in $\\mathbb{C}^{d}$ such\nthat\n\\begin{equation}\nL_{d}(t)=\\left[  X\\right]  (t)-\\left[  Y\\right]  (t). \\label{WHTD\n\\end{equation}\nMoreover, $\\left \\{  \\left[  X\\right]  (t):t\\geq0\\right \\}  $ and $\\left \\{\n\\left[  Y\\right]  (t):t\\geq0\\right \\}  $ are independent processes.\n\\end{theorem}\n\n\\begin{proof}\nFor each $d\\geq1$, $L_{d}$ is an $\\mathbb{H}_{d}$-process of bounded variation\nwith L\\'{e}vy-It\\^{o} decompositio\n\\begin{equation}\nL_{d}(t)=\\Psi t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{H\n_{d(1)}\\backslash \\{0\\}}VJ_{L_{d}}(\\mathrm{d}s,\\mathrm{d}V)\\text{, \nt\\geq0\\text{,} \\label{LIDBV\n\\end{equation}\nwhere $\\Psi \\in \\mathbb{H}_{d}$ and $J_{L_{d}}$ is the Poisson random measure of\n$L_{d}$. Let $Leb\\otimes \\nu_{L_{d}}$ denote the intensity measure of\n$J_{L_{d}}$.\n\nFirst we prove that $L_{d}=L_{d}^{1}-L_{d}^{2}$ where $L_{d}^{1}$ and\n$L_{d}^{2}$ are the L\\'{e}vy processes in $\\overline{\\mathbb{H}}_{d}^{+}$\ngiven by (\\ref{Quad1decomp}) and (\\ref{Quad2decomp}).\\newline Every\n$V\\in \\mathbb{H}_{d\\left(  1\\right)  }$ can be written as $V=\\lambda uu^{\\ast}$\nwhere $\\lambda$ the eigenvalue of $V$ and $u$ is a unitary vector in\n$\\mathbb{C}^{d}$. Let us define $\\left \\vert V\\right \\vert =\\left \\vert\n\\lambda \\right \\vert uu^{\\ast}$ and $V^{+}=\\lambda^{+}uu^{\\ast}$, $V^{-\n=\\lambda^{-}uu^{\\ast}$ where $\\lambda^{+}=\\lambda$ if $\\lambda \\geq0$ and\n$\\lambda^{-}=-\\lambda$ if $\\lambda<0$.\n\nLet $\\varphi_{+}:\\mathbb{R}_{+}\\times \\mathbb{H}_{d(1)}\\rightarrow\n\\mathbb{R}_{+}\\times \\mathbb{H}_{d(1)}^{+}$ and $\\varphi_{-}:\\mathbb{R\n_{+}\\times \\mathbb{H}_{d(1)}\\rightarrow \\mathbb{R}_{+}\\times \\mathbb{H\n_{d(1)}^{+}$ be defined as $\\varphi_{+}\\left(  t,V\\right)  =(t,V^{+})$ and\n$\\varphi_{-}\\left(  t,V\\right)  =(t,V^{-})$ respectively. Let $\\overline\n{\\varphi}_{+}:\\mathbb{H}_{d(1)}\\rightarrow \\mathbb{H}_{d(1)}^{+}$ and\n$\\overline{\\varphi}_{-}:\\mathbb{H}_{d(1)}\\rightarrow \\mathbb{H}_{d(1)}^{+}$ be\ndefined as $\\overline{\\varphi}_{+}(V)=V^{+}$ and $\\overline{\\varphi\n_{-}(V)=V^{-}$ respectively. By Remark \\ref{Decomp} (b) the functions\n$\\varphi_{+},$ $\\overline{\\varphi}_{+},$ $\\varphi_{-}$ and $\\overline{\\varphi\n}_{-}$ are well defined and hence $V=\\overline{\\varphi}_{+}(V)-\\overline\n{\\varphi}_{-}(V)$.\n\nLet us define $J^{+}(\\mathrm{d}s,\\mathrm{d}x)=\\left(  J_{L_{d}}\\circ\n\\varphi_{+}^{-1}\\right)  \\left(  \\mathrm{d}s,\\mathrm{d}x\\right)  $ and\n$J^{-}(\\mathrm{d}s,\\mathrm{d}x)=\\left(  J_{L_{d}}\\circ \\varphi_{-}^{-1}\\right)\n\\left(  \\mathrm{d}s,\\mathrm{d}x\\right)  $ the random measures induced by the\ntransformations $\\varphi_{+}$ and $\\varphi_{-}$ respectively, which are\nPoisson random measures both on $\\mathbb{R}_{+}\\times \\mathbb{H}_{d(1)}^{+}$.\nObserve that $\\mathbb{E}\\left[  J^{+}(t,F)\\right]  =\\mathbb{E[}J_{L_{d}\n\\circ \\allowbreak \\varphi_{+}^{-1}(\\allowbreak \\left \\{  t\\right \\}  \\allowbreak\n\\times \\allowbreak F)]=t\\nu_{L_{d}}\\left(  \\overline{\\varphi}_{+}^{-1}\\left(\nF\\right)  \\right)  =t\\left(  \\nu_{L_{d}}\\circ \\overline{\\varphi}_{+\n^{-1}\\right)  \\left(  F\\right)  $ for $F\\in \\mathcal{B}\\left(  \\mathbb{H\n_{d(1)}^{+}\\backslash \\left \\{  0\\right \\}  \\right)  $ and similarly\n$\\mathbb{E}\\left[  J^{-}(t,F)\\right]  =t\\left(  \\nu_{L_{d}}\\circ\n\\overline{\\varphi}_{-}^{-1}\\right)  \\left(  F\\right)  $. Let us denote\n$\\nu_{L_{d}}^{+}=\\nu_{L_{d}}\\circ \\overline{\\varphi}_{+}^{-1}$ and $\\nu_{L_{d\n}^{-}=\\nu_{L_{d}}\\circ \\overline{\\varphi}_{-}^{-1}$. Note that\\ $\\nu_{L_{d\n}^{+}$ is a L\\'{e}vy measure on $\\mathbb{H}_{d(1)}^{+}$ sinc\n\\begin{align*}\n\\infty &  >\\int_{\\mathbb{H}_{d(1)}\\backslash \\left \\{  0\\right \\}  }\\left(\n1\\wedge \\left \\Vert V\\right \\Vert \\right)  \\nu_{L_{d}}(\\mathrm{d}V)\\geq\n\\int_{\\mathbb{H}_{d(1)}\\backslash \\left \\{  0\\right \\}  }\\left(  1\\wedge\n\\left \\Vert \\overline{\\varphi}_{+}(V)\\right \\Vert \\right)  \\nu_{L_{d\n}(\\mathrm{d}V)\\\\\n&  =\\int_{\\mathbb{H}_{d(1)}^{+}\\backslash \\left \\{  0\\right \\}  }\\left(\n1\\wedge \\left \\Vert W\\right \\Vert \\right)  \\nu_{L_{d}}^{+}(\\mathrm{d}W)\\text{.\n\\end{align*}\nHence $Leb\\otimes \\nu_{L_{d}}^{+}$ is the intensity measure of $J^{+}$.\nSimilarly, one can see that $Leb\\otimes \\nu_{L_{d}}^{-}$ is the intensity\nmeasure of $J^{-}$.\n\nThere exist $\\Psi^{+}$ and $\\Psi^{-}$ in $\\mathbb{H}_{d}^{+}$ such that\n$\\Psi=\\Psi^{+}-\\Psi^{-}$. Let us take the L\\'{e}vy processes $X$ and $Y$ in\n$\\mathbb{C}^{d}\n\\[\nX(t)=\\left \\vert \\Psi^{+}\\right \\vert ^{1\/2}B_{I}(t)+\\int \\nolimits_{\\left[\n0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert \\leq\n1\\}}x\\widetilde{J}^{+}(\\mathrm{d}s,\\mathrm{d}x)+\\int \\nolimits_{\\left[\n0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert >1\\}\nxJ^{+}(\\mathrm{d}s,\\mathrm{d}x)\\text{, }t\\geq0\\text{,\n\\\n\\[\nY(t)=\\left \\vert \\Psi^{-}\\right \\vert ^{1\/2}B_{I}(t)+\\int \\nolimits_{\\left[\n0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert \\leq\n1\\}}x\\widetilde{J}^{-}(\\mathrm{d}s,\\mathrm{d}x)+\\int \\nolimits_{\\left[\n0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert x\\right \\vert >1\\}\nxJ^{-}(\\mathrm{d}s,\\mathrm{d}x)\\text{, }t\\geq0\\text{,\n\\]\nwhere $B_{I}$ is a $\\mathbb{C}^{d}$-valued standard Brownian motion with\nquadratic variation $tI_{d}$.\n\nObserve tha\n\\begin{equation}\n\\left[  X\\right]  (t)=\\Psi^{+}t+\\int \\nolimits_{\\left[  0,t\\right]  \n\\int_{\\mathbb{C}^{d}\\backslash \\{0\\}}xx^{\\ast}J_{+}(\\mathrm{d}s,\\mathrm{d\nx)=\\Psi^{+}t+\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{H}_{d(1)\n^{+}\\backslash \\{0\\}}VJ_{L_{d}}(\\mathrm{d}s,\\mathrm{d}V) \\label{Quad1decomp\n\\end{equation}\nan\n\\begin{align}\n\\left[  Y\\right]  (t)  &  =\\Psi^{-}t+\\int \\nolimits_{\\left[  0,t\\right]  \n\\int_{\\mathbb{C}^{d}\\backslash \\{0\\}}xx^{\\ast}J^{-}(\\mathrm{d}s,\\mathrm{d\nx)=\\Psi^{-}t-\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{C\n^{d}\\backslash \\{0\\}}\\left(  -xx^{\\ast}\\right)  J_{L_{d}}(\\mathrm{d\ns,\\mathrm{d}x)\\nonumber \\\\\n&  =\\Psi^{-}t-\\int \\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{H}_{d(1)\n^{-}\\backslash \\{0\\}}VJ_{L_{d}}(\\mathrm{d}s,\\mathrm{d}V), \\label{Quad2decomp\n\\end{align}\nwhere $\\mathbb{H}_{d(1)}^{-}$ denotes the cone of negative (nonpositive)\ndefinite matrices of rank one in $\\mathbb{H}_{d}$. The first assertion follows\nfrom (\\ref{LIDBV}). Finally, since $J_{L_{d}}$ is a Poisson random measure and\n$\\mathbb{H}_{d(1)}^{+}\\backslash \\{0\\}$ and $\\mathbb{H}_{d(1)}^{-\n\\backslash \\{0\\}$ are disjoint sets, from the last expressions in\n(\\ref{Quad1decomp}) and (\\ref{Quad2decomp}) we have that $\\left[  X\\right]  $\nand $\\left[  Y\\right]  $ are independent processes, although $X$ and $Y$ are not.\n\\end{proof}\n\nNext we consider the matrix L\\'{e}vy processes associated to the BGCD matrix\nensembles $(M_{d})_{d\\geq1}$. We have the following two consequences of the\nformer results.\n\n\\begin{corollary}\n\\label{corBG}Let $M_{d}=\\left \\{  M_{d}(t):t\\geq0\\right \\}  $ be the matrix\nL\\'{e}vy process associated to the BGCD random matrix ensembles.\n\na) Let $\\mu$ be the infinitely divisible distribution with triplet $\\left(\n0,\\psi,\\nu \\right)  $ associated to $M_{d}$ such that\n\\[\n\\int_{\\left \\vert x\\right \\vert \\leq1}\\left(  1\\wedge x\\right)  \\nu\n(\\mathrm{d}x)<\\infty,\\  \\  \\nu((-\\infty,0])=0\\  \\text{ and\\ }\\mathcal{\\psi\n_{0}:=\\mathcal{\\psi}-\\int_{x\\leq1}x\\nu(\\mathrm{d}x)\\geq0.\n\\]\nLet us consider the L\\'{e}vy-It\\^{o} decomposition of $M_{d}(t)$ in\n$\\overline{\\mathbb{H}}_{d}^{+}$\n\\[\nM_{d}(t)=\\mathcal{\\psi}_{0}tdI_{d}+\\int \\nolimits_{\\left[  0,t\\right]  \n\\int_{\\mathbb{H}_{d(1)}^{+}\\backslash \\{0\\}}VJ_{M_{d}}(\\mathrm{d\ns,\\mathrm{d}V).\n\\]\nThen there exists a L\\'{e}vy process $X=\\left \\{  X(t):t\\geq0\\right \\}  $ in\n$\\mathbb{C}^{d}$ such that $M_{d}(t)=\\left[  X\\right]  (t)$, where\n\\[\nX(t)=\\left \\vert \\mathcal{\\psi}_{0}\\right \\vert ^{1\/2}B_{I}(t)+\\int\n\\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert\nx\\right \\vert \\leq1\\}}x\\widetilde{J}(\\mathrm{d}s,\\mathrm{d}x)+\\int\n\\nolimits_{\\left[  0,t\\right]  }\\int_{\\mathbb{C}^{d}\\cap \\{ \\left \\vert\nx\\right \\vert >1\\}}xJ(\\mathrm{d}s,\\mathrm{d}x)\\text{, }t\\geq0\\text{,\n\\]\n$B_{I}$ is a $\\mathbb{C}^{d}$-valued standard Brownian motion with quadratic\nvariation $tI_{d}$, and the Poisson random measure $J$ is given by\n$J=J_{M_{d}}\\circ \\varphi_{+}^{-1}$.\n\nb) If $M_{d}$ has bounded variation then there exist L\\'{e}vy processes\n$X=\\left \\{  X(t):t\\geq0\\right \\}  $ and $Y=\\left \\{  Y(t):t\\geq0\\right \\}  $ in\n$\\mathbb{C}^{d}$ such that $M_{d}(t)=\\left[  X\\right]  (t)-\\left[  Y\\right]\n(t),$ where $\\left \\{  \\left[  X\\right]  (t):t\\geq0\\right \\}  $ and $\\left \\{\n\\left[  Y\\right]  (t):t\\geq0\\right \\}  $ are independent.\n\\end{corollary}\n\n\\section{Covariation matrix processes approximation}\n\nWe now consider approximation of general BGCD ensembles by BGCD matrix\ncompound Poisson processes which are covariation of $\\mathbb{C}^{d}$-valued\nL\\'{e}vy processes.\n\nThe following results gives realizations of BGCD ensembles of compound Poisson\ntype as the covariation of two $\\mathbb{C}^{d}$-valued L\\'{e}vy processes. Its\nproof is straightforward.\n\n\\begin{proposition}\n\\label{BGCDCPP} Let $\\mu$ be a compound Poisson distribution on $\\mathbb{R}$\nwith L\\'{e}vy measure $\\nu$ and drift $\\mathcal{\\psi}\\in \\mathbb{R}$ and let\n$(M_{d})_{d\\geq1}$ be the BGCD matrix ensemble for $\\Lambda(\\mu).$ For each\n$d\\geq1$, assume that\n\ni) $(\\beta_{j})_{j\\geq1}$ is a sequence of i.i.d. random variables with\ndistribution $\\nu\/\\nu \\left(  \\mathbb{R}\\right)  $.\n\nii) $(u_{j})_{j\\geq1}$ is a sequence of i.i.d. random vectors with uniform\ndistribution on the unit sphere of $\\mathbb{C}^{d}$.\n\niii) $\\left \\{  N(t)\\right \\}  _{t\\geq0}$ is a Poisson process with parameter one.\n\nAssume that $(\\beta_{j})_{j\\geq1}$, $(u_{j})_{j\\geq1}$ and $\\left \\{\nN(t)\\right \\}  _{t\\geq0}$ are independent. Then\n\na) $M_{d}$ has the same distribution as $M_{d}(1)$ where\n\\begin{equation}\nM_{d}(t)=\\mathcal{\\psi}tI_{d}+\\sum_{j=1}^{N(t)}\\beta_{j}u_{j}u_{j}^{\\ast\n},\\quad t\\geq0. \\label{BGCP1\n\\end{equation}\n\n\nb) $M_{d}(\\cdot)=[X_{d},Y_{d}](\\cdot)$ where $X_{d}=\\left \\{  X_{d}(t)\\right \\}\n_{t\\geq0},$ $Y_{d}=\\left \\{  Y_{d}(t)\\right \\}  _{t\\geq0}$ are the\n$\\mathbb{C}^{d}$-valued L\\'{e}vy processe\n\\begin{equation}\nX_{d}(t)=\\sqrt{\\left \\vert \\mathcal{\\psi}\\right \\vert }B(t)+\\sum_{j=1\n^{N(t)}\\sqrt{\\left \\vert \\beta_{j}\\right \\vert }u_{j},\\quad t\\geq0,\n\\label{BGCP2\n\\end{equation\n\\begin{equation}\nY_{d}(t)=\\mathrm{sign}\\left(  \\mathcal{\\psi}\\right)  \\sqrt{\\left \\vert\n\\mathcal{\\psi}\\right \\vert }B(t)+\\sum_{j=1}^{N(t)}\\mathrm{sign}\\left(\n\\beta_{j}\\right)  \\sqrt{\\left \\vert \\beta_{j}\\right \\vert }u_{j},\\quad t\\geq0,\n\\label{BGCP3\n\\end{equation}\nand $B=\\left \\{  B(t)\\right \\}  _{t\\geq0}$ is a $\\mathbb{C}^{d}$-valued standard\nBrownian motion independent of $(\\beta_{j})_{j\\geq1}$, $(u_{j})_{j\\geq1}$ and\n$\\left \\{  N(t)\\right \\}  _{t\\geq0}$.\n\\end{proposition}\n\nFor the general case we have the following sample path approximation by\ncovariation processes for L\\'{e}vy processes generated by the BGCD matrix ensembles.\n\n\\begin{theorem}\n\\label{General} Let $\\mu$ be an infinitely divisible distribution on\n$\\mathbb{R}$ with triplet $(a^{2},\\psi,\\nu)$ and let $(M_{d})_{d\\geq1}$ be the\ncorresponding BGCD matrix ensemble for $\\Lambda(\\mu).$ Let $d\\geq1$ fixed and\nassume that for $n\\geq1$\n\ni) $(\\beta_{j}^{n})_{j\\geq1}$ is a sequence of i.i.d. random variables with\ndistribution $\\mu^{\\ast \\frac{1}{n}}$.\n\nii) $(u_{j}^{n})_{j\\geq1}$ is a sequence of i.i.d. random vectors with uniform\ndistribution on the unit sphere of $\\mathbb{C}^{d}$.\n\niii) $N^{n}=\\left \\{  N^{n}(t)\\right \\}  _{t\\geq0}$ is a Poisson process with\nparameter $n$.\n\niv) $B^{n}=\\left \\{  B^{n}(t)\\right \\}  _{t\\geq0}$ is a $\\mathbb{C}^{d}$-valued\nstandard Brownian motion.\n\nv) $(\\beta_{j}^{n})_{j\\geq1}$, $(u_{j}^{n})_{j\\geq1},N^{n}$ and $B^{n}$are independent.\n\nLet\n\\begin{equation}\nX_{d}^{n}(t)=\\sqrt{\\left \\vert \\mathcal{\\psi}\\right \\vert }B^{n}(t)+\\sum\n_{j=1}^{N^{n}(t)}\\sqrt{\\left \\vert \\beta_{j}^{n}\\right \\vert }u_{j}^{n},\\quad\nt\\geq0, \\label{Gen1\n\\end{equation\n\\begin{equation}\nY_{d}^{n}(t)=\\mathrm{sign}\\left(  \\mathcal{\\psi}\\right)  \\sqrt{\\left \\vert\n\\mathcal{\\psi}\\right \\vert }B^{n}(t)+\\sum_{j=1}^{N^{n}(t)}\\mathrm{sign}\\left(\n\\beta_{j}^{n}\\right)  \\sqrt{\\left \\vert \\beta_{j}^{n}\\right \\vert }u_{j\n^{n},\\quad t\\geq0. \\label{Gen2\n\\end{equation}\nThen for each $d\\geq1$ there exist $\\mathbb{M}_{d}$-valued processes\n$\\widetilde{M}_{d}^{n}=\\left \\{  \\widetilde{M}_{d}^{n}(t)\\right \\}  _{d\\geq1}$\nsuch that $\\widetilde{M}_{d}^{n}\\overset{\\mathcal{L}}{=}[X_{d}^{n},Y_{d}^{n\n]$\n\\[\n\\sup_{0<s\\leq t}\\left \\Vert \\widetilde{M}_{d}^{n}(s)-M_{d}(s)\\right \\Vert\n\\underset{n\\rightarrow \\infty}{\\overset{\\Pr}{\\longrightarrow}}0\\text{,\\quad\n}\\forall t\\geq0\\text{,\n\\]\nwhere $\\left \\{  M_{d}(t):t\\geq0\\right \\}  $ is the $\\mathbb{M}_{d}$-valued\nL\\'{e}vy process associated to $(M_{d})_{d\\geq1}$.\n\\end{theorem}\n\n\\begin{proof}\nBy the compound Poisson approximation for infinitely divisible distributions\non $\\mathbb{R}$ (see \\cite[pp 45]{Sato1})$,$ we choose $\\mu_{n}$ an infinitely\ndivisible distribution such that $\\mu_{n}\\longrightarrow \\mu,$ where we take\nthe triplet of $\\mu_{n}$ as $\\left(  0,\\psi^{n},\\nu^{n}\\right)  ,$ $\\psi\n^{n}=\\int \\frac{x}{1+\\left \\vert x\\right \\vert ^{2}}\\nu^{n}\\left(  dx\\right)  $\nand $\\nu^{n}=n\\mu^{\\ast \\frac{1}{n}}$, satisfying (see \\cite[Theorem\n8.7]{Sato1}) that for every bounded continuous function $f$ vanishing in a\nneighborhood of zer\n\\begin{equation}\n\\int_{\\mathbb{R}}f\\left(  r\\right)  \\nu^{n}\\left(  dr\\right)  \\longrightarrow\n\\int_{\\mathbb{R}}f\\left(  r\\right)  \\nu \\left(  dr\\right)  \\text{ as\n}n\\rightarrow \\infty \\text{,} \\label{lev\n\\end{equation}\nfor each $\\varepsilon>0\n\\begin{equation}\n\\int_{\\left \\vert r\\right \\vert \\leq \\varepsilon}r^{2}\\nu^{n}\\left(  dr\\right)\n\\longrightarrow a^{2}\\text{ as }n\\rightarrow \\infty, \\label{gau\n\\end{equation}\nand $\\psi^{n}\\rightarrow \\psi$.\n\nA similar proof as for Proposition \\ref{BGCDCPP} give\n\\[\nM_{d}^{n}(t):=\\left[  X_{d}^{n},Y_{d}^{n\\ast}\\right]  (t)=\\mathcal{\\psi\n}t\\mathrm{I}_{d}+\\sum_{j=0}^{N^{n}(t)}\\beta_{j}^{n}u_{j}^{n}u_{j}^{n\\ast},\n\\]\nwhich is a matrix value compound Poisson process with triplet $\\left(\n\\mathcal{A}_{d}^{n},\\psi_{d}^{n},\\nu_{d}^{n}\\right)  $ given by $\\mathcal{A\n_{d}^{n}=0,\\  \\psi_{d}^{n}=\\psi \\mathrm{I}_{d}$ an\n\\begin{equation}\n\\nu_{d}^{n}\\left(  E\\right)  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})}\\int\n_{0}^{\\infty}1_{E}\\left(  rV\\right)  \\nu_{V}^{n}\\left(  \\mathrm{d}r\\right)\n\\Pi \\left(  \\mathrm{d}V\\right)  ,\\quad E\\in \\mathcal{B}\\left(  \\mathbb{H\n_{d}\\backslash \\left \\{  0\\right \\}  \\right)  \\text{,} \\label{nudn\n\\end{equation}\nwhere $\\nu_{V}^{n}=\\nu^{n}|_{(0,\\infty)}$ or $\\nu^{n}|_{(-\\infty,0)}$\naccording to $V\\geq0$ or\\ $V\\leq0$ and $\\Pi$ is the measure on $\\mathbb{S\n(\\mathbb{H}_{d(1)})$ in (\\ref{pi}).\n\nWe will prove that $M_{d}^{n}\\overset{\\mathcal{L}}{\\longrightarrow}M_{d}$ by\nshowing that the triplet $\\left(  \\mathcal{A}_{d}^{n},\\psi_{d}^{n},\\nu_{d\n^{n}\\right)  $ converges to the triplet $\\left(  \\mathcal{A}_{d},\\psi_{d\n,\\nu_{d}\\right)  $ of the BGCD matrix ensemble in Proposition \\ref{polar} in\nthe sense of Proposition \\ref{convternas}:\n\nWe observe that $\\psi_{d}^{n}=\\psi \\mathrm{I}_{d}$ for each $n$.\n\nLet $f:\\mathbb{H}_{d(1)}\\longrightarrow \\mathbb{R}$ be a continuous bounded\nfunction vanishing in a neighborhood of zero. Using the polar decomposition\n(\\ref{PDBGCD}) for $\\nu_{d}^{n}$ we hav\n\\begin{align}\n\\int_{\\mathbb{H}_{d(1)}}f\\left(  \\xi \\right)  \\nu_{d}^{n}\\left(  d\\xi \\right)\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})}\\int_{0}^{\\infty}f\\left(  rV\\right)\n\\nu_{V}^{n}\\left(  dr\\right)  \\Pi \\left(  dV\\right) \\nonumber \\\\\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H}}_{d}^{+}\n\\int_{\\left \\{  -1,1\\right \\}  }\\int_{0}^{\\infty}f\\left(  trV\\right)  \\nu\n_{V}^{n}\\left(  dr\\right)  \\lambda^{n}\\left(  dt\\right)  \\omega_{d}\\left(\ndV\\right)  . \\label{intfb\n\\end{align}\nFor $V\\in \\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H}}_{d}^{+}$\nfixed,\n\\begin{align*}\n\\int_{\\left \\{  -1,1\\right \\}  }\\int_{0}^{\\infty}f\\left(  trV\\right)  \\nu\n_{V}^{n}\\left(  dr\\right)  \\lambda^{n}\\left(  dt\\right)   &  =\\lambda\n^{n}\\left(  \\left \\{  1\\right \\}  \\right)  \\int_{0}^{\\infty}f\\left(  rV\\right)\n\\nu^{n}\\left(  dr\\right) \\\\\n&  +\\lambda^{n}\\left(  \\left \\{  -1\\right \\}  \\right)  \\int_{-\\infty\n^{0}f\\left(  rV\\right)  \\nu^{n}\\left(  dr\\right)  \\text{.\n\\end{align*}\nAs a function of $r$, $f\\left(  rV\\right)  $ is a real valued continuous\nbounded function vanishing in a neighborhood of zero, hence using (\\ref{lev}\n\\[\n\\lambda^{n}\\left(  \\left \\{  1\\right \\}  \\right)  \\int_{0}^{\\infty}f\\left(\nrV\\right)  \\nu^{n}\\left(  dr\\right)  \\longrightarrow \\lambda \\left(  \\left \\{\n1\\right \\}  \\right)  \\int_{0}^{\\infty}f\\left(  rV\\right)  \\nu \\left(  dr\\right)\n\\]\nan\n\\[\n\\lambda^{n}\\left(  \\left \\{  -1\\right \\}  \\right)  \\int_{-\\infty}^{0}f\\left(\nrV\\right)  \\nu^{n}\\left(  dr\\right)  \\longrightarrow \\lambda \\left(  \\left \\{\n-1\\right \\}  \\right)  \\int_{-\\infty}^{0}f\\left(  rV\\right)  \\nu \\left(\ndr\\right)  .\n\\]\nThen from (\\ref{intfb}\n\\begin{align*}\n\\int_{\\mathbb{H}_{d(1)}}f\\left(  \\xi \\right)  \\nu_{d}^{n}\\left(  d\\xi \\right)\n&  \\longrightarrow d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline\n{\\mathbb{H}}_{d}^{+}}\\int_{\\left \\{  -1,1\\right \\}  }\\int_{0}^{\\infty}f\\left(\ntrV\\right)  \\nu_{V}\\left(  dr\\right)  \\lambda \\left(  dt\\right)  \\omega\n_{d}\\left(  dV\\right) \\\\\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})}\\int_{0}^{\\infty}f\\left(  rV\\right)\n\\nu_{d}\\left(  dr\\right)  \\Pi \\left(  dV\\right)  =\\int_{\\mathbb{H}_{d(1)\n}f\\left(  \\xi \\right)  \\nu_{d}\\left(  d\\xi \\right)  .\n\\end{align*}\n\n\nNext, we verify the convergence of the Gaussian part.\n\nLet us define, for each $\\varepsilon>0$ and $n\\geq1,$ the operator\n$\\mathcal{A}^{n,\\varepsilon}:\\mathbb{H}_{d}\\longrightarrow \\mathbb{H}_{d}$ by\n\\[\n\\mathrm{tr}\\left(  \\Theta \\mathcal{A}^{n,\\varepsilon}\\Theta \\right)\n=\\int_{\\left \\Vert \\xi \\right \\Vert \\leq \\varepsilon}\\left \\vert \\mathrm{tr}\\left(\n\\Theta \\xi \\right)  \\right \\vert ^{2}\\nu_{d}^{n}\\left(  d\\xi \\right)  .\n\\]\nFrom (\\ref{nudn}) we get\n\\begin{align*}\n&  \\int_{\\left \\Vert \\xi \\right \\Vert \\leq \\varepsilon}\\left \\vert \\mathrm{tr\n\\left(  \\Theta \\xi \\right)  \\right \\vert ^{2}\\nu_{d}^{n}\\left(  d\\xi \\right)\n=d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})}\\int_{0}^{\\infty}\\mathbf{1}_{\\left \\{\n\\left \\Vert rV\\right \\Vert \\leq \\varepsilon \\right \\}  }\\left(  rV\\right)\n\\left \\vert \\mathrm{tr}\\left(  r\\Theta V\\right)  \\right \\vert ^{2}\\nu_{V\n^{n}\\left(  dr\\right)  \\Pi \\left(  dV\\right) \\\\\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H}}_{d}^{+}\n\\int_{\\left \\{  -1,1\\right \\}  }\\int_{0}^{\\infty}\\mathbf{1}_{\\left \\{\nr\\leq \\varepsilon \\right \\}  }\\left(  rtV\\right)  r^{2}\\left \\vert \\mathrm{tr\n\\left(  \\Theta V\\right)  \\right \\vert ^{2}\\nu_{V}^{n}\\left(  dr\\right)\n\\lambda \\left(  dt\\right)  \\omega_{d}\\left(  dV\\right) \\\\\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H}}_{d}^{+}\n\\int_{\\mathbb{R}}\\mathbf{1}_{\\left \\{  r\\leq \\varepsilon \\right \\}  }\\left(\nrV\\right)  r^{2}\\left \\vert \\mathrm{tr}\\left(  \\Theta V\\right)  \\right \\vert\n^{2}\\nu^{n}\\left(  dr\\right)  \\omega_{d}\\left(  dV\\right) \\\\\n&  =d\\int_{\\mathbb{S}(\\mathbb{H}_{d(1)})\\cap \\overline{\\mathbb{H}}_{d}^{+\n}\\left \\vert \\mathrm{tr}\\left(  \\Theta V\\right)  \\right \\vert ^{2\n\\int_{\\left \\vert r\\right \\vert \\leq \\varepsilon}r^{2}\\nu^{n}\\left(  dr\\right)\n\\omega_{d}\\left(  dV\\right)  .\n\\end{align*}\nThen using (\\ref{gau})\n\\[\n\\int_{\\left \\Vert \\xi \\right \\Vert \\leq \\varepsilon}\\left \\vert \\mathrm{tr}\\left(\n\\Theta \\xi \\right)  \\right \\vert ^{2}\\nu_{d}^{n}\\left(  d\\xi \\right)\n\\longrightarrow da^{2}E_{u}\\left \\vert \\mathrm{tr}\\left(  \\Theta uu^{\\ast\n}\\right)  \\right \\vert ^{2}\\text{,\n\\]\nwhere $u$ is a uniformly distributed column random vector in the unit sphere\nof $\\mathbb{C}^{d}$. Finally\n\\begin{equation}\nda^{2}E_{u}\\left \\vert \\mathrm{tr}\\left(  \\Theta uu^{\\ast}\\right)  \\right \\vert\n^{2}=\\frac{a^{2}}{d+1}\\left(  \\mathrm{tr}\\left(  \\Theta^{2}\\right)  +\\left(\n\\mathrm{tr}\\left(  \\Theta \\right)  \\right)  ^{2}\\right)  =\\mathrm{tr}\\left(\n\\Theta^{\\ast}\\mathcal{A}_{d}\\Theta^{\\ast}\\right)  , \\label{covgau\n\\end{equation}\n\\newline where $\\mathcal{A}_{d}$ is as in (\\ref{GPBGCD}) and the first\nequality in (\\ref{covgau}) follows from page $637$ in \\cite{CD}. Thus\n$M_{d}^{n}\\overset{\\mathcal{L}}{\\longrightarrow}M_{d}$ and the conclusion\nfollows from Proposition \\ref{convproc}.\n\\end{proof}\n\n\\section{Final remarks}\n\n\\begin{enumerate}\n\\item For the present work we do not have a specific financial application in\nmind. However, infinitely divisible nonnegative definite matrix processes with\nrank one jumps as characterized in Theorem \\ref{Sub}, might be useful in the\nstudy of multivariate high-frequency data using realized covariation, where\nmatrix covariation processes appear; see for example \\cite{BNSh04}. Moreover,\nit seems interesting to explore the construction of financial oriented matrix\nL\\'{e}vy based models as in \\cite{BNSe09} for the specific case of rank one\njumps matrix process of bounded variation.\n\n\\item In the direction of free probability, it is well known that the\nso-called Hermitian Brownian motion matrix ensemble $\\left \\{  B_{d\n(t):t\\geq0\\right \\}  $, $d\\geq1,$ is a realization of the free Brownian motion.\nIt is an open question if the matrix L\\'{e}vy processes from BGCD models\n$\\left \\{  M_{d}(t):t\\geq0\\right \\}  $, $d\\geq1$, are realizations of free\nL\\'{e}vy processes. A first step in this direction would be to prove that the\nincrements of a BGCD ensemble become free independent. A second step, more\nrelated to our work, would be to have an insight of the implication of the\nrank one condition of the matrix L\\'{e}vy BGCD process in Corollary\n\\ref{corBG} as realization of a positive free L\\'{e}vy process. These two\nproblems are the subjects of current research of one of the coauthors.\n\n\\item In \\cite{BG07} a new Bercovici-Pata bijection for certain free\nconvolution $\\boxplus_{c}$ is established and a $d\\times d^{\\prime}$ random\nmatrix model for this bijection which is very close to the one given by the\nBGCD random matrix model is established. It can be seen that the L\\'{e}vy\nmeasures of these rectangular BGCD random matrices are supported in the subset\nof $d\\times d^{\\prime}$ complex matrices of rank one, in a similar way as done\nin \\cite{DRA} for the BGCD case. It would be of interest to have the analogue\nresults on bounded variation of Section 4 for the L\\'{e}vy processes\nassociated to these rectangular BGCD random matrices, considering an\nappropriate nonnegative definite notion for rectangular matrices.\n\\end{enumerate}\n\n\\noindent \\textbf{Acknowledgement}. \\emph{This work was done while Victor\nP\\'{e}rez-Abreu was visiting Universidad Aut\\'{o}noma de Sinaloa in January\nand May of 2012. The authors thank two referees for the very carefully and\ndetailed reading of a previous version of the manuscript and for their\ncomments that improved Theorems \\ref{Sub} and \\ref{Bdv} and the presentation\nof the present version of the manuscript.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n The classical algorithm for multiplying two $n\\times n$ matrices performs\n $2n^3-n^2$ additions and multiplications. Strassen's algorithm~\\citep{strassen1969gaussian}\n does the job with only $\\mathcal{O}(n^{\\log_27})$ additions and multiplications, by\n recursively applying a certain scheme for computing the product of two $2\\times\n 2$-matrices with only 7 instead of the usual 8 multiplications. The discovery\n of Strassen's algorithm has initiated substantial work during the past 50 years\n on finding the smallest exponent $\\omega$ such that matrix multiplication costs\n $\\mathcal{O}(n^\\omega)$ operations in the coefficient ring.\n The current record is $\\omega\\leq 2.3728639$ and was obtained by~\\cite{LeGall:2014:PTF:2608628.2608664}.\n It improves the previous record of~\\cite{Williams:2012:MMF:2213977.2214056} by just $3\\cdot10^{-7}$.\n Extensive background in this direction is available in text books~\\citep{buergisser-book,landsberg2017geometry} and\n survey articles~\\citep{blaeser-survey2013,pan-survey2018}. \n Contrary to wide-spread belief, Strassen's algorithm is not only efficient in theory but also in practice. \n Special purpose software for exact linear algebra, such as the FFLAS and FFPACK packages~\\citep{DGP:2008},\n have been using it since long, and there are also reports that its performance in a numerical context\n is not as bad as its reputation~\\citep{Huang:2016:SAR:3014904.3014983}.\n \n Besides the quest for the smallest exponent, which only concerns the asymptotic\n complexity for asymptotically large~$n$, it is also interesting to know how many\n multiplications are needed for a specific (small) $n$ to compute the product of two $n\\times\n n$-matrices. Thanks to Strassen, we know that the answer is at most 7 for\n $n=2$, and it can be shown~\\citep{winograd1971multiplication} that there is no way to do it with 6\n multiplications. It can further be shown that, in a certain sense, Strassen's scheme\n is the only way of doing it with 7 multiplications~\\citep{de1978varieties}. \n\n Already for $n=3$, the situation is not completely understood. \\cite{laderman1976noncommutative}\n showed that 23 multiplications suffice, and \\cite{blaser2003complexity} showed that at least 19\n multiplications are needed.\n For larger sizes as well as rectangular matrices, many people have been searching\n for new schemes using fewer and fewer coefficient multiplications.\n For $n=4$, the best we know is to apply Strassen's scheme recursively, which requires~49 multiplications.\n For $n=5$, the record of 100 multiplications was held~\\cite{MAKAROV1987205} for 30 years until it was\n improved to 99 by~\\cite{DBLP:journals\/corr\/Sedoglavic17aa}.\n For $n=6$, there is a recent scheme by~\\cite{smirnov2013bilinear} which needs only 160 multiplications.\n For $n=7$, \\cite{DBLP:journals\/corr\/abs-1712-07935} found a way to compute the product with 250 multiplications.\n For larger sizes and rectangular matrices, see the extensive tables compiled by~\\cite{smirnov2013bilinear,smirnovseveral}\n and~\\cite{fastmm}.\n Many of the schemes for larger matrix sizes are obtained by combining\n multiplication schemes for smaller matrices~\\citep{DBLP:journals\/tcs\/DrevetIS11}.\n \n Although nobody knows whether there is a scheme using only 22 multiplications for~$n=3$ (in an exact and\n non-commutative setting), 23 multiplications can be achieved in many different\n ways. \\cite{DBLP:journals\/siamcomp\/JohnsonM86} have in fact found infinitely many ways.\n They presented a family of schemes involving three free parameters.\n However, their families involve fractional coefficients and therefore\n do not apply to arbitrary coefficient rings~$K$. Many others have reported isolated\n schemes with fractional or approximate coefficients. Such schemes can be \n constructed for example by numerically solving a certain optimization problem,\n or by genetic algorithms. In Laderman's multiplication scheme, all coefficients are\n $+1$,~$-1$, or~$0$, which has the nice feature that it works for any\n coefficient ring. As far as we know, there are so far only three other schemes\n with this additional property, they are due to \\cite{smirnov2013bilinear}, \\cite{oh2013inequivalence}, and\n \\cite{DBLP:journals\/corr\/abs-1108-2830}, respectively. We add more than 13\\,000 new schemes to this list. \n\n The isolated scheme presented by Courtois et al. was not found numerically but with the help of\n a SAT solver. SAT~\\citep{DBLP:series\/faia\/2009-185} refers to the decision problem of propositional logic:\n given a Boolean formula in conjunctive normal form, is there an assignment of the Boolean variables\n such that the formula evaluates to true under this assignment? Although SAT is a prototypical example\n of an NP-complete problem, modern SAT solvers are able to solve very large instances. In addition to\n various industrial applications, they have recently also contributed to the solution of difficult\n mathematical problems, see \\cite{DBLP:conf\/sat\/HeuleKM16} and~\\cite{DBLP:conf\/aaai\/Heule18} for two\n examples. SAT solvers also play a central role in our approach. \n As explained in Section~\\ref{sec:searching}, we first use a SAT solver to find multiplication\n schemes for the coefficient ring~$\\set Z_2$, starting from some known solutions. \n In a second step, explained in Section~\\ref{sec:sieving}, we discard solutions that\n are equivalent to solutions found earlier.\n Next, we simplify the new solutions (Sect.~\\ref{sec:switching}), and use them as \n starting points for a new round of searching.\n Altogether about 35 years of computation time were spent in several iterations of this process. \n In the end, we lifted the solutions from $\\set Z_2$ to arbitrary coefficient rings (Sect.~\\ref{sec:signing}),\n and we extracted families with up to 17 free parameters from them (Sect.~\\ref{sec:families}).\n Our 13,000 isolated schemes and our parameterized families are provided in various formats on our website~\\citep{mmr}. \n \n \\section{The Brent Equations}\\label{sec:brent}\n\n The general pattern of a matrix multiplication scheme consists of two sections.\n In the first section, several auxiliary quantities $M_1,M_2,\\dots, M_m$ are computed,\n each of which is a product of a certain linear combination of the entries of the first matrix\n with a certain linear combination of the entries of the second matrix.\n In the second section, the entries of the resulting matrix are obtained as certain\n linear combinations of the auxiliary quantities $M_1,M_2,\\dots, M_m$.\n\n For example, writing\n \\[\n A = \\begin{pmatrix}\n   a_{1,1}&a_{1,2}\\\\\n   a_{2,1}&a_{2,2}\\\\\n \\end{pmatrix},\\quad\n B = \\begin{pmatrix}\n   b_{1,1}&b_{1,2}\\\\\n   b_{2,1}&b_{2,2}\n \\end{pmatrix},\\quad\\text{and}\\quad\n C = \\begin{pmatrix}\n   c_{1,1}&c_{1,2}\\\\\n   c_{2,1}&c_{2,2}\n \\end{pmatrix} := AB,   \n \\]\n Strassen's multiplication scheme proceeds as follows:\n \n \\hangindent=3em\\hangafter=1\\emph{First section.}\\\\\n   $M_1 = (a_{1,1} + a_{2,2}) (b_{1,1} + b_{2,2})$\\\\\n   $M_2 = (a_{2,1} + a_{2,2}) (b_{1,1})$\\\\\n   $M_3 = (a_{1,1}) (b_{1,2} - b_{2,2})$\\\\\n   $M_4 = (a_{2,2}) (b_{2,1} - b_{1,1})$\\\\\n   $M_5 = (a_{1,1} + a_{1,2})(b_{2,2})$\\\\\n   $M_6 = (a_{2,1} - a_{1,1}) (b_{1,1}+ b_{1,2})$\\\\\n   $M_7 = (a_{1,2} - a_{2,2}) (b_{2,1} + b_{2,2})$\n\n \\hangindent=3em\\hangafter=1\\emph{Second section.}\\\\\n   $c_{1,1} = M_1 + M_4 - M_5 + M_7$\\\\\n   $c_{1,2} = M_3 + M_5$\\\\\n   $c_{2,1} = M_2 + M_4$\\\\\n   $c_{2,2} = M_1 - M_2 + M_3 + M_6$.\n\n Observe that the number of multiplications is exactly the number of~$M$'s.\n Also observe that while it is not obvious how to construct such a scheme\n from scratch, checking that a given scheme is correct is an easy and\n straightforward calculation. For example, $c_{2,1}=M_2+M_4=(a_{2,1} + a_{2,2}) (b_{1,1})\n + (a_{2,2}) (b_{2,1} - b_{1,1}) = a_{2,1}b_{1,1} + a_{2,2}b_{2,1}$. \n\n In order to search for a multiplication scheme for a prescribed shape of\n matrices (e.g., $3\\times 3$) and a prescribed number of multiplications (e.g.,\n $23$), we can make an ansatz for the coefficients of the various linear\n combinations,\n \\begin{alignat*}1\n   M_1 &= (\\alpha_{1,1}^{(1)}a_{1,1} + \\alpha_{1,2}^{(1)}a_{1,2}+\\cdots)(\\beta_{1,1}^{(1)}b_{1,1} + \\beta_{1,2}^{(1)}b_{1,2}+\\cdots)\\\\\n   M_2 &= (\\alpha_{1,1}^{(2)}a_{1,1} + \\alpha_{1,2}^{(2)}a_{1,2}+\\cdots)(\\beta_{1,1}^{(2)}b_{1,1} + \\beta_{1,2}^{(2)}b_{1,2}+\\cdots)\\\\\n   &\\vdots\\\\\n   M_{23} &= (\\alpha_{1,1}^{(23)}a_{1,1} + \\alpha_{1,2}^{(23)}a_{1,2}+\\cdots)(\\beta_{1,1}^{(23)}b_{1,1} + \\beta_{1,2}^{(23)}b_{1,2}+\\cdots)\\\\\n   c_{1,1} &= \\gamma_{1,1}^{(1)}M_1 + \\gamma_{1,1}^{(2)}M_2 + \\cdots + \\gamma_{1,1}^{(23)}M_{23}\\\\\n   c_{1,2} &= \\gamma_{1,2}^{(1)}M_1 + \\gamma_{1,2}^{(2)}M_2 + \\cdots + \\gamma_{1,2}^{(23)}M_{23}\\\\\n   &\\vdots\\\\\n   c_{3,3} &= \\gamma_{3,3}^{(1)}M_1 + \\gamma_{3,3}^{(2)}M_2 + \\cdots + \\gamma_{3,3}^{(23)}M_{23}\n \\end{alignat*}\n and then compare coefficients such as to enforce $c_{i,j}=\\sum_k\n a_{i,k}b_{k,j}$. Doing so leads to a system of polynomial equations for the\n undetermined coefficients $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}},\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}},\n \\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$. The equations in this system are known as the Brent\n equations~\\citep{brent1970algorithms}.  For $3\\times 3$-matrices and 23 multiplications, the\n equations turn out to be\n \\[\n   \\sum_{\\iota=1}^{23} \\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} = \\delta_{i_2,j_1}\\delta_{i_1,k_1}\\delta_{j_2,k_2}\n \\]\n for $i_1,i_2,j_1,j_2,k_1,k_2\\in\\{1,2,3\\}$, i.e., there are 621 variables and\n 729 cubic equations. The $\\delta_{u,v}$ on the right refer to the\n Kronecker-delta, i.e., $\\delta_{u,v}=1$ if $u=v$ and $\\delta_{u,v}=0$\n otherwise.\n\n The equations become a bit more symmetric if we connect the matrices $A,B,C$\n through $C^\\top=AB$ rather than $C=AB$. In the version with the transposition,\n which we shall use from now on, and which is also more common in the\n literature, the right hand side has to be replaced with\n $\\delta_{i_2,j_1}\\delta_{j_2,k_1}\\delta_{k_2,i_1}$.\n\n In any case, the problem boils down to finding a solution of the Brent\n equations. In principle, this system could be solved using Gr\\\"obner\n bases~\\citep{buchberger65,cox2013ideals,buchberger10}, but doing so would require an absurd amount of computation\n time. Some of the solutions reported in the literature have been found using\n numerical solvers~\\citep{smirnov2013bilinear,oh2013inequivalence},\n and~\\cite{laderman1976noncommutative} claims that his solution\n was found by solving the Brent equations by hand. He writes that he would explain in a later paper\n how exactly he did this, but apparently this later paper has never been written. Only recently,\n \\cite{DBLP:journals\/corr\/Sedoglavic17} has\n\n given a convincing explanation of how Laderman's scheme can be derived from Strassen's scheme\n for $2\\times 2$ matrices.\n \\cite{DBLP:journals\/corr\/abs-1108-2830} found their solution using a SAT solver.\n We also start our search using SAT solvers.\n\n \\section{SAT Encoding and Streamlining}\\label{sec:searching}\n\n In order to encode the problem as a SAT\n problem, we view the Brent equations as equations for the finite field~$\\set Z_2$, interpret \n its elements as truth values, \n its addition as exclusive {\\sc or} ($\\oplus$), \n and its multiplication as\n conjunction ($\\land$). These propositional formulas cannot be \n directly be processed by most state-of-the-art SAT solvers, because they \n require the formulas in conjunctive normal form (CNF). A formula is in \n CNF if it is a conjunction of clauses, where a clause is a disjunction \n ($\\lor$) of \n literals and a literal is a Boolean variable $x$ or the negation \n of a Boolean \n variable ($\\bar x$). For avoiding an exponential blow-up when transforming an \n arbitrary structured formula to CNF, auxiliary variables are introduced \n that abbreviate certain subformulas. For every $i_1,i_2,j_1,j_2\\in\\{1,2,3\\}$ and\n every $\\iota=1,\\dots,23$, we introduce a fresh variable $s^{\\text{\\tiny (}\\iota\\text{\\tiny )}}_{i_1,i_2,j_1,j_2}$\n and impose the condition\n \\[\n s_{i_1,i_2,j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\leftrightarrow (\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\land \\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}),\n \\]\n whose translation to CNF requires three clauses.\n Similarly, for every $i_1,i_2,j_1,j_2,k_1,k_2\\in\\{1,2,3\\}$ and every\n $\\iota=1,\\dots,23$, we introduce a fresh variable $t_{i_1,i_2,j_1,j_2,k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$\n and impose the condition\n \\[\n t_{i_1,i_2,j_1,j_2,k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\leftrightarrow (s_{i_1,i_2,j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\land \\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}),\n \\]\n whose translation to CNF costs again three clauses.\n\n \\def\\even{\\mathrm{even}}\\def\\odd{\\mathrm{odd}\n For each fixed choice $i_1,i_2,j_1,j_2,k_1,k_2\\in\\{1,2,3\\}$, there is a Brent equation\n which says that the number of $\\iota$'s for which $t_{i_1,i_2,j_1,j_2,k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$\n is set to true should be even (if $\\delta_{i_2,j_1}\\delta_{i_1,k_1}\\delta_{j_2,k_2}=0$)\n or that it should be odd (if $\\delta_{i_2,j_1}\\delta_{i_1,k_1}\\delta_{j_2,k_2}=1$).\n It therefore remains to encode the condition that an even number (or an odd number) of a\n given set of $p$ variables should be true, i.e., we need to construct a formula\n $\\even(x_1,\\dots,x_p)$ which is true if and only if an even number among the\n variables $x_1,\\dots,x_p$ is true. Such a formula can again be constructed using\n auxiliary variables. Note that $\\even(x_1,\\dots,x_p)$ is true if and only if\n $\\even(x_1,\\dots,x_i,y)\\land\\even(x_{i+1},\\dots,x_p,y)$ is true, because this is the\n case if and only if both $\\{x_1,\\dots,x_i\\}$ and $\\{x_{i+1},\\dots,x_p\\}$ contain an\n even number of variables set to true (and then $y$ is set to false) or both sets contain\n an odd number of variables set to true (and then $y$ is set to true). Applying this principle\n recursively for $p = 23$ (the number of summands in each Brent equation), the problem can be broken down to chunks of size four:\n \\begin{alignat*}5\n   &\\even(x_1,x_2,x_3,y_1)&&\\land\n   \\even(x_4,x_5,x_6,y_2)&&\\land\n   \\even(x_7,x_8,x_9,y_3)&&\\land\n   \\even(x_{10},x_{11},x_{12},y_4)\\\\\n   {}\\land{}&\\even(x_{13},x_{14},x_{15},y_5)&&\\land\n   \\even(x_{16},x_{17},x_{18},y_6)&&\\land\n   \\even(x_{19},x_{20},x_{21},y_7)&&\\land\n   \\even(x_{22},x_{23},y_1,y_8)\\\\\n   {}\\land{}&\\even(y_2,y_3,y_4,y_9)&&\\land\n   \\even(y_5,y_6,y_7,y_{10})&&\\land\n   \\even(y_8,y_9,y_{10},y_{11}).\n \\end{alignat*}\n The small chunks can be encoded directly by observing that $\\even(a,b,c,d)$ is equivalent\n to\n \\begin{alignat*}1\n &(a \\lor b \\lor c \\lor \\bar d) \\land\n (a \\lor b \\lor \\bar c \\lor  d) \\land\n (a \\lor \\bar b \\lor c \\lor  d) \\land\n   (\\bar a \\lor b \\lor c \\lor  d) \\land\\\\\n &(a \\lor \\bar b \\lor \\bar c \\lor \\bar d) \\land\n (\\bar a \\lor \\bar b \\lor c \\lor \\bar d) \\land\n (\\bar a \\lor b \\lor \\bar c \\lor \\bar d) \\land\n (\\bar a \\lor \\bar b \\lor \\bar c \\lor d).\n \\end{alignat*}\n For the cases where an odd number of the variables $x_1,\\dots,x_{23}$ must be true, we can\n apply the encoding described above to $\\even(\\bar x_1,x_2,x_3,\\dots,x_{23})$.\n\n The SAT problems obtained in this way are very hard. In order to make the problems more\n tractable, we added further constraints in order to simplify the search performed by the\n solver. This approach is known as streamlining~\\citep{streamlining}. The following\n restrictions turned out to be successful:\n \\begin{itemize}\n \\item Instead of a faithful encoding of the sums in the Brent equations using the\n   even predicate as described above, we also used a more restrictive sufficient condition \n   which instead of requiring an even number of arguments to be true enforces that zero or\n   two arguments should be true. This predicate zero-or-two can be broken\n   into at-most-two and not-exactly-one, which can be efficiently encoded as\n   \\begin{alignat*}1\n     \\textrm{not-exactly-one}(x_1,\\dots,x_p) &= \\bigwedge_{i=1}^p \\Bigl(x_i \\rightarrow \\bigvee_{j\\neq i} x_j\\Bigr)\\\\\n     \\textrm{at-most-two}(x_1,\\dots,x_p) &=\n     (\\bar x_1\\lor\\bar x_2\\lor\\bar x_3)\\land(\\bar x_1\\lor\\bar x_2\\lor\\bar x_4)\\\\\n     &\\quad\\land(\\bar x_1\\lor\\bar x_3\\lor\\bar x_4)\\land(\\bar x_2\\lor\\bar x_3\\lor\\bar x_4)\\\\\n     &\\quad\\land(\\bar x_1\\lor y)\\land(\\bar x_2\\lor y)\\land(\\bar x_1\\lor\\bar x_2\\lor z)\\\\\n     &\\quad\\land(\\bar x_3\\lor z)\\land(\\bar x_4\\lor z)\\land(\\bar x_3\\lor\\bar x_4\\lor y)\\\\\n     &\\quad\\land\\textrm{at-most-two}(y,z,x_5,\\dots,x_p),\n   \\end{alignat*}\n   where $y$ and $z$ are fresh variables. \nThe first two lines of at-most-two assert that at most two variables \nof $x_1, x_2, x_3, x_4$ are true. \nIf two or more of those variables are \ntrue then the new variables \n$y$ and $z$ have to be both true, if one variable is true, \nthen either $y$ or $z$ has to be true, and if all four variables are false, \nthen also $y$ and $z$ can be both false. Encoding this information \nin $y$ and $z$ allows to recursively apply at-most-two with two \narguments less. \nA straightforward direct encoding as in the first two lines \n\t\t is used when $p\\leq 4$.\n \\item We selected a certain portion, say 50\\%, of the variables $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$, $\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$, $\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ and instantiate them with the values they have in one of the known solutions.\n   The SAT solver then has to solve for the remaining variables. It turns out that in\n   many cases, it does not just rediscover the known solution but finds a truly different one\n   that only happens to have an overlap with the original solution.\n \\item Another approach was to randomly set half of the terms $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ with $i_2\\neq j_1$ and $j_2\\neq k_1$ and $k_2\\neq i_1$ to zero. This strategy was\n   motivated by the observation that in most of the known solutions, almost all these\n   terms are zero.\n \\item A third approach concerns the terms $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ with\n   $i_2=j_1$ and $j_2=k_1$ and $k_2=i_1$. Again motivated by the inspection of known solutions,\n   we specified that for each $\\iota$ either one or two such terms should be one.\n   More precisely, we randomly chose a distribution of the 27 terms with $i_2=j_1$ and $j_2=k_1$ and $k_2=i_1$\n   to the 23 summands of the scheme, with the condition that 19 summands should contain one term each\n   and the remaining four summands should contain two terms each.\n \\end{itemize}\n Each of the latter three approaches was used in combination with both the `even' and the `zero-or-two' encoding\n of the Brent equations. The resulting instances were presented to the SAT solver yalsat by \\cite{yalsat}. When it didn't\n find a solution for an instance within a few minutes, the instance was discarded and a new instance with another\n random choice was tried. A detailed analysis of the effect of our optimizations on the performance of the solver\n is provided in a separate paper~\\citep{sls}.\n\n \\section{Recognizing Equivalences}\\label{sec:sieving}\n\n From any given solution of the Brent equations we can generate many equivalent\n solutions.  For example, exchanging $\\alpha$ with $\\beta$ and flipping all\n indices maps a solution to another solution. This operation corresponds to the\n fact that $(AB)^\\top=B^\\top A^\\top$.  It is also clear from the equations\n that replacing $\\alpha$ by~$\\beta$, $\\beta$ by~$\\gamma$, and $\\gamma$\n by~$\\alpha$ maps a solution to another solution, although this operation is\n less obvious in terms of matrix multiplication.  Finally, for any fixed\n invertible matrix~$U$, we can exploit the fact $AB=AUU^{-1}B$ to map solutions\n to other solutions.\n\n The operations just described form a group of symmetries of matrix\n multiplication which was introduced by \\cite{de1978varieties}, who used them\n for showing that Strassen's scheme for $2\\times 2$ matrices is essentially\n unique: it is unique modulo the action of this symmetry group. \n To describe the group more formally, it is convenient to express matrix\n multiplication schemes as tensors,\n \\[\n \\sum_{\\iota=1}^{23}\n \\begin{pmatrix}\n   \\alpha_{1,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{1,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{1,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\alpha_{2,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{2,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{2,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\alpha_{3,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{3,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\alpha_{3,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\n \\end{pmatrix}\\otimes\n \\begin{pmatrix}\n   \\beta_{1,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{1,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{1,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\beta_{2,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{2,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{2,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\beta_{3,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{3,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\beta_{3,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\n \\end{pmatrix}\\otimes\n \\begin{pmatrix}\n   \\gamma_{1,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{1,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{1,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\gamma_{2,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{2,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{2,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} \\\\\n   \\gamma_{3,1}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{3,2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}} & \\gamma_{3,3}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\n \\end{pmatrix}.\n \\]\n A scheme is correct if and only if it is equal, as element of $(K^{3\\times\n   3})^{\\otimes3}$, to $\\sum_{i,j,k=1}^3 E_{i,k}\\otimes E_{k,j}\\otimes E_{j,i}$,\n where $E_{u,v}\\in K^{3\\times3}$ refers to the matrix which has a $1$ at\n position $(u,v)$ and zeros everywhere else.\n\n A permutation $\\pi\\in S_3$ acts on\n a tensor $A\\otimes B\\otimes C$ by permuting the three factors, and transposing\n each of them if $\\operatorname{sgn}(\\pi)=-1$. For example, $(1\\ 2)\\cdot(A\\otimes B\\otimes\n C)=B^\\top\\otimes A^\\top\\otimes C^\\top$ and $(1\\ 2\\ 3)\\cdot(A\\otimes B\\otimes\n C)=B\\otimes C\\otimes A$. A triple $(U,V,W)\\in\\operatorname{GL}(K,3)^3$ of invertible\n matrices acts via\n \\[\n (U,V,W)\\cdot(A\\otimes B\\otimes C)=UAV^{-1}\\otimes VBW^{-1}\\otimes WCU^{-1}.\n \\]\n A tuple $(U,V,W,\\pi)\\in\\operatorname{GL}(K,3)^3\\times S_3$ acts on a tensor $A\\otimes\n B\\otimes C$ by first letting the permutation act as described above, and then\n applying the matrices as described above.  The set $G=\\operatorname{GL}(K,3)^3\\times S_3$ is\n turned into a group by defining the multiplication in such a way that the\n operation described above becomes a group action.  The action of the group $G$\n defined on tensors $A\\otimes B\\otimes C$ is extended to the whole space\n $(K^{3\\times 3})^{\\otimes 3}$ by linearity. In other words, elements of $G$ act\n on sums of tensors by acting independently on all summands.\n\n Two matrix multiplication schemes are called equivalent if they belong to the\n same orbit under the action of~$G$. Whenever a new matrix multiplication scheme\n is discovered, the question is whether it is equivalent to a known scheme, for\n if it is, it should not be considered as new. A common test for checking that\n two schemes are not equivalent proceeds by computing certain invariants of the\n group action. For example, since permutation and multiplication by invertible\n matrices do not change the rank of a matrix, we can count how many matrices of\n rank 1,~2, and~3 appear in the scheme. If the counts differ for two schemes,\n then these schemes cannot be equivalent. For example, \\cite{DBLP:journals\/corr\/abs-1108-2830}\n and \\cite{oh2013inequivalence} proved in this way that their schemes were indeed\n new. Writing a scheme in the form $\\sum_{\\iota=1}^{23}(A_\\iota\\otimes\n B_\\iota\\otimes C_\\iota)$, we can encode this invariant as the polynomial\n $\\sum_{\\iota=1}^{23} (x^{\\operatorname{rank}(A_\\iota)}+x^{\\operatorname{rank}(B_\\iota)}+x^{\\operatorname{rank}(C_\\iota)})$.\n Similarly, also the polynomials\n \\[\n   \\sum_{\\iota=1}^{23} x^{\\operatorname{rank}(A_\\iota)+\\operatorname{rank}(B_\\iota)+\\operatorname{rank}(C_\\iota)}\n   \\quad\\text{and}\\quad\n   x^{\\sum_{\\iota=1}^{23}\\operatorname{rank}(A_\\iota)} +\n   x^{\\sum_{\\iota=1}^{23}\\operatorname{rank}(B_\\iota)} +\n   x^{\\sum_{\\iota=1}^{23}\\operatorname{rank}(C_\\iota)} \n \\]\n are invariants, because changing the order of summation does not affect the\n relative order of the factors in the tensor, and applying a permutation changes\n the relative order of the factors in every summand in the same way.\n\n When we have two schemes for which all three invariants match, they may\n nevertheless be inequivalent. For checking whether a solution found\n by the SAT solver is really new, comparing invariants is useful as a\n first step, but it is not sufficient. In fact, many solutions found\n by the SAT solver were inequivalent although all three invariants stated\n above agreed. Fortunately, it is not too hard to decide the equivalence of two given\n schemes by constructing, whenever possible, a group element that maps one to\n the other. We can proceed as follows. \n\n Suppose we are given two\n multiplication schemes $S,S'$ and we want to decide whether there exists a\n tuple $(U,V,W,\\pi)\\in\\operatorname{GL}(K,3)^3\\times S_3$ such that $(U,V,W,\\pi)\\cdot S=S'$.\n As far as the permutation is concerned, there are only six candidates,\n so we can simply try each of them. Writing $S=\\sum_{\\iota=1}^{23}(A_\\iota\\otimes\n B_\\iota\\otimes C_\\iota)$ and $S'=\\sum_{\\iota=1}^{23}(A_\\iota'\\otimes B_\\iota'\\otimes\n C_\\iota')$, it remains to find $U,V,W$ that map all the summands of $S$ to the\n summands of~$S'$, albeit possibly in a different order. We search for a suitable\n order by the following recursive algorithm, which is initially called with\n $Q$ being full space $K^{3\\times 3}\\times K^{3\\times 3}\\times K^{3\\times\n   3}$.\n\n \\medskip\n \\par\\noindent\n \\emph{Input:} $S,S'$ as above, a basis of a subspace $Q$ of $K^{3\\times 3}\\times K^{3\\times 3}\\times K^{3\\times 3}$\\\\\n \\emph{Output:} A triple $(U,V,W)\\in\\operatorname{GL}(K,3)^3\\cap Q$ with $(U,V,W)\\cdot S=S'$, or $\\bot$ if no such triple exists.\n\n \\step10 if $S$ and $S'$ are empty, then:\n \\step21 return any element $(U,V,W)$ of $Q$ with $\\det(U)\\det(V)\\det(W)\\neq0$, or $\\bot$ if no such element exists.\n \\step30 for all summands $A_\\iota'\\otimes B_\\iota'\\otimes C_\\iota'$ of $S'$, do:\n \\step41 if $\\operatorname{rank}(A_1)=\\operatorname{rank}(A_\\iota')$ and $\\operatorname{rank}(B_1)=\\operatorname{rank}(B_\\iota')$ and $\\operatorname{rank}(C_1)=\\operatorname{rank}(C_\\iota')$, then:\n \\step52 compute a basis of the space $P$ of all $(U,V,W)$ such that $UA_1=A_\\iota'V$, $VB_1=B_\\iota'W$, $WC_1=C_\\iota'U$ by\n    making an ansatz, comparing coefficients, and solving a homogeneous linear system.\n \\step62 compute a basis of $R=P\\cap Q$.\n \\step72 if $R$ contains at least one triple $(U,V,W)$ with $\\det(U)\\det(V)\\det(W)\\neq0$, then:\n \\step83 call the algorithm recursively with the first summand of $S$ and the $\\iota$th summand of $S'$ removed, and with $R$ in place of~$Q$.\n \\step93 if the recursive call yields a triple $(U,V,W)$, return it.\n \\step{10}0 return $\\bot$.\n\n \\medskip\n The algorithm terminates because each recursive call is applied to a sum with\n strictly fewer summands.\n The correctness of the algorithm is clear because it essentially performs an\n exhaustive search through all options. \n In order to perform the check in\n step~7, we can consider a generic linear combination of the basis elements\n of~$R$, with variables as coefficients. Then $\\det(U)\\det(V)\\det(W)$ is a\n polynomial in these variables, and the question is whether this polynomial\n vanishes identically on~$K$. Since we are interested in the case $K=\\set Z_2$,\n we can answer this by an exhaustive search.\n\n The recursive structure of the algorithm with up to 23 recursive calls at every\n level may seem prohibitively expensive.  However, the two filters in lines~4\n and~7 turn out to cut down the number of recursive calls considerably. A\n straightforward implementation in Mathematica needs no more than about one\n second of computation time to decide whether or not two given schemes are\n equivalent. Of course, we first compare the invariants, which is almost for\n free and suffices to settle many cases.\n\n For each scheme found by the SAT solver we have checked whether it is\n equivalent (for $K=\\set Z_2$) to one of the schemes found earlier, or to one\n of the four known schemes found by Laderman, Smirnov, Oh et al., and\n Courtois et al., respectively. From the roughly $270\\,000$ solutions\n found by the SAT solver that were distinct modulo the order of the summands,\n we isolated about $13\\,000$ schemes that were distinct modulo equivalence.\n In the appendix, we list the number of schemes we found separated by\n invariant. \n\n \\section{Simplifying Solutions}\\label{sec:switching}\n\n We can use the symmetries introduced in the previous section not only to\n recognize that a seemingly new scheme is not really new.  We can also use\n them for simplifying schemes. A scheme can for example be regarded\n as simpler than another scheme if the number of terms\n $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ in\n it which evaluate to $1$ is smaller. Calling this number the \\emph{weight} of\n a scheme, we prefer schemes with smaller weight.\n\n Ideally, we would like to replace every scheme $S$ by an equivalent scheme\n with smallest possible weight. In principle, we could find such a minimal\n equivalent element by applying all elements of $G$ to $S$ and taking the\n smallest result. Unfortunately, even for $K=\\set Z_2$, the group $G$ has\n $168^3\\cdot6=28\\,449\\,792$ elements, so trying them all might be feasible if we had\n to do it for a few schemes, but not for thousands of them.\n If we do not insist in the smallest possible weight, we can take a pragmatic\n approach and just spend for every scheme $S$ a prescribed amount of computation\n time (say half an hour) applying random elements of $G$ to~$S$:\n\n \\medskip\n \\par\\noindent\n \\emph{Input:} a multiplication scheme $S$\\\\\n \\emph{Output:} an equivalent multiplication scheme whose weight is less than or equal to the weight of~$S$.\n\n \\step10 while the time limit is not exhausted, do\n \\step21 pick a group element $g$ at random\n \\step31 if $\\mathrm{weight}(g(S))<\\mathrm{weight}(S)$, then set $S = g(S)$\n \\step40 return $S$\n\n \\medskip\n With this algorithm, we were able to replace about 20\\% of the new schemes found by the SAT solver\n by equivalent schemes with smaller weight. It is not too surprising that no\n improvement was found for the majority of cases, because the way we specified the\n problem to the SAT solver already induces a bias towards solutions with a small weight.\n\n The figure below shows the distribution of our $13\\,000$ schemes according to\n weight, after simplification. It is clear that the weight is always odd, hence\n the small gaps between the bars. It is less clear why we seem to have an\n overlay of three normal distributions, but we believe that this is rather an\n artifact of the way we generated the solutions than a structural feature of the\n entire solution set.\n\n \\begin{center}\n \\begin{tikzpicture}[x=2pt,y=2pt,yscale=.1,xscale=1]\n\\foreach \\x\/\\y in {257\/227,259\/129,261\/186,263\/81,265\/153,267\/65,269\/101,271\/36,273\/66,167\/4,275\/31,277\/36,279\/24,281\/21,283\/10,285\/10,287\/11,289\/9,163\/3,293\/4,295\/3,297\/1,299\/2,173\/8,175\/13,177\/14,179\/14,181\/17,183\/38,185\/52,187\/72,189\/90,169\/6,191\/143,193\/165,195\/185,197\/215,199\/252,201\/300,203\/353,205\/331,291\/3,207\/424,209\/379,211\/484,213\/364,215\/509,217\/339,219\/554,221\/315,223\/588,225\/266,227\/557,229\/297,165\/3,231\/533,233\/331,171\/5,235\/484,237\/349,239\/451,241\/330,243\/360,245\/352,247\/278,249\/314,251\/227,253\/290,255\/165}\n\\draw[fill=gray] (\\x,0) rectangle (\\x+1,\\y);\n\\draw[->] (150,0)--(310,0) node[above left] {weight};\n\\draw[->] (150,0)--++(0,610) node[right] {count};\n\\foreach \\x in {160,180,200,220,240,260,280,300} \\draw (\\x.5,0)--++(0,-10) node[below] {\\footnotesize\\x};\n\\foreach \\y in {0,100,200,300,400,500,600} \\draw (150,\\y)--++(0,-1) node[left] {\\footnotesize\\y};\n \\end{tikzpicture}\n \\end{center}\n \n \\section{Generalizing the Coefficient Ring}\\label{sec:signing}\n\n At this point, we have a considerable number of new matrix multiplication\n schemes for the coefficient field $K=\\set Z_2$. The next step is to lift them\n to schemes that work in any coefficient ring. \n The SAT solver presents us with a solution for $\\set Z_2$ in which all\n coefficients are $0$ or~$1$, and in order to lift such a solution, we make the\n hypothesis that this solution originated from a solution for an arbitrary coefficient\n ring in which all coefficients are $+1$, $-1$, or~$0$. The distinction between\n $+1$ and $-1$ gets lost in~$\\set Z_2$, and the task consists in recovering it.\n There is a priori no reason why such a lifting should exist, and indeed, we have\n seen a small number of instances where it fails. One such example is given in the\n appendix. Interestingly however, these examples seem to be very rare. In almost all cases,\n a lifting turned out to exist. \n\n In order to explain the lifting process, we return to the Brent equations discussed in\n Section~\\ref{sec:brent}. We set variables corresponding to coefficients\n that are zero in the SAT solution to zero, which simplifies the system\n considerably. According to the axioms of\n tensor products, we have $(\\lambda A)\\otimes B\\otimes C=A\\otimes(\\lambda B)\n \\otimes C=A\\otimes B\\otimes(\\lambda C)$ for any $A,B,C$ and every\n constant~$\\lambda$. We may therefore select in every summand $A\\otimes\n B\\otimes C$ one variable appearing in $A$ and one variable appearing in $B$ and\n set them to~$+1$. This reduces the number of variables further. However, the\n resulting system is still to hard to be solved directly.\n\n Before calling a general\n purpose Gr\\\"obner bases engine, we apply some simplifications to the system,\n which take into account that we are only interested in solutions whose\n coordinates are $-1$ or~$+1$. In particular, we can replace any exponent $k$\n appearing in any of the polynomials by~$k\\bmod 2$, we can cancel factors that\n clearly do not vanish on the points of interest, and we can replace polynomials\n of the from $xy\\pm1$ by $x\\pm y$.\n These simplifications may bring up some linear polynomials. By triangularizing the linear system\n corresponding to these polynomials, we can eliminate some of the variables. We can then\n simplify again, and possibly obtain new linear equations. The process is repeated until\n no further linear equations appear. We then add for each variable $x$ the polynomial $x^2-1$\n and compute a Gr\\\"obner basis with respect to a degree order. If this leads to new\n linear polynomials, we return to iterating triangularization, elimination,\n and simplification until no further linear equations show up, and then compute again a\n degree Gr\\\"obner basis. The whole process is repeated until we obtain a Gr\\\"obner basis\n that does not contain any new linear equations. If there are more than 15 variables left,\n we next compute a minimal associated prime ideal of an elimination ideal involving only five\n variables, and check whether adding it to the original system and computing a Gr\\\"obner basis\n leads to new linear equations. If it does, we start over with the whole procedure.\n Otherwise, we compute the minimal associated prime ideal of the whole system and return\n the solution corresponding to one of the prime factors. The process is summarized in the following\n listing.\n\n \\medskip\n \\par\\noindent\n \\emph{Input:} A finite subset $B$ of $\\set Q[x_1,\\dots,x_n]$\\\\\n \\emph{Output:} A common root $\\xi\\in\\{-1,1\\}^n$ of all the elements of~$B$, or $\\bot$ if no such common root exists.\n\n \\step10 Replace every exponent $k$ appearing in an element of $B$ by $k\\bmod2$\n \\step20 For every $p\\in B$ and every $i$ with $x_i\\mid p$, replace $p$ by $p\/x_i$\n \\step30 Replace every element of the form $xy-1$ or $-xy-1$ by $x-y$ or $x+y$, respectively.\n \\step40 if $B$ now contains linear polynomials, then:\n \\step51 Use them to eliminate some variables, say $y_1,\\dots,y_k$\n \\step61 Call the procedure recursively on the resulting set of polynomials\n \\step71 if there is a solution, extend it to the eliminated variables $y_1,\\dots,y_k$ and return the result\n \\step81 if there is no solution, return $\\bot$.\n \\step{9}0 Compute a Gr\\\"obner basis $G$ of $B\\cup\\{x_i^2-1:i=1,\\dots,n\\}$ with respect to a degree order\n \\step{10}0 if $G=\\{1\\}$, return $\\bot$\n \\step{11}0 if $G$ contains linear polynomials, then call this procedure recursively and return the result\n \\step{12}0 if $n>15$, then:\n \\step{13}1 Compute a basis $P$ of one of the minimal associated prime ideals of $\\<G>\\cap\\set Q[x_1,\\dots,x_5]$.\n \\step{14}1 Compute a Gr\\\"obner basis $G'$ of $G\\cup P$ with respect to a degree order\n \\step{15}1 if $G'$ contains linear polynomials, then call this procedure recursively and return the result\n \\step{16}0 Compute a basis $P$ of one of the minimal associated prime ideals of $\\<G>\\subseteq\\set Q[x_1,\\dots,x_n]$.\n \\step{17}0 Return the common solution $\\xi$ of~$P$.\n\n \\medskip\n An implementation of this procedure in Mathematica is available on the website of this\n article~\\citep{mmr}. In this implementation, we use Singular \\citep{greuel02} for\n doing the Gr\\\"obner basis calculations and for the computation of minimal associated\n prime ideals. Despite the large number of\n variables, Singular handles the required computations with impressive\n speed, so that the whole signing process takes only about 20 seconds per solution on\n the average. Only a small number of cases, which happen to have a few more\n variables than the others, need much longer, up to a few hours.\n\n \\section{Introducing Parameters}\\label{sec:families}\n\n The idea of instantiating some of the variables based on a known scheme and\n then solving for the remaining variables approach not only applies to SAT\n solving. It also has an algebraic counterpart. Solving the Brent equations with\n algebraic methods is infeasible because the equations are nonlinear, but\n observe that we only have to solve a linear system if we start from a known\n scheme and only replace all $\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ by fresh\n variables. Solving linear systems is of course much easier than\n solving nonlinear ones.\n\n More generally, we can select for each $\\iota\\in\\{1,\\dots,23\\}$ separately\n whether we want to replace all $\\alpha_{i_1,i_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$'s or all\n $\\beta_{j_1,j_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$'s or all $\\gamma_{k_1,k_2}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$'s by fresh\n variables, and we still just get a linear system for these variables. Once we\n make a selection, solving the resulting linear system yields an affine vector\n space. One might expect this affine space will typically consist of a single\n point only, but this is usually not the case.\n\n A solution space with positive dimension can be translated into a\n multiplication scheme involving one or more free parameters. Starting from the\n resulting parameterized scheme, we can play the same game with another\n selection of variables, which may allow us to introduce further parameters. If\n we repeat the procedure several times with random selections of which variables\n are known, we obtain huge schemes involving 40 or more parameters.\n These parameters are however algebraically dependent, or at least it is too\n costly check whether they are dependent or not. We got better results by\n proceeding more systematically, as summarized int in the following listing.\n\n \\medskip\n \\par\\noindent\n \\emph{Input:} A matrix multiplication scheme $S=\\sum_{\\iota=1}^{23}(A_\\iota\\otimes B_\\iota\\otimes C_\\iota)$.\n Write $A_\\iota=((\\alpha_{i,j}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}))$, $B_\\iota=((\\beta_{i,j}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}))$, $C_\\iota=((\\gamma_{i,j}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}))$.\\\\\n \\emph{Output:} A family of matrix multiplication schemes with parameters $x_1,x_2,\\dots$\n\n \\step10 for $\\iota=1,\\dots,23$, do:\n \\step21 for every choice $u,v\\in\\{\\alpha,\\beta,\\gamma\\}$ with $u\\neq v$, do:\n \\step32 replace all entries $u_{i,j}^{\\text{\\tiny (}\\iota\\text{\\tiny )}}$ for $i,j=1,\\dots,3$ in $S$ by fresh variables\n \\step42 replace all entries $v_{i,j}^{(m)}$ for $i,j=1,\\dots,3$ and $m\\neq\\iota$ in $S$ by fresh variables\n \\step52 equate the resulting scheme $S$ to $\\sum_{i,j,k} E_{i,j}\\otimes E_{j,k}\\otimes E_{k,i}$ and compare coefficients\n \\step62 solve the resulting inhomogeneous linear system for the fresh variables introduced in steps 3 and~4\n \\step72 substitute the generic solution, using new parameters $x_i,x_{i+1},\\dots$, into~$S$\n \\step80 return $S$\n\n \\medskip\n With this algorithm and some slightly modified variants (e.g., letting the outer loop run backwards or transposing\n the inner and the outer loop), we were able to obtain schemes with altogether up to 17 parameters.\n Although all new parameters introduced in a certain iteration can only appear\n linearly in the scheme, old parameters that were considered as belonging to the\n ground ring during the linear solving can later appear rationally. However, by manually\n applying suitable changes of variables, we managed to remove all denominators from\n all the families we inspected. Not even integer denominators are needed. \n We can also check using Gr\\\"obner bases\n\n whether the parameters are independent, and for several families with 17 parameters\n they turn out to be. In the language of algebraic geometry, this means that the solution\n set of the Brent equations has at least dimension~17 as an algebraic variety.\n\n One of our families is shown in the appendix, and some further ones are provided\n electronically on our website. These families should be contrasted with the family found by\n Johnson and McLoughlin in the the 1980s~\\citep{DBLP:journals\/siamcomp\/JohnsonM86}. In particular, while they lament\n that their family contains fractional coefficients such as $\\frac12$ and $\\frac13$\n and therefore does not apply in every coefficient ring, our families only involve\n integer coefficients and therefore have no such restriction. Moreover, their family\n has only three parameters, and with the method described above, only $6$ additional\n parameters can be introduced into it. The number of parameters we managed to introduce\n into the known solutions by Laderman, Courtois et al., Oh et al., and Smirnov\n are $0$,~$6$, $10$, and~$14$, respectively.\n \n \\section{Concluding Remarks}\\label{sec:skimming}\n\n Although we have found many new multiplication schemes with 23 multiplications,\n we did not encounter a single scheme with 22 multiplications. We have checked\n all schemes whether some of their summands can be merged together using tensor product\n arithmetic. For doing so, it would suffice if a certain scheme contains some summands\n which share the same $A$'s, say, and where the\n corresponding $B$'s, say, of these rows are linearly independent. We could then\n express one of these $B$'s in terms of the others and eliminate the summand in\n which it appears. For example, if $B_3=\\beta_1B_1+\\beta_2B_2$, then we have $A\\otimes B_1\\otimes C_1\n + A\\otimes B_2\\otimes C_2+A\\otimes B_3\\otimes C_3=A\\otimes\n B_1\\otimes(C_1+\\beta_1C_3)+A\\otimes B_2\\otimes(C_2+\\beta_2C_3)$. Since none of\n our schemes admits a simplification of this kind, it remains open whether a\n scheme with 22 multiplications exists.\n\n Another open question is: how many further schemes with 23 multiplications and coefficients in $\\{-1,0,1\\}$\n are there? We have no evidence that we have found them all. In fact, we rather believe that there\n are many further ones, possibly including schemes that are very different from\n ours. There may also be parametrized families with more than 17 parameters, and it would\n be interesting to know the maximal possible number of parameters, i.e., the actual dimension\n of the solution set of the Brent equations.  \n\n\\bibliographystyle{elsarticle-harv}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Previous Work} \\label{sec:intro}\nDuring the elaboration of~\\cite{BirkvonNeu:36} John von Neumann wrote\nto Garret Birkhoff: ``I would like to make a confession\nwhich may seem immoral: I do not believe absolutely in Hilbert space any more. \nAfter all Hilbert-space (as far as quantum-mechanical things are concerned) \nwas obtained by generalizing Euclidean space, footing on the principle of \n``conserving the validity of all formal rules''. \nThis is very clear, if you consider the axiomatic-geometric definition of\nHilbert-space, where one simply takes Weyl's axioms for a unitary-Euclidean\nspace, drops the condition on the existence of a finite linear basis, and\nreplaces it by a minimum of topological assumptions \n(completeness + separability). Thus Hilbert-space is the straightforward\ngeneralization of Euclidean space, if one considers the {\\em vectors}\nas the essential notions. Now we begin to believe that it is not the \n{\\em vectors} which matter but the {\\em lattice of all linear (closed) \nsubspaces}. Because:\n\\begin{enumerate}\n\\item The vectors ought to represent the physical {\\em states}, but they\ndo it redundantly, up to a complex factor only.\n\\item And besides the {\\em states} are merely a derived notion, the primitive\n(phenomenologically given) notion being the {\\em qualities}, which correspond \nto the {\\em linear closed subspaces}'' \n(see~\\cite{vNeumann_letters}, p. 59, letter dated Nov. 13, Wednesday, 1935).\n\\end{enumerate}\n\nThe goal of this work is to pursue von Neumann's program of describing \nQuantum Logic in terms of closed subspaces and without vectors one step\nfurther. \nThis work presents two original features:\n\\begin{itemize}\n\\item it takes a logical approach to Quantum Physics, where states and\npropositions take the main roles, and\n\\item while it assumes the formalism of Hilbert spaces that fits Quantum \nPhysics, it tries the utmost to use only notions, such as states, propositions,\nprojections, orthogonality and so on, that have a meaning, albeit mostly\ntrivial, in Classical Physics.\nSpecial care will be taken to ensure that the quantic principles proposed\nhold classically.\n\\end{itemize}\n\n\\section{Quantum Logic} \\label{sec:Logic}\nOne may say that Logic is the study of the relation between states of the \nworld and propositions used to talk about those states. Quantum logic must\ntherefore be the study of the relation between quantum states and quantum\npropositions. The accepted view is that both quantum states and quantum\npropositions should be represented by closed subspaces of a Hilbert space.\nQuantum states are one-dimensional subspaces.\nQuantum logic is therefore the study of the relation between one-dimensional \nsubspaces and arbitrary closed subspaces.\nOne obvious topic for Quantum logic is therefore the study of the properties\nof projections in Hilbert spaces: a one-dimensional subspace projects onto\na one-dimensional or zero-dimensional subspace of any closed subspace.\nProjections are also central to Quantum Physics since they correspond to the \nchange brought about by the measurement of a physical property. \nPrevious works~\\cite{LEG:Malg} and~\\cite{AndThen:Leibniz} provided \na first study of some of the properties of such projections: \nthey dealt only with qualitative properties. \nThe present paper inaugurates the quantitative study of the \nprojective geometry of complex Hilbert spaces.\n\nThe purpose of the exercise is to shed light on the notion of measurement \nin Quantum Physics by developing a geometry of Hilbert spaces whose \nentities are physically meaningful: states of physical systems and \nmeasurements on physical systems. Our goal can be understood in considering \nthe history of geometry. Euclidean plane geometry was the starting point. \nIts elements are points and lines. \nMathematical developments (due to Descartes in particular) \nenabled a treatment of geometry in the vector space\n$\\mbox{${\\cal R}$}^{n}$. A new definition of geometry, abstracting from the vector space\nstructure and returning to the basic notions of points and lines, enabled\nthe development of non-Euclidean geometries.\nFor Hilbert spaces, historically the algebraic presentation came first.\nThe purpose of this paper is to extract from the algebraic presentation\na leaner presentation similar in spirit to Euclid's geometry.\nOur basic entities are one-dimensional subspaces and, more\ngenerally, closed subspaces {\\em and not vectors}.\n\n\nIn an obvious way, two elements (vectors) of a Hilbert space \ndefine a number, their inner product. \nWe are looking for numbers that characterize relations between subspaces, not \nvectors. \nThis paper proposes to associate a real number\nwith any pair of one-dimensional subspaces $x, y$: \\mbox{$p(x, y)$} \nand, by extension,\nto any pair of a one-dimensional subspace and a closed \nsubspace $\\alpha$: \\mbox{$p(x, \\alpha)$}. \nThis number is always in the interval $[0,1]$ \nand behaves in many ways like {\\em the probability that the proposition \n$\\alpha$ is\nfound true when it is tested for in state $x$}, in line with the probabilistic\ninterpretation of Quantum Physics. \nIt satisfies further properties that\nare more difficult to interpret and that characterize the linear dependence\nstructure and the structure of projections.\n\nAnother numerical quantity, an angle, $\\theta$, is defined by any triple of \none-dimensional subspaces. It is interpreted as the source of the\ninterference occurring between alternative paths a system could take.\nThis paper is devoted to the study of those aspects of the geometry of\nHilbert spaces related to the numbers $p$ and $\\theta$. \nThe study of those M-algebras \n(see~\\cite{LEG:Malg}) that admit quantities satisfying the properties of $p$,\n$\\theta$ and superpositions is left for further study.\n\n\\section{Background and Notations} \\label{sec:background}\nWe assume a Hilbert space \\mbox{${\\cal H}$}\\ on the field \\mbox{${\\cal C}$}\\ of complex numbers is given.\nThe complex conjugate of a complex number $c$ is $\\overline{c}$.\nFor any complex number $c$, $\\mid \\! c \\! \\mid$ \nrepresents its modulus, which is \na nonnegative real number.\nFor any complex number $c$ different from $0$, $\\arg(c)$ represents its\ncomplex argument: \\mbox{$c = \\, \\mid \\! c \\! \\mid e^{i \\arg(c)}$}.\nElements of \\mbox{${\\cal H}$}\\ will typically be: \\mbox{$\\vec{u}, \\vec{v} \\ldots$}.\nThe zero vector is denoted by $\\vec{0}$. The inner product of $\\vec{u}$\nand $\\vec{v}$ is \\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle$}.\nThe inner product is linear in its first argument and conjugate-linear in\nits second argument.\nTwo vectors $\\vec{u}$ and $\\vec{v}$ are perpendicular, written \n\\mbox{$\\vec{u} \\perp \\vec{v}$}, iff \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle = 0$}.\nThe norm of $\\vec{u}$ is \\mbox{$\\parallel \\vec{u} \\parallel$}.\nA unit vector is a vector of norm $1$.\nWe shall use the notation \\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle > 0$}\nto denote the fact that the inner product is a strictly positive {\\em real} \nnumber.\n\nThe set of all closed subspaces of \\mbox{${\\cal H}$}\\ will be denote by $M$.\nThe elements of $M$ should be thought of representing propositions, \nor, results of physical measurements. Greek letters from the beginning\nof the alphabet will be used to denote elements of $M$. The reader may\nthink of a typical element of $M$, $\\alpha$ as meaning {\\em the spin\nin the $z$-direction is nonnegative}. Note that propositions represent\nmeasurements with a specified result or a set of possible results: \nsuch as measuring the value $1\/2$ for\nthe spin in the $z$-direction or measuring a nonnegative value for this spin.\nTo every \\mbox{$\\alpha \\in M$}\none may associate its orthogonal complement, which will be denoted\n$\\neg \\alpha$. The proposition $\\neg \\alpha$ is interpreted as the measurement\nthat measures the quantity measured by $\\alpha$ but provides a value that is\nnot in the set specified by $\\alpha$. If $\\alpha$ claims that the spin in\nthe $z$-direction is nonnegative, $\\neg \\alpha$ measures the spin along the\nsame direction but finds it negative. \nTwo specific propositions are worth mentioning: falsehood, $0$\nis the null subspace $\\{\\vec{0}\\}$ and truth, $1$ is the whole space \\mbox{${\\cal H}$}.\nAny closed subspace $\\alpha$ of \\mbox{${\\cal H}$}\\ defines the projection of \\mbox{${\\cal H}$}\\ onto\n$\\alpha$. For any \\mbox{$\\vec{u} \\in \\mbox{${\\cal H}$}$} its projection on $\\alpha$ will\nbe denoted $\\alpha(\\vec{u})$. The relation between physical measurements and\nprojections will be explained after we discuss states.\n \nAmong the closed subspaces of \\mbox{${\\cal H}$}\\ particular attention will be paid to\none-dimensional subspaces. The set of one-dimensional subspaces of \\mbox{${\\cal H}$}\\\nis denoted $X$ and the elements of $X$ are typically letters from the end\nof the alphabet: $x$, $y$ and so on. As mentioned just above:\n\\mbox{$X \\subseteq M$}. Elements of $X$ will be called {\\em states}.\nA one-dimensional subspace $x$ represents a possible (pure) state of the \nphysical system. Think of the state in which the spin in the $z$-direction is \n$1\/2$, for example. We assume that states are propositions.\nThe fact that \\mbox{$X \\subseteq M$} reflects\nthe situation in which every pure state has an associated measurement that\ncharacterizes it: one may measure the spin in the $z$-direction and one\nof the possible values is $1\/2$. The proposition ``the spin in the \n$z$-direction is nonnegative'' is not a state.\n\nSince a proposition \\mbox{$\\alpha \\in M$} is a closed subspace of \\mbox{${\\cal H}$},\nfor any \\mbox{$x \\in X$}, either \\mbox{$x \\subseteq \\alpha$} or $\\alpha$ \ncontains no vector of $x$ except the zero vector. Any proposition is the union\nof the states it includes and any proposition can be seen as the set of all\nthe states it includes. We shall indeed prefer the notation \n\\mbox{$x \\in \\alpha$} to \\mbox{$x \\subseteq \\alpha$}.\n\nNote that if \n\\mbox{$\\vec{v} \\in x \\in X$} and \\mbox{$\\vec{u} \\perp \\vec{v}$} then\n\\mbox{$\\vec{u} \\perp \\vec{w}$} for every \\mbox{$\\vec{w} \\in x$}.\nWe denote such a situation by \\mbox{$\\vec{u} \\perp x$}.\nIf every \\mbox{$\\vec{u} \\in \\alpha$} is orthogonal to $x$ we say that\n\\mbox{$x \\perp \\alpha$}. If every \\mbox{$x \\in X$}, \n\\mbox{$x \\in \\alpha$} is orthogonal to $\\beta$ we say that\n\\mbox{$\\alpha \\perp \\beta$}.\nThe image of any \\mbox{$x \\in X$} by any (projection) \\mbox{$\\alpha \\in M$} \nis either a one-dimensional subspace \\mbox{$y \\in X$} or the zero-dimensional\nsubspace. This second possibility occurs exactly when $x$ is orthogonal\nto $\\alpha$. We shall denote by $\\alpha(x)$ the one-dimensional \nor zero-dimensional subspace that is the projection of $x$ onto $\\alpha$.\nNote that \\mbox{$\\alpha(x) = x$} iff \\mbox{$x \\in \\alpha$}.\nWe write \\mbox{$\\alpha(x) = 0$} to denote the case $\\alpha(x)$ \nis zero-dimensional, i.e., the case \\mbox{$x \\perp \\alpha$}.\nThe projection of the zero-dimensional subspace on any $\\alpha$ is the\nzero-dimensional subspace and we shall extend the action of $\\alpha$ by\nsetting \\mbox{$\\alpha(0) = 0$}.\n\nIn Quantum Physics measurements may change\nthe state of the system. The state obtained when measuring $\\alpha$ in state\n$x$ is precisely $\\alpha(x)$, the projection of $x$ on the subspace $\\alpha$.\nIf $x$ is orthogonal to $\\alpha$, then the measurement $\\alpha$ is impossible\nin state $x$: this happens precisely when the quantity measured by $\\alpha$\nhas, in $x$, a well-defined value that is not in the set specified by $\\alpha$.\nEquivalently, this happens precisely when $x$ is in the subspace $\\neg \\alpha$,\nor \\mbox{$(\\neg \\alpha)(x) = x$}. \n\n\\section{Classical Physics} \\label{sec:classical}\nThe notions described in Section~\\ref{sec:background} have been given\na meaning grounded in the Hilbert space formalism of Quantum Mechanics.\nThis seems to preclude their application to Classical Mechanics, since,\nclassically, states are not rays in a Hilbert space.\nNevertheless, the common wisdom is that Quantum Mechanics should apply \neverywhere and that Classical Mechanics should be a limiting case of\nQuantum Mechanics. Indeed, both Classical Mechanics and Quantum Mechanics\ncan be studied in structures that abstract from the concepts\nof Section~\\ref{sec:background}, preserving the properties of states\nand measurements. A full treatment is left for future work, \nbut the following remark explains the main feature of classical systems.\n\nClassically, measurements do not change the state of a system, therefore if\na state $x$ is not orthogonal to a proposition $\\alpha$,\nwe have \\mbox{$\\alpha(x) = x$}, expressing the fact that either $x$ possesses\nthe property $\\alpha$ or it possesses its negation $\\neg \\alpha$. We have:\n\\[\n{\\bf Principle \\ of \\ Classical \\ Physics \\ } \n{\\rm \\ Any \\ two \\ different \\ states \\ are \\ orthogonal.}\n\\]\n\n\\section{The Reciprocity Principle} \\label{sec:reciprocity}\nBefore proceeding to the analysis of the notion of a superposition which is the\ncrux of this paper, we need a simple remark. It will be presented as a \nprinciple, to stress the physical meaning of a fact that is woven so deep in \nthe familiar linear structure of Hilbert spaces that we tend not to reflect \non it anymore.\nIf the measurement $\\neg x$ acting on state $y$ and on state $z$ \nproduces the same state, then $x$, $y$ and $z$ must sit in the same\nplane, and therefore the measurement $\\neg y$ must produce the same state\nwhen acting on $x$ and on $z$.\n\\pagebreak[0]\n\\[\n{\\bf Reciprocity \\ Principle \\ } {\\rm \\ Let \\ } x, y, z \\in X, \n{\\rm \\ be \\ pairwise \\ different}.\n\\]\n\\[\n{\\rm Then \\ } (\\neg x)(y) = (\\neg x)(z) \\: \\Rightarrow \\: \n(\\neg y)(z) = (\\neg y)(x).\n\\]\n\nThe Reciprocity Principle suggests the following definition.\n\\begin{definition} \\label{def:coplanarity}\nWe shall say that states $x$, $y$ and $z$ are {\\em coplanar},\nwritten \\mbox{$coplanar(x, y, z)$} iff either two out of the three\nare equal, or they are pairwise different and \n\\mbox{$(\\neg x)(y) = (\\neg x)(z)$}. \n\\end{definition}\nThe Reciprocity Principle says that \n{\\em coplanarity} is a property of the set \\mbox{$\\{x, y, z\\}$}, \ni.e., for any permutation $x'$, $y'$, $z'$ of $x$, $y$, $z$\n\\mbox{$coplanar(x', y', z')$} is equivalent to \\mbox{$coplanar(x, y, z)$}.\n\nThe Reciprocity Principle is experimentally testable: if the {\\em no} answer\nto a test $x$ gives the same state when performed on $y$ and on $z$, the\n{\\em no} answer on a test $y$ will give the same answer on $z$ and $x$.\n\nIn Hilbert space, indeed, if $y$ and $z$ have the same projection on the \nsubspace orthogonal to $x$, call it $x'$, then all four one-dimensional\nsubspaces: $x$, $x'$, $y$ and $z$ are in the same two-dimensional subspace,\ncall it $\\alpha$,\nand therefore the projections of $z$ and $x$ on the subspace orthogonal\nto $y$ are both the one-dimensional subspace of $\\alpha$ orthogonal to $y$.\n\nIn Classical Physics, the Reciprocity Principle holds trivially,\nsince its assumptions are never satisfied. \nIndeed if \\mbox{$x \\neq y$}, we have \n\\mbox{$(\\neg x)(y) = y$}, and similarly\n\\mbox{$(\\neg x)(z) = z$} and therefore the assumption\n\\mbox{$(\\neg x)(y) = (\\neg x)(z)$} implies \\mbox{$y = z$}, contrary to \nassumption.\n\n\\section{Superpositions: Conceptual Analysis} \\label{sec:conceptual}\nThe concept of a superposition is a revolutionary novelty introduced\nby Quantum Mechanics.\nIf a system may be in any one of two pure states $x$ and $y$, we must consider\nthat it may also be in any one of many {\\em superpositions} of $x$ and $y$.\nThis paper is devoted to an in-depth analysis of superpositions.\n\nThe following remark has resulted in a vast literature: \nthe revolutionary character of quantic superpositions is the consequence\nof the fact no such superpositions have to be considered, or may be \nseen in classical systems. In Schr\\\"{o}dinger's colorful thought experiment:\nthe cat is either dead or alive, but nobody has evidence of a superposition\nof a dead and a live cat. This seems to contradict the principle exposed\nin Section~\\ref{sec:classical}, of the universality of Quantum Mechanics.\nIf everything in the universe is quantic and any two quantic states can be\nsuperposed, then any two classical states, such as a live and a dead cat,\ncan be superposed. Many explanations have been proposed and this is not\nthe place for a survey. Most explanations accept the existence of \nsuperpositions of classical states and explain why such superpositions are\nnot seen. The analysis of the superposition concept to be developed below\nproposes a radically different explanation. It is not the case, it is claimed \nhere, that, in Quantum Mechanics, any two states can be superposed: \non the contrary, no superposition of orthogonal states can ever be considered. \nSince different classical states are orthogonal,\nthe only superpositions of classical states that can ever occur are\ntrivial: superpositions of a state with itself. Trivial superpositions\nare indeed observed and unproblematic.\n\nTo avoid any misunderstanding: if \\mbox{$\\mid + \\rangle$} and \n\\mbox{$\\mid - \\rangle$} are orthogonal states, the state \n\\mbox{$1 \/ \\sqrt{2} (\\mid + \\rangle + \\mid - \\rangle)$} is a perfectly legal\nstate, but it is not a superposition of \\mbox{$\\mid + \\rangle$} and \n\\mbox{$\\mid - \\rangle$}. It is equal, as will be clear, to many different\nsuperpositions of non-orthogonal states (that are themselves linear \ncombinations of the states \\mbox{$\\mid + \\rangle$} and \\mbox{$\\mid - \\rangle$}.\nThe reader will be well advised {\\em not} to think {\\em linear combination} \nwhen {\\em superposition} is read.\n \nTo explain the surprising position above, let us, first, reflect on the \nnature of superpositions and their origin:\nwhat are they and how do they come into consideration, without trying to\ndescribe formally such superpositions. Then, we shall propose a formalization\nand an algebraic structure.\n\nThe reader should notice that the linear combination of vectors of a Hilbert \nspace provides a formal operation, not a conceptual analysis, and also that,\nsince vectors do not represent states, the linear combination of vectors\ncannot offer a proper formalization for the superpositions of states.\nEven though we announced above that orthogonal states cannot be superposed,\nit is clear that orthogonal unit vectors can be combined linearly to form \nunit vectors. This should convince the reader that we shall not formalize\nsuperposition as a straightforward linear combination.\n\n\\subsection{Nature and Origin} \\label{sec:nature}\nSuperpositions must be considered to describe systems about which all we \nknow is that they are the result of one of a number of different possible \npaths (or histories), \ni.e., if we have no way of knowing which history indeed took place.\nIn such a case, we must consider that the system is in some state that is\na superposition, i.e., a {\\em compound} of the states that are the produced\nby each of the possible paths. The term {\\em compound} is used here\nwhere, chemically-speaking, the term {\\em mixture} may be more appropriate\nbecause this last term is used in Quantum Mechanics with a different meaning.\n\nIf one knows which path has been taken, or one could discover \nwhich path has been taken, then one must consider that the system is in the\nstate that results from the path taken, and one must use probability theory\nto describe one's ignorance about the state of the system.\nIf one does not know and cannot know which path has been taken, then one must\nconsider that the system is in some specific superposition of the states \nresulting from the different possible paths. This is a general principle:\nif one cannot know which path has been taken, then those paths {\\em interfere}\nand therefore the system cannot be described using only probability theory,\nbut must be described by a state that is a compound, i.e., a superposition of\nthe states resulting from the different interfering paths.\nThis general principle holds also in Classical Physics, as will be seen in \nSection~\\ref{sec:conditions}. The way in which the different paths may \ninterfere, i.e., the parameters that characterize the different possible\nsuperpositions will be described in Section~\\ref{sec:parameters}.\n\nThe paradigmatic example of such a situation is a the two-slits\nexperiment in which a particle travels through one of two slits and\none does not know which.\n\n\\subsection{Parameters} \\label{sec:parameters}\nTo leave things simple we shall consider only the superpositions of two\nstates, without loss of generality as long as we consider only a finite\nnumber of possible paths. Generalizing to path integrals is beyond the scope\nof this paper. Suppose therefore that we must deal with a system that may\nresult from two different paths. If path $p_{1}$ was taken, the system is \nin state $y$; if path $p_{2}$ was taken, the system is in state $z$.\nIf one cannot know which path was taken, one must consider that \nthe system is in\na state that is some superposition of the two states $y$ and $z$. \nMany such superpositions are possible and the purpose of this section is\nto describe the experimental parameters that influence the superposition\nto be used.\nIn Section~\\ref{sec:conditions}, the question of whether we can know which\npath was taken will be given an unequivocal answer.\n\nIn a situation in which any one of two paths may have been taken, \nthe experimental conditions determine the respective weights to be\ngiven to each one of the possible paths. These relative weights may be \ninterpreted as describing the a-priori probability of each one of the paths,\nor the relative proportions in which each of the paths is taken.  \nA superposition of $y$ and $z$ obtained\nas the result of the interference between the two paths $p_{1}$ and $p_{2}$\nwill therefore be characterized by a single parameter \\mbox{$r \\in [0,1]$}. \nThe proper value to be chosen\nfor this parameter is a function of the experimental setup. \nThe reader should notice that, even though\nwe shall describe such a superposition of states $x$ and $y$ as some sort of \n{\\em compound} or {\\em mixture} of $x$ and $y$, a superposition is a pure\nstate, not what is known in QM as a mixed state.\n\nThe parameter $r$ that characterizes a superposition\ndescribes, in a sense, the respective proportions (ratios) of $y$ and $z$\npresent in the superposition, though this intuitive analogy should not \nbe taken too seriously. The parameter $r$ is therefore a real number:\n\\mbox{$0 \\leq r \\leq 1$} that describes the {\\em weight} of $y$ relative to $z$\nin the superposition.\n\nIn the two-slits experiment, where $y$ represents the state resulting from\nthe electron moving through the upper slit and $z$ the state resulting from\nthe electron moving through the lower slit, the parameter $r$ will depend on\nthe respective widths of the two slits and the respective distance of those\nslits to the origin.\n\nThe superpositions we shall consider are therefore of the\nform \\mbox{$super(y, z, r)$} for states \\mbox{$y, z \\in X$} and\nreal number \\mbox{$0 \\leq r \\leq 1$}.\nThe telling notation \\mbox{$r y \\, + \\, (1 - r) z$} \nwill be used in place of the more austere \\mbox{$super(y, z, r)$},\nbut the reader is warned that $+$ does not mean addition, juxtaposition \ndoes not mean\nmultiplication and some of the properties one would expect from our notation\ndo {\\em not} hold. In particular the composition of superpositions does not \npossess the properties suggested by the notation.\n\n\\subsection{Conditions} \\label{sec:conditions}\nSection~\\ref{sec:parameters} indicated that superpositions of states $y$ and\n$z$ should be considered only if there is no way to know which\none of the paths $p_{1}$ or $p_{2}$ leading to\n$y$ and $z$ respectively has been traveled.\nIt is time to reflect on this condition.\n\nIf the states $y$ and $z$ are orthogonal: \\mbox{$y \\perp z$}, then there is\na way to find out for sure which of the two paths has been traveled:\nperform on the resulting state a measurement testing whether the state is $y$\nor not: a test $y$, $(\\neg y)$. If path $p_{1}$ has been traveled, the result\nwill be a {\\em yes} for sure since the state is $y$. If path $p_{2}$ \nhas been traveled, the result, for sure, will be a {\\em no} since the state\nis $z$, orthogonal to $y$. Similarly, we could have tested for $z$ or for any\nproposition satisfied by one of the states $y$ or $z$ and orthogonal to the\nother one. We see that no superposition of orthogonal states can ever\nbe defined. This is is stark contrast with the linear combination of vectors\nin a Hilbert space.\n\nFurther reflection shows that if the states $y$ and $z$ are not orthogonal,\none can never find out for sure which of the paths $p_{1}$ or $p_{2}$ has\nbeen traveled. Indeed the only situation in which one could find out would be\nto test for some proposition $\\alpha$ satisfied, for sure, by one of the two\nstates $y$ or $z$ and not satisfied, for sure, by the other state.\nIn other terms, a closed subspace $\\alpha$ containing one of $y$ or $z$ and\northogonal to the other one. But this implies \\mbox{$y \\perp z$}.\nWe see that:\n\\[\n{\\bf Principle \\ of \\ Superposition} {\\rm \\ The \\ superposition \\ }\nr y \\, + \\, (1 - r) z \n\\]\n\\[ \n{\\rm \\ is \\ defined \\ if \\ and \\ only \\ if \\ } y \\not \\perp z.\n\\]\n\nIn Section~\\ref{sec:superp_def} a definition of superpositions \nin the formalism of Hilbert spaces will be provided, \nbut, first, we shall discuss two general principles, and justify them\nby considerations independent of the Hilbert space formalism. \n\n\\subsection{Trivial Superpositions} \\label{sec:trivial}\nLet us consider, first, the superpositions of a state $y$ with itself:\n\\mbox{$r y \\, + \\, (1 - r) y$}.\nBy the Principle of Classical Physics, these are the only superpositions\npossible in classical physics.\n\nEvidence from both classical and quantum physics shows that such superpositions\nare trivial:\n\\[\n{\\bf Principle \\ of \\ Triviality} \\ \\forall y \\in X, \\forall r \\in [0 , 1],\nr y \\, + \\, (1 - r) y = y.\n\\]\n\nHaving disposed of the cases \\mbox{$y \\perp z$} and \\mbox{$y = z$}, let us\nstudy the generic case of superpositions.\n\n\\subsection{Principle of Coplanarity} \\label{sec:coplanarity}\n\nA superposition is coplanar with its components.\nAssume \\mbox{$y \\not \\perp z$}.\n\\[\n{\\bf Principle \\ of \\ Coplanarity} \\  \\forall r \\in [0 , 1],\ncoplanar(r y \\, + \\, (1 - r) z, \\, y, \\, z) .\n\\]\nThis principle can be justified in the following way.\nThe superposition\n\\mbox{$x = r y \\, + \\, (1 - r) z$} results from\nour inability to know which of $p_{1}$, resulting in $y$ or $p_{2}$,\nresulting in $z$ has been traveled.\nMeasuring $\\neg y$ on $x$ shows that the path $p_{1}$ has not been traveled\nand therefore $p_{2}$ has been traveled and the current state\n$(\\neg y)(x)$ is in fact $(\\neg y)(z)$.\n\nWe shall propose a precise definition of\nsuperpositions such as \\mbox{$r y \\, + \\, (1 - r) z$} for\n\\mbox{$y \\not \\perp z$} in Section~\\ref{sec:superp_def}.\nThen, in Sections~\\ref{sec:euclidean} and~\\ref{sec:theta}, \nwe shall define fundamental geometric\nquantities in terms of which the properties of superpositions \nwill be studied in Section~\\ref{sec:superp_prop}.\n\n\\section{Definition of Superpositions} \\label{sec:superp_def}\nWe shall now present the definition of the superposition\n\\mbox{$r y \\, + \\, (1 - r) z$}.\nOur definition is taken from the everyday practice of physicists.\n\\begin{definition} \\label{def:superposition}\nFor any \\mbox{$r \\in [0 , 1]$}, \nfor any \\mbox{$y , z \\in X$} such that \\mbox{$y \\not \\perp z$},\nwe shall define\n\\mbox{$r y \\, + \\, (1 - r) z$} in the following way.\n\nChoose some arbitrary unit vector $\\vec{v}$ in $y$.\nSince \\mbox{$y \\not \\perp z$}, there is a unique unit vector $\\vec{w}$\nof $z$ such that \\mbox{$\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle \\, > \\, 0$}.\nDefine, now: \n\\begin{equation} \\label{eq:super}\n\\vec{u} \\: = \\: \n\\sqrt{r} \\, \\vec{v} \\, + \\, \\sqrt{1 - r} \\, \\vec{w}.\n\\end{equation}\nNote that \\mbox{$\\vec{u} \\neq \\vec{0}$}: if \\mbox{$y = z$} then \n\\mbox{$\\vec{v} = \\vec{w}$} and \\mbox{$\\sqrt{r} + \\sqrt{1 - r} > 0$}.\nOtherwise $\\vec{v}$ and $\\vec{w}$ are\nlinearly independent and at least one of $\\sqrt{r}$ or $\\sqrt{1 - r}$ is\nstrictly positive.\nWe may now define \\mbox{$r y \\, + \\, (1 - r) z$} \nto be the one-dimensional subspace generated by $\\vec{u}$.\n\\end{definition}\n\nNote that the vector $\\vec{u}$ above is not a unit vector.\nDefinition~\\ref{def:superposition} squares well with the Dirac notation\nand the way it is used in everyday physics.\nIf $y$ and $z$ are to be compounded in equal parts (\\mbox{$r = 1\/2$}) then \n\\mbox{$1\/2 y + 1\/2 z$} is defined by the vector \n\\mbox{$1 \/ \\sqrt{2} (\\vec{v} + \\vec{w})$}, which is a unit vector in case \n\\mbox{$y \\perp z$}. Notice, though, that the case $y$ and $w$ are orthogonal\nis a case we do not allow.\n\nThe following is expected on general considerations and easily shown to\nfollow from Definition~\\ref{def:superposition}.\n\\begin{lemma} \\label{le:+commu}\nFor any \\mbox{$y, z \\in X$} such that \\mbox{$y \\not \\perp z$}, we have\n\\begin{enumerate}\n\\item \\label{one}\n\\mbox{$1 y \\, + \\, 0 z = y$}, and\n\\item \\label{comm}\nfor any \\mbox{$r \\in [0 , 1]$}\n\\mbox{$r y \\, + \\, (1 - r) z =$}\n\\mbox{$(1 - r) z \\, + \\, r y$}.\n\\end{enumerate}\n\\end{lemma}\n\nWe shall now define two geometrical quantities that will help us understand\nthe structure of superpositions.\n\n\\section{The Geometry of Hilbert Spaces} \\label{sec:geometry}\nFirst, we shall define a geometrical property of two states.\n\\subsection{Quantities from Euclidean Geometry} \\label{sec:euclidean}\n\\subsubsection{The Quantity $a(x, y)$} \\label{sec:a}\nWe shall now define the first geometric quantity we wish to consider.\nWhen considering the geometry of Hilbert spaces it is useful to\nbegin by reflecting on the geometry of Euclidean spaces, \nabout which we know much more and have a much better intuition. \nConsider two lines, i.e, one-dimensional linear (not affine) subspaces, \nin $\\mbox{${\\cal R}$}^{n}$.\nThe only invariant characterizing their relation is their angle.\nTwo lines define a plane and four angles. Those four angles are two pairs\nof equal angles. Therefore only two quantities are defined by two lines.\nMoreover those two angles add up to $\\pi$, therefore there is essentially\nonly one quantity defined. One can take as the fundamental quantity either\nthe acute or the obtuse angle. Let us consider the acute angle as the\nquantity of interest. Two lines in Euclidean space define an angle $\\varphi$\nin the interval \\mbox{$[0 , \\pi \/ 2]$}. Equivalently, they define a real number\nin the interval \\mbox{$[0 , 1]$}, the value of $\\cos(\\varphi)$.\n\nThe same quantity may be defined in Hilbert spaces. \nConsider two states \\mbox{$x, y \\in X$}. We are trying to associate\na numerical quantity to this pair of states. The most natural thing to consider\nis the inner product of two vectors contained in $x$ and $y$ respectively.\nIt is very natural to choose two unit vectors \\mbox{$\\vec{u} \\in x$} and\n\\mbox{$\\vec{v} \\in y$} and consider the inner product \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle$}. This will not do since the \nquantity depends on the choice of the unit vectors $\\vec{u}$ and $\\vec{v}$\nand we are looking for a quantity that depends only on $x$ and $y$.\nThe inner product depends on the choice of the unit vectors, but its modulus\ndoes not.\nConsider therefore the quantity \n\\[\na(x, y) \\: \\stackrel{\\rm def}{=} \\: \\mid \\! \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\! \\mid\n\\]\nfor arbitrary unit vectors $\\vec{u}$ and $\\vec{v}$ of $x$ and $y$ respectively.\nAny unit vector $\\vec{u}'$ of $x$ has the form:\n\\mbox{$\\vec{u}' \\, = \\, e^{i \\theta} \\vec{u}$} and any $\\vec{v}'$ of\n$y$ has the form:\n\\mbox{$\\vec{v}' \\, = \\, e^{i \\varphi} \\vec{v}$}.\nTherefore\n\\mbox{$\\langle \\vec{u}' \\, , \\, \\vec{v}' \\rangle \\, = \\,$}\n\\mbox{$e^{i (\\theta - \\varphi)} \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle$},\nand \n\\mbox{$\\mid \\langle \\vec{u}' \\, , \\, \\vec{v}' \\rangle \\mid \\, = \\,$}\n\\mbox{$\\mid \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\mid$}.\n\nThe following is easily proved.\n\\begin{lemma} \\label{le:a}\nFor any \\mbox{$x, y \\in X$}:\n\\begin{enumerate}\n\\item \\label{r:01}\n$a(x,y)$ is a real number of the interval $[0,1]$,\n\\item \\label{r:one}\n\\mbox{$a(x,y) = 1$} iff \\mbox{$x = y$},\n\\item \\label{r:zero}\n\\mbox{$a(x,y) = 0$} iff \\mbox{$x \\perp y$},\n\\item \\label{r:sym}\n\\mbox{$a(y,x) = a(x,y)$}.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsubsection{Similarity: $p$} \\label{sec:similarity}\nIt turns out that the square of the quantity \\mbox{$a(x, y)$}, akin to the \n$\\cos^2$ of an angle has even more remarkable properties.\n\\begin{definition} \\label{def:p}\nGiven any states \\mbox{$x, y \\in X$}, we shall define their similarity\n$p(x, y)$ by\n\\[\np(x, y) = a^{2}(x, y). \n\\]\n\\end{definition}\n\nThe quantity $p$ will be called {\\em similarity} because it measures\nhow similar, i.e., close, are its arguments $x$ and $y$. Its physical \ninterpretation is straightforward: $p(x,y)$ is the probability that,\nwhen, on state $x$, one tests whether $y$ is the case, one gets a positive \nanswer. With probability $1 - p(x,y)$ one gets the the answer that $y$ is\nnot the case. This physical interpretation is the reason $p = a^{2}$ \nand not $a$ has been chosen as the quantity of reference. \nNote that $p$ can be directly obtained experimentally.\nBelow, we shall extend the definition of $p$ to measure\nthe {\\em similarity} between any state \\mbox{$x \\in X$} and any proposition\n\\mbox{$\\alpha \\in M$}, i.e., the degree to which state $x$ satisfies \nproposition $\\alpha$.\n\nA straightforward result on Hilbert spaces will be recalled now.\n\\begin{lemma} \\label{le:inner}\nLet \\mbox{$\\vec{u}, \\vec{v} \\in \\mbox{${\\cal H}$}$}.\nAssume $\\vec{v}$ is a unit vector and \\mbox{$\\vec{v} \\in x \\in X$}.\nThen the projection \\mbox{$x(\\vec{u})$} of $\\vec{u}$ on $x$ is \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\, \\vec{v}$}.\n\\end{lemma}\n\\begin{proof}\n\\mbox{$\\vec{u} - \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\, \\vec{v}$} \nis indeed \northogonal to $\\vec{v}$ and therefore to $x$.\n\\end{proof}\n\nFirst properties of $p$ are described in the following.\n\\begin{lemma} \\label{le:p}\nFor any \\mbox{$x, y \\in X$}:\n\\begin{enumerate}\n\\item \\label{p:01}\n$p(x,y)$ is a real number in the interval $[0,1]$,\n\\item \\label{p:one}\n\\mbox{$p(x,y) = 1$} iff \\mbox{$x = y$},\n\\item \\label{p:zero}\n\\mbox{$p(x,y) = 0$} iff \\mbox{$x \\perp y$},\n\\item \\label{p:sym}\n\\mbox{$p(y,x) = p(x,y)$},\n\\item \\label{p:inner}\nfor any unit vector \\mbox{$\\vec{u} \\in x$},\n\\mbox{$p(x,y) \\, = \\,$}\n\\mbox{$\\langle \\vec{u} \\, , \\, y(\\vec{u}) \\rangle$} \nwhere $y(\\vec{u})$ is the projection of \n$\\vec{u}$ on $y$,\n\\item \\label{p:proj}\nfor any unit vector \\mbox{$\\vec{u} \\in x$},\n\\mbox{$p(x,y) \\, = \\,$}\n\\mbox{$\\parallel \\! y(\\vec{u}) \\! \\parallel^{2}$}.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFor~\\ref{p:inner}, note that \nfor any unit vector $\\vec{v}$ of $y$, we have, by Lemma~\\ref{le:inner},\n\\mbox{$y(\\vec{u}) \\, = \\, \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\, \\vec{v}$},\nand therefore\n\\mbox{$\\langle \\vec{u} \\, , \\, y(\\vec{u}) \\rangle \\, = \\,$}\n\\mbox{$\\overline{\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle} \\,  \n\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\, = \\,$}\n\\mbox{$\\mid \\! \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle \\! \\mid^{2}$}.\nNote that this implies that the inner product\n\\mbox{$\\langle \\vec{u} \\, , \\, y(\\vec{u}) \\rangle$} is a real number.\nFor~\\ref{p:proj}, note that projections are Hermitian and idempotent, and\ntherefore\n\\mbox{$\\langle y(\\vec{u}) \\, , \\, y(\\vec{u}) \\rangle \\, = \\,$}\n\\mbox{$\\langle \\vec{u} \\, , \\, y(y(\\vec{u})) \\rangle \\, = \\,$}\n\\mbox{$\\langle \\vec{u} \\, , \\, y(\\vec{u}) \\rangle$}.\n\\end{proof}\n\nThe next result is central. \nIt shows that, for any given proposition $\\alpha$, the projection\non $\\alpha$ is determined by the $p$-structure.\n\\begin{theorem} \\label{the:p}\nFor any proposition \\mbox{$\\alpha \\in M$} and any\nstates \\mbox{$x, y \\in X$}, if \\mbox{$x \\not \\perp \\alpha$}\nand \\mbox{$y \\in \\alpha$} then\n\\mbox{$p(x,y) = p(x,\\alpha(x)) \\, p(\\alpha(x), y)$}. \n\\end{theorem}\n\\begin{proof}\nLet $\\vec{u}$ be a unit vector of $x$.\nSince \\mbox{$y \\in \\alpha$}, the projection of any vector on $y$\ncan be obtained by projecting the vector first on $\\alpha$ and then\nprojecting the result on $y$. In particular,\n\\mbox{$y(\\vec{u}) = y(\\alpha(\\vec{u}))$}.\nTherefore\n\\[\np(x,y) = \\parallel \\! y(\\vec{u}) \\! \\parallel^{2} = \n\\parallel \\! y(\\alpha(\\vec{u})) \\! \\parallel^{2} \/ \n\\parallel \\! \\alpha(\\vec{u}))\\! \\parallel^{2} \/: \\times \\:\n\\parallel \\! \\alpha(\\vec{u}) \\! \\parallel^{2}\n\\]\nLet \\mbox{$\\vec{v} =$}\n\\mbox{$ \\alpha(\\vec{u}) \/ \\parallel \\! \\alpha(\\vec{u} \\! \\parallel$}. \nNotice that $\\vec{v}$ is a unit vector of $\\alpha(x)$ and therefore\n\\[\np(x,y) = \\parallel \\! \\vec{v} \\! \\parallel^{2} \\: \\times \\:\n\\parallel \\! \\alpha(\\vec{u}) \\! \\parallel^{2} = \np(\\alpha(x), y) \\: \\times \\: p(x, \\alpha(x))\n\\]\nsince $\\alpha(\\vec{u})$ is the projection of $\\vec{u}$ on $\\alpha(x)$,\nand by Lemma~\\ref{le:p}.\n\\end{proof}\n\n\\begin{corollary} \\label{le:max}\nFor any proposition \\mbox{$\\alpha \\in M$} and any\nstate \\mbox{$x \\in X$}, if \\mbox{$x \\not \\perp \\alpha$}\nthen $\\alpha(x)$ is the unique state $y$ of $\\alpha$ on which\nthe value of $p(x,y)$ is maximal. \n\\end{corollary}\nIn short, there is a unique state of $\\alpha$ that is most similar to $x$,\nthis is $x$'s projection on $\\alpha$.\n\\begin{proof}\nBy Theorem~\\ref{the:p}, since \\mbox{$p(\\alpha(x),y) \\leq 1$} by\nLemma~\\ref{le:p}, \\mbox{$p(x,y) \\leq p(x,\\alpha(x))$} for any\n\\mbox{$y \\in \\alpha$}.\n\nFor uniqueness, suppose \\mbox{$y \\in \\alpha$} and \n\\mbox{$p(x,y) = p(x,\\alpha(x))$}.\nBy Theorem~\\ref{the:p}, \n\\mbox{$p(x,\\alpha(x)) =$}\n\\mbox{$p(x, \\alpha(x)) \\, p(\\alpha(x), y)$}.\nSince $x$ is not orthogonal to $\\alpha$, \\mbox{$p(x,\\alpha(x)) > 0$}\nand therefore \\mbox{$p(\\alpha(x),y) = 1$} and \\mbox{$\\alpha(x) = y$}.\n\\end{proof}\n\nIt is now only natural to extend the definition of $p$ to an arbitrary\nproposition as second argument. \nFor any \\mbox{$x \\in X$} and \n\\mbox{$\\alpha \\in M$}, we define \\mbox{$p(x,\\alpha)$} in the following way:\n\\begin{itemize}\n\\item \\mbox{$p(x,\\alpha) = 0$} if \\mbox{$x \\perp \\alpha$}, and\n\\item \\mbox{$p(x,\\alpha) = p(x,\\alpha(x))$} otherwise.\n\\end{itemize}\n\nThe following is known, in Physics, as Born's rule.\nThe quantity $p(x,\\alpha)$ is the probability of measuring the property\n$\\alpha$ in state $x$.\n\\begin{lemma} \\label{le:Born}\nFor any state $x \\in X$ and any proposition \\mbox{$\\alpha \\in M$}, if\n\\mbox{$\\vec{u} \\neq \\vec{0} \\in x$},\n\\mbox{$p(x, \\alpha) = \\parallel \\alpha(\\vec{u}) \\parallel^{2} \/ \n\\parallel \\vec{u} \\parallel^{2}$}.\n\\end{lemma}\nThe proof is obvious. The following is an obvious consequence of \nCorollary~\\ref{le:max}.\n\\begin{corollary} \\label{le:satisfaction}\nFor any state $x$ and any proposition $\\alpha$,\n\\mbox{$x \\in \\alpha$} iff \\mbox{$\\alpha(x) = x$} iff \n\\mbox{$p(x, \\alpha) = 1$}.\n\\end{corollary}\n\nThe next two sections prove additional properties of the quantity $p$.\nOn a first reading the reader is advised to advance to \nSection~\\ref{sec:theta}. \nSection~\\ref{sec:probas} shows that, \nfor any given $x$ and different $\\alpha$'s, \\mbox{$p(x, \\alpha)$} \nbehaves very much as a probability on the propositions. \nExactly so, for propositions that commute as projections.\nSection~\\ref{sec:num_inter} proves an intriguing inequality that\nprovides a numerical strengthening of the {\\bf Interference} property\nof~\\cite{LEG:Malg}.\n\n\\subsubsection{Similarity as Probability} \\label{sec:probas}\nThe following results will show that, for any fixed \\mbox{$x \\in X$}, \nthe quantities\n$p(x, \\alpha)$ for different measurements $\\alpha$ play the role of a \nprobability on the propositions.\nFor any two propositions \\mbox{$\\alpha, \\beta \\in M$} we shall define, as\ntraditional since~\\cite{BirkvonNeu:36},\ntheir conjunction \\mbox{$\\alpha \\wedge \\beta$} as their intersection\n\\mbox{$\\alpha \\cap \\beta$} (note the intersection of closed subspaces\nis a closed subspace) and their disjunction \\mbox{$\\alpha \\vee \\beta$}\nas the topological closure of their linear sum: \n\\mbox{$cl(\\alpha + \\beta)$}.\nNote that these notations are inconsistent with those of~\\cite{LEG:Malg}\nwhere conjunction and disjunction were defined only for {\\em commuting}\npropositions.\nWe shall demonstrate a particular interest in {\\em commuting} propositions.\nFor the sake of obtaining a straightforward definition of commutation, \nwe shall extend our notation for projections.\n\\begin{definition} \\label{def:commuting}\nLet \\mbox{$\\alpha, \\beta \\in M$} be two propositions. We shall say that\n$\\alpha$ and $\\beta$ {\\em commute} iff for any \\mbox{$x \\in X$} \n\\mbox{$\\alpha(\\beta(x)) = \\beta(\\alpha(x))$}.\n\\end{definition}\n\n\\begin{lemma} \\label{le:commuting}\nAny two propositions \\mbox{$\\alpha, \\beta \\in M$} commute iff\nthere are three pairwise orthogonal propositions \n\\mbox{$\\gamma_{i}, i = 1, \\ldots , 3$} such that\n\\mbox{$\\alpha =$} \\mbox{$\\gamma_{1} \\vee \\gamma_{2}$} and\n\\mbox{$\\beta =$} \\mbox{$\\gamma_{1} \\vee \\gamma_{3}$}.\n\\end{lemma}\nNote that one of the propositions $\\gamma_{i}$ may be the falsehood $\\bot$.\n\\begin{proof}\nThe {\\em if} claim is obvious. The {\\em only if} claim follows \nfrom the fact that \nprojections are Hermitian and that Hermitian operators commute iff they\nhave a joint basis of eigenvectors.\n\\end{proof}\n\n\\begin{corollary} \\label{le:comm}\nFor any \\mbox{$\\alpha, \\beta \\in X$}, if \\mbox{$\\alpha \\subseteq \\beta$}\nor \\mbox{$\\alpha \\perp \\beta$}, then $\\alpha$ and $\\beta$ commute.\n\\end{corollary}\n\\begin{proof}\nIn the first case, take \\mbox{$\\gamma_{1} = \\alpha$}, \n\\mbox{$\\gamma_{2} = \\bot$} and \\mbox{$\\gamma_{3} =$}\n\\mbox{$\\neg \\alpha \\wedge \\beta$}.\nIn the second case, take \\mbox{$\\gamma_{1} = \\alpha$}, \n\\mbox{$\\gamma_{2} = \\bot$} and \\mbox{$\\gamma_{3} = \\beta$}.\n\\end{proof}\n\n\\begin{corollary} \\label{le:comm_neg}\nFor any \\mbox{$\\alpha, \\beta \\in X$}, if $\\alpha$ and $\\beta$ commute\nthen $\\neg \\alpha$ and $\\beta$ commute.\n\\end{corollary}\n\\begin{proof}\nLet \\mbox{$\\alpha =$} \\mbox{$\\gamma_{1} \\vee \\gamma_{2}$} and\n\\mbox{$\\beta =$} \\mbox{$\\gamma_{1} \\vee \\gamma_{3}$}.\nThen \\mbox{$\\neg \\alpha =$} \\mbox{$\\neg \\gamma_{1} \\wedge \\neg \\gamma_{2}$}.\nSince \\mbox{$\\gamma_{3} \\subseteq \\neg \\alpha$}, we have, by the\northomodular property,\n\\mbox{$\\neg \\alpha =$} \n\\mbox{$\\gamma_{3} \\vee \\neg \\gamma_{1} \\wedge \\neg \\gamma_{2} \n\\wedge \\neg \\gamma_{3}$}.\nBut \\mbox{$\\beta =$} \\mbox{$\\gamma_{3} \\vee \\gamma_{2}$} and\n\\mbox{$\\gamma_{2} \\perp \\neg \\gamma_{1} \\wedge \\neg \\gamma_{2}\n\\wedge \\neg \\gamma_{3}$}.\n\\end{proof}\n\nFirst, we shall consider disjunctions of orthogonal propositions.\n\\begin{lemma} \\label{le:orthodisjunction}\nIf \\mbox{$\\alpha \\perp \\beta$} then, for any \\mbox{$x \\in X$},\n\\mbox{$p(x, \\alpha \\vee \\beta) =$}\n\\mbox{$p(x, \\alpha) + p(x, \\beta)$}.\n\\end{lemma}\n\\begin{proof}\nConsider any \\mbox{$\\vec{u} \\neq \\vec{0} \\in x$}.\nNow \\mbox{$(\\alpha \\vee \\beta)(\\vec{u}) =$}\n\\mbox{$\\alpha(\\vec{u}) + \\beta(\\vec{u})$} (see~\\cite{Halmos:Hilbert} \nTheorem 2, page 46). Therefore\n\\mbox{$\\langle \\vec{u} \\, , \\, (\\alpha \\vee \\beta)(\\vec{u}) \\rangle =$}\n\\mbox{$\\langle \\vec{u} \\, , \\, \\alpha(\\vec{u}) \\rangle +$}\n\\mbox{$\\langle \\vec{u} \\, , \\, \\beta(\\vec{u}) \\rangle$}. \n\\end{proof}\n\\begin{corollary} \\label{co:orthodisjunction}\nIf \\mbox{$\\alpha_{i}$} is a family of pairwise orthogonal measurements, then\nfor any \\mbox{$x \\in X$} we have\n\\mbox{$p(x, \\bigvee_{i \\in I} \\alpha_{i}) =$}\n\\mbox{$\\sum_{i \\in I} p(x, \\alpha_{i})$}.\n\\end{corollary}\n\\begin{proof}\nBy induction on the size of $I$, and associativity of disjunction.\n\\end{proof}\n\nThe following lemmas are fundamental characteristics of probabilities.\n\\begin{lemma} \\label{le:neg_sum_one}\nFor any \\mbox{$\\alpha \\in M$} and any \\mbox{$x \\in X$}:\n\\mbox{$p(x, \\alpha) + p(x, \\neg \\alpha) = 1$}.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{le:orthodisjunction}, \n\\mbox{$p(x, \\alpha) + p(x, \\neg \\alpha) = p(x, \\alpha \\vee \\neg \\alpha)$}.\nBut \\mbox{$\\alpha \\vee \\neg \\alpha = \\top$} and therefore \n\\mbox{$(\\alpha \\vee \\neg \\alpha)(x) = x$} and, \nby Corollary~\\ref{le:satisfaction},\n\\mbox{$p(x, \\alpha \\vee \\beta) = 1$}.\n\\end{proof}\n\\begin{lemma} \\label{le:zero_one}\nFor any \\mbox{$\\alpha \\in M$} and any \\mbox{$x \\in X$}:\n\\mbox{$0 \\leq p(x, \\alpha) \\leq 1$}.\n\\end{lemma}\n\\begin{proof}\nBy Lemmas~\\ref{le:Born} and~\\ref{le:neg_sum_one}.\n\\end{proof}\n\\begin{lemma} \\label{le:disj_prob}\nLet \\mbox{$\\alpha, \\beta \\in M$} be any {\\em commuting} measurements.\nFor any \\mbox{$x \\in X$}\n\\mbox{$p(x, \\alpha \\vee \\beta) =$}\n\\mbox{$p(x, \\alpha) + p(x, \\beta) - p(x, \\alpha \\wedge \\beta)$}.\n\\end{lemma}\n\\begin{proof}\nWe know that \\mbox{$\\alpha \\vee \\beta =$}\n\\mbox{$(\\alpha \\wedge \\beta) \\vee (\\alpha \\wedge \\neg \\beta)\n\\vee (\\neg \\alpha \\wedge \\beta)$}. The three parts of the disjunction above\nare pairwise orthogonal, therefore Corollary~\\ref{co:orthodisjunction} implies\nthat \\mbox{$p(x, \\alpha \\vee \\beta) =$}\n\\mbox{$p(x, \\alpha \\wedge \\beta) +$}\n\\mbox{$p(x, \\alpha \\wedge \\neg \\beta) +$}\n\\mbox{$p(x, \\neg \\alpha \\wedge \\beta)$}.\nBut, by Lemma~\\ref{le:orthodisjunction}:\n\\mbox{$p(x, \\alpha \\wedge \\beta) +$}\n\\mbox{$p(x, \\alpha \\wedge \\neg \\beta) =$}\n\\mbox{$p(x, \\alpha)$} and\n\\mbox{$p(x, \\alpha \\wedge \\beta) +$}\n\\mbox{$p(x, \\neg \\alpha \\wedge \\beta) =$}\n\\mbox{$p(x, \\beta)$}.\n\\end{proof}\nThe lemmas above dealt mostly with the properties of disjunction. The next\nresult concerns conjunction and parallels the consideration of conditional\nprobabilities.\n\\begin{lemma} \\label{le:conj}\nLet \\mbox{$\\alpha, \\beta \\in M$} be any {\\em commuting} measurements.\nFor any \\mbox{$x \\in X$}:\n\\mbox{$p(x, \\alpha \\wedge \\beta) =$}\n\\mbox{$p(x, \\alpha) \\: p(\\alpha(x), \\beta)$}.\n\\end{lemma}\n\\begin{proof}\nSince \\mbox{$\\alpha \\wedge \\beta = \\alpha \\circ \\beta$}, by the definition\nof $p$, taking any \\mbox{$\\vec{u} \\neq \\vec{0} \\in x$}:\n\\[\np(x, \\alpha \\wedge \\beta) =\n{{\\parallel (\\alpha \\circ \\beta)(\\vec{u}) \\parallel^{2}} \\over\n{\\parallel \\vec{u} \\parallel^{2}}} =\n{{\\parallel (\\alpha \\circ \\beta)(\\vec{u}) \\parallel^{2}} \\over\n{\\parallel \\alpha(\\vec{u}) \\parallel^{2}}} \\ \n{{\\parallel \\alpha(\\vec{u}) \\parallel^{2}} \\over \n{\\parallel \\vec{u} \\parallel^{2}}} =\np(\\alpha(x), \\beta) \\  p(x, \\alpha).\n\\]\n\\end{proof}\n\\begin{corollary} \\label{co:leq}\nLet \\mbox{$\\alpha, \\beta \\in M$} be any measurements such that \n\\mbox{$\\alpha \\leq \\beta$}.\nThen for any \\mbox{$x \\in X$}, \\mbox{$p(x, \\alpha) \\leq p(x, \\beta)$}.\n\\end{corollary}\n\\begin{proof}\nIf \\mbox{$\\alpha \\leq \\beta$}, the two measurements commute and\n\\mbox{$\\alpha = \\beta \\wedge \\alpha$}.\nBy Lemma~\\ref{le:conj}, then \\mbox{$p(x, \\alpha) =$}\n\\mbox{$p(x, \\beta) \\: p(\\beta(x), \\alpha) \\leq$}\n\\mbox{$p(x, \\beta)$} by Lemma~\\ref{le:zero_one}. \n\\end{proof}\n\\begin{corollary} \\label{co:comp}\nLet \\mbox{$\\alpha, \\beta \\in M$} be any {\\em commuting} measurements.\nThen for any \\mbox{$x \\in X$}, \n\\mbox{$p(x, \\beta) = p(x, \\alpha) \\, p(\\alpha(x), \\beta) \\: + \\: \np(x, \\neg \\alpha) \\, p((\\neg \\alpha)(x), \\beta)$}.\n\\end{corollary}\n\\begin{proof}\nSince $\\alpha$ and $\\beta$ commute, by Theorem~1 of~\\cite{LEG:Malg},\n\\mbox{$\\beta = (\\alpha \\wedge \\beta) \\vee (\\neg \\alpha \\wedge \\beta)$}.\nBy Lemma~\\ref{le:orthodisjunction} we have:\n\\mbox{$p(x, \\beta) = p(x, \\alpha \\wedge \\beta) \\: + \\: p(x, \\neg \\alpha \\wedge \\beta)$}.\nWe conclude, by Lemma~\\ref{le:conj}, that\n\\mbox{$p(x, \\beta) = p(x, \\alpha) \\, p(\\alpha(x), \\beta) \\: + \\: \np(x, \\neg \\alpha) \\, p((\\neg \\alpha)(x), \\beta)$}.\n\\end{proof}\nIn Corollary~\\ref{co:comp} one cannot omit the requirement \nthat $\\alpha$ and $\\beta$ commute. The consideration of a two-dimensional \nEuclidean space where $\\alpha$ is the x-axis and $x$ makes an angle\n$\\theta$ with the x-axis is sufficient. If $\\beta$ is $x$, then\n\\mbox{$p(x, \\beta) = 1$} whereas \n\\mbox{$p(x, \\alpha) =$}\n\\mbox{$\\cos^{2}(\\theta) =$} \\mbox{$p(\\alpha(x), \\beta)$} and\n\\mbox{$p(x, \\neg \\alpha) =$}\n\\mbox{$\\sin^{2}(\\theta) =$} \\mbox{$p((\\neg \\alpha)(x), \\beta)$}.\nAlso taking $\\beta$ orthogonal to $x$ gives\n\\mbox{$p(x, \\beta) = 0$} and \n\\mbox{$p(x, \\alpha) =$} \\mbox{$\\cos^{2}(\\theta) =$}\n\\mbox{$p((\\neg \\alpha)(x), \\beta)$} and\n\\mbox{$p(x, \\neg \\alpha) =$} \\mbox{$\\sin^{2}(\\theta) =$}\n\\mbox{$p(\\alpha(x), \\beta)$}.\nNevertheless the result holds in the following case.\n\\begin{lemma} \\label{le:orthomodular_equality}\nFor any \\mbox{$x \\in X$} and any \\mbox{$\\alpha, \\beta \\in M$} such that\n\\mbox{$\\alpha(x) \\in \\beta$} and \\mbox{$(\\neg \\alpha)(x) \\in \\beta$},\none has \n\\[\np(x, \\beta) = p(x, \\alpha) \\, p(\\alpha(x), \\beta) \\: + \\: \np(x, \\neg \\alpha) \\, p((\\neg \\alpha)(x), \\beta)= 1.\n\\]\n\\end{lemma}\n\\begin{proof}\nBy assumption both $\\alpha(x)$ and $(\\neg \\alpha)(x)$ are subspaces of \n$\\beta$. Given any \\mbox{$\\vec{u} \\in x$}, both \n$\\alpha(\\vec{u})$ and $(\\neg \\alpha)(\\vec{u})$ are in $\\beta$.\nBut $\\beta$ is a subspace and therefore \n\\mbox{$\\alpha(\\vec{u}) + (\\neg \\alpha)(\\vec{u}) =$}\n\\mbox{$\\vec{u} \\in \\beta$}.\n\\end{proof}\n\n\\begin{lemma} \\label{le:local_comp}\nFor any \\mbox{$x \\in X$} and any \\mbox{$\\alpha, \\beta \\in M$} such that\n\\mbox{$(\\alpha \\circ \\beta)(x) =$} \\mbox{$(\\beta \\circ \\alpha)(x)$}, we have\n\\mbox{$p(x, \\beta) =$} \\mbox{$p(x, \\alpha) \\, p(\\alpha(x), \\beta) \\: + \\: \np(x, \\neg \\alpha) \\, p((\\neg \\alpha)(x), \\beta)$}.\n\\end{lemma}\n\\begin{proof}\nAssume that \\mbox{$(\\alpha \\circ \\beta)(x) =$}\n\\mbox{$(\\beta \\circ \\alpha)(x)$}.\nBy Lemma~\\ref{le:comm_neg}, \n\\mbox{$(\\neg \\alpha \\circ \\beta)(x) =$} \n\\mbox{$(\\beta \\circ \\neg \\alpha)(x)$}.\nTake any \\mbox{$\\vec{u} \\neq \\vec{0} \\in x$}.\nThen,\n\\[\np(x, \\beta) \\ = \\  \n\\parallel \\beta(\\vec{u}) \\parallel^{2} \\: \/ \\: \n\\parallel \\vec{u} \\parallel^{2} \\ = \\ \n\\parallel \\alpha(\\beta(\\vec{u}))) + (\\neg \\alpha)(\\beta(\\vec{u})) \\parallel^{2}\n\\: \/ \\: \\parallel \\vec{u} \\parallel^{2} \\ = \n\\]\n\\[\n\\parallel \\alpha(\\beta(\\vec{u}))) \\parallel^{2}\\: \/ \\: \n\\parallel \\vec{u} \\parallel^{2}  \n+ \\parallel (\\neg \\alpha)(\\beta(\\vec{u})) \\parallel^{2}\n\\: \/ \\: \\parallel \\vec{u} \\parallel^{2} \\ = \\ \n\\]\n\\[\n\\parallel \\beta(\\alpha(\\vec{u}))) \\parallel^{2}\\: \/ \\: \n\\parallel \\vec{u} \\parallel^{2}  \n+ \\parallel (\\beta)((\\neg \\alpha)(\\vec{u})) \\parallel^{2}\n\\: \/ \\: \\parallel \\vec{u} \\parallel^{2} \\ = \\ \n\\]\n\\[\n{{\\parallel \\beta(\\alpha(\\vec{u})) \\parallel^{2}} \\over\n{\\parallel \\alpha(x) \\parallel^{2}}} \\ \n{{\\parallel \\alpha(x) \\parallel^{2}} \\over\n{\\parallel \\vec{u} \\parallel^{2}}}\n+ \n{{\\parallel (\\beta)((\\neg \\alpha)(\\vec{u})) \\parallel^{2}} \\over \n{\\parallel (\\neg \\alpha)(x) \\parallel^{2}}} \\ \n{{\\parallel (\\neg \\alpha)(x) \\parallel^{2}} \\over \n{\\parallel \\vec{u} \\parallel^{2}}} \\ = \\ \n\\]\n\\[\np(\\alpha(x),\\beta) \\, p(x, \\alpha) \\: + \\: p((\\neg \\alpha)(x), \\beta) \\, \np(x, \\neg \\alpha). \n\\]\n\\end{proof}\n\n\\subsubsection{An Inequality} \\label{sec:num_inter}\nThe next result strengthens the Interference property of~\\cite{LEG:Malg}\nby presenting a quantitative version of the principle.\n\\begin{theorem} \\label{the:quant_interference}\nFor any \\mbox{$\\alpha, \\beta \\in M$} and any \\mbox{$x \\in X$} such that\n\\mbox{$\\alpha(x) = x$}, \n\\[\np(x, \\beta) \\: (1 - p(\\beta(x), \\alpha))^{2} \\: \\leq \\:\np(\\beta(x), \\alpha) \\: (1 - p(\\alpha(\\beta(x)), \\beta))\n\\]\n\\end{theorem}\nNote that, by Theorem~\\ref{the:p},\n\\mbox{$p(x, \\beta) \\leq p(\\beta(x), \\alpha)$} but\n\\mbox{$(1 - p(\\beta(x), \\alpha)) \\geq (1 - p(\\beta(x), \\alpha))$}.\nThe fact that the quantity \\mbox{$1 - p(\\beta(x), \\alpha)$} \nappears squared seems inevitable. \nAn examination of $\\mbox{${\\cal R}$}^{3}$ shows that it may be the case that\n\\mbox{$p(x, \\beta) \\: (1 - p(\\beta(x), \\alpha)) \\: > \\:$}\n\\mbox{$p(\\beta(x), \\alpha) (1 - p(\\alpha(\\beta(x)), \\beta))$}.\n\\begin{proof}\nAssume \\mbox{$\\vec{t} \\neq \\vec{0} \\in x$}.\nLet \\mbox{$\\vec{u} = \\beta(\\vec{t})$}, \\mbox{$\\vec{v} = \\alpha(\\vec{u})$}\nand \\mbox{$\\vec{w} = \\beta(\\vec{v})$}.\n\nIn a first step we want to show that:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{2} =\n\\langle \\vec{t} \\, , \\, \\vec{v} - \\vec{w} \\rangle.\n\\]\n\nIndeed: \\mbox{$\\parallel \\vec{u} - \\vec{v} \\parallel^{2} =$}\n\\mbox{$\\langle \\vec{u} - \\vec{v} \\, , \\, \\vec{u} - \\vec{v} \\rangle =$}\n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{u} - \\vec{v} \\rangle -$}\n\\mbox{$\\langle \\vec{v} \\, , \\, \\vec{u} - \\vec{v} \\rangle$}.\nBut the last term is null since \\mbox{$\\vec{u} - \\vec{v}$} is orthogonal\nto $\\alpha$ in general and in particular to $\\vec{v}$.\nWe have:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{2} =\n\\langle \\vec{u} \\, , \\, \\vec{u} - \\vec{v} \\rangle.\n\\]\nBut \\mbox{$\\vec{t} - \\vec{u}$} is, similarly, orthogonal\nto $\\vec{u}$ and \\mbox{$\\langle \\vec{u} \\, , \\, \\vec{u} \\rangle =$}\n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{u} \\rangle$}.\nSince \\mbox{$\\vec{u} - \\vec{v}$} is orthogonal to $\\vec{t}$,\n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{u} \\rangle =$}\n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{v} \\rangle$}.\nWe have:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{2} =\n\\langle \\vec{t} \\, , \\, \\vec{v} \\rangle - \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle.\n\\]\nAgain, \\mbox{$\\vec{v} - \\vec{w}$} is orthogonal to $\\vec{u}$ and therefore:\n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle =$}\n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{w} \\rangle$} \nand \\mbox{$\\vec{t} - \\vec{u}$} is orthogonal to $\\vec{w}$ and\nwe have: \\mbox{$\\langle \\vec{u} \\, , \\, \\vec{w} \\rangle =$}\n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{w} \\rangle$}.\nTherefore:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{2} =\n\\langle \\vec{t} \\, , \\, \\vec{v} \\rangle - \\langle \\vec{t} \\, , \\, \\vec{w} \\rangle =\n\\langle \\vec{t} \\, , \\, \\vec{v} - \\vec{w} \\rangle.\n\\]\nBy Cauchy-Schwarz therefore we have:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{2} \\: \\leq \\:\n\\parallel \\vec{t} \\parallel \\: \\parallel \\vec{v} - \\vec{w} \\parallel.\n\\]\nand:\n\\[\n\\parallel \\vec{u} - \\vec{v} \\parallel^{4} \\: \\leq \\:\n\\parallel \\vec{t} \\parallel^{2} \\: \\parallel \\vec{v} - \\vec{w} \\parallel^{2}.\n\\]\nBut: \\mbox{$\\parallel \\vec{u} \\parallel^{2} =$}\n\\mbox{$\\parallel \\vec{v} \\parallel^{2} +$}\n\\mbox{$\\parallel \\vec{u} - \\vec{v} \\parallel^{2}$}, and\n\\mbox{$\\parallel \\vec{v} \\parallel^{2} =$}\n\\mbox{$\\parallel \\vec{w} \\parallel^{2} +$}\n\\mbox{$\\parallel \\vec{v} - \\vec{w} \\parallel^{2}$}.\nTherefore we have:\n\\[\n(\\parallel \\vec{u} \\parallel^{2} - \\parallel \\vec{v} \\parallel^{2})^{2} \n\\: \\leq \\:\n\\parallel \\vec{t} \\parallel^{2} \\: \n(\\parallel \\vec{v} \\parallel^{2} - \\parallel \\vec{w} \\parallel^{2}).\n\\]\nand\n\\[\n{{\\parallel {\\vec{u}} \\parallel^{2}} \\over \n{\\parallel \\vec{t} \\parallel^{2}}}\n\\: ( 1 - \n{{\\parallel \\vec{v} \\parallel^{2}} \\over \n{\\parallel \\vec{u} \\parallel^{2}}})^{2}\n\\: \\leq \\:\n{{\\parallel \\vec{v} \\parallel^{2} - \\parallel \\vec{w} \\parallel^{2}}\n\\over\n{\\parallel \\vec{u} \\parallel^{2}}},\n\\]\n\\[\np(x, \\beta) \\: ( 1 - p(\\beta(x), \\alpha))^{2} \\: \\leq \\:\n{{\\parallel \\vec{v} \\parallel^{2}} \\over {\\parallel \\vec{u} \\parallel^{2}}}\n\\: (1 - {{\\parallel \\vec{w} \\parallel^{2}} \\over \n{\\parallel \\vec{v} \\parallel^{2}}}).\n\\]\nWe conclude that:\n\\[\np(x, \\beta) \\: ( 1 - p(\\beta(x), \\alpha))^{2} \\: \\leq \\:\np(\\beta(x), \\alpha) \\: (1 - p(\\alpha(\\beta(x)), \\beta)).\n\\]\n\\end{proof}\nTheorem~\\ref{the:quant_interference} is a quantitative strengthening of the\n{\\bf Interference} property of projections in Hilbert spaces that plays a \ncentral role in the definition of an M-algebra~\\cite{LEG:Malg}.\nIndeed, assuming that \\mbox{$x \\in \\alpha$}, if \n\\mbox{$\\alpha(\\beta(x)) \\in \\beta$}, then, by Corollary~\\ref{le:satisfaction},\n\\mbox{$p(\\alpha(\\beta(x)), \\beta) = 1$} and by \nTheorem~\\ref{the:quant_interference}, either \\mbox{$p(x, \\beta) = 0$}\nor \\mbox{$p(\\beta(x), \\alpha) = 1$}. In both cases we have \n\\mbox{$p(\\beta(x), \\alpha) = 1$} and, by  Corollary~\\ref{le:satisfaction},\n\\mbox{$\\beta(x) \\in \\alpha$}.\n\n\\subsection{Phases for Triangles: $\\theta(x, y, z)$} \n\\label{sec:theta}\nWe may now proceed to the definition of a second geometric quantity \nrelating three states: $\\theta(x, y, z)$. This quantity does not seem to\nhave been studied previously.\n\nIn section~\\ref{sec:a} a quantity was attached to any pair of states.\nThis quantity was the modulus of some inner product.\nIt seems natural that the argument of a similar inner product represents\nanother important geometrical quantity. But, clearly some thinking must be\ndone to define, out of such an argument, a quantity that does not depend\non the vectors chosen, but only on states.\nA new quantity, \\mbox{$\\theta(x, y, z)$}, \nan angle in the interval $[0, 2 \\pi]$\nwill be attached to triples of states.\nThis quantity can be defined only if no two of the three states\n$x$, $y$ and $z$ are orthogonal.\n\n\\begin{definition} \\label{def:theta}\nLet \\mbox{$x, y, z \\in X$} be such that \\mbox{$x \\not \\perp y$},\n\\mbox{$y \\not \\perp z$} and \\mbox{$z \\not \\perp x$}.\nWe shall define\n\\mbox{$\\theta(x, y, z)$} in the following way.\nChoose arbitrary unit vectors\n$\\vec{u}$, $\\vec{v}$ and $\\vec{w}$ in $x$, $y$ and $z$ respectively \nand let:\n\\[\n\\theta(x, y, z) \\: = \\: \\arg(\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{w} \\, , \\, \\vec{u} \\rangle).\n\\]\n\nNote that each of those three inner products is different from zero, \nby assumption, and therefore the three complex arguments are well-defined. \n\\end{definition}\nWe need to justify the definition by showing that the quantity \n$\\theta(x, y, z)$ depends only on $x$, $y$ and $z$ and does not\ndepend on the vectors $\\vec{u}$, $\\vec{v}$ and $\\vec{w}$.\nFor example, the definition is independent of the vector $\\vec{u}$ \nchosen in $x$ since any unit vector $\\vec{s}$ of $x$ has the form\n\\mbox{$\\vec{s} = e^{i \\varphi} \\vec{u}$} for some\n\\mbox{$\\varphi \\in [0 , 2 \\pi]$}.\nHad we used $\\vec{s}$ instead of $\\vec{u}$ we would have obtained:\n\\[\n\\arg(\\langle e^{i \\varphi} \\vec{u} \\, , \\, \\vec{v} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{w} \\, , \\, e^{i \\varphi} \\vec{u} \\rangle) \\: = \\:\n\\]\n\\[\n\\arg(e^{i \\varphi} \\langle \\vec{u} \\, , \\, \\vec{v} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) \\: + \\:\n\\arg(e^{- i \\varphi} \\langle \\vec{w} \\, , \\, \\vec{u} \\rangle) \\: = \\:\n\\]\n\\[\n\\varphi + \\arg(\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle) \\: + \\:\n\\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) \\: - \\: \\varphi \\: + \\:\n\\arg(\\langle \\vec{w} \\, , \\, \\vec{u} \\rangle).\n\\]\nA similar line shows that the choice of none of $\\vec{v}$ or\n$\\vec{w}$ influences $\\theta(x, y, z)$.\n\nWe shall now prove some properties of $\\theta$.\nFirst, \\mbox{$\\theta(x, y, z)$} is invariant under a circular permutation\nof the arguments and antisymmetric under transpositions.\n\\begin{lemma} \\label{le:cyclic}\nFor any generic states $x$, $y$ and $z$, we have:\n\\mbox{$\\theta(y, z, x) =$} \\mbox{$\\theta(x, y, z)$},\n\\mbox{$\\theta(x, z, y) =$} \\mbox{$- \\theta(x, y, z)$} and\n\\mbox{$\\theta(x, y, w) =$}\n\\mbox{$\\theta(x, y, z) + \\theta(x, z, w) + \\theta(z, y, w)$}.\n\\end{lemma}\n\\begin{proof}\nObvious.\n\\end{proof}\nThe behavior of $\\theta$ under (planar) orthogonal complements is also\nantisymmetric.\n\\begin{lemma} \\label{le:prime}\nAssume \\mbox{$x, y, z \\in X$} are states no two of them are equal\nand no two of them are orthogonal\nand such that \\mbox{$coplanar(x, y, z)$}.\nLet \\mbox{$x' =$} \\mbox{$(\\neg x)(y) =$} \\mbox{$(\\neg x)(z)$},\n\\mbox{$y' =$} \\mbox{$(\\neg y)(z) =$} \\mbox{$(\\neg y)(x)$} and\n\\mbox{$z' =$} \\mbox{$(\\neg z)(x) =$} \\mbox{$(\\neg z)(y)$}.\nThen \\mbox{$\\theta(x', y', z') =$} \\mbox{$- \\theta(x, y, z)$}.\n\\end{lemma}\n\\begin{proof}\nChoose an arbitrary unit vector $\\vec{u}$ in $x$. \nLet $\\vec{v}$ be the unit vector of $y$ such that \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle > 0$}.\nLet $\\vec{u}'$ be the unit vector of $x'$ such that\n\\mbox{$\\langle \\vec{v} \\, , \\, \\vec{u}' \\rangle > 0$}.\nLet us have \\mbox{$\\vec{v} =$}\n\\mbox{$r_{1} \\vec{u} + r_{2} \\vec{u}'$} for positive real numbers\n\\mbox{$r_{i}, i = 1 , 2$}.\nThe vector \\mbox{$r_{2} \\vec{u} - r_{1} \\vec{u}'$} is a unit vector in $y'$.\nLet \\mbox{$\\vec{v}' = r_{2} \\vec{u} - r_{1} \\vec{u}'$}.\nLet $\\vec{w}$ be the unit vector of $z$ such that \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{w} \\rangle > 0$}.\nLet \\mbox{$\\vec{w} = r_{3} \\vec{u} + r_{4} e^{i \\varphi} \\vec{u}'$}\nfor positive $r_{i}$'s $i = 3 , 4$ and some angle $\\varphi$.\nLet \\mbox{$\\vec{w}' = r_{4} e^{- i \\varphi} \\vec{u} - r_{3} \\vec{u}'$},\na unit vector of $z'$.\n\nWe see that:\n\\[\n\\theta(x, y, z) \\, = \\, \\arg(\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle) \\, + \\,\n\\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) \\, + \\,\n\\arg(\\langle \\vec{w} \\, , \\, \\vec{u} \\rangle) \\: = \\:\n0 + \\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle) + 0.\n\\]\nand\n\\[\n\\theta(x', y', z') \\, = \\, \\arg(\\langle \\vec{u}' \\, , \\, \\vec{v}' \\rangle) \\, + \\,\n\\arg(\\langle \\vec{v}' \\, , \\, \\vec{w}' \\rangle) \\, + \\,\n\\arg(\\langle \\vec{w}' \\, , \\, \\vec{u}' \\rangle) \\, = \\,\n\\pi + \\arg(\\langle \\vec{v}' \\, , \\, \\vec{w}' \\rangle) + \\pi.\n\\]\nWe are left to show that \n\\mbox{$\\arg(\\langle \\vec{v}' \\, , \\, \\vec{w}' \\rangle) = $}\n\\mbox{$- \\arg(\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle)$}.\nIn fact, we shall show that\n\\mbox{$\\langle \\vec{v}' \\, , \\, \\vec{w}' \\rangle = $}\n\\mbox{$\\langle \\vec{w} \\, , \\, \\vec{v} \\rangle$}.\nIndeed, \\mbox{$\\langle \\vec{v}' \\, , \\, \\vec{w}' \\rangle = $}\n\\mbox{$r_{2} r_{4} e^{i \\varphi} + r_{1} r_{3}$}\nand\n\\mbox{$\\langle \\vec{w} \\, , \\, \\vec{v} \\rangle = $} \n\\mbox{$r_{1} r_{3} + r_{2} r_{4} e^{i \\varphi}$}.\n\\end{proof}\n\n\\section{Properties of Superpositions} \\label{sec:superp_prop}\n\nA most remarkable novelty of QM is that the components of a superposition\ninterfere. To put this in evidence, let us consider\n\\mbox{$p(r y + (1 - r) z, x)$}. One would expect this quantity to be, \nessentially, equal to \\mbox{$r p(y, x) + (1 - r) p(z, x)$}.\n\\begin{lemma} \\label{le:p_basis}\nIf \\mbox{$x \\not \\perp y$}, \\mbox{$y \\not \\perp z$}, \n\\mbox{$z \\not \\perp x$}, \\mbox{$r \\in [0,1]$}, \n\\mbox{$\\omega(r, y, z) = 1 + 2 \\sqrt{r (1 -r) p(y, z)}$} we have, for any \n\\mbox{$x \\in X$}:\n\\[\np(r y \\, + \\, (1 - r) z, x) = \n\\frac{r p(y, x) + (1 - r) p(z, x) + \n2 \\cos(\\theta(x, y, z)) \\sqrt{r (1 - r) p(y,x) p(z, x)}} \n{\\omega(r, y, z)}\n\\]\n\\end{lemma}\nWe see that, indeed, \\mbox{$p(r y + (1 - r) z, x)$} is almost \nequal to \\mbox{$r p(y, x) + (1 - r) p(z, x)$}. But there are two correction\nterms. The term \n\\mbox{$2 \\cos(\\theta(x, y, z)) \\sqrt{r (1 - r) p(y,x) p(z, x)}$} is\nan interference term, a characteristic of QM.\nThe denominator \\mbox{$\\omega(r, y, z)$} is a normalization factor.\nNote that the interference term contains $\\cos(\\theta(x, y, z))$, not\n$\\sin(\\theta(x, y, z))$. Even if all angles $\\theta$ are equal to zero,\nwhich is the case in a Euclidean space, the term is non-zero. \n\\begin{proof}\nLet $\\vec{v}$ and $\\vec{w}$ be unit vectors of $y$ and $z$\nrespectively with \\mbox{$\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle > 0$}.\nLet \\mbox{$\\vec{u} = \\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w}$}.\nWe have \\mbox{$\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle =$}\n\\mbox{$\\langle \\vec{w} \\, , \\, \\vec{v} \\rangle =$}\n\\mbox{$\\mid \\langle \\vec{v} \\, , \\, \\vec{w} \\rangle \\mid = $}\n\\mbox{$\\sqrt{p(y, z)}$} and\n\\[\n\\mid \\vec{u} \\mid^{2} = \\langle \\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w} \n\\, , \\,\n\\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w} \\rangle = \n\\]\n\\[\nr + 2 \\sqrt{r (1 - r)} \\sqrt{p(y, z)} + (1 - r) = \n\\omega(r, y, z).\n\\]\n\nLet now $\\vec{t}$ be the unit vector of $x$ such that \n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{v} \\rangle > 0$}. \nWe have:\n\\mbox{$\\theta(x, y, z) =$}\n\\mbox{$0 + 0 + \\arg(\\langle \\vec{w} \\, , \\, \\vec{t} \\rangle)$}.\nTherefore \n\\mbox{$\\langle \\vec{w} \\, , \\, \\vec{t} \\rangle =$}\n\\mbox{$\\sqrt{p(x, z)} e^{i \\theta(x, y, z)}$} and\n\\[\n\\langle \\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w} \\, , \\, \\vec{t} \\rangle =\n\\sqrt{r} \\sqrt{p(y, x)} + \\sqrt{1 - r} \\sqrt{p(z, x)} (\\cos(\\theta(x, y, z)) \n+ i \\sin(\\theta(x, y, z))\n\\]\nTherefore\n\\[\n\\mid \\langle \\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w} \\, , \\, \n\\vec{t} \\rangle \\mid^{2}\n=\n(r p(x, y) + (1 -r) p(x, z) \\cos^{2}(\\theta(x, y, z)) + \n\\]\n\\[\n2 \\sqrt{r (1 - r) p(x, y) p(x, z)} cos(\\theta(x, y, z)) +\n(1 - r) p(x, z) \\sin^{2}(\\theta(x, y, z)). \n\\]\n\\end{proof}\n\n\\begin{lemma} \\label{le:prop1}\nIf \\mbox{$y \\not \\perp z$}, \\mbox{$r \\in [0,1]$}, and\n\\mbox{$x = r y \\, + \\, (1 - r) z$} then:\n\\begin{enumerate}\n\\item \\mbox{$coplanar(x, y, z)$},\n\\item \\mbox{$\\theta(x, y, z) = 0$}, \n\\item \\label{form} \n\\mbox{$p(x, y) = 1 - (1 - r) (1 - p(y, z)) \\: \/ \\: \\left( 1 + 2 \\sqrt{r (1 - r) p(y, z)} \\right)$}, and\n\\item for any \\mbox{$0 < r \\leq 1$}, we have\n\\mbox{$p(r y + (1 - r) z, y) > p(y, z)$}.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet \\mbox{$x = r y \\, + \\, (1 - r) y$} and\n\\mbox{$\\vec{u} = \\sqrt{r} \\, \\vec{v} + \\sqrt{1 - r} \\, \\vec{w}$}.\nImmediately, by Definition~\\ref{def:superposition}, \\mbox{$coplanar(x, y, z)$}.\nSince \\mbox{$\\langle \\vec{v} \\, , \\, \\vec{w} \\rangle > 0$},\nwe have \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle > 0$}\nand also \n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{w} \\rangle > 0$}.\nWe conclude that \\mbox{$\\theta(x, y, z) = 0$}.\nFor~\\ref{form}) the value of \\mbox{$p(x, y)$} is straightforward from \nLemma~\\ref{le:p_basis}.\nFrom the same Lemma,\n\\[\np(r y + (1 - r) z, y) = \n\\frac{r + (1 - r) p(y, z) + 2 \\sqrt{r (1 - r) p(y, z)}}\n{1 + 2 \\sqrt{r (1 - r) p(y, z)}} > \n\\]\n\\[\n\\frac{r p(y, z) + (1 - r) p(y, z) + \n2 p(y, z) \\sqrt{r (1 - r) p(y, z)}}\n{1 + 2 \\sqrt{r (1 - r) p(y, z)}} = p(y, z).\n\\]\n\\end{proof}\n\n\\begin{corollary} \\label{le:co_prime}\nIf $x$, $x'$, $y$ and $z$ are coplanar states with \\mbox{$x \\perp x'$},\none has:\n\\[\n\\cos(\\theta(x', y, z)) = \\frac{\\sqrt{p(y, z)} - \\cos(\\theta(x, y, z)) \\,\n\\sqrt{p(x, y) p(x, z)}}  \n{\\sqrt{(1 - p(x, y))(1 - p(x, z))}}.\n\\] \n\\end{corollary}\n\\begin{proof}\nWe have: \\mbox{$p(r y + (1 - r) z, x) +  \np(r y + (1 - r) z, x') = 1$}. By Lemma~\\ref{le:p_basis}:\n\\[\np(r y + (1 - r) z, x) = \\frac{r p(x, y) + (1 - r) p(x, z) +\n2 \\cos(\\theta(x, y, z)) \\sqrt{r (1 - r) p(x, y) p(x, z)}}\n{1 + 2 \\sqrt{r (1 - r) p(y, z)}}\n\\]\nand\n\\[\np(r y + (1 - r) z, x') = \n\\]\n\\[\n\\frac{r (1 - p(x, y)) + (1 - r) (1 - p(x, z)) +\n2 \\cos(\\theta(x', y, z)) \\sqrt{r (1 - r) (1 - p(x, y)) (1 - p(x, z))}}\n{1 + 2 \\sqrt{r (1 - r) p(y, z)}}.\n\\]\nTherefore\n\\[\n1 + 2 \\sqrt{r (1 - r) p(y, z)} =\n\\]\n\\[\n r + (1 - r) + \n2 \\cos(\\theta(x, y, z)) \\sqrt{r (1 - r) p(x, y) p(x, z)} + \n\\]\n\\[\n2 \\cos(\\theta(x', y, z)) \\sqrt{r (1 - r) (1 - p(x, y)) (1 - p(x, z))}\n\\]\nand\n\\[\n\\sqrt{r (1 - r) p(y, z)} = \n\\]\n\\[ \n\\cos(\\theta(x, y, z)) \\sqrt{r (1 - r) p(x, y) p(x, z)} + \n\\]\n\\[\n\\cos(\\theta(x', y, z)) \\sqrt{r (1 - r) (1 - p(x, y)) (1 - p(x, z))}\n\\]\n\\end{proof}\n\nIn parallel with Lemma~\\ref{le:p_basis},\none would like to express \\mbox{$\\theta(ry + (1-r)z, x1, x2)$}\nin terms of $r$ and the $p$'s and $\\theta$'s of $y$, $z$, $x1$ and $x2$, for\ncoplanar states.\nThe formula obtained (by considering some orthonormal basis for the two\ndimensional subspace) is, unfortunately, not very appealing and shall\nnot be presented here.\n\n\\section{Mappings that Preserve Superpositions} \\label{sec:mappings}\nIt is a thesis of this paper that the structure of superpositions is \nthe fundamental structure of Hilbert spaces that is meaningful for \nQuantum Physics. To support this thesis one should, now, analyze the\nfundamental constructions used in Quantum Physics, such as tensor products \nand quotients as universal, i.e., categorical constructions in the\ncategory of superposition preserving mappings.\nSuch an analysis has not been performed yet. Some first reflections on tensor\nproducts may be found in Section~\\ref{sec:conclusion}.\n\nA preliminary step must be the proper definition of the category of\nsuperposition structures and their superposition preserving mappings.\nThis paper does not provide for a proper definition of such a category,\nwhose objects must include both structures defined by Hilbert spaces,\nstudied here, and classical structures in which any two distinct states\nare orthogonal, and all structures in-between.\nWe shall, therefore, consider only superposition structures defined by\nsome Hilbert space. A more general definition abstracting from Hilbert\nspaces and based on the properties of the quantities $p$ and $\\theta$\nis left for future work.\n\nLet \\mbox{${\\cal H}$}\\ be a Hilbert space on the complex field, \nand $X$ be the set of all one-dimensional subspaces of \\mbox{${\\cal H}$}.\nWith any triple \\mbox{$y, z \\in X$},\n\\mbox{$r \\in [0 , 1]$} such that \\mbox{$y \\not \\perp z$}, we can associate\nthe superposition \\mbox{$r y \\: + \\: (1 - r) z$}.\nA function \\mbox{$f : X_{1} \\longrightarrow X_{2}$} between two such sets\nof one-dimensional subspaces $X_{1}$ and $X_{2}$ preserves superpositions\niff for any \\mbox{$y, z \\in X_{1}$}, such that \n\\mbox{$y \\not \\perp z$} and for any\n\\mbox{$r \\in [0 , 1]$} the superposition, in $X_{2}$, \n\\mbox{$r f(y) \\, + \\, (1 - r) f(z)$} is defined,\ni.e., \\mbox{$f(y) \\not \\perp f(z)$} and is equal to\n\\mbox{$f(r y \\, + \\, (1 - r) z)$}.\n\nNote that if \\mbox{$f : X_{1} \\rightarrow X_{2}$} preserves superpositions \nand \\mbox{$x \\not \\perp y$} then \n\\mbox{$f(x) \\not \\perp f(y)$} since the superpositions\n\\mbox{$r f(x) \\, + \\, (1 - r) f(y)$} must be defined.\n \nWe shall now present some preliminary results concerning mappings that preserve\nsuperpositions. First, note that if $\\mbox{${\\cal H}$}_{2}$ is a one-dimensional \nHilbert space, then $X_{2}$ contains one element only and, for any $X_{1}$, \nthe unique mapping \\mbox{$X_{1} \\rightarrow X_{2}$} preserves superpositions.\nSuch a mapping does not preserve $p$ or $\\theta$.\n\nA natural way to obtain a mapping\n\\mbox{$f : X_{1} \\rightarrow X_{2}$} is to start from a \n{\\em linear} map \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$}.\nSuch a map $m$ associates, with every one-dimensional subspace\nof $\\mbox{${\\cal H}$}_{1}$, i.e., every member of $X_{1}$, \na subspace of $\\mbox{${\\cal H}$}_{2}$ that is either one-dimensional or zero-dimensional.\nAny injective, i.e., left-invertible, linear map $m$ defines an application\n\\mbox{$\\overline{m} : X_{1} \\rightarrow X_{2}$} defined by:\n$\\overline{m}(x)$ is the image $m(x)$ of the subspace $x$.\n\\begin{definition} \\label{def:regular}\nA mapping obtained from an injective linear mapping between \nHilbert spaces in the way described just above will be called {\\em regular}.\nIf such a map \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$} is a linear isometry, \ni.e., a unitary map of $\\mbox{${\\cal H}$}_{1}$ onto its image, we shall say that \nthe mapping $\\overline{m}$ is an isometry.\n\\end{definition}\nNote that the mappings preserving superpositions described just above \nthat map into a singleton are not regular unless $\\mbox{${\\cal H}$}_{1}$ is also of \ndimension one.\nNote also that if \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$} is an\ninjective linear map, then, for any complex number $c$ different from zero,\nthe map $c \\, m$ is an injective linear map \n\\mbox{$\\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$}\nand that \\mbox{$\\overline{c \\, m} = \\overline{m} : X_{1} \\rightarrow X_{2}$}.\n\nWe shall now characterize the regular mappings that preserve\nsuperpositions. First, a well-known result from the theory of Hilbert spaces.\n\\begin{theorem} \\label{the:isometry}\nLet \\mbox{$H_{1}, H_{2}$} be Hilbert spaces. If \n\\mbox{$f : H_{1} \\rightarrow H_{2}$} is a linear isometry, i.e., \n\\mbox{$\\parallel f(\\vec{u}) \\parallel =$}\n\\mbox{$\\parallel \\vec{u} \\parallel$} for every \\mbox{$\\vec{u} \\in \\mbox{${\\cal H}$}_{1}$}\nthen it preserves inner products:\n\\mbox{$\\langle f(\\vec{u}) \\, , \\, f(\\vec{v}) \\rangle =$}\n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle$}  for every \n\\mbox{$\\vec{u} \\, , \\, \\vec{v} \\in \\mbox{${\\cal H}$}_{1}$}.\n\\end{theorem}\n\nWe now move to prove that if $\\overline{m}$ is any regular mapping that\npreserves superpositions, then $\\overline{m}$ is an isometry.\n\\begin{lemma} \\label{le:regular}\nLet $\\mbox{${\\cal H}$}_{1}$ and $\\mbox{${\\cal H}$}_{2}$ be Hilbert spaces and let $X_{1}$ and $X_{2}$ \nbe the one-dimensional subspaces of $\\mbox{${\\cal H}$}_{1}$ and $\\mbox{${\\cal H}$}_{2}$ respectively.\nAssume \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$} is an injective linear mapping \nand that \\mbox{$\\overline{m} : X_{1} \\rightarrow X_{2}$} \npreserves superpositions.\nThen there is a strictly positive real constant $c$ such that, for every\n\\mbox{$\\vec{u} \\in \\mbox{${\\cal H}$}_{1}$}, one has \\mbox{$\\parallel m(\\vec{u}) \\parallel =$}\n\\mbox{$c \\parallel \\vec{u} \\parallel$}, and $\\overline{m}$ is an isometry.\n\\end{lemma}\n\\begin{proof}\nNotice first that, if \\mbox{$\\parallel m(\\vec{u}) \\parallel =$}\n\\mbox{$c \\parallel \\vec{u} \\parallel$} for every \\mbox{$\\vec{u} \\in \\mbox{${\\cal H}$}_{1}$},\nthen, if we define \\mbox{$n = m \/ c$} the mapping $n$ is a linear isometry and \none has \\mbox{$\\overline{m} = \\overline{n}$}, proving that $\\overline{m}$ \nis an isometry.\n \nLet $m$ be linear and assume $\\overline{m}$ preserves superpositions.\nLet \\mbox{$x, y \\in X_{1}$} be one-dimensional subspaces of $\\mbox{${\\cal H}$}_{1}$.\nIt is enough to show that there are unit vectors \\mbox{$\\vec{u}, \\vec{v}$} \nin $x$ and $y$ respectively such that \n\\mbox{$\\parallel m(\\vec{u}) \\parallel =$}\n\\mbox{$\\parallel m(\\vec{v}) \\parallel$}. \n\nIf \\mbox{$x = y$} the result follows from the linearity of $m$.\nWe may therefore assume that \\mbox{$x \\neq y$}.\n\nSuppose, first, that \\mbox{$x \\not \\perp y$}\nand let \\mbox{$r \\in ]0, 1[$}.\nThere are unit vectors \\mbox{$\\vec{u} \\in x$}, \\mbox{$\\vec{v} \\in y$},\n\\mbox{$\\vec{t} \\in \\overline{m}(x)$} and \\mbox{$\\vec{w} \\in \\overline{m}(y)$} \nsuch that\n\\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle > 0$} and \n\\mbox{$\\langle \\vec{t} \\, , \\, \\vec{w} \\rangle > 0$}.\nNote that \\mbox{$\\overline{m}(x) \\neq \\overline{m}(y)$} since $m$ is injective.\nThe vector \\mbox{$\\sqrt{r} \\, \\vec{u} + \\sqrt{1 - r} \\, \\vec{v}$} \nis a vector of\nthe superposition \\mbox{$r x \\, + \\, (1 -r) y$}.\nThe vector \\mbox{$\\sqrt{r} \\, \\vec{t} + \\sqrt{1 - r} \\, \\vec{w}$} \nis a vector of\nthe superposition \\mbox{$r m(x) \\, + \\, (1 -r) m(y) =$}\n\\mbox{$\\overline{m}(r x \\, + \\, (1 - r) y)$} since $\\overline{m}$ preserves \nsuperpositions.\nSince $m$ is linear the vector \n\\mbox{$\\sqrt{r} \\, m(\\vec{u}) + \\sqrt{1 - r} \\, m(\\vec{v})$} is a vector of \n\\mbox{$\\overline{m}(r x \\, + \\, (1 - r) y)$}.\nWe conclude that both vectors\n\\mbox{$\\sqrt{r} \\, \\vec{t} + \\sqrt{1 - r} \\, \\vec{w}$} and\n\\mbox{$\\sqrt{r} \\, m(\\vec{u}) + \\sqrt{1 - r} \\, m(\\vec{v})$}\nare members of the same one-dimensional subspace. \nThis implies that \\mbox{$m(\\vec{u}) = d \\vec{t}$} and\n\\mbox{$m(\\vec{v}) = d \\vec{w}$} for some complex number $d$ and\n\\mbox{$\\parallel m(\\vec{u}) \\parallel =$} \n\\mbox{$\\parallel m(\\vec{v}) \\parallel =$}\n\\mbox{$\\mid d \\mid$}.\n\nLet us now assume that \\mbox{$x \\perp y$}. We can find find some \n\\mbox{$z \\in X_{1}$} such that \\mbox{$z \\neq x$}, \\mbox{$z \\neq y$},\n\\mbox{$coplanar(z, x, y)$}. Since \\mbox{$z \\not \\perp x$}, by the above\nwe can find unit vectors \\mbox{$\\vec{u} , \\vec{w}$} in $x$ and $z$ respectively\nsuch that \\mbox{$\\parallel m(\\vec{u}) \\parallel =$}\n\\mbox{$\\parallel m(\\vec{w}) \\parallel$}. \nSimilarly, we can find unit vectors \n\\mbox{$\\vec{v} , \\vec{w'}$} in $y$ and $z$ respectively\nsuch that \\mbox{$\\parallel m(\\vec{v}) \\parallel =$}\n\\mbox{$\\parallel m(\\vec{w'}) \\parallel$}. \nBut \\mbox{$\\parallel m(\\vec{w'}) \\parallel =$}\n\\mbox{$\\parallel m(\\vec{w}) \\parallel$}. \n\\end{proof}\n\nWe shall show now that any isometry preserves superpositions.\n\\begin{lemma} \\label{le:unit_superp}\nLet \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$} be a linear isometry. \nThen $\\overline{m}$ preserves  $p$, $\\theta$ and superpositions.\n\\end{lemma}\n\\begin{proof}\nBy Theorem~\\ref{the:isometry}, $f$ preserves inner products and therefore \npreserves orthogonality, $p$ and $\\theta$.\n\nAssume now that \\mbox{$x \\not \\perp y$} and\n\\mbox{$z = r x + (1 -r) y$}.\nWe have\n\\mbox{$\\overline{m}(x) \\not \\perp \\overline{m}(y)$} and therefore\nthe superposition \\mbox{$r \\overline{m}(x) + (1 - r) \\overline{m}(y)$}\nis defined.\n\nIf \\mbox{$\\vec{u} , \\vec{v}$} are unit vectors of $x$ and $y$ respectively,\nsuch that \\mbox{$\\langle \\vec{u} \\, , \\, \\vec{v} \\rangle > 0$} then\n\\mbox{$m(\\vec{u}) , m(\\vec{v})$} are unit vectors of \n\\mbox{$\\overline{m}(x) , \\overline{m}(y)$} respectively such that\n\\mbox{$\\langle m(\\vec{u}) \\, , \\, m(\\vec{v}) \\rangle > 0$} and therefore\n\\mbox{$r \\overline{m}(x) + (1 - r) \\overline{m}(y)$} is the one-dimensional\nsubspace generated by\n\\mbox{$\\sqrt{r} \\, m(\\vec{u}) + \\sqrt{1 - r} \\, m(\\vec{v}) =$}\n\\mbox{$m(\\sqrt{r} \\, \\vec{u} + \\sqrt{1 - r} \\, \\vec{v})$} which is\n\\mbox{$\\overline{m}(r x \\, + \\, (1 - r) y)$}.\n\\end{proof}\n\nWe can now characterize regular mappings that preserve superpositions. \n\\begin{theorem} \\label{the:char_morph}\nLet \\mbox{$m : \\mbox{${\\cal H}$}_{1} \\rightarrow \\mbox{${\\cal H}$}_{2}$} be any linear injective mapping.\nThe function \\mbox{$\\overline{m} :$}\n\\mbox{$X_{1} \\rightarrow X_{2}$} preserves superpositions \niff it is an isometry.\n\\end{theorem}\n\\begin{proof}\nThe {\\em if} part is Lemma~\\ref{le:unit_superp}.\nThe {\\em only if} part is Lemma~\\ref{le:regular}.\n\\end{proof}\n\n \n\\section{Conclusion and Future Work} \\label{sec:conclusion}\nWe have shown that the properties of superpositions are governed by two\ngeometrical quantities $p$ and $\\theta$ defined, respectively for pairs\nand triples of one-dimensional subspaces in a Hilbert space, \nthus moving forward\nJohn von Neumann's program of focusing on subspaces and not on vectors.\n\nThe most pressing task is probably now to provide an abstract definition of\nstructures admitting a superposition operation, \ngeneralizing those structures provided by Hilbert spaces.   \n\nA quantic system composed of two sub-systems is represented by the tensor\nproduct of the Hilbert spaces representing the two sub-systems.\nProduct states of the form $x_{1} \\otimes x_{2}$ are elements of this tensor\nproduct. On such product states, the quantities $p$ and $\\theta$ \nare easily analyzed:\nwe have \n\\[\np(x_{1} \\otimes x_{2}, y_{1} \\otimes y_{2}) =\np(x_{1}, y_{1}) p(x_{2}, y_{2})\n\\]\nand\n\\[\n\\theta(x_{1} \\otimes x_{2}, y_{1} \\otimes y_{2}, z_{1} \\otimes z_{2}) =\n\\theta(x_{1}, y_{1}, z_{1}) + \\theta(x_{2}, y_{2}, z_{2}).\n\\]\nThe tensor product can be characterized as the closure of the set of product \nstates under superpositions (in our sense) and the operation of taking \nthe state orthogonal to a given state in a given two-dimensional plane. \n\nExtending this definition to superpositions of product states in accordance\nwith the properties of $p$ and $\\theta$ on superpositions provides a \nsuperposition structure that is a original presentation of the tensor product\nand may be found useful to study symmetry properties. \n\n\\section{Acknowledgements} \\label{sec:ack}\nI am most grateful to Kurt Engesser and Dov Gabbay for extremely fruitful \ndiscussions during the elaboration of this paper. I thank Dorit Aharonov and\nJean-Marc L\\'{e}vy-Leblond for their interest and help.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDistributional representations of words have become an indispensable asset in natural language processing (NLP) research due to its wide application in downstream tasks such as parsing \\cite{bansal2014tailoring}, named entity recognition \\cite{lample2016neural}, and sentiment analysis \\cite{tang2014learning}. Of these, ``neural'' word vectors such as Word2Vec \\cite{Mikolov2013}, GloVe \\cite{Pennington2014}, and Paragram \\cite{wieting2015paraphrase} are amongst the most prevalently used and on which we focus in this article. \n\nThere has been a recent thrust in the study of word vector postprocessing methods \\cite{Faruqui2015,Fried2015,Mrksic2016,Mrksic2017,Shiue2017,Mu2018,Liu2019,Tang2019}. These methods directly operate on word embeddings and effectively enhance their linguistic regularities in light-weight fashions. Nonetheless, existing postprocessing methods usually come with a few limitations. For example, some rely on external linguistic resources such as English WordNet \\cite{Faruqui2015,Fried2015,Mrksic2016,Mrksic2017,Shiue2017}, leaving out-of-database word vectors untouched. Others use heuristic methods to flatten the spectrum of word vector embedding matrices \\cite{Mu2018,Liu2019,Wang2019,Tang2019}. Although being effective, these spectral flattening algorithms are primarily motivated by experimental observations but lack of direct interpretability.\n\nIn this paper, we propose a novel word vector postprocessing approach that addresses these limitations. Under a \\emph{causal inference} framework, the proposed method meets the joint desiderata of (1) \\emph{theoretical interpretability}, (2) \\emph{empirical effectiveness}, and (3) \\emph{computational efficiency}. Concretely, the postprocessing pipeline is realized by Half-Sibling Regression (HSR) \\cite{Scholkopf2016}, a method for identifying and removing confounding noise of word vectors. Using a simple linear regression method, we obtain results that are either on-par or outperform state-of-the-art results on a wide battery of lexical-level evaluation tasks and downstream sentiment analysis tasks. More specifically, our contributions are as follows:\n\n \\begin{itemize}\n \n \\item We formulate the word vector postprocessing task as a confounding noise identification problem under a putative causal graph. This formulation brings causal interpretability and theoretical support to our postprocessing algorithm.\n \n\\item The proposed method is data-thrifty and computationally simple. Unlike many existing methods, it does not require external linguistic resources (e.g., synonym relationships); besides, the method can be implemented easily via simple linear regressions.\n\n\\item The proposed postprocessing method yields highly competitive empirical results. For example, while achieving the best performance on 20 semantic textual similarity tasks, on average, our proposed method brings 4.71\\%, 7.54\\%, and 6.54\\% improvement respectively compared to the previously best results, and it achieves 7.13\\%, 22.06\\%, and 9.83\\% improvement compared to the original word embedding when testing on Word2Vec, GloVe, and Paragram.\n\\end{itemize} \n \nThe rest of the paper is organized as follows. We first briefly review prior work on word vector postprocessing. Next, we introduce Half-Sibling Regression as a causal inference framework to remove confounding noise; we then proceed to explain how to apply Half-Sibling Regression to remove noise from word embeddings. Then, we showcase the effectiveness of the Half-Sibling Ridge Regression model on word similarity tasks, semantic textual similarity tasks, and downstream sentiment analysis tasks using three different pre-trained English word embeddings. Finally, we conduct statistical significance tests on all experimental results\\footnote{Our codes are available at \\url{https:\/\/github.com\/KunkunYang\/denoiseHSR-AAAI}}.\n \n\\section{Prior Work} \\label{sec:prior}\n In this section, we review prior art for word vector postprocessing. Modern word vector postprocessing methods can be broadly divided into two streams: (1) lexical and (2) spatial approaches.\n\n\\paragraph{The Lexical Approach} The lexical approach uses lexical relational resources to enhance the quality of word vectors. These lexical relational resources specify semantic relationships of words such as synonym and antonym relationships. For example, \\citet{Faruqui2015} inject synonym lexical information into pre-trained collections of word vectors. \\citet{Mrksic2016} generalize this approach and insert both antonym and synonymy constraints into word vectors. \\citet{Mrksic2017} use constraints from mono- and cross-lingual lexical resources to fine-tune word vectors. \\citet{Fried2015} and \\citet{Shiue2017} propose to use hierarchical semantic relations such as hypernym semantics to enrich word vectors. To make sure that word vectors satisfy the lexical relational constraints, supervised machine learning algorithms are used.\n\n\\paragraph{The Spatial Approach} The spatial approach differs from the lexical approach in that it does not require external knowledge bases. The general principle of this approach is to enforce word vectors to be more ``isotropic'', i.e., more spread out in space. This goal is usually achieved by flattening the spectrum of word vectors. For example, \\citet{Mu2018} propose All-But-The-Top (ABTT) method which removes leading principal components of word vectors; \\citet{Wang2019} extend this idea by softly shrinking principal components of word embedding matrix using a variance normalization method; \\citet{Liu2019} propose the Conceptor Negation (CN) method, which employs regularized identity maps to filter away high-variance latent features of word vectors;  more recently, \\citet{Tang2019} develop SearchBeta (SB) that uses a centralized kernel alignment method to smooth the spectrum of word vectors.\n\n\n\\section{The Causal Inference Approach for Word Vector Postprocessing}\n\nThe lexical and spatial approaches introduced in the previous section have empirically proven to be effective. Nonetheless, they also suffer from a few limitations. A shortcoming of the lexical approach is that it is unable to postprocess out-of-database word vectors. Indeed, lexical relational resources like synonym-antonym relationships are informative for word meaning, in particular word meaning of \\emph{adjectives}. However, many non-adjective words do not have abundant lexical connections with other words, and for this reason, they are not well-represented in lexical-relationship databases. For instance, most nouns (e.g., \\texttt{car}) and verbs (e.g., \\texttt{write}) have few synonyms and even fewer antonyms, making the lexical postprocessing methods inapplicable to these words. The spatial approach favorably avoids this problem by lifting the requirement of lexical relational resources. Yet, one major downside of the spatial approach is its lack of direct interpretability. For example, many spatial approaches propose to completely or softly remove a few leading principal components (PCs) of word vectors. However, it is rather unclear what exactly has been encoded by these leading PCs other than the empirical finding that these leading PCs are somehow correlated with word frequencies \\cite{Mu2018}.\n\nIn this paper, we go beyond the lexical and spatial schemes and introduce a novel \\emph{causal inference approach} for postprocessing word vectors. The method does not seek to infer the causal structure of words or word vectors; instead, in line with \\citet{Scholkopf2012On} and \\citet{Scholkopf2016}, it incorporates causal beliefs and assumptions for empirical objectives -- postprocessing off-the-shelf word vectors in our case. Concretely, this is achieved by identifying and removing confounding noise of word vectors using Half-Sibling Regression (HSR) method \\cite{Scholkopf2016}. Here we first briefly introduce HSR and then explain how to apply HSR to word vectors.\n\n\n\\subsection{Half-Sibling Regression}\n\nIn the passing, we introduce HSR mainly based on the presentation of \\citet{Scholkopf2016}. Consider a hypothetical causal graph, shown in Figure \\ref{fig:causalGraph}, where each vertex labeled by $Q$, $N$, $Y$, and $X$ are random variables defined on an underlying probability space and each directed edge indicates the probabilistic dependency between two random variables. We are mostly interested in quantities taken by the random variable $Q$. Unfortunately, it is not possible to directly observe these quantities. Instead, we are given only the \\emph{corrupted} observations of $Q$, taken value by the random variable $Y$. That is, intuitively $Y$ can be seen as a noisy, lossy version of $Q$. A natural assumption of $Y$ is that it statistically depends on its ``clean'' version $Q$ as well as some noise, whose values are taken by some unobservable random variable $N$ that encodes the noise source. We further assume that the noise source $N$ affects another random variable, $X$, whose quantities are directly observable. Importantly, we require $X$ to be independent of $Q$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{3106_causalGraph.png}\n\\caption{The causal graph for HSR (adapted from \\citet{Scholkopf2016}). Each vertex labeled by $Q$, $N$, $Y$, and $X$ is a random variable defined on an underlying probability space. Directed edges connecting random variables describe probabilistic dependency between random variables.}\n\\label{fig:causalGraph}\n\\end{figure}\n\n\nRecall that we are mostly interested in the unobservable random variable $Q$. Hence the question we aim to answer is: How to reconstruct the quantities taken by $Q$ by leveraging the underlying statistical dependency structure in Figure \\ref{fig:causalGraph}? HSR provides a simple yet effective solution to this question -- It estimates $Q$ via its approximation $\\hat{Q}$, which is defined as\n\n\\begin{equation} \\label{eq:HSR}\n\\hat{Q} \\coloneqq Y - \\EE [Y \\mid X].\n\\end{equation}\n\nThe HSR Equation \\ref{eq:HSR} can be straightforwardly interpreted as follows. Recall that $X$ is independent of $Q$, and therefore $X$ is \\emph{not} predictive to $Q$ or $Q$'s influence on $Y$. However, $X$ is predictive to $Y$, because $X$ and $Y$ are both influenced by the \\emph{same} noise source $N$. When predicting $Y$ based on $X$ realized by the term $\\EE [Y \\mid X]$, since those signals of $Y$ coming from $Q$ cannot be predicted by $X$, only those noise contained in $Y$ coming from $N$ could be captured. To reconstruct $Q$ from $Y$, we can therefore remove the captured noise $\\EE [Y \\mid X]$ from $Y$, resulting in the reconstruction $\\hat{Q} \\coloneqq Y - \\EE [Y \\mid X]$, which is Equation \\ref{eq:HSR}. This procedure is referred to as Half-Sibling Regression because $X$ and $Y$ share one parent vertex $N$. $Y$ is regressed upon its half-sibling $X$ to capture the components of $Y$ inherited from their shared parent vertex $N$.\n\nHSR enjoys a few appealing theoretical properties. In particular, it is possible to show that $\\hat{Q}$ reconstructs $Q$ (up to its expectation $\\EE[Q]$) at least as good as the mean-subtraction $Y - \\EE [Y]$ does. We refer the readers to \\citet{Scholkopf2016} for detailed theoretical discussions.\n\n\n\\subsection{Applying HSR to De-Noise Word Vectors}\n\nWe now explain how we apply HSR to remove noise from word vectors. Before getting into the details, we first recall some linguistic basics of words, which are the key enablers of our approach. Semantically, words can be divided into two basic classes: (1) content or open-class words and (2) function or closed-class words (also known as stop words). Content words are those that have meaning or semantic value, such as nouns, verbs, adjectives, and adverbs. Function words have little lexical meaning; rather, they mainly exist to explain grammatical or structural relationships with other words. In English, examples of function words include \\texttt{a}, \\texttt{to}, \\texttt{for}, \\texttt{of}, \\texttt{the}, and more.\n\n\n\\begin{algorithm}[ht]\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\\Input{(i) $\\{v_i^Y\\}_{i = 1}^K$: a collection of $K$ content-word vectors, each of dimension $n$; $\\mathbf{V}^Y$ is a $n \\times K$ matrix whose columns are from $\\{v_i^Y\\}_{i = 1}^K$. (ii) $\\{v_i^X\\}_{i = 1}^P$: a collection of $P$ function-word vectors, each of dimension $n$; $\\mathbf{V}^X$ is a $n \\times P$ matrix whose columns are from $\\{v_i^X\\}_{i = 1}^P$. (iii) Regression constants $\\alpha_1, \\alpha_2 > 0$.} \n\\textbf{Postprocess content-word vectors}: \\newline\n Step 1.1: \\emph{Identify noise contained in content-word vectors}: Estimate a weight matrix $\\textbf{W}_1$ such that\n\\[ \\mathbf{V}^Y \\approx \\mathbf{V}^X \\mathbf{W}_1, \\]\n with ridge regression\n\\[ \\mathbf{W}_1 = \\left  ( (\\mathbf{V}^X)^\\top \\mathbf{V}^X  + \\alpha_1  \\mathbf{I} \\right )^{-1} (\\mathbf{V}^X)^\\top \\mathbf{V}^Y.\n\\]\n Step 1.2: \\emph{Remove noise contained in content-word vectors}: \n\\[ \\hat{\\mathbf{V}}^Y \\coloneqq \\mathbf{V}^Y  - \\mathbf{V}^X \\mathbf{W}_1. \\]  \\\\\n\n\\textbf{Postprocess stop-word vectors}: \\newline\n Step 2.1: \\emph{Identify noise contained in stop-word vectors}: Estimate a weight matrix $\\textbf{W}_2$ such that\n\\[ \\mathbf{V}^X \\approx \\mathbf{V}^Y \\mathbf{W}_2, \\]\n with ridge regression\n\\[ \\mathbf{W}_2 = \\left  ((\\mathbf{V}^Y)^\\top \\mathbf{V}^Y  + \\alpha_2  \\mathbf{I} \\right )^{-1} (\\mathbf{V}^Y)^\\top \\mathbf{V}^X.\n\\]\n Step 2.2: \\emph{Remove noise contained in stop-word vectors}: \n\\[ \\hat{\\mathbf{V}}^X \\coloneqq \\mathbf{V}^X  - \\mathbf{V}^Y \\mathbf{W}_2. \\]  \\\\\n\\Output{(i) HSR postprocessed content-word vectors $\\{\\hat{v}_i^Y\\}$, which are columns of $\\hat{\\mathbf{V}}^Y$; (ii) HSR postprocessed stop-word vectors $\\{\\hat{v}_i^X\\}$, which are columns of  $\\hat{\\mathbf{V}}^X$.}  \n\\caption{HSR algorithm for word vector postprocessing}\n\\label{alg:hsr}\n\\end{algorithm}\n\n\n\nBased on these basic linguistic facts, we posit that content-word vectors and function-word vectors can be seen as half-siblings as their linguistic properties align well with the HSR foundations. Indeed, since function-word vectors carry little semantic content, they could not be predictive to clean content-word vectors. Additionally, since content-word vectors and function-word vectors are induced from some shared training corpora, we hypothesize that they are subjected to the same noise profile. Using HSR language of Figure \\ref{fig:causalGraph}, this means we can model the off-the-shelf stop-word vectors with $X$, off-the-shelf content-word vectors with $Y$, and ``clean'' yet unseen content-word vectors with $Q$. Under the HSR framework, when we regress content-word vectors upon function-word vectors, only the noise of the former is captured. Once such noises are identified, they can be directly subtracted, so that the clean content-word vectors will be reconstructed.\n\n\nThe above described procedure can be mathematically realized as follows. Let $\\{v^X_i\\}_{i=1}^P$ be a collection of function-word vectors and let $\\{v^Y_i\\}_{i=1}^K$ be a collection of content-word vectors. To postprocess content-word vectors $\\{v^Y_i\\}_{i=1}^K$, we run a simple two-step algorithm. In the first step, we estimate parameters of a linear multiple-output model \\cite[Section 3.2.4]{Hastie2001}, in which we use model inputs $v^X_1, \\cdots, v^X_P$ to predict model outputs $v^Y_1, \\cdots, v^Y_K$. This amounts to estimate each $w_{ij}$ such that $v^Y_j \\approx \\sum_{i = 1}^P w_{ij} v^X_i$ for each $j \\in \\{1, \\cdots, K\\}$. In the second step, we remove the regression result from the target of the regression. That is, we let $\\hat{v}^Y_j \\coloneqq v_j^Y - \\sum_{i = 1}^P w_{ij} v^X_i$ be the postprocessed content-word vector.\n\nSo far, we have described how to postprocess content-word vectors. To postprocess function-word vectors, we can employ a similar pipeline with the predictor and target flipped. That is, to identify confounding noise contained in stop-word vectors, we use off-the-shelf content-word vectors as features to predict off-the-shelf stop-word vectors. The full algorithm is summarized in Algorithm \\ref{alg:hsr}.\n\nWe provide a few remarks on the practical implementations and further generalizations of Algorithm \\ref{alg:hsr}. Our first remark goes to how to identify the function and content words in practice. Throughout our experiments, to identify function words, we use the stop word list provided by Natural Language Toolkit (NLTK) package\\footnote{\\url{https:\/\/www.nltk.org\/}}, which is a list of 179 words. We regard words outside of this list to be content words. A small amount of stop words works efficiently when postprocessing tens of thousands of content-word vectors because in this case, we only have a handful of features. However, when postprocessing stop-word vectors, it is cumbersome because the number of content words as features are too large to be efficiently implemented. For this reason, in practice, we only use a small sample of commonly used content-word vectors as features for postprocessing stop-word vectors. Specifically, borrowing the word list provided by \\citet{Arora2017}, we use the most frequent 1000 content words as features in Step 2.1 and Step 2.2 of Algorithm \\ref{alg:hsr}.\n\nMoreover, while our framework postprocesses both content and function words, we have tried only postprocessing content words and leaving function words unchanged. We discover that the experimental results are still better than the baseline spatial approaches but worse than postprocessing both content and function words. The reason might be that stop words play non-trivial roles in various NLP tasks. As all baseline spatial approaches postprocess both content and function words, we follow this setting.\n\nFinally, we remark that the linear model used in Algorithm \\ref{alg:hsr} can be straightforwardly generalized to non-linear models. For this, we have formulated and tested Multilayer Perceptrons (MLPs) as extensions to the linear model used in Algorithm \\ref{alg:hsr}. The detailed MLP version of Algorithm \\ref{alg:hsr} is postponed to the appendix.\n\n\n\\begin{table*}[t]\n  \\centering\n  \\caption{Spearman's rank correlation coefficient of seven word similarity tasks}\n\\scalebox{0.72}{\n\n             \\begin{tabular}{lrrrrrrrrrrrrrrr}\n             \\toprule\n    \\multirow{2}[0]{*}{} & \\multicolumn{5}{c}{\\textbf{WORD2VEC}} & \\multicolumn{5}{c}{\\textbf{GLOVE}}    & \\multicolumn{5}{c}{\\textbf{PARAGRAM}} \\\\\n    \\cmidrule(r){2-6} \\cmidrule(r){7-11} \\cmidrule(r){12-16}\n          & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} \\\\\n            \\toprule\n    \\textbf{RG65} & 0.7494 & \\underline{0.7869} & \\underline{\\textbf{0.8041}} & \\underline{0.7964} & 0.7569 & 0.7603 & 0.7648 & \\underline{\\textbf{0.7913}} & \\underline{0.7850} & \\underline{0.7694} & 0.7630 & \\underline{0.7683} & 0.7594 & \\underline{\\textbf{0.7898}} & \\underline{0.7760} \\\\\n    \\textbf{WordSim-353} & \\underline{0.6999} & 0.6929 & \\underline{0.6992} & 0.6856 & \\underline{\\textbf{0.7059}} & 0.7379 & \\underline{0.7668} & \\underline{0.7886} & 0.7115 & \\underline{\\textbf{0.7887}} & 0.7302 & \\underline{\\textbf{0.7386}} & \\underline{0.7321} & 0.7196 & \\underline{0.7338} \\\\\n    \\textbf{RW} & 0.5997 & 0.5984 & \\underline{\\textbf{0.6036}} & \\underline{0.5998} & \\underline{0.6033} & 0.5101 & \\underline{0.5716} & \\underline{\\textbf{0.5898}} & 0.4879 & \\underline{0.5580} & 0.5972 & \\underline{\\textbf{0.6038}} & \\underline{0.6006} & 0.5769 & \\underline{0.6023} \\\\\n    \\textbf{MEN} & 0.7706 & \\underline{\\textbf{0.7929}} & \\underline{0.7901} & \\underline{0.7888} & 0.7726 & 0.8013 & \\underline{0.8234} & \\underline{\\textbf{0.8339}} & 0.7853 & \\underline{0.8258} & \\underline{0.7728} & 0.7705 & \\underline{0.7746} & 0.7693 & \\underline{\\textbf{0.7750}} \\\\\n    \\textbf{MTurk} & \\underline{0.6831} & 0.6538 & 0.6610 & \\underline{0.6846} & \\underline{\\textbf{0.6854}} & 0.6916 & \\underline{\\textbf{0.7233}} & \\underline{0.7116} & 0.6731 & \\underline{0.7074} & \\underline{0.6300} & 0.6106 & \\underline{0.6251} & 0.6147 & \\underline{\\textbf{0.6319}} \\\\\n    \\textbf{SimLex-999} & 0.4427 & 0.4629 & \\underline{\\textbf{0.4728}} & \\underline{0.4702} & \\underline{0.4672} & 0.4076 & \\underline{0.4650} & \\underline{\\textbf{0.4858}} & 0.3985 & \\underline{0.4728} & 0.6847 & \\underline{0.6862} & 0.6854 & \\underline{0.6878} & \\underline{\\textbf{0.6903}} \\\\\n    \\textbf{SimVerb-3500} & 0.3659 & 0.3792 & \\underline{0.3868} & \\underline{0.3865} & \\underline{\\textbf{0.3978}} & 0.2842 & \\underline{0.3433} & \\underline{0.3632} & 0.2671 & \\underline{\\textbf{0.3980}} & 0.5411 & \\underline{0.5461} & \\underline{0.5413} & 0.5389 & \\underline{\\textbf{0.5518}} \\\\\n      \\bottomrule\n    \\end{tabular}\n}%\n  \\label{tab:word_sim}%\n\\end{table*}%\n\n\\begin{table*}[htbp]\n  \\centering\n  \\caption{Pearson correlation coefficient of 20 semantic textual similarity tasks}\n\\scalebox{0.70}{%\n\n   \\begin{tabular}{lrrrrrrrrrrrrrrr}\n   \\toprule\n    \\multirow{2}[0]{*}{} & \\multicolumn{5}{c}{\\textbf{WORD2VEC}} & \\multicolumn{5}{c}{\\textbf{GLOVE}}    & \\multicolumn{5}{c}{\\textbf{PARAGRAM}} \\\\\n    \\cmidrule(r){2-6} \\cmidrule(r){7-11} \\cmidrule(r){12-16}\n          & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} & \\multicolumn{1}{c}{\\textbf{Orig.}} & \\multicolumn{1}{c}{\\textbf{ABTT}} & \\multicolumn{1}{c}{\\textbf{CN}} & \\multicolumn{1}{c}{\\textbf{SB}} & \\multicolumn{1}{c}{\\textbf{HSR-RR}} \\\\\n     \\toprule\n    \\textbf{STS-2012-MSRpar} & \\textbf{41.78} & 38.70 & 39.42 & 40.77 & 34.42 & \\textbf{42.06} & 41.41 & 41.27 & 41.15 & 32.49 & 39.32 & 38.84 & 39.84 & 37.72 & \\textbf{41.44} \\\\\n    \\textbf{STS-2012-MSRvid} & 76.27 & 75.60 & 75.32 & 74.98 & \\textbf{79.63} & 65.85 & 67.84 & 62.50 & 64.71 & \\textbf{80.03} & 56.34 & 57.65 & 56.78 & 55.55 & \\textbf{62.31} \\\\\n    \\textbf{STS-2012-surprise.OnWN} & 70.62 & 70.89 & 70.73 & 69.99 & \\textbf{71.27} & 60.74 & 69.48 & 67.87 & 57.02 & \\textbf{72.24} & 62.60 & 64.61 & 63.21 & 60.68 & \\textbf{67.91} \\\\\n    \\textbf{STS-2012-SMTeuroparl} & 31.20 & 35.71 & 35.29 & 33.88 & \\textbf{40.32} & 51.97 & \\textbf{54.36} & 52.58 & 50.06 & 51.60 & 50.64 & 51.64 & 50.63 & 51.34 & \\textbf{51.92} \\\\\n    \\textbf{STS-2012-surprise.SMTnews} & \\textbf{51.07} & 46.24 & 47.34 & 47.10 & 50.09 & 46.35 & 48.19 & 47.69 & 45.18 & \\textbf{54.41} & 52.94 & 50.18 & 52.66 & \\textbf{54.16} & 53.87 \\\\\n    \\hdashline\n    \\textbf{STS-2012} & 54.19 & 53.43 & 53.62 & 53.34 & \\textbf{55.15} & 53.39 & 56.26 & 54.38 & 51.62 & \\textbf{58.15} & 52.37 & 52.58 & 52.62 & 51.89 & \\textbf{55.49} \\\\\n    \\toprule\n    \\textbf{STS-2013-FNWN} & 39.68 & 43.51 & 43.40 & 42.95 & \\textbf{49.09} & 39.48 & 45.81 & 42.03 & 39.15 & \\textbf{46.47} & 35.79 & 36.05 & 35.93 & 34.35 & \\textbf{38.00} \\\\\n    \\textbf{STS-2013-OnWN} & 67.98 & 70.56 & 69.29 & 69.12 & \\textbf{75.57} & 53.75 & 63.86 & 57.45 & 52.36 & \\textbf{74.91} & 48.07 & 48.18 & 48.23 & 48.28 & \\textbf{56.57} \\\\\n    \\textbf{STS-2013-headlines} & 63.29 & 63.24 & 63.62 & 63.22 & \\textbf{63.65} & 63.54 & 66.70 & 67.00 & 60.65 & \\textbf{68.56} & 64.43 & 65.13 & 64.69 & 62.99 & \\textbf{66.90} \\\\\n    \\hdashline\n    \\textbf{STS-2013} & 56.98 & 59.10 & 58.77 & 58.43 & \\textbf{62.77} & 52.26 & 58.79 & 55.49 & 50.72 & \\textbf{63.31} & 49.43 & 49.79 & 49.62 & 48.54 & \\textbf{53.82} \\\\\n    \\toprule\n    \\textbf{STS-2014-OnWN} & 74.85 & 75.92 & 75.27 & 74.43 & \\textbf{81.40} & 61.91 & 70.93 & 66.43 & 60.36 & \\textbf{81.39} & 60.29 & 61.95 & 60.75 & 59.45 & \\textbf{68.30} \\\\\n    \\textbf{STS-2014-deft-forum} & 41.30 & 42.25 & 42.74 & 42.03 & \\textbf{46.73} & 28.82 & 38.90 & 37.57 & 25.91 & \\textbf{45.85} & 35.17 & 37.60 & 35.75 & 33.59 & \\textbf{40.84} \\\\\n    \\textbf{STS-2014-deft-news} & 66.76 & 64.87 & 65.45 & 64.97 & \\textbf{67.88} & 63.41 & 68.72 & 69.08 & 61.27 & \\textbf{70.60} & 62.19 & 63.73 & 62.75 & 61.09 & \\textbf{66.66} \\\\\n    \\textbf{STS-2014-headlines} & 60.87 & 60.61 & \\textbf{61.09} & 60.66 & 60.93 & 59.28 & 61.34 & 61.71 & 56.25 & \\textbf{64.01} & 60.84 & 60.72 & 60.97 & 60.21 & \\textbf{62.83} \\\\\n    \\textbf{STS-2014-tweet-news} & 73.33 & 75.13 & 74.87 & 73.66 & \\textbf{76.00} & 62.43 & 74.62 & \\textbf{75.38} & 58.70 & 75.09 & 69.29 & 72.43 & 70.14 & 66.75 & \\textbf{75.16} \\\\\n    \\textbf{STS-2014-images} & 77.44 & 77.81 & 78.42 & 77.11 & \\textbf{80.55} & 61.89 & 69.40 & 65.81 & 59.03 & \\textbf{78.45} & 53.67 & 58.29 & 54.86 & 51.58 & \\textbf{65.10} \\\\\n    \\hdashline\n    \\textbf{STS-2014} & 65.76 & 66.10 & 66.31 & 65.48 & \\textbf{68.92} & 56.29 & 63.99 & 62.66 & 53.59 & \\textbf{69.23} & 56.91 & 59.12 & 57.54 & 55.45 & \\textbf{63.15} \\\\\n    \\toprule\n    \\textbf{STS-2015-answers-forums} & 52.65 & 54.01 & 53.99 & 50.51 & \\textbf{66.77} & 36.86 & 49.58 & 48.62 & 36.76 & \\textbf{65.46} & 38.79 & 41.19 & 39.25 & 38.35 & \\textbf{48.37} \\\\\n    \\textbf{STS-2015-answers-students} & 70.82 & 70.92 & 71.65 & 69.74 & \\textbf{72.16} & 62.77 & 69.46 & \\textbf{69.68} & 61.84 & 67.38 & 67.52 & 69.46 & 67.96 & 66.80 & \\textbf{71.98} \\\\\n    \\textbf{STS-2015-belief} & 60.11 & 61.91 & 61.62 & 58.10 & \\textbf{77.08} & 44.20 & 61.43 & 59.77 & 41.19 & \\textbf{76.12} & 49.77 & 55.57 & 50.79 & 46.98 & \\textbf{61.32} \\\\\n    \\textbf{STS-2015-headlines} & 68.11 & 68.28 & 68.65 & 68.19 & \\textbf{69.02} & 65.42 & 68.90 & 69.20 & 63.25 & \\textbf{71.41} & 67.85 & 68.40 & 68.09 & 66.92 & \\textbf{70.38} \\\\\n    \\textbf{STS-2015-images} & 80.07 & 80.18 & 80.74 & 79.48 & \\textbf{83.08} & 69.14 & 73.53 & 71.43 & 67.81 & \\textbf{80.58} & 66.55 & 68.29 & 67.08 & 65.55 & \\textbf{73.17} \\\\\n    \\hdashline\n    \\textbf{STS-2015} & 66.35 & 67.06 & 67.33 & 65.20 & \\textbf{73.62} & 55.68 & 64.58 & 63.74 & 54.17 & \\textbf{72.19} & 58.10 & 60.58 & 58.63 & 56.92 & \\textbf{65.04} \\\\\n    \\toprule\n    \\textbf{SICK} & 72.25 & \\textbf{72.49} & 72.40 & 72.32 & 72.02 & 66.64 & 68.12 & 66.42 & 66.03 & \\textbf{71.62} & 64.55 & 64.89 & 64.78 & 64.05 & \\textbf{67.07} \\\\\n    \\bottomrule\n    \\end{tabular}%\n\n    }\n  \\label{tab:STS}%\n\\end{table*}%\n\n\n\\begin{table*}[htbp]\n  \\centering\n  \\caption{Five-fold cross-validation accuracy of four sentiment analysis tasks}\n\\scalebox{0.75}{%\n\n    \\begin{tabular}{lrrrrrrrrrrrrrrr}\n    \\toprule\n          & \\multicolumn{5}{c}{\\textbf{WORD2VEC}} & \\multicolumn{5}{c}{\\textbf{GLOVE}}    & \\multicolumn{5}{c}{\\textbf{PARAGRAM}} \\\\\n          \\cmidrule(r){2-6} \\cmidrule(r){7-11} \\cmidrule(r){12-16}\n          & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{SB}} & \\multicolumn{1}{l}{\\textbf{HSR-RR}} & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{SB}} & \\multicolumn{1}{l}{\\textbf{HSR-RR}} & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{SB}} & \\multicolumn{1}{l}{\\textbf{HSR-RR}} \\\\\n          \\toprule\n    \\textbf{AR} & 0.8375 & 0.8338 & 0.8329 & 0.8302 & \\textbf{0.8377} & 0.8441 & 0.8431 & 0.8444 & 0.8426 & \\textbf{0.8454} & 0.8124 & 0.8129 & 0.8113 & 0.8124 & \\textbf{0.8152} \\\\\n    \\textbf{CR} & 0.7800 & 0.7792 & 0.7718 & 0.7726 & \\textbf{0.7824} & \\textbf{0.7829} & 0.7800 & 0.7808 & 0.7819 & 0.7792 & 0.7657 & 0.7649 & 0.7628 & 0.7644 & \\textbf{0.7673} \\\\\n    \\textbf{IMDB} & 0.8392 & 0.8369 & 0.8370 & 0.8281 & \\textbf{0.8434} & 0.8491 & 0.8453 & \\textbf{0.8493} & 0.8459 & \\textbf{0.8493} & 0.7957 & 0.7960 & 0.7953 & 0.7938 & \\textbf{0.7999} \\\\\n    \\textbf{STS-B} & \\textbf{0.8071} & 0.8062 & 0.8048 & 0.8052 & 0.8056 & 0.8044 & 0.8045 & 0.8049 & 0.8031 & \\textbf{0.8053} & 0.7818 & 0.7819 & 0.7778 & 0.7813 & \\textbf{0.7846} \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n  \\label{tab:sentiment}%\n\\end{table*}%\n\n\\section{Experiments}\n\nWe evaluate the HSR postprocessing algorithm described in Algorithm \\ref{alg:hsr} (denoted by HSR-RR as it is based on ridge regression). We test it on three different pre-trained English word embeddings including Word2Vec\\footnote{\\url{https:\/\/code.google.com\/archive\/p\/word2vec\/}} \\cite{Mikolov2013}, GloVe\\footnote{\\url{https:\/\/nlp.stanford.edu\/projects\/glove\/}} \\cite{Pennington2014}, and Paragram\\footnote{\\url{https:\/\/www.cs.cmu.edu\/~jwieting\/}} \\cite{wieting2015paraphrase}. The original word vectors, as well as word vectors postprocessed by ABTT \\cite{Mu2018}, CN \\cite{Liu2019}, and SB \\cite{Tang2019}, are set as baselines. The performances of these baselines against HSR-RR are compared on word similarity tasks, semantic textual similarity tasks, and downstream sentiment analysis tasks. A statistical significance test is conducted on all experimental results to verify whether our method yields significantly better results compared to the baselines. For ABTT, we set $d = 2$ for GloVe and $d = 3$ for Word2Vec and Paragram as suggested by the original authors. For CN, we fix $d = 2$ for all word embeddings as suggested by the original authors. For HSR, we fix the regularization constants $\\alpha_1, \\alpha_2 = 50$ for HSR-RR. Generally, we recommend using $\\alpha_1, \\alpha_2 = 50$ for HSR-RR and other HSR models. Furthermore, we construct a Multilayer Perceptrons HSR model (denoted by HSR-MLP), and the experimental result of HSR-MLP is shown in the appendix.\n\n\\subsection{Word Similarity} \n\nWe use seven popular word similarity tasks to evaluate the proposed postprocessing method. The seven tasks are RG65 \\cite{Rubenstein1965}, WordSim-353 \\cite{Finkelstein2002}, Rare-words \\cite{Luong2013}, MEN \\cite{Bruni2014}, MTurk \\cite{Radinsky2011}, SimLex-999 \\cite{Hill2015}, and SimVerb-3500 \\cite{Gerz2016}.\n\nFor each task, we calculate the cosine similarity between the vector representation of two words, and the Spearman's rank correlation coefficient \\cite{Myers1995} of the estimated rankings against the human rankings is reported in Table \\ref{tab:word_sim}. In the table, the result marked in bold is the best, and the results underlined are the top three results.\n\n\nFrom the table, we could see that while no postprocessing method performs dominantly better than others, HSR-RR has the best performance overall by performing the best on the most number of tasks for two out of the three word embeddings, which are Word2Vec and Paragram. HSR-RR generally achieves the best on these five tasks: WordSim-353, MEN, MTurk, SimLex-999, and SimVerb-3500. Notably, HSR-RR has the best performance on the task SimVerb-3500 for all three word embeddings, which achieves 8.72\\%, 40.04\\%, and 1.98\\% improvement respectively on SimVerb-3500 dataset relative to the original word embeddings and 2.84\\%, 9.58\\%, and 1.04\\% increase compared to the runner-up method. Since SimVerb-3500 is the state-of-the-art task that contains the highest number of word pairs and distinguishes genuine word similarity from conceptual association \\cite{Hill2015}, the result obtained on SimVerb-3500 is usually deemed to be more telling than those of other tasks \\cite{Liu2019}.\n\n\\subsection{Semantic Textual Similarity} \n\nNext, we test the sentence-level effectiveness of our proposed HSR method on semantic textual similarity (STS) tasks, which measure the degree of semantic equivalence between two texts \\cite{Agirre2012}. The STS tasks we employ include 20 tasks from 2012 SemEval Semantic Related task (SICK) and SemEval STS tasks from 2012 to 2015 \\cite{Marco2014,Agirre2012,Agirre2013,Agirre2014,Agirre2015}.\n\nTo construct the embedding of each sentence in the tasks, we first tokenize the sentence into a list of words, then average the word embedding of all words in the list as the vector representation of the sentence. Following \\citet{Agirre2012}, we calculate the cosine distance between the two sentence embeddings and record the Pearson correlation coefficient of the estimated rankings of sentence similarity against the human rankings.\n\nIn Table \\ref{tab:STS}, we present the result of the 20 STS tasks as well as the average result each year. From the table, we could observe that HSR-RR dominantly outperforms the original word embedding as well as other postprocessing methods, as the average result by year of HSR-RR is the best for all tasks except the SICK task on Word2Vec. On average, HSR-RR improves the Pearson correlation coefficient by 4.71\\%, 7.54\\%, and 6.54\\% respectively over the 20 STS tasks compared to the previously best results, and it achieves 7.13\\%, 22.06\\%, and 9.83\\% improvement respectively compared to the original word embeddings.\n\n\\begin{table*}[htbp]\n  \\centering\n  \\caption{P-value of one-tailed Student's t-test of three experiments}\n\\scalebox{0.77}{%\n            \\begin{tabular}{lrrrrrrrrrrrr}\n            \\toprule\n          & \\multicolumn{4}{c}{\\textbf{Word Similarity}} & \\multicolumn{4}{c}{\\textbf{Semantic Textual Similarity}} & \\multicolumn{4}{c}{\\textbf{Sentiment Analysis}} \\\\\n           \\cmidrule(r){2-5} \\cmidrule(r){6-9} \\cmidrule(r){10-13}\n          & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{SB}} & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{SB}} & \\multicolumn{1}{l}{\\textbf{Orig.}} & \\multicolumn{1}{l}{\\textbf{ABTT}} & \\multicolumn{1}{l}{\\textbf{CN}} & \\multicolumn{1}{l}{\\textbf{SB}} \\\\\n          \\toprule\n    \\textbf{WORD2VEC} & \\textbf{2.51e-02} & 3.56e-01 & 3.29e-01 & 3.38e-01 & \\textbf{2.92e-03} & \\textbf{1.12e-03} & \\textbf{2.22e-03} & \\textbf{1.42e-03} & 9.27e-02 & \\textbf{1.35e-03} & \\textbf{3.84e-03} & \\textbf{2.49e-04} \\\\\n    \\textbf{GLOVE} & \\textbf{6.85e-03} & 1.83e-01 & 2.30e-01 & \\textbf{7.02e-03} & \\textbf{2.88e-05} & \\textbf{1.35e-03} & \\textbf{5.49e-04} & \\textbf{5.51e-06} & 4.02e-01 & 4.58e-01 & 1.25e-01 & 1.28e-01 \\\\\n    \\textbf{PARAGRAM} & \\textbf{4.86e-03} & 7.13e-02 & \\textbf{1.62e-02} & 5.13e-02 & \\textbf{5.35e-07} & \\textbf{1.17e-07} & \\textbf{5.94e-07} & \\textbf{3.69e-07} & \\textbf{1.23e-04} & \\textbf{3.32e-04} & \\textbf{5.62e-04} & \\textbf{1.20e-05} \\\\\n    \\bottomrule\n    \\end{tabular}%\n\n    }\n  \\label{tab:t_test}%\n\\end{table*}%\n\n\\subsection{Downstream Task: Sentiment Analysis} \n\nSince the success of intrinsic lexical evaluation tasks does not imply success on downstream tasks, we test the performance of HSR on four sentiment analysis tasks. The dataset we adopt include Amazon reviews\\footnote{\\url{https:\/\/www.kaggle.com\/bittlingmayer\/amazonreviews\\#train.ft.txt.bz2}} (AR), customer reviews (CR) \\cite{hu2004mining}, IMDB movie reviews (IMDB) \\cite{maas2011learning}, and SST binary sentiment classification (SST-B) \\cite{socher2013recursive}, which are all binary sentence-level sentiment classification tasks. Sentiment analysis is an important task in NLP which has been widely applied in business areas such as e-commerce and customer service. \n\nSimilar to the STS tasks, we first tokenize the sentence, then average the corresponding word embeddings as the vector representation of the sentence. We use a logistic regression model trained by minimizing cross-entropy loss to classify the sentence embeddings into positive or negative emotions. This procedure was adopted in previous studies such as \\citet{zeng2017socialized}. We report the five-fold cross-validation accuracy of the sentiment classification results in Table \\ref{tab:sentiment}.\n\nFrom Table \\ref{tab:sentiment}, we could observe that HSR-RR has the best downstream-task performance among all the tested postprocessing methods. Specifically, for Paragram, HSR-RR achieves the highest classification accuracy on all four tasks; for Word2Vec and GloVe, HSR-RR performs the best on three out of the four tasks.\n\n\\subsection{Statistical Significance Test}\n\nTo show that our proposed method yields significant improvement compared to the baselines, we employ the one-tailed Student's t-test. The p-value of the t-test of HSR-RR against other methods for all three experiments is shown in Table \\ref{tab:t_test} in scientific notation. We use the convention that a p-value is significant if it is smaller than 0.05, and all significant p-values are marked in bold.\n\nFrom Table \\ref{tab:t_test}, we observe that on word similarity and STS tasks, the improvements yielded by HSR are significant when compared to all three original word vectors. On sentiment analysis tasks, the improvement on Paragram is significant. We also test the significance of improvement of results yielded by HSR-RR with those yielded by other state-of-the-art baseline methods (ABTT, CN, and SB). We find that, for STS tasks, improvements against all three baseline methods on all three word vectors are significant; for sentiment analysis, the improvements against all three baseline methods on Word2Vec and Paragram are significant; for word similarity, only two results (SB on GloVe and CN on Paragram) are significant. While in other cases, improvements of HSR-RR over the original word vectors and the baseline algorithms are not significant, conversely, the baseline methods and the original word vectors also fail to surpass the performance of HSR-RR when the null hypothesis and alternative hypothesis are switched. Therefore, we conclude that HSR-RR yields solid improvement when compared to the original word vectors, and it is either significantly better or on-par with other state-of-the-art baseline methods.\n\nWe want to remark that, while statistical significance tests are useful for algorithm comparison, it is mostly excluded in previous word vector evaluation papers \\cite{Bullinaria2007,Levy2015,Faruqui2015,Fried2015,Mrksic2016,Mrksic2017,Shiue2017,Mu2018,Liu2019,tang2014learning}, and there could be a valid reason for this. As pointed out by \\citet{dror2018hitchhiker}, the way how existing NLP datasets are structured tends to cripple those widely adopted significance tests: while most statistical significance tests (e.g., t-test) assume that the test set consists of independent observations, NLP datasets usually violate this hypothesis. For instance, most STS datasets only contain sentences from a certain source (e.g., news or image captions) and word similarity datasets usually contain words of specialized types (e.g., verbs). This makes a proper significance test quite challenging. Some NLP researchers even contend to abandon the null hypothesis statistical significance test approach due to this hard-to-meet assumption \\cite{koplenig2019against,mcshane2019abandon}.\n\n\\section{Conclusion and Future Work}\n\nIn this paper, we present a simple, fast-to-compute, and effective framework for postprocessing word embeddings, which is inspired by the recent development of causal inference. Specifically, we employ Half-Sibling Regression to remove confounding noise contained in word vectors and to reconstruct latent, ``clean'' word vectors of interest. The key enabler of the proposed Half-Sibling Regression is the linguistic fact that function words and content words are lexically irrelevant to each other, making them natural ``half-siblings''. The experimental results on both intrinsic lexical evaluation tasks and downstream sentiment analysis tasks reveal that the proposed method efficiently eliminates noise and improves performance over the existing alternative methods on three different brands of word embeddings.\n\nThe current work has a few limitations, which we wish to address in the future. The first limitation resides in the way we formulate the regression. Note that, when performing the multiple-output regression step in HSR algorithm (Step 1.1 and Step 2.1 of Algorithm \\ref{alg:hsr}), we do not take the correlation of targets into account. Such correlations, however, could be beneficial in some cases. Consider, for instance, the task of predicting content words based on stop words (Step 1.1 of Algorithm \\ref{alg:hsr}). As content words as targets are strongly correlated (e.g., synonyms and antonyms), such correlations can be further employed to facilitate the regression with well-studied methods such as Reduced-rank regression \\cite{Anderson1949}. For a survey of these multiple outcome regression methods taking output into account, please see \\citet{Hastie2001}, Section 3.7.\n\nThe second line of future work concerns how to use a non-linear model for HSR more effectively. Although we have tried neural-network-based HSR algorithms for various tasks (see the appendix for details), empirically they bring marginally improved results, if not slightly worsened. One hypothesis for explaining this phenomenon is that neural networks tend to be highly expressive, overfitting small datasets easily. For future work, we plan to explore more regularization methods which may improve the results of neural-network-based HSR. \n\nThe third line of future work is to develop a unified framework for understanding word vector postprocessing. As various word vector postprocessing algorithms yield (sometimes surprisingly) similar results in a few cases, it is our hope to establish connections between these approaches in the future. The recent work by \\citet{zhou2019getting} points toward this direction.\n\nLast but not least, we believe that there remain ample opportunities for using HSR in other NLP tasks and models. For instance, recently, we have observed that pre-trained language models such as BERT \\cite{devlin2019bert} start to replace word vectors as default feature representations for downstream NLP tasks. The HSR framework, in principle, can be incorporated in language model postprocessing pipelines as well. We would like to explore these possibilities in the future.\n\n\n\\paragraph{Acknowledgement} This work was partially supported by the National Natural Science Foundation of China (grant number 71874197). We appreciate the anonymous reviewers for their detailed and constructive comments. We thank all the people who helped Zekun Yang flee from Hong Kong to Shenzhen on Nov. 12th, 2019 such that she could safely finish writing the camera-ready version of this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nProbabilistic Gaussian processes (GPs)~\\cite{Rasmussen2006} are the method of choice for probabilistic regression: Their non-parametric nature allows for flexible modelling without specifying low-level assumptions (e.g., the degree of a polynomial) in advance. Moreover, for the standard Gaussian likelihood, inference can be performed in closed form in a principled way simply by applying Bayes' theorem. GPs have had substantial impact in various research areas, including geostatistics~\\cite{Cressie1993}, optimisation~\\cite{Jones1998,Brochu2009}, data visualisation~\\cite{Lawrence2005}, robotics and reinforcement learning~\\cite{Deisenroth2014}, spatio-temporal modelling~\\cite{Luttinen2012}, and active learning~\\cite{Krause2008}. A strength of the GP model is that it can be used  fairly reliably as a black-box function approximator, i.e., it produces reasonable predictions without manual parameter tuning.\n\n\nA practical limitation of the GP model is its computational demand: In\nstandard implementations, training and predicting scale in\n$\\mathcal{O}(N^3)$ and $\\mathcal{O}(N^2)$, respectively, where $N$ is\nthe size of the training data set. For large $N$ (e.g., $N>10,000$) we\noften use sparse approximations~\\cite{Williams2001,Quinonero-Candela2005,\n  Hensman2013, Titsias2009, Lazaro-Gredilla2010, Shen2006}. Typically, these sparse approximations lower the computational burden by implicitly (or explicitly)  using a subset of the data. They scale GPs up to multiple tens or hundreds of thousands of data points. However, even with sparse approximations it is inconceivable to apply GPs to data set sizes of tens or hundreds of millions of data points. \n\nAn alternative to sparse approximations is to distribute the computations by using local models. These local models typically require stationary kernels for a notion of ``distance'' and ``locality''. \nShen et al.~\\cite{Shen2006} used KD-trees to recursively partition the data space by means of a multi-resolution tree data structure, which allows to scale GPs up to multiple tens of thousands of training points. However, the approach proposed in~\\cite{Shen2006} does not provide solutions for variance predictions and is limited to stationary kernels. Similarly, \\cite{Nguyen-Tuong2009a} used a heuristic partitioning scheme based on the locality notion of stationary kernels for real-time mean predictions of GPs.\nAlong the lines of exploiting locality, mixture-of-experts (MoE) models~\\cite{Jacobs1991} have been applied to GP regression~\\cite{Meeds2006,Rasmussen2002,Yuan2009}. However, these models have not primarily been  used to speed up GP regression, but rather to allow for heteroscedasticity (input-dependent variations) and non-stationarity. Each local model possesses its own set of hyper-parameters to be optimised. Predictions are made by collecting the predictions of all local expert models, and weighing them using the responsibilities assigned by the gating network. In these MoE models, a Dirichlet process prior is placed on the multinomial ``responsibility'' vector of each local expert, which allows for a data-driven partitioning on the fly. Unfortunately, inference in these models is  computationally intractable, requiring MCMC sampling or variational inference to assign data points to each GP expert, a computationally demanding process.\n\nWithin the context of spatio-temporal models with $10^6$ data points, \\cite{Luttinen2012} propose efficient inference that exploits computational advantages from both separable and compactly supported kernels, leading to very sparse kernel matrices. The authors propose approximate (efficient) sampling methods to deal with both noisy and missing data.\n\nRecently, Gal et al.~\\cite{Gal2014} proposed a distributed approach that scales variational sparse GPs further by exploiting distributed computations. In particular, they derive an exact re-parameterisation of the variational sparse GP model by Titsias~\\cite{Titsias2009}, to update the variational parameters independently on different nodes. This is implemented within a Map-Reduce framework, and the corresponding architecture consists of a central node and many local nodes, i.e., a single-layer architecture.\n\nIn this paper, we follow an approach orthogonal to sparse approximations in order to \\emph{scale full GPs to large data sets} by exploiting massive parallelisation. In particular, we propose a hierarchical mixture of experts model that distributes the computational load amongst a large set of independent computational units.  Our model recursively recombines computations by these independent units to an overall GP inference\\slash training procedure. This idea is related to Tresp's Bayesian Committee Machine~\\cite{Tresp2000}, which combines estimators. A key advantage of our model is that all computations can be performed analytically, i.e., no sampling is required. Given sufficient computing power our model can handle arbitrarily large data sets. In our experiments we demonstrate that our model can be applied easily to data sets of size $2^{24}\\approx 1.7\\times 10^7$, which exceeds the typical data set sizes sparse GPs deal with by orders of magnitude. However, even with limited computing resources, our model is practical in the sense that it can train a full GP with a million training points in less than half an hour on a laptop.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Objective and Set-up}\nWe consider a regression setting $y=f(\\vec x)+\\epsilon$, where $\\vec x\\in\\mathds{R}^d$ and $\\epsilon\\sim\\gauss{0}{\\sigma_\\epsilon^2}$ is i.i.d. Gaussian noise. The objective is to infer the latent function $f$ from a training data set $\\boldsymbol X = \\{\\vec x_i\\}_{i=1}^n, \\vec y = [y_1, \\dotsc, y_N]^\\top$. For small data set sizes $N$, a Gaussian process is one of the methods of choice for probabilistic non-parametric regression.\n\n\nA Gaussian process is a non-parametric Bayesian model, which is often used for (small-scale) regression. A GP is defined as a collection of random variables, any finite number of which is Gaussian distributed. A GP is fully specified by a mean function and a covariance function (kernel). In this paper, we assume that the prior mean function is 0. Furthermore, we consider the Gaussian (squared exponential) kernel\n\\begin{align}\n\\hspace{-2.6mm}k(\\vec x_i, \\vec x_j) = \\sigma_f^2\\exp\\big(-\\tfrac{1}{2}(\\vec x_i - \\vec x_j)^\\top\\boldsymbol\\Lambda^{-1}(\\vec x_i-\\vec x_j)\\big),\n\\end{align}\n$\\vec x_i,\\vec x_j\\in\\mathds{R}^D$, where $\\sigma_f^2$ is the variance of the latent function $f$ and $\\boldsymbol\\Lambda = \\mathrm{diag}(l_1^2,\\dotsc,l_D^2)$ is the diagonal matrix of squared length-scales $l_i$, $i = 1,\\dotsc,D$. Furthermore, we consider a Gaussian likelihood $p(y|f(\\vec x))=\\gauss{f(\\vec x)}{\\sigma_\\epsilon^2}$ to account for the measurement noise. \n\nA GP is typically trained by finding hyper-parameters $\\vec\\theta = \\{\\sigma_f, l_i, \\sigma_\\epsilon\\}$ that maximise the log-marginal likelihood~\\cite{Rasmussen2006}, which is (up to a constant) given by\n\\begin{align}\n\\log p(\\vec y|\\boldsymbol X, \\vec\\theta) &\\stackrel{.}{=} -\\tfrac{1}{2}\\big(\\vec y(\\boldsymbol K+\\sigma_\\epsilon^2\\boldsymbol I)^{-1}\\vec y + \\log|\\boldsymbol K+\\sigma_\\varepsilon^2\\boldsymbol I|\\big)\\,.\n\\label{eq:log-marginal likelihood}\n\\end{align}\nThe computationally demanding computations in~\\eqref{eq:log-marginal likelihood} are the inversion and the determinant of \\mbox{$\\boldsymbol K + \\sigma_\\epsilon^2\\boldsymbol I$}, both of which scale in $\\mathcal{O}(N^3)$ with a standard implementation.\n\nFor a given set of hyper-parameters $\\vec\\theta$, a training set $\\boldsymbol X, \\vec y$ and a test input $\\vec x_*\\in\\mathds{R}^D$, the GP posterior predictive distribution of the corresponding function value $f_* = f(\\vec x_*)$ is Gaussian with mean and variance given by\n\\begin{align}\n\\mathds{E}[f_*] &= m(\\vec x_*) = \\vec k_*^\\top(\\boldsymbol K + \\sigma_\\varepsilon^2\\boldsymbol I)^{-1} \\vec y\\,,\\label{eq:mean GP}\\\\\n\\mathrm{var}[f_*] &= \\sigma^2(\\vec x_*) = k_{**} -\\vec k_*^\\top(\\boldsymbol K + \\sigma_\\varepsilon^2\\boldsymbol I)^{-1} \\vec k_*\\,,\\label{eq:var GP}\n\\end{align}\nrespectively, where $\\vec k_* = k(\\vec x_*, \\boldsymbol X)$ and $k_{**}=k(\\vec x_*, \\vec x_*)$. When we cache $(\\boldsymbol K + \\sigma_\\epsilon^2\\boldsymbol I)^{-1}$ computing the mean and variance in~\\eqref{eq:mean GP} and \\eqref{eq:var GP} requires $\\mathcal{O}(N)$ and $\\mathcal{O}(N^2)$ computations, respectively.\n\nFor $N>10,000$ training and predicting become rather time-consuming procedures, which additionally require large amounts of memory, i.e., $\\mathcal{O}(N(N+D))$. \n\nOur working hypothesis is that a standard GP can model the latent function $f$. However, due to the data set size $N$ the standard GP is not applicable.\n\nInstead of a sparse approximation~\\cite{Quinonero-Candela2005, Titsias2009}, we address both the computational and the memory issues of full GPs by distributing the computational and memory load to many individual computational units that only operate on subsets of the data. This results in an approximation of the full GP, but this approximation can be computed efficiently (time and memory) by exploiting massive parallelisation. \n\n\n\\section{Hierarchical Mixture-of-Experts Model}\n\\begin{figure*}[tb]\n\\subfigure[Single-layer model.]{\n\\includegraphics[height = 2.45cm]{.\/figures\/1layer}\n\\label{fig:1layer}\n}\n\\hspace{15mm}\n\\subfigure[Two-layer model.]{\n\\includegraphics[height = 4cm]{.\/figures\/2layer}\n\\label{fig:2layer}\n}\n\\caption{Hierarchical MoE model. Main computations are at the leaf nodes (black). All other nodes (linearly) recombine information from their direct children, allowing for an arbitrarily deep architecture.}\n\\end{figure*}\n\n\nConsider a GP with a training data set $\\mathcal{D} = \\{\\boldsymbol X, \\vec y\\}$.  We define a set $\\mathcal{S}$ of $c$ subsets (not necessarily a partition) of the data set as $\\mathcal{S} = \\{\\mathcal{D}^\\idx{1},\\dotsc,\\mathcal{D}^\\idx{c}\\}$ where $\\mathcal{D}^{(i)} = \\{ \\boldsymbol{X}^{(i)} , \\vec{y}^{(i)} \\}$. These subsets are from the full training set  $\\mathcal{D}$, and we will use a GP on each of them as a (local) expert\\footnote{The notion of ``locality'' is  misleading as our model does not require similarity measures induced by stationary kernels.}. Each of these local expert models computes means and variances conditioned on their respective training data\\footnote{Both mean and variances are necessary for training and inference.}. These (local) predictions are recombined to a mean\\slash variance by a parent node (see Fig.~\\ref{fig:1layer}), which subsequently can play the role of an expert at the next level of the model architecture. Recursively applying these recombinations, our model results in a tree structured architecture with arbitrarily many layers, see Fig.~\\ref{fig:2layer}.\\footnote{We discuss different architecture choices in Section~\\ref{sec:architectures}.}\nIn our model, all GPs at the leaves of this tree are trained jointly and share a single set of hyper-parameters $\\vec\\theta$.\n\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width = 0.9\\hsize]{.\/figures\/block-diagonal2}\n\\caption{In a single-layer MoE model, partitioning the data leads to a block-diagonal approximation of the kernel matrix (top path). By duplicating data points this clear separation between blocks is smoothed out (bottom path), and the effects of the independence assumption are mitigated.}\n\\label{fig:kernel matrix approx}\n\\end{figure*}\n\nWe train our model and make predictions under the assumption that each expert is independent of the other experts at the same level of the tree, which allows us to parallelise and distribute computations over independent processing units.\nThe assumption that the GPs at each level are independent effectively results in a GP with a block-diagonal covariance matrix, which can be efficiently inverted by distribution. If $\\mathcal{S}$ is a partition of $\\mathcal{D}$, this covariance matrix would be composed of the block-diagonal elements of the original GP's covariance matrix (with zeros on the off block-diagonals). However, with a partition, some information about the structure of the data is lost, which would otherwise be captured by the cross-covariance terms in the full GP.  Since our model aims to replicate a full GP we are interested in mitigating the effects of the independence assumption. We do so by sharing arts of the training set amongst multiple subsets in $\\mathcal{S}$. Thereby, we account for the covariance effects between the points in $\\mathcal{D}$ to some degree. This approach is illustrated in the bottom path of Fig.~\\ref{fig:kernel matrix approx}, where parts of the training set are shared amongst individual nodes. Note that memory consumption can be kept constant since the training set is not modified (read-only access).\n\n\n\\begin{comment}\nWe train our model and make predictions under the assumption that each expert is independent of the other experts at the same level of the tree, which allows us to parallelise and distribute computations over independent processing units. \nIf $\\mathcal{S}$ is a partition of $\\mathcal{D}$ this implies that the data in each subset is independent of the data in every other subset. This effectively results in a block-diagonal approximation of the kernel matrix (which can be efficiently inverted), see top path of Fig.~\\ref{fig:kernel matrix approx}. However, some information about the structure of the data  is lost\\footnote{This information would otherwise be captured by the cross-covariance terms in the full GP}.  Since our model aims to replicate a standard GP we are interested in mitigating the effects of the independence assumption. We do so by having each of the subsets in $\\mathcal{S}$ \\emph{overlap} (i.e., share parts of the training set), thereby allowing the individual child-GPs to account to some degree for the covariance effects between the points in $\\mathcal{D}$. This is illustrated in the bottom path of Fig.~\\ref{fig:kernel matrix approx}, where parts of the training set are shared amongst individual nodes. Note that the duplication of memory can be prevented since the training set is not modified (read-only access). For further details we refer to~\\cite{Ng2014}.\n\\end{comment}\n\n\\subsection{Dividing the Data into Subsets}\nCreating subsets of the training data and using each subset with a GP forms the basis of our hierarchical MoE GP (HGP) model, where we have divided the problem into a number of GPs, each using a smaller training data set.  This can be done recursively, further subdividing the problem until a desired training set size for the leaf-GPs is achieved.\\footnote{\nThe data set sizes assigned to the leaves can be chosen depending on the computational resources available.\n}\nFor an efficient implementation, the number $c$ of data subsets $\\mathcal{D}^\\idx{k}$ should correspond to a multiple of the number of computing units available. For disjoint data sets $\\mathcal{D}^\\idx{k}$, every leaf-GP would possess $N\/c$ data points; if we allow for shared data sets this number scales linearly with the degree of sharing. For instance, if every data point appears in two local experts  in a single-layer model, we would have each $\\mathcal{D}^\\idx{k}$ of size $2N\/c$.\n\nThere are various ways of assigning data points to the experts at the leaves of the tree in Fig.~\\ref{fig:2layer}. For instance, random assignment is fast and can work quite well. Most of our results, however are based on a different approach: First, we use a KD-tree to recursively divide the input space into non-overlapping regions. We terminate the division when the required number of partitions is reached. Second, each region is then partitioned into $p$ disjoint groups of inputs. Third, we construct each data set $\\mathcal{D}^\\idx{k},\\,k = 1,\\dotsc, c$, for the local experts, such that it contains exactly one group from each region.  After this procedure, each data set $\\mathcal{D}^\\idx{k}$ will contain points across the entire input space, rather than being clustered in the same region in the input space. Note that neither method for assigning data points to GP experts relies on any locality notion induced by the kernel.\n\n\n\n\n\\subsection{Training}\nWe train the model by maximising a factorising approximation to the log-marginal likelihood in~\\eqref{eq:log-marginal likelihood}, i.e.,\n\\begin{align}\n\\log p(\\vec y| \\boldsymbol X, \\vec\\theta) \\approx \\sum\\nolimits_k \\log p(\\vec y^\\idx{k}|\\boldsymbol X^\\idx{k}, \\vec\\theta)\n\\label{eq:approximate log-marginal likelihood}\n\\end{align}\nwith respect to the kernel hyper-parameter $\\vec\\theta$, which are shared amongst all individual GPs. The factorising approximation in~\\eqref{eq:approximate log-marginal likelihood} is implied by our independence assumption of the individual (small) GP models. Each term in~\\eqref{eq:approximate log-marginal likelihood} is given by \n\\begin{align}\n\\hspace{-2mm}\\log p(\\vec y^\\idx{k}|\\boldsymbol X^\\idx{k},\\vec\\theta) &= -\\tfrac{1}{2}\\vec y^\\idx{k}(\\boldsymbol K_{\\vec\\theta}^\\idx{k} + \\sigma_\\epsilon^2\\boldsymbol I)^{-1}\\vec y^\\idx{k} \\nonumber\\\\\n&\\quad -\\tfrac{1}{2}\\log |\\boldsymbol K_{\\vec\\theta}^\\idx{k} + \\sigma_\\epsilon^2\\boldsymbol I| + \\text{const}\n\\label{eq:term in approx. LML}\n\\end{align}\nand requires the inversion and determinant of $\\boldsymbol K_{\\vec\\theta}^\\idx{k} +\\sigma_\\epsilon^2\\boldsymbol I$, where $\\boldsymbol K_{\\vec\\theta}^\\idx{k} = k(\\boldsymbol X^\\idx{k}, \\boldsymbol X^\\idx{k})$ is a $p\\times p$ matrix, and $p$ is the size of the data set associated with the GP expert $k$. These computations can be performed in $\\mathcal{O}(p^3)$ time with a standard implementation. Note that $p$ is significantly smaller than $N$, i.e., the  size of the full data set. The memory consumption is $\\mathcal{O}(p^2 + pD)$ for each individual model.\n\nNote that in~\\eqref{eq:approximate log-marginal likelihood} the number of parameters to be optimised is relatively small since we do not consider additional variational parameters or inducing inputs that we optimise. The gradients of~\\eqref{eq:approximate log-marginal likelihood} and~\\eqref{eq:term in approx. LML} with respect to $\\vec\\theta$ can be computed in independently at all $k$ nodes, which allows for massive parallelisation and a significant speed-up of training compared to training a full GP. \n\n\n\\begin{figure}[tb]\n\\centering\n \\includegraphics[width = \\hsize]{.\/figures\/hgp_timings}\n\\caption{Computing time for the log-marginal likelihood and its gradient with respect to the kernel hyper-parameters as a function of the size of the training data. The HGP scales favourably to large-scale data sets. With an increasing number of child-GPs (but fixed computational resources), the HGP scales to more than $10^7$ data points.}\n\\label{fig:hgp timings}\n\\end{figure}\nTo evaluate the training time for our GP model, we computed the amount of time required to compute the log-marginal likelihood and its gradients with respect to the kernel hyper-parameters. A typical optimisation procedure for the kernel hyper-parameters, e.g., conjugate gradients or (L)BFGS, requires these values. The full training time is proportional to the time it takes to compute the log-marginal likelihood and its gradient (it still depends on the number of line-searches). We chose a computer architecture of 64 nodes with 4 cores each. Furthermore, we chose a three-layer model with varying widths (branching factors). For data sets of $\\leq 2^{20}$ data points the leaf nodes possessed 512 data points each, for data set sizes of $>2^{20}$, we chose the number of data points per node to be 128.\n\nFig.~\\ref{fig:hgp timings} shows the time required for computing the log-marginal likelihood and its gradient with respect to the hyper-parameters. The horizontal axis shows the size of the training set (logarithmic scale), the left vertical axis shows the computation time in seconds (logarithmic scale) for our model (HGP, blue-dashed), a full GP (red-dashed) and a sparse GP with inducing inputs~\\cite{Snelson2006} (green-dashed). For the sparse GP model, we chose the number $M$ of inducing inputs to be 10\\% of the size of the training set, i.e., the computation time is of the order of $\\mathcal{O}(NM^2)=\\mathcal{O}(N^3\/100)$, which offsets the curve of the full GP. Taking even fewer inducing inputs (e.g., 1\\% or 0.1\\% of the data) would push the sparse approximation towards multiple-hundred thousand data points. However, this can only be done if the data set possesses a high degree of redundancy. The right vertical axis shows the number of leaf GPs (black-solid), i.e., the number of GP experts amongst which we distribute the computation.  While the training time of the full GP reaches impractical number at data set sizes of about 10,000, the sparse GP model can be reasonably trained up to 50,000 data points.\\footnote{In this comparison we did not include any computational overhead for selecting the inducing inputs (either as variational parameters or as free parameters to be optimised), which is often non-negligible.} The computational time required for the HGP to compute the marginal likelihood and gradients is significantly lower than that of the full GP, and we scaled it up to $2^{24} \\approx 1.7 \\times 10^7$ training data points, which required about the same amount of time ($\\approx 230$\\,s) for training a full GP with $2^{14}\\times 10^4$ and a sparse GP with $2^{15}\\approx 3.2\\times 10^4$ data points. The figure shows that for any problem size, we are able to find an architecture that allows us to compute the marginal likelihood (hence, train the model) within a feasible amount of time. \n\n\n\nEven if a big computing infrastructure is not available, our model is useful in practice: We performed a full GP training cycle (which includes many evaluations of the log-marginal likelihood and its gradient) with $10^6$ data points on a standard laptop in about 20 minutes. This is also a clear indicator that the memory consumption of the HGP is relatively small.\n\n\n\n\n\n\n\\subsection{Predictions\/Inference}\nThe predictive distribution is computed by an iterative recombination of the computations at the leaf-GPs. In particular, the parent nodes compute\n\\begin{align}\np(y_*|\\vec x_*, \\boldsymbol X, \\vec y) &\\propto \\prod\\nolimits_k p(y_*|\\vec x_*, \\boldsymbol X^\\idx{k}, \\vec y^\\idx{k}) \\label{eq:predictive_likelihood}\\\\\n&\\propto\\gaussx{y_*}{\\mu_*}{\\sigma_*^2}\\,,\\label{eq:predictive_distribution} \n\\end{align}\nas the product of all Gaussians passed on by the children. The resulting distribution is also Gaussian with\nmean $\\mu_*$ and variance $\\sigma_*^2$, where\n\\begin{align*}\n\\mu_* = \\sigma_*^{2}\\sum\\nolimits_k \\frac{\\mu_k(\\vec x_*)}{\\sigma_k^2(\\vec x_*)}\\,,\\quad\n\\sigma_*^2 = \\left(\\sum\\nolimits_k \\sigma_k^{-2}(\\vec x_*)\\right)^{-1}.\n\\end{align*}\nBoth $\\mu_*$ and $ \\sigma_*^2$ are computed analytically. We exploit the distributed architecture of our model to ensure small computational overhead. The mean of the HGP predictive distribution can be written as a weighted sum of the means of the child-GPs' predictive distributions. $\\mu_* = \\sum\\nolimits_k w_k \\mu_k(\\vec x_*)$ where\n$w_k = \\sigma_k^{-2}(\\vec x_*)\/\\sum\\nolimits_j \\sigma_j^{-2}(\\vec x_*)$. The weights on the child-GPs' predictions are proportional to the inverse variances of their prediction, which allows more accurate predictions (lower variances) to have bigger weights in the final prediction, and less accurate predictions (higher variances) to have weights closer to zero. This, in general, allows the HGP to remain effective across various methods of assigning data points to the child-GPs.\n\n\\subsection{Architecture Choice}\n\\label{sec:proofs}\nThus far, we have described the single-level version of the HGP model, where\nthe child-GPs are standard GPs. Since the HGP possesses the same ``interface''\n(marginal likelihood, predictive distribution) as a standard GP, we can use\nHGPs themselves as the child-GPs of an HGP. This can be done recursively to build up a tree\nstructure of arbitrary depth and width. \n\nIn the following, we show that a multi-level HGP is\nequivalent to a single-level HGP if its children (experts) are the leaf-GPs (experts) of a\nmulti-level HGP. For this, we show that training and prediction are identical in both models.\n\n\\paragraph{Training} In \\eqref{eq:approximate log-marginal likelihood}, we\nexpressed the log-marginal likelihood of the HGP as a sum of the log marginal\nlikelihoods of its child-GPs. If the child-GPs themselves are also HGPs (``child-HGPs''),\nthen this sum can be expanded and expressed as the sum of the log-marginal\nlikelihood of the child-GPs of the child-HGPs. This generalises to HGPs of\nan arbitrary number of levels, and we can always write the log-marginal likelihood\nof a HGP as a sum of the log-marginal likelihood of its leaf GPs (experts):\n\\begin{align}\n\\log p(\\vec y| \\boldsymbol X, \\vec\\theta) &\\approx \\sum\\nolimits_k \\log p(\\vec y^\\idx{k}|\\boldsymbol X^\\idx{k}, \\vec\\theta) \\nonumber \\\\\n&\\approx \\sum\\nolimits_k \\sum\\nolimits_{i_k} \\log p(\\vec y^\\idx{i_k}|\\boldsymbol X^\\idx{i_k}, \\vec\\theta) \\nonumber \\\\\n&\\approx \\cdots \\nonumber \\nonumber \\\\\n&\\approx \\sum\\nolimits_{l \\in leaves} \\log p(\\vec y^\\idx{l}|\\boldsymbol X^\\idx{l}, \\vec\\theta)\n\\label{eq:hgp_lik_expansion}\n\\end{align}\nEquation~\\eqref{eq:hgp_lik_expansion} shows that the log-marginal likelihood of a multi-level HGP\nis the sum of the log marginal-likelihoods of its leaves, which is equivalent\nto the log-marginal likelihood of a single-level HGP with the (multi-level HGP's) leaves as its child-GPs (experts). Hence, the structure of the HGP above the leaves has no effect on the computation of the log-marginal likelihood.\n\n\\paragraph{Prediction}\nWe now show that the predictions of a multi-level HGP and a single-level HGP are identical. The product in \\eqref{eq:predictive_likelihood} can be factorised\\footnote{This is not strictly a factorisation, since it is only proportional to \\eqref{eq:predictive_likelihood}. While it is not crucial to our application, we can easily recover the normalising constant by integration.} into\n$\\prod\\nolimits_{l \\in leaves} p(y_*|\\vec x_*, \\boldsymbol X^\\idx{l}, \\vec y^\\idx{l})$,\na product of terms involving only the leaf-GPs (experts) and not containing terms relating to the intermediate levels.\n\nIt is, however, not immediately obvious that the predictive distribution in\n\\eqref{eq:predictive_distribution} is equivalent to the predictive distribution of a single-level HGP with the same leaves. To show this, we provide a simple proof that the resulting distribution of the product of an arbitrary number of Gaussians, which are in turn the product of Gaussians (we refer to them as the ``sub-Gaussians''), is Gaussian and equivalent to the distribution resulting from the product of all the sub-Gaussians.\n\nThe product of Gaussians is proportional to a Gaussian, i.e.,  \n$\\prod\\nolimits_k \\gaussx{x}{\\mu_k}{\\sigma_k^2} \\propto\n\\gaussx{x}{\\mu_*}{\\sigma_*^2}\n$ where\n$\\mu_* = \\sigma_*^2\\sum\\nolimits_k \\mu_k\\sigma_k^{-2}$\nand $\\sigma_*^2 = (\\sum\\nolimits_k \\sigma_k^{-2})^{-1} $.\nSuppose each of the component Gaussians are themselves product of Gaussians.\nThat is,\n$\\gaussx{x}{\\mu_k}{\\sigma_k^2}\n\\propto\n\\prod\\nolimits_{i_k} \\gaussx{x}{\\mu_{i_k}}{\\sigma_{i_k}^2}\n$ and\n$\\mu_k = \\sigma_k^2\\sum\\nolimits_{i_k} \\mu_{i_k}\\sigma_{i_k}^{-2}$\nand $\\sigma_k^2 = (\\sum\\nolimits_{i_k} \\sigma_{i_k}^{-2})^{-1} $. Then\n\\begin{align}\n\\mu_* &= \\sigma_*^2\\sum\\nolimits_k \\mu_k\\sigma_k^{-2}\n= \\sigma_*^2\\sum\\nolimits_k \\sum\\nolimits_{i_k} \\mu_{i_k}\\sigma_{i_k}^{-2}\\,,\\\\\n\\sigma_*^2 &= \n\\big(\\sum\\nolimits_k \\sigma_k^{-2}\\big)^{-1} =\n\\big(\\sum\\nolimits_k \\sum\\nolimits_{i_k} \\sigma_{i_k}^{-2} \\big)^{-1}\\,,\n\\end{align}\n(where $i_k$ are the indices corresponding to the children of the child GPs, and so on)\ni.e., the distribution from the product of the Gaussian distributions is equivalent to the distribution from the product of the sub-Gaussians.\nThis result generalises to any number of levels of Gaussian products (if the sub-Gaussians are derived as products of ``sub-sub-Gaussians'' we apply the above again), and completes our proof for the equivalence of a multi-level HGP and a single-level HGP with the same leaves (experts).\n\nTherefore, mathematically it does not matter whether to choose a shallow or deep architecture if the leaf GPs (experts) are the same. However, a multi-level HGP still makes sense from a computational point of view.\n\n\\paragraph{Multi-level HGPs}\nGiven the same leaf-GPs, the depth of the HGP has no effect on the model, for both training and prediction, as shown in Section~\\ref{sec:proofs}. \nAlthough it is mathematically not necessary to construct an HGP of more than one level, in practice, a multi-level HGP allows us to fully utilise a given set of distributed computing hardware. To implement a single-level HGP on a distributed system, we have one ``master'' node, which is responsible for the computational work of the HGP (combining the outputs of the child-GPs). The computational work of the child-GPs is distributed evenly across the other ``slave'' nodes. Such a set-up imposes a heavy communication and computational load on the master node, since it has to manage its communication with all slave nodes, and perform the computations required for combining the child-GPs on its own (during which the slave nodes will idle). This is not an optimal use of resources, and we exploit the fact that the HGP model is invariant to the presence of intermediate layers to propose a better solution, which is illustrated in Fig.~\\ref{fig:arch}. Starting from the top of the HGP tree, divide the number of computational nodes available into $c$ groups where $c$ is the number of child-GPs that the HGP  possesses, and assign each child-GP\/child-HGP to one group. We do this recursively until we reach the leaves of the HGP or until there is only a single node available to the HGP. This approach leads to a more uniform distribution of network communication and computational load amongst all nodes. \n\n\n\\begin{figure}\n\\includegraphics[width = \\hsize]{.\/figures\/arch_new}\n\\caption{The flexibility of choosing amongst equivalent architectures enables the HGP to evenly distribute computational (and network communication) work. Each blue vertical rectangle represents one distributed computing unit while each coloured node denotes an HGP. The coloured rectangles represent the overall responsibility of the corresponding coloured nodes. The overlap between the coloured rectangles and the blue represent the computing resources available for the computational work related to a particular HGP. The main computations are performed at the leaf-experts (black).}\n\\label{fig:arch}\n\\end{figure}\n\n\n\\paragraph{Number of Experts}\nFig.~\\ref{fig:depth_ndata} (top row) illustrates the effect of the number of leaf-GPs (experts) on the accuracy of the HGP. We constructed 3 HGPs with 1, 2, and 3 levels (4, 16, and 64 experts), respectively, on a training data set of size 200. This resulted in each HGP having experts with data sets  of sizes 100, 50, and 25, respectively. As the number of experts increases, the accuracy decreases. Especially with 64 leaves, no expert has enough data points to model the underlying latent function reasonably well. However, with more training data the HGP with 64 experts recovers the prediction accuracy of the full GP. \n\n\n\\begin{figure*}[tb]\n\\begin{center}\n\\includegraphics[width=0.76\\hsize]{figures\/img_02_02_img04_nchild_deep}\n\\caption{Top row: Comparison of HGPs (blue lines) with varying depths (hence, number of experts) with a ground truth GP (red lines). The mean functions and corresponding $2\\sigma$ predictive intervals are shown. The model accuracy decreases with the number of experts. Bottom row: For a fixed depth (and number of experts), the HGP model becomes more and more similar to the ground truth GP model with an increasing number of data points.}\n\\label{fig:depth_ndata}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsection{Implementation: True Concurrency in Python}\nA known issue of the CPython interpreter, which we use in our implementation, is the lack of true concurrency using the in-built threading library. Due to\nthe \\emph{Global Interpreter Lock} (GIL, which is implemented in the interpreter because Python's memory management is not thread safe), only a single thread of Python code can be executed at any point in time. Therefore, the use of threads in the Python context only provides logical concurrency in terms of the flow of programs, but not true simultaneous computations. \n\nThere exists a workaround for the true concurrency problem in Python, via the use of processes instead of threads to perform simultaneous computations. In the POSIX model, threads are lightweight units of computations belonging to the same process, thus, sharing the same memory space. Processes have their own memory space and come with increased system overheads compared to threads. However, on Linux (which we use for this implementation), the creation of duplicate processes (forking) does not incur large memory costs since Linux implements a copy-on-write model. This means that when a process forks into two, the memory is not copied, unless the new process attempts to modify it. In the context of our implementation, we make no modification to the training data, which is shared amongst all child-GPs. In terms of the memory  usage, each child-GP only needs to compute its own kernel matrix and the corresponding Jacobian matrix per hyper-parameter, which have no interaction with any other child-GP. Therefore, computing each child-GP using a separate process does not incur any large, redundant memory costs that would not be present in a true concurrency model implemented by native threads.\n\n\n\n\n\n\n\n\\section{Experimental Results}\n\nIn this section, we apply our model to two real-world data sets. For both the full GP (if applicable) and the HGP model we optimised the kernel hyper-parameters by maximising the log-marginal likelihood using BFGS. \nIn all following experiments, we used a single standard desktop computer with four cores (16 GB RAM, Intel Core-i7). \n\n\\subsection{Robotic-Arm Data}\nWe applied our HGP model to the kin40k data set~\\cite{kin40k}. The kin40k data set is generated with a simulation of the forward dynamics of an 8-link all-revolute robot arm. The task in the data set is to predict the distance of the end-effector from a target using joint positions and twist angles. The training set size is 10,000 (i.e., we can still train a full GP on it for comparison), the test set size is 30,000, and the input dimension is 8. We trained our model with various architectures (fixed branching factor), ranging from a single-layer model with four experts to a model with 7 layers and $4^7=16,384$ experts. We chose different architectures to assess the trade-off between computation time and accuracy with respect to the full GP.\n\n\n\n\\begin{table*}\n\\caption{Overview of the kin40K-data set results.}\n\\label{tab:kin40K}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c|c} \n\\bf \\shortstack{\\-\\\\Model\\\\\\-}  &\n\\bf \\shortstack{\\-\\\\Number of Levels\\\\(HGP)} &\n\\bf \\shortstack{\\-\\\\Number of Leaves\\\\(HGP)} &\n\\bf \\shortstack{\\-\\\\Training Time (s)\\\\ per BFGS Iteration}  &\n\\bf \\shortstack{\\-\\\\Data Points\\\\per Leaf} &\n\\bf \\shortstack{\\-\\\\Likelihood\\\\ Ratio}\n\\\\ \\hline\nGP (ground truth) & - & - & 218.5 &  10,000  & 1 \\\\ \\hline\n\\multirow{7}{*}{HGP}\n       & 1 & $4  $  & 75.6 &  5,000   & 0.992 \\\\ \\cline{2-6}\n       & 2 & $4^2=16$  & 56.5 &  2,500   & 0.978 \\\\ \\cline{2-6}\n       & 3 & $4^3=64$  & 52.0 &  1,250   & 0.956 \\\\ \\cline{2-6}\n       & 4 & $4^4=256$  & 49.4 &  625   & 0.909 \\\\ \\cline{2-6}\n       & 5 & $4^5=1024$  & 32.2 &  313 & 0.875 \\\\ \\cline{2-6}\n       & 6 & $4^6=4096$  & 17.1  & 157  & 0.834 \\\\ \\cline{2-6}\n       & 7 & $4^7=16384$  & 22.0 &  79  & 0.815 \n\\end{tabular}\n\\end{center}\n\\end{table*}\nTable~\\ref{tab:kin40K} summarises the results. We report the training time per BFGS iteration (all models required 50--70 iterations), the number of data points per computational unit, and the likelihood ratio $\\LR{GP}{HGP}$, which tells us how close our model is to the full GP. The likelihood ratio $\\LR{G_1}{G_2}$ of two distributions $G_1$ and $G_2$ is defined as\n\\begin{align*}\n\\LR{G_1}{G_2} &:=\n\\prod_{i=1}^N \\frac{p(y_i|G_2)}{p(y_i|G_1)} \\stackrel{N\\to\\infty}{\\longrightarrow} \\exp{\\big( - \\KL{G_1}{G_2}  \\big)}\n\\end{align*}\nwhere $y_i\\sim G_1$ (see supplementary material for the proof).\nThe basic single-level HGP with only four experts was able to achieve very similar results in a significantly shorter amount of time. The performance of the HGP decreased with increasing depth since the number of data points per expert becomes too small (note that the input space is 8D) as discussed in Fig.~\\ref{fig:depth_ndata}.\n\n\n\\subsection{Airline Delays (US Flight Data)}\nWe considered a data set reporting flight arrival and departure times for every commercial fight in the US from January to April 2008. This dataset contains extensive information about almost 2 million  flights, including the delay (in minutes) in  reaching the destination. For this data set, we followed the procedure described in~\\cite{Hensman2013}\\footnote{The data set can be obtained from \\url{http:\/\/stat-computing.org\/dataexpo\/2009\/}. Thanks to J Hensman for the availability of the pre-processing script.}: We randomly selected 800,000 data points from which we used a random subset of 700,000 samples to train the model and 100,000 to test it.   We chose the same eight input variables $\\vec x$ as in~\\cite{Hensman2013}: age of the aircraft, distance that needs to be covered, airtime, departure and arrival times, day of the week and month, month. This data set has been evaluated in~\\cite{Hensman2013, Gal2014}, both of which use sparse variational GP methods to deal with this training set size. We applied our HGP model, using data duplication (each training instance is used by two experts) and 1,400 experts  with 1,000 data points each. Data was assigned randomly to the expert GPs. \n\nWe repeated the experiment four times. The average training time was 35.6 minutes and 14 BFGS iterations.\nTable~\\ref{tab:airline} reports the average RMSE values for predicting airline delays.\n\\begin{table}[tb]\n\\centering\n\\caption{Average RMSE values for predicting airline delays (700K training data, 100K test data).}\n\\label{tab:airline}\n\\begin{tabular}{c|c|c}\nSVGP~\\cite{Hensman2013} & Dist SVGP~\\cite{Gal2014} & HGP\\\\\n\\hline\n 33.00 & 32.95 & \\textbf{27.45}\n\\end{tabular}\n\\end{table}\nTable~\\ref{tab:airline} also relates our results for predicting airline delays to the corresponding results reported in~\\cite{Gal2014}, where 100 inducing points were used for the sparse variational GP (SVGP)~\\cite{Hensman2013} and for the distributed sparse variational GP (Dist SVGP)~\\cite{Gal2014}, which are in line  with the results reported in~\\cite{Hensman2013}. Compared to the sparse variational GP methods, our HGP achieves a substantially better predictive performance. Additionally, the HGP converged after a few tens of iterations, whereas the sparse variational GP methods~\\cite{Hensman2013,Gal2014} require hundreds or thousands of iterations.\n\n\n\n\n\\section{Discussion}\n\n\nOur approach to scaling up GPs is conceptually straightforward and practical: It recursively recombines computations by independent lower-level experts to an overall prediction. Unlike any other approach to scaling GPs to large data sets our model is not an explicit sparse approximation of the full GP.\nTherefore, the leaf nodes still perform full GP computations, i.e., their computations scale cubically in the number of data points. However, the number of data points at each leaf can be controlled by adjusting the number of leaves.\n\nIn the limit of a single expert, our hierarchical GP model is equivalent to a standard GP. Additionally, even with more than a single expert, our hierarchical mixture-of-experts model is still a Gaussian process: Any finite number of function values is Gaussian distributed, although we make an implicit (smoothed out) block-diagonal approximation of the covariance matrix. Note that the Deep GP model~\\cite{Damianou2013} is actually not a GP. \n\nIn our model, the kernel hyper-parameters are shared amongst all local GP experts. This makes sense as our objective is to reproduce a full ``vanilla'' GP, which, for practical reasons (size of training set) cannot be applied to the problem. Shared hyper-parameters also do not suffer much from overfitting problems: Even if a local expert fits a poor model, its (wrong\/biased) gradient only has a small contribution if the number of local models is high. \nTraining our model is relatively easy: Besides the shared GP hyper-parameters there are no additional parameters, such as inducing inputs~\\cite{Gal2014, Titsias2009, Snelson2006}, i.e.,  compared to these approaches it is less likely to end up in local optima.\n\nThe main purpose of our model is to scale up the vanilla GP by distributing computational load. Therefore, all kinds of variations that are conceivable for standard GPs could also be implemented in our model. In particular, this includes classification, sparse approximations, heteroscedastic models, and non-Gaussian likelihoods. Note that these models would only be necessary at the level of the leaf GPs: All other computations are (linear) recombinations of the computations at the leaves.\n\n \n\nCompared to other mixture-of-experts models, we chose the simplifying assumption that we know the number of experts, which in our case corresponds to the individual computing units. Thus, we do not need a Dirichlet process prior over partitions and, hence, sampling methods, which dramatically simplifies (and speeds up) training and inference in our model.\n\n\n\\section{Conclusion and Future Work}\nWe presented a conceptually straightforward, but effective, hierarchical  model that allows to scale probabilistic Gaussian processes to (in principle) arbitrarily large data sets. The key idea behind our model is to massively parallelise computations by distributing them amongst independent computational units. A recursive (and closed-form) recombination of these independent computations results in a practical hierarchical mixture-of-GP-experts model that is both computationally and memory efficient. Compared to the most recent sparse GP approximations, our model performs very well, learns fast, requires little memory, and does not suffer from high-dimensional variational optimisation of inducing inputs. We have demonstrated that our model scales well to large data sets: (a) Training a GP with a million data points takes less than 30 minutes on a laptop, (b) with more computing power training a GP with more than $10^7$ data points can be done in a few hours.\n\nThe model presented in this paper lays the foundation for a variety of future work in the context of Gaussian processes, including classification, non-Gaussian likelihoods in regression, and the combination with sparse GP methods (for really large data sets and limited computing power) at the level of the leaf nodes of our mixture-of-experts model.\n\n\n\\subsubsection*{Acknowledgements}\nMPD was supported by an Imperial College Junior Research Fellowship.\n\n\\bibliographystyle{abbrv}\n\n\\section{Likelihood Ratio}\nLet $G_1 = \\gauss{\\mu_1}{\\sigma_1^2}$ and $G_2 = \\gauss{\\mu_2}{\\sigma_2^2}$ be two Gaussian distributions.  We compare $G_2$ to $G_1$ by evaluating the ratio of the likelihood of $G_2$ to $G_1$ given observations drawn from $G_1$.\nWe can do this empirically by drawing $N$ independent samples $y_1,\\cdots,y_N$\nfrom $G_1$, and evaluate the likelihood ratio $\\LR{G_1}{G_2} :=\n\\prod_{i=1}^N \\frac{p(y_i|G_2)}{p(y_i|G_1)} =\\exp{\\left\\{-\\sum_{i=1}^N\n\\log{\\left(\\frac{p(y_i|G_1)}{p(y_i|G_2)}\\right)}\\right\\}}$.  Here $y_i$ are the\nindependent observations of $Y$ and $p(\\cdot|G_2)$ is the likelihood function\nof $G_2$.  We write the likelihood ratio as the exponential of\nthe negative sum of log-likelihood ratios to use the Kullback-Leibler (KL)\ndivergence to compute this in closed form, instead of drawing samples, i.e.,\n\\begin{align*}\n\\sum_{i=1}^N \\log{p(y_i|G_j)} &\\propto \\frac{1}{N} \\sum_{i=1}^N \\log{p(y_i|G_j)}\\\\\n&\\stackrel{N\\rightarrow\\infty}{\\approx} \\mathds{E}_{G_1}[\\log[p(Y|G_j)]\n\\end{align*}\nWith this substitution,\n\\begin{align*}\n\\sum_{i=1}^{N} &\\log{\\frac{p(y_i|G_1)}{p(y_i|G_2)}} = \n\\sum_{i=1}^{N} \\log{p(y_i|G_1)} - \\sum_{i=1}^{N} \\log{p(y_i|G_2)}\\\\ &\\propto \\sum_{i=1}^{N} \\frac{1}{N}\\log{p(y_i|G_1)} - \\frac{1}{N}\\sum_{i=1}^{N} \\log{p(y_i|G_2)}  \\\\\n&\\stackrel{N\\rightarrow\\infty}{\\approx} \\mathds{E}_{G_1}[\\log{p(Y|G_1)}] - \\mathds{E}_{G_1}[\\log{p(Y|G_2)}]  \\\\\n&= \\KL{G_1}{G_2}\n\\end{align*}\nthe likelihood ratio becomes\n$$\n\\LR{G_1}{G_2} = \\exp{\\big( - \\KL{G_1}{G_2}  \\big)},\n$$\nwhich we can evaluate in closed form for two Gaussian distributions $G_1$ and\n$G_2$. Since $\\KL{G_1}{G_2} \\in [0,\\infty)$ and is continuous\nin all the parameters of $G_1$ and $G_2$, it follows that $\\LR{G_1}{G_2} \\in (0,1]$. Thus, we can interpret $LR{G_1}{G2}$ as the similarity\nof $G_2$ and $G_1$. \nThe likelihood ratio $\\LR{G_1}{G_2}$ is a monotonic decreasing function of\n$\\KL{G_1}{G_2}$ and therefore not symmetric. In the comparisons we make, we set $G_1$ to be the predictive\ndistribution of the full GP, which we assume is ``correct'' and $G_2$ to be the\npredictive distribution of the HGP. $\\LR{G_1}{G_2}$ then tells us how well the\nHGP models after the full GP.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec_intro}\n\nLow-density parity-check (LDPC) codes, which are collectively called graph-based codes, are a family of error-correcting codes (ECC) that were introduced by Gallager \\cite{gallager} in 1962. Three decades later, with the advances of circuit design and their decoding algorithms, LDPC codes were revisited and new code constructions, including repeat-accumulate (RA) \\cite{divsalar} and irregular repeat-accumulate (IRA) codes \\cite{jin_ira}, were proposed. Today, graph-based codes have applications in many areas including wireless communication and data storage.\n\nIn ECC, unequal error protection (UEP) is used in applications where some of the channel bits are more sensitive to error or where error in a specific feature of the data is more costly than others, i.e., data has unequal value to the user \\cite{calderbank_uep, katsman_uep}. Codes with unequal protection of bits, and higher protection of input information bits over parity bits offer performance gains \\cite{wolf_uep,furzun_uep}. In this paper, we apply UEP with higher protection of the parity bits over the input information bits of an RA code to limit rate loss, and we demonstrate threshold gains.\\footnote{Although in our UEP setup we protect some \\textit{data} bits without any additional error-correction, we stick to the more common nomenclature of unequal \\textit{error} protection. Moreover, error here refers to flips\/erasures.}\n\nThe idea of applying higher protection on parity bits was introduced in \\cite{ahh_loco} in the context of data storage. The authors introduced lexicographically-ordered constrained (LOCO) codes, which significantly mitigate inter-symbol interference (ISI) in magnetic recording (MR) systems. In their model, parity bits of a spatially-coupled (SC) LDPC code are encoded via a LOCO code, and up to $20\\%$ density gain, with limited rate loss, is achieved compared with using the LDPC code only. Protecting parity bits solely via a LOCO code achieves a significant rate-density gain trade-off. The same UEP idea can also be used to limit speed loss in Flash and other applications where the constrained code rate is already high \\cite{ahh_qaloco, ahh_general}. This UEP setup is successful because when parity bits have higher fidelity messages, e.g., log-likelihood ratios (LLRs), those reliable messages are diffused into all bits during message passing \\cite{ahh_loco}. Thus, the decoder effectively experiences a higher signal-to-noise ratio (SNR) compared to the uniform case. The empirical results presented in \\cite{ahh_loco} are the motivation behind this paper, which is to demonstrate the threshold gains of this UEP setup theoretically. We start with the binary erasure channel (BEC) and the binary symmetric channel (BSC), and subject parity bits to lower erasure and crossover probabilities, respectively, compared to input information bits. With this setup, we ensure obtaining the closest model to achieving high reliability on parity bits which was done by constrained coding in \\cite{ahh_loco}.\n\nWe use extrinsic information transfer (EXIT) charts \\cite{brink_cid} for the threshold analysis of RA LDPC codes with our idea of UEP via more reliable parity bits on BEC and BSC. EXIT charts are a tool to visualize the asymptotic performance and predict convergence behavior of iterative decoders \\cite{brink_cid, alexei_bec}. EXIT charts plot \\textit{average extrinsic information} coming out of the decoder as a function of \\textit{average a-priori information} going into the decoder during the iterations. In the literature, EXIT charts were used in the design of RA codes that are capacity-approaching \\cite{brink_ra}. EXIT functions were derived for BEC, and models for the decoding of RA and general LDPC codes were introduced in \\cite{alexei_bec}. In \\cite{sharon_bms}, methods to obtain EXIT functions for binary-input memoryless symmetric channels were developed through an alternative pseudo-MAP decoder.\n\nIn this paper, we formulate and solve a linear programming (LP) problem to find the optimal degree distribution of the RA code, which maximizes the rate given the erasure or crossover probabilities such that decoding convergence is guaranteed on the EXIT chart, for both the UEP and the uniform setups. For our UEP setup with more reliable parity bits, EXIT functions are used to investigate how the change in the mutual information of parity bits affects the behavior of the overall coding scheme. We discuss an alternative derivation of the EXIT functions for BEC from a combined channel perspective, and using the decoding model introduced in \\cite{alexei_bec}, we derive EXIT functions of variable nodes (VNs) and check nodes (CNs) of an RA LDPC code for BSC. Our experimental results demonstrate the effectiveness of this UEP idea as it achieves up to about $17\\%$ and $28\\%$ threshold gains on BEC and BSC, respectively. The ideas and results we present in this paper are the first step towards developing the UEP theoretical framework for modern data storage systems.\n\n\nThe rest of the paper is organized as follows. In Section~\\ref{sec_prelim}, we discuss the preliminaries. In Section~\\ref{sec_steps}, we introduce our theoretical methodology and derive the (LP) problem for our UEP idea. In Section~\\ref{sec_bec}, we apply this methodology to BEC, and show threshold gains. In Section~\\ref{sec_bsc}, we do the same for BSC. Section~\\ref{sec_conc} concludes the paper.\n\n\\section{Preliminaries}\\label{sec_prelim}\n\nWe use the decoding model shown in Fig.~\\ref{fig_1} for EXIT chart analysis. A binary-symmetric source produces bits that take $0$ or $1$ value with equal probability. There exist $P(\\underline{y}|\\underline{x})$ \\textit{communication channel} and $P(\\underline{w}|\\underline{v})$ \\textit{extrinsic channel} with output vectors $\\underline{y}$ and $\\underline{w}$ (noisy versions of the inputs $\\underline{x}$ and $\\underline{v}$) and LLRs $\\underline{c}$ and $\\underline{a}$, respectively. Fig.~\\ref{fig_1} models iterative decoding where the extrinsic channel, which is actually an artificial channel, models extrinsic information coming from the previous decoding iteration \\cite{alexei_bec}. The decoder uses outputs of both channels $\\underline{y}$ and $\\underline{w}$ to calculate \\textit{a-posteriori} and \\textit{extrinsic} LLRs $\\underline{d}$ and $\\underline{e}$ of $\\underline{v}$, respectively. See \\cite{alexei_bec} for more details.\n\n\\begin{figure}\n\\vspace{-0.3em}\n\\centering\n\\includegraphics[trim={2.1in 3.5in 2.2in 1.7in},clip,width=3.7in]{decoding_model_switch.pdf}\n\\vspace{-2.7em}\n\\caption{Decoding model for EXIT analysis}\n\\label{fig_1}\n\\vspace{-0.8em}\n\\end{figure}\n\nWe investigate the threshold gains of higher protection of the~parity bits of an RA code using EXIT chart analysis. When considering VNs of the RA code, the switch in Fig.~\\ref{fig_1} is closed, resulting in $\\underline{u} = \\underline{x}$, $\\underline{u}$ is one bit, and the Encoder is a repetition code with length $d_{\\textup{v}}$ (VN degree). Whereas when considering the CNs of the RA code, the switch in Fig.~\\ref{fig_1} is open, and the Encoder is a single parity-check (SPC) code with length $d_{\\textup{c}}$ (CN degree) \\cite{alexei_bec}. We investigate the effect of UEP on RA codes in two setups. In the first setup, both communication and extrinsic channels are BECs with erasure probabilities $q$ and $p$, respectively. In the second, both channels are BSCs with crossover probabilities $\\epsilon$ and $\\delta$, respectively.\n\nIn the model of iterative decoding, extrinsic LLR $e_{i}$ at the decoder output in one iteration re-enters the decoder as a-priori LLR $a_{i}$ after passing through an interleaver in the next iteration \\cite{alexei_bec, brink_cid}. This is consistent with the  modelling of the extrinsic channel discussed above. Next, EXIT functions are defined. Let $m$ be the length of $\\underline{v}$, $\\underline{w}$, $\\underline{a}$, and $\\underline{e}$. The \\textit{average a-priori information} $I_{\\textup{A}}$ going into the decoder is then \\cite{alexei_bec}:\n\\begin{equation} \\label{IA}\nI_{\\textup{A}} = \\frac{1}{m}\\sum_{i=1}^{m} I(V_{i};A_{i}) = I(V_{1};A_{1}).\n\\end{equation} \nThe second equality follows from the observation that $V_{i}$, for all $i$, have the same distribution and that the extrinsic channel is memoryless and time invariant. The \\textit{average extrinsic information} $I_{\\textup{E}}$ coming out of the decoder is \\cite{alexei_bec}:\n\\begin{equation} \\label{IE}\nI_{\\textup{E}} = \\frac{1}{m}\\sum_{i=1}^{m} I(V_{i};E_{i}) = I(V_{1};E_{1}) = I(V_{1};\\underline{Y},\\underline{A}_{[1]}),\n\\end{equation} \nwhere an a-posteriori probability (APP) decoder is assumed. We write random variables with upper case letters, their realizations with lower case letters. $\\underline{A}_{[i]}$ denotes vector $\\underline{A}$ with the $i$th entry removed. The third equality in (\\ref{IE}) follows from the proposition proved in \\cite{alexei_bec} for APP decoders with extrinsic message passing. This allows the extrinsic information to be defined as mutual information between input of the extrinsic channel and the inputs of the decoder instead of the extrinsic LLR at the output of the decoder. An EXIT chart plots extrinsic information as a function of a-priori information. \\cite{brink_cid}.\n\n\\section{Methodology for UEP Analysis}\\label{sec_steps}\n\nFor our UEP setup, we use EXIT functions to see how a change in the mutual information of parity bits affects the behavior of the overall coding scheme. In this section, we illustrate the code construction and our methodology with steps which are applied to BEC and BSC in Sections~\\ref{sec_bec} and~\\ref{sec_bsc}.\n\nTo guarantee the diffusion of higher fidelity messages from parity to input information bits, a specific property is required in the code construction. That is, for each VN representing an input information bit, there exists at least one VN representing a parity bit connected to the first through a CN. This property is satisfied in RA LDPC codes with parity-check matrix $H$ of the form $H = [J\\textup{ }P]$, where $P$ is $(n-k)\\times(k+1)$ sparse matrix and $J$ is $(n-k)\\times(n-k-1)$ matrix of the form:\n\\begin{equation}\nJ = \n\\begin{bmatrix}\n1 & 0 &   &   & &...& 0 \\\\\n1 & 1 & 0 &   & &...& 0 \\\\\n0 & 1 & 1 & 0  & &...& 0 \\\\\n0 & 0 & 1 & 1 &0 &...& 0 \\\\\n  &   &   & ...  & &&   \\\\\n0 & ...  &  &   & 0 & 1 & 1 \\\\\n0 & ...  &  &   &  & 0 & 1 \n\\end{bmatrix}.\n\\end{equation}\nAssuming that the first column of $P$ has weight $2$, and it is linearly independent from the columns of J, the first $(n-k)$ bits of the RA codeword (corresponding to the columns of $J$ and the first column of $P$) can be considered parity bits, whereas the last $k$ bits can be considered input information bits. Also assuming $d_{\\textup{c}} > 3$, each parity bit (except the first, $(n-k-1)$-th, and the last bits) is connected to another parity bit and at least $d_{\\textup{c}}-3$ input information bits, which satisfies the required property. ($P$ has no $\\underline{0}$ columns.)\n\nWith this RA code construction, we can now derive EXIT functions for our UEP setup via the following steps. Let the communication and extrinsic channels have error probabilities $\\sigma$ and $\\beta$, respectively.\n\n\\noindent\\textbf{Step~1:} Consider the VNs of the RA code. First, derive the extrinsic information $I_{\\textup{E,v}}$ as a function of the a-priori information $I_{\\textup{A,v}}$ for the VNs without considering UEP, i.e., assuming both input information and parity bits are transmitted via a communication channel with fixed error probability $\\sigma$. This step assumes fixed $d_{\\textup{v}}$ for simplicity.\n\n\\noindent\\textbf{Step~2:} Consider the CNs of the RA code. Derive the extrinsic information $I_{\\textup{E,c}}$ as a function of the a-priori information $I_{\\textup{A,c}}$ without considering UEP. Next, derive the inverse EXIT function of CNs.\\footnote{Always $I_{\\textup{A,v}} =  I_{\\textup{A,c}}$. Thus, we use  $I_{\\textup{A}}$ notation in the rest of the paper.} We adopt fixed $d_{\\textup{c}}$ in our RA code construction.\n\n\\noindent\\textbf{Step~3:} We are now ready to apply unequal protection on parity and input information bits. Let all CNs have degree $d_{\\textup{c}}$. Let $\\lambda_i$ be the fraction of branches (edges) connected to VNs of degree $i$, which we refer to as the {\\textit{degree distribution}}\\cite{jin_ira}.\\footnote{In this paper, we refer to an edge that is adjacent to a node as \"connected\" to the node. Here, we mean they are directly connected.} Let $N$ be the total number of branches. Let $\\lambda_2 = a + b$, where $a$ is the fraction of branches connected to $(n-k)$ VNs corresponding to $(n-k)$ parity bits. It follows from the RA code construction discussed earlier that $n-k = N\\cdot \\frac{1}{d_{\\textup{c}}} = N\\cdot \\frac{a}{2} \\implies a = \\frac{2}{d_{\\textup{c}}}$, where $n-k$ is the number of CNs. Thus we have,\n\\begin{equation} \\label{constraint1}\n    \\frac{2}{d_{\\textup{c}}} + b + \\sum_{i \\geq 3}\\lambda_i = 1,\n\\end{equation}\nwhich is the first constraint of the linear programming (LP) problem to be explained in the upcoming step.\n\nLet $\\sigma_{1}$ and $\\sigma_{2}$ be the error probabilities of parity bits and input information bits transmitted through the communication channel, respectively. We now derive $I^{*}_{\\textup{E,v}}$ for the  UEP setup:\n\\begin{align} \\label{general_Iev}\n    I^{*}_{\\textup{E,v}} &= a \\cdot I_{\\textup{E,v}}(\\sigma = \\sigma_1, d_{\\textup{v}} = 2) + b \\cdot I_{\\textup{E,v}}(\\sigma = \\sigma_2, d_{\\textup{v}} = 2) \\nonumber \\\\ &\\hspace{+1.0em}+  \\sum_{i \\geq 3}\\lambda_i \\cdot I_{\\textup{E,v}}(\\sigma = \\sigma_2, d_{\\textup{v}} = i),\n\\end{align}\nwhich is a weighted sum over branches. Note that we extend the arguments of $I_{\\textup{E,v}}$ in (\\ref{general_Iev}) for clarity.\\footnote{The remaining information equations ($I_{\\textup{E,c}}$ and $I_{\\textup{A}}$) are same for unequal and uniform error protection (also same notation). A-priori information for both VNs and CNs depend only on the input and output of extrinsic channel, not the communication channel. Same for extrinsic information when considering CNs due to the open switch in Fig.~\\ref{fig_1}.} Note also that (\\ref{general_Iev}) is used for the uniform protection setup as well by setting $\\sigma_{1} = \\sigma_{2}$.\n\n\\noindent\\textbf{Step~4:} In this step, we formulate an LP problem. Iterative decoding will be successful, i.e., convergence occurs, if the EXIT function of VNs lies above and does not intersect with the inverse of EXIT function of CNs \\cite{alexei_bec}, i.e.,\n\\begin{equation} \\label{constraint2}\n    I^{*}_{\\textup{E,v}}(I_{\\textup{A}}) > I^{-1}_{\\textup{E,c}}(I_{\\textup{A}}),  \\textup{ } I_{\\textup{A}} \\in (0,1), \n\\end{equation}\nwhich is the second constraint of the LP problem.\n\nWe calculate the code rate $R$ as follows:\n\\begin{align} \\label{rate}\n    R = 1- \\frac{\\frac{1}{d_{\\textup{c}}}}{\\frac{a}{2} + \\frac{b}{2} + \\sum_{i \\geq 3} \\frac{\\lambda_i}{i}} = 1- \\frac{\\frac{1}{d_{\\textup{c}}}}{\\frac{1}{d_{\\textup{c}}} + \\frac{b}{2} + \\sum_{i \\geq 3} \\frac{\\lambda_i}{i}}\n\\end{align}\n\nWe then formulate the following LP problem for finding optimal degree distribution that maximizes the rate of the code under code construction and EXIT convergence constraints, (\\ref{constraint1}) and (\\ref{constraint2}), for pre-determined $\\sigma_{1}$, $\\sigma_{2}$ error probabilities:\n\\begin{align} \\label{LP}\n    \\nonumber \\textbf{maximize } &\\; \\frac{b}{2} + \\sum_{i \\geq 3} \\frac{\\lambda_i}{i} \\\\ \\nonumber\n    \\textbf{subject to } &\\; \\frac{2}{d_{\\textup{c}}} + b + \\sum_{i \\geq 3}\\lambda_i = 1, \\nonumber \\\\\n    & I^{*}_{\\textup{E,v}}(I_{\\textup{A}}) > I^{-1}_{\\textup{E,c}}(I_{\\textup{A}}),  \\textup{ } I_{\\textup{A}} \\in (0,1). \n\\end{align}\n\nThis LP problem is derived in order to investigate the gains of unequal error protection. We solve the LP problem numerically using a software program. Given the channel probabilities, the solution of this optimization problem gives the degree distribution that achieves the highest rate. How to use the LP solution to obtain the threshold gains is discussed in the following sections.\n\n\n\\section{Unequal Error Protection on BEC}\\label{sec_bec}\n\nLet the communication and extrinsic channels in Fig.~\\ref{fig_1} be BECs with erasure probabilities $q$ and $p$, respectively. In this section, we apply the steps outlined in Section~\\ref{sec_steps} and discuss the results of applying unequal error protection on parity and input information bits of the RA code for BEC.\n\n\\noindent\\textbf{\\underline{Step~1:}} When considering the VNs of the RA code, $\\underline{u} = \\underline{x}$, and the Encoder is a repetition code with length $d_{\\textup{v}}$. From \\cite{alexei_bec} (see also the intuitive explanation after Step 2), for fixed $d_{\\textup{v}}$,\n\\begin{align}\n    & I_{\\textup{A}} =  I(V_{1};A_{1}) = 1-p, \\\\ \n    & I_{\\textup{E,v}} = 1-qp^{d_{\\textup{v}}-1}, \\label{bec_Iev} \\\\\n    & I_{\\textup{E,v}}(I_{\\textup{A}}) = 1-q(1-I_{\\textup{A}})^{d_{\\textup{v}}-1}. \\label{fcn_bec_Iev}\n\\end{align}\n \n\\noindent\\textbf{\\underline{Step~2:}} When considering the CNs of the RA code, the switch on the top branch is open, and the Encoder is an SPC code with length $d_{\\textup{c}}$. From \\cite{alexei_bec} (see also the intuitive explanation after Step 2) and for fixed $d_{\\textup{c}}$,\n\\begin{align}\n    & I_{\\textup{A}} =  I(V_{1};A_{1}) = 1-p, \\\\\n    & I_{\\textup{E,c}} = (1-p)^{d_{\\textup{c}}-1}, \\label{bec_Iec} \\\\\n    & I_{\\textup{E,c}}(I_{\\textup{A}}) = (I_{\\textup{A}})^{d_{\\textup{c}}-1}, \\label{fcn_bec_Iec} \\\\\n    & I^{-1}_{\\textup{E,c}}(I_{\\textup{A}}) = (I_{\\textup{A}})^{\\frac{1}{d_{\\textup{c}}-1}}. \\label{fcn_bec_Iec_inv}\n\\end{align}\n\nTo intuitively explain (\\ref{bec_Iec}), we can think of a combined BEC setup. Let us consider the CNs side first. In order that a CN sends a correct message to a VN, the messages from all other $d_{\\textup{c}}-1$ VNs must be correct. The probability of getting correct information from a VN straight from a BEC($p$) \\textit{channel} is $1-p$. Note that here, a \\textit{channel} refers to the extrinsic channel, not the communication channel. The probability that all those $d_{\\textup{c}}-1$ VNs send correct messages is $p_{\\textup{correct}} = (1-p)^{d_{\\textup{c}}-1}$. Thus, the probability of receiving wrong (erased) information is $p_{\\textup{wrong}} = 1-p_{\\textup{correct}} = 1-(1-p)^{d_{\\textup{c}}-1}$, which is actually the erasure probability ($p_{\\textup{erasure}}$) of the new combined channel. Hence, the mutual information under equiprobable inputs (the capacity) of the combined BEC is $I_{\\textup{E,c}} = 1-p_{\\textup{erasure}} = 1-(1-(1-p)^{d_{\\textup{c}}-1}) = (1-p)^{d_{\\textup{c}}-1}$, which is (\\ref{bec_Iec}).\n\nThe same logic can be applied to the VNs side to get (\\ref{bec_Iev}), where the combined channel erasure probability will be $qp^{(d_{\\textup{v}}-1)}$ this time. In order that a VN sends a wrong message to a CN, the information at that VN has to be erased by the communication channel, which is the first event, and all messages coming to that VN from the other $d_{\\textup{v}}-1$ CNs are also erased, which is the second event. The probability of the first event is $q$, and the probability of the second event is $p^{d_{\\textup{v}}-1}$. Note that any CN with non-erased information suffices to fix the erasure at that VN (different from BSC, see Section~\\ref{sec_bsc}). Hence, $qp^{(d_{\\textup{v}}-1)}$ is the erasure probability $p_{\\textup{erasure}}$ of the new combined channel. Thus, $I_{\\textup{E,v}} = 1-p_{\\textup{erasure}} = 1-qp^{(d_{\\textup{v}}-1)}$, which is (\\ref{bec_Iev}). \n\n\\begin{table*}\n\\caption{Threshold Gains of UEP at Various Erasure Probabilities When LP is Solved for Uniform and Unequal Error Protection on BEC}\n\\vspace{-0.5em}\n\\centering\n\n\\begin{tabular}{|c|c|c|c|c|c!{\\vrule width 0.85pt}c|c|c|c|c|c|}\n\\hline\n\\multicolumn{6}{|c!{\\vrule width 0.85pt}}{\\makecell{\\textbf{First method:} LP solved for uniform protection}} & \\multicolumn{6}{c|}{\\makecell{\\textbf{Second method:} LP solved for unequal protection}} \\\\\n\\hline\nRate & $q_{\\textup{uniform}}$ & $q'_1$ & {$q'_2$} & $q'_{\\textup{avg}}$ & \\textbf{Gain} & Rate & $q_1$ & {$q_2$} & $q_{\\textup{avg}}$ & $q'_{\\textup{uniform}}$  & \\textbf{Gain} \\\\\n\\hline\n$0.6316$ & $0.28$ & $0.003$ & $0.488$ & $0.3093$ & $10.5\\%$ & $0.6376$ & $0.050$ & $0.500$ & $0.3369$ & $0.2883$ & $16.9\\%$ \\\\\n\\hline\n$0.6430$ & $0.25$ & $0.002$ & $0.426$ & $0.2746$ & $9.9\\%$ & $0.6644$ & $0.080$ & $0.430$ & $0.3126$ & $0.2739$ & $14.1\\%$ \\\\\n\\hline\n$0.7237$ & $0.20$ & $0.003$ & $0.299$ & $0.2172$ & $8.6\\%$ & $0.7347$ & $0.080$ & $0.300$ & $0.2416$ & $0.2180$ & $10.8\\%$ \\\\\n\\hline\n$0.7853$ & $0.14$ & $0.003$ & $0.188$ & $0.1483$ & $5.9\\%$ & $0.7876$ & $0.090$ & $0.210$ & $0.1845$ & $0.1750$ & $5.4\\%$ \\\\\n\\hline\n$0.8527$ & $0.10$ & $0.004$ & $0.122$ & $0.1046$ & $4.6\\%$ & $0.8526$ & $0.004$ & $0.122$ & $0.1046$ & $0.1009$ & $3.7\\%$ \\\\\n\\hline\n\\end{tabular}\n\n\\label{table1}\n\\vspace{-0.25em}\n\\end{table*}\n\n\n\\noindent\\textbf{\\underline{Step~3:}} We now apply the UEP setup explained in the previous section to derive $I^{*}_{\\textup{E,v}}$. Parity and input information bits transmitted through the communication channel face erasure probabilities $q_{1}$ and $q_{2}$, respectively. Using (\\ref{general_Iev}) and (\\ref{fcn_bec_Iev}), the EXIT function is derived to be:\n\\begin{align} \\label{UEP_bec_fcn}\n    I^{*}_{\\textup{E,v}}(I_{\\textup{A}}) &= a \\cdot (1-q_{1}(1-I_{\\textup{A}})) + b \\cdot (1-q_{2}(1-I_{\\textup{A}})) \\nonumber \\\\ &\\hspace{+1.0em}+  \\sum_{i \\geq 3}\\lambda_i \\cdot (1-q_{2}(1-I_{\\textup{A}})^{i-1}).\n\\end{align}\n\\vspace{-0.6em}\n\n\\noindent\\textbf{\\underline{Step~4:}} Substituting (\\ref{fcn_bec_Iec_inv}) and (\\ref{UEP_bec_fcn}) in (\\ref{constraint2}) completes the LP problem in (\\ref{LP}). We now solve this LP problem to find the optimal degree distribution. That is, we maximize the rate of the code in (\\ref{rate}), subject to degree distribution constraint (\\ref{constraint1}) and EXIT convergence constraint (\\ref{constraint2}).\n\\newline\n\nWe can find the solution ($a$, $b$ and $\\lambda$'s) of the LP problem numerically using a software program. We implement two methods to investigate the threshold gains of our UEP idea over uniform protection in our software. In the first method, we mimic the approach adopted in data storage devices. In particular, the graph-based code is designed and optimized assuming all codeword bits will have the same protection \\cite{ahh_loco,ahh_md}. As the device ages, the constrained code can be applied to the parity bits only, resulting in UEP and achieving density\/lifetime gains \\cite{ahh_loco}. Thus, in our first setup, we solve the LP problem for the uniform protection case, i.e, $q_1 = q_2 = q_{\\textup{uniform}}$, and find the optimal degree distribution. An appropriate $d_{\\textup{c}}$ is chosen according to $q_{\\textup{uniform}}$. A code with this degree distribution has \\textit{threshold} $q_{\\textup{uniform}}$. After solving the LP problem for the uniform setup, we apply higher protection on parity bits (with $q'_1<q'_2$) for the same code construction (same $d_{\\textup{c}}$ and degree distribution, thus same rate). We search for the ($q'_1, q'_2$) combination with the highest average that satisfies the EXIT convergence constraint in (\\ref{constraint2}) using (\\ref{fcn_bec_Iec_inv}), and (\\ref{UEP_bec_fcn}) obtained in Step 3. We calculate the average erasure probability ($q'_{\\textup{avg}}$) and the percentage threshold gains ($TG$) as follows: \n\\begin{align}\\label{eqn_qavg}\n    &q'_{\\textup{avg}} = q'_1(1-R) + q'_2(R), \\\\ \\label{eqn_tg}\n    &TG = \\frac{q'_{\\textup{avg}} - q_{\\textup{uniform}}}{ q_{\\textup{uniform}}} \\times 100\\%. \n\\end{align}\nNumerical results demonstrating the UEP threshold gains are illustrated in the left panel of Table~\\ref{table1}.\n\nIn Fig.~\\ref{fig_2}, the top curve, which is the VN curve for uniform protection $I_{\\textup{E,v}}(I_{\\textup{A}})$, is generated by substituting the solution of the LP problem in (\\ref{UEP_bec_fcn}) with $q_{\\textup{uniform}} = 0.28$. The CN inverse curve $I^{-1}_{\\textup{E,c}}(I_{\\textup{A}})$, which is the bottom curve, is generated from (\\ref{fcn_bec_Iec_inv}). As observed, LP constraints are satisfied. The middle curve, which is the VN curve with UEP $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$, is plotted by substituting the solution of the LP problem in (\\ref{UEP_bec_fcn}) with the best $q'_1$ and $q'_2$ values that satisfy the constraints. Fig.~\\ref{fig_2} also demonstrates the decoding convergence for the UEP setup (red\/bottom curve is always below blue\/middle curve).\n\n\\begin{figure}\n\\vspace{-1.0em}\n\\centering\n\\hspace{+0.5em}\\includegraphics[trim={1.3in 0.5in 1.3in 0.5in},width=3.4in]{plot_bec.pdf}\n\\vspace{-1.8em}\n\\caption{EXIT chart on BEC. The LP problem is solved for uniform protection with $q_{\\textup{uniform}}=0.28$, and the top curve represents $I_{\\textup{E,v}}(I_{\\textup{A}})$. For the same code parameters, $q'_1 = 0.003$ and $q'_2 = 0.488,$ are reached for UEP, and the middle curve represents $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$. The bottom curve represents $I^{-1}_{\\textup{E,c}}(I_{\\textup{A}})$.}\n\\label{fig_2}\n\\vspace{-1.5em}\n\\end{figure}\n\nIn our second method of investigating the threshold gains, we do the opposite. We find the optimal degree distribution for given unequal erasure probabilities $q_1$ and $q_2$ with $q_{\\textup{avg}}$ obtained as in (\\ref{eqn_qavg}), where $q_1<q_2$. Next, we find the highest uniform erasure probability $q'_{\\textup{uniform}} = q'_1 = q'_2$ satisfying the constraints given the code parameters. Numerical results demonstrating the UEP gains are illustrated in the right panel of Table~\\ref{table1}.\n\nTable~\\ref{table1} presents the threshold gains we get by applying our UEP idea on BEC. For both methods, as the rate increases, i.e, as the system gets better (lower erasure probabilities), gains decrease as expected. The results of the first method, which optimizes for uniform protection, demonstrate that UEP offers a higher gain of around $10\\%$ for a moderate rate of around $0.64$, which is suitable to wireless communication. There are also gains at higher rates suitable for data storage. This threshold gain means that with our UEP setup, the exact same code can appropriately operate at average erasure probabilities that are up to $10\\%$ higher than the maximum it can operate at under the uniform setup. The second method optimizes for UEP instead of uniform protection, and it is included in this paper as a proof of concept since it slightly favors our UEP idea. The results of this method demonstrate a gain of around $17\\%$ for a rate of around $0.64$. This is a further demonstration that UEP via more reliable parity bits is a promising idea that achieves significant threshold gains.\n\nWe note that BEC is the simplest channel to start with. The gains in Table~\\ref{table1} motivate applying our UEP idea to error (bit flip) channels like BSC, which is discussed in the next section, and to more advanced data storage channels, which is left for our future work.\n\n\\section{Unequal Error Protection on BSC}\\label{sec_bsc}\n\nLet the communication and extrinsic channels in Fig.~\\ref{fig_1} be BSCs with crossover probabilities $\\epsilon$ and $\\delta$, respectively. In this section, we apply the steps outlined in Section~\\ref{sec_steps} and discuss the results of applying unequal error protection on parity and input information bits of the RA code for the BSC.\n\n\\noindent\\textbf{\\underline{Step~1:}} When considering the VNs of the RA code, $\\underline{u} = \\underline{x}$, $\\underline{u}$ is one bit, and the Encoder is a repetition code with length $d_{\\textup{v}}$. It is known that for a BSC with crossover probability $\\delta$,\n\\begin{align}\n    I_{\\textup{A}} =  I(V_{1};A_{1}) = 1-H(\\delta),   \n\\end{align}\nwhere $H(\\delta)$ is the binary entropy function with:\n\\begin{align}\n    H(\\delta) = -\\delta\\textup{log}_2(\\delta) - (1-\\delta)\\textup{log}_2(1-\\delta). \\nonumber\n\\end{align}\n\nWe also use $H(X)$ to represent the entropy of a random variable $X$. Next, we derive $I_{\\textup{E,v}}$ for VNs in Theorem~\\ref{thm_bsc_Iev} for fixed $d_{\\textup{v}}$.\n\n\\begin{theorem}\\label{thm_bsc_Iev}\nConsider the VNs of the RA code. The average extrinsic information coming out of decoder when the communication and extrinsic channels are BSCs with crossover probabilities $\\epsilon$ and $\\delta$, respectively, is:\n\\begin{align} \\label{bsc_Iev}\nI_{\\textup{E,v}} &= 1 - \\sum_{i=0}^{d_{\\textup{v}}-1}\\binom{d_{\\textup{v}}-1}{i}\\big[\\theta^{i}_1+\\theta^{i}_2\\big]H\\bigg(\\frac{\\theta^{i}_2}{\\theta^{i}_1 + \\theta^{i}_2}\\bigg),\n\\end{align}\nwhere\n\\begin{align} \\label{bsc_Iev_terms}\n    \\theta^{i}_1 = (1-\\epsilon)(1-\\delta)^{i}\\delta^{d_{\\textup{v}}-1-i} \\text{ and } \\theta^{i}_2 = \\epsilon \\delta^{i}(1-\\delta)^{d_{\\textup{v}}-1-i}.\n\\end{align}\n\n\\end{theorem}\n\n\\begin{IEEEproof}\nWe prove Theorem~\\ref{thm_bsc_Iev} by directly using (\\ref{IE}) and calculating the mutual information:\n\\begin{align} \\label{rule}\n    I_{\\textup{E,v}} &= I(V_{1};\\underline{Y},\\underline{A}_{[1]}) = H(V_{1})-H(V_{1} | \\underline{Y},\\underline{A}_{[1]}),\n\\end{align}\nwhere $H(V_{1}) = 1$. Next, we calculate the conditional entropy:\n\\begin{align} \\label{cond_entropy}\n    H(V_{1} | \\underline{Y},\\underline{A}_{[1]}) &= - \\sum_{v_1} \\sum_{\\underline{y}} \\sum_{\\underline{a}_{[1]}} p(v_{1})p(\\underline{y},\\underline{a}_{[1]} | v_{1}) \\nonumber \\\\ &\\hspace{+1.0em} \\cdot \\textup{log}_2\\bigg(\\frac{p(v_{1})p(\\underline{y},\\underline{a}_{[1]} | v_{1})}{p(\\underline{y},\\underline{a}_{[1]})}\\bigg),\n\\end{align}\nand $p(\\underline{y},\\underline{a}_{[1]} | v_{1}) = p(\\underline{y}| v_{1})p(\\underline{a}_{[1]} | v_{1})$.\n\nWe start with:\n\\begin{align} \\label{h1}\n    p(v_{1}=0) = p(v_{1}=1) = \\frac{1}{2}.\n\\end{align}\nAdditionally, we know that:\n\\begin{align} \n    &p(\\underline{y}= 0| v_{1}=0) = p(\\underline{y}= 1| v_{1}=1) = (1-\\epsilon), \\label{h2}\\\\\n    &p(\\underline{y}= 0| v_{1}=1) = p(\\underline{y}= 1| v_{1}=0) = \\epsilon. \\label{h22}\n\\end{align}\n\nNext, recall that $\\underline{a}_{[1]}$ is a binary vector of length $d_{\\textup{v}}-1$. Let the superscript $i$ in $\\underline{a}^{i}_{[1]}$ indicate that among the $d_{\\textup{v}}-1$ entries of $\\underline{a}^{i}_{[1]}$, $i$ of them are $0$'s and $d_{\\textup{v}}-1-i$ of them are $1$'s. For example, $\\underline{1} = \\underline{a}^{0}_{[1]}$ and $\\underline{0} = \\underline{a}^{d_{\\textup{v}}-1}_{[1]}$. Observe that due to BSC, $p(\\underline{a}^{i}_{[1]}| v_{1}=0) = p(\\underline{a}^{d_{\\textup{v}}-1-i}_{[1]}| v_{1}=1)$. Thus, for general $i \\in \\{0,1,2,\\dots,d_{\\textup{v}}-1 \\}$,\n\\begin{align} \\label{h3}\n    p(\\underline{a}^{i}_{[1]} | v_{1}=0) = (1-\\delta)^{i}\\delta^{d_{\\textup{v}}-1-i} = p(\\underline{a}^{d_{\\textup{v}}-1-i}_{[1]} | v_{1}=1) .\n\\end{align}\n\nNext, we calculate the joint probability term $p(\\underline{y},\\underline{a}_{[1]})$ in (\\ref{cond_entropy}). We have:\n\\begin{equation} \\label{ex2}\n    p(\\underline{y},\\underline{a}^{i}_{[1]}) = \\sum_{v_1}  p(\\underline{y},\\underline{a}^{i}_{[1]} | v_1) p(v_1).\n\\end{equation}\nObserve that due to BSC , $p(\\underline{y}= 0,\\underline{a}^{i}_{[1]}) = p(\\underline{y}= 1,\\underline{a}^{d_{\\textup{v}}-1-i}_{[1]})$. Thus, for general $i \\in \\{0,1,2,\\dots,d_{\\textup{v}}-1 \\}$,\n\\begin{align} \\label{h4}\n    p(\\underline{y}= 0,\\underline{a}^{i}_{[1]}) &= \\frac{1}{2}\\big[(1-\\epsilon)(1-\\delta)^{i}\\delta^{d_{\\textup{v}}-1-i}  \\nonumber \\\\ &\\hspace{+1.0em} + \\epsilon\\delta^{i}(1-\\delta)^{d_{\\textup{v}}-1-i}\\big] = p(\\underline{y}= 1,\\underline{a}^{d_{\\textup{v}}-1-i}_{[1]}),\n\\end{align}\nwhich provides the joint probability for all pairs of $\\underline{y}$ and $\\underline{a}_{[1]}$. \n\nNow, we substitute equations (\\ref{h1}), (\\ref{h2}), (\\ref{h22}), (\\ref{h3}), and (\\ref{h4}) in (\\ref{cond_entropy}). Note that for each $i$, there exist $\\binom{d_{\\textup{v}}-1}{i}$ distinct $\\underline{a}^{i}_{[1]}$ vectors. For example, if we substitute $(v_{1}=0, \\underline{y}=0, \\underline{a}^{0}_{[1]})$ and $(v_{1}=0, \\underline{y}=1, \\underline{a}^{d_{\\textup{v}}-1}_{[1]})$ in the summations of (\\ref{cond_entropy}), we get:\n\\begin{align} \\label{ex3}\n    \\frac{1}{2}\\binom{d_{\\textup{v}}-1}{0}\\big[\\theta^{0}_1+\\theta^{0}_2\\big]H\\bigg(\\frac{\\theta^{0}_2}{\\theta^{0}_1 + \\theta^{0}_2}\\bigg) \\nonumber\n\\end{align}\nfor the two scenarios combined, where\n\\begin{equation}\n\\theta^{0}_1 = (1-\\epsilon)\\delta^{d_{\\textup{v}}-1} \\text{ and } \\theta^{0}_2 = \\epsilon(1-\\delta)^{d_{\\textup{v}}-1} \\nonumber.   \n\\end{equation}\nWhen $v_{1}=1$, the above result is multiplied by $2$ due to symmetry. Finally, if we compute the summations for all values of $v_1$, $\\underline{y}$, and $\\underline{a}_{[1]}$, we obtain:\n\\begin{align} \\label{cond}\n    H(V_{1} | \\underline{Y},\\underline{A}_{[1]}) &= \\sum_{i=0}^{d_{\\textup{v}}-1}\\binom{d_{\\textup{v}}-1}{i}\\big[\\theta^{i}_1+\\theta^{i}_2\\big]H\\bigg(\\frac{\\theta^{i}_2}{\\theta^{i}_1 + \\theta^{i}_2}\\bigg),\n\\end{align}\nwhere\n\\begin{align} \\label{condd}\n    \\theta^{i}_1 = (1-\\epsilon)(1-\\delta)^{i}\\delta^{d_{\\textup{v}}-1-i} \\text{ and } \\theta^{i}_2 = \\epsilon \\delta^{i}(1-\\delta)^{d_{\\textup{v}}-1-i}.\n\\end{align}\nSubstituting (\\ref{cond}) and (\\ref{condd}) in (\\ref{rule}), completes the proof.\n\\end{IEEEproof}\n\nTo derive the EXIT function $I_{\\textup{E,v}}(I_{\\textup{A}})$ in terms of $I_{\\textup{A}}$, we use (\\ref{bsc_Iev}) and (\\ref{bsc_Iev_terms}) with $\\delta = H^{-1}(1-I_{\\textup{A}})$ for $\\delta \\in (0,0.5)$ ($ I_{\\textup{A}} \\in (0,1)$), where $H^{-1}$ is inverse binary entropy function.\n\n\\noindent \\textbf{\\underline{Step~2:}} When considering the CNs of the RA code, the switch on the top branch is open, and the Encoder is an SPC code with length $d_{\\textup{c}}$. It is clear that:\n\\begin{align}\n    I_{\\textup{A}} =  I(V_{1};A_{1}) = 1-H(\\delta).\n\\end{align}\n\nWe derive $I_{\\textup{E,c}}$ for CNs in Theorem~\\ref{thm_bsc_Iec} for fixed $d_{\\textup{c}}$.\n\\begin{theorem}\\label{thm_bsc_Iec}\nConsider the CNs of the RA code. The average extrinsic information coming out of decoder when the communication and extrinsic channels are BSCs with crossover probabilities $\\epsilon$ and $\\delta$, respectively, is:\n\\begin{align} \\label{bsc_Iec}\nI_{\\textup{E,c}} &= 1 - H\\bigg(\\frac{1}{2}\\Big[1-(1-2\\delta)^{d_{\\textup{c}}-1}\\Big]\\bigg).\n\\end{align}\n\n\\end{theorem}\n\n\\begin{IEEEproof}\nWe prove Theorem~\\ref{thm_bsc_Iec} by directly using (\\ref{IE}) and calculating the mutual information. Recall that the switch on the top branch is open. Thus,\n\\begin{align} \\label{rule2}\n    I_{\\textup{E,c}} &= I(V_{1};\\underline{A}_{[1]}) = 1 -H(V_{1} | \\underline{A}_{[1]}).\n\\end{align}\nNext, we calculate the conditional entropy,\n\\vspace{-0.1em}\\begin{align} \\label{cond_entropy2}\n    H(V_{1} | \\underline{A}_{[1]}) \\negthickspace &= - \\negthickspace \\sum_{v_1} \\sum_{\\underline{a}_{[1]}} p(v_{1})p(\\underline{a}_{[1]} | v_{1})\\textup{log}_2\\Big(\\frac{p(v_{1})p(\\underline{a}_{[1]} | v_{1})}{p(\\underline{a}_{[1]})}\\Big).\n\\end{align}\n\nWe start with:\n\\begin{align} \\label{h1_2}\n    p(v_{1}=0) = p(v_{1}=1) = \\frac{1}{2}.\n\\end{align}\nAdditionally, for any binary vector $\\underline{a}_{[1]}$,\n\\begin{align} \\label{h2_2}\n   p(\\underline{a}_{[1]}) = p(\\underline{v}_{[1]}) = \\frac{1}{2^{d_{\\textup{c}}-1}}.\n\\end{align}\n\nLet $E$ and $O$ be the sets of binary vectors of length $d_{\\textup{c}}-1$ that have even and odd number of $1$'s, respectively. We know that to satisfy the SPC equation, $v_{1}=0$ implies that $\\underline{v}_{[1]}$ has an even number of $1$'s, and $v_{1}=1$ implies that $\\underline{v}_{[1]}$ has an odd number of $1$'s.\\footnote{The event that the SPC equation is satisfied, i.e, $\\underline{v}$ has an even number of $1$'s, holds by default. Additional conditioning on this event does not make any impact on the result.} Thus, for $\\underline{a}_{[1]} \\in E$,\n\\begin{align} \n    p(\\underline{a}_{[1]} | v_{1} = 0) &= \\sum_{\\underline{v}_{[1]}} p(\\underline{a}_{[1]} | \\underline{v}_{[1]}, v_{1} = 0) p(\\underline{v}_{[1]} | v_{1} = 0) \\nonumber \\\\ &= \\frac{1}{2^{d_{\\textup{c}}-2}} \\sum_{\\underline{v}_{[1]}} p(\\underline{a}_{[1]} | \\underline{v}_{[1]}, v_{1} = 0). \\label{lolo_1}\n\\end{align}\nThe summation at the end of (\\ref{lolo_1}) is just the probability of an even number of flips at the extrinsic channel. Consequently,   \n\\begin{align}\n    &p(\\underline{a}_{[1]} | v_{1} = 0) = \\frac{1}{2^{d_{\\textup{c}}-2}} \\hspace{-0.7em} \\sum_{i=0}^{\\left \\lfloor {(d_{\\textup{c}}-1)\/2}\\right \\rfloor} \\negthickspace \\binom{d_{\\textup{c}}-1}{2i} \\delta^{2i}(1-\\delta)^{d_{\\textup{c}}-1-2i}, \\label{h3_2}\n\\end{align}\nUsing the same logic for $\\underline{a}_{[1]} \\in O$ gives:\n\\begin{align}\n    &p(\\underline{a}_{[1]} | v_{1} = 0) = \\frac{1}{2^{d_{\\textup{c}}-2}} \\hspace{-0.7em} \\sum_{i=0}^{\\left \\lfloor {(d_{\\textup{c}}-2)\/2}\\right \\rfloor} \\negthickspace \\binom{d_{\\textup{c}}-1}{2i+1} \\delta^{2i+1}(1-\\delta)^{d_{\\textup{c}}-2-2i}. \\label{h4_2}\n\\end{align}\nLet $\\underline{a}^{\\textup{e}}_{[1]}$ be in $E$ and $\\underline{a}^{\\textup{o}}_{[1]}$ be in $O$. Observe that due to symmetry, $p(\\underline{a}^{\\textup{e}}_{[1]} | v_{1} = 0) = p(\\underline{a}^{\\textup{o}}_{[1]} | v_{1} = 1)$ and $p(\\underline{a}^{\\textup{o}}_{[1]} | v_{1} = 0) = p(\\underline{a}^{\\textup{e}}_{[1]} | v_{1} = 1)$. \n\nNow, we substitute equations (\\ref{h1_2}), (\\ref{h2_2}), (\\ref{h3_2}), and (\\ref{h4_2}) in (\\ref{cond_entropy2}), and we get:\n\\begin{align}\n    H(V_{1} | \\underline{A}_{[1]}) &= H \\bigg(\\sum_{i=0}^{\\left \\lfloor {(d_{\\textup{c}}-2)\/2}\\right \\rfloor} \\negthickspace \\binom{d_{\\textup{c}}-1}{2i+1} \\delta^{2i+1}(1-\\delta)^{d_{\\textup{c}}-2-2i} \\bigg) \\nonumber \\\\ \n    &= H\\bigg(\\frac{1}{2}\\Big[1-(1-2\\delta)^{d_{\\textup{c}}-1}\\Big]\\bigg) \\label{cond2},\n\\end{align}\nwhere second equation is obtained from Gallager's thesis \\cite{gallager}. Finally, substituting (\\ref{cond2}) in (\\ref{rule2}) completes the proof.\n\\end{IEEEproof}\n\nContinuing with this step, the EXIT function $I_{\\textup{E,c}}(I_{\\textup{A}})$ and the inverse EXIT function $I^{-1}_{\\textup{E,c}}(I_{\\textup{A}})$ for $ \\delta \\in (0,0.5)$ are then:\n\\begin{align}\n     &I_{\\textup{E,c}}(I_{\\textup{A}}) = 1- H\\bigg(\\frac{1}{2}\\Big[1-\\Big(1-2H^{-1}(1- I_{\\textup{A}})\\Big)^{d_{\\textup{c}}-1}\\Big]\\bigg), \\\\\n     &I^{-1}_{\\textup{E,c}}(I_{\\textup{A}}) = 1- H\\bigg(\\frac{1}{2}\\Big[1-\\Big(1-2H^{-1}(1- I_{\\textup{A}})\\Big)^{\\frac{1}{d_{\\textup{c}}-1}}\\Big]\\bigg). \\label{fcn_bsc_Iec_inv}\n\\end{align}\n\n\\begin{remark}\nWhen the communication channel is BSC, the extrinsic channel can also be assumed as additive white Gaussian noise (AWGN) channel. In a way conceptually connected to the threshold analysis in \\cite{gallager}, we derive primary conclusions by assuming the extrinsic channel as BSC in this work.\n\\end{remark}\n\n\\noindent\\textbf{\\underline{Step~3:}} We now apply the UEP setup explained in Section~\\ref{sec_steps} to derive $I^{*}_{\\textup{E,v}}$. Parity and input information bits transmitted through the communication channel face crossover probabilities $\\epsilon_{1}$ and $\\epsilon_{2}$, respectively. Substituting (\\ref{bsc_Iev}), (\\ref{bsc_Iev_terms}) with $\\delta = H^{-1}(1-I_{\\textup{A}})$, $\\delta \\in (0,0.5)$, in (\\ref{general_Iev}), we derive the EXIT function $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$.\n\n\\noindent\\textbf{\\underline{Step~4:}} Substituting (\\ref{fcn_bsc_Iec_inv}) and $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$ found in previous step in (\\ref{constraint2}) completes the LP problem in (\\ref{LP}). We now solve this LP problem to find the optimal degree distribution. That is, we maximize the code rate in (\\ref{rate}), subject to degree distribution constraint (\\ref{constraint1}) and EXIT convergence constraint (\\ref{constraint2}).\n\\newline\n\nWe find the solution of the LP problem numerically using a software program. In order to investigate the threshold gains of our UEP idea over uniform protection, we applied the first method in BSC. As explained in detail in Section~\\ref{sec_bec}, we first solve the LP problem for uniform protection case, i.e., $\\epsilon_1 = \\epsilon_2 = \\epsilon_{\\textup{uniform}}$, and find the optimal degree distribution. A code with this degree distribution has \\textit{threshold} $\\epsilon_{\\textup{uniform}}$. For the same code construction, we search for the best ($\\epsilon'_1, \\epsilon'_2$) combination with the highest average that satisfies the EXIT convergence constraint in (\\ref{constraint2}) using (\\ref{fcn_bsc_Iec_inv}) and $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$ found in Step 3. The average crossover probability ($\\epsilon'_{\\textup{avg}}$) and the percentage threshold gains are computed as in (\\ref{eqn_qavg}) and (\\ref{eqn_tg}) by replacing $q$'s with $\\epsilon$'s. Numerical results demonstrating the UEP threshold gains are provided in Table~\\ref{table2}.\n\nIn Fig.~\\ref{fig_3}, the top curve, which is the VN curve for uniform error protection $I_{\\textup{E,v}}(I_{\\textup{A}})$, is generated by substituting the solution of the LP problem in $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$ with $\\epsilon_{\\textup{uniform}} = 0.028$. The CN inverse curve $I^{-1}_{\\textup{E,c}}(I_{\\textup{A}})$, which is the bottom curve, is generated from (\\ref{fcn_bsc_Iec_inv}). As observed, LP constraints are satisfied. Using the solution of the LP problem, VN curve with UEP is generated, and it is the middle curve. As  observed, $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$ with the best $\\epsilon'_1$ and $\\epsilon'_2$ values ($\\epsilon'_1 = 0.001, \\epsilon'_2 = 0.039$), satisfies the constraints. \n\n\\begin{figure}\n\\vspace{-1.1em}\n\\centering\n\\hspace{+0.5em}\\includegraphics[trim={1.3in 0.5in 1.3in 0.5in},width=3.4in]{plot_bsc.pdf}\n\\vspace{-1.9em}\n\\caption{EXIT chart on BSC. The LP problem is solved for uniform protection with $\\epsilon_{\\textup{uniform}}=0.028$, and the top curve represents $I_{\\textup{E,v}}(I_{\\textup{A}})$. For the same code parameters, $\\epsilon'_1 = 0.001$ and $\\epsilon'_2 = 0.039$ are reached for UEP, and the middle curve represents $I^{*}_{\\textup{E,v}}(I_{\\textup{A}})$. The bottom curve represents $I^{-1}_{\\textup{E,c}}(I_{\\textup{A}})$.}\n\\label{fig_3}\n\\vspace{-0.8em}\n\\end{figure}\n\n\n\\begin{table}\n\\caption{Threshold Gains of UEP at Various Crossover Probabilities When LP is Solved for Uniform Error Protection on BSC}\n\\vspace{-0.5em}\n\\centering\n\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{6}{|c|}{\\makecell{LP solved for uniform protection}} \\\\\n\\hline\nRate & $\\epsilon_{\\textup{uniform}}$ & $\\epsilon'_1$ & {$\\epsilon'_2$} & $\\epsilon'_{\\textup{avg}}$ & \\textbf{Gain}   \\\\\n\\hline\n$0.5815$ & $0.0820$ & $0.001$ & $0.1800$ & $0.1051$ & $28.2\\%$ \\\\\n\\hline\n$0.6349$ & $0.0670$ & $0.001$ & $0.1300$ & $0.0829$ & $23.7\\%$ \\\\\n\\hline\n$0.6747$ & $0.0570$ & $0.001$ & $0.1010$ & $0.0685$ & $20.1\\%$ \\\\\n\\hline\n$0.7051$ & $0.0498$ & $0.001$ & $0.0828$ & $0.0587$ & $17.8\\%$ \\\\\n\\hline\n$0.7305$ & $0.0440$ & $0.001$ & $0.0700$ & $0.0514$ & $16.8\\%$ \\\\\n\\hline\n$0.7525$ & $0.0390$ & $0.002$ & $0.0590$ & $0.0449$ & $15.1\\%$  \\\\\n\\hline\n$0.8032$ & $0.0280$ & $0.001$ & $0.0390$ & $0.0315$ & $12.6\\%$  \\\\\n\\hline\n$0.8545$ & $0.0180$ & $0.001$ & $0.0230$ & $0.0198$ & $10.0\\%$ \\\\\n\\hline\n\\end{tabular}\n\n\\label{table2}\n\\vspace{-0.25em}\n\\end{table}\n\nTable~\\ref{table2} presents the threshold gains we obtain by applying our  UEP idea on BSC, when the LP problem is solved for uniform protection.\\footnote{Note that for uniform protection, threshold lines are $3-13\\%$ away from capacity lines (see also \\cite{brink_ra} and \\cite{verdu_ira}).} For the same code construction, the best unequal crossover probabilities satisfying the convergence constraints are found. In a way similar to Table~\\ref{table1}, at moderate and lower rates corresponding to higher crossover probabilities, we obtain higher gains. At a rate of around $0.58$, which is suitable to wireless communication, $28\\%$ gain is obtained with higher protection of parity bits. At data storage rates of around $0.75$ and $0.85$, significant gains of $15\\%$ and $10\\%$ are achieved, respectively. At a moderate rate of around $0.7$, almost $18\\%$ gain is obtained. This gain means that with our UEP setup, the exact same code can appropriately operate at average crossover probabilities that are up to $18\\%$ higher than the maximum it can operate under uniform setup. In other words, the threshold is improved by $18\\%$.\n\nIn data storage terms, as the device deteriorates, lower rate codes are typically used to combat higher error probabilities and increase lifetime. Whereas with our UEP idea, lifetime gains can be achieved with no (or limited) rate loss.\n\n\\begin{remark}\nIn \\cite{ahh_loco}, density gains of up to $20\\%$ in MR channels were achieved when parity bits of an LDPC code are protected by constrained (LOCO) coding. We model the higher protection of parity bits via constrained coding by applying lower crossover (resp., erasure) probability to parity bits in BSC (resp., BEC). The threshold gains illustrated in Table~\\ref{table2} (resp., Table~\\ref{table1}) are consistent with, and thus confirm, the empirical results in \\cite{ahh_loco}. This idea works because of the message passing decoding of LDPC codes. Highly reliable messages, e.g., LLRs, of parity bits are spread to other bits during message passing, which results in threshold gains in our UEP setup. In our future work, we will also simulate finite-length RA codes in BEC and BSC models to further investigate our UEP idea. We will also investigate this UEP idea for advanced channels, including the AWGN channel and channels with interference like MR \\cite{ahh_loco}, Flash \\cite{ahh_qaloco}, and two-dimensional MR (TDMR) channels \\cite{ahh_general, ahh_md}.\n\\end{remark}\n\n\\section{Conclusion}\\label{sec_conc}\n\nWe analyzed threshold gains of applying UEP via parity bits of an LDPC code on BEC and BSC. We modeled UEP using lower erasure\/crossover probabilities for parity bits compared to input information bits on BEC\/BSC for EXIT analysis. We used EXIT functions as a tool to understand the effect of a change in the mutual information of parity bits on the behavior of the overall coding scheme. We described the decoding model we use for EXIT analysis, and constructed a suitable RA code. We proposed a systematic methodology for the EXIT analysis of our UEP setup when both the communication and extrinsic channels are BEC\/BSC. We derived a-priori and extrinsic information functions for VNs and CNs. For both channels, we constructed an LP problem to maximize the rate of the code under code construction and convergence constraints by finding the optimal degree distribution. After determining the code parameters, we made comparisons between the uniform protection setup and our UEP setup for the same code. We demonstrated up to around $17\\%$ and $28\\%$ UEP threshold gains in BEC and BSC, respectively. We plan to extend this work to AWGN channel along with practical data storage channels in our future work. We suggest that this UEP setup can contribute to significant density\/lifetime gains in various data storage devices.\n\n\n\\section*{Acknowledgment}\\label{sec_ack}\n\nThis research was supported by NSF under Grant CCF 1908730 and by AFOSR under Grant FA 8750-20-2-0504.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\nPoint processes are a standard model when the objects of study are the number and repartition of otherwise identical points on a domain, usually time or space. The Poisson Point Process is probably the most commonly used point process. It is fully characterised by an intensity function that is inferred from the data. Gaussian Processes have been successfully used to form a prior over the (log-) intensity function for applications such as astronomy \\cite{Gregory92}, forestry \\cite{heik99}, finance \\cite{basu02}, and neuroscience \\cite{Cunni08b}. We offer extensions to existing work as follows:  we develop an exact non-parametric Bayesian model that enables inference on Poisson processes. Our method scales linearly with the number of data points and does not resort to gridding the domain. We derive a MCMC sampler for core components of the model and show that our approach offers a faster and more accurate solution, as well as producing less correlated samples, compared to other approaches on both real-life and synthetic data.\n\n\\section{RELATED WORK}\nNon-parametric inference on point processes has been extensively studied in the literature. \\cite{Rathbum94} and \\cite{Moller98} used a finite-dimensional piecewise constant log-Gaussian for the intensity function. Such approximations are limited in that the choice of the grid on which to represent the intensity function is arbitrary and one has to trade-off precision with computational complexity and numerical accuracy, with the complexity being cubic in the precision and exponential in the dimension of the input space.  \\cite{Kottas06, Kottas07} used a Dirichlet process mixture of Beta distributions as prior for the normalised intensity function of a Poisson process. \\cite{Cunni08} proposed a model using Gaussian Processes evaluated on a fixed grid for the estimation of intensity functions of renewal processes with log-concave renewal distributions. They turned hyper-parameters inference into an iterative series of convex optimization problems, where ordinarily cubic complexity operations such as Cholesky decompositions are evaluated in $\\mathcal{O}(n\\log{n})$ leveraging the uniformity of the grid and the log-concavity of the renewal distribution. \\cite{Murray09} proposed an exact Markov Chain Monte Carlo (MCMC) inference scheme for the posterior intensity function of a Poisson Process with a Sigmoid Gaussian prior intensity, or equivalently a Cox Process \\cite{Cox55} with Sigmoid Gaussian stochastic intensity. The authors simplified the likelihood of a Cox process by introducing latent \\textit{thinning points}. The proposed scheme has a complexity exponential in the dimension of the input space, cubic in the number of data and thinning points, and performs particularly poorly when the data are sparse. \\cite{gunter} extended this model to structured point processes. \\cite{YWT11} used \\textit{uniformization} to produce exact samples from a non-stationary renewal process whose hazard function is modulated by a Gaussian Process, and consequently proposed an MCMC sampler to sample from the posterior intensity of a unidimensional point process. Although the authors have illustrated that their model is faster than \\cite{Murray09} on some synthetic and real-life data, their method still scales cubically in the number of thinned and data points, and is not applicable to data in dimension higher than 1, such as spatial point processes.\n\n\\section{MODEL}\n\\subsection{Setup}\nWe are tasked with making non-parametric Bayesian inference on the intensity function of a Poisson Point Process assumed to have generated a dataset $\\mathcal{D} =\\{ s_1, ..., s_n\\}$. To simplify the discourse without loss of generality, we will assume that data points take values in $\\mathbb{R}^d$.\n\nFirstly, let us recall that a \\textit{\\textbf{Poisson Point Process}} (PPP) on a bounded domain $\\mathcal{S} \\subset \\mathbb{R}^d$ with non-negative \\textit{\\textbf{intensity function}} $\\lambda$ is a locally finite random collection of points in $\\mathcal{S}$ such that the numbers of points occurring in disjoint parts $B_i$ of $\\mathcal{S}$ are independent and each follows a Poisson distribution with mean $\\int_{B_i} \\lambda(s) ds$.\n\nThe likelihood of a PPP is given by:\n\\begin{align}\n\\label{eq:likelihood_poisson}\nL(\\lambda|s_1, ...,s_n) = \\exp\\left(-\\int_{\\mathcal{S}} \\lambda(s) ds\\right) \\prod_{i=1}^n \\lambda(s_i)\n\\end{align}\n\n\\subsection{Tractability discussion}\nThe approach adopted thus far in the literature to make non-parametric Bayesian inference on Point Process using Gaussian Processes (GP) \\cite{Rassmussen06} consists of putting a \\textit{functional prior} on the intensity function in the form of a positive function of a GP: $\\lambda(s)= f(g(s))$ where $g$ is drawn from a GP and $f$ is a positive function. Examples of such $f$ include the exponential function and a scaled sigmoid function \\cite{Murray09, YWT11}. This approach can be seen as a Cox Process where the stochastic intensity follows the same dynamics as the functional prior. When the Gaussian Process used has almost surely continuous paths, the random vector\n\\begin{align}\n\\label{eq:tractable_joint}\n(\\lambda(s_1),..., \\lambda(s_n), \\int_{\\mathcal{S}} \\lambda(s) ds)\n\\end{align}\nprovably admits a probability density function (pdf). Moreover, we note that any piece of information not contained in the implied pdf over the vector in Equation (\\ref{eq:tractable_joint}) will be lost as the likelihood only depends on those variables. Hence, given a functional prior postulated on the intensity function, the only necessary piece of information to be able to make a full Bayesian treatment is the implied joint pdf over the vector in Equation (\\ref{eq:tractable_joint}). \n\nFor many useful transformations $f$ and covariance structures for the GP, the aforementioned implied pdf might not be available analytically. We note however that there is no need to put a \\textit{functional prior} on the intensity function. In fact, for every \\textit{finite-dimensional prior} over the vector in Equation (\\ref{eq:tractable_joint}), there exists a Cox process with an a.s. $\\mathcal{C}^\\infty$ intensity process that coincides with the postulated prior (see appendix for the proof).\n\nThis approach is similar to that of \\cite{Kottas06}. The author regarded $I =\\int_{\\mathcal{S}} \\lambda(s) ds$ as a random variable and noted that $p(s)=\\frac{\\lambda(s)}{\\int_{\\mathcal{S}} \\lambda(s) ds}$ can be regarded as a pdf whose support is the domain $\\mathcal{S}$. He then made inference on $(I, p(s_1), ..., p(s_n))$, postulating as prior that $I$ and $(p(s_1), ..., p(s_n))$ are independent, $I$ has a Jeffreys prior and $(s_1, ..., s_n)$ are i.i.d. draws from a Dirichlet Process mixture of Beta with pdf $p$. \n\nThe model we present in the following section puts an appropriate \\textit{finite-dimensional} prior on $(\\lambda(s_1),..., \\lambda(s_n), \\lambda(s^\\prime_1),..., \\lambda(s^\\prime_k), \\int_{\\mathcal{S}} \\lambda(s) ds)$ for some inducing points $s^\\prime_j$ rather than putting a \\textit{functional prior} on the intensity function directly.\n\n\\subsection{Our model}\n\\subsubsection{Intuition}\nThe intuition behind our model is that the data are not a `natural grid' at which to infer the value of the intensity function. For instance, if the data consists of 200,000 points on the interval $[0, 24]$ as in one of our experiments, it might not be necessary to infer the value of a function at 200,000 points to characterise it on $[0, 24]$. Instead, we find a small set of inducing points $\\mathcal{D}^\\prime = \\{s^\\prime_1, ..., s^\\prime_k \\}, k \\ll n $ on our domain, through which we will define the prior over the vector in Equation (\\ref{eq:tractable_joint}) augmented with $\\lambda(s^\\prime_1),..., \\lambda(s^\\prime_k)$. The set of inducing points will be chosen so that knowing $\\lambda(s^\\prime_1),..., \\lambda(s^\\prime_k)$ would result in knowing the values of the intensity function elsewhere on the domain, in particular $\\lambda(s_1),..., \\lambda(s_n)$, with `arbitrary certainty'. We will then analytically integrate out the dependency in $\\lambda(s_1),..., \\lambda(s_n)$ from the posterior, thereby reducing the complexity from cubic to linear in the number of data points without `loss of information', and reformulating our problem as that of making exact Bayesian inference on the value of the intensity function at the inducing points. We will then describe how to obtain predictive mean and variance of the intensity function elsewhere on the domain from training.\n\n\\subsubsection{Model specification}\nLet us denote by $\\lambda^{*}$ a positive stochastic process on $\\mathcal{S}$ such that $\\log{\\lambda^{*}}$ is a stationary Gaussian Process  with covariance kernel $\\gamma^{*}: (s_1, s_2) \\to \\gamma^{*}(s_1, s_2)$ and constant mean $m^{*}$. Let us further denote by $\\hat{\\lambda}$ a positive stochastic process on $\\mathcal{S}$ such that $\\log{\\hat{\\lambda}}$ is a  \\textit{\\textbf{Conditional Gaussian Process}} \\textit{coinciding with $\\log{\\lambda^{*}}$ at $k$ inducing points $\\mathcal{D}^\\prime = \\{s^\\prime_1, ..., s^\\prime_k \\}, k \\ll n $}. That is, $\\log{\\hat{\\lambda}}$ is the non-stationary Gaussian Process whose mean function $m$ is defined by\n\\begin{align}\n\\label{eq:cond_mean}\nm(s) = m^{*} + \\Sigma^{*}_{s\\mathcal{D}^\\prime}\\Sigma^{*-1}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}G\n\\end{align}\nwhere $G=\\big(\\log{\\lambda^{*}(s^\\prime_1)} - m^*, ..., \\log{\\lambda^{*}(s^\\prime_k)} - m^*\\big)$ and $\\Sigma^{*}_{XY}$ is the covariance matrix between the vectors $X$ and $Y$ under the covariance kernel $\\gamma^*$. Moreover,  $\\log{\\hat{\\lambda}}$ is such that for every vector $S_1$ of points in $\\mathcal{S}$, the auto-covariance matrix $ \\Sigma_{S_1S_1}$ of the values of process at $S_1$  reads\\footnote{The positive definitiveness of the induced covariance kernel $\\gamma$ is a direct consequence of the positive definitiveness of $\\gamma^{*}$.}\n\\begin{align}\n\\label{eq:cond_cov}\n& \\Sigma_{S_1S_1} = \\Sigma^{*}_{S_1S_1} - \\Sigma^{*}_{S_1\\mathcal{D}^\\prime}\\Sigma^{*-1}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}\\Sigma^{*T}_{S_1\\mathcal{D}^\\prime}.\n\\end{align}\n\nThe prior distribution in our model is constructed as follows: \n\\begin{enumerate}\n  \\item $\\{\\log{\\lambda(s^\\prime_i)}\\}_{i=1}^{k}$ are samples from the stationary GP $\\log{\\lambda^{*}}$ at $\\{s^\\prime_i\\}_{i=1}^{k}$ respectively, with $m^{*} =\\log{\\frac{\\# \\mathcal{D}}{\\mu(\\mathcal{S})}}$, where $\\mu(\\mathcal{S})$ is the size of the domain.\n  \\item $I=\\int_{\\mathcal{S}}\\lambda(s) ds$ and  $\\{\\log{\\lambda(s_j)}\\}_{j=1}^{n}$ are conditionally independent given  $\\{\\log{\\lambda(s^\\prime_i)}\\}_{i=1}^{k}$.\n  \\item Conditional on $\\{\\log{\\lambda(s^\\prime_i)}\\}_{i=1}^{k}$, $\\{\\log{\\lambda(s_j)}\\}_{j=1}^{n}$ are independent, and for each $j \\in [1..n]$ $\\log{\\lambda(s_j)}$ follows the same distribution as $\\log{\\hat{\\lambda}(s_j)}$.\n  \\item Conditional on $\\{\\log{\\lambda(s^\\prime_i)}\\}_{i=1}^{k}$, $I$ follows a Gamma distribution with shape $\\alpha_I$ and scale $\\beta_I$.\n    \\item The mean $\\mu_I = \\alpha_I \\beta_I$ and variance $\\sigma^2_I = \\alpha_I \\beta_I^2$ of $I$ are that of $\\int_{\\mathcal{S}}  \\hat{\\lambda}(s) ds$.\n\\end{enumerate}\nAssertion 3. above is somewhat similar to the FITC model of \\cite{FTCI}. \n\nThis construction yields a prior pdf of the form:\n\\begin{align}\n\\label{eq:tractable_joint_prior_condgp}\n& p\\big(\\log{\\lambda(s_1)},..., \\log{\\lambda(s_n)}, \\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)},  I,  \\theta \\big) \\nonumber \\\\ &=   \\mathcal{N}\\big(\\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)} \\rvert m^{*} 1_{k}, \\Sigma^{*}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}\\big)  \\nonumber \\\\\n& \\times \\mathcal{N}\\big(\\log{\\lambda(s_1)},..., \\log{\\lambda(s_n)} \\rvert  M, \\text{diag}(\\Sigma_{\\mathcal{D}\\mathcal{D}})\\big)  \\nonumber \\\\\n& \\times \\gamma_d\\big(I\\rvert \\alpha_I, \\beta_I\\big) \\times p(\\theta)\n\\end{align}\n\nwhere $\\mathcal{N}(. \\rvert X, C)$ is the multivariate Gaussian pdf with mean $X$ and covariance matrix $C$, $M = (m(s_1), ..., m(s_n))$, $1_k$ is the vector with length k and elements 1, $\\text{diag}(\\Sigma_{\\mathcal{D}\\mathcal{D}})$ is the diagonal matrix whose diagonal is that of $\\Sigma_{\\mathcal{D}\\mathcal{D}}$, $\\gamma_d\\big(x \\rvert \\alpha, \\beta)$ is the pdf of the gamma distribution with shape $\\alpha$ and scale $\\beta$, and where $\\theta$ denotes the hyper-parameters of the covariance kernel $\\gamma^{*}$.\n\nIt follows from the fifth assertion in our prior specification that $\\alpha_I = \\frac{\\mu_I^2}{\\sigma_I^2} \\text{ and } \\beta_I = \\frac{\\sigma_I^2}{\\mu_I}$. We also note that\n\\begin{align}\n\\label{eq:gamma_mean}\n\\mu_I &= \\text{E}\\big(\\int_{\\mathcal{S}}  \\hat{\\lambda}(s) ds\\big)\\nonumber \\\\\n& = \\int_{\\mathcal{S}}  \\text{E}\\big(\\exp(\\log{\\hat{\\lambda}(s)}) \\big)ds \\nonumber \\\\\n& = \\int_{\\mathcal{S}}  \\exp(m(s)+\\frac{1}{2} \\gamma(s,s))ds\n& := \\int_{\\mathcal{S}}  f(s)ds\n\\end{align}\nand \n\\begin{align}\n\\label{eq:gamma_variance}\n\\sigma_I^2 &= \\text{E}\\bigg(\\big(\\int_{\\mathcal{S}}  \\hat{\\lambda}(s) ds\\big)^2\\bigg) -\\mu_I^2\\nonumber \\\\\n&= \\text{E}\\bigg(\\int_{\\mathcal{S}}\\int_{\\mathcal{S}} \\exp(\\log{\\hat{\\lambda}(s_1)}+\\log{\\hat{\\lambda}(s_2)}) ds_1 ds_2\\bigg) -\\mu_I^2\\nonumber \\\\\n&= \\int_{\\mathcal{S}}\\int_{\\mathcal{S}} \\text{E}\\bigg(\\exp\\big(\\log{\\hat{\\lambda}(s_1)}+\\log{\\hat{\\lambda}(s_2)}\\big)\\bigg) ds_1 ds_2 -\\mu_I^2\\nonumber \\\\\n& = \\int_{\\mathcal{S}} \\int_{\\mathcal{S}}  \\exp\\big(m(s_1)+ m(s_2) + \\gamma(s_1, s_2) + \\frac{1}{2} \\gamma(s_1,s_1) \\nonumber \\\\\n& + \\frac{1}{2} \\gamma(s_2,s_2)\\big)ds_1 ds_2 -\\mu_I^2 \\nonumber \\\\\n& := \\int_{\\mathcal{S}} \\int_{\\mathcal{S}} g(s_1, s_2) ds_1 ds_2 -\\mu_I^2.\n\\end{align}\n\nThe integrals in Equations (\\ref{eq:gamma_mean}) and (\\ref{eq:gamma_variance}) can be easily evaluated with numerical methods such as Gauss-Legendre quadrature \\cite{Hilde56}. \n\nIn particular, when $\\mathcal{S}=[a,b]$, \n\\begin{align}\n\\mu_I \\approx \\frac{b-a}{2} \\sum_{i=1}^{p} \\omega_i f\\left(\\frac{b-a}{2} x_i + \\frac{b+a}{2}\\right) \n\\end{align}\nand \n\\begin{align}\n& \\sigma_I^2 \\approx \\frac{(b-a)^2}{4} \\sum_{i=1}^{p} \\sum_{j=1}^{p} \\omega_i \\omega_j g\\Big(\\frac{b-a}{2} x_i + \\frac{b+a}{2}, \\frac{b-a}{2} x_j + \\nonumber \\\\\n& \\frac{b+a}{2}\\Bigg)  -\\mu_I^2\n\\end{align}\nwhere the roots $x_i$ of the Legendre polynomial of order $p$ and the weights $\\omega_i$ are readily available from standard textbooks on numerical analysis such as \\cite{Hilde56} and scientific programming packages (R, Matlab and Scipy). Extensions to rectangles in higher dimensions are straightforward. Moreover, the complexity of such approximations only depends on the number of inducing points and $p$ (see Equations (\\ref{eq:cond_mean}) and (\\ref{eq:cond_cov})), and hence \\emph{scales well} with the data size.\n\nA critical step in the derivation of our model is to analytically integrate out $\\log{\\lambda(s_1)},..., \\log{\\lambda(s_n)}$ in the posterior, to eliminate the cubic complexity in the number of data points.\nTo do so, we note that:\n\\begin{align}\n\\label{eq:moment_gen}\n&\\int_{(\\mathbb{R}^d)^n}  \\prod_{i=1}^{n} \\lambda(s_i) \\mathcal{N}\\big(\\log{\\lambda(s_1)},..., \\log{\\lambda(s_n)}\\rvert  M, \\nonumber \\\\\n& \\text{diag}(\\Sigma_{\\mathcal{D}\\mathcal{D}})\\big)  d\\log{\\lambda(s_1)}...d\\log{\\lambda(s_n)} \n\\nonumber \\\\\n& = \\text{E}\\left(\\exp\\left(\\sum_{i=1}^{n} \\log{\\lambda(s_i)}\\right)\\right) \\nonumber \\\\\n& = \\exp(1_{n}^T M + \\frac{1}{2} \\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}}))\n\\end{align}\nwhere the second equality results from the moment generating function of a multivariate Gaussian.\n\nThus, putting together the likelihood of Equation (\\ref{eq:likelihood_poisson}) and Equation (\\ref{eq:tractable_joint_prior_condgp}), and integrating out $\\big(\\log{\\lambda(s_1)},..., \\log{\\lambda(s_n)}\\big)$, we get:\n\\begin{align}\n\\label{eq:joint}\n& p\\big(\\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)},  I, \\theta \\big \\rvert \\mathcal{D})  \\nonumber \\\\ \n& \\sim p(\\theta)  \\mathcal{N}\\big(\\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)} \\rvert M^{*}, \\Sigma^{*}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}\\big)  \\nonumber \\\\\n& \\times \\exp(1_{n}^T M + \\frac{1}{2}  \\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}}))  \\exp(-I)  \\gamma_d\\big(I \\rvert \\alpha_I, \\beta_I\\big)\n\\end{align}\n\n\nFinally, although our model allows for joint inference on the intensity function and its integral, we restrict our attention to making inference on the intensity function for brevity.  By integrating out  $I$ from Equation (\\ref{eq:joint}), we get the new posterior:\n\\begin{align}\n\\label{eq:joint_int}\n p(\\lambda, \\theta \\rvert \\mathcal{D}) & :=  p\\big(\\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)},\\theta\\big \\rvert \\mathcal{D})  \\\\\n&   \\sim p(\\theta)\\exp(1_{n}^T M +  \\frac{1}{2} \\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}})) (1+\\beta_I)^{-\\alpha_I}  \\nonumber \\\\\n&  \\times \\mathcal{N}\\big(\\log{\\lambda(s^\\prime_1)}, ..., \\log{\\lambda(s^\\prime_k)} \\rvert M^{*}, \\Sigma^{*}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}\\big) \\nonumber\n\\end{align}\nwhere we noted that the dependencies of Equation (\\ref{eq:joint}) in $I$ is of the form $\\exp(-x)\\gamma_d(x \\rvert \\alpha, \\beta)$ which can be integrated out as the moment generating function of the gamma distribution evaluated at $-1$, that is $ (1+\\beta)^{-\\alpha}$.\n\n\\subsubsection{Selection of inducing points}\nInferring the number $k$ and positions of the inducing points $s^\\prime_i$ is critical to our model, as $k$ directly affects the complexity of our scheme and the positions of the inducing points affect the quality of our prediction. Too large a $k$ will lead to an unduly large complexity. Too small a $k$ will lead to loss of information (and subsequently excessively uncertain predictions from training), and might make assertion 2 of our prior specification inappropriate. For a given $k$, if the inducing points are not carefully chosen, the coverage of the domain will not be adapted to changes in the intensity function and as a result, the predictive variance in certain parts of the domain might considerably differ from the posterior variance we would have obtained, had we chosen inducing points in those parts of the domain.\n\nIntuitively, a good algorithm to find inducing points should leverage prior knowledge about the smoothness, periodicity, amplitude  and length scale(s) of the intensity function to optimize for the quality of (post-training) predictions while minimising the number of inducing points. \n\nWe use as utility function for the choice of inducing points:\n\\begin{align}\n\\label{eq:utility_inducing}\n\\mathcal{U}(\\mathcal{D}^\\prime) = \\text{E}_{\\theta}(\\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*}(\\theta) \\Sigma_{\\mathcal{D^\\prime}\\mathcal{D^\\prime}}^{*-1}(\\theta) \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*T}(\\theta)))\n\\end{align}\nwhere $\\theta$ is the vector of hyper-parameters of the covariance kernel $\\gamma^{*}$, and the expectation is taken with respect to the prior distribution over $\\theta$. In other words, the utility of a set of inducing points is the expected total reduction of the (predictive) variances of $\\log{\\lambda(s_1)}, ...\\log{\\lambda(s_n)}$ resulting from knowing $\\log{\\lambda(s^\\prime_1)}, ...,\\log{\\lambda(s^\\prime_k)}$.\n\nIn practice, the expectation in Equation (\\ref{eq:utility_inducing}) might not be available analytically. We can however use the Monte Carlo estimate:\n\\begin{align}\n\\label{eq:utility_inducing_MC}\n\\tilde{\\mathcal{U}}(\\mathcal{D}^\\prime) &= \\frac{1}{N} \\sum_{i=1}^{N} \\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*}(\\tilde{\\theta}_i) \\Sigma_{\\mathcal{D^\\prime}\\mathcal{D^\\prime}}^{*-1}(\\tilde{\\theta}_i) \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*T}(\\tilde{\\theta}_i)).\n\\end{align}\n\nThe algorithm proceeds as follows. We sample $(\\tilde{\\theta}_i)_{i=1}^{N}$ from the prior. Initially we set $k=0$, $\\mathcal{D}^{\\prime} = \\varnothing$ and $u_0=0$. We increment $k$ by one, and consider adding an inducing point. We then find the point $s_{k}^\\prime$ that maximises $\\tilde{\\mathcal{U}}(\\mathcal{D}^{\\prime} \\cup \\{s\\})$\n\\begin{align}\ns_{k}^\\prime := \\underset{s \\in \\mathcal{S}}{\\text{argmax }} \\tilde{\\mathcal{U}}(\\mathcal{D}^{\\prime} \\cup \\{s\\})\n\\end{align}\nusing Bayesian optimisation \\cite{Mockus13}. We compute the utility of having $k$ inducing points as \\[u_{k} = \\tilde{\\mathcal{U}}(\\mathcal{D}^{\\prime} \\cup \\{s_k^\\prime\\}),\\] we update $\\mathcal{D}^\\prime = \\mathcal{D}^\\prime \\cup \\{s_{k}\\} $ and stop when \\[ \\frac{u_{k}-u_{k-1}}{u_{k}} < \\alpha, \\]\nwhere $0< \\alpha \\ll1$ is a convergence threshold.\n\n\\begin{algorithm}[ht]\n   \\caption{Selection of inducing points}\n   \\label{al:inducing}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Inputs:}$0< \\alpha \\ll1$, $N$, $p_{\\theta}$\n   \\STATE {\\bfseries Output:} $u_{f}$, $\\mathcal{D}^{\\prime}$\n   \\STATE $k=0$, $u_0=0$, $\\mathcal{D}^{\\prime} = \\varnothing$, $e=1$;\n   \\STATE  Sample $(\\tilde{\\theta}_i)_{i=1}^{N}$ from $p(\\theta)$;\n   \\WHILE{$e> \\alpha$}\n    \\STATE $k = k +1;$\n    \\STATE $s^\\prime_k =  \\underset{s \\in \\mathcal{S}}{\\text{argmax }} \\tilde{\\mathcal{U}}(\\mathcal{D}^{\\prime} \\cup \\{s\\});$\n    \\STATE $u_k=\\tilde{\\mathcal{U}}(\\mathcal{D}^{\\prime} \\cup \\{s^\\prime_k\\});$\n    \\STATE $\\mathcal{D}^\\prime = \\mathcal{D}^\\prime \\cup \\{s^\\prime_k\\};$\n     \\STATE $e = \\frac{u_{k}-u_{k-1}}{u_{k}};$\n   \\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\\textbf{Proposition}\n\n\\textit{(a) For any $\\mathcal{D}$, $\\alpha$, $N$ and $p_{\\theta}$ Algorithm~\\ref{al:inducing} stops in finite time and the sequence $(u_k)_{k \\in \\mathbb{N}}$ converges at least linearly with rate $1-\\frac{1}{\\#\\mathcal{D}}$}.\n\n\\textit{(b) Moreover, the maximum utility $u_f(\\alpha)$ returned by Algorithm~\\ref{al:inducing} converges to the average total unconditional variance $w_{\\infty} := \\frac{1}{N} \\sum_{i=1}^{N}  \\text{Tr}(\\Sigma_{\\mathcal{D}\\mathcal{D}}^{*}(\\tilde{\\theta}_i))$ as $\\alpha$ goes to $0$}.\n\nThe idea behind the proof of this proposition is that the sequence of maximum utilities $u_k$ is positive, increasing\\footnote{Intuitively, conditioning on a new point increases the reduction of variance from the unconditional variance.}, and upper-bounded by the total unconditional variance $w_{\\infty}$\\footnote{The variance cannot be reduced by more than the total unconditional variance.}. Hence, the sequence $u_k$ converges to a strictly positive limit, which implies that the stopping condition of the while loop will be met in finite time regardless of $\\mathcal{D}$, $\\alpha$, $N$ and $p_{\\theta}$. Finally, we construct a sequence $w_k$ upper-bounded by the sequence $u_k$ and that converges linearly to the average total unconditional variance $w_{\\infty}$ with rate $1-\\frac{1}{\\#\\mathcal{D}}$. As the sequence $u_k$ converges and is itself upper-bounded by $w_{\\infty}$, its limit is $w_{\\infty}$ as well, and it converges at least as fast as $w_k$. \\textit{(See appendix for the full proof)}\n\nOur algorithm is particularly suitable to Poisson Point Processes as it prioritises sampling inducing points in parts of the domain where the data are denser. This corresponds to regions where the intensity function will be higher, thus where the local random counts of the underlying PPP will vary more\\footnote{The variance of the Poisson distribution is its mean.} and subsequently where the posterior variance of the intensity is expected to be higher. Moreover, it leverages prior smoothness assumptions on the intensity function to limit the number of inducing points and to appropriately and sequentially improve coverage of the domain. \n\nAlgorithm~\\ref{al:inducing} is illustrated on a variety of real life and synthetic data sets in section \\ref{sec:5}.\n\n\\section{INFERENCE}\nWe use a Squared Exponential kernel for $\\gamma^{*}$ and \\textit{Scaled Sigmoid Gaussian} priors for the kernel hyper-parameters; that is $\\theta_i = \\frac{\\theta_{i\\text{max}}}{1+\\exp(-x_i)}$ where $x_i$ are i.i.d standard Normal. The problem-specific scales, $\\theta_{i\\text{max}}$, restrict the supports of those distributions using prior knowledge to avoid unlikely extreme values and to improve conditioning.\n\nWe use a Block Gibbs Sampler \\cite{Gibbs84} to sample from the posterior. We sample the hyper-parameters using the Metropolis-Hastings \\cite{Hastings70} algorithm taking as proposal distribution the prior of the variable of interest. We sample the log-intensities at the inducing points using Elliptical Slice Sampling \\cite{Murray09b} with the pdf in Equation (\\ref{eq:joint_int}).\n\n\\textbf{Prediction from training}\n\nTo predict the posterior mean at the data points we note from the law of total expectation that \n\\begin{align}\n\\label{eq:tot_exp}\n& \\forall s_i \\in \\mathcal{D}, ~ \\text{E}(\\log{\\lambda(s_i)}|\\mathcal{D}) \\nonumber \\\\ & = \\text{E}\\left(\\text{E}\\left(\\log{\\lambda(s_i)}|\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k, \\mathcal{D}\\right)|\\mathcal{D}\\right).\n\\end{align}\nAlso, we note from Equations (\\ref{eq:likelihood_poisson}) and  (\\ref{eq:tractable_joint_prior_condgp}) that the dependency of the posterior of $\\log{\\lambda(s_i)}$ conditional on $\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k$ is of the form \\[\\exp(\\log{\\lambda(s_i)}) \\times \\mathcal{N}(\\log{\\lambda(s_i)}|m(s_i), \\gamma(s_i, s_i)),\\] where we recall that $m(s_i)$ is the $i$-th element of the vector $M$ and $\\gamma(s_i, s_i)$ is the $i$-th diagonal element of the matrix $\\Sigma_{\\mathcal{D}\\mathcal{D}}$. Hence, the posterior distribution of $\\log{\\lambda(s_i)}$ conditional on  $\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k$ is Gaussian with mean \n\\begin{align}\n\\label{eq:predictive_mean}\n\\text{E}\\left(\\log{\\lambda(s_i)}|\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k, \\mathcal{D} \\right)=M[i]+\\Sigma_{\\mathcal{D}\\mathcal{D}}[i,i]\n\\end{align}\nand variance \n\\begin{align}\n\\label{eq:predictive_variance}\n\\text{Var}\\left(\\log{\\lambda(s_i)}|\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k, \\mathcal{D} \\right)= \\Sigma_{\\mathcal{D}\\mathcal{D}}[i,i].\n\\end{align}\nFinally, it follows from Equation (\\ref{eq:tot_exp}) that $\\text{E}(\\log{\\lambda(s_i)}|\\mathcal{D})$ is obtained by averaging out $M[i]+\\Sigma_{\\mathcal{D}\\mathcal{D}}[i,i]$ over MCMC samples after burn-in. \n\nSimilarly, the law of total variance implies that\n\\begin{align}\n& \\text{Var}(\\log{\\lambda(s_i)}|\\mathcal{D})  \\nonumber \\\\\n& = \\text{E}\\left(\\text{Var}\\left(\\log{\\lambda(s_i)}|\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k,\\mathcal{D}\\right)|\\mathcal{D}\\right)  \\nonumber \\\\\n& +\\text{Var}\\left( \\text{E}\\left(\\log{\\lambda(s_i)}|\\{\\log{\\lambda^{*}(s^\\prime_j)}\\}_{j=1}^k, \\mathcal{D}\\right)|\\mathcal{D}\\right).\n\\end{align}\n\nHence, it follows from Equations (\\ref{eq:predictive_mean}) and (\\ref{eq:predictive_variance}) that the posterior variance at a data point $s_i$ is obtained by summing up the sample mean  of $\\Sigma_{\\mathcal{D}\\mathcal{D}}[i,i]$ with the sample variance of $M[i]+\\Sigma_{\\mathcal{D}\\mathcal{D}}[i,i]$, where sample mean and sample variance are taken over MCMC samples after burn-in.\n\n\\section{EXPERIMENTS}\n\\label{sec:5}\nWe selected four data sets to illustrate the performance of our model. We restricted ourselves to one synthetic data set for brevity. We chose the most challenging of the synthetic intensity functions of \\cite{Murray09} and \\cite{YWT11}, $\\lambda(t) =2\\exp(-\\frac{t}{15}) + \\exp(-(\\frac{t-25}{10})^2)$, to thoroughly compare our model with competing methods. We also ran our model on a standard 1 dimensional real-life data set (the coal mine disasters dataset used in \\cite{jarrett79}; 191 points) and a standard real-life 2 dimensional data (spatial location of bramble canes \\cite{Diggle83}; 823 points). Finally we ran our model on a real-life data set large enough to cause problems to competing models. This data set consists of the UTC timestamps (expressed in hours in the day) of Twitter updates in English published in the \\cite{Twitter14} on September 1st 2014 (188544 points).\n\n\\subsection{Inducing points selection}\n\nFigure \\ref{fig:convergence_plot} illustrates convergence of the selection of inducing points on the 4 data sets. We ran the algorithm 10 times with $N=20$, and plotted the average normalised utility $\\frac{u_k}{u_{\\infty}} \\pm 1 \\text{ std}$ as a function of the number of inducing points. Table \\ref{table:hyper} contains the maximum hyper-parameters that were used for each data set. Table \\ref{table:num_ind} contains the number of inducing points required to achieve some critical normalised utility values for each of the 4 data sets. We note that just 8 inducing points were required to achieve a 95\\% utility for the Twitter data set (188544 points). In regards to the positions of sampled inducing points, we note from Figures \\ref{fig:expo_gauss_plot} and \\ref{fig:real_data_plot} that when the intensity function was bimodal, the first inducing point was sampled around the argument of the highest mode, and the second inducing point was sampled around the argument of the second highest mode. More generally, the algorithm sampled inducing points where the latent intensity function varies the most, as expected.\n\n\\begin{table}[t]\n\\caption{Maximum output (resp. input) scale $h_{\\text{max}}$ (resp. $l_{\\text{max}}$) used for each data set to select inducing points.}\n\\label{table:hyper}\n\\vskip 0.1in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{  l | l | l | l | l}\n    \\hline\n     & synthetic & coal mine & bramble & twitter \\\\ \\hline\n    $h_{\\text{max}}$ & 10.0 & 10.0& 10.0 & 10.0  \\\\ \\hline\n    $l_{\\text{max}}$ & 25.0 & 50.0 & 0.25 & 5.0  \\\\ \\hline\n    \\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{paper_convergence_plot.eps}\n\\caption{Average normalised utility $\\frac{u_k}{u_{\\infty}}$ of choosing k inducing points using Algorithm~\\ref{al:inducing} $\\pm$ 1 standard deviation as a function of k on the synthetic data set (eg), the coal mine data set (cm), the Twitter data set (t) and the bramble canes data set (b). The average was taken over 10 runs.}\n\\label{fig:convergence_plot}\n\\end{figure}\n\n\\begin{table}\n\\caption{Number of inducing points produced by Algorithm \\ref{al:inducing} required to achieve some critical normalised utility values on the 4 data sets.}\n\\label{table:num_ind}\n\\vskip 0.1in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{ l | l | l | l | l |}\n    \\cline{2-5}\n  & \\multicolumn{4}{c|} {$k$}  \\\\ \\hline\n  $\\frac{u_k}{u_{\\infty}}$ & synthetic & coal mine & bramble & twitter \\\\ \\cline{1-5} \\hline\n    0.75& 2 & 2& 8 & 3  \\\\ \\cline{1-5} \\hline\n    0.90 & 3 & 4 & 17 & 5 \\\\ \\cline{1-5} \\hline\n    0.95 & 4 & 5 & 28 & 8 \\\\ \\cline{1-5} \\hline\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\n\\subsection{Intensity function}\nIn each experiment we generated 5000 samples after burn-in (1000 samples). For each data set we used the set of inducing points that yielded a 95\\% normalized utility. The exact numbers are detailed in Table \\ref{table:num_ind}.\n\nWe ran a Monte Carlo simulation for the stochastic processes considered herein and found that the Legendre polynomial order $p=10$ was sufficient to yield a Quadrature estimate for the standard deviation of the integral less than 1\\% away from the Monte Carlo estimate (using the trapezoidal rule), and a Quadrature estimate for the mean of the integral less than a standard error away from the Monte Carlo average. We took a more conservative stand and used $p=20$.\n\n\\textbf{Inference on synthetic data}\n\nWe generated a draw from a Poisson point process with the intensity function $\\lambda(t) =2\\exp(-\\frac{t}{15}) + \\exp(-(\\frac{t-25}{10})^2)$ of \\cite{Murray09} and \\cite{YWT11}. The draw consisted of 41 points (blue sticks in Figure \\ref{fig:expo_gauss_plot}). We compared our model to \\cite{Murray09} (SGCP) and \\cite{YWT11} (RMP). We ran the RMP model with the renewal parameter $\\gamma$ set to 1 (RMP 1), which corresponds to an exponential renewal distribution or equivalently an inhomogeneous Poisson process. We also ran the RMP model with a uniform prior on $[1,5]$ over the renewal parameter $\\gamma$ (RMP full). Figure \\ref{fig:expo_gauss_plot} illustrates the posterior mean intensity function under each model. Finally we ran the Dirichlet Process Mixture of Beta model of \\cite{Kottas06} (DPMB). As detailed in Table \\ref{table:synth_comp}, our model outperformed that of \\cite{Murray09}, \\cite{YWT11} and \\cite{Kottas06} in terms of accuracy and speed.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{paper_expo_gauss_plot}\n\\caption{Inference on a draw (blue sticks) from a Poisson point process with intensity $\\lambda(t) =2\\exp(-\\frac{t}{15}) + \\exp(-(\\frac{t-25}{10})^2)$ (black line). The red dots are the inducing points generated by our algorithm, labelled in the order they were selected.  The solid blue line and the grey shaded area are the posterior mean $\\pm$ 1 posterior standard deviation under our model. SGCP is the posterior mean under \\cite{Murray09}. RMP full and RMP 1 are the posterior mean intensities under  \\cite{YWT11} with $\\gamma$ inferred and set to $1$ respectively. DPMB is the Dirichlet Process mixture of Beta \\cite{Kottas06}}\n\\label{fig:expo_gauss_plot}\n\\end{figure}\n\n\\begin{table}\n\\caption{Some statistics on the MCMC runs of Figure \\ref{fig:expo_gauss_plot}. RMSE and MAE denote the Root Mean Square Error and the Mean Absolute Error, expressed as a proportion of the average of the true intensity function over the domain. LP denotes the log mean predictive probability on 10 held out PPP draws from the true intensity $\\pm$ 1 std. t(s) is the average time in seconds it took to generate 1000 samples $\\pm$ 1 std and ESS denotes the average effective sample size \\cite{Gelman13} per 1000 samples.}\n\\label{table:synth_comp}\n\\vskip 0.1in\n   \\begin{center}\n   \\begin{small}\n   \\begin{sc}\n\\resizebox{0.5\\textwidth}{!}{\n    \\begin{tabular}{  l | l | l | l | l | l}\n    \\hline\n     & MAE & RMSE & LP & t (s) & ESS\\\\ \\hline\n    SGCP & 0.31 & 0.37 & -45.07 $\\pm$ 1.64  & 257.72 $\\pm$ 16.29 & 6\\\\ \\hline\n    RMP 1 & 0.32 & 0.38 & -45.24 $\\pm$ 1.41 & 110.19 $\\pm$ 7.37 & 23  \\\\ \\hline\n    RMP full & 0.25 & 0.31 & -43.51 $\\pm$ 2.15 & 139.64 $\\pm$ 5.24 & 6 \\\\ \\hline\n    DPMB & 0.23 & 0.32 & -42.95 $\\pm$ 3.58 & 23.27 $\\pm$ 0.94  & \\textbf{47} \\\\ \\hline\n    Us & \\textbf{0.19} & \\textbf{0.27} & \\textbf{-42.84 $\\pm$ 3.07} & \\textbf{4.35 $\\pm$  0.12} & 38\\\\ \\hline\n    \\end{tabular}}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\textbf{Inference on real-life data}\n\nFigure \\ref{fig:real_data_plot} shows the posterior mean intensity functions of the coal mine data set, the Twitter data set and the bramble canes data set under our model. \n\\paragraph{Scalability:} We note that it took only 240s on average to generate 1000 samples on the Twitter data set (188544 points). As a comparison, this is the amount of time that would be required to generate as many samples on a data set that has 50 points (resp. 100 points) under the models of \\cite{Murray09} (resp. \\cite{YWT11}). More importantly, it was not possible to run either of those two competing models on the twitter data set. Doing so would require computing $17\\times10^{10}$ covariance coefficients to evaluate a single auto-covariance matrix of the log-intensity at the data points, which a typical personal computer cannot handle.\n\n\\begin{figure}[!ht]\n(a) \\includegraphics[width=0.45\\textwidth]{coal_mine_mcmc_inference}\n(b) \\includegraphics[width=0.45\\textwidth]{twitter_mcmc_inference}\n(c) \\includegraphics[width=0.5\\textwidth]{brambles_mcmc_inference}\n\n\\caption{Inference on the intensity functions of the coal mine data set (top), the twitter data set (middle), and the bramble canes data set (bottom). Blue dots are data points, red dots are inducing points (labelled in the upper panels in the order they were selected), the grey area is the 1 standard deviation confidence band.}\n\\label{fig:real_data_plot}\n\\end{figure}\n\n\\section{DISCUSSION}\n\\textbf{Scalability of the selection of inducing points}\n\nThe computational bottleneck of the selection of inducing points is in the evaluation of \\[\\text{Tr}( \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*}(\\tilde{\\theta}_i) \\Sigma_{\\mathcal{D^\\prime}\\mathcal{D^\\prime}}^{*-1}(\\tilde{\\theta}_i) \\Sigma_{\\mathcal{D}\\mathcal{D}^{\\prime}}^{*T}(\\tilde{\\theta}_i)).\\]Hence, the complexity and the memory requirement of the selection of inducing points are both linear in the number of data points $n:= \\#\\mathcal{D}$.\n\nThe number of inducing points generated by our algorithm does not increase with the size of the data, but rather as a function of the size of the domain and the resolution implied by the prior over the hyper-parameters. \n\n\\textbf{Comparison with competing models}\n\nWe note that the computational bottleneck of our MCMC inference is in the evaluation of \\[\\text{Tr}(\\Sigma_{\\mathcal{D}\\mathcal{D}}) = \\text{Tr}(\\Sigma^{*}_{\\mathcal{D}\\mathcal{D}}) - \\text{Tr}(\\Sigma^{*}_{\\mathcal{D}\\mathcal{D}^\\prime}\\Sigma^{*-1}_{\\mathcal{D}^\\prime\\mathcal{D}^\\prime}\\Sigma^{*T}_{\\mathcal{D}\\mathcal{D}^\\prime}).\\]\nHence, inferring the intensity function under our model scales computationally in $\\mathcal{O}(nk^2)$ and has a memory requirement  $\\mathcal{O}(nk)$, where the number of inducing points $k$ is negligible. This is considerably better than alternative methods using Gaussian Processes \\cite{Murray09, YWT11} whose complexities are cubic in the number of data points and whose memory requirement is squared in the number of data points. Moreover, the superior accuracy of our model compared to \\cite{Murray09} and \\cite{YWT11} is due to our use of the exponential transformation rather than the scaled sigmoid one. In effect, unlike the inverse scaled sigmoid function that tends to amplify variations, the logarithm tends to smooth out variations. Hence, when the true intensity is uneven, the log-intensity is more likely to resemble a draw from a stationary GP than the inverse scaled sigmoid of the true intensity function, and subsequently a stationary GP prior in the inverse domain is more suitable to the exponential transformation than to the scaled sigmoid transformation.\n\nOur model is also more suitable than that of \\cite{Cunni08} when confidence bounds are needed for the intensity function, or when the input space is of dimension higher than 1. The model is a useful alternative to that of \\cite{Kottas06}, whose complexity is also linear. In effect, Gaussian Processes (GP) are more flexible than a Dirichlet Process (DP) mixture of Beta distributions. This is the result of the large number of known covariance kernels available in the literature and the state-of-the-art understanding of how well a given kernel can approximate an arbitrary function \\cite{Micchelli06, Pillai07}. Moreover, unlike a Dirichlet Process mixture of Beta distributions, Gaussian Processes allow directly expressing practical prior features such as smoothness, amplitude, length scale(s) (memory), and periodicity. \n\nAs our model relies on the Gauss-Legendre quadrature, we would not recommend it for applications with a large input space dimension. However, most interesting point process applications involve modelling temporal, spatial or spatio-temporal events, for which our model scales considerably better with the data size than competing approaches. In effect, the models proposed by \\cite{Kottas06, Cunni08, Cunni08b, YWT11} are all specific to unidimensional input data, whereas the model introduced by \\cite{Kottas07} is specific to spatial data. As for the model of \\cite{Murray09}, it scales very poorly with the input space dimension for its complexity is cubic in the sum of the number of data points and the number of latent thinning points, and the number of thinning points grows exponentially with the input space dimension\\footnote{The expected number of thinning points grows proportionally with the volume of the domain, which is exponential in the dimension of the input space when the domain is a hypercube with a given edge length.}.\n\n\\textbf{Extension of our model}\n\nAlthough the covariance kernel $\\gamma^{*}$ was assumed stationary, no result in this paper relied on that assumption. We solely needed to evaluate covariance matrices under $\\gamma^{*}$. Hence, the proposed model and algorithm can also be used to account for known non-stationarities. More generally, the model presented in this paper can serve as foundation to make inference on the stochastic dependency between multiple point processes when the intensities are assumed to be driven by known exogenous factors, hidden common factor, and latent idiosyncratic factors.\n\n\\section{SUMMARY}\nIn this paper we propose a novel exact non-parametric model to make inference on Poisson Point Processes using Gaussian Processes. We derive a robust MCMC scheme to sample from the posterior intensity function. Our model outperforms competing benchmarks in terms of speed and accuracy as well as in the decorrelation of MCMC samples. A critical advantage of our approach is that it has a numerical complexity and a memory requirement \\emph{linear} in the data size $n$ ($\\mathcal{O}(nk^2)$, and $\\mathcal{O}(nk)$ respectively, with $k \\ll n$). Competing models using Gaussian Processes have a cubic numerical complexity and squared memory requirement. We show that our model readily handles data sizes not yet considered in the literature.\n\n\\section*{Acknowledgments}\nYves-Laurent Kom Samo is supported by the Oxford-Man Institute of Quantitative Finance.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we have proposed a novel semi-supervised learning framework which enables learning contact with monocular videos.\nThe main idea behind this study was to demonstrate that this can successfully be achieved with visual and geometric consistency constraints for pseudo-label generation. \nWe designed an efficient graph-based network for inferring contact maps and shown benefits of combining visual cues and contact consistency constraints to produce more physically-plausible reconstructions.\nIn the future, we would like to explore more consistencies over time and or multiple views to further improve the accuracy.\\\\\n\n\\section{Introduction}\n\n\nUnderstanding hand-object interactions have been an active area of study in recent years \\cite{hasson2019learning,hasson2020leveraging,hasson20_handobjectconsist,cao2020reconstructing,tekin2019h+,Kwon_2021_ICCV,jiang2021hand,yang2021cpf,liu2021semi}. Besides common practical applications in augmented and virtual reality \\cite{han2020megatrack,wang2020rgb2hands,mueller2019real}, it is a key ingredient to advanced human-computer interaction \\cite{ueda2003hand} and imitation learning in robotics \\cite{zhang2018deep}.\nIn this paper, as illustrated in Figure \\ref{fig:highlevel_framework}, we tackle the problem of 3D reconstruction of the hand and manipulated object with the focus on contact map estimation. \n\n\n\\input{Figure\/highlevel_framework\/item}\n\nPrevious works in hand-object interactions typically formulate this as a joint hand and object pose estimation problem. Along with the development of data collection and annotation methods, more accurate 3D annotations for real datasets \\cite{FirstPersonAction_CVPR2018,hampali2020honnotate,chao2021dexycb} are available for learning-based methods \\cite{doosti2020hope,tekin2019h+}. Despite the efforts, there still exist gaps between hand-object pose estimation and contact as ground-truth in datasets are not perfect. Recent works attempt to address this problem with interaction constraints (attraction and repulsion) under an optimisation framework \\cite{hasson20_handobjectconsist,cao2020reconstructing,yang2021cpf}. However, inferred poses continue to exhibit sufficient error to cause unrealistic hand-object contact, making downstream tasks challenging \\cite{grady2021contactopt}. In addition, annotations under constrained laboratory environments rely on strong priors such as limited hand motion which prevents the trained model from generalising to novel scenes and out-of-domain objects.\n\nTo address the problem of hand-object contact modelling,\nBrahmbhatt~\\textit{et al.}~\\cite{brahmbhatt2019contactdb} used thermal cameras observing the heat transfer from hand to object after the grasp to capture detailed ground-truth contact. Their follow-up work contributed a large grasp dataset (\\emph{ContactPose}) with contact maps and hand-object pose annotations.\nRecent works are able to leverage contact maps to refine inaccurate hand-object pose estimations \\cite{grady2021contactopt} and generate grasps given object model \\cite{jiang2021hand}. Therefore, the ability to generate an accurate contact map is one of the key elements to reasoning physical contact. However, the number of annotated objects is incomparable to manipulated objects in real life and insufficient to cover a wide range of human intents. Furthermore, obtaining annotations for contact maps is non-trivial as it requires thermal sensors during data collection.\n\n\nTo enable the wider adoption of contact maps, we propose a unified framework that leverages existing hand-object datasets for generating pseudo-labels in semi-supervised learning. Specifically, we propose to exploit the visual and geometric consistencies of contact maps in hand-object interactions. This is built upon the idea that the poses of the hands and objects are highly-correlated where the 3D pose of the hand often indicates the orientation of the manipulated object. We further extend this by enforcing our contact consistency loss for the contact maps across a video.\n\n\n\nAs the input to contact map estimator are in the form of point clouds, recent related works \\cite{grady2021contactopt,jiang2021hand} typically follow a PointNet-based architectures \\cite{qi2017pointnet,qi2017pointnet++}. This achieves permutation invariance of points by operating on each point independently and subsequently applying a symmetric function to accumulate features \\cite{wang2019dynamic}. However, the network performances are limited as points are treated independently at a local scale to maintain permutation invariance. To overcome this fundamental limitation, many recent approaches adopt graph convolutional networks (GCN) \\cite{defferrard2016convolutional,kipf2016semi} and achieve state-of-the-art performances in 3D representation learning on point clouds for classification, part segmentation and semantic segmentation \\cite{wang2019dynamic,li2019deepgcns,lin2020convolution}. The ability to capture local geometric structures while maintaining permutation invariance is particularly important for estimating contact maps. However, it comes at the cost of high computation and memory usage for constructing a local neighbourhood with $K$-nearest neighbour ($K$-NN) search on point clouds at each training epoch.\nFor this reason, we design a graph-based neural network that demonstrates superior results with less than half the learning parameters and faster convergence.\n\n\nOur contributions are three-fold:\n\\begin{itemize}\n    \\setlength{\\itemsep}{0pt}%\n    \\setlength{\\parskip}{0pt}%\n   \n    \\item We propose a novel semi-supervised learning framework that combines pseudo-label with consistency training. Experimental results demonstrate the effectiveness of this training strategy.\n    \\item We propose a novel graph-based network for processing hand-object point clouds, which is at least two times more efficient than PointNet-based architecture for estimating contact between hand and object.\n    \\item We conduct comprehensive experiments on three commonly-used hand-object datasets. Experiments show that our proposed framework S$^2$Contact outperforms recent semi-supervised methods.\n    \n\\end{itemize}\n\\section{Methodology}\nGiven a noisy estimate of hand and object meshes from an image-based algorithm, we seek to learn a hand-object contact region estimator by exploiting real-world hand and object video datasets without contact ground-truths. Figure \\ref{fig:highlevel_framework} shows an overview of our approach. \nIn the following section, we describe our learned contact map estimation network (GCN-Contact) in Section \\ref{sec:ContactNet} and our newly proposing semi-supervised training pipeline (S$^2$Contact) in Section \\ref{sec:SemiContact} that utilise a teacher-student mutual learning framework.\n\n\\input{Figure\/contactnet\/item}\n\n\\subsection{GCN-Contact: 3D hand-object contact estimation} \\label{sec:ContactNet}\nAs pose estimates from an image-based algorithm can be potentially inaccurate, GCN-Contact learns to infer contact maps $\\mathbf{C}=(\\mathbf{C}_{hand},\\mathbf{C}_{obj})$ from hand and object point clouds $\\mathbf{P}=(\\mathbf{P}_{hand},\\mathbf{P}_{obj})$. We adopted the differential MANO \\cite{romero2017embodied} model from \\cite{hasson2019learning}. It maps pose ($\\boldsymbol{\\theta}\\in\\mathbb{R}^{51}$) and shape ($\\boldsymbol{\\beta}\\in\\mathbb{R}^{10}$) parameters to a mesh with $N=778$ vertices. Pose parameters ($\\boldsymbol{\\theta}$) consists of $45$ DoF ({\\it i.e.}\\hspace{0.1cm} $3$ DoF for each of the $15$ finger joints) plus $6$ DoF for rotation and translation of the wrist joint. Shape parameters ($\\boldsymbol{\\beta}$) are fixed for a given person. We sample $2048$ points randomly from object model to form object point cloud. Following \\cite{grady2021contactopt}, we include $F$-dimensional point features for each point: binary per-point feature indicating whether the point belongs to the hand or object, distances from hand to object and surface normal information. With network input $\\mathbf{P}=(\\mathbf{P}_{hand},\\mathbf{P}_{obj})$ where $\\mathbf{P}_{hand}\\in\\mathbb{R}^{778\\times F}$ and $\\mathbf{P}_{obj}\\in\\mathbb{R}^{2048\\times F}$, GCN-Contact can be trained to infer discrete contact representation ($\\mathbf{C}=(\\mathbf{C}_{hand},\\mathbf{C}_{obj}) \\in [0,1]$) \\cite{Brahmbhatt_2020_ECCV} using binary cross-entropy loss. Similarly to \\cite{grady2021contactopt}, the contact value range $[0,1]$ is evenly split into $10$ bins and the training loss is weighted to account for class imbalance.\n\n\\paragraph{\\textbf{Revisiting PointNet-based methods.}}\nRecent contact map estimators are based on PointNet \\cite{jiang2021hand} and PoinetNet++ \\cite{grady2021contactopt}.\nPointNet \\cite{qi2017pointnet} directly processes unordered point sets using shared multi-layer perceptron (MLP) networks. \nPointNet++ \\cite{qi2017pointnet++} learns hierarchical features by stacking multiple learning stages and recursively capturing local geometric structures. \nAt each learning stage, farthest point sampling (FPS) algorithm is used to re-sample a fixed number of points and $K$ neighbours are obtained from ball query's local neighbourhood for each sampled point to capture local structures. The kernel operation of PointNet++ for point $p_i \\in\\mathbb{R}^{F}$ with $F$-dimensional features can be described as:\n\\begin{align} \\label{eq:pointnet++}\n    \\dot{p_i} = \\sigma\\big(\\Phi(p_{j}| j \\in \\mathcal{N}(i))\\big),\n\\end{align}\nwhere the updated point $\\dot{p_i}$ is formed by max-pooling function $\\sigma(\\cdot)$ and PointNet as the basic building block for local feature extractor $\\Phi(\\cdot)$ around point neighbourhood $\\mathcal{N}(i)$ of point $p_i$. The kernel of the point convolution can be implemented with MLPs.\nHowever, MLPs are unnecessarily performed on the neighbourhood features which\ncauses a considerable amount of latency in PointNet++ \\cite{qian2021assanet}.\nThis motivates us to employ advanced local feature extractors such as convolution \\cite{wu2019pointconv,xu2021paconv}, graph \\cite{wang2019dynamic,li2019deepgcns,lin2020convolution} or self-attention mechanisms \\cite{guo2021pct,zhao2021point}.\n\n\\paragraph{\\textbf{Local geometric information.}}\nWhile contact map estimation can take advantage of detailed local geometric information, they usually suffer from two major limitations. First, the computational complexity is largely increased with delicate extractors which leads to low inference latency. \nFor instance, in graph-based methods, neighbourhood information gathering modules are placed for better modelling of the locality on point clouds. This is commonly established by $K$-nearest neighbour ($K$-NN) search which increases the computational cost quadratically with the number of points and even further for dynamic feature representation \\cite{wang2019dynamic}. For reference on ModelNet$40$ point cloud classification task \\cite{qian2021assanet}, the inference speed of PointNet \\cite{qi2017pointnet} is $41$ times faster than DGCNN \\cite{wang2019dynamic}.\nSecond, Liu \\textit{et al.}'s investigation on local aggregation operators reveals that advanced local feature extractors make surprisingly similar contributions to the network performance under the same network input~\\cite{liu2020closer}. For these reasons, we are encouraged to develop a computationally efficient design while maintaining comparable accuracy for learning contact map estimation.\n\n\\paragraph{\\textbf{Proposed method.}}\nTo overcome the aforementioned limitations, we present a simple yet effective graph-based network for contact map estimation. We use EdgeConv \\cite{wang2019dynamic} to generate edge features that describe the relationships between a point and its neighbours:\n\\begin{align} \\label{eq:edgeconv}\n    \\Phi(p_i,p_j) = \\text{ReLU}\\big(\\text{MLP}(p_j-p_i,p_i)\\big),\\;\\; j\\in \\mathcal{N}(i),\n\\end{align}\nwhere neighbourhood $\\mathcal{N}(i)$ is obtained by $K$-NN search around the point $p_i$. As shown in Figure \\ref{fig:ContactNet}, we only compute $K$-NN search once at each network pass to improve computational complexity and reduce memory usage. \nIn addition, we apply dilation on the $K$-NN results to increase the receptive field without loss of resolution. To better construct local regions when hand and object are perturbed, we propose to perform $K$-NN search on 3D position and point features separately. Note that \\cite{grady2021contactopt} perform ball query on $0.1-0.2m$ \nradius and \\cite{wang2019dynamic} combine both position and features. \nFinally, we take inspirations from the Inception model \\cite{szegedy2015going} in which they extract\nmulti-scale information by using different kernel sizes in different paths of the architecture. Similarly, we process spatial information at various dilation factors and then aggregates.\nThe experiment demonstrates the effectiveness of our proposed method and is able to achieve constant memory access cost regardless of the size of dilation factor $d$ (see Table \\ref{table:contactpose}).\n\n\n\n\\subsection{S$^2$Contact: Semi-supervised training pipeline} \\label{sec:SemiContact}\n\\input{Figure\/pipline\/item}\nCollecting ground-truth contact annotation for hand-object dataset can be both challenging and time-consuming.\nTo alleviate this, we introduce a semi-supervised learning framework to learn 3D hand-object contact estimation by leveraging large-scale unlabelled videos.\nAs shown in Figure~\\ref{fig:highlevel_framework}, our proposed framework relies on two training stages: 1) pre-training stage where the model is pre-trained on the existing labelled data~\\cite{Brahmbhatt_2020_ECCV}; 2) semi-supervised stage where the model is trained by the pseudo-labels from  unlabelled hand-object datasets~\\cite{chao2021dexycb,hampali2020honnotate,FirstPersonAction_CVPR2018}. \nAs pseudo-labels are often noisy, we propose confidence-based filtering with geometric and visual consistency constraints to improve the quality of pseudo-labels.\n\n\\paragraph{\\textbf{Pre-training.}} \nAs good initial contact estimate enables semi-supervision, we pre-train our graph-based contact estimator using a small labelled dataset $\\left\\{\\mathbf{x}^{l},\\mathbf{y}^{l}\\right\\}$. We followed \\cite{grady2021contactopt} and optimise hand-object poses to achieve target contact. Upon convergence, we clone the network to create a pair of student-teacher networks.\n\n\\paragraph{\\textbf{Pseudo-label generation.}}\nTo maintain a reliable performance margin over the student network throughout the training, we adopt an EMA teacher which is commonly used in semi-supervised learning. The output of the student network is the predicted contact map $\\widehat{\\mathbf{C}}$.\nThe teacher network generates pseudo-labels which includes pre-filter contact map $\\widetilde{\\mathbf{C}}$ and refined hand-object pose $\\widetilde{P}_{ref}$. As it is crucial for the teacher network to generate high-quality pseudo-labels under a semi-supervised framework, we propose a confidence-based filtering mechanism that leverages geometric and visual consistency constraints. \n\n\\paragraph{\\textbf{Contact consistency constraint for consistency training.}}\nWe propose a contact consistency loss to encourage robust and stable predictions for unlabelled data $\\mathbf{x}^{u}$.\nAs shown in Figure~\\ref{fig:SemiContact} (a), we first apply stochastic transformations $\\mathcal{T}$ which includes flipping, rotation and scaling on the input hand-object point clouds $\\mathbf{x}^{u}$ for the student network. \nThe predictions of the student network $\\widehat{\\mathbf{y}}^{u}_{s} \\in \\widehat{\\mathbf{C}}$ are compared with the teacher predictions $\\widetilde{\\mathbf{y}}^{u}_{t} \\in \\widetilde{\\mathbf{C}}$ processed by the same transformation $\\mathcal{T}$ using contact consistency loss:\n\\begin{align}\n\\begin{split}\n\\mathcal{L}_{cont}(\\mathbf{x}^{u}) &=\\|\\Omega(\\mathcal{T}(\\mathbf{x}^{u}))-(\\mathcal{T}(\\Omega(\\mathbf{x}^{u})))\\|_{1} \\\\\n&=\\|\\widehat{\\mathbf{y}}^{u}_{s}-\\widetilde{\\mathbf{y}}^{u}_{t}\\|_{1},\n\\end{split}\n\\end{align}\nwhere $\\Omega(\\cdot)$ represents the predicted contact map. \n \n\n\\paragraph{\\textbf{Geometric consistency constraint for pseudo-labels filtering.}}\nAs shown in Figure~\\ref{fig:SemiContact} (b), we propose a geometric consistency constraint to the hand and object pseudo pose label $\\widetilde{P}_{ref}$.\nConcretely, we allow the Chamfer distance $\\mathcal{L}_{Cham}$ between hand and object meshes to be less than threshold $t_{dist}$:\n\\begin{align}\n    \\mathcal{L}_{Cham}(\\widetilde{\\mathbf{v}}_{hand},\\widetilde{\\mathbf{v}}_{obj}) = &\\frac{1}{\\left|\\widetilde{\\mathbf{v}}_{obj}\\right|}\\sum_{x \\in \\widetilde{\\mathbf{v}}_{obj}}d_{\\widetilde{\\mathbf{v}}_{hand}}(x) + \\frac{1}{\\left|\\widetilde{\\mathbf{v}}_{hand}\\right|}\\sum_{y \\in \\widetilde{\\mathbf{v}}_{hand}}d_{\\widetilde{\\mathbf{v}}_{obj}}(y), \n\\end{align}\nwhere $\\widetilde{\\mathbf{v}}_{hand}$ and $\\widetilde{\\mathbf{v}}_{obj}$ refers to hand and object point sets, $d_{\\widetilde{\\mathbf{v}}_{hand}}(x)=\\min_{y \\in \\widetilde{\\mathbf{v}}_{hand}}\\allowbreak\\left\\|x-y\\right\\|_{2}^{2}$, and $d_{\\widetilde{\\mathbf{v}}_{obj}}(y)=\\min_{x \\in \\widetilde{\\mathbf{v}}_{obj}}\\left\\|x-y\\right\\|_{2}^{2}$.\nSimilarly for interpenetration, we use $\\mathcal{L}_{SDF}(\\widetilde{\\mathbf{v}}_{obj})=\\sum_{hand,obj} \\sum_{i} \\Psi_{h}\\left(\\widetilde{\\mathbf{v}}_{obj}^{i}\\right) \\leq t_{pen}$ to ensure object is being manipulated by hand.\n$\\Psi_{h}$ is the Signed Distance Field (SDF) from the\nhand mesh ({\\it i.e.}\\hspace{0.1cm}, $\\Psi_{h}(\\widetilde{\\mathbf{v}}_{obj})=-\\min \\left(\\operatorname{SDF}\\left(\\widetilde{\\mathbf{v}}_{obj}\\right), 0\\right)$) to detect object penetrations.\n\n\n\n\\paragraph{\\textbf{Visual consistency constraint for pseudo-labels filtering.}} \nWe observed that geometric consistency is insufficient to correct hand grasp (see Table \\ref{table:ab_ssl}). \nTo address this, we propose a visual consistency constraint to filter out the pseudo-labels whose rendered hand-object image $\\widetilde{I}_{ho}$ does not match the input image. \nWe first use a renderer~\\cite{kato2018neural} to render the hand-object image from the refined pose $\\widetilde{P}_{ref}$ and obtain the hand-object segment of the input image $I$ by applying the segmentation mask $M_{gt}$. \nThen, the structural similarity (SSIM)~\\cite{wang2004image} between two images can be computed. We keep pseudo-labels when $\\mathcal{L}_{SSIM} \\leq t_{SSIM}$:\n\\begin{equation}\n\\mathcal{L}_{SSIM}(I,M_{gt},\\widetilde{I}_{ho})=1-SSIM\\left(I \\odot M_{gt}, \\widetilde{I}_{ho}\\right),\n\\end{equation}\nwhere $\\odot$ denotes element-wise multiplication. \n\n\\paragraph{\\textbf{Self-training with pseudo-labels.}} \nAfter filtering pseudo-labels, our model is trained with the union set of the human-annotated dataset and the remaining pseudo-labels.\nThe total loss $\\mathcal{L}_{semi}$ can be described as:\n\\begin{equation}\n\\mathcal{L}_{semi}(\\widehat{\\mathbf{C}},\\mathbf{y}^{l},\\widetilde{\\mathbf{C}}^{u},\\mathbf{x}^{u})=\\mathcal{L}_{sup}\\left(\\widehat{\\mathbf{C}},\\mathbf{y}^{l}\\right)+\\mathcal{L}_{unsup}\\left(\\widehat{\\mathbf{C}},\\widetilde{\\mathbf{C}}^{u}\\right)+\\lambda_{c} \\mathcal{L}_{cont}\\left(\\mathbf{x}^{u}\\right),\n\\end{equation}\nwhere $\\mathcal{L}_{sup}$ is a supervised contact loss, $\\mathcal{L}_{unsup}$ is a unsupervised contact loss with pseudo-labels and $\\lambda_{c}$ is a hyperparameter. Note that $\\mathcal{L}_{sup}$ (see Figure \\ref{fig:ContactNet}) and $\\mathcal{L}_{unsup}$ (see Figure \\ref{fig:SemiContact}) are both binary cross-entropy loss.\n\n\n\\section{Related work}\n\nOur work tackles the problem of hand and object reconstruction from monocular RGB videos, exploiting geometric and visual consistencies on contact maps for semi-supervised learning. To the best of our knowledge, we are the first to apply such consistencies on hand-object scenarios. We first review the literature on \\emph{hand-object reconstruction}. Then, we review \\emph{point cloud analysis} with the focus of graph-based methods. Finally, we provide a brief review on \\emph{semi-supervised learning in 3D hand-object pose estimation}.\n\n\\subsection{Hand-object reconstruction}\n\nPrevious works mainly tackle 3D pose estimations on hands \\cite{simon2017hand,zimmermann2017learning,mueller2018ganerated,spurr2018cross,yang2020seqhand,yang2020seqhand,tang2014latent} and objects \\cite{labbe2020,li2018deepim,xiang2017posecnn,chen2020g2l,chen2021fs} separately. Joint reconstruction of hands and objects has been receiving increasing attention \\cite{hasson2019learning,hasson2020leveraging,cao2020reconstructing,hasson20_handobjectconsist}. Hasson \\textit{et al.}~\\cite{hasson2019learning} introduces an end-to-end model to regress MANO hand parameters jointly with object mesh vertices deformed from a sphere and incorporates contact losses which encourages contact surfaces and penalises penetrations between hand and object. A line of works \\cite{tekin2019h+,doosti2020hope,hasson2020leveraging,cao2020reconstructing,hasson20_handobjectconsist,yang2021cpf,grady2021contactopt,huang2020hot} assume known object models and regress a 6DoF object pose instead. Other works focus on grasp synthesis \\cite{corona2020ganhand,karunratanakul2020grasping,taheri2020grab,jiang2021hand}. In contrast, our method is in line with recent optimisation-based approaches for modelling 3D hand-object contact. ContactOpt~\\cite{grady2021contactopt} proposes a contact map estimation network and a contact model to produce realistic hand-object interaction. ContactPose dataset~\\cite{Brahmbhatt_2020_ECCV} is unique in capturing ground-truth thermal contact maps. However, 3D contact labels are seldom available and limited to constrained labratory settings. In this work, we treat contact maps as our primary learning target and leverage unannotated datasets.\n\n\\subsection{Point cloud analysis} \nSince point cloud data is irregular and unordered, early works tend to project the original point clouds to intermediate voxels \\cite{maturana2015voxnet} or images \\cite{you2018pvnet}, {\\it i.e.}\\hspace{0.1cm} translating into a well-explored 2D image problem. As information loss caused by projection degrades the representational quality, PointNet \\cite{qi2017pointnet} is proposed to directly process unordered point sets and PointNet++ \\cite{qi2017pointnet++} is extends on local point representation in multi-scale. As PointNet++ \\cite{qi2017pointnet++} can be view as the generic point cloud analysis network framework, the research focus has been shifted to generating better regional points representation. Methods can be divided into convolution \\cite{wu2019pointconv,xu2021paconv}, graph \\cite{wang2019dynamic,li2019deepgcns,lin2020convolution} and attention \\cite{guo2021pct,zhao2021point} -based. \n\\noindent \\textbf{Graph-based methods.}\nGCNs have been gaining much attention in the last few years. This is due to two reasons: 1) the rapid increase of non-Euclidean data in real-world applications and 2) the limited performance of convolutional neural networks when dealing with such data. As the unstructured nature of point clouds poses a representational challenge in the community, graph-based methods treat points as nodes of a graph and formulate edges according to their spatial\/feature relationships. MoNet \\cite{monti2017geometric} defines the convolution as Gaussian mixture models in a local pseudo-coordinate system. 3D-GCN \\cite{lin2020convolution} proposes a deformable kernels which has shift and scale-invariant properties for point cloud processing. DGCNN \\cite{wang2019dynamic} proposes to gather nearest neighbouring points in feature space and follow by the EdgeConv operators for feature extraction. The EdgeConv operator dynamically computes node adjacency at each graph layer using the distance between point features. In this paper, we propose a computationally efficient network for contact map estimation which requires less than half the parameters of PointNet \\cite{qi2017pointnet} and GPU memory of DGCNN \\cite{wang2019dynamic}.\n\n\\subsection{Semi-supervised learning in 3D hand-object pose estimation}\nLearning from both labelled and unlabelled data simultaneously has recently attracted growing interest\nin 3D hand pose estimation~\\cite{yang2021semihand,spurr2021adversarial,kaviani2021semi,chen2019so,tang2013real}.\nThey typically focus on training models with a small amount of labelled data as well as a relatively larger amount of unlabelled data. \nAfter training on human-annotated datasets, pseudo-labelling and consistency training can be used to train further and a teacher-student network with exponential moving average (EMA) strategy~\\cite{wang20213dioumatch} is common to accelerate the training. \nFor instance, So-HandNet~\\cite{chen2019so} leverages the consistency between the recovered hand point cloud and the original hand point cloud for semi-supervised training.\nSemiHand~\\cite{yang2021semihand} is the first to combine pseudo-labelling and consistency learning for hand pose estimation. \nLiu \\textit{et al.}~\\cite{liu2021semi} is the only prior work on 3D hand-object pose estimation with semi-supervised learning.\nThey proposed spatial and temporal constraints for selecting the pseudo-labels from videos.\nHowever, they are limited to pseudo hand labels and did not account for physical contact with manipulated objects.\nIn contrast, our work is the first to explore pseudo-labelling for 3D hand-object contact map with geometric and visual consistency constraints.\n\n\n\\section{Experiments}\n\n\\paragraph{\\textbf{Implementation details.}}\nWe implement our method in PyTorch~\\cite{PyTorch}. All experiments are run on an Intel i9-CPU @ 3.50GHZ, 16 GB RAM, and one NVIDIA RTX 3090 GPU. \nFor pseudo-labels filtering, $t_{dist}=0.7$, $t_{pen}=6$ and $t_{SSIM}=0.25$ are the constant thresholds and stochastic transformations includes flipping ($\\pm20\\%$), rotation ($\\pm180^{\\circ}$) and scaling ($\\pm20\\%$). We train all parts of the network simultaneously with Adam optimiser \\cite{kingma2014adam} at a learning rate $10^{-3}$ for 100 epochs. We empirically fixed $K=10,d=4$ to produce the best results.\n\n\\paragraph{\\textbf{Datasets and evaluation metrics.}}\n\\emph{ContactPose} is the first dataset \\cite{Brahmbhatt_2020_ECCV} of hand-object contact paired with hand pose, object pose and RGB-D images. It contains 2,306 unique grasps of $25$ household objects grasped with $2$ functional intents by $50$ participants, and more than $2.9$M RGB-D grasp images. For fair comparisons with ContactOpt \\cite{grady2021contactopt}, we follow their \\emph{Perturbed ContactPose} dataset where hand meshes are modified by additional noise to MANO parameters. This results in 22,624 training and 1,416 testing grasps. \\emph{DexYCB} is a recent real dataset for capturing hand grasping of objects \\cite{chao2021dexycb}. It consists of a total of 582,000 image frames on 20 objects from the YCB-Video dataset \\cite{xiang2017posecnn}. We present results on their default official dataset split settings. \\emph{HO-3D}~\\cite{hampali2020honnotate} is similar to \\emph{DexYCB} where it consists of 78,000 images frames on 10 objects. We present results on the official dataset split (version 2). The hand mesh error is reported after procrustes alignment and in $mm$.\n\n\n\n\\begin{itemize}\n\\item \\emph{Hand error:} We report the mean end-point error ($mm$) over 21 joints and mesh error in $mm$.\n\\item \\emph{Object error:} We report the percentage of average object 3D vertices error within $10\\%$ of object diameter (ADD-$0.1$D).\n\\item \\emph{Hand-object interaction:} \nWe report the intersection volume ($cm^{3}$) and contact coverage ($\\%$). Intersection volume is obtained by voxelising the hand and object using a voxel size of $0.5cm$. Contact coverage refers to the percentage of hand points between $\\pm2mm\\%$ of the object surface~\\cite{grady2021contactopt}.\n\\end{itemize}\n\n\n\\paragraph{\\textbf{Baseline.}}\nFor refining image-based pose estimates, we use the baseline pose estimation network from Hasson \\textit{et al.}~\\cite{hasson2020leveraging} and retrain it on the training split of the respecting dataset. We filter out frames where the minimum distance between the ground truth hand and object surfaces is greater than 2 $mm$. We also use the contact estimation network DeepContact from Grady \\textit{et al.}~\\cite{grady2021contactopt} which takes ground-truth object class and pose. For semi-supervised learning, we use the baseline method from Liu \\textit{et al.}~\\cite{liu2021semi}, a semi-supervised learning pipeline for 3D hand-object pose estimation from large-scale hand-object interaction videos.\n\n\n\\subsection{Comparative results}\n\\paragraph{\\textbf{Refining small and large inaccuracies.}}\nWe use \\emph{ContactPose} to evaluate GCN-Contact for refining poses with small (\\emph{ContactPose}) and large (\\emph{Perturbed ContactPose}) inaccuracies. Table \\ref{table:contactpose} shows the results for both cases. For \\emph{Perturbed ContactPose}, the mean end-point error over $21$ joints is $82.947mm$ before refinement. This is aimed at testing the ability to improve hand poses with large errors. In contrast, \\emph{ContactPose} is used to evaluate $mm$-scale refinement. As shown, our method consistently outperforms baseline and DGCNN \\cite{wang2019dynamic}. We attribute the performance gain to multi-scale feature aggregation with dilation. Qualitative comparison with ContactOpt~\\cite{grady2021contactopt} is provided in Figure \\ref{fig:vis_contactpose}.\n\n\\input{Table\/contactpose\/item}\n\\input{Figure\/vis_contactpose\/item}\n\n\n\\paragraph{\\textbf{Refining Image-based pose estimates.}}\nWe evaluate S$^2$Contact in refining poses from an image-based pose estimator. We use the baseline image-based pose estimation network from Hasson \\textit{et al.}~\\cite{hasson2020leveraging} and retrain it on the training split of the respecting dataset. Unlike \\cite{grady2021contactopt}, we do not rely on ground-truth object class and pose. \nIn particular, we compare with the current state-of-the-art~\\cite{liu2021semi} which is also a semi-supervised framework for 3D hand-object pose estimation. \nLiu \\textit{et al.}~\\cite{liu2021semi} proposes spatial-temporal consistency in large-scale hand-object videos to generate pseudo-labels for hand. In contrast, we leverage physical contact and visual consistency constraints to generate pseudo contact labels which can be optimised jointly with hand and object poses. \nAs shown in Table~\\ref{table:semisupervised}, our method outperforms \\cite{liu2021semi} by $11.5\\%$ in average object ADD-$0.1$D score.\nBesides, we also compare with our baseline contact model ContactOpt~\\cite{grady2021contactopt}.\nAs shown in  Table~\\ref{table:semisupervised}, we are able to further improve hand error by $1mm$ and $0.8mm$ over joints and mesh. \nIn addition to hand-object pose performance, our method is able to better reconstruct hand and object with less intersection volume and higher contact coverage.\nThe above demonstrates that our method provides a more practical alternative to alleviate the reliance on heavy dataset annotation in hand-object. In addition, we provide qualitative comparison on \\emph{HO-3D} in Figure \\ref{fig:vis_ho3d}. We also report the cross-dataset generalisation performance of our model\non \\emph{DexYCB} in Table~\\ref{table:crossdataset}. We select three objects ({\\it i.e.}\\hspace{0.1cm}, mustard bottle, potted meat can and bleach cleanser), to be consistent with \\emph{HO-3D}. \nAs shown, our method consistently shown improvements across all metrics.\n\n\n\\input{Table\/semisupervised\/item}\n\\input{Table\/cross_dataset\/item}\n\\input{Figure\/vis_ho3d\/item}\n\n \n\\subsection{Ablation study}\n\\paragraph{\\textbf{Number of $K$ neighbours.}}\nTable \\ref{table:ab_graph}(a) shows the results of varying number of $K$ neighbours without dilation. As shown, increasing $K$ improves immediately at $K=10$ but does not gain performance further. \n\\vspace{-0.2cm}\n\\paragraph{\\textbf{Size of dilation factor $d$.}}\nAs shown above that performance saturates at $K=10$, we now fix $K$ and vary the size of dilation factor $d$ in Table \\ref{table:ab_graph}(b). We find that the combination $K=10, d=4$ produce the best performance and do not improve by further increasing $d$. This demonstrates the effectiveness of increasing the receptive field for contact map estimation.\n\\vspace{-0.2cm}\n\\paragraph{\\textbf{Combining $K$-NN computation.}}\nTo further study the effect of separately computing $K$-NN, we experiment with combined $K$-NN computation with $d=4$ in Table \\ref{table:ab_graph}(c). It can be seen that the performance exceed the lower bound of (a) and similar to DGCNN's performance in Table \\ref{table:contactpose}. This is expected as this is similar to static EdgeConv \\cite{wang2019dynamic} with dilation. This shows that separate $K$-NN is crucial for this framework.\n\\input{Table\/ablation_graph\/item}\n\\paragraph{\\textbf{Impact of our components.}}\nWe study the impact of semi-supervised learning on \\emph{HO-3D}. Since the hand model (MANO) is consistent across datasets, the contact estimator can easily transfer hand contact to new datasets without re-training. \nHowever, it is insufficient to adapt to unlabelled dataset due to diverse object geometries.\nTherefore, we propose a semi-supervised learning method to generate high-quality pseudo-labels. \nAs shown in Table~\\ref{table:ab_ssl}, our method enables performance boost on both hand and object. \nThe hand joint error is $8.74mm$ while it is $9.92mm$ without semi-supervised training. \nAlso, the average object ADD-$0.1$D has a significant $8.56\\%$ improvement under S$^2$Contact.\n\nTable~\\ref{table:ab_ssl} shows a quantitative comparison of S$^2$Contact with various filtering constraints disabled demonstrating that constraints from both visual and geometry domains are essential for faithful training. \nWe also observed that disabling $\\mathcal{L}_{cont}$ can easily lead to unstable training and $5.45\\%$ performance degradation in object error. \nIn contrast, geometric consistencies ($\\mathcal{L}_{Cham}$ and $\\mathcal{L}_{SDF}$) have a comparably smaller impact on hand and object pose.\nDespite that they account for less than $2\\%$ performance drop to object error, geometric consistencies are important for contact ({\\it i.e.}\\hspace{0.1cm}, more than $5\\%$ for contact coverage).\nThe remaining factor, measuring visual similarity, has a more significant impact. \nDisabling visual consistency constraint $\\mathcal{L}_{SSIM}$ results in hand joint error and object error increase by $1mm$ and $8.04\\%$, respectively.\nWe validate that the combination of our pseudo-label filtering constraints are critical for generating high-quality pseudo-labels and improving hand-object pose estimation performance. Finally, we provide qualitative examples on out-of-domain objects in supplementary.\n\n\n\\input{Table\/ablation_ssl\/item}\n\n\n\\paragraph{\\textbf{Computational analysis.}}\nWe report the model parameters and GPU memory cost in Table 4 of supplementary material. For fair comparisons, all models are tested using a batch size of $64$. As shown, our model has $2.4$X less the number of learnable model parameters and $2$X less the GPU memory cost when compared to baseline and DGCNN \\cite{wang2019dynamic}, respectively. We alleviate the need to keep a high density of points across the network (DGCNN) while gaining performance.\n\n\\section{Implementation details of GCN-Contact} \\label{sec:implementation}\nThe network architecture is illustrated in Table \\ref{table:network_architecture}.\n\n\\input{Table\/supplementary\/network\/item}\n\n\n\\section{Ablations on pseudo-labels} \\label{sec:pseudo-labels}\n\\input{Figure\/supplementary\/thresholds1\/item}\n\\subsection{Effects of varying thresholds for pseudo-labels generation}\nWe study the effect of different thresholds of the pseudo-labels filtering mechanism on \\emph{HO-3D} in Figure~\\ref{fig:threthold1} and Figure~\\ref{fig:threshold2}. As shown in Figure~\\ref{fig:threthold1}, the performance\nis higher than the method without Chamfer distance by large margins when $60<t_{dist}\\leq80$. Note that the performance peaks at $t_{dist}=70$. When $t_{dist}>70$, further\nincreasing $t_{dist}$ led to a drastic drop in pseudo-label as hand and object are so far away from each other such that the resulting pseudo-labels are in low-quality. \nOn the contrary, when $t_{dist}\\leq60$, there are less qualifying pseudo-labels leading to performance drop from insufficient training labels.\nWe obtain similar observations for $t_{pen}$ and $t_{SSIM}$.\n\n\\subsection{Amounts of pseudo-labels}\n\\input{Figure\/supplementary\/thresholds2&amounts\/item}\nWe analyse the effect of using different fractions of pseudo-labels in semi-supervised learning on \\emph{HO-3D} in Figure~\\ref{fig:amounts}. We uniformly sample 20\\%, 40\\%, 60\\%, and 80\\% of the collected pseudo-labels for semi-supervised learning. As shown, the performance has been significantly improved after adding 20\\% of pseudo-labels. We observe that the more pseudo-labels used in training, the better the performance the model can achieve.\n\n\n\\section{Comparison with state-of-the-arts} \\label{sec:SOTA}\nWe compare against the state-of-the-art approaches [3,17,18] on \\emph{HO-3D} in Table \\ref{table:semisupervised}. [3] is an optimisation-based method which leverages 2D image cues and 3D contact priors for reconstructing hand-object interactions. [17] uses a feed-forward neural network to predict 3D hand pose and object pose where its single-frame model with full 3D supervision. \n[18] follows a fitting-based approach which builds on estimates from neural network models for detection, object segmentation and 3D hand pose estimation trained with full supervision. \n\\input{Table\/supplementary\/sota\/item}\n\n\\section{Additional qualitative examples} \\label{sec:qualitative}\nIn this section, we show additional qualitative results. In Figure \\ref{fig:sup_vis_contactpose}, we visualise the predictions of ContactOpt~[13] and our method as well as the ground-truth. We can see that our method is able to better reconstruction hand and object with more accurate contact map estimations. Our method performs significantly\nbetter than previous approaches. In Figure~\\ref{fig:sup_vis_ho3d}, we show more examples of our method and [30] on \\emph{HO-3D}. Our method significantly improve the hand-object pose. The last row of Figure~\\ref{fig:sup_vis_ho3d} shows a failure case where the region of hand-object contact was too small for the network to produce a good contact prediction.\n\nWe provide qualitative examples on out-of-domain objects in Figure~\\ref{fig:vis_in_the_wild}.\n\n\\input{Figure\/supplementary\/contactpose\/item}\n\\input{Figure\/supplementary\/ho3d\/item}\n\\input{Figure\/vis_in_the_wild\/item}\n\n\\section{Performances of different GCN-Contact design choices} \\label{sec:ablation_graph}\nComplete results are reported in Table \\ref{table:ab_graph}.\n\\input{Table\/supplementary\/ablation_graph\/item}\n\n\\section{Computational analysis} \\label{sec:computational}\nComplete results are reported in Table \\ref{table:runtime}.\n\\input{Table\/runtime\/item}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Motivation}\nIncreasing  demand for wireless applications on one hand, and the\nlimited available resources on the other hand, provoke more\nefficient usage of such resources. Due to its significance, many\nresearchers have addressed the problem of resource allocation in\nwireless networks.  One major challenge in wireless networks is the\ndestructive effect of multi-user interference, which degrades the\nperformance when multiple users share the spectrum. As such, an\nefficient and low complexity resource allocation scheme that\nmaximizes the quality of service while mitigating the impact of the\nmulti-user interference is desirable.  The existing resource\nallocation schemes are either \\textit{centralized}, i.e., a central\ncontroller manages the resources, or \\textit{decentralized}, where\nresource allocation is performed locally at each node. Due to the\ncomplexity of adapting the centralized schemes to the network\nstructure (e.g. number of active users), these schemes are usually\ndesigned for a fixed network structure. This makes inefficient\nusage of resources because, in most cases, the number of active users may\nbe considerably less than the value assumed in the design process. On\nthe other hand, most of the decentralized resource allocation\nschemes suffer from the complexity, either in the algorithm (e.g.\ngame-theoretic approaches involving iterative methods) or in the\nhardware (e.g. cognitive radio). Therefore, it is of interest to\ndevise an efficient and low-complexity decentralized resource\nallocation scheme, which is the main goal of this paper.\n\n\n\\subsection{Related Works}\n\n\\subsubsection{Centeralized Schemes}\nIn recent years, many centralized power and spectrum allocation\nschemes have been studied in cellular and multihop wireless networks\n\\cite{ KumaranITWC0505,\nElBattITWC0104,HollidayISIT2004,WassermanITWC0603, HanITC0805,\nKatzelaIPC0696, KianiISIT2006, LiangITIT1007}. Clearly, centralized\nschemes perform better than the decentralized (distributed)\napproaches, while requiring extensive knowledge of the network\nconfiguration. In particular, when the number of nodes is large,\ndeploying such centralized schemes may not be practically feasible.\n\nTraditional wireless systems aimed to avoid the interference among\nusers by using orthogonal transmission schemes. The most common\nexample is the Frequency Division (FD) system, in which different\nusers transmit over disjoint frequency sub-bands. The assignment of\nfrequency sub-bands is usually performed by a central controller.\nDespite its simplicity, FD is shown to achieve the highest\nthroughput in certain scenarios. In particular, \\cite{1} proves that\nin a wireless network where interference is treated as noise\n(no multi-user detection is performed), if the crossover gains are\nsufficiently larger than the forward gains, FD is Pareto-rate-optimal. Due to\npractical considerations, such FD systems usually rely on a fixed\nnumber of frequency sub-bands. Hence, if the number of users\nchanges, the system is not guaranteed to offer the best possible\nspectral efficiency  because, most of the time, the majority of the\npotential users may be inactive.\n\n\\subsubsection{Decentralized Schemes}\nIn decentralized schemes, decisions concerning network resources\nare made by individual nodes based on their local information.\nMost of decentralized schemes reported in the literature rely on\neither \\textit{game-theoretic} approaches or \\textit{cognitive\nradios}. Cognitive radios \\cite{2} have the ability to sense the\nunoccupied portion of the available spectrum and use this\ninformation in resource allocation. Fundamental limits of wireless\nnetworks with cognitive radios are studied in \\cite{3,4,5,7}.\nAlthough cognitive radios avoid the use of a central controller,\nthey require sophisticated detection techniques for sensing the\nspectrum holes and dynamic frequency assignment, which add to the\noverall system complexity \\cite{8}. Noting the above points, it is\ndesirable to have a decentralized frequency sharing strategy\nwithout the need for cognitive radios, which allows the users to\ncoexist while utilizing the spectrum efficiently and fairly.\n\n\n\nBeing a standard technique in spread spectrum communications and due\nto its interference avoidance nature, hopping is the\nsimplest spectrum sharing method to use in decentralized networks.\nAs different users typically have no prior information about the\ncodebooks of the other users, the most efficient method is avoiding\ninterference by choosing unused channels. As mentioned earlier,\nsearching the spectrum to find spectrum holes is not an easy task\ndue to the dynamic spectrum usage. As such, Frequency Hopping (FH) is a realization of a\ntransmission scheme without sensing, while avoiding the collisions\nas much as possible.  Frequency Hopping is one of the standard\nsignaling schemes~\\cite{15} adopted in ad-hoc networks. In short\nrange scenarios, bluetooth systems \\cite{19,20,21} are the most\npopular examples of a Wireless Personal Area Network (WPAN). Using\nFH over the unlicensed ISM band, a bluetooth system provides robust\ncommunication to unpredictable sources of interference. A\nmodification of Frequency Hopping, called Dynamic Frequency Hopping\n(DFH), selects the hopping pattern based on interference\nmeasurements in order to avoid dominant interferers. The performance\nof a DFH scheme when applied to a cellular system is assessed in\n\\cite{22,23,24}.\n\nIn \\cite{Jindal}, the authors consider the problem of bandwidth\npartitioning in a decentralized wireless  network where different transmitters are connected to\ndifferent receivers through channels with similar path loss\nexponent. Assuming the transmitters are scattered over the two\ndimensional plane according to a Poisson point process, a fixed\nbandwidth is partitioned into a certain number of sub-bands such\nthat the so-called transmission intensity in the network is\nmaximized while the probability of outage per user is below a\ncertain threshold. The transmission strategy is based on choosing\none sub-band randomly per transmission, which is a special case of\nFH.\n\nFrequency hopping is also proposed in \\cite{7} in the context of\ncognitive radios, where each cognitive transmitter selects a\nfrequency sub-band but quits transmitting if the sub-band is already\noccupied by a primary user.\n\nRecently, Orthogonal Frequency Division Multiplexing (OFDM) has been\nconsidered as a promising technique in many wireless technologies.\nOFDM partitions a wide-band channel to a group of narrow-band\northogonal sub-channels. The popularity of OFDM motivates us to\nconsider a Frequency Hopping scheme operating over $u$ narrow-band\northogonal frequency sub-bands. We note that the results of the\npaper are valid in a general setup where  hopping is performed over\nan arbitrary orthogonal basis. To make the presentation as simple as\npossible, we take the orthogonal basis in frequency, which can be\nrealized in practice using OFDM systems.\n\n\n\\subsection{Contribution}\nIn this paper, we consider a decentralized wireless communication\nnetwork with a fixed number $u$ of frequency sub-bands to be shared\namong $N$ transmitter-receiver pairs. It is assumed that the number\nof active users is a random variable with a given probability mass\nfunction. Moreover, users are unaware of each other's codebooks,\nand hence, no multiuser detection is possible.  We propose a\nrandomized Frequency Hopping scheme in which the $i^{th}$\ntransmitter randomly hops over $v_{i}$ out of $u$ sub-bands from\ntransmission to transmission. Assuming i.i.d. Gaussian signals are\ntransmitted over the chosen sub-bands, the distribution of the noise\nplus interference becomes mixed Gaussian, which makes the calculation\nof the achievable rate complicated. The main contributions of the\npaper are:\n \\begin{itemize}\n\\item We derive lower and upper bounds\non the mutual information between the transmitted and received\nsignals of each user and demonstrate that, for large SNR values, the\ntwo bounds coincide. Thereafter,  we are able to show that the achievable rate of\nthe $i^{th}$ user scales like $\\frac{v_{i}}{2} \\prod_{\\substack{j=1\\\\j\\neq i}}^{N}\\left(1-\\frac{v_{j}}{u}\\right) \\log\\mathsf{SNR}$.\n\n\\item We show that each transmitter only needs the knowledge\nof the number of active users in the network, the forward channel\ngain and the maximum interference at its desired receiver to\nregulate its transmission rate. Knowing these quantities, we\ndemonstrate how the $i^{th}$ user can achieve a multiplexing gain of\n$\\frac{v_{i}}{2} \\prod_{\\substack{j=1\\\\j\\neq\ni}}^{N}\\left(1-\\frac{v_{j}}{u}\\right)$.\n\n\\item\nWe obtain the optimum design parameters $\\{v_{i}\\}_{i=1}^{N}$ in\norder to maximize various performance measures.\n\n\\item\nWe compare the performance of the FH with that of the Frequency\nDivision in terms of the following performance measures:\naverage sum multiplexing gain $(\\eta^{(1)})$, average minimum \nmultiplexing gain per user $(\\eta^{(2)})$, minimum nonzero\nmultiplexing gain per user $(\\eta^{(3)})$ and service capability\n($\\eta^{(4)}$). We show that (depending on the probability mass\nfunction of the number of active users) the FH system can offer a\nsignificant improvement in terms of $\\eta^{(1)}$, $\\eta^{(2)}$, and\n$\\eta^{(4)}$ (implying a more efficient usage of the spectrum). It\nis also shown that\n$\\frac{1}{e}\\leq\\frac{\\eta_{\\mathrm{FH}}^{(3)}}{\\eta_{\\mathrm{FD}}^{(3)}}\\leq\n1$, i.e., the loss incurred in terms of $\\eta^{(3)}$ is not more\nthan $\\frac{1}{e}$.\n\n\\end{itemize}\n\nThe paper outline is as follows. The system model is given in section\nII. Section III offers an analysis of the achievable rates. Upper\nbounds and lower bounds on the achievable rates of users are\npresented in this section. In section IV, based on the results in\nsection III, we discuss how users in the FH system can fairly share\nthe spectrum. Comparison with the FD scheme in terms of different\nperformance measures is discussed in section V. Section VI offers a\ncomparison between two versions of the proposed FH, i.e., the robust\nfrequency hopping and adaptive frequency hopping. Finally, section\nVII states the concluding remarks.\n\n\\subsection{Notation}\nThroughout the paper, we use the notation $\\mathrm{E} \\{.\\}$ for the\nexpectation operator. $\\mathrm{Pr}\\{\\mathcal{E}\\}$ denotes the\nprobability of an event $\\mathcal{E}$, $\\mathbb{1}(\\mathcal{E})$ the\nindicator function of an event $\\mathcal{E}$ and $p_{X}(.)$  the\nprobability density function (PDF) of a random variable $X$. Also,\n$\\mathrm{I} (X;Y)$ denotes the mutual information between random\nvariables $X$ and $Y$ and $\\mathrm{h} (X)$ denotes the differential entropy of\na continuous random variable $X$. Finally, the notation $f(\\gamma)\\sim\ng(\\gamma)$ implies\n$\\lim_{\\gamma\\rightarrow\\infty}\\frac{f(\\gamma)}{g(\\gamma)}=1$.\n\n\\section{System Model and Assumptions}\nWe consider a wireless network with $N$ users\\footnote{Each user\nconsists of a transmitter-receiver pair.} operating over a spectrum\nconsisting of $u$ orthogonal sub-bands. The number of active users\nis assumed to be a random variable with a given distribution, however, it is fixed during the whole transmission once it is set first. The\ntransmission blocks of each user comprise of an arbitrarily large\nnumber of transmission slots. We remark that the results of this paper are valid regardless of having block synchronization among the users, however, we assume synchronization at the symbol level. It is assumed that the $i^{th}$ user\nexploits $v_{i}(\\leq u)$ out of the $u$  sub-bands in each\ntransmission slot and hops randomly to another set of $v_{i}$\nfrequency sub-bands in the next transmission slot.  This user\ntransmits independent real Gaussian signals of variance\n$\\frac{P}{v_{i}}$ over the chosen sub-bands, in which $P$ denotes\nthe total average power for each transmitter. Each receiver is\nassumed to know the hopping pattern of its affiliated transmitter.\nIt is assumed that the users are not aware of each other's\ncodebooks and hence, no multiuser detection or interference\ncancelation is possible at the receiver sides. The\nstatic and non-frequency selective channel gain of the link\nconnecting the $i^{th}$ transmitter to the $j^{th}$ receiver is\nshown by $h_{i,j}$. As it will be shown in (\\ref{hoolooo}), the only\ninformation each transmitter needs  in order to regulate its\ntransmission rate (focusing on the achieved multiplexing gain) is\nits forward channel gain, the maximum interference level at its\nassociated receiver and the number of active users in the network.\nThis information can be obtained at the receiver side by\ninvestigating the interference PDF and provided to the\ncorresponding receiver via a feedback link.\n\nAs all users hop over different portions of the spectrum from\ntransmission slot to transmission slot, no user is assumed to be capable of\ntracking the instantaneous interference level. This assumption makes\nthe interference plus noise PDF at the receiver side of each user \nbe mixed Gaussian. In fact, depending on different choices the other\nusers make to select the frequency sub-bands and values of the\ncrossover gains, the interference on each frequency sub-band at any\ngiven receiver can have up to $2^{N-1}$ power levels. The vector\nconsisting of the received signals on the frequency sub-bands at the\n$i^{th}$ receiver in a typical transmission slot is\n\\begin{equation}\n\\vec{Y}_{i}=h_{i,i}\\vec{X}_{i}+\\vec{Z}_{i},\n\\end{equation}\nwhere $\\vec{X}_{i}$ is the $u\\times 1$ transmitted vector and\n$\\vec{Z_{i}}$ is the noise plus interference vector at the receiver\nside of the $i^{th}$ user. Due to the fact that each transmitter\nhops randomly from slot to slot, one may write $p_{\\vec{X}_{i}}(.)$\nas\n\\begin{eqnarray}\np_{\\vec{X}_{i}}(\\vec{x})=\\sum_{C\\in \\mathfrak{C}}\\frac{1}{{u\\choose v_{i}}}g_{u}(\\vec{x},C),\n\\end{eqnarray}\nwhich corresponds to the mixed Gaussian distribution. In the above\nequation, $g_{u}(\\vec{x},C)$ denotes the PDF of a zero-mean $u\\times\n1$ jointly Gaussian vector of covariance matrix $C$ and the set\n$\\mathfrak{C}$ includes all $u\\times u$ diagonal matrices in which\n$v_{i}$ out of the $u$ diagonal elements are $\\frac{P}{v_{i}}$ and\nthe rest are zero.  Denoting the noise plus interference on the\n$j^{th}$ sub-band at the receiver side of the $i^{th}$ user by\n$Z_{i,j}$ (the $j^{th}$ component of $\\vec{Z}_{i}$), it is clear\nthat $p_{Z_{i,j}}(.)$ is not dependent on $j$. This is due to the\nfact that the crossover gains are not frequency selective and\nthere is no particular interest to a specific frequency sub-band by\nany user. We assume there are $L_{i}+1$ ($L_{i}\\leq 2^{N-1}-1$)\npossible non-zero power levels for $Z_{i,j}$, say\n$\\{\\sigma^{2}_{i,l}\\}_{l=0}^{L_{i}}$. Denoting the occurrence\nprobability of $\\sigma^{2}_{i,l}$ by $a_{i,l}$, $p_{Z_{i,j}}(.)$\nidentifies a mixed Gaussian PDF as\n\\begin{equation}\n\\label{hf}\np_{Z_{i,j}}(z)=\\sum_{l=0}^{L_{i}}\\frac{a_{i,l}}{\\sqrt{2\\pi}\\sigma_{i,l}}\\exp \\left(-\\frac{z^2}{2\\sigma_{i,l}^{2}} \\right),\n\\end{equation}\nwhere  $\\sigma^{2}=\\sigma_{i,0}^{2}< \\sigma_{i,1}^{2}<\n\\sigma_{i,2}^{2}<...<\\sigma_{i,L_{i}}^{2}$ ($\\sigma^{2}$ is the\nambient noise power). We notice that for each $l\\geq 0$, there\nexists a $c_{i,l} \\geq 0$ such that\n$\\sigma_{i,l}^{2}=\\sigma^{2}+c_{i,l}P$ where\n$0=c_{i,0}<c_{i,1}<c_{i,2}<...<c_{i,L_{i}}$. One may write $Z_{i,j}=\n\\sum_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\epsilon_{k,j}h_{k,i}X_{k,j}+\\nu_{i,j}$ where $\\epsilon_{k,j}$ is a Bernoulli random variable showing if the\n$k^{th}$ user has utilized the $j^{th}$ sub-band, $X_{k,j}$ is\nthe signal of the $k^{th}$ user sent on the $j^{th}$ sub-band (assuming it has utilized the $j^{th}$ sub-band),\n and $\\nu_{i,j}$ is\nthe ambient noise which is a zero-mean Gaussian random variable with\nvariance $\\sigma^{2}$. The ratio $\\frac{P}{\\sigma^{2}}$ is taken as\na measure of SNR and is denoted by $\\gamma$ throughout the paper.\n\n\\section{Analysis of the Achievable Rate}\nLet us denote the achievable rate of the $i^{th}$ user by\n$\\mathscr{R}_{i}$. It can be observed that the communication channel\nof this user is a channel with state $S_{i}$, the hopping pattern of\nthe $i^{th}$ user, which is independently changing over different\ntransmission slots, and is known to both the transmitter and the receiver.\nThe achievable rate of such a channel is given by\n\\begin{equation}\n\\mathscr{R}_{i}=\\mathrm{I}(\\vec{X}_{i};\\vec{Y}_{i}\\vert S_{i})=\\sum_{s_{i}\\in \\mathfrak{S}_{i}}\\Pr(S_{i}=s_{i})\\mathrm{I}(\\vec{X}_{i};\\vec{Y}_{i}\\vert S_{i}=s_{i}),\n\\end{equation}\nwhere $\\mathrm{I}(\\vec{X}_{i};\\vec{Y}_{i}\\vert S_{i}=s_{i})$ is the\nmutual information between $\\vec{X_{i}}$ and $\\vec{Y_{i}}$ for the\nspecific sub-band selection corresponding to $S_{i}=s_{i}$. The set\n$\\mathfrak{S}_{i}$ denotes all possible selections of $v_{i}$ out of\nthe $u$ sub-bands. As $p_{\\vec{Z_{i}}}(.)$ is a symmetric density\nfunction, meaning all its components have the same PDF given in\n(\\ref{hf}), we deduce that $\\mathrm{I}(\\vec{X}_{i};\\vec{Y}_{i}\\vert\nS_{i}=s_{i})$ is independent of $s_{i}$. Therefore, to calculate\n$\\mathscr{R}_{i}$, we may assume any specific sub-band selection for\nthe $i^{th}$ user in $\\mathfrak{S}_{i}$, say the first $v_{i}$\nsub-bands. Denoting this specific state by $s^{*}_{i}$, we get\n\\begin{equation}\n\\mathscr{R}_{i}=\\mathrm{I}(\\vec{X}_{i};\\vec{Y}_{i}\\vert S_{i}=s^{*}_{i}).\n\\end{equation}\nIn this case, we denote $\\vec{Y}_{i}$ and $\\vec{X}_{i}$ by\n$\\vec{Y}_{i}(s^{*}_{i})$ and $\\vec{X}_{i}(s^{*}_{i})$, respectively.\nObviously, we have\n\\begin{equation}\n\\label{lala}\n\\mathscr{R}_{i}=\\mathrm{I}(\\vec{X}_{i}(s^{*}_{i});\\vec{Y}_{i}(s^{*}_{i}))=\\mathrm{h}(\\vec{Y}_{i}(s^{*}_{i}))-\\mathrm{h}(\\vec{Z}_{i}).\n\\end{equation}\nBecause $\\vec{Y}_{i}(s^{*}_{i})$ and $\\vec{Z}_{i}$ have mixed Gaussian\ndistribution, there is no closed-form expression for the\ndifferential entropy of these vectors. As such, we provide an upper\nbound and a lower bound on the achievable rate of each user in the\nfollowing subsections and show that these bounds converge in the\nasymptotic high SNR regime.\n\n \\subsection{Upper Bound on The Achievable Rates}\nIn this section, we develop an upper bound\n$\\mathscr{R}_{i,\\mathrm{ub}}$ on the achievable rate of the $i^{th}$\nuser that is tight enough to ensure that\n$\\mathscr{R}_{i,\\mathrm{ub}}-\\mathscr{R}_{i}$ does not increase\nunboundedly as SNR increases. The idea behind this upper bound is\nthe convexity of $\\mathscr{R}_{i}$ in terms of\n$p_{\\vec{Y}_{i}(s^{*}_{i})\\vert \\vec{X}_{i}(s^{*}_{i})}(.|.)$.\n\n\\begin{thm}\nThere exists an upper bound on the achievable rate of the $i^{th}$ user given by\n \\begin{equation}\n\\label{loopy}\n\\mathscr{R}_{i,\\mathrm{ub}}=\\frac{1}{2}v_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{v_{k}}{u}\\right)\\log\\left(1+\\frac{\\vert h_{i,i}\\vert^{2}\\gamma}{v_{i}}\\right)+\\widetilde{\\mathscr{R}}_{i,\\mathrm{ub}}\n\\end{equation}\nwhere\n$\\lim_{\\gamma\\rightarrow\\infty}\\widetilde{\\mathscr{R}}_{i,\\mathrm{ub}}<\\infty$.\nIn particular,\n$\\mathscr{R}_{i,\\mathrm{ub}}\\sim\\frac{1}{2}v_{i}\\prod_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\left(1-\\frac{v_{k}}{u}\\right)\\log\\gamma$.\n\\end{thm}\n\\begin{proof}\nLet $\\vec{W}_{i}$ be the $u\\times 1$ interference vector where its\n$j^{th}$ component $W_{i,j}$ is a random variable showing the\ninterference term on the $j^{th}$ frequency sub-band at the receiver\nof the $i^{th}$ user. We have $W_{i,j}=\\sum_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\epsilon_{k,j}h_{k,i}X_{k,j}$. Clearly, $\\vec{W}_{i}$ is a\nmixed Gaussian random vector where the Gaussian components in its PDF represent\ndifferent choices the other users make in selecting their sub-bands.\nIn fact, we have\n$p_{\\vec{W}_{i}}(\\vec{w})=\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}g_{u}(\\vec{w},D_{i,m})$,\nwhere $M_{i}=\\prod_{k\\neq i}{u\\choose v_{k}}$ and\n$D_{i,m}=\\mathrm{diag}(d_{i,m}^{(1)},\\cdots,d_{i,m}^{(u)})$, in\nwhich $d_{i,m}^{(j)} = \\sum_{\\substack{k=1\\\\k\\neq i}}^{N}\n\\epsilon_{k,j}^2 |h_{k,i}|^{2} \\frac{P}{v_k}$ denotes the variance\nof $W_{i,j}$ for the $m^{th}$ realization\nof $\\{\\epsilon_{k,j}\\}_{k\\neq i}$ out of $M_i$ possible realizations\\footnote{Note that as each\nuser transmits independent Gaussian signals over its chosen\nsub-bands, the matrices $\\{D_{i,m}\\}_{m=1}^{M_{i}}$ are diagonal.}. If the probability density\nfunction of the interference vector consisted only of\n$g_{u}(\\vec{w},D_{i,m})$, the forward link of the $i^{th}$ user\nwould be converted into an additive Gaussian channel. The achievable\nrate of such a virtual channel is given by\n\\begin{eqnarray}\n\\mathscr{R}_{i,m}&=&\\frac{1}{2}\\log\\frac{\\det\\left(\\mathrm{Cov}(\\vec{X}_{i}(s^{*}_{i}))+D_{i,m}+\\sigma^{2}I_{u}\\right)}{\\det\\left(D_{i,m}+\\sigma^{2}I_{u}\\right)} \\notag\\\\\n&=& \\frac{1}{2}\\log\\frac{\\prod_{j=1}^{v_{i}}\\left(\\frac{\\vert h_{i,i}\\vert^{2}P}{v_{i}}+d_{i,m}^{(j)}+\\sigma^{2}\\right)}{\\prod_{j=1}^{v_{i}}(d_{i,m}^{(j)}+\\sigma^{2})} \\notag\\\\&=&\\frac{1}{2}\\sum_{j=1}^{v_{i}}\\log\\bigg(1+\\frac{\\vert h_{i,i}\\vert^{2}P}{v_{i}(d_{i,m}^{(j)}+\\sigma^{2})}\\bigg).\n\\end{eqnarray}\nOne may also state this as follows. Let $T_{i,m}\\triangleq\\left\\{ j\n: 1\\leq j\\leq v_{i}, d_{i,m}^{(j)}=0\\right\\}$. Then,\n\\begin{equation}\n\\mathscr{R}_{i,m}=\\frac{\\vert T_{i,m}\\vert}{2}\\log\\bigg(1+\\frac{\\vert h_{i,i}\\vert^{2}\\gamma}{v_{i}}\\bigg)+\\widetilde{\\mathscr{R}}_{i,m},\n\\end{equation}\nwhere\n\\begin{equation}\n\\widetilde{\\mathscr{R}}_{i,m}=\\frac{1}{2}\\sum_{1\\leq j\\leq v_{i}: d_{i,m}^{(j)}\\neq 0}\\log\\bigg(1+\\frac{\\vert h_{i,i}\\vert^{2}P}{v_{i}(d_{i,m}^{(j)}+\\sigma^{2})}\\bigg)\n\\end{equation}\nand $\\vert T_{i,m}\\vert$ denotes the cardinality of the set\n$T_{i,m}$. As each nonzero $d_{i,m}^{(j)}$ is proportional to $P$,\nit is clear that\n$\\lim_{\\gamma\\rightarrow\\infty}\\widetilde{\\mathscr{R}}_{i,m}<\\infty$.\nWe know that $\\mathscr{R}_{i}$ is convex in terms of\n$p_{\\vec{Y}_{i}(s_{i}^{*})\\vert\n\\vec{X}_{i}(s_{i}^{*})}(\\vec{y}\\vert\\vec{x})=p_{\\vec{Z}_{i}}(\\vec{y}-h_{i,i}\\vec{x})$\n\\cite{53}. Noting this and the fact that\n$p_{\\vec{Z}_{i}}(\\vec{z})=\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}g_{u}(\\vec{z},D_{i,m}+\\sigma^{2}I_{u})$,\nwe have\n\\begin{equation}\n\\label{lal}\n\\mathscr{R}_{i}\\leq \\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\mathscr{R}_{i,m}=\\left(\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\vert T_{i,m}\\vert\\right)\\frac{1}{2}\\log\\bigg(1+\\frac{\\vert h_{i,i}\\vert^{2}\\gamma}{v_{i}}\\bigg)+\\widetilde{\\mathscr{R}}_{i,\\mathrm{ub}},\n\\end{equation}\nwhere\n$\\widetilde{\\mathscr{R}}_{i,\\mathrm{ub}}=\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\widetilde{\\mathscr{R}}_{i,m}$.\nClearly, as each $\\widetilde{\\mathscr{R}}_{i,m}$ saturates by\nincreasing $\\gamma$, one has\n$\\lim_{\\gamma\\rightarrow\\infty}\\widetilde{\\mathscr{R}}_{i,\\mathrm{ub}}<\\infty$.\nThe following Lemma offers an explicit expression for\n$\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\vert T_{i,m}\\vert$.\n\\begin{lem}\n\\begin{equation}\\label{nhjbv}\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\vert T_{i,m}\\vert=v_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{v_{k}}{u}\\right).\\end{equation}\n\\end{lem}\n\\begin{proof}\nDefining $A_{i,j}\\triangleq\\left\\{ m: 1\\leq m\\leq M_{i}, \\vert\nT_{i,m}\\vert=j\\right\\}$ for each $1\\leq i\\leq N$ and $1\\leq j\\leq\nv_{i}$, one may express the left-hand side of (\\ref{nhjbv}) as\n\\begin{equation}\n\\label{lki} \\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\vert\nT_{i,m}\\vert=\\frac{1}{M_{i}}\\sum_{j=1}^{v_{i}}j\\vert\nA_{i,j}\\vert.\\end{equation}\nLet $F_{i}$ be a random variable showing\nthe number of interference-free sub-bands among the $v_{i}$\nsub-bands selected by the $i^{th}$ user. Using (\\ref{lki}) and noting that\n$\\Pr\\{F_{i}=j\\}=\\frac{\\vert A_{i,j}\\vert}{M_{i}}$, \n\\begin{equation}\\frac{1}{M_{i}}\\sum_{m=1}^{M_{i}}\\vert T_{i,m}\\vert=\\sum_{j=1}^{v_{i}}j\\Pr\\{F_{i}=j\\}=\\mathrm{E}\\{ F_{i}\\}.\\end{equation}\nLet us define\n\\begin{equation}F_{i,j}\\triangleq\\left\\{ \\begin{array}{cc}\n1 &  W_{i,j}=0 \\\\\n0 &  W_{i,j}\\neq 0 \\end{array}\\right.\\end{equation}\nfor any $1\\leq i\\leq N$ and $1\\leq j\\leq v_{i}$.\nObviously, $F_{i}=\\sum_{j=1}^{v_{i}}F_{i,j}$. As such,\n\\begin{equation}\\mathrm{E}\\lbrace F_{i}\\rbrace=\\sum_{j=1}^{v_{i}}\\mathrm{E}\\{ F_{i,j}\\}=\\sum_{j=1}^{v_{i}}\\Pr\\{W_{i,j}=0\\}.\\end{equation}\nSince $\\Pr\\{\\epsilon_{k,j}=1\\}=\\frac{v_{k}}{u}$,\n\\begin{eqnarray}\\Pr\\{W_{i,j}=0\\}=\\Pr\\{\\textrm{$Z_{i,j}$ contains no interference}\\}=\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\Pr\\{\\epsilon_{k,j}=0\\}=\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{ v_{i}}{u}\\right).\n\\end{eqnarray}  This yields\n\\begin{equation}\\mathrm{E}\\{ F_{i}\\}=v_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{v_{k}}{u}\\right),\\end{equation}\nwhich completes the proof of Lemma 1.\n\\end{proof}\nBased on (\\ref{lal}) and Lemma 1, the proof of Theorem 1 is complete.\n\\end{proof}\n\n\n\\subsection{Lower Bound on the Achievable Rates}\nIn this section, we derive a lower bound on the achievable rates of\nusers. The idea behind deriving this lower bound is to invoke the\nclassical entropy power inequality (EPI). As we will see, this\ninitial lower bound is not in a closed form as it depends on the\ndifferential entropy of a mixed Gaussian random variable. In\nappendix A,  we obtain an appropriate upper bound on such an entropy\nwhich leads to the final lower bound on $\\mathscr{R}_{i}$.\n\\begin{thm}\nThere exists a lower bound $\\mathscr{R}_{i,\\mathrm{lb}}$ on the\nachievable rate of the $i^{th}$ user which can be written as\n\\begin{equation}\\mathscr{R}_{i,\\mathrm{lb}} = \\frac{1}{2}v_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{v_{k}}{u}\\right)\\log\\gamma + \\widetilde{\\mathscr{R}}_{i,\\mathrm{lb}},\\end{equation}\nsuch that $\\lim_{\\gamma \\to \\infty}\n\\widetilde{\\mathscr{R}}_{i,\\mathrm{lb}} < \\infty$.  In particular,\n$\\mathscr{R}_{i,\\mathrm{lb}}\\sim\\frac{1}{2}v_{i}\\prod_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\left(1-\\frac{v_{k}}{u}\\right)\\log\\gamma$.\n\\end{thm}\n\\begin{proof}\nWe define $\\vec{X}'_{i}$ to be the $v_{i}\\times 1$ signal vector\ncorresponding to the first $v_{i}$ elements of\n$\\vec{X}_{i}(s^{*}_{i})$. Clearly, $\\vec{X}'_{i}$ is a Gaussian\nvector with covariance matrix $\\frac{P}{v_i} I_{v_i}$. Let\n$\\vec{Y}'_{i}=h_{i,i}\\vec{X}'_{i}+\\vec{Z}'_{i}$ where $\\vec{Z}'_{i}$\nis the noise plus interference vector at the receiver side of the\n$i^{th}$ user on the first $v_{i}$ sub-bands. Using entropy power\ninequality, we have\n\\begin{equation}\n2^{\\frac{2}{v_{i}}\\mathrm{h}(\\vec{Y}'_{i})}\\geq 2^{\\frac{2}{v_{i}}\\mathrm{h}(h_{i,i}\\vec{X}'_{i})}+2^{\\frac{2}{v_{i}}\\mathrm{h}(\\vec{Z}'_{i})}.\n\\end{equation}\nDividing both sides by $2^{\\frac{2}{v_i}\\mathrm{h}(\\vec{Z}'_{i})}$, we get\n\\begin{equation}\n\\label{piu}\n\\mathrm{h}(\\vec{Y}'_{i})-\\mathrm{h}(\\vec{Z}'_{i})\\geq \\frac{v_{i}}{2}\\log\\bigg(2^{\\frac{2}{v_{i}}\\left(\\mathrm{h}(h_{i,i}\\vec{X}'_{i})-\\mathrm{h}(\\vec{Z}'_{i})\\right)}+1\\bigg).\n\\end{equation}\nOn the other hand, since $\\vec{Y}'_{i}$ is a subvector of $\\vec{Y}_{i}(s^{*}_{i})$, we have\n\\begin{equation}\n\\label{qiu}\n\\mathscr{R}_{i}=\\mathrm{I}(\\vec{X}_{i}(s^{*}_{i});\\vec{Y}_{i}(s^{*}_{i}))\\geq \\mathrm{I}(\\vec{X}'_{i};\\vec{Y}'_{i})=\\mathrm{h}(\\vec{Y}'_{i})-\\mathrm{h}(\\vec{Z}'_{i}).\n\\end{equation}\nComparing (\\ref{piu}) and (\\ref{qiu}) yields\n\\begin{equation}\n\\label{uuu}\n\\mathscr{R}_{i}\\geq \\frac{v_{i}}{2}\\log\\bigg(2^{\\frac{2}{v_{i}}\\left(\\mathrm{h}(h_{i,i}\\vec{X}'_{i})-\\mathrm{h}(\\vec{Z}'_{i})\\right)}+1\\bigg).\n\\end{equation}\nClearly,\n$\\mathrm{h}(h_{i,i}\\vec{X}'_{i})=\\frac{v_{i}}{2}\\log\\left(2\\pi\ne\\frac{\\vert h_{i,i}\\vert^{2} P}{v_{i}}\\right)$. As $\\vec{Z}'_{i}$\nis a mixed Gaussian random vector, there is no closed-form formula\nfor $\\mathrm{h}(\\vec{Z}'_{i})$. Hence, we have to find an\nappropriate upper bound on $\\mathrm{h}(\\vec{Z}'_{i})$ to further\nsimplify (\\ref{uuu}). Using the chain rule for the differential\nentropy, we obtain\n\\begin{equation}\n\\label{hjk}\n\\mathrm{h}(\\vec{Z}'_{i})\\leq\\sum_{j=1}^{v_{i}}\\mathrm{h}(Z_{i,j}).\n\\end{equation}\nRecalling the definitions of $\\{a_{i,l}\\}_{l=0}^{L_{i}}$ and\n$\\{c_{i,l}\\}_{l=0}^{L_{i}}$ in the system model, the following Lemma\nyields an upper bound on $\\mathrm{h}(Z_{i,j})$ for each $1\\leq j\\leq\nv_{i}$.\n\\begin{lem}\nFor every $1\\leq j\\leq v_{i}$ and for all values of $\\gamma$, there exists an upper bound on $\\mathrm{h}(Z_{i,j})$ given by\n\\begin{equation}\n\\label{uu-u}\n\\mathrm{h}(Z_{i,j})\\leq \\frac{1-a_{i,0}}{2}\\log(c_{i,L_{i}}\\gamma+1)+\\log(\\sqrt{2\\pi e}\\sigma)+\\mathscr{H}_{i}\n\\end{equation}\nwhere $\\mathscr{H}_{i}\\triangleq-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l}$  is the discrete entropy of $\\{a_{i,l}\\}_{l=0}^{L_{i}}$.\n\\end{lem}\n\\begin{proof}\nSee Appendix A.\n\\end{proof}\nBy (\\ref{uu-u}), (\\ref{hjk}) and (\\ref{uuu}),\n\\begin{eqnarray}\n\\label{uuku}\n\\mathscr{R}_{i}\\geq \\mathscr{R}_{i,\\mathrm{lb}}&\\triangleq&\\frac{v_{i}}{2}\\log\\bigg(\\frac{2^{-2\\mathscr{H}_{i}}\\vert h_{i,i}\\vert^{2}\\gamma}{v_{i}(c_{i,L_{i}}\\gamma+1)^{1-a_{i,0}}}+1\\bigg) \\notag\\\\\n&=& \\frac{1}{2} v_{i} a_{i,0}\\log \\gamma + \\frac{v_{i}}{2} \\log \\left( \\frac{2^{-2\\mathscr{H}_{i}}\\vert h_{i,i}\\vert^{2}}{v_{i}(c_{i,L_{i}}+\\gamma^{-1})^{1-a_{i,0}}}+\\gamma^{-a_{i,0}}\\right).\n\\end{eqnarray}\nDefining $\\widetilde{\\mathscr{R}}_{i,\\mathrm{lb}} \\triangleq\n\\frac{v_{i}}{2} \\log \\left( \\frac{2^{-2\\mathscr{H}_{i}}\\vert\nh_{i,i}\\vert^{2}}{v_{i}(c_{i,L_{i}}+\\gamma^{-1})^{1-a_{i,0}}}+\\gamma^{-a_{i,0}}\\right)$,\nwe note that $\\lim_{\\gamma \\to \\infty}\n\\widetilde{\\mathscr{R}}_{i,\\mathrm{lb}} < \\infty $. Combining this\nwith the fact that $a_{i,0} = \\prod_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\left(1-\\frac{v_{k}}{u}\\right)$ completes the proof of\nTheorem 2.\n\\end{proof}\nIn \\cite{kami-1}, we address another approach to propose a lower\nbound on the achievable rate of the $i^{th}$ user with the same SNR\nscaling as $\\mathscr{R}_{i,\\mathrm{lb}}$.\n\nOne may consider the following generalization of the FH scheme.\nLet us assume that the users are not restricted to choose a fixed\nnumber of frequency sub-bands in each transmission slot. In fact, in\neach transmission slot the number of selected sub-bands can be any\ninteger between $0$ and $u$, and the probability of choosing $v\\in[0,u]\\cap\\mathbb{Z}$ sub-bands by the $i^{th}$ user is denoted by $\\mu_{i,v}$.\nTherefore, the $i^{th}$ user has two random generators. The first\nrandom generator selects a number $0\\leq v\\leq u$ according to the\nprobability mass function $\\{\\mu_{i,v}\\}_{v=0}^{u}$, while the other\ngenerator selects $v$ sub-bands among the whole available $u$\nsub-bands. This repeats independently from transmission slot to\ntransmission slot. Based on the arguments made in section II, the\nachievable rate of the $i^{th}$ user can be written as\n\\begin{equation}\n\\label{hfd}\n\\mathscr{R}_{i}=\\sum_{v=0}^{u}\\mu_{i,v}\\mathrm{I}(\\vec{X}_{i}(s^{*}_{i,v});\\vec{Y}_{i}(s^{*}_{i,v})),\n\\end{equation}\nwhere $s^{*}_{i,v}$ denotes the state where the $i^{th}$ user\nselects the first $v$ sub-bands. Clearly,\n$\\mathrm{I}(\\vec{X}_{i}(s^{*}_{i,0});\\vec{Y}_{i}(s^{*}_{i,0}))=0$\nfor any $1\\leq i\\leq N$. Furthermore, \n \\begin{eqnarray} \\label{27}\n a_{i,0} &=& \\Pr\\left\\{\\textrm{A given component of $\\vec{Z}_{i}$ contains no interference}\\right\\} \\notag\\\\\n &=&\\sum_{v_{1}=0}^{u}\\cdots \\sum_{v_{i-1}=0}^{u} \\sum_{v_{i+1}=0}^{u} \\cdots \\sum_{v_{N}=0}^{u}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\mu_{k,v_{k}} \\left(1-\\frac{v_{k}}{u} \\right) \\notag\\\\\n&=& \\prod_{\\substack {k=1\\\\k \\neq i}}^{N}\\sum_{v_{k}=0}^{u}\\mu_{k,v_{k}} \\left(1-\\frac{v_{k}}{u} \\right) \\notag\\\\\n &=& \\prod_{\\substack {k=1\\\\k \\neq i}}^{N}\\sum_{v=0}^{u}\\mu_{k,v} \\left(1-\\frac{v}{u}\\right) \\notag\\\\\n&=& \\prod_{\\substack {k=1\\\\k \\neq i}}^{N} \\left(1-\\frac{\\bar{v}_k}{u}\\right)\n   \\end{eqnarray}\nwhere $\\bar{v}_k\\triangleq\\sum_{v=0}^{u}v\\mu_{k,v}$. Based on the results of\nthis section, we get\n \\begin{equation}\n \\label{poih}\n \\mathrm{I}(\\vec{X}_{i}(s^{*}_{i,v});\\vec{Y}_{i}(s^{*}_{i,v}))\\sim \\frac{1}{2}va_{i,0}\\log\\gamma.\n  \\end{equation}\nUsing (\\ref{hfd}), (\\ref{27}) and (\\ref{poih}) yields\n\\begin{eqnarray}\n\\label{mach}\n \\mathscr{R}_i &\\sim& \\sum_{v=0}^{u} \\frac{1}{2}\\mu_{i,v} v \\prod_{\\substack {k=1\\\\k \\neq i}}^{N}\\left(1-\\frac{\\bar{v}_k}{u}\\right) \\log \\gamma\\notag\\\\\n &=& \\frac{1}{2} \\bar{v}_i \\prod_{\\substack {k=1\\\\k \\neq i}}^{N} \\left(1-\\frac{\\bar{v}_k}{u}\\right) \\log \\gamma.\n\\end{eqnarray}\nIn fact, (\\ref{mach}) demonstrates that the generalized FH scheme is equivalent to\nthe FH scheme through substituting $\\{v_i\\}_{i=1}^{N} $ by\n$\\{\\bar{v}_i\\}_{i=1}^N$. However, it is remarkable that in contrast\nto the FH scheme in which $\\{v_i\\}_{i=1}^{N} $ are integer values,\nin the generalized FH scheme  $\\{\\bar{v}_i\\}_{i=1}^N$ are real\nvalues. This provides more flexibility in system design. The above\nobservation motivates us to use the generalized scenario in the\nsequel and we simply refer to it as the FH scheme. In this scheme,\nthe $i^{th}$ user has a parameter $\\bar{v}_i$, which can be chosen to\nbe any real number in the interval $[0,u]$.\n\n\\section{System Design}\nIn this section, we find the optimum operation point  for the FH scheme. This requires finding\nthe optimum values of $\\{\\bar{v}_i\\}_{i=1}^N$.  Based on the results\nestablished in the previous section, there exist upper and lower\nbounds on the achievable rate of each user that coincide in the\nhigh SNR regime. As such, the achievable rate itself must be\nasymptotically equivalent to each of these bounds, i.e.,\n\\begin{equation}\n\\label{uu}\n\\mathscr{R}_{i}\\sim \\frac{1}{2} \\bar{v}_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{\\bar{v}_{k}}{u}\\right)\\log\\gamma,\n\\end{equation}\nwhere, based on the conclusion made at the end of section III, the\nparameters $\\{\\bar{v}_{i}\\}_{i=1}^{N}$ can be adjusted to be any\nreal number in the range $[0,u]$. By (\\ref{uu}), the network sum-rate can be asymptotically written as\n\\begin{equation}\n\\sum_{i=1}^{N}\\mathscr{R}_{i}\\sim \\mathsf{SMG}\\left(\\bar{v}_{1},\\cdots,\\bar{v}_{N}\\right)\\log\\gamma,\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathsf{SMG}\\left(\\bar{v}_{1},\\cdots,\\bar{v}_{N}\\right)\\triangleq\\sum_{i=1}^{N}\n\\frac{1}{2} \\bar{v}_{i}\\prod_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\left(1-\\frac{\\bar{v}_{k}}{u}\\right).\\end{equation} We call\n$\\mathsf{SMG}\\left(\\bar{v}_{1},\\cdots,\\bar{v}_{N}\\right)$\nthe \\textit{sum multiplexing gain} of the system.\n$\\mathsf{SMG}\\left(\\bar{v}_{1},\\cdots,\\bar{v}_{N}\\right)$\nis a symmetric function of $(\\bar{v}_{1},\\cdots,\\bar{v}_{N})$ and\nhas a saddle point at $\\bar{v}_{i}=\\frac{u}{N}$ for  $1\\leq i\\leq\nN$. In a \\emph{fair} FH system, it is required that $\\bar{v}_{i}=v$ for\nall $1\\leq i\\leq N$ where $v$ is any real number in the interval\n$[0,u]$. Hence, we define\n\\begin{equation}\n\\label{yu}\n\\mathsf{SMG}(v,N)\\triangleq\\mathsf{SMG}\\left(\\bar{v}_{1},\\cdots,\\bar{v}_{N}\\right)\\Big|_{\\forall i:\\bar{v}_{i}=v}=\\frac{N}{2} v\\left(1-\\frac{v}{u}\\right)^{N-1}.\\end{equation}\nMaximizing this in terms of $v$ yields\n\\begin{equation}\n\\label{yuu}\nv_{\\mathrm{opt}}=\\frac{u}{N}.\n\\end{equation}\nComputation of $v_{\\mathrm{opt}}$ requires that all transmitters\nknow the number of active users $N$ in the network. As far as all\nchannel gains are realizations of independent and continuous random\nvariables, the number of power levels in the PDF of noise plus\ninterference on any frequency sub-band at the receiver side of any\nuser is almost surely equal to $2^{N-1}$. Therefore, any receiver\ncan identify $N$ and send it to its corresponding transmitter\nthrough a feedback link.\n\nSetting $v=v_{\\mathrm{opt}}$, the highest sum multiplexing gain\nof the fair  FH scheme is given by\n \\begin{equation}\n \\label{bxxxx}\n\\sup_{v}\\, \\, \\mathsf{SMG}(v,N)=\\frac{1}{2}\nu\\left(1-\\frac{1}{N}\\right)^{N-1}.\\end{equation} It is remarkable\nthat $\\frac{u}{N}$ may not be a positive\ninteger. If we do not adopt the generalized FH scheme, then all\nusers must hop randomly over  sets of $\\tilde{v} =\n\\max\\left\\{\\lfloor\\frac{u}{N}\\rfloor,1 \\right\\}$ frequency\nsub-bands. This results in a sum multiplexing gain of\n$\\frac{N}{2} \\tilde{v} \\left(1-\\frac{\\tilde{v}}{u}\\right)^{N-1}$.\nThis is, in general, less than\n$\\frac{1}{2}u\\left(1-\\frac{1}{N}\\right)^{N-1}$. By adopting the\ngeneralized FH scheme in case $\\frac{u}{N}\\notin\\mathbb{Z}$, each\nuser only needs to hop randomly over different sets of frequency\nsub-bands of cardinality $\\lfloor\\frac{u}{N}\\rfloor$ or\n$\\lceil\\frac{u}{N}\\rceil$. In fact, each user has two random\ngenerators. The first random generator selects one of the numbers\n$\\lfloor\\frac{u}{N}\\rfloor$ and $\\lceil\\frac{u}{N}\\rceil$ with\nprobabilities $\\mu$ and $1-\\mu$, respectively, such that\n$\\mu\\lfloor\\frac{u}{N}\\rfloor+(1-\\mu)\\lceil\\frac{u}{N}\\rceil=\\frac{u}{N}$\nor equivalently $\\mu=\\lceil\\frac{u}{N}\\rceil-\\frac{u}{N}$. Let us\nassume the first random generator has selected a number\n$a\\in\\{\\lfloor\\frac{u}{N}\\rfloor,\\lceil\\frac{u}{N}\\rceil\\}$. Then,\nthe second random generator selects a subset of cardinality\n$a$ among the $u$ frequency sub-bands. Doing this independently from\ntransmission slot to transmission slot, the sum multiplexing gain given\nin (\\ref{bxxxx}) is achieved.\n\n\\textit{Observation 1}- One might suggest another well-known utility\nfunction that is popular in the game theory context, namely the\n\\textit{proportional fair function}, which is defined as $\\sum_{i=1}^{N}\\log\n\\mathscr{R}_{i}$ . We have\n\\begin{eqnarray}\n\\sum_{i=1}^{n}\\log \\mathscr{R}_{i} &\\sim& \\sum_{i=1}^{N}\\log\\left( \\frac{1}{2} \\bar{v}_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N} \\left(1-\\frac{\\bar{v}_{k}}{u} \\right)\\log \\gamma\\right) \\notag\\\\\n&=& \\sum_{i=1}^{N}\\log\\left(\\frac{1}{2} \\bar{v}_{i}\\prod_{\\substack{k=1\\\\k\\neq i}}^{N}\\left(1-\\frac{\\bar{v}_{k}}{u}\\right)\\right)+N\\log\\log\\gamma .\n\\end{eqnarray}\nIt can be easily verified that $\\sum_{i=1}^{N}\\log\\left(\\frac{1}{2}\n\\bar{v}_{i}\\prod_{\\substack{k=1\\\\k\\neq\ni}}^{N}\\left(1-\\frac{\\bar{v}_{k}}{u}\\right)\\right)$ has an absolute\nmaximum at $\\bar{v}_{i}=\\frac{u}{N}$ for $1\\leq i\\leq N$.\n\n\\textit{Observation 2}- As we will discuss in more detail in the\nnext section, the number of active users in the system is in general\na random variable. Although users in the FH system can use their\nknowledge about the number of active users to\nadjust the hopping parameter (as explained earlier in this section),  one may devise a sub-optimal rule to\nfix $v=v^{*}$ given by\n\\begin{equation}\n\\label{pof}\nv^{*}=\\arg\\,\\,\\, \\max_{v\\in[0,u]}\\mathrm{E}\\left\\{\\mathsf{SMG}(v,N)\\right\\},\n\\end{equation}\nwhere the expectation is with respect to the number of active users\nin the network. This selection of $v$ by all users makes the system robust\nagainst changes in the number of active users in the\nnetwork\\footnote{In fact, the transmitters use their knowledge about\nthe instantaneous number of active users only to regulate their\ntransmission rate. This is explained in more details in\n(\\ref{hoolooo}).}. We call this version of the FH system the robust Frequency Hopping. In fact, in the robust FH scenario, there exists a \\emph{global hopping parameter} $v$ where all users hop over a number $v$ of the $u$ frequency sub-bands. We remark that the rule in (\\ref{pof}) is a particular design approach for the robust FH system.  In the next section, we consider another design rule based on maximizing the average of the minimum multiplexing gain per user in term of the number of active users in the network. \n\n\n  \\section{Comparison of the robust FH scenario with the FD scheme}\n\nIn a centralized setup, under the condition that no user is aware of\nthe other users' codebooks and the number of users is fixed and\nknown to the central controller, it is shown in \\cite{1} that if the crossover channel gains are sufficiently larger than the forward channel gains, then every Pareto optimal rate vector is realized by Frequency Division for all ranges of SNR. However,\nin realistic scenarios, the number of active users is not fixed.\nThis degrades the performance of the FD scheme as it is designed for\na specified number of users. In particular, if the number of active users\nis less than the designed target of the FD scheme, a considerable portion\nof the spectrum may remain unused. This encourages us to compare the performance\nof the proposed robust FH scheme with that of the FD scheme in a setup where\nthe number of active users is a random variable with a given distribution\\footnote{Note that, as explicitly mentioned in the system model, the number of users is assumed to be\nfixed for the whole transmission period of interest.}.\n\nTo perform the comparison, we introduce four different performance\nmeasures. In the following definitions, the $\\sup$ operation is over\npossible adjustable parameters in the system, e.g., the hopping\nparameter in the FH scenario.  All expectations are taken with respect to\n$N$. We define $q_{n}\\triangleq\\Pr\\{N=n\\}$ for all $n\\geq 0$.  It is\nassumed that the maximum number of active users in the network is\n$n_{\\max}$, i.e.,  $\\Pr\\{N>n_{\\max}\\}=0$. We usually take\n$q_{0}=0$ unless otherwise stated.\n\\begin{itemize}\n \\item \\textit{Average sum multiplexing gain}, which is defined as\n\\begin{eqnarray}\n \\eta^{(1)} \\triangleq \\sup\\lim_{\\gamma \\to \\infty} \\frac{\\mathrm{E} \\left\\{\\sum_{i=1}^N \\mathscr{R}_i \\right\\}}{\\log \\gamma} = \\sup\\mathrm{E} \\left\\{\\mathsf{SMG}\\right\\},\n\\end{eqnarray}\n where $\\mathsf{SMG}=\\lim_{\\gamma\\to\\infty}\\frac{\\sum_{i=1}^{N}\\mathscr{R}_{i}}{\\log\\gamma}$ is the sum multiplexing gain.\n\\item \\textit{Average minimum multiplexing gain per user}, which is defined as\n\\begin{eqnarray}\n \\eta^{(2)} \\triangleq\\sup \\lim_{\\gamma \\to \\infty} \\frac{\\mathrm{E} \\left\\{\\min_{1\\leq i\\leq N}\\mathscr{R}_{i} \\right\\}}{\\log \\gamma}.\n\\end{eqnarray}\n\\item \\textit{Minimum nonzero multiplexing gain per user}, which is defined as\n\\begin{eqnarray}\n \\eta^{(3)} \\triangleq\\min_{n: q_n \\neq 0} \\quad \\min_{\\substack {N_{\\mathrm{serv}}=n\\\\1 \\leq i \\leq n}} \\quad \\lim_{\\gamma \\to \\infty}  \\frac{\\mathscr{R}_i}{\\log \\gamma}\n\\end{eqnarray}\n where $N_{\\rm{serv}}$ denotes the number of active users receiving service (i.e., their multiplexing gain is strictly positive).\\item \\textit{Service capability}, which is defined as\n\\begin{eqnarray} \\label{eta4}\n \\eta^{(4)} \\triangleq \\sup\\mathrm{E} \\left\\{ \\frac{N_{\\rm{serv}}}{N}\\right\\}.\n\\end{eqnarray}\n\n\n\n\n\\end{itemize}\n \n\nThe FD system is designed to service, at most, a certain number of\nactive users. We denote this design target in the FD scheme by\n$n_{\\mathrm{des}}$. Therefore, the spectrum is divided to\n$n_{\\mathrm{des}}$ \\emph{bands} where each band contains\n$\\frac{u}{n_{\\mathrm{des}}}$ frequency sub-bands. This requires that\n$u$ is divisible by $n_{\\mathrm{des}}$, which is assumed to be the\ncase to guarantee fairness. Each user that becomes active occupies\nan empty band. If there is no empty band, no service is available.\nIn case $n_{\\max}$ is finite, the central controller in the FD\nsystem sets $n_{\\mathrm{des}}=n_{\\max}$ to ensure that all users can\nreceive service upon activation. In case $n_{\\max}$ is not a finite\nnumber, the central controller sets $n_{\\mathrm{des}}=u$ to guarantee\nthat as many users receive service as possible. Therefore, $n_{\\mathrm{des}}=\\min\\{n_{\\max},u\\}$. In fact, we will show that selecting $n_{\\mathrm{des}}=\\min\\{n_{\\max},u\\}$ maximizes the service capability in the FD system.\n\nWe remark that due to the nature of the robust FH scheme, as far as users  hop over a proper subset of size $v$ of the $u$ sub-bands, all\nusers receive service, while if $v=u$ and $N>1$, no user receives\nservice, i.e.,  the multiplexing gain achieved by any active user is\nzero. As such, to get the largest service capability in the FH scenario, we require $v\\in(0,u)$. As an example, if $v^{*}$ in (\\ref{pof}) is equal to $u$, the service capability will be less than $1$. To avoid this, we set the global hopping parameter $v=v^{*}-\\varepsilon=u-\\varepsilon$ for sufficiently small $\\varepsilon$ such that the performance of the robust FH is still above the performance of the FD scenario.   \n\n$\\bullet$ \\textit{ Average sum multiplexing gain}\n\nThis measure\nis a meaningful tool of comparison if $n_{\\max}<\\infty$.\nHence, we assume $n_{\\max}$ is a finite number and $u$ is a multiple\nof $n_{\\max}$ in this subsection. It is easily seen that the\nsum multiplexing gain in the FD scenario is\n \\begin{equation}\n \\label{pooohj}\n \\mathsf{SMG}_{\\mathrm{FD}}(n_{\\mathrm{des}},N)=\\left\\{\\begin{array}{cc }\n    \\frac{N}{2} \\frac{u}{n_{\\mathrm{des}}}  & N\\leq n_{\\mathrm{des}}   \\\\\n       \\frac{u}{2}  &   N>n_{\\mathrm{des}}\n\\end{array}\\right..\n \\end{equation}\n Noting (\\ref{yu}),  $\\mathsf{SMG}_{\\mathrm{FH}}(v,N)$\nis given by\n \\begin{equation}\n \\label{llll}\n \\mathsf{SMG}_{\\mathrm{FH}}(v,N)=\\frac{1}{2}Nv\\left(1-\\frac{v}{u}\\right)^{N-1}. \\end{equation}\nSince the number of active users $N$ is a global knowledge, all\nusers can choose $v=v_{\\mathrm{opt}}=\\frac{u}{N}$ to achieve the\nmaximum sum multiplexing gain.  However, as mentioned earlier, a robust\nhopping strategy against changes in the number of active users is\nthe one given in (\\ref{pof}).\n\nIt is notable that although the value of $v$ is fixed at $v^{*}$,\nall users regulate their rates based on the instantaneous number of\nactive users to avoid transmission failure. Using the lower bound on\nthe achievable rate of the $i^{th}$ user given in (\\ref{uuku}), the\n$i^{th}$ user selects its actual rate $R_{i}$ as\n\\begin{equation}\n\\label{hoolooo}\nR_{i}=\\frac{v^{*}}{2}\\log\\left(\\frac{\\left(\\frac{v^{*}}{u}\\right)^{-\\frac{2(N-1)v^{*}}{u}}\\left(1-\\frac{v^{*}}{u}\\right)^{-2(N-1)\\left(1-\\frac{v^{*}}{u}\\right)}\\vert\nh_{i,i}\\vert^{2}\\gamma}{v^{*}\\left(1+\\frac{\\sum_{j\\neq\ni}|h_{j,i}|^{2}\\gamma}{v^{*}}\\right)^{1-\\left(1-\\frac{v^{*}}{u}\\right)^{N-1}}}+1\\right).\n\\end{equation}\nIt is seen that the quantities the $i^{th}$ transmitter needs to\nevaluate $R_{i}$ are $|h_{i,i}|$, $\\sum_{j\\neq i}|h_{j,i}|^{2}$ and\n$N$. The $i^{th}$ receiver sends these required data to the\ntransmitter via a feedback link.\n\nWe present an example to compare the performance of FH with that of FD in terms of\n$\\eta^{(1)}$.\n\n \\textit{Example 1}-\nLet us consider a network where $n_{\\max}=2$. The central controller\nin the FD system sets $n_{\\mathrm{des}}=2$, and according to\n(\\ref{pooohj}),\n$ \\eta_{\\mathrm{FD}}^{(1)}=\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FD}}(2,N)\\}=q_{1}\\frac{u}{4}+q_{2}\\frac{u}{2}=\\frac{q_{1}+2q_{2}}{4}u$. Based on (\\ref{llll}),\n $\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}=\\frac{1}{2}q_{1}v+q_2v\\left(1-\\frac{v}{u}\\right)$. \nUsing this in (\\ref{pof}),\n\\begin{equation}\nv^{*}=\\arg\\max_{v\\in[0,u]}\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}=\\left\\{\\begin{array}{cc}\n   \\frac{q_{1}+2q_{2}}{4q_{2}}u   & q_{1}\\leq 2q_{2}   \\\\\n    u  &  q_{1}>2q_{2}\n\\end{array}\\right..\n\\end{equation}\nTherefore,\n\\begin{eqnarray}\n\\eta_{\\mathrm{FH}}^{(1)}=\\sup_{v\\in[0,u]}\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}=\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v^{*},N)\\}\n=\\left\\{\\begin{array}{cc}\n    \\frac{(q_{1}+2q_{2})^{2}}{16q_{2}}u  &  q_{1}\\leq 2q_{2}  \\\\\n    \\frac{q_{1}}{2}u  & q_{1}>2q_{2}\n\\end{array}\\right..\n\\end{eqnarray}\nIt is easy to see that\n$\\eta_{\\mathrm{FH}}^{(1)}>\\eta_{\\mathrm{FD}}^{(1)}$ if and only if\n$q_{1}>2q_{2}$, or equivalently, $q_{1}>\\frac{2}{3}$. We note that\nin this case $v^{*}=u$, i.e., all users spread their power on the\nwhole spectrum and no hopping is performed. This makes service capability be strictly less than $1$ because, if both users are active, non of them receive service. As such, we take $v=u-\\varepsilon$. To ensure that the performance of the robust FH scenario is above that of the FD system, we require\n\\begin{equation}\n\\label{lki}\n\\frac{1}{2}q_{1}(u-\\varepsilon)+q_2(u-\\varepsilon)\\left(1-\\frac{u-\\varepsilon}{u}\\right)>\\frac{q_{1}+2q_{2}}{4}u.\n\\end{equation}\nAs far as $\\varepsilon<\\frac{u}{2}$, (\\ref{lki}) is equivalent to\n$q_{1}>2q_{2}\\frac{1-\\frac{2\\varepsilon}{u}\\left(1-\\frac{\\varepsilon}{u}\\right)}{1-\\frac{2\\varepsilon}{u}}$. This is a more restrictive condition than $q_{1} > 2q_{2}$ which is the cost paid for having full service capability. However, for $\\varepsilon \\ll u$ the two regions of $(q_{1},q_{2})$ are almost the same.\n $\\square$\n\nIn \\cite{kami-1}, it is shown that in case $n_{\\max}=3$,\nthere exists a probability set of $(q_{1},q_{2},q_{3})$  on the number of active users that makes  FH achieve a\nhigher performance compared to FD in terms of $\\eta^{(1)}$ while $v^{*}$ is\nstrictly less than $u$.\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$\\bullet$ \\textit{Average minimum multiplexing gain per user}\n\nThis measure  can also be written as\n  \\begin{equation}\n  \\eta^{(2)}=\\sup\\mathrm{E} \\left \\lbrace \\frac{\\mathsf{SMG}}{N}\\mathbb{1}(N_{\\mathrm{serv}}=N)\\right \\rbrace.\n  \\end{equation}\nIn fact, if $N_{\\mathrm{serv}} \\neq N$, there exists at least one\nuser that achieves no multiplexing gain. Therefore, the minimum\nmultiplexing gain per user is zero in this case. However, if\n$N_{\\mathrm{serv}}=N$, all users achieve a nonzero multiplexing\ngain. This measure can be used whether $n_{\\max}$ is finite or\ninfinite.\n\nIn case of the FH scenario, the rule to choose the optimum value of the global hopping parameter\n$v$, denoted by $v^{\\dagger}$, is given by\n \\begin{equation}\n \\label{pof2}\nv^{\\dagger}=\\arg\\,\\,\\, \\max_{v\\in[0,u]}\\mathrm{E}\\left \\lbrace \\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\mathbb{1}(N=N_{\\mathrm{serv}})\\right \\rbrace.\n\\end{equation}\nIn this case, the actual transmission rate of the $i^{th}$ user is given by (\\ref{hoolooo}) where $v^{*}$ is replaced by $v^{\\dagger}$.\n\n\\textit{Example 2}- Considering the same setup in example 1, as\n$n_{\\max}<\\infty$, we have $N_{\\mathrm{serv,FD}}=N$. Hence, we have\n$\\eta_{\\mathrm{FD}}^{(2)}=\\frac{1}{2}\\frac{u}{2}q_1 +\n\\frac{1}{2}\\frac{u}{2} q_{2}=\\frac{u}{4}$. In case of the FH scheme,\n\\begin{equation}\n\\label{ }\n\\mathbb{1}(N_{\\mathrm{serv,FH}}=N)=\\left\\{\\begin{array}{cc}\n    1  &  \\textrm{$N=1$ or ($N>1$ and $v\\neq u$) }  \\\\\n    0  &   \\mathrm{oth.}\n\\end{array}\\right..\n\\end{equation}\n\nHence,\n\\begin{eqnarray}\n\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\mathbb{1}(N_{\\mathrm{serv}}=N)\\right\\}&=& \\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\mathbb{1}(N_{\\mathrm{serv}}=N)\\Big| N=1\\right\\}\\Pr\\{N=1\\}\\notag\\\\&&+\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\mathbb{1}(N_{\\mathrm{serv}}=N) \\Big|N=2\\right\\}\\Pr\\{N=2\\}\\notag \\\\\n&=& \\frac{1}{2}q_{1}v+\\frac{1}{2}q_{2}v\\left(1-\\frac{v}{u}\\right)\\mathbb{1}(v\\neq u)\\notag\\\\\n&=& \\frac{1}{2}q_{1}v+\\frac{1}{2}q_{2}v\\left(1-\\frac{v}{u}\\right).\n\\end{eqnarray}\nHence,\n\\begin{equation}\n\\label{ }\nv^{\\dagger}=\\arg\\max_{v\\in(0,u]}\\left\\{q_{1}v+q_{2}v\\left(1-\\frac{v}{u}\\right)\\right\\},\\end{equation}\nwhich yields\n\\begin{equation}\n\\label{ }\nv^{\\dagger}=\\left\\{\\begin{array}{cc}\n    \\frac{u}{2q_{2}}  &  2q_{2}>1  \\\\\n    u  &   2q_{2}\\leq 1\n\\end{array}\\right..\n\\end{equation}\n\nAs such,\n\\begin{eqnarray}\n\\eta_{\\mathrm{FH}}^{(2)}=\\left\\{\\begin{array}{cc}\n   \\frac{1}{8q_{2}}u &  2q_{2}>1  \\\\\n    \\frac{1}{2}q_{1}u  & 2q_{2}\\leq 1\n\\end{array}\\right..\n\\end{eqnarray}\nIt is easy to see that $\\eta_{\\mathrm{FH}}^{(2)}>\\eta_{\\mathrm{FD}}^{(2)}$\nif and only if $2q_{2}<1$, or equivalently $q_{1}>\\frac{1}{2}$. However, in this\ncase $v^{\\dagger}=u$. Hence, to make the service capability be $1$, we choose the global hopping parameter $v=u-\\varepsilon$. To ensure that FH still outperforms FD in terms of the average minimum multiplexing gain per user, we require, \n\\begin{equation}\n\\label{ }\n \\frac{1}{2}q_{1}(u-\\varepsilon)+\\frac{1}{2}q_{2}(u-\\varepsilon)\\left(1-\\frac{u-\\varepsilon}{u}\\right)>\\frac{u}{4}.\n \\end{equation}\n This is equivalent to $2q_{2}<\\frac{2}{1-\\frac{\\varepsilon}{u}}\\left(1-\\frac{1}{2\\left(1-\\frac{\\varepsilon}{u}\\right)}\\right)$. \n $\\square$\n \nIn the next example, we provide a case where\n$\\eta_{\\mathrm{FH}}^{(2)}>\\eta_{\\mathrm{FD}}^{(2)}$\nwhile $v^{\\dagger}$ is strictly less than $u$.\n\n\n\\textit{Example 3}- Let $n_{\\max}<\\infty$. In this example, we aim to derive a\nsufficient condition on $\\{q_{n}\\}_{n=1}^{n_{\\max}}$ such that\n$\\eta_{\\mathrm{FH}}^{(1)}> \\eta_{\\mathrm{FD}}^{(1)}$ or $\\eta_{\\mathrm{FH}}^{(2)}> \\eta_{\\mathrm{FD}}^{(2)}$.\n\n\\textit{Case 1-} Let us consider the measure $\\eta^{(1)}$.  We have the following result.\n\\begin{proposition}\nAs far as\n\\begin{equation}\\label{goolj}\\mathrm{E}\\{N\\}< \\frac{1}{2}\\ln\\left((e^{2}-1)n_{\\max}\\right),\\end{equation} we have $\\eta_{\\mathrm{FD}}^{(1)}<\\eta_{\\mathrm{FH}}^{(1)}$.\n\\end{proposition}\n\\begin{proof}\nSee Appendix B.\n\\end{proof}\nFor example, if $n_{\\max}=2$, (\\ref{goolj}) gives $\\mathrm{E}\\{N\\} \\leq 1.274$, or equivalently $q_{1} \\geq 0.726$.\nBy example 1, we notice that $\\eta_{\\mathrm{FH}}^{(1)}\\geq \\eta_{\\mathrm{FD}}^{(1)}$ if and only if\n$q_{1} \\geq 0.667$.\n\n\\textit{Case 2-} As for $\\eta^{(2)}$, along the same lines leading\nto (\\ref{goolj}),  a sufficient condition for\n$\\eta_{\\mathrm{FH}}^{(2)}>\\eta_{\\mathrm{FD}}^{(2)}$ is given in the\nfollowing Proposition.\n\\begin{proposition}\nAs far as\n\\begin{equation}\n\\label{vegggg}\n\\frac{1}{\\mathrm{E}\\{N\\}}\\left(1-\\frac{1}{\\mathrm{E}\\{N\\}}\\right)^{\\mathrm{E}\\{N\\}-1}>\\frac{1}{n_{\\max}},\\end{equation}\nwe have $\\eta_{\\mathrm{FD}}^{(2)}<\\eta_{\\mathrm{FH}}^{(2)}$.\\end{proposition}\n\\begin{proof}\nSee Appendix C.\n\\end{proof}\n\n\n\n\n\nFor example, if $n_{\\max}=10$, $q_{1}=0.22$,\n$q_{2}=q_{3}=q_{4}=0.24$ and $q_{5}=q_{6}=\\cdots=q_{10}=0.01$, one\nhas $\\mathrm{E}\\{N\\}=2.78$, which satisfies (\\ref{vegggg}).\nTherefore, we conclude\n$\\eta_{\\mathrm{FH}}^{(2)}>\\eta_{\\mathrm{FD}}^{(2)}$. Computing these\nquantities directly, we get $\\eta_{\\mathrm{FD}}^{(2)}=\\frac{u}{16}$\nand\n\\begin{eqnarray}\n\\label{ }\n\\eta_{\\mathrm{FH}}^{(2)}&=&\\frac{1}{2}\\max_{v\\in[0,u]}\\left\\{v\\sum_{n=1}^{10}q_{n}\\left(1-\\frac{v}{u}\\right)^{n-1}\\right\\}\\notag\\\\\n&\\stackrel{(a)}{=}&\\frac{1}{2}u\\max_{\\omega_{v}\\in[0,1]}(1-\\omega_{v})\\left(0.22+0.24(\\omega_{v}+\\omega_{v}^{2}+\\omega_{v}^{3})+0.01(\\omega_{v}^{4}+\\omega_{v}^{5}+\\omega_{v}^{6}+\\omega_{v}^{7}+\\omega_{v}^{8}+\\omega_{v}^{9})\\right)\\notag\\\\\n&\\stackrel{(b)}{=}&0.1121u\n\\end{eqnarray}\nwhere in $(a)$, we define $\\omega_{v}\\triangleq 1-\\frac{v}{u}$ and $(b)$ is\nobtained by setting $\\omega_{v}=0.28$, or equivalently\n$v=v^{\\dagger}=0.72u$. This yields\n$\\frac{\\eta_{\\mathrm{FH}}^{(2)}}{\\eta_{\\mathrm{FD}}^{(2)}}=1.7936$.\n\n\n\n\n\\textit{Example 4}- In this example, we assume a Poisson\ndistribution on the number of active users, i.e.,\n$q_{n}=\\frac{e^{-\\lambda}\\lambda^{n}}{n!}$, $n\\geq 0$. This\nassumption corresponds to the scenario where potentially a large\nnumber $n_{\\max}$ of users may share the spectrum. However, the\nactivation probability $p$ of each user is very small. One can well\napproximate the number of active users in the network by a Poisson\nrandom variable with parameter $\\lambda = pn_{\\max}$. We have\n\\begin{eqnarray}\n\\label{ }\n\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\mathbb{1}(N_{\\mathrm{serv,FH}}=N)\\right\\}&\\stackrel{(a)}{=}&\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)}{N}\\right\\}\\notag\\\\&=&\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda}\\lambda^{n}}{n!}\\left(v\\left(1-\\frac{v}{u}\\right)^{n-1}\\right)\\notag\\\\\n&=&\\frac{1}{2}\\frac{v}{1-\\frac{v}{u}}\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda}\\lambda^{n}}{n!}\\left(1-\\frac{v}{u}\\right)^{n}\\notag\\\\\n&\\stackrel{(b)}{=}&\\frac{1}{2}\\frac{v}{1-\\frac{v}{u}}\\left(e^{\\lambda\\left(e^{\\ln\\left(1-\\frac{v}{u}\\right)}-1\\right)}-e^{-\\lambda}\\right) \\notag\\\\\n&=& \\frac{e^{-\\lambda}(1-\\omega_{v})\\left(e^{\\lambda\\omega_{v}}-1\\right)}{2\\omega_{v}}u.\n\\end{eqnarray}\nIn the above equation, $(a)$ results from the fact that $\\mathbb{1}(N_{\\mathrm{serv,FH}}=N)=0$ whenever $v=u$ and $N >1$, however, $\\mathsf{SMG}_{\\mathrm{FH}}(v,N) =0$ in this case. $(b)$ follows by the fact that\n$E\\{e^{tN}\\}=e^{\\lambda(e^{t}-1)}$ for any $t$ and\n$\\omega_{v}=1-\\frac{v}{u}$ as defined in example 3. It can be easily\nseen that the optimal $\\omega_{v}$ satisfies the nonlinear equation\n$e^{-\\lambda\\omega_{v}}=1-\\lambda\\omega_{v}+\\lambda\\omega_{v}^{2}$.\nSolving this for $\\omega_{v}$, we find out that $v^{\\dagger}$ is not\nequal to $u$ for all $\\lambda >2$. The following table lists the optimum values of\n$\\omega_{v}$, i.e., $\\omega_{v^{\\dagger}}$, the values of\n$v^{\\dagger}$ and also the corresponding average minimum\nmultiplexing gain per user $\\eta_{\\mathrm{FH}}^{(2)}$ for\n$\\lambda\\in\\{3,\\cdots,10\\}$.\n\\begin{equation}\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n$\\lambda$&$3$& $4$& $5$ &$6$&$7$&$8$&$9$&$10$\\\\\n\\hline\n$\\omega_{v^{\\dagger}}$& $0.4536$&$0.6392$&$0.7347$&$0.7912$&$0.828$&$0.8537$&$0.8727$&$0.8873$\\\\\n\\hline\n$v^{\\dagger}$&$0.5464u$& $ 0.3608u$&   $ 0.2653u$&$ 0.2088u$&   $0.1720u$&  $  0.1463u$&$0.1273u$&$0.1127u$\\\\\n\\hline\n$\\eta_{\\mathrm{FH}}^{(2)}$& $0.0869u$&$0.0615u$&$0.0467u$&$0.0374u$&$0.0311u$&$0.0266u$&$0.0232u$&$0.0206u$\\\\\n\\hline\n\\end{tabular}.\\end{equation}\n\nIn order to provide fairness among the users, the FD system tries to\nserve as many users as it can. Since it is not possible to serve\nmore than $u$ users and $n_{\\max} \\gg u$, the central controller\nsets $n_{\\mathrm{des}}=u$. Therefore, $N_{\\mathrm{serv,FD}}<N$ if\nand only if $N>u$. Using this and  by (\\ref{pooohj}),\n\\begin{eqnarray}\n\\eta_{\\mathrm{FD}}^{(2)}&=&\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FD}}(n_{\\mathrm{des}},N)}{N}\\mathbb{1}(N_{\\mathrm{serv,FD}}=N)\\right\\}\\notag\\\\\n&=&\\mathrm{E}\\left\\{\\frac{\\mathsf{SMG}_{\\mathrm{FD}}(u,N)}{N}\\Bigg|N\\leq u\\right\\}\\Pr\\{N\\leq u\\}\\notag\\\\&=&\\frac{1}{2}\\sum_{n=1}^{u}\\frac{e^{-\\lambda}\\lambda^{n}}{n!}.\n\\end{eqnarray}\nWe have sketched $\\eta_{\\mathrm{FH}}^{(2)}$ and $\\eta_{\\mathrm{FD}}^{(2)}$\nin terms of $\\lambda$ in fig. 1 and fig. 2 for the cases $u=7$ and $u=20$, respectively.\nIt is noticeable that $\\eta_{\\mathrm{FH}}^{(2)}$ scales linearly with $u$.\nHowever, $\\eta_{\\mathrm{FD}}^{(2)}$ is always less than $\\frac{1}{2}$ no matter how\nlarge $u$ is. Thus, as $u$ increases, the advantage of FH over FD becomes more apparent.\n$\\rightmark{\\square}$\n\\begin{figure}[h!b!t]\n  \\centering\n  \\includegraphics[scale=.6] {coex_33.png}\n  \\caption{Curves of $\\eta_{\\mathrm{FH}}^{(2)}$ and $\\eta_{\\mathrm{FD}}^{(2)}$ in terms of $\\lambda$ in a network with $u=7$ sub-bands.}\n  \\label{figapp1}\n \\end{figure}\n \\begin{figure}[h!b!t]\n  \\centering\n  \\includegraphics[scale=.6] {coex_444.png}\n  \\caption{Curves of $\\eta_{\\mathrm{FH}}^{(2)}$ and $\\eta_{\\mathrm{FD}}^{(2)}$ in terms of $\\lambda$ in a network with $u=20$ sub-bands.}\n  \\label{figapp1}\n \\end{figure}\n\\newpage\n\n$\\bullet$ \\textit{Minimum nonzero multiplexing gain per user}\n\n The minimum nonzero multiplexing gain per user is the smallest nonzero\nmultiplexing gain that a user in the network attains for different\nrealizations in terms of the number of active users. Assuming\n$n_{\\max}<\\infty$, this happens when there are exactly $n_{\\max}$\nactive users in the system. As the FD system is already designed to\nhandle the case where $n_{\\max}$ users are present in the network,\nthe minimum multiplexing gain per user is automatically higher in FD\nas compared to FH.  Setting $n_{\\mathrm{des}}=n_{\\max}$, we have\n$\\eta_{\\mathrm{FD}}^{(3)}=\\frac{{\\mathsf{SMG}}_{\\mathrm{FD}}(u,n_{\\max})}{n_{\\max}}=\\frac{u}{2\nn_{\\max}}$. In the case of FH, we assume that all users select\n$v=\\frac{u}{n_{\\max}}$. Hence,\n$\\eta_{\\mathrm{FH}}^{(3)}=\\frac{{\\mathsf{SMG}}_{\\mathrm{FH}}\\left(\\frac{u}{n_{\\max}},n_{\\max}\\right)}{n_{\\max}}=\\frac{u}{2\nn_{\\max}}\\left(1-\\frac{1}{n_{\\max}}\\right)^{n_{\\max}-1}$. Clearly,\n$\\frac{1}{e}\\leq\n\\frac{\\eta_{\\mathrm{FH}}^{(3)}}{\\eta_{\\mathrm{FD}}^{(3)}}\\leq 1$ as\n$\\left(1-\\frac{1}{n_{\\max}}\\right)^{n_{\\max}-1}$ approaches\n$\\frac{1}{e}$ from above by increasing $n_{\\max}$. Therefore, the\nloss incurred in terms of $\\eta^{(3)}$ for the FH system is always less\nthan $\\frac{1}{e}$.\n\n $\\bullet$ \\textit{Service capability}\n\nService capability demonstrates the fraction of users receiving\nservice among the whole active users in the network. As mentioned earlier, a user is said\nto receive service whenever the achieved multiplexing gain of the\nuser is nonzero. In the FD scenario, if $N>u$, then a fraction of\nusers cannot share the spectrum. However, in case $\\Pr\\{N\\leq\nu\\}=1$, the FD scheme achieves the full service capability. As for\nthe FH scheme, we already know that as far as all users hop over\nproper subsets of the sub-bands, every user achieves a nonzero\nmultiplexing gain. The following examples offer comparisons between\nFD and FH in terms of the service capability.\n\n\\textit{Example 5}- In this example, we consider a setup where\n$n_{\\max}<\\infty$. The central controller in FD simply sets\n$n_{\\mathrm{des}}=n_{\\max}$ and the service capability is always\nequal to $1$. The number of served users $N_{\\mathrm{serv,FH}}$ in\nthe FH scenario can be written as\n\\begin{equation}\n\\label{doogh}\nN_{\\mathrm{serv,FH}}=\\left\\{\\begin{array}{cc}\n   N   &  \\textrm{$N=1$ or ($N>1$ and $v\\neq u$)}  \\\\\n    0  &  \\textrm{oth.}\n\\end{array}\\right..\n\\end{equation}\nTherefore, as far as $v\\neq u$, we have $N_{\\mathrm{serv,FH}}=N$ and\nthe service capability is one. This shows that to achieve the\nmaximum service capability in a system where $n_{\\max}>1$, the\nhopping parameter $v$ must be strictly less than $u$.\n$\\rightmark{\\square}$\n\n \n \n\\textit{Example 6}- In this example, we provide a case where\n$n_{\\max}$ is not finite. Let us assume the distribution of the\nnumber of active users in the network is a Poisson distribution with\nparameter $\\lambda$, i.e., $q_{n}=\\frac{\\lambda^n e^{-\\lambda}}{n!}$\nfor $n\\geq 0$ where $\\lambda > 1$. Let us compute $v^{*}$ for the FH\nscenario. We have,\n\\begin{eqnarray}\n\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}&=&\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda}\\lambda^{n}}{n!}\\left(nv\\left(1-\\frac{v}{u}\\right)^{n-1}\\right)\\notag\\\\\n&=&\\frac{1}{2}v\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda}\\lambda^{n}}{(n-1)!}\\left(1-\\frac{v}{u}\\right)^{n-1}\\notag\\\\\n&=&\\frac{1}{2}\\lambda v\\sum_{n=0}^{\\infty}\\frac{e^{-\\lambda}\\lambda^{n}}{n!}\\left(1-\\frac{v}{u}\\right)^{n}\\\\\n&=& \\frac{1}{2}\\lambda ve^{-\\frac{\\lambda v}{u}}. \\label{bil2}\n\\end{eqnarray}\nThus,\n\\begin{equation}\n\\label{pooskooshi}\nv^{*}=\\arg\\max_{v}\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}=\\frac{u}{\\lambda}.\\end{equation}\nSince $\\lambda\\neq 1$, we get $v^{*}\\neq u$. Thus, choosing\n$v=v^{*}$ maximizes\n$\\mathrm{E}\\left\\{\\frac{N_{\\mathrm{serv,FH}}}{N}\\right\\}$ and\n$\\mathrm{E}\\{\\mathsf{SMG}_{\\mathrm{FH}}(v,N)\\}$ simultaneously,\ni.e., $\\eta_{\\mathrm{FH}}^{(4)}=1$.\n\nIn the FD system, $N_{\\mathrm{serv,FD}}$ is given by\n\\begin{equation}\n\\label{pmnbv}\nN_{\\mathrm{serv,FD}}=\\left\\{\\begin{array}{cc}\n    N  & N\\leq n_{\\mathrm{des}}   \\\\\n     n_{\\mathrm{des}} &   N>n_{\\mathrm{des}}\n\\end{array}\\right..\n\\end{equation}\nThus,\n\\begin{eqnarray}\n\\mathrm{E}\\left\\{\\frac{N_{\\mathrm{serv,FD}}}{N}\\right\\}&=&\\Pr\\{ N \\leq n_{\\mathrm{des}}\\} +n_{\\mathrm{des}} \\sum_{n=n_{\\mathrm{des}}+1}^{\\infty}\\frac{q_{n}}{n}\\notag\\\\\n&=&1-\\Pr\\{N\\geq n_{\\mathrm{des}}+1\\}+n_{\\mathrm{des}} \\sum_{n=n_{\\mathrm{des}}+1}^{\\infty}\\frac{q_{n}}{n}\\notag\\\\\n&=&1-\\sum_{n=n_{\\mathrm{des}}+1}q_{n}\\left(1-\\frac{n_{\\mathrm{des}}}{n}\\right)\n\\end{eqnarray}\nBy this expression, it is clear that to maximize\n$\\mathrm{E}\\left\\{\\frac{N_{\\mathrm{serv,FD}}}{N}\\right\\}$, one must\nselect $n_{\\mathrm{des}}$ as large as possible. This basically\njustifies the assumption we made about selecting\n$n_{\\mathrm{des}}=u$ in the FD scheme in the case where $n_{\\max}$\nis not finite. Thus,\n\\begin{equation}\n\\label{ }\n\\eta_{\\mathrm{FD}}^{(4)}=1-\\sum_{n=u+1}q_{n}\\left(1-\\frac{u}{n}\\right)\\end{equation}\nFor instance, in the case of $u=5$ and $\\lambda=3$, we have\n$\\eta_{\\textrm{FD}}^{(4)}=0.9806$. $\\rightmark{\\square}$\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\\section{Adaptive Frequency Hopping}\n\nThe results of the previous section are obtained based on the\nassumption that the hopping parameter $v$ is fixed and is not\nadaptively changed based on the number of active users. The\nperformance of the FH system can be improved by letting the\ntransmitters adapt their hopping parameter based on the number of\nactive users using (\\ref{yuu}). We refer to this scenario as\nAdaptive Frequency Hopping (AFH). In the following example, we study\nthe performance improvement offered by AFH over FH in terms of\n$\\eta^{(1)}$ and $\\eta^{(2)}$.\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\\textit{Example 7}- Let us assume that the number of active users is\na Poisson random variable with parameter  $\\lambda>1$.  We already\nhave\n\\begin{eqnarray} \\label{bil}\n\\eta_{\\mathrm{FH}}^{(1)}=\\frac{u}{2e},\n\\end{eqnarray}\nwhile by (\\ref{bxxxx}),\n\\begin{eqnarray}\n\\label{fag}\n\\eta_{\\mathrm{AFH}}^{(1)}  =  \\frac{u}{2}\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda} \\lambda^{n}}{n!} \\Big(1-\\frac{1}{n}\\Big)^{n-1}.\n\\end{eqnarray}\nFigure \\ref{fig6} shows the plots of $\\eta_{\\mathrm{FH}}^{(1)}$ and\n$\\eta_{\\mathrm{AFH}}^{(1)}$ versus $\\lambda$ for $u=10$. It is\nobserved that $\\eta_{\\mathrm{FH}}^{(1)}$ does not change with\n$\\lambda$, while $\\eta_{\\mathrm{AFH}}^{(1)}$ decreases by increasing\n$\\lambda$. This indicates that in a crowded network (large\n$\\lambda$), AFH does not provide any significant advantage over FH\nin terms of $\\eta^{(1)}$.\n  \\begin{figure}[h!b!t]\n  \\centering\n  \\includegraphics[scale=.6] {castaway.png}\n  \\caption{$\\eta_{\\mathrm{AFH}}^{(1)}$ versus $\\eta_{\\mathrm{FH}}^{(1)}$ for $u=10$. }\n  \\label{fig6}\n \\end{figure}\n \nWe have already calculated $\\eta_{\\mathrm{FH}}^{(2)}$ in example 4\nin a system where $3\\leq\\lambda\\leq10$. However, in case of AFH,\n \\begin{eqnarray}\n\\label{fag}\n\\eta_{\\mathrm{AFH}}^{(2)}= \\frac{u}{2}\\sum_{n=1}^{\\infty}\\frac{e^{-\\lambda} \\lambda^{n}}{n!}\\frac{1}{n}\\Big(1-\\frac{1}{n}\\Big)^{n-1}.\n\\end{eqnarray}\nFigure \\ref{fig7} presents the plots of $\\eta_{\\mathrm{FH}}^{(2)}$\nand $\\eta_{\\mathrm{AFH}}^{(2)}$ versus $\\lambda$ for $u=10$. Both\n$\\eta_{\\mathrm{FH}}^{(2)}$ and $\\eta_{\\mathrm{AFH}}^{(2)}$ decrease\nby increasing $\\lambda$. However, the ratio\n$\\frac{\\eta_{\\mathrm{AFH}}^{(2)}}{\\eta_{\\mathrm{FH}}^{(2)}}$\ndecreases as $\\lambda$ increases. This indicates that for large\nvalues of $\\lambda$, AFH does also not provide any significant advantage\nover FH in terms of $\\eta^{(2)}$. $\\rightmark{\\square}$\n\\begin{figure}[h!b!t]\n  \\centering\n  \\includegraphics[scale=.6] {castaway2.png}\n  \\caption{$\\eta_{\\mathrm{AFH}}^{(2)}$ versus $\\eta_{\\mathrm{FH}}^{(2)}$ for $u=10$.  }\n  \\label{fig7}\n \\end{figure}\n\n\n\n\n\n\n\n\n\n \n \n\n  \n\n\n\n\n\n\n\n \n\n\n  \n  \n \n \n  \n \n \n   \n  \n  \n  \n  \n \n  \n  \n  \n \n \n\n  \n   \n \n  \n  \n\n  \n  \n\n  \n  \n   \n\n\n \n\n\n  \\section{Conclusion}\nWe have addressed a decentralized  wireless communication network\nwith a fixed number $u$ of frequency sub-bands to be shared among\n$N$ transmitter-receiver pairs. It is assumed that the number of\nactive users is a random variable with a given distribution.\nMoreover, users are assumed to be unaware of each other's codebooks and hence, no\nmultiuser detection is possible.  We proposed a randomized Frequency\nHopping (FH) scheme in which each transmitter randomly hops over\nsubsets of the $u$ sub-bands from transmission to transmission.\nAssuming all users transmit Gaussian signals, the distribution of\nnoise plus interference is mixed Gaussian, which makes\nthe calculation of the mutual information between the input and output\nof each user intractable. We derived lower and upper bounds on this\nmutual information and demonstrated that for large SNR values, the\ntwo bounds coincide. This observation enabled us to compute the\nsum multiplexing gain of the system and obtain the optimum\nhopping strategy for maximizing this value. We compared the\nperformance of the  FH with that of the FD in terms of the following\nperformance measures: average sum multiplexing gain\n$(\\eta^{(1)})$, average minimum multiplexing gain per user $(\\eta^{(2)})$,\nminimum nonzero multiplexing gain per user $(\\eta^{(3)})$ and\nservice capability ($\\eta^{(4)}$). We showed that (depending on the\nprobability mass function of the number of active users) the FH\nsystem can offer a significant improvement in terms of $\\eta^{(1)}$\nand $\\eta^{(2)}$ (implying a more efficient usage of the spectrum).\nIt was also shown that\n$\\frac{1}{e}\\leq\\frac{\\eta_{\\mathrm{FH}}^{(3)}}{\\eta_{\\mathrm{FD}}^{(3)}}\\leq\n1$, i.e., the loss incurred in terms of $\\eta^{(3)}$ is not more\nthan $\\frac{1}{e}$. Moreover, computation of the so-called service\ncapability showed that in the FH system any number of users can\ncoexist fairly, while the maximum number of users in the FD system\nis limited by the number of  sub-bands.\n\n\n \\section*{Appendix A; Proof of Lemmas 2}\nLet us consider a general $t\\times 1$ vector mixed Gaussian\ndistribution $p_{\\vec{\\Theta}}(\\vec{\\theta})$ with different\ncovariance matrices $\\{C_{l}\\}_{l=1}^{L}$ and associated\nprobabilities $\\{p_{l}\\}_{l=1}^{L}$ given by\n\\begin{equation}\n\\label{gb}\np_{\\vec{\\Theta}}(\\vec{\\theta})=\\sum_{l=1}^{L}p_{l}g_{t}(\\vec{\\theta},C_{l}),\\end{equation}\nwhere $g_{t}(\\vec{\\theta},C_{l})=\\frac{1}{(2\\pi)^{\\frac{t}{2}}(\\det C_{l})^{\\frac{1}{2}}}\\exp \\left \\{-\\frac{1}{2}\\vec{\\theta}^{T}C_{l}^{-1}\\vec{\\theta} \\right\\}$.\n Hence,\n\\begin{equation} \\label{pteta}\n\\int p_{\\vec{\\Theta}}(\\vec{\\theta})\\log p_{\\vec{\\Theta}}(\\vec{\\theta})d\\vec{\\theta}=\\sum_{l=1}^{L}J_{l}\n\\end{equation}\nwhere $J_{l}\\triangleq p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log p_{\\vec{\\Theta}}(\\vec{\\theta})d\\vec{\\theta}$ for $1\\leq l\\leq L$.\nTo find a lower bound on $J_{l}$, we observe that\n\\begin{eqnarray}\n\\label{gbb}\nJ_{l} &=& p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log\\bigg(\\sum_{m=1}^{L}p_{m}g_{t}(\\vec{\\theta},C_{m})\\bigg)d\\vec{\\theta} \\notag\\\\\n&\\geq& p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log\\big(p_{l}g_{t}(\\vec{\\theta},C_{l})\\big)d\\vec{\\theta} \\notag\\\\\n&=& \\left(p_{l}\\log p_{l}\\right)\\int g_{t}(\\vec{\\theta},C_{l})d\\vec{\\theta}+p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log g_{t}(\\vec{\\theta},C_{l})d\\vec{\\theta}\\notag\\\\\n&=& p_{l}\\log p_{l}+p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log g_{t}(\\vec{\\theta},C_{l})d\\vec{\\theta}\\end{eqnarray}\nUsing this together with (\\ref{pteta}) yields\n\\begin{eqnarray}\n\\mathrm{h}(\\vec{\\Theta}) &=& -\\int p_{\\vec{\\Theta}}(\\vec{\\theta})\\log p_{\\vec{\\Theta}}(\\vec{\\theta})d\\vec{\\theta}\\notag\\\\&=&-\\sum_{l=1}^{L}J_{l} \\notag\\\\\n&\\leq&  -p_{l}\\log p_{l}-p_{l}\\int g_{t}(\\vec{\\theta},C_{l})\\log g_{t}(\\vec{\\theta},C_{l})d\\vec{\\theta}\\notag\\\\\n&\\stackrel{(a)}{=}&-\\sum_{l=1}^{L}p_{l}\\log\np_{l}+\\frac{1}{2}\\sum_{l=1}^{L}p_{l}\\log\\left((2\\pi e)^{t}\\det\nC_{l}\\right)\\end{eqnarray}\nwhere in $(a)$, we have used the fact that\nthe differential entropy of a $t\\times 1$ Gaussian vector with\ncovariance matrix $C_{l}$ is $\\frac{1}{2}\\log\\left((2\\pi e)^{t}\\det\nC_{l}\\right)$.\n\nLet $t=1$ and $\\vec{\\Theta}=Z_{i,j}$. Therefore,\n\\begin{eqnarray}\n \\mathrm{h}(Z_{i,j})&\\leq& \\frac{1}{2}\\sum_{l=0}^{L_{i}}a_{i,l}\\log(2\\pi e\\sigma_{i,l}^{2})-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l} \\notag\\\\\n&=& \\frac{1}{2}\\sum_{l=0}^{L_{i}}a_{i,l}\\log(2\\pi e\\sigma_{i,0}^{2})+\\frac{1}{2}\\sum_{l=0}^{L_{i}}a_{i,l}\\log\\frac{\\sigma_{i,l}^{2}}{\\sigma_{i,0}^{2}}-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l}\\notag\\\\\n&=&\\log(\\sqrt{2\\pi e}\\sigma_{i,0})+\\frac{1}{2}\\sum_{l=1}^{L_{i}}a_{i,l}\\log\\frac{\\sigma_{i,l}^{2}}{\\sigma_{i,0}^{2}}-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l}\\end{eqnarray}\nHowever, for all $l\\geq 1$, we have $\\frac{\\sigma_{i,l}^{2}}{\\sigma_{i,0}^{2}}\\leq \\frac{\\sigma_{i,L_{i}}^{2}}{\\sigma_{i,0}^{2}}=1+c_{i,L_{i}}\\gamma$. Thus,\n\\begin{eqnarray}\n \\mathrm{h}(Z_{i,j})&\\leq& \\frac{1}{2}\\sum_{l=1}^{L_{i}}a_{i,l}\\log(1+c_{i,L_{i}}\\gamma)+\\log(\\sqrt{2\\pi e}\\sigma_{i,0})-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l} \\notag\\\\\n&=& \\frac{1}{2}(1-a_{i,0})\\log(1+c_{i,L_{i}}\\gamma)+\\log(\\sqrt{2\\pi e}\\sigma_{i,0})-\\sum_{l=0}^{L_{i}}a_{i,l}\\log a_{i,l}.\n\\end{eqnarray}\nThis concludes the proof of Lemma 2.\n\n\n\n\n\n\n \\section*{Appendix B; Proof of Proposition 1}\nWe have\n$\\eta_{\\mathrm{FD}}^{(1)}=\\frac{\\mathrm{E}\\{N\\}u}{2n_{\\max}}$ and\n$\\eta_{\\mathrm{FH}}^{(1)}=\\frac{1}{2}\\max_{v}\\left\\{v\\mathrm{E}\\left\\{N\\left(1-\\frac{v}{u}\\right)^{N-1}\\right\\}\\right\\}$.\nLet us define $\\Omega(v,N)\\triangleq N\\omega_{v}^{N-1}$ where\n$\\omega_{v}=1-\\frac{v}{u}$. Thinking of $N$ as a real parameter for\nthe moment, we have $\\frac{\\partial^{2}}{\\partial\nN^{2}}\\Omega(v,N)=\\omega_{v}^{N-1}\\left(N\\left(\\ln\n\\omega_{v}\\right)^{2}+2\\ln\\omega_{v}\\right)$. As $N\\geq 1$, we have\n$\\frac{\\partial^{2}}{\\partial\nN^{2}}\\Omega(v,N)\\geq\\omega_{v}^{N-1}\\left(\\left(\\ln\n\\omega_{v}\\right)^{2}+2\\ln\\omega_{v}\\right)$. But, $\\left(\\ln\n\\omega_{v}\\right)^{2}+2\\ln\\omega_{v}\\geq 0$ if and only if\n$\\omega_{v}\\leq \\frac{1}{e^{2}}$ or $\\omega_{v}\\geq 1$. Since\n$\\omega_{v}\\leq 1$, we get $\\omega_{v}\\leq \\frac{1}{e^{2}}$. This\nimplies that the function $\\Omega(v,N)$ is a convex function of $N$\nas far as $\\omega_{v}\\leq \\frac{1}{e^{2}}$. Therefore, by Jensen's\ninequality,\n\\begin{eqnarray}\n\\mathrm{E}\\left\\{N\\left(1-\\frac{v}{u}\\right)^{N-1}\\right\\}=\\mathrm{E}\\{\\Omega(v,N)\\}\\geq \\Omega\\left(v,\\mathrm{E}\\{N\\}\\right)=\\mathrm{E}\\{N\\}\\left(1-\\frac{v}{u}\\right)^{\\mathrm{E}\\{N\\}-1}\n\\end{eqnarray}\nwhich is valid as far as $v\\geq \\left(1-\\frac{1}{e^{2}}\\right)u$. Hence,\n\\begin{eqnarray}\n\\label{veg1}\n\\eta_{\\mathrm{FH}}^{(1)}&=&\\frac{1}{2}\\max_{v}\\left\\{v\\mathrm{E}\\left\\{N\\left(1-\\frac{v}{u}\\right)^{N-1}\\right\\}\\right\\}\\notag\\\\\n&\\geq&\\frac{1}{2}\\max_{v\\in\\left[\\left(1-\\frac{1}{e^{2}}\\right)u,u\\right]}\\left\\{v\\mathrm{E}\\left\\{N\\left(1-\\frac{v}{u}\\right)^{N-1}\\right\\}\\right\\}\\notag\\\\\n&\\geq&\\frac{1}{2}\\mathrm{E}\\{N\\}\n\\max_{v\\in\\left[\\left(1-\\frac{1}{e^{2}}\\right)u,u\\right]}\\left\\{v\\left(1-\\frac{v}{u}\\right)^{\\mathrm{E}\\{N\\}-1}\\right\\}.\n\\end{eqnarray}\nThe function $v\\left(1-\\frac{v}{u}\\right)^{\\mathrm{E}\\{N\\}-1}$ is a\nconcave function in terms of $v$ that achieves its absolute maximum\nat $\\frac{u}{\\mathrm{E}\\{N\\}}$. Therefore,\n\\begin{equation}\n\\label{veg2}\n\\max_{v\\in\\left[\\left(1-\\frac{1}{e^{2}}\\right)u,u\\right]}\\left\\{v\\left(1-\\frac{v}{u}\\right)^{\\mathrm{E}\\{N\\}-1}\\right\\}=\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\left(1-\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\right)^{\\mathrm{E}\\{N\\}-1}u.\n\\end{equation}\nUsing (\\ref{veg1}) and (\\ref{veg2}),\n\\begin{equation}\n\\label{veg3}\n\\eta_{\\mathrm{FH}}^{(1)}\\geq \\frac{1}{2}\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\left(1-\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\right)^{\\mathrm{E}\\{N\\}-1}\\mathrm{E}\\{N\\}u.\\end{equation}\nHence, a sufficient condition for $\\eta_{\\mathrm{FH}}^{(1)}>\\eta_{\\mathrm{FD}}^{(1)}$ to hold is that\n\\begin{equation}\n\\label{veggg}\n\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\left(1-\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}\\right)^{\\mathrm{E}\\{N\\}-1}>\\frac{1}{n_{\\max}}.\\end{equation}\nIf $\\mathrm{E}\\{N\\}\\geq \\frac{e^{2}}{e^{2}-1}$, we have\n$\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}=1-\\frac{1}{e^{2}}$.\nHence, (\\ref{veggg}) reduces to the inequality\n$\\mathrm{E}\\{N\\}<\\frac{1}{2}\\ln\\left((e^{2}-1)n_{\\max}\\right)$.\nTherefore, if $\\frac{e^{2}}{e^{2}-1}\\leq\\mathrm{E}\\{N\\}<\n\\frac{1}{2}\\ln\\left((e^{2}-1)n_{\\max}\\right)$, then (\\ref{veggg}) is\nsatisfied. On the other hand, if $\\mathrm{E}\\{N\\}\\leq\n\\frac{e^{2}}{e^{2}-1}=1.1565$, we get\n$\\max\\left\\{1-\\frac{1}{e^{2}},\\frac{1}{\\mathrm{E}\\{N\\}}\\right\\}=\\frac{1}{\\mathrm{E}\\{N\\}}$.\nThus, (\\ref{veggg}) reduces to the inequality\n$\\frac{1}{\\mathrm{E}\\{N\\}}\\left(1-\\frac{1}{\\mathrm{E}\\{N\\}}\\right)^{\\mathrm{E}\\{N\\}-1}>\\frac{1}{n_{\\max}}$.\nFor each $n_{\\max}\\geq 2$, this yields an upper bound on\n$\\mathrm{E}\\{N\\}$. Since\n$\\frac{1}{\\mathrm{E}\\{N\\}}\\left(1-\\frac{1}{\\mathrm{E}\\{N\\}}\\right)^{\\mathrm{E}\\{N\\}-1}$\nis a decreasing function of $\\mathrm{E}\\{N\\}$, the smallest of these\nupper bounds is obtained for $n_{\\max}=2$ and is equal to $1.2938$.\nThis means that for $\\mathrm{E}\\{N\\}\\leq1.1565$, (\\ref{veggg}) is automatically satisfied. Thus, (\\ref{veggg}) is equivalent to\n\\begin{equation}\n\\mathrm{E}\\{N\\}< \\frac{1}{2}\\ln\\left((e^{2}-1)n_{\\max}\\right). \\label{dooly}\\end{equation}\n\\section*{Appendix C; Proof of Proposition 2}\nWe have $\\eta_{\\mathrm{FD}}^{(2)}=\\frac{u}{2n_{\\max}}$ and\n$\\eta_{\\mathrm{FH}}^{(2)}=\\frac{1}{2}\\max_{v}\\left\\{v\\mathrm{E} \\left \\lbrace \\left(1-\\frac{v}{u}\\right)^{N-1}\\right \\rbrace \\right\\}$.\nThe function $\\left(1-\\frac{v}{u}\\right)^{N-1}$ is  convex  in terms\nof $N$. Using Jenson's inequality,\n\\begin{equation}\\label{oiuyt}\\eta_{\\mathrm{FH}}^{(2)}\\geq\n\\frac{1}{2}\\max_{v}\\left\\{v\\left(1-\\frac{v}{u}\\right)^{\\mathrm{E}\\{N\\}-1}\\right\\}=\\frac{u}{2\\mathrm{E}\\{N\\}}\\left(1-\\frac{1}{\\mathrm{E}\\{N\\}}\\right)^{\\mathrm{E}\\{N\\}-1}.\\end{equation}\nHence, a sufficient condition for\n$\\eta_{\\mathrm{FH}}^{(2)}>\\eta_{\\mathrm{FD}}^{(2)}$ to hold is\n\\begin{equation}\n\\frac{1}{\\mathrm{E}\\{N\\}}\\left(1-\\frac{1}{\\mathrm{E}\\{N\\}}\\right)^{\\mathrm{E}\\{N\\}-1}>\\frac{1}{n_{\\max}}.\\end{equation}\n\n\n\\section{Acknowledgments}\\\nThe first author is indebted to M. A. Maddah-Ali and S. Oveis Gharan for their invaluable suggestions. \n\n    \\bibliographystyle{IEEEbib}\n      ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\n\n\\blfootnote{\n    %\n   \n    %\n    \\hspace{-0.65cm} \n    This work is licensed under a Creative Commons Attribution 4.0 International License. License details: \\url{http:\/\/creativecommons.org\/licenses\/by\/4.0\/}.\n    %\n   \n    %\n   \n   \n   \n   \n   \n   \n   \n    %\n   \n   \n   \n   \n   \n}\n\nPropaganda techniques need to attach importance to arouse the emotions\nof the receivers, sometimes even by temporarily bypassing the intellectual\ndefenses of the receivers \\newcite{Pearlin1952} . Propaganda uses\npsychological and rhetorical techniques to achieve its purpose. Such\ntechniques include using logical fallacies and appealing to the\nemotions of the audience. Logical fallacies are usually hard to spot\nsince the argumentation, at first, might appear correct and\nobjective \\cite{DaSanMartino2019} . However, careful analysis\nshows that the conclusion cannot be drawn from the premise without\nmisusing logical rules. Another set of techniques uses emotional\nlanguage to induce the audience to agree with the speaker only\nbased on the emotional bond that is being created, provoking the\nsuspension of any rational analysis of the argumentation.\n\nThe traditional NLP task generally classifies and detects propaganda\ntechniques at the article level, which often fails to meet more\ndetailed requirements. This fact has also been confirmed by previous\niterations of the SemEval competition, where leading solutions used\nconvolutional neural networks (CNN), long short-term\nmemory (LSTM)  \\cite{Baziotis2018}  and\ntransfer learning techniques \\cite{Duppada2018} .\nThe main features of an article are extracted by using the feature\ncapture and pooling of the CNN model, but these methods can only be\nused at the article level and are coarse-grained detection\nmethods. However, limited research has focused on text\nclassification \\cite{Lewis1995,Song2010Hierarchical}.\n News articles have also been classified using the Bi-LSTM-CNN\nmodel \\cite{Li2018} .   However, there are often many propaganda\ntechniques in one article, and most of these techniques are efficient\nfor propaganda classification but lack the ability to detect categories\nof propaganda techniques. Thus, they cannot achieve good results and are\nless efficient in practice.\n\nNow the difficulty is to detect propaganda techniques at the fine-grained\nlevel. The SemEval-2020 Task 11, ``Detection of Propaganda Techniques in\nNews Articles'', is designed to promote research on this task. We used\nthe word embedded representation of the pretrained model and LSTM model\nto detect the news article propaganda techniques at a fine-grained\nlevel, and we also evaluate the performance among different neural\nnetwork models on this task.\n\nThe task consists of two subtasks.\n\\begin{enumerate}\n\t\\item Span Identification (SI): Given a plain-text document, identify those specific fragments that contain at least one propaganda technique \\cite{DaSanMartinoSemeval20task11} .  This is a binary sequence tagging task. We need to detect which fragments of the news article belong to the propaganda technique and mark the fragments with begin\\_offset and end\\_offset. The span ranges from begin\\_offset to end\\_offset-1.\n\t\\item Technique Classification (TC): The purpose of this subtask is to identify the category of the propaganda technique. Given a text fragment identified as propaganda and its document context, identify the applied propaganda technique in the fragment. Since there are overlapping spans, formally, this is a multilabel multiclass classification problem. However, whenever a span is associated with multiple techniques, the input file will have multiple copies of such fragments; therefore, the problem can be treated as a multiclass classification problem. The techniques include Appeal\\_to\\_Authority, Appeal\\_to\\_fear-prejudice, Black-and-White\\_Fallacy, and so on.\n\\end{enumerate}\n\nThe rest of the paper is organized as follows. Section 2 describes the details of the LSTM used in our system. Section 3 presents the experimental results. Conclusions and future works are described in Section 4.\n\n\n\n\\section{System Description}\n\nWe implemented LSTM model to accomplish this task. Meanwhile, the representations of input words are trained by using GloVe model \\cite{pennington-etal-2014-glove}.\n\n\n\\subsection{Span Identification (SI)}\n\nFor the SI subtask requirements, we need to detect which fragments\nof the news article utilized a propaganda technique. The SemEval\norganizers provided us with 371 training sets. The data were plain\ntext files, and the SI task was to identify specific pieces that\ncontained at least one propaganda technique. To detect news article\npropaganda techniques, we tested some deep learning model and\nintegration architectures \\cite{Chen2016} . For the SI subtask, we\nalso experimented with GloVe-BiLSTM  \\cite{Li2017,Luo2018,cross-huang-2016-incremental} ,\nBERT-LSTM, GloVe-LSTM and BERT-BiLSTM \\cite{Agirre2016,MacAvaney2018} .\nAs illustrated in Figure 1, our model includes an embedding layer, an LSTM\nlayer, a fully connected layer and an output layer. First, the embedding\nlayer represents every word using pretrained word embeddings. The LSTM\nlayer is implemented to obtain contextual information. The hidden vector\nproceeded by each LSTM cell will be further fed into a dense layer with\nthe   \\emph{sigmoid} activation function. Then, we can discriminate whether a word\nis propaganda or not. Finally, we record the index of propaganda words and\nrecognize the propaganda fragments.\n\n\n\n\n\n  Based on our experimental results, we can be concluded that the\n  LSTM model with GloVe word embeddings obtained the best performance\n  on this task. For the embeddings layer, we implemented GloVe to\n  train the word embeddings \\cite{Papagiannopoulou2018}.  The\n  input tokens were obtained using the NLTK toolkit on the given\n  articles. After the word embedding representation trained by GloVe\n  is obtained, an LSTM layer is connected. In LSTM, recurrent cells\n  are connected in a special way to avoid vanishing and exploding\n  gradients, and the number of hidden nodes in the LSTM layer is set\n  to 150. We find that the Bi-LSTM model \\cite{Ma2016} does not\n  perform well on this task. Next, the features captured by the LSTM\n  layer are flattened and passed to the hidden dense layer, and the\n  parameters of the dense layer are set to 8, which analyzes the\n  interactions among the obtained vectors. The dropout rate of the\n  dense layer is set to 0.2 to prevent model overfitting.\n\n\n\n\n\\subsection{Technique Classification (TC)}\n\nThe TC task is a multiclass classification task representing an\nextension of the SI task, and the TC subtask seeks to classify\nthe various propaganda techniques identified by the SI\nsubtask. Such techniques include the use of logical fallacies\nand appealing to the emotions of the audience. Logical fallacies\nare usually hard to spot since the argumentation, at first, might\nseem correct and objective. For the TC subtask, we obtain the\nsentence representations via two ways. The first is feeding\npretrained word embeddings into the LSTM model and then we treat\nthe last hidden vector as the sentence representation. The second\nway is using a pretrained language model, such as BERT, to directly\nproduce sentence representations. Meanwhile, we also compare the\neffects of  \\emph{softmax} and XGBoost  \\cite{Mitchell2017} on\nthe classification task. Through comparing the experimental results, we\ncan conclude that the BERT-LSTM model can obtain good performance on the\nTC subtask. The detailed analysis of the experimental results of the TC\nsubtask will be introduced in Section 3 of this paper.\n\nWe will introduce the selected model and parameter setting for the\nexperiment. The BERT pretrained language model has been proved\nefficient in many NLP tasks, and the pretraining model used in\nour experiment is BERT-Base. Therefore, we implement BERT to train\nthe word representations for the TC subtask. As illustrated in\nFigure 2, we implement BERT to train the word representations obtained\nthrough the bert-as-service library. The 768-dimensional word embeddings\ntrained by BERT are fed to the LSTM layer, and the number of hidden\nnodes in the LSTM layer is set to 50. Same as the SI task, the features\ncaptured by the LSTM layer are flattened and passed to the hidden dense\nlayer, and the number of parameters for the dense layer is set to 32. The\ndropout rate of the dense layer is set to 0.2 \\cite{Srivastava2014}.\n\n\\begin{figure}[t!]\n  \\centering\n  \\begin{minipage}[t]{0.5\\textwidth}\n  \\centering\n  \\includegraphics[width=7.5cm,height=6.5cm]{figure1.pdf}\n  \\caption{SI subtask model architecture.}\n  \\end{minipage}\n  \\begin{minipage}[t]{0.49\\textwidth}\n  \\centering\n  \\includegraphics[width=7.5cm,height=8cm]{figure2.pdf}\n  \\caption{TC subtask model architecture.}\n  \\end{minipage}\n  \\end{figure}\n\n\n\n\\section{Experimental Results}\n\\label{sec:Results}\n\nIn Section 3, we first introduce the dataset of this task, and\nthen we analyze the performance of different neural networks and\nintegrated learning models for this task.\n\n\\subsection{Dataset}\n\nThe dataset contains 371 news articles for the training\nset, 75 news articles for the development set, and 90 news\narticles for the test set. The articles may contain several\npropaganda spans. The beginning position and the ending position\nwere marked by ``begin\\_offset'' and ``end\\_offset'', respectively. As\nillustrated in Table 1, for the ``111111111'' article, it contains 3\npropaganda spans with a span range from ``begin\\_offset'' to ``end\\_offset'' minus\none because the index of words in the article started from zero.\n\n\\begin{table}[t!]\n  \\begin{center}\n  \\begin{tabular}{|c|c|c|p{4cm}<{\\centering}|}\n  \\hline\n  \\small ID&\\small Begin\\_offset&\\small End\\_offset&\\small Text \\\\\n  \\hline\n  \\small 111111111&\\small 265&\\small 323&\\small The next transmission could be more pronounced or stronger \\\\\n  \\hline\n  \\small 111111111&\\small 1069&\\small 1091&\\small a very, very different \\\\\n  \\hline\n  \\small111111111&\\small1577&\\small1616&\\small but warned that the danger was not over \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ gold label SI file: article111111111.task1-SI. }\n\\end{table}\n\nThe TC subtask requires us to recognize the techniques of a\ncertain propaganda span. The propaganda technique and the\ncorresponding text contents are shown in Table 2. The number\nof propaganda techniques is 18; therefore, the TC subtask is\na multiclass classification task.\n\n\\begin{table}[t!]\n  \\begin{center}\n  \\begin{tabular}{|c|c|c|c|p{4cm}<{\\centering}|}\n  \\hline\n  \\small ID&\\small Technique&\\small Begin\\_offset&\\small End\\_offset&\\small Text \\\\\n  \\hline\n  \\small 111111111&\\small Appeal\\_to\\_Authority&\\small265&\\small323&\\small The next transmission could be more pronounced or stronger \\\\\n  \\hline\n  \\small111111111&\\small Repetition&\\small1069&\\small1091&\\small a very, very different  \\\\\n  \\hline\n  \\small111111111&\\small Appeal\\_to\\_fear-prejudice&\\small1577&\\small1616&\\small but warned that the danger was not over \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ gold label TC file: article111111111.task2-TC. }\n\\end{table}\n\n\n\\subsection{Evaluation Metrics}\n\nFor both subtasks, the participating systems were evaluated using\nstandard evaluation metrics, including the $accuracy$, $precision$,\n$recall$ and $F_{1}$-$score$, which are calculated as follows:\n\\begin{equation}\n  accuracy{\\rm{ }} = \\frac{{true{\\rm{ }} \\ positives + true{\\rm{ }} \\  negatives }}{{total{\\rm{ }}\\  \\ number \\ of \\ instances}}\n\\end{equation}\n\n\\begin{equation}\n  precision{\\rm{ }} = {\\rm{ }}\\frac{{true{\\rm{ }}\\ positives}}{{true{\\rm{ }}\\ positives + {\\rm{ }}false{\\rm{ }}\\ positives}}\n\\end{equation}\n\n\\begin{equation}\n  recall{\\rm{ }} = \\frac{{true{\\rm{ }}\\ positives}}{{true{\\rm{ }}\\ positives + {\\rm{ }}false{\\rm{ }}\\ negatives}}\n\\end{equation}\n\n\\begin{equation}\n  {F_1}{\\rm{ }} = 2 \\cdot \\frac{{precision \\cdot recall}}{{precision + recall}}\n\\end{equation}\n\nThe organizers provided baseline models for each subtask. For the SI subtask, the macro-$F_{1}$-$score$ for the baseline model were 0.011 on the development set and 0.003 on the test set. For the TC subtask, the micro-$F_{1}$-$score$ for the baseline model was 0.265 on the development set and 0.252 on the test set.\n\n\n\\subsection{SI Subtask Results}\n\nAfter fine-tuning the different parameters of the model, we finally\ndecided to use the adaptive moment estimation (Adam)\noptimizer \\cite{Kingma2015} with the learning rate set to 0.01. The\nscores of the fine-tuning process of different learning rates on\nthe LSTM model with GloVe word embeddings are shown in Figure 3.\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[\n    height=6.72cm,width=8.94cm,\n    keepaspectratio]\n  {figure33.pdf}\n  \\caption{The fine-tuning process of different learning rates for the development set on the LSTM model with GloVe word embeddings.}\n  \\end{figure}\n\nOn the development set, the LSTM model with GloVe word embeddings\nobtained a macro-$F_{1}$-$score$ of 0.423. In the test set, this\nmethod achieved an $F_{1}$-$score$ of 0.406. The comparative results\nare presented in Table 3.\n\n\\begin{table}[t!]\n  \\begin{center}\n  \\begin{tabular}{|c|c|c|p{1.5cm}<{\\centering}|}\n  \\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\n  \\small \\bf System&\\small  \\bf  $F_{1}$-$score$&\\small  \\bf $Precision$ &\\small  \\bf $Recall$ \\\\\n  \\hline\n  \\small GloVe+LSTM&\\small  \\bf 0.423&\\small  \\bf 0.321&\\small  \\bf 0.620\t \\\\\n  \\hline\n  \\small GloVe+BiLSTM&\\small 0.404&\\small 0.360&\\small 0.460 \\\\\n  \\hline\n  \\small BERT+LSTM&\\small 0.397&\\small 0.287&\\small 0.643 \\\\\n  \\hline\n  \\small BERT+BiLSTM&\\small 0.360&\\small 0.256&\\small 0.608 \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{  Scores of different models for the SI subtask on the development set. }\n\\end{table}\n\nOur system ranked 17th out of 36 teams. The selected LSTM model with GloVe word embeddings significantly exceeded the system baseline in terms of performance, which proved that the model performed well on this task and could detect the span of propaganda techniques in news articles.\n\n\\subsection{TC Subtask Results}\n\nTC is a multiclass classification task. As\nillustrated in Table 4, the distribution of the\ngolden labels is rather imbalanced. Therefore, the\nofficial evaluation measure for the task is the micro-$F_{1}$-$score$ .\n\n\n\\begin{table}[t!]\n  \\begin{center}\n  \\begin{tabular}{|c|p{5cm}<{\\centering}|}\n  \\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n  \\small \\bf Propaganda Technique&\\small  \\bf Number Of TC Training Sets\\\\\n  \\hline\n  \\small APPEAL\\_TO\\_AUTHORITY&\\small 155\t \\\\\n  \\hline\n  \\small APPEAL\\_TO\\_FEAR-PREJUDICE&\\small 321 \\\\\n  \\hline\n  \\small BANDWAGON,REDUCTIO\\_AD\\_HITLERUM&\\small \t77 \\\\\n  \\hline\n  \\small BLACK-AND-WHITE\\_FALLACY&\\small \t112 \\\\\n  \\hline\n  \\small CAUSAL\\_OVERSIMPLIFICATION&\\small \t212\\\\\n  \\hline\n  \\small DOUBT&\\small \t517\\\\\n  \\hline\n  \\small EXAGGERATION,MINIMISATION&\\small \t493\\\\\n  \\hline\n  \\small FLAG-WAVING\t&\\small 250\\\\\n  \\hline\n  \\small LOADED\\_LANGUAGE&\\small \t2200\\\\\n  \\hline\n  \\small NAME\\_CALLING,LABELING&\\small \t1105\\\\\n  \\hline\n  \\small REPETITION&\\small \t621\\\\\n  \\hline\n  \\small SLOGANS&\\small \t138\\\\\n  \\hline\n  \\small THOUGHT-TERMINATING\\_CLICHES&\\small \t80\\\\\n  \\hline\n  \\small WHATABOUTISM,STRAW\\_MEN,RED\\_HERRING&\\small \t109\\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{The imbalanced data of the propaganda technique labels. }\n\\end{table}\n\nThe number of propaganda techniques is 18, but Table 4 only lists 14\ntechniques because some propaganda techniques are combined due to\ninsufficient data for some propaganda techniques in the\ncorpus \\cite{DaSanMartinoSemeval20task11} . Since there are\noverlapping spans, formally, it is a multilabel and multiclass\nclassification problem. However, whenever a span is associated\nwith multiple techniques, the input file will have multiple copies\nof these fragments; therefore, the problem can be algorithmically\ntreated as a multiclass classification problem. We tried to use GloVe\nand BERT to generate sentence embeddings, but the experimental results\nshowed that the sentence embedding produced by BERT pretraining was\nbetter. After training on the datasets given by the TC\ntask  \\cite{Drissi2019,Deriu2016} , through the model mentioned\nabove, the micro-$F_{1}$-$score$ on the development set was 0.561 and that on\nthe test set was 0.505. Our system ranked 22th out of 31 teams.\n\nIn addition to the LSTM model, we also tested some machine learning\narchitectures and some integrated learning methods. Because the\nperformance of the BERT-LSTM model on this task is better than those\nof other models, we adopted the BERT-LSTM model as our final model. The\nexperimental results of the different models are shown in Table 5.\n\n\n\\begin{table}[t!]\n  \\begin{center}\n  \\begin{tabular}{|c|p{2cm}<{\\centering}|}\n  \\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\n  \\small \\bf System&\\small  \\bf $F_{1}$-$score$ \\\\\n  \\hline\n  \\small BERT+LSTM&\\small  \\bf 0.561\t \\\\\n  \\hline\n  \\small BERT+BiLSTM&\\small 0.520 \\\\\n  \\hline\n  \\small BERT&\\small 0.438 \\\\\n  \\hline\n  \\small BERT+XGBoost&\\small 0.476 \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Scores of different models for the TC subtask on the development set. }\n\\end{table}\n\n\n\\section{Conclusion}\n\nIn this paper, we presented our system for the SemEval-2020 Task 11, which\nleverages LSTM and pretrained word embeddings without using human-engineered\nfeatures for representation learning. Our experimental results show that the LSTM model with GloVe word\nembeddings can get better performance according to the scores of different\nneural network models and integration models on this task. The main goal\nof this task is to detect propaganda techniques in news articles at a\nfine-grained level, not just to make coarse judgments about whether the\nnews articles use propaganda techniques.\\cite{Zhong2019ntuerAS}\n\nIt is known that neural networks perform well on large training sets, but sometimes a large, accurately labeled dataset cannot be obtained. For future work, the development of propaganda technology detection in news articles can be greatly improved in the pretraining model and the integrated model architecture.\n\n\\section*{Acknowledgements}\nThis work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61966038, 61702443 and 61762091. The authors would like to thank the anonymous reviewers for their constructive comments.\n\\bibliographystyle{coling2020}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{he} Symmetricom test set has become a useful instrument for phase noise and frequency stability measurements that allows one to compare a signal from a device under test (DUT)  against a reference signal and as a result measure the relative phase noise and frequency stability between the two signals~\\cite{Stein2010}. The manufacturer supplies a few models but currently the maximum input bandwidth  is 400 MHz (model 5125A) while the highest performance unit (model 5120-01), with the lowest phase noise measurement noise floor, only has a measurement bandwidth of 30 MHz. So how would one measure the additive or residual phase noise of a device under test which has an operating frequency much higher than 30 MHz or  even 400 MHz? \n\n\n\nOne method to achieve this is to take a beat note between two oscillators, and, by engineering a relatively low noise reference through frequency division of a signal derived from one of the oscillators, one can make a measurement on oscillators operating at X-band frequencies~\\cite{Hartnett2012}. If the oscillators have similar phase noise 3 dB can be subtracted for their individual performance.  However this method is limited to the case where the beat note between the signals from two oscillators being compared falls within the measurement bandwidth of the test set being used.  \n\nThis paper outlines a method to use the Symmetricom test set outside its normal area of application (oscillator noise measurements). We show that it can be used for measuring phase fluctuations in passive and active microwave components (circulators, amplifiers, voltage controlled phase shifters etc) and at frequencies well outside its normal operational bandwidth. \n\nThis method is quite different to that normally used for phase noise measurements where a homodyne technique uses baseband measurements and calibration of the mixer is required to calculate the phase noise from voltage noise measurements. It  allows one to simply measure the phase noise and\/or frequency stability of practically any device at any frequency, provided the frequency is sufficiently stable. \n\nThe DUT could be an active device where only phase noise is of interest or alternatively two oscillators where both phase noise and frequency stability are measured.\n\n\n  \n  \n\\section{Measurement technique}  \nThe essential elements of the measurement technique are shown in Fig. 1. Here the DUT is a microwave amplifier operating at frequency $f_0$. The microwave signal from a stable source is divided into two arms. One arm passes through the DUT and picks up the intrinsic phase fluctuations due to the amplifier. An attenuator (att1) is adjusted to give the desired input power and a second attenuator (att2) is used to provide the suitable LO drive level for the first mixing stage.  At this point an auxiliary source is used to provide a  signal at a frequency $f_a$ that is within the range of the test set being used. The important feature of this technique is to impose the DUT phase noise onto the low frequency $f_a$ auxiliary  signal, which is done via the use of two mixing stages.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig1.pdf}\n\\caption{A block diagram of the measurement system 1(a) and 1(b). DUT = Device Under Test. LPF = Low Pass Filter; BPF = Band Pass Filter;  attX = coaxial attenuator.  Both microwave synthesizer and the function generator were referenced by 10 MHz from a Kvarz CH1-75A hydrogen maser.  The dashed line rectangles indicate two implementations: 1(a) uses the BPF and 1(b) replaces the BPF with a (mechanical) phase shifter.}\n\\label{fig1}\n\\end{figure}\n\nIf necessary, attenuators (att3 and att4) are used to adjust the LO drive level and the RF power level to the second mixing stage. Alternatively an isolator could be used instead of attenuator att4. A band pass filter (BPF) is used to reject one of the sidebands, either $f_0 + f_a$ or $f_0- f_a$,  at the output of the first mixing stage. (An image rejection or single side band mixer could also be used.)  The second mixing stage recovers the  frequency ($f_a$) of the auxiliary signal, after low pass filtering (LPF) but it carries all the phase fluctuations of the device under test. The signal is then amplified and input to the Symmetricom test set. The reference to the test set is taken directly from the auxiliary oscillator.  This combination of two mixing stages with the test set is integral to this technique and allows one to essentially use the test set to measure the phase noise of any device operating at any frequency within the range of the components used.\n\n\\subsection{Measurement system with a band pass filter}\n\nConsidering the configuration of Fig. 1(a), we model the input signal to the second mixing stage, after filtering, as proportional to, $$cos[2 \\pi (f_0 - f_a) t + \\delta \\varphi(t)+ \\varphi_{LO1}],$$  where the upper sideband has been rejected,  and where $\\delta \\varphi(t)$ is the phase spectrum of the noise added by the DUT to the $f_0$ signal before being mixed with the $f_a$ auxiliary signal. The parameter  $\\varphi_{LO1}$ represents the initial phase of the signal at LO1. After the second mixing stage  the resulting signal frequency dependence  can be modeled by the product of the  latter with the original $f_0$ microwave pump signal as, \n\\begin{eqnarray}\n&cos[2 \\pi (f_0 - f_a) t + \\delta \\varphi(t)+ \\varphi_{LO1}] cos[2 \\pi f_0 t + \\varphi_{LO2}] = \\nonumber \\\\\n&\\frac{1}{2} cos[ 2 \\pi f_a t + \\delta \\varphi(t)+ \\Phi],\n\\label{eqn1}\n\\end{eqnarray}\nwhere $\\Phi = \\varphi_{LO2}-\\varphi_{LO1}$ is a phase constant and the high frequency $2 f_0$ mixing product has been rejected by the low pass filter. The $2 f_0$  signal is well suppressed by the mixer itself  if $f_0$ is in the microwave range. Here $\\varphi_{LO2}$ represents the initial phase of the signal at LO2. This and the initial phase at LO1 are a fixed phase with respect to the initial phase of the auxiliary signal at the IF port of the first mixing stage. Therefore the last term in Eq. (\\ref{eqn1}) is a constant phase factor and as a result  the $f_a$ signal input to the test set contains the phase noise of the DUT. \n\nThe correlated phase fluctuations of the pump microwave signal ($f_0$) are rejected at the level of the second mixer. Phase fluctuations of the auxiliary oscillator are rejected by the test set because it measures fluctuations of the phase difference between two input signals (it is essentially a ``digital'' phase bridge).  The measurement system white noise floor is then determined primarily by the final low frequency RF amplifier used to provide sufficient signal power to the input of the test set.\n\n\n\nThe two mixers down converting the microwave signal, that passes through the DUT, to the frequency of the auxiliary signal, and the rejection of one of the sidebands, are the main elements in this technique. All variations on this design contain this important aspect. Ideas similar to this have been developed before under the name of ``transposed gain'' where a signal is down converted from a much higher microwave frequency to a low frequency where the amplifier, used as the important gain stage, has much lower intrinsic noise and after which the signal is up converted again to the higher microwave frequency~\\cite{Driscoll1995, Everard1995a, Everard1995b, Everard1995c}. This technique is quite different as it does not actually use a transpose gain but transfers the noise of the DUT to a much lower frequency signal that is accessible to the phase noise test set used to measure the device's phase noise and\/or frequency stability.\n\n\nIt is worth noting that the  DUT could have, alternately, been located on the other branch of the power splitter before the mixer input LO2 but we would get the same result as Eq. (\\ref{eqn1}).\n\nCaution needs to be taken to ensure that the LO power on the mixers is maintained at their optimum drive level and at the same level for all measurements. One might be measuring the DUT as a function of input power, as in Fig. 1, which will change the power at LO1. The mixer's contribution to the measurement system noise floor depends on its input power. It was observed that when the mixer power was allowed to drop well below optimum drive levels the measurement system noise floor was elevated and limited the measurement. \n\n\\subsection{Measurement system without a band-pass filter}\n\nHowever as indicated in Fig. 1(b) the band-pass filter could also be replaced with a (mechanical) phase shifter but it will be shown that this does not result in the correct measurement of the phase noise of the DUT. \n\nWe model this configuration with a signal at the first mixing stage as proportional to, $$cos[2 \\pi (f_0 \\pm f_a)t + \\delta \\varphi(t)+ \\varphi_{LO1} + \\varphi_p],$$ where both sidebands are present, the term $\\delta \\varphi(t)$ represents the flicker phase fluctuations of the DUT. (The treatment of the white phase noise of the DUT is given below).  The term $\\varphi_p$ represents the added phase from the variable (mechanical) phase shifter. When this is mixed at the second mixing stage the resulting signal frequency dependence  can be modeled by the product of the  latter with the original $f_0$ microwave pump signal as,\n\\begin{eqnarray}\n&cos[2 \\pi (f_0 \\pm f_a)t + \\delta \\varphi(t) + \\varphi_{LO1} + \\varphi_p] \\times \\nonumber \\\\\n&cos[2 \\pi f_0 t + \\varphi_{LO2} ] = \\nonumber \n\\\\\n&cos[2 \\pi f_a t ]cos[\\delta \\varphi(t)- \\Phi+\\varphi_p],\n\\label{eqn2}\n\\end{eqnarray}\nwhere the high frequency $2 f_0$ mixing product has been low pass filtered out and the phase noise sidebands are added to the phase constants. In contrast to the previous case, shown by Eq. (1), the phase noise of the DUT is not imposed on the output signal at frequency $f_a$. Therefore, the measurement system without the band-pass filter (Fig. 1(b)) is incapable of measuring \\textit{flicker} phase fluctuations of the DUT. \n\n\n\n\n\\subsection{Broadband white noise}\n\nIn this section we show how the measurement system in Fig. 1(b) treats white phase noise of the DUT. In this case the output signal voltage from the DUT with effective temperature $T_{eff}$ as given by,\n\\begin{equation}\nu_{in} = U_1 |T| cos[2 \\pi f_0 t + \\delta \\varphi(t)+ \\varphi_{LO1}] + \\nu(t),\n\\end{equation}\nwhere $U_1$ is the amplitude of the signal input to the DUT, $|T|$ is the DUT transmission coefficient and $\\nu(t)$ is the white noise from the DUT with power spectral density,\n\\begin{equation}\nS_{\\nu} =k_B T_{eff} \\, \\,  \\verb1[W\/Hz], 1\n\\end{equation}\nwhere $k_B$ is Boltzmann's constant. Therefore the power spectral density of phase fluctuations due to this thermal noise source  is,\n\\begin{equation}\nS_{\\phi} = \\frac{k_B T_{eff}}{2 P |T|^2},\n\\end{equation}\nwhere $P$ is the power of the DUT input signal.\n\nAfter the reference phase shifter is adjusted to maximize the output signal from the second mixer, the input signal to the Symmetricom test set, can be modeled by,\n\\begin{equation}\nu_{out} = \\chi U_1 |T| cos[2 \\pi f_a t ]+ \\xi(t),\n\\label{eqn3}\n\\end{equation}\nwhere $\\chi$ is the mixer's conversion coefficient. The first term in Eq. (\\ref{eqn3}) is the same as Eq. (\\ref{eqn2}) but with the phase shifter tuned to maximize the $f_a$ signal, and the second term is the broadband white phase noise term, which unlike the first term is not dependent on the chosen reference phase shift. The power spectral density of $\\xi(t)$ can be expressed as, \n\\begin{equation}\nS_{\\xi} = \\chi^2 k_B T_{eff} \\,\\, \\verb1[W\/Hz]. 1\n\\label{eqn3-2}\n\\end{equation}\nThe effective power spectral density of phase fluctuations of the output signal due to thermal noise is then, \n\\begin{equation}\nS_{\\phi} = \\frac{k_B T_{eff}}{2 P |T|^2}.\n\\end{equation}\nThis expression results from combining Eqs (\\ref{eqn3}) and (\\ref{eqn3-2}). It is power dependent and models what is observed. Equations (8) and (5) coincide. This proves that the measurement system in Fig. 1(b), while insensitive to flicker phase noise of the DUT, can correctly measure its white phase noise. Hence the technique using the phase shifter, though it \\textit{does not} allow us to measure the DUT narrow band phase noise on the auxiliary signal, it \\textit{does correctly} measure its broadband white noise.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig2.pdf}\n\\caption{A block diagram of the measurement system used to measure the intrinsic phase noise of oscillators. The same key elements are employed as in Fig. 1(a): the two mixing stages and the band-pass filter that down convert the frequency of the oscillator under test to that of the auxiliary oscillator. A band-pass filter is essential between the two mixers.  \n}\n\\label{fig2}\n\\end{figure}\n\n\n\\subsection{Oscillator comparisons}\n\nThe same key elements of the configuration of Fig. 1(a), using the band-pass filter, can be used to measure the phase noise and frequency stability of a pair of  oscillators at different frequencies, where we assign $f_{01}$ and $f_{02}$ to the frequencies of the two oscillators. The same aspects apply to this configuration as discussed above, so we'll only consider the situation where a band-pass filter is used. If the phase shifter is used instead of the band-pass filter it can be shown, like in Eq. (2), that the relative phase fluctuations of the  oscillators are not recovered on the auxiliary signal.\n\nFig. 2 describes the basic features for the measurement method for two oscillators where a band-pass filter is used after the first mixing stage. The main difference with Fig. 1 is that two devices are needed. One may be used as a DUT and the other as a reference to obtain the measured signal frequency where, \n\\begin{equation}\nf_a \\pm  |f_{01} - f_{02}| < 400 \\,\\, \\verb1MHz, 1\n\\label{eqn400}\n\\end{equation}\nwith the $\\pm$ depending on the choice of band-pass filter.  This ensures a signal at a frequency measureable by the test set containing the phase fluctuations of the two oscillators. Here we have assumed a 400 MHz bandwidth test set is being used. Therefore Eq. (\\ref{eqn400}) means the frequency difference between the oscillators  must be less than twice the bandwidth of the test set otherwise additional mixing stages are needed.\n\nIn the case where a reference oscillator has much lower phase noise than the DUT oscillator the final result is that of the DUT but if they are nominally identical oscillators then we get the relative phase fluctuations of the two devices. This means 3 dB must be subtracted for the phase noise of a single oscillator.\n\nFrom one of the oscillators an intermediate frequency is generated with an auxiliary oscillator in the first mixing stage, which is filtered to reject one of the sidebands with the band-pass filter, and then mixed in the second stage with the signal from the other  oscillator, low pass filtered, amplified and measured on the Symmetricom test set.   The phase noise is read directly from the test set, but to find the Allan deviation of the signal at frequency $f_{01}$ or $f_{02}$ one must scale by the ratio $(f_a \\pm |f_{01}-f_{02}|)\/f_{0}$ where $f_0$ is chosen as equal to either $f_{01}$ or $f_{02}$. \n\nIn the case of Fig. 2 we can model the results of the first mixing stage as proportional to, $$cos[2 \\pi (f_{01} - f_a)t + \\delta \\varphi_1 (t)+ \\varphi_{LO1}],$$ where upper $f_a$ sideband has been filtered out by the band-pass filter. The phase noise of the oscillator is represented by $\\delta \\varphi_1 (t)$. When this is mixed at the second mixing stage with the output of the second oscillator with its own phase noise $\\delta\\varphi_2 (t)$ at the  frequency $f_{02}$, and modeled as proportional to, $$cos[2 \\pi f_{02} t +\\delta \\varphi_2 (t) + \\varphi_{LO2} ],$$ the resulting signal frequency dependence can be modeled by the product of the  latter with the former as,\n\\begin{eqnarray}\n&cos[2 \\pi (f_{01} - f_a)t +\\delta \\varphi_1(t) + \\varphi_{LO1}] \\times \\nonumber \\\\\n&cos[2 \\pi f_{02} t +\\delta \\varphi_2 (t) + \\varphi_{LO2} ] = \\nonumber \\\\\n&\\frac{1}{2}cos[2 \\pi (f_{01}-f_{02}+f_a) t +\\delta \\varphi_1(t)+\\delta \\varphi_2 (t) + \\Phi],\n\\label{eqn4}\n\\end{eqnarray}\nwhere $\\Phi = \\varphi_{LO2}-\\varphi_{LO1}$ is a phase constant and the high frequency  mixing product of order $f_{01}+f_{02}$ has been filtered out. The power of phase fluctuations of the resulting signal at frequency $f_{01}-f_{02}+f_a$ is equal to the  combined power of phase fluctuations of the individual microwave oscillators.\n\nOf course the low pass filter must now pass the signal with frequency $f_{01}-f_{02}+f_a$, and this frequency must be within the bandwidth of the measurement test set. This then necessarily affects the frequency at which the phase noise is measured and the scaling ratio to calculate the Allan deviation becomes $(f_{01}-f_{02}+f_a)\/f_{0}$. If the oscillators are nominally identical an additional $1\/\\sqrt2$ factor must be applied to get the stability of a single oscillator.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig3.pdf}\n\\caption{The SSB phase noise, as a function of input power of the Endwave amplifier model no. JCA812-5001, operating at 11.2 GHz, measured  using the measurement method  of Fig. 1(a) with $f_a = 20$ MHz. The legend indicates the input power to the amplifier and the measurement noise floor (curve 4) which was measured by removing the DUT from the circuit. RF amplifier = MiniCircuits ZFL-500LN+.}\n\\label{fig3}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig4.pdf}\n\\caption{After replacing the BPF with a mechanical phase shifter (Fig. 1(b)) the SSB phase noise of the output signal at $f_a$ = 20 MHz was measured, where a AML amplifier model no. AML812PNA5402, operating at 11.2 GHz, was used with different input power levels. The legend indicates the input power to the amplifier and the measurement noise floor (curve 4) which was measured by removing the DUT from the circuit. A smoothing by taking a 50 point average was applied to data of curve 4. RF amplifier = MiniCircuits ERA-5+. See discussion in text.}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig5.pdf}\n\\caption{The SSB phase noise of an Agilent E8257C synthesizer at 1 GHz without external reference (curve 1), and phase locked to a 10 Mz signal from the frequency doubled output of an Oscilloquartz 8607 quartz oscillator (curve 2). The latter was compared with a 1 GHz signal derived from an ultra-low phase noise cryogenic sapphire oscillator and measured using the measurement method of Fig. 2(a).  Curve 3 is the measurement system noise floor. Curve 4 is the phase noise of the  locked synthesizer at 1.2 GHz but referred to 1 GHz. See text for details.}\n\\label{fig5}\n\\end{figure}\n\n\n\\section{Results}\n\nUsing the measurement method of Fig. 1(a) we measured the phase noise of an Endwave amplifier (model number JCA812-5001) as a function of input power. The band-pass filter is tunable home made cavity filter and the low-pass filter a  MiniCircuits SLP-21.4+.  The microwave mixers are Watkins Johnson model number M14A. The function generator is a 20 MHz Agilent model 33220A. The final RF amplifier used was either a MiniCircuits ZFL-500LN+ or an ERA-5+ amplifier.\n\nThe results are shown in Fig. 3. Where these are above the measurement noise floor, they are consistent with those expected for the amplifier and indicate a flicker law for Fourier frequencies $10 \\leq f \\leq 100$ Hz. The thermal noise floor is dependent on input power to the amplifier as expected. A carrier frequency of $f_0 = 11.2$ GHz and an auxiliary  frequency  $f_a = 20$ MHz were used in these measurements. \n\nThe measurement system noise floor was measured by removing the DUT and it was found that it limits the measurements for Fourier frequencies $f < 10$ Hz. The phase noise of the final RF amplifier was determined to be the principle limitation of the measurement system especially at Fourier frequencies $f > 100$ Hz. By using either a MiniCircuits ERA-5+ or a ZFL-500LN+ amplifier the measurement system white noise floor was limited to about -163 dBc\/Hz and about -158 dBc\/Hz  (on the 20 MHz carrier) respectively. The former is very close to white noise floor of the Symmetricom 5125A test set itself. \n\nThe additive phase noise of the RF amplifiers used here were separately measured by splitting the 20 MHz signal of the auxiliary oscillator and passing one signal through the amplifier to the input channel of the test set and the other directly to the reference channel. In both cases, the same input power to the amplifier was used as in the measurement system, and the results matched the thermal noise floor of the measurement system when using these RF amplifiers. \n\nFinally, when using the ERA-5+ RF amplifier,  a pair of MiniCircuits ZX05-153+ mixers were substituted for the Watkins Johnson M14A mixers and this raised the measurement system white noise floor 3 dB to about -160 dBc\/Hz, indicating that a small contribution also may come from the choice of mixers used.\n\nUsing the method of Fig. 1(b) and  a very low phase noise AML X-band microwave amplifier we measured essentially the phase noise floor of the measurements system as a function of input power to the X-band amplifier. In this configuration where a mechanical phase shifter (an Arra 9428B in this case) is used instead of the band-pass filter one does not get the phase noise of the DUT (nor that for the oscillator(s) if a phase shifter is used in the configuration of Fig. 2). The  results are shown in Fig. 4. However the measurements of Fig. 4 indicate that as one decreases the input power to the X-band amplifier (the DUT) one observes rising thermal noise. At input power levels $\\geq -20$ dBm (curve 1) the measured noise is at the measurement system noise floor. At input power levels less than this we observe an increase in white noise approximately proportional to the change in input power. \n\nThe measurement system noise floors for both configurations shown in Fig. 1 were determined by removing the DUT, and ensuring the power to the mixers and the RF amplifier were the same as used when making the DUT phase noise measurements. Looking at Figs 3 and 4 there is a clear difference in the frequency dependence and level of the measurement system phase noise floor. The difference is that in the method of Fig 1.(a), with the band-pass filter, the mixer noise is present for $f < 10^3$ Hz (curve 4 in Fig. 3) but in the method of  Fig 1.(b), without the band-pass filter, the mixer noise is not present (curve 4 in Fig. 4) and we only measure the test set noise floor for $f < 10^2$ Hz.  At higher offset frequencies it is the RF amplifier that sets the thermal noise floor.\n\nAs shown in Eq. (\\ref{eqn2}), the DUT phase noise is not added to the auxiliary signal, when the phase shifter was substituted for the band-pass filter, so also the mixers in the circuit are devices under test and their phase noise is not added but shows up as amplitude noise on the signal input to the test set. By contrast, from Eq. (\\ref{eqn1}), when the band-pass filter is used, the  phase noise of the mixers is, in addition to the phase noise of the DUT, added to the signal input to the test set. Therefore with the DUT removed we measure a system noise floor that includes the phase noise of the two mixers and the RF amplifier (at the higher offset frequencies).\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.0in]{Fig6.pdf}\n\\caption{Allan deviation of an Agilent E8257C synthesizer at 1 GHz without external reference (curve 1), and phase locked to a 10 Mz signal from the frequency doubled output of an Oscilloquartz 8607 quartz oscillator (curve 2). The latter was compared with a 1 GHz signal derived from an ultra-low phase noise cryogenic sapphire oscillator and measured using the measurement method of Fig. 2(a). Curve 3 is the Allan deviation of the signal frequency of the locked synthesizer at 1.2 GHz after the appropriate scaling was applied. See text for details.}\n\\label{fig6}\n\\end{figure}\n\nUsing the measurement method of Fig. 2 we measured the phase noise of an Agilent E8257C synthesizer generating a 1 GHz signal compared to a 1 GHz signal synthesized from an ultra-low phase noise cryogenic sapphire oscillator. The phase noise spectra are shown in Fig. 5, and the frequency stability in Fig. 6. Curve 1 represents the phase noise of the synthesizer without external reference, and curve 2 when it was referenced by a 10 Mz signal from the frequency doubled output of an Oscilloquartz 8607 quartz oscillator. \n\nThe measurement system used two MiniCircuits ZX05-10L-S+ mixers, a ZVBP-909-S+ band-pass filter, a SLP-100+ low pass filter and a ZFL-500LN+ amplifier. Its measurement system phase noise floor is shown in curve 3 in Fig. 5. This was determined by using the same low phase noise oscillator on both input ports (for Oscillators 1 and 2 in Fig. 2). The auxiliary oscillator frequency used in these measurements was $f_a = 100$ MHz, derived from the  ultra-low phase noise cryogenic sapphire oscillator, with a phase noise of -130 dBc\/Hz at 1 Hz offset  \\cite{Nand2010, Hartnett2012}. In this case it was necessary to scale resulting stability generated by the test set by $f_a\/f_{0} = 100$ MHz$\/1.0$ GHz $= 1\/10$ where $f_{0} = f_{01}= f_{02} = 1$ GHz.\n\nWe also compared the same oscillators where we raised the output signal frequency of the  Agilent  E8257C synthesizer to $f_{02}=1.2$ GHz.  The auxiliary oscillator was still at $f_a = 100$ MHz, which was mixed with the signal at $f_{01} =1$ GHz from the ultra-low phase noise cryogenic sapphire oscillator, then we filtered out the upper sideband with the band-pass filter  leaving the lower sideband at 900 MHz to be mixed with the 1.2 GHz signal. After low-pass filtering (using a MiniCircuits SLP-400 filter) and amplification this resulted in a 300 MHz signal  that was measured by the test set. Curve 4 in Fig. 5 is the result where a factor of $20 log(1.2)$ has been subtracted to compare the results all at 1 GHz. It is essentially identical with curve 2 as expected.  And in Fig 6 we show the Allan deviation of the 1.2 GHz synthesizer signal where the correct scaling has been applied to the output of the test set.  In this case $(f_{01}-f_{02}+f_a)\/f_{02} = 300$ MHz$\/1.2$ GHz $= 1\/4$. Only Allan deviation data with a noise equivalent bandwidth (NEQ) of 0.5 Hz is shown.\n\nThe cryogenic sapphire oscillator's phase noise and frequency stability \\cite{Hartnett2012} are orders of magnitude lower than that of the Agilent E8257C synthesizer. At 1 GHz its phase noise is approximately equal to the measurement system noise floor (curve 3 or Fig. 5), hence it does not contribute to these results. Figs 5 and 6 therefore show only the performance of the  E8257C synthesizer.\n\nIt should be noted that when the frequency difference of the two oscillators falls within the bandwidth of the test set the additive phase noise can be measured by taking the beat note of the two signals and comparing it to a previously characterized low noise reference. To confirm the validity of our transposed gain technique we used this direct comparison method to repeat the measurements in figures 5 and 6 and found no discrepancy in the results.\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nA technique has been developed that allows one to measure the phase noise of active components operating at frequencies well outside the bandwidth of the phase noise measurement test set being used. \n\nDigital measurements of phase fluctuations offer several advantages over the usual homodyne technique. Here is no need for the calibration of the mixer nor is there any requirement to  phase lock two oscillators to maintain zero voltage at the output of the mixer during the measurement process. The latter feature  permitted, for the first time, the phase noise measurement of ultra-stable cryogenic sapphire oscillators. \n\nThis method is only limited by the operational bandwidth of the components used, and the noise floor imposed on the measurements by the two mixing stages and the final RF amplifier used to provide sufficient signal power to the test set. Using the band-pass filter and X-band mixers a measurement system noise floor at low offset frequencies of $10^{-12.7}\/f$ was obtained, while with 1 GHz mixers $10^{-12.0}\/f$ was achieved. Using a MiniCircuits ERA-5+ RF amplifier at the input to the test set a measurement system  white noise floor of -163 dBc\/Hz was obtained.\n\nHence one may achieve a measurement white noise floor at the limits of the test set itself, near -165 dBc\/Hz for the 5125A model and -175 dBc\/Hz for 5120A-01 model with ultra-low-noise option. Provided the final RF amplifier is sufficiently low noise one can reach this white noise floor and potentially measure the phase noise of extremely low noise components. \n\n\n\\section{Acknowledgments}\nThis work was supported by ARC grant LP110200142.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{section:introduction}\n\nLocal probes of the cuprate superconductors reveal signatures of\nelectronic inhomogeneity both at the microscopic scales of lattice constants and at somewhat larger mesoscopic scales \\cite{Chang:1992,Pan:2001,Howald:2001,Gomes:2007,Koshaka:2007,Pasupathy:2008,Parker:2010}.\nThe inhomogeneity is generally seen as separated regions with either a large or a small gap, which\nhave been attributed to local variations in the doping level with respect to the half filled Mott insulator\\cite{Parker:2010}.\n\nExperiments which probe global properties indicate that the average doping level has two direct effects on the superconducting properties. First, the pairing gap is seen to decrease with hole doping away from half filling\\cite{Ding:2001,Ino:2002}. Second, the superfluid stiffness extracted from penetration depth measurements, increases with doping\\cite{Uemura:1989,Boyce:2000}. This interplay between two energy scales relevant to superconductivity is thought to give rise to the dome shaped dependence of $T_c$ on hole doping\\cite{Emery:1995}. Doping inhomogeneity is therefore expected to  lead to spatial modulations of the pairing amplitude along with variations of the charge carrier density.\n\nIn this paper we shall investigate how inhomogeneity in the doping level affects global superconducting properties of the material. Specifically we address the effect of inhomogeneity on the temperature dependent thermodynamic stiffness and, ultimately, on the transition temperature. To this end we employ a semi-phenomenological model of a $d$-wave superconductor that takes into account the\nthe proximity to the Mott insulator through a strong on-site repulsion. Furthermore we consider\nvarious scales of inhomogeneities, ranging from the microscopic scale of a lattice constant to mesoscopic scales, somewhat larger than the coherence length (see Fig \\ref{fig:overview}). An important question for practical applications is whether the transition temperature can be enhanced significantly by judicious design of the inhomogeneity. The idea is to gain from an optimal combination of large pairing gap in the low doping regions and large carrier density in the highly doped ones\\cite{Kivelson:2002}.\n\nEnhancement of $T_c$ due to a similar mechanism was predicted in cuprate heterostructures composed of an underdoped  superconducting layer coupled to an overdoped metallic one.\\cite{Berg:2008, Okamoto:2008,Goren:2009} The underdoped layer induces a proximity gap in the overdoped layer, which then contributes to the zero temperature phase stiffness of the system and considerably enhances it compared with the suppressed stiffness of the underdoped layer. On the other hand, the $d$-wave proximity gap which is induced on the metallic layer is small, and thus results in a sharp reduction of the stiffness with the temperature\\cite{Goren:2009}. We found in Ref. \\onlinecite{Goren:2009} that the combined effect can in principle lead to enhancement of $T_c$ compared with an optimally doped layer. However to attain such enhancement the coupling between layers needs to be much larger than the realistic coupling between the cooper-oxide planes. It is therefore unlikely that these simplified models provide a satisfactory explanation for the $T_c$ enhancement observed in various experiments on heterostructures.\\cite{Yuli:2008,Gozar:2008,Jin:2011} However, if there is doping inhomogeneity within a plane the coupling between the overdoped and underdoped regions would naturally be large since they are connected by the in-plane rather than the c-axis tunneling. As we shall see this situation can indeed give rise to enhancement of the maximal critical temperature compared to a pure system.\n\nSpecific kinds of in-plane inhomogeneity and their effect on superconductivity have been previously investigated theoretically. For example a weak-coupling BCS theory of the\n{\\em attractive} Hubbard model showed that $T_c$ can be enhanced by periodic modulations of the weak attraction.\\cite{Martin:2005} A density matrix renormalization group (DMRG) study of the {\\em repulsive} Hubbard model on a two leg ladder showed that modulations of the hopping matrix element along the ladder can enhance the pairing correlations and thereby possibly increase the $T_c$ of a coupled ladder system.\\cite{Karakonstantakis:2011} A direct study of the two dimensional Hubbard model using contractor renormalization (CORE) also indicated that there is an optimal modulation of the hopping matrix element, which maximizes the pairing correlations.\\cite{Baruch:2010} Finally, dynamical mean field and cluster Monte Carlo calculations find increased pairing gap, and possibly $T_c$, in a state with charge modulation near $1\/8$ doping.\\cite{Maier:2010,Okamoto:2010}\n\nThe above studies focus on the effect of periodic commensurate charge modulations on the pairing order parameter. We complement and extend the analysis in several ways.\nFirst, we use an effective theory, amenable to analytic treatment that allows to identify the physical origin of the various effects. Second we compute the temperature dependent superfluid stiffness, which at least in the underdoped cuprates is a more complete measure of superconductivity than the pairing amplitude and allows us to directly estimate $T_c$. Third, in addition to the stripe model treated in previous work we also consider random doping variations, which appear to be the\nmore generic situation in samples of doping above $1\/8$. Both for the stripe model\nand the random inhomogeneity we asses the possibility of enhancing $T_c$ by tuning the\nmagnitude of characteristic doping modulations and their length scale.\n\nWe implement the inhomogeneity in the form of inclusions of a highly overdoped phase, already in the metallic regime, embedded in a background of underdoped or optimally doped material. The case of mesoscopic inhomogeneity, where the metallic inclusions are of the size of the superconducting coherence length or larger is sketched in Fig. \\ref{fig:overview}(a). This is treated within an effective stripe model of the metallic regions, where we average over stripe orientations to obtain an isotropic macroscopic stiffness. Another case we consider, is where the metallic regions are much smaller than the coherence length and are modeled as point impurities. This case is depicted in Fig. \\ref{fig:overview}(b).\n\nIn both cases we include the crucial effects of strong coulomb repulsion and of the $d$-wave symmetry of the order parameter. The former is the reason for the low superfluid density $\\rho_s$ at low doping, while the second  is responsible for the linear suppression of $\\rho_s$ with $T$ at low temperatures.\\cite{Lee:1997} These effects are taken into account within a slave boson mean field theory of the $t-J$ model.\\cite{Zhang:1988,Kotliar:1988} Furthermore, we include Fermi-liquid-like corrections phenomenologically, to the description of low energy quasiparticles.\\cite{Millis:1998,Wen:1998,Paramekanti:2002}\n\nRegardless of the model for the metallic regions we find an increase of the zero temperature stiffness and for a wide range of doping levels, also higher critical temperature compared to the pure system with the same average doping. Furthermore, in the case of microscopic impurities we even predict that a higher $T_c$ can be attained even compared to the maximal $T_c$ at optimal doping of the pure system.\n \nThe paper is structured as follows: In Sec. \\ref{section:overview} we give a general overview of the models used, of the assumptions that underlie our choice of models, and of the main results obtained in the different regimes.  Section \\ref{section:stripes} gives a detailed treatment of a model representing mesoscopic inhomogeneity, while in section \\ref{section:disorder} we consider a model with point impurities. Section \\ref{section:summary} is a summary and discussion of the results.\n\n\n\\section{Overview}\\label{section:overview}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics{overview_xfig}\n\\end{center}\n\\caption{(a) An illustration of a mesoscopic-scale inhomogeneous layer. The typical size of the metallic regions is equal or larger to the superconducting coherence length. (b) Microscopic-scale inhomogeneous layer. the metallic regions are point-like impurities. }\n\\label{fig:overview}\n\\end{figure}\nIn this section we introduce the framework for treating the inhomogeneous cuprate layer within a slave boson mean field theory. We describe the essential ingredients of the theory for the case of  mesoscopic inhomogeneity as well as for point impurities. Finally we summarize the main results that are derived in detail in later sections.\n\nIn order to describe doping inhomogeneity in cuprate materials we make use of models that can account for the effects of doping of the Mott insulating parent compound.\nA simple theoretical framework that captures many of the important effects is the renormalized mean field theory (RMFT)\\cite{Zhang:1988} or slave boson mean field theory (SBMFT)\\cite{Kotliar:1988} of the $t-J$ Hamiltonian,\n\\begin{equation}\nH_{\\rm tJ}=-P_G\\sum_{ij{\\sigma}}t_{ij}c_{i{\\sigma}}^\\dag c_{j{\\sigma}} P_G+ h.c + \\sum_{\\langle ij\\rangle}J_{ij}{\\bf S}_i {\\bf S}_j.\n\\end{equation}\nHere $J_{ij}=4t_{ij}^2\/U$ is the super-exchange interaction, ${\\bf S}_i=c^\\dagger_{is}{\\bf{\\sigma}}_{ss'}c{^{\\vphantom{\\dagger}}}_{is'}$ and $P_G=\\Pi_i (1-n_{i{\\uparrow}}n_{i{\\downarrow}})$ implements the Gutzwiller constraint, which prohibits double occupancy of sites.\n\nThe standard mean field treatment of the $t-J$ model includes two approximations. The first is to account for the projection only through renormalization of the hopping $t_{ij}\\to g_{ij} t_{ij}$, while working in the full rather than the projected Hilbert space \\cite{Zhang:1988}. The second approximation consists of a standard decoupling of the quartic term in both the Fock and BCS channels. The resulting mean field Hamiltonian is given by\n\\begin{eqnarray}\\label{HRMFT}\nH_{\\rm MF}&=&-\\sum_{i,j,{\\sigma}} (g_{ij}^t t + \\chi_{ij} ) c_{i{\\sigma}}^\\dag c_{j{\\sigma}}+h.c -\\sum_{i,s} \\mu_i c_{i{\\sigma}}^\\dag c_{i{\\sigma}} \\nonumber\\\\\n&+& \\sum_{\\langle ij\\rangle } {\\Delta}_{ij}c_{i{\\uparrow}}^\\dag c_{j{\\downarrow}}^\\dag + h.c\n\\end{eqnarray}\nwhere $\\chi_{ij}=3J_{ij}\\sum_{\\sigma} \\langle c_{i{\\sigma}}^\\dag c_{j{\\sigma}} \\rangle\/4$, ${\\Delta}_{ij}=3J_{ij}\\langle c_{i{\\uparrow}}^\\dag c_{j{\\downarrow}}^\\dag-c_{i{\\downarrow}}^\\dag c_{j{\\uparrow}}^\\dag\\rangle\/4$ and\n$g_{ij}$ are doping dependent renormalization factors that account for the effect of the no-double-occupancy constraint. In a uniform system of doping ${p}$, $g=2{p}\/(1+{p})$, $\\chi_{ij}=\\chi$ for all nearest neighboring $i,j$, and ${\\Delta}_{i,i+\\hat{x}}=-{\\Delta}_{i,i+\\hat{y}}={\\Delta}$ such that the pairing has a $d_{x^2-y^2}$ symmetry.\n\nThe mean field theory of the $t-J$ model captures the crucial fact that the zero temperature superfluid stiffness of underdoped cuprates scales linearly with the hole doping, $\\rho_0\\propto p$.\\cite{Uemura:1989,Lee:1997} It also accounts for the $d$-wave symmetry of the gap that gives rise to a low energy quasiparticle spectrum of the form $E_{\\bf k}=[({\\bf v}_f {\\bf k}_{||})^2+ ({\\bf v}_{\\Delta}{\\bf k}_{\\perp})^2 ]^{1\/2}$. This form of the spectrum explains the observed linear reduction of the superfluid stiffness with temperature, $\\rho_s(T)=\\rho_0-b_0 T$ with $b_0=2\\log{2} (2\\sqrt{2}Zt)^2\/(\\pi v_f v_{\\Delta})$.\\cite{Lee:1997}\nHowever, the mean field theory does not give the correct value of $Z$. This can be viewed as a Fermi liquid correction that may be strongly renormalized at low energies due to quasi-particle interactions not included in the mean field theory.\\cite{Millis:1998,Wen:1998,Paramekanti:2002} Therefore $Z$ is best taken as a phenomenological parameter to be extracted from experiments.\\cite{Wen:1998,Ioffe:2002}\n\n\nIn this paper we extend the analysis of the stiffness and the critical temperature to\nthe case of an inhomogeneous system. Specifically we describe an underdoped system in the bulk ($0.1<p_1<0.15$) embedded with highly overdoped  metallic regions. We consider two regimes of inhomogeneity as illustrated in Fig.~\\ref{fig:overview}. First is when the metallic regions are of the order or larger than the superconducting coherence length and second when they are of the order of one lattice constant. As discussed in the introduction a pertinent question we wish to address is whether such inhomogeneity can lead to enhanced $T_c$.\n\n\n\\subsection{Mesoscopic Inhomogeneity}\nIn the first model, described in Sec.~\\ref{section:stripes}, we assume a 2D mixture of a superconducting underdoped phase (of doping $p_1$) and an extremely overdoped metallic phase (of doping $p_2>0.3$).\nThe doping level varies considerably only across a length scale of the order of the coherence length $\\xi\\sim v_f\/{\\Delta}$, which is typically around 5 lattice spacings, such that the 2D regions are of intermediate size $\\ge \\xi$ as depicted in Fig.~\\ref{fig:overview}(a). This scenario is reminiscent of various experiments that find gap variations on a similar scale, of $5-10 nm$ \\cite{Chang:1992,Pan:2001,Howald:2001,Gomes:2007,Parker:2010}.\n\nWe model this system as a mixture of striped domains, each one with alternating underdoped and overdoped stripes along the $x$ or $y$ direction, such that on a macroscopic scale the system is fourfold rotationally invariant [see Fig.~\\ref{fig:inhomsketch}(a)]. This allows us to obtain an expression for the superfluid stiffness of the entire system. The superconducting stripes are described by the $t-J$ Hamiltonian and the metallic stripes are modeled by free fermions. We vary the widths of the stripes in order to explore the superconducting properties in various geometries. To calculate the critical temperature of the inhomogeneous mixture, we solve self consistently the Bogoliubov de Gennes equations for Hamiltonian (\\ref{HRMFT}) allowing for position dependent $g_{ij}$, ${\\Delta}_{ij}$ and $\\chi_{ij}$.\nWe derive a general formula for the superfluid stiffness $\\rho_s(T)$ of a striped superconductor in terms of response kernels that can be directly calculated from the Bogoliubov de Gennes solution [see Eqns. (\\ref{Kyy}),(\\ref{Kxx})].\nWe then use the Kosterlitz-Thouless criterion $\\rho_s(T_c)=2T_c\/\\pi$ to determine $T_c$ of the mixed system.\n\nWe show that there exist optimal configurations which allow for an enhanced zero temperature superfluid stiffness in the inhomogeneously doped layer, compared with the homogeneous superconducting one. This is a consequence of proximity effect that leads to a gap in the metallic regions. The metallic regions, having a large density of charge carriers, can then contribute significantly to the superfluid stiffness of the inhomogeneous layer at $T=0$. On the other hand, since the proximity gap is much smaller than the original superconducting gap, the reduction of the stiffness at finite temperature is sharper than in the uniform superconductor.\nIt therefore does not immediately follow that the interplay of these two effects can lead to an enhancement of the critical temperature. Previously  we found that such an enhancement is possible in a bilayer of underdoped and overdoped cuprates, under appropriate conditions\\cite{Goren:2009}.\nIn the present scenario, however, we find that $T_c$ of the inhomogeneously doped layer\nis lower than the one of a homogeneous underdoped superconductor of doping $p_1$. The reason is that already at $T=0$ the enhancement of the stiffness due to enlarged carrier density is counteracted to a large extent by a significant paramagnetic suppression of the stiffness which is inevitable in inhomogeneous superconductors. Consequently, the zero temperature stiffness is enhanced compared with the uniform case, but not enough to allow for an enhancement of $T_c$.\n\nNonetheless, we find that the critical temperature of the system increases with the reduction of the relative width of the metallic stripes. This allows for a large proximity gap in the metallic regions, manifested in a relatively small reduction of the stiffness at finite temperature. In order to maximize the proximity effect, but  at the same time allow for a significant contribution of carriers from the metallic region, an optimal configuration should have small but relatively dense metallic regions. In the following we consider the effect of small metallic regions.\n\n\n\n\\subsection{Microscopic Inhomogeneity}\nIn this model, described in Sec.~\\ref{section:disorder} we assume microscopic overdoped regions (doping $p_2$)  which are placed in a low doping superconducting background (doping $p_1$), see Fig.\\ref{fig:overview}(b).\nThe microscopic overdoped regions are modeled as single site impurities with zero or very weak local Hubbard repulsion ($U\\sim0$), which induces modified hopping and pairing amplitudes along their neighboring bonds, as depicted in Fig.~\\ref{fig:dis-sketch}. The hopping amplitude along these bonds is the bare $t$ rather than the renormalized value of SBMFT, and the local pairing strength there is suppressed to zero.\n\nIn the presence of the bond disorder we compute the temperature dependent superfluid stiffness using a perturbative expansion to second order in the impurity strength for disorder averaging (second order Born approximation).\nThen we determine the transition temperature using the Kosterlitz-Thouless criterion as before.\n\n Since the variations in doping generates unconventional bond disorder, the calculation bears several important differences from the standard impurity averaging. The most important difference is that the bond disorder introduces local modulations in the current operator thus renormalizing the coupling to the external vector potential. As a result, the superfluid response obtains vertex corrections which have no counterpart in standard (on-site) impurity averaging but play a crucial role in our case. One important effect of these corrections is to allow for an enhancement of the zero temperature diamagnetic stiffness of the disordered system compared with the pure one.\n A second effect of the vertex corrections is to introduce a paramagnetic reduction of the stiffness at zero temperature, similarly to the mesoscopic inhomogeneous scenario.\n In addition, the disorder introduces self-energy corrections which amount to an anti-proximity effect that acts to reduce the average pairing gap and contributes to the suppression of the stiffness at finite temperature.\n\n The net effect that we find is an enhancement of the superfluid stiffness and a concomitant increase in the critical temperature for a given bulk doping level $p_1$. Interestingly, we even find an overall enhancement of the maximal $T_c$, that is at optimal doping, compared to the maximal $T_c$ of the homogeneous system.\n\n\n\\section{Mesoscopic scale inhomogeneity}\\label{section:stripes}\n\\subsection{The Model}\\label{subsec:Model1}\nIn this section we consider a stripe model. The inhomogeneity is of mesoscopic scale in the sense that the width of the stripes is of the order or somewhat larger than the coherence length associated with the superconducting regions.\nThe superfluid response of such a striped system is of course anisotropic. However we envision that  it becomes isotropic on macroscopic scales due to mixing of striped domains with random orientations as sketched in Fig~\\ref{fig:inhomsketch}(a). The doping level of the stripes alternates between $p_1 $ in underdoped superconducting (SC) stripes of width $l$, and $p_2$ in metallic (M) stripes of width $d$.\n\n\nAs the Hamiltonian of a single domain we take the $t-J$ model\n\\begin{eqnarray}\\label{H:stripes}\nH&=&P_G\\sum_{i,j,{\\sigma}}  t_{ij}c_{i{\\sigma}}^\\dag c_{j{\\sigma}} P_G + \\sum_{\\langle ij\\rangle } J_{ij}{\\bf s}_i {\\bf s}_j\n\\end{eqnarray}\nwhere $P_G$ is the Gutzwiller projection that eliminates double occupancy of sites in the superconducting stripes, but does not affect the metallic stripes. The magnetic exchange coupling is $J_{ij}=J$ in the superconducting stripes and it vanishes in the metallic stripes.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics{inhomsketch_xfig}\n\\end{center}\n\\caption{(a) A model of the inhomogeneous layer as an array of striped domains, which on average is macroscopically fourfold rotationally invariant. (b) The self consistent gap profile, solved for $J=t\/3, t'=0, x=0.1, d=3a$. $SC$ and $M$ denote the underdoped and overdoped regions respectively.}\n\\label{fig:inhomsketch}\n\\end{figure*}\nWe treat the space dependent projection and exchange interaction within slave boson mean field theory (SBMFT) \\cite{ Kotliar:1988,Zhang:1988}. The resulting Hamiltonian is of the form (\\ref{HRMFT}), with space dependent $\\mu_i,g^t_{ij}$, $\\chi_{ij}$ and ${\\Delta}_{ij}$.\nThe electro-chemical potential $\\mu_i$ is determined such that the doping levels of the superconducting and the metallic regions are $p_1$ and $p_2$ respectively. Due to the spatial variations in doping the renormalization of the hopping varies in space too and equals $g^t_{ij}={2p_1}\/(p_1+1) $ in the superconducting stripes and $g^t_{ij}=1$ in the metallic stripes, while the tunneling at the interface between the two regions is renormalized by $g^t_{ij}=\\sqrt{{2p_1}\/(p_1+1)}$).\n\nGiven all the parameters of the mean field model, the fields $\\chi_{ij}$ and ${\\Delta}_{ij}$ can now be determined by the self consistency conditions:\n\\begin{eqnarray}\\label{eqn:selfconsistent}\n \\chi_{ij}&=&\\frac{3J_{ij}}{8}\\sum_{\\sigma} \\langle c_{i{\\sigma}}^\\dag c_{j{\\sigma}}  \\rangle \\nonumber\\\\\n{\\Delta}_{ij}&=&\\frac{3J_{ij}}{8}\\langle c_{i{\\uparrow}}^\\dag c_{j{\\downarrow}}^\\dag - c_{i{\\downarrow}}^\\dag c_{j{\\uparrow}}^\\dag \\rangle\n\\end{eqnarray}\n An example of the resulting profile of the pairing amplitudes is plotted in Fig. \\ref{fig:inhomsketch}(b), where ${\\Delta}_x$ and ${\\Delta}_y$ denote the pairing amplitudes on bonds along the $x$ and the $y$ directions respectively.\nBecause the pairing amplitude in the metallic regions is non zero, these regions contribute to the superfluid stiffness at low temperatures.\n\n\n\n\\subsection{Calculation of the Superfluid Stiffness}\\label{subsec:calculation1}\nIn a striped system the superfluid response depends on the direction of the applied phase twist. However, we assume that the system consists of many striped domains with random orientations along the principal axes. Under this assumption the superfluid response is homogeneous on large scales. It was shown in Ref.~\\onlinecite{Carlson:2000} that the superfluid stiffness of the mixed domains is given by the geometric mean of the $x$ and $y$ components of the stiffness of a single domain $\\rho_s=\\sqrt{K_{xx}K_{yy}}$. Here $K_{aa}$ ($a=x,y$) are the diagonal components of the response tensor,\n \\begin{equation}\\label{stiffness:definition}\n K_{ab}=\\frac{I_a}{{\\Delta} \\theta_b}\n \\end{equation}\n where ${\\Delta} \\theta_b$ is the static phase difference applied across the system in the $\\hat{b}$ direction and $I_a=\\int  J_a ds_a$ is the total current measured in the $\\hat{a}$ direction.\n\n In an inhomogeneous system, we express the stiffness tensor using the microscopic response kernel $\\kappa_{ab}(r,r')$ defined through\n \\begin{equation}\\label{localresponse:definition}\nJ_a(r)=\\int_{t<t'}dr' \\ \\kappa_{ab}(r,r')\\partial_b \\theta(r').\n \\end{equation}\nThe response kernel can then be calculated using the standard Kubo formalism.\nIn the $y$ direction, parallel to the stripes, the stiffness $K_{yy}=\\int dx J_y\/{\\Delta} \\theta_y$ is simply an algebraic sum of the response kernels along the $x$ direction\n\\begin{equation}\\label{Kyy}\nK_{yy} =  \\int dx \\int dx' \\kappa_{yy}(x,x',q_y=0),\n\\end{equation}\nwhere we used the uniformity along the $y$ direction to express it in terms of the $q_y=0$ Fourier component of the response kernel.\n\nTo derive an analogous relation between $K_{xx}$  and $\\kappa_{xx}(x,x')$, we follow the arguments presented in Ref.~\\onlinecite{Carlson:2000}. It is convenient to use the lattice formulation and express all convolution integrals as matrix products.\nThe response kernel is then defined by (summation over repeating indices implied),\n\\begin{equation}\\label{localresponse:definition:lattice}\nJ_a(i)=\\ \\kappa_{ab}(ij)\\partial_b\\theta(j)\n\\end{equation}\nwhere the $y$ dependence is suppressed and we denote by $i,j$ the position in the $x$ direction only.\n\nA static current is divergenceless $\\nabla\\cdot {\\bf J}=0$ and therefore derived from a potential, $J_a(i)={\\epsilon}_{ab}\\partial_b \\phi(i)$. Plugging this back into (\\ref{localresponse:definition:lattice}) we obtain\n\\begin{equation}\\label{Phi:definition}\n{\\epsilon}_{ab}\\partial_b\\phi(i)= \\ \\kappa_{ab}(ij)\\partial_b\\theta(j).\n\\end{equation}\nWe can now derive a second relation between $\\phi$ and $\\theta$ if we multiply by ${\\epsilon} \\kappa^{-1}$ from the left. Defining $\\kappa^D_{ab} = {\\epsilon}_{ac} \\kappa^{-1}_{cd}{\\epsilon}_{db}$ we arrive at\n\\begin{equation}\\label{derivedrelation}\n\\kappa^D_{ab}(ji)\\partial_b\\phi(i)={\\epsilon}_{ab}\\partial_b\\theta(j).\n\\end{equation}\nEquations (\\ref{Phi:definition}) and (\\ref{derivedrelation}) establish a duality relation $\\theta \\leftrightarrow \\phi$ and $\\kappa \\leftrightarrow \\kappa^D$. We make use of this duality in the calculation of $K_{xx}$.\nThe response $K_{xx}$ is obtained by applying a phase difference ${\\Delta}\\theta_x$ along the $x$ direction and measure the resulting current $J_x$, equivalent to a difference in $\\phi$ in the transverse direction, ${\\Delta} \\phi_y=\\int dy J_x$. The response is then $K_{xx}={\\Delta} \\phi_y\/{\\Delta}\\theta_x$.\n\nUsing Eqn.~(\\ref{derivedrelation}) we deduce that $\\partial_x\\theta=-\\kappa_{yy}^D\\partial_y \\phi$, which allows us to apply relation (\\ref{Kyy}) with $\\theta \\leftrightarrow \\phi$ and $\\kappa\\leftrightarrow \\kappa^D$. Doing so, we obtain the response ${\\Delta} \\theta_x\/{\\Delta}\\phi_y=-\\int dx dx' \\kappa^D_{yy}=K_{xx}^{-1}$. This gives the result\n\\begin{equation}\\label{Kxx}\nK_{xx}[\\kappa] = \\frac{1}{\\int dx\\int dx'\\kappa^{-1}_{xx}(x,x')}.\n\\end{equation}\nWhen the stripes are macroscopic we can take the response functions to be translationally invariant within a stripe. Then (\\ref{Kxx}) reduces to the well known fact that  the stiffness of macroscopic objects in series adds like resistors in parallel.\n\n\nThe Superfluid stiffness is now expressed in terms of the local response Kernel which can be computed using the standard Kubo formalism. The diamagnetic contribution of the response to a transverse vector potential is\n\\begin{equation}\\label{kernel:dia}\n \\kappa_{aa}^{\\rm dia} (x,x',q_y {\\rightarrow} 0)=e^2\\delta_{xx'}\\langle -K_a(x,q_y {\\rightarrow} 0)\\rangle.\n \\end{equation}\n where $K_a(j)=-t(j)\\sum_{\\sigma}( c_{j+\\hat{x}_a,{\\sigma}}^\\dag c_{j,{\\sigma}} + c_{j,{\\sigma}}^\\dag c_{j+\\hat{x}_a,{\\sigma}})$\n  is the local kinetic energy operator. The paramagnetic contribution is\n \\begin{eqnarray}\\label{kernel:para}\n \\mathcal{\\kappa}_{ab}^{\\rm para}(x,x',q_y,i\\omega_n)&=&-\\int_0^{\\beta} d\\tau e^{i\\omega_n \\tau}\\times\\\\\n &\\times& \\langle T_\\tau j_a^p(x,q_y ,\\tau)  j_b^p(x',-q_y,0)\\rangle \\nonumber,\n \\end{eqnarray}\n in the limit  $\\omega=0, q_y\\ra0$. Here $ j_a^p$ is the paramagnetic current operator:\n\\begin{equation}\n  j_a^p(j)=it(j) \\sum_{\\sigma}\\left( c_{j+\\hat{x}_a,{\\sigma}}^\\dag c_{j,{\\sigma}} - c_{j,{\\sigma}}^\\dag c_{j+\\hat{x}_a,{\\sigma}} \\right).\n \\end{equation}\n In order to calculate (\\ref{kernel:dia}) and (\\ref{kernel:para}) we diagonalize the Hamiltonian using the Bogoliubov transformation\n \\begin{eqnarray}\\label{bogoliubov}\nc_{j{\\uparrow}}^\\dag(k_y) &=& \\sum_{\\alpha} u^*_{{\\alpha}}(j,k_y){\\gamma}_{\\alpha}^\\dag(k_y) \\nonumber\\\\\n  c_{j{\\downarrow}}(-k_y) &=& \\sum_{\\alpha} v^*_{{\\alpha}}(j,k_y){\\gamma}_{\\alpha}^\\dag(k_y).\n \\end{eqnarray}\n We then solve the self consistent equations (\\ref{eqn:selfconsistent}) and express $K_a(j)$ and $ j_a^p$ in the new basis.\n As an example, the $x$ component of the static paramagnetic response kernel is\n\\begin{eqnarray}\\label{xx-kernel:result}\n\\kappa_{xx}^{\\rm para}(j,j', q_y)&=&\\sum_{k_y{\\alpha}{\\beta}} \\mathcal{J}_{{\\alpha}{\\beta}}^{(x)}(j,k_y,q_y) \\mathcal{J}_{{\\beta}{\\alpha}}^{(x)}(j',k_y,-q_y)\\times\\nonumber\\\\&\\times&\\mathcal{P} \\left[ \\frac{f(E_{{\\alpha},k_y})-f(E_{{\\beta},k_y-q_y})}{E_{{\\alpha},k_y}-E_{{\\beta},k_y-q_y}}\\right]\n\\end{eqnarray}\nwhere $\\mathcal{P}$ denotes the principal part, $E_{{\\alpha},k_y}$  are the eigen-energies, $f({\\epsilon})=[1+e^{{\\beta}{\\epsilon}}]^{-1}$ is the Fermi-Dirac distribution, and\n\\begin{eqnarray}\n\\mathcal{J}^{(x)}_{{\\alpha}{\\beta}}(j,k,q)&\\equiv& i t(j)\n\\bigl[\nu^*_{{\\alpha}}(j+1,k)u_{{\\beta}}(j,k-q) \\nonumber\\\\\n&+&v^*_{{\\alpha}}(j+1,k)v_{{\\beta}}(j,k-q)\n\\bigr]\n + \\rm{c.c.}\\nonumber\n\\end{eqnarray}\nTo solve the periodic problem we introduce an additional superlattice momentum, whose index is suppressed here for simplicity.\n\nFinally, the superfluid stiffness $\\rho_s(T)=\\sqrt{K_{xx}K_{yy}}$ is computed from the response kernels $\\kappa_{aa}(x,x')= \\kappa_{aa}^{\\rm dia}(x,x') + \\kappa_{aa}^{\\rm para}(x,x')$ using Eqns. (\\ref{Kyy}) and (\\ref{Kxx}).\n\n\n\n\\subsection{Results and Discussion}\\label{subsec:results1}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics{DOSrhoTc_eps2xfig}\n\\end{center}\n\\caption[stiffness and Tc - stripes]{ (a) Temperature dependent superfluid stiffness of a striped versus a homogeneous superconducting layer. In this example the width of metallic stripes (doping $p_2=0.35$) in this example is $d=3$ and that of the superconducting stripes ($J=t\/3$ and doping $p_1=0.1$) is $l=17$ (in lattice constants). The critical temperature estimated from the Kosterlitz-Thouless criterion, is seen to be higher in the homogeneous system (circle marks).\n(b) Zero temperature stiffness vs the relative size of the metallic segments, $d\/L$ for  a fixed $L=20$. The diamagnetic (dashed) and paramagnetic (dot-dashed) contributions to the stiffness $K_{yy}$ are plotted for comparison. All results are normalized by the zero temperature stiffness $\\rho_{\\rm SC}$ of the uniform superconductor. (c) Critical temperature, normalized by the uniform value, vs $d\/L$ for the same parameters as in (b) and different unit cells $L$ (in units of the lattice spacing $a$). (d) The density of states for a system with the same parameters as (a). At low energies the density of states (DOS) of the striped system (black) is identical to the DOS of a uniform underdoped superconductor of the same size (gray). The dashed line denotes the calculated slope $d\\nu\/dE$ at $E=0$ for the uniform superconductor. }\n\\label{fig:rho_Tc}\n\\end{figure*}\n\n\nWe first discuss the superfluid stiffness at zero temperature, and then turn to an analysis of the temperature dependence of the stiffness, in order to estimate the critical temperature of the striped system.\n\n The superfluid stiffness at zero temperature $\\rho_s(T=0)$ is plotted in Fig.~\\ref{fig:rho_Tc}(b)\nas function of the relative width of the metallic stripes, $\\zeta=d\/(d+l)$ at fixed width $l+d=20a$ ($a $ is the lattice spacing). The doping levels of the superconducting and the metallic stripes are $p_1=0.1$ and $p_2=0.35$ respectively.\nNote the enhancement of the stiffness compared with the uniform superconductor $\\rho_{\\rm SC}$ which is maximized for $\\zeta\\sim 0.5$.\n\n\nThe enhancement of the zero temperature stiffness and the optimal volume fraction $\\zeta$ are determined by the interplay of two effects. First, the metallic regions are gapped by the proximity effect, and contribute their large number of carriers  to the diamagnetic superfluid density $K_{yy}^{\\rm dia}$ which increases with $\\zeta$. However, this increase is partially countered by a zero temperature paramagnetic term $K_{yy}^{\\rm para}$ special to inhomogeneous superconductors. A similar effect was noted by us in a bilayer heterostructure \\cite{Goren:2009}.\n\nTo see if the moderate net increase of the zero temperature stiffness will facilitate enhancement of the transition temperature we compute the full temperature dependence of the stiffness. As an example Fig.~\\ref{fig:rho_Tc}(a) shows the result for a specific ratio $d\/(d+l)=3\/20$ with $p_1=0.1$ and $p_2=0.35$.\n The macroscopic stiffness $\\rho_s$ is seen to decrease approximately linearly with temperature, as in a uniform $d$-wave superconductor, but with a larger slope $d\\rho_s\/dT$. As a result, the transition temperature, determined using the Kosterlitz-Thouless criterion $\\rho_s(T_c)=2T_c\/\\pi$, is found to be lower in the inhomogeneous layer despite the increased stiffness at zero temperature. This remains the case in all possible stripe geometries, as shown in Fig.~\\ref{fig:rho_Tc}(c).\n\nThe slope $d\\rho_s\/dT$ is affected by two main factors: the first is the density of states (DOS) of low energy quasiparticles that carry the paramagnetic current and the second is the effective charge of these quasiparticle (or the effective current renormalization). The DOS of the system is plotted in Fig.~\\ref{fig:rho_Tc}(d). Below a threshold energy of $E\\sim 0.05 t$ the DOS is the same as in the uniform superconductor of $p_1=0.1$. This agrees with experimental results in inhomogeneous cuprate superconductors \\cite{Howald:2001}. Note that the limit of very narrow metallic stripes ($d=3a$) preserves the low energy DOS up to a relatively high energy, compared with the critical temperature. As the metallic stripes get wider, there are more low energy states that contribute to the reduction of the stiffness with the temperature.\n\n\n Despite the fact that the low energy DOS is the same as in the uniform superconductor, the slope $|d\\rho_s\/dT|$ is still larger than in the uniform case. This is a consequence of the difference in the effective charge of quasiparticles in the two systems: In the underdoped superconducting regions the quasiparticle charge is renormalized down by a factor of $g_t=2p_1\/(1+p_1)$, whereas in the metallic regions there is no such renormalization and the current is carried by electrons. As a result, at finite temperature the stiffness reduction in the inhomogeneous system is steeper than in the uniform underdoped superconductor.\n\n Here we should note again that, in general, the renormalization of the current carried by a quasi-particle that enters the low temperature dependence of the stiffness is a Fermi-liquid parameter that may be  renormalized compared to the SBMFT value of $g_t=2p_1\/(1+p_1)$. Indeed measurements of the temperature dependent stiffness give a renormalization factor is independent of doping over a wide range of doping in contradiction to the mean field prediction. However for an inhomogeneous system there is no unambiguous way to replace the mean field value of the current renormalization by a single phenomenological parameter. Moreover the existence of a length scale $d$ of the superconducting regions may introduce a cutoff that prevents this parameter from flowing far from its mean field value at low energies.\n\nThe stripes model shows that doping inhomogeneity on a mesoscopic scale can lead to an increased superfluid stiffness at zero temperature. This is a consequence of a proximity gap that opens in the metallic stripes which then contribute their high carrier density to the stiffness. On the other hand the metallic stripes also give rise to low energy states that  hasten the reduction of stiffness with temperature. In addition there is a paramagnetic reduction of the stiffness even at zero temperature due to the impurities. For these reasons the transition temperature of the striped system is found to be always lower than that of the homogeneous system.\nThe highest $T_c$ is obtained for the narrowest metallic stripes because then the proximity coupling to the bulk is high and the Andreev bound state are only slightly below the gap.\nIt is therefore tempting to consider the case of even smaller metallic regions by reducing the length of the stripes in addition to their width to a microscopic scale. In the following section we examine a model that takes a step in this direction.\n\n\n\\section{Microscopic scale inhomogeneity}\\label{section:disorder}\n\n\\subsection{The Model}\\label{subsec:Model2}\nIn this section we consider a scenario in which the metallic regions embedded in the underdoped superconductor are nearly point like. We model these metallic impurities as cross vertices of the square lattice (see Fig. ~\\ref{fig:dis-sketch}) on which the average doping $p_2$  is higher than the bulk average $p_1$. The effective hopping matrix elements and the pairing amplitudes on these links naturally take different values than the bulk. Specifically, in the mean field model of Eqn.~(\\ref{HRMFT}) the parameters $g^t_{ij}$, ${\\Delta}_{ij}$ and $\\chi_{ij}$ take a different value on the impurity bonds.\n\nWe analyze two scenarios: in the first, the impurities are metallic, with doping $p_2>0.3$, such that on the impurity bonds $g_{ij}=1$ and ${\\Delta}_{ij}=\\chi_{ij}=0$. In the other scenario the excess doping on the impurity sites $p_2-p_1$ is small, leading to $g_{ij}=g(p_2)$ and ${\\Delta}_{ij}={\\Delta}(p_2)$ with the doping dependence of SBMFT. In this case we assume that $\\chi$, which has a very weak doping dependence, is uniform throughout the system.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{dis_sketch}\n\\end{center}\n\\caption{\\emph{Doping inhomogeneity on a microscopic scale.} An illustration of the model. Nearest neighbouring bonds to the impurity sites (solid lines) are characterized by enhanced hopping amplitude $gt+\\delta_t$ and reduced pairing ${\\Delta}_0+\\delta_{\\Delta}$, with respect to the superconducting background of $gt$ and ${\\Delta}_0$ respectively.}\n\\label{fig:dis-sketch}\n\\end{figure}\n\nThe Hamiltonian $H=H_0+H_{\\rm imp}$ consists of a uniform part and an impurity contribution. Written in momentum space, the uniform Hamiltonian is the Fourier transform of (\\ref{HRMFT}),\n\\begin{equation}\nH_0=\\sum_{\\bf k} \\Psi_{\\bf k}^\\dag (\\xi_{\\bf k} {\\sigma}_3 + {\\Delta}_{\\bf k} {\\sigma}_1)\\Psi_{\\bf k}.\n\\end{equation}\nHere $\\Psi_{\\bf k}^\\dag= \\{c_{{\\bf k }{\\uparrow}}^\\dag, \\  c_{{\\bf -k }{\\downarrow}}\\}$, ${\\sigma}_a$ are Pauli matrices, $\\xi_{\\bf k}=-2t_{\\rm eff} (\\cos{k_x}+\\cos{k_y})-\\mu$ and ${\\Delta}_{\\bf k}={\\Delta}_0 (\\cos{k_x}-\\cos{k_y})$ with $t_{\\rm eff}=g(p_1)t+\\chi$. The impurity Hamiltonian is\n\\begin{equation}\\label{disorderpotential}\nH_{\\rm imp}=\\sum_{\\bf kk'} \\Psi_{\\bf k}^\\dag  \\rho_{\\bf k'-k} \\left[{U}_{\\bf kk'}{\\sigma}_3  + {V}_{\\bf kk'}{\\sigma}_1 \\right] \\Psi_{\\bf k'}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nU_{\\bf kk'}&=&-2\\delta_t (\\cos{k_x}+\\cos{k_y}+\\cos{k'_x}+\\cos{k'_y})\\nonumber\\\\\nV_{\\bf kk'}&=&\\delta_{\\Delta}(\\cos{k_x}-\\cos{k_y}+\\cos{k'_x}-\\cos{k'_y}) \\nonumber\\\\\n \\hat{\\rho}_{{\\bf k'}-{\\bf k}}  &=& \\sum_{\\alpha} e^{-i({\\bf k'}-{\\bf k}){\\bf r}_{\\alpha}}\n\\end{eqnarray}\nHere ${\\delta}_{\\Delta}={\\Delta}(p_2)-{\\Delta}(p_1)$, and $\\{{\\bf r}_{\\alpha}\\}$ are the impurity sites. The excess hopping at the impurity sites is ${\\delta}_t=t_{\\rm eff}(p_2)-t_{\\rm eff}(p_1)$. In the case of metallic impurities we set $t_{\\rm eff}(p_2)=t$. \n\nThe above terms result from the shift in doping level from $p_1$ to $p_2$ near the impurity. We should in principle include also the direct impurity potential, which caused the change in hole concentration. Such a potential that acts locally on the impurity as $U({\\bf r})=U_0{\\delta}({\\bf r}-{\\bf r}_{\\bf{\\alpha}})$, can be regarded as a $k$ independent contribution to $U_{\\bf kk'}$. \nThe magnitude of this term can be estimated from the observed change in hole concentration through $U_0 \\simeq (p_2-p_1)\/\\kappa$, where $\\kappa$ is the local compressibility. \nWe omit this term from the calculations described below. Then, at the end of  Sec.~\\ref{subsec:results2} we quantify the contribution of the direct impurity potential and explain why it can be neglected.\n\nOur goal is to compute the temperature dependent superfluid stiffness and estimate the transition temperature of the inhomogeneous layer compared to a uniform layer. To this end we use the Born approximation to perform the disorder average.\nThis is strictly valid in the limit of dilute uncorrelated impurities and weak disorder. We expand to first order in the impurity concentration $n_i$ and second order in the strength of a single impurity $\\delta_t\/t_{\\rm eff}$ and $\\delta_{\\Delta}\/{\\Delta}_0 $. In practice we will allow ${\\delta}_\\Delta\/{\\Delta}_0$ to be close to $-1$ which is the case when the overdoped inclusions are already in or close to the metallic regime.\n\n\n\n\n\n\\subsection{Calculation of the Superfluid Stiffness}\\label{subsec:calculation2}\n\n\nThe stiffness is the linear response of the system to an externally applied vector potential ${\\bf A}({\\bf r})$.\nIn order to calculate it in the disordered system, it is convenient to resort to the real space Hamiltonian (\\ref{HRMFT}) and include a vector potential through a Peierls substitution, $g_{ij}t{\\rightarrow} g_{ij}t\\exp{[ieA_{ij}]}=g_{ij}t\\exp{[ieA_x({\\bf r}_i)]}$ in the case of a vector potential in the $x$ direction.\nFor the linear response calculation we expand the Hamiltonian to second order in $A_x$ \\cite{Scalapino:1993},\n\n\n\\begin{equation}\nH(A_x)=H(0)-\\sum_{\\bf r} \\left[e j_x({\\bf r})A_x({\\bf r}) + \\frac{e^2}{2}K_x({\\bf r}) A_x^2({\\bf r})\\right]\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{jK}\nj_x({\\bf r})&=&i\\sum_{{\\bf r},{\\sigma}} t_x({\\bf r})(c_{{\\bf r}+x,{\\sigma}}^\\dag c_{{\\bf r},{\\sigma}}-c_{{\\bf r},{\\sigma}}^\\dag c_{{\\bf r}+x,{\\sigma}})\\nonumber\\\\\nK_x({\\bf r})&=&-\\sum_{{\\bf r},{\\sigma}} t_x({\\bf r})(c_{{\\bf r}+x,{\\sigma}}^\\dag c_{{\\bf r},{\\sigma}}+c_{{\\bf r},{\\sigma}}^\\dag c_{{\\bf r}+x,{\\sigma}})\\nonumber.\n\\end{eqnarray}\nHere $ t_x({\\bf r})=g(p_1)t+\\delta_t^{c}\\sum_{\\alpha} {\\delta}({\\bf r}-{\\bf r}_{\\alpha})$ is the coupling to the external vector potential, in the presence of the modified bonds around sites ${\\bf r}_{\\alpha}$. The excess local current on the impurity sites is $\\delta_t^{c}=t[g(p_2)-g(p_1)]$. In the case of highly overdoped impurities ($p_2>0.3$) we take $g(p_2)=1$. Note that this impurity contribution is different from ${\\delta}_t$ that appears in the impurity Hamiltonian (\\ref{disorderpotential}). The reason is that the external vector potential couples only to the hopping $g_{ij}t$, and not to the Fock term proportional to $\\chi$, which originated from the magnetic exchange interaction.\n\nThe superfluid stiffness is now given by\\cite{Scalapino:1993}\n\\begin{equation}\n\\rho_s=\\overline{\\langle -K_x \\rangle} +\\lim_{{\\bf q}{\\rightarrow} 0} \\overline{\\Pi_{xx}}({\\bf q},\\omega=0)\n\\end{equation}\nwhere $\\overline{X}$ denotes the average over disorder realizations and,\n\\begin{eqnarray}\n \\overline{\\Pi_{xx}}({\\bf q},\\omega_n)\\! &\\!=\\!&\\!-\\!\\int_0^{\\beta} d\\tau e^{i\\omega_n\\tau} \\overline{\\langle j_x({\\bf q},\\tau) j_x({\\bf -q'},0)\\rangle} \\nonumber\n\\end{eqnarray}\nNote that after disorder averaging the RHS is proportional to $\\delta_{\\bf qq'}$.\nThe different contributions to $\\rho_s$ are presented as diagrams in Fig.~\\ref{fig:diagrams}, where we denote diamagnetic terms by $D_\\alpha$ and paramagnetic terms by $\\Pi_{\\alpha}$.\n\n\nOne type of correction to the stiffness stems from standard renormalization of the electron self-energy by the impurities. Such corrections are given by diagrams $D_0$ and $\\Pi_0$ in Fig.~\\ref{fig:diagrams}. Similar terms would arise in the common case of point (on-site) impurities. We note that the vertex correction $\\Pi_1$ vanishes due to inversion symmetry of the impurity potential.\n\nA second type of correction to the stiffness is special to the bond disorder we consider here. The disorder in the hopping amplitude introduces modulations in the local current operator and kinetic energy, proportional to ${\\delta}_t$. This causes a direct renormalization of the coupling to the external vector potential, as represented in diagrams $D_1, D_2, \\Pi_2,\\Pi_3$ and $\\Pi_4$ in Fig.~\\ref{fig:diagrams}.\\\\\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics{selfenergy_diagrams}\n\\end{center}\n\\caption{\\emph{Diagrammatic calculation of the superfluid stiffness in the disordered system.} Top: diamagnetic contributions. Middle: Paramagnetic contributions. Bottom: Green's function renormalization and the definition of the self energy within Born approximation. A dashed line corresponds to a scattering event and $\\times$ denotes a single impurity. Note that the scattering is a matrix in Nambu space.}\n\\label{fig:diagrams}\n\\end{figure}\n\n {\\bf Self energy corrections.} The disorder in the hopping and pairing strength introduces renormalizations to the spectrum parameters or to the electronic Green's function, which in turn, affect the superfluid stiffness. Such corrections are represented by diagrams $D_0$ and $\\Pi_0$ in Fig.~\\ref{fig:diagrams}. In order to calculate these diagrams we first compute the renormalized Green's function using the Born approximation.\n\nThe Dyson equation for the disorder averaged Green's function is depicted in Fig.~\\ref{fig:diagrams} and given by\n\\begin{equation}\nG_{{\\bf k},\\omega_n}^{-1}=(G^0_{{\\bf k},\\omega_n})^{-1} + \\Sigma_{{\\bf k},\\omega_n},\n\\end{equation}\nwhere the bare Green's function is\n\\begin{equation}\n(G^0_{{\\bf k},\\omega_n})^{-1}  = i\\omega_n{\\sigma}_0 - \\xi_{\\bf k} {\\sigma}_3 - {\\Delta}_{\\bf k}{\\sigma}_1,\n\\end{equation}\nand $\\Sigma_{{\\bf k},\\omega_n}\\equiv \\sum_{a=0}^3\\Sigma_a{\\sigma}_a $ is the self-energy after disorder averaging.\nTo calculate the self energy explicitly in the limit of small $\\omega_n$, we use the fact that the main contributions arise from the vicinity of the nodal points ${\\Delta}_{\\bf k}=\\xi_{\\bf k}=0$. We expand around these points and solve self consistently for the decay rate $\\Sigma_0$ in the limit of $\\omega\\ra0, {\\bf k}{\\rightarrow} {\\bf k}_{\\rm node}$ similarly to Ref.~\\onlinecite{Lee:1993}. The high energy cutoff for this approximation is defined as $p_0$. This calculation gives (see Appendix \\ref{app:selfenergy}) ,\n\\begin{eqnarray}\\label{selfenergy}\n\\Sigma_0&\\simeq& -ip_0 e^{-\\frac{2 \\pi v_f v_{\\Delta} t_{\\rm eff}^2}{n_i(2\\mu\\delta_t)^2}}\n\\\\\n\\Sigma_1({\\bf k})&\\simeq& 2n_i \\frac{\\delta_{\\Delta}}{{\\Delta}_0} \\left(1-\\eta\\right){\\Delta}_{\\bf k}\\nonumber\\\\\n\\Sigma_3({\\bf k})&\\simeq&\\nonumber  2n_i \\frac{\\delta_t}{t_{\\rm eff}}\\left(1-\\eta\\right)\\xi_{\\bf k} + \\delta_\\mu \\nonumber\n\\end{eqnarray}\nwhere $\\eta\\equiv S_1 {\\delta_{\\Delta}}\/{{\\Delta}_0} +(1-S_1){\\delta_t}\/{t_{\\rm eff}} $, $S_1\\equiv \\sqrt{2{\\Delta}_0\/t_{\\rm eff}}\/\\pi=2\\sqrt{v_{\\Delta}\/ v_f}\/\\pi$, and $\\delta_\\mu$ is a ${\\bf k}$ independent constant that renormalizes the chemical potential.\n\n\nThe low energy limit of $\\Sigma_0$ is exponentially small close to zero doping, and is further suppressed by the large number $t_{\\rm eff}^2\/(n_i \\delta_t^2)$. We solve  for the other components of the self energy under the self consistent assumption that any ${\\bf k}$ dependence  of $\\Sigma_0$ is negligible and indeed get that the entire effect of the decay rate $i \\Sigma_0$ is negligible. For more details about the calculation the reader should turn to Appendix \\ref{app:selfenergy}.\n\nIn the absence of decay, no zero energy states are introduced by the disorder. The effect of $\\Sigma_1$ and $\\Sigma_3$ is to renormalize the gap and the hopping, leading to a corrected spectrum $\\tilde{E}_{\\bf k}=[\\tilde{\\xi}_{\\bf k}^2 + \\tilde{{\\Delta}}_{\\bf k}^2]^{1\/2}$. In the low energy limit this is equivalent to a renormalization of the effective values of $v_f$ and $v_{\\Delta}$ which we find to be,\n\\begin{eqnarray}\\label{vfvd-ren}\n\\tilde{v}_f &=&v_f\\left[1+ 2n_i \\frac{\\delta_t}{t_{\\rm eff}}\\left(1-\\eta\\right) \\right]\\nonumber\\\\\n\\tilde{v}_{\\Delta}&=& v_{\\Delta} \\left[1+2n_i \\frac{\\delta_{\\Delta}}{{\\Delta}_0} \\left(1-\\eta\\right)\\right].\n\\end{eqnarray}\nThe renormalization of $v_\\Delta$ is the anti-proximity effect due to the metallic inclusions, which gives rise to a modified coefficient of the linear DOS compared with the pure system. These modifications primarily affect the low temperature physics in the disordered system.\n\n\nWith the Green's function at hand we can calculate the leading contributions to the superfluid stiffness. Details of the calculations appear in Appendices \\ref{app:diamagnetic} and \\ref{app:paramagnetic}.\nThe contributions to the superfluid stiffness, to second order in the disorder strength, can be separated into zero temperature and finite temperature contributions.\n\n\\emph{  Zero temperature.--} The contribution to the zero temperature stiffness due to self energy corrections is the diamagnetic response expressed in diagram $D_0$. This is a non-universal contribution which turns out to differ only very slightly from the bare diamagnetic stiffness of the pure system (see Appendix \\ref{app:diamagnetic} for details),\n\\begin{eqnarray}\\label{D0}\nD_0 &=& 2gt \\sum_{\\bf k} \\cos{k_x} \\left(1-\\frac{\\tilde{\\xi}_{\\bf k}}{\\sqrt{\\tilde{\\xi}^2_{\\bf k} + \\tilde{\\Delta}^2_{\\bf k}} }\\right)\\nonumber\\\\\n&=&gt \\mathcal{D}_0\\left(\\frac{\\tilde{v}_{\\Delta}}{\\tilde{v}_f}\\right)\n\\end{eqnarray}\nwhere $\\mathcal{D}_0(X)$ is an order unity slowly decreasing function of its argument in the relevant range of parameters. Note that in practice, this function may include a weak dependence on the chemical potential which we neglect, assuming low doping.\nTo conclude, the renormalization of the spectrum parameters due to the self energy corrections have a negligible effect on the diamagnetic stiffness.\\\\\n\n\\emph{  Finite temperature.--} The finite temperature contribution to the stiffness due to self energy corrections arises from diagram $\\Pi_0$. The effect of disorder here is to modify the low energy density of states through a renormalization of the effective values of $v_f$ and $v_{\\Delta}$.  This affects the superfluid stiffness through the paramagnetic contribution $\\Pi_0$ leading to faster reduction of the stiffness with temperature. More precisely\n\\begin{eqnarray}\\label{Pi0}\n\\Pi_0 &=& - \\frac{2\\log{2}}{\\pi }\\ \\frac{8(Zt)^2}{\\tilde{v}_f \\tilde{v}_{\\Delta}}\\ T\\\\\n&\\simeq&-\\left[1-4n_i \\left(\\frac{\\delta_{\\Delta}}{{\\Delta}_0} + \\frac{\\delta_t}{t_{\\rm eff}} \\right)\\left( 1-\\eta\\right)\\right]b_0 T\\nonumber.\n\\end{eqnarray}\nHere $b_0=-d\\rho_s\/dT$ is the slope in the clean system and $Z$ is the renormalization of the quasiparticle current by interactions. The low $T$ behavior is dominated by low energy quasiparticles, which may be altered by Fermi-liquid renormalization not included in the mean field theory. Therefore $Z$ should be taken as a phenomenological Fermi-liquid parameter\\cite{Millis:1998,Ioffe:2002} and not as the value $g(p)$ dictated by the microscopic mean field theory.\n\nIn our case the disorder acts to induce faster decrease of the superfluid stiffness with temperature. This is because when the inclusions are highly overdoped with nearly zero gap then ${\\delta}_{\\Delta}\/{\\Delta}_0\\gtrsim -1$, while $0<{\\delta}_t\/t_{\\rm eff}\\ll1$.\\\\\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{Tcrhophase_metal}\n\\end{center}\n\\caption{\\emph{Superfluid stiffness and critical temperature in the inhomogeneously doped layer with point impurities. } In panels (a) and (b) the impurities have an average doping charge $p_2=0.35$ while the bulk doping $p_1$ varies, with impurity concentration $n_i=0.1$. (a) Zero temperature stiffness (Solid) compared to that of the clean case (dashed). The Dashed-dotted line marks the diamagnetic contribution. (b) $T_c$ with impurities compared to $T_c^{0}$ without.  We used a doping independent quasiparticle current renormalization of $Z=0.5$. Results are plotted only within the validity range of the diagrammatic expansion. (c) Relative change in $T_c$ for the case of small excess doping on the impurity, with impurity concentration $n_i=0.2$. Contours map the relative change $T_c\/T_c^{0}$ as a function of the base doping $p_1$ and the excess doping on the impurities $p_2-p_1$.}\n\\label{fig:disorder_rhoTc}\n\\end{figure}\n\n\n\n {\\bf Current operator renormalizations.} The second type of corrections to the stiffness have no counterpart in systems with standard on-site disorder. The disorder in the hopping amplitude introduces renormalizations of the kinetic energy $K_x$  and the current operator $j_x$, proportional to $\\delta_t^{c}$. This leads to corrections of $O(\\delta_t^{c}), O[(\\delta_t^{c})^2]$ to the stiffness, that are represented as vertex corrections in diagrams $D_1,D_2,\\Pi_2,\\Pi_3,\\Pi_4$ of Fig.~\\ref{fig:diagrams}. We again distinguish between zero temperature and finite temperature contributions to the stiffness.\\\\\n\n\n\n\\emph{  Zero temperature.--}\nThe most intuitive effect of the vertex correction is the increase of the diamagnetic stiffness at the impurity sites due to the extra charge carriers they contribute. This effect is reflected in the diagram $D_1$ with each impurity bringing an additional $2\\delta_t^{c}$ to the average kinetic energy\n\n\\begin{eqnarray}\\label{D1}\nD_1 &=& 4n_i\\delta_t^{c} \\sum_{\\bf k} \\cos{k_x} \\left(1-\\frac{{\\xi}_{\\bf k}}{\\sqrt{{\\xi}^2_{\\bf k} +{\\Delta}^2_{\\bf k}} }\\right)\\nonumber\\\\\n&=&2 n_i\\delta_t^{c} \\mathcal{D}_0\\left(\\frac{v_{\\Delta}}{v_f}\\right)\\simeq 2 n_i\\frac{\\delta_t^{c}}{gt} \\rho_0.\n\\end{eqnarray}\nThis expression reveals a small parameter, $n_i{\\delta}_t^{c}\/gt $, that did not appear in the Hamiltonian.\nThe perturbative correction inevitably becomes large upon underdoping towards the Mott insulator where $n_i{\\delta}_t^{c}\/gt \\to \\infty$. This signals the breakdown of the Born approximation at doping levels smaller than $p_1^*\\simeq g(p_2) n_i\/(1+2n_i)$.\n\nThe second significant contribution to the zero temperature stiffness stems from the paramagnetic diagram $\\Pi_4$, which is seen to be\n \\begin{eqnarray}\\label{Pi4}\n\\Pi_4 &=& - 2 n_i \\frac{(\\delta_t^{c})^2}{t_{\\rm eff}}\\mathcal{P}_0\\left(\\frac{v_{\\Delta}}{v_f}\\right).\n\\end{eqnarray}\nHere $\\mathcal{P}_0(X)$ is an order unity decreasing function of its argument in the relevant parameter range. This term is closely analogous to the zero temperature paramagnetic reduction in the stripe model of sec.~\\ref{section:stripes}. Here as in the stripe model, The effect acts to moderate the  enhancement of the stiffness at zero temperature.\n\nAnother correction to the zero temperature stiffness is given by the diagram $D_2$. This diagram, which represents a combined renormalization of the vertex and the spectrum, is calculated to be\n\\begin{eqnarray}\\label{D2}\nD_2 &=&4n_i\\delta_t^{c} \\left[\\frac{{\\delta} D}{{\\Delta}_0} \\mathcal{D}_1\\left(\\frac{v_{\\Delta}}{v_f}\\right) +  \\frac{\\delta_t}{t_{\\rm eff}} \\mathcal{D}_2\\left(\\frac{v_{\\Delta}}{v_f}\\right)\\right]\\nonumber.\n\\end{eqnarray}\nHere $\\mathcal{D}_1(X)$ is an increasing function and $\\mathcal{D}_2(X)$ is slowly decreasing, and their weak dependence on the chemical potential is again neglected.\nThis diagram turns out to give a negligible numerical contribution to the overall stiffness.\n\n\\emph{Finite temperature.--}\nThe finite temperature contributions to the stiffness that arise from current renormalization are shown in diagrams $\\Pi_2$ and $\\Pi_3$. An explicit calculation gives\n\\begin{eqnarray}\\label{Pi2Pi3}\n\\Pi_2 &=& - 4 n_i \\frac{\\delta_t^{c}}{gt}\\frac{2\\log{2}}{\\pi }\\ \\frac{(2\\sqrt{2}Zt)^2}{\\tilde{v}_f \\tilde{v}_{\\Delta}}\\ T\\nonumber\\\\\n\\Pi_3 &=& 4 n_i\\  \\eta\\  \\frac{\\delta_t^{c}}{gt}\\frac{2\\log{2}}{\\pi }\\ \\frac{(2\\sqrt{2}Zt)^2}{\\tilde{v}_f \\tilde{v}_{\\Delta}}\\ T.\n\\end{eqnarray}\nWithin SBMFT the current renormalization $Z=g(p)$ depends strongly on the doping. However, it is known that this strong doping dependence leads to a disagreement with the experimentally measured slope $d\\rho_s\/dT$, which is seen to be almost independent of doping.\\cite{Wen:1998}\n\nHere we adopt a phenomenological approach, with an effective paramagnetic current renormalization $Z$ which is independent of doping.\\cite{Paramekanti:2002, Millis:1998} This holds at finite low temperature, when the physics is dominated by the effective theory of low energy Dirac quasiparticles. In this case, the entire contribution $\\Pi_2+\\Pi_3$ is negligible because it stems precisely from the difference in the local current operator between the $p_1$ superconductor and the $p_2$ impurities. Therefore, when summing up the finite $T$ contributions to the stiffness we neglect these two diagrams.\n\n\n\n\n\\subsection{Results and Discussion}\\label{subsec:results2}\nWe can summarize the results of this section by putting together the various contributions to the superfluid stiffness. This gives the temperature dependent stiffness\n\\begin{equation}\n\\rho_s(T)=\\rho_0-b_0 T +2 n_i({\\delta}\\rho_s(0)-{\\delta} b\\  T)\n\\label{rho-final}\n\\end{equation}\nHere the first two terms constitute the usual expression for the temperature dependent superfluid stiffness of a uniform $d$-wave superconductor\\cite{Lee:1997}, as reviewed in sec. \\ref{section:overview}.\nThe second term is\n\\begin{equation}\n{\\delta}\\rho_s(0)=\\frac{\\delta_t^{c}}{gt}\\rho_0-\\frac{(\\delta_t^{c})^2}{t_{\\rm eff}}\\mathcal{P}_0\\left(\\frac{v_{\\Delta}}{v_f}\\right)\n\\end{equation}\nThe leading order correction to ${\\delta}\\rho_s(0)$ in the impurity strength is due to the added charge carriers donated by the impurities. The negative\nsecond order term is a paramagnetic correction to the zero temperature stiffness analogous to\nthe paramagnetic correction that we derived previously for a bilayer heterostructure. In the latter case\nthis correction was proportional to $(J_1-J_2)^2$, the square of the difference of the quasi-particle currents on the two layers. Here similarly this contribution scales as $(\\delta_t^{c})^2\\propto[g(p_2)-g(p_1)]^2$,\nwhich is the square of the difference between the local current renormalization in the bulk and near the impurity.\n\nThe last term in Eq. \\ref{rho-final} is the change of the linear reduction of the stiffness with temperature\ndue to the presence of impurities. It is given by\n\\begin{equation}\\label{stiffness:disorder}\\nonumber\n{\\delta} b=2b_0 \\left(-\\frac{\\delta_{\\Delta}}{{\\Delta}_0} - \\frac{\\delta_t}{t_{\\rm eff}} \\right)  \\left(1-S_1 \\frac{\\delta_{\\Delta}}{{\\Delta}_0} +(1+S_1) \\frac{\\delta_t}{t_{\\rm eff}} \\right)\n\\end{equation}\nwhere $S_1=2\\sqrt{v_{\\Delta}\/ v_f}\/\\pi$ and $b_0$ is the parameter for the uniform superconductor given in sec. \\ref{section:overview} $b_0=2\\log{2} (2\\sqrt{2}Zt)^2\/(\\pi v_f v_{\\Delta})$\\cite{Lee:1997}. Note that the expression in the first bracket is positive because ${\\delta}{\\Delta} <0$ on the impurities. Hence the superfluid stiffness is reduced faster as a function of temperature than in the uniform superconductor. We estimate the parameters of the uniform system using SBMFT, so that $v_f=2\\sqrt{2}t_{\\rm eff}$ and $v_{\\Delta}=\\sqrt{2}{\\Delta}$. Taking $J=t\/3$, the effective hopping and gap parameters are given by $t_{\\rm eff}(p)=g(p) t+\\chi$ and ${\\Delta}(p)=\\chi[1-4 p]$, where $\\chi$ is the value of the mean fields (both pairing and Fock field) at zero doping.\n\n\n\nFigure~\\ref{fig:disorder_rhoTc}(a) displays the calculated zero temperature stiffness as function of the doping $p_1$ with the impurities fixed to a high doping level $p_2=0.35$, which corresponds to zero pairing amplitude, and a hopping amplitude of $t$. We plot the total stiffness $\\rho_s(T=0)$ as well as the diamagnetic contribution $\\rho_{\\rm dia}=D_0+D_1+D_2$. Note that the diamagnetic contribution in the disordered system $\\rho_{\\rm dia}$ is significantly increased with respect to the pure case, $\\rho_0$. However, the total zero temperature stiffness $\\rho_s(T=0)$ is only moderately increased compared to the uniform case (where $\\rho_0$ is the total stiffness at $T=0$). The reason for this is the zero temperature paramagnetic contribution of the impurities $\\Pi_4$.\n\nIn panel (b) of the same figure we plot the critical temperature as a function of the bulk doping $p_1$, estimated from Eq. (\\ref{stiffness:disorder}) using the Kosterlitz-Thouless criterion $\\rho_s(T_c)=2T_c\/\\pi$. Again this is for a fixed value of $p_2=0.35$ and $n_i=0.1$ and the result is compared against $T_{c}^0$ of the pure system. The critical temperature of the disordered system is significantly enhanced, above the maximal $T_c$ of the clean superconductor. The maximum of $T_c$ is shifted to the underdoped regime. These results are reminiscent of experiments by Yuli et al\\cite{Yuli:2008} that show a $T_c$ enhancement in an underdoped-overdoped bilayer. We can relate our results to the experiment if we assume that the interface between the two layers is in fact an inhomogeneous mixture of underdoped and overdoped regions. Our results suggest that an optimal configuration for $T_c$ enhancement can be achieved by placing point-like metallic inclusions inside a slightly underdoped superconductor.\n\n Figure \\ref{fig:disorder_rhoTc}(c) shows the relative change in the critical temperature with respect to $T_c^{0}$ as function of $p_1$ and ${\\delta} p=p_2-p_1$, for $n_i=0.2$. The critical temperature is enhanced relative to the clean system by up to $15\\%$, in a broad range of $p_1$ and ${\\delta} p$. Here the excess doping on the impurities ${\\delta} p$ is small, and there is no enhancement of $T_c$ above the maximal $T_c$ of the clean superconductor. The main reason for this is the zero temperature paramagnetic reduction of the stiffness due to the impurities. Without this effect we could have obtained an absolute enhancement of $T_c$ in the disordered system, also in the small ${\\delta} p$ limit. We have checked and found that whether we use the microscopic or phenomenological parameter $Z$ to renormalize the quasiparticle current makes very little difference to the final result of $T_c$.\\\\\n\nIt is instructive to look at the behavior of the stiffness and $T_c$, for small values of ${\\delta} p\\equiv p_2-p_1$, for which we can neglect second order contributions in $\\delta_t\/t_{\\rm eff}$ and $|\\delta_{\\Delta}\/{\\Delta}_0|$, such as the paramagnetic effect. Here we use the SBMFT doping dependence for both the bulk and the impurity, such that $\\delta_t=\\delta_t^{c}=t[g(p_2)-g(p_1)]$. In this regime there is a simple expression for the superfluid stiffness,\n\\begin{equation}\n\\rho_s(T)\\simeq \\rho_{\\rm cl}(T)+ 2n_i\\frac{\\delta_t^{c}}{gt}\\rho_0-4n_i \\left(\\left|\\frac{\\delta_{\\Delta}}{{\\Delta}_0} \\right|- \\frac{\\delta_t}{t_{\\rm eff}}\\right)b_0 T\\nonumber\n\\end{equation}\nwhere $\\rho_{\\rm cl}(T)=\\rho_0-b_0T$ is the stiffness of the clean superconductor. The zero temperature stiffness is always enhanced, whereas the slope $|d\\rho_s\/dT|$ is increased. The latter is easily seen by expressing $|\\delta_{\\Delta}\/{\\Delta}_0|-\\delta_t\/t_{\\rm eff}$ as function of $p_1$ and ${\\delta} p$.\nUsing the Kosterlitz-Thouless criterion as above we can estimate the change in transition temperature $T_c$ with respect to the critical temperature $T_c^{0}$ of the clean superconductor,\n\\begin{equation}\n\\frac{T_c}{T_c^{0}}\\simeq1+ 2n_i \\left[\\frac{\\delta_t^{c}}{gt}+\\frac{2b_0}{b_0+2\/\\pi} \\left(\\frac{\\delta_t}{t_{\\rm eff}} -\\left|\\frac{\\delta_{\\Delta}}{{\\Delta}_0} \\right|\\right) \\right].\n\\end{equation}\nThis can be expressed in terms of the doping level $p_1$ of the clean superconductor and the difference in doping ${\\delta} p=p_2-p_1$ between the background and the impurities. We obtain an expression of the form\n\\begin{equation}\n\\nonumber\n\\frac{T_c}{T_c^{0}}\n=1+ 2n_i\n\\frac{{\\delta} p}{p_1}(p^*-p_1)\\mathcal{F}(p_1).\n\\label{Tcdp}\n\\end{equation}\nwhere $\\mathcal{F}(p_1)$ is positive for $p_1<0.25$. This implies that for $p_1<p^*$, the critical temperature of the disordered system is enhanced compared with the clean superconductor with doping $p_1$.\nUnder the assumptions of SBMFT, with $Z=0.5$ and $\\chi=0.4t$, we get $p^*\\sim0.125$ and\n\\begin{equation}\n\\mathcal{F}(p_1)\\simeq\\frac{(1\\!-\\!p_1)(0.08\\!+\\!p_1)(0.6+\\!p_1)}{(0.5\\!-\\!p_1)(0.25\\!-\\!p_1)(0.17\\!+\\!p_1)(0.29+p_1)}.\n\\end{equation}\nWe note that $\\mathcal{F}(0.1)\\simeq 18$.\n\nIt seems from Eq. (\\ref{Tcdp}) that $T_c$ can be further enhanced by increasing the impurity concentration. However, by doing\nthis we would quickly violate the Born approximation. In particular, the superfluid stiffness in this non perturbative regime should be calculated as the resistance of an effective resistor network with $\\rho_s^{-1}$ of the various puddles playing the role of the resistance.\\\\\n\nWe now remark on the nature of our perturbative approach and the small parameters involved in it. The scattering from individual impurities is taken into account within the Born approximation to second order in the impurity strength as measured by the parameters $\\delta_t\/t_{\\rm eff}$ and $|\\delta_{\\Delta}\/{\\Delta}_0|$. We found that the second order correction to both $\\rho_s$ and $T_c$ was always negligible compared to the first order contribution. This was the case even when we took the parameter $|{\\delta} {\\Delta}\/{\\Delta}|\\simeq 1$.\nAn additional small parameter $n_i{\\delta}_t^c\/gt$ appeared through the effect of the impurities on the coupling to the electromagnetic field rather than the scattering on the impurity potential.\n\nWe would like to contrast our approach with the commonly used self consistent $T$-matrix approximation (SCTMA)\\cite{comment-SCTMA}, which treats the single impurities exactly. This turns out to be important to describe the effect of in-plane ion substitutions such as Zn impurities that act as unitary scatterers and give rise to strong bound states. However in our case the SCTMA is not analytically solvable because of the strong momentum dependence of the impurity potential and the fact that it acts as a matrix in Nambu space (${\\delta}_t$ is the diagonal component and ${\\delta}_{\\Delta}$ off-diagonal). Fortunately, the disorder potential that interests us is generated by dopants, which reside outside the CuO plane.\\cite{comment-Yttrium} \\nocite{MacKenzie:1994} Indeed we can show that the fact that such impurities induces only a small change in the local doping level ($p_2-p_1\\ll 1$) implies that the impurity scattering is far from the unitary limit and does not give rise to a bound state. To see this consider\nthe strength of the local impurity potential $U_0\\simeq (p2-p1)\/\\kappa$. Since the compressibility $\\kappa$ is approximately the density of states at the Fermi level $\\nu_0$, the dimensionless impurity strength is just $U_0\\nu_0\\simeq p_2-p1\\ll 1$. This is far from satisfying the condition for formation of a bound state.\\cite{Balatsky:1995} In this limit the direct impurity scattering can be taken within the Born approximation and lead to negligible contributions to the low energy DOS.\\cite{Durst:2000,Sharapov:2002} Hence it leads to a concomitantly small correction to the stiffness. \n\n\\section{Conclusions}\\label{section:summary}\nWe investigated the effects of doping inhomogeneity on the superconducting properties of the cuprates using the slave boson mean field theory\\cite{Kotliar:1988} supplemented by phenomenological Fermi liquid parameters to account for the low energy quasiparticle properties\\cite{Ioffe:2002}. In particular the superfluid stiffness and the critical temperature was calculated within two different models of the inhomogeneity.\n\nThe first model described doping variations on mesoscopic scales, comparable to or larger than the superconducting coherence length. Technically we computed the transverse electromagnetic response tensor of a model system with metallic stripes embedded in an underdoped superconducting bulk. This was done by solving the appropriate Bogoliubov-de Gennes equations within the renormalized mean field theory. We then averaged over the different stripe orientations to obtain an isotropic superfluid stiffness.\n\nIn the second model we considered microscopic impurities that carried an excess doping charge.\nThe temperature dependent stiffness in this case was calculated using a perturbative expansion  expansion assuming both dilute and weak impurities (Born approximation).\n\nIn both models, the regions of higher doping add to the total carrier density and hence increase the\nsuperfluid stiffness at zero temperature. On the other hand the impurity regions give rise to low energy states that lead to a faster reduction of the superfluid stiffness with temperature. Nevertheless we found that for a range of doping levels in the underdoped regime a higher $T_c$ than a uniform superconductor of the same doping can be attained. Moreover, in the case of microscopic impurities it is even possible to attain a higher critical temperature than the maximal $T_c$ obtained in the pure system, that is, higher than $T_c$ at optimal doping.\n\nThe last result can help to understand the enhancement of $T_c$ seen at the interface between an underdoped and a highly overdoped LSCO film.\\cite{Yuli:2008} We have previously noted that such an increase in $T_c$ due to coupling between two homogeneous layers with different doping requires unrealistically strong coupling between the two CuO planes.\\cite{Goren:2009} However if, due to the structure of the interface, overdoped and underdoped layers interpenetrate each other, then the proximity coupling can be induced by the strong in-plane hopping and the situation becomes equivalent to the one considered here.\n\nFinally we remark that our main result for the case of microscopic impurities was obtained within a perturbative expansion in the impurity strength. It would be interesting to compare this to a numerical solution that takes into account scattering, at least from individual impurities, exactly.\nIf indeed excess dopants concentrated at random locations can lead to increase of the maximal $T_c$, this opens up intriguing possibilities for further enhancement of $T_c$. For example through design of an optimal ordered arrangement of the highly doped regions.\n\n\\section{Acknowledgments}\nWe thank H. Bary-Soroker, E. Berg, E. Demler, S. Huber, Y. Kraus, K. Michaeli, and Z. Ringel for valuable discussions. This work was supported by grants from the Israeli Science Foundation and the Minerva foundation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec: intro}\n\nDusty Star-Forming Galaxies (DSFGs) provide an enormous contribution to the total energy density emitted by galaxies and hence represent a vital stage in their formation and evolution. In the last couple of decades, there has been a profound development in our understanding of the nature of DSFGs and their role in galaxy formation and evolution, particularly those selected at sub-mm wavelengths (typically 850\\,$\\mu$m). These sources have subsequently come to be known as sub-mm galaxies or SMGs \\citep{2002PhR...369..111B}. The redshift distribution of SMGs peaks at $z \\sim 2 - 3$ with a significant tail extending to higher redshifts \\citep{2005ApJ...622..772C,2011MNRAS.415.1479W,2014ApJ...788..125S,2014MNRAS.444..117K,2017MNRAS.471.2453S,2017ApJ...840...78D,2019MNRAS.487.4648S}. SMGs are typically massive \\cite[$M_{\\star} \\sim 10^{10} - 10^{11}$\\,M$_{\\odot}$:][]{2004ApJ...617...64S,2011ApJ...740...96H,2012A&A...541A..85M,2013MNRAS.432...23G,2014ApJ...788..125S,2020MNRAS.494.3828D,2021MNRAS.504..928P}), rich in both gas and dust \\citep{2002MNRAS.331..495S,2005MNRAS.359.1165G,2006ApJ...650..592K,2006ApJ...640..228T,2008ApJ...680..246T,2010ApJ...714.1407C,2010MNRAS.403..274C,2013MNRAS.429.3047B,2020MNRAS.494.3828D,2021MNRAS.504..928P} and extremely luminous \\cite[with total infrared luminosities $L_{\\text{TIR}} \\sim 10^{11} - 10^{13}$\\,L$_{\\odot}$:][]{2005ApJ...622..772C,2012A&A...548A..22M,2014MNRAS.438.1267S,2015MNRAS.451.3419G,2017MNRAS.468.4006M,2017MNRAS.469..492M,2018A&A...619A.169R,2018MNRAS.477.2042H,2019MNRAS.490.3840C,2020MNRAS.496.2315G}, corresponding to dust-obscured star formation rates (SFRs) of hundreds to thousands of solar masses per year and hence contributing significantly to the Cosmic Star Formation Rate Density (CSFRD) at these redshifts \\citep{2005ApJ...622..772C,2011MNRAS.415.1479W,2012ApJ...761...89B,2014MNRAS.438.1267S,2017MNRAS.471.2453S,2020MNRAS.494.3828D}. There is still some debate surrounding the morphology and origin of DSFGs and they are likely a diverse population in this respect \\cite[see e.g.][]{2014PhR...541...45C}. Some studies suggest that many DSFGs may represent isolated, gas-rich, disk galaxies undergoing a burst of star formation \\citep{2010ApJ...714.1407C,2010MNRAS.404.1355D,2013MNRAS.432.2012T,2017MNRAS.469..492M}, while other studies suggest that they are primarily merger-driven \\citep{10.1111\/j.1365-2966.2004.08553.x,2008ApJ...680..246T,2010ApJ...724..233E,2012MNRAS.425.1320I,2012ApJ...757...23K,2015ApJ...799..194C,2021MNRAS.503.2622C} and potentially represent a key, but short-lived  \\cite[typically of order a few hundred Myr:][]{2016ApJ...824...36C,2021MNRAS.504..928P} phase in an evolutionary sequence, followed by a bright quasar phase which then decays to leave a massive elliptical galaxy \\citep{2005ApJ...632..736A,2006ApJ...641L..17F,2008ApJS..175..356H,2010MNRAS.402.2113C,2014ApJ...782...68T,2017MNRAS.464.1380W,2020MNRAS.494.3828D}. Whilst SMGs are important objects to study in their own right, a number of studies have found numerous DSFGs residing within high redshift protoclusters \\citep{2009ApJ...694.1517D, 2011Natur.470..233C, 2012Natur.486..233W, 2014MNRAS.439.1193C, 2014A&A...570A..55D, 2015ApJ...815L...8U, 2015ApJ...808L..33C, 2016MNRAS.461.1719C, 2016ApJ...828...56W, 2018MNRAS.476.3336G, 2018ApJ...856...72O, 2018Natur.556..469M, 2019A&A...625A..96K, 2019ApJ...872..117G, 2020arXiv201002909W, 2021A&A...646A.174A, 2021ApJ...916...46J} thought to be the progenitors of local, massive galaxy clusters. SMGs therefore additionally have the potential to be signposts for these overdense regions in the early Universe, whose study can be used to test cosmological models and investigate the formation and evolution of large scale structure \\cite[see e.g.][]{2018Natur.553...51M}. \n\nDespite this immense progress, our understanding of the nature of SMGs remains incomplete, particularly for those objects residing at $z > 4$ where small samples and a lack of spectroscopic confirmations obstructs further progress. The \\textit{Herschel Space Observatory} discovered a surprisingly large population ($3.3 \\pm 0.8$\\,deg$^{-2}$) of extremely luminous, candidate $z > 4$ DSFGs, selected as sources with a rising Spectral Energy Distribution (SED) in the three \\textit{Herschel} SPIRE \\citep{2010A&A...518L...3G} bands \\cite[i.e. $S_{250} < S_{350} < S_{500}$:][]{2014ApJ...780...75D, 2016MNRAS.462.1989A,2016ApJ...832...78I} and hence typically known as `500\\,$\\mu$m risers'. Numerous 500\\,$\\mu$m risers have been confirmed at $z > 4$ \\cite[e.g.][]{2011ApJ...740...63C, 2013Natur.496..329R, 2017ApJ...850....1R, 2021ApJ...907...62R, 2017MNRAS.472.2028F} with corresponding SFRs in excess of 1000\\,M$_{\\odot}$yr$^{-1}$, making them extreme objects, potentially at the high luminosity end of a larger population of high-$z$ DSFGs \\citep{2020MNRAS.496.2315G}. Simulations have significant difficulty in reproducing this population whilst simultaneously satisfying other observational constraints \\cite[see e.g.][and references therein]{2021MNRAS.502.2922H}. Accumulating observations of high redshift DSFGs will therefore be vital in gaining a complete understanding of this population. \nVery few studies have focused on investigating the nature of these 500\\,$\\mu$m risers, and as such the majority of their properties remain poorly constrained. Simulations suggest that $\\sim 40\\%$ of faint (S$_{500} < 60$\\,mJy) \\textit{Herschel} sources should be comprised of multiple DSFGs, while the brightest should be exclusively strongly lensed single galaxies \\citep{2017A&A...607A..89B}. Additionally, some fraction of these multiple systems will be chance line-of-sight alignments of DSFGs, rather than physically associated structures. \\cite{2017arXiv170904191O} provide follow-up observations of a sample of 44 500\\,$\\mu$m risers with the Atacama Large Millimeter Array \\cite[ALMA:][]{2004AdSpR..34..555B}, finding that $\\sim 61\\%$ resolve into a single sub-mm bright source, while the remainder break up into multiple DSFGs (four break up into $\\geq 3$ sources, with one resolving into five separate sources in the synthesised $\\sim 0.12^{\\prime\\prime}$ ALMA beam). Additionally, 18 of their sources show some evidence of gravitational lensing. \\cite{2020MNRAS.496.2315G}, hereafter known as G20, selected a sample of thirty-four 500\\,$\\mu$m risers from the \\textit{Herschel} Multi-tiered Extragalactic Survey \\cite[HerMES:][]{2012MNRAS.424.1614O}, and carried out interferometric follow-up observations with the Submillimeter Array \\cite[SMA:][]{2004ApJ...616L...1H} between 2010 and 2015. They found that 4 break up into two individual sources and 18 resolve into a single source, with the remaining 12 maps containing no apparent SMA counterpart. G20 suggested that non-detections were likely a result of the bright \\textit{Herschel} sources breaking up into multiple faint counterparts clustered within the \\textit{Herschel} SPIRE beamsize ($\\sim 17.6^{\\prime\\prime}$ for the 250\\,$\\mu$m band) but separated by distances greater than the SMA beamsize ($\\sim2^{\\prime\\prime}$) and lying below their detection threshold, with simple flux calculations indicating that these multiple systems should contain $\\geq 3$ individual DSFGs. Based on this assumption, G20 estimated that $\\sim 60\\%$ of faint (S$_{500} < 60$\\,mJy) 500\\,$\\mu$m risers and $\\sim 35\\%$ of bright (S$_{500} > 60$\\,mJy) 500\\,$\\mu$m risers should be blends, indicating that this population is likely much more diverse than predicted. Similarly, based on LMT\/AzTEC 1.1\\,mm follow-up observations of their sample of bright \\textit{Herschel} 500\\,$\\mu$m risers, \\cite{2021MNRAS.505.5260M} find that $45\/93$ have no counterpart in the higher resolution imaging, with some of these non detections expected to be multiple systems and others expected to be individual, resolved sources that are faint at 1.1\\,mm due to a large dust spectral emissivity index $\\beta$. The multiplicity of their sample could therefore be $\\sim 9\\%$ in the most conservative scenario, or $\\sim 50-60\\%$ in the most extreme scenario. Any systems with high multiplicities would naturally be candidate high redshift protocluster cores containing numerous DSFGs, structures discovered in only a handful of studies to date \\cite[e.g.][]{2018Natur.556..469M, 2018ApJ...856...72O}.\n\nIn order to determine a more robust multiplicity fraction for samples of 500\\,$\\mu$m risers, deeper follow-up observations of 500\\,$\\mu$m risers without apparent cross-matches in higher resolution data are required. We therefore followed on from G20 and targeted four of their 500\\,$\\mu$m risers without an apparent SMA counterpart for additional, deeper FIR\/sub-mm observations. Given the significant improvements to the SMA since 2015 (particularly in terms of the improved bandwidth provided by the SWARM correlator), we were able to obtain much deeper, high-resolution SMA 345\\,GHz continuum imaging of these 500\\,$\\mu$m risers. We additionally obtained complementary observations at 850\\,$\\mu$m with the Submillimeter Common User Bolometer Array 2 \\cite[SCUBA-2:][]{2013MNRAS.430.2513H} on the James Clerk Maxwell Telescope (JCMT), which provide an integrated flux density with a coarser resolution at a similar wavelength. The primary aim of this paper is to determine the multiplicities of these sources, allowing us to comment further on the diversity of the 500\\,$\\mu$m riser population. The SCUBA-2 observations also allow us to investigate the wider field surrounding these sources and hence evaluate the environments in which they reside.\n\nThis paper is structured as follows: In Section \\ref{sec: sample} we describe the selection of the 500\\,$\\mu$m riser sample, followed by a discussion of the observations and data reduction in Section \\ref{sec: bootes_observations}. In Sections \\ref{sec: results} and \\ref{sec: discussion} we present the results of this study and discuss the properties of our 500\\,$\\mu$m risers, followed by the conclusions and summary in Section \\ref{sec: conclusions}. Throughout this work, we adopt the standard flat $\\Lambda$CDM cosmology: $\\Omega_{m} = 0.3$, $\\Omega_{\\Lambda}= 0.7$, and $H_{0} = 70$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$.\n\n\\input{tables\/integrated_fir}\n\n\\section{Sample Selection}\n\\label{sec: sample}\n\nA full description of the selection of the original sample of 500$\\mu$m risers is provided in G20. Here we briefly outline the main points of this selection. G20 select a heterogenous sample of thirty-four 500$\\mu$m risers based only on their \\textit{Herschel} SPIRE flux densities and colours ($S_{250} < S_{350} < S_{500}$) from HerMES. The sample has an average 500$\\mu$m flux density of $ 67 \\pm 29$\\,mJy and all sources are detected to $>4\\sigma$. G20 note that recent refinements to the data reduction process means that the flux densities of the sources in the sample vary somewhat in the most up-to-date maps and catalogues, such that $\\sim 7$ of the original 34 sources cannot now be strictly defined as 500$\\mu$m risers. These sources are still included in the G20 sample as flux boosting is known to introduce some variation into the \\textit{Herschel} colours of 500$\\mu$m risers, and typical star-forming SED shapes indicate that these sources could still reside at high redshift. G20 used the ALESS \\citep{2015ApJ...806..110D} average SED at redshifts between 4 and 6 and normalised to a 500\\,$\\mu$m flux density of 60\\,mJy to estimate that, given their observed sub-mm flux densities from HerMES and assuming that a single source is responsible for the total \\textit{Herschel} flux density, these sources should have flux densities of $\\sim 24-38$\\,mJy at 345\\,GHz. However, despite reaching detection thresholds of $\\sim 7 - 10$\\,mJy, twelve of these maps showed no detections. Assuming in the most conservative case that any multiple systems are comprised of sources with similar flux densities, G20 predicted that these 500\\,$\\mu$m risers must break up into $\\geq 3$ individual sources, and are therefore potential indications of forming compact cluster cores at $z > 4$ similar to those discovered by \\cite{2018ApJ...856...72O} and \\cite{2018Natur.556..469M}. \n\nWe selected the three brightest sources at 500\\,$\\mu$m from the twelve sources in the G20 sample without SMA counterparts (Bootes15, Bootes24 and Bootes27) for multi-wavelength follow-up observations. An additional 500\\,$\\mu$m riser (XMM-M5) was observed as part of a separate SMA project, but will also be presented in this paper. The positions and \\textit{Herschel} SPIRE photometry for these sources are presented in Table \\ref{table: integrated_fir}, and in Figure \\ref{fig: sample_colours} we show their \\textit{Herschel} SPIRE colours. We find that our 500\\,$\\mu$m risers are quite typical of the majority of the global 500\\,$\\mu$m riser population, although potentially residing towards\nthe less extreme end. \n\n\\begin{figure}\n\\hspace{-0.2cm}\n\\includegraphics[width=0.45\\textwidth]{figures\/sample_colours.pdf}\n\\caption{\\textit{Herschel} SPIRE colours for our 500\\,$\\mu$m risers alongside those from a selection of studies in the literature. Note that the \\citet{2019ApJS..244...30M} and \\citet{2021MNRAS.505.5260M} colours are based on deblended \\textit{Herschel} photometry using high resolution follow-up observations and so contain a number of sources that are no longer strictly 500\\,$\\mu$m risers. We find that our 500\\,$\\mu$m risers are towards the less extreme end of the global 500\\,$\\mu$m riser population.}\n\\label{fig: sample_colours}\n\\end{figure}\n\n\\section{Observations and Data Reduction}\n\\label{sec: bootes_observations}\n\n\\subsection{SMA Observations}\n\\label{sec: SMA_obs}\n\nThe SMA observations were taken in three tracks between 31$^{\\text{st}}$ January and 26$^{\\text{th}}$ February 2020 as part of the SMA program 2019B-S016 (PI: J. Cairns). The conditions for these observations were exquisite, with $\\tau$ values of $\\sim 0.02 - 0.04$. We used the SMA in its COM configuration with 7 antennae, resulting in a beamsize of $\\sim 1.7^{\\prime\\prime} \\times 2.1^{\\prime\\prime}$. The SMA uses two receivers, each with two sidebands of 8\\,GHz separated by a gap of 8\\,GHz, and with each sideband separated into four chunks. We tuned these two receivers to 337\\,GHz and 345\\,GHz respectively to produce 32\\,GHz of continuous bandwidth for our observations. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figures\/SMA_maps_SCUBA_contours_v2.pdf}\n\\caption{SMA maps overlaid with SCUBA-2 850\\,$\\mu$m contours from $3\\sigma$ to $7\\sigma$ significance for the four 500\\,$\\mu$m risers. Bootes15, Bootes24 and Bootes27 each break up into two individual sources in the SMA maps, while XMM-M5 resolves into one faint source. Bootes24 and Bootes27 are well detected by SCUBA-2, while Bootes15 and XMM-M5 are marginally detected to $\\sim 3\\sigma$ significance.} \n\\label{fig: sma_maps}\n\\end{figure*}\n\nWe reduced our SMA maps using the standard IDL-based SMA data reduction package MIR\\footnote{\\url{https:\/\/github.com\/qi-molecules\/sma-mir}}. Firstly, we manually inspected the data, flagging any regions that contained phase jumps and fixing any channels that contained significant noise spikes, before applying the system temperature correction. The passband calibration was completed using the sources 3c454.3, 3c279, 0927+390 or 3c84, while for flux calibration, we used either 0854+201 or Callisto. Gain calibration on the SMA is carried out by periodically observing nearby brights quasars with known flux densities alternately to the desired source. For these purposes, we used the quasars 1506+426 and 3c345. The error for this calibration process is estimated to be $\\sim 10\\%$ \\citep[see e.g.][]{2018A&A...612A..54L}.  \n\nWe then used the Common Astronomy Software Applications package\\footnote{\\url{https:\/\/casa.nrao.edu\/index.shtml}} \\cite[CASA:][]{2007ASPC..376..127M} to complete the imaging. Following the calibration process in MIR, each chunk of each sideband on each receiver is exported into CASA, before being concatenated together to produce one large visibility dataset. We manually flagged the $\\sim 300$ channels on the edge of each chunk in order to avoid including poor data in the final image, inspecting the result to ensure that we had flagged enough of the edge channels and that there were no spikes or strong lines in the data. We generated the continuum by averaging all of the remaining spectral channels together. In order to clean the SMA maps, we first identified the positions of likely sources based on the dirty maps. We followed G20 who found that a detection threshold of $3.75\\sigma$ (where $\\sigma$ represents the global noise in the dirty map) without any corrections for the response of the primary beam produced a good balance between minimising the number of fake sources extracted and minimising the signal-to-noise threshold for extraction. We therefore found all sources above this $3.75\\sigma$ detection threshold in the dirty maps and considered them to be real sources. We used the CASA command \\code{tclean} to complete interactive CLEANing of our SMA maps. We selected natural weighting which maximises the signal-to-noise and used $<100$ CLEAN iterations for each map, placing CLEAN windows in the images to preferentially select flux from the positions of the $>3.75\\sigma$ detections. The resulting CLEANed maps reached 1$\\sigma$ RMS values of 0.17, 0.29 and 0.27\\,mJy\/beam for Bootes15, Bootes24 and Bootes27 respectively, around an order of magnitude deeper than the maps obtained by G20. \n\nXMM-M5 was observed as part of the SMA project 2019A-S004 (PI: D. L. Clements). The observations and subsequent data reduction will be discussed in a forthcoming paper (Clements et al. in prep) and we simply highlight the important features here. The SMA was used in its COM configuration with the two receivers tuned to 198\\,GHz and 206\\,GHz respectively, resulting in a beamsize of $\\sim 3.6^{\\prime\\prime} \\times 3.2^{\\prime\\prime}$. The reduced SMA map for XMM-M5 has a $1\\sigma$ RMS sensitivity of $\\sim 0.28$\\,mJy\/beam. The CLEANed SMA maps for all four sources are shown in Figure \\ref{fig: sma_maps}.\n\n\\input{tables\/sma}\n\n\\subsection{SCUBA-2 Observations}\n\\label{sec: SCUBA_obs}\n\nThe JCMT\/SCUBA-2 450\\,$\\mu$m and 850\\,$\\mu$m observations for Bootes15, Bootes24 and Bootes27 were taken between 1$^{\\text{st}}$ December 2020 and 20$^{\\text{th}}$ January 2021 as part of the SCUBA-2 program M20BP036 (PI: J. Cairns). Each source was observed with several pointings using the CV Daisy mode, with a total of $\\sim 2.7$ hours on Bootes15, $\\sim 1.4$ hours on Bootes24 and $\\sim 4.1$ hours on Bootes27. Weather conditions were typically between Band 2 and Band 3, with $\\tau_{225}$ values in the range $0.08 - 0.14$. This allowed us to reach 1$\\sigma$ RMS sensitivities of 2.2, 2.8 and 1.7\\,mJy\/beam in our SCUBA-2 850\\,$\\mu$m maps of Bootes15, Bootes24 and Bootes27 respectively. We reduced our observations using the Sub-Millimetre User Reduction Facility (\\texttt{SMURF}) package \\citep{2013MNRAS.430.2545C} and the \\texttt{ORAC-DR} data reduction pipeline \\citep{1999ASPC..172...11E}. \n\nFor the first reduction we completed each step of the data reduction process manually following \\citet{2019MNRAS.490.3840C}. We first used the \\texttt{MAKEMAP} command on the individual scans from SCUBA-2, with the \\texttt{METHOD} parameter set to the default \\texttt{ITERATE} method, which uses an iterative technique to fit a number of models for noise and instrumental behaviour. We then co-added the individual scans using the \\texttt{PICARD} recipe \\texttt{MOSAIC\\_JCMT\\_IMAGES} to produce a single map. This step also removes a number of contaminant signals. For this first reduction, we assumed that the individual sources detected in the SMA images would be contained within the much larger SCUBA-2 beam, and so would appear as a single point-like source. We applied a matched filter using the \\texttt{PICARD} recipe \\texttt{SCUBA2\\_MATCHED\\_FILTER}, which subtracts the background by convolving the\nmaps and the PSF with a 30$^{\\prime\\prime}$ FWHM Gaussian kernel, before convolving the maps with the PSF to produce the matched-filtered signal map. This\nprocess gives an effective beam FWHM of 14.6$^{\\prime\\prime}$ surrounded by a shallow negative ring, and is commonly used for finding sources with angular scales of a similar size or smaller to the beamsize of the SCUBA-2 instrument. The signal maps produced using this method are in units of pW, and so must be calibrated using a Flux Correction Factor (FCF). We used the standard FCF value\\footnote{\\url{https:\/\/www.eaobservatory.org\/jcmt\/instrumentation\/continuum\/scuba-2\/calibration\/}} of $537 \\pm 43$\\,Jy\\,beam$^{-1}$\\,pW$^{-1}$. \n\nFor our second data reduction method, we made use of the \\texttt{ORAC-DR} data reduction pipeline. We first used the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} recipe which employs a similar method to our manual data reduction process above. Raw data are passed to the map maker to produce an image calibrated in mJy\/beam. The pipeline then estimates the RMS noise in the image and calculates the Noise Equivalent Flux Density (NEFD). Once all the individual observations have been processed, the pipeline co-adds individual scans together, which it then convolves with a matched filter to enhance the signal-to-noise ratio of point sources. The RMS noise and NEFD are calculated for this co-added image, and a signal-to-noise map is produced. We find that the pipeline produces essentially identical results to our manual data reduction.\n\nFinally, we completed a third data reduction process using the \\texttt{ORAC-DR} pipeline with the \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} recipe. Given that the SCUBA-2 beamsize is smaller than that of \\textit{Herschel} SPIRE at 250\\,$\\mu$m but larger than that of the SMA, it is possible that any multiple systems may be partially resolved in the SCUBA-2 850\\,$\\mu$m maps. If this is the case then flux density estimates should be calculated based on SCUBA-2 maps reduced without the matched filter to avoid missing flux for the partially resolved sources lying outside of the SCUBA-2 beam. The \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} recipe passes the raw data for the individual scans to the map maker which processes them to produce a Frame image, which is then calibrated in units of mJy\\,arcsec$^{-2}$. Individual scans are then co-added together and the noise properties of this image are calculated. The analysis of the SCUBA-2 maps in the following sections is based on the two pipeline reductions. While the SCUBA-2 instrument simultaneously provides 450\\,$\\mu$m and 850\\,$\\mu$m photometry, the noise levels in the 450\\,$\\mu$m maps are too large to detect our 500\\,$\\mu$m risers based on their \\textit{Herschel} SPIRE flux densities.\n\nWe additionally used archival 850\\,$\\mu$m SCUBA-2 maps for XMM-M5 from the Canadian Astronomy Data Center. XMM-M5 was observed as part of the SCUBA-2 Large eXtragalactic Survey (S2LXS\\footnote{\\url{https:\/\/www.eaobservatory.org\/jcmt\/science\/large-programs\/s2lxs\/}}: Geach et al. M17BL001) and is present in two overlapping pointings. We re-reduced each of these pointings using the \\texttt{ORAC-DR} pipeline as outlined above, and co-added the two pointings to obtain a deeper SCUBA-2 850\\,$\\mu$m image of XMM-M5. After this reduction process, we find that the RMS noise in an aperture of radius 350$^{\\prime\\prime}$ centered on the SMA position of XMM-M5 is $\\sim3.3$\\,mJy\/beam. In Figure \\ref{fig: sma_maps} we show the SMA maps for the 500\\,$\\mu$m risers with 850\\,$\\mu$m contours based on the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} reduced SCUBA-2 maps. The SCUBA-2 maps with the matched filter were chosen for Figure \\ref{fig: sma_maps} as this reduction maximises the signal-to-noise ratio in the map.\n\n\\subsection{Extracting Flux Densities}\n\\label{sec: extracting_flux}\n\nWe then extracted flux densities from the SMA and SCUBA-2 maps. G20 found that the most robust method for extracting the flux density of point-like SMA sources is to simply extract the peak flux density directly from the CLEANed SMA map. We therefore extracted peak flux densities from the CLEANed SMA maps for each source detected to $>3.75\\sigma$ in the dirty SMA maps. These flux densities were then corrected for the primary beam response. The response of the primary beam for the SMA can be described as a Gaussian function with a size determined by the wavelength of the observations\\footnote{\\url{https:\/\/lweb.cfa.harvard.edu\/sma\/miriad\/manuals\/SMAuguide\/smauserhtml\/node130.html}}. We used the \\texttt{pbplot} command in the MIRIAD \\citep{2011ascl.soft06007S} package to determine the FWHM of the SMA primary beam at 345\\,GHz for our three sources in the Bootes field, and at 210\\,GHz for XMM-M5. We then found the primary beam response at the position of the peak flux density value for each source, and divided the extracted flux density by this value, propagating the errors accordingly. In Table \\ref{tab: sma} we present the corrected SMA photometry alongside the corresponding correction factor for the primary beam response, where the uncertainties include the 10\\% calibration error added in quadrature with the RMS noise in the SMA map. We also extracted flux densities from the dirty SMA maps using a similar method, finding that they differ by no more than $\\sim 3\\%$ from those extracted from the CLEANed maps.\n\nFor the SCUBA-2 850\\,$\\mu$m maps, we extracted two separate flux densities, one for each of the two pipeline reductions. For the 850\\,$\\mu$m map reduced using the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} recipe, we located the highest signal-to-noise pixel associated with the detection and extracted the corresponding flux density. For the 850\\,$\\mu$m map reduced using the \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} recipe, we laid down a series of apertures in the image with increasing radii from 0$^{\\prime\\prime}$ to 10$^{\\prime\\prime}$, and calculated the sum of the pixel values within these apertures. This produces a flux density in units of mJy\\,arcsec$^{-2}$, which we then multiplied by 16 in order to convert to mJy (based on the 4$^{\\prime\\prime}$ pixel size of the SCUBA-2 850\\,$\\mu$m images). For the final flux density measurement, we selected the aperture with the largest flux density value, as this aperture should contain the maximum amount of flux density from the source without including too much contaminating background (we also manually checked each final aperture to make sure that it was a reasonable size). \n\nSCUBA-2 850\\,$\\mu$m photometry must be corrected for flux boosting. \\cite{2017MNRAS.465.1789G} investigated the effects of flux boosting in the SCUBA-2 Cosmology Legacy Survey (S2CLS), finding that the level of flux boosting is consistent across the whole survey and well described by the power law\n\\begin{eqnarray}\n    B = 1 + 0.2 \\left( \\frac{\\text{SNR}}{5} \\right)^{-2.3}\n\\end{eqnarray}\n\nwhere $B$ is the flux boosting factor and SNR is the signal-to-noise ratio of the detection. We estimated errors for our SCUBA-2 flux density measurements based on the combination of instrumental noise, confusion noise and the typically assumed 5\\% calibration error.\n\n\n\\subsection{Ancillary Data}\n\\label{sec: ancillary_data}\n\nDue to their position towards the edge of the field, our three 500\\,$\\mu$m risers in Bootes only benefit from sporadic ancillary data. All three are covered by the DESI Legacy Imaging Surveys\\footnote{\\url{http:\/\/legacysurvey.org\/}} which combines the Dark Energy Camera Legacy Survey, the Beijing\u2013Arizona Sky Survey, and the Mayall $z$-band Legacy Survey, mapping $\\sim 14,000$\\,deg$^{2}$ of the sky using the Blanco telescope at the Cerro Tololo Inter-American Observatory, as well as the Mayall and 2.3\\,m Bart Bok Telescopes at the KPNO down to AB magnitudes of 23.72, 22.87 and 22.29 in the $g$, $r$ and $z$ bands respectively \\citep{2019AJ....157..168D}. There is a tenuous detection in $g$, $r$ and $z$ for one of the resolved objects associated with Bootes15, but the remainder are undetected in these images.\n\nThe Infrared Bootes Imaging Survey \\cite[IBIS:][]{2010AAS...21641513G} provides deep NIR images and catalogues in the $J$, $H$ and $K_{s}$ bands, down to limiting AB magnitudes of 22.0, 21.5, and 20.8  respectively, and we extract 3.4, 4.6, 12, and 22\\,$\\mu$m \\textit{Wide-Field Infrared Survey Explorer} \\cite[\\textit{WISE}:][]{2010AJ....140.1868W} images and catalogues from the ALLWISE database, which combines the \\textit{WISE} cryogenic and NEOWISE \\citep{2011ApJ...731...53M} post-cryogenic phases. Additionally, the \\textit{Spitzer} Deep Wide-Field Survey \\cite[SDWFS:][]{2009ApJ...701..428A} provides deep 3.6, 4.5, 5.8 and 8.0\\,$\\mu$m imaging for 10\\,deg$^{2}$ in the Bootes field. Bootes24 resides just within the footprint of SDWFS, although it only benefits from data at 4.5 and 8.0\\,$\\mu$m. For Bootes15 and Bootes27, we rely on archival IRAC imaging. Bootes15 is covered at 3.6 and 4.5\\,$\\mu$m as part of \\textit{Spitzer} Program 80156 (PI: A. Cooray) aiming to study lensed sub-mm galaxies from the \\textit{Herschel}-ATLAS \\cite[H-ATLAS:][]{2010PASP..122..499E} and HerMES surveys. Additionally, Bootes27 is covered at 4.5 and 8.0\\,$\\mu$m as part of \\textit{Spitzer} Program 30134 (PI: G. Fazio) aiming to study a statistically complete sample of star-forming dwarf galaxies. All three 500\\,$\\mu$m risers remain undetected in the IBIS catalogue, and only one of the resolved sources associated with Bootes15 is detected in the \\textit{WISE} images (although this \\textit{WISE} detection appears to be a blend of two IRAC sources). The two sources associated with Bootes15 are both detected in the two IRAC bands. Those associated with Bootes24 are well detected at 4.5\\,$\\mu$m but not at 8.0\\,$\\mu$m. There are numerous bright 8.0\\,$\\mu$m IRAC sources associated with Bootes27, and a more comprehensive, multi-wavelength analysis of this source will be presented in a forthcoming paper (Cairns et al. in prep). Given the sparse ancillary data in the optical and NIR for our Bootes sources, it is not discussed further in this paper.  \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figures\/Herschel_SPIRE250_SCUBA_SMA_contours.pdf}\n\\caption{\\textit{Herschel} SPIRE 250\\,$\\mu$m maps with SCUBA-2 850\\,$\\mu$m (brown) and SMA (red) contours from $3\\sigma$ to $7\\sigma$ significance for the four 500\\,$\\mu$m risers. Bootes15, Bootes24 and Bootes27 each break up into two individual sources in the SMA maps, while XMM-M5 resolves into one faint source. Bootes24 and Bootes27 are well detected by SCUBA-2, while Bootes15 and XMM-M5 are marginally detected to $\\sim 3\\sigma$ significance. Bootes27 shows two partially resolved SCUBA-2 sources associated with the bright \\textit{Herschel} source.}\n\\label{fig: herschel_map}\n\\end{figure*}\n\nBy contrast, XMM-M5 benefits from a wealth of multi-wavelength data, including `forced photometry' at optical and NIR wavelengths from \\cite{2017ApJS..230....9N}. This includes photometry in the $u$, $g$, $r$, $i$ and $z$ bands from the Canada-France-Hawaii Telescope Legacy Survey \\cite[CFHTLS:][]{2012AJ....143...38G}, in the $Z$, $Y$, $J$, $H$ and $K_{s}$ bands from the VISTA Deep Extragalactic Observations \\cite[VIDEO:][]{2013MNRAS.428.1281J} survey, and in the 3.6 and 4.5\\,$\\mu$m IRAC bands from the \\textit{Spitzer} Extragalactic Representative Volume Survey \\cite[SERVS:][]{2012PASP..124..714M}. The reader is directed to \\cite{2017ApJS..230....9N} for a description of how this photometry was extracted. XMM-M5 is also included in the Subaru XMM Deep Survey \\citep[SXDS:][]{2008ApJS..176....1F} which covers five broadband filters reaching limiting AB magnitudes of $B = 28.4$, $V = 27.8$, $R_{c} = 27.7$, $i^{\\prime} = 27.7$, and $z^{\\prime} = 26.6$, as well as the Hyper Suprime-Cam Subaru Strategic Program \\citep[HSC SSP:][]{2018PASJ...70S...4A} Survey which, in its `Deep' layer, reaches limiting AB magnitudes of 27.5, 27.1, 26.8, 26.3 and 25.3 in the $g$, $r$, $i$, $z$ and $y$ bands respectively. A more comprehensive, multi-wavelength analysis of XMM-M5 will be presented in Clements et al. (in prep).\n\n\\section{Results}\n\\label{sec: results}\n\n\\subsection{SMA and SCUBA-2 Detections}\n\\label{sec: SMA_SCUBA_detection}\n\nIn Figure \\ref{fig: sma_maps}, we present the SMA maps for the four 500\\,$\\mu$m risers, overlaid with SCUBA-2 850\\,$\\mu$m contours. In Figure \\ref{fig: herschel_map}, we present a similar plot showing the 250\\,$\\mu$m \\textit{Herschel} SPIRE map overlaid with both SCUBA-2 850\\,$\\mu$m and SMA contours. We find that Bootes15 and Bootes24 each break up into two bright sources, both of which are detected to $>5\\sigma$ in both the dirty and CLEANed SMA maps. Bootes27 also breaks up into two resolved sources, but while the brighter source is similarly detected to $>5\\sigma$, the fainter source is a more marginal detection at $\\sim3.9\\sigma$ in the dirty map and $\\sim3.1\\sigma$ in the CLEANed map. Moreover, this fainter source is offset by $\\sim 10^{\\prime\\prime}$ from the edge of the observed SCUBA-2 $3\\sigma$ emission region, and so, while it may contribute to the integrated \\textit{Herschel} SPIRE flux density, it is unlikely to contribute to the observed 850\\,$\\mu$m flux density. After correcting for the primary beam, we find that this fainter source has a flux density of $1.7 \\pm 0.5$\\,mJy and, as a result, we would not expect to detect it given the 1.7\\,mJy\/beam RMS noise in the SCUBA-2 map and the $\\sim 10^{\\prime\\prime}$ offset from the main emission region. There is another marginal $\\sim3.3\\sigma$ detection in the dirty SMA map coinciding with the Northern component of the SCUBA-2 contours (Figure \\ref{fig: herschel_map}), but as this falls below our $3.75\\sigma$ detection threshold we do not consider this detection robust. The SMA map for XMM-M5 shows a single $\\sim 4.7\\sigma$ peak coincident with the bright \\textit{Herschel} 500\\,$\\mu$m riser (Figure \\ref{fig: herschel_map}).\n\nThe SCUBA-2 850\\,$\\mu$m contours in Figures \\ref{fig: sma_maps} and \\ref{fig: herschel_map} demonstrate that XMM-M5 is marginally detected as a point-like source with a signal-to-noise ratio (SNR) of 3.65. Bootes15 is also marginally detected with a signal-to-noise ratio of 3.06, but is likely to be extended as the SCUBA-2 flux density peaks between the two SMA detections and is elongated along the direction of their separation. By comparison, Bootes24 and Bootes27 are both well detected to $>5\\sigma$ and appear partially resolved in the SCUBA-2 contours. Bootes27 in particular is clearly comprised of two partially resolved sources, one of which has no apparent SMA cross-match. After correcting for flux boosting we obtain flux densities of $2.9 \\pm 1.8$\\,mJy, $9.7 \\pm 2.2$\\,mJy, $6.9 \\pm 1.3$\\,mJy and $8.1 \\pm 3.3$\\,mJy for Bootes15, Bootes24, Bootes27 and XMM-M5 respectively based on the 850\\,$\\mu$m maps reduced using the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} recipe. Using the 850\\,$\\mu$m maps reduced using the  \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} recipe, we obtain flux densities of $2.6 \\pm 1.5$\\,mJy, $11.3 \\pm 2.6$\\,mJy and $10.1 \\pm 1.6$\\,mJy for Bootes15, Bootes24 and Bootes27 respectively, while no reliable flux density could be extracted for XMM-M5 using this reduction pipeline. The flux density estimates from the two pipeline reduction methods agree within the errors for all sources except for Bootes27, where the significantly larger flux density based on the \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} recipe reflects the more extended nature of the object. For the analysis in this paper, we will use the flux density from the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} reduction for XMM-M5 which appears as a point source in the SCUBA-2 maps, and the flux densities from the \\texttt{REDUCE\\_SCAN\\_EXTENDED\\_SOURCES} reduction for the three sources in the Bootes field which appear as extended sources. We additionally note that, while Bootes15 and XMM-M5 are detected to $>3\\sigma$ in the SCUBA-2 850\\,$\\mu$m maps, after including the instrumental, confusion and calibration uncertainties, the photometry is constrained to an accuracy of $\\sim1.7\\sigma$ and $\\sim2.5\\sigma$ respectively.\n\nIn Table \\ref{table: integrated_fir}, we present the integrated FIR\/sub-mm photometry for the four 500\\,$\\mu$m risers, where we have estimated the total SMA flux density for each source by adding together the flux densities of the resolved sources in the SMA continuum maps from Table \\ref{tab: sma}. We find that, despite being observed at similar wavelengths, the integrated 850\\,$\\mu$m flux densities from SCUBA-2 are larger than the combined flux densities of the detected SMA sources at 870\\,$\\mu$m for Bootes24 and Bootes27 (note that XMM-M5 is observed at a lower frequency and so we exclude it from this analysis). This disparity is particularly evident in Bootes27 - if we exclude the fainter SMA source that does not appear to contribute to the observed SCUBA-2 flux density, we measure integrated flux densities of $10.1 \\pm 1.6$\\,mJy and $1.9 \\pm 0.3$\\,mJy from the SCUBA-2 and SMA maps respectively, indicating a $>4\\sigma$ disparity between the two measurements. We investigate this discrepancy by plotting the SCUBA-2 850\\,$\\mu$m flux density against the SMA continuum flux density at $\\sim 870$\\,$\\mu$m for each of the 500\\,$\\mu$m risers alongside similar studies which obtained both interferometric and single-dish sub-mm photometry, where for Bootes27 we include only the SMA flux density for the resolved source coincident with the SCUBA-2 emission. \\cite{2018MNRAS.477.2042H} targeted 70 of the brightest 850\\,$\\mu$m sources from S2CLS down to a limiting flux density of $S_{850} \\sim 8$\\,mJy with high-resolution SMA follow-up observations at $860$\\,$\\mu$m. Similarly, \\cite{2019MNRAS.487.4648S} present the ALMA survey of the SCUBA-2 Cosmology Legacy Survey UKIDSS\/UDS field (AS2UDS), providing high resolution, ALMA Band 7 follow-up observations of SCUBA-2 850\\,$\\mu$m sources in the UKIDSS\/UDS field. We combined the SMA, ALMA and SCUBA-2 photometry from these surveys and, for any SCUBA-2 sources which break up into multiple components in the higher resolution interferometric observations, we add the flux densities of the resolved sources together in order to compare with our own photometry. The result of this comparison is shown in Figure \\ref{fig: interferometer_vs_single-dish}. We find that the SMA and SCUBA-2 flux densities for Bootes15 are consistent.  Bootes24 is $\\sim1.7$ times fainter in the SMA maps than the SCUBA-2 maps and lies towards the edge of the regions probed by the \\cite{2018MNRAS.477.2042H} and \\cite{2019MNRAS.487.4648S} samples in Figure \\ref{fig: interferometer_vs_single-dish}, but the two values are within $3\\sigma$ of each other. For Bootes27, the single resolved SMA source coincident with the SCUBA-2 contours is roughly five times fainter than the integrated flux density from the SCUBA-2 maps, placing it well outside of the region probed by the \\cite{2018MNRAS.477.2042H} and \\cite{2019MNRAS.487.4648S} samples. We interpret this disparity as evidence of additional faint sources residing below the SMA detection limit, but contributing to the integrated flux density at 850\\,$\\mu$m.\n\n\\begin{figure}\n\\hspace{-0.2cm}\n\\includegraphics[width=0.45\\textwidth]{figures\/interferometer_vs_single_dish.pdf}\n\\caption{The combined 870\\,$\\mu$m flux density for the resolved sources in either SMA or ALMA maps as a function of their integrated 850\\,$\\mu$m flux density from SCUBA-2. Orange squares and blue triangles show the \\citet{2019MNRAS.487.4648S} and \\citet{2018MNRAS.477.2042H} samples respectively, while the green points show our sample of 500\\,$\\mu$m risers. The black dashed line shows where the two flux density measurements are equal. We do not include XMM-M5 in this plot as the SMA observations are tuned to a shorter wavelength (210\\,GHz rather than 345\\,GHz).}\n\\label{fig: interferometer_vs_single-dish}\n\\end{figure}\n\nMoreover, we find that the integrated SCUBA-2 850\\,$\\mu$m flux densities of our 500\\,$\\mu$m risers are somewhat lower than we might expect based on their \\textit{Herschel} SPIRE colours. In Figure \\ref{fig: 850v500} we compare the integrated 500\\,$\\mu$m and 850\\,$\\mu$m flux densities of our 500\\,$\\mu$m risers, alongside the S2CLS and STUDIES \\cite[SCUBA-2 Ultra Deep Imaging EAO Survey:][]{2017ApJ...850...37W} samples. While the FIR\/sub-mm photometry for Bootes24, Bootes27 and XMM-M5 is reasonably consistent with the brighter end of S2CLS and STUDIES, Bootes15 is significantly fainter at 850\\,$\\mu$m than sources with similar 500\\,$\\mu$m flux densities. Additionally, we know that there is certainly 850\\,$\\mu$m flux associated with Bootes27 that is missed by the SCUBA-2 observations, as the fainter resolved SMA counterpart is significantly offset from the observed SCUBA-2 flux and has no apparent cross-match at 850\\,$\\mu$m. \n\n\\begin{figure}\n\\hspace{-0.2cm}\n\\includegraphics[width=0.45\\textwidth]{figures\/850_vs_500.pdf}\n\\caption{SCUBA-2 850\\,$\\mu$m flux density vs. \\textit{Herschel} SPIRE 500\\,$\\mu$m flux density for our 500\\,$\\mu$m risers (green points) as well as for the S2CLS (orange squares) and STUDIES (blue triangles) surveys. Bootes15 is significantly fainter than expected at 850\\,$\\mu$m based on its 500\\,$\\mu$m flux density.}\n\\label{fig: 850v500}\n\\end{figure}\n\n\\subsection{FIR\/Sub-mm Colours}\n\\label{sec: colours}\n\nWe have found that $3\/4$ of our 500\\,$\\mu$m risers suffer from blending, which likely influences their FIR\/sub-mm SEDs in numerous ways. Firstly, rather than being intrinsically red due to high redshifts ($z > 4$), it is likely that our 500\\,$\\mu$m risers are artificially reddened in the \\textit{Herschel} SPIRE bands due to the fact that the successively larger beamsizes at longer wavelengths exacerbate the effects of blending \\cite[see e.g.][]{2015ApJ...812...43B,2018MNRAS.477.1099D}. For example, \\cite{2019ApJS..244...30M} and \\cite{2021MNRAS.505.5260M} estimate that $\\sim 20\\%$ and $\\sim 25\\%$ of their samples respectively would not pass the 500\\,$\\mu$m riser selection criterion after accounting for the effects of blending. We additionally find that our SCUBA-2 observations may miss a significant fraction of the integrated flux density at 850\\,$\\mu$m (particularly for Bootes15 and Bootes27) and that there may be additional faint sources contributing to the integrated FIR\/sub-mm flux density that remain undetected in our SMA observations (particularly for Bootes27). The combination of missing flux at 850\\,$\\mu$m and stronger blending at 500\\,$\\mu$m would produce an artificially steep Rayleigh-Jeans tail, leading to significant difficulties in interpreting the resulting galaxy properties. Therefore, for a rigorous SED analysis, de-blending of the \\textit{Herschel} SPIRE flux densities will be necessary. However, due to the dearth of multi-wavelength ancillary data in this region of the Bootes field and the uncertainty as to whether we have recovered all of the sub-mm sources associated with each region, accurate de-blending of the \\textit{Herschel} SPIRE flux density would be extremely challenging and could potentially produce misleading results. Given these difficulties, we do not attempt a rigorous SED fitting in this paper, and instead we simply estimate the photometric redshifts of our 500\\,$\\mu$m risers based on their integrated $S_{250} \/ S_{350}$ vs. $S_{350} \/ S_{850}$ colours (Figure \\ref{fig: 500_riser_colours}). The purple and blue areas highlight regions in the colour-colour plot where $z \\geq 2$ and $1 < z < 2$ DSFGs likely lie based on photometry from typical DSFG templates measured at various redshifts (see \\cite{2019MNRAS.490.3840C} for more details). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figures\/herschel_colours_500_risers.pdf}\n\\caption{$S_{250} \/ S_{350}$ vs. $S_{350} \/ S_{850}$ colour-colour diagram for our four 500\\,$\\mu$m risers. The purple region in the lower left corner, and the blue region surrounding it, indicate likely colours for $z \\geq 2$ and $1 < z < 2$ DSFGs respectively. Bootes24, Bootes27 and XMM-M5 are all consistent with residing at $z \\geq 2$, while Bootes15 remains unconstrained due to the poor SCUBA-2 photometry.}\n\\label{fig: 500_riser_colours}\n\\end{figure}\n\nAs an independent estimate of the photometric redshift, we additionally run the {\\sc MMpz} algorithm\\footnote{\\url{http:\/\/www.as.utexas.edu\/~cmcasey\/mmpz.html}} \\citep{2020ApJ...900...68C} which finds the most likely redshift at which a galaxy resides by determining where its FIR\/mm SED is most consistent with the observed $L_{\\text{IR}} - \\lambda_{\\text{peak}}$ relation, where $\\lambda_{\\text{peak}}$ is the rest-frame wavelength at which the FIR\/mm SED peaks \\citep{2013ApJ...778..131L,2016ApJ...822...80S,2018ApJ...862...77C,2020ApJ...900...68C,2021arXiv211006930C}. We use the integrated FIR photometry for each source, including the \\textit{Herschel} SPIRE and SCUBA-2 photometry, and obtain a photometric redshift PDF for each of our 500\\,$\\mu$m risers. The full PDF for each source can be found in Appendix \\ref{sec: pdfs}. Based on the upper and lower 68\\% credible intervals, we find likely photometric redshifts of $1.83^{+1.33}_{-1.80}$ for Bootes15, $3.03^{+1.02}_{-0.90}$ for Bootes24, $2.53^{+0.83}_{-0.70}$ for Bootes27 and $2.88^{+1.02}_{-0.90}$ for XMM-M5. These values are in good agreement with the FIR\/sub-mm colours, and we infer that Bootes24, Bootes27 and XMM-M5 likely lie at $z \\geq 2$. We note that the photometric redshift of Bootes15 remains essentially unconstrained due to the poor SCUBA-2 photometry.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{figures\/cumulative_number_counts.pdf}\n\\caption{Cumulative number counts for the four 500\\,$\\mu$m riser fields, plotted alongside the expected cumulative number counts in the field from S2CLS. The Bootes15, Bootes24 and XMM-M5 fields are overdense in bright ($S_{850} \\gtrsim 8$\\,mJy) sub-mm sources.}\n\\label{fig: number_counts}\n\\end{figure*}\n\n\\subsection{Wider Environment}\n\n\\subsubsection{Overdensities of 850\\,$\\mu$m Sources}\n\\label{sec: number_counts}\n\nWe then investigate the wider environment surrounding our four 500\\,$\\mu$m risers using the SCUBA-2 850\\,$\\mu$m maps reduced using the \\texttt{REDUCE\\_SCAN\\_FAINT\\_POINT\\_SOURCES} recipe (as this reduction provides the highest signal-to-noise). We first crop the SCUBA-2 maps to a circle of diameter 700$^{\\prime\\prime}$ centred on the position of the SCUBA-2 detection, removing the edges of the map where the variance becomes much larger. We take each SNR map and find any pixel with $\\text{SNR} > 4$, with connected pixels considered part of the same source. We then manually examine the extracted sources to ensure that only $\\text{SNR} > 4$ detections are included and that there is no evidence of shredding (i.e. a single source being detected as multiple sources in the maps). In Appendix \\ref{sec: scuba_maps} we present the reduced SCUBA-2 850\\,$\\mu$m maps and the positions of the $\\text{SNR} > 4$ detected sources. For each source we extract deboosted flux densities and errors following the method outlined in Section \\ref{sec: extracting_flux}, and construct cumulative number counts for our maps in steps of 1\\,mJy. We note that the 500\\,$\\mu$m risers themselves are excluded from these cumulative number count calculations as they were selected to lie within the maps. We apply an effective area correction to account for the variable noise in the SCUBA-2 maps. For each of our detected SCUBA-2 sources, we take the extracted flux density prior to deboosting and find the threshold RMS noise value above which the source could no longer be detected with $\\text{SNR} > 4$. We then calculate the number of pixels in the map with a standard deviation below this threshold value and multiply by the area of one pixel to calculate the effective area ($A_{e}$) in square arcseconds over which the source could be detected to SNR $ > 4$. We convert this effective area into units of `map size', where one map size is equal to the area of the 700$^{\\prime\\prime}$ diameter circular region. Each source then effectively contributes $1\/A_{e}$ to the cumulative number counts \\citep{2017ApJ...850...37W}. \n\nIn order to determine whether there are overdensities of SMGs in our SCUBA-2 maps, we follow the analysis of \\cite{2019MNRAS.490.3840C} and compare our cumulative number counts to the expected number of field counts from S2CLS. We first convert the cumulative number counts per square degree from Table 4 in \\cite{2017MNRAS.465.1789G} to cumulative number counts per map size by multiplying by our map area in square degrees. In Figure \\ref{fig: number_counts} we compare the cumulative number counts from S2CLS with our SCUBA-2 maps. We then quantify the overdensity at each flux density level using Poisson statistics. We first take the observed cumulative number counts above each flux density level and calculate the probability of observing that number of sources using the Poisson probability mass function\n\n\\begin{eqnarray}\n    f(k) = \\exp(-\\mu)\\frac{\\mu^{k}}{k!},\n\\end{eqnarray}\n\nwhere $k$ represents the number of observed counts and $\\mu$ represents the expected number of counts from S2CLS. We then calculate the overdensity level using the equation\n\n\\begin{eqnarray}\n    \\delta = \\frac{k - \\mu}{\\sqrt{\\mu}},\n\\end{eqnarray}\n\n\n\\input{tables\/scuba_integrated_fir}\n\nas well as calculating an uncertainty in this overdensity by propagating the errors on the observed and expected counts.\n\nWe test the reliability in our SCUBA-2 maps using two separate methods. For the first method, we invert our SCUBA-2 point source reduced maps and extract the $>4\\sigma$ negative noise peaks using the same method as outlined above. Assuming that there are roughly equal numbers of positive and negative spurious peaks in our SCUBA-2 maps, taking the ratio of these negative peaks to the total number of extracted positive sources should give us a reasonable estimate of how many of our sources are spurious. For the Bootes15 and XMM-M5 fields, the reliability is $\\gtrsim 80\\%$ at $4\\sigma$, similar to other analyses of this nature \\citep{2017MNRAS.468.4006M, 2019MNRAS.490.3840C}. For the Bootes24 and Bootes27 fields, the reliability drops to $\\sim 60\\%$ and $\\sim 50\\%$ respectively. The second method makes use of `jackknife' maps to estimate the reliability. First, we used the \\texttt{MAKEMAP} command in the \\texttt{SMURF} package to create an individual map for each of the available SCUBA-2 scans. We then invert half of these maps before co-adding all scans together to produce a single map with the sources removed (i.e. containing just noise). As with the previous reductions, we then applied a matched filter and subsequently produced a signal-to-noise map for each field. We find that the Bootes15 and Bootes24 fields both have a 100\\% reliability at 4\\,$\\sigma$ based on the jackknife maps, while the Bootes27 field has a reliability of $\\sim 88\\%$. XMM-M5 has a somewhat lower reliability of 60\\% using this method. The reliability estimates based on the jackknife maps are likely more robust as they do not rely on the assumption that there are the same number of spurious positive and negative noise spikes in the maps. We therefore include the reliability estimates from the jackknife maps in the uncertainties associated with the overdensity estimates.  We additionally tested the reliability for numerous different reductions in which we co-added various combinations of the available SCUBA-2 integrations for each field. The aforementioned reliability values, and the following discussions, are based on those combinations of integrations with the highest reliabilities.\n\nTo test the completeness of our SCUBA-2 maps we inserted fake sources at random positions within the SCUBA-2 map, re-ran the source extraction method and observed how many of the fake sources were recovered. The details of this completeness calculation are discussed in Appendix \\ref{sec: completeness}. We find that Bootes15 and Bootes27 both reach a completeness of $\\sim 50\\%$ at $\\sim 10$\\,mJy, and rise to a completeness of $\\gtrsim 80\\%$ at 14\\,mJy. By comparison, Bootes24 and XMM-M5 have a slightly poorer completeness, reaching $\\sim 80$\\% at $\\sim 17$\\,mJy and $\\sim 16$\\,mJy respectively. \nWe find that Bootes15 hosts an $8.6 \\pm 3.5$\\,$\\sigma$ overdensity of SMGs brighter than 8\\,mJy, while Bootes24 shows a more marginal $5.8 \\pm 3.5$\\,$\\sigma$ overdensity of SMGs brighter than 8\\,mJy. The probabilities of observing these numbers of sources in a random field are $2.8 \\times 10^{-7}$ and $1.1 \\times 10^{-4}$ respectively. While Bootes27 appears to harbour a large overdensity of faint ($S_{850} > 4$\\,mJy) sources, this overdensity is driven by a single source with a deboosted flux density of $\\sim 4.9$\\,mJy that would only be detected to $\\text{SNR} > 4$ in $\\sim 3\\%$ of the total map area. This results in a large effective area correction and hence an artificially high overdensity value. If this single source is removed, then we do not observe any overdensity in the Bootes27 region. For XMM-M5, we find an overdensity of $9.3 \\pm 6.5$\\,$\\sigma$ of sources brighter than 9\\,mJy compared to the field counts, with a probability of observing this number of counts in a blank field of $\\sim 2 \\times 10^{-6}$. This analysis therefore identifies Bootes15, Bootes24 and XMM-M5 as regions plausibly containing overdensities of bright SCUBA-2 850\\,$\\mu$m sources.\n\n\\subsubsection{Cross-matching with HerMES Sources}\n\nTo investigate these potential overdensities further, we then cross-match the SCUBA-2 850\\,$\\mu$m detected sources with the HerMES catalogues. We use the optimum search radius of 9$^{\\prime\\prime}$ identified by \\cite{2019MNRAS.490.3840C},  matching half of the FWHM of the \\textit{Herschel} SPIRE 250\\,$\\mu$m beam, such that any SCUBA-2 source within 9$^{\\prime\\prime}$ of a source in the HerMES catalogue is considered a potential companion. Through this cross-matching process we find that $3\/9$ (33\\%), $2\/4$ (50\\%), $3\/7$ (43\\%) and  $3\/5$ (60\\%) SCUBA-2 detections have \\textit{Herschel} counterparts for the Bootes15, Bootes24, Bootes27 and XMM-M5 fields respectively. These numbers exclude the 500\\,$\\mu$m risers themselves. The positions and FIR\/sub-mm photometry for each of these sources can be found in Table \\ref{tab: scuba_integrated_fir}. In Figure \\ref{fig: colours} we present the $S_{250} \/ S_{350}$ vs. $S_{350} \/ S_{850}$ colours for both the 500\\,$\\mu$m risers and the SCUBA-2 detections, while in Figure \\ref{fig: mmpz_redshifts} we plot their photometric redshifts and associated errors as estimated by {\\sc MMpz} based on their \\textit{Herschel} and SCUBA-2 photometry. The full photometric redshift PDFs from {\\sc MMpz} can be found in Appendix \\ref{sec: pdfs}.\n\nThe photometric redshift of Bootes15 is essentially unconstrained due to its poor SCUBA-2 photometry. However, the three SCUBA-2 detected sources with HerMES counterparts in the Bootes15 field have FIR\/sub-mm colours consistent with $z \\geq 2$, although {\\sc MMpz} has trouble constraining the redshifts of two of them. Both Bootes24 and its companion SCUBA-2 detections have colours consistent with $z \\geq 2$, and {\\sc MMpz} places these sources at $z \\sim 2-3$ (although Bootes24 itself suffers from larger error bars, placing it between $z \\sim 2-4$). The FIR\/sub-mm colours of sources in the Bootes27 region suggest that they are spread over a large range in redshift and so are unlikely to be physically associated, and this is supported by the results of {\\sc MMpz}. The XMM-M5 field is plausibly overdense in FIR\/sub-mm bright sources, and their colours suggest that they all reside at $z \\geq 2$. The {\\sc MMpz} results, however, offer a somewhat more complicated picture, placing two of the SCUBA-2 detections at $z \\sim 1.5 - 2$ and the third (XMM-M5.SCUBA1) at $z \\sim 4$. This is consistent with the \\textit{Herschel} SPIRE photometry for XMM-M5.SCUBA1 which satisfies the colour criteria for a 500\\,$\\mu$m riser (see Table \\ref{tab: scuba_integrated_fir}). \n\nFinally, we note that, for Bootes27,  one of the $\\text{SNR} > 4$ SCUBA detections lies within $\\sim 31^{\\prime\\prime}$ of the Bootes27 500\\,$\\mu$m riser, placing it just within the FWHM of the \\textit{Herschel} SPIRE beamsize at 500\\,$\\mu$m ($35.2^{\\prime\\prime}$) but outside of the 250\\,$\\mu$m ($17.6^{\\prime\\prime}$) and 350\\,$\\mu$m ($23.9^{\\prime\\prime}$) beamsize. This SCUBA-2 source does not have a separate counterpart in the HerMES catalogues, indicating that it may contribute to making Bootes27 artificially red. For the remaining 500\\,$\\mu$m risers, the closest $\\text{SNR} > 4$ detected SCUBA-2 source is too distant to be blended within the \\textit{Herschel} SPIRE beamsize.  \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{figures\/herschel_colours.pdf}\n\\caption{The $S_{250} \/ S_{350}$ vs. $S_{350} \/ S_{850}$ FIR\/sub-mm colours for our four 500\\,$\\mu$m risers (dark points) alongside the $\\text{SNR} > 4$ detections in the SCUBA-2 850\\,$\\mu$m maps with \\textit{Herschel} counterparts (light points). The purple region in the lower left corner, and the blue region surrounding it, indicate likely colours for $z \\geq 2$ and $1 < z < 2$ DSFGs respectively. These plots demonstrate that the overdensities associated with Bootes24 and XMM-M5 are potentially physically associated at $z \\geq 2$.}\n\\label{fig: colours}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{figures\/mmpz_redshifts.pdf}\n\\caption{The photometric redshifts for our 500\\,$\\mu$m risers and SCUBA-2 $\\text{SNR} > 4$ detected sources estimated by {\\sc MMpz}. Similarly to the FIR\/sub-mm colours, we find that the Bootes24 region is consistent with containing a physically associated overdensity of bright DSFGs, with well constrained photometric redshifts placing the surrounding SMGs at $z \\approx 2 - 3$, and Bootes24 itself at $z \\approx 2 - 4$. While the photometric redshift of XMM-M5 does overlap with those of the bright SCUBA-2 sources, there is a larger range of photometric redshifts, indicating that this region may be somewhat more complex. The DSFGs associated with Bootes15 do not have well constrained photometric redshifts, and the bright SCUBA-2 sources in the Bootes27 region are likely unassociated with each other.}\n\\label{fig: mmpz_redshifts}\n\\end{figure*}\n\n\n\n\\section{Discussion}\n\\label{sec: discussion}\n\n\\subsection{Multiplicities}\n\\label{sec: multiplicities}\n\nWe find evidence of multiplicity in three of our four 500\\,$\\mu$m risers, with Bootes15, Bootes24 and Bootes27 each breaking up into two resolved sources in the high resolution SMA maps, and all three showing extended emission in the SCUBA-2 maps. This builds on the work of G20, who found that $\\sim 35\\%$ of bright ($S_{500} > 60$\\,mJy) 500\\,$\\mu$m risers are multiples, while fainter 500\\,$\\mu$m risers are also blends more often than expected. These conclusions were reached based on the assumption that the twelve 500\\,$\\mu$m risers without SMA detections in their follow-up observations are comprised of multiple DSFGs that were individually too faint to be detected in their SMA maps, but all of which contribute to the bright \\textit{Herschel} source. By identifying evidence of multiplicity in three out of four of these 500\\,$\\mu$m risers, our observations support this conclusion. It is worth noting that, while we do observe evidence of multiplicity, we currently do not detect the high multiplicities of $\\geq 3$ that were predicted by G20 for these 500\\,$\\mu$m risers. However, given the discrepancies between the \\textit{Herschel}, SCUBA-2 and SMA photometry, we cannot rule out the possibility of additional faint counterparts lying below our current detection limits. In the following sections, we discuss Bootes15 and Bootes27, which are particularly interesting cases, in more detail. \n\n\\subsubsection{Bootes15}\n\\label{sec: bootes15_multiplicity}\n\nFigure \\ref{fig: interferometer_vs_single-dish} demonstrates that the SMA and SCUBA-2 photometry for Bootes15 are in good agreement within the rather large 850\\,$\\mu$m error bars, but Figure \\ref{fig: 850v500} indicates that its 850\\,$\\mu$m flux density is much lower than we would expect based on its 500\\,$\\mu$m flux density. The SCUBA-2 observations reach a depth of 2.2\\,mJy, and so it is reasonable to expect that at least part of this discrepancy could be due to the SCUBA-2 observations missing a significant fraction of the total 850\\,$\\mu$m flux density in the system. This missing flux density could be in the form of extended, diffuse emission or additional point-like sources lying below our SCUBA-2 detection threshold. However, if we assume that the 850\\,$\\mu$m flux density is underestimated and in reality is more in line with similarly bright \\textit{Herschel} sources in the S2CLS and STUDIES samples (e.g. $\\gtrsim 10$\\,mJy at 850\\,$\\mu$m), this would introduce a significant discrepancy between the SMA and SCUBA-2 photometry, similar to the discrepancy we observe in Bootes27. We therefore suggest that this discrepancy is caused not only by missing 850\\,$\\mu$m flux density in the SCUBA-2 observations, but also in part by faint sources lying below both our SMA and SCUBA-2 detection limits. Additional undetected sources would not only account for the faint SMA and SCUBA-2 flux densities, but the effects of blending would also artificially redden the \\textit{Herschel} SPIRE photometry. \\cite{2015ApJ...812...43B} find that blending can have a significant effect on the FIR\/sub-mm colours of bright \\textit{Herschel} sources. For example, blends of three sources can boost the $S_{500} \/ S_{350}$ colours of bright \\textit{Herschel} sources by as much as $\\sim 1.8$ times the deblended colours. This interpretation can therefore simultaneously account for the bright \\textit{Herschel} SPIRE source associated with Bootes15, its red \\textit{Herschel} colours and its unexpectedly faint SCUBA-2 and SMA photometry.\n\n\n\\subsubsection{Bootes27}\n\\label{sec: bootes27_multiplicity}\n\nBootes27 is another interesting case study. While the bright \\textit{Herschel} 500\\,$\\mu$m riser does resolve into two individual SMA sources, one of these is much fainter than the other, remains undetected in the SCUBA-2 map and is offset from the detected SCUBA-2 850\\,$\\mu$m emission by $\\sim 10^{\\prime\\prime}$. The SCUBA-2 contours show two partially resolved sources, a southern component coincident with the brighter SMA source, and a northern component with no apparent counterpart in the SMA images. Moreover, even excluding the partially resolved northern SCUBA-2 component, the brighter southern component has a peak, deboosted flux density of $\\sim 6.9$\\,mJy, significantly larger than the $1.9 \\pm 0.3$\\,mJy resolved SMA source associated with it. We also note that the single resolved SMA source is slightly offset from the brightest pixel in the SCUBA-2 maps. We interpret this tension between the SCUBA-2 and SMA observations as evidence of additional faint sources lying below even our deeper SMA detection threshold.\n\nWe can make a rough estimate of the number of sources that likely remain undetected in Bootes27 by comparing the integrated flux density from the SCUBA-2 observations to the resolved flux density from the SMA observations. The dirty map for Bootes27 that we used for source extraction has a $1\\sigma$ RMS noise of $\\sim 0.29$\\,mJy\/beam, meaning that any source with a flux density of less than $\\sim 1.1$\\,mJy would fall below the required $3.75\\sigma$ detection threshold. The single resolved SMA source has a flux density of $1.9 \\pm 0.3$\\,mJy. Based on the integrated SCUBA-2 flux density of $10.1 \\pm 1.6$\\,mJy, we require at least 2.5\\,mJy of additional flux density in our SMA maps in order to avoid a 3$\\sigma$ tension between the SMA and SCUBA-2 photometry, corresponding to at least three additional faint sources lying below our SMA detection threshold. Alternatively, if we take the flux density estimates at face value, the SMA observations miss $8.2 \\pm 1.6$\\,mJy of flux density, and we would therefore require $\\sim 6 - 9$ undetected SMA sources contributing to the total SCUBA-2 flux density. Such a high multiplicity would make Bootes27 a candidate high redshift protocluster core, potentially similar to those discovered by \\cite{2018Natur.556..469M} and \\cite{2018ApJ...856...72O}. We note that the ancillary \\textit{Spitzer} IRAC data for Bootes27 shows $\\sim 7$ individual sources detected to $>3\\sigma$ at 8.0\\,$\\mu$m, $\\sim 3$ of which remain completely undetected at 4.5\\,$\\mu$m, indicating that there may be additional, extremely red sources contributing to the SCUBA-2 flux density that are undetected in the SMA map. A more comprehensive, multi-wavelength analysis of Bootes27 will be presented in Cairns et al. (in prep).\n\n\\subsection{Photometric Redshifts}\n\\label{sec: photo-z}\n\nBased on their integrated FIR\/sub-mm colours in Figure \\ref{fig: 500_riser_colours}, we find that our 500\\,$\\mu$m risers likely lie at $z \\geq 2$, excluding Bootes15 whose FIR\/sub-mm colours remain poorly constrained due to the uncertain 850\\,$\\mu$m photometry. This result is corroborated by {\\sc MMpz}, which indicates that these 500\\,$\\mu$m risers likely lie at $z \\sim 2 - 4$. These photometric redshifts are consistent with our interpretation of these sources as blends of multiple DSFGs. As well as being effective at selecting intrinsically red, high redshift DSFGs, the 500\\,$\\mu$m riser selection criterion will also naturally include some fraction of intermediate redshift systems that are artificially reddened by the blending of multiple sources at moderately large separations. For example, \\cite{2021MNRAS.505.5260M} find in their sample of $\\sim 100$ 500\\,$\\mu$m risers that the photometric redshift distribution of those that resolve into multiple components is skewed towards lower redshifts, with a slightly lower median redshift ($z _{\\text{med}} = 3.5$ vs. $z _{\\text{med}} = 3.8$) and a higher fraction residing at lower redshifts (27\\% at $z_{\\text{phot}} < 3$ vs. 10\\%) compared to single systems. We find that our 500\\,$\\mu$m risers typically fall into the category of multiple systems at moderate redshifts and, based on their FIR\/sub-mm flux densities and intermediate photometric redshifts, we would expect them to be examples of Hyper-Luminous Infrared Galaxies (HLIRGS: $L_{\\text{FIR}} > 10^{13}$\\,L$_{\\odot}$), similar to those discussed in G20.\n\n\\subsection{Environments}\n\\label{sec: environments}\n\nOur SCUBA-2 850\\,$\\mu$m observations map out sub-mm emission over a $>10'$ region surrounding our 500\\,$\\mu$m risers, and are therefore extremely useful for characterising their environments. DSFGs have the potential to be used as signposts for overdensities of galaxies in the early Universe, allowing us to identify the potential progenitors of massive, local galaxy clusters. We find that, while the Bootes27 region is consistent with the expected number of field counts from S2CLS, Bootes15, Bootes24 and XMM-M5 are all consistent with lying in overdensities of SMGs with 850\\,$\\mu$m flux densities greater than $\\sim 8$\\,mJy. We do note, however, that the error bars on these overdensity values for Bootes24 and XMM-M5 are rather large, with only a very marginal overdensity at the lower limit. This result is interesting as we would not necessarily expect our 500\\,$\\mu$m risers to be effective tracers of high-redshift galaxy overdensities - based on the clustering analysis of SMGs from the ALESS survey, \\cite{2020ApJ...904....2G} find that only the brightest SMGs ($S_{870} > 5 - 6$\\,mJy) should trace massive structures at $z \\sim 2$, while our sources are significantly fainter than this in the resolved SMA maps. Without spectroscopic redshifts for the SCUBA-2 sources we cannot say with certainty whether these overdensities are physically associated structures or simply chance line of sight projections of bright sub-mm sources. In the absence of spectroscopic data, we must rely on photometric redshifts estimated from FIR\/sub-mm colour-colour diagrams and the {\\sc MMpz} algorithm. From Figures \\ref{fig: colours} and \\ref{fig: mmpz_redshifts}, we infer that the marginal overdensity associated with Bootes24 could be physically associated, with the FIR\/sub-mm colours placing the sources at $z \\geq 2$, and {\\sc MMpz} placing them at $z \\sim 2-4$. For the Bootes15 field, improvements in the FIR\/sub-mm photometry are required to better constrain the photometric redshifts of the detected sources, while the sources in the Bootes27 field likely reside at very different redshifts. The SCUBA-2 detections in the XMM-M5 field could represent an overdensity of SMGs, and their colours suggest that they are all at $z \\geq 2$, but {\\sc MMpz} places two of these sources at $z \\sim 1.5 - 2$, and the third at $z \\sim 4$. This overdensity is therefore less likely to be a physically associated structure than Bootes24. Additional observations providing improved photometry and\/or spectroscopic redshifts for these sources will be required to conclude with certainty whether they are physically associated structures.\n\n\\subsection{Comparison to Predictions for Bright \\textit{Herschel} Sources}\n\\label{sec: simulations}\n\n\\subsubsection{Multiplicity Fraction}\n\nOur results support the conclusions of G20, who find that $\\sim 35\\%$ of bright ($S_{500} > 60$\\,mJy) 500\\,$\\mu$m risers, and $\\sim 60\\%$ of faint ($S_{500} < 60$\\,mJy) 500\\,$\\mu$m risers are multiples, although we note that these observations were somewhat inhomogeneous, making use of a variety of frequencies (ranging from 231\\,GHz to 346\\,GHz) and configurations, with a mixture of full tracks, partial tracks and track sharing, resulting in a range of angular resolutions (with an average beam FWHM of $2.5 \\pm 0.8$ arcseconds) and depths ($0.5-2.9$\\,mJy\/beam).\n\n\\cite{2017arXiv170904191O} provide high-resolution ($\\sim 0.12^{\\prime\\prime}$) ALMA follow-up observations at 870\\,$\\mu$m for 44 ultra-red DSFGs from HerMES and H-ATLAS with $S_{500} \/ S_{250} > 1.5$ and $S_{500} \/ S_{350} > 1.0$, reaching $1\\sigma$ RMS sensitivities of 0.1\\,mJy\/beam. They find that, in total, $\\approx 39\\%$ resolve into multiple components and, more specifically, $\\approx 22\\%$ of bright ($S_{500} > 60$\\,mJy) and $\\approx 65\\%$ of faint ($S_{500} \\leq 60$\\,mJy) sources in their sample resolve into multiple components.\n\n\\cite{2017A&A...607A..89B} produce a FIR to sub-millimeter simulation of the extragalactic sky based on a galaxy evolution model with two star-formation modes. They extract sources from maps based on these observations to investigate the effects that the limited angular resolution of single-dish instruments has on observations in the FIR\/sub-mm. They predict that all \\textit{Herschel} sources brighter than 60\\,mJy at 500\\,$\\mu$m should be individual, lensed sources, while $\\sim40\\%$ of sources fainter than 60\\,mJy at 500\\,$\\mu$m should be blended. \n\nIn their compilation of 63 500\\,$\\mu$m risers, \\cite{2019ApJS..244...30M} estimate a multiplicity fraction of $\\sim 27\\%$ based on numerous follow-up observations with NOEMA, ALMA and the SMA, although these studies reach various depths and angular resolutions, as well as probing much fainter 500\\,$\\mu$m flux densities. This multiplicity fraction rises to $\\sim 39\\%$ when sources showing evidence of strong gravitational lensing are removed.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{figures\/flux_distribution_v2.pdf}\n\\caption{The relative contributions of each of our SMA detections to the total integrated SCUBA-2 850\\,$\\mu$m flux density. The orange and purple lines show the flux density distributions of the DRC and SPT2349-56 protoclusters respectively, while the black dashed lines show the $3.75\\sigma$ detection limits in our dirty SMA maps.  The contribution of each SMA source to the integrated SCUBA-2 flux density of Bootes15 remains poorly constrained. The SMA sources associated with Bootes24 and Bootes27 recover $60 \\pm 15\\%$ and $19 \\pm 4\\%$ of the total integrated SCUBA-2 flux density and so there may be additional sources residing below our SMA detection limits. However, if further sources do reside below our detection threshold, they likely follow a different flux density distribution to the DRC and SPT2349-56 protoclusters.}\n\\label{fig: flux_distribution}\n\\end{figure*}\n\n\\cite{2021MNRAS.505.5260M} carry out 1.1\\,mm LMT\/AzTEC observations of $\\sim 100$ 500\\,$\\mu$m risers with $S_{500} \\approx 35 - 80$\\,mJy, achieving an average beam FWHM of $9.6 \\pm 0.5$ arcseconds and $1\\sigma$ RMS sensitivities of $\\approx 0.7-2.8$\\,mJy. They find that $\\sim 9\\%$ show direct evidence of multiplicity, rising to $\\sim23\\%$ if 500\\,$\\mu$m risers which display evidence of multiplicity but are slightly below their detection threshold are included, and up to $\\sim50\\%$ if 500\\,$\\mu$m risers without counterparts in their higher resolution observations are assumed to be multiple sources. This is considered an extreme scenario as a higher dust spectral emissivity index $\\beta$ could also render some of their sources undetectable at 1.1\\,mm, but our observations of 500\\,$\\mu$m risers without counterparts in previous, high resolution follow-up observations indicate that a significant fraction of these non-detections could be multiple systems. For the twelve sources in their sample with $S_{500} > 60$\\,mJy, two break up into multiple components and three show no counterparts in their higher resolution images, indicating a multiplicity of $\\approx 17-42\\%$. However, the larger beamsize of their observations compared to \\cite{2017arXiv170904191O} and G20 probe multiplicity on a significantly different scale, such that smaller scale multiplicities would likely be classified as individual sources. They estimate a correction factor of $\\sim 10\\%$ accounting for multiple systems separated by distances much smaller than their $9.6 \\pm 0.5$ arcsecond beamsize, bringing the multiplicity fraction up to  $\\gtrsim 18\\%$ in the conservative case and $\\sim 60\\%$ in the extreme case.\n\nWe find that our three Bootes sources all show some evidence of multiplicity, with all three resolving into two sources in the high-resolution SMA maps and showing an extended morphology in the SCUBA-2 maps. This result supports the conclusions of G20 and, by extension, is in good agreement with the multiplicity fractions for 500\\,$\\mu$m risers quoted by \\cite{2017arXiv170904191O}, despite their ALMA observations achieving significantly better sensitivities and angular resolutions. These results are also in reasonably good agreement with the more extreme scenario in \\cite{2021MNRAS.505.5260M}, and well within the range of potential multiplicity fractions of the brightest ($S_{500} > 60$\\,mJy) sources in their sample. These multiplicity fractions indicate that 500\\,$\\mu$m risers are a much more diverse population than previously predicted. \n\n\\subsubsection{Brightest-Galaxy Fraction}\n\nWe can also compare the flux distributions of the resolved SMA sources in our 500\\,$\\mu$m risers to the literature. \\cite{2017A&A...607A..89B}, based on their simulations, predict that the brightest galaxy should contribute $\\sim 60\\%$ of the total 500\\,$\\mu$m flux density. Similarly, \\cite{2018A&A...614A..33D} find that the brightest galaxy inside the \\textit{Herschel} beam for their simulated SPIRE data contributes on average $75\\%$ and $64\\%$ at 250\\,$\\mu$m and 500\\,$\\mu$m respectively. \\cite{2021MNRAS.505.5260M} find that the brightest component contributes on average $50 - 75\\%$ of the total flux density at 1.1\\,mm, while \\cite{2019ApJS..244...30M} find that the brightest ALMA source contributes on average $41-80\\%$ of the total ALMA flux density and $15-59\\%$ of the total SCUBA-2\/LABOCA flux density at a similar wavelength. \n\nWe find that the brightest SMA component in Bootes15 contributes $56 \\pm 12\\%$ to the total SMA flux density, and $69 \\pm 41\\%$ of the integrated SCUBA-2 850\\,$\\mu$m flux density. However, as discussed in Section \\ref{sec: SMA_SCUBA_detection}, it is likely that we do not recover the total integrated FIR\/sub-mm flux density of Bootes15 in our observations. For Bootes24 and Bootes27, the brightest SMA component contributes $50 \\pm 9\\%$ and $53 \\pm 18\\%$ respectively to their total SMA flux densities, but just $30 \\pm 8\\%$ and $19 \\pm 4\\%$ respectively to their integrated SCUBA-2 flux densities at a similar wavelength. These values are in good agreement with \\cite{2019ApJS..244...30M}, and indicate that for Bootes24 and Bootes27 we may not recover the total sub-mm flux density. \n\n\\subsubsection{Comparison to Known Protocluster Cores}\n\nWe can also compare these values to the flux density distributions of known high-$z$ protocluster cores, such as the Distant Red Core \\cite[DRC:][]{2018ApJ...856...72O} and SPT2349-56 \\citep{2018Natur.556..469M}. For the DRC, we focus on the ALMA 2\\,mm follow-up observations carried out by \\cite{2018ApJ...856...72O}, reaching depths of $\\sim 6$\\,$\\mu$Jy\/beam with a synthesised beam FWHM of $1.68^{\\prime\\prime} \\times 1.54^{\\prime\\prime}$. For SPT2349-56, we use the flux densities extracted from the ALMA 850\\,$\\mu$m continuum maps based on the [{\\sc Cii}] map channels with no line emission \\citep{2020MNRAS.495.3124H}. In Figure \\ref{fig: flux_distribution}, we then plot the contribution of each of the ten brightest resolved sources associated with the protocluster cores to the total flux density in the aforementioned bands. For our 500\\,$\\mu$m risers, we plot the contribution of each of the resolved SMA sources to the integrated SCUBA-2 850\\,$\\mu$m flux density, and estimate the uncertainties in these contributions by adding in quadrature the errors in the resolved SMA and integrated SCUBA-2 flux densities. Note that we exclude XMM-M5 from this plot, as the SMA and SCUBA-2 observations were taken at very different wavelengths. We additionally exclude the fainter SMA source associated with Bootes27, as it is offset from the observed SCUBA-2 emission and so likely does not contribute to the observed 850\\,$\\mu$m flux density.\n\nFor Bootes15, the contribution of each SMA source to the total SCUBA-2 flux density is poorly constrained, primarily due to the large uncertainties in the 850\\,$\\mu$m photometry. In total, we estimate that the SMA observations recover $123 \\pm 73\\%$ of the SCUBA-2 observations. However, as previously discussed, it is likely that some fraction of the total SCUBA-2 flux density is missed by our observations. The two SMA sources associated with Bootes24 recover $60 \\pm 15\\%$ of the total SCUBA-2 flux density, shared equally between the two resolved sources.  Therefore, the combined contribution from the two brightest sources is in good agreement with the two brightest sources in the DRC, which similarly contribute $\\sim 60\\%$ to the total flux density of the system. However, the flux density is shared much more unequally between the two brightest sources in the DRC. If there were additional sub-mm sources contributing to Bootes24, we would expect to detect one extra source in our SMA observations if they were to follow a DRC-like flux density distribution. Alternatively, we would expect to detect the brightest $4-5$ sources if they were to follow a SPT2349-56-like flux density distribution. For Bootes27, the single resolved SMA source associated with the observed SCUBA-2 emission contributes just $19 \\pm 4\\%$ to the integrated 850\\,$\\mu$m flux density. This is similar to the $\\sim 22\\%$ contribution to the total system from the brightest source in SPT2349-56. However, if Bootes27 were to represent a protocluster core with a similar flux density distribution to SPT2349-56, we would similarly expect to detect the brightest $4-5$ components, whereas we only detect a single source associated with the 850\\,$\\mu$m emission. \n\nWe infer that Bootes24 and Bootes27 may contain additional, faint sources lying below our SMA detection threshold that contribute to the total integrated SCUBA-2 flux density. However, it seems that this flux density must be distributed between its members differently to known high redshift protocluster cores, with each of the fainter members individually contributing significantly less to the total sub-mm flux density than expected. \n\n\n\\section{Conclusions}\n\\label{sec: conclusions}\n\nWe have presented \\textit{Herschel}, SMA and SCUBA-2 observations of four 500\\,$\\mu$m risers from the HerMES survey. These sources were selected based on their red \\textit{Herschel} SPIRE colours, but remained undetected in previous high-resolution follow-up observations conducted by G20. The previous results indicated that they were likely comprised of multiple DSFGs that individually were faint enough to remain undetected in their high resolution SMA observations, but each contributing to a large integrated flux density when smoothed over the \\textit{Herschel} SPIRE beamsize. In our deeper FIR\/sub-mm observations, each source is detected to $>3\\sigma$ in the SCUBA-2 maps, and there is at least one $>3.75\\sigma$ detection in each of the SMA maps. Our findings can be summarised as follows.\n\n\\begin{enumerate}\n    \\item We find evidence of multiplicity in three out of four 500\\,$\\mu$m risers. Bootes15, Bootes24 and Bootes27 each break up into two faint sources in the high-resolution SMA images, while XMM-M5 resolves into a single source. Bootes27 additionally displays a bright, partially resolved SCUBA-2 detection without any apparent SMA counterparts, indicating that there may be further faint DSFGs lying below the SMA detection threshold. This is in line with the results of G20 who found (by assuming that their 500\\,$\\mu$m risers without SMA counterparts are sources with high multiplicities) that $\\sim35\\%$ of bright ($S_{500} > 60$\\,mJy) 500\\,$\\mu$m risers and $\\sim 60\\%$ of faint ($S_{500} < 60$\\,mJy) 500\\,$\\mu$m risers are multiples. By providing deeper SMA follow-up observations of these 500\\,$\\mu$m risers, we confirm that they are likely comprised of multiple faint components. These results indicate that the 500\\,$\\mu$m riser population is significantly more diverse than expected.\n    \n    \\item For the three 500\\,$\\mu$m risers in the Bootes field, the SMA observations recover $123 \\pm 73\\%$, $60 \\pm 15\\%$ and $19 \\pm 4\\%$ of the integrated SCUBA-2 flux density respectively indicating that, for Bootes24 and Bootes27, there may be additional, faint sources below the current SMA detection limit, but contributing to the integrated SCUBA-2 flux density. In particular, for Bootes27, there is a $>3\\sigma$ disparity between the resolved SMA and integrated SCUBA-2 flux densities. We estimate that at least three additional faint sources are required to recover the total SCUBA-2 flux density. This could make Bootes24 and Bootes27 examples of dense, protocluster cores of DSFGs, only a handful of which have been discovered to date.\n    \n    \\item It is likely that our SCUBA-2 observations miss some sub-mm flux density, particularly for Bootes15, either in the form of extended diffuse emission or additional, faint, point-like sources. For Bootes27, there is a resolved SMA source offset from the observed SCUBA-2 emission and without an apparent SCUBA-2 counterpart.\n    \n    \\item We estimate the photometric redshifts of our 500\\,$\\mu$m risers using their FIR\/sub-mm colours, as well as the {\\sc MMpz} algorithm. While the photometric redshift of Bootes15 remains poorly constrained, Bootes24, Bootes27 and XMM-M5 all likely lie at $z \\geq 2$. This is consistent with the interpretation that the 500\\,$\\mu$m riser selection criterion recover both intrinsically red sources at $z > 4$ and artificially red, multiple systems at more moderate redshifts. Based on their FIR flux densities and photometric redshifts, we expect our sources to be HLIRGs, similar to those in the G20 sample.\n\n    \\item By comparing the cumulative number counts of $\\text{SNR} > 4$ detections in the SCUBA-2 850\\,$\\mu$m maps with the expected number of field counts from the SCUBA-2 Cosmology Legacy Survey, we find that Bootes15, Bootes24 and XMM-M5 lie in regions that are $8.6 \\pm 3.5$\\,$\\sigma$, $5.8 \\pm 3.5$\\,$\\sigma$ and $9.3 \\pm 6.5$\\,$\\sigma$ overdense in bright ($\\gtrsim 8$\\,mJy) SMGs respectively. For those SCUBA-2 detections that have \\textit{Herschel} SPIRE counterparts, we estimate photometric redshifts based on their FIR\/sub-mm photometry. We find that the overdensity associated with Bootes24 could feasibly be physically associated, while Bootes15 and XMM-M5 are more likely to be chance line-of-sight arrangements of sources.\n    \n    \\item We compare the inferred multiplicity fractions of 500\\,$\\mu$m risers from G20 to similar studies in the literature, finding that they are in good agreement with \\cite{2017arXiv170904191O} and with the more extreme scenarios in \\cite{2021MNRAS.505.5260M}. We also find that the brightest SMA component of Bootes24 and Bootes27 contributes a similar amount to the total integrated SCUBA-2 flux density as the brightest resolved source associated with the 500\\,$\\mu$m risers in the  \\cite{2019ApJS..244...30M} sample.\n    \n    \\item We also compare the flux density distributions of our resolved SMA sources to those of known protocluster cores. We find that, while the SMA sources associated with Bootes24 and Bootes27 only recover a fraction of the total sub-mm flux density, we would have expected to detect the brightest $4-5$ components if the multiple systems followed a similar flux density distribution to SPT2349-56, or the brightest three components if they followed a similar flux density distribution to the DRC. Therefore, if these 500\\,$\\mu$m risers are examples of dense protocluster cores, they likely follow a different flux density distribution to known, high-$z$ protocluster cores.\n    \n\\end{enumerate}\n\nWe therefore find that our 500\\,$\\mu$m risers are typically comprised of multiple DSFGs and may represent examples of high redshift protocluster cores similar to those found by \\cite{2018Natur.556..469M} and \\cite{2018ApJ...856...72O}. However, additional follow-up observations will be required to confirm the nature of these sources, and there are a number of available avenues that these observations could take. Deeper sub-mm continuum\/spectroscopic observations with NOEMA will likely be required to determine if there are any additional, faint, point-like sources below our current SMA detection limits that contribute to the integrated SCUBA-2 flux (and if these sources are physically associated), particularly for Bootes24 and Bootes27 where our SMA observations currently only recover a fraction of the total 850\\,$\\mu$m flux density. Alternatively, deep optical\/NIR follow-up observations would allow us to accurately deblend the \\textit{Herschel} SPIRE flux density, from which we could attempt a rigorous SED fitting for each of the resolved components and estimate galaxy properties such as SFR and stellar mass. Finally, additional spectroscopic follow-up observations for sources in the wider field would allow us to conclude whether any larger scale overdensities in these regions represent physically associated structures.\n\n\\section*{Acknowledgements}\n\nThe authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.\n\nThe Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. \n\nThe James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; Center for Astronomical Mega-Science (as well as the National Key R\\&D Program of China with No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities and organizations in the United Kingdom and Canada. \n\nThis research has made use of data from HerMES project (\\url{http:\/\/hermes.sussex.ac.uk\/}). HerMES is a Herschel Key Programme utilising Guaranteed Time from the SPIRE instrument team, ESAC scientists and a mission scientist.The HerMES data was accessed through the Herschel Database in Marseille (HeDaM - \\url{http:\/\/hedam.lam.fr}) operated by CeSAM and hosted by the Laboratoire d'Astrophysique de Marseille.\n\nThis research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.\n\nThis research made use of APLpy, an open-source plotting package for Python hosted at \\url{http:\/\/aplpy.github.com}. This research made use of Astropy, a community-developed core Python package for Astronomy \\citep{2013A&A...558A..33A,2018AJ....156..123A}. This research made use of Photutils, an Astropy package for\ndetection and photometry of astronomical sources \\citep{larry_bradley_2021_5796924}. This research made use of the Starlink Table\/VOTable Processing Software \\texttt{TOPCAT} \\citep{2005ASPC..347...29T}.\n\nJ.C. acknowledges support from the Science and Technology Facilities Council [grant number ST\/S505432\/1]. I.P.-F. acknowledges support from the Spanish State Research Agency (AEI) under grant number PID2019-105552RB- C43. \n\n\\section*{Data Availability}\nThe data underlying this article will be shared on reasonable request to the corresponding author.\n\n\n\n\\balance\n\\bibliographystyle{mnras}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider the problem of estimating a random vector $\\mathbf{x}^0$\nfrom linear measurements $\\mathbf{y}$ of the form\n\\begin{equation} \\label{eq:yAx}\n    \\mathbf{y} = \\mathbf{A} \\mathbf{x}^0 + \\mathbf{w}, \\quad\n    \\mathbf{w} \\sim {\\mathcal N}(\\mathbf{0},\\theta_2^{-1} \\mathbf{I}),\n    \\quad\n    \\mathbf{x}^0 \\sim p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1),\n\\end{equation}\nwhere $\\mathbf{A}\\in{\\mathbb{R}}^{M\\times N}$ is a known matrix,\n$p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ is a density on $\\mathbf{x}^0$ with\nparameters ${\\boldsymbol{\\theta}}_1$,\n$\\mathbf{w}$ is additive white Gaussian noise (AWGN) independent of $\\mathbf{x}^0$,\nand $\\theta_2 > 0$ is the noise precision (inverse variance).\nThe goal is to estimate $\\mathbf{x}^0$,\nwhile simultaneously learning the unknown parameters\n${\\boldsymbol{\\theta}} := ({\\boldsymbol{\\theta}}_1,\\theta_2)$, from the data $\\mathbf{y}$ and $\\mathbf{A}$.\nThis problem arises in Bayesian forms\nof linear inverse problems in signal processing,\nas well as in linear regression in statistics.\n\nExact estimation of the parameters ${\\boldsymbol{\\theta}}$ via maximum likelihood or other methods is generally intractable.\nOne promising class of approximate methods combines\napproximate message passing (AMP) \\cite{DonohoMM:09} with expectation-maximization (EM)\\@.\nAMP and its generalizations (e.g., \\cite{Rangan:11-ISIT})\nconstitute a powerful, relatively new class of algorithms based on expectation propagation \\cite{Minka:01} (EP)-type techniques.\nThe resulting AMP algorithms are computationally fast\nand have been successfully applied to a wide range of problems,\ne.g., \\cite{FletcherRVB:11,Schniter:11,SomS:12,ZinielS:13b,Vila:TSP:14,Schniter:TSP:15,Ziniel:TSP:15,fletcher2014scalable}.\nMost importantly, for large, zero-mean, sub-Gaussian i.i.d.\\ random matrices $\\mathbf{A}$,\nthe performance of these AMP methods can be exactly predicted by a scalar \\emph{state evolution}\n(SE) \\cite{BayatiM:11,javanmard2013state} that provides testable conditions\nfor optimality, even for non-convex priors.  When the parameters ${\\boldsymbol{\\theta}}$ are unknown,\nAMP can be easily combined with EM for joint learning of the parameters ${\\boldsymbol{\\theta}}$ and\nvector $\\mathbf{x}$~\\cite{krzakala2012statistical,vila2013expectation,KamRanFU:12-IT}.\nUnder more general conditions on $\\mathbf{A}$, however, not only do the theoretical results\nnot hold, but standard AMP techniques often diverge and require a variety of modifications for stability\n\\cite{RanSchFle:14-ISIT,Vila:ICASSP:15,manoel2015swamp,rangan2017inference}.\nFor example, when $\\mathbf{A}$ has nonzero mean, its largest singular value grows with the\nproblem size, making $\\mathbf{A}$ arbitrarily poorly conditioned.\n\nA recent work \\cite{fletcher2016emvamp} combined EM with the so-called Vector AMP (VAMP)\nmethod of \\cite{rangan2016vamp}. Similar to AMP, VAMP is based on\nEP-like \\cite{Minka:01} approximations of belief propagation \\cite{rangan2016vamp} and can also be considered as a special case\n of expectation consistent (EC) approximate inference \\cite{opper2004expectation,OppWin:05,fletcher2016expectation}.\nVAMP's key attraction is that it applies to a larger class of matrices $\\mathbf{A}$ than\nthe original AMP method.\nIt\nhas provable SE analyses and convergence guarantees that apply to all\nright-rotationally invariant matrices $\\mathbf{A}$ \\cite{rangan2016vamp,takeuchi2017rigorous}~--\na significantly larger class of matrices than i.i.d.\\ sub-Gaussian.\nUnder further mild conditions, the mean-squared error (MSE)\nof VAMP matches the replica prediction for\noptimality~\\cite{tulino2013support,barbier2016mutual,reeves2016replica}.\nWhen the distributions on $\\mathbf{x}$ and $\\mathbf{w}$ are unknown,\nthe work \\cite{fletcher2016emvamp} proposed to combine EM and VAMP using the approximate inference\nframework of \\cite{heskes2004approximate}.\nWhile \\cite{fletcher2016emvamp} provided\nnumerical simulations suggesting excellent performance for EM-VAMP on several\nsynthetic problems, there were no provable convergence guarantees.\n\nThe contributions of this work are as follows:\n\\begin{itemize}\n\\item \\emph{Rigorous state evolution analysis:}  We provide a rigorous analysis of\na generalization of EM-VAMP that we call Adaptive VAMP\\@.\nSimilar to the analysis of VAMP, we consider a certain large-system limit (LSL)\nwhere the matrix $\\mathbf{A}$ is random and right-rotationally invariant.\nImportantly, this class of matrices includes much more than i.i.d.\\ Gaussian, as used in the\noriginal LSL analysis of\n Bayati and Montanari \\cite{BayatiM:11}.\nIt is shown (Theorem~\\ref{thm:em-se}) that,\nin this LSL, the parameter estimates at each iteration converge to deterministic limits ${\\overline{\\boldsymbol{\\theta}}}_k$\nthat can be computed from a set of SE equations that extend those of VAMP\\@.\nThe analysis also exactly characterizes the asymptotic joint distribution of the\nestimates $\\widehat{\\mathbf{x}}$ and the true vector $\\mathbf{x}^0$.\nThe SE equations depend only on the initial parameter estimate\n, the adaptation function (to be discussed), and statistics on the matrix $\\mathbf{A}$,\nthe vector $\\mathbf{x}^0$, and the noise $\\mathbf{w}$.\n\n\\item \\emph{Asymptotic consistency}:\nIt is also shown (Theorem~\\ref{thm:theta1cons}) that, under an additional identifiability condition\nand a simple auto-tuning procedure,\nAdaptive VAMP can yield provably consistent parameter estimates in the LSL\\@.\nThis approach is inspired by an ML-estimation approach from \\cite{KamRanFU:12-IT}.\nRemarkably, the result is true under very general problem formulations.\n\n\\item \\emph{Bayes optimality}:  In the case when the parameter estimates converge to the\ntrue value, the behavior of adaptive VAMP matches that of VAMP\\@.  In this case,\nit is shown in \\cite{rangan2016vamp} that, when the SE equations have a unique fixed point,\nthe MSE of VAMP matches the MSE of the Bayes optimal estimator, as predicted by the replica\nmethod~\\cite{tulino2013support,barbier2016mutual,reeves2016replica}.\n\\end{itemize}\n\nIn this way, we have developed a computationally efficient inference scheme for\na large class of linear inverse problems.\nIn a certain high-dimensional limit, our scheme guarantees that\n(i) the performance of the algorithm can be exactly characterized,\n(ii) the parameter estimates ${\\widehat{\\boldsymbol{\\theta}}}$ are asymptotically consistent,\nand\n(iii) the algorithm has testable conditions for when\nthe signal estimates $\\widehat{\\mathbf{x}}$ match the replica prediction of Bayes optimality.\n\n\\section{VAMP with Adaptation}\n\nWe assume that the prior density on $\\mathbf{x}$ can be written as\n\\begin{equation} \\label{eq:pxf}\n    p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1) = \\frac{1}{Z_1({\\boldsymbol{\\theta}}_1)}\\exp\\left[ -f_1(\\mathbf{x}|{\\boldsymbol{\\theta}}_1) \\right],\n    \\quad  f_1(\\mathbf{x}|{\\boldsymbol{\\theta}}_1) = \\sum_{n=1}^N f_{1}(x_n|{\\boldsymbol{\\theta}}_1),\n\\end{equation}\nwhere $f_1(\\cdot)$ is a separable penalty function, ${\\boldsymbol{\\theta}}_1$ is a parameter vector and\n$Z_1({\\boldsymbol{\\theta}}_1)$ is a normalization constant.  With some abuse of notation,\nwe have used $f_1(\\cdot)$ for the function on the vector $\\mathbf{x}$ and its components $x_n$.\nSince $f_1(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ is separable,  $\\mathbf{x}$ has i.i.d.\\ components\nconditioned on ${\\boldsymbol{\\theta}}_1$.\nThe likelihood function under the AWGN model \\eqref{eq:yAx} can be written as\n\\begin{equation} \\label{eq:pyxf}\n    p(\\mathbf{y}|\\mathbf{x},\\theta_2)\n    := \\frac{1}{Z_2(\\theta_2)}\n    \\exp\\left[-f_2(\\mathbf{x},\\mathbf{y}|\\theta_2)\\right], \\quad\n    f_2(\\mathbf{x},\\mathbf{y}|\\theta_2)\n    := \\frac{\\theta_2}{2}\\|\\mathbf{y}-\\mathbf{A}\\mathbf{x}\\|^2,\n\\end{equation}\nwhere $Z_2(\\theta_2) = (2\\pi\/\\theta_2)^{N\/2}$.\nThe joint density of $\\mathbf{x},\\mathbf{y}$ given parameters\n${\\boldsymbol{\\theta}}=({\\boldsymbol{\\theta}}_1,\\theta_2)$ is then\n\\begin{equation} \\label{eq:pxy}\n    p(\\mathbf{x},\\mathbf{y}|{\\boldsymbol{\\theta}}) = p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)p(\\mathbf{y}|\\mathbf{x},\\theta_2).\n\\end{equation}\nThe problem is to estimate the parameters ${\\boldsymbol{\\theta}}$ along with\nthe vector $\\mathbf{x}^0$.\n\n\\begin{algorithm}[t]\n\\caption{Adaptive VAMP}\n\\begin{algorithmic}[1]  \\label{algo:em-vamp}\n\\REQUIRE{Matrix $\\mathbf{A}\\in{\\mathbb{R}}^{M\\times N}$, measurement vector $\\mathbf{y}$,\ndenoiser function $\\mathbf{g}_1(\\cdot)$, statistic function $\\phi_1(\\cdot)$,\nadaptation function $T_1(\\cdot)$ and number of iterations $N_{\\rm it}$.  }\n\\STATE{ Select initial $\\mathbf{r}_{10}$, $\\gamma_{10}\\geq 0$, ${\\widehat{\\boldsymbol{\\theta}}}_{10}$, ${\\widehat{\\theta}}_{20}$.}\n\\FOR{$k=0,1,\\dots,N_{\\rm it}-1$}\n\n    \\STATE{\/\/ Input denoising }\n    \\STATE{$\\widehat{\\mathbf{x}}_{1k} = \\mathbf{g}_1(\\mathbf{r}_{1k},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k}))$,\n        \\qquad\n        $\\eta_{1k}^{-1} = \\gamma_{1k}\/\\bkt{ \\mathbf{g}_1'(\\mathbf{r}_{1k},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k})}$}\n        \\label{line:x1}\n    \\STATE{$\\gamma_{2k} = \\eta_{1k} - \\gamma_{1k}$}\n        \\label{line:gam2}\n    \\STATE{$\\mathbf{r}_{2k} = (\\eta_{1k}\\widehat{\\mathbf{x}}_{1k} - \\gamma_{1k}\\mathbf{r}_{1k})\/\\gamma_{2k}$}\n        \\label{line:r2}\n    \\STATE{ }\n\n    \\STATE{\/\/ Input parameter update }\n    \\STATE{ ${\\widehat{\\boldsymbol{\\theta}}}_{1,k\\! + \\! 1} = T_1(\\mu_{1k})$,\n        \\qquad\n        $\\mu_{1k} = \\bkt{\\phi_1(\\mathbf{r}_{1k},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k})}$ }\n        \\label{line:theta1}\n    \\STATE{}\n\n    \\STATE{\/\/ Output estimation }\n    \\STATE{$\\widehat{\\mathbf{x}}_{2k} = \\mathbf{Q}_k^{-1}( {\\widehat{\\theta}}_{2k}\\mathbf{A}^{\\text{\\sf T}}\\mathbf{y} + \\gamma_{2k}\\mathbf{r}_{2k})$,\n        \\qquad\n        $\\mathbf{Q}_k = {\\widehat{\\theta}}_{2k}\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}$}\n        \\label{line:x2}\n    \\STATE{$\\eta_{2k}^{-1} = (1\/N)\\Tr(\\mathbf{Q}_k^{-1})$}\n        \\label{line:eta2}\n    \\STATE{$\\gamma_{1,k\\! + \\! 1} = \\eta_{2k} - \\gamma_{2k}$}\n        \\label{line:gam1}\n    \\STATE{$\\mathbf{r}_{1,k\\! + \\! 1} = (\\eta_{2k}\\widehat{\\mathbf{x}}_{2k} - \\gamma_{2k}\\mathbf{r}_{2k})\/\\gamma_{1,k\\! + \\! 1}$}\n        \\label{line:r1}\n    \\STATE{}\n\n\n    \\STATE{\/\/ Output parameter update }\n    \\STATE{ ${\\widehat{\\theta}}_{2,k\\! + \\! 1}^{-1} =\n        (1\/N)\\{\\|\\mathbf{y}-\\mathbf{A}\\widehat{\\mathbf{x}}_{2k}\\|^2 + \\Tr(\\mathbf{A}\\mathbf{Q}_k^{-1}\\mathbf{A}^{\\text{\\sf T}} )\\}$ }\n        \\label{line:theta2}\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nThe steps of the proposed adaptive VAMP algorithm\nthat performs this estimation are shown in Algorithm~\\ref{algo:em-vamp}.\nAdaptive VAMP is a generalization of the EM-VAMP method in \\cite{fletcher2016emvamp}.\nAt each iteration $k$, the algorithm produces, for $i=1,2$,\nestimates ${\\widehat{\\boldsymbol{\\theta}}}_{ik}$ of the parameter ${\\boldsymbol{\\theta}}_i$, along with estimates $\\widehat{\\mathbf{x}}_{ik}$\nof the vector $\\mathbf{x}^0$.  The algorithm is tuned by selecting three key functions:\n(i) a \\emph{denoiser function} $\\mathbf{g}_1(\\cdot)$;\n(ii) an \\emph{adaptation statistic} $\\phi_1(\\cdot)$; and\n(iii) a \\emph{parameter selection function} $T_1(\\cdot)$.\nThe denoiser is used to produce the estimates $\\widehat{\\mathbf{x}}_{1k}$, while the adaptation statistic\nand parameter estimation functions produce the estimates ${\\widehat{\\boldsymbol{\\theta}}}_{1k}$.\n\n\\paragraph*{Denoiser function}\nThe denoiser function $\\mathbf{g}_1(\\cdot)$ is discussed in detail in \\cite{rangan2016vamp}\nand is generally based on the prior $p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$.  In the original EM-VAMP\nalgorithm~\\cite{fletcher2016emvamp},\n$\\mathbf{g}_1(\\cdot)$ is selected as the so-called minimum mean-squared error (MMSE) denoiser.\nSpecifically, in each iteration, the variables $\\mathbf{r}_i$, $\\gamma_i$ and ${\\widehat{\\boldsymbol{\\theta}}}_i$\nwere used to construct \\emph{belief estimates},\n\\begin{equation} \\label{eq:bidef}\n    b_i(\\mathbf{x}|\\mathbf{r}_i,\\gamma_i,{\\widehat{\\boldsymbol{\\theta}}}_i) \\propto \\exp\\left[ -f_i(\\mathbf{x},\\mathbf{y}|{\\widehat{\\boldsymbol{\\theta}}}_i) -\n        \\frac{\\gamma_i}{2}\\|\\mathbf{x}-\\mathbf{r}_i\\|^2 \\right],\n\\end{equation}\nwhich represent estimates of the posterior density $p(\\mathbf{x}|\\mathbf{y},{\\boldsymbol{\\theta}})$.\nTo keep the notation symmetric, we have written\n$f_1(\\mathbf{x},\\mathbf{y}|{\\widehat{\\boldsymbol{\\theta}}}_1)$ for $f_1(\\mathbf{x}|{\\widehat{\\boldsymbol{\\theta}}}_1)$ even though the first penalty function\ndoes not depend on $\\mathbf{y}$.  The EM-VAMP method then selects $\\mathbf{g}_1(\\cdot)$ to be the\nmean of the belief estimate,\n\\begin{equation} \\label{eq:gmmse}\n    \\mathbf{g}_1(\\mathbf{r}_1,\\gamma_1,{\\boldsymbol{\\theta}}_1) := \\mathbb{E}\\left[ \\mathbf{x} |\\mathbf{r}_1,\\gamma_1,{\\boldsymbol{\\theta}}_1 \\right].\n\\end{equation}\nFor line~\\ref{line:x1} of Algorithm~\\ref{algo:em-vamp}, we define\n$[\\mathbf{g}_1'(\\mathbf{r}_{1k},\\gamma_{1k},{\\boldsymbol{\\theta}}_1)]_n\n:= \\partial [\\mathbf{g}_{1}(\\mathbf{r}_{1k},\\gamma_{1k},{\\boldsymbol{\\theta}}_1)]_n\/\\partial r_{1n}$\nand we use $\\bkt{\\cdot}$ for the empirical mean of a vector,\ni.e., $\\bkt{\\mathbf{u}} = (1\/N)\\sum_{n=1}^N u_n$.\nHence, $\\eta_{1k}$ in line~\\ref{line:x1} is a scaled inverse divergence.\nIt is shown in \\cite{rangan2016vamp} that, for the MMSE denoiser\n\\eqref{eq:gmmse}, $\\eta_{1k}$ is the inverse average posterior variance.\n\n\n\\paragraph{Estimation for ${\\boldsymbol{\\theta}}_1$ with finite statistics}\nFor the EM-VAMP algorithm~\\cite{fletcher2016emvamp},\nthe parameter update for ${\\widehat{\\boldsymbol{\\theta}}}_{1,k\\! + \\! 1}$ is performed via a maximization\n\\begin{equation} \\label{eq:thetaEM}\n    {\\widehat{\\boldsymbol{\\theta}}}_{1,k\\! + \\! 1} = \\mathop{\\mathrm{arg\\,max}}_{{\\boldsymbol{\\theta}}_1} \\mathbb{E}\\left[ \\ln p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)\n        \\left| \\mathbf{r}_{1k},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k} \\right. \\right],\n\\end{equation}\nwhere the expectation is with respect to the belief estimate $b_i(\\cdot)$ in\n\\eqref{eq:bidef}.  It is shown in~\\cite{fletcher2016emvamp} that using \\eqref{eq:thetaEM}\nis equivalent to approximating the M-step in the standard EM method.\nIn the adaptive VAMP method in Algorithm~\\ref{algo:em-vamp}, the M-step maximization\n\\eqref{eq:thetaEM} is replaced by line~\\ref{line:theta1}.  Note that line~\\ref{line:theta1}\nagain uses $\\bkt{\\cdot}$ to denote empirical average,\n\\begin{equation}\n    \\mu_{1k} = \\bkt{{\\boldsymbol{\\phi}}_1(\\mathbf{r}_{1k},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k})}\n        := \\frac{1}{N} \\sum_{n=1}^N \\phi_1(r_{1k,n},\\gamma_{1k},{\\widehat{\\boldsymbol{\\theta}}}_{1k}) \\in {\\mathbb{R}}^d,\n\\end{equation}\nso $\\mu_{1k}$ is the empirical average of some $d$-dimensional\nstatistic $\\phi_1(\\cdot)$ over the components\nof $\\mathbf{r}_{1k}$.  The parameter estimate update ${\\widehat{\\boldsymbol{\\theta}}}_{1,k\\! + \\! 1}$ is then computed from some\nfunction of this statistic, $T_1(\\mu_{1k})$.\n\nWe show in Appendix~\\ref{sec:finite} that there are two important cases where the EM update\n\\eqref{eq:thetaEM} can be computed from a finite-dimensional statistic as in line~\\ref{line:theta1}:\n(i) The prior $p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ is given by an exponential family,\n$f_1(\\mathbf{x}|{\\boldsymbol{\\theta}}_1) = {\\boldsymbol{\\theta}}_1^{\\text{\\sf T}} \\varphi(\\mathbf{x})$ for some sufficient statistic $\\varphi(\\mathbf{x})$;\nand (ii) There are a finite number of values for the parameter ${\\boldsymbol{\\theta}}_1$.\nFor other cases, we can approximate more general parameterizations via discretization of the\nparameter values ${\\boldsymbol{\\theta}}_1$.  The updates in line~\\ref{line:theta1} can also incorporate other\ntypes of updates as we will see below.\nBut, we stress that it is preferable to compute the estimate for ${\\boldsymbol{\\theta}}_1$ directly from the maximization\n\\eqref{eq:thetaEM}~-- the use of a finite-dimensional statistic is for the sake of analysis.\n\n\\paragraph{Estimation for $\\theta_2$ with finite statistics}\nIt will be useful to also write the adaptation of $\\theta_2$\nin line~\\ref{line:theta2} of Algorithm~\\ref{algo:em-vamp}\nin a similar form as line~\\ref{line:theta1}.\nFirst, take a singular value decomposition (SVD) of $\\mathbf{A}$ of the form\n\\begin{equation} \\label{eq:ASVD}\n    \\mathbf{A}=\\mathbf{U}\\mathbf{S}\\mathbf{V}^{\\text{\\sf T}}, \\quad \\mathbf{S} = \\mathop{\\mathrm{Diag}}(\\mathbf{s}),\n\\end{equation}\nand define the transformed error and transformed noise,\n\\begin{equation} \\label{eq:qerrdef}\n    \\mathbf{q}_k := \\mathbf{V}^{\\text{\\sf T}}(\\mathbf{r}_{2k}-\\mathbf{x}^0), \\quad {\\boldsymbol \\xi} := \\mathbf{U}^{\\text{\\sf T}}\\mathbf{w}.\n\\end{equation}\nThen, it is shown in Appendix~\\ref{sec:theta2TPf} that ${\\widehat{\\theta}}_{2,k\\! + \\! 1}$ in line~\\ref{line:theta2}\ncan be written as\n\\begin{equation} \\label{eq:theta2T}\n    {\\widehat{\\theta}}_{2,k\\! + \\! 1} = T_2(\\mu_{2k}) := \\frac{1}{\\mu_{2k}},\n    \\quad \\mu_{2k} = \\bkt{{\\boldsymbol{\\phi}}_2(\\mathbf{q}_2,{\\boldsymbol \\xi},\\mathbf{s},\\gamma_{2k},{\\widehat{\\theta}}_{2k})}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:phi2def}\n    \\phi_2(q,\\xi,s,\\gamma_2,{\\widehat{\\theta}}_2) := \\frac{\\gamma_2^2}{(s^2{\\widehat{\\theta}}_2+\\gamma_2)^2}(sq + \\xi)^2\n        + \\frac{s^2}{s^2{\\widehat{\\theta}}_2+\\gamma_2}.\n\\end{equation}\nOf course, we cannot directly compute $\\mathbf{q}_k$ in \\eqref{eq:qerrdef} since we do not know\nthe true $\\mathbf{x}^0$.  Nevertheless, this form will be useful for analysis.\n\n\\section{State Evolution in the Large System Limit}\n\n\\subsection{Large System Limit} \\label{sec:lsl}\n\nSimilar to the analysis of VAMP in \\cite{rangan2016vamp},\nwe analyze Algorithm~\\ref{algo:em-vamp} in a certain large-system limit (LSL).\nThe LSL framework was developed by Bayati and Montanari in \\cite{BayatiM:11} and\nwe review some of the key definitions in Appendix~\\ref{sec:empConv}.\nAs in the analysis of VAMP, the LSL considers a sequence of problems indexed by the vector\ndimension $N$.  For each $N$, we assume that there is a ``true'' vector $\\mathbf{x}^0\\in{\\mathbb{R}}^N$\nthat is observed through measurements of the form\n\\begin{equation} \\label{eq:yAxslr}\n    \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w} \\in {\\mathbb{R}}^N, \\quad \\mathbf{w} \\sim {\\mathcal N}(\\mathbf{0}, \\theta_2^{-1}\\mathbf{I}_N),\n\\end{equation}\nwhere $\\mathbf{A}\\in{\\mathbb{R}}^{N\\times N}$ is a known transform, $\\mathbf{w}$ is white Gaussian noise with ``true'' precision $\\theta_2$.  The noise precision $\\theta_2$ does not change with $N$.\n\nIdentical to \\cite{rangan2016vamp}, the transform $\\mathbf{A}$\nis modeled as a large, \\emph{right-orthogonally invariant} random matrix.\nSpecifically, we assume that it has an SVD of the form \\eqref{eq:ASVD},\nwhere $\\mathbf{U}$ and $\\mathbf{V}$ are $N\\times N$ orthogonal matrices\nsuch that $\\mathbf{U}$ is deterministic and\n$\\mathbf{V}$ is Haar distributed (i.e.\\ uniformly distributed on the set of orthogonal matrices).\nAs described in \\cite{rangan2016vamp},\nalthough we have assumed a square matrix $\\mathbf{A}$, we can consider general rectangular $\\mathbf{A}$\nby adding zero singular values.\n\nUsing the definitions in Appendix~\\ref{sec:empConv},\nwe assume that the components of the singular-value vector $\\mathbf{s}\\in{\\mathbb{R}}^N$ in \\eqref{eq:ASVD}\nconverge empirically with second-order moments as\n\\begin{equation} \\label{eq:Slim}\n    \\lim_{N \\rightarrow \\infty} \\{ s_n \\} \\stackrel{PL(2)}{=} S,\n\\end{equation}\nfor some non-negative random variable $S$ with $\\mathbb{E}[S] > 0$ and $S \\in [0,S_{\\rm max}]$\nfor some finite maximum value $S_{\\rm max}$.\nAdditionally, we assume that\nthe components of the true vector, $\\mathbf{x}^0$, and the initial input to the denoiser, $\\mathbf{r}_{10}$,\nconverge empirically as\n\\begin{equation} \\label{eq:RX0lim}\n    \\lim_{N \\rightarrow \\infty} \\{ (r_{10,n}, x^0_n) \\} \\stackrel{PL(2)}{=} (R_{10},X^0), \\quad\n    R_{10} = X^0 + P_0, \\quad P_0 \\sim {\\mathcal N}(0,\\tau_{10}),\n\\end{equation}\nwhere $X^0$ is a random variable representing the \\emph{true\ndistribution} of the components $\\mathbf{x}^0$; $P_0$ is an initial error and $\\tau_{10}$ is an initial error variance.\nThe variable $X^0$ may be distributed as $X^0 \\sim p(\\cdot|{\\boldsymbol{\\theta}}_1)$ for some\ntrue parameter ${\\boldsymbol{\\theta}}_1$.  However, in order to incorporate under-modeling, the existence of such a true\nparameter is not required.\nWe also assume that the initial second-order term and parameter estimate converge almost surely as\n\\begin{equation} \\label{eq:gam10lim}\n    \\lim_{N \\rightarrow \\infty} (\\gamma_{10},{\\widehat{\\boldsymbol{\\theta}}}_{10},{\\widehat{\\theta}}_{20})\n        = (\\overline{\\gamma}_{10},{\\overline{\\boldsymbol{\\theta}}}_{10},{\\overline{\\theta}}_{20})\n\\end{equation}\nfor some $\\overline{\\gamma}_{10} > 0$ and $({\\overline{\\boldsymbol{\\theta}}}_{10},{\\overline{\\theta}}_{20})$.\n\n\n\\subsection{Error and Sensitivity Functions}\nWe next need to introduce parametric forms of\ntwo key terms from \\cite{rangan2016vamp}:  error functions and sensitivity functions.\nThe error functions describe MSE\nof the denoiser and output estimators under AWGN measurements.\nSpecifically, for the denoiser $g_1(\\cdot,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$, we define the error function as\n\\begin{align}\n    {\\mathcal E}_1(\\gamma_1,\\tau_1,{\\widehat{\\boldsymbol{\\theta}}}_1)\n    := \\mathbb{E}\\left[ (g_1(R_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)-X^0)^2 \\right], \\quad\n     R_1 = X^0 + P, \\quad P \\sim {\\mathcal N}(0,\\tau_1), \\label{eq:eps1}\n\\end{align}\nwhere $X^0$ is distributed according to the true distribution of the components $\\mathbf{x}^0$ (see above).\nThe function ${\\mathcal E}_1(\\gamma_1,\\tau_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$ thus represents the MSE of the\nestimate $\\widehat{X} = g_1(R_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$ from a measurement $R_1$\ncorrupted by Gaussian noise of variance $\\tau_1$ under the parameter estimate ${\\widehat{\\boldsymbol{\\theta}}}_1$.\nFor the output estimator, we define the error function as\n\\begin{align}\n    \\MoveEqLeft {\\mathcal E}_2(\\gamma_2,\\tau_2,{\\widehat{\\theta}}_2)\n    := \\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\mathbb{E} \\| \\mathbf{g}_2(\\mathbf{r}_2,\\gamma_2,{\\widehat{\\theta}}_2) -\\mathbf{x}^0 \\|^2, \\nonumber \\\\\n    & \\mathbf{x}^0 = \\mathbf{r}_2 + \\mathbf{q}, \\quad \\mathbf{q} \\sim {\\mathcal N}(0,\\tau_2 \\mathbf{I}), \\quad\n    \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w}, \\quad \\mathbf{w} \\sim {\\mathcal N}(0,\\theta_2^{-1} \\mathbf{I}),\n    \\label{eq:eps2}\n\\end{align}\nwhich is the average per component error of the vector estimate under Gaussian noise.\nThe dependence on the true noise precision, $\\theta_2$, is suppressed.\n\nThe sensitivity functions describe the expected divergence of the estimator.\nFor the denoiser, the sensitivity function is defined as\n\\begin{align}\n    A_1(\\gamma_1,\\tau_1,{\\widehat{\\boldsymbol{\\theta}}}_1)\n    := \\mathbb{E}\\left[ g_1'(R_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1) \\right], \\quad\n     R_1 = X^0 + P, \\quad P \\sim {\\mathcal N}(0,\\tau_1), \\label{eq:sens1}\n\\end{align}\nwhich is the average derivative under a Gaussian noise input.  For the\noutput estimator, the sensitivity is defined as\n\\begin{align}\n     A_2(\\gamma_2,\\tau_2,{\\widehat{\\theta}}_2)\n    := \\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\Tr\\left[ \\frac{\\partial \\mathbf{g}_2(\\mathbf{r}_2,\\gamma_2,{\\widehat{\\theta}}_2)}{\\partial \\mathbf{r}_2}\n            \\right],\n\\end{align}\nwhere $\\mathbf{r}_2$ is distributed as in \\eqref{eq:eps2}.\nThe paper \\cite{rangan2016vamp} discusses the error and sensitivity functions in detail\nand shows how these functions can be easily evaluated.\n\n\\subsection{State Evolution Equations}\nWe can now describe our main result, which are the SE equations for Adaptive VAMP\\@.\nThe equations are an extension of those in the VAMP paper \\cite{rangan2016vamp},\nwith modifications for the parameter estimation.\nFor a given iteration $k \\geq 1$, consider the set of components,\n\\[\n     \\{ (\\widehat{x}_{1k,n},r_{1k,n},x^0_n), ~ n=1,\\ldots,N \\}.\n\\]\nThis set represents the components of the true vector $\\mathbf{x}^0$,\nits corresponding estimate $\\widehat{\\mathbf{x}}_{1k}$ and the denoiser input\n$\\mathbf{r}_{1k}$.  We will show that, under certain assumptions,\nthese components converge empirically as\n\\begin{equation} \\label{eq:limrx1}\n    \\lim_{N \\rightarrow \\infty} \\{ (\\widehat{x}_{1k,n},r_{1k,n},x^0_n) \\}\n    \\stackrel{PL(2)}{=} (\\widehat{X}_{1k},R_{1k},X^0),\n\\end{equation}\nwhere the random variables $(\\widehat{X}_{1k},R_{1k},X^0)$ are given by\n\\begin{equation} \\label{eq:RX0var}\n    R_{1k} = X^0 + P_k, \\quad P_k \\sim {\\mathcal N}(0,\\tau_{1k}), \\quad\n    \\widehat{X}_{1k} = g_1(R_{1k},\\overline{\\gamma}_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k}),\n\\end{equation}\nfor constants $\\overline{\\gamma}_{1k}$, ${\\overline{\\boldsymbol{\\theta}}}_{1k}$ and $\\tau_{1k}$ that will be defined below.\nWe will also see that ${\\widehat{\\boldsymbol{\\theta}}}_{1k} \\rightarrow {\\overline{\\boldsymbol{\\theta}}}_{1k}$, so ${\\overline{\\boldsymbol{\\theta}}}_{1k}$ represents the\nasymptotic parameter estimate.\nThe model \\eqref{eq:RX0var} shows that each component $r_{1k,n}$ appears as the true component $x^0_n$ plus\nGaussian noise.  The corresponding estimate $\\widehat{x}_{1k,n}$ then appears as the\ndenoiser output with $r_{1k,n}$ as the input and ${\\overline{\\boldsymbol{\\theta}}}_{1k}$ as the parameter estimate.\nHence, the asymptotic behavior\nof any component $x^0_n$ and its corresponding $\\widehat{x}_{1k,n}$ is identical to\na simple scalar system.  We will refer to \\eqref{eq:limrx1}-\\eqref{eq:RX0var} as the denoiser's \\emph{scalar equivalent model}.\n\nWe will also show that these transformed errors $\\mathbf{q}_k$ and noise ${\\boldsymbol \\xi}$ in \\eqref{eq:qerrdef}\nand singular values $\\mathbf{s}$ converge\nempirically to a set of independent random variables $(Q_k,\\Xi,S)$ given by\n\\begin{equation} \\label{eq:limqxi}\n    \\lim_{N \\rightarrow \\infty} \\{ (q_{k,n},\\xi_n,s_n) \\}\n    \\stackrel{PL(2)}{=} (Q_k,\\Xi,S), \\quad\n    Q_k \\sim {\\mathcal N}(0,\\tau_{2k}), \\quad \\Xi \\sim {\\mathcal N}(0,\\theta_2^{-1}),\n\\end{equation}\nwhere $S$ has the distribution of the singular values of $\\mathbf{A}$,\n$\\tau_{2k}$ is a variance that will be defined below and $\\theta_2$\nis the true noise precision in the measurement model \\eqref{eq:yAxslr}.\nAll the variables in \\eqref{eq:limqxi} are independent.\nThus \\eqref{eq:limqxi} is a scalar equivalent model for the output estimator.\n\nThe variance terms are defined recursively through the \\emph{state evolution}\nequations,\n\\begin{subequations} \\label{eq:se}\n\\begin{align}\n    \\overline{\\alpha}_{1k} &= A_1(\\overline{\\gamma}_{1k},\\tau_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k}), \\quad\n    \\overline{\\eta}_{1k} = \\frac{\\overline{\\gamma}_{1k}}{\\overline{\\alpha}_{1k}}, \\quad\n    \\overline{\\gamma}_{2k} = \\overline{\\eta}_{1k} - \\overline{\\gamma}_{1k} \\label{eq:eta1se} \\\\\\\n    {\\overline{\\boldsymbol{\\theta}}}_{1,k\\! + \\! 1} &= T_1(\\overline{\\mu}_{1k}), \\quad \\overline{\\mu}_{1k} =\n        \\mathbb{E}\\left[ \\phi_1(R_{1k},\\overline{\\gamma}_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k}) \\right] \\label{eq:theta1se} \\\\\n    \\tau_{2k} &= \\frac{1}{(1-\\overline{\\alpha}_{1k})^2}\\left[\n        {\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k}) - \\overline{\\alpha}_{1k}^2\\tau_{1k} \\right],\n            \\label{eq:tau2se} \\\\\n    \\overline{\\alpha}_{2k} &= A_2(\\overline{\\gamma}_{2k},\\tau_{2k},{\\overline{\\theta}}_{2k}), \\quad\n    \\overline{\\eta}_{2k} = \\frac{\\overline{\\gamma}_{2k}}{\\overline{\\alpha}_{2k}}, \\quad\n    \\overline{\\gamma}_{1,k\\! + \\! 1} = \\overline{\\eta}_{2k} - \\overline{\\gamma}_{2k} \\label{eq:eta2se} \\\\\n    {\\overline{\\theta}}_{2,k\\! + \\! 1} &= T_2(\\overline{\\mu}_{2k}), \\quad \\overline{\\mu}_{2k} =\n        \\mathbb{E}\\left[ \\phi_2(Q_{k},\\Xi,S,\\overline{\\gamma}_{2k},{\\overline{\\theta}}_{2k}) \\right] \\label{eq:theta2se} \\\\\n    \\tau_{1,k\\! + \\! 1} &= \\frac{1}{(1-\\overline{\\alpha}_{2k})^2}\\left[\n        {\\mathcal E}_2(\\overline{\\gamma}_{2k},\\tau_{2k}) - \\overline{\\alpha}_{2k}^2\\tau_{2k} \\right],\n            \\label{eq:tau1se}\n\\end{align}\n\\end{subequations}\nwhich are initialized with $\\tau_{10} = \\mathbb{E}[(R_{10}-X^0)^2]$ and the\n$(\\overline{\\gamma}_{10},{\\overline{\\boldsymbol{\\theta}}}_{10},{\\overline{\\theta}}_{20})$\ndefined from the limit \\eqref{eq:gam10lim}.  The expectation in \\eqref{eq:theta1se} is\nwith respect to the random variables \\eqref{eq:limrx1} and the expectation in \\eqref{eq:theta2se} is\nwith respect to the random variables \\eqref{eq:limqxi}.\n\n\\begin{theorem} \\label{thm:em-se}\nConsider the outputs of Algorithm~\\ref{algo:em-vamp}.\nUnder the above assumptions and definitions, assume additionally that for all iterations $k$:\n\\begin{enumerate}[(i)]\n\\item The solution $\\overline{\\alpha}_{1k}$ from the SE equations \\eqref{eq:se} satisfies\n$\\overline{\\alpha}_{1k} \\in (0,1)$.\n\n\\item The functions $A_i(\\cdot)$, ${\\mathcal E}_i(\\cdot)$ and $T_i(\\cdot)$\nare continuous at $(\\gamma_i,\\tau_i,{\\widehat{\\boldsymbol{\\theta}}}_i,\\mu_i)=(\\overline{\\gamma}_{ik},\\tau_{ik},{\\overline{\\boldsymbol{\\theta}}}_{ik},\\overline{\\mu}_{ik})$.\n\n\\item The denoiser function $g_1(r_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$ and its derivative\n $g_1'(r_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$\nare uniformly Lipschitz in $r_1$ at $(\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)=(\\overline{\\gamma}_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k})$.\n(See  Appendix~\\ref{sec:empConv} for a precise definition of uniform Lipschitz continuity.)\n\n\\item The adaptation statistic $\\phi_1(r_1,\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)$ is\n uniformly pseudo-Lipschitz of order 2 in $r_1$ at\n $(\\gamma_1,{\\widehat{\\boldsymbol{\\theta}}}_1)=(\\overline{\\gamma}_{1k},{\\overline{\\boldsymbol{\\theta}}}_{1k})$.\n\\end{enumerate}\nThen, for any fixed iteration $k \\geq 0$,\n\\begin{equation} \\label{eq:aglim}\n    \\lim_{N \\rightarrow \\infty} (\\alpha_{ik},\\eta_{ik},\\gamma_{ik},\\mu_{ik},{\\widehat{\\boldsymbol{\\theta}}}_{ik}) =\n    (\\overline{\\alpha}_{ik},\\overline{\\eta}_{ik}, \\overline{\\gamma}_{ik},\\overline{\\mu}_{ik},{\\overline{\\boldsymbol{\\theta}}}_{ik})\n\\end{equation}\nalmost surely.\nIn addition, the empirical limit \\eqref{eq:limrx1} holds almost surely for all $k > 0$,\nand \\eqref{eq:limqxi} holds almost surely for all $k \\geq 0$.\n\\end{theorem}\n\nTheorem~\\ref{thm:em-se} shows that, in the LSL, the parameter estimates ${\\widehat{\\boldsymbol{\\theta}}}_{ik}$ converge to\ndeterministic limits ${\\overline{\\boldsymbol{\\theta}}}_{ik}$ that can be precisely predicted by the\nstate-evolution equations.  The SE equations incorporate the true distribution of\nthe components on the prior $\\mathbf{x}^0$, the true noise precision $\\theta_2$,\nand the specific parameter estimation and denoiser functions used by the Adaptive VAMP method.\nIn addition, similar to the SE analysis of VAMP in \\cite{rangan2016vamp},\nthe SE equations also predict the asymptotic joint distribution of $\\mathbf{x}^0$ and\ntheir estimates $\\widehat{\\mathbf{x}}_{ik}$.  This joint distribution can be used to measure\nvarious performance metrics such as MSE~-- see \\cite{rangan2016vamp}.\nIn this way, we have provided a rigorous and precise characterization of a class of adaptive\nVAMP algorithms that includes EM-VAMP\\@.\n\n\\section{Consistent Parameter Estimation with Variance Auto-Tuning} \\label{sec:autotune}\n\nBy comparing the deterministic limits ${\\overline{\\boldsymbol{\\theta}}}_{ik}$ with the true parameters ${\\boldsymbol{\\theta}}_i$,\none can determine under which problem conditions\nthe parameter estimates of adaptive VAMP are asymptotically consistent.\nIn this section, we show with a particular choice of parameter estimation functions, one\ncan obtain provably\nasymptotically consistent parameter estimates under suitable identifiability conditions.\nWe call the method \\emph{variance auto-tuning}, which\ngeneralizes the approach in \\cite{KamRanFU:12-IT}.\n\n\\begin{definition} \\label{def:ident1}  Let $p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ be a parameterized\nset of densities.  Given a finite-dimensional statistic\n$\\phi_1(r)$, consider the mapping\n\\begin{equation} \\label{eq:phi1map}\n    (\\tau_1,{\\boldsymbol{\\theta}}_1) \\mapsto \\mathbb{E}\\left[ \\phi_1(R) | \\tau_1, {\\boldsymbol{\\theta}}_1 \\right],\n\\end{equation}\nwhere the expectation is with respect to the model \\eqref{eq:RX0cons}.\nWe say the $p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ is \\emph{identifiable} in Gaussian noise\nif there exists a finite-dimensional statistic $\\phi_1(r) \\in {\\mathbb{R}}^d$\nsuch that (i) $\\phi_1(r)$ is pseudo-Lipschitz continuous of order 2; and\n(ii) the mapping \\eqref{eq:phi1map} has a continuous inverse.\n\\end{definition}\n\n\\begin{theorem} \\label{thm:theta1cons}\nUnder the assumptions of Theorem~\\ref{thm:em-se},\nsuppose that $X^0$ follows $X^0 \\sim p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1^0)$ for some true parameter\n${\\boldsymbol{\\theta}}_1^0$.  If $p(\\mathbf{x}|{\\boldsymbol{\\theta}}_1)$ is identifiable in Gaussian noise,\nthere exists an adaptation rule\nsuch that, for any iteration $k$, the estimate ${\\widehat{\\boldsymbol{\\theta}}}_{1k}$ and noise estimate\n${\\widehat{\\tau}}_{1k}$ are\nasymptotically consistent in that $\\lim_{N \\rightarrow\\infty} {\\widehat{\\boldsymbol{\\theta}}}_{1k} = {\\boldsymbol{\\theta}}_1^0$\nand $\\lim_{N \\rightarrow\\infty} {\\widehat{\\tau}}_{1k} = \\tau_{1k}$ almost surely.\n\\end{theorem}\n\nThe theorem is proved in Appendix~\\ref{sec:autotuneDetails} which also provides\ndetails on how to perform the adaptation.\nThe Appendix also provides a similar result for\nconsistent estimation of the noise precision $\\theta_2$.\nThe result is remarkable as it shows that a simple variant\nof EM-VAMP can provide provably consistent parameter estimates under extremely general\ndistributions.\n\n\\section{Numerical Simulations}\n\n\\paragraph*{Sparse signal recovery}\nWe consider a sparse linear inverse problem of\nestimating a vector $\\mathbf{x}$ from measurements $\\mathbf{y}$ from \\eqref{eq:yAx} without knowing the signal parameters ${\\boldsymbol{\\theta}}_1$ or the noise precision $\\theta_2>0$.\nThe paper \\cite{fletcher2016emvamp} presented several\nnumerical experiments to assess the performance of EM-VAMP relative\nto other methods.\nTo support the results in this paper,\nour goal is to demonstrate that state evolution can correctly predict\nthe performance of EM-VAMP, and to\nvalidate the consistency of EM-VAMP with auto-tuning.\nDetails are given in Appendix~\\ref{sec:sim}.\nBriefly, to model the sparsity, $\\mathbf{x}$ is drawn as an i.i.d.\\\nBernoulli-Gaussian (i.e., spike and slab) prior with unknown sparsity level, mean and variance.\nThe true sparsity is $\\beta_x=0.1$.\nFollowing \\cite{RanSchFle:14-ISIT,Vila:ICASSP:15}, we take $\\mathbf{A} \\in {\\mathbb{R}}^{M \\times N}$ to be a random right-orthogonally invariant matrix with dimensions under $M=512$, $N=1024$\nwith the condition number set to  $\\kappa = 100$ (high condition number\nmatrices are known to be problem for conventional AMP methods).  The left panel of Fig.~\\ref{fig:sim} shows the normalized mean square error (NMSE) for various algorithms.\nAppendix~\\ref{sec:sim} describes the algorithms in details and also shows similar results for $\\kappa=10$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\columnwidth]{figures_EM_GEC\/sim_cond100.png}\n\\hfill\n\\includegraphics[width=0.45\\columnwidth]{figures_EM_GEC\/sat_nmse.png}\n\\caption{Numerical simulations.  Left panel:  Sparse signal recovery:  NMSE versus iteration for condition number for a random matrix\nwith a condition number $\\kappa=100$. Right panel:  NMSE for sparse image recovery (in AWGN at 40~dB SNR) as a function of the condition number $\\kappa$.}\n\\label{fig:sim}\n\\end{figure}\n\nWe see several important features.\nFirst, for all variants of VAMP and EM-VAMP, the SE equations\nprovide an excellent prediction of the per iteration performance of the algorithm.\nSecond, consistent with the simulations in \\cite{rangan2016vamp},\nthe oracle VAMP converges remarkably fast ($\\sim$ 10 iterations).\nThird, the performance of EM-VAMP with\nauto-tuning is virtually indistinguishable from oracle VAMP, suggesting that the parameter estimates\nare near perfect from the very first iteration.  Fourth,\nthe EM-VAMP method performs initially worse than the oracle-VAMP, but these errors are exactly\npredicted by the SE\\@.  Finally, all the VAMP and EM-VAMP algorithms exhibit much faster convergence\nthan the EM-BG-AMP\\@.  In fact, consistent with observations in \\cite{fletcher2016emvamp},\nEM-BG-AMP begins to diverge at higher condition numbers.  In contrast, the VAMP algorithms are\nstable.\n\n\\paragraph*{Compressed sensing image recovery}\nWhile the theory is developed on theoretical signal priors, we demonstrate\nthat the proposed EM-VAMP algorithm can be effective on natural images.  Specifically,\nwe repeat the experiments in\n\\cite{vila2014empirical} for recovery of a sparse image.  Again, see Appendix~\\ref{sec:sim}\nfor details including a picture of the image and the various reconstructions.\nAn $N=256\\times 256$ image of a satellite with $K=6678$ non-zero pixels is observed\nthrough a linear transform $\\mathbf{A}=\\mathop{\\mathrm{diag}}(\\mathbf{s})\\mathbf{P}\\mathbf{H}\\mathop{\\mathrm{diag}}(\\mathbf{d})$ in AWGN at 40~dB SNR,\nwhere\n$\\mathbf{H}$ is the fast Hadamard transform,\n$\\mathbf{P}$ is a random sub-selection to $M<N$ measurements,\n$\\mathbf{s}$ is a scaling to adjust the condition number, and\n$\\mathbf{d}$ is a random $\\pm 1$ vector.\nAs in the previous example,\nthe image vector $\\mathbf{x}$ is modeled as sparse Bernoulli-Gaussian and the EM-VAMP\nalgorithm is used to estimate the sparsity ratio, signal variance, and noise variance.\nThe transform is set to have measurement ratio $M\/N = 0.5$.\nWe see from the right panel of Fig.~\\ref{fig:sim} that the EM-VAMP algorithm\nis able to reconstruct the images with improved performance over the standard basis pursuit denoising\nmethod SPGL1 \\cite{van2008probing} and the EM-BG-GAMP method from \\cite{vila2013expectation}.\n\n\\section{Conclusions}  Due to its analytic tractability, computational simplicity,\nand potential for Bayes optimal inference, VAMP is a promising technique\nfor statistical linear inverse problems.\nImportantly, it does not suffer the convergence issues of standard AMP methods\nwhen $\\mathbf{A}$ is ill conditioned.\nHowever, a key challenge in using VAMP and related\nmethods is the need to precisely specify the\nproblem parameters.\nThis work provides a rigorous foundation for analyzing VAMP in combination with various\nparameter adaptation techniques including EM\\@.  The analysis reveals that VAMP,\nwith appropriate tuning, can also provide consistent parameter estimates under very\ngeneral settings, thus yielding a powerful approach for statistical linear inverse problems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n \\noindent This paper is concerned with the {oneparameter} family of  holomorphic correspondences {acting on the complex plane} \\begin{equation}\\label{woekg} (w-c)^q=z^p,\\end{equation} where $p>q \\geq 2$ {are integers.}  {A holomorphic correspondence, {as defined in this paper,} is a relation $z\\mapsto w$ determined by a polynomial equation $P(z,w)=0$ in two complex variables. The family \\eqref{woekg} {has bidegree $(p,q)$; this means  that \\eqref{woekg} defines a multifunction {that maps} every $z\\neq0$ to $q$ different values of $w$ and its inverse maps every $w\\neq c$ to $p$ values of $z.$} {We shall establish an upper bound for the Hausdorff dimension of certain Julia sets in this family (Theorem A), thereby showing that the Julia set of \\eqref{woekg} has zero Lebesgue measure for parameters $c$ close to zero when $q^2 <p$ (Theorem B).}\n\n\n\n\nThe family  \\eqref{woekg} can be regarded as a generalisation of the quadratic family $z^2+c$, for if $\\beta=p\/q$ in lowest terms, then \\eqref{woekg} is the multifunction $\\mathbf{f}_c(z)=z^{\\beta}+c$, in the sense that $\\mathbf{f}_c(z)$ is the set of all $w$ satisfying  \\eqref{woekg}. There is another way of generalising the quadratic family in the context of holomorphic correspondences using matings. Indeed, the space of holomorphic correspondences includes  all possible Kleinian groups and rational maps, and some correspondences concentrate in a single expression $P(z,w)=0$ the dynamics of a polynomial map and a Kleinian group.   This is the case of the oneparameter family of holomorphic correspondences $\\mathcal{F}_a$\n introduced by Bullett and Penrose nearly thirty years ago in \\cite{Bullett1994}, when they discovered that   if $4\\leq a \\leq 7$, then the correspondence $\\mathcal{F}_a$ is a mating between some quadratic map $z^2 +c$ and the modular group $\\operatorname{PSL}(2, \\mathbb{Z}).$ In the same paper \\cite{Bullett1994}, the authors conjectured that the connectedness locus of the family $\\mathcal{F}_a$ is homeomorphic to the Mandelbrot set. Recently, Bullett and Lomonaco  \\cite{BL19}  have shown that  {\\it $\\mathcal{F}_a$   is a mating between some parabolic map and the modular group, for every parameter $a$ in the connectedness locus of the family $\\mathcal{F}_a$.} Moreover, they have developed a strategy to prove that the connectedness locus of the family of matings is indeed homeomorphic to the Mandelbrot set, see \\cite[page 4]{BL17} and \\cite{BL20}.  Holomorphic correspondences also appear in many other contexts; indeed,  Lee, {Lyubich,} Makarov and Mukherjee \\cite{Mukherjee} have  investigated the dynamics {oneparameter} families of Schwarz reflections which give rise to anti-holomorphic correspondences that are, in a suitable sense, matings of anti-rational maps with the abstract modular group $\\mathbb{Z}_2 * \\mathbb{Z}_3.$ \n\n\n\\subsection{Hausdorff dimension}\n  The Julia set $J(\\mathbf{f}_c)$ of $\\mathbf{f}_c(z)=z^{\\beta} +c$, when $\\beta>1$,  is defined as the closure of all repelling cycles (see section \\ref{qpdkg});  \nit is always the projection of a solenoid in $\\mathbb{C}^2$ when the parameter $c$ is close to zero, as described by Siqueira and Smania in \\cite{SS17}.\nSimon \\cite{Simon97} has derived an explicit formula for the Hausdorff dimension of the Smale-Williams solenoid which relies on the zero of the pressure function, see \\cite[page 1224]{Simon97}.  Before Simon,  the pioneer work of Bowen  \\cite{Bowen1979} on quasi-Fuchsian groups was the first to establish the formula $P(t\\phi)=0$, relating  the Hausdorff dimension to the unique zero of the pressure function.    Similarly, in the early eighties Ruelle \\cite{Ruelle}  proved that  if the Julia set {$J(f)$} of a rational function $f$ is hyperbolic, then the Hausdorff dimension of {$J(f)$} depends real analytically on $f.$ The strategy used by Ruelle consists of: \n\n(I) showing that the Hausdorff dimension is given by the \\emph{Bowen's formula:}  $P(t\\phi)=0,$ where $t=\\dim_H J$ and  $\\phi(z)=-\\log |f'(z)|$ is the geometric potential; and\n\n (II) proving that {$f \\mapsto t(f)$} is real analytic, where {$t=t(f)$} is implicitly given by $P(t\\phi)=0.$\n \n \n In the context of holomorphic correspondences, we have the following result. \n \n \n \\begin{thm}[Theorem \\ref{kfieg}] \\label{bjdq}Suppose $J(\\mathbf{f}_c)$ is a locally eventually onto hyperbolic repeller and let $t_c$ be the unique zero of the pressure function. Then $\\operatorname{dim_{H}} J(\\mathbf{f}_c) \\leq t_c. $\n\n \n \\end{thm}\n \n\n \n The Bowen parameter $t_c$ in Theorem \\ref{bjdq} comes from an expanding and topologically mixing map $f_c: J(f_c) \\to J(f_c)$ acting on a `Julia set' $J(f_c) \\subset \\mathbb{C}^2$ -- see Theorem \\ref{igdg}.\n\n  The shape of $J(f_c)$ is similar to that of the Smale-Williams solenoid for parameters close to zero, and the projection of $J(f_c)$ is always the Julia set $J(\\mathbf{f}_c)$  in the plane \\cite{SS17}.  The sets $J(f_c)$ are related by a holomorphic motion in $\\mathbb{C}^2$ (section \\ref{bndhw}). \n  Since $J(f_c) \\subset \\mathbb{C}^2$ moves holomorphically, we believe that the Hausdorff dimension of $J(f_c)$  depends real analytically, or at least continuously on $c.$ However, this problem is still unsolved. (It should be noticed that even though  $t_c$ comes from the dynamics of $f_c$ on $J(f_c) \\subset \\mathbb{C}^2,$ in this paper the parameter $t_c$  is used to estimate the Hausdorff dimension of the Julia set $J(\\mathbf{f}_c)$ \\emph{in the plane}). \n  \n  \n   The estimate provided by Theorem \\ref{bjdq} can be used to derive the following result.\n  \n  \n  \\begin{thm}[Corollary \\ref{hjeeh}] \\label{qpoig}If $c$ is sufficiently close to zero and $q^2<p$, then the solenoidal Julia set of $z^{p\/q} +c$ has zero Lebesgue measure.\n\n\\end{thm}\n\nSee \\cite[page 3106]{SS17} for a typical figure of the solenoidal Julia set of Theorem \\ref{qpoig}.\n \n \n \n\\subsection{Hyperbolic components} The \\emph{connectedness locus} of the family $z^{\\beta}+c$, denoted by $\\mathcal{M}_{\\beta}$ (for $\\beta >1$ and $\\beta \\in \\mathbb{Q}),$ is the set of all parameters $c$ for which the Julia set of $z^{\\beta}+c$ is connected.   We define $\\mathcal{M}_{\\beta,0}$  as the set of $c$ for which zero has at least one bounded forward orbit under $z^{\\beta}+c.$ {Siqueira has shown \\cite[Theorem 2.3]{Rigidity}} that $\\mathcal{M}_{\\beta}$  contains $\\mathcal{M}_{\\beta,0}.$ Therefore,   the Julia set of $z^{\\beta}+c$ is connected whenever the critical point  has at least one bounded forward orbit under $\\mathbf{f}_c.$ \n\n\\noindent For the quadratic family we have $\\beta=2$ and the definitions  of $\\mathcal{M}_{2}$ and $\\mathcal{M}_{2,0}$ coincide with the Mandelbrot set. For some non-integer values of $\\beta$, the parameter space $\\mathcal{M}_{\\beta} - \\mathcal{M}_{\\beta,0}$ {is known to be nonempty } and generates an intriguing class of Julia sets named Carpets: they are hyperbolic, connected, and {have infinitely many holes.} In spite of being hyperbolic, Carpets seem to have positive area. See \\cite[section 3]{BLS} for more details and figures.\n\nA {special} version of the Fatou conjecture for polynomial maps states that the interior of $\\mathcal{M}_d$ consists of hyperbolic parameters. One implication of this conjecture is: \\emph{the Julia set of every polynomial $z^d+c$ has zero Lebesgue measure if $c$ is in the interior of $\\mathcal{M}_d$.} The Hausdorff dimension $d(c)$ of the Julia set of the polynomial $z^d +c$ is real analytic on every hyperbolic component; in particular, it is real analytic on $\\mathbb{C} - \\mathcal{M}_d.$  In the boundary of $\\mathcal{M}_d,$ the Hausdorff dimension $d(c)$  is not even continuous at semihyperbolic parameters. However, Rivera-Letelier  \\cite{rivera-letelier2001} has established some sort of continuity of $d(c)$ at semihyperbolic parameters $c_0$ in the boundary of the multibrot set $\\mathcal{M}_d$, proving that {$d(c_n) \\to d(c_0),$} whenever $c_n$ converges to $c_0$ in an appropriate way (see \\cite{rivera-letelier2001} for more details). \n\nLittle is known about the connectedness locus $\\mathcal{M}_{\\beta}$ of the family $z^{\\beta}+c.$ One possible way to start the investigation has been presented in \\cite{Rigidity}: study the dynamics of $z^{\\beta}+c$ when $c$ is close to a centre. Recall from Douady and Hubbard \\cite{DH84, DH85}  that the  Mandelbrot set has infinitely many hyperbolic components  $U$, each of which encoded by a centre $c\\in U$; the centre is the only parameter in $U$ for which  the orbit of the critical point of $z^2+c$ is a cycle. For the correspondence $z^{\\beta}+c,$ the parameter $a$ is a \\emph{simple centre} if only  one forward orbit of zero $(0, z_1, z_2, \\ldots)$  is periodic, and any other orbit $(0, w_1, w_2, \\ldots)$ diverges to infinity ({see Definition \\ref{hgdw};} it is not necessary to compute all orbits to test if $a$ is a simple centre. Indeed, the basin of attraction of $\\infty$ contains a forward invariant  disk $|z|>R$, and therefore we have to check only finitely many iterates).\n\nThere exists an open set $H_{\\beta}$  containing the complement of $\\mathcal{M}_{\\beta,0}$ and every simple centre such that, for any $c$ in $H_{\\beta},$  $z^{\\beta}+c$ is hyperbolic and its Julia set is stable by means of branched holomorphic motions; moreover, $J(\\mathbf{f}_c)$ is a locally eventually onto (LEO) hyperbolic repeller, whenever $c\\in H_{\\beta}.$  { (See Definition 4.2, Corollary 5.7.1 and Theorems 5.8 and 5.10 of \\cite{Rigidity}).   }\n\nCombining these facts with Theorem \\ref{bjdq} we have the following result. \n\n\\begin{thm}[Corollaries \\ref{abc} and \\ref{abcxz}] If $c$ is sufficiently close to a simple centre, or if $c$ belongs to the complement of $\\mathcal{M}_{\\beta,0}$, then $$\\dim_H J(\\mathbf{f}_c) \\leq t_c.$$\n\\end{thm}\n\n\n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Hyperbolic sets}  \\label{qpdkg}\n\n\nConsider the  holomorphic correspondence \\eqref{woekg}, i.e., the relation $z\\mapsto w$ determined by the polynomial equation $(w-c)^{q}=z^p.$ This correspondence shall be denoted by $\\mathbf{f}_c.$ The integers $p> q \\geq 2$ are fixed. {Therefore, $(\\mathbf{f}_c)_{c\\in \\mathbb{C}}$ is a oneparameter family of holomorphic correspondences.  } \n\n\\subsection{Preliminary definitions}  {If $z$ and $w$ are related by \\eqref{woekg}, then we say that $w$ is an \\emph{image} of $z$ and write $z \\mapsto w.$}   Every sequence $(z_i)_0^{\\infty}$ of the plane where $z_{i+1}$ is an image of $z_i$ is an \\emph{orbit of the correspondence.} An orbit is a \\emph{cycle of period $n$} if $z_n=z_0,$  and $z_i \\neq z_0$ if $0<i<n.$ \n\nThe set of images of a point $z$ under the correspondence \\eqref{woekg}  is denoted by $\\mathbf{f}_{c}(z).$ Similarly, $\\mathbf{f}_c(A)$ is the union of all $\\mathbf{f}_c(z)$ when $z$ belongs $A.$ The inverse image set $\\mathbf{f}_{c}^{-1}(A)$ is defined by the set of all $z$  which has at least one image in $A.$\n\n\n\n\n\n If $z_1$ is an image of $z_0$ under the correspondence \\eqref{woekg} and $z_0\\neq 0,$ then  there exists a unique univalent function $w=g(z)$ defined on a neighbourhood $U$ of $z_0$,  implicitly defined by the equation  \\eqref{woekg}  and $g(z_0) =z_1.$  If $(z_i)_0^{n}$ is a finite orbit {not containing zero,} we denote \\begin{equation}\\label{gopihqe} g_{z_0, z_1}(z) = g(z)   \\ \\textrm{and} \\   g_{z_0,z_1,\\ldots, z_n}(z) = (g_{z_{n-1}, z_{n}} \\circ \\cdots \\circ g_{z_0,z_1})(z). \\end{equation} The domain of $g_{z_0,z_1,\\ldots, z_n}$ is  an unspecified neighbourhood of $z_0.$ A cycle $(z_i)_0^{n}$  with period $n$ is \\emph{repelling}  if the absolute value of the derivative of $g_{z_0, z_1, \\ldots, z_n}$ at $z_0$ is strictly greater than $1.$  \n Therefore, no point of a repelling cycle is allowed to be zero. \n \n \n\\begin{defi}[Julia set] The closure of the union of all repelling cycles is the \\emph{Julia set} $J(\\mathbf{f}_c)$ of the correspondence $\\mathbf{f}_c.$ \n\\end{defi}\n\n\n\n\n\\begin{defi}[Hyperbolic repeller]\\label{gjege} A compact set $\\Lambda \\subset \\mathbb{C}^*$ is a {\\emph{hyperbolic repeller}} if $\\mathbf{f}_c^{-1}(\\Lambda)=\\Lambda$ and there exist $C>0$ and $\\lambda >1$ such that \\begin{equation} \\label{gjcmwe}|g_{z_0\\cdots z_n}'(z_0)| \\geq C\\lambda^n, \\end{equation} for every finite orbit $(z_i)_0^{n}$ contained in $\\Lambda.$  \\end{defi}\n\nSince every hyperbolic repeller $\\Lambda$  is backward invariant, its complement is forward invariant.  { We define   $B_{r}(a) =\\{z\\in \\mathbb{C}: |z-a|<r\\}.$ }\n\n\\begin{lem}[Koebe's distortion] \\label{dgrwbff} Let $\\mathcal{F}$ denote the family of all univalent functions $\\varphi:B_r(a) \\to \\mathbb{C},$ where $r>0$ and $a\\in \\mathbb{C}$ are allowed to vary arbitrarily.  \n{There exists a constant $K>0$  such that, for any $\\varphi$ in  $\\mathcal{F}$,} we have\n\n\\begin{equation}\\label{igdw} K^{-1} |\\varphi'(a)| \\cdot |z-w|\\leq {|\\varphi(z)-\\varphi(w)|} \\leq K |\\varphi'(a)| \\cdot |z-w|, \\end{equation}\nwhenever $z$ and $w$ belong to $B_{r\/4}(a),$ where $r$ is the radius of the domain $B_{r}(a)$ of $\\varphi.$ (Note that $K$ depends neither on $r$ nor $a$). \n\n\n\\end{lem}\n\n\n\\begin{defi}[Expansive constant for correspondences] \\normalfont Let $\\omega \\in \\mathbb{N}_{0}\\cup \\{\\infty\\}.$ {A pair of orbits $(z_i)_0^{\\omega}$ and $(w_i)_0^\\omega$ is said to be $\\epsilon$-close if $|z_i -w_i|< \\epsilon$ for $0\\leq i \\leq \\omega.$ } We say that $\\epsilon>0$ is an \\emph{expansive constant} for a hyperbolic repeller $\\Lambda$ if every pair of $\\epsilon$-close orbits $(z_i)_0^\\infty$ and $(w_i)_0^\\infty$ {in} $\\Lambda$  must actually coincide. \n\\end{defi}\n\n\n\n\\begin{lem}\\label{gjecb} Every hyperbolic repeller $\\Lambda$ of $\\mathbf{f}_c$ has an expansive constant $\\epsilon>0$ satisfying the following property: if {$(z_i)_0^n$} and $(w_i)_0^n$ is a pair of finite $\\epsilon$-close orbits contained in $\\Lambda,$ then both maps $(g_{z_0 \\cdots z_n})^{-1}$ and $(g_{w_0 \\cdots w_n})^{-1}$ are well defined and  coincide on $B_{\\epsilon}(z_n).$  In particular, $(g_{z_0 \\cdots z_n})^{-1}$  maps $w_n$ to $w_0.$ \n\\end{lem} \n\n\\begin{proof} \n\nLet $X_{n,c}$ denote the space of all finite orbits $(z_i)_0^n$  contained in $\\Lambda.$ We know that $X_{n,c}$ is a closed subset of the product space $\\Lambda^n$, and therefore $X_n,c$ is a compact metric space with the induced metric from $\\Lambda^n$. (We have used this fact repeatedly  in \\cite{SS17}; see, for example,  the beginning of \\cite[proof of Lemma 2.3]{SS17}).\n\n\nThe proof is divided into four steps.\n\n\n\\noindent  {\\bf Claim A:} {\\it  given a finite orbit $(z_i)_0^n \\in X_{n,c}$ there exist $\\epsilon>0$ and $\\rho$ with $\\epsilon <\\rho$ such that for any $(w_i)_0^n$ in the open set $V_0 \\subset X_{n,c}$ consisting of all finite orbits which are $\\epsilon$-close to $(z_i)_0^n,$ we have:\n\n\n $(i)$ the univalent branches $\\varphi=g_{z_0 \\cdots z_n}$ and $\\psi=g_{w_0 \\cdots w_n}$ coincide on $B_{\\rho}(z_0)$; and \n \n $(ii)$ the image of $B_{\\rho}(z_0)$ under $\\varphi$  contains $B_{\\epsilon}(z_n).$  Consequently, $\\varphi^{-1}$ and $\\psi^{-1}$ coincide on $B_{\\epsilon}(z_n)$ and both send $w_n$ to $w_0.$ } \n \n \n Let us prove the case $n=1.$   Recall that if $z_0\\neq 0$, then there exist $q$ univalent branches $g_i: U \\to \\mathbb{C}$ defined on a neighbourhood of $z_0$ such that $\\mathbf{f}_c(U)$ is completely determined by the union of all $g_i(U).$ We may assume, without loss of generality, that for some $\\delta>0,$ the diameter $|g_i(U)|$ is less then $\\delta$ and $|g_i(z_0) - g_j(z_0)| > 9\\delta$, whenever $i \\neq j.$ (Roughly speaking, the sets $g_i(U)$ are very small and away from each other).  Hence, for some $0<\\epsilon_1< \\delta$,  if $w_0 \\mapsto w_1$ is $\\epsilon_1$-close to $z_0 \\mapsto z_1$, it follows that $g_{z_0, z_1}$ and $g_{w_0, w_1}$ must be determined by the same univalent branch $g_{i_0}: U \\to \\mathbb{C},$ otherwise $z_1$ and $w_1$ would not belong to the same $g_{i_0}(U)$, and consequently, $|z_1-w_1| > \\epsilon_1,$ which is impossible, since we have a pair of $\\epsilon_1$-close orbits. There exists $\\rho>0$ such that $B_{\\rho}(z_0) \\subset U.$  Hence $g_{z_0, z_1}|_{B_{\\rho}(z_0)} = g_{w_0, w_1}|_{B_{\\rho}(z_0)}=g_{i_0}|_{B(z_0, \\rho)},$ and it follows that $z_1$ belongs to $g_{i_0}(B_{\\rho}(z_0)).$ There exists  $\\epsilon> 0$  such that $B_{\\epsilon}(z_1) \\subset g_{i_0}(B_{\\rho}(z_0))$, with $\\epsilon < \\epsilon_1$ and $\\epsilon <\\rho.$  Then $(i)$ and $(ii)$ of Claim A follow with this choice of $\\epsilon$ and $\\rho.$ This proves Claim A for  $n=1.$  The general case $n\\geq 1$ follows by induction.\n  \n   Our next step is to eliminate the dependence of $\\epsilon$ on $(z_i)_0^n.$ \n \n \\noindent  {\\bf Claim B:}  {\\it for a fixed $n\\geq 1$, there exists $0<\\epsilon<1$ such that for any pair of $\\epsilon$-close orbits $(z_i)_0^n$ and $(w_i)_0^n$ in $X_{n,c}$,  both maps $(g_{z_0 \\cdots z_n})^{-1}$ and $(g_{w_0 \\cdots w_n})^{-1}$ are well defined and  coincide on $B_{\\epsilon}(z_n).$     }\n \n  Using Claim A, we construct a finite  covering $\\{V_i\\}$ of $X_{n,c}$ and finitely many $ \\rho_i > \\epsilon_i>0$ for which $(i)$ and $(ii)$ of Claim A hold, whenever $(z_i)_0^n$ and $(w_i)_0^n$ are in $V_i.$ Then chose $0<\\epsilon<1$ as being  a Lebesgue number of the covering such that $\\epsilon<\\epsilon_i$, for every $i$. Any pair of $\\epsilon$-close orbits are within the same $V_i;$ by Claim A $(ii)$ the corresponding inverses coincide on $B_{\\epsilon_i}(z_n).$ Since $\\epsilon<\\epsilon_i$, they coincide on $B_{\\epsilon}(z_n).$  \n \n (The following $\\epsilon$ does not depend on $n$).\n \n  \n\\noindent   {\\bf Claim C:} {\\it there exists  $0<\\epsilon<1$ such that, for every $n\\geq 1$ and every $(z_i)_0^{n}$ in $X_{n,c}$, the domain of $(g_{z_0 \\ldots z_n})^{-1}$ contains $B_{\\epsilon}(z_n).$  Moreover, if $(w_i)_0^n$ is $\\epsilon$-close to $(z_i)_0^n$ then $(g_{w_0 \\cdots w_n})^{-1}$ is defined on $B_{\\epsilon}(z_n)$ and coincide with $(g_{z_0 \\cdots z_n})^{-1}$ on $B_{\\epsilon}(z_n).$    } \n\nWe shall postpone the proof of this claim.    \n  \n\n     \n\\noindent {\\bf Claim D:}  {\\it $\\epsilon_{0}=\\epsilon\/4$ is an expansive constant, where $\\epsilon$ is given by Claim C. \n}\n\n\n { (Therefore, Lemma \\ref{gjecb} follows from Claim D with $\\epsilon$ replaced by $\\epsilon\/4.$)}\n\n  \n  {\\it Proof of Claim D.} Suppose $(z_i)_0^\\infty$ is $\\epsilon_{0}$-close to $(w_i)_0^\\infty$. We need to show that $z_n=w_n$, for every $n.$ Fix $n$ and let $\\{\\varphi_m\\}$ denote the family of univalent maps $(g_{z_n \\cdots z_m})^{-1}$; by Claim C they are all defined on $B_{\\epsilon}(z_m),$ for any $m>n$, and each $\\varphi_{m}$ maps $w_m$ to $w_n.$  By \\eqref{gjcmwe}, the absolute value of the derivative of $(g_{z_n \\cdots z_m})^{-1}$ at $z_m$ is at most $C^{-1}\\lambda^{-(m-n)}$, where $C>0$ and $\\lambda>1$ are the constants of \\eqref{gjcmwe}. Using \\eqref{gjcmwe} and Koebe's Lemma \\ref{dgrwbff} applied to the family of all $\\varphi_m:B_{\\epsilon}(z_m) \\to \\mathbb{C}$     we have\n  \\begin{equation} \\label{bjkdfe}\n\\begin{split}\n|z_n -w_n| =|\\varphi_m(z_m) - \\varphi_{m}(w_m)| & \\leq K C^{-1}\\lambda^{-(m-n)}|z_m -w_m| \\\\  & \\leq KC^{-1}\\lambda^{-(m-n)} 2\\epsilon_0 \\to 0, \n\\end{split}\n\\end{equation}\n  as $m\\to \\infty.$ Notice that $K$ is the constant of Koebe's Lemma \\ref{dgrwbff} and we have used the fact $|z_m -w_m|< \\epsilon\/4,$ which is fundamental for the application of Lemma \\ref{dgrwbff}. We conclude that  $z_n=w_n$, for every $n.$\n\n\n\n  {\\it Proof of Claim C.}  Let $K$ be the constant of Lemma \\ref{dgrwbff}; let $C>0$ and $\\lambda>1$ as in  \n  \\eqref{gjcmwe}. There exists $n>0$ such that $KC^{-1} \\lambda^{-n} <1\/2.$ Keep $n$ fixed and find a corresponding $\\epsilon'>0$ satisfying the properties of Claim B.  Let $k>0$ be an integer. Suppose  $(w_i)_{0}^{nk}$  and $(z_i)_0^{nk}$ are finite orbits contained in $\\Lambda$ which are $\\epsilon'$-close to one another. By Claim B, for any $0< i \\leq k$, the maps $\\varphi_i=(g_{z_{n(i-1)} \\cdots z_{ni}})^{-1}$ and  $\\psi_i=(g_{w_{n(i-1)} \\cdots w_{ni}})^{-1}$ are well defined and coincide on $B_{\\epsilon'}(z_{ni}).$ Using the same argument of \\eqref{bjkdfe} in the proof of Claim D,  we conclude that the maps $\\varphi_i$ and $\\psi_i$ are well defined contractions by factor $1\/2$ on $B_{\\epsilon'\/4}(z_{ni})$ and $B_{\\epsilon'\/4}(w_{ni}),$ respectively. Moreover, they define the same inverse branch on  $B_{\\epsilon'\/4}(z_{ni})$ and  $\\varphi_i$ maps $B_{\\epsilon'\/4}(z_{ni})$  into $B_{\\epsilon'\/8}(z_{n(i-1)}).$ This ball of radius $\\epsilon'\/8$ is obviously contained in the domain $B_{\\epsilon'}(z_{n(i-1)})$ of $(g_{z_{n(i-2)} \\cdots z_{n(i-1)}})^{-1};$ so that we are allowed to perform the compositions:  $$(g_{z_0 \\cdots z_{nk}})^{-1}= (g_{z_0 \\cdots z_n})^{-1} \\circ \\cdots  \\circ (g_{z_{n(k-1)} \\cdots z_{nk}})^{-1},$$       \n  $$(g_{w_0 \\cdots w_{nk}})^{-1}= (g_{w_0 \\cdots w_n})^{-1} \\circ \\cdots  \\circ (g_{w_{n(k-1)} \\cdots w_{nk}})^{-1},$$ thereby showing that $(g_{w_0 \\cdots w_{nk}})^{-1}$ and  $(g_{z_0 \\cdots z_{nk}})^{-1}$ are well defined and coincide on $B_{\\epsilon'\/4}(z_{nk}).$ Since $k>0$ is arbitrary, Claim C follows with $\\epsilon=\\epsilon'\/4.$  \n  \n  \n\n\n This completes the proof. \\end{proof}\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n \n \n\n \n \n\n\\begin{defi}[LEO hyperbolic repellers] The correspondence $\\mathbf{f}_c$ is called \\emph{locally eventually onto (LEO)} on a hyperbolic repeller $\\Lambda$ if for every  relatively open subset $U$ of $\\Lambda$  there exists $n>0$ such that $\\mathbf{f}_c^n(U)$ contains $\\Lambda. $\n\n\\end{defi}\nAccording to Theorem 4.3 of \\cite{SS17}, every LEO hyperbolic repeller $\\Lambda$ of $\\mathbf{f}_c$ is contained in the Julia set $J(\\mathbf{f}_c).$\n\n  \\begin{defi}[Simple centre]\\label{hgdw} The critical point $z=0$ has infinitely many forward orbits under $\\mathbf{f}_c.$ If precisely one forward orbit of the critical point of $\\mathbf{f}_c$ is periodic and the others diverge to $\\infty$, we say that $c$ is a \\emph{simple centre.}  \n  \n  \\end{defi}\n\nThe following result is stated as Theorem 5.4 in \\cite{Rigidity}.\n\n\\begin{thmm} If $a$ is a simple centre for the family of holomorphic correspondences $\\mathbf{f}_c$, then $J(\\mathbf{f}_c)$ is a LEO hyperbolic repeller for every $c$ in a neighbourhood of $a.$\n\n\\end{thmm}\n\n\n\n\nWe shall now  summarise some results of \\cite{SS17}   concerning $\\mathbb{C}^2$-extensions.  \n\n\\subsection{The $\\mathbb{C}^2$-extension} \\label{bndhw} Suppose  $J(\\mathbf{f}_{c_0})$ is a LEO hyperbolic repeller for the correspondence $\\mathbf{f}_{c_0}.$ There exists  a family of holomorphic maps \\begin{equation}\\label{gjemqe}f_c: V \\to \\mathbb{C}^2\\end{equation} defined on a open subset $V$ of $\\mathbb{C}^2$ and parameterised on a neighbourhood $U$ of $c_0,$ such that   the closure of the periodic points of $f_c,$ denoted by $J(f_c)$, is a subset of $\\mathbb{C}^2$ which is completely invariant under $f_c.$ (See \\cite[Lemma 2.3]{SS17}). Moreover, {$f_c$ is a $p$-to-$1$ map } on $J(f_c).$   The  notation  $J(f_c)$ suggests the definition of a Julia set, or that every periodic point of $f_c$ is repelling.  As a matter of fact, the Jacobian determinant of $f_c^n$ at every periodic point of period $n$ is strictly greater than $1.$ This fact is explained in Remark 2.4 of \\cite{SS17} and somehow justifies the notation $J(f_c).$ In spite of this analogy, for technical reasons $J(f_c)$ shall not be referred to as the Julia set of $f_c.$\n\n\n The family \\eqref{gjemqe} is defined for parameters in a neighbourhood of $c_0$ and {enjoys some important properties.} \n\n\n \\begin{itemize} \\item[$(A)$]\n{There exists} a holomorphic motion $h_c: J(f_{c_0}) \\to J(f_c)$  parameterised on $U$ and {based at $c_0$,} given by a family of conjugacies $h_c$ from $J(f_{c_0})$ to $J(f_c).$ {(See Theorems B and C of \\cite{SS17}).}\n \n\\item[$(B)$] The projection $\\pi(z,w)=z$ establishes a semiconjugacy $\\pi: J(f_c) \\to J(\\mathbf{f}_c)$ {between $f_c$ and $\\mathbf{f}_c$,}  in the sense that $J(\\mathbf{f}_c)=\\pi J(f_c)$ is also a LEO hyperbolic repeller for $\\mathbf{f}_c$ and $\\pi f_c(x)$ is an image of $\\pi(x)$ under $\\mathbf{f}_c,$ for every $x$ in $J(f_c).$  Hence $$\\pi(x) \\to \\pi f(x) \\to \\pi f^2(x) \\to  \\cdots$$ is a forward orbit under $\\mathbf{f}_c.$ This is, by definition, the \\emph{projected orbit of} $x.$ {(See Lemma 2.3 of \\cite{SS17}).}\n \n\n \n\n\n\n\\end{itemize}\n\n\\section{The Bowen parameter}\n\n\n\nRecall that a continuous surjective map $f: X \\to X $ of a compact metric space is \\emph{expanding} if there exists $\\ell >1$ such that every point in $X$ has a neighbourhood $V$  such that $f^{-1}(V)$ is a finite union of disjoint open sets $U_j,$ each of which is mapped homeomorphically  onto $V$  and  $$d(f(x), f(y)) > \\ell d(x,y)$$  for $x$ and $y$ in $ U_j.$ \n\n\nThe LEO property may be defined for every  $f:X \\to X.$ It means that every nonempty open set of $X$ is eventually mapped onto $X.$  The following result is stated as Theorem 4.3 in \\cite{SS17}. \n\n\n\\begin{thmm}\\label{gjegde} {Let $c_0$ be a parameter in $\\mathbb{C}$ such that $J(\\mathbf{f}_{c_0})$ is a LEO hyperbolic repeller for $\\mathbf{f}_{c_0}.$} {For every $c$ in a neighbourhood of $c_0,$}  the map $f_c$ of the family \\eqref{gjemqe} {is  LEO on $J(f_c)$ and expanding}  with respect to the metric\n$$ d_s(x,y)=  \\sum_{n=0}^{\\infty} s^{-n}|\\pi f_c^n(x) - \\pi f_c^n(y) |$$ where $s>1$ is arbitrary  and $x,y$ belong to $J(f_c).$\n\\end{thmm}\n\nThe \\emph{dynamic ball of radius $\\epsilon$, time $n$ and centre $x$} is defined by  \n  $$B(x, n, \\epsilon)=\\{y\\in X: d(f^jx, f^jy)< \\epsilon, \\ 0 \\leq j \\leq n\\}.$$    Every expanding map $f: X \\to X$ has an \\emph{expansive constant} $\\epsilon>0$ characterised by the fact that if  $d(f^nx,f^ny)<\\epsilon$ for every $n\\geq 0,$ then $x=y.$  (It is clear that any other positive real number $\\epsilon_0 < \\epsilon$ is also an expansive constant.) \n  \n  A continuous map $f: X \\to X$ is \\emph{topologically mixing}  if for every pair of nonempty open subsets $U$ and $V$ there exists $n_0$ such that $f^n(U)\\cap V$ is nonempty, for every $n\\geq n_0.$ LEO maps are  topologically mixing. \n  \n  \n   Suppose  $f$ is an expanding map of a compact metric space $(X,d)$, and  $\\phi: X\\to \\mathbb{R}$ is a potential (i.e, a real valued continuous function).   {The topological pressure of $\\phi$ with respect to the system $f$ is denoted by $P(\\phi),$ and $S_n\\phi(x)$ denotes a Birkhoff sum.} Since $f$ is expanding, there exists an expansive constant $\\epsilon>0.$ If $\\mu$ is a probability measure on $X$ and there exists $C_{\\epsilon}>0$ such that $$C_{\\epsilon}^{-1} \\leq \\frac{\\mu(B(x,n,\\epsilon))}{ \\exp (S_n\\phi(x) - n P(\\phi))} \\leq C_{\\epsilon}, $$ for every $x$ in $X$ and $n>0,$  then $\\mu$ is a \\emph{Gibbs measure} of $f$ and $\\phi.$ (See \\cite[Chap. 4]{ConformalF}).\n\n\n\\subsection{Transfer operators.}  \nLet $f: X \\to X$ be an expanding map of a compact metric space $(X,d).$ The  \\emph{transfer operator} $\\mathcal{L}_{\\phi}: C(X) \\to C(X)$ with potential $\\phi$ acts on the space $C(X)$ of continuous complex valued functions $g: X \\to \\mathbb{C},$ and is defined by $$ \\mathcal{L}_{\\phi}g(x) = \\sum_{f(y)=x} e^{\\phi(y)} g(y).$$ \n\n\\noindent (See \\cite[Chap. 4]{ConformalF}). The iterate $\\mathcal{L}_{\\phi}^n$ is precisely the transfer operator $\\mathcal{L}_{S_n\\phi}$ with respect to $f^n:X\\to X.$\n\n The dual operator $\\mathcal{L}_{\\phi}^*: M(X) \\to M(X)$ acts on the space of complex measures defined on the Borel $\\sigma$-algebra of $X.$ It is defined by $\\langle \\mathcal{L}_{\\phi}^*\\mu, g\\rangle = \\langle \\mu,  \\mathcal{L}_{\\phi} g\\rangle, $ for every $\\mu$ in $M(X)$ and $g$ in $C(X),$ where $\\langle \\mu, g\\rangle = \\int_X g  d\\mu.$ The following theorem summarises some well known results concerning the transfer operator, and  the first three sentences are often referred to as the \\emph{Ruelle-Perron-Frobenius theorem} \\cite[Chap. 4]{ConformalF}. \n\n\n\n\\begin{thmm} \\label{gjdcs}  Suppose $f: X \\to X$ is an expanding and topologically mixing map of a compact metric space $X.$ If $\\phi: X \\to \\mathbb{R}$ is H\\\"older continuous and $\\lambda_{\\phi}=\\exp P(\\phi),$ then: \n\n\n\n\\begin{itemize}\\item[$(a)$] there exists a unique probability measure $\\nu$ on $X$ such that  $\\mathcal{L}_{\\phi}^*(\\nu)=\\lambda_{\\phi} \\nu;$ \n\n\\item[$(b)$] there exists a unique real valued continuous function $h>0$ on $X$ such that $\\mathcal{L}_{\\phi}(h)=\\lambda_{\\phi} h$ and $\\int_X hd\\nu=1;$\n\n\\item[$(c)$] for any continuous real valued function $g$ on $X,$  $$\\lambda_{\\phi}^{-n} \\mathcal{L}_{\\phi}^{n}(g) \\to  h \\int_X g d\\nu,$$ {as $n\\to \\infty$, }and the convergence is uniform on $X;$  \n\\item[$(d)$] the measure $\\mu(A) = \\int_A hd\\nu$  is the unique invariant Gibbs measure of $f$ and $\\phi.$\n\n\n\\end{itemize}\n\n\\end{thmm}\n\n\n\n\\begin{remark} \\label{qodwe}\\normalfont\nThe number $P(0)$ is also known as the topological entropy of $f.$ \nSuppose that $f: X\\to X$ is a $d$-to-$1$ map (i.e., every point has exactly $d$ preimages). Then $f^n$ is $d^n$-to-$1.$ By   Theorem \\ref{gjdcs},  $$\\lambda_{\\phi}^{-n}\\mathcal{L}_0^n(1) = \\lambda_{\\phi}^{-n} \\sum_{f^n(y)=x} e^{S_n \\phi(y)} \\cdot 1 = \\lambda_{\\phi}^{-n}d^n$$ converges to $h>0$, where $\\phi=0.$ It follows that $\\lambda_{\\phi}= d,$ and the topological entropy of $f$ must be $\\log(d).$\n\\end{remark}\n\n\nThe following result introduces the Bowen parameter $t_c.$ \n\nLet $z_0(x)$ and $z_1(x)$ denote the first two elements of the projected orbit of an element $x$ of  $J(f_c).$  We define $\\varphi_c: J(f_c)  \\to \\mathbb{R}$ by   $ \\varphi_c(x)= -\\log|g_{z_0, z_1}'(z_0)|,$ where $z_0$ and $z_1$ are $z_0(x)$ and $z_1(x),$ respectively.   The map $\\varphi_c$ is therefore a potential on $J(f_c)$, and  the zero of the pressure function $t\\mapsto P(t\\varphi_c) $  is defined as the \\emph{Bowen parameter $t_c.$} However, we need to show that there exists only one zero of the pressure function. This is established in the following theorem. Note that $\\varphi_c(x)$ is defined directly by  \\eqref{lkmfg}, for then  $g_{z_0, z_1}'(z_0)$ equals $p(z_1 -z_0)\/(qz_0).$ \n\n\n\n\n\\begin{thmm}[Bowen parameter] \\label{igdg}   If the Julia set  $J(\\mathbf{f}_c) \\subset \\mathbb{C}$ of the holomorphic correspondence $(w-c)^q=z^p$ is a LEO hyperbolic repeller, then the topological entropy of the $\\mathbb{C}^2$-extension $$f_c: J(f_c) \\to J(f_c)$$ is strictly positive. Moreover, the   potential  \n\\begin{equation} \\label{lkmfg}\\varphi_{c}(x) = - \\log|p(z_1 -z_0)\/(qz_0)|\\end{equation}   is  H\\\"older continuous with respect to the metric $d_s$ on $J(f_c),$ and   there exists a unique zero $t_c>0$ of the pressure function $t\\mapsto P(t\\varphi_c)$ defined on $[0, \\infty).$ \n\\end{thmm}\n\n\\begin{proof} According to  \\cite[Chap. 2B]{Bowen1979},  the topological pressure of a potential $\\phi$ can be calculated by \\begin{equation} \\label{eidhg} P(\\phi) = \\lim_{|\\mathcal{C}|\\to 0} \\lim_{n\\to \\infty} \\frac{1}{n} \\log (\\inf_{\\mathcal{C}_n} \\sum_{U\\in \\mathcal{C}_n} \\exp {S_n\\phi(U))},\\end{equation}\n\n\\noindent where $\\mathcal{C}$ is any finite covering of $J(f_c);$  $|\\mathcal{C}|$ is the greatest diameter of an element of $\\mathcal{C};$ and  $S_n\\varphi(U)$ is given by the supremum of all $S_n\\phi(x)$ when $x\\in U.$  The infimum in  \\eqref{eidhg} is taken over all possible covers $\\mathcal{C}_n$ of $J(f_c)$ whose elements  can be written as $$U=\\left \\{x\\in J(f_c): f_c^{j}(x) \\in U_j, \\ 0\\leq j \\leq n\\right \\},$$ for some finite sequence  $(U_j)_1^n$ of elements of $\\mathcal{C}.$ \n\n\n\n\nSince $t\\mapsto  P(t\\varphi_c)$ is continuous, in order to prove the existence of a unique zero it suffices to show that the pressure function  is strictly decreasing with $P(0) >0$ and $P(t\\varphi_c) \\to -\\infty$ as $t\\to \\infty.$   \n\n\n\n\\noindent Indeed,  $J(\\mathbf{f}_c)$ is a hyperbolic repeller, and therefore $|g_{z_0 \\cdots z_n}'(z_0)| \\geq C\\lambda^n$, where $\\lambda>1$ and   $(z_j)_0^{\\infty}$ denotes the projected orbit of any $x.$ By Theorem \\ref{igdg} and a  simple computation involving the logarithm, \n\\begin{equation} \\label{nbfwrh}\\exp S_n (t\\varphi_c)(x) = |g_{z_0\\cdots z_n}'(z_0)|^{-t}.\\end{equation} Hence,\n \\begin{align*} P((t+\\tau)\\varphi_c) & = \\lim_{|\\mathcal{C}|\\to 0} \\lim_{n\\to \\infty} \\frac{1}{n} \\log (\\inf_{\\mathcal{C}_n} \\sum_{U\\in \\mathcal{C}_n} \\exp S_n(t\\varphi_c)(U) \\cdot \\exp S_n(\\tau \\varphi_c)(U)) \\\\ \n& = \\lim_{|\\mathcal{C}|\\to 0} \\lim_{n\\to \\infty} \\frac{1}{n} \\log ( \\inf_{\\mathcal{C}_n} \\sum_{U\\in \\mathcal{C}_n} \\sup_{x\\in U} |g_{z_0 \\cdots z_n}'(z_0)|^{-t} \\cdot \\exp S_n(\\tau \\varphi_c)(U)) \\\\\n& \\leq  \\lim_{|\\mathcal{C}|\\to 0} \\lim_{n\\to \\infty} \\frac{1}{n} \\log ( C^{-t} \\lambda^{-n t}   \\inf_{\\mathcal{C}_n} \\sum_{U\\in \\mathcal{C}_n}  \\exp S_n(\\tau \\varphi_c)(U))   \\\\\n&= - t\\log(\\lambda) + P(\\tau \\varphi_c). \\end{align*}\n\n\n\\noindent Since $\\log(\\lambda) >0$, we conclude that the pressure function is indeed strictly decreasing and $P(t\\varphi_c) \\to -\\infty$ as $t \\to \\infty.$  Since $f_c$ is a $p$-to-one map, it follows from Remark \\ref{qodwe} that  $P(0) = \\log(p)>0.$   Hence, there exists a unique zero $t_c$ of the pressure function in $(0,\\infty).$   \n\n   The proof of the H\\\"older continuity of the potential with respect to the metric $d_s$ is a straightforward application of the mean value theorem. \n  \n\n\\end{proof}\n\n\\begin{remark} \\normalfont {\\it The value of $t_c$ does not depend on the particular choice of $f_c.$} \n\n{In fact,} if we consider the space $X_c$ of all forward orbits $(z_i)_0^\\infty$ under $\\mathbf{f}_c$ which are contained in  $J(\\mathbf{f}_c),$  then $g_c(x)= (\\pi f_c^n(x))_{n=0}^{\\infty}$ defines a homeomorphism from $J(f_c)$ onto $X_c$ which conjugates the dynamics of $f_c$ to the dynamics of  the (left) one-sided shift  $\\sigma$ on $X_c$  \\cite[pages 3110-3112]{SS17}.  Since the pressure is a topological invariant, this is sufficient to show that $t_c$ depends only on the dynamics of the shift map on the space of orbits.\n   \nIndeed,  \\begin{equation}\\label{ghewdgd} P(t\\varphi_c; f_c) = P(t\\varphi_c \\circ g_c^{-1}; \\sigma).\\end{equation}  The notation $P(\\phi; f)$ makes explicit the dependence of the pressure on the dynamics of $f.$  By definition, $g_c^{-1}$ maps every $(z_i)_0^\\infty$ in $X_c$ to the unique $x\\in J(f_c)$ such that $\\pi f_c^n(x)=z_n,$ for every $n.$  Since $\\varphi_c$ is defined by \\eqref{lkmfg} and $(z_i)_0^{\\infty}$ is the projected orbit of $x,$\n$$(t\\varphi_c\\circ g_c^{-1})\\left((z_i)_{0}^\\infty\\right)= -\\log | p(z_1 -z_0)\/(qz_0)|.$$  This shows that the potential  on the right side of \\eqref{ghewdgd}   does not depend on $g_c$ {or}  $f_c.$ Hence the same is true for the unique zero of the pressure function, as desired.  \\end{remark}\n\\section{Hausdorff dimension}\n\nThe \\emph{$s$-dimensional Hausdorff outer measure} $\\mathcal{H}^s$ of a set $\\Lambda \\subset \\mathbb{C}$ is defined by\n\n$$\\mathcal{H}^s (\\Lambda)= \\lim_{\\delta \\to 0}  \\inf \\sum_{i=1}^{\\infty} |U_i|^s,$$ where $\\inf$ is taken over all countable coverings $\\{U_i\\}_0^{\\infty}$ of $\\Lambda$ with diameter $|U_i|\\leq \\delta.$\nThere exists a unique  nonnegative real number $d$ characterised by the following properties:  $\\mathcal{H}^s(\\Lambda) = 0$ if $s>d$  and $\\mathcal{H}^s(\\Lambda) =\\infty$ if $0\\leq s <d.$ The number $d$ is, by definition, the \\emph{Hausdorff dimension} of $\\Lambda, $ denoted by $\\operatorname{dim_{H}} \\Lambda.$\n\n\n\n\n\\begin{thmm}\\label{kfieg} Suppose $J(\\mathbf{f}_c)$ is a LEO hyperbolic repeller and let $t_c$ be the unique zero of the associated pressure function. {Then  $\\mathcal{H}^{t_c}(J(\\mathbf{f}_c))$ is finite,} and therefore $$\\operatorname{dim_{H}} J(\\mathbf{f}_c) \\leq t_c .$$ \n\n\\end{thmm}\n\n\n\n\\begin{proof} {Fix a parameter $c$ for which {$\\Lambda=J(\\mathbf{f}_c)$} is a hyperbolic repeller. By Lemma \\ref{gjecb}, $\\Lambda$ has an expansive constant $\\epsilon>0.$ }  Cover $J(f_c) \\subset \\mathbb{C}^2$ with finitely many dynamic balls  of fixed time $n$ and radius $\\epsilon:$ \n  $$B(x_i, n, \\epsilon) =\\{ y\\in J(f_c):  d_s(f_c^j(x_i), f_c^{j}(y))< \\epsilon, \\ 0\\leq j \\leq n \\}.$$ \n\n\n\n  \n  The finite covering is possible because $J(f_c)$ is compact. The covering is called \\emph{minimal} if   $x_j \\not \\in B(x_i, n, \\epsilon)$ whenever $i\\neq j.$\n  We may assume  that the covering is minimal by removing $B(x_i, n,  \\epsilon)$ if $x_i$ is contained in some other $B(x_j, n, \\epsilon).$  Since the covering is minimal, the corresponding dynamic balls with radius $\\epsilon\/2$  are pairwise disjoint, for if $B(x_i, n, \\epsilon\/2)$ intersects $B(x_j, n, \\epsilon\/2),$ then by the triangle inequality  $x_j$ must belong to $B(x_i, n, \\epsilon).$ {But   $x_j$ should not belong to $B(x_i, n, \\epsilon) $ when $i\\neq j$ (the covering is minimal). } \n\nIt will be convenient to denote such covering of $J(f_c)$  by $\\mathcal{V}_n$, and its elements by $V_i= B(x_i, n, \\epsilon).$ Now we are going to construct a covering of $J(\\mathbf{f}_c).$ \n\n\\vspace{0.1cm}\n\\noindent {\\bf Claim:} {\\it the family of  all sets $U_i = (g_{z_0\\cdots z_n})^{-1}(B_{\\epsilon}(z_n)), $ where $(z_k)_{0}^{\\infty}$ is the projected orbit  of the centre $x_i$ of $V_i$,  is a covering of $J(\\mathbf{f}_c)$. We shall denote this covering by $\\mathcal{U}_n.$  }\n\\vspace{0.1cm}\n\n{By Lemma \\ref{gjecb}, each set $U_i$ is well defined since the domain of $(g_{z_0\\cdots z_n})^{-1}$ contains $B_{\\epsilon}(z_n).$}    As we shall see, the sets $U_i$ do in fact cover $J(\\mathbf{f}_c)$ because each $U_i$ contains $\\pi V_i$, $J(\\mathbf{f}_c)$ is the projection of $J(f_c)$, and $\\{V_i\\}$ is a covering of $J(f_c).$ In order to check that $\\pi V_i$ is contained in $U_i$ it suffices to show that $\\pi(y)$ is in $U_i,$ whenever $y\\in V_i.$  Since $y$ is in $B(x_i, n, \\epsilon),$ by definition the distance with respect to the metric $d_s$ between $f_c^{j}(x_i)$ and $f_c^{j}(y)$ is strictly less than $\\epsilon$, for $0\\leq j \\leq n.$ This means \n$$\\sum_{k=0}^{\\infty} s^{-k} |\\pi f_c^{k+j}(x_i) - \\pi f_c^{k+j}(y) | < \\epsilon, $$ \nfor $0\\leq j\\leq n.$ In particular, the first term of the above series is less than $\\epsilon$, and   it follows that $|z_j  - w_j| < \\epsilon$ for $0\\leq j \\leq n,$ where $(z_j)_0^{\\infty}$ and $(w_j)_0^{\\infty}$ are the projected orbits  of $x_i$ and $y$, respectively. \n{By Lemma \\ref{gjecb}, $(g_{z_0 \\cdots z_n})^{-1}$ maps $w_n$ to $w_0.$ Hence $w_0 =\\pi(y)$ belongs to $U_i,$ as desired. }\n\n\n\n\nThe diameter of $\\mathcal{U}_n$ is defined as the supremum of all diameters $| U_i |.$ \n\n\\vspace{0.1cm}\n\\noindent {\\bf Claim}:  {\\it the diameter of $\\mathcal{U}_n$ tends to zero as $n\\to \\infty.$} \n\\vspace{0.1cm}\n\n\n\\noindent By  Definition \\ref{gjege}  and Koebe's Lemma \\ref{dgrwbff}, for every element $U_i$ of $\\mathcal{U}_n$ we have \n$$|U_i| = |(g_{z_0\\cdots z_n})^{-1}(B_{\\epsilon}(z_n))| \\leq K |g_{z_0 \\cdots z_n}'(z_0)|^{-1} 2\\epsilon \\leq  (2\\epsilon K C^{-1}) \\lambda^{-n} \\to 0,    $$\nas $n\\to \\infty.$ The claim is proved.\n\n\n\nBy Theorem \\ref{gjegde}, $f_c:J(f_c) \\to J(f_c)$ is LEO and expanding. Therefore, for every H\\\"older continuous potential there corresponds a unique invariant Gibbs measure. Let $\\phi=t_c\\varphi_c.$ Since $P(\\phi)=0$ and $\\epsilon\/2$ is also an expansive constant, the corresponding Gibbs measure $\\mu$ for the system  satisfies:\n\\begin{equation} \\label{jhjewer} {C_{\\epsilon\/2}^{-1}\\leq \\frac{\\mu(B(x,n,\\epsilon\/2))}{\\exp {S_n\\phi(x)}} \\leq C_{\\epsilon\/2},} \\end{equation} for any $x$ in $J(f_c)$ and $n,$ where $C_{\\epsilon\/2}$ is a constant independent of $x$ and $n.$ {By Koebe's Lemma \\ref{dgrwbff}, \\eqref{nbfwrh} and \\eqref{jhjewer}  we have}\n\n\n\\begin{equation}\n\\begin{split}\n |U_i|^{t_c} = |(g_{z_0\\cdots z_n})^{-1}(B_{\\epsilon}(z_n)) |^{t_c} & \\leq (K |g_{z_0 \\cdots z_n}'(z_0)  |^{-1}2\\epsilon)^{t_c} \\\\  & \\leq (2\\epsilon K)^{t_c}  C_{\\epsilon\/2} \\mu(B(x_i, n, \\epsilon\/2)),\n\\end{split}\n\\end{equation}\n\n\n\n\\noindent for every element $U_i$ of $\\mathcal{U}_n.$ Since $\\mu$ is a probability measure and the dynamic balls $B(x_i, n, \\epsilon\/2)$ are pairwise disjoint, we conclude that $ \\sum_{i=0}^{\\infty} |U_i|^{t_c} \\leq (2\\epsilon K)^{t_c}C_{\\epsilon\/2}. $  Since $\\mathcal{U}_n$ is a covering of $J(\\mathbf{f}_c)$ whose diameter  tends to zero as $n\\to \\infty,$ the $t_c$-dimensional Hausdorff measure of $J(\\mathbf{f_c})$ is finite. Hence, $\\operatorname{dim_{H}}J(\\mathbf{f}_c) \\leq t_c.$\n\\end{proof}\n\n\n\\begin{cor} \\label{abc} For any parameter $c$ sufficiently close to a simple centre,\n$$\\operatorname{dim_{H}} J(\\mathbf{f}_c) \\leq t_c. $$\n\n\\end{cor}\n\n\n\\begin{proof}\nBy Theorem 5.4 of \\cite{Rigidity}, the Julia set $J(\\mathbf{f}_c)$ is a LEO hyperbolic repeller, for every $c$ in a certain subset $H_{\\beta}$ of the parameter space. By definition 4.1 of \\cite{Rigidity}, the set $H_{\\beta}$ includes a sufficiently small neighbourhood of every simple centre. \\end{proof}   \n\n\\begin{cor} \\label{nmcs} Suppose $J(\\mathbf{f}_c)$ is a LEO hyperbolic repeller and  $|g'_{z,w}(z) | \\geq  \\kappa >1,$ for every $z,w$ in $J_c$ such that $w$ is an image of $z$ under $\\mathbf{f}_c.$ Then \\begin{equation} \\label{dgew} \\operatorname{dim_{H}} J(\\mathbf{f}_c)  \\leq (\\log p)\/(\\log \\kappa). \\end{equation}\n\n\\end{cor}\n\n\\begin{proof}  Let $\\phi= t_c \\varphi_c.$ Since $P(\\phi)=0$, by the Ruelle-Perron-Frobenius Theorem \\ref{gjdcs}, $ \\mathcal{L}_{\\phi}^{n}(1) \\to h$ uniformly on $J(f_c),$ where $h$ is a continuous function from $J(f_c)$ to $(0,\\infty).$ \nBy \\eqref{nbfwrh}, we have  \\begin{equation} \\label{gherw}\\mathcal{L}_{\\phi}^n(1)(x) = \\sum_{f^n_c(y)=x} e^{S_n\\phi(y)}\\cdot 1 = \\sum_{f_c^n(y)=x} |g_{w_0\\cdots w_n}'(w_0)|^{-t_c} \\leq  p^n \\kappa^{-nt_c},\\end{equation}\nwhere $(w_i)_0^{\\infty}$ is the projected orbit of $y.$ In the above estimate we have used the hypothesis and the fact that  $f_c^n$ is a $p^n$-to-$1$ map. Since $\\mathcal{L}_{\\phi}^n(1)$ converges to a positive function, we have $p\\kappa^{-t_c} \\geq 1,$ otherwise $p^n\\kappa^{-n t_c}$ would converge to zero.\n Solving the inequality $p\\kappa^{-t_c} \\geq 1$ for $t_c$ yields \\eqref{dgew}. \\end{proof}\n \n\n\n\n\\begin{cor}[\\bf Zero area] \\label{hjeeh}If $q^2<p$ and $c$ is sufficiently close to zero, then   \\begin{equation}\\label{qpde} \\operatorname{dim_{H}} J(\\mathbf{f}_c) < 2.\\end{equation} \n\n\n \\end{cor}\n\n\\begin{proof}   The derivative of a branch $g(z)$ of $\\mathbf{f}_c$ is given by $(p\/q)(w-c)\/z$ where $w=g(z).$ {The set function $c\\mapsto J(\\mathbf{f}_c)$  is continuous with respect to the Hausdorff topology for parameters $c$ in an open set $H_{\\beta}$  which contains zero and every simple centre (see Theorem 5.11 of \\cite{Rigidity}).}  Since $J(\\mathbf{f}_0)$ is the unit circle $\\mathbb{S}^1,$ we conclude that $|g_{z,w}'(z)| \\geq \\kappa$ for $z$ and $w$ in $J(\\mathbf{f}_c)$, where  $\\kappa>1$ can be chosen arbitrarily close to $p\/q$ as $c$ tends to zero. It is easy to check that $\\log(p)\/\\log (p\/q) <2$ if, and only if, $q^2<p.$ In particular, $\\log(p)\/\\log(\\kappa) <2$ for $\\kappa$ close to $p\/q.$ From Corollary \\ref{nmcs} we conclude \\eqref{qpde}, for every $c$ sufficiently close to zero. \n\\end{proof}\n\n\\begin{remark} \\normalfont {\\it For arbitrary integers $p>q\\geq 2$, we have $t_0 >1,$ and therefore the estimate of Theorem \\ref{kfieg} is not sharp if $c=0.$ }\n\n{\\it Proof.} Indeed,  in \\eqref{gherw} $|g'_{w_0 \\cdots w_n}(w_0)|$ equals $(p\/q)^n$  when all the elements of the projected orbit $(w_i)_0^{\\infty}$ are contained in  $\\mathbb{S}^1.$  It follows from \\eqref{gherw} that $\\mathcal{L}_{\\phi}^n(1)$ is simply $(p\/q)^{-nt_0}p^n$ when $\\phi$ is the potential $t_0\\varphi_0.$ In particular, $\\mathcal{L}_{\\phi}(1)$ is a constant function $\\omega \\cdot 1$, where $\\omega$ is a real number and $1$ is the function which is constantly one on $J(f_0).$ Since $\\mathcal{L}_{\\phi}^n(1)$ is $\\omega^n\\cdot 1$, which converges to a positive function $h>0$ on $J(f_0)$, it follows that $\\omega=1$ and consequently $h=1.$ Since $\\mathcal{L}_{\\phi}(1)$ is $(p\/q)^{-t_0}p$ and $1$ is a fixed point of $\\mathcal{L}_{\\phi}$, it follows that $(p\/q)^{-t_0}p=1.$ Since $p>q\\geq 2$, we have $t_0 >1$. \n\\end{remark}\n\n\\begin{cor} \\label{abcxz}If every orbit of zero  under $\\mathbf{f}_c$ diverges to infinity, then $$\\dim_{H} J(\\mathbf{f}_c) \\leq t_c.$$\n\\end{cor}\n\n\\begin{proof}  According to \\cite[Theorem 5.4]{Rigidity}, if every orbit of zero under $\\mathbf{f}_c$ diverges to infinity, then $J(\\mathbf{f}_c)$ is a LEO hyperbolic repeller. It follows from Theorem \\ref{kfieg} that $\\dim_{H} J(\\mathbf{f}_c) \\leq t_c.$  \\end{proof}\n\n\n\n\\subsection*{Acknowledgments}\nResearch partially supported by the grants 2016\/16012-6 S\\~ao Paulo Research Foundation and CNPq 232706\/2014-0.\n  The author would like to thank Daniel Smania for many discussions and key insights which led to some preliminary results of this paper, already described in the author's PhD thesis \\cite{Carlos}.\n\n\n\n\n\n\n\n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe rare-earth nitrides (RN, R = rare-earth atom) have gained\nattention recently as simple (NaCl) structures for which the\ninfluence of strong correlations on the electronic band structures\ncan be treated with some confidence \\cite{larson2007esr}. In\nparallel with theoretical advances there has developed an\nexperimental interest in the growth and passivation of thin films\n\\cite{leuenberger2005gtf,granville2006sgs}. This interplay of theory\nand experiment has revealed a number of interesting properties of\nboth fundamental and technological importance. Firstly the\nambient-temperature paramagnetic phase has a narrow indirect gap\nthat varies systematically across the series. Secondly in the\nferromagnetic state (which may be the ground state for them all)\nthey are predicted to have spin-polarised carriers, opening the\npotential to doped spintronic structures. Early data on these\ncompounds were plagued by a lack of stoichiometric reproducibility\nand a rapid degradation of the RN under atmosphere\n\\cite{hullinger1978hot,vogt1993hot}. The magnetic properties in\nparticular are very sensitive to nitrogen vacancies and oxygen\nimpurities, which are difficult to control in these materials. Even\nthe exchange interactions between the rare earth $4f$ spins are not\nwell understood, though a number of theoretical models have been\nproposed \\cite{duan2007}. The exchange is usually described to\noperate within two competitive channels of superexchange via the\nnitrogen atom. The nearest-neighbor (nn) interaction is configured\nat $90^\\circ$ and is believed to be ferromagnetic. It strongly\ndepends on the carrier concentration and becomes dominant when RKKY\nindirect interactions take place via the polarisation of conduction\nelectrons. The next-nearest-neighbor (nnn) interaction is configured\nat $180^\\circ$, is antiferromagnetic and in principle dominant for\nnon-metals. Note that the existence of a ferromagnetic order in\nsemiconducting rare-earth nitrides implies that a ferromagnetic\ninteraction dominates even in the absence of free carriers\n\\cite{duan2007}.\n\nThe most thoroughly studied of these compounds is GdN, which has a\nhalf-filled $4f$ shell with the maximum $\\frac{7}{2}$ net spin and\nzero net orbital angular momentum. It is ferromagnetic below $70$~K,\nwith a saturation moment of $7 \\mu_B$ and is an indirect-gap\nsemiconductor with an optical gap of $1.3$~eV in the paramagnetic\nphase, reduced to $0.9$~eV below the Curie temperature\n\\cite{trodahl2007fro}. For the lighter rare-earths, Hund's rules\nspecify that $L$ is anti-parallel to $S$ in the ground state. Within\nthat scenario SmN is of special interest; Sm$^{+3}$, with two\nelectrons below half-filling, has $S = \\frac{5}{2}$ and $L = 5$ and\na net magnetic moment given by $M \\approx (L_z+2S_z)\\mu_B \\ll\n\\mu_B$. There is thus the potential for Sm compounds to condense\ninto a ferromagnetic phase in which the spins are ferromagnetically\nordered, but with their spin moment nearly cancelled by an opposing\norbital moment. Moments substantially smaller than the free-ion\nmoment of $0.71 \\mu_B$ are not unknown in ferromagnetic Sm compounds\n\\cite{givord1979uff,stewart1974fos,wijn1973eoc,adachi1999sot,adachi2001zmf,ahn2007pra},\nbut its occurrence in a semiconductor has to our knowledge not been\nreported previously. Such a material offers special advantages for\nspintronics: (i) it can inject spin-polarised electrons into a\nconventional semiconductor without the deleterious effects of a\nfringe magnetic field \\cite{nie2008phm}, and (ii) in principle it\ncan form field-free, fully spin-polarised electronically active\nstructures.\n\nEarly magnetic measurements suggested that SmN was antiferromagnetic\nbelow $20$~K \\cite{hullinger1978hot,vogt1993hot}, but this was not\nconfirmed by neutron diffraction \\cite{moon1979mpo}, suggesting that\nit might indeed be ferromagnetic but with near-cancellation between\nthe spin and orbital moments. More recently we reported clear\nferromagnetism in GdN \\cite{granville2006sgs, trodahl2007fro} and\nDyN \\cite{preston2007bso}, but somewhat weaker ferromagnetic\nevidence in SmN \\cite{preston2007bso}. The resistivities of the\nfilms show them all to be semiconductors, with the expected anomaly\nat $T_C$ signalling a narrowed gap in the ferromagnetic state. In\nthe present work we report magnetic experiments performed on thicker\nSmN films, seeking to resolve the uncertainties concerning the\nmagnetic state of this unusual material, and clear picture of\nnear-zero moment ferromagnetism emerges.\n\n\n\\section{Experimental details}\n\n\nSmN films were grown in a vacuum chamber pumped to a base pressure\nin the $10^{-9}$~mbar range. Sm metal was evaporated in the presence\nof an atmosphere of pure N$_2$ gas at a pressure of $10^{-4}$~mbar;\nthe growth conditions, the structure and stoichiometry have been\nreported previously \\cite{granville2006sgs, trodahl2007fro}. For the\npresent measurements the films were deposited on Si substrates\ncovered by their natural oxide. X-ray diffraction exhibits the Bragg\npeaks of only the rock salt cubic structure and establishes the\nfilms as untextured polycrystalline with an average crystal grain\nsize of $10$~nm. The lattice parameter ($5.07$~{\\AA}) is consistent\nwith the previous data for the rare earth nitride series, confirming\nthat samarium is trivalent. All ex-situ measurements are performed\non films protected by a cap layer of nanocrystalline GaN.\nConductivity and x-ray spectroscopies on the films have established\nthem to be semiconductors in both the ambient-temperature state and\nto $4$~K in the magnetically ordered low temperature\nstate~\\cite{preston2007bso}.\n\nThree films of differing thickness ($300-400$~nm) were used for\nmagnetic measurements reported here. These magnetic properties were\ninvestigated with a SQUID magnetometer (Quantum Design MPMS) working\nup to a maximum applied field of $6$~Tesla. All experiments were\nperformed with the magnetic field applied parallel to the film\nplane. The films were prepared in parallel on both thick\n($400~\\mu$m) and thin ($100~\\mu$m) Si substrates in order to apply\nmore reliable corrections for substrate signals, which are of the\nsame order of magnitude as the SmN signal. Si is diamagnetic and the\nsusceptibility is supposed to be temperature independent. The\nsusceptibility measured on the uncovered Si substrates is in\nagreement with the theoretical susceptibility $\\chi(Si)=-3.4\\times\n10^{-6}$ at room temperature within a $5\\%$ error. This Si signal is\ncharacterized well enough to permit a trouble-free correction in the\ndata shown below.\n\n\nThe magnetic signal from the capping layer is somewhat more\nproblematic. GaN is weakly paramagnetic mainly because self-doping\nis provided by N vacancies, for which we collected reference data\nfrom a measurement on a Si substrate covered with a GaN layer of the\nsame thickness. In a phenomenological approach, we have fit the\ntemperature variation of the magnetisation using a theoretical\ncalculation by Sonder and Schweinler \\cite{sonder1960} predicting a\nmodified Curie law for the susceptibility of interacting donor\ncentres in doped semi-conductors: $\\chi=\\frac{C}{T^{(1-\\alpha)}}$.\nIn this model the parameter $\\alpha$ is proportional to the donor\nconcentration $n_d$. We find $C=4.05\\times 10^{-3}$ and\n$\\alpha=0.86$, but uncertainty about the donor concentration\nprevents further analysis. We note that this approach has been used\nto explain the susceptibility of a zinc-blende GaN thin film\n\\cite{fanciulli95}. The authors derive $\\alpha\\simeq0.8$ for which\nthey estimate $n_d=2\\times 10^{17}$ cm$^{-3}$, in agreement with the\nexperimental value attributed to nitrogen vacancies.\n\nHowever, the susceptibility of GaN is very sensitive to N vacancy\nconcentration \\cite{fanciulli95}. There is some variation in the\nsusceptibility of the GaN capping layers associated with minor\ndifferences in stoichiometry resulting from the ion-assisted\ndeposition process. Nevertheless we are able to subtract the capping\nlayer signal to reasonable accuracy, so the evidence of the\nremarkable magnetic behaviour of SmN is not strongly affected by\nthis uncertainty. Below we present the data both uncorrected and\nafter correction for the substrate\/capping layer signals.\n\n\n\n\\section{Results}\n\n\nFigure \\ref{fig1} shows the temperature variation of the\nmagnetisation of sample I in an applied field of $0.5$~T, after\ncooling in zero field (ZFC) or in $0.5$~T (FC). The curves are\nsuperimposed down to $30$~K, below which a sharp increase of the FC\ncurve denotes ferromagnetic order with a spontaneous magnetic\nmoment. The same behaviour has been confirmed in an applied field as\nlow as $5$~mT. In the ZFC curve the increase of the magnetisation\nbelow $20$~K is assigned to the GaN cap layer. The transition\ntemperature estimated from the maximum of the ZFC curve is found at\n$T_C = 30~\\pm~2$~K. The other two films gave Curie temperatures of\n$24~\\pm~2$~K and $26~\\pm~2$~K. We have not been able to relate these\nsmall differences to the films' compositions or structures, so we\nquote $T_C = 27 \\pm 3$~K. Note in this regard that it is known that\nN vacancies lower the Curie temperature in GdN \\cite{cutler1975sam}.\nOur films are close to stoichiometric, but absolute measurements of\nthe composition have an accuracy of about $5\\%$. It is notable that\nearlier heat capacity \\cite{stutius1969tsh} and magnetisation\n\\cite{busch1965ioc} measurements suggested Curie temperatures in the\n$15-20$~K range; they were likely performed on N deficient samples.\n\n\n\\begin{figure}\n\\centering{\n  \\includegraphics[width=8cm]{Figure1.eps}\n} \\caption{ Temperature dependence of the magnetisation of SmN after\ncooling without (ZFC) and with an applied field (FC) of $0.5$~T\n(sample \\textit{I}). The inset shows the result of the fit of the FC\nsusceptibility ($\\chi = M\/H$) in a Van~Vleck approach.} \\label{fig1}\n\\end{figure}\n\n\n\n\\subsection{Ferromagnetic state}\n\nAbove $T_C$ the magnetisation $M$ is linear in field $B$ with a\nparamagnetic susceptibility that includes both Curie-Weiss and\ntemperature-independent Van~Vleck contributions as will be discussed\nin the following section. At $T_C$ $M(B)$ becomes nonlinear, and an\nhysteresis loop develops. The evolution of the loop as the\ntemperature is lowered is shown in Figure \\ref{fig2}. Here the left\nside [Fig.~\\ref{fig2}(a, b, c)] represents the loops uncorrected for\nGaN while in the right side [Fig.~\\ref{fig2}(d, e, f)] the\ncorrection is included. We first focus on the loops measured after\ncooling the films in zero field (red full dots). At $15$~K\n[Fig.~\\ref{fig2}(a)] the saturation is achieved at $3$ T. Beyond the\nirreversibility point, the magnetisation is linear in field, due to\nthe paramagnetic contribution of GaN. After correction for GaN the\nloop exhibits a coercive field of $0.9$~T [Fig.~\\ref{fig2}(d)]. At\n$10$~K [Fig.~\\ref{fig2}(b)] the reversible part at high field is\nmissing; the closing field lies above the $6$~T maximum available.\nThis behaviour is exactly as expected when the maximum applied field\nis insufficient to achieve the reversal of the moments, so that only\nminor loops are measured, signaling a magnetocrystalline anisotropy\nthat grows at lower temperature \\cite{callen1960am}. At $5$~K the\nloop shows only a very small opening and at $2$~K\n[Fig.~\\ref{fig2}(c)] the hysteresis has completely disappeared, so\nthat only the GaN paramagnetic contribution is seen in the ZFC data.\nClearly the magnetic field necessary to even initiate the reversal\nof ferromagnetic domains is higher than $6$~T at these temperatures,\nand the magnetisation process is dominated by the reversible\nparamagnetic contribution. Hysteretic behaviour can nonetheless be\nconfirmed at these temperatures by cooling in the presence of the\nmaximum field of $6$~T to prepare the film in a magnetised state.\nThus Figure \\ref{fig2} compares the hysteresis patterns obtained at\n$15$~K, $10$~K and $2$~K after zero-field cooling (red full dots)\nand after cooling in $6$~T (black open squares). The loops are\nsuperimposed at $15$~K [Fig.~\\ref{fig2}(a,d)], but at the lower\ntemperatures the patterns are shifted from one another\n[Fig.~\\ref{fig2}(b,e) and Fig.~\\ref{fig2}(c,f)]. Exactly the same\nbehaviour is observed when cooling the system under $-6$~T, though\nwith the shifts found in the opposite sense. The results confirm\nthat the coercive and saturation fields are larger than $6$~T at\nthese temperatures.\n\n\\begin{figure}\n\\centering{\n  \\includegraphics[width=10cm]{Figure2.eps}\n} \\caption{ (color online) Magnetisation loops of SmN\\textit{II}\nafter cooling the film in zero-field (red full circles) and after\ncooling under $6$~T (open black squares). Left: correction for the\nSi substrate only at (a)~$15$~K, (b)~$10$~K and (c)~$2$~K. Right:\nadditional correction for the paramagnetic signal of the GaN cap\nlayer according to the curve shown on Figure \\ref{fig3} at\n(d)~$15$~K, (e)~$10$~K and (f)~$2$~K .} \\label{fig2}\n\\end{figure}\n\n\nIn addition a very small spontaneous moment is observed on the\nsaturated loops; the temperature dependence of the magnetic moment\ndown to $0$~K is interesting to evaluate. At $15$~K, $M_S = 0.012\n\\mu_B\/$Sm, compared to $0.008 \\mu_B\/$Sm at remanence when the field\nis reduced to zero. The difference is due in part to single-domain\ncrystallites relaxing to an easy-axis magnetisation in the remanent\nstate, leaving the moments distributed in a cone about the field\ndirection. The easy axis in SmN is as yet unknown, but if it lies\nalong one of the high-symmetry directions $\\langle100\\rangle$,\n$\\langle110\\rangle$ or $\\langle111\\rangle$, the field-parallel\ncomponent of the moment is reduced by about $15\\%$, explaining about\none half of the measured reduction from the saturation to the\nremanence. We conclude that the remnant moment provides a reasonable\nlower limit for the single-domain spontaneous magnetisation. To\nrefine the estimate we have performed experiments in which the\nmaterial is prepared in the magnetised state by cooling in $6$~T,\nfollowed by measurements of the magnetisation in a small field of\n$20$~mT. The resulting remnant magnetisation drops to zero at the\nCurie temperature from a zero-temperature magnetisation of $0.030\n\\pm 0.006 \\mu_B\/$Sm averaged over the three films. Assuming that\nthis value represents somewhat less than $85\\%$ of the spontaneous\nmoment we quote that moment as $0.035 \\pm 0.010 \\mu_B\/$Sm at the\nlowest temperature. Such a small moment explains the null result in\nthe early neutron search for ferromagnetic order \\cite{moon1979mpo},\nand is in agreement with the near cancellation of spin and orbital\nmoments suggested by Larson~\\textit{et~al.}~\\cite{larson2007esr}.\nThe saturation field is large, rising above our $6$~T maximum\navailable field below $15$~K. It is important in this regard to note\nthat the ferromagnetic state, with its very small moment, couples\nrelatively weakly with the magnetic field.\n\n\n\\subsection{Paramagnetic state}\n\nRemarkably, the small moment found in the ferromagnetic phase is not\ncarried across the transition, rather the paramagnetic behaviour of\nSmN can be fully understood within the established description of Sm\nin the crystalline environment. We start by recalling that the\nground state configuration of the Sm$^{3+}$ free ion is\n$^{6}H_{\\frac{5}{2}}$ with $L=5$, $S=\\frac{5}{2}$, and\n$J=\\frac{5}{2}$. The Land\\'e factor is $g_J = \\frac{2}{7}$ and the\nmagnetic moment is $\\mu = 0.71 \\mu_B$. The first excited multiplet\n$J=\\frac{7}{2}$ is not thermally populated, but a paramagnetic\nmoment will partly arise from an admixture of the multiplets induced\nby the applied magnetic field. As usually observed in trivalent Sm\ncompounds, the reciprocal susceptibility is therefore not linear in\ntemperature, preventing a Curie-Weiss-like analysis.\n\nWith the inclusion of the $J = \\frac{7}{2}$ admixture in the\n$\\frac{5}{2}$ ground state, the paramagnetic susceptibility of\nSm$^{3+}$ compounds is reasonably well described by the Van~Vleck\napproach, with \\cite{stewart1972pso}\n\n\\begin{equation}\n    \\chi = \\chi_0 + \\frac{C}{T-\\Theta_p}.\n  \\label{eq:1}\n\\end{equation}\n\nThe second term is the conventional Curie-Weiss susceptibility\ninvolving the effective magnetic moment $\\mu_{\\mathrm{eff}}$ for\n$J=\\frac{5}{2}$, with $C = \\mu_0 N \\mu_{\\mathrm{eff}}^2 \/ 3 k_B$,\nwhere $N$ is the Sm ion density. The Van~Vleck term $\\chi_0$ depends\non the energy difference $\\Delta E \\simeq 1500$~K between the two\nlowest $J (\\frac{5}{2}, \\frac{7}{2})$ multiplets \\cite{wijn1973eoc,\nstewart1972pso}:\n\n\\begin{equation}\n    \\chi_0 = \\frac{\\mu_0 N {\\mu_B}^2}{k_B} \\cdot \\frac{20}{7 \\Delta E}.\n    \\label{eq:2}\n\\end{equation}\n\nWe have used Equation~(\\ref{eq:1}) to fit the measured paramagnetic\nsusceptibility curves above $T_C$ as shown in the inset of\nFig.~\\ref{eq:1} for the magnetisation curve obtained under $0.5$~T.\nIt can be seen that the free-ion Van~Vleck term, with no adjustable\nparameters, provides an excellent fit to the temperature-independent\ntail at high temperatures. The diverging contribution below $100$~K\nyields a Curie temperature $\\theta_P = 28 \\pm 1$~K, in excellent\nagreement with the value of $30 \\pm 2$~K where this film showed a\ncusp in the ZFC magnetisation curve. Similar agreement has been\nfound also with the other two films, as can be seen on Figure\n\\ref{fig3} for sample \\textit{II} . A sense of the relative strength\nof the contributions to the susceptibility can be obtained by\ncomparing the Van Vleck term $\\chi_{VV}\\sim\\frac{60}{7 \\Delta E}=\n5.7\\times10^{-3}$ obtained from Equation~\\ref{eq:2} to the\nCurie-Weiss term $\\chi_{CW}\\sim\n\\mu_B^{-2}\\mu_{eff}^{2}\/(T-\\Theta_p)$. For example with\n$\\mu_{eff}=0.845\\mu_B$, at 300 K $\\chi_{CW}= 2.6 \\times10^{-3}$ and\nat 100 K $\\chi_{CW}= 9.9 \\times10^{-3}$.\n\n\n\\begin{figure}\n\\centering{\n  \\includegraphics[width=8cm]{Figure3.eps}\n} \\caption{ (color online) Susceptibility ($\\chi = M\/H$) versus\ntemperature (sample\\textit{II}), before (full black circles) and\nafter (open red circles)the correction for the GaN cap layer. The\nGaN signal used for the correction is shown in small blue dots. The\nfull black line is the Van Vleck fit of the susceptibility to the\nfully corrected data, setting the multiplet separation $\\Delta\nE=1500$~K.} \\label{fig3}\n\\end{figure}\n\n\n\nThe effective magnetic moment derived from the Curie constant $C$\nfor the three films is $\\mu_{\\mathrm{eff}} = 0.45 \\pm 0.1 \\mu_B$ per\nSm ion, somewhat smaller than the free ion effective paramagnetic\nmoment of $g_J [J(J+1)]^{\\frac{1}{2}} \\mu_B = 0.845 \\mu_B$, though\nstill larger than the value of $0.035\\mu_B$ in the ferromagnetic\nphase. The paramagnetic moment is understood by noting that the\n$J=\\frac{5}{2}$ level is decomposed by the cubic octahedral crystal\nfield into a doublet $\\Gamma_7$ and a quartet $\\Gamma_8$. Specific\nheat data \\cite{stutius1969tsh} quoted in \\cite{moon1979mpo} report\nthat the $\\Gamma_7$ sublevel is the ground state with a separation\nof $225$~K to the $\\Gamma_8$ sublevel for the bonding configuration\nof SmN. We can therefore assume that at $30$~K only the ground\ndoublet is significantly populated and calculate an approximate\nvalue of the low temperature susceptibility. The $\\Gamma_7$ Kramers\ndoublet \\cite{lea1962tro} can be described with the equivalent wave\nfunctions $\\vert\\pm\\frac{1}{2}\\rangle$ for a fictitious spin\n$S'=\\frac{1}{2}$, for which we have calculated a Land\\'e factor,\n$g'$, of $\\frac{10}{21}$. We obtain $\\mu_{\\mathrm{eff}}(\\Gamma_7) =\n[g'^2 \\mu_B^2 S'(S'+1)]^{\\frac{1}{2}} = 0.41 \\mu_B$, in good\nagreement with the experimental effective moment. It is important to\nnote that these results establish quite clearly that the entire Sm\npopulation in the films participates in the paramagnetic Curie-Weiss\nsignal diverging at $T_C$, emphasising that the full population also\nparticipates in the ferromagnetic order.\n\n\n\n\n\\section{Conclusion}\n\n\nThe present work gives strong evidence for a ferromagnetic state in\nSmN. The magnetic moment in the ferromagnetic phase is an order of\nmagnitude smaller than in the paramagnetic state, confirming a\nnearly-zero-moment ferromagnet below $27$~K. The magnetic behaviour\nin the paramagnetic phase is in quantitative agreement with the\nexpected moments of Sm, showing the effects of both the excited\nspin-orbit    state and the constraints imposed by the crystal\nfield. The reduced ferromagnetic moment is established quite clearly\nby the experiments, but remains a theoretical challenge. The\nnear-zero moment ferromagnetic state in a semiconductor has clear\npotential for various fundamental studies and devices involving\ncontrol of spin- and charge-degrees of freedom without the\nperturbing effects of a fringe magnetic field. For device\napplications it would clearly be interesting to investigate the\nprospect of raising the Curie temperature of this material, either\nby alloying or strain. However, there is also considerable\npossibility that specific high technology spintronics devices that\nrun at $77$~K (and even He temperature) will be used in the future.\n\n\n\n\\begin{acknowledgments}\nThe MacDiarmid Institute is supported by the New Zealand Centre of\nResearch Excellence Fund and the research reported here by a grant\nfrom the New Zealand New Economy Research Fund. C.M. is grateful to\nthe staff of the School of Chemical and Physical Sciences for their\nhospitality, and thanks the financial support of the MacDiarmid\nInstitute and of the Royal Society of New Zealand.\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn the last few years, ITU\/MPEG video coding standards---HEVC\n\\citep{Sullivan:2012:OHE:2709080.2709221} and VVC \\citep{VVC_Ref}---have been\nchallenged by learning-based codecs. The learned image coding framework\nintroduced by \\citet{DBLP:conf\/iclr\/BalleLS17,DBLP:conf\/iclr\/BalleMSHJ18} eases\nthe design process and improves the performance by jointly optimizing all steps\n(encoder, decoder, entropy coding) given a rate-distortion objective. The best\nlearned coding system \\citep{cheng2020learned} exhibits performance on par with\nthe image coding configuration of VVC. In video coding, temporal redundancies\nare removed through motion compensation. Motion information between\nframes are transmitted and used to interpolate reference frames to obtain a\ntemporal prediction. Then, only the residue (prediction error) is sent, reducing\nthe rate. Frames coded using references are called \\textit{inter} frames, while others\nare called \\textit{intra} frames.\n\nAlthough most learning-based video coding systems follow the framework of\nBall\\'{e} et al., the end-to-end character of the training is often overlooked.\nThe coders introduced by \\citet{DBLP:conf\/cvpr\/LuO0ZCG19} or\n\\citep{DBLP:journals\/corr\/abs-1912-06348} rely on a dedicated pre-training to\nachieve efficient motion compensation. Dedicated training requires proxy\nmetrics not necessary in line with the real rate-distortion objective, leading\nto suboptimal systems. Due to the presence of both intra and inter frames,\nlearned video coding methods transmit two kinds of signal: image-domain signal\nfor intra frames and residual-domain for inter frames. Therefore, most works\n\\citep{Agustsson_2020_CVPR} adopt a \\textit{two-coder} approach, with separate\ncoders for intra and inter frames, resulting in heavier and less factorizable\nsystems.\n\n\nThis paper addresses these shortcomings by introducing a novel framework for\nend-to-end learned video coding, based on a single coder for both intra and\ninter frames. Pursuing the work of \\citet{LaduneMMSP20}, the coding scheme is\ndecomposed into two sub-networks: MOFNet and CodecNet. MOFNet conveys motion\ninformation and a coding mode, which arbitrates between transmission with\nCodecNet or copy of the temporal prediction. MOFNet and CodecNet use conditional\ncoding to leverage information from the previously coded frames while being\nresilient to their absence. This allows to process intra and inter frames with\nthe same coder. The system is trained as a whole with no pre-training or\ndedicated loss term for any of the components. It is shown that the system is\nflexible enough to be competitive with HEVC under three coding configurations.\n\n\\vspace{-0.03cm}\n\n\\section{Proposed system}\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{.\/arxiv_figure\/RA_4-crop.pdf}\n    \\caption{Random Access configuration, GOP size is set to 4 to have concise diagrams.}\n    \\label{fig:gopstruct}\n\\end{figure}\n\n\nLet $\\left\\{\\mathbf{x}_i, i \\in \\mathbb{N}\\right\\}$ be a video sequence, each\nframe $\\mathbf{x}_i$ being a vector of $C$ color channels\\footnote{Videos are in\nYUV 420. For convenience, a bilinear upsampling is used to obtain YUV 444 data.}\nof height $H$ and width $W$. Video codecs usually process Groups Of Pictures\n(GOP) of size $N$, with a regular frame organization. Inside a GOP, all frames\nare inter-coded and rely on already sent frames called references: B-frames use\ntwo references while P-frames use a single one. The first frame of the GOP\nrelies either on a preceding GOP or on an intra-frame (I-frame) denoted as\n$\\mathbf{x}_0$. This work primarily targets the \\textit{Random Access}\nconfiguration (Fig. \\ref{fig:gopstruct}), because it features I, P and B-frames.\nHere, we consider the rate-distortion trade-off, weighted by $\\lambda$, of a\n\\textit{single} GOP plus an initial I-frame $\\mathbf{x}_0$:\n\\begin{equation}\n    \\mathcal{L}_\\lambda = \\sum_{t=0}^{N} \\mathrm{D}(\\hat{\\mathbf{x}}_t, \\mathbf{x}_t) +\n    \\lambda \\mathrm{R}(\\hat{\\mathbf{x}}_t), \\text{ with }\\mathrm{D} \\text{ the MSE and } \\mathrm{R} \\text{ the rate.}\n    \\label{eq:loss}\n\\end{equation}\n\n\n\\subsection{B-frame Coding}\n\nThe proposed architecture processes the entire GOP (I, P and B-frames) using a\nunique neural-based coder. B-frames coding is detailed here. Thanks to\nconditional coding, I and P-frames are processed by simply bypassing some steps\nof the B-frame coding process as explained in Section\n\\ref{subsec:conditionalcoding}.\n\nLet $\\mathbf{x}_t$ be the current B-frame and $(\\hat{\\mathbf{x}}_{p},\\hat{\\mathbf{x}}_{f})$ two\nreference frames. Figure \\ref{fig:overalldiagram} depicts the coding process of\n$\\mathbf{x}_t$. First, $(\\mathbf{x}_t,\\hat{\\mathbf{x}}_{p},\\hat{\\mathbf{x}}_{f})$ are fed to MOFNet\nwhich computes and conveys---at a rate  $R_m$---two optical flows $(\\mathbf{v}_{p},\n\\mathbf{v}_{f})$, a pixel-wise prediction weighting $\\boldsymbol{\\beta}$ and a pixel-wise\ncoding mode selection $\\boldsymbol{\\alpha}$. The optical flow $\\mathbf{v}_{p}$ (respectively\n$\\mathbf{v}_{f}$) represents a 2D pixel-wise motion from $\\mathbf{x}_t$ to\n$\\hat{\\mathbf{x}}_{p}$ (resp. $\\hat{\\mathbf{x}}_{f}$). It is used to interpolate the reference through a\nbilinear warping $w$. The pixel-wise weighting $\\boldsymbol{\\beta}$ is applied to obtain the\nbi-directional weighted prediction $\\tilde{\\mathbf{x}}_t$:\n\\begin{equation}\n    \\tilde{\\mathbf{x}}_t = \\boldsymbol{\\beta} \\odot w(\\hat{\\mathbf{x}}_{p}; \\mathbf{v}_{p}) + (1 - \\boldsymbol{\\beta}) \\odot w(\\hat{\\mathbf{x}}_{f}; \\mathbf{v}_{f}),\n    \\left\\{ \\begin{array}{l}\n        \\odot \\text{ is a pixel-wise multiplication,} \\\\\n        \\mathbf{v}_{p} \\text{ and } \\mathbf{v}_{f} \\in \\mathbb{R}^{2 \\times H \\times W},\\ \\boldsymbol{\\beta}  \\in \\left[0, 1\\right]^{H \\times W}\n    \\end{array}\n    \\right.\n    \\label{eq:pred}\n\\end{equation}\nThe coding mode selection $\\boldsymbol{\\alpha} \\in \\left[0, 1\\right]^{H \\times W}$ arbitrates\nbetween transmission of $\\mathbf{x}_t$ using CodecNet versus \\textit{Skip mode},\na direct copy of $\\tilde{\\mathbf{x}}_t$. CodecNet sends areas of $\\mathbf{x}_t$ selected by\n$\\boldsymbol{\\alpha}$, using information from $\\tilde{\\mathbf{x}}_t$ to reduce its rate $R_c$. The total\nrate required for $\\mathbf{x}_t$ is $R = R_m + R_c$ and the decoded frame\n$\\hat{\\mathbf{x}}_t$ is the sum of both contributions: $\\hat{\\mathbf{x}}_t = \\underbrace{(1 -\n\\boldsymbol{\\alpha}) \\odot \\tilde{\\mathbf{x}}_t}_{\\text{Skip}} + \\underbrace{c(\\boldsymbol{\\alpha} \\odot \\mathbf{x}_t,\n\\boldsymbol{\\alpha} \\odot \\tilde{\\mathbf{x}}_t)}_{\\text{CodecNet}}$.\n\n\n\n\\subsection{Conditional Coding}\n\\label{subsec:conditionalcoding}\n\nConditional coding \\citep{LaduneMMSP20} allows to exploit decoder-side information\nmore efficiently than residual coding. Its architecture is similar to an\nauto-encoder \\citep{DBLP:conf\/iclr\/BalleMSHJ18}, with one additional\n\\textit{shortcut} transform (Fig. \\ref{fig:overalldiagram}). It\ncan be understood through the description of its 3 transforms.\\\\\n\\textbf{Shortcut transform} $g^\\prime_a$ (\\textit{Decoder})---Its role is to extract information\nfrom the reference frames available at the decoder (\\textit{i.e.} at no rate). The\ninformation is computed as latents $\\mathbf{y}^\\prime$.\\\\\n\\textbf{Analysis transform} $g_a$ (\\textit{Encoder})---It estimates and conveys\nthe information not available at the decoder \\textit{i.e.} the unpredictable\npart. The information is computed as latents $\\hat{\\mathbf{y}}$.\\\\\n\\textbf{Synthesis transform} $g_s$ (\\textit{Decoder})---Latents from the analysis and shortcut\ntransforms are concatenated and synthesized to obtain the desired output.\n\nUnlike residual coding, conditional coding leverages decoder-side information in\nthe latent domain. As noted by \\citet{DBLP:conf\/iccv\/DjelouahCSS19}, this makes\nthe system more resilient to the absence of information at the decoder\n(\\textit{i.e.} for I-frames). Thus, MOFNet and CodecNet implement\nconditional coding to be able to process I, P and B-frames as well as lowering\ntheir rate. I and P-frames are compressed using the B-frames coding scheme, with\nthe same parameters, and ignore the unavailable elements.\\\\\n\\textbf{I-frame}---Motion compensation is not available. As such,\nMOFNet is ignored, $\\boldsymbol{\\alpha}$ is set to 1 and CodecNet conveys the whole frame, with its shortcut latents\n$\\mathbf{y}^\\prime_c$ set to $0$.\\\\\n\\textbf{P-frame}---Bi-directional motion compensation is not available. $\\boldsymbol{\\beta}$\nis set to 1 to only rely on the prediction from $\\hat{\\mathbf{x}}_{p}$. MOFNet shortcut\nlatents $\\mathbf{y}^\\prime_m$ are set to $0$.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\linewidth]{.\/arxiv_figure\/GOP_Skip_Code_v3-crop.pdf}\n    \\caption{Diagram of the system. A detailed version can be found in appendix\n    \\ref{app:detailarchitecture}. Arithmetic coding uses hyperpriors\n    \\citep{DBLP:conf\/iclr\/BalleMSHJ18} omitted for clarity. Attention modules\n    are implemented as proposed by \\citet{cheng2020learned} and $f = 128$. There\n    are 20 millions learnable parameters $\\left\\{\\boldsymbol{\\phi},\\boldsymbol{\\theta}\\right\\}$.}\n    \\label{fig:overalldiagram}\n\\end{figure}\n\n\\section{Training}\n\nThe training aims at learning to code I, P and B-frames. As such, it considers\nthe smallest coding configuration featuring all 3 types of frame: a GOP of size\n2 plus the preceding I-frame. Each training iteration consists in the coding of\nthe 3 frames, followed by a single back-propagation to minimize the\nrate-distortion cost of \\eqref{eq:loss}. Unlike previous works, the entire\nlearning process is achieved through this rate-distortion loss. No element of\nthe system requires a pre-training or a dedicated loss term. Moreover, coding the\nentire GOP in the forward pass enables the system to model the dependencies\nbetween coded frames, leading to better coding performance.\n\nThe training set is made of 400~000 videos crops of size $256 \\times 256$, with\nvarious resolutions (from 540p to 4K) and framerates (from 24 to 120 fps). The\noriginal videos are from several datasets: KonViD-1k \\citep{hosu2017konstanz},\nCLIC20 P-frame and Youtube-NT \\citep{yang2020hierarchical}. The batch size is 4\nand the learning rate is set to $10^{-4}$ and decreased to $10^{-5}$ during the\nlast epochs. Rate-distortion curves are obtained by training systems for\ndifferent $\\lambda$.\n\n\n\\section{Visual Illustrations}\n\\label{sec:visualization}\n\nThis section shows the different quantities at stakes when coding a B-frame\n$\\mathbf{x}_t$ (Fig. \\ref{subfig:code}). First, MOFNet outputs two optical\nflows $(\\mathbf{v}_{p},\\mathbf{v}_{f})$ (Fig. \\ref{subfig:flow}), the prediction\nweighting $\\boldsymbol{\\beta}$ (Fig. \\ref{subfig:beta}) and the coding mode selection\n$\\boldsymbol{\\alpha}$. The temporal prediction is then computed following \\eqref{eq:pred}.\nMost of the time, $\\boldsymbol{\\beta} \\simeq 0.5$, mitigating the noise from both bilinear\nwarpings. When the background is disoccluded by a moving object (\\textit{e.g.}\nthe woman), $\\boldsymbol{\\beta}$ equals $0$ on one side of the object and $1$ on the other\nside. This allows to retrieve the background from where it is available. The\ncompetition between Skip mode and CodecNet is weighted by $\\boldsymbol{\\alpha}$. Here, most\nof $\\hat{\\mathbf{x}}_t$ comes from the Skip mode\\footnote{Video frames are in\nYUV format. Thus zeroed areas appear green.} (Fig. \\ref{subfig:skipmode}).\nHowever, the less predictable parts, \\textit{e.g.} the woman, are sent by\nCodecNet.\n\nTo illustrate the conditional coding, $\\mathbf{v}_{f}$ is computed by the MOFNet\nsynthesis transform using only the shortcut latents $\\mathbf{y}^\\prime_m$ (Fig.\n\\ref{subfig:flowshortcut}), the transmitted ones $\\hat{\\mathbf{y}}_m$ (Fig.\n\\ref{subfig:flowsent}) or both (Fig. \\ref{subfig:flow}). The shortcut transform\ncaptures the nature of the motion in $\\mathbf{y}^\\prime_m$, which allows to\nsynthesize most of $\\mathbf{v}_{f}$ without any transmission involved. In contrast,\n$\\hat{\\mathbf{y}}_m$ consists in a refinement of the flow magnitude. The\nrate of $\\hat{\\mathbf{y}}_m$ is reduced by using a low spatial resolution,\nunlike $\\mathbf{y}^\\prime_m$ which keeps all the spatial accuracy.\n\n\\newcommand{\\imagepath\/}{.\/arxiv_figure\/}\n\\begin{figure}[htb]\n    \\centering\n\\begin{subfigure}[t]{0.24\\textwidth}\n  \\includegraphics[width=\\linewidth]{\\imagepath\/\/frame_1.png}\n  \\caption{Frame to code $\\mathbf{x}_t$}\n  \\label{subfig:code}\n\\end{subfigure}\\hfil\n\\begin{subfigure}[t]{0.265\\textwidth}\n    \\includegraphics[width=\\linewidth]{\\imagepath\/\/ModeNet_beta.png}\n    \\caption{Prediction weighting $\\boldsymbol{\\beta}$}\n    \\label{subfig:beta}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[t]{0.24\\textwidth}\n    \\includegraphics[width=\\linewidth]{\\imagepath\/\/png_copy_part.png}\n    \\caption{Skip mode $(1-\\boldsymbol{\\alpha}) \\odot \\tilde{\\mathbf{x}}_t$}\n    \\label{subfig:skipmode}\n    \\end{subfigure}\n\n   \n    \\vspace{-0.1cm}\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\includegraphics[width=\\linewidth]{\\imagepath\/\/v_next_all_optical_flow.png}\n        \\caption{Optical flow $\\mathbf{v}_{f}$}\n       \n        \\label{subfig:flow}\n    \\end{subfigure}\\hfil\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\includegraphics[width=\\linewidth]{\\imagepath\/\/v_next_shortcut_optical_flow.png}\n        \\caption{$\\mathbf{v}_{f}$ from $g_s(\\mathbf{y}^\\prime_m; \\boldsymbol{\\theta}_m)$}\n        \\label{subfig:flowshortcut}\n      \\end{subfigure}\\hfil\n      \\begin{subfigure}[t]{0.23\\textwidth}\n        \\includegraphics[width=\\linewidth]{\\imagepath\/\/v_next_sent_optical_flow.png}\n        \\caption{$\\mathbf{v}_{f}$ from $g_s(\\hat{\\mathbf{y}}_m; \\boldsymbol{\\theta}_m)$}\n        \\label{subfig:flowsent}\n      \\end{subfigure}\n\\caption{B-frame coding from the \\textit{BQMall} sequence featuring\nmoving people on a static background. This crop PSNR is $31.57$ dB, MOFNet\nrate is $322$ bits and CodecNet rate is $2~240$ bits. Second\nrow shows $\\mathbf{v}_{f}$ computed by MOFNet synthesis transform from both latents\n$\\texttt{cat}(\\hat{\\mathbf{y}}_m,\\mathbf{y}^\\prime_m)$, from shortcut latents\n$\\mathbf{y}^\\prime_m$ and from the transmitted latent $\\hat{\\mathbf{y}}_m$.}\n\\end{figure}\n\n\\section{Rate-Distortion Results}\n\nThe proposed system is assessed against \\texttt{x265}\\footnote{Preset medium,\nthe exact command line can be found in appendix \\ref{subsec:seqbyseqbd}.}, an\nimplementation of HEVC. The quality is measured with the PSNR and the BD-rate\n\\citep{Bjontegaard} indicates the rate difference for the same distortion\nbetween two coders. The test sequences are from the HEVC Common Test Conditions\n\\citep{HEVC_CTC}. The system flexibility is tested under three coding\nconfigurations: All Intra (AI) \\textit{i.e.} coding only the first I-frame,\nLow-delay P (LDP) \\textit{i.e.} coding one I-frame plus 8 P-frames and Random\nAccess (RA) \\textit{i.e.} coding one I-frame plus a GOP of size 8. BD-rates of\nthe proposed coder against HEVC are presented in the Table \\ref{table:bdrate}.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{BD-rate of the proposed coder against HEVC.\n    Negative results indicate that the proposed coder requires less rate than HEVC for equivalent quality.\n    }\n    \\begin{tabular}{l||rrrrr|r}\n        \\multirow{2}{*}{Coding configuration}              & \\multicolumn{5}{c|}{Class (Resolution)}                                & \\multirow{2}{*}{Average}\\\\\n                            &    A (1600p)       &    B (1080p)      &     C (480p)       &     D (240p)       &     E (720p)       &                         \\\\\n        \\hline \n        All Intra (AI)      & $\\mathbf{-11.3}$\\% & $\\mathbf{-9.6}$\\% & $\\mathbf{-14.8}$\\% & $\\mathbf{-45.6}$\\% & $\\mathbf{-25.8}$\\% & $\\mathbf{-21.4}$\\%      \\\\\n        Low-delay P (LDP)   & $\\mathbf{-4.7}$\\%  & $29.1$\\%          & $14.3$\\%           & $\\mathbf{-9.5}$\\%  & $10.0$\\%           & $7.8$\\%                 \\\\\n        Random Access (RA)  & $5.3$\\%            & $29.9$\\%          & $7.0$\\%            & $\\mathbf{-27.2}$\\% & $\\mathbf{-18.7}$\\% & $\\mathbf{-0.7}$\\%       \\\\\n    \\end{tabular}\n\\label{table:bdrate}\n\\end{table}\n\nThe proposed system outperforms HEVC in AI configuration, proving that it\nproperly handles I-frames. It is on par with HEVC for RA coding and slightly\nworse than HEVC for LDP coding. This shows that the same coder is also able to\nefficiently code P and B-frames, without affecting the I-frames performance. To\nthe best of our knowledge, this is the first system to achieve compelling\nperformance under different coding configurations with a single end-to-end\nlearned coder for the three types of frame.\n\n\\section{Conclusion}\n\nThis paper proposes a new framework for end-to-end video coding. It is based on\nMOFNet and CodecNet, which use conditional coding to leverage the information\npresent at the decoder. Thanks to conditional coding, all types of frame (I, P\n\\& B) are processed using the same coder with the same parameters, offering a great\nflexibility in the coding configuration. The entire training process is\nperformed through the minimization of a unique rate-distortion cost. Its flexibility is\nillustrated under three coding configurations: All Intra, Low-delay P and Random\nAccess, where the system achieves performance competitive with HEVC.\\\\\nThe main focus of this work is not in the internal design of the networks architecture\n(MOFNet and CodecNet). Future work will investigate more advanced architectures,\nfrom the optical flow estimation or the learned image coding literature, which\nshould bring performance gains.\n\n\\newpage\n\\bibliographystyle{iclr2021_conference}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Details and basic definitions}\n\n\n\nIn the present work, we use the {\\em $O^*$ notation}: we write $f(X)=O^*(h(X))$, as $X\\to a$ to indicate that $|f(X)|\\leq h(X)$ in a neighborhood of $a$, where, in absence of precision, $a$ corresponds to $\\infty$.\nWe also consider the {\\em Euler $\\varphi_s$ and Kappa $\\kappa_s$} functions: let $s$ be any complex number, we define $\\varphi_s:\\mathbb{Z}_{>0}\\to\\mathbb{C}$ as $q\\mapsto q^s\\prod_{p|q}\\left(1-\\frac{1}{p^s}\\right)$ and $\\kappa_s:\\mathbb{Z}_{>0}\\to\\mathbb{C}$ as $q\\mapsto q^s\\prod_{p|q}\\left(1+\\frac{1}{p^s}\\right)$. \n\n{\\em Computational details.} Every constant in this article has been estimated using interval arithmetic. \nEarly numerical analysis was carried out using the ARB implementation, under the SageMath commands RBF and RIF, implemented in Python. We decided, however, to use Platt's implementation in C\\texttt{++}, used for example in \\cite{Pla16}, as it provides results with double precision, when compared to ARB, and at higher performance and faster speed. \n\nThroughout our calculations, we have set a precision order equal to $6\\cdot 10^9$ and run a .cpp  script compiled with C\\texttt{++}. We have also written a .ipynd  script (compiled by SageMath) to verify some of our results.\n\n\\section{Introduction}\\label{Int}\n\nThe convolution method terminology was made popular by Ramar\\'e in 1995, particularly in \\cite[Lemma 3.2]{RA95}, where it was given in a somewhat hidden version with respect to the one we present in this article. It is a technique, already present in \\cite{Mot78} and \\cite{W1927}, among many other places, that relies upon a convolution identity and helps obtaining explicit estimations of averages of arithmetic functions, under some conditions. It is particularly meaningful when these arithmetic functions are supported on the square-free numbers, having a sufficiently regular behavior on all large prime numbers.  \n\nWhile the convolution method provides the main term of a asymptotic expansion for the average of an arithmetic function with ease, it is at the remainder term where it shows its true potential, as it succeeds in giving a good enough estimation, explicit, for the error term: if the average is performed for the range $(0,X]$, where $X>0$, then the convolution methods gives error term explicit estimations of magnitude $X^{-\\delta}$ when $\\delta$ belongs to a maximal real open and positive interval $I$. \n\nNevertheless, the nature of the convolution method does not allow one to obtain an error term estimation of magnitude $X^{-\\delta_0}$ where $\\delta_0$ is the right endpoint of $I$. Since it is usually a subject of interest in the explicit theory of numbers to improve error term magnitudes of expressions of interest, it is thus natural to ask whether or not one can provide, necessarily by a different method, an error term of critical order $\\delta_0$ so that the overall estimation is qualitatively improved, going thus to the edge of the method of convolution.\n\nWe first present in \\S\\ref{dirichlet}, a special form of the convolution method involving sufficiently regular square-free supported functions, as shown in Theorem \\ref{general}. As it relies upon some complex analytic facts, this method is related to a typical complex analytic approach for estimating the asymptotic expansion for the average of an arithmetic function by means of residue theory. \n\nOur main result, presented in \\S\\ref{improvement}, differs from complex analysis. In \\S\\ref{achieving}, we see how the use of some very particular estimations given in \\S\\ref{particular}, constitute the main ingredient to obtain reasonable explicit estimations of critical exponent in almost all cases where the convolution method may be applied with some conditions. Indeed, since our technique also relies upon the convergence of infinite products, some extra conditions on the regularity of the arithmetic function that is being averaged are needed, as Theorem \\ref{general++} tells, and therefore there is a small range of functions that are not considered in our improvements, namely when the values of $\\alpha$ and $\\beta$ defined in Theorem \\ref{general++} have a difference of absolute value smaller or equal than $\\frac{1}{2}$. However, as most of the applications we mention throughout this article do not involve that missing case, we then claim that every one of these ones are improved up to their critical exponent.\n\nPrevious work towards the obtention of error terms of critical exponent can be found, on some particular averages, in \\cite{Bu14} and \\cite{W1927}. In \\cite{RA13} and \\cite{RA19}, the obtention of the critical exponent is carried out by a completely different approach, using some results known as \\emph{the covering remainder lemma} and \\emph{the unbalanced Dirichlet hyperbola formula} as well as strong explicit bounds on some summatory functions involving the M\\\"obius functions that, unlike our case of study, do oscillate. Furthermore, it is important to point out that whereas a similar path as in \\cite{RA13} or \\cite{RA19} could have been followed, these results consider specific properties of the functions that are being averaged and they are thus not easy to generalize to a broader class of functions. This is the reason why \\cite[Thm. 1.2]{RA13} improves on the classic convolution method result presented in Corollary \\ref{corollary} $\\mathbf{(a)}$ but still requires the convolution method to estimate related averages of less simple arithmetic functions; for example, with the result we present in Theorem \\ref{general++}, one can now immediately derive stronger estimations for \\cite[Lemmas 7.1, 7.2, 7.6, 7.7, 7.8, 7.9]{RA13} that may lead to further improvements on the cited article of Ramar\\'e--Akhilesh. In that aspect, our result might help as a reference for further improvements on many places where the convolution method is employed; it read as follows. \n\n\\begin{theorem*} \n  Let $X>0$, be a real number and $q$ a positive integer. Consider a multiplicative function $f:\\mathbb{Z}^+\\to\\mathbb{C}$ such that for every prime number $p$ satisfying $(p,q)=1$, we have $f(p)=\\frac{1}{p^{\\alpha}}+O\\left(\\frac{1}{p^{\\beta}}\\right)$, where $\\alpha$, $\\beta$ are real numbers satisfying $\\beta>\\alpha$, $\\beta-\\alpha>\\frac{1}{2}$. Then there exists a constant $\\mathrm{W}_{\\alpha}^q>0$ such that\n \\begin{align*}\n \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})=F_\\alpha^{q}(X)+\\begin{cases}\nO^*\\left(\\mathrm{W}_\\alpha^q\\ X^{\\frac{1}{2}-\\alpha}\\right),\\quad&\\text{ if }\\alpha\\neq\\frac{1}{2},\\\\\nO^*\\left(\\mathrm{W}_\\alpha^q\\ \\log(X)\\right),\\quad&\\text{ if }\\alpha=\\frac{1}{2},\\\\\n\\end{cases}\n \\end{align*} \n where  \n \\begin{align*} \n F_\\alpha^q(X)&=\\frac{M_{\\alpha}^{q}\\zeta(\\alpha)\\varphi_\\alpha(q)}{q^\\alpha}-\\frac{N_{\\alpha}^q\\varphi(q)}{(\\alpha-1)q}\\frac{1}{X^{\\alpha-1}},\\quad&&\\text{if\\ }\\alpha>\\frac{1}{2},\\ \\alpha\\neq 1,\\\\\n F_1^q(X)&=\\frac{M_{1}^{q}\\varphi(q)}{q}\\left(\\log\\left(X\\right)+T_f^q+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p-1}\\right),\\\\\n&\\phantom{xxxxxxxxx}T_{f}^{q}=\\sum_{p\\nmid q}\\frac{\\log(p)(1-(p-2)f(p))}{(f(p)+1)(p-1)},\\\\\nF_\\alpha^q(X)&=\\frac{M_{\\alpha}^{q}\\varphi(q)}{(1-\\alpha)q}X^{1-\\alpha},\\quad&&\\text{if\\ }\\alpha\\leq\\frac{1}{2},\n \\end{align*}\nand, \n\\begin{align*}\n M_{\\alpha}^{q}&=\\begin{cases}\n\\prod_{p\\nmid q}\\left(1-\\frac{1-f(p)p^\\alpha+f(p)}{p^{\\alpha}}  \\right),\\quad&\\text{ if }\\alpha>\\frac{1}{2},\\\\\nN_{\\alpha}^{q},\\quad&\\text{ if }\\alpha\\leq\\frac{1}{2},\n\\end{cases}\\\\\nN_{\\alpha}^{q}&=\\prod_{p\\nmid q}\\left(1-\\frac{p^{1-\\alpha}-f(p)p+f(p)}{p^{2-\\alpha}} \\right).\n\\end{align*}\n \\end{theorem*}\n\nAs an application of the above theorem, we deduce how the improvement on the convolution method produces better savings on the error term constant of $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\varphi(\\ell)}, X>0, q\\in\\mathbb{Z}_{>0}$ than the one in \\cite[Thm. 1.1]{RA13} , when prime coprimality conditions are introduced. This situation is examined in \\S\\ref{Cop}, and we have for instance the improvement on the constant \\sage{Trunc(5.9*21\/25,3)}, given in \\cite[Thm. 1.1]{RA13}, by $\\sage{Upper(CONSTANT_RAM*Prod_Ram_Upper*CONSTANT2\/CONSTANT_RAM,digits)}$, according to the following result.\n\n\\begin{lemma*}\nLet $X>0$, then\n\\begin{equation*}\n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\varphi(\\ell)}=\\frac{1}{2}\\left(\\log\\left(X\\right)+\\mathfrak{a}_2\\right)+O^*\\left(\\frac{\\sage{Upper(CONSTANT_RAM*Prod_Ram_Upper*CONSTANT2\/CONSTANT_RAM,digits)}}{\\sqrt{X}}\\right),\n\\end{equation*}\nwhere $\\mathfrak{a}_2=\\sage{Trunc(C_sum1+log(2)\/2,digits)}\\ldots $. \n\\end{lemma*}\n\n\n\n\n\\section{A special version of the method of convolution}\\label{dirichlet}\n\n\n \nIn the convolution method, it is crucial to preserve regularity conditions, that is, conditions that do not impose specific ranges other than the variable itself being a positive integer, under, perhaps, some coprimality restrictions. \n\nTo put an example, when one carries out a summation on a variable $e\\in\\mathbb{Z}_{>0}$ such that $e\\leq \\frac{X}{d}$ for certain real number $X>0$ and a positive integer $d$, it is often implicitly assumed that $\\frac{X}{d}\\geq 1$, so that the set $\\{e\\in\\mathbb{Z}_{>0}, \\ e\\leq\\frac{X}{d}\\}$ is not empty. If $d$ is itself a variable, that means that we have the range condition $\\{d\\leq X\\}$ on the variable $d$. Hence, if we are able to estimate asymptotically a summation on the variable $e\\in\\mathbb{Z}_{>0}$ such that $e\\leq \\frac{X}{d}$, regardless of whether or not an empty condition sum is performed, that is an \\emph{empty sum}, then the range condition on the variable $d$ will be absent.\n\n\n\n\\subsection{Regularity conditions: estimating empty summations}\n\n\\begin{lemma}\\label{recip} \nLet $\\alpha\\in \\mathbb{R}^+\\setminus \\{1\\}$ and $X>0$. Then\n\\begin{equation*}\n\\sum_{n\\leq X}\\frac{1}{n^\\alpha}=\\zeta(\\alpha)-\\frac{1}{(\\alpha-1)X^{\\alpha-1}}+O^*\\left(\\frac{1}{X^{\\alpha}}\\right).\n\\end{equation*}\n\\end{lemma} \n\n\\begin{proof}\n  By definition of $\\zeta(s)$ for $\\Re(s)>1$, and by analytic continuation\n  for all $s\\ne 1$ with $\\Re(s)>0$,\n  \\begin{equation}\\label{EulerMac}\n \\zeta(s) - \\frac{1}{(s-1) X^{s-1}} - \\sum_{n\\leq X} \\frac{1}{n^s} =\n  \\sum_{n=1}^\\infty \n  \\left(\\int_{n-1}^{n} \\frac{dx}{(X+x)^s} - \\frac{1}{(\\lfloor X\\rfloor+n)^s}\\right).\\end{equation}\n  Set $s=\\alpha$; clearly $(\\lfloor X\\rfloor+n)^{-\\alpha} \\geq (X+n)^{-\\alpha}$ and by convexity of\n  $t\\mapsto \\frac{1}{t^\\alpha}$,\n  $\\int_{n-1}^{n} \\frac{dx}{(X+x)^\\alpha} \\leq \\frac{1}{2} \\left(\n  \\frac{1}{(X+n-1)^\\alpha} + \\frac{1}{(X+n)^\\alpha}\\right)$.\n  Hence, the right hand side of \\eqref{EulerMac} is at most \n\\begin{equation*}\\sum_{n=1}^\\infty \\frac{1}{2} \\left(\n  \\frac{1}{(X+n-1)^\\alpha} - \\frac{1}{(X+n)^\\alpha}\\right) \n  \\leq  \\frac{1}{2X^\\alpha}.\n\\end{equation*}\nOn the other hand, by the mean value theorem, for any $n\\in\\mathbb{Z}_{>0}$, there exists $r\\in[n-1,n]$ such that\n$\\int_{n-1}^{n}\\frac{dx}{(X+x)^{\\alpha}}-\\frac{1}{(\\lfloor X\\rfloor+n)^{\\alpha}}=\\frac{1}{(X+r)^{\\alpha}}-\\frac{1}{(\\lfloor X\\rfloor+n)^{\\alpha}}$. Thus, by the monotonicity of $t\\mapsto\\frac{1}{t^{\\alpha}}$ and the fact that $X+r$ and $\\lfloor X\\rfloor+n$ are both contained in $[X+n-1,X+n]$, we have that \nthe right hand side of \\eqref{EulerMac} is at least\n\\begin{equation*}\n\\sum_{n=1}^{\\infty}\\left(\\frac{1}{(X+n)^{\\alpha}}-\\frac{1}{(X+n-1)^{\\alpha}}\\right)=-\\frac{1}{X^{\\alpha}}.\n\\end{equation*}\n\\end{proof}\n\nThe following lemma estimates asymptotically some sums even when they have an empty condition.\n\n \\begin{lemma}\\label{SumEstimations} \n Let $X>0$ and $\\alpha>0$. If $0<\\delta\\leq 1$, we have\n \\begin{align}\\label{harmonic}\n \\sum_{n\\leq X}\\frac{1}{n}&=\\log(X)+\\gamma+O^*\\left(\\frac{\\Delta_{1}^{\\delta}}{X^\\delta}\\right);\n \\end{align}\n if $\\max\\{0,\\alpha-1\\}<\\delta\\leq\\alpha$ and $\\alpha\\neq 1$, we have\n \\begin{equation}\\label{generalz}\n \\sum_{n\\leq X}\\frac{1}{n^\\alpha}=\\zeta(\\alpha)-\\frac{1}{(\\alpha-1)X^{\\alpha-1}}+O^*\\left(\\frac{\\Delta_{\\alpha}^{\\delta}}{X^\\delta}\\right),\n \\end{equation} \nwhere $\\Delta_{1}^{\\delta}=\\max\\left\\{\\gamma,\\frac{1}{\\delta e^{\\gamma\\delta+1}}\\right\\}$\nand, for $\\alpha\\neq 1$,\n \\begin{align*}\n \\Delta_{\\alpha}^{\\delta}&=\n     \\begin{cases}\n     \\max\\left\\{1,\\left(\\frac{1}{\\delta^{\\delta}}\\left(\\frac{(\\delta-\\alpha+1)}{|\\zeta(\\alpha)(\\alpha-1)|}\\right)^{\\delta-\\alpha+1}\\right)^{\\frac{1}{\\alpha-1}},\\zeta(\\alpha)-\\frac{1}{\\alpha-1}\\right\\},&\\quad\\text{ if }\\delta\\neq\\alpha,\\\\\n     1,&\\quad\\text{ if }\\delta=\\alpha.\n     \\end{cases}\n \\end{align*}\n \\end{lemma}\n  \\begin{proof} By \\cite[Lemma 2.1]{RA13} and Lemma \\ref{recip}, for $X>0$, we have\n  \\begin{align}\n \\sum_{n\\leq X}\\frac{1}{n}&=\\log(X)+\\gamma+O^*\\left(\\frac{\\gamma}{X}\\right),\\label{Ha}\\\\\n \\sum_{n\\leq X}\\frac{1}{n^\\alpha}&=\\zeta(\\alpha)-\\frac{1}{(\\alpha-1)X^{\\alpha-1}}+O^*\\left(\\frac{1}{X^\\alpha}\\right),\\quad\\text{if\\ }\\alpha> 0\\text{\\ and\\ }\\alpha\\neq 1,\\label{Ge}\n \\end{align}  \n  respectively. \n Thus, if $X\\geq 1$, the result holds trivially as $\\delta'\\mapsto X^{\\delta'}$ is increasing and $\\delta<\\alpha$. Otherwise, when $0<X<1$ the above summations are empty; write $X=\\frac{1}{Y}$ with $Y>1$ and           observe first that the function $f:Y\\geq 1\\mapsto\\frac{\\log(Y)-\\gamma}{Y^\\delta}$ has a single critical point at $y_0=e^{\\frac{1}{\\delta}+\\gamma}>1$ taking the value $f(y_0)=\\frac{1}{\\delta e^{\\gamma\\delta+1}}>0$. As $f(1)=-\\gamma$ and $\\lim_{Y\\to\\infty}f(Y)=0$, $f$ is increasing in $[1,y_0]$ and decreasing in $[y_0,\\infty)$, and hence $\\sup_{\\{Y>1\\}}|f(Y)|=\\max\\left\\{\\gamma,\\frac{1}{\\delta e^{\\gamma\\delta+1}}\\right\\}$. \n\n  Secondly, by \\cite[Cor. 1.14]{MV07}, we have that $\\zeta(\\alpha)>\\frac{1}{\\alpha-1}$ and $\\zeta(\\alpha)(\\alpha-1)>0$ for all $\\alpha\\geq 0$ and $\\alpha\\neq 1$. Moreover, the function $g:Y>0\\mapsto\\frac{1}{Y^\\delta}\\left(\\zeta(\\alpha)-\\frac{Y^{\\alpha-1}}{\\alpha-1}\\right)$ has a critical point $y_0$ satisfying $y_0^{\\alpha-1}=\\frac{\\zeta(\\alpha)(\\alpha-1)\\delta}{\\delta-\\alpha+1}>0$, since $\\delta>\\alpha-1$ and $\\delta>0$ and in this case, we have that $\\lim_{Y\\to\\infty}g(Y)=0$ and, thus, $|g|$ is decreasing in $[y_0,\\infty)$. We conclude then that $\\max_{[y_0,\\infty)}|g(Y)|$ $=|g(y_0)|$, where \n \\begin{equation*}\n |g(y_0)|=\\left(\\frac{1}{\\delta^{\\delta}}\\left(\\frac{(\\delta-\\alpha+1)}{|\\zeta(\\alpha)(\\alpha-1)|}\\right)^{\\delta-\\alpha+1}\\right)^{\\frac{1}{\\alpha-1}}.\n \\end{equation*} \n If $y_0\\leq 1$, then $|g(1)|=g(1)\\leq|g(y_0)|$ and $\\sup_{\\{Y>1\\}}|g(Y)|=g(1)$; otherwise, if $y_0>1$, as $g$ is also monotonic between $1$ and $y_0$, we derive that $\\sup_{\\{Y>1\\}}|g(Y)|=\\max\\{g(1),|g(y_0)|\\}$, which gives us the desired result.\n  \\end{proof}\n\nIt is important to point out that in case that $\\alpha>1$, it would have been possible to give an error term expression even if $\\delta=\\alpha-1>0$, whereas, if $\\delta<\\alpha-1$, then $|g|$ would have been unbounded in $[1,\\infty)$.\n\nOn the other hand, as pointed out at the beginning of \\S\\ref{dirichlet}, it is essential to have an estimation of the above summations when they have actually an empty condition, that is when $X\\in(0,1)$. Indeed, this will provide regularity for some sum conditions during the proof of Theorem \\ref{general} that otherwise would impose some variables to be at least $1$ and some sums to be non-empty. It should be expected, though, that the fact of imposing regularity conditions, or rather asking for estimations of sums up to the variable $X$ with $X>0$, will worsen a bit the constants on the involved error terms; for instance, when $\\alpha=1$ and when we are restricted to the range $X\\geq 1$, the value of $\\gamma=\\sage{Trunc(gamma,5)}\\ldots$ given in \\eqref{Ha} can be improved to $2(\\log(2)+\\gamma-1)=\\sage{Trunc(2*(log(2)+euler_gamma-1),5)}\\ldots$ (refer to \\cite[Lemma 2.1]{RA13} ). \n\n\n\\subsection{The convolution method} \\label{TCM}\n\n\n\nThe following theorem will help us to state Corollary \\ref{corollary}. Although inspired by \\cite[Lemma 3.2]{RA95}, it is presented in a much general framework, in an attempt to understand and deduce with ease the order of averages of sufficiently regular square-free supported arithmetic functions. By sufficiently regular, we mean an arithmetic function having a specific constant dominant term on all sufficiently large prime numbers. As it turns out, it is precisely the regularity of an arithmetic function that helps to derive the asymptotic expansion of its average under the method of convolution.\n \\begin{theorem}\\label{general}    \n  Let  $q$ a positive integer and let $X$, $\\alpha$, $\\beta$ be real numbers such that $X>0$, $\\beta>1$ and $\\beta>\\alpha>\\frac{1}{2}$. Consider a multiplicative function $f:\\mathbb{Z}^+\\to\\mathbb{C}$ such that \n $f(p)=\\frac{1}{p^{\\alpha}}+O\\left(\\frac{1}{p^{\\beta}}\\right)$, for every sufficiently large prime number $p$ coprime to $q$. Then for any real number $\\delta>0$ such that $\\max\\{0,\\alpha-1\\}<\\delta<\\min\\{\\beta-1,\\alpha-\\frac{1}{2}\\}$ we have the estimation \n \\begin{equation*}\n \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})=F_\\alpha^{q}(X)+O^*\\left(\\Delta_{\\alpha}^{\\delta}\\frac{\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{\\overline{H}_{f}^{\\phantom{.}q}(-\\delta)}{X^\\delta}\\right),\n \\end{equation*} \n where, if $\\alpha\\neq 1$,\n \\begin{align*} \n F_\\alpha^q(X)&=\\frac{H_{f}^{q}(0)\\zeta(\\alpha)\\varphi_\\alpha(q)}{q^\\alpha}-\\frac{H_{f}^{q}(1-\\alpha)\\varphi(q)}{(\\alpha-1)q}\\frac{1}{X^{\\alpha-1}},\n\\end{align*}\nand, if $f(p)= -1$ for some prime number $p$,  $F_1^q(X)=-\\sum_{d}\\frac{h_{f}^{q}(d)\\log(d)}{d^\\alpha}$, whereas, if $f(p)\\neq -1$ for any prime number $p$,\n\\begin{align*}\n F_1^q(X)&=\\frac{H_{f}^{q}(0)\\varphi(q)}{q}\\left(\\log\\left(X\\right)+T_f^q+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p-1}\\right),\\\\\nT_{f}^{q}&=\\sum_{p\\nmid q}\\frac{\\log(p)(1-(p-2)f(p))}{(f(p)+1)(p-1)}.\n \\end{align*}\n Here, $\\Delta_{\\alpha}^{\\delta}$ is defined as in Lemma \\ref{SumEstimations} and  $H_{f}^{q}:\\{s\\in\\mathbb{C},\\ \\Re(s)>\\frac{1}{2}-\\alpha\\}\\to\\mathbb{C}$ is an analytic function satisfying  \n \\begin{align*}\n H_{f}^{q}(s)&=\\prod_{p\\nmid q}\\left(1-\\frac{1-f(p)p^\\alpha}{p^{s+\\alpha}}-\\frac{f(p)}{p^{2s+\\alpha}}  \\right)=\\sum_{\\substack{d\\\\(d,q)=1}}\\frac{h_{f}^{q}(d)}{d^{s+\\alpha}},\\\\\n \\overline{H}_{f}^{\\phantom{.}q}(s)&=\\prod_{p\\nmid q}\\left(1+\\frac{|1-f(p)p^\\alpha|}{p^{\\Re(s)+\\alpha}}+\\frac{|f(p)|}{p^{2\\Re(s)+\\alpha}}  \\right)=\\sum_{\\substack{d\\\\(d,q)=1}}\\frac{|h_{f}^{q}(d)|}{d^{\\Re(s)+\\alpha}}.\n \\end{align*} \n \\end{theorem}\n\n  \\begin{proof} \n  By the asymptotic condition on $f$ in the statement, the Dirichlet series $D_{f}^{q}$ associated with $\\ell\\mapsto\\mu^2(\\ell)f({\\ell})\\mathds{1}_q(\\ell)$, where $\\mathds{1}_q$ is defined as the multiplicative function $\\ell\\mapsto\\mathds{1}_{\\{(\\ell,q)=1\\}}(\\ell)$, converges absolutely for any $s\\in\\mathbb{C}$ such that $\\Re(s)>1-\\alpha$. Thus, in the set $\\{s\\in\\mathbb{C},  \\ \\Re(s)>1-  \\alpha\\}$, the equality  \n  \\begin{align}\n  D_{f}^{q}(s)=\\sum_{\\substack{\\ell\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)f(\\ell)}{\\ell^{s}}=\\prod_{p\\nmid q}\\left(1+\\frac{f(p)}{p^s}\\right)\n  \\end{align}\n  holds and the function $s\\mapsto\\zeta(s+\\alpha)$ can be expressed by an Euler product.  For any $s$ such that $\\Re(s)>1-\\alpha$, we have then\n  \\begin{align}\\label{D}\n  &\\frac{D_{f}^{q}(s)}{\\zeta(s+\\alpha)}=\\prod_{p\\nmid q}\\left(1+\\frac{f(p)}{p^s}\\right)\\left(1-\\frac{1}{p^{s+\\alpha}}\\right)\\cdot\\prod_{p|q}\\left(1-\\frac{1}{p^{s+\\alpha}}\\right)\\nonumber\\\\\n  &\\phantom{xxxx}=\\frac{\\varphi_{s+\\alpha}(q)}{q^{s+\\alpha}}\\cdot\\prod_{p\\nmid q}\\left(1-\\frac{1-f(p)p^\\alpha}{p^{s+\\alpha}}-\\frac{f(p)}{p^{2s+\\alpha}}\\right)\\ \\ =\\ \\frac{\\varphi_{s+\\alpha}(q)}{q^{s+\\alpha}}\\cdot H_{f}^{q}(s).\\nonumber\n  \\end{align}\n  Also, we have that $\\frac{1-f(p)p^{\\alpha}}{p^{s+\\alpha}}=O\\left(\\frac{1}{p^{\\Re(s)+\\beta}}\\right)$ and $\\frac{f(p)}{p^{2s+\\alpha}}=O\\left(\\frac{1}{p^{2\\Re(s)+2\\alpha}}\\right)$. Since $\\beta>\\alpha$, we have that $H$ can be extended analytically from $\\{s\\in\\mathbb{C},\\ \\Re(s)>1-\\alpha\\}$ onto $\\{s\\in\\mathbb{C},\\ \\Re(s)>\\max\\{1-\\beta,\\frac{1}{2}-\\alpha\\}\\}$. Further, as $0>1-\\beta$ and $0>\\frac{1}{2}-\\alpha$, $H_{f}^{q}(0)$ exists and, if $f(p)\\neq -1$ for any prime number $p$, it is different from $0$, since each factor defining it can be expressed as $(1+f(p))\\left(1-\\frac{1}{p^\\alpha}\\right)$ and $\\alpha\\neq 0$.\n\n  Now, the formal equality $D_{f}^{q}(s)=H_{f}^{q}(s)\\cdot\\prod_{p\\nmid q}\\left(1+\\frac{1}{p^{s+\\alpha}}+\\frac{1}{p^{2(s+\\alpha)}}+\\ldots\\right)$ hides the convolution product\n  \\begin{equation}\\label{identity1}\n  \\ell^\\alpha \\mu^2(\\ell)f(\\ell)\\mathds{1}_{(\\ell,q)=1}(\\ell)=(h_{f}^{q}\\star\\mathds{1}_q)\\ (\\ell)=\\sum_{\\substack{d|\\ell}}h_{f}^{q}(d)\\mathds{1}_q\\left(\\frac{\\ell}{d}\\right), \n  \\end{equation}\n  where $h$ is a multiplicative function defined on the prime numbers as\n  \\begin{align}\n  &h_{f}^{q}(p)=(f(p)p^\\alpha-1)\\cdot\\mathds{1}_q(p),\\qquad h_{f}^{q}(p^2) = -f(p)p^\\alpha\\cdot \\mathds{1}_q(p),\\label{hfq}\\\\\n  &\\phantom{xxxxxxxxxxxxxx} h_{f}^{q}(p^k) = 0, \\quad k>2.\\nonumber\n  \\end{align}\n  Therefore, from \\eqref{identity1} we conclude that\n  \\begin{align}\\label{DirichletIdentity}\n  \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})=\\sum_{\\substack{\\ell\\leq X}}\\frac{(h_{f}^{q}\\star\\mathds{1}_q)\\ (\\ell)}{\\ell^\\alpha}=\\sum_{\\substack{d}}\\frac{h_{f}^{q}(d)}{d^\\alpha}    \\sum_{\\substack{e\\leq\\frac{X}{d}\\\\(e,q)=1}}  \\frac{1}{e^\\alpha}\\phantom{xxxxxxxxxxxxx}&\\nonumber\\\\\n=\\sum_{\\substack{d}}\\frac{h_{f}^{q}(d)}{d^\\alpha}\\sum_{\\substack{e\\leq\\frac{X}{d}}}\\frac{1}{e^\\alpha}\\sum_{d'|e,d'|q}\\mu(d')=\\sum_{\\substack{d}}\\frac{h_{f}^{q}(d)}{d^\\alpha}\\sum_{d'|q}\\frac{\\mu(d')}{d'^\\alpha}\\sum_{\\substack{e\\leq\\frac{X}{dd'}}}\\frac{1}{e^\\alpha}&,\n  \\end{align}\n  where there is no upper bound conditions on the variables $d$ and $d'$ present in the outer sums above, their being encoded by the innermost  sum of  \\eqref{DirichletIdentity}, which, in order to continue our analysis, we must estimate regardless of whether or not it is empty: Lemma \\ref{SumEstimations} allow us to handle this situation. \n\nHence, as $\\max\\{0,\\alpha-1\\}<\\delta<\\min\\{\\beta-1,\\alpha-\\frac{1}{2}\\}<\\alpha$, we derive that the second sum in \\eqref{DirichletIdentity} can be expressed as\n  \\begin{align}\\label{Sum:alpha neq 1}\n  \\sum_{d'|q}\\frac{\\mu(d')}{d'^\\alpha}\\sum_{\\substack{e\\leq\\frac{X}{dd'}}}\\frac{1}{e^\\alpha}=\\sum_{d'|q}\\frac{\\mu(d')}{d'^\\alpha}\\left(\\zeta(\\alpha)-\\frac{(dd')^{\\alpha-1}}  {(\\alpha-1)X^{\\alpha-1}}+O^*\\left(\\Delta_{\\alpha}^{\\delta}\\frac{(dd')^\\delta}{   X^{\\delta}}\\right)\\right)&\\nonumber\\\\\n =\\ \\frac{\\zeta(\\alpha)\\varphi_\\alpha(q)}{q^\\alpha}-\\frac{\\varphi(q)}{(\\alpha-1)q}\\cdot\\frac{d^{\\alpha-1}}{X^{\\alpha-1}}+O^*\\left(\\Delta_{\\alpha}^{\\delta}\\frac{\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{d^\\delta}{X^\\delta}\\right)&,\n  \\end{align}\n  if $\\alpha\\neq 1$, or as\n  \\begin{align}\\label{Sum:alpha=1}\n  \\sum_{d'|q}\\frac{\\mu(d')}{d'^\\alpha}\\sum_{\\substack{e\\leq\\frac{X}{dd'}}}\\frac{1}{e^\\alpha}=\\sum_{d'|q}\\frac{\\mu(d')}{d'^\\alpha}\\left(\\log\\left(\\frac{X}{dd'}\\right)+  \\gamma+O^*\\left(\\frac{\\Delta_1^{\\delta}(dd')^\\delta}{X^\\delta}\\right)\\right)\\phantom{xxxxxx}&  \\nonumber\\\\\n  =\\frac{\\varphi_\\alpha(q)}{q^\\alpha}\\left(\\log\\left(\\frac{X}{d}\\right)+\\gamma\\right)-\\sum_{d'|q}\\frac{\\mu(d')\\log(d')}{d'^\\alpha}+O^*\\left(\\frac{\\Delta_1^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{d^\\delta}  {X^\\delta}\\right)&\\nonumber\\\\\n  =\\frac{\\varphi_\\alpha(q)}{q^\\alpha}\\left(\\log\\left(\\frac{X}{d}\\right)+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p^\\alpha-1}\\right)+O^*\\left(\\frac{\\Delta_1^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{d^\\delta}{X^\\delta}  \\right),&\n  \\end{align}\n  if $\\alpha=1$, where we have used that\n  \\begin{align}-\\sum_{d'|q}\\frac{\\mu(d')\\log(d')}{d'^\\alpha}=\\left(\\frac{\\varphi_{s+\\alpha}(q)}{q^{s+\\alpha}}\\right)'_{s=0}=\\frac{\\varphi_\\alpha(q)}{q^\\alpha}\\sum_{p|q}  \\frac{\\log(p)}{p^\\alpha-1}.\\label{derivated}\n  \\end{align}\n  On the other hand, observe that $H_{f}^{q}(1-\\alpha)$ and $\\overline{H}_{f}^{\\phantom{.}q}(-\\delta)$ are well-defined, as $\\min\\{1-\\alpha,-\\delta\\}>\\max\\{1-\\beta,\\frac{1}{2}-\\alpha\\}$. Therefore, from \\eqref{DirichletIdentity}, the sum $\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})$ can be estimated either as\n  \\begin{align}\\label{DirichletIdentity:alpha=1} \n  \\sum_{\\substack{d}}\\frac{h_{f}^{q}(d)}{d^\\alpha}\\left(\\frac{\\zeta(\\alpha)\\varphi_\\alpha(q)}{q^\\alpha}-\\frac{\\varphi(q)}{(\\alpha-1)q}\\cdot\\frac{d^{\\alpha-1}}{X^{\\alpha-1}}  +O^*\\left(\\frac{\\Delta_{\\alpha}^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}  \\cdot\\frac{d^\\delta}{X^\\delta}\\right)\\right)&\\\\\n  =H_{f}^{q}(0)\\ \\frac{\\zeta(\\alpha)\\varphi_\\alpha(q)}{q^\\alpha}-\\frac{\\varphi(q)}{(\\alpha-1)q}\\cdot\\frac{H_{f}^{q}(1-\\alpha)}{X^{\\alpha-1}}+O^*\\left(\\frac{\\Delta_{\\alpha}^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{\\overline{H}_{f}^{\\phantom{.}q}(-\\delta)}{X^\\delta}  \\right)\\nonumber&,\n  \\end{align}\n  if $\\alpha\\neq 1$, by using \\eqref{Sum:alpha neq 1}, or \n  \\begin{align}\\label{DirichletIdentity:alphaneq1}\n  \\sum_{\\substack{d}}\\frac{h_{f}^{q}(d)}{d^\\alpha}\\left(\\frac{\\varphi_\\alpha(q)}{q^\\alpha}\\left(\\log\\left(\\frac{X}{d}\\right)+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p^\\alpha-1}\\right)+O^*\\left(\\frac{\\Delta_{1}^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}\\cdot\\frac{d^\\delta}{X^\\delta}\\right)\\right)&\\\\\n  = H_{f}^{q}(0)\\ \\frac{\\varphi_\\alpha(q)}{q^\\alpha}\\left(\\log\\left(X\\right)+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p^\\alpha-1}\\right)+H_{f}^{q}\\phantom{}'(0)\\phantom{xx}\\nonumber&\\\\\n + O^*\\left(\\frac{\\Delta_{1}^{\\delta}\\kappa_{\\alpha-\\delta}(q)}{q^{\\alpha-\\delta}}  \\cdot\\frac{\\overline{H}_{f}^{\\phantom{.}q}(-\\delta)}{X^\\delta}\\right),\\phantom{xx}\\nonumber&\n  \\end{align}\n  if $\\alpha=1$, by using \\eqref{Sum:alpha=1} and that $-\\sum_{d}\\frac{h_{f}^{q}(d)\\log(d)}{d^\\alpha}=H_{f}^{q}\\phantom{}'(0)$. The result is thus obtained by noticing that if $H_{f}^{q}(0)\\neq 0$, then $\\frac{H_{f}^{q}\\phantom{}'(0)}{H_{f}^{q}(0)}$ equals\n  \\begin{align*}\n  \\left(\\prod_{p\\nmid q}\\left(1-\\frac{1-f(p)p^\\alpha}{p^{s+\\alpha}}-\\frac{f(p)}{p^{2s+\\alpha}}\\right)\\right)'_{s=0}=\\sum_{p\\nmid q}\\frac{\\log(p)(1-f(p)p^\\alpha+2f(p))}{(f(p)+1)(p^\\alpha-1)}.\n  \\end{align*} \n  \\end{proof}\n\n \n \\begin{corollary}\\label{corollary}\n\n Let $X>0$ and $q\\in\\mathbb{Z}_{>0}$. The following estimations hold\n \\begin{align} \n \\mathbf{(a)}\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\varphi(\\ell)}&=\\frac{\\varphi(q)}{q}\\left(\\log\\left(X\\right)+\\mathfrak{a}_q\\right)+O^*\\left(\\frac{     \\sage{Error_sum1}     \\cdot\\mathpzc{A}_q}{X^{\\sage{delta}}}\\right)\\label{sum1:eq}, \\\\ \n \\mathbf{(b)} \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell}&=\\frac{6}{\\pi^2}\\frac{q}{\\kappa(q)}\\left(\\log\\left(X\\right)+\\mathfrak{b}_q\\right)+O^*\\left(\\frac{\\sage{Upper(Error_sum2,digits)}  \\cdot\\mathpzc{B}_q}{X^{\\sage{delta}}}\\right),\\label{sum2:eq}\n  \\end{align}  \n where\n \\begin{align*}\n  \\mathpzc{A}_q= \\prod_{p|q}\\left(1+\\frac{p-p^{\\sage{delta}}-2}{(p-1)p^{\\sage{1-delta}}+p^{\\sage{delta}}+1}\\right),\\mathpzc{B}_q =\\prod_{p|q}\\left(1+\\frac{p^{\\sage{1-delta}}-1}{p^{\\sage{2-2*delta}}+1}\\right),\n  \\end{align*} \nand\n \\begin{align*} \n \\mathfrak{a}_q&=\\sum_{p}\\frac{\\log(p)}{p(p-1)}+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p},\\sum_{p}\\frac{\\log(p)}{p(p-1)}+\\gamma=   \\sage{C_sum1}    \\ldots,\\\\\n\\mathfrak{b}_q &=\\sum_{p}\\frac{2\\log(p)}{p^2-1}+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p+1}, \n\\sum_{p}\\frac{2\\log(p)}{p^2-1}+\\gamma=\\sage{C_sum2}\\ldots .\n\\end{align*} \n \\end{corollary}\n\n  \\begin{proof}\n  For the case $\\mathbf{(a)}$ (respectively $\\mathbf{(b)}$), apply Theorem \\ref{general} with $f(p)=\\frac{1}{\\varphi(p)}=\\frac{1}{p-1}$ (respectively $f(p)=\\frac{1}{p}$), $\\alpha=1$, $\\beta=2$ and $0\\leq\\delta=\\sage{delta}<\\frac{1}{2}$. \n\nThe infinite products that participate in the main and error terms as well as the infinite summation that participates in the main term can be estimated by using a rigorous implementation of interval arithmetic, and some techniques for accelerating convergence.\n  \\end{proof}\n\n\n\\noindent\\textbf{Remarks.} Conditions $\\alpha>\\frac{1}{2}$ and $\\beta>1$ in Theorem \\ref{general} are necessary to ensure the existence of $H_{f}^{q}(0)$. Nonetheless, we can derive an analogous result for any multiplicative arithmetic function $f$ satisfying the conditions $f(p)=\\frac{1}{p^{\\alpha}}+O\\left(\\frac{1}{p^{\\beta}}\\right)$, for every sufficiently large prime number $p$ coprime to $q$, where $\\alpha\\leq\\frac{1}{2}$ and $\\beta>\\alpha$ by using of Theorem \\ref{general} and summation by parts. In this instance, there will not be any secondary term appearing and the error term magnitude will be $O\\left(X^{1-\\alpha-\\delta}\\right)$ for any $0<\\delta<\\min\\{\\beta-\\alpha,\\frac{1}{2}\\}$\n\nUpon having Theorem \\ref{general} at our disposal, the asymptotic estimation of averages $\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})$ satisfying conditions of that theorem becomes an automatized, but not uninteresting task, that involves each time a choice of parameters: a value for $\\delta$ and a precision value in order to obtain a rigorous estimation of some infinite products.\n\nIn general, we have freedom to choose the error term parameter $\\delta$ described in \\S\\ref{dirichlet} but some of them are not optimal. For instance, if $\\alpha=1$, then in terms of Theorem \\ref{general} and Lemma \\ref{SumEstimations}, $\\Delta_1^\\delta\\to\\infty$ as $\\delta\\to 0^{+}$. Since $\\overline{H}_f^{\\phantom{.}q}(-\\delta)$ converges, that makes the expression $\\Delta_1^\\delta\\overline{H}_f^{\\phantom{.}q}(-\\delta)$ tending to $\\infty$ as well, thus not providing a numerical acceptable value. On the other hand,  when $\\delta\\to\\frac{1}{2}^-$, the infinite product given by $\\overline{H}_f^{\\phantom{.}q}(-\\delta)$ tends to $\\infty$, whereas $\\Delta_1^\\delta\\to\\Delta_1^{\\frac{1}{2}}$, thus bounded, so that one also derives that the expression $\\Delta_1^\\delta\\overline{H}_f^{\\phantom{.}q}(-\\delta)$ becomes too big to be practical. The search looks for a value of $\\delta$ not too close to the boundaries of $(0,\\frac{1}{2})$, and in almost all cases it seems acceptable to set $\\delta=\\frac{1}{3}$. \n\nA natural question is whether or not we can improve on the error term estimation given in Theorem \\ref{general}, mandatorily with a different method, of exponent $\\delta=\\min\\{\\beta-1,\\alpha-\\frac{1}{2}\\}$.  \nWhen $\\beta-\\alpha>\\frac{1}{2}$, then $\\delta=\\alpha-\\frac{1}{2}$ and the answer to that question is given in \\S\\ref{improvement}: it is positive and constitutes our main result. We provide in addition explicit estimations for those \\textit{ critical exponents }.\n\nOut of the results above, the sum \\eqref{sum1:eq} is classical and it has been thoroughly studied by Ramar\\'e and Akhilesh in \\cite{RA13}, by Ramar\\'e in \\cite[Thm. 3.1]{RA19}, \\cite[Lemma 3.4]{RA95} and given in our simpler form by Helfgott in \\cite[\\S 6.1.1]{Hel19}. \n\n\n\n\\section{Improvements on the convolution method}\\label{improvement}\n\n\n \nDuring the proof of Theorem \\ref{general}, it was crucial to have an empty sum estimation for the inner sum given in \\eqref{DirichletIdentity} so that, thanks to the regularity on the variable $d$ we find convergent main and error term coefficients, as shown in \\eqref{DirichletIdentity:alpha=1} and \\eqref{DirichletIdentity:alphaneq1}.  \n\nThis general idea misses the fact that the function $h_f^q$ defined in \\eqref{hfq} vanishes on all non cube-free numbers, and that the particular function $h_f^q:p,(p,q)=1\\mapsto\\frac{1}{p^\\alpha}$, with $\\alpha>\\frac{1}{2}$, satisfies $h_f^q(p)=0$. Moreover, the fact that that particular function is meaningful only on the square of the prime numbers, will allow us to achieve the critical exponent $\\delta=\\frac{1}{2}$, if $\\alpha=1$ or $\\delta=\\alpha-\\frac{1}{2}$, if $\\alpha\\neq 1$ and $\\alpha>\\frac{1}{2}$, when $f$ is an arithmetic function satisfying the conditions of Theorem \\ref{general} with $\\beta-\\alpha>\\frac{1}{2}$.\n\n\n\n\\subsection{A particular case} \\label{particular}\n\n\n\nLet us see how we can improve the estimation $\\mathbf{(b)}$ given in Corollary \\ref{corollary}.\n\\begin{lemma}\\label{sum2:critic:1} \nLet $X>0$.\nThen\n\\begin{align}\\label{sum2:v1} \n\\sum_{\\ell\\leq X}\\frac{\\mu^2(\\ell)}{\\ell}&=\\frac{6}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_1\\right)+O^*\\left(\\frac{\\sage{Upper(CONSTANT1,digits)}}{\\sqrt{X}}\\right),\\\\\n\\label{sum2:v2}\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\ell}&=\\frac{4}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_2\\right)+O^*\\left(\\frac{\\sage{Upper(CONSTANT_ALT,digits)}}{\\sqrt{X}}\\right),\n\\end{align}  \nwhere $\\mathfrak{b}_1=\\gamma+\\sum_{p}\\frac{2\\log(p)}{p^2-1}=\\sage{C_sum2}\\ldots$, $\\mathfrak{b}_2=\\mathfrak{b}_1+\\frac{\\log(2)}{3}=\\sage{C_sum2_22}\\ldots$. \n\nIf we restraint ourselves to the range $X\\geq 1$,\nthen $\\sage{Upper(CONSTANT1,digits)}$ may be replaced by $\\sage{Upper(Ram_new_cst,2)}$ and $\\sage{Upper(CONSTANT_ALT,digits)}$ may be replaced by $\\sage{Upper(max(C2_v2_1,C2_v2_2),digits)}$.\n\\end{lemma} \n\n\\begin{proof} Equation \\eqref{sum2:eq} gives the  main term of \\eqref{sum2:v2} and from that, we can conclude by summation by parts that for all $X\\geq 1$, $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\ell}$ equals\n\\begin{align}\\label{eqq:v=2}\n\\frac{4(\\log(X)+\\mathfrak{b}_2)}{\\pi^2}+\\left(\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\mu^2(\\ell)-\\frac{4}{\\pi^2}X\\right)\\frac{1}{X}-\\int_X^{\\infty}\\left(\\sum_{\\substack{\\ell\\leq t\\\\(\\ell,2)=1}}\\mu^2(\\ell)-\\frac{4}{\\pi^2}t\\right)\\frac{dt}{t^2}.\n\\end{align}\nMoreover, by \\cite[Lemma 5.2]{Hel19}, we have \n\\begin{align}\n\\sup_{\\{X\\geq 1573\\}}\\frac{1}{\\sqrt{X}}\\left|\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\mu^2(\\ell)-\\frac{4}{\\pi^2}X\\right|&\\leq\\frac{9}{70}\\label{bound:1},\n\\end{align}\nso that, by \\eqref{eqq:v=2}, \n\\begin{align*}\n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\ell}=\\frac{4}{\\pi^2}(\\log(X)+\\mathfrak{b}_2)+O^*\\left(\\frac{27}{70}\\frac{1}{\\sqrt{X}}\\right),\\quad\\text{ if }X\\geq 1573,\n\\end{align*}\nwhere $\\frac{27}{70}=\\sage{Trunc(C2_v2_1,digits)}\\ldots$. We further verify by interval arithmetic that\n\\begin{align}\\label{intt:v=2}\n\\sup_{\\{1\\leq X\\leq 1573\\}}\\sqrt{X}\\left|\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{4}{\\pi^2}(\\log(X)+\\mathfrak{b}_2)\\right|\\leq\\sage{Upper(C2_v2_2,digits)}\n\\end{align} \nthe above upper bound being almost achieved when $X\\to 3^-$. On the other hand \\cite[Cor. 1.2]{RA19} tells us that\n\\begin{align}\n\\sup_{\\{X\\geq 1\\}}\\sqrt{X}\\left|\\sum_{\\ell\\leq X}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{6}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_1\\right)\\right|&\\leq\\sage{Upper(Ram_new_cst,2)}\\label{sum2:bound:X>=1}.\n\\end{align}\nHence, by using \\eqref{bound:1}, \\eqref{intt:v=2} and \\eqref{sum2:bound:X>=1}, when $v\\in\\{1,2\\}$, we have the bounds\n\\begin{align}\\label{BB}\n\\sup_{\\{X\\geq 1\\}}\\sqrt{X}\\left|\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{v}{\\kappa(v)}\\frac{6}{\\pi^2}(\\log(X)+\\mathfrak{b}_v)\\right|\\leq\n\\begin{cases}\n\\sage{Upper(Ram_new_cst,2)}&,\\quad\\text{ if }v=1,\\\\\n\\sage{Upper(max(C2_v2_1,C2_v2_2),digits)} &,\\quad\\text{ if }v=2.\n\\end{cases}&\n\\end{align} \nIn order to derive the result, it is sufficient to obtain bounds for \\eqref{BB} when $X\\in(0,1)$, in which case the above summation vanishes. By defining $Y=\\frac{1}{X}>1$ and $t_v:Y\\mapsto\\frac{6v(\\log(Y)-\\mathfrak{b}_v)}{\\kappa(v)\\pi^2\\sqrt{Y}}$, we need to find $\\sup_{\\{Y>1\\}}|t_v(Y)|$. By calculus, the function $t_v$ has a critical point at $y_0=e^{2+\\mathfrak{b}_v}$, with value $t_v(y_0)=\\frac{12v}{\\kappa(v)\\pi^2e^{1+\\frac{\\mathfrak{b}_v}{2}}}$, and it is monotonic in $[1,y_0]$ and in $[y_0,\\infty)$. As $\\lim_{Y\\to \\infty}t_v(Y)=0$ and $t_v(y_0)>0$, we conclude that $t_v$ is decreasing in $[y_0,\\infty)$. Similarly, as $t_v(1)=-\\frac{6v\\mathfrak{b}_v}{\\kappa(v)\\pi^2}<0$, $t_v$ is increasing in $[1,y_0]$. Therefore \n\\begin{align}\\label{BB(0,1)}\n\\sup_{\\{0<X<1\\}}\\sqrt{X}\\left|\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{v}{\\kappa(v)}\\frac{6}{\\pi^2}(\\log(X)+\\mathfrak{b}_v)\\right|=\\max\\{|t_v(1)|,|t_v(y_0)|\\}\\nonumber&\\\\\n=\\frac{6v\\mathfrak{b}_v}{\\kappa(v)\\pi^2}=\\begin{cases}\n\\sage{Upper(6*C_sum2\/pi^2,digits)}, &\\quad\\text{ if }v=1,\\\\\n\\sage{Upper(4*C_sum2_22\/pi^2,digits)},&\\quad\\text{ if }v=2.\n\\end{cases}&\n\\end{align} \nFinally, whenever either $v=1$ or $v=2$, the constant in the error term is obtained by taking the maximum between the bounds \\eqref{BB} and \\eqref{BB(0,1)}.\n\\end{proof}\n\n\\begin{lemma}\\label{sum2:critic}   \nLet $X>0$ and $\\alpha>\\frac{1}{2}$. If $\\alpha\\neq 1$, then\n\\begin{align*}\n\\sum_{\\ell\\leq X}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}&=\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+O^*\\left(\\frac{\\mathrm{E}_\\alpha^{(1)}}{X^{\\alpha-\\frac{1}{2}}}\\right),\\\\\n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,2)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}&=\\frac{2^\\alpha}{(2^{\\alpha}+1)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{4}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+O^*\\left(\\frac{\\sqrt{2}}{\\varphi_{\\frac{1}{2}}(2)}\\frac{\\mathrm{E}_\\alpha^{(2)}}{X^{\\alpha-\\frac{1}{2}}}\\right),\n\\end{align*}\nwhere, for $v\\in\\{1,2\\}$, we have\n\\begin{align*}\n\\mathrm{E}_\\alpha^{(v)}=\\max\\left\\{\\mathrm{D}_v\\left(1+\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\right),\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\left|\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\right|,\\phantom{xxx}\\right.&\\\\\n\\left.\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\left(\\frac{3\\kappa_\\alpha(v)\\zeta(2\\alpha)}{\\left(\\alpha-\\frac{1}{2}\\right)v^{\\alpha-1}\\kappa(v)\\pi^2|\\zeta(\\alpha)(\\alpha-1)|}\\right)^{\\frac{2}{\\alpha-1}}\\right\\}\\phantom{x}&\n\\end{align*}\nand $\\mathrm{D}_1=\\sage{Upper(Ram_new_cst,2)}$, $\\mathrm{D}_2=\\sage{Upper((1-1\/sqrt(2))*max(C2_v2_1,C2_v2_2),digits)}.$\n\nIf $X\\geq 1$, and $\\alpha\\neq 1$ then we can replace $\\mathrm{E}_\\alpha^{(v)}$ by $\\mathrm{D}_v\\left(1+\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\right)$. \n\\end{lemma}\n\n\\begin{proof} \nIf $X\\geq 1$, by summation by parts, we can write $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}$ as\n\\begin{align}\n&\\left(\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{v}{\\kappa(v)}\\frac{6\\left(\\log(X)+\\mathfrak{b}_v\\right)}{\\pi^2}\\right)\\frac{1}{X^{\\alpha-1}}-\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+\\nonumber\\\\\n&\\frac{v}{\\kappa(v)}\\frac{6(\\mathfrak{b}_v(\\alpha-1)+1)}{\\pi^2(\\alpha-1)}+(\\alpha-1)\\int_1^X\\left(\\sum_{\\substack{\\ell\\leq t\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{v}{\\kappa(v)}\\frac{6\\left(\\log(t)+\\mathfrak{b}_v\\right)}{\\pi^2}\\right)\\frac{dt}{t^{\\alpha}}.\\nonumber\\\\\\label{step:alpha}\n\\end{align}\nBy Theorem \\ref{general}, when $\\alpha>\\frac{1}{2}$, the main term in the asymptotic expression of the above summation is $\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{v}{\\kappa(v)}\\frac{6}{\n(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}\n$. By using Lemma \\ref{sum2:critic:1} and by making $X\\to\\infty$, we conclude from  \\eqref{step:alpha} that $\\frac{v}{\\kappa(v)}\\frac{6(\\mathfrak{b}(\\alpha-1)+1)}{\\pi^2(\\alpha-1)}+(\\alpha-1)\\int_1^\\infty\\left(\\sum_{\\substack{\\ell\\leq t\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell}-\\frac{6}{\\pi^2}\\left(\\log(t)+\\mathfrak{b}_v\\right)\\right)\\frac{dt}{t^{\\alpha}}=\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}$. Further, by equation \\eqref{BB}, we conclude that, for all $X\\geq 1$, $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}$ is equal to\n\\begin{align*}\n\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+O^*\\left(\\frac{\\sqrt{v}\\ \\mathrm{D}_v}{\\varphi_{\\frac{1}{2}}(v)}\\left(1+\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\right)\\frac{1}{X^{\\alpha-\\frac{1}{2}}}\\right), \n\\end{align*}\nwhere $\\mathrm{D}_1=\\sage{Upper(Ram_new_cst,2)}$ and $\\frac{\\varphi_{\\frac{1}{2}}(2)}{\\sqrt{2}}\\sage{Upper(max(C2_v2_1,C2_v2_2),digits)}\\leq\\mathrm{D}_2=\\sage{Upper((1-1\/sqrt(2))*max(C2_v2_1,C2_v2_2),digits)}$. \n\nSuppose now that $X\\in(0,1)$. Define $g:X>0\\mapsto\\frac{v^{\\alpha-1}\\kappa(v)\\pi^2\\zeta(\\alpha)(\\alpha-1)}{6\\kappa_\\alpha(v)\\zeta(2\\alpha)}X^{\\alpha-\\frac{1}{2}}$ $-\\sqrt{X}$. We have by \\cite[Cor. 1.14]{MV07} that $1<\\zeta(\\alpha)(\\alpha-1)<\\alpha$. If $\\alpha>1$, we derive that $\\frac{\\zeta(\\alpha)(\\alpha-1)}{\\zeta(2\\alpha)}>\\frac{1}{\\zeta(2)}$. As $\\frac{v^{\\alpha-1}\\kappa(v)}{\\kappa_\\alpha(v)}=\\frac{1+\\frac{1}{v}}{1+\\frac{1}{v^\\alpha}}>1$ we conclude that $g(1)>0$ and $g$ has a critical point $x_0$ satisfying $0<x_0^{\\alpha-1}=\\frac{3\\kappa_\\alpha(v)\\zeta(2\\alpha)}{\\left(\\alpha-\\frac{1}{2}\\right)v^{\\alpha-1}\\kappa(v)\\pi^2|\\zeta(\\alpha)(\\alpha-1)|}<1$, with value $g(x_0)=\\frac{1-\\alpha}{\\alpha-\\frac{1}{2}}\\sqrt{x_0}<0$. As $g(0)=0$, we conclude that if $\\alpha>1$, then $\\sup_{\\{0<X<1\\}}|g(X)|=\\max\\{g(1),|g(x_0)|\\}$.\n\nOn the other hand, if $\\frac{1}{2}<\\alpha<1$, then $2\\alpha-1<1$, $\\zeta(\\alpha)(\\alpha-1)<\\alpha<1<\\frac{\\zeta(2\\alpha)}{\\zeta(2)}$ and $\\frac{v^{\\alpha-1}\\kappa(v)}{\\kappa_\\alpha(v)}<1$, so that $g(1)<0$. Moreover, the critical point $x_0$ of $g$ satisfies $x_0^{1-\\alpha}<1$, so that $x_0<1$, and $g(x_0)>0$. Therefore, if $\\frac{1}{2}<\\alpha<1$, then $\\sup_{\\{0<X<1\\}}|g(X)|=\\max\\{|g(1)|,g(x_0)\\}$.\n\nAll in all, we derive\n\\begin{align}\n\\sup_{\\{0<X<1\\}}X^{\\alpha-\\frac{1}{2}}\\left|\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}-\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}+\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}\\right|&\\nonumber\\\\\n=\\frac{v}{\\kappa(v)}\\frac{6}{|\\alpha-1|\\pi^2}\\max\\{|g(1)|,|g(x_0)|\\},\\label{boundAlpha}&\n\\end{align}\nwhere,\n\\begin{align*}\n&\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\frac{v}{\\kappa(v)}\\frac{6|g(1)|}{|\\alpha-1|\\pi^2}=\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\left|\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\right|,\\\\\n&\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\frac{v}{\\kappa(v)}\\frac{6|g(x_0)|}{|\\alpha-1|\\pi^2}=\\\\\n&\\phantom{xxxxxxxxxxxxxx}\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\left(\\frac{3\\kappa_\\alpha(v)\\zeta(2\\alpha)}{\\left(\\alpha-\\frac{1}{2}\\right)v^{\\alpha-1}\\kappa(v)\\pi^2|\\zeta(\\alpha)(\\alpha-1)|}\\right)^{\\frac{2}{\\alpha-1}}.\\nonumber\n\\end{align*}\nThe result is obtained by defining $\\mathrm{E}_\\alpha^{(v)}, v\\in\\{1,2\\}$, as the maximum between $\\mathrm{D}_v\\left(1+\\frac{|\\alpha-1|}{\\alpha-\\frac{1}{2}}\\right)$ and the expression \\eqref{boundAlpha}.\n\\end{proof} \n \n\\begin{remark}\\label{newdef} With the goal of obtaining a similar error shape in Lemma \\ref{sum2:critic:1} to the one given in Lemma \\ref{sum2:critic}, we extend the definition of $\\mathrm{E}_\\alpha^{(v)}$, $v\\in\\{1,2\\}$ for $\\alpha>\\frac{1}{2}$, \n$\\alpha\\neq 1$, to the case $\\alpha=1$ by defining $\\mathrm{E}_1^{(1)}=\\sage{Upper(CONSTANT1,digits)}$ and, upon observing that $\\frac{\\varphi_{\\frac{1}{2}}(2)}{\\sqrt{2}}\\sage{Upper(4*C_sum2_22\/pi^2,digits)}\\leq\\sage{Upper((1-1\/sqrt(2)) * 4*C_sum2_22\/pi^2,digits)}$, defining $\\mathrm{E}_1^{(2)}=\\sage{Upper(CONSTANT2,digits)}$.\n\\end{remark}\n\n\\begin{lemma}\\label{seekfor}  Let $X>0$ and $q\\in\\mathbb{Z}_{>0}$. Then $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)}}\\frac{\\mu^2(\\ell)}{\\ell}$ equals\n\\begin{align*}\n\\frac{q}{\\kappa(q)}\\frac{6}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_q\\right)+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\mathrm{E}_1^{(1)}\\prod_{2|q}\\frac{\\mathrm{E}_1^{(2)}}{\\mathrm{E}_1^{(1)}}}{\\sqrt{X}}\\right), \n\\end{align*}\nwhere $\\mathfrak{b}_q$ is defined in Lemma \\ref{corollary} and, if $\\alpha>\\frac{1}{2}$, $\\alpha\\neq 1$, $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}$ equals\n\\begin{align*}\n\\frac{q^\\alpha}{\\kappa_{\\alpha}(q)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{q}{\\kappa(q)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\mathrm{E}_\\alpha^{(1)}\\prod_{2|q}\\frac{\\mathrm{E}_\\alpha^{(2)}}{\\mathrm{E}_\\alpha^{(1)}}}{X^{\\alpha-\\frac{1}{2}}}\\right),\n\\end{align*}\n where $\\mathrm{E}_\\alpha^{(v)}, v\\in\\{1,2\\}$, is defined as in Lemma \\ref{sum2:critic}.\n\\end{lemma}\n\\begin{proof} Proceed as in \\cite[Lemma 2.17]{MV07}. Define $\\mathcal{D}_{r}=\\{p\\text{ prime }, p|d\\implies p|r\\}\\subset\\mathbb{Z}_{\\geq 0}$. Consider $v\\in\\{1,2\\}$ and write $q=v^kr, k\\in\\mathbb{Z}_{>0}$, with $(v,r)=1$ (where, if $v=1$, then $k=0$). Then for all $s\\in\\mathbb{C}$ such that $\\Re(s)>1-\\alpha$, we have the identity \n\\begin{align*}\n\\sum_{\\substack{\\ell\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^{s+\\alpha}}=\\prod_{p|r}\\left(1+\\frac{1}{p^{s+\\alpha}}\\right)^{-1}\\cdot\\sum_{\\substack{\\ell\\\\(\\ell,v)=1}}\\frac{\\mu^2(\\ell)}{\\ell^{s+\\alpha}}=\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d^{s+\\alpha}}\\cdot\\sum_{\\substack{e\\\\(e,v)=1}}\\frac{\\mu^2(e)}{e^{s+\\alpha}},\n\\end{align*}\nwhere $\\lambda$ corresponds to the Liouville function:  the completely multiplicative function taking the value $-1$ at every prime number.\nHence\n\\begin{equation}\\label{nice}\n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}=\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d^\\alpha}\\sum_{\\substack{e\\leq\\frac{X}{d}\\\\(e,v)=1}}\\frac{\\mu^2(e)}{e^\\alpha},\n\\end{equation}\nwhich, as in Lemma \\ref{SumEstimations}, does not require the condition $\\{d\\leq X\\}$. We are considering thus an infinite range of values of $d$ for the above outer sum, which can be estimated as long as the inner sum is expressed asymptotically with an error term valid even when it has an empty condition plus the fact that the series of error terms for this expression, formed by the outer sum, converges.\n\nIf $\\alpha=1$, by using Lemma \\ref{sum2:critic:1} in \\eqref{nice}, we derive the same main term as the one given in Corollary \\ref{corollary} $\\mathbf{(b)}$, but a better error term magnitude, since $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell}$ can be written as \n\\begin{align*}\n&\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d}\\left(\\frac{6}{\\pi^2}\\frac{v}{\\kappa(v)}\\left(\\log\\left(\\frac{X}{d}\\right)+\\mathfrak{b}_v\\right)+O^*\\left(\\frac{\\sqrt{v}}{\\varphi_{\\frac{1}{2}}(v)}\\frac{\\mathrm{E}_{1} ^{(v)}\\sqrt{d}}{\\sqrt{X}}\\right)\\right)\\\\\n&=\\frac{vr}{\\kappa(vr)}\\frac{6}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_v\\right)-\\frac{v}{\\kappa(v)}\\frac{6}{\\pi^2}\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)\\log(d)}{d}\\\\\n&\\phantom{xxxxxxxxxxxxxxxxxxll}+O^*\\left(\\frac{\\sqrt{v}}{\\varphi_{\\frac{1}{2}}(v)}\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\mathrm{E}_{1} ^{(v)}}{\\sqrt{d}}\\cdot\\frac{1}{\\sqrt{X}}\\right)\\\\\n&=\\frac{q}{\\kappa(q)}\\frac{6}{\\pi^2}\\left(\\log(X)+\\mathfrak{b}_q\\right)+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\mathrm{E}_1^{(1)}\\prod_{2|q}\\frac{\\mathrm{E}_1^{(2)}}{\\mathrm{E}_1^{(1)}}}{\\sqrt{X}}\\right),\n\\end{align*}\nwhere we have used that\n\\begin{align*}\n\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{-\\lambda(d)\\log(d)}{d}=\\frac{r}{\\kappa(r)}\\left(\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d^s}\\right)^{-1}_{s=1}\\cdot\\left(\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d^s}\\right)'_{s=1}\\phantom{xxxxxxx}&\\\\\n=\\frac{r}{\\kappa(r)}\\sum_{p|r}\\left[\\left(\\left(1+\\frac{1}{p^s}\\right)^{-1}\\right)'\\left(1+\\frac{1}{p^s}\\right)\\right]_{s=1}=\\frac{r}{\\kappa(r)}\\sum_{p|r}\\frac{\\log(p)}{p+1},&\n\\end{align*}\nand that $\\frac{vr}{\\kappa(vr)}=\\frac{q}{\\kappa(q)}$, $\\frac{\\sqrt{vr}}{\\varphi_{\\frac{1}{2}}(vr)}=\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}$, $\\sum_{p|v}\\frac{\\log(p)}{p+1}+\\sum_{p|r}\\frac{\\log(p)}{p+1}=\\sum_{p|q}\\frac{\\log(p)}{p+1}$.\n\nFinally, if $\\alpha\\neq 1$, then by using Lemma \\ref{sum2:critic} in \\eqref{nice} and by noticing that $\\frac{(vr)^\\alpha}{\\kappa_\\alpha(vr)}=\\frac{q^\\alpha}{\\kappa_\\alpha(q)}$, we derive that $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}$ can be expressed as \n\\begin{align*}\n&\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d^\\alpha}\\left(\\frac{v^\\alpha}{\\kappa_\\alpha(v)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{v}{\\kappa(v)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{d^{\\alpha-1}}{X^{\\alpha-1}}+O^*\\left(\\frac{\\sqrt{v}}{\\varphi_{\\frac{1}{2}}(v)}\\frac{\\mathrm{E}_\\alpha^{(v)} d^{\\alpha-\\frac{1}{2}}}{X^{\\alpha-\\frac{1}{2}}}\\right)\\right)\\\\\n&=\\frac{q^\\alpha}{\\kappa_{\\alpha}(q)}\\frac{\\zeta(\\alpha)}{\\zeta(2\\alpha)}-\\frac{q}{\\kappa(q)}\\frac{6}{(\\alpha-1)\\pi^2}\\frac{1}{X^{\\alpha-1}}+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\mathrm{E}_\\alpha^{(1)}\\prod_{2|q}\\frac{\\mathrm{E}_\\alpha^{(2)}}{\\mathrm{E}_\\alpha^{(1)}}}{X^{\\alpha-\\frac{1}{2}}}\\right),\n\\end{align*}\nwhich, again, has the expected main term according to Theorem \\ref{general} but an error term of lower magnitude.\n\\end{proof}\n\nLet us recall that the requirement of the empty sum estimation, as in Lemma \\ref{SumEstimations}, worsens a bit the error term constants with respect to the ones under condition $X\\geq 1$, say, as shown in lemmas \\ref{sum2:critic:1} and \\ref{sum2:critic}, but we gain regularity in our expressions in the variable $d$. It is precisely that regularity that allows us to derive the coprimality restrictions products in a simpler manner: for example, we derive immediately that $\\sum_{\\substack{d\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d}=\\frac{r}{\\kappa(r)}$, whereas condition $\\frac{X}{d}\\geq 1$ would have imposed us to analyze $\\sum_{\\substack{d\\leq X\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d}$ or, rather, $\\sum_{\\substack{d>X\\\\d\\in\\mathcal{D}_r}}\\frac{\\lambda(d)}{d}$. This last observation is key for the work carried out in \\cite{RA13} and \\cite{RA19}. \n\n\\begin{corollary} Let $X>0$. Then\n\\begin{align*}\n\\sum_{\\substack{\\ell> X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^2}=\\frac{q}{\\kappa(q)}\\frac{6}{\\pi^2}\\frac{1}{X}+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\sage{Upper(E(2,1),digits)}}{X^{\\frac{3}{2}}}\\right),&\\quad\\text{ if }2\\nmid q,\\\\\n\\phantom{\\sum_{\\substack{\\ell> X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^2}}=\\frac{q}{\\kappa(q)}\\frac{6}{\\pi^2}\\frac{1}{X}+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\sage{Upper(E(2,2),digits)}}{X^{\\frac{3}{2}}}\\right),&\\quad\\text{ if }2|q.\n\\end{align*} \n\\begin{proof} By applying Lemma \\ref{seekfor} with $\\alpha=2$, we have\n\\begin{align*}\n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^2}&=\\frac{q^2}{\\kappa_{2}(q)}\\frac{\\zeta(2)}{\\zeta(4)}-\\frac{q}{\\kappa(q)}\\frac{6}{\\pi^2}\\frac{1}{X}+O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\frac{\\mathrm{E}_2^{(1)}\\prod_{2|q}\\frac{\\mathrm{E}_2^{(2)}}{\\mathrm{E}_2^{(1)}}}{X^{\\frac{3}{2}}}\\right),\n\\end{align*}\nwhere, for $v\\in\\{1,2\\}$, $\\mathrm{E}_2^{(v)}$ is defined as\n\\begin{align*}\n\\max\\left\\{\\frac{5\\ \\mathrm{D}_v}{3},\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\left|\\frac{v^2}{\\kappa_2(v)}\\frac{\\zeta(2)}{\\zeta(4)}-\\frac{v}{\\kappa(v)}\\frac{6}{\\pi^2}\\right|,\\frac{\\varphi_{\\frac{1}{2}}(v)}{\\sqrt{v}}\\frac{2}{3}\\left(\\frac{2\\kappa_2(v)\\zeta(4)}{v\\kappa(v)\\pi^2\\zeta(2)}\\right)^{2}\\right\\}&\\\\\n\\leq\\begin{cases}\n\\sage{Upper(E(2,1),digits)},&\\quad\\text{ if }v=1,\\\\\n\\sage{Upper(E(2,2),digits)},&\\quad\\text{ if }v=2.\n\\end{cases}&\n\\end{align*}\nWe obtain the result by observing that\n\\begin{equation*}\n\\sum_{\\substack{\\ell> X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^2}=\\frac{q^2}{\\kappa_{2}(q)}\\frac{\\zeta(2)}{\\zeta(4)}\n-\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^2}.\n\\end{equation*} \n\\end{proof} \n\\end{corollary}\n\n\n\n\\subsection{Achieving the critical exponent}\\label{achieving}  \n\n\n\nWe present a new method to achieve the critical exponent for estimations of averages of the form studied in Theorem \\ref{general} provided that the difference between $\\beta$ and $\\alpha$ defined therein is strictly bigger than $\\frac{1}{2}$: in this case, we go to the edge of the special form of the convolution method given in \\S\\ref{TCM}; moreover, no extra conditions on $\\beta$ are needed but $\\beta-\\alpha>\\frac{1}{2}$. Nonetheless, if $\\beta-\\alpha\\leq\\frac{1}{2}$, then we should still refer to Theorem \\ref{general} and its choice of parameter (or indirectly to it, as shown by summation by parts in Theorem \\ref{general++} $\\mathbf{(B)}$, $\\mathbf{(C)}$).   \n\n\\begin{theorem}\\label{general++}  \n  Let $X>0$, be a real number and $q$ a positive integer. Consider a multiplicative function $f:\\mathbb{Z}^+\\to\\mathbb{C}$ such that for every prime number $p$ satisfying $(p,q)=1$, we have $f(p)=\\frac{1}{p^{\\alpha}}+O\\left(\\frac{1}{p^{\\beta}}\\right)$, where $\\alpha$, $\\beta$ are real numbers satisfying $\\beta>\\alpha$, $\\beta-\\alpha>\\frac{1}{2}$. We have the following\n\n \\noindent $\\mathbf{(A)}$ If $\\alpha>\\frac{1}{2}$ then \n \\begin{align*}\n \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})=F_\\alpha^{q}(X)+O^*\\left(\\mathrm{p}_\\alpha(q)\\cdot\\frac{\\mathrm{w}_\\alpha^q\\ \\mathrm{P}_\\alpha}{X^{\\alpha-\\frac{1}{2}}}\\right),\n \\end{align*} \n where $F_\\alpha^q(X)$ is defined as in Theorem \\ref{general}, and, if $2|q$, $\\mathrm{w}_{\\alpha}^q=\\mathrm{E}_\\alpha^{(2)}$, whereas, if $2\\nmid q$, \n\\begin{align*}\n\\mathrm{w}_\\alpha^q&=\\left(\\frac{\\sqrt{2}-1}{\\sqrt{2}-1+|2^\\alpha f(2)-1|}\\right)\\left(\\mathrm{E}_\\alpha^{(1)}+\\frac{|2^\\alpha f(2)-1|\\ \\mathrm{E}_\\alpha^{(2)}}{\\varphi_{\\frac{1}{2}}(2)}\\right).\n\\end{align*}\nHere $\\mathrm{E}_\\alpha^{(v)}, v\\in\\{1,2\\}$ is defined in Lemma \\ref{sum2:critic} and Remark \\ref{newdef}, and we have \n\\begin{align*}\n\\mathrm{p}_\\alpha(q)=\\prod_{p|q}\\left(1+\\frac{1-|f(p)p^\\alpha-1|}{\\sqrt{p}-1+|f(p)p^{\\alpha}-1|}\\right),\\mathrm{P}_\\alpha=\\prod_{p}\\left(1+\\frac{|f(p)p^\\alpha-1|}{\\sqrt{p}-1}\\right),\n \\end{align*}    \nfor all $\\alpha$, where $\\mathrm{P}_\\alpha$ is a convergent infinite product. \n\n \\noindent $\\mathbf{(B)}$ If $\\alpha<\\frac{1}{2}$ then $ \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})$ can be expressed as\n \\begin{equation*}\n\\frac{H_{f'}^q(0)\\varphi(q)}{(1-\\alpha)q}X^{1-\\alpha}+O^*\\left(\\mathrm{p}_{\\alpha}(q)\\cdot\\left(1+\\frac{2-2\\alpha}{1-2\\alpha}\\right)\\mathrm{w'}_{\\alpha}^q\\mathrm{P}_{\\alpha} \\ X^{\\frac{1}{2}-\\alpha}\\right),\n \\end{equation*} \nwhere $\\mathrm{p}_{\\alpha}(q)$ and $\\mathrm{P}_{\\alpha}$ are as in $\\mathbf{(A)}$ and for $\\alpha\\leq\\frac{1}{2}$,\n \\begin{align*}\nH_{f'}^q(0)&=\\prod_{p\\nmid q}\\left(1-\\frac{p^{1-\\alpha}-f(p)p+f(p)}{p^{2-\\alpha}} \\right),\\\\\n\\mathrm{w'}_{\\alpha}^q&= \n\\begin{cases}\\mathrm{E}_1^{(2)}=\\sage{Upper(CONSTANT2,digits)},&\\quad\\text{ if }2|q,\\\\\n\\left(\\frac{\\sqrt{2}-1}{\\sqrt{2}-1+|2^\\alpha f(2)-1|}\\right)\\left(\\mathrm{E}_1^{(1)}+\\frac{|2^\\alpha f(2)-1|\\ \\mathrm{E}_1^{(2)}}{\\varphi_{\\frac{1}{2}}(2)}\\right),&\\quad\\text{ if }2\\nmid q.\\\\\n\\end{cases}\n\\end{align*} \n\n \\noindent $\\mathbf{(C)}$ If $\\alpha=\\frac{1}{2}$ then $\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})$ can be written as\n \\begin{equation*}\n\\frac{H_{f'}^q(0)\\varphi(q)}{(1-\\alpha)q}X^{1-\\alpha}+O^*\\left(\\mathrm{C}+\\mathrm{p}_{\\alpha}(q)\\mathrm{w'}_{\\alpha}^q\\mathrm{P}_{\\alpha} \\ \\left(1+\\frac{1}{2}\\log(X)\\right)\\right),\n \\end{equation*} \nwhere $\\mathrm{p}_{\\alpha}(q)$ and $\\mathrm{P}_{\\alpha}$ are as in $\\mathbf{(A)}$, $H_{f'}^q(0)$ and $\\mathrm{w'}_{\\alpha}^q$ are as in $\\mathbf{(B)}$ and\n\\begin{align*}\n\\mathrm{C}&=\\left|\\frac{H_{f'}^{q}(0)\\varphi(q)}{q}\\left(\\sum_{p\\nmid q}\\frac{\\log(p)(\\sqrt{p}-(p-2)f(p))}{(f(p)+\\sqrt{p})(p-1)}+\\gamma+\\sum_{p|q}\\frac{\\log(p)}{p-1}-2\\right)\\right|.\n\\end{align*}\n \\end{theorem}\n\n\\begin{proof} Let us derive $\\mathbf{(A)}$. Consider the arithmetic function $i_f$ defined on each prime as $p\\mapsto f(p)p^\\alpha-1$. Observe that  \n\\begin{align}\n \\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})=\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}\\cdot f(\\ell)\\ell^{\\alpha}=\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}\\cdot \\prod_{p|\\ell}(1+i_f(p))&\\nonumber\\\\\n=\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\ell^\\alpha}\\sum_{d|\\ell}\\mu^2(d)i_f(d)=\\sum_{\\substack{d\\\\(d,q)=1}}\\frac{\\mu^2(d)i_f(d)}{d^{\\alpha}}\\sum_{\\substack{e\\leq\\frac{X}{d}\\\\(e,qd)=1}}\\frac{\\mu^2(e)}{e^\\alpha}&\\label{followup},\n \\end{align} \nwhere we have not imposed upper bound conditions on the variable $d$. \n\nIn order to continue our estimation, we must be able to estimate the innermost summation in \\eqref{followup} regardless of whether or not it has an empty condition, so that their remainder terms converge upon effecting the corresponding outermost summation. As $\\alpha>\\frac{1}{2}$, this situation can be treated with the help of Lemma \\ref{seekfor}; we distinguish two cases.\n\n\\noindent $\\mathbf{i)}$ $2|q$. Then continuing from \\eqref{followup}, along with the ideas of the proof of Theorem \\ref{general} and Lemma \\ref{seekfor}, it is not difficult to see, as expected, that for all $\\alpha>\\frac{1}{2}$, the main term of $\\sum_{\\substack{\\ell\\leq X\\\\ (\\ell,q)=1}}\\mu^2(\\ell)f({\\ell})$ is $F_\\alpha^q(X)$. As for the error term, it corresponds to\n\\begin{align*}\\sum_{\\substack{d\\\\(d,q)=1}}\\frac{\\mu^2(d)|i_f(d)|}{d^\\alpha}O^*\\left(\\frac{\\sqrt{qd}}{\\varphi_{\\frac{1}{2}}(qd)}\\frac{\\mathrm{E}_\\alpha^{(2)}\\ d^{\\alpha-\\frac{1}{2}}}{X^{\\alpha-\\frac{1}{2}}}\\right)\\phantom{xxxxxxxxx}&\\\\\n=O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\prod_{p\\nmid q}\\left(1+\\frac{|i_f(p)|}{\\sqrt{p}-1}\\right)\\cdot\\frac{\\mathrm{E}_\\alpha^{(2)}}{X^{\\alpha-\\frac{1}{2}}}\\right),&\n\\end{align*}\nwhere, for any $\\alpha>\\frac{1}{2}$, $\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\prod_{p\\nmid q}\\left(1+\\frac{|i_f(p)|}{\\sqrt{p}-1}\\right)$ may be expressed as\n\\begin{align*}\n\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\prod_{p|q}\\left(1+\\frac{|f(p)p^\\alpha-1|}{\\sqrt{p}-1}\\right)^{-1}\\cdot\\mathrm{P}_\\alpha=\\mathrm{p}_\\alpha(q)\\cdot\\mathrm{P}_\\alpha.\n\\end{align*}\nwhere $\\mathrm{p}_\\alpha(q)$ and  $\\mathrm{P}_\\alpha$ are defined in the statement. Observe that $\\mathrm{P}_\\alpha$ converges, as $\\frac{|i_f(p)|}{\\sqrt{p}-1}=\\frac{|f(p)p^\\alpha-1|}{\\sqrt{p}-1}=O\\left(\\frac{1}{p^{\\beta-\\alpha+\\frac{1}{2}}}\\right)$ and $\\beta-\\alpha+\\frac{1}{2}>1$.\n\n\\noindent $\\mathbf{ii)}$ $2\\nmid q$. Then we can write \\eqref{followup} as\n\\begin{align*}\n &\\sum_{\\substack{d\\\\(d,2q)=1}}\\frac{\\mu^2(d)i_f(d)}{d^{\\alpha}}\\sum_{\\substack{e\\leq\\frac{X}{d}\\\\(e,qd)=1}}\\frac{\\mu^2(e)}{e^\\alpha}+\\frac{i_f(2)}{2^\\alpha}\\sum_{\\substack{d\\\\(d,2q)=1}}\\frac{\\mu^2(d)i_f(d)}{d^{\\alpha}}\\sum_{\\substack{e\\leq\\frac{X}{2d}\\\\(e,2qd)=1}}\\frac{\\mu^2(e)}{e^\\alpha}\\\\\n&=S_\\alpha^q(X)+\\frac{i_f(2)}{2^\\alpha}T_\\alpha^q(X).\n\\end{align*}\nAgain, it is not difficult to see that, for any $\\alpha>\\frac{1}{2}$, the main term of $S_\\alpha^q(X)+\\frac{i_f(2)}{2^\\alpha}T_\\alpha^q(X)$ is $F_\\alpha^q(X)$, defined in Theorem \\ref{general}. On the other hand, the error term of $S_1^q(X)+\\frac{i_f(2)}{2}T_1^q(X)$, it can be expressed as\n\\begin{align*}\n &\\sum_{\\substack{d\\\\(d,2q)=1}}\\frac{\\mu^2(d)|i_f(d|)}{d^{\\alpha}}O^*\\left(\\frac{\\sqrt{qd}}{\\varphi_{\\frac{1}{2}}(qd)}\\frac{\\mathrm{E}_{\\alpha}^{(1)}\\ d^{\\alpha-\\frac{1}{2}}}{X^{\\alpha-\\frac{1}{2}}}\\right)\\\\\n&\\phantom{xxxx}+\\frac{|i_f(2)|}{2^{\\alpha}}\\sum_{\\substack{d\\\\(d,2q)=1}}\\frac{\\mu^2(d)|i_f(d)|}{d^{\\alpha}}O^*\\left(\\frac{\\sqrt{2qd}}{\\varphi_{\\frac{1}{2}}(2qd)}\\frac{\\mathrm{E}_{\\alpha}^{(2)}\\ (2d)^{\\alpha-\\frac{1}{2}}}{X^{\\alpha-\\frac{1}{2}}}\\right)=\\\\\n&O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\prod_{p\\nmid 2q}\\left(1+\\frac{|i_f(p)|}{\\sqrt{p}-1}\\right)\\left(\\mathrm{E}_\\alpha^{(1)}+\\frac{|i_f(2)|\\ \\mathrm{E}_\\alpha^{(2)}}{\\varphi_{\\frac{1}{2}}(2)}\\right)\\cdot\\frac{1}{X^{\\alpha-\\frac{1}{2}}}\\right)=\\\\\n&O^*\\left(\\mathrm{p}_\\alpha(q)\\left(\\frac{\\sqrt{2}-1}{\\sqrt{2}-1+|2^\\alpha f(2)-1|}\\right)\\left(\\mathrm{E}_\\alpha^{(1)}+\\frac{|2^\\alpha f(2)-1|\\ \\mathrm{E}_\\alpha^{(2)}}{\\varphi_{\\frac{1}{2}}(2)}\\right)\\cdot\\frac{\\mathrm{P}_\\alpha}{X^{\\alpha-\\frac{1}{2}}}\\right),\n\\end{align*}\nwhence the first case.\n\nCondition $\\alpha>\\frac{1}{2}$ in the case $\\mathbf{(A)}$ is necessary, as we have used Lemma \\ref{seekfor}. Nonetheless, we can readily derive an analogous result for the cases $\\mathbf{(B)}$  and $\\mathbf{(C)}$. Indeed, we can write $f(p)=p^{1-\\alpha}f'(p)$, where $A(t)=\\sum_{\\substack{\\ell\\leq t\\\\(\\ell,q)=1}}\\mu^2(\\ell)f'(\\ell)$ can be estimated by the case $\\mathbf{(A)}$ with $\\alpha'=1$, $\\beta'=1-\\alpha+\\beta$. We can then estimate $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\mu^2(\\ell)f(\\ell)=\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\mu^2(\\ell)f'(\\ell)\\ell^{1-\\alpha}$ by means of a summation by parts, obtaining the result.\n\\end{proof}\n\nNote that the error term improvement from Theorem \\ref{general}, when $\\alpha=\\frac{1}{2}$ and under conditions of Theorem \\ref{general++}, is of logarithmic nature with respect to $O(X^{\\frac{1}{2}-\\delta})$ for any $\\delta\\in(0,\\frac{1}{2})$.\n\nConcerning the error term in Theorem \\ref{general++}, in some particular cases one can do much better in terms of error constants. For instance, it is known, by  \\cite[Lemmas 5.1-5.2]{Hel19}\nthat if $f(p)=1$ and $v\\in\\{1,2\\}$, we have that for any $X>0$ that\n\\begin{equation}\\label{squarefree}  \n\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,v)=1}}\\mu^2(\\ell)=\\frac{6}{\\pi^2}\\frac{v}{\\kappa(v)}X+O^*(\\mathrm{H}_v\\sqrt{X}),\n\\end{equation}\nwhere\n\\begin{equation}\\mathrm{H}_v=\n\\begin{cases}\n\\sqrt{3}\\left(1-\\frac{6}{\\pi^2}\\right)&\\quad\\text{ if }v=1,\\label{HH2}\\\\\n1-\\frac{4}{\\pi^2}&\\quad\\text{ if }v=2,\n\\end{cases}\n\\end{equation}\nwhereas Corollary \\ref{general++}  provides only an explicit error term of the form $O^*\\left(\\frac{\\sqrt{q}}{\\varphi_{\\frac{1}{2}}(q)}\\cdot\\sage{Upper(cst_crucial*3,digits)}\\sqrt{X}\\right)$.\n\n\n\n\\subsection{Consequences}\\label{Cop}  \n\n\n\n\\begin{lemma}\\label{consequences}\nLet $X>0$, then the sum $\\sum_{\\substack{\\ell\\leq X\\\\(\\ell,q)=1}}\\frac{\\mu^2(\\ell)}{\\varphi(\\ell)}$ may be estimated as\n\\begin{equation}\\label{summi} \n\\frac{\\varphi(q)}{q}\\left(\\log\\left(X\\right)+\\mathfrak{a}_q\\right)+O^*\\left(\\prod_{p|q}\\left(1+\\frac{p-2}{p^{\\frac{3}{2}}-p-\\sqrt{p}+2}\\right)\\cdot\\frac{\\sage{Upper(CONSTANT_RAM*Prod_Ram_Upper,digits)}\\prod_{2|q}\\sage{Upper(CONSTANT2\/CONSTANT_RAM,digits)}}{\\sqrt{X}}\\right),\n\\end{equation}\nwhere $\\mathfrak{a}_q$ is defined in Corollary \\ref{corollary}.\n\\end{lemma}\n\\begin{proof} We already know the main term of the asymptotic expression of the above sum, thanks to Corollary \\ref{corollary} $\\mathbf{(a)}$; obtaining it again from Theorem \\ref{general++} is an exercise. On the other hand, by Theorem \\ref{general++} with $f(p)=\\frac{1}{p-1}$, $\\alpha=1$, $\\beta=2$, its error term can be expressed as $O^*\\left(\\mathrm{p}(q)\\cdot\\frac{\\mathrm{w}^q\\  \\mathrm{P}}{\\sqrt{X}}\\right)$, where\n\\begin{align*}\n\\mathrm{p}(q)&=\\prod_{p|q}\\left(1+\\frac{p-2}{p^{\\frac{3}{2}}-p-\\sqrt{p}+2}\\right),\\\\\n \\mathrm{P}&=\\prod_{p}\\left(1+\\frac{1}{(p-1)(\\sqrt{p}-1)}\\right)\\in[\\sage{Trunc(Prod_Ram_Lower,dlong)},\\sage{Trunc(Prod_Ram_Upper,dlong)}],\\\\\n\\mathrm{w}^q&=\\begin{cases}\n\\sage{Trunc(CONSTANT2,digits)},&\\text{ if }2|q,\\\\\n\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\left(\\mathrm{E}_1^{(1)}+\\frac{\\mathrm{E}_1^{(2)}}{\\varphi_{\\frac{1}{2}}(2)}\\right)=\\sage{Trunc(CONSTANT_RAM,digits)}\\ldots,&\\text{ if }2\\nmid q\n\\end{cases}\\leq\\sage{Upper(CONSTANT_RAM,digits)}\\prod_{2|q}\\sage{Upper(CONSTANT2\/CONSTANT_RAM,digits)},\n\\end{align*} \nand where $\\mathrm{E}_1^{(v)}$, $v\\in\\{1,2\\},$ is defined in \\S\\ref{particular}.\n\\end{proof}\n\nWhen there is no coprimality conditions, we have obtained an error constant equal to $\\sage{Upper(CONSTANT_RAM*Prod_Ram_Upper,digits)}$, that held under condition $X>0$. Ramar\\'e and Akhilesh in \\cite[Thm. 1.2]{RA13} have given the constant $3.95$ under the condition $X\\geq 1$, later improved by Ramar\\'e himself in \\cite{RA19} to $2.44$ under the condition $X>1$. From these last two bounds, it is not difficult to extend the range of estimation to $X>0$, as we have done for example throughout Lemma \\ref{SumEstimations}, and these bounds continue to be better than the value $\\sage{Upper(CONSTANT_RAM*Prod_Ram_Upper,digits)}$.\n\nNonetheless, the above lemma improve considerably \\cite[Thm. 1.1]{RA13} when coprimality conditions given by $q\\geq 2$ are involved. For example, we have\n\\begin{align}\\label{values}\n\\sage{Upper(crux(v0)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v0})\\leq\\sage{Upper(pp(v0)*crux(v0)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v0),digits)}\\leq 5.9\\cdot j(\\sage{v0}),\\nonumber\\\\\n\\sage{Upper(crux(v1)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v1})\\leq\\sage{Upper(pp(v1)*crux(v1)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v1),digits)}\\leq 5.9\\cdot j(\\sage{v1}),\\nonumber\\\\\n\\sage{Upper(crux(v2)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v2})\\leq\\sage{Upper(pp(v2)*crux(v2)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v2),digits)}\\leq 5.9\\cdot j(\\sage{v2}),\\\\\n\\sage{Upper(crux(v1_2)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v1_2})\\leq\\sage{Upper(pp(v1_2)*crux(v1_2)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v1_2),digits)}\\leq 5.9\\cdot j(\\sage{v1_2}),\\nonumber\\\\\n\\sage{Upper(crux(v3_2)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v3_2})\\leq\\sage{Upper(pp(v3_2)*crux(v3_2)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v3_2),digits)}\\leq 5.9\\cdot j(\\sage{v3_2}),\\nonumber\\\\\n\\sage{Upper(crux(v7_2)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(\\sage{v7_2})\\leq\\sage{Upper(pp(v7_2)*crux(v7_2)*Prod_Ram_Upper,digits)}&\\leq\\sage{Lower(5.9*ramp(v7_2),digits)}\\leq 5.9\\cdot j(\\sage{v7_2}),\\nonumber\n\\end{align}\nwhere $j$ is the error term arithmetic function defined in \\cite[Thm. 1.1]{RA13} as $2\\mapsto\\frac{21}{25}$ and $p\\geq 3\\mapsto 1+\\frac{p-2}{p^{\\frac{3}{2}}-\\sqrt{p}+1}$. Furthermore, the estimation given in Lemma \\ref{consequences} is better than the one in \\cite[Thm. 1.1]{RA13} for all $q=p$ prime. Indeed, we observe in \\eqref{values} that it is better when $p\\in\\{2,3,5\\}$; now, since\n\\begin{align*}\n\\frac{p-2}{p^{\\frac{3}{2}}-p-\\sqrt{p}+2}&<\\frac{1}{\\sqrt{p}}&&\\text{ for all }p\\geq 3,\\\\\n\\frac{p-2}{p^{\\frac{3}{2}}-\\sqrt{p}+1}&>\\frac{1}{2\\sqrt{p}}&&\\text{ for all }p\\geq 5,\n\\end{align*}\nwe have, for all $p\\geq 3$, that\n\\begin{align*}\n\\sage{Upper(crux(v1)*Prod_Ram_Upper,digits)}\\cdot\\mathrm{p}(p)\\leq\\sage{Upper(crux(v1)*Prod_Ram_Upper,digits)}\\cdot\\left(1+\\frac{1}{\\sqrt{p}}\\right)\\leq5.9\\cdot\\left(1+\\frac{1}{2\\sqrt{p}}\\right)\\leq 5.9\\cdot j(p),\n\\end{align*}\nwhence the conclusion.\n\n\nAs a final remark, observe that that the main contribution to the product $\\mathrm{P}$ given in Lemma \\ref{consequences} is precisely when $p=2$. This is the reason why, in the present work, we have distinguished if $q$ is either odd or even. Further, as the second main contribution to the product $\\mathrm{P}$ is given by its factor at $p=3$ (the subsequent factors when $p>3$ being rather small, as $\\frac{1}{\\sqrt{p}-1}<1$), the interested reader may study the behavior of the error term bounds given in Theorem \\ref{general++}, and therefore the error term in Lemma \\ref{consequences}, by distinguishing whether or not $(6,q)=1$: this procedure will require an extension of Lemma \\ref{sum2:critic:1} to the cases $(3,q)=1$ and, by using the inclusion-exclusion principle, to the case $(6,q)=1$; afterwards, the analysis will continue exactly as in the current version of Theorem \\ref{general++}. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction} \\label{sec:introduction}\nMechanical search -- extracting a desired object from a heap of objects -- is a fundamental task for robots in unstructured e-commerce warehouse environments or for robots in home settings. It remains challenging due to uncertainty in perception and actuation as well as lack of models for occluded objects in the heap. \n\n\n\nData-driven methods are promising for grasping unknown objects in clutter and bin picking~\\cite{mahler2019learning,pinto2016supersizing,kalashnikov2018qt,morrison2018closing,gualtieri2016high}, and can reliably plan grasps on the most accessible object without semantic knowledge of the target object. Some reinforcement learning~\\cite{yang2019deep,jang2017end} or hierachical~\\cite{danielczuk2019mechanical} mechanical search policies use semantics, but have so far been limited to specific objects or heuristic policies. \n\nIn this paper, we draw on recent work on shape completion to reason about occluded objects~\\cite{varley2017shape,price2019inferring} and work on predicting multiple pose hypotheses~\\cite{manhardt2018explaining,rupprecht2017learning}. X-Ray combines occlusion inference and hypothesis predictions to estimate an occupancy distribution for the bounding box most similar to the target object to estimate likely poses -- translations and rotations in the image plane. X-Ray can efficiently extract the target object from a heap where it is fully occluded or partially occluded (Figure~\\ref{fig:splash}).\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{splash-v4.png}\n\\caption{Mechanical search with a fully occluded target object (top row) and a partially occluded target object (bottom row). We predict the target object occupancy distribution, which depends on the target object's visibility and the heap (second column). Each pixel value in the distribution image corresponds to the likelihood of that pixel containing part of the target object. X-Ray plans a grasp on the object that minimizes the estimated support of the resulting occupancy distribution to minimize the number of actions to extract the target object. We show two nearly-identical heaps; in the fully occluded case, X-Ray grasps the mustard bottle whereas in the partially occluded case, the policy grasps the face lotion (third column), resulting in the respective next states (fourth column).}\n\\label{fig:splash}\n\\vspace{-8pt}\n\\end{figure}\n\n\nThis paper provides four contributions:\n\\begin{enumerate}\n\\item X-Ray (maXimize Reduction in support Area of occupancY distribution): a mechanical search policy that minimizes support of learned occupancy distributions.\n\\item An algorithm for estimating target object occupancy distributions using a set of neural networks trained on a dataset of synthetic images that transfers seamlessly to real images.\n\\item A synthetic dataset generation method and 100,000 RGBD images of heaps labeled with occupancy distributions for a single partially or fully occluded target object, constructed for transfer to real images.\n\\item Experiments comparing the mechanical search policy against two baselines in 1,000 simulated and 20 physical heaps that suggest the policy can reduce the median number of actions needed to extract the target object by 20\\% with a simulated success rate of 87\\% and physical success rate of 100\\%.\n\\end{enumerate}\n\n\\section{Related Work} \\label{sec:relwork}\n\n\\subsection{Pose Hypothesis Prediction}\nThere is a substantial amount of related work in computer vision on 3D and 6D pose prediction of both known and unknown objects in RGB, depth, and RGBD images~\\cite{kehl2017ssd,xiang2017posecnn,li2018deepim,hinterstoisser2012model}. Many of these papers assume that the target objects are either fully visible or have minor occlusions.  In addition, many assume that there is no ambiguity in object pose due to self-occlusion or rotational symmetry of the object, as these factors can significantly decrease performance for neural network-based approaches~\\cite{corona2018pose}. Recent work has attempted to address the pose ambiguity that results from object geometry or occlusions by restricting the range of rotations~\\cite{rad2017bb8} predicting multiple hypotheses for each detected object~\\cite{rupprecht2017learning, manhardt2018explaining}. \\citet{rupprecht2017learning} find that refining multiple pose hypotheses to a 6D prediction outperforms single hypothesis predictions on a variety of vision tasks, such as human pose estimation, object classification, and frame prediction. \\citet{manhardt2018explaining} note that directly regressing to a rotation for objects with rotational symmetries can result in an averaging effect where the predicted pose does not match any of the possible poses; thus, they predict multiple pose hypotheses for objects with pose ambiguities to better predict the underlying pose and show Bingham distributions of the predicted hypotheses. However, only minor occlusions are considered and since ground truth pose distributions are not available for these images and objects, comparisons for continuous distributions can only be made qualitatively. Predicting multiple hypotheses or a distribution to model ambiguity has also been applied to gaze prediction from facial images~\\cite{prokudin2018deep}, segmentation~\\cite{kohl2018probabilistic}, and monocular depth prediction~\\cite{yang2019inferring}. In contrast to these works, we learn occupancy distributions in a supervised manner.\n\n\\subsection{Object Search}\nThere has been a diverse set of approaches to grasping in cluttered environments, including methods that use geometric knowledge of the objects in the environment to perform wrench-based grasp metric calculations, nearest-neighbor lookup in a precomputed database, or template matching~\\cite{berenson2008grasp,moll2017randomized,mahler2016dex}, as well as methods using only raw sensor data~\\cite{katz2014perceiving,saxena2008learning}, commonly leveraging convolutional neural networks~\\cite{kalashnikov2018qt,jang2017end,lenz2015deep}. While multi-step bin-picking techniques have been studied, they do not take a specific target object into account~\\cite{mahler2017learning}.\n\n\\citet{kostrikov2016end} learn a critic-only reinforcement learning policy to push blocks in a simulated environment to uncover an occluded MNIST block. \\citet{zeng2018learning} train joint deep fully-convolutional neural networks to predict both pushing and grasping affordances from heightmaps of a scene containing multicolored blocks, then show that the resulting policy (VPG) can separate and grasp novel objects in cluttered heaps. The policy can be efficiently trained on both simulated and physical systems, and can quickly learn elegant pushes to expand the set of available grasps in the scene. \\citet{yang2019deep} train similar grasping and pushing networks as well as separate explorer and coordinator networks to address the exploration\/exploitation tradeoff for uncovering a target object. Their policy learns to push through heaps of objects to find the target and then coordinate grasping and pushing actions to extract it, outperforming a target-centered VPG baseline in success rate and number of actions. Both approaches can generalize to objects outside the training distribution, although they are evaluated on a limited set of novel objects, and Yang \\textit{et al.} separate the cases where the target object is partially occluded and fully occluded. Additionally, we focus only on grasping actions, as some mechanical search environments may be constrained or objects may be fragile.\n\nRecently, several approaches to the mechanical search problem have been proposed, both in tabletop and bin picking environments. \\citet{price2019inferring} propose a shape completion approach that predicts occlusion regions for objects to guide exploration in a tabletop scene, while \\citet{xiao2019online} implement a particle filter approach and POMDP solver to attempt to track all visible and occluded objects in the scene. However, 75\\% of the objects in Price \\textit{et al.}'s evaluation scenes are seen in training and Xiao \\textit{et al.}'s method requires models of each of the objects in the scene. We benchmark our policy on a variety of non-rigid, non-convex household objects not seen in training and require no object models. In previous work, \\citet{danielczuk2019mechanical} proposed a general mechanical search problem formulation and introduced a two-stage perception and search policy pipeline. In contrast, we introduce a novel perception network and policy based on minimizing support of occupancy distributions that outperforms the methods introduced in~\\cite{danielczuk2019mechanical}.\n\\section{Problem Statement} \\label{sec:problem}\nWe consider an instance of the mechanical search problem where a robot must extract a known target object from a heap of unknown objects by iteratively grasping to remove non-target objects. The objective is to extract the target object using the fewest number of grasps.\n\n\\subsection{Assumptions}\n\\begin{itemize}\n    \\item One known target object, fully or partially occluded by unknown objects in a heap on a planar workspace.\n    \\item A robot with a gripper, an overhead RGBD sensor with known camera intrinsics and pose relative to the robot.\n    \\item A maximum of one object is grasped per timestep.\n    \\item A target object detector that can return a binary mask of visible target object pixels when queried.\n\\end{itemize}\n\n\\begin{figure*}[th!]\n\\vspace{1.5mm}\n\\includegraphics[width=\\textwidth]{dataset_generation-v2.png}\n\\caption{Training dataset generation for learning the occupancy distribution function. Each dataset image is generated by sampling $N = 14$ object models from a dataset of 1296 CAD models. The target object (colored red) is dropped, followed by the $N$ other objects (colored gray), into a planar workspace using dynamic simulation. Camera intrinsics and pose are sampled from uniform distributions centered around their nominal values and an RGBD image is rendered of the scene. The augmented depth image (top right), consisting of a binary target object modal mask and a two-channel depth image, is the only input used for training for seamless transfer from simulation to real images. The ground truth target object distribution is generated by summing all shifted amodal target object masks whose modal masks correspond with the target object modal mask.}\n\\label{fig:datagen}\n\\vspace{-8pt}\n\\end{figure*}\n\n\\subsection{Definitions} \\label{subsec:defs}\nWe define the problem as a partially-observable Markov decision process (POMDP) with the 7-tuple $(S, A, T, R, \\Omega, O, \\gamma)$ and a maximum horizon $H$:\n\\begin{itemize}\n    \\item \\textbf{States} $(S)$: A state $\\mathbf{s}_k$ at timestep $k$ consists of the robot, a static overhead RGBD camera, and a static bin containing $N+1$ objects, target object $\\mathcal{O}_t$ and distractor objects $\\lbrace \\mathcal{O}_{1,k}, \\mathcal{O}_{2,k}, \\ldots, \\mathcal{O}_{N,k}\\rbrace$. No prior information is known about the $N$ distractor objects. \n    \\item \\textbf{Actions} $(A)$: A grasp action $\\mathbf{a}_k$ at timestep $k$ executed by the robot's gripper.\n    \\item \\textbf{Transitions} $(T)$: In simulation, the transition model $T(\\mathbf{s}_{k+1} \\ | \\ \\mathbf{a}_k, \\mathbf{s}_k)$ is equivalent to that used by Mahler et al.~\\cite{mahler2017learning} and uses pybullet~\\cite{coumans2017bullet} for dynamics. On the physical system, next states are determined by executing the action on a physical robot and waiting until objects come to rest.\n    \\item \\textbf{Rewards} $(R)$: The reward $r_k = R(\\mathbf{s}_k, \\mathbf{a}_k, \\mathbf{s}_{k+1}) \\in \\lbrace 0, 1 \\rbrace$ is 1 if the target object is successfully grasped and lifted from the bin, otherwise the reward is 0.\n    \\item \\textbf{Observations} $(\\Omega)$: An observation $\\mathbf{y}_k \\in \\mathbb{R}_+^{h \\times w \\times 4}$ at timestep $k$ consists of an RGBD image with width $w$ and height $h$ taken by the overhead camera.\n    \\item \\textbf{Observation Model} $(O)$: A deterministic observation model $O(\\mathbf{y}_k \\ | \\ \\mathbf{s}_k)$ is defined by known camera intrinsics and extrinsics.\n    \\item \\textbf{Discount Factor} $(\\gamma)$: To encourage efficient extraction of the target object, $0 < \\gamma < 1$.\n\\end{itemize}\n\nWe also define the following terms:\n\\begin{itemize}\n    \\item Modal Segmentation Mask $(\\mathcal{M}_{m,i})$: the region(s) of pixels in an image corresponding to object $\\mathcal{O}_i$ which are visible~\\cite{kanizsa1979organization}.\n    \\item Amodal Segmentation Mask $(\\mathcal{M}_{a,i})$: the region(s) of pixels in an image corresponding to object $\\mathcal{O}_i$ which are visible or invisible (occluded by other objects in the image)~\\cite{kanizsa1979organization}.\n    \\item The oriented minimum bounding box is the 3D box with the minimum volume that encloses the object, subject to no orientation constraints. We use this box to determine scale and aspect ratio for a target object.\n    \\item The \\textit{occupancy distribution} $\\rho \\in \\mathcal{P}$ is the unnormalized distribution describing the likelihood that a given pixel in the observation image contains some part of the target object's amodal segmentation mask.\n\\end{itemize}\n\n\\subsection{Objective} \\label{subsec:objective}\nGiven this problem definition and assumptions, the objective is to find a policy $\\pi_\\theta^*$ with parameters $\\theta$ that maximizes the expected discounted sum of rewards:\n\\begin{align*}\n    \\theta^* = \\arg \\max_\\theta \\ \\mathbb{E}_{p(\\tau | \\theta)} \\left[\\sum_{k=0}^{H-1} \\gamma^k R(\\mathbf{s}_k, \\pi_\\theta(\\mathbf{y}_k), \\mathbf{s}_{k+1}) \\right]\n\\end{align*}\nwhere $p(\\tau \\ | \\ \\theta) = \\mathbb{P}(s_0) \\Pi_{k=0}^{H-1} T(\\mathbf{s}_{k+1} \\ | \\ \\pi_\\theta(\\mathbf{y}_k), \\mathbf{s}_k) O(\\mathbf{y}_k \\ | \\ \\mathbf{s}_k)$ is the distribution of state trajectories $\\tau$ induced by a policy $\\pi_\\theta$~\\cite{mahler2017learning}. Maximizing this objective corresponds to removing the target object in the fewest number of actions.\n\n\\subsection{Surrogate Reward} \\label{subsec:surrogate-reward}\nBecause the reward defined in Section~\\ref{subsec:defs} is sparse and the transition function relies on complex inter-object and grasp contact dynamics, it is difficult to directly optimize for $\\pi_\\theta$. Thus, we instead introduce a dense surrogate reward $\\Tilde{R}$ describing the reduction of the support of the target object's occupancy distribution:\n\\begin{align*}\n    \\Tilde{R}(\\mathbf{y}_k, \\mathbf{y}_{k+1}) = |\\textrm{supp}(f_\\rho(\\mathbf{y}_{k}))| - |\\textrm{supp}(f_\\rho(\\mathbf{y}_{k+1}))|,\n\\end{align*}\nwhere $f_\\rho : \\Omega \\longrightarrow \\mathcal{P}$ is a function that takes an observation $\\mathbf{y}_k$ and produces the corresponding occupancy distribution $\\rho_k$ for a given bounding box and $\\textrm{supp}(\\rho) = \\lbrace (i, j) \\in \\lbrace 0, \\ldots, h-1 \\rbrace \\times \\lbrace 0, \\ldots, w-1 \\rbrace \\ | \\ \\rho(i,j) \\neq 0$ is the \\textit{support} of the occupancy distribution. Then, $|\\textrm{supp}(\\rho)|$ is the number of nonzero pixels in $\\rho$. Section~\\ref{sec:perception} discusses a data-driven approximation for the function $f_\\rho$ while Section~\\ref{sec:xray-policy} discusses a greedy policy using the learned $f_\\rho$ and $\\Tilde{R}$.\n\\section{Learning Occupancy Distributions} \\label{sec:perception}\nWe describe a method for estimating the function $f_\\rho$ via a deep neural network.  Each pixel in the occupancy distribution $\\rho \\in [0, 1]^{h \\times w}$ has a value representing the likelihood of it containing part of the target object's amodal segmentation mask, or the likelihood that some part of the object, in some planar translation or rotation, would occupy that pixel without any occlusions from other objects. We train this pixelwise distribution network on a dataset of augmented depth images and ground-truth occupancy distributions.\n\n\\begin{table*}[th!]\n\t\\centering\n\t\\vspace{1.5mm}\n\t\\begin{tabu} to \\textwidth {X[2c]X[c]X[c]X[c]X[c]X[c]X[c]X[c]X[c]} \\toprule\n\t\t  & \\multicolumn{2}{c}{\\textbf{Test}} & \\multicolumn{2}{c}{\\textbf{Lid}} & \\multicolumn{2}{c}{\\textbf{Domino}} & \\multicolumn{2}{c}{\\textbf{Flute}}  \\\\\n\t\t\\textbf{Aspect Ratio} & Bal. Acc. & IoU & Bal. Acc. & IoU & Bal. Acc. & IoU & Bal. Acc. & IoU \\\\\\midrule\n\t\t1:1 & $98\\%$ & $0.91$ & $\\bm{93\\%}$ & $\\bm{0.70}$ & $92\\%$ & $0.74$ & $71\\%$ & $0.30$\\\\\n\t\t2:1 & $97\\%$ & $0.90$ & $79\\%$ & $0.44$ & $\\bm{96\\%}$ & $0.81$ & $84\\%$ & $0.44$\\\\\n\t\t5:1 & $97\\%$ & $0.90$ & $66\\%$ & $0.23$ & $96\\%$ & $\\bm{0.83}$ & $\\bm{86\\%}$ & $\\bm{0.49}$\\\\\n\t\t10:1 & $97\\%$ & $0.87$ & $84\\%$ & $0.49$ & $82\\%$ & $0.58$ & $82\\%$ & $0.41$ \\\\\\bottomrule\n\t\\end{tabu}\n\t\\caption{Balanced accuracy (Bal. Acc.) and Intersection over Union (IoU) metrics for networks trained on various aspect ratio target boxes. The first column is the respective set of 2,000 test images for the network's training dataset. The other columns show how the networks can generalize to unseen objects outside the training distribution. Each dataset contains 1,000 test images for the lid, domino, and flute objects, respectively. These objects are shown in Figure~\\ref{fig:perceptionbenchmark} and have approximate aspect ratios of 1:1, 2:1, and 5:1, respectively. Each network performs very well when estimating distributions for its training target object and makes reasonable predictions for target objects with similar bounding box aspect ratios, even for novel target objects at different scales and in the presence of new occluding objects. However, a network trained on a small aspect ratio does not generalize well to higher aspect ratio objects, as it tends to overestimate the occupancy distribution.}\n\t\\label{tab:perceptionbenchmark}\n\t\\vspace{-6pt}\n\\end{table*}\n\n\\subsection{Dataset Generation} We generate a dataset of 10,000 synthetic augmented depth images labeled with target object occupancy distributions for a rectangular box target object. We choose 10 box targets of various dimensions ranging from $3 cm \\times 3 cm \\times 5 mm$ to $9.5 cm \\times 0.95 cm \\times 5 mm$ (aspect ratios varying from 1:1 to 10:1) with equal volume and generate a dataset for each, resulting in a total of 100,000 dataset images. We choose a relatively small thickness for the target so that it is more likely to be occluded in heaps of objects, as it tends to lie flat on the workspace. We sample a state $\\mathbf{s}_0$ by uniformly sampling a set of $N$ 3D CAD models as well as a heap center and 2D offsets for each object from a 2D truncated gaussian. First, $\\mathcal{O}_t$ is dropped from a fixed height above the workspace, then the other $N$ objects are dropped one by one from a fixed height and dynamic simulation is run until all objects come to rest (all velocities are zero). Any objects that fall outside of the workspace are removed. $N$ is drawn from a Poisson distribution ($\\lambda = 12$) truncated such that $N \\in [10, 15]$. The 3D CAD models are drawn from a dataset of 1296 models available on Thingiverse, including ``packaged\" models, where the original model has been augmented with a rectangular backing, as in~\\cite{mahler2019learning}. The camera position is drawn from a uniform distribution over a viewsphere and camera intrinsics are sampled uniformly from a range around their nominal values. We use the Photoneo Phoxi S datasheet intrinsics and a camera pose where the camera points straight down at the heap at a height of $0.8 m$ for the nominal values. An RGBD image is rendered and augmented depth images are created by concatenating a binary modal mask of the target object with the depth image. Note that if the target object is not visible, the image is equivalent to a two-channel depth image, as the first channel is all zeros. We find that training on these images, as opposed to training on RGBD images directly, allows for seamless transfer between simulated and real images.\n\nTo generate the ground-truth occupancy distribution, we find the set of translations and rotations in the image plane for the target object such that an image rendered from the same camera pose with all other objects in the scene in the same respective poses will yield the same target object modal segmentation mask. Thus, when the object is fully visible, the distribution's support collapses to the pixels of the target object modal segmentation mask. However, when the object is partially or fully occluded, then multiple target object translations or rotations may result in the same image and the distribution will spread to reflect where the target could hypothetically be hiding. In practice, we generate this distribution by discretizing the set of possible translations into a $64 \\times 48$ grid (every 8 pixels in the image) and rotations into 16 bins, then shifting and rotating a target-only depth image to each point on the grid, offsetting by the depth of the bottom of the workspace at that point. By comparing the depths for the set of these shifted and rotated depth images to original depth image, we can determine the modal segmentation mask for the target object as if it were at each location. Any location for which there is intersection-over-union (IoU) greater than 0.9 (or, in cases where the target object has a blank modal mask due to full occlusion, any location for which the modal mask is also blank) is considered to result in the same image. Then, the amodal target object masks from all locations resulting in the same image are summed and the resulting normalized single-channel image is the ground truth occupancy distribution. A visualization of this process is shown in Figure~\\ref{fig:datagen}. Dataset generation for 10,000 images took about 5 hours on an Ubuntu 16.04 machine with a 12-core 3.7 GHz i7-8700k processor.\n\n\\subsection{Occupancy Distribution Model} We split each dataset of 10,000 images image-wise and object-wise into training and test sets (8,000 training images and 2,000 test images, where objects are also split such that training objects only appear in training images and test objects only appear in test images). We train a fully-convolutional network with a ResNet-50 backbone~\\cite{long2015fully} using a pixelwise mean-squared-error loss for 40 epochs with a learning rate of $10^{-5}$, momentum of 0.99, and weight decay of 0.0005. The input images were preprocessed by subtracting the mean pixel values calculated over the dataset and transposing to BGR. Training took approximately 2.5 hours on an NVIDIA V100 GPU and a single forward pass took 6 ms on average as compared to 1.5 s for generating the ground-truth distribution.\n\n\\subsection{Simulation Experiments for Occupancy Distributions}\nWe benchmark the trained model on the full set of 2,000 test images as well as on 1,000 images with three other simulated target objects shown in Figure~\\ref{fig:perceptionbenchmark} - a lid, a domino, and a flute - to test generalization to object shapes, aspect ratios and scales not seen during training. We chose these target objects due to their diversity in scale and object aspect ratio (e.g., the flute is longer, thinner, and deeper, while the lid is nearly square and flat). We report two metrics: balanced accuracy, the mean of pixelwise accuracies on positive and negative pixel labels, and intersection-over-union, the sum of positive pixels in both the ground truth and predicted distribution divided by the sum of total positive pixels in either distribution. We consider true positives as the ground truth pixel having normalized value greater than 0.1 and the predicted value being within 0.2 of the ground truth value. Similarly, we consider true negatives as the ground truth pixel having normalized value less than 0.1 and the predicted value being within 0.2 of the ground truth value. Results are shown in Table~\\ref{tab:perceptionbenchmark}. \n\n\\textbf{Target Object Scale.} For objects of different scale than the training target object, we scale the input image by a factor equal to the difference in scale between the box target object and the other target object, feed it through the network, and then rescale the output distribution. We find that this scaling dramatically improves performance with minimal preprocessing of the input image; for example, when testing on the lid object, which is about twice as large as the training box object, we increase balanced accuracy and IoU from $63.0\\%$ and $0.186$ to $93.1\\%$ and $0.697$, respectively.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{1.5mm}\n    \\includegraphics[width=0.75\\linewidth]{aspect_ratios.png}\n    \\caption{The ground truth occupancy distributions for a target object of various aspect ratios for the same heap image.}\n    \\label{fig:aspectratios}\n    \\vspace{-8pt}\n\\end{figure}\n\n\\textbf{Target Aspect Ratios.} We found that, while our network performed well on objects with similar aspect ratios, longer and thinner objects with higher aspect ratios resulted in the model overestimating the support of the distribution. This effect can be seen in Figure~\\ref{fig:aspectratios}, which shows ground truth occupancy distributions for target objects of different aspect ratios in the same heap image. Table~\\ref{tab:perceptionbenchmark} suggests that the trained networks can accurately predict occupancy distributions for target objects that have similar aspect ratios to the training boxes, but do not perform as well when tasked with predicting a distribution for objects with dramatically different aspect ratios. In particular, the network trained with a 1:1 box target object tends to overestimate the support for target objects with high aspect ratios, leading to a drop in metrics. This effect is especially visible along corners of occluding objects, where more rotations of a low aspect ratio object are possible, while only one or two rotations of a high aspect ratio object are possible.\n\n\\begin{figure}\n    \\centering\n    \\vspace{1.5mm}\n    \\includegraphics[width=\\linewidth]{benchmark_perception-v3.png}\n    \\caption{Example predicted target object occupancy distributions for three target objects, a lid, domino, and flute, unseen during training (far left). Warmer colors indicate a higher likelihood of that pixel containing part of the target object's amodal mask. The network is able to accurately predict a distribution across many objects, a collapsed distribution when the object is partially visible, and multimodal distributions when there are gaps between objects (top three rows). The final row shows a failure mode where the network spuriously predicts an extra mode for the distribution when the target object is partially occluded.}\n    \\label{fig:perceptionbenchmark}\n    \\vspace{-8pt}\n\\end{figure}\n\n\\begin{figure*}[th!]\n\\centering\n\\vspace{1.5mm}\n\\includegraphics[width=\\textwidth]{policy.png}\n\\caption{The perception stage takes as input an RGBD image of the scene and outputs an occupancy distribution prediction using a network based on the target object bounding box dimensions and the created augmented depth image. The perception stage also produces a set of segmentation masks. The X-Ray mechanical search policy then finds the mask that has the most overlap with the occupancy distribution (colored yellow in the grasp scores image) and plans a grasp on that mask.}\n\\label{fig:policy}\n\\vspace{-8pt}\n\\end{figure*}\n\nFigure~\\ref{fig:perceptionbenchmark} shows occupancy distribution predictions with ground truth distributions for the three unseen objects using the network trained on the closest aspect ratio target object and scaled appropriately. Results suggest that the network is able to accurately predict diverse distributions when occluding objects not seen in training are present. Figure~\\ref{fig:perceptionbenchmark} suggests not only that the network can predict the correct distribution spanning multiple occluding objects in unimodal and multimodal cases when the target object is fully occluded, but also that it can correctly collapse the distribution to a small area around the visible part of the target object when it is only partially occluded.\n\n\\section{X-Ray: Mechanical Search Policy} \\label{sec:xray-policy}\n\nUsing the learned occupancy distribution function $f_\\rho$, we propose X-Ray, a mechanical search policy that optimizes for the objective and surrogate reward $\\Tilde{R}$ defined in Section~\\ref{sec:problem}. We create both simulated and physical object heaps and generate overhead camera images using an observation model based on the Photoneo PhoXi S depth camera. The heap RGBD image and target object are inputs to the perception system, which uses the network trained on the most similar bounding box to the target object to predict an occupancy distribution for the target. The policy takes the predicted distribution and a set of modal segmentation masks for the scene and computes a grasping action that would maximally reduce the support of the subsequent distribution. Specifically, the policy takes an element-wise product of each segmentation mask with the predicted occupancy distribution and sums over all entries in the resulting image, leading to a score for each of the segmentation masks. The policy then plans a grasp on the object mask with the highest score and executes it, as shown in Figure~\\ref{fig:policy}. \n\n\\subsection{Simulation Experiments with X-Ray}\nWe first evaluate the mechanical search policy with simulated heaps of novel objects. To further test the ability of the learned network to generalize to unseen occluding objects, we use a set of objects unseen in training and validation: 46 YCB objects~\\cite{calli2015benchmarking} and 13 ``packaged'' YCB objects (augmented in the same way as described in Section~\\ref{sec:perception}). Initial states were generated as explained in Section~\\ref{sec:perception}, first dropping the target object, followed by the other $N$ objects. We use $N=14$ so each heap initially contained 15 total objects, color{red}a similar or larger size to previous bin-picking work~\\cite{mahler2017learning,morrison2018closing}. As the focus of this work was not instance segmentation or target detection, we use ground truth segmentation masks and target binary masks in simulation, although we note that any class-agnostic instance segmentation network~\\cite{kuo2019shapemask,danielczuk2019segmenting} or object detection network~\\cite{zhao2019object} can be substituted. For each grasp, either a parallel jaw or suction cup grasp, we use wrench space analysis to determine whether it would result in the object being lifted from the workspace under quasi-static conditions~\\cite{prattichizzo2008grasping,mahler2016dex, mahler2017dex}. If the grasp is collision-free and the object can be lifted, the object is lifted until the remaining objects come to rest using dynamic simulation implemented in pybullet, resulting in the next state. Otherwise the state remains unchanged.\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabu} to \\linewidth {XX[2c]X[c]X[c]X[c]}\n\t\\toprule\n\t\t\\textbf{Policy} & \\textbf{Success Rate} & \\multicolumn{3}{c}{\\textbf{Number of Actions Quartiles}} \\\\\\midrule\n\t\tRandom & $42\\%$ & $4$ & $7$ & $9$ \\\\\n\t\tLargest & $67\\%$ & $4$ & $\\mathbf{5}$ & $7$ \\\\\n\t\tX-Ray & $\\bm{82\\%}$ & $\\mathbf{3}$ & $\\mathbf{5}$ & $\\mathbf{6}$ \\\\\\bottomrule\n\t\\end{tabu}\n\t\\caption{Evaluation metrics for each policy over 1,000 simulated rollouts. The lower quartiles, medians, and upper quartiles for number of actions are reported for successful rollouts. X-Ray extracts the target at a higher success rate with significantly fewer actions.}\n\t\\label{tab:simresults}\n\t\\vspace{-6pt}\n\\end{table}\n\nIn addition to the policy proposed here, we evaluate two previously proposed baseline policies, \\textbf{Random} and \\textbf{Largest}~\\cite{danielczuk2019mechanical}. The \\textbf{Random} policy that first attempts to grasp the target object, and, if no grasps are available on the target object, grasps an object chosen uniformly at random from the bin. The \\textbf{Largest} policy that first attempts to grasp the target object, and, if no grasps are available on the target object, iteratively attempts to grasp the objects in the bin according to the size of their modal segmentation mask.\n\nEach policy was rolled out on 1,000 total heaps until either the target object was grasped (successful rollout) or the horizon $H=10$  was reached (failed rollout). We benchmark each policy using two metrics: success rate of the policy and mean number of actions taken to extract the target object in successful rollouts. Table~\\ref{tab:simresults} and Figure~\\ref{fig:simresults} show these metrics and the distribution of successful rollouts over the number of actions taken to extract the target object, respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{sim_results_max_10.png}\n\\caption{Histogram of the number of actions taken to extract the target object over the 1,000 simulated rollouts for the three policies tested. The median number of actions for each policy is shown by the corresponding vertical line.}\n\\label{fig:simresults}\n\\vspace{-8pt}\n\\end{figure}\n\nWhile the Random and Largest policies occasionally are able to quickly extract the target object, X-Ray consistently extracts the target in fewer actions and succeeds in 15\\% more heaps than the best-performing baseline. Largest is a reasonable heuristic for these heaps, as shown in~\\cite{danielczuk2019mechanical}, as large objects typically have a greater chance of occluding the target, but X-Ray combines this intuition with superior performance when the object is partially occluded. X-Ray outperforms the Largest policy on heaps where the target object is partially occluded by a thin or small object (such as a fork or dice) at some point during the rollout. In these scenarios, a robust grasp is often not available on  the target object, and while X-Ray can correctly identify that the occluding object should be removed, the Largest policy will often grasp a larger object further from the target object. In scenarios where there are many large objects, but some are lying to the side, X-Ray will typically grasp objects that are in the more cluttered area of the bin, since they are more likely to reveal the target object. This behavior is a function of weighting the object area by the predicted distribution, which encourages the policy to ignore solitary objects.\n\n\\subsection{Physical Experiments with X-Ray}\nWe also evaluate X-Ray with heaps of novel household objects on a physical ABB YuMi robot with a suction cup and parallel jaw gripper, using two target objects. Some examples of the objects used can be seen in Figures~\\ref{fig:splash} and~\\ref{fig:policy}. Initial states were generated by placing the target object on the workspace, filling a bin with the $N$ other objects, and then dumping the bin on top of the target object. In these heaps, $N=24$ was used so that each heap initially contained 25 total objects. We chose 25 total objects because it has been commonly used in cluttered bin-picking environments~\\cite{mahler2019learning} and objects tend to disperse further on the physical setup. For segmentation masks, we used the class-agnostic instance segmentation network from~\\cite{danielczuk2019segmenting}, and for grasp quality analysis, we used FC-GQCNN~\\cite{satish2019policy}. To generate binary target masks, we use HSV color segmentation from OpenCV and use red target objects. While we make this assumption for simplicity, we note that we could substitute this process with a target object segmentation method that uses visual features, semantics and shape, such as the one described in~\\cite{danielczuk2019segmenting}.\n\nWe perform 20 rollouts for each of the three policies. Each policy was rolled out until either the target object was grasped (successful rollout) or the horizon $H=10$  was reached (failed rollout). We report the same metrics as in the simulated experiments in Table~\\ref{tab:physicalresults}.\n\nWe find that X-Ray outperforms both baselines, extracting the target object in a median 5 actions over the 20 rollouts as compared to 6 actions for the Largest and Random policies while succeeding in extracting the target object within 10 actions in each case. These results suggest that X-Ray not only can extract the target more efficiently than the baseline policies, but also has lower variance. The Largest policy performed comparatively worse with more objects in the heap than in simulation, as it relies heavily on accurate segmentation masks. However, when objects are densely clustered together, segmentation masks are often merged, leading to grasps on smaller objects that do not uncover the target. In this case or in the case of spurious segmentation masks that do not cover objects, X-Ray reduces this reliance on accurate segmentation masks, as the occupancy distribution and segmentation are combined to create a score for the mask. This property of X-Ray causes it to compare favorably to a policy that directly scores segmentation masks based on their relationship to the target object geometry. X-Ray also reduces reliance on the target object binary mask being accurate; if the detector cannot see enough of the target object to generate a detection even when it is partially visible, X-Ray will continue to try and uncover it according to the fully occluded occupancy distribution until more of the target is revealed.\n\n\\begin{table}\n\\vspace{2mm}\n\t\\centering\n\t\\begin{tabu} to \\linewidth {XX[2c]X[c]X[c]X[c]} \\toprule\n\t\t\\textbf{Policy} &\\textbf{Success Rate} &  \\multicolumn{3}{c}{\\textbf{Number of Actions Quartiles}} \\\\\\midrule\n\t\tRandom & $85\\%$ & $\\mathbf{4}$ & $6$ & $7$  \\\\\n\t\tLargest & $85\\%$ & $4$ & $6$ & $7$ \\\\\n\t\tX-Ray & $\\bm{100\\%}$ & $\\mathbf{4}$ & $\\mathbf{5}$ & $\\mathbf{5.25}$ \\\\\\bottomrule\n\t\\end{tabu}\n\t\\caption{Evaluation metrics for each policy over 20 physical rollouts. The lower quartiles, medians, and upper quartiles for the number of actions are reported across successful rollouts. X-Ray extracts the target with significantly fewer actions, always extracting it within 10 actions.}\n\t\\label{tab:physicalresults}\n\t\\vspace{-6pt}\n\\end{table}\n\n\n\\section{Discussion and Future Work} \\label{sec:discussion}\nWe present X-Ray, a mechanical search algorithm that minimizes support of a learned occupancy distribution. We showed that a model trained only on a synthetic dataset of augmented depth images labeled with ground truth distributions learns to accurately predict occupancy distributions for target objects unseen in training. We benchmark X-Ray in both simulated and physical experiments, showing that it can efficiently extract the target object from challenging heaps containing 15-25 objects that fully occlude the target object in 82\\% - 100\\% of heaps using a median of just 5 actions.\n\nIn future work, we will address some of the failure modes of the system, especially for objects that are significantly non-planar. Currently, the assumption that the object is flat can result in incorrect occupancy distributions for taller objects. Additionally, we will look to add memory to the policy so that if objects shift into previously free space, the distribution will not cover that area, and explore reinforcement learning policies based on a reward of target object visibility.\n\n\\section*{Acknowledgments}\n\\footnotesize\nThis research was performed at the AUTOLAB at UC Berkeley in affiliation with the Berkeley AI Research (BAIR) Lab. The authors were supported in part by donations from Google. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1752814. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Sponsors. We thank our colleagues and collaborators who provided helpful feedback, code, and suggestions, especially Julian Ibarz, Brijen Thananjeyan, Andrew Li, Andrew Lee, Andrey Kurenkov, Roberto Mart\\'in Mart\\'in, Animesh Garg, Matt Matl, and Ashwin Balakrishna.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nAtomic nuclei in condensed phases behave, in many cases, as quantum objects. For instance, Nuclear Quantum Effects are responsible for the {\\em heat capacity problem}, i.e., the deviation from the classical Dulong and Petit law for the heat capacity of solids at low temperatures. The solution of this issue eventually led to the development of the harmonic theory of solids, an accurate quantum theory that lets us to compute their thermal properties at temperatures lower than the Debye temperature, and can be corrected to account for anharmonic effects~\\cite{BornHuang,AshcroftMermin}. By reducing the description of an insulating solid to a set of independent harmonic oscillators, the {\\em phonons}, weakly interacting through anharmonic couplings, this theory also provides a framework for the computation of transport properties, in particular of heat conductivity. In contrast to the very high accuracy that can be achieved for thermal properties, however, the computation of transport properties is sensibly more delicate and often requires ad hoc approximations for the lifetime of phonons, which is limited by phonon-phonon scattering processes and the presence of defects. \n\nThe general framework of the harmonic theory of solids, originally developed for crystals, can be adapted to {\\em disordered solids}. This is at the expenses of employing a numerical approach to characterize the harmonic eigenmodes, that replace phonons and are no longer determined by symmetries. Again, this procedure can be efficiently employed to determine thermal properties, while its application to transport is much more limited. Very often these properties are indeed calculated via classical statistical mechanics approaches (based on classical Molecular Dynamics simulations), whose results are next empirically corrected to account for quantum effects (see, among others, ~\\cite{mizuno2016relation}). We also note that, in systems (ordered or disordered) involving light nuclei (e.g., hydrogen in solid ice), the large wavelength associated with light atoms makes the harmonic approximation itself  inappropriate. Therefore, an exact calculation should in general be considered even for thermal properties, or for the determination of phase boundaries~\\cite{Bronstein2016}.\n\nThe harmonic theory of crystalline solids undoubtedly constitutes a remarkable achievement, as many results can be obtained based on an almost fully analytical approach. However, the above limitations in computing transport properties or in applying the theory to disordered structures, point to the necessity of numerical approaches. It would therefore be highly desirable to develop a numerical methodology that could fully take into account the quantum nature of atomic nuclei, allowing us to determine without approximations both thermal and transport properties of any insulating solid. \n\nWhen interested in thermal properties, an exact numerical method that encompasses all quantum aspects and is valid at any temperature, independently of the strength of anharmonic effects, involves the path integral representation of the partition function~\\cite{Barker1979,Chandler1981,Herman1982,Pollock1984}. In the absence of exchange effects (a reasonable hypothesis in most common solids), the determination of thermodynamic properties at the inverse temperature $\\beta =(k_B T)^{-1}$  involves the sampling of an equivalent system where each quantum particle is replaced by a discretized \"path\" consisting of $M$ \"imaginary time slices\". The method becomes exact in the limit of large $M$, and the sampling of $N$ quantum degrees  of freedom at temperature $T$ turns out to be equivalent to that of $N\\times M$ classical degrees of freedom at temperature $M\\times T$. This sampling can be achieved efficiently using Monte Carlo or Molecular Dynamics methods, leading to the PIMC and PIMD methods, respectively.\n\nComputation of transport properties is more problematic. The standard Green and Kubo statistical mechanics approach to transport coefficients~\\cite{Green1952,Kubo1957,Luttinger1964}, obtains the heat conductivity tensor $\\kappa$ in a system of volume $V$ at temperature $T$ from a time correlation function of the energy current operator $\\bf{J}$ as,\n\\begin{equation}\n\\kappa_{\\alpha \\beta } = \\frac{1}{Vk_BT^2}\\int_0^\\infty dt \\langle J_\\alpha(t)J_\\beta(0)\\rangle.\n\\label{eq:k_ab}\n\\end{equation}\nUnfortunately, the path integral method provides directly  static (time-independent) quantities only. A possible solution to this problem has been identified long ago~\\cite{Thirumalai1983}, by noting that the PIMC approach can rather supply the analytical continuation of the correlation functions on the {\\em imaginary time} axis, simply by computing the correlation between two imaginary time slices along the path. The power spectrum, $S_{AB}(\\omega)$, of a real time correlator, $C_{AB}$, between two operators $A$ and $B$ can then be obtained in an apparently straightforward manner by using the identity,\n\\begin{equation}\nC_{AB}(i\\tau) = \\int_0^\\infty d\\omega \\left[S_{AB}(\\omega) e^{-\\hbar\\omega\\tau} + S_{BA}(\\omega)e^{-\\hbar\\omega(\\beta-\\tau)}\\right].\n\\label{eq:inversion}\n\\end{equation}\n\nWhile Eq.~(\\ref{eq:inversion}) in principle allows one to obtain $S$ based on the data for $C(i\\tau)$, with $\\tau$ in $[0,\\beta]$, it is well known that the inversion problem is ill-posed, in the sense that determining $S$ with high precision is an extremely difficult task, even if $C$ is known with excellent accuracy. For this reason, the approach pioneered by a few groups in the eighties within the framework of path integral calculations did not spread widely. Many recent studies obtained  in various fields~\\cite{PhysRevB.57.10287,Bertaina2017,LEVY2017149,PhysRevB.95.014102,PhysRevB.98.134509}, however, indicate that the present computing capabilities should by now allow us to carry out this program satisfactorily, by addressing the two major (and related) difficulties: {\\em i)} to obtain with high accuracy the imaginary time correlation, in particular for current operators which suffer from the well known issue of diverging variance~\\cite{Herman1982} in the limit of large $M$; and {\\em ii)} to solve the ill-posed problem of obtaining the frequency spectrum from the imaginary time correlation functions. \n\nHere we address these two issues based on numerical and analytical calculations of very simple examples, namely a single harmonic oscillator or an ensemble of oscillators with a continuum distribution of frequencies. The interest of this choice is twofold. First, due to its simplicity, we can obtain exact analytical expressions for most quantities of interest, including all time dependent correlations and exact expressions for the discretized path integrals. The availability of these expressions enables a precise control of the different sources of error, which can be both of statistical origin or associated with the discretization itself. Second, the harmonic oscillator is at the heart of the harmonic theory of solids, the natural starting point for any calculation of transport in insulating solids. Completely controlling this case is, therefore, crucial for any serious step forward in this direction. \n\nThe manuscript is organized as follows: in Sect.~\\ref{sec:PIMC} we introduce the general formalism of the path integral and imaginary time correlations, while in Sect.~\\ref{sec:inversion} we present the procedure that we have developed to cope with the inversion problem. In Sect.~\\ref{sec:estimators} we next describe a new approach that circumvents the issue of the diverging variance for current-current correlators. Finally, in Sects~\\ref{sec:case1} and~\\ref{sec:case2} we illustrate the application of these methods to a single harmonic oscillator, followed by the case of a collection of oscillators with a continuum distribution of frequencies, mimicking the density of states of a crystalline solid. In Sect.~\\ref{sec:conclusion} we draw our conclusions.\n\\section{\\label{sec:PIMC}The path integral formalism for time correlations}\n\nThe path integral Monte Carlo method provides a numerically exact route to the evaluation of thermodynamic properties of quantum systems at finite temperature, $T$. If we consider, for simplicity, a system described by a single degree of freedom $X$ of mass $m$, with Hamiltonian $ \\hat{H} = \\hat{P}^2\/2m + U(\\hat{X})$, the average value of an observable $\\hat{A}$ is \n\\begin{equation}\n\\langle\\hat{A} \\rangle =\\frac{1}{Z(\\beta)} \\text{Tr} [\\hat{A}\\, e^{-\\beta \\hat{H} }  ],\n\\end{equation}\nwhere $Z(\\beta)=\\text{Tr}[e^{-\\beta\\hat{H}}]$. In the PIMC approach, the trace is evaluated by expressing the density operator as $e^{-\\beta\\hat{H}}= (e^{-\\beta\\hat{H}\/M})^M$. In the position representation $\\vert X\\rangle$, and using the notation $\\rho(X,Y,\\tau) = \\langle X \\vert e^{-\\tau\\hat{H}}  \\vert   Y \\rangle $, we can write\n\\begin{multline}\n\\langle  \\hat{A} \\rangle  = \\frac{1}{Z(\\beta)}\n\\int dX_0\\ldots dX_M \\\\\n\\langle X_0 \\vert \\hat{A} \\vert   X_1 \\rangle \\rho(X_1,X_2,\\beta\/M)\\ldots\\rho(X_M,X_0,\\beta\/M).\n\\label{eq:average1}\n\\end{multline}\nIf an expression for $\\rho(X,Y,\\tau)$ is known, the observable can be evaluated by sampling the \"path\" $\\{X_0\\ldots X_M\\}$ with a statistical weight proportional to $\\rho(X_1,X_2,\\beta\/M)\\ldots\\rho(X_M,X_0,\\beta\/M)$. As the matrix element $\\langle X_0 \\vert \\hat{A} \\vert X_1 \\rangle$ of a {\\em local} operator $\\hat{A}$ involves in general a term $\\delta(X_0-X_1)$, the sampling is actually performed over a closed path of $M$ points. In the following we will repeatedly consider the \"primitive\" approximation, based on the factorisation of the kinetic and potential parts of the density operator and valid in the limit of small $\\tau$\\cite{Chandler1981},\n\\begin{equation}\n\\rho_p(X,Y,\\tau) \\simeq \\sqrt{\\frac{m}{2\\pi\\hbar^2\\tau}}\\exp\\left\\{-m\\frac{ (X-Y)^2}{2\\hbar^2\\tau} -\\frac{\\tau}{2}\\left[U(X)+U(Y)\\right]\\right\\}.\n\\label{eq:primitive-approx}\n\\end{equation}\nThis simplified expression can be replaced by a more accurate one if needed, and if the exact value of $\\rho$ is known, as it is the case for the harmonic oscillator, the latter can be used to sample the path  more efficiently~\\cite{feynman1998statistical}.\n\nHere, we are interested in equilibrium time correlation functions that determine the linear response properties of the system. A time correlation involving the observables $A$ at time $t$ and $B$ at time $t=0$ is the equilibrium average of the product of the operators $\\hat{A}(t)= e^{itH\/\\hbar}\\hat{A} e^{-itH\/\\hbar}$, and $\\hat{B}(0)=\\hat{B}$, which we can write as,\n\\begin{equation}\nC_{AB}(t\/\\hbar) = \\langle\\hat{A}(t)\\hat{B}(0)\\rangle = \\frac{1}{Z(\\beta)}\\text{Tr}[\\hat{A}(t)\\hat{B}(0)e^{-\\beta\\hat{H}}].\n\\end{equation}\nObviously, the splitting method could be applied to the operators $\\exp(it\\hat{H}\/\\hbar)$. Unfortunately, the statistical weight associated with the resulting path is imaginary, and therefore it is not suitable for usual sampling methods. If, however, the real time $t$ is replaced by an imaginary time $t=i\\tau \\hbar$, we can write,\n\\begin{multline}\nC_{AB}(i\\tau) =  \\frac{1}{Z(\\beta)}\\text{Tr}  [\\hat{A}e^{-\\tau\\hat{H}}\\hat{B}e^{-(\\beta-\\tau)\\hat{H}}]  \\\\\n=\\frac{1}{Z(\\beta)} \\int dX dX'dY dY' \\\\\n\\langle X \\vert \\hat{A} \\vert   X' \\rangle\n\\rho(X',Y,\\tau)  \\langle Y \\vert \\hat{B} \\vert   Y' \\rangle\n\\rho(Y',X,\\beta -\\tau),\n\\end{multline}\nwhich is defined for $0 \\le \\tau \\le  \\beta$, and verifies $ C_{AB}(i\\tau)=  C_{BA}(i(\\beta-\\tau))$. \n\nPartitioning again the interval $[0,\\beta]$ into $M$ slices of width $\\Delta\\tau= \\beta\/M$, the correlation function can be sampled for discrete values of $\\tau$ of the form $\\tau_k=k\\Delta\\tau$, with $k=0\\ldots M-1$, at a computational cost that is similar to that needed to calculate the thermodynamic observables of Eq.~(\\ref{eq:average1}), obtaining\n\\begin{multline}\n C_{AB}(i\\tau_k)= \n \\frac{1}{Z(\\beta)}\n  \\int dXdY dX_1... dX_M \\langle X \\vert \\hat{A} \\vert   X_1 \\rangle \\rho(X_1,X_2,\\Delta \\tau)...\\\\ \\rho(X_{k-1},X_k,\\Delta\\tau)\n  \\langle X_k \\vert \\hat{B} \\vert   Y  \\rangle\n  \\rho(Y,X_{k+1},\\Delta\\tau)...\\rho(X_M,X,\\Delta\\tau).\n\\end{multline}\nAs in Eq.~(\\ref{eq:average1}), here the sampling must be performed over the $\\{X_1\\ldots X_M\\}$ coordinates of the path, the $X$ and $Y$ variables being eliminated by the $\\delta$-functions contained in the matrix elements of $\\hat{A}$ and $\\hat{B}$.\n\n\n\\section{A statistical approach to the inversion problem}\n\\label{sec:inversion}\n\n\nOnce the imaginary time correlations, denoted by $C(\\tau)$ from now on,  have been obtained for a set of $M$ discrete values $\\{\\tau_0...\\tau_{M-1}\\}$ in the interval $[0,\\beta]$, the real time correlation functions relevant to describe the system physical response can, in principle, be obtained by inverting Eq.~(\\ref{eq:inversion}). This is common to many studies of quantum systems, and generally described as the \"analytical continuation\" procedure. It is, however, ill-posed, in the sense that if the spectrum $S(\\omega)$~\\footnote{In this paragraph we drop the $AB$ subscripts in Eq.~(\\ref{eq:inversion})} is described by a set of parameters (such as the values of $S$ on a discrete $\\omega$-grid, or the coefficients of an expansion in terms of some basis set), and the $C(\\tau_k)$ are affected by statistical errors, a very large number of solutions for $S$ compatible with the original data will be found.\n\nThis topic is the subject of a vast literature, and it is fair to conclude that no single method emerges as a preferred solution. Generally speaking, most current solutions employ some particular version of a \"maximum entropy\" approach~\\cite{JARRELL1996,Boninsegni1996}. The spectral function, $S_{ME}$, is therefore obtained as an average over the possible $S(\\omega)$'s (defined by some finite set of parameters), weighted by the probability that they are the exact model given the data set $(\\textbf{C},\\sigma^2)$,\n\\begin{equation}\nS(\\omega)_{ME} = \\int \\mathcal{D}S \\ p(S|\\text{C},\\sigma^2)S(\\omega).\n\\label{eq:sme1}\n\\end{equation}\nHere $\\mathcal{D}S$ indicates the phase space element associated with the parametrization of $S(\\omega)$, $\\textbf{C} = (C(\\tau_1), C(\\tau_2), \\dots, C(\\tau_M))^{\\dagger} \\equiv (C_1, C_2, \\dots C_M)^{\\dagger}$ is a line vector that contains the data points, and $\\sigma^2$ describes the statistical uncertainty of these data in the form of a covariance matrix. By using the Bayes formula,\n\\begin{equation}\np(S|\\text{C},\\sigma^2) = \\frac{p(\\textbf{C},\\sigma^2|S)}{p(\\textbf{C}, \\sigma^2)}p(S),\n\\end{equation}\nand  making the assumption of  Gaussian statistics for the likelihood, we can write,\n\\begin{equation}\np(\\textbf{C}|S,\\sigma) \\propto e^{-\\frac{1}{2}(\\textbf{C} - \\textbf{C}[S])(\\sigma^2)^{-1} (\\textbf{C} - \\textbf{C}[S])} = e^{-\\frac{1}{2}\\chi^2[S]}\\label{likelihood},\n\\end{equation}\nwhich we can interpret as the definition of $\\chi^2[S]$. Here $\\textbf{C}[S]$ is the expression of the vector $C$, obtained by inserting a known spectrum $S$ into the r.h.s. of Eq.~(\\ref{eq:inversion}) and computing the resulting $M$ correlation values. In the case of a spectrum defined by the amplitudes $A(\\omega_p)$ for a set of $N_\\omega$ discrete frequencies on  a regular grid, using Eq.~(\\ref{eq:inversion}) we obtain, \n\\begin{equation}\n\\tilde{C}[S](\\tau_\\alpha) = \\sum_{p=1}^{N_\\omega} A(\\omega_p) \\left( e^{-\\omega_p \\tau_\\alpha} + e^{-(\\beta - \\tau_\\alpha)\\omega_p}\\right) \\label{eq::correlation fit}.\n\\end{equation}\n\nIn traditional maximum entropy methods, Eq.~(\\ref{eq:sme1}) is solved at the saddle point level, by minimizing the functional $\\mathcal{F}=\\frac{1}{2}\\chi^2[S] - H[S]$. Here, $H[S]$ is an entropic functional, which assigns a penalty to irregular solutions that would lead to an overfitting of the statistical errors contained in the data. For a positive spectrum, $H[S]$ is usually chosen as the associated Shannon entropy, with a coefficient controlling the strength of the regularisation. \n\nIn this work we employ the so-called \"stochastic analytical inference\" or \"stochastic maximum entropy\"~\\cite{Fuchs2010} method, where Eq.~(\\ref{eq:sme1}) is sampled by Monte-Carlo methods over $\\mathcal{D}S$, which can be constrained to positive values of $S$ through the prior probability $p(S)$. The term $\\frac{1}{2}\\chi^2[S]$ can hence be considered as an effective energy functional, and the method can be refined by introducing an additional parameter in the form of an effective inverse temperature $\\Theta$ as,\n\\begin{equation}\nS(\\omega,\\Theta)_{ME}= {Z(\\Theta)^{-1}}\\int \\mathcal{D}S \\   S(\\omega) e^{-\\frac{1}{2}\\Theta \\chi^2[S]}.\n\\label{eq:sme2}\n\\end{equation}\nHere the normalisation $Z(\\Theta)= 1 \/ \\exp{\\{\\Theta F(\\Theta)\\}}$ is an effective partition function. Note that the traditional maximum entropy approach corresponds to a mean field version of Eq.~(\\ref{eq:sme2}), where one uses as an estimate of the spectrum the minimum of the mean field free energy $F_{MF}(\\theta)= \\frac{1}{2}\\chi^2[S]- \\Theta^{-1} H[S] $. In view of the following analysis, we make the simplifying assumption of uncorrelated data points, so that the covariance matrix is diagonal. As a result, we can write the energy functional $\\chi^2[S]$ in the form, \n\\begin{equation}\n\\chi^2 = \\sum_{\\alpha=0}^{M-1}\\frac{[C(\\tau_\\alpha) - \\tilde{C}[S](\\tau_\\alpha)]^2}{\\sigma^2(\\tau_\\alpha)},\n\\label{eq:chi2}\n\\end{equation}\nwith $\\sigma^2(\\tau_\\alpha)$ the statistical uncertainty on the data point $\\alpha$. Several arguments~\\cite{Fuchs2010} have been evoked for fixing $\\Theta=1$. In contrast, in~\\cite{Fuchs2010} it has been proposed to pick for $\\Theta$ the value $\\Theta^*$ that maximises $Z(\\Theta)$, which is argued to also maximise the posterior probability $P(\\theta | C)$. This possibility, which corresponds to a balance between energy and entropy dominated solutions, requires however a full free energy calculation. \n\nAt variance with these proposals, we optimise the value of $\\Theta$ employing the following procedure. An initial data set, $C(\\tau_\\alpha)$, is generated with known statistical uncertainty $\\sigma^2(\\tau_\\alpha)$ by using, for instance, a path integral simulation of the considered model. In cases were $C(\\tau)$ is known analytically, synthetic data could also be generated starting from the exact solution, and introducing a controlled uncertainty. Starting from these data, the spectrum $S_{ME}(\\Theta)$, described by $P$ degrees of freedom  $A(\\omega_p)$, is obtained through a Monte-Carlo sampling of Eq.~(\\ref{eq:sme2}) for a given value of $\\Theta$. Note that a well converged Monte-Carlo average will lead to a spectrum $S_{ME}(\\Theta)$ with an associated $\\chi^2\\sim \\mathcal{O} (M\\epsilon)$, where $\\epsilon$ is a residual error, while the average $\\langle \\chi^2 \\rangle \\sim\\mathcal{O}(M\\epsilon+P\/\\Theta)$. We denote $\\bar{C}_\\Theta(\\tau_\\alpha)$ the correlation function associated with this average spectrum.\n\nIn order to determine the optimal choice of $\\Theta$, therefore discriminating among different models for $S(\\omega)$ (e.g., different finite discretizations on an $\\omega$-gird), we combine the maximum entropy approach with a validation procedure borrowed from the  statistical learning theory~\\cite{MEHTA20191}. We, therefore, generate $P'$ new sets of validation data, $C_{\\mathrm{val}, i}(\\tau_\\alpha)$ ($i=1,\\ldots, P'$), by using the same technique (even not necessarily with the same accuracy) that we use to produce the original data set, and determine the associated,\n\\begin{equation}\n\\chi^2_{\\mathrm{val}} = \\frac{1}{P'}\\sum_{i=1}^{P'} \\sum_{\\alpha=0}^{M-1}[\\bar{C}_\\Theta(\\tau_\\alpha) - C_{\\mathrm{val},i}(\\tau_\\alpha)]^2 .\n\\label{eq::chi2 validation}\n\\end{equation}\nWe can show that this can be interpreted as a measure of the difference between the estimate $\\bar{C}_\\Theta(\\tau_\\alpha)$ and the exact correlation function, denoted by ${C}_{\\mathrm {exact}}(\\tau_\\alpha)$. Indeed, by writing \n\\begin{equation}\n \\chi^2_{\\mathrm{val}}= \\frac{1}{P'} \\sum_{i=1}^{P'} \\sum_{\\alpha=0}^{M-1} [\\bar{C}_\\Theta(\\tau_\\alpha) - {C}_{\\mathrm{exact}}(\\tau_\\alpha) + {C}_{\\mathrm{exact}}(\\tau_\\alpha) - C_{val,i}(\\tau_\\alpha)]^2 ,\n\\end{equation}\nin the limit of large $P'$ and assuming that the average over the validation data returns the exact correlation function, we obtain\n\\begin{equation}\n \\chi^2_{\\mathrm{val}}=  \\sum_{\\alpha=0}^{M-1} [\\bar{C}_\\Theta(\\tau_\\alpha) - {C}_{\\mathrm{exact}}(\\tau_\\alpha)]^2 + \\sum_{\\alpha=0}^{M-1} \\sigma^2_{\\mathrm{val}}(\\tau_\\alpha).\n\\end{equation}\nHere, the first term is the distance of the estimate to the exact data, while the second is the variance of the validation data, which is independent of $\\Theta$. The choice of $\\Theta$ will therefore be eventually dictated by the behaviour of the first term.\n\\section{\\label{sec:estimators}Improved estimators for current correlations}\nThe computation of transport coefficients typically implies correlation functions involving the momentum operator, a prototypical one being $C_{pp}(\\tau) = \\langle p(\\tau) p(0) \\rangle $. In the path integral approach and within the primitive approximation of Eq.~(\\ref{eq:primitive-approx}), the momentum operator is expressed as a difference of coordinates, so that the correlation function for $\\tau \\ne 0$ takes the form $C_{pp}(\\tau_k) = -\\frac{1}{\\Delta\\tau^2}\\langle (x_{k+1}-x_k)(x_1-x_0)\\rangle$, where $x_k \\equiv x(\\tau_k)$, and $\\tau_k = k\\Delta\\tau \\equiv k\\frac{\\beta}{M}$ is the discretized imaginary time. The MC evaluation of $C_{pp}(\\tau_k)$ is hampered by the fact that, when $\\Delta\\tau$ gets small, relative fluctuations in $(x_{i+1}-x_i)$ become large and the variance of the measured observable grows rapidly (in fact it diverges for $\\Delta\\tau\\rightarrow0$). As the uncertainty $\\delta_{MC}$ of the MC estimate of an observable $A$ is related to its variance $\\sigma_A^2$ by $\\delta_{MC} \\propto \\sigma_A\/\\sqrt{\\tau_{sim}}$, one is therefore forced to increase the simulation time, $\\tau_{sim}$, in order to achieve a given precision. \n\nThis problem was identified early in the development of PIMC, when trying to estimate the atoms kinetic energy, which is $\\propto C_{pp}(\\tau=0)$. A solution was proposed in~\\cite{Herman1982}: instead of directly using the above expression for  $C_{pp}(\\tau_k)$, the integrals entering the correlation function can be rearranged obtaining a new estimator for $C_{pp}(\\tau_k)$, with identical average but smaller variance. The new expression, known in the case of the kinetic energy as the \"virial estimator\", does not depend explicitly on $\\Delta\\tau$, and therefore does not suffer from the diverging variance associated with the \"naive\" estimator. \n\nWe now show that the strategy used to obtain the virial estimator can be generalized to any correlation function involving the momentum operator~\\cite{PhysRevLett.111.050406}. Specifically, we consider correlation functions of the general form involved in calculation of transport coefficients, e.~g., $C_{pF}(\\tau) = \\langle ( \\hat{p}(\\tau)\\hat{F}(\\tau))_s (\\hat{p}(0)\\hat{F}(0))_s \\rangle$. Here $\\hat{F}(\\tau)$ is a shorthand notation for a generic local function $F(\\hat{X}(\\tau))$, which in the case of heat transport would be related to the potential energy. The subscript $s$ indicates that the operator product, which represents an observable quantity, is by convention made Hermitian by symmetrizing the operator, as $( \\hat{p}\\hat{F})_s = \\frac{1}{2}(\\hat{p}\\hat{F}+\\hat{F}\\hat{p})$. \n\nWithin the primitive approximation and following this definition one obtains,\n\\begin{multline}\nC_{pF}(\\tau_k) = - \\frac{1}{\\Delta\\tau^2}m^2 \\langle (x_{k+1} - x_{k}) F(x_{k}) (x_{1} - x_{0}) F(x_0) \\rangle \\\\\n+ \\frac{1}{2\\Delta\\tau}m \\langle (x_{k+1} - x_{k}) F(x_{k}) F'(x_0)\\rangle - \\\\\n+ \\frac{1}{2\\Delta\\tau}m \\langle (x_{1} - x_{0}) F(x_{0}) F'(x_{k})\\rangle\n- \\frac{1}{4} \\langle F'(x_k) F'(x_0) \\rangle, \n\\label{eq::pF_correlation}\n\\end{multline}\nThis expression is valid for $k\\ge 1$, while the case $k=0$ must be treated separately, along similar lines. \n\nThe MC calculation of Eq.~(\\ref{eq::pF_correlation}) suffers from the same numerical problem as the momentum correlations, the variance of the leading term in $1\/\\Delta\\tau$ diverging as $\\Delta\\tau$ approaches zero. In order to improve the estimator, we have generalized the procedure originally used for the kinetic energy calculations ($C_{pp}(0)$), and obtain a new estimator with reduced variance for general correlation functions. We start from the first term in Eq.~(\\ref{eq::pF_correlation}), which has the strongest dependence on $\\Delta \\tau$, and can be expressed as,\n\\begin{multline}\n\\frac{1}{\\Delta\\tau^2}\\langle F(x_k)(x_{k+1}-x_k)F(x_0)(x_1-x_0)\\rangle =\\\\=\\frac{1}{\\hbar^2\\Delta\\tau^2 Z} \\int dx_0 \\int dx_1 \\dots \\int dx_M F(x_k) (x_{k+1}-x_k) F(x_0) (x_1-x_0)\\\\ \\rho_0(x_1-x_0; \\Delta\\tau)\\dots \\rho_0(x_M- x_{M-1}; \\Delta\\tau) \\exp\\left[-\\Delta\\tau \\sum _{j=0}^{M} V(x_i)\\right].\n\\end{multline}\nWe now transform the set of coordinates $\\{x_0, x_i\\}$ to $\\{x_0, y_i\\}$, such that $y_i = x_{i+1}-x_i$. The constraint $x_{M} \\equiv x_0$ is accounted for by introducing a term $\\delta\\left(\\sum_{i=0}^{M-1} y_i\\right)$, leading to\n\\begin{multline}\n\\frac{1}{\\Delta\\tau^2}\\langle F(x_k)(x_{k+1}-x_k)F(x_0)(x_1-x_0)\\rangle =\\\\=\\frac{1}{\\Delta\\tau^2 Z} \\int dx_0 \\int dy_0 \\dots \\int dy_{M-1} \\delta\\left(\\sum_{i=0}^{M-1} y_i\\right) F\\left(\\sum_{i=0}^{k-1}y_i +x_0\\right) \\\\ y_k F(x_0)y_0 \\rho_0(y_0; \\Delta \\tau)\\dots \\rho_0(y_{M-1};\\Delta \\tau) \\exp[-\\Delta\\tau W],\n\\end{multline}\nwith\n\\begin{equation}\n W = \\sum _{j=0}^{M-1} V\\left(\\sum_{i = 0}^j y_i + x_0\\right).\n\\end{equation}\nBy using the  identity:\n\\begin{equation}\n\\frac{1}{\\hbar\\Delta\\tau}y_k \\rho(y_k;\\Delta\\tau)= -\\partial _{y_k} \\rho(y_k, \\Delta\\tau),\n\\end{equation}\nwe can integrate by parts for the integration over $y_k$. Our next step is based on the observation that the derivative of the $\\delta$ function w.~r.~t. to $y_0$ can be distributed over all coordinates, i.e., $\\partial_{y_k}\\delta\\left(\\sum y_j\\right) = \\frac{1}{M} \\sum_i \\partial_{y_i}\\delta\\left(\\sum y_j\\right)$. A second integration by parts over each of the $y_i$ variables eventually leads to\n\\begin{multline}\n\\frac{1}{\\Delta\\tau^2}\\langle F(x_k)(x_{k+1}-x_k)F(x_0)(x_1-x_0)\\rangle\n =  \\\\ \\left\\langle F(x_k)(x_1-x_0)F(x_0)\\left[\\frac{1}{M}\\sum_{j=1}^{M-1} j \n V'(x_j)- \n \\sum_{j=k+1}^{M-1} V'(x_j)\\right]\n \\right\\rangle-\\\\- \\frac{k}{(\\Delta\\tau M)}\\langle F'(x_k)(x_1-x_0)F(x_0)\\rangle\n-\\frac{1}{(\\Delta\\tau M)}\\langle F(x_k)F(x_0)\\rangle. \\label{eq::virial_expression}\n\\end{multline}\nFor the special case $F(x) \\equiv 1$, we can show that Eq.~(\\ref{eq::virial_expression}) reduces to a virial-like formula for the momenta correlations $C_{pp}(\\tau_k)=\\langle x_k V'(x_0)\\rangle$ (see App.~\\ref{sec:appendixA}). Repeating the procedure for the terms linear in $\\frac{1}{\\Delta\\tau}$, such as the second term in Eq.~(\\ref{eq::virial_expression}), we can write the correlation in a form that apparently does not depend on $\\Delta \\tau$ (recall that $M\\Delta \\tau =\\beta$ is a constant). The calculations, together with the expressions appropriate for the special case $k=0$, are sketched in App.~\\ref{sec:appendixA}.\n\nIn contrast with the initial expression Eq.~(\\ref{eq::pF_correlation}), all terms are now  well-defined as $\\Delta\\tau \\rightarrow 0$. We note, however, that the number of terms involved in the first part of Eq.~(\\ref{eq::virial_expression}) increases linearly with $M=\\beta\/\\Delta \\tau$, so that the gain following our manipulation is not immediately obvious. The  argument that Eq.~(\\ref{eq::virial_expression}) indeed leads to a variance reduction is the following: If all the $M$ contributions to the first term were independent, its variance would scale as $\\Delta \\tau\\times M$, where $\\Delta \\tau$ comes from the term $\\langle \\vert x_1-x_0\\vert \\rangle$, and the factor $M$ accounts for the $M$ contributions in the sum. As the segments in the path are correlated, even if this estimate is only approximate it still indicates that the variance remains finite even for $\\Delta \\tau \\rightarrow 0$. We explicitly verify the variance reduction  numerically for the harmonic oscillator in the following section.\n\nWe conclude this section by stressing that the above derivation to improve generic estimators involving momentum operators is by no means limited to the harmonic oscillator, but remains valid in general, in particular for the case of interacting particles and also beyond the use of the primitive approximation in the path integral.\n\\section{\\label{sec:case1}Case study I: the single harmonic oscillator}\n\\subsection{Computing correlation functions}\n\\label{sec:computing}\nWe now apply the methods described above to our test cases. We start by considering the canonical example of a single quantum harmonic oscillator of frequency $\\omega_0$ in one dimension, with potential energy $V=\\frac{1}{2} m\\omega_0^2 X^2$, and focus on the time correlation function of an operator with the structure of an energy current, e.~g., $C_{pV}(\\tau) = \\langle (p(\\tau)V(\\tau))_s (p(0)V(0))_s \\rangle$. The PIMC approach within the primitive approximation allows us to extract the values of the imaginary time correlation function $C_{pV}(\\tau_k)$, at $M$ discrete time values, $\\tau_k= (k-1)\\beta\/M$. Two main sources of inaccuracy are associated to this procedure: a systematic error, associated to the use of the primitive approximation for the density matrix, and the statistical uncertainty due to finite sampling. In the following we show how to plainly control these issues.\n\nFor an harmonic oscillator, the systematic deviation due to the discretization of the imaginary time $\\Delta\\tau=\\beta\/M$ can be assessed directly, by comparing the result expected from the PIMC approach (which in this case can be obtained exactly) with the analytical expression for the correlation function $C_{pV}(\\tau)$, which corresponds to the continuous limit $M\\rightarrow \\infty$. By applying the canonical formalism for the harmonic oscillator, we indeed obtain,\n\\begin{multline}\nC^{\\text{exact}}_{pV}(\\tau) =\\left( \\frac{m\\hbar^3\\omega_0^3}{256}\\right) \\frac{1}{\\sinh^3(\\beta \\omega_0\/2)}\\times \\\\\n\\left[12\\cosh\\left(\\frac{3\\beta\\omega_0}{2}-3\\omega_0 \\tau\\right)\\right. \\\\\n\\left.+2\\left(4e^{-\\beta\\omega_0}+e^{-2\\beta\\omega_0}+1\\right)e^{\\beta\\omega_0} \\cosh\\left(\\frac{\\beta\\omega_0}{2} -\\omega_0 \\tau\\right) \\right].\n\\label{eq::exact pv correlation}\n\\end{multline}\nIn order to calculate the exact expression of the correlation function within the primitive approximation of the discretized path integral, we first note that all the integrals involved in the calculation are Gaussian. By using the discretized representation for the momentum operator, one writes $C_{pV}(\\tau)$ as a thermodynamic average of products of the variables $x$. Wick's theorem allows to recast such correlations $\\langle x_1 \\dots x_{2n}\\rangle$ into products of pair correlation functions $\\langle x_ix_j\\rangle$ which are easily accessible as $\\langle x_ix_j\\rangle = A_{ij}^{-1}$. $\\mathbf{A}$ is a symmetric $M \\times M$ matrix, and we can write, $\\langle x_ix_j\\rangle =\\int dX x_ix_j\\text{e}^{-X^T\\mathbf{A}X}$. We can therefore use numerical methods to calculate the matrix elements, as discussed in App.~\\ref{sec:appendixB}. The relative difference between the two calculations is illustrated in Fig.~\\ref{fig::discretization_error_pV}.\nWe observe that, for a sufficiently small value of $\\beta\/M$, the deviation is virtually not affected by a change of $\\beta$.\n\\begin{figure}[t]\n\\center{\\includegraphics[width=1. \\linewidth]{fig01.pdf}}\n\\caption{\nRelative discretization error, $1-C_{pV}^{\\mathrm{PI}}(\\tau)\/ C_{pV}^{\\mathrm{exact}}(\\tau)$, between the path integral, $C_{pV}^{\\mathrm{PI}}(\\tau)$,  and the exact results, $C_{pV}^{\\mathrm{exact}}(\\tau)$, for the energy current correlation function, as a function of $\\beta\/M$. We show the data corresponding to the imaginary times $\\tau=0$ and $\\tau=\\hbar\\beta\/2$, and indicate with symbols and solid lines the results for $\\beta=3$ and $10$, respectively.\n}\n\\label{fig::discretization_error_pV}\n\\end{figure}\n\\begin{figure}[b]\n\\center{\\includegraphics[width=1. \\linewidth]{fig02.pdf}}\n\\caption{\nDifference between the exact correlation function $C_{pV}^{\\mathrm{exact}}(\\tau)$ and the values obtained by Monte Carlo sampling, $C_{pV}^{\\mathrm{MC}}(\\tau)$, of a path with $M=100$ time slices, for $\\beta=1$, illustrating the variance reduction obtained by the improved\nestimator discussed in Sect.~\\ref{sec:estimators}. We show with line-points the primitive estimator and with the continuous line the improved estimator, both using the same\nMonte Carlo data.\n}\n\\label{fig::pv_correlation_beta1}\n\\end{figure}\n\\begin{figure}[t]\n\\center{\\includegraphics[width=1. \\linewidth]{fig03.pdf}}\n\\caption{\nReconstruction of the spectral function associated to $C_{pV}(\\tau)$ at $\\beta=10$ corresponding to the indicated values for the number of delta functions in the model, $N_\\omega$, and effective temperature $\\Theta=1$. The area of the filled rectangles indicate the weight of the two delta-functions of the exact spectrum centered at $\\omega_0 $ and $3\\omega_0$, corresponding to the $\\Delta\\omega=1$ discretization.\n}\n\\label{fig::sp function b10 discretization}\n\\end{figure}\n\\begin{figure}[b]\n\\center{\\includegraphics[width=1. \\linewidth]{fig04.pdf}}\n\\caption{\nReconstructed spectra for the energy current correlation function $C_{pV}(\\tau)$ at $\\beta=10$, with $N_\\omega=25$ and at the indicated values of $\\Theta$. The filled rectangles are centered at the positions of the two delta-functions of the exact spectrum, with an area corresponding to their respective weights.} \n\\label{fig::sp function b10 theta}\n\\end{figure}\n\nIn addition to this quantitative estimate, it is important to note that, for this system, the discretization preserves the qualitative shape of the correlation functions. One can show (see App.~\\ref{sec:appendixC}) that the calculation using a finite but large $M$ corresponds to the exact calculation ($M\\to\\infty$) for slightly shifted oscillator strength and inverse temperature. The Trotter error therefore only introduces small quantitative deviations in the spectral density, but does not give rise to spurious qualitative features such as a broadening of the spectral lines.\n\nWe next focus on the second source of error affecting the PIMC calculation, limited sampling. Indeed, error bars corresponding to average values are obtained by estimating the variance of the observable, which decreases as $\\tau_\\text{sim}^{-1\/2}$, with $\\tau_\\text{sim}$ the simulation time. For a given simulation time, the quality of the result therefore crucially depends on the variance of the estimator. We illustrate this point in Fig.\\ref{fig::pv_correlation_beta1}, by comparing calculations for the energy current correlation function, $C_{pV}$, using the  naive estimator, Eq.~(\\ref{eq::pF_correlation}), and the improved version of Eq.~(\\ref{eq::virial_expression}). The data of Fig.~\\ref{fig::pv_correlation_beta1} clearly show that the virial estimator leads to a spectacular improvement compared to the naive one, with a statistical error that is now comparable to the systematic one resulting from the discretization. \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1. \\linewidth]{fig05.pdf}\n\\caption{\n$N_\\omega$-dependence of the $\\chi^2_{\\mathrm{val}}$ extracted from the validation step of the reconstructed spectral functions for $C_{pV}(\\tau)$, at $\\beta=10$ and with $\\Theta=1$. Squares and triangles correspond to shifted grids: for $N_\\omega=5$ red square shows the shift $\\delta\\omega=0.25$ and the green one $\\delta\\omega=0.5$, for $N_\\omega=10$ plot shows $\\delta\\omega=0.1$ as a red triangle and $\\delta\\omega=0.25$ as a green one.\n}\\label{fig::chi valid b10 discretization}\n\\end{figure}\n\\begin{figure}[b]\n\\center{\\includegraphics[width=1. \\linewidth]{fig06.pdf}}\n\\caption{ \nMain panel: Comparison of the $\\chi^2_{\\mathrm{val}}$ obtained from our validation for various values of $\\Theta$ and $N_\\omega$. The area of the circles is proportional to the corresponding value of $ \\chi^2_{\\mathrm{val}}$. Inset: $\\chi^2_{\\mathrm{val}}$ as a function of $\\Theta$, at the indicated values of $N_\\omega$.\n}\n\\label{chi2 valid b10 table}\n\\end{figure}\n\\begin{figure}[t]\n\\center{\\includegraphics[width=1. \\linewidth]{fig07.pdf}}\n\\caption{\nSpectral reconstruction for $C_{pV}(\\tau)$ at $\\beta=3$, obtained at the indicated values of the discretization, $N_\\omega$, for a fixed $\\Theta=1$. The filled rectangles are centered at the positions of the two delta-functions of the exact spectrum for $\\Delta\\omega=1$, with an area corresponding to their respective weights.\n}\n\\label{fig::sp function b3 discretization}\n\\end{figure}\n\\begin{figure}[b]\n\\center{\\includegraphics[width=1. \\linewidth]{fig08.pdf}}\n\\caption{\nSpectral reconstructions from $C_{pV}(\\tau)$ at $\\beta=3$ for $N_\\omega=5$ using different values of $\\Theta$. The filled rectangles are centered at the positions of the two delta-functions of the exact spectrum, with an area corresponding to their respective weights.\n}\n\\label{fig::sp function b3 theta}\n\\end{figure}\n\\begin{figure}[t]\t\\center{\\includegraphics[width=1. \\linewidth]{fig09.pdf}}\n\\caption{\nMain panel: $\\chi^2_{\\mathrm{val}}$ from the validation procedure at the corresponding values $\\Theta$ and $N_\\omega$. The area of the circles is proportional to the value of $ \\chi^2_{\\mathrm{val}}$. Inset: $\\chi^2_{\\mathrm{val}}$ as a function of the effective temperature $\\Theta$, at the indicated values of $N_\\omega$.\n}\n\\label{chi2 valid b3 table}\n\\end{figure}\n\\subsection{The inversion problem}\nWe now use the reconstruction procedure outlined in Sect.~\\ref{sec:inversion} to extract the frequency spectrum for the correlation functions obtained in Sect.~\\ref{sec:computing}. In order to perform a reconstruction one needs both to define the set of parameters that expresses the spectral density in Eq.~(\\ref{eq::correlation fit}) and in the integration measure of Eq.~(\\ref{eq:sme2}), and to chose the effective inverse temperature $\\Theta$. In the following, we use a discretized model of the spectral density, which is described as a sum of $N_\\omega$ delta-functions in the $\\omega$-space, see Eq.~(\\ref{eq:sme1}). Specifically, we consider a regular grid of $\\omega$-values defined on the interval $[0, 5]$, with a fixed spacing between points, $\\Delta\\omega=5\/N_\\omega$. In addition, we will consider the possibility of a global shift of the grid by $\\delta\\omega < \\Delta \\omega$. Unless specified otherwise, $\\delta\\omega=0$, and we fix the origin of the grid in $\\omega=0$. \n\nThe exact expression for the time correlation function, Eq.~(\\ref{eq::exact pv correlation}), implies that $C_{pV}(\\tau)$ decays exponentially with $\\tau$ in the interval $[0,\\beta\/2]$, with a decay rate $\\mathcal{O}(\\omega_0)$. Larger values of $\\beta$ therefore lead to a larger amplitude in the decay, with the consequence that the contribution of different frequencies can be more easily resolved for larger $\\beta$'s. In short, a correlation function of the form $[\\exp(-\\omega_0\\tau) +\\exp(-3\\omega_0\\tau)]$ will be hard to distinguish from $2\\exp(-2\\omega_0\\tau)$ if data are only available in the interval $ [0,1\/\\omega_0]$. Resolving the two frequencies $\\omega_0$ and $3\\omega_0$ is therefore essentially impossible if $\\beta\/2 <1$.\n\nIn order to illustrate this point, we calculate and analyse the spectral function for the energy current correlation functions at the two inverse temperatures $\\beta=3$ and $10$, with an imaginary time discretization $\\Delta\\tau = 0.1$. With this value of $\\Delta\\tau$, the systematic discretization error is smaller than the statistical error for our simulation time, so it can be safely neglected. The main constraint for the reconstruction comes from the imaginary time interval $[0, 1\/\\omega_0]$. The relative error of the MC data corresponding to these values of $\\tau$ is of $\\mathcal{O}(10^{-2})$. For larger $\\tau$ the relative error becomes comparable with the data due to the fact that $C_{pV}(\\tau)$ approaches 0 with $\\tau \\rightarrow\\beta\/2$.\n\nWe start by considering the case $\\beta=10$. First, we evaluate the effect of the grid size, $N_\\omega$, on the reconstruction. In Fig.~\\ref{fig::sp function b10 discretization} we show the spectra obtained for various values of $N_\\omega$, keeping a fixed $\\Theta=1$. As  mentioned above, there is no {\\em a-priori} argument guiding the most appropriate parametrization of the spectrum. In the following we analyze the accuracy of the spectral reconstruction by comparing the values of $\\chi^2_{\\mathrm{val}}$ defined in Eq.~(\\ref{eq::chi2 validation}), using an independent test data set. This is obtained within an additional MC simulation of the correlation function, with the same parameters as the original one. We also consider a data set of the same size, $P'$, as the one that was used to produce $C_{pV}(\\tau_k)$.\n\\begin{figure}[t]\n\\center{\\includegraphics[width=1. \\linewidth]{fig10.pdf}}\n\\caption{\nSpectral reconstruction of $C_{pV}^{\\text{cont}} (\\tau)$ for the continuous distribution of oscillator frequencies, at the indicated values of the discretization $N_\\omega$, at fixed $\\Theta=1$. The shaded area indicates the exact spectral function.\n}\n\\label{fig::contin spectrum discr}\n\\end{figure}\n\\begin{figure}[b]\n\\center{\\includegraphics[width=1. \\linewidth]{fig11.pdf}}\n\\caption{\nSpectral reconstruction of $C_{pV}^{\\text{cont}} (\\tau)$ for the continuous distribution of oscillators, for $N_\\omega=10$ and $\\Theta=1$ and $10$, respectively. Here we compare the results pertaining to a grid shifted by $\\delta\\omega= 0.25$ to those with $\\delta\\omega=0$, the usual (not shifted) case. The shaded area indicates the exact spectral function.\n}\n\\label{fig::contin spectrum shift theta=1 and theta=10}\n\\end{figure}\n\nIn Fig.~\\ref{fig::chi valid b10 discretization} we show $\\chi^2_{\\mathrm{val}}$ as a function of the number of grid points. Clearly, increasing the number of coefficients $A(\\omega_i)$ of Eq.~(\\ref{eq:sme1}) does not lead to a better spectral reconstruction. In contrast, by introducing more degrees of freedom, one increases the entropy, and the spectral weight is smeared out excessively. In Fig.~\\ref{fig::chi valid b10 discretization} we also show the effect on $\\chi^2_{\\mathrm{val}}$ of a shift $\\delta \\omega$. As expected, shifting the nodes away from $\\omega_1=\\omega_0$ and $\\omega_2=3\\omega_0$, which are the only frequencies present in the exact spectrum determined by Eq.~(\\ref{eq::exact pv correlation}), deteriorates the accuracy of the spectrum obtained through the validation step. \n\nThe second parameter determining the quality of the statistical maximum entropy reconstruction is the effective temperature, $\\Theta$. In Fig.~\\ref{fig::sp function b10 theta} we show the behaviour of the spectral function for a chosen $\\omega$-grid at the indicated values of $\\Theta$. As expected from Eq.~(\\ref{eq:sme1}), by increasing $\\Theta$ the result approaches the most probable configuration that describes the correlation function $C_{pV}(\\tau)$, reducing entropic effects. In Fig.~\\ref{chi2 valid b10 table} we combine the above results for different pairs of parameters ($\\Theta$, $N_\\omega$), and plot the corresponding $\\chi^2_{\\mathrm{val}}$. Our validation procedure therefore strongly points to using models with a smaller number of delta functions combined with large values of $\\Theta \\gg 1$ for the spectral reconstruction. Based on the comparison with the exact spectrum, this choice is also clearly the one that leads to the description of the spectrum in closest agreement with the exact prediction. We conclude that the use of $\\chi^2_{\\mathrm{val}}$ indeed seems to provide an unbiased estimate of the quality of the reconstruction.\n\nWe now consider the spectral reconstruction for $C_{pV}(\\tau)$ at $\\beta=3$, again clarifying the influence of $\\Theta$ and of the lattice discretization $N_\\omega$. In Figs.~\\ref{fig::sp function b3 discretization} and~\\ref{fig::sp function b3 theta} we show selected examples of the resulting spectra. In contrast to the case $\\beta=10$, we now observe in general a much stronger broadening of the peaks, which prevents us from resolving the two peak structure for $\\Theta=1$, even for sparse $\\omega$-grids. However, when combining sparse grids with sufficiently large $\\Theta$ in the inversion, one improves towards the correct two peaks structure, as can be seen in Fig.~\\ref{fig::sp function b3 theta}. The data shown in Fig.~\\ref{chi2 valid b3 table} also indicate that this choice indeed corresponds to the lowest values of $\\chi^2_{\\mathrm{val}}$, confirming the validity of this indicator.\n\n\\begin{figure}[t]\n\\center{\\includegraphics[width=1. \\linewidth]{fig12.pdf}}\n\\caption{\nMain panel: $\\chi^2_{\\mathrm{val}}$ from the validation procedure at the corresponding values $\\Theta$ and $N_\\omega$. The area of the circles is proportional to the value of $ \\chi^2_{\\mathrm{val}}$. Blue circles correspond to the results for an $\\omega$-grid shifted by $\\delta\\omega=\\Delta\\omega \/ 2$. Inset: $\\chi^2_{\\mathrm{val}}$ as a function of the effective temperature $\\Theta$, at the indicated values of $N_\\omega$.\n}\n\\label{fig::cont spectrum valid}\n\\end{figure}\n\n\\section{\\label{sec:case2}Case study II: continuum distribution of oscillators}\nWe now move to our second test model, and study the potential energy current correlation function of a system containing a large number of independent, non interacting harmonic oscillators. Considering the $C_{pV}$ of Eq.~(\\ref{eq::exact pv correlation}) as a function of $\\omega_0$, the correlation function for an ensemble of oscillators with a continuum of frequencies can be  written as,\n\\begin{equation}\nC_{pV}^{\\text{cont}} (\\tau) = \\int_0^{\\omega_{cut}} d\\omega_0\\; C^{\\text{exact}}_{pV}(\\tau;\\omega_0) g(\\omega_0).\n\\label{eq::exact continous pv correlation}\n\\end{equation}\nThe form of the density of states, $g(\\omega_0)$, and the value of the frequency cutoff, $\\omega_{cut}$, are arbitrary. In the following we consider a Debye-like form, $g(\\omega_0)\\propto\\omega_0^2$, with $\\omega_{cut}=1$, and fix $\\beta=10$. With this choice, the exact spectrum  for the energy current correlation is a superposition of two functions with a compact support, assuming non zero values in the range $[0,\\omega_{cut}]$ and $[0,3\\omega_{cut}]$, respectively. As a result, it will display two sharp discontinuities, at $\\omega_{cut}$ and $3\\omega_{cut}$, respectively. \nContrary to the single oscillator case, here we do not generate the data by Monte Carlo simulation, but we rather employ the exact analytical expression, subsequently adding a Gaussian random noise with a variance proportional to the data themselves, $\\sigma_k = 10^{-2}\\times C_{pV}^{\\text{cont}} (\\tau_k)$. This variance is also used as the uncertainty to compute the $\\chi^2$ of Eq.~ (\\ref{eq:chi2}).\n\nBy following the same workflow discussed above for the single oscillator, we reconstruct the spectral densities for different values of $\\Theta$ and number of delta functions in the model, $N_\\omega$. In Fig.~\\ref{fig::contin spectrum discr}, we show the influence of the discretization $N_\\omega$ by fixing the canonical value $\\Theta=1$. Following the same procedure than above, we calculate again $\\chi^2_{\\mathrm{val}}$ for the validation set by generating test correlation function from the exact result of Eq.~(\\ref{eq::exact continous pv correlation}), with the same variance $\\sigma_k$. The values of $\\chi^2_{\\mathrm{val}}$, shown in Fig.~\\ref{fig::cont spectrum valid}, indicate again a more statistically sound reconstruction corresponding to sparse grids. Unfortunately, none of the curves of Fig.~\\ref{fig::contin spectrum discr}, convincingly captures the sharp edges of the exact spectral density, which rather resemble two symmetrically broadened peaks. Considering shifted grids (Fig.~\\ref{fig::contin spectrum shift theta=1 and theta=10}), however, as also quantitatively supported by the validation procedure, results in contrast in more asymmetric features, clearly improving the reconstruction towards the exact spectrum. Note, however, that employing sparse $\\omega$-grids considerably limits frequency resolution, so that the reconstruction in the case of the continuous spectrum with its sharp discontinuities remains quite difficult.\n\n\\section{\\label{sec:conclusion}Conclusion and outlook}\nIn this paper, we have examined the reconstruction of spectral functions for transport coefficients, starting from imaginary time correlation functions obtained by path integral Monte Carlo simulations. In particular, we have described a general strategy for wisely expressing improved estimators with reduced statistical variance for imaginary time correlation functions involving current or momentum operators. We have next introduced an inversion procedure based on a stochastic maximum entropy method, a Bayesian approach commonly used for such problems. The outcome of these procedures is in general strongly dependent on the involved parameters, as we have illustrated in the case of the harmonic oscillator spectra employing different values for the effective inverse temperature, $\\Theta$, as well as different choices for the grid discretization, $N_\\omega$, or offset, $\\delta \\omega$. Despite their apparent simplicity, the oscillator models studied here provide\nchallenging benchmarks for the spectral reconstruction due to the sharp undamped delta-functions they contain.\n\n\nPure Bayesian approaches suggest to eliminate the parameters dependence by using a flat prior with the most general and flexible model for the spectral density, e.~g., a large value for $N_\\omega$, together with $\\Theta=1$ to encompass all possible solutions consistent with the data. In contrast, in our case studies we have shown that the spectra corresponding to these standard choices exceedingly suffer from the usual problems of all maximum entropy reconstructions: broadening or merging of peaks, smoothing out any sharp features in the underlying exact spectrum. \n\n\nIndeed, in practice, path integral Monte Carlo data are strongly correlated in imaginary time, undermining a true justification of the Bayesian choice $\\Theta=1$. Different values of $\\Theta$ may be therefore considered to approximate efficiently the true, unknown likelihood function. On the other hand, the use of flexible models for the spectral function, containing a large number of parameters, possibly introduces a large amount of entropy into the Bayesian inversion, such that different parametrizations (linear or logarithmic grids in regions where spectral densities are flat, for instance) in general strongly modify the results. The representation of a model must therefore be considered itself as a \"parameter\", making illusory in our view a \"parameter-free\" Bayesian inversion.\n\nIn this paper we have addressed exactly the above difficulties, and developed a validation procedure to quantitatively control any parameter dependence of the Bayesian inversion. Our proposal is based on the quantity $\\chi^2_{\\mathrm{val}}$ constructed from independent data not involved in the  maximum entropy inversion, which provides an efficient and readily applicable method to select the optimal choice of parameters, corresponding to the lowest value of $\\chi^2_{\\mathrm{val}}$.\n\nWe have shown explicitly that the new validation step clearly identifies a discrete set of two delta functions in the case study of the single harmonic oscillator, and provides unambiguous indications towards the correct asymmetric sharp edges in the case of an underlying continuous frequency spectrum. Also, in both cases, our validation procedure eventually selects models containing just a limited number of parameters, which intrinsically limits the resolution of the reconstruction. Overall, combining in a consistent workflow Bayesian inversion together with an efficient validation procedure able to select model parameters and effective temperature dependence, indeed seems to offer promising perspectives for capturing qualitative and quantitative features in spectral reconstruction.\n\nWe conclude by noting that the Green-Kubo method, combined with the harmonic theory of solids and a numerical perturbative treatment of anharmonic effects, has recently proven to be remarkably effective for the determination of heat conductivity at low temperature in systems such as amorphous silicon~\\cite{Isaeva2019,Simoncelli2019}. Our hope is to extend those works to arbitrary temperatures and stronger anharmonic effects, on one hand employing path integrals to relax the assumptions underlying the perturbative treatment of anharmonicity, and on the other hand using the strategies for the spectral reconstruction developed in the present paper.\n\\acknowledgements\n{This work has been supported by the project Heatflow (ANR-18-CE30-0019-01) funded by the french \"Agence Nationale de la Recherche\".}\n\\section*{Data Availability Statement}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStarting with a question of Erd\\H{o}s and Rothschild~\\cite{erdos}, there has been substantial interest in the problem of characterizing graphs that admit the largest number of $r$-edge-colorings avoiding a fixed pattern of a graph $F$, where the number $r$ of colors and the pattern of $F$ are given. To be precise, fix an integer $r\\geq 2$ and a graph $F$. We say that $P$ is an \\emph{$r$-pattern} of $F$ if it is a partition of the edge set of $F$ into at most $r$ classes. An $r$-edge-coloring (or $r$-coloring, for short) of a graph $G$ is said to be \\emph{$P$-free} if $G$ does not contain a copy of $F$ such that the partition of the edge set induced by the coloring is isomorphic to $P$. We write $c_{r,P}(G)$ for the number of $P$-free $r$-colorings of a graph $G$ and we define \n$$c_{r,P}(n)=\\max\\{c_{r,P}(G)\\colon |V(G)|=n\\}.$$\nAn $n$-vertex graph $G$ such that $c_{r,P}(G)=c_{r,P}(n)$ is said to be \\emph{$(r,P)$-extremal}. \n\nWe focus on the case when $F$ is the triangle $K_3$. There are three possible patterns: the monochromatic pattern $K_3^M$, the rainbow pattern $K_3^R$, and the pattern $K_3^{(2)}$ with two classes, one containing two edges and one containing a single edge, depicted in the figure below. \n\n\n\\begin{figure}[H]\n\n\t\t\t\\begin{tikzpicture}[line cap=round,line join=round]\n\t\\begin{axis}[\nwidth=0.9\\textwidth,\nheight=0.25\\textwidth,\nxmin=-1, xmax=9,\nymin=0, ymax=0.5,\nminor tick num=3,\ngrid=none,\naxis lines=none]\n\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=1pt,blue,samples=1] coordinates {(2,0)(1,0.5)(0,0)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=2pt,blue,samples=1] coordinates {(0,0)(2,0)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=1pt,blue,samples=1] coordinates {(5,0)(4,0.5)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=2pt,green,samples=1] coordinates {(3,0)(5,0)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=1pt,red,samples=1] coordinates {(3,0)(4,0.5)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=2pt,blue,samples=1] coordinates {(6,0)(8,0)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=1pt,blue,samples=1] coordinates {(8,0)(7,0.5)};\n\n\\addplot[mark=*,mark size=3,mark options={draw=black,fill=black},line width=1pt,red,samples=1] coordinates {(6,0)(7,0.5)};\n\n\\end{axis}\n\\end{tikzpicture} \\\\\n\n\n\t\n\t\\label{pmpa}\n\t\\caption{The patterns $K_3^M$, $K_3^R$, and $K_3^{(2)}$, respectively.}\n\\end{figure}\n\nRegarding the monochromatic pattern $K_3^M$, and assuming that $n$ is sufficiently large, it is known that the balanced, complete bipartite Tur\\'an graph $T_2(n)$ is the unique $(r,K_3^M)$-extremal graph if $r\\in \\{2,3\\}$~\\cite{alon,yuster}, and that $T_2(n)$ is not $(r,K_3^M)$-extremal for $r\\geq 4$. Extremal configurations are known for $r=4$~\\cite{yilma}, for $r\\in \\{5,6\\}$~\\cite{botler} and for $r=7$~\\cite{PS2022}. We should note that, for $r=5$, this problem admits several non-isomorphic configurations that achieve the extremal value in an asymptotic sense. The exact extremal graphs are not known for all sufficiently large $n$. Moreover, even though the extremal configurations are not known for larger values of $r$, structural properties of these configurations have been studied in~\\cite{PS2021,PS2022}. \n\nRegarding the rainbow pattern $K_3^R$, again assuming that $n$ is sufficiently large, it is known that the complete graph $K_n$ is $(r,K_3^M)$-extremal for $r\\in \\{2,3\\}$~\\cite{baloghli}, and that the Tur\\'{a}n graph $T_2(n)$ is $(r,K_3^M)$-extremal for $r \\geq 4$~\\cite{baloghli,rainbow_triangle}. We also refer the reader to~\\cite{BBH20} for related results.\n\nThe first results about the pattern $K_3^{(2)}$ have been obtained by two of the current authors~\\cite{2coloredlagos}, who showed that $T_2(n)$ is $(r,K_3^{(2)})$-extremal for $2 \\leq r \\leq 12$ and sufficiently large $n$. They also observed that this conclusion cannot be extended to $r \\geq 27$, as the following example illustrates. Let $n$ be a positive integer (for the sake of the argument, assume that it is divisible by $4$), and consider the complete, balanced $4$-partite Tur\\'an graph $T_4(n)$ with vertex partition $V_1\\cup\\cdots\\cup V_4$, where each class has size $n\/4$. Let $C$ be a set of 27 colors and partition $C$ as $C_1\\cup C_2\\cup C_3$, where $|C_i|=9$ for $1\\leq i\\leq 3$. Consider the colorings of $T_4(n)$ that assign colors in $C_1$ to edges between $V_1$ and $V_2$ and to edges between $V_3$ and $V_4$; colors in $C_2$ to edges between $V_1$ and $V_3$ and to edges between $V_2$ and $V_4$; colors in $C_3$ to edges between $V_1$ and $V_4$ and to edges between $V_2$ and $V_3$. Note that no copy of $K_3^{(2)}$ may appear in such a coloring (indeed, all triangles are rainbow), and that the number of such colorings is equal\n$$9^{\\frac{6n^2}{16}}=27^{\\frac{n^2}{4}}=c_{27,K_3^{(2)}}(T_2(n)).$$\nMoreover, other colorings of $T_4(n)$ may be produced, for instance by choosing a different partition $C_1\\cup C_2\\cup C_3$. This construction shows that $T_2(n)$ is not $(r,K_3^{(2)})$-extremal for any $r\\geq 27$.\n\nIn this paper, we prove that the result in~\\cite{2coloredlagos} may be extended all the way to $r=26$, as we now state.\n\\begin{theorem}\\label{theorema:main}\nGiven a number $r$ of colors satisfying $2 \\leq r \\leq 26$, there exists $n_0$ such that, for every $n \\geq n_0$ and every $n$-vertex graph $G$, we have\n\\begin{equation}\\label{eq_main}\nc_{r,K_3^{(2)}}(G) \\leq r^{\\ex(n,K_3)}.\n\\end{equation}\nMoreover, equality holds in~\\eqref{eq_main} for $n \\geq n_0$ if and only if $G$ is isomorphic to the bipartite Tur\\'{a}n graph $T_2(n)$.\n\\end{theorem}\n\nThe work in \\cite[Lemma~4.4]{eurocomb15} implies that to prove Theorem~\\ref{theorema:main}, it suffices to show a related stability result establishing that any $n$-vertex graph with a `large' number of $r$-colorings must be `almost bipartite'. For a formal statement of this stability result, given a graph $G=(V,E)$ and a set $W \\subseteq V$, we write $e_G(W)$ for the number $|E(G[W])|$ of edges in the subgraph of $G$ induced by $W$. \n\\begin{lemma} \\label{lemma:main_result}\nLet $2 \\leq r \\leq 26$.  For any fixed $\\delta > 0$, there exists $n_0$ such that the following holds for all $n\\geq n_0$. If $G=(V,E)$ is an $n$-vertex graph such that\n$$ c_{r,K_3^{(2)}}(G) \\geq r^{\\ex(n, K_3)},$$  \nthen there is a partition $V = W_1 \\cup  W_2$ of its vertex set such that $e_G(W_1) + e_G(W_2) \\leq \\delta n^2$.\n\\end{lemma} \n\nThe proof of our results combines the regularity method used in \\cite{alon} and~\\cite{eurocomb15} with linear programming. In some previous applications of this method, the general bounds provided by linear programming were not strong enough to extend the conclusion of Lemma~\\ref{lemma:main_result} to the entire range of $r$. In this paper we use an inductive component in the proof, which allows us to better exploit local restrictions and to extend the result in~\\cite{2coloredlagos} to all values of $r$ for which it was conjectured to hold. The remainder of the paper is organized as follows. In the next section, we introduce the basic preliminary results and notation required. In Section~\\ref{sec:main_proof}, we prove Lemma~\\ref{lemma:main_result}.\n\n\\section{Notation and auxiliary tools}\n\nIn this section, we introduce some notation and auxiliary tools that will be useful for our purposes.\n\n\\subsection{Stability for Erd\\H{o}s-Rothschild type problems} As mentioned in the introduction, results in~\\cite{eurocomb15} ensure that if we prove the stability result stated in Lemma~\\ref{lemma:main_result}, we immediately obtain our main result Theorem~\\ref{theorema:main}. To describe why this is the case, we start with a definition.\n\n\\begin{definition}\\label{colored_stability}\nLet $F$ be a graph with chromatic number $\\chi(F)=k \\geq 3$ and let $P$ be a pattern of $F$. We say that the pair $(F,P)$ satisfies the Color Stability Property for a positive integer $r$ if, for every $\\delta>0$, there exists $n_0$ with the following property. If $n>n_0$ and $G$ is an $n$-vertex graph such that $c_{r,P}(G) \\geq r^{\\ex(n,F)}$, then there exists a partition $V(G)=V_1 \\cup \\cdots \\cup V_{k-1}$ such that $\\sum_{i=1}^{k-1} e(V_i) < \\delta n^2$.\n\\end{definition}\nThe authors of~\\cite{eurocomb15} have proved that, under the technical conditions below, if we show that a pair $(F,P)$ satisfies the Color Stability Property for a positive integer $r$, then we may immediately conclude that the Tur\\'{a}n graph $T_{k-1}(n)$ is the unique $(r,P)$-extremal graph for sufficiently large $n$. In the next statement, a pattern $P$ of $K_{k}$ is \\emph{locally rainbow} if there is a vertex that is incident with edges in $k-1$ different classes of $P$. For $K_3$, the patterns $K_3^R$ and $K_3^{(2)}$ are locally rainbow patterns, but $K_3^M$ is not.\n\\begin{theorem}\\cite[Lemma 4.4]{eurocomb15} \\label{exact} \nLet  $k \\geq 3$ and let $P$ be a locally rainbow pattern of $K_{k}$ such that $(K_{k},P)$ satisfies the Color Stability Property of Definition~\\ref{colored_stability} for a positive integer \n\\begin{equation*}\nr \\geq\n\\begin{cases}\n3 & \\textrm{ if $k=3$}\\\\\n\\lceil ek \\rceil & \\textrm{ if $k\\geq 4$},\n\\end{cases}\n\\end{equation*}\nwhere $e$ denotes Euler's number. Then there is $n_0$ such that every graph of order $n > n_0^2$ has at most $r^{\\ex(n,K_{k})}$ distinct $(K_{k},P)$-free $r$-edge colorings. Moreover, the only graph on $n$ vertices for which the number of such colorings is $r^{\\ex(n,K_{k})}$ is the Tur\\'{a}n graph $T_{k}(n)$.\n\\end{theorem}\nThis justifies why we only need to prove Lemma~\\ref{lemma:main_result} to derive our exact result.  \n\n\\subsection{Regularity and embeddings} Our strategy relies on a colored version of the celebrated Szemer\\'{e}di Regularity Lemma. To state it, we need some terminology. Let $G = (V,E)$ be a graph, and let $A$ and $B$ be two subsets of $V(G)$. If $A$ and $B$ are non-empty, define the edge density between $A$ and $B$ by\n$$d_G(A,B) = \\frac{e_G(A,B)}{|A||B|},$$\nwhere $e_G(A,B)$ is the number of edges with one vertex in $A$ and the other in $B$.\nFor $\\eps > 0$ the pair $(A,B)$ is called \\emph{$\\eps$-regular} if, for all subsets $X \\subseteq A$ and $Y \\subseteq B$ satisfying $|X| \\geq \\eps|A|$ and $|Y| \\geq \\eps|B|$, we have\n$$|d_G(X,Y) - d_G(A,B)| < \\eps.$$\nAn \\emph{equitable partition} of a set $V$ is a partition of $V$ into pairwise disjoint classes $V_1,\\ldots,V_m$ satisfying $\\arrowvert |V_i| - |V_j| \\arrowvert \\leq 1$ for all pairs $i,j$. An equitable partition of the vertex set $V$ of $G$ into classes $V_1,\\ldots,V_m$ is called \\emph{$\\eps$-regular} if at most $\\eps \\binom{m}{2}$ of the pairs $(V_i, V_j)$ are not $\\eps$-regular.\n\nWe are now ready to state a colored version of the Regularity Lemma, whose proof may be found in \\cite{kosi}. For a positive integer $r$, we use the standard notation $[r] = \\{1, \\ldots , r\\}$.\n\\begin{lemma} \\label{lemma:regularity}\nFor every $\\eps > 0$ and every positive integer $r$, there exists $M = M(\\eps,r)$ such that the following holds. If the edges of any graph $G$ of order $n > M$ are $r$-colored $E(G) = E_1 \\cup \\cdots \\cup E_r$, then there is a partition of the vertex set $V(G) = V_1 \\cup \\cdots \\cup V_m$, with $1\/\\eps \\leq m \\leq M$, which is $\\eps$-regular simultaneously with respect to the graphs $G_i = (V,E_i)$ for all $i \\in [r]$.\n\\end{lemma}\nA partition $V_1 \\cup \\cdots \\cup V_m$ of $V(G)$ as in Lemma \\ref{lemma:regularity} will be called a \\textit{multicolored $\\eps$-regular partition}. For $\\eta>0$, we may define a \\textit{multicolored cluster graph} $\\mathcal{H}(\\eta)$ associated with this partition, where the vertex set is $[m]$ and $e = \\{i,j\\}$ is an edge of $\\mathcal{H}(\\eta)$ if $\\{V_i,V_j\\}$ is a regular pair in $G$ \\textit{for every} color $c \\in [r]$ and the edge density of the partition is at least $\\eta$ for some color $c \\in [r]$. Each edge $e$ in $\\mathcal{H}(\\eta)$ is assigned the list $L_{e}$ containing all colors for which its edge density is at least $\\eta$, so that $|L_e| \\geq 1$ for every edge in the multicolored cluster graph $\\mathcal{H}(\\eta)$. Given an (edge)-coloring $\\widehat{F}$ of a graph $F$, we say that a multicolored cluster graph $\\mathcal{H}$ contains $\\widehat{F} $ if $\\mathcal{H}$ contains a copy of $F$ for which the color of each edge of $\\widehat{F} $ is contained in the list of the corresponding edge in $\\mathcal{H}$.  More generally, if $F$ is a graph with color pattern $P$, we say that $\\mathcal{H}$ contains $(F, P)$ if it contains some coloring $\\widehat{F}$ of $F$ with pattern $P$.\n\nIn connection with this definition, we shall use the following standard embedding result, which is a special case of~\\cite[Lemma~2.4]{eurocomb15}.\n\\begin{lemma} \\label{lemma:colored_subgraph} \nFor every $\\eta > 0$ and all positive integers $k$ and $r$, there exist $\\eps = \\eps (r, \\eta, k) > 0$ and a positive integer $n_0(r, \\eta, k)$ with the following property. Suppose that $G$ is an $r$-colored graph on $n > n_0$ vertices with a multicolored $\\eps$-regular partition $V = V_1 \\cup \\cdots \\cup V_m$ which defines the multicolored cluster graph $\\mathcal{H} = \\mathcal{H}(\\eta)$. Let $F$ be a fixed $k$-vertex graph with a prescribed color pattern $P$ on $t \\leq r$ classes. If $\\mathcal{H}$ contains $(F, P)$, then the graph $G$ also contains $(F, P)$.\n\\end{lemma}\n\n\\subsection{Stability}\nAnother basic tool in our paper is a stability result for graphs.\nWe shall use the following theorem by F\\\"uredi \\cite{fu15}.\n\\begin{theorem} \\label{theorem:stability_furedi} \nLet $G = (V,E)$ be a $K_{k}$-free graph on $n$ vertices. If $|E| = \\ex(n, K_{k}) - t$, then there exists a partition  $V= V_1 \\cup \\cdots \\cup  V_{k-1}$ with $\\sum_{i = 1}^{k-1} e(V_i) \\leq  t$. \n \\end{theorem}\n \nWe also use the following version of a simple lemma due to Alon and Yuster \\cite{AY}. For completeness, we include its proof, that relies on the well known fact that any graph on $m\\geq 1$ edges contains a bipartite subgraph with more than $m\/2$ edges.\n\\begin{lemma} \\label{lemma:AY} \nLet $0 < t < n^2\/16$ and let $G$ be a $K_3$-free graph with $n$ vertices and with $\\ex(n,K_{3})-t$ edges. If we produce a new graph $G'$ by adding at least $5t$ new edges to the graph $G$, then $G'$ contains a copy of $K_{3}$ with exactly one new edge.\n\\end{lemma}\n\\begin{proof}\nLet $G=(V,E)$ be a $K_3$-free graph with $n$ vertices and with $\\ex(n,K_{3})-t$ edges and let $F$ be a set of at least $5t$ new edges. Let $E' \\subset E$ be a set with maximum cardinality such that $G'=(V,E')$ is bipartite. Let $V=V_1 \\cup V_2$ be the bipartition of $G'$ and let $E''=E \\setminus E'$. By Theorem~\\ref{theorem:stability_furedi}, we have $|E''| \\leq t$. \n\nThe number of edges of $F$ with one end in $V_1$ and the other  in $V_2$ is at most \n$$|V_1||V_2|-|E'| \\leq \\ex(n,K_3)-|E'| \\leq |E|+t-|E'| = t+|E''|.$$\nSo, if $F'$ denotes the subset of $F$ containing all edges with both ends in $V_1$ or with both ends in $V_2$, we have\n$$|F'| \\geq 5t - (t+|E''|)=4t-|E''|.$$ \nTo conclude our proof, consider a maximum subset $F'' \\subset F'\\cup E''$ such that $G''=(V,F'')$ is bipartite. It is well known that \n$$|F''|>\\frac{|F'\\cup E''|}{2}\\geq \\frac{(4t-|E''|)+|E''|}{2}=2t.$$  \n\nTo conclude the proof, consider the graph $G^\\ast=(V,E' \\cup F'')$. Observe that the edges of $G^\\ast$ with one end in $V_1$ and the other in $V_2$ are in $E'$, and the others are in $F''$. By the above considerations,  \n$$|E'|+|F''| > \\ex(n,K_{3})-(t+|E''|)+2t \\geq \\ex(n,K_{3}),$$ \nso that $G^\\ast$ contains a triangle by Tur\\'{a}n's Theorem. We claim that this triangle contains exactly one edge in $F''$. It is obvious that it must contain at least one edge in $F''$ and that it cannot contain an edge with both ends in $V_1$ and another with both ends in $V_2$.  Moreover, if the triangle contained two edges with all ends in one of the sides of the bipartition, say in $V_1$, the third edge of the triangle would also connect vertices in $V_1$, so that the three edges would lie in $F''$, contradicting the fact that $G''=(V,F'')$ is bipartite.\n\\end{proof}\n\n\n\\section{Proof of Lemma~\\ref{lemma:main_result}}\\label{sec:main_proof}\n\nIn this section, we will prove Lemma~\\ref{lemma:main_result}. To this end, fix $r \\in \\{2,\\ldots,26\\}$ and let $\\delta>0$. With foresight, we consider auxiliary constants $\\alpha$, $\\xi$ and $\\eta > 0$ such that\n\\begin{center}\n\\begin{equation}\\label{eq:quantification}\n\\alpha=\\frac{1}{1000}, \\ \\   \\xi < \\dfrac{\\delta}{22}, \\ \\ \\xi > 10^4\\cdot H((r+1)\\eta) + (10^4+1)\\cdot  (r+1)\\cdot \\eta \\ \\ \\text{ and } \\ \\ \\eta < \\dfrac{\\delta}{2r},\n\\end{equation}\n\\end{center}\nwhere $H \\colon [0,1] \\to [0,1]$ is the \\emph{entropy function} given by $H(0) = H(1) = 0$ and by $H(x) = -x \\log_2 x - (1-x) \\log_2(1-x)$ for $x \\in (0,1)$. \n\nLet $\\varepsilon = \\varepsilon(r,\\eta, 3) > 0$ and $n_0 = n_0(r,\\eta,3)$ satisfy the assumptions of Lemma~\\ref{lemma:colored_subgraph}, and assume without loss of generality that $\\varepsilon < \\min\\{\\eta\/2,1\/n_0\\}$. Fix $M = M(r,\\varepsilon)$ given by Lemma~\\ref{lemma:regularity}.\n\nGiven an $n$-vertex graph $G$ such that $n \\geq n_0$, let $\\mathcal{C}=\\mathcal{C}(G)$ the set of all $K_3^{(2)}$-free $r$-colorings of $G$. By Lemma \\ref{lemma:regularity} and the discussion following it, each coloring $\\Phi \\in \\mathcal{C}$ is associated with a multicolored $\\eps$-regular partition $V= V_1 \\cup \\cdots \\cup V_m$, where $1 \/ \\eps \\leq m \\leq M$. This partition is in turn is associated with a multicolored cluster graph $\\mathcal{H}=\\mathcal{H}(\\eta)$. Our choice of parameters implies that $\\mathcal{H}$ must be $K_3^{(2)}$-free, otherwise the coloring of $G$ leading to it would contain a copy of $(K_3,\\leq 2)$ by Lemma~\\ref{lemma:colored_subgraph}.\n\nTowards an upper bound on the size of $\\mathcal{C}$, we determine an upper bound on the number of colorings that give rise to a fixed partition $ V_1 \\cup \\cdots \\cup V_m$ and to a fixed multicolored cluster graph $\\mathcal{H}$. We first consider the edges of $G$ whose colors are not captured by the lists $L_e$ associated with edges $e\\in E(\\mathcal{H})$. Lemma~\\ref{lemma:regularity} ensures that, for each color in $[r]$, there are at most $\\eps \\binom{m}{2}$ irregular pairs with respect to the partition $V = V_1 \\cup \\dots \\cup V_m$, hence at most \n\\begin{align}\nr \\cdot \\eps \\cdot \\binom{m}{2} \\cdot \\left(\\frac{n}{m}\\right) ^ 2 \\leq r \\eps \\cdot n ^ 2 \\leq \\frac{r \\eta}{2} \\cdot n ^ 2 \\label{eq:irregular}\n\\end{align}\nedges of $G$ are contained in an irregular pair with respect to one of the colors. Moreover, there are at most\n\\begin{align}\nm \\cdot \\left(\\frac{n}{m}\\right) ^ 2 = \\frac{n ^ 2}{m} \\leq \\eps n ^ 2 \\leq \\frac{\\eta}{2} \\cdot n ^ 2 \\label{eq:inside}\n\\end{align}\nedges with both ends in the same class $V_i$. Finally, we consider edges $f$ whose endpoints are in distinct classes $V_i$ and $V_j$ and such that the edge density of the edges with the color of $f$ is less than $\\eta$ with respect to this pair. The number of edges of this type is at most\n\\begin{align}\nr \\cdot \\eta \\cdot \\binom{m}{2} \\cdot \\left(\\frac{n}{m}\\right) ^ 2 \\leq \\frac{r \\eta}{2} \\cdot n ^ 2. \\label{eq:density}\n\\end{align}\nUsing \\eqref{eq:irregular}, \\eqref{eq:inside} and \\eqref{eq:density} gives at most $(r+1) \\eta n^2$ edges of these three types. \n\nClearly, the remaining edges of $G$ have endpoints in pairs that are regular for every color and must be assigned a color\nthat is dense with respect to the pair where its endpoints lie, i.e., their color must lie in the list of the corresponding edge of $\\mathcal{H}$. This means that the number of elements of $\\mathcal{C}$ that can be associated with a given multicolored partition $V_1 \\cup \\cdots \\cup V_m$ and a given $m$-vertex multicolored cluster graph $\\mathcal{H}$ is bounded above by \n\\begin{align} \\label{eq:multicolor_bound}\n \\binom{n^2}{(r+1) \\eta n^2} \\cdot r^{(r+1) \\eta n^2} \\cdot \\left( \\prod_{j=1}^{r} j^{e_j(\\mathcal{H})} \\right)^{\\left( \\frac{n}{m} \\right)^2},\n\\end{align}\nwhere $e_j(\\mathcal{H})$ denotes the number of edges of $\\mathcal{H}$ whose lists have size equal to $j$.\nHere, we assume that $m$ divides $n$ to avoid dealing with lower order terms that can be absorbed into the error term $r^{(r+1) \\eta n^2}$.\nThere are at most $M^n$ partitions of $V$ on $m \\leq M$ classes and at most $2^{r\\binom{m}{2}}$ multicolored cluster graphs with vertex set $[m]$.  Moreover, it is well-known that the entropy function satisfies\n$$\\binom{n^2}{(r+1) \\eta n^2} \\leq 2^{H((r+1)\\eta n^2)}.$$\n\nThus, summing the upper bound~\\eqref{eq:multicolor_bound} over all partitions and all corresponding multicolored cluster graphs, the number of $K_3^{(2)}$-free edge colorings of $G$ is at most\n\\begin{align}\nM^n \\cdot 2^{r M^2\/2} \\cdot 2 ^ {H((r+1) \\eta) n^2} \\cdot r^{(r+1) \\eta n^2} \\cdot \\max_{\\mathcal{H}} \\left( \\prod_{j=1}^{r} j ^ {\\frac{e_j(\\mathcal{H})}{|V(\\mathcal{H})|^2}} \\right)^{n^2}. \\label{eq:coloring_of_g}\n\\end{align}\n\nOur aim is to find an upper bound on \\eqref{eq:coloring_of_g}. The term $j=1$ in the product in \\eqref{eq:coloring_of_g} does not affect the result. So, we define $\\mathcal{S}=\\mathcal{S}(G)$ to be the set of all subgraphs of multicolored cluster graphs $\\mathcal{H}$ of $G$ such that all edges are associated with lists of size at least two. Abusing the terminology, we also call the subgraph given by edges whose lists have size at least 2 the multicolored cluster graph associated with a coloring of $G$. Note that $\\mathcal{H} \\in \\mathcal{S}$ is $K_3^{(2)}$-free if and only if all lists associated with edges on a triangle are mutually disjoint. Given $\\mathcal{H} \\in \\mathcal{S}$, we let \n\\begin{equation}\\label{def_cH}\nc(\\mathcal{H})=\\prod_{e\\in E(\\mathcal{H})}|L_e|^{\\frac{1}{|V(\\mathcal{H})|^2}}.\n\\end{equation}\nWe wish to find $\\max_{\\mathcal{H} \\in \\mathcal{S} } c(\\mathcal{H})$ to bound~\\eqref{eq:coloring_of_g}.\n\nAs discussed in~\\cite{2coloredlagos}, this is easy to do for the case of $r\\leq 12$ colors (for completeness, we present the proof for $r\\leq 12$ as an appendix). In the remainder of the proof, we focus on the remaining values of $r$ and consider the functions\n\\begin{eqnarray*}\nr_0&=&r_0(r)=\n\\begin{cases} \n6& \\textrm{ if } r=13\\\\\n\\lfloor r-2\\sqrt{r}\\rfloor& \\textrm{ if } 14\\leq r \\leq 26.\n\\end{cases} \\\\%\\label{def:r0}\\\\\nr_1&=&r_1(r)=r_0+1. \n\\end{eqnarray*}\nFor $r\\geq 13$, the crucial property in the definition of $r_1$ is that \n\\begin{equation}\\label{eq:Ar}\nA(r)=\\left\\lfloor (r-r_1)\/2 \\right\\rfloor \\cdot \\left\\lceil (r-r_1)\/2\\right\\rceil < r.\n\\end{equation}\nNote that,  $A(r)\\leq (r-r_1)^2\/4$ and, as both factors of $A(r)$ are integers, we have $A(r) \\leq r-1$.\nThe validity of~\\eqref{eq:Ar}  may be verified directly for $r=13$, and if $r \\geq 14$ we have $r-r_1=r-\\lfloor r-2\\sqrt{r}\\rfloor-1<r-(r-2\\sqrt{r}-1)-1=2\\sqrt{r}$. \n\nAnother important property of $r_1$ is that $2r_1 \\geq r$. This may be verified directly for $r\\in\\{13,14,15\\}$. For $r\\geq 16$, we have \n$$2r_1 \\geq 2(r-2\\sqrt{r}) \\geq r,$$\nwhere the last inequality is equivalent to $\\sqrt{r}\\geq 4$. \n\nOur conclusion is a consequence of the following. \n\\begin{claim}\\label{claim13}\nThere is a multicolored cluster graph $\\mathcal{H}$ such that\n\\begin{align*}\ne_{r_1}(\\mathcal{H}) + \\dots + e_{r}(\\mathcal{H}) \\geq \\ex(m, K_3) - \\xi m ^ 2.\n\\end{align*}\n\\end{claim}\n\nBefore addressing the proof of Claim~\\ref{claim13}, we show that it implies the desired result. Let $\\mathcal{H}$ be an $m$-vertex multicolored cluster graph such that  $e_{r_1}(\\mathcal{H}) + \\dots + e_{r}(\\mathcal{H}) \\geq \\ex(m, K_3) - \\xi m ^ 2$. Let $\\mathcal{H}^{\\prime}$ be the subgraph of $\\mathcal{H}$ with edge set $E_{r_1} (\\mathcal{H})\\cup \\dots \\cup E_{r}(\\mathcal{H})$, where $E_j(\\mathcal{H})$ denotes the set of edges of $\\mathcal{H}$ whose lists have size $j$. We claim that $\\mathcal{H}^{\\prime}$ is $K_3$-free. Indeed, if  $\\mathcal{H}^{\\prime}$ contained a triangle with edges $e_1,e_2,e_3$, then \n$$|L_{e_1}|+|L_{e_2}|+|L_{e_3}| \\geq 3r_1> 2r_1+1>r$$ \nimplies that two of the lists have non-empty intersection, leading to a $K_3^{(2)}$ in $G$ by Lemma~\\ref{lemma:colored_subgraph}. \n\nBy Theorem~\\ref{theorem:stability_furedi}, there is a partition $U_1 \\cup U_{2} = [m]$ with\n\\begin{align*}\ne_{\\mathcal{H}^{\\prime}}(U_1)  + \\ e_{\\mathcal{H}^{\\prime}}(U_{2}) \\leq \\xi m^2,\n\\end{align*}\nwhere $e_{\\mathcal{H}^{\\prime}}(U_i)$ is the number of edges of $\\mathcal{H}^{\\prime}$ with both endpoints in $U_i$.  The bipartite subgraph $\\widehat{\\mathcal{H}}$ obtained from $\\mathcal{H}^{\\prime}$ obtained by removing all edges with both endpoints in the same class satisfies\n\\begin{align*}\ne(\\widehat{\\mathcal{H}}) \\geq  (\\ex(m, K_3) - \\xi m ^ 2) - \\xi m^2=  \\ex(m, K_3) - 2 \\xi m^2.\n\\end{align*}\n\nWe claim that, even if we add arbitrary edges with lists of size $1$ to $\\mathcal{H}$ (while preserving its property of being $K_3^{(2)}$-free),  $e_1(\\mathcal{H}) + \\cdots + e_{r_0}(\\mathcal{H})  \\leq 10\\xi m^2$. Otherwise, by our choice of $\\xi$in~\\eqref{eq:quantification}, Lemma~\\ref{lemma:AY} can be applied and the graph obtained by adding the edges in $E_1 \\cup \\cdots \\cup E_{r_0}$ to $\\widehat{\\mathcal{H}}$ would contain a $K_3$ such that exactly one of the edges, say $f_1$, is in some set $U_i$. Let $f_2,  f_{3}$ be the other edges of the copy of $K_3$, which lie in $E_{r_1} \\cup \\cdots \\cup E_{r}$. By construction, we have \n$$|L_{f_1}|+|L_{f_2}|+|L_{f_3}| \\geq 1+2r_1>r,$$\na contradiction.\n\nAs a consequence, the number of edges of $\\mathcal{H}$ with both ends in the same set $U_i$ is at most $11 \\xi m^2$. Let $W_i = \\cup_{j \\in U_i} V_j$ for $i \\in \\{1, 2\\}$. Then, by our choice of $\\eta$ and $\\xi$ in~\\eqref{eq:quantification}, we have\n\\begin{align*}\ne_G(W_1) +  e_G(W_{2}) \\leq r \\eta n^2 + (n\/m)^2 \\cdot (e_\\mathcal{H}(U_1) + e_\\mathcal{H}(U_{2})) < \\delta n^2,\n\\end{align*} \nas required. This proves Lemma~\\ref{lemma:main_result}.\n\nWe now move to the actual proof of Claim~\\ref{claim13}.  Given a cluster graph $\\mathcal{H}$, let $E_b(\\mathcal{H})$ be the set of all edges whose color lists have sizes between $r_1$ and $r$. We refer to them as the \\emph{blue} edges of $\\mathcal{H}$. Let $E_g(\\mathcal{H})$ be the set of all edges whose color lists have sizes between $2$ and $r_0$, the \\emph{green} edges of $\\mathcal{H}$.  The main ingredient in the proof of Claim~\\ref{claim13} is the following auxiliary lemma.\n\\begin{lemma}\\label{lemma:claim13}\nLet $r$ be an integer such that $2\\leq r \\leq 26$ and let $\\mathcal{H}$ be a $(K_3,\\leq 2)$-free multicolored cluster graph for which all edges are green. Then, for all $0 < \\alpha \\leq \\frac{1}{1000}$ it is\n$$c(\\mathcal{H})\\leq r^{\\frac14-\\alpha} <r^{\\frac14}.$$\n\\end{lemma}\n\nBefore proving this lemma, we show that it implies the validity of Claim~\\ref{claim13}. To this end, assume for a contradiction that any coloring of $G$ avoiding $K_{3}^{(2)}$ leads to a multicolored cluster graph $\\mathcal{H}$, where $|V(\\mathcal{H})|=m$, for which\n\\begin{align}\\label{eq:contradiction13}\ne_{r_1}(\\mathcal{H}) + \\dots + e_{r}(\\mathcal{H}) < \\ex(m, K_3) - \\xi m ^ 2. \n\\end{align}\n\nLet $\\mathcal{H}_0 = \\mathcal{H}$. If $\\mathcal{H}_0$ contains a blue edge, let $e_1 = \\{ u, v \\}$ be one such edge, so that $|L_{e_1}|\\geq r_1$. Let $w$ be a vertex $V\\setminus\\{u,v\\}$. If at most one of the pairs $\\{u,w\\}$ and $\\{v,w\\}$ is an edge of $\\mathcal{H}_0$, the list of colors on this edge cannot produce a $K_3^{(2)}$ involving $e_1$. If $w$ is a joint neighbor of $u$ and $v$, the sum of the sizes of $L_{\\{u, z\\}}$  and  $L_{\\{v, z\\}}$  is at most $r-|L_{e_1}| \\leq r-r_1$, otherwise we obtain a copy of $K_3^{(2)}$. Thus the product of the sizes of $L_{\\{u, z\\}}$  and  $L_{\\{v, z\\}}$ is at most $A(r)=\\left\\lfloor (r-r_1)\/2 \\right\\rfloor \\cdot \\left\\lceil (r-r_1)\/2\\right\\rceil < r$ by~\\eqref{eq:Ar}. We say that a vertex $w$ has \\emph{type 1} if there is a single edge connecting it to $\\{u,v\\}$, and this edge is blue. Otherwise it is said to have \\emph{type 2}. If $w$ is a type 2 vertex, then either the list of the edge connecting it to $u$ or $v$ has size at most $r_0$ (in case $w$ is adjacent to at most one vertex among $u$ and $v$), or the product of the sizes of the lists is at most $A(r)$ (in case $w$ is adjacent to $u$ and $v$). This means that the product of the sizes of the lists of edges connecting a type 2 vertex to $\\{u,v\\}$ is at most $B(r)=\\max\\{r_0,A(r)\\}<r$.    We know that $B(r) \\leq r-1$. Note that\n\\begin{equation}\\label{bound:B}\n\\frac{B(r)}{r} \\leq \\frac{r-1}{r} \\leq r^{-\\alpha}\n\\end{equation}\nholds if $e^{-1\/r} \\leq  r^{-\\alpha}$, which is equivalent to $\\alpha \\ln r - 1\/r \\leq 0$. The derivative of $f(r)=\\alpha \\ln r - 1\/r$ is positive, which means that~\\eqref{bound:B} is satisfied for all $r\\leq 26$ if $\\alpha \\ln 26 - 1\/26 \\leq 0$, which holds for $\\alpha \\leq 0.0118$. \n\nLet $n_1(e_1)$ and $n_2(e_1)$ denote the number of vertices of type 1 and 2 with respect to $e_1$. Now remove vertices $u$  and $v$ and all incident edges from $\\mathcal{H}_0$, and call the remaining multicolored graph on $(m-2)$ vertices $\\mathcal{H}_1$. If $\\mathcal{H}_1$ contains a blue edge $e_2$, we repeat this argument for $\\mathcal{H}_1$ and for subsequent graphs until we reach a graph $\\mathcal{H}_{k_1}$ on $(m-2k_1)$ vertices that does not contain any blue edge, that is, such that every edge in $\\mathcal{H}_{k_1}$ is green. By construction, we have \n\\begin{equation}\\label{eq_number}\nk_1+\\sum_{i=1}^{k_1} (n_1(e_i)+n_2(e_i)) =  \\sum_{i=1}^{k_1}  (m-2i+1) = k_1m -k_1^2.\n\\end{equation}\nWe conclude that the number of $K_3^{(2)}$-free colorings of $\\mathcal{H}$ is at most\n\\begin{eqnarray*}\n && r^{k_1+n_1(e_1)+\\cdots+n_1(e_{k_1})} \\cdot (B(r))^{n_2(e_1)+\\cdots+n_2(e_{k_1})} \\cdot c(\\mathcal{H}_{k_1})^{(m-2k_1)^2}\\label{eq:26colors-1}   \\\\\n &=& \\left(\\frac{B(r)}{r}\\right)^{n_2(e_1)+\\cdots+n_2(e_{k_1})} \\cdot  r^{k_1m -k_1^2} \\cdot c(\\mathcal{H}_{k_1})^{(m-2k_1)^2}. \\nonumber\n\\end{eqnarray*}\n\nAs all edges in the graph $\\mathcal{H}_{k_1}$ are green, using  the upper bound given in Lemma~\\ref{lemma:claim13}, we obtain \n\\begin{eqnarray} \\label{eq:26colors1}\nc(\\mathcal{H})^{m^2} &\\leq &   \\left(\\frac{B(r)}{r}\\right)^{n_2(e_1)+\\cdots+n_2(e_{k_1})} \\cdot  r^{k_1m -k_1^2} \\cdot r^{\\left(\\frac{1}{4}-\\alpha\\right)(m-2k_1)^2}\\nonumber \\\\\n&=& \\left(\\frac{B(r)}{r}\\right)^{n_2(e_1)+\\cdots+n_2(e_{k_1})}\\cdot r^{\\frac{m^2}{4}-\\alpha(m-2k_1)^2}.\n\\end{eqnarray}\nRecall that, by~(\\ref{eq:contradiction13}), we are assuming that the number of blue edges of $\\mathcal{H}$ is at most $\\ex(m, K_3) - \\xi m ^ 2$. We shall consider two scenarios. First suppose that $k_1m -k_1^2 \\leq \\ex(m, K_3) - \\xi m ^ 2$, so that $k_1 \\leq m\/2 - \\sqrt{\\xi}m$. In this case, all of the vertices in the above construction may have type 1 and~\\eqref{eq:26colors1} is at most \n\\begin{eqnarray*}\nr^{\\frac{m^2}{4}-\\alpha \\xi m^2} \\leq r^{\\frac{m^2}{4}-\\frac{1}{10^4}\\xi m^2}.\n\\end{eqnarray*}\n\nIn the second scenario, assume that $k_1 = m\/2 - \\sqrt{\\xi}m+q$, for $q>0$ and $q \\leq \\sqrt{\\xi}m$, so that, by~\\eqref{eq_number} and our restriction on the number of blue edges, i.e., $k_1+\\sum_{j=1}^{k_1} n_1(e_{j}) \\leq \\ex(m, K_3) - \\xi m ^ 2$, we have\n\\begin{eqnarray*}\n\\sum_{j=1}^{k_1} n_2(e_{j})&=& k_1m-k_1^2-\\left(k_1+\\sum_{j=1}^{k_1} n_1(e_{j})\\right)\\\\\n&\\geq& q(2\\sqrt{\\xi}m-q).\n\\end{eqnarray*}\nEquation \\eqref{eq:26colors1} is at most\n\\begin{eqnarray}\\label{eq:26colors2} \n&& \\left(\\frac{B(r)}{r}\\right)^{q(2\\sqrt{\\xi}m-q)} \\cdot r^{\\frac{m^2}{4}-\\alpha(2\\sqrt{\\xi}m-2q)^2}.\n\\end{eqnarray}\nIf $q \\leq 3\\sqrt{\\xi} m\/4$, equation~\\eqref{eq:26colors2} is at most\n$$ r^{\\frac{m^2}{4}-\\alpha(2\\sqrt{\\xi}m-2q)^2} \\leq r^{\\frac{m^2}{4}- \\frac{\\alpha\\xi m^2}{4}} \\leq  r^{\\frac{m^2}{4}-\\frac{1}{10^4}\\xi m^2}$$\nfor  $\\alpha \\geq 1\/10^3$.\n\nIf $q \\geq 3\\sqrt{\\xi} m\/4$, equation~\\eqref{eq:26colors2} is at most\n$$\\left(\\frac{B(r)}{r}\\right)^{\\frac{15\\xi m^2}{16}} \\cdot r^{\\frac{m^2}{4}} \\stackrel{\\eqref{bound:B}}{\\leq} r^{\\frac{m^2}{4}-\\frac{1}{10^4}\\xi m^2},$$\nas for $2 \\leq r \\leq 26$\n$$\n\\left(\\frac{B(r)}{r}\\right)^{\\frac{15}{16}}  \\cdot r^{\\frac{1}{10^4}} \\leq \\left(\\frac{r-1}{r}\\right)^{\\frac{15}{16}}  \\cdot r^{\\frac{1}{10^4}} \\leq  e^{-\\frac{15}{16r} + \\frac{1}{10^4} \\ln r } \\leq 1.\n$$\n\n\nCombining the above cases, and using the upper bound~\\eqref{eq:coloring_of_g}, we conclude that the number of $K_3^{(2)}$-free colorings of the graph $G$ satisfies\n\\begin{eqnarray*}\n|\\mathcal{C}_{r,(K_3,\\leq 2)}(G)| &\\leq &  M^n \\cdot  2^{(H((r+1) \\eta)) n^2 + r M^2 \/ 2} \\cdot r^{(r+1) \\eta n^2} \\cdot\\left(  r^{\\frac{m^2}{4}-\\frac{1}{10^4}\\xi m^2}\\right)^{\\left( \\frac{n}{m} \\right) ^ 2} \\\\\n &\\stackrel{n \\gg 1}{\\ll}& r^{\\ex(n, K_3)},  \\label{eq:result014}\n\\end{eqnarray*}\nas $\\xi > (10^4+1) \\cdot (r+1) \\cdot \\eta + 10^4 \\cdot H((r+1)\\eta)$, which is a contradiction to the hypothesis that $ |\\mathcal{C}_{r,(K_3,\\leq 2)}(G)|  \\geq r^{\\ex(n,K_3)}$ and proves Claim~\\ref{claim13}. \n\nTo conclude the proof of Claim~\\ref{claim13} (and thus of Lemma~\\ref{lemma:main_result}), we still need to prove Lemma~\\ref{lemma:claim13}. To this end, fix an integer $r$ such that  $13 \\leq r \\leq 26$, and let $\\mathcal{H}$ be a $K^{(2)}_3$-free multicolored cluster graph for which all edges are green. Recall that we are assuming that no edges of $\\mathcal{H}$ have lists of size less than two. \n\n\\subsection{Proof of Lemma~\\ref{lemma:claim13}}\n\nThe proof of Lemma~\\ref{lemma:claim13} will be by induction, and we shall split the set $\\mathcal{S}$ of $K_3^{(2)}$-free cluster graphs for which all edges are green into two classes. One such class, called $\\mathcal{S}_1$ contains all $K_4$-free $\\mathcal{H}\\in \\mathcal{S}$ such that there is no copy of $K_3$ whose three edges have color lists of size at least four\n\\begin{lemma}\\label{lemma26:s1<27}\nGiven $\\mathcal{H} \\in \\mathcal{S}_1$ with $m$ vertices, the following holds for $13 \\leq r \\leq 26$ and $0 < \\alpha \\leq \\frac{1}{1000}$:\n\\begin{eqnarray*}\nc(\\mathcal{H})  \\leq r^{\\frac{1}{4} - \\alpha}. \n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nFix $\\mathcal{H} \\in \\mathcal{S}_1$ with $m$ vertices. If $m=1$, there is nothing to prove. If $m=2$ and $r=13$, we have $c(\\mathcal{H}) \\leq r_0 =6 \\leq 13^{1-\\alpha}$ for $\\alpha \\leq  0.3014 \\leq (\\log{13}-\\log{6})\/\\log{13}$. For $r\\geq 14$, we have \n$$c(\\mathcal{H}) \\leq r_0 \\leq r-2\\sqrt{r} = r\\left(1-\\frac{2}{\\sqrt{r}} \\right) \\leq r e^{-2\/\\sqrt{r}} \\leq r^{1-\\alpha}$$ for \n$\\alpha \\leq 0.1203 <\\frac{2}{\\sqrt{26} \\ln 26} \\leq \\frac{2}{\\sqrt{r} \\ln r}$.\n \n Next assume $m\\geq 3$. By the definition of $\\mathcal{S}_1$,  the set $E_4\\cup \\cdots \\cup E_{r_0}$ cannot induce a copy of $K_3$. Thus, by Tur\\'{a}n's Theorem, we have\n\\begin{equation*}\n|E_4| +\\cdots + |E_{r_0}| \\leq \\ex(m, K_3) \\leq \\frac{1}{4}  m^2.\n\\end{equation*}\n\nAgain by the definition of $\\mathcal{S}_1$,  the set $E_2\\cup \\cdots \\cup E_{r_0}$ cannot contain a copy of $K_4$, leading to\n\\begin{equation*}\n|E_2| +\\cdots + |E_{r_0}| \\leq \\ex(m, K_4) \\leq \\frac{1}{3}  m^2.\n\\end{equation*}\n\nIf we set $x_i=|E_i|\/m^2$ for $i \\in \\{2,\\ldots,r_0\\}$, by~\\eqref{def_cH}, we have\n$$c(\\mathcal{H}) = \\prod_{j=2}^{r_0} j^{x_j},$$ so that the logarithm $\\ln c(\\mathcal{H})$ is bounded above by the solution to the linear program\n\\begin{eqnarray*}\n&\\max& \\sum_{j=2}^{r_0} x_j \\ln j\\\\\n&\\textrm{s.t.}&   \\sum_{j=4}^{r_0} x_j\\leq \\frac{1}{4},~~\n\\sum_{j=2}^{r_0} x_j \\leq \\frac{1}{3}\\\\\n&&x_j \\geq 0,~j\\in \\{2,\\ldots,r_0\\}.\n\\end{eqnarray*}\nSolving this linear program gives the optimal solution $x_{r_0}=1\/4$, $x_3=1\/12$ and $x_j=0$ for the remaining values of $j$. For $14\\leq r\\leq 26$, we have\n\\begin{eqnarray}\n&& r_0^{\\frac14} \\cdot 3^{\\frac{1}{12}} \\leq \\left( r - 2 \\sqrt{r}\\right)^{\\frac{1}{4}}\\cdot 3^{\\frac{1}{12}} \\leq r^{\\frac{1}{4} - \\alpha}\\nonumber \\\\\n&\\Longrightarrow&  r^{\\alpha} \\cdot \\left( 1 - \\frac{2}{\\sqrt{r}} \\right)^{\\frac{1}{4}} \\cdot 3^{\\frac{1}{12}} \\leq  26^{\\alpha} \\cdot \\left( 1 - \\frac{2}{\\sqrt{26}} \\right)^{\\frac{1}{4}} \\cdot 3^{\\frac{1}{12}} \\leq  1. \\label{eq_final}\n\\end{eqnarray}\nFor $r=13$, we get\n\\begin{equation}\\label{eq_final2}\n13^{\\alpha} \\cdot \\left( \\frac{6}{13} \\right)^{\\frac{1}{4}} \\cdot 3^{\\frac{1}{12}} \\leq 1.\n\\end{equation}\n Equations~\\eqref{eq_final} and~\\eqref{eq_final2} hold for $0 < \\alpha \\leq 0.0101$. This leads to $c(\\mathcal{H}) \\leq  r^{1\/4 -\\alpha}$.\n\\end{proof}\n\nTo complete the proof of Claim~\\ref{claim13}, we consider the cluster graphs that are not considered in Lemma~\\ref{lemma26:s1<27}. \nIn our arguments, we use the following optimization problem for given positive integers $p \\geq 2$ and $L$:\n\\begin{eqnarray}\\label{maxlemma}\n&\\max& \\prod_{j=1}^c x_j \\\\\n&\\textrm{s.t.}& c, x_1,\\dots , x_c\\in {\\mathbb N} = \\{1,2,\\ldots \\} \\nonumber \\\\\n&& x_1+\\cdots+x_c \\leq p\\nonumber\\\\\n&& c \\leq L.\\nonumber\n\\end{eqnarray}\n\n\\begin{definition}\\label{def:cs}\nGiven positive integers $k \\geq 2$ and $r \\geq 2$, let\n\\begin{itemize}\n\\item[(i)] $c_k(r)$ be the maximum of the optimization problem~\\eqref{maxlemma} with $p=r\\cdot \\lfloor k\/2 \\rfloor$ and $L=\\binom{k}{2}$; \n\n\\item[(ii)] $c_k^*(r)$ be the maximum of the optimization problem~\\eqref{maxlemma} with $p=r$ and $L=k$. \n\\end{itemize}\n\\end{definition}\n\nWe shall use the following three straightforward lemmas.\n \\begin{lemma}\\label{gen_claim}\nLet $k \\geq 3$, let $\\mathcal{H}$ be a $K_3^{(2)}$-free multicolored cluster graph such that $|L_e| \\geq 2$ for all $e \\in E(\\mathcal{H})$, and assume that $A \\subset  V(\\mathcal{H})$ is such that $\\mathcal{H}[A]$ is isomorphic to $K_k$. For a vertex $v \\in V(\\mathcal{H})$ let $E'(v)=\\{\\{v,x\\} \\in E(\\mathcal{H}) \\colon x \\in A\\}$. For any $v \\in V(\\mathcal{H}) \\setminus A$, it holds that\n \\begin{equation}\\label{gen_UB}\n \\prod_{e \\in E'(v)} |L_e| \\leq c^*_k(r) \\leq \\overline{c}_k(r)=\\max\\left\\{\\left(\\frac{r}{j}\\right)^{j} \\colon j \\in \\{1,\\ldots,k\\}\\right\\}.\n \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor each edge $e \\in E'(v)$, set $x_e=|L_e|$. Because $A$ induces a clique and $\\mathcal{H}$ is $K_3^{(2)}$-free, the lists associated with edges between $v$ and $A$ are mutually disjoint, so that $\\sum_{e \\in E'(v)} x_e \\leq r$. Let $j\\leq k$ be the number of edges between $v$ and $A$. It is clear that \n$$\\prod_{e \\in E'(v)} |L_e| \\leq \\max\\left\\{\\prod_{i=1}^j a_i \\colon 1 \\leq j \\leq k, a_1,\\ldots,a_j>0, a_1+\\cdots+a_j \\leq r\\right\\}=c_k^*(r).$$\nThe result follows because for $a_1+\\cdots+a_j \\leq r$ it is\n$$\\prod_{i=1}^j a_i\\leq \\left(\\frac{r}{j}\\right)^j.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:UC}\nLet $r$ and $k \\geq 3$ be positive integers. For $j \\geq 1$, consider a partition $E(K_{k}) = E_1 \\cup \\cdots \\cup E_j$ of the edge set of the complete graph $K_k$ and integers $1 \\leq s_1, \\ldots , s_{j} \\leq r$ such that\n $$r \\left\\lfloor \\frac{k}{2} \\right\\rfloor < \\sum_{i=1}^j |E_i| s_i.$$ \nThen, for any assignment of color lists in $[r]$ to the edges of $K_k$ such that, for each $i$, all edges $e \\in E_i$ have list size at least $s_i$, there exists a copy of $K_3$ for which two of the lists have non-empty intersection.  \n \\end{lemma}\n \n\\begin{proof}\nAssume that  is an assignment of lists as in the statement such that, for all copies of $K_3$ in $K_k$, the lists associated with any two of its edges are disjoint. This means that, for every color $\\alpha$, the edges whose lists contain $\\alpha$ form a matching in $K_k$. Since a maximum matching in $K_k$ has size $\\lfloor k\/2 \\rfloor$, we must have\n$$ r   \\left\\lfloor \\frac{k}{2} \\right\\rfloor \\geq \\sum_{e \\in E(K_k)} |L_e| \\geq  \\sum_{i=1}^j |E_i| s_i,$$\ncontradicting our assumption about $r$ and $k$.\n \\end{proof}\n\n\\begin{lemma}\\label{lemma(ck)}\nLet $r\\geq 2$ and $k\\geq 3$ be integers. Let $\\mathcal{H}$ be a $K_3^{(2)}$-free multicolored cluster graph whose underlying graph is $K_k$ and whose edge lists are contained in $[r]$ and have size at least two. Then \n$$\\prod_{e \\in E(\\mathcal{H})} |L_e| \\leq \\tilde{c}_k(r)=\\left(\\frac{r}{\\binom{k}{2}}\\left\\lfloor \\frac k2 \\right\\rfloor \\right)^{\\binom{k}{2}}.$$\n\\end{lemma}\n\n\\begin{proof}\nGiven an edge $e \\in E(\\mathcal{H})$, let $x_e=|L_e|$. Let $E_i$ denote the set of edges of $\\mathcal{H}$ whose lists have size $i$. By Lemma~\\ref{lemma:UC}, $\\sum_{e \\in E(\\mathcal{H})} x_e = \\sum_{i=2}^r i \\cdot |E_i| \\leq r \\lfloor k\/2 \\rfloor$, since $\\mathcal{H}$ is $K_3^{(2)}$-free.\n\nIn particular, the vector $(x_e)_{e \\in E(\\mathcal{H})}$ is a feasible solution to the optimization problem~\\eqref{maxlemma} with $p=r \\lfloor k\/2 \\rfloor$ and $L=\\binom{k}{2}$. For the inequality, observe that for any choice of $j$ positive real numbers such that $a_1+ \\cdots + a_j \\leq r\\lfloor k\/2\\rfloor$, we have \n$$\\prod_{i=1}^{\\binom{k}{2}} a_i \\leq \\left(\\frac{r}{j}\\left\\lfloor \\frac k2 \\right\\rfloor \\right)^{\\binom{k}{2}}.$$\nThis concludes the proof.\n\\end{proof}\n\nWe are now ready to prove the desired result.\n\\begin{lemma}\\label{lemma26:s2<27}\nFix an integer $r$ such that $13 \\leq r \\leq 26$. Given $\\mathcal{H} \\in \\mathcal{S} \\setminus \\mathcal{S}_1$ and $0 < \\alpha\\leq  \\frac{1}{1000}$, we have\n\\begin{eqnarray*}\\label{eq:lemma26}\nc(\\mathcal{H}) \\leq r^{\\frac14 - \\alpha}.\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nLet $r \\in \\{13,\\ldots,26\\}$. For a contradiction, assume that the result is false and choose a counterexample $\\mathcal{H} \\in \\mathcal{S} \\setminus \\mathcal{S}_1$ with the minimum number of vertices. Recall that the edges of $\\mathcal{H}$ have lists with sizes between $2$ and $r_0$. Let $m$ be the number of vertices of $\\mathcal{H}$. We first show that $\\mathcal{H}$ is not isomorphic to a clique $K_m$ such that $3 \\leq m \\leq 6$.\n\nFor $\\mathcal{H}$ isomorphic to $K_3$, Lemma~\\ref{lemma(ck)} tells us that $c(\\mathcal{H})^{9} \\leq \\tilde{c}_3(r) = (r\/3)^3$. Lemma~\\ref{lemma26:s2<27} holds in this case because $c(\\mathcal{H}) \\leq {((r\/3)^3)}^{\\frac{1}{9}} \\leq r^{\\frac{1}{4} - \\alpha}$ for $r^{\\frac{1}{12}+\\alpha} \\leq 26^{\\frac{1}{12}+\\alpha} \\leq 3^{\\frac13}$, which in turn holds for\n$$\\alpha \\leq 0.02906<  \\frac{4\\ln{3}-\\ln{26}}{12 \\ln 26} .$$\n\nFor $\\mathcal{H}$ isomorphic to $K_4$, Lemma~\\ref{lemma(ck)} gives $c(\\mathcal{H})^{16} \\leq \\tilde{c}_4(r) =(r\/3)^6$. Therefore, we need $c(\\mathcal{H}) \\leq {((r\/3)^6)}^{\\frac{1}{16}} \\leq  r^{\\frac{1}{4} - \\alpha}$, which holds for \n$$\n\\alpha \\leq 0.00144< \\frac{3\\ln{3}-\\ln{26}}{8 \\ln 26}.\n$$\n\nIf $\\mathcal{H}$ is isomorphic to $K_5$, Lemma~\\ref{lemma(ck)} gives $c(\\mathcal{H})^{25} \\leq \\tilde{c}_5(r) = (r\/5)^{10}$. Therefore, $c(\\mathcal{H}) \\leq ((r\/5)^{10})^{\\frac{1}{25}} ={(r\/5)^{2\/5}}  < r^{\\frac{1}{4} - \\alpha}$ if\n$$\n\\alpha \\leq 0.04759 < \\frac{8 \\ln 5 - 3 \\ln 26}{20 \\ln 26}.\n$$\n\nFinally, if $\\mathcal{H}$ is isomorphic to $K_6$, by  Lemma~\\ref{lemma(ck)} the product of the sizes of the color lists of $\\mathcal{H}$ is at most  $\\tilde{c}_6(r)$. We get \n\\begin{eqnarray*}\nc(\\mathcal{H})^{36} &\\leq& (\\tilde{c}_6(r))^{\\frac{1}{36}} = \\left( \\frac{r}{5} \\right)^{\\frac{15}{36}} \n\\leq r^{\\frac{1}{4} - \\alpha},\n\\end{eqnarray*}\nwhich holds for \n$$ \\alpha \\leq 0.03915<  \\frac{15 \\ln 5 - 6 \\ln 26}{36 \\ln 26}.\n$$\n\nHaving established that $\\mathcal{H}$ is not isomorphic to a clique on $3\\leq m \\leq 6$ vertices, let $\\omega(\\mathcal{H})\\geq 3$ denote the size of a maximum clique in $\\mathcal{H}$. If $\\omega(\\mathcal{H})\\geq 6$, then the fact that $\\mathcal{H}\\neq K_6$ implies that $m>6$. Fix $k=6$ and choose a set $A$ of vertices such that $A$ induces a copy of $K_6$ in $\\mathcal{H}$. Otherwise, let $k=\\omega(\\mathcal{H})$ and fix a set $A$ of vertices of size $k$ that induces a copy of $K_k$ in $\\mathcal{H}$. By the above, we know that $m>k$ in this case. Given a vertex $v \\in V(\\mathcal{H})\\setminus A$, let $c_v$ be the product of the sizes of the lists on edges connecting $v$ to $A$. Clearly,\n\\begin{equation}\\label{eq_UB}\nc(\\mathcal{H})^{m^2} = c(\\mathcal{H}[A])^{k^2} \\cdot \\left(\\prod_{v \\in V(\\mathcal{H})\\setminus A} c_v \\right) \\cdot  c(\\mathcal{H}[V(\\mathcal{H})\\setminus A])^{(m-k)^2}.\n\\end{equation}\nWe know that $c(\\mathcal{H}[A]),c(\\mathcal{H}[V(\\mathcal{H})\\setminus A]) \\leq  r^{\\frac{1}{4} - \\alpha} $ by the minimality of $\\mathcal{H}$. \n\nIf $k=6$, we have $c_v \\leq \\overline{c}_{6}(r)$ by Lemma~\\ref{gen_claim}. Then~\\eqref{eq_UB} leads to\n\\begin{eqnarray*}\nc(\\mathcal{H})^{m^2} &\\leq &  r^{(\\frac{1}{4}- \\alpha)6^2} \\cdot (\\overline{c}_{6}(r))^{m-6} \\cdot r^{(\\frac{1}{4}- \\alpha)(m-6)^2}\\nonumber \\\\\n&=& \\left(\\frac{\\overline{c}_6(r)}{r^{3 - 12\\alpha}}\\right)^{m-6} \\cdot  r^{(\\frac{1}{4} - \\alpha)m^2}.\n\\end{eqnarray*}\nWe conclude that $c(\\mathcal{H})^{m^2} \\leq  r^{(\\frac{1}{4} - \\alpha)m^2}$ because $\\overline{c}_6(r)\/r^{3-12\\alpha} <1$ for $13 \\leq r \\leq 26$. Here, it suffices to verify that the quantity $\\overline{c}_6(r)$ defined in~\\eqref{gen_UB} satisfies $\\overline{c}_6(r)<r^{3-12\\alpha}$. We have \n$$\\overline{c}_6(r)=\n\\begin{cases}\n(r\/6)^6& \\textrm{ if } r\\geq 15\\\\\n(r\/5)^5& \\textrm{ if } 13\\leq r\\leq 14.\\\\\n\\end{cases}\n$$ \nAs it turns out, $\\overline{c}_6(r)<r^{3-12\\alpha}$ for \n$$\\alpha \\leq 0.0249< \\frac{2\\ln{6}-\\ln{26}}{4 \\ln{26}}.$$\n\nFor $k<6$, the maximality of the clique implies that any vertex  $v \\in V(\\mathcal{H})\\setminus A$ has at most $k-1$ neighbors in $A$, so that~\\eqref{eq_UB} becomes\n\\begin{eqnarray}\\label{UB_otherk}\nc(\\mathcal{H})^{m^2} &\\leq &  r^{(\\frac{1}{4}- \\alpha)k^2} \\cdot (c^*_{k-1}(r))^{m-k} \\cdot r^{(\\frac{1}{4}- \\alpha)(m-k)^2}\\nonumber \\\\\n&=& \\left(\\frac{c^*_{k-1}(r)}{r^{\\frac{k}{2} - 2k\\alpha}}\\right)^{m-k} \\cdot  r^{(\\frac{1}{4} - \\alpha)m^2}\n\\end{eqnarray}\n To obtain our result, we show that $c^*_{k-1}(r)<r^{\\frac{k}{2} - 2k\\alpha}$ for all suitable $k$ and $r$.\n \n If $k=5$, we use that $c^*_{4}(r)\\leq \\overline{c}_{4}(r)=(r\/4)^4\\stackrel{(i)}<r^{\\frac{5}{2} - 10\\alpha}$ for $2 \\leq r\\leq 26$. Indeed, (i) is equivalent to $r^{10\\alpha+\\frac{3}{2}}\\leq 4^{4}$, which holds if \n $$\\alpha \\leq 0.0201<\\frac{8\\ln{4}-3\\ln{26}}{20 \\ln{26}}.$$ \n\nIf $k=4$, we use that $c^*_{3}(r)\\leq \\overline{c}_{3}(r)=(r\/3)^3\\stackrel{(ii)}<r^{2 - 8\\alpha}$ for $2 \\leq r\\leq 26$. Indeed, (ii) holds for\n$$\\alpha \\leq 0.0014 <\\frac{3\\ln{3}-\\ln{26}}{8 \\ln{26}}.$$ \nWe observe that, for $r=27$, there is no $\\alpha > 0$ for which~(ii) holds.\n\nIf $k<4$, we know by the hypothesis that $\\mathcal{H} \\notin \\mathcal{S}_1$ that $\\mathcal{H}$ is $K_4$-free, but contains a copy of $K_3$ whose edges have lists of size at least four. We fix such a $3$-vertex set $A \\subset V(\\mathcal{H})$ that induces a copy of $K_3$ whose edges have lists of size at least four. If $v \\in V(\\mathcal{H}) \\setminus A$, then $v$ has at most two neighbors in $A$. If $v \\in V(\\mathcal{H}) \\setminus A$ has at most one neighbor in $A$, then its list has size at most $r_0$. If $v \\in V(\\mathcal{H}) \\setminus A$ has exactly two neighbors in $A$, say $v_1$ and $v_2$, then these edges form a triangle with an edge in the copy of $K_3$. Since the list of the edge $\\{v_1, v_2 \\}$ has size  at least four, the product $c_v$ of the sizes of the lists associated with the two edges between $A$ and $v$ is at most $(r-4)^2\/4=s^*(r)$. We observe that $r_0 < s^*(r)$ for all $r\\geq 13$. With this, the inequality~\\eqref{UB_otherk} may be sharpened as\n$$c(\\mathcal{H})^{m^2} \\leq  \\left(\\frac{(r-4)^2}{4r^{\\frac{3}{2} - 6\\alpha}}\\right)^{m-k} \\cdot  r^{(\\frac{1}{4} - \\alpha)m^2}.$$ \nNote that $(r-4)^2 \\leq 4 \\cdot r^{\\frac{3}{2} - 6 \\alpha}$ is equivalent to \n$$r^{\\frac{1}{2}+6\\alpha}\\left(1-\\frac{4}{r}\\right)^2\\leq 4.$$ For $\\alpha>0$, the left-hand side is increasing as a function of $r$, so this holds for $13 \\leq r \\leq 26$ because $$26^{\\frac{1}{2}+6\\alpha}\\left(1-\\frac{4}{26}\\right)^2 \\leq 4$$ \nholds for $\\alpha \\leq 0.0046$. This concludes the induction step and proves Lemma~\\ref{lemma26:s2<27}.\n\\end{proof}\n\n\\section{Final remarks and open problems}\n\nThe objective of this paper was to characterize the values of $r \\geq 2$ for which the bipartite Tur\\'{a}n graph $T_2(n)$ is the unique $(r,K_3^{(2)})$-extremal graph for all sufficiently large $n$. With Theorem~\\ref{theorema:main}, we established that this holds precisely for $2\\leq r\\leq 26$. In this section, our aim is to put this result in a more general perspective and to discuss a few open problems.\n\nLet $P$ be a pattern of a complete graph $K_k$. The following facts are known to hold (see~\\cite{eurocomb15}):\n\\begin{itemize}\n\n\\item[(a)] If $P=K_k^R$, the rainbow pattern of $K_k$, then there exists $r_0$ such that the following holds for all $r\\geq r_0$. There exists $n_0$ such that, for all $n \\geq n_0$, the unique $n$-vertex $(r,P)$-extremal graph is the Tur\\'an graph $T_{k-1}(n)$.\n\n\\item[(b)] If $P\\neq K_k^R$, then there exists $r_1$ such that the following holds for all $r\\geq r_1$. There exists $n_0$ such that, for all $n \\geq n_0$, the Tur\\'{a}n graph $T_{k-1}(n)$ is \\emph{not} $(r,P)$-extremal.\n\\end{itemize}\nThis raises natural questions.\n\\begin{prob}\nGiven $k\\geq 3$, let $r_0(k)$ be the least value of $r_0$ such that (a) holds. Determine $r_0(k)$.\n\\end{prob}\nIt is known that $r_0(3)=4$~\\cite{baloghli}. For $k\\geq 4$, upper and lower bounds on $r_0(k)$ have been provided in~\\cite{multipattern,eurocomb15}, but we believe that the upper bounds in these papers are much larger than the actual value of this parameter. \n\n\\begin{prob}\nGiven $k\\geq 3$, characterize the $n$-vertex $(r,K_k^R)$-extremal graphs for $r<r_0(k)$. \n\\end{prob}\nTrivially, any such extremal graph must be isomorphic to $K_n$ for $r<\\binom{k}{2}$, but it is not clear what happens for the remaining values of $r<r_0(k)$. In the case $k=3$, the only remaining value is $r=3$, and it is known that $K_n$ is the unique extremal graph in this case (for $n$ large.) To the best of our knowledge, no pair $(k,r)$ has been found for which there are arbitrarily large $(r,K_k^R)$-extremal graphs that are neither isomorphic to $T_{k-1}(n)$ nor to $K_n$.\n\n\\begin{prob}\nGiven $k\\geq 3$ and a pattern $P$ of $K_k$ such that $P\\neq K_k^R$, let $r_1(P)$ be the greatest value of $r_1$ such that (b) holds. Determine $r_1(P)$ and \n$$r_1(k)=\\max \\{r_1(P) \\colon  P \\textrm{ pattern of }K_k,~P\\neq K_k^R\\}.$$\n\\end{prob}\nIn this paper, we showed that $r_1(K_3^{(2)})=2$. It is known that, for any monochromatic pattern $K_k^M$, we have $r_1(K_k^M)=4$. In particular $r_1(3)=27$.\n\n\\begin{prob}\\label{prob4}\nGiven $k\\geq 3$ and a pattern $P$ of $K_k$ such that $P\\neq K_k^R$, characterize the $n$-vertex $(r,P)$-extremal graphs for $r>r_1(P)$. \n\\end{prob}\nFor $P=K_3^{(2)}$ and $r=27$, we believe that $T_4(n)$ is the unique $(r,P)$-extremal graph for all sufficiently large $n$, but we do not have a proof of this. Moreover, the work in~\\cite{BHS17} implies that, except for the patterns $K_k^M$ and $K_3^{(2)}$, the $(r,P)$-extremal graphs mentioned in Problem~\\ref{prob4} must be complete multipartite graphs. However, it is not known whether the partition must always be equitable. Recent work by Botler et al.~\\cite{botler} and by Pikhurko and Staden~\\cite{PS2022} shows that, for some monochromatic patterns of complete graphs, unbalanced complete multipartite graphs can be very close to extremal, and may even be extremal in some cases.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}  \n    As quantitatively pioneered by those such as \n\\cite{Hoyle}~(1953), \\cite{Silk77}~(1977) and  \\cite{ReesO78}~(1978) and more\nrecently noted by authors such as Frenk \\mbox{et\\,al.}\\ (1995), hierarchical galaxy\nformation models in $\\Omega_0=1$ cold dark matter (CDM) universes \ntypically combine assumptions on up to six distinct physical\nprocesses: (1) the non-linear growth \nphase of matter density peaks (known as ``haloes''),\n(2) cooling gas dynamics, (3) star formation, (4) star-to-gas\nenergy feedback, (5) stellar evolution, (6) galaxy mergers. In principle,\nif there are more free parameters describing these processes\nthan independent observational galaxy statistics, \nthen the observations should provide little constraint on\n galaxy formation ``recipes''.\nFortunately, the contrary is\npresently the case for the ``semi-analytical {\\em ab initio}'' models which \nmake various \nanalytical estimates of process (1), \ncombine semi-empirical and simple scaling \nparametrisations to represent processes (2)-(4) and (6) and\nuse evolutionary stellar population synthesis for process (5). \nSince each of these models have problems explaining at least some \nof the observations\nmeans, the models are better constrained than might\nhave been hoped for.\n\n\tThese models can be considered to be semi-analytical because rather\nthan calculating what is possibly the most important process, \nthe non-linear formation and merging history \nof collapsed objects [process (1)], via N-body simulations, various \nstatistical analytical approximations are used.\nThe models of Lacey \\mbox{et\\,al.}\\ (1993) use an approximation developed \nin \\cite{LS91}~(1991) from the BBKS peaks formalism \n(\\cite{BBKS}~1986),\nKauffmann et {al.} (\\cite{KW93}~1993; \\cite{KWG93}~1993)\nuse a probabilistic method (\\cite{Bower91}~1991) \nbased on the Press-Schecter formalism \n(e.g., \\cite{PS74}~1974; White \\& Frenk 1991) and \nexcursion set mass function calculations (\\cite{Bond91}~1991) while \n\\cite{ColeGF94}~(1994) use a spatially quantised ``block'' method\ndescribed in \\cite{ColeKais}~(1988). \n\nFurther semi-analytical developments \ninclude those adding spatial auto-correlation information \nto a Press-Schechter formalism (\\cite{MoWh96}~1996) or to the\n``block'' model (\\cite{RT96}~1996; \\cite{Naga97}~1997) and a technique \nof separately treating global, weakly non-linear and local, strongly \nnon-linear dynamics (the ``peak-patch'' formalism, \\cite{BoMy96}~1996).\n\nEach of the models which has been compared \nto observational statistics has difficulty in \nsimultaneously explaining the flatness of the\npresent-day (surface-brightness limited) galaxy luminosity function\n(e.g., Loveday \\mbox{et\\,al.}\\ 1992), the steepness of the faint galaxy counts\n(e.g., \\cite{TSei88}~1988; \\cite{Tys88}~1988), the shape of the moderately\nfaint galaxy ($B\\ltapprox23$) spectroscopic redshift distributions \n(e.g., \\cite{Coll90}~1990; \\cite{Coll93}~1993), the Tully-Fisher\nrelation and the colour distributions of present-day galaxies, \nin a CDM $\\Omega_0=1$ universe. Even though the \\cite{ColeGF94}~(1994) model\nis better than the previous models in allowing big $z=0$ galaxies to\nbe at least as red as higher $z$ galaxies, it \nshares the problem of the other models in lacking big red\nellipticals. It also shares with the \\cite{KWG93} model the problem\nthat if the large number of small haloes predicted by CDM models at \n$z\\approx0$\nfollow the IR Tully-Fisher relation (e.g., \\cite{PTully92}~1992), \nthen the slope of \nthe faint end of the general galaxy luminosity function should be\nsteeper than that estimated locally (e.g.,\nLoveday \\mbox{et\\,al.}\\ 1992). Changing the cosmology in the \n\\cite{ColeGF94} models \n(\\cite{Heyl95}~1995: low $H_0$, low $\\Omega_0$, non-zero $\\lambda_0$ and\nCHDM models) is insufficient to match the observations.\nAnother way of allowing these models to fit the\nobservations is to make a strong assumption for process (6)---to\nsuppose that\ngalaxies can merge as fast as galaxy haloes merge, or even faster---but \nsimple present-day \nconstraints on the products of the mergers \n(\\cite{Dalc93}~1993) and the relative weakness of the faint galaxy \nangular auto-correlation function (\\cite{RY93}~1993) strongly\nrestrict this possibility.\n\nIn order to avoid problems which may be due to the approximation of \nnon-linear gravitational collapse and evolution by the \nsemi-analytical techniques \nmentioned above, an alternative technique is to calculate both processes (1)\nand (2) from first principles \nin numerical N-body simulations, folding in the other physical \nprocesses as simple scaling formulae or using stellar population synthesis\nfor (5). Several authors \n(e.g., \\cite{Ev88} 1988; \\cite{NavB91}~1991; \n\\cite{CO92cdmhydro}~1992; \\cite{Ume93}~1993; \\cite{StM95}~1995) \nhave \nexperimented with these techniques, but resolution limits\non present-day computers make the results hard to interpret. For example,\n\\cite{WeinbHK97}~(1997) point out that although low resolution \ngravito-hydrodynamic simulations suggest that a UV photoionisation \nbackground can suppress galaxy formation (by heating the gas so that\nit is unable to cool and form stars),  higher resolution simulations\nshow that this is a numerical artefact: the higher resolution \nsimulations show little sensitivity to either the details of\nphotoionisation or star formation.\n\n\tIn this article, rather than claiming a global ``recipe'' for galaxy\nformation, our primary purpose is to concentrate on \nprocess (1) in a way complementary to that of other techniques. \nThis is unlikely to be sufficient to solve all the observational conflicts.\nOn the contrary, this method should increase the ability of modellers \nto verify the extent to which model predictions are sensitive to the \nprecision of modelling of gravity.\n\n\tThe method presented here is \nto derive merging history trees of dark matter haloes \ndirectly from N-body simulations. \nRather than just investigating virialised haloes for\na particular dark matter model (e.g., CDM), (a) both $n=-2$ and $n=0$\ninitial perturbation spectrum simulations \n(where $n$ is the index of the power spectrum)\nare examined, and (b) since the halo-to-galaxy relationship may be\nmore complex than a simple one-to-one mapping, two significantly different\ndensity thresholds are used for halo detection. \nThis reveals the sensitivity of halo\nmerger history trees and halo statistics to these parameters. \nThe N-body simulations used are presented in \\S\\ref{s-N-body}, \nthe choice of a group-finding algorithm in \\S\\ref{s-peakalg} and \nthe defining criterion and algorithm for calculating the merging history trees\nin \\S\\ref{s-tree}.\n\n\tProperties of the haloes detected are discussed in \\S\\ref{s-halostats}.\nIn particular, the resulting merging history trees are presented in graphical\nform in \\S\\ref{s-mhtrees}, enabling patterns of halo\nmerging calculated from fully non-linear simulations to be visualised directly.\n\n\tIf processes (2)-(5) are simple enough, and if process (6), \ngalaxy merging, corresponds in a one-to-one way with halo merging,\nthen these halo merger history trees would lead directly to galaxy\nmerger history trees. We therefore examine an example application\nof the merger history trees by \nmaking minimal assumptions for processes (2)-(4), using stellar evolutionary\npopulation synthesis for process (5), and for process (6), \nassuming maximal galaxy merging (every halo merger corresponds to\na galaxy merger). \n\\S\\ref{s-bursts} presents \n(6)+(3) merger-induced star formation and \\S\\ref{s-geps} explains how\nprocess (5) is modelled.\n\nIn order for these processes to have an effect on\nthe luminosity function, an option is considered in which\neach merger induces a burst of star formation, \nscaled according to the appropriate halo and gas masses\nand the dynamical time scale. \nApart from this star formation rate option, \nwe do not explore parameter space for non-gravitational processes \nin this paper; we merely adopt simple observationally normalised \nscaling laws.\nResulting luminosity functions are presented in \\S\\ref{s-galstats}.\n\n\tApplications of N-body derived \nhalo merger trees with more complex assumptions for processes (2)-(6)\nare of course possible, and indeed to be welcomed. The galaxy formation\n``recipe'' explored here is only one simple example. \n\n\tCosmological conventions adopted for this paper are a Hubble constant \nof $H_0=50 \\,$km\\,s$^{-1}$Mpc$^{-1}$, comoving units (at $t=t_0$) and \nan $\\Omega_0=1\\.0, \\Lambda=0$ universe is assumed, \nexcept where otherwise specified.\n\n\n\\section{Halo Merger Histories (Gravity)}\n\n\\subsection{Method}\n\n\\subsubsection{N-body Models of Matter Density} \\label{s-N-body} \n\tThe non-linear gravitational evolution of matter \ndensity is modelled by \nN-body cosmological simulations run by \n\\cite{Warr92}~(1992). \nThese simulations use a 128$^3$ initial \nparticle mesh, of side length 10~Mpc. \n[The simulations analysed here \nare for power law initial perturbation spectra ($n=-2$ and $n=0$), so this is \nsimply a default choice for the scaling of units. This default scaling \nis used hereafter except where otherwise specified.]\nParticles are placed on this mesh, making a cube of $\\sim 2 {\\scriptstyle \\times} 10^{6}$\nparticles. \n\nAn initial perturbation spectrum is imposed on this cube by Fourier transforming\nthe initial complex amplitudes from the perturbation spectrum and using \nthe Zel'dovich\ngrowing mode method (\\cite{Warr92}~1992)\non this Fourier transform and the $128^3$ particle mesh.\nThe amplitude of the perturbation spectrum is chosen\nsuch that linear perturbation growth implies that \n$(\\delta M\/M) (r=0\\.5 \\mbox{$h^{-1}\\mbox{\\rm Mpc}$})=2\\.0$ at $z=0,$ where $(\\delta M\/M)(r)$ \nis the r.m.s. value of the excess mass (over uniform\ndensity) in spheres of radius $r$ (\\cite{Warr92}~1992).\nThis choice ensures that the haloes which collapse are about the same size\nfor different values of $n,$ so that the dependence on $n$ of properties \nof halo dynamics---or merging histories---can easily be studied. \nThe absolute normalisation of the spatial correlation function \nof the haloes cannot be directly interpreted in terms of observational\nquantities. The relative amount of power on different scales \n(or slope of the spatial correlation function), and the halo detection\nthreshold, are the parameters which may affect the rates and ways in\nwhich haloes merge with one another.\n\nThe initial cube of perturbed particles is trimmed to a sphere,\ni.e., particles more than $5000\\,$kpc from the centre of the \ncube are removed, resulting in a sphere of $\\sim 1\\.1 {\\scriptstyle \\times} 10^{6}$ particles.\n\nThis is then evolved forward gravitationally via a tree-code \n(e.g., see \\cite{BHut86}~1986), initially\nwith roughly logarithmic time steps up to $t=0\\.3 \\,$Gyr, after which equal \ntime steps of $0\\.03 \\,$Gyr are used. Every hundredth time\nstep is stored on disk; these are the time steps available for\nhalo analysis (hereafter ``time stages''). A vacuum boundary condition \nis used and the softening parameter is 5~kpc (proper units).\n\n\n\\subsubsection{Group-Finding Algorithm} \\label{s-peakalg} \n      The simulation data are searched for density peaks at each \ntime step by an algorithm which \nuses the ``oct-tree'' method to find all overdense regions without \noverwhelming computer memory, followed by an iterative means of joining \ntogether contiguous overdense regions. \n\n\tAlternative group-finding methods which could be used\ninclude the ``friends-of-friends'' (FOF) algorithm \n(e.g., \\cite{WDEF87}~1987), the algorithm used by \\cite{Warr92}~(1992) \nor the DENMAX algorithm (\\cite{GB94}~1994).\n\nThe FOF group-finder has the advantage of low memory requirements and\nan obvious relation between the mean particle separation and the \ngroup-finding resolution, but has the disadvantages that if the link\nparameter $l$ is too low, then low density haloes---or the low density \nenvelopes of haloes---are missed, while if $l$ is higher, small but \ndistinct haloes may be erroneously joined together as single objects.\n\n\\cite{Warr92}'s (1992) \nmethod, based on the accelerations of individual particles,\nand the DENMAX algorithm, which includes a de-binding procedure to separate\nhaloes which are only temporarily close to one another, are both more\nphysically motivated than FOF. However, for a first investigation of \nthe use of N-body generated merging history trees in galaxy formation \nmodels, the use of the simple method outlined below seems prudent. Since \ntwo different density detection thresholds are used, the implications \nof having either a low or a high fixed \ndensity threshold (which are similar to the cases of high or low $l$ \nrespectively in FOF) can be seen. For further \ndevelopment, it would certainly be useful to consider\nuse of a more complex algorithm such as DENMAX.\n\n\tDetails of the method are as follows.\n\n      Conceptually, a cube concentric to the sphere of particles, \nhaving as side length the diameter of the sphere, is divided into\neight equally sized subcubes.\nAny of these subcubes\ncontaining more than one particle is itself subdivided into eight\nsubcubes. By not subdividing cubes \nwith only one or zero particles, computer memory is not wasted on\nanalysing ``empty'' space.\nThe subdividing process is iterated to a depth of $n_{\\mbox{\\rm \\small levels}}$\n levels below the original cube,\nunless at some level \nall the cubes have one or zero particles in them, in which \ncase subdividing stops (this would happen at $n_{\\mbox{\\rm \\small levels}}=8$\nfor this $1\\.1 {\\scriptstyle \\times} 10^{6}$-particle model for a uniform particle\ndistribution).\nThe side length of the smallest cube is $174\\,$kpc and $20 \\,$kpc\nfor $n_{\\mbox{\\rm \\small levels}}=6$ and $n_{\\mbox{\\rm \\small levels}}=9$ \nrespectively at $z=0.$ \n\nThe ``primary'' list of density peaks is then simply the list of \neach cube at the deepest level (i.e., of size $2^{-\\nlev_tiny}$ times the\nsimulation sphere diameter)  whose density is at or \nabove $r_{\\protect\\mbox{\\rm \\small thresh}}$ times the mean density. The list of particles in\neach of these peaks is recorded.\n\nThe results presented here are for $r_{\\protect\\mbox{\\rm \\small thresh}} =5$ and $r_{\\protect\\mbox{\\rm \\small thresh}} =1000.$ \nFor a flat rotation curve of the Galaxy of $220 \\mbox{\\rm\\,km\\,s$^{-1}$},$ \nthe cumulative mass to a radius $r$ is $M(<r)\\propto r,$\nand the density is\n$\\rho(r)= 1\\.2 {\\scriptstyle \\times} 10^{7} \\rho_c  r^{-2}$ for $H_0$ as above and $r$ in kpc.\nSo, detection densities \nof $r_{\\protect\\mbox{\\rm \\small thresh}} =5$ and $r_{\\protect\\mbox{\\rm \\small thresh}} =1000$\ncorrespond to the total (baryonic plus nonbaryonic) matter\ndensity at galactocentric distances of about 1500~kpc and \n110~kpc respectively. The latter is a reasonable\nvalue for the halo radius, but the former is several times \ngreater than the largest radii claimed for the halo of the Galaxy.\n\nThe key to a simple way to join together contiguous ``primary'' peaks\nis to order the primary density peak list \nby mass, from largest to smallest, so that each ``secondary'' peak\ncan be created by starting from its nucleus (densest region) and successively\njoining on regions of lower and lower density which are adjacent\nto the region which has already been \naggregated.\\footnote{The idea to order the ``primary'' peaks by mass \nwas inspired by the group-finding algorithm of \\cite{Warr92}~(1992).}\n\n\nThe parameters used to decide on adjacency are \nthe radius $r$ (from centre to outermost particle) of each secondary peak \nand an ``incremental radius'', \n$r_{\\mbox{\\rm \\small inc}}$, defined as $1\\.1$ times half of the largest diagonal of the\nsmall cube used in finding the primary density peaks. \nA primary peak is\nconsidered adjacent to a secondary peak if its centre is within $r+r_{\\mbox{\\rm \\small inc}}$\nof the secondary peak. \nThe radius $r$ is re-evaluated each time a primary peak is \njoined to a secondary peak.\nThe nucleus of the first secondary peak is the\nfirst (i.e., most massive) primary peak; each following secondary peak \nstarts with the most massive primary peak not previously included in a \nsecondary peak. \n\nThe final secondary peak list, \n corresponding to all separated regions inside isodensity contours\nof $r_{\\protect\\mbox{\\rm \\small thresh}}$ times the mean density, is hereafter simply termed\na peak list, since the list of primary peaks is not of astrophysical\ninterest. \t\nThe members of this list are considered to be dark matter haloes.\n\n\\subsubsection{Creation of History Tree} \\label{s-tree} \n      A peak (halo) merging history tree is obtained as follows. \n\n      Peak lists for a series of time stages of the model are obtained by \nthe algorithm just described, each obtained with the same \nvalues of $n_{\\mbox{\\rm \\small levels}}$ and $r_{\\protect\\mbox{\\rm \\small thresh}} .$ \nFor each pair of successive output \ntimes, $t_i, t_{i+1},$ the peaks at the \ntwo times are compared. Two arrays, $a_i, a_{i+1},$ each \nwith as many elements as the number of particles in the simulation, are \ncreated. For each element $j$ of array $a_i,$ \nthe integer $k$ identifying\nthe peak that particle $j$ is a member of is assigned to $a_i(j)$, where this\nis a peak according to the peak list for $t_i$. \nThe array $a_{i+1}$ is evaluated in the same way using the peak list for\n$t_{i+1}.$\nIf the particle is not a member\nof any peak, a null value is assigned. A simultaneous sort \nis performed on arrays $a_i$ and $a_{i+1}$, \npermuting both in the same way in order that \n$a_{i+1}$ is a non-decreasing arithmetical sequence. \n\nThe result is that with the new ordering of $a_i$ and $a_{i+1},$ \n(1) a peak $k$ at $t_{i+1}$ is represented by a contiguous list \n$a_{i+1}(j)$ to $a_{i+1}(j')$ (each containing the peak number $k$) \nand \n(2) $a_{i}(j)$ to $a_{i}(j')$ (for the same $j,$ $j'$) represent the\nsame particles and contain values $(k_1, k_2, ...)$ indicating the peaks\n(at $t_i$) of which the particles were members.\nIn other words, the peak \nmembership at $t_i$ of particles in a single peak at $t_{i+1}$ \nis listed in $a_{i}(j)$ to $a_{i}(j').$\n\n      For any peak at $t_{i+1}$, if more than 50\\% of the \nparticles in any of the peaks at $t_i$  \nare present in the peak at $t_{i+1}$, \nthen the peak at $t_{i+1}$ is \nconsidered a ``descendant'' of \nthe peak at $t_i$ and the peak at $t_i$ is \na ``progenitor'' of the peak at $t_{i+1}.$\nThese links are represented by appropriate arrays. Due to the nature of this\nalgorithm, no peak can have more than one descendant, though it can certainly \nhave more than one progenitor, which is allowed for by using what are, \nin effect, pointers. \n\n      By applying this comparison \nacross each pair of successive times $t_i,t_{i+1}$, a \nrepresentation of the peak merging history is obtained.\n\n\n\\subsection{Results} \\label{s-halostats} \n\tThe method described above has been applied to\nboth an $n=0$ and\nan $n=-2$ power law initial perturbation spectrum N-body model \n(labelled ``n0b'', ``n-2b'' by \\cite{Warr92}~1992). \nTable~\\ref{t-redsh} shows redshifts and cosmological times for the \noutput timesteps\nfor these two models. \nThe negative redshifts correspond to\nfuture times according to the default time scaling. \nIf the time unit chosen were different to the default, \nthen these latter time stages could be moved into the past\nor the present.\n\n\\begin{table\n\\caption{\\label{t-redsh} Parameters of Time Stages Used}\n$$\\begin{array}{c ccc}\n\\hline \\multicolumn{1}{ c}{redshift} &\n      \\multicolumn{1}{c }{t (\\mbox{\\rm Gyr})} &\n      \\multicolumn{2}{ c }{time step} \\\\\n      \\hline  \n\\multicolumn{2}{ c }{} & \\multicolumn{1}{ c }{n=0 } &\n      \\multicolumn{1}{c }{n=-2 } \\\\ \n      \\cline{3-4}\n       11\\.2 & 0\\.31 &  40  &(15)  \\\\ \n      3\\.2 & 1\\.51    & -    &55 \\\\\n      1\\.5 & 3\\.3    &  140&  115\\\\\n      0\\.62 & 6\\.3   &  240 & 215\\\\\n      0\\.25 &9\\.3    &  340 & 315\\\\\n       0\\.039 & 12\\.3& 440 &  415\\\\\n       -0\\.10 & 15\\.3 & 540 &  515\\\\\n       -0.203 & 18\\.3& 640 &  615\\\\ \\hline\n\\end{array}$$\n\\end{table}\n\n\n Haloes are detected in both of these models at thresholds of both\n$r_{\\protect\\mbox{\\rm \\small thresh}} =5$ and $r_{\\protect\\mbox{\\rm \\small thresh}} =1000$\ntimes the mean universe density\n(for an $\\Omega_0=1\\.0, h=0\\.5, \\lambda_0=0$\nuniverse.)\nThe former density threshold for detection should result in\nhaloes which have only just turned around from following the\nsmooth Hubble flow, while the latter should result in well-virialised haloes.\nThese thresholds should span most cases of interest.\nStatistics of the haloes detected and comparison of these to \nresults from other methods \nare presented in \\S\\ref{s-peaks} and \\S\\ref{s-cfother}; examples of halo\nmerging histories are displayed in \\S\\ref{s-mhtrees}; \nthe spatial two-point autocorrelation functions of the haloes  \nare discussed in \\S\\ref{s-nbcorr}; and a suggestion for further \ndevelopment by interpolation of merger times between time steps \nis outlined in \\S\\ref{s-interp}.\n\n\n\\subsubsection{Basic Halo Statistics} \\label{s-peaks}\n   The numbers of haloes are \nshown in Table~\\ref{t-npks}. In the $n=-2$ model,\nmatter has not yet collapsed into haloes\nat $t=0\\.31 \\,$Gyr, so the $t=1\\.5 \\,$Gyr time step was used instead. \n\n\\begin{table\n\\caption{\\label{t-npks} Number of haloes found for the different power\nspectra and detection thresholds.}\n$$\\begin{array}{ c cc cc }\n\\hline \\multicolumn{1}{ c }{t (\\mbox{\\rm Gyr})} \n\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =5}\n\t\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =1000} \n\t\\rthstrut \\\\\n\\cline{2-5}  \n      & n=0  & n=-2  & n=0  & n=-2 \\\\ \\hline\n0\\.3  & 3959 & -    & 238  & - \\\\\n1\\.5  & -    & 2086 & -    & 412 \\\\\n3\\.3  & 1539 & 1890 & 4214 & 1421 \\\\\n6\\.3  & 1053 & 1121 & 2695 & 1516 \\\\\n9\\.3  &  836 &  804 & 2121 & 1176 \\\\\n12\\.3 &  712 &  637 & 1891 & 923 \\\\\n15\\.3 &  629 &  492 & 1674 & 790 \\\\\n18\\.3 &  590 &  433 & 1524 & 672 \\\\ \\hline\n\\end{array}$$\n\n\\end{table}\n\n\tThe reality of these haloes is verified visually by rectilinear \nprojections of a sample of the points for each halo,  \nby radial particle count profiles  \nand by an interactive program which plots a sampling\nof all the points on a computer screen. The program offers   \noptional colouring of a range of haloes in a \ncolour different to both the particle\ncolour and the black background \nand allows real time rotation of the image in order to give\nan intuitive feel for the three-dimensional shape of the data. \nExamples of haloes are given in \nFig.~\\ref{f-profile}, which shows the radial \nparticle count \nprofiles for four of the biggest haloes (by number) \ndetected using \n$r_{\\protect\\mbox{\\rm \\small thresh}}=5$ in the final time stage of the $n=-2$ model.\nThe profiles are simply numbers of particles per spherical shell, so the\nrapid decrease to zero shows that the density falls off \nfaster than $r^2.$ Note that one profile has  \ntwo prominent maxima, \nneither at $r=0$. This is because, as a closer visual investigation\nof the haloes shows, a small proportion of the ``haloes''\nare in fact fairly close binary haloes rather than single haloes. \nThese binaries are usually quite uneven in size, so consideration of the \nhalo as a single halo is still a good approximation. \n\nAs noted above, \nuse of a group-finder such as DENMAX (\\cite{GB94}~1994) would be an \nelegant way to avoid the adoption of ``binary'' haloes as single haloes. \nAlternatively, \na positional proximity based group-finder could separate bound overlapping\nbound groups of particles by requiring an additional layer of analysis,\nsuch as detection of ``haloes within haloes'', in which either $r_{\\protect\\mbox{\\rm \\small thresh}}$ \nor $l$ (for a friends-of-friends group-finder) would have to again be \nspecified.\n\n\\fprof\n\nAs described in \\S\\ref{s-tree}, for each time stage, a halo at time $t_i$ \nis considered to merge into a halo at the following\ntime stage $t_{i+1}$ (or retain its identity) \nif and only if more than $50\\%$ of the particles of the halo at time $t_i$ are\npresent in the halo at time $t_{i+1}.$\nTable~\\ref{t-percm} shows the means and standard deviations of the \nfraction of a halo at time $t_i$ present in a halo at time $t_{i+1}.$ \nBy definition, these fractions are \nconstrained to be greater than $50\\%.$\n\n\\begin{table\n\\caption{\\label{t-percm} Statistics of fraction of halo at time stage listed\nhere contained in halo at following time stage.}$$\\begin{array}{ c cc cc }\n\\hline \\multicolumn{1}{ c }{t (\\mbox{\\rm Gyr})} \n\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =5}\n\t\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =1000} \n             \\rthstrut \\\\\n\\cline{2-5}  & n=0  & n=-2  & n=0  & n=-2 \\\\ \\hline\n\n0\\.3 & 96\\pm8\\% & -              & 90\\pm11\\,\\%  & - \\\\\n1\\.5 &  -       & 87\\pm15\\% &  -      & 76\\pm12\\,\\% \\\\\n3\\.3 & 92\\pm9\\% & 84\\pm14\\% & 78\\pm12\\,\\%  & 74\\pm12\\,\\% \\\\\n6\\.3 & 92\\pm9\\% & 81\\pm14\\%  & 79\\pm12\\,\\%  & 71\\pm11\\,\\% \\\\\n9\\.3 & 93\\pm8\\% & 82\\pm13\\% & 81\\pm12\\,\\%   & 72\\pm10\\,\\% \\\\\n12\\.3 & 93\\pm8\\% & 82\\pm13\\% & 82\\pm11\\,\\%   & 72\\pm11\\,\\% \\\\\n15\\.3 & 94\\pm8\\% & 84\\pm12\\% & 84\\pm11\\,\\%   & 74\\pm10\\,\\% \\\\\n18\\.3 & -   & -              &               &           \\\\ \\hline\n\\end{array}$$\n\n\\end{table}\n\n\\fmfns\n\n   Figs~\\ref{f-mfn-2br5}-\\ref{f-mfn0br1000}\nshow the mass functions of these\nhaloes and comparable mass functions from various semi-analytical \nmethods. Our N-body derived mass functions are interpreted in terms of merging \nrates in the following paragraphs and compared with \nother mass function calculations in \\S\\ref{s-cfother}.\n\nThe mass function figures suggest that with \nthe detection threshold of $r_{\\protect\\mbox{\\rm \\small thresh}} =5,$ \nfor either \nspectral index the overall halo merging rate\nfrom  $t= 3\\.3 \\,$Gyr \n(i.e., for $z \\approx 1\\.5$), to the present  \nis little more than about\n$ 3 - 10$ for galaxies below about $10^{10} M_{\\odot}.$ While this\nmerging goes on, the number of large\nhaloes in the largest bin in the $n=0 $ model increases somewhat until\nthe last time step. Depending on the average number of small haloes which \nmerge into a single large one, the increase in the number of large haloes would\nappear at first sight to be explained by the decrease in the number of smaller\nones, consistent with a merging ratio of about $3-10.$ Though the high mass\nend of the $n=-2 $ mass function is noisy,\nsimilar interpretation could be made.\n\n\tThese plots show a significant dependence on detection threshold\nand a weak dependence on $n.$\n\tFor the haloes detected at \n$r_{\\protect\\mbox{\\rm \\small thresh}} =1000,$ the merging is much weaker than for $r_{\\protect\\mbox{\\rm \\small thresh}} =5.$ \nIn a given simulation, objects detected at the higher threshold \nconsist of the dense cores of the objects detected at the lower threshold.\nHence, a simple explanation for the weaker merging is that \nif the low density envelopes merge, the cores don't necessarily do so,\nbut if the cores merge, the large low density envelopes are almost \ncertainly going to merge. \n\n\tHowever, some simple statistics show that the merging history\nis not as simple as an $N$:1 ratio\napplying equally to all haloes.\n\n\tTable~\\ref{t-nnodesc} shows the fraction of the haloes\n at each time stage that have no descendants, i.e., the \nfraction of the haloes for\nwhich no more than $50\\%$ of their particles appear in any single halo\nat the following time stage. The fact that these are nonzero \n(from about $5\\%$\nfor $n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$ to $30\\%-50\\%$ for $n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =1000$) \nshows that many haloes are destroyed\nin the sense that more than $50\\%$ of their particles may have been\npulled into an ``atmosphere'' of a large halo at a density lower than the\nthreshold density or possibly thrown out of the halo \nor pulled into another halo. This means that the halo number density does\nnot only decrease by merging, it also decreases by this halo destruction.\nFor example, if the overall number ratio between two time stages is {4:1}, \nbut one in four haloes \nterminates, then the underlying ratio of haloes actually merging is only\n{3:1}. Of course, this distinction is dependent on the definition of halo\ndetection identity as described above.\n\n\\begin{table\n\\caption{\\label{t-nnodesc} Fraction of haloes which have no descendants at \nfollowing time stage.}\n$$\\begin{array}{ c cc cc }\n\\hline \\multicolumn{1}{ c }{t (\\mbox{\\rm Gyr})} \n\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =5}\n\t\t& \\multicolumn{2}{c }{r_{\\protect\\mbox{\\rm \\small thresh}} =1000} \\rthstrut \\\\\n\\cline{2-5}  & n=0  & n=-2  & n=0  & n=-2 \\\\ \\hline\n\n0\\.3 & 5\\% & -    &  11\\%  & - \\\\\n1\\.5 & -   & 15\\% & -      & 32\\% \\\\\n3\\.3 & 8\\% & 24\\% & 32\\%  & 40\\% \\\\\n6\\.3 & 8\\% & 30\\% & 26\\%  & 49\\% \\\\\n9\\.3 & 7\\% & 29\\% & 23\\%   & 46\\% \\\\\n12\\.3 & 7\\% & 24\\% & 20\\%   & 44\\% \\\\\n15\\.3 & 4\\% & 21\\% & 15\\%   & 36\\% \\\\\n18\\.3 & -   & -    &   -    &   -   \\\\ \\hline\n\\end{array}$$\n\\end{table}\n\n\n\tMore direct statistics are those of the histories of the\nhaloes detected at the final time stage. The mean (and standard deviation) \nof the overall number of haloes which \ncollapse to above the threshold density \n(either at the first time stage or at a later time stage) and end up in\na final halo is shown in Table~\\ref{t-nances}.\n\n\\begin{table\n\\caption{ \\label{t-nances} Numbers of original haloes which end up in a halo\ndetected at the final time stage (mean$\\pm$st.deviation).}\n$$\\begin{array}{ c cc }\n\\hline r_{\\protect\\mbox{\\rm \\small thresh}} \\rthstrut  & n=0 & n=-2 \\\\ \\hline\n5   &  7\\.4 \\,\\pm\\, 20\\.7  & 5\\.0 \\,\\pm\\, 16\\.9 \\\\\n1000 & 3\\.2 \\,\\pm\\,  6\\.5  & 2\\.6 \\,\\pm\\,  6\\.2 \n \\\\ \\hline\n\\end{array}$$\n\\end{table} \n\nWhile these mean values are in the range $3-10$ estimated above, the \nstandard deviations show that many final haloes come from as many as 20 or more\noriginal haloes. In fact, the maximum number of haloes that any final halo\noriginates from is $233$ for the $n=0$ model and $259$ for the $n=-2$\nmodel (for $r_{\\protect\\mbox{\\rm \\small thresh}} =5$). For $r_{\\protect\\mbox{\\rm \\small thresh}} =1000,$ the overall rate is lower,\nand the maximum numbers of haloes per any final halo are $88$ and $95$ for \n$n=0$ and $n=-2$ respectively.\n\n\\begin{table\n\\caption{ \\label{t-rainfrac} Mass fraction of a final halo which comes from\nmatter directly accreted from the ``background'' (mean$\\pm$st.deviation).}\n$$\\begin{array}{ c cc }\n\\hline  r_{\\protect\\mbox{\\rm \\small thresh}} \\rthstrut  & n=0 & n=-2 \\\\ \\hline\n5   &  32 \\,\\pm\\, 26  & 23 \\,\\pm\\, 28 \\\\\n1000 & 36 \\,\\pm\\,  25  & 29 \\,\\pm\\,  32\n\\\\ \\hline\n\\end{array}$$\n\\end{table} \n\n\tAs already suggested by the number of haloes which terminate, throwing\nmatter back out into the background,\nthe amount of matter which ``rains'' or accretes onto haloes directly rather\nthan first collapsing into smaller density haloes is non-negligible. \nTypically 30\\% of a final halo's mass comes from matter directly \naccreted from the background, but this fraction varies widely between \nindividual haloes. Moreover, these fractions are little dependent on \n$n$ and $r_{\\protect\\mbox{\\rm \\small thresh}},$ as can be seen in the statistics listed in \nTable~\\ref{t-rainfrac}.\n\n\n\\subsubsection{Comparison with Other Methods} \\label{s-cfother}\n\nWhat advantages and disadvantages does this method have relative to \nothers, e.g., the BBKS peaks formalism method used in \n\\cite{LGRS93}~(1993) or the peak-patch formalism of \\cite{BoMy96}~(1996)?\n\nThe advantages are that the simplifications made in \nthe semi-analytical methods are not made in the N-body simulation.\nFor example:  \n\\begin{list}{(\\roman{enumi})}{\\usecounter{enumi}}\n\\item Most of the semi-analytical methods assume spherically symmetric \ncollapse (though \\cite{BoMy96}~1996 also consider the collapse of homogeneous\nellipsoids). \\cite{Warr92}~(1992) show that the haloes in the simulations \nanalysed in this paper are in fact triaxial. \n\n\\item Only the most recent (e.g., \\cite{RT96}~1996) \nmethods include spatial two-point \nauto-correlation function information. Again, this\nis automatically included in the N-body derived merging history trees \n(see \\S\\ref{s-nbcorr} for halo correlation functions). The same applies \nfor higher order correlation information.\n\n\\item Characteristic dynamics of halo mergers such as tidal tails and\nparticles thrown out into low-density ``atmospheres'' (\\S\\ref{s-peaks}) \nare modelled in the N-body simulation but \nnot taken into account in semi-analytical models.\n\\end{list}\n\nDisadvantages include scientific problems such as \nresolution limits and practical problems such as managing large data files \nwhich occupy large amounts of disk space. \n\nA first order illustration of the similarities and differences between \nthis method and others is obtained by examining the mass function plots\n(Figs.~\\ref{f-mfn-2br5}-\\ref{f-mfn0br1000}). \nThe Press-Schechter (PS) formula \nderivation of \\cite{LC93}~(1993), using their Eqs.(2.11) and \n(2.1)\\protect\\footnote{The factor \nof $2\/20$ in Eq.(2.1) of \\protect\\cite{LC93}~(1993) should read $3\/20$.}] \nprovides a simple analytical description of both the shape and \nthe time evolution of the mass functions. \nPS functions for two early and two later time \nsteps are shown for comparison in each mass function plot, and the values\nof $\\delta_{c0}\\equiv\\delta_c(z=0)$ \nare increased (from the default $\\delta_{c0}=\\delta_{\\mbox{\\rm \\small vir}} \\equiv 1\\.686$) \nin some plots in order to show ways in which \nthe PS formula could be used to interpret the mass functions. \n\nFor the $n=-2$ mass functions, our results bracket the \nsemi-analytical model results for CDM \ninitial fluctuation spectra, which have roughly this slope on galaxy \nscales. \nShown on these plots are \\cite{LS91}'s~(1991) mass function \n(Fig.~1; collapsed peaks) and \\cite{BoMy96}'s (1996, Fig.~12) mass functions. \n(The latter are calculated for clusters, so \nmasses and $\\sigma_8$ are rescaled according to a length scaling \n$L \\rightarrow L\/20$.)\n\nThe time evolution of the PS functions, both in normalisation and shape, \nis similar to that of our haloes. This is best seen in \nFig.~\\ref{f-mfn0br5}, where the decrease in normalisation in proportion\nto $(1+z)$ [expected from Eq.(2.1) of \\cite{LC93}~(1993)] \nis clearly seen. \n\nThe absolute normalisation of the PS functions is only \nabout $1-3$ times that of the $r_{\\protect\\mbox{\\rm \\small thresh}}=5$ mass functions. Since the\nPS formula is not intended to model $r_{\\protect\\mbox{\\rm \\small thresh}}=5$ or $r_{\\protect\\mbox{\\rm \\small thresh}}=1000$ \nhaloes, better agreement should not be expected.\n\nIndeed, Fig.~\\ref{f-mfn-2br1000} shows that the \nhaloes detected at $r_{\\protect\\mbox{\\rm \\small thresh}}=1000$ have a considerably lower \nnormalisation than that of the PS formula, but similar to that of\n\\cite{BoMy96}'s prediction (rescaled to galaxy scales) for \nellipsoidal internal halo dynamics. Since the N-body derived haloes\nanalysed here have a variety of triaxialities, it is not surprising that\nthis provides the best agreement. However, as remarked upon by \n\\cite{BoMy96}, their method may not be ideal for small haloes, since\nthe low mass end ``must find the nooks and the crannies'' left over \nfrom the analysis devoted to high mass haloes, so an N-body method with\nsufficient resolution may be better for obtaining the low mass slope of\nmass functions.\n\n\\cite{LS91}'s~(1991) peaks formalism based mass functions are somewhat steeper \nthan the present-day PS mass functions and do not provide the optimal \nfit to our mass functions.\n\nAlthough the early epoch high-mass ends of the N-body mass functions \nfor $r_{\\protect\\mbox{\\rm \\small thresh}}=5$ (Figs~\\ref{f-mfn-2br5}, \\ref{f-mfn0br5}) are\nsimilar to the PS cutoffs (for $\\delta_{c0}=\\delta_{\\mbox{\\rm \\small vir}}$), the \ncorresponding cuvres for the high density haloes ($r_{\\protect\\mbox{\\rm \\small thresh}}=1000$) can only be\nmatched to PS curves by increasing $\\delta_{c0}$ \n(Figs~\\ref{f-mfn-2br1000}, \\ref{f-mfn0br1000}). Since \n$\\delta_{c0}$ signifies a critical overdensity \n(according to a linear growth rate), \nit is not surprising that the high density halo high-mass cutoffs can be fit.\n(However, this would not enable the PS formulae to \nfit the later time steps, since the normalisation increases in proportion \nto $\\delta_{c0},$ which would worsen the disagreement for the high density\nhaloes relative to the PS prediction.)\n\n\\fmratios\t\n\nOf course, what is potentially\nmost interesting for explaining the flatness in observed \n(high surface brightness galaxy) luminosity \nfunctions (e.g., \\cite{EEP88}~1988) is the agreement in the \nN-body derived ($r_{\\protect\\mbox{\\rm \\small thresh}}=5$) \nand PS slopes for $n=0.$ \n(The PS function has the \nform $\\log_{10} dn\/dM = [(n-3)\/6] \\log_{10}M + $const, \nfor haloes of mass $M$).\nAn initial \nfluctuation spectrum slope of $n=0$ implies a less steep low mass (faint)\nend of the mass (luminosity) functions than for $n=-2.$\n\nMore subtle merging properties, which can be compared with \n\\cite{KW93}'s~(1993) PS derived Monte Carlo simulations, \nare the \nmass ratios of haloes and their most massive progenitors. The \nstatistics of these ratios \ncan be used to constrain the relation between galaxies \nand haloes via implications for the survival of disk galaxies. \n\nFig.~\\ref{f-rntn1_r5} \nshows that the mass ratios of merging haloes found\nin \\cite{KW93}'s~(1993) PS derived Monte Carlo simulations are \nsimilar to those found in our analysis for haloes detected \nat $r_{\\protect\\mbox{\\rm \\small thresh}}=5,$ as for the mass functions. Since the conditions of the \nsimulations are somewhat different \n(PS vs N-body, CDM vs $n=-2$), \nthe agreement suggests that these ratios are quite\nrobust with respect to different modelling techniques. \n\nComparison with Fig.~\\ref{f-rntn1_r1000} shows that at any given\ntime, the fraction of a high density halo which has already collapsed \nor accreted\nis statistically lower than for low density haloes.\n\n\\mhplots\t\n\n\\subsubsection{Merging History Trees}  \\label{s-mhtrees}\nMerging history tree plots \n(Figs~\\ref{f-mhn-2b.r5.1_5}--\\ref{f-mhn-2b.r5.400_410})\nare obtained by choosing a range of haloes at the final time\nstage and tracing back all the progenitors of these haloes. \nThe line segments joining the circles are the key feature of the plots. \nThese indicate that the halo at the earlier time is considered to merge into \n(or be ``identical'' to) \nthe halo at the later time according to the 50\\% criterion \nexplained in \\S\\ref{s-tree}.\nThe aim of the plots is to show connectivity over time. So, the\nhorizontal axis is designed to separate the haloes according to their\nfuture merging activity. It  \ndoesn't directly indicate space positions, although there\nshould be some correlation between how close two haloes are in the plot and\nhow close they are in space (since haloes need to be close \nin order to merge together later on).\n\n      Much information on the merging process is represented in these tree\nplots.  However, they do not show the entire complexity of \nthe merging process (or ``graph'' in mathematical terminology). \nSince the halo merger history trees presented here \nstart with a range of final haloes and trace backwards, \nhaloes which have no descendant at the final time\nstage are not shown.\nAs summarised in Table~\\ref{t-nnodesc}, a significant fraction \nof the haloes at a given time stage have no descendants at the \nfollowing time stage. A simple explanation is that a large\nfraction of a halo can evaporate in the merging process and contribute \nless than 50\\% of the mass of the final halo, \nin which case the merging\/identity criterion adopted fails to find \na descendant.\nTypically, about $25\\% \\pm 25\\%$ of\na halo evaporates (e.g., in tails) or forms low-density ``atmospheres''\nin N-body simulations of $N\\sim 2$ \ninteracting galaxies (Quinn, 1992). These tails or\natmospheres are likely to fall below the density detection threshold.\n\nFigs~\\ref{f-mhn-2b.r5.1_5} and \\ref{f-mhn-2b.r1000.1_5} \nshow that merging ratios of up to around 10:1 occur for many of the most massive \nhaloes at low redshifts, while as indicated by the maximum number of original\nhaloes for any final halo, the merging ratios between early\ntime stages can be much higher, as high as a few hundred to one \nin several cases for $r_{\\protect\\mbox{\\rm \\small thresh}} =5.$ \n\nThe difference in detection thresholds \nappears clearly in the difference between the early time stages of \nFigs~\\ref{f-mhn-2b.r5.1_5} and \\ref{f-mhn-2b.r1000.1_5}. At the nominal redshift \n$z=11$, many low mass haloes have reached the turnaround density and are thus\ndetected above $r_{\\protect\\mbox{\\rm \\small thresh}}=5,$ after which they merge rapidly. In contrast, \nhaloes detected above $r_{\\protect\\mbox{\\rm \\small thresh}}=1000$ are mostly detected at \n$z=1\\.5$; the few which form earlier do not merge in ratios as high as for\nthe $r_{\\protect\\mbox{\\rm \\small thresh}}=5$ haloes. Depending on the assumptions one wishes to make\nregarding the relationship between galaxies and haloes, either of these\nlimiting cases could be interesting. \n\nFor merging history trees of the haloes having lower masses at the final\ntime stage (Figs~\\ref{f-mhn-2b.r5.50_60} and \\ref{f-mhn-2b.r5.400_410}), \nvery little merging occurs apart from the earliest few time stages.  \nIndeed, Fig.~\\ref{f-mhn-2b.r5.400_410} shows that many of the smallest haloes \neither have only recently \ncollapsed or are unmerged objects which have formed well after the first\ntime stage.  \n\tThis can be generalised by stating that \nthe larger a galaxy halo is, the more original haloes it is likely to have\nbeen created from, and at any time in general, the more massive a galaxy halo\nis, the more haloes are likely to be merging into it. (This effect can also\nbe seen in Figs~\\ref{f-rntn1_r5} and \\ref{f-rntn1_r1000}.)\n\n   That the lowest mass peaks may form fairly recently or may\nform early yet undergo no merging at all is to be expected \nin a ``hierarchical galaxy formation'' scenario. Bottom-up\ngravitational collapse \ndoes not only imply that small haloes form first and successively merge\nto form more and more massive haloes, but also that \nsome low mass haloes (e.g., from\nlow amplitude short length-scale perturbations, which must exist if the\ninitial perturbation amplitudes are part of a zero-mean Gaussian distribution)\ncontinue to form at late epochs. \n\n\tThe recently forming small haloes in our \nhalo merger trees could be used to model the \n(observed) existence of low mass, young, low redshift\ngalaxies, such as the dwarf Spheroidals. An interesting case is \n the SBS\\-~0335-052(W) pair of dwarf galaxies at\na redshift of $z=0\\.014$, which have a (stellar) age of not more than $10^8$~yr\nand have a metallicity (O\/H) around 1\/40 of the solar value \n(\\cite{Lipo97}~1997).  \nThese two dwarfs are \nextremely hard to understand as anything other than young, low redshift,\nlow mass galaxies. For such galaxies to provide a constraint on galaxy \nformation models, precise estimations of number densities and age \ndistributions would be useful---though obviously difficult to obtain without \nstrong observational selection effects.\n\n\\fcorr\n\n\\subsubsection{Halo Correlation Functions}  \\label{s-nbcorr}\nAnother important statistic of the haloes \nis their spatial two-point\nauto-correlation function, $\\xi(r).$ The natural inclusion of $\\xi$ in\nN-body simulations is something not usually present in semi-analytical\ngalaxy formation modelling. Indeed, \\cite{YNG96}'s (1996)\napplication of \\cite{Jdzk}'s (1995) approach to the ``cloud-in-cloud'' \nproblem of the Press-Schechter formalism (\\cite{PS74}~1974) shows that\nthe mass function of the latter suffers significantly from the lack of\ninclusion of $\\xi.$ \\cite{Naga97}~(1997) confirm this in semi-analytical\nmerging history tree simulations using \n\\cite{RT96}'s (1996) \nmodification of the Block model (\\cite{ColeKais}~1988).\n\n  Fig.~\\ref{f-corr.n0br5} shows the correlation functions for the\nhaloes detected for $n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5.$ These functions have \nslopes $-\\gamma\\approx -1\\.8,$ where \n\\begin{equation}\n\\xi= (r_0\/r)^{\\gamma} (1+z)^{-(3+\\epsilon-\\gamma)},\n\\label{e-defneps}\n\\end{equation}\n$r$ and $r_0$ are expressed in comoving coordinates, \nand $\\epsilon$ parametrises the growth rate of $\\xi_0(z)\\equiv\\xi(r_0,z)$, \ndeduced either from observation or theory (\\cite{GroP77}~1977).\n\n The slopes of $\\xi$ for the other combinations of $n$ and $r_{\\protect\\mbox{\\rm \\small thresh}}$ \nare similar, so  \nare consistent with that observed for galaxies, i.e., \n$1\\.7 \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  \\gamma \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  1\\.8$ (e.g., \\cite{Bible}~1980; \n\\cite{DavP83}~1983; \\cite{Love92}~1992). Some observations indicate that \n$\\gamma$\nevolves with time (\\cite{IP95}~1995), but others do not detect this \n(\\cite{HudLil}~1996). N-body simulation derived correlation \nfunction slopes vary around \nthe same values, but are sensitive to whether the correlation\nis that of particles (identified as galaxies), of haloes or of mass-density\n(\\cite{DEFW85}~1985; \\cite{EFWD88}~1988;   \n\\cite{SutoS91}~1991; \\cite{SugiS91}~1991; \\cite{Melo92}~1992).\n\nThe haloes of our N-body derived merging history trees therefore have realistic \ncorrelation slopes and do not suffer from the lack \nof a correlation function of the traditional semi-analytical methods.\n\nAnother interesting property of the correlation function is the growth \nrate of $\\xi_0,$ parametrised by $\\epsilon.$ This is shown \nin Fig.~\\ref{f-xi0evol}. \nSince the simulations used here are not normalised \nconventionally, the values of $\\xi_0(z=0)$\\footnote{These \ncould be reinterpreted via \na change of time or length scale (earlier time or larger length scale),\nor the region studied could be considered to be a denser than average region\nin a low density (e.g., $\\Omega_0=0\\.1, \\lambda_0=0$) universe model.}\nare less useful than the change of $\\xi_0$ with redshift.\n\n\\def$z_{\\mbox{\\rm \\small med}}${$z_{\\mbox{\\rm \\small med}}$}\n\nTo the extent that Eq.~\\ref{e-defneps} is a good approximation, \ndirect observational estimates of $\\epsilon$ at varying \n{$z_{\\mbox{\\rm \\small med}}$} (median redshift) include \n$\\epsilon=1\\.6\\pm0\\.5$ (\\cite{WIHS93}~1993, {$z_{\\mbox{\\rm \\small med}}$}$=0\\.4$), \n$\\epsilon=-2\\.0\\pm2\\.7$ (\\cite{CEBC}~1994, {$z_{\\mbox{\\rm \\small med}}$}$=0\\.16$), \n$\\epsilon\\approx2\\.2$ (\\cite{Shep97}~1997, {$z_{\\mbox{\\rm \\small med}}$}$=0\\.37$) \nand\n$\\epsilon=2\\.8$ [cf. \\S 4.1(3), \\cite{CFRS-VIII}~(1996), \nadopting $r_0=5\\.0 h^{-1}$~Mpc and $c${$z_{\\mbox{\\rm \\small med}}$}(Stromlo-APM)$=15,200$\\mbox{\\rm\\,km\\,s$^{-1}$}  \n(\\cite{Love92}~1992, 1995), {$z_{\\mbox{\\rm \\small med}}$}(CFRS)$=0\\.56$].\nBy hypothesising major\nchanges in visible galaxy populations from low to high redshifts, \nthese values of $\\epsilon$ could be lowered.\n\nSimple theoretical values of $\\epsilon$ include\nthose expected for clustering fixed in comoving coordinates \n($\\epsilon=\\gamma-3 \\approx -1\\.2$, \nso that the index in Eq.~\\ref{e-defneps} is zero); \nclustering fixed in proper coordinates on small\nscales [``stable clustering'', $\\epsilon=0$; since the numbers of \n``clusters'' changes by $(1+z)^3$ and the number of galaxy pairs \nby $(1+z)^6$, the factor in Eq.~\\ref{e-defneps} for $r$ in proper \ncoordinates is $(1+z)^{-3}$]; \nand linear growth of initial fluctuations for an $\\Omega_0=1$ \nuniverse ($\\epsilon=\\gamma-1 \\approx 0\\.8$).  \n\n\\begin{table\n\\caption{ \\label{t-eps} Low redshift spatial correlation function growth rates \n$\\epsilon$ (see Eq.~\\protect\\ref{e-defneps}). Only the four lowest redshift \npoints are used for $n=-2.$}\n$$\\begin{array}{ c cc }\n\\hline  r_{\\protect\\mbox{\\rm \\small thresh}} \\rthstrut  & n=0 & n=-2 \\\\ \\hline\n5    & -0\\.6\\pm0\\.1   & -0\\.6\\pm0\\.6 \\\\\n1000 &  -0\\.4\\pm0\\.3  & -0\\.3\\pm0\\.3\n\\\\ \\hline\n\\end{array}$$\n\\end{table} \n\nMore detailed analyses include those of \\cite{Peac96}~(1996) and\n\\cite{Matar97}~(1997), who discuss \ndepartures from the power law of Eq.~\\ref{e-defneps}, bias factors \nand differences between the linear, quasi-linear and non-linear overdensity\nregimes. In particular, the transition between the linear and non-linear\nregimes can give rise to $\\epsilon \\approx 2\\.8$ [for $\\gamma=1\\.8$ in \nEq.~(29) of \\cite{PD96}~(1996) and $\\Omega_0=1$].\n\nBy contrast to the above estimates, the values of $\\epsilon$ for our\nhaloes are those for which {\\em haloes merge between time steps}. This means\nthat as haloes approach each other and merge, halo pairs are replaced by \nsingle haloes, so that the correlation decreases more slowly than if the\nhaloes' identities were kept separate. Because of this, the values of \n$\\epsilon$ derived here (Table~\\ref{t-eps}) lie between those \nexpected for clustering fixed in comoving and \nproper coordinates respectively, and \nare lower than the more precise of the observational estimates.\n\nImplications of the effect of merging, in particular on the \nangular correlation function, have been presented in more detail in \n\\cite{RY93}~(1993).\n\nAnother property to note is that for the $n=-2$ model, \n$\\xi_0$ initially {\\em decreases} until $z\\sim 1,$ after which it increases \nto the present.\nThis is because during the transition from linear perturbation amplitudes to\nnon-linear collapse, small length-scale perturbations on top of large\nlength-scale perturbations have their density boosted, and so can collapse\nwell before small length-scale perturbations in low density regions. \nDuring this\nperiod, perturbations in the low-density regions are not \nrepresented by collapsed\nobjects.  Hence, the mean number density of collapsed objects used to normalise\nthe correlation function represents\nthe numbers of haloes in only the high-density regions, \nbut includes the volume in both low\nand high-density regions. The mean density is therefore biased low, \nso the correlation function amplitude is biased high. As a larger fraction of\nperturbations collapse, this initially high bias in $\\xi$ decreases until\nnormal correlation growth takes place.\n\n\tThis effect is not seen for $n=0$ because large length-scale \nperturbations have relatively less power for $n=0$ than for $n=-2.$\nPossible consequences of such a ``Decreasing Correlation Period'' \n(also reported by \\cite{Rouk93}~1993 and \\cite{BV94}~1994) on \nobservations of the angular correlation function are discussed in\n\\cite{ORY97}~(1997). From Fig.~\\ref{f-xi0evol}, \nthe rate of correlation decrease \nis $d(\\log_{10}\\xi_0)\/d[\\log_{10}(1+z)] \\sim 0\\.3\\pm 0\\.15$ to \n$z\\sim 0\\.5-1.$\n\n\\finterp\n\n\\subsubsection{Merging Between Output Time Steps} \\label{s-interp}\nThe merging history trees of haloes could in principle be analysed with\na higher time resolution than in the figures presented here, merely by\nusing many more output time steps from the N-body simulation. The only\ndrawback would be practical handling of computer disk space. \nA single time step for the \\cite{Warr92}~(1992) simulations normally\noccupies 32~Megabytes. \n\nHere we note an alternative to using more output times. \nInterpolation between time steps, making analytical \nassumptions for the trajectories of the haloes, can give interpolated\nmerger times which are statistically consistent with the more accurate \ntrajectories calculated in the full N-body simulation. \nFig.~\\ref{f-interp} shows that the approximation of point particles falling\ninto isothermal potentials gives an approximately decreasing merger rate,\nwhich is consistent with the overall merging rate.  \nA small fraction of objects have estimated\nmerger times greater than the output time step interval, i.e., greater\nthan the time required for a merger \naccording to the full N-body simulation. This is a limitation \nto (at least this) analytical interpolation, but the fraction of such \ncases is small enough that this interpolation technique may \nbe a fair approximation statistically.\n\nHowever, N-body analyses of few-galaxy systems (e.g., \n\\cite{PrugC92}~1992; \\cite{HernW89}~1989) \nsuggest that merger times may be difficult to predict by any simple\nanalytical method, so caution is needed. Although the results should\nnot differ significantly from those of the authors just cited, use\nof such interpolations for the calculation of N-body merger history\ntrees could be verified by comparison with the intermediate time steps\nof the N-body systems.\tInterpolation is not adopted for this paper.\n\n\\section{Galaxy Formation Application (Star Formation and Evolution)} \n\\label{s-other}\n\nThe example of a galaxy formation recipe adopted here combines \n(a) gravity: halo merging history trees are derived from \ndensity peaks detected above a given overdensity threshold \nin $1\\.1$ million particle pure gravity N-body simulations;\nand\n(b) other physical processes: \none galaxy is assumed to form in each halo; \nwe insert an observationally\ninspired star formation rate due to merger-induced starbursts (and\/or\na quiescent star formation rate); and we\ncombine this with the initial mass function and stellar evolution\nmodelled in Bruzual's (1983) evolutionary population synthesis code.\n\n\n\\subsection{Modelling Starbursts to Occur on \n\t\tMerging} \\label{s-bursts}  \n\tStar formation physics is represented by an initial mass function\n(IMF) and a star formation rate (SFR). \nThe separability of these two functions \nappears to be a reasonable assumption according to both \nobservation and theory, e.g., the recent ISO (Infrared Satellite Observatory)\nobservations of both infrared-ultra-luminous and ``normal'' star-forming\ngalaxies are consistent with existence of a universal IMF \n(\\cite{Lutz96}~1996; \\cite{Vigr96}~1996; \\cite{GG96}~1996).\nThe IMF is discussed in \\S\\ref{s-geps},\nwhile this section describes the modelling \nof the SFR, based on starbursts.\n\n      For the purpose of this first-order examination of the effect of\nmerger-induced starbursts on galaxy luminosity evolution, we only use very\nsimple models of the starbursts. Observational and \ntheoretical motivations are as follows.\n\n      An early observational model of a starburst (not necessarily \ncaused by a merger), is that of Rieke \\mbox{et\\,al.} \\ (1980).\n\n      Rieke \\mbox{et\\,al.} \\ (1980) model the starburst in the nucleus of M82\nvia evolutionary population synthesis. They find that instantaneous\n(Dirac delta function) and constant star formation rate models fail to produce\nthe observed spectrum, but exponential decay SFR models with an IMF with\na low-mass cutoff well above a solar mass are necessary. Their best models\n(say, D and F) have the e-folding time in the SFR $t_0=2 {\\scriptstyle \\times} 10^{7} $yr and \n$t_0=1 {\\scriptstyle \\times} 10^{8}$yr and run for $t=5 {\\scriptstyle \\times} 10^{7} $yr and $t=1\\.6 {\\scriptstyle \\times} 10^{7} $yr respectively.\nBoth have IMF's with $\\alpha=2$ and the mass range $3\\.5 - 31 M_{\\odot}.$\nThe mass turned into stars is $\\approx 1\\.5 - 2  {\\scriptstyle \\times} 10^{8} M_{\\odot},$ this being\nconstrained to be less than the total mass in the nucleus, estimated as\n$3 {\\scriptstyle \\times} 10^{8} M_{\\odot}$ by Rieke \\mbox{et\\,al.} \\ This constraint is the\nmajor reason for the\nneed of a high low-mass cutoff. If a normal\nlow-mass cutoff is used, then the mass actually present\nin the nucleus is insufficient to generate the observed luminosity. \n\n      Rieke (1991) describes more recent observational constraints\non the models for M82, finding that the above conclusion still holds using\nmore modern stellar evolutionary tracks in the models.\n\n      Scoville \\& Soifer (1991) argue from IRAS far-infrared data that\n``virtually all of the strong {\\it global} starbursts occur in ...\nstarburst-infrared galaxies,'' where the latter are \ndefined as ``those with $L_{IR}\/M_{H_2}$ \nsignificantly higher than in normal galaxies,'' and that \nstarburst-infrared galaxies correlate\nhighly with ``the occurrence of a recent [galaxy-galaxy] interaction.''\nThey argue that such global starbursts require the progenitor galaxies \nto be of\ncomparable mass in order to generate such activity.\n\nWhile this result \ndoesn't necessarily imply the converse, i.e., that all mergers of\nsimilarly sized galaxies\ninduce major global starbursts, the converse is a fair hypothesis.\nWith the assumption of this converse, \nScoville \\& Soifer find that the high luminosity end\nof the infrared (galaxy) luminosity function from the IRAS survey is \nconsistent with $0\\.2\\%$ of all spiral galaxies undergoing global starbursts\nat the present with lifetimes equal to the dynamical times of large galaxies,\n$\\sim 1-2 {\\scriptstyle \\times} 10^{8} $yr. For the most IR-luminous\ngalaxies they find SFR's of $10-100 M_{\\odot} $yr$^{-1}.$ They don't find a\nhigh low-mass cutoff necessary for their models to fit the observations.\n\n      Norman (1991), citing the models of merging gas-rich disk galaxies of\nHernquist and Barnes, (\\cite{Barn90}~1990, equal mass galaxies; \nHernquist 1989, differing mass galaxies), describes a qualitative\nthree-phase model to\ntake into account gas falling into the galaxy centre. The three phases \nessentially\ncorrespond to proportions of the gas having fallen to the centre. Published\nstar formation models following these three phases separately are not\ncited in the article, but would obviously be of interest.\n\n      Norman (1991) also argues that constant SFR models of starbursts\nsatisfy an observed\n sparsity of post-starburst galaxies relative to starburst galaxies,\nbut that instantaneous SFR models predict too many post-starbursts.\n\n      Hence, for this example application of an N-body derived\nhalo merger history tree, a starburst with a constant SFR\nis chosen.\n\n        For pairwise mergers, the following canonical values are chosen.\nWe normalise the rate of the starburst for a ``typical''\nlarge galaxy merger product\nto be an SFR of \n$\\psi_0= 50 M_{\\odot} $yr$^{-1}$ \nas in the models of Scoville \\& Soifer (1991).\nThe low-mass cutoff in the IMF used here ($0\\.08 M_{\\odot}$, \nsee Eqn~\\ref{e-imf}) is consistent with Scoville \\& Soifer's\nvalue of $0\\.1 M_{\\odot}.$\n\nWe consider the merger product to be the remnant of two large galaxies each of\n gas mass \n$M^{\\mbox{\\rm \\small gas}}_0= 10^{10}M_{\\odot},$ \ntotal mass $M^{\\mbox{\\rm \\small halo}}_0= 10^{12} M_{\\odot}$ and halo \nradius $50 \\,$kpc.\nThis gives a dynamical time $t_{\\mbox{\\rm \\small dyn}}\\equiv (G\\rho_0)^{-1\/2} \\approx 2 {\\scriptstyle \\times} 10^{8} $yr,\nwhere the mean density of either galaxy to its\nhalo radius is $\\rho_0= 8 {\\scriptstyle \\times} 10^{6} M_{\\odot}$kpc$^{-3}.$\n      The modelling by \n\\cite{Barn90}~(1990), Hernquist (1989) or earlier non-gaseous N-body models\nsuch as those of Quinn \\& Goodman (1986) find that the merger takes place over\nonly a few dynamical times. \nSo the choice of progenitor galaxies here gives a $t_{\\mbox{\\rm \\small dyn}}$ matching\nScoville \\& Soifer's \nburst duration of $t^{\\mbox{\\rm \\small burst}}_0= 2 {\\scriptstyle \\times} 10^{8} $yr, which is chosen\nas the canonical burst duration.\n\n      In this canonical case, $50\\%$ of the total gas mass is used up in \nthe burst. To sum up, we have\n\n\\begin{eqnarray}\n\\psi_0 &=& 50 M_{\\odot} \\mbox{\\rm yr}^{-1} \\cr\nM^{\\mbox{\\rm \\small gas}}_0 &=& 10^{10}M_{\\odot}               \\cr\n M^{\\mbox{\\rm \\small halo}}_0 &=& 10^{12} M_{\\odot}             \\cr\n  \\rho_0 &=& 8 {\\scriptstyle \\times} 10^{6} M_{\\odot} \\mbox{\\rm kpc}^{-3}    \\cr\n t^{\\mbox{\\rm \\small burst}}_0 &=& 2 {\\scriptstyle \\times} 10^{8} \\mbox{\\rm yr.}\n\\label{e-canon}\n\\end{eqnarray}\n\nThe canonical values are then scaled for haloes of arbitrary mass. \nTo first order\nit seems reasonable that the kinetic energy \navailable for generating star formation should be proportional to the mass\nof the smaller halo and that the SFR should also be proportional to the\ntotal amount of gas available. \nSo, the SFR is scaled by the mass of the smaller halo times the ratio of\ncombined gas mass to $ 2 M^{\\mbox{\\rm \\small gas}}_0$. \n\nSince we consider the \nduration of the starburst to be the order of a dynamical time, \n$t^{\\mbox{\\rm \\small burst}}$ is scaled\nby $\\rho^{-1\/2},$ where $\\rho$ is the density of the larger halo.\n\nGiven a coarse time resolution in the merging histories, each merger is \nidentified by the code as a multiple merger---e.g., seven haloes merge to\none---instead of as a series of several individual pairwise mergers.\nIf we consider the approximation that each of the pairwise mergers involves\nthe largest progenitor halo, then \na compromise can be carried out as follows.\n\n      (a) Have a single burst with\nthe above normalisation.\nScale the SFR by the sum of the masses of each of the smaller\nhaloes (i.e., all but the largest) and by the ratio of the combined gas\nmass from all progenitor haloes \n(i.e., including the largest) to $2 M^{\\mbox{\\rm \\small gas}}_0,$ giving\n\\begin{equation}\n   \\psi(t)=\\psi_0  \n      {\\sum_{i\\neq  i^{\\mbox{\\rm \\tiny max}}} M^{\\mbox{\\rm \\small gal}}_i \\over M^{\\mbox{\\rm \\small halo}}_0}\n        {\\sum_{{\\rm all}\\,i} M^{\\mbox{\\rm \\small gas}}_i \\over 2 M^{\\mbox{\\rm \\small gas}}_0}\n\\label{e-psi}\n\\end{equation}\nwhere $\\psi(t)$ is the star formation rate (in $M_{\\odot} $yr$^{-1}$), \nthe progenitor\nhaloes are labelled by $i$, and $i^{\\mbox{\\rm \\small max}}$ is the label of the progenitor\nhalo of greatest mass. (For the group-finding algorithm \nused in this article, $i^{\\mbox{\\rm \\small max}}=1.$)\n\n(b) Scale the starburst duration as above,\nby $\\rho^{-1\/2},$ where $\\rho$ is the density of the largest halo,\ngiving\n\\begin{equation}\n  t^{\\mbox{\\rm \\small burst}}=t^{\\mbox{\\rm \\small burst}}_0 \\left({\\rho_{\\mbox{\\rm \\small halo}}\\over\\rho_0}\\right)^{-1\/2}.\n\\label{e-tburst}\n\\end{equation}\nTwo modifications may have to be applied in some situations.\nFirstly, where the starburst at this rate uses up more gas than is actually \navailable, it is truncated at the point of time when zero gas mass is left.\nSecondly, if the duration of the starburst is longer than\nthe time interval to the next time step, it is truncated at that next\ntime step.\n\n While this modelling of multiple\nmerger-induced starbursts with large time steps\nmay make the luminosity evolution more temporally discrete than it should be,\nit does conserve physical quantities in line with the \nobservational and theoretical constraints discussed above, and \nshould be sufficient for this demonstration of the use of N-body based\nmerger history trees.\n\n\n\n\\subsection{Connection with Stellar Evolutionary Population \nSynthesis}  \\label{s-geps}  \n\tWe use stellar evolutionary population synthesis to combine\nstar formation and stellar evolutionary physics.\n      A version of Bruzual's stellar evolutionary population\nsynthesis code (Bruzual 1983) which is \nessentially that of 1983, but with some updating\nand conversion to Unix, is the primary population synthesis \ncode used, but the code of \\cite{GRV87}~(1987) and \n\\cite{RVG88}~(1988) (1993 version) gives similar results.  \nIn the interactions of the programs \npresented above with this\ncode, the SFR history \nand masses of galaxies and gas are determined outside of the\nBruzual routines or by amended versions of the Bruzual routines.\nThe return of gas from supernovae to a galaxy was turned off for test \npurposes but otherwise left on. The loss of this supernova gas from a galaxy\nwas not invoked, neither was the option allowing \ninfall of gas to a galaxy. \n\n\tThe initial mass function (IMF) used was the default\nIMF chosen by the code, after Scalo (1986) (e.g., Fig.~16 in Scalo, 1986).\nWhere $f(m) \\propto m^{-(1+x)}$ is the number of stars born per \nunit (linear) mass in a given mass range, the IMF slopes used are\n\\begin{equation}\n       x = \\left\\{ \n\t\\begin{array}{lll}\n\t-2\\.60  , &  0\\.05 \\le M \\le    0\\.08 M_{\\odot} \n\t\t&\\mbox{(brown dwarfs)} \\\\\n       -2\\.60  ,  & 0\\.08 \\le M \\le    0\\.18 M_{\\odot} \\\\\n        0\\.01  ,  & 0\\.18 \\le M \\le    0\\.42 M_{\\odot} \\\\\n        1\\.75  ,  & 0\\.42 \\le M \\le    0\\.62 M_{\\odot} \\\\\n        1\\.08  ,  & 0\\.62 \\le M \\le    1\\.18 M_{\\odot} \\\\\n        2\\.50  ,  & 1\\.18 \\le M \\le    3\\.5 M_{\\odot} \\\\\n        1\\.63  ,  & 3\\.50 \\le M \\le   75 M_{\\odot}. \n\t\\end{array}\n\t\\right.\n\\label{e-imf}\n\\end{equation}\n\nThe Bruzual code normally works \nby using simple analytical expressions for the SFR history, so\nthat no numerical effects \n(e.g., rounding errors) can be introduced at this stage.\n Numerical effects can of course be present \nwhen the galaxy spectral energy distribution is calculated, since only\nfinitely many points representing different stellar ages are present for each\nof the finitely many mass tracks. \n\n      However, this has the disadvantage that one\ncannot simply stop the code after a certain time step, \nsave the population data,\nstart up the program from scratch, read in the saved population data and \ncontinue on as if the program had never stopped.\nThe population data could be stored and later read back in, \nbut this would round each star's\nage to the appropriate stellar evolution track age at every time step, making\ncumulative errors. \n\nBruzual's code was therefore modified slightly in order \nto allow use of numerical SFR \nhistories. For each time step and each halo, \nan array of time points from that time step\nto the following time step and the corresponding\narray of integrals of the SFR are stored. \nThese integrals of the SFR are the total number of \nstars created since the first star\nformation in any of the progenitor peaks which end up in the present peak\nbeing worked on. \nBecause\nthe integral of the SFR is used,\nthe errors are not cumulative, and in the present\nversion are $\\sim 0\\.1\\%.$ \n\n      The program which applies \nstellar evolutionary population synthesis to the merging histories\nstores an SFR history \nfor each peak as it is evolved to\nthe next time step, adds these together for merged peaks, and from that point\non evolutionary population synthesis applies just as it would for \nan isolated galaxy, except that the merging history tree information is\ncontained implicitly in the complex shape of the SFR history.\n\n      For the present model, population synthesis is applied \nby optionally having an exponentially decreasing SFR between mergers\nand optionally having starburst SFR's commencing at each merger.\nGas masses and total masses are by default\nconserved, i.e., the gas and total masses of a galaxy are the\nrespective sums of the gas and total masses of predecessor galaxies,\nexcept that  matter accreting directly onto the density peaks\nis added as gaseous mass. \nIf both exponential and burst\nstar formation are turned on simultaneously (probably the most realistic\nmodel)\nthe SFR's are simply added together, conserving the number of stars created.\n\n\n\n\\subsection{Luminosity Functions} \\label{s-galstats}\n\nThis combination of N-body generated merging\nhistory trees with other elements of a simple galaxy formation recipe \nis sufficient to generate \nluminosity functions which are comparable to the estimated \npresent-day luminosity function, and so could be extended to \nan exploration of \ngalaxy formation parameter space, in which the parameters are \nfree rather than motivated by observation as in the present work.\n\nThe merging history trees presented above are derived from power law\ninitial spectrum N-body simulations intended for comparison of \nthe relative properties of haloes of different masses, so need to\nbe renormalised in a cosmological context. For simplicity, \nan increase in the length scale by a factor of two is the \nrenormalisation adopted. \nMasses scale as the cube of the length in order to\nleave the numerical operations in the N-body simulations\nand the merger history tree analysis unchanged and time is not rescaled.\nThis brings the {\\em number} density of the haloes close to that of\nobserved galaxies. \n\n\\fsfr\n\n\\fmlratio\n\nA baryon fraction of 10\\% is adopted. \nNucleosynthesis results (e.g., \\cite{Walk91}~1991) and \n$H_0$ estimates (\\cite{Tanv95}~1995; \\cite{Kundic97}~1997) make this \na reasonable round number for a critical density universe with a null \ncosmological constant. [Some other estimates of similar baryon fractions\ninclude Galaxy estimates of $0\\.7$ out to a Holmberg radius, or down to possibly \n$0\\.07$ for the whole Galaxy (\\cite{Free87}~1987); $0\\.25$ for galaxy \nclusters (e.g., \\cite{Sara87}~1987); and $\\sim 0\\.1$ over the whole range \nof cosmological structures from dwarf spheroidals to rich clusters \n(\\cite{Blum88}~1988).]\n\n\\flfevsb\n\nAll baryonic matter is assumed to be potentially star-forming material.\n \nAn example of an SFR \nhistory (for the galaxy resident in the most massive final halo \nin the $n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$ exponential-plus-burst model)\nis shown in Fig.~\\ref{f-n0b.r5n6.eb.sfr}. Both the exponentially declining \n``quiescent'' star formation rates and the merger-induced bursts are\nclearly visible. Bruzual's \nSFR parameter has the value $\\mu=0\\.15$ \n[$\\mu$ is the proportion of gas\nturned into stars in a (non-merging) galaxy within $1 \\,$Gyr].\n\nMass-to-light ratios, $\\MM_{\\mbox{\\rm \\small lum}}\/L_{\\mbox{\\rm \\small IIIaJ}},$ \n(using rest frame values of $L_{\\mbox{\\rm \\small IIIaJ}}$, i.e., no K-corrections) \nfor the galaxies in this model at different time steps \nare shown in Fig.~\\ref{f-mln0b.eb}. These mass-to-light ratios are\nsomewhat high relative to optical galaxy estimates \n(e.g., $2 \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  {\\cal M}_{\\mbox{\\rm \\small disk}}\/{L_{B_{\\mbox{\\rm \\small disk}}}} \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  7 \\,M_\\odot L_\\odot^{-1}$\nderived from the inner part of the optical\/HI rotation curves of disk galaxies\nwithout bulges, \\cite{Free87}~1987; \nstellar population values, \\cite{LT78}~1978) but similar to those estimated \nfrom X-ray emission in ellipticals \n($20 \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  \\MM_{\\mbox{\\rm \\small tot}}\/L_B \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  30 \\,M_\\odot L_\\odot^{-1} $ within a radius $r\\sim30-40$~kpc,\n\\cite{Caniz87}~1987).\n\nThe corresponding luminosity functions for the galaxies in this \nmodel are shown in Fig.~\\ref{f-lfn-2b.eb}, while luminosity functions \nfor either exponentially\ndecaying or burst SFR's only (not both) are shown in \nFigs~\\ref{f-lfn-2b.e} and \\ref{f-lfn-2br5.b} (for $n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$).\nA local luminosity function parametrised as a Schechter function \n\\begin{equation} \\label{e-schech}\ndN\/dL = \\phi^* (L\/L^*)^\\alpha \\exp(-L\/L^*) d(L\/L^*) \n\\end{equation}\n(\\cite{Sch76}~1976) where \n$\\phi^*= 1\\.56 {\\scriptstyle \\times} 10^{-2} h^3$~Mpc$^{-3}$, \n$M^* =-19\\.6+5\\log_{10}h$ and $\\alpha =-1\\.1$\n(\\cite{EEP88}~1988), is shown for comparison. \n[This estimate is similar to the more recent Stromlo-APM \nestimate of \\cite{Love92}~(1992), while the CfA2 estimate of \n\\cite{Marz94a}~(1994) differs from both in having a value of\n$M^*$ about $0\\.7$~mag fainter.]\n\nThe luminosity functions for the \n$n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$ exponential-plus-burst and exponential-only \nSFR's are similar to one another \n(Figs~\\ref{f-lfn-2b.eb}, \\ref{f-lfn-2b.e}),\nas the rate of the exponential SFR alone is enough to use up most of the gas,\nleaving little possibility \nfor the bursts to make any difference between galaxies of differing merging\nhistories. \nThe luminosity function \nslopes are steeper than the observational faint end slopes (for field galaxies), \nand are similar to the \nsteep slopes expected from the mass functions\n(Figs~\\ref{f-mfn-2br5} - \\ref{f-mfn-2br1000}). \nThe other three combinations of $n$ and $r_{\\protect\\mbox{\\rm \\small thresh}}$ for the \nexponential-plus-burst and exponential-only models give \nsimilarly steep luminosity functions. \nThis result is similar to that of other semi-analytical models \n(e.g., \\cite{KWG93}~1993). \n\nThe ``bump'' at the bright end of some of the later luminosity functions \n(particularly in Fig.~\\ref{f-lfn-2b.e}) combined with the steep slope is\nsuggestive of luminosity functions in clusters, but a cluster environment\nis one in which merger-induced star bursts could be expected to be the more \nimportant rather than the less important star formation mode, \nso it is not clear whether or not \nthis is a useful characteristic of the models.\n\t\nThe faint end sudden drop \n(at $M \\,\\lower.6ex\\hbox{$\\buildrel <\\over \\sim$} \\,  -16$ in these two figures) is simply due \nto the resolution limit of the models.\n\n  The $n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$ burst-only model (Fig.~\\ref{f-lfn0br5.b}) \nis quite striking in having a ``faint end'' slope almost precisely \nas shallow as the slope of the observational luminosity function.\nIn addition, \nthe bright ends of the earlier of these luminosity functions \nhave steep drops as in\nthe observational luminosity function, though this occurs \nfor galaxies a few magnitudes too bright. \n\nHow significant are the luminosity functions of the \n$n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$ burst-only model? \nFor a flat rotation curve for the Galaxy with \ndensity falling off as $r^{-2},$ \nthe detection threshold of $r_{\\protect\\mbox{\\rm \\small thresh}} =5$ corresponds to \na Galaxy halo radius of about $1500 \\,$kpc, larger than\nany claimed value. Apart from the recent \nobservational suggestion \nof \\cite{HonSof96}~(1996) that the halo is only 15~kpc in radius, a \nconventional estimate of $r_{halo}\\sim 100-300$~kpc would still imply that the\ndetection threshold used here overestimates the mass of the Galaxy by\nan order of magnitude [since $\\rho \\propto r^{-2} \\Rightarrow M(<r) \\propto r$].\nAlso, members of the Local Group would missed, since one galaxy per halo\nand maximal merging are assumed.\n\nThese problems \ncould conceivably be corrected by a reduction in galaxy masses, which would \nreduce the luminosities, \nand by an increase in the normalisation to account for the missing Local Group \ngalaxies, whose analogues would presumably be detected (in the model) in \nisolation in other regions of space. However, while \nthe former would improve the fit, \nthe latter would worsen it.\n\n\tThe $n=0, r_{\\protect\\mbox{\\rm \\small thresh}} =1000$ burst-only \nmodel does include these smaller galaxies \n(``halo substructure'') and results in a luminosity function which is \nslightly too steep (Fig.~\\ref{f-lfn0br1000.b}). \nThe $n=-2$ models (Figs~\\ref{f-lfn-2br5.b}, \\ref{f-lfn-2br1000.b}) \nalso have slopes which are relatively shallow, but none as shallow \nas that for the $n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}}=5$ model. \n\nThe reason why \nthe burst-only models have shallower slopes can be explained \nfairly simply: smaller mass haloes typically undergo less merging, so \nbecome relatively underluminous relative to \nhigher mass haloes. {\\em Star formation dominated \nby merger-induced bursts could therefore provide a simple way of reducing \nthe faint-end slopes expected from the Press-Schechter function.}\n\n\\flfnrthr\n\nAs might be expected from the similarity\nin the mass functions (Figs~\\ref{f-mfn-2br5}, \\ref{f-mfn-2br1000}, \n\\ref{f-mfn0br1000})\nfor three of the burst-only models, the\nluminosity functions are also similar to one another. \nHowever, a characteristic of the\nmass function of the $n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}}=5$ model appears (marginally) to\nbe shared by the corresponding luminosity function: the slope seems\nto become shallower with time. If this were significant, it could be of\nconsiderable interest in reconciling faint galaxy counts \n(e.g., \\cite{TSei88}~1988; \\cite{Tys88}~1988) with the flatness\nof the locally estimated luminosity function, though it would also need\nto show evolution in the bright end of the luminosity function as\ndeduced from recent redshift surveys \n(\\cite{CFRS-VI}~1995, \\cite{Ell96LF}~1996 and \\cite{Cow96}~1996). \n\nThe evolution in slope could be attributed to \nthe evolution in the Press-Schechter mass function shown in \nFig.~\\ref{f-mfn-2br5}, with the addition of a systematic reduction \nin slope induced by our burst-only SFR formalism. \n\n\tBoth the flatness of the $n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}}=5$ burst-only \nluminosity functions and the decreasing slope of the\n$n=-2,\\, r_{\\protect\\mbox{\\rm \\small thresh}}=5$ model suggest that these parameter combinations \nshould be interesting for future explorations of \ngalaxy formation models using N-body derived merger history trees.\n\n\\section{Conclusion} \n\tThe algorithms and results discussed above show that this \nmethod of deriving halo merging history trees directly from N-body simulations\nis feasible and easily combined with evolutionary stellar population synthesis\nfor exploration of a simple galaxy formation model. \nThe method has been applied to \\cite{Warr92}'s (1992) N-body simulations\nwith $n=0$ and $n=-2$\npower law initial perturbation power spectra, using $r_{\\protect\\mbox{\\rm \\small thresh}} =5$ and \n$r_{\\protect\\mbox{\\rm \\small thresh}} =1000$ overdensity detection thresholds to detect dark\nmatter haloes.\n\nSeveral subsets of the merging history\ntrees have been plotted, directly showing quantitative \npatterns of halo merging from fully non-linear calculations.\nThe merging history trees were \ncombined with simple, observationally normalised proportionality assumptions\nfor star formation rates and stellar \nevolutionary population synthesis in order to\ndemonstrate a simple galaxy formation ``recipe'', in which star formation\nis either exponentially decreasing, induced by mergers, or both.\nIn addition, interesting properties of halo formation and evolution\nwere noted.\n\nThe galaxy formation recipe adopted shows that \nif most star formation occurs as merger-induced bursts, then \nsome flattening of the faint end \nof the galaxy luminosity function relative to \nthat expected from the mass functions may be obtained. Indeed, \nfor the analysis using \n(a) a white noise perturbation spectrum slope ($n=0$)\nand \n(b)  a detection threshold typical of that for a perturbation\njust reaching the turn-around density\n($r_{\\protect\\mbox{\\rm \\small thresh}} =5$),\nthe resulting present-day luminosity function  \nhas a faint-end slope similar to $\\alpha=-1\\.1,$ i.e., \nthat estimated for local, field, high surface brightness galaxies. \nMoreover, for the same value of\n$r_{\\protect\\mbox{\\rm \\small thresh}}$ and a perturbation slope similar to that of CDM, there marginally\nappears to be a steepening of the faint end slope for increasing redshift.\n\nHowever, since this is only a simple test application of merging history\ntrees to galaxy formation modelling, and the N-body simulations are\ndesigned for the study of relative rather than absolute halo properties,\nthese results should be taken as indicative of worthwhile galaxy \nformation ``recipes'' to explore further, rather than as a\ndefinite explanation for a luminosity function with a flat faint end.\n\n\tOther interesting by-products from this method of analysing\nnon-linear gravitational collapse \nare properties of hierarchical halo formation.\n\n(1) Individual merger rates\ncan be very different from average merger rates and the fraction\nof mass coming from accretion can be quite high. \nFor example, for $n=0,\\, r_{\\protect\\mbox{\\rm \\small thresh}} =5$, \nthe mean number of haloes which collapse \nat any time stage and end up in a halo at the final time stage \nis $7\\.4,$ the standard deviation in this quantity is \n$20\\.7$ (Table~\\ref{t-nances}) and \nthe maximum is 233.\nFor the same halo detection parameters, \na mean fraction of 32\\% of the mass in the final haloes\ncomes directly from accretion of uncollapsed matter (Table~\\ref{t-rainfrac}), \nbut this fraction varies widely between different haloes.\n(For the other three parameter choices, \nthe merger ratios are about a factor of two lower, \nwhile the accretion percentages are about the same.)\n\n(2) Low mass haloes may either form at very recent cosmological epochs\nor may undergo no mergers at all throughout a Hubble time. The former\nmay be counter-intuitive, but can be \nsimply understood as being due to the existence of \nsmall amplitude, small length-scale perturbations at early epochs, \nexpected for a zero-mean Gaussian amplitude distribution. \nSome (if not many) perturbations have small initial\namplitudes: these only become\nnon-linear recently. Thus, observations such as that of \na young, low redshift, low metallicity dwarf galaxy pair (\\cite{Lipo97}~1997)\ncan be explained naturally within a hierarchical halo formation model.\n\n(3) If there is sufficiently more power on large than small scales, \ne.g., $n=-2,$ then the first fluctuations to collapse (forming haloes) \nare only those \ninside of large length-scale\nperturbations, so that the spatial distribution of the haloes is initially\nmuch less uniform than that of the linear (and non-linear) fluctuations\nin general. In other words, the amplitude of\nthe spatial correlation function of the haloes \nmay start from a highly biased value (relative to the linear perturbation \ntheory expectation), \nand decrease in bias until a transition redshift when ``normal'' correlation \ngrowth commences. Observational \nconsequences of such a ``Decreasing Correlation Period''\nare discussed in \\cite{ORY97}~(1997).\n\nMany further developments of the use of N-body derived \nmerger history trees are obvious: different assumptions for\nprocesses (2)-(5) cited in the introduction could be adopted; a mechanism\n(e.g., dissipation) could be chosen \nto change (6), so that galaxies would not\nmerge every time their parent haloes merged; or at the expense of using\nmore computer disk space, N-body simulations with higher $N$ or \nlarger numbers of output \ntime steps could be used. Use of more physically complex \ngroup-finders (e.g., \\cite{GB94}~1994) should also make some differences \nto detailed properties of the merger histories.\nSystematic comparisons between recent analytical predictions \n(e.g., \\cite{YNG96}~1996), \nMerging-Cell-Models models (\\cite{RT96}~1996) and N-body derived merger history\ntrees could be made. An example of a different assumption for processes\n(1)-(5) would be to replace the observationally inspired\nparameters adopted here by free parameters which would\nbe varied to see how sensitive the reduction in the \nslope of the faint end of the luminosity\nfunction is to these parameters. \n\n\\section{Acknowledgements}\n  We would like to thank Carlos Frenk, George Efstathiou, Cedric Lacey, \nTim de Zeeuw, Walter Jaffe, \nSimon White, Gary Mamon, Peter Thomas, Masahiro Nagashima and \nNaoteru Gouda for several interesting and \nuseful comments and suggestions. Thank you also to Bob Mann\nfor making many constructive recommendations.\n  This research has been carried out with support from the \nAustralian National University, from the \nInstitut d'Astrophysique de Paris, CNRS, from an ANTARES grant, \nfrom a PPARC Research Fellowship, from a Centre of Excellence Foreign \nVisiting Fellowship at the National Astronomical Observatory of Japan,\nand has made use of Starlink computing resources. \n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec::Introduction}\nStellar feedback is a crucial feature of realistic galaxy simulations, from the cosmological scales of galaxy formation and evolution down to the sub-galactic scales of molecular cloud formation and dispersal. Comparable quantities of energy and momentum are injected into the interstellar medium by the explosive destruction of stars at the end of their lifetimes (supernovae), and by the expansion of the heated, ionised gas surrounding massive stars~\\citep[HII regions, see][for details]{Matzner02,2013MNRAS.432..455D}. Smaller but significant quantities are injected by the outflows of gas generated within the stellar atmosphere (stellar winds). This energy and momentum is responsible for the formation of a three-phase interstellar medium~\\citep{1977ApJ...218..148M}, for the generation of galactic outflows~\\citep[e.g.][]{2014MNRAS.442.3013K,2020ApJ...903L..34K}, and for the dispersal of molecular gas surrounding young stars~\\citep[e.g.][]{2019Natur.569..519K,2020arXiv201013788C,Chevance20}. Its interplay with gravity determines the gas disc scale-heights of galaxies and affects the turbulent velocity dispersions of their cold neutral media and molecular components~\\citep{OstrikerShetty2011}. As such, stellar feedback plays an important role in limiting the star formation efficiencies of individual molecular clouds, and so the star formation rates of entire galaxies.\n\nThe implementation of supernova feedback in galactic and cosmological simulations has historically presented a number of problems. The inefficient conversion of thermal energy to momentum when the momentum-generating phase of blast-wave expansion is unresolved (the `overcooling problem') leads to an underestimation of the kinetic energy injected into the interstellar medium~\\citep[e.g.][]{Katz92,1993MNRAS.265..271N,2003MNRAS.339..289S,2006MNRAS.371.1125S}. Overcooling can be most-effectively resolved by explicitly injecting the terminal momentum of the blast-wave~\\citep[e.g.][]{KimmCen14}, and several different parametrisations of the terminal momentum are available, derived from high-resolution simulations of supernova remnants~\\citep[e.g.][]{Gentry17,2020MNRAS.492.1243G}. Within clusters of stars, the distribution of supernova frequencies, energies and `delay times' between stellar birth and the first supernova detonation depends sensitively on the massive end of the initial stellar mass function (IMF) from which the stellar population is drawn~\\citep{2020arXiv200911309S,2020arXiv201010533S,2020arXiv200403608K}. Variations in the methods used to stochastically-sample the IMF~\\citep[e.g.][]{2010A&A...512A..79H,2013NewAR..57..123C,daSilva14,Krumholz15,2020arXiv201010533S} can therefore introduce large variations in the final energy and momentum injected by supernova feedback. For example, erroneously-long delays will lead to overly-violent supernova explosions due to an increased degree of supernova clustering, as well as to the overly-slow removal of gas from around young stars~\\citep[e.g.][]{Fujimoto19}. These two factors will directly influence the giant molecular cloud lifetime, the star formation efficiency within individual clouds, and so the star formation efficiencies of galaxies.\n\nIn simulations of entire galaxies~\\citep{2013MNRAS.428..129S,Agertz13,Hopkins18,2019MNRAS.489.4233M,2020arXiv200911309S}, supernova feedback on its own has been found inadequate to reproduce the observed properties of the three-phase interstellar medium. This has led to the inclusion of `pre-supernova feedback' mechanisms in most such state-of-the-art hydrodynamical simulations, which inject energy and momentum into the interstellar medium before the detonation of the first supernovae. The mechanisms these recipes are intended to model include the energy provided by direct and dust-reprocessed radiation pressure, stellar winds and photoionisation. The explicit injection of momentum from pre-supernova feedback, at mass resolutions between $10^3$ and $10^4$ solar masses, is found to make a qualitative difference to the large-scale properties of the interstellar medium~\\citep[e.g.][]{2013MNRAS.434.3142A,Agertz13,2015ApJ...804...18A,2020arXiv200911309S}. However, it is likely that these treatments substantially over-estimate the effectiveness of direct radiation pressure in driving gas away from young stars, because they do not resolve the radius of the dust destruction front~\\citep{2018MNRAS.480.3468K}. In isolated disc galaxy simulations including purely-thermal energy injection from HII regions~\\citep[e.g.][]{Goldbaum16,Fujimoto19}, pre-supernova feedback is unable to drive dense gas away from young star particles, or to destroy molecular clouds, implying that HII region feedback may suffer from the same `overcooling problem' as does supernova feedback, if the Str\\\"{o}mgren radii of the HII regions are unresolved or only marginally-resolved.\n\nDisentangling the relative roles of supernovae and pre-supernova feedback mechanisms from the spurious numerical variations of each individual feedback mechanism is a work in progress. Here we examine the relative effects of HII region feedback and supernova feedback when the Str\\\"{o}mgren radii of the HII regions are marginally-resolved, at mass resolutions spanning the range from $10^3$ down to $10^5~{\\rm M}_\\odot$ per gas cell. To this end, we develop a novel, physically-motivated sub-grid model for the thermal energy and momentum injected by HII regions at these resolutions, based on the analytic theory of~\\cite{KrumholzMatzner09}. This explicitly includes the momentum from both radiation and gas pressure, and we ensure that our prescription is as close to convergence as possible, by introducing a grouping prescription for simulated star particles with overlapping ionisation front radii. We investigate the effect of our feedback prescription on the interstellar medium and on its giant molecular cloud population, relative to the case of supernova feedback only. We also test the effect of thermal vs.~kinetic energy injection from the HII regions, and of the influence of HII-region shape (spherical, or `blister-type', sitting at the edge of a giant molecular cloud).\n\nThe remainder of the paper is organised as follows. Our `control' isolated galaxy simulation, without HII region feedback, is described in Section~\\ref{Sec::SNe-only}. The analytic theory and numerical implementation of our HII region feedback model is detailed in Section~\\ref{Sec::HII-region-fb}. This section also includes convergence tests for the individual components of the model. In Section~\\ref{Sec::results} we describe the impact of HII region feedback on the large-scale properties of the interstellar medium and on the cloud-scale properties of its cold molecular gas, across a range of mass resolutions from $10^3$ to $10^5~{\\rm M}_\\odot$ per gas cell. Section~\\ref{Sec::discussion} describes the caveats of our model and compares our results to existing studies in the literature. Finally, a summary of our conclusions is given in Section~\\ref{Sec::conclusion}.\n\n\\section{Basic simulation setup (no HII region feedback)} \\label{Sec::SNe-only}\nWe run the simulations presented in this work using the moving-mesh hydrodynamics code {\\sc Arepo}~\\citep{Springel10}. Within {\\sc Arepo}, the gaseous component of each simulation is modelled by an unstructured moving mesh, defined by the Voronoi tessellation about a discrete set of points. The mesh moves with the fluid flow in a way similar to that of smoothed particle hydrodynamics codes, however its spatial resolution can be refined locally and automatically to arbitrarily-high levels without requiring large over-densities to be present in the regions of interest, as in Eulerian codes. As such, a high level of accuracy can be maintained in dealing with shock fronts and low-density flows. \n\nWe use the isolated disc initial condition generated for the Agora comparison project~\\citep{Kim14}. This initial condition is designed to resemble a Milky Way-like galaxy at redshift $z \\sim 0$. It has a~\\cite{Navarro97} dark matter halo of mass $M_{200} = 1.07 \\times 10^{12}~{\\rm M}_\\odot$, a virial radius of $R_{200} = 205~{\\rm kpc}$, a halo concentration parameter of $c = 10$ and a spin parameter of $\\lambda = 0.04$. The stellar bulge is of~\\cite{Hernquist90} type and has a mass of $3.437 \\times 10^9~{\\rm M}_\\odot$, while the exponential disc has a mass of $4.297 \\times 10^{10}~{\\rm M}_\\odot$, a scale-length of $3.43~{\\rm kpc}$, and a scale-height of $0.34~{\\rm kpc}$. The bulge to stellar disc ratio is 0.125 and the overall gas fraction is $0.18$. The lowest-resolution disc (LOW) has a star particle mass (for the initial disc and bulge) of $3.437 \\times 10^5~{\\rm M}_\\odot$, and a dark matter particle mass of $1.254 \\times 10^7~{\\rm M_\\odot}$. The medium-resolution (MED) and high-resolution (HI) discs have stellar and dark matter particle masses that are smaller by factors of $0.1$ and $0.01$, respectively. All discs were generated using the MakeNewDisc code~\\citep{Springel05}. During runtime, we set a softening length of $80~{\\rm pc}$ for the LOW disc, $40~{\\rm pc}$ for the MED disc and $20~{\\rm pc}$ for the HI disc, with median gas cell masses of $8.59 \\times 10^4~{\\rm M}_\\odot$, $8.59 \\times 10^3~{\\rm M}_\\odot$ and $859~{\\rm M}_\\odot$ respectively, following~\\cite{Kim14}. We set the initial gas temperature to be $10^4~{\\rm K}$, however this re-equilibriates within the first $\\leq 10~{\\rm Myr}$ of run-time, according to the state of thermal balance between line-emission cooling and heating due to photoelectric emission from dust grains and polycyclic aromatic hydrocarbons (PAHs), as modelled by our chemical network.\n\nOur star formation prescription locally reproduces the observed relation between the star formation rate and gas surface density~\\citep{Kennicutt98}. The star formation rate density in a gas cell $i$ with volume density $\\rho_i$ is given by\n\\begin{equation}\n\\label{Eqn::starformation}\n\\frac{\\mathrm{d} \\rho_{*,i}}{\\mathrm{d} t} = \n\\begin{cases}\n      \\frac{\\epsilon_{\\rm ff} \\rho_i}{t_{{\\rm ff},i}}, \\; \\rho_i \\geq \\rho_{\\rm thres} \\\\\n      0, \\; \\rho_i < \\rho_{\\rm thres}\\\\\n   \\end{cases},\n\\end{equation}\nwhere $t_{{\\rm ff}, i} = \\sqrt{3\\pi\/(32 G\\rho_i)}$ is the local free-fall time-scale and we set the density threshold $\\rho_{\\rm thresh}$ for star formation at $1~{\\rm cm}^{-3}$, $100~{\\rm cm}^{-3}$ and $1000~{\\rm cm}^{-3}$ for the LOW, MED and HI discs, respectively. Our choice of $\\rho_{\\rm thresh}$ ensures that the densest gas in each simulation is Jeans-unstable at the corresponding mass resolution. We set the star formation efficiency at $\\epsilon_{\\rm ff} = 0.1$, consistent with the upper end of the observed range in dense, molecular gas~\\citep{1974ApJ...192L.149Z,Krumholz&Tan07,2012ApJ...745...69K,2014ApJ...782..114E,2016A&A...588A..29H}. We also tested a value of $\\epsilon_{\\rm ff} = 0.01$ at the lower end of the observed range, however found that this produced much higher turbulent velocity dispersions in the galactic mid-plane of up to $40~{\\rm km~s}^{-1}$, inconsistent with observations of the Milky Way's disc~\\citep[e.g.][]{2009AJ....137.4424T}.\n\nWe follow the stellar evolution of each live star particle formed during the simulation using the Stochastically Lighting up Galaxies code (SLUG), described in~\\cite{daSilva12,daSilva14,Krumholz15}. Each star particle of birth mass $m_{\\rm birth}$ is assigned a `cluster' that is populated with $N$ stars drawn randomly from a~\\cite{2005ASSL..327...41C} initial stellar mass function (IMF), where $N$ is chosen from a Poisson distribution with an expectation value of $m_{\\rm birth}\/\\overline{M}$, with $\\overline{M}$ the mean mass of a single star. For a large number of star particles, this procedure ensures that the IMF is fully-sampled across the stellar population, and that the total mass of populated stars equals the combined mass of the star particles. Within each cluster, SLUG evolves individual stars along Padova solar metallicity tracks~\\citep{Fagotto94a,Fagotto94b,VazquezLeitherer05} with starburst99-like spectral synthesis~\\citep{Leitherer99}.\n\nIn addition to the model for HII region feedback described in the following section, we include feedback from supernovae and mass ejection from stellar winds. Our stellar evolution model provides the number of supernovae $N_{*, {\\rm SN}}$ produced by every star particle, along with the mass loss $\\Delta m_*$ due either to supernovae or to stellar winds. For $N_{*, {\\rm SN}} = 0$, we treat the mass loss as resulting from stellar winds, and simply deposit $\\Delta m_*$ into the nearest gas cell. If $N_{*, {\\rm SN}} > 0$, we make the approximation that all mass loss results from supernovae. We do not resolve the energy-conserving, momentum-generating phase of supernova blast-wave expansion in our simulations, such that we must calculate the terminal momentum of the blast-wave explicitly to prevent over-cooling, following the prescription of~\\cite{KimmCen14}. We use the (unclustered) parametrisation of the terminal momentum injected into the gas cells $k$ neighbouring a central cell $j$, derived from the high-resolution simulations of~\\cite{Gentry17}, and given by\n \\begin{equation} \\label{Eqn::gentry17}\n \\frac{p_{{\\rm t}, k}}{{\\rm M}_\\odot {\\rm kms}^{-1}} = 4.249 \\times 10^5 N_{j, {\\rm SN}} \\Big(\\frac{n_k}{{\\rm cm}^{-3}}\\Big)^{-0.06},\n \\end{equation}\nwhere $N_{j, {\\rm SN}}$ is the cumulative number of supernovae received by a gas cell $j$ from all of the star particles for which it is the nearest neighbour. This terminal momentum is then spread into the cells surrounding the central cell, as in~\\cite{2020arXiv200403608K,2020MNRAS.498..385J,Jeffreson21a}, with an upper limit set by kinetic energy conservation as the shell sweeps up the mass in the cells surrounding the central one~\\citep[see also][for similar prescriptions]{Hopkins18,Smith2018}. A convergence test for a single supernova explosion, implemented via the above method, is presented in Appendix~\\ref{App::res-tests}.\n\nThe chemical composition of the gas in our simulations evolves according to the simplified network of hydrogen, carbon and oxygen chemistry described in~\\cite{GloverMacLow07a,GloverMacLow07b} and in~\\cite{NelsonLanger97}. For each Voronoi gas cell, fractional abundances are computed and tracked for the chemical species ${\\rm H}$, ${\\rm H}_2$, ${\\rm H}^+$, ${\\rm He}$, ${\\rm C}^+$, ${\\rm CO}$, ${\\rm O}$ and ${\\rm e}^-$. The chemistry is coupled to the heating and cooling of the interstellar medium via the atomic and molecular cooling function of~\\cite{Glover10}. The full list of heating and cooling processes is given in their Table 1. As such, the heating and cooling rates in our simulations depend not only on the gas density and temperature, but also on the strength of the interstellar radiation field, the cosmic-ray ionisation rate, the dust fraction and temperature, and on the set of chemical abundances tracked for each gas cell. We assign a value of $1.7$~Habing fields to the UV component of the ISRF according to~\\cite{Mathis83}, a value of $3 \\times 10^{-17}$~s$^{-1}$ to the cosmic ionisation rate~\\citep{2000A&A...358L..79V}, and assume the solar value for the dust-to-gas ratio.\n\n\\section{HII region feedback} \\label{Sec::HII-region-fb}\nIn this section, we derive the momentum per unit time provided by a single HII region to the surrounding interstellar medium. We develop a novel sub-grid model for injecting this momentum in simulations that do not resolve the median Str{\\\"o}mgren radius. We also describe our prescription for heating the interstellar medium within this radius. Parts of the following prescription are used in the simulations `HII heat', `HII spherical mom.', `HII beamed mom.' and `HII heat \\& beamed mom.', as listed in Table~\\ref{Tab::sims}.\n\n\\begin{table}\n\\begin{center}\n\\label{Tab::sims}\n  \\caption{The five simulations run in this work and their feedback prescriptions.}\n  \\begin{tabular}{@{}l m{4.5cm}@{}}\n  \\hline\n   Simulation name & Feedback prescription \\\\\n  \\hline\n   SNe (control) & Supernovae only \\\\\n   HII heat & Supernovae plus thermal HII regions \\\\\n   HII spherical mom. & Supernovae plus spherical HII region momentum \\\\\n   HII beamed mom. & Supernovae plus beamed HII region momentum \\\\\n   HII heat \\& beamed mom. & Supernovae plus thermal HII regions plus beamed HII region momentum \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\subsection{Analytic theory} \\label{Sec::Theory}\nWe consider the momentum injected into a molecular cloud by a massive star or stellar cluster that produces ionising photons of energy $\\epsilon_0 \\sim 13.6$~eV at a rate $S$. We choose a system of co-ordinates that is centred on the ionising source, and following~\\citealt{KrumholzMatzner09} (hereafter KM09), we parametrize the density profile of the surrounding gas as a power-law of the form $\\rho(r) = \\rho_0(r\/r_0)^{-k_\\rho}$. A source may be fully `embedded' within the host cloud, such that it is surrounded on all sides by dense gas, or it may be located at the edge of a molecular cloud, such that it produces a `blister-type', hemispherical HII region, for which we take $\\rho = 0$ for the outward-facing hemisphere. The photons from the source transfer their energy to the surrounding gas via two primary mechanisms. Firstly, kinetic energy is carried away by the products of ionisation (free electrons, hydrogen nuclei and helium nuclei), heating the ionised material to a temperature of $T_{\\rm II}$~\\citep{Spitzer78}. As the sound speed $c_{\\rm II}$ in the ionised gas is much higher than that in the surrounding neutral gas, the initial expansion of the HII region sweeps up a thin shell of neutral material that separates the ionised region from its surroundings. Secondly, photons may be absorbed by dust grains and hydrogen atoms, delivering a momentum kick that accelerates the particles away from the ionising source. The radiative acceleration will always be highest closest to the source (KM09), again contributing to the production of a thin shell bounding the HII region. As such, the momentum delivered to the dense gas outside the HII region is very well approximated by the momentum of this bounding shell. The momentum equation for this shell may be written in the form of~\\cite{Matzner02} as\n\\begin{equation}\n\\label{Eqn::momentumequation_orig}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} = \\frac{\\mathrm{d}}{\\mathrm{d} t}(M_{\\rm sh} \\dot{r}_{\\rm II}) = A_{\\rm sh}\\Big[P_{\\rm gas} + P_{\\rm rad}\\Big],\n\\end{equation}\nwhere $M_{\\rm sh} = (2,4)\\pi r_{\\rm II}^3 \\overline{\\rho}(r_{\\rm II})$ is the mass of neutral material swept into the shell of ionisation front radius $r_{\\rm II}$ during its initial rapid expansion, $A_{\\rm sh} = (2,4)\\pi r_{\\rm II}^2$ is its surface area, and $\\overline{\\rho}(r) = 3\/(3-k_\\rho) \\rho_0 (r\/r_0)^{-k_\\rho}$ is the mean volume density inside a radius $r$ in the initial molecular cloud. The first value in the above parentheses corresponds to the case of a blister-type HII region, in which the gas pressure at the HII-cloud interface is augmented by a thrust of equal magnitude and direction, due to the flux of gas through the opposing hemisphere.  The equality of the pressure and thrust terms depends on the assumption that the ejected gas can escape freely from the HII region, such that its velocity relative to the velocity $\\dot{r}_{\\rm II}$ of the ionisation front tends towards the speed of sound within the HII region~\\citep{Kahn54}. The second value in parentheses then corresponds to the case of an embedded HII region, in which no thrust is produced. The momenta delivered by thermal heating and radiative acceleration are given in terms of a gas pressure $P_{\\rm gas}$ and a radiation pressure $P_{\\rm rad}$, respectively. The gas pressure is given by\n\\begin{equation}\n\\label{Eqn::Pgas}\nP_{\\rm gas} = (2,1) \\rho_{\\rm II} c_{\\rm II}^2,\n\\end{equation}\nwhere $\\rho_{\\rm II}$ refers to the density of the heated, ionised gas inside the swept-up shell. The radiation pressure term in Equation (\\ref{Eqn::momentumequation_orig}) can be written in the form of KM09 as\n\\begin{equation}\n\\label{Eqn::Prad}\nP_{\\rm rad} = \\frac{f_{\\rm trap} \\epsilon_0 S}{4\\pi c r_{\\rm II}^2},\n\\end{equation}\nwhere the factor $f_{\\rm trap}$ quantifies the enhancement of the radiative force via the trapping of photons and stellar winds in the expanding shell, and $c$ is the speed of light.\n\n\\begin{figure*}\n  \\label{Fig::FoF-convergence-test}\n    \\includegraphics[width=\\linewidth]{figs\/FoF-convergence-test.pdf}\n    \\caption{Distributions of birth masses (left), ionising luminosities (centre) and momentum injection rates (right) for the star particles (thin lines) and FoF groups of star particles with overlapping ionisation front radii (bold lines) in our three `HII heat \\& beamed mom.' simulations at time $t=600$~Myr. Each line colour corresponds to a different numerical mass resolution: $10^5~{\\rm M}_\\odot$ (LOW, dark blue), $10^4~{\\rm M}_\\odot$ (MED, grey), and $10^3~{\\rm M}_\\odot$ (HI, yellow). The arrows in the left-hand panel point to the median mass resolution in each galaxy.}\n\\end{figure*}\n\n\\begin{figure}\n  \\label{Fig::rst0}\n    \\includegraphics[width=\\linewidth]{figs\/rst0.pdf}\n    \\caption{Distributions of ionisation front radii (solid lines) and Str\\\"{o}mgren radii (dotted lines) for the star particles in our three `HII heat \\& beamed mom.' simulations at time $t=600$~Myr. Each line colour corresponds to a different numerical mass resolution: $10^5~{\\rm M}_\\odot$ (LOW, dark blue), $10^4~{\\rm M}_\\odot$ (MED, grey), and $10^3~{\\rm M}_\\odot$ (HI, yellow). The arrows point to the median gas cell radius in each galaxy.}\n\\end{figure}\n\nTo obtain the momentum injected into the cloud per unit time, we must solve the equation of motion for the expansion of the shell. We assume that the contribution of $P_{\\rm gas}$ is well-approximated by its value in a gas pressure-dominated HII region, for which $P_{\\rm gas} \\gg P_{\\rm rad}$. In the case that $P_{\\rm gas} \\ll P_{\\rm rad}$, the inaccuracy associated with this assumption will be small in comparison to the radiation pressure. Once the rate of expansion $\\dot{r}_{\\rm II}$ slows to the speed of sound in the ionised gas, the density inside the HII region equilibriates to a uniform value on the sound-crossing time, and is described by the condition of photoionisation balance~\\citep{Spitzer78}, such that\n\\begin{equation}\n\\label{Eqn::photoionisationbalance}\n\\frac{4}{3} \\pi r_{\\rm II}^3 \\alpha_{\\rm B} \\Big(\\frac{\\rho_{\\rm II} c_{\\rm II}^2}{k_{\\rm B} T_{\\rm II}}\\Big)^2 \\eta^2 = \\phi S,\n\\end{equation}\nwhere $\\alpha_{\\rm B}$ is the case-B recombination coefficient and $\\phi$ is a dimensionless constant that quantifies the effect of photon absorption by dust grains.\\footnote{This accounts for the 27~per~cent of photons at Milky Way metallicity~\\protect\\citep{McKee&Williams97} that are absorbed by dust grains and so do not contribute to the gas pressure. Following KM09, we assume that the gas and dust are well-coupled by the ambient magnetic field, so that the direct radiation pressure does not depend on the gas-to-dust ratio. For further details, see the Appendix of KM09.} The parameter $\\eta$ is given by\n\\begin{equation}\n\\label{Eqn::eta}\n\\eta = \\frac{\\mu}{\\sqrt{\\mu_e \\mu_{\\rm H^+}}},\n\\end{equation}\nwhere $\\mu_{\\rm H^+}$ is the mean mass per proton in the ionised region, $\\mu_{e}$ is the mean mass per free electron, and $\\mu$ is the mean mass per free particle. Combining Equations (\\ref{Eqn::Pgas}) and (\\ref{Eqn::photoionisationbalance}) to rewrite the gas pressure term in Equation (\\ref{Eqn::momentumequation_orig}) gives the momentum delivered to the host cloud per unit time as\n\\begin{align}\n\\label{Eqn::momentumequation_rdim}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} &= \\frac{\\mathrm{d}}{\\mathrm{d} t} (M_{\\rm sh} \\dot{r}_{\\rm II}) = \\frac{f_{\\rm trap} \\epsilon_0 S}{(2,1)c} \\Big[1 + x_{\\rm II}^{\\tfrac{1}{2}}\\Big] \\\\\n&\\approx (1.2, 2.4) \\times 10^3 \\: S_{49} \\: M_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\Big[1 + x_{\\rm II}^{\\tfrac{1}{2}}\\Big],\n\\end{align}\nwhere we define the dimensionless scale parameter as $x_{\\rm II} = r_{\\rm II}\/r_{\\rm ch}$ with\n\\begin{align}\n\\label{Eqn::r_ch}\nr_{\\rm ch} &= \\frac{\\alpha_{\\rm B}}{12(4,1)\\pi \\phi} \\Big(\\frac{\\epsilon_0 \\eta}{k_{\\rm B} T_{\\rm II}}\\Big)^2 \\Big(\\frac{f_{\\rm trap}}{c}\\Big)^2 S \\\\\n&\\approx (0.5, 1.9) \\times 10^{-2} S_{49} \\; {\\rm pc},\n\\end{align}\nwhich is the characteristic radius at which the gas and radiation pressure make equal contributions to the rate of momentum injection. To obtain the order-of-magnitude estimates for $\\mathrm{d} p\/\\mathrm{d} t$ and $r_{\\rm ch}$ in Equations (\\ref{Eqn::momentumequation_rdim}) and (\\ref{Eqn::r_ch}), we have used the same fiducial values as in KM09, setting $T_{\\rm II} = 7000$~K, $\\alpha_{\\rm B} = 3.46 \\times 10^{-13}$~cm$^3$~s$^{-1}$ and $\\phi = 0.73$, consistent with the Milky-Way dust-to-gas ratio~\\citep{McKee&Williams97}. We set $f_{\\rm trap} = 8$, consistent with observations of the pressure inside young HII regions by~\\cite{2021ApJ...908...68O}. We take $\\eta = 0.48$\\footnote{The condition for photo-ionisation balance given in Equation (\\ref{Eqn::photoionisationbalance}) differs from Equation (2) of KM09 by a factor of $\\eta^2 \\sim 0.2$, and therefore the value of $r_{\\rm ch}$ that we obtain is smaller than theirs by the same factor.}, corresponding to a ten-to-one ratio of hydrogen to helium atoms in the neutral gas, with the helium atoms singly-ionised. The ionising luminosity has been rescaled as $S_{49} = S\/10^{49} {\\rm s}^{-1}$. We assume that the volume density of the gas swept up by the HII region is approximately uniform, so that $k_\\rho = 0$. The time-evolution of $\\mathrm{d} p\/\\mathrm{d} t$ can be computed by writing Equation (\\ref{Eqn::momentumequation_rdim}) in the following non-dimensional form\n\\begin{equation}\n\\label{Eqn::momentumequation_nondim}\n\\frac{\\mathrm{d}}{\\mathrm{d} \\tau}\\Big(x_{\\rm II}^{\\rm 3-k_\\rho} \\frac{\\mathrm{d}}{\\mathrm{d} \\tau} x_{\\rm II}\\Big) = 1 + x_{\\rm II}^{1\/2},\n\\end{equation}\nwhere $\\tau = t\/t_{\\rm ch}$ for a characteristic time $t_{\\rm ch}$ at which the gas and radiation pressure are equal, given by\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::t_ch}\nt_{\\rm ch} &= \\sqrt{\\frac{4\\pi c \\overline{\\rho}(r_{\\rm st,0})}{3 f_{\\rm trap} \\epsilon_0 S} r_{\\rm ch}^4 \\Big(\\frac{r_{\\rm ch}}{r_{\\rm st,0}}\\Big)^{-k_\\rho}} \\\\\n&\\approx (45,333) \\: \\overline{n}_{\\rm H,2}^{1\/6} S_{49}^{7\/6} \\: {\\rm yr}\n\\end{split}\n\\end{equation}\nwhere $r_{\\rm st,0}$ is the initial Str{\\\"o}mgren radius and $\\overline{\\rho}(r_{\\rm st,0})$ is the mean density inside $r_{\\rm st,0}$ in the initial molecular cloud. These two quantities are related by Equation (\\ref{Eqn::photoionisationbalance}) as\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::r_st0}\nr_{\\rm st,0} &= \\Big(\\frac{3\\phi S}{4\\pi \\alpha_{\\rm B}}\\Big)^{1\/3} \\Big(\\frac{\\mu m_{\\rm H}}{\\overline{\\rho}(r_{\\rm st,0})\\eta}\\Big)^{2\/3} \\\\\n&\\approx 2.5 \\: {\\rm pc} \\: S_{49}^{1\/3} \\overline{n}_{\\rm H,2}^{-2\/3}.\n\\end{split}\n\\end{equation}\nTo obtain Equation (\\ref{Eqn::t_ch}), we have used the radial scaling of the mean volume density to write $M_{\\rm sh} = (2,4) \\pi r_{\\rm II}^3 \\overline{\\rho}(r_{\\rm st,0}) (r_{\\rm II}\/r_{\\rm st,0})^{-k_\\rho}$. The numerical value of $t_{\\rm ch}$ is obtained by writing $\\overline{\\rho}_{\\rm st,0} = 100 \\overline{\\mu} m_{\\rm H} \\overline{n}_{\\rm H,2}$ as in KM09, where $\\overline{\\mu} \\sim 1.4$ is the mean molecular weight in the ionised gas, $m_{\\rm H}$ is the proton mass, and $\\overline{n}_{\\rm H,2}$ is the number density of hydrogen atoms inside $r_{\\rm st, 0}$, in units of $100$~cm$^{-3}$.\n\nEquation (\\ref{Eqn::momentumequation_nondim}) has an approximate analytic solution that interpolates between the gas-dominated and radiation-dominated cases to an accuracy of better than $5$~per~cent~\\citep{KrumholzMatzner09}, given by\n\\begin{equation}\n\\label{Eqn::xapprox}\nx_{\\rm II, approx} = \\Big[\\frac{3}{2}\\tau^2 + \\Big(\\frac{25}{28} \\tau^2\\Big)^{6\/5}\\Big]^{1\/3}.\n\\end{equation}\nWith this solution, we may finally write the momentum equation as\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::momentumeqn_final}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} \\approx (1.2, 2.4) \\: &S_{49} \\times 10^3 M_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\times \\\\\n&\\left\\{ 1 + \\Big[\\frac{3}{2}\\frac{t^2}{t_{\\rm ch}^2} + \\Big(\\frac{25}{28} \\frac{t^2}{t_{\\rm ch}^2}\\Big)^{6\/5}\\Big]^{1\/6}\\right\\},\n\\end{split}\n\\end{equation}\nwith $t_{\\rm ch}$ given by Equation (\\ref{Eqn::t_ch}). An HII region will deposit momentum at this rate into the surrounding dense gas until its expansion stalls (see Section~\\ref{Sec::stalling}). For a Galactic giant molecular cloud with $\\overline{n}_{\\rm H,2} \\sim 1$ and a star cluster with an ionising luminosity of $S_{49} \\sim 100$, this characteristic time is around $10,000$ years, indicating that for the majority of its life-span (of order a few Myr), the momentum output from such an HII region is dominated by gas pressure. The gas pressure contributes over 90~per~cent of the final injected momentum. Only for the most luminous clusters (such as M82 L$^{\\rm a}$ in KM09, with $t_{\\rm ch} \\sim 10$~Myr) does radiation pressure dominate the momentum budget. We note that other sub-grid models for HII region feedback at similar resolutions~\\citep[e.g.][]{2011MNRAS.417..950H,2013MNRAS.434.3142A,Agertz13,2014MNRAS.445..581H,2015ApJ...804...18A} consider only the part of $\\mathrm{d} p\/\\mathrm{d} t$ due to radiation pressure (the first term in the curly brackets in Equation~\\ref{Eqn::momentumeqn_final}). In order to inject significant quantities of momentum from HII regions, they therefore inflate $f_{\\rm trap}$ above observed values. This is discussed further in Section~\\ref{Sec::SNe-fb}.\n\n\\subsection{Numerical implementation of HII region momentum} \\label{Sec::num-methods}\n\\subsubsection{Grouping of star particles} \\label{Sec::grouping}\nThe rate of momentum injection given in Equation~(\\ref{Eqn::momentumeqn_final}) does not scale linearly with the cluster luminosity $S_{49}$. This means that the total momentum injected by the star particles in a numerical simulation will not trivially converge with increasing mass resolution. As shown in the left-hand panel of Figure~\\ref{Fig::FoF-convergence-test}, the maximum stellar particle mass in {\\sc Arepo} is equal to twice the simulation mass resolution (the median gas cell mass, given by the solid vertical lines). Upon reaching the star formation threshold $\\rho_{\\rm thresh}$, larger gas cells are decremented in mass and `spawn' star particles at twice the simulation resolution, while smaller gas cells are deleted and replaced by stars of equal mass. At mass resolutions of $\\sim 900~{\\rm M}_\\odot$, the largest stellar clusters are made up of hundreds of star particles with overlapping ionisation-front radii $r_{\\rm II,*}$, each of which is given by\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::rII}\nr_{{\\rm II}, *}(t) &= r_{\\rm ch} x_{\\rm II, approx} \\\\\n&\\approx 0.5 \\times 10^{-2} S_{49,*} \\: {\\rm pc} \\: \\times \\\\\n&\\left\\{\\frac{3}{2} \\Big(\\frac{t_*}{t_{{\\rm ch},*}}\\Big)^2 + \\Big[\\frac{25}{28} \\Big(\\frac{t_*}{t_{{\\rm ch},*}}\\Big)^2\\Big]^{6\/5}\\right\\}^{1\/3},\n\\end{split}\n\\end{equation}\nwhere $t_{\\rm ch,*}$ is the characteristic time for an individual star particle, (see Equation~\\ref{Eqn::t_ch}). Physically, a group of star particles whose ionisation fronts overlap should be treated as a single HII region with a single birth density $\\overline{n}_{\\rm H,2}$ and luminosity $S_{49}$, given that the density of the ionised gas inside the bounding shell of a subsonic HII region equilibriates on its sound-crossing time and becomes uniform. We therefore substantially improve the resolution-convergence of our momentum deposition (in a physically-motivated sense) by using a Friends-of-Friends (FoF) linking algorithm between star particles, with a linking length of $r_{\\rm II, *}$.\\footnote{Note that this method cannot produce perfect resolution convergence because the ionisation front radii also depend weakly on both the stellar luminosity $S_{49}$ and the stellar birth density $\\overline{n}_{\\rm H,2}$.} A sample of the FoF groups produced by this algorithm in the low-resolution run is shown in Figure~\\ref{Fig::FoF-schematic}.\n\nIn practice, the total momentum injected by an FoF-grouped HII region during a numerical time-step $\\Delta t_{\\rm HII}$ is then given by\n\\begin{equation}\n\\Delta p_{\\rm HII} = {\\Big(\\frac{\\mathrm{d} p}{\\mathrm{d} t}\\Big)_{\\rm FoF} \\Delta t_{\\rm HII}},\n\\end{equation}\nwith a momentum injection rate of\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::momentumequation_fof}\n\\Big( \\frac{\\mathrm{d} p}{\\mathrm{d} t} &\\Big)_{\\rm FoF} = \\; (1.2, 2.4) \\times 10^3 \\times \\\\ &\\sum_{*=1}^N{S_{49,*}} \\: {\\rm M}_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\times \\\\\n &\\left\\{1 + \\Big[\\frac{3}{2} \\Big(\\frac{\\langle t \\rangle_S}{t_{\\rm ch, FoF}}\\Big)^2 + \\Big(\\frac{25}{28} \\frac{\\langle t \\rangle_S}{t_{\\rm ch,FoF}}\\Big)^{6\/5}\\Big]^{1\/6} \\right\\},\n\\end{split}\n\\end{equation}\nand a characteristic time of\n\\begin{equation}\n\\label{Eqn::t_ch_fof}\nt_{\\rm ch,FoF} = \\sqrt{(0.6, 33) \\Big(\\sum^N_{*=1}{S_{49,*}}\\Big)^{7\/3} \\langle \\overline{n}_{\\rm H,2} \\rangle_S^{1\/3} \\: {\\rm pc} \\: {\\rm s}^2}\n\\end{equation}\nfor each FoF group. The angled brackets $\\langle ... \\rangle_S$ denote ionising luminosity-weighted averages over the star particles $*=1 ... N$ in the group, such that $\\langle t \\rangle_S$ is the luminosity-averaged age of the star particles. The momentum is injected at the luminosity-weighted centre of the group, given by\n\\begin{equation}\n\\label{Eqn::centre_fof}\n\\langle \\bm{x} \\rangle_S = \\frac{\\sum_{*=1}^N{S_{49,*}\\bm{x}_*}}{\\sum_{*=1}^N{S_{49,*}}}.\n\\end{equation}\nTo ensure that all star particles in a single group have their ionising luminosities, ionisation front radii and ages updated on the same time-step, we set $\\Delta t_{\\rm HII}$ to be the global time-step for the simulation. This means that we inject HII region feedback on global time-steps only, which have a maximum value of $0.1$~Myr for the simulations presented in this work.\\footnote{Due to the hierarchical time-stepping procedure used in {\\sc Arepo}~\\citep[see][for details]{Springel10}, the ionising properties of different star particles from the same FoF group would otherwise be updated at different time intervals. Our global time-step has a maximum value of $0.1$~Myr, and the peak ionisation-front expansion rate for the most massive star particles in our high-resolution simulation ($\\sim 2 \\times 10^3$~M$_\\odot$, see Figure~\\ref{Fig::FoF-convergence-test}) is $\\sim 20$~pc Myr$^{-1}$, so in the very worst case, our FoF groups may be affected by an error of order $\\sim 2$~pc.}\n\n\\begin{figure}\n  \\label{Fig::FoF-schematic}\n    \\includegraphics[width=\\linewidth]{figs\/FoF-schematic.pdf}\n    \\caption{FoF groups of HII regions with overlapping ionisation front radii in a $(950~{\\rm pc})^3$ box within the LOW-resolution simulation with spherical HII region feedback (`HII spherical mom.') at $\\sim 600~{\\rm Myr}$ (the galaxy is centred at the origin). The coloured circles represent the ionisation front radii of individual HII regions, while the small black stars mark the positions of unlinked HII regions (FoF group size of one) and the coloured stars mark the luminosity-weighted centres of the linked HII regions (FoF group size greater than one). The colour bar gives the z-position of the HII regions.}\n\\end{figure}\n\n\\begin{figure}\n  \\label{Fig::2HII-convergence}\n    \\includegraphics[width=\\linewidth]{figs\/2HII-convergence.pdf}\n    \\caption{Energy (upper panel) and radial momentum (lower panel) of the gas cells in a box of size $(950~{\\rm pc})^3$ and density $100~{\\rm cm}^3$ containing two star particles of mass $1 \\times 10^4~{\\rm M}_\\odot$, separated by a distance of $5$~pc, as a function of time. The star particles both inject spherical momentum from HII regions (no thermal energy injection) according to the prescription outlined in Section~\\ref{Sec::num-methods}. In the lower panel, the combined momentum from the individual star particles (dashed lines) is compared to the momentum injected by the FoF-grouped pair (solid lines). The resolution varies from a lowest value of $10^7~{\\rm M}_\\odot$ per gas cell (dark blue) up to a highest resolution of $10^3~{\\rm M}_\\odot$ per gas cell (yellow).}\n\\end{figure}\n\n\\begin{figure*}\n  \\label{Fig::blister-projections}\n    \\includegraphics[width=\\linewidth]{figs\/blister-projections.pdf}\n    \\vspace{-0.5cm} \\caption{Column density projections of the gas cells surrounding an HII region of stellar mass $10^4$~M$_\\odot$ in a $(950~{\\rm pc})^3$ box containing $128^3$ gas cells at a volume density of $100~{\\rm cm}^{-3}$. The top row shows a spherical\/embedded HII region, while the bottom row shows a beamed\/blister-type HII region with momentum injected accordering to Equation (\\ref{Eqn::jet-profile}), using an opening angle of $\\Theta = \\pi\/12$ and axis aligned along $(x=1\/\\sqrt{2}, y=1\/\\sqrt{2})$. All projections are computed parallel to the simulation $z$-axis.}\n\\end{figure*}\n\nIn Figure~\\ref{Fig::FoF-convergence-test} we show the effect of grouping on the HII region masses (left-hand panel), the ionising luminosities (centre panel), and the momentum injection rate (right-hand panel) in the low-resolution (dark blue lines), medium-resolution (grey lines) and high-resolution (yellow lines) Agora disc simulations. Comparison of the bold lines (FoF groups) and thin lines (star particles) demonstrates that the distributions of stellar birth masses $m_{\\rm birth}$, luminosities $S_{49}$, and momentum injection rates $\\mathrm{d} p\/\\mathrm{d} t$, are brought closer to convergence by the FoF grouping. The ionisation front radii $r_{\\rm II}$ used to compute the groups are displayed for each simulation in Figure~\\ref{Fig::rst0}. We might also consider using the Str{\\\"o}mgren radius $r_{\\rm st,0}$ (dotted lines, left-hand panel) as the FoF linking-length, however $r_{\\rm st,0}$ is more heavily-dependent on the stellar birth density than is $r_{\\rm II}$ (see Equations~\\ref{Eqn::r_st0} and~\\ref{Eqn::rII}), and so is more heavily-dependent on the simulation resolution, making it a less favourable choice.\n\nWhile our FoF grouping corrects for the non-linear dependence of Equation~(\\ref{Eqn::momentumeqn_final}) on the ionising luminosity, and reduces the spurious cancellation of the momentum injected between adjacent star particles, it \\textit{does not} address resolution-dependent variations in the spatial distribution of stellar mass in our simulations, caused partly by the variation in star particle mass, and partly by the suppressed clustering of star particles at lower resolutions. These effects change the spatial distribution of the energy injected by stellar feedback. As discussed in~\\cite{2020arXiv200911309S,2020arXiv200403608K}, an increase in the clustering of supernovae leads to burstier feedback with larger outflows perpendicular to the galactic mid-plane. We discuss these effects further in Section~\\ref{Sec::discussion}.\n\n\\begin{figure}\n  \\label{Fig::heat-convergence}\n    \\includegraphics[width=\\linewidth]{figs\/heat-convergence.pdf}\n    \\caption{Ionising luminosity emitted by star particles ($S_{49, *}$, solid lines) and absorbed by surrounding gas cells ($S_{49, {\\rm g}}$, thin lines) at a density of $100~{\\rm cm}^{-3}$ in a box of size $(950~{\\rm pc})^3$ containing 100 star particles of mass $1 \\times 10^4~{\\rm M}_\\odot$, as a function of time. The star particles inject HII region heating feedback according to the prescription outlined in Section~\\ref{Sec::num-methods}. The resolution varies from a lowest resolution of $10^7~{\\rm M}_\\odot$ per gas cell (dark blue) up to a highest resolution of $10^3~{\\rm M}_\\odot$ per gas cell (yellow). Note that lines for the $10^3~{\\rm M}_\\odot$ and $10^4~{\\rm M}_\\odot$ overlap.}\n\\end{figure}\n\n\\begin{figure*}\n  \\label{Fig::morphology}\n    \\includegraphics[width=\\linewidth]{figs\/morphology.pdf}\n    \\caption{Column density maps of the gas at each resolution (LOW = $10^5~{\\rm M}_\\odot$ per cell, MED = $10^4~{\\rm M}_\\odot$, HI = $10^3~{\\rm M}_\\odot$) and with each feedback prescription, at $t=600$~Myr. Momentum injection from HII regions changes the phase structure of the interstellar medium in the MED- and HI-resolution cases only. It also improves the numerical convergence of the gas-disc scale-height. Heating from HII regions makes no difference to the disc morphology at any resolution.}\n\\end{figure*}\n\n\\begin{figure}\n  \\label{Fig::tuning-forks}\n    \\includegraphics[width=\\linewidth]{figs\/tuning-forks.pdf}\n    \\caption{The gas-to-SFR flux ratio relative to the galactic average value as a function of aperture size, for each of our high-resolution galaxies at $t = 600$~Myr. The upper branch represents apertures focussed on molecular gas peaks, while the lower branch represents apertures focussed on the peaks of surface density of `young stars' (ages $0$-$5$~Myr). The error bars on each data point represent the 1$\\sigma$ uncertainty on the value of the gas-to-SFR flux ratio. The shaded areas on each data point indicate the effective $1\\sigma$ uncertainty range that accounts for the covariance between the data points. The grey-shaded region shows the result of applying the same analysis to observations of the Milky Way-like galaxy NGC 628~\\protect\\citep{Chevance20}. The dashed coloured lines represent the best-fit model of~\\protect\\cite{Kruijssen18a}.}\n\\end{figure}\n\n\\begin{figure*}\n  \\label{Fig::outflows}\n    \\includegraphics[width=\\linewidth]{figs\/outflow-rate.pdf}\n    \\caption{Global galactic star formation rate (top row), gas outflow rate (see Section~\\protect\\ref{Sec::outflows}) from the galactic mid-plane (middle row) and mass-loading of outflows (bottom row) as a function of simulation time for the simulation with both thermal and beamed momentum from HII regions (HII thermal \\& beamed mom.), at low-resolution (LOW, dark blue), at medium-resolution (MED, grey), and high-resolution (HI, yellow).}\n\\end{figure*}\n\n\\begin{figure*}\n  \\label{Fig::vss}\n    \\includegraphics[width=\\linewidth]{figs\/vss.pdf}\n    \\caption{Azimuthally- and temporally-averaged cold-gas ($\\leq 10^4~{\\rm K}$) gas-disc scale-height (top row), molecular gas (dashed lines) and total gas (solid lines) surface density (second row), thermal (dashed lines) and total (thermal plus turbulent, solid lines) velocity dispersion (third row), and the Toomre $Q_{\\rm g}$ parameter of the total gas distribution (bottom row) as a function of the galactocentric radius for each simulated galaxy at each resolution, across the time interval $t=300$-$600$~Myr.}\n\\end{figure*}\n\n\\subsubsection{Injection of momentum from HII regions} \\label{Sec::injection}\nWe inject the radial momentum $\\Delta p_{{\\rm HII}, *}$ from each star particle at the luminosity-weighted centre $\\langle \\bm{x} \\rangle_S$ of its FoF group via the same procedure as used for supernovae, described in~\\cite{2020arXiv200403608K,2020MNRAS.498..385J}. Briefly, the algorithm proceeds as follows.\n\\begin{enumerate}\n  \\item For each FoF group, find the nearest-neighbour gas particle $j$ to the luminosity-weighted centre of mass.\n  \\item Increment the total radial momentum $\\Delta p_{j, {\\rm HII}}$ received by cell $j$ from all of the FoF groups it hosts, such that\n\\begin{equation}\n\\Delta p_{j, {\\rm HII}} = \\sum_{{\\rm FoF}=1}^N {\\Big(\\frac{\\mathrm{d} p}{\\mathrm{d} t}\\Big)_{\\rm FoF}} \\Delta t_{\\rm HII}.\n\\end{equation}\n  \\item For each gas cell $j$ that has received HII-region momentum, find the set of neighbouring gas cells $k$ with which it shares a Voronoi face. Compute the fraction of the radial momentum received by each facing cell according to\n\\begin{equation}\n\\label{Eqn::j-to-k-inj}\n\\Delta \\bm{p}_{k, {\\rm HII}} = w_k \\hat{\\bm{r}}_{j \\rightarrow k} \\Delta p_{j, {\\rm HII}},\n\\end{equation}\nwhere $\\hat{\\bm{r}}_{j \\rightarrow k}$ is the unit vector from the centre of the host cell to the centre of the cell receiving the feedback, and the weight factor $w_k$ is the fractional Voronoi face area $A_{j \\rightarrow k}$ shared between these cells, such that\n\\begin{equation}\n\\label{Eqn::weight-fn}\nw_k = \\frac{A_{j \\rightarrow k}}{\\sum_k{A_{j \\rightarrow k}}}.\n\\end{equation}\nEquation (\\ref{Eqn::j-to-k-inj}) ensures that the momentum injection is perfectly isotropic, regardless of the distribution over the volumes of the cells $k$.\n  \\item Ensure conservation of linear momentum by subtracting the sum of the injected momenta $\\sum_k{\\Delta p_{k, {\\rm HII}}}$ from the momentum of the central cell $j$.\\footnote{We note that we do not account for a full tensor renormalisation of the injected momentum, as in~\\cite{Hopkins18b,Smith2018,2020arXiv200911309S}, for example.}\n\\end{enumerate}\n\nIn Figure~\\ref{Fig::2HII-convergence}, we check the numerical convergence of the momentum and energy injected according to the algorithm described above, at mass resolutions varying between $10^7~{\\rm M}_\\odot$ and $10^3~{\\rm M}_\\odot$ per gas cell. We take a box of side-length $950~{\\rm pc}$ and uniform gas density $100~{\\rm cm}^{-3}$, containing a single pair of star particles of mass $10^4~{\\rm M}_\\odot$ each. We record the radial momentum of the gas cells in the box as a function of time when the stars inject momentum from their individual HII regions (dashed lines, lower panel), and when the stars are grouped via the FoF procedure described in Section~\\ref{Sec::grouping} (solid lines, lower panel). We also record the kinetic and total energies of the gas cells in the FoF-grouped case, represented by the thin and bold lines, respectively, in the top panel of Figure~\\ref{Fig::2HII-convergence}. The bottom panel demonstrates that the radial momentum injected is converging to within 1.1 dex in momentum per 3 dex in mass resolution in both the FoF-grouped (solid lines) and ungrouped (dashed lines) cases, for the mass resolutions between $10^3~{\\rm M}_\\odot$ and $10^5~{\\rm M}_\\odot$ spanned by our isolated disc galaxies. At lower resolutions the injected momentum does not persist, but rather begins to drop steeply after about $10$~Myr of evolution. This is because the ionisation front bounding the HII region is never resolved at mass resolutions of $>10^5~{\\rm M}_\\odot$, and so the neighbouring gas cells $k$ have a combined mass much larger than that of the swept-up shell. This greatly reduces their final velocities\/kinetic energies (shown in the top panel of Figure~\\ref{Fig::2HII-convergence}), and so the injected momentum is quickly lost. This behaviour is not inaccurate, as entirely-unresolved feedback processes should not have any impact on the simulated interstellar medium.\n\n\\subsubsection{Directional injection for blister-type HII regions}\nThe weight factor $w_k$ in Section~\\ref{Sec::injection} results in isotropic momentum injection, appropriate for embedded HII regions. To mimic the directional outflow from a blister-type HII region along an axis $\\hat{\\bm{z}}_{\\rm FoF}$, we instead weight the momenta $\\Delta p_{k, {\\rm HII}}$ by the following axisymmetric factor,\n\\begin{equation}\n\\label{Eqn::jet-profile}\n\\begin{split}\nw(\\theta_k, A_k) &= \\frac{A_{j \\rightarrow k} f(\\theta_k)}{\\sum_k{A_{j \\rightarrow k} f(\\theta_k)}} \\\\\nf(\\theta_k) &= \\Big[\\log{\\Big(\\frac{2}{\\Theta}\\Big)(1+\\Theta^2-\\cos^2{\\theta_k})}\\Big]^{-1}\n\\end{split}\n\\end{equation}\nwhere $\\Theta$ controls the width of the beam and $\\theta_k$ is the angle between the beam-axis and the unit vector $\\hat{\\bm{r}}_{j \\rightarrow k}$ connecting cells $j$ and $k$, defined by\n\\begin{equation}\n\\cos{\\theta_k} = \\frac{\\hat{\\bm{r}}_{j \\rightarrow k} \\cdot \\hat{\\bm{z}}_{\\rm FoF}}{|\\hat{\\bm{z}}_{\\rm FoF}|}.\n\\end{equation}\nThe opening angle is set to $\\Theta = \\pi\/12$ in our simulations, and the beam-axis vector $\\hat{\\bm{z}}_*$ for each star particle is drawn randomly from a uniform distribution over the spherical polar angles about the star's position, $\\phi_*$ and $\\theta_*$.  This value is fixed throughout the star particle's lifetime, and the beam-axis $\\hat{\\bm{z}}_{\\rm FoF}$ of each FoF group is calculated as a luminosity-weighted average of $\\hat{\\bm{z}}_*$ across the constituent star particles. In Figure~\\ref{Fig::blister-projections} we compare the density profiles for spherical- (top row) and blister-type (bottom row) momentum injection, at simulation times $1$~Myr, $10$~Myr and $30$~Myr after the birth of the stellar cluster in a uniform medium of density $100~{\\rm cm}^{-3}$. Qualitatively, the blister-type momentum injection results in a faster and wider ejection of gas away from the cluster centre than does the spherical momentum injection, despite the fact that the ionisation front radius (solid white lines) is only marginally larger. We note that in this uniform-density box, the number of Voronoi cells surrounding the star particle is relatively small, resulting in a deviation from perfect spherical symmetry when the feedback is injected isotropically (top row of Figure~\\ref{Fig::blister-projections}, the momentum propagates along rays joining the star particle to the centroids of the neighbouring cells). This effect will be less marked in the highly-overdense star-forming regions of isolated disc galaxies.\n\n\\subsection{Stalling of HII regions} \\label{Sec::stalling}\nIn computing the FoF groups via the method presented in Section~\\ref{Sec::grouping}, we must be careful to exclude star particles whose ionisation fronts have stalled, and which are no longer depositing significant quantities of momentum into the surrounding gas. Stalling occurs when the rate of HII region expansion becomes comparable to the velocity dispersion of the host cloud, at which point the ionised and neutral gas are able to intermingle and the swept-up shell loses its coherence~\\citep{Matzner02}. After this transition, it no longer makes sense to include the stalled HII region in an FoF group of expanding HII regions, as its radius and internal density are no longer well-defined. In particular, we want to avoid the case where such an HII region links together two active HII regions, spuriously shifting the origin of their momentum ejection to a position halfway between the two particles. Before the FoF groups are calculated, we therefore compute the rate of HII region expansion $\\dot{r}_{{\\rm II},*}$ for each star particle, and if this is found to be smaller than the velocity dispersion of the surrounding gas at the same scale, we flag the particle as `stalled'. Star particles with stalled ionisation fronts are not allowed to be FoF group members, but are still allowed to contribute to HII region feedback with what little remains of their ionising luminosity. Following KM09, we approximate the ambient velocity dispersion by considering a blister-type HII region centred at the origin of a cloud with an average density of $\\overline{\\rho}(r) = 3\/(3-k_\\rho) \\rho_0 (r\/r_0)^{-k_\\rho}$ and a virial parameter $\\alpha_{\\rm vir}$ as measured on the scale of the HII region. This gives a cloud velocity dispersion of\n\\begin{equation}\n\\label{Eqn::sigma}\n\\begin{split}\n\\sigma_{\\rm cl}(r_{\\rm II}) &= \\sqrt{\\frac{\\alpha_{\\rm vir} G M(<r_{\\rm II})}{5r_{\\rm II}}} \\\\\n&= \\sqrt{\\frac{2\\pi}{15} \\alpha_{\\rm vir} G \\overline{\\rho}(r_{\\rm st,0}) r_{\\rm ch}^{2-k_\\rho} r_{\\rm st,0}^{k_\\rho} x_{\\rm II}^{2-k_\\rho}},\n\\end{split}\n\\end{equation}\nwhere we assume the cloud is in approximate virial balance with $\\alpha_{\\rm vir}=1$, and we again take $k_{\\rho} = 0$. Equations (\\ref{Eqn::r_ch}), (\\ref{Eqn::r_st0}) and (\\ref{Eqn::sigma}) then imply that the radius at which expansion stalls is given by\n\\begin{equation}\n\\label{Eqn::stalling_velocity}\n\\begin{split}\n\\dot{r_{\\rm II}} &= \\frac{r_{\\rm ch}}{t_{\\rm ch}} \\frac{\\mathrm{d} x_{\\rm II}}{\\mathrm{d} \\tau} \\\\\n&= 8.8 \\times 10^{-3} \\: {\\rm km} \\: {\\rm s}^{-1} S_{49} \\: \\overline{n}_{\\rm H,2}^{-1\/6} \\: {\\rm cm}^{1\/2} x_{\\rm II}^{1\/2}.\n\\end{split}\n\\end{equation}\nWe calculate the value of $\\dot{r}_{{\\rm II},*}=\\Delta r_{{\\rm II}, *}\/\\Delta t_*$ numerically for each star particle, where $\\Delta r_{{\\rm II},*}$ is the increment in the ionisation front radius during the particle's time-step $\\Delta t_*$. When $\\dot{r}_{{\\rm II},*}$ crosses $\\sigma_{\\rm cl}(r_{{\\rm II},*})$, the HII region is considered to have stalled.\n\n\\subsection{Heating from HII regions} \\label{Sec::HII-heat}\nTo examine the influence of momentum injection relative to thermal energy injection for the HII region feedback, we run one simulation with thermal HII region feedback (without HII region momentum, `HII thermal'), and one simulation with both thermal and kinetic HII region feedback (`HII thermal + beamed mom.'). In these simulations, we inject thermal energy associated with photo-ionisation heating up to a temperature of $7000$~K. To do this, we piggy-back on the injection procedure for the HII region momentum, depositing thermal energy into the nearest-neighbour gas cell and its immediate neighbours. We do not need to access gas cells beyond these immediate neighbours because the Str{\\\"o}mgren radii of the FoF groups in our simulations are at best marginally-resolved. We also do not need to deal with overlapping Str{\\\"o}mgren spheres once the FoF groups have already been computed. We therefore proceed as follows:\n\\begin{enumerate}\n\t\\item For each FoF group, find the nearest-neighbour gas particle $j$ for its luminosity-weighted centre of mass.\n\t\\item Increment the total number of photons $S_{\\rm in}$ per unit time received by this gas cell from all the FoF group centres it hosts, such that the final value is $S_{j, {\\rm in}} = \\sum_{{\\rm FoF} = 1}^N{S_{\\rm FoF}}$, where $S_{\\rm FoF}$ is the total ionising luminosity of all stars in the FoF group, and $N$ is the number of FoF group centres hosted by $j$.\n\t\\item Compute the number of photons $S_{j, {\\rm cons}} = \\alpha_B N_{j, {\\rm H}} n_{j, e}$ per unit time that can be consumed via the ionisation of the material in gas cell $j$, where $N_{j, {\\rm H}}$ is the number of hydrogen atoms in the cell and $n_{j, e}$ is the number density of electrons.\n\t\\item If $S_{j, {\\rm in}} < S_{j, {\\rm cons}}$, then ionise cell $j$ with a probability $S_{j, {\\rm in}}\/S_{j, {\\rm cons}}$. This ensures that over a large number of gas cells, the number of injected photons converges to $S_{j, {\\rm in}}$.\n\t\\item If $S_{j, {\\rm in}} > S_{j, {\\rm cons}}$, ionise cell $j$ and compute the `residual' ionisation rate $S_{j, {\\rm res}}$ to be spread to the facing cells $k$, such that $S_{j, {\\rm res}} = S_{j, {\\rm in}} - S_{j, {\\rm cons}}$. Each ionised cell is heated to a temperature of $7000$~K.\n\\end{enumerate}\nIn the case that $S_{j, {\\rm in}} < S_{j, {\\rm cons}}$, the algorithm ends here. Otherwise we continue as follows.\n\\begin{enumerate}\n\t\\setcounter{enumi}{6}\n\t\\item For each gas cell $j$ with $S_{j, {\\rm res}} > 0$, find the set of neighbouring cells $k$ with which it shares a Voronoi face. Compute the fraction of photons it receives according to\n\\begin{equation}\nS_{k, {\\rm in}} = w_k S_{j, {\\rm res}},\n\\end{equation}\nwhere $w_k = A_{j \\rightarrow k}\/\\sum_k{A_{j \\rightarrow k}}$, as for the injection of HII region momentum in Section~\\ref{Sec::injection}.\n\t\\item Ionise each facing cell $k$ with a probability of $S_{k, {\\rm in}}\/S_{k, {\\rm cons}}$. Summed over the set of facing cells for many HII regions, this ensures that the expectation value for the rate of ionisation converges to $S_{j, {\\rm res}}$.\n\\end{enumerate}\nSubsequent to the above procedure for thermal energy injection, the chemistry and cooling for each gas cell is computed using {\\sc SGChem}, as described in Section~\\ref{Sec::SNe-only}. During this computation, we impose a temperature floor of $7000$~K, which is enforced until the next HII-region update. We rely on the chemical network to collisionally-ionise the gas cells in a manner that is self-consistent with their temperatures. This will only produce an ionisation fraction of $10^{-5}$ when cold gas is heated to $7000$~K, but in the non-equilibrium case, whereby gas cools from much higher temperatures to a floor of $7000$~K, much higher ionisation fractions can be achieved. After the chemistry computation, the ionised cells are unflagged and are ready to absorb more photons.\n\nIn Figure~\\ref{Fig::heat-convergence}, we check that at mass resolutions between $10^3$ and $10^7~{\\rm M}_\\odot$, the above method ensures convergence of the quantity of photoionised gas. We consider a box of side-length $950~{\\rm pc}$ containing a gas of uniform density $100~{\\rm cm}^{-3}$, along with $100$ star particles of mass $10^4~{\\rm M}_\\odot$ each. We record the total cumulative value of $S_{49, *}$ emitted by these particles as a function of time (solid lines), as well as the total cumulative $S_{49, {\\rm g}}$ absorbed by the surrounding gas cells (dashed lines). Cooling and chemistry are switched on. We see that the bold and solid lines match at all resolutions, indicating that none of the emitted photons are `wasted' by our restriction of photon injection to the set of facing cells $k$ surrounding each star particle. The offset for the lowest-resolution ($10^7~{\\rm M}_\\odot$ per gas cell) case is due to the stochastic procedure for choosing the gas cells to ionise: in the limit of a very large number of HII regions ($\\gg 100$), we would expect this offset to approach zero. The star particle mass used in this test is in the 99th percentile for FoF grops in the highest-resolution isolated disc simulation used in this work ($10^3~{\\rm M}_\\odot$ per gas cell), and the gas density is ten times lower than the birth density of these star particles. If all photons are absorbed in this case, then the algorithm described above is valid in its modelling of the heating due to the vast majority of our marginally-resolved HII regions.\n\n\\section{Results} \\label{Sec::results}\nIn this section, we analyse the properties of the four simulated disc galaxies with thermal HII region feedback (`HII heat'), spherically-injected HII region momentum (`HII spherical mom.'), blister-type HII region momentum with $\\Theta = \\pi\/12$ (`HII beamed mom.'), and a combination of blister-type momentum and thermal energy (`HII heat \\& beamed mom.'), relative to our control simulation with supernova feedback on its own (`SNe only'). The simulations are summarised in Table~\\ref{Tab::sims}. We consider the morphology, stability, global star formation rate and phase structure of the interstellar medium (Section~\\ref{Sec::disc-props}), and the distribution of the lifetimes, masses, star formation rate densities, and velocity dispersions of its molecular clouds (Section~\\ref{Sec::molecular-props}).\n\nIn this section, whenever we compare to observed quantities involving molecular hydrogen, we use synthetic $^{12}{\\rm CO}(J=1\\rightarrow 0)$ maps obtained by post-processing the simulations using the {\\sc Despotic} code~\\citep{Krumholz14}, rather than using the ${\\rm CO}$ or ${\\rm H}_2$ abundances determined from the {\\sc SGChem} during run-time. We convert these ${\\rm CO}$ maps back to synthetic ${\\rm H}_2$ maps using a constant ${\\rm H}_2$-to-${\\rm CO}$ conversion factor $\\alpha_{\\rm CO} = 4.3~{\\rm M}_\\odot ({\\rm K}~{\\rm kms}^{-1}{\\rm pc}^{-2})^{-1}$, mimicking the procedures used in observations~\\citep{Bolatto13}. This allows a direct comparison of our results in Section~\\ref{Sec::molecular-props} to observed molecular cloud populations. Our motivation for this method is that, while {\\sc SGChem} produces fully time-dependent chemical abundances, it does not calculate ${\\rm CO}$ excitation or line emission, whereas {\\sc Despotic} includes a full treatment of the ${\\rm CO}$ emission, out of local thermal equilibrium. This allows us to capture the effects of local variations in the ${\\rm CO}$ luminosity per unit ${\\rm H}_2$ mass, which may be important for comparing to observations. Full details of the post-processing procedure are provided in Appendix~\\ref{App::postproc}.\n\n\\subsection{Galactic-scale properties of the interstellar medium} \\label{Sec::disc-props}\n\\subsubsection{Disc morphology} \\label{Sec::morphology}\nThe face-on and edge-on gas column densities across all simulation resolutions and feedback prescriptions are displayed in Figure~\\ref{Fig::morphology}. In the medium- (centre row) and high-resolution (top row) cases, the addition of momentum from HII regions visibly reduces the sizes of the largest voids in the gas of the interstellar medium, blown by supernova feedback. This corresponds to a qualitative reduction in the amount of outflowing gas from the galactic mid-plane, as seen in the edge-on view, and so to a visible reduction in the gas disc scale-height. The introduction of thermal energy from HII regions without momentum (`HII heat') has no effect on the interstellar medium. In the low-resolution case (bottom row), the difference between the simulations with and without HII region momentum is eradicated. This can likely be attributed to the reduction in supernova clustering with decreasing resolution, as discussed in Section~\\ref{Sec::discussion}.\n\nFigure~\\ref{Fig::tuning-forks} quantifies the structure of the multi-scale molecular gas distribution in our simulations, relative to the distribution of young stars. This is the result of measuring the gas-to-stellar flux ratio enclosed in apertures centred on ${\\rm H}_2$ peaks (top branch) and SFR peaks using `young stars' with ages in the range $0$ to $5$~Myr (bottom branch), and for aperture sizes ranging between $50$~pc and $4000$~pc, following~\\cite{Kruijssen2014} and~\\cite{Kruijssen18a}. The deviation of the lower branch from the top branch, which sets in at around the gas-disc scale-height~\\citep[see also][]{2019Natur.569..519K,Jeffreson21a}, indicates how effectively (on average) molecular gas is removed from around young star clusters in each simulation. If the regions surrounding young stars are effectively cleared of dense gas, then the lower branch drops significantly below the galactic average gas-to-stellar flux ratio at small scales. By contrast, if the young stars remain embedded for long periods of time, then the lower branch remains close to the galactic average value. This is seen in the simulations of~\\cite{Fujimoto19}, who find a duration of $23 \\pm 1$~Myr, nearly an order of magnitude longer than observed~\\citep{2014ApJ...795..156W,2015MNRAS.449.1106H,2018MNRAS.481.1016G,2019MNRAS.483.4707G,2019MNRAS.490.4648H,2019Natur.569..519K,Chevance20,2020arXiv201107772K,2021ApJ...909..121M}. In our simulations, this time-scale ranges from $4.4$~Myr (HII region momentum runs) up to $>5$~Myr (runs without HII region momentum; representing a lower limit, because the duration of co-existence cannot exceed the adopted duration of the young stellar phase, which is $5$~Myr). All of the above numbers are comparable to those obtained for the galaxies with the highest gas surface densities (appropriate for the Agora initial conditions) in the observational sample of~\\cite{Chevance20}, who used the same diagnostic to infer time-scales. This provides a qualitative indication that our feedback implementation broadly matches observed feedback-driven dispersal rates of molecular clouds. Indeed, we see that our HII region momentum feedback moves the morphology of the molecular gas and stellar distribution towards that observed in NGC 628. The qualitative result that the top branch is flatter than the bottom branch indicates a cloud lifetime that is longer than the lifetimes of the young stellar groups (here chosen to be 5 Myr). In Section~\\ref{Sec::GMC-lifetimes}, we further discuss the influence of our feedback prescription on molecular cloud lifetimes and cloud properties.\n\n\\subsubsection{Galactic outflows} \\label{Sec::outflows}\nThe top row of Figure~\\ref{Fig::outflows} shows the total galactic star formation rate as a function of the simulation time $t$ at each simulation resolution and for each feedback prescription. At the beginning of the simulation, the disc collapses vertically and a burst of star formation is produced, after which the interstellar medium settles into a state of dynamical equilibrium. In our simulations, equilibrium is achieved after around $200$~Myr. In the medium- and high-resolution cases, the introduction of HII region momentum suppresses the initial starburst at earlier times and so decreases its magnitude. No such effect is seen for the thermal HII regions (`HII heat'), or in any of the low-resolution simulations, mirroring the qualitative results presented in Section~\\ref{Sec::morphology}. At $\\sim 600$~Myr the star formation rate is consistent with current observed values in the Milky Way~\\citep{Murray&Rahman10,Robitaille&Whitney10,Chomiuk&Povich,Licquia&Newman15}. The feedback prescription does not have a perceivable effect on the global star formation rate after the galaxies have equilibriated.\n\nIn the centre row of Figure~\\ref{Fig::outflows}, we show the rate of gas outflow from each galaxy. The outflow rates are calculated as the total momentum of the gas moving away from the disc, summed over two planar slabs of thickness $500~{\\rm pc}$, located at $\\pm 5$~kpc above and below the galactic disc. This is the same definition used in~\\cite{2014MNRAS.442.3013K,2020arXiv200403608K}. In the medium- and high-resolution simulations, the outflow rate is decreased by around an order of magnitude upon the introduction of HII region momentum feedback. This is again consistent with a reduced level of supernova clustering, which decreases the effectiveness of supernova feedback in driving outflows~\\citep{2020arXiv200911309S,2020arXiv200403608K}. The mass-loading $\\eta$ of the stellar feedback in our model (bottom row of Figure~\\ref{Fig::outflows}) divides the outflow rate by the star formation rate. We note that there is a clear resolution-dependence of the feedback-induced outflow rates and mass-loadings for all feedback prescriptions, likely due to the increased clustering of supernovae at higher resolutions. This is discussed further in Section~\\ref{Sec::SNe-fb}.\n\n\\subsubsection{Resolved disc stability} \\label{Sec::vss}\nThe presence of momentum feedback from HII regions makes a significant difference to the velocity dispersion $\\sigma_{\\rm g}$ and gravitational stability $Q_{\\rm g}$ of the cold gas ($T \\leq 10^4$~K) in our high- and medium-resolution simulations (left and centre columns in Figure~\\ref{Fig::vss}, respectively), as well as to the scale-height $h_{\\rm g}$ of the total gas distribution. We calculate the line-of-sight turbulent velocity dispersion as\n\\begin{equation}\n\\sigma_{\\rm los, g}^2 = \\frac{\\langle |\\bm{v}_i - \\langle \\bm{v}_i \\rangle|^2 \\rangle}{3},\n\\end{equation}\nwhere $\\{\\bm{v}_i\\}$ are the velocity vectors of the gas cells in each radial bin, and angled brackets denote mass-weighted averages over these cells. The~\\cite{Toomre64} $Q$ parameter of the cold gas is then defined as\n\\begin{equation}\nQ_{\\rm g} = \\frac{\\kappa \\sigma_{\\rm g}}{\\pi G \\Sigma_{\\rm g}},\n\\end{equation}\nwith $\\kappa$ the epicyclic frequency of the galactic rotation curve and $\\sigma_{\\rm g} = \\sqrt{c_s^2 + \\sigma_{\\rm los, g}^2}$ for gas sound speed $c_s$. In the top row of Figure~\\ref{Fig::vss}, we quantitatively show the result for the disc scale-height that was demonstrated qualitatively in Figure~\\ref{Fig::morphology}: the reduction in the violence of feedback-induced outflows perpendicular to the galactic mid-plane leads to a smaller disc scale-height when momentum from HII regions is incorporated. In the second row, we demonstrate that for galactocentric radii $R < 8$~kpc in the high-resolution simulation, the amount of cold gas is increased by up to 50~per~cent when HII region momentum is included (solid lines) and that the amount of molecular gas is almost doubled (dashed lines). This is due to two effects: (1) the overall mass of the interstellar medium is larger in the simulations with HII region momentum, due to the suppression of the initial `starburst' (see Section~\\ref{Sec::outflows}), and (2) the fraction of the interstellar medium in the cold and molecular phases is increased (see Section~\\ref{Sec::phases}). We also find that the cold gas has a lower velocity dispersion by $\\sim 5~{\\rm kms}^{-1}$ at all galactocentric radii. Accordingly, the Toomre $Q$ factor ($Q_{\\rm g}$, bottom row of panels) is suppressed by a factor of $\\sim 2$ out to $R \\sim 8~{\\rm kpc}$. The HII region momentum causes the interstellar medium to become clumpier and less gravitationally-stable, leading to the formation of more molecular clouds, as will be discussed in Section~\\ref{Sec::molecular-props}. This is again consistent with the idea that the HII region feedback reduces the momentum injected by supernova feedback, likely by reducing its clustering. In the low-resolution case, none of the observables associated with galactic disc stability are altered by the addition of HII region momentum, consistent with the results presented in Sections~\\ref{Sec::morphology} and~\\ref{Sec::outflows}.\n\n\\begin{figure}\n  \\label{Fig::phases}\n    \\includegraphics[width=\\linewidth]{figs\/phases.pdf}\n    \\caption{{\\it Upper panel:} Density-temperature phase diagram for the HI-resolution simulation with both thermal and beamed momentum from HII regions (HII thermal + beamed mom.), at $t=600$~Myr. Dashed lines delineate the regions of phase space corresponding to the warm neutral medium (WNM), the thermally-unstable phase (Unstable), the cold neutral median (CNM), gas heated by HII regions (HII), and gas heated by supernovae (SN). {\\it Lower panel:} Partitioning of the gas mass in each HI-resolution simulation into five ISM phases from warmest to coolest, as a fraction of the total gas mass in the simulation: hot gas that has received thermal energy from stellar feedback ($M_{\\rm SN+HII}$), the warm neutral medium ($M_{\\rm WNM}$), the unstable phase ($M_{\\rm unstable}$), the cold neutral medium ($M_{\\rm CNM}$), and the star-forming gas in the molecular phase ($M_{\\rm H_2}$, as computed using {\\sc Despotic}, see Appendix~\\protect\\ref{App::postproc}).}\n\\end{figure}\n\n\\begin{figure*}\n  \\label{Fig::KS-rln}\n    \\includegraphics[width=\\linewidth]{figs\/KS-rln.pdf}\n    \\caption{Pixel density as a function of $\\Sigma_{\\rm H_2}$ and $\\Sigma_{\\rm SFR}$ each disc, at a spatial resolution of $750$~pc, corresponding to the resolved molecular star-formation relation of~\\protect\\cite{Kennicutt98}. Gas depletion times of $10^8$, $10^9$ and $10^{10}$~Myr are given by the black solid, dashed and dotted lines respectively. The contours encircle 90~per~cent (dotted), 50~per~cent (dashed) and 10~per~cent (solid) of the observational data for nearby galaxies from~\\protect\\cite{Bigiel08}. All maps are computed at $t=600$~Myr.}\n\\end{figure*}\n\n\\begin{figure}\n  \\label{Fig::cloud-lifetimes}\n    \\includegraphics[width=\\linewidth]{figs\/cloud-lifetimes.pdf}\n    \\caption{\\textit{Top:} Cumulative distribution of trajectory lifetimes $t_{\\rm life}$ in each of the high-resolution simulations. The characteristic cloud lifetimes, obtained from the exponential distributions by fitting a function $\\exp{(-t\/\\tau_{\\rm life})}$ according to Equation~(\\ref{Eqn::lifetime-dstbn}), are annotated according to the legend colours. \\textit{Bottom:} Characteristic cloud lifetime for each simulation (transparent solid lines) as a function of the cloud mass, with the cloud mass PDFs below. The mean values with and without HII region momentum feedback are given by the solid and dashed black lines, respectively. The error-bars correspond to the errors associated with the exponential fits to the distributions $D(t_{\\rm life}>t)$ in each mass bin. Three regimes are annotated: $(i)$ for clouds destroyed preferentially by HII region feedback, $(ii)$ for clouds destroyed preferentially by supernovae, and $(iii)$ for clouds dominated by interactions.}\n\\end{figure}\n\n\\begin{figure}\n  \\label{Fig::node-connectivity}\n    \\includegraphics[width=\\linewidth]{figs\/node-connectivity.pdf}\n    \\caption{Fraction of nodes in the cloud evolution network (molecular clouds observed at an instant in time) that split into two or more children ($\\theta_{\\rm child}>1$) or are the result of a merger between two or more parents ($\\theta_{\\rm par}>1$), as a function of cloud mass. We see that the connectivity of the network increases exponentially from $\\sim 10$~per~cent of multiply-connected nodes for cloud masses $M \\sim 5\\times 10^5 M_\\odot$ up to $\\sim 70$~per~cent at $M \\sim 3\\times 10^7 M_\\odot$. This is the same mass range over which cloud lifetimes decrease with mass in the lower panel of Figure~\\protect\\ref{Fig::cloud-lifetimes}, and cease to depend on the feedback prescription used. The three mass regimes are annotated as in Figure~\\protect\\ref{Fig::cloud-lifetimes}.}\n\\end{figure}\n\n\\subsubsection{ISM phase structure} \\label{Sec::phases}\nIn the top panel of Figure~\\ref{Fig::phases} we display the mass-weighted distribution of gas temperature as a function of the gas volume density (the phase diagram) for the high-resolution simulation including both thermal and beamed HII region momentum (`HII heat \\& beamed mom'). The gas cells cluster around a state of thermal equilibrium in which the rate of cooling (dominated in our simulations by line emission from ${\\rm C}^+$, ${\\rm O}$ and ${\\rm Si}^+$) balances the rate of heating due to photoelectric emission from PAHs and dust grains. The thin horizontal line of particles at high volume densities and $T \\sim 7000$~K contains the particles that are heated by the thermal feedback from HII regions. The dashed black lines delineate the partitioning of the interstellar medium into the feedback-heated phases (SN and HII) the warm neutral medium (WNM), the unstable phase, the cold neutral medium (CNM) and the set of gas cells that are predominantly molecular (${\\rm H}_2$). We have chosen the partitioning of the WNM and CNM gas by eye, according to the major regions of gas accumulation along the thermal equilibrium curve in the phase diagram. The region bridging the WNM and CNM is then classified as `unstable' following~\\cite{Goldbaum16}, and material that is lifted above the equilibrium curve is attributed to feedback-related heating. In the lower panel of Figure~\\ref{Fig::phases} we show the fraction of the total gas mass in each of these phases for the five high-resolution simulations. The mass of molecular hydrogen we use is that which would be inferred by an observer from the CO luminosity, as computed by {\\sc Despotic} (see Appendix~\\ref{App::postproc}). The addition of thermal feedback from HII regions does nothing to the phase structure of the interstellar medium, relative to the case of supernovae only. By contrast, explicit injection of momentum from HII regions leads to almost double the mass of molecular gas and $\\sim 50$~per~cent more cold gas overall ($T \\leq 10^4$~K). The masses of warm and hot, feedback-heated gas are correspondingly reduced. We also note that the overall gas mass remaining in the galaxy at $t=600$~Myr is larger by around $0.8 \\times 10^9~{\\rm M}_\\odot$. This is because the initial `bursts' of star formation, as the galaxy settles into equilibrium, are smaller in the case of effective pre-supernova feedback, as discussed in Section~\\ref{Sec::outflows}.\n\n\\subsubsection{Star formation in molecular gas} \\label{Sec::molecular-SF}\nAlthough the global star formation rate in our simulations appears insensitive to the feedback prescription applied (top row of Figure~\\ref{Fig::outflows}), a slightly greater level of variation is revealed when we look explicitly at the star formation rate surface density $\\Sigma_{\\rm SFR}$ as a function of the molecular gas surface density $\\Sigma_{\\rm H_2}$ (the molecular Kennicutt-Schmidt relation) in Figure~\\ref{Fig::KS-rln}. We find that with the addition of momentum feedback from HII regions, the gradient of the slope in the $\\Sigma_{\\rm H_2}$-$\\Sigma_{\\rm SFR}$ plane is flattened slightly and the molecular gas surface densities are increased by a factor of around two. This means that they fall closer to the observed values delineated by the closed black contours. However this fact should not be over-interpreted,  given that the size of the shift in surface density is smaller than the uncertainty in the ${\\rm H}_2$-to-CO conversion factor used to compute the molecular gas abundances (see Appendix~\\ref{App::postproc}). Again, the addition of thermal HII region feedback on its own has no effect.\n\n\\subsection{Properties of molecular clouds} \\label{Sec::molecular-props}\nIn this section, we analyse the molecular clouds identified at the native spatial resolution ($6$~pc) of the column-density projections for our high-resolution simulations. These clouds span a size range from $18$~pc up to $200$~pc and a mass range from $100~{\\rm M}_\\odot$ up to $10^8~{\\rm M}_\\odot$. We identify clouds by taking a threshold of $\\log{(\\Sigma_{\\rm H_2}\/{\\rm M}_\\odot~{\\rm pc}^{-2})} = -3.5$ on the molecular gas column density, as calculated using {\\sc Despotic} (see the beginning of Section~\\ref{Sec::results} and Appendix~\\ref{App::postproc}). The clouds themselves are identified using the {\\sc Astrodendro} package for Python. This procedure is described in detail in Appendix~\\ref{App::postproc}, and is discussed at length in Section 2.9 of~\\cite{2020MNRAS.498..385J}, where we also show that the molecular clouds identified by this method have properties in agreement with observations of clouds in Milky Way-like galaxies, including their masses, sizes, velocity dispersions, surface densities, pressures and star formation rate surface densities. Similarly to~\\cite{2020MNRAS.498..385J}, we discard clouds spanning fewer than 9 pixels ($324~{\\rm pc}^2$), or containing fewer than 20 Voronoi gas cells.\n\nOnce the molecular clouds in our simulations have been identified at every simulation time-step, we follow their evolution as a function of time according to the procedure described in Section 3.2 of~\\cite{Jeffreson21a}. Briefly, we take the two-dimensional pixel masks associated with the sets of molecular clouds in two consecutive snapshots at times $t=t_1$ and $t=t_2$. We project the mask positions of the clouds at $t=t_1$ using the positions and velocities of the gas cells that they span, such that $(x_1, y_1) \\rightarrow (x_1 + v_x \\Delta t, y_1 + v_y \\Delta t)$. We then compare the projected masks to the true pixel masks of the clouds at $t=t_2$. If the projected and true pixel maps overlap by one pixel, then the clouds are indistinguishable at the spatial resolution ($6$~pc) and temporal resolution ($1$~Myr) of the snapshots, and so each cloud at $t=t_1$ is assigned as a parent of the children at $t=t_2$. A given cloud can spawn multiple children (\\textit{cloud splits}) or have multiple parents (\\textit{cloud mergers}). We store the network of parents and children using the {\\sc NetworkX} package for python~\\citep{NetworkX}, and `prune away' unphysical nodes produced by regions of faint CO background emission in our astrochemical post-processing, which do not contain sufficient quantities of CO-luminous gas. We find that these nodes can be removed by taking a cut of $\\sigma \\sim 0.03~{\\rm kms}^{-1}$ on the cloud velocity dispersion, as described in~\\cite{Jeffreson21a}.\\footnote{The mass cut applied in~\\cite{Jeffreson21a} is not required here, as we discard clouds containing fewer than 20 Voronoi cells.}\n\n\\subsubsection{Molecular cloud lifetimes} \\label{Sec::GMC-lifetimes}\nUsing the \\textit{cloud evolution network} described above, we calculate the lifetime $t_{\\rm life}$ of each distinct molecular cloud identified at a given time in our simulations, by performing a Monte Carlo (MC) walk through the network. At the beginning of each MC iteration, walkers are initialised at every \\textit{formation node} in the network (nodes corresponding to a net increase in cloud number). The walkers step along time-directed edges of length $1$~Myr between consecutive nodes, until an \\textit{interaction node} is reached. An interaction may be a merger, a split, or a transient meeting. A random number from a uniform distribution is used to choose between the possible subsequent trajectories for each walker, including the possibility of cloud destruction, if it exists at that node. If the cloud is destroyed, the final lifetime $t_{\\rm life}$ is returned. This algorithm satisfies the requirements of:\n\\begin{enumerate}\n\t\\item \\textit{Cloud uniqueness:} Edges between nodes in the network represent time-steps in the evolution of a single cloud, so must not be double-counted.\n\t\\item \\textit{Cloud number conservation:} The number of cloud lifetimes retrieved from the network must be equal to the number of cloud formation events and cloud destruction events, as each cloud can be formed and destroyed just once.\n\\end{enumerate}\nSeventy MC iterations are performed to reach convergence of the characteristic molecular cloud lifetime $\\tau_{\\rm life}$ for the cloud population of the entire galaxy.\n\nIn the top panel of Figure~\\ref{Fig::cloud-lifetimes}, we show the cumulative distributions $D(t_{\\rm life} > t)$ of lifetimes $t_{\\rm life}$ for the molecular clouds in our high-resolution simulations. These distributions have an exponential form, as expected if the formation and destruction of clouds has reached a steady state. The simulations with HII region momentum feedback do not appear significantly different to those without. We have annotated the \\textit{characteristic cloud lifetime} $\\tau_{\\rm life}$ for each simulation by fitting an exponential profile to each distribution, and assuming the steady-state proportionality\n\\begin{equation} \\label{Eqn::lifetime-dstbn}\n\\ln{D(t_{\\rm life})} \\propto -\\frac{t}{\\tau_{\\rm life}},\n\\end{equation}\nas in~\\cite{Jeffreson21a}. We find only a marginal increase of $4$~Myr in the overall value of $\\tau_{\\rm life}$ upon the introduction of HII region momentum feedback (an average of $37 \\pm 2$~Myr with HII region momentum vs. $33 \\pm 2$~Myr without). However, in the lower panel of Figure~\\ref{Fig::cloud-lifetimes}, we see that the the influence of the feedback prescription is dependent on the cloud mass. Its influence can be divided into three regimes as follows:\n\\begin{enumerate}\n\t\\item $M\/{\\rm M}_\\odot \\la 5.6 \\times 10^4$: HII region momentum depresses the cloud lifetime by $\\sim 10$~Myr.\n\t\\item $5.6 \\times 10^4 \\la M\/{\\rm M}_\\odot \\la 5 \\times 10^5$: HII region momentum increases the cloud lifetime by $\\sim 7$~Myr.\n\t\\item $5 \\times 10^5 \\la M\/{\\rm M}_\\odot$: HII region momentum has no effect on the cloud lifetime.\n\\end{enumerate}\nThis result is consistent with the following scenario: the least massive molecular clouds in $(i)$ are less likely to contain the massive stellar clusters required for the fastest and most efficient injection of supernova energy. This results in an uptick of the characteristic cloud lifetime for the simulations without HII region momentum feedback (blue and purple lines in Figure~\\ref{Fig::cloud-lifetimes}) at small masses. However, the least-massive clouds are also the easiest to disperse, and so the relatively-small amount of momentum injected by HII regions can truncate the cloud lifetime in the absence of efficient supernova feedback. At larger cloud masses $(ii)$, the HII region momentum is too puny to cause disruption, so its main influence is to reduce supernova clustering and thus decrease the efficacy of the supernova feedback, consistent with its effect on the large-scale properties of the interstellar medium, presented in Section~\\ref{Sec::disc-props}. This increases the characteristic cloud lifetime. Finally, the most massive molecular clouds in $(iii)$ are often unresolved cloud complexes, and undergo increasingly more mergers and splits as the cloud mass is increased from $10^{5.8}$ through $10^{7.5}~{\\rm M}_\\odot$, as shown in Figure~\\ref{Fig::node-connectivity}. Across this mass regime, the fraction of multiply-connected nodes increases from 10~per~cent up to 70~per~cent, elevating the number of short MC trajectories containing high-mass nodes. The trajectory lifetimes returned by the MC walk are therefore likely to be determined by the level of graph connectedness, rather than by the feedback-induced destruction of the molecular clouds. This also explains the drop in the cloud lifetime for the most massive clouds. In order to determine the effects of stellar feedback on these high-mass clouds, we will need to develop the algorithm put forward in~\\cite{Jeffreson21a}, to distinguish between cloud mergers of varying mass ratio. Overall, the cloud lifetimes across masses span the range from $10$-$40$~Myr, similar to observations~\\citep{Engargiola03,Blitz2007,Kawamura09,Murray11,Meidt15,Corbelli17,Chevance20}.\n\nFinally, we note that the number of molecular clouds generated per unit mass of the interstellar medium in our simulations is increased by the presence of HII region momentum. In the `SNe only' and `HII heat' simulations, the average number of clouds identified is $3.7$ per $10^7~{\\rm M}_\\odot$; this increases to $6.2$ per $10^7~{\\rm M}_\\odot$ for the `HII spherical mom.', `HII beamed mom.' and `HII heat \\& beamed mom.' simulations. Combined with the reduced number of high-mass clouds, this result indicates that the molecular interstellar medium is slightly more fragmented in the case of the HII region feedback.\n\n\\begin{figure*}\n  \\label{Fig::cloud-veldisp-surfdens}\n    \\includegraphics[width=\\linewidth]{figs\/HII_veldisp-surfdens.pdf}\n    \\caption{Distributions of molecular cloud surface densities $\\Sigma$ and velocity dispersions $\\sigma$ in each of the high-resolution simulations. Each value is a median over a trajectory in the cloud evolution network. \\textit{Left\/Centre:} Cumulative distribution of the surface density\/velocity dispersion, with medians indicated by the vertical dashed lines. \\textit{Right:} Scaling relation of the surface density with the velocity dispersion. The contours enclose 90~per~cent of the identified molecular clouds. The grey-shaded histogram contains the full cloud distribution for the `HII heat \\& beamed mom.' simulation. Virial parameters of $\\alpha_{\\rm vir}=0.5$ and $1$ for spherical clouds of size $6$~pc are given by the dashed and dot-dashed lines, respectively. We see that the introduction of HII region momentum predominantly (but hardly) affects the cloud velocity dispersion.}\n\\end{figure*}\n\n\\subsubsection{Cloud velocity dispersions and surface densities} \\label{Sec::GMC-veldisp-surfdens}\nWe now turn to the physical properties of the molecular clouds in our simulations: first to the scaling relation between the cloud surface density $\\Sigma$ and velocity dispersion $\\sigma$. Each value corresponds to an average (median) taken over the cloud lifetime $t_{\\rm life}$ (i.e.~along a unique trajectory in the cloud evolution network). The right-hand panel of Figure~\\ref{Fig::cloud-veldisp-surfdens} shows the scaling relation itself, for which the clouds fall along a line of constant virial parameter, as observed in nearby Milky Way-like galaxies~\\citep[e.g.][]{Sun18}. Lines representing virial parameters of $1$ and $2$ for spherical clouds at a fixed size of $6$~pc (our native resolution) are given by the dashed and dot-dashed lines, respectively. The coloured contours enclose 90~per~cent of the clouds for each high-resolution simulation, while the grey-shaded histogram displays the entire cloud population for the `HII heat \\& beamed mom.' simulation. In the left and central panels of Figure~\\ref{Fig::cloud-veldisp-surfdens} we show the cumulative distributions of the cloud surface density and velocity dispersion separately. We see that the introduction of HII region momentum makes little difference to the distribution of surface densities, and reduces the median cloud velocity dispersion by only $0.5~{\\rm kms}^{-1}$. This reduction is consistent with the drop in the bulk velocity dispersion of the cold gas in our simulations, presented in Figure~\\ref{Fig::vss}.\n\n\\subsubsection{Cloud masses and star formation rates} \\label{Sec::GMC-mass-SFR}\nThe influence of the stellar feedback prescription on the masses and star formation rate surface densities of our molecular clouds is shown in Figure~\\ref{Fig::cloud-mass-SFR}. In the right-hand panel, the number of clouds $N(>M)$ with mass greater than $M$ is compared to the power-law form $\\mathrm{d} N\/\\mathrm{d} M \\propto M^{-\\beta}$ observed for clouds in the Milky Way over the mass range of $\\log{M} \\in [4.8, 6.5]$, with $\\beta \\in [1.6, 1.8]$~\\citep{Solomon87,Williams&McKee97,Heyer+09,Roman-Duval+10,Miville-Deschenes17,Colombo+19}. When we fit corresponding powerlaws to the PDF of each mass spectrum (via simple linear regression in the mass range $\\log{(M\/M_\\odot)} \\in [4.8, 6.5]$), we find a slope of $\\beta = 1.75 \\pm 0.09$ for the `SNe only' simulation and a slope of $\\beta = 1.80 \\pm 0.12$ for the `HII heat \\& beamed mom.' simulation. That is, the number of the most massive clouds is reduced slightly by the presence of HII region feedback.\n\nWe note that this result (a steeper mass function with HII region momentum) is the opposite of that expected if the characteristic rates $\\xi_{\\rm form}$ and $1\/\\tau_{\\rm life}$ for cloud formation and destruction in each galaxy are independent of the cloud mass, as assumed in~\\cite{2017ApJ...836..175K}. These authors use an analytic rate equation for the number of clouds $N$, explicitly accounting for the process of cloud coagulation, to derive a mass function slope of $\\mathrm{d} N\/\\mathrm{d} M \\propto -(\\xi_{\\rm form} \\tau_{\\rm life})^{-1}$. In the steady-state approximation of Equation (\\ref{Eqn::lifetime-dstbn}), the number of clouds present in the galaxy at a given time approaches $N \\rightarrow \\tau_{\\rm life} \\xi_{\\rm form}$, so the predicted slope goes as $\\mathrm{d} N\/\\mathrm{d} M \\propto -1\/N$. We find $N$ to be higher in the simulations with HII region momentum, but $\\mathrm{d} N\/\\mathrm{d} M$ to be steeper, in contradiction with this work. We attribute this to the fact that $\\tau_{\\rm life}$ is manifestly dependent on the cloud mass (see Figure~\\ref{Fig::cloud-lifetimes}) and that the mass-dependence of $\\xi_{\\rm form}$ is not studied here, but likely non-negligible.\n\nIn the central panel of Figure~\\ref{Fig::cloud-mass-SFR}, we show the star formation rate $\\Sigma_{\\rm SFR}$ per unit area of the molecular clouds in each simulation. The introduction of HII region momentum causes a three-fold drop in the value of $\\Sigma_{\\rm SFR}$. In the right-hand panel, we show that this drop in the star formation rate occurs across the whole range of cloud masses. This result agrees broadly with the results from high-resolution simulations of resolved HII regions~\\citep[e.g.][]{2016ApJ...829..130R,2018MNRAS.478.4799H,2018MNRAS.475.3511G,2019MNRAS.489.1880H,2020MNRAS.497.3830F,2020MNRAS.499..668G,2020MNRAS.492..915G,2020arXiv201107772K}, which show that HII region feedback can efficiently suppress the overall star formation efficiency within individual molecular clouds.\n\n\\begin{figure*}\n  \\label{Fig::cloud-mass-SFR}\n    \\includegraphics[width=\\linewidth]{figs\/HII_CDFs_mass-SFR.pdf}\n    \\caption{Distributions of molecular cloud masses $M$ and star formation rates per unit area of the galactic mid-plane $\\Sigma_{\\rm SFR}$ in each of the high-resolution simulations. \\textit{Left:} Normalised number of clouds $N(>M)$ with mass larger than or equal to $M$. The solid black and dashed black lines denote the range of power-law slopes for the observed cloud mass distribution in the Milky Way, given by $\\mathrm{d} N\/\\mathrm{d} M \\propto M^{-\\beta}$ with $\\beta \\in [1.6, 1.8]$. \\textit{Centre:} Cumulative distribution of the star formation rate surface density, with medians indicated by the vertical dashed lines. \\textit{Right:} Scaling relation of the star formation rate surface density with the cloud mass. The contours enclose 90~per~cent of the identified molecular clouds. The grey-shaded histogram contains the full cloud distribution for the `HII heat \\& beamed mom.' simulation. We see that the introduction of HII region momentum reduces the star formation rates of all clouds, and slightly decreases the number of massive clouds.}\n\\end{figure*}\n\n\\subsection{Beamed vs.~spherical HII region momentum} \\label{Sec::beamed-vs-spherical}\nAside from the finding that thermal feedback from marginally-resolved HII regions is ineffective in transferring energy to the surrounding interstellar medium, a recurring theme in the preceding sub-sections is that there is no discernible difference between our simulations with spherical HII region feedback and beamed HII region feedback. The morphology, phase structure and stability of the interstellar medium are identical in these cases, and the properties of the molecular clouds are unaffected. This might be surprising, considering the qualitative difference in the appearance of HII regions in the blistered and spherical cases (see Figure~\\ref{Fig::blister-projections}) and the difference in their ionisation-front and Str\\\"{o}mgren radii. We find that it is only the quantity of momentum injected in our simulations that matters (this is roughly equivalent in the spherical and beamed cases), and not the direction in which it is injected. However, we might expect that if the direction of momentum injection were not chosen randomly for each FoF group and star particle, but rather preserved over the evolution of each molecular cloud, the blistered HII region feedback might be more effective in removing the gas from around star particles.\n\n\\section{Discussion} \\label{Sec::discussion}\nWe have shown in Section~\\ref{Sec::results} that the injection of momentum from HII regions, according to a novel numerical model based on the analytic framework of KM09, reduces the mass-loading of outflows perpendicular to the mid-plane of isolated disc galaxies, and increases the fraction of cold gas within these discs, while decreasing its velocity dispersion and scale-height. The resolved molecular clouds formed from this cold gas reservoir suffer alterations in their lifetimes, masses, star formation rates and velocity dispersions. We find that these results apply across a mass resolution range of $10^3$-$10^4~{\\rm M}_\\odot$ in the moving-mesh code {\\sc Arepo.}\n\nIt is important to note that all of the above results depend not just on our modelling of HII regions, but on a number of other assumptions made during the construction of our stellar population and its feedback, including its supernova feedback. In Section~\\ref{Sec::caveats}, we outline the key caveats of our model, their possible effects, and how these could in the future be disentangled from the relative roles of HII region and supernova feedback in isolated galaxy simulations. In Section~\\ref{Sec::literature-sims} we compare our results to studies of HII region feedback in the literature.\n\n\\subsection{Caveats of our model} \\label{Sec::caveats}\n\\subsubsection{Photon trapping and escape} \\label{Sec::comp-highres}\nWithin the model for HII region feedback, we have used a value of\n\\begin{equation}\nf_{\\rm trap} = 1 + f_{\\rm trap, w} + f_{\\rm trap, IR} + f_{{\\rm trap, Ly}\\alpha} \\sim 8\n\\end{equation}\nto account for the enhancement of pressure inside the ionisation front, due to the trapping of energy from stellar winds ($f_{\\rm trap, w}$), infrared photons ($f_{\\rm trap, IR}$) and Lyman-$\\alpha$ ($f_{{\\rm trap, Ly}\\alpha}$) photons. The value of $f_{\\rm trap}$ is constrained by~\\cite{2021ApJ...908...68O} using infrared observations to infer the pressures inside young HII regions in the Milky Way. By using $f_{\\rm trap} \\sim 8$, we therefore implicitly assume that $f_{{\\rm trap, Ly}\\alpha} \\approx 0$ and $f_{\\rm trap, w} \\approx 0$, because an estimation of the effects of Lyman-$\\alpha$ photons and winds would require observations of the dust and diffuse gas surrounding the sources, in the optical and the X-ray, respectively. Our model therefore does not account for the absorption of Lyman-$\\alpha$ photons, or for the trapping of stellar winds. In addition, the interaction of stellar winds with radiation pressure is a complex problem: numerous high-resolution, radiative-transfer numerical studies of HII regions inside individual clouds~\\citep[e.g.][]{2017MNRAS.467.1067D,2017ApJ...850..112R,2018ApJ...859...68K,2019ApJ...883..102K} have shown that stellar winds (along with inhomogeneities in the gas surrounding HII regions) can lead to the escape of radiation through holes in the shell bounding the ionised gas, and so to a reduction in the overall radiation pressure by factors of $\\sim 5$-$10$~\\citep{2019ApJ...883..102K}. The HII regions in our model are assumed perfectly spherical or hemi-spherical, and we have not accounted for stellar winds. It is therefore possible that we have under-estimated the strength of radiation pressure by ignoring Lyman-$\\alpha$ photon and wind trapping, or have over-estimated it by ignoring photon escape. However, this is unlikely to have a large effect on the total amount of momentum injected by our HII regions, because for the ionising luminosities between $S_{49} \\sim 1$ and $100$ spanned by the FoF groups in our simulations, the momentum contribution made by the gas pressure is around ten times that made by the radiation pressure, according to our Equation~(\\ref{Eqn::momentumeqn_final}).\n\n\\subsubsection{Resolution-dependence of supernova feedback} \\label{Sec::SNe-fb}\nAs noted throughout Section~\\ref{Sec::results}, the differences between the simulations with and without HII region momentum do not persist down to resolutions of $10^5~{\\rm M}_\\odot$ per gas cell. In addition, when supernova feedback is used on its own, the outflow rates and their mass-loadings, as well as the gravitational stabilities of the gas in the galactic discs, are substantially different for the low-, medium- and high-resolution simulations. This may be due to a decrease in the effectiveness of supernova clustering at resolutions of $<10^5~{\\rm M_\\odot}$ (i.e. the resolution is too low for clustering to be resolved). As discussed by~\\cite{2020arXiv200911309S}, early feedback from HII regions and stellar winds affects the interstellar medium by reducing the degree of supernova clustering and so the violence of the resulting explosions, decreasing the sizes of galactic outflows and the mid-plane gas velocity dispersion. Therefore, if clustering is not resolved, the effect of our HII region feedback on the large-scale properties of the interstellar medium may be spuriously-weakened at low resolutions.~\\cite{2020arXiv201010533S} also discuss the non-convergence of various stellar feedback prescriptions due to an under-sampling the IMF at high mass resolutions. However, this is not a problem in our simulations, due to the use of the Poisson sampling procedure from~\\cite{Krumholz15}. By this procedure, the number of stars assigned to a given star particle\/cluster depends on the star particle mass, but the form of the resulting distribution of stellar masses is not affected.\n\n\\subsection{Comparison to the literature} \\label{Sec::literature-sims}\n\\subsubsection{High-resolution simulations of molecular clouds}\nThe molecular cloud sample in our simulations has yielded two key results: (1) the lifetimes of the least-massive clouds are truncated by HII region feedback (while those of intermediate-mass clouds are extended), and (2) HII region feedback suppresses the star formation rate within molecular clouds by a factor of three. These findings can be qualitatively compared to high-resolution simulations of resolved HII regions in individual molecular clouds. In particular,~\\cite{2012MNRAS.424..377D,2013MNRAS.430..234D,2017ApJ...851...93K} find that only the least-massive molecular clouds are prone to dispersal by HII region feedback, and that this dispersal occurs on time-scales of $<10$~Myr, as we have found in Section~\\ref{Sec::GMC-lifetimes}. Larger clouds can only be disrupted by supernovae. Across molecular clouds of all masses,~\\cite{2016ApJ...829..130R,2018MNRAS.478.4799H,2018MNRAS.475.3511G,2019MNRAS.489.1880H,2020MNRAS.497.3830F,2020MNRAS.499..668G,2020MNRAS.492..915G,2020arXiv201107772K} show that the star formation efficiency per free-fall time is suppressed by the presence of HII region feedback, as we have discussed in Section~\\ref{Sec::GMC-mass-SFR}. Although it will be important to check the convergence of our sub-grid model with high-resolution simulations such as these, it is encouraging to note that the main results for our molecular cloud sample echo existing results in single-cloud studies.\n\n\\subsubsection{Isolated disc simulations}\nWe may also compare our results with other implementations of radiation\/thermal pressure from HII regions in isolated disc galaxies at similar mass resolutions.~\\cite{2020arXiv200911309S} investigate the role of pre-supernova feedback in suppressing supernova clustering in dwarf galaxies in {\\sc Arepo}, reaching mass resolutions of $20~{\\rm M}_\\odot$. The Str\\\"{o}mgren radii of the HII regions in their simulations are well-resolved, allowing for the explicit ionisation and heating of gas cells to be converted to momentum. Using this prescription, the authors find that supernova clustering is decreased by the presence of HII region feedback. This leads to a significant suppression of outflows and their mass-loadings, as well as a reduction in the sizes of supernova-blown voids within the interstellar medium, in agreement with our results.~\\cite{Fujimoto19} investigate molecular clouds in an isolated disc galaxy at a comparable resolution to ours, but with only thermal HII region feedback~\\citep[see also][]{Goldbaum16}. These authors find that both the pre-supernova and supernova feedback in their simulations are inefficient at disrupting the parent molecular clouds around young stars, resulting in a flat scale-dependence of the gas-to-stellar flux ratio when apertures are centred on young stellar peaks (by comparison to the diverging branch we find in our Figure~\\ref{Fig::tuning-forks}). This leads to a much longer duration of the embedded phase of star formation, as derived via the method of~\\cite{Kruijssen18a}: $23$~Myr in~\\cite{Fujimoto19} vs. $4.4$~Myr in our simulations.\n\nAt mass resolutions of $10^3$-$10^4$ solar masses per gas cell, ~\\cite{2011MNRAS.417..950H,2013MNRAS.434.3142A,2014MNRAS.445..581H,Agertz13,2015ApJ...804...18A} inject HII region momentum via a similar prescription to ours, but in the analytic form of a `direct radiation pressure' during the radiation-dominated phase of HII region expansion. As discussed in KM09 and in our Section~\\ref{Sec::Theory}, radiation pressure dominates the expansion of only the largest HII regions, while those with ionising luminosities in the range of $1 < S_{49} < 100$ (as for the FoF groups in our high-resolution simulation) suffer a factor of ten or more reduction in the momentum injected, if the gas-pressure term in Equation~(\\ref{Eqn::momentumeqn_final}) is ignored. Despite this, the above works find that their radiation pressure prescriptions are necessary to achieve a realistic interstellar medium. This can be attributed to their use of an $f_{\\rm trap}$ factor far exceeding that found in observations~\\citep[e.g.][]{2021ApJ...908...68O}. In~\\cite{2013MNRAS.434.3142A,Agertz13,2015ApJ...804...18A} a value of $f_{\\rm trap} \\sim 25$ is used, and in~\\cite{2011MNRAS.417..950H,2014MNRAS.445..581H} this value is further increased within the range $f_{\\rm trap} \\sim 10$-$100$. By contrast, later works of~\\cite{Hopkins18,2019MNRAS.489.4233M} reduce the value of $f_{\\rm trap}$ back to order one, and the authors find that in this case, the direct radiation pressure has a negligible effect~\\citep[see Figure 36 of][]{Hopkins18}. In summary, the above works agree with our results in the sense that a pre-supernova momentum injection of $\\sim 10$-$100 \\times L\/c$ has a substantial influence on the intermediate- and large-scale properties of the interstellar medium. This momentum injection is needed to achieve an interstellar medium consistent with observations. However, according to the calculations presented in KM09 and in our Section~\\ref{Sec::Theory}, the vast majority of this momentum comes from the gas pressure inside the HII region, and not from the radiation pressure.\n\nFinally, we note that an identical feedback prescription (but using $f_{\\rm trap} = 2$ rather than $f_{\\rm trap} = 8$) was adopted in~\\cite{2020MNRAS.498..385J} and in~\\cite{Jeffreson21a} to investigate molecular cloud properties in three isolated disc galaxies with external, analytic galactic potentials. The molecular cloud population at the native resolution in these studies was on average less massive (maximum mass of $\\sim 10^7~{\\rm M}_\\odot$ vs. $\\sim 10^8~{\\rm M}_\\odot$ here) and had a shorter median cloud lifetime ($\\sim 20$~Myr vs. $\\sim 35$~Myr here). This can be attributed to the fact that the mid-plane turbulent pressure in the Agora disc used here is approximately eight times that of the discs introduced in~\\cite{2020MNRAS.498..385J}, i.e.~the mid-plane gas surface density and velocity dispersion are both doubled. This may be due to the use of a live dark matter and stellar potential, which allows for a greater degree of baryon clustering. The star formation efficiency per free-fall time of $\\epsilon_{\\rm ff} = 10$~per~cent used in this work is also ten times the value of $1$~per~cent used in~\\cite{2020MNRAS.498..385J}, because we have found that the lower star formation efficiency results in unphysically-bursty star formation, and an unphysically-high turbulent velocity dispersion of the cold gas on kpc-scales.\n\n\\section{Conclusions} \\label{Sec::conclusion}\nIn this work, we have developed a novel model for the momentum imparted by marginally-resolved HII regions in simulations with mass resolutions between $10^3$ and $10^5~{\\rm M}_\\odot$ per gas cell. The model can be applied in a spherical or a beamed configuration, where the latter corresponds to the directed momentum injected from blister-type HII regions on the edges of molecular clouds. We have compared simulations with only supernova feedback to simulations with supernova and thermal HII region feedback, spherical HII region momentum, beamed HII region momentum, and a combination of beamed momentum and thermal injection, across the mass resolution range $10^3$-$10^5~{\\rm M}_\\odot$. In general, we find that:\n\\begin{enumerate}\n\t\\item Thermal feedback from marginally-resolved HII regions has no influence on the interstellar medium, at any scale or resolution.\n\t\\item The geometry of momentum injection (spherical or beamed) from HII regions similarly has very little effect.\n\\end{enumerate}\nWhen HII region momentum is introduced at mass resolutions between $10^3$ and $10^4~{\\rm M}_\\odot$, the large-scale interstellar medium responds in the following ways:\n\\begin{enumerate}\n\t\\item The mass-loading and magnitude of galactic outflows are reduced by an order of magnitude.\n\t\\item The gas-disc scale-height is reduced by $0.5$~dex for galactocentric radii $>5$~kpc.\n\t\\item The velocity dispersion of the cold gas is supressed by $\\sim 5~{\\rm kms}^{-1}$, and the gravitational stability of the gas disc is correspondingly decreased  by a factor of around two in the~\\cite{Toomre64} $Q$ parameter.\\footnote{We recall that the factors of decrease in the velocity dispersion and Toomre $Q$ parameter do not match exactly because of the difference in gas density at the simulation time of comparison.}\n  \\item The mass fraction of cold gas is increased by $\\sim 50$~per~cent and the mass fraction of cold molecular gas is approximately doubled.\n\\end{enumerate}\nThe above results are consistent with the idea that HII region feedback (and pre-supernova feedback in general) reduces the clustering of supernovae and therefore their efficacy in depositing large quantities of momentum into the interstellar medium, as studied by~\\cite{2020arXiv200911309S}. The results do not persist down to resolutions of $10^5~{\\rm M}_\\odot$, although the HII regions are still marginally-resolved. This non-convergence is likely due to a decrease in the effectiveness of supernova clustering.\n\nFor molecular clouds specifically, we find the following results:\n\\begin{enumerate}\n\t\\item The lifetimes of the least massive molecular clouds ($M\/{\\rm M}_\\odot \\la 5.6 \\times 10^4$) are reduced from $\\sim 18$~Myr to $<10$~Myr. That is, HII region momentum is able to disperse low-mass clouds that do not contain supernovae from massive stellar clusters.\n\t\\item The lifetimes of intermediate-mass ($5.6 \\times 10^4 \\la M\/{\\rm M}_\\odot \\la 5 \\times 10^5$) clouds are increased by $\\sim 7$~Myr. That is, HII region momentum decreases the efficiency of supernovae in dispersing intermediate-mass clouds.\n\t\\item The molecular cloud star formation rate surface density is suppressed by a factor of three.\n\t\\item The molecular cloud velocity dispersion is reduced by $\\sim 0.5~{\\rm kms}^{-1}.$\n\\end{enumerate}\n\nIn summary, we find that the large- and intermediate-scale properties of the simulated interstellar medium, as well as the properties of its molecular clouds, are significantly altered by the introduction of momentum from HII regions at numerical mass resolutions from $10^3~{\\rm M}_\\odot$ to $10^5~{\\rm M}_\\odot$ per gas cell. More than 90~per~cent of the injected momentum is due to the thermal expansion of the HII regions, rather than radiation pressure. The injection of thermal energy without momentum has no discernible effect on the simulated galaxies.\n\n\\section*{Acknowledgements}\nWe thank an anonymous referee for a constructive report, which improved the paper. We thank Volker Springel for providing us access to Arepo. SMRJ is supported by Harvard University through the ITC. We gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through an Emmy Noether Research Group (SMRJ, MC, JMDK; grant number KR4801\/1-1) and the DFG Sachbeihilfe (MC, JMDK; grant number KR4801\/2-1), as well as from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme via the ERC Starting Grant MUSTANG (SMRJ, BWK, JMDK; grant agreement number 714907). SMRJ, MRK, YF, MC and JMDK acknowledge support from a UA-DAAD grant. BWK acknowledges funding in the form of a Postdoctoral Research Fellowship from the Alexander von Humboldt Stiftung. MRK acknowledges support from the Australian Research Council through Future Fellowship FT80100375 and Discovery Projects award DP190101258. The work of LA was partly supported by the Simons Foundation under grant no.~510940. The work was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI; award jh2), which is supported by the Australian Government. We are grateful to Jeong-Gyu Kim, Eve Ostriker and Vadim Semenov for helpful discussions.\n\n\\section*{Data Availability Statement}\nThe data underlying this article are available in the article and in its online supplementary material.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n Scattering amplitudes as well as further quantities in Quantum Field Theory contain a rich mathematical structure, whose understanding has frequently expanded our calculational reach -- benefiting both phenomenological tests of the Standard Model of Particle Physics at the LHC as well as more formal studies.\n \n At one-loop order, and in certain cases also at higher loop orders, the functions that occur in Feynman integrals and thus in Quantum Field Theory are multiple polylogarithms (MPLs) \\cite{Chen:1977oja,G91b,Goncharov:1998kja,Remiddi:1999ew,Borwein:1999js,Moch:2001zr}, which are by now very well understood.\n Increasing the loop order, the next class of functions we encounter are elliptic multiple polylogarithms (eMPLs), on which there has been much recent progress \\cite{Laporta:2004rb,MullerStach:2012az,brown2011multiple,Bloch:2013tra,Adams:2013nia,Adams:2014vja,Adams:2015gva,Adams:2015ydq,Adams:2015ydq,Adams:2016xah,Adams:2017ejb,Adams:2017tga,Bogner:2017vim,Broedel:2017kkb,Broedel:2017siw,Remiddi:2017har,Chen:2017soz,Bourjaily:2017bsb,Adams:2018yfj,Broedel:2018iwv,Broedel:2018qkq,Honemann:2018mrb,Bogner:2019lfa,Broedel:2019hyg,Duhr:2019rrs,Walden:2020odh,Weinzierl:2020fyx,Kristensson:2021ani,Frellesvig:2021hkr},\n in particular by studying the two-loop massive sunrise integral in two dimensions \\cite{SABRY1962401,Broadhurst:1993mw,Laporta:2004rb,Muller-Stach:2011qkg,Adams:2013nia,Bloch:2013tra,Remiddi:2013joa,Bloch:2014qca,Adams:2014vja,Adams:2015gva,Adams:2015ydq,Adams:2015ydq,Bloch:2016izu,Remiddi:2016gno,Adams:2017ejb,Broedel:2017siw} (see figure \\ref{subfig:sunrise}). More recently, also more complicated Feynman integrals are starting to be understood,\n  in particular the \n two-loop ten-point double-box integral with massless internal propagators in four dimensions \\cite{Bourjaily:2017bsb,Kristensson:2021ani} (see figure \\ref{subfig:doublebox}).\n Beyond eMPLs, also integrals over more complicated geometries than elliptic curves occur \\cite{Brown:2009ta,Brown:2010bw,Bourjaily:2018yfy,Bourjaily:2018ycu,Festi:2018qip,Broedel:2019kmn,Besier:2019hqd,Bourjaily:2019hmc,Vergu:2020uur,mirrors_and_sunsets,\nBroadhurst:1987ei,Adams:2018kez,Adams:2018bsn,Huang:2013kh,Klemm:2019dbm,Bonisch:2020qmm,Bonisch:2021yfw,Muller:2022gec,Chaubey:2022hlr}; an understanding of the corresponding spaces of functions is still in its infancy.\n For a recent review on functions in scattering amplitudes beyond MPLs, see \\cite{Bourjaily:2022bwx}.\n \nOur good understanding of MPLs is to a large extend due to the Hopf algebra structure underlying these functions \\cite{Gonch2,Goncharov:2010jf,Brown:2011ik,Duhr:2011zq,Duhr:2012fh}, and in particular the symbol \\cite{Goncharov:2010jf}.\nThe symbol map associates to each MPL $f$ a simple tensor product, $\\mathcal{S}(f)=\\sum \\log(\\phi_{i_1})\\otimes\\dots\\otimes \\log(\\phi_{i_k})$.    \nThe entries in this tensor product, called symbol letters, are logarithms of rational or algebraic functions $\\phi_i$ of the kinematic invariants.\nSince a tensor product is easy to manipulate, and the identities of the symbol letters, $\\log(a)+\\log(b)=\\log(ab)$, are well understood, the symbol provides a powerful way of finding identities between MPLs and for simplifying expressions. \nThe symbol moreover manifests physical properties of the corresponding function. For example, the first entry of the symbol describes the discontinuities, which are heavily restricted in particular in massless theories resulting in so-called first-entry conditions \\cite{Gaiotto:2011dt}.\nMoreover, discontinuities in overlapping channels are forbidden by the so-called Steinmann conditions \\cite{Steinmann,Steinmann2}, restricting which symbol letter in the second entry can follow a particular letter in the first entry.\nThe symbol has made enormous progress possible for quantities consisting of MPLs, both in relation to phenomenology and more formal studies, including in particular powerful bootstrap techniques \\cite{Dixon:2011pw,Dixon:2011nj,Dixon:2013eka,Dixon:2014voa,Dixon:2014iba,Drummond:2014ffa,Dixon:2015iva,Caron-Huot:2016owq,Dixon:2016apl,Dixon:2016nkn,Drummond:2018caf,Caron-Huot:2019vjl,Li:2016ctv,Almelid:2017qju,Dixon:2020bbt,Dixon:2022rse,Abreu:2020jxa,Chicherin:2020umh,Dixon:2012yy,Chestnov:2020ifg,He:2021fwf,Drummond:2017ssj,Chicherin:2017dob,Caron-Huot:2018dsv,Henn:2018cdp,He:2021non,He:2021eec,Heller:2019gkq,Heller:2021gun,Duhr:2021fhk}.\n\nWhile the symbol for eMPLs was defined in \\cite{BrownNotes,Broedel:2018iwv}, it has so far not been put to much use, and is still much less understood than its analog for MPLs.\nOne reason is that the symbol letters $\\Omega^{(i)}$ for eMPLs are themselves elliptic functions of the kinematic invariants. The simplest letters $\\Omega^{(0)}$ satisfy simple identities as the consequence of the group law on the elliptic curve.\nIn \\cite{Kristensson:2021ani}, some identities for the elliptic letters $\\Omega^{(i)}$ with $i=1,2$ were observed numerically in the study of the symbol of the ten-point elliptic double-box integral. \nUsing these identities, it was found that the elliptic letters in the first two entries combine to logarithms, manifesting the same first-entry condition as for polylogarithmic amplitudes as well as the Steinmann conditions.\nMoreover, the last entries were found to be given by simple elliptic integrals $\\Omega^{(0)}$, with $\\Omega^{(2)}$ only occurring in the third entry preceding the modular parameter $\\tau$ in the last entry.\n\nIn this paper, we show how the identities observed in \\cite{Kristensson:2021ani} for $\\Omega^{(1)}$ are a consequence of Abel's theorem \\cite{abel1841}.\nMoreover, we demonstrate that the identities observed in \\cite{Kristensson:2021ani} for $\\Omega^{(2)}$ are a consequence of the elliptic Bloch relation \\cite{Zagier2000,bloch2011higher} for the elliptic dilogarithm, which generalizes the Bloch relation for the classic dilogarithm and which have also been studied in the context of finding identities between elliptic multiple polylogarithms \\cite{Broedel:2019tlz,Bolbachan:2019dsu}.\nWhile the identities for $\\Omega^{(1)}$ can be reduced to three-term identities similar to $\\log(a)+\\log(b)=\\log(ab)$ in the case of the logarithm, the elliptic Bloch relation, and thus the identities for $\\Omega^{(2)}$, are five-term identities similar to the Bloch relation for the classical dilogarithm, which are made manifest only by the symbol.\nThus, we introduce a symbol prime $\\mathcal{S}'$ for the symbol letters $\\Omega^{(2)}$\n(and similarly for $\\Omega^{(n>2)}$)\nin analogy to the symbol for MPLs and eMPLs, which makes the identities due to the elliptic Bloch relation manifest. \n\nIn general, eMPLs transform under modular transformations of $\\tau$ in a complicated way \\cite{Duhr:2019rrs,Weinzierl:2020fyx}, and results given in terms of eMPLs are not manifestly double periodic.\nHowever, in the examples we studied, we find that the symbol prime makes both double periodicity as well as a simple behaviour under modular transformations manifest.\nFinally, it makes also part of the integrability conditions manifest, which result from the requirement that partial derivatives commute.\n\nTo illustrate the use of the symbol for eMPLs, the application of the identities of elliptic letters as well as the symbol prime, we study two concrete examples.\nThe first example is the two-loop sunrise integral in two dimensions with all internal masses being unequal (see figure \\ref{subfig:sunrise}). \nThe second example is the ten-point two-loop double-box integral in four dimensions with massless internal propagators (see figure \\ref{subfig:doublebox}).\nIn addition to the aforementioned properties and techniques, we also demonstrate how the (elliptic) symbol reduces to polylogarithmic symbol in kinematic limits where the elliptic curve degenerates. \n\n\n\n\\begin{figure}\n \\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\begin{tikzpicture}[scale=1.5,label distance=-1mm]\n\\clip (0,14.6) rectangle (3,16.4);\n\t\t\\node (0) at (0, 15.5) {};\n\t\t\\node (4) at (3.0, 15.5) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (1.5,15.5) circle (0.75);\n\\end{tikzpicture}  \n  \\caption{\\textcolor{white}{.}}\n  \\label{subfig:sunrise}\n\\end{subfigure}\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\begin{tikzpicture}[scale=1.5,label distance=-1mm]\n\\clip (0,14) rectangle (3,16);\n\t\t\\node (0) at (0, 15.5) {};\n\t\t\\node (1) at (1.5, 16) {};\n\t\t\\node (2) at (0.5, 16) {};\n\t\t\\node (3) at (2.5, 16) {};\n\t\t\\node (4) at (3.0, 15.5) {};\n\t\t\\node (5) at (0.5, 14) {};\n\t\t\\node (6) at (0, 14.5) {};\n\t\t\\node (7) at (1.5, 14) {};\n\t\t\\node (8) at (3, 14.5) {};\n\t\t\\node (9) at (2.5, 14) {};\n        \\node (11) at (5, 15) {};\n\t\t\\node (12) at (6, 15) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (6.center) to (8.center);\n\t\t\\draw[thick] (2.center) to (5.center);\n\t\t\\draw[thick] (1.center) to (7.center);\n\t\t\\draw[thick] (3.center) to (9.center);\n\\end{tikzpicture}\n  \\caption{\\textcolor{white}{.}}\n  \\label{subfig:doublebox}\n\\end{subfigure}\n\\caption{The sunrise integral in two dimensions with unequal internal masses (\\subref{subfig:sunrise}) as well as the ten-point double-box integral in four dimensions with massless internal propagators (\\subref{subfig:doublebox}).}\n\\label{fig: diagrams intro} \n\\end{figure}\n\n\n\nThe remainder of this paper is organized as follows. We review \n elliptic multiple polylogarithms in section \\ref{sec:2}. In section \\ref{sec:3}, we derive identities for elliptic symbol letters -- based on Abel's addition theorem for $\\Omega^{(1)}$'s and by introducing the symbol prime map for $\\Omega^{(2)}$'s. We illustrate the use of these techniques for the unequal-mass sunrise integral in section \\ref{sec: example 1} and for the ten-point double-box integral in section \\ref{sec: example 2}.\n In particular, we provide analytic results for the non-elliptic nine-point double-box integral and its symbol, which result from taking the soft limit of the ten-point double-box integral.\n We conclude with a summary and an  outlook on open questions in section \\ref{sec:5}.\nIn appendix \\ref{app:sunrise}, we review the calculation of the unequal-mass sunrise integral via Feynman parameters using our conventions and notation.  \nThe details of simplifying the symbols for the sunrise integral, as well as the expressions of the functions and \nsymbols for the ten-point elliptic integral and its soft limit, are included as ancillary files (\\texttt{sunrise\\_symbol.nb}, \\texttt{doublebox\\_omega2} and \\texttt{doublebox\\_soft}).%\n\\footnote{In this article, \nwe only provide the expression for the ten-point double box in the normalization by the period $-\\omega_2$ since the corresponding expressions in the normalization by the period $\\omega_1$ can be found in the ancillary files of \\cite{Kristensson:2021ani}.} \n\n\n\n\\section{Review of Elliptic Multiple Polylogarithms}  \\label{sec:2}\n\n\nLet us first review several elementary facts about elliptic multiple polylogarithms; see \\cite{Broedel:2017kkb,Broedel:2017siw,Broedel:2018iwv,Broedel:2018qkq} for further details. We follow the notations and conventions of \\cite{Kristensson:2021ani}, which differ slightly from those of \\cite{Broedel:2017kkb,Broedel:2017siw,Broedel:2018iwv,Broedel:2018qkq}.\n\n\\subsection{Elliptic multiple polylogarithms on the elliptic curve}\n\nIn this paper, the elliptic curves $\\mathcal{C}$ are described by monic quartic equations:\n\\begin{equation}\n    y^{2} =P_{4}(x)= x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} \\:. \\label{quarticEcurve}\n\\end{equation}\nSuch elliptic curves can be converted to the standard Weierstrass form \n\\begin{equation}\n   Y^{2}=4X^{3}-g_{2}X-g_{3}                    \\end{equation}\nusing the rational point at $(x,y)=(+\\infty,+\\infty)$, where $(X,Y)$ are related to $(x,y)$ by \n\\begin{equation} \\label{XYtoxy}\n    \\begin{aligned}\n        X &= \\frac{1}{12}\\bigl( a_{2} + 3a_{3}x + 6x^{2}+6y\\bigr) \\:,  \\\\\n        Y &= \\frac{1}{4}\\bigl( a_{1} + 2a_{2}x + 3a_{3}x^{2} + 4x^{3} +a_{3}y+4xy\\bigr) \\:.\n    \\end{aligned}    \n\\end{equation}\n\n\nOn the curve $\\mathcal{C}$, we can introduce elliptic multiple polylogarithms $\\mathrm{E}_{4}$, which are recursively defined as \\cite{Broedel:2017kkb}%\n\\footnote{The subscript ``$4$'' indicates that the elliptic curve is given by a quartic polynomial. Analogous functions for a cubic polynomial were also defined in \\cite{Broedel:2017kkb}.}\n\\begin{equation}\n\\label{Eiterateddefinition}\n    \\Ef{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}=\n    \\int_{0}^{x}\\dif  x'\\,\\psi_{n_{1}}(c_{1},x')\\Ef{n_{2} & \\ldots & n_k}{c_{2} & \\ldots& c_k}{x'}\n\\end{equation}\nwith $\\mathrm{E}_{4}(;x)=1$, where\n\\begin{equation} \\label{psibasis}\n    \\begin{split}\n        &\\psi_{0}(0,x)=\\frac{1}{y} \\,,\\qquad  \\:\\:\\psi_{-1}(\\infty,x)=\\frac{x}{y}\\,, \\\\\n        &\\psi_{1}(c,x)=\\frac{1}{x-c}\\,,\\quad \n        \\psi_{-1}(c,x)=\\frac{y_{c}}{y(x-c)}\\,,\n    \\end{split} \n\\end{equation}\nwith $y_{c}=y\\vert_{x=c}$. The definitions of $\\psi_{n}(c,x)$ for $n=\\pm2,\\pm3,\\dots$ can be found in \\cite{Broedel:2017kkb}; the kernels \\eqref{psibasis} are sufficient for the purpose of this paper, though. \n\nThe class of elliptic multiple polylogarithms $\\Ef{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}$ contains in particular all non-elliptic (Goncharov) multiple polylogarithms, defined by \n\\begin{equation}\n G(c_1,\\dots,c_n;x)=\\int_0^x\\frac{\\dif x'}{x'-c_1}G(c_2,\\dots,c_n;x')\n\\end{equation}\nwith $G(;x)=1$,\nsince by definition $\\Ef{1 & \\ldots & 1}{c_1 & \\ldots& c_k}{x}\\equiv G(c_1,\\dots,c_k;x)$.\n\n\nIn general, any integral of the form\n\\begin{equation}\n    \\int\\frac{\\dif x}{y} \\:\\mathcal{G}(x,y)\\,, \\label{1dIntofPolylog}\n\\end{equation}\nwhere $\\mathcal{G}$ is a polylogarithm whose letters are rational functions of $x$ and $y$, can be converted to $\\mathrm{E}_{4}$ functions with only the four kinds of integration kernels defined in \\eqref{psibasis}.%\n\\footnote{Since $\\mathcal{G}$ is a polylogarithm, the integration kernels have only simple poles, in addition to being rational in $x$ and $y$. While all integration kernels $\\psi_n$ have only simple poles, only $\\psi_{-1,0,+1}$ are rational functions of $x,y$.}\n In particular, this is the case for the (unequal-mass) sunrise integral and the ten-point double-box integral, which we will study as examples in sections \\ref{sec: example 1}--\\ref{sec: example 2}.\n \n \\subsection{From the elliptic curve to the torus}\n\nThe functions $\\mathrm{E}_{4}$ are closely related to Feynman integrals since they are defined in an algebraic way. However, the purity of some elliptic Feynman integrals, such as integrals of the form \\eqref{1dIntofPolylog}, is hidden in terms of $\\mathrm{E}_{4}$ functions since taking the total derivative of a $\\mathrm{E}_{4}$ function does not necessarily decrease its length.%\n\\footnote{This can be seen concretely by how the integration kernels $\\psi_{-1}(\\infty,x)\\dif x$ and $\\psi_{-1}(c,x) \\dif x$ are related to the kernels of pure functions given below in \\eqref{psitog1}.\n}\nOn the other hand, iterated integrals defined on the torus, such as the $\\tilde{\\Gamma}$ functions we will review below, are manifestly pure and hence allow a symbol map defined via the total derivative. \n\nTo connect both sides, we first need a bijection between the elliptic curve $\\mathcal{C}$ and the torus $\\mathbb{C}\/\\Lambda$, where $\\Lambda$ is the lattice generated by the periods $\\omega_{1}$ and $\\omega_{2}$ of the elliptic curve. For an elliptic curve of the form \\eqref{quarticEcurve}, one can find such a map through the birationally \nequivalent curve in the Weierstrass normal form: first solve $(x,y)$ in terms of $(X,Y)$ from  \\eqref{XYtoxy}, then replace $X$ and $Y$ with the Weierstrass elliptic function $\\wp(z)$ and its derivative $\\wp'(z)$, respectively. \nThis gives \\begin{equation}\n            z\\mapsto (x,y)=(\\kappa(z),\\kappa'(z))\\,,\n           \\end{equation}\nwhere\n \\begin{equation}\n     \\kappa(z)=\\frac{6a_{1}-a_{2}a_{3}+12 a_{3}\\wp(z)-24\\wp'(z)}{3a_{3}^{2}-8a_{2}-48\\wp(z)}.\n \\end{equation} \nIt is easy to see that $\\kappa(0)=\\infty$, and hence all lattice points are mapped to the infinity point in the $(x,y)$ space. Furthermore, each point $c$ in the $x$-space corresponds to two points $(c,\\pm y_{c})$ on the elliptic curve $\\mathcal{C}$ and hence to two images on the torus $\\mathbb{C}\/\\Lambda$, which we denote by $z_{c}^{\\pm}$;\n these two images satisfy \n \\begin{equation} \\label{wpwmrelation}\n     z_{c}^{+}+z_{c}^{-}= z^{-}_{\\infty} + z^{+}_{\\infty}\\equiv z^{-}_{\\infty} \\:  \\operatorname{mod} \\Lambda \\:,\n \\end{equation}\n since the corresponding points $(X_{c}^{\\pm},Y_{c}^{\\pm})$ and\n \\begin{equation}\n    (X_{\\infty}^{-},Y_{\\infty}^{-})  = \\bigl(\\tfrac{1}{48}(3a_{3}^{2}-8a_{2}),\\tfrac{1}{32}(-8a_{1}+4a_{2}a_{3}-a_{3}^{3}) \\bigr)  \n \\end{equation}\n are on the same line. \n \n\n\n\\tikzset{cross\/.style={cross out, thick, draw=black, fill=none, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt}, cross\/.default={2pt}}\n\n\\begin{figure}\n   \\centering\n   \\begin{tikzpicture}\n           \\draw[->, thick] (0,0) to (0,3);\n           \\draw[->, thick] (-3,1.5) to (1,1.5);\n           \\draw[<-,line width=0.23mm,red] (-0.8,0.0) to (-0.8,3.0);\n           \\draw[->,line width=0.23mm,blue] (-3,1.4) .. controls (-1,1.4) ..   (-1.0,0.0) ; \n           \\node[cross,label=above:$r_{1}$] at (-1.5, 2.0) {};\n           \\node[cross,label=below:$r_{4}$] at (-1.5, 1.0) {};\n           \\node[cross,label=above:$r_{2}$] at (-0.5, 0.8) {};\n           \\node[cross,label=below:$r_{3}$] at (-0.5, 2.2) {};\n           \\node at (-0.5,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n           \\node at (-1.3,0.1) {\\textcolor{blue}{$\\gamma_{-}$}};\n           \\node at (0.4,3) {$\\Im x$};\n           \\node at (1.3,1.5) {$\\Re x$};\n   \\end{tikzpicture}\n   \\caption{The distribution of the four roots of $y^{2}(x)$ and the two integration contours $\\gamma_1$ and $\\gamma_-$ defining the period $\\omega_1$ and $z_\\infty^-$. The contour $\\gamma_2$ which defines the period $\\omega_2$ runs along the real axis.\n   }\n   \\label{fig: contours}\n   \\end{figure}\n\nThe inverse map from the torus to the elliptic curve is simply given by the Abel-Jacobi map. We assume that the four roots of $y^{2}(x)$ come in complex conjugate pairs as shown in figure \\ref{fig: contours}.%\n\\footnote{For a discussion of other possible distributions of roots, see \\cite{Broedel:2017kkb}.}\nThen, the torus image $z_{c}^{+}$ for any real $c$ is given by\n\\begin{equation}\n    z_{c}^{+}=\\int_{-\\infty}^{c} \\frac{\\dif x}{y}\\,.\n\\end{equation} \nHence, $z_{\\infty}^{+}$ is one period of the torus, and we choose it to be $\\omega_{2}$. The image $z_{c}^{-}$ can be obtained by \\eqref{wpwmrelation} together with $z_{\\infty}^{-}=\\int_{\\gamma_{-}} \\dif x\/y$, and the other period is $\\omega_{1}=\\int_{\\gamma_{1}} \\dif x\/y$; see figure \\ref{fig: contours} for the definitions of the integration contours $\\gamma_{-}$ and $\\gamma_{1}$.\nDue to the distribution of roots, $\\omega_{2}$ and $\\mi\\omega_{1}$ are \\emph{positive} reals.\n\n\n\\subsection{Elliptic multiple polylogarithms on the torus}\n\n\nDue to the equivalence between the elliptic curve and the torus,  \nanother way to define elliptic multiple polylogarithms is via iterated integrals on a torus. Such iterated integrals can be formulated in several ways. \nIn this paper, we use the so-called $\\tilde{\\Gamma}$ functions \\cite{Broedel:2017kkb,Broedel:2018iwv}, which are defined as\\footnote{Another variant of such iterated integrals extensively used in one-loop string amplitudes  are the so-called $\\Gamma$ functions, whose integration kernels $f^{(n)}$ are double periodic but \\emph{not} meromorphic; see e.g.\\ \\cite{Broedel:2014vla}.} \n\\begin{equation}\n    \\gamt{n_1 & \\ldots & n_k}{w_1 & \\ldots& w_k}{w|\\tau}=\n    \\int_{0}^{w} \\dif w'\\,g^{(n_{1})}(w'{-}w_{1},\\tau)\\gamt{n_{2} & \\ldots & n_k}{w_{2} & \\ldots& w_k}{w'|\\tau}\n\\end{equation}\nwith $\\tilde{\\Gamma}(;w|\\tau)=1$; we will frequently suppress the dependence on $\\tau$ for ease of notation. Such an iterated integral is said to have length $k$ and weight $\\sum_kn_k$, and in contrast to the case of MPLs both quantities are not necessarily equal. \nThe integration kernels $g^{(n)}(w,\\tau)$ are generated by the \\emph{Eisenstein-Kronecker series}\n\\begin{equation}\n    \\frac{\\partial_{w}\\theta_{1}(0|\\tau)\\theta_{1}(w+\\alpha|\\tau)}{\\theta_{1}(w|\\tau)\\theta_{1}(\\alpha|\\tau)} = \\sum_{n\\geq 0}\\alpha^{n-1}g^{(n)}(w,\\tau)\\:,\n\\end{equation}\nwhere $\\theta_{1}(w|\\tau)$ is the odd Jacobi theta function. \nAll the integration kernels $g^{(n)}$ except $g^{(0)}=1$ are quasi double periodic,\n\\begin{equation}\n    g^{(n)}(w+1)=g^{(n)}(w)\\:, \\qquad  g^{(n)}(w+\\tau)=\\sum_{j=0}^{n}\\frac{(-2\\pi \\mi)^{j}}{j!}g^{(j)}(w)\\:,\n\\end{equation}\nbut meromorphic with only logarithmic poles at most \\cite{Broedel:2017kkb,Broedel:2018iwv}. \n\nNote that the functions $\\tilde\\Gamma$ and the integration kernels $g^{(n)}$ are defined on the normalized torus, and that the torus with periods $(\\omega_{1},\\omega_{2})$ has the two normalizations $[1:\\tau=\\omega_{2}\/\\omega_{1}]$ and $[1:\\tau'=-\\omega_{1}\/\\omega_{2}]$, which are related by the modular $S$-transformation $\\tau\\to -\\tau^{-1}$. We denote the images of $c$ on $[1:\\tau=\\omega_{2}\/\\omega_{1}]$ as $w^{\\pm}_{c}=z^{\\pm}_{c}\/\\omega_1$ and the images on $[1:\\tau'=-\\omega_{1}\/\\omega_{2}]$ as $\\xi_{c}^{\\pm}=z^{\\pm}_{c}\/(-\\omega_2)$. The two are related by $\\xi_c^\\pm=\\tau' w_c^\\pm$. In what follows, most of the results are written in terms of $w$-coordinates, but it should be understood that the analogous results also hold in terms of $\\xi$-coordinates unless otherwise indicated.\n\n\nThe integration kernels $\\psi_{n}$ can be identified as combinations of $g^{(j)}$'s by matching poles on both sides. On the torus $[1:\\tau=\\omega_{2}\/\\omega_{1}]$, one can easily find the following relations between $g^{(j)}$'s and $\\psi_{n}$'s,\n\\begin{subequations} \\label{psitog1}\n    \\begin{align}\n        \\psi_{1}(c,x)\\dif x &=\\Bigl(g^{(1)}(w-w_{c}^{+})+g^{(1)}(w-w_{c}^{-}) \\nonumber \\\\ \n                            &\\qquad \\qquad \\qquad \\qquad -g^{(1)}(w-w_{\\infty}^{+}) -g^{(1)}(w-w_{\\infty}^{-})\\Bigr)\\dif w \\,, \n                            \\label{psitog1a}\\\\\n        \\psi_{-1}(c,x)\\dif x &= \n        \\Bigl( g^{(1)}(w-w_{c}^{+})-g^{(1)}(w-w_{c}^{-}) + g^{(1)}(w_{c}^{+}) -g^{(1)}(w_{c}^{-}) \\Bigr) \\dif w  \\,,  \\label{psitog1b}\\\\\n        \\psi_{-1}(\\infty,x)\\dif x &=   \\Bigl(g^{(1)}(w-w_{\\infty}^{-})-g^{(1)}(w) + g^{(1)}(w_{\\infty}^{-})-\\omega_{1}a_{3}\/4 \\Bigr)\\dif w\\,, \\\\\n        \\psi_{0}(0,x)\\dif x&= \\omega_{1}\\dif w \\,.\n    \\end{align}    \n\\end{subequations}\nOn the torus $[1:\\tau']$, the corresponding relations can be obtained by replacing $w\\to\\xi$ and $\\omega_1\\to -\\omega_2$ in \\eqref{psitog1}.\n\nSometimes, it is more convenient to combine $\\tilde{\\Gamma}$ functions into the so-called $\\mathcal{E}_{4}$ functions \\cite{Broedel:2018qkq}, especially if the $\\tilde{\\Gamma}$ functions originally arose from an expression of $\\mathrm{E}_{4}$ functions. The elliptic multiple polylogarithms $\\mathcal{E}_{4}$ are defined in complete analogy to \\eqref{Eiterateddefinition}:\n\\begin{equation}\n\\label{curlyEiterateddefinition}\n    \\cEf{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}=\n    \\int_{0}^{x}\\dif  x'\\,\\Psi_{n_{1}}(c_{1},x')\\cEf{n_{2} & \\ldots & n_k}{c_{2} & \\ldots& c_k}{x'}\n\\end{equation}\nwith $\\mathcal{E}_{4}(;x)=1$,\n\\begin{align}\n \\Psi_{\\pm (n>0)}(c,x) \\dif  x=\\Bigl(&g^{(n)}(w-w_{c}^{+})\\pm g^{(n)}(w-w_{c}^{-})  \n\\nonumber \\\\\n    &\n    - \\delta_{\\pm n,1}\\bigl( g^{(1)}(w-w_{\\infty}^{+})+g^{(1)}(w-w_{\\infty}^{-} )\\bigr) \\Bigr) \\dif w  \n\\label{Psikernels} \n\\end{align}\nand $\\Psi_0(x)\\dif x=\\dif w$, as well as analogous  expressions in terms of $\\xi$.\nThe weight of a function $\\cEf{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}$ is defined as $\\sum_i |n_i|$.\n\n\\subsection{Symbol}\n\nBy construction,\n the total derivative of $\\tilde{\\Gamma}$ admits a recursive structure \\cite{Broedel:2018iwv},\n\\begin{align} \\nonumber\\label{devoftG}\n    &\\quad \\dif\\tilde{\\Gamma}(A_{1},\\ldots,A_{k};w)  \\\\\n    &= \\sum_{p=1}^{k-1}(-1)^{n_{p+1}}\\tilde{\\Gamma}(A_{1},\\ldots,A_{p-1},\\vec{0},A_{p+2},\\ldots,A_{k};w) \n    \\times\\omega^{(n_{p}+n_{p+1})}(w_{p+1,p})\n      \\\\\n    &\\quad+ \\sum_{p=1}^{k}\\sum_{r=0}^{n_{p}+1} \\Biggl[ \n        \\binom{n_{p-1}{+}r{-}1}{n_{p-1}{-}1}\\tilde{\\Gamma}(A_{1},\\ldots, A_{p-1}^{[r]},A_{p+1},\\ldots,A_{k};w) \n         \\times\\omega^{(n_{p}-r)}(w_{p-1,p}) \\nonumber \\\\\n    &\\qquad \\qquad  - \\binom{n_{p+1}{+}r{-}1}{n_{p+1}{-}1} \\tilde{\\Gamma}(A_{1},\\ldots, A_{p-1},A_{p+1}^{[r]},\\ldots,A_{k};w) \n   \\times \\omega^{(n_{p}-r)}(w_{p+1,p})\n    \\Biggr],\\nonumber\n\\end{align}\nwhere $\\vec{0}\\equiv\\bigl(\\begin{smallmatrix}\n    0 \\\\\n    0       \n    \\end{smallmatrix}\\bigr)$, $w_{i,j}\\equiv w_{i}-w_{j}$, as well as \n\\begin{equation}\n    A_{i}^{[r]}\\equiv\\bigl(\\begin{smallmatrix}\n    n_{i}+r \\\\\n    w_{i}       \n    \\end{smallmatrix}\\bigr)\\:, \\qquad A_{i}^{[0]}\\equiv A_{i}\\:.\n\\end{equation}\nThe forms $\\omega^{(j)}(w)$ are exact, and we can thus write them as \n\\begin{equation}\n \\omega^{(j)}(w,\\tau)=(2\\pi i)^{j-1}\\dif\\Omega^{(j)}(w,\\tau)\\,,\n\\end{equation}\nwith\n\\begin{align} \\label{wdef}\n    \\Omega^{(-1)}(w,\\tau) &= -2\\pi\\mi\\tau\\:, \\quad \\Omega^{(0)}(w,\\tau) =2\\pi\\mi w\\:, \\quad \\Omega^{(1)}(w,\\tau)=\\log \\frac{\\theta_{1}(w|\\tau)}{\\eta(\\tau)} \\:, \\nonumber \\\\\n    \\Omega^{(\\text{odd } j>1)}(w,\\tau)&=- \\frac{2j\\zeta_{j+1}\\tau}{(2\\pi \\mi)^{j}} +\n    \\frac{1}{(j{-}1)!}\\sum_{n=1}^{\\infty}\n    n^{j-1}\\log\\bigl((1-\\me^{2\\pi \\mi (n\\tau-w)})(1-\\me^{2\\pi \\mi (n\\tau+w)})\\bigr), \\nonumber \\\\ \n    \\Omega^{(\\text{even } j)}(w,\\tau)&=-\\frac{2\\zeta_{j}w}{(2\\pi i)^{j-1}} \n    + \\frac{1}{(j{-}1)!}\\sum_{n=1}^{\\infty} n^{j-1}\\log\\frac{1-\\me^{2\\pi \\mi (n\\tau+w)}}{1-\\me^{2\\pi \\mi (n\\tau-w)}} ,\n\\end{align}\nwhere $\\eta(\\tau)$ is the Dedekind eta function and $\\zeta_{j}=\\sum_{n\\in\\mathbb{Z}_{+}} n^{-j}$ are the Riemann zeta values.\\footnote{Recall that $\\zeta_{2n}=\\frac{(-1)^{n+1}B_{2n}(2\\pi)^{2n}}{2(2n)!}$ with $B_{2n}$ being the $(2n)^{\\rm th}$ Bernoulli number, such that the first terms in \\eqref{wdef} can equivalently be written in terms of Bernoulli numbers.}\nThe functions $\\Omega^{(j)}$ satisfy   \n\\begin{equation}\n    g^{(j)}(w,\\tau)=(2\\pi\\mi)^{j-1}\\partial_{w}\\Omega^{(j)}(w,\\tau)=\\frac{(2\\pi\\mi)^{j-1}}{j-1}\\partial_{\\tau}\\Omega^{(j-1)}(w,\\tau).\n\\end{equation}\nThe sum representation \\eqref{wdef} can be derived using the sum representation of the $g^{(n)}$ functions given in \\cite{Broedel:2018iwv}.\nIn particular, \n\\begin{equation}\n (2\\pi i)^{1-n}\\gamt{n}{0}{w}=\\Omega^{(n)}(w)-\\Omega^{(n)}(0)\\,,\n\\end{equation}\nwhere $\\Omega^{(n)}(0)$ vanishes for even $n$ and is the primitive of the Eisenstein series for odd $n$; see \\cite{Broedel:2018iwv}.\nWe will see below that the functions $\\Omega^{(j)}$ play the role of elliptic symbol letters.\nAs can be seen from \\eqref{wdef}, $\\Omega^{(1)}$ has a logarithmic singularity at all lattice points, while \n$\\Omega^{(j>1)}$ has a logarithmic singularity at all lattice points except for the origin \\cite{Broedel:2018iwv}.\n\n\nFor a function $\\widetilde{\\Gamma}_{k}^{(n)}$ of weight $n$ and length $k$, we can define \n$ \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}=(2\\pi i)^{k-n}\\widetilde{\\Gamma}_{k}^{(n)}.$\nSchematically, the differential of $\\widetilde{\\underline{\\Gamma}}_{k}^{(n)}$ then takes the form \n\\begin{equation}\n     \\dif \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}=\\sum_i   \\widetilde{\\underline{\\Gamma}}^{(n-j_{i})}_{k-1} \\dif \\Omega^{(j_i)}(w_i) \\,,\n\\end{equation}\nThus, it is natural to define the symbol of $\\tilde{\\underline{\\Gamma}}_{k}^{(n)}$ as \n\\begin{equation}\n\\label{eq: elliptic symbol}\n     \\mathcal{S}(\\widetilde{\\underline{\\Gamma}}_{k}^{(n)})=\\sum_i \\mathcal{S}(\\tilde{\\underline{\\Gamma}}_{k-1}^{(n-j_{i})})\\otimes \\Omega^{(j_i)}(w_i) \\,.\n\\end{equation}\n\n\n\nNote that in contrast to \\cite{Broedel:2018iwv} we have included additional factors of $(2\\pi\\mi)$ in the definition of the elliptic letters $\\Omega^{(n)}$ and consider the symbol of $ \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}$ rather than $\\widetilde{\\Gamma}_{k}^{(n)}$.\\footnote{In \\cite{Broedel:2018iwv}, there is also a projection operator $\\pi_{k}$ in the definition of the symbol for $\\tilde{\\Gamma}$ functions due to the fact that some eMPLs of weight $0$ evaluate to rational numbers, such as $\\gamt{0}{0}{1}=1$. Here we exclude it by introducing these $2\\pi \\mi$ factors.} This is such that the elliptic letters and symbols degenerate to logarithms and polylogarithmic symbols without additional factors of $(2\\pi i)$ in the limit where the elliptic curve degenerates, see sections \\ref{sec: example 1}--\\ref{sec: example 2}.\n\n\n\\subsection{Shuffle regularization}\n\nLet us close this section by remarking on shuffle regularization. One can easily see that $\\gamt{1}{0}{z}=\\Omega^{(1)}(z)-\\Omega^{(1)}(0)$ is divergent since $\\Omega^{(1)}(0)$ is singular according to \\eqref{wdef}. The shuffle regularization used in  \\cite{brown2011multiple,Broedel:2018iwv} takes $\\Omega^{(1)}(0)\\equiv2\\log\\eta(\\tau)$.\nHowever, that regularization leads to an issue if we start with an integral of the form \\eqref{1dIntofPolylog} since it is inconsistent with the usual shuffle regularization for polylogarithms, $G(0;x)\\equiv\\log x$. To see this, we apply \\eqref{psitog1} to $G(0;1)=\\log 1=0$ and find\n\\begin{equation}\n    0 \\stackrel{?}{=} \\gamt{1}{0}{w_{1}^{+}-w_{0}^{+}} + \\gamt{1}{w_{0}^{-}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}} - \\gamt{1}{w_{\\infty}^{+}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}} - \\gamt{1}{w_{\\infty}^{-}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}}  \\:,\n\\end{equation}\nwhich is in general not true if we use $\\Omega^{(1)}(0)\\equiv2\\log\\eta(\\tau)$.\nTo reconcile both sides, we expand $\\gamt{1}{w'}{w}=\\Omega^{(1)}(w-w')-\\Omega^{(1)}(-w')$ to arrive at the \nfollowing regularization for elliptic multiple polylogarithms:\n\\begin{align}\n    \\Omega^{(1)}(0)&\\equiv\\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{-})+\\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{+}) -\\Omega^{(1)}(w_{0}^{+}-w_{0}^{-}) \\nonumber \\\\\n    &\\quad +\\Omega^{(1)}(w_{1}^{+}-w_{0}^{-}) +\\Omega^{(1)}(w_{1}^{+}-w_{0}^{+}) -\\Omega^{(1)}(w_{1}^{+}-w_{\\infty}^{-}) - \\Omega^{(1)}(w_{1}^{+}-w_{\\infty}^{+})  \\nonumber \\\\\n    &=2\\log \\eta(\\tau)+\\log\\frac{2\\pi\\mi}{\\omega_{1}}-\\log y_{0} = \\frac{1}{12}\\log \\Delta -\\log y_{0}\\:, \\label{shuffreg2}\n\\end{align} \nwhere $\\Delta= g_{2}^{3}-27g_{3}^{2}$ is the discriminant of the elliptic curve. The second equality will be explained in the next section and the third equality shows that this regularization is actually independent of the normalization of the torus. \n\n\n\n\n\n\\section{Identities of Elliptic Symbol Letters and the Symbol Prime} \n\\label{sec:3}\n\nWe have briefly reviewed several elementary facts about elliptic multiple polylogarithms, and we saw that the symbol letters of elliptic multiple polylogarithms are the functions $\\Omega^{(n)}(w,\\tau)$. These functions stand in the way of analyzing the elliptic symbols. For one thing, the relations among $\\Omega^{(n)}$'s are much more complicated than the manipulation rules $\\log a+\\log b=\\log ab$ for the symbol letters of multiple polylogarithms. For another, they depend on the kinematics in a rather indirect way -- their arguments $w$ and $\\tau$ are (ratios) of elliptic integrals involving kinematics. \n\n\nIn this section, we investigate the identities of the elliptic letter $\\Omega^{(n)}(w,\\tau)$. \nThe most trivial identities these letters satisfy are the following: \n\\begin{align}\n    \\text{Parity :}&\\quad \\Omega^{(n)}(-w)=(-1)^{n+1} \\Omega^{(n)}(w) \\:, \n    \\label{parity}\\\\\n    \\text{Quasi periodicity :} &\\quad \\Omega^{(n)}(w+\\tau) = \\sum_{j=0}^{n+1}\\frac{(-1)^{j}}{j!}\\Omega^{(n-j)}(w) \\:. \\label{quasiperiodicity}\n\\end{align}\nThey immediately follow from \\eqref{wdef}.\nOur investigation of more non-trivial identities will be focussed on the cases $n=0,1,2$ since -- for the two examples considered in this paper, namely the sunrise integral and the double-box integral -- the identities among $\\Omega^{(n\\leq 2)}$ are sufficient to simplify the symbols after using \\eqref{parity} and \\eqref{quasiperiodicity}. We comment on a generalization to identities among $\\Omega^{(n>2)}$ at the end of this section.\n\nLet us start with the slightly trivial identity\n\\begin{align} \\label{identity1}\n    \\quad \\log\\frac{c-b}{c-a} &= \\sum_{\\sigma\\in \\pm} \\Bigl(\\Omega^{(1)}(w_{c}^{\\sigma}-w_{b}^{+})-\\Omega^{(1)}(w_{c}^{\\sigma}-w_{a}^{+}) \n     \\nonumber \n     \\\\\n     &\\qquad \\qquad \\qquad \n    -\\Omega^{(1)}(w_{\\infty}^{\\sigma}-w_{b}^{+})+\\Omega^{(1)}(w_{\\infty}^{\\sigma}-w_{a}^{+})\\Bigr)\\,,\n\\end{align}\nwhich is a simple consequence of applying \\eqref{psitog1a} to $\\int_{a}^{b}\\psi_{1}(c,x)\\dif x = \\log\\frac{c-b}{c-a} $. \nThe identity \\eqref{identity1} has two important special cases. One is obtained by taking $a\\to \\infty$, giving\n\\begin{align} \\label{identity2}\n    \\sum_{\\sigma\\in\\pm} \\Omega^{(1)}(w_{c}^{\\sigma}-w_{b}^{+}) \n    &=\\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+})+\\sum_{\\sigma\\in\\pm}\\left[\\Omega^{(1)}(w_{c}^{\\sigma}-w_{\\infty}^{+})+\\Omega^{(1)}(w_{b}^{\\sigma}-w_{\\infty}^{+}) \\right] \\nonumber \\\\ \n    &\\quad - \\log\\biggl(\\frac{2\\pi\\mi}{\\omega_{1}}\\biggr)  -2\\log\\eta(\\tau)+\\log(c-b)\\,;\n\\end{align}\nthe other one is obtained by further taking $b\\to c$, which yields\n\\begin{align} \\label{identity3}\n    \\Omega^{(1)}(w_{c}^{-}{-}w_{c}^{+}) &= 2\\biggl(\\Omega^{(1)}(w_{c}^{-}-w_{\\infty}^{+})+\\Omega^{(1)}(w_{c}^{+}-w_{\\infty}^{+})\n    -2\\log\\eta(\\tau) -\\log\\frac{2\\pi \\mi}{\\omega_{1}}  \\biggr) \\nonumber \\\\ \n    &\\quad  - \\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+}) - \\log y_{c} \\:.\n\\end{align}\nNow one can easily see that the second equality in \\eqref{shuffreg2} is the consequence of applying \\eqref{identity2} and \\eqref{identity3}. The two special cases are particularly useful since the letters $\\Omega^{(1)}$ on their right-hand sides always involve $w^{\\pm}_{\\infty}$ and hence can serve as a basis.\n\n\\subsection{Abel's addition theorem} \\label{sec:3.1}\n\nSurprisingly, a very classical and powerful theorem, Abel's addition theorem \\cite{abel1841}, \nyields other identities for $\\Omega^{(0)}$ and $\\Omega^{(1)}$.\\footnote{See e.g.\\ \\cite{GriffithsHarris} for a textbook treatment of Abel's addition theorem and \\cite{Tarasov:2017yyd,Tarasov:slides} for previous applications of it to Feynman integrals.}\n\nLet us first spell out this theorem: Let $\\mathcal{C}$ and $\\mathcal{C}'$ be curves given by two polynomial equations\n\\begin{align}\n    \\mathcal{C}:& \\quad  F(x,y)=0 \\:, \\\\\n    \\mathcal{C}':&\\quad  Q(x,y)=0\\:,\n\\end{align}\nwhere $\\mathcal{C}$ is viewed as a \\emph{fixed} curve and $\\mathcal{C}'$ as a \\emph{variable} curve with coefficients collectively denoted as $\\{b_{i}\\}$. \nSuppose that these two curves intersect at $n$ points $(x_{1},y_{1})$, ..., $(x_{n},y_{n})$. Let $R(x,y)$ be a rational function defined on $\\mathcal{C}$. Then the following holds.\n\\begin{theorem}[Abel]\n    The integral \n    \\begin{equation}\n        I(\\{b_{i}\\}) =\\sum_{i=1}^{n} \\int_{x_{\\ast}}^{x_{i}}R(x,y)\\:\\dif x\\:,\n    \\end{equation}\n    where $x_{\\ast}$ is an arbitrary base point, contains at most rational functions and logarithms of $\\{b_{i}\\}$. \n\\end{theorem}\n\\noindent This theorem can be proven by showing that $\\partial_{b_{\\nu}}I$ is always a rational function of $\\{b_{i}\\}$ for all $b_{\\nu}$.\n\n\nIf a symbol letter $\\phi(u)$ can be expressed as $\\int^{u} R(x,y)\\dif x$, one can try to find the composition rule of $\\phi(u)$ through Abel's addition theorem.\nOf all applications of this theorem, we are most interested in the cases that $\\mathcal{C}'$ has only \\emph{two} degrees of freedom and intersects $\\mathcal{C}$ at \\emph{three} points. \nFor this case, Abel's addition theorem gives \n\\begin{equation}\n    \\phi(u)+\\phi(v)=\\phi\\bigl(T(u,v)\\bigr)+\\cdots,\n\\end{equation}\nwhere $T(u,v)$ is an algebraic function of $u$ and $v$ and `$\\cdots$' denotes simpler objects, like logarithms. \n\nAn example is the composition rule $\\log(x_1)+\\log(x_2)=\\log(x_1x_2)$ for logarithms,\\footnote{This example can e.g.\\ be found in \\cite{Tarasov:slides}.} which is given by choosing \n\\begin{align}\n    \\mathcal{C}:&\\quad y= \\frac{1}{x} \\:, \\\\\n    \\mathcal{C}': &\\quad y=x^{2}+b_{1}x+b_{2} \\:.\n\\end{align}\nThese two curves intersect at the three points  $x_1,x_2,x_3$ that solve \n\\begin{equation}\n x^3+b_1x^2+b_2x-1=0,\n\\end{equation}\nand hence satisfy $x_1 x_2 x_3=1$.\nNow consider\n \\begin{equation}\n I=\\sum_{i=1}^3\\int_1^{x_i}\\frac{\\dif x}{x},\n  \\end{equation}\n  which satisfies\n\\begin{equation}\n \\partial_{b_j}I=\\sum_{i=1}^3\\frac{1}{x_i}\\frac{\\partial x_i}{\\partial b_j}=\\frac{1}{x_1 x_2 x_3}\\frac{\\partial}{\\partial b_j}x_1x_2x_3=0 ,\n\\end{equation}\nsince $x_1 x_2 x_3=1$. Thus, $I$ is a  constant. To fix this constant, we can pick $b_1=-3,b_2=3$, such that $x_1=x_2=x_3=1$, yielding $I=0$.\nAgain using $x_1 x_2 x_3=1$, we thus have \n\\begin{equation}\n 0=I=\\log(x_1)+\\log(x_2)+\\log(x_3)=\\log(x_1)+\\log(x_2)-\\log(x_1 x_2),\n\\end{equation}\nas claimed.\n\n\n\nFor the case we are most interested in, the fixed curve $\\mathcal{C}$ is given by  \\eqref{quarticEcurve}, and we find that a convenient choice for $\\mathcal{C}'$ is \n\\begin{equation}\n    y=-x^{2}+b_{1}x+b_{2} \\:.\n\\end{equation} \nOne can easily check that these two curves intersects at three point at most.\nSuppose that two intersection points are $(x_{1},y_{1}=\\sqrt{P_{4}(x_{1})})$ and $(x_{2},y_{2}=\\sqrt{P_{4}(x_{2})})$, then\n\\begin{align}\n    b_{1}&=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}+x_{1}+x_{2} \\:, &&\n    b_{2}= \\frac{x_{1}y_{2}-x_{2}y_{1}}{x_{1}-x_{2}}-x_{1}x_{2} \\:, && \\text{for }x_{1}\\neq x_{2} \\:, \\\\\n    b_{1} &= \\frac{P_{4}'(x_{1})}{2 y_{1}}+2x_{1} \\:, && b_{2}=y_{1}+x_{1}^{2}-b_{1}x_{1} \\:, && \\text{for }x_{1}= x_{2} \\:,\\intertext {and}    \n    x_{3} &= \\frac{b_{1}^{2}-2b_{2}-a_{2}}{2b_{1}+a_{3}} - x_{1}-x_{2} \\:, && \n    y_{3} = -\\sqrt{P_{4}(x_{3})} \\:.&&\n\\end{align}\nSince $z_{c}^{+}=\\int_{-\\infty}^{c} \\dif x\/y$, Abel's addition theorem tells us \n\\begin{subequations} \\label{zIdentity}\n    \\begin{align}\n        z_{x_{1}}^{+}+ z^{+}_{x_{2}} &\\equiv z^{+}_{x_{3}} \\operatorname{mod} \\Lambda \\:, && \\text{for }b_{1}\\neq -a_{3}\/2 \\:,  \\\\ \n        z_{x_{1}}^{+}+ z^{+}_{x_{2}} &\\equiv 0 \\operatorname{mod} \\Lambda \\:, && \\text{for }b_{1}= -a_{3}\/2  \\:,\n        \\end{align} \n\\end{subequations}\nwhich is the well-known group law on the elliptic curve.\nFurthermore, if we take $b_{2}=(a_{3}^{2}-4 a_{2})\/8$ aside $b_{1}=-a_{3}\/2$, then $\\mathcal{C}$ and $\\mathcal{C}'$ only intersect at one point,\n\\begin{equation}\n    \\chi=\\frac{a_{3}^{4}-8a_{2}a_{3}^{2}+16a_{2}^{2}-64a_{0}}{8\\bigl(a_{3}^{3}-4a_{2}a_{3}+8a_{1}\\bigr)} \\:.\n\\end{equation}\nTogether with a little divisor theory, \nthis gives\\footnote{For any meromorphic function $F$ on a torus, by using $\\oint  \\dif \\log F(z)=0$ and $\\oint z \\:\\dif \\log F(z)=0$, one can conclude that the number and the sum of its poles are the same as its zeros, where poles and zeros of order $n$ are counted $n$ times. Now consider the function \n\\[ \n    F=-\\kappa'(z)-\\kappa(z)^{2}-a_{3}\\kappa(z)\/2+(a_{3}^{2}-4a_{2})\/8\\:,\n\\]\nwhich has poles at lattice points but vanishes only at $z_{\\chi}^{-}$ and $z_{\\infty}^{-}$, two intersection points of the curve $ y=-x^{2}-a_{3}x\/2+(a_{3}^{2}-4a_{2})\/8$ and the elliptic curve. We then obtain \\eqref{minfid} by using \\eqref{wpwmrelation}.}\n\\begin{equation}\n    2z_{\\infty}^{-} \\equiv \\omega_{1}+z_{\\chi}^{+} \\operatorname{mod} \\omega_{2} \\:. \\label{minfid}\n\\end{equation}\n\nSimilarly, for the integral $\\int \\psi_{-1}(c,x)\\dif x$, the same procedure gives \n\\begin{align} \\label{eq1}\n    \\int_{\\ast}^{x_{1}}\\frac{y_{c}\\,\\dif x}{y(x-c)}+\\int_{\\ast}^{x_{2}}\\frac{y_{c}\\,\\dif x}{y(x-c)}-\\int_{\\ast}^{x_{3}}\\frac{y_{c}\\,\\dif x}{y(x-c)} &= \\log\\frac{c^{2}-b_{1} c-b_{2}+y_{c}}{c^{2}-b_{1}c-b_{2}-y_{c}} + \\text{const.} \n\\end{align}\nIf $z_{x_{1}}^\\pm+z_{x_{2}}^\\pm=z_{x_{3}}^\\pm$, applying \\eqref{psitog1b} to \\eqref{eq1} gives%\n\\begin{align}\n    &\\quad \\sum_{i=1}^{2}\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{-}) \\label{Abel1}\\\\\n    &=\\Omega^{(1)}(w_{c}^{+}{-}w_{x_{3}}^{+})-\\Omega^{(1)}(w_{c}^{+}{-}w_{x_{3}}^{-}) \n   +\\Omega^{(1)}(w_{c}^{+})-\\Omega^{(1)}(w_{c}^{-})+\\log\\frac{c^{2}-b_{1}c-b_{2}+y_{c}}{c^{2}-b_{1}c-b_{2}-y_{c}}. \\nonumber \n\\end{align}\nIf $z_{x_{1}}^\\pm+z_{x_{2}}^\\pm\\equiv z_{x_{3}}^\\pm \\mod \\Lambda$, a corresponding identity can be found from \\eqref{Abel1} using the quasi double periodicity of $\\Omega^{(1)}$ \\eqref{quasiperiodicity}.\n\nThree boundary cases of \\eqref{Abel1} require special care: \\\\\n(i) taking $c\\to\\infty$ gives\n\\begin{align}\n    \\sum_{i=1}^{2}\\Omega^{(1)}(w_{x_{i}}^{+})-\\Omega^{(1)}(w_{x_{i}}^{-})\n    &=\\Omega^{(1)}(w_{x_{3}}^{+})-\\Omega^{(1)}(w_{x_{3}}^{-})  \\nonumber \\\\\n    &\\quad-\\Omega^{(1)}(\\omega_{\\infty}^{-}-\\omega_{\\infty}^{+})-\\log\\frac{2b_{1}+a_{3}}{4}+\\frac{1}{12}\\log\\Delta \\:, \\label{Abel2} \n\\end{align}\n(ii) taking $x_{3}\\to \\infty$ gives \n\\begin{align}\n    & \\quad \\sum_{i=1}^{2}\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{-}) \\label{Abel3} \\\\\n    &=\\Omega^{(1)}(w_{c}^{+}{-}w_{\\infty}^{+})-\\Omega^{(1)}(w_{c}^{+}{-}w_{\\infty}^{-})  \n    +\\Omega^{(1)}(w_{c}^{+})-\\Omega^{(1)}(w_{c}^{-})+\\log\\frac{c^{2}+a_{3}c\/2-b_{2}+y_{c}}{c^{2}+a_{3}c\/2-b_{2}-y_{c}} \\:.  \\nonumber \n\\end{align} \n(iii) taking $c\\to\\infty$ and $x_{3}\\to \\infty$ gives \n\\begin{align}\n    \\sum_{i=1}^{2}\\left[\\Omega^{(1)}(w_{\\infty}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{\\infty}^{+}-w_{x_{i}}^{-})\n   \\right] &=-2\\Omega^{(1)}(w_{\\infty}^{+}-w_{\\infty}^{-})+\\Omega^{(0)}(w_{\\infty}^{-}-w_{\\infty}^{+})  \\nonumber \\\\\n    &\\quad+\\frac{1}{6}\\log\\Delta-\\log\\frac{4a_{2}-a_{3}^{2}+8b_{2}}{16} \\:. \\label{Abel4}\n\\end{align} \n\nEqs.\\ \\eqref{zIdentity}, \\eqref{identity1}--\\eqref{identity3} as well as \\eqref{Abel1}--\\eqref{Abel4} explain the subset of the identities numerically found in \\cite{Kristensson:2021ani} which only involve $\\Omega^{(0)}$'s and $\\Omega^{(1)}$'s.\n\nNote that the identities we presented in this subsection can be equivalently formulated in terms of divisor theory, see e.g.\\ \\cite{Broedel:2019tlz}.\n\n\\subsection{Elliptic Bloch relation and the symbol prime}\n\\label{subsec: symbol prime}\n\nIn \\cite{Kristensson:2021ani}, also five identities involving $\\Omega^{(2)}$'s were observed which are much lengthier than the other identities; each of these five identities contains at least 100 terms in the form that they were found. It turns out all these identities are consequences of the so-called elliptic Bloch relation \\cite{bloch2011higher,Zagier2000},  \nan elliptic generalization of the five-term identity for dilogarithms,\n\\begin{equation}\n\\label{eq: five term identity}\n    D(x)+D(y)+D\\biggl(\\frac{1-x}{1-xy}\\biggr)+D(1-xy)+D\\biggl(\\frac{1-y}{1-xy}\\biggr)=0 \\:,\n\\end{equation}\nwhere $D(z)=\\Im(\\Li_{2}(z))+\\arg(1-z)\\log |z|$ is the Bloch-Wigner function.%\n\\footnote{To show this concretely, one  would need to do the divisor-theory analog of finding a curve that intersects the elliptic curve at precisely the points given by the more than 100 terms in the identities.\nAn algorithm for doing this is given in \\cite{Bolbachan:2019dsu}.}\n\nIn practice, it is difficult to simplify even expressions containing dilogarithms by using the above five-term identity directly. Instead, we introduce the symbol map \\cite{Goncharov:2010jf} for polylogarithms as an assistance; we associate to each polylogarithm a tensor product -- the so-called symbol -- whose entries satisfy simpler identities. We then exploit that the symbol of a combination of polylogarithms vanishes if that combination of polylogarithms vanishes. \n\n\n\nA similar strategy can be used for the elliptic letters $\\Omega^{(2)}(w)=(2\\pi \\mi)^{-1}\\gamt{2}{0}{w}$, although they already serve as entries of the symbol for elliptic multiple polylogarithms. Inspired by the proof of the elliptic Bloch relation for $\\gamt{2}{0}{w}$ in \\cite{Broedel:2019tlz}, we associate to $\\Omega^{(2)}(w)$ a rank-two tensor through the \\emph{symbol prime} map,\n\\begin{equation} \\label{symbolp1}\n    \\mathcal{S}'\\bigl(\\Omega^{(2)}(w)\\bigr) = \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w).\n\\end{equation}  \nThis map has a property similar to that of the symbol map:\n\\begin{equation}\n\\label{eq: symbol prime property}\n    \\sum_{j}c_j \\Omega^{(2)}(w_{j})=0  \\quad \n    \\text{``$\\Rightarrow$''}\n    \\quad\n    \\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))\\equiv\n     \\sum_{j}c_j\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w_{j})=0\n\\end{equation}\nfor some rational coefficients $c_j$.\nTo show this, consider the sum $\\sum_{j}c_j\\gamt{1&0}{0&0}{w_{j}}$.\nAccording to \\eqref{devoftG}, \n\\begin{align}\n    \\label{dev_tG2}\n\\mathcal{S}(2\\pi i \\gamt{1&0}{0&0}{w})= \\Omega^{(0)}(w) \\otimes \\Omega^{(1)}(w)- \\Omega^{(2)}(w) \\otimes (2\\pi i\\tau)\\:,\n\\end{align}\nwhere we used that $\\gamt{0}{0}{w}=w$, $\\gamt{2}{0}{w}=2\\pi \\mi\\Omega^{(2)}(w)$ and $\\Omega^{(-1)}=-2\\pi i \\tau$.\nIf the arguments $w_j$ and coefficients $c_j$ are such that $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ due to an elliptic Bloch relation, $\\sum_{j}c_j\\gamt{1&0}{0&0}{w_{j}}=0$ according to an analogous elliptic Bloch relation \\cite{Broedel:2019tlz}, which in turn implies that the second term in \\eqref{dev_tG2} drops out in the sum, i.e.\\ $\\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))$.\nIn this sense, the symbol prime makes the elliptic Bloch relations manifest.\n\nNote that we have assumed that $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ vanishes \\emph{due to an elliptic Bloch relation} here in order to show that $\\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))=0$. We have indicated this in \\eqref{eq: symbol prime property} as ``$\\Rightarrow$''.\nHowever, we currently have no way of proving that all identities $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ are due to an elliptic Bloch relation \\cite{bloch2011higher,Zagier2000}.\nThis is similar to the case of dilogarithms, where \\eqref{eq: five term identity} is only conjectured but not proven to generate all functional identities among dilogarithms.%\n\nThe symbol map itself has a kernel, and the same is true for the symbol prime.\nIf $\\mathcal{S}'(\\sum_j \\Omega^{(2)}(w_{j}))=0$, the first term in \\eqref{dev_tG2} drops out in the sum. This implies that $\\sum_{j}\\gamt{1&0}{0&0}{w_{j}}$ and thus $\\sum_j \\Omega^{(2)}(w_{j})$ is a function \\emph{only} of $\\tau$. However, not all functions only of $\\tau$ are in the kernel of the symbol prime; for example, $\\Omega^{(2)}(\\tau\/n)$ with some positive integer $n$ only depend on $\\tau$ but has a non-vanishing symbol prime.%\n\\footnote{However, the appearance of such a letter means that the kinematic image $\\kappa(\\omega_1\\tau\/n)$ relates to the physical problem and hence is algebraic in general. This is not the case for the examples of the unequal-mass sunrise integral and the ten-point double-box integral studied in sections \\ref{sec: example 1}--\\ref{sec: example 2}, but it is the case for the \\emph{equal}-mass sunrise integral.}\n\n\nOne can find the action of the symbol prime map on the letters $\\Omega^{(n<2)}$ by expressing them in terms of $\\Omega^{(2)}$ using the quasi periodicity \\eqref{quasiperiodicity} of $\\Omega^{(n)}$:\n\\begin{align}\n    \\Omega^{(1)}(w) &= \\tfrac{1}{6}\\Omega^{(2)}(w+2\\tau)-\\Omega^{(2)}(w+\\tau)+\\tfrac{1}{2}\\Omega^{(2)}(w)\n    +\\tfrac{1}{3}\\Omega^{(2)}(w-\\tau) \\:, \\\\\n    \\Omega^{(0)}(w)&=\\Omega^{(2)}(w+\\tau)+\\Omega^{(2)}(w-\\tau)-2\\Omega^{(2)}(w) \\:, \\\\\n    \\Omega^{(-1)}&=-\\Omega^{(2)}(w+2\\tau)+3\\Omega^{(2)}(w+\\tau)-3\\Omega^{(2)}(w)+\\Omega^{(2)}(w-\\tau)\n    \\:.\n\\end{align}\nThis yields \n\\begin{subequations} \\label{symbolp2}\n    \\begin{align} \n        \\mathcal{S}'\\bigl(\\Omega^{(1)}(w)\\bigr)&= \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(0)}(w)\n        +\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w) \\:, \\\\\n        \\mathcal{S}'\\bigl(\\Omega^{(0)}(w)\\bigr) &= \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(-1)}\n         + 2 \\Omega^{(-1)} \\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(0)}(w) \\:, \\\\\n         \\mathcal{S}'\\bigl(\\Omega^{(-1)}\\bigr)  &= 3\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(-1)} \\:,\n    \\end{align}    \n\\end{subequations}\nwhere we have moreover used quasi periodicity to simplify the entries of the symbol prime.\nIn particular, by expressing a logarithm in terms of $\\Omega^{(1)}$'s and $\\Omega^{(0)}$'s either through \\eqref{identity1} or \\eqref{Abel1}, one finds\n\\begin{equation} \\label{symbolp3}\n    \\mathcal{S}'\\bigl(\\log c\\bigr) = \\Omega^{(-1)} \\otimes'}%{\\otimes_{\\mathrm{p}} \\log c\\:.\n\\end{equation}\nThus, for a combination of $\\Omega^{(n\\leq 2)}$'s and logarithms, one can compute its symbol prime. \nIt involves only $\\Omega^{(n\\leq1)}$ and can thus be simplified using the techniques discussed in subsection \\ref{sec:3.1}. If the symbol prime is not zero, one may search for a simpler combination of $\\Omega^{(n\\leq 2)}$'s and logarithms with the same symbol prime according to \\eqref{symbolp1}, \\eqref{symbolp2} and \\eqref{symbolp3}.%\n\\footnote{In particular, if the first entry of the symbol prime is only $\\tau$, then the function is the sum of logarithms and a function of $\\tau$.} The difference of these two combinations has to be a function of $\\tau$ only, which can be fixed by sending the independent $w$-variables to any values, say $0$. In this way, we have proven the five identities involving $\\Omega^{(2)}$'s found in \\cite{Kristensson:2021ani}.\n\n\nThe current definitions for $\\Omega^{(n\\leq 2)}$ are sufficient for the two examples treated in this paper. \nFor $n>2$, one might similarly define the symbol prime for $\\Omega^{(n)}$'s as \n\\begin{equation}\n    \\mathcal{S}^{(n-1)}\\bigl(\\Omega^{(n)}(w)\\bigr) = \\frac{1}{n-1}\\Omega^{(0)}(w)\\otimes^{(n-1)} \\Omega^{(n-1)}(w),\n\\end{equation}\ndue to the fact that,\n\\begin{equation}\n\\mathcal{S}\\bigl((2\\pi i)^{2-n} \\gamt{n-1&0}{0&0}{w}\\bigr)=\\Omega^{(0)}(w) \\otimes \\Omega^{(n-1)}(w)- (n-1)\\bigl( \\Omega^{(n)}(w)-\\Omega^{(n)}(0)\\bigr)\\otimes (2\\pi i\\tau)\\:,\n\\end{equation}\nwhere $\\Omega^{(n)}(0)$ is either zero or a function only depending on $\\tau$ for even or odd $n$, respectively.\nWith the knowledge of the identities among $\\Omega^{(n-1)}$, we can then find identities among $\\Omega^{(n)}$ recursively. \nWe leave the exploration of the symbol prime for $\\Omega^{(n>2)}$ to future work.\n\n\\section{Example I: Unequal-Mass Sunrise Integral}\n\\label{sec: example 1}\n\nTwo particularly interesting cases of elliptic Feynman integrals are the unequal-mass sunrise integral in two dimensions and the double-box integral in four dimensions. We will investigate these two integrals through the tools developed so far. The main focus will be on the sunrise integral treated in this section, since this integral is simple enough such that the main results can be written within a couple of lines. After applying the symbol prime map, we will see that several properties, such as double periodic invariance, modular invariance (covariance) and part of  integrability are manifest. \n\n\n\nWe calculate the unequal-mass sunrise integral in terms of elliptic multiple polylogarithms $\\mathcal{E}_{4}$ in appendix \\ref{app:sunrise}.\nThis integral was originally calculated in terms of iterated integrals on the moduli space $\\overline{\\mathcal{M}}_{1,3}$  in \\cite{Bogner:2019lfa}. \nWe closely follow the Feynman-parameter approach of \\cite{Broedel:2017siw} for the equal-mass case.\n\nThe resulting expression when normalizing the torus by the period $\\omega_1$ is \n\\begin{equation}\n\\label{eq: sunrise general}\n    I_{\\sr} = \\frac{\\omega_{1}}{2\\pi\\mi m_{1}^{2}} (2\\pi \\mi T^{(1)}_{\\sr})\n    \\,,\n\\end{equation}\nwhere the periods were defined in figure \\ref{fig: contours} and $T_{\\sr}^{(1)}$ is a pure combination of elliptic multiple polylogarithms of weight one and length two,\n\\begin{align}\n    T_{\\sr}^{(1)} &= \\cEf{0&-1}{0&-1}{\\infty|\\tau}-\\cEf{0&-1}{0&0}{\\infty|\\tau} + \\cEf{0&-1}{0&r}{\\infty|\\tau} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau} \\nonumber  \\\\\n    &\\quad +4\\pi \\mi\\cEf{0&0}{0&0}{\\infty|\\tau}  -\\cEf{0}{0}{\\infty|\\tau} \\log\\frac{t^{2}_{2}}{t_{3}^{2}} \\:, \\label{eq:sunrise normalization 1}\n    \\end{align}\nwhere we introduced $t_{i}^{2}=m_{i}^{2}\/p^{2}$ and $r=-t_{3}^{2}\/t^{2}_{1}$. \nNote that we have included seemingly redundant factors of $(2\\pi i)$ in the numerator and denominator of \\eqref{eq: sunrise general} that ensure that the prefactor degenerates to an algebraic function in the limit where the elliptic curve degenerates, and the term in parentheses degenerates to a pure logarithm of transcendental weight two; see subsection \\ref{sec:4.1}. \n\nHowever, we can also normalize the torus by the period $\\omega_2$, finding\n\\begin{equation}\n\\label{eq: sunrise general 2}\n    I_{\\sr}     = \\frac{-\\omega_{2}}{2\\pi\\mi m_{1}^{2}} (2\\pi \\mi T^{(2)}_{\\sr})\\,,\n\\end{equation}\nwith\n\\begin{align}\n    T_{\\sr}^{(2)} &= \\cEf{0&-1}{0&-1}{\\infty|\\tau'}-\\cEf{0&-1}{0&0}{\\infty|\\tau'} + \\cEf{0&-1}{0&r}{\\infty|\\tau'} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau'} \\nonumber \\\\ \n    &\\quad -\\cEf{0}{0}{\\infty|\\tau'} \\log\\frac{t^{2}_{2}}{t_{3}^{2}} \\:,\\label{eq:sunrise normalization 2}\n\\end{align}\n(Recall that $\\tau'=-\\omega_1\/\\omega_2$.)\n\nAccording to \\eqref{eq: sunrise general} and \\eqref{eq: sunrise general 2}, the values of $T_{\\sr}^{(1)}$ and $T_{\\sr}^{(2)}$ are related by $T_{\\sr}^{(1)}= -\\tau T_{\\sr}^{(2)}$, but this relation is not obvious from their expressions in terms of eMPLs. \nIn general, eMPLs transform non-trivially under the modular $S$-transformation $\\tau\\to \\tau'=-1\/\\tau$; for example,\n\\begin{equation}\n\\cEf{-1}{c}{x|\\tau}=    \\cEf{-1}{c}{x|\\tau'}-\\frac{2\\pi \\mi(\\xi_{c}^{+}-\\xi^{-}_{c})}{\\tau'}  \\cEf{0}{0}{x|\\tau'} \\:.\n\\end{equation}\nSee \\cite{Duhr:2019rrs} for the cases of iterated integrals of modular forms. The same is true for the symbol, as we will see soon. \nHowever, we will see that the application of the symbol prime map makes the behavior under the modular $S$-transformation manifest.\n\n\\subsection{Symbol of the sunrise integral} \\label{sec:4.1}\n\nThe symbol of $ T_{\\sr}^{(1,2)}$ can be calculated by first rewriting $\\mathcal{E}_4$'s in terms of $\\tilde\\Gamma$'s via \\eqref{Psikernels} and then applying \\eqref{devoftG}--\\eqref{eq: elliptic symbol}.\nFor example,\n\\begin{equation}\n \\cEf{0&-1}{0&c}{x}= \\gamt{0&1}{0&w_{c}^{+}-w_{0}^{+}}{w_{x}^{+}-w_{0}^{+}}-  \\gamt{0&1}{0&w_{c}^{-}-w_{0}^{+}}{w_{x}^{+}-w_{0}^{+}}\n\\end{equation}\nand \n\\begin{align}\n \\mathcal{S}\\bigl(2\\pi i\\gamt{0&1}{0&w_{1}}{w_{2}}\\bigr)&= \\Bigl(\\Omega^{(2)}(-w_{1})-\\Omega^{(2)}(w_{2}-w_{1})\\Bigr)\\otimes \\Omega^{(-1)}\n -\\Omega^{(0)}(w_{2})\\otimes \\Omega^{(1)}(w_{1}) \\nonumber\\\\ \n &\\quad + \\Bigl(\\Omega^{(1)}(w_{2}-w_{1})-\\Omega^{(1)}(-w_{1})\\Bigr)\\otimes \\Omega^{(0)}(w_{2}-w_{1}) \\,.\n\\end{align}\n\n\nThe simplification of the symbols in this case \nis slightly non-trivial: it involves some non-trivial relations of $\\Omega^{(1)}$'s, $\\Omega^{(0)}$'s and logarithms as described in \nsection \\ref{sec:3.1}; for instance, \n\\begin{align}\n    \\log \\frac{t_{1}}{t_{3}} &= \\Omega^{(1)}(w_{-1}^{+}-w_{0}^{+})-\\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{+}) \n    +\\Omega^{(1)}(w_{-1}^{+}-w_{0}^{-})-\\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{-}) \\:,  \\\\\n    \\log \\frac{t_{2}}{t_{3}} &= \\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{+}) - \\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{+}) \n    +\\Omega^{(1)}(w_{-1}^{+}-w_{0}^{-})-\\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+}) \\:.\n\\end{align}\n(Recall that $w_{c}^{+}=\\omega_{1}^{-1}\\int_{-\\infty}^{c} \\dif x\/y$). \nAll the relations involving $\\Omega^{(2)}$ in this case are relatively trivial; they are the consequences of \\eqref{quasiperiodicity}. \nWe present the full simplification in the attached file \\texttt{sunrise\\_symbol.nb}.\n\nThe final result is\n\\begin{align}\n    \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(1)} \\bigr) &= \\log \\frac{t_{2}^{2}}{t_{1}^{2}} \\otimes \\Omega^{(0)}(w_{0}^{+})\n    + \\log \\frac{t_{1}^{2}}{t_{3}^{2}}\\otimes \\Omega^{(0)}(w_{-1}^{+}) \\nonumber \\\\\n    &\\quad+\\biggl[\\frac{1}{2\\pi\\mi} \\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr)+\\log\\frac{t_{3}^{2}}{t_{2}^{2}} \\biggr]\\otimes (2\\pi\\mi\\tau)\\:. \\label{symbolw1} \n\\end{align}\nwhere we have moreover used \n\\begin{align}\n\\label{eq:Ecal-gammat-relation}\n \\frac{\\cEf{-n}{c}{\\infty}}{(2\\pi\\mi)^{n-1}}=\\Omega^{(n)}(w_{\\infty}^{+}-w_{c}^{+})-\\Omega^{(n)}(w_{0}^{+}-w_{c}^{+})\n -\\Omega^{(n)}(w_{\\infty}^{+}-w_{c}^{-})+\\Omega^{(n)}(w_{0}^{+}-w_{c}^{-}) \\:.\n\\end{align}\nSimilarly, \n \\begin{align}\n    \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(2)} \\bigr) &= \\log \\frac{t_{2}^{2}}{t_{1}^{2}} \\otimes \\Omega^{(0)}(\\xi_{0}^{+})\n    + \\log \\frac{t_{1}^{2}}{t_{3}^{2}}\\otimes \\Omega^{(0)}(\\xi_{-1}^{+}) \\nonumber \\\\\n    &\\quad+ \\biggl[\\frac{1}{2\\pi \\mi}\\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr)\\nonumber \\\\\n    &\\qquad \\qquad +\\log\\frac{t_{1}}{t_{3}} +\n    \\Omega^{(1)}(\\xi_{\\infty}^{+}-\\xi_{-1}^{+})-\\Omega^{(1)}(\\xi_{-1}^{+}-\\xi_{0}^{+})\\biggr] \\otimes (2\\pi\\mi\\tau^{\\prime}) \\:, \\label{symbolw2}\n\\end{align}\nwhere we used \\eqref{eq:Ecal-gammat-relation} in terms of $\\xi$-coordinates. \n\nAt this point, the symbols of the sunrise integral partially show some desired properties; for example, the first entries of the first two terms in \\eqref{symbolw1} and \\eqref{symbolw2} indicate the physical first-entry conditions known from the massless case, and their last entries are related by simple $S$-transformations $w\\to\\xi$.\nHowever, the first two terms on their own are neither double periodic nor integrable.\n\nThe first entries of the last terms in \\eqref{symbolw1} and \\eqref{symbolw2}, i.e., $\\partial_\\tau T_{\\sr}^{(1)}$ and $\\partial_{\\tau'} T_{\\sr}^{(2)}$,\nare relatively complicated and the main obstacles to understanding the entire symbols, since it is hard to see how they render the whole symbol double periodic and integrable. In this respect, it is instructive to consider the symbol primes of these entries:\n\\begin{align}\n    \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &=   \\Omega^{(0)}(w_{0}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}+\\Omega^{(0)}(w_{-1}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}} \\:, \\label{spofsr1}\\\\\n    \\mathcal{S'}\\bigl(\\partial_{\\tau'} T_{\\sr}^{(2)} \\bigr) &=\\Omega^{(0)}(\\xi_{0}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}+\\Omega^{(0)}(\\xi_{-1}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}} \\:.\\label{spofsr2}\n\\end{align}\nThey have the following advantageous properties:\n\\begin{enumerate}\n    \\item It is obvious that $\\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)} \\bigr)$ differs from $\\mathcal{S'}\\bigl(\\partial_{\\tau'} T_{\\sr}^{(2)} \\bigr) $ only by a modular $S$-transformation $w\\to\\xi$.\n    \\item If we shift $w_{-1}^{+}$ or $w_{0}^{+}$ by $\\tau$, then $\\partial_\\tau T_{\\sr}^{(1)}$ changes by $\\log\\frac{t_{3}^2}{t_{1}^{2}} $ or  $\\log\\frac{t_{1}^2}{t_{2}^{2}}$, respectively, (and similarly for $\\partial_{\\tau'} T_{\\sr}^{(2)}$) since\n        \\begin{align}\n            \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)}|_{w_{-1}^{+}\\to w_{-1}^{+}+\\tau}-\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &=   (2\\pi\\mi\\tau)\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}}=\\mathcal{S'}\\biggl(-\\log\\frac{t_{1}^2}{t_{3}^{2}}\\biggr) \\:, \\\\\n            \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)}|_{w_{0}^{+}\\to w_{0}^{+}+\\tau}-\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &=   (2\\pi\\mi\\tau)\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}=\\mathcal{S'}\\biggl(-\\log\\frac{t_{2}^2}{t_{1}^{2}}\\biggr) \\:.\n        \\end{align}\n        The first two terms in the symbol change by corresponding terms with opposite sign that cancel these.  Thus, $\\mathcal{S}(2\\pi i T_{\\sr}^{(1,2)})$ are \\emph{double periodic}.\n        \\item Moreover, the symbol prime also makes integrability with respect to $\\tau$ manifest. This is slightly trivial in the case of the length-two sunrise integral, and will thus be discussed in full generality for the case of the double-box integral in section \\ref{sec:5.1}.\n\\end{enumerate}\n\nFinally, note that the equal-mass case can be obtained smoothly by taking $t_1=t_2=t_3$; we will briefly comment on this case in section \\ref{sec:5}. \n\n\n\\subsection{Degeneration at \\texorpdfstring{$p^{2}=0$}{p**2=0} and pseudo-thresholds}\n\nNext, let us see how the symbol of the unequal-mass sunrise integral behaves in kinematic limits where the elliptic curve degenerates.\n\nThe kinematic configurations where the elliptic curve degenerates can be easily read off from the discriminant\n\\begin{equation} \\label{Discriminant}\n    \\Delta_{\\sr}=\\frac{t_{2}^{4}t_{3}^{4}}{t_{1}^{20}}\\Bigl((t_{1}+t_{2}+t_{3})^{2}-1\\Bigr)\\Bigl((t_{1}+t_{2}-t_{3})^{2}-1\\Bigr)\\Bigl((t_{1}-t_{2}+t_{3})^{2}-1\\Bigr)\\Bigl((-t_{1}+t_{2}+t_{3})^{2}-1\\Bigr) \\:,\n\\end{equation}\nwhere $t_i^2=m_i^2\/p^2$ as before. \nIn particular, the sunrise integral remains finite at $p^{2}=0$, at the pseudo-thresholds $p^{2}=\\{(m_{1}+m_{2}-m_{3})^{2},(m_{1}+m_{3}-m_{2})^{2},(m_{2}+m_{3}-m_{1})^{2}\\}$ and at the threshold $p^{2}=(m_{1}+m_{2}+m_{3})^{2}$, while it diverges for $m_i=0$.\nThe values at $p^{2}=0$ and the pseudo-thresholds were given in terms of MPLs in \\cite{Bloch:2013tra}.  \nIn what follows, we will show how the symbols $\\mathcal{S}(2\\pi iT_{\\sr}^{(1,2)})$ reproduce the corresponding symbols in these two limits.%\n\\footnote{The threshold can be treated in a similar way.}\n\n\n\n\n\\begin{figure} \n      \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n        \\begin{tikzpicture}[scale=1.5]\n         \n                    \\draw[->, thick] (0,0) to (0,3);\n                    \\draw[->, thick] (-3,1.5) to (1,1.5);\n                     \\draw[<-,line width=0.23mm,red] (-1.0,0.2) to (-1.0,2.8);\n                    \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n                    \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n                    \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n                    \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n                     \\draw[->] (-1.65,2.0) to (-1.1,2.1);\n                     \\draw[->] (-1.65,1.0) to (-1.1,0.9);\n                     \\draw[->] (-0.35,0.8) to (-0.9,0.87);\n                     \\draw[->] (-0.35,2.2) to (-0.9,2.13);\n                    \\node at (-1.1,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n                    \\node at (0.4,3) {$\\Im x$};\n                    \\node at (1.3,1.5) {$\\Re x$};\n            \\end{tikzpicture}\n         \\caption{\\phantom{.}}\n         \\label{Fig:nullmomentumlimit}\n     \\end{subfigure}\n     \\hfill\n      \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n            \\begin{tikzpicture}[scale=1.5]\n             \n                        \\draw[->, thick] (0,0) to (0,3);\n                        \\draw[->, thick] (-3,1.5) to (1,1.5);\n                       \\draw[<-,line width=0.23mm,red] (-1.0,0.2) to (-1.0,2.8);\n                        \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n                        \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n                        \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n                        \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n                         \\draw[->] (-1.7,1.95) to (-1.7,1.6);\n                         \\draw[->] (-1.7,1.05) to (-1.7,1.4);\n             \\node at (-1.1,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n                        \\node at (0.4,3) {$\\Im x$};\n                        \\node at (1.3,1.5) {$\\Re x$};\n                        \\node[cross] at (-1.7, 1.5) {};\n                        \\node[cross,label=above:$-1$] at (-1.3, 1.5) {};\n                \\end{tikzpicture}\n         \\caption{\\phantom{.}}\n         \\label{Fig:pseudothreshold}\n     \\end{subfigure}\n    \\caption{The roots of $y^{2}(x)$ for the sunrise integral coincide in the null-momentum limit (\\subref{Fig:nullmomentumlimit}) and pseudo-thresholds (\\subref{Fig:pseudothreshold}). In the latter case, the position of the coinciding roots relative to $-1$ is shown for the case $t_3>t_1$.}\n    \\label{fig: sr_degeneration}\n\\end{figure}\n\n\\paragraph{Null-momentum limit} As $p^{2}\\to 0$, $t_{1}$, $t_{2}$ and $t_{3}$ approach infinity while their ratios remain finite. The elliptic curve degenerates in a way that $r_{1} \\to r_{3}$ and $r_{2}\\to r_{4}$; cf.\\ figure \\ref{Fig:nullmomentumlimit}.\nIn this case, $\\omega_{1}\\to \\infty$ since the roots pinch the corresponding integration contour $\\gamma_1$, while  \n    \\begin{equation}\n      \\omega_{2} \\to \\int_{-\\infty}^{\\infty} \\frac{\\dif x}{x^{2}+\\bigl(1+(t_{3}\/t_{1})^{2}-(t_{2}\/t_{1})^{2}\\bigr)x+(t_{3}\/t_{1})^{2}}\n        = \\frac{2\\pi \\mi}{\\sqrt{\\Bigl(1-\\frac{t_{2}^{2}}{t_{1}^{2}}-\\frac{t_{3}^{2}}{t_{1}^{2}}\\Bigr)^{2}-4\\frac{t_{2}^{2}t_{3}^{2}}{t_{1}^{4}}}} \\:.\n    \\end{equation}\n    Then $-(2\\pi \\mi)^{-1}m_{1}^{-2}\\omega_{2}$ reproduces the same normalization factor as in \\cite{Bloch:2013tra} (up to a sign).\n        Thus, we should expect that $\\mathcal{S}(2\\pi iT_{\\sr}^{(2)})$ reduces to the corresponding symbol.\n        In this limit,  $q=\\exp(2\\pi \\mi \\tau')$ vanishes, and hence all $\\cEf{-2}{c}{x}$ in \\eqref{symbolw1} vanish; cf.\\ \\eqref{eq:Ecal-gammat-relation} and \\eqref{wdef}. Furthermore, \n    \\begin{align}\n        &\\Omega^{(0)}(\\xi_{c}^{+} )\\to \\log \\frac{2c+1-u+v+\\sqrt{(1-u-v)^{2}-4uv}}{2c+1-u+v-\\sqrt{(1-u-v)^{2}-4uv}} \\:, \\\\\n        &\\log\\frac{t_{1}}{t_{3}} +\n    \\Omega^{(1)}(\\xi_{\\infty}^{+}-\\xi_{-1}^{+})-\\Omega^{(1)}(\\xi_{-1}^{+}-\\xi_{0}^{+}) \n    \\to 0\\,,\n    \\end{align}\n    where we have introduced $u = (t_{2}\/t_{1})^{2}=z\\bar{z}$ and $v=(t_{3}\/t_{1})^{2}=(1-z)(1-\\bar{z})$. \n    Then,\n\\begin{align}\n    \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(2)} \\bigr) \\to \\log u \\otimes \\log\\frac{1-\\bar{z}}{1-z}-\\log v \\otimes \\log \\frac{\\bar{z}}{z} \\:.\n\\end{align}\nwhich is the symbol of $-4i$ times the Bloch-Wigner dilogarithm $D(z)$, in perfect agreement with \\cite{Bloch:2013tra}.\n\n\n    \\paragraph{Pseudo-thresholds} Without loss of generality, we consider the pseudo-threshold $p^{2}=(m_{1}+m_{2}-m_{3})^{2}$. In terms of $t_{i}$, this pseudo-threshold is equal to the condition $(t_{1}+t_{2}-t_{3})^{2}=1$. We only consider the solution $t_{3}=t_{1}+t_{2}-1$ since the treatment for the other solution is similar. At $t_{3}=t_{1}+t_{2}-1$, the roots $r_{1}$ and $r_{4}$ pinch the real axis; cf.\\ figure \\ref{Fig:pseudothreshold}. Thus $\\omega_2$ diverges and we should consider the normalization of the torus by $\\omega_1$, $T_{\\sr}^{(1)}$. We can then close the contour $\\gamma_{1}$ with a large semi-circle in the left half-plane and evaluate the integral via residues:\n    \\begin{align}\n        \\int_{\\gamma_{1}}\\frac{t_{1}\\dif x}{\\Bigl(x+\\frac{t_{3}}{t_{1}}\\Bigr)\\sqrt{\\Bigl(t_{1}^{2}x^{2}+\\bigl(t_{1}^{2}+t_{3}^{2}-(t_{2}+1)^{2}\\bigr)x+t_{3}^{2}\\Bigr)}} =-2\\pi \\mi\\sqrt{\\frac{t_{1}^{3}}{4t_{2}t_{3}}} \\:.\n    \\end{align}\n    Again $(2\\pi\\mi)^{-1}m_{1}^{-2}\\omega_{1}$ reproduces the same normalization factor as in \\cite{Bloch:2013tra} at $p^{2}=(m_{1}+m_{2}-m_{3})^{2}$. Thus, we should expect that $\\mathcal{S}(2\\pi i T_{\\sr}^{(1)})$ reduces to the corresponding symbol. \n    At the pseudo-threshold, $q=\\exp(2\\pi \\mi \\tau)$ vanishes. Furthermore, we assume $t_{3}>t_{1}$;\\footnote{The case $t_{1}<t_{3}$ can be obtained by an analytic continuation.} then all $\\cEf{-2}{c}{x}$ appearing in \\eqref{symbolw1} also vanish in this limit. \n    Each of the three last entries in \\eqref{symbolw1} is divergent in this limit since the roots pinch the integration contour, but all these divergences cancel in the following combinations \n    \\begin{align}\n        \\Omega^{(0)}(w_{\\infty}^{+}-w_{-1}^{+}) &=2\\pi i\\int_{-1}^{\\infty} \\frac{\\dif x}{y}\\to \\log\\frac{1+\\sqrt{\\frac{t_{3}}{{t_{1}t_{2}}}}}{1-\\sqrt{\\frac{t_{3}}{{t_{1}t_{2}}}}} \\:, \\\\\n        \\Omega^{(0)}(w_{\\infty}^{+}-w_{0}^{+}) & =2\\pi i\\int_{0}^{\\infty} \\frac{\\dif x}{y}\\to\\log\\frac{1+\\sqrt{\\frac{t_{2}}{{t_{1}t_{3}}}}}{1-\\sqrt{\\frac{t_{2}}{{t_{1}t_{3}}}}} \\:, \\\\ \n        \\Omega^{(0)}(w_{0}^{+}-w_{-1}^{+}) &=2\\pi i\\int_{-1}^{0} \\frac{\\dif x}{y}\\to \\log\\frac{1+\\sqrt{\\frac{t_{1}}{{t_{2}t_{3}}}}}{1-\\sqrt{\\frac{t_{1}}{{t_{2}t_{3}}}}} \\:;\n    \\end{align}\ncf.\\ figure \\ref{Fig:pseudothreshold}.\nCorrespondingly, the symbol is reduced to\n\\begin{align}\n    \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(1)} \\bigr)  &\\to \\log \\frac{t_{1}}{t_{2}t_{3}} \\otimes \\log\\frac{1-\\sqrt{\\frac{t_{1}}{{t_{2}t_{3}}}}}{1+\\sqrt{\\frac{t_{1}}{{t_{2}t_{3}}}}} \n    +(t_1\\leftrightarrow t_2)-(t_1\\leftrightarrow t_3)\n\\:,\n\\end{align}\nwhere each term is the symbol of $\\Li_{2}(z)-\\Li_{2}(-z)+\\log(z)\\log\\bigl((1-z)\/(1+z)\\bigr)$ -- in perfect agreement with \\cite{Bloch:2013tra}.\n\n\n\\section{Example II: Double-Box Integral}   \\label{sec: example 2}\n\n\nAs a second example of Feynman integrals that evaluate to  elliptic multiple polylogarithms, we consider the two-loop ten-point double-box integral:\n\\begin{align} \\label{dbIntegrand}\n    I_{\\db}=\n\\begin{aligned}\n\\begin{tikzpicture}[scale=0.85,label distance=-1mm]\n\t\t\\node[label=left:$8$] (0) at (0, 15.5) {};\n\t\t\\node[label=above:$10$] (1) at (1.5, 16) {};\n\t\t\\node[label=above:$9$] (2) at (0.5, 16) {};\n\t\t\\node[label=above:$1$] (3) at (2.5, 16) {};\n\t\t\\node[label=right:$2$] (4) at (3.0, 15.5) {};\n\t\t\\node[label=below:$6$] (5) at (0.5, 14) {};\n\t\t\\node[label=left:$7$] (6) at (0, 14.5) {};\n\t\t\\node[label=below:$5$] (7) at (1.5, 14) {};\n\t\t\\node[label=right:$3$] (8) at (3, 14.5) {};\n\t\t\\node[label=below:$4$] (9) at (2.5, 14) {};\n        \\node(11) at (5, 15) {};\n\t\t\\node (12) at (6, 15) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (6.center) to (8.center);\n\t\t\\draw[thick] (2.center) to (5.center);\n\t\t\\draw[thick] (1.center) to (7.center);\n\t\t\\draw[thick] (3.center) to (9.center);\n        \\node[label=left:\\textcolor{blue!50}{$x_8$}] (10) at (0.2, 15) {};\n        \\node[label=above:\\textcolor{blue!50}{$x_{10}$}] (13) at (1.0, 15.8) {};\n        \\node[label=above:\\textcolor{blue!50}{$x_1$}] (14) at (2.0, 15.8) {};\n        \\node[label=right:\\textcolor{blue!50}{$x_3$}] (15) at (2.8, 15) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_5$}] (16) at (2.0, 14.3) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_6$}] (17) at (1.0, 14.3) {};\n\t\t \\draw[very thick,blue!50] (10.center) to (15.center);\n\t\t \\draw[very thick,blue!50] (13.center) to (17.center);\n\t\t \\draw[very thick,blue!50] (14.center) to (16.center);\n\\end{tikzpicture} \n\\end{aligned}\n\\hspace{-0.14\\textwidth}\n&=\\int \\dif^4 x_{l} \\dif^4x_{k}\\frac{(x_1-x_5)^2}{(x_1-x_l)^2(x_3-x_l)^2(x_5-x_l)^2}\\nonumber \n\\\\[-1.5\\baselineskip]\n&\\quad \\times \\frac{(x_3-x_8)^2(x_6-x_{10})^2}{(x_l-x_k)^2(x_6-x_k)^2(x_8-x_k)^2(x_{10}-x_k)^2} \\:,\n\\end{align}\nwhere we have introduced the dual coordinates $x_{i}$ defined as $x_{i}-x_{i+1}=p_{i}$ and the notation $x_{i,j}=x_{i}-x_{j}$. \nThis integral is a particular component of the two-loop ten-point N$^{3}$MHV amplitude in planar maximally  supersymmetric Yang-Mills theory ($\\mathcal{N}=4$ sYM theory) \\cite{CaronHuot:2012ab}, and was recently integrated in terms of elliptic multiple polylogarithms in \\cite{Kristensson:2021ani}. It depends on seven dual conformal cross-ratios\n\\begin{gather}  \\label{uvdef}\n    u_{1}=\\frac{x_{1,3}^{2}x_{5,8}^{2}}{x_{1,5}^{2}x_{3,8}^{2}}\\:, \\qquad \n    u_{2}=\\frac{x_{3,6}^{2}x_{8,10}^{2}}{x_{3,8}^{2}x_{6,10}^{2}} \\:, \\qquad \n    v_{1}=\\frac{x_{1,8}^{2}x_{3,5}^{2}}{x_{1,5}^{2}x_{3,8}^{2}} \\:, \\qquad \n    v_{2}=\\frac{x_{3,10}^{2}x_{6,8}^{2}}{x_{3,8}^{2}x_{6,10}^{2}} \\:, \\qquad  \\nonumber \\\\\n    u_{3}=\\frac{x_{1,3}^{2}x_{5,10}^{2}}{x_{1,5}^{2}x_{3,10}^{2}} \\:, \\qquad \n    u_{4}=\\frac{x_{1,6}^{2}x_{3,5}^{2}}{x_{1,5}^{2}x_{3,6}^{2}} \\:, \\qquad \n    u_{5}= \\frac{x_{1,5}^{2}x_{6,10}^{2}}{x_{1,6}^{2}x_{5,10}^{2}}\n\\end{gather}\nand contains the elliptic curve defined by\n\\begin{align}\n    y^{2}&=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}  \\nonumber  \\\\\n&=\\biggl(\\frac{v_{1}}{u_{4}}\\bigl((1{-}u_{4})(x{+}1{-}v_{2}){-}u_{1}{+}u_{3}v_{2}\\bigr){+}h_{1}{+}h_{2}\\biggr)^{2}{-}4h_{1}h_{2}, \\label{ellcurve}\n\\end{align}\nwhere\n\\begin{equation}\n \\begin{aligned}\n    h_{1}&=\\frac{u_{2}u_{4}}{v_{1}}\\bigl(x^{2}+(1{-}u_{1}{+}v_{1})x+v_{1}\\bigr),  \\\\\n    h_{2}&=\\Bigl(x{+}\\frac{v_{1}}{u_{4}}\\Bigr)\\Bigl((1{+}x{-}u_{1})\\Bigl(\\frac{u_{2}u_{4}}{v_{1}}-1\\Bigr)+(1{-}u_{3})v_{2}\\Bigr). \n\\end{aligned}\n\\end{equation}\n\n\n\n\n\nAs shown in \\cite{Kristensson:2021ani}, the elliptic double-box integral can be evaluated in terms of $\\mathcal{E}_{4}$ functions \nwhose arguments make up the set\n\\begin{align} \\label{arguset}\n   \\{c_i\\}= \\biggl\\{&0,-1,\\infty,-u_{2},-v_{1},-\\frac{v_{1}}{u_{4}},-1+\\frac{u_{1}}{u_{3}},-u_{2}u_{4}u_{5}, \n   {-}u_{2}(1{-}u_{4}){-}v_{1},  \\nonumber \\\\\n   &\\frac{u_{2}(u_{3}{+}u_{4}{-}1){-}v_{1}}{1-u_{3}},\n   \\frac{u_{2}u_{3}u_{4}u_{5}{-}v_{1}}{1-u_{3}}, \n   \\frac{u_{2}u_{3}u_{4}u_{5}{-}v_{1}}{u_{4}(1-u_{3}u_{5})},\\frac{u_{2}(u_{3}u_{4}u_{5}v_{2}{-}u_{1})}{u_{3}v_{2}-u_{1}},\n   \\nonumber \\\\\n   &\\frac{v_{1}(u_{3}u_{4}u_{5}v_{2}{-}u_{1})}{u_{4}(u_{1}{-}u_{3}u_{5}v_{2})}, \n   \\frac{u_{4}u_{5}(u_{2}(u_{4}{-}1){-}v_{1}){+}v_{1}}{u_{4}(u_{5}-1)},\\frac{u_{1}u_{2}(u_{4}{-}1){-}v_{1}(u_{1}{-}u_{3}v_{2})}{u_{1}-u_{3}v_{2}}, \\nonumber \\\\\n   &z_{1}{-}1,\\bar{z}_{1}{-}1,z_{1,3,6,8}{-}1,\\bar{z}_{1,3,6,8}{-}1, {-}z_{3,5,8,10},{-}\\bar{z}_{3,5,8,10},{-}z_{2},{-}\\bar{z}_{2},\\nonumber \\\\\n   & \\frac{u_{2}u_{3}u_{4}u_{5}{-}v_{1}+ r_{+}}{1-u_{3}} ,\\frac{u_{2}u_{3}u_{4}u_{5}{-}v_{1}+ r_{-}}{1-u_{3}} \\biggr\\}\\,,\n\\end{align} \nwhere $z_{a,b,c,d}\\bar{z}_{a,b,c,d}=x_{a,b;c,d}$, $(1-z_{a,b,c,d})(1-\\bar{z}_{a,b,c,d})=x_{d,a;b,c}$, with the abbreviation\n\\[\nx_{a,b;c,d}=\\frac{x_{a,b}^{2}x_{c,d}^{2}}{x_{a,c}^{2}x_{b,d}^{2}}\\:.     \n\\]\nNote that $u_{1}=x_{1,3;5,8}, v_{1}=x_{3,5;8,1}$ and $ u_{2}=x_{3,6;8,10}, v_{1}=x_{6,8;10,3}$, and we thus have abbreviated $z_1\\equiv z_{1,3,5,8}$ and $z_2\\equiv z_{3,6,8,10}$.\nThe expressions for $r_{\\pm}$ are slightly lengthy,\n\\begin{equation}\n    r_{\\pm} = \\frac{\\mathbf{G}^{-1}_{45}\\operatorname{det}\\mathbf{G}\\pm\\sqrt{\\operatorname{det}\\mathbf{G}^{(45)}}\\sqrt{-\\operatorname{det}\\mathbf{G}}}{2(1-u_{5})x_{1,5}^{2}x_{3,10}^{2}x_{1,6}^{2}x_{3,8}^{2}x_{5,10}^{2}} \\:,\n\\end{equation}\nwhere $\\mathbf{G}$ denotes the Gram matrix $(x_{a,b}^{2})$ with $a$ and $b$ running over the index set of dual coordinates $\\{1,3,5,6,8,10\\}$, $\\mathbf{G}^{-1}_{ij}$ denotes the elements of the inverse of the Gram matrix $(\\mathbf{G}^{-1})_{ij}$, and $\\mathbf{G}^{(ij)}$ denotes the matrix obtained from $\\mathbf{G}$ by deleting the $i$'th and $j$'th rows and columns. \nBy using \\eqref{psitog1}, the double-box integral was also expressed in terms of $\\tilde{\\Gamma}$ functions on the torus $[1:\\tau=\\omega_{2}\/\\omega_{1}]$ \\cite{Kristensson:2021ani}.\n\n\nThe last two arguments in \\eqref{arguset} can be written in a slightly more compact form if we introduce the \\emph{momentum twistor} variables~\\cite{Hodges:2009hk},\n\\begin{equation}\n    Z_{i}^{a}=(\\lambda_{i}^{\\alpha},x_{i}^{\\alpha\\dot{\\alpha}}\\lambda_{i \\alpha}) \\:,\\qquad \\alpha,\\dot{\\alpha}=1,2\\,,\n\\end{equation}\nas well as the $\\mathrm{SL}(4)$-invariant $\\langle ijkl\\rangle=\\epsilon_{abcd}Z_{i}^{a}Z_{j}^{b}Z_{k}^{c}Z_{l}^{d}$, where $\\lambda_{i}$ is the usual spinor-helicity variable $p_{i}^{\\mu}\\sigma_{\\mu}^{\\alpha\\dot{\\alpha}}=\\lambda^{\\alpha}_{i}\\tilde{\\lambda}_{i}^{\\dot{\\alpha}}$. \nIn terms of these variables, the last two arguments in \\eqref{arguset} can be written as \n\\begin{equation}\n \\frac{\\langle 9,10,1,(7,8)\\cap (2,3,5)\\rangle}{\\langle 1,5,9,10\\rangle\\langle 2,3,7,8\\rangle}, \n   \\frac{\\langle 4,5,6, (2,3)\\cap(7,8,10) \\rangle}{ \\langle 2,3,7,8\\rangle \\langle 4,5,6,10\\rangle}\\,,\n\\end{equation}\nwhere $  (ab)\\cap(ijk)=Z_{a}\\langle bijk\\rangle+Z_{b}\\langle ijka\\rangle $ is the intersection point of the line $(ab)$ and the plane $(ijk)$~\\cite{ArkaniHamed:2010kv}.\n\n\n\nIn the present paper, we compute  $I_{\\db}$ in terms of \n$\\tilde{\\Gamma}$ functions on the torus normalized as $[1:\\tau'=-\\omega_{1}\/\\omega_{2}] $, as well as the corresponding symbol. \nThe computation is straightforward; however, both expressions are too lengthy to be recorded in the main text and are given in the ancillary file \\texttt{doublebox\\_omega2}.\nIn what follows, we will give a schematic expression of the symbol as well as the symbol prime and comment on several interesting aspects, such as the integrability condition and the soft limit $p_{10}\\to0$. \n\n\n\\subsection{Symbol of the double-box integral} \\label{sec:5.1}\n\n\nLet us first review some results and notations from \\cite{Kristensson:2021ani}. After normalization by $\\omega_{1}$, $T^{(1)}_{\\db}=I_{\\db}\/\\omega_{1}$ is a pure combination of $\\tilde{\\Gamma}$ functions of length four and weight three. The corresponding symbol $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ satisfies the same physical first-entry conditions as in the MPL cases~\\cite{Gaiotto:2011dt}; namely, the first entries are given by $\\log u_i$ and $\\log v_i$. The first two entries are $\\Li_{2}(1-x_{a,b;c,d})$, $\\log x_{a,b;c,d} \\log x_{a',b';c',d'}$ \nor four-mass-box functions,\n\\begin{equation}\n -\\Li_{2}(z_{a,b,c,d})+\\Li_{2}(\\bar{z}_{a,b,c,d})-\n\\frac{1}{2}\\log (x_{a,b;c,d})\\log\\biggl(\\frac{1-z_{a,b,c,d}}{1-\\bar{z}_{a,b,c,d}}\\biggr) \\:.\n\\end{equation}\nThe last entries of this symbol consist of seven letters of elliptic type $\\Omega^{(0)}$. According to these last entries, the symbol can be organized as\\footnote{Here we use a slightly different notation for $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ than the one in \\cite{Kristensson:2021ani} for bookkeeping.} \n\\begin{align}\n    \\mathcal{S}(2\\pi i T_{\\db}^{(1)})&= \\mathcal{S}(I_{\\text{hex}})\\otimes \\Omega^{(0)}(w_{c_{25}}^{+}) +\n    \\mathcal{S}(F_{-})\\otimes \\Omega^{(0)}(w_{\\infty}^{-}) + \\mathcal{S}(F_{\\tau})\\otimes (2\\pi i\\tau) \\nonumber \\\\\n    &\\quad + \\mathcal{S}(F_{z_{1}-1})\\otimes \\Omega^{(0)}(w_{z_{1}-1}^{+})+\n \\mathcal{S}(F_{\\bar{z}_{1}-1})\\otimes \\Omega^{(0)}(w_{\\bar{z}_{1}-1}^{+}) \\nonumber  \\\\\n  &\\quad + \\mathcal{S}(F_{-z_{2}})\\otimes \\Omega^{(0)}(w_{-z_{2}}^{+})+\n \\mathcal{S}(F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(w_{-\\bar{z}_{2}}^{+}) \\:, \\label{symbol_db_1}\n\\end{align}\nwhere $I_{\\text{hex}}$ is the 6D hexagon integral (normalized to be\npure), $c_{25}=\\frac{\\langle 9,10,1,(7,8)\\cap (2,3,5)\\rangle}{\\langle 1,5,9,10\\rangle\\langle 2,3,7,8\\rangle}$ is the $25$'th element of the set \\eqref{arguset} and we used $\\Omega^{(0)}(w_{\\infty}^{+})=2\\pi i\\tau$. In particular, the symbols of all weight-three functions except $F_{\\tau}$, \nthat are $I_{\\text{hex}}$, $F_{-}$, etc., are \\emph{polylogarithmic}; their symbol entries are logarithms of kinematics. Furthermore, $I_{\\text{hex}}$ and $F_{-}$ are invariant up to a sign under the two reflections\n\\begin{align}\n    & R_{1}:\\quad p_{i}\\to p_{15-i} \\:,\\\\\n    & R_{2}:\\quad p_{i}\\to p_{10-i} \\:,\n\\end{align}\nwhere $p_{i}\\equiv p_{i+10}$, while $F_{z_{1}-1}$, $F_{\\bar{z}_{1}-1}$, $F_{-z_{2}}$ and $F_{-\\bar{z}_{2}}$ form an orbit (up to a sign) under both reflections. %\nBesides the seven elliptic last entries, elliptic letters only appear at the third entry of $\\mathcal{S}(F_{\\tau})$ and come in only 13 linear independent combinations of $\\Omega^{(2,1,0)}$'s.\nFor the full symbol alphabet and a form that is manifestly invariant under the two reflections, see \\cite{Kristensson:2021ani}.\n\nIn the other normalization, $T^{(2)}_{\\db}=-I_{\\db}\/\\omega_{2}$, the structure of the symbol is almost the same as \\eqref{symbol_db_1},\n\\begin{align}\n    \\mathcal{S}(2\\pi i T_{\\db}^{(2)})&= \\mathcal{S}(I_{\\text{hex}})\\otimes \\Omega^{(0)}(\\xi_{c_{25}}^{+}) +\n    \\mathcal{S}(F_{-})\\otimes \\Omega^{(0)}(\\xi_{\\infty}^{-}) + \\mathcal{S}(F_{\\tau'})\\otimes (2\\pi i\\tau') \\nonumber \\\\\n    &\\quad + \\mathcal{S}(F_{z_{1}-1})\\otimes \\Omega^{(0)}(\\xi_{z_{1}-1}^{+})+\n \\mathcal{S}(F_{\\bar{z}_{1}-1})\\otimes \\Omega^{(0)}(\\xi_{\\bar{z}_{1}-1}^{+})  \\label{symbol_db_2} \\\\\n  &\\quad + \\mathcal{S}(F_{-z_{2}})\\otimes \\Omega^{(0)}(\\xi_{-z_{2}}^{+})+\n \\mathcal{S}(F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(\\xi_{-\\bar{z}_{2}}^{+}) \\:. \\nonumber\n\\end{align}\nBesides the last entries, which are simple modular $S$-transformations of the last entries of $ \\mathcal{S}(2\\pi i T_{\\db}^{(1)})$, the only difference arises from the third entry of $F_{\\tau'}$ which consist of \\emph{only} $\\Omega^{(2)}$'s. Meanwhile, the $\\Omega^{(2)}$'s appearing in the third entry of $\\mathcal{S}(F_\\tau')$ also come in 13 linear independent combinations. \n\nIt is not surprising that all weight-three functions appearing in $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ except $F_{\\tau}$ are \nthe same as in $\\mathcal{S}(2\\pi i T_{\\db}^{(2)})$. Actually, for any elliptic multiple polylogarithm $I$ given by an integral of the form \\eqref{1dIntofPolylog}, its two normalizations, $T^{(1)}=I\/\\omega_{1}$ and $T^{(2)}=-I\/\\omega_{2}$, are related by $T^{(2)}=\\tau' T^{(1)}$. \nWith the last entry of $T^{(1)}$ being of elliptic type $\\Omega^{(0)}$, i.e.,\n\\begin{equation}\n    \\dif T^{(1)} = \\sum_{i} F_{i} \\dif w_{i} + F_{\\tau}\\dif \\tau ,\n\\end{equation}\nwhere $F_{i}$ and $F_{\\tau}$ are elliptic multiple polylogarithms of lower length, a straightforward computation gives\n\\begin{equation}\n    \\dif T^{(2)} = \\sum_{i}F_{i}\\dif \\xi_{i} + \\underbrace {\\biggl(T^{(1)}-\\sum_{i}F_{i}w_{i}-\\tau F_\\tau\\biggr)}_{F_{\\tau'}}\\dif \\tau' \\:.\n    \\label{eq: S transformation Ftau}\n\\end{equation}\nWhile it is not manifest from \\eqref{eq: S transformation Ftau}, this particular expression for $F_{\\tau'}$ has length three.\n\nOnce we evaluate the symbol prime for the third entry of $F_{\\tau}$ and $F_{\\tau}'$, a simple connection emerge. Both $ \\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ and $ \\mathcal{S}(2\\pi i T_{\\db}^{(2)})$ have similar structures as in the case of the sunrise integral: \n\\begin{align}\n    \\mathcal{S}(2\\pi i  T_{\\db}^{(1)})&= \\sum_{ij} \\mathcal{S}(f_{i})\\otimes \\Bigl( \\log a_{ij} \\otimes \\Omega^{(0)}(w_{j}) + \\bm{\\Omega}_{i}\\otimes (2 \\pi i\\tau)   \\Bigr), \\label{symbol_prime_db}\n    \\intertext{with}\n    \\mathcal{S}'(\\bm{\\Omega}_{i})&=\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}\\,,\n\\end{align}\nand a similar expression for $\\mathcal{S}(2\\pi i  T_{\\db}^{(2)})$ with $\\xi_{j}$ and $\\tau'$ in place of $w_{j}$ and $\\tau$. Here $w_{j}$ are the six last entries except $\\tau$, $a_{ij}$ are some algebraic functions of kinematics, $f_{i}$ are of the form $\\Li_{2}(1-x_{ab;cd})$, $\\log x_{ab;cd} \\log x_{a'b';c'd'}$ or four-mass-box functions, and $\\bm{\\Omega}_{i}$ are combinations of $\\Omega^{(2,1,0)}$ that occur in the last entry of $F_{\\tau}$ associated with $\\mathcal{S}(f_{i})$.\n\nLet us close this subsection by remarking some advantages of the form \\eqref{symbol_prime_db}. Firstly, the symbol prime is manifestly double-periodic due to the same argument used in section \\ref{sec:4.1}: under the translation $w_{j}\\to w_{j}+\\tau$,\n\\begin{align}\n    & \\underbrace{[\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}}\\log a_{ij}]}_{\\mathcal{S}'(\\log a_{ij})} \\otimes \\Omega^{(0)}(w_{j}) + \\underbrace{[\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}]}_{\\mathcal{S}'(\\bm{\\Omega}_{i})} \\otimes (2 \\pi i\\tau) \\nonumber\\\\\n    &\\to [\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}}\\log a_{ij}] \\otimes \\Omega^{(0)}(w_{j}+\\tau) + [\\Omega^{(0)}(w_{j}+\\tau)\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}] \\otimes (2 \\pi i\\tau) \\\\\n    & ={}\\,[\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}}\\log a_{ij}] \\otimes \\Omega^{(0)}(w_{j}) + [\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}] \\otimes (2 \\pi i\\tau) \\:,\\nonumber\n\\end{align}\nwhere we have used \\eqref{symbolp3} and $\\Omega^{(-1)}=-\\Omega^{(0)}(\\tau)=-2\\pi i\\tau$.\nSecondly, this form makes a part of the integrability conditions manifest and sheds light on the (13 linearly independent) combinations of $\\Omega^{(2,1,0)}$ that occur in the last entry of $F_{\\tau}$. \nThe integrability condition requires \n\\begin{equation}\n\\partial_{w_j}\\bm{\\Omega}_{i} + \\partial_{\\tau} \\log a_{ij} =0\\,,\n\\end{equation}\nwhich is a consequence of \n\\begin{equation}\n    \\mathcal{S}'(\\bm{\\Omega}_{i})= \\sum_{j}\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}\\:,\n\\end{equation}\nsince \n\\begin{equation}\n    \\bm{\\Omega}_{i}\\otimes \\tau - \\sum_{j}\\Omega^{(0)}(w_{j})\\otimes \\log a_{ij} =\n    \\bm{\\Omega}_{i}\\otimes \\tau +\\sum_{j} \\log a_{ij} \\otimes \\Omega^{(0)}(w_{j}) - \\mathcal{S}(\\log a_{ij} \\Omega^{(0)}(w_{j}))\n\\end{equation}\nis an integrable symbol by the definition of the symbol prime.\nWe will discuss the remaining integrability conditions in upcoming work \\cite{integrability_in_progress}.\n\n\n\\subsection{Soft limit}\n\nIn this subsection, we will consider the soft limit $p_{10}\\to 0$, in which the elliptic double-box integral remains finite and becomes polylogarithmic:\n\\begin{equation}\n\\begin{aligned}\n\\begin{tikzpicture}[scale=0.85,label distance=-1mm]\n\t\t\\node[label=left:$8$] (0) at (0, 15.5) {};\n\t\t\\node[label=above:$10$] (1) at (1.5, 16) {};\n\t\t\\node[label=above:$9$] (2) at (0.5, 16) {};\n\t\t\\node[label=above:$1$] (3) at (2.5, 16) {};\n\t\t\\node[label=right:$2$] (4) at (3.0, 15.5) {};\n\t\t\\node[label=below:$6$] (5) at (0.5, 14) {};\n\t\t\\node[label=left:$7$] (6) at (0, 14.5) {};\n\t\t\\node[label=below:$5$] (7) at (1.5, 14) {};\n\t\t\\node[label=right:$3$] (8) at (3, 14.5) {};\n\t\t\\node[label=below:$4$] (9) at (2.5, 14) {};\n        \\node(11) at (5, 15) {};\n\t\t\\node (12) at (6, 15) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (6.center) to (8.center);\n\t\t\\draw[thick] (2.center) to (5.center);\n\t\t\\draw[thick] (1.center) to (7.center);\n\t\t\\draw[thick] (3.center) to (9.center);\n        \\node[label=left:\\textcolor{blue!50}{$x_8$}] (10) at (0.2, 15) {};\n        \\node[label=above:\\textcolor{blue!50}{$x_{10}$}] (13) at (1.0, 15.8) {};\n        \\node[label=above:\\textcolor{blue!50}{$x_1$}] (14) at (2.0, 15.8) {};\n        \\node[label=right:\\textcolor{blue!50}{$x_3$}] (15) at (2.8, 15) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_5$}] (16) at (2.0, 14.3) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_6$}] (17) at (1.0, 14.3) {};\n\t\t \\draw[very thick,blue!50] (10.center) to (15.center);\n\t\t \\draw[very thick,blue!50] (13.center) to (17.center);\n\t\t \\draw[very thick,blue!50] (14.center) to (16.center);\n\\end{tikzpicture} \n\\end{aligned}\n\\hspace{-0.1\\linewidth}\n\\xrightarrow{p_{10}\\to 0}\\qquad\n\\begin{aligned}\n\\begin{tikzpicture}[scale=0.85,label distance=-1mm]\n\t\t\\node[label=left:$8$] (0) at (0, 15.5) {};\n\t\t\\node[] (1) at (1.5, 15.5) {};\n\t\t\\node[label=above:$9$] (2) at (0.5, 16) {};\n\t\t\\node[label=above:$1$] (3) at (2.5, 16) {};\n\t\t\\node[label=right:$2$] (4) at (3.0, 15.5) {};\n\t\t\\node[label=below:$6$] (5) at (0.5, 14) {};\n\t\t\\node[label=left:$7$] (6) at (0, 14.5) {};\n\t\t\\node[label=below:$5$] (7) at (1.5, 14) {};\n\t\t\\node[label=right:$3$] (8) at (3, 14.5) {};\n\t\t\\node[label=below:$4$] (9) at (2.5, 14) {};\n        \\node(11) at (5, 15) {};\n\t\t\\node (12) at (6, 15) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (6.center) to (8.center);\n\t\t\\draw[thick] (2.center) to (5.center);\n\t\t\\draw[thick] (1.center) to (7.center);\n\t\t\\draw[thick] (3.center) to (9.center);\n        \\node[label=left:\\textcolor{blue!50}{$x_8$}] (10) at (0.2, 15) {};\n        \\node[label=above:\\textcolor{blue!50}{$x_{1}$}] (13) at (1.5, 16) {};\n        \\node[label=right:\\textcolor{blue!50}{$x_3$}] (15) at (2.8, 15) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_5$}] (16) at (2.0, 14.3) {};\n        \\node[label=below:\\textcolor{blue!50}{$x_6$}] (17) at (1.0, 14.3) {};\n\t\t \\draw[very thick,blue!50] (10.center) to (15.center);\n\t\t \\draw[very thick,blue!50] (13.center) to (1,15) to (17.center);\n\t\t \\draw[very thick,blue!50] (13.center) to (2,15) to (16.center);\n\\end{tikzpicture} \n\\end{aligned}\n\\hspace{-0.1\\linewidth}\n\\end{equation}\n\n\nIn terms of momentum twistors, this limit amounts to first setting \n\\begin{equation}\n    Z_{10} \\to Z_{9}+\\alpha Z_{1}+\\epsilon (Z_{8}+\\beta Z_{2})  \\label{soft_limit_mt}\n\\end{equation}\nwith finite $\\alpha$ and $\\beta$, and then taking the limit $\\epsilon\\to 0$~\\cite{Drummond:2010mb}. In this limit, \nthe seven cross-ratios become\n\\begin{gather}  \\label{uvdefsoft}\n    u_{1}=\\frac{x_{1,3}^{2}x_{5,8}^{2}}{x_{1,5}^{2}x_{3,8}^{2}}\\:, \\qquad \n    u_{2}=\\frac{x_{3,6}^{2}x_{8,1}^{2}}{x_{3,8}^{2}x_{6,1}^{2}} \\:, \\qquad \n    v_{1}=\\frac{x_{1,8}^{2}x_{3,5}^{2}}{x_{1,5}^{2}x_{3,8}^{2}} \\:, \\qquad \n    v_{2}=\\frac{x_{3,1}^{2}x_{6,8}^{2}}{x_{3,8}^{2}x_{6,1}^{2}} \\:, \\qquad  \\nonumber \\\\\n    u_{3}=1 \\:, \\qquad \n    u_{4}=\\frac{x_{1,6}^{2}x_{3,5}^{2}}{x_{1,5}^{2}x_{3,6}^{2}}=\\frac{v_{1}}{u_{2}} \\:, \\qquad \n    u_{5}=1 \\:.\n\\end{gather}\nCorrespondingly, the elliptic curve \\eqref{ellcurve} degenerates to \n\\renewcommand{\\rho}{r}\n\\begin{equation}\n    y^{2}=\\bigl(x^{2}+(1-u_{1}+u_{2})x+u_{2}-u_{1}u_{2}+v_{1}v_{2}\\bigr)^{2}=\\bigl((x-\\rho)(x-\\bar{\\rho})\\bigr)^{2} \\:, \\label{ellcurvesoft}\n\\end{equation}\nwhere \n\\begin{align}\n    \\rho       &=-\\frac{1}{2}\\bigl(1-u_{1}+u_{2}+\\sqrt{(1-u_{1}-u_{2})^{2}-4v_{1}v_{2}}\\bigr)  \\:,\\\\\n    \\bar{\\rho} &=-\\frac{1}{2}\\bigl(1-u_{1}+u_{2}-\\sqrt{(1-u_{1}-u_{2})^{2}-4v_{1}v_{2}}\\bigr)  \\:.\n\\end{align}\n\nThe double-box integral in this soft limit can be easily integrated to multiple polylogarithms starting from the one-fold \nintegral representation in \\cite{Kristensson:2021ani} by using, for instance, \n{\\sc HyperInt} \\cite{Panzer:2014caa} or {\\sc PolyLogTools} \\cite{Duhr:2019tlz}.\nWe record both the function and the symbol result in the ancillary file \\texttt{doublebox\\_soft}. \nIn the following, we demonstrate how the same result for the symbol is obtained as a limit of the elliptic symbol of the ten-point double-box integral \\eqref{symbol_db_1}.\n\n\nWe work in the region given by positive momentum-twistor kinematics~\\cite{Arkani-Hamed:2013jha}, where the four roots of $y^{2}(x)$ in \\eqref{ellcurve} come in complex conjugate pairs as shown in figure \\ref{fig: contours}, but both $\\rho$ and $\\bar{\\rho}$ are negative real numbers, i.e., \n$r_{1},r_{4}\\to \\rho$, while $r_{2},r_{3}\\to\\bar{\\rho}$ in the soft limit. Therefore, $\\omega_{2}\\to\\infty$ in this limit, \nand \n\\begin{equation}\n  \\omega_{1}\\to  \\int_{\\gamma_{1}} -\\frac{\\dif x}{(x-\\rho)(x-\\bar{\\rho})} = \\frac{2\\pi i}{\\rho-\\bar{\\rho}} \n\\end{equation}\nwhere we made the following choice for the degenerated curve \\eqref{ellcurvesoft} in the different regions:\n\\begin{align}\n  y = \\begin{cases}\n    -(x-\\rho)(x-\\bar{\\rho}) \\:, & \\text{for} \\quad \\rho<\\operatorname{Re} x <\\bar{\\rho} \\\\\n     (x-\\rho)(x-\\bar{\\rho}) \\:, & \\text{otherwise} \\,.\n   \\end{cases}\n\\end{align}\nThe reason for this choice is that we want to keep $y$ non-negative on the real $x$-axis for the degenerated curve, following the convention we chose for the original curve in \\eqref{ellcurve}.\nAgain, $(2\\pi \\mi)^{-1} \\omega_{1}$ gives the correct normalization factor for this nine-point double-box integral, and we should expect that $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ will reproduce the correct symbol in this soft limit.\n\n\n\nThe immediate problem we encounter in taking the soft limit for $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ is that \nnot all of the seven last entries in \\eqref{symbol_db_1} have smooth, definite and finite limits.\nWhile the soft limit of $c_{25}$ depends on the arbitrary parameters $\\alpha$ and $\\beta$ in \\eqref{soft_limit_mt}, \nthe term $\\mathcal{S}(I_{\\text{hex}})\\otimes \\Omega^{(0)}(w_{c_{25}}^{+})$ does not introduce a problem since $\\mathcal{S}(I_{\\text{hex}})$ vanishes in the soft limit.\nThe integration contours for $w_{\\infty}^{-}$, $w_{\\bar{z}_{1}-1}^{+}$ as well as $w_{-\\bar{z}_{2}}^{+}$ go through the pole at $x=\\rho$, and the integration contours for $w_{z_{1}-1}^{+}$ as well as $w_{\\infty}^{+}=\\tau$ go through both poles at $x=\\rho$ and $x=\\bar{\\rho}$; see figure \\ref{fig: degeneration}. \n\\begin{figure}\n   \\centering\n   \\begin{tikzpicture}[scale=2]\n\n           \\draw[->, thick] (0,0) to (0,3);\n           \\draw[->, thick] (-3,1.5) to (1,1.5);\n           \\draw[->,line width=0.23mm,blue] (-3,1.4) .. controls (-1,1.4) ..   (-1.0,0.0) ; \n           \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n           \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n           \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n           \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n            \\draw[->] (-1.7,1.95) to (-1.7,1.7);\n            \\draw[->] (-1.7,1.05) to (-1.7,1.3);\n            \\draw[->] (-0.3,0.85) to (-0.3,1.45);\n            \\draw[->] (-0.3,2.15) to (-0.3,1.55);\n           \\node at (-1.3,0.1) {\\textcolor{blue}{$\\gamma_{-}$}};\n           \\node at (0.4,3) {$\\Im x$};\n           \\node at (1.3,1.5) {$\\Re x$};\n           \\node[cross,label=above:${-z_{2}}$] at (-2.5, 1.5) {};\n           \\node[cross,label=above:${\\bar{z}_{1}-1}$] at (-1.3, 1.5) {};\n           \\node[cross,label=above:${-\\bar{z}_{2}}$] at (-0.7, 1.5) {};\n           \\node[cross,label=above:${z_{1}-1}$] at (0.5, 1.5) {};\n   \\end{tikzpicture}\n   \\caption{In the soft limit $p_{10}\\to0$, the roots of $y^{2}(x)$ pairwise pinch the integration contours for $w^+_{\\bar{z}_{1}-1}, w^+_{-\\bar{z}_{2}}$ and $w^+_{z_{1}-1}$, which run along the real axis.\n   By subtracting $w^-_{\\infty}$ and $w^+_{\\infty}$, respectively, we obtain integration contours that can be deformed such they are not pinched, thus resulting in finite integrals in the soft limit.  \n   }\n   \\label{fig: degeneration}\n   \\end{figure}\n To cancel the resulting singularities, we reorganize $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ as \n\\begin{align}\n    \\mathcal{S}(2\\pi i T_{\\db}^{(1)}) &= \\mathcal{S}(F_{-z_{2}})\\otimes \\Omega^{(0)}(w_{-z_{2}}^{+})+\n    \\mathcal{S}(F_{z_{1}-1})\\otimes \\Omega^{(0)}(w_{z_{1}-1}^{+}-w_{\\infty}^{+})  \\nonumber \\\\\n    &\\quad + \\mathcal{S}(F_{\\bar{z}_{1}-1})\\otimes \\Omega^{(0)}(w_{\\bar{z}_{1}-1}^{+}-w_{\\infty}^{-}) \n    + \\mathcal{S}(F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(w_{-\\bar{z}_{2}}^{+}-w_{\\infty}^{-}) \\nonumber \\\\\n    &\\quad \n    +\\mathcal{S}(F_{\\tau}+F_{z_{1}-1}) \\otimes (2\\pi i\\tau) +\\mathcal{S}(F_{-}+F_{\\bar{z}_{1}-1}+F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(w_{\\infty}^{-}) \\nonumber \\\\\n    &\\quad+\\mathcal{S}(I_{\\text{hex}}) \\otimes \\Omega^{(0)}(w_{c_{25}}^{+}) \\:,\n\\end{align}\ncf.\\ figure \\ref{fig: degeneration}.\nOne can easily check that not only the last term but the  last three terms do not contribute in the soft limit since the three weight-three symbols making up their first three entries vanish in the soft limit. The first four terms yield the correct polylogarithmic symbol in the soft limit with last entries\n\\begin{equation}\n\\begin{aligned}\n    \\Omega^{(0)}(w_{-z_{2}}^{+}) &\\to (\\rho-\\bar{\\rho}) \\int_{-\\infty}^{-z_{2}}\\frac{\\dif x}{(x-\\rho)(x-\\bar{\\rho})}\\:= \n    \\log \\frac{\\rho+z_{2}}{\\bar{\\rho}+z_{2}} \\:, \\\\\n    \\Omega^{(0)}(w_{z_{1}-1}^{+}-w_{\\infty}^{+})& \\to (\\rho-\\bar{\\rho}) \\int_{+\\infty}^{z_{1}-1}\\frac{\\dif x}{(x-\\rho)(x-\\bar{\\rho})} = \\log\\frac{1+\\rho-z_{1}}{1+\\bar{\\rho}-z_{1}} \\:, \\\\\n    \\Omega^{(0)}(w_{\\bar{z}_{1}-1}^{+}-w_{\\infty}^{-}) & \\to (\\rho-\\bar{\\rho}) \\int_{-i\\infty}^{\\bar{z}_{1}-1} \n    \\frac{-\\dif x}{(x-\\rho)(x-\\bar{\\rho})}= \\log\\frac{1+\\bar{\\rho}-\\bar{z}_{1}}{1+\\rho-\\bar{z}_{1}} \\:, \\\\\n    \\Omega^{(0)}(w_{-\\bar{z}_{2}}^{+}-w_{\\infty}^{-})&\\to (\\rho-\\bar{\\rho}) \\int_{-i\\infty}^{-\\bar{z}_{2}}\\frac{-\\dif x}{(x-\\rho)(x-\\bar{\\rho})} \\: = \\log \\frac{\\bar{\\rho}+\\bar{z}_{2}}{\\rho+\\bar{z}_{2}} \\:.\n\\end{aligned} \n\\end{equation}\nNote that in the soft limit $z_{1}\\equiv z_{1,3,5,8}$ and $z_{2}\\equiv z_{3,6,8,1}$ and the reflection symmetry $R_{1}$ is broken while $R_{2}$ survives; \nthus, the symbol for this nine-point double-box integrals can be expressed as\n\\begin{align}\n    \\mathcal{S}\\Bigl((\\rho-\\bar{\\rho})I_{\\softdb}\\Bigr) &= \\mathcal{S}(F_{-z_{2}}\\vert_{p_{10}\\to 0})\\otimes \\log \\frac{\\rho+z_{2}}{\\bar{\\rho}+z_{2}} +\\mathcal{S}(F_{-\\bar{z}_{2}}\\vert_{p_{10}\\to 0})\\otimes \\log \\frac{\\bar{\\rho}+\\bar{z}_{2}}{\\rho+\\bar{z}_{2}} \\nonumber \\\\\n    & \\quad \n    + (\\text{images under }R_2\n    ) \\:,\n\\end{align}\nwhere $R_2$ acts on the last entries via  $\\rho\\leftrightarrow -(1{+}\\bar{\\rho})$, $z_{1}\\leftrightarrow z_{2}$ and $\\bar{z}_{1}\\leftrightarrow \\bar{z}_{2}$.\nFurthermore, $F_{-z_{2}}\\vert _{p_{10}\\to 0}$ and $F_{-\\bar{z}_{2}}\\vert _{p_{10}\\to 0}$ are related by exchanging $z_{2}$ and $\\bar{z}_{2}$, same as the corresponding last entries.\nThe reason is that $z$ and $\\bar{z}$ occur symmetrically in their definition $\\{z\\bar{z}=u, (1-z)(1-\\bar{z})=v\\}$ , and thus have to occur symmetrically in the symbol as well.\n\nThe symbol alphabet of the nine-point double-box integral consists of 10 rational letters and 11 algebraic letters:\n\\begin{enumerate}\n    \\item Rational letters:\n\\begin{equation}\n    \\begin{gathered}\n        u_{1}\\,,\\:\\: u_{2} \\,,\\:\\:  v_{1} \\,,\\:\\: v_{2} \\,,\\:\\: u_{1}-v_{2} \\,,\\:\\: v_{1}-u_{2} \\,,\\:\\: \n        u_{1}u_{2}-v_{1}v_{2} \\,,\\:\\: \\Delta_{1} \\,,\\:\\:  \\Delta_{2}\\:, \\\\\n        \\frac{\\langle 5(91)(23)(78)\\rangle \\langle \\bar{5}(91)(23)(78)\\rangle\\langle 1239\\rangle \\langle 1789\\rangle}{ \\langle 1459\\rangle^{2}\\langle 1569\\rangle^{2}\\langle 2378\\rangle^{3}} \\:,\n    \\end{gathered} \n    \\end{equation}\n    where we introduced the following notations:\\footnote{Here we use $(\\bar{a})\\equiv Z_{a-1}{\\wedge}Z_{a}{\\wedge} Z_{a+1}$ to denote the dual plane of $Z_{a}$. Then a vanishing $\\langle \\bar{a} (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle$ means that the three intersection points $(i\\,i{+1}){\\cap} (\\bar{a})$, $(j\\,j{+1}){\\cap} (\\bar{a})$ and $(k\\,k{+1}){\\cap} (\\bar{a})$ are on the same line, which is the dual picture of the vanishing of $\\langle a (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle$. We are grateful to Cristian Vergu for pointing this out.}\n    \\begin{align}\n        \\langle a(bc)(de)(fg)\\rangle &=\\langle abde\\rangle \\langle acfg \\rangle-\\langle acde\\rangle \\langle abfg \\rangle , \\\\\n    \\langle \\bar{a} (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle &= \\langle (i\\,i{+}1)\\cap (\\bar{a}) \\,j\\,j{+}1\\,(k,k{+}1)\\cap (\\bar{a})\\rangle \n    \\end{align}\nand \n\\begin{align}\n    \\Delta_{i}=(1-u_{i}-v_{i})^{2}-4u_{i}v_{i}=(z_{i}-\\bar{z}_{i})^{2} \\:, \\qquad i=1,2\\,.\n\\end{align}\n    \\item Algebraic letters:\n    \\begin{itemize}\n        \\item $\\displaystyle \\frac{z_{1}}{\\bar{z}_{1}}$, $\\displaystyle \\frac{1-z_{1}}{1-\\bar{z}_{1}}$, $\\displaystyle \\frac{1+\\bar{\\rho}-z_{1}}{1+\\rho-z_{1}}$, $\\displaystyle \\frac{1+\\bar{\\rho}-\\bar{z}_{1}}{1+\\rho-\\bar{z}_{1}}$, \n        $\\displaystyle \\frac{\\frac{\\langle 5(23)(46)(78)\\rangle \\langle 1239\\rangle }{\\langle 5(19)(23)(78)\\rangle \\langle 2378\\rangle}-z_{1}}{\\frac{\\langle 5(23)(46)(78)\\rangle \\langle 1239\\rangle }{\\langle 5(19)(23)(78)\\rangle \\langle 2378\\rangle}-\\bar{z}_{1}}$, \\\\\n         and five others generated by the reflection $R_2$. \n        \\item $\\displaystyle \\frac{(z_{1}-1+\\bar{z}_{2})(\\bar{z}_{1}-1+z_{2})}{(z_{1}-1+z_{2})(\\bar{z}_{1}-1+\\bar{z}_{2})}$.\n    \\end{itemize}\n\\end{enumerate}\nWe find that there are three different square roots in this alphabet; two of them are of four-mass-box type and the other, that is the square root in $\\rho$ and $\\bar{\\rho}$, arises from the leading singularity of the whole Feynman diagram. Furthermore, the new type of square root \\emph{only} appears in the last entries.\nThe symbol alphabet is organized such that the symbol is manifestly invariant (up to a sign) under the reflection $R_{2}$, as well as under each of the three transformations $z_{1} \\leftrightarrow \\bar{z}_{1}$, \n$z_{2} \\leftrightarrow \\bar{z}_{2}$, and $\\rho \\leftrightarrow \\bar{\\rho}$.\nFor an analysis of these letters through Schubert problems, see \\cite{Yang:2022gko}.\n\n\\section{Conclusion and Outlook} \\label{sec:5}\n\nIn this paper, we have investigated various techniques for manipulating and simplifying the symbol of Feynman integrals that evaluate to elliptic multiple polylogarithms. In particular, we study identities between the elliptic symbol letters $\\Omega^{(i)}$.\n\nIn contrast to ordinary multiple polylogarithms, the length of an elliptic multiple polylogarithm is not necessarily equal to its weight. A symbol letter $\\Omega^{(i)}$, whose length is by definition one, can have weight $i\\neq 1$.\nIdentities for $\\Omega^{(0)}$ follow from the well-known group law on the elliptic curve.\nMoreover, we found that various identities for $\\Omega^{(1)}$ can be derived from Abel's theorem, which generalize the identity $\\log(a)+\\log(b)=\\log(ab)$ in the polylogarithmic case.\nThe higher-weight letters $\\Omega^{(2)}$ satisfy significantly more intricate identities, closer to those of $\\Li_{2}(a)$ than those of $\\log(a)$, which are harder to exploit in a direct fashion.\nWe thus introduce the \\emph{symbol prime} $\\mathcal{S}'$ for elliptic symbol letters $\\Omega^{(2)}$, which plays the same role the symbol $\\mathcal{S}$ plays for  $\\Li_{2}(a)$. We also introduced a symbol prime for $\\Omega^{(i>2)}$ but leave its exploration for future work.\n\nWe studied two concrete examples at two-loop order, namely the sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. In particular, we provided proofs for the identities between symbol letters numerically found in \\cite{Kristensson:2021ani}.\n\nIn addition to identities between symbol letters, we also studied how the symbol behaves under kinematic limits in which the elliptic curve degenerates. We recover the known symbols of the sunrise integral in the null-momentum limit $p^{2}\\to 0$ and the pseudo threshold $p^2\\to(m_{1}+m_{2}-m_{3})^{2}$ \\cite{Bloch:2013tra}, as well as the nine-point limit of the double box, which has not previously appeared in the literature.\n\nThe numeric values of elliptic Feynman integrals are of course independent of whether we normalize the torus by the period $\\omega_1$ or $-\\omega_2$; the corresponding modular parameters $\\tau=\\omega_{2}\/\\omega_{1}$ and $\\tau'=-1\/\\tau$ are related by a modular $S$-transformation. In particular, for an elliptic integral of the form $\\int \\mathcal{G}(x,y)\\, \\dif x\/y$ as in \\eqref{1dIntofPolylog}, its two normalizations $T^{(1)}$ and $T^{(2)}$, which are obtained by dividing by $\\omega_{1}$ and $-\\omega_{2}$ respectively, are simply related by $T^{(2)} = \\tau' T^{(1)}$. However, this property is not manifest when expressed in terms of elliptic multiple polylogarithms or their symbols. \nInstead, we find that the application of the symbol prime to the two examples in this paper yields symbols of the form\n\\begin{align}\n    & \\sum_{ij} \\mathcal{S}(f_{i})\\otimes \\Bigl( \\log a_{ij} \\otimes \\Omega^{(0)}(w_{j}) + \\bm{\\Omega}_{i}\\otimes (2 \\pi i\\tau)   \\Bigr), \n    \\intertext{with}\n    &\\qquad\\mathcal{S}'(\\bm{\\Omega}_{i})=\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}\\,.\n\\end{align}\nNot only the modular covariance is manifest in this form, but also the double-periodic invariance and the integrability conditions involving $\\tau$. However, one could \\emph{not} expect that the application of the symbol prime to the one-fold integral of general polylogarithms, of the form $\\int \\mathcal{G}(x,y) \\,\\dif x\/y$, yields such a structure.\nAs a simple counter-example, consider the symbol of the integral $\\int_{0}^{c} \\log(x+a)\\,\\dif x\/y$, with arbitrary values of $a$ and $c$ as well as the elliptic curve given by \\eqref{ellcurve}, which does \\emph{not} follow the above structure after applying the symbol prime map.\nIt would be very interesting to investigate why the two elliptic Feynman integrals we considered in this paper turn out to exhibit such a structure after applying the symbol prime map, and to study whether this property extends to further Feynman integrals.\n\n\n\nAlready the polylogarithmic symbol has a kernel, which is given by $i\\pi$, multiple zeta values (MZVs) and their products with MPLs.\nSimilarly, also the symbol prime has a kernel. \nAs discussed in section \\ref{subsec: symbol prime}, all functions in the kernel necessarily depend only on the modular parameter $\\tau$, but not all functions that depend only on $\\tau$ are in the kernel. \nTo see that the kernel can be non-trivial, \nconsider the symbol of the \\emph{equal-mass} sunrise integral, which is an iterated integral of modular forms and \\emph{only} depends on $\\tau$:\n\\begin{align}\n    \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(1)} \\bigr) &= \\biggl[\\frac{1}{2\\pi\\mi} \\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr) \\biggr]\\otimes (2\\pi\\mi\\tau)\\:. \\label{eq:symbol equal mass sunrise} \n\\end{align}\nwhere the $\\mathcal{E}_4$ are specific combinations of $\\Omega^{(2)}$'s given in \\eqref{eq:Ecal-gammat-relation}.\nThe application of the symbol prime map to the first entry in \\eqref{eq:symbol equal mass sunrise} yields $0$.\nWe leave a \ncomprehensive treatment of the kernel of the symbol prime map to a future study.\n\nAnother interesting problem is to lift simplified symbols to simplified functions for elliptic multiple polylogarithms. \nAs a primary example, let us consider how to lift the simplified symbol \\eqref{symbolw1} to a simplified function for the sunrise integral. By rewriting the logarithms in \\eqref{symbolw1} in terms of $\\Omega^{(1)}$'s, we find a (slightly) simpler expression  for $T^{(1)}_{\\sr}$,\n\\begin{equation}\n\\begin{aligned}\n    T^{(1)}_{\\sr} &= 2\\cEf{0&-1}{0&-1}{\\infty|\\tau}-\\cEf{0&-1}{0&0}{\\infty|\\tau} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau} \\\\ \n      &\\quad  - \\biggl(2\\log\\frac{t_{2}}{t_{3}}+\\cEf{-1}{-1}{\\infty}\\biggr)\\cEf{0}{0}{\\infty|\\tau}  \\:,\n\\end{aligned}\n\\end{equation}\n(and a similar expression for $T^{(2)}_{\\sr}$), in which the $\\mathcal{E}_4$ functions only contain $c=0,-1,\\infty$ but not the fourth argument $r$ that occurs in  \\eqref{eq:sunrise normalization 1}. \nHowever, the general prescription of uplifting more complicated elliptic symbols to functions is still underexplored.%\n\\footnote{In particular, there are symbols that can be written in terms of only logarithms as letters that can nevertheless not be lifted to polylogarithmic functions, only to elliptic ones \\cite{Duhr:2020gdd}.\n}\n\nAlthough the simplified symbols of the elliptic Feynman integrals manifest some desired properties, such as double periodicity and modular covariance, after applying the symbol prime map, the singularity structures are not completely manifest. For example, the sunrise integral becomes singular at $m_{i}=0$ as well as at the threshold $p^{2}=(m_{1}+m_{2}+m_{3})^{2}$, as can be seen through a Landau analysis~\\cite{OKUN1960261}. One can see the branch cut at $m_{i}=0$ explicitly from eq.\\ \\eqref{symbolw1} or \\eqref{symbolw2}; however, the branch cut at the threshold is not manifest from the symbol. \nIn general, the logarithmic letter $\\log(a)$ has a logarithmic singularity if $a=0$ or $a=\\infty$.\nIn contrast, the elliptic letter $\\Omega^{(1)}(w)$ has a logarithmic singularity at all lattice points, while $\\Omega^{(j\\geq2)}(w)$ has a logarithmic singularity at all lattice points except for the origin.\nHowever, $w$ is a function of the kinematics; typically, $w=w_c^+=1\/\\omega_1 \\int_{-\\infty}^c\\dif x\/y$, where $c$ is an algebraic function of the kinematics. If the configuration of roots in $y$ does not change as we vary $c$, $w_c^+=0$ if $c=-\\infty$ and $w_c^+=\\tau$ if $c=+\\infty$. However, the configuration of roots may also vary as we vary $c$;  \nwe leave a comprehensive analysis to a future study. \n\n\n\nIt would be interesting to apply the techniques used in this paper to bootstrap the symbol of scattering amplitudes or Feynman integrals that can be expressed in terms of elliptic multiple polylogarithms, such as the twelve-point elliptic double box. \nOn top of the integrability condition for the final entry $\\tau$, which is made manifest by the symbol prime, this requires understanding the integrability condition for the other last entries  \\cite{integrability_in_progress}.\nMoreover, it requires an educated guess for the alphabet of symbol letters that occur in them.\nFor six- and seven-point amplitudes in $\\mathcal{N}=4$ sYM theory, the symbol alphabet was shown to be given by cluster algebras \\cite{Golden:2013xva,Golden:2014pua,Drummond:2017ssj,Drummond:2018dfd,Drummond:2018caf}, and similar techniques have recently been extended to Feynman integrals and amplitudes~\\cite{Zhang:2019vnm,He:2020vob,He:2020lcu,Li:2021bwg} containing symbol letters that are given by logarithms of algebraic functions of the kinematics \\cite{Chicherin:2020umh,He:2021esx,He:2021non,Drummond:2019cxm,Arkani-Hamed:2019rds,Henke:2019hve,Herderschee:2021dez,Henke:2021ity,Ren:2021ztg}.\nIt would be interesting to extend these techniques also to the elliptic case.\n\n\n\n\n\n\n\\acknowledgments\n\n\nWe thank Johannes Br\u00f6del, Simon Caron-Huot, Claude Duhr, Zhenjie Li, Robin Marzucca, Andrew McLeod, Cristian Vergu, Matthias Volk, Matt von Hippel and Stefan Weinzierl for interesting discussions as well as Andrew McLeod, Mark Spradlin and  Stefan Weinzierl for comments on the manuscript.\nMW thanks the organizers of the conference ``Elliptics '21'', where part of this work was presented.\nThis work was supported by the research grant  00025445 from Villum Fonden and the ERC starting grant 757978.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe study of Interplanetary Shocks associated with major solar eruptions is very important  not only from the theoretical point of view, but also because of potential impacts on human technologies. First because shocks are, as well as solar flares, optimal locations for the acceleration of Solar Energetic Particles (SEPs; i.e. electrons, protons and He ions with energies from a few KeV to some GeV) that constitute an important hazard for satellites and astronauts, and may affect the ionosphere around polar caps. Moreover, as the shocks reach the Earth, significant southward components of the interplanetary magnetic field associated with them can magnetically reconnect with the magnetosphere, thus disturbing the system and producing severe geomagnetic storms \\citep[see e.g. review by][]{schwenn2006}. Hence, understanding the origin, propagation and physical properties of interplanetary shocks is also crucial for future developments of our capabilities of forecasting possible Space Weather effects of solar activity. For these reasons, over the last decades huge efforts have been devoted in order to improve our knowledge of these phenomena and of the associated Coronal Mass Ejections (CMEs), by using different instrumentation taking remote sensing as well as in situ data. In particular, over the last few years, the most recent space based missions, such as the twin STEREO satellites, the Hinode and SDO observatories, provided significant new insights, thus allowing to investigate shocks from the early phases of their formation at the base of the corona out to their propagation into the interplanetary space.\n\nA clear signature of the formation and propagation of interplanetary shocks associated with CME expansion and\/or flare explosions is the detection of type-II radio bursts \\citep[see][for a review of the problem of type-II sources]{vrsnak2008}. Combination of radio data with images acquired at different wavelengths is able to provide unique new information on these phenomena. Recently, combined analysis of EUV images and radio dynamic spectra were used to demonstrate \\citep{cho2013,chen2014} that type-II bursts may be excited in the lower corona through interaction between CMEs and nearby dense structures such as streamers \\citep[see also][]{classenaurass2002, reiner2003, ma04}. A similar result was also obtained with the use of a new radio triangulation technique exploiting radio data acquired by different spacecraft \\citep{magdalenic2014}. Hence, type-II radio bursts are likely to be excited during the early propagation phase of the shocks (that is, at heliocentric distances $r < 1.5$~$R_\\odot$), around the expected location of the local minimum of $v_\\text{A}(r)$ profile \\citep{gopalswamy2012a, gopalswamy2013}. Thanks to the high cadence, good sensitivity and spatial resolution now available in EUV with SDO\/AIA, it has been shown \\citep{kouloumvakos2014} also that the sole analysis of EUV images can provide by itself an estimate of the density compression ratio $X$ (an important shock parameter given by the ratio between the downstream and the upstream plasma densities, $X = n_\\text{d} \/ n_\\text{u}$) and that this estimate is in agreement with the one derived from radio data in sheat regions. The above results clearly have important implications for the identification of SEP source regions.\n\nOver the last decade it also became clear that a significant number of information on interplanetary shocks can be derived from White Light (WL) coronagraphs data alone, as first shown by \\citet{vourlidas03}. Analysis of these data allowed to verify that shocks form when their propagation velocity $v_\\text{sh}$ (measured in a reference system at rest with the solar wind, moving at velocity $v_\\text{sw}$) is larger than the local Alfv\\'en velocity $v_\\text{A}$ ($|\\mathbf{v_\\text{sh}} - \\mathbf{v_\\text{sw}}| > v_\\text{A} = B\/ \\sqrt{4 \\pi \\rho}$). Hence, the lower is the velocity of the driver, the larger are the distances where shock front forms \\citep{eselevich2011}. Moreover, combination of EUV and WL data shows that the shock thickness $\\delta$ is of the same order as the proton mean free path $\\lambda_p$ only for heliocentric distances $r < 6$~$R_\\odot$\\, while higher up in the corona $\\delta << \\lambda_p$. Hence, during its propagation, the shock regime changes from collisional to collisionless \\citep{eselevich2012}. These information are crucial for our understanding of the physics at the base of the shock. Also, at larger heliocentric distances, the analysis of WL data provided by heliospheric imagers have demonstrated that the driver (CME) and the shock undergo different magnetic drag deceleration during their interplanetary expansion, with the shock propagating faster than the ejecta, thus leading to possible CME-shock decouplings \\citep[][]{hesszhang2014}. Statistically, the coupling has been found to be stronger for faster CMEs \\citep{mujiber2013}. Studies of interplanetary propagation of shocks have tremendous implications for Space Weather prediction capabilities as well.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f1.pdf}\n\\caption{Top: sequence of SDO\/AIA 304 and SOHO\/LASCO-C2 images acquired on June 7, 2011 during the eruptive event analyzed here. The LASCO-C2 images are shown in inverted color scale (brighter features are darker and vice-versa) and after the application of a filter to enhance the visibility of CME structures (images created with JHelioviewer). Bottom: sequence of LASCO-C2 and -C3 images showing the CME propagation at higher altitudes; again the images are shown in inverted color scale and after the application of a filter to enhance the visibility of CME structures (images created with JHelioviewer).}\\label{fig:event}\n\\end{figure*}\n\nSignificant advances were also made from comparisons between observations and numerical simulations. At heliocentric distances $r > 2$~$R_\\odot$\\, coronal protons and electrons are no more coupled by Coulomb collisions. This leads to different temperatures for these two species, with slightly larger proton than electron temperatures (by a factor depending on the relevant altitude and coronal structure) as demonstrated by coronal UV spectra acquired by the UV coronagraph Spectrometer \\citep[UVCS; see reviews by][]{antonucci2006,kohl06}. Protons, however, being much heavier than electrons, have much smaller microscopic velocities (by a factor of 42.85). CME-driven shocks are thus supersonic only with respect to the proton thermal speed, implying that only protons are expected to be significantly heated by the transit of the shock. This was recently confirmed from both observations and simulations: in particular, \\citet{manchester2012, jin2013} demonstrate that the WL appearances of CME-driven shocks are better reproduced by 2-temperature (2T) MHD simulations with respect to 1-temperature (1T) simulations, where 2T plasma protons are heated up to $\\sim 90$~MK, and 2T shocks have larger Alfv\\'enic Mach numbers $M_\\text{A}$ (by a factor $\\sim 1.25$--1.4) with respect to the 1T plasma case. Very similar results were recently obtained by the combined analysis of UV and WL observations of a CME driven shock performed by \\citet{bem14}.\n\nThe latter work was the result of a sequence of previous researches performed on CME-driven shocks and based on the combined analyses of UV spectra acquired by UVCS and WL images acquired by the LASCO coronagraph. As first demonstrated by \\citet{bem10}, this unique combination allows to measure not only the plasma compression ratio $X$, but also the pre- and post-shock plasma temperatures. Moreover, once these informations are combined with the Rankine-Hugoniot equations written for the general case of oblique shocks, and by measuring geometrical (inclination) and kinematical (velocity) properties of the shock from WL data, it is even possible to determine both the pre- and post-shock magnetic and velocity field vectors projected on the plane of the sky. This technique allowed \\citet{bem11,bem13} to conclude that, for a few specific events, radio-loud (radio-quiet) CMEs are more likely associated with super- (sub-) critical shocks, and that only a small region around the shock center is super-critical in the early evolution phases, while higher up (i.e. later on) the whole shock becomes sub-critical. Moreover, the same technique applied to different points located along the same shock front allowed \\citet{bem14} to demonstrate that the transit of shock leads to a significant deflection of the magnetic field close to the shock nose, and a smaller deflection at the flanks, implying a draping of field lines around the expanding CME, in nice agreement with the post-shock magnetic field rotations obtained by \\citet{liu2011} with 3D MHD numerical simulations.\n\nIn this paper the above results are further extended: in particular we demonstrate here that, under some specific hypotheses, the analysis of WL coronagraphic data alone not only can provide the density compression ratios at different times and locations along the shock front, but also the $M_\\text{A}$ numbers and the pre-shock coronal magnetic fields, allowing us to derive a 2D map of magnetic field strength covering an heliocentric distance interval by $\\sim 10$~$R_\\odot$\\ and a latitude interval by $\\sim 110^\\circ$. Moreover, the combined analysis of WL and radio data allows us to derive the possible location of the source for the type-II radio burst. The paper is organized as follows: after a general description of the event being analyzed here (Section \\ref{sec:obs}), we describe the analysis of data (Section \\ref{sec:datanal}), focusing in particular on LASCO\/C2 and C3 WL coronagraphic images (Section \\ref{sec:wldata}) and WAVES\/RAD1-RAD2 radio dynamic spectra (Section \\ref{sec:radiodata}). Then, the obtained results are summarized and discussed (Section \\ref{sec:concl}).\n\n\n\\section{Observations} \\label{sec:obs}\n\nOn June 7th 2011, a GOES M2.6 class flare from AR 11226 (located in the southwest quadrant at 22$^\\circ$ S and 66$^\\circ$ W) occurred between 06:16 and 06:59 UT, peaking around 06:16 UT. This soft X-ray flare was associated with significant HXR emission and even $\\gamma-$ray emission lasting for about 2 hours \\citep{ackermann2014}. The impressive eruption associated with this flare has been extensively studied by many previous authors who focused on different physical phenomena related with the event. They focused on several aspects of this event, such as the early evolution of the released CME bubble and compression front \\citep{cheng2012}, the propagating EUV wave \\citep{li2012}, the magnetic reconnections driven by the CME expansion \\citep{vandriel2014}, the flare emission \\citep{inglisgilbert2013}, and the associated type-II radio burst \\citep{dorovskyy2013, dorovskyy2015}. Moreover, this spectacular eruption was followed by the ejection of huge radial columns of chromospheric plasma, reaching the field of view of LASCO and COR1 coronagraphs, and then falling back to the sun. Thus, other authors focused also on the dynamics and plasma properties of returning plasma blobs \\citep{innes2012, williams2013, carlyle2014, dolei2014}, as well as on the energy release from falling material impact on the sun \\citep{gilbert2013, reale2013, reale2014}.\n\nIn this work we study the evolution of the shock wave associated with this eruption as observed by white light coronagraphic images. As reported by \\citet{cheng2012}, immediately after the flare onset (around 06:26 UT) a circular plasma CME bubble was observed in the SDO\/AIA images expanding at $\\sim 960$~km~s$^{-1}$; in the early phases, due to the small standoff distance, the compression front and the front of the driver (i.e. the CME bubble) cannot be discerned. The two fronts started to separate only later on, when a deceleration of the CME bubble is observed; at the same time, a type II radio burst started (as well as a type-III burst), suggesting that the compression wave had just turned itself into a shock wave. Later on, the CME enters in the field of view of the SOHO\/LASCO-C2 coronagraph starting from the frame acquired at 06:49 UT (Figure 1, top row), and then enters in the field of view of the LASCO C3 coronagraph starting form the frame acquired at 07:11 UT (Figure 1, bottom row). The LASCO C2 frames clearly show the propagation of the shock wave associated with the event, as well as the CME front and the circular flux rope, while this latter part becomes hardly discernible in the LASCO C3 frames (see \\ref{fig:event}).\n\nIn what follows we describe how the sequence of white light images acquired by LASCO C2 and C3 has been analyzed to derive the pre-CME coronal density and the different physical parameters of the shock wave.\n\n\n\\section{Data analysis} \\label{sec:datanal}\n\n\\subsection{WL coronagraphic images} \\label{sec:wldata}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f2.pdf}\n\\caption{Appearance of the white light corona as observed on June 4, 02:48 UT (left) before the acquisition of the pB image used for the coronal density determination, and on June 7, 06:04 UT (right) before the occurrence of the eruption.}\\label{fig:corona}\n\\end{figure*}\n\n\\subsubsection{Pre-CME coronal densities}\nFor the density calculation we use SOHO\/LASCO C2 polarized brightness (pB) images. It is well known that the K-corona brightness originates from Thomson scattering of photospheric light by free electrons in the solar corona \\citep[e.g.,][]{bil66}. Because the emission is optically thin, the observer sees a contribution from electrons located all along the line of sight. In addition to the K-corona, observations will contain a component due to scattering of photospheric light from interplanetary dust (the so-called F-corona). This component must be eliminated from the data to derive the coronal electron density; however, in the case of pB observations at small altitudes ($\\lesssim 5$~$R_\\odot$), the F corona can be assumed unpolarized and thus does not contribute to the pB \\citep[][]{hay01}.\n\nThe intensity of the scattered light depends on the number of scattering electrons and several geometric factors, as was first outlined by \\citet[][]{min30}. In the absence of F corona, the polarized brightness observed on the plane of the sky is given by the following equation:\n\\begin{equation}\\label{eq:vandehulst}\n\\text{pB}(\\varrho)=C\\int_\\varrho^\\infty n_e(r)\\left[A(r)-B(r)\\right]\\frac{\\varrho^2\\,dr}{r\\sqrt{r^2-\\varrho^2}},\n\\end{equation}\nwhere $C$ is a unit conversion factor, $n_e$ is the electron density, $A$ and $B$ are geometric factors \\citep[][]{vdh50,bil66}, $\\varrho$ is the projected heliocentric distance of the point (impact distance), and $r$ is the actual heliocentric distance from Sun center. The integration is performed along the line of sight through the considered point. \\citet[][]{vdh50} developed a well known method for estimating the electron density by the inversion of Equation (\\ref{eq:vandehulst}) under the assumptions that: (1) the observed polarized brightness along a single radial can be expressed in the polynomial form $\\text{pB}(r)=\\sum_k \\alpha_k r^{-k}$ and (2) that the coronal electron density is axisymmetric. We apply this method to the latest LASCO C2 pB image acquired before the June 7th CME, in order to determine the pre-CME electron density distribution in the corona.\n\nThe pB image considered here is obtained from the polarization sequence of observations recorded on June~4th 2011, starting at 02:54~UT, i.e. about three days before the occurrence of the June 7 CME. During this three-day time lag, three other much smaller CMEs occurred having a central propagation direction in the same latitudinal sector crossed by the June 7th CME ($70^\\circ$S--$40^\\circ$N), as reported in the SOHO\/LASCO CME catalog: on June~4th, at 06:48~UT and 22:05~UT, and on June~6th, at 07:30~UT. Nevertheless, despite these smaller scale events and coronal evolution, a direct comparison between the LASCO C2 white-light images acquired on June~4th at 02:48~UT and on June~7th immediately before the eruption at 06:04~UT shows that the overall density structure of the corona above the west limb of the Sun is quite similar even after more than three days (Figure \\ref{fig:corona}), hence the electron density estimated from the inversion of the June~4th pB data can be considered at least a first order approximation of the real pre-CME coronal density configuration.\n\n\\begin{figure*}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.27\\textwidth]{f3a.pdf}}\n\\subfigure[]{\\includegraphics[width=0.72\\textwidth]{f3b.pdf}}\n\\caption{LASCO C2 polarized-brightness image of the solar corona above the west limb, acquired on June 4th 2011 at 02:57~UT (a) and the corresponding 2D electron density map derived from the inversion of the pB data (b).}\\label{fig:density}\n\\end{figure*}\n\nThe electron density radial profiles obtained at different latitudes from the pB image (Fig.~\\ref{fig:density}a) are combined into a 2D map in polar coordinates, shown in Fig.~\\ref{fig:density}b. The map shows the density distribution in the latitudinal region being crossed later on by the shock, for heliocentric distances ranging between 2 and 12~$R_\\odot$; electron densities at distances from the Sun larger than 6~$R_\\odot$\\ (the outer limit of the LASCO C2 field of view) are obtained through a power-law extrapolation of the density profiles assuming a radial dependence proportional to $r^{-2}$. The presence of the coronal streamer centered around $50^\\circ$S, that is persistent till June~7, is very clear as it is associated with a local electron density maximum. Notice that in general coronal features are much less evident in the pB image and in the density map (Figure \\ref{fig:density}) with respect to the regular LASCO frames (Figure \\ref{fig:corona}) because the latter are obtained after subtraction of a monthly minimum background average to enhance the visibility of fainter structures.\n\n\\subsubsection{Shock position and kinematics}\n\nWhite-light coronagraphic images can be used to identify the shock front location at different times and to distinguish between the shock-compressed plasma and the CME material, as extensively demonstrated by several works \\citep[e.g.,][]{vourlidas03,ont09,bem10,bem11}. The CME-driven shock front can be identified as a weak brightness increase located above the expanding CME front, that is generally interpreted as the visible signature of the downstream plasma compression and density enhancement caused by the transit of the shock; for this reason, the shock front becomes visible only when the intensity scale of WL images is adjusted to bring out the fainter structures.\n\nIn this work, we determine the location of the shock front in both LASCO C2 and C3 total brightness images using a common procedure that consists of three steps: (1) we compute excess-mass (or base-difference) images by subtracting from each calibrated LASCO frame an average pre-event image that is representative of the quiescent background corona \\citep[see][]{vou00,ont09}; (2) we apply a Normalizing Radial Graded Filter (NRGF), as described by Morgan et al. (2006), in order to reveal faint emission features at high heliocentric distances in the corona (this is particularly useful for the identification of the shock front in LASCO C3 images); (3) we measure the projected altitude of the shock by locating the intensity jump at the front in the radial direction. With this technique the location of the shock can be identified with an estimated uncertainty of $\\pm 3$ pixels on average and $\\pm 5$ pixels for LASCO C2 and C3 images, respectively. Larger uncertainties could be related with the applied procedure of background subtraction, in the possible locations where the pre-eruption corona significantly changed during the event.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f4.pdf}\n\\caption{Cartesian plot showing the locations of the shock front identified at different times in LASCO C2 and C3 white-light images.}\\label{fig:positions}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f5.pdf}\n\\caption{Base-difference LASCO C3 image showing the location of the shock front (solid white line) at 07:39~UT and a schematic representation of selected vectors normal to the shock surface (white arrows) and corresponding radial directions in the same points (red arrows).}\\label{fig:lasco_norm}\n\\end{figure}\n\nWe apply this procedure to seven consecutive images where we could identify signatures of the shock: two from LASCO C2, acquired at 06:47 and 07:01~UT, and five from LASCO C3, acquired at 07:09, 07:24, 07:39, 07:54, and 08:09~UT, respectively (see Figure \\ref{fig:event}). Later on, we were not able to locate the shock front with a significant accuracy in LASCO C3 images. The curves giving the position the shock fronts identified in the considered WL images are plotted in Figure~\\ref{fig:positions}. The shock appears to propagate almost symmetrically and to exhibit only a moderate latitudinal displacement, since the center of the shock (i.e., the highest point along the front) has a latitudinal location which is always in the range 21--25$^\\circ$S. We notice here that around a latitude of about 12$^\\circ$S the identified location of the shock surface shows a clear discontinuity, which is likely due to the Northward displacement of a the pre-event coronal streamer, leading to an overestimate (underestimate) of the shock projected altitude Northward (Southward) of the streamer itself.\n\nThese curves can be easily employed to derive, all along each shock front, the angle $\\theta_\\text{sh}$ between the normal to the shock front and the radial direction, as well as the latitudinal distribution of the average shock speed, $v_\\text{sh}$. These quantities are essential for the determination of the Alfv\\'enic Mach number and the upstream plasma velocity distribution, as discussed in the following section. As an example, Figure~\\ref{fig:lasco_norm} shows the relative orientation of vectors parallel with the radial direction and those normal to the shock surface at different positions along the front as we identified in the LASCO C3 image acquired at 07:39~UT. It is evident from this Figure that $\\theta_\\text{sh}$ angles are in general larger at the flanks of the shock, and smaller near the shock center (or ``nose''). This result confirms what we already found in recent works \\citep[see, e.g.,][]{bem14} and suggests that we may expect the prevalence of quasi-perpendicular shock conditions at the flanks and quasi-parallel shock conditions at the center of the shock.\n\n\\begin{figure*}\n\\centering\n\\vspace{-4cm}\n\\includegraphics[width=\\textwidth]{f6.pdf}\n\\caption{Compression ratios $X \\equiv \\rho_\\text{d}\/\\rho_\\text{u}$ as measured along the shock fronts identified in LASCO observations and reported in Fig.~\\ref{fig:positions}. Each profile is shown as a thick shaded area representing the uncertainty in the derived $X$ values.}\\label{fig:ratios}\n\\end{figure*}\n\nThe radial component of the average shock speed is obtained at each latitude simply as $v_\\text{r}=\\Delta\\varrho\/\\Delta t$, where $\\Delta\\varrho$ is the variation of the projected heliocentric distance of the shock measured in the radial direction between two consecutive shock curves. The true shock velocity can be then derived simply as $v_\\text{sh}=v_\\text{r}\\cdot\\cos\\theta_\\text{sh}$. Note that, as in \\citet[][]{bem14}, this corresponds to assume isotropic self-similar expansion of the front in the range of common latitudes between consecutive curves, but taking into account the correction for the latitudinal shock propagation. A 2D polar map of radial velocity distribution $v_\\text{r}$ in the region where the shock propagates is obtained by interpolating with polynomial fitting the heliocentric distance values at each latitude and altitude along the shock fronts, and is shown in Figure~\\ref{fig:results1} (top-left panel). The resulting radial shock speed is (as expected) larger at the center of the shock at all altitudes, then it decreases toward the shock flanks; at a heliocentric distance of $2.5$~$R_\\odot$\\ it reaches a value as high as $\\sim 1200$~km~s$^{-1}$ near the center and $\\sim 800-900$~km~s$^{-1}$ $\\sim 20^\\circ$ away from it. The shock also appears to decelerate during its propagation, since the velocity at higher altitudes is progressively smaller: for instance, at 12~$R_\\odot$\\ $v_\\text{sh}\\simeq 1000$~km~s$^{-1}$ at the shock center. This means that the shock is losing its energy as it expands; this is also supported by the results we obtain for the compression ratio and the Alfv\\'enic Mach number, as discussed in the following section.\n\n\\subsubsection{Compression ratio, Alfv\\'enic Mach number, and Alfv\\'en speed}\n\nThe shock compression ratio $X$, defined as the ratio between the downstream (i.e., post-shock) and the upstream (i.e., pre-shock) plasma densities, $X\\equiv \\rho_\\text{d}\/\\rho_\\text{u}$, is determined here as described in \\citet[][]{bem11}. For each pixel along an identified shock front, we measure the total white-light brightness of the compressed downstream plasma, tB$_\\text{d}$, from the corresponding LASCO C2 or C3 image, and, at the same locations in the corona, the upstream brightness tB$_\\text{u}$ from the last image acquired before the arrival of the shock. This provides us with the observed ratio $(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{obs}$.\n\nOn the other hand, the upstream total brightness tB$_\\text{u}(\\varrho)$ expected at a projected altitude $\\varrho$ in the corona can be evaluated through the line-of-sight integration of the upstream electron density profile, $n_e(r)$, multiplied by a geometrical factor $K$ that includes all the geometrical parameters for Thomson scattering:\n\\begin{equation}\n\\text{tB}_\\text{u}(\\varrho)=\\int_{\\varrho}^{\\infty}{K(r,\\varrho)\\cdot n_e(r)\\,dr},\n\\end{equation}\nwhere $r$ is the heliocentric distance of the scattering point along the line of sight. The expected downstream total brightness tB$_\\text{d}$ is similarly given by the sum of two integrals: one performed over the unshocked corona (with density $n_e$) and the other over a length $L$ across the shocked plasma with density $X\\cdot n_e$ ($X \\geq 1$):\n\\begin{eqnarray}\n&&\\text{tB}_\\text{d}(\\varrho)=\\int_{\\varrho}^{\\infty}{K(r,\\varrho)\\cdot n_e(r)\\,dr} +\\\\ \\nonumber \n&&\\int_{\\varrho}^{r_\\text{sh}}{K(r,\\varrho)\\cdot (X-1)\\cdot n_e(r)\\,dr},\n\\end{eqnarray}\nwhere $r_\\text{sh}=\\sqrt{\\varrho^2+L^2}$ and $X$ is precisely the unknown compression ratio. The shock depth $L$ is estimated as in \\citet[][]{bem10}, i.e., by assuming that the shock surface has the three-dimensional shape of an hemispherical shell with thickness equal to the 2D projected thickness $d$ of the white-light intensity jump across the shock, corrected for the shock motion during the LASCO C2 or C3 exposure time. For each frame we estimated an average value of the shock depth $L$, and applied the same value to the whole shock front. Given $L$ and by adopting the radial density profiles derived from the analysis of the LASCO C2 pB, the shock compression ratio $X$ can be inferred directly from the comparison between the observed and the expected total brightness ratios: $(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{obs}=(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{exp}$.\n\nThe corresponding curves for the compression ratio $X$ measured along the shock fronts with different LASCO C2 and C3 frames are reported in Figure~\\ref{fig:ratios}. The uncertainties in $X$ values shown in this Figure are due to the uncertainty in the identification of the exact location of the shock in C2 and C3 images (see above). The compression ratio reaches the maximum value of $\\sim 2.1$ at 06:47~UT in a point that is very close to center of the shock front at that time located around a latitude of -20$^\\circ$S; this $X$ value is quite lower than the upper limit adiabatic compression of 4 expected for a monoatomic gas. In all cases, the latitudinal dependence is similar: $X$ has a maximum around the center of the shock front, progressively but not monotonically decreasing toward the flanks. As the shock expands, the $X$ values decrease on average all along the shock fronts: for instance, at 08:09~UT the maximum value is of $\\sim 1.5$; as already pointed out in the previous section, this indicates that the shock is dissipating its energy while propagating in the corona. These results are in agreement with those reported by \\citet[][]{bem11} in their analysis of a different CME-driven shock. We notice here that, as explained above, the $X$ values have been not derived after background subtraction, but from the ratio between the total brightnesses observed at the shock location and those observed at the same pixels in the frame acquired just before the arrival of the shock. This method allows to remove in the ratio any possible uncertainty due to the instrumental calibration; moreover, because the shock is the faster feature propagating outward, no significant changes occurred in the corona aligned along the LOS between the two frames other than the compression due to the shock.\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7a.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7b.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7c.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7d.pdf}}\n\\caption{2D maps showing the distribution of the radial shock velocity $v_\\text{r}$ (a), the Alfv\\'enic Mach number $M_\\text{A}$ (b), the Alfv\\'en speed $v_\\text{A}$ (c), and as a reference the pre-shock coronal densities $n_e$ (d). The $M_\\text{A}$ and $v_\\text{A}$ values are derived by assuming a negligible solar wind speed, as described in the text. In each panel real measurements were obtained only in the region between the two dotted lines, while values shown out of these region have been extrapolated at higher and lower altitudes.}\\label{fig:results1}\n\\end{figure*}\n\nThe Alfv\\'enic Mach number is defined as the ratio between the upstream plasma velocity $v_\\text{u}$ (i.e., the velocity of the plasma flowing toward the shock surface in the reference frame at rest with the shock itself) and the Alfv\\'en speed $v_\\text{A}$, $M_\\text{A} \\equiv v_\\text{u}\/v_\\text{A}$. $M_\\text{A}$ can be estimated from the compression ratio $X$ and the angle $\\theta_\\text{sh}$ under two assumptions: (1) the plasma $\\beta \\ll 1$ ($\\beta$ is the ratio between the thermal and magnetic plasma pressures) and (2) the upstream magnetic field is radially directed, so that the angle between the shock normal and the magnetic field vector can be assumed to be equal to $\\theta_\\text{sh}$ on the plane of the sky. These are not strong assumptions, as discussed in \\citet[][]{bem11}, and can be considered fairly verified also in our case. Under these hypotheses, as we verified observationally in \\citet[][]{bem14} and theoretically in \\citet[][]{bacchini2015}, the Alfv\\'enic Mach number is well approximated in the general case of oblique shock by the following semi-empirical formula:\n\\begin{equation}\\label{eq:mach}\nM_{\\text{A}\\angle}=\\sqrt{M_{\\text{A}\\parallel}^2\\cos^2\\theta_\\text{sh}+M_{\\text{A}\\perp}^2\\sin^2\\theta_\\text{sh}},\n\\end{equation}\nwhere $M_{\\text{A}\\parallel}=\\sqrt{X}$ and $M_{\\text{A}\\perp}=\\sqrt{\\frac{1}{2}X(X+5)\/(4-X)}$ are the expected Mach numbers for parallel and perpendicular shocks, respectively, for a $\\beta \\ll 1$ plasma. The validity of Eq.~(\\ref{eq:mach}) has been confirmed by the analysis of \\citet[][]{bem14} which takes advantage of both white-light and ultraviolet data from the \\emph{Ultra-Violet Coronagraph Spectrometer} (UVCS) on board SOHO (see discussion therein) and has been recently tested with MHD numerical simulations by \\citet[][]{bacchini2015}. This equation allowed us to derive, from different values of $X$ and $\\theta_\\text{sh}$ parameters, 2D polar maps of $M_{\\text{A}\\angle}$ values, as shown in Figure~\\ref{fig:results1} (top right panel). This map clearly shows that in the early phases the shock was super-Alfv\\'enic at all latitudes (with larger $M_\\text{A}$ values at the shock nose), while later on (i.e. higher up) keeps super-Alfv\\'enic numbers only at the nose.\n\nThe Alfv\\'en speed can be derived, in turn, from $M_\\text{A}$ values once the upstream plasma velocity is known or estimated. The upstream velocity is given by $v_\\text{u}=|\\mathbf{v_{sw}}-\\mathbf{v_{sh}}|$, where $\\mathbf{v_{sw}}$ is the outflow solar wind speed, assumed to be radial, and $\\mathbf{v_{sh}}$ is the shock speed. In our case, we have no direct measurements of the wind flows in the corona, hence we must adopt a model for the solar wind expansion in order to infer the Alfv\\'en speed from the Alfv\\'enic Mach number. To this end, a first-order approximation can be obtained by assuming $\\mathbf{v_{sw}}=\\mathbf{0}$ in the previous equation, i.e., by neglecting the solar wind at all. This is not a realistic assumption, but it is rather reasonable, considering that at low altitudes in the corona ($\\lesssim 5$~$R_\\odot$) and in the early phase of propagation, the shock speed may be up to one order of magnitude larger than typical wind velocities measured outside coronal holes \\citep[$\\approx 100$--300~km~s$^{-1}$; see, e.g.,][]{sus08}. Under this hypothesis, the estimated Alfv\\'en speed can be considered as an upper limit to the real values. Possible consequences of this assumption will be discussed in the last Section.\n\n\\begin{figure*}\n\\centering\n\\vspace{-11.5cm}\n\\includegraphics[width=0.9\\textwidth]{f8.pdf}\n\\vspace{-2cm}\n\\caption{Lower panel: Dynamic spectrum of the Wind\/WAVES radio data in the frequency range between 20 KHz and 13.8 MHz from 6 to 14 UT on 2011 June 7, showing, showing the decametric to kilometric type II radio emissions associated with the CME. The upper panel at the left shows details of the radio emission associated with the emission excited earlier at the southern flank of the shock. The curves on this plot are also explained in the text.}\\label{fig:radio}\n\\end{figure*}\n\n2D polar maps of the Alfv\\'en speed are shown in Figure~\\ref{fig:results1} (bottom-left panels); these maps have been obtained again with polynomial (third-order) interpolation of the Alfv\\'en speeds measured at different locations (i.e. latitudes and altitudes) of the shock front at different times (Figure~\\ref{fig:positions}). Results plotted in Figure ~\\ref{fig:results1} clearly show that the Alfv\\'en speed has not only radial, but also significant latitudinal modulations. The Alfv\\'en speed reaches the highest value ($\\sim 1000$~km~s$^{-1}$) at the lowest altitudes in the equatorial belt. The latitudinal dependence is rather complex, with an alternation of local minima and maxima ranging between $\\sim 600$ and $\\sim 1000$~km~s$^{-1}$. At increasing altitudes, $v_\\text{A}$ generally decreases, with values that never exceed 800~km~s$^{-1}$ at 12~$R_\\odot$. Interestingly, the regions characterized by the slowest decrease in electron density (around $\\sim 50^\\circ$S and around $\\sim 10^\\circ$N; see Fig.~\\ref{fig:density}) are also those where the Alfv\\'en speed decreases more steeply, reaching values below $\\sim 500$~km~s$^{-1}$ already at 5~$R_\\odot$. As a consequence, in the early propagation phase (i.e., at low altitudes) the shock is significantly super-Alfv\\'enic not only at the nose but also in several regions distributed in the flanks of the shock surface. These high-density and high-Mach number regions are very probable candidates as sources of particle acceleration and type-II radio bursts; we discuss in the next section possible correlations with the sources of radio emission identified from radio dynamic spectra, while the determination of the magnetic field strength is discussed in the last Section.\n\n\\subsection{Radio dynamic specrum} \\label{sec:radiodata}\n\nAs it is well known, shock waves are able to accelerate electron beams to suprathermal energies, which in turn can produce Langmuir waves that are converted by means of nonlinear wave-wave interactions into electromagnetic waves near the fundamental and\/or harmonic of the local electron plasma frequency $f_{pe}$. Since the coronal density $n_e$ decreases with increasing heliocentric distance and $f_{pe} \\propto {n_e^{1\/2}}$, the expanding shock surface produces type-II radio emissions at decreasing frequencies as it propagates through space and the measured frequency drift rate at a given time is directly related to the shock speed. The observed frequency drift rate provides therefore information on the shock dynamics through the corona, while its onset depends on the local magnetosonic speed.\n\nThe dynamic spectrum in the lower panel of Figure \\ref{fig:radio} shows the intensity of the radio data from 06:00 to 14:00~UT on 2011 June 7 in the frequency range between 20~KHz and 13.8~MHz measured by the RAD1 and RAD2 radio receivers of the WAVES experiment on the Wind spacecraft. A very intense complex type-III-like radio emissions was observed beginning at 6:24~UT. This fast-drifting radio emission can be interpreted as the first radio signature indicating the lift-off of the CME on the Sun \\citep[e.g.,][]{reinerkaiser1999} and is probably originated by the reconfiguration of the magnetic field in the lower corona that allows the energetic electrons produced by the flare to escape into the interplanetary medium \\citep{reiner2000}. Two slowly-drifting episodes of strong type-II emission were also observed in the decametric range around 07:00 UT (clearly visible in the expanded upper left panel of Figure \\ref{fig:radio}) and after 09:00 UT, abruptly intensifying between 13:00 and 14:00~UT (lower panel of Figure \\ref{fig:radio}). We interpret these bands of emissions, as usually assumed when only one band is visible, as second harmonics. The origin of the second harmonic emission in type-II bursts is well understood as a result of coalescence of two plasma waves into a transverse one at twice the plasma frequency. Less intense, additional slow-drifting, type-II-like radio emissions at different times and frequencies are also visible, probably originating from different portions of the super-Alfv\\'enically expanding shock surface.\n\n\\begin{figure*}\n\\centering\n\\vspace{-2.5cm}\n\\includegraphics[width=0.9\\textwidth]{f9.pdf}\n\\caption{Comparison between the 2D maps of coronal magnetic field strengths derived by assuming negligible wind speed (top left, upper limit for the field values) and fast wind speed at all latitudes (bottom left, lower limit for the field values). The right panel shows a comparison between the latitudinal average of magnetic fields obtained under the assumption of negligible wind speed (blue line) and assuming fast wind speed at all latitudes (red line).}\\label{fig:fastslowwind}\n\\end{figure*}\n\nIn order to model the observed complex type-II radio emissions displayed in Figure \\ref{fig:radio}, we need to know the coronal electron density profile at the time of the CME event. In fact, the density profile allows to convert the height measurements related to the shock surface dynamics to corresponding values of the coronal density as the frequencies $f$ are simply obtained as $f \\approx f_{pe} \\approx 9 \\sqrt{n_e[\\rm cm^{-3} ]}$~KHz. Instead of relying on a generic coronal electron density model, as usually done in the literature, we used the coronal electron density at different heliocentric distances and latitudes provided by the LASCO pB measurements discussed in the previous section. These density estimates, obtained for heliocentric distances greater than about 2~$R_\\odot$, correspond to radio frequencies below about 14~MHz, i.e., the range of radio emissions observed in the Wind\/WAVES dynamic spectrum. By assuming, as usual, second harmonic type-II emission and using the coronal density distribution inferred from the available LASCO pB observations to relate the type-II frequencies to their heliocentric heights, we identified, knowing the shock's surface height from the previous analysis, a set of synthetic type-II profiles that were superimposed (as dashed lines in Figure \\ref{fig:radio}) to the radio dynamic spectrum for comparison with the actual type-II emissions. This comparison allowed to characterize all observed type-II features and, in particular, two distinct regions (assuming radial propagation) along the shock's surface where the brightest radio emissions were most likely generated. An accurate estimate of the model radio profiles could only be obtained considering the coronal parameters outward from the flare longitude of 66$^\\circ$ W and not from 90$^\\circ$ W (plane of the sky). Unfortunately, at the time when the CME occurred, the STEREO-A and -B spacecraft were located at 94.9$^\\circ$ and 93.0$^\\circ$ from the Sun-Earth line, respectively. Hence, coronagraphic images acquired by the STEREO coronagraphs would not provide any useful information about the corona lying on the meridional plane at 66$^\\circ$ W. Said that, although we assume that no significant temporal and longitudinal variations are present between the density profile we inferred on the plane of the sky and the density really met by the shock, this assumption is undoubtedly much more realistic with respect to the one that involves the adoption of a generic power-law density profile, as usually done in the literature for this kind of studies \\citep[see e.g.][]{ra99, pojo2006, liu2009, kong2015, dorovskyy2015}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f10.pdf}\n\\vspace*{-6cm}\n\\caption{Comparison between radial magnetic field profiles derived in this work at different latitudes (solid black lines), other magnetic field radial profiles provided in the literature (in particular: \\citet{dulkmclean1978} - solid blue line, \\citet{patzold1987} - dash-dotted dark green line, \\citet{vrsnak2004} - dash-dotted orange line, \\citet{gopalyashi2011} - dashed green line, and \\citet{mancusogarzelli2013a} - solid red line) together with a compilation of other measurements (in particular: \\citet{sakurai1994} - blue boxes, \\citet{spangler2005} - red boxes, \\citet{ingleby2007} - green boxes, \\citet{feng2011} - orange boxes, \\citet{you2012} - cyan boxes, and \\citet{bem10} - brown boxes).}\\label{fig:bmag_literature}\n\\end{figure}\n\nWith the above caveat in mind, we show that the two strong type-II bursts in this event are probably generated by two different portions of the shock (see upper right panel of Figure \\ref{fig:radio}), one driven near the CME front and the other one at the southern flank region of the CME. We point out that the angular ranges specified in Figure \\ref{fig:radio} are not intended to designate the accuracy of our results, but that they are simply meant to illustrate the angular location of the models that better fit the observed type II features. This result supports the scenario of type-II shock generation typically\narising at the CME flank due to interaction with a nearby streamer \\citep[e.g.][]{ma04,ch08}. In this case, the type-II-emitting shock front may be quasi-perpendicular and thus apt to accelerate electrons by the shock drift acceleration mechanism \\citep{ho83}.\n\n\n\\section{Discussion and Conclusions} \\label{sec:concl}\n\nThe actual limitations in our understanding of many physical phenomena occuring in the solar corona is due in first place to our limited knowledge of the coronal magnetic fields. Knowledge of its strength and orientation is primarily based on extrapolations from observations of magnetic fields in the photosphere, where the magnetic field is strong and the Zeeman effect produces a detectable splitting of atomic levels and a subsequent polarization of the emitted light. Nevertheless, extrapolations from photospheric fields are model-dependent, static (no eruptive events) and fail to reproduce accurately complex coronal topologies. For these reasons, many different techniques have been developed to measure magnetic fields in the extended corona using radio observations and taking advantage of Faraday rotation \\citep[e.g.,][]{maspan1999, mancusogarzelli2013a, mancusogarzelli2013b} and circular polarization in radio bursts \\citep[e.g.,][]{hariharan2014}, or in the lower corona with EUV images using coronal seismology \\citep[e.g.,][]{west2011} and field extrapolations bounded to 3D reconstructions \\citep[e.g.,][]{aschwanden2014}. The recent development of spectro-polarimetric measurements of magnetic field strength and orientation is now providing very promising results \\citep[e.g.,]{tomczyk2007, dove2011}, even if (due to the required polarimetric sensitivities) these techniques can be applied only in the lower corona ($h < 0.4$~$R_\\odot$).\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\includegraphics[width=0.7\\textwidth]{f11.pdf}\n\\caption{Comparison between the pre-shock coronal white light structures observed by LASCO C2 coronagraph (left) and the magnetic field strengths derived in this work in the LASCO C2 field of view (right). The dashed lines show the location where latitudinal profiles of the WL intensity and field strength have been extracted to be plotted in Figure \\ref{fig:intensitycut}. }\\label{fig:nicefig}\n\\end{figure*}\n\nRecently, an interesting technique to measure coronal fields with CME-driven shocks was proposed by \\citet{gopalyashi2011}. This technique takes advantage of the relationship derived by \\citet{russel2002} between the standoff distance of an interplanetary shock and the radius of curvature of its driver, and is applied to derive the strength of coronal fields just above the shock nose during its propagation. This technique has been applied to images obtained from white light coronagraphic observations and, recently, to CME-driven shocks observed with EUV disk imagers \\citep{gopalswamy2012b} and white light heliospheric images \\citep{poomvises2012} allowing for the first time the derivation of magnetic field strengths up to an heliocentric distance of $\\sim 200$~$R_\\odot$. Notwithstanding the above, this technique has some limitations, in particular: 1) it can be only applied to shocks driven by CMEs, and 2) it is able to provide magnetic field measurements only along the radial located at the position of the shock nose.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f12.pdf}\n\\caption{Comparison between the normalized pre-shock coronal white light structures observed by LASCO C2 coronagraph (dashed line) and the magnetic field strengths (solid line) at the constant altitude of 2.75 R$_\\odot$.}\\label{fig:intensitycut}\n\\end{figure}\n\nOn the other hand, the technique we developed here and in our previous works is able to provide measurements of the pre-shock coronal magnetic field strengths from white light observations of shock waves over all altitudes and latitudes crossed by the shock, independently of any hypothesis on the nature of the shock driver. In fact, once a 2D map for the Alfv\\'en speed and for the electron density $n_e$ are derived, the determination of the 2D coronal magnetic field strength is straightforward and is given by $B = v_\\text{A} \\sqrt{4 \\pi n_e m_p}$. The resulting 2D map of the magnetic field strength is shown in Figure \\ref{fig:fastslowwind} (top left panel) under the assumption that the solar wind speed is negligible with respect to the shock speed. Nevertheless, because the shock speed is decreasing with altitude ($v_\\text{sh}\\simeq 1200$~km~s$^{-1}$ at 2.5~$R_\\odot$\\ and $v_\\text{sh}\\simeq 1000$~km~s$^{-1}$ at 12~$R_\\odot$\\ as we measured at the shock center), while the wind speed is increasing, higher up in the corona the field will be more and more overestimated, leading to larger uncertainties. In order to quantify these uncertainties, lower limit estimates for the Alfv\\'en speed, and thus for the magnetic field, have been derived by assuming that the whole corona is pervaded at all latitudes by fast solar wind; in particular, here we assumed the fast solar wind radial profile provided by \\citet{hu1997}. The resulting 2D map for the lower limit estimate of the magnetic field strength is shown in Figure \\ref{fig:fastslowwind} (bottom left panel). Comparison between the two maps clearly shows that no significant differences are present in the lower corona, while larger differences may exist higher up. In particular, by averaging all the magnetic field radial profiles obtained at different latitudes, we conclude that the maximum difference between the upper and the lower limit estimates is on the order of a factor $\\sim 2.7$ at 12~$R_\\odot$, and smaller factors at lower altitudes (see Figure \\ref{fig:fastslowwind}, right panel).\n\nThe magnetic field values we derived here are in very good agreement with previous measurements provided in the literature at different altitudes and latitudes and obtained with many different techniques, as shown in Figure \\ref{fig:bmag_literature}. Hence, not only the radial variation of the field strength is comparable to other estimates obtained with completely different techniques, but the latitudinal modulation we derived in this work is reliable as well. We remind that the technique applied in this work for the determination of field strengths was only based on the analysis of white light coronagraphic images, which have been analyzed to derive 2D maps (projected on the plane of the sky) of the pre-shock coronal densities, shock compression ratios, shock velocities and inclination of the shock surface with respect to the radial. Then, some assumptions were needed in order to derive the magnetic field strengths: first, we assumed that above the lower boundary of the LASCO C2 occulter ($\\sim 2$~$R_\\odot$) the coronal field is radial, so that the shock inclination with respect to the radial also provides its inclination with respect to the upstream magnetic field. This is not a strong assumption, because it is well known that coronal structures (outlining the magnetic field orientation) are nearly radial above heliocentric distances of $\\sim 2$~$R_\\odot$. Second, we assumed an empirical formula for the determination of the Alfv\\'enic Mach number for the general case of an oblique shock starting from the measured shock compression ratios and shock inclination angles. The validity of this formula has been verified in a previous work \\citep{bem14} where the Alfv\\'enic Mach number was derived independently also form the analysis of white light and UV data; the verification of the same formula with MHD numerical simulations has been also recently provided by another work \\citep{bacchini2015}. Third, in order to convert the derived Alfv\\'enic Mach numbers in estimates for the Alfv\\'en speed, we assumed that the solar wind speed ahead of the shock is negligible with respect to the shock speed; as discussed above, this leads to an overestimate of the magnetic field by a factor no more than $\\sim 2.7$ at 12~$R_\\odot$, decreasing with altitude. For comparison with the white light coronal structures, the magnetic field values derived in this work are shown again in Figure \\ref{fig:nicefig}, plotted in the field of view of the LASCO C2 coronagraph (right panel), together with the original pre-CME coronal white light intensity (left panel). We also notice that the latitudinal distribution of coronal field strength is, in first approximation, anti-correlated with the white light intensity. This result is also better shown in Figure \\ref{fig:intensitycut} providing the latitudinal distribution of the normalized WL intensity and the magnetic field strength at a constant altitude of 2.75 R$_\\odot$. The observed anti-correlation is in nice agreement with what we could expect around the vertical axis of each coronal streamer, where the neutral current sheet corresponds to a region of minimum magnetic field strength.\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\includegraphics[width=0.85\\textwidth]{f13.pdf}\n\\vspace{-0.5cm}\n\\caption{Left: LASCO C2 base difference image acquired on June 7, 2011 at 07:01 UT and with the contrast of faint features enhanced using the filter provided by the JHelioviewer software. The overplot shows the location of the shock (solid white line), the center of the CME flux rope (plus symbol), and the CME propagation direction (dashed black line). Right: same frame shown in the left panel, where the overplot provides again the location of the shock (solid white line), the center of the CME flux rope (plus symbol), and the location of the shock driver (black dotted line) as derived by assuming that the relationship between the Mach number at the shock nose and the $\\Delta\/R$ ratio holds also at different latitudes away from the shock nose (see text).}\\label{fig:shock_driver}\n\\end{figure*}\n\nIn order to further support the correctness of our measurements of coronal magnetic fields, we also applied the same technique proposed by \\citet{gopalyashi2011} and based on the measurement of the shock standoff distance. In order to perform the comparison between the two techniques, we selected the LASCO C2 frame where the circular shape of the CME flux rope is better visible, shown in Figure~\\ref{fig:shock_driver}. For this frame we determined the position of the center of the flux rope (plus symbol in the left plot) and (looking at previous and subsequent frames) the CME propagation direction (dashed line in the left plot). This provides us with the identification of the shock nose, as well as a measurement of the sum between the shock standoff distance $\\Delta$ and the radius $R$ of the flux rope, which turns out to be $\\Delta + R = 1.48$ R$_\\odot$. We thus used the value of the Mach number derived as decribed above at the shock nose ($M_A = 1.50$) and derived the expected $\\Delta \/ R$ ratio, which turns out to be \\citep[see][]{gopalyashi2011}\n\\begin{equation}\n\\frac{\\Delta}{R} = K \\frac{\\left( \\gamma -1 \\right)\\, M_A^2 +2}{\\left( \\gamma +1 \\right)\\, M_A^2} \\simeq 0.45,\n\\end{equation}\nwhere $K = 0.78$ for a circular shape of the shock driver, and $\\gamma = 5\/3$. With the above numbers it turns out that $\\Delta = 0.46$ R$_\\odot$ and $R = 1.02$ R$_\\odot$. The corresponding circumference (plotted in the left panel of Figure~\\ref{fig:shock_driver}) shows a quite nice agreement with the location of the CME flux rope, thus demonstrating that our results are in good agreement with those that could be derived for the same event with the technique described by \\citet{gopalyashi2011}. Moreover, since in this work we derived measurements of the shock Mach number $M_A$ not only at the shock nose, but also at different latitudes, it is interesting to test what happens by assuming that the above relationship relating $M_A$ and the $\\Delta \/ R$ ratio holds also away from the shock nose. In particular, the right plot of Figure~\\ref{fig:shock_driver} shows the locations of the shock driver (black dotted line) as inferred by assuming different values of $M_A$ away from the shock nose along each radial starting from the same position of the center of the flux rope (plus symbol). The resulting curve shows a surprisingly nice agreement with some white light features visible between the CME flux rope and shock. This may suggest that at this time a decoupling between the flux rope and the shock is already occurring away from the shock nose, or alternatively that the side parts of shock are driven at some latitudes by the expansion of other loop-like plasma features surrounding the CME flux rope and embedded within the same CME.\n\nThe analysis performed here provides not only a new technique to derive coronal field strengths with unprecedent radial and latitudinal extension, but also very important insights into the physical relation between the type-II emitting regions and the shock front. In fact, the difference between the 2D maps we derived for the shock and the Alv\\'en speed clearly show that in the early phases (2--4~$R_\\odot$) the whole shock surface is super-Alfv\\'enic, while later on (i.e. higher up) becomes super-Alfvenic only at the nose. For a better understanding of the acceleration regions of SEP, this result has also to be considered  together with our previous finding that in the early propagation phases shocks are super-critical only at the nose and becomes sub-critical later on \\citep[e.g.][]{bem11}. At the same time, we demonstrate here with analysis of radio dynamic spectra that the emission near the front was generated later than the one produced by the flanks, in agreement with the conclusion we derived from the analysis of white light data. This suggests that the acceleration of SEP leading to gradual events could also involve at different times coronal regions located not only at different altitudes, but also at different latitudes and\/or longitudes along the shock front, as recently simulated for instance by \\citet{rodriguez2014}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum error correcting codes have been introduced as an alternative to classical\ncodes for use in quantum communication channels.\nSince the landmark papers~\\cite{shor} and~\\cite{steane96}, this field of research has grown rapidly.\nClassical codes have been used to construct good quantum codes~\\cite{calderbank96}.\nRecently, Lisonek and Singh~\\cite{singh} gave a variant of Construction X that\nproduces binary stabilizer quantum codes from arbitrary linear codes.\nIn their construction, the requirement on the\nduality of the linear codes was relaxed.\nIn this paper, we extend their work on construction X to obtain quantum error-correcting codes over finite fields of order $p^2$\nwhere $p$ is a prime number.\nWe apply our results to the dual of Hermitian repeated root cyclic codes to generate new quantum\nerror-correcting codes.\n\nThe remainder of the paper is organized as follows.\nIn Section 2, we present our main result on the extension of the quantum construction X.\nSection 3 characterizes the generator polynomial of the Hermitian dual of a repeated root cyclic code.\nWe also give the structure of cyclic codes of length $3p^s$ over\n$\\ensuremath{\\mathbb{F}_{p^2}}$ as well as the structure of the dual codes.\nOur interest in this class of codes comes from the importance of relaxing the\ncondition $(n,p)=1$, which allows us to consider codes other than the simple root codes.\n\\section{Extending Construction X for $\\f{p}$}\n\nLet $\\f{p}$ denote the finite field with $p$ elements and\n$\\fzero{p} = \\f{p} \\backslash \\{ 0\\}$.\nFor $x \\in \\f{p^2}$ we denote the \\emph{conjugate} of $x$ by $\\conj{x} = x^p$.\nLet $\\hip{x}{y} = \\sum^n_{i=1} x_i \\conj{y_i} $ be the Hermitian inner product.\nThen the \\emph{norm} of $x$ is defined as $\\norm{x} = \\hip{x}{x} = \\sum_{i=1}^n x^{p+1}$,\nand the \\emph{trace} of $x$ as $\\trace(x) = x + \\conj{x}$.\nBoth the trace and norm are mappings from $\\ensuremath{\\mathbb{F}_{p^2}}$ to $\\f{p}$.\n\nThe following lemmas will be used later.\n\\begin{lemma} \\label{l:existenceZ}\nLet $S$ be a subspace of $\\f{p^2}^n$ such that there exist $x,y$ with $\\hip{x}{y} \\neq 0$.\nThen for all $k \\in \\f{p}$, there exists $z\\in S$ with $\\norm{z} = k$.\n\\end{lemma}\n\\begin{proof}\nThis is a non-constructive proof of the existence of the required element $z$.\nWith the assumption on $x$ and $y$, let $g(c) = \\norm{cx+y} =\n(cx+y)^{p+1}$ be a polynomial of degree $p+1$ in $c$.\nWe claim that as $c$ ranges over the elements of $\\ensuremath{\\mathbb{F}_{p^2}}$, the rhs\nwill range over all elements of $\\f{p}$.\n\nAssume now that there exists some $k\\in \\ensuremath{\\mathbb{F}_{p^2}}$ such that $\\forall c\\in \\ensuremath{\\mathbb{F}_{p^2}}, g(c) \\neq k$.\nFor each $i \\in \\f{p} \\backslash k$, let $S_{i} = \\{c \\in \\ensuremath{\\mathbb{F}_{p^2}};\\, g(c)=i \\}$.\nSince the polynomial $g$ has degree $p+1$, $g$ can have at most $p+1$ roots in any field.\nThen $|S_i| \\leq p+1$, as the polynomial $g(c)-i$ can have at\nmost $p+1$ roots, and the $S_i$ partition the set $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThen $|\\ensuremath{\\mathbb{F}_{p^2}}| = p^2  \\leq \\sum_{i \\in \\f{p} \\backslash {k}} |S_i| \\leq (p+1)(p-1) = p^2-1$,\nwhich is a contradiction.\nHence the result follows.\n\\end{proof}\n\n\\begin{lemma}\\label{l:existenceB}\nLet $D$ be a subspace of $\\f{p^2}^n$ and assume that $M$ is a basis\nfor $D \\ensuremath{\\cap} D^\\ensuremath{{\\perp_h}}$. Then there exists an orthonormal set $B$\nsuch that $M \\ensuremath{\\cup} B$ is a basis for $D$.\n\\end{lemma}\n\\begin{proof}\nThe proof given here is a generalization of the proof for the\nanalogous case presented in \\citep[Theorem~2]{singh}.\nLet $W$ be a subspace of $\\f{p^2}^n$ such that\n\\begin{equation}\n\\label{e:3.1}\nD = (D \\ensuremath{\\cap} D^\\ensuremath{{\\perp_h}} ) \\oplus W,\n\\end{equation}\nand let $l = \\dim(W)$.\nFor each $0 \\leq i \\leq l$, we can construct an\northonormal set $S_i$ that is a basis for an $i$-dimensional\nsubspace $T_i$ of $W$ such that\n\\begin{equation}\n\\label{e:3.2Ti}\nW = T_i \\oplus (T_i^\\ensuremath{{\\perp_h}}\\ensuremath{\\cap} W).\n\\end{equation}\nThe process is iterative.\nDefine $S_0 := \\phi$ and suppose that for some $0 \\leq i < l$, the set $S_i$ is an orthonormal basis for $T_i$\nsuch that $dim(T_i) = i$ and (\\ref{e:3.2Ti}) holds.\nLet $x$ be a non-zero vector in $T^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$.\nThen there exists $y \\in T^\\ensuremath{{\\perp_h}}\\ensuremath{\\cap} W$ such that $\\hip{x}{y} \\neq 0$.\nIf no such $y$ exists, then $x\\in D^\\ensuremath{{\\perp_h}}$, which would\ncontradict (\\ref{e:3.1}) because the intersection of $D$ and $D^\\ensuremath{{\\perp_h}}$ is $\\{0\\}$.\nHence by Lemma \\ref{l:existenceZ}, there must exist a $z \\in T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$ such that $\\norm{z} = 1$.\nSet $S_{i+1}=S_i\\ensuremath{\\cup} \\{z\\}$.\nClearly all the elements in $S_{i+1}$ are orthogonal to each other.\nIn addition, $\\norm{s} = 1$ for all $s\\in S_{i+1}$.\n\nLet $T_{i+1}$ be the subspace spanned by $S_{i+1}$.\nAs $z \\not \\in T_{i}$ we have that $\\dim(T_{i+1}) = i+1$.\nTo show that\n\\begin{equation}\n\\label{3.1Ti+1} W = T_{i+1} \\oplus (T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W),\n\\end{equation}\nwe must first show that $T_{i+1}\\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W = {0}$.\nLet $v \\in T_{i+1} \\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$.\nAs $v \\in T_{i+1}$, we have $v = u + cz$ where $u \\in T_i$ and $c \\in \\f{p^2}$.\nSince $v \\in T_{i+1}^\\ensuremath{{\\perp_h}}$, we have for each $w \\in T_i$\nand each $d \\in \\f{p^2}$ that\n\\begin{equation*}\n0 = \\hip{u + cz}{w + dz} = \\hip{u}{w} + \\conj{d}\\hip{u}{z} +\nc\\hip{z}{w} + c\\conj{d}\\norm{z} = \\hip{u}{w} + c\\conj{d}.\n\\end{equation*}\nWe must have $c = 0$ or else $\\hip{u}{w} + cd$ would not remain\nconstant as $d$ runs over the elements of $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThus $\\hip{u}{w}=0$ for all $w \\in T_{i}$, and hence $u \\in T_{i}^\\ensuremath{{\\perp_h}}$. As $u \\in T_i$ and $T_i\n\\ensuremath{\\cap} T_{i}^\\ensuremath{{\\perp_h}} = {0}$, we obtain that $u=0$.\nHence $v$ is also $0$ and $T_{i+1}\\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W = {0}$.\n\nNext we show that $W = T_{i+1}  + (T_{i+1} \\ensuremath{\\cap} W)$.\nLet $w \\in W$.\nBy assumption $W = T_i + (T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W)$, so\nthere exist vectors $x \\in T_i$ and $y \\in T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$\nsuch that $w = x + y$.\nNow it is shown that $W=T_{i+1}+(T_{i+1}^{\\ensuremath{{\\perp_h}}}\\ensuremath{\\cap} W)$.\nBy assumption $W=T_{i}+(T_{i}^{\\ensuremath{{\\perp_h}}}\\bigcap W)$, so there\nexist vectors $x\\in T_{i}$ and $y\\in T_{i}\\ensuremath{\\cap} W$.\nClearly $x\\in T_{i+1}$ and for any $u+dz\\in T_{i+1}$ (where $u\\in\nT_{i}$ and $d\\in\\ensuremath{\\mathbb{F}_{p^2}}$), we have\n\\begin{eqnarray}\n\\hip{y-\\hip{y}{z}z}{u+dz} &=& \\hip{y}{u}+\\conj{d}\\hip{y}{z}-\\hip{y}{z}\\hip{z}{u}-\\conj{d}\\hip{y}{z}\\norm{z} \\nonumber \\\\\n&=&\\conj{d}\\hip{y}{z}-\\conj{d}\\hip{y}{z} \\nonumber \\\\\n&=&0.\n\\end{eqnarray}\nThus $y\\in T_{i+1}\\ensuremath{\\cap} W$, and hence $W=T_{i+1}+(T_{i+1}\\ensuremath{\\cap} W)$.\nThis completes the proof that (\\ref{e:3.2Ti}) implies\n(\\ref{3.1Ti+1}) assuming that the vector $z$ is chosen as described above.\n\\end{proof}\n\n\\begin{theorem}\n\\label{th:main}\nFor an $[n,k]_{p^2}$ linear code $C$, let $e = n-k-\\dim(C \\cap C^\\ensuremath{{\\perp_h}})$.\nThen there exists a quantum code with parameters\n$\\qecc{n+e}{2k-n}{d}{p}$ with $d \\geq \\min(\\wt(C), \\wt(C + C^\\ensuremath{{\\perp_h}})+1)$.\n\\end{theorem}\n\\begin{proof}\nWe start with the observation that the equation $x^2+1=0$ always has\na solution in $\\ensuremath{\\mathbb{F}_{p^2}}$. This can be proven using the fact that\n$\\ensuremath{\\mathbb{F}_{p^2}}^\\star$ is a cyclic group. Let $\\beta $ be a generator of $\\ensuremath{\\mathbb{F}_{p^2}}^*$.\nThen $\\beta^k = -1$ for some $k$, and it is also known that $-1^2 = 1$.\nHence $\\beta^{2k}=1$ and $p^2-1 | 2k$, so that $k$ is even.\nThus, $\\beta^\\frac{k}{2}$ is the required solution.\n\nAs defined previously\n\\[\ne=\\dim(C^{\\ensuremath{{\\perp_h}}})-\\dim(C\\text{\\ensuremath{\\cap}}C^{\\ensuremath{{\\perp_h}}})=\\dim(C+C^{\\ensuremath{{\\perp_h}}})-\\dim(C).\n\\]\nLet $s=\\dim(C\\cap C^{\\ensuremath{{\\perp_h}}})$, and $G$ be the matrix\n\\begin{equation}\n\\label{m:generatorG}\nG=\\begin{pmatrix}M_{s\\times n} & 0_{s\\times e}\\\\\nA_{(n-e-2s)\\times n} & 0_{(n-e-2s)\\times e}\\\\\nB_{e\\times n} & \\beta^{k\/2}I_{e\\times e}\n\\end{pmatrix},\n\\end{equation}\nwhere the size of the matrix is indicated by the subscripts, and\n$0$ and $I$ denote the zero matrix and identity matrix, respectively.\n\nFor a matrix $P$, let $r(P)$ denote the set of rows of $P$.\nThe matrix $G$ is constructed such that $r(M)$ is a basis for $C\\cap\nC^{\\ensuremath{{\\perp_h}}}$, $r(M)\\cup r(A)$ is a basis for $C$, $r(M)\\cup r(B)$ is a\nbasis for $C$, and $r(B)$ is an orthonormal set.\nThe existence of such a matrix $B$ follows from Lemma \\ref{l:existenceB}.\nNote that $r(M)\\cup r(A)\\cup r(B)$ is a basis for $C+C^{\\ensuremath{{\\perp_h}}}$ .\n\nLet $E$ be the linear code for which $G$ is a generator matrix.\nFurther, let $S$ denote the union of the first $s$ rows of $G$ and the last $e$\nrows of $G$, i.e., $S$ is the set of rows of the matrix\n\\begin{equation}\n\\label{m:generatorS}\nS=\\begin{pmatrix}M_{s\\times n} & 0_{s\\times e}\\\\\nB_{e\\times n} & \\beta^{k\/2}I_{e\\times e}\n\\end{pmatrix}.\n\\end{equation}\nWe observe that each row of $S$ is orthogonal to each row of $G$\nbecause any row from the first $s$ rows of $S$ represents a vector in\n$C\\cap C^{\\ensuremath{{\\perp_h}}}$, and hence is orthogonal with all codewords in $C+C^{\\ensuremath{{\\perp_h}}}$,\nthe code represented by $G$.\n\nConsider a row from the last $e$ rows in $S$. This row is orthogonal\nto the first $n-e-s$ rows of $G$ because they represent the code $C$\nwhile the matrix $B$ represents codewords from $C^{\\ensuremath{{\\perp_h}}}$. The rows\nof the matrix are orthogonal. Because in the case they are different\nrows in the matrix, then they are orthogonal and the $\\beta^{k\/2}I$\nmatrix part will contribute a $0$. Any row $z$ is self-orthogonal\nsince from the construction $\\norm{z}=1$ and the identity matrix\nwill contribute $-1$, giving an inner product of $0$. This completes\nthe proof of the observation. Thus, each vector from $S$ belongs to\n$E^{\\ensuremath{{\\perp_h}}}$, and the vectors in $S$ are linearly independent because\n\\[\n\\dim(E^{\\ensuremath{{\\perp_h}}})=n+e-(n-s)=e+s=|S|.\n\\]\nHence $S$ is a basis for $E^{\\ensuremath{{\\perp_h}}}$.\nSince $S$ is a subset of $G$ by construction, it follows that $E^{\\ensuremath{{\\perp_h}}}\\subseteq E$.\n\nLet $x$ be a non-zero vector in $E$ and due to the\nvertical block structure of $G$, we can write $x=(x^{1}|x^{2})$\nwhere $x^{1}\\in\\ensuremath{\\mathbb{F}_{p^2}}^{n}$ and $x^{2}\\in\\ensuremath{\\mathbb{F}_{p^2}}^{e}$.\nThus $x$ is a linear combination of rows of $G$.\nIf none of the last $e$ rows of $G$ are contained in this linear combination with a non-zero coefficient,\nthen $x^{1}\\in C\\backslash{0}$, and so $\\wt(x)=\\wt(x^{1})\\ge\\wt(C)$.\nIf some of the last $e$ rows of $G$ are in this linear combination with\na non-zero coefficient, then $x^{1}\\in C+C^{\\ensuremath{{\\perp_h}}}$ and\n$\\wt(x)=\\wt(x^{1})+\\wt(x^{2})\\ge\\wt(C+C^\\ensuremath{{\\perp_h}})+1$.\nThus $E$ is an $[n+e,k+e,d]_{p^{2}}$ code with\n$d\\ge\\min(\\wt(C),\\wt(C+C^{\\ensuremath{{\\perp_h}}})+1)$ and $E^{\\ensuremath{{\\perp_h}}}\\subseteq E$.\nThe code $E$ satisfies the required conditions, and thus the proof is complete.\n\\end{proof}\n\nMany constructions of quantum codes use self-orthogonal\ncodes~\\cite{G2010,G-G2013}, which corresponds to the case when $e=0$.\nThe results of the next section are required to construct the quantum codes in subsequent sections.\nNote that many of the results in the next section can easily be generalized to constacyclic codes.\n\n\\section{The Hermitian Dual of Repeated Roots Cyclic Codes}\nLet $p$ is a prime number and $C$ be a cyclic code of length $n$ over the finite field $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThen $C$ is given by the principal ideal $g(x)$ in $\\dfrac{\\ensuremath{\\mathbb{F}_{p^2}}[x]}{\\langle x^{n}-1 \\rangle}$,\nand so $g(x)$ is called the generator polynomial for $C$.\nWhen the length $n$ divides $p$, $C$ is called a repeated root cyclic code.\n\nIn this section, we obtain the generator polynomial of the Hermitian\ndual of a repeated root cyclic code.\nWe also give the structure of the cyclic codes of length $3p^s$ over $\\ensuremath{\\mathbb{F}_{p^2}}$ as well as the\nstructure of the dual code.\nOur interest in this class of codes comes from the importance of relaxing the condition $(n,p)=1$,\nwhich allows us to consider codes other than simple root codes.\n\nLet $f(x)=a_0+a_1 x+\\ldots + a_rx^r$ be a polynomial in\n$\\mathbb{F}_{q^2}[x]$, and $\\conj{f(x)} = \\conj{a_0} + \\conj{a_1}x + \\ldots + \\conj{a_r}x^r$.\nThe polynomial inverse of $f$ is denoted by $f^\\star(x) = x^rf(x^{-1}) = a_r+a_{r-1} x+\\ldots + a_0x^r$,\nso then $f^{\\bot}(x) = \\overline{a_r} + \\overline{a_{r-1}}x + \\ldots + \\overline{a_0}x^r$.\n\nThe following properties can easily be verified.\n\n\\begin{lemma}\n\\label{l:propConjInv}\nLet $f(x)$ and $g(x)$ be polynomials over $\\f{p^m}$.\nThen\n\\begin{enumerate}\n\\item conjugation is additive: $\\conj{f(x) + g(x)} =\\conj{f(x)} + \\conj{g(x)}$;\n\\item conjugation is multiplicative: $\\conj{f(x)g(x)} =\\conj{f(x)}\\, \\conj{g(x)}$;\n\\item polynomial inversion is additive if the polynomials have the same degree:\\\\\n${(f(x) + g(x))}^\\star ={f(x)}^\\star + {g(x)}^\\star $;\n\\item polynomial inversion is multiplicative: ${(f(x)g(x))}^\\star ={f(x)}^\\star \\,{g(x)}^\\star$;\n\\item inversion and conjugation commute with each other: $\\conj{(f(x)^\\star)} = (\\conj{f(x)})^\\star$; and\n\\item both operations are self-inverses: $(f(x)^\\star)^\\star = f(x)$ and $\\conj{\\conj{f(x)}} = f(x)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{lemma}\n\\label{l:herDualCondition1} Let $a(x) = a_0 + a_1x + \\ldots + a_{n-1}x^{n-1}$\nand\n$b(x) = b_0 + b_1x + \\ldots + b_{n-1}x^{n-1}$ be polynomials in $\\dfrac{F_{p^2}[x]}{x^n-1}$.\nThen $a(x)\\conj{b(x)} = 0$ in $\\dfrac{F_{p^2}[x]}{x^n-1}$\nif and only if $(a_0, a_1, \\ldots, a_{n-1})$ is orthogonal to\n$(\\conj{b_{n-1}}, \\conj{b_{n-2}}, \\ldots, \\conj{b_0})$ and all its cyclic shifts.\nThat is $\\hip{a}{\\conj{b^\\star}}=0  \\iff a(x)b(x)^\\bot =0$.\n\\end{lemma}\n\\begin{proof}\nIt well known (see for example~\\cite{huffman03}), that if $a(x) = a_0 + a_1x + \\ldots + a_{n-1}x^{n-1}$\nand $b(x) = b_0 + b_1x + \\ldots + b_{n-1}x^{n-1}$ are\npolynomials in $\\dfrac{F_{p^2}[x]}{x^n-1}$, then $a(x)b(x) = 0$ in\n$\\dfrac{F[x]}{x^n-1}$ if and only if $(a_0, a_1, \\ldots, a_{n-1})$\nis orthogonal to $(b_{n-1}, b_{n-2},\\ldots, b_0)$ and all its cyclic shifts.\nHence by applying this fact to $a(x)$ and\n$\\conj{b(x)}$ and noting that $\\conj{\\conj{b(x)}} = b(x)$, the result follows.\n\\end{proof}\n\nWe now use Lemma~\\ref{l:herDualCondition1} to derive an expression for the Hermitian dual\nof a cyclic code.\nLet $S\\subseteq R$ and let the annihilator be $\\ann(S) = \\{ g\\in R | fg=0, \\; \\forall f\\in S\\}$.\nThen $\\ann(S)$ is also an ideal of the ring and hence is generated by a polynomial.\n\\begin{lemma}\n\\label{l:hdToAnnihilator}\nIf $g(x)$ generates the code $C$, then $C^\\ensuremath{{\\perp_h}} = \\ann(\\conj{g(x)}^\\star)$.\n\\end{lemma}\n\\begin{proof}\nAssume that $g(x)$ generates the code $C$.\nThen each codeword in $C$ has the form  $a(x) = g(x)c(x)$.\nLet a codeword $b(x)$ lie in the Hermitian dual $C^\\ensuremath{{\\perp_h}}$.\nThen by Lemma \\ref{l:herDualCondition1} we have that\n\\[\n     a(x)b^\\bot(x)=0,\n\\]\nand by Lemma \\ref{l:propConjInv}, this is equivalent to\n\\begin{equation}\n\\label{l:annihilatorDerivation}\n b(x) (\\conj{g(x)}^\\star) =0.\n\\end{equation}\nThen by (\\ref{l:annihilatorDerivation}), we have that for a\ncodeword $b(x)$, $b(x)\\in C^\\ensuremath{{\\perp_h}} \\iff b(x)\\in \\ann(\\conj{g(x)}^\\star)$,\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:annihilatorToGeneratorPoly}\nAssume that $C=\\codegenerated{g(x)} $ is a cyclic code of length $n$ over\n$\\mathbb{F}_{p^2}$ with generator polynomial $g(x)$.\nDefine $h(x)=\\frac{x^n-1}{g(x)}$.\nThen we have that $C^{\\ensuremath{{\\perp_h}}}= \\codegenerated{h^{\\bot}(x)}$.\n\\end{lemma}\n\\begin{proof}\nFrom Lemma \\ref{l:hdToAnnihilator} it is known that $C^\\ensuremath{{\\perp_h}} = \\ann(g(x)^\\bot)$.\nThus, we must show that $\\ann(g^\\bot(x)) = \\codegenerated{h^\\bot(x)}$.\nOne way containment is easy since $\\codegenerated{h^\\bot(x)} \\subseteq \\ann(g^\\bot(x))$,\nwhich is true because $h^\\bot(x)g^\\bot(x) = (h(x)g(x))^\\bot = (x^n-1)^\\bot = 0$ by Lemma \\ref{l:propConjInv}.\nFor containment the other way, we observe that since\n$\\ann(g^\\bot(x))$ is an ideal of the polynomial ring\n$\\dfrac{\\ensuremath{\\mathbb{F}_{p^2}}[x]}{x^n-1}$, it is generated by a polynomial, say\n$b^\\bot(x)$. Then $b^\\bot(x)g^\\bot(x) = x^n -1 =\n\\lambda(x^n-1)^\\bot$ (because $b(x)$ is the smallest polynomial, so it is an equality).\nHence $b(x)g(x) = x^n -1$, so it must be that\n$b(x)= h(x)$ since both are unitary polynomials.\nThis completes the proof.\n\\end{proof}\n\n\\begin{theorem}\nLet $p > 3$ be a prime.\nThen\n\\begin{enumerate}\n\\item there exists $\\omega \\in \\ensuremath{\\mathbb{F}_{p^2}}$ such that $\\omega^3=1$ and the factorization of $x^{3p^s}-1$ into irreducible factors over $\\ensuremath{\\mathbb{F}_{p^2}}[x]$ is\n\\[\nx^{3p^s}-1 = (x-1)^{p^s}(x-\\omega)^{p^s}(x-\\omega^2)^{p^s};\n\\]\n\\item the cyclic codes of length $3p^s$ are always of the form\n\\[ \\codegenerated{(x-1)^{i}(x-\\omega)^{j}(x-\\omega^2)^{k}},\\] where $0\n\\leq i,j,k \\leq p^s$, and the code has $p^{2(3p^s-i-j-k)}$ codewords; and\n\\item the Hermitian dual of the codes have the form\n\\begin{equation} \\label{e:hdGeneratorForm}\nC^\\ensuremath{{\\perp_h}} =\n    \\begin{cases}\n        \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}} & \\text{ if } p\\equiv 1\\mod 3, \\\\\n        \\codegenerated{(x-1)^{p^s-i}(x-\\omega^2)^{p^s-j}(x-\\omega)^{p^s-k}} & \\text{ if } p\\equiv 2 \\mod 3.\n    \\end{cases}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n    \\begin{enumerate}\n    \\item Since $p$ is a prime number, $p \\neq 0 \\mod 3$, and $p^2-1 = (p+1)(p-1)$, so either\n     $p+1 = 0 \\mod 3$ or  $p-1 = 0 \\mod 3$. Therefore an element of order 3 exists in $\\ensuremath{\\mathbb{F}_{p^2}}$.\n     Let this element be $\\omega$, so then $(x-1)(x-\\omega)(x-\\omega^2) = x^3-1$.\n    In a field of characteristic $p$, it is known that $x^n-1 = (x^m-1)^p$ if $n=mp$.\n    Therefore we have that $x^{3p^s}-1 = (x^3-1)^{p^s} = ((x-1)(x-\\omega)(x-\\omega^2))^{p^s}.$\n    \\item\n        From the previous part we know that the irreducible factors are $(x-1)$, $(x-\\omega)$ and $(x-\\omega^2)$,\n        each of multiplicity $p^s$.\n        As the generator polynomial divides $x^{3p^s}-1$, the statement follows.\n    \\item\n        We know from Lemma \\ref{l:annihilatorToGeneratorPoly} that\n        \\[C^\\ensuremath{{\\perp_h}} = \\codegenerated{h^\\bot(x)}, \\]\nhence\n\\begin{align}\nC^\\ensuremath{{\\perp_h}} &= \\conj{ \\codegenerated{\\dfrac{(x-1)^{p^s}(x-\\omega)^{p^s}(x-\\omega^2)^{p^s}}{(x-1)^{i}(x-\\omega)^{j}(x-\\omega^2)^{k}}}^\\star } \\nonumber \\\\\n    &= \\conj{ \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}}^\\star } \\nonumber \\\\\n    &= \\conj{\\codegenerated{[(x-1)^{p^s-i}]^\\star[(x-\\omega)^{p^s-j}]^\\star [(x-\\omega^2)^{p^s-k}]^\\star} }  \\nonumber \\\\\n    &= \\conj{\\codegenerated{[-(x-1)^{p^s-i}][- \\omega(x-\\omega^{-1})^{p^s-j}]^\\star [- \\omega^2 (x-\\omega^{-2})^{p^s-k}]^\\star} } \\nonumber \\\\\n    & \\text{Since, } (x-1)^\\star = -x + 1 = -(x-1) , (x-\\omega)^\\star = -\\omega x+1 = -\\omega (x-\\omega^2) \\nonumber \\\\\n    &= \\conj{\\codegenerated{[(x-1)^{p^s-i}][(x-\\omega^2)^{p^s-j}][(x-\\omega)^{p^s-k}]} } \\nonumber \\\\\n    &= \\codegenerated{[(\\conj{x-1})^{p^s-i}][(\\conj{x-\\omega^2})^{p^s-j}][(\\conj{x-\\omega})^{p^s-k}]}  \\nonumber \\\\\n    &= \\codegenerated{[({x-1})^{p^s-i}][({x-\\omega^{2p}})^{p^s-j}][({x-\\omega^p})^{p^s-k}]}  \\nonumber \\\\\n    &= \\begin{cases}\n        \\codegenerated{(x-1)^{p^s-i}(x-\\omega^2)^{p^s-j}(x-\\omega)^{p^s-k}} & \\text{ if } p\\equiv 1\\mod 3, \\\\\n        \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}} & \\text{ if } p\\equiv 2 \\mod 3.\n    \\end{cases}          \\\\\n    & \\text{Since } \\omega^{p} = \\omega \\text{ if } p \\equiv 1 \\mod 3 \\text{, and } \\omega^{p} = \\omega^2 \\text{ if } p \\equiv 2 \\mod 3.\n\\end{align}\n    \\end{enumerate}\nThis completes the proof.\n\\end{proof}\n\n\\section{Extension to Simple Root Cyclic Codes}\nThis section considers cyclic codes of length $n$ over $\\ensuremath{\\mathbb{F}_{p^2}}$ such that $(p,n)=1$.\nIn this case, a cyclic code can be represented by its defining set $Z$.\nIf $m$ has order $p^{2}$ modulo $n$, then $\\f{p^{2m}}$ is the splitting field of\n$x^n-1$ containing a primitive $n$th root of unity.\nConsider a primitive root $\\beta$.\nThen $\\{ k|g(\\beta^{k})=0,\\; 0\\leq k<n \\}$ is a defining set of $C$.\nNote that this set depends on the choice of $\\beta$.\nWe can make a canonical choice for $\\beta$ by fixing a primitive element $\\alpha$ of $\\f{q^m}$ and\nletting $\\beta=\\alpha^{\\frac{q^{m}-1}{n}}$.\nLet this elements be $\\alpha$ as defined by the \\textsf{PrimitiveElement} function in Magma.\n\nFor $n$ and $m$ as above and $a\\in\\{0, \\ldots, n-1\\}$, the set $\\{aq^j\n\\mod n | 0 \\leq j < m \\}$ is called a \\emph{cyclotomic coset modulo} $n$.\nIt is well known that a defining set of a cyclic code of length $n$\nis the union of \\emph{cyclotomic cosets modulo} $n$.\nLet $\\ensuremath{\\mathbb{Z}}_{n}$ denote the set of integers modulo $n$.\nClearly defining sets can be considered as subsets of $\\ensuremath{\\mathbb{Z}}_{n}$.\nFor $S \\subset \\ensuremath{\\mathbb{Z}}_n$, denote $\\conj{S}=\\ensuremath{\\mathbb{Z}}_{n}\n\\backslash S$ and $-qS = \\{-qs \\mod n | s \\in S \\}$.\n\n\nWe now prove the following lemma.\n\\begin{lemma}\nIf $C$ is a linear cyclic code with defining set $Z$, then\n$\\dim(C^\\ensuremath{{\\perp_h}}) - \\dim (C \\ensuremath{\\cap} C^\\ensuremath{{\\perp_h}})  = |Z \\cap -pZ|$.\n\\end{lemma}\n\\begin{proof}\nLet $C$ be a linear cyclic code of length $n$, and $\\prod_{k\\in\nZ}(x-\\beta^{k})$ be the generator polynomial for $C$.\nThen from\nLemma~\\ref{l:annihilatorToGeneratorPoly} the generator polynomial\nfor $C^{\\ensuremath{{\\perp_h}}}$ is $\\prod_{k\\in-p\\conj{Z}}(x-\\beta^{k})$, and the\ngenerator polynomial for $C\\cap C^{\\ensuremath{{\\perp_h}}}$ is $\\prod_{k\\in\nZ\\cap-p\\conj{Z}}(x-\\beta^{k})$, which gives that\n\\[\n\\dim(C^{\\ensuremath{{\\perp_h}}})-dim(C\\cap C^{\\ensuremath{{\\perp_h}}})=n-|-p\\conj{Z}|-(n-|Z\\cup-p\\conj{Z}|)=|Z\\cup-p\\conj{Z}|-|-p\\conj{Z}|=|Z\\cap-pZ|.\n\\]\n\\end{proof}\n\n\\begin{theorem}\nAssume $n$ be divisible by $p^{2}-1$ and let $C$ be an $[n,k]_{4}$ cyclic\ncode with defining set $Z$ such that\n$(Z\\cap-pZ) \\subseteq T = \\{\\frac{nk}{p^{2}-1}|k\\in\\{1,\\ldots, p^{2}-1\\}\\}$.\nIf $e=|Z\\cap-pZ|$, then there exists an $[[n+e,2k-n+e,d]]_{p}$ quantum\ncode with $d\\geq\\min\\{\\wt(C),\\wt(C_{u})+1,\\wt(C+C^{\\ensuremath{{\\perp_h}}})+2\\}$ where\nthe minimum is taken over the cyclic codes $C_{u}$ with defining set\n$Z\\backslash \\{u \\}$ for each $u\\in Z\\cap-pZ$.\n\\end{theorem}\n\\begin{proof}\nThe proof requires a modification to the proof of Theorem \\ref{th:main}, in particular\nthe set of orthonormal vectors used is changed.\nFirst we observe that each of the elements in $T$ is a cyclotomic\ncoset and contains only one element.\nLet $q = p^2-1$, $n = (p^2-1)l = ql$, and $\\omega$ be a $p^2-1$-th root of unity.\nConsider the polynomials\n\\[\nb_t(x) = \\dfrac{x^n-1}{x-\\omega^t} = \\sum_{i=0}^{l-1} (x^{qi+q-1} + \\omega^t x^{qi+q-2} + \\ldots + \\omega^{(q-1)t}x^{qi}).\\]\nFor convenience, we let $\\{b_i | i\\in {0,1,\\dots, l} \\}$ also denote the corresponding codewords.\nThis is an orthonormal set because\n\\[\n\\hip{b_u}{b_v} = q\\sum_{i=0}^{l-1} (\\omega^{i (u + vp) }) =q\\sum{i=0}^{l-1} (\\omega^{i (u - v) }) = \\begin{cases} 0 & u \\neq v \\\\\n                                    ql & u = v \\end{cases}.\n\\]\nTo mitigate the $ql$ factor, we can multiply each element by a constant.\nThus, to add the rows for $B$ to the matrix, we add $U=\\{b_t | \\frac{tn}{q} \\in Z \\cap -pZ \\}$.\n\nTo prove the claim about the distance, we have 3 cases: no row from\nB is a linear combination, exactly one row from $U$ is a linear combination\nwith a non-zero coefficient, and at least two rows are a combination.\nThe proof of the first and the last cases is the same as in the proof of Theorem \\ref{th:main}.\nFor the second case, let $b_t$ be the row with non-zero coefficient.\nThen the code generated would be $span(C, b_t)$, which is precisely the cyclic code with defining set\n$Z \\backslash \\{\\frac{tn}{3}\\}$.\nThis completes the proof.\n\\end{proof}\n\n\\section{Examples of Codes Generated}\nIn this section, a comprehensive table of codes generated is\npresented. Many of these codes have parameters better than the best\nknown quantum codes.\n\\begin{longtable}{| p{.25\\textwidth} | p{.50\\textwidth} |  p{.25\\textwidth}|}\n\\hline\nNew Codes & Generator Polynomial& Best Known Binary QECC \\\\\n\\hline $[[33, 31, 2]]_{3}$ & $x^{13} + \\alpha^5*x^{12} +\n\\alpha^7*x^{11} + \\alpha^2*x^{10} + 2*x^9 + 2*x^8 + \\alpha^3*x^7 +\n\\alpha^6*x^6 + 2*x^5 + \\alpha^3*x^4 + \\alpha^3*x^3 + \\alpha^6*x^2 +\n\\alpha^2$\n &$[[33, 31, 1]]_2$ \\\\\n$[[35, 33, 2]]_{3}$ & $x + 1$ &$[[35, 33, 1]]_2$ \\\\\n$[[39, 37, 2]]_{3}$ & $x+2$ &$[[39, 37, 1]]_2$ \\\\\n$[[40, 26, 5]]_{3}$ & $x^7 + \\alpha*x^6 + \\alpha*x^5 + \\alpha^6*x^4 + x^3 + \\alpha^7*x^2 + \\alpha^5*x + \\alpha$ &$[[40, 26, 4]]_2$ \\\\\n$[[40, 24, 6]]_{3}$ & $x^8 + \\alpha^3*x^7 + \\alpha*x^6 + \\alpha^7*x^5 + 2*x^4 + x^3 + \\alpha^2*x^2 + \\alpha*x + \\alpha^2$ &$[[40, 24, 5]]_2$ \\\\\n$[[41, 39, 2]]_{3}$ & $x + 1$ &$[[41, 39, 1]]_2$ \\\\\n$[[41, 9, 11]]_{3}$ & $x^{16} + \\alpha^5*x^{15} + \\alpha^5*x^{14} + \\alpha*x^{13} + \\alpha^6*x^{12} + \\alpha^2*x^{11} + 2*x^9 + \\alpha^7*x^7 + 2*x^6 + \\alpha^3*x^5 + 2*x^4 + \\alpha^6*x^3 + \\alpha^7*x^2 + \\alpha^7$ &$[[41, 9, 8]]_2$ \\\\\n$[[41, 19, 8]]_{3}$ & $x^{11} + \\alpha*x^{10} + \\alpha*x^9 + \\alpha^5*x^8 + 2*x^7 + \\alpha^3*x^6 + \\alpha^2*x^5 + \\alpha^3*x^4 + x^2 + \\alpha^2*x + 2$ &$[[41, 19, 6]]_2$ \\\\\n$[[40, 20, 7]]_{3}$ & $x^{10} + \\alpha^{6}*x^{9} + \\alpha^{7}*x^{6} + \\alpha^{3}*x^{5} + {2}*x^{4} + \\alpha^{3}*x^{3} + \\alpha*x + \\alpha^{2}$ &$[[40, 20, 6]]_2$ \\\\\n$[[41, 25, 6]]_{3}$ & $x^{8} + \\alpha^{6}*x^{7} + \\alpha^{7}*x^{6} + x^{4} + \\alpha^{5}*x^{3} + \\alpha^{3}*x^{2} + {2}*x + \\alpha^{7}$ &$[[41, 25, 4]]_2$ \\\\\n$[[40, 10, 10]]_{3}$ & $x^{15} + \\alpha^{6}*x^{14} + x^{13} +\n\\alpha^{5}*x^{{12}} + \\alpha^{2}*x^{11} + x^{10} + \\alpha^{3}*x^{9}\n+ \\alpha^{5}*x^{8} + x^{7} + {2}*x^{6} + \\alpha^{7}*x^{5} +\n\\alpha^{7}*x^{4} + x^{3} + \\alpha*x^{2} + \\alpha^{5}*x + \\alpha^{5}\n$ &$[[40, 10, 8]]_2$ \\\\\n$[[40, 16, 8]]_{3}$ & $x^{12} + \\alpha^{3}*x^{11} +\n\\alpha^{3}*x^{10} + x^{9} + \\alpha^{5}*x^{8} + \\alpha^{5}*x^{7} +\n\\alpha^{5}*x^{6} + \\alpha^{5}*x^{5} + {2}*x^{4} + x^{3} +\n\\alpha^{6}*x + \\alpha^{2}\n$ &$[[40, 16, 6]]_2$ \\\\\n$[[41, 13, 9]]_{3}$ & $x^{14} + {2}*x^{13} + \\alpha*x^{12} +\n{2}*x^{10} + \\alpha^{2}*x^{9} + x^{8} + \\alpha^{5}*x^{7} +\n\\alpha^{5}*x^{6} + x^{5} + \\alpha^{3}*x^{4} + \\alpha^{6}*x^{3} +\n{2}*x^{2} + \\alpha^{3}*x + \\alpha^{5}\n$ &$[[41, 13, 7]]_2$ \\\\\n$[[41, 21, 7]]_{3}$ & $x^{10} + \\alpha^{7}*x^{9} + \\alpha^{6}*x^{7} + \\alpha^{6}*x^{6} + {2}*x^{5} + \\alpha^{7}*x^{4} + x^{3} + \\alpha^{3}*x + \\alpha^{5}$ &$[[41, 21, 6]]_2$ \\\\\n$[[41, 27, 5]]_{3}$ & $x^{7} + {2}*x^{6} + \\alpha^{3}*x^{5} + \\alpha^{7}*x^{4} + \\alpha^{7}*x^{3} + {2}*x^{2} + \\alpha^{3}*x + \\alpha^{2}$ &$[[41, 27, 4]]_2$ \\\\\n$[[33, 31, 2]]_{3}$ & $ x + {1}$ &$[[33, 31, 1]]_2$ \\\\\n$[[40, 26, 5]]_{3}$ & $x^{7} + \\alpha*x^{6} + \\alpha*x^{5} + \\alpha^{6}*x^{4} + x^{3} + \\alpha^{7}*x^{2} + \\alpha^{5}*x + \\alpha$ &$[[40, 26, 4]]_2$ \\\\\n$[[40, 24, 6]]_{3}$ & $x^{8} + \\alpha^{3}*x^{7} + \\alpha*x^{6} + \\alpha^{7}*x^{5} + {2}*x^{4} + x^{3} + \\alpha^{2}*x^{2} + \\alpha*x + \\alpha^{2}$ &$[[40, 24, 5]]_2$ \\\\\n$[[41, 39, 2]]_{3}$ & $ x + {1}$ &$[[41, 39, 1]]_2$ \\\\\n$[[41, 9, 11]]_{3}$ & $x^{16} + \\alpha^{5}*x^{15} + \\alpha^{5}*x^{14} + \\alpha*x^{13} + \\alpha^{6}*x^{12} + \\alpha^{2}*x^{11} + {2}*x^{9} + \\alpha^{7}*x^{7} + {2}*x^{6} + \\alpha^{3}*x^{5} + {2}*x^{4} + \\alpha^{6}*x^{3} + \\alpha^{7}*x^{2} + \\alpha^{7}$ &$[[41, 9, 8]]_2$ \\\\\n$[[41, 19, 8]]_{3}$ & $x^{11} + \\alpha*x^{10} + \\alpha*x^{9} +\n\\alpha^{5}*x^{8} + {2}*x^{7} + \\alpha^{3}*x^{6} + \\alpha^{2}*x^{5} +\n\\alpha^{3}*x^{4} + x^{2} + \\alpha^{2}*x + {2}\n$ &$[[41, 19, 6]]_2$ \\\\\n$[[40, 20, 7]]_{3}$ & $x^{10} + \\alpha^{6}*x^{9} + \\alpha^{7}*x^{6} + \\alpha^{3}*x^{5} + {2}*x^{4} + \\alpha^{3}*x^{3} + \\alpha*x + \\alpha^{2}$ &$[[40, 20, 6]]_2$ \\\\\n$[[41, 25, 6]]_{3}$ & $x^{8} + \\alpha^{6}*x^{7} + \\alpha^{7}*x^{6} +\nx^{4} + \\alpha^{5}*x^{3} + \\alpha^{3}*x^{2} + {2}*x + \\alpha^{7}\n$ &$[[41, 25, 4]]_2$ \\\\\n$[[40, 10, 10]]_{3}$ & $x^{15} + \\alpha^{6}*x^{14} + x^{13} +\n\\alpha^{5}*x^{12} + \\alpha^{2}*x^{11} + x^{10} + \\alpha^{3}*x^{9} +\n\\alpha^{5}*x^{8} + x^{7} + {2}*x^{6} + \\alpha^{7}*x^{5} +\n\\alpha^{7}*x^{4} + x^{3} + \\alpha*x^{2} + \\alpha^{5}*x + \\alpha^{5}\n$ &$[[40, 10, 8]]_2$ \\\\\n$[[40, 16, 8]]_{3}$ & $x^{12} + \\alpha^{3}*x^{11} + \\alpha^{3}*x^{10} + x^{9} + \\alpha^{5}*x^{8} + \\alpha^{5}*x^{7} + \\alpha^{5}*x^{6} + \\alpha^{5}*x^{5} + {2}*x^{4} + x^{3} + \\alpha^{6}*x + \\alpha^{2}$ &$[[40, 16, 6]]_2$ \\\\\n$[[41, 13, 9]]_{3}$ & $x^{14} + {2}*x^{13} + \\alpha*x^{12} +\n{2}*x^{10} + \\alpha^{2}*x^{9} + x^{8} + \\alpha^{5}*x^{7} +\n\\alpha^{5}*x^{6} + x^{5} + \\alpha^{3}*x^{4} + \\alpha^{6}*x^{3} +\n{2}*x^{2} + \\alpha^{3}*x + \\alpha^{5}\n$ &$[[41, 13, 7]]_2$ \\\\\n$[[41, 21, 7]]_{3}$ & $x^{10} + \\alpha^{7}*x^{9} + \\alpha^{6}*x^{7} + \\alpha^{6}*x^{6} + {2}*x^{5} + \\alpha^{7}*x^{4} + x^{3} + \\alpha^{3}*x + \\alpha^{5}$ &$[[41, 21, 6]]_2$ \\\\\n$[[41, 27, 5]]_{3}$ & $x^{7} + {2}*x^{6} + \\alpha^{3}*x^{5} + \\alpha^{7}*x^{4} + \\alpha^{7}*x^{3} + {2}*x^{2} + \\alpha^{3}*x + \\alpha^{2}$ &$[[41, 27, 4]]_2$ \\\\\n$[[41, 9, 11]]_{5}$ & $x^{16} + \\alpha*x^{15} + \\alpha^{23}*x^{14} + \\alpha^{3}*x^{13} + {4}*x^{12} + \\alpha^{15}*x^{11} + {3}*x^{10} + \\alpha^{10}*x^{8} + \\alpha^{3}*x^{7} + \\alpha^{14}*x^{5} + \\alpha^{8}*x^{4} + \\alpha^{19}*x^{3} + {4}*x^{2} + \\alpha^{17}*x + \\alpha^{8}$ &$[[41, 9, 8]]_2$ \\\\\n$[[40, 2, 12]]_{5}$ & $x^{19} + \\alpha^{20}*x^{18} + \\alpha^{22}*x^{17} + \\alpha^{10}*x^{16} + \\alpha^{3}*x^{15} + \\alpha^{20}*x^{14} + \\alpha^{21}*x^{13} + \\alpha^{22}*x^{12} + \\alpha^{20}*x^{11} + \\alpha^{8}*x^{10} + {3}*x^{9} + \\alpha^{22}*x^{8} + {4}*x^{7} + \\alpha^{11}*x^{6} + \\alpha^{23}*x^{5} + \\alpha^{22}*x^{4} + \\alpha^{8}*x^{3} + \\alpha^{5}*x^{2} + \\alpha^{9}*x + \\alpha^{4}$ &$[[40, 2, 10]]_2$ \\\\\n$[[41, 5, 12]]_{5}$ & $x^{18} + \\alpha^{15}*x^{17} +\n\\alpha^{19}*x^{16} + \\alpha^{23}*x^{15} + \\alpha^{13}*x^{14} +\n\\alpha^{5}*x^{13} + \\alpha^{7}*x^{12} + x^{11} + \\alpha^{17}*x^{10}\n+ \\alpha^{3}*x^{9} + \\alpha^{19}*x^{8} + \\alpha^{19}*x^{7} +\n\\alpha^{5}*x^{6} + \\alpha^{11}*x^{5} + \\alpha^{8}*x^{4} +\n\\alpha*x^{3} + \\alpha^{10}*x^{2} + \\alpha^{5}*x + {1}\n$ &$[[41, 5, 9]]_2$ \\\\\n$[[40, 6, 11]]_{5}$ & $x^{17} + \\alpha^{10}*x^{16} +\n\\alpha^{4}*x^{15} + \\alpha^{22}*x^{14} + \\alpha^{9}*x^{13} +\n\\alpha*x^{12} + {3}*x^{11} + {4}*x^{10} + \\alpha^{5}*x^{9} +\n\\alpha^{16}*x^{8} + \\alpha^{19}*x^{7} + \\alpha^{22}*x^{6} +\n\\alpha^{9}*x^{5} + \\alpha^{4}*x^{4} + {4}*x^{3} + \\alpha^{17}*x^{2}\n+ \\alpha^{16}*x + {4}\n$ &$[[40, 6, 8]]_2$ \\\\\n$[[39, 15, 9]]_{5}$ & $x^{12} + {2}*x^{11} + \\alpha^{5}*x^{10} +\n\\alpha^{16}*x^{9} + \\alpha^{3}*x^{8} + \\alpha^{3}*x^{7} +\n\\alpha^{13}*x^{6} + \\alpha^{15}*x^{5} + \\alpha^{22}*x^{4} + x^{3} +\n\\alpha^{9}*x^{2} + {2}*x + \\alpha^{16}\n$ &$[[39, 15, 7]]_2$ \\\\\n$[[39, 23, 5]]_{5}$ & $x^{8} + \\alpha^{21}*x^{7} + {3}*x^{6} + \\alpha*x^{5} + \\alpha^{16}*x^{4} + \\alpha^{17}*x^{3} + \\alpha^{2}*x^{2} + \\alpha^{21}*x + \\alpha^{16}$ &$[[39, 23, 4]]_2$ \\\\\n$[[40, 22, 6]]_{5}$ & $x^{9} + \\alpha^{7}*x^{8} + \\alpha^{8}*x^{7} + \\alpha^{2}*x^{6} + \\alpha^{21}*x^{5} + \\alpha^{9}*x^{4} + \\alpha^{14}*x^{3} + \\alpha^{20}*x^{2} + \\alpha^{19}*x + {4}$ &$[[40, 22, 5]]_2$ \\\\\n$[[41, 21, 7]]_{5}$ & $x^{10} + \\alpha^{3}*x^{9} + x^{8} + \\alpha^{10}*x^{7} + \\alpha^{2}*x^{6} + \\alpha^{22}*x^{5} + \\alpha^{23}*x^{4} + \\alpha*x^{3} + \\alpha^{22}*x^{2} + \\alpha^{15}*x + {1}$ &$[[41, 21, 6]]_2$ \\\\\n$[[39, 11, 10]]_{5}$ & $x^{14} + \\alpha^{15}*x^{12} + \\alpha^{21}*x^{11} + \\alpha^{16}*x^{10} + \\alpha^{16}*x^{9} + {4}*x^{8} + \\alpha^{3}*x^{7} + {4}*x^{5} + {4}*x^{4} + \\alpha^{22}*x^{3} + \\alpha^{19}*x^{2} + \\alpha^{9}*x + \\alpha^{8}$ &$[[39, 11, 8]]_2$ \\\\\n$[[39, 19, 7]]_{5}$ & $x^{10} + \\alpha^{14}*x^{8} + \\alpha^{14}*x^{7} + \\alpha^{4}*x^{6} + \\alpha*x^{5} + {4}*x^{4} + \\alpha^{8}*x^{3} + \\alpha^{3}*x^{2} + \\alpha^{14}*x + \\alpha^{8}$ &$[[39, 19, 5]]_2$ \\\\\n$[[40, 18, 8]]_{5}$ & $x^{11} + x^{10} + \\alpha^{13}*x^{9} + \\alpha^{17}*x^{8} + {2}*x^{7} + \\alpha^{14}*x^{6} + \\alpha^{17}*x^{5} + {3}*x^{4} + \\alpha^{15}*x^{3} + \\alpha^{21}*x^{2} + \\alpha^{23}*x + {4}$ &$[[40, 18, 6]]_2$ \\\\\n$[[31, 13, 6]]_{5}$ & $x^{9} + {3}*x^{8} + x^{6} + x^{5} + {4}*x^{4} + x^{3} + {3}*x^{2} + x + {4}$ &$[[31, 13, 5]]_2$ \\\\\n$[[32, 0, 11]]_{5}$ & $x^{16} + {3}*x^{15} + {2}*x^{14} + x^{13} + x^{11} + {2}*x^{10} + x^{9} + x^{8} + {4}*x^{7} + x^{6} + x^{5} + {3}*x^{4} + {2}*x^{3} + x + {1}$ &$[[32, 0, 10]]_2$ \\\\\n$[[31, 7, 8]]_{5}$ & $x^{12} + {4}*x^{11} + {4}*x^{10} + {2}*x^{9} + {4}*x^{8} + {2}*x^{7} + x^{6} + {3}*x^{5} + x^{4} + x^{3} + {2}*x + {1}$ &$[[31, 7, 7]]_2$ \\\\\n$[[32, 12, 7]]_{5}$ & $x^{10} + {3}*x^{7} + x^{6} + x^{5} + x^{4} + {3}*x^{2} + {4}*x + {1}$ &$[[32, 12, 6]]_2$ \\\\\n$[[31, 25, 3]]_{5}$ & $ x^{3} + x^{2} + {3}*x + {4}$ &$[[31, 25, 2]]_2$ \\\\\n$[[32, 18, 5]]_{5}$ & $x^{7} + {3}*x^{5} + {3}*x^{3} + {4}*x^{2} + {4}$ &$[[32, 18, 4]]_2$ \\\\\n$[[32, 6, 9]]_{5}$ & $x^{13} + {2}*x^{11} + x^{10} + x^{9} +\n{4}*x^{8} + {3}*x^{6} + {2}*x^{5} + {4}*x^{3} + {4}*x^{2} + {4}*x +\n{4}\n$ &$[[32, 6, 8]]_2$ \\\\\n$[[33, 31, 2]]_{5}$ & $x + {4}$ &$[[33, 31, 1]]_2$ \\\\\n$[[35, 33, 2]]_{5}$ & $x + {4}$ &$[[35, 33, 1]]_2$ \\\\\n$[[37, 35, 2]]_{5}$ & $x + \\alpha^{16}$ &$[[37, 35, 1]]_2$ \\\\\n$[[37, 35, 2]]_{5}$ & $x + \\alpha^{16}$ &$[[37, 35, 1]]_2$ \\\\\n$[[25, 23, 2]]_{5}$ & $x + \\alpha^{16}$ &$[[25, 23, 1]]_2$ \\\\\n$[[24, 20, 3]]_{5}$ & $x^{2} + \\alpha^{8}*x + \\alpha^{17}$ &$[[24, 20, 2]]_2$ \\\\\n$[[25, 21, 3]]_{5}$ & $x^{2} + \\alpha^{13}*x + \\alpha^{17}\n$ &$[[25, 21, 2]]_2$ \\\\\n$[[24, 18, 4]]_{5}$ & $x^{3} + \\alpha^{10}*x^{2} + \\alpha^{16}*x +\n{3}\n$ &$[[24, 18, 2]]_2$ \\\\\n$[[25, 19, 4]]_{5}$ & $x^{3} + \\alpha^{19}*x^{2} + \\alpha^{10}*x + \\alpha^{21}$ &$[[25, 19, 2]]_2$ \\\\\n$[[25, 17, 5]]_{5}$ & $x^{4} + \\alpha^{7}*x^{3} + {4}*x^{2} + \\alpha^{16}*x + {3}$ &$[[25, 17, 3]]_2$ \\\\\n$[[33, 31, 2]]_{5}$ & $x + {4}$ &$[[33, 31, 1]]_2$ \\\\\n$[[32, 0, 13]]_{7}$ & $x^{16} + {2}*x^{15} + {3}*x^{14} + {4}*x^{12} + x^{11} + {4}*x^{10} + x^{9} + {5}*x^{8} + {4}*x^{7} + {6}*x^{6} + {5}*x^{5} + {2}*x^{4} + {3}*x^{3} + {3}*x^{2} + {4}*x + {1}$ &$[[32, 0, 10]]_2$ \\\\\n$[[31, 1, 12]]_{7}$ & $x^{15} + {3}*x^{14} + {6}*x^{13} + {6}*x^{12} + {3}*x^{11} + {4}*x^{10} + x^{9} + {2}*x^{8} + {4}*x^{6} + {3}*x^{5} + x^{4} + {3}*x^{3} + {6}*x^{2} + {2}*x + {6}$ &$[[31, 1, 11]]_2$ \\\\\n$[[33, 21, 5]]_{7}$ & $x^{6} + \\alpha^{42}*x^{5} + \\alpha^{33}*x^{4} + \\alpha^{20}*x^{3} + \\alpha^{30}*x^{2} + \\alpha^{6}*x + \\alpha^{15}$ &$[[33, 21, 4]]_2$ \\\\\n$[[33, 31, 2]]_{7}$ & $x + {6}$ &$[[33, 31, 1]]_2$ \\\\\n\\hline\n\\caption{Comparison of the codes obtained using Theorem 3 presented here and the best known binary QECC}\n\\label{tab:AgainstBinaryQECC}\n\\end{longtable}\n\n\\section{Codes Generated from Repeated Root Cylic Codes}\n Codes of length $3p^s$ on field of size $p^2$.\n\\begin{longtable}{| p{.33\\textwidth} | p{.33\\textwidth} |  p{.34\\textwidth}|}\n\\hline\n Codes & Codes & Codes \\\\ \\hline\n$[[15,9,2]]_{25}$ & $[[15,7,3]]_{25}$ & $[[16,6,4]]_{25}$ \\\\\n$[[75,69,2]]_{25}$ &$[[75,59,3]]_{25}$ &$[[75,49,4]]_{25}$ \\\\\n$[[82,26,5]]_{25}$ & & \\\\\n$[[375,369,2]]_{25}$ & $[[375,319,3]]_{25}$ &$[[375,269,4]]_{25}$ \\\\\n$[[21,15,2]]_{49}$ & $[[21,13,3]]_{49}$ & $[[21,11,4]]_{49}$ \\\\\n$[[21,7,5]]_{49}$ & $[[22,8,5]]_{49}$ &$ [[21,5,6]]_{49}$ \\\\\n$[[23,1,7]]_{49}$ & & \\\\\n$[[147,141,2]]_{49}$ &$[[147,127,3]]_{49}$ &$[[147,113,4]]_{49}$ \\\\\n$[[147,85,5]]_{49}$ &\n$[[147,71,6]]_{49}$ & \\\\\n\\hline\n\n\\hline\n\\caption{Quantum codes obtained from repeated root cyclic codes using Theorem 8}\n\\label{tab:ConstAgainstBinaryQECC}\n\\end{longtable}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Photonic logic}\n\nPhotonic systems are an attractive choice for quantum information processing\nbecause photons form a natural interface with optical telecommunications and\nexisting telecoms technology. However, making this choice raises a number of\nissues which have to be addressed. It becomes important to be able to\ngenerate single photons, a challenge, but there are now various schemes available\n\\cite{Lounis, Yamamoto, Matthias}.  A key issue, however, concerns how to process\nsingle photons as qubits and this is the issue addressed in this paper along\nwith the fact that single photons are very \\emph{fragile}. On the other hand\nsome aspects of photonic computation are very straightforward. For example\nthe wires of quantum circuits can be represented by optical paths and the\ndelay of a photonic qubit is a simple way of achieving changes of phase.\n\nIn the following we will use the optical language of photons but in the\ndiscussion of the experimental realisation of the models of this work note\nthat we will also consider microwave photons.\n\nA simple way to realise photonic qubits would be if the presence of a photon\nsignifies a state $\\k{1}$ and the absence of the photon indicates the state\n$\\k{0}$. To achieve an entangling quantum gate we could try to arrange for\nthese qubits to interact through the action of a beam-splitter:\n\\begin{eqnarray}\n  \\label{eq:map-Q-SBS}\n    \\k{00} &\\longrightarrow&  \\k{00} \\nonumber\\\\\n    \\k{01} &\\longrightarrow&  \\frac{1}{\\sqrt{2}}(\\k{01} + i\\k{10})  \\nonumber\\\\\n    \\k{10} &\\longrightarrow&  \\frac{1}{\\sqrt{2}}(\\k{10} + i\\k{01}) \\nonumber\\\\\n    \\k{11} &\\longrightarrow&  \\frac{1}{\\sqrt{2}}(\\k{20} +  \\k{02}) .\n\\end{eqnarray}\nHowever, this unfortunately produces states like $\\k{20}$ which are not\nqubit encodings in the original simple scheme.\nAn alternative approach would be to try a non-linear cross-Kerr interaction\n\\cite{Imoto, Yamamoto2} where\n$$\n  H_{x}  = \\chi_x \\hat a^\\dagger_1 \\hat a_1 \\hat a^\\dagger_2 \\hat a_2 .\n$$\nThen the time evolution $\\exp(-i H_{x} t \/\\hbar)$, acting on $\\k{00},\n\\k{01}$ and $ \\k{10}$, results in no change to the state, and the input\nstate $ \\k{11}$ is modified by a phase factor $\\exp(-i \\chi_x t \/\\hbar )$.\nIf $ \\chi_x t \/\\hbar = \\pi$, the CZ gate (controlled-$Z$ gate), could be\nrealised where: \\index{CZ gate}\\index{gate!CZ}\n\\begin{eqnarray}\n  \\label{eq:map-Q-CZ}\n    \\k{00} &\\longrightarrow&  \\k{00} \\nonumber\\\\\n    \\k{01} &\\longrightarrow&  \\k{01} \\nonumber\\\\\n    \\k{10} &\\longrightarrow&  \\k{10} \\nonumber\\\\\n    \\k{11} &\\longrightarrow& -\\k{11} .\n\\end{eqnarray}\nThis can be represented by the table\n\\begin{equation}\n  \\label{eq:mat-Q-CZ}\n  \\mbox{CZ} \\equiv \\left[ \\begin{array}{rrrr}%\n1& & & \\\\\n &1& & \\\\\n & &1& \\\\\n & & &-1\\\\\n\\end{array}\\right]  .\n\\end{equation}\nHere, and in the following we will assume that such tables are expressed in\nthe standard qubit basis $\\{\\k{00}, \\k{01}, \\k{10}, \\k{11}\\}$.  The CZ-gate\nis a universal quantum gate when combined with single qubit rotations.\nHowever, for the Hamiltonian $H_{x} = \\chi_x \\hat a^\\dagger_1 \\hat a_1 \\hat a^\\dagger_2 \\hat a_2$ the\nnon-linearity is too weak to achieve this simple gate in this way\n\\cite{Schmidt, Kok2}.\n\nIn the linear optical quantum computing vision \\cite{Kok} non-linear optics\ncan be avoided in qubit processing by including photon detection in the\ncomputation process. Measurement projection is an interruptive and hence\nnon-linear process which can be configured for quantum information\nprocessing and which includes heralding to indicate the successful arrival\nof photons through the processor. In achieving this the KLM scheme (Knill,\nLaflame, Milburn \\cite{KLM}) was a breakthrough. Here the undesirable output\nstates $\\k{02}$ and $\\k{20}$ are avoided by the use of a special non-linear\nsign gate indicated by NLS. The NLS gate has the special property that for\nthe general input state $ \\alpha\\k{0} + \\beta\\k{1} + \\gamma\\k{2}\n\\longrightarrow \\alpha\\k{0} + \\beta\\k{1} - \\gamma\\k{2}$ so that there is a\nsign change for the amplitude of the $\\k2$ state. This means that this gate\nhas no effect on the three basic input states $\\{\\k{00}, \\k{01}, \\k{10}\\}$\nwhich work well with a beam-splitter, and thus the gate acts as a\nMach-Zehnder interferometer. For the $\\k{11}$ input, however, the change of\nsign from the NLS components ensures that the output is a sign flipped\n$\\k{11}$ state.  However, the NLS gate has a maximum theoretical probability\nof 1\/4 for success \\cite{Rudolph}. This can start to be an issue if very\nlarge numbers of gate operations are required.\n\nFor these reasons we will move away from flying photonic qubits in the\nfollowing and examine the possibilities for stationary photonic qubits in a\ncavity.  The advantage is that we already know we can realise an extremely\nstrong non-linear interaction between photonic qubits, i.e.\\ through a\nmultimode Jaynes-Cummings model (JCM). We already know in the case of the\nsingle-mode JCM that the interaction can be so strong that the system\nexhibits Rabi oscillations \\cite{Jaynes}. However, a critical issue to\nexamine is the decay of the system, both through cavity decay and through\natomic decay (cavity modified spontaneous emission).\n\nIn the following we consider some multimode, multi-$\\Lambda$ cavity QED\nsystems for quantum information processing. We start with a simple two-mode,\nsingle $\\Lambda$ system in section~\\ref{sec:Larson}. After examining this\nsystem's weaknesses we introduce a four-mode double-$\\Lambda$ system in\nsection~\\ref{sec:MultiphotonLogic}. Section~\\ref{sec:FourModesVariants}\nexamines the dynamics of the four mode resonance with and without intermediate resonant\nstates: intermediate resonant states can achieve a speed-up. This study includes an examination of\ndecoherence effects. Section~\\ref{sec:ThreeAndOneQubitGates} examines some\nthree-qubit gates and single qubit rotations. The paper concludes in\nsection~\\ref{sec:Conclude} and several Appendices follow with details of\nsome of the calculations of effective Hamiltonians for these multi-state,\nmultimode systems.\n\n\n\\section{A two-mode $\\Lambda$-system}\n\\label{sec:Larson}\n\nIn Ref.\\ \\cite{Barry2}, a system involving two EM modes interacting with a\nthree level atomic $\\Lambda$-system (see Fig.~\\ref{fig:Larson}) has been\nexamined.\n\\begin{figure}\\centering\n  \n   \\includegraphics[width=\\bmgstdfigwidthshorter]{figure1}\n\n  \\caption{A three-level atom in $\\Lambda$-configuration, with lower levels\n    $|a\\rangle$, $|b\\rangle$ and upper level $|c\\rangle$, interacts with two\n    cavity modes $\\omega_1$ and $\\omega_2$. The states $|b\\rangle$ and\n    $|a\\rangle$ couple to $|c\\rangle$ through a dipole interaction with modes\n    one and two respectively.} \n\n\\label{fig:Larson}\n\\end{figure}\nThe Hamiltonian for this system, in the Rotating Wave Approximation (RWA), is then given by\n\\begin{align}\\label{eq:LarsonHamileff}\n H_I = &-\\Delta_b\\OP{n}_1 + \\Gee{cb}{1}\\left(\\Ann_1\\Sig{bc} + \\Cre_1\\Sig{cb}\\right)\n\\nonumber       \\\\\n&-\\Delta_a \\OP{n}_2 + \\Gee{ac}{2}\\left(\\Ann_2\\Sig{ac} +\n         \\Cre_2\\Sig{ca}\\right)\\ ,\n\\end{align}\nwhere $\\hat{a}_{1}$ and $\\hat{a}_{2}$ are the boson operators for the two\nmodes, the atomic operators have the form $\\sigma_{\\mu \\nu}\\equiv\n|\\mu\\rangle \\langle \\nu|$, and $g_{1,2}$ are the atom-field coupling\nconstants. The definitions of the system detunings are\n$\\Delta_a=(\\omega_c-\\omega_{a})-\\omega_2$ and\n$\\Delta_b=(\\omega_c-\\omega_b)-\\omega_1$.\n\nWe can consider the limit for a two-photon process when the detuning\n$\\Delta_a = \\Delta_b \\rightarrow \\Delta$. In this limit we can perform a\nstandard adiabatic elimination \\cite{Gerry} to realise an effective\nHamiltonian\n\\begin{equation}\n  H=g(\\hat a_2^{\\dagger}\\hat a_1\\hat \\sigma^-+\\hat a_2 \\hat\n  a_1^{\\dagger}\\sigma^+)\\ ,\n\\label{Larsoneffham}\n\\end{equation}\nwhere we have introduced operators that effectively make direct transitions\nbetween the two ground states, i.e.\\ $\\hat\\sigma^+=|b\\rangle\\langle a|$,\n$\\hat\\sigma^-=|a\\rangle\\langle b|$.  The adiabatic elimination results in an\neffective coupling\n\\begin{equation}\n g=g_{ac}g_{bc}\/\\Delta \\ .\\label{eq:Larsongeff}\n\\end{equation}\nThis system makes transitions between the two ground states if it is\npossible to extract a photon from one of the modes and transfer it to the\nother one.\nWe can quickly find a general solution to the Hamiltonian\n(\\ref{Larsoneffham}) which is reminiscent of the Jaynes-Cummings model\nitself, but with two-index terms in the Rabi frequency because of the two\nmodes \\cite{Barry2}: \n\\begin{align*}\n|\\Psi(t)\\rangle \\simeq &\n\\sum_{n,m}C_n^{(1)}C_m^{(2)}\\left\\{c_a\\left[\\cos(gt\\sqrt{(n+1)m})|n,m,a\\rangle\\right.\\right.\n\\\\ &  \\left.-i\\sin(gt\\sqrt{(n+1)m})|n+1,m-1,b\\rangle\\right]\n\\\\ &\n+ c_b \\left[\\cos(gt\\sqrt{(m+1)n})|n,m,b\\rangle\\right. \\\\ &\n\\left.\\left.-i\\sin(gt\\sqrt{(m+1)n})|n-1,m+1,a\\rangle\\right]\\right\\}.\n\\end{align*}\nWe have considered the potential for logic operations, but note that for\nsome obvious input states we have mappings where state $\\k{1,1,a}\\rightarrow\n\\k{1,1,a},\\k{2,0,b}$ at frequency $g\\sqrt{2}$, and state\n$\\k{0,1,a}\\rightarrow \\k{0,1,a},\\k{1,0,b}$ at frequency $g$. These\nfrequencies are non-commensurate which makes it awkward to eliminate\n$\\k{2,0}$ and obtain a standard gate.\nHowever, for higher photon numbers the state $\\k{3,3,a}\\rightarrow\n\\k{3,3,a},\\k{4,2,b}$ at frequency $2g\\sqrt{3}$ and the state\n$\\k{0,3,a}\\rightarrow \\k{0,3,a},\\k{1,4,b}$ at frequency $g\\sqrt{3}$, which\nare commensurate frequencies.\nThus, when $gt\\sqrt{3}=\\pi$ the atom becomes an ancilla and \n\\begin{equation}\\label{phgate}\n\\begin{array}{lcl}\n|0,0,a\\rangle\\rightarrow|0,0,a\\rangle & &\n|0,0,b\\rangle\\rightarrow|0,0,b\\rangle \\\\ \n|0,3,a\\rangle\\rightarrow-|0,3,a\\rangle & &\n|0,3,b\\rangle\\rightarrow|0,3,b\\rangle \\\\ \n|3,0,a\\rangle\\rightarrow|3,0,a\\rangle & &\n|3,0,b\\rangle\\rightarrow-|3,0,b\\rangle \\\\ \n|3,3,a\\rangle\\rightarrow|3,3,a\\rangle & &\n|3,3,b\\rangle\\rightarrow|3,3,b\\rangle.\n\\end{array}\n\\end{equation}\nIn this way we can realise the CZ gate for photons (Eq.~(\\ref{eq:mat-Q-CZ}))\nprovided we identify the logical qubits:\n\\begin{align*}\n  \\text{mode 1:}&&\\k0 &\\rightarrow \\text{`1',} &\\k3 &\\rightarrow \\text{`0'}\\\\\n  \\text{mode 2:}&&\\k0 &\\rightarrow \\text{`0',} &\\k3 &\\rightarrow \\text{`1'}\n\\,.\n\\end{align*}\nHowever, this immediately raises a number of issues relating to the use of a\nhigher Fock state $\\k3$, such as decoherence, and how to initialise the\nqubits efficiently and how to perform single qubit rotations? For these\nreasons we take a different approach. \n\n\n\\section{Multiphoton logic}\\label{sec:MultiphotonLogic}\n\nTo bring about solutions, or near solutions to these problems mentioned at the\nend of the previous section we will make an adaptation to the scheme\npresented there. This adaptation is to encode our cavity qubits\nas dual-mode cavity qubits \\cite{Barry1}. The more familiar term ``dual-rail''\nseems inappropriate here as the qubits are not flying and there is no\nrail. The dual mode qubits are formulated, for example, as shown in\nTable~\\ref{tab:qubit-encoding}. \n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{ccc}\n EM Modes && Logical qubit\\\\\n \\hline\n $\\ket{1}\\ket{0}$ &$\\mapsto$& $\\ket{1}$\\\\\n $\\ket{0}\\ket{1}$ &$\\mapsto$& $\\ket{0}$\\\\\n \\end{tabular}\n \n\n \\caption{Dual-mode qubit coding of the type used in this section.}\n\\label{tab:qubit-encoding}\n\\end{table}\nIn this approach a qubit state is always encoded with a single excitation, as\nin the dual rail approach \\cite{Yamamoto2, Yamamoto3, Kok} and this\nensures that if a cavity decay process takes place, we have: $\\ket{1}\\ket{0}\n\\rightarrow \\ket{0}\\ket{0}$, and then the result is not a valid qubit in this\nencoding. The difference between the two logical qubit states is where the\nexcitation of the two modes is located, as indicated in\nFig.~\\ref{fig:mode-picture}. \n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\bmgstdfigwidthshorter]{figure2}\n\n  \\caption{Schematic showing the relationship of logical qubits to cavity mode\n    excitations (indicated as a darker colour). The case of two cavity modes\n  is illustrated.}\n\\label{fig:mode-picture}\n\\end{figure}\n\nA consequence of using the dual-mode approach is that in order to have a\ntwo-qubit gate we need to have an interaction of an atom with \\emph{four}\ncavity modes, which is quite demanding (as we shall see), and it is this\nwhich we investigate below. As in the simple two-mode system of section\n\\ref{sec:Larson} and Hamiltonian~(\\ref{eq:LarsonHamileff}), we will achieve\nthis by having a sequence of off-resonant interactions with a multilevel\natom (Fig.~\\ref{fig:FourLevels}), and, as in section \\ref{sec:Larson}, that\natom will be an ancilla which will not be entangled with the final result of\na gate operation. The aim of the off-resonant interactions is to reduce\nspontaneous emission from the ancilla atom. In order that we do not extract\n``information'' from the system via the ancilla atom it is important for the\nsequence of atomic transitions to start and end on the atomic state $\\k{a}$\n(see Fig.~\\ref{fig:FourLevels}).  Multiphoton resonance will also require\nthe final detuning, $\\Delta_4$ to ensure resonance. This will mean\n$\\Delta_4\\approx 0$, but it has to be optimised for level shifts as we see\nbelow.\n\n\n\\begin{figure}\n\n \\includegraphics[width=0.35\\textwidth]{figure3}\n \n  \\caption{Scheme of two-qubit i\\textsc{swap} gate, where $\\omega_i$ (with $i=1,2,3,4$) are\n    modes of the four-mode high $Q$ cavity, and $|a\\rangle$, $|b\\rangle$,\n    $|c\\rangle$, and $|d\\rangle$ are four energy levels of the atom.}\n\\label{fig:FourLevels}\n\\end{figure}\n\n\nThe full Hamiltonian of the four-level system is then \\cite{Jaynes, Shore2}:\n\\begin{eqnarray} \nH&=&\\sum \\limits_{i=a,b,c,d}\\ \\omega_{i}\\ \\hat{\\sigma}_{ii}+\\sum\n\\limits_{j=1}^4 \\ \\omega_j\\ \\hat{a}^{\\dag}_j \\hat{a}_j \\\\\n\\nonumber&&+ [\\Gee{ab}{1}\\ \\hat{a}_1\\ \\hat{\\sigma}^{ba}+\\Gee{bc}{2}\\\n\\hat{\\sigma}^{cb}\\ \\hat{a}^{\\dag}_2+\\Gee{cd}{3} \\hat{a}_3\\\n\\hat{\\sigma}^{dc}+\\Gee{da}{4}\\ \\hat{\\sigma}^{ad}\\ \\hat{a}^{\\dag}_4\\\\\n\\nonumber&&+\\rm{h.c.}], \\label{eq:iSWAP Hamiltonian}\n\\end{eqnarray} \nwhere the first two terms represent the non-coupling Hamiltonians, and the\nremaining terms describe the atom-field interaction Hamiltonian. \nThe four key initial states of the system and their mappings can be\nsummarised as \n\\begin{align} \\label{eeq:key-states1}\n\\nonumber \\text{physical}&\\qquad\\text{logical}\\\\\n\\nonumber \\ket{a,\\ 0110}&\\mapsto\\ket{a,\\ 00}\\\\\n          \\ket{a,\\ 0101}&\\mapsto\\ket{a,\\ 01}\\\\\n\\nonumber \\ket{a,\\ 1010}&\\mapsto\\ket{a,\\ 10}\\\\\n\\nonumber \\ket{a,\\ 1001}&\\mapsto\\ket{a,\\ 11} \\ .\n\\end{align}\n\\begin{figure}[h]\\centerline{\n\\subfigure[]{\\includegraphics[width=0.2\\textwidth]{figure4a}}\n\\subfigure[]{\\includegraphics[width=0.2\\textwidth]{figure4b}}}\n\\centerline{\n\\subfigure[]{\\includegraphics[width=0.2\\textwidth]{figure4c}}\n\\subfigure[]{\\includegraphics[width=0.2\\textwidth]{figure4d}}}\n\\caption{The four key initial states of the system with possible\n  interactions. Empty red circles represent empty cavity modes and filled\n  red circles represent cavity modes with a single excitation. The filled\n  black circles indicate where the atomic population is located in this\n  simplified analysis of the gate process. Starting from atomic state $\\k a$ we can see that\n(a) and (b) can progress, but (c) and (d) are blocked.} \\label{fig:4level-blocked}\n\\end{figure}\nIn the case of the initial state $\\ket{a,\\ 1010}$, the system (atom+field) is\ngoverned by the Hamiltonian $H^{\\prime}$ which can be expressed, in the matrix\nrepresentation and with $\\ket{a,\\ 1010}$ to be the zero-point energy, as\n\n\\begin{eqnarray}\nH^{\\prime}=\\left(\\begin{array}{ccccc}0&\\Gee{ab}{1}&0&0&0\\\\\n                                    \\Gee{ab}{1}&\\Delta_1&\\Gee{bc}{2}&0&0\\\\\n                                    0&\\Gee{bc}{2}&\\Delta_2&\\Gee{cd}{3}&0\\\\\n                                    0&0&\\Gee{cd}{3}&\\Delta_3&\\Gee{da}{4}\\\\\n                                    0&0&0&\\Gee{da}{4}&\\Delta_4 \\end{array}\\right)\\\n                                ,\\label{eq:iSWAP_int_Ham} \n\\end{eqnarray}\nin the basis states $\\{|a\\ 1010\\rangle,|b\\\n0010\\rangle,|c\\ 0110\\rangle,|d\\ 0100\\rangle,\\\\ |a\\ 0101\\rangle \\}$.\n\n\nShould we start with different logical qubits, i.e. different\narrangements of the excited modes, the multiphoton process is\nblocked in the case of $|a\\ 0110\\rangle$. In Fig.~\\ref{fig:4level-blocked}(c) \nthere is no cavity photon available in the first and last modes to raise the\natomic state from $\\k{a}$. When the initial state is $|a\\ 1001\\rangle$, on the\nother hand, the evolution of the system is governed by the Hamiltonian\n$H^{''}$ given as \n\\begin{eqnarray}\nH^{''}=\\left(\\begin{array}{ccccc}\\Delta_2&\\Gee{bc}{2}     &0       &0       &0     \\\\\n                             \\Gee{bc}{2}    &\\Delta_1&\\Gee{ab}{1}     &0       &0       \\\\\n                             0      &\\Gee{ab}{1}     &0       &\\Gee{da}{4}     &0       \\\\\n                             0      &0       &\\Gee{da}{4}     &(\\Delta_3-\\Delta_4)&\\Gee{cd}{3}     \\\\\n                             0      &0       &0       &\\Gee{cd}{3}     &(\\Delta_2-\\Delta_4)\n        \\end{array} \\right)\\ ,\\label{eq:a1001 Hamiltonian}\n\\end{eqnarray}\nacting in the basis states $\\{|c\\ 0101\\rangle,|b\\ 0001\\rangle,|a\\\n1001\\rangle,\\\\ |d\\ 1000\\rangle,|c\\ 1010\\rangle\\}$.\n\n\\begin{figure}\n\n  \\includegraphics[width=0.23\\textwidth]{figure5a}\\llap{\n \\parbox[b]{3.5in}{(a)\\\\\\rule{0ex}{.8in}}}\n\n  \\includegraphics[width=0.23\\textwidth]{figure5b}\\llap{\n \\parbox[b]{3.5in}{(b)\\\\\\rule{0ex}{.7in}}}\n\n  \\includegraphics[width=0.23\\textwidth]{figure5c}\\llap{\n \\parbox[b]{3.5in}{(c)\\\\\\rule{0ex}{.6in}}}\n\n  \\includegraphics[width=0.23\\textwidth]{figure5d}\\llap{\n \\parbox[b]{3.5in}{(d)\\\\\\rule{0ex}{.5in}}}\n\n  \\includegraphics[width=0.23\\textwidth]{figure5e}\\llap{\n \\parbox[b]{3.5in}{(e)\\\\\\rule{0ex}{.4in}}}\n  \n  \\caption{Sequence of steps for the shuffling excitation amongst four cavity\n    modes. This shuffling realises the swapping part of a gate operation,\n    i.e.\\ $|a\\ 1010\\rangle \\leftrightarrow |a\\ 0101\\rangle$.} \n\\label{fig:4level-sequence}\n\\end{figure}\n\n\n\nThe way the multiphoton process works to shuffle cavity excitation can be\nseen from Fig.~\\ref{fig:4level-sequence}. An initial state is set up in\nFig.~\\ref{fig:4level-sequence}(a) where the first and third modes are excited\nand the second and fourth are not and the atom is in state $\\k{a}$. Moving to\nFig.~\\ref{fig:4level-sequence}(b) the atom state moves to $\\k{b}$ by means of\nthe absorption of a photon from the first mode. Subsequently this can be\nemitted into the second mode, Fig.~\\ref{fig:4level-sequence}(c), as the atom\napproaches the off-resonant state $\\k{c}$.\nFigures~\\ref{fig:4level-sequence}(d) and \\ref{fig:4level-sequence}(e) show how\nthe atomic state returns to $\\k{a}$ by means of a similar process via the\noff-resonant level $\\k{d}$.\n\n\n\\section{Variants of 4 modes}\n\\label{sec:FourModesVariants}\n\nIn this section we will examine several alternate approaches to carrying out\nthe swapping part of the gate. Because the time evolution of the i\\textsc{swap} gate\nexhibited in the following Sec.~\\ref{sec:iSWAP2level} is extremely slow, we\ncan achieve a speed-up by allowing some of the intermediate states in\nFig.~\\ref{fig:FourLevels} to become resonant. However there are a wide range\nof possibilities which are shown in Fig.~ \\ref{fig:Models3and4}.  Our choice\nof system needs to be informed by which configurations can keep the qubit\nstate $|a 11\\rangle$ in its initial state at the appropriate interaction\ntime.\n\nTo easily identify the different model systems we will use a binary type\nnotation to indicate which levels in the sequence of states are to be\nresonant, and which are not. Thus the Model(10001) in\nFig.~\\ref{fig:Models3and4} has only two resonant states in the chain, the\nfirst and last ones, and will be treated in Eq.~(\\ref{eq:Heff-iSWAP-slow})\nIf we took a fully resonant model, where every transition is resonant, the\nmodel would be Model(11111) also shown in Fig.~\\ref{fig:Models3and4}. In the\nnext section we will focus on Model(11001), as one of the cases with three\nresonant levels, i.e.\\ it has two intermediate resonant levels.\n\n\\begin{figure}\n\n  \\renewcommand{\\arraystretch}{2.0}\n  \\begin{tabular}[t]\n{rm{0.2\\textwidth}}\n     \\multicolumn{2}{c}{2 resonant states}\\\\\n    Model(10001)&\\includegraphics[width=0.2\\textwidth]{figure6a}\\\\\n   \n     \\multicolumn{2}{c}{3 resonant states}\\\\\n    Model(10101)&\\includegraphics[width=0.2\\textwidth]{figure6b}\\\\\n    Model(11001)&\\includegraphics[width=0.2\\textwidth]{figure6c}\\\\\n   \n\n   \n     \\multicolumn{2}{c}{4 resonant states}\\\\\n     %\n    Model(11011)&\\includegraphics[width=0.2\\textwidth]{figure6d}\\\\\n    Model(10111)&\\includegraphics[width=0.2\\textwidth]{figure6e}\\\\\n   \n   \n     \\multicolumn{2}{c}{5 resonant states}\\\\\n     %\n    Model(11111)&\\includegraphics[width=0.2\\textwidth]{figure6f}\\\\\n   \n  \\end{tabular}\n\n\\caption{Linkage schemes for 4 modes N resonant states (with\n  N$=1,2,3,4,5$).} \\label{fig:Models3and4} \n\\end{figure}\n\n\n\\subsection{An effective two-level system} \n\\label{sec:iSWAP2level} \n\n\nFor the initial state $|a\\ 1010\\rangle$, the effective wave-function of the\nsystem can be expressed as:\n\\begin{eqnarray}\\label{eq:fourmodebasis}\n\\nonumber\n|\\Psi(t)\\rangle&=&c_1 |a1010\\rangle +c_2|b0010\\rangle+c_3|c0110\\rangle\n\\\\\n&&+ c_4|d0100\\rangle+c_5  |a0101\\rangle \\ .\n\\end{eqnarray}\nwhich is a superposition of the key states used for single photon swapping. \n\nIn Appendix~\\ref{sec:TheoryAdiabaticElimination} we will utilize a theory\nfor the adiabatic elimination of the unwanted (off-resonant) levels as shown\nin Fig.~\\ref{fig:FourLevels}. Thus using the basis of\nEq.~(\\ref{eq:fourmodebasis}) we will develop an effective two-level\nHamiltonian which takes the form\n\\begin{equation}\\label{eq:Heff-iSWAP-slow}\n\\eff{H}=\\begin{bmatrix}\n0 & \\eff{g}\\\\\n\\eff{g} & \\eff{\\Delta}\n\\end{bmatrix} \\ ,\n\\end{equation}\nwhere ($g$$\\ll$ $\\Delta$) is required (see\nappendix~\\ref{app:TheIswapGate1}). The effective coupling is\nfound to be \n\\begin{align}\\label{eq:geff-ISWAP-slow}\n\\eff{g}\\approx-\\frac{\\Gee{ab}{1}\\Gee{bc}{2}\\Gee{cd}{3}\\Gee{da}{4}}{\\Delta_1\\Delta_2\\Delta_3}\\ ,\n\\end{align}\nand the effective detuning of the two-level system is \n\\begin{eqnarray} \\label{eq:Delta-ISWAP-slow}\n\\Delta_{\\rm{eff}}\\approx \\Delta_4+\\frac{(\\Gee{ab}{1})^2}{\\Delta_1}-\\frac{(\\Gee{da}{4})^2}{\\Delta_3} \\ .\n\\label{eq:DeltaEffIn2LevelIniSWAP}\n\\end{eqnarray}\nThe time evolution of this system is given by the following equations for\nthe logical states,\n\\begin{align}\\label{eq:iSWAP-slow-2Eqns}\n\\nonumber\n\\ket{a, 10}&\\mapsto\\cos(\\eff{g}t)\\ket{a, 10}-\\rm{i}\\sin(\\eff{g}t)\\ket{a, 01}\\\\\n\\ket{a, 01}&\\mapsto\\cos(\\eff{g}t)\\ket{a, 01}-\\rm{i}\\sin(\\eff{g}t)\\ket{a, 10}\\ .\n\\end{align}\n\nThis two-level system undergoes swapping of the states $|a1010\\rangle$ and\n$|a0101\\rangle$ when the resonance condition is achieved by setting\n$\\Delta_{\\rm{eff}}$ to zero in equation (\\ref{eq:DeltaEffIn2LevelIniSWAP}).\nThis is illustrated in Fig.~\\ref{fig:iSWAPslow} where excellent agreement\nbetween an exact numerical calculation and the analytic treatment of\nequations~(\\ref{eq:iSWAP-slow-2Eqns}) is presented. Because of the finite\nvalue of $\\Delta$ chosen in Fig.~\\ref{fig:iSWAPslow} a fine high frequency\noscillation can be seen. For larger detuning this oscillation becomes\nsmaller as the two-level approximation is realised more accurately.\n\n\\begin{figure}\\centering\n\n  \\includegraphics[width=0.5 \\textwidth]{figure7}\n                                                                        \n  \\caption{Population swapping in the model (10001). All\n    couplings $g_j$ ($j=1,2,3,4$) are set to $g$, and the detunings\n    $\\Delta_i$ with $i=1,2,3$ are set to $\\Delta$. The value of $\\Delta_4$ is\n    determined by Eq.~(\\ref{eq:DeltaEffIn2LevelIniSWAP}).}\\label{fig:iSWAPslow}\n\\end{figure}\n\nThe effective Hamiltonian~(\\ref{eq:Heff-iSWAP-slow}) only connects the\nstates $|a\\ 1010\\rangle$ and $|a\\ 0101\\rangle$; the state $|a\\ 1001\\rangle$\nis effectively governed instead by the full Hamiltonian~(\\ref{eq:a1001\n  Hamiltonian}), and the other logical state of the system, $\\ket{00}$, is\nunchanged. Considering the parameters in Fig.~\\ref{fig:iSWAPslow} and at the\nspecial time $|\\eff{g}t|=\\pi\/2$, we obtain the following outputs for the\nbasic inputs (see Fig.~\\ref{fig:M(10001)_fidelities}):\n\\begin{equation}\n\\begin{tabular}{ccc}\nInput && Output\\\\\n\\hline\n$\\ket{00}$ && $\\ket{00}$\\\\\n$\\ket{01}$ && $\\rm{i}\\ket{10}$\\\\\n$\\ket{10}$ && $\\rm{i}\\ket{01}$\\\\\n$\\ket{11}$ && $\\ket{11}$\n\\end{tabular}\\label{eq:iSWAP-slow-logic-table}\n\\end{equation}\nThis realises an i\\textsc{swap} gate and we can represent the above mappings as\nthe table\n\\begin{equation}\n  \\mbox{i\\textsc{swap}} \\equiv \\left[ \\begin{array}{rrrr}%\n1& & & \\\\\n &0&\\rm{i}& \\\\\n &\\rm{i}&0& \\\\\n & & &1\\\\\n\\end{array}\\right] \\ .\n\\end{equation}\nThe i\\textsc{swap} gate is a universal gate when combined with single qubit\nrotations. For example it can be directly related to the \\textsc{cnot} gate (also a\nuniversal gate) by means of a quantum circuit as shown in \\cite{Schuch}.\nRecently, it is shown that the i\\textsc{swap} gate can be very useful for applications \nin quantum information process QIP and quantum computing. For example, it is\nreported that the replacement of the standard \\textsc{cnot} gate by the i\\textsc{swap} gate\nprovides more efficient, simpler, and faster way of generating cluster states \n\\cite{Tanamoto}, which  play a crucial role in the so-called one-way quantum\ncomputation approach \\cite{Raussendorf2, Raussendorf3}. \n\n\n\\begin{figure}\n\n  \\includegraphics[width=0.45\\textwidth]{figure8}\\llap{\n \\parbox[b]{.005in}{(b)\\\\\\rule{0ex}{.6in}}}\\llap{\n \\parbox[b]{.005in}{(a)\\\\\\rule{0ex}{1.8in}}}\n  \n  \\caption{(a) The fidelity (black line) of swapping the logical states $|a\\ 10\\rangle$ and $|a\\\n    01\\rangle$ in the model (10001). Red-dashed line shows the conditional\n    fidelity. (b) The fidelity of keeping the system initially in the logical state $|a\\\n    11\\rangle$. Parameters of the detunings $\\Delta$ and the coupling\n    strengths $g$ are defined in Fig.~\\ref{fig:iSWAPslow}.} \\label{fig:M(10001)_fidelities} \n\\end{figure}\n\n\\subsection{A three-level behaviour}\n\\label{sec:3LevelBehavior}\n\nModel (11001) has an intermediate resonant state. The system takes the form of\na double $\\Lambda$ system with a common initial and final state. Since\n$|b\\ 0010\\rangle$ is taken to be resonant, we apply the adiabatic\nelimination theory (as shown in appendix~\\ref{app:TheIswapGate2}) to obtain\nan effective three-level system in the reduced space of states\n$\\{|a\\ 1010\\rangle,|b\\ 0010\\rangle,|a\\ 0101\\rangle\\}$.  The effective\ncouplings are found to be\n\\begin{eqnarray}\ng_{\\rm{eff}}^{(1)}=g_1^{\\rm{ab}} ,\\qquad\ng_{\\rm{eff}}^{(2)}\n\\approx\\frac{g^{\\rm{bc}}_2g^{\\rm{cd}}_3g^{\\rm{da}}_4}{\\Delta_2\\Delta_3} \\ \n, \\label{eq:effective coupling in 11001} \n\\end{eqnarray}\nand the effective detunings\n\\begin{eqnarray}\n\\nonumber\n\\Delta_{\\rm{eff}}^{(1)}&\\approx&\\Delta_1-\\frac{(\\Gee{bc}{2})^2}{\\Delta_2}\\ ,\\\\\n\\Delta_{\\rm{eff}}^{(2)}&\\approx&\\Delta_4-\\frac{(\\Gee{da}{4})^2}{\\Delta_3}\\\n.\\label{eq:resonance conditions in 11001} \n\\end{eqnarray}\nThe resulting Hamiltonian takes the form \n\\begin{equation}\nH_{\\rm{eff}}=\\left[\\begin{array}{ccc}0&g_{\\rm{eff}}^{(1)}&0\\\\\n                                      g_{\\rm{eff}}^{(1)}&\\Delta_{_{\\rm{eff}}}^{(1)}&g_{\\rm{eff}}^{(2)}\\\\\n                                    0&g_{\\rm{eff}}^{(2)}&\\Delta_{_{\\rm{eff}}}^{(2)}  \n                 \\end{array}\\right] \\ .\n\\label{eq:Three-level eff_Ham}\n\\end{equation}\n\nThe time evolution of this gate is shown in\nFig.~\\ref{fig:M11001_swapping_fidelities}(a) and it is immediately apparent\nthat the population swapping happens much faster than the evolution seen in\nFig.~\\ref{fig:iSWAPslow} for Model (10001). The time evolution of the\nswapping follows the equations (for the initial state $|a,1010\\rangle$)\n\\begin{eqnarray}\n\\nonumber|a,10\\rangle\\longrightarrow && \n[\\frac{(\\Gee{(1)}{\\rm{eff}})^2}{\\overline{g}^2}+\\frac{(\\Gee{(2)}{\\rm{eff}})^2}{\\overline{g}^2}\\ \n\\cos(\\overline{g}t)]\\ |a,10\\rangle\\\\\n&-&\\rm{i}\\frac{\\Gee{(1)}{\\rm{eff}}}{\\overline{g}}\\ \\sin(\\overline{g}t)\\\n|\\Phi\\rangle \\label{eq:three_level_analytical_sol}\\\\\n\\nonumber&+&\\frac{\\Gee{(1)}{\\rm{eff}} \\Gee{(2)}{\\rm{eff}}}{\\overline{g}^2} \\\n[\\cos(\\overline{g}t)-1]\\ e^{i\\eta \n  t}\\ |a,01\\rangle\\ ,\n\\end{eqnarray}\nwhere\n$\\overline{g}=\\sqrt{(\\Gee{(1)}{\\rm{eff}})^{2}+(\\Gee{(2)}{\\rm{eff}})^{2}}$ and\n$|\\Phi\\rangle\\equiv|\\rm{b}\\ 0010\\rangle$. \n\nThe spin-$J$ model predicts the proper values for the effective coupling\nconstants so that a complete qubit swapping can be achieved. That is, in the\nprevious three-level system, the general scaling expression \\cite{Cook}\n\\begin{eqnarray}\n\\Gee{(n)}{\\rm{eff}}=g_0\\sqrt{n(N-n)} \\label{eq:SequenceOfRabiFreq.}\n\\end{eqnarray} \n(where $g_0$ is a constant and in our case $N=3$ and $n=1,2$) suggests that\n$|\\Gee{(1)}{\\rm{eff}}t|=|\\Gee{(2)}{\\rm{eff}}t|=\\pi\/\\sqrt{2}$. Then, the\ntransformation $|a\\ 1010\\rangle \\rightarrow$ $\\exp(\\rm{i}\\eta t)|a\\\n0101\\rangle$ can take place by setting $\\overline{g}t=\\pi$.  \nA global phase $\\eta t$ in the previous two- and three-level systems can be \nproduced by following different proposals. Some examples will be suggested\nwhen we discuss the single-qubit gate in Sec.~\\ref{SingleQubitGates}. \n\nBy substituting the parameters of the coupling constants and detunings in\nthe model (11001) into the Hamiltonian~(\\ref{eq:a1001 Hamiltonian}) which\ndescribes the time evolution of the initial state $|a\\ 1001\\rangle$, it is\nnoticeable that this state can be sufficiently forced to stay in its\ninitial state, as demonstrated by the red-dashed line in\nFig.~\\ref{fig:M11001_swapping_fidelities}(b).\n\\begin{figure}\\centering\n\n  \\includegraphics[width=.4\\textwidth]{figure9a}\\llap{\n \\parbox[b]{6in}{(a)\\\\\\rule{0ex}{1.1in}}}\n\n\\includegraphics[width=.4\\textwidth]{figure9b}\\llap{\n \\parbox[b]{6in}{(b)\\\\\\rule{0ex}{1in}}}\n  \n  \\caption{[Color online] A three-level behaviour with the states $|c\\rangle$\n    and $|d\\ \\rangle$ chosen to be highly-detuned (see \n    Fig.~\\ref{fig:FourLevels}). Parameters: the coupling constants $g_{2,3,4}$\n    are all set to $g$, and the detunings $\\Delta_{2,3}$ are set to\n    $\\Delta$. (a) the probability of the states $|a\\ 1010\\rangle$, $|b\\\n    0010\\rangle$, and $|a\\ 0101\\rangle$ with $\\Delta=20 g$. (b) the fidelity of\n    mapping $|a\\ 10\\rangle$ to $|a\\ 01\\rangle$ (blue-solid line), and of\n    keeping $|a\\ 11\\rangle$ in its initial state(red dotted-dashed line). The\n    black-dashed and green-dotted lines represent the conditioned fidelity and\n    the interaction time $gt_{\\rm{int}}$,\n    respectively.} \\label{fig:M11001_swapping_fidelities}\n\\end{figure}\n\n\n\\subsection{Enhancement conditional on measurements}\n\nIn the scheme we have developed so far the atom plays the role of an ancilla\nwhich simply ``enables'' the shuffling of energy between cavity modes.\nHowever, if the proposed gate is slightly imperfect there will be a small\nadmixture of other atomic states. The role of a conditional measurement can\nbe to improve the quality of the final state (i.e.\\ to improve the\nfidelity). For example, let us suppose the state of the system is\n\\begin{align}\\label{eq:cond-example-all}\n|\\Psi(t)\\rangle&=c_1  |a\\ 1010\\rangle+ c'|c\\ 0110\\rangle\n+c_2 |a\\ 0101\\rangle  + ...\n\\end{align}\nThen if the atom exits the system in state $\\k a$ we must project the\nstate (\\ref{eq:cond-example-all}) onto the atomic state $\\k a$ to\nobtain the conditioned result \n\\begin{align}\\label{eq:cond-example-measured}\n|\\Psi(t)\\rangle^{\\prime}&\\rightarrow \\frac{c_1}{\\sqrt{|c_1|^2 + |c_2|^2}}|a\\ 1010\\rangle\n             + \\frac{c_2}{\\sqrt{|c_1|^2 + |c_2|^2}} |a\\ 0101\\rangle. \n\\end{align}\nBecause of the renormalisation that takes place this conditionally enhances a\ndesired result (such as $|a\\ 0101\\rangle$). \n\nTo a limited extent this makes the scheme probabilistic. However, the\nprobability of success in such a measurement is expected to be high and the\nintent is that this measurement process simply enhances the result and\ncleans up the wave-function. Red-dashed line in\nFig.~\\ref{fig:M(10001)_fidelities}(a) and black-dashed line in\nFig.~\\ref{fig:M11001_swapping_fidelities}(b) show significant improvement of\nfidelity in the models (10001) and (11001).\n\n\n\\subsection{Decoherence process}\n\\label{sec:M10101-decoherence}\n\n\nThe time evolution of the previous systems can be governed by Liouville's\nequation\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}\\ \\rho=-\\rm{i} \\left[H,\\ \\rho\\right]+\\mathcal{L}\\\n\\rho\\ ,\\label{eq:MasterEquation}\n\\end{eqnarray}\nwhere $\\rho$ is the density operator of the atom-field system \nand the so-called Liouvillian operator $\\mathcal{L} \\rho$ describes the\ndissipative mechanisms in the system. The general Lindblad form of the\nLiouvillian operator $\\mathcal{L} \\rho$ can be expressed as\n\\cite{Lindblad}\n\\begin{eqnarray}\n\\mathcal{L}\\ \\rho&=&\\sum \\limits_{i} \\frac{\\eta_{(i)}}{2}\\\n([L_{(i)}\\ \\rho\\ ,\\ L^{\\dag}_{(i)}]+[L_{(i)}\\ ,\\ \\rho \\ L^{\\dag}_{(i)}])\\ ,\n\\label{eq:Liouville operator}\n\\end{eqnarray} \nwhere $\\eta$ represents the loss of population which can be due to either the \nspontaneous emission $\\Gamma$ or to the cavity field rate $\\kappa$. The\noperators $L$ and $L^{\\dag}$ are the corresponding system operators. More\nexplicitly, in the presence of the atomic decay $L$ and $L^{\\dag}$ can be\nreplaced by the atomic operators $\\sigma_{-}$ and $\\sigma_{+}$, and in the\ncase of the cavity decay they are represented by the field operators $a$ and\n$a^{\\dag}$. \n\nGiven the initial states to be either $|a\\ 1010\\rangle$ or $|a\\\n0101\\rangle$, we can also investigate the influence of the atomic and\nphotonic relaxations by considering the following Hamiltonian\n\\begin{eqnarray}\nH^{\\prime}=H-\\rm{i}\\frac{\\kappa}{2}\\sum \\limits_{i=1}^{4}\\\na^{\\dag}_{i}a_{i}\\ -\\rm{i}\\frac{\\Gamma}{2} (|b\\rangle \\langle b|+|d\\rangle \n\\langle d|)\\ ,\n\\label{eq:ConditionalHamiltonian} \n\\end{eqnarray}\nwhere $H$ is the original Hamiltonian of the system in the absence of any\ndecay. \n\n\\begin{figure}\n\n\\subfigure[]{\\includegraphics[width=0.5 \\textwidth]{figure10a}}\n\\subfigure[]{\\includegraphics[width=0.5 \\textwidth]{figure10b}}\n\\caption{The influence of the dissipative mechanisms on the qubit state $|a\\\n  10\\rangle$ or $|a\\ 01\\rangle$ in the models (10001) and (11001). (a) in the\n  model (10001), the three curves corresponding to the values of photonic\n  decays $\\kappa \\sim (0, 0.033, 0.24)g_{\\rm{eff}}$ with $\\Delta=10 g$. The\n  solutions by theory (dashed lines) and simulation (solid lines) are highly\n  matched. (b) in the model (11001) The three curves corresponding to the\n  values of the atomic decay rates $\\Gamma \\sim (0, 0.1, 0.75)\n  \\bar{g}\/\\sqrt{2}$ with $\\kappa=0$, or to the values of photonic decays\n  $\\kappa \\sim (0, 0.025, 0.185)\\bar{g}\/\\sqrt{2}$ when no atomic decay rate is \n  considered. The three curves in all these plots represent (100, 90, 50)\\%\n  fidelity.}\\label{fig:CavityDecayIn10001}\n\\end{figure}\n\nThis procedure is valid under the condition that no photon is\ndetected \\cite{Dalibard, Molmer, Zoller}, and the Shore's method \\cite{Shore}\ncan be reapplied to produce the damped N-level configurations. We have seen\npreviously that the models (10001) and (11001) are capable to realise the\ni\\textsc{swap} gate. Starting with the model (10001), the time evolution of the state\n$|a\\ 0101\\rangle$ in the strong coupling regime is \n\\begin{eqnarray}\nc_{a01}(t)&=&-\\rm{i}\\frac{g_{\\rm{eff}}}{\\tilde{g}} \\ \\ e^{-\\kappa t}\\\n\\sin(\\tilde{g}t)\\ e^{-\\rm{i}\\Delta_{\\rm{eff}}t\/2}\\ ,\\label{eq:iSWAP (10001) \n  exact sol}\n\\end{eqnarray}\nwhere $\\tilde{g}=\\sqrt{g_{\\rm{eff}}^2+(\\Delta_{\\rm{eff}}\/2)^2}$. The norm of \nthe system shows that it decays with the rate\n$(2\\kappa)$. Figure~\\ref{fig:CavityDecayIn10001}(a) demonstrates\nthe fidelity for different values of $\\kappa$. As we have mentioned before, \nthe i\\textsc{swap} gate formed from the model (10001) is very slow gate and, therefore, it is\nvery sensitive to photonic decay rates.\n\nIn the case of the model (11001) and taking into account the atomic and\ncavity decay rates in the effective Hamiltonian~(\\ref{eq:Three-level eff_Ham}),\nthe eigenvalues for this effective Hamiltonian can be determined, under the\ncondition $4\\bar{g}\\geq (\\kappa-2\\Gamma)$ and with vanishing effective\ndetunings, as  \n\\begin{eqnarray}\n\\lambda_1=-\\kappa\\ ,\\ \\ \n\\lambda_{2,3}=-(\\frac{3\\kappa+2\\Gamma}{4})\\pm \\rm{i}\\ {\\lambda}\\ ,\n\\end{eqnarray}\nwhere $\\lambda=\\{ \\bar{g}^2 -\\frac{1}{4}(\\kappa-2\\Gamma)^2 \\}^{1\/2}$, \n$\\bar{g}=\\sqrt{(\\Gee{(1)}{\\rm{eff}})^2+(\\Gee{(2)}{\\rm{eff}})^2}$, and\n$\\Gee{(1)}{\\rm{eff}}$ and $\\Gee{(2)}{\\rm{eff}}$ are the effective coupling\nconstants. The corresponding eigenvectors can be found and then the time\nevolution of the logical state $|a\\ 01\\rangle$ can be expressed as \n\\begin{widetext}\n\\begin{eqnarray}\nc_{a01}(t)=\\frac{\\Gee{(1)}{\\rm{eff}}\\Gee{(2)}{\\rm{eff}}}{{\\bar{g}}^2}  \\\n\\exp(-\\kappa\\ t)\\\n\\left\\{-1+\\left[\\cos({\\lambda}t)-(\\frac{\\kappa-2\\Gamma}{4{\\lambda}}) \\\n  \\sin({\\lambda}t) \\right] \\ \\exp(\\frac{\\kappa-2\\Gamma}{4}\\ t) \\right\\}\\\n,\\label{eq:FastIswapPopulationLoss} \n\\end{eqnarray}\nfor the initial condition that the system is completely set in the initial state\n$|a\\ 1010\\rangle$ at $t=0$, i.e.\\ $c_{a10}(0)=1$.\n\\end{widetext}\n\nIn Fig.~\\ref{fig:CavityDecayIn10001}(b) we consider the model (11001) and\nmeasure the fidelity at different values of $\\kappa$.\nFigure~\\ref{fig:CavityDecayIn10001} shows that the impact of cavity\nfield relaxation is less in the model (11001) when compared to the model\n(10001), and this is because of the improvement in the qubit states speed.\n\nIn the case of the initial state to be $|a\\ 0110\\rangle$, this state decays\ndue to the cavity relaxation and the population loss follows\n$|c_{a00}|^2=\\exp(-2\\kappa t_{\\rm{int}})$, where the interaction time\n$t_{\\rm{int}}$ is same to the models (10001) and (11001). In\nthe final state $|a\\ 1001\\rangle$ we directly use the master\nequation~(\\ref{eq:MasterEquation}) to investigate the effect of atomic and\nphotonic dampings in either the model (10001) or the model (11001).\n\n\nWe are now in the position to subject all qubit states in the i\\textsc{swap} \ngate to experimental values including the coupling strength $g$, the atomic\ndecay $\\Gamma$, and the photonic decay rate $\\kappa$ so that the performance\nof the two-qubit dual-rail CQED gate can be practically tested. Considering the\nmicrowave cavity-QED experiment in \\cite{Haroche5}, highly excited\nRydberg atoms (typically $^{85}$Rb) with a radiative time $\\tau_{rad}\\sim 30$\nms have been used to interact with a superconducting cavity with $Q\\mapsto\n4\\times 10^{10}$. The photon lifetime inside the cavity is in order\n$\\tau_{ph}\\sim 130$ ms, and the coupling strength is around $g\/2\\pi\\sim 50$\nkHz. By setting $\\Delta=10$ g, this corresponds to\ncavity-atom interaction time $t_{int}\\mapsto 5$ ms in the configuration\n(10001), and $t_{int} \\mapsto \\frac{1}{\\sqrt{2}}$ ms in the configurations\n(11001). The quantity $\\tau_{ph}\/t_{int}$ shows that the last configuration\nis much better for QIP applications with the present cavity QED techniques. \nPlots in Fig.~\\ref{fig:iSWAP_GatesWithExper.Parameters} show the population\nloss in the i\\textsc{swap} gate realised by the configurations (10001) and (11001) with\nconsidering the above values of the parameters $g$, $\\Gamma$, and $\\kappa$.\n\n\n\\begin{figure}\n  \\centering\n  \\subfigure[]{\\includegraphics[width=\\bmgstdfigwidth]{figure11a}}\n  \\subfigure[]{\\includegraphics[width=\\bmgstdfigwidth]{figure11b}}\n  \\caption{Truth table of the numerically simulated i\\textsc{swap} gate with\n    $\\Delta=10\\ g$ in the presence of decoherence processes. (a)\n  in the configuration (10001). (b) in the model (11001). Parameters: the\n  coupling constant is approximate $g\/2\\pi=50$ kHz, $\\Gamma\/g \\sim 10^{-4}$,\n  and $\\kappa\/g \\sim\n  2.5\\times10^{-5}$.} \\label{fig:iSWAP_GatesWithExper.Parameters} \n\\end{figure}\n\n\n\n\n\\section{Three qubit gates and rotations} \n\\label{sec:ThreeAndOneQubitGates}\n\nSo far we have described a two-qubit entangling gate. However, to have\na universal set of gate operations it is necessary to have single\nqubit rotations (or single qubit gates). We also present below a three\nqubit gate, where full details will be presented\nin appendices~\\ref{sec:FastFredkinGate} and~\\ref{NotGate}. \n\n\\subsection{Three qubit gate}\n\nThe three qubit scheme of Fig.~\\ref{fig:FredkinLevels} contains\ndual-mode qubits with a repeated mode at $\\omega_1$. The partner mode\nfor $\\omega_1$ is not shown as it is unaffected by the logic process.\nThe remaining pairs are $\\omega_{2,3}$ and $\\omega_{5,6}$, (a mode\nwith the label $\\omega_4$ is omitted in the following analysis to\navoid confusion). The consequence of the repeated mode~1 is that the\nchain only completes if mode~1 is present (in which case the\nexcitation is absorbed and then re-emitted) and if modes~3 and 5 have\nexcitations present (which implies necessary empty modes~2 and~6 in the\ndual-mode qubit basis). The result of all this is that the qubit~1 acts as\na control qubit which swaps the qubits present in logical qubits 2 and\n3: i.e.\\ we have the logic needed for a Fredkin gate \\cite{Fredkin}. In a\nFredkin gate we aim for logical qubits to be mapped as follows:\n\n\\begin{equation}\n\\begin{tabular}{ccc}\nInput && Output\\\\\n\\hline\n$\\ket{000}$ && $\\ket{000}$\\\\\n$\\ket{001}$ && $\\ket{001}$\\\\\n$\\ket{010}$ && $\\ket{010}$\\\\\n$\\ket{011}$ && $\\ket{011}$\\\\\n$\\ket{100}$ && $\\ket{100}$\\\\\n$\\ket{110}$ && $\\ket{101}$\\\\\n$\\ket{101}$ && $\\ket{110}$\\\\\n$\\ket{111}$ && $\\ket{111}$\\\\\n\\end{tabular}\n\\end{equation}\n\n\nThe Fredkin gate is a universal gate owning important\nproperties which set up the general principles of logic gates and circuits in\nboth classical and quantum computing \\cite{Fredkin}. This gate swaps the\nsecond and the third qubits if the first qubit is $|1\\rangle$, otherwise, all\nqubits remain unchanged. Two examples of quantum circuits generating this gate\ncan be considered. Firstly, it is observable that the Fredkin gate is nothing but the\ncontrolled \\textsc{swap} gate \\cite{Nielsen}. Secondly, since the \\textsc{swap} gate is equivalent to three\n\\textsc{cnot} gates, we can see that the Fredkin gate is a combination of the Toffoli\ngate together with two \\textsc{cnot} gates \\cite{Barnett3}.\n\\subsection{Two- and three-level configurations}\n\n  \\begin{figure}\n    \n \\subfigure[]{\\includegraphics[width=\\bmgstdfigwidth]{figure12a}}\n \\subfigure[]{\\includegraphics[width=\\bmgstdfigwidth]{figure12b}}\n  \\caption{Fidelity for the fast Fredkin gate shown as a function of the\n    detuning $\\Delta$ where we again take $\\Delta_j \\rightarrow \\Delta $ for\n    $j=1, 2, 4, 5$ with $\\Delta_3=0$ and $\\Delta_6\\sim 0$ (given by\n    equation~(\\ref{eq:Fredkin-Delta-eff}) set to zero). \n    The initial state is (a) $|a\\ 101\\rangle$, (b) $|a\\ 100\\rangle$.\n    For the couplings we\n    take $ g_j \\rightarrow g $. The dashed line in (a) shows the interaction time\n    $\\bar{g}t_\\mathrm{int} =\\pi$ as found from the three-state\n    model~\\ref{sec:3LevelBehavior}.} \n\n\\label{fig:FastFredkinFidelity}\n\\end{figure}\n\n  For a de-excited atom interacting with multimode\n  cavities, the realisation of the Fredkin gate at the interaction time\n  $gt_{\\rm{int}}$ can be achieved when the transformation $|a\\ 101\\rangle\n  \\leftrightarrow |a\\ 110\\rangle$ is made and the remaining logical qubits\n  are in their initial states. Once again, Shore's method plus the spin-$J$ \n  model provide useful tools to find out several configurations that are\n  capable of swapping the states $|a\\ 101\\rangle$ and $|a\\\n  110\\rangle$. However, to meet the truth table of the Fredkin gate, and\n  since there is a presence of a repeated mode~1 in addition to certain over-shot\n  states (see appendix~\\ref{sec:FastFredkinGate}) it is noticed that the only\n  possible configurations realising the gate are the models $(1000001)$ and\n  $(1001001)$. In the model $(1000001)$ we set all states other than the\n  states $|a\\ 101\\rangle$ and $|a\\ 110\\rangle$ to be far-resonance, and in the\n  model $(1001001)$ a further atomic state $|d\\rangle$ is allowed to be resonant. \n  The full details of the analysis of the former model (1000001) are not\n  presented here, it can be found in \\cite{Barry1} where we also show that an\n  excellent speed-up is obtained by chosen an intermediate resonant\n  energy level, i.e.\\ the model (1001001). In this case the effective\n  Hamiltonian reduces to a three-level system again, where (see\n  appendix~\\ref{sec:FastFredkinGate}) the effective coupling\n  constants are \n\\begin{align}\\label{eq:Fredkin-g-eff}\n\\Gee{(1)}{\\rm{eff}} &=\n\\frac{\\Gee{ab}{1}\\Gee{bc}{2}\\Gee{cd}{3}}{\\Delta_1\\Delta_2},& \\Gee{(2)}{\\rm{eff}}\n& = \\frac{\\Gee{de}{1}\\Gee{ef}{5}\\Gee{fa}{6}}{\\Delta_4\\Delta_5}\\ ,\n\\end{align}\nand the effective detunings are\n\\begin{equation}\\label{eq:Fredkin-Delta-eff}\n\\begin{split}\n&\\Delta_1^{\\mathrm{eff}}\\approx\\Delta_3 +\n\\frac{\\left(\\Gee{ab}{1}\\right)^2}{\\Delta_1} -\n\\frac{\\left(\\Gee{cd}{3}\\right)^2}{\\Delta_2}-\\frac{\\left(\\Gee{de}{1}\\right)^2}{\\Delta_4}\\,,\\\\\n&\\Delta_2^{\\mathrm{eff}}\\approx\\Delta_6 -\n\\frac{\\left(\\Gee{af}{6}\\right)^2}{\\Delta_5}\\ .\n\\end{split}\n\\end{equation}\n\nGiven the initial state to be either the logic $|a\\ 101\\rangle$ or $|a\\\n110\\rangle$, equations of motion in~(\\ref{eq:three_level_analytical_sol})\nand the sequence of effective Rabi frequencies\nin~(\\ref{eq:SequenceOfRabiFreq.}) can be reused to swap the qubit states\n$|a\\ 10\\rangle$ and $|a\\ 01\\rangle$. A high fidelity result can be obtained\nfor sufficient detunings, Fig.~\\ref{fig:FastFredkinFidelity}(a) shows how this\nincreases, and also how the use of measurement of the ancilla atom strongly\nimproves the result for the quite low value of $\\Delta\/g\\gtrsim 3$. The same\nFig.~shows the interaction time which is fairly high for large $\\Delta$\n($gt\\sim 500$ for $\\Delta=15g$) but reduces substantially near $\\Delta=5g$.\nThe fidelity for keeping the qubit state $|a\\ 100\\rangle$ in its initial\nstate at the same interaction time above has been illustrated in\nFig.~\\ref{fig:FastFredkinFidelity}(b). For other qubit states, large detunings\nfor all atomic levels except $|a\\rangle$ and $|d\\rangle$ ensure an efficient\nconfinement of the populations in desired levels so that the three-qubit\nFredkin gate is built.\n\n\n\\subsection{Decoherence in fast Fredkin gate}\nConsidering the parameters of the coupling constants and the detunings in the\nmodel (1001001) given by\nEqs.~(\\ref{eq:Fredkin-g-eff}, \\ref{eq:Fredkin-Delta-eff}), we can use the\nmaster equation~(\\ref{eq:MasterEquation}) to address the influence \nof atomic and photonic relaxations on the Fredkin gate. Furthermore, we can\nuse the conditional Hamiltonian~(\\ref{eq:ConditionalHamiltonian}) to find\nanalytic solutions for certain qubit states in Fredkin gate. For example,\nthe damping due to atomic and photonic decays can be studied as follows. \nUnder the strong coupling regime, the eigenvalues of the damped three-level\nbehaviour (1001001) can be given as \n\\begin{eqnarray}\n\\lambda_1=-3\\kappa\/2\\ ,\\ \\ \\lambda_{2,3}=-(\\frac{5\\kappa+2\\Gamma}{4})\\pm\n\\rm{i} {\\lambda}\\ ,\n\\end{eqnarray}\nwhere $\\lambda=\\bigg(\\bar{g}^2-\\frac{1}{4}(\\kappa-2\\Gamma)^2\\bigg)^{1\/2}$, \n$\\bar{g}=\\sqrt{(\\Gee{(1)}{\\rm{eff}})^2+(\\Gee{(2)}{\\rm{eff}})^2}$, and\n$\\Gee{(1)}{\\rm{eff}}$ and $\\Gee{(2)}{\\rm{eff}}$ are given by\nEq.~(\\ref{eq:Fredkin-g-eff}) and the effective detunings all are set to\nzero. The time evolution of the coefficient $c_{a110}$, with the initial\nconditions $c_{a101}(t=0)=1$, $c_{d}(0)=0$, and $c_{a110}(0)=0$, reads\n\\begin{widetext}\n\\begin{eqnarray}\nc_{a110}=\\frac{\\Gee{(1)}{\\rm{eff}}\\Gee{(2)}{\\rm{eff}}}{{\\bar{g}}^2}\\\n\\exp(-\\frac{3\\kappa}{2}\\ t)\\ \\{-1+\\left[\\cos({\\lambda}t)-(\\frac{\\kappa-2\\Gamma}{4{\\lambda}})\\\n  \\sin({\\lambda}t)\\right]\\ \\exp(\\frac{\\kappa-2\\Gamma}{4}\\ t)\\} \\ \n.\\label{eq:FastFredkinPopulationLoss} \n\\end{eqnarray}\nFigure~\\ref{fig:DampedFredkinGate} demonstrates the population loss by either\nthe spontaneous emission $\\Gamma$ or the cavity field decay $\\kappa$, and the\ndamped Fredkin gate under the recent QIP techniques. \n\\begin{figure}\n  \\centerline{  \n  \\subfigure[]{\\includegraphics[width=.35\\textwidth]{figure13a}}\n  \\subfigure[]{\\includegraphics[width=.32\\textwidth]{figure13b}}\n  \\subfigure[]{\\includegraphics[width=.38\\textwidth]{figure13c}}}\n  \n  \\caption{In (a) and (b) the probability of the qubit state $|a\\ 110\\rangle$\n    in the model (1001001) with non-vanishing photonic and atomic\n    decays. Solid and dashed curves show the numerical and theoretical\n    solutions, respectively. The coupling constants $g_{i}$ ($i=1,2,3,5,6$)\n    all are set to $g$, and the detunings $\\Delta_{1,2,4,5}=\\Delta$ with \n    $\\Delta=20 g$. The effective couplings of the truncated system\n    $\\Gee{(1)}{\\rm{eff}}$ and $\\Gee{(2)}{\\rm{eff}}$ are given by\n    Eq.~(\\ref{eq:Fredkin-g-eff}), and the detunings $\\Delta_3$ and $\\Delta_6$\n    are defined by the resonance conditions in\n    Eq.~(\\ref{eq:Fredkin-Delta-eff}). The black, blue, and red \n    lines respectively represent (at the interaction time $gt_{int}$) the\n    system probability $P\\sim (100, 90, 50) \\%P_0$ with a maximum probability\n    $P_0 \\sim 0.9950$. The three curves corresponding to the values of: (a)\n    the photonic decay rates $\\kappa \\sim (0,\\ 0.0174,\\\n    0.1186)\\bar{g}\/\\sqrt{2}$ with $\\Gamma \\mapsto 0$, and (b) the atomic decay\n    rates $\\Gamma \\sim (0,\\ 0.0976,\\ 0.764)\\bar{g}\/\\sqrt{2}$ with $\\kappa\n    \\mapsto 0$. (c) a truth table of the numerically simulated Fredkin gate\n    with $\\Delta=10\\ g$ in the presence of decoherence processes. Parameters:\n    the coupling constant is approximately $g\/2\\pi=50$ kHz, $\\Gamma\/g \\sim\n    10^{-4}$, and $\\kappa\/g \\sim\n    2.5\\times10^{-5}$.} \\label{fig:DampedFredkinGate} \n\\end{figure}\n\\end{widetext}\nIn the absence of $\\Gamma$ it is shown by\nEqs.~(\\ref{eq:FastIswapPopulationLoss}, \\ref{eq:FastFredkinPopulationLoss})\nthat the Fredkin gate is more sensitive to cavity field decay rate, and the\ni\\textsc{swap} and the Fredkin gates in the previous three-level configurations have\nthe same sensitivity to the spontaneous emissions when no photonic decay is\nconsidered.\n\n\n\\subsection{Single qubit gates} \n\\label{SingleQubitGates}\n\n\\begin{figure}\\centering\n  \n  \\includegraphics[width=.3\\textwidth]{figure14}\n  \n  \\caption{The model for a single-qubit Pauli-X gate. A three-level atom with\n    $\\Lambda$ configuration of levels interacting with the optical cavity\n    modes $\\omega_1$ and $\\omega_2$ on the transitions $|a\\rangle\n    \\leftrightarrow |b\\rangle$ and $|b\\rangle \\leftrightarrow |c\\rangle$. The\n    transition $|c\\rangle \\leftrightarrow |a\\rangle$ is coupled by the\n    classical field $\\omega_{class}$.}\n\\label{fig:QubitRotations}\n\\end{figure}\n We turn briefly to single qubit rotations which are required to make\n a universal set of gates. Within the dual-mode scheme a\n $\\Lambda$-atom-mode system, such as seen in Fig.~\\ref{fig:Larson}\n cannot be used because for an arbitrary rotation of the mode\n excitation between the cavity modes the atomic state changes as\n well. For that reason we add a classical field to allow a return to\n the original atomic state $\\k a$ (see\n Fig.~\\ref{fig:QubitRotations}). Otherwise the scheme is similar to\n Fig.~\\ref{fig:Larson} in that we have a lambda atom which has two\n transitions  coupled to two cavity modes that make up a qubit. We will\n adiabatically eliminate levels $\\ket{b}$ and $\\ket{c}$ from the interaction \n under the conditions $\\Delta_1, \\, \\Delta_2  \\gg  \\Gee{ab}{1}, \\,\n \\Gee{bc}{2}, \\,\\Omega\/2, \\, \\Delta_3$ (see appendix~\\ref{NotGate}) to find\n the effective detuning\n\\begin{equation}\\label{xRot-Detuning}\n\\eff{\\Delta} =  \\Delta_3 + \\frac{(\\Gee{ab}{1})^2}{\\Delta_1}\\ ,\n\\end{equation}\nand the effective coupling \n\\begin{equation}\\label{xRot-g}\n\\eff{g} = \\frac{\\Gee{ab}{1} \\Gee{bc}{2}\\Omega}{2 \\Delta_1\\Delta_2}\\ .\n\\end{equation}\nWe can then ensure a qubit rotation in the form\n\\begin{equation}\n\\mathrm{\\hat R}_x(g_\\mathrm{eff}t) = \\cos(g_\\mathrm{eff}t\/2) \\rm{\\hat I}\n            -\\mathrm{i}\\sin(g_\\mathrm{eff}t\/2) \\ \\hat\\sigma_x\\ .\n\\end{equation}\n\nThe Pauli Z gate can be easily realized by our qubits, too. Generally\nspeaking, the atom-cavity interaction in the Jaynes-Cummings model shows that \nfor an atom in the ground state $|g\\rangle$ interacting with a single mode\nhaving $n$ photons:\n\\begin{equation*}\n |g,n\\rangle \\mapsto \\cos(g\\sqrt{n}t)|g,n\\rangle -\\rm{i} \\sin(g \\sqrt{n}t)\n |e,n-1\\rangle\\ .\n\\end{equation*}\nThis is the case when the frequencies of the atomic transitions and the mode\nare equally matched (i.e. the resonance case $\\omega_{eg}=\\omega$). In the\ncase of very large detuning ($\\Delta \\gg g$), on the other hand, the system\nremains in its initial state and a phase shift can be produced as\n\\begin{eqnarray}\n|g,n\\rangle \\mapsto e^{\\rm{i}\\Phi(n)}|g,n\\rangle \\ ,\n\\end{eqnarray}\nwith $\\Phi(n)$ can be expressed as \\cite{Walther5}\n\\begin{eqnarray}\n\\Phi(n)=\\frac{\\Delta}{2v} \\int_0^L\ndz\\bigg[\\sqrt{1+n\\bigg(\\frac{g(z)}{\\Delta\/2}\\bigg)^2}-1 \\bigg] \\ ,\n\\end{eqnarray}\nwhere $v$ is the velocity of the atom passing through a cavity, $L$ is the\ncavity length, and $g(z)$ is the coupling constant which in our case is\nindependent of z. By setting $n=0$ nothing happens, but with a cavity being\ninitially in the number state $|1\\rangle$ (i.e.\\ there is $n=1$ photon) a\nset of phase gates can be realised and the rotation operator $R_{z}$ can be\nproduced. In the case of two modes inside the cavity interacting with\nan atom in the ground state $|a\\rangle$, such a case in the dual-rail qubits\n$|a\\ 10\\rangle$ and $|a\\ 01\\rangle$, the previous argument can be followed to\nintroduce a phase shift. That is, we can set a large detuning\nbetween the atom and, say the first mode, and set a very high detuning between\nthe atom and the second mode. In this case, if the excitation is in the first\nmode, one finds $|a\\ 10\\rangle \\mapsto e^{\\rm{i}g^2 t\/\\Delta}|a\\ 10\\rangle$;\notherwise, $|a\\ 01\\rangle \\mapsto |a\\ 01\\rangle$.\n\nDifferent ways can be considered to add a global phase\n$\\eta t$ to the two- or three-qubit gates already discussed in the previous\nsections. For instance, the single-qubit phase gate above can be employed for\nthis purpose. That is, in the case of the previous i\\textsc{swap} gate and after\nproducing the transformations in Sec.~\\ref{sec:FourModesVariants}, an atom\ninitially in the ground state $|a\\rangle$ sent to the first two-mode cavity can\nintroduce a phase to the logical state $|a\\ 10\\rangle$ when the atom is\ndetuned from the first mode $\\hat{n}_1$ of a qubit, and far detuned from the\nmode $\\hat{n}_2$. Then, another atom in $|a\\rangle$ interacting with the\nsecond two-mode cavity, where the atom is detuned from $\\hat{n}_4$ and far\ndetuned from $\\hat{n}_3$, can add a phase to the logic $|a\\ 01\\rangle$. \n\n\n\\section{Conclusion}\n\\label{sec:Conclude}\n\nA multiphoton resonance can be a very useful technique for applications\nin quantum information processing as it involves a process conditional on\nthe presence of various photons. Here we use the theory of\nmultiphoton resonance with a multilevel multiphoton Jaynes-Cummings model. \nInformation is stored in photonic qubits and we produce a set of practical\none-qubit, two-qubit, and even three-qubit gates. This works because CQED\noffers a high non-linearity with low losses. \n\nUp to date, the strong interaction between a multilevel atom with a\nmultimode field (such an interaction proposed in our scheme)\nremains an experimental challenge. In fact, with the remarkable progress in\nnanotechnology, this kind of interaction might be possible in near\nfuture \\cite{Zubairy3}. It is reported in \\cite{Benson} that a transfer of\nenergy between two individual nanoparticles strongly coupled to high-Q\nwhispering-gallery modes in a microsphere resonator is experimentally\nachieved. This achievement gives a great hope of finding experiments that\nproceed an interaction between a single multimode cavity interacting with a\nmultilevel atom in the limit of strong coupling. \n\n\nMMA would like to acknowledge support from King Khalid University (KKU).\nMSE thanks the Japanese Society for the Promotion of Science.  BMG\nacknowledges the support of the Leverhulme Trust and thanks Michael Hartmann\nand Bruce W.\\ Shore for comments. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe development of the rapid-frame-capture detector array sensors based on CCD (charge-coupled device) and CMOS (complementary metal oxide semiconductor) technology has revolutionized imaging \\cite{Shimadzu2012}. \nRecently, there has also been interest in new methods that leverage the massive bandwidth of optical signals to perform imaging using a single pixel detector. One such approach uses the broadband spectrum of ultrashort optical pulses \\cite{Jalali_Nature2009} and works by mapping different optical frequencies of the broadband spectrum into distinct spatial locations using spatial dispersers such as demonstrated in the technique of femtosecond pulse shaping \\cite{Weiner_RSI2000}. To create a 2-dimensional (2D) map, a conventional grating disperses the spectrum in one spatial dimension, while a virtually imaged phase array (VIPA) disperses light into the other spatial dimension.  As shown in Fig. \\ref{Fig1Scheme}(a), the grating and the VIPA create a `2D spectral shower' in which distinct optical frequencies have a one-to-one (spectral-spatial) correspondence with spatial coordinates in 2-dimensions \\cite{Weiner_OE2008TwoD,Diddams_Nature2007,Jalali_Nature2009,Yasui_Optica2018}. To recover the image, the spectrum can be measured by the time-stretch method, which converts the spectrally-encoded spatial information into a temporal waveform measured on the single-pixel photodetector \\cite{Jalali_Nature2009}. This approach measures the image on a shot-by-shot basis and 6 MHz frame rates have been demonstrated \\cite{Jalali_Nature2009}.\n\n\\begin{figure*}[t]\n\\begin{centering}\n\\includegraphics[width=0.9\\linewidth]{fig1_v25rev.pdf}\n\\caption{{\\bf Dual-comb imaging using microresonator solitons.} (a) A conceptual diagram showing the operational principle for spectral-spatial-mapping and dual-microcomb imaging. Two soliton microcombs (signal and reference) having slightly different repetition rates are generated using two on-chip microresonators. A 2D disperser (VIPA+grating) maps frequencies from the signal microcomb into a 2D grid of spatial locations (spectral shower) that are reflected by a target. The reflected signal spectrum is measured by multi-heterodyne detection with the reference microcomb. The chip is shown with small (high rate) and larger (low rate) comb pairs in both the signal and reference arms. These can enable different operational modes for the imaging system.\n(b) Dual-comb imaging proceeds by illuminating the target (right panel) with the 2D spectral shower formed as shown in panel a. As shown in the left panel, the target reflection amplitude is encoded onto the signal comb (Optical comb \\textrm{II}). The signal comb is then heterodyned with the reference comb (Optical comb \\textrm{I}) to generate the RF comb. $f_\\text{{rep1}}$, $f_\\text{{rep2}}$, and $\\Delta f_\\text{{rep}}$ are the frequency line spacing of the reference comb, the signal comb, and the signal RF comb. (c) Dual comb imaging concept based on integrated waveguide grating antennas. Microcomb outputs are divided into multiple waveguides that drive the grating antennas. Comb light is   dispersed by a corresponding waveguide grating antenna (eliminates VIPA and grating) to create one imaging dimension in the spectral shower (right). The second imaging dimension is provided by the spatial location of each grating antenna. This approach combines spectral parallelism of photonics with spatial parallelism of detector arrays to greatly magnify performance. A single (shared) pump is shown, but the microcombs could also be individually pumped so as to create frequency combs that are spectrally displaced.}\n\\label{Fig1Scheme}\n\\end{centering}\n\\end{figure*}\n\nAn alternative image recovery technique based on two frequency-combs has also been recently demonstrated  \\cite{Yasui_Optica2018}. This approach, termed here dual-comb imaging, parallels the technique of dual-comb spectroscopy \\cite{Newbury_Optica2016,Picque_NP2019} by converting an optical spectrum into a radio-frequency (RF) electrical signal. In effect the method maps the optical signal comb with target information into these radio-frequency components (see Fig. \\ref{Fig1Scheme}(b)). If the signal and reference comb are phase locked then both amplitude and phase information about the target can be retrieved, enabling acquisition of 3 dimensional information \\cite{Yasui_Optica2018}. Line-scan spectral-spatial imaging using dual frequency combs has also been recently reported \\cite{Zeng_OL2018,ZhangXL_OL2018,Yasui_JQE2019}. To generate broadband optical pulses for imaging, table-top mode-locked lasers have so far been used. A recent advance in optical pulse and frequency comb generation is based on dissipative Kerr soliton mode locking in optical microcavities \\cite{Kippenberg_NP2014,Vahala_Optica2015,Gaeta_Ol2016Thermal,Weiner_OE2016,Kippenberg_review2018,Gaeta_NP2019Review}. The devices provide high repetition rate soliton streams and their associated optical frequency combs feature smooth spectral envelopes. These miniature frequency combs or microcombs \\cite{Kippenberg_Science2011} are considered a possible way to dramatically reduce the form factor of conventional frequency comb systems. Accordingly, they are being studied for several applications including dual-comb spectroscopy \\cite{Vahala_Science2016,Lipson_SA2018}, ranging \\cite{Vahala_Science2018Range,Kippenberg_Science2018Range},\noptical communications \\cite{Kippenberg_Nature2017Commun}, optical frequency synthesis \\cite{Diddams_arXiv2017Synthesis}, and exoplant detection in astronomy \\cite{obrzud2019microphotonic,suh2019searching}.\n\n\nThe application of microcombs to the dual comb imaging method is considered here. These devices offer a system-on-a-chip architecture that eliminates fiber optics (i.e., that required for the time-stretch image recovery method and to generate mode locked pulses). A fully integrated platform that avoids the free space grating and VIPA elements is also possible (see Fig. \\ref{Fig1Scheme}(c)). This work explores soliton microcomb dual-comb imaging by measuring a USAF1951 test target and by monitoring microparticles in a flow-cell. An important feature of microcombs is their very high repetition rates as compared to conventional combs (typically microwave to THz rates as compared to radio frequency rates). The impact of such high repetition rates on future dual-comb imaging system performance is also considered.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[t]\n\\begin{centering}\n\\includegraphics[width=0.95\\linewidth]{Fig3DualComb16.pdf}\n\\caption{{ \\bf Dual-microcomb imaging of static targets.} (a) Typical optical spectrum of the 9.39 GHz soliton microcombs showing sech$^2$ spectral envelope fit (red) with 3 dB bandwidth of 1.2 THz. The inset is the electrical spectrum of the photodetected soliton pulse stream and gives the repetition rate. (b) An example of the measured interferogram in a 5 $\\mu$s time window from the reference arm and the signal arm (see Fig 1(a)). (c) RF spectrum of the 5 $\\mu$s signal interferogram in panel b. (d) Image of three vertical bars constructed from the measured interferogram in panel b. (e) Typical optical spectrum of the 1.86 GHz soliton microcombs showing sech$^2$ spectral envelope fit (red) with 3 dB bandwidth of 0.44 THz. A notch near the spectral maximum is produced by narrow-band filtering of the optical pump.  The inset is the electrical spectrum of the photodetected soliton pulse stream. (f) Examples of the recorded interferograms for reference and signal arms using the 1.86 GHz microcombs. (g) RF spectrum of the signal interferogram in panel f. The spectral hole around 380 MHz results from the notch filter used to suppress the optical pump.\n(h) Image of 3 horizontal bars constructed from the measured 10 $\\mu$s interferogram in panel e. (i) Image of number `4' on the USAF target produced using the 1.86 GHz microcombs. The dark discontinuities shown by the arrows in panels (h, i) result from the spectral notch produced by filtering the optical pump (see panel e). As an aside, a similar discontinuity is located within a dark region of the image in panel d and is therefore not visible.}\n\\label{Fig3DualComb}\n\\end{centering}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\begin{centering}\n\\includegraphics[width=0.95\\linewidth]{Fig3FLOW7.pdf}\n\\caption{{ \\bf Monitoring flowing particles.} (a) An illustration of the microparticle monitoring experiment. Microparticles are suspended in water and flow inside the cell. When a particle passes through the 2D spectral shower, the particle can be imaged using the dual-comb interferogram. (b) Measured interferogram shows varying amplitude when the microparticle flows through the 2D spectral shower. (c) A snapshot of the measured microparticle, which is constructed from a 5 $\\mu$s duration interferogram (shaded bar in panel b). The dashed circle suggests the microparticle size ($\\sim$100 $\\mu$m). The dark vertical band results from filtering of comb lines around the pump. (d) Center position of the microparticle plotted versus time. A linear fit gives a flow velocity of 0.21 m\/s in reasonable agreement with the set water flow velocity of 0.25 m\/s.}\n\\label{Flow}\n\\end{centering}\n\\end{figure*}\n\n\\section{Experimental setup}\nHigh-$Q$ silica-on-silicon wedge microresonators \\cite{Vahala_NP2012Wedge} are used to generate the dual soliton streams. The devices feature repetition rates of 1.86 GHz and 9.39 GHz \\cite{Vahala_Optica2018GHz}.\nDetails on methods used to trigger and stabilize the soliton microcombs are presented elsewhere \\cite{Vahala_Optica2015}. The microcombs are coupled directly to optical fibers using tapered-fiber couplers \\cite{Cai2000,Spillane2003}. The signal comb and reference comb are conveyed along the optical train as shown in Fig. \\ref{Fig1Scheme}(a). The tapered-fiber couplers can be replaced by integrated waveguides \\cite{Vahala_NP2018}. Moreover, fully integrated soliton microcombs with an on-chip pump have also been reported \\cite{stern2018battery}.\n\nThe VIPA and grating act together to create the 2D spectral shower with the VIPA dispersing the spectrum along the vertical direction and the grating dispersing the spectrum along the horizontal direction. More specifically, the VIPA disperses light only within its free-spectral-range (FSR) which means that optical frequencies $\\nu$ and $\\nu+mf_\\text{VIPA}$ (where $m$ is an integer and $f_\\text{VIPA}$ is the VIPA FSR) will overlap in space. By adding a grating, frequencies $\\nu$ and $\\nu+mf_\\text{VIPA}$ can be further dispersed to create the 2D spectral shower as illustrated in Figs. \\ref{Fig1Scheme}(a, b). The VIPA used in our experiment has an FSR of 60 GHz (LightMachinery) and this limits pixel count along the vertical direction to 6 pixels (9.39 GHz microcomb) and 32 pixels (1.86 GHz microcomb). Analysis of more optimal designs is provided in the Discussion. The spectral shower is reflected by the object, coupled back into the fiber for return to a photodetector where it is heterodyned with the reference microcomb. Figure \\ref{Fig1Scheme}(b) illustrates how the image reflection amplitude is transferred from the spectral shower to the signal RF spectrum produced by dual-comb heterodyne.\n\nAlso shown in Fig. \\ref{Fig1Scheme}(a) are a collimator and a cylindrical lens (focal length of 150 mm) that focuses the collimated comb onto the VIPA. Additionally, the 2D spectral shower is focused onto the target by a spherical lens (focal length of 30 mm). The targets are placed at the focal plane of the spherical lens and aligned to provide maximum reflection coupled into the fiber. Because the dual-comb measurement can resolve single comb lines, the spatial resolution is set by the imaging system only. This is in contrast to the time-stretch method where spatial resolution can also be limited by the ability to resolve the frequency components \\cite{Jalali_Nature2009}. The spot diameter of a dispersed comb line is $\\sim$15 $\\mu$m in the current setup and a finer spot size can be achieved by expanding the beam size before the focusing spherical lens.\n\n\\section{Imaging a static target}\nTo demonstrate this approach, a USAF 1951 test target (negative) is imaged.\nIn a first measurement, two independent free-running silica microcombs with repetition rates close to 9.39 GHz are used. Dual combs can also be generated from a single microresonator \\cite{Vahala_NP2017Counter,Gaeta_OL2018CP,Kippenberg_NP2018CP} which can result in strong common noise suppression but it also limits the freedom of choosing the repetition rate difference. Therefore, two independent microresonators were used. The spectrum of one of the microcombs is shown in Fig. \\ref{Fig3DualComb}(a) and features a sech$^2$-shaped spectral envelope and 3 dB bandwidth of 1.2 THz. The spectral spurs in the spectrum result from the dispersive wave emission induced by avoided-mode-crossings, which also assists single soliton generation \\cite{Weiner_Optica2017}. The detected electrical spectrum for this comb is shown in the inset of Fig. \\ref{Fig3DualComb}(a). The other microcomb has a similar spectrum, but its repetition rate differs from the first microcomb by $\\Delta f_{\\text{rep}}\\sim$1.5 MHz. The repetition rate difference was chosen to maintain the interferogram bandwidth within the 1 GHz bandwidth of the digitizer used in the experiment. The close matching in the selected repetition rates is possible by good microfabrication control of the resonator diameters using a common mask size and calibration of etch rates \\cite{Vahala_NP2012Wedge}. In Fig. \\ref{Fig3DualComb}(b), typical examples of the heterodyned dual-comb interferograms measured over a 5 $\\mu$s interval from the reference arm (upper panel) and signal arm (lower panel) are shown. While the reference interferogram contains a readily identifiable periodic signal resulting from the difference in repetition rates of the signal and reference microcombs, the signal interferogram contains complex structure associated with the image.\n\nTo construct an image, an RF spectrum is first calculated by taking the fast Fourier transform (FFT) of the signal interferogram produced by illuminating a patterned region of the target (Fig. \\ref{Fig3DualComb}(c)). Even though free running microcombs were used, the signal-to-noise-ratio (SNR) is at least 10 dB over most of the spectrum and exceeds 30 dB over a substantial fraction of the spectrum. Furthermore, the RF spectrum is compressed into a bandwidth less than 400 MHz, a rate that is much smaller than, for example, that required in pulsed imaging work \\cite{Jalali_Nature2009}. The signal spectrum is then normalized using the FFT of the reference interferogram. Following this calculation, the same procedure is applied except using a non-patterned (uniform) region of the target. Finally, this non-patterned RF spectrum is used to normalize the RF spectrum of the patterned region,\nand the resulting normalized spectrum is sorted into the 2D image matrix with each column containing one VIPA FSR. An example of the constructed image is shown in Fig. \\ref{Fig3DualComb}(d). The bars appear slightly tilted because the 2D spectral shower generated by the VIPA and grating is usually titled \\cite{Diddams_Nature2007,Weiner_OE2008TwoD}.\n\nThe low pixel number ($\\sim$6$\\times$34) limits the resolution of the image, especially along the vertical direction which is set by the combination of the 9.39 GHz microcombs with the fixed VIPA FSR. To illustrate possible improvement in the vertical direction, two 1.86 GHz soliton microcombs ($\\Delta f_{\\text{rep}}$=0.7 MHz) were also tested in the imaging setup (see Fig. \\ref{Fig3DualComb}(e) for the optical and electrical spectra of one of the microcombs). The pixel number using the 1.86 GHz microcombs is $\\sim$15$\\times$32. The field of view (shown in Fig. \\ref{Fig3DualComb}) is smaller than the 9.39 GHz microcombs due to the narrower comb bandwidth. Reference and signal dual-comb interferograms can be obtained as before (Fig. \\ref{Fig3DualComb}(f)) with a corresponding signal FFT (Fig. \\ref{Fig3DualComb}(g)). Fig. \\ref{Fig3DualComb}(h, i) show the resulting images of three horizontal bars and the number `4' (test target), respectively. Figs. \\ref{Fig3DualComb}(h, i) are recorded within a time interval of 10 $\\mu$s.\n\nAs an aside and as noted earlier, locking of the signal and reference combs can allow for phase retrieval enabling 3 dimensional imaging similar to ref. \\cite{Yasui_Optica2018}. Such locking could also leverage the mutual locking of counter-propagating solitons \\cite{Vahala_NP2017Counter}, however, this will lower the frame rate due to the relatively small repetition rate difference in counter-propagating solitons. Also, even though the 9.39 GHz microcombs and 1.86 GHz microcombs were generated from different chips in the current experiments, multiple comb pairs can be integrated on a single chip to enable agile switching between different operating modes (see Fig. \\ref{Fig1Scheme}(a) for the concept).\n\n\\medskip\n\n\\section{Monitoring a flowing microparticle}\n\nTo demonstrate measurement of a rapidly-changing scene, two 9.39 GHz soliton microcombs are used to monitor a microparticle moving in a high-speed flow-cell (Fig. \\ref{Flow}(a)). For this purpose a $\\sim$ 0.25 m\/s laminar flow cell (cross section 0.2 mm $\\times$ 8 mm) was set up and microparticles with diameter of $\\sim$ 100 $\\mu$m were suspended in water to flow through the cell. To improve signal-to-noise a mirror was placed behind the cell. When a microparticle passes through the spectral shower, it modulates lines in the spectral shower, which results in the amplitude varying interferogram shown in Fig. \\ref{Flow}(b). An image of a recorded flowing microparticle is constructed from the 5 $\\mu$s portion of the interferogram (shaded bar) in Fig. \\ref{Flow}(b) and is shown in Fig. \\ref{Flow}(c). The size of the reconstructed microparticle (dashed circle in figure) is consistent with the particle's actual size. The particles are not well resolved on account of the limited pixel number. To measure the motion of the microparticle, its center position is plotted versus time in Fig. \\ref{Fig3DualComb}(d). A linear fit gives a flow velocity of 0.21 m\/s, consistent with the flow-cell set-point velocity of 0.25 m\/s. A 200 kHz frame-rate was used in this measurement and $\\sim$7 frames of the heterodyne interferogram were averaged to produce the image. A comparison of frame rate, pixel count and fillrate for this result versus work using fiber lasers \\cite{Yasui_Optica2018} is provided in Table 1.\n\n\n\n\\medskip\n\n\n\n\n\n\\section{Discussion and outlook}\n\\subsection{Repetition rate and imaging performance}\n\nComb repetition rate affects pixel count, fillrate and frame rate in the dual comb imaging system. Pixel count determines target resolution and is equal to the number of comb lines,\n\\begin{equation}\n\\label{Pixelcount}\nM_1 \\times M_2 = B \/ f_{\\text{rep}}\n\\end{equation}\nwhere $M_1$ and $M_2$ are the pixel count along the two axes of the spectral shower, $B$ is the comb bandwidth and $f_{\\text{rep}}$ is the comb repetition rate.\n\nBeyond pixel count, the image processing rate is also important. To better understand the constraints inherent in the dual comb approach, note that the $M_1 \\times M_2$ comb line pixels, once mapped into the radio frequency domain, must fit within a radio frequency bandwidth that is less than $f_\\text{rep}\/2$. If this condition is not satisfied, then the optical to radio-frequency mapping would result in spectral folding of comb components \\cite{Newbury_Optica2016}.\nThis constraint gives $M_1 \\times M_2 \\times \\Delta f_{\\text{rep}} < f_{\\text{rep}}\/2$. The rate $\\Delta f_{\\text{rep}}$ also sets the maximum frame rate of the imaging system. This can be understood by considering the interference of the the signal and reference combs in the time domain where their different repetition rates causes the combs to strobe one another on the photodetector. Each strobed signal in the detected current contains the complete image information so that the strobing rate (i.e., $\\Delta f_{\\text{rep}}$) is the maximum image frame rate. In practice, several strobe periods ($C$ periods) must be averaged to improve the signal-to-noise so that the practical frame rate is $f_\\text{frame} = \\Delta f_{\\text{rep}} \/ C$. As a result of this relationship, the number of pixels that can be detected per second by the dual comb imaging system is given by,\n\\begin{equation}\n\\label{Fillrate}\nF_1 \\equiv M_1 \\times M_2 \\times f_{\\text{frame} } = f_{\\text{rep}}\/ (2 C)\n\\end{equation}\nwhere $F_1$ is the fillrate (pixels per second) of a VIPA+grating imager (Fig. \\ref{Fig1Scheme}(a)) or a single waveguide antenna in Fig. \\ref{Fig1Scheme}(c). Fillrate is widely used to characterize video cards, but here it is used to assess the combined space and time resolution of the dual-comb imaging system. Finally, it is also possible to eliminate $M_1 \\times M_2$ in the above expression to arrive at,\n\\begin{equation}\n\\label{Framerate}\nf_{\\text{frame}} < f_{\\text{rep}}^2 \/ (2 B C)\n\\end{equation}\n\nIn summary, Eqns. (1), (2) and (3) show that even while the high repetition rates of microcombs degrade space resolution by reduction of pixel count, they simultaneously improve the frame rates and fillrate of the dual-comb imaging system. Indeed, the fillrates of the current demonstrations are higher than fiber laser based systems (see Table \\ref{Design}). Moreover, as discussed in the next section, it is possible to restore pixel count by combining the spatial parallelism of conventional detector arrays with the spectral parallelism of single-pixel photonic imaging as illustrated in Fig. \\ref{Fig1Scheme}(c).\n\n\\subsection{Design comparisons}\n\nFor the current VIPA+grating based system, the assignment of spectral shower pixels to horizontal and vertical axes is controlled through the VIPA FSR ($f_{\\text{VIPA}}$) such that $M_1 = f_{\\text{VIPA}}\/f_{\\text{rep}} $ and $M_2 = B \/ f_{\\text{VIPA}} $.\nAs examples, a 25 THz comb bandwidth would enable a horizontal pixel number of 50 using a 500 GHz VIPA (custom design available at LightMachinery). For the vertical direction, the pixel number could be increased to over 50 using 10 GHz repetition rate microcombs. This configuration would provide 2500 pixels at a fillrate of 1 GigaPixels\/s (see Table \\ref{Design}).\n\n\n\n\n\n\n\\begin{table*}[hbtp]\n\\centering\n\\caption{Towards optimal design of dual microcomb imaging systems. $^a$ This is the comb bandwidth that is available for the image retrieval and can be greater than the 3 dB bandwidth. $^b$ The 1.2 kHz frame rate for ref. \\cite{Yasui_Optica2018} uses a single period of the interferogram which could limit the image quality. Another frame rate (12 Hz) averages 100 periods and was also reported in ref. \\cite{Yasui_Optica2018}. $^c$ In the current work, averaging over 5 periods of the interferogram is assumed. $^d$ The array consists of 10 signal microcombs with a 10:1 waveguide grating antenna fanout.}\n\\begin{tabular}{ccccccc}\n\\hline\n~~ & VIPA FSR &  Comb bandwidth$^a$ & $\\Delta f_\\text{rep}$ & Frame rate & Pixels &  Fillrate \\\\\n\\hline\n100 MHz fiber lasers \\cite{Yasui_Optica2018} & 15 GHz & 1.2 THz & 1.2 kHz &  1.2 kHz (12 Hz)$^b$ & 12,382 & 15 (0.15) MegaPixels\/s \\\\\n9.4 GHz microcombs & 60 GHz & 1.8 THz & 1.5 MHz & 200 kHz & 204 & 40 MegaPixels\/s \\\\\n1.9 GHz microcombs & 60 GHz & 0.9 THz & 0.7 MHz & 100 kHz & 480 & 48 MegaPixels\/s \\\\\nPossible 10 GHz microcombs & 500 GHz & 25 THz & 2 MHz & 400 kHz$^c$ & 2,500 & 1 GigaPixels\/s \\\\\n100 GHz microcomb array $^d$ & N\/A & 10 THz & 500 MHz & 100 MHz$^c$ & 10,000 & 1 TeraPixels\/s \\\\\n\\hline\n\\end{tabular}\n  \\label{Design}\n\\end{table*}\n\n\nIt is also possible to eliminate the discrete VIPA and grating components shown in Fig. \\ref{Fig1Scheme}(a) using the architecture in Fig. \\ref{Fig1Scheme}(c) so that the imaging system (with the exception lenses) can be monolithically integrated. In this design, a waveguide splitter allows fan-out of a single comb into multiple waveguide grating emitters and thereby multiplies pixel count and fillrate by the fan-out number $N$  (i.e., $F_N=NF_1$). The chip-based nature of microcombs can be further leveraged here to implement a monolithic microcomb array (as shown) to provide additional multiplication of the pixel count in cases where fan-out might be limited on account of comb power. For example, by using an array of one-hundred waveguide elements driven by ten, 100 GHz repetition-rate microcombs (fanout of 10:1 for each microcomb) a fillrate of 1 TeraPixels\/s is possible with 10,000 pixels (see Table \\ref{Design}). 100 GHz microcombs can be generated from silica microresonators with a diameter of $\\sim$600 $\\mu$m or silicon nitride microresonators with a diameter of $\\sim$400 $\\mu$m. Thus, ten microcombs may be accommodated using less than 1 cm along one linear dimension and far less along the remaining direction. On-chip grating antenna arrays are widely used for beam steering and Lidar \\cite{Watts_Nature2013}. Different from beam steering, the antennas in the present application can be well separated. The spectral shower can be focused on the imaging target by two orthogonal cylindrical lenses, similar to the control of the dispersed beams in ref. \\cite{Weiner_OE2008TwoD}. Then, the encoded spectral shower can be either transmitted or reflected to another receiver photonic chip with a receiver antenna array, a reference microcomb array and a photodetector array for dual-microcomb heterodyne and image retrieval. In principle, the reference microcomb array and the detector array can be integrated on the same chip with the signal microcomb array. Overall, such a design of monolithic micrcombs and grating antennas could be used to resolve transient scenes with excellent space and time resolution in the future.\n\nAs an aside, achieving wide comb bandwidth at higher repetition rates using microcombs is straightforward (e.g., 100 GHz microcombs in the 5th row design in Table \\ref{Design}). However, wider bandwidth operation at reduced rates (e.g., 100 GHz microcombs in the 4th row design in Table \\ref{Design}) is more challenging on account of the way continuous-wave pumping efficiency scales with repetition rate \\cite{Vahala_Optica2015,Vahala_OL2016Raman}. However, pulsed pumping can be used to dramatically improve pumping efficiency and bandwidth \\cite{Herr_NP2017Pulse}. For example, a 28 GHz microcomb spanning 60 THz has been demonstrated using pulse pumping \\cite{Kippenberg_CLEO2019}. External (chip-based) broadeners have also be used to achieve octave-span spectral coverage for 15 GHz microcombs \\cite{Diddams_PRApp2018}.\n\n\n\\section{Conclusion}\n\nIn conclusion, soliton microcombs have been applied to image static and moving targets, thereby validating the feasibility of using microcombs for dual-comb imaging. Further integration of microcombs with fanout waveguide grating arrays and integrated detector arrays would improve fillrate performance and image resolution. This direction of work would leverage both the spatial parallelism of conventional imaging arrays and the spectral parallelism of single-pixel optical imaging. Such systems can also eliminate discrete VIPA and grating components by bringing these functions onto the compact imaging chip. The current demonstration was in the 1.55 $\\mu$m band, but can be readily shifted to the 1 $\\mu$m band which could be more suitable for biological applications \\cite{Vahala_NC2017Visible,Kippenberg_NC2018bio}. The method can find widespread applications in fundamental science as well as in industrial production.\n\n\n\\noindent\n\n\\vspace{3 mm}\n\\section{Funding}\nThis work was supported by the Air Force Office of Scientific Research (award no. FA9550-18-1-0353), the Resnick Institute and the Kavli Nanoscience Institute at Caltech.\n\n\\section{Acknowledgments}\nThe authors thank Taeyoon Jeon and Chengmingyue Li for helpful discussions on the flow-cell experiment. CB gratefully acknowledges support from a Postdoctoral Fellowship from the Resnick Institute at Caltech.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The map-graded Hochschild complex}\n\n\n\\section{Introduction}\n\nHochschild cohomology originated as a cohomology theory for algebras $A$. While the degree $0$ cohomology is the center of $A$ and the degree $1$ cohomology corresponds to the derivations of $A$, the degree $2$ cohomology group parametrizes first order deformations of $A$. The relation between Hochschild cohomology and deformation theory is in fact more profound, and can be understood in terms of the structured Hochschild complex $\\mathbf{C}(A)$. In the mean time, Hochschild cohomology and Hochschild complexes have been defined for a wide range of objects of algebro-geometric nature, from schemes \\cite{swan} and presheaves of algebras \\cite{gerstenhaberschack2} to differential graded, exact \\cite{kellerdih} and abelian categories \\cite{lowenvandenberghhoch}, and various links with deformation theory have been established in these contexts \\cite{gerstenhaberschack1} \\cite{lowenvandenberghab}, \\cite{lowenmap}.\n\nAn important shortcoming of Hochschild cohomology when compared, for instance, to Hochschild homology, is its lack of functoriality. Even for algebras, a morphism $f: A \\longrightarrow B$ does not naturally give rise to a map between the Hochschild cohomologies $HH^{\\ast}(A)$ and $HH^{\\ast}(B)$ in either direction. When we turn from algebras to linear categories, i.e. algebras with several objects in the sense of \\cite{mitchell}, the situation becomes somewhat better. More precisely, the inclusion of a full subcategory $\\mathfrak{b} \\subseteq \\mathfrak{a}$ naturally gives rise to a ``restriction'' map $\\mathbf{C}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}(\\mathfrak{b})$ between the Hochschild complexes. Furthermore, this observation of \\emph{limited functoriality} provides a way to relate the Hochschild complexes of categories related by a bimodule, through an intermediate \\emph{arrow category}. In \\cite{kellerdih}, the \\emph{arrow category argument} for structured ($B_{\\infty}$-algebra) Hochschild complexes is developed by Keller in the context of dg categories. The main applications of this argument can be divided into two types:\n\\begin{enumerate}\n\\item Relating the Hochschild complex of a more involved object (the arrow category) to the Hochschild complexes of it's easier building blocks (two smaller categories and a bimodule relating them).\n\\item Proving that two objects have isomorphic Hochschild complexes in the homotopy category of $B_{\\infty}$-algebras, by relating them via a suitable bimodule.\n\\end{enumerate}\nApplications of type (1) can be seen as generalizations of results on the Hochschild cohomology of triangular matrix algebras, a topic which, since the work of Happel \\cite{happel}, has received a lot of attention \\cite{platzeck, guccione, green, snashall, cibils, guin, saorin}. An application of type (2) is the relation between the Hochschild complexes of Koszul dual algebras obtained in \\cite{kellerdih}. The argument is also extensively used in \\cite{lowenvandenberghhoch} to compare various candiate Hochschild complexes of abelian categories and ringed spaces. The paper \\cite{lowenvandenberghhoch} also contains some results more in the spirit of (1), like the existence of Mayer-Vietoris sequences for Hochschild cohomology of ringed spaces.\n\n\nThe main aim of the current work is to provide a comprehensive treatment of the natural tools for ``breaking down'' complicated Hochschild complexes into easier pieces. \nIn subsequent work, we will apply our results both to improve our understanding of deformation theory, for instance of schemes, and to the computation of Hochschild cohomology groups of various origins, for instance for singular schemes in the absence of the classical HKR decomposition.\n\nThe framework we choose for our exposition is that of map-graded categories \\cite{lowenmap}. Although a more powerful theory can be obtained in the combined context of ``map-graded differential graded categories'', for simplicity we present our work in the context of linear map-graded categories. A map-graded category $\\mathfrak{a}$ can be viewed as a group-graded algebra with several objects. It has an underlying grading category $\\ensuremath{\\mathcal{U}}$, object sets $\\mathfrak{a}_U$ for $U \\in \\ensuremath{\\mathcal{U}}$, and morphism modules $\\mathfrak{a}_u(A,A')$ for $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$, $A \\in \\mathfrak{a}_U$ and $A' \\in \\mathfrak{a}_{U'}$. Thus, the grading category $\\ensuremath{\\mathcal{U}}$ can be seen as prescribing a certain shape for $\\mathfrak{a}$. The following are examples of naturally map-graded categories:\n\\begin{enumerate}\n\\item[(i)] For a presheaf of $k$-algebras $\\ensuremath{\\mathcal{A}}: \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Alg}} (k)$, there is an associated $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{a}$ obtained as a kind of Grothendieck construction from $\\ensuremath{\\mathcal{A}}$. The structured Hochschild complex $\\mathbf{C}(\\mathfrak{a})$ controls the deformation theory of $\\ensuremath{\\mathcal{A}}$ as a \\emph{twisted} presheaf of algebras \\cite{lowenmap} and calculates the Hochschild cohomology of $\\ensuremath{\\mathcal{A}}$ from \\cite{gerstenhaberschack} as shown in \\cite{lowenvandenberghCCT}.\n\\item[(ii)] For two linear categories $\\mathfrak{a}$ and $\\mathfrak{b}$ and an $\\mathfrak{a}$-$\\mathfrak{b}$-bimodule $M$, the arrow category $(\\mathfrak{b} \\rightarrow_M \\mathfrak{a})$ is naturally graded over the path category of $\\bullet \\rightarrow \\bullet$.\\end{enumerate}\nOur point of view is that limited functoriality is determined by grading categories in a fundamental way. In \\S \\ref{parmapgraded}, we first endow the category $\\ensuremath{\\mathsf{Cat}} $ of small categories with the Grothendieck pretopology of $n$-covers (for $n \\in \\mathbb{N}$) for which a collection of functors $(\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\longrightarrow \\ensuremath{\\mathcal{U}})_{i \\in I}$ is an $n$-cover provided that it induces a jointly surjective collection of maps $(\\ensuremath{\\mathcal{N}}(\\varphi_i): \\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{V}}_i) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}))_{i \\in I}$ between $n$-simplices of the simplicial nerves. We show that the category $\\ensuremath{\\mathsf{Map}} $ of map-graded categories is naturally fibered over $\\ensuremath{\\mathsf{Cat}} $ (Proposition \\ref{propcart}) and constitutes a stack for $n \\geq 3$ (Corollary \\ref{corstack}). Let $\\ensuremath{\\mathsf{Map}} _c \\subseteq \\ensuremath{\\mathsf{Map}} $ denote the full subcategory of cartesian morphisms  with respect to the fibered category $\\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} $.  In \\S \\ref{parfunct}, we show that taking Hochschild complexes defines a functor (Proposition \\ref{propfunct})\n$$\\mathbf{C}: \\ensuremath{\\mathsf{Map}} _c \\longrightarrow B_{\\infty}: (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}).$$\nNow endow $\\ensuremath{\\mathsf{Map}} _c$ with the pretopology of $\\infty$-covers for which the collection $((\\varphi_i, F_i): (\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}))_{i \\in I}$ is an $\\infty$-cover provided that the collection $(\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\longrightarrow \\ensuremath{\\mathcal{U}})_{i \\in I}$ is an $\\infty$-cover in $\\ensuremath{\\mathsf{Cat}} $, i.e an $n$-cover for every $n \\geq 0$. Our theorem \\ref{mainsheaf} implies:\n\n\\begin{theorem}\nThe functor $\\mathbf{C}: \\ensuremath{\\mathsf{Map}} _c \\longrightarrow B_{\\infty}$ is a sheaf for the pretopology of $\\infty$-covers on $\\ensuremath{\\mathsf{Map}} $.\n\\end{theorem}\n\nAs an application of the theorem, in \\S \\ref{parMV} we obtain a Mayer-Vietoris sequence of Hochschild complexes for a $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{a}$ and two subcategories $\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\subseteq \\ensuremath{\\mathcal{U}}$ for $i \\in \\{1,2\\}$ that constitute an $\\infty$-cover of $\\ensuremath{\\mathcal{U}}$:\n$$0 \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1}(\\mathfrak{a}^{\\varphi_1}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi_2}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi}) \\longrightarrow 0.$$\nHere $(\\ensuremath{\\mathcal{V}}_i, \\mathfrak{a}^{\\varphi_i}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and $(\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2, \\mathfrak{a}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ are chosen to be cartesian.\n\nIn \\S \\ref{parhochsup}, we start from a single subcategory $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$, and a cartesian functor $(\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$.  This gives rise to a surjective morphism between Hochschild complexes $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$ of which we investigate the kernel $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a})$. The results we obtain depend upon the assumption that the $\\ensuremath{\\mathcal{U}}$-morphisms outside of $\\ensuremath{\\mathcal{V}}$ constitute an ideal $\\ensuremath{\\mathcal{Z}}$ in $\\ensuremath{\\mathcal{U}}$. In this case we show in Proposition \\ref{propidsubcat} that \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, (1_{\\mathfrak{a}})_{\\ensuremath{\\mathcal{Z}}})$$\nwhere $(1_{\\mathfrak{a}})_{\\ensuremath{\\mathcal{Z}}}$ is the natural restriction of $1_{\\mathfrak{a}}$ to an $\\mathfrak{a}$-bimodule supported on $\\ensuremath{\\mathcal{Z}}$ (i.e. with zero values outside of $\\ensuremath{\\mathcal{Z}}$).\nAn example where our setup applies is the situation where $\\ensuremath{\\mathcal{U}}$ is the category associated to a collection of open subsets of a topological space $X$ ordered by inclusion, $\\ensuremath{\\mathcal{V}}$ is the full subcategory of subsets $U \\subseteq V$ for a fixed subset $V$, and $\\ensuremath{\\mathcal{Z}}$ contains the inclusions $U' \\subseteq U$ with $U \\nsubseteq V$.  \n\nIn \\S \\ref{pararrow}, we revisit the arrow category construction from \\cite{kellerdih} in the map-graded context. For a $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$-$(\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$-bimodule $(S, M)$, we take the natural inclusion $\\ensuremath{\\mathcal{V}} \\coprod \\ensuremath{\\mathcal{U}} \\longrightarrow (\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}})$ and corresponding cartesian functor \n$$(\\ensuremath{\\mathcal{V}} \\coprod \\ensuremath{\\mathcal{U}}, \\mathfrak{b} \\coprod \\mathfrak{a}) \\longrightarrow (\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}}, \\mathfrak{b} \\rightarrow_M \\mathfrak{a})$$\nas starting point for obtaining map-graded analogues of some of the main results from \\cite{kellerdih}. Sections \\S \\ref{parconn} and \\S \\ref{parderived} are entirely modelled upon the treatment in \\cite{kellerdih}, and mainly formulate results from \\cite{kellerdih} in the map-graded context, making use of the natural Hom and tensor functors from \\S \\ref{parbimod}. Further, in \\S \\ref{pararrowthin}, we give an intrinsic characterization of arrow categories based upon the \\emph{thin ideals} introduced in \\S \\ref{parthin}. \n\nIn \\S \\ref{pargroth}, we present a unified framework for constructing map-graded categories and deconstructing their Hochschild complexes. Our main observation is that both examples (i) and (ii) that we gave of map-graded categories can be viewed as special cases of a generalized Grothendieck construction for map-graded categories. The classical Grothendieck construction from \\cite{SGA1} takes a pseudofunctor $\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ as input and turns it into a category fibered over $\\ensuremath{\\mathcal{U}}$. The construction from \\cite{lowenmap} of which we gave an example in (i) is clearly a $k$-linearized version of this construction, using the category $\\ensuremath{\\mathsf{Cat}} (k)$ of $k$-linear categories instead of $\\ensuremath{\\mathsf{Cat}} $. If we relax $\\ensuremath{\\mathsf{Cat}} (k)$ to the bicategory $\\underline{\\ensuremath{\\mathsf{Cat}} }(k)$ of $k$-linear categories and bimodules, we can in fact describe \\emph{any} $\\ensuremath{\\mathcal{U}}$-graded category as a kind of Grothendieck construction of a naturally associated pseudofunctor \n$$\\mathfrak{a}: \\ensuremath{\\mathcal{U}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }(k): U \\longmapsto \\mathfrak{a}_U.$$\nIn the paper, we go yet another step further and start from a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$$\nwhere $\\ensuremath{\\mathcal{C}}$ is a small category and $\\underline{\\ensuremath{\\mathsf{Map}} }$ is the bicategory of map-graded categories and bimodules described in \\S \\ref{parbimod}. Allowing arbitrary bimodules rather than functors between map-graded categories allows us to capture the arrow category with respect to a bimodule from (ii). In general, we can now deconstruct the Hochschild complex of the Grothendieck construction $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ of $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ based upon the internal structure of $\\ensuremath{\\mathcal{C}}$. Here, the strategy is to cover $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ by other Grothendieck constructions, based upon base change for pseudofunctors from \\S \\ref{parbase}. For instance, in Proposition \\ref{propcstar}, we prove the following:\n\n\\begin{proposition}\\label{propintro}\nSuppose $\\ensuremath{\\mathcal{C}}$ has finite products. Let $\\ensuremath{\\mathcal{C}}^{\\ast}$ be the category $\\ensuremath{\\mathcal{C}}$ with terminal object $\\ast$ adjoined. Put $(\\tilde{\\ensuremath{\\mathcal{U}}}|_{\\ast}, \\tilde{\\mathfrak{a}}|_{\\ast}) = (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ and let $(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C})$ be the Grothendieck construction of the restriction of $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ to $\\ensuremath{\\mathcal{C}}\/C$.\nLet $(C_i)_{i \\in I}$ be a collection of objects in $\\ensuremath{\\mathcal{C}}$ such that for every $C \\in \\ensuremath{\\mathcal{C}}$ there exists a map $C \\longrightarrow C_i$ for some $i$. There is a natural functor\n$$\\mathbf{C}: \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow B_{\\infty}: C \\longmapsto \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_C}(\\tilde{\\mathfrak{a}}|_C)$$\nwhich satisfies the sheaf property with respect to the collection of maps $(C_i \\longrightarrow \\ast)_{i \\in I}$ in $\\ensuremath{\\mathcal{C}}^{\\ast}$.\n\\end{proposition}\n\nLet us now restrict our attention to the case where $\\ensuremath{\\mathcal{C}}$ is the category associated to a poset. Then it is not hard to see that $\\ensuremath{\\mathcal{C}}$ can be covered by path categories of:\n$$\\bullet \\rightarrow \\bullet \\rightarrow \\dots \\rightarrow \\dots \\rightarrow \\bullet \\rightarrow \\bullet$$\nIn \\S \\ref{pargenarrow}, we consider the Grothendieck constructions of such categories as ``generalized arrow categories'', and deconstruct them using iterated arrow category constructions.\nAs such, we see in \\S \\ref{parcoverarrow} how the sheaf property for Hochschild complexes on the one hand, and the arrow category construction on the other hand, can be seen as complementary tools for deconstructing Hochschild cohomology. \n\nIn the final section \\S \\ref{parcomp}, we consider a pseudofunctor\n$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}} _c$ with some extension $(\\ensuremath{\\mathcal{U}}^{\\star}, \\ensuremath{\\mathcal{C}}^{\\star}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _c$, which we compare to the natural pseudofunctor of Grothendieck constructions\n$$(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _c: C \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C}).$$\nOur main Theorem \\ref{thmmaincomp} encompasses the following comparison result:\n\n\\begin{theorem}\\label{thmintro}\nThere is a morphism of pseudofunctors landing in the homotopy category of $B_{\\infty}$-algebras\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\ast}}(\\mathfrak{a}^{\\ast}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\star}}(\\mathfrak{a}^{\\star})$$\nin which every component morphism is a quasi-isomorphism except possibly the top one.\nThe situation where the top morphism is a quasi-isomorphism too can be characterized by the usual bimodule criterion from \\cite{kellerdih}.\n\\end{theorem}\n\nThe proof of the theorem is heavily based upon Keller's arrow category argument in the case of a fully faithful functor $\\mathfrak{b} \\longrightarrow \\mathfrak{a}$, which is in fact a special case of our theorem.\nA combination of Proposition \\ref{propintro} and Theorem \\ref{thmintro} allows us to pull a deconstruction of Hochschild complexes of Grothendieck constructions back to a deconstruction of the Hochschild complexes of the categories $(\\ensuremath{\\mathcal{U}}^{\\star}_{\\ast}, \\mathfrak{a}^{\\star}_{\\ast})$ and $(\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$ in which we are primarily interested - at least in the homotopy category of $B_{\\infty}$-algebras. In particular, exact sequences of Mayer-Vietoris type now naturally give rise to Mayer-Vietoris exact triangles, inducing the desired long exact cohomology sequences.\nAs an application, we recover the Mayer-Vietoris triangles for ringed spaces from \\cite[\\S 7.9]{lowenvandenberghhoch}.\n\n\\vspace{0,5cm}\n\n\\noindent \\emph{Acknowledgement.} The author is very grateful to Michel Van den Bergh for the original idea of introducing Hochschild complexes of map-graded categories, as a tool for obtaining a Hochschild cohomology local-to-global spectral sequence for arbitrary ringed spaces, dating back to one of her research stays at the Mittag-Leffler institute in 2004. A write up of the envisaged approach, which involves the theory of hypercoverings, remains a joint work in progress \\cite{lowenvandenberghlocglob}.\n\n\n\n\n\n\\section{Map-graded categories} \\label{parmapgraded}\n\nLet $k$ be a commutative ground ring.\nIn this section we introduce the category $\\ensuremath{\\mathsf{Map}} $ of small $k$-linear map-graded categories and functors, which is naturally fibered over the category $\\ensuremath{\\mathsf{Cat}} $ of small categories and functors. We also introduce the intermediate category $\\ensuremath{\\mathsf{Mas}} $ of map-graded sets. A map-graded set $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ consists of a small category $\\ensuremath{\\mathcal{U}}$ and for every object $U \\in \\ensuremath{\\mathcal{U}}$, a set $\\mathfrak{a}_U$. A map-graded category $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is a map-graded set with additionally, for every morphism $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$, $A \\in \\mathfrak{a}_U$ and $A' \\in \\mathfrak{a}_{U'}$, a $k$-module $\\mathfrak{a}_u(A, A')$. These modules are endowed with category-like composition and identity morphism \\cite{lowenmap}. Every map-graded set or category $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ has an associated object $(\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp})$ of the same kind, with ``ungrouped object sets'', i.e. with $\\mathrm{Ob}(\\ensuremath{\\mathcal{U}}^{\\sharp}) = \\coprod_{U \\in \\ensuremath{\\mathcal{U}}}\\mathfrak{a}_U$ and $(\\mathfrak{a}^{\\sharp})_{A} = \\{A\\}$. We define the \\emph{nerve} $\\ensuremath{\\mathcal{N}}(\\mathfrak{a})$ of $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ to be the simplicial nerve $\\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$. For small categories, map-graded sets, and map-graded categories, we define pretopologies of $n$-covers by declaring a collection of functors to be an $n$-cover provided that the induced collection of maps between $n$-simplices of the nerves is jointly surjective (Definition \\ref{defncover}).\nWe prove that for $n \\geq 0$, $\\ensuremath{\\mathsf{Mas}} $ is a stack over $\\ensuremath{\\mathsf{Cat}} $ (Theorem \\ref{thmmasstack}) and for $n \\geq 3$, $\\ensuremath{\\mathsf{Map}} $ is a stack over $\\ensuremath{\\mathsf{Mas}} $ (Theorem \\ref{thmmapstack}) and over $\\ensuremath{\\mathsf{Cat}} $ (Corollary \\ref{corstack}).\n\n\\subsection{Fibered categories and stacks}\\label{parfibstack}\n\nIn this section we recall the basic concepts concerning fibered categories and stacks \\cite{SGA1} \\cite{vistoli}, and give some results we will use later on.\n\nLet $\\ensuremath{\\mathcal{U}}$ be a category. A \\emph{category over $\\ensuremath{\\mathcal{U}}$} is a functor $F: \\ensuremath{\\mathcal{A}} \\longrightarrow \\ensuremath{\\mathcal{U}}$. The functor is percieved from the point of view of its fibers. That is, for every $U \\in \\ensuremath{\\mathcal{U}}$ we consider the fiber of objects over $U$:\n$$\\ensuremath{\\mathcal{A}}_U = \\{ A \\in \\ensuremath{\\mathcal{A}} \\,\\, |\\,\\, F(A) = U\\}$$\nand for $u: V \\longrightarrow U$, $A \\in \\ensuremath{\\mathcal{A}}_U$, $B \\in \\ensuremath{\\mathcal{A}}_{V}$,  we consider the fiber of morphisms over $u$:\n$$\\ensuremath{\\mathcal{A}}_{u}(B,A) = \\{ a \\in \\ensuremath{\\mathcal{A}}(B,A) \\,\\, |\\,\\, F(a) = u \\}.$$\n\nA morphisms $a: B \\longrightarrow A$ in $\\ensuremath{\\mathcal{A}}$ with $F(a) = u: V \\longrightarrow U$ is \\emph{cartesian (with respect to $F$)} provided that for every $v: W \\longrightarrow V$ in $\\ensuremath{\\mathcal{U}}$ and $C \\in \\ensuremath{\\mathcal{A}}_W$, the composition map\n$$a-: \\ensuremath{\\mathcal{A}}_v(C,B) \\longrightarrow \\ensuremath{\\mathcal{A}}_{uv}(C,A)$$\nis an isomorphism.\n\nThe category $\\ensuremath{\\mathcal{A}}$ is called \\emph{fibered over $\\ensuremath{\\mathcal{U}}$} provided that for every $u: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{U}}$ and $A \\in \\ensuremath{\\mathcal{A}}_U$, there is an object $u^{\\ast}A \\in \\ensuremath{\\mathcal{A}}_V$ and a cartesian morphism $\\delta^{u, A} \\in \\ensuremath{\\mathcal{A}}_u(u^{\\ast}A, A)$. The choice, for every $u$ and $A$, of such a cartesian morphism $\\delta^{u,A}$ is called a \\emph{choice of cartesian morphisms}. For a fibered category with a choice of cartesian morphisms, there is an associated pseudofunctor\n$$\\ensuremath{\\mathcal{A}}: \\ensuremath{\\mathcal{U}}^{^{\\mathrm{op}}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} : U \\longmapsto \\ensuremath{\\mathcal{A}}(U)$$\nwhere $\\ensuremath{\\mathsf{Cat}} $ denotes the category of small categories. Here $\\ensuremath{\\mathcal{A}}(U)$ is the category with object set given by $\\ensuremath{\\mathcal{A}}_U$ and $\\ensuremath{\\mathcal{A}}(U)(B,A) = \\ensuremath{\\mathcal{A}}_{1_U}(B,A)$. For a map $u: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{U}}$, the corresponding functor $u^{\\ast}: \\ensuremath{\\mathcal{A}}(U) \\longrightarrow \\ensuremath{\\mathcal{A}}(V)$ is naturally determined by the chosen cartesian morphisms.\n\n\\begin{proposition}\\label{funchoice}\nLet $F: \\ensuremath{\\mathcal{A}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be a fibered category over $\\ensuremath{\\mathcal{U}}$ with a choice of cartesian morphisms. Suppose for $v: W \\longrightarrow V$, $u: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{U}}$ and $A \\in \\ensuremath{\\mathcal{A}}_U$, and for the cartesian morphisms $\\delta^{u,A}: u^{\\ast}A \\longrightarrow A$, $\\delta^{v, u^{\\ast}A}: v^{\\ast} u^{\\ast} A \\longrightarrow u^{\\ast} A$ and $\\delta^{uv, A}: (uv)^{\\ast} A \\longrightarrow A$, we have $v^{\\ast} u^{\\ast} A = (uv)^{\\ast} A$ and $\\delta^{u,A} \\delta^{v, u^{\\ast} A} = \\delta^{uv, A}$. Then the associated pseudofunctor $\\ensuremath{\\mathcal{A}}: \\ensuremath{\\mathcal{U}}^{^{\\mathrm{op}}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ is actually a functor.\n\\end{proposition}\n\nNext we recall the definition of a pretopology. Let $\\ensuremath{\\mathcal{U}}$ be a category with pullbacks. A \\emph{pretopology} on $\\ensuremath{\\mathcal{U}}$ consists of the notion of a \\emph{covering collection} of maps $(U_i \\longrightarrow U)_{i \\in I}$, also called a \\emph{cover} of the object $U$, satisfying the following axioms:\n\\begin{enumerate}\n\\item The collection consisting of $1_{U}: U \\longrightarrow U$ is a cover of $U$.\n\\item If $(U_i \\longrightarrow U)_{i \\in I}$ is a cover of $U$ and $u: V \\longrightarrow U$ an arbitrary map, the the collection of pullbacks $(V \\times_{U} U_i \\longrightarrow V)_{i \\in I}$ is a cover of $V$.\n\\item If $(U_i \\longrightarrow U)_{i \\in I}$ is a cover of $U$ and for every $i \\in I$, $(U_{ij} \\longrightarrow U_i)_{j \\in J_i}$ is a cover of $U_i$, then the collection of compositions $(U_{ij} \\longrightarrow U_i \\longrightarrow U)_{i\\in I, j \\in J_i}$ is a cover of $U$.\n\\end{enumerate}\n\nConsider a fibered category $\\ensuremath{\\mathcal{A}}$ over $\\ensuremath{\\mathcal{U}}$ with a choice of cartesian morphisms and associated pseudofunctor. Let there be given a pretopology on $\\ensuremath{\\mathcal{U}}$ and let $S = (U_i \\longrightarrow U)_{i \\in I}$ be a cover of $U \\in \\ensuremath{\\mathcal{U}}$. The \\emph{descent category} $\\ensuremath{\\mathrm{Des}}(S, \\ensuremath{\\mathcal{A}})$ is defined in the following way. An object, called a \\emph{descent datum}, consists of a collection $(A_i)_{i \\in I}$ of objects with $A_i \\in \\ensuremath{\\mathcal{A}}(U_i)$, together with for every $i, j \\in I$ an isomorphism $\\alpha_1^{\\ast}(A_i) \\cong \\alpha_2^{\\ast}(A_j)$ in $\\ensuremath{\\mathcal{A}}(U_{ij})$ for the pullback\n$$\\xymatrix{ {U_i} \\ar[r] & {U} \\\\ {U_{ij}} \\ar[u]^{\\alpha_1} \\ar[r]_{\\alpha_2} & {U_j.} \\ar[u] }$$\nThese isomorphisms have to satisfy the natural compatibility requirement on triple pullbacks.\nA morphism between descent data $(A_i)_i \\longrightarrow (B_i)_i$ consists of compatible morphisms $A_i \\longrightarrow B_i$ in $\\ensuremath{\\mathcal{A}}(U_i)$.\nThe fibered category $\\ensuremath{\\mathcal{A}}$ is called a \\emph{stack} (resp. a \\emph{prestack}) provided that the natural comparison functor\n$$\\ensuremath{\\mathcal{A}}(U) \\longrightarrow \\ensuremath{\\mathrm{Des}}(S, \\ensuremath{\\mathcal{A}}): A \\longmapsto (u_i^{\\ast}A)_i$$\nis an equivalence of categories (resp. fully faithful) for every cover $S = (u_i: U_i \\longrightarrow U)_{i \\in I}$.\n\nNext we collect some useful facts concerning composable functors  $G: \\ensuremath{\\mathcal{A}} \\longrightarrow \\ensuremath{\\mathcal{X}}$ and $F: \\ensuremath{\\mathcal{X}} \\longrightarrow \\ensuremath{\\mathcal{U}}$. We suppose all three categories have pullbacks and the functors $F$ and $G$ preserve pullbacks.\n\n\\begin{proposition}\\label{propfibtrans}\n\\begin{enumerate}\n\\item If a morphism $a: A \\longrightarrow A'$ in $\\ensuremath{\\mathcal{A}}$ is cartesian with respect to $G$ and $G(a)$ is cartesian with respect to $F$, then $a$ is cartesian with respect to $FG$.\n\\item If both $F$ and $G$ are fibered, then so is $FG$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\nSuppose $\\ensuremath{\\mathcal{U}}$ is endowed with a pretopology. There is a pretopology on $\\ensuremath{\\mathcal{X}}$ for which $(x_i: X_i \\longrightarrow X)_i$ is a cover of $X$ if and only if $(F(x_i): F(X_i) \\longrightarrow F(X))_i$ is a cover of $F(X)$.\n\\end{proposition}\n\n\\begin{proposition}\nConsider morphisms in $\\ensuremath{\\mathcal{X}}$:\n$$\\xymatrix{ {X_2} \\ar[r]_{x_2} & {X_1} \\ar[r]_{x_1} & {X}}$$\n\\begin{enumerate}\n\\item If $x_2$ and $x_1$ are cartesian, then so is $x_1x_2$.\n\\item If $x_1$ and $x_1 x_2$ are cartesian, then so is $x_2$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition} Consider a commutative square in $\\ensuremath{\\mathcal{X}}$:\n$$\\xymatrix{ {X_1} \\ar[r]^{x_1} & {X} \\\\ {X_{12}} \\ar[u]^{y_1} \\ar[r]_{y_2} & {X_2} \\ar[u]_{x_2} }$$\n\\begin{enumerate}\n\\item If the square is a pullback and $x_2$ is cartesian, then so is $y_1$.\n\\item If the image of the square under $F$ is a pullback, and the morphisms $x_1$, $x_2$, $y_1$, $y_2$ are cartesian, then the square is a pullback.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\nThere is a pretopology on $\\ensuremath{\\mathcal{X}}$ for which $(x_i: X_i \\longrightarrow X)_i$ is a cover of $X$ if and only if the morphisms $x_i$ are cartesian with repect to $F$.\n\\end{proposition}\n\n\\begin{proposition}\\label{proppretopx}\nSuppose $\\ensuremath{\\mathcal{U}}$ is endowed with a pretopology. There is a pretopology on $\\ensuremath{\\mathcal{X}}$ for which $(x_i: X_i \\longrightarrow X)_i$ is a cover of $X$ if and only if the following two conditions hold:\n\\begin{enumerate} \n\\item $(F(x_i): F(X_i) \\longrightarrow F(X))_i$ is a cover of $F(X)$;\n\\item every $x_i$ is cartesian with respect to $F$.\n\\end{enumerate}\n\\end{proposition}\n\nNow suppose both $F$ and $G$ are fibered, and consider a choice of cartesian morphisms for both functors. For a morphism $u: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{U}}$ and $A \\in \\ensuremath{\\mathcal{A}}_U$ with respect to $FG$, we obtain the cartesian morphism $x = \\delta^{u, G(A)}: u^{\\ast}G(A) \\longrightarrow G(A)$ in $\\ensuremath{\\mathcal{X}}$. Next we obtain the cartesian morphism $\\delta^{x, A}: x^{\\ast}A \\longrightarrow A$. We make a choice of cartesian morphisms for $FG$ by putting $u^{\\ast}A = x^{\\ast}A$ and $\\delta^{u,A} = \\delta^{x, A}$.\nLet $\\ensuremath{\\mathcal{U}}$ be endowed with a pretopology and endow $\\ensuremath{\\mathcal{X}}$ with the pretopology described in Proposition \\ref{proppretopx}.\nLet $\\ensuremath{\\mathcal{X}}^F$ be the pseudofunctor associated to $F: \\ensuremath{\\mathcal{X}} \\longrightarrow \\ensuremath{\\mathcal{U}}$, $\\ensuremath{\\mathcal{A}}^{G}$ the one associated to $G: \\ensuremath{\\mathcal{A}} \\longrightarrow \\ensuremath{\\mathcal{X}}$ and $\\ensuremath{\\mathcal{A}}^{FG}$ the one associated to $FG: \\ensuremath{\\mathcal{A}} \\longrightarrow \\ensuremath{\\mathcal{U}}$.\n\n\\begin{proposition}\\label{stacktrans}\nIf $\\ensuremath{\\mathcal{X}}^F$ and $\\ensuremath{\\mathcal{A}}^G$ are stacks (resp. prestacks), then so is $\\ensuremath{\\mathcal{A}}^{FG}$.\n\\end{proposition}\n\n\n\n\n\\subsection{Map-graded categories and functors}\\label{mapgraded}\n\nLet $\\ensuremath{\\mathcal{U}}$ be a base category.\nA \\emph{$\\ensuremath{\\mathcal{U}}$-graded set} $\\mathfrak{x}$ consists of the datum, for every $U \\in \\ensuremath{\\mathcal{U}}$, of a set $\\mathfrak{x}_U$.\nA \\emph{($k$-linear) $\\ensuremath{\\mathcal{U}}$-graded category} (see \\cite{lowenmap}) $\\mathfrak{a}$ consists of the following data:\n\n\\begin{itemize}\n\\item For every $U \\in \\ensuremath{\\mathcal{U}}$, a set $\\mathfrak{a}_U$ of \\emph{objects over $U$}.\n\\item For every $u: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{U}}$, $B \\in \\mathfrak{a}_V$, $A \\in \\mathfrak{a}_U$, a $k$-module $\\mathfrak{a}_u(B,A)$ of \\emph{morphisms over $u$}.\n\\item For a further $v: W \\longrightarrow V$ in $\\ensuremath{\\mathcal{U}}$ and $C \\in \\mathfrak{a}_W$, a \\emph{composition} map\n$$\\mathfrak{a}_u(B,A) \\otimes_k \\mathfrak{a}_v(C,B) \\longrightarrow \\mathfrak{a}_{uv}(C,A).$$\n\\item For $A \\in \\mathfrak{a}_U$, an \\emph{identity element}\n$$1_A \\in \\mathfrak{a}_{1_U}(A,A).$$\n\\end{itemize}\n\nThese data should satisfy the obvious category-type axioms, i.e. the composition has to be associative and the identity elements have to act identically under composition.\nThere is an associated $k$-linear category $\\tilde{\\mathfrak{a}}$ with $\\mathrm{Ob}(\\tilde{\\mathfrak{a}}) = \\coprod_{U \\in \\ensuremath{\\mathcal{U}}} \\mathfrak{a}_U$ and $\\mathfrak{a}(A_V, A_U) = \\oplus_{u \\in \\ensuremath{\\mathcal{U}}(V,U)} \\mathfrak{a}_u(A_V, A_U)$.\nClearly, a $\\ensuremath{\\mathcal{U}}$-graded category has an underlying $\\ensuremath{\\mathcal{U}}$-graded set given by the object sets.\n\n\\begin{remark}\\label{remlin}\\label{remgraded}\nThe definition of a $k$-linear $\\ensuremath{\\mathcal{U}}$-graded category is a $k$-linearized version of the notion of a category over $\\ensuremath{\\mathcal{U}}$ from \\S \\ref{parfibstack}. Indeed, if we drop $k$-linearity from the definition, a $\\ensuremath{\\mathcal{U}}$-graded category $\\ensuremath{\\mathcal{A}}$ now has a natural associated category $\\tilde{\\ensuremath{\\mathcal{A}}}$ with $\\tilde{\\ensuremath{\\mathcal{A}}}(A_V, A_U) = \\coprod_{u \\in \\ensuremath{\\mathcal{U}}(V,U)} \\mathfrak{a}_u(A_V, A_U)$. Hence, there is a natural functor $\\tilde{\\ensuremath{\\mathcal{A}}} \\longrightarrow \\ensuremath{\\mathcal{U}}$, and the datum of this functor is equivalent to the datum of $\\ensuremath{\\mathcal{A}}$.\nTo avoid confusion, for us a \\emph{$\\ensuremath{\\mathcal{U}}$-graded category} will impicitely mean a $k$-linear $\\ensuremath{\\mathcal{U}}$-graded category over some fixed $k$, whereas we will refer to the non-linear variant explicitely as a \\emph{non-linear $\\ensuremath{\\mathcal{U}}$-graded category}.\n\\end{remark}\n\n\\begin{example}\\label{exmonoid}\nSuppose $\\ensuremath{\\mathcal{U}}$ has a single object $\\ast$ and $\\mathfrak{a}_{\\ast} = \\{\\ast\\}$. Then $G = \\ensuremath{\\mathcal{U}}(\\ast, \\ast)$ is a monoid, and $\\mathfrak{a}$ corresponds to a $G$-graded algebra $A$ with $A_g = \\mathfrak{a}_g(\\ast, \\ast)$. Thus, in the spirit of \\cite{mitchell}, we can view map-graded categories as monoid-graded algebras with several objects.\n\\end{example}\n\n\\begin{example}\\label{exstandgr}\nA $k$-linear category $\\mathfrak{a}$ can be made into a graded category in several natural ways:\n\\begin{enumerate} \n\\item Let $\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}$ be the category with $\\mathrm{Ob}(\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}) = \\mathrm{Ob}(\\mathfrak{a})$ and $\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}(B,A) = \\{\\ast \\}$ for all $B, A \\in \\mathfrak{a}$. Then we can make $\\mathfrak{a}$ into a $\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}$-graded category $\\mathfrak{a}_{st}$ with $(\\mathfrak{a}_{st})_A = \\{ A \\}$ and $(\\mathfrak{a}_{st})_{\\ast}(B,A) = \\mathfrak{a}(B,A)$. We refer to this grading as the \\emph{standard grading} on $\\mathfrak{a}$.\n\\item Let $e$ be the category with one object $\\ast$ and one morphism $1_{\\ast}$. Then we can make $\\mathfrak{a}$ into an $e$-graded category $\\mathfrak{a}_{tr}$ with $(\\mathfrak{a}_{tr})_{\\ast} = \\mathrm{Ob}(\\mathfrak{a})$ and $(\\mathfrak{a}_{tr})_{1_{\\ast}}(B,A) = \\mathfrak{a}(B,A)$. We refer to this grading as the \\emph{trivial grading} on $\\mathfrak{a}$.\n\\item Clearly, every partition of $\\mathrm{Ob}(\\mathfrak{a})$ will give rise to a graded incarnation of $\\mathfrak{a}$ ``in between'' the two extremes described in (1) and (2).\n\\end{enumerate}\n\\end{example}\n\n\\begin{example}\\label{exfreegr}\nLet $\\ensuremath{\\mathcal{U}}$ be a category. The \\emph{free $\\ensuremath{\\mathcal{U}}$-graded category} $k\\ensuremath{\\mathcal{U}}$ is defined by putting $k\\ensuremath{\\mathcal{U}}_U = \\{ U \\}$ and $k\\ensuremath{\\mathcal{U}}_u(V,U) = k$ for all $u: V \\longrightarrow U \\in \\ensuremath{\\mathcal{U}}$. Compositions are defined by means of the multiplication of $k$, and identity elements are provided by the unit of $k$. Instead of $k$, we can actually use an arbitrary $k$-algebra $A$ and perform a similar construction.\n\\end{example}\n\n\\begin{example}\\label{exsharp}\nLet $\\mathfrak{x}$ be a $\\ensuremath{\\mathcal{U}}$-graded set. There is an associated small category $\\ensuremath{\\mathcal{U}}^{\\sharp}$ and a $\\ensuremath{\\mathcal{U}}^{\\sharp}$-graded set $\\mathfrak{x}^{\\sharp}$ involving only singleton sets above the objects of $\\ensuremath{\\mathcal{U}}^{\\sharp}$. Precisely, we put $\\mathrm{Ob}(\\ensuremath{\\mathcal{U}}^{\\sharp}) = \\coprod_{U \\in \\ensuremath{\\mathcal{U}}} \\mathfrak{x}_U$ and $\\ensuremath{\\mathcal{U}}^{\\sharp}(X_U, X_{U'}) = \\ensuremath{\\mathcal{U}}(U, U')$, and $\\mathfrak{x}^{\\sharp}_{X_U} = \\{ X_U \\}$.  If $\\mathfrak{a}$ is a $\\ensuremath{\\mathcal{U}}$-graded category, we further obtain the $\\ensuremath{\\mathcal{U}}^{\\sharp}$-graded category $\\mathfrak{a}^{\\sharp}$ with in addition $\\mathfrak{a}^{\\sharp}_u(A_U, A_{U'}) = \\mathfrak{a}_u(A_U, A_{U'})$.\n\\end{example}\n\n\\begin{example}\\label{extrst}\nWith the notations of Example \\ref{exstandgr}, for a $k$-linear category, we have $(\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}, \\mathfrak{a}_{st}) = (e, \\mathfrak{a}_{tr})^{\\sharp}$.\n\\end{example}\n\n\nLet $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be a functor, $\\mathfrak{x}$ a $\\ensuremath{\\mathcal{U}}$-graded set and $\\mathfrak{y}$ a $\\ensuremath{\\mathcal{V}}$-graded set. A \\emph{$\\varphi$-graded map} $F$ consists of maps $F_V: \\mathfrak{y}_V \\longrightarrow \\mathfrak{x}_{\\varphi(V)}$ for every $V \\in \\ensuremath{\\mathcal{V}}$.\n\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and $\\mathfrak{b}$ a $\\ensuremath{\\mathcal{V}}$-graded category.\nA \\emph{$\\varphi$-graded functor} $F$ consists of the following data:\n\n\\begin{itemize}\n\\item For every $V \\in \\ensuremath{\\mathcal{V}}$, a map $F_V: \\mathfrak{b}_V \\longrightarrow \\mathfrak{a}_{\\varphi(V)}$.\n\\item For every $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $B \\in \\mathfrak{b}_V$ and $B' \\in \\mathfrak{b}_{V'}$, a $k$-linear map $$\\mathfrak{b}_v(B,B') \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(F(B), F(B')).$$\n\\end{itemize}\n\nThese data should satisfy the obvious functoriality-type axioms, i.e. $F$ respects compositions and identity elements.\nClearly, a $\\varphi$-graded functor has an underlying $\\varphi$-graded map between object sets.\nLet an underlying ground ring $k$ be fixed. Map-graded categories (resp. sets) and functors (resp. maps) constitute a category $\\ensuremath{\\mathsf{Map}} $ (resp. $\\ensuremath{\\mathsf{Mas}} $) in the following way. An object of $\\ensuremath{\\mathsf{Map}} $ (resp. $\\ensuremath{\\mathsf{Mas}} $) is given by a small category $\\ensuremath{\\mathcal{U}}$ and a $\\ensuremath{\\mathcal{U}}$-graded category (resp. set) $\\mathfrak{a}$. A morphism $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto (\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$ is given by a functor $\\varphi: \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathcal{V}}$ and a $\\varphi$-graded functor (resp. map) $F: \\mathfrak{a} \\longrightarrow \\mathfrak{b}$. \n\n\\begin{example}\nLet $\\mathfrak{a}$ be a linear category. Let $(\\mathfrak{a}_{st}, \\ensuremath{\\mathcal{U}}_{\\mathfrak{a}})$ be the graded category resulting from the standard grading on $\\mathfrak{a}$ as in Example \\ref{exstandgr} (1) and let $(\\mathfrak{a}_{tr}, e)$ be the graded category resulting from the trivial grading on $\\mathfrak{a}$ as in Example \\ref{exstandgr} (2). Let $\\varphi: \\ensuremath{\\mathcal{U}}_{\\mathfrak{a}} \\longrightarrow e$ be the unique functor. The map $(\\mathfrak{a}_{st})_A = \\{ A \\} \\longrightarrow \\mathrm{Ob}(\\mathfrak{a}) = (\\mathfrak{a}_{tr})_{\\ast}: A \\longmapsto A$ and the maps $1: (\\mathfrak{a}_{st})_{\\ast}(A,A') = \\mathfrak{a}(A,A') \\longrightarrow \\mathfrak{a}(A,A') = (\\mathfrak{a}_{tr})_{\\ast}(A,A')$ define a $\\varphi$-graded functor.\n\\end{example}\n\n\\begin{example}\\label{exadj}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and $\\mathfrak{b}$ a linear category with associated $e$-graded category $\\mathfrak{b}_{tr}$ with trivial grading as in Example \\ref{exstandgr} (2) and let $\\varphi: \\ensuremath{\\mathcal{U}} \\longrightarrow e$ be the unique functor. There is a one-one correspondence between $\\varphi$-graded functors $\\mathfrak{a} \\longrightarrow \\mathfrak{b}_{tr}$ and $k$-linear functors $\\tilde{\\mathfrak{a}} \\longrightarrow \\mathfrak{b}$.\n\\end{example}\n\n\\begin{example}\\label{exfunsharp}\nWith the notations of Example \\ref{exsharp}, let $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded set resp. category and let $\\varphi_{\\mathfrak{a}}: \\ensuremath{\\mathcal{U}}^{\\sharp} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be the forgetful functor. There is a $\\varphi_{\\mathfrak{a}}$-graded map resp. functor $F_{\\mathfrak{a}}: \\mathfrak{a}^{\\sharp} \\longrightarrow \\mathfrak{a}$ given by $F(A_U) = A_U \\in \\mathfrak{a}_U$ (and $1: \\mathfrak{a}^{\\sharp}_u(A,A') \\longrightarrow \\mathfrak{a}_u(A,A')$ in the category case).\nA graded map resp. functor $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ gives rise to a commutative square\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}, \\mathfrak{b})} \\ar[r]^{(\\varphi, F)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})} \\\\ {(\\ensuremath{\\mathcal{V}}^{\\sharp}, \\mathfrak{b}^{\\sharp})} \\ar[u]^{(\\varphi_{\\mathfrak{b}}, F_{\\mathfrak{b}})} \\ar[r]_{(\\varphi^{\\sharp}, F^{\\sharp})} & {(\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp}).} \\ar[u]_{(\\varphi_{\\mathfrak{a}}, F_{\\mathfrak{a}})} }$$\nWe thus obtain a natural functor \n$$(-)^{\\sharp}: \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Map}} : (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto (\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp})$$\nand similarly in the case of $\\ensuremath{\\mathsf{Mas}} $.\n\\end{example}\n\n\\begin{example}\nDenote the category of small $k$-linear categories by $\\ensuremath{\\mathsf{Cat}} (k)$. There are natural functors $\\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} (k): \\mathfrak{a} \\longmapsto \\tilde{\\mathfrak{a}}$ and $\\ensuremath{\\mathsf{Cat}} (k) \\longrightarrow \\ensuremath{\\mathsf{Map}} : \\mathfrak{a} \\longmapsto \\mathfrak{a}_{tr}$. By Example \\ref{exadj}, the first functor is left adjoint to the second functor.\n\\end{example}\n\n\\subsection{Two fibered categories of map-graded categories}\\label{parfibmap}\n\nLet $k$ be a fixed commutative ground ring. We consider the categories $\\ensuremath{\\mathsf{Map}} $ and $\\ensuremath{\\mathsf{Mas}} $ as defined in \\S \\ref{mapgraded}, as well as the category $\\ensuremath{\\mathsf{Cat}} $ of small categories.\nThere are natural forgetful functors\n$$\\Psi_1: \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Mas}} : (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto (\\ensuremath{\\mathcal{U}}, \\mathfrak{x}_{\\mathfrak{a}})$$\nwhere $\\mathfrak{x}_{\\mathfrak{a}}$ is the underlying $\\ensuremath{\\mathcal{U}}$-graded set of objects of $\\mathfrak{a}$ and\n$$\\Psi_0: \\ensuremath{\\mathsf{Mas}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} : (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\ensuremath{\\mathcal{U}}.$$\nFurther, all three categories $\\ensuremath{\\mathsf{Map}} $, $\\ensuremath{\\mathsf{Mas}} $ and $\\ensuremath{\\mathsf{Cat}} $ have pullbacks and the two functors $\\Psi_1$ and $\\Psi_0$ preserve them. We first look at $\\ensuremath{\\mathsf{Cat}} $. For functors $\\varphi_1: \\ensuremath{\\mathcal{V}}_1 \\longrightarrow \\ensuremath{\\mathcal{U}}$ and $\\varphi_2: \\ensuremath{\\mathcal{V}}_2 \\longrightarrow \\ensuremath{\\mathcal{U}}$, the pullback\n\\begin{equation}\\label{pb}\n\\xymatrix{{\\ensuremath{\\mathcal{V}}_1} \\ar[r]^{\\varphi_1} & {\\ensuremath{\\mathcal{U}}} \\\\ {\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2} \\ar[u]^{\\alpha_1} \\ar[r]_{\\alpha_2} & {\\ensuremath{\\mathcal{V}}_2} \\ar[u]_{\\varphi_2}}\n\\end{equation}\nis given by the category $\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2$ where\n$$\\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2) = \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_1) \\times_{\\mathrm{Ob}(\\ensuremath{\\mathcal{U}})} \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_2)$$ and\n$$(\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2)((V_1, V_2), (V_1', V_2')) = \\ensuremath{\\mathcal{V}}_1(V_1, V_1') \\times_{\\ensuremath{\\mathcal{U}}(\\varphi_1(V_1), \\varphi_2(V_2))} \\ensuremath{\\mathcal{V}}_2(V_2, V_2')$$\nare given by the pullbacks in the category of sets.\nFor a collection of functors $\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\longrightarrow \\ensuremath{\\mathcal{U}}$, we similarly obtain a limit category $\\prod_{i, \\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_i$. \n\n\\begin{example}\\label{exint}\nConsider subcategories $\\ensuremath{\\mathcal{V}}_1 \\subseteq \\ensuremath{\\mathcal{U}}$ and $\\ensuremath{\\mathcal{V}}_2 \\subseteq \\ensuremath{\\mathcal{U}}$. We define the category $\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2$ with $\\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2) = \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_1) \\cap \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}_2)$ and $(\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2)(V,V') = \\ensuremath{\\mathcal{V}}_1(V,V') \\cap \\ensuremath{\\mathcal{V}}_2(V,V')$. The canonical inclusion functors $\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2 \\subseteq \\ensuremath{\\mathcal{V}}_1$ and $\\ensuremath{\\mathcal{V}}_2 \\subseteq \\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2$ induce an isomorphism of categories $\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2 \\cong \\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2$.\n\\end{example}\n\n\nPullbacks in $\\ensuremath{\\mathsf{Mas}} $ and $\\ensuremath{\\mathsf{Map}} $ are described in a similar fashion. For instance, the pullback of graded categories\nis described by\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}_1, \\mathfrak{b}_1)} \\ar[r]^{(\\varphi_1, F_1)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})}\\\\   {(\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2, \\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)} \\ar[u]^{(\\alpha_1, G_1)} \\ar[r]_-{(\\alpha_2, G_2)} & {(\\ensuremath{\\mathcal{V}}_2, \\mathfrak{b}_2)} \\ar[u]_-{(\\varphi_2, F_2)} }$$\nwith underlying pullback of categories described by \\eqref{pb} and with\n$$(\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)_{(V_1, V_2)} = (\\mathfrak{b}_1)_{V_1} \\times_{\\mathfrak{a}_{\\varphi_1(V_1)}} (\\mathfrak{b}_2)_{V_2}$$\nand\n$$(\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)_{(v_1, v_2)}((B_1, B_2), (B_1', B_2')) = (\\mathfrak{b}_1)_{v_1}(B_1, B_1') \\times_{\\mathfrak{a}_{\\varphi_1(v_1)}(F_1(B_1), F_1(B_1'))} (\\mathfrak{b}_2)_{v_2}(B_2, B_2').$$\n\n\\begin{example}\\label{expullsharp}\nThe functors $(-)^{\\sharp}: \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Map}} $ and $(-)^{\\sharp}: \\ensuremath{\\mathsf{Mas}}  \\longrightarrow \\ensuremath{\\mathsf{Mas}} $ from Example \\ref{exfunsharp} preserves pullbacks.\n\\end{example}\n\n\n\n\n\nThus, the results of \\S \\ref{parfibstack} apply to the composable functors $\\Psi_0$ and $\\Psi_1$.\nIn the remainder of this section we show that both these functors are fibered.\n\nWe start with $\\Psi_0$. Let $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be a functor and $\\mathfrak{x}$ a $\\ensuremath{\\mathcal{U}}$-graded set. We define the $\\ensuremath{\\mathcal{V}}$-graded set $\\mathfrak{x}^{\\varphi}$ with $$\\mathfrak{x}^{\\varphi}_V = \\mathfrak{x}_{\\varphi(V)}$$\nand the morphism of graded sets $(\\varphi, \\delta^{\\varphi, \\mathfrak{x}}): (\\ensuremath{\\mathcal{V}}, \\mathfrak{x}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})$ with $\\delta^{\\varphi, \\mathfrak{x}}_V: \\mathfrak{x}^{\\varphi}_V = \\mathfrak{x}_{\\varphi(V)} \\longrightarrow \\mathfrak{x}_{\\varphi(V)}$ given by the identity morphism.\n\n\n\\begin{proposition}\nA morphism $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})$ of graded sets is cartesian with respect to $\\Psi_0$ if and only if for every $V \\in \\ensuremath{\\mathcal{V}}$ the map $F_V: \\mathfrak{y}_V \\longrightarrow \\mathfrak{x}_{\\varphi(V)}$ is an isomorphism of sets.\n\\end{proposition}\n\n\\begin{proposition}\\label{propfib0}\n\\begin{enumerate}\n\\item The morphisms $(\\varphi, \\delta^{\\varphi, \\mathfrak{x}})$ are cartesian with respect to $\\Psi_0$.\n\\item The category $\\ensuremath{\\mathsf{Mas}} $ is fibered over $\\ensuremath{\\mathsf{Cat}} $ through $\\Psi_0$.\n\\end{enumerate}\n\\end{proposition}\n\nNext we look at $\\Psi_1$. \nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})$ be a morphism of graded sets, and let $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category with underlying graded set $\\mathfrak{x}$. We define the $\\ensuremath{\\mathcal{V}}$-graded category $\\mathfrak{a}^{\\varphi, F}$ with\n$$\\mathfrak{a}^{\\varphi, F}_V = \\mathfrak{y}_V$$\nand with, for $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $Y \\in \\mathfrak{y}_V$, $Y' \\in \\mathfrak{y}_{V'}$:\n$$\\mathfrak{a}^{\\varphi, F}_v(Y, Y') = \\mathfrak{a}_{\\varphi(v)}(F(Y), F(Y'))$$\nand with, for another $v': V' \\longrightarrow V''$ in $\\ensuremath{\\mathcal{V}}$ and $Y'' \\in \\mathfrak{y}_{V''}$, the composition in $\\mathfrak{a}^{\\varphi, F}$\n$$\\mathfrak{a}_{\\varphi(v')}(F(Y'), F(Y'')) \\otimes \\mathfrak{a}_{\\varphi(v)}(F(Y), F(Y')) \\longrightarrow \\mathfrak{a}_{\\varphi(v' v)}(F(Y), F(Y''))$$\ndefined by the composition in $\\mathfrak{a}$, and similarly for the identity elements in $\\mathfrak{a}^{\\varphi, F}_{1_V}(Y,Y) = \\mathfrak{a}_{1_{\\varphi(V)}}(F(Y), F(Y))$.\nWe define the morphism of graded categories\n$(\\varphi, \\delta = \\delta^{\\varphi, F, \\mathfrak{a}}): (\\ensuremath{\\mathcal{V}}, \\mathfrak{a}^{\\varphi, F}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ by the morphisms $\\delta_V = F_V: \\mathfrak{y}_V \\longrightarrow \\mathfrak{a}_{\\varphi(V)}$ and the identity morphisms\n$$\\mathfrak{a}^{\\varphi, F}_v(Y, Y') = \\mathfrak{a}_{\\varphi(v)}(F(Y), F(Y')) \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(F(Y), F(Y')).$$\n\n\\begin{proposition}\\label{lemcart1}\nA morphism $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ of graded categories is cartesian with respect to $\\Psi_1$ if and only if for every $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $B \\in \\mathfrak{b}_V$, $B' \\in \\mathfrak{b}_{V'}$, the map $\\mathfrak{b}_v(B,B') \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(F(B), F(B'))$ is an isomorphism. \n\\end{proposition}\n\n\\begin{proposition}\\label{propfib1}\n\\begin{enumerate}\n\\item The morphisms $(\\varphi, \\delta^{\\varphi, F, \\mathfrak{a}})$ are cartesian with respect to $\\Psi_1$.\n\\item The category $\\ensuremath{\\mathsf{Map}} $ is fibered over $\\ensuremath{\\mathsf{Mas}} $ through $\\Psi_1$.\n\\end{enumerate}\n\\end{proposition}\n\nBy Proposition \\ref{propfibtrans}, as a consequence of Propositions \\ref{propfib0} and \\ref{propfib1}, the category $\\ensuremath{\\mathsf{Map}} $ is fibered over $\\ensuremath{\\mathsf{Cat}} $ through $\\Psi$. We end this section by describing the choice of cartesian morphisms which follows from the higher choices for $\\Psi_0$ and $\\Psi_1$.\n\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$ graded category and let $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be a functor. We define the $\\ensuremath{\\mathcal{V}}$-graded category $\\mathfrak{a}^{\\varphi}$ with \n$$\\mathfrak{a}^{\\varphi}_V = \\mathfrak{a}_{\\varphi(V)}$$ and with, for $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $A \\in \\mathfrak{a}^{\\varphi}_V$ and $A' \\in \\mathfrak{a}^{\\varphi}_{V'}$\n$$\\mathfrak{a}^{\\varphi}_v(A, A') = \\mathfrak{a}_{\\varphi(v)}(A,A').$$\nFor another $v': V' \\longrightarrow V''$ in $\\ensuremath{\\mathcal{V}}$ and $A'' \\in \\mathfrak{a}^{\\varphi}_{V''}$, the composition in $\\mathfrak{a}^{\\varphi}$\n$$\\mathfrak{a}_{\\varphi(v')}(A', A'') \\otimes \\mathfrak{a}_{\\varphi(v)}(A,A') \\longrightarrow \\mathfrak{a}_{\\varphi(v'v)}(A,A'')$$\nis defined by the composition in $\\mathfrak{a}$, and similarly for the identity elements in $\\mathfrak{a}_{\\varphi(1_V)}(A,A) = \\mathfrak{a}_{1_{\\varphi(V)}}(A,A)$.\nWe further define the morphism of graded categories $(\\varphi,\\delta^{\\varphi, \\mathfrak{a}}): (\\ensuremath{\\mathcal{V}}, \\mathfrak{a}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$\nfor which the maps\n$$\\mathfrak{a}^{\\varphi}_V \\longrightarrow \\mathfrak{a}_{\\varphi(V)}$$ are given by identities and the maps\n$$\\mathfrak{a}^{\\varphi}_v(A, A') \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(A,A')$$\nas well. \n\n\\begin{proposition}\\label{propcart}\n\\begin{enumerate}\n\\item The morphisms $(\\varphi, \\delta^{\\varphi, \\mathfrak{a}})$ are cartesian with respect to $\\Psi_1$.\n\\item The category $\\ensuremath{\\mathsf{Map}} $ is fibered over $\\ensuremath{\\mathsf{Cat}} $ through $\\Psi$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{lemcart}\nA morphism $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ of graded categories is cartesian with respect to $\\Psi$ if and only if the following hold:\n\\begin{enumerate}\n\\item For $V \\in \\ensuremath{\\mathcal{V}}$, the map $F_V: \\mathfrak{b}_V \\longrightarrow \\mathfrak{a}_{\\varphi(V)}$ is an isomorphism.\n\\item For $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $B \\in \\mathfrak{b}_V$, $B' \\in \\mathfrak{b}_{V'}$, the map $\\mathfrak{b}_v(B,B') \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(F(B), F(B'))$ is an isomorphism. \n\\end{enumerate}\n\\end{proposition}\n\nTo distinguish between the notions arrising in Propositions \\ref{lemcart1} and \\ref{lemcart}, we make the following\n\n\\begin{definition}\\label{defcart}\nConsider a morphism $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ between graded categories.\n\\begin{enumerate}\n\\item $(\\varphi, F)$ is called \\emph{cartesian} if it is cartesian with respect to $\\Psi$.\n\\item $(\\varphi, F)$ is called \\emph{subcartesian} if it is cartesian with respect to $\\Psi_1$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item If $(\\varphi, F)$ is cartesian, then it is subcartesian.\n\\item $(\\varphi,F)$ is subcartesian if and only if $(\\varphi^{\\sharp}, F^{\\sharp})$ as defined in Example \\ref{exfunsharp} is cartesian.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{example}\\label{exfff}\nLet $\\mathfrak{b} \\longrightarrow \\mathfrak{a}$ be a fully faithful $k$-linear functor between $k$-linear categories. With the notations from Example \\ref{exstandgr} the natural graded functor $(e, \\mathfrak{b}_{tr}) \\longrightarrow (e, \\mathfrak{a}_{tr})$ is subcartesian and the functor\n$(\\ensuremath{\\mathcal{U}}_{\\mathfrak{b}}, \\mathfrak{b}_{st}) \\longrightarrow (\\ensuremath{\\mathcal{U}}_{\\mathfrak{a}}, \\mathfrak{a}_{st})$ is cartesian (see also Example \\ref{extrst}).\n\\end{example}\n\n\n\\subsection{Sites of categories, graded sets and graded categories}\\label{parsite}\n\nIn this section, we will introduce pretopologies on the categories $\\ensuremath{\\mathsf{Cat}} $, $\\ensuremath{\\mathsf{Mas}} $ and $\\ensuremath{\\mathsf{Map}} $.\nWe start by introducing appropriate nerves. For a small category $\\ensuremath{\\mathcal{U}}$, we denote by $\\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}})$ the simplicial nerve of $\\ensuremath{\\mathcal{U}}$. Concretely, for $n \\geq 1$, $\\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{U}})$ consists of the $n$-simplices\n$$\\xymatrix{ {U_0} \\ar[r]_{u_0} & {U_1} \\ar[r]_{u_1} & {\\dots} \\ar[r]_{u_{n-1}} & {U_n}}$$\nwith $U_i \\in \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$ and $u_i \\in \\mathrm{Mor}(\\ensuremath{\\mathcal{U}})$, and $\\ensuremath{\\mathcal{N}}_0(\\ensuremath{\\mathcal{U}}) = \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$. Note that $\\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{U}}) = \\mathrm{Mor}(\\ensuremath{\\mathcal{U}})$.\nFor a functor $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$, we obtain an induced map $\\ensuremath{\\mathcal{N}}(\\varphi): \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{V}}) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}})$ between the  nerves, with components $\\ensuremath{\\mathcal{N}}_n(\\varphi): \\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{V}}) \\longrightarrow \\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{U}})$.\n\nFor a $\\ensuremath{\\mathcal{U}}$-graded category or set $\\mathfrak{a}$, we consider the associated category $\\ensuremath{\\mathcal{U}}^{\\sharp}$ from Example \\ref{exsharp} with $\\mathrm{Ob}(\\ensuremath{\\mathcal{U}}^{\\sharp}) = \\coprod_{U \\in \\ensuremath{\\mathcal{U}}} \\mathfrak{a}_U$ and $\\ensuremath{\\mathcal{U}}^{\\sharp}(A_U, A'_{U'}) = \\ensuremath{\\mathcal{U}}(U, U')$. We define the \\emph{nerve} of $\\mathfrak{a}$ to be $\\ensuremath{\\mathcal{N}}(\\mathfrak{a}) = \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) = \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$. \nBy Example \\ref{exfunsharp}, a graded map or functor $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ induces an associated map $\\ensuremath{\\mathcal{N}}(F): \\ensuremath{\\mathcal{N}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\mathfrak{a})$ given by $\\ensuremath{\\mathcal{N}}(\\varphi^{\\sharp}): \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{V}}^{\\sharp}) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$.\n\n\n\n\\begin{definition}\\label{defncover}\nLet $n \\in \\mathbb{N} \\cup \\{\\infty\\}$.  A collection of functors (resp. graded maps, resp. graded functors) $(F_i: \\mathfrak{b}_i \\longrightarrow \\mathfrak{a})_{i \\in I}$ is an \\emph{$n$-cover} of $\\mathfrak{a}$ in $\\ensuremath{\\mathsf{Cat}} $ (resp. in $\\ensuremath{\\mathsf{Mas}} $, resp. in $\\ensuremath{\\mathsf{Map}} $) if for all $k \\in \\mathbb{N}$ with $k \\leq n$, the collection of maps $(\\ensuremath{\\mathcal{N}}_k(F_i))_{i \\in I}$ is jointly surjective.\n\\end{definition}\n\nThus, a collection of graded maps (resp. graded functors) $(\\varphi_i, F_i): (\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is an $n$-cover of $\\mathfrak{a}$ in $\\ensuremath{\\mathsf{Mas}} $ (resp. in $\\ensuremath{\\mathsf{Map}} $) if and only if the collection of functors $(\\varphi^{\\sharp}: \\ensuremath{\\mathcal{V}}_i^{\\sharp} \\longrightarrow \\ensuremath{\\mathcal{U}}^{\\sharp})_{i \\in I}$ is an $n$-cover in $\\ensuremath{\\mathsf{Cat}} $.\n\n\\begin{lemma}\\label{lemcover}\nLet $n \\in \\mathbb{N}$. \nConsider a collection of functors (resp. graded maps, resp. graded functors) $\\ensuremath{\\mathcal{S}} = (F_i: \\mathfrak{b}_i \\longrightarrow \\mathfrak{a})_{i \\in I}$. If the collection of maps $(\\ensuremath{\\mathcal{N}}_n(F_i))_{i \\in I}$ is jointly surjective, then $\\ensuremath{\\mathcal{S}}$ is an $n$-cover in $\\ensuremath{\\mathsf{Cat}} $ (resp. in $\\ensuremath{\\mathsf{Mas}} $, resp. in $\\ensuremath{\\mathsf{Map}} $).\n\\end{lemma}\n\n\\begin{proof}\nLet $k < n$. Let us look at the graded cases, and let $\\mathfrak{a}$ be graded over $\\ensuremath{\\mathcal{U}}$ and $\\mathfrak{b}$ over $\\ensuremath{\\mathcal{V}}_i$. Every $k$-simplex $u$ for $\\ensuremath{\\mathcal{U}}^{\\sharp}$ gives rise to an $n$-simplex $u'$ by adding $n - k$ identity maps $1_{U_0}: U_0 \\longrightarrow U_0$. If $u' = \\varphi_i(v')$ for some $i$ and $v' \\in \\ensuremath{\\mathcal{V}}_i^{\\sharp}$, then removing the first $n -k$ maps from $v'$ yields a $k$-simplex $v$ with $\\varphi(v) = u$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemnervepb}\nLet $n \\in \\mathbb{N}$. Consider a pullback in $\\ensuremath{\\mathsf{Cat}} $, $\\ensuremath{\\mathsf{Mas}} $ or $\\ensuremath{\\mathsf{Map}} $:\n$$\\xymatrix{ {\\mathfrak{b}_1} \\ar[r]^{F_1} & {\\mathfrak{a}} \\\\ {\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2} \\ar[u]^{G_1} \\ar[r]_-{G_2} & {\\mathfrak{b}_2} \\ar[u]_{F_2} }$$\nWe have $\\ensuremath{\\mathcal{N}}_n(\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2) \\cong \\ensuremath{\\mathcal{N}}_n(\\mathfrak{b}_1) \\times_{\\ensuremath{\\mathcal{N}}(\\mathfrak{a})} \\ensuremath{\\mathcal{N}}(\\mathfrak{b}_2).$\n\\end{lemma}\n\n\n\\begin{proposition}\\label{pretop}\nLet $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$ be fixed. The $n$-covers from Definition \\ref{defncover} define pretopologies on $\\ensuremath{\\mathsf{Cat}} $, $\\ensuremath{\\mathsf{Mas}} $ and $\\ensuremath{\\mathsf{Map}} $. \n\\end{proposition}\n\n\\begin{proof}\nThe identity and glueing properties are immediate. Concerning the pullback property, we note that in $\\ensuremath{\\mathsf{Set}} $, the pullback of a jointly surjective collection of maps is jointly surjective. Hence, the result follows from Example \\ref{expullsharp}.\n\\end{proof}\n\n\\begin{example}\\label{excover}\n\\begin{enumerate}\n\\item A functor $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ which is full and surjective on objects constitutes an $\\infty$-cover of $\\ensuremath{\\mathcal{U}}$ in $\\ensuremath{\\mathsf{Cat}} $.\n\\item Consider a collection of objects $(U_i)_{i \\in I}$ in the category $\\ensuremath{\\mathcal{U}}$ and the accociated collection of functors $(\\varphi_i: \\ensuremath{\\mathcal{U}}\/U_i \\longrightarrow \\ensuremath{\\mathcal{U}})_{i \\in I}$. The collection $(\\varphi_i)_i$ is a $1$-cover in $\\ensuremath{\\mathsf{Cat}} $ if for every object $U \\in \\ensuremath{\\mathcal{U}}$, there exists a morphism $U \\longrightarrow U_i$ for some $i$. In this case, the collection $(\\varphi_i)_i$ is automatically an $\\infty$-cover of $\\ensuremath{\\mathcal{U}}$. \n\\end{enumerate}\n\\end{example}\n\n\\subsection{The stack of map-graded sets}\\label{parmasstack}\n\nConsider the functor $\\Psi_0: \\ensuremath{\\mathsf{Mas}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ and consider the pretopology $\\ensuremath{\\mathcal{T}}_n$ of $n$-covers from Proposition \\ref{pretop} on $\\ensuremath{\\mathsf{Cat}} $ for some $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$.\n\n\\begin{proposition}\nThe pseudofunctor $\\ensuremath{\\mathsf{Mas}} $ associated to $\\Psi_0$ is a functor.\n\\end{proposition}\n\nLet $\\ensuremath{\\mathcal{S}} = (\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\longrightarrow \\ensuremath{\\mathcal{U}})_i$ be an $n$-cover of $\\ensuremath{\\mathcal{U}}$ in $\\ensuremath{\\mathsf{Cat}} $. We introduce some notation to be able to describe the descent category $\\ensuremath{\\mathrm{Des}}(\\ensuremath{\\mathcal{S}}, \\ensuremath{\\mathsf{Mas}} )$. \nWe put $\\ensuremath{\\mathcal{V}}_{ij} = \\ensuremath{\\mathcal{V}}_{i} \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_{j}$ and $\\ensuremath{\\mathcal{V}}_{ijk} = \\ensuremath{\\mathcal{V}}_i \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_j \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_k$. \nWe denote the canonical maps from a $k$-fold pullback by $\\alpha_1, \\dots, \\alpha_k$ to avoid confusion when some of the indices $i, j, \\dots$ coincide. For example, we have $\\alpha_1: \\ensuremath{\\mathcal{V}}_{ij} \\longrightarrow \\ensuremath{\\mathcal{V}}_i$ and $\\alpha_2: \\ensuremath{\\mathcal{V}}_{ij} \\longrightarrow \\ensuremath{\\mathcal{V}}_j$.\nFor $\\mathfrak{b}_i \\in \\ensuremath{\\mathsf{Mas}} (\\ensuremath{\\mathcal{V}}_i)$, we use notations like ${\\mathfrak{b}_i}|^1_{ij} = \\mathfrak{b}_i^{\\alpha_1} \\in \\ensuremath{\\mathsf{Map}} (\\ensuremath{\\mathcal{V}}_{ij})$ and ${\\mathfrak{b}_j}|^2_{ijk} = \\mathfrak{b}_j^{\\alpha_2} \\in \\ensuremath{\\mathsf{Map}} (\\ensuremath{\\mathcal{V}}_{ijk})$.\nThe descent category $\\ensuremath{\\mathrm{Des}}(\\ensuremath{\\mathcal{S}}, \\ensuremath{\\mathsf{Mas}} )$ has objects given by $(\\mathfrak{b}_{i})_{i}$ with $\\mathfrak{b}_{i} \\in \\ensuremath{\\mathsf{Mas}} (\\ensuremath{\\mathcal{V}}_{i})$ along with compatible isomorphisms \n$$\\rho_{ij}: {\\mathfrak{b}_i}|^1_{ij} \\cong {\\mathfrak{b}_j}|^2_{ij}.$$\nPrecisely, we require that the cocycle condition\n\\begin{equation} \\label{cocyc}\n(\\rho_{jk}|^{23}_{ijk})(\\rho_{ij}|^{12}_{ijk}) = \\rho_{ik}|^{13}_{ijk}\n\\end{equation}\non triple pullbacks holds. Consider $(V_1, V_2, V_3) \\in \\ensuremath{\\mathcal{V}}_{ijk}$. We have $(\\rho_{jk}|^{23}_{ijk})_{(V_1, V_2, V_3)} = (\\rho_{jk})_{(V_2, V_3)}: (\\mathfrak{b}_j)_{V_2} \\longrightarrow (\\mathfrak{b}_k)_{V_3}$, $(\\rho_{ij}|^{12}_{ijk})_{(V_1, V_2, V_3)} = (\\rho_{ij})_{(V_1, V_2)}: (\\mathfrak{b}_i)_{V_1} \\longrightarrow (\\mathfrak{b}_j)_{V_2}$, $(\\rho_{ik}|^{13}_{ijk})_{(V_1, V_2, V_3)} = (\\rho_{ik})_{(V_1, V_3)}: (\\mathfrak{b}_i)_{V_1} \\longrightarrow (\\mathfrak{b}_k)_{V_3}$. hence, from \\eqref{cocyc} we obtain\n\\begin{equation} \\label{cocyc2}\n(\\rho_{jk})_{(V_2, V_3)} (\\rho_{ij})_{(V_1, V_2)} = (\\rho_{ik})_{(V_1, V_3)}.\n\\end{equation}\nIn particular, we have\n\\begin{equation} \\label{cocyc3}\n(\\rho_{ii})_{(V,V)} = 1: (\\mathfrak{b}_i)_V \\longrightarrow (\\mathfrak{b}_i)_V.\n\\end{equation}\n\n\n\n\\begin{theorem}\\label{thmmasstack}\nThe functor $\\ensuremath{\\mathsf{Mas}} $ is a stack.\n\\end{theorem}\n\n\\begin{proof}\nWe may take $n = 0$. Let $\\ensuremath{\\mathcal{S}} = (\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\longrightarrow \\ensuremath{\\mathcal{U}})_i$ be a $0$-cover of $\\ensuremath{\\mathcal{U}}$ in $\\ensuremath{\\mathsf{Cat}} $.\n\nFirst, consider $\\mathfrak{a}, \\mathfrak{b} \\in \\ensuremath{\\mathsf{Mas}} (\\ensuremath{\\mathcal{U}})$ and a compatible collection of morphism $F_i: \\mathfrak{b}^{\\varphi_i} \\longrightarrow \\mathfrak{a}^{\\varphi_i}$. For the pullback\n$$\\xymatrix{ {\\ensuremath{\\mathcal{V}}_i} \\ar[r]^{\\varphi_i} & {\\ensuremath{\\mathcal{U}}} \\\\ {\\ensuremath{\\mathcal{V}}_{ij}} \\ar[u]^{\\alpha_1} \\ar[r]_{\\alpha_2} & {\\ensuremath{\\mathcal{V}}_j} \\ar[u]_{\\varphi_j}}$$\nwe have $(\\mathfrak{a}^{\\varphi_i})^{\\alpha_1} = (\\mathfrak{a}^{\\varphi_j})^{\\alpha_2}$ and similarly for $\\mathfrak{b}$, and compatibility amounts to the fact that $(F_i)^{\\alpha_1} = (F_j)^{\\alpha_2}$. We are to define a unique glueing $F: \\mathfrak{b} \\longrightarrow \\mathfrak{a}$. This consists of maps $F_U: \\mathfrak{b}_U \\longrightarrow \\mathfrak{a}_U$ for every $U \\in \\ensuremath{\\mathcal{U}}$ such that for every $i$ and $V \\in \\ensuremath{\\mathcal{V}}_i$ we have $(F_i)_V = (F^{\\varphi_i})_V = F_{\\varphi_i(V)}: \\mathfrak{b}_{\\varphi_i(V)} \\longrightarrow \\mathfrak{a}_{\\varphi_i(V)}$. Since we have a $0$-cover, for $U \\in \\ensuremath{\\mathcal{U}}$ there is some $i$ and some $V \\in \\ensuremath{\\mathcal{V}}_i$ with $\\varphi_i(V) = U$, so we put\n$F_U = (F_i)_V$. If for another $j$ and $V' \\in \\ensuremath{\\mathcal{V}}_j$ we also have $\\varphi_j(V') = U$, then $(V, V') \\in \\ensuremath{\\mathcal{V}}_{ij}$ and $(F_i)_V = (F_i)^{\\alpha_1}_{(V,V')} = (F_j)^{\\alpha_2}_{(V,V')} = (F_j)_{V'}$.\n\nNext we consider a descent datum $(\\mathfrak{b}_i)_i$ with $\\mathfrak{b}_i \\in \\ensuremath{\\mathsf{Mas}} (\\ensuremath{\\mathcal{V}}_i)$ with compatible isomorphisms $\\rho_{ij}: {\\mathfrak{b}_i}|^1_{ij} \\cong {\\mathfrak{b}_j}|^2_{ij}$. We define the $\\ensuremath{\\mathcal{U}}$-graded set $\\mathfrak{b}$ with \n$$\\mathfrak{b}_U = \\coprod_{\\varphi_i(V) = U} (\\mathfrak{b}_i)_{V} \/ \\sim$$\nwhere for $X_1 \\in (\\mathfrak{b}_i)_{V_1}$ and $X_2 \\in (\\mathfrak{b}_j)_{V_2}$ we have $X_1 \\sim X_2$ if and only if the two elements correspond through the isomorphism $(\\rho_{ij})_{(V_1, V_2)}: (\\mathfrak{b}_i)_{V_1} \\longrightarrow (\\mathfrak{b}_j)_{V_2}$. \nThe relation $\\sim$ is obviously symmetric, it is transitive by \\eqref{cocyc2} and reflexive by \\eqref{cocyc3}.  For every $i$ and $V \\in \\ensuremath{\\mathcal{V}}_i$ with $\\varphi_i(V) = U$, the canonical map\n$(\\mathfrak{b}_i)_V \\longrightarrow \\mathfrak{b}_U$ is an isomorphism, giving rise to a compatible collection of isomorphisms $F_i: \\mathfrak{b}_i \\longrightarrow \\mathfrak{b}^{\\varphi_i}$ constituting an isomorphism $F: (\\mathfrak{b}_i)_i \\longrightarrow (\\mathfrak{b}^{\\varphi_i})_i$ in $\\ensuremath{\\mathrm{Des}}(\\ensuremath{\\mathcal{S}}, \\ensuremath{\\mathsf{Mas}} )$.\n\\end{proof}\n\n\\subsection{The stack of map-graded categories}\\label{parstackmap}\n\nConsider the functor $\\Psi_1: \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Mas}} $ and consider the pretopology $\\ensuremath{\\mathcal{T}}_n$ of $n$-covers from Proposition \\ref{pretop} on $\\ensuremath{\\mathsf{Mas}} $ for some $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$.\n\n\\begin{proposition}\nThe pseudofunctor $\\ensuremath{\\mathsf{Map}} $ associated to $\\Psi_1$ is a functor.\n\\end{proposition}\n\nLet $\\ensuremath{\\mathcal{S}} = ((\\varphi_i,F_i): (\\ensuremath{\\mathcal{V}}_i,\\mathfrak{y}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x}))_i$ be an $n$-cover of $\\mathfrak{x}$ in $\\ensuremath{\\mathsf{Mas}} $. We use notation that is similar to the notation introduced in \\S \\ref{parmasstack}.\nPullbacks are denoted $\\mathfrak{y}_{ij} = \\mathfrak{y}_i \\times_{\\mathfrak{x}} \\mathfrak{y}_j$ and $\\mathfrak{y}_{ijk} = \\mathfrak{y}_i \\times_{\\mathfrak{x}} \\mathfrak{y}_j \\times_{\\mathfrak{x}} \\mathfrak{y}_k$ and for $\\mathfrak{b}_i \\in \\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_i)$, restrictions are denoted $\\mathfrak{b}_i|^1_{ij} \\in \\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_{ij})$ etc.. The descent category $\\ensuremath{\\mathrm{Des}}(\\ensuremath{\\mathcal{S}}, \\ensuremath{\\mathsf{Map}} )$ has objects given by $(\\mathfrak{b}_i)_i$ with $\\mathfrak{b}_i \\in \\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_i)$ along with compatible isomorphisms\n$$\\rho_{ij}: \\mathfrak{b}_i|^1_{ij} \\cong \\mathfrak{b}_j|^2_{ij}$$\nfor which the cocycle condition \\eqref{cocyc} holds. To unravel this condition, we now consider maps $v_{\\kappa}: (V_{\\kappa}, Y_{\\kappa}) \\longrightarrow (V'_{\\kappa}, Y'_{\\kappa})$ in $\\ensuremath{\\mathcal{V}}_{\\kappa}^{\\sharp}$ for $\\kappa \\in \\{ 1, 2, 3 \\}$ Suppose $(v_1, v_2, v_3) \\in \\mathfrak{y}_{ijk}$. We have $(\\rho_{ij})_{v_1, v_2}: (\\mathfrak{b}_i)_{v_1}(Y_1, Y_1') \\longrightarrow (\\mathfrak{b}_j)_{v_2}(Y_2, Y_2')$, $(\\rho_{jk})_{v_2, v_3}: (\\mathfrak{b}_i)_{v_2}(Y_2, Y_2') \\longrightarrow (\\mathfrak{b}_j)_{v_3}(Y_3, Y_3')$, $(\\rho_{ik})_{v_1, v_3}: (\\mathfrak{b}_i)_{v_1}(Y_1, Y_1') \\longrightarrow (\\mathfrak{b}_k)_{v_3}(Y_3, Y_3')$ and by \\eqref{cocyc} \n\\begin{equation} \\label{cocycbis2}\n(\\rho_{jk})_{v_2, v_3} (\\rho_{ij})_{v_1, v_2} = (\\rho_{ik})_{v_1, v_3}.\n\\end{equation}\nIn particular, we have\n\\begin{equation}\\label{cocycbis3}\n(\\rho_{ii})_{v_1,v_1} = 1: (\\mathfrak{b}_i)_{v_1}(Y_1,Y_1') \\longrightarrow (\\mathfrak{b}_i)_{v_1}(Y_1,Y_1').\n\\end{equation}\n\n\\begin{theorem}\\label{thmmapstack}\n\\begin{enumerate}\n\\item If $n \\geq 2$, then $\\ensuremath{\\mathsf{Map}} $ is a prestack.\n\\item If $n \\geq 3$, then $\\ensuremath{\\mathsf{Map}} $ is a stack.\n\\end{enumerate}\\end{theorem}\n\n\\begin{proof}\nLet $\\ensuremath{\\mathcal{S}} = ((\\varphi_i,F_i): (\\ensuremath{\\mathcal{V}}_i,\\mathfrak{y}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x}))_i$ be a $2$-cover of $\\mathfrak{x}$ in $\\ensuremath{\\mathsf{Mas}} $. \n\nFirst, consider $\\mathfrak{a}, \\mathfrak{b} \\in \\ensuremath{\\mathsf{Map}} (\\mathfrak{x})$ and a compatible family of morphisms $G_i: \\mathfrak{b}|_i \\longrightarrow \\mathfrak{a}|_i$ in $\\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_i)$. Thus, for every $Y \\in (\\mathfrak{y}_i)_V$, $Y' \\in (\\mathfrak{y}_i)_{V'}$ and $v \\in \\ensuremath{\\mathcal{V}}_i(V,V')$, we have a map\n$(G_i)_{v, Y, Y'}: (\\mathfrak{b}|_i)_v(Y, Y') \\longrightarrow (\\mathfrak{a}|_i)_v(Y,Y')$, i.e a map $(G_i)_{v, Y, Y'}: \\mathfrak{b}_{\\varphi_i(v)}(F_i(Y), F_i(Y')) \\longrightarrow \\mathfrak{a}_{\\varphi_i(v)}(F_i(Y), F_i(Y'))$. To define a unique glueing $G: \\mathfrak{b} \\longrightarrow \\mathfrak{a}$ in $\\ensuremath{\\mathsf{Map}} (\\mathfrak{x})$, we consider $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$, $X \\in \\mathfrak{x}_U$, $X' \\in \\mathfrak{x}_{U'}$. Since $\\ensuremath{\\mathcal{S}}$ is a $1$-cover, there is an $i$ and $v: V \\longrightarrow V$ in $\\ensuremath{\\mathcal{V}}_i$, $Y \\in (\\mathfrak{y}_i)_V$, $Y' \\in (\\mathfrak{y}_i)$ with $\\varphi_i(v) = u$, $F_i(Y) = X$, $F_i(Y') = X'$. We put\n$$G_{u, X, X'} = (G_i)_{v, Y, Y'}: \\mathfrak{b}_u(X,X') \\longrightarrow \\mathfrak{a}_u(X,X').$$\nCompatibility of the $G_i$ ensures that $G$ is well defined and the fact that $\\ensuremath{\\mathcal{S}}$ is a $2$-cover can be used to show that $G$ is a $\\ensuremath{\\mathcal{U}}$-graded functor.\n\nNext we consider a descent datum $(\\mathfrak{b}_i)_i$ with $\\mathfrak{b}_i \\in \\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_i)$ with compatible isomorphisms $\\rho_{ij}: {\\mathfrak{b}_i}|^1_{ij} \\cong {\\mathfrak{b}_j}|^2_{ij}$ in $\\ensuremath{\\mathsf{Map}} (\\mathfrak{y}_{ij})$. We are to define a $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{b}$ with underlying graded set $\\mathfrak{x}$. \nFor $u: (U, X) \\longrightarrow (U', X')$ in $\\ensuremath{\\mathcal{U}}^{\\sharp}$, we first define the set\n$$\\mathfrak{x}_u(X,X') = \\coprod (\\mathfrak{b}_i)_v(B,B')$$\nwhere the coproduct is taken over all $v: (V, B) \\longrightarrow (V', B')$ in $\\ensuremath{\\mathcal{V}}_i^{\\sharp}$ with $\\varphi_i(v) = u$, $X = F_i(B)$ and $X' = F_i(B')$.\nFor $w: (W, C) \\longrightarrow (W', C')$ in $\\ensuremath{\\mathcal{V}}_j^{\\sharp}$ with $\\varphi_j(w) = u$, $X = F_j(C)$ and $X' = F_j(C')$, we use the isomorphism\n$$(\\rho_{ij})_{v,w}: (\\mathfrak{b}_i)_v(B,B') \\cong (\\mathfrak{b}_j)_w(C,C')$$\nto declare when morphisms are equivalent. We thus obtain an equivalence relation on $\\mathfrak{x}_u(X,X')$ such that the quotient $\\mathfrak{b}_u(X,X')$ has a natural $k$-module structure, and the canonical morphisms\n$$(\\mathfrak{b}_i)_v(B,B') \\longrightarrow \\mathfrak{b}_u(X,X')$$\nare $k$-linear isomorphisms. To define the composition on $\\mathfrak{b}$, we use the fact that $\\ensuremath{\\mathcal{S}}$ is a $2$-cover to choose, for $u: (U, X) \\longrightarrow (U', X')$ in $\\ensuremath{\\mathcal{U}}^{\\sharp}$ and $u': (U', X') \\longrightarrow (U'', X'')$ a single $(\\varphi_i, F_i): (\\ensuremath{\\mathcal{V}}_i, \\mathfrak{y}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})$ and $v: (V, B) \\longrightarrow (V', B')$, $v': (V', B') \\longrightarrow (V'', B'')$ with $\\varphi^{\\sharp}_i(v) = u$, $\\varphi^{\\sharp}_i(v') = u'$ and to use \nthe composition\n$$(\\mathfrak{b}_i)_{v'}(B', B'') \\otimes (\\mathfrak{b}_i)_v(B,B') \\longrightarrow (\\mathfrak{b}_i)_{v'v}(B,B'').$$\nFinally, to show that the composition is associative, we use the fact that $\\ensuremath{\\mathcal{S}}$ is a $3$-cover.\n\\end{proof}\n\n\\begin{lemma}\\label{lemtopocomp}\nConsider the functor $\\Psi_0: \\ensuremath{\\mathsf{Mas}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} $. On $\\ensuremath{\\mathsf{Cat}} $, consider the pretopology of $n$-covers for some $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$. On $\\ensuremath{\\mathsf{Mas}} $, consider the pretopology of collections consisting of cartesian morphisms with respect to $\\Psi_0$, that are mapped to an $n$-cover under $\\Psi_0$ (as described in Proposition \\ref{proppretopx}).\nEvery cover for this pretopology is an $n$-cover in the sense of Proposition \\ref{pretop}.\n\\end{lemma}\n\n\\begin{corollary}\\label{corstack}\nConsider the functor $\\Psi = \\Psi_0 \\Psi_1: \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ and consider the pretopology of $n$-covers from Proposition \\ref{pretop} on $\\ensuremath{\\mathsf{Cat}} $ for some $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$.\nThe pseudofunctor $\\ensuremath{\\mathsf{Map}} '$ associated to $\\Psi$ is a functor.\n\\begin{enumerate}\n\\item If $n \\geq 2$, then $\\ensuremath{\\mathsf{Map}} '$ is a prestack.\n\\item If $n \\geq 3$, then $\\ensuremath{\\mathsf{Map}} '$ is a stack.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nThis follows from Theorems \\ref{thmmasstack}, \\ref{thmmapstack}, Lemma \\ref{lemtopocomp} and Proposition \\ref{stacktrans}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Bimodules}\\label{parparbimod}\n\nIn this section we introduce the bicategory $\\underline{\\ensuremath{\\mathsf{Map}} }$ of map-graded categories and bimodules between them. We develop the usual machinary of tensor (\\S \\ref{partensor}) and Hom (\\S \\ref{parhom}) functors in the map-graded context. Some attention is given to the fact that in this context, the notion of bimodule is very natural, while there seems to be no natural notion of module available which does not implicitly or explicitly use bimodules.\n\n\\subsection{Bifunctors}\\label{parbifun}\nLet $\\ensuremath{\\mathcal{U}}$ and $\\ensuremath{\\mathcal{V}}$ be small categories. A \\emph{$\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor} $S$ is by definition a functor\n$$\\ensuremath{\\mathcal{V}}^{^{\\mathrm{op}}} \\times \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Set}} : (V,U) \\longmapsto S(V,U).$$\nFunctoriality translates into the existence of action maps\n$$\\ensuremath{\\mathcal{U}}(U, U') \\times S(V,U) \\times \\ensuremath{\\mathcal{V}}(V',V) \\longrightarrow S(V',U')$$\nsatisfying the natural associativity and identity axioms. For $\\ensuremath{\\mathcal{U}}$, we have the \\emph{identity $\\ensuremath{\\mathcal{U}}$-bifunctor}\n$$1_{\\ensuremath{\\mathcal{U}}}: \\ensuremath{\\mathcal{U}}^{^{\\mathrm{op}}} \\times \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Set}} : (V,U) \\longmapsto \\ensuremath{\\mathcal{U}}(V,U).$$\nA \\emph{morphism} between $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctors $S$ and $S'$ is a natural transformation $S \\longrightarrow S'$.\nThere is a natural category $\\mathsf{Bifun}(\\ensuremath{\\mathcal{U}}, \\ensuremath{\\mathcal{V}})$ of $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctors and their morphisms.\n\n\\begin{example}\\label{exbifun}\n\\begin{enumerate}\n\\item Let $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ be a functor between small categories. There is an associated $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S_{\\varphi}$ with $S_{\\varphi}(V,U) = \\ensuremath{\\mathcal{U}}(\\varphi(V), U)$.\n\\item Let $\\varphi: \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathcal{V}}$ be a functor between small categories. There is an associated $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S^{\\varphi}$ with $S^{\\varphi}(V,U) = \\ensuremath{\\mathcal{U}}(V, \\varphi(U))$.\n\\end{enumerate}\n\\end{example}\n\nBifunctors can be composed in the following way. Consider an additional small category $\\ensuremath{\\mathcal{W}}$ and a $\\ensuremath{\\mathcal{V}}$-$\\ensuremath{\\mathcal{W}}$ bifunctor $T$. We define $S \\circ T$ to be the $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{W}}$-bifunctor with \n$$S \\circ T(W,U) = \\coprod_{V \\in \\ensuremath{\\mathcal{V}}} S(V,U) \\times T(W,V)\/ \\sim$$\nwhere for $s \\in S(V,U)$, $v \\in \\ensuremath{\\mathcal{V}}(V',V)$ and $t \\in T(W, V')$ we have $(sv, t) \\sim (s, vt)$.\nThere are canonical isomorphisms $(R \\circ S) \\circ T \\cong R \\circ (S \\circ T)$, $1_{\\ensuremath{\\mathcal{U}}} \\circ S \\cong S$ and $S \\circ 1_{\\ensuremath{\\mathcal{V}}} \\cong S$. These give rise to a bicategory $\\underline{\\ensuremath{\\mathsf{Cat}} }$ of categories, bifunctors and natural transformations.\n\n\n\\subsection{Bimodules and tensor functors}\\label{parbimod}\\label{partensor}\n\nConsider a $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{a}$, a $\\ensuremath{\\mathcal{V}}$-graded category $\\mathfrak{b}$ and a $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S$. An \\emph{$\\mathfrak{a}$-$S$-$\\mathfrak{b}$-bimodule} consists of $k$-modules $M_s(B,A)$ for $s \\in S(V,U)$, $B \\in \\mathfrak{b}_V$, $A \\in \\mathfrak{a}_U$ with actions\n$$\\mathfrak{a}_u(A, A') \\otimes M_s(B,A) \\otimes \\mathfrak{b}_v(B',B) \\longrightarrow M_{usv}(B',A')$$\nsatisfying the natural associativity and identity axioms.\n\n\n\\begin{example}\\label{exbimod}\n\\begin{enumerate}\n\\item Let $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a graded functor. Let $S_{\\varphi}$ be the $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor from Example \\ref{exbifun}(1). For $s \\in S_{\\varphi}(V,U) = \\ensuremath{\\mathcal{U}}(\\varphi(V), U)$ and  $B \\in \\mathfrak{b}_V$, $A \\in \\mathfrak{a}_U$, we put $(M_F)_s(B,A) = \\mathfrak{a}_{s}(F(B), A)$. This defines an $\\mathfrak{a}$-$S_{\\varphi}$-$\\mathfrak{b}$-bimodule $M_F$.\n\n\\item Let $(\\varphi, F): (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longrightarrow (\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$ be a graded functor. Let $S^{\\varphi}$ be the $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor from Example \\ref{exbifun}(2). For $s \\in S^{\\varphi}(V,U) = \\ensuremath{\\mathcal{V}}(V,\\varphi(U))$ and $B \\in \\mathfrak{b}_V$, $A \\in \\mathfrak{a}_U$, we put $(M^F)_s(B,A) = \\mathfrak{b}_{s}(B, F(A))$. This defines an $\\mathfrak{a}$-$S^{\\varphi}$-$\\mathfrak{b}$-bimodule $M^F$.\n \n \\end{enumerate}\n\\end{example}\n\nThe $\\mathfrak{a}$-$S$-$\\mathfrak{b}$-bimodules form an abelian category $\\ensuremath{\\mathsf{Bimod}} _{S}(\\mathfrak{a}, \\mathfrak{b})$ with the natural choice of morphisms. If $\\ensuremath{\\mathcal{U}} = \\ensuremath{\\mathcal{V}}$, we can take $S = 1_{\\ensuremath{\\mathcal{U}}}$ and  $\\mathfrak{a}$-$1_{\\ensuremath{\\mathcal{U}}}$-$\\mathfrak{b}$-bimodules are simply called $\\mathfrak{a}$-$\\mathfrak{b}$-bimodules. The corresponding category is denoted $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, \\mathfrak{b})$. As usual, $\\mathfrak{a}$-$\\mathfrak{a}$-bimodules are called $\\mathfrak{a}$-bimodules and the corresponding category is denoted $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$. In $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$, we have the \\emph{identity $\\mathfrak{a}$-bimodule} $1_{\\mathfrak{a}}$ with\n${(1_{\\mathfrak{a}})}_u(A,A') = \\mathfrak{a}_u(A,A')$.\n\nSimilar to \\cite[Proposition 2.11]{lowenmap}, the category $\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b})$ can be described as a module category over a linear category. To do so we define the linear category $\\mathfrak{a}^{^{\\mathrm{op}}} \\otimes_S \\mathfrak{b}$ with\n$$\\mathrm{Ob}(\\mathfrak{a}^{^{\\mathrm{op}}} \\otimes_S \\mathfrak{b}) = \\coprod_{s \\in S(V,U)} \\mathfrak{a}_V \\times \\mathfrak{b}_U$$\nand\n$$\\mathrm{Hom}((s, A, B), (s', A', B')) = \\oplus_{s' = usv} \\mathfrak{a}_u(A,A') \\otimes \\mathfrak{b}_v(B',B).$$\n\n\n\\begin{proposition}\nThere is an isomorphism of linear categories\n$$\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a},\\mathfrak{b}) \\cong \\ensuremath{\\mathsf{Mod}} (\\mathfrak{a}^{^{\\mathrm{op}}} \\otimes_S \\mathfrak{b}).$$\n\\end{proposition}\n\nConsider an additional $\\ensuremath{\\mathcal{W}}$-graded category $\\mathfrak{c}$ and a $\\ensuremath{\\mathcal{V}}$-$\\ensuremath{\\mathcal{W}}$-bifunctor $T$. There is a natural tensor product \n$$\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\times \\ensuremath{\\mathsf{Bimod}} _T(\\mathfrak{b}, \\mathfrak{c}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{S \\circ T}(\\mathfrak{a}, \\mathfrak{c}): (M, N) \\longmapsto M \\otimes_{\\mathfrak{b}} N$$\nwith\n$$(M \\otimes_{\\mathfrak{b}} N)_{r}(C,A) = \\oplus_{[(s,t)] = r,B} M_s(B,A) \\otimes_k N_t(C,B)\/\\sim$$\nwhere for $s \\in S(V,U)$, $v \\in \\ensuremath{\\mathcal{V}}(V',V)$ and $t \\in T(W, V')$, $B\\in \\mathfrak{b}_V$, $B' \\in \\mathfrak{b}_{V'}$, $m \\in M_s(B,A)$, $b \\in \\mathfrak{b}_v(B,B')$ and $n \\in N_t(C,B')$ we have $(mb, n) \\sim (m, bn)$.\n\nAs an application of the tensor product, we obtain tensor actions\n$$\\otimes_{\\mathfrak{b}}: \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\times \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}): (M, X) \\longmapsto M \\otimes_{\\mathfrak{b}} X$$\nand\n$$\\otimes_{\\mathfrak{a}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\times \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}): (X, M) \\longmapsto X \\otimes_{\\mathfrak{a}} M.$$\n\n\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a graded functor. There is an induced functor\n$$F^{\\ast}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}): M \\longmapsto F^{\\ast}M$$\nwith $(F^{\\ast}M)_v(B,B') = M_{\\varphi(v)}(F(B), F(B'))$.\n\n\n\n\n\n\\begin{example}\nFor a functor $\\varphi: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{U}}$ and a $\\ensuremath{\\mathcal{U}}$-graded $\\mathfrak{a}$, consider the cartesian $\\varphi$-graded functor $\\delta^{\\varphi, \\mathfrak{a}}: \\mathfrak{a}^{\\varphi} \\longrightarrow \\mathfrak{a}$. We obtain the induced functor\n$$(-)^{\\varphi} = (\\delta^{\\varphi, \\mathfrak{a}})^{\\ast}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{\\varphi}): M \\longmapsto M^{\\varphi}$$\nwith $M^{\\varphi}_v(A,A') = M_{\\varphi(v)}(A,A')$.\n\\end{example}\n\nLetting $S$ vary, we obtain a category $\\ensuremath{\\mathsf{Bimod}} (\\mathfrak{a}, \\mathfrak{b})$ of $\\mathfrak{a}$-$\\mathfrak{b}$ bimodules in the following way. An $\\mathfrak{a}$-$\\mathfrak{b}$ bimodule consist of a $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S$ and an $\\mathfrak{a}$-$S$-$\\mathfrak{b}$-bimodule $N$. A morphism of bimodules $(\\phi, F): (S, M) \\longrightarrow (T, N)$ consists of a natural transformation $\\phi: S \\longrightarrow T$ of bifunctors and, for every $A \\in \\mathfrak{a}_U$, $B \\in \\mathfrak{b}_V$ and $s \\in S(V,U)$, a morphism $$F_{s,B,A}: M_s(B,A) \\longrightarrow N_{\\phi(s)}(B,A)$$ such that the morphism $F_{s,B,A}$ are compatible with the actions of $\\mathfrak{a}$ and $\\mathfrak{b}$.\n\nCombining the composition of bifunctors and the tensor product of bimodules, we obtain a tensor product\n$$\\ensuremath{\\mathsf{Bimod}} (\\mathfrak{a}, \\mathfrak{b}) \\times \\ensuremath{\\mathsf{Bimod}} (\\mathfrak{b}, \\mathfrak{c}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} (\\mathfrak{a}, \\mathfrak{c}): ((S,M), (T,N)) \\longmapsto (S \\circ T, M \\otimes_{\\mathfrak{b}} N).$$\nIf we consider a furter $\\ensuremath{\\mathcal{Z}}$-graded category $\\mathfrak{z}$ and $(R,P) \\in \\ensuremath{\\mathsf{Bimod}} (\\mathfrak{z}, \\mathfrak{a})$, then there are natural isomorphisms\n$$((R \\circ S) \\circ T, (P \\otimes_{\\mathfrak{a}} M) \\otimes_{\\mathfrak{b}} N) \\cong (R \\circ (S \\circ T), P \\otimes_{\\mathfrak{a}} (M \\otimes_{\\mathfrak{b}} N))$$\nand $(1_{\\ensuremath{\\mathcal{U}}} \\circ S, 1_{\\mathfrak{a}} \\otimes_{\\mathfrak{a}} M) \\cong (S,M) \\cong (S \\circ 1_{\\ensuremath{\\mathcal{V}}}, M \\otimes_{\\mathfrak{b}} 1_{\\mathfrak{b}})$.\nThese give rise to a bicategory $\\underline{\\ensuremath{\\mathsf{Map}} }$ of map-graded categories, bimodules and bimodule morphisms.\n\n\n\\subsection{One sided bimodules and Hom functors}\\label{parhom}\nBimodules are the natural notion when working with graded categories, but some bimodules can be considered to be more ``one-sided'' than others. Let $\\ensuremath{\\mathcal{U}}$ and $\\ensuremath{\\mathcal{V}}$ be categories with an $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bimodule $S$. Consider a $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{a}$ and a $\\ensuremath{\\mathcal{V}}$-graded category $\\mathfrak{b}$. Furthermore, consider the free $\\ensuremath{\\mathcal{U}}$-graded category $k\\ensuremath{\\mathcal{U}}$ and the free $\\ensuremath{\\mathcal{V}}$-graded category $k\\ensuremath{\\mathcal{V}}$ as in example \\ref{exfreegr}. Then there are natural bimodule categories\n$$\\ensuremath{\\mathsf{Mod}} _S(\\mathfrak{a}) = \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, k\\ensuremath{\\mathcal{V}})$$ and\n$$\\ensuremath{\\mathsf{Mod}} _S(\\mathfrak{b}) = \\ensuremath{\\mathsf{Bimod}} _S(k\\ensuremath{\\mathcal{U}}, \\mathfrak{b}).$$\n\n\\begin{example}\\label{exmodmod}\nIf we take, in the first case, $\\ensuremath{\\mathcal{V}} = e$, then there is a unique $\\ensuremath{\\mathcal{U}}$-$e$-bimodule $S$ with $S(\\ast, U) = \\{ \\ast \\}$ for every $U \\in \\ensuremath{\\mathcal{U}}$. Thus we obtain the category of left $\\mathfrak{a}$-modules\n$$\\ensuremath{\\mathsf{Mod}} ^l(\\mathfrak{a}) = \\ensuremath{\\mathsf{Mod}} _S(\\mathfrak{a}).$$\nSimilarly, taking $\\ensuremath{\\mathcal{U}} = e$ there  is a unique $e$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S$ with $S(V, \\ast) = \\{ \\ast \\}$ for every $V \\in \\ensuremath{\\mathcal{V}}$, yielding the category of right $\\mathfrak{b}$-modules\n$$\\ensuremath{\\mathsf{Mod}} ^r(\\mathfrak{b}) = \\ensuremath{\\mathsf{Mod}} _S(\\mathfrak{b}).$$\n\\end{example}\n\nNow we return to the general situation of an underlying $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bimodule $S$. Then fixing one argument in an $\\mathfrak{a}$-$\\mathfrak{b}$-bimodule $M$ yields one-sided bimodules in the following sense.\nFix $U \\in \\ensuremath{\\mathcal{U}}$ and $A \\in \\mathfrak{a}_U$. Then $S$ yields an $e$-$\\ensuremath{\\mathcal{V}}$-bimodule $S_U$ with\n$$S_U(V, \\ast) = S(V,U)$$\nand $M$ yields a $ke$-$\\mathfrak{b}$-bimodule $M_A \\in \\ensuremath{\\mathsf{Bimod}} _{S_U}(ke, \\mathfrak{b})$ with\n$$(M_A)_s(B, \\ast) = M_s(B,A).$$\nFurthemore, the categories $\\ensuremath{\\mathsf{Bimod}} _{S_U}(ke, \\mathfrak{b})$ are connected in the following way. Consider a morphism $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$. Then there is an associated functor\n$$u^{\\ast}: \\ensuremath{\\mathsf{Bimod}} _{S_{U'}}(ke, \\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{S_{U}}(ke, \\mathfrak{b}): M \\longmapsto u^{\\ast}M$$\nwith $$(u^{\\ast}M)_s(B, \\ast) = M_{us}(B, \\ast).$$\nNow we are ready to define the Hom functor\n$$\\mathrm{Hom}_{\\mathfrak{b}}: \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\times \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}): (M, N) \\longmapsto \\mathrm{Hom}_{\\mathfrak{b}}(M,N)$$\nwhere for $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$ and $A \\in \\mathfrak{a}_U$, $A' \\in \\mathfrak{a}_{U'}$\n$$\\mathrm{Hom}_{\\mathfrak{b}}(M,N)_u(A,A') = \\ensuremath{\\mathsf{Bimod}} _{S_U}(ke, \\mathfrak{b})(M_A, u^{\\ast} N_{A'}).$$\nThere are natural morphisms\n$$1_{\\mathfrak{a}} \\longrightarrow \\mathrm{Hom}_{\\mathfrak{b}}(M,M)$$\ngiven by the natural action maps\n$$\\mathfrak{a}_{u}(A,A') \\otimes M_s(B,A) \\longrightarrow M_{us}(B, A').$$\n\nIn a similar way we define\n$$\\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}: \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\times \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}): (M,N) \\longmapsto \\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M,N).$$\n\n\\begin{proposition}\\label{propadj}\n\\begin{enumerate}\n\\item The functor $- \\otimes_{\\mathfrak{a}} M: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b})$ is left adjoint to $\\mathrm{Hom}_{\\mathfrak{b}}(M,-): \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\longrightarrow  \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$.\n\\item The functor $M \\otimes_{\\mathfrak{b}} -: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b})$ is left adjoint to $\\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M, -): \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b}) \\longrightarrow  \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\\section{Functoriality of map-graded Hochschild complexes} \\label{parfunct}\n\nThe Hochschild complex $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ of a map-graded category $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is defined in analogy with the Hochschild complex of an algebra \\cite{lowenmap}, naturally making use of the simplicial structure of the nerve $\\ensuremath{\\mathcal{N}}(\\mathfrak{a}) = \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$. In this section we investigate the functoriality properties of map-graded Hochschild complexes. A morphism of map-graded categories is called \\emph{subcartesian} if it is cartesian with respect to $\\ensuremath{\\mathsf{Map}}  \\longrightarrow \\ensuremath{\\mathsf{Mas}} $. This amounts to the fact that all maps occuring between $\\mathrm{Hom}$-modules are isomorphisms. For instance, if we consider a fully faithful functor $\\mathfrak{b} \\longrightarrow \\mathfrak{a}$ between $k$-linear categories as a graded functor beteen trivially graded categories, it is subcartesian (Example \\ref{exfff}).\nIn Proposition  \\ref{propfunct}, we show that taking Hochschild complexes is functorial with respect to subcartesian functors. This constitutes a natural generalization of the \\emph{limited functoriality} of Hochschild complexes of linear categories (see \\cite{kellerdih} in the differential graded context). Let $\\ensuremath{\\mathsf{Map}} _{sc} \\subseteq \\ensuremath{\\mathsf{Map}} $ denote the full subcategory of subcartesian morphisms. We endow $\\ensuremath{\\mathsf{Map}} _{sc}$ with the pretopology of $n$-covers from Definition \\ref{defncover}. In Theorem \\ref{mainsheaf}, we show that the functor\n$$\\mathbf{C}^n: \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$$\nis a sheaf. As an application of the theorem, in \\S \\ref{parMV} we obtain a Mayer-Vietoris sequence of Hochschild complexes\n$$0 \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1}(\\mathfrak{a}^{\\varphi_1}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi_2}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi}) \\longrightarrow 0$$\nfor a map-graded category $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and two cartesian morphisms $(\\ensuremath{\\mathcal{V}}_i, \\mathfrak{a}^{\\varphi_i}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and $(\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2, \\mathfrak{a}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ associated to subcategories $\\varphi_i: \\ensuremath{\\mathcal{V}}_i \\subseteq \\ensuremath{\\mathcal{U}}$ constituting an $n$-cover of $\\ensuremath{\\mathcal{U}}$ for all $n \\geq 0$. Finally, in \\S \\ref{parcensor}, we discuss censoring subcategories as a natural generalization of the censoring relations from \\cite[\\S 4.3]{lowenvandenberghhoch}.\n\n\n\n\\subsection{The map-graded Hochschild complex}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category. Let $\\ensuremath{\\mathcal{U}}^{\\sharp}$ be the category defined in Example \\ref{exsharp}. \nRecall from \\S \\ref{parsite} that the nerve of $\\mathfrak{a}$ is defined to be the simplicial set $\\ensuremath{\\mathcal{N}}(\\mathfrak{a}) = \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$ with $n$-simplices $\\sigma = (u, A)$ given by data\n$$\\xymatrix{ {A_0} & {A_1} & {\\dots} & {A_n}\\\\ {U_0} \\ar[r]_{u_0} & {U_1} \\ar[r]_{u_1} & {\\dots} \\ar[r]_{u_{n-1}} & {U_n}}$$\nwith $u_i \\in \\ensuremath{\\mathcal{U}}, A_i \\in \\mathfrak{a}_{U_i}$.\nFor $u \\in \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}})_n$, we will use the notation\n$$|u| = u_{n-1}\\cdots u_1 u_0.$$\nA graded functor $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ induces a functor $\\ensuremath{\\mathcal{V}}^{\\sharp} \\longrightarrow \\ensuremath{\\mathcal{U}}^{\\sharp}$ and hence a map $\\ensuremath{\\mathcal{N}}(F): \\ensuremath{\\mathcal{N}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\mathfrak{a})$.\n\n\nLet $M$ be an $\\mathfrak{a}$-bimodule.\nThe \\emph{Hochschild complex of $\\mathfrak{a}$ with values in $M$} naturally arises from this simplicial structure as the complex $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a},M)$ with\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}^n(\\mathfrak{a}) = \\prod_{(u, A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{a})} \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_n))$$\nwith the simplicial Hochschild differential.\nWe put $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) = \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, 1_{\\mathfrak{a}})$. This complex is in fact a $B_{\\infty}$-algebra \\cite{lowenmap}.\n\n\\begin{example}\nLet $\\mathfrak{a}$ be a linear category. For all the $\\ensuremath{\\mathcal{U}}$-gradings on $\\mathfrak{a}$ of Example \\ref{exstandgr}, the corresponding Hochschild complexes $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ are canonically isomorphic to $\\mathbf{C}(\\mathfrak{a})$. This results from the fact that all the nerves of these graded categories are canonically isomorphic to $\\ensuremath{\\mathcal{N}}(\\mathfrak{a})$.\n\\end{example}\n\n\n\n\\subsection{Limited functoriality}\\label{parlimfun}\nIt is well known that the Hochschild complex of linear categories satisfies so called ``limited functoriality'' with respect to inclusions of full subcategories, see \\cite{kellerdih} for the more general statement for differential graded categories. In this section we discuss a limited functoriality property for map-graded categories.\n\nRecall that by Proposition \\ref{lemcart1}, a graded functor $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is subcartesian in the sense of Definition \\ref{defcart} provided that for every $v: V \\longrightarrow V'$ in $\\ensuremath{\\mathcal{V}}$, $B \\in \\mathfrak{b}_V$, $B' \\in \\mathfrak{b}_{V'}$, the map\n$$\\mathfrak{b}_v(B,B') \\longrightarrow \\mathfrak{a}_{\\varphi(v)}(F(B), F(B'))$$\nis an isomorphism.\n\n\\begin{proposition}\\label{propfunct}\nConsider a graded functor $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$. Let $M$ be an $\\mathfrak{a}$-bimodule with induced $\\mathfrak{b}$-bimodule $F^{\\ast}M$. \n\\begin{enumerate}\n\\item There is a canonical map\n$${\\prod_{(u,A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{a})_n} \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_n))}$$\n$$\\xymatrix{ {} \\ar[r]_-{(F^{\\ast}_M)^n} & {}}$$\n$${\\prod_{(v,B) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{b})_n} \\mathrm{Hom}_k(\\mathfrak{b}_{v_{n-1}}(B_{n-1}, B_n) \\otimes \\dots \\otimes \\mathfrak{b}_{v_0}(B_0, B_1), (F^{\\ast} M)_{|v|}(B_0, B_n))}$$\ngiven by\n$$(F^{\\ast}_M)^n((\\phi_{(u,A)})_{(u,A)}) = (\\phi_{(\\varphi(v), F(B))} \\circ F^{\\otimes n})_{(v,B)}.$$\n\\item The maps $(F^{\\ast}_M)^n$ determine a morphism of complexes\n$$F^{\\ast}_M: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M).$$\n\\item If $(\\varphi, F)$ is subcartesian, we have $F^{\\ast}1_{\\mathfrak{a}} \\cong 1_{\\mathfrak{b}}$ and the maps $(F^{\\ast}_{1_{\\mathfrak{a}}})^n$ determine a morphism of $B_{\\infty}$-algebras\n$$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$$\nwith\n$$(F^{\\ast})^n((\\phi_{(u,A)})_{(u,A)}) = (F^{-1} \\circ \\phi_{(\\varphi(v), F(B))} \\circ F^{\\otimes n})_{(v,B)}.$$\n\\end{enumerate}\n\\end{proposition}\n\nClearly, graded categories with subcartesian graded functors consitute a subcategory $\\ensuremath{\\mathsf{Map}} _{sc} \\subseteq \\ensuremath{\\mathsf{Map}} $. Let $B_{\\infty}$ denote the category of $B_{\\infty}$-algebras and morphisms. By Proposition \\ref{propfunct} (3), we obtain a contravariant functor\n\\begin{equation} \\label{CC}\n\\mathbf{C}: \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow B_{\\infty}: (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}).\n\\end{equation}\n\n\\begin{definition}\\label{defsurjinj}\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a graded functor between graded categories (resp. a graded map between graded sets). Let $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$.\n\\begin{enumerate}\n\\item $(\\varphi, F)$ is \\emph{$n$-surjective} if the canonical\n$\\ensuremath{\\mathcal{N}}_k(F): \\ensuremath{\\mathcal{N}}_k(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathcal{N}}_k(\\mathfrak{a})$ is surjective for $k \\leq n$.\n\\item $(\\varphi, F)$ is \\emph{$n$-injective} if $\\ensuremath{\\mathcal{N}}_k(F): \\ensuremath{\\mathcal{N}}_k(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathcal{N}}_k(\\mathfrak{a})$ is injective for $k \\leq n$. \n\\item $(\\varphi, F)$ is \\emph{injective} if $\\ensuremath{\\mathcal{N}}_1(\\varphi): \\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{V}}) \\longrightarrow \\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{U}})$ is injective and every $F: \\mathfrak{b}_V \\longrightarrow \\mathfrak{a}_{\\varphi(V)}$ is injective.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\\label{remsurjinj}\n\\begin{enumerate}\n\\item $(\\varphi, F)$ is $n$-surjective if and only if the collection containing $(\\varphi, F)$ as single element is an $n$-cover in $\\ensuremath{\\mathsf{Map}} $ (resp. $\\ensuremath{\\mathsf{Mas}} $) in the sense of Definition \\ref{defncover}.\n\\item If $(\\varphi, F)$ is $1$-injective, then it is $\\infty$-injective.\n\\item If $(\\varphi, F)$ is injective, then it is $1$-injective.\n\\item $(\\varphi,F)$ is $1$-injective if and only if $(\\varphi^{\\sharp}, F^{\\sharp})$ defined as in Example \\ref{exfunsharp} is injective.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{proposition}\\label{propsurjinj}\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a subcartesian graded functor and let $M$ be an $\\mathfrak{a}$-bimodule. Let $n \\in \\mathbb{N}$.\n\\begin{enumerate}\n\\item If $(\\varphi, F)$ is $n$-injective then $(F^{\\ast})^n: \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M)$ is surjective.\n\\item If $(\\varphi, F)$ is $n$-surjective then $(F^{\\ast})^n: \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M)$ is injective.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} \n(1) If $\\ensuremath{\\mathcal{N}}_n(F): \\ensuremath{\\mathcal{N}}_n(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathcal{N}}_n(\\mathfrak{a})$ is injective, the map $(F^{\\ast})^n$ is isomorphic to a projection on a subproduct, whence surjective.\n(2) Looking at the prescription for $(F^{\\ast})^n$, if $\\phi \\in \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M)$ is such that $\\phi_{(\\varphi(v), F(B))} = 0$ for every $(v,B) \\in \\ensuremath{\\mathcal{N}}_n(\\mathfrak{b})$, then using $n$-surjectivity we can write every $(u,A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{a})$ as $(u, A) = (\\varphi(v), F(B))$ for some $(v,B)$ whence $\\phi_{(u,A)} = 0$.\n\\end{proof}\n\n\\begin{example}\\label{exisosharp}\nConsider the graded functor $(\\varphi, F): (\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ as in Example \\ref{exfunsharp}. Clearly, $(\\varphi, F)$ is subcartesian and $\\ensuremath{\\mathcal{N}}(F): \\ensuremath{\\mathcal{N}}(\\mathfrak{a}^{\\sharp}) = \\ensuremath{\\mathcal{N}}((\\ensuremath{\\mathcal{U}}^{\\sharp})^{\\sharp}) \\cong \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp}) \\longrightarrow \\ensuremath{\\mathcal{N}}(\\ensuremath{\\mathcal{U}}^{\\sharp})$ is an isomorphism. Consequently,\n$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\sharp}}(\\mathfrak{a}^{\\sharp})$ is an isomorphism.\n\\end{example}\n\n\\subsection{The sheaf of Hochschild complexes}\\label{parsheafhoch}\nConsider the presheaf\n$$\\mathbf{C}: \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow B_{\\infty}: (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$$\nfrom \\eqref{CC}.\nLet $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$ be fixed. We endow $\\ensuremath{\\mathsf{Map}} _{sc} \\subseteq \\ensuremath{\\mathsf{Map}} $ with the pretopology of $n$-covers, as described in Proposition \\ref{pretop}.\n\n\\begin{theorem}\\label{mainsheaf}\nThe presheaf $\\mathbf{C}^n: \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ is a sheaf.\nIf $n = \\infty$, then $\\mathbf{C}: \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow B_{\\infty}$ is a sheaf of $B_{\\infty}$-algebras.\n\\end{theorem}\n\n\\begin{proof}\nLet $((\\varphi_i, F_i): (\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}))_i$ be an $n$-cover in $\\ensuremath{\\mathsf{Map}} _{sc}$. A compatible family of elements for this cover consists of $n$-cocycles\n$$\\phi_i \\in \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_i}(\\mathfrak{b}_i)$$\nsuch that for every pullback diagram\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i)} \\ar[r]^{(\\varphi_i, F_i)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})}\\\\   {(\\ensuremath{\\mathcal{V}}_i \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_j, \\mathfrak{b}_i \\times_{\\mathfrak{a}} \\mathfrak{b}_j)} \\ar[u]^{(\\alpha_1, G_1)} \\ar[r]_-{(\\alpha_2, G_2)} & {(\\ensuremath{\\mathcal{V}}_j, \\mathfrak{b}_j)} \\ar[u]_-{(\\varphi_j, F_j)} }$$\nwe have $$G_i^{\\ast}(\\phi_i) = G_j^{\\ast}(\\phi_j).$$\nTo define the unique glueing of this family on $\\mathfrak{a}$, we must define for every $(u, A) \\in \\ensuremath{\\mathcal{N}}_n(\\mathfrak{a})$ a corresponding cocycle \n$$\\phi_{(u,A)} \\in \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), \\mathfrak{a}_{|u|}(A_0, A_n)).$$\nSince the collection $(\\ensuremath{\\mathcal{N}}_n(F_i))_i$ is jointly surjective, there in an $i$ and $(v, B) \\in \\ensuremath{\\mathcal{N}}_n(\\mathfrak{b}_i)$ for which $\\varphi_i(v) = u$, $F_i(B) = A$. We thus have isomorphisms\n$F_i: (\\mathfrak{b}_i)_{v_i}(B_i, B_{i+1}) \\longrightarrow \\mathfrak{a}_{u_i}(A_i, A_{i+1})$\nand\n$F_i: (\\mathfrak{b}_i)_{|v|}(B_0, B_{n}) \\longrightarrow \\mathfrak{a}_{|u|}(A_0, A_{n})$.\nWe put\n$$\\phi_{(u,A)} = F_i \\circ {\\phi_i}_{(v,B)} \\circ (F_i^{-1})^{\\otimes n}.$$\nIt remains to show that this is well defined. Suppose there is another $j$ and $(w, C) \\in \\ensuremath{\\mathcal{N}}_n(\\mathfrak{b}_j)$ for which $\\varphi_j(w) = u$, $F_j(C) = A$. We are to show that \n$F_i \\circ {\\phi_i}_{(v,B)} \\circ (F_i^{-1})^{\\otimes n} = F_j \\circ {\\phi_j}_{(w,C)} \\circ (F_j^{-1})^{\\otimes n}$.\nNow $G_1^{\\ast}$ maps the collection $\\phi_i$ to a collection $\\psi_i$ with\n$${\\psi_i}_{((v, w), (B, C))} = (G_1)^{-1} \\circ {\\phi_i}_{(v,B)} \\circ (G_1)^{\\otimes n}$$\nand $G_2^{\\ast}$ maps the collection $\\phi_j$ to a collection $\\psi_j$ with\n$${\\psi_j}_{((v, w), (B,C))} = (G_2)^{-1} \\circ {\\phi_j}_{(w,C)} \\circ (G_2)^{\\otimes n}.$$\nWe thus have\n$$F_i \\circ {\\phi_i}_{(v,B)} \\circ (F_i^{-1})^{\\otimes n} = (F_i G_1) \\circ {\\psi_i}_{((v, w), (B, C))} \\circ ((F_i G_1)^{-1})^{\\otimes n}$$\nand\n$$F_j \\circ {\\phi_j}_{(w,C)} \\circ (F_j^{-1})^{\\otimes n} = (F_j G_2) \\circ {\\psi_j}_{((v, w), (B, C))} \\circ ((F_j G_2)^{-1})^{\\otimes n}.$$\nBy the commutativity of the pullback square and the compatibility assumption $\\psi_i = \\psi_j$, these two expressions are equal as desired.  \n\\end{proof}\n\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category with underlying graded set $\\mathfrak{x}$. There are natural functors\n$$\\Phi_1: \\ensuremath{\\mathsf{Mas}} \/\\mathfrak{x} \\longrightarrow \\ensuremath{\\mathsf{Map}} : ((\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\rightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})) \\longmapsto \\mathfrak{a}^{(\\varphi, F)}$$\nand\n$$\\Phi_2: \\ensuremath{\\mathsf{Cat}} \/\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Map}} : (\\varphi: \\ensuremath{\\mathcal{V}} \\rightarrow \\ensuremath{\\mathcal{U}}) \\longmapsto \\mathfrak{a}^{\\varphi}.$$\n\nLet $n \\in \\mathbb{N} \\cup {\\infty}$. Endow $\\ensuremath{\\mathsf{Mas}} \/ \\mathfrak{x}$ and $\\ensuremath{\\mathsf{Cat}} \/ \\ensuremath{\\mathcal{U}}$ with the pretopologies of $n$-covers induced from the ones on $\\ensuremath{\\mathsf{Mas}} $ and $\\ensuremath{\\mathsf{Cat}} $, and also endow $\\ensuremath{\\mathsf{Map}} $ with the pretopology of $n$-covers.\n\n\\begin{proposition}\nThe functors $\\Phi_1$ and $\\Phi_2$ both map covers to covers. \n\\end{proposition}\n\n\\begin{corollary}\\label{corsheafhoch}\nThe presheaves\n$$\\mathbf{C}^n_1 = \\mathbf{C}^n \\circ \\Phi_1: \\ensuremath{\\mathsf{Mas}} \/ \\mathfrak{x} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): ((\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\rightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{(\\varphi, F)})$$\nand\n$$\\mathbf{C}^n_2 = \\mathbf{C}^n \\circ \\Phi_2:  \\ensuremath{\\mathsf{Cat}} \/\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): (\\varphi: \\ensuremath{\\mathcal{V}} \\rightarrow \\ensuremath{\\mathcal{U}}) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{\\varphi})$$\nare sheaves.\n\\end{corollary}\n\nWe end this section by noting that for a fixed $\\mathfrak{a}$-bimodule $M$, one similarly has:\n\n\\begin{proposition}\\label{sheafbimod}\nThere are natural sheaves\n$$\\mathbf{C}^n_{1,M}: \\ensuremath{\\mathsf{Mas}} \/\\mathfrak{x} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): ((\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\rightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{x})) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{(\\varphi, F)}, F^{\\ast}M)$$\nand\n$$\\mathbf{C}^n_{2, M} = \\mathbf{C}^n \\circ \\Phi_2:  \\ensuremath{\\mathsf{Cat}} \/\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Mod}} (k): (\\varphi: \\ensuremath{\\mathcal{V}} \\rightarrow \\ensuremath{\\mathcal{U}}) \\longmapsto \\mathbf{C}^n_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{\\varphi}, M^{\\varphi}).$$\n\\end{proposition}\n\n\\begin{remark}\nIt is possible to formulate a version of Theorem \\ref{mainsheaf} taking map graded categories endowed with a bimodule as input data for $\\mathbf{C}$, such that Proposition \\ref{sheafbimod} is obtained as a corollary. The details are left to the reader.\n\\end{remark}\n\n\n\n\n\n\n\\subsection{$\\infty$-surjections}\n\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be an $\\infty$-surjective morphism in $\\ensuremath{\\mathsf{Map}} $. To formulate the sheaf property of $\\mathbf{C}^n$ for the corresponding $\\infty$-cover, we look at the pullback\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}, \\mathfrak{b})} \\ar[r]^{(\\varphi, F)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})}\\\\   {(\\ensuremath{\\mathcal{V}} \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}, \\mathfrak{b} \\times_{\\mathfrak{a}} \\mathfrak{b})} \\ar[u]^{(\\alpha_1, G_1)} \\ar[r]_-{(\\alpha_2, G_2)} & {(\\ensuremath{\\mathcal{V}}, \\mathfrak{b})} \\ar[u]_-{(\\varphi, F)} }$$\nAccording to Theorem \\ref{mainsheaf}, we obtain an exact sequence of $B_{\\infty}$-algebras\n$$\\xymatrix{ 0 \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_{F^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})} \\ar[r]_-{G_2^{\\ast} - G_1^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}} \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}}(\\mathfrak{b} \\times_{\\mathfrak{a}} \\mathfrak{b}).} }$$\nDefine the complex $$\\mathbf{C}_{\\ensuremath{\\mathcal{V}}\/\\ensuremath{\\mathcal{U}}}(\\mathfrak{b}\/\\mathfrak{a}) = \\mathrm{Im}(G_2^{\\ast} - G_1^{\\ast}).$$\nThen we obtain a long exact cohomology sequence\n$$\\xymatrix{ {\\dots} \\ar[r] & {HH^i_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_-{F^{\\ast}} & {HH^i_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})} \\ar[r]_-{G_2^{\\ast} - G_1^{\\ast}} & {HH^i_{\\ensuremath{\\mathcal{V}}\/\\ensuremath{\\mathcal{U}}}(\\mathfrak{b}\/\\mathfrak{a})} \\ar[r] & {\\dots}}$$\n\n\n\n\n\n\\subsection{Mayer-Vietoris sequences}\\label{parMV}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category. Consider two subcartesian $1$-injections\n$(\\varphi_1, F_1): (\\ensuremath{\\mathcal{V}}_1, \\mathfrak{b}_1) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and $(\\varphi_2, F_2): (\\ensuremath{\\mathcal{V}}_2, \\mathfrak{b}_2) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ that together constitute an $\\infty$-cover of $\\mathfrak{a}$. It is our aim to formulate the sheaf property, so we have to look at all possible pullbacks between $(\\varphi_1, F_1)$ and $(\\varphi_2, F_2)$. For $i \\in \\{1,2\\}$, we first look at the pullback\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i)} \\ar[r]^{(\\varphi_i, F_i)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})}\\\\   {(\\ensuremath{\\mathcal{V}}_i \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i \\times_{\\mathfrak{a}} \\mathfrak{b}_i)} \\ar[u]^{(\\alpha_1, G_1)} \\ar[r]_-{(\\alpha_2, G_2)} & {(\\ensuremath{\\mathcal{V}}_i, \\mathfrak{b}_i).} \\ar[u]_-{(\\varphi_i, F_i)} }$$\nThe functor $(-)^{\\sharp}$ from Example \\ref{exfunsharp} maps this pullback to the pullback of $(\\varphi_i^{\\sharp}, F_i^{\\sharp}): (\\ensuremath{\\mathcal{V}}_i^{\\sharp}, \\mathfrak{b}_i^{\\sharp}) \\longrightarrow (\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp})$ with itself. Now $(\\varphi_i^{\\sharp}, F_i^{\\sharp})$ is an injection by Remark \\ref{remsurjinj} (4), whence it is easily seen to be a monomorphism in $\\ensuremath{\\mathsf{Map}} $. Thus, the pullback of this morphism with itself is given by identity morphisms, inducing identity morphisms between Hochschild complexes. Using Example \\ref{exisosharp}, we conclude that\nan element $\\phi \\in \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_i}(\\mathfrak{b}_i)$ satisfies\n$$G^{\\ast}_1(\\phi) = G^{\\ast}_2(\\phi) \\in \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_i \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_i}(\\mathfrak{b}_i \\times_{\\mathfrak{a}} \\mathfrak{b}_i).$$\nIt thus remains to look at the pullback\n$$\\xymatrix{ {(\\ensuremath{\\mathcal{V}}_1, \\mathfrak{b}_1)} \\ar[r]^{(\\varphi_1, F_1)} & {(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})}\\\\   {(\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2, \\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)} \\ar[u]^{(\\alpha_1, G_1)} \\ar[r]_-{(\\alpha_2, G_2)} & {(\\ensuremath{\\mathcal{V}}_2, \\mathfrak{b}_2)} \\ar[u]_-{(\\varphi_2, F_2)} }$$\nfrom which we obtain an exact sequence of complexes:\n$$\\xymatrix{ 0 \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_-{\\begin{pmatrix} F_1^{\\ast} \\\\ F_2^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1}(\\mathfrak{b}_1) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_2}(\\mathfrak{b}_2)} \\ar[r]_-{\\begin{pmatrix} - G_1^{\\ast} & G_2^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2}(\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)} \\ar[r] & 0.}$$\nHere, the sequence is left exact by the sheaf property Theorem \\ref{mainsheaf} and moreover right exact by Proposition \\ref{propsurjinj} (2).\nWe thus obtain a long exact cohomology sequence\n$$\\xymatrix{ {\\dots} \\ar[r] & {HH^i_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_-{\\begin{pmatrix} F_1^{\\ast} \\\\ F_2^{\\ast} \\end{pmatrix}} & {HH^i_{\\ensuremath{\\mathcal{V}}_1}(\\mathfrak{b}_1) \\oplus HH^i_{\\ensuremath{\\mathcal{V}}_2}(\\mathfrak{b}_2)} \\ar[r]_-{\\begin{pmatrix} - G_1^{\\ast} & G_2^{\\ast} \\end{pmatrix}} & {HH^i_{\\ensuremath{\\mathcal{V}}_1 \\times_{\\ensuremath{\\mathcal{U}}} \\ensuremath{\\mathcal{V}}_2}(\\mathfrak{b}_1 \\times_{\\mathfrak{a}} \\mathfrak{b}_2)} \\ar[r] & {\\dots}}$$\n\n\\begin{example}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and let $\\varphi_1: \\ensuremath{\\mathcal{V}}_1 \\subseteq \\ensuremath{\\mathcal{U}}$ and $\\varphi_2: \\ensuremath{\\mathcal{V}}_2 \\subseteq \\ensuremath{\\mathcal{U}}$ be subcategories that together constitute an $\\infty$-cover of $\\ensuremath{\\mathcal{U}}$ in $\\ensuremath{\\mathsf{Cat}} $. Then the induced cartesian morphisms $(\\ensuremath{\\mathcal{V}}_1, \\mathfrak{a}^{\\varphi_1}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and $(\\ensuremath{\\mathcal{V}}_2, \\mathfrak{a}^{\\varphi_2}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ in $\\ensuremath{\\mathsf{Map}} $ constitute an $\\infty$-cover by $1$-injections. Put $\\varphi: \\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2 \\subseteq \\ensuremath{\\mathcal{U}}$ the inclusion. By Example \\ref{exint} we obtain an exact sequence of complexes\n$$\\xymatrix{ 0 \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_-{\\begin{pmatrix} \\varphi_1^{\\ast} \\\\ \\varphi_2^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1}(\\mathfrak{a}^{\\varphi_1}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi_2})}\n \\ar[r]_-{\\begin{pmatrix} - \\alpha_1^{\\ast} & \\alpha_2^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}_1 \\cap \\ensuremath{\\mathcal{V}}_2}(\\mathfrak{a}^{\\varphi})} \\ar[r] & 0.}$$\nBased upon Proposition \\ref{sheafbimod}, one obtains a version of this sequence involving bimodules.\n\\end{example}\n\n\n\\begin{example}\nLet $(X, \\ensuremath{\\mathcal{O}})$ be a ringed space with an acyclic basis $\\ensuremath{\\mathcal{B}}$ of open sets in the sense of \\cite{lowenvandenberghhoch}. Consider open sets $U_0, U_1$ and $U_2$ with $U_0 = U_1 \\cup U_2$ and put $U_{12} = U_1 \\cap U_2$. Put \n$$\\ensuremath{\\mathcal{B}}_{\\star} = \\{ B \\in \\ensuremath{\\mathcal{B}} \\,\\, |\\,\\, B \\subseteq U_{\\star} \\}$$\nfor $\\star = 1, 2, 12$. Then $\\ensuremath{\\mathcal{B}}_{\\star}$ is an acyclic basis for $U_{\\star}$ and $\\ensuremath{\\mathcal{B}}_{12} = \\ensuremath{\\mathcal{B}}_1 \\cap \\ensuremath{\\mathcal{B}}_2$. Futher, $\\ensuremath{\\mathcal{B}}_0 = \\ensuremath{\\mathcal{B}}_1 \\cup \\ensuremath{\\mathcal{B}}_2$ is an acyclic basis for $U_0$ (which is smaller that the basis $U_0$ immediately inherits form $\\ensuremath{\\mathcal{B}}$). \nNow consider all the posets $(\\ensuremath{\\mathcal{B}}_{\\star}, \\subseteq)$ as small categories. Then we have a cover of $\\ensuremath{\\mathcal{B}}_0$ consisting of the injections $\\varphi_i: \\ensuremath{\\mathcal{B}}_i \\longrightarrow \\ensuremath{\\mathcal{B}}_0$ for $i = 1,2$ and we have $\\ensuremath{\\mathcal{B}}_{12} = \\ensuremath{\\mathcal{B}}_1 \\cap \\ensuremath{\\mathcal{B}}_2$ as categories. \nOn each of these categories, we consider the corresponding $\\ensuremath{\\mathcal{B}}_{\\star}$-graded category $\\mathfrak{b}_{\\star}$ with $(\\mathfrak{b}_{\\star})_V = \\{V\\}$ and\n$$\\mathfrak{b}_{\\star}(W,V) = \\ensuremath{\\mathcal{O}}(W)$$\nfor $W \\subseteq V$, i.e. $\\mathfrak{b}_{\\star}$ is the map-graded category associated to the restricted structure sheaf on $\\ensuremath{\\mathcal{B}}_{\\star}$.\nClearly, $\\mathfrak{b}_1 = (\\mathfrak{b}_0)^{\\varphi_1}$ and similarly for the other injections. According to \\cite{lowenvandenberghhoch}, $\\mathbf{C}_{\\ensuremath{\\mathcal{B}}_{\\star}}(\\mathfrak{b}_{\\star})$ computes the Hochschild cohomology of the ringed space $(U_{\\star}, \\ensuremath{\\mathcal{O}}|_{U_{\\star}})$.\nFrom the above, we obtain an exact sequence\n$$\\xymatrix{ 0 \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{B}}_0}(\\mathfrak{b}_0)} \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{B}}_1}(\\mathfrak{b}_1) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{B}}_2}(\\mathfrak{b}_2)} \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{B}}_{12}}(\\mathfrak{b}_{12})} \\ar[r] & 0}$$\nand an induced long exact cohomology sequence\n$$\\xymatrix{ {\\dots} \\ar[r] & {HH^i(U_0)} \\ar[r] & {HH^i(U_1) \\oplus HH^i(U_2)} \\ar[r] & {HH^i(U_{12})} \\ar[r] & {\\dots}}$$\nThe more subtle problem of defining a sheaf of Hochschild complexes on quasi-compact opens of a quasi-compact separated scheme was solved in \\cite{lowensheafhoch}. The existence of Mayer-Vietoris sequences for arbitrary ringed spaces was shown in \\cite[\\S 7.9]{lowenvandenberghhoch} making use of the definition of the Hochschild complex of $X$ as the Hochschild complex of the linear category of injectives in the category of sheaves $\\ensuremath{\\mathsf{Mod}} (X)$. We come back to this approach in \\S \\ref{parcomp}.\n\\end{example}\n\n\n\\subsection{$1$-injections}\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a $1$-injective subcartesian graded functor and let $M$ be an $\\mathfrak{a}$-bimodule. We thus have injections\n$\\ensuremath{\\mathcal{N}}_n(\\varphi^{\\sharp}): \\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{V}}^{\\sharp}) \\longrightarrow \\ensuremath{\\mathcal{N}}_n(\\ensuremath{\\mathcal{U}}^{\\sharp})$ for all $n \\in \\mathbb{N}$. By Proposition \\ref{propsurjinj}, we obtain a surjective morphism\n$$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M).$$\nDefine the complex\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M) = \\mathrm{Ker}(F^{\\ast}).$$\nThen $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}^n(\\mathfrak{a}, M)$ is isomorphic to\n\\begin{equation}\\label{eqsupport}\n\\prod_{(u,A) \\notin \\mathrm{Im}(\\ensuremath{\\mathcal{N}}_n(\\varphi^{\\sharp}))} \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_n)).\n\\end{equation}\n\n\\begin{proposition}\nThe following are equivalent:\n\\begin{enumerate}\n\\item $(u, A) \\in \\mathrm{Im}(\\ensuremath{\\mathcal{N}}_n(\\varphi^{\\sharp}))$.\n\\item For every $u_i: A_i \\longrightarrow A_{i+1}$ in $\\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{U}}^{\\sharp})$ occuring in $(u, A)$, we have $u_i \\in \\mathrm{Im}(\\ensuremath{\\mathcal{N}}_1(\\varphi^{\\sharp}))$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nSuppose (2) holds. Let $u_i = \\varphi^{\\sharp}(v): A_i \\longrightarrow A_{i+1}$ for $v: B \\longrightarrow B'$ in $\\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{V}}^{\\sharp})$ and let  $u_{i+1} = \\varphi^{\\sharp}(w): A_{i+1} \\longrightarrow A_{i+2}$ for $w: C \\longrightarrow C'$ in $\\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{V}}^{\\sharp})$. Then $\\varphi^{\\sharp}(1_{B'}) = 1_{A_{i+1}} = \\varphi^{\\sharp}(1_{C})$ whence $B' = C$.\n\\end{proof}\n\n\nWe call $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M)$ the \\emph{Hochschild complex of $\\mathfrak{a}$ with support outside $\\ensuremath{\\mathcal{V}}^{\\sharp}$ (and values in $M$)}.\nWe thus obtain an exact sequence of complexes\n\\begin{equation}\\label{eqkern}\n0 \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M) \\longrightarrow 0.\n\\end{equation}\nand a long exact cohomology sequence\n$$\\xymatrix{{\\dots} \\ar[r] & {HH^i_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M)} \\ar[r] & {HH^i_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M)} \\ar[r]_-{F^{\\ast}} & {HH^i(\\mathfrak{b}, F^{\\ast} M)} \\ar[r] & {\\dots} }$$\n\n\\begin{example}\\label{exuminv}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ a subcategory. Consider the cartesian morphism $\\delta^{\\varphi, \\mathfrak{a}}: \\mathfrak{a}^{\\varphi} \\longrightarrow \\mathfrak{a}$. Then $(\\varphi, \\delta^{\\varphi, \\mathfrak{a}}): (\\ensuremath{\\mathcal{V}}, \\mathfrak{a}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is injective whence $1$-injective. In this case the following are equivalent:\n\\begin{enumerate}\n\\item $(u, A) \\in \\mathrm{Im}(\\ensuremath{\\mathcal{N}}_n(\\varphi^{\\sharp}))$.\n\\item For every $u_i: U_i \\longrightarrow U_{i+1}$ in $\\ensuremath{\\mathcal{N}}_1(\\ensuremath{\\mathcal{U}})$ occuring in $u$, we have $u \\in \\mathrm{Im}(\\ensuremath{\\mathcal{N}}_1(\\varphi))$.\n\\end{enumerate}\nConsequently, we will denote $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}, M) = \\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M)$ and call this complex the \\emph{Hochschild complex of $\\mathfrak{a}$ with support outside $\\ensuremath{\\mathcal{V}}$}.\n\\end{example}\n\n\n\n\n\\subsection{Censoring subcategories}\\label{parcensor}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category. A subcategory $\\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ is called \\emph{censoring} if for all $u: U \\longrightarrow U'$, $A \\in \\mathfrak{a}_U$, $A' \\in \\mathfrak{a}_{U'}$ with $u \\notin \\ensuremath{\\mathcal{V}}$, we have\n$\\mathfrak{a}_u(A,A') = 0$. \nThe terminology is taken from \\cite[\\S 4.3]{lowenvandenberghhoch}, cfr. the following example:\n\n\\begin{example}\\label{extrans}\nLet $\\mathfrak{a}$ be a linear category and let $\\ensuremath{\\mathcal{R}}$ be a transitive relation on $\\mathrm{Ob}(\\mathfrak{a})$. In \\cite{lowenvandenberghhoch}, the relation $\\ensuremath{\\mathcal{R}}$ is called \\emph{censoring} if $\\mathfrak{a}(B, A) = 0$ for $(B,A) \\notin \\ensuremath{\\mathcal{R}}$. Clearly, this yields a special case of a censoring subcategory $\\ensuremath{\\mathcal{U}}_{\\ensuremath{\\mathcal{R}}}$ of the standard grading category $\\ensuremath{\\mathcal{U}}$ of $\\mathfrak{a}$. Precisely, we let $\\ensuremath{\\mathcal{U}}_{\\ensuremath{\\mathcal{R}}}$ be the category with $\\mathrm{Ob}(\\ensuremath{\\mathcal{U}}_{\\ensuremath{\\mathcal{R}}}) = \\mathrm{Ob}(\\mathfrak{a})$ and \n$$\\ensuremath{\\mathcal{U}}_{\\ensuremath{\\mathcal{R}}}(B,A) = \\begin{cases} \\ast & \\text{if} \\,\\, (B,A) \\in \\ensuremath{\\mathcal{R}} \\\\ \\varnothing & \\text{otherwise.} \\end{cases}$$\n\\end{example}\nA $1$-injection $(\\varphi, F) : (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ of graded sets is called \\emph{censoring} if for all $u: A \\longrightarrow A'$ in $ \\ensuremath{\\mathcal{U}}^{\\sharp}$ which is not in the image of $\\ensuremath{\\mathcal{N}}_1(F): \\ensuremath{\\mathcal{N}}_1(\\mathfrak{y}) \\longrightarrow \\ensuremath{\\mathcal{N}}_1(\\mathfrak{a})$, we have $\\mathfrak{a}_u(A,A') = 0$. \n\n\\begin{remark}\n\\begin{enumerate}\n\\item A subcategory $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ is censoring if and only if the cartesian map $(\\varphi, \\delta^{\\varphi, \\mathfrak{a}}): (\\ensuremath{\\mathcal{V}}, \\mathfrak{a}^{\\varphi}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is a censoring $1$-injection.\n\\item A $1$-injection $(\\varphi, F) : (\\ensuremath{\\mathcal{V}}, \\mathfrak{y}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ is censoring if and only if $\\varphi^{\\sharp}: \\ensuremath{\\mathcal{V}}^{\\sharp} \\longrightarrow \\ensuremath{\\mathcal{U}}^{\\sharp}$ defines a censoring subcategory for the graded category $(\\ensuremath{\\mathcal{U}}^{\\sharp}, \\mathfrak{a}^{\\sharp})$.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{proposition}\\label{propcens}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and let $M$ be an $\\mathfrak{a}$-bimodule.\n\\begin{enumerate}\n\\item Let $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ be a censoring subcategory. Then the map\n$$\\varphi^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{a}^{\\varphi}, M^{\\varphi})$$\nis an isomorphism of complexes.\n\\item Let $(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a subcartesian, censoring $1$-injection. Then the map\n$$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, F^{\\ast}M)$$ is an isomorphism of complexes. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nBy the higher remarks, it suffices to prove either one of (1), (2). Let us prove (2). \nBy the assumption, every component \n$$\\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_n))$$\nin \\eqref{eqsupport} contains some $u_i: A_i \\longrightarrow A_{i+1}$ not contained in $\\mathrm{Im}(\\ensuremath{\\mathcal{N}}_1(\\varphi^{\\sharp}))$, which thus has $\\mathfrak{a}_{u_i}(A_i, A_{i+1}) = 0$. Consequently $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}^{\\sharp}}(\\mathfrak{a}, M) = 0$.\n\\end{proof}\n\nProposition \\ref{propcens} has a useful corollary, which says that a censoring subcategory essentially censors possible uncontrollable parts of bimodules. This was precisely the original intuition behind the terminology in \\cite{lowenvandenberghhoch}. \n\n\\begin{corollary}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and let $\\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ be a censoring subcategory. Let \n$$f: M \\longrightarrow N$$\nbe a morphism of $\\mathfrak{a}$-bimodules. If \n$$f^{\\varphi}: M^{\\varphi} \\longrightarrow N^{\\varphi}$$\nis a quasi-isomorphism, then\nso is\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, f): \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a},M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, N).$$\n\\end{corollary}\n\n\n\n\n\\section{Hochschild cohomology with support}\\label{parhochsup}\n\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category, $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ a subcategory, and $(\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ a cartesian functor.\nIn this section we investigate some cases where the cohomology of the complex $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}, M)$ (see Example \\ref{exuminv}) has a nice cohomological interpretation. The main point is that we need some control over the ``complement'' of $\\ensuremath{\\mathcal{V}}$ in $\\ensuremath{\\mathcal{U}}$. Precisely, we assume that this complement (the $\\ensuremath{\\mathcal{U}}$-morphisms not in $\\ensuremath{\\mathcal{V}}$) constitutes an ideal $\\ensuremath{\\mathcal{Z}}$ in $\\ensuremath{\\mathcal{U}}$. In this case we show in Proposition \\ref{propidsubcat} that \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}, M) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{Z}}})$$\nwhere $M_{\\ensuremath{\\mathcal{Z}}}$ is the natural restriction of $M$ to an $\\mathfrak{a}$-bimodule supported on $\\ensuremath{\\mathcal{Z}}$ (i.e. with zero values outside of $\\ensuremath{\\mathcal{Z}}$).\nOur setup applies in the situation where $\\ensuremath{\\mathcal{U}}$ is the category associated to a collection of open subsets of a topological space $X$ ordered by inclusion, $\\ensuremath{\\mathcal{V}}$ is the full subcategory of subsets $U \\subseteq V$ for a fixed subset $V$, and $\\ensuremath{\\mathcal{Z}}$ contains the inclusions $U' \\subseteq U$ with $U \\nsubseteq V$.  \n\nIn \\S \\ref{pararrow}, we revisit the arrow category construction from \\cite{kellerdih} in the map-graded context. For an $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$-$(\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$-bimodule $(S, M)$, we take the natural inclusion $\\ensuremath{\\mathcal{V}} \\coprod \\ensuremath{\\mathcal{U}} \\longrightarrow (\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}})$ and corresponding cartesian functor \n$$(\\ensuremath{\\mathcal{V}} \\coprod \\ensuremath{\\mathcal{U}}, \\mathfrak{b} \\coprod \\mathfrak{a}) \\longrightarrow (\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}}, \\mathfrak{b} \\rightarrow_M \\mathfrak{a})$$\nas starting point for obtaining map-graded analogues of some of the main results from \\cite{kellerdih}. Sections \\S \\ref{parconn} and \\S \\ref{parderived} are entirely modelled upon the treatment in \\cite{kellerdih}, and mainly formulate results from \\cite{kellerdih} in the map-graded context, making use of the natural Hom and tensor functors from \\S \\ref{parbimod}. Further, in \\S \\ref{pararrowthin}, we give an intrinsic characterization of arrow categories based upon the \\emph{thin ideals} introduced in \\S \\ref{parthin}. \n\n\\subsection{Ideals in categories}\nLet $\\ensuremath{\\mathcal{U}}$ be an arbitrary category. Recall from \\S \\ref{parbifun} that a $\\ensuremath{\\mathcal{U}}$-bifunctor $S$ consists of sets $S(V,U)$ for $U, V \\in \\ensuremath{\\mathcal{U}}$ and actions\n$$\\ensuremath{\\mathcal{U}}(U,U') \\times S(V,U) \\times \\ensuremath{\\mathcal{U}}(V',V) \\longrightarrow S(V', U').$$\nDenote the category of $\\ensuremath{\\mathcal{U}}$-bimodules by $\\ensuremath{\\mathsf{Bimod}} (\\ensuremath{\\mathcal{U}})$.\nA \\emph{(two sided) ideal} in $\\ensuremath{\\mathcal{U}}$ is a subfunctor\n$$\\ensuremath{\\mathcal{Z}} \\subseteq 1_{\\ensuremath{\\mathcal{U}}}$$\nof the identity bifunctor in the category $\\mathsf{Bifun}(\\ensuremath{\\mathcal{U}})$ of $\\ensuremath{\\mathcal{U}}$-bifunctors. More precisely, it consists of subsets $\\ensuremath{\\mathcal{Z}}(V,U) \\subseteq \\ensuremath{\\mathcal{U}}(V,U)$ for all $U, V \\in \\ensuremath{\\mathcal{U}}$ such that the composition of $\\ensuremath{\\mathcal{U}}$ restricts to\n$$\\ensuremath{\\mathcal{U}}(U,U') \\times \\ensuremath{\\mathcal{Z}}(V,U) \\times \\ensuremath{\\mathcal{U}}(V',V) \\longrightarrow \\ensuremath{\\mathcal{Z}}(V', U').$$\nWe put the \\emph{morphisms in $\\ensuremath{\\mathcal{Z}}$} equal to $\\mathrm{Mor}(\\ensuremath{\\mathcal{Z}}) = \\coprod_{V, U} \\ensuremath{\\mathcal{Z}}(V,U)$.\n\n\\subsection{Bimodules over ideals}\\label{parbimideal}\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and let $\\ensuremath{\\mathcal{Z}}$ be an ideal in $\\ensuremath{\\mathcal{U}}$. An \\emph{$\\mathfrak{a}$-bimodule $M$ on $\\ensuremath{\\mathcal{Z}}$} consists of:\n\\begin{itemize}\n\\item $k$-modules $M_z(B, A)$ for all $z: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{Z}}$ and $A \\in \\mathfrak{a}_U$, $B \\in \\mathfrak{a}_{V}$,\n\\item actions $\\mathfrak{a}_u(A,A') \\otimes M_z(B, A) \\otimes \\mathfrak{a}_v(B',B) \\longrightarrow M_{uzv}(B',A')$ for all additional $u: U \\longrightarrow U'$, $v: V' \\longrightarrow V$ in $\\ensuremath{\\mathcal{U}}$ and $A' \\in \\mathfrak{a}_{U'}$, $B' \\in \\mathfrak{a}_{V'}$\n\\end{itemize}\nwith the natural axioms. The $\\mathfrak{a}$-bimodules on $\\ensuremath{\\mathcal{Z}}$ form an abelian category $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a})$. There are obvious exact funcors\n$$(-)|^{\\ensuremath{\\mathcal{U}}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}): M \\longmapsto M|^{\\ensuremath{\\mathcal{U}}}$$\nwhere $M|^{\\ensuremath{\\mathcal{U}}}$ is the extension of $M$ by zero values outside of $\\ensuremath{\\mathcal{Z}}$, and\n$$(-)|_{\\ensuremath{\\mathcal{Z}}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a}): M \\longmapsto M|_{\\ensuremath{\\mathcal{Z}}}$$\nwhere $M|_{\\ensuremath{\\mathcal{Z}}}$ is the restriction of $M$ to $\\ensuremath{\\mathcal{Z}}$. Clearly, $(-)|_{\\ensuremath{\\mathcal{Z}}}$ is right adjoint to $(-)|^{\\ensuremath{\\mathcal{U}}}$, $$(-)|_{\\ensuremath{\\mathcal{Z}}}(-)|^{\\ensuremath{\\mathcal{U}}} = 1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a})}$$\nand the canonical \n$$(-)_{\\ensuremath{\\mathcal{Z}}} = (-)|^{\\ensuremath{\\mathcal{U}}}(-)|_{\\ensuremath{\\mathcal{Z}}} \\longrightarrow 1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})}$$\ncorresponds to inclusions of $\\mathfrak{a}$-modules\n$$M_{\\ensuremath{\\mathcal{Z}}} \\subseteq M$$\nwhere $M_{\\ensuremath{\\mathcal{Z}}}$ is obtained by changing the values of $M$ to zero outside of $\\ensuremath{\\mathcal{Z}}$. \nWe can thus identify $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a})$ with the full subcategory of $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ of bimodules \\emph{supported on $\\ensuremath{\\mathcal{Z}}$}.\n\n\\subsection{Ideal-subcategory decomposition}\nWe now turn to our main situation of interest, which is summarized as follows:\n\\begin{itemize}\n\\item $\\ensuremath{\\mathcal{U}}$ is an arbitrary category;\n\\item $\\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ is a subcategory;\n\\item $\\ensuremath{\\mathcal{Z}} \\subseteq 1_{\\ensuremath{\\mathcal{U}}}$ is an ideal;\n\\item $\\mathrm{Mor}(\\ensuremath{\\mathcal{U}}) = \\mathrm{Mor}(\\ensuremath{\\mathcal{V}}) \\coprod \\mathrm{Mor}(\\ensuremath{\\mathcal{Z}})$.\n\\end{itemize}\nIn this case, we call $(\\ensuremath{\\mathcal{Z}}, \\ensuremath{\\mathcal{V}})$ an \\emph{ideal-subcategory decomposition} of $\\ensuremath{\\mathcal{U}}$.\n\n\\begin{example}\nLet $X$ be a topological space, $V$ an open subset and $Z = X \\setminus V$ its closed complement. Let $\\ensuremath{\\mathcal{U}} \\subseteq \\mathrm{open}(X)$ be a sub-poset of the open sets of $X$. Let $\\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ be the full subcategory with\n$$\\mathrm{Ob}(\\ensuremath{\\mathcal{V}}) = \\{ U \\in \\ensuremath{\\mathcal{U}} \\,\\, |\\,\\, U \\subseteq V \\}$$\nand $\\ensuremath{\\mathcal{Z}} \\subseteq 1_{\\ensuremath{\\mathcal{U}}}$ the ideal with $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{Z}}$ if and only if $U' \\nsubseteq V$.\nClearly, $(\\ensuremath{\\mathcal{Z}}, \\ensuremath{\\mathcal{V}})$ is an ideal-subcategory decomposition of $\\ensuremath{\\mathcal{U}}$.\n\\end{example}\n\n\n\n\n\n\n\\subsection{The localization sequence of bimodule categories}\\label{parlocbimod} \nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category and $(\\ensuremath{\\mathcal{Z}}, \\ensuremath{\\mathcal{V}})$ an ideal-subcategory decomposition of $\\ensuremath{\\mathcal{U}}$. Put $\\mathfrak{b} = \\mathfrak{a}|_{\\ensuremath{\\mathcal{V}}}$.\nIn this section we take a closer look at the sequence\n$$\\xymatrix{{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a})} \\ar[r]_-{(-)|^{\\ensuremath{\\mathcal{U}}}_{(\\ensuremath{\\mathcal{Z}})}} & {\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_{(-)|_{\\ensuremath{\\mathcal{V}}}} & {\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}).}}$$ \nClearly $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a})$ is the kernel of $(-)|_{\\ensuremath{\\mathcal{V}}}$, and we already know from \\S \\ref{parbimideal} that $(-)|^{\\ensuremath{\\mathcal{U}}}_{(\\ensuremath{\\mathcal{Z}})}$ has a right adjoint $(-)|_{\\ensuremath{\\mathcal{Z}}}$. Since $(-)|_{\\ensuremath{\\mathcal{V}}}$ is induced by an underlying $k$-linear functor\n$\\mathfrak{b} \\otimes_{\\ensuremath{\\mathcal{V}}} \\mathfrak{b} \\longrightarrow \\mathfrak{a} \\otimes_{\\ensuremath{\\mathcal{U}}} \\mathfrak{a}$, both adjoints of $(-)|_{\\ensuremath{\\mathcal{V}}}$ exist and can be described explicitely. Moreover, since $\\ensuremath{\\mathcal{Z}}$ is an ideal, the right adjoint has a particularly easy description. Define the functor\n$$(-)|_{(\\ensuremath{\\mathcal{V}})}^{\\ensuremath{\\mathcal{U}}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}): M \\longmapsto M|^{\\ensuremath{\\mathcal{U}}}$$\nin the following way. Put\n$$(M|^{\\ensuremath{\\mathcal{U}}})_u(B,A) = \\begin{cases} M_u(B,A) &\\text{if}\\,\\, u \\in \\ensuremath{\\mathcal{V}} \\\\ 0 & \\text{otherwise}\\end{cases}.$$\nThen there is a unique way to let $\\mathfrak{a}$ act on $M|^{\\ensuremath{\\mathcal{U}}}$ so that \n$$\\mathfrak{a}_{u'}(A,A') \\otimes (M|^{\\ensuremath{\\mathcal{U}}})_u(B,A) \\otimes \\mathfrak{a}_{u''}(B',B) \\longrightarrow (M|^{\\ensuremath{\\mathcal{U}}})_{u'uu''}(B',A')$$\nis given by the action of $\\mathfrak{b}$ on $M$ if $u$, $u'$, $u''$ are in $\\ensuremath{\\mathcal{V}}$, and is zero otherwise. This is a well defined action thanks to the fact that $\\ensuremath{\\mathcal{Z}}$ is an ideal. The functor $(-)|_{(\\ensuremath{\\mathcal{V}})}^{\\ensuremath{\\mathcal{U}}}$ is right adjoint to $(-)|_{\\ensuremath{\\mathcal{V}}}$ and we have\n$$(-)|_{\\ensuremath{\\mathcal{V}}}(-)|_{(\\ensuremath{\\mathcal{V}})}^{\\ensuremath{\\mathcal{U}}} = 1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})}.$$\nThe canonical\n$$1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\longrightarrow (-)|_{(\\ensuremath{\\mathcal{V}})}^{\\ensuremath{\\mathcal{U}}}(-)|_{\\ensuremath{\\mathcal{V}}} = (-)_{\\ensuremath{\\mathcal{V}}}$$\ncorresponds to quotients of $\\mathfrak{a}$-bimodules\n$$M \\longrightarrow M_{\\ensuremath{\\mathcal{V}}}$$\nwhere $M_{\\ensuremath{\\mathcal{V}}}$ is obtained from $M$ by changing the values of $M$ outside $\\ensuremath{\\mathcal{V}}$ to zero.\n\nSummarizing, we have an exact sequence of functors\n$$0 \\longrightarrow (-)_{\\ensuremath{\\mathcal{Z}}} \\longrightarrow 1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\longrightarrow (-)_{\\ensuremath{\\mathcal{V}}} \\longrightarrow 0$$\ncorresponding to exact sequences of bimodules\n\\begin{equation} \\label{eqbimod}\n0 \\longrightarrow M_{\\ensuremath{\\mathcal{Z}}} \\longrightarrow M \\longrightarrow M_{\\ensuremath{\\mathcal{V}}} \\longrightarrow 0.\n\\end{equation}\n\n\\subsection{The localization sequence for Hochschild cohomology}\\label{parlochoch}\nWe can now use \\eqref{eqbimod} to obtain the following short exact sequence of Hochschild complexes:\n\\begin{equation}\\label{eqhoch2}\n0 \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{Z}}}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{V}}}) \\longrightarrow 0.\n\\end{equation}\n\n\\begin{proposition}\\label{propidsubcat}\nThe sequences \\eqref{eqkern} and \\eqref{eqhoch2} are canonically isomorphic.\n\\end{proposition}\n\n\\begin{proof}\nThis follows from the fact that $(\\ensuremath{\\mathcal{Z}}, \\ensuremath{\\mathcal{V}})$ is an ideal-subcategory decomposition of $\\ensuremath{\\mathcal{U}}$. Indeed, let us first compare $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{V}}})$ to $\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, M|_{\\ensuremath{\\mathcal{V}}})$. In every product\n$$\\prod_{(u, A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{a})} \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), (M_{\\ensuremath{\\mathcal{V}}})_{|u|}(A_0, A_n))$$\nturning up in $\\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{V}}})$ we may clearly remove the pieces with $|u| \\in \\ensuremath{\\mathcal{Z}}$ since then $(M_{\\ensuremath{\\mathcal{V}}})_{|u|}(A_0, A_n) = 0$. But since $\\ensuremath{\\mathcal{Z}}$ is an ideal and $\\ensuremath{\\mathcal{V}}$ a subcategory, the remaining pieces are precisely the ones with all the $u_i \\in \\ensuremath{\\mathcal{V}}$, and we recover $\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, M|_{\\ensuremath{\\mathcal{V}}})$. Similarly, consider a product\n$$\\prod_{(u, A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{a})} \\mathrm{Hom}_k(\\mathfrak{a}_{u_{n-1}}(A_{n-1}, A_n) \\otimes \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), (M_{\\ensuremath{\\mathcal{Z}}})_{|u|}(A_0, A_n))$$\nin $\\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{Z}}})$.\nThis time, all the pieces with $|u| \\in \\ensuremath{\\mathcal{V}}$ can be removed. What remains are the pieces in which at least one $u_i$ belongs to $\\ensuremath{\\mathcal{Z}}$, which corresponds precisely to $\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}, M)$.\n\\end{proof}\n\nIn fact, the isomorphism $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{V}}}) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, M|_{\\ensuremath{\\mathcal{V}}})$ is easily understood on the derived level. Indeed, the localization between $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ and $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$ of \\S \\ref{parlocbimod} yields:\n$$\\mathrm{RHom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})}(1_{\\mathfrak{a}}, (M|_{\\ensuremath{\\mathcal{V}}})|^{\\ensuremath{\\mathcal{U}}}) \\cong \\mathrm{RHom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})}(1_{\\mathfrak{b}}, M|_{\\ensuremath{\\mathcal{V}}}).$$\nThe more mysterious part in the isomorphic sequences \\eqref{eqkern} and \\eqref{eqhoch2} remains the Hochschild complex with support\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\setminus \\ensuremath{\\mathcal{V}}}(\\mathfrak{a}, M) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{Z}}}),$$\nbut at least it now has an interpretation as an ordinary Hochschild complex of $\\mathfrak{a}$ with values in the bimodule $M_{\\ensuremath{\\mathcal{Z}}}$ supported on $\\ensuremath{\\mathcal{Z}}$. \nSo far, the most meaningful incarnation of the sequences \\eqref{eqkern} and \\eqref{eqhoch2} is perhaps\n\\begin{equation}\\label{eqhoch3}\n0 \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M_{\\ensuremath{\\mathcal{Z}}}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, M) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, M|_{\\ensuremath{\\mathcal{V}}}) \\longrightarrow 0.\n\\end{equation}\n\n\\subsection{Thin ideals}\\label{parthin}\nConsider $\\varphi: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{U}}$ with $\\mathrm{Ob}(\\ensuremath{\\mathcal{V}}) = \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$ and let $(\\ensuremath{\\mathcal{Z}}, \\ensuremath{\\mathcal{V}})$ be an ideal-subcategory decomposition of $\\ensuremath{\\mathcal{U}}$. Let $\\mathfrak{b}$ be a $\\ensuremath{\\mathcal{V}}$-graded category. We define a \\emph{$\\mathfrak{b}$-bimodule on $\\ensuremath{\\mathcal{Z}}$} to consist of the following data:\n\n\\begin{itemize}\n\\item for all $z: V \\longrightarrow U$ in $\\ensuremath{\\mathcal{Z}}$ and $B \\in \\mathfrak{b}_V$, $A \\in \\mathfrak{b}_U$, a $k$-module $M_z(B,A)$;\n\\item actions $\\mathfrak{b}_v(A, A') \\otimes M_z(B,A) \\otimes \\mathfrak{b}_{v'}(B' B) \\longrightarrow M_{vzv'}(B',A')$ for all additional $v: U \\longrightarrow U'$, $v': V' \\longrightarrow V$ in $\\ensuremath{\\mathcal{V}}$ and $A' \\in \\mathfrak{b}_{U'}$, $B' \\in \\mathfrak{b}_{V'}$.\n\\end{itemize}\nWe thus obtain the abelian category $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{b})$ of \\emph{$\\mathfrak{b}$-bimodules on $\\ensuremath{\\mathcal{Z}}$}.\n\nNow consider a $\\ensuremath{\\mathcal{U}}$-graded category $\\mathfrak{a}$ and suppose $\\mathfrak{b} = \\mathfrak{a}|_{\\ensuremath{\\mathcal{V}}}$.\nThere is a functor\n$$(-)|^{\\mathfrak{a}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{b}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a}): M \\longmapsto M|^{\\mathfrak{a}}$$\nwhere the actions $\\mathfrak{a}_{z'}(A,A') \\otimes M_z(B,A) \\longrightarrow M_{z'z}(B,A')$ and $M_z(B,A) \\otimes \\mathfrak{a}_{z''}(B',B) \\longrightarrow M_{zz''}(B',A)$ are zero for $z$, $z'$ and $z''$ in $\\ensuremath{\\mathcal{Z}}$. This yields a well defined action since $\\ensuremath{\\mathcal{Z}}$ is an ideal.\nObviously, there is also a functor\n$$(-)|_{\\mathfrak{b}}: \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a}) \\longrightarrow \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{b}): M \\longmapsto M|_{\\mathfrak{b}}$$ which restricts the action to $\\mathfrak{b}$. Clearly,\n$$(-)|_{\\mathfrak{b}}(-)|^{\\mathfrak{a}} = 1_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{b})}.$$\n\n\\begin{definition}\n\\begin{enumerate}\n\\item An ideal $\\ensuremath{\\mathcal{Z}} \\subseteq 1_{\\ensuremath{\\mathcal{U}}}$ is called \\emph{thin} if it contains no consecutive morphisms, i.e. if we consider $z \\in \\ensuremath{\\mathcal{Z}}(V,U)$, $u' \\in \\ensuremath{\\mathcal{U}}(V',V)$, $u \\in \\ensuremath{\\mathcal{U}}(U,U')$, then it follows that $u' \\notin \\ensuremath{\\mathcal{Z}}$ and $u \\notin \\ensuremath{\\mathcal{Z}}$. \n\\item An $\\mathfrak{a}$-bimodule $M$ on $\\ensuremath{\\mathcal{Z}}$ is called \\emph{$\\ensuremath{\\mathcal{Z}}$-thin} if it receives no non trivial actions from $\\ensuremath{\\mathcal{Z}}$, i.e if we have $z \\in \\ensuremath{\\mathcal{Z}}(V,U)$, $z' \\in \\ensuremath{\\mathcal{Z}}(V',V)$, $z'' \\in \\ensuremath{\\mathcal{Z}}(U,U')$, then it follows that both $\\mathfrak{a}_{z''}(A, A') \\otimes M_z(B,A) \\longrightarrow M_{z''z}(B,A')$ and $M_z(B,A) \\otimes \\mathfrak{a}_{z'}(B',B) \\longrightarrow M_{zz'}(B',A)$ are zero for all $A$, $A'$, $B$, $B'$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{example}\\label{exthin}\nLet $\\ensuremath{\\mathcal{U}}$ be a category and $\\mathrm{Ob}_1 \\subseteq \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$ and $\\mathrm{Ob}_2 \\subseteq \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$ be two classes of objects with no morphisms going from $\\mathrm{Ob}_2$ to $\\mathrm{Ob}_1$. Then the $\\ensuremath{\\mathcal{U}}$-morphisms starting in $\\mathrm{Ob}_1$ and landing in $\\mathrm{Ob}_2$ form a thin ideal in $\\ensuremath{\\mathcal{U}}$.\n\\end{example}\n\nObviously, if $\\ensuremath{\\mathcal{Z}}$ is thin, then every $M$-bimodule on $\\ensuremath{\\mathcal{Z}}$ is $\\ensuremath{\\mathcal{Z}}$-thin. In general, we make the following easy observations:\n\\begin{proposition}\nFor an $\\mathfrak{a}$-bimodule $M$ on $\\ensuremath{\\mathcal{Z}}$, the following are equivalent:\n\\begin{enumerate}\n\\item $M$ is $\\ensuremath{\\mathcal{Z}}$-thin;\n\\item $M \\cong (M|_{\\mathfrak{b}})|^{\\mathfrak{a}}$;\n\\item $M$ is in the image of $(-)|^{\\mathfrak{a}}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{corollary}\\label{isobimod}\nIf $\\ensuremath{\\mathcal{Z}}$ is thin, then $(-)|^{\\mathfrak{a}}$ and $(-)|_{\\mathfrak{b}}$ constitute inverse isomorphisms\n$$\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{b}) \\cong \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{Z}}}(\\mathfrak{a}).$$\n\\end{corollary}\n\n\n\\subsection{Arrow categories}\\label{pararrow}\nIn this section, we introduce the arrow category construction from \\cite{kellerdih} in the map-graded setting.\nLet $\\mathfrak{a}$ be a $\\ensuremath{\\mathcal{U}}$-graded category, $\\mathfrak{b}$ a $\\ensuremath{\\mathcal{V}}$-graded category, $S$ a $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor and $M$ and $\\mathfrak{a}$-$S$-$\\mathfrak{b}$-bimodule. To these data we associate the \\emph{arrow category} $\\mathfrak{b} \\rightarrow_M \\mathfrak{a}$ which is a $\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}}$ graded category. Here, $\\ensuremath{\\mathcal{W}} = \\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}}$ is the underlying arrow category with\n$$\\mathrm{Ob}(\\ensuremath{\\mathcal{W}}) = \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}) \\coprod \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$$\nand \n$$\\begin{aligned}\n\\ensuremath{\\mathcal{W}}(V', V) = \\ensuremath{\\mathcal{V}}(V',V) &&& \\ensuremath{\\mathcal{W}}(U, U') = \\ensuremath{\\mathcal{U}}(U,U')\\\\\n\\ensuremath{\\mathcal{W}}(V,U) = S(V,U) &&& \\ensuremath{\\mathcal{W}}(U,V) = \\varnothing\n\\end{aligned}$$\nfor $U, U' \\in \\ensuremath{\\mathcal{U}}$ and $V, V' \\in \\ensuremath{\\mathcal{V}}$.\nSimilarly, $\\mathfrak{c} = \\mathfrak{b} \\rightarrow_M \\mathfrak{a}$ is the $\\ensuremath{\\mathcal{W}}$-graded category with\n$$\\mathrm{Ob}(\\mathfrak{c}) = \\mathrm{Ob}(\\mathfrak{a}) \\coprod \\mathrm{Ob}(\\mathfrak{b})$$\nand\n$$\\begin{aligned}\n\\mathfrak{c}_v(B', B) = \\mathfrak{b}_v(B',B) &&& \\mathfrak{c}_u(A, A') = \\mathfrak{a}_u(A,A')\\\\\n\\mathfrak{c}_s(B,A) = M_s(B,A) &&& \n\\end{aligned}$$\nfor $v: V' \\longrightarrow V$ in $\\ensuremath{\\mathcal{V}}$, $B' \\in \\mathfrak{b}_{V'}$, $B \\in \\mathfrak{b}_V$, $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$, $A \\in \\mathfrak{a}_U$, $A' \\in \\mathfrak{a}_{U'}$.\n\n\\begin{remark}\nEven when $\\mathfrak{a}$ and $\\mathfrak{b}$ are linear categories with standard grading, and $M$ is an ordinary $\\mathfrak{a}$-$\\mathfrak{b}$-bimodule, the resulting arrow category $\\mathfrak{b} \\rightarrow_M \\mathfrak{a}$ is naturally graded in a non-standard way since there are no morphisms going from $\\mathfrak{a}$ to $\\mathfrak{b}$. \n\\end{remark}\n\nConsider the natural inclusions\n$\\varphi_{\\ensuremath{\\mathcal{U}}}: \\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathcal{W}}$, $\\varphi_{\\ensuremath{\\mathcal{V}}}: \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{W}}$ and $\\varphi_{\\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}}}: \\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}} \\longrightarrow \\ensuremath{\\mathcal{W}}$. Clearly, we have\n$\\mathfrak{c}^{\\varphi_{\\ensuremath{\\mathcal{U}}}} = \\mathfrak{a}$, $\\mathfrak{c}^{\\varphi_{\\ensuremath{\\mathcal{V}}}} = \\mathfrak{b}$ and $\\mathfrak{c}^{\\varphi_{\\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}}}} = \\mathfrak{a} \\coprod \\mathfrak{b}$ and we obtain the induced surjections\n$$\\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}),$$\n$$\\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$$\nand\n$$\\varphi_{\\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}}}^{\\ast} \\cong \\begin{pmatrix} \\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast} \\\\ \\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast} \\end{pmatrix} : \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}}}(\\mathfrak{a} \\coprod \\mathfrak{b}) \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}).$$\nFurther, $S$ defines a thin ideal $\\ensuremath{\\mathcal{S}}$ in $\\ensuremath{\\mathcal{W}}$ with\n$$\\ensuremath{\\mathcal{S}}(V,U) = \\begin{cases} S(V,U) & \\text{if}\\,\\, U \\in \\ensuremath{\\mathcal{U}}, V \\in \\ensuremath{\\mathcal{V}} \\\\ \\varnothing & \\text{else} \\end{cases}$$\nand $(\\ensuremath{\\mathcal{S}}, \\ensuremath{\\mathcal{U}} \\coprod \\ensuremath{\\mathcal{V}})$ is an ideal-subcategory decomposition in $\\ensuremath{\\mathcal{W}}$.\nFor the category of $\\mathfrak{b} \\coprod \\mathfrak{a}$-bimodules on $\\ensuremath{\\mathcal{S}}$ in the sense of \\S \\ref{parthin}, we clearly have an isomorphism of categories\n$$\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{S}}}(\\mathfrak{b} \\coprod \\mathfrak{a}) \\cong \\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{b}, \\mathfrak{a})$$\nand hence, by Corollary \\ref{isobimod}, an isomorphism of categories\n\\begin{equation}\\label{isobimod2}\n\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{b}, \\mathfrak{a}) \\cong \\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{S}}}(\\mathfrak{c}).\n\\end{equation}\n\n\n\\subsection{Arrow categories and thin ideals}\\label{pararrowthin}\nIn this section, we characterize the situation that occurs from the arrow category construction.\nLet $\\mathfrak{c}$ be a $\\ensuremath{\\mathcal{W}}$-graded category and let $\\ensuremath{\\mathcal{S}}$ be a thin ideal in $\\ensuremath{\\mathcal{W}}$. We define the full subcategories $\\varphi_{\\ensuremath{\\mathcal{U}}}: \\ensuremath{\\mathcal{U}} \\subseteq \\ensuremath{\\mathcal{W}}$ and $\\varphi_{\\ensuremath{\\mathcal{V}}}: \\ensuremath{\\mathcal{V}} \\subseteq \\ensuremath{\\mathcal{W}}$ in the following way. An object $W \\in \\ensuremath{\\mathcal{W}}$ belongs to $\\ensuremath{\\mathcal{V}}$ if there exists a path\n$$W \\longrightarrow W_1 \\longrightarrow \\dots \\longrightarrow W_n \\longrightarrow W_{n+1}$$\nfor which the last map $W_n \\longrightarrow W_{n+1}$ belongs to $\\ensuremath{\\mathcal{S}}$. Similarly, $W$ belongs to $\\ensuremath{\\mathcal{U}}$ if there exists a path\n$$W_1 \\longrightarrow W_2 \\longrightarrow \\dots \\longrightarrow W$$\nfor which $W_1 \\longrightarrow W_2$ belongs to $\\ensuremath{\\mathcal{S}}$.\nWe put $\\mathfrak{a} = \\mathfrak{c}^{\\varphi_{\\ensuremath{\\mathcal{U}}}}$ and $\\mathfrak{b} = \\mathfrak{c}^{\\varphi_{\\ensuremath{\\mathcal{V}}}}$.\nLet $S$ denote the restriction of $\\ensuremath{\\mathcal{S}}$ to a $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor, and let $M$ be the restriction of $1_{\\mathfrak{c}}$ to an $\\mathfrak{a}$-$\\mathfrak{b}$-bimodule.\n\n\\begin{proposition}\\label{propthinarrow}\nFor $U \\in \\ensuremath{\\mathcal{U}}$ and $V \\in \\ensuremath{\\mathcal{V}}$, we have $\\ensuremath{\\mathcal{W}}(U, V) = \\varnothing$. In particular, the categories $\\ensuremath{\\mathcal{U}}$ and $\\ensuremath{\\mathcal{V}}$ are disjoint, and there is an injection $\\varphi: (\\ensuremath{\\mathcal{V}} \\rightarrow_{S} \\ensuremath{\\mathcal{U}}) \\longrightarrow \\ensuremath{\\mathcal{W}}$ for which $$(\\mathfrak{b} \\rightarrow_M \\mathfrak{a}) = \\mathfrak{c}^{\\varphi}.$$\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\varphi$ is an isomorphism;\n\\item We have $\\mathrm{Ob}(\\ensuremath{\\mathcal{W}}) = \\mathrm{Ob}(\\ensuremath{\\mathcal{V}}) \\cup \\mathrm{Ob}(\\ensuremath{\\mathcal{U}})$ and $\\ensuremath{\\mathcal{S}}(V,U) = \\ensuremath{\\mathcal{W}}(V,U)$ for $V \\in \\ensuremath{\\mathcal{V}}$ and $U \\in \\ensuremath{\\mathcal{U}}$;\n\\end{enumerate}\nIn this case, we have $(\\mathfrak{b} \\rightarrow_M \\mathfrak{a}) = \\mathfrak{c}$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose there is a morphism $U \\longrightarrow V$ with $U \\in \\ensuremath{\\mathcal{U}}$ and $V \\in \\ensuremath{\\mathcal{V}}$. Then there is a path $$W_1 \\longrightarrow W_2 \\longrightarrow \\dots \\longrightarrow U \\longrightarrow V \\longrightarrow \\dots \\longrightarrow W_n \\longrightarrow W_{n+1}$$ with the two morphisms at the ends belonging to $\\ensuremath{\\mathcal{S}}$. But then since $\\ensuremath{\\mathcal{S}}$ is an ideal, also $W_2 \\longrightarrow \\dots \\longrightarrow W_{n+1}$ belongs to $\\ensuremath{\\mathcal{S}}$. With $W_1 \\longrightarrow W_2$ in $\\ensuremath{\\mathcal{S}}$, this contradicts the thinness of $\\ensuremath{\\mathcal{S}}$.\n\nThe functor $\\varphi$ is clearly injective on objects and on morphisms in $\\ensuremath{\\mathcal{U}}$ or $\\ensuremath{\\mathcal{V}}$, and since $\\ensuremath{\\mathcal{S}}(V,U) \\subseteq \\ensuremath{\\mathcal{W}}(V,U)$, the functor is indeed injective. The description of $\\mathfrak{c}^{\\varphi}$ clearly follows.\n\n It remains to show the equivalence of (1) and (2). Condition (2) is clearly necessary for $\\varphi$ to be surjective. Conversely, surjectivity readily follows from (2) taking into account that for $U \\in \\ensuremath{\\mathcal{U}}$ and $V \\in \\ensuremath{\\mathcal{V}}$, we have $\\ensuremath{\\mathcal{W}}(U, V) = \\varnothing$.\n\\end{proof}\n\nRecall that a delta is a category in which the arrows go only one way, i.e.\n\n\\begin{definition}\\label{defdelta}\nA \\emph{delta} is a category $\\ensuremath{\\mathcal{W}}$ such that for all $W \\neq W' \\in \\ensuremath{\\mathcal{W}}$ we have $\\ensuremath{\\mathcal{W}}(W,W') = \\varnothing$ or $\\ensuremath{\\mathcal{W}}(W', W) = \\varnothing$. \n\\end{definition}\n\n\\begin{example}\\label{exdelta}\nLet $\\ensuremath{\\mathcal{W}}$ be a delta with a terminal object $\\ast \\in \\ensuremath{\\mathcal{W}}$. For $W \\in \\ensuremath{\\mathcal{W}}$, let $\\ast_W: W \\longrightarrow \\ast$ denote the unique morphism.  Let $\\ensuremath{\\mathcal{S}}$ consist of all morphsims $\\ast_W : W \\longrightarrow \\ast$ for $W \\neq \\ast$. Since $\\ensuremath{\\mathcal{W}}$ is a delta and $\\ast$ is terminal, for all $W \\neq \\ast$ we have $\\ensuremath{\\mathcal{W}}(\\ast, W) = \\varnothing$ and we have $\\ensuremath{\\mathcal{W}}(\\ast,\\ast) = \\{1_{\\ast} \\} = \\{ \\ast_{\\ast} \\}$.\nThen $\\ensuremath{\\mathcal{S}}$ is a thin ideal by Example \\ref{exthin}. In the above notations, $\\ensuremath{\\mathcal{U}}$ consists of $\\ast$ and $\\ensuremath{\\mathcal{V}}$ consists of all other objects. Then condition (2) in Proposition \\ref{propthinarrow} is fulfilled whence $\\ensuremath{\\mathcal{W}} \\cong (\\ensuremath{\\mathcal{V}} \\rightarrow_S \\ensuremath{\\mathcal{U}})$.\n\\end{example}\n\n\\subsection{Connecting homomorphism}\\label{parconn}\n\nIn the notations of \\S \\ref{pararrow}, we now investigate the complex\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})$$\nwhich fits into the following exact sequence from \\eqref{eqhoch3}:\n$$\\xymatrix{ {0} \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})} \\ar[r] & {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c})} \\ar[r]_-{\\begin{pmatrix} \\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast} \\\\ \\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r] & 0.}$$\nSince we are interested in identifying when $\\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast}$ and $\\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast}$ are quasi-isomorphisms, we will describe the components of the connecting homomorphism explicitely. In fact, this morphism is determined by two maps on the chain level which we will describe next:\n$$\\alpha: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})[1]$$ and\n$$\\beta: \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})[1].$$\nTo do this, we first make $\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})$ explicit. By definition, $\\mathbf{C}^n_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})$ is given by\n$$\\prod_{(u,A) \\in \\ensuremath{\\mathcal{N}}(\\mathfrak{c}) \\exists u_i \\in \\ensuremath{\\mathcal{S}}}  \\mathrm{Hom}_k( \\mathfrak{b}_{u_{n-1}}(A_{n-1},A_n) \\otimes \\dots \\otimes M_{u_i}(A_i,A_{i+1})  \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_n)).$$\nLet $\\phi \\in \\mathbf{C}^n_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ be a Hochschild cocycle.\nFollowing \\cite{kellerdih} we define $\\alpha(\\phi)$ with non-zero components in \n$$\\mathrm{Hom}_k(M_{u_{n}}(A_n,A_{n+1})  \\dots \\otimes \\mathfrak{a}_{u_0}(A_0, A_1), M_{|u|}(A_0, A_{n+1})).$$\ngiven by \n$$\\alpha(\\phi)(x, a_{n-1}, \\dots, a_0) = x\\phi(a_{n-1}, \\dots, a_0),$$\nand similarly for $\\beta$.\n\n\\begin{lemma}\nThe maps $\\alpha$ and $\\beta$ are chain maps for which\n$$\\begin{pmatrix} \\beta & \\alpha \\end{pmatrix}: \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})[1]$$\ninduces the connecting homomorphisms.\n\\end{lemma}\n\nThe triangle \n\\begin{equation}\\label{triangle}\n\\xymatrix{ {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c})} \\ar[r]_-{\\begin{pmatrix} \\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast} \\\\ \\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast} \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\oplus \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[r]_-{\\begin{pmatrix} \\beta & \\alpha \\end{pmatrix}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})[1]} \\ar[r] &}\\end{equation}\ncorresponds to a homotopy bicartesian square\n$$\\xymatrix{ {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c})} \\ar[d]_{\\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast}} \\ar[r]^{\\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[d]^{\\alpha} \\\\ {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})} \\ar[r]_-{\\beta} & {\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_{\\ensuremath{\\mathcal{S}}})[1].}}$$\n\nWe thus have:\n\n\\begin{proposition}\n\\begin{enumerate}\n\\item If $\\alpha$ is a quasi-isomorphism, then so is $\\varphi^{\\ast}_{\\ensuremath{\\mathcal{V}}}$.\n\\item If $\\beta$ is a quasi-isomorphism, then so is $\\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\subsection{Derived interpretation}\\label{parderived}\n\nFollowing \\cite{kellerdih}, we now give a derived interpretation of the morphisms $\\alpha$ and $\\beta$. \nWe start by giving various derived interpretations of $\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_S$.\n\nLet $B(\\mathfrak{a}) \\longrightarrow 1_{\\mathfrak{a}}$ and $B(\\mathfrak{b}) \\longrightarrow 1_{\\mathfrak{b}}$ be the Bar resolutions in $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$ and $\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})$ respectively, as defined in \\cite[\\S 3.2]{lowenmap}.\nWe have \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\cong \\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})}(B(\\mathfrak{a}), 1_{\\mathfrak{a}}) = \\mathrm{RHom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})}(1_{\\mathfrak{a}}, 1_{\\mathfrak{a}})$$ and \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\cong \\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})}(B(\\mathfrak{b}), 1_{\\mathfrak{b}}) = \\mathrm{RHom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})}(1_{\\mathfrak{b}}, 1_{\\mathfrak{b}}).$$\nSimilarly, we have\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_S)[1] \\cong \\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b})}(B(\\mathfrak{a}) \\otimes_{\\mathfrak{a}} M \\otimes_{\\mathfrak{b}} B(\\mathfrak{b}), M)$$\nand $$\\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{a}, \\mathfrak{b})}(B(\\mathfrak{a}) \\otimes_{\\mathfrak{a}} M \\otimes_{\\mathfrak{b}} B(\\mathfrak{b}), M) = \\mathrm{RHom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{S}}}(\\mathfrak{a}, \\mathfrak{b})}(M,M).$$\nIn particular, from the triangle \\eqref{triangle} we obtain a long exact cohomology sequence\n$$\\dots \\longrightarrow HH_{\\ensuremath{\\mathcal{W}}}^i(\\mathfrak{c}) \\longrightarrow HH_{\\ensuremath{\\mathcal{V}}}^i(\\mathfrak{b}) \\oplus HH_{\\ensuremath{\\mathcal{U}}}^i(\\mathfrak{a}) \\longrightarrow \\mathrm{Ext}^{i}_{\\ensuremath{\\mathsf{Bimod}} _S(\\mathfrak{b}, \\mathfrak{a})}(M,M) \\longrightarrow \\dots$$\n\nBy Proposition \\ref{propadj}, we further have\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_S)[1] \\cong \\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})}(B(\\mathfrak{a}), \\mathrm{Hom}_{\\mathfrak{b}}(M \\otimes_{\\mathfrak{b}} B(\\mathfrak{b}), M))$$\nwith $\\mathrm{Hom}_{\\mathfrak{b}}(M \\otimes_{\\mathfrak{b}} B(\\mathfrak{b}), M) = \\mathrm{RHom}_{\\mathfrak{b}}(M,M).$\nSimilarly, \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{W}}}(\\mathfrak{c}, (1_{\\mathfrak{c}})_S)[1] \\cong \\mathrm{Hom}_{\\ensuremath{\\mathsf{Bimod}} _{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})}(B(\\mathfrak{b}), \\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(B(\\mathfrak{a}) \\otimes_{\\mathfrak{a}} M, M))$$\nwith $\\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(B(\\mathfrak{a}) \\otimes_{\\mathfrak{a}} M, M) = \\mathrm{RHom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M,M)$.\n\nConsider the natural morphisms \n$$\\lambda: 1_{\\mathfrak{a}} \\longrightarrow \\mathrm{Hom}_{\\mathfrak{b}}(M,M) \\longrightarrow \\mathrm{RHom}_{\\mathfrak{b}}(M,M)$$\nand \n$$\\omega: 1_{\\mathfrak{b}} \\longrightarrow \\mathrm{Hom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M,M) \\longrightarrow \\mathrm{RHom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M,M)$$\nof bimodules and the induced \n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, \\lambda): \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, \\mathrm{RHom}_{\\mathfrak{b}}(M,M))$$\nand\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, \\omega): \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, \\mathrm{RHom}_{\\mathfrak{a}}(M,M)).$$\n\nWe have the following map-graded version Theorem \\cite[Theorem 4.1.1]{lowenvandenberghhoch} extracted from \\cite[\\S 4]{kellerdih}:\n\n\\begin{theorem}\\label{thmarrowqis}\n\\begin{enumerate}\n\\item We have $\\alpha \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, \\lambda)$ and $\\beta \\cong \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, \\omega)$.\n\\item If $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}, \\lambda)$ is a quasi-isomorphism (hence, in particular, if $\\lambda$ is a quasi-isomorphism), then so is $\\varphi_{\\ensuremath{\\mathcal{V}}}^{\\ast}$.\n\\item If $\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}, \\omega)$  is a quasi-isomorphism (hence, in particular, if $\\omega$ is a quasi-isomorphism), then so is $\\varphi_{\\ensuremath{\\mathcal{U}}}^{\\ast}$.\n\\end{enumerate}\n\\end{theorem}\n\nThe analogue of \\cite[\\S 4.6, Theorem]{kellerdih} in the map-graded context can be formulated and proven in a similar fashion. In particular, we mention the following compatibility results explicitly.\n\nLet $(\\varphi, F): (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}) \\longrightarrow (\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$ be a subcartesian graded functor with associated $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S^{\\varphi}$ and $\\mathfrak{a}$-$S^{\\varphi}$-$\\mathfrak{b}$-bimodule $M^F$ as in Example \\ref{exbimod}.\nOn the one hand, since $(\\varphi, F)$ is subcartesian, we have the induced morphism of $B_{\\infty}$-algebras\n$$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})$$\nfrom Proposition \\ref{propfunct}.\n\n\\begin{proposition}\\label{propcompat}\n\\begin{enumerate}\n\\item $\\lambda: 1_{\\mathfrak{a}} \\longrightarrow \\mathrm{RHom}_{\\mathfrak{b}}(M^F, M^F)$ and hence $\\varphi^{\\ast}_{\\ensuremath{\\mathcal{V}}}$  is a quasi-isomorphism.\n\\item In the homotopy category of $B_{\\infty}$-algebras, we have $F^{\\ast} = \\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}} {\\varphi^{\\ast}_{\\ensuremath{\\mathcal{V}}}}^{-1}$.\n\\end{enumerate}\n\\end{proposition}\n\nLet $(\\varphi,F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ be a subcartesian graded functor with associated $\\ensuremath{\\mathcal{U}}$-$\\ensuremath{\\mathcal{V}}$-bifunctor $S_{\\varphi}$ and $\\mathfrak{a}$-$S_{\\varphi}$-$\\mathfrak{b}$-bimodule $M_F$ as in Example \\ref{exbimod}. This time we have the functor\n$$F^{\\ast}: \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}).$$\n\n\\begin{proposition}\\label{propcompat2}\n\\begin{enumerate}\n\\item $\\omega: 1_{\\mathfrak{b}} \\longrightarrow \\mathrm{RHom}_{\\mathfrak{a}^{^{\\mathrm{op}}}}(M_F, M_F)$ and hence $\\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}}$ is a quasi-isomorphism.\n\\item In the homotopy category of $B_{\\infty}$-algebras, we have $F^{\\ast} = \\varphi^{\\ast}_{\\ensuremath{\\mathcal{V}}} {\\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}}}^{-1}$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\n\\section{Grothendieck construction}\\label{pargroth}\n\nIn this section, we present a unified framework for constructing map-graded categories and deconstructing their Hochschild complexes. The classical Grothendieck construction from \\cite{SGA1} takes a pseudofunctor $\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ as input and turns it into a category fibered over $\\ensuremath{\\mathcal{U}}$. The main construction from \\cite{lowenmap} is a $k$-linearized version of this construction. Now, we go a step further and start from a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$$\nwhere $\\ensuremath{\\mathcal{C}}$ is a small category and $\\underline{\\ensuremath{\\mathsf{Map}} }$ is the bicategory of map-graded categories and bimodules described in \\S \\ref{parbimod}. Allowing arbitrary bimodules rather than functors between map-graded categories allows us to capture the arrow category with respect to a bimodule from \\S \\ref{pararrow}.\nIn general, we can now deconstruct the Hochschild complex of the Grothendieck construction $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ of $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ based upon the internal structure of $\\ensuremath{\\mathcal{C}}$. Here, the strategy is to cover $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ by other Grothendieck constructions, using base change for pseudofunctors from \\S \\ref{parbase}. For instance, in \\S \\ref{pargenarrow}, we consider ``generalized arrow categories'', and deconstruct them using iterated arrow category constructions.\nIn \\S \\ref{parcoverarrow}, we observe how, in the case where $\\ensuremath{\\mathcal{C}}$ is a poset, the sheaf property for Hochschild complexes on the one hand, and the arrow category construction on the other hand, can be seen as complementary tools for deconstructing Hochschild cohomology. \nIn the final section \\S \\ref{parcomp}, we start from a pseudofunctor $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$, which we compare to the natural pseudofunctor of Grothendieck constructions\n$$(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}: C \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C})$$\nbuilt from the restriction $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ of $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star})$ to $\\ensuremath{\\mathcal{C}}$.\nOur main Theorem \\ref{thmmaincomp} is heavily based upon Keller's arrow category argument in the case of a fully faithful functor $\\mathfrak{b} \\longrightarrow \\mathfrak{a}$, which is in fact a special case of our theorem. As an application, we recover the Mayer-Vietoris triangles for ringed spaces from \\cite[\\S 7.9]{lowenvandenberghhoch}.\n\n\\subsection{Diagrams in the bicategory of categories}\\label{pargroth1}\nLet $\\underline{\\ensuremath{\\mathsf{Cat}} }$ be the bicategory of categories, bifunctors and natural transformations described in \\S \\ref{parbifun}. Let $\\ensuremath{\\mathcal{C}}$ be an arbitrary small category and consider a pseudofunctor\n$$\\ensuremath{\\mathcal{U}}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }: C \\longmapsto \\ensuremath{\\mathcal{U}}_C.$$\nThe pseudofunctor maps a map $c: C \\longrightarrow C'$ to a $\\ensuremath{\\mathcal{U}}_{C'}$-$\\ensuremath{\\mathcal{U}}_{C}$-bifunctor $\\ensuremath{\\mathcal{U}}(c) = S_c$ and for an additional $c': C' \\longrightarrow C''$, there is an isomorphism\n$$\\phi_{c', c}: S_{c'} \\circ S_c \\longrightarrow S_{c'c}.$$\nFor a third map $c'': C'' \\longrightarrow C'''$, we have a commutative diagram\n\\begin{equation}\\label{coh}\n\\xymatrix{ {(S_{c''} \\circ S_{c'}) \\circ S_c} \\ar[d]_{\\alpha} \\ar[r]^-{\\phi_{c'. c} \\circ 1} & {S_{c'' c'} \\circ S_c} \\ar[r]^-{\\phi_{c'' c', c}} & {S_{c'' c' c}} \\ar[d]^1 \\\\\n{S_{c''} \\circ (S_{c'} \\circ S_c)} \\ar[r]_-{1 \\circ \\phi_{c', c}} & {S_{c''} \\circ S_{c' c}} \\ar[r]_-{\\phi_{c'', c' c}} & {S_{c'' c' c}} }\n\\end{equation}\nwhere $\\alpha$ is the isomorphism from the bicategory $\\underline{\\ensuremath{\\mathsf{Cat}} }$. We suppose moreover that $S_{1_{C}} = 1_{\\ensuremath{\\mathcal{U}}_C}$.\n\nWe define the \\emph{Grothendieck construction} of $\\ensuremath{\\mathcal{U}}$ to be the non-linear $\\ensuremath{\\mathcal{C}}$-graded category $\\ensuremath{\\mathcal{U}}$ with $\\ensuremath{\\mathcal{U}}_C$ as prescribed and, for $c: C \\longrightarrow C'$, $U \\in \\ensuremath{\\mathcal{U}}_C$, $U' \\in \\ensuremath{\\mathcal{U}}_{C'}$:\n$$\\ensuremath{\\mathcal{U}}_c(U,U') = S_c(U, U').$$\nRecall that $$S_{c'} \\circ S_c(U,U') = \\coprod_{V \\in \\ensuremath{\\mathcal{U}}_{C'}} S_{c'}(V,U'') \\times S_{c}(U,V)\/ \\sim.$$\nThe composition on $\\ensuremath{\\mathcal{U}}$ is defined as the natural map\n$$\\xymatrix{ {S_{c'}(U', U'') \\times S_{c}(U, U')} \\ar[r] & {S_{c'} \\circ S_{c}(U, U'')} \\ar[r]_-{\\phi_{c', c}} & {S_{c'c}(U, U'')}}$$\nand we denote the image of $(s', s)$ under composition by $s's$.\nThere is a corresponding category $\\tilde{\\ensuremath{\\mathcal{U}}}$ over $\\ensuremath{\\mathcal{C}}$ with $\\mathrm{Ob}(\\tilde{\\ensuremath{\\mathcal{U}}}) = \\coprod_{C \\in \\ensuremath{\\mathcal{C}}}\\ensuremath{\\mathcal{U}}_C$ (see Remark \\ref{remgraded}).\n\n\n\\subsection{Diagrams in the bicategory of map-graded categories}\\label{pargroth2}\nThe construction from \\S \\ref{pargroth1} can be extended in the following way. Let $\\underline{\\ensuremath{\\mathsf{Map}} }$ be the bicategory of map-graded categories, bimodules and their morphisms described in \\S \\ref{parbimod}. Let $\\ensuremath{\\mathcal{C}}$ be an arbitrary small category and consider a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C).$$\nThe pseudofunctor maps $c: C \\longrightarrow C'$ to an $\\mathfrak{a}_{C'}$-$\\mathfrak{a}_{C}$-bimodule $(\\ensuremath{\\mathcal{U}}(c) = S_c, \\mathfrak{a}(c) = M_c)$ and for an additional $c': C' \\longrightarrow C''$ there is and isomorphism\n$$(\\phi_{c', c}, F_{c',c}): (S_{c'} \\circ S_{c}, M_{c'} \\otimes_{\\mathfrak{a}_{C'}} M_c) \\longrightarrow (S_{c' c}, M_{c' c}).$$ \nThese isomorphisms satisfy the natural coherence axiom similar to \\eqref{coh} and $(S_{1_C}, M_{1_C}) = (1_{\\ensuremath{\\mathcal{U}}_C}, 1_{\\mathfrak{a}_C})$.\n\nClearly, there is an underlying pseudofunctor\n$$\\ensuremath{\\mathcal{U}}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }: C \\longmapsto \\ensuremath{\\mathcal{U}}_C$$\nwhich determines a non-linear $\\ensuremath{\\mathcal{C}}$-graded category $\\ensuremath{\\mathcal{U}}$ and associated category $\\tilde{\\ensuremath{\\mathcal{U}}}$. We define the \\emph{Grothendieck construction} of $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ to be the following $\\tilde{\\ensuremath{\\mathcal{U}}}$-graded category $\\tilde{\\mathfrak{a}}$. For $U \\in \\ensuremath{\\mathcal{U}}_C$, we define $$\\tilde{\\mathfrak{a}}_U = (\\mathfrak{a}_C)_U.$$ For a morphism $(c: C \\longrightarrow C', s \\in S_c(U,U'))$ in $\\tilde{\\ensuremath{\\mathcal{U}}}$, with $U \\in \\ensuremath{\\mathcal{U}}_C$, $U' \\in \\ensuremath{\\mathcal{U}}_{C'}$, and for $A \\in (\\mathfrak{a}_C)_U$, $A' \\in (\\mathfrak{a}_{C'})_{U'}$, we put\n$$\\tilde{\\mathfrak{a}}_{(c,s)}(A, A') = (M_c)_s(A,A').$$\nRecall that $$(M_{c'} \\otimes_{\\mathfrak{a}_{C'}} M_c)_r(A,A'') = \\oplus_{r = [(s',s)], B} (M_{c'})_{s'}(B,A'') \\otimes (M_c)_s(A,B)\/\\sim.$$\nThe composition on $\\tilde{\\mathfrak{a}}$ is defined as the composition of the natural map\n$$(M_{c'})_{s'}(A', A'') \\otimes (M_c)_s(A,A') \\longrightarrow (M_{c'} \\otimes_{\\mathfrak{a}_{C'}} M_c)_{[(s', s)]}(A,A'')$$\nfollowed by the map\n$$\\xymatrix{ {(M_{c'} \\otimes_{\\mathfrak{a}_{C'}} M_c)_{[(s', s)]}(A,A'')} \\ar[r]_-{F_{c', c}} & {(M_{c' c})_{s' s}(A,A'').} }$$\n\n\n\\subsection{Diagrams in the bicategory of linear categories}\\label{pardiaglin}\nIn many applications, we are in a somewhat simplified situation from \\S \\ref{pargroth2}. Let $\\underline{\\ensuremath{\\mathsf{Cat}} }(k)$ be the bicategory of $k$-linear functors and bimodules. There is a natural map $\\underline{\\ensuremath{\\mathsf{Cat}} }(k) \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: \\mathfrak{a} \\longmapsto \\mathfrak{a}_{tr}$ where $\\mathfrak{a}_{tr}$ is the trivially graded category over $e$ from Example \\ref{exstandgr} (2).\nConsider a pseudofunctor\n$$\\mathfrak{a}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }(k): C \\longmapsto \\mathfrak{a}_C.$$\nIf we consider the corresponding $(\\ensuremath{\\mathcal{E}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: C \\longmapsto (e, \\mathfrak{a}_C)$ we see that there is a canonical isomorphism $\\tilde{\\ensuremath{\\mathcal{E}}} \\cong \\ensuremath{\\mathcal{C}}$ and carrying out the construction from \\S \\ref{pargroth2}, we thus obtain a $\\ensuremath{\\mathcal{C}}$-graded category $\\tilde{\\mathfrak{a}}$ with\n$$\\tilde{\\mathfrak{a}}_C = \\mathfrak{a}_C.$$\nFor a morphism $c: C \\longrightarrow C'$ in $\\ensuremath{\\mathcal{C}}$ and for $A \\in \\mathfrak{a}_C$, $A' \\in \\mathfrak{a}_{C'}$, we have\n$$\\tilde{\\mathfrak{a}}_c(A,A') = M_c(A,A')$$\nand composition is defined in the obvious way.\n\nConversely, let $\\mathfrak{a}$ be an arbitrary $\\ensuremath{\\mathcal{U}}$-graded category. We define the naturally associated pseudofunctor\n$$\\ensuremath{\\mathcal{A}}: \\ensuremath{\\mathcal{U}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }(k): U \\longrightarrow \\ensuremath{\\mathcal{A}}_U$$\nwhere $\\ensuremath{\\mathcal{A}}_U$ is the category with $\\mathrm{Ob}(\\ensuremath{\\mathcal{A}}_U) = \\mathfrak{a}_U$ and $\\ensuremath{\\mathcal{A}}(A,A') = \\mathfrak{a}_{1_U}(A,A')$ and where for $u: U \\longrightarrow U'$ in $\\ensuremath{\\mathcal{U}}$, $\\ensuremath{\\mathcal{A}}(u) = M_u$ is the natural $\\ensuremath{\\mathcal{A}}_{U'}$-$\\ensuremath{\\mathcal{A}}_U$-bimodule with\n$$M_u(A,A') = \\mathfrak{a}_u(A,A').$$\nThen $(\\ensuremath{\\mathcal{U}},\\mathfrak{a}) \\cong (\\ensuremath{\\mathcal{U}}, \\tilde{\\ensuremath{\\mathcal{A}}})$ and hence the correspondence between \\emph{fibered} $\\ensuremath{\\mathcal{U}}$-graded categories and pseudofunctors $\\ensuremath{\\mathcal{U}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} (k)$ naturally extends to a correspondence between \\emph{all} $\\ensuremath{\\mathcal{U}}$-graded categories and pseudofunctors $\\ensuremath{\\mathcal{U}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }(k)$.\n\n\n\\subsection{Diagrams in the category of map-graded categories}\nAnother specification of the setup from \\S \\ref{pargroth2} occurs if we consider a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}}^{^{\\mathrm{op}}} \\longrightarrow \\ensuremath{\\mathsf{Map}} : C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$$\nmapping a morphism $c: C \\longrightarrow C'$ in $\\ensuremath{\\mathcal{C}}$ to a map-graded functor\n$$(\\varphi_c, F_c): (\\ensuremath{\\mathcal{U}}_{C'}, \\mathfrak{a}_{C'}) \\longrightarrow (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C).$$\nBy Example \\ref{exbimod}(2), this gives rise to an $\\ensuremath{\\mathcal{U}}_{C'}$-$\\ensuremath{\\mathcal{U}}_C$-bimodule $S^{\\varphi_c}$ and a $\\mathfrak{a}_{C'}$-$S^{\\varphi_c}$-$\\mathfrak{a}_C$-bimodule $M^{F_c}$ with\n$$\\begin{aligned}\nS^{\\varphi_c}(U,U') = \\ensuremath{\\mathcal{U}}_{C}(U, \\varphi_c(U')); &&& M^{F_c}_s(A,A') = {\\mathfrak{a}_C}_s(A, F_c(A')).\n\\end{aligned}$$\nIn fact, there are natural pseudofunctors $\\ensuremath{\\mathsf{Cat}} ^{^{\\mathrm{op}}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }$ and $\\ensuremath{\\mathsf{Map}} ^{^{\\mathrm{op}}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$ so that we obtain a composed pseudofunctor \n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }: C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C).$$\nFor $c: C \\longrightarrow C'$, $c': C' \\longrightarrow C''$ in $\\ensuremath{\\mathcal{C}}$, we are given natural isomorphisms\n$$(\\eta, \\theta): (\\varphi_c \\varphi_{c'}, F_c F_{c'}) = (\\varphi_c, F_c)(\\varphi_{c'}, F_{c'}) \\longrightarrow (\\varphi_{c' c}, F_{c' c})$$\nsatisfying the natural coherence axiom similar to \\eqref{coh}.\nFrom this we obtain natural isomorphisms\n$$(S^{\\varphi_{c'}} \\circ S^{\\varphi_c}, M^{F_{c'}} \\otimes_{\\mathfrak{a}_{C'}} M^{F_c}) \\longrightarrow (S^{\\varphi_{c' c}}, M^{F_{c' c}})$$\nin the following way. \nOn the level of bifunctors, we have\n$$S^{\\varphi_{c'}} \\circ S^{\\varphi_c}(U, U'') = \\coprod_{V \\in \\ensuremath{\\mathcal{U}}_{C'}}\\ensuremath{\\mathcal{U}}_{C'}(V, \\varphi_{c'}(U'')) \\times \\ensuremath{\\mathcal{U}}_{C}(U, \\varphi_c(V))\/ \\sim$$\n$$\\xymatrix{ {} \\ar[r]_{\\varphi_c \\times 1} & {}}$$\n$$\\coprod_{V \\in \\ensuremath{\\mathcal{U}}_{C'}} \\ensuremath{\\mathcal{U}}_{C}(\\varphi_c(V), \\varphi_c \\varphi_{c'}(U'')) \\times \\ensuremath{\\mathcal{U}}_{C}(U, \\varphi_c(V))\/ \\sim$$\n$$\\xymatrix{ {} \\ar[r]_-{m_{\\ensuremath{\\mathcal{U}}_{C}}} && {\\ensuremath{\\mathcal{U}}_C(U, \\varphi_c \\varphi_{c'}(U''))}}$$\n$$\\xymatrix{ {} \\ar[r]_-{m_{\\ensuremath{\\mathcal{U}}_C}(\\eta_{U''}, -)} && {\\ensuremath{\\mathcal{U}}_C(U, \\varphi_{c' c}(U'')) = S^{\\varphi_{c' c}}(U, U'').}}$$\nSimilarly, on the level of bimodules we have\n$${M^{F_{c'}} \\otimes_{\\mathfrak{a}_{C'}} M^{F_c}_r(A, A'') = \\oplus_{r = [(s', s)], B} (\\mathfrak{a}_{C'})_{s'}(B, F_{c'}(A'')) \\otimes (\\mathfrak{a}_C)_s(A, F_c(B))\/ \\sim}$$\n$$\\xymatrix{ {} \\ar[r]_{F_c \\otimes 1} & {}}$$\n$${\\oplus_{r = [(s', s)], B} (\\mathfrak{a}_{C})_{\\varphi_c(s')}(F_c(B), F_c F_{c'}(A'')) \\otimes (\\mathfrak{a}_C)_s(A, F_c(B))\/ \\sim}$$\n$$\\xymatrix{ {} \\ar[r]_-{m_{\\mathfrak{a}_C}} && {(\\mathfrak{a}_C)_{\\varphi_c(s') s}(A, F_c F_{c'}(A''))}}$$\n$$\\xymatrix{ {} \\ar[r]_-{m_{\\mathfrak{a}_C}(\\theta_{A''}, -)} && {(\\mathfrak{a}_C)_{\\varphi_C(s') s}(A, F_{c' c}(A'')) = M^{F_{c' c}}_{s' s}(A, A'')}}$$\nThese maps determine the composition on the Grothendieck construction $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$.\n\n\\begin{remark}\nNote that the non-linear $\\ensuremath{\\mathcal{C}}$-graded category $\\tilde{\\ensuremath{\\mathcal{U}}}$ is the original Grothendieck construction of $\\ensuremath{\\mathcal{U}}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} $ in the sense of Grothendieck. \n\\end{remark}\n\n\\begin{remark}\\label{remcatk}\nFor a pseudofunctor $\\mathfrak{a}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} (k)$ landing in the $2$-category of $k$-linear categories, functors and natural transformations, composing with $\\ensuremath{\\mathsf{Cat}} (k) \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }(k)$ brings us in the situation of \\S \\ref{pardiaglin}. The Grothendieck construction $\\tilde{\\mathfrak{a}}$ is the one we used in \\cite{lowenmap} in order to turn a presheaf of $k$-algebras, or more generally a pseudofunctor $\\mathfrak{a}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Cat}} (k)$, into a $\\ensuremath{\\mathcal{C}}$-graded category and define its structured Hochschild complex $\\mathbf{C}_{\\ensuremath{\\mathcal{C}}}(\\tilde{\\mathfrak{a}})$. By \\cite{lowenvandenberghCCT}, this complex computes the natural Hochschild cohomology of $\\mathfrak{a}$ generalized from Gerstenhaber and Schack's Hochschild cohomology of presheaves of algebras \\cite{gerstenhaberschack2} \\cite{gerstenhaberschack1}.\n\\end{remark}\n\n\\begin{remark}\nLet $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}} $ be a pseudofunctor. In a completely similar fashion, the $\\ensuremath{\\mathcal{U}}_{C'}$-$\\ensuremath{\\mathcal{U}}_{C}$-bifunctors $S_{\\varphi_c}$ and $\\mathfrak{a}_{C'}$-$S_{\\varphi_c}$-$\\mathfrak{a}_{C}$-bimodules $M_{F_c}$ from Example \\ref{exbimod}(1) give rise to a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}', \\mathfrak{a}'): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}}  \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }.$$\n\\end{remark}\n\n\\subsection{Base change}\\label{parbase}\n\nLet $\\ensuremath{\\mathcal{U}}: \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }$ be a pseudofunctor as in \\ref{pargroth1} and let $\\Phi: \\ensuremath{\\mathcal{D}} \\longrightarrow \\ensuremath{\\mathcal{C}}$ be an arbitrary functor. There is a resulting composed pseudofunctor\n$$\\ensuremath{\\mathcal{U}}^{\\Phi}: \\ensuremath{\\mathcal{D}} \\longrightarrow \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Cat}} }: D \\longmapsto \\ensuremath{\\mathcal{U}}_{\\Phi(D)}$$\nwith an associated category $\\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi}$.\nThe natural maps\n$$\\mathrm{Ob}(\\tilde{\\ensuremath{\\mathcal{U}}}^{\\phi}) = \\coprod_{D \\in \\ensuremath{\\mathcal{D}}} \\ensuremath{\\mathcal{U}}_{\\Phi(D)} \\longrightarrow \\coprod_{C \\in \\ensuremath{\\mathcal{C}}} \\ensuremath{\\mathcal{U}}_C = \\mathrm{Ob}(\\tilde{\\ensuremath{\\mathcal{U}}})$$\nand\n$$\\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi}((D,U), (D', U')) \\longrightarrow  \\tilde{\\ensuremath{\\mathcal{U}}}((\\Phi(D), U), (\\Phi(D'), U))$$ given by\n$$\\coprod_{d \\in \\ensuremath{\\mathcal{D}}(D,D')} S_{\\Phi(d)}(U,U') \\longrightarrow \\coprod_{c \\in \\ensuremath{\\mathcal{C}}(\\Phi(D), \\Phi(D'))} S_c(U,U')$$\ngive rise to a functor \\label{phi}\n\\begin{equation}\n\\varphi: \\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi} \\longrightarrow \\tilde{\\ensuremath{\\mathcal{U}}}.\n\\end{equation}\n\n\\begin{proposition}\\label{proppseudoinj}\nLet $\\ensuremath{\\mathcal{U}}$ and $\\Phi$ be as above. If $\\ensuremath{\\mathcal{N}}_1(\\Phi)$ is injective, then so is $\\ensuremath{\\mathcal{N}}_1(\\varphi)$.\n\\end{proposition}\n\n\\begin{proposition}\\label{proppseudocover}\nLet $\\ensuremath{\\mathcal{U}}$ be as above and consider a collection $(\\Phi_i: \\ensuremath{\\mathcal{D}}_i \\longrightarrow \\ensuremath{\\mathcal{C}})_{i \\in I}$ of functors. If, for $n \\in \\mathbb{N} \\cup \\{ \\infty \\}$, the collection constitutes an $n$-cover in $\\ensuremath{\\mathsf{Cat}} $, then so does the collection $(\\tilde{\\Phi}_i: \\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi_i} \\longrightarrow \\tilde{\\ensuremath{\\mathcal{U}}})_{i \\in I}$.\n\\end{proposition}\n\nNow let $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$ be a pseudofunctor as in \\S \\ref{pargroth2} and $\\Phi: \\ensuremath{\\mathcal{D}} \\longrightarrow \\ensuremath{\\mathcal{C}}$ a functor. We now obtain a composed pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}^{\\Phi}, \\mathfrak{a}^{\\Phi}): \\ensuremath{\\mathcal{D}} \\longrightarrow \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$$\nwith an associated $\\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi}$-graded category $\\tilde{\\mathfrak{a}}^{\\Phi}$.\n\n\\begin{proposition}\nWith the notations of Proposition \\ref{propcart}, we have $$\\tilde{\\mathfrak{a}}^{\\Phi} = (\\tilde{\\mathfrak{a}})^{\\varphi}.$$\nIn particular there is a natural cartesian map-graded functor $$\\tilde{\\Phi} = (\\varphi, \\delta^{\\varphi, \\tilde{\\mathfrak{a}}}) : (\\tilde{\\ensuremath{\\mathcal{U}}}^{\\Phi}, \\tilde{\\mathfrak{a}}^{\\Phi}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}).$$\n\\end{proposition}\n\nNext we use base changes to transform a pseudofunctor $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$ into a functor\n$$(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} $$\nof Grothendieck constructions.\nWe first define the category $\\ensuremath{\\mathcal{C}}^{\\ast}$ to be the arrow category $\\ensuremath{\\mathcal{C}} \\rightarrow_S e$ for the unique $e$-$\\ensuremath{\\mathcal{C}}$-bifunctor $S$ with $S(C, \\ast) = \\{ \\ast \\}$. \nFor every $C \\in \\ensuremath{\\mathcal{C}}$, we obtain a composition\n$$(\\ensuremath{\\mathcal{U}}|_C, \\mathfrak{a}|_C): \\ensuremath{\\mathcal{C}}\/C \\longrightarrow \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$$\nand a cartesian map-graded functor\n$(\\tilde{\\ensuremath{\\mathcal{U}}}|_C, \\tilde{\\mathfrak{a}}|_C) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$. For every $c: C' \\longrightarrow C$ in $\\ensuremath{\\mathcal{C}}$ we have a natural functor $\\ensuremath{\\mathcal{C}}\/C' \\longrightarrow \\ensuremath{\\mathcal{C}}\/C$ and an induced cartesian map-graded functor\n$(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C'}, \\tilde{\\mathfrak{a}}|_{C'}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C})$ and it is not hard to organize these data into a functor\n$$(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}: C \\longmapsto (\\tilde{\\ensuremath{\\mathcal{U}}}|_C, \\tilde{\\mathfrak{a}}|_C)$$\nwhere we define $(\\tilde{\\ensuremath{\\mathcal{U}}}|_{\\ast}, \\tilde{\\mathfrak{a}}|_{\\ast}) = (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$.\n\n\\begin{proposition} \\label{propcstar} Suppose $\\ensuremath{\\mathcal{C}}$ has finite products.\nLet $(C_i)_{i \\in I}$ be a collection of objects in $\\ensuremath{\\mathcal{C}}$ such that for every $C \\in \\ensuremath{\\mathcal{C}}$ there exists a map $C \\longrightarrow C_i$ for some $i$. The composed functor\n$$\\mathbf{C}: \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc} \\longrightarrow B_{\\infty}: C \\longmapsto \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_C}(\\tilde{\\mathfrak{a}}|_C)$$\nsatisfies the sheaf property with respect to the collection of maps $(C_i \\longrightarrow \\ast)_{i \\in I}$ in $\\ensuremath{\\mathcal{C}}^{\\ast}$.\n\\end{proposition}\n\n\\begin{proof}\nBy Example \\ref{excover}(2), the collection $(\\ensuremath{\\mathcal{C}}\/C_i \\longrightarrow \\ensuremath{\\mathcal{C}})_{i \\in I}$ constitutes an $\\infty$-cover of $\\ensuremath{\\mathcal{C}}$ in $\\ensuremath{\\mathsf{Cat}} $. By proposition \\ref{proppseudocover}, the induced restriction maps in $\\ensuremath{\\mathsf{Map}} _{sc}$ also form an $\\infty$-cover. Further, it is readily seen that the pullback of $C_i \\longrightarrow \\ast$ and $C_j \\longrightarrow \\ast$ in $\\ensuremath{\\mathcal{C}}^{\\ast}$ is given by $C_i \\times C_j \\longrightarrow \\ast$ for the product $C_i \\times C_j$ in $\\ensuremath{\\mathcal{C}}$. The pullback of $(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C_i}, \\tilde{\\mathfrak{a}}|_{C_i}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ and $(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C_j}, \\tilde{\\mathfrak{a}}|_{C_j}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ is given by\n$(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C_i \\times C_j}, \\tilde{\\mathfrak{a}}|_{C_i \\times C_j}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}).$ Thus, the result follows from Theorem \\ref{mainsheaf}.\n\\end{proof}\n\n\n\n\n\\subsection{Generalized arrow categories}\\label{pargenarrow}\nWe can cast the arrow category construction from \\S \\ref{pararrow} in the setup from \\S \\ref{pargroth2}. Consider the  path category \n$$\\ensuremath{\\mathcal{C}}' = \\langle \\xymatrix{ {0} \\ar[r]_{\\leq} & {1} } \\rangle$$\nWith the notations of \\S \\ref{pararrow}, we obtain a pseudofunctor\n$$(\\ensuremath{\\mathcal{W}}', \\mathfrak{c}'): \\ensuremath{\\mathcal{C}}' \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$$\nwith \n$$\\begin{aligned} \\ensuremath{\\mathcal{W}}'_0 = \\ensuremath{\\mathcal{V}} &&& \\ensuremath{\\mathcal{W}}'_1 = \\ensuremath{\\mathcal{U}} &&& \\ensuremath{\\mathcal{W}}'(0 \\leq 1) = S \\\\ \\mathfrak{c}'_0 = \\mathfrak{b} &&& \\mathfrak{c}'_1 = \\mathfrak{a} &&& \\mathfrak{c}'(0 \\leq 1) = M \\end{aligned}$$\nThen the Grothendieck constructions amounts to arrow categories:\n$$\\begin{aligned} \\tilde{\\ensuremath{\\mathcal{W}}}' = (\\ensuremath{\\mathcal{V}} \\rightarrow_{S} \\ensuremath{\\mathcal{U}}); &&& \\tilde{\\mathfrak{c}}' = (\\mathfrak{b} \\rightarrow_{M} \\mathfrak{a}). \\end{aligned}$$\n\nWe can generalize the arrow category in the following way. Consider the path category\n$$\\ensuremath{\\mathcal{C}} = \\langle \\xymatrix{ 0 \\ar[r]_{\\leq} & 1 \\ar[r]_{\\leq} & \\dots \\ar[r]_-{\\leq} & {n-1} \\ar[r]_-{\\leq} & n} \\rangle.$$\nWe will call a category isomorphic to a such a category $\\ensuremath{\\mathcal{C}}$ for some $n$ a \\emph{chain category}.\nThen a pseudofunctor\n$$(\\ensuremath{\\mathcal{W}}, \\mathfrak{c}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$$\ntakes values \n$$\\begin{aligned} \\ensuremath{\\mathcal{W}}_i &&& \\ensuremath{\\mathcal{W}}(i \\leq j) = S_{ij} \\\\ \\mathfrak{c}_i &&& \\mathfrak{c}(i \\leq j) = M_{ij} \\end{aligned}$$\nand is equipped with isomorphisms\n$$(\\phi_{kji}, F_{kji}): (S_{jk} \\circ S_{ij}, M_{jk} \\otimes_{\\mathfrak{a}_j} M_{ij}) \\longrightarrow (S_{ik}, M_{ik}).$$\n\n\\begin{example}\nChoosing arbitrary graded categories $(\\ensuremath{\\mathcal{W}}_i, \\mathfrak{c}_i)$, bifunctors $S_{i, i+1}$ and bimodules $M_{i, i+1}$, it is possible to define the remaining data as generalized compositions and tensor products\n$$S_{ij} = S_{j-1,j} \\circ \\dots \\circ S_{i, i+1}$$and\n$$M_{ij} = M_{j-1,j} \\otimes \\dots \\otimes M_{i, i+1}.$$\n\\end{example}\n\nNow let $\\Phi^{ij}: \\ensuremath{\\mathcal{C}}^{ij} \\subseteq \\ensuremath{\\mathcal{C}}$ be the subcategory \n$$\\ensuremath{\\mathcal{C}}^{ij} = \\langle \\xymatrix{ i \\ar[r]_-{\\leq} & i+1 \\ar[r]_{\\leq} & \\dots \\ar[r]_{\\leq} & {j-1} \\ar[r]_-{\\leq} & j} \\rangle$$\nand put $(\\ensuremath{\\mathcal{W}}^{ij}, \\mathfrak{c}^{ij}) = (\\ensuremath{\\mathcal{W}}^{\\Phi^{ij}}, \\mathfrak{c}^{\\Phi^{ij}})$.\nWe thus obtain natural injective cartesian map-graded functors $\\tilde{\\Phi}^{ij}: (\\tilde{\\ensuremath{\\mathcal{W}}}^{ij}, \\tilde{\\mathfrak{c}}^{ij}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}})$ as in \\S \\ref{parbase}. \n\n\nOn $\\ensuremath{\\mathcal{C}}^{01}$, we can define a new pseudofunctor \n$$(\\ensuremath{\\mathcal{W}}', \\mathfrak{c}'): \\ensuremath{\\mathcal{C}}^{01} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$$\nwith\n$$\\begin{aligned} \\ensuremath{\\mathcal{W}}'_0 =  \\ensuremath{\\mathcal{W}}_0 &&& \\ensuremath{\\mathcal{W}}'_1 = \\ensuremath{\\mathcal{W}}^{1n} &&& \\ensuremath{\\mathcal{W}}(0 \\leq 1) = S \\\\ \\mathfrak{c}'_0 = \\mathfrak{c}_0 &&& \\mathfrak{c}'_1 = \\mathfrak{c}^{1n} &&& \\mathfrak{c}'(0 \\leq 1) = M \\end{aligned}$$\nwhere $S$ is the natural $\\tilde{\\ensuremath{\\mathcal{W}}}^{1n}$-$\\ensuremath{\\mathcal{W}}_0$-bimodule and $M$ the natural $\\tilde{\\mathfrak{c}}^{ij}$-$S$-$\\mathfrak{c}_0$ bimodule determined by $\\tilde{\\Phi}^{1n}$, $\\tilde{\\Phi}^{00}$ and the identity bimodule on $(\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}})$ (we have $(\\tilde{\\ensuremath{\\mathcal{W}}}^{00}, \\tilde{\\mathfrak{c}}^{00}) \\cong (\\ensuremath{\\mathcal{W}}_0, \\mathfrak{c}_0)$). Thus, we have\n$$S(0, i) = S_{0i}; \\hspace{1,5cm} M(0,i) = M_{0i}.$$\nIt is readily seen, for instance using the analysis from \\S \\ref{pararrowthin}, that\n$$(\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}}') \\cong (\\tilde{\\ensuremath{\\mathcal{W}}}', \\tilde{\\mathfrak{c}}').$$\nBy induction on $n$, we can thus apply the results of \\S \\ref{pararrow}.\n\nAlternatively, we can take a global approach. Let $\\Phi: \\mathrm{ob}\\ensuremath{\\mathcal{C}} \\subseteq \\ensuremath{\\mathcal{C}}$ be the subcategory consisting of the objects and identity morphisms of $\\ensuremath{\\mathcal{C}}$. \nThere is a corresponding injective cartesian morphism $\\tilde{\\Phi}: (\\tilde{\\ensuremath{\\mathcal{W}}}^{\\Phi}, \\tilde{\\mathfrak{c}}^{\\Phi}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}})$. For every individual object $i \\in \\ensuremath{\\mathcal{C}}$ we have a futher subcategory $\\Phi^i: i \\subseteq \\mathrm{ob}\\ensuremath{\\mathcal{C}}$ corresponding to $\\tilde{\\Phi}^i: (\\ensuremath{\\mathcal{W}}_i, \\mathfrak{c}_i) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{W}}}^{\\Phi}, \\tilde{\\mathfrak{c}}^{\\Phi})$.\nClearly, the complement of $\\tilde{\\ensuremath{\\mathcal{W}}}^{\\Phi} \\subseteq \\tilde{\\ensuremath{\\mathcal{W}}}$ constitutes an ideal which we denote by $\\ensuremath{\\mathcal{S}}$. \nFrom \\eqref{eqhoch3} we thus obtain an exact sequence of Hochschild complexes:\n$$\\xymatrix{ 0 \\ar[r] & {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}}(\\tilde{\\mathfrak{c}}, (1_{\\tilde{\\mathfrak{c}}})_{\\ensuremath{\\mathcal{S}}})} \\ar[r] & {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}}(\\tilde{\\mathfrak{c}})} \\ar[r] & {\\oplus_{i = 0}^n \\mathbf{C}_{\\ensuremath{\\mathcal{W}}_i}(\\mathfrak{c}_i)} \\ar[r] & 0}$$\n\nNow consider a general pseudofunctor $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$. We can use the internal structure of $\\ensuremath{\\mathcal{C}}$ in order to describe $(\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ as an arrow category. \nLet $\\ensuremath{\\mathcal{Z}}$ be a thin ideal in $\\ensuremath{\\mathcal{C}}$ with associated subcategories $\\ensuremath{\\mathcal{C}}_0 \\subseteq \\ensuremath{\\mathcal{C}}$ of ``objects below $\\ensuremath{\\mathcal{Z}}$'' and $\\ensuremath{\\mathcal{C}}_1$ of ``objects above $\\ensuremath{\\mathcal{Z}}$'' as described in \\S \\ref{pararrowthin} and suppose $\\mathrm{Ob}(\\ensuremath{\\mathcal{C}}) = \\mathrm{Ob}(\\ensuremath{\\mathcal{C}}_0) \\cup \\mathrm{Ob}(\\ensuremath{\\mathcal{C}}_1)$ such that $\\ensuremath{\\mathcal{C}} \\cong (\\ensuremath{\\mathcal{C}}_0 \\rightarrow_Z \\ensuremath{\\mathcal{C}}_1)$ for the natural restriction of $1_{\\ensuremath{\\mathcal{C}}}$ to a $\\ensuremath{\\mathcal{C}}_1$-$\\ensuremath{\\mathcal{C}}_0$-bifunctor $Z$.\nThe inclusions $\\ensuremath{\\mathcal{C}}_0 \\subseteq \\ensuremath{\\mathcal{C}}$ and $\\ensuremath{\\mathcal{C}}_1 \\subseteq \\ensuremath{\\mathcal{C}}$ give rise to injective cartesian graded functors\n$$(\\tilde{\\ensuremath{\\mathcal{U}}}_0, \\tilde{\\mathfrak{a}}_0) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}); \\hspace{1,5cm} (\\tilde{\\ensuremath{\\mathcal{U}}}_1, \\tilde{\\mathfrak{a}}_1) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}).$$\nConsider the  natural restriction of $1_{\\tilde{\\ensuremath{\\mathcal{U}}}}$ to a  $\\tilde{\\ensuremath{\\mathcal{U}}}_1$-$\\tilde{\\ensuremath{\\mathcal{U}}}_0$-bifunctor $T$ and the natural restriction of $1_{\\tilde{\\mathfrak{a}}}$ to a $\\tilde{\\mathfrak{a}}_1$-$T$-$\\tilde{\\mathfrak{a}}_0$-bimodule $N$. \nPut $\\tilde{\\ensuremath{\\mathcal{Z}}} = \\coprod S_z(U',U)$ running over $z: C' \\longrightarrow C$ in $\\ensuremath{\\mathcal{Z}}$, $U \\in \\ensuremath{\\mathcal{U}}_C$, $U' \\in \\ensuremath{\\mathcal{U}}_{C'}$.\n\n\\begin{proposition}\\label{proparrowc}\nThe thin ideal $\\tilde{\\ensuremath{\\mathcal{Z}}}$ in $\\tilde{\\ensuremath{\\mathcal{U}}}$ gives rise to isomorphisms\n$$\\tilde{\\ensuremath{\\mathcal{U}}} \\cong (\\tilde{\\ensuremath{\\mathcal{U}}}_0 \\rightarrow_T \\tilde{\\ensuremath{\\mathcal{U}}}_1); \\hspace{1,5cm} \\tilde{\\mathfrak{a}} \\cong (\\tilde{\\mathfrak{a}}_0 \\rightarrow_N \\tilde{\\mathfrak{a}_1}).$$\n\\end{proposition}\n\n\\begin{remark}\nNote that a generalized arrow category over\n$$\\ensuremath{\\mathcal{C}} = \\langle \\xymatrix{ 0 \\ar[r]_{\\leq} & 1 \\ar[r]_{\\leq} & 2} \\rangle.$$\nis used in the proof of \\cite[\\S 4.6, Theorem (d)]{kellerdih}.\n\\end{remark}\n\n\n\\subsection{Covers by arrow categories}\\label{parcoverarrow}\nThe example from \\S \\ref{pargenarrow} suggests how we can view some map-graded categories as being assembled from the primary arrow category construction, which can be seen as a certain way of glueing categories by means of a bimodule. Another way to glue map-graded categories is along covers of of underlying grading categories as described in \\S \\ref{parstackmap}. More generally, we obtained a sheaf of Hochschild complexes on $\\ensuremath{\\mathsf{Map}} _{sc}$ in \\S \\ref{parsheafhoch} and in particular Mayer-Vietoris sequences in \\S \\ref{parMV}. In this section we explain how these different  tools can be combined.\n\nLet $(\\ensuremath{\\mathcal{C}}, \\leq)$ be (the category associated to) a finite poset. For elements $a, b \\in \\ensuremath{\\mathcal{C}}$, we denote $a \\sqsubset b$ if $a \\leq b$ and if $a \\leq c \\leq b$ for $c \\in \\ensuremath{\\mathcal{C}}$ implies $a = c$ of $c = b$. Let $t_0, \\dots, t_n$ be the maximal elements of $\\ensuremath{\\mathcal{C}}$ and $s_0, \\dots, s_m$ the minimal elements. For every composition chain \n$$s_i = c_0 \\sqsubset c_1 \\sqsubset \\dots c_k \\sqsubset c_{k+1} \\dots \\sqsubset c_p = t_j.$$\nwe obtain a generated subcategory which is a chain category in the sense of \\S \\ref{pargenarrow}. \nEvery chain of elements\n$$a_0 \\leq a_1 \\leq a_2 \\leq \\dots \\leq a_l$$\ncan be refined and fitted into a composition chain\n$$s_i = c_0 \\sqsubset \\dots \\sqsubset c_n = a_0  \\sqsubset  \\dots  \\sqsubset c_{m} = a_l \\sqsubset c_{m+1} \\dots \\sqsubset c_p = t_j$$\ncontaining all the elements $a_k$. Thus, the collection of subcategories $\\ensuremath{\\mathcal{D}} \\subseteq \\ensuremath{\\mathcal{C}}$ generated by composition chains of $\\ensuremath{\\mathcal{C}}$ constitutes an $\\infty$-cover of $\\ensuremath{\\mathcal{C}}$.\nClearly, the intersection of two such categories is itself a chain category, of strictly smaller length.\n\nLet $(\\Phi_i: \\ensuremath{\\mathcal{D}}_i \\subseteq \\ensuremath{\\mathcal{C}})_{i \\in I}$ be the collection of chain categories generated by composition chains of $\\ensuremath{\\mathcal{C}}$. Put $\\ensuremath{\\mathcal{D}}_{ij} = \\ensuremath{\\mathcal{D}}_i \\cap \\ensuremath{\\mathcal{D}}_j$ and $\\Phi_{ij}: \\ensuremath{\\mathcal{D}}_{ij} \\longrightarrow \\ensuremath{\\mathcal{C}}$, $\\Phi_{ij}^i: \\ensuremath{\\mathcal{D}}_{ij} \\longrightarrow \\ensuremath{\\mathcal{D}}_i$.\n\nNow consider a pseudofunctor $$(\\ensuremath{\\mathcal{W}}, \\mathfrak{c}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }.$$\nPut $(\\ensuremath{\\mathcal{W}}_{i}, \\mathfrak{c}_{i}) = (\\ensuremath{\\mathcal{W}}^{\\Phi_i}, \\mathfrak{c}^{\\Phi_i})$ and $(\\ensuremath{\\mathcal{W}}_{ij}, \\mathfrak{c}_{ij}) = (\\ensuremath{\\mathcal{W}}^{\\Phi_{ij}}, \\mathfrak{c}^{\\Phi_{ij}})$. We obtain a pullback diagram of injective cartesian map-graded functors \n\\begin{equation}\\label{MVsquare}\n\\xymatrix{ {(\\tilde{\\ensuremath{\\mathcal{W}}}_{i}, \\tilde{\\mathfrak{c}}_{i})} \\ar[r]^{\\tilde{\\Phi}_{i}} & {(\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}})} \\\\ {(\\tilde{\\ensuremath{\\mathcal{W}}}_{ij}, \\tilde{\\mathfrak{c}}_{ij})} \\ar[u]^{\\tilde{\\Phi}^{i}_{ij}} \\ar[r]_{\\tilde{\\Phi}^j_{ij}} & {(\\tilde{\\ensuremath{\\mathcal{W}}}_{j}, \\tilde{\\mathfrak{c}}_{j}).} \\ar[u]_{\\tilde{\\Phi}_j} }\n\\end{equation}\nBy Proposition \\ref{proppseudocover}, the $\\tilde{\\Phi}_i$ constitute an $\\infty$-cover in $\\ensuremath{\\mathsf{Map}} $ and thus we can use Theorem \\ref{mainsheaf} to relate the different Hochschild complexes. \nBy the sheaf property, we obtain an exact sequence\n$$0 \\longrightarrow \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}}(\\tilde{\\mathfrak{c}}) \\longrightarrow \\oplus_{i \\in I} \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}_i}(\\tilde{\\mathfrak{c}}_i) \\longrightarrow \\oplus_{\\{i,j\\}} \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}_{ij}}(\\tilde{\\mathfrak{c}}_{ij}).$$\nAlternatively, we can proceed inductively by first isolating one composition chain category $\\ensuremath{\\mathcal{D}}_0$ and defining $\\ensuremath{\\mathcal{D}}_1$ to be the subcategory generated by all the other composition chains. Both $\\ensuremath{\\mathcal{D}}_1$ and $\\ensuremath{\\mathcal{D}}_{01} = \\ensuremath{\\mathcal{D}}_0 \\cap \\ensuremath{\\mathcal{D}}_1$ are covered by strictly fewer composition chain categories than $\\ensuremath{\\mathcal{D}}$, and the occuring chains do not increase in length. Proceeding in a similar fashion as before, we then obtain a Mayer-Vietoris sequence as in \\S \\ref{parMV}:\n$$0 \\longrightarrow \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}}(\\tilde{\\mathfrak{c}}) \\longrightarrow \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}_0}(\\tilde{\\mathfrak{c}}_0) \\oplus \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}_1}(\\tilde{\\mathfrak{c}}_1) \\longrightarrow \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}_{01}}(\\tilde{\\mathfrak{c}}_{01}) \\longrightarrow 0.$$\n\n\n\\begin{remark}\nThe approach discussed in this section can be extended from a poset to a \\emph{delta}, i.e a category $\\ensuremath{\\mathcal{C}}$ in which the arrows go only one way: for $C, C' \\in \\ensuremath{\\mathcal{C}}$, we have $\\ensuremath{\\mathcal{C}}(C, C') = \\varnothing$ or $\\ensuremath{\\mathcal{C}}(C', C) = \\varnothing$.\n\\end{remark}\n\n\\begin{example}\\label{expreinj}\nPut $\\ensuremath{\\mathcal{C}} = \\{ s, t_0, t_1 \\}$ with $s \\leq t_0$ and $s \\leq t_1$ and consider $(\\ensuremath{\\mathcal{W}}, \\mathfrak{c}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\underline{\\ensuremath{\\mathsf{Map}} }$ with\n$$\\begin{aligned}\n\\ensuremath{\\mathcal{W}}(s) = \\ensuremath{\\mathcal{W}}_s &&& \\ensuremath{\\mathcal{W}}(t_0) = \\ensuremath{\\mathcal{W}}_0 &&& \\ensuremath{\\mathcal{W}}(t_1) = \\ensuremath{\\mathcal{W}}_1 &&& \\ensuremath{\\mathcal{W}}(s \\leq t_0) = S_0 &&& \\ensuremath{\\mathcal{W}}(s \\leq t_1) = S_1 \\\\\n\\mathfrak{c}(s) = \\mathfrak{c}_s &&& \\mathfrak{c}(t_0) = \\mathfrak{c}_0 &&& \\mathfrak{c}(t_1) = \\mathfrak{c}_1 &&& \\mathfrak{c}(s \\leq t_0) = M_0 &&& \\mathfrak{c}(s \\leq t_1) = M_1\n\\end{aligned}$$\nThe category $\\ensuremath{\\mathcal{C}}$ is covered by the two chain categories\n$$\\ensuremath{\\mathcal{D}}_0 = \\langle s \\leq t_0 \\rangle \\hspace{1,5cm}\\ensuremath{\\mathcal{D}}_1 = \\langle s \\leq t_1 \\rangle$$\nand we have $\\ensuremath{\\mathcal{D}}_{01} = \\ensuremath{\\mathcal{D}}_0 \\cap \\ensuremath{\\mathcal{D}}_1 = \\langle s \\rangle$. We denote the inclusions by $\\Phi_i$, $\\Phi_{ij}$, $\\Phi_{ij}^i$ as before. The diagram \\eqref{MVsquare} is given by\n\\begin{equation}\n\\xymatrix{ {(\\ensuremath{\\mathcal{W}}_s \\rightarrow_{S_0} \\ensuremath{\\mathcal{W}}_0, \\mathfrak{c}_s \\rightarrow_{M_0} \\mathfrak{c}_0)} \\ar[r]^-{\\tilde{\\Phi}_{0}} & {(\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}})} \\\\ {(\\tilde{\\ensuremath{\\mathcal{W}}}_{s}, \\tilde{\\mathfrak{c}}_{s})} \\ar[u]^{\\tilde{\\Phi}^{0}_{01}} \\ar[r]_-{\\tilde{\\Phi}^1_{01}} & {(\\ensuremath{\\mathcal{W}}_s \\rightarrow_{S_1} \\ensuremath{\\mathcal{W}}_1, \\mathfrak{c}_s \\rightarrow_{M_1} \\mathfrak{c}_1)} \\ar[u]_{\\tilde{\\Phi}_1} }\n\\end{equation}\nand we obtain a Mayer-Vietoris sequence\n$$0 \\longrightarrow \\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{W}}}}(\\tilde{\\mathfrak{c}}) \\longrightarrow \\oplus_{i = 0}^1\\mathbf{C}_{\\ensuremath{\\mathcal{W}}_s \\rightarrow_{S_i} \\ensuremath{\\mathcal{W}}_i}(\\mathfrak{c}_s \\rightarrow_{M_i} \\mathfrak{c}_i) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{W}}_s}(\\mathfrak{c}_s) \\longrightarrow 0.$$\nNote that this example also fits into the setup from Proposition \\ref{propcstar}, with $C_0 = t_0$, $C_1 = t_1$ and $C_0 \\times C_1 = s$.\n\\end{example}\n\n\n\n\\subsection{Comparison of pseudofunctors}\\label{parcomp}\n\nLet $\\ensuremath{\\mathcal{C}}$ be a delta (see Definition \\ref{defdelta}). Now let us start from a pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}, \\mathfrak{a}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}.$$\nFor every $c: C' \\longrightarrow C$ we have an associated subcartesian graded functor $$(\\varphi_c, F_c): (\\ensuremath{\\mathcal{U}}_{C'}, \\mathfrak{a}_{C'}) \\longrightarrow (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$$ with associated $\\ensuremath{\\mathcal{U}}_C$-$\\ensuremath{\\mathcal{U}}_{C'}$-bifunctor $S_c = S_{\\varphi_c}$ and\n$\\mathfrak{a}_C$-$S_{\\varphi_c}$-$\\mathfrak{a}_{C'}$-bimodule $M_c = M_{\\varphi_c}$ from Example \\ref{exbimod}.\nLet $\\ensuremath{\\mathcal{C}}^{\\ast}$ be the category $\\ensuremath{\\mathcal{C}}$ with terminal object $\\ast$ and morphisms $\\ast_C: C \\longrightarrow \\ast$ attached as described in \\S \\ref{parbase} and let $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$ be any pseudofunctor such that the restriction to $\\ensuremath{\\mathcal{C}}$ equals $(\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$.\n\n\\begin{example}\\label{exstar}\nWe can consider $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$ with $(\\ensuremath{\\mathcal{U}}^{\\star}_C, : C \\longmapsto (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$\nwith $(\\ensuremath{\\mathcal{U}}^{\\star}_{\\ast}, \\mathfrak{a}^{\\star}_{\\ast}) = (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ and the graded functors $(\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}})$ induced by the natural functors $e \\longrightarrow \\ensuremath{\\mathcal{C}}^{\\ast}: \\ast \\longrightarrow C$.\n\\end{example}\nAs described in \\S \\ref{parbase}, we also obtain an associated pseudofunctor\n$$(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$$\nwith subcartesian graded functors\n$$(\\tilde{\\varphi}_c, \\tilde{F}_c): (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C'}, \\tilde{\\mathfrak{a}}|_{C'}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C})$$\nand\n$$(\\tilde{\\varphi}_{\\ast_C}, \\tilde{F}_{\\ast_C}): (\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C}) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}).$$\nIn this section we compare the Hochschild complexes associated to $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star})$ and $(\\ensuremath{\\mathcal{U}}^{\\ast}, \\mathfrak{a}^{\\ast})$.\nFor every $C \\in \\ensuremath{\\mathcal{C}}$, the functor $e \\longrightarrow \\ensuremath{\\mathcal{C}}\/C: \\ast \\longmapsto (1_C: C \\rightarrow C)$ gives rise to a natural graded functor\n$$(\\varphi_C, F_C): (\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C) \\longrightarrow (\\tilde{\\ensuremath{\\mathcal{U}}}|_C, \\tilde{\\mathfrak{a}}|_C).$$\nNote that $(\\tilde{\\varphi}_c, \\tilde{F}_c)(\\varphi_{C'}, F_{C'}) \\neq (\\varphi_C, F_C)(\\varphi_c, F_c)$.\nLet $(\\theta, H): (\\tilde{\\ensuremath{\\mathcal{U}}}, \\tilde{\\mathfrak{a}}) \\longrightarrow (\\ensuremath{\\mathcal{U}}^{\\star}_{\\ast}, \\mathfrak{a}^{\\star}_{\\ast})$ be the natural subcartesian functor with\n$$(\\theta, H)( \\tilde{\\varphi}_{\\ast_C}, \\tilde{F}_{\\ast_C})(\\varphi_C, F_C)  = (\\varphi^{\\star}_{\\ast_C},  F^{\\star}_{\\ast_C}).$$\nBy Example \\ref{exdelta}, the Grothendieck construction of $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star})$ can be described as an arrow category, yielding a commutative diagram\n\\begin{equation} \\label{eqpar}\n\\xymatrix{ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}^{\\star}}(\\tilde{\\mathfrak{a}}^{\\star})} \\ar[r]^{\\cong} \\ar[d]_{\\varphi_{\\tilde{\\ensuremath{\\mathcal{U}}}}^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\star}_{\\ast}}(\\mathfrak{a}^{\\star}_{\\ast})} \\ar[ld]^{H^{\\ast}} \\ar[d]^{(F^{\\star}_{\\ast_C})^{\\ast}} \\\\\n{\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}}(\\tilde{\\mathfrak{a}})} \\ar[d]_{(\\tilde{F}_{\\ast_C})^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}_C}(\\mathfrak{a}_C)} \\\\ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_C}(\\tilde{\\mathfrak{a}}|_C)} \\ar[ru]_{F^{\\ast}_C} }\n\\end{equation}\n\nWe will prove the following:\n\n\\begin{theorem}\\label{thmmaincomp}\n\\begin{enumerate}\n\\item For every $c: C' \\longrightarrow C$ in $\\ensuremath{\\mathcal{C}}$, the natural diagram\n\\begin{equation} \\label{eqcompat}\n\\xymatrix{ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_C}(\\tilde{\\mathfrak{a}}|_C)} \\ar[d]_{\\tilde{F}_c^{\\ast}} \\ar[r]^{F_C^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}_C}(\\mathfrak{a}_C)} \\ar[d]^{F_c^{\\ast}} \\\\\n{\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_{C'}}(\\tilde{\\mathfrak{a}}|_{C'})}  \\ar[r]_{F_{C'}^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}_{C'}}(\\mathfrak{a}_{C'})}}\n\\end{equation}\nis commutative in the homotopy category of $B_{\\infty}$-algebras, and the horizontal arrows are quasi-isomorphism.\n\\item The horizontal arrows naturally give rise to a morphism between pseudofunctors $\\ensuremath{\\mathcal{C}} \\longrightarrow B_{\\infty}$:\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\ast}|_{\\ensuremath{\\mathcal{C}}}}(\\mathfrak{a}^{\\ast}|_{\\ensuremath{\\mathcal{C}}}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a}).$$\n\\item If the canonical morphism\n$$\\omega: 1_{\\mathfrak{a}^{\\star}_{\\ast}} \\longrightarrow \\mathrm{RHom}_{\\tilde{\\mathfrak{a}}^{^{\\mathrm{op}}}}(M_H, M_H)$$ induces a quasi-isomorphism $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\star}_{\\ast}}(\\mathfrak{a}^{\\star}_{\\ast}, \\omega)$, then $H^{\\ast}$ is a quasi-isomorphism and we obtain a morphism of pseudofunctors $\\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow \\mathsf{ho}(B_{\\infty})$:\n$$\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\ast}}(\\mathfrak{a}^{\\ast}) \\longrightarrow \\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\star}}(\\mathfrak{a}^{\\star})$$\nin which the top horizontal arrow is given by $(H^{\\ast})^{-1}$. This holds in particular is we choose $(\\ensuremath{\\mathcal{U}}^{\\star}, \\mathfrak{a}^{\\star})$ as in Example \\ref{exstar}.\n\\end{enumerate}\n\\end{theorem}\n\n\n\nAs a consequence of the Theorem, suppose the conditions of Proposition \\ref{propcstar} are fulfilled and $\\mathbf{C}_{\\ensuremath{\\mathcal{U}}^{\\ast}}(\\mathfrak{a}^{\\ast}): \\ensuremath{\\mathcal{C}}^{\\ast} \\longrightarrow B_{\\infty}$ is a sheaf of Hochschild complexes. Then exact sequences of Hochschild complexes following from the sheaf property for $\\mathbf{C}^{\\bullet}_{\\ensuremath{\\mathcal{U}}^{\\ast}}(\\mathfrak{a}^{\\ast})$ naturally translate to exact triangles in terms of  $\\mathbf{C}^{\\bullet}_{\\ensuremath{\\mathcal{U}}^{\\star}}(\\mathfrak{a}^{\\star})$ in which we were originally interested. In particular, we naturally obtain Mayer-Vietoris exact triangles and their induced long exact cohomology sequences. Before proving the theorem, we start with a key example.\n\n\\begin{example}\\label{excomp}\nThe first example is in the setup for the original arrow category, as described in the beginning of \\S \\ref{pargenarrow}. So $\\ensuremath{\\mathcal{C}} = \\{ 0, 1\\}$ with $0 \\leq 1$. We consider $(\\ensuremath{\\mathcal{W}}, \\ensuremath{\\mathcal{C}}): \\ensuremath{\\mathcal{C}} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$ determined by $(\\ensuremath{\\mathcal{W}}_0, \\mathfrak{c}_0) = (\\ensuremath{\\mathcal{V}}, \\mathfrak{b})$, $(\\ensuremath{\\mathcal{W}}_1, \\ensuremath{\\mathcal{C}}_1) = (\\ensuremath{\\mathcal{U}}, \\mathfrak{a})$ and a single subcartesian graded functor\n$$(\\varphi, F): (\\ensuremath{\\mathcal{V}}, \\mathfrak{b}) \\longrightarrow (\\ensuremath{\\mathcal{U}}, \\mathfrak{a}).$$\nUsing the bifunctor $S_{\\varphi}$ and the bimodule $M_F$, we obtain the arrow category\n$$(\\tilde{\\ensuremath{\\mathcal{W}}}, \\tilde{\\mathfrak{c}}) = (\\ensuremath{\\mathcal{V}} \\rightarrow_{\\varphi} \\ensuremath{\\mathcal{U}}, \\mathfrak{b} \\rightarrow_F \\mathfrak{a}).$$\nObviously, $\\ensuremath{\\mathcal{C}}\/1 \\cong \\ensuremath{\\mathcal{C}}$, $\\ensuremath{\\mathcal{C}}\/0 \\cong e$,  and the diagram \\eqref{eqcompat} reduces to\n$$\\xymatrix{ {\\mathbf{C}_{\\ensuremath{\\mathcal{V}} \\rightarrow_{\\varphi} \\ensuremath{\\mathcal{U}}}(\\mathfrak{b} \\rightarrow_F \\mathfrak{a})} \\ar[r]^-{\\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}}} \\ar[d]_{\\varphi^{\\ast}_{\\ensuremath{\\mathcal{V}}}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}}(\\mathfrak{a})} \\ar[d]^{F^{\\ast}} \\\\\n{\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b})} \\ar[r]_1 & {\\mathbf{C}_{\\ensuremath{\\mathcal{V}}}(\\mathfrak{b}).}}$$\nThe fact that $\\varphi^{\\ast}_{\\ensuremath{\\mathcal{U}}}$ is a quasi-isomorphism and the diagram commutes in the homotopy category of $B_{\\infty}$-algebras is the content of Proposition \\ref{propcompat2}.\n\\end{example}\n\n\\begin{proof}\nWe will concentrate on the proof of (1). The proof of (3) then follows from diagram \\eqref{eqpar} in which $\\varphi_{\\tilde{\\ensuremath{\\mathcal{U}}}}^{\\ast}$ becomes a quasi-isomorphism by Theorem \\ref{thmarrowqis}.\n\nLet $C \\in \\ensuremath{\\mathcal{C}}$ be fixed.\nThe category $\\ensuremath{\\mathcal{C}}\/C$ is a delta with terminal object $1_C: C \\longrightarrow C$. Let $e_C \\subseteq \\ensuremath{\\mathcal{C}}\/C$ be the category with single object $1_C$ and single morphism $1_C$. Let $\\ensuremath{\\mathcal{D}}_C \\subseteq \\ensuremath{\\mathcal{C}}\/C$ be the full subcategory consisting of all objects except $1_C$. For every $c: C' \\longrightarrow C$, there is a unique morphism $c: c \\longrightarrow 1_C$ in $\\ensuremath{\\mathcal{C}}\/C$. According to Example \\ref{exdelta}, we obtain a thin ideal\n$\\ensuremath{\\mathcal{Z}} = \\{ c: c \\longrightarrow 1_C \\,\\, |\\,\\, 1_C \\neq c: C' \\longrightarrow C \\}$ and we have\n$$\\ensuremath{\\mathcal{C}}\/C \\cong \\ensuremath{\\mathcal{D}}_C \\rightarrow_Z e_C.$$\nwith $Z(c, 1_C) = \\{c\\}$.\nThe restriction of $(\\ensuremath{\\mathcal{U}}|_C, \\mathfrak{a}|_C)$ along $e_C \\subseteq \\ensuremath{\\mathcal{C}}\/C$ corresponds to the constant category $(\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)$. Denote the restriction of $(\\ensuremath{\\mathcal{U}}|_C, \\mathfrak{a}|_C)$ along $\\ensuremath{\\mathcal{D}}_C \\subseteq \\ensuremath{\\mathcal{C}}\/C$ by $(\\ensuremath{\\mathcal{V}}_C, \\mathfrak{b}_C)$. \nNow consider $1_C \\neq c: C' \\longrightarrow C$. Since $\\ensuremath{\\mathcal{C}}$ is a delta,  there is a natural factorization $\\ensuremath{\\mathcal{C}}\/C' \\longrightarrow \\ensuremath{\\mathcal{D}}_C \\longrightarrow \\ensuremath{\\mathcal{C}}\/C$ from which we obtain a natural factorization of $(\\tilde{\\varphi}_c, \\tilde{F}_c)$:\n$$\\xymatrix{ {(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C'}, \\tilde{\\mathfrak{a}}|_{C'})}  \\ar[r]_{(\\tilde{\\varphi}^0_c, \\tilde{F}^0_c)} & {(\\tilde{\\ensuremath{\\mathcal{V}}}_C, \\tilde{\\mathfrak{b}}_C)} \\ar[r]_{(\\varphi_{\\ensuremath{\\mathcal{V}}}, F_{\\mathfrak{b}})} & {(\\tilde{\\ensuremath{\\mathcal{U}}}|_{C}, \\tilde{\\mathfrak{a}}|_{C}).}}$$\nWe further obtain a natural subcartesian graded functor $(\\psi, G)$ fitting into commutative diagrams\n\\begin{equation} \\label{eqdiagaux}\n\\xymatrix{ {(\\tilde{\\ensuremath{\\mathcal{V}}}_C, \\tilde{\\mathfrak{b}}_C)} \\ar[r]^{(\\psi, G)} & {(\\ensuremath{\\mathcal{U}}_C, \\mathfrak{a}_C)} \\\\ {(\\ensuremath{\\mathcal{U}}_{C'}, \\mathfrak{a}_{C'})} \\ar[u]^{(\\tilde{\\varphi}_c^0, \\tilde{F}_c^0)(\\varphi_{C'}, F_{C'})} \\ar[ru]_{(\\varphi_c, F_c)} & }\n\\end{equation}\nfor $1_C \\neq c: C' \\longrightarrow C$. By Proposition \\ref{proparrowc}, we have\n$$\\tilde{\\ensuremath{\\mathcal{U}}}|_C \\cong (\\tilde{\\ensuremath{\\mathcal{V}}}_C \\rightarrow_{\\psi} \\ensuremath{\\mathcal{U}}_C) \\hspace{1,5cm} \\tilde{\\mathfrak{a}}|_C \\cong (\\tilde{\\mathfrak{b}}_C \\rightarrow_G \\mathfrak{a}_C)$$\nwhere we use the bimodules $S_{\\psi}$ and $M_{G}$. \nBy Proposition \\ref{propcompat2} (see also Example \\ref{excomp}), we have a commutative diagram\n$$\\xymatrix{ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_C}(\\tilde{\\mathfrak{a}}|_C)} \\ar[r]^{F^{\\ast}_C} \\ar[d]_{F^{\\ast}_{\\mathfrak{b}}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}_C}(\\mathfrak{a}_C)} \\ar[dl]^{G^{\\ast}} \\\\ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{V}}}_C}(\\tilde{\\mathfrak{b}}_C)} & }$$\nin which $F^{\\ast}_C$ is a quasi-isomorphism. Composing this diagram with \n$$\\xymatrix{ {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{V}}}_C}(\\tilde{\\mathfrak{b}}_C)} \\ar[r]_-{(\\tilde{F}^0_c)^{\\ast}} & {\\mathbf{C}_{\\tilde{\\ensuremath{\\mathcal{U}}}|_{C'}}(\\tilde{\\mathfrak{a}}|_{C'})} \\ar[r]_{F_{C'}^{\\ast}} & {\\mathbf{C}_{\\ensuremath{\\mathcal{U}}_{C'}}(\\mathfrak{a}_{C'})}}$$\nand making use of \\eqref{eqdiagaux}, we obtain the commutative diagram \\eqref{eqcompat} as desired.\n\\end{proof}\n\n\\begin{example}\\label{exinj}\nAs an application, we look at the setup from \\cite[\\S 7.6]{lowenvandenberghhoch}. We consider the poset $\\tilde{\\Delta} = \\{ I \\,\\, |\\,\\, I \\subseteq \\{ 1, 2, \\dots, n\\}$ ordered by reversed inclusion and the subposet $\\Delta \\subseteq \\tilde{\\Delta}$ of all $J \\neq \\varnothing$. We consider a stack $\\tilde{\\ensuremath{\\mathcal{S}}}$ of Grothendieck categories on $\\tilde{\\Delta}$ with exact restriction functors and fully faithful right adjoints, and denote $\\ensuremath{\\mathcal{S}}$ for its restriction to $\\Delta$. We further assume that the conditions (C1) and (C2) listed in \\cite[\\S 7.6]{lowenvandenberghhoch} are satisfied. Let $\\ensuremath{\\mathsf{Mod}} (\\ensuremath{\\mathcal{S}})$ denote the category of presheaf objects in $\\ensuremath{\\mathcal{S}}$. The examples to have in mind are the stack of categories of sheaves of modules on a cover of a ringed space, and the stack of quasi-coherent sheaf categories on a finite affine cover of a separated scheme.\n\nFrom $\\tilde{\\ensuremath{\\mathcal{S}}}$ we obtain a pseudofunctor of categories of injectives\n$$\\mathfrak{e}: \\tilde{\\Delta} \\longmapsto \\ensuremath{\\mathsf{Cat}} (k): I \\longmapsto \\mathfrak{e}_I = \\ensuremath{\\mathsf{Inj}}(\\tilde{\\ensuremath{\\mathcal{S}}}(I)).$$\nFor $J \\subseteq I$, we have a corresponding fully faithful functor\n$$F_{IJ}: \\mathfrak{e}_I \\longrightarrow \\mathfrak{e}_J.$$\nSince these functors are fully faithful, there is a corresponding $(e, \\mathfrak{e}): \\tilde{\\Delta} \\longrightarrow \\ensuremath{\\mathsf{Map}} _{sc}$ and the results from this section apply. \nThe collection of objects $\\{1\\}$, $\\{2\\}$, $\\dots$, $\\{n\\}$ in $\\Delta$ satisfies the conditions in Proposition \\ref{propcstar}, where further $\\Delta^{\\ast} \\cong \\tilde{\\Delta}$ and the product of subsets $I$ and $J$ is given by $I \\cap J$. We denote the pseudofunctor\n$$\\mathfrak{e}^{\\ast} = (\\mathfrak{e}|_{\\Delta})^{\\ast}: \\Delta^{\\ast} \\longrightarrow \\ensuremath{\\mathsf{Cat}} (k): I \\longmapsto \\tilde{\\mathfrak{e}}|_I$$\nwhere $\\tilde{\\mathfrak{e}}|_I$ is the Grothendieck construction of the restriction of $\\mathfrak{e}$ to $\\Delta\/I = \\{ J \\in \\Delta \\,\\, |\\,\\, I \\subseteq J\\}$ and $\\tilde{\\mathfrak{e}}|_{\\ast} = \\tilde{\\mathfrak{e}}|_{\\Delta}$ is the Grothendieck construction of $\\mathfrak{e}|_{\\Delta}$. \nThus, we conclude that $\\mathbf{C}(\\mathfrak{e}^{\\ast})$ satisfies the sheaf property with respect to the collection of maps $\\{i\\} \\longrightarrow \\ast$ for $i \\in \\{ 1, \\dots, n\\}$. \nThe first part of the proof of \\cite[Theorem 7.7.1]{lowenvandenberghhoch} amounts to the verification of the condition in Theorem \\ref{thmmaincomp} (3). \nWe thus obtain from Theorem \\ref{thmmaincomp} a morphism of pseudofunctors $\\Delta^{\\ast} \\longrightarrow \\mathsf{ho}(B_{\\infty})$:\n$$\\mathbf{C}(\\mathfrak{e}^{\\ast}) \\longrightarrow \\mathbf{C}(\\mathfrak{e})$$\nin which all component maps are quasi-isomorphisms.  The Mayer-Vietoris exact triangle for ringed spaces proved in \\cite[Theorem 7.9.1]{lowenvandenberghhoch} becomes an immediate corollary of our theorem, and the proof given in \\cite{lowenvandenberghhoch} is in fact a special case of the proof of Theorem \\ref{thmmaincomp}.\n\\end{example}\n\n\\begin{remark}\nThe sheaves of Hochschild complexes we obtain in this paper naturally give rise to hypercohomology spectral sequences. However, if we start for instance from a ringed space $(X, \\ensuremath{\\mathcal{O}}_X)$, the site on which an associated sheaf of Hochschild complexes of Grothendieck constructions of categories of injectives lives is fundamentally different from the standard site associated to $X$. In Example \\ref{exinj} this difference is ``brigded'' by a bimodule between the categories associated to the different suprema associated to a cover of a ringed space (on the one hand, the ringed space and on the other hand, the downset of the cover). A global approach along these lines (possibly combined with techniques from \\cite{lowensheafhoch}) should lead to new Hochschild cohomology spectral sequences.\n\nThe construction of a Hochschild cohomology local-to-global spectral sequence for general ringed spaces based upon map-graded Hochschild cohomology and hypercoverings remains work in progress \\cite{lowenvandenberghlocglob}.\n\\end{remark}\n\n\\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum decoherence gains importance  both in the foundational studies such as quantum measurements and in the practical studies such as the open quantum systems that are applicable to modern quantum technologies. An important recent development in quantum decoherence, known as Quantum Darwinism \\cite{Z03,OPZ1,OPZ2,BZ05,BZ06,BZ08,QD,ZQZ09,ZQZ10,RZ10,RZ11,RZZ12,ZRZ14,ZRZ16,ZZ17,ULZZJ19}, explains how  classical objectivity emerges from the quantum formalism. Basically, in Quantum Darwinism, one treats the information transmission from the system $S$ to a fragment $F$ of the environment $E$ as a quantum channel, so that the accessible classical information as bounded by the Holevo information for different fragments $F_i$ of the environment $E$ can be compared. If these fragments agree upon the accessible information  to some \nreasonable extent, then we say the classical information thus obtained is objective. In this sense, Quantum Darwinism tells us how the environment witnesses the system in the process of quantum decoherence as an effective measurement channel.\n\nAs the Holevo bound involves an optimization procedure, it is difficult to calculate it. To circumvent this computational difficulty,  an alternative formulation of Quantum Darwinism is proposed in the recent works \\cite{22a,22b,22c}, where one makes measurements on the fragments $F_i$ of $E$ and asks for the maximal information about $S$  that one can extract from these measurements. In contrast, in the original framework of Quantum Darwinism one makes measurements on the systems $S$ and considers the Holevo information defined by the conditional entropy conditioned on $S$. Such a change of measurements is reasonable in the sense that the two settings lead to the same scaling behavior of the Holevo information.\n It turns out that such a change of measurements not only makes the computations easier, but also reveals deep relations between the correlations in the environment $E$ and the information extracted from measuring $F_i$. For example, the Koashi-Winter monogamy relation can be invoked in this new framework to show the trade-off relation for the quantum correlations between $S$ and $F_i$ and the classical information of the $F_i$'s.\n\nInterestingly, in this new framework of Quantum Darwinism, one can consider also the scaling of the conditional mutual information $I(S:F_l|F_k)$ \\cite{22b}. This form of conditional mutual information also makes appearance in our recent work on quantum non-Markovianity \\cite{HG21,HG22}. \nWe therefore expect to study the connections between Quantum Darwinism and quantum (non-)Markovianity from a new perspective, in light of these new results. Previous studies on the connection between Quantum Darwinism and quantum (non-)Markovianity relies on specific models. In these model studies \\cite{DM1,DM2,DM3,DM4,DM5,MS21}, there are strong evidences showing that  the presence of non-Markovianity would suppress Quantum Darwinism. However, there are  some models  \\cite{DM7, OdD19,DM9} in which the  non-Markovian nature of  environment does not  prevent  the appearance of Quantum Darwinism. That the presence of non-Markovianity would hinder Quantum Darwinism is expected from the general understanding that quantum non-Markovianity is related to the information backflow from environment to system, but the absence of relations in several models calls upon further study in this direction.\n\nIn this note, we consider the relations between Quantum Darwinism and approximate   quantum Markovianity in a general setting. This is  operationally meaningful since the Holevo information decreases under  quantum operations and hence the quantum operations can only be operated approximately \\cite{Cai03}.\n\nWe first use straightforward arguments from the approximately complete positive  maps \\cite{BDW16} to explain in \\S\\ref{S3} that for  approximately Markovian processes the scaling property of Quantum Darwinism  still holds, meaning that for the quantum non-Markovianity to hinder Quantum Darwinism a large non-Markovianity quantifier should be required. In \\S\\ref{S4}, we study the backflow of correlation as quantified by the conditional mutual information and find such  backflow of correlation is upper bounded by both the classical information and the quantum discord. If the suppressions as in \\cite{22b} hold, then the backflow of information, which is one way of characterizing quantum non-Markovianity, is also greatly suppressed, thereby corroborating the approximately Markovian results discussed in \\S\\ref{S3}.\n\n\\section{Quantum Darwinism: Decohering the system or eavesdropping the environment}\\label{S2}\nWe start   by reviewing briefly both the standard formulation of Quantum Darwinism \\cite{Z03,OPZ1,OPZ2,BZ05,BZ06,BZ08,QD,ZQZ09,ZQZ10,RZ10,RZ11,RZZ12,ZRZ14,ZRZ16,ZZ17} and the recent formulation given in \\cite{22a,22b,22c}. \n\nIn Quantum Darwinism one considers the setting  of an open quantum system  $S$ interacting with an environment $E$ which consists of $N$ fragments (or subenvironments) $F_i$. The interaction between $S$ and $E$, or the quantum decoherence of $S$ by $E$, is considered as a quantum communication channel $\\Lambda_{S\\rightarrow F_i}$ such that information is transmitted from $S$ to $F_i$. Namely, correlation is created between $S$ and $F_i$ as the quantum  mutual information\n\\begin{equation}\nI(S:F_i)=H_S+H_{F_i}-H_{SF_i}\\neq0\n\\end{equation}\nwhere $H_S=-\\text{tr}\\rho_S\\log_2\\rho_S$ is the von Neumann entropy.\n\nTo extract classical information about $S$ from $F_i$, one considers the quantum decoherence of   $S$ by projecting it to the pointer basis, $\\{\\ket{\\hat{s}}\\}$ \\cite{Z03,PBasis}, and hence one obtains the Holevo information\n\\begin{equation}\\label{2}\n\\chi(\\hat{\\Pi}_S:F_i)=H\\Bigl(\\sum_{\\hat{s}}p_{\\hat{s}}\\rho_{F_i|\\hat{s}}\\Bigr)-\\sum_{\\hat{s}}p_{\\hat{s}}H(\\rho_{F_i|\\hat{s}}),\n\\end{equation}\nwhere $\\hat{\\Pi}_S=\\sum_{\\hat{s}}\\pi_{\\hat{s}}\\ket{\\hat{s}}\\bra{\\hat{s}}$ and $\\rho_{F_i|\\hat{s}}=\\braket{\\hat{s}|\\rho_{SF_i}|\\hat{s}}\/p_{\\hat{s}}$ is the fragment state conditioned on system's pointer state with the $p_{\\hat{s}}$ being the probabilities for projecting to the pointer basis. The Holevo information is an upper bound on the classical  information transmittable in the quantum channel $\\Lambda_{S\\rightarrow F_i}$. For the successful extraction of information about $S$, one sets the condition\n\\begin{equation}\\label{3}\n\\braket{\\chi(\\hat{\\Pi}_S:F_i)}_{\\# F_\\delta}\\approx(1-\\delta) H_S\n\\end{equation}\nwhere $\\delta$ is a small number called the information deficit, and $\\braket{}_{\\# F_\\delta}$  denotes the average over the fragments of size ${\\# F_\\delta}$.\nHere, $H_S=H(\\hat{\\Pi}_S)$ quantifies the missing information about $S$ computed in the pointer basis, i.e. the classical information content of the pointer state when the quantum state is decohered. \nThe condition \\eqref{3} says that an observer only needs a number ${\\# F_\\delta}$ of fragments $F_i$ to retrieve $(1-\\delta) H_S$ classical bits of information about $S$, and therefore the rest of the environment is redundant. Let $\\# E=N$ be the size of the environment $E$, then we define the redundancy as\n\\begin{equation}\nR_\\delta=\\frac{\\# E}{\\# F_\\delta}.\n\\end{equation}\nWhen the fragments $F_i$ are of the same size, we have $R_\\delta=1\/f_\\delta$ with $f_\\delta$ the fraction of the relevant fragments.\n\nBy using the complementary relation for the Holevo information and the quantum discord \\cite{ZZ13}, we can rewrite the condition \\eqref{3} as\n\\begin{equation}\\label{5}\nI(S:F_i)\\approx(1-\\delta) H_S.\n\\end{equation}\nThis is because the Holevo information is the upper bound on the transmittable classical information in the pointer state,\n\\begin{equation}\\label{6}\n\\chi(\\hat{\\Pi}_S:F_i)=\\max_{{s}}I({\\Pi}_S:F_i)=J(\\hat{\\Pi}_S:F_i)\n\\end{equation}\nwhere the maximum is taken over all possible eigenbases, and\n$J(\\hat{\\Pi}_S:F_i)$ is the classical correlation used in the definition of quantum discord. When the quantum correlation is small, we obtain \\eqref{5}, with the understanding that it has been averaged, i.e. $I(S:F_i)=\\braket{I(S:F_i)}_{\\# F_\\delta}$.\n\nThe most important property of $I(S:F_i)$ for Quantum Darwinism is  its {\\it scaling property} under the increase of $\\# F_\\delta$. The study of various models confirms the general behavior of $I(S:F_i)$: At the initial stage, $I(S:F_i)$ follows a steep rise  with increasing $\\# F_\\delta$ as the classical correlation increases; then it saturates a long and flat plateau meaning that at this stage the added fragments are redundant, i.e. when \n\\begin{equation}\\label{7}\nI(S:F_i)\\geqslant(1-\\delta) H_S;\\end{equation}\nfinally, $I(S:F_i)$ follows again a steep rise to reach its maximum as the quantum correlation increases.\n\nNow from \\eqref{6} we see that it involves an optimization, which is generically hard. Also, since the Holevo information is only an upper bound, the above conditions for information retrieval can in fact be relaxed. To improve upon these limitations, the recent work \\cite{22a} considers instead the measurements on $F_i$. Let us denote the part being measured by a check, e.g. the Holevo information in \\eqref{2} becomes the asymmetric mutual information $J(\\check{S}:F_i)$. When  $F_i$ is measured,  one has similarly \n\\begin{equation}\nJ(S:\\check{F}_i)=\\chi(S:\\check{F}_i),\n\\end{equation}\nwhich is the maximal information about $S$ one can extract by measuring $F_i$. In fact, this $J(S:\\check{F}_i)$ also upper bounds the classical accessible information \\cite{22c}.\nNumerical results  show that the scaling property for Quantum Darwinism still holds for $J(S:\\check{F}_i)$. In particular, the plateau condition can be written as\n\\begin{equation}\\label{9}\nJ(S:\\check{F}_i)\\geqslant(1-\\delta) H_S.\n\\end{equation}\n\nUsing the symmetric mutual information $I(S:F_i)$ and the asymmetric mutual information $J(S:\\check{F}_i)$, one can define the fragmentary discord, $D(S:\\check{F}_i)=I(S:F_i)-J(S:\\check{F}_i)$. If we consider each fragment $F_i$ has its own information deficit $\\delta_i$, and consider the averaged discord $\\bar{D}(S:\\check{F}_i)=\\frac{1}{N}\\sum_{i=1}^ND(S:\\check{F}_i)$, then the following bound holds \\cite{22b},\n\\begin{equation}\\label{10}\n\\bar{D}(S:\\check{F}_i)\\leqslant\\delta H_S\n\\end{equation}\nwhere $\\delta=\\sum_{i=1}^N\\delta_i\/N$. Similarly, for the $\\# F_\\delta\\leqslant\\frac{N}{2}$ fragments, the bound can be rewritten as\n\\begin{equation}\\label{11}\n\\bar{D}(S:\\check{F}_i)\\leqslant\\bigl[1-(1-\\delta)\\frac{R_\\delta}{N}\\bigr] H_S.\n\\end{equation}\nClearly, since the information deficit is chosen to be small, the quantum correlation as quantified by the quantum discord is greatly suppressed. Importantly, the plateau condition \\eqref{9} implies \n\\begin{equation}\\label{12}\nI(S:F_l|F_k)\\leqslant2\\delta H_S,\n\\end{equation}\nfor $k\\geqslant\\#F_\\delta,~k+l\\leqslant N-\\# F_\\delta$. With \\eqref{12}, one can directly test Quantum Darwinism by inspecting the scaling of the conditional mutual information $I(S:F_l|F_k)$, without facing the problem of optimization.\n\\section{Relation to approximately quantum Markovian processes}\\label{S3}\nWe consider in this section the case in which the quantum channels involved deviate a little  from complete positivity. The complete positivity of the open quantum system dynamics is related to the quantum data processing inequality  \\cite{Bus14}, and the deviation from the   quantum data processing inequality  indicates quantum non-Markovianity. A small deviation from the   quantum data processing inequality can give rise to an approximately complete positive map, and the converse is also true \\cite{BDW16}.\n\nTo apply these deviation results to Quantum Darwinism, we consider the following novel scenario: Instead of focusing on the system $S$, we take the measurement made on the fragment $F_k$ as a quantum channel $\\Lambda_{F_k\\rightarrow F_k'}$. After the system $S$ has decohered and the information about $S$ has been transmitted to $F_k$, we assume that the fragment $F_k$ no longer interacts with $S$ but the interactions between different  $F_i$'s are still possible. With this assumption,\nthe quantum channel $\\Lambda_{F_k\\rightarrow F_k'}$ can be considered as an $F_k$-$F_l$ open system where $F_k$ is the new open ``system'' of an observer's concern and another fragment $F_l$ with $l\\neq k$ is the new environment for $F_k$. On the other hand,  the system $S$ becomes now a {\\it steerable } extension of the $F_k$-$F_l$ open system, because the states $\\rho_{F_kF_l}$ can be engineered according to the pointer basis of $S$. This way, we have adapted the new setting of Quantum Darwinism to that of \\cite{Bus14}.\n\nNow suppose that the   quantum channel $\\Lambda_{F_k\\rightarrow F_k'}$  is approximately CPTP, in the sense that \n\\begin{equation}\n\\frac{1}{2}\\norm{\\sigma_{SF'_k}-\\Lambda_{F_k\\rightarrow F_k'}(\\rho_{SF_k})}_1\\leqslant\\epsilon\n\\end{equation}\nwhere $\\epsilon\\in[0,1]$ and $\\sigma_{SF_kF_l}=U_{F_kF_l}(\\rho_{SF_kF_l})U^\\dag_{F_kF_l}$ is the partially evolved state. \nThen by the Alicki-Fannes-Winter inequality \\cite{AF04,W16}, we have \n\\begin{equation}\nI(S:F'_k)_\\sigma\\leqslant I(S:F_k)+2\\epsilon\\log\\abs{S}+(1+\\epsilon)h[\\frac{\\epsilon}{1+\\epsilon}]\n\\end{equation}\nwhere $h[x]=-x\\log x-(1-x)\\log(1-x)$ is the binary entropy. Furthermore, by \nTheorem 8 of \\cite{BDW16}, we also have the  bound on the  conditional mutual information\n\\begin{equation}\\label{15}\nI(S:F_l|F_k)\\leqslant2\\epsilon\\log\\abs{S}+(1+\\epsilon)h[\\frac{\\epsilon}{1+\\epsilon}].\n\\end{equation}\n\n\nThe bound in \\eqref{15} has the same qualitative structure as \\eqref{12}: two bounds both depends on a small parameter and  the ``entropy'' of $S$.\\footnote{In \\eqref{15}, the term $\\log\\abs{S}$ can be roughly understood as the statistical-mechanical entropy in the Boltzmann sense.} \nWe therefore see that, with a small deviation from complete positivity, the scaling behavior \\eqref{12}  of conditional mutual information for Quantum Darwinism {\\it  can still hold}, if we choose proper parameters $\\epsilon,\\delta$, and\n assume that the fragment $F_k$ no longer interacts with the system $S$. In other words, the  approximately Markovian quantum processes will not hinder the presence of Quantum Darwinism. This understanding is consistent with the strict  bounds on the deviation from the quantum Markovianity  \\cite{FR15,FMP19}.\n \nNext, let us consider the Holevo information of the ensemble of states $\\mathcal{S}=\\{p_{\\hat{s}},\\rho_{\\hat{s}}\\}$,\n\\begin{equation}\n\\chi(\\mathcal{S})=H\\Bigl(\\sum_{\\hat{s}}p_{\\hat{s}}\\rho_{\\hat{s}}\\Bigr)-\\sum_{\\hat{s}}p_{\\hat{s}}H(\\rho_{\\hat{s}}).\n\\end{equation}\nThen the Holevo information \\eqref{2} of the quantum channel $\\Lambda_{S\\rightarrow F_i}$ is the Holevo information $\\chi(\\mathcal{F})$ of the output ensemble $\\mathcal{F}=\\{p_{\\hat{s}},\\rho_{F_i|\\hat{s}}\\}$. The monotonicity of the Holevo information under quantum channels give $\\chi(\\mathcal{S})\\geqslant\\chi(\\mathcal{F})$, and when the equality holds there exists a (Petz) recovery channel $\\Phi_{F_i\\rightarrow S}$ such that $\\Phi_{F_i\\rightarrow S}\\circ\\Lambda_{S\\rightarrow F_i}(\\rho_{\\hat{s}})=\\rho_{\\hat{s}}$ \\cite{HJPW04}. The existence of recovery channel is equivalent to the  backflow of {\\it all} information from $F_i$ to $S$, and we see that this requires the Holevo information of the ensemble of states is intact under the quantum channel $\\Lambda_{S\\rightarrow F_i}$, i.e. \n\\begin{equation}\n\\chi(\\mathcal{S})=\\chi(\\mathcal{F})=\\chi(\\hat{\\Pi}_S:F_i).\\end{equation}\nA small deviation from this equality is allowable, if the following bound holds (cf. Theorem 6 of \\cite{BDW16})\n\\begin{equation}\\label{18}\n\\chi(\\mathcal{S})-\\chi(\\mathcal{F})\\geqslant-2\\log\\sum_{\\hat{s}}p_{\\hat{s}}\\sqrt{F}(\\rho_{\\hat{s}},\\Phi\\circ\\Lambda(\\rho_{\\hat{s}}))\n\\end{equation}\nwhere $\\sqrt{F}(\\rho,\\sigma)=\\norm{\\sqrt{\\rho}\\sqrt{\\sigma}}_1$ is the root of the fidelity. The inequality \\eqref{18} is the constraint for the existence of the approximate recovery channel or approximately full information backflow. \n\nAs far as the quantum decoherence or emergence of classicality is concerned, we expect a large violation of the condition \\eqref{18} to reach an irreversible information transmission. However, when strong non-Markovianity is present, it is still  possible for such an approximate recovery channel to exist.\n\n\\section{Relation to backflow of correlation}\\label{S4}\n\\begin{CJK*}{UTF8}{gbsn}\nIt is well-known by now that  the quantum non-Markovianity  of open quantum dynamics can be quantified by the information backflow  from environment to system, or the increase in distinguishability of system's states \\cite{BLP09}. Recently, we have shown in \\cite{HG21} that the backflow of quantum mutual information can be equivalently reformulated as the backflow of conditional mutual information in a system-environment-ancilla setting. Here, we shall consider the backflow of conditional mutual information in the new setting of Quantum Darwinism.\n\nWe consider first the pure state $\\rho_{SE}$\nfor the system-environment total system. Since for pure states both the quantum discord and the classical correlation takes the maximal value of $H_S$, we have \n\\begin{equation}\nI(S:E)=D(S:\\check{E})+J(S:\\check{E})=2H_S.\n\\end{equation}\nIn the system-environment-ancilla setting of \\cite{HG21}, we require the ancilla-system initial state is maximally entangled, i.e. $I(A:S)_{\\rho_i}=2H_A=2\\ln d_A$. In other words, initially the ancillary $A$ possesses all the information about $S$ (and vice versa). After the information of $S$ is spread into $E$, we have in general\n$I(A:S)\\neq2H_A$ and $I(A:SE)=2H_A$. \nThe memory effect is reflected by the non-monotonicity of the  conditional mutual information $I(A:E|S)=I(A:SE)-I(A:S)$. \n\nNotice that  the ancillary $A$ can be considered either as a fragment $F_i$ of the environment or  a special observer who makes measurements on $S$. \nWe first consider the former interpretation of $A$ as fragment of $E$.\nTo avoid confusions, let us label the   $(S^*,E^*,A^*)$ in the system-environment-ancilla setting with a $*$.\nWe can take the $A^*$ as the system $S$ in Quantum Darwinism setting and take $S^*$ as a marked fragment $F_1$ in Quantum Darwinism setting. Let us consider the following\n\\begin{equation}\nI(A^*:E^*_{sub}|S^*)\\equiv I(S:{F}_l|{F}_1),\n\\end{equation}\nwhere the RHS is understood as the information transmission from $S$ to $F_1$ in the setting of Quantum Darwinism, which can be alternatively understood as in the LHS as information flow from $A^*$ to $S^*$. \nIf we adopt the interpretation of $A^*$ as a fragment of environment, this information transmission is the backflow of information. \nSo we have\n    \\begin{align}\n        &I(A^*:E^*_{sub}|S^*)= I(S:{F}_l|{F}_1)\\notag \\\\\n        =&I(S:F_{l+1})-I(S:F_1)\\notag \\\\\n        =&I(S:F_{\\#F_\\delta})-I(S:F_1)+I(S:F_{l+1})-I(S:F_{\\#F_\\delta})\\notag \\\\\n        =&I(S:{F}_{\\#F_\\delta-1}|\\mathcal{F}_1)+I(S:{F}_{l-\\#F_\\delta+1}|{F}_{\\#F_\\delta})\\notag \\\\\n        \\leqslant&I(S:F_{\\#F_\\delta})-I(S:F_1)+2\\delta H_S \\label{20}\\\\\n        \\approx&J(S;F_{\\#F_\\delta})+D(S;F_{\\#F_\\delta})-(1-2\\delta-\\delta')H(S)\\label{21}\n    \\end{align}\nwhere in \\eqref{20} we have used the scaling \\eqref{12}, and in \\eqref{21} we have used \\eqref{5} and $\\delta'$ is the information deficit for this particular case, $I(S:F_1)\\approx(1-\\delta')H_S$. By expressing \\eqref{5} in terms of $J$,  we obtain\n\\begin{equation}\nI(A^*:E^*_{sub}|S^*)\\lessapprox(\\delta'+\\delta)H_S +D(S;F_{\\#F_\\delta})\\label{22}.\n\\end{equation}\nThis result \\eqref{22} shows that the conditional mutual information $I(A^*:E^*_{sub}|S^*)$ is  bounded both by the classical information  and the quantum correlation. This clearly shows that  the backflow of information contains both classical and quantum correlations. Now, in the  parameter range $ l\\leqslant N-2\\#F_\\delta$, by further using the bound \\eqref{11}, we see that the conditional mutual information, which is to be backflowed from $A^*$ to $S^*$, is greatly suppressed. \n\nNotice that the backflow of information from $A^*$ to $S^*$ is understood by the standard setting of Quantum Darwinism as the channel from $S$ to $F_i$. Although a little bit strange, this is allowable by the general results on the emergence of Quantum Darwinism in a general many-body quantum system \\cite{BPH15,KTPA18,CLAT21,QR21,RZZ16,Rie17,Oll22}.  In a sense, we have modeled the emergence of Quantum Darwinism from $A^*$ to $\\{E_{sub}^*\\}_{i}\\cup\\{S^*\\}$.\n\nNext, we consider an alternative setting by taking $A$ as an observer who measures $S$. Suppose the initial system-environment state  is\n    \\begin{equation}\\label{SODAW0}\n        \\ket{\\psi_{SE}(0)}=\\Bigl(\\sum_{{s}} \\ket{{s}}_S\\Bigr)\\otimes \\ket{\\phi}_E\n    \\end{equation}\nwhere the $\\ket{s}$ is the eigenbasis of $S$ and the normalization is hidden. \nAfter $A$ measured $S$ by projections $\\ket{\\hat{s}}\\bra{\\hat{s}}$ to the pointer basis $\\ket{\\hat{s}}$, the  joint state is then\n    \\begin{equation}\\label{SOH0}\n        \\ket{\\psi'_{ASE}(0)}=\\Bigl(\\sum_{\\hat{s}} \\sqrt{P_{\\hat{s}}}\\ket{\\hat{s}}_A\\otimes\\ket{\\hat{s}}_S\\Bigr)\\otimes \\ket{\\phi}_E\n    \\end{equation}\n where $\\sqrt{P_{\\hat{s}}}=\\braket{\\hat{s}|s}$. In this case, $AS$-state is  entangled, and the observer $A$ keeps all the classical information about $S$. We have in fact  returned to the system-environment-ancilla setting of \\cite{HG21}.\nAfter the time evolution of $SE$ by $U^{SE}$ (or under the quantum channel $\\Lambda_{S\\rightarrow E}$), we have \n \\begin{equation}\\label{SOH}\n    \\ket{\\psi'_{ASE}}=U^{SE}  \\ket{\\psi'_{ASE}(0)}=\\sum_{\\hat{s}} \\sqrt{P_{\\hat{s}}}\\ket{\\hat{s}}_A\\otimes\\ket{\\hat{s}}_S\\otimes \\ket{\\phi_{\\hat{s}}}_E.\n\\end{equation}\nThe $SE$-part is now a branching state\n\\begin{equation}\\label{SOH}\n    \\ket{\\psi_{SE}}=U^{SE}  \\ket{\\psi_{SE}(0)}=\\sum_{\\hat{s}} \\sqrt{P_{\\hat{s}}}\\ket{\\hat{s}}_S\\otimes \\ket{\\phi_{\\hat{s}}}_E.\n\\end{equation}\n\nTo proceed, we observe that joint state $\\ket{\\psi'_{ASE}}$ can be obtained from the state $\\ket{0}_A\\otimes\\ket{\\psi_{SE}}$, with $\\ket{0}_A$ a pure state, by a unitary $U^{AS}: \\ket{\\hat{s}0}\\to \\ket{\\hat{s}\\hat{s}}$,\n\\begin{equation}\n    \\ket{\\psi'_{ASE}}=U^{AS} \\ket{0}_A\\otimes \\ket{\\psi_{SE}}.\n\\end{equation}\nSince the pure state has zero von Neumann entropy and the unitary does not increase von Neumann entropy, we have in this case,\n\\begin{equation}\\label{DCMI1}\n    I(AS:E_{sub})_{\\ket{\\psi'_{ASE}}}=  I(S:E_{sub})_{ \\ket{\\psi_{SE}}}.\n\\end{equation}\nLet us take partial trace of \\eqref{SOH}\n\\begin{equation}\\label{27}\n    \\text{Tr}_{A\\overline{E}_{sub}} \\Pi_{\\psi'}^{ASE}=\\sum_{\\hat{s}} P_{\\hat{s}} \\Pi_{\\hat{s}}^S \\otimes  ( \\text{Tr}_{\\overline{E}_{sub}} \\Pi_{\\phi_{\\hat{s}}}^{E}).\n\\end{equation}\nwhere $\\overline{E}_{sub}E_{sub}=E$.\nNow we can assume the condition of  good decoherence \\cite{ZQZ10}  that the\noff-diagonal terms of $\\rho_{S}$  are negligible, so that  \\eqref{27} is a classical-quantum state with zero quantum discord, i.e.\n    \\begin{equation}\\label{DCMI2}\n        I(S:E_{sub})_{\\ket{\\psi'_{ASE}}}=  \\chi(\\check{S}:E_{sub})_{ \\ket{\\psi_{SE}}}.\n    \\end{equation}\nFinally, combining \\eqref{DCMI1} and \\eqref{DCMI2}, we obtain\n    \\begin{align}\n        I(A:E_{sub}|S)_{\\ket{\\psi'}}&=I(AS:E_{sub})_{\\ket{\\psi'}}-I(S:E_{sub})_{\\ket{\\psi'}}\\notag \\\\\n       & =I(S:\\mathcal{F}_l)- \\chi(\\check{S}:\\mathcal{F}_l)=D(\\check{S}:\\mathcal{F}_l).\\label{29}\n    \\end{align}\nSince we have assumed the good decoherence, we have indeed $D(\\check{S}:\\mathcal{F}_l)\\approx0$ \\cite{22a}.\n\nIn the present case, the $I(A:E_{sub}|S)_{\\ket{\\psi'}}$ contains the part that can be possibly  backflowed, and any backflow of conditional mutual information is also bounded as above. Note that, in comparison to \\eqref{22}, the classical information is not present in the bound \\eqref{29}; this is because the interpretations of $A$ differ in two cases. Here, $A$ contains all the classical information (i.e. the pointer observables) of $S$, while for \\eqref{22} this classical information needs to be transmitted.\n\n\\end{CJK*}\n\\section{Conclusion}\\label{IV}\nWe have discussed the relation between Quantum Darwinism and approximate quantum non-Markovianity. In \\S\\ref{S3}, we have showed that given an approximately Markovian quantum process with an approximate CPTP map, the conditional mutual information still satisfies the scaling property for Quantum Darwinism as is recently given in \\cite{22b}, while in \\S\\ref{S4} we have proved the converse that the existence of Quantum Darwinism greatly suppresses the backflow of correlation. In this sense, we have actually proved, under some physically reasonable assumptions, the following statement:\n\\begin{quote}\n{\\it Quantum Darwinism is present in an open quantum system if and only if the open system is Markovian or approximately Markovian. }\n\\end{quote}\n\nThis result is consistent with the numerical works \\cite{DM1,DM2,DM3,DM4,DM5,MS21} where\nthe highest redundancy is reached when the dynamics is fully Markovian,  the redundancy tends to\nzero when the dynamics is non-Markovian. (Particularly, \\cite{MS21} also considers the information flow from the environment-point-of-view.)\nWith the explicit bounds obtained in this paper, we can further understand the boundary between these two regimes. \nBut the numerical results supporting the irrelevance between Quantum Darwinism and non-Markovianity \\cite{DM7, OdD19,DM9} cannot be understood easily from the point of view of present work. These  irrelevance results involve other aspects of classical objectivity such as the spectrum broadcast structures \\cite{SBS}, which is worthy of further investigations.\n\n\n\\begin{acknowledgments}\nWe would like to thank Diogo O. Soares-Pinto for helpful comments. \nZH  is supported by the National Natural Science Foundation of China under Grant Nos. 12047556, 11725524 and the Hubei Provincial Natural Science Foundation of China under Grant No. 2019CFA003.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFetal lung parenchyma maldevelopment may lead to life-threatening physiologic dysfunction due to pulmonary hypoplasia and pulmonary hypertension \\cite{lakshminrusimha2015persistent}. Accurate assessment of lung maturation before delivery is critical in determining pre-natal and post-natal care and potential interventions  \n \\cite{ahlfeld2014assessment}.\nIn current clinical practice, the non-invasive assessment of fetal lung parenchyma development relies upon fetal lung volume estimation from either ultrasonography \\cite{Moeglin2005FetalUltrasound}, or anatomical magnetic resonance imaging (MRI) \\cite{deshmukh2010mr} data. However, these modalities are limited in providing an insight into lung function and are therefore suboptimal in assessing fetal lung maturity and in providing relevant biomarkers for fetal lung parenchyma maldevelopment.\n\nDiffusion-weighted MRI (DWI) is a non-invasive imaging technique sensitive to the random movement of individual water molecules. The displacement of individual water molecules results in signal attenuation in the presence of a magnetic field encoding gradient pulses. This attenuation increases with the degree of sensitization-to-diffusion of the MRI pulse sequence (the \"b-value\") \\cite{Afacan2016FetalAge}. Quantitative analysis of the DWI data (qDWI) has been suggested previously for functional assessment of fetal lung parenchyma development \\cite{moore2001vivo,Afacan2016FetalAge}. \nThe qDWI analysis is performed by a least-squares fitting of a signal decay model describing the DWI signal attenuation as a function of the b-value to the acquired DWI data. Commonly, a mono-exponential signal decay model of the form of:  \n\\begin{align}\n  S_{i} = {S_{0}}\\cdot e^{-b_i\\cdot ADC} \\;\n  \\label{eq:MonoModel}\n\\end{align}\nWhere $S_i$ is the signal at b-value $b_i$, $S_0$ is the signal without sensitizing the diffusion gradients, and $ADC$ is the apparent diffusion coefficient, which represents the overall diffusivity, is used in clinical practice \\cite{koh2007diffusion}. \n\nHowever, qDWI analysis is intrinsically highly susceptible to gross motion between the acquisition of the different b-value images \\cite{Afacan2016FetalAge}. Hence, the irregular and unpredictable motion of the fetus, in addition to the maternal respiratory and abdominal motion, causes misalignment between image volumes, acquired at multiple b-values, and impairs the accuracy and robustness of the signal decay parameter estimation \\cite{Kurugol2017Motion-robustEstimation}.\n\nSpecifically, Afacan et al. \\cite{Afacan2016FetalAge} examined the influence of the presence of motion during the DWI data acquisition on the correlation between the overall diffusivity characterized by the ADC parameter and the gestational age (GA). They found that the dependence of the ADC parameter on the GA is of an exponential saturation type, with an R-squared $(R^2)$ value of 0.71 for fetal DWI data that was not affected by fetus motion. However, when fetal DWI data with gentle motion were included, the $R^2$ dropped to 0.232, and when fetal DWI data with severe motion were included, the $R^2$ dropped to 0.102.\n\nWe introduce qDWI-Morph, an unsupervised deep-neural-network model for motion-compensated qDWI analysis. Our specific contributions are: 1) unsupervised deep-neural-network (DNN) model for simultaneous motion compensation and qDWI analysis, 2) Introduction of a bio-physically-informed loss function incorporating registration loss with qDWI model fitting loss, and 3) improved correlation between qDWI analysis by means of the ADC parameter and GA demonstrated using in-vivo DWI data of fetal lungs.\n\n\\subsection{Background}\n\\subsubsection{Fetal lung development}\n\nThe normal fetal lung parenchyma development starts in the second trimester and progresses through multiple phases before becoming fully functional at full term. The first phase of development, the embryonic and pseudo glandular stage, is followed by the canalicular phase, which starts at 16 weeks, then the saccular stage, which starts at 24 weeks, and ends with the alveolar stage, which starts at 36 weeks gestation. \\cite{schittny2017development}. The formation of a dense capillary network and a progressive increase in pulmonary blood flow, leading to reduced extra-cellular space on the one hand and increased perfusion on the other hand, characterizes the progression through these phases \\cite{Ercolani2021IntraVoxelMaturation}. Hence, changes in the overall tissue diffusivity are thought to serve as a functional indicator of lung development.  \n\n\\subsubsection{Diffusion-weighted MRI}\n\nDWI is a non-invasive imaging technique sensitive to the random movement of individual water molecules. The movement of water molecules depends on tissue micro-environments and results from two phenomena: diffusion and perfusion \\cite{Ercolani2021IntraVoxelMaturation}. The water molecule mobility attenuates the DWI signal according to the degree of sensitivity-to-diffusion, the \"b-value\", used in the acquisition. Typically, DWI images are acquired at multiple b-values, and the signal decay rate parameters are quantified per voxel by least-squares model fitting  \\cite{Kurugol2017Motion-robustEstimation}. DWI  can potentially be a functional biomarker for lung maturity since it results from diffusion and perfusion.\nAccording to the mono-exponential model, the signal of a particular voxel decreases exponentially with an increasing b-value (Eq.~\\ref{eq:MonoModel}), with a decay rate that depends on the overall diffusivity of the voxel, the ADC, encapsulating both pure diffusion and perfusion influence on the DWI signal. Quantifying the ADC out of the measurements is done by applying $log(\\cdot)$ on both sides of Eq.~\\ref{eq:MonoModel}:\n\\begin{align}\n  log(S_{i}) = log(S_{0})-b_i\\cdot ADC \\;\n  \\label{eq:MonoModelLog}\n\\end{align}\n\nA set of B equations is obtained by scanning with B different b-values. This linear set of equations can be solved by linear least-squares regression (LLS) \\cite{LinearLeastSquares}. The solution is given by:\n\n\\begin{align}\n\\hat{x}_{LS}  & = \\argmin_{S_0,ADC}\\sum_{i\\in B}|\\log{(S_i)}-\\log{(S_0)}+b_{i}ADC|^2 \\; \\\\\n& = (A^TA)^{-1}A^T\\beta\n\\label{eq:LLS}\n\\end{align}\n\nwhere: \n\\begin{align}\nA = \\begin{pmatrix}\n1 & -b_0\\\\ \n\\vdots  & \\vdots\\\\ \n1 & -b_B\n\\end{pmatrix},\n&& \\beta = \\begin{bmatrix}\nlog(S_0), & \\cdots,  & log(S_B) \n\\end{bmatrix}^T,\n&& x = \\begin{bmatrix}\nlog(S_0), & ADC \n\\end{bmatrix}^T\n\\end{align}\n\nIn order to be stable against outliers in the acquired DWI signals, a more robust solution can be obtained by using an iterative-least squares algorithm \\cite{Burrus2018IterativeRL}. This is an iterative method in which each step involves solving a weighted least squares problem:\n\\begin{align}\n\\hat{x}_{IRLS}^{(t)} & = \\argmin_{S_0,ADC}\\sum_{i\\in B}w_i^{(t)}|\\log{(S_i)}-\\log{(S_0)}+b_{i}ADC|\\\\\n& = (A^TW^{(t)}A)^{-1}A^TW^{(t)}b\n\\label{eq:IRLS}\n\\end{align}\n\nwhere $W^{(t)}$ is a diagonal matrix of weights, with all elements set initially to:\n\\begin{align}\n\\left \\{w_i^{(0)}\\right \\}_{i\\in B} =  1\n\\end{align}\n\nand updated after each iteration to:\n\\begin{align}\n\\left \\{w_i^{(t)}\\right \\}_{i \\in B} =  \\frac{1}{\\max \\left \\{|Ax^{(t-1)}-b|_i, 0.0001 \\right \\}} \n\\end{align}\n\nAs shown In Fig.~\\ref{fig:LS_IRLS}, the IRLS is more accurate and ignores outliers by setting them a low weight. On the other hand, the IRLS algorithm is computationally expensive and time-consuming.\n\n\\begin{figure}[t]\n\t\\centering\n    \\includegraphics[height=6cm]{figures\/LS_IRLS_Case697.pdf}\n    \\caption{LLS Vs. IRLS fitting of DWI Signal.}\n\t\\label{fig:LS_IRLS}\n\\end{figure}\n\n\\subsection{Fetal Lung Maturity Assessment with qDWI}\nAssessing fetal lung maturation using qDWI was first suggested by Moore et al. \\cite{moore2001vivo} in 2001. More recently, Afacan et al. \\cite{Afacan2016FetalAge} demonstrated a strong association between ADC and GA in fetuses with normal lung development. The ADC is quantified per voxel by LLS regression (Eq.~\\ref{eq:LLS}). Then, the averaged ADC in the lung can indicate the GA of the fetus according to the saturation-exponential model suggested by \\cite{Afacan2016FetalAge}:\n\\begin{align}\n  ADC = ADC_{sat}(1-e^{-\\alpha \\cdot GA}) \\;\n  \\label{eq:GA_ADC}\n\\end{align}\n\nSince the ADC quantification is done voxel-wise, it is sensitive to potential misalignment between image volumes acquired at different b-values, which accrue from fetal motion. Previous techniques for motion compensation include post-acquisition motion compensation based on image registration to bring the volumes acquired at different b-values into the same physical coordinate space before fitting a signal decay model \\cite{guyader2015influence,mazaheri2012motion}. However, each b-value image has a different contrast. As a result, independent registration of different b-value images to a b = 0 s\/mm$^2$ image may result in suboptimal registration, especially for high b-value images where the signal is significantly attenuated and the signal-to-noise ratio is low \\cite{Kurugol2017Motion-robustEstimation}.\n\n\\section{Method}\nWe hypothesize that simultaneous qDWI analysis and fetal motion compensation will result in improved quantification of the ADC, which will correlate better with fetal lung maturity compared to qDWI analysis without compensating for motion and to motion compensated approach which does not account for the signal decay model fit quality.\n\nWe define the simultaneous qDWI analysis and motion compensation problem as follows:\n\\begin{align}\n    \\widehat{\\Phi},\\widehat{ADC} = \\argmin_{\\Phi,ADC} \\sum_{i \\in B}\\|\\phi_i \\circ S_i-S_0\\exp(-b_i\\cdot ADC)\\|^2 \n    \\label{eq:problem_formulation_base}\n\\end{align}\nwhere $\\Phi=\\{\\phi\\}_{i \\in B}$ is the set of transformations that align the different b-value images to the images predicted by the model, and $B$ is the number of b-values used during the DWI acquisition.\n\nTaking an unsupervised DNN-based perspective, the optimization problem seeks to find the DNN weights that minimize the following:\n\\begin{align}\n    \\widehat{\\Theta} = \\argmin_{\\Theta} \\sum_{i \\in B}\\|f_{\\Theta}(\\{b_i,S_i\\}_{i \\in B})[0] \\circ S_i-S_0\\exp(-b_i\\cdot f_{\\Theta}(\\{b_i,S_i\\}_{i \\in B})[1])\\|^2 \n    \\label{eq:problem_formulation_deep}\n\\end{align}\nwhere $\\Theta$ are the DNN parameters to be optimized,  $f_{\\Theta}(\\{b_i,S_i\\}_{i=0}^{B})[0]$ is the first output of the  DNN model, represents the set of transformations $\\Phi$ between the different b-value images $\\{S_i\\}_{i \\in B}$, and $f_{\\Theta}(\\{b_i,S_i\\}_{i \\in B})[1])$ is the second output of the DNN model represents the predicted ADC.\n\n\\subsection{DNN architecture}\nFig.~\\ref{Arictecture} presents our qDWI-Morph architecture used for the optimization of eq.~\\ref{eq:problem_formulation_deep}. The qDWI-Morph model is composed of two sub-networks: a qDWI model fitting sub-network and an image registration sub-network.\nOur method is iterative, where the motion-corrected images from the previous iteration enter as input to the next iteration until convergence. We normalize the input in each iteration $k$ by the maximal value of the signal at the previous iteration $S_0^{(k-1)}$.\nThe input to the model at each iteration $k$ is a set of 3D images from the same patient, each scanned with a different b-value along with their corresponding b-values $\\left \\{b_i,S_i \\right \\}_{i \\in B}^{(k)}$. \nThe output of our model is composed of the set of motion-corrected images forced pixel-wise to the mono-exponential model (Eq.~\\ref{eq:MonoModel}). \n\n\\begin{figure}[t]\n\t\\centering\n    \\includegraphics[width=\\textwidth]{figures\/Arictecture.pdf}\n    \\caption{Overall architecture of our proposed framework.}\n\t\\label{Arictecture}\n\\end{figure}\n\n\\subsubsection{Quantitative DWI Model Fitting Sub-network} first layer is a LLS layer for quantifying the ADC at each pixel using Eq.~\\ref{eq:LLS}. The second layer is a reconstruction layer that generates a new set of 3D images, $\\left \\{ R_b \\right \\}_{b \\in B}^{(k)}$, by calculating the model pixel-wise using Eq.~\\ref{eq:MonoModel}.\nThe reconstructed set, $\\left \\{ R_b \\right \\}_{b \\in B}^{(k)}$, is retreated as fixed images and is an input for the second sub-network, the registration network.\n\n\\subsubsection{Registration Sub-network} is based on the publicly available deep learning framework for deformable medical image registration, VoxelMorph \\cite{balakrishnan2019voxelmorph}. The moving images are $\\left \\{ S_b \\right \\}_{b \\in B}^{(k)}$, and the fixed one are  $\\left \\{ R_b \\right \\}_{b \\in B}^{(k)}$. The output of the VoxelMorph network is a registration field, $\\phi$, that can be applied to the moving images to get the next motion-compensated images, $\\left \\{ S_b \\right \\}_{b \\in B}^{(k+1)} $. The registration is done between corresponding b-values images from the moving and fixed images. For example, the moving image $S_0$ will be registered to the fixed image $R_0$. \nThis way, we overcome the differences in contrast between b-values and use prior knowledge that the resulting compensated image needs to follow the mono-exponential model.\n  \n\n\n\\subsection{Convergence Criteria}\nIn each iteration, we calculate the averaged ADC of the lung at the given region of interest (ROI). We quantify the ADC by the IRLS algorithm from Eq.~\\ref{eq:IRLS}.\nWe define convergence of the model after five iterations without change in the calculated ADC.\nWe leverage our prior knowledge of the model (Eq.~\\ref{eq:MonoModel}) to choose the best iteration as the iteration with the highest $R^2$ from the IRLS fitting.\n\n\n\\subsection{Bio-physically-informed Loss Function}\nOur bio-physically-informed loss function consists of 3 terms.\n\\begin{align}\nL = L_{similarity} + \\alpha_1 L_{smooth} +\\alpha_2 L_{Model\\: fit} \n\\end{align}\n\n\\subsubsection{L$_{similarity}$} is a $L_1$ loss applied to corresponding b-values images from fixed and wrapped moving images, averaged over the $B$ b-values images:\n \\begin{align}\n L(R^{(k)},S^{(k)}\\circ \\phi_i^{(k)}) = \\frac{1}{B} \\frac{1}{|\\Omega|}\\sum_{i\\in B}\\sum_{p\\in \\Omega}|R_i^{(k)}(p)-[S_i^{(k)}\\circ \\phi_i^{(k)}](p)|\n \\end{align}\n where $\\Omega$ is the image dimension. Minimizing $L_{similarity}$ will encourage  $\\left \\{ S_i \\right \\}_{b \\in B}^ {(k)} \\circ \\phi_i^{(k)}$ to approximate  $\\left \\{ R_i \\right \\}_{b \\in B}^{(k)}$, but may generate a non-smooth $\\phi$ that is not physically realistic \\cite{balakrishnan2019voxelmorph}.\n  \n \\subsubsection{L$_{smooth}$} is a regularization term introduced by Balakrishnan et al. \\cite{balakrishnan2019voxelmorph}.\n This loss term encourages a smooth displacement field $\\phi$ using a diffusion regularize on the spatial gradients of displacement $u$: \n \\begin{align}\n L_{smooth}(\\Phi) = \\sum_{p\\in\\Omega} \\left \\|\\nabla u(p)  \\right \\|^2\n \\end{align}\n  \n\n \\subsubsection{L$_{model\\ fit}$} is the loss that responsible for forcing each voxel to act according to the mono-exponential model from Eq.~\\ref{eq:MonoModelLog}. This loss is an MSE loss on the least square residual: \n \n \n \n \\begin{align}\n L_{model\\ fit}(ADC) =  \\frac{1}{B}\\frac{1}{\\left | \\Omega^* \\right |}\\sum_{i\\in B}\\sum_{p \\in \\Omega^*} (log(S_0)-b_i\\cdot ADC-log(S_i))^2\n \\end{align}\n\nWhere $\\Omega^*$ is a subset of the image, including only voxels from the lung, as we assume the mono-exponential model can describe them, contrary to the background, which doesn't present a mono-exponential decay of the DWI signal.\n\n\n\\section{Experiments}\n\\subsection{Clinical Data-set}\nLegacy fetal DWI data was used in the study. The data acquisition was performed on a Siemens 3T Skyra scanner equipped with an 18-channel body matrix coil. Each patient was scanned with a multi-slice, single shot, echo-planar imaging (EPI) sequence that was used to acquire diffusion-weighted scans of the lungs. The in-plane resolution was 2.5mm \u00d7 2.5mm for each study, and the slice thickness was set at 3mm. Echo time (TE) was 60ms, whereas repetition time (TR) ranged from 2s to 4.4s depending on the number of slices required to cover the lungs. Each patient was scanned with 6 different b-values (0, 50, 100, 200, 400, 600 sec\/mm$^2$) in axial and coronal planes. A region of interest (ROI) was manually drawn for each case in the right lung \\cite{Afacan2016FetalAge}. \nWe cropped each image to a shape: $96 \\times 96 \\times 16$ and normalized it by the maximal value at $S_0$. \nThe data includes 38 cases with a range of minor misalignment between the different b-values image volumes.  \n\n\n\n\n\n\\subsection{Experiment Goals}\nThe objectives of the experiment are:\n1) To analyze the effect of quantitative DWI motion-compensation on the correlation between the ADC parameter and GA. 2) To analyze the contribution of our proposed model fitting quality loss in addition to the registration loss in improving the observed correlation between the ADC parameter and GA.\n\n\n\\subsection{Experimental Setup}\n\nOur baseline method is ADC quantification without motion compensation. We compared the baseline method to our suggested method with: 1) Registration loss solely, and 2) Our hybrid loss combining both registration loss and a model fitting quality loss.  \nFor each of the three methods, we quantified the averaged ADC in the right lung of each subject by IRLS algorithm. We used an exponential saturation model suggested by Afacan et al. \\cite{Afacan2016FetalAge} described in Eq.~\\ref{eq:GA_ADC} above.\n\n\n\n\\subsection{Implementation details}\nWe implemented our models on PyCharm 2021.2.3, Python 3.9.12 with PyTorch 1.11.0.\n\n\nWe applied our suggested methods with a batch size of one, meaning that each batch is data from one patient with size: $n_b \\times n_x \\times n_y \\times n_z $, where $n_b$ is the number of b-values used for scanning the patient, and $n_x \\times n_y \\times n_z$ is the image shape. \nWe trained the networks in two epochs so that the earlier cases will be contributed from the later ones. We experimented the affect of training in a wide range of epochs and found that the network converged after the second epoch, and there was no further improvement after that.\nWe used an Adam optimizer with an initial learning rate of 10-4. If the total loss increases compared to the previous iteration, we reduce the learning rate by 10. We limited the number of iterations for each case to 50.\nThe hyper-parameter we used are: $\\alpha_1 = 0.01$, $\\alpha_2 = 1000$. We choose $\\alpha_2$ by a search-grid of the best $\\alpha_2$ in terms of correlation between ADC and GA. For testing the qDWI-Morph method without fit model loss term, we set $\\alpha_2 = 0$. \n\n\\section{Results}\n\n\n\n\n\n\n\n\n\nFig.~\\ref{fig:Case_691_Original} presents an example of DWI data before accounting for motion by the registration. Fig.~\\ref{fig:Case_691_Summery} presents our iterative motion-compensation and model estimation. \nThe top row is the motion-compensation b=0 s\/mm$^2$ images from different iterations, where the top-left image is the input image. The second row is the ADC map result from the LLS layer, and the bottom row is the corresponded reconstructed b=0 s\/mm$^2$ image. The right column is the convergence iteration. The ADC map (middle row) becomes sharper and more clinically-meaningful as the motion is correct during the iterations. However, the final motion-compensated b-value images become more blurry as a result of the registration wrapping process by the linear interpolation. \n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b0.jpeg}\n         \\caption{}\n         \\label{fig:Original b0}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b1.jpeg}\n         \\caption{}\n         \\label{fig:Original b1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b2.jpeg}\n         \\caption{}\n         \\label{fig:Original b2}\n     \\end{subfigure}\n     \\hfill\n          \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b3.jpeg}\n         \\caption{}\n         \\label{fig:Original b3}\n     \\end{subfigure}\n     \\hfill\n          \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b4.jpeg}\n         \\caption{}\n         \\label{fig:Original b4}\n     \\end{subfigure}\n     \\hfill\n          \\begin{subfigure}[b]{0.31\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/Case_867_Original_img_b5.jpeg}\n         \\caption{}\n         \\label{fig:Original b5}\n     \\end{subfigure}\n        \\caption{Sample case with ROI. Top row left to right: b = 0, 50, 100 s\/mm$^2$. Bottom row left to right: b-value = 200, 400, 600 s\/mm$^2$.}\n        \\label{fig:Case_691_Original}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n    \\includegraphics[height=9cm]{figures\/Case_867_Summery.pdf}\n    \\caption{The process done by our method, qDWI-Morph, demonstrated over time. Top row: the motion-compensation b=0 s\/mm$^2$ images. Second row: ADC map result from the LLS layer. Bottom row: the model-reconstructed b=0 s\/mm$^2$ images. \n}\n\t\\label{fig:Case_691_Summery}\n\\end{figure}\n\nFig.~\\ref{fig:ROI 867} demonstrates the averaged signal in the ROI as a function of the b-value at different iterations. The lines are the corresponding fit to Eq.~\\ref{eq:MonoModel}. At the last iteration, after the motion compensation process (\\ref{fig:ROI iter 38}), the dots behave according to the expected model and are less scattered.\n\n\\begin{figure}[t]\n     \\hspace*{\\fill}\n     \\begin{subfigure}[b]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ROI_iter_Case867_0.pdf}\n         \\caption{}\n         \\label{fig:ROI iter 0}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ROI_iter_Case867_19.pdf}\n         \\caption{}\n         \\label{fig:ROI iter 19}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ROI_iter_Case867_38.pdf}\n         \\caption{}\n         \\label{fig:ROI iter 38}\n     \\end{subfigure}\n     \\hspace*{\\fill}\n        \\caption{The averaged signal in the ROI for one case, as a function of b-value at different iterations, with a curve fit to the mono-exponential model: (a) Original data, (b) iteration 19, and (c) iteration 38.}\n        \\label{fig:ROI 867}\n    \n\\end{figure}\n\n\n\n\nFig.~\\ref{fig:ADC_GA_qDWIMorph} shows the ADC parameter plotted against GA for all three methods. The data is color-coded by the degree of conformity to the mono-exponential model from Eq.~\\ref{eq:MonoModel} in terms of $R^2$. The Darker the dot, the higher $R^2$. Our approach achieved the best correlation ($R^2=0.32$) between the ADC and the GA compared to both motion compensation by our qDWI-Morph without adding the model fit loss ($R^2=0.28$) and to baseline approach without motion compensation ($R^2=0.13$).\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.325\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ADC_GA_NoMotion.pdf}\n         \\caption{}\n         \\label{fig:ADC_GA_NoMotion}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.325\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ADC_GA_NoFitLoss.pdf}\n         \\caption{}\n         \\label{fig:ADC_GA_NoFit}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.325\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ADC_GA_qDWI-morph.pdf}\n         \\caption{}\n         \\label{fig:ADC_GA_qDWIMorph}\n     \\end{subfigure}\n     \\caption{Averaged ADC in the right lung vs. GA with the three methods: (a) Baseline; W.O motion compensation. (b) qDWI-Morph W.O model fit loss. (c) qDWI-Morph.The data is color-coded by the degree of conformity to the mono-exponential model from Eq.~\\ref{eq:MonoModel} in terms of $R^2$.}\n        \\label{fig:ADC_GA_qDWIMorph}\n\\end{figure}\n\n\n\\setlength{\\tabcolsep}{2pt}\n\\begin{table}[tp]\n\\begin{center}\n\\caption{The correlation of the averaged ADC in the right lung vs. GA by the three methods: W.O M.C; the baseline method, ADC quantification without motion compensation. W.O F.L; qDWI-Morph without model fit loss, and our method: q-DWI-Morph.}\n\\label{table:GA_ADC_corr}\n\\begin{tabular}{lcccc}\n\\hline\\noalign{\\smallskip}\nMethod & $R^2$ & \\multicolumn{1}{p{2cm}}{\\centering A $\\times 10^{-3}$ \\\\ $(mm^2\/s)$} \n&  \\multicolumn{1}{p{2cm}}{\\centering B $\\times 10^{-3}$ \\\\ $(mm^2\/s)$} & C\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nW.O. M.C & 0.13 & 3.2 & 0.005 & 0.07 \\\\\nW.O. F.L  & 0.28 & 3.17 & 0.006 & 0.07 \\\\\nqDWI-Morph  & {\\bf 0.32} & 3.19 & 0.006 & 0.07 \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\setlength{\\tabcolsep}{1.4pt}\n\n\nTable~\\ref{table:GA_ADC_corr} summarizes the fitted ADC saturation model (Eq.~\\ref{eq:GA_ADC}) using the different approaches along with the correlation between the ADC and GA in terms of $R^2$. The fitted model parameters are bout the same for all methods. However, our qDWI-Morph achieved the best correlation between the ADC and the GA, since the average ADC vs. GA is less sparse, as seen in Fig.~\\ref{fig:ADC_GA_qDWIMorph}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\nWe introduced qDWI-Morph, an unsupervised deep-neural-networks approach for Motion-compensated quantitative Diffusion-Weighted MRI analysis with application in fetal lung maturity assessment. Our model coupled a registration sub-network with a model fitting sub-network to simultaneously estimate the motion between the different b-value images and the signal decay model parameters. The optimization of the model weights is driven by minimizing a bio-physically-informed loss representing both the registration loss and the fit quality loss. The integration of the fit quality loss encourages the model to produce deformation fields that will lead to bio-physically expected behavior of the signal along the b-value axis, in addition to the standard registration loss, which encourages similarity and smoothness within the deformation field of each b-value image. Our experiments demonstrated an added-value of adding the model-fitting loss in addition to the registration loss for fetal lung maturity assessment. The proposed approach can potentially improve our ability to quantify DWI signal decay model parameter in cases with motion.\n \n\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\nQuiescent galaxies (QGs) are defined as galaxies with their star formation rates (SFR) lower than the average SFR of star-forming galaxies (SFGs) of similar stellar masses at similar redshifts \\citep[i.e., below the ``main sequence'';][]{Daddi2007,Elbaz2007,Noeske2007,Wuyts2011,Ciambur2013}. The quenching mechanism of QGs is an important topic, especially for high-redshift ones. Observations showed that populations of massive QGs exist at $z \\sim$ 1.0--2.0 \\citep{Belli2017, Carnall2019, Newman2018}, and that half of the most massive QGs were formed at $z \\sim$ 1.5 \\citep{Ilbert2013, Muzzin2013}. In recent studies, the QG population has been extended to $z \\sim$ 4.0 \\citep{Straatman2014, Glazebrook2017, Merlin2018, Schreiber2018a, Carnall2020, Forrest2020a, Forrest2020b, Valentino2020}. Among these, \\citet{Valentino2020} reported three QGs with stellar masses around $10^{11}$ M$_{\\odot}$ and with SFR ranging from 1.1 to 24.0 $M_{\\odot}$ year$^{-1}$ (1.0 to 2.1 $\\sigma$ below the main sequence) at $z$ = 3.775, 4.012, and 3.767.  How these high-$z$ QGs can increase their mass and quench the star formation in a short period is still not fully understood. \n\nActive galactic nucleus (AGN) feedback is one of the proposed scenarios to explain the rapid quenching \\citep{Bower2006, Croton2006, Somerville2008, Fabian2012, Man2018}. AGN activities may either provide kinetic energy to the interstellar medium in the host galaxies and reduce the star formation efficiency, or heat up the gas to prevent the gas from cooling \\citep[radio mode AGN feedback;][]{Croton2006, Bower2006, Fabian2012, Somerville2008}. AGN activities may also remove the gas and terminate the star formation \\citep[quasar mode AGN feedback;][]{Fabian2012, Somerville2008}.\n\nHowever, there are also other theoretical scenarios for the quenching mechanism. For example, the existence of turbulence may also provide kinetic energy to the system and result in morphological quenching \\citep{Martig2009,Dekel2009}. Another example would be positive AGN feedback. In this case, AGN activities enhance star formation but rapidly consume the gas, resulting in a quenched galaxy \\citep{Ishibashi2012, Zhuang2020, Shangguan2020}. Other theoretical scenarios include mergers, stellar feedback, virial shock heating, etc \\citep[see discussion in][and references therein]{Man2018}, and the quenching of massive galaxies could be a combination of some of these scenarios. Which of the mentioned scenarios is the dominant channel of the quenching mechanism remains unclear. \n\nHere we would like to first focus on the foundation of the QG studies, the selection criteria of QGs. Rest-frame color-color diagrams are widely applied to selecting QGs. They are often composed of two rest-frame colors: a UV-to-optical color (typically as the $y$-axis) to distinguish blue SFGs from red QGs using the strong UV emission from young stars, and an optical-to-near-infrared color (typically as the $x$-axis) to distinguish old passive stellar populations from dusty\/reddened young stellar populations. Such photometric selections are very convenient since only photometric data are needed. There are various kinds of color-color diagrams proposed for QG selections based on rest-frame absolute magnitudes, such as the $U$--$V$--$J$ diagram \\citep{Wuyts2007,Williams2009,Muzzin2013}, the $NUV$--$r$--$J$ diagram \\citep{Ilbert2013}, and the $NUV$--$r$--$K$ diagram \\citep{Arnouts2013}. \n\nDespite the great success of using color-color diagrams to select large samples of QGs, there are potential issues in such selections. First, the selection boundary that separates QGs and SFGs in a color color diagram was decided empirically \\citep[e.g.,][]{Ilbert2013, Muzzin2013, Williams2009}. Whether a set of selection criteria are still applicable to different datasets should be examined. Second, either because of the intrinsic properties of SFGs and QGs, or because of photometric errors, the distribution of the two groups of galaxies can have unknown levels of overlap around the selection boundaries. It was found that adjusting the position of the boundary by $\\pm$0.1 mag could greatly change the selection efficiency and the subsequent analyses based on the selected samples \\citep[][Appendix B therein]{Muzzin2013}.\n\nOne important factor here is that our understanding of dusty galaxies is limited. The spectral energy distribution (SED) of these high-$z$ dusty galaxies may be more complicated than what was initially assumed to setup color selection.  The selected QG ``candidates'' may still suffer from contamination by red dusty SFGs at high $z$. For instance, a $z$ = 3.717 QG candidate \\citep{Straatman2014,Glazebrook2017} was detected at 450 and 870 $\\mu$m \\citep{Simpson2017}. This implies that the target is a dusty SFG, or is an interacting system consisting of a QG and a dusty galaxy \\citep[e.g.,][]{Schreiber2018a}, or at least contains a significant dusty star-forming component. Such contamination may lead to an overestimated number density of QGs, especially at high $z$, which may have consequences in our pursuit of an understanding of the quenching mechanism.\n\nTherefore, in this paper, we would like to revisit the selection of QGs using color-color diagrams. We will examine the quiescence of our color-selected QG candidates using submillimeter observations. Various previous studies were carried out to analyze the star formation of color-selected QGs. Some studies measured the SFR by either applying SED fittings from the UV to mid-infrared bands \\citep{Fumagalli2014, Merlin2018, Carnall2019, Toba2019}, using spectroscopic data \\citep{Schreiber2018,Belli2017}, or measuring H$\\alpha$ emission \\citep{Belli2017b}. Others searched for far-infrared dust emissions using $Spitzer$ observations \\citep{Fumagalli2014, Man2016, Gobat2018, Magdis2021}, $Herschel$ observations \\citep{Viero2013, Man2016,Straatman2014, Merlin2018, Gobat2018, Magdis2021} or Atacama Large Millimeter\/submillimeter Array (ALMA) observations \\citep{Santini2019, Schreiber2018, Simpson2017}. Others also searched for gas content in QG candidates \\citep{Sargent2015,Young2011}. In particular, \\citet{Man2016} measured the dust emission of their color-selected QGs by stacking $Herschel$ SPIRE data at 250, 350, and 500 $\\mu$m (FWHM $\\simeq$ $18\\arcsec$.2, $24\\arcsec$.9, and $36\\arcsec$.3), and they claimed that contamination of dusty SFG is $\\sim$ 15 \\% among their QG candidates. We will follow the approach in \\citet{Man2016} but re-examine the issue with higher angular resolutions using JCMT SCUBA-2 450 and 850 $\\mu$m data (FWHM = $7\\arcsec$.9 and $13\\arcsec$) and ALMA data.\n\nIn this study, we selected 18,304 QG candidates using the $NUV$--$r$--$J$ diagram with deep galaxy samples from the COSMOS field \\citep{Laigle2016} and analyzed the properties of the selected QG candidates. In the first part of our study, we estimated the contamination of dusty SFGs among the QG candidates by cross-matching them to the multi-wavelength catalogs as well as performing stacking analyses in the submillimeter images. We also estimated the effect of chance projection in the cross-matching and estimated the degrees of small-scale clustering. In the second part of our study, we further investigated the AGN feedback as a potential quenching mechanism among QG candidates. We examined the relation between various AGNs and the QG candidates in our data by calculating the QG fractions in the AGN samples.\n\nWe describe our data in Section~\\ref{sec:data} and introduce the QG color-color selection in Section~\\ref{sec:QG_selection}. In Sections~\\ref{sec:bright_SMG} and \\ref{sec:faint_SMG}, we analyze the contamination of dusty SFGs among the $NUV$--$r$--$J$ selected QG candidates. We also examine the small-scale clustering between dusty SFGs and QG candidates in Section~\\ref{subsec:blind_matching}. In Section~\\ref{sec:AGN_properites}, we discuss the QG fractions among our AGN samples. Section~\\ref{sec:summary} gives a summary of our results. We use the \\citet{Chabrier2003} initial mass function (IMF) and an H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\\Omega_\\Lambda$ = 0.7, and $\\Omega_m$ = 0.3 cosmology throughout this study.\n\n\\section{Multi-wavelength Data} \\label{sec:data}\n\n\\subsection{COSMOS2015 Catalog} \\label{subsec:COSMOS2015}\n\nWe selected galaxies from the multi-wavelength band-merged COSMOS2015 catalog \\citep{Laigle2016}. To use the rest-frame absolute magnitudes for our color selection, we excluded those labeled as failure in SED fitting to avoid bad absolute magnitudes.  We also excluded samples that are labeled as stars.  We excluded galaxies with extreme values in the catalog ($NUV$, $r$, and $J$ absolute magnitudes that are $<-30$ or $>0$, and negative redshifts), which are likely caused by either catastrophic failures in SED fitting or problematic photometry. These selection criteria reject $\\sim57\\%$ (677,085\/1,182,108) of the initial sample.\n\n We further limited the errors of the magnitudes in $K_S$ band to be lower than 0.2. This uniform selection ensures that our sample has a robust set of photometry and avoids biasing against high-$z$ sample. The limiting magnitude is 23.7 for $K_S$ band. The selection criterion of the $K_S$ band magnitude error further rejects $\\sim29\\%$ (344,806\/1,182,108) of the initial sample. Overall, the majority of the rejections are caused by their faintness. They either are not detected at $K_S$ or have $K_S>24$.\n\nWith the above selection criteria, we obtained a total sample size of 160,217 galaxies from the COSMOS2015 catalog. They all have high-quality SED fitting results; all of them have SED fitting based on at least nine filters, and 98\\% of them more than 28 filters. The sample covers an area of 1.58 deg$^2$ in the COSMOS field (Fig.~\\ref{fig:footprint}). We used stellar mass $M_*$, photometric redshift $z$, and rest-frame absolute magnitudes $M_{NUV}$, $M_r$, and $M_J$ from the COSMOS2015 catalog, which were derived from SED fittings. The sample has stellar masses up to $M_* = 10^{12}~M_{\\odot}$ and redshifts over $z\\sim4$ (Fig.~\\ref{fig:data} (b) and (c)). The absolute magnitudes will be applied for our QG selection in the next section.\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{footprint.png}\n\\caption{\\label{fig:footprint}Coverage maps of the COSMOS field. The background shows the S2COSMOS 850 $\\mu$m image. The black polygon corresponds to the coverage of our $K_s$ band selected COSMOS2015 sample, while the white circle corresponds to the 151-arcmin$^2$ coverage of STUDIES 450 $\\mu$m image. The MIPS 24 $\\mu$m and VLA 3 GHz catalogs cover the whole area of the black polygon and are not shown in this figure.}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n\\epsscale{1.15}\n\\plottwo{NUVrJ_1.png} {mass_z.png}\n\\plottwo{hist_all.png}{hist_QG.png} \n\\caption{\\label{fig:data}The full galaxy sample selected from the COSMOS2015 catalog. Panel (a) shows the distribution in the $NUV$--$r$--$J$ diagram, and the QG sub-sample is selected in upper-left corner. The fringe structure in the QG population may be caused by either the lack of QG template or by certain procedures in the SED fitting, but the structure does not affect our result. The reddening vector derived from \\citet{Calzetti2000} extinction is shown in the lower-right corner, while the typical (median) error in the two colors for all the sources is shown in the lower-left corner. Panel (b) and (c) show the stellar mass and the redshift distributions of the full galaxy sample (colored in blue), while panel (b) and (d) show those of the QG sub-sample (colored in black).}\n\\end{figure*}\n\n\n\\subsection{Submillimeter Data} \\label{subsec:submm}\n\nWe used submillimeter data in the COSMOS field from JCMT SCUBA-2 \\citep{Holland2013, Holland1999} at 450 $\\mu$m (STUDIES, \\citealt{Wang2017}; final data release in Gao et al.\\ 2021, in prep.) and 850 $\\mu$m (S2COSMOS, \\citealt{Simpson2019}), in order to search for dusty SFGs that contaminate the QG sample. The 450 $\\mu$m map covers the central 151 arcmin$^2$ of COSMOS, while the 850 $\\mu$m map covers the whole COSMOS field (Fig.~\\ref{fig:footprint}). The 450 $\\mu$m and 850 $\\mu$m maps have detection limits of about 3.5 mJy and 2 mJy, respectively. The detection limits are all substantially higher than the confusion noise ($\\sigma_{\\rm c}\\sim0.7$ mJy at 450 $\\mu$m, e.g., \\citealp{Lim2020}; $\\sigma_{\\rm c}\\sim0.5$ mJy at 850 $\\mu$m, e.g., \\citealt{Simpson2019}).\n\nIn total, we selected 357 objects with 450 $\\mu$m detection and 1,147 objects with 850 $\\mu$m detection from the SCUBA-2 maps. Four of the 450 $\\mu$m sources and 166 of the 850 $\\mu$m sources located outside the region occupied by our optically selected sample, because of the difference in area coverage and the masks in the COSMOS2015 catalog (Fig.~\\ref{fig:footprint}). Among the remaining 353 objects with 450 $\\mu$m detection and 981 objects with 850 $\\mu$m detection, 77 and 370, respectively, have ALMA observations from the AS2COSMOS and A3COSMOS catalogs (Section~\\ref{subsec:auxiliary}).\n\nSince the SCUBA-2 maps have relatively low angular resolution, we could not reliably identify the optical counterparts to the submillimeter sources. We therefore include the auxiliary data in section \\ref{subsec:auxiliary} for the process of cross matching COSMOS2015 galaxies to SCUBA-2 sources, and the details will be described in section \\ref{subsec:traditional_matching}.\n\n\n\n\\subsection{Auxiliary Data} \\label{subsec:auxiliary}\n\nWe included $Spitzer$ MIPS 24 $\\mu$m, VLA 3 GHz, and ALMA catalogs in our study for their better astrometry when our submillimeter data do not have sufficient angular resolution and for analyzing QG properties.\n\nFor 24 $\\mu$m data, we used the $Spitzer$ MIPS S-COSMOS image from \\citet{Sanders2007}. In order to generate a catalog deeper than the archival MIPS catalog of \\citet{Sanders2007}, 24 $\\mu$m sources were extracted  using \\texttt{SExtractor} \\citep{Bertin1996}, and their fluxes were re-calibrated to their $Spitzer$ General Observer Cycle 3 total fluxes. Our MIPS 24 $\\mu$m catalog has a 3.5$\\sigma$ detection limit of  57 $\\mu$Jy, in contrast to the flux cut at 150 $\\mu$Jy in \\citet{Sanders2007}. Our catalog is very similar to the catalog of \\citet{LeFloch2009} in terms of total numbers of detections. The fluxes are also consistent within 6\\% \\citep{Lim2020} as we calibrated our fluxes to that of  \\citet{Sanders2007}.\n\nWe cross-matched the COSMOS2015 catalog with our MIPS 24 $\\mu$m catalog using a search radius of $2\\arcsec$, which corresponds to about 1\/3 of the beam size at 24 $\\mu$m. 26,999 galaxies (16.9\\%) are matched to the MIPS 24 $\\mu$m sources.\n\nFor 3 GHz data, we directly adopted the identification of the COSMOS2015 objects in the VLA catalog of \\citet{Smolcic2017a}, which used a search radius of $0\\farcs8$. The 5$\\sigma$ detection limit of the VLA catalog of \\citet{Smolcic2017a} is 2.3 $\\mu$Jy beam$^{-1}$. 6,002 galaxies (3.7\\%) are matched to the VLA 3-GHz sources.\n\nWe also used catalogs derived from ALMA observations, including the AS2COSMOS \\citep{Simpson2020} and A3COSMOS \\citep{Liu2019} catalogs. The AS2COSMOS catalog was derived from the follow-up 343 GHz observations of 186 bright 850 $\\mu$m sources in the S2COSMOS catalog. The AS2COSMOS sources are essentially complete for the S2COSMOS sources above 6.2 mJy; only one S2COSMOS source does not have ALMA detection. The A3COSMOS catalog collects ALMA archival data in the COSMOS field, at wavelengths from 671 to 90.2 GHz. We cross-matched our optical sample with the ALMA catalogs using a search radius of $1\\arcsec$. This search radius should allow us to overcome the intrinsic offsets between starlight and submillimeter emission from dusty galaxies (e.g., 1$\\sigma$ offset of $0\\farcs55$ in \\citealp{Chen2015}).\n\n\n\\subsection{AGN Sample} \\label{subsec:AGN_samples}\n\nWe also examined the AGN properties of our sample. We cross-matched our sample with radio AGNs from the VLA catalog of \\citet{Smolcic2017}, color-selected mid-IR AGNs from \\citet{Chang2017}, and X-ray AGNs selected from $Chandra$ data by \\citet{Civano2016} and \\citet{Marchesi2016}.\n\nThe radio AGNs were selected by comparing the observed radio emission to the expected radio emission from IR-derived SFR. Those exceeding 3$\\sigma$ in $log(L_{1.4GHz}\/\\mathrm{SFR}_{IR})$ are classified as radio AGNs \\citep[see the details in][]{Smolcic2017, Delvecchio2017}. The mid-IR AGNs were selected in the rest-frame mid-IR color-color diagram. Those which exhibit red power-law SEDs in the mid-IR are classified as mid-IR AGNs \\citep[see the details in][]{Chang2017,Lacy2004, Lacy2007, Donley2012}. The X-ray AGNs were selected with X-ray luminosity of $L_{X(2-10keV)}>10^{42}$ ergs s$^{-1}$ \\citep{Zezas1998,Ranalli2003,Szokoly2004}. We note that if such an X-ray luminosity is produced purely by X-ray binaries rather than an AGN, the inferred SFR would be $>200$~$M_{\\odot}$~yr$^{-1}$ using the conversion between SFR and $L_{X}$ \\citep[e.g.,][]{Ranalli2003}. Such a high SFR would be detect in our submillimeter analyses, but we do not observe it. Therefore, the majority of the $L_{X(2-10keV)}>10^{42}$ ergs s$^{-1}$ sources in our samples should be AGN-dominated.\n\n\\section{Color-Color Diagram} \\label{sec:QG_selection}\n\nWe applied the rest-frame $NUV$--$r$--$J$ color-color diagram to our sample in order to select QG candidates. Various color-color diagrams were used for QG selection, including the $U$--$V$--$J$ diagram \\citep{Williams2009} and the $NUV$--$r$--$J$ diagram \\citep{Ilbert2013}. Although the $U$--$V$--$J$ diagram is more widely used than the $NUV$--$r$--$J$ diagram, there are advantages of using $NUV$ and $r$ bands instead of $U$ and $V$ bands \\citep{Ilbert2013}. The $NUV$ band is at a shorter wavelength, so it is more sensitive to emission from young stars and extinction. The $NUV-r$ color has a wider wavelength span than the $U-V$ color, so it is less vulnerable to photometric errors. The rest-frame $NUV$ band can be obtained from optical data toward higher redshifts, around $z >$ 2, where $U$ band starts to enter the near-IR.  This leads to better sensitivities. Because of the above, we adopted the $NUV$--$r$--$J$ diagram in this study. We note that the selection results of using the two color-color diagrams are similar to each other. About 85\\% of our $NUV$--$r$--$J$ selected QG candidates overlap with the $U$--$V$--$J$ selected QG candidates, and the overlapping fraction slightly varies with redshift and the position of the selection boundary.\n\nOn the $NUV$--$r$--$J$ color-color diagram (Fig.~\\ref{fig:data} (a)), a blue color in $y$-axis indicates the starlight from young stars, while the color in $x$-axis breaks the degeneracy between age and dust reddening. QGs tend to locate in the upper-left corner of the diagram. We adopted the criteria proposed by \\citet{Ilbert2013}:\n$$M_{NUV}-M_{r}> 3(M_{r}-M_{J})+1,$$ \n$$M_{NUV}-M_{r}> 3.1.$$\n\nWe selected 18,304 galaxies to be our QG candidates, which are 11.4$\\pm$0.1 \\% of the total (Fig.~\\ref{fig:data}(a)). The selected QG candidates have a redshift distribution peaking at $z\\sim1.0$ and extending to $\\sim3.0$ (Fig. \\ref{fig:data} (d)). Our selection result is well consistent with the flag ``CLASS=0'' in the COSMOS2015 catalog \\citep{Laigle2016}, which applied the same $NUV$--$r$--$J$ selection method. Among the 24 $\\mu$m detected galaxies, 5.9$\\pm$0.1 \\% enter the QG selection region and therefore are QG candidates (Fig. \\ref{fig:NUVrJ243} (a)). Among the 3 GHz detected galaxies, 17.8$\\pm$0.5 \\% are QG candidates (Fig. \\ref{fig:NUVrJ243} (b)). The redshift distributions of the 24 $\\mu$m and 3 GHz detected QGs also peak at $z\\sim1.0$ but have larger fraction of QGs at high $z$  (Fig. \\ref{fig:NUVrJ243} (c)). The QG selection of submillimeter-detected galaxies will be described in Section \\ref{subsec:traditional_matching}. The numbers of selected QG candidates are summarized in Table \\ref{tab:data}.\n\nTo better understand the diagrams, we show the reddening vector and the typical (median) errors of the two colors in Fig.~\\ref{fig:data} (a), Fig.~\\ref{fig:NUVrJ243} (a), and Fig.~\\ref{fig:NUVrJ243} (b). The reddening vector is derived from \\citet{Calzetti2000} extinction. For the magnitude errors, unfortunately the COSMOS2015 catalog does not provide errors in the absolute magnitudes. To have a rough idea of the errors, we followed the COSMOS2015 procedure (O.\\ Ilbert \\& I.\\ Davidzon, private communication) to select the nearest broad-band filter in the rest-frame that has a photometric error of $<0.3$. And we use the photometric error of that filter band as the absolute magnitude error. This clearly does not account for the errors in the $K$-corrections derived from the fitted SEDs, nor the errors propagated from the photo-$z$ errors, but should still include a substantial part of the error budget.\n\nWith the above-estimated photometric errors, we could further estimate the fraction among the QG candidates that may originate from the SFG color space and scattered into the QG color space by the photometric errors. For each QG candidate, we generated 1,000 randomly perturbed $NUV$, $r$, and $J$ band absolute magnitudes that follow Gaussian distributions according to the photometric errors. We then calculated the percentage of the perturbed colors that are located in the SFG region, i.e., the probability of the QG candidate to be selected as a SFG if there were no photometric errors. The average probability among our QG candidates is $\\sim$7.5\\%, meaning that $\\sim$7.5\\% of our selected QG may have moved from the SFG color space across the boundary into the QG color space due to their photometric errors.  The probabilities can help to understand the nature of dusty SFG contamination in the color-selected QG population. We will further discuss this in Sections \\ref{sec:bright_SMG} and \\ref{sec:faint_SMG}.\n\n\\begin{figure}[ht!]\n\\epsscale{1.05}\n\\plotone{NUVrJ_24.png}\n\\epsscale{1.05}\n\\plotone{NUVrJ_3.png}\n\\epsscale{1.05}\n\\plotone{hist_243.png}\n\\caption{\\label{fig:NUVrJ243}Distribution of the 24-$\\mu$m detected sample (a) and the 3-GHz detected sample (b) in the $NUV$--$r$--$J$ diagram. The reddening vector derived from \\citet{Calzetti2000} extinction and the typical errors in the two colors are also shown, as in Fig.~\\ref{fig:data} (a). The typical errors are smaller than that of the full galaxy sample (Fig.~\\ref{fig:data}(a)). This is resulted from the higher fractions of bright galaxies among the two subgroups (median $r$ $\\sim0.5$ magnitudes brighter).} The QG candidates are selected in upper-left corner of the panels, and the redshift distribution of the two QG subgroups are shown in (c). We note that the two subgroups have partial overlap between each other.\n\\end{figure}\n\n\\begin{deluxetable*}{l|llccc}\n\\tablecaption{\\label{tab:data}Sample sizes and results of multi-wavelength cross-matching.}\n\\tablehead{\n\\colhead{} & \\colhead{SFGs+QGs} & \\colhead{QGs} & \\colhead{QGs\/(all SFGs+QGs)} & \\colhead{QGs\/(all QGs)} & \\colhead{QGs by chance projection}\n}\n\\startdata\ntotal in the COSMOS field & 160217 & 18304 & 11.4$\\pm$0.1 \\% & 100 \\% & - \\\\\n24 um detected & 26999 & 1596 & 5.9$\\pm$0.1 \\% & 8.72$\\pm$0.22 \\% & 382 \\\\\n3 GHz detected & 6002 & 1066 & 17.8$\\pm$0.5 \\% & 5.82$\\pm$0.18 \\% & - \\\\\n850 $\\mu$m detected & 653 & 30 & 4.6$\\pm$0.8 \\% & 0.16$\\pm$0.03 \\% & 7.0\\\\\n~~~850 $\\mu$m + ALMA & 289 & 11 & 3.8$\\pm$1.1 \\% & - & 1.3\\\\\n~~~850 $\\mu$m + 24 $\\mu$m + 3 GHz & 364 & 19 & 5.2$\\pm$1.2 \\% & - & 5.8\\\\\n\\hline\ntotal in the STUDIES field & 15296 & 1846 & 12.1$\\pm$0.3 \\% & 100 \\% & - \\\\\n450 $\\mu$m detected & 239 & 8 & 3.3$\\pm$1.2 \\% & 0.43$\\pm$0.15 \\% & 2.5\\\\\n~~~450 $\\mu$m + ALMA & 58 & 2 & 3.4$\\pm$2.4 \\% & - & 0.3\\\\\n~~~450 $\\mu$m + 24 $\\mu$m + 3 GHz & 181 & 6 & 3.3$\\pm$1.4 \\% & - & 2.1\\\\\n\\hline\nradio AGN & 1378 & 563 & 40.9$\\pm$1.7 \\% & 3.08$\\pm$0.13 \\% & - \\\\\nmid-IR AGN & 791 & 95 & 12.0$\\pm$1.2 \\% & 0.52$\\pm$0.05 \\% & - \\\\\nX-ray AGN & 2267 & 413 & 18.2$\\pm$0.9 \\% & 2.26$\\pm$0.11 \\% & - \\\\\n\\enddata\n\\tablecomments{The errors are set to be Poissonian, and only reflect the uncertainties caused by the finite sample sizes. The 850 $\\mu$m and 450 $\\mu$m detected samples are determined through both the low-resolution SCUBA-2 data and the high resolution auxiliary data. The auxiliary data are either ALMA data, or 24 $\\mu$m and 3 GHz data (see Section \\ref{subsec:traditional_matching} for details).}\n\\end{deluxetable*}\n\nFrom Fig. \\ref{fig:NUVrJ243} (a), we can see that most of the 24 $\\mu$m detected QG candidates tend to distribute close to the selection boundary in the diagram. They can be either dusty galaxies entering the QG color space because of atypical SED shapes, simply regular SFGs scattered into the QG color space because of photometric errors (cross in Fig. \\ref{fig:NUVrJ243} (a)), or chance projections in the cross matching. By measuring the search area of matching through $2\\arcsec$ search radius, we estimated that 382$\\pm$20 out of the 1596 matches (23.9$\\pm$1.2\\%) can be chance projections. In Table \\ref{tab:data}, 24 $\\mu$m detected galaxies have lower QG fractions than that of all the COSMOS2015 sample. 24 $\\mu$m sources are sensitive to dust emission, and the low fraction suggests a low dusty-galaxy contamination in the QG color selection.\n\nOn the other hand, the 3 GHz detected QG candidates distribute well into the QG selection region in Fig. \\ref{fig:NUVrJ243} (b). If we calculate their vertical distances to the selection boundary, we obtain median values of 0.4 and 0.7 for the 24 $\\mu$m and 3 GHz detected QG candidates, respectively. The median distance for the 3 GHz detected QGs is much larger than the typical photometric error (cross in Fig. \\ref{fig:NUVrJ243} (b)), so they are not SFGs scattered into the QG color space. A large fraction of them should be real QGs harboring radio AGNs (see Section \\ref{sec:AGN_properites} and Fig.~\\ref{fig:NUVrJ_AGN} for further evidence). In Table \\ref{tab:data}, the QG fraction among them is considerably higher than those of all the other subgroups. This gives us a hint about the correlation between radio AGN and QG candidates, which will be discussed in Section \\ref{sec:AGN_properites}.\n\n\\section{Bright Submillimeter Galaxies Among QG Candidates} \\label{sec:bright_SMG}\n\nIn this section, we conduct a thorough analysis on the contamination of bright submillimeter galaxies among our QG candidates. In Section \\ref{subsec:traditional_matching}, we cross-matched our sample with the SCUBA-2 450 $\\mu$m and 850 $\\mu$m catalogs using the positions of MIPS 24 $\\mu$m, VLA 3 GHz, and ALMA submillimeter sources. In Section \\ref{subsec:blind_matching}, we further performed a blind cross-matching and reported the finding of small-scale clustering between QG candidates and SCUBA-2 sources.\n\n\\subsection{Traditional Cross Matching} \\label{subsec:traditional_matching}\n\n\\subsubsection{Counterpart Identification using Auxiliary Data} \\label{subsubsec:traditional_matching_process}\n\nWe have searched for MIPS 24 $\\mu$m, VLA 3 GHz, and ALMA counterparts in the COSMOS2015 catalog with data presented in Section \\ref{subsec:auxiliary}. We can therefore search for the optical counterparts to the low-resolution SCUBA-2 submillimeter sources by including the high-resolution multi-wavelength information. Such a two-step counterpart identification method is traditionally used on SCUBA-2 sources. In general, this method was shown to be able to pick up some 2\/3 of SCUBA-2 source counterparts \\citep[e.g.,][]{Casey2013,Koprowski2016,Cowie2017,Michalowski2017,An2018,Simpson2020,Lim2020}, but the exact fractions depend on the sensitivity of the high-resolution observations in the mid-IR, submillimeter, or radio. \n\nWe first cross-matched our optical sample with the SCUBA-2 450 $\\mu$m and 850 $\\mu$m sources using a search radius of $4\\arcsec$ and $7\\arcsec$, respectively. The search radii are approximately half of the full width at half maximum of the beams (FWHM = $7\\arcsec$.9 at 450 $\\mu$m and $13\\arcsec$ at 850 $\\mu$m). Such larger search radii (cf.\\ 1\/3 FWHM for the 24 $\\mu$m matching) are required as the SCUBA-2 positional accuracy is more impacted by confusion effects and telescope pointing errors, rather than just the beam sizes. Then, from the matched sample, we narrowed down the optical counterparts by searching for ALMA detected galaxies from the AS2COSMOS and A3COSMOS catalogs (described in Section \\ref{subsec:auxiliary}). For the remaining sources without ALMA detection, we identified their optical counterparts by searching for 24 $\\mu$m and 3 GHz detected galaxies (described in Section \\ref{subsec:auxiliary}). Those without MIPS and VLA counterparts are likely to be at higher redshifts \\citep[$z\\gtrsim3$, see Section 3.3 in][]{Lim2020} and are not the main targets of interest in this paper given the redshift distributions in Fig.~\\ref{fig:data}. We note that when there are multiple sources within the search radius, we consider all of the sources and narrow down the possible counterparts only with multi-wavelength information without considering their distances to the SCUBA-2 position.\n\nThe results of the cross-matching are summarized in Table \\ref{tab:data}. For the SCUBA-2 450 $\\mu$m sources, we matched 58 COSMOS2015 galaxies through ALMA observations and 181 through the MIPS and VLA catalogs. We defined them as 450 $\\mu$m detected galaxies. Two out of the 58 galaxies and six out of the 181 galaxies are selected as QG candidates in the $NUV$--$r$--$J$ diagram (Fig.~\\ref{fig:NUVrJ_submm} (a)). For the SCUBA-2 850 $\\mu$m sources, there are 289 and 364 matches when using the ALMA catalogs and using the MIPS and VLA catalogs. We defined them as 850 $\\mu$m detected galaxies. 11 out of the 289 galaxies and 19 out of the 364 galaxies are selected as QG candidates in the $NUV$--$r$--$J$ diagram (Fig.~\\ref{fig:NUVrJ_submm} (b)).\n\nOne thing worth noting is the distribution of 450 $\\mu$m and 850 $\\mu$m detected QG candidates in the $NUV$--$r$--$J$ diagrams in Fig.~\\ref{fig:NUVrJ_submm}. Although the sample sizes are small here, these QG candidates do not appear to have a tendency of locating near the selection boundaries (cf.\\ the 24 $\\mu$m case in Fig.~\\ref{fig:NUVrJ243} (a)), comparing to the typical color errors (crosses in Fig.~\\ref{fig:NUVrJ_submm}). This suggests that most of them are systems that consist of a quiescent component that dominates the rest-frame UV\/optical emission and a dusty component that shows up in the far-IR. This can be either an interacting system like the one in \\citet{Simpson2017} and \\citet{Schreiber2018a}, or a foreground quiescent galaxy which lenses a background dusty galaxy. Indeed, one of the QG candidate is matched to both 450 and 850 $\\mu$m sources through ALMA observations. This target is likely to be a lensed system (star symbol in Fig.~\\ref{fig:NUVrJ_submm}), and we will further discuss it in the end of this section and in Appendix~\\ref{appendix:lensed}.\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{NUVrJ_450.png}\n\\plotone{NUVrJ_850.png}\n\\caption{Distributions of the 450 $\\mu$m (a) and 850 $\\mu$m (b) detected sample in the $NUV$--$r$--$J$ diagram. The filled circles are samples matched to ALMA sources, while the empty circles are samples matched to 24 $\\mu$m or 3 GHz sources. The star symbols show the position of the lensed system described in Appendix~\\ref{appendix:lensed}. The reddening vector derived from \\citet{Calzetti2000} extinction and the typical errors in the two colors for the submillimeter sources are also shown with arrows and crosses, respectively.  The color errors of the 850 $\\mu$m sources are larger because these sources are generally fainter in the optical than 450 $\\mu$m sources. \\label{fig:NUVrJ_submm}}\n\\end{figure}\n\nFrom the numbers of QG candidates that have 450 or 850 $\\mu$m detections, we could estimate the fraction of the bright submillimeter galaxies among our QG candidates. The results show that 0.43$\\pm$0.15\\% (8\/1,846) and 0.16$\\pm$0.03\\% (30\/18,304) of our QG candidates are bright 450 $\\mu$m and 850 $\\mu$m sources, respectively (Table \\ref{tab:data}). The fraction of 450 $\\mu$m detected QGs is slightly ($\\sim1.8\\sigma$) larger than that of 850 $\\mu$m ones. This may be a result of either the better luminosity sensitivity or the higher source density at 450 $\\mu$m.  The former allows us to detect more QGs at 450 $\\mu$m, while the latter increases the probability of chance projection between unrelated QGs and 450 $\\mu$m sources. If we remove the expected number of chance projections (Section~\\ref{subsubsec:chance_projection}), then the difference reduces to $\\sim1.3\\sigma$. So the reason of the difference between the fractions at 450 and 850 $\\mu$m remains unclear under our sample sizes.\n\nWe further spilt the populations into redshift bins (Table~\\ref{tab:brightSMG} and Fig.~\\ref{fig:BSMGfraction}). We can see that the fraction of 850 $\\mu$m detected QG candidates increases with redshift and rises up to 3.51$\\pm$2.48\\% at $z>$ 2. This higher contamination rate at $z>2$ could come from either a real redshift evolution, or simply larger photometric uncertainties on high-redshift sources. Nevertheless, this few-percent contamination rate is still quite low. In conclusion, our QG candidates could be contaminated by bright dusty SFGs at a 0.16\\% to 0.43\\% level, and the contamination rises up to $\\sim$ 1.7\\% to 3.5\\% at higher redshift. We note that the contamination rates may be underestimated since we may not pick up all SCUBA-2 source counterparts in the two-step counterpart identification. We will perform a ``blind'' cross-matching in Section \\ref{subsec:blind_matching} to provide a different estimate of the contamination.\n\nWe analyze the role of photometric errors in the bright SMG contamination. In Section~\\ref{sec:QG_selection}, we estimated the probability of intrinsically being in the SFG color space but scattered into the QG color space by photometric errors for each QG candidate. The mean probabilities are 9.6\\% and 6.2\\% for 450 and 850 $\\mu$m detected QGs, respectively. These both account for less than 10\\% of the SMG contaminations. Therefore, the bright SMG contamination is mainly due to intrinsic properties of the QGs rather than photometric errors.\n\n\\begin{deluxetable*}{l|ccc|ccc}\n\\tablecaption{\\label{tab:brightSMG}Percentage of bright submillimeter galaxies (sub-mm detected QGs) among COSMOS2015 QGs.}\n\\tablehead{\n\\colhead{} & \\colhead{total in the} & \\colhead{850 $\\mu$m detected} & \\colhead{percentage} & \\colhead{total in the} & \\colhead{450 $\\mu$m detected} & \\colhead{percentage} \\\\\n\\colhead{} & \\colhead{COSMOS field} & \\colhead{} & \\colhead{} & \\colhead{STUDIES field} & \\colhead{} & \\colhead{}\n}\n\\startdata\nall          & 18304 & 30 & 0.16$\\pm$0.03 \\% & 1846 & 8 & 0.43$\\pm$0.15 \\% \\\\\n$z\\leq$ 1    & 11562 & 8  & 0.07$\\pm$0.02 \\% & 1314 & 5 & 0.38$\\pm$0.17 \\% \\\\\n1$< z\\leq$ 2 & 6045  & 10 & 0.17$\\pm$0.05 \\% & 475  & 1 & 0.21$\\pm$0.21 \\% \\\\\n$z>$ 2       & 697   & 12 & 1.72$\\pm$0.50 \\% & 57   & 2 & 3.51$\\pm$2.48 \\% \\\\\n\\enddata\n\\tablecomments{The errors are set to be Poissonian. The 850 $\\mu$m and 450 $\\mu$m detected samples are determined through both the low-resolution SCUBA-2 data and the high resolution auxiliary data.}\n\\end{deluxetable*}\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{BSMGfraction.png}\n\\caption{Percentage of bright submillimeter galaxies (450 and 850 $\\mu$m detected QGs) among COSMOS2015 QGs (Table \\ref{tab:brightSMG}) in  logarithmic scale. The data points of 450 $\\mu$m detected QGs are slightly offset along $x$-axis for clarity. The error bars of the 450 $\\mu$m detected QGs are larger because of the smaller coverage of the STUDIES map. The errors are Poissonian.\\label{fig:BSMGfraction}}\n\\end{figure}\n\nIn the above, we used both the AS2COSMOS and the A3COSMOS catalogs during the cross-matching. We can also estimate the contamination by matching QG candidates to only the AS2COSMOS catalog, which contains a homogeneous selection and complete observations of SCUBA-2 850 $\\mu$m sources with $S_{850 \\mu m}>$ 6.2 mJy in the S2COSMOS map. If we match our QG candidates to 850 $\\mu$m sources through only the AS2COSMOS catalog, we found 7 galaxies to be 850 $\\mu$m detected QG candidates. If we assume the same QG fraction for all SCUBA-2 850 $\\mu$m sources, we estimate that there should be 36.9$\\pm$14.0 QG candidates. This accounts for 0.2$\\pm$0.1\\% among all the QG candidates. This agrees with the 0.16$\\pm$0.03\\% contamination mentioned above.  \n\nFurthermore, the SCUBA-2 catalog has a detection limit of 2 mJy but is not complete for sources above 2 mJy.  The complete number of 850 $\\mu$m sources can be estimated from the sources counts corrected for completeness, from \\citet{Simpson2019}.  If we estimate the complete number of sources above 2 mJy and assume the same QG fraction in the AS2COSMOS catalog, we obtain a dusty galaxy contamination rate of 0.6$\\pm$0.3\\%. The relative uncertainty here is slightly larger than that simply propagated from the number of QGs in the AS2COSMOS catalog since the source counts also contain an uncertainty. Nevertheless, this value is larger than the above-estimated value of 0.16$\\pm$0.03\\% and is probably a more realistic estimate if we do have a deeper and more complete survey at 850 $\\mu$m.\n\nWe note that one out of the 11 QG candidates (ID = 659416 in the COSMOS2015 catalog) that are matched to SCUBA-2 850 $\\mu$m sources through ALMA catalogs is likely to be a lensed system because of its unusual submillimeter\/radio flux ratio (see Appendix~\\ref{appendix:lensed} for details). This example demonstrates that when matching QG candidates to the submillimeter sources, a match does not imply the QG candidate and the long-wavelength source to be the same object. They could be physical associations such as the lensed system here, or an interacting galaxy pair consisting of a QG and a dusty object \\citep[e.g.,][]{Schreiber2018a}. Based on our small sample size, the probability for such association is about 9\\% (1\/11). Such spatial correlation effects caused by lensing or galaxy interaction will be further discussed in Section \\ref{subsec:blind_matching}.\n\n\\subsubsection{Effect of Chance Projection}\n\\label{subsubsec:chance_projection}\n\nGiven the small numbers of matched objects in the previous section, we would like to examine whether the matches between our QG candidates and bright submillimeter sources are caused by chance projection or by real spatial correlation. We could estimate the effect of chance projection by simple calculations.\n\nFirst, we calculated the search area in our 2-step cross matching. For the SCUBA-2 sources with ALMA observation, we used a search radius of $1\\arcsec$ for the ALMA sources. For the remaining ones, we used search radii of $2\\arcsec$ and $0.8\\arcsec$ for 24 $\\mu$m and 3 GHz sources, respectively. If a 3 GHz source located within the $2\\arcsec$ search radius of a 24 $\\mu$m source, we only adapted the search area of the 3 GHz source. We then calculated the expected fraction of randomly distributed QG candidates locating in the search area with $1-e^{-na\/A}$, where $A$ is the survey area, $n$ is the number of searched sources in the high-resolution catalogs, and $a$ is the search area per source. The estimated numbers of chance projections are given in the last column of Table \\ref{tab:data}. When we matched through ALMA catalogs, the numbers of chance projections is significantly lower. When we matched through 24 $\\mu$m and 3GHz catalogs, the probability of chance projections is about 1\/3; 2.1 out of 6 (35.0\\%) and 5.8 out of 19 (30.5\\%) matches can be chance projections for 450 and 850 $\\mu$m detected QG candidates.\n\nWe conclude that among the submillimeter detected QG candidates mentioned in the previous section, accounting for 0.16\\% to 0.43\\% among our QG candidates, the majority are real physical associations. The estimated bright dusty SFG contamination is not mainly driven by chance projections.\n\n\\subsection{Blind Cross-Matching} \\label{subsec:blind_matching}\n\n\\subsubsection{Matching with Large Radii and Estimate of Chance Projection}\n\nIn the cross-matching aided by 3 GHz and 24 $\\mu$m astrometry described in Secton \\ref{subsec:traditional_matching}, there is a possibility that the real optical counterparts of the submillimeter sources are undetected at 3 GHz and\/or 24 $\\mu$m. The different redshift dependences of sensitivities in the submillimeter, radio, and mid-IR may introduce such a bias.  To avoid this, we can perform a ``blind'' cross-matching to the SCUBA-2 sources. We directly match the QG candidates with SCUBA-2 450 $\\mu$m and 850 $\\mu$m sources using $4\\arcsec$ and $7\\arcsec$ search radii, respectively, without relying on radio and mid-IR positions. The large matching radii here will unavoidably lead to larger numbers of chance projections, so we need to more precisely estimate the number of chance projections.\n\nTo do this, we simulated the matching results using SCUBA-2 submillimeter sources with random positions. Here we do not apply the $1-e^{-na\/A}$ method because the distribution of QGs may not be random at the scale of the relatively large search radii for SCUBA-2 sources and therefore the dispersion in the mean cannot be estimated.  We calculated the expected number of QG candidates located within a search radius from the randomly distributed submillimeter sources and compared the results with the actual number of matched QG candidates. The simulation is repeated 1,000 times. The estimated number of matches and its error are set to be the mean and the 68\\% interval of the 1,000 results.  The results are summarized in Table \\ref{tab:spatial}, and the fractional difference between the expected matches (chance projections) and the actual matches are also shown in Fig. \\ref{fig:spatial_plot}. We note that the detection limit of the SCUBA-2 450 $\\mu$m, 850 $\\mu$m, and ALMA sources are different. We also estimated the probability that the expected number is equal to or larger than the actual number (Table \\ref{tab:spatial}). We show the results of SFGs for comparison in Table \\ref{tab:spatialSFG}.\n\n\\begin{deluxetable*}{cccc|ccccc}\n\\tablecaption{Cross-Matches and Expected Chance Projections between QGs and Submillimeter Sources \\label{tab:spatial}}\n\\tablehead{\n\\colhead{SCUBA-2} & \\colhead{Group} & \\colhead{Match} & \\colhead{Number}\n& \\colhead{Expected} & \\colhead{Actual}\n& \\colhead{Difference} & \\colhead{Fractional} & \\colhead{Probability\\tablenotemark{b}}\\\\\n\\colhead{sources} & \\colhead{} & \\colhead{Radius\\tablenotemark{a}} & \\colhead{}\n& \\colhead{Matched QG} & \\colhead{Matched QG}\n& \\colhead{} & \\colhead{Difference} & \\colhead{}\n}\n\\startdata\n           & all          & $4\\arcsec$  & 353 & \n20.3$^{+ 4.7 }_{- 4.3 }$ &  29  &  8.7$^{+ 4.7 }_{- 4.3 }$ &  42.6$^{+ 22.9 }_{- 21.3 }$\\% & 0.05 \\\\\n450 $\\mu$m & without ALMA & $4\\arcsec$  & 276 & \n16.1$^{+ 3.9 }_{- 4.1 }$ &  24  &  7.9$^{+ 3.9 }_{- 4.1 }$ &  48.9$^{+ 24.1 }_{- 25.5 }$\\% & 0.046 \\\\\nsources    & with ALMA    & $4\\arcsec$  & 77  &\n4.6$^{+ 2.4 }_{- 2.6 }$ &  5  &  0.4$^{+ 2.4 }_{- 2.6 }$ &  9.6$^{+ 53.5 }_{- 56.1 }$\\% & 0.475 \\\\\n           & ALMA sources & $1\\arcsec$  & 85  &\n0.3$^{+ 0.7 }_{- 0.3 }$ &  2  &  1.7$^{+ 0.7 }_{- 0.3 }$ &  534.9$^{+ 217.5 }_{- 100.0 }$\\% & 0.042 \\\\\n\\hline\n           & all          & $7\\arcsec$  & 981 &\n135.0$^{+ 12.0 }_{- 12.0 }$ &  206  &  71.0$^{+ 12.0 }_{- 12.0 }$ &  52.6$^{+ 8.9 }_{- 8.9 }$\\% & 0 \\\\\n850 $\\mu$m & without ALMA & $7\\arcsec$  & 611 &\n84.6$^{+ 10.4 }_{- 9.6 }$ &  120  &  35.4$^{+ 10.4 }_{- 9.6 }$ &  41.8$^{+ 12.2 }_{- 11.4 }$\\% & 0 \\\\\nsources    & with ALMA    & $7\\arcsec$  & 370 &\n50.6$^{+ 7.4 }_{- 7.6 }$ &  86  &  35.4$^{+ 7.4 }_{- 7.6 }$ &  69.8$^{+ 14.5 }_{- 15.1 }$\\% & 0 \\\\\n           & ALMA sources & $1\\arcsec$  & 452 &\n1.3$^{+ 0.7 }_{- 1.3 }$ &  11  &  9.7$^{+ 0.7 }_{- 1.3 }$ &  771.6$^{+ 58.5 }_{- 100.0 }$\\% & 0 \\\\\n\\enddata\n\\tablenotetext{a}{The radius of $1\\arcsec$ to $7\\arcsec$ correspond to 8--56 kpc at $z=1$ and 8--57 kpc at $z=2.5$.}\n\\tablenotetext{b}{The probability that the expected number of matches (based on random spatial distribution) is equal to or larger than that of the actual matches.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{cccc|ccccc}\n\\tablecaption{Cross-Matches and Expected Chance Projections between SFGs and Submillimeter Sources \\label{tab:spatialSFG}}\n\\tablehead{\n\\colhead{SCUBA-2} & \\colhead{Group} & \\colhead{Match} & \\colhead{Number}\n& \\colhead{Expected} & \\colhead{Actual}\n& \\colhead{Difference} & \\colhead{Fractional} & \\colhead{Probability\\tablenotemark{b}}\\\\\n\\colhead{sources} & \\colhead{} & \\colhead{Radius\\tablenotemark{a}} & \\colhead{}\n& \\colhead{Matched SFG} & \\colhead{Matched SFG}\n& \\colhead{} & \\colhead{Difference} & \\colhead{}\n}\n\\startdata\n           & all          & $4\\arcsec$  & 353 & \n149.7$^{+ 12.3 }_{- 12.7 }$ &  466  &  316.3$^{+ 12.3 }_{- 12.7 }$ &  211.3$^{+ 8.2 }_{- 8.5 }$\\% & 0 \\\\\n450 $\\mu$m & without ALMA & $4\\arcsec$  & 276 & \n116.4$^{+ 10.6 }_{- 10.4 }$ &  366  &  249.6$^{+ 10.6 }_{- 10.4 }$ &  214.3$^{+ 9.1 }_{- 9.0 }$\\% & 0 \\\\\nsources    & with ALMA    & $4\\arcsec$  & 77  &\n32.5$^{+ 5.5 }_{- 5.5 }$ &  100  &  67.5$^{+ 5.5 }_{- 5.5 }$ &  207.6$^{+ 16.9 }_{- 16.9 }$\\% & 0 \\\\\n           & ALMA sources & $1\\arcsec$  & 85  &\n2.2$^{+ 1.8 }_{- 1.2 }$ &  56  &  53.8$^{+ 1.8 }_{- 1.2 }$ &  2443.1$^{+ 81.7 }_{- 54.6 }$\\% & 0 \\\\\n\\hline\n           & all          & $7\\arcsec$  & 981 &\n1045.7$^{+ 35.5 }_{- 35.9 }$ &  2087  &  1041.3$^{+ 35.5 }_{- 35.9 }$ &  99.6$^{+ 3.4 }_{- 3.4 }$\\% & 0 \\\\\n850 $\\mu$m & without ALMA & $7\\arcsec$  & 611 & \n652.9$^{+ 28.1 }_{- 27.9 }$ &  1204  &  551.1$^{+ 28.1 }_{- 27.9 }$ &  84.4$^{+ 4.3 }_{- 4.3 }$\\% & 0 \\\\\nsources    & with ALMA    & $7\\arcsec$  & 370 &\n394.0$^{+ 23.0 }_{- 23.0 }$ &  883  &  489.0$^{+ 23.0 }_{- 23.0 }$ &  124.1$^{+ 5.8 }_{- 5.8 }$\\% & 0 \\\\\n           & ALMA sources & $1\\arcsec$  & 452 &\n9.9 $^{+ 3.1 }_{- 2.9 }$ &  277  &  267.1$^{+ 3.1 }_{- 2.9 }$ &  2687.3$^{+ 30.8 }_{- 29.6 }$\\% & 0 \\\\\n\\enddata\n\\tablenotetext{ab}{~The parameters follow those in Table \\ref{tab:spatial}.}\n\\end{deluxetable*}\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{figures\/spatial_4.png}\n\\plotone{figures\/spatial_7.png}\n\\plotone{figures\/spatial_1.png}\n\\caption{Fractional differences between the actual matches and the expected matches based on random spatial distributions (see also Table~\\ref{tab:spatial}). Panels (a), (b), and (c) show results for SCUBA-2 450 $\\mu$m, 850 $\\mu$m, and ALMA sources, where $4\\arcsec'$, $7\\arcsec$, and $1\\arcsec$ search radii are used respectively. It can be seen that the QG results for the SCUBA-2 850 $\\mu$m sources and ALMA sources are all significantly above zero, showing that these extra matches in the actual sample are caused by real physical connection between the QG candidates and the submillimeter sources, rather than chance projection.\\label{fig:spatial_plot}}\n\\end{figure}\n\nThe first and fifth rows of Table \\ref{tab:spatial} and \\ref{tab:spatialSFG} presents simple cross-matching between the QGs\/SFGs and single-dish submillimeter samples without information from ALMA.  For SCUBA-2 450 $\\mu$m sources, the actual matches are 42.6$^{+22.9}_{-21.3}$\\% larger than the expected random matches. Although the significance is only about 2$\\sigma$ (Fig. \\ref{fig:spatial_plot} (a)), the estimated probability that the expected number equals to or is larger than the actual number is 0.05, which is quite low. This result suggests that there are $8.7^{+4.7}_{-4.3}$ QGs physically associated with the 353 450 $\\mu$m sources. This corresponds to $0.47^{+0.25}_{-0.23}\\%$ of the 1846 QGs in the 450 $\\mu$m map area. The statistically derived number of $8.7^{+4.7}_{-4.3}$ matches also nicely agrees with the 8 matches found with high-resolution data (Section~\\ref{subsec:traditional_matching} and Table~\\ref{tab:data}). Also, for comparison, $316.3^{+12.3}_{-12.7}$ SFGs are physically associated with the 353 450 $\\mu$m sources.  This is as expected, since dusty submillimeter sources should be dominated by SFGs.\n\nFor SCUBA-2 850 $\\mu$m sources, the actual matches are 52.6$^{+8.9}_{-8.9} \\%$ larger than the expected random matches (see also \\ref{fig:spatial_plot} (b)), which is significant ($\\sim6\\sigma$). The estimated probability that the expected number equals to or is larger than the actual number is nearly zero. This means that among the 206 matches between the QGs and 850 $\\mu$m sources, $71\\pm12$ are real physical associations. These $\\sim71$ sources account for $0.39\\pm0.07\\%$ of the 18304 QGs. We note that the number of 71 is significantly different from the number of 30 quoted in Table~\\ref{tab:data}, or 23 after removing the expected number of chance projections.  This is because here we do not require high-resolution multi-wavelength data to pin down the cross-matching.  This suggests that a significant fraction of QG--850 $\\mu$m associations in our data do not have 24 $\\mu$m, 3 GHz, or ALMA counterparts. This is perhaps partially because of the insufficient sensitivity (24 $\\mu$m and 3 GHz cases) and incomplete coverage (ALMA cases). However, we will soon show that a large fraction of these 71 sources are clustered around the SCUBA-2 sources at a $\\sim7\\arcsec$ scale, but they are not the submillimeter, mid-IR, or radio emitters. Finally, as in the case for 450 $\\mu$m cross-matching, the overlap between 850 $\\mu$m sources and SFGs is much larger than that for QG, which is expected.\n\n\\subsubsection{Verification and Comparison with ALMA Sources}\n\nThe above ``blind'' matching between QG candidates and SCUBA-2 sources using large matching radii (1\/2 of the single-dish beam FWHM) and the comparison between actual matches and simulations allows us to statistically assess the numbers of real physical associations.  Now with the ALMA data, we can pin down the cross-matching with a much smaller matching radius, with a smaller subsample.  We split the SCUBA-2 sources into those with and without ALMA observations. The results are listed in the remaining rows of Table \\ref{tab:spatial} and \\ref{tab:spatialSFG}.\n\nAs a simple sanity check, we ran cross-matching with large matching radii over the subsamples. The fractional differences between actual and expected random matches do not change significantly between the ALMA and non-ALMA subsamples for QGs (Fig. \\ref{fig:spatial_plot} (a) and (b)). This is even true for the 450 $\\mu$m sample, albeit the small sample sizes and therefore the large errors.  This implies that there is no special selection bias in the ALMA observations regarding their QG-submillimeter properties.\n\nFor the SCUBA-2 sources with ALMA observations, the expected numbers for matches under a $1\\arcsec$ search radius and random spatial distributions are always small comparing to the actual matches. This is reflected on the large fractional differences between the actual matches and expected random matches, which are 534.9$^{+217.5}_{-100.0}$\\% for the 450 $\\mu$m sources and 771.6$^{+58.5}_{-100.0}$\\% for the 850 $\\mu$m sources (Fig.~\\ref{fig:spatial_plot}(c), the fourth and eighth rows).  This means that the majority of the observed matches between QGs and ALMA sources under a $1\\arcsec$ matching radius are real physical associations.\n\nAn interesting comparison is to see if the $1\\arcsec$ matching pinned down by ALMA agrees with the statistical estimates of real physical associations derived from the large-radius blind matching. Table~\\ref{tab:spatial} and \\ref{tab:spatialSFG} show that if we match the 77 SCUBA-2 450 $\\mu$m sources to QGs and SFGs using a $4\\arcsec$ matching radius, we expect $0.4^{+2.4}_{-2.6}$ out of the 5 QG matches and $67.5^{+5.5}_{-5.5}$ out of the 100 SFG matches to be real associations.  These can be compared with the ALMA results for the same sub-sample: 1.7$^{+ 0.7 }_{- 0.3 }$ and 53.8$^{+ 1.8 }_{- 1.2 }$ real associations for the QGs and the SFGs.  The values for QG--SMG associations agree nicely, albeit the small sample size.  This probably validates the statistical method for estimating the number of real associations and chance projections using simulations and random distributions.  On the other hand, the values for SFG--SMG associations (67.5 and 53.8) differ by 25\\% and the difference is about $2\\sigma$. The excess in the number of SFGs around SMGs within $4\\arcsec$ comparing to the number of true associations pinpointed by ALMA suggests a weak clustering of SFGs around SMGs.  This excess is only $2\\sigma$ and is not statistically significant. However, if we look at the 850 $\\mu$m values, the excesses for both QG--SMG and SFG--SMG associations become highly significant.\n\nWe make a similar comparison on the 850 $\\mu$m ALMA subsample.  The expected numbers of real associations under a $7\\arcsec$ matching radius for the 370 850 $\\mu$m sources are $35.4^{+7.4}_{-7.6}$ and $489.0^{+23.0}_{-23.0}$ for QGs and SFGs, respectively. However, the numbers revealed by ALMA observations are much smaller: 9.7$^{+ 0.7 }_{- 1.3 }$ and 267.1$^{+ 3.1 }_{- 2.9 }$.  The differences between the two sets of numbers are both significant.  This implies that once we increase the matching radius from $1\\arcsec$ to $7\\arcsec$ ($\\lesssim 60$ kpc at $z=1$--2), additional clustering effects kick in, i.e., there are QGs and SFGs physically associated with the submillimeter sources at such large scales, but they are not the submillimeter sources themselves nor arcsec-scale galaxy-galaxy lensing pairs. This effect becomes undetectable (QGs) or much weaker (SFGs) under the $4\\arcsec$ matching for the 450 $\\mu$m sources, either because of the small sample sizes for the 450 $\\mu$m analysis or because of the different spatial distribution for low-dust-luminosity sources.\n\nPrevious studies of QG autocorrelation functions found that QGs show stronger clusting signal than SFGs at arcminute scales \\citep{Williams2009}, but there do not exist QG-SFG or QG-SMG cross-correlation analyses. Our results suggest that QGs and SMGs are clustered, and detailed cross-correlation studies between these two distinct populations will be an interesting future topic.\n\nIn summary, with direct cross-matching to SCUBA-2 sources and statistical analyses of chance projection effects, we do not find evidence for a different dusty galaxy contamination rate among QGs comparing to what we found with counterpart identifications using ALMA, 24 $\\mu$m, and 3 GHz data. Instead, we found a clustering effect between the bright submillimeter sources and our QG candidates at scales from $1\\arcsec$ to $7\\arcsec$ ($\\sim8$--60 kpc at $z=1$--2).\n\nWe note that our studies in Section \\ref{subsubsec:traditional_matching_process} and \\ref{subsec:blind_matching} thus far imply several possibilities for the submillimeter detected QG candidates obtained from the cross-matching process in Section \\ref{subsubsec:traditional_matching_process}, depending on the angular scales to which the observations are sensitive. They could be the correct submillimeter counterparts to the QG candidates. There are also situations where the QG candidates are not submillimeter emitters, but are physically associated with submillimeter galaxies through effects like galaxy-galaxy lensing (e.g., Fig.~\\ref{fig:lensing}), galaxy interaction, or clustering effects at scale of a few arcsec.\n\n\\section{Faint Submillimeter Galaxies Among QG Candidates} \\label{sec:faint_SMG}\n\nIn Section \\ref{sec:bright_SMG}, we matched our QGs candidates to submillimeter sources and demonstrated that fractions of the matched QG candidates are physically related to the submillimeter sources. However, the 450 $\\mu$m and 850 $\\mu$m sources have a detection limit of about 3.5 mJy and 2 mJy, respectively, which correspond to SFR of roughly 60 and 180 $M_{\\odot}$ year$^{-1}$ at $z = 1$. Therefore, we further perform stacking analysis in order to search for fainter submillimeter emissions among the QG candidates.\n\n\\subsection{Stacking Analysis}\n\nWe measured the submillimeter emission from the SCUBA-2 maps at the positions of our selected QG candidates and calculated the error-weighted average of their fluxes. As our sources are point-like under JCMT's resolution and the SCUBA-2 maps were beam-matched to produce maximum-likelihood flux for point sources, fluxes are measured by directly reading the map values in Jy beam$^{-1}$ at the positions of the QGs. We excluded QG candidates that we matched to the bright submillimeter sources in Section \\ref{subsec:traditional_matching}, as well as QG candidates whose measured SCUBA-2 fluxes exceed $3\\sigma$, in order to prevent our results from being biased by the small number of bright submillimeter sources. To estimate the bias and uncertainty in such a stacked flux, we then stacked at 1,000 random positions and repeated this 10,000 times. In this process, bright submillimeter sources are removed according to the same criteria as above.  The mean from these random stacks is considered as the bias in stacking. It is consistent with zero, because of the zero-sum nature of the match-filtered SCUBA-2 maps. Nevertheless, this small bias is subtracted from the mean of the QGs. The dispersion among the 10,000 measurements of the random samples is considered as the uncertainty of stacking 1,000 sources. It is scaled by 1\/$\\sqrt{N}$ to be the uncertainty of the QG stacking.  We also stacked different numbers of random sources to verify this 1\/$\\sqrt{N}$ dependence.\n\nThe stacking results are shown in Table \\ref{tab:450stack} and Table \\ref{tab:850stack}.  The first rows of the two tables show that we can reach a $6.3\\sigma$ statistical detection at 850 $\\mu$m if we simply stack all QG candidates, but not a significant detection at 450 $\\mu$m. The non-detection at 450 $\\mu$m may be due to the smaller coverage of the STUDIES map.  Furthermore, we can divide the QG candidates into subgroups according to their properties, to see if there is a particular group of QGs that contributes to the majority of the stacked signal. First, we classified QG candidates either with 24 $\\mu$m counterparts or with 3 GHz counterparts labeled with SFG flags in the VLA catalog \\citep{Smolcic2017} as ``IR-radio-bright'' QGs, and the rests as ``IR-radio-faint'' QGs. Our terminology is similar but slightly different from that in \\citet{Man2016}. \\citet{Man2016} defined QG candidates with SFR derived from 24 $\\mu$m over 100 $M_{\\odot}$ year$^{-1}$ as ``IR-bright'' QGs, and the rests as ``IR-faint'' QGs. They used 24 $\\mu$m data and SFR constraints to classify the subgroups, while we used 24 $\\mu$m data, 3 GHz data, and radio AGN classification in our work. Then, for the 850 $\\mu$m stacking, because of the larger area of the SCUBA-2 map and therefore more available QGs, we can further divide the QG sample into various redshift and stellar mass bins.\n\n\\begin{deluxetable*}{lcccccccc}\n\\tablecaption{450 $\\mu$m QG Stacking Results\\label{tab:450stack}}\n\\tablehead{\n\\colhead{Groups} & \\colhead{log$M_*$\\tablenotemark{a}} & \\colhead{$z$\\tablenotemark{b}} & \n\\colhead{Number} & \\colhead{$S_{450\\rm \\mu m}$} & \\colhead{SNR}      &\n\\colhead{log($L_{\\rm IR}$)} & \\colhead{SFR$_{450\\rm \\mu m}$} & \\colhead{SFR$_{\\rm optical}$\\tablenotemark{c}}\\\\\n& \\colhead{(log($M_{\\odot}$))} & \\colhead{} & \\colhead{} & \\colhead{(mJy)} & \\colhead{} &  \\colhead{(log($L_{\\odot}$))} &  \\colhead{($M_{\\odot}$ yr$^{-1}$)} &  \\colhead{($M_{\\odot}$ yr$^{-1}$)}\n}\n\\startdata\nAll QG                 & 10.7$^{+0.3}_{-1.1}$ & 0.9 & 1799 & 0.06$\\pm$0.05 & 1.3 & 9.3$^{+0.3}_{-0.7}$  & 0.2$\\pm$0.2 & 2.5$^{+0.0}_{-0.0}$  \\\\\n24-$\\mu$m counterpart  & 10.9$^{+0.3}_{-0.7}$ & 0.8 & 155  & 0.66$\\pm$0.16 & 4.1 & 10.6$^{+0.1}_{-0.1}$ & 3.6$\\pm$0.9 & 11.2$^{+0.4}_{-0.3}$ \\\\\n3-GHz counterpart      & 11.2$^{+0.2}_{-0.4}$ & 0.9 & 103  & 0.60$\\pm$0.20 & 3.0 & 10.5$^{+0.1}_{-0.2}$ & 3.5$\\pm$1.2 & 9.5$^{+0.4}_{-0.4}$  \\\\\n3-GHz counterpart: SFG & 11.1$^{+0.2}_{-0.4}$ & 0.9 & 45   & 0.85$\\pm$0.30 & 2.8 & 10.8$^{+0.1}_{-0.2}$ & 6.4$\\pm$2.3 & 10.3$^{+0.8}_{-0.7}$ \\\\\n3-GHz counterpart: AGN & 11.2$^{+0.2}_{-0.3}$ & 0.9 & 58   & 0.39$\\pm$0.26 & 1.5 & 10.2$^{+0.2}_{-0.5}$ & 1.5$\\pm$1.0 & 8.9$^{+0.5}_{-0.6}$  \\\\\nIR-radio-faint QG      & 10.6$^{+0.3}_{-1.1}$ & 0.9 & 1620 & 0.00$\\pm$0.05 & 0.0 & 0.0$^{+9.3}_{-0.0}$  & 0.0$\\pm$0.2 & 1.7$^{+0.0}_{-0.0}$  \\\\\nIR-radio-bright QG     & 11.0$^{+0.3}_{-0.6}$ & 0.8 & 179  & 0.65$\\pm$0.15 & 4.3 & 10.6$^{+0.1}_{-0.1}$ & 3.6$\\pm$0.8 & 10.2$^{+0.4}_{-0.2}$ \\\\\n\\enddata\n\\tablenotetext{a}{Mean and 68\\% interval of stellar mass in logarithmic scale.}\n\\tablenotetext{b}{Median of redshift.}\n\\tablenotetext{c}{Mean of SFRs from COSMOS2015. The error shows the typical error in COSMOS2015 scaled by 1\/$\\sqrt{N}$. Uncertainty of template fitting is not included, which may be large for QG population.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lcccccccc}\n\\tablecaption{850 $\\mu$m QG Stacking Results\\label{tab:850stack}}\n\\tablehead{\n\\colhead{Groups} & \\colhead{log$M_*$\\tablenotemark{a}} & \\colhead{$z$\\tablenotemark{b}} & \n\\colhead{Number} & \\colhead{$S_{850\\rm \\mu m}$} & \\colhead{SNR}      &\n\\colhead{log($L_{\\rm IR}$)} & \\colhead{SFR$_{850\\rm \\mu m}$} & \\colhead{SFR$_{\\rm optical}$\\tablenotemark{c}}\\\\\n& \\colhead{(log($M_{\\odot}$))} & \\colhead{} & \\colhead{} & \\colhead{(mJy)} & \\colhead{} &  \\colhead{(log($L_{\\odot}$))} &  \\colhead{($M_{\\odot}$ yr$^{-1}$)} &  \\colhead{($M_{\\odot}$ yr$^{-1}$)}\n}\n\\startdata\nAll QG                 & 10.7$^{+0.3}_{-1.0}$ & 0.9 & 18011 & 0.06$\\pm$0.01 & 6.3 & 10.0$^{+0.1}_{-0.1}$ & 1.0$\\pm$0.2  & 3.0$^{+0.0}_{-0.0}$\\\\\n24-$\\mu$m counterpart  & 10.9$^{+0.3}_{-0.6}$ & 0.8 & 1538  & 0.27$\\pm$0.03 & 8.2 & 11.1$^{+0.0}_{-0.1}$ & 11.7$\\pm$1.4 & 6.2$^{+0.1}_{-0.1}$\\\\\n3-GHz counterpart      & 11.1$^{+0.2}_{-0.4}$ & 0.9 & 1028  & 0.14$\\pm$0.04 & 3.5 & 10.8$^{+0.1}_{-0.1}$ & 6.8$\\pm$1.9  & 6.1$^{+0.1}_{-0.1}$\\\\\n3-GHz counterpart: SFG & 11.1$^{+0.2}_{-0.5}$ & 0.9 & 473   & 0.30$\\pm$0.06 & 5.0 & 11.1$^{+0.1}_{-0.1}$ & 13.9$\\pm$2.7 & 7.6$^{+0.2}_{-0.1}$\\\\\n3-GHz counterpart: AGN & 11.2$^{+0.2}_{-0.4}$ & 0.9 & 555   & 0.01$\\pm$0.05 & 0.1 & 9.2$^{+1.0}_{-9.2}$  & 0.2$\\pm$1.4  & 4.9$^{+0.1}_{-0.1}$\\\\\n\\hline\nIR-radio-faint QG      & 10.7$^{+0.3}_{-1.0}$ & 0.9 & 16242 & 0.04$\\pm$0.01 & 3.8 & 9.8$^{+0.1}_{-0.1}$ & 0.7$\\pm$0.2 & 2.6$^{+0.0}_{-0.0}$\\\\\n~~~$z\\leq$ 0.5         &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 9.8$^{+0.4}_{-1.6}$  & 0.3 & 2399  &-0.01$\\pm$0.03 &-0.4 & 0.0$^{+9.3}_{-0.0}$ & 0.0$\\pm$0.2 & 0.0$^{+0.0}_{-0.0}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.1}_{-0.3}$ & 0.4 & 632   & 0.07$\\pm$0.05 & 1.4 & 9.8$^{+0.2}_{-0.6}$ & 0.7$\\pm$0.5 & 0.1$^{+0.0}_{-0.0}$\\\\\n~~~0.5 $< z\\leq$ 1.0   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.1$^{+0.3}_{-0.6}$ & 0.8 & 3739  & 0.01$\\pm$0.02 & 0.4 & 9.2$^{+0.6}_{-9.2}$ & 0.2$\\pm$0.4 & 0.4$^{+0.0}_{-0.0}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.2}_{-0.3}$ & 0.8 & 3375  & 0.05$\\pm$0.02 & 2.2 & 9.9$^{+0.2}_{-0.3}$ & 0.8$\\pm$0.4 & 0.3$^{+0.0}_{-0.0}$\\\\\n~~~1.0 $< z\\leq$ 1.5   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.2$^{+0.2}_{-0.3}$ & 1.2 & 1461  &-0.02$\\pm$0.03 &-0.5 & 0.0$^{+9.8}_{-0.0}$ & 0.0$\\pm$0.6 & 2.1$^{+0.0}_{-0.0}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.2}_{-0.3}$ & 1.2 & 2351  & 0.09$\\pm$0.03 & 3.4 & 10.6$^{+0.1}_{-0.2}$ & 3.6$\\pm$1.0 & 1.1$^{+0.0}_{-0.0}$\\\\\n~~~1.5 $< z\\leq$ 2.0   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.2}_{-0.2}$ & 1.7 & 469   & 0.07$\\pm$0.06 & 1.3 & 10.3$^{+0.3}_{-0.7}$ & 2.1$\\pm$1.7 & 12.6$^{+0.5}_{-0.3}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.2}_{-0.3}$ & 1.7 & 1234  & 0.08$\\pm$0.04 & 2.3 & 10.6$^{+0.2}_{-0.2}$ & 3.7$\\pm$1.6 & 8.9$^{+0.1}_{-0.1}$\\\\\n~~~2.0 $< z\\leq$ 2.5   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.2}_{-0.2}$ & 2.3 & 107   & 0.11$\\pm$0.12 & 0.9 & 10.5$^{+0.3}_{-10.5}$ & 2.9$\\pm$3.2 & 25.5$^{+2.5}_{-1.5}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.2}_{-0.3}$ & 2.3 & 260   & 0.25$\\pm$0.08 & 3.2 & 11.1$^{+0.1}_{-0.2}$ & 12.8$\\pm$4.0 & 29.1$^{+1.3}_{-0.8}$\\\\\n~~~$z>$ 2.5            &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.1}_{-0.1}$ & 2.7 & 47    &-0.05$\\pm$0.19 &-0.3 & 0.0$^{+11.0}_{-0.0}$ & 0.0$\\pm$10.0 & 25.5$^{+4.4}_{-2.1}$\\\\\n~~~log$M_*>$ 10.5      & 10.9$^{+0.2}_{-0.3}$ & 2.7 & 168   & 0.11$\\pm$0.10 & 1.1 & 10.8$^{+0.3}_{-0.9}$ & 6.0$\\pm$5.3 & 37.3$^{+2.5}_{-1.4}$\\\\\n\\hline\nIR-radio-bright QG     & 11.0$^{+0.2}_{-0.6}$ & 0.9 & 1769 & 0.26$\\pm$0.03 & 8.6 & 11.1$^{+0.0}_{-0.1}$ & 11.7$\\pm$1.4 & 6.4$^{+0.1}_{-0.1}$\\\\\n~~~$z\\leq$ 0.5         &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.1$^{+0.3}_{-1.5}$ & 0.3 & 90  & 0.28$\\pm$0.13 & 2.0 & 10.8$^{+0.2}_{-0.3}$ & 6.8$\\pm$3.3 & 0.2$^{+0.0}_{-0.0}$\\\\\n~~~log$M_*>$ 10.5      & 11.1$^{+0.2}_{-0.4}$ & 0.4 & 214 & 0.14$\\pm$0.09 & 1.6 & 10.3$^{+0.2}_{-0.4}$ & 2.0$\\pm$1.2 & 0.4$^{+0.0}_{-0.0}$\\\\\n~~~0.5 $< z\\leq$ 1.0   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.2$^{+0.2}_{-0.3}$ & 0.8 & 220 & 0.24$\\pm$0.09 & 2.8 & 11.0$^{+0.1}_{-0.2}$ & 10.5$\\pm$3.8 & 5.1$^{+0.1}_{-0.1}$\\\\\n~~~log$M_*>$ 10.5      & 11.1$^{+0.2}_{-0.4}$ & 0.8 & 730 & 0.13$\\pm$0.05 & 2.7 & 10.5$^{+0.1}_{-0.2}$ & 3.0$\\pm$1.1 & 1.7$^{+0.0}_{-0.0}$\\\\\n~~~1.0 $< z\\leq$ 1.5   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.2}_{-0.3}$ & 1.2 & 44  & 0.39$\\pm$0.19 & 2.0 & 11.3$^{+0.2}_{-0.3}$ & 21.1$\\pm$10.5 & 8.7$^{+1.0}_{-0.5}$\\\\\n~~~log$M_*>$ 10.5      & 11.1$^{+0.2}_{-0.4}$ & 1.2 & 260 & 0.40$\\pm$0.08 & 5.1 & 11.4$^{+0.1}_{-0.1}$ & 22.6$\\pm$4.5 & 3.6$^{+0.2}_{-0.1}$\\\\\n~~~1.5 $< z\\leq$ 2.0   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.1}_{-0.1}$ & 1.6 & 27  & 0.62$\\pm$0.25 & 2.5 & 11.9$^{+0.1}_{-0.2}$ & 80.9$\\pm$32.3 & 28.6$^{+4.9}_{-2.2}$\\\\\n~~~log$M_*>$ 10.5      & 11.0$^{+0.2}_{-0.4}$ & 1.7 & 108 & 0.81$\\pm$0.12 & 6.6 & 12.0$^{+0.1}_{-0.1}$ & 102.3$\\pm$15.6 & 21.1$^{+0.8}_{-0.7}$\\\\\n~~~2.0 $< z\\leq$ 2.5   &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.3$^{+0.1}_{-0.2}$ & 2.2 & 6   & 1.25$\\pm$0.52 & 2.4 & 12.1$^{+0.2}_{-0.2}$ & 128.8$\\pm$54.0 & 65.8$^{+14.3}_{-15.2}$\\\\\n~~~log$M_*>$ 10.5      & 11.1$^{+0.2}_{-0.4}$ & 2.2 & 39  & 0.48$\\pm$0.20 & 2.4 & 11.6$^{+0.2}_{-0.2}$ & 38.6$\\pm$16.3 & 61.1$^{+5.7}_{-3.5}$\\\\\n~~~$z>$ 2.5            &&&&&&&\\\\\n~~~log$M_*\\leq$ 10.5   & 10.5$^{+0.0}_{-0.0}$ & 2.7 & 2   & 0.42$\\pm$0.90 & 0.5 & 11.4$^{+0.5}_{-11.4}$ & 28.1$\\pm$60.4 & 27.8$^{+17.4}_{-9.9}$\\\\\n~~~log$M_*>$ 10.5      & 11.0$^{+0.2}_{-0.4}$ & 2.8 & 27  & 0.44$\\pm$0.25 & 1.8 & 11.5$^{+0.2}_{-0.4}$ & 30.8$\\pm$17.1 & 62.5$^{+11.6}_{-6.6}$\\\\\n\\enddata\n\\tablenotetext{abc}{~~~The parameters follow those in Table \\ref{tab:450stack}.}\n\\end{deluxetable*}\n\nOverall, we see that QG candidates with 24 $\\mu$m counterparts and QG candidates with 3 GHz counterparts that are not radio AGNs (i.e., IR-radio-bright QGs) exhibit the strongest stacking signal at both 450 $\\mu$m and 850 $\\mu$m.  These IR-radio-bright QGs account for 9.7$\\pm$0.2\\% (1769\/18304) of all the QG candidates.  In general, we do not reach significant detections of IR-radio-faint QGs.  However, even with the low SNR, the stacked 850 $\\mu$m fluxes for high-mass ($>10^{10.5}~M_{\\odot}$) IR-radio-faint QGs are consistently higher than those for low-mass IR-radio-faint QGs.  This suggests that even the IR-radio-faint QGs have dust emission in the rest-frame far-IR, or they are clustered around dusty objects (see below). On the other hand, among IR-radio-bright QGs, it is not apparent that the high-mass ones show consistently higher 850 $\\mu$m fluxes than the low-mass ones. This suggests that we are not seeing a population of well-behaved galaxies who follow the star-formation main sequence. This is expected for QGs.\n\nWe can compare our stacked 450 $\\mu$m fluxes with the \\emph{Herschel} 500 $\\mu$m stacked fluxes in \\citet{Man2016}.  Our mean 450 $\\mu$m flux of IR-radio-faint QGs is 0.00$\\pm$0.02 mJy, whose 1$\\sigma$ upper limit is over 10 times lower than their stacked 500 $\\mu$m fluxes of IR-faint QGs, which range from 0.2 to 2.5 mJy in different mass and redshift bins. The difference of defining the two QG subsamples (SFR derived from 24 $\\mu$m to be under or over 100 $M_{\\odot}$ year$^{-1}$ in \\citet{Man2016}) may be one of the possible explanations. However, our mean 450 $\\mu$m flux of IR-radio-bright QGs, 0.65$\\pm$0.15 mJy, is still lower than most of the above mean 500 $\\mu$m fluxes of IR-faint QGs (0.2 to 2.5 mJy) except for some of those with $M_*<10^{10.6}~M_{\\odot}$.\n\nOur stacked 450 $\\mu$m fluxes are also lower compared with results in \\citet{Magdis2021}. They selected QG candidates with several color-color diagrams and stacked samples without 24 $\\mu$m detection, so we compare our stacked fluxes of IR-radio-faint QGs with their results. Their stacked \\emph{Herschel} 500 $\\mu$m fluxes, ranging from 0.12 to 0.59 mJy in various redshift bins, are also much higher than our stacked 450 $\\mu$m flux, 0.00$\\pm$0.02 mJy. On the other hand, their stacked SCUBA-2 850 $\\mu$m fluxes, ranging from 0.04 to 0.1 mJy, are at a similar level as our stacked 850 $\\mu$m fluxes.\n\nA possible explanation for the differences between the SCUBA-2 450 $\\mu$m and \\emph{Herschel} 500 $\\mu$m stacked fluxes is that the stacked \\emph{Herschel} fluxes were biased by source clustering at the scale of the large $35\\arcsec$ \\emph{Herschel} 500 $\\mu$m beam (e.g., \\citealt{Viero2013}, also see discussion in \\citealt{Bethermin2017}). Although \\citet{Magdis2021} modeled the emission of all the stacked images to separate surrounding sources, it appears that their 500 $\\mu$m stacked fluxes are higher.  Although \\citet{Man2016} applied \\texttt{SIMSTACK} to stack and deblend simultaneously, it remains possible that the effects of source blending and clustering were not completely removed.\n\nThe above comparison confirms the well known bias in submillimeter stacking analysis: when source are clustered at scales comparable to the beam size, the stacked flux would be overestimated. This bias becomes quite severe under \\emph{Herschel}'s large beams in the two longest wavebands.  How about our SCUBA-2 stacked fluxes?  In the previous section, we showed that QG candidates are clustered around 850 $\\mu$m sources at scales of SCUBA-2's beam.  So if we blindly stack these QGs in the 850 $\\mu$m image, the stacked flux will be overestimated.  Fortunately, the majority of our 850 $\\mu$m stacking signal comes from the IR-radio-bright subsample.  Their 24 $\\mu$m and 3 GHz counterparts are likely to be 850 $\\mu$m sources themselves, and the bias caused by clustering should therefore be negligible. On the other hand, the IR-radio-faint sample does not have deep high-resolution data to confirm that the QGs are responsible for the detected 850 $\\mu$m fluxes.  Therefore, strictly speaking, our stacked 850 $\\mu$m fluxes for IR-radio-faint QGs should be considered as upper-limits.  Even if a detection is reached on certain subsample of IR-radio-faint QGs, the detected flux should be only an indication that these QG candidates are physically related to faint submillimeter emitters, rather than evidence for in situ star formation in the QG candidates.\n\nFinally, we can examine if the strong 850 $\\mu$m detection (8.6$\\sigma$) of the IR-radio-bright QGs really come from galaxies in the QG color-color space in the $NUV$--$r$--$J$ diagram, or from galaxies originally in the SFG color space scattered by photometric errors across the selection boundary. In Section \\ref{sec:QG_selection}, we show that such SFG contamination caused by photometric errors account for about 7.5\\% of the selected QGs.  With the same method, the estimated fraction for misidentified IR-radio-bright QGs caused by photometric errors is slightly higher, 8.9\\%.  Moreover, we identified individual IR-radio-bright QGs whose probability of being scattered from the SFG color space to be $>0.05$. These sources account for 33\\% of our IR-radio-bright QGs (584\/1769). We excluded them and re-did the stacking on the remaining IR-radio-bright QGs, and still obtained a strong detection of $0.22\\pm0.04$ mJy (5.9$\\sigma$) despite the very generous probability cut of $>0.05$. These results imply that misidentified QG candidates due to photometric errors account for only $<$10\\% of our estimated dusty SFG contamination, and this minor population does not dominate our stacking results. The majority of the dusty SFG contamination is caused by their intrinsic properties rather than photometric errors.\n\n\n\n\\subsection{Examining the Quiescence}\n\nTo examine if our sub-samples are consistent with a quiescent population, we need to derive their SFRs and compare with their stellar masses.\n\nWe calculated the IR luminosity from the mean submillimeter fluxes and the median redshift of each group of the stacking sample. Since we only conducted measurements at 450 and 850 $\\mu$m, we performed single-band SED ``fitting'' by assuming that there is a unique relation between SED shape and IR luminosity. To do so, we adopt the luminosity-dependent dust SED templates of J.\\ K.\\ Chu et al.\\ (in preparation), which are based on the latest \\emph{WISE} and \\emph{Herschel} photometry for 201 local IR-selected galaxies \\citep{Chu2017}.  This set of templates covers IR luminosity of $7\\times10^9$ to $1.7\\times10^{12} L_{\\odot}$.  We further supplement the submillimeter galaxy SED from the zLESS program \\citep{Danielson2017}, which has an IR luminosity of $5.2\\times10^{12} L_{\\odot}$. We redshift these SEDs to the redshifts of our targets and calculated their observed 450 $\\mu$m or 850 $\\mu$m fluxes.  We picked the templates with redshifted fluxes closest to our stacked flux and interpolate between the template fluxes to obtain the IR luminosity of our targets. We scaled the IR luminosity by 1\/SNR to estimate $1\\sigma$ error of the IR luminosity. For groups with negative mean flux, we calculated the corresponding IR luminosity of flux error to estimate $1\\sigma$ upper limit of the IR luminosity. The results are are presented in the seventh columns of Tables \\ref{tab:450stack} and \\ref{tab:850stack}.\n\nTo verify the results based on the local SEDs of Chu et al., we also repeated the calculations using the SED library of \\citet{Schreiber2018c}. Overall, we find no systematic differences if we assume main sequence galaxies ($R_{\\rm SB}=1$) for the Schreiber et al.\\ library. The mean difference in the calculated $L_{\\rm IR}$ is less than 0.1 dex for the non-zero entries in Tables \\ref{tab:450stack} and \\ref{tab:850stack}, while the rms dispersion is within 0.25 dex. This small difference can be further reduced if we assume a sub-main-sequence $R_{\\rm SB}$ for the IR-radio-faint subsamples in Table~\\ref{tab:850stack} and a starburst $R_{\\rm SB}$ for the IR-radio-bright subsamples. This tuning of the $R_{\\rm SB}$ parameter is consistent with our interpretation of these two subgroups (see below). In our subsequent analyses, we adopt the calculations based on the SEDs of Chu et al.\n\nAfter calculating the IR luminosity, we followed the $L_{\\rm IR}$--SFR calibration applied in \\citet{Man2016}. We estimate the SFR by applying the relation applicable for SFGs \\citep{Kennicutt1998}:\n$$\\mathrm{SFR} (M_{\\odot}\\, \\mathrm{yr^{-1}})=1.7\\times10^{-10} L_{\\rm IR} (L_{\\odot}).$$\nWe then adjusted the obtained SFR to the \\citet{Chabrier2003} IMF by applying the calibration used in \\citet{Man2016}:\n$$\\mathrm{SFR}_{\\rm Chabrier}=\\mathrm{SFR}_{\\rm Salpeter}\/1.7$$\n\nThe results are presented in the eighth columns of Table \\ref{tab:450stack} and Table \\ref{tab:850stack} as SFR$_{\\rm 450 \\mu m}$ and SFR$_{\\rm 850 \\mu m}$, respectively.  A few observations can be made here. First, the 850-$\\mu$m derived SFR is in general higher than the 450-$\\mu$m derived SFR. This is partly caused by the much deeper in luminosity sensitivity of the SCUBA-2 450 $\\mu$m imaging and the $<3\\sigma$ thresholds we imposed in the stacking procedure. If we remove this threshold in the 450 $\\mu$m imaging, the difference reduces to within a factor of 2, which is not very significant if we consider the overall low S\/N of the 450 $\\mu$m stacked fluxes and the small number of available sources for the 450 $\\mu$m stacking.\n\nWe compare our results with SFRs derived from optical SED fitting in COSMOS2015 (last column in Table \\ref{tab:450stack} and  \\ref{tab:850stack}). Their mean SFRs of IR-radio-faint QGs are higher than ours at high $z$ but their mean SFRs of IR-radio-bright QGs are lower.  This can be explained with the age-extinction degeneracy in the SED fitting when there is an absence of far-IR photometry.\n\nWe compare the submilimeter-derived SFRs with the stellar masses of the galaxies in Fig.~\\ref{fig:SSFR}. We show the star-formation ``main sequence'' of \\citet{Speagle2014} with black solid lines and the $\\pm$0.9 dex ranges with shaded areas.  \n\nWe show the results from \\citet{Man2016} for comparison (Fig.~\\ref{fig:SSFR}). Our results are in broad agreement with theirs but tend to have slightly lower SFR for low-$z$ samples. In the $0.5<z\\leq1.0$ and $1.0<z\\leq1.5$ redshift bins, their massive IR-bright QGs are located on or $\\sim1$ to $2\\sigma$ above the main sequence, while our IR-radio-bright QGs are located $\\sim2$ to $3\\sigma$ below the main sequence. Their IR-faint QGs are located $\\sim3\\sigma$ below the main sequence, while our IR-radio-faint QGs are located $>3\\sigma$ below the main sequence.  Other than these, our derived SFRs are fairly consistent. We note that the SFR of \\citet{Man2016} was derived from SED fitting using stacked fluxes across the entire far-IR range.  This explains why their 500 $\\mu$m stacked fluxes are much higher than ours, but their SFRs are not. \n\nWe also show the results from \\citet{Magdis2021} in Fig.~\\ref{fig:SSFR}. Their SFRs are about the same as or slightly lower than our SFRs, different from the comparison with \\citet{Man2016}. We note that their IR luminosities were derived from SED fitting using stacked fluxes from mid-IR to radio data. They then also obtained SFRs by applying the relation in \\citet{Kennicutt1998}, but they used a Salpeter IMF and added SFR derived from the optical photometry. We converted their IR luminosity to SFR by the same process in \\citet{Man2016} and this work for a fair comparison.  We also show their SFRs obtained by the original conversion in their work for reference, which are in general closer to the SFRs from \\citet{Man2016}.\n\nThe conclusion we can draw from Fig.~\\ref{fig:SSFR} is that the IR-radio-faint QGs are in general below the star-formation main sequence, while the majority of the IR-radio-bright QGs are consistent with the main sequence, probably except for the high-mass end in the two low-redshift bins and in the highest redshift bin.\n\n\\begin{figure*}[ht!]\n\\epsscale{1.15}\n\\plotone{SFRvsM_IMF.png}\n\\caption{SFR derived from 850 $\\mu$m fluxes versus stellar mass. The purple diamonds represent IR-radio-bright QGs, while the red circles represent IR-radio-faint QGs. The smaller semi-transparent symbols are results from other works. The purple diamonds, red circles, and blue squares represent the IR-bright QGs, IR-faint QGs, and SFGs in \\citet{Man2016}, respectively. The red triangles represent QGs in \\citet{Magdis2021}. IR-radio-bright QGs in our work are defined as QG candidates either with 24 $\\mu$m counterparts or with 3 GHz counterparts labeled with SFG flags in the VLA catalog \\citep{Smolcic2017}, while IR-bright QGs in \\citet{Man2016} are defined as QG candidates with SFR derived from 24 $\\mu$m over 100 $M_{\\odot}$ year$^{-1}$. QGs in \\citet{Magdis2021} are defined as QG candidates without 24 $\\mu$m detection. The filled triangles are derived by the same $L_{IR}$--$\\mathrm{SFR}$ conversion with the other two works, while the open triangles are derived by the conversion described in their work.} The black solid line shows the SFR of the redshift-dependent main sequence with a 1--3 $\\times$ 0.3 dex scatters in \\citet{Speagle2014}.\\label{fig:SSFR}\n\\end{figure*}\n\nTo sum up, our stacking results show that only the IR-radio-bright QGs have SFR similar to main-sequence galaxies. These are likely to be faint dusty SFGs that contaminate the QG color selection. However, the population of the IR-radio-bright QGs is small, which accounts for 9.7$\\pm$0.7\\% (179\/1846) and 9.7$\\pm$0.2\\% (1769\/18304) of all the  QG candidates, respectively, in the 450 and 850 $\\mu$m images. The fractions range from 7\\% to 12\\% in different redshift bins and do not have a strong redshift dependence. We conclude that the contamination of dusty SFGs is of $\\sim$ 10\\% among the color-selected QG candidates, and that the contamination can be removed using multi-wavelength data such as the 24 $\\mu$m and 3 GHz data for the COSMOS field.\n\nFor comparison, \\citet{Man2016} suggested that the maximum contamination is 15\\% and could be removed by using 24 $\\mu$m observations. In this study, we used submillimeter data with better sensitivities and resolutions, and our estimate of contamination is somewhat tighter (10\\%) than that in \\citet{Man2016}. \n\nLike what we did in Section~\\ref{subsubsec:traditional_matching_process}, if we assume the same fraction of QGs among AS2COSMOS sources, we can estimate the number of faint submillimeter sources that are QGs.  For the number of faint submillimeter sources, we again applied the 850 $\\mu$m number count in \\citet{Simpson2019} and extrapolated it to a flux level of S$_{850 \\mu m}= 0.5$ mJy.  This leads to 707$\\pm$462 QGs among faint submillimeter sources, and a dusty galaxy contamination rate among QGs of 3.9$\\pm$2.5\\%.  We can further extrapolate the counts to 0.26 mJy, the stacked 850 $\\mu$m flux of IR-radio-bright QGs. This will further increase the estimated contamination rate. However, such an extrapolation is is probably too aggressive given the uncertainty in the faint-end counts. Nevertheless, considering the unknown uncertainty of extrapolating the number count to a flux level lower than the detection limit, we concluded that the above estimation is not inconsistent with the $\\sim10\\%$ contamination derived from the stacking of IR-radio-bright QGs.\n\nFinally, using either 24 $\\mu$m or 3 GHz data to pinpoint star-forming contaminants among color-selected QG candidates may not work well at high redshifts ($z>3$ or 4). This is because mid-IR and radio suffer from the strong $K$-correction and are not sensitive to high-redshift SFGs.  Our $3.4\\sigma$ detection in the 850 $\\mu$m stacking of the IR-radio-faint QGs at $z>2$ seems to agree with this, i.e., there may exist dusty galaxy contamination that are faint in the mid-IR and radio.  In other words, the effectiveness of QG color selection at high redshift remains untested in this framework.  Since the formation of QGs at higher redshifts require both rapid growth of the stellar population and rapid quenching, identifications of high-redshift QGs are of great interest \\citep{Merlin2018,Straatman2014,Carnall2020, Valentino2020}.  Removing dusty contaminants among high-redshift QGs is beyond the sensitivities of \\emph{Spitzer}, \\emph{Herschel}, and the current VLA, and will require deep ALMA data.\n\n\\section{AGN Properties} \\label{sec:AGN_properites}\n\nIn this section, we discuss the properties of the AGNs among our QG candidates. Fig.~\\ref{fig:NUVrJ_AGN} shows the distribution of three different classes of AGN samples in the $NUV$--$r$--$J$ diagram, including radio AGNs, mid-IR AGNs, and X-ray AGNs (Section~\\ref{subsec:AGN_samples}) in two mass bins. The stellar mass of the samples were limited to above $10^{10.5}~M_{\\odot}$. This is because the samples are likely to be incomplete below $10^{10.5}~M_{\\odot}$ at $z\\sim2$ (see Fig.~\\ref{fig:data} (b)).  This mass limit is also consistent with the 90\\% completeness limit found by \\citet{Laigle2016} for QGs at high redshifts. We therefore applied this stellar mass cut for fair comparisons of the QG fractions. In Fig.~\\ref{fig:NUVrJ_AGN}, we can see that the distribution of the radio AGNs is different from those of the other two. A similar distinction also exists between radio selected sources and 24 $\\mu$m selected sources in Fig.~\\ref{fig:NUVrJ243}. Fig.~\\ref{fig:NUVrJ_AGN} provides evidence that the difference in Fig.~\\ref{fig:NUVrJ243} is driven by radio AGNs.\n\n\\begin{figure*}[!ht]\n\\epsscale{1.15}\n\\plotone{NUVrJAGN.png}\n\\caption{The distribution of radio AGNs (left penals), mid-IR AGNs (middle penals), and  X-ray AGNs (right penals) on the $NUV$--$r$--$J$ diagram with $M_*>10^{11}~M_{\\odot}$ (top penals) and $10^{10.5}~M_{\\odot}<M_*\\leq10^{11.0}~M_{\\odot}$ (lower penals).\\label{fig:NUVrJ_AGN}}\n\\end{figure*}\n\nWe calculated the QG fractions in different classes of the AGN samples in different redshift and mass bins (Fig~ \\ref{fig:QG_fraction}). The QG fraction of the radio AGNs is ($\\sim0.1$ to $1.9\\sigma$) higher than that of the non-AGN samples at $z<1.5$ in both two mass bins, while the QG fractions of the other two AGN classes are significantly lower. The trend does not persist beyond $z\\sim1.5$, and this may be due partially to the detection limit of the radio AGNs, or a real redshift evolution. The high QG fraction among radio AGNs implies a correlation between radio jets and our QG candidates.\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{QGfraction_zbin.png}\n\\epsscale{1.15}\n\\plotone{QGfraction_zbin_lessmass.png}\n\\caption{The QG fraction among different classes of AGN samples in redshift bins with $M_*>10^{11}~M_{\\odot}$ (a) and $10^{10.5}~M_{\\odot}<M_*\\leq10^{11.0}~M_{\\odot}$ (b). The data points are slightly offset along x-axis for clarity. The errors are Poissonian.\\label{fig:QG_fraction}}\n\\end{figure}\n\nWe also calculated the AGN fractions in our COSMOS2015 sample (Fig~ \\ref{fig:AGN_fraction}). The radio AGN fraction among QG candidates is ($\\sim0.2$ to $1.7\\sigma$) higher than that among the COSMOS2015 QG and SFG sample, also at $z<1.5$ and in both two mass bins. The situations are the opposite for both the mid-IR and X-ray AGNs, where the AGN fractions among the QG+SFG sample are higher. The higher radio AGN fraction among QGs also suggests a correlation between radio jets and our QG candidates.\n\n\\begin{figure}[ht!]\n\\epsscale{1.15}\n\\plotone{AGNfraction_zbin_massive.png}\n\\epsscale{1.15}\n\\plotone{AGNfraction_zbin_lessmass.png}\n\\caption{Different classes of AGN fraction among COSMOS2015 optical sample in redshift bins with $M_*>10^{11}~M_{\\odot}$ (a) and $10^{10.5}~M_{\\odot}<M_*\\leq10^{11.0}~M_{\\odot}$ (b). The solid lines represent AGN fraction among QG candidates, while the dotted line represent AGN fraction among all the galaxies (QGs and SFGs). The data points are slightly offset along x-axis for clarity. The errors are Poissonian.\\label{fig:AGN_fraction}}\n\\end{figure}\n\nThe above correlation between radio AGNs and QGs may provide some clues about the quenching mechanism. Our data suggest that the formation of radio jets and quenching of star formation are connected, but the exact causality is unclear. One possible scenario is that radio-mode AGN feedback, particularly radio jets with high kinetic energy, could have impacts on the host galaxy and its environment \\citep{Croton2006, Bower2006, Fabian2012, Somerville2008, Wagner2011}. Either the interstellar medium in the host galaxy may be disrupted by the jets \\citep[e.g.,][]{Fabian2012,Somerville2008}, or the gas in the halo can be heated up and be prevented from cooling and falling onto the host galaxy \\citep[e.g.,][]{Croton2006, Bower2006, Somerville2008}. Although the radio AGNs fraction is only $\\sim$ 10 to 20\\% among QG candidates (Fig.~\\ref{fig:AGN_fraction}), this can be explained by the AGN duty cycle in the massive galaxies.\n\nOn the other hand, it is also possible that radio AGNs are not directly linked to the quenching process.  Instead, it may be just easier to trigger radio AGNs in galaxies that are already quenched.  In the local universe, it is well known that there is a dominant tendency for radio galaxies to reside in early-type hosts \\citep[e.g.,][]{Govoni2000,Hamilton2002}, which is consistent with what we found on QGs and radio AGNs.  The lack of cool gas in early-type (quenched) galaxies can lead to radiatively inefficient accretion that is associated with the radio mode AGN feedback \\citep{Croton2006, Bower2006, Somerville2008}.\n\nObservationally, the stacking result in \\citet{Man2016} also showed a similar trend. The authors compared the radio stacked fluxes and the \\emph{Herschel} stacked fluxes of their sample. Their SFRs derived from the two are consistent for SFGs, but there exists a radio excess for massive IR-faint QGs with $M_*\\geq10^{11}~M_{\\odot}$ at $z<$ 1.5. They further measured the FIR-to-radio luminosity ratio and concluded that most of their massive QGs host low-luminosity radio AGNs. Also, \\citet{Smolcic2009} found that the radio AGN fraction among red galaxies increases from $z \\sim$ 0.5 to 1, showing that radio AGNs play an important role at $z \\sim$ 1. It is also known that SFR is correlated with AGN accretion rate \\citep{Zhuang2020}, and that low-accretion rate AGNs are radio-loud \\citep{Ho2002, Ho2008}. This implies that QGs tends to be radio AGNs. All these results suggest that there is a correlation between QGs and synchrotron radiation from radio AGNs.\n\nWe could also see a weak correlation between X-ray AGNs and QG candidates at $z>2.5$. In Fig.~\\ref{fig:QG_fraction}, the QG fractions among X-ray AGNs are $\\sim0.6$ to $1.5\\sigma$ larger than those among non AGNs in $z>2.5$ redshift bins, with respect to their own error bars. In Fig.~\\ref{fig:AGN_fraction}, the AGN fraction among QGs is $\\sim0.9\\sigma$ larger than that among the full sample in the $z>2.5$ and $M_*>10^{11}~M_{\\odot}$ bin. This may be caused by either selection bias or a real evolution trend. The evolution trend could be explained by the role of quasar-mode AGN feedback \\citep{Fabian2012, Somerville2008}, gas inflow in X-ray AGNs that removes gas and quenches star formation. One possibility is that the mode of AGN quenching may change from quasar-mode to radio-mode from high $z$ to low $z$. Another possibility could be that X-ray AGNs are related to the initial quenching, while radio AGNs are responsible for the maintenance of the quiescence. This could also explain the increasing radio AGN fraction among QG candidates in Fig.~\\ref{fig:AGN_fraction} at lower redshift. Nevertheless, the rises in the X-ray AGNs in Fig.~\\ref{fig:QG_fraction} and \\ref{fig:AGN_fraction} only occur in the highest redshift bins where the sample sizes are the smallest and the selection completeness is less well understood. This has to be further tested with more data and careful examination of various selection biases in the high-redshift ends.\n\nTo sum up, our data show a strong correlation between radio AGNs and QGs but do not point to the right scenario. Our data also do not show whether radio AGNs are related to the initial quenching, or just related to the maintenance of the quiescence.\n\n\\section{Summary} \\label{sec:summary}\n\nIn this study, we examined the submillimeter properties of $NUV$--$r$--$J$ selected QG candidates at $z\\lesssim3$. We cross-matched the QG candidates with bright submillimeter sources detected by JCMT SCUBA-2 and ALMA. For the former, we used \\emph{Spitzer} 24 $\\mu$m and VLA 3 GHz data to refine their positions to overcome the low angular resolution of JCMT.  This way, we found that 0.16$\\pm$0.03\\% to 0.43$\\pm$0.15\\% among our QG candidates are likely to be bright 850 and 450 $\\mu$m submillimeter galaxies, respectively. The contamination increases to 1.72$\\pm$0.50\\% to 3.51$\\pm$2.48\\% at $z>$ 2.  We further performed stacking analysis of QG candidates in the JCMT 450 and 850 $\\mu$m images.  We can obtain strong stacking detections on a subsample of QGs with \\emph{Spitzer} 24 $\\mu$m and VLA 3 GHz counterparts that are not radio AGNs.  This special class of ``IR-radio-bright'' QGs account for about 10\\% of the entire QG sample and they are likely to be faint submillimeter sources with SFRs of a few tens to about a hundred $M_\\sun$ yr$^{-1}$.  These results are broadly consistent with the contaminate rates derived from a small sample of ALMA detected QGs and the 850 $\\mu$m number counts.  We conclude that the dusty star-forming galaxy contamination rate among $NUV$--$r$--$J$ selected QG candidates is up to $\\sim10\\%$, but such contamination can be removed by 24 $\\mu$m, submillimeter, or 3 GHz observations at current sensitivity levels.\n\nWhen we cross-matched the QG candidates with JCMT SCBUA-2 850 $\\mu$m SMGs without relying on high-resolution data, we adopted a large matching radius of $7\\arcsec$ because of the large SCUBA-2 beam size.  This leads to a large fraction of chance projections among the matched QGs. We estimated the number of chance projections with simulations by assuming random spatial distributions for SCUBA-2 sources. After statistically subtracting the chance projections, we found that on average, 0.096 (35.4\/370) QG is physically related to an 850 $\\mu$m selected SMG, while ALMA observations indicate that only 0.026 (9.7\/370) QG really coincides with an SMG within $1\\arcsec$. This implies a clustering between these two populations at a scale of $1\\arcsec$ to $7\\arcsec$, and should be a future topic of investigation.\n\nFinally, we examined the QG fractions among our AGN samples and found a correlation between our QG candidates and radio AGNs. When we limited our studies to galaxies with stellar masses larger than $10^{10.5}M_\\sun$, we found that the QG fraction of radio AGNs are larger than those of the non-AGN samples, IR AGNs, and X-ray AGNs at $z<$ 1.5. This suggests a connection between the radio jets and the quenching or the maintenance of the quiescence of the QGs, or the so-called radio-mode AGN feedback. However, our data do not rule out the possibility that radio AGNs are just more easily triggered in quenched galaxies, rather than being responsible for the initial quenching.\n\n\\acknowledgments\n\nThe authors thank Bau-Ching Hsieh, Ian Smail, Iary Davidzon, and Olivier Ilbert for the discussion and comments, the anonymous referee for the comments that greatly improve the manuscript, and JCMT staff for the observational support. Y.H.H., W.H.W., Y.Y.C., C.F.L., and Z.K.G. acknowledge grant support from the Ministry of Science and Technology of Taiwan (MoST, 105-2112-M-001-029-MY3, 108-2112-M-001-014-, and 109-2112-M-001-011-).  C.C.C. acknowledges MoST grant 109-2112-M-001-016-MY3. M.J.M. acknowledges the support of the National Science Centre, Poland through the SONATA BIS grant 2018\/30\/E\/ST9\/00208. M.P.K. acknowledges support from the First TEAM grant of the Foundation for Polish Science No. POIR.04.04.00-00-5D21\/18-00. L.C.H. was supported by the National Science Foundation of China (11721303, 11991052) and the National Key R\\&D Program of China (2016YFA0400702). Y.G. acknowledges National Science Foundation of China (NSFC) grants \\#11861131007, 12033004, and 11420101002, and Chinese Academy of Sciences Key Research Program of Frontier Sciences (Grant No. QYZDJ-SSW-SLH008). The submillimeter data used in this work include observations from the JCMT Large and Legacy Programs: S2COSMOS (M16AL002), STUDIES (M16AL006), and S2CLS (MJLSC01), the JCMT PI program of Casey et al.\\ (M11BH11A, M12AH11A, and M12BH21A), the ALMA program AS2COSMOS (ADS\/JAO.ALMA \\#2016.1.00463.S), and various ALMA archival data.  The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of the National Astronomical Observatory of Japan; the Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; and the Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and Facility Instruments, budgeted from the Ministry of Finance (MOF) of China and administrated by the Chinese Academy of Sciences (CAS), as well as the National Key R\\&D Program of China (No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO, and NAOJ.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Details on Small-world Inter-clique Topology}\n\\label{app:small_world}\n\n We present a more detailed and precise explanation of the algorithm to establish a small-world\n inter-clique topology (Algorithm~\\ref{Algorithm:Smallworld}). Algorithm~\\ref{Algorithm:Smallworld} instantiates the function \n\\texttt{inter} with a\nsmall-world inter-clique topology as described in Section~\\ref{section:interclique-topologies}. It adds a\nlinear number of inter-clique edges by first arranging cliques on a ring. It then adds a logarithmic number of ``finger'' edges to other cliques on the ring chosen such that there is a constant number of edges added per set, on sets that are exponentially bigger the further away on the ring. ``Finger'' edges are added symmetrically on both sides of the ring to the cliques in each set that are closest to a given set. ``Finger`` edges are added for each clique on the ring, therefore adding in total a linear-logarithmic number of edges.\n\n\\begin{algorithm}[h]\n   \\caption{$\\textit{smallworld}(DC)$:  adds $O(\\# N \\log(\\# N))$ edges}\n   \\label{Algorithm:Smallworld}\n   \\begin{algorithmic}[1]\n        \\STATE \\textbf{Require:} set of cliques $DC$ (set of set of nodes)\n        \\STATE ~~size of neighborhood $ns$ (default 2)\n        \\STATE ~~function $\\textit{least\\_edges}(S, E)$ that returns one of the nodes in $S$ with the least number of edges in $E$\n        \\STATE $E \\leftarrow \\emptyset$ \\COMMENT{Set of Edges}\n        \\STATE $L \\leftarrow [ C~\\text{for}~C \\in DC ]$ \\COMMENT{Arrange cliques in a list}\n        \\FOR{$i \\in \\{1,\\dots,\\#DC\\}$}\n          \\FOR{$\\textit{offset} \\in \\{ 2^x~\\text{for}~x~\\in \\{ 0, \\dots, \\lceil \\log_2(\\#DC) \\rceil \\} \\}$} \n             \\FOR{$k \\in \\{0,\\dots,ns-1\\}$}\n                 \\STATE $n \\leftarrow \\textit{least\\_edges}(L_i, E)$\n                 \\STATE $m \\leftarrow \\textit{least\\_edges}(L_{(i+\\textit{offset}+k) \\% \\#DC}, E)$\n                 \\STATE $E \\leftarrow E \\cup \\{ \\{n,m\\} \\}$\n                 \\STATE $n \\leftarrow \\textit{least\\_edges}(L_i, E)$\n                 \\STATE $m \\leftarrow \\textit{least\\_edges}(L_{(i-\\textit{offset}-k)\\% \\#DC} , E)$\n                 \\STATE $E \\leftarrow E \\cup \\{ \\{n,m\\} \\}$\n             \\ENDFOR\n           \\ENDFOR\n        \\ENDFOR\n         \\RETURN E\n   \\end{algorithmic}\n\\end{algorithm}\n\nAlgorithm~\\ref{Algorithm:Smallworld}  expects a set of cliques $DC$, previously computed by \nAlgorithm~\\ref{Algorithm:greedy-swap}; a size of neighborhood $ns$,\nwhich is the number of finger edges to add per set of cliques, and a function \n\\textit{least\\_edges}, which given a set of nodes $S$ and an existing set of\nedges $E =  \\{\\{i,j\\}, \\dots \\}$, returns one of the nodes in $E$ with the least number of edges. It returns a new set of edges $\\{\\{i,j\\}, \\dots \\}$ with all edges added by the small-world topology.\n\nThe implementation first arranges the cliques of $DC$ in a list, which\nrepresents the ring. Traversing the list with increasing indices is equivalent\nto traversing the ring in the clockwise direction, and inversely. Then, for every clique $i$ on the ring from which we are computing the distance to others, a number of edges are added. All other cliques are implicitly arranged in mutually exclusive sets, with size and at offset exponentially bigger (doubling at every step). Then for every of these sets, $ns$ edges are added, both in the clockwise and counter-clockwise directions, always on the nodes with the least number of edges in each clique. The ring edges are implicitly added to the cliques at offset $1$ in both directions.\n \n\n\n\n\n\n     \n     \n     \n\n\n\n\n\n\n\\section{Additional Experiments on Scaling Behavior with Increasing Number of\nNodes}\n\\label{app:scaling}\n\nSection~\\ref{section:scaling} compares the convergence speed of various inter-clique topologies at a scale of 1000 nodes. In this section, we show the effect of scaling the number of nodes, by comparing the convergence speed with 1, 10, 100, and 1000 nodes, and adjusting the batch size to maintain a constant number of updates per epoch. We present results for Ring, Fractal, Small-world, and Fully-Connected inter-clique topologies.\n \nFigure~\\ref{fig:d-cliques-mnist-scaling-fully-connected} shows the results for\nMNIST. For all topologies, we notice a perfect scaling up to 100 nodes, i.e.\nthe accuracy curves overlap, with low variance between nodes. Starting at 1000\nnodes, there is a significant increase in variance between nodes and the\nconvergence is slower, only marginally for Fully-Connected but\nsignificantly so for Fractal and Ring. Small-world has higher variance between nodes but maintains a convergence speed close to that of Fully-Connected.\n\n\n\n\n\\begin{figure}[htbp]\n         \\centering     \n             \n             \n              \n      \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-scaling-fully-connected-cst-updates}\n         \\caption{Fully-Connected}\n     \\end{subfigure}\n     \\quad\n      \n      \n      \n      \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-scaling-smallworld-cst-updates}\n         \\caption{Small-world}\n     \\end{subfigure}\n     \\quad\n\n         \n         \n      \n         \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-scaling-fractal-cliques-cst-updates}\n         \\caption{Fractal}\n     \\end{subfigure}  \n     \\quad\n    \n    \n      \n         \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-scaling-ring-cliques-cst-updates}\n         \\caption{Ring}\n     \\end{subfigure}  \n     \n     \\caption{\\label{fig:d-cliques-mnist-scaling-fully-connected} MNIST:\n     D-Cliques scaling behavior (constant updates per epoch and 10 nodes per clique) for different\n     inter-clique topologies.} \n\\end{figure}\n     \nFigure~\\ref{fig:d-cliques-cifar10-scaling-fully-connected} shows the results\nfor CIFAR10. When increasing from 1 to 10 nodes (resulting in a single\nfully-connected clique), there is actually a small increase both in final\naccuracy and convergence speed. We believe this increase is due to the\ngradient being computed with better representation of examples from all\nclasses with 10 fully-connected non-IID nodes, while the gradient for a single\nnon-IID node may have a slightly larger bias because the random sampling \nmay allow more bias in the representation of classes in each batch. At a\nscale of 100 nodes, there is no difference between Fully-Connected and\nFractal, as the connections are the same; however, a Ring already shows a\nsignificantly slower convergence. At 1000 nodes, the convergence significantly\nslows down for Fractal and Ring, while remaining close, albeit with a larger\nvariance, to Fully-Connected. Similar to MNIST, Small-world has\nhigher variance and slightly lower convergence speed than Fully-Connected but\nremains very close.\n\nWe therefore conclude that Fully-Connected and Small-world have good scaling\nproperties in terms of convergence speed, and that the\nlinear-logarithmic number of edges of Small-world makes it the best compromise\nbetween convergence speed and connectivity, and thus the best choice for\nefficient large-scale decentralized learning in practice.\n\n\n\\begin{figure}[htbp]\n         \\centering\n    \n    \n      \n      \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-scaling-fully-connected-cst-updates}\n         \\caption{Fully-Connected}\n     \\end{subfigure}\n     \\quad\n    \n    \n      \n      \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-scaling-smallworld-cst-updates}\n         \\caption{Small-world}\n     \\end{subfigure}\n     \n     \n    \n    \n      \n         \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-scaling-fractal-cst-updates}\n         \\caption{Fractal}\n     \\end{subfigure}  \n     \\quad\n\n    \n    \n      \n         \\begin{subfigure}[b]{0.35\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-scaling-ring-cst-updates}\n         \\caption{Ring}\n     \\end{subfigure}  \n     \n     \\caption{\\label{fig:d-cliques-cifar10-scaling-fully-connected} CIFAR10: D-Cliques scaling behavior (constant updates per epoch and 10 nodes per clique) for different\n     inter-clique topologies.} \n\\end{figure}\n\n\\section{Additional Experiments with Extreme Label Skew}\n\\label{app:extreme-local-skew} \n\nIn this section, we present additional results for similar experiments as in\nSection~\\ref{section:evaluation} but in the presence of\n \\textit{extreme label distribution skew}: we consider that each node only has examples from a single class. This extreme partitioning case provides an upper bound on the effect of label distribution skew suggesting that D-Cliques should perform similarly or better in less extreme cases, as long as a small-enough average skew can be obtained on all cliques. In turn, this helps to provide insights on why D-Cliques work well, as well as to quantify the loss in convergence speed\nthat may result from using construction algorithms that generate cliques with higher skew.\n\n\\subsection{Data Heterogeneity Assumptions}\n\\label{section:non-iid-assumptions}\n\nTo isolate the effect of label distribution skew from other potentially compounding\nfactors, we make the following simplifying assumptions: (1) All classes are\nequally represented in the global dataset; (2) All classes are represented on\nthe same number of nodes; (3) All nodes have the same number of examples.\n\nWhile less realistic than the assumptions used Section~\\ref{section:evaluation}, \nthese assumptions are still reasonable because: (1) Global class imbalance equally\naffects the optimization process on a single node and is therefore not\nspecific to the decentralized setting; (2) Our results do not exploit specific\npositions in the topology;  (3) Imbalanced dataset sizes across nodes can be\naddressed for instance by appropriately weighting the individual loss\nfunctions.\n\nThese assumptions do make the construction of cliques slightly easier by \nmaking it easy to build cliques that have zero skew, as shown in \nSection~\\ref{section:ideal-cliques}. \n\n\\subsection{Constructing Ideal Cliques}\n\\label{section:ideal-cliques}\n \n Algorithm~\\ref{Algorithm:D-Clique-Construction} shows the overall approach\n for constructing a D-Cliques topology under the assumptions of Section~\\ref{section:non-iid-assumptions}.\\footnote{An IID\n version of D-Cliques, in which each node has an equal number of examples of\n all classes, can be implemented by picking $\\#L$ nodes per clique at random.}\n It expects the following inputs: $L$, the set of all classes present in the global distribution $D = \\bigcup_{i \\in N} D_i$; $N$, the set of all nodes; a function $classes(S)$, which given a subset $S$ of nodes in $N$ returns the set of classes in their joint local distributions ($D_S = \\bigcup_{i \\in S} D_i$); a function $intra(DC)$, which given $DC$, a set of cliques (set of set of nodes), creates a set of edges ($\\{\\{i,j\\}, \\dots \\}$) connecting all nodes within each clique to one another; a function $inter(DC)$, which given a set of cliques, creates a set of edges ($\\{\\{i,j\\}, \\dots \\}$) connecting nodes belonging to different cliques; and a function $weigths(E)$, which given a set of edges, returns the weighted matrix $W_{ij}$.  Algorithm~\\ref{Algorithm:D-Clique-Construction} returns both $W_{ij}$, for use in D-SGD (Algorithm~\\ref{Algorithm:D-PSGD} and~\\ref{Algorithm:Clique-Unbiased-D-PSGD}), and $DC$, for use with Clique Averaging (Algorithm~\\ref{Algorithm:Clique-Unbiased-D-PSGD}).\n \n   \\begin{algorithm}[h]\n   \\caption{D-Cliques Construction}\n   \\label{Algorithm:D-Clique-Construction}\n   \\begin{algorithmic}[1]\n        \\STATE \\textbf{Require:} set of classes globally present $L$, \n        \\STATE~~ set of all nodes $N = \\{ 1, 2, \\dots, n \\}$,\n        \\STATE~~ fn $\\textit{classes}(S)$ that returns the classes present in a subset of nodes $S$,\n        \\STATE~~ fn $\\textit{intra}(DC)$ that returns edges intraconnecting cliques of $DC$,\n        \\STATE~~ fn $\\textit{inter}(DC)$ that returns edges interconnecting cliques of $DC$ (Sec.~\\ref{section:interclique-topologies})\n         \\STATE~~ fn $\\textit{weights}(E)$ that assigns weights to edges in $E$ \n         \n        \\STATE $R \\leftarrow \\{ n~\\text{for}~n \\in N \\}$ \\COMMENT{Remaining nodes}\n        \\STATE $DC \\leftarrow \\emptyset$ \\COMMENT{D-Cliques}\n        \\STATE $\\textit{C} \\leftarrow \\emptyset$ \\COMMENT{Current Clique}\n        \\WHILE{$R \\neq \\emptyset$}\n       \\STATE $n \\leftarrow \\text{pick}~1~\\text{from}~\\{ m \\in R | \\textit{classes}(\\{m\\}) \\subsetneq \\textit{classes}(\\textit{C}) \\}$\n       \\STATE $R \\leftarrow R \\setminus \\{ n \\}$\n       \\STATE $C \\leftarrow C \\cup \\{ n \\}$\n       \\IF{$\\textit{classes}(C) = L$}\n           \\STATE $DC \\leftarrow DC \\cup \\{ C \\}$\n           \\STATE $C \\leftarrow \\emptyset$\n       \\ENDIF\n        \\ENDWHILE\n        \\RETURN $(weights(\\textit{intra}(DC) \\cup \\textit{inter}(DC)), DC)$\n   \\end{algorithmic}\n\\end{algorithm}\n \nThe implementation builds a single clique by adding nodes with different\nclasses until all classes of the global distribution are represented. Each\nclique is built sequentially until all nodes are parts of cliques.\nBecause all classes are represented on an equal number of nodes, all cliques\nwill have nodes of all classes. Furthermore, since nodes have examples\nof a single class, we are guaranteed a valid assignment is possible in a greedy manner. \nAfter cliques are created, edges are added and weights are assigned to edges, \nusing the corresponding input functions.\n\n\\subsection{Evaluation}\n\\label{section:ideal-cliques-evaluation}\n\nIn this section, we provide figures analogous to those of the main text using the partitioning \nscheme of Section~\\ref{section:non-iid-assumptions}.\n\n\\subsubsection{Data Heterogeneity is Significant at Multiple Levels of Node Skew} \n\n\\autoref{fig:iid-vs-non-iid-problem-1-class-per-node} is consistent  with \\autoref{fig:iid-vs-non-iid-problem} albeit\nwith slower convergence speed and higher variance.  On the one hand, \\autoref{fig:iid-vs-non-iid-problem-1-class-per-node} shows that an extreme  skew amplifies the difficulty of learning. On the other hand, \\autoref{fig:iid-vs-non-iid-problem} shows that the problem is not limited to the most extreme cases and is therefore worthy of consideration in designing decentralized federated learning solutions.\n\n\n\n\\begin{figure*}[htbp]\n     \\centering\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ring-IID-vs-non-IID-eq-classes-1-class-per-node}\n\\caption{\\label{fig:ring-IID-vs-non-IID-eq-classes-1-class-per-node} Ring topology}\n     \\end{subfigure}\n     \\quad\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/grid-IID-vs-non-IID-eq-classes-1-class-per-node}\n\\caption{\\label{fig:grid-IID-vs-non-IID-eq-classes-1-class-per-node} Grid topology}\n     \\end{subfigure}\n     \\quad\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/fc-IID-vs-non-IID-eq-classes-1-class-per-node}\n\\caption{\\label{fig:fully-connected-IID-vs-non-IID-eq-classes-1-class-per-node} Fully-connected topology}\n     \\end{subfigure}\n        \\caption{Convergence speed of decentralized SGD with and without label distribution skew for different topologies on MNIST (Variation of \\autoref{fig:iid-vs-non-iid-problem} using balanced classes and skewed with 1 class\/node).\n        \\label{fig:iid-vs-non-iid-problem-1-class-per-node}}\n\\end{figure*}\n\n\\subsubsection{D-Cliques Match the Convergence Speed of Fully-Connected with a Fraction of the Edges}\n\n\\autoref{fig:convergence-speed-dc-vs-fc-1-class-per-node} shows consistent\nresults with \\autoref{fig:convergence-speed-dc-vs-fc-2-shards-per-node}:\nD-Cliques work equally well in more extreme skew. It should therefore work\nwell for other levels of label distribution skew commonly encountered in\npractice.\n\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-mnist-dc-fc-vs-fc-1-class-per-node}\n    \\caption{\\label{fig:convergence-speed-mnist-dc-fc-vs-fc-1-class-per-node} MNIST}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-dc-fc-vs-fc-1-class-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-dc-fc-vs-fc-1-class-per-node} CIFAR10 (with momentum)}\n    \\end{subfigure}\n\\caption{\\label{fig:convergence-speed-dc-vs-fc-1-class-per-node} Comparison on 100 heterogeneous nodes\nbetween a fully-connected network and D-Cliques (fully-connected) constructed with Greedy Swap (10 cliques of 10 nodes) using\nClique Averaging. (Variation of \\autoref{fig:convergence-speed-dc-vs-fc-2-shards-per-node} with 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n\n\\subsubsection{Clique Averaging and Momentum are Beneficial and Sometimes Necessary}\n\n\\autoref{fig:d-clique-mnist-clique-avg-1-class-per-node} and \\autoref{fig:cifar10-c-avg-momentum-1-class-per-node} show that, compared respectively to \\autoref{fig:d-clique-mnist-clique-avg} and \\autoref{fig:cifar10-c-avg-momentum}, Clique Averaging increases in importance the more extreme the skew is and provides consistent convergence speed at multiple levels.\n\n\\begin{figure}[htbp]\n         \\centering\n         \\includegraphics[width=0.23\\textwidth]{figures\/convergence-speed-mnist-dc-no-c-avg-vs-c-avg-1-class-per-node}\n\\caption{\\label{fig:d-clique-mnist-clique-avg-1-class-per-node} MNIST: Effect of Clique Averaging on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes). Y axis starts at 89. (Variation of \\autoref{fig:d-clique-mnist-clique-avg} with balanced classes and 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-wo-c-avg-no-mom-vs-mom-1-class-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-wo-c-avg-no-mom-vs-mom-1-class-per-node} Without Clique Averaging }\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-w-c-avg-no-mom-vs-mom-1-class-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-w-c-avg-no-mom-vs-mom-1-class-per-node} With Clique Averaging}\n    \\end{subfigure}\n\\caption{\\label{fig:cifar10-c-avg-momentum-1-class-per-node} CIFAR10: Effect of Clique Averaging, without and with\nmomentum, on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes) (variation of \\autoref{fig:cifar10-c-avg-momentum} with 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n\n\\subsubsection{D-Cliques Clustering is Necessary}\n\\label{section:d-cliques-clustering-is-necessary}\n\nIn this experiment, we compare D-Cliques to different variations of random graphs,\nwith additional variations compared to the experiments of Section~\\ref{section:d-cliques-vs-random-graphs}, \nto show it is actually necessary. Compared to a random graph, D-Cliques enforce additional constraints \nand provide additional mechanisms: they ensure\na diverse representation of all classes in the immediate neighbourhood of all nodes; they enable\n Clique Averaging to debias gradients; and they provide a high-level of clustering, i.e. neighbors \n of a node are neighbors themselves, which tends to lower variance.\nIn order to distinguish the effect of the first two from the last, we compare D-Cliques to other variations \nof random graphs: (1) with the additional constraint that all classes should be represented in the immediate neighborhood of all nodes \n(i.e. 'diverse neighbors'), and (2) in combination with unbiased gradients computed using \nthe average of the gradients of a subset of neighbors of a node such that the skew of that subset is 0.\n\nThe partitioning scheme we use (Section~\\ref{section:non-iid-assumptions}) makes the construction of both D-Cliques and diverse random graphs easy and ensures that in both cases the skew of the cliques or neighborhood subset is exactly 0. This removes the challenge of designing topology optimization algorithms for both D-Cliques and random graphs that would guarantee reaching the same level of skews in both cases to make results comparable.\n\n\n\n\\begin{figure}[htbp]\n     \\centering     \n         \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/convergence-mnist-random-vs-d-cliques-1-class-per-node}\n                  \\caption{MNIST}\n         \\end{subfigure}\n                 \\hfill                      \n        \\begin{subfigure}[b]{0.23\\textwidth}\n        \\centering\n         \\includegraphics[width=\\textwidth]{figures\/convergence-cifar10-random-vs-d-cliques-1-class-per-node}\n         \\caption{CIFAR10}\n     \\end{subfigure} \n \\caption{\\label{fig:convergence-random-vs-d-cliques-1-class-per-node} Comparison to variations of Random Graph with 10 edges per node on 100 nodes (variation of \\autoref{fig:convergence-random-vs-d-cliques-2-shards} with 1 class\/node instead of 2 shards\/node as well as additional random graphs with more constraints).} \n\\end{figure}\n\n\\autoref{fig:convergence-random-vs-d-cliques-1-class-per-node} compares the convergence speed of D-Cliques with all the variations of random graphs on both MNIST and CIFAR10. In both cases,\nD-Cliques converge faster than all other options. In addition, in the case of CIFAR10, the clustering appears to be critical\nfor good convergence speed: even a random graph with diverse neighborhoods and unbiased gradients \nconverges significantly slower.\n\n\n\n\n\n\\subsubsection{D-Cliques Scale with Sparser Inter-Clique Topologies}\n\n\\autoref{fig:d-cliques-scaling-mnist-1000-1-class-per-node} and \\autoref{fig:d-cliques-scaling-cifar10-1000-1-class-per-node} are consistent with \\autoref{fig:d-cliques-scaling-mnist-1000} and \\autoref{fig:d-cliques-scaling-cifar10-1000}. The less extreme skew enables a slightly faster convergence rate in the case of CIFAR10 (\\autoref{fig:d-cliques-scaling-cifar10-1000}).\n\n\n\\begin{figure}[htbp]\n     \\centering\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n            \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-mnist-1000-linear-1-class-per-node}\n             \\caption{\\label{fig:d-cliques-scaling-mnist-1000-linear-1-class-per-node} Linear}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-mnist-1000-super-linear-1-class-per-node}\n\\caption{\\label{fig:d-cliques-scaling-mnist-1000-super-linear-1-class-per-node}  Super- and Quasi-Linear}\n     \\end{subfigure}\n\\caption{\\label{fig:d-cliques-scaling-mnist-1000-1-class-per-node} MNIST: D-Cliques convergence speed with 1000 nodes (10 nodes per clique, same number of updates per epoch as 100 nodes, i.e. batch-size 10x less per node) with different inter-clique topologies. (variation of \\autoref{fig:d-cliques-scaling-mnist-1000} with 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n\n\\begin{figure}[htbp]\n     \\centering\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n            \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-cifar10-1000-linear-1-class-per-node}\n             \\caption{\\label{fig:d-cliques-scaling-cifar10-1000-linear-1-class-per-node} Linear}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-cifar10-1000-super-linear-1-class-per-node}\n\\caption{\\label{fig:d-cliques-scaling-cifar10-1000-super-linear-1-class-per-node}  Super- and Quasi-Linear}\n     \\end{subfigure}\n\\caption{\\label{fig:d-cliques-scaling-cifar10-1000-1-class-per-node} CIFAR10: D-Cliques convergence speed with 1000 nodes (10 nodes per clique, same number of updates per epoch as 100 nodes, i.e. batch-size 10x less per node) with different inter-clique topologies (variation of \\autoref{fig:d-cliques-scaling-cifar10-1000} with 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n\n\\subsubsection{Full Intra-Clique Connectivity is Necessary}\n\n\n\n\\begin{figure}[htbp]\n     \\centering\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering   \n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-ideal-wo-clique-avg-impact-of-edge-removal}     \n\\caption{\\label{fig:d-cliques-ideal-wo-clique-avg-impact-of-edge-removal} Without Clique Averaging }\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-ideal-w-clique-avg-impact-of-edge-removal}\n\\caption{\\label{fig:d-cliques-ideal-w-clique-avg-impact-of-edge-removal} With Clique Averaging}\n\\end{subfigure}\n\\caption{\\label{fig:d-cliques-ideal-mnist-intra-connectivity} MNIST: Impact of intra-clique edge removal on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes) (variation of \\autoref{fig:d-cliques-mnist-intra-connectivity} with 1 class\/node instead of 2 shards\/node). Y axis starts at 89.}\n\\end{figure}\n\n\\autoref{fig:d-cliques-ideal-mnist-intra-connectivity} and \\autoref{fig:d-cliques-ideal-cifar10-intra-connectivity} show higher variance than \\autoref{fig:d-cliques-mnist-intra-connectivity} and \\autoref{fig:d-cliques-cifar10-intra-connectivity}, with a significantly lower convergence speed in the case of CIFAR10 (\\autoref{fig:d-cliques-ideal-cifar10-intra-connectivity}).\n\n\n\n\n\\begin{figure}[t]\n     \\centering\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering   \n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-ideal-cifar10-wo-clique-avg-impact-of-edge-removal}     \n\\caption{\\label{fig:d-cliques-ideal-cifar10-wo-clique-avg-impact-of-edge-removal} Without Clique Averaging }\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-ideal-cifar10-w-clique-avg-impact-of-edge-removal}\n\\caption{\\label{fig:d-cliques-ideal-cifar10-w-clique-avg-impact-of-edge-removal} With Clique Averaging}\n\\end{subfigure}\n\\caption{\\label{fig:d-cliques-ideal-cifar10-intra-connectivity} CIFAR10: Impact of intra-clique edge removal (with momentum) on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes) (variation of \\autoref{fig:d-cliques-cifar10-intra-connectivity} with 1 class\/node instead of 2 shards\/node).}\n\\end{figure}\n     \n \n\\section{Conclusion}\n\\label{section:conclusion}\n\nWe proposed D-Cliques, a sparse topology that obtains similar convergence\nspeed as a fully-connected network in the presence of label distribution skew.\nD-Cliques is based on assembling subsets of nodes into cliques such\nthat the clique-level class distribution is representative of the global\ndistribution, thereby locally recovering homogeneity of data. Cliques are\nconnected together by a\nsparse inter-clique topology so that\nthey quickly converge to the same model. We proposed Clique\nAveraging to remove the bias in gradient computation due to non-homogeneous\naveraging neighborhood by averaging gradients only with other nodes within the clique. Clique Averaging\ncan in turn be used to implement an effective momentum.\nThrough our extensive set of experiments, we\nshowed that the clique structure of D-Cliques is critical in obtaining these\nresults and that a small-world inter-clique topology with only $O(n \\log n)$ \nedges achieves a very good compromise between\nconvergence speed and scalability with the number of nodes.\n\nD-Cliques thus appears to be very promising to reduce bandwidth\nusage on FL servers and to implement fully decentralized alternatives in a\nwider range of applications where global coordination is impossible or costly.\nFor instance, the relative frequency of classes in each node\ncould be computed using PushSum~\\cite{kempe2003gossip}, and the topology could\nbe constructed in a decentralized and adaptive way with\nPeerSampling~\\cite{jelasity2007gossip}. This will be investigated in future work.\nWe also believe that our ideas can be useful to deal\nwith more general types of data heterogeneity beyond the important case\nof\nlabel distribution skew on which we focused in this paper. An important\nexample is\ncovariate shift or feature distribution skew \\cite{kairouz2019advances}, for\nwhich local density estimates could be used as basis to construct cliques that\napproximately recover the global distribution.\n\\section{D-Cliques}\n\\label{section:d-cliques}\n\nIn this section, we introduce D-Cliques, a topology\ndesigned to compensate for data heterogeneity. We also present some\nmodifications of D-SGD that leverage some properties of the proposed\ntopology and allow to implement a successful momentum scheme.\n\n\\subsection{Intuition}\n\nTo give the intuition behind\nour approach, let us consider the neighborhood of a single node in a grid\ntopology represented\non Figure~\\ref{fig:grid-iid-vs-non-iid-neighbourhood}.\nNodes are distributed randomly in the grid and the colors of a node represent\nthe proportion of each class in its local dataset. In the homogeneous\nsetting, the label distribution is the same across\nnodes: in the example shown in Figure~\\ref{fig:grid-iid-neighbourhood}, all classes\nare represented in equal proportions on all nodes. This is not the case in the\nheterogeneous setting: Figure~\\ref{fig:grid-non-iid-neighbourhood} shows an\nextreme case of label distribution skew where each\nnode holds examples of a single class only.\n\nFrom the point of view of the center node in\nFigure~\\ref{fig:grid-iid-vs-non-iid-neighbourhood}, a single training step of\nD-SGD is\nequivalent to sampling a mini-batch five times larger from the union of the\nlocal distributions of neighboring nodes.\nIn the homogeneous case, since gradients are computed from examples of all\nclasses,\nthe resulting averaged gradient  points in a direction that tends to reduce\nthe loss across all classes. In contrast, in the heterogeneous case, the\nrepresentation of classes in the immediate neighborhood of the node is\ndifferent from the global label distribution\n(in Figure~\\ref{fig:grid-non-iid-neighbourhood}, only a\nsubset of classes are represented), thus the gradients will\nbe biased.\nImportantly, as the distributed averaging process takes several steps to\nconverge, this variance persists across iterations as the locally computed\ngradients are far from the global average.\\footnote{One could perform a\nsufficiently large number of\naveraging steps between each gradient step, but this is too costly in\npractice.} This can significantly slow down\nconvergence speed to the point of making decentralized optimization\nimpractical.\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.18\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/grid-iid-neighbourhood}\n\\caption{\\label{fig:grid-iid-neighbourhood} Homogeneous data}\n     \\end{subfigure}\n     \\hspace*{.5cm}\n     \\begin{subfigure}[b]{0.18\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/grid-non-iid-neighbourhood}\n\\caption{\\label{fig:grid-non-iid-neighbourhood}  Heterogeneous data}\n     \\end{subfigure}\n        \\caption{Neighborhood in a grid.}\n        \\label{fig:grid-iid-vs-non-iid-neighbourhood}\n\\end{figure}\n\nWith D-Cliques, we address label distribution skew by\ncarefully designing a\nnetwork topology composed of \\textit{locally representative cliques} while \nmaintaining \\textit{sparse inter-clique connections} only.\n\n\\subsection{Constructing Locally Representative Cliques}\n\nD-Cliques construct a topology in which each node is part of a \\emph{clique} \n(i.e., a subset of nodes whose induced subgraph is fully connected)\nsuch that the label distribution in each clique is\nclose to the global label distribution. Formally, for a label $y$ and a\nclique composed of nodes $C\\subseteq N$, we denote by $p_C(y)=\n\\frac{1}{|C|}\\sum_{i\\in C} p_i(y)$ the distribution of $y$ in $C$\nand by $p(y)=\\frac{1}{n}\\sum_{i\\in N} p_i(y)$ its global distribution.\nWe measure the \\textit{skew} of $C$ by the sum\nof the absolute differences of $p_C(y)$ and $p(y)$:\n\\begin{equation}\n\\label{eq:skew}\n        \\textit{skew}(C) =\\\n    \\sum_{l=1}^L | p_C(y = l) - p(y = l) |.\n\\end{equation}\n\n\n\nTo efficiently construct a set of cliques with small skew, we propose\nGreedy-Swap (Algorithm~\\ref{Algorithm:greedy-swap}). The parameter\n$M$ is the maximum size of cliques and controls the\nnumber of intra-clique edges. We start by initializing cliques at\nrandom. Then, for\na certain number of steps $K$, we randomly pick two cliques and swap two of\ntheir nodes so as to decrease the sum of skews of the two cliques. The swap is\nchosen randomly among the ones that decrease the skew, hence\nthis algorithm can be seen as a form of randomized greedy algorithm.\nWe note that this algorithm only requires\nthe knowledge of the label distribution $p_i(y)$ at each node $i$. For the\nsake of\nsimplicity, we assume that D-Cliques are constructed from the global\nknowledge of these distributions, which can easily be obtained by\ndecentralized averaging in a pre-processing step \\citep[e.g.,][]\n{jelasity2005largegossip}.\n\n\\begin{algorithm}[t]\n   \\caption{D-Cliques Construction via Greedy Swap}\n   \\label{Algorithm:greedy-swap}\n   \\begin{algorithmic}[1]\n        \\STATE \\textbf{Require:} maximum clique size $M$, max steps $K$, set\n        of all nodes $N = \\{ 1, 2, \\dots, n \\}$,\n       \n        procedure $\\texttt{inter}(\\cdot)$ to create intra-clique connections\n        (see Sec.~\\ref{section:interclique-topologies})\n       \n         \\STATE $DC \\leftarrow []$\n         \\WHILE {$N \\neq \\emptyset$}\n         \\STATE $C \\leftarrow$ sample $M$ nodes from $N$ at random\n         \\STATE $N \\leftarrow N \\setminus C$; $DC.\\text{append}(C)$\n         \\ENDWHILE\n         \\FOR{$k \\in \\{1, \\dots, K\\}$}\n        \\STATE $C_1,C_2 \\leftarrow$ random sample of 2 elements from $DC$\n          \\STATE $s \\leftarrow \\textit{skew}(C_1) + skew(C_2)$\n        \\STATE $\\textit{swaps} \\leftarrow []$\n        \\FOR{$i \\in C_1, j \\in C_2$}\n          \\STATE $s' \\leftarrow \\textit{skew}(C_1\\setminus\\{i\\}\\cup\\{j\\})\n          + \\textit{skew}(C_2 \\setminus\\{i\\}\\cup\\{j\\})$\\hspace*{-.05cm}\n          \\IF {$s' < s$}\n            \\STATE \\textit{swaps}.append($(i, j)$)\n          \\ENDIF\n        \\ENDFOR\n        \\IF {len(\\textit{swaps}) $> 0$}\n          \\STATE $(i,j) \\leftarrow$ random element from $\n          \\textit{swaps}$ \n          \\STATE $C_1 \\leftarrow C_1 \\setminus\\{i\\}\\cup\\{j\\}; C_2 \\leftarrow C_2 \\setminus\\{j\\}\\cup\\{i\\}$\n        \\ENDIF\n         \\ENDFOR\n        \\STATE $E\\leftarrow \\{(i,j) : C\\in DC, i,j\\in C, i\\neq j\\}$\n        \n        \\RETURN topology $G=(N,E \\cup \n        \\texttt{inter}(DC))$\n   \\end{algorithmic}\n\\end{algorithm}\n\n\nThe key idea of D-Cliques is to ensure the clique-level label distribution\n$p_C(y)$\n matches closely the global distribution $p(y)$. As a consequence,\nthe local models of nodes across cliques remain rather close. Therefore, a\nsparse inter-clique topology can be used, significantly reducing the total\nnumber of edges without slowing down the convergence. We discuss some possible\nchoices for this inter-clique topology in the next section.\n\n\\subsection{Adding Sparse Inter-Clique Connections}\n\\label{section:interclique-topologies}\n\nTo ensure a global consensus and convergence, we introduce\n\\textit{inter-clique connections} between a small number of node pairs that\nbelong to different cliques, thereby implementing the \\texttt{inter}\nprocedure called at the end of Algorithm~\\ref{Algorithm:greedy-swap}.\nWe aim to ensure that the degree of each node remains low and balanced so as\nto make the network topology well-suited to decentralized federated learning.\nWe consider several choices of inter-clique topology, which offer\ndifferent scalings for the number of required edges and the average distance\nbetween nodes in the resulting graph.\n\nThe \\textit{ring} has (almost) the fewest possible number of edges for the\ngraph to be connected: in this case, each clique is connected to exactly\ntwo other cliques by a single edge. This topology requires only $O(\\frac{n}\n{M})$ inter-clique edges but suffers an $O(n)$ average distance between nodes.\n\nThe\n\\textit{fractal} topology\nprovides a logarithmic bound on the average distance. In this\nhierarchical scheme, cliques are arranged in larger groups of $M$ cliques that\nare connected\ninternally with one edge per\npair of cliques, but with only one edge between pairs of larger groups. The\ntopology is built recursively such that $M$ groups will themselves form a\nlarger group at the next level up. This results in at most $M$ edges per node \nif edges are evenly distributed: i.e., each group within the same level adds \nat most $M-1$ edges to other groups, leaving one node per group with $M-1$ \nedges that can receive an additional edge to connect with other groups at the next level.\nSince nodes have at most $M$ edges, the total number of inter-clique edges\nis at most $nM$ edges.\n\nWe can also design an inter-clique topology in which the number of edges\nscales in a log-linear fashion by following a\nsmall-world-like topology~\\cite{watts2000small} applied on top of a\nring~\\cite{stoica2003chord}. In this scheme, cliques are first arranged in a\nring. Then each clique adds symmetric edges, both clockwise and\ncounter-clockwise on the ring, with the $c$ closest cliques in sets of\ncliques that are exponentially bigger the further they are on the ring (see\nAlgorithm~\\ref{Algorithm:Smallworld} in Appendix~\\ref{app:small_world} for\ndetails on the construction). This topology ensures a good connectivity with\nother cliques that are close on the ring, while keeping the average\ndistance small. This scheme uses $O(c\\frac{n}{M}\\log\\frac{n}{M})$ edges,\ni.e.\nlog-linear in $n$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.20\\textwidth]{figures\/fully-connected-cliques}\n    \\caption{\\label{fig:d-cliques-figure} D-Cliques with $n=100$, $M=10$ and a\nfully connected inter-clique topology on a problem with 1 class\/node.}\n\\end{figure}\n\nFinally, we can consider a \\emph{fully connected} inter-clique topology\n such that each clique has exactly\none edge with each of the other cliques, spreading these additional edges\nequally among the nodes of a clique, as illustrated in Figure~\\ref{fig:d-cliques-figure}. \nThis has the advantage of\nbounding the distance between any pair of nodes to $3$ but requires\n$O(\\frac{n^2}{M^2})$ inter-clique edges, i.e. quadratic in $n$.\n\n\n\n\n\n\n\n\\subsection{Optimizing over D-Cliques with Clique Averaging and Momentum}\n\\label{section:clique-averaging-momentum}\n\n\n\nWhile limiting the number of inter-clique connections reduces the\namount of messages traveling on the network, it also introduces a form of\nbias.\nFigure~\\ref{fig:connected-cliques-bias} illustrates the problem on the\nsimple case of two cliques connected by one inter-clique edge (here,\nbetween the green node of the left clique and the pink node of the right\nclique). In this example, each node holds example of a single class. Let us\nfocus on node A. With weights computed as in \\eqref{eq:metro},\nnode A's self-weight is $\\frac{12}\n{110}$, the weight between A and the green node connected to B is\n$\\frac{10}{110}$, and\nall other neighbors of A have a weight of $\\frac{11}{110}$. Therefore, the\ngradient at A is biased towards its own class (pink) and against the green\nclass. A similar bias holds for all other nodes\nwithout inter-clique edges with respect to their respective classes. For node\nB, all its edge weights (including its self-weight) are equal to $\\frac{1}\n{11}$. However, the green class is represented twice (once as a clique\nneighbor and once from the inter-clique edge), while all other classes are\nrepresented only once. This biases the gradient toward the green class. The\ncombined effect of these two sources of bias is to increase the variance\nof the local models across nodes.\n\n\\begin{figure}[t]\n         \\centering\n         \\includegraphics[width=0.3\\textwidth]{figures\/connected-cliques-bias}\n\\caption{\\label{fig:connected-cliques-bias} Illustrating the bias induced by\ninter-clique connections (see main text for details).}\n\\end{figure}\n\n\\paragraph{Clique Averaging.} \nWe address this problem by adding \\emph{Clique\nAveraging} to D-SGD\n(Algorithm~\\ref{Algorithm:Clique-Unbiased-D-PSGD}), which essentially\ndecouples gradient averaging from model averaging. The idea is to use only the\ngradients of neighbors within the same clique to compute the average gradient\nso as to remove the bias due to inter-clique edges. In contrast, all\nneighbors' models (including those in different cliques)\nparticipate in model averaging as in the original version. Adding Clique Averaging\nrequires gradients to be sent separately from the model parameters: the number\nof messages\nexchanged between nodes is therefore twice their number of edges.\n\n\\begin{algorithm}[t]\n   \\caption{D-SGD with Clique Averaging, Node $i$}\n   \\label{Algorithm:Clique-Unbiased-D-PSGD}\n   \\begin{algorithmic}[1]\n        \\STATE \\textbf{Require} initial model $\\theta_i^{(0)}$, learning\n        rate $\\gamma$, mixing weights $W$, mini-batch size $m$, number of\n        steps $K$\n        \\FOR{$k = 1,\\ldots, K$}\n          \\STATE $S_i^{(k)} \\gets \\text{mini-batch of $m$ samples drawn\n          from~} D_i$\n          \\STATE $g_i^{(k)} \\gets \\frac{1}{|\\textit{Clique}(i)|}\\sum_{j \\in \n          \\textit{Clique(i)}}  \\nabla F(\\theta_j^{(k-1)}; S_j^{(k)})$\n          \\STATE $\\theta_i^{(k-\\frac{1}{2})} \\gets \\theta_i^{(k-1)} - \\gamma g_i^{(k)}$ \n          \\STATE $\\theta_i^{(k)} \\gets \\sum_{j \\in N} W_{ji}^{(k)} \\theta_j^{(k-\\frac{1}{2})}$\n        \\ENDFOR\n   \\end{algorithmic}\n\\end{algorithm}\n\n\n\\paragraph{Implementing momentum with Clique Averaging.}\nEfficiently training high capacity models usually requires additional\noptimization techniques. In particular, momentum~\\cite{pmlr-v28-sutskever13}\nincreases the magnitude of the components of the gradient that are shared\nbetween several consecutive steps, and is critical for deep convolutional networks like\nLeNet~\\cite{lecun1998gradient,quagmire} to converge quickly. However, a direct\napplication of momentum in data heterogeneous settings can\nactually be very detrimental and even fail to converge, as we will show in\n our experiments (Figure~\\ref{fig:cifar10-c-avg-momentum} in\n Section~\\ref{section:evaluation}).\nClique Averaging allows us to reduce the bias in the momentum by using the\nclique-level average gradient $g_i^{(k)}$ of\nAlgorithm~\\ref{Algorithm:Clique-Unbiased-D-PSGD}:\n\\begin{equation}\nv_i^{(k)} \\leftarrow m v_i^{(k-1)} +  g_i^{(k)}.\n\\end{equation}\nIt then suffices to modify the original gradient step to apply momentum:\n\\begin{equation}\n\\theta_i^{(k-\\frac{1}{2})} \\leftarrow \\theta_i^{(k-1)} - \\gamma v_i^{(k)}.\n\\end{equation}\n\n\n\n\\section{Evaluation}\n\\label{section:evaluation}\n\nIn this section, we first compare D-Cliques to alternative topologies to\nshow the benefits and relevance of our main design choices. Then, \nwe evaluate different inter-clique topologies to further reduce the number of\ninter-clique connections so as to gracefully scale with the number of\nnodes. Then, we show the impact of removing intra-clique edges.\n Finally, we show that Greedy Swap\n(Alg.~\\ref{Algorithm:greedy-swap}) \nconstructs cliques efficiently with consistently lower skew than\nrandom cliques.\n\n\\subsection{Experimental Setup}\n\\label{section:experimental-settings}\n\nOur main goal is to provide a fair comparison of the convergence speed across\ndifferent topologies and algorithmic variations, in order to\nshow that D-Cliques\ncan remove much of the effects of label distribution skew.\n\nWe experiment with two datasets: MNIST~\\cite{mnistWebsite} and\nCIFAR10~\\cite{krizhevsky2009learning}, which both have $L=10$ classes.\nFor MNIST,  we use 50k and 10k examples from the original 60k training \nset for training and validation respectively. We use all 10k examples of \nthe test set to measure prediction accuracy.  The validation set preserves the\noriginal unbalanced ratio of the classes in the test set, and the remaining\nexamples become the training set.\nFor CIFAR10, classes are evenly balanced: we initially used 45k\/50k images \nof the original training set for training, 5k\/50k for validation, and all 10k examples \nof the test set for measuring prediction accuracy. After tuning hyper-parameters\non initial experiments, we then used all 50k images of the original training set\nfor training for all experiments, as the 45k did not split evenly in 1000 nodes\nwith the partitioning scheme explained in the next paragraph.\n\nFor both MNIST and CIFAR10, we use the heterogeneous data partitioning scheme\nproposed by~\\citet{mcmahan2016communication} \nin their seminal FL work: \nwe sort all training examples by class, then split the list into shards of\nequal size, and randomly assign two shards to each node. When the number of\nexamples of one class does not divide evenly in shards, as is the case for MNIST, some shards may have examples of more than one class and therefore nodes may have examples\nof up to 4 classes. However, most nodes will have examples of 2 classes.  The varying number \nof classes, as well as the varying distribution of examples within a single node, makes the task \nof creating cliques with low skew nontrivial.\n\nWe\nuse a logistic regression classifier for MNIST, which\nprovides up to 92.5\\% accuracy in the centralized setting.\nFor CIFAR10, we use a Group-Normalized variant of LeNet~\\cite{quagmire}, a\ndeep convolutional network which achieves an accuracy of $74.15\\%$ in the\ncentralized setting.\nThese models are thus reasonably accurate (which is sufficient to\nstudy the effect of the topology) while being sufficiently fast to train in a\nfully decentralized setting and simple enough to configure and analyze.\nRegarding hyper-parameters, we jointly optimize the learning rate and\nmini-batch size on the\nvalidation set for 100 nodes, obtaining respectively $0.1$ and $128$ for\nMNIST and $0.002$ and $20$ for CIFAR10.\nFor CIFAR10, we additionally use a momentum of $0.9$.\n\nWe evaluate 100- and 1000-node networks by creating multiple models \nin memory and simulating the exchange of messages between nodes.\nTo ignore the impact of distributed execution strategies and system\noptimization techniques, we report the test accuracy of all nodes (min, max,\naverage) as a function of the number of times each example of the dataset has\nbeen sampled by a node, i.e. an \\textit{epoch}. This is equivalent to the classic \ncase of a single node sampling the full distribution.\nTo further make results comparable across different number of nodes, we lower\nthe batch size proportionally to the number of nodes added, and inversely,\ne.g. on MNIST, 128 with 100 nodes vs. 13 with 1000 nodes. This\nensures the same number of model updates and averaging per epoch, which is\nimportant to have a fair comparison.\\footnote{Updating and averaging models\nafter every example can eliminate the impact of label distribution skew. However, the\nresulting communication overhead is impractical.}\n\nFinally, we compare our results against an ideal baseline:\na fully-connected network topology with the same number of nodes. \nThis baseline is essentially equivalent to a centralized (single) IID node using a batch size\n$n$ times bigger, where $n$ is the number of nodes.  Both a fully-connected network and a single IID node\n effectively optimize a single model and sample\nuniformly from the global distribution: both therefore remove entirely the\neffect of label distribution skew and of the network topology on the\noptimization. In practice, we prefer a\nfully-connected network because it\n converges slightly faster and obtains slightly \nbetter final accuracy than a single node sampling randomly from the global\ndistribution.\\footnote{We \nconjecture that an heterogeneous data partition in a fully-connected network may force \nmore balanced representation of all classes in the union of all mini-batches, leading to better convergence.}\n\n\\subsection{D-Cliques Match the Convergence Speed of Fully-Connected with a\nFraction of the Edges}\n\\label{section:d-cliques-vs-fully-connected}\n\nIn this first experiment, we show that D-Cliques with Clique Averaging (and\nmomentum when mentioned) converges \nalmost as fast as a fully-connected network on both MNIST and CIFAR10. Figure~\\ref{fig:convergence-speed-dc-vs-fc-2-shards-per-node} \nillustrates the convergence speed of D-Cliques with $n=100$ nodes on MNIST (with Clique Averaging) \nand CIFAR10 (with Clique Averaging and momentum). Observe that the convergence speed is\nvery close to that of a fully-connected topology, and significantly better than with\na ring or a grid (see Figure~\\ref{fig:iid-vs-non-iid-problem}). \nIt also has less variance than both the ring and grid. \n\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-mnist-dc-fc-vs-fc-2-shards-per-node}\n    \\caption{\\label{fig:convergence-speed-mnist-dc-fc-vs-fc-2-shards-per-node} MNIST}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-dc-fc-vs-fc-2-shards-per-node}\n    \\caption{\n    \\label{fig:convergence-speed-cifar10-dc-fc-vs-fc-2-shards-per-node} CIFAR10 (w\/ momentum)}\n    \\end{subfigure}\n\\caption{\\label{fig:convergence-speed-dc-vs-fc-2-shards-per-node} Comparison on 100 heterogeneous nodes (2 shards\/node)\nbetween a fully-connected network and D-Cliques (fully-connected) constructed with Greedy Swap (10 cliques of 10 nodes) using\nClique Averaging. Bold line is the average accuracy over\nall nodes. Thinner upper and lower lines are maximum and minimum accuracy over\nall nodes.}\n\\end{figure}\n\n\n\\subsection{Clique Averaging is Beneficial and Sometimes Necessary}\n\\label{sec:exp:clique_avg}\n\nIn this experiment, we perform an ablation study of the effect of Clique Averaging.\nFigure~\\ref{fig:d-clique-mnist-clique-avg} shows that Clique Averaging\n(Algorithm~\\autoref{Algorithm:Clique-Unbiased-D-PSGD})\n reduces the variance of models across nodes and slightly accelerates the\nconvergence on MNIST. Recall that Clique Averaging induces a small\nadditional cost, as gradients\nand models need to be sent in two separate rounds of messages. \nNonetheless, compared to fully connecting all nodes, the total number \nof messages per round for 100 nodes is reduced by $\\approx 80\\%$.\n\n\\begin{figure}[htbp]\n         \\centering\n         \\includegraphics[width=0.23\\textwidth]{figures\/convergence-speed-mnist-dc-no-c-avg-vs-c-avg-2-shards-per-node}\n\\caption{\\label{fig:d-clique-mnist-clique-avg} MNIST: Effect of Clique Averaging on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes). Y axis starts at 89.}\n\\end{figure}\n\n\n\nThe effect of Clique Averaging is much more pronounced on CIFAR10, as can\nbe seen in\nFigure~\\ref{fig:cifar10-c-avg-momentum}, especially when used in combination with momentum.\nWithout Clique Averaging,\nthe use of momentum is actually detrimental. With Clique Averaging, the \nsituation reverses and momentum is again beneficial. The combination\nof both has the fastest convergence speed and the lowest variance among all\nfour possibilities. We believe that the gains obtained with Clique\nAveraging are larger on CIFAR10 than on MNIST because the model we train on\nCIFAR10 (a deep convolutional network) has much higher capacity than the\nlinear model used for MNIST. The resulting highly nonconvex objective increases the\nsensitivity of local updates to small differences in the gradients, making\nthem point in different directions, as observed by \\citet{consensus_distance}\neven in the homogeneous setting.\nClique Averaging helps to reduce this effect by reducing the bias in\nlocal gradients.\n\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-wo-c-avg-no-mom-vs-mom-2-shards-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-wo-c-avg-no-mom-vs-mom-2-shards-per-node} Without Clique Averaging }\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-w-c-avg-no-mom-vs-mom-2-shards-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-w-c-avg-no-mom-vs-mom-2-shards-per-node} With Clique Averaging}\n    \\end{subfigure}\n\\caption{\\label{fig:cifar10-c-avg-momentum} CIFAR10: Effect of Clique Averaging, without and with\nmomentum, on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes).}\n\\end{figure}\n\n\n\\subsection{D-Cliques Converge Faster than Random Graphs}\n\\label{section:d-cliques-vs-random-graphs}\n\nIn this experiment, we compare D-Cliques to a random graph that has a similar \nnumber of edges (10) per node to determine\nwhether a simple sparse topology could work equally well. \nTo ensure a fair comparison, because a random graph does not support \nClique Averaging, we do not use it for D-Cliques either.\n\\autoref{fig:convergence-random-vs-d-cliques-2-shards} \nshows that even \\textit{without} Clique Averaging, D-Cliques converge faster and with\nlower variance. Furthermore, the use of momentum in a random graph\nis detrimental, similar to D-Cliques without the use of Clique Averaging \n(see \\autoref{fig:convergence-speed-cifar10-wo-c-avg-no-mom-vs-mom-2-shards-per-node}).\nThis shows that a careful design of the topology is indeed necessary.\n\nD-Cliques converge faster even if we were to create diverse neighborhoods \nin a random graph with lower skew and used those to unbias gradients in an analogous \nway to Clique Averaging (details in Annex~\\ref{section:d-cliques-clustering-is-necessary}, as \nthe experiments require a different partitioning scheme for a fair comparison).\nThe clustering provided by D-Cliques therefore provides faster convergence.\n\n\n\n\\begin{figure}[htbp]\n     \\centering     \n         \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/convergence-mnist-random-vs-d-cliques-2-shards}\n                  \\caption{MNIST}\n         \\end{subfigure}\n                 \\hfill                      \n        \\begin{subfigure}[b]{0.23\\textwidth}\n        \\centering\n         \\includegraphics[width=\\textwidth]{figures\/convergence-cifar10-random-vs-d-cliques-2-shards}\n         \\caption{CIFAR10}\n     \\end{subfigure} \n \\caption{\\label{fig:convergence-random-vs-d-cliques-2-shards} Comparison on 100 heterogeneous nodes between D-Cliques (fully-connected) with 10 cliques of size 10 and a random graph with 10 edges per node \\textit{without} Clique Averaging or momentum.} \n\\end{figure}\n\n\n\n\\subsection{D-Cliques Scale with Sparser Inter-Clique Topologies}\n\\label{section:scaling}\n\nIn this experiment, we explore the trade-offs between scalability and\nconvergence speed induced by the several sparse inter-clique topologies\nintroduced in Section~\\ref{section:interclique-topologies}.\n\\autoref{fig:d-cliques-scaling-mnist-1000} and \\autoref{fig:d-cliques-scaling-cifar10-1000}  \nshow the convergence speed  respectively on MNIST and CIFAR10 on a larger network of 1000 nodes, \ncompared to the ideal baseline of a\nfully-connected network representing\nthe fastest convergence speed achievable if topology had no impact. Among the linear schemes, the ring\ntopology converges but is much slower than our fractal scheme. Among the super-linear schemes, the small-world\ntopology has a convergence speed that is almost the same as with a\nfully-connected inter-clique topology but with 22\\% less edges\n(14.5 edges on average instead of 18.9). \n\n\n\n\n\\begin{figure}[htbp]\n     \\centering\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n            \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-mnist-1000-linear}\n             \\caption{\\label{fig:d-cliques-scaling-mnist-1000-linear} Linear}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-mnist-1000-super-linear}\n\\caption{\\label{fig:d-cliques-scaling-mnist-1000-super-linear}  Super- and Quasi-Linear}\n     \\end{subfigure}\n\\caption{\\label{fig:d-cliques-scaling-mnist-1000} MNIST: D-Cliques convergence\nspeed with 1000 nodes (10 nodes per clique, same number of updates per epoch as 100 nodes, i.e. batch-size 10x less per node) and different inter-clique topologies.}\n\\end{figure}\n\n\\begin{figure}[htbp]\n     \\centering\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n            \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-cifar10-1000-linear}\n             \\caption{\\label{fig:d-cliques-scaling-cifar10-1000-linear} Linear}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-scaling-cifar10-1000-super-linear}\n\\caption{\\label{fig:d-cliques-scaling-cifar10-1000-super-linear}  Super- and Quasi-Linear}\n     \\end{subfigure}\n\\caption{\\label{fig:d-cliques-scaling-cifar10-1000} CIFAR10: D-Cliques\nconvergence speed with 1000 nodes (10 nodes per clique, same number of updates per epoch as 100 nodes, i.e. batch-size 10x less per node) and different inter-clique topologies.}\n\\end{figure}\n\nWhile the small-world inter-clique topology shows promising scaling behavior, the\nfully-connected inter-clique topology still offers\nsignificant benefits with 1000 nodes, as it represents a 98\\% reduction in the\nnumber of edges compared to fully connecting individual nodes (18.9 edges on\naverage instead of 999) and a 96\\% reduction in the number of messages (37.8\nmessages per round per node on average instead of 999). \nWe refer to Appendix~\\ref{app:scaling} for additional results comparing the convergence speed across different number of nodes. \nOverall, these results show that D-Cliques can gracefully scale with the\nnumber of nodes.\n  \n\n\\subsection{Full Intra-Clique Connectivity is Necessary}\n\nIn this experiment, we measure the impact of removing intra-clique edges \n to assess how critical full connectivity is within cliques. We choose edges to remove\n among the 45 undirected edges present in cliques of size 10. The removal of\n an edge removes the connection in both directions. We remove 1 and 5 edges\n randomly, respectively 2.2\\% and 11\\% of intra-clique edges. \\autoref{fig:d-cliques-mnist-intra-connectivity} \n shows that for  MNIST, when not using Clique Averaging, \nremoving edges decreases slightly the convergence speed and increases \nthe variance between nodes. When using Clique Averaging, removing up to 5\nedges does not noticeably affect\nthe convergence speed and variance.\n\n\\begin{figure}[htbp]\n     \\centering\n\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering   \n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-wo-clique-avg-impact-of-edge-removal}     \n\\caption{\\label{fig:d-cliques-mnist-wo-clique-avg-impact-of-edge-removal} Without Clique Averaging }\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-mnist-w-clique-avg-impact-of-edge-removal}\n\\caption{\\label{fig:d-cliques-mnist-w-clique-avg-impact-of-edge-removal} With Clique Averaging}\n\\end{subfigure}\n\\caption{\\label{fig:d-cliques-mnist-intra-connectivity} MNIST: Impact of\nintra-clique edge removal on D-Cliques (fully-connected) with 10\ncliques of 10 heterogeneous nodes (100 nodes). Y axis starts at 89.}\n\\end{figure}\n\nIn contrast, \\autoref{fig:d-cliques-cifar10-intra-connectivity} shows that for CIFAR10, the impact is stronger. We show the results with and without Clique Averaging\nwith momentum in both cases, as momentum is critical for obtaining the best\nconvergence speed on CIFAR10. Without Clique Averaging,\nremoving edges has a small effect on convergence speed and variance, but the convergence speed is too slow to be practical.\nWith Clique Averaging, removing a single edge has a small but noticeable\neffect. Strikingly, removing 5 edges per clique significantly damages the\nconvergence and yields a sharp increase in the variance across nodes.\nTherefore, while D-Cliques can tolerate the removal of some intra-clique edges\nwhen training simple linear models and datasets as in MNIST, fast\nconvergence speed and low variance requires full or nearly full connectivity\nwhen using high-capacity models and more difficult datasets. This is\nin line with the observations made in Section~\\ref{sec:exp:clique_avg}\nregarding the effect of Clique Averaging. Again, these results show the\nrelevance of our design choices, including the choice of constructing fully\nconnected cliques.\n\n\\begin{figure}[htbp]\n     \\centering\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering   \n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-wo-clique-avg-impact-of-edge-removal}     \n\\caption{\\label{fig:d-cliques-cifar10-wo-clique-avg-impact-of-edge-removal} Without Clique Averaging }\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[htbp]{0.23\\textwidth}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures\/d-cliques-cifar10-w-clique-avg-impact-of-edge-removal}\n\\caption{\\label{fig:d-cliques-cifar10-w-clique-avg-impact-of-edge-removal} With Clique Averaging}\n\\end{subfigure}\n\\caption{\\label{fig:d-cliques-cifar10-intra-connectivity} CIFAR10: Impact of intra-clique edge removal (with momentum) on D-Cliques (fully-connected) with 10 cliques of 10 heterogeneous nodes (100 nodes).}\n\\end{figure}\n\n\\subsection{Greedy Swap Improves Random Cliques at an Affordable Cost}\n\\label{section:greedy-swap-vs-random-cliques}\n\nIn the next two sub-sections, we compare cliques built with Greedy Swap (Alg.~\\ref{Algorithm:greedy-swap})\nto Random Cliques, a simple and obvious baseline, on their quality (skew),  the cost \nof their construction, and their convergence speed.\n\n\\subsubsection{Cliques with Low Skew can be Constructed Efficiently with Greedy Swap}\n\\label{section:cost-cliques}\n\nWe compared the final average skew of 10 cliques with 10 nodes each (for\n$n=100$) created either randomly or with Greedy Swap,\nover 100 experiments after 1000 steps. \\autoref{fig:skew-convergence-speed-2-shards}, in the form of an histogram,\n shows that Greedy Swap generates cliques of significantly lower skew, close to 0 in a majority of cases for both MNIST and CIFAR10.\n\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.2\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/final-skew-distribution-mnist}\n    \\caption{\\label{fig:final-skew-distribution-mnist} MNIST }\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.2\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/final-skew-distribution-cifar10}\n    \\caption{\\label{fig:final-skew-distribution-cifar10} CIFAR10}\n    \\end{subfigure}\n\\caption{\\label{fig:final-skew-distribution} Final quality of cliques (skew) with a maximum size of 10 over 100 experiments in a network of 100 nodes.}\n\\end{figure}\n\n\\autoref{fig:skew-convergence-speed-2-shards} shows such a low skew can be achieved \nin less than 400 steps for both MNIST and CIFAR10. In practice it takes less\nthan 6 seconds in Python 3.7 on a \nMacbook Pro 2020 for a network of 100 nodes and cliques of size 10. Greedy Swap \nis therefore fast and efficient. Moreover, it illustrates the fact that a\nglobal imbalance in the number of examples\nacross classes makes the construction of cliques with low skew harder and\nslower.\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.25\\textwidth]{figures\/skew-convergence-speed-2-shards}\n    \\caption{\\label{fig:skew-convergence-speed-2-shards} Skew decrease during clique construction of 10 cliques of 10 heterogeneous nodes (100 nodes). Bold line is the average over 100 experiments. Thin lines are respectively the minimum and maximum over all experiments. In wall-clock time, 1000 steps take less than 6 seconds in Python 3.7 on a MacBook Pro 2020.}\n\\end{figure}\n\n\\subsubsection{Cliques built with Greedy Swap Converge Faster than Random Cliques}\n\n\\autoref{fig:convergence-speed-dc-random-vs-dc-gs-2-shards-per-node} compares\nthe convergence speed of cliques optimized with Greedy Swap for 1000 steps with cliques built randomly \n(equivalent to Greedy Swap with 0 steps). For both MNIST and CIFAR10, convergence speed\nincreases significantly and variance between nodes decreases dramatically. Decreasing the skew of cliques\nis therefore critical to convergence speed.\n\n\n\\begin{figure}[htbp]\n    \\centering        \n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-mnist-dc-random-vs-dc-gs-2-shards-per-node}\n    \\caption{\\label{fig:convergence-speed-mnist-dc-random-vs-dc-gs-2-shards-per-node} MNIST}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.23\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/convergence-speed-cifar10-dc-random-vs-dc-gs-2-shards-per-node}\n    \\caption{\\label{fig:convergence-speed-cifar10-dc-random-vs-dc-gs-2-shards-per-node} CIFAR10}\n    \\end{subfigure}\n\\caption{\\label{fig:convergence-speed-dc-random-vs-dc-gs-2-shards-per-node} Convergence speed of D-Cliques constructed randomly vs Greedy Swap with 10 cliques of 10 heterogeneous nodes (100 nodes).}\n\\end{figure}\n\n\\subsection{Additional Experiments on Extreme Label Distribution Skew}\n\nIn Appendix~\\ref{app:extreme-local-skew}, we replicate experimental\nresults on an extreme case of label distribution skew where each node only has\nexamples of a single class. These results consistently show that our\napproach remains effective even for extremely skewed label distributions\nacross nodes.\n\n\\section{Introduction}\n\nMachine learning is currently shifting from a \\emph{centralized}\nparadigm, where training data is located on a single\nmachine or\nin a data center, to \\emph{decentralized} ones in which data is processed\nwhere it was naturally produced.\nThis shift is illustrated by the rise of Federated\nLearning\n(FL)~\\cite{mcmahan2016communication}. FL allows\nseveral parties (hospitals, companies, personal\ndevices...) to collaboratively train machine learning models\non their joint\ndata without centralizing it. Not only does FL\navoid the costs of moving data, but it also  mitigates privacy and\nconfidentiality concerns~\\cite{kairouz2019advances}.\nYet, working with natural data distributions introduces new challenges for\nlearning systems, as\nlocal datasets\nreflect the usage and production patterns specific to each participant: in\nother words, they are\n\\emph{heterogeneous}. An important type of data heterogeneity encountered in\nfederated classification problems, known as \\emph{label distribution skew} \n\\cite{kairouz2019advances,quagmire}, occurs when the frequency of different\nclasses of examples varies significantly across local datasets.\nOne of the key challenges in FL is to design algorithms that\ncan efficiently deal with such heterogeneous data distributions\n\\cite{kairouz2019advances,fedprox,scaffold,quagmire}.\n\nFederated learning algorithms can be classified into two categories depending\non the underlying network topology they run on. In server-based FL, the\nnetwork is organized according to a star topology: a central server orchestrates the training process by\niteratively aggregating model updates received from the participants\n(\\emph{clients}) and sending back the aggregated model \\cite{mcmahan2016communication}. In contrast,\nfully decentralized FL algorithms operate over an arbitrary network topology\nwhere participants communicate only with their direct neighbors\nin the network. A classic example of such algorithms is Decentralized\nSGD (D-SGD) \\cite{lian2017d-psgd}, in which participants alternate between\nlocal SGD updates and model averaging with neighboring nodes.\n\nIn this paper, we focus on fully decentralized algorithms as they can\ngenerally scale better to the large number of participants seen in ``cross-device''\napplications \\cite{kairouz2019advances}. Effectively, while a central\nserver may quickly become a bottleneck as the number of participants increases, the topology used in fully decentralized algorithms can remain sparse\nenough such that all participants need only to communicate with a small number of other participants, i.e. nodes have small (constant or logarithmic) degree \n\\cite{lian2017d-psgd}. In the homogeneous setting where data is\nindependent and identically distributed (IID) across nodes, recent work\nhas shown both empirically\n\\cite{lian2017d-psgd,Lian2018} and theoretically \\cite{neglia2020} that sparse\ntopologies like rings or grids\ndo not significantly affect the convergence\nspeed compared to using denser topologies.\n\n\n\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/ring-IID-vs-non-IID-uneq-classes}\n\\caption{\\label{fig:ring-IID-vs-non-IID-uneq-classes} Ring topology}\n     \\end{subfigure}\n     \\quad\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/grid-IID-vs-non-IID-uneq-classes}\n\\caption{\\label{fig:grid-IID-vs-non-IID-uneq-classes} Grid topology}\n     \\end{subfigure}\n     \\quad\n     \\begin{subfigure}[b]{0.25\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/fc-IID-vs-non-IID-uneq-classes}\n\\caption{\\label{fig:fully-connected-IID-vs-non-IID-uneq-classes} Fully-connected topology}\n     \\end{subfigure}\n        \\caption{Convergence speed of decentralized\n        SGD with and without label distribution skew for different topologies.\n        The task is logistic regression on MNIST (see\n        Section~\\ref{section:experimental-settings} for details on\n        the experimental setup). Bold lines show the\n        average test\n        accuracy across nodes\n        while thin lines show the minimum\n        and maximum accuracy of individual nodes. While the effect of topology\n        is negligible for homogeneous data, it is very significant in the\n        heterogeneous case. On a fully-connected network, both cases converge\n        similarly.}\n        \\label{fig:iid-vs-non-iid-problem}\n\\end{figure*}\n\nIn contrast to the homogeneous case however, our experiments demonstrate that \n\\emph{the impact of topology is extremely significant for heterogeneous data}.\nThis phenomenon is illustrated in Figure~\\ref{fig:iid-vs-non-iid-problem}: we observe that under\nlabel distribution skew, using a\nsparse topology (a ring or\na grid) clearly jeopardizes the convergence speed of decentralized SGD.\nWe stress the fact\nthat, unlike in centralized FL\n\\cite{mcmahan2016communication,scaffold,quagmire}, this\nhappens even when nodes perform a single local update before averaging the\nmodel with their neighbors. In this paper, we thus address the following\nquestion:\n\n\\textit{Can we design sparse topologies with  convergence\n  speed similar to a  fully connected network for problems involving\n  many participants with label distribution skew?}\n\nSpecifically, we make the following contributions:\n(1) We propose D-Cliques, a sparse topology in which nodes are organized in\ninterconnected cliques (i.e., locally fully-connected sets of nodes) such that\nthe joint label distribution of each clique is close to that of the global \ndistribution; (2) We design Greedy Swap, a randomized greedy algorithm for\nconstructing such cliques efficiently;\n (3) We introduce Clique Averaging, a modified version of \nthe standard D-SGD algorithm which decouples gradient averaging, used for\noptimizing local models, from distributed averaging, used to ensure that all\nmodels converge, thereby reducing the bias introduced by inter-clique\nconnections; \n(4) We show how Clique Averaging can be used to implement unbiased momentum\nthat would otherwise be detrimental in the heterogeneous setting; (5) We \ndemonstrate\nthrough an extensive experimental study that our approach  removes the effect\nof label distribution skew when training a linear\nmodel and a deep\nconvolutional network on the MNIST\nand CIFAR10\ndatasets respectively;  (6) Finally, we demonstrate the scalability of our\napproach by considering  up to 1000-node networks, in contrast to most\nprevious work on fully decentralized learning which performs empirical\nevaluations on networks with\nat most a few tens\nof nodes\n\\cite{tang18a,neglia2020,momentum_noniid,cross_gradient,consensus_distance}.\n\nFor instance, our results show that under strong label distribution shift,\nusing D-Cliques in a 1000-node network\nrequires 98\\% less edges ($18.9$ vs $999$ edges per participant on average) to obtain a similar convergence speed as a fully-connected topology,\nthereby yielding a 96\\% reduction in the total number of required messages \n(37.8 messages per round per node on average instead of 999). Furthermore an additional 22\\% improvement\nis possible when using a small-world inter-clique topology, with further\npotential gains at larger scales through a quasilinear $O(n\n\\log n)$ scaling in the number of nodes $n$.\n\nThe rest of this paper is organized as follows.\nWe first describe the problem setting in Section~\\ref{section:problem}. We\nthen present the design of D-Cliques in Section~\\ref{section:d-cliques}.\nSection~\\ref{section:evaluation}\ncompares D-Cliques to different topologies \nand algorithmic variations to demonstrate their benefits, constructed with and without Greedy Swap\nin an extensive experimental study. Finally, we review some related work\nin Section~\\ref{section:related-work}, and conclude with promising directions\nfor future work in Section~\\ref{section:conclusion}.\n\n\\section{Related Work}\n\\label{section:related-work}\n\nIn this section, we review some related work on dealing with heterogeneous\ndata in federated learning, and on the role of topology in fully decentralized\nalgorithms.\n\n\\paragraph{Dealing with heterogeneity in server-based FL.}\nData heterogeneity is not much of an issue in server-based FL if\nclients send their parameters to the server after each gradient update.\nProblems arise when one seeks to reduce\nthe number of communication rounds by allowing each participant to perform\nmultiple local updates, as in the popular FedAvg algorithm \n\\cite{mcmahan2016communication}. Indeed, data heterogeneity can prevent\nsuch algorithms from\nconverging to a good solution \\cite{quagmire,scaffold}. This led to the design\nof algorithms that are specifically designed to mitigate the impact\nof heterogeneity while performing\nmultiple local updates, using adaptive client sampling \\cite{quagmire}, update\ncorrections \\cite{scaffold} or regularization in the local objective \n\\cite{fedprox}. Another direction is to embrace the heterogeneity by\nlearning personalized models for each client \n\\cite{smith2017federated,perso_fl_mean,maml,moreau,Marfoq2021a}.\nWe note that recent work explores rings of server-based topologies \n\\cite{tornado}, but the focus is not on dealing with heterogeneous data but\nto make server-based FL more scalable to a large number of clients.\n\n\\paragraph{Dealing with heterogeneity in fully decentralized FL.}\nData heterogeneity is known to negatively impact the convergence speed\nof fully decentralized FL algorithms in practice \\cite{jelasity}. Aside from approaches that aim to learn personalized models \\cite{Vanhaesebrouck2017a,Zantedeschi2020a}, this\nmotivated the design of algorithms with modified updates based on variance\nreduction \\cite{tang18a}, momentum correction \\cite{momentum_noniid},\ncross-gradient\naggregation \\cite{cross_gradient}, or multiple averaging steps\nbetween updates \\citep[see][and references therein]{consensus_distance}. These\nalgorithms\ntypically require significantly more communication and\/or computation, and\nhave only been evaluated on small-scale networks with a few tens of\nnodes.\\footnote{We\nalso observed that \\cite{tang18a} is subject to numerical\ninstabilities when run on topologies other than rings. When\nthe rows and columns of $W$ do not exactly\nsum to $1$ (due to finite precision), these small differences get amplified by\nthe proposed updates and make the algorithm diverge.}\nIn contrast, D-Cliques focuses on the design of a sparse topology which is\nable to compensate for the effect of heterogeneous data and scales to large\nnetworks. We do not modify the simple\nand efficient D-SGD\nalgorithm \\cite{lian2017d-psgd} beyond removing some neighbor\ncontributions\nthat otherwise bias the gradient direction.\n\n\\paragraph{Impact of topology in fully decentralized FL.} It is well\nknown\nthat the choice of network topology can affect the\nconvergence of fully decentralized algorithms. In theoretical convergence\nrates, this is typically accounted\nfor by a dependence on the spectral gap of\nthe network, see for instance \n\\cite{Duchi2012a,Colin2016a,lian2017d-psgd,Nedic18}.\nHowever, for homogeneous (IID) data, practice contradicts these classic\nresults as fully decentralized algorithms have been observed to converge\nessentially as fast\non sparse topologies like rings or grids as they do on a fully connected\nnetwork \\cite{lian2017d-psgd,Lian2018}. Recent work \n\\cite{neglia2020,consensus_distance} sheds light on this phenomenon with refined convergence analyses based on differences between gradients or parameters across nodes, which are typically\nsmaller in the homogeneous case. However, these results do not give any clear insight\nregarding the role of the topology in the presence of heterogeneous data. \nWe note that some work\nhas gone into designing efficient topologies to optimize the use of\nnetwork resources \\citep[see e.g.,][]{marfoq}, but the topology is chosen\nindependently of how data is distributed across nodes. In summary, the role\nof topology in the heterogeneous data scenario is not well understood and we are not\naware of prior work focusing on this question. Our work is the first\nto show that an\nappropriate choice of data-dependent topology can effectively compensate for\nheterogeneous data.\n\\section{Problem Setting}\n\n\\label{section:problem}\n\n\\paragraph{Objective.} We consider a set $N = \\{1, \\dots, n \\}$ of $n$ nodes\nseeking to\ncollaboratively solve a classification task with $L$ classes. We denote a\nlabeled data point by a tuple $(x,y)$ where $x$ represents the data point \n(e.g., a feature vector) and $y\\in\\{1,\\dots,L\\}$ its label.\nEach\nnode has\naccess to a local dataset that\n follows its own local distribution $D_i$ which may differ from that of other\n nodes.\nIn this work, we tackle \\emph{label distribution skew}: formally, this means\nthat the\nprobability of $(x,y)$ under the local distribution $D_i$ of node $i$, denoted\nby $p_i(x,y)$,\ndecomposes as $p_i(x,y)=p(x|y)p_i(y)$, where $p_i(y)$ may vary across nodes.\nWe\nrefer to \n\\cite{kairouz2019advances,quagmire} for concrete examples of problems\nwith label distribution skew.\n\nThe objective is to find the parameters\n$\\theta$ of a global model that performs well on the union of the local\n distributions by\n minimizing\n the average training loss:\n\\begin{equation}\n\\min_{\\theta} \\frac{1}{n}\\sum_{i=1}^{n} \\mathds{E}_\n{(x_i,y_i) \\sim D_i} [F_i(\\theta;x_i,y_i)],\n\\label{eq:dist-optimization-problem}\n\\end{equation}\nwhere $(x_i,y_i)$ is a data point drawn from $D_i$ and $F_i$ is the loss\nfunction\non node $i$. Therefore, $\\mathds{E}_{(x_i,y_i) \\sim D_i} F_i(\\theta;x_i,y_i)$\ndenotes \nthe\nexpected loss of model $\\theta$ over $D_i$.\n\n\n\n\nTo collaboratively solve Problem \\eqref{eq:dist-optimization-problem}, each\nnode can exchange messages with its neighbors in an undirected network graph\n$G=(N,E)$ where $\\{i,j\\}\\in E$ denotes an edge (communication channel)\nbetween nodes $i$ and $j$.\n\n\\paragraph{Training algorithm.}\nIn this work, we use the popular Decentralized Stochastic\nGradient Descent algorithm, aka D-SGD~\\cite{lian2017d-psgd}. As\nshown in Algorithm~\\ref{Algorithm:D-PSGD},\na single iteration of D-SGD at node $i$ consists in sampling a mini-batch\nfrom its local distribution\n$D_i$, updating its local model $\\theta_i$ by taking a stochastic gradient\ndescent\n(SGD) step according to the mini-batch, and performing a weighted average of\nits local model with those of its\nneighbors.\nThis weighted average is defined by a\nmixing matrix $W$, in which $W_{ij}$ corresponds to the weight of\nthe outgoing connection from node $i$ to $j$ and $W_{ij} = 0$ for $\n\\{i,j\\}\\notin\nE$. To ensure that the local models converge on average to a stationary\npoint\nof Problem\n\\eqref{eq:dist-optimization-problem}, $W$\nmust be doubly\nstochastic ($\\sum_{j \\in N} W_{ij} = 1$ and $\\sum_{j \\in N} W_{ji} = 1$) and\nsymmetric, i.e. $W_{ij} = W_{ji}$~\\cite{lian2017d-psgd}.\nGiven a network topology $G=(N,E)$, we generate a valid $W$ by computing\nstandard\nMetropolis-Hasting weights~\\cite{xiao2004fast}:\n\\begin{equation}\n  W_{ij} = \\begin{cases}\n    \\frac{1}{\\max(\\text{degree}(i), \\text{degree}(j)) + 1} & \\text{if}~i \\neq\n    j \\text{ and } \\{i,j\\}\\in E,\\\\\n   1 - \\sum_{j \\neq i} W_{ij} & \\text{if } i = j, \\\\\n   0 & \\text{otherwise}.\n  \\end{cases}\n  \\label{eq:metro}\n\\end{equation}\n\n\\begin{algorithm}[t]\n   \\caption{D-SGD, Node $i$}\n   \\label{Algorithm:D-PSGD}\n   \\begin{algorithmic}[1]\n        \\STATE \\textbf{Require:} initial model $\\theta_i^{(0)}$,\n        learning rate $\\gamma$, mixing weights $W$, mini-batch size $m$,\n        number of steps $K$\n        \\FOR{$k = 1,\\ldots, K$}\n          \\STATE $S_i^{(k)} \\gets \\text{mini-batch of $m$ samples drawn\n          from~} D_i$\n          \\STATE $\\theta_i^{(k-\\frac{1}{2})} \\gets \\theta_i^{(k-1)} - \\gamma\n          \\nabla F(\\theta_i^{(k-1)}; S_i^{(k)})$ \n          \\STATE $\\theta_i^{(k)} \\gets \\sum_{j \\in N} W_{ji}^{(k)} \\theta_j^{(k-\\frac{1}{2})}$\n        \\ENDFOR\n   \\end{algorithmic}\n\\end{algorithm}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{Introduction}}\n\nThe processes of interaction of charged particles with matter, in particular, crystalline\nsolids, have been long studied both experimentally and theoretically. \nThe goal of these studies is to determine such characteristics of the interaction as the mean free path traveled\nby particles in the material, their energy losses, emission spectra, and others.\n\nChanneling in crystals, when charged particles falling into a potential channel shaped by electrostatic forces propagate along\ncrystallographic planes or axes, has become the focus of much attention in recent years.\nThe particles trapped in a channel of a straight crystal can travel long distances exceeding the mean free path in an \namorphous target, since such particles lose considerably less energy along their path \\cite{Lindhard_KDan_v34_p1_1965}. \nFor electrons, the channel lies along atomic rows or ion chains of the crystal, while for positrons it lies in the space between\natomic rows. \nThe stability of particle motion along the channels depends on the energy of the transverse motion that is low compared\nwith the height of the potential barrier.\n\nA particle trapped in the channel experiences oscillations in a plane transverse\nto the direction of the particle's propagation, inducing radiation during its channeling \\cite{ChRad:Kumakhov1976}.\nThis radiation is determined by the transverse energy of the channeled particle, and its intensity\nvaries depending on the type of crystal and its orientation. \nOscillatory radiation is incoherent and has a broad energy spectrum\n\\cite{ChRad:AndersenEtAl1983,BakEtAl_NPB_v254_p491_1985,\nBakEtAl_NPB_v525_p302_1988,BazylevZhevago:Uspekhi-v28-p565-1982,KumakhovKomarov-AIP}.\n\nChanneling can also occur in bent crystals, which are often used to bend charged particle\nbeams accelerated to relativistic energies \\cite{Tsyganov_TM-682_1976}. \nThe motion of a particle consists of two components: its oscillatory motion in the\nchannel and its propagation along the centerline of the bent channel. \nThe stability of the second component of motion in such a bent channel is provided by an additional condition, namely,\nthat the bending radius $R$ should significantly exceed the critical value $R_c$ determined by the\nenergy of the particle  \\cite{Tsyganov_TM-682_1976}. \nThis motion of a relativistic particle trapped in a bent channel induces additional synchrotron radiation. \nThe intensity and frequency of synchrotron radiation depend on the type and energy of the channeled particles, as well as on\nthe characteristics of the crystal \\cite{KaplinVorobev1978,Bashmakov1981,TaratinVorobiev1989,ArutyunovEtAl_NP_1991,\nTaratin_PhysPartNucl_v29_p1063_1998-English,KSG1998,KSG_review_1999,ChannelingBook2014}.\n\nUndulator radiation is certainly an interesting subject to explore in connection with the concept of the crystal undulator (see,\nfor example, Ref. \\cite{ChannelingBook2014} and references therein). \nChanneling of charged relativistic particles in a periodically bent crystal (a crystal undulator)\ncan produce a new source of monochromatic radiation with energies ranging from hundreds\nof keV to several MeV.\n\nThere has been a number of experiments in the recent years with a view to create the crystal\nundulator, measuring the channeling parameters and the characteristics of the emission spectra\nof ultrarelativistic positrons \\cite{BaranovEtAl_CU_2006,Backe_EtAl_NuovoCimC_v34_p175_2011,Backe_EtAl_2008}\nand electrons \\cite{Backe_EtAl_2011,Backe_EtAl_2013} \nin straight and bent crystals of silicon and diamond. \nTheoretical studies on channeling in these crystals are carried out using the newly developed MBN Explorer\npackage \\cite{MBN_Explorer_Paper,MBN_Explorer_Site}. \nSimulations for amorphous and crystalline silicon have verified that this package is applicable for describing \nthe channeling of electrons and positrons \n\\cite{MBN_ChannelingPaper_2013,Sub_GeV_2013,PolozkovEtAl:NTV_v1_p212_2015,Korol_EtAl_NIMB_v387_p41_2016}. \n\nSince experiments are currently being carried out to measure the emission spectra of electrons in a periodically bent \ndiamond crystal \\cite{BadEms_p58}, theoretical interpretation of the experimental results is clearly an interesting problem.\n\nIn view of the above, the goal of this study is theoretical analysis of channeling of ultrarelativistic \nelectrons and positrons with an energy of 270 MeV both in a straight diamond crystal oriented along the (110) \ncrystallographic plane and in a periodically bent diamond crystal.\n\nWe have performed simulations of electron and positron channeling in straight, bent and\nperiodically bent channels using the versatile MBN Explorer software package.\n\n\n\\section{Simulation procedure with the MBN Explorer package \\label{Procedure}}\n\nThree-dimensional simulation of ultrarelativistic particles passing through a\ncrystalline medium is carried out using a molecular dynamics algorithm implemented in\nthe MBN Explorer software package \\cite{MBN_ChannelingPaper_2013}. \nThe characteristics of the motion of high-energy particles inside the crystal were obtained by\nintegrating the relativistic equations of motion.\nStep-by-step dynamic simulation of the crystalline medium was performed to construct\nthe particle trajectory.\n\nA quasi-classical approximation is applicable to describing the motion of ultrarelativistic\nparticles, and, since the quantum corrections are small, it is sufficient to use the equations of\nclassical relativistic mechanics:\n\\begin{eqnarray}\n\\dot{\\bfp} = q \\bfE(\\bfr)\n\\label{eq.01}\n\\end{eqnarray}\nHere $\\bfE(\\bfr)$ is the external electrostatic field, \n$q$ is the particle charge, and $\\bfp$ is its relativistic\nmomentum $\\bfp = m\\gamma \\bfv$,\nwhere $m$ and $v$ are the mass and velocity of the particle, respectively,\n$\\gamma = \\left(1 -v^2\/c^2\\right)^{-1\/2} \\gg1 $  is\nthe relativistic factor  ($c$ is the speed of light).\n\nInitial values of the coordinates and velocity of the particle\nare used to integrate Eq. (\\ref{eq.01}).\n\nIn the MBN Explorer channeling module, the force $q \\bfE(\\bfr)$ is calculated as the gradient of \nthe electrostatic potential $U(\\bfr)$ equal to the sum of atomic potentials $U_{\\rm at}$:\n\\begin{eqnarray}\nU(\\bfr) = \\sum_{j} U_{\\rm at}(\\bfrho)\n\\label{eq.02}\n\\end{eqnarray}\nwhere $\\bfrho_j = \\bfr - \\bfR_j$ with $\\bfR_j$ standing for the position vector of a $j$th atom.\n\nFormally, the sum in (\\ref{eq.02}) accounts for all crystal atoms. \nHowever, given a rapid decrease of $U_{\\rm at}(\\bfrho)$  with distance, one can introduce the maximum \ndistance $\\rho_{\\max}$, beyond which the contribution of the atomic potential is negligible. \nTherefore, for a given observation point $\\bfr$, the sum can be limited to the atoms located inside a \nsphere with the radius $\\rho_{\\max}$. \nThe linked cell algorithm implemented in the MBN Explorer is used to search for such atoms. \nThis algorithm involves dividing the crystal into cells and considering only\nthe atoms closest to the particle. \nThe described scheme is used to calculate the force $q \\bfE(\\bfr)$ acting on the projectile \nat each step of integration.\n\nThe motion of particles along a crystallographic plane with the Miller indices\n$(k l m)$ is simulated by the following procedure \\cite{ChannelingBook2014,MBN_ChannelingPaper_2013}. \nA simulation box with the dimensions $L_x\\times L_y \\times L_z$ \nis introduced, containing a crystal lattice. \nThe $z$ axis is oriented along the incident beam and is parallel to the\n$(k l m)$ plane, the $y$ axis is perpendicular to this plane. \nThe position vectors of the lattice sites are generated in accordance with the type of the\nBravais cell of the crystal, using predefined values of \u200b\u200b the translation vectors.\n\nOnce the nodes inside the simulation box are determined, the position vectors of the\natomic nuclei are generated taking into account the thermal vibrations of these nuclei\nresulting in a random displacement from the nodal positions; these displacements are determined\nby the normal distribution with respect to the root-mean-square amplitude of thermal vibrations \\cite{Gemmel}.\n\nIntegration of the equations of motion begins at instant $t = 0$, when the particle enters the crystal \nat $z = 0$.\nA random number generator is used to choose \nThe initial transverse coordinates $x_0$ and $y_0$ are generated randomly.\nFor a beam with zero emittance, the initial velocity $\\bfv_0$ is oriented along the $z$ axis.\nParticle propagation through a crystal with a finite thickness $L$ is simulated in MBN Explorer using \nthe so-called dynamic simulation box \\cite{ChannelingBook2014,MBN_ChannelingPaper_2013} as a new type of boundary conditions. \nA particle moving inside the box interacts with atoms lying inside the cutoff sphere.\nTo optimize the numerical procedure, the dimensions of the box are chosen to be\n3 to 5 times larger than $\\rho_{\\max}$.\nAt the instant when the distance $l$ from the particle to the nearest face of the box becomes close to $\\rho_{\\max}$, \na new simulation box of the same size is generated, with its geometric center approximately coinciding with the position of\nthe particle. \nThe atoms located at the intersection of the old and the new simulation boxes are left intact.\nThe positions of the atoms in the rest of the new box are generated anew.\nSimulation is interrupted when the $z$ coordinate of the particle\nbecomes equal to the crystal thickness $L$.\n\n\\section{Simulation of electron and positron trajectories \\label{Trajectories}}\n\nThe MBN Explorer package was used to simulate the trajectories of 270 MeV electrons\nand positrons incident on diamond crystals along the (110) crystallographic planes. \nThe calculations were performed for a straight crystal and for a crystal with periodical cosine-like bending. \nIn both cases the crystal length was set to $L=20$ $\\mu$m. \nThe periodical bending was considered with the amplitude $a=2.5$ \\AA{} and \nperiod $\\lambda_{\\rm u}=5$ $\\mu$m. \nEach set of calculations included simulation of  6000 trajectories of projectiles \nwhich were analyzed further to calculate the channeling parameters and radiation emission.\n\nAn ordinary diamond crystal has straight channels due to the periodic arrangement of its atoms. \nThe width of the channel is determined by the interatomic distance which is $d = 1.26$\\AA{}. \nParticles trapped in straight channels with a low transverse energy leave such\nchannels less often. \nSince the crystal is short enough, positrons most often move through the entire\nstraight crystal while staying in the channel, and electrons are more likely to collide with\nlattice atoms and leave the channel. \nThis is because positrons move between the crystal atoms, where they are confined by repulsive\ninteraction with the lattice ions. \nOn the other hand, electrons move along helical trajectories in the immediate vicinity of the nuclei, so they\nare much more likely to collide with them and escape the channel.\n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[width=7.7cm,clip]{Figure1a_v02.eps}\n\\includegraphics[width=7.7cm,clip]{Figure1b_v02.eps}\n\\caption{\nRepresentative trajectories of electrons (left) and positrons (right) with energies of 270 MeV\nin a periodically bent 20 $\\mu$m thick oriented diamond(110) crystal. \nChanneling (curves 1), dechanneling (2) and rechanneling (3) modes are indicated.\n}\n\\label{Figure1.fig}\n\\end{figure}\n\nThe trajectories of charged particles channeled in bent crystals become more complex and diverse. \nAs an example, Fig. \\ref{Figure1.fig} shows several typical trajectories of electrons (left panel) \nand  positrons (right panel) in periodically bent diamond. \nThin solid lines in the figure indicate the boundaries of the channels; the distance\n$y$ is plotted along the vertical axis in a plane perpendicular to the direction of motion\n(the distance is measured in units of the interatomic spacing $d$). \nThe main features and characteristics of particle motion in a crystal, such as the channeling, \ndechanneling, and rechanneling modes, are shown in the figures. \nRechanneling is a process when a particle moving outside a channel can experience a\ncollision and get trapped into some channel as a result.\n\nFigure \\ref{Figure1.fig} left presents the trajectory of the only electron that propagated through \na crystal staying in the same channel. \nStatistically, such trajectories are an exception, as the rest of the trajectories\npresented correspond to the more typical motion of electrons in dechanneling and\nirregular rechanneling modes in short segments of different channels.\nComparison of the trajectories shown Fig. 1 left and right, indicates that positrons channel\nmuch better than electrons, and this pattern is observed for both straight and bent crystals.\nOnly a small part of the positrons originally trapped in the channel escapes it, while most\nof them move through the entire crystal while staying in one channel.\nTherefore, the intensity of synchrotron radiation should be higher in periodically bent crystal.\n\nNotably, positrons may have different oscillation amplitudes inside the channel,\nbut transverse oscillations are practically isochronous and their period remains almost\nunchanged, which corresponds to harmonic oscillations. \nConsequently, all positrons emit energy at approximately the same wavelength,\nand their channeling radiation peak is narrower and more intense, in contrast to the maximum\nradiation intensity for electrons.\n\nStatistical analysis of the calculated trajectories allowed to obtain the main\nparameters characterizing the channeling of charged particles (given in the table).\n\nThe particle trapping coefficient $A$ (acceptance) is the ratio of the number $N_{\\rm acc}$\nof the particles trapped in the channel upon entering the crystal to the number $N_0$ of all\nincident particles: $A=N_{\\rm acc}\/N_0$.\n\nThe values given in the table refer to the acceptance for the particles falling along the $z$\naxis.\nThe remaining parameters are related to the mean distances or the times during which\nthe charged particles stay in one or several channels. \nThe channeling length $L_{\\rm ch}$ is defined as the mean total distance traveled by a particle in\nthe channeling mode throughout the crystal. \nThe rechanneling length $L_{\\rm rech}$ is the mean distance covered by a particle from the moment \nwhen it dechannels until the opposite event of rechanneling, i.e. capture into the channeling mode\nas a result of collisions with the crystal atoms.\nTo more parameters are listed in the Table.\nThese are so-called penetration lengths \\cite{ChannelingBook2014,MBN_ChannelingPaper_2013}. \nThe first one, denoted as $L_{\\rm p1}$, is the mean distance traveled by a particle, accepted into the \nchanneling mode at the entrance, until it dechannels at some point in the bulk.\nThe penetration length $L_{\\rm p2}$ is calculated as the arithmetic mean of all channeling segments \n(initial and secondary) with respect to the total number channeling segments in all simulated trajectories.\nThus, it characterizes the average distance traveled by a particle in the channeling mode.\n\n\\begin{table}\n\\caption{\nChanneling parameters of $855$ MeV positrons ($e^+$) and electrons ($e^-$) in\nstraight and periodically bent (PB) $20$ $\\mu$m thick oriented diamond(110) crystal:\nacceptance $A$, \nchanneling length $L_{\\rm ch}$,\nrechanneling length $L_{\\rm rech}$,\npenetration lengths $L_{\\rm p1}$ and $L_{\\rm p2}$\n(all in $\\mu$m). \n}\n\\footnotesize\\rm\n\\begin{tabular}{@{}rrrrrrr}\n\\br\nParameter      & \\multicolumn{2}{c}{straight crystal}& \\ & \\multicolumn{2}{c}{PB crystal}\\\\ \n               &     $e^-$     &    $e^+$            & \\ &    $e^-$     &    $e^+$        \\\\\n\\br  \n $A$           & 0.70          &  0.96               & \\ &    0.51      &   0.89          \\\\\n$L_{\\rm ch}$   & 9.04          &  18.7               & \\ &    6.06      &  17.2           \\\\\n$L_{\\rm rech}$ & 4.18          &  6.08               & \\ &    5.98      &  7.53           \\\\\n$L_{\\rm p1}$   & 5.43          & 19.1                & \\ &    4.30      & 18.8            \\\\\n$L_{\\rm p2}$   & 4.55          & 18.0                & \\ &    3.60      & 16.4            \\\\\n\\br  \n\\end{tabular}\n\\label{Table_ep-data.C}\n\\end{table}\n\nSince the crystal is rather short (20 $\\mu$m), the positrons accepted in the channeling mode travel through\nalmost the entire crystal staying in the same channel, and, thus, they have greater penetration,\nchanneling and rechanneling lengths.\nElectrons experience collisions with lattice ions at a higher rate, since their trajectories\npass in the immediate vicinity of the ions, and thus the dechanneling events are more frequent.\n\n\\section{Emission spectra of electrons and positrons}\n\nFor each projectile, the simulated dependences $\\bfr = \\bfr(t)$ and $\\bfv = \\bfv(t)$ \nallow one to calculate the spectral characteristics of the radiation emitted by the particle.\n\nThe spectral angular distribution of the radiated energy $\\d^3 E \/ (\\d\\hbar\\om \\d \\Om)$ \n($\\om$ and $\\Om$ stand for the frequency of radiation and the emission solid angle, respectively) is\ncalculated following the general formula derived within the quasi-classical approximation \\cite{Baier}:\n\\begin{eqnarray} \n\\fl\n{\\d^3 E \\over \\hbar\\d\\om\\, \\d \\Om}\n=\n\\alpha \\,\n{ q^2\\omega^2  \\over 8\\pi^2 }\n\\int\\limits_{-\\infty}^{\\infty} \\d t_1\\!\n\\int\\limits_{-\\infty}^{\\infty} \\d t_2\\,\n\\ee^{\\i \\,\\omega^{\\prime} \\left(\\psi(t_1) -\\psi(t_2)\\right)}\n\\left[\n\\left( 1+(1+u)^2 \\right)\n\\left(\n{\\bfv_1\\cdot\\bfv_2 \\over c^2}  -1\n\\right)\n+{u^2 \\over \\gamma^2}\n\\right]\\,.\n\\label{eq.03} \n\\end{eqnarray}\nHere $\\alpha= e^2\/ \\hbar\\, c$ is the fine structure constant,\n$q$ is measured in units of the elementary charge, $\\bfv_{1,2} =\\bfv(t_{1,2})$, \nand\nthe $\\psi(t) = t - \\bfn\\cdot\\bfr(t)\/ c$, with $\\bfn$ being the unit vector in the \ndirection of radiation emission.\nOther quantities, which account for the radiative recoil, are as follows:\n$\\om^{\\prime} = (1+u)\\, \\om$ and $u = \\hbar \\om\/(\\E - \\hbar \\om)$.\n\nFor each individual trajectory $j$, the spectral distribution is calculated by \nnumerically integrating the values of $\\d^3 E_j \/ (\\d\\hbar\\om \\d \\Om)$ \nover the ranges $\\phi=[0,2\\pi]$ and $\\theta=[0,\\theta_0]$, where \n$\\theta_0$ is related to the detector aperture.\nThe resulting distribution is calculated averaging $\\d^3 E_j$  \nover the ensemble of the trajectories.\n\nThe results presented below refer to the emission within the cone\n$\\theta_0 \\leq 0.2$ mrad. \n\nFigure 2 left shows the emission spectra of electrons in the straight and periodically bent crystal. \nThe broad peak (curve 1) at $\\hbar \\om \\geq 0.4$ MeV is due to the channeling radiation \n(ChR). \nThe decrease in the intensity of this peak in a periodically bent crystal (curve 2)\nis associated with the decrease in the number of channeling electrons.\n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[width=7.7cm,clip]{Figure2a.eps}\n\\includegraphics[width=7.7cm,clip]{Figure2b.eps}\n\\caption{\nRepresentative trajectories of electrons (left) and positrons (right) with energies of 270 MeV\nin a periodically bent 20 $\\mu$m thick oriented diamond(110) crystal. \nChanneling (curves 1), dechanneling (2) and rechanneling (3) modes are indicated.\n}\n\\label{CLS.fig}\n\\end{figure}\n\nFigure 2 right presents the corresponding emission spectra of positrons.\nHere, the ChR maximum (curve 1) is narrower and higher because the channeling oscillations\nof positrons is much more harmonic that of electrons and, thus, the radiation emitted is \nconcentrated in the narrower bandwidth $\\Delta \\om$.\n\nIt can be seen from Fig. 2 left and right (curves 2) that a radiation intensity peak is observed for\nchanneling in the PB crystal at a photon energy of the order of 130 keV, which is absent in\nthe straight crystal. \nThis peak appears due to motion of channeling particles along\nthe centerline of the periodically bent channel. \nThe particle radiation frequency is related to the period of  the channel curvature and the \nlongitudinal energy of the charged particle. \nThis radiation, termed as a crystalline undulator radiation, has a narrow spectral width and bears the features of \nradiation emitted by projectiles moving in magnetic undulators.\nSince the study deals with electrons and positrons with the same energy, the position\nof the undulator peak on the emission spectra is the same. \nHowever, radiation intensity is higher for positrons than for electrons by an order of magnitude, because positrons\nexperience harmonic oscillations and longer channeling.\n\n\\section{Conclusion}\n\nWe have numerically simulated the trajectories of ultrarelativistic charged particles\nin straight and bent diamond crystals, with electrons and positrons incident on the (110)\ncrystallographic plane, using the MBN Explorer software package \\cite{MBN_Explorer_Paper,MBN_Explorer_Site}.\nThe coordinates of the particles upon entering the crystal in the transverse plane were chosen with a random\nnumber generator. \nStatistical processing of the obtained trajectories made it possible to determine the channeling \nparameters of electrons and positrons with an energy of 270 MeV in a $20$ $\\mu$m thick diamond crystal.\nWe have established that channeled positrons have a larger acceptance and run substantially\nlonger distances in the crystalline channel as compared to electrons.\n\nThe calculated emission spectra of electrons and positrons channeled in a periodically bent crystal \ncontain two main regions. \nThe high-energy intensity peak is associated with ChR induced by oscillatory motion of the\nparticles in the channel; the same peak was obtained under channeling in a straight crystal.\n\nA low-energy peak in the 130 keV region occurs when particles move in a periodically\nbent channel and has an undulatory nature.\nThis radiation is coherent and, even though the bent crystal has a small number of periods\n(only 4), the radiation is characterized by a noticeable intensity, which is significant for\npotential applications in lasers \\cite{KSG_review_1999,ChannelingBook2014,KSG_review2004}.\n\nThe obtained channeling parameters and the calculated emission spectra are of interest in\nview of the experiments on electron channeling in straight and bent crystals currently under way\nat the University of Mainz (Germany) \\cite{BadEms_p58}.\n\n\\ack\n\nThis work has been supported by the European Commission (the PEARL Project within the H2020-MSCA-RISE-2015 call, GA 690991).\nWe acknowledge the Supercomputing Center of Saint Petersburg Polytechnic University \n(SPbPU) for providing the opportunities to carry out large-scale simulations.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:1}\nThis paper is dedicated to the comparison \nof multivariate probability \ndistributions with respect to extreme portfolio losses. \nA new notion of stochastic ordering  \nnamed \\emph{asymptotic portfolio loss order} ($\\mathrel{\\preceq_{\\mathrm{apl}}}$) is introduced.   \nSpecially designed for the ordering of stochastic risk models \nwith respect to extreme portfolio losses, \nthis notion allows to compare the inherent extreme portfolio risks \nassociated with different model parameters such as correlations, \nother kinds of dependence coefficients, or diffusion parameters.  \n\\par\nIn a recent paper of \\cite{Mainik\/Rueschendorf:2010} the notion of \n\\emph{extreme risk index} has been introduced in the framework of \nmultivariate regular variation. This index, denoted by $\\gamma_\\xi$, \nis a functional of the vector $\\xi$ of portfolio weights and of the \ncharacteristics of the multivariate regular variation of $X$ given \nby the tail index $\\alpha$ and the spectral measure $\\Psi$. \nIt measures the sensitivity of the portfolio loss\nto extremal \nevents and characterizes the probability distribution of extreme losses. \nIn particular, it serves to determine the optimal portfolio diversification \nwith respect to extreme losses. \nWithin the framework of multivariate regular variation \nthe notion of asymptotic portfolio loss ordering introduced in this paper \nis tightly related to model comparison in terms of \nthe extreme risk index $\\gamma_\\xi$. \nThus this paper can be seen as a supplement of the previous one, \nallowing to order multivariate risk models with respect to    \ntheir extremal portfolio loss behaviour. \n\\par\nIn Section \\ref{sec:2} of the present paper we introduce the \nasymptotic portfolio loss order $\\mathrel{\\preceq_{\\mathrm{apl}}}$ and highlight some relationships \nto further well-known ordering notions. \nIt turns out that even strong dependence and convexity \norders do not imply the asymptotic portfolio loss order in general. \nWe present counter-examples, based on the the inversion of diversification \neffects in models with infinite loss expectations. \nAnother example of particular interest discussed here is given \nby the elliptical distributions. \nIn this model family we establish a precise criterion for the asymptotic \nportfolio loss order, which perfectly accords with the classical \nresults upon other well-known order relations. \nSection \\ref{sec:3} is devoted to multivariate regularly varying models. \nWe discuss the relationship between the asymptotic portfolio loss order \nand the comparison of the extreme risk index and characterize $\\mathrel{\\preceq_{\\mathrm{apl}}}$ \nin terms of a suitable ordering of the canonical spectral measures. \nThese findings allow to establish sufficient conditions for \n$\\mathrel{\\preceq_{\\mathrm{apl}}}$ in terms of spectral measures, which can be verified \nby analytical or numerical methods.  \nIn particular, we characterize the dependence structures that yield \nthe best and the worst possible diversification effects for a multivariate \nregularly varying risk vector $X$ in $\\Rplus^{d}$ with tail index $\\alpha$.\nFor $\\alpha\\ge1$ the best case is given by the asymptotic independence and  \nthe worst case is the asymptotic comonotonicity. \nThe result for $\\alpha\\le 1$ is exactly the opposite \n(cf.\\ Theorem~\\ref{thm:3.8} and Corollary~\\ref{cor:3.10}).\nRestricting $X$ to $\\Rplus^{d}$ means that $X$ represents only the losses, whereas \nthe gains are modelled separately. \nThis modelling approach is particularly suitable for applications in \ninsurance, operational risk, and credit risk.\nIf $X$ represents both losses and gains, these results remain valid if the \nextremal behaviour of the gains is weaker than that of the losses, so \nthat there is no loss-gain compensation for extremal events.\nIn Section \\ref{sec:4} we discuss the interconnections between $\\mathrel{\\preceq_{\\mathrm{apl}}}$ \nor ordered canonical spectral measures and other well-known \nnotions of stochastic ordering. \nOrdering of canonical spectral measures allows to conclude $\\mathrel{\\preceq_{\\mathrm{apl}}}$\nfrom the (directionally) \nconvex or the supermodular order. \nIt is not obvious how to obtain this implication in a general setting.\nFinally, in Section~\\ref{sec:5} we present  a series of examples \nwith graphics illustrating the numerical results upon the ordering \nof spectral measures.\nThe relationship to spectral measures provides a useful numerical tool to \nestablish $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in practical applications.\n\\section{Asymptotic portfolio loss ordering}\\label{sec:2}\nTo compare stochastic risk models with respect to extreme portfolio losses, \nwe introduce the asymptotic portfolio loss order $\\mathrel{\\preceq_{\\mathrm{apl}}}$. \nThis order relation is designed\nfor the analysis of the asymptotic \ndiversification effects and the identification of models that generate \nportfolio risks with stronger extremal behaviour. \n\\par\nBefore stating the definition, some basic notation is needed. \nFocusing on risks, let $X$ be a \\emph{random loss vector} with values in \n$\\R^{d}$, i.e., let positive values of the components $X^{(i)}$, $i=1,\\ldots,d$,\nrepresent losses and let negative values of $X^{(i)}$ represent gains \nof some risky assets. \nFollowing the intuition of diversifying a unit capital over several assets,\nwe restrict the set of portfolios to the unit simplex in $\\R^{d}$:  \n\\[\n\\Simp^d := \\cubrfl{\\xi\\in\\Rplus^{d}: \\sum_{i=1}^{d}\\xi_i=1 }\n\\ldotp\n\\]\nThe portfolio loss resulting from a random vector $X$ and the portfolio $\\xi$ \nis given by the scalar product of $\\xi$ and $X$. \nIn the sequel it will be denoted by  $\\xi^{\\top} X$.\n\\begin{definition}\nLet $X$ and $Y$ be $d$-dimensional random vectors. \nThen $X$ is called smaller than $Y$ in \n\\emph{asymptotic portfolio loss order}, $X\\mathrel{\\preceq_{\\mathrm{apl}}} Y$, \nif \n\\begin{equation}\\label{eq:2.1}\n\\forall\\xi\\in\\Simp^d\n\\quad\n\\limsup_{t\\to\\infty}  \n\\frac{\\mathrm{P}\\cubr{\\xi^{\\top} X> t}}{\\mathrm{P}\\cubr{\\xi^{\\top} Y\\ge t}} \\le 1\n\\ldotp\n\\end{equation}\nHere, $\\frac00$ is defined to be 1.\n\\end{definition}\n\\begin{remark}\\label{rem:2.1}\n\\begin{enumerate}[(a)]\n\\item\nAlthough designed for random vectors, $\\mathrel{\\preceq_{\\mathrm{apl}}}$ is also defined for \nrandom variables. In this case, the portfolio set has only one element, \n$\\Sigma^1=\\cubr{1}$.   \n\\item\\label{item:rem:2.1.b}\nIt is obvious that  $\\mathrel{\\preceq_{\\mathrm{apl}}}$ is invariant under componentwise rescaling. \nLet $vx$ denote the componentwise product of $v,x\\in \\R^{d}$: \n\\begin{equation}\\label{eq:apl.2}\nvx:= (v^{(i)} x^{(i)},\\dots,v^{(d)} x^{(d)}),\n\\end{equation}\nThen it is easy to see that $ X\\mathrel{\\preceq_{\\mathrm{apl}}} Y$ implies $vX \\mathrel{\\preceq_{\\mathrm{apl}}} vY$ for \nall $v\\in\\Rplus^{d}$.\nHence condition~\\eqref{eq:2.1} can be equivalently stated for $\\xi\\in\\Rplus^{d}$. \n\\end{enumerate}\n\\end{remark}\n\\par\nThe ordering statement $X\\mathrel{\\preceq_{\\mathrm{apl}}} Y$ means that for all portfolios \n$\\xi\\in\\Simp^d$\nthe portfolio loss $\\xi^{\\top} X$ is asymptotically smaller $\\xi^{\\top} Y$. \nThus $\\mathrel{\\preceq_{\\mathrm{apl}}}$ concerns only the extreme portfolio losses. \nIn consequence, this order relation is weaker than the (usual) \nstochastic ordering $\\mathrel{\\preceq_{\\mathrm{st}}}$ of the portfolio losses:\n\\begin{equation}\\label{eq:2.2}\n\\xi^{\\top} X \\mathrel{\\preceq_{\\mathrm{st}}} \\xi^{\\top} Y \\text{ for all } \\xi\\in\\Simp^d \\text{ implies } X\\mathrel{\\preceq_{\\mathrm{apl}}} Y.\n\\end{equation}\nHere, for real random variables $U$, $V$ the \\emph{stochastic ordering} \n$U \\mathrel{\\preceq_{\\mathrm{st}}} V$ is defined by \n\\begin{equation}\\label{eq:2.3a}\n\\forall t\\in\\R\\quad \\mathrm{P}\\cubr{U>t}\\le\\mathrm{P}\\cubr{V>t}.\n\\end{equation}\n\\par\nSome related, well-known stochastic orderings  \n\\citep[cf.][]{Mueller\/Stoyan:2002,Shaked\/Shanthikumar:1997} \nare collected in the following list. Remind that $f:\\R^{d}\\to \\R$ \nis called \\emph{supermodular} if\n\\begin{equation}\\label{eq:2.3b}\n\\forall x,y\\in\\R^{d}\n\\quad\nf(x\\wedge y)+f(x\\vee y)\\ge f(x)+f(y)\n\\ldotp\n\\end{equation}\n\\begin{definition}\\label{def:2.2}\nLet $X$, $Y$ be random vectors in $\\R^{d}$. Then $X$ is said to be smaller than $Y$ in\n\\begin{enumerate}[(a)]\n\\item \\emph{(increasing) convex order}, \n$X \\mathrel{\\preceq_{\\mathrm{cx}}} Y$ ($X\\mathrel{\\preceq_{\\mathrm{icx}}} Y$), if $\\mathrm{E} f(X) \\le \\mathrm{E} f(Y)$ for all (increasing) convex functions $f:\\R^{d}\\mapsto \\R$ such that the expectations exist; \n\\item\n\\emph{linear convex order}, $X \\mathrel{\\preceq_{\\mathrm{lcx}}} Y$, if \n$\\xi^{\\top} X \\mathrel{\\preceq_{\\mathrm{cx}}} \\xi^{\\top} Y$ for all $\\xi\\in\\R^{d}$;\n\\item\n\\emph{positive linear convex order}, $X \\mathrel{\\preceq_{\\mathrm{plcx}}} Y$, \nif $\\xi^{\\top} X \\mathrel{\\preceq_{\\mathrm{cx}}} \\xi^{\\top} Y$ for all $\\xi\\in\\Rplus^{d}$;\n\\item \\emph{supermodular order} $X\\mathrel{\\preceq_{\\mathrm{sm}}} Y$,  if \n$\\mathrm{E} f(X) \\le \\mathrm{E} f(Y)$ for all supermodular functions $f:\\R^{d}\\to\\R$ such\nthat the expectations exist;\n\\item\n\\emph{directionally convex order}, $X\\mathrel{\\preceq_{\\mathrm{dcx}}} Y$, if \n$\\mathrm{E} f(X) \\le \\mathrm{E} f(Y)$ for all directionally  convex, i.e., supermodular and componentwise convex functions  \n$f:\\R^{d}\\to\\R$ such that the expectations exist.\n\\end{enumerate}\n\\end{definition}\n\\par\nThe stochastic orderings listed in Definition \\ref{def:2.2} are useful \nfor describing the risk induced by larger diffusion (convex risk) as well as \nthe risk induced by positive dependence \n(supermodular and directionally convex). \nThe following implications are known to hold generally for random \nvectors $X$, $Y$ in $\\R^{d}$: \n\\begin{enumerate}[(a)]\n\\item $(X\\mathrel{\\preceq_{\\mathrm{sm}}} Y)_{\\phantom{icx}\\kern-2ex} \\Rightarrow\n(X\\mathrel{\\preceq_{\\mathrm{dcx}}} Y)_{\\phantom{l}\\kern-.5ex} \\Rightarrow (X\\mathrel{\\preceq_{\\mathrm{plcx}}} Y)$\n\\item $(X\\mathrel{\\preceq_{\\mathrm{cx}}} Y)_{\\phantom{ism}\\kern-2ex} \\Rightarrow %\n(X\\mathrel{\\preceq_{\\mathrm{lcx}}} Y)_{\\phantom{d}\\kern-.5ex} \\Rightarrow (X\\mathrel{\\preceq_{\\mathrm{plcx}}} Y)$\n\\item $(X\\mathrel{\\preceq_{\\mathrm{icx}}} Y)_{\\phantom{sm}\\kern-2ex} \\Rightarrow %\n(X\\mathrel{\\preceq_{\\mathrm{plcx}}} Y)$\n\\end{enumerate}\n\\begin{remark}\\label{rem:apl.1}\n\\begin{enumerate}[(a)]\n\\item\\label{item:apl.1}\nIt is easy to see that the usual stochastic order $\\mathrel{\\preceq_{\\mathrm{st}}}$ implies \n$\\mathrel{\\preceq_{\\mathrm{apl}}}$ in the univariate case.\n\\item\\label{item:apl.2}\nIn spite of being strong risk comparison orders, the order relations \noutlined in Definition~\\ref{def:2.2} do not imply $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in general. \nFor instance, it is known that the comonotonic dependence structure is \nthe worst case with respect to the strong supermodular ordering $\\mathrel{\\preceq_{\\mathrm{sm}}}$,\nwhereas it is not necessarily the worst case with respect to $\\mathrel{\\preceq_{\\mathrm{apl}}}$\n(cf.\\ Examples~\\ref{ex:6} and \\ref{ex:2}). \n\\end{enumerate}\n\\end{remark}\n\\par\nThe following proposition helps to establish sufficient criteria \nfor $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in the univariate case. \nTo obtain multivariate results, \nit can be separately applied to each portfolio loss $\\xi^{\\top} X$ for \n$\\xi\\in\\Simp^d$.   \n\\par\n\\begin{proposition}\\label{prop:2.3}\nLet $R_1$, $R_2\\ge 0$ be real random variables and let $V$ be a real random variable independent of $R_i$, $i=1,2$.\n\\begin{enumerate}[(a)]\n\\item\n\\label{item:prop2.3b} \nIf $R_1\\mathrel{\\preceq_{\\mathrm{apl}}} R_2$ and $V < K$ \nfor some constant $K$, then \n\\begin{equation}\\label{eq:2.7}\nR_1V\\mathrel{\\preceq_{\\mathrm{apl}}} R_2V\n\\ldotp\n\\end{equation}\n\\item  \\label{item:prop2.3a} \nIf $R_1\\mathrel{\\preceq_{\\mathrm{st}}} R_2$, then\n\\begin{equation}\n\\label{eq:2.5}\n\\robr{R_1V}_{+}  \\mathrel{\\preceq_{\\mathrm{st}}} \\robr{R_2V}_{+} \n\\quad \n\\text{and}\n\\quad\n\\robr{R_2V}_{-} \\mathrel{\\preceq_{\\mathrm{st}}} \\robr{R_1V}_{-}\n\\ldotp\n\\end{equation}\nIn addition, if $V$ and $R_i$ are integrable and $EV \\ge 0$, then \n\\begin{equation}\\label{eq:2.6}\nR_1V \\mathrel{\\preceq_{\\mathrm{icx}}} R_2V\n\\ldotp\n\\end{equation}\nMoreover, if $EV=0$, then $R_1V \\mathrel{\\preceq_{\\mathrm{cx}}} R_2V$.\n\\end{enumerate}\n\\end{proposition}\n\\par\n\\begin{myproof}\n\\par Part~(\\ref{item:prop2.3b}). \nSince $R_1V \\mathrel{\\preceq_{\\mathrm{apl}}} R_2V$ is trivial for $V \\le 0$, we \nassume that $\\mathrm{P}\\cubr{V>0}>0$. \nHence $V\\le K$ implies for all $t>0$\n\\begin{align} \n\\mathrm{P}\\cubrfl{R_1V>t}  \n&= \\nonumber\n\\int_{(0,K)} \\mathrm{P}\\cubrfl{R_1>{t}\/{v}} \\mathrm{d}\\mathrm{P}^V(v) \\\\\n&=\\label{eq:2.10b}\n\\int_{(0,K)} f\\robrfl{{t}\/{v}} \\mathrm{P}\\cubrfl{R_2>{t}\/{v}} \\mathrm{d}\\mathrm{P}^V(v), \n\\end{align}\nwhere \n\\[\nf(z):=\\frac{\\mathrm{P}\\cubr{R_1>z}}{\\mathrm{P}\\cubr{R_2>z}}\n\\ldotp\n\\]\nAn obvious consequence of \\eqref{eq:2.10b} is the inequality \n\\begin{equation}\n\\mathrm{P}\\cubr{R_1V > t} \n\\le\\label{eq:2.10c}\n\\sup \\cubrfl{ f(z): z > {t}\/{K}} \\cdot \\mathrm{P}\\cubrfl{R_2V > t}\n\\end{equation}\nSince $R_1\\mathrel{\\preceq_{\\mathrm{apl}}} R_2$ is equivalent to $\\limsup_{z\\to\\infty} f(z)\\le 1$,\nwe obtain\n\\[\n\\limsup_{t\\to\\infty}\\frac{\\mathrm{P}\\cubr{R_1V>t}}{\\mathrm{P}\\cubr{R_2V>t}}\\le 1\n\\ldotp\n\\] \n\\par\nPart (\\ref{item:prop2.3a}). \nBy the well-known coupling principle for the stochastic ordering $\\mathrel{\\preceq_{\\mathrm{st}}}$\nwe may assume without loss of generality that $R_1 \\le R_2$ \npointwise on the underlying probability space.\nThis implies \n\\[\n\\mathrm{P}\\cubr{R_1 V > t} \\le \\mathrm{P}\\cubr{R_2 V > t}\n,\\quad t \\ge 0,\n\\]\nand, similarly,\n\\[\n\\mathrm{P}\\cubr{R_1 V \\le t} \\le \\mathrm{P}\\cubr{R_2 V \\le t}\n,\\quad t \\le 0\n\\ldotp\n\\]\nIn consequence we obtain~\\eqref{eq:2.5}.\n\\par\nFrom the proof of \\eqref{eq:2.5}\nit follows that the distribution functions\nof the products $R_iV$, $i=1,2$, satisfy the cut criterion of Karlin--Novikov \n(cf.\\ \\citealp{Shaked\/Shanthikumar:1994}, Theorem 2.A.17 and \n\\citealp{Mueller\/Stoyan:2002}, Theorem 1.5.17) \nHence we obtain\n\\begin{equation}\\label{eq:2.9}\nR_1V \\mathrel{\\preceq_{\\mathrm{icx}}} R_2V\n\\ldotp\n\\end{equation}\nIf $EV=0$, then $E\\sqbr{R_1V}=E\\sqbr{R_2V}$ and therefore\n\\begin{equation}\\label{eq:2.10a}\nR_1V\\mathrel{\\preceq_{\\mathrm{cx}}} R_2V\n\\ldotp\n\\end{equation}\n\\end{myproof}\n\\begin{remark}\\label{rem:2.3}\n\\begin{enumerate}[(a)]\n\\item %\nNote that\n\\eqref{eq:2.5}\nimplies (without assuming the existence of moments) \nthat $\\robr{R_2V}_{+} \\mathrel{\\preceq_{\\mathrm{decx}}} \\robr{R_1V}_{+}$ \nwhere $\\mathrel{\\preceq_{\\mathrm{decx}}}$ denotes the \\emph{decreasing convex order}. \nSimilarly one obtains \n$\\robr{R_2V}_{-} \\mathrel{\\preceq_{\\mathrm{icx}}} \\robr{R_1V}_{-}$\n\\item \nIf $f(t):={\\mathrm{P}\\cubr{R_1>t}}\/{\\mathrm{P}\\cubr{R_2>t}} \\le C < \\infty$ and \n$R_1\\mathrel{\\preceq_{\\mathrm{apl}}} R_2$, then $R_1V\\mathrel{\\preceq_{\\mathrm{apl}}} R_2V$. \n\\item \nA related problem is the ordering of products $RV_i$ for $R\\ge 0$ with  \n$V_1$ and $V_2$ independent of $R$.\nIn the special case when $R$ is \\emph{regularly varying} with \\emph{tail index} \n$\\alpha>0$, i.e., \n\\begin{equation}\\label{eq:apl.5}\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubr{R>tx}}{\\mathrm{P}\\cubr{R>t}} \n=x^{-\\alpha}\n,\\quad \nx>0,\n\\end{equation}\nexact criteria for $\\mathrel{\\preceq_{\\mathrm{apl}}}$ can be obtained from Breiman's Theorem \n\\citep[cf.][Proposition 7.5]{Resnick:2007}. \nIf $\\mathrm{E} \\robr{V_i}_{+}^{\\alpha+\\varepsilon} <\\infty$ for $i=1,2$ and \nsome $\\varepsilon>0$, then \n\\[\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubr{RV_i>t}}{\\mathrm{P}\\cubr{R>t}} \n= \nE\\sqbrfl{\\robr{V_i}_{+}^{\\alpha}}\n\\ldotp\n\\]\nThis yields \n\\[\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubr{RV_1>t}}{\\mathrm{P}\\cubr{RV_2>t}} \n= \n\\frac\n{\\mathrm{E}\\sqbrfl{\\robr{V_1}_{+}^{\\alpha}}}\n{\\mathrm{E}\\sqbrfl{\\robr{V_2}_{+}^{\\alpha}}}\n\\ldotp\n\\]\n\\end{enumerate}\n\\end{remark}\nAn important class of stochastic models with various applications are \n\\emph{elliptical distributions}, \nwhich are natural generalizations of multivariate \nnormal distributions. \nA random vector $X\\in\\R^{d}$ is called elliptically distributed, \nif there exist $\\mu\\in\\R^{d}$ and a $d\\times d$ matrix $A$ such that  \n$X$ has a representation of the form\n\\begin{equation}\\label{eq:2.11}\nX \\mathrel{\\stackrel{\\mathrm{d}}{=}} \\mu + RAU, \n\\end{equation}\nwhere $U$ is uniformly distributed on the Euclidean unit sphere $\\Sbb^d_2$,  \n\\[\n\\Sbb^d_2=\\cubrfl{x\\in\\R^{d} :  \\norm{x}_2 =1},\n\\] \nand $R$ is a non-negative random variable independent of $U$. \nBy definition we have \n\\begin{equation}\\label{eq:2.12}\nE\\norm{X}_2^2 <\\infty \\Leftrightarrow E R^2<\\infty,\n\\end{equation}\nand in this case   \n\\begin{equation}\\label{eq:2.13}\n\\mathrm{Cov}(X)=\\mathrm{Var}(R) A A^{\\top}\n\\ldotp \n\\end{equation}\nThe matrix $C:= A A^{\\top}$ is unique except for a constant factor and \nis also called the \\emph{generalized covariance matrix} of $X$. \nWe denote the elliptical distribution constructed \naccording to~\\eqref{eq:2.11} by $\\Ecal(\\mu,C,F_R)$, \nwhere $F_R$ is the distribution of $R$.\n\\par\nA classical stochastic ordering result going back to \n\\cite{Anderson:1955} and \\cite{Fefferman\/Jodeit\/Perlman:1972} \n\\citep[cf.][p.~70]{Tong:1980} \nsays that \\emph{positive semidefinite ordering}\nof the generalized covariance matrices $C_1 \\mathrel{\\preceq_{\\mathrm{psd}}} C_2$, defined as\n\\begin{equation}\n\\label{eq:2.13a}\n\\forall \\xi\\in\\R^{d} \\quad \\xi^{\\top} C_1 \\xi \\le \\xi^{\\top} C_2 \\xi,\n\\end{equation}\nimplies symmetric convex ordering if  \nthe location parameter $\\mu$ and the distribution $F_R$ of the radial \nfactor are fixed:  \n\\begin{equation}\\label{eq:2.14}\n\\Ecal(\\mu, C_1,F_R) \\mathrel{\\preceq_{\\mathrm{symmcx}}} \\Ecal(\\mu, C_2, F_R)\n\\ldotp\n\\end{equation}\nIt is also known that for elliptical random vectors $X\\sim \\Ecal(\\mu,C,F_R)$ \nthe multivariate distribution function \n$F(x):=\\mathrm{P}\\cubr{X_1\\le x_1,\\dots,X_d\\le x_d}$ is increasing in $C_{i,j}$ for $i\\not=j$, where $C=(C_{i,j})$ \\citep[see, e.g.,][Theorem 2.21]{Joe:1997}.\n\\par\nThe following result is concerned \nwith the asymptotic portfolio loss ordering $\\mathrel{\\preceq_{\\mathrm{apl}}}$ for elliptical \ndistributions. \n\\begin{theorem}\\label{theo:2.4}\nLet $X\\mathrel{\\stackrel{\\mathrm{d}}{=}} \\mu_1+R_1A_1U$,  $Y\\mathrel{\\stackrel{\\mathrm{d}}{=}}\\mu_2+R_2A_2U$ be elliptically distributed \nwith generalized covariances $C_i:=A_iA_i^{\\top}$. If \n\\begin{equation}\\label{eq:2.15}\n\\mu_1\\le \\mu_2, \\enskip \nR_1\\mathrel{\\preceq_{\\mathrm{apl}}} R_2,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:2.15a}\n\\forall \\xi\\in\\Simp^d \\quad \\xi^{\\top} C_1\\xi \\le \\xi^{\\top} C_2 \\xi,\n\\end{equation}\nthen\n\\begin{equation}\\label{eq:2.16}\nX\\mathrel{\\preceq_{\\mathrm{apl}}} Y.\n\\end{equation}\n\\end{theorem}\n\\par\n\\begin{myproof}\nIt suffices to show that $\\xi^{\\top} Y \\mathrel{\\preceq_{\\mathrm{apl}}} \\xi^{\\top} Y$ for an \narbitrary portfolio $\\xi\\in\\Simp^d$. \nFurthermore, without loss of generality we can assume $\\mu_1=\\mu_2=0$.\nFor $i=1,2$ and $\\xi\\in\\Simp^d$ denote \n\\[\na_i = a_i(\\xi):=  \\robrfl{\\xi ^{\\top} C_i \\xi}^{1\/2}\n\\] \nand \n\\[\nv_i = v_i(\\xi) := \\frac{\\xi^{\\top} A_i}{a_i}\n\\ldotp\n\\]\nThen, by definition of elliptical distributions, we have\n\\begin{equation}\\label{eq:2.17}\n\\xi^{\\top} X \\mathrel{\\stackrel{\\mathrm{d}}{=}} R_1 a_1 v_1 U\n\\quad\\text{and}\\quad\n\\xi^{\\top} Y \\mathrel{\\stackrel{\\mathrm{d}}{=}} R_2 a_2 v_2 U\n\\ldotp\n\\end{equation}\nSince the vectors $v_i=v_i(\\xi)$ have unit length by construction, \nthe random variables $v_i U$ are orthogonal projections of $U\\sim \\mathrm{unif}(S_2^d)$ \non vectors of unit length. \nSymmetry arguments yield that the distribution of $v_i U$ is independent of \n$v_i$ and that $v_iU \\mathrel{\\stackrel{\\mathrm{d}}{=}} (1,0,\\ldots,0)^{\\top} U=U^{(1)}$. \n\\par\nThus we have \n\\[\n\\xi^{\\top} X \\mathrel{\\stackrel{\\mathrm{d}}{=}} a_1 R_1 V\n\\quad\\text{and}\\quad\n\\xi^{\\top} Y\\mathrel{\\stackrel{\\mathrm{d}}{=}} a_2 R_2 V\n\\] \nwith $V:=U^{(1)}$.\nBy assumption we have $a_1\\le a_2$ and $R_1\\mathrel{\\preceq_{\\mathrm{apl}}} R_2$. \nApplying Proposition \\ref{prop:2.3}(\\ref{item:prop2.3b}) \nwe obtain $\\xi^{\\top} X\\mathrel{\\preceq_{\\mathrm{apl}}} \\xi^{\\top} Y$. \n\\end{myproof}\n\\par\n\\begin{remark}\\label{rem:2.6}\n\\begin{enumerate}[(a)]\n\\item\\label{item:rem:2.6.a}\nIt should be noted that condition~\\eqref{eq:2.15a} is indeed weaker than \n\\eqref{eq:2.13a}. \nLet $-1 < \\rho_1 < \\rho_2 <1$ and consider covariance matrices \n\\[\nC_i:=\n\\robrfl{\n\\begin{array}{cc}\n1 & \\rho_i\\\\\n\\rho_i & 1\n\\end{array}\n}\n,\\quad \ni=1,2\n\\ldotp\n\\]\nStraightforward calculations show that $C_i$ satisfy~\\eqref{eq:2.15a}, but \nnot~\\eqref{eq:2.13a}.\n\\item\nFor subexponentially distributed \n$R_i$ the assumption $\\mu_1\\le \\mu_2$ in \\eqref{eq:2.15} can be omitted.\n\\end{enumerate}\n\\end{remark}\n\\section{Multivariate regular variation: $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in terms of spectral measures}\n\\label{sec:3}\n\\par\nThis section is concerned with the characterization of the asymptotic \nportfolio loss order $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in the framework of multivariate regular \nvariation. The results obtained here highlight the influence of the tail \nindex $\\alpha$ and the spectral measure $\\Psi$ on $\\mathrel{\\preceq_{\\mathrm{apl}}}$,  \nwith primary focus put on  dependence structures captured by $\\Psi$. \nIt is shown that $\\mathrel{\\preceq_{\\mathrm{apl}}}$ corresponds to a family of order relations  \non the set of canonical spectral measures and that these order relations \nare intimately related to the extreme risk index $\\gamma_\\xi$ introduced \nin \\citet{Mainik\/Rueschendorf:2010} and \\citet{Mainik:2010}. \n\\par\nThe main result of this section is stated in Theorem~\\ref{theo:3.4}, \nproviding criteria for $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$ in terms of componentwise ordering \n$X^{(i)} \\mathrel{\\preceq_{\\mathrm{apl}}} Y^{(i)}$ for $i=1,\\ldots,d$ \nand ordering of canonical spectral measures. \nA particular consequence of these criteria is the \ncharacterization of the dependence structures that \nyield the best and the worst possible diversification effects for \nrandom vectors in $\\Rplus^{d}$ \n(cf.\\ Theorem~\\ref{thm:3.8} and Corollary~\\ref{cor:3.10}).\nAnother application concerns elliptical distributions. Combining  \nTheorem~\\ref{theo:3.4} with results on $\\mathrel{\\preceq_{\\mathrm{apl}}}$ obtained in \nTheorem~\\ref{theo:2.4}, we obtain ordering of the corresponding \ncanonical spectral measures. \n\\par\nRecall the notions of regular variation. \nIn the univariate case it can be defined separately for the lower \nand the upper tail of a random variable via~\\eqref{eq:apl.5}. \nA random vector $X$ taking values in $\\R^{d}$ is called \n\\emph{multivariate regularly varying} with tail index $\\alpha\\in(0,\\infty)$\nif  there exist a sequence $a_n\\to\\infty$ and a (non-zero) Radon measure $\\nu$ on \nthe Borel $\\sigma$-field $\\mathcal{B}\\robr{[-\\infty,\\infty]^d\\setminus\\cubr{0}}$ \nsuch that $\\nu\\robr{[-\\infty,\\infty]^d \\setminus \\R^{d}}=0$ and, \nas $n\\to\\infty$,\n\\begin{equation}\n\\label{eq:29}\n\\index{$\\nu$}\nn \\mathrm{P}^{\\,a_n^{-1} X} \\stackrel{\\mathrm{v}}\\rightarrow \\nu\n\\text{ on }\\mathcal{B}\\robr{[-\\infty,\\infty]^d\\setminus\\cubr{0}},\n\\end{equation}\nwhere $\\stackrel{\\mathrm{v}}\\rightarrow$ denotes the \n\\emph{vague convergence} of Radon measures \nand $\\mathrm{P}^{\\,a_n^{-1} X}$ is the probability distribution of\n$a_n^{-1} X$.\n\\par\nIt should be noted that random vectors with non-negative components \nyield limit measures $\\nu$ that are concentrated on \n$[0,\\infty]^d\\setminus\\cubr{0}$. \nTherefore multivariate regular variation in this special case can also \nbe defined by vague convergence on $\\mathcal{B}([0,\\infty]^d\\setminus\\cubr{0})$.\n\\par\nMany popular distribution models are multivariate regularly  varying. \nIn particular, according to \\citet{Hult\/Lindskog:2002}, \nmultivariate regular variation of an elliptical distribution \n$\\Ecal\\robr{\\mu,C,F_R}$ is equivalent to the regular variation of the \nradial factor $R$ and the tail index $\\alpha$ is inherited from $R$.\nOther popular examples are obtained by endowing regularly varying margins \n$X^{(i)}$ with an appropriate copula \n\\citet[cf.][]{Wuethrich:2003, Alink\/Loewe\/Wuethrich:2004, Barbe\/Fougeres\/Genest:2006}\n\\par\nFor a full account of technical details related to the notion of \nmultivariate regular variation, vague convergence, and \nthe Borel $\\sigma$-fields on the punctured spaces \n$[-\\infty,\\infty]^d\\setminus\\cubr{0}$ and $[0,\\infty]^d\\setminus\\cubr{0}$  \nthe reader is referred to \\citet{Resnick:2007}.\n\\par\nIt is well known that the limit measure $\\nu$ obtained in~\\eqref{eq:29}\nis unique except for a constant factor, has a singularity in the origin\nin the sense that \n$\\nu\\robr{(-\\varepsilon,\\varepsilon)^d}=\\infty$ for any $\\varepsilon>0$,  \nand exhibits the scaling property \n\\begin{equation}\n\\label{eq:30}\n\\nu(tA)=t^{-\\alpha}\\nu(A)\n\\end{equation} \nfor all sets $A\\in\\mathcal{B}\\robrfl{[-\\infty,\\infty]^d\\setminus\\cubr{0}}$ that\nare bounded away from $0$. \n\\par\nIt is also well known that~\\eqref{eq:29} implies that the random variable \n$\\norm{X}$ with an arbitrary norm $\\norm{\\cdot}$ on $\\R^{d}$ is \nunivariate regularly varying with tail index $\\alpha$.\nMoreover,\nthe sequence $a_n$ can always be chosen as \n\\begin{equation}\n\\label{eq:181}\na_n:=F_{\\norm{X}}^{\\leftarrow}(1-1\/n),\n\\end{equation}\nwhere $F_{\\norm{X}}^{\\leftarrow}$ is the quantile function of\n$\\norm{X}$. The resulting limit measure $\\nu$ \nis normalized on the set $A_{\\norm{\\cdot}}:=\\cubr{x\\in\\R^{d}: \\norm{x}>1}$ by \n\\begin{equation}\n\\label{eq:182}\n\\nu\\robrfl{A_{\\norm{\\cdot}}}=1\n\\ldotp\n\\end{equation}\n\\par\nThus, after normalizing $\\nu$ by~\\eqref{eq:182}, \nthe scaling relation~\\eqref{eq:30} yields an equivalent rewriting of \nthe multivariate regular variation condition~\\eqref{eq:29} \nin terms of weak convergence:\n\\begin{equation}\n\\label{eq:34}\n\\mathcal{L}\\cubrfl{t^{-1} X\\,|\\,\\norm{X}>t}\n\\stackrel{\\mathrm{w}}{\\rightarrow}\n\\nu|_{A_{\\norm{\\cdot}}}\n\\text{ on } \n\\mathcal{B}\\robrfl{A_{\\norm{\\cdot}}}\n\\end{equation}\nfor $t\\to\\infty$,\nwhere $\\nu|_{A_{\\norm{\\cdot}}}$ is the restriction of $\\nu$ to the set $A_{\\norm{\\cdot}}$. \n\\par\nAdditionally to~\\eqref{eq:29}\nit is assumed that the limit measure $\\nu$ is \nnon-degen\\-erate\nin the \nfollowing sense:\n\\begin{equation}\n\\label{eq:4}\n\\nu\\robrfl{\\cubrfl{x\\in\\R^{d}: \\absfl{x^{(i)}}> 1}} >0\n,\\quad i=1,\\ldots,d\n\\ldotp\n\\end{equation}\nThis assumption ensures that\nall asset losses $X^{(i)}$ are relevant for the extremes of the portfolio loss \n$\\xi^{\\top} X$. If~\\eqref{eq:4} is satisfied in the upper tail region, i.e., if \n\\begin{equation}\n\\label{eq:4a}\n\\nu\\robrfl{\\cubrfl{x\\in\\R^{d}: x^{(i)}> 1}} >0 \n,\\quad i=1,\\ldots,d,\n\\end{equation} \nthen  $\\nu$ also characterizes the asymptotic distribution\nof the componentwise maxima\n$M_n:=\\robr{M^{(1)},\\ldots,M^{(d)}}$ with \n$M^{(i)}:=\\max\\cubr{X_1^{(i)},\\ldots,X_n^{(i)}}$\nby the limit relation \n\\begin{equation}\n\\label{eq:164}\n\\mathrm{P}\\cubrfl{a_n^{-1} M_n\\in[-\\infty,x]} \\stackrel{\\mathrm{w}}{\\rightarrow}\n\\exp\\robrfl{-\\nu\\robrfl{[-\\infty,\\infty]^d\\setminus[-\\infty,x]}}\n\\end{equation}\nfor $x\\in(0,\\infty]^d$. \nTherefore $\\nu$ is called \n\\emph{exponent measure}. \nFor more details concerning the asymptotic distributions of maxima\nthe reader is referred to~\\citet{Resnick:1987} \nand \\citet{de_Haan\/Ferreira:2006}.\n\\par\nAnother consequence of the scaling property~\\eqref{eq:30} is the  \nproduct representation of $\\nu$ in polar coordinates \n\\[\n(r,s):=\\tau(x):=(\\norm{x},\\norm{x}^{-1} x)\n\\] \nwith respect to an arbitrary norm $\\norm{\\cdot}$ on $\\R^{d}$.\nThe induced \nmeasure $\\nu^\\tau:=\\nu\\circ\\tau^{-1}$ necessarily satisfies\n\\begin{equation}\n\\label{eq:28}\n\\nu^\\tau=c\\cdot\\rho_\\alpha\\otimes\\Psi\n\\end{equation}\nwith the constant factor \n\\[\nc=\\nu\\robrfl{A_{\\norm{\\cdot}}}\n>0,\n\\]  \nthe measure $\\rho_\\alpha$ on $(0,\\infty]$ defined by \n\\begin{equation}\n\\label{eq:176}\n\\rho_\\alpha((x,\\infty]):=x^{-\\alpha},\n\\quad \nx\\in(0,\\infty],\n\\end{equation} \nand a probability measure $\\Psi$ on the unit sphere $\\Sbb^d_{\\norm{\\cdot}}$\nwith respect to $\\norm{\\cdot}$,\n\\[ \n\\Sbb^d_{\\norm{\\cdot}}:=\\cubrfl{s\\in\\R^{d} : \\norm{s} = 1}\n\\ldotp\n\\]\nThe measure $\\Psi$ is called \n\\emph{spectral measure} \nof $\\nu$ or $X$.\nSince the term \\enquote{spectral measure} is already used in other areas, \n$\\Psi$ is also referred to as \n\\emph{angular measure}.\nIn the special case of $\\Rplus^{d}$-valued random vectors $X$ it\nmay be convenient to reduce the domain of $\\Psi$ to \n$\\Sbb^d_{\\norm{\\cdot}}\\cap\\Rplus^{d}$. \n\\par\nAlthough the domain of the spectral measure $\\Psi$ depends on the\nnorm $\\norm{\\cdot}$ underlying the polar coordinates, the \nrepresentation~\\eqref{eq:28} is norm-independent in the following sense:\nif~\\eqref{eq:28} holds for some norm $\\norm{\\cdot}$, then it also holds for \nany other norm $\\norm{\\cdot}_\\diamond$ that is equivalent to $\\norm{\\cdot}$. \nThe tail index $\\alpha$ is the same and the spectral measure $\\Psi_\\diamond$ \non the unit sphere $\\Sbb^d_\\diamond$ corresponding to $\\norm{\\cdot}_\\diamond$ \nis obtained from $\\Psi$ by the following transformation:\n\\[\n\\Psi_\\diamond=\\Psi^T,\\quad T(s):=\\norm{s}_\\diamond^{-1} s\n\\ldotp\n\\]\n\\par\nFinally, it should be noted that multivariate regular variation of \nthe loss vector $X$ is intimately related with the univariate regular variation \nof portfolio losses $\\xi^{\\top} X$.\nAs shown in \\citet{Basrak\/Mikosch\/Davis:2002}, \nmultivariate regular variation of $X$ \nimplies existence of a portfolio vector $\\xi_0\\in\\R^{d}$ such that $\\xi_0 ^{\\top} X$ \nis regularly varying with tail index $\\alpha$ and any \nportfolio loss $\\xi^{\\top} X$ satisfies\n\\begin{equation}\n\\label{eq:192}\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\xi_0^{\\top} X >t}} \n=c(\\xi,\\xi_0)\n\\in [0,\\infty)\n\\ldotp\n\\end{equation}\nThis means that all portfolio losses $\\xi^{\\top} X$ are either regularly\nvarying with tail index $\\alpha$ or asymptotically negligible \ncompared to $\\xi_0^{\\top} X$. \n\\par\nMoreover, it is also worth a remark \nthat for $\\Rplus^{d}$-valued random vectors $X$ \nthe converse implication is true in the sense that~\\eqref{eq:192} \nand univariate regular variation of $\\xi_0^{\\top} X$ \nimply multivariate regular variation of the random vector $X$.\nThis sort of Cram\\'er-Wold theorem was established in \n\\citet{Basrak\/Mikosch\/Davis:2002} and \\citet{Boman\/Lindskog:2009}.\n\\par\nUnder the assumption of multivariate regular variation of $X$ \nthe \\emph{extreme risk index} $\\gamma_\\xi = \\gamma_\\xi(X)$ \nis defined as \n\\begin{equation}\\label{eq:3.2}\n\\gamma_\\xi(X)=\\lim_{t\\to\\infty} \\frac{\\mathrm{P}\\cubr{\\xi^{\\top} X>t}}{\\mathrm{P}\\cubr{\\norm{X}_1>t}}.\n\\end{equation}\nIn \\citet{Mainik\/Rueschendorf:2010} the random vector $X$ is restricted to \n$\\Rplus^{d}$ and the portfolio vector $\\xi$ is restricted to $\\Simp^d$. \nThe general case with $X$ in $\\R^{d}$ and possible negative portfolio \nweights, i.e., short positions, is considered in \\citet{Mainik:2010}. \nNormalizing the exponent measure $\\nu$ by~\\eqref{eq:182},\none obtains \n\\begin{equation}\\label{eq:3.1}\n\\gamma_\\xi(X)=\\nu\\robrfl{\\cubrfl{x\\in\\R^{d}: \\xi^{\\top} x > 1}}\n\\ldotp\n\\end{equation}\nRewriting this representation in terms of the spectral measure $\\Psi$ \nand the tail index $\\alpha$ yields\n\\begin{equation}\\label{eq:apl.1}\n\\gamma_\\xi\n=\n\\int_{\\Sbb^d_1}\\robrfl{\\xi^{\\top} s}_{+}^{\\alpha} \\mathrm{d} \\Psi(s)\n\\ldotp\n\\end{equation}\nDenoting the integrand by $f_{\\xi,\\alpha}$, we will write this representation \nas $\\gamma_\\xi=\\Psif_{\\xi,\\alpha}$.   \n\\par\nThe extreme risk index $\\gamma_\\xi(X)$ allows to compare the risk of different \nportfolios. It is easy to see that \\eqref{eq:3.2} implies\n\\begin{equation}\\label{eq:3.4}\n\\lim_{t\\to\\infty} \\frac{\\mathrm{P}\\cubr{\\xi_1^{\\top} X>t}}{\\mathrm{P}\\cubr{\\xi_2^{\\top} X>t}} = \\frac{\\gamma_{\\xi_1}(X)}{\\gamma_{\\xi_2}(X)}.\n\\end{equation}\nThus, by construction, \nordering of the extreme risk index $\\gamma_\\xi$ is related to the \nasymptotic portfolio loss order $\\mathrel{\\preceq_{\\mathrm{apl}}}$. \n\\par\nHowever, designed for the comparison of different portfolio risks within one \nmodel, the extreme risk index $\\gamma_\\xi$ cannot be directly applied \nto the comparison of different models. \nThe major problem is the standardization by $\\mathrm{P}\\cubr{\\norm{X}_1>t}$ in \n\\eqref{eq:3.2}. Indeed, since $\\mathrm{P}\\cubr{\\norm{X}_1>t}$ also depends on the \nspectral measure $\\Psi_X$ of $X$, criteria for $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in terms of \n$\\gamma_\\xi$  demand the specification of the limit\n\\[\n\\lim_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubr{\\norm{X}_1>t}}{\\mathrm{P}\\cubr{\\norm{Y}_1>t}}\n\\ldotp\n\\]\n\\par\nAnother technical issue arises from the invariance of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ under \ncomponentwise rescalings. Since the spectral measure $\\Psi$ does not exhibit \nthis property, ordering of spectral measures needs additional normalization \nof margins that makes it consistent with $\\mathrel{\\preceq_{\\mathrm{apl}}}$. To solve these problems,\n we use an alternative representation of $\\gamma_\\xi$ in terms of the \nso-called canonical spectral measure $\\Psi^\\ast$, \nwhich has standardized marginal weights.\n\\par\nThis representation is closely related to the asymptotic risk aggregation \ncoefficient discussed by \\cite{Barbe\/Fougeres\/Genest:2006}. \nFurthermore, the link between the canonical spectral measure and \nextreme value copulas \nallows to transfer ordering results for copulas into the $\\mathrel{\\preceq_{\\mathrm{apl}}}$ \nsetting. These results are presented in Section~\\ref{sec:4}.\n\\par\nTo reduce the problem to the essentials,  \nwe start with the observation that $\\mathrel{\\preceq_{\\mathrm{apl}}}$ is trivial for  \nmultivariate regularly varying random vectors with different \ntail indices and non-degenerate portfolio losses. \n\\par\n\\begin{proposition}\\label{prop:3.1}\nLet $X$ and $Y$ be multivariate regularly varying on $\\R^{d}$ and assume that $\\gamma_\\xi(Y)>0$ for all $\\xi\\in\\Simp^d$. \n\\begin{enumerate}[(a)]\\label{item:prop.3.1a}\n\\item If\n\\begin{equation}\\label{eq:3.5}\n\\lim_{t\\to\\infty} \\frac{\\mathrm{P}\\cubr{\\norm{X}_1>t}}{\\mathrm{P}\\cubr{\\norm{Y}_1>t}} = 0,\n\\end{equation}\nthen $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\vspace{0.5em\n\\item If $\\alpha_X>\\alpha_Y$, then $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\end{enumerate}\n\\end{proposition}\n\\par\n\\begin{myproof}\n\\begin{enumerate}[(a)]\n\\item %\nUsing relation \\eqref{eq:3.2} we obtain \n\\begin{align*}\n\\hspace{2em}&\\hspace{-2em}\n\\limsup_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y > t}}\\\\\n&=\n\\limsup_{t\\to\\infty}\n\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t}}{\\mathrm{P}\\cubrfl{\\norm{X}_1>t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\norm{Y}_1>t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y > t}} \n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\norm{X}_1 > t}}{\\mathrm{P}\\cubrfl{\\norm{Y}_1>t}}\n}\\\\\n&=\n\\frac{\\gamma_\\xi(X)}{\\gamma_\\xi(Y)}\n\\cdot \n\\limsup_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\norm{X}_1 > t}}{\\mathrm{P}\\cubrfl{\\norm{Y}_1>t}}\\\\\n&=0\n\\ldotp\n\\end{align*}\n\\item %\nRecall that multivariate regular variation of $X$ implies regular variation of $\\norm{X}_1$ with tail index $\\alpha_X$. Analogously, $\\norm{Y}_1$ is regularly varying with tail index $\\alpha_Y$. Finally, $\\alpha_X > \\alpha_Y$ yields \\eqref{eq:3.5} and by~(\\ref{item:prop.3.1a}) we obtain $X\\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\qed\n\\end{enumerate}\n\\end{myproof}\n\\par\nThus the primary setting for studying the influence of dependence \nstructures on the ordering of extreme portfolio losses is the case of \nrandom variables $X$ and $Y$ with equal tail indices:\n\\[\n\\alpha_X=\\alpha_Y=:\\alpha\n\\ldotp\n\\] \nIn the framework of multivariate regular variation, asymptotic dependence in the tail region \nis characterized by the spectral measure $\\Psi$  or its canonical version $\\Psi^\\ast$.  \nThe \\emph{canonical exponent measure} $\\nu^{\\ast}$ of $X$ is obtained from the exponent \nmeasure $\\nu$ as \n\\[\n\\nu^{\\ast}=\\nu\\circ T\n\\]\nwith the transformation $T:\\R^{d}\\to\\R^{d}$ defined by\n\\begin{equation}\n\\quad\nT(x)\n:=\\label{eq:3.7}\n\\robrfl{T_\\alpha\\robrfl{\\nu\\robr{B_1}\\cdot x^{(1)}},\\ldots, T_\\alpha\\robrfl{\\nu\\robr{B_d}\\cdot x^{(d)}}},\n\\end{equation}\nwhere \n\\begin{equation} \nT_\\alpha(t)\n:=\\label{eq:3.8}\n\\robrfl{t_{+}^{1\/\\alpha} - t_{-}^{1\/\\alpha}}  \\text{ and }\nB_i := \\cubrfl{x\\in\\R^{d}: \\absfl{x^{(i)}} > 1}\n\\ldotp\n\\end{equation}\nFurthermore, $\\nu^{\\ast}$ exhibits the scaling property \n\\[\n\\nu^{\\ast}(tA)=t^{-1}\\nu^{\\ast}(A),\n\\quad t>0,\n\\] \nand, analogously to~\\eqref{eq:28}, has a product structure in polar \ncoordinates:\n\\begin{equation}\\label{eq:apl.3}\n\\nu^{\\ast}\\circ\\tau^{-1} = \\rho_1 \\otimes \\Psi^\\ast,\n\\end{equation}\nThe measure $\\Psi^\\ast$ is the \\emph{canonical spectral measure} of $X$. \n\\par  \nSince $\\mathrel{\\preceq_{\\mathrm{apl}}}$ and $\\Psi^\\ast$ are invariant under componentwise rescalings, \nthe canonical spectral measure $\\Psi^\\ast$ is more suitable for the \ncharacterization of $\\mathrel{\\preceq_{\\mathrm{apl}}}$. \nThe following lemma provides a representation of the extreme risk index \n$\\gamma_\\xi$ in terms of $\\Psi^\\ast$.  Note that the formulation makes use of \nthe componentwise product notation~\\eqref{eq:apl.2}.\n\\par\n\\begin{proposition}\n\\label{prop:3.2}\nLet $X$ be multivariate regularly varying on $\\R^{d}$ with tail index \n$\\alpha\\in(0,\\infty)$. \nIf $X$ satisfies the non-degeneracy condition~\\eqref{eq:4},  \nthen \n\\begin{equation}\\label{eq:3.9}\n\\gamma_\\xi(X)=\\int_{\\Sbb^d_1}g_{\\xi,\\alpha}\\robrfl{v s} \\, \\mathrm{d} \\Psi^\\ast(s),\n\\end{equation}\nwhere $\\Psi^\\ast$ denotes the canonical spectral measure of $X$, \nthe rescaling vector $v=\\robr{v^{(1)},\\ldots,v^{(d)}}$ \nis defined by \n\\begin{equation}\\label{eq:3.10}\nv^{(i)}:=\\robr{\\gamma_{\\ei}(X)+\\gamma_{-\\ei}(X)},\n\\end{equation}  \nand the function $g_{\\xi,\\alpha}:\\R^{d}\\to\\R$ is defined as \n\\begin{equation}\\label{eq:3.11}\ng_{\\xi,\\alpha}(x):=%\n\\robrfl{\\sum_{i=1}^d\\xi^{(i)}\\cdot\\robrfl{\\robrfl{x^{(i)}}_{+}^{1\/\\alpha} - \\robrfl{x^{(i)}}_{-}^{1\/\\alpha}}}_{+}^{\\alpha}\n\\ldotp\n\\end{equation}\n\\end{proposition}\n\\par\n\\begin{myproof}\nDenote $A_{\\xi,1}:=\\{x\\in\\R^{d}: \\xi^{\\top} x\\ge 1\\}$. Then, by definition of $\\nu^{\\ast}$,\n\\begin{align}\n\\gamma_\\xi(X)\n&=\\nonumber\n\\nu(A_{\\xi,1})\\\\\n&=\\nonumber\n\\nu^{\\ast}\\robr{T^{-1}(A_{\\xi,1})}\\\\\n&=\\nonumber\n\\nu^{\\ast}\\cubrfl{x\\in\\R^{d}: T(x) \\in A_{\\xi,1}}\\\\\n&=\\label{eq:3.12}\n\\int_{\\Sbb^d_1}\\int_{(0,\\infty)} \n1\\cubrfl{\\xi^{\\top} T(rs) > 1} \\, \\mathrm{d} \\rho_1(r) \\, \\mathrm{d} \\Psi^\\ast(s)\n\\ldotp\n\\end{align}\nIt is easy to see that~\\eqref{eq:3.8} implies \n$T_\\alpha(rt)=r^{1\/\\alpha}T_\\alpha(t)$ for $r>0$ and $t\\in\\R$. \nConsequently, \\eqref{eq:3.7} yields\n\\begin{equation}\n\\label{eq:3.13}\nT(rx) = r^{1\/\\alpha} T(x)\n\\end{equation}\nfor $r>0$ and $x\\in\\R^{d}$. \nApplying~\\eqref{eq:3.13} to~\\eqref{eq:3.12}, one obtains  \n\\begin{align}\n\\gamma_\\xi(X)\n&=\\nonumber\n\\int_{\\Sbb^d_1}\\int_{(0,\\infty)} \n1\\cubrfl{r^{1\/\\alpha} \\xi^{\\top} T(s) > 1} \\, \\mathrm{d} \\rho_1(r) \\, \\mathrm{d} \\Psi^\\ast(s)\\\\\n&=\\nonumber\n\\int_{\\Sbb^d_1}\\int_{(0,\\infty)} \n1\\cubrfl{\\xi^{\\top} T(s) > 0}\n1\\cubrfl{r > \\robrfl{\\xi^{\\top} T(s)}^{-\\alpha}} \\, \\mathrm{d} \\rho_1(r) \\, \\mathrm{d} \\Psi^\\ast(s)\\\\\n&=\\nonumber\n\\int_{\\Sbb^d_1}\n1\\cubrfl{\\xi^{\\top} T(s) > 0} \\robrfl{\\xi^{\\top} T(s)}^{\\alpha} \\, \\mathrm{d} \\Psi^\\ast(s)\\\\\n&=\\label{eq:3.14}\n\\int_{\\Sbb^d_1}\n\\robrfl{\\xi^{\\top} T(s)}^{\\alpha}_{+} \\, \\mathrm{d} \\Psi^\\ast(s)\n\\ldotp\n\\end{align}\nFinally, consider the sets $B_i$ defined in~\\eqref{eq:3.8}. \nIt is easy to see that \n\\begin{equation*}\n\\nu(B_i) \n= \n\\gamma_{\\ei}(X)+\\gamma_{-\\ei}(X) = v^{(i)}\n\\ldotp \n\\end{equation*}\nHence \n\\begin{align*}\n\\robrfl{\\xi^{\\top} T(s)}_{+}^{\\alpha} \n&= \n\\robrfl{\n\\sum_{i=1}^d\\xi^{(i)}\\cdot\\robrfl{T_\\alpha\\robrfl{v^{(i)} s^{(i)}}}\n}  _{+}^{\\alpha}\\\\\n&=\ng_{\\xi,\\alpha}\\robrfl{vs}\n\\ldotp\n\\qed\n\\end{align*}\n\\end{myproof}\n\n\nAs already mentioned above, $\\mathrel{\\preceq_{\\mathrm{apl}}}$ and $\\Psi^\\ast$ are invariant under \nrescaling of components. \nConsequently, characterization of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ can be reduced to the case when the marginal weights $v^{(i)}=\\gamma_{\\ei}(X)+\\gamma_{-\\ei}(X)$ in~\\eqref{eq:3.9} are standardized by\n\\begin{equation}\n\\label{eq:3.15}\n\\forall i,j\\in\\cubr{1,\\ldots,d}\n\\quad \n\\lim_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubr{\\abs{X^{(i)}}> t}}{\\mathrm{P}\\cubr{\\abs{X^{(j)}}>t}}\n=1\n\\ldotp\n\\end{equation}\nThis condition will be referred to as the \n\\emph{balanced tails condition}. \nThe following result shows that this condition significantly simplifies the representation~\\eqref{eq:3.9}. \n\\par\n\\begin{proposition}\n\\label{prop:3.3}\nSuppose that $X$ is  multivariate regularly varying on $\\R^{d}$ with tail index\n$\\alpha\\in(0,\\infty)$. \n\\begin{enumerate}[(a)]\n\\item \n\\label{item:L39.3}%\nIf $X$ has balanced tails in the sense of~\\eqref{eq:3.15}, then \n\\begin{equation}\n\\label{eq:3.17}\n\\frac{\\gamma_\\xi(X)}{\\gamma_{e_1}(X) + \\gamma_{-e_1}(X)} = \\Psi^\\ast g_{\\xi,\\alpha}\n\\ldotp\n\\end{equation}\n\\item\n\\label{item:L39.1}%\nThe non-degeneracy condition~\\eqref{eq:4} is equivalent to \nthe existence of a vector $w\\in(0,\\infty)^d$ \nsuch that $wX$ has balanced tails.  \n\\item\n\\label{item:L39.2}%\nThe extreme risk index $\\gamma_\\xi$ of the rescaled vector $wX$ obtained \nin part~(\\ref{item:L39.1})  satisfies\n\\begin{equation}\n\\label{eq:3.18}\n\\frac{\\gamma_\\xi(wX)}{\\gamma_{e_1}(wX) + \\gamma_{-e_1}(wX)} = \\Psiast_X g_{\\xi,\\alpha}\n\\ldotp\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\\par\n\\begin{myproof}\nPart~(\\ref{item:L39.3}). \nConsider the integrand $g_{\\xi,\\alpha}(vs)$ in the representation~\\eqref{eq:3.9}:\n\\[\ng_{\\xi,\\alpha}(vs)\n=\n\\robrfl{\\sum_{i=1}^{d} \\xi^{(i)} \\cdot \n\\robrfl{\\robrfl{v^{(i)} s^{(i)}}_{+}^{1\/\\alpha} - \\robrfl{v^{(i)} s^{(i)}}_{-}^{1\/\\alpha}}\n}_{+}^{\\alpha}\n\\ldotp\n\\]\nThe balanced tails condition~\\eqref{eq:3.15} implies that $X$ is \nnon-degenerate in the sense of~\\eqref{eq:4}. \nFurthermore, all weights $v^{(i)}$ in the representation~\\eqref{eq:3.9} \nare equal: \n\\begin{align*}\n1\n&=\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}}>t} \/ \\mathrm{P}\\cubrfl{\\normfl{X}_1 >t}} \n{\\mathrm{P}\\cubrfl{\\absfl{X^{(j)}}>t} \/ \\mathrm{P}\\cubrfl{\\normfl{X}_1 >t}}\n=\n\\frac{\\gamma_{\\ei}(X) + \\gamma_{-\\ei}(X)}{\\gamma_{\\ej}(X) + \\gamma_{-\\ej}(X)}\\\\\n&=\n\\frac{v^{(i)}}{v^{(j)}}\n,\\quad i,j\\in\\cubr{1,\\ldots,d}\n\\ldotp\n\\end{align*}\nHence $g_{\\xi,\\alpha}(vs)$ simplifies to   \n\\begin{align*}\ng_{\\xi,\\alpha}(vs)\n&=\nv^{(1)} g_{\\xi,\\alpha}(s)\\\\\n&=\n\\robrfl{\\gamma_{e_1}(X) + \\gamma_{-e_1}(X)} g_{\\xi,\\alpha}(s)\n\\ldotp\n\\end{align*} \n\\par\nPart~(\\ref{item:L39.1}). \nSuppose that $X$ satisfies~\\eqref{eq:4}. Then the sets $B_i$ defined in~\\eqref{eq:3.8} satisfy $\\nu(B_i)>0$ for $i=1,\\ldots,d$. Consequently, the random variables $\\abs{X^{(i)}}$ are regularly varying with tail index $\\alpha$. Denoting \n\\begin{equation}\n\\label{eq:3.19}\nw^{(i)}:=\\robr{\\nu(B_i)}^{-1\/\\alpha},\n\\end{equation}\none obtains \n\\begin{align*}\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\absfl{w^{(i)} X^{(i)}} > t}}{\\mathrm{P}\\cubrfl{\\norm{X}_1>t}} \n&=\n\\lim_{t\\to\\infty}\n\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t\/w^{(i)}}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t}}{\\mathrm{P}\\cubrfl{\\norm{X}_1>t}} \n}\\\\\n&= \\robrfl{w^{(i)}}^{\\alpha}\\cdot\\nu(B_i)\\\\\n&= 1\n\\end{align*}\nfor $i=1,\\ldots,d$. Hence, for any $i,j\\in\\cubr{1,\\ldots,d}$,\n\\begin{align*}\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{w^{(i)} X^{(i)}} > t}}{\\mathrm{P}\\cubrfl{\\absfl{w^{(j)} X^{(j)}} > t}}\n&=\n1\n\\ldotp\n\\end{align*}\n\\par\nTo prove the inverse implication, suppose that $Z:=wX$ has balanced tails \nfor some $w\\in(0,\\infty)^d$. Then the the exponent measure $\\nu$ of $X$\nsatisfies \n\\begin{align*}\n\\frac{\\nu(B_i)}{\\nu(B_1)}\n&=\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}} > t}}\\\\\n&=\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\absfl{Z^{(i)}}>w^{(i)} t}}{\\mathrm{P}\\cubrfl{\\absfl{Z^{(1)}} > w^{(1)} t}}\\\\\n&=\n\\lim_{t\\to\\infty}\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Z^{(i)}}> w^{(i)} t}}{\\mathrm{P}\\cubrfl{\\absfl{Z^{(i)}} >t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Z^{(1)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{Z^{(1)}} > w^{(1)} t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Z^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{Z^{(1)}} >t}}\n}\\\\\n&=\n\\robrfl{\\frac{w^{(i)}}{w^{(1)}}}^{-\\alpha}\n\\in(0,\\infty)\n,\\quad i\\in\\cubrfl{1,\\ldots,d}\n\\ldotp\n\\end{align*}\nSince multivariate regular variation of $X$ implies $\\nu(B_j)>0$ for at least \none index $j\\in\\cubr{1,\\ldots,d}$, this yields $\\nu(B_i)>0$ for all $i$.\n\\par\n\nPart~(\\ref{item:L39.2}). This is an immediate consequence of \npart (\\ref{item:L39.3}) and the invariance of canonical spectral measures \nunder componentwise rescaling.\n\\end{myproof}\n\\par\nRepresentation~\\eqref{eq:3.17} suggests that ordering of \nthe normalized extreme risk indices $\\gamma_\\xi\/(\\gamma_{e_1} + \\gamma_{-e_1})$ \nin the balanced tails setting can be considered \nas an \\emph{integral order relation} \nfor canonical spectral measures with respect to the function class\n\\begin{equation}\\label{eq:3.20}\n\\Gcal_{\\alpha}:=\\cubrfl{g_{\\xi,\\alpha}:\\xi\\in\\Simp^d}\n\\ldotp\n\\end{equation}\nThis justifies the following definition.\n\\par\n\\begin{definition}\n\\label{def:3.4}\nLet $\\Psi^\\ast$ and $\\Phi^\\ast$ be canonical spectral measures on $\\Sbb^d_1$\nand let $\\alpha>0$. \nThen the order relation $\\Psi^\\ast \\mathrel{\\preceq_{\\Gcalalpha}} \\Phi^\\ast$ is defined by\n\\begin{equation}\\label{eq:3.21}\n\\forall g\\in\\Gcal_{\\alpha}\n\\quad\n\\Psi^\\ast g \\le\\Phi^\\ast g\n\\ldotp\n\\end{equation}\n\\end{definition}\n\\par\n\\begin{remark}\n\\label{rem:3.1}\n\\begin{enumerate}[(a)]\n\\item\n\\label{item:r14.1}%\nFor $\\alpha=1$ and spectral measures on $\\Simp^d$ the extreme risk index \n$\\gamma_\\xi(X)$ is linear in $\\xi$ \\citep[cf.][Lemma~3.2]{Mainik\/Rueschendorf:2010}.\nConsequently, $\\mathrel{\\preceq_{\\Gcalalpha}}$ is indifferent in this case, \ni.e., \nany $\\Psi^\\ast$ and $\\Phi^\\ast$ on $\\mathcal{B}\\robr{\\Simp^d}$ satisfy \n\\begin{equation}\n\\label{eq:3.22}\n\\Psi^\\ast \\mathrel{\\preceq}_{\\mathcal{G},1} \\Phi^\\ast \n\\quad\\text{and}\\quad\n\\Phi^\\ast \\mathrel{\\preceq}_{\\mathcal{G},1} \\Psi^\\ast\n\\ldotp\n\\end{equation}\n\\item\n\\label{item:r14.2}%\nThe order relation $\\mathrel{\\preceq_{\\Gcalalpha}}$ is mixing invariant  \nin the sense that \nuniform ordering of two parametric families \n$\\cubr{\\Psi^\\ast_\\theta:\\theta\\in\\Theta}$ and \n$\\cubr{\\Phi^\\ast_\\theta:\\theta \\in\\Theta}$,  \n\\[\n\\forall \\theta\\in\\Theta\n\\quad\n\\Psi^\\ast_\\theta\\mathrel{\\preceq_{\\Gcalalpha}}\\Phi^\\ast_\\theta\n,\n\\]\nimplies \n\\[\n\\int_\\Theta\\Psi^\\ast_\\theta \\, \\mathrm{d} \\mu(\\theta) \n\\mathrel{\\preceq_{\\Gcalalpha}} \n\\int_\\Theta\\Phi^\\ast_\\theta \\, \\mathrm{d} \\mu(\\theta)\n\\]\nfor any probability measure $\\mu$ on $\\Theta$. \n\\end{enumerate}\n\\end{remark}\n\\par\nThe following theorem states that $\\mathrel{\\preceq_{\\mathrm{apl}}}$ is in a certain sense  \nequivalent to the ordering of canonical spectral measures \nand allows to reduce the verification of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ to the verification \nof $\\mathrel{\\preceq_{\\Gcalalpha}}$. \nSome exemplary applications are given in  Section~\\ref{sec:5}.    \nFurthermore, given explicit representations of spectral measures or their \ncanonical versions,  \nthis result allows to verify $\\mathrel{\\preceq_{\\mathrm{apl}}}$ numerically, which is very \nuseful in practice. \n\\par\n\\begin{theorem}\n\\label{theo:3.4}\nLet $X$ and $Y$ be multivariate regularly varying random vectors on $\\R^{d}$ with tail index $\\alpha\\in(0,\\infty)$ and canonical  spectral measures $\\Psiast_X$ and $\\Psiast_Y$. Further, suppose that $X$ and $Y$ satisfy the balanced tails condition~\\eqref{eq:3.15}. \n\\begin{enumerate}[(a)]\n\\item\n\\label{item:t4.1}%\nIf $\\absfl{X^{(1)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\absfl{Y^{(1)}}$,\nthen $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$ implies $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\vspace{0.5em\n\\item\n\\label{item:t4.2}%\nIf \n$\\absfl{X^{(1)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\absfl{Y^{(1)}}$ \nand \n$\\absfl{Y^{(1)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\absfl{X^{(1)}}$,\nthen \n$\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$ is equivalent to  $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\end{enumerate}\n\\end{theorem}\n\\par\n\\begin{myproof}\n(\\ref{item:t4.1})\nSince $X$ has balanced tails, Proposition \\ref{prop:3.3}(\\ref{item:L39.3}) yields\n\\begin{align*}\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\n&=\n\\lim_{t\\to\\infty}\n\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\norm{X}_1>t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\norm{X}_1>t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\n}\\\\\n&=\n\\frac{\\gamma_\\xi(X)}{\\gamma_{e_1}(X) + \\gamma_{-e_1}(X)}\\\\\n&=\n\\Psiast_X g_{\\xi,\\alpha}\n\\ldotp\n\\end{align*}\nAnalogously one obtains\n\\[\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}\n=\n\\Psiast_Y g_{\\xi,\\alpha}\n\\ldotp\n\\]\nMoreover, $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$ implies \n\\begin{equation}\n\\label{eq:3.23}\n\\frac{\\PsiastXg_{\\xi,\\alpha}}{\\PsiastYg_{\\xi,\\alpha}}\n\\le\n1\n\\ldotp\n\\end{equation}\nConsequently, \n\\begin{align}\n\\hspace{2em}&\\hspace{-2em}\\nonumber\n\\limsup_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}\\\\\n&=\\nonumber\n\\limsup_{t\\to\\infty}\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}\n}\\\\\n&=\\label{eq:3.24}\n\\frac{\\PsiastXg_{\\xi,\\alpha}}{\\PsiastYg_{\\xi,\\alpha}}\n\\cdot\n\\limsup_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}}>t}}\\\\\n&\\le\\nonumber\n1\n\\end{align}\ndue to~\\eqref{eq:3.23} and $\\abs{X^{(i)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\abs{Y^{(i)}}$.\n\\par \n\\medskip\n\\noindent %\n(\\ref{item:t4.2})\nBy part~(\\ref{item:t4.1}), it suffices to show that \n$X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$ implies $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$. \nBy assumption $\\abs{X^{(1)}}$ and $\\abs{Y^{(1)}}$ have asymptotically equivalent tails,\n\\[\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}} >t}} \n= \n1\n\\ldotp\n\\]\nThus~\\eqref{eq:3.24} yields\n\\[\n\\frac{\\PsiastXg_{\\xi,\\alpha}}{\\PsiastYg_{\\xi,\\alpha}}\n=\n\\limsup_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}\n\\]\nand $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$ implies $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$.\n\\end{myproof}\n\\par\nThe following result answers the question for dependence structures \ncorresponding to the best and the worst possible diversification effects \nfor multivariate regularly varying random vectors in $\\Rplus^{d}$.\nAccording to Theorem~\\ref{theo:3.4}, it suffices to find the upper and \nthe lower elements with respect to $\\mathrel{\\preceq_{\\Gcalalpha}}$ in the set \nof all canonical spectral measures on $\\Simp^d$. \nIt turns out that for $\\alpha > 1$  \nthe best diversification effects are obtained in case of asymptotic \nindependence, i.e., the $\\mathrel{\\preceq_{\\Gcalalpha}}$-maximal element is given by   \n\\begin{equation}\\label{eq:apl.8}\n\\Psi^\\ast_0 := \\sum_{i=1}^{d} \\Dirac{e_i},\n\\end{equation}\nwhereas the worst diversification effects are obtained in case of \nthe asymptotic comonotonicity, represented by \n\\begin{equation}\\label{eq:apl.9}\n\\Psi^\\ast_1 := d \\cdot \\Dirac{(1\/d,\\ldots,1\/d)}\n\\ldotp\n\\end{equation}\nFor $\\alpha < 1$ the situation is inverse. \n\\begin{theorem}\\label{thm:3.8}\nLet $\\Psi^\\ast$ be an arbitrary canonical spectral measure on $\\Simp^d$ and let \n$\\Psi^\\ast_0$ and $\\Psi^\\ast_1$ be defined according to~\\eqref{eq:apl.8} \nand~\\eqref{eq:apl.9}. Then\n\\begin{enumerate}[(a)]\n\\item\\label{item:thm:3.8.a}\n$\\Psi^\\ast_0  \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_1$ \nfor $\\alpha \\ge 1$.\n\\vspace{0.5em}\n\\item\\label{item:thm:3.8.b}\n$\\Psi^\\ast_1  \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_0$ \nfor $\\alpha \\in (0,1]$. \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nLet $X$ be multivariate regularly varying on $\\Rplus^{d}$ with canonical spectral \nmeasure $\\Psi^\\ast$. Without loss of generality we can assume that $X$ \nsatisfies the balanced tails condition~\\eqref{eq:3.15}. \nThen, according to~\\eqref{eq:3.17}, we have \n\\begin{equation}\n\\label{eq:apl.6}\n\\Psiastg_{\\xi,\\alpha}=\\frac{\\gamma_\\xi(X)}{\\gamma_{e_1}(X)}\n\\ldotp\n\\end{equation}\nFurthermore, \nwe have $\\Psi^\\ast g_{e_i,\\alpha} = 1$ for $i=1,\\ldots,d$.  \nRecall that the mapping $\\xi\\mapsto\\gamma_\\xi$ is convex for $\\alpha \\ge 1$ \n\\citep[cf.][Lemma~3.2]{Mainik\/Rueschendorf:2010}. \nDue to~\\eqref{eq:apl.6} this behaviour is inherited by the mapping \n$\\xi\\mapsto\\Psiastg_{\\xi,\\alpha}$. \nThus for $\\alpha \\ge 1$  \nwe have $\\Psiastg_{\\xi,\\alpha} \\le 1 = \\Psi^\\ast_1g_{\\xi,\\alpha}$ for all $\\xi\\in\\Simp^d$, \nwhich exactly means $\\Psi^\\ast \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_1$ for $\\alpha \\ge 1$. \n\\par\nTo complete the proof of part~(\\ref{item:thm:3.8.a}), note that \nthe normalization of canonical spectral measures yields \n\\begin{equation}\\label{eq:apl.7}\n\\forall\\xi\\in\\Simp^d\n\\quad\n\\Psi^\\ast_0g_{\\xi,\\alpha} \n= \n\\sum_{i=1}^{d}\\robrfl{\\xi^{(i)}}^{\\alpha}\n=\n\\int_{\\Simp^d} \\sum_{i=1}^{d} \\robrfl{\\xi^{(i)}}^{\\alpha} s ^{(i)}  \n\\,\\Psi^\\ast(\\mathrm{d} s)\n\\end{equation}\nComparing the integrand on the right side of \\eqref{eq:apl.7} with \nthe function $g_{\\xi,\\alpha}(s)=\\robr{\\xi^{\\top} s^{1\/\\alpha}}^{\\alpha}$, \nwe see that \n\\[\n\\sum_{i=1}^{d} \\robrfl{\\xi^{(i)}}^{\\alpha} s^{(i)}\n=\ng_{\\xi,\\alpha}(s)\n\\cdot\n\\sum_{i=1}^{d} z_i^{\\alpha}\n\\]\nwith\n\\[\nz_i := \\frac{\\xi^{(i)} \\cdot\\robrfl{s^{(i)}}^{1\/\\alpha}} \n{\\xi^{\\top} s^{1\/\\alpha}}\n\\ldotp\n\\]\nThus it suffices to demonstrate that $\\sum_{i=1}^{d} z_i^{\\alpha} \\le 1$, \nwhich follows from $z_i\\in[0,1]$, $z_i^{\\alpha} \\le z_i$ for $\\alpha \\ge 1$, and \n$\\sum_{i=1}^{d} z_i=1$.\n\\par\nThe inverse result for $\\alpha\\in(0,1]$ stated in~(\\ref{item:thm:3.8.b})  \nfollows from \nthe concavity of the mapping $\\xi\\mapsto\\Psiastg_{\\xi,\\alpha}$ \nand the inequality $z_i^{\\alpha} \\ge z_i$.  \n\\end{proof}\n\\par\nDue to Theorem~\\ref{theo:3.4}, an analogue of the foregoing \nresult for $\\mathrel{\\preceq_{\\mathrm{apl}}}$ is straightforward.\n\\begin{corollary}\\label{cor:3.10}\nLet $X$ be multivariate regularly varying in $\\Rplus^{d}$ with tail index \n$\\alpha\\in(0,\\infty)$ and identically \ndistributed margins $X^{(i)}\\sim F$, $i=1,\\ldots,d$.\nFurther, let $Y$ be a random vector with independent margins \n$Y ^{(i)}\\sim F$, and let $Z$ be a random vector with totally dependent \nmargins $Z^{(i)}=Z^{(1)}$ $\\mathrm{P}$-a.s.\\ and $Z^{(1)}\\sim F$. \nThen\n\\begin{enumerate}[(a)]\n\\item\n$Y \\mathrel{\\preceq_{\\mathrm{apl}}} X \\mathrel{\\preceq_{\\mathrm{apl}}} Z$ for $\\alpha \\ge 1$\n\\item\n$Z \\mathrel{\\preceq_{\\mathrm{apl}}} X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$ for $\\alpha \\in (0,1]$.\n\\end{enumerate}\n\\end{corollary}\n\\par\n\\begin{remark}\nThe strict assumptions of Corollary~\\ref{cor:3.10} are chosen for clearness and simplicity.\nThe independence of $Y^{(i)}$ and the total dependence of $Z^{(i)}$ \nare needed only in the tail region, i.e., it suffices for $Y$ and $Z$ \nto be multivariate regularly varying with canonical spectral measures \n$\\Psi^\\ast_0$ and $\\Psi^\\ast_1$, respectively.\nFurthermore, the assumption of identically distributed margins \ncan be replaced by equivalent tails:\n\\[\n1=\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{Y^{(i)} >t}}{\\mathrm{P}\\cubrfl{X^{(i)} >t}}\n=\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{Z^{(i)} >t}}{\\mathrm{P}\\cubrfl{X^{(i)} >t}}\n,\\quad i=1,\\ldots,d\n\\ldotp\n\\]\nFinally, the non-negativity of $X^{(i)}$, $Y^{(i)}$, and $Z^{(i)}$  \nis needed only in the asymptotic sense. The ordering results remain true \nif the spectral measure of $X$ is restricted to the unit simplex $\\Simp^d$.  \n\\end{remark}\n\\par\nCombining Theorem~\\ref{theo:3.4} with Theorem~\\ref{theo:2.4}, one obtains an \nordering result for the canonical spectral measures of multivariate regularly \nvarying elliptical distributions. \nThe notation $\\Psi^\\ast=\\Psi^\\ast(\\alpha,C)$ is justified by the fact that \nspectral measures of elliptical distributions depend only on  the tail \nindex $\\alpha$ and the generalized covariance matrix $C$. \nAn explicit representation of spectral densities for bivariate elliptical \ndistributions was obtained by \\citet{Hult\/Lindskog:2002}. \nAlternative representations that are valid for all dimensions $d\\ge2$ \nare given in \\citet{Mainik:2010}, Lemma~2.8.\n\\par\n\\begin{proposition}\n\\label{prop:3.7}\nLet $C$ and $D$ be $d$-dimensional covariance matrices satisfying \n\\begin{equation}\n\\label{eq:3.25}\nC_{i,i} = D_{i,i} > 0\n,\\quad\ni = 1,\\ldots,d,\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:3.26}\n\\forall \\xi\\in\\Simp^d \n\\quad\n\\xi^{\\top} C \\xi \\le \\xi^{\\top} D \\xi\n\\ldotp\n\\end{equation}\nThen \n\\[\n\\forall \\alpha>0 \n\\quad\n\\Psi^\\ast\\robr{\\alpha,C} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast\\robr{\\alpha,D}\n\\ldotp\n\\]\n\\end{proposition}\n\\par\n\\begin{myproof}\nFix $\\alpha\\in(0,\\infty)$ and consider random vectors \n\\[\nX\\mathrel{\\stackrel{\\mathrm{d}}{=}} R A U\n,\\quad\nY\\mathrel{\\stackrel{\\mathrm{d}}{=}} R  B  U, \n\\]\nwhere $A$ and $B$ are square roots of the matrices $C$ and $D$ \nin~\\eqref{eq:3.26}, i.e., \n\\[ \nC=A  A^{\\top}\n,\\quad \nD=B B^{\\top},\n\\]\nand $R$ is an arbitrary regularly varying non-negative \nrandom variable with tail index $\\alpha$.\n\\par\nAs a consequence of Theorem \\ref{theo:2.4} one obtains $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$. \nFurthermore, invariance of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ under componentwise rescaling \nyields $wX \\mathrel{\\preceq_{\\mathrm{apl}}} wY$ for $w=\\robr{w^{(1)},\\ldots,w^{(d)}}$ with   \n\\[\nw^{(i)}:={C_{i,i}}^{-1\/2}={D_{i,i}}^{-1\/2}\n,\\quad \ni=1,\\ldots,d\n\\ldotp\n\\]\nMoreover, as a particular consequence of arguments \nunderlying \\eqref{eq:2.17}, one obtains \n\\[\nw^{(i)} X^{(i)} \\mathrel{\\stackrel{\\mathrm{d}}{=}} w^{(j)} Y^{(j)}\n,\\quad\ni,j\\in\\cubr{1,\\ldots,d}\n\\ldotp\n\\]\nHence the random vectors $wX$ and $wY$ satisfy the balanced tails condition \\eqref{eq:3.15}, whereas their components are mutually ordered with respect to $\\mathrel{\\preceq_{\\mathrm{apl}}}$.\nFinally, Theorem~\\ref{theo:3.4}(\\ref{item:t4.2}) and invariance of canonical spectral measures under componentwise rescalings yield\n\\begin{equation*}\n\\Psi^\\ast(\\alpha,C) = \\Psi^\\ast_{wX} \n\\mathrel{\\preceq_{\\Gcalalpha}} \n\\Psi^\\ast_{wY} = \\Psi^\\ast(\\alpha,D)\n\\ldotp\n\\qed\n\\end{equation*}\n\\end{myproof}\n\\par\nThe subsequent result extends Theorem~\\ref{theo:3.4} to random vectors that do not have balanced tails.\n\\par\n\\begin{theorem}\n\\label{theo:3.8}\nLet $X$ and $Y$ be multivariate regularly varying random vectors on $\\R^{d}$ \nwith tail index $\\alpha\\in(0,\\infty)$  \nand canonical spectral measures $\\Psiast_X$ and $\\Psiast_Y$. \nFurther, assume that $\\abs{X^{(i)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\abs{Y^{(i)}}$ with \n\\begin{equation}\n\\label{eq:3.27}\n\\lambda_i\n:=\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}}>t}} \n\\in(0,1]\n\\end{equation}\nfor $i=1,\\ldots,d$  and that the vector $v=\\robr{v^{(1)},\\ldots,v^{(d)}}$ defined by\n\\begin{equation}\n\\label{eq:3.28}\nv^{(i)}:=\\lambda_i^{-1\/\\alpha}\n\\end{equation} \nsatisfies \n\\begin{equation}\n\\label{eq:3.29}\nX \\mathrel{\\preceq_{\\mathrm{apl}}} v X\n\\quad\\text{or}\\quad\nv^{-1} Y \\mathrel{\\preceq_{\\mathrm{apl}}} Y\n\\ldotp\n\\end{equation}\nThen $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$ implies $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\end{theorem}\n\\par\n\\begin{myproof}\nAccording to Proposition~\\ref{prop:3.3}(\\ref{item:L39.1}), there exists $w\\in\\Rplus^{d}$ \nsuch that $wY$ satisfies the balanced tails condition~\\eqref{eq:3.15}. \nFurthermore, the tails of the random vector \n\\[\nvwX:=\\robrfl{v^{(1)} w^{(1)} X^{(1)} ,\\ldots, v^{(d)} w^{(d)} X^{(d)}}\n\\]\nwith $v$ defined in~\\eqref{eq:3.27} are also balanced. Indeed, it is easy to see that \n\\[\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{w^{(i)} Y^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}} >t}} \n=\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} w^{(i)} X^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} X^{(i)}} >t}} \n=\n\\robrfl{w^{(i)}}^{\\alpha}\n\\]\nfor $i=1,\\ldots,d$. Analogously one obtains \n\\[\n\\lim_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} X^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} >t}} \n=\n\\robrfl{v^{(i)}}^{\\alpha} = \\lambda_i^{-1}\n\\]\nand, as a result,\n\\begin{align*}\n\\hspace{2em}&\\hspace{-2em}\n\\lim_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} w^{(i)} X^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{w^{(i)} Y^{(i)}} > t}}\\\\\n&=\n\\lim_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} X^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}} > t}}\\\\\n&=\n\\lim_{t\\to\\infty}\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{v^{(i)} X^{(i)}} >t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}} > t}}\n}\\\\\n&=\n\\lambda_i^{-1} \n\\cdot\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}} > t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}} > t}}\\\\\n&=\n1\n\\end{align*}\nfor $i=1,\\ldots,d$. Hence the balanced tails condition for $wY$ implies that the tails of $vwX$ are also balanced.\n\\par\nFurthermore, invariance of canonical spectral measures under \ncomponentwise rescaling yields\n\\[\n\\Psi^\\ast_{vwX} = \\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y = \\Psi^\\ast_{wY}\n\\ldotp\n\\]\nThus, applying Theorem~\\ref{theo:3.4}(\\ref{item:t4.1}), one obtains\n\\begin{equation}\n\\label{eq:3.30}\nvwX \\mathrel{\\preceq_{\\mathrm{apl}}} wY\n\\ldotp\n\\end{equation}\nSince $v^{(i)}=\\lambda_i^{-1\/\\alpha} > 0$ for $i=1,\\ldots,d$, \ncondition~\\eqref{eq:3.30} is equivalent to \n\\begin{equation}\n\\label{eq:3.31}\nwX \\mathrel{\\preceq_{\\mathrm{apl}}} v^{-1} wY\n\\ldotp\n\\end{equation}\nMoreover, assumption~\\eqref{eq:3.29} implies\n\\begin{equation}\n\\label{eq:3.32}\nwX \\mathrel{\\preceq_{\\mathrm{apl}}} vwX\n\\quad\\text{or}\\quad \nv^{-1}wY\\mathrel{\\preceq_{\\mathrm{apl}}} wY.\n\\end{equation}\nCombining this ordering statement \nwith \\eqref{eq:3.30} and~\\eqref{eq:3.31}, \none obtains \n\\[\nwX \\mathrel{\\preceq_{\\mathrm{apl}}} wY\n\\ldotp\n\\]\nFinally, \ninvariance of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ with respect to componentwise rescaling yields \n$X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\end{myproof}\n\\par\nIn the special case of random vectors in $\\Rplus^{d}$ Theorem~\\ref{theo:3.8} \ncan be simplified to the following result. \n\\par\n\\begin{corollary}\n\\label{cor:3.9}  \nLet $X$ and $Y$ be multivariate regularly varying random vectors on $\\Rplus^{d}$ \nwith tail index $\\alpha\\in(0,\\infty)$ and canonical \nspectral measures $\\Psiast_X$ and $\\Psiast_Y$. \nFurther, suppose that \n\\begin{equation}\n\\label{eq:3.33}\n\\lambda_i\n:=\n\\limsup_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(i)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(i)}}>t}} \\in (0,1], \n\\quad\ni = 1,\\ldots,d\n\\ldotp\n\\end{equation}\nThen $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$ implies $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.\n\\end{corollary}\n\\par\n\\begin{myproof}\nAssumption~\\eqref{eq:3.33} yields that the rescaling vector $v$ \ndefined in \\eqref{eq:3.28} is an element of $[1,\\infty)^d$. \nThus $v-(1,\\ldots,1)\\in\\Rplus^{d}$ and, since $X$ takes values in $\\Rplus^{d}$, \nwe have \n\\[\nX \\mathrel{\\preceq_{\\mathrm{apl}}} X + \\robr{v - (1,\\ldots,1)}X = vX\n\\ldotp\n\\]\nSimilar arguments yield $v^{-1} Y\\mathrel{\\preceq_{\\mathrm{apl}}} Y$. \nHence condition~\\eqref{eq:3.29} of Theorem~\\ref{theo:3.8} is satisfied.\n\\end{myproof}\n\\par\nThe final result of this section is due to the indifference of \n$\\mathrel{\\preceq_{\\Gcalalpha}}$ for $\\alpha=1$ mentioned  in \nRemark~\\ref{rem:3.1}(\\ref{item:r14.1}).\nThis special property of spectral measures on $\\Simp^d$ allows to reduce \n$\\mathrel{\\preceq_{\\mathrm{apl}}}$ to the ordering of components. \nIt should be noted that this result cannot be extended to the \ngeneral case of spectral measures on $\\Sbb^d_1$. \n\\par\n\\begin{lemma}\n\\label{lem:3.10}\nLet $X$ and $Y$ be multivariate regularly varying on $\\Rplus^{d}$ with tail \nindex $\\alpha=1$. \nFurther, suppose that $Y$ satisfies the non-degeneracy \ncondition~\\eqref{eq:4} and that \n$X^{(i)} \\mathrel{\\preceq_{\\mathrm{apl}}} Y^{(i)}$ for $i=1,\\ldots,d$. Then $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$.   \n\\end{lemma}\n\\par\n\\begin{myproof}\nAccording to Proposition \\ref{prop:3.3}(\\ref{item:L39.1}), \nthere exists $w\\in(0,\\infty)^d$ such that $wY$ satisfies the balanced tails \ncondition~\\eqref{eq:3.15}. \nFurthermore, due to the invariance of $\\mathrel{\\preceq_{\\mathrm{apl}}}$ under componentwise \nrescaling, $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$ is equivalent to $wX \\mathrel{\\preceq_{\\mathrm{apl}}} wY$.  \n\\par\nThus it can be assumed without loss of generality that $Y$ has balanced tails. \nThis yields \n\\[\n\\lambda_i\n:=\n\\limsup_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{X^{(i)} >t}}{\\mathrm{P}\\cubrfl{Y^{(i)} >t}}\n=\n\\limsup_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{X^{(i)} >t}}{\\mathrm{P}\\cubrfl{Y^{(1)} >t}}\n,\\quad\ni=1,\\ldots,d\n\\ldotp\n\\]\nHence the  assumption $X^{(i)} \\mathrel{\\preceq_{\\mathrm{apl}}} Y^{(i)}$ for $i=1,\\ldots,d$ \nimplies $\\lambda_i\\in[0,1]$ for all $i$.\nMoreover, the balanced tails condition for $Y$ yields\n\\begin{equation}\n\\label{eq:3.34}\n\\gamma_{e_1}(Y)=\\ldots=\\gamma_{e_d}(Y)\n\\ldotp\n\\end{equation}\n\\par\n\nNow consider the random vector $X$ and denote \n\\[\nj:=\\mathop{\\mathrm{arg\\,max}}\\displaylimits_{i\\in\\cubr{1,\\ldots,d}} \\gamma_{\\ei}(X)\n\\ldotp\n\\]\nRecall that $\\gamma_{\\ei}(X)=\\nu_X\\robr{\\cubr{x\\in\\Rplus^{d}: x^{(i)}>1}}$ \nwith $\\nu_X$ \ndenoting the exponent \nmeasure of $X$ and that $\\nu_X$ is non-zero. This yields $\\gamma_{\\ej}(X)>0$ \neven if $X$ does not satisfy the non-degeneracy condition~\\eqref{eq:4}. \nMoreover, for $\\alpha=1$, the mapping \n$\\xi\\mapsto\\gamma_\\xi(X)$ is linear. This implies\n\\begin{equation}\n\\label{eq:3.35}\n\\gamma_\\xi(X)\n=\n\\sum_{i=1}^{d} \\xi^{(i)} \\cdot  \\gamma_{\\ei}(X)\n\\le \n\\gamma_{\\ej}(X)\n,\\quad \n\\xi\\in\\Simp^d\n\\end{equation}\nand \\eqref{eq:3.34} yields\n\\begin{equation}\n\\label{eq:3.36}\n\\gamma_\\xi(Y) = \\sum_{i=1}^{d} \\xi^{(i)} \\cdot \\gamma_{\\ei}(Y) = \\gamma_{e_1}(Y)\n,\\quad\n\\xi\\in\\Simp^d\n\\ldotp\n\\end{equation}\nHence\n\\begin{align*}\n\\hspace{2em}&\\hspace{-2em}\n\\limsup_{t\\to\\infty}\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}\\\\\n&=\n\\limsup_{t\\to\\infty}\\robrfl{\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t}}{\\mathrm{P}\\cubrfl{X^{(j)} > t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{X^{(j)} > t}}{\\mathrm{P}\\cubrfl{Y^{(1)} > t}} \n\\cdot\n\\frac{\\mathrm{P}{\\cubrfl{Y^{(1)} > t}}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y >t}}\n}\\\\\n&=\n\\frac{\\gamma_\\xi(X)}{\\gamma_{\\ej}(X)}\n\\cdot\n\\lambda_j\n\\cdot\n\\frac{\\gamma_{e_1}(Y)}{\\gamma_\\xi(Y)} \\\\\n&\\le\n1\n\\end{align*}\ndue to $\\lambda_j\\le 1$, \\eqref{eq:3.35}, and \\eqref{eq:3.36}.\n\\end{myproof}\n\\section{Relations to convex and supermodular orders}\\label{sec:4}\nAs mentioned in Remark~\\ref{rem:apl.1}(\\ref{item:apl.2}), \ndependence orders $\\mathrel{\\preceq_{\\mathrm{sm}}}$, $\\mathrel{\\preceq_{\\mathrm{dcx}}}$ \nand convexity orders $\\mathrel{\\preceq_{\\mathrm{cx}}}$, $\\mathrel{\\preceq_{\\mathrm{icx}}}$, $\\mathrel{\\preceq_{\\mathrm{plcx}}}$ \ndo not imply $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in general. \nHowever, it turns out that the relationship between $\\mathrel{\\preceq_{\\mathrm{apl}}}$ and\nthe ordering of canonical spectral measures by $\\mathrel{\\preceq_{\\Gcalalpha}}$ allows \nto draw conclusions of this type  \nin the special case of multivariate regularly varying models.  \nThe core result of this section is stated in Theorem~\\ref{thm:5}. \nIt entails \na collection of sufficient \ncriteria for $\\mathrel{\\preceq_{\\mathrm{apl}}}$ in terms of convex and supermodular order relations,   \nwith particular interest paid to the \ninversion of diversification effects for $\\alpha<1$.   \nAn application to copula based models is given in Proposition~\\ref{prop:4.4}.\n\\par \nThis approach was applied by \\citet{Embrechts\/Neslehova\/Wuethrich:2009} \nto the ordering of risks for the portfolio vector \n$\\xi=(1,\\ldots,1)$ and for a specific family of multivariate \nregularly varying models with identically distributed, non-negative margins \n$X^{(i)}$ (cf. Example~\\ref{ex:2} in Section~\\ref{sec:5}). \n\\par\nThe next theorem is the core element of this section. \nIt generalizes the arguments of \\citet{Embrechts\/Neslehova\/Wuethrich:2009} \nto multivariate regularly varying random vectors in $\\R^{d}$ with balanced \ntails and tail index $\\alpha\\ne 1$. \nThe case $\\alpha=1$ is not included for two reasons.\nFirst, this case is partly trivial due to the indifference of $\\mathrel{\\preceq_{\\Gcalalpha}}$ \nfor spectral measures on $\\Simp^d$ (cf.\\ Remark~\\ref{rem:3.1}(\\ref{item:r14.1})).\nSecond, Karamata's theorem  used in the proof of the integrable case $\\alpha>1$ does not yield the desired result for random variables with tail index $\\alpha=1$. \n\\par\n\\begin{theorem}\n\\label{thm:5}\nLet $X$ and $Y$ be multivariate regularly varying on $\\R^{d}$ with identical  \ntail index $\\alpha\\ne 1$. Further, assume that $X$ and $Y$ satisfy \nthe balanced tails condition~\\eqref{eq:3.15}.  \n\\begin{enumerate}[(a)]\n\\item\n\\label{item:t5.1}\nFor $\\alpha>1$ let\n\\begin{equation}\n\\label{eq:309}\n\\limsup_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}\n= 1\n\\end{equation}\nand let there exist $u_0>0$ such that with $h_u(t):=\\robr{t-u}_{+}$ \n\\begin{equation}\n\\label{eq:292a}\n\\forall u \\ge u_0 \\ \\forall \\xi\\in\\Simp^d \n\\quad\n\\mathrm{E}  h_u\\robrfl{\\xi^{\\top} X} \n\\le \n\\mathrm{E}  h_u\\robrfl{\\xi^{\\top} Y}\n\\ldotp \n\\end{equation}\nThen $\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$. \n\\vspace{0.5em}\n\\item\n\\label{item:t5.2}\nFor $\\alpha<1$ suppose that $\\abs{X^{(1)}}$ and $\\abs{Y^{(1)}}$ are \nequivalent with respect to $\\mathrel{\\preceq_{\\mathrm{apl}}}$, i.e.,\n\\begin{equation}\n\\label{eq:310a}\n\\absfl{X^{(1)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\absfl{Y^{(1)}}\n\\quad\\text{and}\\quad \n\\absfl{Y^{(1)}} \\mathrel{\\preceq_{\\mathrm{apl}}} \\absfl{X^{(1)}}, \n\\end{equation}\nand let there exist $u_0 >0$ such that with $f_u(t):=-(t \\wedge u)$, \n\\begin{equation}\n\\label{eq:292b}\n\\forall u \\ge u_0 \\ \\forall \\xi\\in\\Simp^d \n\\quad\n\\mathrm{E} f_u\\robrfl{\\robrfl{\\xi^{\\top} X}_{+}} \n\\le \n\\mathrm{E} f_u \\robrfl{\\robrfl{\\xi^{\\top} Y}_{+}}\n\\ldotp \n\\end{equation}\nThen $\\Psiast_Y \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_X$. \n\\end{enumerate}\n\\end{theorem}\n\\par\nThe proof will be given after some conclusions and remarks. \nIn particular, it should be noted that the relation between \n$\\mathrel{\\preceq_{\\Gcalalpha}}$ and $\\mathrel{\\preceq_{\\mathrm{apl}}}$ established in Theorem~\\ref{theo:3.4} \nimmediately yields the following result.\n\\begin{corollary}\n\\label{cor:8}\n\\begin{enumerate}[(a)]  \n\\item \n\\label{item:c8.1}%\nIf random vectors $X$ and $Y$ satisfy conditions of Theorem~\\ref{thm:5}(\\ref{item:t5.1}), \nthen $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$;\n\\item\n\\label{item:c8.2}%\nIf $X$ and $Y$ satisfy conditions of Theorem~\\ref{thm:5}(\\ref{item:t5.2}), \nthen $Y \\mathrel{\\preceq_{\\mathrm{apl}}} X$.\n\\end{enumerate} \n\\end{corollary}\n\\par\nIt should also be noted that \nconditions \\eqref{eq:292a} and \\eqref{eq:292b} are asymptotic forms of \nthe increasing convex ordering $\\xi^{\\top} X \\mathrel{\\preceq_{\\mathrm{icx}}} \\xi^{\\top} Y$ \nand the decreasing convex ordering $\\xi^{\\top} X \\mathrel{\\preceq_{\\mathrm{decx}}} \\xi^{\\top} Y$, \nrespectively. \nThe consequences can be outlined as follows.  \n\\begin{remark}\n\\label{rem:12}\n\\begin{enumerate}[(a)]\n\\item \nThe following criteria are sufficient for~\\eqref{eq:292a} \nand \\eqref{eq:292b} to hold: \n\\begin{enumerate}[(i)]\n\\vspace{0.5em\n\\item \n$\\robr{\\xi^{\\top} X}_{+} \\mathrel{\\preceq_{\\mathrm{cx}}} \\robr{\\xi^{\\top} Y}_{+}$ \nfor all $\\xi\\in\\Simp^d$,\n\\item\n$X$ and $Y$ are restricted to $\\Rplus^{d}$ and \n$X \\mathrel{\\preceq} Y$ with $\\mathrel{\\preceq}$ denoting either \n$\\mathrel{\\preceq_{\\mathrm{plcx}}}$, $\\mathrel{\\preceq_{\\mathrm{lcx}}}$,  $\\mathrel{\\preceq_{\\mathrm{cx}}}$, $\\mathrel{\\preceq_{\\mathrm{dcx}}}$, or $\\mathrel{\\preceq_{\\mathrm{sm}}}$.\n\\end{enumerate}\n\\vspace{0.5em}\n\\item\nAdditionally, condition~\\eqref{eq:292a} follows from  \n$X \\mathrel{\\preceq} Y$ with $\\mathrel{\\preceq}$ denoting either \n$\\mathrel{\\preceq_{\\mathrm{plcx}}}$, $\\mathrel{\\preceq_{\\mathrm{lcx}}}$,  $\\mathrel{\\preceq_{\\mathrm{cx}}}$, $\\mathrel{\\preceq_{\\mathrm{dcx}}}$, or $\\mathrel{\\preceq_{\\mathrm{sm}}}$.\n\\end{enumerate}\n\\end{remark}\nFinally, a comment should be made upon   \nconvex ordering of non-integrable random variables and  \ndiversification for $\\alpha<1$.\nThe so-called \\emph{phase change} at $\\alpha=1$, \ni.e., the inversion of diversification effects \ntaking place when the tail index $\\alpha$ crosses this critical value, \ndemonstrates that the implications of convex ordering are essentially different \nfor integrable and non-integrable random variables.\nIndeed, it is easy to see that if a random variable $Z$ on $\\R$ \nsatisfies $\\mathrm{E} \\sqbr{Z_{+}} = \\mathrm{E} \\sqbr{Z_{-}}=\\infty$,\nthen the only integrable convex functions of $Z$ are the constant ones.\nMoreover, if $Z$ is restricted to $\\R_{+}$ and $\\mathrm{E} Z =\\infty$,\nthen any integrable convex function of $Z$ is necessarily non-increasing. \n\\vspace{0.5em}\n\\par\n\\begin{myproofx}{of Theorem~\\ref{thm:5}}(\\ref{item:t5.1})\nConsider the expectations in~\\eqref{eq:292a}.\nIt is easy to see that for $u>0$ \n\\begin{align*}\n\\oneby{u}\\mathrm{E} h_u\\robrfl{\\xi^{\\top} X} \n&=\n\\oneby{u}\\int_{(u,\\infty)}\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t} \\mathrm{d} t\\\\\n&=\n\\int_{(1,\\infty)} \\mathrm{P}\\cubrfl{\\xi^{\\top} X > tu} \\mathrm{d} t\n\\end{align*}\nand, as a consequence, \n\\[\n\\frac{u^{-1}\\mathrm{E} h_u\\robrfl{\\xi^{\\top} X}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}\n=\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > u}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}\n\\int_{(1,\\infty)} \n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > tu}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > u}}\n\\, \\mathrm{d} t\n\\ldotp\n\\]\nMoreover, Proposition~\\ref{prop:3.3}(\\ref{item:L39.3}) implies\n\\begin{equation}\n\\label{eq:313}\n\\lim_{u\\to\\infty} \n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > u}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}\n=\n\\frac{\\gamma_\\xi(X)}{\\gamma_{e_1}(X)+\\gamma_{-e_1}(X)} \n=\n\\PsiastXg_{\\xi,\\alpha}\n\\end{equation}\nand Karamata's theorem \n\\citep[cf.][Theorem B.1.5]{de_Haan\/Ferreira:2006}  \nyields\n\\[\n\\lim_{u\\to\\infty} \n\\int_{(1,\\infty)} \n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > tu}}{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > u}}\n\\, \\mathrm{d} t\n=\n\\int_{(1,\\infty)}\nt^{-\\alpha} \\mathrm{d} t\n=\n\\oneby{\\alpha-1}\n\\ldotp\n\\]\nAs a result one obtains\n\\begin{equation*}\n\\lim_{u\\to\\infty}\n\\frac{u^{-1} \\mathrm{E} h_u\\robrfl{\\xi^{\\top} X}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}\n=\n\\oneby{\\alpha-1}\\Psiast_X g_{\\xi,\\alpha}\n\\end{equation*}\nand, analogously,\n\\begin{equation*}\n\\lim_{u\\to\\infty}\n\\frac{u^{-1}\\mathrm{E} h_u\\robrfl{\\xi^{\\top} Y}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>u}}\n=\n\\oneby{\\alpha-1}\\Psiast_Y g_{\\xi,\\alpha}\n\\ldotp\n\\end{equation*}\nHence \\eqref{eq:292a} and \\eqref{eq:309} yield\n\\begin{align*}\n1\n&\\ge\n\\limsup_{u\\to\\infty}\\frac{u^{-1}\\mathrm{E}h_u\\robrfl{\\xi^{\\top} X}}{u^{-1}\\mathrm{E} h_u\\robrfl{\\xi^{\\top} Y}}\\\\\n&=\n\\limsup_{u\\to\\infty}\\robrfl{\n\\frac{u^{-1} \\mathrm{E}h_u\\robrfl{\\xi^{\\top} X}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>u}}{u^{-1} \\mathrm{E} h_u\\robrfl{\\xi^{\\top} Y}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>u}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>u}}\n}\\\\\n&=\n\\frac{\\PsiastXg_{\\xi,\\alpha}}{\\PsiastYg_{\\xi,\\alpha}}\n\\end{align*}\nfor all $\\xi\\in\\Simp^d$, which exactly means \n$\\Psiast_X \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_Y$. \n\\par\n\\medskip\n(\\ref{item:t5.2})\nNote that~\\eqref{eq:310a} implies \n\\begin{equation}\n\\label{eq:310}\n\\lim_{t\\to\\infty} \n\\frac{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}\n= 1\n\\end{equation}\nand that~\\eqref{eq:292b} yields\n\\begin{equation}\n\\label{eq:311}\n\\forall u>u_0 \\ \\forall v\\ge 0\n\\quad\n\\mathrm{E} f_{u+v}\\robrfl{\\xi^{\\top} X}  - \\mathrm{E} f_{u+v}\\robrfl{\\xi^{\\top} Y}  \\le 0\n\\ldotp\n\\end{equation}\nFurthermore, it is easy to see that any random variable $Z$ in $\\R_{+}$ \nsatisfies\n\\begin{align*}\n\\mathrm{E}\\sqbrfl{Z\\wedge u} \n&= \n\\int_{(0,\\infty)} \\robrfl{t \\wedge u} \\mathrm{d} \\mathrm{P}^Z(t)\\\\\n&=\n\\int_{(0,\\infty)}\\int_{(0,\\infty)} 1\\cubr{s<t} \\cdot 1\\cubr{s<u} \\, \\mathrm{d} s \\, \\mathrm{d} \\mathrm{P}^Z(t)\\\\\n&=\n\\int_{(0,\\infty)}1\\cubr{s<u} \\int_{(0,\\infty)} 1\\cubr{s<t} \\, \\mathrm{d} \\mathrm{P}^Z(t) \\, \\mathrm{d} s\\\\\n&=\n\\int_{(0,u)}\\mathrm{P}\\cubrfl{Z>s} \\mathrm{d} s\n\\ldotp\n\\end{align*}\nThis implies\n\\[\n\\mathrm{E} f_{u+v}(Z) = \\mathrm{E} f_u(Z) - \\int_{(u,u+v)} \\mathrm{P}\\cubrfl{Z>t} \\mathrm{d} t\n\\ldotp\n\\]\nConsequently, \\eqref{eq:311} yields\n\\begin{equation}\n\\label{eq:312}\n\\forall u\\ge u_0\\ \\forall v>0\n\\quad \n\\mathrm{E} f_u\\robrfl{\\xi^{\\top} X} - \\mathrm{E}f_u\\robrfl{\\xi^{\\top} Y} \n\\le \nI(u,v)\n\\end{equation}\nwhere \n\\begin{align*}\nI(u,v)\n&:=\n\\int_{(u, u+v)}\n\\robrfl{\\mathrm{P}\\cubrfl{\\xi^{\\top} X >t } - \\mathrm{P}\\cubrfl{\\xi^{\\top} Y > t}} \n\\, \\mathrm{d} t\\\\\n&\\phantom{:}=\n\\int_{(u,u+v)} \n\\phi(t) \\cdot \\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}\n\\,\\mathrm{d} t\n\\end{align*}\nwith  \n\\[\n\\phi(t)\n:=\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t} - \\mathrm{P}\\cubrfl{\\xi^{\\top} Y > t}}\n{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\n\\ldotp\n\\]\nMoreover, \\eqref{eq:310}, \\eqref{eq:313}, and an \nanalogue of~\\eqref{eq:313} for $Y$ yield \n\\begin{align}\n\\phi(t)\n&=\\nonumber\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} X > t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\n-\n\\frac{\\mathrm{P}\\cubrfl{\\xi^{\\top} Y > t}}{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}\n\\cdot\n\\frac{\\mathrm{P}\\cubrfl{\\absfl{Y^{(1)}}>t}}{\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}>t}}\\\\\n&\\to\\label{eq:314}\n\\PsiastXg_{\\xi,\\alpha} - \\PsiastYg_{\\xi,\\alpha}\n,\\quad\nt\\to\\infty\n\\ldotp\n\\end{align}\n\\par\nNow suppose that $\\Psiast_Y \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_X$ is not satisfied, \ni.e., there exists $\\xi\\in\\Simp^d$ such that \n$\\PsiastYg_{\\xi,\\alpha} > \\PsiastXg_{\\xi,\\alpha}$.\nThen~\\eqref{eq:314} yields $\\phi(t) \\le - \\varepsilon$ \nfor some $\\varepsilon>0$ and sufficiently large $t$. \nThis implies  \n\\begin{equation} \n\\label{eq:315}\nI(u,v)\n\\le \n-\\varepsilon \n\\int_{(u, u+v)}\n\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}> t} \\, \\mathrm{d} t\n\\end{equation}\nfor sufficiently large $u$ and all $v\\ge0$. \nMoreover, regular variation of $\\absfl{X^{(1)}}$ with tail index \n$\\alpha<1$ implies $\\mathrm{E}\\absfl{X^{(1)}}=\\infty$. \nConsequently, the integral on the right side of~\\eqref{eq:315} tends to \ninfinity for $v\\to\\infty$:\n\\[\n\\forall u>0\n\\quad\n\\lim_{v\\to\\infty}\n\\int_{(u, u+v)}\n\\mathrm{P}\\cubrfl{\\absfl{X^{(1)}}> t} \\, \\mathrm{d} t\n=\n\\infty\n\\ldotp\n\\]\nHence, choosing $u$ and $v$ sufficiently large, one can achieve \n$I(u,v)<c$ for any $c\\in\\R$. In particular, $u$ and $v$ can be \nchosen such that   \n\\[\nI(u,v) < \\mathrm{E} f_u\\robrfl{\\xi^{\\top} X} - \\mathrm{E}f_u\\robrfl{\\xi^{\\top} Y}, \n\\]\nwhich contradicts~\\eqref{eq:312}. Thus  \n$\\PsiastYg_{\\xi,\\alpha} > \\PsiastXg_{\\xi,\\alpha}$ cannot be true \nand therefore it necessarily holds that $\\Psiast_Y \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_X$.\n\\end{myproofx}\n\\par\nNow let us return to the ordering criterion in terms of the supermodular \norder $\\mathrel{\\preceq_{\\mathrm{sm}}}$ stated in Remark~\\ref{rem:12}. The invariance of $\\mathrel{\\preceq_{\\mathrm{sm}}}$ \nunder non-decreasing component transformations \nallows to transfer these criteria to copula models. \nFurthermore, since we are interested in the ordering of the asymptotic \ndependence structures represented by the canonical spectral measures, \n$\\Psiast_1$ and $\\Psiast_2$, we can take \nany copulas that yield $\\Psiast_1$ and $\\Psiast_2$ as asymptotic \ndependence structures. \n\\par\nA natural choice is given by the \\emph{extreme value copulas}, defined \nas the copulas of \\emph{simple max-stable distributions} corresponding to \n$\\Psi^\\ast_i$, i.e., the distributions \n\\begin{equation}\\label{eq:apl.4}\nG^\\ast_i(x):=\\exp\\robrfl{-\\nu^{\\ast}_i\\robrfl{-[\\infty,x]^\\mathrm{c}}}\n,\\quad\nx\\in\\Rplus^{d}\n\\end{equation}\nwhere $\\nu^{\\ast}_i$ is the canonical exponent associated with $\\Psi^\\ast_i$ \nvia~\\eqref{eq:apl.3}. For further details on max-stable and simple max-stable \ndistributions we refer to \\citet{Resnick:1987}.\nSince extreme value copulas and canonical spectral measures can be \nconsidered as  \nalternative parametrizations of the same asymptotic dependence structures, \nwe obtain the following result. \n\\par\n\\begin{proposition}\n\\label{prop:4.4}\nLet $\\Psiast_1$ and $\\Psiast_2$ be canonical spectral measures on $\\Simp^d$. \nFurther, for $i=1,2$, let $C_i$ denote the copula of the \nsimple max-stable distribution $G^\\ast_i$ induced by $\\Psi^\\ast_i$ \naccording to~\\eqref{eq:apl.4} and~\\eqref{eq:apl.3}. \nThen $C_1 \\mathrel{\\preceq_{\\mathrm{sm}}} C_2$ implies\n\\begin{enumerate}[(a)]\n\\item\n$\\Psiast_1 \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_2$ for $\\alpha\\in(1,\\infty)$;\n\\vspace{0.5em}\n\\item\n$\\Psiast_2 \\mathrel{\\preceq_{\\Gcalalpha}} \\Psiast_1$ for $\\alpha\\in(0,1)$.\n\\end{enumerate}\n\\end{proposition}\n\\par\n\\begin{myproof}\nLet $\\nu^{\\ast}_i$ denote the canonical exponent measures corresponding\nto $\\Psi^\\ast_i$ and $G^\\ast_i$.\nIt is easy to see that the transformed measures \n\\[\n\\nu_{\\alpha,i}:=\\nu^{\\ast}_i\\circ T^{-1}\n,\\quad i=1,2,\n\\]\nwith $\\alpha>0$ and the transformation $T$ defined as\n\\[\nT : x \\mapsto \n\\robrfl{\\robrfl{x^{(i)}}^{1\/\\alpha},\\ldots,\\robrfl{x^{(d)}}^{1\/\\alpha}}\n,\n\\quad\nx\\in\\Rplus^{d}, \n\\]\nexhibit the scaling property with index $-\\alpha$:\n\\begin{align*}\n\\nu_{\\alpha,i}\\robr{tA} \n&=\nt^{-\\alpha} \\nu_{\\alpha,i}(A)\n,\\quad\nA\\in\\mathcal{B}\\robr{\\Rplus^{d}\\setminus\\cubr{0}}\n\\ldotp\n\\end{align*}\nHence the transformed distributions \n\\begin{equation}\n\\label{eq:320}\nG_{\\alpha,i}(x)\n:=\nG^\\ast_i\\circ T^{-1}(x) \n= \n\\exp\\robrfl{-\\nu_{\\alpha,i}\\robrfl{[0,x]^c}}\n\\end{equation}\nare max-stable with exponent measures $\\nu_{\\alpha,i}$.\n\\par\nIt is well known that max-stable distributions with identical heavy-tailed margins are multivariate regularly varying \\citep[cf.][]{Resnick:1987}.\nMoreover, the limit measure $\\nu$ in the multivariate regular variation condition can be chosen equal to the exponential measure associated with the property of max-stability.  \nConsequently, the probability distributions $G_{\\alpha,i}$ for $i=1,2$ and $\\alpha>0$ are multivariate regularly varying with tail index $\\alpha$ and canonical spectral measures $\\Psi^\\ast_i$. \n\\par\nFurthermore, it is easy to see that $X\\simG_{\\alpha,1}$ and $Y\\simG_{\\alpha,2}$ \nhave identical margins:\n\\[\nX^{(i)} \\mathrel{\\stackrel{\\mathrm{d}}{=}} Y^{(j)}\n,\\quad\ni,j\\in\\cubr{1,\\ldots,d}\n\\ldotp\n\\]\nMoreover, due to the invariance of $\\mathrel{\\preceq_{\\mathrm{sm}}}$ under non-decreasing marginal \ntransformations, $C_1 \\mathrel{\\preceq_{\\mathrm{sm}}} C_2$ implies \n\\[\nG_{\\alpha,1} \\mathrel{\\preceq_{\\mathrm{sm}}} G_{\\alpha,2}\n\\]\nfor all $\\alpha>0$. \nThus an application of the ordering criteria from Remark~\\ref{rem:12} \nto $X\\simG_{\\alpha,1}$ and $Y\\simG_{\\alpha,2}$ completes the proof.\n\\end{myproof}\n\\section{Examples}\n\\label{sec:5}\nThis section concludes the paper by a series of examples with parametric models illustrating the results from the foregoing sections.\nExamples~\\ref{ex:6} and \\ref{ex:2} demonstrate application of Proposition~\\ref{prop:4.4} to copula based models and the phenomenon of phase change for random vectors in $\\Rplus^{d}$.\nThe fact that the phase change does not necessarily occur in the general case is demonstrated by multivariate Student-t distributions in Example~\\ref{ex:3}.\n\\par\n\\begin{example}\n\\label{ex:6}\nRecall the family of Gumbel copulas given by\n\\begin{equation}\nC_\\theta(u)\n:=\n\\exp\\robrfl{- \\robrfl{\\sum_{i=1}^{d}\\robrfl{-\\log u^{(i)}}^\\theta}^{1\/\\theta}}\n,\\quad\n\\theta\\in[1,\\infty)\n\\ldotp\n\\end{equation}\nGumbel copulas are extreme value copulas, i.e., they are copulas of simple max-stable distributions. \nAccording to \\citet{Hu\/Wei:2002}, Gumbel copulas with dependence parameter $\\theta\\in[1,\\infty)$ are ordered by $\\mathrel{\\preceq_{\\mathrm{sm}}}$:\n\\begin{equation}\n\\label{eq:322}\n\\forall \\theta_1,\\theta_2\\in[1,\\infty) \n\\quad\n\\theta_1 \\le \\theta_2\n\\Rightarrow\nC_{\\theta_1} \\mathrel{\\preceq_{\\mathrm{sm}}} C_{\\theta_2}\n\\ldotp\n\\end{equation}\nConsequently, Proposition \\ref{prop:4.4} applies to the family of \ncanonical spectral measures $\\Psi^\\ast_\\theta$ corresponding to \nthe Gumbel copulas $C_\\theta$.\nThus $1\\le\\theta_1\\le\\theta_2<\\infty$ implies \n$\\Psi^\\ast_{\\theta_1} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_{\\theta_2}$ for $\\alpha>1$ \nand there is a phase change when $\\alpha$ crosses the value $1$, i.e., \nfor $\\alpha\\in(0,1)$ there holds \n$\\Psi^\\ast_{\\theta_2} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_{\\theta_1}$.\n\\par\nApplying Theorem~\\ref{theo:3.4}, one obtains ordering with respect to $\\mathrel{\\preceq_{\\mathrm{apl}}}$ for random vectors $X$ and $Y$ on $\\Rplus^{d}$ that are multivariate regularly varying with canonical spectral measures \nof Gumbel type and have balanced tails ordered by $\\mathrel{\\preceq_{\\mathrm{apl}}}$. \nIn particular, \nthis is the case if $X$ and $Y$ have identical \nregularly varying marginal distributions and \nArchimedean copulas that satisfy appropriate \nregularity conditions \n\\citep[cf.][]{Genest\/Rivest:1989, Barbe\/Fougeres\/Genest:2006}.\n\\par \nMoreover, it is also worth a remark that  \nmultivariate regularly varying random vectors with Archimedean copulas \ncan only induce extreme value copulas of Gumbel type \n\\citep[cf.][]{Genest\/Rivest:1989}.  \n\\par\nFigure~\\ref{figure:32} illustrates the resulting diversification effects \nin the bivariate case, \nincluding indifference to portfolio diversification for $\\alpha=1$ and \nthe phase change occurring when $\\alpha$ crosses this critical value. \nThe graphics show the function \n$\\xi^{(1)}\\mapsto\\Psi^\\ast_\\theta\\,g_{\\xi,\\alpha}$ for selected values of \n$\\theta$ and $\\alpha$. \nDue to $X\\in\\Rplus^{d}$, representation \n$\\Psi^\\ast_\\theta\\,g_{\\xi,\\alpha}=\\gamma_\\xi\/(\\gamma_{e_1} + \\gamma_{-e_1})$ simplifies to \n$\\Psi^\\ast_\\theta\\,g_{\\xi,\\alpha}=\\gamma_\\xi\/\\gamma_{e_1}$\nand therefore \n\\[\n\\Psi^\\ast_\\theta \\, g_{e_1,\\alpha} = \\Psi^\\ast_\\theta \\, g_{e_2,\\alpha} = 1\n\\ldotp\n\\]\n\\par\n\\begin{figure\n\\centering\n\\subfigure[Varying $\\alpha$ for $\\theta=1.4$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig1a-gumbel-eri-1}}\n\\subfigure[Varying $\\alpha$ for $\\theta=2$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig1b-gumbel-eri-2}}\n\\\\\n\\subfigure[Varying $\\theta$ for $\\alpha>1$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig1c-gumbel-eri-3}}\n\\subfigure[Varying $\\theta$ for $\\alpha<1$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig1d-gumbel-eri-4}}\n\\caption{Bivariate Gumbel copulas: Diversification effects represented by functions $\\xi^{(1)}\\mapsto\\Psi^\\ast_\\theta \\,g_{\\xi,\\alpha}$ for selected values of $\\theta$ and $\\alpha$.}\n\\label{figure:32\n\\end{figure}\n\\end{example}\nAs already mentioned above, Theorem~\\ref{thm:5} generalizes some \narguments from \\citet{Embrechts\/Neslehova\/Wuethrich:2009}. \nThe next example concerns Galambos copulas as addressed in \nthat original publication. \n\\par\n\\begin{example}\n\\label{ex:2}\nAnother family of extreme value copulas that are ordered by $\\mathrel{\\preceq_{\\mathrm{sm}}}$ \nis the family of $d$-dimensional \n\\emph{Galambos copulas} \nwith parameter $\\theta\\in(0,\\infty)$:\n\\begin{equation}\nC_\\theta(u)\n:=\n\\exp\\robrfl{\\sum_{I\\subset\\cubr{1,\\ldots,d}} \n(-1)^{\\abs{I}}\\robrfl{\\sum_{i\\in I} \\robrfl{-\\log u^{(i)}}^{-\\theta}}^{-1\/\\theta}\n}\n\\ldotp\n\\end{equation}\n\nAccording to \\citet{Hu\/Wei:2002}, \n$\\theta_1 \\le \\theta_2$ implies $C_{\\theta_1} \\mathrel{\\preceq_{\\mathrm{sm}}} C_{\\theta_2}$.\nThus Proposition~\\ref{prop:4.4} yields ordering of the corresponding canonical spectral measures $\\Psi^\\ast_\\theta$ with respect to $\\mathrel{\\preceq_{\\Gcalalpha}}$. \nSimilarly to the case of Gumbel copulas, $\\theta_1\\le\\theta_2$ implies $\\Psi^\\ast_{\\theta_1} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_{\\theta_2}$ for $\\alpha>1$  \nand $\\Psi^\\ast_{\\theta_2} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_{\\theta_1}$ for $\\alpha\\in(0,1)$. \n\\par\nFinally, it should be noted that Galambos copulas correspond to the \ncanonical exponent measures of random vectors $X$ in $\\Rplus^{d}$ with \nidentically distributed regularly varying margins $X^{(i)}$ and \ndependence structure of $-X$ given by an Archimedean copula with a \nregularly varying generator $\\phi(1-1\/t)$. Models of this type were \ndiscussed in recent studies of aggregation effects for extreme risks %\n\\citep[cf.][]{Alink\/Loewe\/Wuethrich:2004, %\nAlink\/Loewe\/Wuethrich:2005, %\nEmbrechts\/Neslehova\/Chavez-Demoulin:2006, %\nBarbe\/Fougeres\/Genest:2006, %\nEmbrechts\/Lambrigger\/Wuethrich:2008, %\nEmbrechts\/Neslehova\/Wuethrich:2009}.\n\\end{example}\n\\par\nThe final example illustrates results established in Proposition~\\ref{prop:3.7} and Theorem~\\ref{theo:2.4}. In particular, it shows that elliptical distributions do not exhibit a phase change at $\\alpha=1$. \n\\par\n\\begin{example}\n\\label{ex:3}\nRecall multivariate Student-t distributions and \nconsider the case with equal degrees of freedom, i.e., \n\\begin{equation}\nX\\mathrel{\\stackrel{\\mathrm{d}}{=}} \\mu_X + R  A_X  U,\n\\quad\nY\\mathrel{\\stackrel{\\mathrm{d}}{=}} \\mu_Y + R A_Y U,\n\\end{equation}\nwhere $R\\mathrel{\\stackrel{\\mathrm{d}}{=}}\\abs{Z}$ for a Student-t distributed random variable $Z$ with degrees of freedom equal to $\\alpha\\in(0,\\infty)$. \nFurther, let the generalized covariance matrices $C_X=C(\\rho_X)$ and $C_Y=C(\\rho_Y)$ be defined as \n\\begin{equation}\n\\label{eq:146}\nC(\\rho):=\n\\left(\n\\begin{array}{cc}\n1 &\\rho\\\\\n\\rho & 1\n\\end{array}\n\\right)\n\\end{equation} \nand assume that $\\rho_X \\le \\rho_Y$.\n\\par\nAs already mentioned in Remark~\\ref{rem:2.6}(\\ref{item:rem:2.6.a}), \n$C_X$ and $C_Y$ satisfy condition~\\eqref{eq:3.26} and \nProposition~\\ref{prop:3.7} yields $X \\mathrel{\\preceq_{\\mathrm{apl}}} Y$. Moreover, \nProposition~\\ref{prop:3.7} implies a uniform ordering of diversification \neffects in the sense that \n\\[\n\\Psiast_X=\\Psi^\\ast_{\\alpha,\\rho_X} \\mathrel{\\preceq_{\\Gcalalpha}} \\Psi^\\ast_{\\alpha,\\rho_Y}=\\Psiast_Y\n\\]\nfor all $\\alpha\\in(0,\\infty)$.\n\\par\nFigure~\\ref{figure:7} shows functions $\\xi^{(1)} \\mapsto \\Psi^\\ast_{\\alpha,\\rho}\\,g_{\\xi,\\alpha}$ for selected parameter values $\\rho$ and $\\alpha$ that illustrate the ordering of asymptotic portfolio losses by $\\rho$ and the missing phase change at $\\alpha=1$. The indifference to portfolio diversification for $\\alpha=1$ is also absent. \nMoreover, symmetry of elliptical distributions implies $\\gamma_{-e_1} = \\gamma_{e_1}$ and, as a result,\n\\[\n\\Psi^\\ast_{\\alpha,\\rho}\\, g_{e_1,\\alpha} \n= \n\\Psi^\\ast_{\\alpha,\\rho}\\, g_{e_2,\\alpha}  \n=\n1\/2\n\\ldotp\n\\] \nThus the standardization of the plots in Figure~\\ref{figure:7} is different from that in Figure~\\ref{figure:32}. \n\\par\n\\begin{figure\n\\centering\n\\subfigure[Varying $\\alpha$ for $\\rho>0$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig2a-ellipt-1}}\n\\subfigure[Varying $\\alpha$ for $\\rho<0$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig2b-ellipt-2}}\n\\\\\n\\subfigure[Varying $\\rho$ for $\\alpha>1$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig2c-ellipt-3}}\n\\subfigure[Varying $\\rho$ for $\\alpha<1$]\n{\\includegraphics[width=.45\\textwidth]{Graphics\/Fig2d-ellipt-4}}\n\\caption{Bivariate elliptical distributions with generalized covariance matrices defined in~\\eqref{eq:146}: Diversification effects represented by functions $\\xi^{(1)}\\mapsto\\Psi^\\ast_{\\alpha,\\rho}\\, g_{\\xi,\\alpha}$ for selected values of $\\rho$ and $\\alpha$.}\n\\label{figure:7\n\\end{figure}\n\\end{example}\n\\par\n\\begin{remark}\n\\label{rem:15}%\nAll examples the authors are aware of suggest that the diversification \ncoefficient $\\Psiastg_{\\xi,\\alpha}$ is decreasing in $\\alpha$.\nThis means that risk diversification is stronger for lighter component tails \nthan for heavier ones.\n\\par\nHowever, it should be noted that the influence of the tail index $\\alpha$ \non risk aggregation is different from that. The asymptotic risk aggregation coefficient\n\\[\nq_d := \\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{X^{(1)}+\\ldots+X^{(d)} > t}}{\\mathrm{P}\\cubrfl{X^{(1)}>t}}\n\\]\nintroduced by %\n\\citet{Wuethrich:2003} is known to be increasing in $\\alpha$ when the loss components \n$X^{(i)}$ are non-negative %\n\\citep[cf.][]{Barbe\/Fougeres\/Genest:2006}. \nIt is easy to see that the restriction to non-negative $X^{(i)}$ implies \n\\[\nq_d\n=\n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\norm{X}_1 > t}}{\\mathrm{P}\\cubrfl{X^{(1)}>t}}\n=\n\\oneby{\\gamma_{e_1}}\n\\ldotp\n\\]\nMoreover, denoting the uniformly diversified portfolio by $\\eta$, \n\\[\n\\eta:=d^{-1}\\robr{1,\\ldots,1}\n,\n\\]\none obtains \n\\[\nq_d \n= \n\\lim_{t\\to\\infty}\\frac{\\mathrm{P}\\cubrfl{\\eta^{\\top} X > d^{-1} t}}{\\mathrm{P}\\cubrfl{X^{(1)}>t}}\n=\nd^{\\alpha} \\frac{\\gamma_\\eta}{\\gamma_{e_1}}\n\\ldotp\n\\]\nThus $q_d$ is a product of the factor $d^{\\alpha}$, which is \nincreasing in $\\alpha$, and the ratio $\\gamma_\\eta\/\\gamma_{e_1}$, \nwhich is closely related to to the diversification coefficient \n$\\Psiastg_{\\xi,\\alpha}$. \n\\par\nIn particular, given equal marginal weights, i.e., \n\\[\n\\gamma_{e_1}=\\ldots=\\gamma_{e_d},\n\\]\nProposition~\\ref{prop:3.3}(\\ref{item:L39.3}) yields \n\\[\n\\frac{\\gamma_\\eta}{\\gamma_{e_1}} = \\Psi^\\ast g_{\\eta,\\alpha}\n\\ldotp\n\\]\nAs already mentioned above, the coefficients $\\Psiastg_{\\xi,\\alpha}$ \nwith $\\xi\\in\\Simp^d$ are decreasing in all examples considered here.\nThis means that the aggregation and the diversification of risks are \ninfluenced by the tail index $\\alpha$ in different, maybe even always \ncontrary ways.\n\\par\nThe question for the generality of this contrary influence is currently open. \nOne can easily prove that the extreme risk index  $\\gamma_\\xi= \\Psif_{\\xi,\\alpha}$ is decreasing in $\\alpha$ for $\\xi\\in\\Simp^d$. However, this result cannot be extended to $\\Psiastg_{\\xi,\\alpha}$ directly  since $\\Psiastg_{\\xi,\\alpha}$ is related to $\\Psif_{\\xi,\\alpha}$ by the normalizations~\\eqref{eq:3.17} and~\\eqref{eq:3.18}. \nThe question \nwhether $\\Psiastg_{\\xi,\\alpha}$ with arbitrary $\\xi\\in\\Simp^d$  or at least \n$\\Psi^\\ast g_{\\eta,\\alpha}$ is generally decreasing in $\\alpha$ \nis an interesting subject for further research. \n\\end{remark}\n\\section{Acknowledgements}\nThe research underlying this paper was done at the University of Freiburg. \nGeorg Mainik would also like to thank RiskLab for financial support. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nThere are several notions which describe some relative position between two subalgebras of operator algebras. \nAs one of such  notions  between two subalgebras of finite von Neumann \nalgebras, Popa introduced the notion of {\\it mutually orthogonal subalgebras } (definition below) in \\cite{Po}. \nBy the terminology {\\it complementarity}, \nthe same notion is investigated in the theory of quantum systems (see  \\cite{Pe2} for example).  \n\nThe most primary interest would be the case where two subalgebras of some full matrix algebra, both of  \nwhich  are either maximal abelian or isomorphic to also some full matrix algebra. \nIn such the  cases, two subalgebras are connected by a unitary. \n\nOur motivation for this work arises from the following fact:   \n\nIn the previous paper \\cite{Ch1}, we defined a constant $h(A|B)$ \nfor two subalgebras $A$ and $B$ of  a finite von Neumann algebra, \nand showed  the relative position between maximal abelian subalgebras $A$ and $B$ of $M_n(\\mathbb{C})$ \nby using the values of  $h(A|B)$. \nThis  $h(A|B)$ is  a slight modification of  Connes-St\\o rmer  relative entropy $H(A|B)$ in \\cite{CS} (cf. \\cite{NS}). \nIf $A_1$ and $A_2$ are maximal abelian subalgebras of $M_n(\\mathbb{C}),$ \nthen there exists a unitary $u$ such that $A_2 = uA_1u^*,$ and then \n$h(A_1| A_2)$ coincides with the entropy $H(b(u))$  defined in \\cite{ZSKS} of the unistochastic matrix $b(u)$ \ninduced by the unitary $u$. \nAs a consequence, we showed that $A_1$ and $A_2$ are  mutually orthogonal if and if $h(A_1| A_2) = H(b(u)) = \\log n.$ \nThis means that $A_1$ and $A_2$ are  mutually orthogonal if and if $h(A_1| A_2)$ is maximal and  \nequals to the  logarithm of the dimension of the subalgebras. \nRelated results in the case of subfactors of the type II$_1$ factors are obtained in \\cite {Ch2}.  \nHere, it does not hold in general that  $H(A_1| A_2) = H(b(u))$ \n(see, for example \\cite [Appendix] {PSW} ). \n\nOn the other hand, Petz showed in \\cite{Pe2} for subalgebras $A$ and $B$ of $M_n(\\mathbb{C})$ \nthat if $A$ is homogeneous and abelian, then  $H(A|B)$  is \nmaximal if and only if $A$ and $B$ are complementary. \nHere  homogeneous means that all minimal projections of $A$ have the same trace. \nAlso he remarked  that  Connes-St\\o rmer relative entropy cannot characterize the complementarity of \nsubalgebras in the general case. \n\\smallskip \n\nIn this paper, we study the case when the subalgebras  $A$ and $B$ in question are  isomorphic to \nsome $M_n(\\mathbb{C})$. \nWe introduce  some density matrix arising from  the pair $\\{A, B\\},$ and \nwe show that the von Neumann entropy of the density matrix gives  a  characterization of \nthe mutual orthogonality (that is the complementarity). \n\\smallskip \n\nIn order  to define the  entropy for automorphisms of operator algebras, \ntwo kind of notion of {\\it a finite partition of unity} played an important role. \nOne was used by Connes-St\\o rmer, and it corresponds to a  finite measurable partition of \na given space in the ergodic theory (see \\cite {NS} \\cite {OP} for example). \nThe other was used by Alicci and Fannes in \\cite{AF} and it is called  a {\\it finite operational \npartition of unity}. \nHere, we apply the latter, that is operational partition of unity, \nand we give a numerical characterization for pairs of mutually orthogonal \n subalgebras which are both isomorphic to some full matrix algebra of the same size. \n\\smallskip \n\nThe paper is organized as follows. After preliminaries on \nbasic notions in Section 2, in Section 3  we define  some density matrix \nwhich is closely related to subfactors $A$ and $B$  which are both isomorphic to some \n$M_n(\\mathbb{C})$, and  we   show that $A$ and $B$ are  mutually orthogonal \nif and only if the von Neumann entropy of  \nthe density matrix is the maximum value $2\\log n$,  \nwhich is the logarithm of the dimension of the subfactors. \n\\vskip 0.3cm\n\n\\section{Preliminaries}\nHere we summarize notations, terminologies and \nbasic facts. \n\nLet $M$ be a finite von  Neumann algebra acting on a separable Hilbert space,  \nand $\\tau$ be a fixed normal faithful tracial state. \nIn the case where $M$ is the algebra $M_n(\\mathbb{C})$ of $n \\times n$ matrices, \n$\\tau (x) = {\\rm Tr}(x)\/ n, $ where  {\\rm Tr} the usual standard trace on $M_n(\\mathbb{C})$. \nThe norm $\\Vert x \\Vert_\\tau$ is given by $\\Vert x \\Vert_\\tau = \\tau(x^*x)^{1\/2}$ for all $x \\in M$.  \nBy a von Neumann subalgebra $A$ of $M,$  we mean that $A$ is a $*$-subalgebra closed \nin the weak operator topology, the unit of which is the same with the unit of $M$.  \nA conditional expectation of $M$  onto \na von Neumann subalgebra  $A$ of $M$ is  \na completely positive linear map $E_A : M \\to A$ with \n$E_A(axb) = aE_A(x)b$ for all $x \\in M$ and $a, b \\in A$.  \nIn the case of a finite von Neumann algebra $M$ with \na  faithful normal tracial state $\\tau$, \nthere exists always a unique faithful normal conditional expectation $E_A$ \nof $M$ onto a von Neumann subalgebra  $A$ of $M$ \nsuch that $\\tau(xa) = \\tau(E_A(x)a)$ for all $x \\in M$ and $a \\in A$. \nIt is called the conditional expectation with respect to $\\tau$.  \n\n\\subsection{\\bf Mutually orthogonal (or complementary) subalgebras.} \nLet $A$ and $B$ be von Neumann subalgebras of $M$. \nIn \\cite [Lemma 2.1] {Po}, Popa showed that the following conditions are equivalent.  \n\\begin{enumerate}\n\\item$\\tau(ab) = 0$ for $a \\in A, b \\in B$ with $\\tau(a) = \\tau(b) = 0$;\n\\item $\\tau(ab) = \\tau(a) \\tau(b)$ for all $a \\in A, b \\in B$;\n\\item $\\Vert ab \\Vert_{\\tau} = \\Vert a \\Vert_{\\tau} \\Vert b \\Vert_{\\tau} $ for all $a \\in A, b \\in B$;\n\\item $E_A E_B(x)  = \\tau(x) 1_M,$ for all $x \\in M$; \n\\item $E_A (B) \\subset \\mathbb{C} 1_M$.  \n\\end{enumerate}\nMoreover (1) - (5) are equivalent with the analogue conditions obtained by interchanging $A$ with $B$. \n\n\\smallskip\nTwo  von Neumann subalgebras $A$ and $B$ of $M$ are called {\\it mutually orthogonal} \nif one of the above conditions (1) - (5) is satisfied (\\cite [Definition 2.2] {Po}). \n\nMutually orthogonal subalgebras are also called  {\\it complementary  subalgebras}, \n(see, \\cite{Pe1} \\cite{Pe2} for example). \n\n\\subsection{\\bf Density matrix and von Neumann entropy.} \nBy a density matrix, we mean a positive semidefinite matrix $\\rho$ \nsuch that ${\\rm Tr}(\\rho) = 1$. \nTo a density matrix  $\\rho$, the von Neumann entropy $S(\\rho)$ is given by \n$S(\\rho) = {\\rm Tr}(\\eta(\\rho))$. Here, $\\eta$ is   defined on the interval $[0,1]$ by \n$$\\eta(t) = -t \\log t \\quad  (0 < t \\leq 1) \\quad \\text{and} \\quad \\eta(0) = 0.$$ \n\\smallskip \n\n\\section{Main results} \nLet $ M_n(\\mathbb{C}) $  be the algebra of $n\\times n$ complex  matrices,  and let \n${\\rm Tr}$ be the trace of $ M_n(\\mathbb{C}) $ with ${\\rm Tr}(p) = 1 $ for every minimal projection $p$. \nLet $L$ be a finite von Neumann algebra, and \n let $\\tau_L$ be a fixed normal  faithful tracial state. \n \nWe let $M = M_n(\\mathbb{C}) \\otimes L,$ and let $\\tau_M = {\\rm Tr}\/ n \\otimes \\tau_L$. \n\\smallskip\n\n\\subsection{} We consider the subalgebra $N = M_n(\\mathbb{C}) \\otimes 1_L$ of $M$. \nIn this case, the conditional expectation $E_N$ with respect to $\\tau_M$ satisfies that \n$$E_N(x \\otimes y) = \\tau_L(y)x \\otimes 1_L, \\quad x \\in M_n(\\mathbb{C}), \\quad y \\in L.$$\n\nThe following lemma is an easy consequence from the definition, and it is essential to our study. \n\n\\subsubsection{\\bf Lemma} \n{\\it \nLet $N = M_n(\\mathbb{C}) \\otimes 1_L$ and let $u \\in M$ be a unitary operator. Then \n$N$ and $uNu^*$ are mutually orthogonal if and only if \n$$E_N(u^*(a \\otimes 1_L)u) = \\tau_M(a \\otimes 1_L) 1_M \n = \\frac{{\\rm Tr}(a)} n 1_M,\\quad \\text{for all} \\quad a \\in  M_n(\\mathbb{C}).$$} \n\n\\begin{proof} \nAssume that $N$ and $uNu^*$ are mutually orthogonal, that is, \n$$E_NE_{uNu^*}(x) = E_{uNu^*}E_N(x) = \\tau_M(x) 1_M, \\quad \\text{for all} \\quad x \\in M.$$ \nThen \n$uE_N(u^*xu)u^* = E_{uNu^*}(E_N(x)) =  \\tau_M(x) 1_M$, for all $x \\in N$. \nThis implies that \n$$E_N(u^*xu) =  \\tau_M(x) 1_M, \\quad \\text{for \\ all} \\quad x \\in N.$$\n\nConversely, \nassume that $E_N(u^*xu) =  \\tau_M(x) 1_M$, for all $x \\in N$. Then \n$$E_{uNu^*}(x) = uE_N(u^*xu)u^* = \\tau_M(x) 1_M$$ \nfor all $x \\in N$. \nHence \n$$E_{uNu^*}E_N(x) = \\tau_M(x) 1_M \\quad \\text{ for \\ all} \\ x \\in M$$ \nso that $N$ and $uNu^*$ are mutually orthogonal.\n\\end{proof}\n\\smallskip\n\n\\subsection{}\nLet $\\{e_{ij} ; i,j = 1, \\cdots, n \\}$ be a system of matrix units of $M_n(\\mathbb{C}),$ \nso that \n$$e_{ij}^* = e_{ji}, \\quad e_{ij} e_{st} = \\delta_{js} e_{it}, \\quad \\sum_{i = 1}^n e_{ii} = 1_{M_n(\\mathbb{C})}.$$ \nThen each  $x$ in $M = M_n(\\mathbb{C}) \\otimes L$ is written in the unique form: \n$$x = \\sum_{i,j = 1}^n e_{ij} \\otimes x_{ij}, \\quad x_{ij} \\in L,$$\nand  $u = \\sum_{i,j = 1}^n e_{ij} \\otimes u_{ij}$  is a unitary in $M$ if and only if \n$$\\sum_{j=1}^n u_{ij} u_{kj}^* = \\delta_{ik} 1_{L}\\quad \\text{and} \\quad \n\\sum_{i=1}^n u_{ij}^* u_{ik} = \\delta_{jk} 1_L . $$\n\\smallskip\n\nWe give a characterization for a unitary $u \\in M$ to satisfy that $N$ and $uNu^*$ are mutually orthogonal. \n\\smallskip\n\n\\subsubsection{\\bf Theorem.}  \n{\\it  \nAssume that a von Neumann subalgebra $N$ of $M$ is given by  $N =  M_n(\\mathbb{C}) \\otimes 1_L$ \nand let $ u \\in M$ be  unitary.  \nThen $N$ and $uNu^*$ are mutually orthogonal if and only if \n$$\\tau_L(u_{ij}^* u_{kl}) = \\delta_{ik}\\delta_{jl} \\frac 1n, \\quad \\text{for  all} \\quad i,j,k,l = 1, \\cdots, n.$$\n}\n\n\\begin{proof}  \nAssume that  $N$ and $uNu^*$ are mutually orthogonal. \nThen by Lemma 3.1.1 \n$$ E_N(u^*(e_{ij} \\otimes 1_L) u ) = \\delta_{ij}\\frac 1n 1_M.$$ \nOn the other hand, since\n$$u^*(e_{ij} \\otimes 1_L) u = \\sum_{l, t = 1}^n e_{lt} \\otimes u_{il}^* u_{jt}, \\ \\text{for \\ all} \\ i, j  = 1, \\cdots, n,$$ \nby applying  that $E_N(x \\otimes y) = \\tau_L(y)x \\otimes 1_L,$ \nwe have that \n$$ E_N(u^*(e_{ij} \\otimes 1_L) u )\n = \\sum_{l, t = 1}^n \\tau_L( u_{il}^* u_{jt}) e_{lt} \\otimes 1_L.$$\nHence  \n$\\sum_{l, t = 1}^n \\tau_L( u_{il}^* u_{jt}) e_{lt} = \\delta_{ij}\\frac 1n 1_{M_n(\\mathbb{C})}$. \nThis means that \n$$\\tau_L(u_{ij}^* u_{kl}) = \\delta_{ik}\\delta_{jl} \\frac 1n \\quad \\text{for  all} \\quad i,j,k,l = 1, \\cdots, n.$$\n\nConversely, assume that $\\tau_L(u_{ij}^* u_{kl}) = \\delta_{ik}\\delta_{jl} \\frac 1n$ for all $i,j,k,l = 1, \\cdots, n$. \nThen we have that \n$${E_N (u^*(e_{ij} \\otimes 1_L)u) } =  \\sum_{l, t = 1}^n e_{lt} \\otimes \\tau_L(u_{il}^*u_{jt}) 1_L \n =  \\sum_{l = 1}^n e_{ll} \\otimes \\delta_{ij} \\frac 1n 1_L \\nonumber = \\delta_{ij} \\frac 1n 1_M\n$$\nfor all $i,j,k,l = 1, \\cdots, n.$ \nHence   $N$ and $uNu^*$ are mutually orthogonal, by Lemma 3.1.1. \n\\end{proof}\n\n\\subsubsection{\\bf Note}  \nTheorem 3.2.1 implies that if $N =  M_n(\\mathbb{C}) \\otimes 1_L$  and if $N$ and $uNu^*$ are mutually orthogonal \nfor some unitary $u \\in M = M_n(\\mathbb{C}) \\otimes L,$ then \nthe set $\\{  u_{ij}  \/ {\\sqrt n} \\ ; i,j = 1, \\cdots, n\\} \\subset L$ has to be an orthonormal system with respect to \nthe inner product induced by $\\tau_L$ so that $\\dim(L) \\geq n^2$.\n\n\\subsection{ Entropy associated to an inner conjugate pair of subfactors} \nIn order to give a numerical characterization for mutually orthogonal subalgebras which are all isomorphic to \n$M_n(\\mathbb{C})$, \nwe  apply the notion of a  finite operational partition $X$ of unity of size $k$  and the density matrix $\\rho_\\phi[X]$ \nwhich were introduced by Alicki and Fannes in \\cite {AF}. \n\n\\subsubsection{\\bf Finite  operational partition} \nLet $A$ be a unital $C^*$-algebra. A {\\it finite  operational partition of unity of size $k$ } is a set \n$X = \\{x_1, ..., x_k \\}$ of elements of $A$ satisfying \n$$\\sum_i^k x_i^* x_i = 1_A.$$\n\nWe remark that a similar terminology  \"finite partition\" is usually used  in the different following form: \nA finite subset $ \\{x_1, ..., x_k \\}$ in  $A$ is called  a finite partition of unity if \nthey are nonnegative operators in $A$ such that $1_A = \\sum_{i=1}^n  x_i$. See \\cite{NS} or \\cite{OP}. \n\n\\subsubsection{{\\bf Density matrix} $\\rho[X]$} \n\nLet $\\phi$ be a state of $A$. \nTo a finite operational partition $X$  of unity of size $k$,   \nwe associate a $k \\times k$ density matrix  $\\rho_\\phi[X]$ such that \nthe $(i,j)$-coefficient $\\rho_\\phi[X] (i,j)$ of $\\rho_\\phi[X]$ is given by \n$$ \\rho_\\phi[X] (i,j) = \\phi(x_j^*x_i), \\quad i,j = 1, \\cdots, k.$$ \nIn the case that $A$ is a finite von Neumann algebra and that $\\phi$ is a given tracial state $\\tau$ of $A,$  \nthen we denote $\\rho_\\tau[X]$ simply by $\\rho[X].$\n\\smallskip\n\n\\subsubsection{{\\bf Finite operational partition  induced by a unitary} $u$} \nNow let $ M_n(\\mathbb{C}) $  be the algebra of $n\\times n$ complex  matrices  and let \n${\\rm Tr}$ be the trace with ${\\rm Tr}(p) = 1 $ for every minimal projection $p$. \nLet $L$ be a finite von Neumann algebra, and\n let $\\tau_L$ be a fixed normal  faithful tracial state. \nLet $M = M_n(\\mathbb{C}) \\otimes L,$ and let $\\tau_M = {\\rm Tr}\/ n \\otimes \\tau_L$. \nLet $u$ be a unitary in $M_n(\\mathbb{C}) \\otimes L,$ \nand let $u = \\sum_{i,j} e_{ij} \\otimes u_{ij}, \\ (u_{ij} \\in L),$ \nwhere $\\{e_{ij}\\}_{i,j = 1, \\cdots, n}$ is a set of matrix units of $M_n(\\mathbb{C})$. \nWe consider the set \n$$U = \\{\\frac 1{\\sqrt n} {u_{ij}} \\ ; \\ i, j = 1, \\cdots, n\\}.$$ \nIt is not so essential, but \nwe renumber the elements of $U$ for the sake of convenience. \nFor example, if $kn+1 \\leq i \\leq (k+1)n, $ for some $k = 0, 1, \\cdots, n-1,$ then we put \n$$u_i = \\frac 1{\\sqrt n} {u_{i-kn \\ k+1}}.$$\nIt is clear the correspondence $i \\longleftrightarrow (i-kn,  k+1)$ for some $k = 0, 1, \\cdots, n-1$ \nis one to one. \nSince $u$ is a unitary, clearly  the set $U$ is a finite  operational partition  of unity of size $n^2$. \nWe call this set $U$ the  {\\it finite operational partition  of unity induced by} $u$. \n\\smallskip\n\n\\subsubsection{{\\bf von Neumann entropy} $S(\\rho[U] )$} \nWe consider \nthe von Neumann entropy $S(\\rho_\\phi[U] )$ of the density operator $\\rho_\\phi[U] $ \nin order to characterize the mutual orthogonality  for subfactors. \nSo,  we assume that our state $\\phi$ is the  given normalized trace and \n$$ S(\\rho[U] ) = {\\rm Tr}(\\eta(\\rho[U] )).$$  \n\\smallskip\n\n\\subsubsection{\\bf Theorem.}  \n{\\it \nLet $L$ be a finite von Neumann algebra and let $\\tau_L$ be a normalized trace of  $L$. \nWe let  $ M =  M_n(\\mathbb{C}) \\otimes L$ and  $\\tau = {\\rm Tr} \/ n  \\otimes \\tau_L$.\nAssume that $N =  M_n(\\mathbb{C}) \\otimes 1_L$   \nand that $u$ is a unitary operator in $M$.  \nThen  the following conditions are equivalent: \n\\begin{enumerate}\n \\item $N$ and $uNu^*$ are mutually orthogonal; \n \\item $n^2 \\rho[U] $ is the $n^2 \\times n^2$ identity matrix; \n \\item $ S(\\rho[U] ) = 2 \\log n = \\log\\dim  N. $  \n\\end{enumerate}\nHere $U$ is the  finite operational partition  of unity induced by $u$. \n}\n\\smallskip\n\n\\begin{proof}  \nFirst we remark that \n$$\\rho[U](i,j) = \\frac 1n \\tau(u_{j-ln, \\ l+1}^* u_{i-kn, \\  k+1})$$ \nwhere  $u_i = (1 \/ {\\sqrt n}) u_{i-kn, \\ k+1}  $,  \nfor some $k = 0, 1, \\cdots, n-1$ with $kn+1 \\leq i \\leq (k+1)n, $ \nand \n$u_j = ( 1 \/ {\\sqrt n}) {u_{j-ln \\ l+1}}$,  \nfor some $l = 0, 1, \\cdots, n-1$ with \n$ln+1 \\leq j \\leq (lk+1)n $. \n\\smallskip\n\n(1) $\\Rightarrow$ (2):   \nAssume that $N$ and $uNu^*$ are mutually orthogonal. \nThen by Theorem 3.2.1 and by the definition of $\\rho[U],$ \nthe $n^2 \\times n^2$ density matrix $\\rho[U]$ is the diagonal \nmatrix such that \n$$\\rho[U] (i,i) = \\frac 1{n^2} \\quad \\text{for} \\ i = 1, 2, \\cdots, n^2.$$\n\\smallskip\n\n(2) $\\Rightarrow$ (3):    \nClearly, the  the von Neumann entropy $S(\\rho [U] ) = 2 \\log n$ and it is the dimension of $N$. \n\\smallskip\n\n(3) $\\Rightarrow$ (2):  \nAssume that $S(\\rho[U] ) =  \\log n^2$. \nLet $(\\lambda_1, \\cdots, \\lambda_{n^2})$ be  an eigenvalue sequence of $\\rho[U] $ and let \n$(p_1,  \\cdots, p_{n^2})$ be  the corresponding sequence of the minimal projections. \nThen there exists a  $n^2 \\times n^2$  unitary  matrix $w$ so that \n$$w\\rho[U] w^* = \\sum_{i = 1} ^{n^2} \\lambda_i p_i.$$ \nSince \n$$ \\log n^2 = S(\\rho[U] )= \\sum_{i = 1} ^{n^2} \\eta(\\lambda_i ), $$\nit implies that, by the concavity of the function $\\eta$, \n$$\\lambda_i = \\frac 1{n^2} \\quad \\text{for all} \\quad i = 1, 2, \\cdots, n^2$$\nso that \n$$w\\rho[U] w^* = \\frac 1{n^2} 1_{M_{n^2}(\\mathbb{C})}.$$\nHence (2) holds. \n\\smallskip\n\n(2) $\\Rightarrow$ (1):   \nBy the definition of $\\rho[U]$ and the condition (2), we have that \n$$\\delta_{ij} \\frac 1{n^2} = \\rho[U](i,j) = \\frac 1n \\tau(u_{j-ln \\ l+1}^* u_{i-kn \\  k+1}).$$ \n\nThis relation corresponds that   \n$\\tau_L(u_{ij}^* u_{kl}) = \\delta_{ik}\\delta_{jl} \\frac 1n$. \nHence by Theorem 3.2.1,   $N$ and $uNu^*$ are mutually orthogonal. \n\\end{proof}\n\\vskip 0.3cm\n\n\\subsubsection{\\bf Note.}  \nTheorem 3.3.5 means that the mutually orthogonality for inner conjugate \nsubfactors are characterized by  the maximum value \n$\\log \\dim$ of the subfactors.  \n\nIn fact, since the density matrix $\\rho[U]$ is a $n^2 \\times n^2$ matrix and  \nthe function $\\eta$ is operator concave, the value $2 \\log n$ is the maximum. \n\n\\subsubsection{\\bf Note.}  \nThe proof  shows that the statement of Theorem 3.3.5 does not depend on any choice of a matrix units. \n\n\n\\subsection{\\bf Subfactors of matrix algebras}\nLet $A$ and $B$ be subalgebras of  $M_k(\\mathbb{C})$ and assume that both subalgebras are isomorphic to \n$M_n(\\mathbb{C})$. Then $k = mn$. We can assume that \n$M_k(\\mathbb{C}) = M_n(\\mathbb{C}) \\otimes M_m(\\mathbb{C})$ and \n$A = M_n(\\mathbb{C}) \\otimes \\mathbb{C}1$.   \nThere exists a unitary matrix $u \\in M_k(\\mathbb{C})$ such that  $B = uAu^*$.    \nWe denote by $u(A,B)$ this unitary  and also by $U(A,B)$  the  finite operational \npartition  of unity induced by $u(A,B)$.  \nThen we have the followings:  \n\n\\subsubsection{} \nPetz's  characterization of complementarity was given in (\\cite [Theorem 4] {Pe1}): \nThe subalgebra $u(1 \\otimes M_m(\\mathbb{C}) ) u^*$ is complementary to $1 \\otimes M_m(\\mathbb{C})$ \nif and only if \n$$\\frac mn \\sum_{i,j = 1}^n |u_{ij} >< u_{ij}| = 1.$$ \nWhen n = m this condition means that $\\{u_{ij}\\}_{ij}$  is an orthonormal basis in $M_n(\\mathbb{C})$ \nwith respect to the inner product by ${\\rm Tr}$. \n\\vskip 0.3cm\n\n\\smallskip\nOur characterization is the following Corollary of Theorem 3.3.6 by letting $L =  M_m(\\mathbb{C})$. \n\n\\subsubsection{\\bf Corollary.}\n{\\it\nLet $A$ and $B$ be subalgebras of  $M_{k}(\\mathbb{C})$ and assume that both subalgebras are isomorphic to \n$M_n(\\mathbb{C})$. Then $A$ and $B$ are mutually orthogonal  if and only if \n$$ S(\\rho[U(A,B)] ) = 2 \\log n = \\log(\\dim A). $$ \n}\n\\smallskip\n\\subsubsection{\\bf Note} \nIn the above 3.4.1 and 3.4.2, the numbers $m$ and $n$ should be  $m \\geq n.$ \n\\smallskip\n\n\\subsubsection{\\bf Comparison with the case of maximal abelian subalgebras.} \nWe remark that  Corollary 3.4.2  corresponds to \\cite [Corollary 3.2, Corollary 3.3] {Ch1}: \n\\smallskip\n\nAssume that $A$ and $B$ are maximal abelian subalgebras of  $M_{n}(\\mathbb{C})$. \nThen there exists  a unitary $u$ in $M_{n}(\\mathbb{C})$ with $uAu^* = B$, \nand we have that \n\\begin{enumerate}\n \\item $h(A \\mid B) = H(b(u)).$ \n  \\item \n  $A$ and $B$ are mutually orthogonal  if and only if \n    $$h(A \\mid B) = \\log n = \\log(\\dim A).$$ \n\\end{enumerate}\n\nHere, $h(A \\mid B) $ is the conditional relative entropy for $A$ and $B$ in \\cite{Ch1}  and \n$ H(b(u)) $ is the entropy  for the unistochastic operator $b(u)$ induced by the unitary $u$  in \\cite{ZSKS}. \n\\smallskip\n\nThis means that $A$ and $B$ are mutually orthogonal  if and only if \n$h(A \\mid B)$ takes the maximum value $\\log(\\dim A)$, \nbecause $\\log n$ is the maximum value by the definition of $H(b(u))$ and by \nthe property of the function $\\eta$.  \n\\smallskip\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{ Introduction }\nDepression is a significant health concern worldwide, and its early-stage symptom monitoring, detection, and prediction are becoming crucial for us to mitigate this disease. With considerable attentions devoted to this field, traditional diagnosis and monitoring procedures usually rely on subjective measurements. It is desirable to develop more biomarkers that can be automatically extracted from objective measurements. Depression will leave recognizable markers in patient's vocal acoustic, linguistic, and facial patterns, all of which have demonstrated increasing promise on evaluating and predicting patient's mental condition in an unobtrusive way \\cite{kachele2014fusion}. In this work, we aim to extend the existing body of related work and investigate the performances of each of the biomarker modalities (audio, linguistic, and facial) for the task of depression severity evaluation, and further boost our results by using a confidence based fusion mechanism to combine all three modalities. Experiments on the recently released AVEC 2017 \\cite{AVEC2017} depression dataset have verified the promising performance of the proposed model.\n\n\\section{Feature Engineering}\n\nThe AVEC 2017 dataset includes audio and video recordings, as well as extensive questionnaire responses in text formats, collected from (nearly) real-world settings. We will next introduce how we developed feature engineering techniques based on given data and features in each modality. \n\nThe original audio datasets were pre-extracted features using the COVAREP toolbox.  We further extracted descriptors: fundamental frequency (F0), voicing (VUV), normalized amplitude quotient (NAQ), quasi open quotient (QOQ), the first two harmonics of the differentiated glottal source spectrum (H1, H2), parabolic spectral parameter (PSP), maxima dispersion quotient (MDQ), spectral tilt\/slope of wavelet responses (peak\/slope), shape parameter of the Liljencrants-Fant model of the glottal pulse dynamic (Rd), Rd conf, Mel cepstral coefficient (MCEP 0-24), harmonic model and phase distortion mean (HMPDM 0-24) and deviations (HMPDD 0-12), and the first 3 formants. The top $10$ largest discrete cosine transformation (DCT) coefficients were computed for each descriptor to balance between information loss and efficiency. Delta and Delta-Delta features known as differential and acceleration coefficients were calculated as additional features to capture the spectral domain dynamic information. In addition, a series of statistical descriptors such as mean, median, std, peak-magnitude to rms ratio, were calculated. Overall, a total of 1425 audio features were extracted.\n\n2D coordinates of 68 points on the face, estimated from raw video data were provided. To develop visual features from this data-limited setting, we chose stable regions between eyes and mouth due to minimal involvement in facial expression. We calculated the mean shape of 46 stable points not confounding with gender. The pairwise Euclidean distance between coordinates of the landmarks were calculated as well as the angles (in radians) between the points, resulting in 92 features. Finally, we split the facial landmarks into three groups of different regions: the left eye and left eyebrow, the right eye and right eyebrow, and the mouth. We calculated the difference between the coordinates of the landmarks and finally calculated the Euclidean distances ($\\ell_2$-norm) between the points for each group, resulting in 41 features. Overall, we obtained 133 features.\n\nThe transcript file includes translated communication content between each participant and the animated virtual interviewer 'Ellie'. Basic statistics of words or sentences from the transcription file including number of sentences over the duration, number of the words, ratio of number of the laughters over the number of words were calculated. The depression related words were identified from a dictionary of more than 200 words downloaded from online resources\\footnote{\\url{https:\/\/myvocabulary.com\/word-list\/depression-vocabulary\/}}. The ratio of depression-related words over the total number of words over the duration was calculated.  \n\nIn addition, we introduced a new set of \\textbf{text sentiment} features, obtained using the tool of AFINN sentiment analysis \\cite{nielsen2011new}, that would represent the valence of the current text by comparing it to an exiting word list with known sentiment labels. The outcome of AFINN is an integer between minus five (negative) and plus five (positive), where negative and positive number number shows negative and positive positive sentiment subsequently. The mean, median, min, max, and standard deviation of the sentiment analysis outcomes (as a time series) were used. A total of 8 features were extracted. The new set of sentiment features was found to be highly helpful in experiments.\n\n\\section{Multi-Modal Fusion Framework}\n\nWe adopted an input-specific classifier for each modality, followed by a decision-level fusion module to predict the final result. In detail, for each modality biomarker we used a random forest to translate features into predictive scores, while these scores were further combined in a confidence based fusion method to make final prediction on the PHQ8. To fuse the modalities, we implemented a decision-level fusion method. Rather than simple averaging, we recognized that each modality itself might be noisy. Therefore, for each modality we calculated the standard deviation for the outcomes of all trees, defined as the modality-wise \\textbf{confidence score}. After trying several different strategies, the \\textit{winner-take-all} strategy, i.e., picking the single-modality prediction with the highest confidence score as the final result seems to be the most effective and reliable in our setting. In most cases, we observed that audio modality tends to dominate during the prediction. We conjectured that it implies the imbalanced (or say, complementary) informativeness of three modalities, and one modality often tends to dominate in each time of prediction. An overview of the confidence based decision-level fusion method is shown in Figure \\ref{fig:framework}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{framework}\n\\caption{Overview of the confidence based decision-level fusion method}\n\\label{fig:framework}\n\\end{figure}\n\n\\section{Preliminary Result and Future Work }\nBaseline scripts provided by AVEC have been made available in the data repositories where depression severity was computed using random forest regressor.\nTable~\\ref{tab:perf_fusion} reports the performance of the baseline and our model for development and training sets. For both models, we reported the performance of single modality and multi-modal fusion methods. Comparing to the baseline, confidence based fusion could achieve comparable or even marginally better performance than the baseline in terms of both RMSE and MAE. \n\n\n\\begin{table}[thp]\n\\caption{Performance comparison among single modality and confidence based fusion model}\n\\centering\n\\label{tab:perf_fusion}\n\\begin{tabular}{  c| c c | c c }\n\\hline\nFeature used & \\multicolumn{2}{c}{`development'} & \\multicolumn{2}{c}{`train'} \\\\\n& \\textbf{RMSE} & MAE & RMSE & MAE \\\\\n\\hline \\hline\n\\multicolumn{5}{c}{The baseline provided by AVEC organizer} \\\\\n\\hline \nVisual only & 7.13 & 5.88 & 5.42 & 5.29 \\\\\nAudio only & 6.74 & 5.36 & 5.89 & 4.78 \\\\\nAudio \\& Video & 6.62 & 5.52 & 6.01 & 5.09 \\\\\n\\hline \n\\multicolumn{5}{c}{Our model that doesn't include gender variable} \\\\\n\\hline\nVisual only & 6.67 & 5.64 & 6.13 & 5.08 \\\\\nAudio only & 5.45 & 4.52 & 5.21 & 4.26 \\\\\nText only & 5.59 & 4.78 & 5.29 & 4.47 \\\\\nFusion model & 5.17 & 4.47 & 4.68 & 4.31 \\\\\n\\hline\n\\multicolumn{5}{c}{Our model that includes the gender variable} \\\\\n\\hline\nVisual only & 5.65 & 4.87 & 4.99 & 4.46 \\\\\nAudio only & 5.11 & 4.69 & 4.84 & 4.23 \\\\\nText only & 5.51 & 4.87 & 5.13 & 4.28 \\\\\nFusion model & 4.81 & 4.06 & 4.23 & 3.89 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe plan to enhance our methodology in the following directions. First, to improve decision rules, we will use Rule ensemble models to exhaustively search interactions among features and scale up the high-dimensional feature space. In addition, we are interested to perform vowel formants analysis to allow a straightforward detection of high arousal emotions. Second, we found that with more relevant features refined, the overall performance could be improved (e.g.,  silence detection). Finally, we plan to implement our model to a more general clinical environment (e.g., routine patient-provider communication) to characterize social interactions to support clinicians in predicting depression severity. \n\n \\pdfoutput=1\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA natural question in Riemannian geometry is: When does a closed manifold $X$ admit a Riemannian metric with positive scalar curvature? (See \\cite{rosenberg1994manifolds} for a survey on this problem. We call such manifolds ``psc-manifolds''.) The answer is fully understood in the following two cases:\n\\begin{itemize}\n\\item $X$ is $3$-dimensional or less \\cite{perelman2003ricci};\n\\item $X$ is simply connected and $5$-dimensional or more \\cite{gromov-Lawson,stolz1992simply}.\n\\end{itemize}\n\nNow consider the case that $X$ is a $4$-dimensional psc-manifold. Then we have the following three constrains on the topology of $X$:\n\\begin{enumerate}[(i)]\n\\item Suppose $X$ is spin. Then the signature of $X$ (denoted by $\\operatorname{sign}(X)$) must be zero. Similar result holds for its covering spaces \\cite{hitchin1974harmonic,lichnerowicz1963spineurs};\n\\item Suppose $b_{3}(X)>0$. Then up to a nonzero multiple, any element of $H_{3}(X;\\mathds{R})$ can be represented by an embedded, oriented psc $3$-manifold. Similar result holds for its covering spaces \\cite{schoen1979structure};\n\\item Suppose $b_{2}^{+}(X)>1$. Then the Seiberg-Witten invariant $SW(X,\\hat{\\mathfrak{s}})$ must equal $0$ for any spin$^{c}$ structure $\\hat{\\mathfrak{s}}$. Similar result holds for its covering spaces \\cite{Witten}.\n\\end{enumerate}\n\nIn the current paper, we consider the following case:\n\\begin{assum}\\label{homology S3timesS1}\n $X$ is a $4$-manifold with the same homology as $S^{1}\\times S^{3}$; the homology group $H_{3}(X;\\mathds{Z})$ is generated by an embedded $3$-manifold $Y$ with $b_{1}(Y)=0$.\n\\end{assum}\n\nFor such $X$, condition (i) tells nothing interesting and condition (ii) provides a cobordism between $Y$ and a psc $3$-manifold. As for condition (iii), it can not be applied because the Seiberg-Witten invariants are not well defined (since $b_{2}^{+}(X)=0$).\n\nThe first purpose of the current paper is to obtain a new obstruction of positive scalar curvature in the direction of (iii). Recall that for $X$ satisfying Assumption \\ref{homology S3timesS1}, although the original Seiberg-Witten invariant is not well defined, there are two other invariants from the Seiberg-Witten theory:\n\\begin{itemize}\n\\item The $4$-dimensional Casson-type invariant $\\lambda_{SW}(X)$, defined by Mrowka-Ruberman-Saveliev \\cite{MRS};\n\\item The Fr{\\o}yshov invariant $\\operatorname{h}(Y,\\mathfrak{s})$, defined by Fr{\\o}yshov \\cite{Froyshov}, where $\\mathfrak{s}$ is the unique spin structure on $Y$ that can be extended to a spin structure on $X$. (It was proved in \\cite{Froyshov} that this invariant does not depend on the choice of $Y$.)\n\\end{itemize}\nHere is the main theorem of the paper:\n\\begin{thm}\\label{new obstruction}\nSuppose $\\lambda_{SW}(X)+\\operatorname{h}(Y,\\mathfrak{s})\\neq 0$. Then $X$ admits no Riemannian metric with positive scalar curvature.\n\\end{thm}\n\\begin{rmk}\nWe conjecture that one should be able to recover Theorem \\ref{new obstruction} as a special case of Schoen-Yau's result \\cite{schoen1979structure} using monopole Floer homology.\n\\end{rmk}\n\n\nSince it was proved in \\cite{MRS} that the mod-$2$ reduction of $\\lambda_{SW}(X)$ is always $\\rho(Y,\\mathfrak{s})$ (the Rohlin invariant of $(Y,\\mathfrak{s})$), we have the following corollary:\n\\begin{cor}\\label{rohlin=froyshov}\nSuppose $X$ is a homology $S^{3}\\times S^{1}$ with $H_{3}(X;\\mathds{Z})$ generated by an embedded rational homology sphere $Y$ satisfying $$\n\\operatorname{h}(Y,\\mathfrak{s}) \\not\\equiv \\rho(Y,\\mathfrak{s})\\ (\\textrm{mod}\\ 2).\n$$\nThen $X$ admits no Riemannian metric with positive scalar curvature.\n\\end{cor}\nThis corollary gives a large family of interesting examples of $4$-manifolds (with $b_{2}=0$) admitting no positive scalar curvature metric.\n\\begin{ex}\nLet $X$ be obtained by furling up any homology cobordism from $Y=\\Sigma(2,3,7)$ (the Brieskorn sphere) to itself. Then $X$ admits no Riemannian metric with positive scalar curvature because $\\rho(Y)=1$ and $\\operatorname{h}(Y)=0$.\n\\end{ex}\n\nWe summarize the idea in the proof of Theorem \\ref{new obstruction} as follows: Let $W$ be the cobordism from $Y$ to itself obtained by cutting $X$ along $Y$. We consider the manifold\n$$\nZ_{+}=((-\\infty,0]\\times Y)\\cup_{Y}W\\cup_{Y}W\\cup_{Y}...\n$$\nThis non-compact manifold has two ends: one is cylindrical and the other one is periodic. (The word ``periodic'' indicates the fact that we are gluing togegher infinitely many copies of the same manifold $W$. See \\cite{Taubes} for the precise definition.) For a Riemannian metric $g_{X}$ on $X$, we can construct, using a cut-off function, a metric on $Z_{+}$ that equals the a lift of $g_{X}$ over the periodic-end and restricts to the product metric on the cylindrical end. Now consider the (suitably perturbed) Seiberg-Witten equations on $Z_{+}$. More specifically, let $[\\mathfrak{b}]$ be a critical point of the Chern-Simons-Dirac functional with certain absolute grading.  We consider the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ of gauge equivalent classes of solutions that approaches $[\\mathfrak{b}]$ on the cylindrical end and has exponential decay on the periodic end. By adding end points to the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$, which correspond to ``broken solutions'' on $Z_{+}$, we get the moduli space $\\mathcal{M}^{+}([\\mathfrak{b}],Z_{+})$, which is a $1$-manifold with boundary. Now we use the assumption that $g_{X}$ has positive scalar curvature. Under this assumption, we can prove that $\\mathcal{M}^{+}([\\mathfrak{b}],Z_{+})$ is compact. Therefore, the number of points in $\\partial\\mathcal{M}^{+}([\\mathfrak{b}],Z_{+})$, counted with sign, should be $0$. This actually implies that a certain reducible critical point $[\\mathfrak{a}_{0}]$ can not be ``killed by the boundary map'' and hence survives in the monopole Floer homology. By this argument, we show that $-2\\operatorname{h}(Y,\\mathfrak{s})\\leq 2\\lambda_{\\textnormal{SW}}(X)$. By the same argument on $-X$, we can also prove $-2\\operatorname{h}(Y,\\mathfrak{s})\\geq 2\\lambda_{\\textnormal{SW}}(X)$, which completes the proof of Theorem \\ref{new obstruction}.\n\nAs can be seen from the above discussion, the study of Seiberg-Witten equations on end-periodic manifolds plays a central role in our argument. We note that the first application of gauge theory on end-periodic manifolds was given by Taubes \\cite{Taubes} in the context of Donaldson theory, where he proved that the Euclidean space $\\mathds{R}^{4}$ admits uncountable many exotic smooth structures. However, the Seiberg-Witten theory on end-periodic manifold is still not well developed. One major difficulty in this direction is finding a reasonable substitution for the assumption $\\pi_{1}(W)=1$ (which was used in \\cite{Taubes}) and prove the compactness theorem under this new assumption. In the current paper, we use the positive scalar curvature assumption, which tells something interesting but still not general enough. This motivates the second purpose of the paper: we try to develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds. Actually, all the results (except Lemma \\ref{orientation reversal 2}) in Section 2, Section 3 and the appendix are stated and proved without the positive scalar curvature assumption.\n\nWe note that many of the results and proofs in the current paper follow the same line as Kronheimer-Mrowka's book \\cite{KM}. The idea is that: by working with suitably weighted Sobolev spaces, one can treat the non-compact manifold $$X_{+}=W\\cup_{Y} W\\cup_{Y}...$$ as a compact manifold whose signature equals the correction term $-w(X,0,g_{X})$ (see Subsection 2.4).\n\nThe precise statements of all the results used in the current paper will be given. However, to keep the length of the paper somehow under control, we will omit the proofs that are word by word translations from the corresponding parts of \\cite{KM}. In order to help the reader to follow the argument, we will always give the precise reference of the omitted details. From now on, we will refer to \\cite{KM} as \\textbf{the book}.\n\nThe paper is organized as follows: In Section 2, we  briefly recall the definition of the monopole Floer homology, the Fr\\o yshov invariant $\\operatorname{h}(Y,\\mathfrak{s})$ and the $4$-dimensional Casson invariant $\\lambda_{\\textnormal{SW}}(X)$. We will also review and prove some results about linear analysis on end-periodic manifolds. In Section 3, we start setting up the gauge theory on end-periodic manifolds and define the moduli spaces. In Section 4, we prove the compactness result under the positive scalar curvature assumption. In Section 5, we will put all the pieces together and finish the proof of Theorem \\ref{new obstruction}. In the appendix, we prove (using Fourier-Laplace transformation) Proposition \\ref{laplace equation}, which states the uniqueness and existence of the solution of the Laplace equation on end-periodic manifolds. This may be of independent interest for some readers.\n\\\\\n\\\\\n\\textbf{Acknowledgement.} The author wishes to thank\nPeter Kronheimer, Tomasz Mrowka, Ciprian Manolescu, Daniel Ruberman, Nicolai Saviliev and Richard Schoen for sharing their expertise in several inspiring discussions. The author is especially grateful to Clifford Taubes for suggesting the idea of proof of Lemma \\ref{exp decay} (the key estimate in Section 4) and Terence Tao for providing an alternative proof of Lemma \\ref{Solving laplace equation on covering space}. Corollary \\ref{rohlin=froyshov} was also proved by Daniel Ruberman \\cite{Ruberman} using Schoen-Yau's minimal surface result.\n\nDuring the preparation of the current paper, the author noticed that a different version of compactness theorem for Seiberg-Witten equations over manifolds with periodic ends of positive scalar curvature was proved earlier in Diogo Veloso's thesis \\cite{Diogo}. A different type of Hodge decomposition for such manifolds was also studied there.\n\n\n\\section{Preliminaries}\n\\subsection{The set up and the notations}\nLet $X$ connected, oriented, smooth 4-manifold satisfying the condition $$H_{1}(X;\\mathds{Z})\\cong \\mathds{Z},\\  H_{2}(X;\\mathds{Z})\\cong 0.$$ In other words, $X$ is a homology $S^{1}\\times S^{3}$. We further assume that $H_{3}(X;\\mathds{Z})$ is generated by an embedded rational homology 3-sphere $Y$. (This is not always the case.)  We fix a homology orientation of $X$ by fixing a generator $[1]\\in H_{1}(X;\\mathds{Z})$. This induces an orientation on $Y$ by requiring that $[1]\\cup [Y]=[X]$. Let $W$ be the cobordism from $Y$ to itself obtained from cutting $X$ open along $Y$. The infinite cyclic covering space of $X$ has a decomposition\n$$\n\\tilde{X}=...\\cup_{Y} W_{-1}\\cup_{Y} W_{0}\\cup_{Y} W_{1}\\cup ... \\ \\text{with all }W_{n}\\cong W.\n$$\n We choose a lift of $Y$ to $\\tilde{X}$ and still call it $Y$. We let\n$$X_{+}=W_{0}\\cup_{Y} W_{1}\\cup_{Y} W_{2}\\cup ... $$\nbe one of the two components of $\\tilde{X}\\setminus Y$.\n\\begin{nota}\nIn the current paper, we will use $\\cup$ to denote the disjoint union and use $\\cup_{Y}$ to denote the result of gluing two manifolds along their common boundary $Y$.\n\\end{nota}\nThere are two spin structures on $X$. We pick one of them and denote it by $\\hat{\\mathfrak{s}}$. It induces spin structures on the various manifolds we constructed so far. In particular, we have an induced spin structure on $Y$ and we denote it by $\\mathfrak{s}$. It is not hard to see that $\\mathfrak{s}$ does not depend on the choice of $\\hat{\\mathfrak{s}}$. These spin structures will be fixed through out the paper and we will suppress them from most of our notations. We denote by $S^{+}$ and $S^{-}$ the positive and negative spinor bundles over various 4-manifold. The spin connection over $4$-manifolds are all denoted by $A_{0}$. For the 3-manifold $Y$, we denote the spinor bundle by $S$ and the spin connection by $B_{0}$. In both dimensions, we write $\\rho$ for the Clifford multiplication.\n\nOther than $\\tilde{X}$ and $X_{+}$, we also consider the following two (non-compact) spin $4$-manifolds\n$$\nM_{+}:=M\\cup_{Y} X_{+}\\ \\text{and }Z_{+}:=Z\\cup_{Y} X_{+},\n$$\nwhere $Z=(-\\infty,0]\\times Y$ and $M$ is a compact spin $4$-manifold bounded by $(Y,\\mathfrak{s})$. By doing surgeries along loops in $M$, we can assume that $b_{1}(M)=0$. We denote by $\\bar{M}$ the orientation reversal of $M$.\n\n\nNow we specify Riemannian metrics on these manifolds: Let $g_{X}$ be a metric on $X$. We consider a harmonic map\n\\begin{equation}\\label{harmonic function}f:X\\rightarrow S^{1}\\cong \\mathds{R}\/\\mathds{Z}\\end{equation}\nsatisfying\n$$\nf^{*}(d\\theta)=[1]\\in H^{1}(X;\\mathds{Z}).\n$$\nIt was proved in \\cite{ruberman2007dirac} that for a generic choice of $g_{X}$, the Dirac operator\n$$\\slashed{D}^{+}_{A}:L^{2}_{1}(X;S^{+})\\rightarrow  L^{2}(X;S^{-}),$$\nassociated to the connection $A=A_{0}+i a\\cdot f^{*}(d\\theta)$ for any $a\\in \\mathds{R}$, has trivial kernel. We call such metric ``admissible metric''.\n\\begin{assum}\\label{admissible metric}\nThroughout this paper, we fix a choice of admissible metric $g_{X}$.\n\\end{assum}\n\n\\begin{rmk}\nBy the Weitzenb\\\"ock formula, any metric with positive scalar curvature is admissible. However, we will not impose this positive scalar curvature condition until Section 4.\n\\end{rmk}\n\nLet $g_{\\tilde{X}}$ be the lift of $g_{X}$ on $\\tilde{X}$ and $g_{Y}$ be an arbitrary metric on $Y$. Using a cut-off function, we can construct a metric $g_{X_{+}}$ on $X_{+}$ which is isomorphic to the product metric $[0,3]\\times g_{Y}$ near the boundary (with $\\{0\\}\\times Y$ identified with $\\partial X_{+}$)\n and whose restriction on $X_{+}\\setminus W_{0}$ equals $g_{\\tilde{X}}$. Let $g_{M}$ be a metric on $M$ isomorphic to the product metric near the boundary. By gluing $g_{M}$ and $g_{X_{+}}$ together, we get a metric $g_{M_{+}}$ on $M_{+}$. Similarly, we obtain the metric\n$g_{Z^{+}}$ on $Z^{+}$ by gluing the metric $g_{X_{+}}$ together with the product metric on $Z$.\n\n\n\\subsection{The monopole Floer homology and the Fr\\o yshov invariant }\nIn this subsection, we briefly review the definition of the monopole Floer homology and the Fr\\o yshov invariant. For details, we refer to the book and \\cite{Froyshov}.\n\nLet $k\\geq 3$ be an integer fixed throughout the paper. To begin with, we define\n$$\n\\mathcal{A}_{k-1\/2}(Y)=\\{B_{0}+a| a\\in L^{2}_{k-1\/2}(Y;i\\mathds{R})\\}\n$$\nas the space of spin$^{\\text{c}}$ connections over $Y$ of class $L^{2}_{k-1\/2}$. Consider the configuration space:\n$$\n\\mathcal{C}_{k-1\/2}(Y)=\\mathcal{A}_{k-1\/2}(Y)\\times L^{2}_{k-1\/2}(Y;S).\n$$\nThe pair $(B,\\Psi)\\in \\mathcal{C}_{k-1\/2}(Y)$ is called reducible if $\\Psi=0$. Denote by $\\mathcal{C}^{\\text{red}}_{k-1\/2}(Y)$ the space of reducible pairs. We will also consider the blown-up configuration space:\n\\begin{equation}\n\\begin{split}\n\\mathcal{C}^{\\sigma}_{k-1\/2}(Y)=\\{&(B,s,\\Psi)|\\  B\\in \\mathcal{A}_{k-1\/2}(Y),\\\\\n&s\\in \\mathds{R}_{\\geq 0} \\text{ and } \\Psi\\in L^{2}_{k-1\/2}(Y;S) \\text{ satisfies }\\|\\Psi\\|_{L^{2}}=1\\}.\\end{split}\n\\end{equation}\nThe gauge group\n$$\\mathcal{G}_{k+1\/2}(Y)=\\{u:Y\\rightarrow S^{1}|\\  \\|u\\|_{L^{2}_{k+1\/2}}<\\infty\\}$$ acts on both $\\mathcal{C}_{k-1\/2}(Y)$ and $\\mathcal{C}^{\\sigma}_{k-1\/2}(Y)$. Denote the quotient spaces by $\\mathcal{B}_{k-1\/2}(Y)$ and $\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$ respectively. It was proved in the book that $\\mathcal{C}_{k-1\/2}(Y)$ and $\\mathcal{B}_{k-1\/2}(Y)$ are Hilbert manifolds without boundary, while\n$\\mathcal{C}_{k-1\/2}(Y)$ and $\\mathcal{B}_{k-1\/2}(Y)$ are Hilbert manifolds with boundary.\n\nWe define the Chern-Simons-Dirac functional $\\mathcal{L}$ (with $B_{0}$ as the preferred reference connection) on $C_{k-1\/2}(Y)$ as\n\\begin{equation}\\label{CSD}\n\\mathcal{L}(B,\\Psi)=-\\frac{1}{8}\\int_{Y}(B^{t}-B^{t}_{0})\\wedge (F_{B^{t}}+F_{B^{t}_{0}})+\\frac{1}{2}\\int_{Y}\\langle \\slashed{D}_{B}\\Psi,\\Psi\\rangle\\,d\\text{vol},\n\\end{equation}\nwhere $B^{t}$ and $B_{0}^{t}$ denote the induced connections on the determine bundle $\\text{det}(S)$ and $F_{B^{t}}$, $F_{B_{0}^{t}}$ denote their curvatures. We denote by $\\operatorname{grad}\\mathcal{L}$ the formal gradient of $\\mathcal{L}$. This is a section of the $L^{2}_{k-3\/2}$-completed tangent bundle of $\\mathcal{C}_{k-1\/2}(Y)$. In order to get the transversality condition, we need to add a perturbation $\\mathfrak{q}$ on $\\operatorname{grad}\\mathcal{L}$. The sum $\\operatorname{grad}\\mathcal{L}+\\mathfrak{q}$ is gauge invariant and gives rise to a ``vector field'' $$v_{\\mathfrak{q}}^{\\sigma}:\\mathcal{B}_{k-1\/2}^{\\sigma}(Y)\\rightarrow \\mathcal{T}_{k-3\/2}(Y),$$ where $\\mathcal{T}_{k-3\/2}(Y)$ denotes the $L^{2}_{k-3\/2}$ completion of the tangent bundle of $\\mathcal{B}_{k-1\/2}^{\\sigma}(Y)$. (We put the quotation marks here because $v_{\\mathfrak{q}}^{\\sigma}$ is not a section of the actual tangent bundle). We call the perturbation $\\mathfrak{q}$ admissible if all critical points of $v^{\\sigma}_{\\mathfrak{q}}$ are nondegenerate and the moduli spaces of flow lines connecting them are regular. (See Page 411 of the book for an exact definition.) Under this admissibility condition, the set $\\mathfrak{C}$ of critical points of $v^{\\sigma}_{\\mathfrak{q}}$ is discrete and can be decomposed into the disjoint union of three subsets:\n\\begin{itemize}\n\\item $\\mathfrak{C}^{o}$: the set of  irreducible critical points;\n\\item $\\mathfrak{C}^{s}$: the set of reducible, boundary stable critical points (i.e., reducible critical points where $v^{\\sigma}_{\\mathfrak{q}}$ points outside the boundary);\n\\item $\\mathfrak{C}^{u}$: the set of reducible, boundary unstable critical points (i.e., reducible critical points where $v^{\\sigma}_{\\mathfrak{q}}$ points inside the boundary).\n\\end{itemize}\nThe monopole Floer homologies $\\widebar{HM}(Y,\\mathfrak{s};\\mathds{Q})$,\n$\\widecheck{HM}(Y,\\mathfrak{s};\\mathds{Q})$ and $\\widehat{HM}(Y,\\mathfrak{s};\\mathds{Q})$ are defined as the homology of the chain complexes freely generated by $\\mathfrak{C}^{o}$, $\\mathfrak{C}^{o}\\cup \\mathfrak{C}^{s}$ and $\\mathfrak{C}^{o}\\cup \\mathfrak{C}^{u}$ respectively.\n\nOur main concern will be $\\widebar{HM}(Y,\\mathfrak{s};\\mathds{Q})$ and $\\widecheck{HM}(Y,\\mathfrak{s};\\mathds{Q})$. To give the precise definitions, we first recall that a two-element set $\\Lambda([\\mathfrak{b}])$ (called the orientation set) can be associated to each $[\\mathfrak{b}]\\in \\mathfrak{C}$ (see Section 20.3 of the book). After making a choice of preferred element $\\chi([\\mathfrak{b}])\\in\\Lambda([\\mathfrak{b}])$ for each $[\\mathfrak{b}]$, we can canonically orient the moduli spaces of trajectories connecting them. Now let $C^{o}$ (resp. $C^{u}$ and $C^{s}$) be a vector space over $\\mathds{Q}$ with basis $\\{e_{[\\mathfrak{b}]}\\}$ indexed by elements $[\\mathfrak{b]}$ in $\\mathfrak{C}^{o}$ (resp. $\\mathfrak{C}^{s}$ and  $\\mathfrak{C}^{u}$). We define the linear maps\n$$\n\\partial^{o}_{o}:C^{o}\\rightarrow C^{o},\\ \\ \\ \\ \\partial^{o}_{s}:C^{o}\\rightarrow C^{s},\n$$\n$$\n\\partial^{u}_{o}:C^{u}\\rightarrow C^{o},\\ \\ \\ \\ \\partial^{u}_{s}:C^{u}\\rightarrow C^{s}.\n$$\nby the formulae\n$$\n\\partial^{o}_{o}e_{[\\mathfrak{b}]}=\\mathop{\\sum}\\limits_{[\\mathfrak{b}']\\in \\mathfrak{C}^{o}} \\#\\breve{\\mathcal{M}}([\\mathfrak{b}],[\\mathfrak{b}'])\\cdot e_{[\\mathfrak{b}']}\\ \\ \\ \\ ([\\mathfrak{b}]\\in \\mathfrak{C}^{o})\n$$\nand so on, where the integer $\\#\\breve{\\mathcal{M}}([\\mathfrak{b}],[\\mathfrak{b}'])$ counts (with sign) the number of points in $\\breve{\\mathcal{M}}([\\mathfrak{b}],[\\mathfrak{b}'])$ (the moduli space of Seiberg-Witten trajectories going from $[\\mathfrak{b}]$ to $[\\mathfrak{c}]$) that has dimension $0$.\n\nBy considering the number $\\#\\breve{\\mathcal{M}}^{\\text{red}}([\\mathfrak{b}],[\\mathfrak{b}'])$ instead (i.e., only counting reducible trajectories), we can similarly define the linear maps\n$$\n\\bar{\\partial}^{s}_{s}:C^{s}\\rightarrow C^{s},\\ \\ \\ \\\n\\bar{\\partial}^{s}_{u}:C^{s}\\rightarrow C^{u},$$\n$$\n\\bar{\\partial}^{u}_{s}:C^{u}\\rightarrow C^{s},\\ \\ \\ \\ \\bar{\\partial}^{u}_{u}:C^{u}\\rightarrow C^{u}.\n$$\n(We note that $\\bar{\\partial}^{u}_{s}$ is different with $\\partial^{u}_{s}$.)\n\nThe following definition was given as Definition 22.1.7 of the book.\n\\begin{defi}\\label{monopole Floer}\nThe monopole Floer homology groups $\\widebar{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$ and $\\widecheck{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$ are defined as the homology groups of the chain complexes $\\bar{C}=C^{s}\\oplus C^{u}$ and  $\\check{C}=C^{o}\\oplus C^{s}$ with the differentials\n\\begin{equation}\\label{differential for hm-bar}\n\\bar{\\partial}=\\left(\\begin{array} {cc}\n \\bar{\\partial}^{s}_{s}  & \\bar{\\partial}^{u}_{s}  \\\\\n \\bar{\\partial}^{s}_{u}  & \\bar{\\partial}^{u}_{u}\n\\end{array}\\right)\\text{ and }\\check{\\partial}:=\\left(\\begin{array} {cc}\n \\partial^{o}_{o}  & -\\partial^{u}_{o}\\bar{\\partial}^{s}_{u} \\\\\n \\partial^{o}_{s} & \\bar{\\partial}^{s}_{s}-\\partial^{u}_{s}\\bar{\\partial}^{s}_{u}\n\\end{array}\\right)\n\\end{equation}\nrespectively. There is a natural map $i_{*}:\\widebar{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})\\rightarrow \\widecheck{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$  induced by the chain map $i:\\bar{C}\\rightarrow \\check{C}$ defined as\n\\begin{equation}\\label{chain map}\n\\left(\\begin{array} {cc}\n0  & -\\partial^{u}_{o}\\\\\n 1 & -\\partial^{u}_{s}\n\\end{array}\\right).\n\\end{equation}\n\\end{defi}\nTo each $[\\mathfrak{b}]\\in \\mathfrak{C}$, we can assign a rational number $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])$ (called the absolute grading) as follows (see Definition 28.3.1 of the book): Let $\\operatorname{gr}(M,[\\mathfrak{b}])$ be the ``relative $M$-grading'' of $[\\mathfrak{b}]$. This number describes the expected dimension of the Seiberg-Witten moduli space on the manifold $M^{*}=M\\cup_{Y} ([0,+\\infty)\\times Y)$ with limit $[\\mathfrak{b}]$. It was proved in the book that the quantity\n\\begin{equation}\\label{absolute grading}\n-\\operatorname{gr}(M,[\\mathfrak{b}])-b_{2}^{+}(M)-\\frac{1}{4}\\operatorname{sign}(M)-1\n\\end{equation}\ndoes not depend on the choice of $M$ and we define it as $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])$. This grading induces absolute gradings on $\\widebar{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q}),\\  \\widehat{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$ and $\\widecheck{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$. The map $i_{*}$ in Definition \\ref{monopole Floer} preserves this grading.\n\\begin{rmk}\nIn (\\ref{absolute grading}), we use $\\operatorname{gr}(M,[\\mathfrak{b}])$ instead of $\\operatorname{gr}([\\mathfrak{a}_{0}],M\\setminus B^{4},[\\mathfrak{b}])$ as in the book. Here $[\\mathfrak{a}_{0}]$ denotes the first boundary stable critical point in $\\mathcal{B}^{\\sigma}_{k-1\/2}(S^{3})$. These two gradings satisfy the relation (see Lemma 27.4.2 of the book)\n$$\n\\operatorname{gr}(M,[\\mathfrak{b}])=\\operatorname{gr}(B^{4},[\\mathfrak{a}_{0}])+\\operatorname{gr}([\\mathfrak{a}_{0}],M\\setminus B^{4},[\\mathfrak{b}])=-1+\\operatorname{gr}([\\mathfrak{a}_{0}],M\\setminus B^{4},[\\mathfrak{b}]).\n$$\nThis explains the extra term ``$-1$'' in our formula.\n\\end{rmk}\n\n\n\\begin{rmk}\nIn general, one needs to specify a connected component of $\\mathcal{B}^{\\sigma}_{k}(M)$ (the blown-up quotient configuration space of $M$) to define the relative $M$-grading. However, in our case the space $\\mathcal{B}^{\\sigma}_{k}(M)$ is connected since $b_{1}(Y)=0$.\n\\end{rmk}\n\n\\begin{defi}\\cite{Froyshov}\nThe Fr\\o yshov invariant is defined as\n$$\\operatorname{h}(Y,\\mathfrak{s}):=-\\frac{1}{2}\\cdot\\inf\\{\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])|[\\mathfrak{b}]\\text{ represents a nonzero elements in }\\operatorname{im}i_{*}\\}.$$\n\\end{defi}\nThe following lemma was proved in \\cite{Froyshov} (in a (possibly) different version of monopole Floer homology). The proof can be easily adapted to the version used in the book.\n\\begin{lem}\\label{orientation reversal}\nFor any rational homology sphere $Y$ and any  spin$^{\\text{c}}$ structure $\\mathfrak{s}$ on $Y$, we have $\\operatorname{h}(-Y,\\mathfrak{s})=-\\operatorname{h}(Y,\\mathfrak{s})$.\n\\end{lem}\n\n\n\\begin{defi}\nAn admissible perturbation $\\mathfrak{q}$ is called a ``nice perturbation'' if $\\mathfrak{q}=0$ when restricted to ${\\mathcal{C}^{\\operatorname{red}}_{k-1\/2}(Y)}$.\n\\end{defi}\n\n\\begin{rmk}\\label{component of perturbation}\nSince the tangent bundle of $\\mathcal{C}_{k-1\/2}(Y)$ is trivial with fiber $$L^{2}_{k-1\/2}(Y;i\\mathds{R})\\oplus L^{2}_{k-1\/2}(Y;S),$$ we can write the perturbation $\\mathfrak{q}$ as $(\\mathfrak{q}^{0},\\mathfrak{q}^{1})$, where $\\mathfrak{q}^{0}$ denotes the connection component and $\\mathfrak{q}^{1}$ denotes the spinor component. Note that by the gauge invariance, the restriction of $\\mathfrak{q}^{1}$ to $\\mathcal{C}^{\\operatorname{red}}_{k-1\/2}(Y)$ is always $0$. Therefore, an admissible perturbation $\\mathfrak{q}$ is nice if and only if $\\mathfrak{q}^{0}=0$ when restricted to ${\\mathcal{C}^{\\operatorname{red}}_{k-1\/2}(Y)}$.\n\\end{rmk}\n\nUnder the assumption that $\\mathfrak{q}$ is nice, there is only one reducible critical point downstairs (up to gauge transformation), which is just $(B_{0},0)$. As for the critical points upstairs, the sets $\\mathfrak{C}^{u}$ and $\\mathfrak{C}^{s}$ can be described explicitly as follows:\nConsider the self-adjoint operator\n\\begin{equation}\\label{perturbed dirac}\n\\slashed{D}_{\\mathfrak{q},B_{0}}:L^{2}_{k-1\/2}(Y;S)\\rightarrow L^{2}_{k-3\/2}(Y;S)\n\\end{equation}\n$$\n\\Psi\\mapsto \\slashed{D}_{B_{0}}\\Psi+\\mathcal{D}_{(B_{0},0)}\\mathfrak{q}^{1}(0,\\Psi) .\n$$\nSince $\\mathfrak{q}$ is admissible, $0$ is not an eigenvalue of $\\slashed{D}_{\\mathfrak{q},B_{0}}$ and all eigenvalues have multiplicity $1$ (see Proposition 12.2.5 of the book). We arrange the eigenvalues $\\lambda_{*}$ so that\n$$\n...\\lambda_{-2}<\\lambda_{-1}<0<\\lambda_{0}<\\lambda_{1}<...\n$$\nFor each $i$, we pick an eigenvector $\\psi_{i}$  with eigenvalue $\\lambda_{i}$ and $\\|\\psi_{i}\\|_{L^{2}}=1$. We let $[\\mathfrak{a}_{i}]=[(B_{0},0,\\psi_{i})]$.  By Proposition 10.3.1 of the book, we have\n$$\n\\mathfrak{C}^{s}=\\{[\\mathfrak{a}_{i}]|\\,i\\geq 0\\},\\ \\mathfrak{C}^{u}=\\{[\\mathfrak{a}_{i}]|\\,i< 0\\}.\n$$\nFrom now on, we always use $[\\mathfrak{a}_{*}]$ to denote these reducible critical points. Note that $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{a}_{i}])-\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{a}_{i-1}])$ equals $1$ when $i=0$ and equals $2$ otherwise.\n\n\\begin{defi}\nLet $\\mathfrak{\\mathfrak{q}}$ be a nice pertrubation. The height of $\\mathfrak{q}$ is defined as $$\\operatorname{ht}(\\mathfrak{q})=\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{a}_{0}]).$$\nIn other words, the height is defined to be the absolute grading of the lowest boundary stable critical point.\n\\end{defi}\nConsdier the operator\n$$\nD_{\\mathfrak{q}}:L^{2}_{k}(M;S^{+})\\rightarrow L^{2}_{k-1}(M;S^{-})\\oplus (L^{2}_{k-1\/2}(Y;S)\\cap H_{1}^{-})\n$$\n$$\n\\Phi\\mapsto (\\slashed{D}^{+}_{\\hat{\\mathfrak{q}},A_{0}}\\Phi,\\pi^{-}(\\Phi|_{Y}))\n$$\nwhere $\\slashed{D}^{+}_{\\hat{\\mathfrak{q}},A_{0}}$ is a perturbed Dirac operator over $M$ which equals $\\frac{d}{dt}+\\slashed{D}_{\\mathfrak{q},B_{0}}$ near the boundary; $H^{-}_{1}$ (resp. $H^{+}_{1}$) is the closure in $L^{2}(Y;S)$ of the eigenvectors of $\\slashed{D}_{\\mathfrak{q},B_{0}}$ with negative (resp. positive) eigenvalue; $\\pi^{-}$ is the projection to $L^{2}_{k-1\/2}(Y;S)\\cap H_{1}^{-}$ with kernel $H^{+}_{1}$.\n\\begin{lem}\\label{height as index}\nFor any nice perturbation $\\mathfrak{q}$, we have\n\\begin{equation}\\label{height as eta invariant}\n\\operatorname{ht}(\\mathfrak{q})=-2\\operatorname{ind}_{\\mathds{C}}D_{\\mathfrak{q}}-\\tfrac{\\operatorname{sign}(M)}{4}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nBy the same argument as Page 508 of the book, we can identify $\\operatorname{grad}(M,[\\mathfrak{a}_{0}])$ with the index of the Fredholm operator (24.41) in the book. A further deformation identifies this index with the index of the operator  $D_{\\mathfrak{q}}\\oplus B$, where $B$ is the Fredholm operator\n$$\nL^{2}_{k}(M;iT^{*}M)\\rightarrow L^{2}_{k-1}(M;i\\mathds{R}\\oplus i\\wedge^{2}_{+}T^{*}M)\\oplus L^{2}_{k-1\/2}(Y;i\\mathds{R})\\oplus C^{-}\n$$\n$$\n\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha,\\langle \\alpha,\\vec{v}\\rangle,\\alpha^{-}).\n$$\nHere $C^{-}\\subset (\\operatorname{ker}d^{*}\\cap L^{2}_{k-1\/2}(Y;iT^{*}Y))$ denotes the negative eigenspace of the operator $*d$ and $\\alpha^{-}\\in C^{-}$ denotes projection of $\\alpha|_{Y}$.  By Lemma 24.8.1 of the book, we have $\\operatorname{ind}_{\\mathds{R}}B=-b^{+}_{2}(M)-1$. Therefore, we get\n$$\n\\operatorname{grad}(M,[\\mathfrak{a}_{0}])=2\\operatorname{ind}_{\\mathds{C}}D_{q}-b_{2}^{+}(M)-1.\n$$\nBy (\\ref{absolute grading}), this implies the lemma.\n\\end{proof}\nNow consider the following subset of $\\mathds{Q}$\n$$\n\\mathfrak{m}(Y,\\mathfrak{s})=\\{a\\in \\mathds{Q}|\\, a=[-\\frac{\\operatorname{sign}(M)}{8}]\\in \\mathds{Q}\/\\mathds{Z}\\}.\n$$\n\\begin{rmk}$\\mathfrak{m}(Y,\\mathfrak{s})$ is actually determined by the Rohlin invariant $\\rho(Y,\\mathfrak{s})$ and hence independent with the choice of $M$.\\end{rmk}\n\n\n\\begin{pro}\\label{height of nice perturbation}\nFor any $e\\in \\mathfrak{m}(Y,\\mathfrak{s})$, there exists a nice perturbation $\\mathfrak{q}$ with $\\tfrac{\\operatorname{ht}(\\mathfrak{q})}{2}=e$.\n\\end{pro}\n\\begin{proof}\nLet $\\{\\psi_{n}|\\,n\\in \\mathds{Z}_{\\geq 0}\\}$ be a complete, orthonormal set of eigenvectors of  $\\slashed{D}_{B_{0}}$. Let the eigenvalue of $\\psi_{n}$ be $\\lambda_{n}'$. For each $n$, we consider the the function\n$$\nf_{n}:\\mathcal{C}_{k-1\/2}(Y)\\rightarrow \\mathds{R}\n$$\n$$\n(B_{0}+a,\\Psi)\\mapsto |\\langle e^{i\\xi}\\Psi,\\psi_{n}\\rangle_{L^{2}}|^{2}\n$$\nwhere $\\xi:Y\\rightarrow\\mathds{R}$ is the unique solution of\n$$\ni\\Delta \\xi=d^{*}da,\\ \\int_{Y}\\xi=0.\n$$\nOne can prove that $f_{n}$ is invariant under the action of $\\mathcal{G}_{k+1\/2}(Y)$. We denote by $\\mathfrak{q}_{n}$ the formal gradient of $f_{n}$. A simple calculation shows that\n$$\n\\mathcal{D}_{(B_{0},0)}q_{n}^{1}(0,\\Psi)=2\\langle \\Psi,\\psi_{n}\\rangle_{L^{2}}\\cdot \\psi_{n}.\n$$\nWe let $\\mathfrak{q}'=\\mathop{\\sum}\\limits_{n=0}^{+\\infty}c_{n}\\mathfrak{q}_{n}$, where $\\{c_{n}\\}$ is a sequence of real numbers. We require $|c_{n}|$ decreasing to $0$ fast enough so that $\\mathfrak{q}'$ is a tame-perturbation (see Definition 10.5.1 of the book). Now consider the perturbed Dirac operator $\\slashed{D}_{\\mathfrak{q}',B_{0}}$ (see (\\ref{perturbed dirac})). Its eigenvalues are of the form $\\lambda'_{n}+2c_{n}$ and the corresponding eigenvector is just $\\psi_{n}$. By choosing a generic sequence $\\{c_{n}\\}$, we can assume\n$$\n\\lambda'_{n}+2c_{n}\\neq \\lambda'_{m}+c_{m},\\ \\forall n\\neq m\\text{ and } \\lambda'_{n}+2c_{n}\\neq 0,\\ \\forall n\\in\\mathds{Z}_{\\geq 0}.\n$$\nNote that the number $\n-\\operatorname{ind}_{\\mathds{C}}D_{\\mathfrak{q}'}-\\tfrac{\\operatorname{sign}(M)}{8}\n$\nalways belongs to $\\mathfrak{m}(Y,\\mathfrak{s})$. Moreover, as we varies $\\{c_{n}\\}$, this number changes by the spectral flow of $\\slashed{D}_{\\mathfrak{q}',B_{0}}$. Therefore, by choosing suitable $\\{c_{n}\\}$, we may assume that\n$$\ne=-\\operatorname{ind}_{\\mathds{C}}D_{\\mathfrak{q}'}-\\frac{\\operatorname{sign}(M)}{8}.\n$$\nUnder this perturbation $\\mathfrak{q}'$, the reducible critical points are just $[(B_{0},0,\\psi_{n})]$ with $n\\geq 0$. All of them are non-degenerate by \\cite[Proposition 12.2.5]{KM}. Therefore, by the compactness result of the critical points, we can find $\\epsilon>0$ such that the gauge invariant open subset\n$$\nU(\\epsilon)=\\{(B,\\Phi)| \\|\\Phi\\|_{L^{2}}<\\epsilon\\} \\subset \\mathcal{C}_{k-1\/2}(Y)\n$$\ncontains no irreducible critical point. Now consider the Banach space\n$$\\mathcal{P}(U(\\epsilon)):=\\{\\mathfrak{q}''\\in \\mathcal{P}|\\  \\mathfrak{q}''|_{U(\\epsilon)}= 0\\},$$\nwhere $\\mathcal{P}$ is the large Banach space of tame perturbations constructed in Theorem 11.6.1 of the book. By repeating the proof of Theorem 15.1.1 of the book, we can find a perturbation $\\mathfrak{q}''\\in \\mathcal{P}(U(\\epsilon))$ such that the perturbation $\\mathfrak{q}=\\mathfrak{q}''+\\mathfrak{q}'$ is admissible. Since both $\\mathfrak{q}''$ and $\\mathfrak{q}'$ vanishes on $\\mathcal{C}_{k-1\/2}^{\\text{red}}(Y)$, the perturbation $\\mathfrak{q}$ is nice. Moreover, since $\\mathfrak{q}''$ vanishes on $U(\\epsilon)$, we have $D_{\\mathfrak{q}}=D_{\\mathfrak{q}'}$. By Lemma \\ref{height as eta invariant}, we have\n$$\n\\frac{\\operatorname{ht}(\\mathfrak{q})}{2}=-\\operatorname{ind}_{\\mathds{C}}D_{\\mathfrak{q}}-\\tfrac{\\operatorname{sign}(M)}{8}=-\\operatorname{ind}_{\\mathds{C}}D_{\\mathfrak{q}'}-\\tfrac{\\operatorname{sign}(M)}{8}=e.\n$$This finishes the proof.\\end{proof}\n\n\\begin{lem}\\label{alternative defi of Froyshov}\nSuppose $\\mathfrak{q}$ is a nice perturbation with $\\operatorname{ht}(\\mathfrak{q})<-2\\operatorname{h}(Y,\\mathfrak{s})$. Then we have\n\\begin{equation}\\label{Froyshov 1}\n\\begin{split}\n-2\\operatorname{h}(Y,\\mathfrak{s})=&\\inf\\{\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{a}_{j}])|\\,j\\geq 0;\\   \\nexists\\ n,m_{1},...,m_{l}\\in \\mathds{Z}_{\\neq 0} \\text{ and }[\\mathfrak{b}_{1}],...,[\\mathfrak{b}_{l}]\\in \\mathfrak{C}^{o} \\text{ s.t. }\\\\\n&\\partial^{o}_{o}(m_{1}[\\mathfrak{b}_{1}]+...+m_{j}[\\mathfrak{b}_{l}])=0 \\text{ and } \\partial^{o}_{s}(m_{1}[\\mathfrak{b}_{1}]+...+m_{j}[\\mathfrak{b}_{l}])=n[\\mathfrak{a}_{j}]\\}.\n\\end{split}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nFor the grading reason, all the maps $\\bar{\\partial}^{*}_{*}$ vanish. As a result, the set\n$$\\{[e_{[\\mathfrak{a}_{j}]}]|\\,j\\in \\mathds{Z}\\}$$ is a basis of $\\widebar{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$. For $j\\geq 0$, the map $i_{*}$ sends $$[e_{[\\mathfrak{a}_{j}]}]\\in \\widebar{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})$$ to $$[e_{[\\mathfrak{a}_{j}]}]\\in \\widecheck{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q}).$$ Since we have $\\operatorname{ht}(\\mathfrak{q})<-2\\operatorname{h}(Y;\\mathfrak{s})$, the set\n$$\nS=\\{j|j\\geq 0,\\ [e_{[\\mathfrak{a}_{j}]}]\\neq 0\\in \\widecheck{HM}_{*}(Y,\\mathfrak{s};\\mathds{Q})\\}\n$$\ndoes not equals $\\mathds{Z}_{\\geq 0}$ and we have\n\\begin{equation}\\label{Froyshov 2}-2\\operatorname{h}(Y,\\mathfrak{s})=\\inf\\{\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{a}_{j}])|j\\in S\\}.\\end{equation}\nSince we have\n$$\n\\check{\\partial}=\\left(\\begin{array} {cc}\n \\partial^{o}_{o}  & 0\\\\\n \\partial^{o}_{s} & 0\n\\end{array}\\right).\n$$ in the current case, (\\ref{Froyshov 1}) and (\\ref{Froyshov 2}) coincide with each other. This finishes the proof of the lemma.\n\\end{proof}\n\n\n\\subsection{Linear analysis on end-periodic manifolds}\n\nIn this subsection, we will set up the appropriate Sobolev spaces on end-periodic manifolds and review the related Fredholm theory. Our construction is inspired from \\cite{Taubes} and \\cite{MRS}.\n\nLet $E$ be an end-periodic bundle (over $\\tilde{X},X_{+},M_{+} \\text{ or }Z_{+}$) equipped with an end-periodic metric  $|\\cdot|$ and an end-periodic connection $\\nabla$ (see \\cite{Taubes} for definition). For any $j,p\\in \\mathds{Z}_{\\geq 0}$,  we can define the unweighted Sobolev norm of a smooth section $s$ in the usual way:\n\\begin{equation}\\label{unweighted sobolev space}\n\\|s\\|_ {L^{p}_{j}}:=(\\mathop{\\Sigma}\\limits_{i=0}^{j}\\int|\\nabla^{(i)}s|^{p} d\\operatorname{vol})^{\\frac{1}{p}}.\n\\end{equation}\n(We can also define the $L^{p}_{j}$ norm for negative $j$ using integration.)\n\\begin{rmk}\nOther then a trivial real or complex line bundle, which we denote by $\\mathds{R},\\mathds{C}$ respectively, two other types of end-periodic bundle will be considered: the spinor bundle $S^{\\pm}$ (associated to spin structures) and the bundle of differential forms. Both of them have a canonical metric. As for the connection, we use the spin connection for the former and the Levi-Civita connection for the latter.\n\\end{rmk}\n\nIn general, the differential operators that we will consider do not have Fredholm properties under the norms defined in \\ref{unweighted sobolev space}. Therefore, we need to use the weighted Sobolev norms instead. To define them, recall that we have a harmonic map $f:X\\rightarrow S^{1}$ corresponding to a generator of $H^{1}(X;\\mathds{Z})$. We lift $f$ to a function $\\tilde{f}:\\tilde{X}\\rightarrow \\mathds{R}$ satisfying\n$$\nf^{-1}([-1,1])\\subset \\mathop{\\cup}\\limits_{n=-N}^{N}W_{n}\n\\text{ for some }N\\gg 0.$$\nNow consider the following smooth cut-off functions:\n\\begin{itemize}\n\\item $\\tau_{0}:\\tilde{X}\\rightarrow [0,+\\infty)$: a function that equals $|f|$ on $\\tilde{X}\\setminus \\mathop{\\cup}\\limits_{n=-N}^{N} W_{n}$;\n\\item $\\tau_{1}:X_{+}\\rightarrow [0,+\\infty)$: the restriction of $\\tau_{0}$;\n\\item $\\tau_{2}:M_{+}\\rightarrow [0,+\\infty)$: an extension of $\\tau_{1}$;\n\\item $\\tau_{3}:Z_{+}\\rightarrow [0,+\\infty)$: an extension of $\\tau_{1}$ with the property that $$\\tau_{2}(t,y)=|t|,\\ \\forall (t,y)\\in (-\\infty,-1]\\times Y.$$\n\\end{itemize}\n\\begin{defi}\nFor $\\delta\\in \\mathds{R},j\\in \\mathds{Z},p\\in \\mathds{Z}_{\\geq 0}$, we define the weighted Sobolev norm of a smooth section $s$ of $E$ in different ways depending on the underlying manifold:\n\\begin{itemize}\n\\item Over $X_{+}$, we set $\\|s\\|_ {L^{p}_{j,\\delta}}=\\|e^{\\delta\\cdot \\tau_{1}}\\cdot s\\|_ {L^{p}_{j}}$;\n\\item Over $M_{+}$, we set $\\|s\\|_ {L^{p}_{j,\\delta}}=\\|e^{\\delta\\cdot \\tau_{2}}\\cdot s\\|_ {L^{p}_{j}}$;\n\\item Over $\\tilde{X}$, we set $\\|s\\|_ {L^{p}_{j;-\\delta,\\delta}}=\\|e^{\\delta\\cdot \\tau_{0}}\\cdot s\\|_ {L^{p}_{j}}$;\n\\item Over $Z_{+}$, we set $\\|s\\|_ {L^{p}_{j;-\\delta,\\delta}}=\\|e^{\\delta\\cdot \\tau_{3}}\\cdot s\\|_ {L^{p}_{j}}$.\n\\end{itemize}\n(Note that we use two weight indices for manifolds $\\tilde{X}$ and $Z_{+}$ because they both have two ends.) We denote the corresponding Sobolev space respectively by $$L^{2}_{j,\\delta}(X_{+};E),\\ L^{2}_{j,\\delta}(M_{+};E),\\ L^{2}_{j;-\\delta,\\delta}(\\tilde{X};E) \\text{ and } L^{2}_{j;-\\delta,\\ \\delta}(Z_{+};E).$$\nWe remove $j$ from our notations when it equals $0$.\nWe sometimes also suppress the bundle $E$ when it is clear from the context.\n\\end{defi}\n\n\n\nThe following lemma is a straightforward corollary of \\cite[Lemma 5.2]{Taubes}. It asserts that one can control the weighted Sobolev norm of a function using the weighted Sobolev norm of its derivative. (Although \\cite{Taubes} only stated the result for smooth functions, we can prove the general case easily using standard arguments, i.e., approximating a Sobolev function by smooth functions.)\n\n\\begin{lem} \\label{Taubes's lemma}\nFor any $\\delta>0,j\\geq 0$, we can find a positive constant $C$ with the following significance:\n\\begin{enumerate}\n\\item For any $u\\in L^{2}_{1,\\operatorname{loc}}(X_{+};\\mathds{R})$ with $\\|du\\|_{L^{2}_{j,\\delta}}<\\infty$, there exists a unique number $\\bar{u}\\in \\mathds{R}$ such that $\\|u-\\bar{u}\\|_{L^{2}_{j+1,\\delta}}<\\infty$. Moreover, in this case we have $$\\|u-\\bar{u}\\|_{L^{2}_{j+1,\\delta}}\\leq C\\|d\\bar{u}\\|_{L^{2}_{j,\\delta}}.$$\n\\item Fix a smooth function $$\\tau_{4}:Z_{+}\\rightarrow [0,1] \\text{ with } \\tau_{4}|_{Z}=0, \\tau_{4}|_{W_{i}}=1\\  \\forall i\\geq 1.$$Then for any $u\\in L^{2}_{1,\\operatorname{loc}}(Z_{+};\\mathds{R})$ with $\\|du\\|_{L^{2}_{j;-\\delta,\\delta}}<\\infty$, there exists unique numbers $\\bar{u},\\bar{\\bar{u}}\\in \\mathds{R}$ such that $\\|u-\\bar{u}-\\bar{\\bar{u}}\\cdot\\tau_{4}\\|_{L^{2}_{j+1;-\\delta,\\delta}}<\\infty$. In this case we have $$\\|u-\\bar{u}-\\bar{\\bar{u}}\\cdot\\tau_{4}\\|_{L^{2}_{j+1;-\\delta,\\delta}}\\leq C\\|du\\|_{L^{2}_{j;-\\delta,\\delta}}.$$\n\\end{enumerate}\n\\end{lem}\n\n\nNext, we summarize the Sobolev embedding and multiplication theorems. We focus on the manifold $X_{+}$ (although similar results holds other manifolds) because that will be our main concern. The proofs are straightforwardly adapted from the unweighted case (Theorem 13.2.1 and Theorem 13.2.2 of the book) and the cylindrical end case\n(\\cite[Proposition 2.9, Proposition 2.10]{francescolin}) so we omit them.\n\\begin{pro}\\label{Sobolev embedding}\nLet $E$ be an end-periodic bundle over $X_{+}$. There is a continuous inclusion\n$$\nL^{p}_{j,\\delta}(X_{+};E)\\rightarrow L^{q}_{l,\\delta'}(X_{+};E)\n$$\nfor $j\\geq l,\\ \\delta\\geq \\delta'\\geq 0,\\  p \\leq q$ and $(j-4\/p)\\geq (l-4\/q)$. This embedding is compact when $j>l,\\ \\delta>\\delta'$ and  $(j-4\/p)> (l-4\/q)$.\n\\end{pro}\n\\begin{pro}\\label{Sobolev multiplication}\nLet $E,F$ be two end-periodic bundles over $X_{+}$.\nSuppose $\\delta+\\delta'\\geq \\delta'',\\ j,l\\geq m$ and $1\/p+1\/q\\geq 1\/r$, with $\\delta,\\delta',\\delta''\\geq 0$ and $p,q,r>1$. Then the multiplication\n$$\nL^{p}_{j,\\delta}(X_{+};E)\\times L^{q}_{l,\\delta'}(X_{+};F)\\rightarrow L^{r}_{m,\\delta''}(X_{+};E\\otimes F)\n$$\nis continuous in any of the following three cases:\n\\begin{enumerate}\n\\item \\begin{enumerate}\n\\item $(j-4\/p)+(l-4\/q)\\geq m-4\/r,$ and\n\\item $j-4\/p<0,$ and\n\\item $l-4\/q<0$;\n\\end{enumerate}\n\\hspace{-4mm} or\n\\item \\begin{enumerate}\n\\item $\\min \\{j-4\/p,l-4\/q\\}\\geq m-4\/r,$ and\n\\item either $j-4\/p>0$ or $l-n\/q>0$;\n\\end{enumerate}\n\\hspace{-4mm} or\n\\item \\begin{enumerate}\n\\item $\\min \\{j-4\/p,l-4\/q\\}> m-4\/r,$ and\n\\item either $j-4\/p=0$ or $l-4\/q=0$.\n\\end{enumerate}\n\n\\end{enumerate}\nWhen the map is continuous, it is a compact operator as a function of second variable for fixed first variable provided $l>m$ and $l-4\/q>m-4\/r$.\n\\end{pro}\n\n\nThe following corollary will be very useful because the differential operators we are going to consider can often be composed into the sum of a first-order, linear operator with a zeroth-order, quadratic operator.\n\\begin{cor}\nFor any $j>2,\\delta>0$, the multiplication map $$L^{2}_{j,\\delta}(X_{+};E)\\times L^{2}_{j,\\delta}(X_{+};F)\\rightarrow L^{2}_{j-1,\\delta}(X_{+};E\\otimes F)$$ is compact.\n\\end{cor}\n\\begin{proof}\nBy Proposition \\ref{Sobolev multiplication}, this map factors through the natural inclusion $$L^{2}_{j,2\\delta}(X_{+};E\\otimes F)\\rightarrow L^{2}_{j-1,\\delta}(X_{+};E\\otimes F),$$\nwhich is compact by Proposition \\ref{Sobolev embedding}.\n\\end{proof}\nNow we start discussing the related Fredholm theory.\n\\begin{pro}\\label{laplace equation}\nThere exists a small $\\delta_{0}>0$ such that for any $j\\in \\mathds{Z}_{\\geq 0}$ and $\\delta\\in (0,\\delta_{0})$, we have the following results:\n\\begin{enumerate}[(i)]\n\\item The operator\n$$\n\\Delta(\\tilde{X};-\\delta,\\delta):L^{2}_{j+2;-\\delta,\\delta}(\\tilde{X};\\mathds{R})\\rightarrow L^{2}_{j;-\\delta,\\delta}(\\tilde{X};\\mathds{R})$$\n$$u \\mapsto \\Delta u\n$$\nis a Fredholm operator with trivial kernel and two dimensional cokernel. The same result holds for the manifold $Z_{+}$.\n\\item The operator\n$$\n\\Delta(M_{+};\\delta):L^{2}_{j+2,\\delta}(M_{+};\\mathds{R})\\rightarrow L^{2}_{j,\\delta}(M_{+};\\mathds{R})$$\n$$u\\mapsto \\Delta u\n$$\nis a Fredholm operator with trivial kernel and 1-dimensional cokernel.\n\\item The operator\n$$\n\\Delta(X_{+};\\delta):L^{2}_{j+2,\\delta}(X_{+};\\mathds{R})\\rightarrow L^{2}_{j,\\delta}(X_{+};\\mathds{R})\\oplus L^{2}_{j+1\/2}(Y;\\mathds{R})$$ $$u\\mapsto (\\Delta u,\\langle du,\\vec{v}\\rangle)$$\nis Fredholm with trivial kernel and $1$-dimensional cokernel, where $\\vec{v}$ denotes the inward normal vector on the boundary.\n\\end{enumerate}\n\\end{pro}\nProposition \\ref{laplace equation} will be proved in the appendix.\n\\begin{lem}\\label{half De rham complex}\nThere exists a constant $\\delta_{1}\\in (0,\\delta_{0})$ such that for any $j\\in \\mathds{Z}_{\\geq 0}$ and $\\delta\\in(0,\\delta_{1})$, we have the following results:\n\\begin{enumerate}[(i)]\n\\item For any $w\\in L^{2}_{j;-\\delta,\\delta}(Z_{+};\\mathds{R})$ with $\n\\int_{Z_{+}} w\\,d\\operatorname{vol}=0,\n$\nwe can find $u\\in L^{2}_{j+2,\\operatorname{loc}}(Z_{+};\\mathds{R})$ satisfying\n$$\n|du|_{L^{2}_{j+1;-\\delta,\\delta}}<\\infty,\\ \\Delta u=w.\n$$\n\\item The operator $$\nD(M_{+}):L^{2}_{j+1,\\delta}(M_{+};T^{*}M_{+})\\rightarrow  L^{2}_{j,\\delta}(M_{+};\\mathds{R}\\oplus \\wedge_{2}^{+}T^{*}M_{+} ):\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha)\n$$ is Fredholm with index  $-(b_{2}^{+}(M)+1)$;\n\\item The operator $$D(Z_{+}):L^{2}_{j+1;-\\delta,\\delta}(Z_{+};T^{*}Z_{+})\\rightarrow L^{2}_{j;-\\delta,\\delta}(Z_{+};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}Z_{+}):\n\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha)\n$$ is Fredholm with trivial kernel and $1$-dimensional cokernel. Its image equals $$\\{(w,\\beta)|\\ \\int_{Z_{+}}w \\,d\\operatorname{vol}=0\\}.$$\n\\item The operator $$D(X_{+}):L_{j+1,\\delta}^{2}(X_{+};T^{*}X_{+})\\rightarrow L^{2}_{j,\\delta}(X_{+};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}X_{+})\\oplus  L^{2}_{j+1\/2}(Y;\\mathds{R})\\oplus C^{+}$$\ngiven by\n\\begin{equation}\\label{half De Rham with boundary}\n\\alpha \\mapsto (d^{*}\\alpha,d^{+}\\alpha,\\langle \\alpha,\\vec{v}\\rangle,\\pi^{+}(\\alpha|_{Y}))\n\\end{equation}\nis Fredholm with trivial kernel and one dimensional cokernel, which can be canonically identified with $\\mathds{R}$. Here $C^{+}$ (resp. $C^{-}$) is the closure in $ L^{2}_{j+1\/2}(Y;T^{*}Y)\\cap \\operatorname{ker} d^{*}$ of the space spanned by the eigenvectors of $*d$ with positive (resp. negative) eigenvalues and $$\\pi^{+}:L^{2}_{j+1\/2}(Y;iT^{*}Y)\\rightarrow C^{+}$$ is the projection with kernel $C^{-}$.\n\\end{enumerate}\n\n\\end{lem}\n\\begin{proof}\n(i) We consider two vector spaces:\n$$\nV_{1}=\\{u\\in L^{2}_{j+2,\\text{loc}}(Z_{+};\\mathds{R})|\\ \\|du\\|_{L^{2}_{j+1;-\\delta,\\delta}}<\\infty\\}\n$$\n$$\nV_{2}=\\{w\\in L^{2}_{j;-\\delta,\\delta}(Z_{+};\\mathds{R})|\\ \\int_{Z_{+}} w\\,d\\text{vol}=0\\}.\n$$\nNow assume $\\delta \\in (0,\\delta_{0})$, where $\\delta_{0}$ is the constant in Proposition \\ref{laplace equation}. By Lemma \\ref{Taubes's lemma}, we also have \\begin{equation}\\label{equivalent defi}V_{1}=L^{2}_{j+2;-\\delta,\\delta}(Z_{+};\\mathds{R})\\oplus\\mathds{R}\\oplus \\mathds{R}\\tau_{4}.\n\\end{equation}\nUsing this identification and integration by part, we can show that $\\Delta u\\in V_{2}$ for any $u\\in V_{1}$. In other words, we have a well defined operator\n$$\n\\Delta: V_{1}\\rightarrow V_{2}.\n$$\nComparing the domain and target of this operator with the one in Proposition \\ref{laplace equation} (1), we see that it is a Fredholm operator with index $1$. To finish the proof, we just need to prove kernel of $\\Delta$ consists only of constant functions. This is a simple consequence of the maximum principle, noticing that all functions in $V_{1}$ are bounded (because of (\\ref{equivalent defi})).\n\n(ii) Consider the operator\n$$\nd^{+}:L^{2}_{j+1,\\delta}(M_{+};T^{*}M_{+})\\rightarrow L^{2}_{j,\\delta}(M_{+};\\wedge_{+}^{2}T^{*}M_{+} ).\n$$\nBy \\cite[Proposition 5.1]{Taubes}, when $\\delta_{1}>0$ is small enough, both the kernel and the image of this operator  (which we denote by $V_{3}$ and $V_{4}$ respectively) are closed with the following properties:\n\\begin{equation}\\label{1st homology vanish}\nV_{3}\\cong L^{2}_{j+2,\\delta}(M_{+};\\mathds{R}):du\\leftrightarrow u;\n\\end{equation}\n\\begin{equation}\\label{2nd homology}\n\\text{dim}(L^{2}_{j,\\delta}(M_{+};\\wedge_{2}^{+}T^{*}M_{+} )\/V_{4})=b_{2}^{+}(M).\n\\end{equation}\n\nBy (\\ref{1st homology vanish}), the operator\n$$\nV_{3}\\rightarrow L^{2}_{j-1,\\delta}(M_{+};\\mathds{R}): \\alpha\\mapsto d^{*}\\alpha.\n$$\nis essentially the same with the operator $\\Delta(M_{+},\\delta)$ in Proposition \\ref{laplace equation}, which is Fredholm with index $-1$. This  implies that the operator\n$$\nL^{2}_{j,\\delta}(M_{+};T^{*}M_{+})\\rightarrow L^{2}_{j-1,\\delta}(M_{+};\\mathds{R})\\oplus V_{4}:\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha)\n$$\nis also Fredholm with the same index. Therefore, by (\\ref{2nd homology}), the operator\n$$\nL^{2}_{j+1,\\delta}(M_{+};T^{*}M_{+})\\rightarrow L^{2}_{j,\\delta}(M_{+};\\mathds{R})\\oplus L^{2}_{j,\\delta}(M_{+};\\wedge^{2}_{+}T^{*}M_{+} ):\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha)\n$$\nis Fredholm with index $-(b_{2}^{+}(M)+1)$.\n\n(iii) To apply the excision principle of the index, we consider the manifold $M_{-}=Z\\cup_{Y}\\bar{M}$. (Recall that $\\bar{M}$ is the orentation reversal of $M$.) We choose a  function\n$$\\tau:M_{-}\\rightarrow [0,+\\infty)\\text{ with }\\tau(t,y)=|t|,\\ \\forall(t,y)\\in (-\\infty,-1]\\times Y$$\nand define the weighted Sobolev norm of a section $s$ over $M_{-}$ as\n$$\n\\|s\\|_{L^{2}_{j,-\\delta}}:=\\|e^{\\delta\\tau}s\\|_{L^{2}_{j}}\n.$$\nBy similar argument as (ii), one can show that the operator\n$$\nL^{2}_{j+1,-\\delta}(M_{-};T^{*}M_{-})\\rightarrow L^{2}_{j,-\\delta}(M_{-};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}M_{-} ) :\\alpha\\mapsto (d^{*}\\alpha,d^{+}\\alpha)\n$$\nis Fredholm with index $-(b_{2}^{+}(\\bar{M})+1)$. Notice that we have the decompositions\n$$\nM_{+}=M\\cup_{Y}X_{+},\\ M_{-}=Z\\cup_{Y}\\bar{M},\\ Z_{+}= Z\\cup_{Y}X_{+}.\n$$\nBy an exision argument, we see that the operator\n$$\n(d^{*},d^{+}):L^{2}_{j+1;-\\delta,\\delta}(Z_{+},T^{*}Z_{+})\\rightarrow L^{2}_{j;-\\delta,\\delta}(Z_{+},\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}Z_{+})\n$$\nis Fredholm with index\n$$\n-(1+b_{2}^{+}(M))-(1+b_{2}^{+}(\\bar{M}))+(1+b_{2}^{+}(M\\cup_{Y}\\bar{M}))=-1.\n$$\nHaving proved this fact, we are left to show that the kernel is trivial. Suppose we have $$\\alpha\\in L^{2}_{j+1;-\\delta,\\delta}(Z_{+};T^{*}Z_{+})\\text{ with }d^{*}\\alpha=0,d^{+}\\alpha=0.$$ Integrating by part, we get $d\\alpha=0$. Since $H^{1}(Z_{+};\\mathds{R})=0$, we have $\\alpha=du$ for some harmonic function $u$. Notice that $\\|du\\|_{L^{2}_{j+1,-\\delta,\\delta}}<\\infty$. By Lemma \\ref{Taubes's lemma}, the function $u$ is bounded. By the maximal principle, $u$ is a constant, which implies $\\alpha=du=0$.\n\n(iv) Consider the operator\n$$D(\\bar{M}):L_{j+1}^{2}(\\bar{M};T^{*}\\bar{M})\\rightarrow L^{2}_{j}(\\bar{M};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}\\bar{M})\\oplus  L^{2}_{j+1\/2}(Y;\\mathds{R})\\oplus C^{+}$$\ndefined by the same formula as (\\ref{half De Rham with boundary}). By Lemma 24.8.1 of the book, $D(\\bar{M})$ is a Fredholm operator with index $-b^{+}(\\bar{M})-1$. We note that the boundary of $\\bar{M}$ is $-Y$ while the boundary of the manifold in that Lemma is $Y$, this explains the reason we use $C^{+}$ while the book use $C^{-}$. We also note that the additional term ``$-1$'' in our index formula comes from the $1$-dimensional cokernel of the map\n$$D(\\bar{M}):L_{j+1}^{2}(\\bar{M};T^{*}\\bar{M})\\rightarrow L^{2}_{j}(\\bar{M};\\mathds{R}\\oplus i\\wedge^{2}_{+}T^{*}\\bar{M})\\oplus  L^{2}_{j+1\/2}(Y;\\mathds{R})$$\n$$\\alpha\\mapsto (d^{*}\\alpha, d^{+}\\alpha,\\langle\\alpha,\\vec{v}\\rangle ).$$ By an excision argument involving the operators $D(X_{+}),D(\\bar{M}),D(M_{+})$ and the operator\n$$d^{*}\\oplus d^{+}:L^{2}_{j+1}(M\\cup_{Y}\\bar{M};T^{*}(M\\cup_{Y}\\bar{M}))\\rightarrow L^{2}_{j}(M\\cup_{Y}\\bar{M};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}(M\\cup_{Y}\\bar{M})),$$\nwe can prove that  $D(X_{+})$ is Fredholm with index $-1$. Now suppose $\\alpha\\in\\operatorname{ker}D(X_{+})$. Then by the integration by part argument on page 502 of the book, we can prove $d\\alpha=0$. Since $H^{1}(X_{+};\\mathds{R})=0$, we have $\\alpha=df$ for some local $L^{2}_{j+1}$-function $f$. By Lemma \\ref{Taubes's lemma}, we can assume $\\|f\\|_{L^{2}_{j+1,\\delta}}<\\infty$ after adding some constant function. Then $f$ satisfies\n$\\Delta f=0,\\ \\langle df,\\vec{v}\\rangle=0$.\nBy Lemma \\ref{laplace equation}, we see that $f$ (hence also $\\alpha$) equals $0$. We have proved that the kernel is trivial, which implies that the cokernel is $1$-dimensional. Using integration by part again, one can easily see that a necessary condition for an element $$(w_{1},\\beta,w_{2},\\alpha' )\\in  L^{2}_{j,\\delta}(X_{+};\\mathds{R}\\oplus \\wedge^{2}_{+}T^{*}X_{+})\\oplus  L^{2}_{j+1\/2}(Y;\\mathds{R})\\oplus C^{+}$$ belonging to $\\operatorname{im}D(X_{+})$ is $$\\int_{X_{+}}w_{1}d\\text{vol}+\\int_{Y}w_{2}d\\text{vol}=0.$$ Since the cokernel is $1$-dimensional, we see that this is also a sufficient condition. Moreover, we have a canonical isomorphism $$\\operatorname{coker}D(X_{+})\\cong \\mathds{R}:\\ [(w_{1},\\beta,w_{2},\\alpha' )]\\leftrightarrow \\int_{X_{+}}w_{1}d\\text{vol}+\\int_{Y}w_{2}d\\text{vol}.$$\n\\end{proof}\n\nNow we study the Fredholm properties related to the linearized Seiberg-Witten equations. Recall that we chose an ``admissible metric'' $g_{X}$ on $X$ (see Assumption \\ref{admissible metric}). Under this assumption, we have the following proposition.\n\n\\begin{pro}[\\cite{MRS}]\\label{Dirac operator is Fredholm}There exists a number $\\delta_{2}>0$ such that for any $\\delta\\in (-\\delta_{2},\\delta_{2}),j\\in \\mathds{Z}_{\\geq 0}$, the end-periodic Dirac operator\n$$\n\\slashed{D}^{+}_{A_{0}}:L^{2}_{j+1,\\delta}(M^{+};S^{+})\\rightarrow L^{2}_{j,\\delta}(M^{+};S^{-})\n$$\nis Fredholm. Moreover, the number $$\\operatorname{ind}_{\\mathds{C}}(\\slashed{D}^{+}_{A_{0}}(M_{+}))+\\frac{\\operatorname{sign}(M)}{8}$$ is an invariant of the pair $(X,g_{X})$, which we denote by $w(X,g_{X},0)$.\n\\end{pro}\n\nTo end this subsection, let us consider the Atiyah-Patodi-Singer boundary problem on the end-periodic manifold $X_{+}$. This will be essential in our study of local structure of the Seiberg-Witten moduli space. To simplify our notation,  we denote the following bundles over $X_{+}$\n$$iT^{*}X_{+}\\oplus S^{+}\\text{ and }i(\\mathds{R}\\oplus \\wedge_{+}^{2}T^{*}X_{+})\\oplus S^{-}$$ respectively by $E_{1}$ and $E_{2}$. We also write $F_{1}$ for the bundle $i(\\mathds{R}\\oplus T^{*}Y)\\oplus S$ over $Y$.\n\nRecall that $k$ is a fixed integer greater than $2$. First consider the linear operator\n$$D=D_{0}+K:L^{2}_{k,\\delta}(X_{+};E_{1})\\rightarrow L^{2}_{k-1,\\delta}(X_{+};E_{2}),$$\nwhere $D_{0}=(d^{*},d^{+},\\slashed{D}_{A_{0}})$ and $K$ is an operator that can be extended to a bounded operator\n$$\nK:L^{2}_{j,\\delta}(X_{+};E_{1})\\rightarrow L^{2}_{j,2\\delta}(X_{+};E_{2})\n$$\nfor any integer $j\\in [-k,k]$. Next, we define the restriction map\n$$\nr:L^{2}_{k,\\delta}(X_{+};E_{1})\\rightarrow L^{2}_{k-1\/2}(Y;F_{1})$$\n$$(a,\\phi)\\mapsto (\\langle a,\\vec{v}\\rangle,a|_{Y},\\phi|_{Y}).$$\n Let $H_{0}^{+}$ (resp. $H_{0}^{-}$) be the closure in $L^{2}_{1\/2}(Y;F_{1})$ of the span of the eigenvectors eigenvalues of operator\n$$\nL_{0}: C^{\\infty}(Y;F_{1})\\rightarrow C^{\\infty}(Y;F_{1})\n$$\n$$\n(u,\\alpha,\\phi)\\mapsto (d^{*}\\alpha,*d\\alpha-du,\\slashed{D}_{A_{0}}\\phi).\n$$\nwith positive (resp. non-positive) eigenvalues. We write $\\Pi_{0}$ for the projection\n$$\nL^{2}_{1\/2}(Y;F_{1})\\rightarrow L^{2}_{1\/2}(Y;F_{1})\n$$\nwith image $H_{0}^{-}$ and kernel $H_{0}^{+}$. It also maps $L^{2}_{s}(Y;F_{1})$ to $L^{2}_{s}(Y;F_{1})$ for all $s$. Consider another projection\n$$\n\\Pi: L^{2}_{1\/2}(Y;F_{1})\\rightarrow L^{2}_{1\/2}(Y;F_{1})$$\nsatisfying\n$$\\Pi (L^{2}_{s}(Y;F_{1}))\\subset L^{2}_{s}(Y;F_{1})$$ for any $s$. We say that $\\Pi$ and $\\Pi_{0}$ are $k$-commonmensurate if the difference $$\n\\Pi-\\Pi_{0}: L^{2}_{j-1\/2}(Y;F_{1})\\rightarrow L^{2}_{j-1\/2}(Y;F_{1})\n$$\nis a compact operator, for all $1\\leq j\\leq k$. We write $H^{-}$ for $\\operatorname{im}(\\Pi)\\subset L^{2}_{1\/2}(Y;F_{1})$ and $H^{+}$ for $\\operatorname{im}(1-\\Pi)\\subset L^{2}_{1\/2}(Y;F_{1}).$\n\\begin{pro}\\label{APS}\nLet $\\delta_{1},\\delta_{2}$ be the constant provided by\nLemma \\ref{half De rham complex} and Proposition \\ref{Dirac operator is Fredholm} respectively. Then for any $\\delta\\in (0,\\min(\\delta_{1},\\delta_{2}))$ and any $1\\leq j\\leq k$, the operator\n$$\nD\\oplus ((1-\\Pi)\\circ r): L^{2}_{j,\\delta}(X_{+};E_{1})\\rightarrow L^{2}_{j-1,\\delta}(X;E_{2})\\oplus (H^{+}\\cap L^{2}_{j-1\/2}(Y;F_{1}))\n$$\nis Fredholm. In addition, if $u_{i}$ is a bounded sequence in $L^{2}_{j,\\delta}(X_{+};E_{1})$ and $Du_{i}$ is Cauchy in $L^{2}_{j-1,\\delta}(X_{+};E_{2})$, then $\\Pi\\circ r(u_{i})$ has a convergent subsequence in $L^{2}_{j-1\/2}(Y;F_{1})$. In particular, the maps $\\Pi\\circ r$ and $(1-\\Pi)\\circ r$ restricted to the kernel of $D$, are respectively, compact and Fredholm.\n\\end{pro}\n\\begin{proof}\nWe consider the following two operators:\n\\begin{itemize}\n\\item The operator over $\\bar{M}$\n$$\n(d^{*},d^{+},\\slashed{D}_{A_{0}})\\oplus (1-\\Pi)\\circ r_{\\bar{M}}: L^{2}_{j}(\\bar{M})\\rightarrow L^{2}_{j-1}(\\bar{M})\\oplus   (H^{+}\\cap L^{2}_{j-1\/2}(Y)),$$ where $r_{\\bar{M}}$ is the restriction map defined similarly as $r$;\n\\item The operator over $M_{+}$\n$$\n(d^{*},d^{+},\\slashed{D}_{A_{0}}):  L^{2}_{j,\\delta}(M_{+})\\rightarrow L^{2}_{j-1,\\delta}(M_{+}).$$\n\\end{itemize}\n By Proposition 17.2.5 of the book, Lemma \\ref{half De rham complex} and Proposition \\ref{Dirac operator is Fredholm}, both of these two operators are Fredholm. Note that they correspond to the operator $D_{0}\\oplus((1-\\Pi)\\circ r)$ on $X_{+}$. We can prove the Fredholm property of $D_{0}\\oplus((1-\\Pi)\\circ r)$ using standard parametrix patching argument (see Page 245 of the book). Since the embedding $L^{2}_{j,2\\delta}\\rightarrow L^{2}_{j-1,\\delta}$ is compact, the operator $D\\oplus((1-\\Pi)\\circ r)$ is a compact perturbation of $D_{0}\\oplus((1-\\Pi)\\circ r)$ and we conclude that $D\\oplus((1-\\Pi)\\circ r)$ is also Fredholm. To prove the second part of the Proposition, we multiply the sequence $\\{u_{i}\\}$ by a bump function $\\beta$ supporting near $\\partial X_{+}$ and follow the argument on Page 304 of the book.\n\\end{proof}\n\\subsection{The invariant $\\lambda_{\\textnormal{SW}}(X)$} Now we review the definition of $\\lambda_{\\textnormal{SW}}(X)$. By \\cite[Lemma 2.1]{MRS}, for a generic pair $(g_{X},\\beta)$ with $\\beta\\in L^{2}_{k+1}(X;iT^{*}X)$, the blown-up Seiberg-Witten moduli space $\\mathcal{M}(X,g_{X},\\beta)$ consisting of the gauge equivalence classes of the triples $$(A,s,\\phi)\\in \\mathcal{A}_{k}(X)\\times \\mathds{R}_{\\geq 0}\\times L^{2}_{k}(X;S^{+}),\\ \\|\\phi\\|_{L^{2}}=1$$ that solve the Seiberg-Witten equation\n\\begin{equation}\\label{blown up seiberg-witten}\n\\left\\{\\begin{array} {cc}\n F_{A}^{+}-s^{2}(\\phi\\phi^{*})_{0}=d^{+}\\beta  \\\\\n \\slashed{D}_{A}^{+}(X,g_{X})(\\phi)=0\n\\end{array}\\right.\n\\end{equation}\nis an oriented manifold of dimension $0$ and contains no reducible points (i.e. triples with $s=0$). We call such $(g_{X},\\beta)$ a regular pair. Now consider the end-periodic (perturbed) Dirac operator\n$$\n\\slashed{D}^{+}_{A_{0}}(M_{+},g_{M_{+}})+\\rho(\\beta'): L^{2}_{1}(M_{+};S^{+})\\rightarrow  L^{2}(M_{+};S^{-}).\n$$\nwhere $\\beta'$ is an imaged valued one form on $M_{+}$ that equals the pull back of $\\beta$ when restricted to $X_{+}$. As proved in \\cite{MRS}, this operator is Fredholm and the quantity\n$$\n\\operatorname{ind}_{\\mathds{C}}(\\slashed{D}^{+}_{A_{0}}(M_{+},g_{M_{+}})+\\rho(\\beta'))+\\frac{\\text{sign}(M)}{8}\n$$\nis an invariant of $(X,g_{X},\\beta)$, which we denote by $w(X,g_{X},\\beta)$.\n\\begin{thm}[\\cite{MRS}] The number $\n\\#\\mathcal{M}(X,g_{X},\\beta)-w(X,g_{X},\\beta)\n$\ndoes not depend on the choice of regular pair $(g_{X},\\beta)$ and hence is an invariant of the manifold of $X$, which we define as $\\lambda_{\\textnormal{SW}}(X)$; morveover, the reduction of $\\lambda_{\\textnormal{SW}}(X)$ modulo $2$ is the Rohlin invariant of $X$.\n\\end{thm}\n\\begin{lem}\\label{casson for psc}\nSuppose $g_{X}$ is a metric with positive scalar curvature. Then the pair $(g_{X},0)$ is regular and $\\lambda_{SW}(X)=-\\omega(X,g_{X},0)$.\n\\end{lem}\n\\begin{proof}\nThis is a simple consequence of the Weitzenb\\\"ock formula.\n\\end{proof}\n\n\\begin{lem}\\label{orientation reversal 2}\nSuppose $X$ admits a metric $g_{X}$ with positive scalar curvature. Then we have\n$\n\\lambda_{\\textnormal{SW}}(X)=-\\lambda_{\\textnormal{SW}}(-X)\n$.\n\\end{lem}\n\\begin{proof}\n By Lemma \\ref{casson for psc}, we have $\\lambda_{\\textnormal{SW}}(X)=w(X,g_{X},0)$. Similarly, $\\lambda_{\\textnormal{SW}}(-X)=w(-X,g_{X},0)$. Notice that $$\\text{sign}(M)+\\text{sign}(\\bar{M})=\\text{sign}(M\\cup_{Y}\\bar{M})=0.$$ By an excision argument relating indices of the Dirac operator on  $M_{+}\\cup \\bar{M}_{+}$ (where $\\bar{M}_{+}$ denotes the orientation reversal of $M_{+}$) and the Dirac operator on $(M\\cup_{Y}\\bar{M})\\cup \\tilde{X}$, we get\n\\begin{equation}\n\\begin{split}\nw(X,g_{X},0)+w(-X,g_{X},0)=&\\operatorname{ind}_{\\mathds{C}}\\slashed{D}^{+}_{A_{0}}(M_{+},g_{M_{+}})+\\operatorname{ind}_{\\mathds{C}}\\slashed{D}^{+}_{A_{0}}(\\bar{M}_{+},g_{M_{+}})\\\\=&\\operatorname{ind}_{\\mathds{C}}\\slashed{D}^{+}_{A_{0}}(\\tilde{X},g_{\\tilde{X}}),\n\\end{split}\n\\end{equation}\nwhere $$\\slashed{D}^{+}_{A_{0}}(\\tilde{X},g_{\\tilde{X}}):L^{2}_{1}(\\tilde{X};S^{+})\\rightarrow L^{2}(\\tilde{X};S^{-})$$\nis the (unperturbed) Dirac operator on $\\tilde{X}$. As in the proof of \\cite[Proposition 5.4]{MRS}, this operator has index $0$. Therefore, we have proved the lemma.\n\\end{proof}\n\\begin{rmk}\nIt was conjectured in \\cite{MRS} that the relation $\n\\lambda_{\\textnormal{SW}}(X)=-\\lambda_{\\textnormal{SW}}(-X)$ holds for a general homology $S^{3}\\times S^{1}$ (without any assumption on the metric). This conjecture is still open. \\end{rmk}\n\n\\section{Gauge theory on end-periodic manifolds}\nIn this section, we study the gauge theory on the end-periodic manifolds. First, we will carefully set up the (blown up) configuration space, the gauge group and the moduli spaces. Once this was done correctly, the arguments in Section 24 and 25 of the book can be repeated without essential difficulty. For this reason, some proofs in this section will only be sketched and we refer to the book for complete details.\n\n\n\n Let $\\delta$ be a positive number smaller than $\\min(\\delta_{1},\\delta_{2})$, where $\\delta_{1},\\delta_{2}$ are constants provided by Lemma \\ref{half De rham complex} and Proposition \\ref{Dirac operator is Fredholm} respectively. We let\n $$\n \\mathcal{A}_{k,\\delta}(X_{+})=\\{A_{0}+a| a\\in L^{2}_{k,\\delta}(X_{+};iT^{*}X_{+})\\}\n $$\n be the space of spin$^{\\text{c}}$ connections of class $L^{2}_{k,\\delta}$. The configuration spaces are defined as\n\\begin{equation}\\label{configuration space}\n\\begin{split}\n\\mathcal{C}_{k,\\delta}(X_{+})=\\mathcal{A}_{k,\\delta}(X_{+})\\times L^{2}_{k,\\delta}(X_{+};S^{+});\\\\\n\\mathcal{C}_{k,\\delta}^{\\sigma}(X_{+})=\\{(A,s,\\phi)| A\\in\\mathcal{A}_{k,\\delta}(X_{+}),\\phi\\in L_{k,\\delta}^{2}(X_{+};S^{+}), \\|\\phi\\|_{L^{2}}=1, s\\in \\mathds{R}_{\\geq 0}\\}.\\end{split}\n\\end{equation}\nIt is easy to see that $\\mathcal{C}_{k,\\delta}(X_{+})$ is a Hilbert manifold without boundary, while $\\mathcal{C}_{k,\\delta}^{\\sigma}(X_{+})$ is a Hilbert manifold with boundary. There is a map $\\boldsymbol{\\pi}:\\mathcal{C}_{k,\\delta}^{\\sigma}(X_{+})\\rightarrow \\mathcal{C}_{k,\\delta}(X_{+})$ given by\n$$\n\\boldsymbol{\\pi}(A,s,\\phi)=(A,s\\phi).\n$$\nNext, we define the gauge groups\n$$\n\\mathcal{G}^{0}_{k+1,\\delta}(X_{+})=\\{u:X_{+}\\rightarrow S^{1}| (1-u)\\in L^{2}_{k+1,\\delta} (X_{+};\\mathds{C}) \\};\n$$\n$$\n\\mathcal{G}_{k+1,\\delta}(X_{+})=\\mathcal{G}_{c}\\times \\mathcal{G}^{0}_{k+1,\\delta}(X_{+}),\n$$\nwhere $\\mathcal{G}_{c}\\cong S^{1}$ denotes the group of constant gauge transformations. Note that we impose the $L^{2}_{k+1,\\delta}$-topology on $\\mathcal{G}^{0}_{k+1,\\delta}(X_{+})$ and the product topology on $\\mathcal{G}_{k+1,\\delta}(X_{+})$. Using the equality $$1-uv=(1-u)+(1-v)-(1-u)(1-v)$$ together with the Sobolev multiplication theorem, one can prove that $\\mathcal{G}^{0}_{k+1,\\delta}$ (and hence $\\mathcal{G}_{k+1,\\delta}$) is a group. A standard argument (see \\cite{Taubes} and \\cite{Freed-Uhlenbeck} for the non-abelian case) shows that they are actually  Hilbert Lie groups. The Lie algebra of $\\mathcal{G}_{k+1,\\delta}$ is given by \\begin{equation}\\label{lie algebra}T_{e}\\mathcal{G}_{k+1,\\delta}(X_{+})=\\mathds{R}\\oplus L^{2}_{k+1,\\delta}(X_{+};i\\mathds{R}).\\end{equation}\n\\begin{rmk}\nOur main concern will be the group $\\mathcal{G}_{k+1,\\delta}(X_{+})$, while the group $\\mathcal{G}^{0}_{k+1,\\delta}(X_{+})$ is introduced to smooth the arguments.\n\\end{rmk}\nThe actions of $\\mathcal{G}_{k+1,\\delta}(X_{+})$  on  $\\mathcal{C}_{k,\\delta}(X_{+})$ and $\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ are respectively given by\n$$\nu\\cdot (A,\\Phi)=(A-u^{-1}du,u\\Phi)$$ and $$u\\cdot (A,s,\\phi)=(A-u^{-1}du,s,u\\phi).\n$$\nNote that the latter action is free. We denote the quotient spaces by $\\mathcal{B}_{k,\\delta}(X_{+})$ and $\\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})$ respectively.\n\n\n\n\n\\begin{lem}\n$\\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})$ is Hausdorff.\n\\end{lem}\n\\begin{proof}\nBy standard argumet, the proof is reduced to the following claim:\n\n\\begin{claim}: Suppose we have sequences $\\{u_{n}\\}\\subset \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+}),\\ \\{g_{n}\\}\\subset \\mathcal{G}_{k+1,\\delta}(X_{+})$ such that $u_{n}\\rightarrow u_{\\infty}\\text{ and } g_{n}u_{n}\\rightarrow v_{\\infty}$\nfor some $u_{\\infty},v_{\\infty}\\in \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$. Then we can find $g_{\\infty}\\in \\mathcal{G}_{k+1,\\delta}(X_{+})$ such that $g_{\\infty}u_{\\infty}=v_{\\infty}$.\\end{claim}\nTo prove the claim, we let $u_{n}=(A_{n},s_{n},\\phi_{n})$. Then both $A_{n}$ and $A_{n}-g_{n}^{-1}dg_{n}$ converges in $L^{2}_{k+1,\\delta}$ norm, which implies that the sequence  $\\{g_{n}^{-1}dg_{n}\\}$ is Cauchy in $L^{2}_{k,\\delta}(X_{+};i\\mathds{R})$. Let $g_{n}=e^{\\xi_{n}}$. Then $\\{d\\xi_{n}\\}$ is Cauchy in $L^{2}_{k,\\delta}(X_{+};i\\mathds{R})$. By Lemma \\ref{Taubes's lemma}, we can find numbers $\\bar{\\xi}_{n}\\in i\\mathds{R}$ such that $\\{\\xi_{n}-\\bar{\\xi}_{n}\\}$ is a Cauchy sequence in $L^{2}_{k+1,\\delta}(X_{+};i\\mathds{R})$. Using the fact that the exponential map\n$$\n L^{2}_{k+1,\\delta}(X_{+};iT^{*}X_{+})\\rightarrow \\mathcal{G}^{0}_{k+1,\\delta}(X_{+}):\\ \\xi\\mapsto e^{\\xi}\n$$\nis well defined and continuous (which is a consequence of the Sobolev multiplication theorem). We see that $\\{e^{\\xi_{n}-\\bar{\\xi}_{n}}\\}$ is a Cauchy sequence in $\\mathcal{G}^{0}_{k+1,\\delta}(X_{+})$.\n\nOn the other hand, by replacing $\\xi_{n}$ with $\\xi_{n}-2m_{n}\\pi i$ for $m_{n}\\in \\mathds{Z}$. We can assume $\\bar{\\xi}_{n}\\in [0,2\\pi)$. After passing to a subsequence, we may assume $\\bar{\\xi}_{n}$ converges to some number $\\bar{\\xi}_{\\infty}$, which implies $e^{\\bar{\\xi}_{n}}$ converges to $e^{\\bar{\\xi}_{\\infty}}$ as elements of $\\mathcal{G}_{c}$.\n\nNow we see that $g_{n}=e^{\\bar{\\xi}_{n}}\\cdot e^{\\xi_{n}-\\bar{\\xi}_{n}}$ has a subsequencial limit $g_{\\infty}$ in $\\mathcal{G}_{k+1,\\delta}(X_{+})$. Since the action of $\\mathcal{G}_{k+1,\\delta}(X_{+})$ is continuous, we get $g_{\\infty}\\cdot u_{\\infty}=v_{\\infty}$.\\end{proof}\n\nNext, we define the local slice $S^{\\sigma}_{k,\\delta,\\gamma}$ of the gauge action at $\\gamma=(A_{0},s_{0},\\phi_{0})\\in \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$. By taking derivative on gauge group action, we get a map\n$$d^{\\sigma}_{\\gamma}:T_{e}\\mathcal{G}_{k+1,\\delta}(X_{+})\\rightarrow T_{\\gamma}\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$$\n$$\n\\xi\\mapsto(-d\\xi,0,\\xi\\phi_{0}).\n$$\nWe denote the image of $d^{\\sigma}_{\\gamma}$ by $\\mathcal{J}^{\\sigma}_{k,\\delta,\\gamma}$, which is the tangent space of the gauge orbit. To define its complement, we consider the subspace $\\mathcal{K}^{\\sigma}_{k,\\delta,\\gamma}\\subset T_{\\gamma}C^{\\sigma}_{k,\\delta}(X_{+})$ as the kernel of the operator (c.f. formula (9.12) of the book)\n\\begin{equation}\\label{d-sigma-flat}\\begin{split}\nd^{\\sigma,\\flat}_{\\gamma}:L^{2}_{k,\\delta}(X_{+};i\\mathds{R})\\oplus \\mathds{R}\\oplus L^{2}_{k,\\delta}(X_{+};S^{+})\\rightarrow L^{2}_{k-1\/2}(Y;i\\mathds{R})\\oplus L^{2}_{k-1,\\delta}(X_{+};i\\mathds{R})\\\\\n(a,s,\\phi)\\mapsto (\\langle a, \\vec{v} \\rangle, -d^{*}a+is_{0}^{2}\\text{Re}\\langle i\\phi_{0},\\phi\\rangle +i|\\phi_{0}|^{2}\\cdot \\int_{X_{+}} \\text{Re}\\langle i\\phi_{0},\\phi\\rangle\\,d\\text{vol})\n\\end{split}\\end{equation}\n\\begin{rmk}To motivate this construction, we note that when $s_{0}>0$,  $\\mathcal{K}^{\\sigma}_{k,\\delta,\\gamma}$ is obtained by lifting the $L^{2}$-orthogonal complement of the tangent space of the gauge orbit (through $\\boldsymbol{\\pi}(\\gamma)$) in $\\mathcal{C}_{k,\\delta}(X_{+})$.\n\\end{rmk}\n\\begin{rmk}\nWe also note that in the book, the integral in the formula corresponding to (\\ref{d-sigma-flat}) is divided by the total volume of the $4$-manifold. However, this difference is not essential because the kernel is not affected.\n\\end{rmk}\n\\begin{lem}\\label{decomposition of the tangent space}\nFor any $\\gamma$, we have a decomposition $T_{\\gamma}\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})=\\mathcal{J}^{\\sigma}_{k,\\delta,\\gamma}\\oplus \\mathcal{K}^{\\sigma}_{k,\\delta,\\gamma}$.\n\\end{lem}\n\\begin{proof}\nWe want to show that for any $(a,s,\\phi)\\in T_{\\gamma}\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$, there exists a unique $\\xi\\in T_{e}\\mathcal{G}_{k+1,\\delta}(X_{+})$ such that\n$$(a,s,\\phi)-d^{\\sigma}_{\\gamma}\\xi\\in \\mathcal{K}^{\\sigma}_{k,\\delta,\\gamma}.$$\nThis is equivalent to the condition\n\\begin{equation}\\label{perturbed laplace}\nD\\xi=(\\langle a, \\vec{v}\\rangle,-is_{0}^{2}\\operatorname{Re}\\langle i\\phi_{0},\\phi\\rangle -i|\\phi_{0}|^{2}\\int_{X_{+}} \\operatorname{Re}\\langle i\\phi_{0},\\phi\\rangle d\\text{vol} +d^{*}a)\n\\end{equation}\nwhere the operator  $$D:T_{e}\\mathcal{G}_{k+1,\\delta}(X_{+})\\rightarrow L^{2}_{k-1\/2}(Y;i\\mathds{R})\\oplus L^{2}_{k-1,\\delta}(X_{+};i\\mathds{R})$$\nis given by\n$$\n\\xi\\mapsto (\\langle d\\xi, \\vec{v}\\rangle, \\Delta \\xi +s_{0}^{2}|\\phi_{0}|^{2}\\xi +i |\\phi_{0}|^{2} \\int_{X_{+}} (-i\\xi)|\\phi_{0}|^{2}d\\text{vol})\n$$\nNotice that the map\n$$\n\\xi \\mapsto s_{0}^{2}|\\phi_{0}|^{2}\\xi +i |\\phi_{0}|^{2} \\int_{X_{+}} (-i\\xi)|\\phi_{0}|^{2}d\\text{vol}\n$$\nactually factors through the space $L^{2}_{k,2\\delta}(X;i\\mathds{R})$. Therefore, the operator $D$ is a compact perturbation of the operator $D'$ given by\n$$\n\\xi\\mapsto   (\\langle d\\xi, \\vec{v}\\rangle, \\Delta \\xi).\n$$\nThe index of $D'$ (hence $D$) equals $0$ by Proposition \\ref{laplace equation} (iii). Here the index is increased by $1$ because we have an additional summand $\\mathds{R}$ in the domain (see (\\ref{lie algebra})). As in the proof of Proposition 9.3.5 of the book, we can show that $D$ has trivial kernel using integration by part.  Therefore, $D$ is an isomorphism and (\\ref{perturbed laplace}) has a unique solution. \\end{proof}\n\\begin{rmk}\nThe integration by part argument over the noncompact manifold $X_{+}$ is justified by the following fact (which can be proved using bump function): For any $\\delta>0$ and $\\theta\\in L^{2}_{k}(X_{+};\\wedge^{3}T^{*}X_{+})$, we have\n$$\n\\int_{X^{+}}d\\theta=\\int_ {\\partial{X_{+}}}\\theta.\n$$\n\\end{rmk}\nWe define the local slice $\\mathcal{S}^{\\sigma}_{k,\\delta,\\gamma}\\subset \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ (at $\\gamma$) as the set of points $(A,s,\\phi)$ satisfying\n$$\nd^{\\sigma,\\flat}_{\\gamma}(A-A_{0},s,\\phi)=0\n$$\nBy Lemma 9.3.2 of the book, Lemma \\ref{decomposition of the tangent space} has the following corollary.\n\\begin{cor}\n$\\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})$ is a Hilbert manifold with boundary. For any $\\gamma\\in C^{\\sigma}_{k,\\delta}(X_{+})$, there is an open neighborhood of $\\gamma$ in the slice\n$$\nU\\subset \\mathcal{S}^{\\sigma}_{k,\\delta,\\gamma}\n$$\nsuch that $U$ is a diffeomorphism onto its image under the natural projection from  $\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ to  $\\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})$, which is an open neighborhood of $[\\gamma]$ in $\\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})$.\n\\end{cor}\n\n\nNow we study the Seiberg-Witten equations on the manifold $X_{+}$. Let $\\mathcal{V}^{\\sigma}_{k,\\delta}(X_{+})$ be the trivial bundle $\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ with fiber $L^{2}_{k-1,\\delta}(i\\mathfrak{su}(S^{+})\\oplus S^{-})$, where $\\mathfrak{su}(S^{+})$ denotes the bundle of skew-hermitian, trace-$0$ automorphisms on $S^{+}$. We consider a smooth section $$\\mathfrak{F}^{\\sigma}:C^{\\sigma}_{k,\\delta}(X_{+})\\rightarrow \\mathcal{V}^{\\sigma}_{k,\\delta}(X_{+})$$ given by\n$$\n\\mathfrak{F}^{\\sigma}(A,s,\\phi)=(\\frac{1}{2}\\rho(F^{+}_{A^{t}})-s^{2}(\\phi\\phi^{*})_{0},\\slashed{D}_{A}^{+}\\phi)\n$$\nThe zero locus of $\\mathfrak{F}^{\\sigma}$ describes the solution of the (blown-up) Seiberg-Witten equations.\n\nTo obtain the transversality condition, we introduce a perturbation on $\\mathfrak{F}^{\\sigma}$. This was done in the same way as the book: Recall that the boundary $\\partial X_{+}$ has a neighborhood $N$ which is isomorphic to $[0,3]\\times Y$ (with $\\{0\\}\\times Y$ identified with $\\partial X_{+}$). Pick two $3$-dimensional tame perturbations $\\mathfrak{q}$ and $\\mathfrak{p}_{0}$. We impose the following assumption on $\\mathfrak{q}$:\n\\begin{assum}\\label{3 dimensional perturbation}\n$\\mathfrak{q}$ is a nice perturbation with $\\operatorname{ht}(\\mathfrak{q})=-2w(X,g_{X},0)$. Such perturbation exists by Proposition \\ref{height of nice perturbation}.\n\\end{assum}\nThese two perturbations induce, in a canonical way, 4-dimensional perturbations $\\hat{\\mathfrak{q}}^{\\sigma},\\hat{\\mathfrak{p}}_{0}^{\\sigma}$ on $N$ (see Page 153 and 155 of the book). Pick a cut-off function  $\\beta$ that equals $1$ near $\\{0\\}\\times Y$ and equals $0$ near $\\{3\\}\\times Y$ and a bump function $\\beta_{0}$ supported in $(0,-3)\\times Y$. Then the sum\n\\begin{equation}\\label{mixed perturbation}\n\\hat{\\mathfrak{p}}^{\\sigma}=\\beta\\cdot \\hat{\\mathfrak{q}}^{\\sigma}+\\beta_{0}\\cdot \\hat{\\mathfrak{p}}_{0}^{\\sigma}\n\\end{equation} is a section of $\\mathcal{V}^{\\sigma}_{k,\\delta}(X_{+})$ with the property that: $\\hat{\\mathfrak{p}}^{\\sigma}(A,s,\\phi)\\in L^{2}_{k-1,\\delta}(i\\mathfrak{su}(S^{+})\\oplus S^{-})$ is supported in $N$ and only depends on $(A|_{N},s,\\phi|_{N})$.\n\nWe denote by $\\mathfrak{p}$ the $4$-dimensional perturbation given by the section $\\hat{\\mathfrak{p}}^{\\sigma}$. Let $\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}=\\mathfrak{F}^{\\sigma}+\\hat{\\mathfrak{p}}^{\\sigma}$.\nWe can define the moduli spaces $$\\mathcal{M}(X_{+})=\\{(A,s,\\phi)|\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(A,s,\\phi)=0\\}\/\\mathcal{G}_{k+1,\\delta}(X_{+})\\subset \\mathcal{B}_{k,\\delta}^{\\sigma}(X_{+})$$\n$$\\mathcal{M}^{\\text{red}}(X_{+})=\\{[(A,s,\\phi)]\\in \\mathcal{M}(X_{+})|\\ s=0 \\}$$\nas the set of gauge equivalent classes of the solutions of the perturbed Seiberg-Witten equations. ( For simplicity, we do not include $\\mathfrak{p}$ in our notations of moduli spaces.)\n\\begin{lem}\nFor any choice of perturbations $\\mathfrak{q},\\mathfrak{p}_{0}$, the moduli space $\\mathcal{M}(X_{+})$ is always a Hilbert manifold with boundary $\\mathcal{M}^{\\operatorname{red}}(X_{+})$.\n\\end{lem}\n\\begin{proof}\nThe proof is essentially identical with Proposition 24.3.1 in the book. Just replace the manifold $X$ there with $X_{+}$ and use weighted Sobolev space through out the argument.\n\\end{proof}\nBecause of the unique continuation theorem (see Proposition 10,8.1 of the book), we have $\\phi|_{\\partial X_{+}}\\neq 0$ for any $[(A,s,\\phi)]\\in \\mathcal{M}(X_{+})$. Therefore, we have a well defined map\n\\begin{equation}\\label{restriction from the right}\nR_{-}:\\mathcal{M}(X_{+})\\rightarrow \\mathcal{B}_{k-1\/2}^{\\sigma}(Y)\n\\end{equation}\ngiven by\n$$\n(A,s,\\phi)\\mapsto (A|_{\\partial X_{+}},s\\|\\phi|_{\\partial X_{+}}\\|_{L^{2}},\\frac{\\phi|_{\\partial X_{+}}}{\\|\\phi|_{\\partial X_{+}}\\|_{L^{2}}}).\n$$\n\nNow we attach the cylindrical end $(-\\infty,0]\\times Y$ on $X_{+}$ and consider the Seiberg-Witten equations on the manifold $Z_{+}$. We define the configuration space  as\n$$\\mathcal{C}_{k;\\text{loc},\\delta}(Z_{+})=\\{(A_{0}+a,\\Phi)| (a,\\Phi)\\in L^{2}_{k,\\text{loc}}(Z_{+};iT^{*}Z_{+}\\oplus S^{+}),\\ \\|(a,\\Phi)|_{X_{+}}\\|_{L^{2}_{k,\\delta}}<\\infty\\}$$\nand gauge group as\n$$\n\\mathcal{G}_{k+1;\\text{loc},\\delta}(Z_{+})=\\{u:Z_{+}\\rightarrow S^{1}|\\ u\\in L^{2}_{k+1,\\text{loc}}(Z_{+};\\mathds{C}),\\ u|_{X_{+}}\\in \\mathcal{G}_{k+1,\\delta}(X_{+})\\}.\n$$\nNote that in the above definitons, we only impose the exponential decay condition over the periodic end. As before, the action of $\\mathcal{G}_{k+1;\\text{loc},\\delta}(Z_{+})$ on $\\mathcal{C}_{k;\\text{loc},\\delta}(Z_{+})$ is not free. Therefore, we need to blow up the configuration space. Since  $\\mathcal{C}_{k;\\text{loc},\\delta}(Z_{+})$ is not a Banach manifold now, the blown-up configuration space should be defined in the following manner: Let $\\mathds{S}$ be the topological quotient of the space\n$$\n\\{\\Phi\\in L^{2}_{k,\\text{loc}}(Z_{+};S^{+})|\\|\\Phi|_{X_{+}}\\|_{L^{2}_{k,\\delta}}<\\infty\\}\\setminus \\{0\\}\n$$\nby the action of $\\mathds{R}_{>0}$. The blown-up configuration configuration space is defined as\n$$\n\\mathcal{C}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})=\\{(A,\\Phi,\\phi)|(A,\\Phi)\\in\\mathcal{C}_{k;\\text{loc},\\delta}(Z_{+}),\\ \\phi \\in \\mathds{S},\\ \\Phi\\in \\mathds{R}_{\\geq 0}\\phi\\}.\n$$\nNow we define the blown-up quotient configuration space as $$\\mathcal{B}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})=\\mathcal{C}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})\/\\mathcal{G}_{k+1;\\text{loc},\\delta}(Z_{+}).$$\n\nThe bundle $\\mathcal{V}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})$ and its section $\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(Z_{+})$ are defined similarly as the book. The section $\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(Z_{+})$ is invariant under the action of $\\mathcal{G}_{k+1;\\text{loc},\\delta}(Z_{+})$. We omit the detail here because the specific definition is not important for us. Just keep in mind that the perturbation equals $\\hat{\\mathfrak{q}}^{\\sigma}$ over the cylindrical end $Z$, equals $\\hat{\\mathfrak{p}}$ over $[0,3]\\times Y$ and equals 0 on $Z_{+}\\setminus (-\\infty,3]\\times Y$. We call $(A,\\phi,\\Phi)$ a ``$Z_{+}$-trajectory'' if $\n \\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(Z_{+})(A,\\phi,\\Phi)=0$. This is equivalent to the condition that $(A,\\Phi,\\phi)$ satisfies the blown-up perturbed Seiberg-Witten equations\n$$\n\\left\\{\\begin{array} {cc}\n F_{A}^{+}-(\\Phi\\Phi^{*})_{0}=\\hat{\\mathfrak{p}}^{0,\\sigma}_{Z_{+}}(A,\\Phi)  \\\\\n \\slashed{D}_{A}^{+}\\phi=\\hat{\\mathfrak{p}}^{1,\\sigma}_{Z_{+}}(A,\\phi)\n\\end{array}\\right.\n$$\nwhere $\\hat{\\mathfrak{p}}^{0,\\sigma}_{Z_{+}}(A,\\Phi)$ and $\\hat{\\mathfrak{p}}^{1,\\sigma}_{Z_{+}}(A,\\phi)$ are certain perturbation terms supported on $(-\\infty,3]\\times Y$. The second equation should be thought as a homogeneous equation in $\\phi$, i.e., both sides of the equation will be rescaled by the same factor as we change the representative of $\\phi$. By the unique continuation theorem, we have $\\phi|_{\\{t\\}\\times Y}\\neq 0$ for any $t\\leq 0$. As a result, the triple\n$\n(A|_{\\{t\\}\\times Y},\\|\\Phi_{t\\times Y }\\|,\\tfrac{\\phi}{\\|\\phi|_{\\{t\\}\\times Y }\\|_{L^{2}}})\n$\ngives a point of $\\mathcal{C}^{\\sigma}_{k-1\/2}(Y)$, which we define to be the restriction $(A,\\Phi,\\phi)|_{\\{t\\}\\times Y}$. By restricting to $(-\\infty,0]\\times Y$, a gauge equivalent class $[(A,\\Phi,\\phi)]\\in \\mathcal{B}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})$ of $Z_{+}$-trajectory gives a path  $(-\\infty,0]\\rightarrow B^{\\sigma}_{k-1\/2}(Y)$.\n\nLet $[\\mathfrak{b}]\\in \\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$ be a critical point of $\\mathfrak{F}^{\\sigma}_{\\mathfrak{q}}(Y)$. We consider the moduli space $$\\mathcal{M}([\\mathfrak{b}],Z_{+})=\\{[\\gamma]\\in \\mathcal{B}^{\\sigma}_{k;\\text{loc},\\delta}(Z_{+})|\\  \\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(Z_{+})(\\gamma)=0,\\ \\mathop{\\lim}\\limits_{t\\rightarrow -\\infty}[\\gamma|_{\\{t\\}\\times Y}]=[\\mathfrak{b}]\\}.$$\nIt consists of $Z_{+}$-trajectories that are asymptotic to $[\\mathfrak{b}]$ over the cylindrical end. By restricting to the submanifolds $Z$ and $X_{+}$, we get a map\n\\begin{equation}\\label{restriction}\n\\rho:\\mathcal{M}([\\mathfrak{b}],Z_{+})\\rightarrow \\mathcal{M}([\\mathfrak{b}],Z)\\times \\mathcal{M}(X_{+}).\n\\end{equation}\nHere $\\mathcal{M}([\\mathfrak{b}],Z)$ denotes moduli space of Seiberg-Witten half-trajectories with limit $[\\mathfrak{b}]$. In other words, $\\mathcal{M}([\\mathfrak{b}],Z)$ consists of gauge equivalent classes of paths\n$$\n\\gamma:(-\\infty,0]\\rightarrow \\mathcal{C}^{\\sigma}_{k-1\/2}(Y)\\text{ with }\\frac{d}{dt}\\gamma(t)+\\mathfrak{F}^{\\sigma}_{\\mathfrak{q}}(Y)(\\gamma(t))=0,\\ \\mathop{\\lim}_{t\\rightarrow -\\infty}\\gamma(t)=\\mathfrak{b}.\n$$\nJust like $\\mathcal{M}(X_{+})$, the moduli space $\\mathcal{M}([\\mathfrak{b}],Z)$ is always a Hilbert manifold with boundary $\\mathcal{M}^{\\text{red}}([\\mathfrak{b}],Z)$ (the moduli space of reducible half-trajectories) for arbitary perturbation. Note that we have a well defined restriction map\\begin{equation}\\label{restriction from the left} R_{+}: \\mathcal{M}(Z,[\\mathfrak{b}])\\rightarrow \\mathcal{B}^{\\sigma}_{k-1\/2}(Y) \\text{ given by }[\\gamma]\\mapsto [\\gamma(0)].\\end{equation}\nThe proof of the following lemma is identical with Lemma 24.2.2 in the book.\n\\begin{lem}\\label{fiber sum}\nThe map $\\rho$ is a homeomorphism from $\\mathcal{M}(Z_{+},[\\mathfrak{b}])$ to its image, which equals the fiber product $\\operatorname{Fib}(R_{-},R_{+})$. (The maps $R_{\\pm}$ are defined in (\\ref{restriction from the right}) and (\\ref{restriction from the left}) respectively.)\n\\end{lem}\n\nNow we start discussing the regularity of the moduli spaces. Recall that for any point $[\\mathfrak{c}]\\in \\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$, we have a decomposition\n$$\nT_{[\\mathfrak{c}]}\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)\\cong \\mathcal{K}^{+}_{\\mathfrak{c}}\\oplus \\mathcal{K}^{-}_{\\mathfrak{c}}\n$$\ngiven by the spectral decomposition of the Hessian operator $\\text{Hess}^{\\sigma}_{\\mathfrak{q}}(\\mathfrak{c})$ (see Page 313 of the book).\n\\begin{lem}\\label{APS for moduli space}\nFor any $([\\gamma_{1}],[\\gamma_{2}])\\in \\operatorname{Fib}(R_{+},R_{-})$. Let $[\\mathfrak{c}]$ be the common restriction of $[\\gamma_{j}]\\ (j=1,2)$ on the boundary $Y$.  Denote by $\\pi$ the projection from $T_{[c]}\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$ to $\\mathcal{K}^{-}_{\\mathfrak{c}}$ with kernel $\\mathcal{K}^{+}_{\\mathfrak{c}}$. Then we have the following results.\n\\begin{enumerate}[(i)]\n\\item The linear operators $$(1-\\pi)\\circ\\mathcal{D}R_{+}:T_{[\\gamma_{1}]}\\mathcal{M}([\\mathfrak{b}],Z)\\rightarrow \\mathcal{K}_{\\mathfrak{c}}^{+} \\text{ and }\\pi\\circ\\mathcal{D}R_{+}:T_{[\\gamma_{1}]}\\mathcal{M}([\\mathfrak{b}],Z)\\rightarrow \\mathcal{K}_{\\mathfrak{c}}^{-}$$\nare respectively compact and Fredholm.\n\\item The linear operators $$(1-\\pi)\\circ\\mathcal{D}R_{-}:T_{[\\gamma_{2}]}\\mathcal{M}(X_{+})\\rightarrow \\mathcal{K}_{\\mathfrak{c}}^{+} \\text{ and }\\pi\\circ\\mathcal{D}R_{-}:T_{[\\gamma_{2}]}\\mathcal{M}(X_{+})\\rightarrow \\mathcal{K}_{\\mathfrak{c}}^{-}$$\nare respectively Fredholm and compact.\n\\item The linear operator $$\\mathcal{D}R_{+}+\\mathcal{D}R_{-}:T_{[\\gamma_{1}]}\\mathcal{M}([\\mathfrak{b}],Z)\\oplus T_{[\\gamma_{2}]}\\mathcal{M}(X_{+})\\rightarrow T_{[\\mathfrak{c}]}\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$$\nis Fredholm.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n(i) was proved in Theorem 17.3.2 of the book.  With Proposition \\ref{APS} in place, (ii) can be proved similarly (see also Proposition 24.3.2 of the book). (iii) is directly implied by (i) and (ii).\n\\end{proof}\n\nThe following definition is parallel to Definition 24.4.2 of the book.\n\n\\begin{defi}\nLet $[\\gamma]\\in \\mathcal{M}([\\mathfrak{b}],Z_{+})$. If $[\\gamma]$ is irreducible, we say the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ is regular at $[\\gamma]$ if the maps of Hilbert manifolds\n$$ R_{+}:\\mathcal{M}([\\mathfrak{b}],Z)\\rightarrow  \\mathcal{B}^{\\sigma}_{k-1\/2}(Y)\\text{ and } R_{-}:\\mathcal{M}(X_{+})\\rightarrow  \\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$$are transverse at $[\\gamma]$. If $[\\gamma]$ is reducible, we say the moduli space $\\mathcal{M}(Z_{:};[c])$ is regular at $[\\gamma]$ if the maps of Hilbert manifolds\n$$ R_{+}:\\mathcal{M}^{\\operatorname{red}}([\\mathfrak{b}],Z)\\rightarrow  \\partial\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)\\text{ and } R_{-}:\\mathcal{M}^{\\operatorname{red}}(X_{+})\\rightarrow  \\partial\\mathcal{B}^{\\sigma}_{k-1\/2}(Y)$$are transverse at $\\rho([\\gamma])$ (see (\\ref{restriction})). We say the moduli space is regular if it is regular at all points.\n\\end{defi}\nRecall that the perturbation $\\mathfrak{p}$ on $Z_{+}$ is determined a pair of $3$-dimensional perturbations $(\\mathfrak{q},\\mathfrak{p}_{0})$ (see (\\ref{mixed perturbation})), where $\\mathfrak{q}$ is a nice perturbation that is fixed throughout our argument (see Assumption \\ref{3 dimensional perturbation}). We want to obtain the transversality condition by varying the second perturbation $\\mathfrak{p}_{0}$. To do this, let $\\mathcal{P}(Y)$ be the large Banach space of $3$-dimensional tame perturbations provided by Theorem 11.6.1 of the book. We have the following result.\n\\begin{pro}\\label{transversality}\nThere exists a residual subset $U_{1}$ of  $\\mathcal{P}(Y)$ such that for any $\\mathfrak{p}_{0}\\in U_{1}$, the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ corresponding to  $(\\mathfrak{q},\\mathfrak{p}_{0})$ is regular for any critical point $[\\mathfrak{b}]\\in\\mathfrak{C}$.\n\\end{pro}\n\\begin{proof}\nThe proof follows the standard argument as in the proof of Proposition 24.4.7 of the book: We consider parametrized moduli space\n$$\\mathfrak{M}(X_{+})\\subset \\mathcal{B}^{\\sigma}_{k,\\delta}(X_{+})\\times \\mathcal{P}(Y)$$\n$$\n\\mathfrak{M}(X_{+})=\\{(A,s,\\phi,\\mathfrak{p}_{0})|\\,\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}=0\\}\/\\mathcal{G}_{k+1,\\delta}(X_{+}).\n$$\nFor any $[\\mathfrak{b}]\\in \\mathfrak{C}$, we can prove that the map\n$$\nR_{+}\\times \\mathfrak{R}_{-}: \\mathcal{M}([\\mathfrak{b}],Z)\\times  \\mathfrak{M}(X_{+})\\rightarrow \\mathcal{B}_{k-1\/2}^{\\sigma}(Y)\\times  \\mathcal{B}_{k-1\/2}^{\\sigma}(Y)\n$$\nis transverse to the diagonal by the same argument as the book. Here the map $\\mathfrak{R}_{-}$ is defined similarly with $R_{-}$ (but with larger domain). Now we apply the Sard-Smale lemma (Lemma 12.5.1 of the book) to finish the proof. We note that Lemma \\ref{APS for moduli space} (iii) is used essentially in this last step.\n\\end{proof}\n\n\nThe proof of the following proposition is by standard transversility argument and we omit it. (Compare Proposition 24.4.3 of the book.)\n\\begin{pro}\\label{moduli space is a manifold}\nSuppose the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ is regular and non-empty. Then the moduli space is\n\\begin{enumerate}[(i)]\n\\item a smooth manifold consisting only of irreducibles, if $[\\mathfrak{b}]$ is irreducible;\n\\item a smooth manifold consisting only of reducibles, if $[\\mathfrak{b}]$ is reducible and boundary-stable;\n\\item a smooth manifold with (possibly empty) boundary, if $[\\mathfrak{b}]$ is reducible and boundary-unstable.\n\\end{enumerate}\nIn the last case, the boundary consists of the reducible elements of the moduli space (i.e., we have $\\partial\\mathcal{M}([\\mathfrak{b}],Z_{+})=\\mathcal{M}^{\\text{red}}([\\mathfrak{b}],Z_{+})$).\n\\end{pro}\nRecall that we associated a rational number  $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])$ to each critical point $[\\mathfrak{b}]$. We have the following result.\n\\begin{pro}\\label{expected dimension}\nSuppose the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ is regular. Then the moduli space is\n\\begin{enumerate}[(i)]\n\\item the empty set, if $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])+2w(X,g_{X},0)<0$;\n\\item a manifold with dimension  $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])+2w(X,g_{X},0)$, if  $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])+2w(X,g_{X},0)\\geq 0$.\n\\end{enumerate}\n\\end{pro}\n\\begin{proof}\nWe just need to show that the expected dimension of $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ (which we denote by $\\operatorname{gr}(Z_{+};[\\mathfrak{b}])$) can be expressed as\n $$\\operatorname{gr}(Z_{+};[\\mathfrak{b}])=\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])+2w(X,g_{X},0).$$ This can be done by direct computation. But we follow an alternative argument here. Recall that $M$ is a spin manifold with bounded by $(Y,\\mathfrak{s})$ with $b_{1}(M)=0$. We let $M^{*}=M\\cup _{Y}([0,+\\infty)\\times Y)$.  As discussed before, the $M$-grading of $[\\mathfrak{b}]$ (which we denoted by $\\operatorname{gr}(M;[\\mathfrak{b}])$) equals the expected dimension of the moduli space consisting of solutions on $M^{*}$ that are asymptotic to $[\\mathfrak{b}]$. Since one can deform the linearized Seiberg-Witten equations over the manifold $M^{*}\\cup Z_{+}$ first to the corresponding equations over the manifold $$M\\cup_{Y}([-T,T]\\times Y)\\cup_{Y}X_{+}\\text{ with }T\\gg 0$$\n and then to the manifold $M_{+}$. We see that the grading is additive in the sense that the sum $\\operatorname{gr}(M;[\\mathfrak{b}])+\\operatorname{gr}(Z_{+};[\\mathfrak{b}])$ equals the expected dimension $\\mathcal{M}(M_{+})$, the moduli space consisting of gauge equivalent classes of solutions over $M_{+}$ that decay exponentially on the periodic end. The linear operator that determines the local structure of $\\mathcal{M}(M_{+})$ is a compact perturbation of the operator\n $$\n L^{2}_{k,\\delta}(M_{+};iT^{*}M_{+}\\oplus S^{+})\\rightarrow L^{2}_{k-1,\\delta}(M_{+};i\\mathds{R}\\oplus i\\wedge^{2}_{+}T^{*}M_{+}\\oplus S^{-})\n $$\n $$\n (a,\\Phi)\\mapsto (d^{*}a,d^{+}a,\\slashed{D}_{A_{0}}\\Phi).\n $$\n By Lemma \\ref{half De rham complex} and Proposition \\ref{Dirac operator is Fredholm}, the (real) index of this operator equals\n $$\n -\\frac{\\text{sign}(M)}{4}+2w(X,g_{X},0)+b^{+}_{2}(M)-1.\n $$\n By (\\ref{absolute grading}), this implies\n \\begin{equation}\\begin{split}\\operatorname{gr}(Z_{+};[\\mathfrak{b}])&=-\\frac{\\text{sign}(M)}{4}+2w(X,g_{X},0)+b^{+}_{2}(M)-1-\\operatorname{gr}(M;[\\mathfrak{b}])\\\\&=\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])+2w(X,g_{X},0).\\end{split}\\end{equation}\n \\end{proof}\n\nRecall that we denote the lowest boundary stable reducible critical point by $[\\mathfrak{a}_{0}]$. Recall that the absolute grading  $[\\mathfrak{a}_{0}]$ equals the height of the nice perturbation $\\mathfrak{q}$, which has been chosen to be $-2w(X,g_{X},0)$ (see Assumption \\ref{3 dimensional perturbation}). By Proposition \\ref{moduli space is a manifold} and Proposition \\ref{expected dimension}, for any $\\mathfrak{p}\\in U_{1}$ (the residue set provided by Lemma \\ref{transversality}), the moduli space $\\mathcal{M}([\\mathfrak{a}_{0}],[Z_{+}])$ consists of discrete elements, all of which are reducible because $[\\mathfrak{a}_{0}]$ is boundary stable. The moduli spaces $\\mathcal{M}([\\mathfrak{a}_{i}],[Z_{+}])\\ (i< 0)$ are all empty.\n\\begin{pro}\\label{only one reducible}\nThere exists an open neighborhood $U_{2}\\subset \\mathcal{P}(Y)$ of $0$ such that for any $\\mathfrak{p}_{0}\\in U_{2}$, the moduli space $\\mathcal{M}([\\mathfrak{a}_{0}],[Z_{+}])$ corresponding to  $(\\mathfrak{q},\\mathfrak{p}_{0})$ contains a single point.\n\\end{pro}\n\\begin{proof}\nSince the moduli space only consists of reducibles, we do not need to consider the nice perturbation $\\mathfrak{q}$ since it vanishes on the reducibles. Moreover, we can describe the moduli space explicitly: each gauge equivalent class of solutions of the downstairs equation\n\\begin{equation}\\label{downstairs reducible equation}\nd^{+}a-\\beta_{0}\\cdot \\rho^{-1}(\\mathfrak{\\hat{p}}^{0}_{0}(A_{0}+a,0))=0,\\ a\\in L^{2}_{k+1;\\text{loc},\\delta}(Z_{+};i\\mathds{R})\n\\end{equation}\ncontributes a copy of $\\mathds{CP}^{0}$ in $\\mathcal{M}([\\mathfrak{a}_{0}],[Z_{+}])$. (Here $\\beta_{0}$ is the bump function in (\\ref{mixed perturbation}) and $\\hat{\\mathfrak{p}}^{0}_{0}$ is a component of the $4$-dimensional, downstairs perturbation $\\hat{\\mathfrak{p}}_{0}$ induced by the $3$-dimensional perturbation $\\mathfrak{p}_{0}$.) We want to show that when $\\mathfrak{p}_{0}$ (hence $\\hat{\\mathfrak{p}}_{0}^{0}$) is small enough, (\\ref{downstairs reducible equation}) has exactly one solution up to gauge equivalence. By the exponential decay result Theorem 13.3.5 of the book (applied to $a|_{Z}$) and Lemma \\ref{half De rham complex} (i), we see that each equivalent class contains a unique representative satisfying\n$$\n\\|a\\|_{L^{2}_{k;-\\delta,\\delta}}<\\infty,\\ d^{*}a=0.\n$$\nIn other words, we just need to prove (\\ref{downstairs reducible equation}) has a unique solution satisfying the above gauge fixing condition when the perturbation is small. To do this, we consider the map\n$$\n\\mathfrak{P}:\\mathcal{P}(Y)\\times L^{2}_{k;-\\delta,\\delta}(Z_{+};iT^{*}Z_{+})\\rightarrow V\\oplus L^{2}_{k;-\\delta,\\delta}(Z_{+}; i\\wedge^{2}_{+}T^{*}Z_{+}),\n$$\nwhere $V=\\{\\xi\\in L^{2}_{k,-\\delta,\\delta}(Z_{+};i\\mathds{R})|\\int_{Z_{+}}\\xi d\\text{vol}=0\\}$, given by\n$$\n(\\mathfrak{p}_{0},a)\\mapsto (d^{*}a,d^{+}a-\\beta_{0}\\cdot \\rho^{-1}(\\mathfrak{\\hat{p}}_{0}^{0}(A_{0}+a,0))).\n$$\nBy Lemma \\ref{half De rham complex} (iii), the restriction of $\\mathfrak{P}$ to $\\{0\\}\\times L^{2}_{k;-\\delta,\\delta}(Z_{+};iT^{*}Z_{+})$ is a (linear) isomorphism.  Therefore, by the implicit function theorem, there exists a neighborhood $U$ of $0\\in  L^{2}_{k;-\\delta,\\delta}(Z_{+};iT^{*}Z_{+})$ and a neighborhood $U'$ of $0\\in \\mathcal{P}(Y)$ with the property that: for any $\\mathfrak{p}_{0}\\in U'$, there exists a unique solution of the equation $\\mathfrak{P}(\\mathfrak{p}_{0},a)=0$ with $a\\in U$. Now we claim that we can find another neighborhood $U''$ of $0\\in \\mathcal{P}(Y)$\nsuch that for any $\\mathfrak{p}_{0}\\in U''$,  $\\mathfrak{P}(\\mathfrak{p}_{0},a)=0$ implies $a\\in U$. This will finish the proof because we can set $U_{2}=U'\\cap U''$. Now we prove our claim by contradiction. Suppose there exist $\\mathfrak{p}_{0,n}\\rightarrow 0$ and $a_{n}\\notin U$ such that $\\mathfrak{P}(\\mathfrak{p}_{0,n},a_{n})=0$ for each $n$. Integrating by part on $(-\\infty,-0]\\times Y$ and $X_{+}\\setminus [3,+\\infty)$ respectively, we see that\n$$\n\\operatorname{CSD}((A_{0}+a_{n})|_{Y\\times\\{0\\}},0)<0,\\ \\operatorname{CSD}((A_{0}+a_{n})|_{Y\\times\\{3\\}},0)>0.\n$$\nUsing these energy estimates, one can easily adapt the proof of Theorem 10.7.1 of the book (from the single perturbation case to the case of a convergent sequence of perturbations) and prove that: after passing to a subsequence and applying suitable gauge transformations $u_{n}$, the sequence $\nu_{n}\\cdot((A_{0}+a_{n})|_{Y\\times[1,2]},0)\n$ converges smoothly. Notice that the gauge invariant term $\\beta_{0}\\cdot \\rho^{-1}(\\mathfrak{\\hat{p}}_{0,n}^{0}(A_{0}+a_{n},0))$ is supported on $Y\\times [1,2]$ and only depends on  $(A_{0}+a_{n})|_{Y\\times[1,2]}$ (because the bump function $\\beta_{0}$ is supported on $[1,2]\\times Y$). We see that $$\\|(d^{*}a_{n},d^{+}a)\\|_{L^{2}_{k-1;-\\delta,\\delta}} =\\|\\beta_{0}\\cdot \\rho^{-1}(\\mathfrak{\\hat{p}}_{0,n}^{0}(A_{0}+a_{n},0))\\|_{L^{2}_{k-1;-\\delta,\\delta}}\\rightarrow 0\\text{ as }n\\rightarrow\\infty$$ since $\\mathfrak{p}_{0,n}\\rightarrow 0$. By Lemma \\ref{half De rham complex} (iii) again, we get $\\|a_{n}\\|_{L^{2}_{k;-\\delta,\\delta}}\\rightarrow 0$. This contradicts with our assumption $a_{n}\\notin U$ and completes our proof.\\end{proof}\n\\begin{assum}\\label{4 dim perturbation}\nFrom now on, we fix a choice of perturbation $\\mathfrak{p}_{0}\\in U_{1}\\cap U_{2}$, where $U_{1}$, $U_{2}$ are subsets of $\\mathcal{P}(Y)$ provided by Proposition \\ref{transversality} and Proposition \\ref{only one reducible} respectively.\n\\end{assum}\nAs in the cylindrical case, a sequence of $Z_{+}$-trajectories (even with unifomly bounded energy) can converge to a broken trajectory. For this reason, we have to introduce the moduli space of broken trajectories before discussing the compactness property. Although our construction can be generalized to moduli space of higher dimension without essential difficulty, we focus on $1$-dimensional moduli spaces for simplicity. This will be enough for our application.\n\nWe start with recalling the ``$\\tau$-module'' for blow up. (See Section 6.3 of the book for details.) Let $I\\subset \\mathds{R}$ be an interval. Denote the product manifold $I\\times Y$ by $Z_{I}$. There are two cases:\n\\begin{itemize}\n\\item Suppose $I$ is compact, we define the configuration space \\begin{equation}\\label{tau-module}\\begin{split}\n\\mathcal{C}^{\\tau}_{k}(Z_{I})=&\\{(A_{0}+a,s,\\phi)|(a,\\phi)\\in L^{2}_{k}(Z_{I};iT^{*}Z_{I}\\oplus S^{+}),\\ s\\in L^{2}_{k}(I;\\mathds{R})\\\\& \\text{satisfies }s(t)\\geq 0,\\ \\|\\phi|_{Y\\times \\{t\\}}\\|_{L^{2}(Y)}=1\\text{ for any }t\\in I \\}\\end{split}\\end{equation}\nThe gauge group $\n\\mathcal{G}_{k+1}(Z_{I})$ acts on $\\mathcal{C}^{\\tau}_{k}(Z_{I})$ as\n$$\nu\\cdot(A_{0}+a,s,\\phi)= (A_{0}+a-u^{-1}du,s, u\\phi).\n$$ We denote the quotient space by $\\mathcal{B}^{\\tau}_{k}(Z_{I})$.\n\\item Suppose $I$ is non-compact, we define $\\mathcal{C}^{\\tau}_{k,\\text{loc}}(Z_{I})$ by replacing  $L^{2}_{k}$ with $L^{2}_{k,\\text{loc}}$ in (\\ref{tau-module}). We let $\\mathcal{B}^{\\tau}_{k,\\text{loc}}(Z_{I})=\\mathcal{C}^{\\tau}_{k,\\text{loc}}(Z_{I})\/\\mathcal{G}_{k+1,\\text{loc}}(Z_{I})$.\n\\end{itemize}\nIn both cases, we impose the quotient topology on the quotient configuration space. For any $[\\mathfrak{b}],[b']\\in \\mathfrak{C}$, the moduli space $\\mathcal{M}([\\mathfrak{b}],[b'])$ is a subset of $\\mathcal{B}^{\\tau}_{k,\\text{loc}}(Z_{(-\\infty,+\\infty)})$ and consists of the non-constant Seiberg-Witten trajectories going from $[\\mathfrak{b}]$ to $[b']$. We let $\\breve{\\mathcal{M}}([\\mathfrak{b}],[b'])=\\mathcal{M}([\\mathfrak{b}],[b'])\/\\mathds{R}$, where $\\mathds{R}$ acts as translation (reparametrization).\n\nNow we define the moduli space of broken trajectories. Let $[\\mathfrak{b}_{0}]$ be a critical point with $\\operatorname{gr}^{\\mathds{Q}}([b_{0}])=2w(X,g_{X},0)+1$. By our assumption about $\\operatorname{ht}(\\mathfrak{q})$, $[\\mathfrak{b}_{0}]$ must be irreducible. We consider the set\n$$\n\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})=\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})\\cup(\\mathop{\\cup}\\limits_{[\\mathfrak{b}]\\in \\mathfrak{C}}\\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],[\\mathfrak{b}])\\times \\mathcal{M}([\\mathfrak{b}],Z_{+})).\n$$\nBy our regularity assumption,  $\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})$ is a $1$-dimensional manifold (without boundary). The set $\\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],[\\mathfrak{b}])\\times \\mathcal{M}([\\mathfrak{b}],Z_{+})$ is nonempty only if $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])=2w(X,g_{X},0)$, in which case it is a discrete set.\n\nTo define the topology on $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$, we need to specify a neighborhood base for each point. For those points in $\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})$, we just use their neighborhood basis inside $\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})$. For a broken trajectory $([\\gamma_{-1}],[\\gamma_{0}])\\in \\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],[\\mathfrak{b}])\\times \\mathcal{M}([\\mathfrak{b}],Z_{+})$, we let $[\\gamma_{-1}]$ be represented by a parametrized trajectory\n$$\n\\gamma_{-1}\\in \\mathcal{M}([\\mathfrak{b}_{0}],[\\mathfrak{b}]).\n$$\nLet $U_{0}$ be a neighborhood of $[\\gamma_{0}]$ in $\\mathcal{B}^{\\sigma}_{k,\\operatorname{loc},\\delta}(Z_{+})$ and let $I\\subset\\mathds{R}$ be a compact interval and $U_{-1}\\subset  \\mathcal{B}^{\\tau}_{k}(Z_{I})$ be a neighborhood of $[\\gamma_{-1}|_{I}]$. For any $T\\in \\mathds{R}_{>0}$ with the property that $I-T$ (the translation of $I$ by $-T$) is contained in $\\mathds{R}_{\\leq 0}$, we define $\\Omega(U_{-1},U_{0},T)$ to be the subset of  $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$ consisting of the broken $Z_{+}$-trajectory $([\\gamma_{-1}],[\\gamma_{0}])$ and (unbroken) $Z_{+}$-trajectories $[\\gamma] \\in \\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})$ satisfying the following conditions:\n\\begin{itemize}\n\\item $[\\gamma]\\in U_{0};$\n\\item There exists  $T_{-1}>T$ such that $[\\tau_{T_{-1}}(\\gamma|_{I-T_{-1}})]\\in U_{-1}$, where $\\tau_{T_{-1}}(\\gamma|_{I-T_{-1}})$ denotes the translation of $\\gamma|_{I-T_{-1}}$ by $T_{-1}$ (in the positive direction).\n\\end{itemize}\nWe put the sets of the form $\\Omega(U_{-1},U_{0},T)$ form a neighborhood basis for $([\\gamma_{-1}],[\\gamma_{0}])$.\nWith the topology on $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$ defined, we have the following gluing theorem, whose proof is a word by word translation from the proof of Theorem 24.7.2 in the book and we omit.\n\n\\begin{thm}\\label{gluing}\nFor each broken $Z_{+}$-trajectory $([\\gamma_{-1}],[\\gamma_{0}])\\in \\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$, we can find its open neighborhood $U$ with $U\\setminus ([\\gamma_{-1}],[\\gamma_{0}])\\subset\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+})$ and a homeomorphism $f: (0,1]\\times ([\\gamma_{-1}],[\\gamma_{0}]) \\rightarrow U$ that sends $\\{1\\}\\times ([\\gamma_{-1}],[\\gamma_{0}]) $ to $([\\gamma_{-1}],[\\gamma_{0}])\\in U$.\n\\begin{rmk}\nTheorem 24.7.2 in the book actually contains the two parts: the boundary obstructed case and the boundary unobstructed case. The second case is much easier than the first case.  Theorem \\ref{gluing} here corresponds to the second case with the additional assumption that the moduli space is $1$-dimensional and the boundary of the $4$-manifold is connected. This further simplifies the statement of the result.\n\\end{rmk}\n\\end{thm}\n\nNow we consider the orientation of the moduli spaces. As mentioned in Subsection 2.2, a choice of $\\chi([\\mathfrak{b}])$ in the orientation set $\\Lambda([\\mathfrak{b}])$ for each $[\\mathfrak{b}]$ canonically induces an orientation of the moduli space $\\breve{\\mathcal{M}}([\\mathfrak{b}],[\\mathfrak{b}])$ for any critical points $[\\mathfrak{b}],[\\mathfrak{b}']$. It was also proved in Threorem 24.8.3 of the book that a choice of $\\chi([\\mathfrak{b}])$ and a homology orientation of $M$ determines an orientation of $\\mathcal{M}(M^{*},[\\mathfrak{b}])$ (the moduli space of gauge equivalent classes consisting of solutions on $M^{*}=M\\cup_{Y}[0,+\\infty)\\times Y$ that are asymptotic to $[\\mathfrak{b}]$). By replacing the compact manifold $M$ with the non-compact manifold $X_{+}$ and working with the weighted Sobolev spaces instead of the unweighted ones, one can repeat the argument there and prove the following similar result. Note that we do not need any homology orientation of $X_{+}$. This is essentially because of Lemma \\ref{half De rham complex} (iv) (compare Lemma 24.8.1 of the book). An alternative viewpoint is that $H^{1}(X_{+};\\mathds{R})=H^{2}(X_{+};\\mathds{R})=0$.\n\\begin{thm}\\label{orientation}\nA choice of $\\{\\chi([\\mathfrak{b}])|\\,[\\mathfrak{b}]\\in\\mathfrak{C}\\}$ canonically induces an orientation on the moduli space $\\mathcal{M}([\\mathfrak{b}],Z_{+})$ for any critical point $[\\mathfrak{b}]$. These orientations are compatible with the gluing map in the following sense: the map $f$ provided by Theorem \\ref{gluing} is orientation preserving when restricted to $(0,1)\\times ([\\gamma_{-1}],[\\gamma_{0}])$, if we orient the moduli spaces $\\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],[\\mathfrak{b}])$, $\\mathcal{M}([\\mathfrak{b}],Z_{+}))$ and $\\mathcal{M}([\\mathfrak{b}_{0}],Z_{+}))$ by the same choice $\\{\\chi([\\mathfrak{b}])|\\,[\\mathfrak{b}]\\in\\mathfrak{C}\\}$ and use the positive orientation on the interval $(0,1)$.\n\\end{thm}\n\\section{Compactness}\nIn the current and the next section, we impose the following assumption:\n\\begin{assum}\nThe scalar curvature $\\operatorname{scal}$ of $g_{X}$ to be everywhere positive. In other words, we have\n$$\ns_{0}=\\mathop{\\operatorname{inf}}\\limits_{x\\in X}\\operatorname{scal}(x)>0.\n$$\n\\end{assum}\nThis assumption implies that the restriction of $g_{Z_{+}}$ on $\\mathop{\\cup}\\limits_{n\\geq 1}W_{n}$, which is a lift of $g_{X}$, has uniformly positive scalar curvature. Under this assumption, we will prove the following compactness theorem:\n\\begin{thm}\\label{compactness}\nFor any $[\\mathfrak{b}_{0}]\\in \\mathfrak{C}$ with $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}_{0}])=-2w(X,g_{X},0)+1$, the moduli space $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$ is compact.\n\\end{thm}(Again, the result can be generalized to arbitrary $[\\mathfrak{b}_{0}]$. But we focus on the current case because that is all we need.)\n\\subsection{The topological energy $\\mathcal{E}^{\\text{top}}$ and the quantity $\\Lambda_{\\mathfrak{q}}$} We start with some standard definitions in the book, which will be useful in our proof of compactness theorem. Let $\\hat{X}$ be a general spin$^{c}$ $4$-manifold and $(A,\\Phi)$ be a point of the configuration space (i.e., $A$ is a spin$^{c}$ connection and $\\Phi$ is a positive spinor over $\\hat{X}$). Its topological energy is defined as\n\\begin{equation}\\label{topolocial energy}\n\\mathcal{E}^{\\text{top}}(A,\\Phi)=\\frac{1}{4}\\int_{\\hat{X}}F_{A^{t}}\\wedge F_{A^{t}}-\\int_{\\partial\\hat{X}}\\langle \\Phi|_{\\partial\\hat{X}},\\slashed{D}_{B}(\\Phi|_{\\partial\\hat{X}})\\rangle d\\text{vol}+\\int_{\\partial\\hat{X}}(H\/2)|\\Phi|^{2}d\\text{vol}\n\\end{equation}\nwhere $B=A|_{\\partial\\hat{X}}$ and $H$ denotes the mean curvature of the boundary, which will be vanishing if we use the product metric near the boundary.\nNote that in our situation, the integrals in (\\ref{topolocial energy}) are always convergent (even if $\\hat{X}$ is not compact) because $F_{A^{t}}$ decays exponentially over the end of $\\hat{X}$.\n We also talk about the topological energy of a point in the blown-up configuration space (i.e., a triple $(A,e,\\phi)$ with $e\\geq 0$ and $|\\phi|_{L^{2}}=1$). In this case, we define $\\mathcal{E}^{\\text{top}}(A,s,\\phi)$ to be  $\\mathcal{E}^{\\text{top}}(\\boldsymbol\\pi(A,s,\\phi))$ where  $$\\boldsymbol{\\pi}(A,s,\\phi)=(A,s\\phi)$$\nas before. Since the topological energy is invariant under gauge transformation, it also makes sense to talk about the topological energy of a gauge equivalent class.\n\nNow we return to our end-periodic manifold $X_{+}$. Recall that $\\mathfrak{q}$ is a nice perturbation (of height $-2w(X,g_{X},0)$). After choosing a gauge invariant function\n\\begin{equation}\\label{perturbation}\nv:\\mathcal{C}_{k-1\/2}(Y)\\rightarrow \\mathds{R}.\n\\end{equation} whose formal gradient equals $\\mathfrak{q}$. We can define the perturbed topological energy of a point $\\gamma\\in \\mathcal{C}^{\\sigma}_{k,\\text{loc}}(X_{+})$  as\n$$\n\\mathcal{E}^{\\text{top}}_{\\mathfrak{q}}(\\gamma)=\\mathcal{E}^{\\text{top}}(\\gamma)-2v(\\boldsymbol\\pi(\\gamma)|_{Y}).$$\n\nLet $\\epsilon$ be a number lying in $(0,\\frac{1}{2})$. We consider two other manifolds:\n$$\nX^{'}_{+}=X_{+}\\setminus ([0,2\\epsilon)\\times Y),\\\nX^{''}_{+}=X_{+}\\setminus ([0,\\epsilon)\\times Y)\n$$ We can define the blown-up configuration space $\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+}^{'})$ similarly as $\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$. There is a partially defined restriction map $$\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})\\dashrightarrow \\mathcal{C}^{\\sigma}_{k,\\delta}(X^{'}_{+})$$\n$$\n(A,s,\\phi)\\rightarrow (A|_{X^{'}_{+}},s\\|\\phi|_{X^{'}_{+}}\\|_{L^{2}},\\frac{\\phi|_{X^{'}_{+}}}{\\|\\phi|_{X^{'}_{+}}\\|_{L^{2}}})\n$$\nwhose domain contains triples $(A,s,\\phi)$ with $\\phi|_{X^{'}_{+}}\\neq 0.$ We denote by $(A,s,\\phi)|_{X'_{+}}$ the image of $(A,s,\\phi)$ under this map. Under the assumption $\\phi|_{Y\\times \\{\\epsilon\\}}\\neq 0$, we can define $(A,s,\\phi)|_{Y\\times \\{\\epsilon\\}}\\in \\mathcal{C}^{\\sigma}_{k-1\/2}(Y)$ in a similar vein. Note that since we are considering the solution of the perturbed Seiberg-Witten equations, these conditions are always satisfied by the unique continuation theorem.\n\nOther than the (perturbed) topological energy, there is another quantity that will be useful when dealing with the blown-up configuration space. Let $(B,r,\\psi)$ be a point of $C^{\\sigma}_{k-1\/2}(Y)$. We define the quantity  $$\\Lambda_{\\mathfrak{q}}(B,r,\\psi)=\\operatorname{Re}\\langle \\psi,\\slashed{D}_{B}\\psi+\\tilde{\\mathfrak{q}}^{1}(B,r,\\psi)\\rangle_{L^{2}}$$\nwhere $\\tilde{\\mathfrak{q}}^{1}(B,r,\\psi)$ is defined as (see Remark \\ref{component of perturbation})\n$$\n\\tilde{\\mathfrak{q}}^{1}(B,r,\\psi)=\\int_{0}^{1}\\mathcal{D}_{(B,sr\\psi)}\\mathfrak{q}^{1}(0,\\psi)ds.\n$$\n(Recall that $\\mathfrak{q}^{1}$ denotes the spinor component of the perturbation $\\mathfrak{q}$.)\n\n\\subsection{Compactness: local results}\nIn this subsection, we will prove the compactness results for solutions on the manifold $X_{+}=W_{0}\\cup_{Y}W_{1}\\cup_{Y}...$. To simplify the notation, we denote by $W_{n,n'}$ the manifold\n$$\nW_{n}\\cup_{Y}W_{n+1}\\cup_{Y}...\\cup_{Y}W_{n'}\\subset X_{+}\n,$$\nand write $\\|\\cdot\\|_{L^{2}_{j}(W_{n,n'})}$ for the $L^{2}_{j}$ norm of the restriction to $W_{n,n'}$. We will use similar notation for other manifolds.\n\nLet us start with the following lemma, which was communicated to the author by Clifford Taubes.\n\n\\begin{lem}\\label{exp decay}\nThere exists uniform constants $C,\\delta_{3}>0$ with the following significance: for any $\\delta\\in (0,\\delta_{3})$ and any solution $\\gamma=(A,s,\\phi)\\in \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ of the equation  $\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(\\gamma)=0$, we have\n$$\n\\|\\phi\\|_{L^{2}(W_{n})}\\leq Ce^{-\\delta_{3}n},\\ \\ \\forall n\\geq 0.\n$$\n\\end{lem}\n\\begin{proof}\nWe first consider $W_{n}$ for $n\\geq 1$. Over these manifolds, the perturbation $\\mathfrak{p}$ equals $0$ and hence we have\n\\begin{equation}\\begin{split}\n\\rho(F^{+}_{A^{t}})-2s^{2}(\\phi\\phi^{*})_{0}=0\\\\\\slashed{D}^{+}_{A}\\phi=0.\\end{split}\n\\end{equation}\nWe choose an integer $N$ large enough such that there exists a bump function\n$$\n\\tau:W_{1,3N}\\rightarrow[0,1]\n$$\nwith the following properties: i) $\\tau$ is supported on $W_{2,3N-1}$; ii) $\\tau$ equals $1$ when restricted to $W_{N+1,2N}$; iii) $|d\\tau(x)|^{2}<s_{0}\/10$ for any $x\\in W_{1,3N}$. By the covering tranformations, $\\tau$ induces a bump function on\n$$\\tau_{m}:W_{mN+1,(m+3)N}\\rightarrow[0,1]\\ \\ (m\\geq 0).$$\nLet $\\phi_{m}=\\tau_{m}\\cdot \\phi$. Then $\n\\slashed{D}^{+}_{A}\\phi_{m}=\\rho(d\\tau_{m})\\phi\n$.\nNotice that $d\\tau_{m}$ is supported on $$W_{mN+1,mN+N}\\cup W_{mN+2N+1,mN+3N}.$$\nTherefore, for any $m\\geq 1$, we have\n\\begin{equation}\\label{31}\n\\begin{split}\\|\\slashed{D}^{+}_{A}\\phi_{m}\\|^{2}_{L^{2}(X_{+})}&\\leq \\|d\\tau_{m}\\|^{2}_{C^{0}}\\cdot (\\|\\phi\\|^{2}_{L^{2}(W_{mN+1,mN+N})}+\\|\\phi\\|^{2}_{L^{2}(W_{mN+2N+1,mN+3N})})\\\\&\\leq \\frac{s_{0}}{10}(\\|\\phi_{m-1}\\|^{2}_{L^{2}(X_{+})}+\\|\\phi_{m+1}\\|^{2}_{L^{2}(X_{+})}).\\end{split}\\end{equation}\nOn the other hand, since $\\phi_{m}$ is supported on $W_{mN+1,mN+3N-1}$, by the Weitzenb\\\"ock formula, we have\n\\begin{equation}\\label{32}\\begin{split}\n\\|\\slashed{D}_{A}\\phi_{m}\\|^{2}_{L^{2}(X_{+})}&=\\int_{X_{+}}\\langle \\slashed{D}_{A}^{-}\\slashed{D}^{+}_{A}\\phi_{m},\\phi_{m}\\rangle\\\\&=\\int_{X_{+}}\\langle (\\nabla_{A}^{*}\\nabla_{A}+\\frac{1}{2}\\rho(F_{A^{t}}^{+})+\\frac{\\operatorname{scal}(x)}{4})\\phi_{m},\\phi_{m}\\rangle \\\\&= \\int_{X_{+}}\\langle \\nabla_{A}^{*}\\nabla_{A}\\phi_{m},\\phi_{m}\\rangle +2s^{2}\\int_{X_{+}}\\tau_{m}^{2}\\langle(\\phi\\phi^{*})_{0}\\phi,\\phi\\rangle+\\int_{X_{+}}\\frac{\\operatorname{scal(x)}}{4}\\langle\\phi_{m},\\phi_{m}\\rangle\\\\\n&\\geq\\frac{s_{0}}{4}\\|\\phi_{m}\\|_{L^{2}(X_{+})}^{2}\n\\end{split}\\end{equation}\nLet $a_{m}=\\|\\phi_{m}\\|^{2}_{L^{2}(X_{+})}$. By (\\ref{31}) and (\\ref{32}), we have\n$$\n5a_{m}\\leq 2(a_{m-1}+a_{m+1}),\\ \\forall m\\geq 1,\n$$\nwhich is the same as\n$$\n2(2a_{m}-a_{m-1})\\leq 2a_{m+1}-a_{m},\\ \\forall m\\geq 1.\n$$\nNotice that $a_{m}\\leq\\|\\phi\\|^{2}_{L^{2}(X_{+})}=1$ for any $m$. We must have $2a_{m}-a_{m-1}\\leq 0$ for any $m\\geq 1$ because otherwise $2a_{m}-a_{m-1}$ (and hence $a_{m}$) will increase exponentially. Therefore, we get\n$a_{m}\\leq 2^{-m}a_{0}\\leq 2^{-m}$ for any $m\\geq 0$. For any $n\\geq N$, we have\n$$\n\\|\\phi\\|^{2}_{L^{2}(W_{n})}\\leq a_{\\lfloor n\/N \\rfloor}\\leq 2\\cdot (2^{1\/N})^{-n}.\n$$\nSince $\\|\\phi\\|^{2}_{L^{2}(W_{n})}\\leq \\|\\phi\\|^{2}_{L^{2}(X_{+})}=1$ for any $n\\geq 0$. We can set $C=2^{1\/2}$ and $\\delta_{3}=(\\ln 2)\/2N$.\\end{proof}\n\n\n\n\\begin{thm}\\label{local compactness}\nFor any $\\delta\\in (0,\\delta_{3})$, we have the following compactness result: Let $\\gamma_{n}\\in\\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ $(n\\geq 1)$ be a sequence solutions of the perturbed equation $\\mathfrak{F}^{\\sigma}_{\\mathfrak{p}}(\\gamma)=0$. Suppose that there is a uniform bound on the perturbed topological energy:\n$$\n\\mathcal{E}^{\\textnormal{top}}_{\\mathfrak{q}}(\\gamma_{n})\\leq C_{1},\n$$\nand a uniform upper bound\n$$\n\\Lambda_{\\mathfrak{q}}(\\gamma_{n}|_{\\{\\epsilon\\}\\times Y})\\leq C_{2}.\n$$\nThen there is a sequence of gauge transformations $u_{n}\\in \\mathcal{G}_{k+1,\\delta}(X_{+})$ such that the sequence $u_{n}(\\gamma_{n})|_{X'}$ converge in the topology of $\\mathcal{C}^{\\sigma}_{k,\\delta}(X')$ to a solution $\\gamma_{\\infty} \\in \\mathcal{C}^{\\sigma}_{k,\\delta}(X')$.\n\\end{thm}\n\nThe proof of this theorem uses Lemma \\ref{exp decay} and the bootstraping argument. To make it easier to follow, we break it into several lemmas. First, we let $\\gamma_{n}=(A_{n},s_{n},\\phi_{n})$ and $\\Phi_{n}=s_{n}\\phi_{n}$. The topological energy of $\\gamma_{n}$ can be broken into three parts (we treat the last two terms as one part)\n\\begin{equation}\\label{energy decomposition}\n\\begin{split}\\mathcal{E}^{\\text{top}}_{\\mathfrak{q}}(\\gamma_{n})=&\\mathcal{E}^{\\text{top}}((A_{n},\\Phi_{n})|_{X_{+}\\setminus W_{0}})+\\mathcal{E}^{\\text{top}}((A_{n},\\Phi_{n})|_{W_{0}\\setminus ([0,3]\\times Y)})\\\\&+(2\\mathcal{L}_{\\mathfrak{q}}((A_{n},\\Phi_{n})|_{\\{0\\}\\times Y})-2\\mathcal{L}((A_{n},\\Phi_{n})|_{\\{3\\}\\times Y}),\\end{split}\\end{equation}\nwhere $\\mathcal{L}_{\\mathfrak{q}}=\\mathcal{L}-v$ (see (\\ref{perturbation})). We denote the first, second and third part on the right hand side of (\\ref{energy decomposition}) by $\\mathcal{E}_{1,n}$, $\\mathcal{E}_{2,n}$ and $\\mathcal{E}_{3,n}$ respectively.\n\n\\begin{lem}\\label{energy bound}\nThe energy terms $\\mathcal{E}_{1,n}$, $\\mathcal{E}_{2,n}$ and $\\mathcal{E}_{3,n}$ are all uniformly bounded above.\n\\end{lem}\n\\begin{proof}\nSince the restriction of $(A_{n},\\Phi_{n})$ on $X_{+}\\setminus ([0,3]\\times Y)$ is a solution of the unperturbed Seiberg-Witten equations. By the relation between the topological energy and analytical energy (see Page 96 of the book) and Lemma 24.5.1 of the book, we have the following estimates\n $$\\mathcal{E}_{1,n}=\\frac{1}{4}\\int_{X_{+}\\setminus W_{0}}(|F_{A^{t}_{n}}|^{2}+4|\\nabla_{A_{n}}\\Phi_{n}|^{2}+|\\Phi|^{4}+\\operatorname{scal}(x)|\\Phi|^{2})$$\n$$\\mathcal{E}_{2,n}=\\frac{1}{4}\\int_{W_{0}\\setminus ([0,3]\\times Y)}(|F_{A^{t}_{n}}|^{2}+4|\\nabla_{A_{n}}\\Phi_{n}|^{2}+(|\\Phi|^{2}+\\operatorname{scal}(x)\/2)^{2})-\\int_{W_{0}\\setminus ([0,3]\\times Y)}\\frac{\\operatorname{scal}(x)^{2}}{16}$$\n$$\n\\mathcal{E}_{3,n}\\geq\n\\frac{1}{2}\\int_{[0,3]\\times Y}(|F_{A^{t}_{n}}|^{2}+4|\\nabla_{A_{n}}\\Phi_{n}|^{2}+(|\\Phi_{n}|^{2}-D_{1})^{2})-D_{2}\n$$\nwhere $D_{1},D_{2}$ are certain uniform constants. Note that $\\operatorname{scal}(x)$ is positive on $X_{+}\\setminus W_{0}$. It is easy to see that $\\mathcal{E}_{1,n}$, $\\mathcal{E}_{2,n}$ and $\\mathcal{E}_{3,n}$ are all uniformly bounded below. Since the sum of these three terms is bounded above, each of them should also be bounded above.\n\\end{proof}\n\nFor each $m\\geq 0,n\\geq 1$, we let $u_{m,n}:W_{m,m+6}\\rightarrow S^{1}$ be a gauge transformation with the following properties (recall that $A_{0}$ denotes the spin connection):\n\\begin{enumerate}[(i)]\n\\item $d^{*}(A_{n}-u^{-1}_{m,n}du_{m,n}-A_{0})=0;$\n\\item $\\langle A_{n}-u^{-1}_{m,n}du_{m,n}-A_{0},\\vec{v}_{m}\\rangle=0;$\n\\item $u_{1,n}(o_{3})=u_{0,n}(o_{3}),\\ u_{m+2}(o_{m+4})=u_{m}(o_{m+4}),\\ \\forall m\\geq1;$\n\\end{enumerate}\nwhere $\\vec{v}_{m}$ is the normal vector on $\\partial W_{m,m+6}$ and $o_{m}\\in W_{m}$ corresponds to a fixed base point $o\\in \\operatorname{int}(W)$. Such $u_{m,n}$ can always be found by solving the Laplace equation (with Neumann boundary condition) on $W_{m,m+6}$ (see Page 101 of the book). We let $$A_{m,n}=A_{n}-u^{-1}_{m,n}du_{m,n},\\ \\Phi_{m,n}=u_{m,n}\\Phi_{n},\\ \\phi_{m,n}=u_{m,n}\\phi_{n}.$$\n\n\n\\begin{lem}\\label{bounded results}\nThere exist a constant $C$ such that the following bounded results hold:\n\\begin{enumerate}[(i)]\n\\item $s_{n}\\leq C,\\ \\forall n\\geq 1$;\n\\item $\\|(A_{m,n}-A_{0},\\Phi_{m,n})\\|_{L^{2}_{k+1}(W_{m+1,m+5})}\\leq C,\\ \\forall n\\geq 1, m\\geq 1;$\n\\item $\\|(A_{0,n}-A_{0},\\Phi_{0,n})\\|_{L^{2}_{k+1}(W_{0,5}\\setminus([0,\\epsilon]\\times Y))}\\leq C,\\ \\forall n\\geq 1;$\n\\item $\\|(A_{m,n}-A_{0},\\phi_{m,n})\\|_{L^{2}_{k+1}(W_{m+2,m+4})}\\leq Ce^{-\\delta_{3}m},\\ \\forall n\\geq 1,m\\geq 1;$\n\\item $\\|(A_{0,n}-A_{0},\\phi_{0,n})\\|_{L^{2}_{k+1}(W_{0,4}\\setminus([0,2\\epsilon]\\times Y))}\\leq C,\\ \\forall n\\geq 1.$\n\\end{enumerate}\n\\end{lem}\n\n\n\n\\begin{proof}\n(i) By Lemma \\ref{energy bound}, the energy terms $\\mathcal{E}_{1,n},\\mathcal{E}_{2,n},\\mathcal{E}_{3,n}$ are all bounded above. They provide upper bounds on $\\|\\Phi_{n}\\|_{L^{2}(X_{+}\\setminus W_{0})},\\  \\|\\Phi_{n}\\|_{L^{4}(W_{0}\\setminus ([0,3]\\times Y))}$ and $\\|\\Phi_{n}\\|_{L^{4}([0,3]\\times Y)}$ respectively. Hence we get an upper bound $\\|\\Phi_{n}\\|_{L^{2}(X_{+})}$, which is exactly $s_{n}$.\n\n(ii) Suppose we do not have a uniform bound. Then there exists sequences of positive numbers  $\\{m_{l}\\},\\{n_{l}\\}$ such that\n$$\\|(A_{m_{l},n_{l}}-A_{0},\\Phi_{m_{l},n_{l}})\\|_{L^{2}_{k+1}(W_{m_{l}+1,m_{l}+5})}\\rightarrow +\\infty.$$\nNotice that the upper bound on $\\mathcal{E}_{1,n}$ gives a uniform upper bound on $$\\mathcal{E}^{\\operatorname{top}}((A_{n},\\Phi_{n})|_{W_{m,m+6}})$$ for all $m,n\\geq 1$. Suppose we treat the restriction of $A_{m_{l},n_{l}}$ (resp. $\\Phi_{m_{l},n_{l}})$) on $W_{m_{l}+1,m_{l}+5}$ as a connection (resp. spinor) on a fixed manifold $W\\cup_{Y}W\\cup_{Y}...\\cup_{Y}W$ for each $l$. Then by Theorem 5.1.1 of the book,\nthey converge in $C^{\\infty}$ topology after passing to a subsequence, which is a contradiction.\n\n(iii) The proof is similar with (ii): just use  $\\mathcal{E}_{2,n}+\\mathcal{E}_{3,n}$ to control the topological energy on $W_{0}$ then apply Theorem 5.1.1 of the book.\n\n(iv) On the submanifold $W_{m+1,m+5}$, the triple $(A_{m,n},s_{n},\\phi_{m,n})$ satisfies the (unperturbed) Seiberg-Witten equations and the gauge fixing condition\n$$\n\\left\\{\\begin{array} {cc}\n d^{+}(A_{m,n}-A_{0})-s_{n}\\rho(\\Phi_{m,n}\\phi^{*}_{m,n})_{0}=0  \\\\\n d^{*}(A_{m,n}-A_{0})=0\\\\\n \\slashed{D}_{A_{0}}^{+}\\phi_{m,n}=-\\rho(A_{m,n}-A_{0})\\phi_{m,n}\n\\end{array}\\right.\n$$\nThe result can be proved using bootstraping argument involving (i),(ii) and Lemma \\ref{exp decay}.\n\n(v) The proof is similar with (iv): just use (iii) instead of (ii). Note that in this case we have an extra term in the Seiberg-Witten equations coming from the perturbation $\\hat{\\mathfrak{p}}$. But since the perturbation is tame, the bootstraping argument still works. (See the proof Theorem 10.9.2 in the book for a similar but more detailed argument.)\\end{proof}\n\n\\begin{lem}\\label{regularity for solutions} Let $\\delta_{3}$ be the constant in Lemma \\ref{exp decay} and let $\\gamma_{n}\\in \\mathcal{C}^{\\sigma}_{k,\\delta}(X_{+})$ be the solutions in Theorem \\ref{local compactness}. For any $\\delta'\\in (\\delta,\\delta_{3})$, we can find gauge transformations $u_{n}\\in \\mathcal{G}_{k+1,\\delta}$ such that the restrictions $u_{n}\\cdot\\gamma_{n}|_{X'}=(A'_{n},s'_{n},\\phi'_{n})$ is a bounded sequence in $\\mathcal{C}^{\\sigma}_{k+1,\\delta'}(X')$. In other words, we have $$\\mathop{\\sup}\\limits_{n} \\|(A'_{n}-A_{0},\\phi'_{n})\\|_{L^{2}_{k+1,\\delta'}}<\\infty\\text{ and } \\mathop{\\sup}\\limits_{n} s'_{n}<\\infty.$$\n\\end{lem}\n\\begin{proof}\nThe idea is to obtain $u_{n}$ by gluing $u_{0,n}|_{W_{0,3}}$ and $u_{m,n}|_{W_{m+2,m+4}}\\ (m=1,3,5...)$ together using cutoff functions. Recall that $W$ is a manifold with boundary $(-Y)\\cup Y$. We choose a cutoff function $\\tau:W\\rightarrow [0,1]$ that equals $0$ near the left boundary and equals $1$ near the right boundary. We also use $\\tau$ for the induced cutoff function on $W_{m}$. For $m,n\\geq 1$, we consider the function $\\xi_{m,n}:W_{m+4}\\rightarrow [0,1]$ with the property that $$\\xi_{m,n}(o_{m+4})=0,\\ e^{i\\xi_{m,n}}\\cdot(u_{m,n}|_{W_{m+4}})=u_{m+2,n}|_{W_{m+4}}.$$\nWe also define $\\xi^{0}_{n}:W_{0,3}\\rightarrow [0,1]$ by the condition\n$$\n\\xi_{0,n}(o_{3})=0,\\ e^{i\\xi_{0,n}}(u_{0,n}|_{W_{3}})=u_{1,n}|_{W_{3}}.\n$$\nThen we have $d\\xi_{0,n}=A_{0,n}|_{W_{3}}-A_{1,n}|_{W_{3}}$ and $d\\xi_{m,n}=A_{m+2,n}|_{W_{m+4}}-A_{m,n}|_{W_{m+4}}$ for $m\\geq 1$. By Lemma \\ref{bounded results} (iv) and (v), there exists a uniform constant $C$ such that $$\\|\\xi_{m,n}\\|_{L^{2}_{k+2}(W_{m+2})}\\leq Ce^{-\\delta_{3}m},\\ \\forall m\\geq 0,n\\geq1,$$ which implies similar bounds for $\\tau \\xi_{m,n}$ and $(1-\\tau)\\xi_{m,n}$. We consider the gauge transformations $$ \\tilde{u}_{0,n}(x)=\\left\\{\\begin{array} {ll}\n\n 1 & x\\in W_{0,2}\\\\\n e^{i\\tau(x)\\xi_{0,n}(x)}& x\\in W_{3}\n\\end{array}\\right.$$\nand\n$$\\tilde{u}_{m,n}(x)=\\left\\{\\begin{array} {ll}\n e^{i(\\tau(x)-1)\\xi_{m-1,n}(x)}& x\\in W_{m+2}  \\\\\n 1&x\\in W_{m+3}\\\\\n e^{i\\tau(x)\\xi_{m,n}(x)}& x\\in W_{m+4}\n\\end{array}\\right.\\ (m\\geq1)$$\nBy the Sobolev multiplication theorem, there exists a uniform constant $C'$ such that\n\\begin{itemize}\n\\item $\n\\|\\tilde{u}_{0,n}^{-1}d\\tilde{u}_{0,n}\\|_{L^{2}_{k+1}(W_{0,3})}\\leq C'$;\n\\item $\\|1-\\tilde{u}_{0,n}\\|_{L^{2}_{k+1}(W_{0,3})}\\leq C'\n$;\n\\item $\\|\\tilde{u}_{m,n}^{-1}d\\tilde{u}_{m,n}\\|_{L^{2}_{k+1}(W_{m+2,m+4})}\\leq C'e^{-\\delta_{3}m}$ for any $m\\geq 1$;\n\\item $\\|1-\\tilde{u}_{m,n}\\|_{L^{2}_{k+1}(W_{m+2,m+4})}\\leq C'e^{-\\delta_{3}m}$ for any $m\\geq 1$.\n\\end{itemize}\nSince it is easy to check that $$u_{0,n}\\cdot \\tilde{u}_{0,n}=u_{1,n}\\cdot \\tilde{u}_{1,n} \\text{ on }W_{3};$$ $$u_{m,n}\\cdot \\tilde{u}_{m,n}=u_{m+2,n}\\cdot \\tilde{u}_{m+2,n}\\text{ on } W_{m+4} \\text{ for } m=1,3,5...,$$ we can glue $$\\{(u_{0,n}\\cdot \\tilde{u}_{0,n})|_{W_{0,3}}\\}\\cup \\{(u_{m,n}\\cdot\\tilde{u}_{m,n})|_{W_{m+2,m+4}}|\\ m=1,3,5...\\}$$ together to get a guage transformation $u_{n}:X_{+}\\rightarrow S^{1}$. Denote $u_{n}(A_{n},\\phi_{n})$ by $(\\tilde{A}_{n},\\tilde{\\phi}_{n})$. Then we have\n$$(\\tilde{A}_{n},\\tilde{\\phi}_{n})=\\left\\{\\begin{array} {cc}\n \\tilde{u}_{0,n}(A_{0,n},\\phi_{0,n})\\ \\text{ on } W_{0,3}\\\\\n \\tilde{u}_{m,n}(A_{m,n},\\phi_{m,n})\\ \\text{ on } W_{m+2,m+4} \\text{ for }m=1,3,5,...\n\\end{array}\\right.$$ By Lemma \\ref{bounded results} (iv) and (v) and the above estimates on $\\tilde{u}_{m,n}$, we can prove that there exists a uniform constant $C''$ such that\n$$\n\\|(\\tilde{A}_{n}-A_{0},\\tilde{\\phi}_{n})\\|_{L^{2}_{k+1}(W_{0,4}\\setminus ([0,2\\epsilon]\\times Y))}\\leq C''\n$$\nand\n$$\n\\|(\\tilde{A}_{n}-A_{0},\\tilde{\\phi}_{n})\\|_{L^{2}_{k+1}(W_{m+2,m+4})}\\leq C''e^{-\\delta_{3}m}\\text{ for } m=1,3,5,...\n$$\nSince $\\delta'<\\delta_{3}$, we get $$\\|(\\tilde{A}_{n}-A_{0},\\tilde{\\phi}_{n})\\|_{L^{2}_{k+1,\\delta'}(X'_{+})}<C'''$$\nfor some constant $C'''$. The relation between $(A_{n}',s'_{n},\\phi_{n}')$ and $(\\tilde{A}_{n},s_{n},\\tilde{\\phi}_{n}')$ is given by\n$$\nA_{n}'=\\tilde{A}_{n}|_{X'_{+}},\\ s_{n}'=s_{n}\\cdot \\|\\tilde{\\phi}_{n}\\|_{L^{2}(X'_{+})}\\text{ and }\\phi_{n}'=\\frac{\\tilde{\\phi}_{n}|_{X'_{+}}}{\\|\\tilde{\\phi}_{n}\\|_{L^{2}(X'_{+})}}.\n$$\nAs in the proof of Theorem 24.5.2 in the book, the condition $\n\\Lambda_{\\mathfrak{q}}(\\gamma_{n}|_{\\{\\epsilon\\}\\times Y})\\leq C_{2}$ ensures that the norm $\\|\\tilde{\\phi}_{n}\\|_{L^{2}_{X_{+}'}}$ (which is always less than $1$) is bounded away from $0$. Therefore, we have proved the estimate in the lemma.\n\nWe are left to check that $u_{n}\\in\\mathcal{G}_{k+1,\\delta}(X_{+})$. We write $u_{n}$ as $e^{i\\xi_{n}}$. Then $d\\xi_{n}=\\tilde{A}_{n}-A_{n}\\in L^{2}_{k,\\delta}(X_{+};i\\mathds{R})$. By Lemma \\ref{Taubes's lemma}, we can find $\\bar{\\xi}_{n}\\in i\\mathds{R}$ such that $\\xi_{n}-\\bar{\\xi}_{n}\\in L^{2}_{k+1,\\delta}(X_{+},i\\mathds{R})$. Then we have\n$$\nu_{n}=e^{i\\bar{\\xi}_{n}}\\cdot e^{i(\\xi_{n}-\\bar{\\xi}_{n})}\\in \\mathcal{G}_{k+1,\\delta}(X_{+}).$$\n\\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{local compactness}]\nBy Lemma \\ref{regularity for solutions}, we can find  $u_{n}\\in \\mathcal{G}_{k+1,\\delta}(X_{+})$ such that $u_{n}(\\gamma_{n})|_{X'_{+}}$ is a bounded sequence in $C^{\\sigma}_{k+1,\\delta'}(X')$. Since $\\delta'>\\delta$, the natural inclusion $C^{\\sigma}_{k+1,\\delta'}(X')\\rightarrow C^{\\sigma}_{k,\\delta}(X')$ maps a bounded closed set to a compact set. Therefore, we can find a subsequence that converges in $C^{\\sigma}_{k,\\delta}(X')$.\\end{proof}\n\\subsection{Compactness: broken trajectories} With  Theorem \\ref{local compactness} (compare Theorem 24.5.2 in the book) proved, the proof of Theorem \\ref{compactness} is essentially the same with the proof of Theorem 24.6.4 in the book. For completeness, we sketch it as follows:\n\\begin{proof}[Proof of Theorem \\ref{compactness}](Sketch)\nWe first consider a sequence $[\\gamma_{n}]\\in \\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],Z_{+})$ $(n\\geq 1)$ represented by unbroken $Z_{+}$-trajectories $\\gamma_{n}$. Using integration by part, it is easy to see that  $\\mathcal{E}_{\\mathfrak{p}}^{\\text{top}}(\\gamma_{n}|_{X_{+}})=\\mathcal{L}_{\\mathfrak{q}}(\\gamma_{n}|_{\\{0\\}\\times Y})$ for any $n$, which implies $\\mathcal{E}_{\\mathfrak{p}}^{\\text{top}}(\\gamma_{n}|_{X_{+}})<\\mathcal{L}_{\\mathfrak{q}}([\\mathfrak{b}_{0}])$ (because $\\gamma|_{Z}$ is a flow line with limit $[\\mathfrak{b}_{0}]$). By similar decomposition as in the proof of Lemma \\ref{energy bound}, we can prove that\n$$\n\\mathcal{L}_{\\mathfrak{q}}(\\gamma_{n}|_{\\{\\epsilon\\}\\times Y})=\\mathcal{E}_{\\mathfrak{p}}^{\\text{top}}(\\gamma_{n}|_{X''_{+}})>C,\\ \\mathcal{L}_{\\mathfrak{q}}(\\gamma_{n}|_{\\{2\\epsilon\\}\\times Y})=\\mathcal{E}_{\\mathfrak{p}}^{\\text{top}}(\\gamma_{n}|_{X'_{+}})>C.\n$$\nfor some uniform constant $C$. This implies both \\begin{equation}\\label{energy control1}\\mathcal{E}_{\\mathfrak{q}}^{\\text{top}}(\\gamma_{n}|_{(-\\infty,\\epsilon]\\times Y})<\\mathcal{L}_{\\mathfrak{q}}([\\mathfrak{b}_{0}])-C\\end{equation} and \\begin{equation}\\label{energy control2}\\mathcal{E}_{\\mathfrak{q}}^{\\text{top}}(\\gamma_{n}|_{(-\\infty,2\\epsilon]\\times Y})<\\mathcal{L}_{\\mathfrak{q}}([\\mathfrak{b}_{0}])-C.\\end{equation}\nBy the same argument as proof of Lemma 16.3.1 in the book, condition (\\ref{energy control1}) actually implies $$\\Lambda_{\\mathfrak{q}}(\\gamma_{n}|_{\\{t\\}\\times Y})\\leq C',\\ \\forall t\\in (-\\infty,\\epsilon]$$ for some constant $C'$. Now we apply Theorem \\ref{local compactness} to show that after applying suitable gauge transformations $u_{n}:X_{+}\\rightarrow S^{1}$ and passing to a subsequence, the restriction $u_{n}(\\gamma_{n}|_{X_{+}})|_{X'_{+}}$ has a limit $C^{\\sigma}_{k,\\delta}(X')$. Since $\\Lambda_{\\mathfrak{q}}(\\cdot)$ is gauge invariant, we get $$\\Lambda_{\\mathfrak{q}}(\\gamma_{n}|_{\\{2\\epsilon\\}\\times Y})=\\Lambda_{\\mathfrak{q}}(u_{n}(\\gamma_{n})|_{\\{2\\epsilon\\}\\times Y})\\geq C''$$ for some uniform constant $C''$. Another application of Lemma 16.3.1 in the book provides a uniform lower bound\n$$\n\\Lambda_{\\mathfrak{q}}(\\gamma_{n}|_{\\{t\\}\\times Y})\\geq C''',\\ \\forall t\\in (-\\infty,2\\epsilon].\n$$\nNow the proof proceed exactly as in the book: We can show that after passing to a further subsequence, $\\gamma_{n}|_{(-\\infty,2\\epsilon]}$ converges to a (possibly broken) half trajectory. Putting the two pieces $\\gamma_{n}|_{(-\\infty,2\\epsilon]\\times Y}$ and $\\gamma_{n}|_{X'_{+}}$ together, we see that after passing to a subsequence and composing with suitable gauge transformations, $\\gamma_{n}$ converges to a (possibly broken) $Z_{+}$-trajectory $\\gamma_{\\infty}$. By our regularity assumption, $\\gamma_{\\infty}$ can have a most one breaking point, whose absolute grading must be $2w(X,g_{X},0)$. In other words, the limit $\\gamma_{\\infty}$ represents a point of $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$.\n\nWe have shown that any sequence $[\\gamma_{n}]\\in \\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],Z_{+})$ contains convergent subsequence in  $\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})$. By a similar argument, we see that $\\breve{\\mathcal{M}}([\\mathfrak{b}],Z_{+})$ contains at most finitely many elements for any $[\\mathfrak{b}]$ with $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])=-2w(X,g_{X},0)$. Since there are only finitely many critical points $[\\mathfrak{b}]$ with $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])=-2w(X,g_{X},0)$ and $\\breve{\\mathcal{M}}([b_{0}],[\\mathfrak{b}])$ is a finite set for each of them, we see that $$\\mathcal{M}^{+}([\\mathfrak{b}_{0}],Z_{+})\\setminus\\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],Z_{+}) =(\\mathop{\\cup}\\limits_{ \\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}])=-2w(X,g_{X},0)}\\breve{\\mathcal{M}}([\\mathfrak{b}_{0}],[\\mathfrak{b}])\\times \\mathcal{M}([\\mathfrak{b}],Z_{+}))$$\nis a finite set. This finishes the proof of the theorem.\n\\end{proof}\n\n\n\\section{Proof of the theorem \\ref{new obstruction}}\n Suppose $g_{X}$ has positive scalar curvature everywhere. We first prove that $-2\\operatorname{h}(Y,\\mathfrak{s})\\leq 2\\lambda_{SW}(X)$. Suppose this is not the case. Recall that $\\lambda_{SW}(X)=-\\omega(X,g_{X},0)$ by Lemma \\ref{casson for psc}. By Assumption \\ref{3 dimensional perturbation}, the perturbation $\\mathfrak{q}$ is chosen so that the condition of Lemma \\ref{alternative defi of Froyshov} is satisfied. As a result, we can find nonzero integers\n$ n,m_{1},...,m_{l}$ and irreducible critical points $[\\mathfrak{b}_{1}],...,[\\mathfrak{b}_{l}]\\in \\mathfrak{C}^{o}$ with $\\operatorname{gr}^{\\mathds{Q}}([\\mathfrak{b}_{l}])=-2w(X,g_{X},0)+1$ such that\n\\begin{equation}\\label{b0 killed}\n\\partial^{o}_{o}(m_{1}[\\mathfrak{b}_{1}]+...+m_{j}[\\mathfrak{b}_{l}])=0 \\text{ and } \\partial^{o}_{s}(m_{1}[\\mathfrak{b}_{1}]+...+m_{j}[\\mathfrak{b}_{l}])=n[\\mathfrak{a}_{0}].\\end{equation}\nNow consider the manifold $$\\mathcal{M}=(\\mathop{\\cup}\\limits_{\\{l|m_{l}>0\\}}m_{l}\\cdot\\mathcal{M}^{+}([\\mathfrak{b}_{l}],Z_{+}))\\cup( \\mathop{\\cup}\\limits_{\\{l|m_{l}<0\\}}m_{l}\\cdot \\bar{\\mathcal{M}}^{+}([\\mathfrak{b}_{l}],Z_{+})$$\nwhere $m_{l}\\cdot*$ means the disjoint union $m_{l}$ copies and $\\bar{\\mathcal{M}}^{+}([\\mathfrak{b}_{l}],Z_{+}))$ denotes the orientation reversal of $\\mathcal{M}^{+}([\\mathfrak{b}_{l}],Z_{+})$. By Theorem \\ref{gluing}, Theorem \\ref{orientation}, Theorem \\ref{compactness} and condition (\\ref{b0 killed}), $\\mathcal{M}$ is an oriented, compact $1$-dimensional manifold with $$\\#\\partial\\mathcal{M}=n\\cdot\\#\\breve{\\mathcal{M}}([\\mathfrak{a}_{0}],Z_{+}),$$\nwhere as before, $\\#*$ denotes the number of points, counted with sign, in an oriented $0$-dimensional manifold. By Assumption \\ref{4 dim perturbation} and Proposition \\ref{only one reducible}, we get\n$$\\#\\partial\\mathcal{M}=n\\cdot \\pm 1=\\pm n\\neq 0,$$\nwhich is impossible because we know that the number, counted with sign, of boundary points in any compact $1$-manifold should be $0$.\nThis contradiction finishes the proof of the inequality $\\operatorname{h}(Y,\\mathfrak{s})\\leq 2\\lambda_{SW}(X)$.\n\nBy applying the same argument to the manifold $-X$, we also get $-2\\operatorname{h}(-Y,\\mathfrak{s})\\leq 2\\lambda_{\\textnormal{SW}}(-X)$, which implies $-2\\operatorname{h}(Y,\\mathfrak{s})\\geq 2\\lambda_{\\textnormal{SW}}(X)$ by Lemma \\ref{orientation reversal} and Lemma \\ref{orientation reversal 2}. Therefore,  we have $-2\\operatorname{h}(Y,\\mathfrak{s})= 2\\lambda_{\\textnormal{SW}}(X)$  and the theorem is proved.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection{ObjectNav}\n\nIn ObjectNav, the agent is tasked with navigating to one of a set of target object types (\\eg navigate to the bed) given ego-centric sensory inputs. The sensory input can be an RGB image, a depth image, or combination of both. At each time step the agent must issue one of the following actions: \\emph{Move Forward}, \\emph{Rotate Right}, \\emph{Rotate Left}, \\emph{Look Up}, \\emph{Look Down}, and \\emph{Done}. The \\emph{Move Forward} action moves the agent by 0.25m and the rotate and look actions are performed in $30^\\circ$ increments.\n\nEpisodes are considered successful if (1) the object is visible in the camera's frame (2) the distance between the agent and the target object is within 1 meter and (3) the agent issues the Done action. The starting location of the agent is a random location in the scene.\n\nOur workshop has held 2 ObjectNav challenges: the RoboTHOR ObjectNav Challenge~\\cite{deitke2020robothor} and the Habitat ObjectNav Challenge~\\cite{habitatchallenge2022, savva2019habitat}. Both challenges use the mentioned action and observation space, as well as a simulated LoCoBot robotic agent. In comparison:\n\\begin{itemize}\n    \\item \\textit{Scenes.} The RoboTHOR Challenge\\footnote{\\url{https:\/\/ai2thor.allenai.org\/robothor\/challenge}} includes 89 room-sized dorm-like scenes. The Habitat 2021 Challenge\\footnote{\\url{https:\/\/aihabitat.org\/challenge\/2021\/}} 90 houses from the Matterport3D dataset~\\cite{Chang3DV2017Matterport} and the Habitat 2022 Challenge\\footnote{\\url{https:\/\/aihabitat.org\/challenge\/2022\/}} uses 120 houses from the HM3D Semantics dataset~\\cite{ramakrishnan2021habitat}. Both iterations of the Habitat Challenge use scenes collected from real-world scans. In contrast, RoboTHOR scenes were hand-built by 3D artists to be accessible in AI2-THOR \\cite{ai2thor} in the Unity game engine. Habitat houses are significantly larger than those in RoboTHOR, often consisting of multiple floors.\n    \\item \\textit{Target Objects.} The RoboTHOR Challenge uses 13 relatively small objects as target object types (\\eg Alarm Clock, Basketball, Laptop). The Habitat 2021 Challenge used 21 target objects types and the Habitat 2022 Challenge used 6 target object types. The target object types in both Habitat Challenges typically represent larger objects (\\eg Bed, Fireplace, Sofa).\n\\end{itemize}\n\nFor the RoboTHOR Challenge, state-of-the-art is currently held by ProcTHOR~\\cite{deitke2022procthor}, which has a test SPL~\\cite{anderson_arxiv18} of 0.2884 and a success rate of 65\\% on unseen scenes during training. ProcTHOR uses a fairly simple model that embeds images with CLIP, feeds it through a GRU, and uses an actor-critic output optimized with DD-PPO. Its novelty is pre-training on 10K procedurally generated houses (ProcTHOR-10K). It then fine tunes in RoboTHOR. For the Habitat 2022 Challenge, state-of-the art by SPL is also held by ProcTHOR, achieving 0.32 SPL and a success rate of 54\\% on unseen scenes. For the Habitat 2022 Challenge, ProcTHOR pre-trains on ProcTHOR-10K and fine-tunes on the HM3D Semantics scenes. When sorting the Habitat 2022 Challenge entries by success rate, imitation learning with Habitat-Web~\\cite{ramrakhya2022habitat}, fine-tuned with RL, achieves a state-of-the-art 60\\% success rate and an SPL of 0.30 on unseen scenes. Habitat-Web built a web interface to collect human demonstrations of ObjectNav with Amazon Mechanical Turk. It also achieved state-of-the-art in the Habitat 2021 Challenge, with an SPL of 0.146 and a success rate of 34\\%.\n\n\n\n\n\n\\section{Introduction}\n\n\nWithin the last decade, advances in deep learning, coupled with the creation of massive datasets and high-capacity models, have resulted in remarkable progress in computer vision, audio, NLP, and the broader field of AI. This progress has enabled models to obtain superhuman performance on a wide variety of passive tasks (\\eg image classification). However, this progress has also enabled a paradigm shift towards  embodied agents (\\eg robots) which learn, through interaction and exploration, to creatively solve challenging tasks within their environments. The field of embodied AI focuses on how intelligence emerges from an agent's interactions with its environment. An interaction in the environment involves an agent taking an action that affects its future state. For instance, the agent may perform navigation actions to move around the environment or take manipulation actions to open or pick up objects within reach. Embodied AI is a focus of a growing collection of researchers and research challenges.\n\n\nConsider asking a robot to \\myquote{Clean my room} or \\myquote{Drive me to my favorite restaurant}. To succeed at these tasks in the real world, the robots need skills like \\textit{visual perception} (to recognize scenes and objects), \\textit{audio perception} (to receive the speech spoken by the human), \\textit{language understanding} (to translate questions and instructions into actions), \\textit{memory} (to recall how items should be arranged or to recall previously encountered situations), \\textit{physical intuition} (to understand how to interact with other objects), \\textit{multi-agent reasoning} (to predict and interact with other agents), and \\textit{navigation} (to safely move through the environment). The study of embodied agents both provides a challenging testbed for building intelligent systems and tries to understand how intelligence emerges through interaction with an environment. As such, it involves many disciplines, such as computer vision, natural language processing, acoustic learning, reinforcement learning, developmental psychology, cognitive science, neuroscience, and robotics.\n\nIn this paper, we present a retrospective on the state of embodied AI, focusing on the challenges highlighted at the 2020--2022 CVPR embodied AI workshops. The challenges presented in the workshop have focused on benchmarking progress in navigation, rearrangement, and embodied vision-and-language. The navigation challenges include Habitat PointNav~\\cite{habitat2020sim2real} and ObjectNav~\\cite{batra2020objectnav}, Interactive and Social Navigation with iGibson~\\cite{xia2020interactive}, RoboTHOR ObjectNav~\\cite{deitke2020robothor}, MultiON~\\cite{wani2020multion}, RVSU Semantic SLAM~\\cite{hall2020robotic}, and Audio-Visual Navigation with SoundSpaces~\\cite{chen_soundspaces_2020}; rearrangement challenges include AI2-THOR Rearrangement~\\cite{weihs2021rearrangement}, TDW-Transport~\\cite{gan2022threedworld}, and RVSU Scene Change Detection~\\cite{hall2020robotic}; and embodied vision-and-language challenges include RxR-Habitat~\\cite{rxr}, ALFRED~\\cite{ALFRED20}, and TEACh~\\cite{teach}. We discuss the setup of each challenge and its state-of-the-art performance, analyze common approaches between winning entries across the challenges, and conclude with a discussion of promising future directions in the field.\n\n\n\n\n\n\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/reduced\/header_v3.pdf}\n    \\vspace{-12mm}\n    \\caption{Passive AI tasks are based on predictions over independent samples of the world, such as images collected without a closed loop with a decision-making agent. In contrast, embodied AI tasks include an active artificial agent, such as a robot, that must perceive and interact with the environment purposely to achieve its goals, including in unstructured or even uncooperative settings. \n    Enabled by the progress in computer vision and robotics, embodied AI represents the next frontier of challenges to study and benchmark intelligent models and algorithms for the physical world.}\n    \\label{fig:passive-vs-embodied-ai}\n\\end{figure*}\n\n\\section{What is Embodied AI?}\n\n\n\n\\emph{Embodied AI} studies artificial systems that express intelligent behavior through bodies interacting with their environments. The first generation of embodied AI researchers focused on robotic embodiments \\cite{pfeifer2004embodied}, arguing that robots need to interact with their noisy environments with a rich set of sensors and effectors, creating high-bandwidth interaction that breaks the fundamental assumptions of clean inputs, clean outputs, and static world states required by \\textit{classical AI} approaches \\cite{wilkins1988practical}. More recent embodied AI research has been empowered by rich simulation frameworks, often derived from scans of real buildings and models of real robots, to recreate environments more closely resembling the real world than those previously available. These environments have enabled both discoveries about the properties of intelligence \\cite{partsey2022mapping} and systems which show excellent sim-to-real transfer \\cite{8968053,DBLP:journals\/corr\/abs-1804-10332}.\n\nAbstracting away from real or simulated embodiments, embodied AI can be defined as the study of intelligent agents that can\n\\begin{inparaitem}[]\n    \\item \\emph{see} (or more generally perceive their environment through vision, audition, or other senses),\n    \\item \\emph{talk} (\\ie hold a natural language dialog grounded in the environment),\n    \\item \\emph{listen} (\\ie understand and react to audio input anywhere in a scene.),\n    \\item \\emph{act} (\\ie navigate their environment and interact with it to accomplish goals), and \n    \\item \\emph{reason} (\\ie consider the long-term consequences of their actions).\n\\end{inparaitem}\nEmbodied AI focuses on tasks which break the clean input\/output formalism of passive tasks such as object classification and speech understanding, and require agents to interact with - and sometimes even modify - their environments over time (Fig. \\ref{fig:passive-vs-embodied-ai}).\nFurthermore, embodied AI environments generally violate the clean dynamics of structured environments such as games and assembly lines, and require agents to cope with noisy sensors, effectors, dynamics, and other agents, which creates unpredictable outcomes.\n\n\\paragraph{Why is Embodiment Important?}\nEmbodied AI can be viewed as a reaction against extreme forms of the \\emph{mind-body duality} in philosophy, which some perceive to view intelligence as a purely mental phenomenon. The mind-body problem has faced philosophers and scientists for millennia \\cite{crane2012history}: humans are simultaneously ``physical agents'' with mass, volume and other bodily properties, and at the same time ``mental agents'' that think, perceive, and reason in a conceptual domain which seems to lack physical embodiment. Some scholars argue in favor of a strict mind-body duality in which intelligence is a purely mental quality only loosely connected to bodily experience \\cite{ryle2009concept}. Other scholars, across philosophy, psychology, cognitive science and artificial intelligence, have challenged this mind-body duality, arguing that intelligence is intrinsically connected to embodiment in bodily experience, and that separating them has distorting effects on research \\cite{brooks1990elephants, paul2021extended, varela2017embodied, mehta2011mind, ryle2009concept}.\n\nThe history of research in artificial intelligence has mirrored this debate over mind and body, focusing first on computational solutions for symbolic problems which appear hard to humans, a strategy often called GOFAI (\"Good Old Fashioned AI\", \\cite{boden20144,mcdermott2015gofai}). The computational theory of mind argued that if intelligence was reasoning operations in the mind, computers performing similar computations could also be intelligent \\cite{piccinini2004first,Sclar2022}. %\nPurely symbolic artificial intelligence were often disconnected from the physical world, requiring symbolic representations as input, creating problems with grounding symbols in perception \\cite{harnad1990symbol,steels2008symbol} and often leading to brittleness \\cite{mccarthy2007here,lohn2020estimating,cummings2020surprising}. However, symbolic reasoning problems themselves often proved to be relatively easy, whereas the physical problems of perceiving the environment or acting in it were actually the most challenging: what is unconscious for humans often requires surprising intelligence, often known as Moravec's Paradox \\cite{goldberg2015robotics,agrawal2010study}. \nSome researchers challenged this approach, arguing that for machines to be intelligent, they must interact with noisy environments via rich sets of sensors and effectors, creating high-bandwidth interactions that break the assumption of clean inputs and outputs and discrete states required by \\textit{classical AI} \\cite{wilkins1988practical}; %\nthese ideas were echoed by roboticists already concerned with connecting sensors and actuators more directly \\cite{arkin1998behavior,brooks1990elephants,moravec2000ripples}.\nMuch as neural network concepts hibernated through several AI winters before enjoying a renaissance, embodied AI ideas have now been revived by new interest from fields such as computer vision, machine learning and robotics - often in combination with neural network ideas.\nNew generations of artificial neural networks are now able to digest raw sensor signals, generate commands to actuators, and autonomously learn problem representations, linking \"classical AI\" tasks to embodied setups.\n\nThus, embodied AI is more than just the study of agents that are active and situated in their environments: it is an exploration of the properties of intelligence. Embodied AI research has demonstrated that intelligent systems that perform well at embodied tasks often look different than their passive counterparts \\cite{fu2022coupling} - but, conversely, that highly performing passive AI tasks can often contribute greatly to embodied systems as components \\cite{shridhar2022cliport}. Furthermore, the control over embodied agents provided by modern simulators and deep learning libraries enables ablation studies that reveal fine-grained details about the properties needed for individual embodied tasks \\cite{partsey2022mapping}.\n\n\n\\paragraph{What is \\emph{not} Embodied AI?}\n\nEmbodied AI overlaps with many other fields, including robotics, computer vision, machine learning, artificial intelligence, and simulation.\nHowever, there are differences in focus which make embodied AI a research area in its own right.\n\nAll \\textit{robotic} systems are embodied; however, not all embodied systems are robots (e.g., AR glasses), and robotics requires a great deal of work beyond purely trying to make systems intelligent. Embodied AI also includes work that focuses on exploring the properties of intelligence in realistic environments while abstracting some of the details of low-level control.\nFor example, the ALFRED~\\cite{ALFRED20} benchmark uses simulation to abstract away low-level robotic manipulation (\\eg moving a gripper to grasp an object) to focus on high-level task planning. Here, the agent is tasked with completing a natural language instruction, such as \\textit{rinse the egg to put it in the microwave}, and it can open or pickup an object by issuing a high-level \\textit{Open} or \\textit{Pickup} action that succeeds if the agent is looking at the object and is sufficiently close to it.\nAdditionally, \\cite{partsey2022mapping} provides an example of studying properties of intelligence, where they attempt to answer whether mapping is strictly required for a form of robotic navigation.\nConversely, robotics includes work that focuses directly on the aspects of the real world, such as low-level control, real-time response, or sensor processing.\n\n\\textit{Computer vision} has contributed greatly to embodied AI research; however, computer vision is a vast field, much of which is focused purely on improving performance on passive AI tasks such as classification, segmentation, and image transformation. Conversely, embodied AI research often explores problems that require other modalities with or without vision, such as navigation with sound \\cite{chen_soundspaces_2020} or pure LiDAR images.\n\n\\textit{Machine learning} is one of the most commonly used techniques for building embodied agents. However, machine learning is a vast field encompassing primarily passive tasks, and most embodied AI tasks are formulated in such a way that they are learning agnostic. For example, the iGibson 2020 challenge \\cite{shen2020igibson} allowed training in simulated environments but deployment in holdout environments in both real and simulation; nothing required the solutions to use a learned approach as opposed to a classical navigation stack (though learned approaches were the ones deployed).\n\n\\textit{Artificial intelligence} is written into the name of embodied AI, but the field of embodied AI was created to address the perceived limitations of classical artificial intelligence \\cite{pfeifer2004embodied}, and much of artificial intelligence is focused on problems like causal reasoning or automated programming which are hard enough without introducing the messiness of real embodiments. More recently, techniques from more traditional artificial intelligence domains like natural language understanding have been applied to embodied problems with great success \\cite{ahn2022can}.\n\n\\textit{Simulation} and embodied AI are intimately intertwined; while simulations of real-world systems go far beyond the topics of robotics, and the first generation of embodied AI focused on robotic embodiments \\cite{pfeifer2004embodied}, much of modern embodied AI research has expanded to simulated benchmarks, emulating or even scanned from real environments, which provide challenging problems for traditional AI approaches, with or without physical embodiments. Despite not starting with robots, systems that have resulted from this work have nevertheless found success in real-world environments \\cite{8968053,DBLP:journals\/corr\/abs-1804-10332}, providing hope that simulated benchmarks will prove a fruitful way to develop more capable real-world intelligent systems.\n\n\n\n\n\n\\paragraph{Why focus on real-world environments?}\nMany researchers are exploring intelligence in areas such as image recognition or natural language understanding where at first blush interaction with an environment appears not to be required. Genuine discoveries about intelligent systems appear to have been made here, such as the role of convolutions in image processing and the role of recurrent networks and attention in language processing. So a reasonable question is, why do we need to focus on interactive and realistic (if not real-world) environments if we want to understand intelligence?\n\nFocusing on interactive environments is important because each new modality of intelligence we consider - classification, image processing, natural language understanding, and so on - has required new architectures for learning systems \\cite{goodfellow2016deep}, \\cite{chollet2021deep}. Interacting with an environment over time requires the techniques of reinforcement learning. Deep reinforcement learning has made massive strides in creating learning systems for synthetic environments, including traditional board games, Atari games, and even environments with simulated physics such as the Mujoco environments.\n\nHowever, embodied AI research focuses on environments that are either more realistic \\cite{habitatchallenge2022} or which require actual deployments in the real world \\cite{habitat2020sim2real,shen2020igibson}). This shift in emphasis has two primary reasons. First, many embodied AI researchers\nbelieve that the challenges of realistic environments are critical for developing systems that can be deployed in the real world. Second, many embodied\nAI researchers believe that there are genuine discoveries to be made about the properties of intelligence needed to handle real world environments that can only be made by attempting to solve problems in environments that are as close to the real world as is feasible at this time.\n\n\n\n\n\n\\section{Challenge Details}\n\nIn this section, we discuss the 13 challenges present at our Embodied AI Workshop between 2020--2022. The challenges are partitioned into navigation challenges, rearrangement challenges, and embodied vision-and-language challenges. Most challenges present a distinctive tasks, metrics and training datasets, though many challenges share similar observation spaces, action spaces, and environments.\n\n\\subsection{Navigation Challenges}\n\nOur workshop has featured a number of challenges relating to embodied visual navigation. At a high-level, the tasks consist of an agent operating in a simulated 3D environment (\\eg a household), where its goal is to move to some target. For each task, the agent has access to an egocentric camera and observes the environment from a first-person's perspective. The agent must learn to navigate the environment from its visual observations.\n\nThe challenges primarily differ based on how the target is encoded (\\eg ObjectGoal, PointGoal, AudioGoal), how the agent is expected to interact with the environment (\\eg static navigation, interactive navigation, social navigation), the training and evaluation scenes (\\eg 3D scans, video-game environments, the real world), the observation space (\\eg RGB vs. RGB-D, whether to provide localization information), and the action space (\\eg outputting discrete high-level actions or continuous joint movement actions).\n\n\\noindent\n\\begin{minipage}[l]{0.46\\textwidth}\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/pointnav_task_v2.pdf}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\captionof{figure}{The \\emph{PointNav} task requires an agent to navigate to a goal coordinate in a novel environment (potentially with noisy sensory inputs), without access to a pre-built map of the environment.}\n    \\vspace{-0.1in}\n\\end{minipage}\n\n\n\\subsubsection{PointNav}\n\nIn PointNav, the agent's goal is to navigate to target coordinates in a novel environment that are relative to its starting location (\\eg navigate 5m north, 3m west relative to its starting pose), without access to a pre-built map of the environment. The agent has access to egocentric sensory inputs (RGB images, depth images, or both), and an egomotion sensor (sometimes referred to as GPS+Compass sensor) for localization. The action space for the robot consists of: \\emph{Move Forward 0.25m}, \\emph{Rotate Right $30^\\circ$}, \\emph{Rotate Left $30^\\circ$}, and \\emph{Done}. An episode is considered successful if the agent issues the \\emph{Done} command within 0.2 meters of the goal and within 500 maximum steps. The agent is evaluated using the Success Rate (SR) and \"Success\nweighted by Path Length\" (SPL) \\cite{anderson_arxiv18} metrics, which measures the success and efficiency of the path taken by the agent. For training and evaluation, challenge participants use the train and val splits from the Gibson 3D dataset \\cite{zamir_cvpr18}. \n\nIn 2019, AI Habitat hosted its first challenge on PointNav.\\footnote{\\url{https:\/\/aihabitat.org\/challenge\/2019\/}} The winning submission \\cite{chaplot2020learning} utilized a combination of classical and learning-based methods, and achieved a high test SPL of 0.948 in the RGB-D track, and 0.805 in the RGB track. In 2020 and 2021, the PointNav challenge was modified to emphasize increased realism and on sim2real predictivity (the ability to predict performance on a real robot from its performance in simulation) based on findings from Kadian et al. \\cite{habitatsim2real20ral}. Specifically, the challenge (PointNav-v2) introduced (1) no GPS+Compass sensor, (2) noisy actuation and sensing, (3) collision dynamics and `sliding', and (4) minor changes to the robot embodiment\/size, camera resolution, height to better match the LoCoBot robot. These changes proved to be much more challenging, with the winning submission in 2020 \\cite{ramakrishnan2020occant} achieving a SPL of 0.21 and SR of 0.28. In 2021, there was a major breakthrough with a 3$\\times$ performance improvement over the winners in 2020; the winning submission achieved a SPL of 0.74 and SR of 0.96 \\cite{habitat2020sim2real}. Since an agent with perfect GPS + Compass sensors in this PointNav-v2 setting can only achieve a maximum of 0.76 SPL and 0.99 SR, the PointNav-v2 challenge was considered solved, and discontinued in future years. \n\n\n\n\n\\subsubsection{Interactive and Social PointNav}\n\nIn Interactive and Social Navigation, the agent is required to reach a PointGoal in dynamic environments that contain dynamic objects (furniture, clutter, etc) or dynamic agents (pedestrians). Although robot navigation achieves remarkable success in static, structured environments like warehouses, it still remains a challenging research question in dynamic environments like homes and offices. In 2020 and 2021, the Stanford Vision and Learning Lab in collaboration with Robotics@Google hosted challenges on Interactive and Social (Dynamic) Navigation\\footnote{\\url{https:\/\/svl.stanford.edu\/igibson\/challenge2021.html}}. These challenges used the simulation environment iGibson~\\cite{shen2020igibson, li2021igibson} with a number of realistic indoor scenes, as illustrated in Fig.~\\ref{fig:iGibsonChallenges}. The 2020 Challenge\\footnote{\\url{https:\/\/svl.stanford.edu\/igibson\/challenge2020.html}} also featured a Sim2Real component where the participants trained their policies in the iGibson simulation environment and deployed in the real world.\n\n\n\\begin{figure}[ht!]\n     \\centering\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=1\\textwidth]{fig\/igibson\/ig_interactive_nav_2022.jpg}\n         \\caption{Interactive Navigation}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.23\\textwidth}\n         \\centering\n         \\includegraphics[width=1\\textwidth]{fig\/igibson\/ig_social_nav_2022.jpg}\n         \\caption{Social Navigation}\n     \\end{subfigure}\n    \\caption{\\emph{Interactive Navigation} (left) requires the agent to push aside small obstacles (\\eg shoes, boxes) whereas \\emph{Social Navigation} (right) requires the agent to navigate among pedestrians and respect their personal space.}\n    \\label{fig:iGibsonChallenges}\n\\end{figure}\n\nIn \\textit{Interactive Navigation}, we challenge the notion that navigating agents are to avoid collision at any cost. We argue for the contrary -- in clutter-filled real environments, such as homes, an agent will have to interact and push away objects to achieve meaningful navigation. Note that all objects in the scenes are assigned realistic physical weight and are  interactable. As in the real world, while some objects are light and movable by the robot, others are not. Along with the furniture objects originally in the scenes, additional objects (e.g. shoes and toys) from the Google Scanned Objects dataset~\\cite{downs2022google} are added to simulate real-world clutter. The performance of the agent is evaluated using a novel Interactive Navigation Score (INS)~\\cite{xia2020interactive} that measures both navigation success as well as the level of disturbance to the scene an agent has caused along the way.\n\nIn \\textit{Social Navigation}, the agent navigates among walking humans in a home environment. The humans in the scene move towards randomly sampled locations, and their 2D trajectories are simulated using the model of Optimal Reciprocal Collision Avoidance (ORCA)~\\cite{berg2011reciprocal} integrated in iGibson~\\cite{shen2020igibson, li2021igibson, darpino2021socialnav}. The agent shall avoid collisions or proximity to pedestrians beyond a threshold (distance <0.3 meter) to avoid episode termination. It should also maintain a comfortable distance to pedestrians (distance <0.5 meter), beyond which the score is penalized but episodes are not terminated. Social Navigation Score (SNS), which is the average of STL (Success weighted by Time Length) and PSC (Personal Space Compliance), is used to evaluate performance of the agent.\n\nThe agent takes in the current RGB-D images, the target coordinates in its local frame, and current velocities as observations, and outputs a continuous twist command (desired linear and angular velocities) as actions. The dataset includes eight training scenes, two validation scenes and five testing scenes. All scenes are fully interactive.\n\nIn the 2020 edition we saw 4 submissions while in the subsequent 2021 edition we had 6 submissions. The current state-of-the-art learning based methods achieved some level of success for Interactive and Social Navigation tasks (around $0.5$ INS and $0.45$ SNS), but they are still far from being solved. In both competitions participants improved over navigation success rate while keeping environment disturbance relatively constant. The common failure cases include the agent being too conservative and not being able to clear the obstacles in time, and the agent being too aggressive and colliding with the other moving pedestrians. \n\nOne of the challenges for the Social Nav part was the difficulty in simulating the trajectories of the human agents, including reactivity and interaction between agents. Often times, getting to the goal requires negotiation of the space or the agent would require to go over the desired personal space threshold; or the simulated human agents behave erratically due to limitations on the behavior models and the space constraints. For future editions, we are to emphasize on the importance of high fidelity simulation of navigation with human-like behaviors.\n\nFor the Sim2Real component of the 2020 Challenge, a significant performance drop was observed during the Sim2Real transfer, due to the reality gap in visual sensor readings, dynamics (\\eg motor actuation), and 3D modeling (\\eg soft carpets). More analysis of the takeaways can be found in the iGibson Challenge 2020\\footnote{\\url{https:\/\/www.youtube.com\/watch?v=0BvUSjcc0jw}} and 2021\\footnote{\\url{https:\/\/www.youtube.com\/watch?v=1uSsds7HSrQ}} videos, along with the winning entry paper~\\cite{yokoyama2021benchmarking}.\n\n\n\\input{challenges\/robothor-objectnav}\n\n\\noindent\n\\begin{figure}[ht!]\n    \\vspace{0.1in}\n    \\centering\n    \\textbf{Task:} Find the Bed\\\\[-0.2in]\n    \\includegraphics[width=0.475\\textwidth]{fig\/objectnav.png}\n    \\vspace{-0.1in}\n    \\caption{\n    \\emph{ObjectNav} tasks the agent with navigating to a given object type in the scene. This example shows the agent tasked with navigating to the \\emph{Bed} in the scene. The house is curtousy of the ArchitecTHOR dataset \\cite{deitke2022procthor}.\n    }\n    \\vspace{-0.1in}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[ht!]\n     \\centering\n     \\begin{subfigure}[b]{0.46\\textwidth}\n         \\centering\n         \\includegraphics[width=1\\textwidth]{fig\/multion-cyl.png}\n         \\caption{}\n         \\label{fig:multion_cyl}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=1\\textwidth]{fig\/multion-real.png}\n         \\caption{}\n         \\label{fig:multion_real}\n     \\end{subfigure}\n    \\caption{\\emph{Multi-ObjectNav}: (a) Top-down visualization of a MultiON episode with 5 target cylinder objects in a particular sequence; (b) Top-down visualization of a MultiON episode with 5 target real objects in a particular sequence.}\n\\end{figure}\n\n\\subsubsection{Multi-ObjectNav}\nIn Multi-ObjectNav (MultiON)~\\cite{wani2020multion}, the agent is initialized at a random starting location in an environment and asked to navigate to an ordered sequence of objects placed within realistic 3D interiors (Figures~\\ref{fig:multion_cyl},~\\ref{fig:multion_real}). The agent must navigate to each target object in the given sequence and call the \\emph{Found} action to signal the object's discovery.\nThis task is a generalized variant of ObjectNav, whereby the agent must navigate to a sequence of objects rather than a single object.\nMultiON explicitly tests the agent's navigation capability in locating previously observed goal objects and is, therefore, a suitable test bed for evaluating memory-based architectures for Embodied AI.\n\nThe agent is equipped with an RGB-D camera and a (noiseless) GPS+Compass sensor.\nThe GPS+Compass sensor provides the agent's current location and orientation relative to its initial location and orientation in the episode.\nIt is not provided with a map of the environment. The action space comprises of \\emph{Move Forward} by 0.25 meters, \\textit{Rotate Left} by 30$^\\circ$, \\textit{Rotate Right} by 30$^\\circ$ and \\emph{Found}.\n\nThe MultiON dataset is created by synthetically adding objects in the Habitat-Matterport 3D (HM3D)~\\cite{ramakrishnan_arxiv21} scenes. The objects are either cylinder-shaped or natural-looking (real) objects. As shown in Figure~\\ref{fig:multion_cyl},  the cylinder objects are of the same height and radius, with different colors. However, such objects do not appear realistic in the indoor scenes of Matterport houses. Furthermore, detecting the same object with different colors might be easy for the agent to learn. This has led us to include realistic-looking objects that can naturally occur in houses (Figure~\\ref{fig:multion_real}). These objects are of varying sizes and shapes and pose a more demanding detection challenge. \nThere are 800 HM3D scenes and 8M episodes in the training split, 30 unseen scenes and 1050 episodes in the validation split, and 70 unseen scenes and 1050 episodes in the test split.\nThe episodes are generated by sampling random navigable points as start and goal locations, such that the locations are on the same floor and a navigable path exists between them. Next, five goal objects are randomly sampled from the set of Cylinder or Real objects to be inserted between the start and the goal, maintaining a minimum pairwise geodesic distance between them to avoid cluttering. Furthermore, to make the task even more realistic and challenging, three distractor objects (which are not goals) are inserted in each episode. The presence of distractors will encourage new agents to distinguish between goal objects and other objects in the environment. An episode is considered successful if the agent is able to reach within 1 meter of every goal in the specified order and generate the \\textit{FOUND} action at each goal object. Apart from the standard evaluation metrics used in ObjectNav, such as Success Rate (SR) and Success weighted by path length (SPL)~\\cite{anderson_arxiv18}, we additionally use Progress and Progress weighted by path length (PPL) to measure agent performance. The leaderboard for the challenge is based on the PPL metric. MultiON challenge was hosted on evalAI, an open-source platform for evaluating and comparing artificial intelligence methods. The participants implemented their methods in docker images and submitted them to evalAI. The docker images were evaluated on evaluation servers, and the results were uploaded to evalAI.\n\nThe MultiON task is similar to ObjectNav, but at the same time, it tries to solve different challenges. Notably, it aims to inject long-term planning capabilities into the agents. In the ObjectNav task, the object detection task takes on a fundamental role. Still, the agent does not have to remember all the objects (and their semantic information) encountered in the past. In MultiON, on the other hand, we assume a more limited part of the detection (e.g., detecting cylinders or a set of a limited number of natural objects). Parallelly, the agent must be able to remember the objects already seen. Thus, this task is more tailored to the real world than ObjectNav. In fact, the agents operate in the same environment for a very long time and, therefore, must be able to remember what has already been seen. For this reason, the approaches developed for MultiON, unlike those for ObjectNav, always add a component that stores the semantic information obtained through exploration.\n\nFor the 2021 challenge, a simpler setup was used. The distractors were absent, the objects were only the cylinders, and the dataset was developed on Matterport3D \\cite{chang2017matterport3d}. The Proj-Neural model was used as Baseline \\cite{wani2020multion}. This model takes advantage of an egocentric map that is used as an input for an end-to-end model that achieved 29\\% Progress and 12\\% Success. Surprisingly, two models based on mapping and path planning, SgoLAM (64\\% progress, 52\\% Success) and Memory Augmented SLAM (Mem-SLAM) (57\\% Progress, 36\\% Success), exceeded the results obtained from the Baseline by a large margin demonstrating that this type of model works well on long-horizon tasks. Instead, the model proposed in \\cite{marza2021teaching} won the 2021 challenge, with a progress of 67\\% and a success of 55\\%. This model is an evolution of Proj-Neural, where three auxiliary tasks were used to inject information about the map and objects into the agent's internal representation.\n\nIn the 2022 challenge, instead, we noticed some similarities between the Baseline method, Mem-SLAM, and the winning entry in the 2022 MultiON challenge, Exploration and Semantic Mapping for Multi Object-Goal Navigation (EXP-MAP). Both the methods are modular, consisting of detection (identifying objects from raw RGB images), Mapping (incrementally building a top-down map of the environment using Depth observations and relative poses), and Planning (navigating to a detected goal object by generating low-level actions) modules. All these models record previously seen objects in some memory (e.g., semantic map of the environment). The EXP-MAP can achieve 70\\% Progress and 60\\% Success in the Test-Challenge split of the Cylinder objects track of the challenge while achieving 55\\% Progress and 40\\% Success in the Real objects track. These results show that episodes with natural objects are more challenging to detect than the cylinders.\n\n\n\\noindent\n\\begin{minipage}[l]{0.46\\textwidth}\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=\\textwidth, height=1.75in]{fig\/rvsu\/challenge_hero_image_small.jpg}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\captionof{figure}{In the \\emph{RVSU Semantic SLAM} task, an autonomous agent explores environment to create a semantic 3D cuboid map of objects.}\n    \\vspace{-0.1in}\n\\end{minipage}\n\n\\subsubsection{Navigating to Identify All Objects in a Scene}\n\nThe RVSU semantic SLAM challenge tasks participants with exploring a simulation environment to map out all objects of interest therein. \nThis challenge asks a robot agent the question, ``what objects are where?'' within the scene.\nRobot agents traverse a scene, create an axis aligned 3D cuboid semantic map of the objects within that scene, and are evaluated based on their map's accuracy.\nProviding a semantic understanding of objects can assist a robot's ability to interpret attributes of its environment such as knowing how to interact with objects and understanding what type of room it might be in.\nThis semantic understanding is typically viewed as a semantic simultaneous localization and mapping (SLAM) problem.\nThe task of semantic SLAM has already seen great investigation using static datasets such as KITTI~\\cite{Geiger2013IJRR}, Sun RGBD~\\cite{song2015sun} and SceneNet~\\cite{McCormac:etal:ICCV2017}.\nHowever, these static datasets ignore the active capabilities of robots and forego searching the physical action space for the actions that best explore and understand an environment.\nAddressing this limitation, the RVSU semantic SLAM challenge~\\cite{hall2020robotic} helps bridge the gap between passive and active semantic SLAM systems by providing a framework and simulation environments for repeatable, quantitative comparison of both passive and active approaches.\n\nParticipation in the challenge is conducted through simulated environments, accessed and controlled using the BenchBot framework~\\cite{talbot2020benchbot}.\nThe environments used are a version of the BenchBot environments for active robotics (BEAR)~\\cite{hall2022bear} rendered using the NVIDIA Omniverse Isaac Simulator\\footnote{\\url{https:\/\/developer.nvidia.com\/isaac-sim}}.\nBEAR provides 25 high-fidelity indoor environments comprising of five base environments with five variations thereof.\nBetween variations, objects are added and removed, and lighting conditions are changed.\nAcross environments there are 25 object classes of interest to be mapped within the challenge.\nThe challenge splits BEAR into 2 base environments for algorithm development and 3 for final testing and evaluation.\nThe BenchBot framework enables a simulated robot to explore BEAR using either passive or active control  through discretised actions that are pre-defined or actively chosen by the agent respectively.\nThe action space for robot agents is \\textit{MOVE\\_NEXT} for passive mode and \\textit{MOVE\\_DISTANCE} and \\textit{MOVE\\_ANGLE} for active mode with magnitude of movement being defined by users with a minimum distance of 0.01 m and a minimum angle of 1\u00b0.\nBenchBot provides the robot agent access to RGB-D camera, laser, and either ground-truth or estimated pose information for the robot immediately after completing any given action.\nThe progression of passive control with ground-truth pose data, through to active control with estimated pose data is designed to gradually bridge the gap from passive to active semantic SLAM.\nThe final cuboid map created by the agent within the challenge is evaluated using the new object map quality (OMQ) measure outlined in~\\cite{hall2020robotic}.\nThis evaluation measure considers the quality of every provided object cuboid, in terms of both geometric and semantic accuracy, when compared to its best match in the ground-truth map, as well as the number of provided cuboids with no matching ground-truth equivalent and vice verse.\nThe final OMQ score is between 0 and 1 with 1 being the best score.\n\nCurrent results from the RVSU Semantic SLAM challenge have shown that while the challenge is simple in concept, there is still room for improvement from current state-of-the-art methods.\nThe highest result for semantic SLAM achieved was 0.39 OMQ when using ground-truth pose data and passive control.\nWhen digging deeper into the results provided, we can see that although the quality of matching cuboids is often good (pairwise quality of up to 0.72) there are too many unmatched cuboids to get a high score.\nWhen competitors bridge the gap from passive to active control, we also commonly see a drop in OMQ of approximately 0.06 despite having more control of the robot's observations.\nThose who participated in both passive and active control versions of the semantic SLAM task, focused their research in how to map a scene given a sequence of inputs, rather than how to actively explore to maximize understanding of the scene.\nThese results suggest that potentially the most fruitful areas for future research lie in better filtering out cuboids that do not match any true object, and in how to best exploit active robot control to improve scene understanding.\nThere is also yet to be an attempt at solving this challenge using active control and noisy pose estimation which adds further difficulty to the challenge.\n\n\n\\subsubsection{Audio-Visual Navigation}\n\n\nMoving around in the real world is a multi-sensory experience, and an intelligent agent should be able to see, hear and move to successfully interact with its surroundings. While current navigation models tightly integrate seeing and moving, they are deaf to the world around them, motivated by these factors, the audio-visual navigation task was introduced~\\cite{gan2019look,chen_soundspaces_2020}, where an embodied agent is tasked to navigate to a sounding object in an unknown unmapped environment with its egocentric visual and audio perception (Figure~\\ref{fig:soundspaces_concept}). This audio-visual navigation task can find applications in assistive and mobile robotics, e.g., robots for search and rescue operations and assistive home robots. Along with the task, the SoundSpaces platform was also introduced, a first-of-its-kind audio-visual simulator where an embodied agent could move around in the simulated environment while seeing and hearing.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/soundspaces.pdf}\n    \\caption{\\emph{AudioGoal} tasks an autonomous agent to find an audio source in an unmapped 3D environment by navigating to a goal. Here the top down map is overlaid with the acoustic pressure field heatmap. While audio provides rich directional information about the goal, and audio intensity variation is correlated with the shortest path distance, vision reveals the surrounding geometry in the form of obstacles and free space. An AudioGoal navigation agent should intelligently leverage the synergy of these two complementary signals to successfully navigate in the environment.}\n    \\label{fig:soundspaces_concept}\n\\end{figure}\n\n\nAudio-visual navigation is a challenging task because the agent not only needs to perceive the surrounding environment, but also to reason about the spatial location of the sound emitter in the environment via the received sound. This new multimodal embodied navigation task has gained attention over the past few years and different methods have been proposed to solve this task, including learning hierarchical policies~\\cite{chen_waypoints_2020}, training robust policies with adversarial attack~\\cite{YinfengICLR2022saavn} or data augmentation for generalization to novel sounds~\\cite{dynamic_av_nav}. However, the performance of SOTA audio-visual navigation models is still not perfect, and thus we organized the SoundSpaces Challenge~\\footnote{\\url{https:\/\/soundspaces.org\/challenge}} at CVPR 2021 and 2022, which aims to promote research in the field of developing autonomous embodied agents that are capable of navigating to sounding objects of interest using audio and vision. \n\nMore specifically, in an AudioGoal navigation episode, a sound source is placed at a random location in the environment, and the agent is also positioned with a random pose (location and orientation) at the start of the episode. The agent is tasked to navigate to the sounding object with one of the four actions from the action space: \\emph{Move Forward}, \\emph{Rotate Left}, \\emph{Rotate Right}, and \\emph{Done}. At each episode step, the agent receives egocentric (noiseless) RGB-D images captured with a $90^\\circ$ field-of-view (FoV) camera, the binaural audio received by the agent. The episode terminates when the agent executes the \\emph{Done} action, or it runs out of a pre-specified time budget. The agent is evaluated using standard embodied navigation metrics, such as Success Rate (SR) and SPL~\\cite{anderson_arxiv18}. We use SPL as the metric for ranking challenge participants. \n\nWe set up the AudioGoal navigation task on the Matterport3D (MP3D)~\\cite{Chang3DV2017Matterport} scene dataset, split into train\/val\/test splits in 59\/10\/12 for this challenge due to its large scale. SoundSpaces provides audio renderings for MP3D in the form of pre-rendered room impulse responses (RIRs), which are transfer functions that characterize how sound propagates from one point in space to another point in space. For all MP3D scenes, SoundSpaces discretizes them into grids of spatial resolution 1 meter $\\times$ 1 meter and provide RIRs for all pairs of grid points. For the source sound, we use 73\/11\/18 disjoint sounds in our train\/val\/test splits, respectively. Each sound clip is 1 second long. The received sound at every step is the result of convolution between the source sound and the RIR corresponding to the source location and current agent pose in the scene. While the \\emph{Move Forward} action takes the agent forward by 1 meter in the direction it's currently facing if there is a navigable node in the scene grid in that direction, \\emph{Rotate Left} and \\emph{Rotate Right} rotate the agent by $90^\\circ$ in the clockwise and anti-clockwise directions, respectively. The episode terminates when the agent issues the \\emph{Done} action, or it exceeds a budget of 500 steps. \n\nIn SoundSpaces Challenge 2021 and 2022, a total of 25 teams showed interest and 8 teams participated. For SoundSpaces Challenge 2022's leading teams, we observed some similarities between the model design of the top two teams. Both models used a hierarchical navigation architecture (inspired by AV-WaN~\\cite{chen_waypoints_2020}), where a high-level (long-term) planner predicts a navigation waypoint in the local neighborhood of the agent at each step, and a low-level (short-term) planner executes atomic actions, such as \\emph{Move Forward} and \\emph{Rotate Left}, to take the agent to the predicted waypoint. Further, agents that leverage the audio-visual cues from the full $360^\\circ$ FoV and train a separate model for stopping are more successful and efficient than the others. Moreover, training an AudioGoal navigation agent in the presence of distractor sound sources also results in learning robust navigation policies that boost navigation performance. The presentation videos from the leading teams can be found on the challenge website.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/ss2.pdf}\n    \\caption{SoundSpaces 2.0, a continuous, configurable, and generalizable audio-visual simulation platform. It models various acoustic phenomena and renders visual and audio observations with spatial and acoustic correspondence.}\n    \\label{fig:ss2}\n\\end{figure}\n\nOne of the limitations of the SoundSpaces platform is that it provides pre-rendered RIRs for fixed grid points and does not allow users to render sounds for arbitrary locations or environments. To tackle this issue, we have introduced SoundSpaces 2.0~\\cite{chen22soundspaces2} (Fig.~\\ref{fig:ss2}, a continuous, configurable and generalizable simulator. This new simulator has enabled continuous audio-visual navigation as well as many other embodied audio-visual tasks. We believe this simulator will take the audio-visual navigation task to the next step. Another important direction for future research is for the agent to reason about the semantics between the sound and objects (\\eg semantic audio-visual navigation~\\cite{chen2021savi} and finding fallen objects~\\cite{gan2022finding}). If the agent could leverage the semantics of sounding objects, it could navigate faster by reasoning where the object is located in space based on its category information.\n\nWe believe studying audio-visual embodied AI is of vital importance for building truly autonomous robots with rich perception modalities in the real world.\n\n\\subsection{Rearrangement Challenges}\n\nThis section discusses rearrangement challenges. Rearrangement is described as a canonical task in Embodied AI that may lead to learning representations that are for many downstream tasks~\\cite{batra2020rearrangement}. Here, the agents goal is to move or detect the changes from one state of the scene to another. For examples, several objects, such as an apple and a banana may move, and the agent is tasked with detecting that they moved and putting them back to their correct locations.\n\n\\noindent\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=.46\\textwidth]{fig\/rvsu\/scd_example.png}\n    \\caption{Example of the scene change detection challenge. Between two scenes some objects are added (blue) and removed (orange) and these need to be identified and mapped out.}\n\\end{figure}\n\n\\subsubsection{Scene Change Detection} \n\nThe RVSU scene change detection (SCD) challenge, an extension to the RVSU semantic SLAM challenge, requires identification and mapping of objects which have been added and removed between two traversals of the same base scene~\\cite{hall2020robotic}.\nHuman environments are inherently non-static with objects frequently being added, removed, or shifted.\nIn order to operate within said environments whilst utilising object maps, it becomes important to be able to identify when these changes have occurred.\nThis challenge examines perhaps the simplest of such scenarios, where some objects are added or removed while all others remain fixed in place.\n\nThe setup for SCD is similar to that shown for the RVSU Semantic SLAM challenge described previously but with some differences in challenge setup within BenchBot.\nThe SCD challenge also uses the BEAR dataset~\\cite{hall2022bear} which already has multiple variants of a set of base scenes.\nVariants differ in some objects are added and some removed (as desired for the SCD task), and also there are some lighting variations to increase challenge difficulty.\nBenchBot enables the switching between environment variants within one SCD submission as soon as the robot agent determines it is finished with its first traversal.\nBenchBot supplies the same robot control that progresses from passive to active control via discrete actions and wherein each action is followed by an observation containing RGB-D images, laser scan, and either ground-truth or estimated robot pose data.\nThe SCD challenge utilises a variant of the OMQ evaluation measure~\\cite{hall2020robotic} which evaluates the final 3D object cuboid map output as part of the challenge.\nThis variant introduces the necessity for the map to provide an estimate for the likelihood that an object has been added or removed from the scene between traversals.\nThis state estimation for the object is then combined with the estimation of the label and location of the object to make up the object-level quality score.\nAs before, the best OMQ score possible is 1 and the worst is 0.\n\nThere has been limited engagement with the SCD challenge and there is much room for improvement.\nIn the CVPR 2022 iteration of this challenge the highest OMQ  score achieved was 0.25.\nThis is quite lower than the best OMQ score of the semantic SLAM challenge which was able to reach OMQ of 0.39.\nThis can be attributed somewhat to the approach that competitors used in solving SCD.\nAll SCD submissions performed semantic SLAM on the two different traversals and did a naive comparison of the resultant cuboid maps.\nThis led to an accumulation of the errors seen across the maps for both traversals.\nThis simple beginning shows that there are many directions that can still be experimented with in order to improve SCD in future years.\nThis may include more targeted approaches to navigation and\/or mapping within the second traversal which utilises the scene knowledge from the first traversal.\nThere is still much research to be done in how to reliably identify and map out changes between scenes.\n\n\\noindent\n\\begin{figure}[ht!]\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{fig\/rearrangement.jpg}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\caption{\\textbf{AI2-THOR Visual Room Rearrangement Challenge.} An agent must change pose and attributes of objects in a household environment to restore the environment to an initial state.}\n    \\vspace{-0.1in}\\label{fig:rearrangement}\n\\end{figure}\n\n\\subsubsection{Interactive Rearrangement}\n\n\n\n\nWhile the PointNav and ObjectNav tasks have led to substantial advances in embodied AI, performance on these tasks has steadily improved with PointNav being nearly solved~ \\cite{ramakrishnan_arxiv21}. In light of this fast progress, researchers from nine institutions proposed the \\emph{rearragement} as the next frontier for research in embodied AI~ \\cite{batra2020rearrangement}. At a high level, in rearrangement, an embodied agent must interact with its environment to transform the environment from it's initial state $s^{\\text{init}}$ to a goal state $s^{\\text{goal}}$. This general formulation of rearrangement leaves much unspecified, namely: (1) which environment? (2) what affectors\/actions are available to agent? (3) how are states $s^{\\text{init}},s^{\\text{goal}}$ specified? Given the EAI community's focus on building agents capable of assisting humans in everyday tasks, all existing instantiations of the rearrangement task embody agents in household environments and focus on object-based rearrangement: the difference between goal and initial environment states is confined to objects pose (position\/rotation) and attributes (e.g. is the object opened or closed?). Successful rearrangement in these tasks requires agents to flexibly encode environment environment states, to dynamically update these encodings as they interact with their environment, and also to making long-term plans (frequently of the traveling-salesman variety) to maximize the efficiency of rearrangement. We now detail the two rearrangement challenges, AI2-THOR Visual Room Rearrangement and TDW-Transport, held at the EAI workshop in past years.\n\nThe AI2-THOR Visual Room Rearrangement (RoomR) task~\\cite{weihs2021rearrangement} occurs in two phases, see Figure~\\ref{fig:rearrangement}. In the \\emph{Walkthrough phase} the the agent explores a room and builds an internal representation of the room's configuration ($s^{\\text{goal}}$). Then, in the \\emph{Unshuffle phase}, the agent is placed within the same environment but objects within this environment have been randomly moved to different locations and opened\/closed ($s^{\\text{init}}$), the agent must now restore objects back to their original states. As this 2-phase RoomR is quite challenging, a 1-phase variant was also proposed where the agent enacts the Walkthrough and Unshuffle phases simultaneously, receiving egocentric RGB-D images of the environment in both the $s^{\\text{init}}$ and $s^{\\text{goal}}$ states at each step. In the 2021 RoomR challenge, no participants were able to outperform the baseline model, which used a 2D semantic mapping approach along with imitation learning from a heuristic expert agent. In 2022 however, several exciting approaches were released resulting in dramatic improvements in performance. For the 1-phase variant, performance leapt from ${\\approx}9\\%$ to ${\\approx}24\\%$ on the \\textsc{FixedStrict} metric on the test-set. Advances making this possible included (1) the use of CLIP-pretrained visual encoders~\\cite{khandelwalEtAl2021embodiedclip} and (2) large-scale pre-training using procedurally generated environments~\\cite{deitke2022procthor}. Unlike the end-to-end approaches used for the 1-phase variant, the most successful methods for the 2-phase variant used powerful inductive biases in the form of semantic mapping and planning algorithms. In a yet unpublished work, 2022 2-phase challenge winner used voxel-based 3D semantic map and shortest path planners to bulid an agent attaining ${\\approx}15\\%$ \\textsc{FixedStrict} on the test-set (dramatically beating the baseline performance of $<1\\%$). The differences between the approaches used in the 1- and 2-phase variants is striking: it seems that new algorithms are required to bring fully end-to-end methods to the challenging 2-phase setting.\n\n\n\\noindent\n\\begin{figure}[ht!]\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{fig\/transport_comp.jpg}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\caption{\\textbf{TDW-Transport Challenge.} In this example task, the agent must transport two objects on the table in one room and place them on the bed in the bedroom. The agent can first pick up the container, put two objects into it, and then transport them to the target location.}\n    \\vspace{-0.1in}\\label{fig:TDW-transport}\n\\end{figure}\n\nTDW-Transport Challenge~\\cite{gan2022threedworld} is an object-goal driven interactive navigation task (see Figure ~\\ref{fig:TDW-transport}). In this challenge, an embodied agent is spawned randomly in a house and is required to find a small set of objects scattered around the house and transport them to a desired final location.\nWe also position various containers around the house; the agent can find these containers and place some objects into them. Without using a container as a tool, the agent can only transport up to two objects at a time. However, using a container, the agent can collect several objects and then transport them together. While the containers help the agent transport more than two items, it also takes some time to find them. Therefore, the agent has to decide to use containers or not.\n\nThe embodied agent is equipped with an RGB-D camera. There are two types of actions of the agent: navigation and interactive actions. Navigation actions include Move Forward($\\alpha$ meters), Turn Left($\\theta$ degrees), Turn Right($\\theta$ degrees). Interactive actions include Reach to object, Put into container, Grasp, and Drop.  The objective of this challenge is to transport the maximum number of objects in fixed steps as efficiently as possible. We use the transport rate as an evaluation metric, which measure the fraction of the objects successfully transported to the desired position within a given budget.\n\n\\subsection{Embodied Vision-and-Language}\n\nThis section discusses the embodied vision-and-language challenges. In each challenge, natural language is used to convey the goal to the agent. For example, the agent may be tasked with following instructions to complete a task. Since language is the primary means of human communication, advances in embodied vision-and-language research will make it easier for a human to naturally interact with the trained agents. Additionally, language imposes a data-sparse regime, as examples cannot be created automatically, as precision in language is directly tied to a specific scene layout (e.g. ``on the left\/right\"), and it is an open challenge as to if unimodal representations can be leveraged in this embodied space \\cite{bisk2020}.\n\n\\noindent\n\\begin{minipage}[l]{0.46\\textwidth}\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=\\textwidth]{challenges\/rxr-habitat\/rxr-habitat-challenge.pdf}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\captionof{figure}{The Room-Across-Room Habitat Challenge (RxR-Habitat) is a multilingual instruction-following task set in simulated indoor environments requiring realistic navigation over long action sequences.}\n    \\vspace{-0.1in}\n    \\label{fig:rxr_habitat_challenge}\n\\end{minipage}\n\n\\subsubsection{Navigation Instruction Following}\nNavigation guided by natural language has long been a desired foundational ability of intelligent agents. In Vision-and-Language Navigation (VLN), an agent is given egocentric vision in a realistic, previously-unseen environment and tasked with following a path described in natural language, \\eg, \\textit{Move toward the dining table. Go down the hallway toward the kitchen and stop at the sink}. The Room-Across-Room Habitat Challenge (RxR-Habitat) instantiates VLN in simulated indoor environments, provides multilingual instructions, and requires agents to navigate via long action sequences in a realistic, continuous 3D world (Figure~\\ref{fig:rxr_habitat_challenge}). Solving RxR-Habitat would have applications in many domains, such as personal robotic assistants, and lead to a better scientific understanding of the connection between language, vision, and action.\n\nThe RxR-Habitat Challenge takes place in 3D reconstructions of Matterport3D scenes \\cite{matterport3d} and interacts with those scenes using the Habitat Simulator \\cite{savva2019habitat}. We model the agent embodiment after a robot of radius 0.18m and height 0.88m with a camera mount at 0.88m. An episode is specified by a scene, a start location, a language instruction, and the implied path.\nAt each time step, the agent observes egocentric vision in the form of a single forward-facing, noiseless 480x640 RGB-D image with a 79$^{\\circ}$ HFOV. The agent also receives the natural language instruction from one of three languages: English, Hindi, or Telugu. The action space is discrete and noiseless, consisting of actions $\\{$\\texttt{MOVE\\_FORWARD}, \\texttt{TURN\\_LEFT}, \\texttt{TURN\\_RIGHT}, \\texttt{STOP}, \\texttt{LOOK\\_UP}, \\texttt{LOOK\\_DOWN}$\\}$. Forward movement is 0.25m and turning and looking actions are performed in 30$^{\\circ}$ increments. Actions that result in collision terminate upon collision, \\ie, no wall sliding. An episode ends when the agent calls \\texttt{STOP}.\n\nThe dataset used in RxR-Habitat is the Room-Across-Room (RxR) dataset \\cite{rxr} ported from high-level discrete VLN environments \\cite{anderson_cvpr18} to the continuous VLN-CE environments \\cite{krantz_vlnce_2020} used in Habitat. The dataset is split into training (Train: 60,300 episodes, 59 scenes), validation in environments seen during training (Val-Seen: 6,746 episodes, 57 scenes), validation in environments not seen during training (Val-Unseen: 11,006 episodes, 11 scenes), and testing in environments not seen during training (Test-Challenge: 9,557 episodes, 17 scenes), each with a roughly equal distribution between English, Hindi, and Telugu instructions. To submit to the RxR-Habitat leaderboard~\\footnote{\\url{https:\/\/ai.google.com\/research\/rxr\/habitat}}, participants run inference on the Test-Challenge split and submit the inferred agent paths. The leaderboard evaluates these paths against held-out ground-truth paths. Agent performance is reported as the average of episodic performance. The official comparison metric between the agent's path and the ground truth path is normalized dynamic time warping (nDTW) \\cite{magalhaes2019effective} which scores path alignment between 0 and 1 with 1 indicating identical paths. Additional metrics reported for analysis include path length (PL), navigation error (NE), success rate (SR) and success weighted by inverse path length (SPL)~\\cite{anderson_arxiv18}.\n\nRxR-Habitat is incredibly difficult; the interplay between perception, control, and language understanding makes instruction-following an interdisciplinary problem. Realistic environments and unconstrained natural language lead to a long tail of vision and language grounding, and the low-level action space makes learning the relationship between instructions and actions highly implicit. The RxR-Habitat Challenge took place in 2021 and again in 2022. The baseline model is a cross-modal attention (CMA) model \\cite{krantz_vlnce_2020} that attends between vision and language encodings, predicts actions end-to-end from observation, and is trained with behavior cloning (nDTW: 0.3086). In the first year, teams failed to surpass the performance of this baseline. However, a significant improvement in SOTA was attained in 2022; the top submission (Reborn \\cite{an20221st}) produced an nDTW of 0.5543 --- an 80\\% relative improvement over the baseline. This was enabled by an effective hierarchy of waypoint candidate prediction, waypoint selection (the discrete VLN task), and waypoint navigation. For waypoint selection, a history-aware transformer was trained in discrete VLN with augmentations including synthetic instructions, environment editing, and ensembling. It was then transferred and tuned in continuous environments. Despite this remarkable improvement, a performance gap still exists between SOTA in continuous versus discrete environments, with human performance even higher. Evidently, this direction of research is still far from saturated.\n\n\\noindent\n\\begin{minipage}[l]{0.46\\textwidth}\n    \\vspace{0.1in}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/inst_teaser_v9.pdf}\n    \\vspace{-0.1in}\n    \\captionsetup{type=figure}\n    \\captionof{figure}{ALFRED involves interactions with objects, keeping track of state changes, and references to previous instructions. The dataset consists of 25k language directives corresponding to expert demonstrations of household tasks.\n    We highlight several frames corresponding to portions of the accompanying language instruction.}\n    \\vspace{-0.1in}\n\\end{minipage}\n\n\\subsubsection{Interactive Instruction Following.} \n\nALFRED is a benchmark for connecting human language to \\textit{actions}, \\textit{behaviors}, and \\textit{objects} in interactive visual environments.\nPlanner-based expert demonstrations are accompanied by both high- and low-level human language instructions in 120 indoor scenes in AI2-THOR. \nThese demonstrations involve partial observability, long action horizons, underspecified natural language, and irreversible actions. \n\nThe dataset includes over 25K English language directives describing 8K expert demonstrations averaging 50 steps each, resulting in >428K image-action pairs.\nMotivated by work in robotics on segmentation-based grasping, agents in ALFRED interact with objects visually, specifying a pixelwise interaction mask of the target object.\nThis inference is more realistic than simple object class prediction, where localization is treated as a solved problem.\nExisting beam-search and backtracking solutions are infeasible due to the larger action and state spaces, long horizon, and inability to undo certain actions. Agents are evaluated on their ability to achieve directives in both seen and unseen rooms. Evaluation metrics include: success rate (SR), success weighted by path-length (SPL), and Goal-Condition success which measures completed subtasks.\n\nCurrent state-of-the-art approaches in ALFRED use spatial-semantic mapping  \\cite{blukis2022persistent,min2021film} to explore and build persistent representations of the environment before grounding instructions. These representations have also been coupled with symbolic planners and modular policies for better generalization to unseen rooms. Currently, the best performing agent achieves 40\\% success in seen rooms and 36\\% in unseen rooms.\n\n\\noindent\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=.45\\textwidth]{fig\/teach\/teaser.pdf}\n    \\caption{\n        In the TEACh Two-Agent Task Completion challenge, the \\textit{Commander}\\ has oracle task details (a), object locations (b), a map (c), and egocentric views from both agents, but cannot act in the environment, only communicate.\n        The \\textit{Follower}\\ carries out the task and asks questions (d).\n        The agents can only communicate via text.}\n    \\label{fig:teach_teaser}\n\\end{figure}\n\n\\subsubsection{Interactive Instruction Following with Dialog}\n\\textbf{T}ask-driven \\textbf{E}mbodied \\textbf{A}gents that \\textbf{C}hat (TEACh) is a dataset of over 3,000 human--human, interactive dialogues and demonstrations of household task completion in the AI2-THOR simulator.\nRobots operating in human spaces must be able to engage in such natural language interaction with people, both understanding and executing instructions and using conversation \\cite{thomason:corl19,Roman2020} to resolve ambiguity \\cite{Nguyen2022} and recover from mistakes.\nA \\textit{Commander}\\ with access to oracle information about a task communicates in natural language with a \\textit{Follower}. \nThe \\textit{Follower}\\ navigates through and interacts with the environment to complete tasks varying in complexity from \\texttt{Make Coffee} to \\texttt{Prepare Breakfast}, asking questions and getting additional information from the \\textit{Commander}\\ (Figure~\\ref{fig:teach_teaser}).\n\nThere are 12 task types in TEACh with 438 unique combinations of task parameters (e.g., \\texttt{Make Salad} with 1 versus 2 slices of \\texttt{Tomato}) in 109 AI2-THOR environments.\nOn average, there are more than 13 utterances in each cooperative dialogue, with tasks taking an average of 131 \\textit{Follower}\\ actions to complete compared to ALFRED's 50 due to both task complexity and non-optimal planning.\nA major difference between the TEACh and ALFRED challenge is edge cases in the environments due to ALFRED's rejection sampling: if a PDDL planner could not resolve an ALFRED task given an initial scene configuration, it was rejected from data, where TEACh scene configurations are rejected only when a \\textit{human} cannot resolve them.\nThis decision results in many ``corner cases'' in TEACh that require human ingenuity, for example filling a pot with water using a cup as an intermediate vessel when the pot itself is too large to fit in the sink basin.\n\nThe Two-Agent Task Completion (TATC) challenge is based on the TEACh data, and involves modeling \\textit{both} the \\textit{Commander}\\ and \\textit{Follower}\\ agents, which have distinct action and observation spaces but a common household task goal.\nThe \\textit{Commander}\\ agent has access to a structured representation of the goal and its component parts, as well as search functions to identify the locations and physical appearance of objects in the environment by class or id.\nThe \\textit{Follower}\\ is analogous to an ALFRED agent, but with a wider action space that includes, for example, pouring liquids from one container to another.\nFurther, object interactions are done via individual $(x,y)$ coordinate predictions, rather than the full object masks used in ALFRED, analogous to the click inputs of human users who provided demonstrations.\nThe agents both have a \\texttt{communicate} action that adds to a mutually-visible dialogue history, and requires generating text.\n\nTATC agents are evaluated via SR and SPL, similar to ALFRED agents.\nRule-based, planning agents for TATC achieve about 24\\% SR, with planning corner cases dominating failures.\nA learned \\textit{Follower}\\ based on the Episodic Transformer~\\cite{pashevich:et} with a rule-based, simple \\textit{Commander}\\ that simply reports the raw text of the next task subgoal as a communication action achieves nearly 0\\%.\nWe are eager to see whether mapping-based approaches like those succeeding at ALFRED can adapt to the wider space of tasks and environment corner cases in TEACh.\n\n\n\\section{Common Approaches}\n\nThis section presents common approaches used by the winners of the challenges. We discuss large-scale training by scaling up datasets and compute, leveraging visual pre-trained models such as CLIP, the use of inductive biases such as maps, goal embeddings to represent different tasks, and visual and dynamic augmentation to make simulators more noisy and closer to reality.\n\n\\subsection{Large-Scale Training}%\n\\input{directions\/large-scale-training}\n\n\\subsection{Visual Pre-Training}\nInitial successes in deep reinforcement learning were largely focused on graphically simplistic environments, \\eg Atari games, for which complex visual processing was, in large part, unnecessary. For instance, the seminal work of Mnih et al.~\\cite{mnih_nature15} achieved human-level performance on dozens of Atari games using used a model with only three convolutional layers. Several initial works in embodied AI, in part due to computational constraints when training RL agents, adopted this mindset; for instance, Savva et al.~\\cite{habitat19iccv} trained models using a 3-layer image processing CNN for Point Navigation. As embodied agents are, ostensibly, meant to be embodied in the real-world, one might expected the they would benefit from image processing architectures designed for use with real images and, indeed, this has proven to be the case. A recent work has shown that modifying existing embodied baseline models by replacing their visual backbones with a CLIP-pretrained ResNet-50 can result in dramatic improvements~\\cite{khandelwalEtAl2021embodiedclip}. The top performing models of 1-Phase Rearrangement, RoboTHOR ObjectNav leaderboards, and Habitat ObjectNav leaderboard, use variants of this ``EmbCLIP'' architecture~\\cite{deitke2022procthor}. Several other top performing models to other challenges use pretrained vision models for object detection and semantic segmentation (RVSU Semantic SLAM, MultiON, and Two-Phase Rearrangement).\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table*}[]\n\\small   \n\\begin{tabular}{lllllclllc}\n\\hline\n\\textbf{}                         & \\textbf{}          & \\textbf{} & \\multicolumn{3}{c}{\\textbf{Best End-to-end}}       & \\textbf{} & \\multicolumn{3}{c}{\\textbf{Best Modular}}          \\\\\n\\textbf{Challenge}                & \\textbf{Simulator} & \\textbf{} & \\textbf{Method} & \\textbf{Success} & \\textbf{Rank} & \\textbf{} & \\textbf{Method} & \\textbf{Success} & \\textbf{Rank} \\\\ \\hline\nObjectNav                         & Habitat            &           & Habitat-Web     & 60               & 2             &           & Stretch         & 60               & 1             \\\\\nAudio-Visual Navigation           & SoundSpaces        &           & Freiburg Sound  & 73               & 2             &           & colab\\_buaa     & 78               & 1             \\\\\nMulti-ON                          & Habitat            &           & -               & -                & -             &           & exp\\_map        & 39               & 1             \\\\\nNavigation Instruction Following  & VLN-RxR            &           & CMA Baseline    & 13.93            & 10            &           & Reborn          & 45.82            & 1             \\\\\nInteractive Instruction Following & AI2-THOR           &           & APM             & 15.43            & 14            &           & EPA             & 36.07            & 1             \\\\\nRearrangement                     & AI2-THOR           &           & ResNet18 + ANM  & 0.5              & 6             &           & TIDEE           & 28.94            & 1             \\\\ \\hline\n\\end{tabular}\n\\vspace{-6pt}\n\\caption{Table summarizing the performance of best end-to-end and best modular methods across various challenges.}\n\\label{tab:e2e_modular}\n\\end{table*}\n\n\\subsection{End-to-end vs Modular} \nIn the last few years, two classes of methods have emerged for various embodied AI tasks: (1) end-to-end and (2) modular. The end-to-end methods learn to predict low-level actions directly from input observations. They typically use a deep neural network consisting of a visual encoder followed by a recurrent layer for memory and are trained using imitation learning or reinforcement learning. Earliest application of end-to-end methods on embodied AI tasks include \\cite{ lample2016playing, zhu2017target, mirowski2016learning, chaplot2017arnold, savva2017minos, hermann2017grounded, chaplot2017gated}. End-to-end RL methods have also been scaled to train with billions of samples using distributed training~\\cite{wijmans2019decentralized} or using tens of thousands of procedurally generated scenes~\\cite{deitke2022procthor}. Researchers have also introduced some structure in end-to-end policies such using spatial representations~\\cite{gupta2017cognitive, parisotto2017neural, chaplot2018active, henriques2018mapnet, gordon2018iqa} and topological representations~\\cite{yang2018visual, savinov2018semi, savinov2018episodic}. \n\nModular methods use multiple modules to break down the embodied AI tasks. Each module is trained for a specific subtask using direct supervision. The modular decomposition typically includes separate modules for perception (mapping, pose estimation, SLAM), encoding goals, global waypoint selection policies, planning and local obstacle avoidance policies. Rather than training all modules end-to-end, each module is trained separately using direct supervision, which also allows use of non-differentiable classical modules within the embodied AI pipeline. Earliest learning-based modular methods include ~\\cite{chaplot2020learning, chaplot2020neural, chaplot2020object} which show their effectiveness on various navigation tasks such as Exploration, ImageNav and ObjectNav. Variants of these methods include improvements in mapping by anticipating unseen parts~\\cite{ramakrishnan2020occant} or by using density-based maps~\\cite{bigazzi2022focus}; and learning global waypoint selection policies in ObjectNav and ImageNav entirely using offline or passive datasets to improve sample and compute efficiency~\\cite{ramakrishnan2022poni,hahn_nrns_2021,mezghani2021memory,wassermanlast}. Recently, modular methods have also been applied to longer horizon tasks such as Navigation Instruction Following in VLN-CE~\\cite{Krantz_2021_ICCV,an20221st,raychaudhuri2021language}, Interactive Instruction Following in ALFRED~\\cite{min2021film, liu9planning, murray2022following} and Rearrangement in AI2 Thor~\\cite{sarch2022tidee, trabucco2022simple}.\n\nIn Table~\\ref{tab:e2e_modular}, we show the performance of best end-to-end and modular methods in various 2022 Embodied AI challenges. The table shows that while end-to-end method performance is comparable to modular methods on easier and relatively shorter horizon tasks such as ObjectNav and Audio-Visual Navigation, the performance gap increases as the complexity of the task increases such as in Interactive Navigation and Rearrangement. This is likely because as the task horizon increases, the exploration complexity increases exponentially when training end-to-end with just reinforcement learning.\n\n\n\n\\subsection{Visual and Dynamic Augmentation}\nVisual and dynamic augmentation of real-world datasets has proven to be a key technique for enabling robotic systems trained in simulation to transfer to unseen environments and even to reality. For years in the robotics and learning community, a prevalent attitude has been that simulation transfers poorly to reality. One justification for this perspective is that the dynamics models of most simulations are not good enough to reveal problems that typically occur in real robotic deployments, such as wheel slippage, odometry drift, floor irregularities, nonlinear motor and dynamic responses, and component breakage and burnout. Another justification is that simulated evaluation can reveal problems with systems, but cannot validate them: validation tests for robotic systems must ultimately be performed on-robot.\n\nNevertheless, many existing systems have shown successful transfer to novel and to real-world environments by augmenting training datasets with noise, static obstacles, dynamic obstacles, and changes to visual appearance. \nMany approaches add noise to sensors, actions and even environment dynamics, effectively making each episode occur in a  distinctive environment; these techniques have proved useful for translating LiDAR-based policies trained in simulation to the real world \\cite{faust2018prm,francis2020long} and for estimating the safety of plans prior to deployment \\cite{xiao2021toward}. %\nOther approaches improve performance by adding static obstacles to the environment in simulation, also effectively increasing the space of environments trained on \\cite{xiao2021toward}.\nAn interesting example of this presented at the workshop involves training in a simulated environment with variable dynamics and using an adaptation module to perform system identification in real environments \\cite{kumar2021rma}, \\cite{fu2022coupling}.\n\nHowever, visual policies present other difficulties: a policy trained on one set of objects and lighting conditions is unlikely to transfer to other objects and conditions~\\cite{deitke2022procthor}. Adding noise has been used to improve robustness \\cite{fang2019scene}, and the RSVU challenges add distractor objects to reduce the effects of distractors~\\cite{hall2020robotic}. The RL-CycleGan approach uses style transfer to make simulated environments appear more like the real world \\cite{rao2020rl}. Most recently, ProcTHOR~\\cite{deitke2022procthor} attempts to address the visual diversity issue by generating large numbers of synthetic environments.\n\nFinally, while the pandemic disrupted many plans for real-world  deployments, both the iGibson, RoboTHOR, and Habitat challenges included tests of simulation-trained policies in real deployments~\\cite{xia2020interactive,deitke2020robothor,batra2020objectnav}. These environments proved challenging for many policies; nevertheless, many policies were still able to function, and going forward tests in the real will be an important validation step for embodied AI agents. As datasets collected from real evaluations increase, the opportunity exists to train policies directly over this real-world data, which has already proved useful in a grasping and manipulation context~\\cite{bahl2022human} and for legged locomotion~\\cite{smith2022walk}. \n\n\\section{Future Directions}\n\nIn this section, we discuss promising future directions for embodied AI, including further leveraging pre-trained models, world models and inverse graphics, simulation and dataset advances, sim2real approaches, procedural generation, generalist agents, and multi-agent and human interaction.\n\n\\subsection{Pre-training}\n\nPre-training has powered impressive results from visual recognition~\\cite{girshick2014rich}, natural language~\\cite{radford2019language,devlin2018bert}, and audio~\\cite{oord2016wavenet}.\nPre-trained models can be repurposed through fine-tuning, zero-shot generalization, or prompting to perform diverse tasks.\nHowever, pre-training has not yet found such levels of success in embodied AI. \nRecent work has begun to explore this direction, showing that pre-trained models can help improve performance, efficiency and expand the scope of solvable tasks.\nThis section discusses how pre-training can help embodied AI with visual pre-training objectives, the role of scale in pre-training, pre-training for task specification, and pre-trained behavioral priors.\n\nOne promising area is new pre-training objectives for visual representations in embodied AI.\nPrior work shows supervised pre-training is effective for navigation and manipulation tasks \\cite{yen2020learning, shah2021rrl, sax2018mid}.\nHowever, a large-scale study \\cite{wijmans2019dd} showed that at scale, supervised pre-training visual representations from ImageNet could hurt downstream performance in PointNav.\nEmbCLIP shows that unsupervised pre-training with a pre-trained CLIP visual encoder is effective for various embodied AI tasks \\cite{khandelwalEtAl2021embodiedclip}.\nOther works explore pre-training with masked auto-encoders~\\cite{xiao2022masked}, contrastive learning~\\cite{du2021curious,nair2022r3m,sermanet2018time}, or other SSL objectives~\\cite{yadav2022offline}.\nFuture work may explore tailoring pre-training objectives specifically for control.\nFor example, pre-training may account for the temporal aspect of decision making~\\cite{gregor2018temporal}, be embodiment agnostic~\\cite{stadie2017third},  curiosity-driven~\\cite{du2021curious},or avoid pixel reconstruction~\\cite{zhang2020learning}. Analogous to pretrained visual representation for visual navigation, audio-visual representations~\\cite{alwassel2020self,Morgado2021AudioVisualID,mittal2022learning} can be adopted for tasks with multi-modal inputs~\\cite{gan2019look,chen_soundspaces_2020} in future work.\n\nAnother way pre-training may benefit embodied AI is with scaling model and dataset size.\nCurrently, works use a variety of datasets for pre-training such as Epic Kitchens \\cite{damen2022rescaling,damen2018scaling,VISOR2022}, YouTube 100 days of hands \\cite{shan2020understanding}, Something-Something \\cite{goyal2017something}, Ego4D \\cite{grauman2022ego4d}, and RealEstate10k \\cite{46965} datasets.\nThe curation of data for pre-training matters, with pre-training on unlabeled curated datasets outperforming labeled datasets on downstream tasks \\cite{xiao2022masked}.\nIncreasing model size also promises benefits, with larger ResNet showing better performance \\cite{wijmans2020train}.\nPrior work pre-trains ResNet-50 \\cite{nair2022r3m,khandelwalEtAl2021embodiedclip,yadav2022offline}, CLIP \\cite{khandelwalEtAl2021embodiedclip}, or ViT models \\cite{xiao2022masked}.\nWith the success of neural scaling laws~\\cite{kaplan2020scaling} in vision and language, future work in embodied AI may translate these lessons to pre-training larger models with larger datasets.\n\nPre-training also provides a way to specify diverse tasks for agents easily.\nOpen-world agents must be able to flexibly complete tasks with unseen goals or task specifications.\nPrior work shows that pre-trained models can provide dense reward supervision \\cite{cui2022can, shao2021concept2robot, chen2021learning}. \nOther work shows that pre-trained models can be leveraged for open-world object detection, allowing for zero-shot generalization to new goals in navigation tasks \\cite{al2022zero,gadre2022clip,majumdar2022zson}. \nFinally, some methods explore generalization to new language instructions by employing pre-trained models \\cite{shridhar2022cliport}. \nThere are further opportunities to use such models for zero-shot generalization to completing new tasks, new goals, or flexibly specifying goals in different input modalities.\n\nFinally, pre-training can learn behavioral priors for interaction.\nThe previously discussed pre-training objectives primarily focus on learning representations of input modalities.\nHowever, this leaves out a critical part of embodied AI, interacting with the environment.\nRather than pre-training representations, pre-training can also learn models of behavior that account for agent actions.\nOne line of work pre-trains models with supervised learning to predict actions from sensor inputs on large interaction datasets and then fine-tune this model to specific downstream tasks~\\cite{baker2022video}.\nOther work learns skills or reusable behaviors from offline datasets that can adapt to downstream tasks~\\cite{pertsch2020accelerating,gupta2019relay}.\nFuture work may explore how scaling dataset size, model size, and compute can pre-train behavioral policies better suited for fine-tuning on downstream tasks.\n\n\n\\subsection{World models and inverse graphics}\nAs previously discussed, semantic and free-space maps have been hugely successful in enabling high performance and efficient learning across embodied-AI tasks (\\eg in navigation~\\cite{chaplot2020learning} and rearrangement~\\cite{trabucco2022mass}). These mapping approaches are successful as they provide a simple, highly-structured, model of the agent's environment that enables explicit planning. The simplicity of existing mapping approaches is also one of their major limitations: as embodied tasks become more complex they require agents to reason about new semantic categories and new types of interaction (\\eg arm-based manipulation).\nExtending existing approaches to include new capabilities is generally possible but non-trivial, often requiring substantive human effort. For instance, a 2D free-space mapping approach successful for PointGoal Navigation~\\cite{chaplot2020learning} was explicitly extended to include semantic mapping channels so as to enable training agents for ObjectGoal Navigation~\\cite{chaplot2020object}. \nThese challenges in mapping raise an important question: how can we build flexible models of an agent's environment that can be used for general purpose task planning? We identify two exciting directions toward answering this question: end-to-end trainable world models and game-engine simulation via inverse-graphics.\n\nAt a high-level, a world models $W$ is a function that, given the state of the environment $s_t$ at time $t$ and an agent action $a$, produces a prediction $W(s,a)=\\widehat{s}_{t+1}$ of the state of the world at time $t+1$ if the agent were to take action $a$~\\cite{ha2018worldmodels}. Iterative applications of the world model can thus be used to simulate agent trajectories and, thus, for model-based planning. As may be expected, building and training world models made challenging by several factors: (1) generally full state information ($s_t$) is not available as agent's have access only to partial, egocentric, observations, (2) the dynamics of an environment are frequently stochastic and thus cannot be predicted deterministically, (3) many details encoded in a state are irrelevant to task completion (\\eg minor color or texture variations of objects) and attempting to predict these details needlessly complicates training, and (4) collecting high-quality training data for the end-to-end training of world models may require the design of increasingly complex physical states (\\eg a tower of plates to be knocked over). While more work is needed before world models will become a ubiquitous tool for embodied AI agents, recent work has shown that world models can be successfully used to training agents to play Atari games~\\cite{hafner2021discreteworldmodels} and to build navigation-only models of embodied environments~\\cite{koh2021pathdreamer}.\n\nAs world models are meant to be broadly applicable and learned from data, they frequently eschew inductive biases and use general purpose architectures. The disadvantage of this approach is clear: we have well-understood models of physics that should not have to be re-learned from data for every task. Moreover, we have simulators designed explicitly to simulate 3D objects and their physical interactions, video game engines. These observations suggest another approach: rather than learning an implicit world model, can we use techniques from inverse-graphics to back-project an agent's observations to 3D assets within a scene in a game engine? Once this back-projection is complete, the game engine can be used to perform physical simulations and planning. This approach, which can be thought of as world modeling with strong inductive biases, has used successfully to build models of intuitive physics in constrained settings~\\cite{wu2017learningtoseephysics}. While this approach appears very promising it does present some challenges: (1) the problem of inverse graphics is especially challenging in this setting as de-rendered objects must be in physically plausible relationships with one another for simulation to be meaningful and (2) game-engines are, generally, non-differentiable and can be slow. Nevertheless, this approach of explicitly bringing our understanding of physical laws to world models seems a promising direction toward building embodied models that can physically reason and plan.\n\n\\subsection{Simulation and Dataset Advances}\n\n\nOne factor towards improving the reliability and scope of embodied AI research in the future will be the continued improvement of simulation capabilities and realism, and increase in the scale and quality of 3D assets used in simulation.\nRepeatable, quantitative analysis of embodied AI systems at scale has been made possible through the use of simulation.\nAs research in embodied AI continues to grow and tackle increasingly complex problems within increasingly complex scenes, the needs placed on simulation environments and assets  will increase. \n\nOne important area of improvement for simulation environments is physics realism during agent-object interaction. Past simulation environments have solidly supported both abstracted~\\cite{ai2thor,puig2018virtualhome} and rigid-body physics-based agent interactions~\\cite{shen2020igibson, szot2021habitat, gan2020threedworld, ehsani2021manipulathor}. There has been quite some progress in physics simulation of flexible material (rope, cloth, soft body)~\\cite{lin2020softgym, seita_bags_2021}, fluids~\\cite{fu2022rfuniverse}, and contact-rich interaction (e.g. nut-and-bot)~\\cite{narang2022factory}, leveraging state-of-the-art physics engines like PyBullet~\\cite{coumans2021} and NVIDIA's PhysX\/FleX. Some environment like iGibson 2.0~\\cite{li2021igibson} even attempts to go beyond kinodynamic simulation and use approximate models to simulate more complex physical processes such as thermodynamics. However, all of these simulations are still far from perfect and oftentimes face a grim trade-off between fidelity and efficiency. More efficient and realistic simulation of physical interaction of agents with all elements of their environment can greatly assist in the applicability of embodied AI trained using simulation, to solving real-world problems.\n\nWith the prevalence of vision sensors for solving problems, the need for increased visual realism has also become imperative for research that is to translate to the real world. This has been aided in recent years through aspects like new graphics technology like real-time ray tracing.\nAn example of how these advances can improve visual realism can be found within iterations of the RVSU challenge~\\cite{hall2020robotic}  %\nthat recently migrated to NVIDIA's Isaac Omniverse\\footnote{see  \\url{https:\/\/developer.nvidia.com\/blog\/making-robotics-easier-with-benchbot-and-isaac-sim\/} for details}. Yet, the rendering speed can still become a bottleneck as the number of objects and light sources increase in the scenes. \n\nAside from advances in computer graphics, visual realism also relies on high-quality 3D assets of scenes and objects. It has been a standard practice for embodied AI researchers to benchmark navigation agents in large-scale static scene datasets like Matterport3D~\\cite{matterport3d}, Gibson~\\cite{xiazamirhe2018gibsonenv}, and HM3D~\\cite{ramakrishnan2021habitat}. On the other hand, interactive scenes have been quite limited. iGibson 2.0~\\cite{li2021igibson} provides fifteen fully interactive scenes with added clutter that aim to capture the messiness of the real world, and Habitat 2.0~\\cite{szot2021habitat} also similarly converts a subset of an existing static dataset~\\cite{replica19arxiv} to become fully interactive. ProcTHOR~\\cite{deitke2022procthor} recently attempted to scale up the effort and procedurally generate fully interactive scenes with realistic room structures and object layout.\n\nMany object datasets have been proposed and heavily utilized by embodied AI researchers in the past years~\\cite{chang2015shapenet,mo2019partnet,xiang2020sapien,calli2017yale,srivastava2022behavior,downs2022google,collins2022abo}. Although increased scale and quality has been the general trend for these datasets, it still remains extremely costly to make them useable for interactive tasks. For example, most of the objects in these datasets do not support interaction, such as the ability to open cabinets. Such work not only requires modifying meshes, but also requires a tremendous amount of annotation to provide part-level and articulation annotation, as was done in the PartNet and PartNet-Mobility datasets~\\cite{mo2019partnet, xiang2020sapien}. Similarly, it requires additional annotation and mesh editing to support object states (\\eg whether the object is cookable, sliceable) for the BEHAVIOR dataset~\\cite{srivastava2022behavior} or in AI2-THOR~\\cite{ai2thor}. Yet, these annotations are essential as we ramp up the complexity of embodied AI tasks.\n\nAnother important aspect of realistic simulation is its multimodal nature, one of the most important ones is auditory perception. Existing acoustic simulation like SoundSpaces~\\cite{chen_soundspaces_2020} allows the agent to move around in the environment with both visual and auditory sensing to search for a sounding object. However, it pre-computes the room impulse response (RIR) based on scene geometry and can't be configured. Recent work like SoundSpaces 2.0~\\cite{chen22soundspaces2} (Fig.~\\ref{fig:ss2}) extended the simulation to make it continuous, configurable and generalizable to arbitrary scene datasets, which enables the agent to explore the acoustics of the space even further.\n\nIn addition, tactile sensing is also super important to future simulation environments. As these sensors become more cost-efficient, robots will likely be equipped with these new sensing capabilities in the foreseeable future. Researchers have made tremendous progress in tactile simulation~\\cite{narang2021sim, agarwal2021simulation} in the past years, which can unlock tremendous potential for multi-modal embodied AI research. \n\n\n\n\\subsection{Sim2Real Approaches}\n\nAs the embodied AI community grows, and benchmarks in simulation continue to improve, a fundamental question that remains is: how well does this progress translate to the real world? Towards answering this question, the embodied AI community has made significant efforts in 1) building infrastructure to facilitate sim2real transfer on hardware, 2) providing support for researchers across the world to evaluate policies in the real-world, and 3) developing sim2real adaptation techniques. \n\nSignificant advances have been made in recent years on real-world hardware targets, with the emergence of low-cost robots for evaluation \\cite{pyrobot2019,kemp2022design} and open-source infrastructure for sim2robot deployment \\cite{habitat2020sim2real, talbot2020benchbot, deitke2020robothor}. These advances have lowered the barrier to entry for robotics, and enable the embodied AI community to evaluate the performance of various research algorithms both in simulation and on real-world robots. Currently, each approach is limited to a specific simulator or a limited set of robot platforms. A key future direction is for these translation technologies to become ubiquitous interfaces, with support for any simulator or physical robot platform required by the researcher.\n\nBy comparing the performance of policies in simulation and the real-world, researchers are able to identify flaws in the simulator design that lead to poor sim2real transfer \\cite{habitat2020sim2real}, and develop novel methods to overcome the sim2real gap. Common approaches for bridging the sim2real gap include domain randomization \\cite{tobin2017domain, anderson2020sim}, or domain adaptation, a technique in which data from a source domain is adapted to more closely resemble data from a target domain. Prior works leveraged GAN techniques to adapt the visual appearance of objects from sim-to-real \\cite{rao2020rl}, and other works built models \\cite{truong2021bi, truong2022kin2dyn, deitke2020robothor}, or learned latent embeddings of the robot's dynamics \\cite{truong2020learning, kumar2021rma, yu2017preparing} to adapt to the actuation noise found in the real world. Models of real-world camera and actuation noises have since been integrated into simulators, and included as part of the Habitat, RoboTHOR, RVSU and iGibson Challenges, thereby improving the realism of the challenge and decreasing the sim2real gap. Continuing this close integration between real-world evaluation and improving simulators and benchmarks will help accelerate the speed of progress in robotics research.\n\nA final future direction, is in addressing the differences between simulated and real-world sensorimotor interfaces. It is common currently for actuation to be broken into discretised chunks, and simulated sensor inputs treated the same as real-world inputs. While simulators and datasets will continue to advance, there will likely always be a difference between emulated and real-world sensorimotor experiences. Research approaches that leverage simulated data to learn policies, then embrace the limitations of these policies when transferring to real-world scenarios, have begun to emerge in recent years \\cite{rana2021zero}. This is a start, but approaches like these will need to be expanded upon in the future. \n\n\n\\subsection{Procedural Generation}\n\\input{directions\/procedural-generation}\n\n\\subsection{Generalist Agents}\n\\input{directions\/generalist-agents}\n\n\n\\subsection{Multi-Agent \\& Human Interaction}\n\n\nAnalogous to the social learning in humans, it is desirable that embodied agents can observe, learn from, and collaborate with other agents (including humans) in their environment. The advanced and realistic simulated environments being developed for Embodied AI research will serve as virtual worlds for agent-agent and human-agent interaction. The two pillars for social, multi-agent, and human-in-the-loop embodied agents are (1) accurately simulating a subset of agent and human behavior relevant to a given embodied task and (2) creating realistic benchmarks for multi-agent and human-AI collaboration.\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{fig\/multi-agent\/furnmove.pdf}\n    \\caption{Furniture Moving~\\cite{jain2020cordial} is a collaborative multi-agent task for agents to move a heavy furniture item.}\n\\end{figure}\n\nImmersing humans in simulation creates an opportunity for a new class of experiences and user studies that involve human-virtual agent interaction, data collection of human demonstrations at scale in controlled environments, and creation of events and visualizations that are impossible or irreproducible in real scenarios. Some examples towards this goal include VirtualHome~\\cite{puig_cvpr18} where programs are collected and created to model human behaviors along with animated atomic actions such as walk\/run, grab, switch-on\/off, open\/close, place, look-at, sit\/standup, touch. \nTEACh~\\cite{teach} collects both human instructions, demonstrations, and question answers from human who interact with the simulator through a web interface~\\cite{teach}, while BEHAVIOR uses virtual reality to collect high-fidelity human demonstrations directly in the action space of a simulated robot agent~\\cite{srivastava2022behavior}.\nTo train policies, modeling the task-relevant aspects of human behavior is of prime focus. \nIn challenges such as SocialNav, human agents are simulated following a simple interaction model that considers interactions between agents. \nLooking forward, with robust motion solutions models~\\cite{rong2021frankmocap,lugaresi2019mediapipe} and human behavior animation~\\cite{won2020scalable,2021-TOG-AMP}, emulating from large-scale human-activity datasets~\\cite{gu2018ava,smaira2020short,damen2022rescaling,grauman2022ego4d} is an exciting prospect for modeling human behaviors in simulation. To  train and transfer these policies to the real world, we must develop low-shot approaches and realistic benchmarks to learn socially intelligent agents.\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{fig\/multi-agent\/wah.pdf}\n    \\caption{Watch and Help encourages social intelligence where an agent learns in the presence of human-like teachers (image credits: Puig~\\etal~\\cite{puig2021watchandhelp}).}\n\\end{figure}\n\nSeveral benchmarks have helped make progress within the space of multi-agent and social learning in embodied AI. Within AI2-THOR, collaborative task completion~\\cite{jain2019two} and furniture moving~\\cite{jain2020cordial} were one of the first benchmarks for multi-agent learning in embodied AI, focussed on task that cannot be done by a single agent. While abstracted gridworlds~\\cite{jain2021gridtopix} provide a faster training ground for such tasks, efficiently going beyond 2-3 agents models with high visual fidelity is challenging. Emergent communication~\\cite{Patel_2021_ICCV} and emergent visual representations~\\cite{weihs2020learning} show  examples of learning heterogeneous agents possessing specialized skills. SocialNav in iGibson presents early steps towards robot learning for mobility around humans and other moving objects within the environment. Within VirtualHome, the watch-and-help~\\cite{puig2021watchandhelp} benchmark will enabled few-shot learning of policies that can interact with a human-like agent to replicate demonstrations in an unseen environment.\n\nOverall, simulated environments offer a scalable platform for procedural training and testing of interactive policies, potentially addressing some of the limiting challenges inherent to research on human interaction: scaling up with safety and speed, standardize environments to support reproducible research, and procedural testing and benchmarking of a minimum set of tests before deploying on real robots. Progress on all these fronts requires the integration and convergence of contributions from diverse fields such as graphics, animation, and simulation, towards fully functional, realistic and interactive virtual environments. \n\n\\subsection{Impact of Embodied AI}\n\nWhether in simulation or reality, embodied AI research focuses on embodied tasks in the hope of delivering on the fundamental promise of AI: the creation of embodied agents, such as robots, which learn, through interaction and exploration, to creatively solve challenging tasks within their environments.\nMany embodied AI researchers believe that creating intelligent agents that can solve embodied tasks will produce outsized real-world impacts.\nIncreasingly capable robotic platforms and effective sim-to-real techniques make it easier to transfer learned policies to the real world.\nEven small advances at interesting embodied tasks could serve as the foundation for technologies that could improve the lives of people with disabilities or free able-bodied humans from mundane tasks.\nHowever, these advances, as with all automation, could result in disruptions such as the elimination of jobs or disempowerment of individuals.\nWe must be careful to ensure that the benefits of embodied AI become available to all and do not reinforce inequality.\nTherefore, the embodied AI community has promoted discussion of these issues in the hope that it will guide us towards more equitable solutions.\n\n\n\n\\section{Conclusion}\n\nIn this paper, we presented a retrospective on the state of Embodied AI research. We discussed 13 different challenges that make up a testbed for a suite of embodied navigation, interaction, and vision-and-language tasks. Over the past 3 years, we observed large-scale training, visual pre-training, modular and end-to-end training, and visual \\& dynamic augmentation as common approaches to many of the top challenge entries. We discuss improvements to pre-training, world models and inverse graphics, simulation and dataset advances, sim2real, procedural generation, generalist agents, and multi-agent \\& human interaction as promising future directions in the field.\n\n\n\n\n\\section*{Contributions}\n\n\\paragraph{Matt Deitke} led the planning, outline, and coordination of the paper; worked on the abstract, introduction, \\& conclusion and worked on the ObjectNav section, the large-scale training section, the procedural generation, and the generalist agents section.\n\n\\paragraph{Yonatan Bisk} said we should do this, attended a few planning meetings, but then delegated and Matt really ran with it.\n\n\\paragraph{Tommaso Campari} co-wrote the section on Multi-ObjectNav challenge.\n\n\\paragraph{Devendra Singh Chaplot} worked on Habitat Challenge sections and the end-to-end vs modular subsection.\n\n\n\\paragraph{Changan Chen} worked on the audio-visual navigation section and the simulation and dataset advances section.\n\n\\paragraph{Claudia P\\'{e}rez-D'Arpino} worked on the Introduction, Interactive and Social PointNav, and the Multi-Agent \\& Human Interaction sections.\n\n\\paragraph{Anthony Francis} worked on the Introduction, What Is Embodied AI, and Sim to Real sections, and edited other sections.\n\n\\paragraph{Chuang Gan} worked on the rearrangement challenges section.\n\n\\paragraph{David Hall} worked on the RVSU challenge sections and provided some editing on the simulation and dataset advances section.\n\n\\paragraph{Winson Han} created the Figure 1 cover graphic.\n\n\\paragraph{Unnat Jain} worked on audio-visual navigation, multi-object navigation, and multi-agent sections.\n\n\\paragraph{Jacob Krantz} worked on the challenge section on Navigation Instruction Following.\n\n\\paragraph{Chengshu Li} worked on the Interactive and Social PointNav section and the Simulation and Dataset Advances section in Future Directions.\n\n\\paragraph{Sagnik Majumder} worked on the audio-visual navigation section.\n\n\\paragraph{Roberto Mart\\'{i}n-Mart\\'{i}n} worked on the What Is Embodied AI section, and the Interactive and Social PointNav section.\n\n\\paragraph{Sonia Raychaudhuri} co-wrote the section on Multi-ObjectNav challenge.\n\n\\paragraph{Mohit Shridhar} worked on the interactive instruction following section for challenge details and the generalist agents section for future directions. \n\n\\paragraph{Niko S\\\"{u}nderhauf} worked on the RVSU challenge sections.\n\n\\paragraph{Andrew Szot} worked on the pre-training section for future directions.\n\n\\paragraph{Ben Talbot} worked on the RVSU challenge sections and Sim2Real Approaches advances section.\n\n\\paragraph{Jesse Thomason} worked on the interactive instruction following and interactive instruction following with dialog sections of the challenge details, and the multi-agent \\& human interaction section of future directions.\n\n\\paragraph{Alexander Toshev} worked on Social and Interactive Navigation section.\n\n\\paragraph{Joanne Truong} worked on the PointNav section for challenge details, and the Sim2Real approaches section for future directions.\n\n\\paragraph{Luca Weihs} worked on the rearrangement section for challenge details, the visual pre-training section for common approaches, and the world models and inverse graphics section for future directions.\n\n\\paragraph{Dhruv Batra, Angel X. Chang, Kiana Ehsani, Ali Farhadi, Li Fei-Fei, Kristen Grauman, Aniruddha Kembhavi, Stefan Lee, Oleksandr Maksymets, Roozbeh Mottaghi, Mike Roberts, Manolis Savva, Silvio Savarese, Joshua B. Tenenbaum, Jiajun Wu} advised and provided feedback on the draft, workshop, and\/or challenges.\n\n\n\n\n\n\\section*{References}\n\n\\singlespace\n\\renewcommand{\\section}[2]{}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. Experimental Methodology} \\label{sec:experimental_methodology}\n\n\\subsection{Preparation of lipid-coated particles}\n\nFluorescently labeled, lipid-coated particles were created by coating silica micro-beads with a supported lipid bilayer (SLB) containing a minority fraction of fluorescently tagged lipid. \n1,2-dioleoyl-sn-glycero-3-phos-phocholine (DOPC) and 1,2-dioleoyl-sn-glycero-3-phospho-L-serine (DOPS) were purchased from Avanti Polar Lipids.\nAtto 647-1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE-Atto 647) was purchased from ATTO-TEC GmbH.\nSilica microspheres (diameter 2.5 $\\mu$m; catalog code: SS05000) were purchased from Bangs Laboratories.\nSmall unilamellar vesicles (SUVs) were formed using an established sonication method \\cite{bakalar2018size}.\nIn brief, a lipid film containing DOPC, 5\\% DOPS, and 0.5\\% DOPE-Atto 647 was dried under nitrogen and then under vacuum for 30 minutes.\nThe film was rehydrated in Milli-Q (MQ) water to 0.2 mg\/mL lipids, sonicated at low power using a tip sonicator (Branson SFX250 Sonifier) at 20\\% of maximum, 1 s\/2 s on\/off, for three minutes. \nMOPS buffer was added at a final concentration of 50 mM MOPS, pH 7.4, 100 mM NaCl to the resulting SUV mixture.\n\nSilica microspheres were cleaned using a 3:2 mixture of sulfuric acid:hydrogen peroxide (Piranha) for 30 minutes in a bath sonicator, spun at 1000 g, and washed 3 times before being resuspended in MQ water. \nTo form SLBs on the beads, 50 $\\mu$L of SUV solution was mixed with 10 $\\mu$L of the cleaned bead suspension. \nThe bead\/SUV mixture was incubated for 15 minutes at room temperature while allowing the beads to sediment to the bottom of the centrifuge tube. \nBeads were washed 5 times with MQ water by gently adding\/removing the liquid without resuspending the beads into solution. \nThe fluidity of the SLB was verified by imaging beads on a glass coverslip at high laser intensity, where the diffusion of labeled lipids was visible after photo-bleaching a small region. \nLipid-coated beads were deposited into a chamber containing MQ water and sealed off to eliminate drift.\nThe beads settled down to the bottom of the chamber and all experiments were conducted in 2D.\n\n\n\n\n\n\\subsection{Optical tweezer setup and calibration}\n\nAn array of moving harmonic traps was generated using optical tweezers (Tweez 305, Aresis Ltd; Ljubljana, Slovenia), using an IR laser (1064 nm) with a maximum power of 5 W continuous wave (CW).\nWe selected a trap-to-trap switching rate of 100 kHz to ensure that the particles will effectively feel a continuous harmonic potential.\nWe used a 16 $\\times$ 16 array of traps, which results in $\\approx 2.5$ ms time delay to illuminate all trap positions.\nThis time delay is significantly smaller than the Brownian and oscillatory convection timescales in our system, ensuring that the particles experience a continuous harmonic potential.\nA custom MATLAB script was written to construct a time trajectory of oscillatory trap positions for each cell lattice position and incorporated into the tweezer software.\nThe trap focus was adjusted to the mid-plane of the colloids sitting above the substrate.\nLaser powers were adjusted from 0.05-0.5 W to vary the trap stiffness from $\\kappa = 0.5$-6 $kT\/\\mu\\mathrm{m}^2$.\n\n\nThe trap stiffness $\\kappa$ was calibrated by measuring the equilibrium probability distribution of the particles in a stationary array of traps.\nFor each laser power, $\\kappa$ was obtained by binning particles by their radial position $r$ from the center of the trap and fitting the binned data to a Boltzmann distribution, $P(r) = (\\kappa\/2\\pi)\\mathrm{e}^{-\\kappa r^{2}\/(2kT)}$.\nAn example of a distribution and fit is shown in Fig.~\\ref{Fig:SI1}.\nWe verified that there are no variations in trip stiffness between different lattice positions in the array.\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\linewidth]{Figs\/SI\/SIFig_Trap_Stiffness.jpg}\n\t\t\\caption{%\n\t\t\tMeasurement of trap stiffness $\\kappa$ from the equilibrium probability distribution of particles diffusing in a harmonic well generated by optical tweezers.  \n\t\t\tData are fit to a Boltzmann distribution to obtain $\\kappa$ ($\\kappa = 4~kT \/\\mu\\mathrm{m}^2$ in the case shown).\n\t\t\tThis measurement was averaged over all 16 $\\times$ 16 trap positions in the lattice array and repeated for every laser power used in this study.\n\t\t}\n\t\t\\label{Fig:SI1}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\nThe trap width $W_{\\text{trap}}$ was determined from a separate set of experiments.\nTwo traps were placed side-by-side with center-to-center separation distance $W$.\nThe first trap, containing a trapped particle, was held fixed while the position of the second trap was varied; the average position $\\langle x_{i}(t) \\rangle$ of the particle was measured as a function of the separation distance $W$ (Fig.~\\ref{Fig:SI2}).\nWhen the second trap is placed far away, no interference is observed on the average position of the particle.\nHowever, as the second trap is moved closer, $W < 3$ $\\mu$m for a particle of radius $a = 1.25$ $\\mu$m, the average position drifts towards the second trap.\nWe found that the average particle position remains approximately constant within the range of separation distances of $W =$ 3-3.5 $\\mu$m, giving an approximate trap width $W_{\\text{trap}} \\approx 3.2$ $\\mu$m.\n\n\n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\linewidth]{Figs\/SI\/SIFig_Trap_Width.jpg}\n\t\t\\caption{%\n\t\t\tMeasurement of trap width $W_{\\text{trap}}$.\n\t\t\tA second trap was placed at varying separation distances from the first trap containing a trapped bead. \n\t\t\tWe measured the time-averaged position of the trapped bead, $\\langle x_i(t) \\rangle$, for varying separation distances at fixed trap stiffness.\n\t\t\tWe found that the average position is pulled towards the second trap at distances $W < 3$ $\\mu$m and is approximately constant in the range $W = 3$-3.5 $\\mu$m. This gives an average trap width $W_{\\text{trap}} \\approx 3.2$ $\\mu$m.\n\t\t}\n\t\t\\label{Fig:SI2}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\n\\subsection{Measurement of diffusivity}\n\nThe long-time self diffusivity was determined by particle tracking. All imaging was carried out on an inverted Nikon Ti2-Eclipse microscope (Nikon Instruments) using a water-immersion objective (Plan Apochromat VC 60x, numerical aperture 1.2, water). \nLumencor SpectraX Multi-Line LED Light Source was used for excitation (Lumencor, Inc).\nFluorescent light was spectrally filtered with an emission filter (680\/42; Semrock, IDEX Health and Science) and imaged on a Photometrics Prime 95 CMOS Camera (Teledyne Photometrics).\nIn order to achieve satisfactory long-time statistics, particle trajectories were measured for times much larger than all other timescales in the system (including the diffusive timescale $\\gamma L^{2}\/kT$, oscillation period $2\\uppi\/\\omega$, and trapping timescale $\\gamma\/\\kappa$). A modified MATLAB script, based on the IDL code by Crocker and Grier \\cite{crocker1996methods,crockerweeksIDL,blairdufresneMATLAB}, was used to track the individual particles by identifying each particle center and tracking its trajectory over time using an image stack with one frame taken every 1-2 s. \nParticles that were immobile (due to defects) were filtered out so as not to be considered during image post-processing.\n\n\n\n\nThe average diffusivity tensor is classically defined in terms of the long-time derivative of the mean squared displacements (MSD) of the particles: \n\\begin{equation}\n    \\overline{\\tens{D}} = \\lim_{t \\rightarrow \\infty} \\frac{1}{2}\n    \\frac{\\mathrm{d}}{\\mathrm{d}t}\n    \\langle \\Delta \\bm{R}(t) \\Delta \\bm{R}(t) \\rangle\n    ,\n    \\label{eq:SI_diffusivity_tensor}\n\\end{equation}\nwhere $\\bm{R}$ denotes the {\\it global} position vector [related to the {\\it local} position vector $\\bm{r}$ by Eq. \\eqref{eq:SI_coordinate_conversion}, below] and the angle brackets $\\langle\\,\\cdot\\,\\rangle$ denote an {\\it ensemble} average (not to be confused with the {\\it cell} average defined in the main text). The MSD tensor over a time interval $t$ is computed from the formula,\n\\begin{equation}\n\t\\braket{\n\t\\Delta \\bm{R} (t)\n\t\\Delta \\bm{R} (t)\n\t}\n\t=\n\t\\frac{1}{N_{\\text{p}}}\n\t\\sum_{i=1}^{N_{\\text{p}}}\n\t\\lim_{\\tau\\rightarrow\\infty}\n\t\\frac{1}{\\tau-t}\n\t\\int_{0}^{\\tau-t} \n\t\\left[ \\bm{R}_{i}(s + t) - \\bm{R}_{i}(s)  \\right]\n\t\\left[ \\bm{R}_{i}(s + t) - \\bm{R}_{i}(s)  \\right]\n\t\\, \\mathrm{d} s\n\t,\n\t\\label{eq:SI_msd_experiments}\n\\end{equation}\nwhere $\\bm{R}_{i}(t)$ denotes the global position of the $i$th particle at time $t$.\nIn Eq.~\\eqref{eq:SI_msd_experiments}, the squared displacement of a particle with index $i$ is first averaged over all time windows of duration $t$ within the interval $\\tau$ of the particle's trajectory. This ``time average'' for each $i$th particle, evaluated in the limit as $\\tau \\rightarrow \\infty$, is subsequently averaged over all particles $i = 1, 2, \\dots, N_{\\text{p}}$ to approximate the ensemble average of all squared displacements with satisfactory statistics.\nAt long times, the MSD tensor $\\langle \\Delta \\bm{R}(t) \\Delta \\bm{R}(t) \\rangle$ oscillates with fixed amplitude about a steady, linear growth. Thus, the long-time derivative of the MSD can be measured by simply dividing by time, leading to the relation,\n\\begin{equation}\n    \\overline{\\tens{D}} = \\lim_{t \\rightarrow \\infty} \\frac{1}{2 t}\n     \\langle \\Delta \\bm{R}(t) \\Delta \\bm{R}(t) \\rangle\n     .\n     \\label{eq:SI_diffusivity_experiments}\n\\end{equation}\nEquation \\eqref{eq:SI_diffusivity_experiments} was used to measure the diffusivity from the measured particle trajectories (see Fig.~\\ref{Fig:SI3}). Trajectories were averaged over a sufficiently long time interval $\\tau$ to ensure linear growth, and the time integral in Eq.~\\eqref{eq:SI_msd_experiments} was discretized using the left Riemann sum.\nStatiscal errors in the MSD were calculated using a bootstrap algorithm \\cite{ross2020introduction}.\n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\linewidth]{Figs\/SI\/SIFig_MSD_Example.jpg}\n\t\t\\caption{%\n\t\t\tRepresentative mean squared displacements $\\langle \\Delta x(t) \\Delta x(t) \\rangle \/ (2t)$ (black symbols) and $\\langle \\Delta y(t) \\Delta y(t) \\rangle \/ (2t)$ (blue symbols) measured using Eq.~\\eqref{eq:SI_msd_experiments} for Brownian particles diffusing through an oscillating array of harmonic traps.\n\t\t\tDiffusivities reported in the main text were computed from the long-time plateaus of these curves, using Eq.~\\eqref{eq:SI_diffusivity_experiments}. Statistical errors were calculated using the bootstrap algorithm \\cite{ross2020introduction} over the entire observation time window.\n\t\t}\n\t\t\\label{Fig:SI3}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\nThe particle resistivity $\\gamma$ used in all theoretical calculations was calibrated by measuring the Stokes-Einstein-Sutherland diffusivity $D_{0} = kT\/\\gamma \\approx 0.105$ $\\mu$m$^2$\/s of particles diffusing in the absence of a harmonic potential. For a spherical particle of radius $a$ in a fluid of viscosity $\\eta$, the particle resistivity is given by $\\gamma = 6 \\uppi \\eta a K_{D}$, where $K_{D}$ is a drag-correction factor to account for the hydrodynamic interaction with a nearby wall (in our case, the substrate floor). For our system with $a = 1.25$ $\\mu$m and $\\eta = 1$ cP, we estimate the drag-correction factor to be $K_{D} = kT\/(6\\uppi \\eta a D_{0}) \\approx 1.63$, corresponding to a particle-to-wall spacing of about 0.5 $\\mu$m according to Fax\\'en's formula \\cite{happel2012low}. This gives a particle resistivity of $\\gamma \\approx 9.49$ $kT\\cdot\\text{s}\/\\mu\\text{m}^2$.\n\n\n\n\n\n\n\n\n\\section{2. Taylor-Dispersion Theory} \\label{sec:dispersion_theory}\n\n\\subsection{Derivation of Eqs. (3)-(5): governing equations for the probability density and displacement}\n\n\nThe starting point for deriving the basic equations in the main text is the single-particle Smoluchowski equation,\n\\begin{equation}\n\t\\frac{\\partial P(\\bm{R}, t)}{\\partial t}\n\t=\n\t-\n\t\\bm{\\nabla}_{\\bm{R}} \\cdot \\bm{J} (\\bm{R}, t)\n\t,\n\t\\label{eq:SI_smoluchowski_eqn}\n\\end{equation}\nwhere $P(\\bm{R}, t)$ is the probability density of finding a Brownian particle at a {\\it global} position $\\bm{R}$ and time $t$ and\n\\begin{equation}\n\t\\bm{J} (\\bm{R}, t)\n\t=\n\t\\bm{u} (t) P\n\t-\n\t\\frac{1}{\\gamma}\n\t[\n\tkT \\bm{\\nabla}_{\\bm{R}} P \n\t+\n\tP \\bm{\\nabla}_{\\bm{R}} V (\\bm{R})\n\t]\n\t\\label{eq:SI_probability_flux}\n\\end{equation}\nis the probability flux. The spatial periodicity of the potential-energy field allows us to convert the ``global'' position $\\bm{R}$ to the ``local'' position $\\bm{r}$ via the transformation,\n\\begin{equation}\n\t\\bm{R} = \\bm{n} L + \\bm{r},\n\t\\label{eq:SI_coordinate_conversion}\n\\end{equation}\nwhere $\\bm{n}$ contains the lattice indices of a given periodic cell. In terms of lattice and local coordinates, $V(\\bm{R}) \\equiv V(\\bm{r})$, $P(\\bm{R},t) \\equiv P_{\\bm{n}} (\\bm{r}, t)$, and $\\bm{J} (\\bm{R}, t) \\equiv \\bm{J}_{\\bm{n}} (\\bm{r},t)$. \n\n\nIn the following, we employ the ``flux-averaging'' approach of Brady and coworkers \\cite{morris1996self,Zia2010,Takatori2014,burkholder2017tracer,burkholder2019fluctuation,peng2020upstream}.\nFirst, we define the continuous wavevector $\\bm{k}$ and apply the discrete Fourier transform $\\hat{(\\,\\cdot\\,)} \\equiv \\sum_{\\bm{n}} (\\, \\cdot\\,) \\mathrm{e}^{\\mathrm{i} \\bm{k} \\cdot \\bm{n} L}$ to Eqs.~\\eqref{eq:SI_smoluchowski_eqn}-\\eqref{eq:SI_probability_flux}, obtaining\n\\begin{equation}\n\t\\frac{\\partial \\hat{P}(\\bm{k}, \\bm{r}, t)}{\\partial t}\n\t=\n\t-\n\t(\\mathrm{i} \\bm{k} + \\bm{\\nabla}_{\\bm{r}})  \\cdot \\hat{\\bm{J}} (\\bm{k}, \\bm{r}, t)\n\t,\n\t\\label{eq:SI_smoluchowski_eqn_ft}\n\\end{equation}\n\\begin{equation}\n\t\\hat{\\bm{J}} (\\bm{k}, \\bm{r}, t)\n\t=\n\t\\bm{u} (t) \\hat{P}\n\t-\n\t\\frac{1}{\\gamma}\n\t[\n\tkT (\\mathrm{i} \\bm{k} + \\bm{\\nabla}_{\\bm{r}}) \\hat{P} \n\t+\n\t\\hat{P} \\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]\n\t.\n\t\\label{eq:SI_probability_flux_ft}\n\\end{equation}\nNext, we spatially average Eqs.~\\eqref{eq:SI_smoluchowski_eqn_ft}-\\eqref{eq:SI_probability_flux_ft} over one periodic cell according to $\\braket{\\,\\cdot\\,} \\equiv L^{-2}\\int_{L^{2}} (\\,\\cdot\\,) \\, \\mathrm{d} \\bm{r}$, apply the divergence theorem, and invoke periodic boundary conditions to obtain the continuity equation,\n\\begin{equation}\n\t\\frac{\\partial \\hat{\\rho}(\\bm{k}, t)}{\\partial t}\n\t=\n\t-\n\t\\mathrm{i} \\bm{k} \\cdot \\braket{\\hat{\\bm{J}}} (\\bm{k}, t)\n\t,\n\t\\label{eq:SI_smoluchowski_eqn_ft_avg}\n\\end{equation}\n\\begin{equation}\n\t\\braket{\\hat{\\bm{J}}} (\\bm{k}, t)\n\t=\n\t\\bm{u} (t) \\hat{\\rho}\n\t-\n\t\\frac{1}{\\gamma}\n\t[\n\tkT \\mathrm{i} \\bm{k} \\hat{\\rho} \n\t+\n\t\\braket{\\hat{P} \\bm{\\nabla}_{\\bm{r}}V}\n\t]\n\t,\n\t\\label{eq:SI_probability_flux_ft_avg}\n\\end{equation}\nwhere $\\hat{\\rho} (\\bm{k}, t) \\equiv \\braket{\\hat{P}} (\\bm{k}, t)$ is the Fourier-transformed number density. Eqs.~\\eqref{eq:SI_smoluchowski_eqn_ft_avg}-\\eqref{eq:SI_probability_flux_ft_avg} represent the macroscopic transport equations for the periodic lattice.\n\n\nNext, we define the structure function $\\hat{G}(\\bm{k}, \\bm{r}, t)$ as\n\\begin{equation}\n\t\\hat{P}(\\bm{k}, \\bm{r}, t) \n\t=\n\t\\hat{\\rho} (\\bm{k}, t)\n\t\\hat{G}(\\bm{k}, \\bm{r}, t)\n\t.\n\t\\label{eq:SI_conditional_probability}\n\\end{equation}\nMultiplying Eq.~\\eqref{eq:SI_smoluchowski_eqn_ft_avg} by $\\hat{G}$, subtracting from Eq.~\\eqref{eq:SI_smoluchowski_eqn_ft}, and dividing through by $\\hat{\\rho}$ then gives\n\\begin{flalign}\n\t\\frac{\\partial \\hat{G}(\\bm{k}, \\bm{r}, t)}{\\partial t}\n\t&=\n\t-\n\t\\hat{\\rho}^{-1} [ \n\t\\mathrm{i} \\bm{k} \\cdot ( \\hat{\\bm{J}} - \\braket{\\hat{\\bm{J}}} \\hat{G} )\n\t+\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \\hat{\\bm{J}}\n\t]\n\t\\nonumber\\\\\n\t&=\n\t- \\bm{u} (t) \\cdot \\bm{\\nabla}_{\\bm{r}} \\hat{G}\n\t+\n\t\\frac{kT}{\\gamma}\n\t\\nabla_{\\bm{r}}^{2} \\hat{G}\n\t+\n\t\\frac{1}{\\gamma}\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t[\n\t\\hat{G} \\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]\n\t+\n\t\\mathrm{i} \\bm{k} \\cdot \n\t\\left(\n\t\\frac{2kT}{\\gamma} \\bm{\\nabla}_{\\bm{r}} \\hat{G} \n\t+\n\t\\frac{1}{\\gamma}\n\t[\n\t\\hat{G} \\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t-\n\t\\braket{\\hat{G} \\bm{\\nabla}_{\\bm{r}}V} \\hat{G} \n\t]\n\t\\right)\n\t,\n\t\\label{eq:SI_structure_field_eqn}\n\\end{flalign}\nwhere in the last line we have substituted Eqs.~\\eqref{eq:SI_probability_flux_ft}, \\eqref{eq:SI_probability_flux_ft_avg}, and \\eqref{eq:SI_conditional_probability}. Taylor-expanding $\\hat{G}$ about $\\bm{k} = \\bm{0}$,\n\\begin{equation}\n\t\\hat{G}(\\bm{k}, \\bm{r}, t)\n\t=\n\tg(\\bm{r}, t)\n\t+\n\t\\mathrm{i} \\bm{k} \\cdot \\bm{d} (\\bm{r}, t)\n\t+\n\t\\cdots\n\t,\n\t\\label{eq:SI_taylor_series}\n\\end{equation}\nsubstituting the expansion into Eq.~\\eqref{eq:SI_structure_field_eqn}, and collecting terms of like order in $\\mathrm{i} \\bm{k}$ yields the ordered set of equations,\n\\begin{flalign}\n\t\\frac{\\partial g(\\bm{r}, t)}{\\partial t}\n\t+\n\t\\bm{u} (t) \\cdot \\bm{\\nabla}_{\\bm{r}} g\n\t-\n\t\\frac{kT}{\\gamma}\n\t\\nabla_{\\bm{r}}^{2} g\n\t-\n\t\\frac{1}{\\gamma}\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t[\n\tg \\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]\n\t=\n\t0\n\t,\n\\end{flalign}\n\\begin{flalign}\n\t\\frac{\\partial \\bm{d}(\\bm{r}, t)}{\\partial t}\n\t+ \n\t\\bm{u} (t) \\cdot \\bm{\\nabla}_{\\bm{r}} \\bm{d}\n\t-\n\t\\frac{kT}{\\gamma}\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}\n\t-\n\t\\frac{1}{\\gamma}\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t[\n\t\\bm{d}\\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]^{\\dag}\n\t=\n\t\\frac{2kT}{\\gamma} \\bm{\\nabla}_{\\bm{r}} g\n\t+\n\t\\frac{1}{\\gamma}\n\t[\n\tg\\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t- \n\t\\braket{g \\bm{\\nabla}_{\\bm{r}}V} g\n\t]\n\t.\n\\end{flalign}\nThe last two equations are exactly Eqs.~\\eqref{eq:g_eqn} and \\eqref{eq:d_eqn} from the main text. Conservation of probability requires the $g$- and $\\bm{d}$-fields to satisfy periodic boundary conditions as well as the normalization conditions $\\braket{g} = 1$ and $\\braket{\\bm{d}} = \\bm{0}$.\n\n\n\n\n\n\\subsection{Derivation of Eqs. (6)-(7): effective drift velocity and diffusivity}\n\n\nThe effective drift velocity $\\bm{U}(t)$ and diffusivity $\\tens{D} (t)$ of the Brownian particle are related to the Fourier-transformed, average flux $\\braket{\\hat{\\bm{J}}}$ via the large-wavelength expansion,\n\\begin{equation}\n\t\\braket{\\hat{\\bm{J}}} (\\bm{k}, t)\n\t=\n\t\\hat{\\rho}\n\t\\left[  \n\t\t\\bm{U} (t)\n\t\t-\n\t\t\\mathrm{i} \\bm{k} \\cdot \\tens{D} (t)\n\t\t+\n\t\t\\cdots\n\t\\right]\n\t.\n\t\\label{eq:SI_average_flux_ft_eqn1}\n\\end{equation}\nIn order to derive expressions for $\\bm{U}$ and $\\tens{D}$, we insert Eqs.~\\eqref{eq:SI_conditional_probability} and \\eqref{eq:SI_taylor_series} into \\eqref{eq:SI_probability_flux_ft_avg}, obtaining\n\\begin{flalign}\n\t\\braket{\\hat{\\bm{J}}} (\\bm{k}, t)\n\t&=\n\t\\hat{\\rho}\n\t\\left(\n\t\\bm{u} (t)\n\t-\n\t\\frac{1}{\\gamma}\n\t[\n\tkT \\mathrm{i} \\bm{k} \n\t+\n\t\\braket{\\hat{G} \\bm{\\nabla}_{\\bm{r}}V}\n\t]\n\t\\right)\n\t\\nonumber\\\\\n\t&=\n\t\\hat{\\rho}\n\t\\left[\n\t\\bm{u} (t) - \\frac{1}{\\gamma} \\braket{g \\bm{\\nabla}_{\\bm{r}}V}\n\t-\n\t\\mathrm{i} \\bm{k} \n\t\\left(\n\t\\frac{kT}{\\gamma} \\tens{I} \n\t+\n\t\\frac{1}{\\gamma}\\braket{\\bm{d} \\bm{\\nabla}_{\\bm{r}}V}\n\t\\right)\n\t+\n\t\\cdots\n\t\\right]\n\t.\n\t\\label{eq:SI_average_flux_ft_eqn2}\n\\end{flalign}\nEquating terms of like order in $\\mathrm{i} \\bm{k}$ in Eqs.~\\eqref{eq:SI_average_flux_ft_eqn1} and \\eqref{eq:SI_average_flux_ft_eqn2} furnishes the expressions,\n\\begin{equation}\n\t\\bm{U}(t)\n\t=\n\t\\bm{u} (t)\n\t-\n\t\\frac{1}{\\gamma} \\braket{g \\bm{\\nabla}_{\\bm{r}} V} (t)\n\t,\n\\end{equation}\n\\begin{equation}\n\t\\tens{D}(t)\n\t=\n\t\\frac{kT}{\\gamma} \\tens{I} \n\t+\n\t\\frac{1}{\\gamma}\\braket{\\bm{d} \\bm{\\nabla}_{\\bm{r}}V} (t)\n\t,\n\\end{equation}\nwhich are exactly Eqs.~\\eqref{eq:effective_drift}-\\eqref{eq:effective_diffusivity} in the main text.\n\n\n\n\n\n\n\n\n\n\\section{3. Numerical Method} \\label{sec:numerical_method}\n\nEqs.~\\eqref{eq:g_eqn} and \\eqref{eq:d_eqn} were solved using the finite-element method in COMSOL Multiphysics$^\\text{\\textregistered}$ (Version 5.5) with the ``Coefficient Form PDE'' physics interface. An $L \\times L$ square cell was set up and discretized into triangular elements (Fig.~\\ref{Fig:SIMeshes}). Periodic boundary conditions were applied to the $g$- and $\\bm{d}$-fields at the edges of the cell. Studies were run using both time-dependent ($\\bm{u}\\not=\\bm{0}$) and stationary ($\\bm{u}=\\bm{0}$) solvers. For the time-dependent studies, the $g$- and $\\bm{d}$-fields were initialized to uniform values $1$ and $\\bm{0}$, respectively, and time-advanced using the backward differentiation formula with a timestep $\\Delta t = 0.001(2\\uppi\/\\omega)$ until a periodic steady state was achieved. The number of periods needed to reach steady state generally increased with the oscillation frequency. For the stationary studies, the equations were solved iteratively using Newton's method and the normalization conditions $\\braket{g} = 1$ and $\\braket{\\bm{d}} = \\bm{0}$ were implemented as weak-form constraints. Upon solving for the $g$- and $\\bm{d}$-fields, Eqs.~\\eqref{eq:effective_drift} and \\eqref{eq:effective_diffusivity} were evaluated using a fourth-order domain integration method and (in the time-dependent studies) subsequently time-averaged over the final oscillation period.\n\n\n\n\n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\linewidth]{Figs\/SI\/SIFig_Meshes.png}\n\t\t\\caption{%\n\t\t\tTriangular meshes used for the finite-element calculations. Meshes containing (a) 1132 elements (for the time-dependent studies) and (b) 29,018 elements (for the stationary studies) were used. Coarser meshes were used in the time-dependent calculations to save computational time.\n\t\t}\n\t\t\\label{Fig:SIMeshes}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{4. Asymptotic Limits} \\label{sec:asymptotic_limits}\n\n\n\n\\subsection{Derivation of Eq. (8): stationary traps with shallow potential wells}\n\nIf the harmonic traps held in a fixed configuration, $\\bm{u} = \\bm{0}$ and the $g$- and $\\bm{d}$-fields achieve a steady state. Equations \\eqref{eq:g_eqn} and \\eqref{eq:d_eqn} then simplify to\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} g (\\bm{r})\n\t+\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t[\n\tg(\\bm{r}) \\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]\n\t=\n\t0\n\t,\n\t\\label{eq:SI_g_eqn_steady}\n\\end{flalign}\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}(\\bm{r})\n\t+\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t[\n\t\\bm{d}(\\bm{r})\\bm{\\nabla}_{\\bm{r}} V (\\bm{r})\n\t]^{\\dag}\n\t=\n\t-\n\t2kT \\bm{\\nabla}_{\\bm{r}} g\n\t-\n\tg\\bm{\\nabla}_{\\bm{r}} V \n\t+ \n\t\\braket{g \\bm{\\nabla}_{\\bm{r}}V} g\n\t.\n\t\\label{eq:SI_d_eqn_steady}\n\\end{flalign}\nEq.~\\eqref{eq:SI_g_eqn_steady} may be solved subject to the constraint $\\braket{g} = 1$ to get the Boltzmann distribution,\n\\begin{equation}\n\tg (\\bm{r})\n\t=\n\t\\frac{\\mathrm{e}^{-V(\\bm{r})\/kT}}{\\braket{\\mathrm{e}^{-V\/kT}}}\n\t.\n\t\\label{eq:SI_g_field_steady}\n\\end{equation}\nThe governing equation for the $\\bm{d}$-field, Eq.~\\eqref{eq:SI_d_eqn_steady}, then simplifies to\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}\n\t+\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t(\n\t\\bm{d}\\bm{\\nabla}_{\\bm{r}} V \n\t)^{\\dag}\n\t=\n\t-\n\tkT \\bm{\\nabla}_{\\bm{r}} g\n\t.\n\t\\label{eq:SI_d_eqn_steady_2}\n\\end{flalign}\n\n\nEq.~\\eqref{eq:SI_d_eqn_steady_2} cannot be solved analytically in general. However, for ``shallow'' potential wells, $\\Delta V \\ll kT$, we may Taylor-expand Eq.~\\eqref{eq:SI_g_field_steady} as\n\\begin{equation}\n\tg\n\t=\n\t1 \n\t- \n\t\\frac{V - \\braket{V}}{kT} \n\t+\n\t\\frac{V^{2} - \\braket{V^{2}} - 2 \\braket{V} (V - \\braket{V})}{2(kT)^{2}} \n\t+ \n\t\\cdots\n\t,\n\\end{equation}\nso that Eq.~\\eqref{eq:SI_d_eqn_steady_2} becomes\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}\n\t+\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t(\n\t\\bm{d}\\bm{\\nabla}_{\\bm{r}} V \n\t)^{\\dag}\n\t=\n\t\\left( 1 - \\frac{V - \\braket{V}}{kT} + \\cdots \\right)\\bm{\\nabla}_{\\bm{r}} V\n\t.\n\t\\label{eq:SI_d_eqn_steady_3}\n\\end{flalign}\nTo solve Eq.~\\eqref{eq:SI_d_eqn_steady_3}, we expand the $\\bm{d}$-field in a perturbation series,\n\\begin{equation}\n\t\\bm{d} (\\bm{r})\n\t=\n\t\\bm{d}^{(0)} (\\bm{r})\n\t+\n\t\\bm{d}^{(1)} (\\bm{r})\n\t+\n\t\\cdots\n\t,\n\t\\label{eq:SI_d_field_perturbation_series}\n\\end{equation}\nwhere $\\bm{d}^{(0)} = O(\\Delta V\/kT)$, $\\bm{d}^{(1)} = O[\\Delta V^{2}\/(kT)^{2}]$, and so on. Inserting Eq.~\\eqref{eq:SI_d_field_perturbation_series} into \\eqref{eq:SI_d_eqn_steady_3} and collecting terms of like order in $\\Delta V \/ kT$ yields the ordered set of equations,\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}^{(0)}\n\t=\n\t\\bm{\\nabla}_{\\bm{r}} V\n\t,\n\t\\label{eq:SI_d_eqn_steady_4a}\n\\end{flalign}\n\\begin{flalign}\n\tkT\n\t\\nabla_{\\bm{r}}^{2} \\bm{d}^{(1)}\n\t=\n\t-\n\t\\bm{\\nabla}_{\\bm{r}} \\cdot \n\t(\n\t\\bm{d}^{(0)}\\bm{\\nabla}_{\\bm{r}} V \n\t)^{\\dag}\n\t-\n\t\\frac{V - \\braket{V}}{kT} \\bm{\\nabla}_{\\bm{r}} V\n\t,\n\t\\label{eq:SI_d_eqn_steady_4b}\n\\end{flalign}\nsubject to the constraints $\\braket{\\bm{d}^{(0)}} = \\bm{0}$, $\\braket{\\bm{d}^{(1)}} = \\bm{0}$, etc. Since $V$ and $\\bm{d}$ are spatially periodic, Eqs.~\\eqref{eq:SI_d_eqn_steady_4a}-\\eqref{eq:SI_d_eqn_steady_4b} may be sequentially solved by means of Fourier series:\n\\begin{equation}\n\t\\bm{d}^{(0)} (\\bm{r})\n\t=\n\t-\n\t\\frac{1}{kT}\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\frac{\\mathrm{i} \\bm{q}}{q^{2}} \n\tV_{\\bm{q}} \\mathrm{e}^{\\mathrm{i} \\bm{q} \\cdot \\bm{r}}\n\t,\n\t\\label{eq:SI_d0_soln}\n\\end{equation}\n\\begin{equation}\n\t\\bm{d}^{(1)} (\\bm{r})\n\t=\n\t\\frac{1}{2(kT)^{2}}\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\sum_{\\bm{q}' \\not= \\bm{0}}\n\t\\cdot\n\t\\left( \\frac{\\mathrm{i} \\bm{q}}{q^{2}} + \\frac{2 \\mathrm{i} \\bm{q} \\cdot (\\bm{q} - \\bm{q}')\\bm{q}'}{q^{2} q'^{2}} \\right)\n\tV_{\\bm{q} - \\bm{q}'} V_{\\bm{q}'}\n\t\\mathrm{e}^{\\mathrm{i} \\bm{q} \\cdot \\bm{r}}\n\t,\n\t\\label{eq:SI_d1_soln}\n\\end{equation}\nwhere $\\bm{q}$ is the discrete wavevector and $V_{\\bm{q}} \\equiv L^{-2} \\int_{L^{2}} [V(\\bm{r}) - \\braket{V}] \\mathrm{e}^{- \\mathrm{i} \\bm{q} \\cdot \\bm{r}} \\, \\mathrm{d} \\bm{r}$ denotes the Fourier integral of $V$. \n\nBy use of Eqs.~\\eqref{eq:effective_diffusivity} and \\eqref{eq:SI_d_field_perturbation_series}, the effective diffusivity of the Brownian particle is given by\n\\begin{flalign}\n\t\\tens{D}\n\t&=\n\t\\frac{kT}{\\gamma} \\tens{I}\n\t+\n\t\\frac{1}{\\gamma} \\braket{\\bm{d} \\bm{\\nabla}_{\\bm{r}} V}\n\t\\nonumber\\\\\n\t&=\n\t\\frac{kT}{\\gamma} \\tens{I}\n\t+\n\t\\frac{1}{\\gamma} \\braket{\\bm{d}^{(0)} \\bm{\\nabla}_{\\bm{r}} V}\n\t+\n\t\\frac{1}{\\gamma} \\braket{\\bm{d}^{(1)} \\bm{\\nabla}_{\\bm{r}} V}\n\t+\n\t\\cdots\n\t.\n\t\\label{eq:SI_diffusivity_tensor_smallV_expansion}\n\\end{flalign}\nMultiplying Eqs.~\\eqref{eq:SI_d0_soln} by $\\bm{\\nabla}_{\\bm{r}} V = \\sum_{\\bm{q}\\not=\\bm{0}} \\mathrm{i} \\bm{q} V_{\\bm{q}} \\mathrm{e}^{\\mathrm{i} \\bm{q} \\cdot \\bm{r}}$ and averaging over an $L \\times L$ cell yields the force-displacement dyads,\n\\begin{equation}\n\t\\braket{\\bm{d}^{(0)} \\bm{\\nabla}_{\\bm{r}} V}\n\t=\n\t-\n\t\\frac{1}{kT}\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\frac{\\bm{q}\\bm{q}}{q^{2}} \n\t|V_{\\bm{q}}|^{2}\n\t,\n\t\\label{eq:SI_d0_gradV_avg}\n\\end{equation}\n\\begin{equation}\n\t\\braket{\\bm{d}^{(1)} \\bm{\\nabla}_{\\bm{r}} V}\n\t=\n\t\\frac{1}{2(kT)^{2}}\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\sum_{\\bm{q}' \\not= \\bm{0}}\n\t\\left( \\frac{\\bm{q}\\bm{q}}{q^{2}} + \\frac{2 \\bm{q} \\cdot (\\bm{q} - \\bm{q}')\\bm{q}'\\bm{q}}{q^{2} q'^{2}} \\right)\n\tV_{\\bm{q} - \\bm{q}'} V_{\\bm{q}'} V_{-\\bm{q}}\n\t,\n\t\\label{eq:SI_d1_gradV_avg}\n\\end{equation}\nwhere $|V_{\\bm{q}}|^{2} \\equiv V_{\\bm{q}} V_{-\\bm{q}}$. Thus, the diffusivity tensor $\\tens{D}$ admits the Fourier-series representation,\n\\begin{flalign}\n\t\\tens{D}\n\t&=\n\t\\frac{kT}{\\gamma} \\,\n\t\\left[\n\t\\tens{I}\n\t-\n\t\\frac{1}{(kT)^{2}}\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\frac{\\bm{q}\\bm{q}}{q^{2}} \n\t|V_{\\bm{q}}|^{2}\n\t+\n\t\\frac{1}{2(kT)^{3}}\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\t\\sum_{\\bm{q}' \\not= \\bm{0}}\n\t\\left( \\frac{\\bm{q}\\bm{q}}{q^{2}} + \\frac{2 \\bm{q} \\cdot (\\bm{q} - \\bm{q}')\\bm{q}'\\bm{q}}{q^{2} q'^{2}} \\right)\n\tV_{\\bm{q} - \\bm{q}'} V_{\\bm{q}'} V_{-\\bm{q}}\n\t+\n\t\\cdots\n\t\\right]\n\t.\n\t\\label{eq:SI_diffusivity_smallV_1}\n\\end{flalign}\n\n\n\nAn alternative expression for $\\tens{D}$ can be obtained by writing leading-order displacement field as the negative gradient of a potential, \n\\begin{equation}\n\t\\bm{d}^{(0)} (\\bm{r})\n\t=\n\t- \\frac{1}{kT} \\bm{\\nabla}_{\\bm{r}} \\varPhi(\\bm{r})\n\t,\n\t\\label{eq:SI_diffusivity_as_gradient_of_potential}\n\\end{equation}\nwhere $\\varPhi(\\bm{r})$ satisfies the Poisson equation,\n\\begin{equation}\n\t\\nabla^{2}_{\\bm{r}} \\varPhi(\\bm{r}) = - [V(\\bm{r}) - \\braket{V}],\n\t\\label{eq:SI_poisson_eqn}\n\\end{equation}\nsubject to the closure $\\braket{\\varPhi} = 0$. The Fourier-series solution of Eq.~\\eqref{eq:SI_poisson_eqn} is\n\\begin{equation}\n\t\\varPhi(\\bm{r}) \n\t=\n\t\\sum_{\\bm{q} \\not= \\bm{0}}\n\tq^{-2}\n\tV_{\\bm{q}} \\mathrm{e}^{\\mathrm{i} \\bm{q} \\cdot \\bm{r}}\n\t.\n\t\\label{eq:SI_poisson_field_soln}\n\\end{equation}\nBy use of Eqs.~\\eqref{eq:SI_d0_soln}, \\eqref{eq:SI_d1_gradV_avg}, and the convolution theorem, it can be shown that\n\\begin{equation}\n\t\\braket{\\bm{d}^{(1)} \\bm{\\nabla}_{\\bm{r}} V}\n\t=\n\t\\frac{1}{2kT} \n\t\\braket{(V - \\braket{V})^{2} \\bm{\\nabla}_{\\bm{r}} \\bm{d}^{(0)}}\n\t+\n\t\\braket{(\\bm{\\nabla}_{\\bm{r}} \\bm{d}^{(0)} \\cdot \\bm{\\nabla}_{\\bm{r}} V)\\bm{d}^{(0)}} \n\t.\n\t\\label{eq:SI_d1_gradV_avg_identity}\n\\end{equation}\nThen, by Eqs.~\\eqref{eq:SI_d0_gradV_avg}, \\eqref{eq:SI_diffusivity_as_gradient_of_potential}, \\eqref{eq:SI_poisson_field_soln}, and \\eqref{eq:SI_d1_gradV_avg_identity}, it follows that\n\\begin{equation}\n\t\\braket{\\bm{d}^{(0)} \\bm{\\nabla}_{\\bm{r}} V}\n\t=\n\t- \\frac{1}{kT} \\braket{\\bm{\\nabla}_{\\bm{r}} \\varPhi \\bm{\\nabla}_{\\bm{r}} V}\n\t,\n\t\\label{eq:SI_d0_gradV_avg_2}\n\\end{equation}\n\\begin{equation}\n\t\\braket{\\bm{d}^{(1)} \\bm{\\nabla}_{\\bm{r}} V}\n\t=\n\t\\frac{1}{2(kT)^{2}} \n\t\\left[\n\t-\n\t\\braket{(V - \\braket{V})^{2} \\bm{\\nabla}_{\\bm{r}} \\bm{\\nabla}_{\\bm{r}} \\varPhi}\n\t+\n\t2 \\braket{(\\bm{\\nabla}_{\\bm{r}} \\bm{\\nabla}_{\\bm{r}} \\varPhi \\cdot \\bm{\\nabla}_{\\bm{r}} V)\\bm{\\nabla}_{\\bm{r}} \\varPhi} \n\t\\right]\n\t.\n\t\\label{eq:SI_d1_gradV_avg_2}\n\\end{equation}\nSubstituting Eqs.~\\eqref{eq:SI_d0_gradV_avg_2}-\\eqref{eq:SI_d1_gradV_avg_2} into \\eqref{eq:SI_diffusivity_tensor_smallV_expansion} then gives the alternative representation,\n\\begin{flalign}\n\t\\tens{D}\n\t&=\n\t\\frac{kT}{\\gamma} \n\t\\left(\n\t\\tens{I}\n\t-\n\t\\frac{1}{(kT)^{2}}\n\t\\braket{\\bm{\\nabla}_{\\bm{r}} \\varPhi \\bm{\\nabla}_{\\bm{r}} V}\n\t+\n\t\\frac{1}{2(kT)^{3}}\n\t\\left[\n\t-\n\t\\braket{(V - \\braket{V})^{2} \\bm{\\nabla}_{\\bm{r}} \\bm{\\nabla}_{\\bm{r}} \\varPhi}\n\t+\n\t2 \\braket{(\\bm{\\nabla}_{\\bm{r}} \\bm{\\nabla}_{\\bm{r}} \\varPhi \\cdot \\bm{\\nabla}_{\\bm{r}} V)\\bm{\\nabla}_{\\bm{r}} \\varPhi} \n\t\\right]\n\t+\n\t\\cdots\n\t\\right)\n\t.\n\t\\label{eq:SI_diffusivity_smallV_2}\n\\end{flalign}\n\n\n\nSince $V(\\bm{r})$ is isotropic, only the trace of the steady diffusivity tensor need be computed: $D \\equiv \\tfrac{1}{2} \\tens{D} : \\tens{I}$. Using Eq.~\\eqref{eq:SI_diffusivity_tensor_smallV_expansion}, the scalar diffusivity $D$ is given by\n\\begin{equation}\n\tD\n\t=\n\t\\frac{kT}{\\gamma}\n\t+\n\t\\frac{1}{2\\gamma} \\braket{\\bm{d}^{(0)} \\cdot \\bm{\\nabla}_{\\bm{r}} V}\n\t+\n\t\\frac{1}{2\\gamma} \\braket{\\bm{d}^{(1)} \\cdot \\bm{\\nabla}_{\\bm{r}} V}\n\t+\n\t\\cdots\n\t.\n\t\\label{eq:SI_diffusivity_scalar_smallV_expansion}\n\\end{equation}\nTaking the trace of Eqs.~\\eqref{eq:SI_d0_gradV_avg_2}-\\eqref{eq:SI_d1_gradV_avg_2}, integrating by parts, and applying Eq.~\\eqref{eq:SI_poisson_eqn} then gives\n\\begin{flalign}\n\t\\braket{\\bm{d}^{(0)} \\cdot \\bm{\\nabla}_{\\bm{r}} V}\n\t&=\n\t- \\frac{1}{kT} \\braket{\\bm{\\nabla}_{\\bm{r}} \\varPhi \\cdot \\bm{\\nabla}_{\\bm{r}} V}\n\t\\nonumber\\\\\n\t&=\n\t-\\frac{1}{kT} \\braket{(V - \\braket{V})^{2}}\n\t,\n\t\\label{eq:SI_d0_gradV_avg_3}\n\\end{flalign}\n\\begin{flalign}\n\t\\braket{\\bm{d}^{(1)} \\cdot \\bm{\\nabla}_{\\bm{r}} V}\n\t&=\n\t\\frac{1}{2(kT)^{2}} \n\t\\left[\n\t-\n\t\\braket{(V - \\braket{V})^{2} \\nabla^{2}_{\\bm{r}} \\varPhi}\n\t+\n\t2 \\braket{(\\bm{\\nabla}_{\\bm{r}} \\bm{\\nabla}_{\\bm{r}} \\varPhi \\cdot \\bm{\\nabla}_{\\bm{r}} V) \\cdot\\bm{\\nabla}_{\\bm{r}} \\varPhi} \n\t\\right]\n\t\\nonumber\\\\\n\t&=\n\t\\frac{1}{2(kT)^{2}} \n\t\\left[\n\t\\braket{(V - \\braket{V})^{3}}\n\t+\n\t\\braket{\\bm{\\nabla}_{\\bm{r}} (|\\bm{\\nabla}_{\\bm{r}} \\varPhi|^{2}) \\cdot \\bm{\\nabla}_{\\bm{r}} V} \n\t\\right]\n\t.\n\t\\label{eq:SI_d1_gradV_avg_3}\n\\end{flalign}\nInserting Eqs.~\\eqref{eq:SI_d0_gradV_avg_3}-\\eqref{eq:SI_d1_gradV_avg_3} into \\eqref{eq:SI_diffusivity_scalar_smallV_expansion} then gives\n\\begin{equation}\n\tD\n\t=\n\t\\frac{kT}{\\gamma}\n\t\\left(\n\t1\n\t-\n\t\\frac{1}{2(kT)^{2}}\n\t\\braket{(V - \\braket{V})^{2}}\n\t+\n\t\\frac{1}{4(kT)^{3}}\n\t\\left[\n\t\\braket{(V - \\braket{V})^{3}}\n\t+\n\t\\braket{\\bm{\\nabla}_{\\bm{r}} (|\\bm{\\nabla}_{\\bm{r}} \\varPhi|^{2}) \\cdot \\bm{\\nabla}_{\\bm{r}} V} \n\t\\right]\n\t+\n\t\\cdots\n\t\\right)\n\t.\n\t\\label{eq:SI_diffusivity_scalar_smallV_expansion}\n\\end{equation}\nThe last expression is exactly Eq.~\\eqref{eq:diffusivity_soft_traps} from the main text.\n\n\n\n\\subsection{Derivation of Eq. (9): stationary traps with deep potential wells}\n\n\nFor stationary, ``deep'' potential wells, $\\Delta V \\gg kT$, the small-potential perturbation series \\eqref{eq:SI_d_field_perturbation_series} fails to converge. Unfortunately, no exact analytical solution of Eq.~\\eqref{eq:SI_d_eqn_steady_2} is readily available. However, one can take advantage of the fact that, for deep potential wells, the probability density is strongly localized near the origin $\\bm{r} = \\bm{0}$ of the lattice cell where the potential-energy field $V(\\bm{r})$ is minimized. Then, a useful {\\it approximation} of the $\\bm{d}$-field is\n\\begin{flalign}\n\t\\bm{d} (\\bm{r})\n\t&\\approx\n\t- \\bm{r} g (\\bm{r})\n\t\\nonumber\\\\\n\t&=\n\t- \\frac{\\bm{r} \\mathrm{e}^{-V(\\bm{r})\/kT}}{\\braket{\\mathrm{e}^{-V\/kT}}}\n\t.\n\t\\label{eq:SI_d_field_largeV_approx}\n\\end{flalign}\nEq.~\\eqref{eq:SI_d_field_largeV_approx} is the particular solution of Eq.~\\eqref{eq:SI_d_eqn_steady_2} and conserves probability, $\\braket{\\bm{d}} = \\bm{0}$. However, this particular solution clearly violates the periodic boundary conditions at the edges of the lattice cell $x = \\pm L\/2$, $y=\\pm L\/2$, incurring an error of $O(L\\mathrm{e}^{-\\Delta V\/kT}\/\\braket{\\mathrm{e}^{-V\/kT}})$ that decreases in magnitude with increasing trap stiffness. Fig.~\\ref{Fig:SI_StrongTrap_dx_vs_x} compares the approximation, Eq.~\\eqref{eq:SI_d_field_largeV_approx}, against the ``exact'' numerical solution for the displacement field, showing very good agreement. The slight error in the approximation is due to the neglect of the homogeneous solution of Eq.~\\eqref{eq:SI_d_eqn_steady_2}, which is complicated by the 2D potential-energy field given by Eq.~\\eqref{eq:harmonic_potential}. It will be shown that the error in this approximation for the $\\bm{d}$-field quantitatively (though not qualitatively) impacts the prediction for the effective diffusivity.\n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{Figs\/SI\/SIFig_StrongPotentialLimit_Dx_vs_X.png}\n\t\t\\caption{\n\t\tComparison of numerical solution for the steady displacement field density $d_{x}(x,y)$ against the particular solution [see Eq.~\\eqref{eq:SI_d_field_largeV_approx}] for a stiff trap, $\\kappa=5$ $kT\/\\mu$m$^2$. (a) 2D contour plot of $d_{x}$ with line traces at four distinct values of $y$. (b) Plot of $d_{x}$ against $x$ for each line trace shows favorable agreement to Eq.~\\eqref{eq:SI_d_field_largeV_approx}. \n\t\t}\n\t\t\\label{Fig:SI_StrongTrap_dx_vs_x}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\nUsing Eq.~\\eqref{eq:harmonic_potential} for $V(\\bm{r})$ and Eq.~\\eqref{eq:SI_d_field_largeV_approx} for $\\bm{d}(\\bm{r})$, the force-displacement dyad that appears in Eq.~\\eqref{eq:diffusivity_soft_traps} can now be approximated as\n\\begin{flalign}\n\t\\braket{\\bm{d} \\bm{\\nabla}_{\\bm{r}} V}\n\t&\\approx\n\t-\n\t\\frac{1}{\\kappa}\n\t\\frac{\\braket{\\mathrm{e}^{-V\/kT}\\bm{\\nabla}_{\\bm{r}} V \\bm{\\nabla}_{\\bm{r}} V}}{\\braket{\\mathrm{e}^{-V\/kT}}}\n\t,\n\t\\label{eq:SI_d_gradV_avg_largeV}\n\\end{flalign}\nwhere we've used the fact that $\\bm{\\nabla}_{\\bm{r}} V = \\kappa \\bm{r}$ for $r \\le \\frac{1}{2} W_{\\text{trap}}$ and $=\\bm{0}$ otherwise. Defining the well depth as $\\Delta V =\\tfrac{1}{8} \\kappa W_{\\text{trap}}^{2}$, the cell averages in Eq.~\\eqref{eq:SI_d_gradV_avg_largeV} become\n\\begin{flalign}\n\t\\braket{\\mathrm{e}^{-V\/kT}}\n\t&=\n\t\\frac{2\\uppi kT}{\\kappa L^{2}}\\left( 1 - \\mathrm{e}^{-\\Delta V\/kT} \\right)\n\t+\n\t\\bigg( 1- \\frac{2\\uppi \\Delta V}{\\kappa L^{2}} \\bigg) \\mathrm{e}^{-\\Delta V\/kT}\n\t,\n\\end{flalign}\n\\begin{flalign}\n\t\\braket{\\mathrm{e}^{-V\/kT}\\bm{\\nabla}_{\\bm{r}} V \\bm{\\nabla}_{\\bm{r}} V}\n\t&=\n\t\\frac{2\\uppi (kT)^{2}}{L^{2}}\n\t\\left[\n\t\t1\n\t\t-\n\t\t\\left( 1 + \\frac{\\Delta V}{kT} \\right)\\mathrm{e}^{-\\Delta V\/kT}\n\t\\right]\n\t\\tens{I}\n\t.\n\\end{flalign}\nSubstitution into Eq.~\\eqref{eq:SI_d_gradV_avg_largeV} then gives, upon simplification,\n\\begin{flalign}\n\t\\braket{\\bm{d} \\bm{\\nabla}_{\\bm{r}} V}\n\t&\\approx\n\tkT\n\t\\left\\{\n\t\t-1\n\t\t+\n\t\t\\left[  \n\t\t1\n\t\t+\n\t\t\\frac{2\\uppi kT}{\\kappa L^{2}}\n\t\t\\left(\n\t\t\t\\mathrm{e}^{\\Delta V\/kT}\n\t\t\t-\n\t\t\t\\frac{\\Delta V}{kT}\n\t\t\t-\n\t\t\t1\n\t\t\\right)\n\t\t\\right]^{-1}\n\t\\right\\}\n\t\\tens{I}\n\t\\nonumber\\\\\n\t&\\approx\n\t\\left(\n\t- k T\n\t+\n\t\\frac{\\kappa L^{2}}{2\\uppi}\n\t\\mathrm{e}^{-\\Delta V \/ kT} \n\t\\right)\n\t\\tens{I}\n\t\\quad\n\t\\text{for}\n\t\\quad\n\t\\Delta V \\gg kT.\n\\end{flalign}\nSubstitution into Eq.~\\eqref{eq:effective_diffusivity} and replacing $\\kappa \\tens{I}$ by $(\\bm{\\nabla}_{\\bm{r}}\\bm{\\nabla}_{\\bm{r}}V)|_{\\bm{r}=\\bm{0}}$ then gives the following approximation for the diffusivity tensor:\n\\begin{equation}\n\t\\tens{D}\n\t\\approx\n\t\\frac{L^{2}}{2\\uppi \\gamma}\n\t\\mathrm{e}^{-\\Delta V \/ kT}\n\t(\\bm{\\nabla}_{\\bm{r}}\\bm{\\nabla}_{\\bm{r}}V)|_{\\bm{r}=\\bm{0}}\n\t,\n\\end{equation}\nor, upon taking one-half the trace,\n\\begin{equation}\n\tD\n\t\\approx\n\t\\frac{L^{2}}{4\\uppi \\gamma}\n\t\\mathrm{e}^{-\\Delta V \/ kT}\n\t(\\nabla_{\\bm{r}}^{2}V)|_{\\bm{r}=\\bm{0}}\n\t.\n\t\\label{eq:SI_diffusivity_largeV_approx}\n\\end{equation}\nThis is exactly the form that would be predicted by Kramers' theory for the escape of a Brownian particle from a deep potential well \\cite{kramers1940brownian,brinkman1956brownian,brinkman1956brownian2}.\nComparison of Eq.~\\eqref{eq:SI_diffusivity_largeV_approx} to numerical calculations of $D$ indicates the qualitatively correct dependence on the trapping strength, but quantitative discrepancies due to errors in the approximation \\eqref{eq:SI_d_field_largeV_approx} for the $\\bm{d}$-field (see Fig.~\\ref{Fig:SI_StrongTrap_D_vs_K_LogLinearPlot}). Quantitative agreement can be obtained by renormalizing the above result by a factor that depends upon the ratio $W_{\\text{trap}} \/ L$. Therefore, we write\n\\begin{equation}\n\tD\n\t\\propto\n\t\\frac{L^{2}}{4\\uppi \\gamma}\n\t\\mathrm{e}^{-\\Delta V \/ kT}\n\t(\\nabla_{\\bm{r}}^{2}V)|_{\\bm{r}=\\bm{0}}\n\t\\label{eq:SI_diffusivity_largeV_approx_proportionality}\n\\end{equation}\nup to a proportionality constant. Eq.~\\eqref{eq:SI_diffusivity_largeV_approx_proportionality} is identical to Eq.~\\eqref{eq:diffusivity_stiff_traps} from the main text.\nFor traps of diameter $W_{\\text{trap}} = 3.2$ $\\mu$m spaced a distance $L=6$ $\\mu$m apart, a proportionality constant of 1.5 gives quantitative agreement with the exact dispersion theory (see Fig.~\\ref{Fig:SI_StrongTrap_D_vs_K_LogLinearPlot}).\n\n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\linewidth]{Figs\/SI\/SIFig_StrongPotentialLimit_ExponentialTrends.png}\n\t\t\\caption{%\n\t\tLog-linear plot of diffusivity $D$ against trap stiffness $\\kappa$. The full numerical solution (solid curve) is compared against Eq.~\\eqref{eq:diffusivity_stiff_traps} (dashed curves) using two different constants of proportionality. Irrespective of the numerical prefactor, Eq.~\\eqref{eq:diffusivity_stiff_traps} demonstrates the appropriate scaling with the trapping strength and is consistent with Kramers' theory of activated escape. A proportionality constant of 1.5 gives quantitative agreement with the exact solution for the specific geometry considered in this study.\n\t\t}\n\t\t\\label{Fig:SI_StrongTrap_D_vs_K_LogLinearPlot}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\subsection{Derivation of Eq. (11): oscillating traps in the high-frequency limit}\n\nIn the high-frequency limit, the potential-energy field is cycled in the $x$-direction at a rate much faster than the response time of the Brownian particle. A reasonable model for this system is a quasi-steady, uniform convection in the $x$-direction, for which we make the ansatz $g = g(y)$ and $d_{y} = d_{y} (y)$ (for the time being, we will ignore the $d_{x}$-field). Eqs.~\\eqref{eq:g_eqn} and \\eqref{eq:d_eqn} then simplify to\n\\begin{equation}\n\tkT\n\t\\frac{\\mathrm{d}^{2}g}{\\mathrm{d} y^{2}}\n\t+\n\t\\frac{\\partial V}{\\partial y} \\frac{\\mathrm{d} g}{\\mathrm{d} y}\n\t+\n\t\\left( \\frac{\\partial^{2} V}{\\partial x^{2}} + \\frac{\\partial^{2} V}{\\partial y^{2}} \\right) g\n\t=\n\t0\n\t,\n\t\\label{eq:SI_g_eqn_1d}\n\\end{equation}\n\\begin{equation}\n\tkT\n\t\\frac{\\mathrm{d}^{2} d_{y}}{\\mathrm{d} y^{2}}\n\t+\n\t\\frac{\\partial V}{\\partial y} \\frac{\\mathrm{d} d_{y}}{\\mathrm{d} y}\n\t+\n\t\\left( \\frac{\\partial^{2} V}{\\partial x^{2}} + \\frac{\\partial^{2} V}{\\partial y^{2}} \\right) d_{y}\n\t=\n\t-\n\t2 kT \\frac{\\mathrm{d} g}{\\mathrm{d} y}\n\t- \n\tg \\frac{\\partial V}{\\partial y}\n\t+\n\t\\bigg\\langle g \\frac{\\partial V}{\\partial y} \\bigg\\rangle g\n\t.\n\t\\label{eq:SI_dy_eqn_1d}\n\\end{equation}\nAveraging Eqs.~\\eqref{eq:SI_g_eqn_1d}-\\eqref{eq:SI_dy_eqn_1d} over the $x$-direction only and defining the modified potential,\n\\begin{equation}\n\tv(y) \n\t=\n\t\\frac{1}{L} \\int_{-L\/2}^{L\/2} V(x,y) \\, \\mathrm{d} x\n\t,\n\\end{equation}\nthen gives\n\\begin{equation}\n\tkT\n\t\\frac{\\mathrm{d}^{2}g}{\\mathrm{d} y^{2}}\n\t+\n\t\\frac{\\mathrm{d}}{\\mathrm{d} y} \\left( g\\frac{\\partial v}{\\partial y} \\right)\n\t=\n\t0\n\t,\n\t\\label{eq:SI_g_eqn_1d_2}\n\\end{equation}\n\\begin{equation}\n\tkT\n\t\\frac{\\mathrm{d}^{2} d_{y}}{\\mathrm{d} y^{2}}\n\t+\n\t\\frac{\\mathrm{d}}{\\mathrm{d} y} \\left( d_{y} \\frac{\\partial v}{\\partial y} \\right)\n\t=\n\t-\n\t2 kT \\frac{\\mathrm{d} g}{\\mathrm{d} y}\n\t- \n\tg \\frac{\\mathrm{d} v}{\\mathrm{d} y}\n\t+\n\t\\bigg\\langle g \\frac{\\mathrm{d} v}{\\mathrm{d} y} \\bigg\\rangle g\n\t,\n\t\\label{eq:SI_dy_eqn_1d_2}\n\\end{equation}\nwhere we have applied the conditions $V(L\/2,y)=V(-L\/2,y)$ and $(\\partial V \/ \\partial x)|_{x = \\pm L\/2} = 0$. Here, it is understood that the cell average of a one-dimensional (1D) function $f(y)$ simplifies to a 1D average in the $y$-direction, $\\braket{f} = L^{-1} \\int_{-L\/2}^{L\/2} f(y)\\, \\mathrm{d}y$.\n\nEqs.~\\eqref{eq:SI_g_eqn_1d_2}-\\eqref{eq:SI_dy_eqn_1d_2} are the 1D versions of Eqs.~\\eqref{eq:SI_g_eqn_steady}-\\eqref{eq:SI_d_eqn_steady}. The solution of Eq.~\\eqref{eq:SI_g_eqn_1d_2} for the $g$-field, subject to the constraint $\\braket{g} = 1$, is the 1D analog of Eq.~\\eqref{eq:SI_g_field_steady}:\n\\begin{equation}\n\tg(y)\n\t=\n\t\\frac{\\mathrm{e}^{-v(y)\/kT}}{\\braket{\\mathrm{e}^{-v\/kT}}}\n\t.\n\\end{equation}\nEq.~\\eqref{eq:SI_dy_eqn_1d_2} then simplifies to\n\\begin{flalign}\n\tkT\n\t\\frac{\\mathrm{d}^{2} d_{y}}{\\mathrm{d} y^{2}}\n\t+\n\t\\frac{\\mathrm{d}}{\\mathrm{d} y} \\left( d_{y} \\frac{\\partial v}{\\partial y} \\right)\n\t&=\n\t-\n\tkT \\frac{\\mathrm{d} g}{\\mathrm{d} y}\n\t\\nonumber \\\\\n\t&=\n\t\\frac{\\mathrm{e}^{-v\/kT}}{\\braket{\\mathrm{e}^{-v\/kT}}}\n\t\\frac{\\mathrm{d} v}{\\mathrm{d} y}\n\t,\n\t\\label{eq:SI_dy_eqn_1d_3}\n\\end{flalign}\nwhich is the 1D analog of Eq.~\\eqref{eq:SI_d_eqn_steady_2}. Unlike the 2D problem, the 1D problem admits an exact analytical solution:\n\\begin{flalign}\n\td_{y} (y)\n\t&=\n\t- y g (y)\n\t+\n\tc_{1} \\mathrm{e}^{- v(y)\/kT}\n\t\\int_{0}^{y}\n\t\\mathrm{e}^{v(\\eta)\/kT} \\, \\mathrm{d} \\eta\n\t+\n\tc_{2} L \\mathrm{e}^{-v(y) \/ kT}\n\t\\nonumber\\\\\n\t&=\n\t- \\frac{y \\mathrm{e}^{-v(y)\/kT}}{\\braket{\\mathrm{e}^{-v\/kT}}} \n\t+\n\tc_{1} \\mathrm{e}^{- v(y)\/kT}\n\t\\int_{0}^{y}\n\t\\mathrm{e}^{v(\\eta)\/kT} \\, \\mathrm{d} \\eta\n\t+\n\tc_{2} L \\mathrm{e}^{-v(y) \/ kT}\n\t.\n\t\\label{eq:SI_dy_field_1d_general_soln}\n\\end{flalign}\nThe first term on the right-hand side of Eq.~\\eqref{eq:SI_dy_field_1d_general_soln} is simply the particular solution of Eq.~\\eqref{eq:SI_dy_eqn_1d_3}; it is the 1D analog of Eq.~\\eqref{eq:SI_d_field_largeV_approx}, which was used to approximate the full solution in the strong-potential limit. The remaining terms in Eq.~\\eqref{eq:SI_dy_field_1d_general_soln} are the homogeneous solutions, with constants $c_{1}$, $c_{2}$ that must be determined from the periodicity and normalization conditions,\n\\begin{subequations}\n\\begin{gather}\n\td_{y} (L\/2) - d_{y} (-L\/2) = 0,\n\t\\label{eq:SI_dy_1d_bc1}\n\t\\\\\n\t\\braket{d_{y}}\n\t=\n\t\\frac{1}{L} \\int_{-L\/2}^{L\/2} d_{y}(y) \\, \\mathrm{d}y\n\t=\n\t0.\n\t\\label{eq:SI_dy_1d_bc2}\n\\end{gather}\n\t\\label{eq:SI_dy_1d_bcs}%\n\\end{subequations}\nInserting Eq.~\\eqref{eq:SI_dy_field_1d_general_soln} into \\eqref{eq:SI_dy_1d_bcs}, setting $v(L\/2) = v(-L\/2)$, and solving for the two unknowns $c_{1}$ and $c_{2}$ gives\n\\begin{subequations}\n\\begin{flalign}\n\tc_{1} &= \\braket{\\mathrm{e}^{-v\/kT}}^{-1} \\braket{\\mathrm{e}^{v\/kT}}^{-1}\n\t,\n\t\\\\\n\tc_{2} &= \n\t\\frac{1}{L} \\braket{\\mathrm{e}^{-v\/kT}}^{-2}\n\t\\left(\n\t\t\\braket{y\\mathrm{e}^{-v\/kT}}\n\t\t-\n\t\t\\braket{\\mathrm{e}^{v\/kT}}^{-1}\n\t\t\\bigg\\langle \\mathrm{e}^{-v\/kT}\\int_{0}^{y} \\mathrm{e}^{v(\\eta)\/kT} \\mathrm{d} \\eta \\bigg\\rangle\n\t\\right)\n\t.\n\\end{flalign}\n\t\\label{eq:SI_dy_1d_constants}%\n\\end{subequations}\n\n\nWith the solution for $d_{y}(y)$ fully specified, it remains to compute the effective diffusivity along the $y$-axis. Multiplying Eq.~\\eqref{eq:SI_dy_field_1d_general_soln} by $\\mathrm{d} v \/ \\mathrm{d}y$, applying the inverse chain rule, and averaging over the $y$-direction gives\n\\begin{flalign}\n\t\\bigg\\langle d_{y} \\frac{\\mathrm{d} v}{\\mathrm{d}y}\\bigg\\rangle\n\t&=\n\tkT\n\t\\left(\n\t\\braket{\\mathrm{e}^{-v\/kT}}^{-1}\n\t\\bigg\\langle y \\frac{\\mathrm{d}\\mathrm{e}^{-v\/kT}}{\\mathrm{d}y} \\bigg\\rangle \n\t-\n\tc_{1} \n\t\\bigg\\langle \\frac{\\mathrm{d}\\mathrm{e}^{-v\/kT}}{\\mathrm{d}y}\n\t\\int_{0}^{y}\n\t\\mathrm{e}^{v(\\eta)\/kT} \\, \\mathrm{d} \\eta\n\t\\bigg\\rangle \n\t-\n\tc_{2} L \n\t\\bigg\\langle \\frac{\\mathrm{d}\\mathrm{e}^{-v\/kT}}{\\mathrm{d}y} \\bigg\\rangle \n\t\\right).\n\t\\label{eq:SI_dy_gradv_1d_avg}\n\\end{flalign}\nInserting Eqs.~\\eqref{eq:SI_dy_1d_constants} into \\eqref{eq:SI_dy_gradv_1d_avg} and integrating by parts then gives, after some simplification,\n\\begin{flalign}\n\t\\bigg\\langle d_{y} \\frac{\\mathrm{d} v}{\\mathrm{d}y}\\bigg\\rangle\n\t&=\n\tk T\n\t\\left( \n\t\t-1\n\t\t+\n\t\t\\braket{\\mathrm{e}^{-v\/kT}}^{-1} \\braket{\\mathrm{e}^{v\/kT}}^{-1}\n\t\\right)\n\t.\n\\end{flalign}\nSince $d_{y}$ is independent of $x$, $\\braket{d_{y} (\\partial V \/ \\partial y)} = \\braket{d_{y} (\\mathrm{d} v \/ \\mathrm{d} y)}$. Thus, the $yy$ component of Eq.~\\eqref{eq:effective_diffusivity} simplifies to\n\\begin{flalign}\n\t\\overline{D}_{yy}\n\t&=\n\t\\frac{kT}{\\gamma}\n\t+\n\t\\frac{1}{\\gamma}\n\t\\bigg\\langle d_{y} \\frac{\\mathrm{d} v}{\\mathrm{d}y}\\bigg\\rangle\n\t\\nonumber\\\\\n\t&=\n\t\\frac{kT}{\\gamma}\n\t\\braket{\\mathrm{e}^{-v\/kT}}^{-1} \\braket{\\mathrm{e}^{v\/kT}}^{-1}\n\t,\n\t\\label{eq:SI_Dyy_1d}\n\\end{flalign}\nwhere an overbar is used to denote the long-time average over one periodic cycle.\nThis is the classical result for diffusion of a Brownian particle in a 1D periodic potential \\cite{lifson1962self,Festa1978}.\n\nUp until now, we have neglected the $d_{x}$-field, which appears in the $xx$-component of Eq.~\\eqref{eq:effective_diffusivity} and, therefore, influences the effective diffusivity along the $x$-axis. To a first approximation, we assume that the gradients in the $x$-direction have been ``smeared out'' so that dispersion in that direction is negligible: $\\braket{d_{x} (\\partial V \/ \\partial x)} \\approx 0$. This approximation is consistent with a model of dispersion in an effectively 1D potential. Therefore, the $xx$-component of Eq.~\\eqref{eq:effective_diffusivity} (time-averaged) is simply the Stokes-Einstein-Sutherland diffusivity:\n\\begin{equation}\n\t\\overline{D}_{xx} = \\frac{kT}{\\gamma}\n\t.\n\t\\label{eq:SI_Dxx_1d}\n\\end{equation}\nEqs.~\\eqref{eq:SI_Dyy_1d} and \\eqref{eq:SI_Dxx_1d} are exactly the same as Eq.~\\eqref{eq:diffusivity_high_frequency} from the main text.\n\n\n\n\n\\section{5. Brownian Dynamics Simulations} \\label{sec:brownian_dynamics_simulations}\n\n\nThe Langevin equation of motion corresponding to Eqs.~\\eqref{eq:SI_smoluchowski_eqn}-\\eqref{eq:SI_coordinate_conversion} is given by\n\\begin{equation}\n\t\\frac{\\mathrm{d} \\bm{r}_{i} (t)}{\\mathrm{d} t}\n\t=\n\t-\\bm{u}(t)\n\t-\n\t\\frac{1}{\\gamma} \\bm{\\nabla}_{\\bm{r}} V[\\bm{r}_{i}(t)]\n\t+\n\t\\sqrt{\\frac{2kT}{\\gamma}}\\bm{B}_{i}(t)\n\t,\n\t\\qquad\n\ti = 1, 2, \\dots, N_{\\text{p}}\n\t,\n\t\\label{eq:SI_langevin_eqn}\n\\end{equation}\nwhere $i$ is the particle index, $N_{\\text{p}}$ is the total number of particles in the system, and $\\bm{B}_{i} (t)$ is a white-noise source with statistics,\n\\begin{equation}\n\t\\braket{\\bm{B}_{i}(t)}\n\t=\n\t\\bm{0},\n\t\\qquad\n\t\\braket{\\bm{B}_{i}(t)\\bm{B}_{i}(t')}\n\t=\n\t\\delta(t - t')\n\t\\tens{I}\n\t.\n\t\\label{eq:SI_fluctuation_dissipation_theorem}\n\\end{equation}\n[Note that the angle brackets $\\langle\\,\\cdot\\,\\rangle$ appearing in Eq.~\\eqref{eq:SI_fluctuation_dissipation_theorem} denote {\\it ensemble} averages and are not to be confused with the {\\it cell} average defined in the main text.]\nThe potential-energy field $V(\\bm{r})$ and convective velocity $\\bm{u}(t)$ appearing in Eq.~\\eqref{eq:SI_langevin_eqn} are given by Eqs.~\\eqref{eq:harmonic_potential} and \\eqref{eq:trap_velocity}, respectively. Interactions between particles have been neglected, so the $N_{\\text{p}}$ equations of motion are uncoupled.\nFor the purpose of numerically time-advancing Eq.~\\eqref{eq:SI_langevin_eqn}, it is convenient to shift to the laboratory frame in which the position of each particle is measured as $\\bar{\\bm{r}}_{i}(t) = \\bm{r}_{0}(t) + \\bm{r}_{i}(t)$, where $\\bm{r}_{0}(t) = \\int_{0}^{t} \\bm{u} (\\tau) \\, \\mathrm{d} \\tau = \\hat{\\bm{e}}_{x} A \\sin{(\\omega t)}$ denotes the time-dependent position of the moving traps. In this frame, Eq.~\\eqref{eq:SI_langevin_eqn} becomes\n\\begin{equation}\n\t\\frac{\\mathrm{d} \\bar{\\bm{r}}_{i}(t)}{\\mathrm{d} t}\n\t=\n\t-\n\t\\frac{1}{\\gamma} \\bm{\\nabla}_{\\bar{\\bm{r}}} V[\\bar{\\bm{r}}_{i}(t)-\\bm{r}_{0}(t)]\n\t+\n\t\\sqrt{\\frac{2kT}{\\gamma}}\\bm{B}_{i}(t)\n\t,\n\t\\qquad\n\ti = 1, 2, \\dots, N_{\\text{p}}\n\t.\n\t\\label{eq:SI_langevin_eqn_relative}\n\\end{equation}\nHere, the convective term has been eliminated and the potential-energy field oscillates in time. \n\n\n\n\nIn our Brownian dynamics simulations, we numerically advanced Eq.~\\eqref{eq:SI_langevin_eqn_relative} using the GPU-enabled HOOMD-blue software package \\cite{anderson2020hoomd}. A system of $N_{\\text{p}} = 10,000$ particles was initialized at random positions within a periodically replicated $L\\times L$ cell and advanced for $\\tau = 10,000$ s (2.78 h) using a time step $\\Delta t = 1$ ms. Fig.~\\ref{Fig:SI_Gfield_Theory_vs_Simulation} shows that the simulated probability density shows excellent agreement with the deterministic solution of the corresponding Smoluchowski equation [Eq.~\\eqref{eq:g_eqn}]. The MSD and effective diffusivity of the particles were then computed exactly as in the experiments using Eqs.~\\eqref{eq:SI_msd_experiments}-\\eqref{eq:SI_diffusivity_experiments}, wherein the time integral was discretized using the left Riemann sum. \n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\\includegraphics[width=1\\linewidth]{Figs\/SI\/SIFig_Gfield_TheoryVsSimulation.png}\n\t\t\\caption{%\n\t\tComparison of the convected probability density $g(x,y,t)$ for a stiff trap near the critical frequency ($\\kappa=5$ $kT\/\\mu$m$^2$, $\\omega\/2\\uppi=18.33$ mHz) from\n\t\t(a) deterministic solution of the Smoluchowski equation [Eq.~\\eqref{eq:g_eqn}] and (b) stochastic simulation of the Langevin equation [Eq.~\\eqref{eq:SI_langevin_eqn}].\n\t\t}\n\t\t\\label{Fig:SI_Gfield_Theory_vs_Simulation}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\subsection{Derivation of Eq. (12): convective escape of a Brownian particle from a harmonic well}\n\nWe wish to estimate the critical oscillation frequency $\\omega_{\\text{max}}$ at which a Brownian particle rattling around the bottom of a potential-energy well is convected near the edge of the well with ample probability for escape. To make such an estimate, we start with the Langevin equation, Eq.~\\eqref{eq:SI_langevin_eqn}, simplified for a single particle in a harmonic well $V(\\bm{r}) = \\tfrac{1}{2} \\kappa r^{2}$:\n\\begin{equation}\n\t\\frac{\\mathrm{d} \\bm{r} (t)}{\\mathrm{d} t}\n\t=\n\t-\n\t\\frac{\\kappa}{\\gamma} \\bm{r}(t)\n\t-\n\t\\bm{u}(t)\n\t+\n\t\\sqrt{\\frac{2kT}{\\gamma}}\\bm{B}(t)\n\t.\n\t\\label{eq:SI_langevin_eqn_harmonic_well}\n\\end{equation}\nEq.~\\eqref{eq:SI_langevin_eqn_harmonic_well} may be straightforwardly integrated with the initial condition $\\bm{r}(0) = \\bm{0}$ to give the fluctuating particle position,\n\\begin{flalign}\n\t\\bm{r} (t)\n\t&=\n\t\\mathrm{e}^{-\\kappa t \/ \\gamma}\n\t\\int_{0}^{t}\n\t\\mathrm{e}^{\\kappa s \/ \\gamma}\n\t\\left(\n\t\t-\\bm{u}(s)\n\t\t+\n\t\t\\sqrt{\\frac{2kT}{\\gamma}} \\bm{B}(s)\n\t\\right)\n\t\\mathrm{d} s\n\t.\n\t\\label{eq:SI_langevin_eqn_harmonic_well_soln}\n\\end{flalign}\nSubstituting Eq.~\\eqref{eq:trap_velocity} into \\eqref{eq:SI_langevin_eqn_harmonic_well_soln} for the convective velocity then gives, upon integration,\n\\begin{flalign}\n\t\\bm{r} (t)\n\t&=\n\t-\\hat{\\bm{e}}_{x} A \n\t\\left( \\frac{\\gamma \\omega \/\\kappa}{1 + (\\gamma \\omega \/ \\kappa)^{2}} \\right)\n\t\\left(\n\t\t\\cos{(\\omega t)}\n\t\t+\n\t\t\\frac{\\gamma \\omega}{\\kappa}\n\t\t\\sin{(\\omega t)}\n\t\t-\n\t\t\\mathrm{e}^{-\\kappa t \/ \\gamma}\n\t\\right)\n\t+\n\t\\sqrt{\\frac{2kT}{\\gamma}} \n\t\\mathrm{e}^{-\\kappa t \/ \\gamma}\n\t\\int_{0}^{t}\n\t\\mathrm{e}^{\\kappa s \/ \\gamma}\n\t\\bm{B}(s)\n\t\\, \\mathrm{d} s\n\t.\n\t\\label{eq:SI_langevin_eqn_harmonic_well_soln_2}\n\\end{flalign}\nThe first term on the right-hand side of Eq.~\\eqref{eq:SI_langevin_eqn_harmonic_well_soln_2} is the deterministic part of the fluctuating particle particle position, which is driven by oscillatory convection and attenuated by the trapping force. The second term is the stochastic part due to Brownian motion. The mean displacement and mean squared displacement of the particle respectively capture strength of these deterministic and stochastic elements:\n\\begin{flalign}\n\t\\braket{\\bm{r}(t)}\n\t=\n\t-\\hat{\\bm{e}}_{x} A \n\t\\left( \\frac{\\gamma \\omega \/\\kappa}{1 + (\\gamma \\omega \/ \\kappa)^{2}} \\right)\n\t\\left(\n\t\t\\cos{(\\omega t)}\n\t\t+\n\t\t\\frac{\\gamma \\omega}{\\kappa}\n\t\t\\sin{(\\omega t)}\n\t\t-\n\t\t\\mathrm{e}^{-\\kappa t \/ \\gamma}\n\t\\right)\n\t,\n\t\\label{eq:SI_drift_in_harmonic_well}\n\\end{flalign}\n\\begin{flalign}\n\t\\braket{(\\bm{r}(t) - \\braket{\\bm{r}(t)})(\\bm{r}(t) - \\braket{\\bm{r}(t)})}\n\t=\n\t\\frac{kT}{\\kappa} \\left( 1 - \\mathrm{e}^{-2\\kappa t \/ \\gamma} \\right) \\tens{I}\n\t,\n\t\\label{eq:SI_variance_in_harmonic_well}\n\\end{flalign}\nwhere we have applied the white-noise statistics, Eq.~\\eqref{eq:SI_fluctuation_dissipation_theorem}, of the fluctuating $\\bm{B}$-field.\n\nAfter waiting a long enough time $t \\gg \\gamma \/\\kappa$, the exponential terms in Eqs.~\\eqref{eq:SI_drift_in_harmonic_well}-\\eqref{eq:SI_variance_in_harmonic_well} die off and we are left with an oscillating particle probability with variance $k T \/ \\kappa$ given by Eq.~\\eqref{eq:SI_variance_in_harmonic_well}. The amplitude of these oscillations are found from the extrema of the particle drift, Eq.~\\eqref{eq:SI_drift_in_harmonic_well}:\n\\begin{equation}\n\t\\sup_{t\\ge 0}\n\t| \\langle \\bm{r}(t)\\rangle |\n\t=\n\t\\frac{\\gamma \\omega A\/\\kappa}{\\sqrt{1+ (\\gamma \\omega \/ \\kappa)^{2}}}\n\t\\approx\n\t\\frac{\\gamma \\omega A}{\\kappa}\n\t\\quad\n\t\\text{for}\n\t\\quad\n\t\\frac{\\gamma \\omega}{\\kappa} \\ll 1\n\t.\n\\end{equation}\nThus, the basin of probability of size $\\sim \\sqrt{kT\/\\kappa}$ oscillates with amplitude $\\sim \\gamma \\omega A \/ \\kappa$ about the center of the potential-energy well. As the frequency $\\omega$ is increased, the oscillations become more pronounced. The particle is expected to escape a well of finite width $W_{\\text{trap}}$ when the spatial extent of the particle probability density crosses the edge of the well, at a critical frequency $\\omega_{\\text{max}}$:\n\\begin{equation}\n\t\\tfrac{1}{2} W_{\\text{trap}}\n\t\\approx\n\t\\frac{\\gamma \\omega_{\\text{max}} A}{\\kappa}\n\t+\n\t\\sqrt{\\frac{kT}{\\kappa}}\n\t,\n\\end{equation}\nor, solving for $\\omega_{\\text{max}}$,\n\\begin{equation}\n\t\\omega_{\\text{max}}\n\t\\approx\n\t\\frac{\\kappa}{\\gamma A}\n\t\\left(\n\t\t\\tfrac{1}{2} W_{\\text{trap}}\n\t\t-\n\t\t\\sqrt{\\frac{kT}{\\kappa}}\n\t\\right)\n\t.\n\\end{equation}\nThe last expression is exactly Eq.~\\eqref{eq:critical_frequency} from the main text. \n\n\n\n\n\\section{6. Additional Data}\n\n\n\nIn addition to measuring the effective diffusivity $\\overline{\\tens{\\bm{D}}}$ as a function of the oscillation frequency $\\omega$, we also varied the amplitude $A$ while holding the frequency fixed.\nThe strength of the convective velocity $\\bm{u}(t) = \\hat{\\bm{e}}_{x} \\omega A \\cos{(\\omega t)}$ may be modified by varying either the amplitude $A$ or the frequency $\\omega$. \nFig.~\\ref{Fig:SI4} plots $\\overline{D}_{xx}$ and $\\overline{D}_{yy}$ against $A$ for a fixed trap stiffness $\\kappa = 5~kT \/ \\mu\\mathrm{m}^2$ and frequency $\\omega\/2\\uppi = 18.3$ mHz.\nThis frequency corresponds to the critical frequency $\\omega_{\\text{max}}$ (for which $\\overline{D}_{xx}$ is maximized) for $\\kappa = 5~kT \/ \\mu\\mathrm{m}^2$ and $A = 5$ $\\mu$m, as shown in the main text (see Fig.~\\ref{fig:Fig3}).\nWe find that the $\\overline{D}_{xx}$ is non-monotonic and achieves a maximum at $A = 5$ $\\mu$m.\nFor amplitudes $A > 5$ $\\mu$m, the convective motion is fast compared to the particle response time. Consequently, the particles sample regions outside of the harmonic well and their average diffusivity along the convection axis is reduced. \n\n\n\n\\begin{figure}[!h]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\linewidth]{Figs\/SI\/SiFig_AmplitudeSweep.png}\n\t\t\\caption{%\n\t\t\tEffective diffusivity as a function of oscillation amplitude at a fixed trap stiffness $\\kappa = 5~kT \/ \\mu\\mathrm{m}^2$ and frequency $\\omega\/2\\uppi = 18.3~\\mathrm{mHz}$.\n\t\t\tLike Fig.~\\ref{fig:Fig3} in the main text, $\\overline{D}_{xx}$ is non-monotonic and reaches a maximum when the convection strength balances the harmonic trap strength.\n\t\t\tAt very large amplitudes, the particle cannot quickly respond to the rapidly oscillating trap and explores regions outside of the harmonic well.\n\t\t}\n\t\t\\label{Fig:SI4}\n\t\\end{center}\n\t\\vspace{-18pt}\n\\end{figure}\n\n\n\\pagebreak\n\\section{7. Supplemental Movies}\nBelow, we describe the Supplemental Movies associated with this manuscript. All time stamps corresponds to hours:minutes:seconds.\n\n\\begin{enumerate}\n\t\\item[] \\textbf{S1.} Experimental micrographs of silica particles with radius $a = 1.25~\\mu$m diffusing through a stationary array of harmonic traps (6$\\times$6 grid shown) with varying trap stiffness.\n\t\n\t\\item[] \\textbf{S2.} Microscopic Brownian dynamics simulations of a small sample of particles diffusing through a stationary array of harmonic traps (6$\\times$6 grid shown) with varying trap stiffness (same parameters as in S1).\n\n\t\\item[] \\textbf{S3.} Experimental micrographs of silica particles with radius $a = 1.25~\\mu$m diffusing through an oscillating array of stiff traps (6$\\times$6 grid shown) with varying oscillation frequency and fixed trap stiffness $\\kappa = 5~kT\/\\mu\\mathrm{m}^2$. The second part of the movie shows the trajectories of several tagged particles.\n\t\n\t\\item[] \\textbf{S4.} Microscopic Brownian dynamics simulations of a small sample of particles diffusing through an oscillating array of stiff traps (6$\\times$6 grid shown) with varying oscillation frequency and fixed trap stiffness $\\kappa = 5~kT\/\\mu\\mathrm{m}^2$ (same parameters as in S3).\n\t\n\t\\item[] \\textbf{S5.} Macroscopic Brownian dynamics simulations of 10,000 particles diffusing through a stationary array of harmonic traps (60$\\times$60 grid shown) over long length and time scales, varying the trap stiffness.\n\t\n\t\\item[] \\textbf{S6.} Macroscopic Brownian dynamics simulations of 10,000 particles diffusing through an oscillating array of stiff traps (60$\\times$60 grid shown) over long length and time scales, varying the oscillation frequency at a fixed trap stiffness $\\kappa = 5~kT\/\\mu\\mathrm{m}^2$.\n\t\n\t\\item[] \\textbf{S7.}  2D contour plots of the displacement field density $d_{x}(x,y,t)$ in an $L\\times L$ periodic cell containing an oscillating harmonic trap, varying the oscillation frequency at a fixed trap stiffness $\\kappa = 5~kT\/\\mu\\mathrm{m}^2$ (same parameters as in S6). Bottom row plots the long-time average over one periodic cycle.\n\t\n\\end{enumerate}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Generalities on Poisson algebras}\n\n\\subsection{Poisson algebras}\nLet $\\K$ be a  field of characteristic $0$. A $\\K$-Poisson algebra is a $\\K$-vector space  $\\p$ equipped with two bilinear products denoted by $x\\cdot y$  and $\\{x ,y \\}$, \nhaving the following properties:\n\\begin{enumerate}\n\\item The couple $(\\p,\\cdot)$ is an associative commutative $\\K$-algebra.\n\n\\item The couple $(\\p, \\{ , \\})$ is  a $\\K$-Lie algebra.\n\n\\item The products $\\cdot$ and $\\{, \\}$ satisfy the Leibniz rule:\n$$\\{x\\cdot y,z\\}=x \\cdot\\{y,z\\}+\\{x,z\\}\\cdot y,$$\nfor any $x,y,z \\in \\p.$\n\\end{enumerate}\nThe product $\\{,\\}$ is usually called Poisson bracket and the Leibniz identity means that the Poisson bracket acts as a derivation of the associative product.\n\nIn \\cite{markl-eli-poisson}, one proves that any Poisson structure on a $\\K$-vector space is also given by a nonassociative product, denoted by $xy$ and satisfying the non associative identity\n\\begin{eqnarray}\n\\label{associator} 3A(x,y,z)=(xz)y+(yz) x-(y x)z-(z x) y.\n\\end{eqnarray}\nwhere $A(x,y,z)$ is the associator $A(x,y,z)=(xy)z-x(yz)$. In fact, if $\\p$ is a Poisson algebra given by the associative product $x\\cdot y$ and the Poisson bracket $\\{x,y\\}$, \nthen $xy$ is given by $$xy=\\{x,y\\}+x \\cdot y.$$ Conversely, the Poisson bracket and the associative product of $\\p$ are the skew-symmetric part and the symmetric part of \nthe product $xy$. Thus it is equivalent to present a Poisson algebra classically or by this nonassociative product. In \\cite{Mic-Eli-Poisson}, we have studied algebraic \nproperties of the nonassociative algebra $\\p$. \nIn particular we have proved that this algebra is flexible, power-associative and admits a Pierce decomposition.\n\n\\medskip\n\nIf $\\p$ is a Poisson algebra given by the nonassociative product (\\ref{associator}), \nwe denote by $\\g_{\\p}$ the Lie algebra on the same vector space $\\p$ whose Lie bracket is $$\\{x,y\\}=\\displaystyle\\frac{xy-yx}{2}$$ and by $\\mathcal{A}_{\\p}$ \nthe commutative associative algebra, on the same vector space, whose product is $$x \\cdot y=\\displaystyle\\frac{xy+yx}{2}.$$\nAn important problem in mathematical physics and more precisely in Quantum Field theory is the deformation of Poisson algebras. \nThe classical deformations of Poisson algebras consist of deformations of the Poisson brackets, that is, deformations of $\\g_\\mathcal{P}$ which let the\nassociatif multiplication of $\\mathcal{A}_\\mathcal{P}$ unchanged and satisfying the Leibniz identity \\cite{Pich}. In \\cite{remm-deformPoisson} \nthis type of deformation has been generalized  by using a nonassociative multiplication defining the Poisson stucture. The deformations of this nonassociative\nmultiplication provides general Poisson deformations.\n\n\n\n\\section{Poisson superalgebra\n }\nBy a $\\K$-super vector space, we mean a $\\Z_2$-graded vector space $V=V_0 \\oplus V_1$. The vectors of $V_0$ and $V_1$ are called homogeneous vectors of degree respectively equal to $0$ and $1$. For an homogeneous vector $x$, we denote by $\\mid x\\mid$ its degree.\nA  $\\K$-Poisson superalgebra\n is a $\\K$-super vector space  $\\p=\\p_0 \\oplus \\p_1$ equipped with two bilinear products denoted by $x\\cdot y$  and $\\{x ,y \\}$, having the following properties:\n\\begin{itemize}\n\\item The couple $(\\p,\\cdot)$ is a associative super commutative $\\K$-algebra,\nthat is, $$x \\cdot y=(-1)^{\\mid x \\mid \\mid y \\mid}y \\cdot x.$$ \n\\item The couple $(\\p, \\{ , \\})$ is  a $\\K$-Lie super algebra, that is,\n$$\\{x ,y \\}=-(-1)^{\\mid x \\mid \\mid y \\mid}\\{y ,x \\}$$\nand satisfying the super Jacobi condition:\n$$(-1)^{\\mid z \\mid \\mid x \\mid}\\{x , \\{y ,z \\}\\}+(-1)^{\\mid x \\mid \\mid y \\mid}\\{y , \\{z ,x \\}\\}+\n(-1)^{\\mid y \\mid \\mid z \\mid}\\{z , \\{x ,y \\}\\}=0.$$\n\\item The products $\\cdot$ and $\\{, \\}$ satisfy the super Leibniz rule:\n$$\\{x,y\\cdot z\\}=\\{x,y\\} \\cdot z+(-1)^{\\mid x \\mid \\mid y \\mid}y \\cdot \\{x,z\\}.$$\nwhere $x,y$ and $z$ are homogeneous vectors.\n\\end{itemize}\n\n\\bigskip\n\n\n\\begin{theorem}\nLet $\\p$ a $\\K$-super vector space. Thus $\\p$ is a Poisson superalgebra\n if and only if there exists on $\\p$ a nonassociative product $x  y$ satisfying\n \\begin{equation}\\label{superPoisson}\n\\left\\{\n \\begin{array}{l}\n\\medskip\n 3(xy)z-3x(yz)  + (-1)^{\\mid x \\mid \\mid y \\mid}(yx)z -(-1)^{\\mid y \\mid \\mid z \\mid} (xz)y\n - (-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}(yz)x \\\\\n + (-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}(zx)y =0 \n\\end{array}\n\\right.\n\\end{equation}\nfor any homogeneous vectors $x,y,z \\in\\p$.\n\\end{theorem}\n\n\\noindent{\\it Proof.} Assume that $(\\p,\\cdot,\\{,\\})$ is a Poisson superalgebra\n.\nConsider the multiplication\n$$\\begin{array}{l}\nxy=x \\cdot y + \\{ x,y \\}.\n\\end{array}$$\nWe deduce that\n$$\\displaystyle x \\cdot y=\\frac{1}{2}(xy+(-1)^{\\mid x \\mid \\mid y \\mid}yx).$$\nThus\nthe associativity condition writes for homogeneous vectors\n$$\\begin{array}{rl}\n\\medskip\nv_1(x,y,z)= & A(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}A(z,y,x)\n +(-1)^{\\mid x \\mid \\mid y \\mid}(yx)z\\\\\n & -(-1)^{\\mid y \\mid \\mid z \\mid}x(zy)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z\\mid}(yz)x+(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}z(xy)\\\\\n = &0\n\\end{array}$$\nwhere $A(x,y,z)=(xy)z-x(yz)$. \nLikewise, the Poisson bracket writes for homogeneous vectors\n$$\\displaystyle \\{x , y\\}=\\frac{1}{2}(xy-(-1)^{\\mid x \\mid \\mid y \\mid}yx)$$\nand the super Jacobi condition\n$$\\begin{array}{rl}\n\\medskip\nv_2(x,y,z)=& (-1)^{\\mid x \\mid \\mid z \\mid}A(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}A(y,x,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid y \\mid \\mid z \\mid}A(z,y,x)\\\\\n& -(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}A(x,z,y)+(-1)^{\\mid x \\mid \\mid y \\mid}A(y,z,x)+(-1)^{\\mid y \\mid \\mid z \\mid}A(z,x,y) \\\\\n=&0\n\\end{array}$$\nThe super Leibniz writes\n$$\\begin{array}{rl}\n\\medskip\nv_3(x,y,z)=& A(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid}A(y,x,z)+(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}A(z,y,x)\\\\\n& +(-1)^{\\mid y \\mid \\mid z \\mid}A(x,z,y)\n +(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}A(y,z,x)-(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}A(z,x,y) \\\\\n =& 0.\n\\end{array}$$\nLet us consider the vector\n\n \n$$\\begin{array}{ll}\n\\medskip\nv(x,y,z)  = \n & \\frac{1}{3}\\left[ (-1)^{\\mid x \\mid \\mid y \\mid}(yx)z -(-1)^{\\mid y \\mid \\mid z \\mid} (xz)y\n - (-1)^{\\mid x \\mid (\\mid y \\mid +\\mid z \\mid}(yz)x\n  + (-1)^{(\\mid x \\mid +\\mid y \\mid )\\mid z \\mid}(zx)y\\right] \\\\\n& +(xy)z-x(yz) .\\\\\n\\end{array}$$\nThen \n$$\\begin{array}{l}\nv(x,y,z)= \\frac{1}{6}\\left( 2v_1(x,y,z)+(-1)^{\\mid x \\mid \\mid z \\mid}v_2(x,y,z)+v_3(x,y,z)+2(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}v_3(z,x,y) \\right).\n\\end{array}\n$$\nWe deduce that the product $xy$ satisfies\n$$v(x,y,z)=0$$\nfor any homogeneous vectors $x,y,z$.\n\n\\noindent Conversely, assume that the product of the non associative product $\\p$ satisfies $v(x,y,z)=0$ for any homogeneous vestors $x,y,z.$ Let \n$v_1(x,y,z), v_2(x,y,z), v_3(x,y,z)$ be the vectors of $\\p$ defined in the first part respectively in relation with the associativity, the super Jacobi and super Leibniz relations. \nWe have\n$$\\begin{array}{ll}\n\\medskip\nv_1(x,y,z)=&v(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}v(z,y,x)\n+(-1)^{\\mid y \\mid \\mid z \\mid}v(x,z,y)\\\\\n\\medskip\n& -(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}v(z,x,y)\\\\\n\\medskip\nv_2(x,y,z) =&(-1)^{\\mid x \\mid \\mid z \\mid}v(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}v(y,x,z)-(-1)^{\\mid x \\mid \\mid y \\mid+\\mid y \\mid \\mid z \\mid}v(z,y,x)\\\\\n\\medskip\n& -(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}v(x,z,y)+(-1)^{\\mid x \\mid \\mid y \\mid}v(y,z,x)+(-1)^{\\mid y\\mid \\mid z \\mid}v(z,x,y)\\\\\n\\medskip\nv_3(x,y,z)=&v(x,y,z)-(-1)^{\\mid x \\mid \\mid y \\mid}v(y,x,z)+(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}v(z,y,x)\\\\\n& +(-1)^{\\mid y \\mid \\mid z \\mid}v(x,z,y)+(-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}v(y,z,x)-(-1)^{\\mid x\\mid \\mid z \\mid+\\mid y\\mid \\mid z \\mid}v(z,x,y)\n\\end{array}$$\n\n\\medskip\n\\noindent{\\bf Examples.} Any $2$-dimensional superalgebra $\\p=V_0 \\oplus V_1$ with an homogeneous basis $\\{e_0,e_1\\}$ is defined\n$$\\left\\{\n\\begin{array}{l}\ne_0e_0=ae_0,\\\\\n e_0e_1=be_1, \\ e_1e_0=ce_1,\\\\\ne_1e_1=de_0.\n\\end{array}\n\\right.\n$$\nThis is a super Poisson  multiplication if and only if we have\n$$\\left\\{\n\\begin{array}{l}\nd=0,\\\\\n3(a-b)b+ab-2bc+c^2=0,\\\\\n3(a-c)c+ab-2bc+c^2=0,\n\\end{array}\n\\right.\n$$\nor\n$$\\left\\{\n\\begin{array}{l}\na=0,\\\\\na=b=c.\n\\end{array}\n\\right.\n$$\nWe obtain the following $2$-dimensional Poisson superalgebras\n$$\\left\\{\n\\begin{array}{ll}\n\\mathcal{SP}_{2,1} & e_0e_0=ae_1\\\\\n\\mathcal{SP}_{2,2} & e_0e_0=ae_1, e_0e_1=e_1e_0=ae_1\\\\\n\\mathcal{SP}_{2,3} &  e_0e_1=-e_1e_0=be_1\\\\\n\\mathcal{SP}_{2,4} &  e_1e_1=de_0,\\\\\n\\end{array}\n\\right.\n$$\nthe non written product being considered equal to $0$.\nLet us note that these $2$-dimensional algebras correspond to the algebras $(\\mu_{16},\\beta_2=0),(\\mu_{16},\\beta_2=1),(\\mu_{9},\\alpha_2=0,\\beta_4=0),(\\mu_{5},\\alpha_2=0)$ in the classification \\cite{goze-remm-2algebras}.\n\\section{Properties of Poisson superalgebras }\n\\begin{definition}\nA nonassociative superalgebra  is called super flexive if  the multiplication $xy$ satisfy\n$$A(x,y,z) + (-1)^{(|x||z|+|x||y|+|y||z|)}A(z,y,x)=0$$\nfor any homogeneous elemnts $x,y,z$, where $A(x,y,z)=(xy)z-x(yz)$ is the associator of the multiplication.\n\\end{definition}\n\\begin{proposition}\nLet $\\p$ be a Poisson superalgebra. Then the non associative product defining the super Poisson structure is super flexive.\n\\end{proposition}\n\\noindent{\\it Proof.} In fact, let \n$$B(x,y,z)=3(A(x,y,z) + (-1)^{(|x||z|+|x||y|+|y||z|)}A(z,y,x)) .\n$$\nWe  have\n$$\n\\begin{array}\n{rl}\n\\medskip\nB(x,y,z) =& -(-1)^{\\mid x \\mid \\mid y \\mid}(yx)z +(-1)^{\\mid y \\mid \\mid z \\mid} (xz)y\n + (-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid}(yz)x \n - (-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}(zx)y \\\\\n \\medskip\n \n &+ (-1)^{(|x||z|+|x||y|+|y||z|)}(-(-1)^{\\mid z \\mid \\mid y \\mid}(yz)x +(-1)^{\\mid y \\mid \\mid x \\mid} (zx)y \\\\\n \\medskip\n \n &+ (-1)^{\\mid z \\mid \\mid y \\mid+\\mid z \\mid \\mid x \\mid}(yx)z \n - (-1)^{\\mid z \\mid \\mid x \\mid+\\mid y \\mid \\mid x \\mid}(xz)y )\\\\\n \\medskip\n \n =& (-(-1)^{\\mid x \\mid \\mid y \\mid} +(-1)^{\\mid x \\mid \\mid y \\mid})(yx)z + ((-1)^{\\mid y \\mid \\mid z \\mid} -(-1)^{\\mid y \\mid \\mid z \\mid}) (xz)y \\\\\n \\medskip\n &+((-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid} - (-1)^{\\mid x \\mid \\mid y \\mid+\\mid x \\mid \\mid z \\mid})(yz)x \n +( - (-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid}\\\\\n \\medskip\n & +(-1)^{\\mid x \\mid \\mid z \\mid+\\mid y \\mid \\mid z \\mid})(zx)y \\\\\n \\medskip\n =& 0.\n \\end{array}\n $$\n \n \\medskip\n \n \\noindent{\\bf Remark : On the power associativity.} Recall that a nonassociative algebra is power associative if every element\ngenerates an associative subalgebra. Let $\\p$ be a Poisson superalgebra\n provided with its non associative product $xy$. If $V_0$ is its the even homogeneous part, then the \nrestriction of the product $xy$ is a multiplication in \nthis homogeneous vector space satisfying Identity (\\ref{superPoisson}). Since all the vectors of $V_0$ are of degree $0$, Identity (\\ref{superPoisson}) is reduced to Identity\n (\\ref{associator}). We deduce that $V_0$ is a Poisson algebra and any vector $x$ in $V_0$ generates an associative subalgebra of $V_0$ and of $\\p$.\n  \n  Assume now that $y$ is an odd vector. We have\n  $$y\\cdot y=\\displaystyle\\frac{1}{2}(yy+(-1)yy)=0,$$\n  and\n  $$\\{y,y\\}=\\displaystyle\\frac{1}{2}(yy-(-1)yy)=yy.$$\n  If we write $y^2=yy$, then $$y^2=\\{y,y\\}.$$\n  This implies\n  $$yy^2=y\\{y,y\\}=y \\cdot \\{y,y\\}+\\{y,\\{y,y\\}\\}.$$\n  But from the super identity of Jacobi, $\\{y,\\{y,y\\}\\}=0.$ Thus we have\n  $$yy^2= y \\cdot \\{y,y\\}=\\{y,y\\} \\cdot y=y^2y.$$\n  We can write\n  $$y^3=yy^2=y^2y.$$\n  Now\n  $$y^2y^2=\\{y,y\\}\\{y,y\\}=\\{y,y\\}\\cdot\\{y,y\\}+\\{\\{y,y\\},\\{y,y\\}\\}.$$\n  We have also\n  $$yy^3=y\\cdot y^3+\\{y,y^3\\}=y\\cdot y\\cdot\\{y,y\\}+\\{y,y\\cdot \\{y,y\\}\\}.$$\n  But $y \\cdot y=0$. Thus, from the Leibniz rule,\n  $$yy^3=\\{y,y\\cdot \\{y,y\\}\\}=-y\\cdot \\{y,\\{y,y\\}\\}+\\{y,y\\}\\cdot\\{y,y\\}=\\{y,y\\}\\cdot \\{y,y\\}.$$\n  We deduce\n  $$y^2y^2-yy^3= \\{\\{y,y\\},\\{y,y\\}\\}.$$\n  Since $\\{y,y\\}$ is of degree $0$, we obtain\n$$y^2y^2-yy^3=0.$$\nWe can write\n$$y^4=y^2y^2=yy^3=y^3y$$\nthe last equality results of $\\{y,y\\cdot \\{y,y\\}\\}=\\{y\\cdot \\{y,y\\},y\\}.$ Now, using  Identity (\\ref{superPoisson}) to the triple $(y^i, y^j,y^k)$ with $i+j+k=5$, we obtain a \nlinear system \non the vectors $y^iy^j$ with $i+j=5$, which admits as solutions\n$$yy^4=y^2y^3= y^3y^2=y\u2074y.$$\nThus $y^5$ is well determinated. By induction, using Identity (\\ref{superPoisson}) on the triple $(y^i, y^j,y^k)$ with $i+j+k=n$, using induction hypothesis $y^py^{n-1-p}=y^{n-1}$, we obtain that\n$$y^n=y^py^{n-p}$$\nfor any $p=1,\\cdots,n-1$. Thus any homogeneous element of edd degree generates an associative algebra.\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\begin{note}\n\t\\begin{itemize}\n\t\t\\item Let $\\Z$ and $\\P$ be the set of integers and prime numbers, respectively.\n\t\t\\item Let $\\N$ be $\\{1,2,3,\\cdots\\}$, and its elements are called natural numbers.\n\t\t\\item For real $x$, $\\floor{x}$ means to the unique integer such that $\\floor{x}\\leq x<\\floor{x}+1$; called the integer part of $x$.\n\t\t\\item Unless stated otherwise, the greatest common divisor of two integers $m$ and $n$ is denoted by $(m,n)$.\n\t\\end{itemize}\n\\end{note}\n\nIt is well known that arithmetical progressions are related to prime numbers. For example, the Dirichlet prime number theorem in $1837$ \\cite{Dirichlet} and the Green-Tao theorem in $2004$ \\cite{Green-Tao}. However, the following conjecture of Dickson has not been well studied.\n\n\\begin{cjt}[Dickson \\cite{Dickson}]\\label{cjt:Dickson}\n\tLet $k$ be a natural number. Suppose that $a_i$ and $b_i$ are integers with $b_i\\geq1$ and $(a_i,b_i)=1$ for all $i=1,\\ldots,k$. Then, there are infinitely many integers $n$ such that $a_1+b_1n,\\cdots,a_k+b_kn$ are simultaneously primes.\n\\end{cjt}\n\nIf $k=1$, then this is nothing but the Dirichlet prime number theorem. However, all the other cases are unsolved. In $1962$, Bateman and Horn generalized this conjecture to irreducible polynomials.\n\n\\begin{cjt}[Bateman-Horn \\cite{Bateman-Horn}]\\label{cjt:Bateman-Horn}\n\tSuppose that $f_i(x)$ is an irreducible polynomial whose leading coefficient is a positive integer. Let $N$ be a natural number, and let $d_i$ be the degree of $f_i(x)$. Then, the number of $0<n\\leq N$ such that $f_1(n),\\cdots,f_k(n)$ are simultaneous by prime is given by\n\t\\begin{align}\\label{cjt:Bateman-Horn-1}\n\t\t\\int_2^N\\prod_{i=1}^k\\frac{1}{\\log f_i(x)}dx\\cdot\\prod_{p\\in\\P}\\frac{1-w(p)\/p}{\\pas{1-p^{-1}}^k} \\sim \\frac{N}{d_1\\cdots d_k\\log^kN}\\prod_{p\\in\\P}\\frac{1-w(p)\/p}{\\pas{1-p^{-1}}^k}\n\t\\end{align}\n\twhere $w(p)$ is the number of $x\\in\\{0,1,\\cdots, p-1\\}$ which is a solution for $f_1(x)\\times\\cdots\\times f_k(x)\\equiv0\\pmod p$.\n\\end{cjt}\n\nCaldwell applied Conjecture.\\ref{cjt:Bateman-Horn} to various progressions in \\cite{Caldwell}. One of them is the Cunningham chain.\n\n\\begin{dfn}\n\tLet $p$ be a prime number. A sequence\n\t\\[\n\t\\Pas{p,2p+1,\\cdots,\\pas{p+1}2^{l_1(p)-1}-1}\\;\\pas{\\textup{resp.}\\Pas{p,2p-1,\\cdots,\\pas{p-1}2^{l_{-1}(p)-1}+1}}\n\t\\]\n\tis called the first (\\textup{resp.} second) Cunningham chain generated by $p$ if all elements are prime numbers and $(p+1)2^{l_1(p)-1}-1$ (\\textup{resp.} $(p-1)2^{l_{-1}(p)-1}+1$) is composite. As we will see later, $l_1(p)$ and $l_{-1}(p)$ are finite. Also when we discuss the first case and the second case simultaneously, we write $l_1(p)$ and $l_{-1}(p)$ as $l(p)$.\n\\end{dfn}\n\nIf $p=2$, then\n\\begin{align*}\n\t2\\times2+1=5\\in\\P,\\;5\\times2+1=11\\in\\P,\\;11\\times2+1&=23\\in\\P,\\;23\\times2+1=47\\in\\P,\\;47\\times2+1=95\\not\\in\\P,\\\\\n\t2\\times2-1=3\\in\\P,\\;3\\times2-1&=5\\in\\P,\\;5\\times2-1=9\\not\\in\\P.\n\\end{align*}\nThus $l_1(2)=5, l_{-1}(2)=3$. If $p\\not=2$, we have $l(p)\\leq\\textup{ord}\\pas{p;2}$ by Fermat`s little theorem. Here, $\\textup{ord}\\pas{p;2}$ is the order of $2$ modulo $p$. In fact, in the case of the first Cunningham chain, we have\n\\[\n\\pas{p+1}2^{\\textup{ord}\\pas{p;2}}-1\\equiv0\\pmod p,\n\\]\nand hence $(p+1)2^{\\textup{ord}\\pas{p;2}}-1$ is a composite number. The second case is similar. If we assume Artin`s conjecture (\\text{cf.} \\cite{Artin} pp.viii-x), there exist infinitely many primitive roots of $2$. Thus we only have $l(p)\\leq p-1$ in this way. However, if we use the theorem of Euler, we obtain the following statement.\n\n\\begin{rem}\n\t$l(p)<p\/2$ for all prime $p\\geq7$.\n\\end{rem}\n\\begin{proof}\n\tFirst, we consider the case first Cunningham chain. If $p\\equiv3\\pmod4$, then $(p-1)\/2$ is odd. Let $k=\\varphi((p-1)\/2)\\geq2$. Here $\\varphi$ denotes the Euler function. We have\n\t\\[\n\t\\pas{p+1}2^{k-1}\\equiv2^k\\equiv1 \\pmod{\\frac{p-1}{2}}.\n\t\\]\n\tThus $(p+1)2^{k-1}-1$ is a composite number for $p>3$, and we get $l_1(p)<k=\\varphi((p-1)\/2)<p\/2$. If $p\\equiv1\\pmod4$,\n\t\\[\n\t\\pas{p+1}2^{k-1}\\equiv2^{k+1}\\equiv1\\pmod{\\frac{p-3}{2}}\n\t\\]\n\twhere $k=\\varphi((p-3)\/2)-1\\geq2$, and hence $l_1(p)<\\varphi((p-3)\/2)-1<p\/2$. Therefore $l_1(p)<p\/2$ for all $p\\geq7$. Let us next show the second case. If $p\\equiv3\\pmod4$, setting $k=\\varphi((p+3)\/2)-1\\geq2$, we have\n\t\\[\n\t\\pas{p-1}2^{k-1}\\equiv-2^{k+1}\\equiv -1\\pmod{\\frac{p+3}{2}}.\n\t\\]\n\tThus,\n\t\\[\n\tl_{-1}(p)<\\varphi\\pas{\\frac{p+3}{2}}-1\\leq \\frac{p+3}{2}-2<\\frac{p}{2}.\n\t\\]\n\tSimilarly, $(p-1)2^{\\varphi((p+1)\/2)-1}+1$ is composite if $p\\equiv1\\pmod4$.\n\\end{proof}\n\nIf $2p+1$ is also prime, then this prime number is called a safe prime. Its name comes from the fact that it is a prime number useful for Pollard`s $p-1$ algorithm (\\textup{cf.}\\cite{Pollard}) that suggests the vulnerability of the $RSA$ cryptosystem. It is useful if $p-1$ is decomposed into the product of small prime factors. Since $(2p+1)-1$ has $2$ and $p$ as prime factors and $p$ is big, for two safe primes $2p+1, 2q+1$, it is difficult to factorize $(2p+1)(2q+1)$ by the $p-1$ algorithm. However, the Lenstra elliptic-curve factorization in \\cite{Lenstra} can overcome this weakness in the probabilistic sense. Thus the meaning of safe primes as the $RSA$ cryptosystem has faded. In $2004$, whose first draft is in $2002$, Agrawal, Kayal and Saxena found the $AKS$ primality test\\cite{Agrawal-Kayal-Saxena}. In the present paper, they show that if in the case $k=2, f_1(n)=n, f_2(n)=2n+1$ of $(\\ref{cjt:Bateman-Horn-1})$, the Sophie Germain prime density conjecture is true, the order of the complexity can be reduced to $\\tilde{O}(\\log^6 n)$. Here, $F(n)=\\tilde{O}(G(n))$ is defined by $F(n)=O(G(n)h(\\log G(n)))$ for some polynomial $h(x)$. But, in $2005$, Lenstra and Pomerance found a variant of $AKS$ that runs in $\\tilde{O}(\\log^6n)$ without any conjectures \\cite{Lenstra-Pomerance}. Thus, now it is not necessary to use safe primes on the $AKS$ primality test. Safe primes can be effectively used in the Diffie-Hellman key exchange \\cite{Diffie-Hellman}. Since this is the key to the strength of the discrete logarithm problem, we need to consider the question of how long and how well distributed Cunningham chains are.\n\nWe now let $k$ be a natural number. We consider the number $0<n\\leq N$ such that $n,2n+1,\\cdots,(n+1)2^{k-1}-1$ are simultaneously primes. Since\n\\[\nM(n):=n(2n+1)\\cdots\\pas{(n+1)2^{k-1}-1}\\equiv n\\pmod 2,\n\\]\n$n=0$ is the only solution of $M(n)\\equiv0\\pmod2$ for $n\\in\\{0,1\\}$. Thus $w(2)=1$. If $p>2$, then $(n+1)2^{k-1}-1\\pmod p$ has period $\\textup{ord}\\pas{p;2}$ with respect to $k$. Letting $v(p)=\\min\\{k,\\textup{ord}\\pas{p;2}\\}$, this implies that $w(p)$ is equal to the number $w^\\prime(p)$ of solutions of\n\\[\nn\\pas{2n+1}\\cdots \\pas{(n+1)2^{v(p)-1}-1}\\equiv0\\pmod p.\n\\]\nWe observe that each of $n,2n+1,\\cdots,(n+1)2^{v(p)-1}-1$ has one solution in $\\{0,\\cdots,p-1\\}$. That is, $w^\\prime(p)\\geq v(p)$. Assume that there exist $1\\leq a\\leq b\\leq v(p)$ and $n\\in\\{0,\\cdots,p-1\\}$ satisfy\n\\[\n\\pas{n+1}2^{a-1}\\equiv1\\pmod p \\quad\\textup{ and } \\quad \\pas{n+1}2^{b-1}\\equiv1\\pmod p.\n\\]\nThen, since $n+1$ and $p$ are relatively prime, $2^{a-1}\\equiv2^{b-1}\\pmod p$ and it implies $a=b$. We have $w(p)=w^\\prime(p)=v(p)$. The same argument can be applied to the second Cunningham chains. From this argument and Conjecture.\\ref{cjt:Bateman-Horn}, we can expect the following.\n\n\\begin{cjt}[Caldwell \\cite{Caldwell}]\\label{cjt:Caldwell}\n\tFix a natural number $k$. Then\n\t\\[\n\t\\sum_{\\substack{p\\leq N\\\\ l(p)\\geq k}}1\\sim B_k\\int_2^N\\frac{dx}{\\pas{\\log x}\\pas{\\log 2x}\\cdots\\pas{\\log2^{k-1}x}}\\sim B_k\\frac{N}{\\log^kN}\n\t\\]\n\twhere the partial sum on the left hand side runs through all prime numbers $p\\leq N$ satisfying $l(p)\\geq k$. Here,\n\t\\[\n\tB_k:=\\prod_{p\\in\\P}\\frac{1-w(p)\/p}{\\pas{1-p^{-1}}^k}=2^{k-1}\\prod_{p>2}\\frac{1-\\min\\Pas{k,\\textup{ord}\\pas{p;2}}\/p}{\\pas{1-p^{-1}}^k}.\n\t\\]\n\\end{cjt}\n\nIf we examine the size of $B_k$, we can get some upper bound of $l(p)$.\n\n\\begin{prop}\\label{prop:order of Bk}\n\tLet $\\gamma$ be the Euler constant $\\simeq0.57722$. Then for $k\\geq2$,\n\t\\[\n\t\\log\\log k+\\gamma-2+O\\pas{\\frac{1}{\\log k}}\\leq \\frac{\\log B_k}{k}\\leq \\log k+\\gamma+\\log\\log2-1+O\\pas{\\frac{\\log k}{k}}.\n\t\\]\n\\end{prop}\n\\begin{proof}\n\tThe following facts in \\cite{Rosser-Schoenfeld} will be used in the proof without mentioning explicitly:\n\t\\begin{itemize}\n\t\t\\item[$(a)$] $\\displaystyle \\log\\log x+B-\\frac{1}{2\\log^2x}<\\sum_{p\\leq x}\\frac{1}{p}<\\log\\log x +B+\\frac{1}{\\log^2x},$\n\t\t\\item[$(b)$] $\\displaystyle \\prod_{p\\leq x}\\pas{1-\\frac{1}{p}}=O\\pas{\\frac{1}{\\log x}},$\n\t\t\\item[$(c)$] $\\displaystyle \\prod_{p\\leq x}\\pas{1-\\frac{1}{p}}^{-1}=e^\\gamma\\log x+O\\pas{\\frac{1}{\\log x}},$\n\t\t\\item[$(d)$] $\\displaystyle \\sum_{p\\leq x}\\log p<\\pas{1+\\frac{1}{2\\log x}}x,$\n\t\\end{itemize}\n\twhere $x>2$ and $B=\\gamma-\\sum_{p\\in\\P}\\sum_{k=2}^\\infty k^{-1}p^{-k}$. For $k\\geq2$ and a sufficiently large real number $x$, we define\n\t\\[\n\tB_k(x) := 2^{k-1}\\prod_{2<p\\leq x}\\frac{1-\\min\\Pas{k, \\textup{ord}\\pas{p;2}}\/p}{\\pas{1-p^{-1}}^k}.\n\t\\]\n\tFirst, we estimate the lower bound of $B_k(x)$ in $k$. Write\n\t\\begin{align*}\n\t\tB_k(x)=2^{k-1}\\pas{\\prod_{2<p\\leq x}\\pas{1-\\frac{1}{p}}^{-k}}\\pas{\\prod_{2<p\\leq k}\\times\\prod_{k<p\\leq x}}\\pas{1-\\frac{\\min\\Pas{k, \\textup{ord}\\pas{p;2}}}{p}} =:2^{k-1}\\Pi_1\\times \\Pi_2\\times\\Pi_3,\n\t\\end{align*}\n\tsay. Then $\\log\\Pi_1$, $\\log\\Pi_2$ and $\\log\\Pi_3$ are evaluated as\n\t\\begin{align*}\n\t\t\\log\\Pi_1&=k\\log\\pas{\\frac{1}{2}\\prod_{p\\leq x}\\pas{1-\\frac{1}{p}}^{-1}}=k\\log\\pas{\\frac{e^\\gamma}{2}\\log x+O\\pas{\\frac{1}{\\log x}}}\\\\\n\t\t&=k\\log\\frac{e^\\gamma}{2}+k\\log\\log x+O\\pas{\\frac{k}{\\log^2x}},\\\\\n\t\t\\log\\Pi_2&\\geq\\log\\prod_{2<p\\leq k}\\pas{1-\\frac{p-1}{p}}=\\log2-\\sum_{p\\leq k}\\log p=-k-\\frac{k}{\\log k}+\\log2,\\\\\n\t\t\\log\\Pi_3&\\geq\\prod_{k<p\\leq x}\\pas{1-\\frac{k}{p}}=-\\sum_{k<p\\leq x}\\sum_{n=1}^\\infty\\frac{1}{n}\\pas{\\frac{k}{p}^n},\n\t\\end{align*}\n\trespectively. Since\n\t\\begin{align*}\n\t\tk\\sum_{k<p\\leq x}\\frac{1}{p}&\\leq\\pas{\\log\\pas{\\frac{\\log x}{\\log k}}+\\frac{1}{\\log^2x}+\\frac{1}{2\\log^2k}}k\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\t\t\\sum_{k<p\\leq x}\\sum_{n=2}^\\infty\\frac{1}{n}\\pas{\\frac{k}{p}}^n\n\t\t&=\\sum_{n=2}^\\infty\\frac{k^n}{n}\\sum_{k<p\\leq x}\\frac{1}{p^n}<\\sum_{n=2}^\\infty\\frac{k^n}{n}\\sum_{m=k+1}^\\infty\\frac{1}{m^n}\\\\\n\t\t&<\\sum_{n=2}^\\infty\\frac{k^n}{n}\\pas{\\frac{1}{\\pas{k+1}^n}+\\frac{1}{n-1}\\frac{1}{\\pas{k+1}^{n-1}}}\\\\\n\t\t&<k\\sum_{n=2}^\\infty\\frac{1}{n}\\pas{\\frac{1}{n+k}+\\frac{1}{n-1}}\\\\\n\t\t&=k+O\\pas{\\log k},\n\t\\end{align*}\n\twe have\n\t\\begin{align*}\n\t\t\\log\\Pi_3\\geq-k\\pas{\\log\\pas{\\frac{\\log x}{\\log k}}+\\frac{1}{\\log^2x}+\\frac{1}{2\\log^2k}}-k+O\\pas{\\log k}.\n\t\\end{align*}\n\tTherefore\n\t\\begin{align*}\n\t\t\\frac{1}{k}\\log B_k(x)\n\t\t&=\\frac{1}{k}\\pas{\\pas{k-1}\\log2+\\log\\Pi_1+\\log\\Pi_2+\\log\\Pi_3}\\\\\n\t\t&>\\frac{k-1}{k}\\log2++\\log\\frac{e^\\gamma}{2}+\\log\\log x+O\\pas{\\frac{1}{\\log^2x}}+\\frac{\\log2}{k}-1-\\frac{1}{2\\log k}\\\\\n\t\t&\\quad-\\log\\pas{\\frac{\\log x}{\\log k}}-\\frac{1}{\\log^2x}-\\frac{1}{2\\log^2k}-1+O\\pas{\\frac{\\log k}{k}}\\\\\n\t\t&=\\log\\log k+\\gamma-2-\\frac{1}{2\\log k}-\\frac{1}{2\\log^2k}+O\\pas{\\frac{1}{\\log^2x}}+O\\pas{\\frac{\\log k}{k}}\n\t\\end{align*}\n\tand we have\n\t\\begin{align*}\n\t\t\\frac{1}{k}\\log B_k\\geq \\log\\log k+\\gamma-2-\\frac{1}{2\\log k}-\\frac{1}{2\\log^2k}+O\\pas{\\frac{\\log k}{k}}\n\t\\end{align*}\n\tas $x\\to\\infty$. We next consider the upper bound of $B_k(x)$ in $k$. Write\n\t\\begin{align*}\n\t\tB_k(x)\n\t\t&=2^{k-1}\\pas{\\prod_{2<p\\leq x}\\pas{1-\\frac{1}{p}}^{-k}}\\pas{\\prod_{2<p\\leq k}\\times\\prod_{k<p\\leq 2^k}\\times\\prod_{2^k<p\\leq x}}\\pas{1-\\frac{\\min\\Pas{k,\\textup{ord}\\pas{p;2}}}{p}},\\\\\n\t\t&=:2^{k-1}\\Pi_1\\times\\Pi_2\\times\\Pi_3\\times\\Pi_4\n\t\\end{align*}\n\tsay. Then $\\log\\Pi_1, \\log\\Pi_2$ and $\\log\\Pi_4$ are evaluated as\n\t\\begin{align*}\n\t\t\\log\\Pi_1&\n\t\t=k\\log\\frac{e^\\gamma}{2}+k\\log\\log x+O\\pas{\\frac{k}{\\log^2x}},\\\\\n\t\t\\log\\Pi_2&\n\t\t\\leq\\log\\prod_{2<p\\leq k}\\pas{1-\\frac{1}{p}}=O\\pas{\\log\\log k},\\\\\n\t\t\\log\\Pi_4&\n\t\t\\leq\\log\\prod_{2^k<p\\leq x}\\pas{1-\\frac{k}{p}}=-\\sum_{2^k<p\\leq x}\\sum_{n=1}^\\infty\\frac{1}{n}\\pas{\\frac{k}{p}}^n<-k\\sum_{2^k<p\\leq x}\\frac{1}{p}\\\\\n\t\t&<-k\\pas{\\log\\pas{\\frac{\\log x}{k\\log2}}-\\frac{1}{k^2\\log2}-\\frac{1}{2\\log^2x}}\\\\\n\t\t&=k\\log\\pas{\\frac{k\\log2}{\\log x}}+\\frac{k}{2\\log^2x}+O\\pas{\\frac{1}{k}},\n\t\\end{align*}\n\tsince $\\textup{ord}\\pas{p;2}\\geq k$ where $p>2^k$. If $p>2^{j-1}$ for some $j\\in\\N$, then $\\textup{ord}\\pas{p;2}\\geq j$. Thus, putting $y=\\log k\/\\log2$, we obtain\n\t\\begin{align*}\n\t\t\\log\\Pi_3\n\t\t&\\leq\\log\\prod_{y<j\\leq k}\\prod_{2^{j-1}<p\\leq 2^j}\\pas{1-\\frac{j}{p}}\\\\\n\t\t&<\\log\\prod_{y<j\\leq k}\\pas{\\frac{j-1}{j}}^j\\exp\\pas{\\frac{1}{2j\\log^22}+\\frac{j}{\\pas{j-1}^2\\log^22}}\\\\\n\t\t&=\\log\\pas{\\frac{\\floor{y}^{\\floor{y}}}{\\pas{\\floor{y}-1}!}\\cdot\\frac{\\pas{k-1}!}{k^k}}+\\frac{1}{\\log^22}\\sum_{y<j\\leq k}\\pas{\\frac{1}{2j}+\\frac{j}{\\pas{j-1}^2}}.\n\t\\end{align*}\n\tApplying $\\log(n^n\/n!)=n+O\\pas{\\log n}$, we find that the right-hand side of the above is\n\t\\begin{align*}\n\t\t=\\log\\frac{k!}{k^k}+O\\pas{\\log k}=-k+O\\pas{\\log k}.\n\t\\end{align*}\n\tTherefore, we have\n\t\\begin{align*}\n\t\t\\frac{1}{k}\\log B_k(x)\n\t\t&=\\frac{1}{k}\\pas{\\pas{k-1}\\log2+\\log\\Pi_1+\\log\\Pi_2+\\log\\Pi_3+\\log\\Pi_4}\\\\\n\t\t&\\leq\\frac{k-1}{k}\\log2+\\log\\frac{e^\\gamma}{2}+\\log\\log x+O\\pas{\\frac{1}{\\log^2x}}+O\\pas{\\frac{\\log\\log k}{k}}\\\\\n\t\t&\\quad-1+O\\pas{\\frac{\\log k}{k}}+\\log\\pas{\\frac{k\\log2}{\\log x}}+\\frac{1}{2\\log^2x}+O\\pas{\\frac{1}{k^2}}\\\\\n\t\t&=\\log k+\\gamma+\\log\\log2-1+O\\pas{\\frac{1}{\\log^2x}}+O\\pas{\\frac{\\log k}{k}}.\n\t\\end{align*}\n\\end{proof}\n\n\nLet $k(N)$ be the maximum of the length of the Cunningham chain which is generated by $p\\leq N$, that is,\n\\[\nk(N)=\\max\\Pas{l(p)\\colon p\\leq N}.\n\\]\nLet $f(k)=\\frac{1}{k}\\log B_k$. Then we have\n\\[\nB_{k(N)}\\frac{N}{\\pas{\\log N}^{k(N)}}=\\pas{N^{\\frac{1}{k(N)}}\\frac{f(k(N))}{\\log N}}^{k(N)}=:F(N)^{k(N)}.\n\\]\nSuppose that there exists a sequence $\\{N_n\\}_{n=1}^\\infty$ satisfying\n\\[\n\\pas{1-F(N_n)}k(N_n)\\gg1.\n\\]\nThen, there exists a constant $C>0$ which satisfies $F(N_n)\\leq1-C\/k(N_n)$. Thus\n\\begin{align*}\n\t\\limsup_{n\\to\\infty}B_{k(N_n)}\\frac{N_n}{\\pas{\\log N_n}^{k(N_n)}}\n\t&\\leq \\lim_{n\\to\\infty}\\pas{1-\\frac{C}{k(N_n)}}^{k(N_n)}\\\\\n\t&=\\begin{cases}\n\t\te^{-C}\t\t&\\text{if}\t\\quad\t\\lim_{n\\to\\infty}k(N_n)=+\\infty,\\\\\n\t\t\\max_{n\\in\\N}\\pas{1-\\frac{C}{k(N_n)}}^{k(N_n)}\t\t&\\text{if}\t\\quad\t\\lim_{n\\to\\infty}k(N_n)<+\\infty\n\t\\end{cases}\\\\\n\t&<1.\n\\end{align*}\nHowever, unless the order with respect to $k$ of the terms that vanish by approximation of Conjecture.\\ref{cjt:Caldwell} is small, this result contradicts Conjecture.\\ref{cjt:Caldwell} and the maximality of $k(N)$. Therefore we may expect that $\\lim_{N\\to\\infty}(1-F(N))k(N)=0$. In particular, since $\\lim_{N\\to\\infty}F(N)=1$, we have $F(N)<2$ for sufficiently large $N$. Thus\n\\[\n\\frac{\\log N}{k(N)}+\\log f(k(N))-\\log\\log N<\\log2.\n\\]\nFrom Proposition.\\ref{prop:order of Bk} and $l(2)\\geq3$, we have\n\\[\n\\log f(k(N))>\\log\\log k(N)-2>\\log\\log3-2.\n\\]\nTherefore, we obtain\n\\[\nk(N)\\geq\\pas{1+o(1)}\\frac{\\log N}{\\log\\log N}.\n\\]\nFrom this, we may conjecture the following:\n\n\\begin{cjt}\\label{cjt:omega order of Cunningham chains}\n\t\\[\n\tl(p)=\\Omega\\pas{\\frac{\\log p}{\\log\\log p}}\\quad \\textup{ on }\\P.\n\t\\]\n\\end{cjt}\n\nIn \\cite{Augustin} it is reported that, $p_1:=2759832934171386593519$ is the first term of the longest first Cunningham chain in the data up to $2020$. Its length is $17$. And $p_2=42008163485623434922152331$ is the first term of the longest second whose length is $19$. Thus\n\\begin{align*}\n\tl_1(p_1)=17, \\;\\frac{\\log p_1}{\\log\\log p_1}\\simeq12.661,\\\\\n\tl_{-1}(p_2)=19, \\;\\frac{\\log p_2}{\\log\\log p_2}\\simeq14.470.\n\\end{align*}\nIn this way, a better estimation of $B_k$ implies a better bound of the length of Cunningham chains. However, it is still unknown whether $\\limsup_{p\\to\\infty}l(p)\/p=0$ or not.\n\nIn this paper, by using a generalized Fibonacci sequence, we get $l(p)\\ll\\log p$ under a certain condition $(Theorem.\\ref{thm:related with CC})$. It seems that this sufficient condition is plausible by numerical test. The condition can be extended from prime numbers to natural numbers $(Corollary.\\ref{crl:related with CC})$. This implies that the problem of upper estimation of $l(p)$ is reduced to that on natural numbers. One of the benefit of this is that we can use methods of number theory to solve it.\n\n\\textbf{Acknowledgement.} I wish to express my gratitude to Professor Kohji Matsumoto and Dr. Sh\\={o}ta Inoue for their advice towards the present research. I am also thankful to Mr. Yusei Ishida for assisting in writing computer programs. And I thank Dr. Kenta Endo, Dr. Kota Saito, Mr. Yuichiro Toma, Mr. Hideki Hunakura and Mr. Tomohiro Fukada for many useful discussions.\n\n\\section{$\\mathcal{F}_\\alpha$ numbers}\n\\begin{dfn}\n\tLet $\\alpha\\geq1$ be a natural number. A sequence $\\mathcal{F}_\\alpha$:\n\t\\[\n\tF_0=0, \\;F_1=1, \\;F_{n+1}=\\alpha F_n+F_{n-1}\\quad \\pas{n\\in\\Z}\n\t\\]\n\tis called the generalized Fibonacci sequence, and its elements are called $\\mathcal{F}_\\alpha$ numbers.\n\tFor example, enumerating $F_{-5}$ through $F_5$ we have\n\t\\[\n\t\\cdots,\\alpha^4+3\\alpha^2+1,-\\alpha^3-2\\alpha,\\alpha^2+1,-\\alpha,1,0,1,\\alpha,\\alpha^2+1,\\alpha^3+2\\alpha,\\alpha^4+3\\alpha^2+1,\\cdots.\n\t\\]\n\\end{dfn}\n\nThe following facts on $\\mathcal{F}_\\alpha$ numbers are well known. For example, in \\cite{Koshy}, T. Koshy showed the case $\\alpha=1$ of $(a)$ in Section 5.2 pp.88-90, $(b)$ in Section 5 p.82, $(c)$ in Section 5.6 p.103, $(d)$ in Section 20.1 p.397, $(e)$ in Section 10.1 pp.171-173, and $(f)$ in Section 10.1 pp.173-174. We can similarly show the case $\\alpha>1$.\n\n\\begin{fact}\\label{fact:fact of Fa numbers}\n\tLet $m,n$ be integers, and put $\\varphi_\\alpha=(\\alpha+\\sqrt{\\alpha^2+4})\/2$. We have:\n\t\\begin{itemize}\n\t\t\\item[$(a)$] $\\displaystyle F_n=\\frac{1}{\\sqrt{\\alpha^2+4}}\\pas{\\varphi_\\alpha^n-\\pas{-\\varphi_\\alpha}^{-n}},$\n\t\t\\item[$(b)$] $\\displaystyle \\sum_{i=1}^{n}F_i=\\pas{1+\\frac{1}{\\alpha}}F_n+\\frac{F_{n-1}-1}{\\alpha}<\\pas{1+\\frac{2}{\\alpha}}F_n\\quad \\pas{n\\geq1},$\n\t\t\\item[$(c)$] $\\displaystyle F_{-n}=(-1)^{n+1}F_n,$\n\t\t\\item[$(d)$] $\\displaystyle F_{m+n}=F_{m+1}F_n+F_mF_{n-1},$\n\t\t\\item[$(e)$] $\\displaystyle m\\mid n \\iff F_m\\mid F_n,$\n\t\t\\item[$(f)$] $\\displaystyle (F_m,F_n)=F_{(m,n)}.$\n\t\\end{itemize}\n\\end{fact}\n\nIf $\\alpha=1$, then $m$ should not be $2$. Indeed, $F_2=1, F_3=2$ thus $F_2\\mid F_3$, but $2\\nmid3$. We next define the divisor function on $\\mathcal{F}_\\alpha$.\n\n\\begin{dfn}\n\tIn this article, a natural number $d$ is called a $\\mathcal{F}_\\alpha$ divisor of $n$ if $d\\in\\mathcal{F}_\\alpha$ and $d\\mid n$. A map ${}_{\\Falpha}\\sigma:\\N\\to \\C$ is defined by\n\t\\[\n\t\\FDF{n}:= \\sum_{\\substack{d\\mid n \\\\ 0<d\\in\\mathcal{F}_\\alpha}}d\n\t\\]\n\tfor $n\\in\\N$, and called the $\\mathcal{F}_\\alpha$ divisor function. Let ${}_{\\Falpha}\\sigma^1={}_{\\Falpha}\\sigma$, and ${}_{\\Falpha}\\sigma^k={}_{\\Falpha}\\sigma\\circ{}_{\\Falpha}\\sigma^{k-1}$ for $k>1$.\n\\end{dfn}\n\nIn Section.$3$, we will investigate the relationship between the iteration of ${}_{\\Falpha}\\sigma$ and Cunningham chains.\n\n\\begin{ex}\n\tSuppose $\\alpha=3$. Since\n\t\\[\n\t\\mathcal{F}_\\alpha: \\cdots, 0, 1, 3, 10, 33, 109, 360, \\cdots,\n\t\\]\n\twe get\n\t\\begin{align*}\n\t\t\\FDF{2}&=1, \\;\\FDF{3}=4, \\;\\FDF{4}=1, \\\\\n\t\t\\FDFk{3}{109}&=\\FDFk{2}{110}=\\FDF{11}=1.\n\t\\end{align*}\n\\end{ex}\n\nWe consider the Dirichlet series associated with ${}_{\\Falpha}\\sigma$. Put $\\zeta_\\alpha(s)=\\sum_{0<n\\in\\mathcal{F}_\\alpha}n^{-s}$ for $\\alpha\\geq1$. Then $\\zeta_\\alpha(s)$ converges for all $s$ with $\\Re(s)>0$. Suppose $f(n)$ is $n$ if $n\\in\\mathcal{F}_\\alpha$, and is $0$ otherwise. Then\n\\begin{align}\\label{eq:Dirichlet product}\n\t\\zeta(s)\\zeta_\\alpha(s-1)=\\pas{\\sum_{m=1}^\\infty\\frac{1}{m^s}}\\pas{\\sum_{0<n\\in\\mathcal{F}_\\alpha}\\frac{1}{n^{s-1}}}=\\sum_{m,n=1}^\\infty\\frac{f(n)}{(mn)^s}=\\sum_{n=1}^\\infty\\frac{1}{n^s}\\sum_{\\substack{d\\mid n \\\\ 0<d}}f(d)=\\sum_{n=1}^\\infty\\frac{\\FDF{n}}{n^s}\n\\end{align}\nfor $\\Re(s)>1$. As we can see from this expression, the research of ${}_{\\Falpha}\\sigma$ will be useful for the study of $\\zeta_\\alpha$. In particular, $\\zeta_1$ is called the Fibonacci zeta function by Egami\\cite{Egami} and Navas\\cite{Navas}, and it is a meromorphic function on $\\C$. It is the famous unsolved problem whether $\\zeta_1(1)$ is transcendental or not.\n\nHereafter, we suppose that $\\alpha\\geq3$ unless explicitly stated otherwise. Let $\\ind{n}$ be the index of the maximal $\\mathcal{F}_\\alpha$ number $\\leq n$ for $n\\in\\N$, that is, if we take $k$ satisfying $F_k\\leq n<F_{k+1}$, then $\\ind{n}=k$. Since the sequences $\\mathcal{F}_\\alpha$ are increasing for arbitrary $\\alpha\\geq3$, $\\ind{n}$ is determined uniquely for all $n$. In addition, $\\ind{n}$ is not decreasing for $n$.\n\n\\begin{lem}\\label{lem:there exists a k s.t. FDFk(n)=1}\n\tLet $k^\\prime:=\\ind{\\FDF{n}}$. Then $k^\\prime\\leq\\ind{n}$ for all $n$, and if $\\FDF{n}\\neq1$, then\n\t\\[\n\tF_{k^\\prime}<\\FDF{n}<\\pas{1+\\frac{2}{\\alpha}}F_{k^\\prime}.\n\t\\]\n\tIn particular, $F_{k^\\prime}\\nmid\\FDF{n}$.\n\\end{lem}\n\\begin{proof}\n\t$k^\\prime=1\\leq\\ind{n}$ if $\\FDF{n}=1$. Suppose that $\\FDF{n}>1$. Let $i_0$ be the maximal $i$ with $F_i\\mid n$. Then $i_0>1$ and $n$ has at least two $\\mathcal{F}_\\alpha$ divisors $F_{i_0}$ and $F_1$. Thus we estimate that\n\t\\begin{align*}\n\t\tF_{i_0}<F_{i_0}+F_1\\leq\\FDF{n}\\leq \\sum_{i=1}^{i_0}F_i<\\pas{1+\\frac{2}{\\alpha}}F_{i_0}<2F_{i_0}<F_{i_0+1}\n\t\\end{align*}\n\tfrom Fact.\\ref{fact:fact of Fa numbers} $(b)$. This implies that $k^\\prime=i_0=\\ind{F_{i_0}}\\leq\\ind{n}$.\n\\end{proof}\n\nLet $\\FDF{\\N}$ be the set of all $\\FDF{n}$ for $n\\in\\N$.\n\n\\begin{thm}\\label{thm:there exists a k s.t. FDFk(n)=1}\n\tFor every natural number $n$, there exists a non-negative integer $k\\leq\\log n \/\\log\\varphi_\\alpha +2$ such that $\\FDFk{k}{n}=1$.\n\\end{thm}\n\nIn Section.$4$, we will improve this bound to $k\\ll \\log\\log n$.\n\n\\begin{proof}\n\tLet $a$ be an element of $\\FDF{\\N}$, and let $k=\\ind{a}$. If $k=1$, then $\\FDFk{0}{a}=a=1$. In addition, since\n\t\\[\n\tF_2+F_1=\\alpha+1\\not\\equiv0\\pmod\\alpha,\n\t\\]\n\twe get $\\FDF{a}=1$ if $k=2$. Suppose that $k\\geq3$. Since $F_k\\nmid a$ from Lemma.\\ref{lem:there exists a k s.t. FDFk(n)=1}, we have\n\t\\[\n\t\\FDF{a}\\leq \\sum_{i=1}^{k-1}F_i<\\pas{1+\\frac{2}{\\alpha}}F_{k-1}<F_k.\n\t\\]\n\tThus $\\ind{\\FDF{a}}\\leq k-1$. By iterating this argument, we have $\\ind{\\FDFk{k-2}{a}}\\leq2$. Therefore $\\FDFk{k-1}{a}=1$. This also holds for $k=1,2$. Let $k^\\prime=\\ind{\\FDF{n}}$ for arbitrary $n$. We have\n\t\\[\n\t\\FDFk{k^\\prime}{n}=\\FDFk{k^\\prime-1}{\\FDF{n}}=1.\n\t\\]\n\tSince $k^\\prime\\leq\\ind{n}$ from Lemma.\\ref{lem:there exists a k s.t. FDFk(n)=1}, this leads to $\\FDFk{\\ind{n}}{n}=1$. On the other hand, since\n\t\\[\n\t\\varphi_\\alpha^{\\ind{n}}=F_{\\ind{n}}\\sqrt{\\alpha^2+4}+\\pas{-\\varphi_\\alpha}^{-\\ind{n}}\n\t\\]\n\tfrom Fact.\\ref{fact:fact of Fa numbers} $(a)$ and $F_{\\ind{n}}\\leq n$, we obtain that\n\t\\[\n\t\\ind{n}\\leq \\frac{1}{\\log\\varphi_\\alpha}\\pas{\\log n+\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{\\pas{-\\varphi_\\alpha}^{-\\ind{n}}}{n}}}.\n\t\\]\n\tNow, $n$ is a natural number and $\\varphi_\\alpha=(\\alpha+\\sqrt{\\alpha^2+4})\/2$ is bigger than $\\alpha\\geq3$. Thus\n\t\\begin{align*}\n\t\t\\frac{1}{\\log\\varphi_\\alpha}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{\\pas{-\\varphi_\\alpha}^{-\\ind{n}}}{n}}\n\t\t&<\\frac{1}{\\log\\alpha}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\alpha}}\\\\\n\t\t&=\\frac{1}{\\log\\alpha}\\log\\pas{\\sqrt{1+\\frac{4}{\\alpha^2}}+\\frac{1}{\\alpha^2}}+1\\\\\n\t\t&<\\flog{\\alpha}\\log\\pas{1+\\frac{5}{\\alpha^2}}+1<2.\n\t\\end{align*}\n\tThis implies $\\ind{n}<\\log n\/\\log\\varphi_\\alpha+2$.\n\\end{proof}\n\nHere, we define\n\\[\n\\ord{n}=\\min\\Pas{k\\in\\Z_{\\geq0}\\colon {}_{\\Falpha}\\sigma^k(n)=1}.\n\\]\nIn Section.$4$, we will find that a better estimation of $\\ord{n}$ for $n$ implies a better bound of the length of Cunningham chains.\n\n\n\\section{The relation between $\\mathcal{F}_\\alpha$ numbers and the Cunningham chains}\nIn this section, let $m, n$ be natural numbers. From Fact.\\ref{fact:fact of Fa numbers} $(c)$ and $(d)$,\n\\begin{align*}\n\tF_{m+n}+F_{m-n}\n\t&=F_{m+1}F_n+F_mF_{n-1}+F_mF_{-n+1}+F_{m-1}F_{-n}\\\\\n\t&=\\pas{1+(-1)^n}F_mF_{n-1}+\\pas{F_{m+1}+(-1)^{n-1}F_{m-1}}F_n\\\\\n\t&=\\begin{cases}\n\t\tF_n\\pas{F_{m+1}+F_{m-1}} & \\pas{n \\textup{ is odd}},\\\\\n\t\t2F_mF_{n-1}+\\pas{F_{m+1}-F_{m-1}}F_n & \\pas{n \\textup{ is even}}.\n\t\\end{cases}\n\\end{align*}\nSince $2F_{m+1}F_n=F_{m+1}F_n+\\alpha F_mF_n+F_{m-1}F_n$, we have\n\\begin{align}\\label{eq:summation formula-1}\n\tF_{m+n}+F_{m-n}=\n\t\\begin{cases}\n\t\tF_n\\pas{F_{m+1}+F_{m-1}} & \\pas{n \\textup{ is odd}},\\\\\n\t\tF_m\\pas{F_{n-1}+F_{n+1}} & \\pas{n \\textup{ is even}}.\n\t\\end{cases}\n\\end{align}\nNote that $F_{m+1}+F_{m-1}=F_{2m}\/F_m$ from Fact.\\ref{fact:fact of Fa numbers} $(d)$.\n\n\\begin{lem}\\label{lem:divisors of F(m+1)+F(m-1)}\n\tThe $\\mathcal{F}_\\alpha$ divisors of $F_{m+1}+F_{m-1}$ are at most $1$ and $\\alpha$. Moreover, $\\alpha\\mid F_{m+1}+F_{m-1}$ if and only if $m\\equiv1\\pmod2$, that is,\n\t\\begin{align*}\n\t\t\\FDF{F_{m+1}+F_{m-1}}=\n\t\t\\begin{cases}\n\t\t\t1               & (m\\textup{ is even}), \\\\\n\t\t\t\\alpha+1       & (m\\textup{ is odd}).\n\t\t\\end{cases}\n\t\\end{align*}\n\\end{lem}\n\\begin{proof}\n\tIf $\\alpha\\mid2F_m$, then\n\t\\[\n\t3\\leq\\alpha=\\pas{\\alpha,2F_m}\\leq2\\pas{\\alpha,F_m}=2F_{\\pas{2,m}},\n\t\\]\n\tand hence $2\\mid n$. Conversely, $\\alpha\\mid F_m$ if $m$ is even from Fact.\\ref{fact:fact of Fa numbers} $(e)$. Therefore\n\t\\[\n\t\\alpha\\mid 2F_m \\iff m \\equiv0 \\pmod2.\n\t\\]\n\tAlso since $F_{m+1}+F_{m-1}=\\alpha F_m+2F_{m-1}$, we find that\n\t\\[\n\t\\alpha\\mid F_{m+1}+F_{m-1}\\iff \\alpha\\mid2F_{m-1}\\iff m\\equiv1\\pmod 2.\n\t\\]\n\tHere, take a natural number $d$ that satisfies $F_d\\mid F_{m+1}+F_{m-1}(=F_{2m}\/F_m)$. Since $F_d\\mid F_{2m}$, there exists a natural number $q$ such that $2m=qd$. Suppose that $d$ is odd. Then $q$ is even and $F_d\\mid2F_{\\frac{q}{2}d-1}$ since $F_d\\mid F_{\\frac{q}{2}d}$ from Fact.\\ref{fact:fact of Fa numbers} $(e)$. In addition,\n\t\\[\n\t\\pas{F_d,F_{\\frac{q}{2}d-1}}=F_{\\pas{d,\\frac{q}{2}d-1}}=F_1=1.\n\t\\]\n\tThus, we get $F_d\\mid2$. This and $\\alpha\\geq3$ imply $d=1$. Let us next suppose that $d$ is even. Then $F_{\\frac{d}{2}}\\mid F_d$ from Fact.\\ref{fact:fact of Fa numbers} $(e)$, and this implies $F_{\\frac{d}{2}}\\mid2F_{q\\frac{d}{2}-1}$. Similarly, since $(F_{\\frac{d}{2}}, F_{q\\frac{d}{2}-1})=1$, we have $F_{\\frac{d}{2}}\\mid2$, and hence $d=2$. From these results, we conclude that there is no $d\\geq3$ which satisfies $F_d\\mid F_{m+1}+F_{m-1}$.\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:divisors of F(m+1)+F(m-1)}\n\tLet $p$ be a prime number. If $p\\not=m<2p$, then\n\t\\begin{align*}\n\t\t\\FDF{F_{m+p}+F_{m-p}}=\n\t\t\\begin{cases}\n\t\t\t\\begin{cases}\n\t\t\t\t1  & \\pas{m=1}\\\\\n\t\t\t\tF_3+1  & \\pas{m=3}\n\t\t\t\\end{cases}  & \\pas{p=2},\\\\\n\t\t\tF_p+\n\t\t\t\\begin{cases}\n\t\t\t\t1       & (m\\textup{ is even})\\\\\n\t\t\t\t\\alpha+1 & (m\\textup{ is odd})\n\t\t\t\\end{cases}.  & \\pas{p\\neq2}.\n\t\t\\end{cases}\n\t\\end{align*}\n\\end{thm}\n\\begin{proof}\n\tNote that $F_{m+p}+F_{m-p}=F_p\\cdot F_{2m}\/F_m$ from $(\\ref{eq:summation formula-1})$. If $p=2$,\n\t\\begin{align*}\n\t\tF_{m+p}+F_{m-p}=\n\t\t\\begin{cases}\n\t\t\t\\alpha^2+2 \t&\t\t(m=1),\\\\\n\t\t\t\\pas{\\alpha^2+2}F_3.\t&\t(m=3).\n\t\t\\end{cases}\n\t\\end{align*}\n\tSuppose that $p>2$. We have\n\t\\[\n\t\\pas{F_p,\\frac{F_{2m}}{F_m}}\\leq\\pas{F_p,F_{2m}}=F_{\\pas{p,2m}}=F_1=1\n\t\\]\n\tthat is $(F_p, F_{2m}\/F_m)=1$. Here, let $D_\\alpha(n)$ be the set of all $\\mathcal{F}_\\alpha$ divisors of $n$. We find that\n\t\\[\n\tD_\\alpha(F_p)\\cap D_\\alpha\\pas{\\frac{F_{2m}}{F_m}}=\\Pas{1}.\n\t\\]\n\tAnd in general,\n\t\\[\n\tD_\\alpha\\pas{F_p\\frac{F_{2m}}{F_m}}\\supset D_\\alpha(F_p)\\cup D_\\alpha\\pas{\\frac{F_{2m}}{F_m}}.\n\t\\]\n\tWe will show the inclusion relation of the reverse direction. Take an arbitrary $F_s$ in $D_\\alpha(F_p\\cdot F_{2m}\/F_m)$. Then there exists $a$ and $b$ satisfying $a\\mid F_p, b\\mid F_{2m}\/F_m$ and $ab=F_s$. Since $(b,F_p)=1$,\n\t\\[\n\ta=\\pas{a,F_p}=\\pas{a,F_p}\\pas{b,F_p}=\\pas{ab,aF_p,bF_p,F_p^2}=\\pas{ab,\\pas{a,b,F_p}F_p}=\\pas{ab,F_p}=F_{\\pas{s,p}}=1\\textup{ or }F_p.\n\t\\]\n\tIn the case $a=1$, $F_s=b\\in D_\\alpha(F_{2m}\/F_m)$ holds. Thus we have $ab\\in\\FDF{F_{2m}\/F_m}$. Suppose that $a=F_p$. Then $k:=s\/p$ is a natural number from Fact.\\ref{fact:fact of Fa numbers} $(e)$, and $F_{kp}\/F_p\\mid F_{2m}\/F_m$ holds. If $m<p$, then we have $k=1$ since $F_{kp}\/F_p>F_{2m}\/F_m$ for $k\\geq2$. We consider the case $p<m<2p$. Note that\n\t\\[\n\t\\frac{F_{2n}}{F_n}=F_{n+1}+F_{n-1}\\leq F_{n+2}+F_n=\\frac{F_{2(n+1)}}{F_{n+1}}\n\t\\]\n\tfor all $n\\in\\N$. Thus we obtain that\n\t\\begin{align*}\n\t\t\\frac{F_{2m}}{F_m}\\leq \\frac{F_{2(m+1)}}{F_{m+1}}\\leq \\cdots\\leq \\frac{F_{2(2p-1)}}{F_{2p-1}}\n\t\t&=F_{2p}+F_{2p-2}\\\\\n\t\t&<F_{2p}+F_{2p-2}+F_{p-1}\\frac{F_{2p}}{F_p}+F_{2p-3}\\\\\n\t\t&=\\frac{\\pas{F_p+F_{p-1}}F_{2p}+F_p\\pas{F_{2p-2}+F_{2p-3}}}{F_p}\\\\\n\t\t&<\\frac{F_{p+1}F_{2p}+F_pF_{2p-1}}{F_p}\\\\\n\t\t&=\\frac{F_{3p}}{F_p}.\n\t\\end{align*}\n\tThis implies that $k\\leq2$. We suppose $k=2$ and let $r=m-p$. Then $1\\leq r<p$ and\n\t\\begin{align*}\n\t\t\\frac{F_{2m}}{F_m}\n\t\t&=F_{m+1}+F_{m-1}=F_{p+r+1}+F_{p+r-1}=F_{p+1}F_{r+1}+2F_pF_r+F_{p-1}F_{r-1}\\\\\n\t\t&=\\pas{F_{p+1}+F_{p-1}}F_{r+1}+2F_pF_r+F_{p-1}F_{r-1}-F_{p-1}F_{r+1}\\\\\n\t\t&=\\frac{F_{2p}}{F_p}F_{r+1}+F_pF_r+\\pas{\\alpha F_{p-1}+F_{p-2}}F_r+F_{p-1}F_{r-1}-\\pas{\\alpha F_{p-1}F_r+F_{p-1}F_{r-1}}\\\\\n\t\t&=\\frac{F_{2p}}{F_p}F_{r+1}+\\frac{F_{2(p-1)}}{F_{p-1}}F_r.\n\t\\end{align*}\n\tSince $F_{2p}\/F_p\\mid F_{2m}\/F_m$, we have $F_{2p}\/F_p\\mid F_{2(p-1)}\/F_{p-1}\\cdot F_r$. Thus\n\t\\begin{align*}\n\t\t\\frac{F_{2p}}{F_p}\n\t\t&=\\pas{\\frac{F_{2p}}{F_p},\\frac{F_{2(p-1)}}{F_{p-1}}F_r}\\leq\\pas{\\frac{F_{2p}}{F_p},\\frac{F_{2(p-1)}}{F_{p-1}}}\\pas{\\frac{F_{2p}}{F_p},F_r}\\\\\n\t\t&\\leq\\pas{F_{2p},F_{2(p-1)}}\\pas{F_{2p},F_r}=F_{2(p,p-1)}F_{\\pas{2p,r}}\\leq F_2^2=\\alpha^2.\n\t\\end{align*}\n\tThis implies $F_{2p}\\leq \\alpha^2 F_p$ whose right-hand side does not exceed $F_{p+2}$, and hence $p\\leq2$. However, this contradicts $p>3$, and we get $k=1$. In other words, $b=1$ in the case $a=F_p$, and then $ab\\in D_\\alpha(F_p)$. From these result, we have\n\t\\begin{align*}\n\t\tD_\\alpha(F_p)\\cap D_\\alpha\\pas{\\frac{F_{2m}}{F_m}}=\\Pas{1} \\text{ and } D_\\alpha\\pas{F_p\\frac{F_{2m}}{F_m}}=D_\\alpha(F_p)\\cup D_\\alpha\\pas{\\frac{F_{2m}}{F_m}},\n\t\\end{align*}\n\tand hence\n\t\\begin{align*}\n\t\t\\FDF{F_{m+p}+F_{m-p}}\n\t\t=\\sum_{d\\in D_\\alpha\\pas{F_p\\frac{F_{2m}}{F_m}}}d\n\t\t=\\pas{\\sum_{d\\in D_\\alpha\\pas{F_p}}+\\sum_{d\\in D_\\alpha\\pas{\\frac{F_{2m}}{F_m}}}}d-1\n\t\t=\\FDF{F_p}+\\FDF{\\frac{F_{2m}}{F_m}}-1.\n\t\\end{align*}\n\tApply Lemma.\\ref{lem:divisors of F(m+1)+F(m-1)} to complete the proof.\n\\end{proof}\n\nFrom this theorem, we find the following corollary. That is a relationship between the iteration of ${}_{\\Falpha}\\sigma$ and Cunningham chains.\n\n\\begin{crl}\\label{crl:divisors of F(2p+1)+1}\n\tFor every odd prime $p$,\n\t\\[\n\t\\FDF{F_{2p\\pm1}+1}=F_p+1.\n\t\\]\n\tIn particular, if $2p\\pm1$ is also prime, then\n\t\\begin{align}\\label{crl:divisors of F(2p+1)+1-1}\n\t\t\\FDFk{2}{F_{2p\\pm1}}=\\FDF{F_p}.\n\t\\end{align}\n\tFurther, by iterating this argument, we obtain\n\t\\[\n\tl_{\\pm1}(p)-1=\\ord{F_{(p\\pm1)2^{l_{\\pm1}(p)-1}\\mp1}}-\\ord{F_p}.\n\t\\]\n\\end{crl}\n\\begin{proof}\n\tIt follows $m=p\\pm1$ in Theorem.\\ref{thm:divisors of F(m+1)+F(m-1)}. Further\n\t\\[\n\t\\ord{F_{(p\\pm1)2^{l_{\\pm1}(p)-1}\\mp1}}=1+\\ord{F_{(p\\pm1)2^{l_{\\pm1}(p)-2}\\mp1}}=\\cdots=l_{\\pm1}(p)-1+\\ord{F_p}.\n\t\\]\n\\end{proof}\n\nIn Section.$4$, we will show the converse of $(\\ref{crl:divisors of F(2p+1)+1-1})$.\n\n\n\\section{The upper bound of $\\ord{n}$}\nIn this section, our aim is to prove Theorem.\\ref{thm:converse of cor.3.4} and Theorem.\\ref{thm:related with CC}. Those are important results that suggest the relationship between ${}_{\\Falpha}\\sigma$ and Cunningham chains. In the process of the proof, we will use the following theorem which is called the generalized Zeckendorf`s theorem by Hoggatt \\cite{Hoggatt} and Keller \\cite{Keller}.\n\n\\begin{dfn}[Zeckendorf-Hoggatt-Keller]\n\tEvery natural number $n$ has the unique representation:\n\t\\[\n\tn=\\sum_{i=1}^ra_iF_{c_i}\n\t\\]\n\twhere $r$ is a natural number and sequences $\\{a_i\\}_{i=1}^r, \\{c_i\\}_{i=1}^r\\subset\\N$ satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\displaystyle 0<a_i\\leq\\alpha\\;\\pas{i=1,\\ldots,r}, \\;a_1<\\alpha$,\n\t\t\\item[(ii)] $\\displaystyle 1\\leq c_1<\\cdots<c_r$,\n\t\t\\item[(iii)] $\\displaystyle c_{i+1}=c_i+1\\Longrightarrow a_{i+1}<\\alpha \\;\\pas{i=1,\\ldots,r-1}$.\n\t\\end{enumerate}\n\tSuch a sum is called the Zeckendorf representation of $n$.\n\\end{dfn}\n\n\\begin{rem}\n\tBy definition, $\\sum_{\\substack{d\\mid n \\\\ 0<d\\in\\mathcal{F}_\\alpha}}d$ is the Zeckendorf representation of $\\FDF{n}$.\n\\end{rem}\n\n\\begin{lem}\\label{lem:fractional part of Falpha}\n\tFor every $k>i\\geq0$,\n\t\\begin{align}\n\t\tF_{k+i}&\\equiv \\pas{-1}^{i+1}F_{k-i} \\pmod{F_k}, \\label{lem:fractional part of Falpha-1}\\\\\n\t\tF_{2k+i}&\\equiv \\pas{-1}^kF_i \\pmod{F_k}, \\label{lem:fractional part of Falpha-2}\\\\\n\t\tF_{3k+i}&\\equiv \\pas{-1}^{k+i+1}F_{k-i} \\pmod{F_k}, \\label{lem:fractional part of Falpha-3}\\\\\n\t\tF_{4k+i}&\\equiv F_i \\pmod{F_k}. \\label{lem:fractional part of Falpha-4}\n\t\\end{align}\n\tMore generally, for every two non-negative integers $a,b$ with $a\\equiv b\\pmod4$,\n\t\\begin{align}\\label{lem:fractional part of Falpha-5}\n\t\tF_{ak+i}\\equiv F_{bk+i} \\pmod{F_k}.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tFix a natural number $k$. First, we prove $(\\ref{lem:fractional part of Falpha-1})$. From Fact.\\ref{fact:fact of Fa numbers} $(d)$,\n\t\\[\n\tF_{k+i}=F_kF_{i+1}+F_{k-1}F_i\\equiv F_{k-1}F_i \\pmod{F_k}.\n\t\\]\n\tThus $(\\ref{lem:fractional part of Falpha-1})$ holds for $i=0,1$. Suppose that $(\\ref{lem:fractional part of Falpha-1})$ holds for all natural numbers less than $i\\geq2$. The right hand side is\n\t\\begin{align*}\n\t\t\\pas{-1}^{i+1}F_{k-i}=\\pas{-1}^{i+1}\\pas{F_{k-i+2}-\\alpha F_{k-i+1}}=\\pas{-1}^{i-1}F_{k-i+2}+\\alpha \\pas{-1}^iF_{k-i+1}.\n\t\\end{align*}\n\tUsing the assumptions of induction, we have\n\t\\[\n\t\\pas{-1}^{i-1}F_{k-i+2}+\\alpha \\pas{-1}^iF_{k-i+1}\\equiv F_{k+i-2}+\\alpha F_{k+i-1}=F_{k+i} \\pmod{F_k}.\n\t\\]\n\tThe proof of $(\\ref{lem:fractional part of Falpha-2})$ runs as\n\t\\begin{align*}\n\t\tF_{2k+i}=F_{2k}F_{i+1}+F_{2k-1}F_i\\equiv F_{2k-1}F_i\\equiv \\pas{-1}^kF_i \\pmod{F_k}\n\t\\end{align*}\n\tfrom the case of $i=k-1$ of $(\\ref{lem:fractional part of Falpha-1})$. Similarly, $(\\ref{lem:fractional part of Falpha-3})$ and $(\\ref{lem:fractional part of Falpha-4})$ is obtained from\n\t\\begin{align*}\n\t\tF_{3k+i}\n\t\t&=F_{3k}F_{i+1}+F_{3k-1}F_i\\equiv F_{2k+(k-1)}F_i\\equiv \\pas{-1}^kF_{k-1}F_i\\\\\n\t\t&\\equiv\\pas{-1}^k\\pas{F_kF_{i+1}+F_{k-1}F_i}=\\pas{-1}^kF_{k+i}\\equiv \\pas{-1}^{k+i+1}F_{k-i} \\pmod{F_k},\\\\\n\t\tF_{4k+i}\n\t\t&\\equiv F_{4k-1}F_i\\equiv\\pas{-1}^{k+(k-1)+1}F_1F_i=F_i \\pmod{F_k}.\n\t\\end{align*}\n\tNext, we observe that $(\\ref{lem:fractional part of Falpha-5})$ holds for every non-negative $a,b$ with $a\\equiv b\\pmod4$ if and only if for every non-negative $a$,\n\t\\begin{align}\\label{lem:fractional part of Falpha-6}\n\t\tF_{ak+i}\\equiv F_{\\delta k+i} \\pmod{F_k}\n\t\\end{align}\n\tholds where $a\\equiv\\delta\\pmod4$ with $\\delta\\in\\Pas{0,1,2,3}$. We prove $(\\ref{lem:fractional part of Falpha-6})$ by using induction with respect to $a$. The case $a=0,1,2,$ and $3$ are trivial since $a=\\delta$. And $(\\ref{lem:fractional part of Falpha-4})$ is nothing but the case of $a=4$. Suppose that $(\\ref{lem:fractional part of Falpha-6})$ holds also all natural numbers less than $a\\geq5$, and we take $\\delta^\\prime\\in\\{1,2,3,4\\}$ satisfying $\\delta^\\prime\\equiv a\\pmod4$. Then\n\t\\begin{align*}\n\t\tF_{ak+i}=F_{ak}F_{i+1}+F_{ak-1}F_i\\equiv F_{ak-1}F_i=F_{(a-1)k+(k-1)}F_i.\n\t\\end{align*}\n\tUsing the assumption of induction, we have\n\t\\begin{align*}\n\t\t\\equiv F_{(\\delta^\\prime-1)k+(k-1)}F_i=F_{\\delta^\\prime-1}F_i\\equiv F_{\\delta^\\prime k}F_{i+1}+F_{\\delta^\\prime k-1}F_i=F_{\\delta^\\prime k+i} \\pmod{F_k}.\n\t\\end{align*}\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:best estimation of FDF(n)}\n\tSuppose $a\\in\\FDF{\\N}$, and put $k=\\ind{a}$. If $F_i\\mid a$, then $i\\leq(k+1)\/2$, that is\n\t\\[\n\t\\ind{\\FDF{a}}\\leq\\frac{k+1}{2}.\n\t\\]\n\\end{thm}\n\\begin{proof}\n\tLet $i_0$ be the maximal $i$ satisfying $F_i\\mid a$. The case $k=1$ is $i_0=1$ since $a=1$, and hence the claim holds. Assume that $i_0>(k+1)\/2$ for $k\\geq2$. In particular, $i_0\\geq2$. Then $a>1$. Thus, there exist natural numbers $c_1\\ldots,c_n$ such that\n\t\\begin{align*}\n\t\ta=F_{i_0+c_1}+F_{i_0+c_2}+\\cdots+F_{i_0+c_m}+F_{c_{m+1}}+\\cdots+F_{c_n},\\\\\n\t\tk=i_0+c_1>i_0+c_2>\\cdots>i_0+c_m>c_{m+1}>\\cdots>c_n=1.\n\t\\end{align*}\n\tNote that $c_1\\leq i_0-2$. Since $F_{i_0}\\mid a$,\n\t\\begin{align*}\n\t\ta=\\sum_{j=1}^mF_{i_0+c_j}+\\sum_{j=m+1}^nF_{c_j}\\equiv 0 \\pmod{F_k}\n\t\\end{align*}\n\tFrom Lemma.\\ref{lem:fractional part of Falpha} $(\\ref{lem:fractional part of Falpha-1})$,\n\t\\begin{align*}\n\t\t\\sum_{j=1}^mF_{i_0+c_j}\\equiv \\sum_{j=1}^m\\pas{-1}^{c_j+1}F_{i_0-c_j} \\pmod{F_k}\n\t\\end{align*}\n\tIt is enough to consider only the fractional part, and hence we suppose $c_{m+1}<i_0$ without loss of generality. Since $\\alpha\\geq3$ and $i_0\\geq2$, we estimate\n\t\\[\n\t\\abs{\\sum_{j=1}^m\\pas{-1}^{c_j+1}F_{i_0-c_j}+\\sum_{j=m+1}^nF_{c_j}}\\leq2\\sum_{j=1}^{i_0-1}F_j<F_{i_0}.\n\t\\]\n\tTherefore, we have\n\t\\[\n\t\\sum_{j=m+1}^nF_{c_j}=\\sum_{j=1}^m\\pas{-1}^{c_j}F_{i_0-c_j}.\n\t\\]\n\tAssume that there exist some $j\\in\\Pas{1,\\ldots,m}$ such that $c_j$ is odd, and denote all of them by $c_{d_1},\\ldots,c_{d_h}$. Then\n\t\\[\n\t\\sum_{j=m+1}^nF_{c_j}+\\sum_{i=1}^hF_{i_0-c_{d_i}}=\\sum_{\\substack{j=1\\\\j\\neq c_{d_i}(i=1,\\ldots,h)}}^mF_{i_0-c_j}.\n\t\\]\n\tIn both sides, each coefficient of $\\mathcal{F}_\\alpha$ numbers are $\\leq2$. Thus, they are Zeckendorf representations of a natural number. However, the left hand side has $F_{i_0-c_{d_h}}$ and the right-hand side does not. This is a contradiction to the uniqueness of the Zeckendorf representation. Therefore $c_j$ is even for all $j=1,\\ldots,m$, and\n\t\\[\n\t\\sum_{j=m+1}^nF_{c_j}=\\sum_{j=1}^mF_{i_0-c_j}\n\t\\]\n\tholds. This both sides are Zeckendorf representations. But the left hand side has $F_1$ and the right-hand side does not have it since $c_1\\leq i_0-2$. This is also a contradiction to the uniqueness, and hence we obtain that $i_0\\leq(k+1)\/2$.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem:best estimation of FDF(n)}\n\tThe estimate given by Theorem.\\ref{thm:best estimation of FDF(n)} is best-possible. Let $a=F_{2p-1}+1$ with $p,2p-1\\in\\P$. For example, $p=3$. From Theorem.\\ref{thm:best estimation of FDF(n)}, $i\\leq p$ if $F_i\\mid a$. On the other hand, we get $F_p\\mid a$ if $p$ is odd from Corollary.\\ref{crl:divisors of F(2p+1)+1} $(\\ref{crl:divisors of F(2p+1)+1-1})$.\n\\end{rem}\n\nWe can obtain that the converse of Corollary.\\ref{crl:divisors of F(2p+1)+1} from this theorem.\n\n\\begin{thm}\\label{thm:converse of cor.3.4}\n\tFor two odd prime numbers $p,q$, the following are equivalent:\n\t\\begin{enumerate}\n\t\t\\item[\\textup{(i)}] $\\displaystyle p=2q+1\\text{ or }2q-1$,\n\t\t\\item[\\textup{(ii)}] $\\displaystyle \\FDFk{2}{F_p}=\\FDF{F_q} \\text{ for some }\\alpha\\geq3$,\n\t\t\\item[\\textup{(iii)}] $\\displaystyle \\FDFk{2}{F_p}=\\FDF{F_q} \\text{ for all }\\alpha\\geq3$.\n\t\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n\tFrom Corollary.\\ref{crl:divisors of F(2p+1)+1}, $(\\text{i})$ implies $(\\text{iii})$. Also it is clear that $(\\text{ii})$ follows from $(\\text{iii})$. Suppose that $(\\text{ii})$ is true, and let $p=2q^\\prime+1$. Since\n\t\\begin{align*}\n\t\tF_p+1=F_{(q^\\prime+1)+q^\\prime}+F_{(q^\\prime+1)-q^\\prime}=\n\t\t\\begin{cases}\n\t\t\tF_{q^\\prime}\\pas{F_{q^\\prime+2}+F_{q^\\prime}}\t&\t\\pas{q^\\prime\\text{ is odd}},\\\\\n\t\t\tF_{q^\\prime+1}\\pas{F_{q^\\prime+1}+F_{q^\\prime-1}}\t&\t\\pas{q^\\prime\\text{ is even}}\n\t\t\\end{cases}\n\t\\end{align*}\n\tfrom $(\\ref{eq:summation formula-1})$,\n\t\\begin{align*}\n\t\t\\FDFk{2}{F_p}=\\FDF{F_p+1}\\geq\n\t\t\\begin{cases}\n\t\t\t\\FDF{F_{q^\\prime}}\t&\t\\pas{q^\\prime \\text{ is odd}},\\\\\n\t\t\t\\FDF{F_{q^\\prime+1}}\t&\t\\pas{q^\\prime \\text{ is even}}.\n\t\t\\end{cases}\n\t\\end{align*}\n\tThus\n\t\\[\n\tq=\\ind{F_q+1}=\\ind{\\FDF{F_q}}=\\ind{\\FDFk{2}{F_p}}\\geq q^\\prime+\\frac{1+\\pas{-1}^{q^\\prime}}{2}.\n\t\\]\n\tOn the other hand,\n\t\\[\n\t\\ind{\\FDFk{2}{F_p}}=\\ind{\\FDF{F_p+1}}\\leq\\frac{p+1}{2}=q^\\prime+1\n\t\\]\n\tfrom Theorem.\\ref{thm:best estimation of FDF(n)}. If $q^\\prime$ is even, then we have $q=q^\\prime+1$, and hence $p=2q-1$. If $q^\\prime$ is odd, then we get $q^\\prime=q$ since $q^\\prime\\leq q\\leq q^\\prime+1$ and $q$ is odd. Consequently, we obtain that $p=2q+1$.\n\\end{proof}\n\n\\begin{thm}\\label{thm:estimatation of ord(n) with ind(n)}\n\tFor every natural number $n$,\n\t\\[\n\t\\ord{n}\\leq \\frac{1}{\\log2}\\log\\pas{\\ind{n}}+2.\n\t\\]\n\\end{thm}\n\\begin{proof}\n\tLet $a$ be an arbitrary element in $\\FDF{\\N}$, and take a natural number $k$ such that $2^k\\leq\\ind{a}<2^{k+1}$. From Theorem.\\ref{thm:best estimation of FDF(n)}, we estimate that\n\t\\[\n\t\\ind{\\FDFk{k}{a}}\\leq \\frac{1}{2}\\ind{\\FDFk{k-1}{a}}+\\frac{1}{2}\\leq\\cdots\\leq\\pas{\\frac{1}{2}}^k\\ind{a}+\\sum_{m=1}^{k-1}\\frac{1}{2^m}<3.\n\t\\]\n\tThat is $\\ind{\\FDFk{k}{a}}\\leq2$. For $b\\in\\FDF{\\N}$, if $\\ind{b}=2$, then $b$ is $F_2+F_1$. Since $\\alpha$ does not divide them, we get $\\FDF{b}=1$. Therefore $\\ind{\\FDFk{k+1}{a}}=1$. Since $k\\leq \\log(\\ind{a})\/\\log2$, we find that\n\t\\[\n\t\\ord{a}\\leq\\flog{2}\\log\\pas{\\ind{a}}+1.\n\t\\]\n\tFrom this, for every $n$,\n\t\\[\n\t\\ord{n}=\\ord{\\FDF{n}}+1\\leq\\flog{2}\\log\\pas{\\ind{\\FDF{n}}}+2\\leq \\flog{2}\\log\\pas{\\ind{n}}+2.\n\t\\]\n\\end{proof}\n\nLet us remember that $\\ind{n}$ is the maximal index of $\\mathcal{F}_\\alpha$ numbers $\\leq n$. Since $\\mathcal{F}_\\alpha$ is defined by the linear recurrence sequence whose coefficients are $\\geq1$, it diverges exponentically. Therefore, we expect that $\\ord{n}\\ll\\log\\log n$. In fact, we obtain the following more explicit inequalities.\n\n\\begin{thm}\\label{thm:estimation of ord(n)}\n\tLet $\\alpha\\geq3$ be an integer. Then we have\n\t\\begin{align*}\n\t\t\\ord{n}\n\t\t\\begin{cases}\n\t\t\t=0                       & (n=1),\\\\\n\t\t\t<\\frac{1}{\\log2}\\log\\log n+3 & (n\\geq2).\n\t\t\\end{cases}\n\t\\end{align*}\n\tIn particular, if $\\alpha\\geq55$ and\n\t\\[\n\tn>\\exp\\pas{\\frac{4}{\\log\\varphi_\\alpha-4}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\varphi_\\alpha}}},\n\t\\]\n\tthen we get\n\t\\begin{align}\\label{thm:estimation of ord(n)-1}\n\t\t\\ord{n}<\\frac{1}{\\log 2}\\log\\log n.\n\t\\end{align}\n\tMoreover,\n\t\\begin{align*}\n\t\t\\ord{n}\n\t\t\\begin{cases}\n\t\t\t=0 & (n=1),\\\\\n\t\t\t=1 & (2\\leq n\\leq7),\\\\\n\t\t\t<\\frac{1}{\\log2}\\log\\log n & (n\\geq8)\n\t\t\\end{cases}\n\t\\end{align*}\n\tholds at least in the case $\\alpha\\geq2981$.\n\\end{thm}\n\\begin{proof}\n\tWe see that $\\ord{1}=0$ by definition. Hereafter let $n\\geq2$ and $k=\\ind{n}\\geq1$. Since $F_k\\leq n$, we have\n\t\\[\n\tk\\leq\\flog{\\varphi_\\alpha}\\log\\pas{n\\sqrt{\\alpha^2+4}+\\pas{-\\varphi_\\alpha}^{-k}}\n\t\\]\n\tby the argument in the proof of Theorem.\\ref{thm:there exists a k s.t. FDFk(n)=1}. From Theorem.\\ref{thm:estimatation of ord(n) with ind(n)}, we estimate that\n\t\\begin{align}\\label{thm:estimation of ord(n)-2}\n\t\t\\ord{n}\n\t\t&\\leq\\flog{2}\\pas{\\log\\log\\pas{n\\sqrt{\\alpha^2+4}+\\pas{-\\varphi_\\alpha}^{-k}}-\\log\\log\\varphi_\\alpha}+2\\notag\\\\\n\t\t&<\\flog{2}\\log\\log n+\\flog{2}\\pas{\\log\\pas{1+\\flog{n}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\varphi_\\alpha}}}-\\log\\log\\varphi_\\alpha}+2.\n\t\\end{align}\n\tHere, we define\n\t\\begin{align*}\n\t\tf_\\alpha(x)&:=\\log\\pas{1+\\flog{x}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\varphi_\\alpha}}}-\\log\\log\\varphi_\\alpha,\\\\\n\t\tg_\\alpha(x)&:=\\exp\\pas{f_\\alpha(x)}=\\flog{\\varphi_\\alpha}\\pas{1+\\flog{x}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\varphi_\\alpha}}}\n\t\\end{align*}\n\twith $x\\geq2$. For real $y>(\\log\\varphi_\\alpha)^{-1}$, we have\n\t\\[\n\tx>\\exp\\pas{\\frac{1}{y\\log\\varphi_\\alpha-1}\\log\\pas{\\sqrt{\\alpha^2+4}+\\frac{1}{\\varphi_\\alpha}}}.\n\t\\]\n\tDenote by $A_\\alpha(y)$ the right-hand side of this inequality. Then $A_\\alpha(y)$ is decreasing with respect to $\\alpha$. (We will prove this in Remark.\\ref{rem:estimation of ord(n)}.) Since $\\varphi_3=(3+\\sqrt{13})\/2>3.3$, we have\n\t\\[\n\t\\log A_\\alpha(2)\\leq\\log A_3(2)=\\frac{1}{2\\log\\varphi_3-1}\\log\\pas{\\sqrt{13}+\\varphi_3^{-1}}<1\n\t\\]\n\twith $\\alpha\\geq3$, that is $A_\\alpha(2)<3$. Therefore, we have $g_\\alpha(x)<2$ for $\\alpha\\geq3$ and $x\\geq3$. Thus for every $\\alpha\\geq3$,\n\t\\[\n\t\\ord{n}<\\flog{2}\\log\\log n+\\flog{2}\\log g_\\alpha(n)+2<\\flog{2}\\log\\log n+3\n\t\\]\n\twith $n\\geq3$. In addition, this also holds the case $n=2$ since $\\ord{2}=1$ and $\\log\\log2\/\\log2+3\\simeq2.47$.\n\n\tLet us next prove $(\\ref{thm:estimation of ord(n)-1})$. It is enough to consider the domain of $\\alpha\\geq3$ and $a\\geq2$ which satisfies $f_\\alpha(x)\/\\log2+2<0$. Since this condition can be transformed into $g_\\alpha(s)<1\/4$ it is enough to assume the condition $x>A_\\alpha(1\/4)$ for all sufficient large $\\alpha$. Then $\\log\\varphi_\\alpha>4$, that is $\\alpha\\geq55$. Thus for every $\\alpha\\geq55$, we have\n\t\\[\n\t\\ord{n}<\\flog{2}\\log\\log n\n\t\\]\n\twith $x>A_\\alpha(1\/4)$. Since $A_\\alpha(y)$ is decreasing in $\\alpha$, $\\alpha>A_\\alpha(1\/4)$ holds for all sufficiently large $\\alpha\\geq55$. The lower bound is $\\alpha\\geq2981$ from computer calculations, and hence for every $\\alpha\\geq2981$ we get\n\t\\[\n\t\\ord{n}<\\flog{2}\\log\\log n\n\t\\]\n\twith $n\\geq\\alpha$. $\\ord{n}=1$ if $2\\leq n<\\alpha$, and $\\log\\log n\/\\log2>1$ for $n\\geq8$. This implies\n\t\\begin{align*}\n\t\t\\ord{n}\n\t\t\\begin{cases}\n\t\t\t=1  & \\pas{2\\leq n\\leq 7},\\\\\n\t\t\t<\\flog{2}\\log\\log n  & \\pas{n\\geq8}.\n\t\t\\end{cases}\n\t\\end{align*}\n\\end{proof}\n\n\\begin{rem}\\label{rem:estimation of ord(n)}\n\tWe show that $A_\\alpha(y)$ is decreasing monotonically in $\\alpha$. By definition,\n\t\\[\n\t\\log A_\\alpha(y)=\\frac{\\log\\sqrt{\\alpha^2+4}}{y\\log\\varphi_\\alpha-1}+\\frac{1}{y\\log\\varphi_\\alpha-1}\\log\\pas{1+\\frac{1}{\\varphi_\\alpha\\sqrt{\\alpha^2+4}}}=:B(\\alpha)+C(\\alpha)\n\t\\]\n\tsay, with $y\\log\\varphi_\\alpha>1$. It is clear that $C(\\alpha)$ is decreasing, and hence it is sufficient to discuss on $B(\\alpha)$. For real $\\alpha$ with $y\\log\\varphi_\\alpha>1$,\n\t\\begin{align*}\n\t\t\\frac{d}{d\\alpha}B(\\alpha)=\\frac{1}{\\pas{y\\log\\varphi_\\alpha-1}^2}\\pas{\\frac{2y\\alpha}{\\alpha^2+4}\\log\\varphi_\\alpha-\\frac{2\\alpha}{\\alpha^2+4}-\\frac{y\\varphi_\\alpha^\\prime}{\\varphi_\\alpha}\\log\\pas{\\alpha^2+4}}.\n\t\\end{align*}\n\tSince $\\varphi_\\alpha<\\sqrt{\\alpha^2+4}$, we estimate that the right-hand side of the above is\n\t\\begin{align*}\n\t\t&<\\frac{1}{\\pas{y\\log\\varphi_\\alpha-1}^2}\\pas{\\frac{2y\\alpha}{\\alpha^2+4}\\log\\sqrt{\\alpha^2+4}-\\frac{y\\varphi_\\alpha^\\prime}{\\varphi_\\alpha}\\log\\pas{\\alpha^2+4}}\\\\\n\t\t&=\\frac{y\\log\\pas{\\alpha^2+4}}{\\pas{y\\log\\varphi_\\alpha-1}^2}\\pas{\\frac{\\alpha}{\\alpha^2+4}-\\frac{\\varphi_\\alpha^\\prime}{\\varphi_\\alpha}}\\\\\n\t\t&=\\frac{y\\log\\pas{\\alpha^2+4}}{\\pas{y\\log\\varphi_\\alpha-1}^2\\pas{\\alpha^2+4}\\pas{\\alpha+\\sqrt{\\alpha^2+4}}}\\pas{\\alpha\\pas{\\alpha+\\sqrt{\\alpha^2+4}}-\\pas{1+\\frac{\\alpha}{\\sqrt{\\alpha^2+4}}}\\pas{\\alpha^2+4}}\\\\\n\t\t&=-\\frac{4y\\log\\pas{\\alpha^2+4}}{\\pas{y\\log\\varphi_\\alpha-1}^2\\pas{\\alpha^2+4}\\pas{\\alpha+\\sqrt{\\alpha^2+4}}}<0.\n\t\\end{align*}\n\\end{rem}\n\nLet us consider the case $\\alpha=55$.\n\n\\begin{rem}\n\tSince $\\log A_{55}\\pas{1\/4}\\simeq 2091.79$,\n\t\\[\n\t\\textup{ord}_{55}\\pas{n}<\\frac{1}{\\log2}\\log\\log n\n\t\\]\n\tholds at least for $n>e^{2092}$.\n\\end{rem}\n\nIn addition, we obtain the following corollary from Theorem.\\ref{thm:estimation of ord(n)}.\n\n\\begin{crl}\n\tFor every $\\alpha\\geq3$,\n\t\\[\n\t\\limsup_{n\\to\\infty}\\frac{\\ord{n}}{\\log\\log n}\\leq \\frac{1}{\\log2}.\n\t\\]\n\\end{crl}\n\n\\begin{crl}\n\t\\begin{align*}\n\t\t\\lim_{\\alpha\\to\\infty}\\liminf_{n\\to\\infty}\\pas{\\flog{2}\\log\\log n-\\ord{n}}=+\\infty.\n\t\\end{align*}\n\\end{crl}\n\\begin{proof}\n\tFrom $(\\ref{thm:estimation of ord(n)-2})$, we have\n\t\\[\n\t\\liminf_{n\\to\\infty}\\pas{\\flog{2}\\log\\log n-\\ord{n}}\\geq\\flog{2}\\log\\log\\varphi_\\alpha-2.\n\t\\]\n\\end{proof}\n\nFrom Theorem.\\ref{thm:estimatation of ord(n) with ind(n)}, we have $\\ord{F_p}\\leq \\log p\/\\log2 +2$ for prime $p$. In fact, there is quite a big difference between them, which can be observed by numerical tests $(\\textup{FIGURE}\\ref{FIGURE 1},\\textup{FIGURE}\\ref{FIGURE 2})$. Thus, the author believes that the following theorem will be useful in the future.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\columnwidth]{FIGURE1.png}\n\t\\caption{$\\textup{ord}_3\\pas{F_n}$ and $\\log n\/\\log2+2\\;(n\\leq80000)$}\n\t\\label{FIGURE 1}\n\\end{figure}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\columnwidth]{FIGURE2.png}\n\t\\caption{$\\textup{ord}_3\\pas{F_n}\/\\log (n+1)$ and $1\/\\log2\\;(n\\leq80000)$}\n\t\\label{FIGURE 2}\n\\end{figure}\n\n\\newpage\n\n\\begin{thm}\\label{thm:related with CC}\n\tSuppose $\\alpha\\geq3$, and put $C_\\alpha:=\\limsup_{p\\to\\infty}\\frac{\\ord{F_p}}{\\log p}$. If $C_\\alpha<1\/\\log2$ for some $\\alpha$, then\n\t\\[\n\t\\limsup_{p\\to\\infty}\\frac{l(p)}{\\log p}\\leq\\frac{C_\\alpha}{1-C_\\alpha\\log2}.\n\t\\]\n\\end{thm}\n\\begin{proof}\n\tSuppose that $C_\\alpha<1\/\\log2$. For every $0<\\varepsilon<1\/\\log2-C_\\alpha$,\n\t\\begin{align}\\label{thm:related with CC-1}\n\t\t\\ord{F_p}<\\pas{C_\\alpha+\\varepsilon}\\log p\n\t\\end{align}\n\twith sufficiently large $p$. And\n\t\\[\n\t\\log\\pas{\\pas{p\\pm1}2^{l_{\\pm1}(p)-1}\\mp1}=\\pas{l_{\\pm1}(p)-1}\\log2+\\log p+O\\pas{\\frac{1}{p}}.\n\t\\]\n\tWe replace $p$ by $\\pas{p\\pm1}2^{l_{\\pm1}(p)-1}\\mp1$ in $(\\ref{thm:related with CC-1})$. Then\n\t\\[\n\tl(p)-1+\\ord{F_p}<\\pas{C_\\alpha+\\varepsilon}\\pas{\\pas{l_{\\pm1}(p)-1}\\log2+\\log p+O\\pas{\\frac{1}{p}}}\n\t\\]\n\tfrom Corollary.\\ref{crl:divisors of F(2p+1)+1}, and hence we have\n\t\\[\n\tl(p)<1+\\frac{C_\\alpha+\\varepsilon}{1-(C_\\alpha+\\varepsilon)\\log2}\\log p+O\\pas{\\frac{1}{p}}.\n\t\\]\n\tDivide the both sides by $\\log p$, and take limit superior with respect to $p$. Then we find that\n\t\\[\n\t\\limsup_{p\\to\\infty}\\frac{l(p)}{\\log p}\\leq\\frac{C_\\alpha+\\varepsilon}{1-\\pas{C_\\alpha+\\varepsilon}\\log2}.\n\t\\]\n\\end{proof}\n\nThe sufficient condition of Theorem.\\ref{thm:related with CC}, written in terms of prime numbers, can be replaced by the condition written in terms of natural numbers.\n\n\\begin{crl}\\label{crl:related with CC}\n\tSuppose that $\\alpha\\geq3$, and put $D_\\alpha:=\\limsup_{n\\to\\infty}\\frac{\\ord{n}}{\\log\\log n}$. If $D_\\alpha<1\/\\log2$ for some $\\alpha$, then\n\t\\[\n\t\\limsup_{p\\to\\infty}\\frac{l(p)}{\\log p}\\leq\\frac{D_\\alpha}{1-D_\\alpha\\log2}.\n\t\\]\n\\end{crl}\n\\begin{proof}\n\tFor all natural $n$,\n\t\\[\n\t\\log F_n=n\\log\\varphi_\\alpha+\\log\\pas{\\frac{1-\\pas{-\\varphi_\\alpha^2}^{-n}}{\\sqrt{\\alpha^2+4}}}\n\t\\]\n\tfrom Fact.\\ref{fact:fact of Fa numbers} $(a)$. In particular, $\\log\\log F_n\\sim \\log n$. Then, for every $\\varepsilon>0$, we have $\\log p\/\\log\\log F_p>1-\\varepsilon$ with any sufficiently large $p$. Thus we estimate that\n\t\\[\n\tD_\\alpha\\geq\\limsup_{p\\to\\infty}\\frac{\\ord{F_p}}{\\log\\log F_p}=\\limsup_{p\\to\\infty}\\frac{\\ord{F_p}}{\\log p}\\cdot\\frac{\\log p}{\\log\\log F_p}\\geq\\pas{1-\\varepsilon}\\limsup_{p\\to\\infty}\\frac{\\ord{F_p}}{\\log p}.\n\t\\]\n\tNow, since $\\varepsilon>0$ is arbitrary, we get\n\t\\[\n\t\\limsup_{p\\to\\infty}\\frac{\\ord{F_p}}{\\log p}\\leq D_\\alpha.\n\t\\]\n\tHere we suppose $D_\\alpha<1\/\\log2$ and apply Theorem.\\ref{thm:related with CC}, then the proof is complete.\n\\end{proof}\n\nThe advantage of this corollary is that the problem of upper estimation of $l(p)$ is reduced to the situation that we can use number theoretic methods which cannot be applied to prime numbers.\n\n\n\\section{Remaining Problems}\nIn this paper, an experimentally reliable sufficient condition for $l(p)\\ll\\log p$ was obtained using elementary methods that do not involve differentiation and integration. If we could successfully use analytical methods, perhaps we would obtain better estimation. For example, it is describable to find some analogy of $(\\ref{eq:Dirichlet product})$ with respect to ${}_{\\Falpha}\\sigma^2,{}_{\\Falpha}\\sigma^3,\\cdots$, or some non-trivial order of $\\sum\\ord{n}$. However, the difficulty lies in the fact that $\\ord{n}$ is defined by the iterations of ${}_{\\Falpha}\\sigma$. The iterations of the divisor function $\\sigma(n)$ and the Euler function $\\varphi(n)$ have been considered in \\cite{Erdos},\\cite{Erdos-Granville-Pomerance-Spiro},\\cite{Erdos-Subbarao} and so on; however those researches seem to be possible because $\\sigma,\\varphi$ are number theoretically easier to treat. Even though ${}_{\\Falpha}\\sigma$ is not multiplicative, $\\mathcal{F}_\\alpha$ numbers has some nice properties related to multiplication, such as Fact.\\ref{fact:fact of Fa numbers} $(e),(f)$.\nIn the future, it is expected that such properties will be used well to obtain further results on the function $\\textup{ord}_\\alpha$.\n\nFinally, we list the problems to be solved.\n\\begin{problem}\n\t\\[\n\t\\limsup_{n\\to\\infty}\\frac{\\ord{n}}{\\log\\log n}\\overset{?}{<}\\flog{2}\n\t\\]\n\\end{problem}\nIt is shown in Theorem.\\ref{crl:related with CC} that $l(p)\\ll\\log p$ holds if this inequality is true. Here we recall Conjecture.\\ref{cjt:omega order of Cunningham chains}, which is not so far from the above inequality.\n\\begin{problem}\n\tIs there a sequence that is different from $\\mathcal{F}_\\alpha$ with ``similar properties\"?\n\\end{problem}\nThe term ``similar properties\" means those that are related to a chain of prime numbers.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn pseudoscalar meson photoproduction, the reaction is completely\ndescribed by four amplitudes that are functions of hadronic mass $W$\nand center of mass scattering angle $\\theta_{CM}$ (or, equivalently\n$s$ and $t$). If one were able to extract these amplitudes (allowing\nof course for an overall phase) at $\\left\\{ W,\\cos\\theta_{CM} \\right\\} $ or $\\left\\{ s,t\\right\\} $ points,\nthere is nothing else one could measure that would alter how one could\ninterpret the physics of the reaction. \n\nThis observation is especially important in the study of the spectrum\nof baryon resonances. Despite several decades of investigation, it\nis still not clear whether certain states that are predicted by quark\nmodels exist or not. The signatures of any hitherto undiscovered states\nmust be very subtle, to the extent that they are not readily apparent\nfrom cross section measurements alone. If one could unpick the reaction\namplitudes from suitable observables, that would constitute the most\ncomprehensive test for models. In the case of establishing $s$-channel resonances, extraction of the four amplitudes may not even be enough. Partial-wave analyses will be required, and these can lead to finite ambiguities that require additional information to resolve. In any event, a potential new physical effect would have\nto manifest itself clearly, or be declared unproven.\n\nIn order to extract the amplitudes, it is necessary to measure several\npolarization observables. In addition to the cross section, there\nare three single-spin observables %\n\\footnote{This departure from the usual conventions is to avoid confusion between\n$\\Sigma$-particle and $\\Sigma$-beam asymmetry, as well as $P$-photon\npolarization and $P$-recoil.%\n}: $B$ (photon beam asymmetry), $R$ (recoil polarization) and $T$\n(target polarization), which can be labelled as ${\\cal S}$-type measurements.\nThere are also four beam-recoil (${\\cal BR}$-type), four beam-target\n(${\\cal BT}$-type) and four recoil-target (${\\cal RT}$-type) observables.\nAll these observables are bilinear combinations of the four reaction\namplitudes, and are not independent. In principle, therefore, it is\nnot necessary to measure all of them to be able to infer the amplitudes.\nAs we have now entered an era in which single- and double-polarization\nmeasurements can be made, there exists a real opportunity for progress\nin understanding pseudoscalar meson photoproduction reactions, and\nfor potential discovery of new states.\n\nThe problem of finding a minimum set of measurements that allows the\nunambiguous extraction of amplitudes was addressed by Barker, Donnachie\nand Storrow \\cite{Barker1975}). They found that, in addition to the\nsingle polarization set, five more double polarization observables\nwere needed to remove all ambiguities in the quadrants for each relative\nphase angle. More recently, Chiang and Tabakin \\cite{Chiang:1996em}\ncarried out a detailed analysis of the algebra of observables using\nFierz identities, and showed that the selection of just four suitably\nchosen double polarization observables was sufficient to remove the\nambiguities. Such sets have been designated as {}``complete'' sets.\n\nThe Fierz identity analysis led to a large number of identities among observables. Work by Artru et al.~\\cite{Artru:2006xf,Artru20091} extended this by using positivity constraints to derive many \\emph{inequalities}. This means that the measurement of a subset of observables places limits on the possible values of the undetermined observables, so the inequalities  provide useful guides to whether the values of experimental data are physical.\n\nLabelling sets of observables as {}``complete'', implies\nsomehow that one has reached an ultimate state of knowledge. However,\nthe reality is that all experimental measurements of observables carry\nwith them a finite uncertainty, so the concept of completeness is\nnot well defined. One might be tempted to regard this as an experimental\nfailing, but in practice any experiment has to be performed within\nconstraints of time and technological feasibility; the experiment\nwith zero uncertainty can only be accomplished in an infinite time.\nThe alternative is to embrace experimental uncertainty and include\nit in the interpretation of results. \n\n\n\nThe problem of uncertainty due to noise in communication channels\nled Shannon to develop the foundations of information theory \\cite{Shannon1948}.\nIn that seminal work, the concept of entropy was used as a means of\nquantifying an amount of information. One can also apply this to measurements.\nTo introduce the idea with a concrete example, suppose one measured\na quantity $X$ and obtained a measured value $x$ with an uncertainty\n$\\delta_{x}$. The reporting of this measurement would usually be\nin the form $x\\pm\\delta_{x}$, but this is really shorthand for a\nGaussian probability density function (PDF) $p\\left(x\\right)$. The\nentropy is then\\begin{equation}\nH=-\\int p\\left(x\\right)\\log p\\left(x\\right)dx,\\label{eq:entropy}\\end{equation}\nwhich for a Gaussian PDF is $H=\\log\\sqrt{2\\pi e}+\\log\\delta_{x}$.\nIf a more accurate measurement were to be made, resulting in a reduced\nuncertainty $\\delta_{x}^{\\prime}$, the gain in information can be\nquantified as\\[\nI=H-H^{\\prime}=\\log\\left(\\frac{\\delta_{x}}{\\delta_{x}^{\\prime}}\\right).\\]\n\n\nBy extending this idea to the uncertainty in the reaction amplitudes,\nit is possible to quantify how much information is gained following\nthe measurement of one or more observables. This article represents\na preliminary study of information entropy as applied to pseudoscalar\nmeson photoproduction. Section \\ref{sec:Measuring-Information} develops\nthe idea encapsulated by Eq. (\\ref{eq:entropy}) for the reaction\namplitudes, and introduces a means of calculating it. In section \\ref{sec:Results}\nexamples of hypothetical measurements are given, which show how the\nmagnitudes and relative phases of the amplitudes can be determined.\nIn addition to this, section \\ref{sec:Comparison-of-Models} briefly considers\nhow the information content of measured data can be used as a guide to estimating whether the measurement could in principle reduce uncertainty in derived physical quantities.\n\n\n\\section{\\label{sec:Measuring-Information}Measuring Information}\n\n\n\\subsection{Reduced Amplitudes}\n\nA full analysis of reactions will involve measurements over all scattering\nangles and cover the mass range of interest. To develop the concept\nof information content, however, we restrict ourselves to considering\none region (or {}``bin'') in $\\left\\{ W,\\theta_{CM}\\right\\} $ space.\nThe ideas can be straightforwardly extended to include many regions,\nsince entropies are additive. The issue of whether different experiments\n(measuring different observables) have covered the same $\\left\\{ W,\\theta_{CM}\\right\\} $\nspace has been avoided.\n\nThe choice of basis for amplitudes is arbitrary; information content\nis derived from the measured observables, so it cannot depend on the\nchoice. In this work, the transversity basis is used, where the spin\nof the target nucleon and recoiling baryon is projected onto the normal\nto the scattering plane, and the linear polarization of the photon\nis either normal or parallel to the scattering plane. \n\nIt is assumed that differential cross section measurements have been\nperformed to a level of accuracy of, say, a few percent, so that further\nmeasurement would be unlikely significantly to improve knowledge of\nthe amplitudes. The information gain to be studied here is solely\ndue to an increased accuracy in the knowledge of the polarization\nobservables. Since all these observables are asymmetries, no generality\nis lost if we rescale the amplitudes $b_{i}\\rightarrow a_{i}$ such\nthat\\[\na_{i}=\\frac{b_{i}}{\\sqrt{\\left|b_{1}\\right|^{2}+\\left|b_{2}\\right|^{2}+\\left|b_{3}\\right|^{2}+\\left|b_{4}\\right|^{2}}},\\]\nso that the cross section provides an overall scale factor. Applying\nthis rescaling, we have\\begin{equation}\n\\left|a_{1}\\right|^{2}+\\left|a_{2}\\right|^{2}+\\left|a_{3}\\right|^{2}+\\left|a_{4}\\right|^{2}=1.\\label{eq:7-sphere}\\end{equation}\n Since these reduced amplitudes $a_{i}$ are complex, this represents\nthe equation of a unit 7-sphere, i.e. the eight numbers that are the\nreal and imaginary parts are constrained to be on the surface of a\nunit hypersphere in 8 dimensions (a unit 8-ball). \n\nThe definitions of the observables in terms of the reduced amplitudes\nare given in appendix \\ref{sec:Definitions-of-Observables}. One side-effect\nof choosing transversity amplitudes is that measurement of the ${\\cal S}$-type\nobservables leads to the extraction of the magnitudes, leaving just\nthe relative phases to be determined. There is often a tacit assumption\nthat it is easier to perform single-spin asymmetry measurements. For\nthat reason many analyses \\cite{Barker1975,Chiang:1996em} start from\na point where values of the ${\\cal S}$-type observables have been\ndetermined.\n\n\n\\subsection{Entropy}\n\nThe entropy associated with the state of knowledge of the amplitudes\nis an multidimensional extension of Eq. (\\ref{eq:entropy}):\\begin{equation}\nH=-\\int p\\left(\\left\\{ x_{i}\\right\\} \\right)\\log p\\left(\\left\\{ x_{i}\\right\\} \\right)d\\left\\{ x_{i}\\right\\} ,\\label{eq:nd-entropy}\\end{equation}\nwhere $\\left\\{ x_{i}\\right\\} $ represents the values of the real\nand imaginary parts of the amplitudes. Before the measurement of any\npolarization observable, there is no knowledge of $\\left\\{ x_{i}\\right\\} $,\nother than the constraint imposed by Eq. (\\ref{eq:7-sphere}). To\nencode this as a PDF, we can spread the probability uniformly over\nthe surface area of the unit 7-sphere to give\\[\np\\left(\\left\\{ x_{i}\\right\\} \\right)=\\frac{3}{\\pi^{4}},\\]\nwhich results in a pleasingly simple entropy of \\begin{equation}\nH_{7-sphere}=-\\int\\frac{3}{\\pi^{4}}\\log\\left(\\frac{3}{\\pi^{4}}\\right)d\\left\\{ x_{i}\\right\\} =4\\log\\pi-\\log3.\\label{eq:7-sphere-1}\\end{equation}\n\n\nThe act of measurement can be viewed as a compression of this {}``uniform''\nPDF into as small a region of $\\left\\{ x_{i}\\right\\} $ space as possible.\nAs a rough example, consider a set of measurements that results in\na multi-dimensional Gaussian PDF in amplitude space. The entropy of\nan $n$-dimensional Gaussian is \\cite{Shannon1948}\\begin{equation}\nH_{g}=\\frac{n}{2}\\log\\left(2\\pi e\\right)+\\frac{1}{2}\\log\\left(\\left|c_{ij}\\right|\\right),\\label{eq:entropy-nd-gaussian}\\end{equation}\nwhere $\\left|c_{ij}\\right|$ is the determinant of the covariance\nmatrix. While the four complex amplitudes have eight numbers in total,\nrepresenting real and imaginary parts, all observable quantities are\ninvariant to the choice on an overall phase angle, so the effective\nnumber of numbers to extract is seven. In this case, a 7-dimensional\nGaussian is used to estimate information gain. The projection of the\nGaussian onto the 7-sphere will induce off-diagonal correlations in\n$c_{ij}$, but for simplicity we ignore any correlations and take\nthe standard deviation in each of the $\\left\\{ x_{i}\\right\\} $ to\nbe the same ($\\sigma$, say). The resulting approximate expression\nis\\begin{equation}\nH_{measured}=\\frac{7}{2}\\log\\left(2\\pi e\\right)+7\\log\\sigma.\\label{eq:entropy-7D-gaussian}\\end{equation}\n The gain in information is the difference between this and the initial\nuniform PDF over the 7-sphere:\\begin{equation}\nI=H_{7-sphere}-H_{measured}=4\\log\\pi-\\log3-\\frac{7}{2}\\log\\left(2\\pi e\\right)-7\\log\\sigma.\\label{eq:info-gain}\\end{equation}\nA plot of this quantity as function the size of standard deviation\nis shown in Fig. (\\ref{fig:Rough-guide-to}). The choice of logarithm\nbase is arbitrary, but for this work we select it to be 2. This means\nthat the unit of information is the {}``bit'' (i.e. knowing whether\na quantity is 1 or 0). This unit system is convenient for considering\nquantities related to polarization; determining whether an asymmetry\nis positive or negative is equivalent to a gain of one bit of information,\nwhereas determining a phase angle quadrant is a gain of two bits.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\textwidth]{InfoPlot}\\caption{\\label{fig:Rough-guide-to}(Color online) Rough guide to information gain as a function\nof the standard deviation $\\sigma$ in the real and imaginary parts\nof the amplitudes. }\n\n\n\n\\end{figure}\n\n\nFrom Fig. (\\ref{fig:Rough-guide-to}) it can be seen that if one wants\nto have a measured accuracy of the amplitudes to a value $\\sigma=0.05$,\nthe gain in information is roughly 21 bits (see dashed vertical line\non graph). Attempting to achieve much better accuracy than this from\nexperiments is not likely to be practical, so we should therefore\nregard the 21-bit information gain as a target figure to aim for,\nif we want to be able to say that we have extracted amplitudes. Furthermore,\nif two models differ by only a few percent in the values of their\namplitudes, it is not reasonable to expect that comparison with data\nwould ever lead to being able to differentiate them. \n\n\n\\subsection{Numerical calculation of entropies. }\n\nWhile the calculation sketched out above is a useful rough guide,\nwhen an actual set of observables have been measured, Eq. (\\ref{eq:nd-entropy})\nwill need to be evaluated numerically. The number of dimensions in\nthis system indicates the use of Monte Carlo techniques, and a simple\nimplementation of this is as follows.\n\nSample points are generated randomly in amplitude space with uniform\ndensity on the surface of the unit 7-sphere. The number of points\n$N_{0}$ needs to be sufficiently large to minimize Monte Carlo sampling\nuncertainty. For each point, the observables are evaluated according\nto the algebra of table \\ref{tab:Definition-of-observable} in the appendix. The use of random values of amplitudes  was described in \\cite{Artru20091} in order to establish,\nfor combinations of observables, the limits of regions in observable\nspace that are allowed by postivity constraints, and using this a a guide for deriving inequalities. The present work goes further by not only taking into account these positivity constraints, but also estimating the PDFs of the combinations of observables. One can then simulate\nthe process of measuring an observable by weighting all the points\nby another PDF representing the measured observable. \n\nIn practice, the PDF of an asymmetry is likely to be something like\na beta distribution (or a Gaussian approximation thereof). For illustrative\npurposes, however, we can use a simple top-hat function, which for\na single observable is equivalent to reducing the range of values\nfrom $\\left[-1,1\\right]$ to $\\left[r-\\delta,r+\\delta\\right]$, where\n$r$ is the measured result with some uncertainty $\\pm\\delta$. If\nthe uniform probability density on a multi-dimensional surface $S$\nis $p\\left(x_{i}\\right)d\\left\\{ x_{i}\\right\\} =d\\left\\{ x_{i}\\right\\} \/S$.\nThe entropy of a uniform distribution in a volume $S$ is then\\[\nH=-\\int\\frac{1}{S}\\log\\left(\\frac{1}{S}\\right)d\\left\\{ x_{i}\\right\\} =\\log S,\\]\nas illustrated by the value for the 7-sphere in Eq. (\\ref{eq:7-sphere-1}). \n\nIf the surface is reduced by a cut, say from $S_{0}$ to $S_{1}$,\nthe probability density will be uniform in $S_{1}$ and zero otherwise,\nso the gain in information is simply the log of the ratio of the two\nsurface areas:\\[\nI=\\log S_{0}-\\log S_{1}\\]\n\n\nWhen cuts representing the measurement of a combination of observables\nare imposed, the number of remaining points $N_{1}$ is an estimate\nof the remaining volume, so\\[\nI=\\log N_{0}-\\log N_{1}.\\]\n So in order to gain the 21 bits of information, the surface area\nin amplitude space (and hence the number of points) needs to be reduced\nby a factor of $2^{21}\\approx10^{6}$.\n\nThis is best illustrated with a simple example, such as the measurement\nof one polarization observable, recoil polarization, say. Figure \\ref{fig:Distribution-of-values}\nshows in the light shade the distribution of $10^{6}$ points when\nsampling is done uniformly in amplitude space. The dark shaded region\nshows 126045 points selected when a simulated measurement of $R=-0.4\\pm0.1$\nis selected. The result is an information gain of $6\\log_{2}10-\\log_{2}126045=2.988\\pm0.003$\nbits, where the uncertainty is an estimate of the Monte Carlo error.\nSo we can expect that a measurement of one polarization observable\nto an accuracy of $\\pm$10\\% will give us about 3 bits of information.\n\n\\begin{figure}\n\n\n\\includegraphics[width=0.8\\columnwidth]{RecoilPlot}\\caption{\\label{fig:Distribution-of-values}(Color online) Distribution of values of recoil\npolarization from the uniform PDF in amplitude space. Shaded region\nrepresent the possible values remaining after a {}``measurement''. }\n\n\\end{figure}\n\n\nNote that the {}``uncut'' or prior distribution is quadratic in\nshape, not only for recoil polarization, but for all observables.\nThis is a consequence of the observables being bilinear combinations\nof the amplitudes. \n\n\n\\section{\\label{sec:Results}Simulating Combinations of Measurements}\n\n\n\\subsection{Measuring all ${\\cal S}$-type observables}\n\nFor the extraction of amplitudes, it is usually assumed that the ${\\cal S}$-type\nobservables ($B$, $R$ and $T$) have to be measured. Let us examine\nhow much information one gains by making such measurements. \n\nAs shown in \\cite{Artru:2006xf,Artru20091}, the constraints among\nobservables \\begin{equation}\n\\left|T-R\\right|\\leq1-B;\\quad\\left|T+R\\right|\\leq1+B\\label{eq:BRT-constraints}\\end{equation}\ncarve out a tetrahedron inside a cube $\\left[-1,+1\\right]^{3}$ in\n$BRT$-space. To approximate a measurement of $B$, $R$ and $T$,\nwe define a spherical region, of radius $r$, i.e.\\[\n\\left(B-x\\right)^{2}+\\left(R-y\\right)^{2}+\\left(T-z\\right)^{2}\\leq r^{2},\\]\nwhere $(x,y,z)$ are the coordinates of the sphere centre. This spherical\ncut can be moved to various points within the tetrahedron, and the\neffect on the distributions of magnitudes and phases studied. \n\nA typical example is depicted in Fig. \\ref{fig:Bottom-left-panel}.\nThe bottom left panel shows a projection of the $BRT$ distributions,\nwhich highlights the tetrahedral region. Recall that the points in\nthe light shaded region have been initially sampled over amplitude\nspace, so this represents a projection into $BRT$-space, and affirms\nthe constraints defined by Eq. (\\ref{eq:BRT-constraints}). The points\nin the dark sphere are those selected by the choice of cut region.\nThe radius of the spherical cut is 0.1, which is equivalent in information\ngain to a measured accuracy in each observable of better that $\\pm0.05$\n(see later). It is unlikely, when statistical and systematic uncertainties\nare taken into account, that experiments will be able to determine\nobservables to much greater accuracy than this.\n\nIn the example of Fig. \\ref{fig:Bottom-left-panel}, the spherical\ncut is just touching the midpoint of one of the tetrahedron faces.\nThe top row shows the magnitudes of the amplitudes, and it is clear\nthat values for each one can now be estimated. Note, however, that\nthere is much greater uncertainty in $\\left|a_{2}\\right|$ than in\nthe other ones. \n\nThe relative phase angles are displayed in the remaining panels. While\nonly three relative angles are independent, all six possibilities\nare shown. This is because, for situations in which the magnitudes\nof two amplitudes $a_{i}$ and $a_{j}$ are almost equal (as in this\ncase), very small uncertainties in the relative phase of the two amplitudes\nwith respect to a third ($\\theta_{ik}$ and $\\theta_{jk}$) could\nlead to very large uncertainties in their relative phase $\\theta_{ij}$.\nIt is to be expected that there should be no relative phase information\nfor transversity amplitudes if only ${\\cal S}$-type measurements\nare made, and this is apparent from the distributions in Fig. \\ref{fig:Bottom-left-panel}.\nThe observed increase towards $\\theta_{ij}=0^{\\circ}$ is due to the\nfact that the relative angles are formed from the difference of two\nuniform random variables.\n\n\\begin{figure}\n\n\n\\includegraphics[width=0.8\\textwidth]{TetraMidPlane}\\caption{\\label{fig:Bottom-left-panel}(Color online) Light shade - uniform sample of amplitude space; dark shade - region surviving cut. Panel (a) is the projection of BRT tetrahedron, (b)-(e) show the magnitudes of the amplitudes and the other panels are the distributions of relative phase angles (in degrees).}\n\n\n\n\\end{figure}\n\n\nBy examining the variations in the distributions of magnitudes and\nphases for different positions in the $BRT$ tetrahedron, one can\ndeduce some general heuristics governing the relation between what\nwe shall call a $BRT$ measurement and the magnitudes $\\left|a_{i}\\right|$.\nThese are listed in table \\ref{tab:Guide-to-relative}.\n\n\\begin{table}\n\\begin{tabular}{l|l}\nPosition in $BRT$ tetrahedron & Magnitude information\\tabularnewline\n\\hline\nCenter & All magnitudes equal\\tabularnewline\nMid-point of face & One magnitude small, others large and equal\\tabularnewline\nMid-point of edge & Two magnitudes small and equal, other two large and equal\\tabularnewline\nVertex & One magnitude large, others small and equal\\tabularnewline\n\\end{tabular}\n\n\\caption{\\label{tab:Guide-to-relative}Guide to relative size of magnitudes\nfor various positions within the $BRT$ tetrahedron}\n\n\\end{table}\n\n\nReturning to the information gain from a $BRT$ measurement, if we\nassume that the sampled points in amplitude space project into a uniformly\ndense $BRT$ tetrahedron, the entropy before a measurement is\\[\nH_{tetra}=\\log8-\\log3,\\]\ni.e. the volume is a third of the cube $\\left[-1,+1\\right]^{3}$.\nA 3D gaussian, with symmetric widths $\\sigma$ has entropy\\begin{equation}\nH_{3DGaussian}=\\frac{3}{2}\\log\\left(2\\pi e\\right)+3\\log\\sigma,\\label{eq:3D-Gaussian}\\end{equation}\nfrom Eq. (\\ref{eq:entropy-nd-gaussian}). To establish an equivalent\nspherical cut, one can use the entropy of a sphere of radius $r$\n(inside tetrahedron),\\begin{equation}\nH_{sphere}=\\log\\left(\\frac{4}{3}\\pi r^{3}\\right),\\label{eq:sphere}\\end{equation}\nand equate Eq. (\\ref{eq:3D-Gaussian}) and Eq. (\\ref{eq:sphere})\nto establish a relationship between $r$ and $\\sigma$:\\[\n\\log r-\\log\\sigma=\\frac{1}{2}\\log\\left(2\\pi e\\right)-\\frac{1}{3}\\log\\left(\\frac{4}{3}\\pi\\right),\\]\nfrom which we have $r\\approx2.564\\sigma$, so a spherical cut of radius\n0.1 is equivalent to Gaussian errors on $B$, $R$ and $T$ with $\\sigma=0.039$.\nUsing these figures, the predicted information gain is 9.31 bits for\nany position of the spherical cut within the tetrahedron. For the\ncase depicted in Fig. \\ref{fig:Bottom-left-panel}, the estimated\ninformation gain is 9.28 $\\pm$ 0.02. \n\nIt can be readily demonstrated that the estimates of information gain\nare equal to the predicted value of 9.31 (to within sampling errors)\nfor all the classes of position listed in table \\ref{tab:Guide-to-relative},\nhence verifying the assumption that the $BRT$ tetrahedron is uniformly\nsampled. So for real experiments, knowing the \\emph{uncertainties}\nof the measured values of $B$, $R$ and $T$ allows a calculation\nof information gain, irrespective of the \\emph{size} of the measured\nvalues.\n\n\n\\subsection{Towards Extraction of Amplitudes}\n\nCompared to the original guideline of 21 bits, we can see that the\nmeasurement of just ${\\cal S}$-type observables leaves a lot of information\nto be gained. With just under 10 bits, the magnitudes can be determined\nto roughly 10\\% accuracy, but to determine relative phases to better\nthan, say, a 16th ($=2^{-4}$) of the full angular range will require\nan additional 4 bits for each one. Adding this information together\nbrings us to 22 bits, one greater than the original estimate. One\nmight imagine that measurement of an additional four double polarization\nobservables would now be sufficient, given that individual measurements\ncan gain about 3 bits (see section \\ref{sec:Measuring-Information}).\nHowever, the complicated relations among observables now conspire\nagainst this. \n\nChiang and Tabakin \\cite{Chiang:1996em} systematically listed the\npossible combinations of observables that would lead to a {}``complete''\nset; there are a large number of them. They took one example set which\nshowed a counter-example to the claim in \\cite{Barker1975} that completeness\ncould only be attained if five observables are measured, of which\nfour should not be from the same ${\\cal BR}$-, ${\\cal BT}$- or ${\\cal RT}$-set.\nIn that example, $F$, $G$ and $L_{x}$ were taken to be measured,\nand whereas Ref. \\cite{Barker1975} claimed that $E$ and $L_{z}$\nwere needed, Ref. \\cite{Chiang:1996em} asserted that only $T_{x}$\nwas necessary.\n\nUsing the scheme already outlined, we may examine what happens when\nsimulated measurements are made of the same sets of observables. The\n$BRT$ measurements are all assumed to have been made, but to study\nwhether the results for information gain depend on the measure values,\nfour possible cases of position in the measured $BRT$ tetrahedron\nhave been used: center, mid-face, mid-edge and vertex. They give a\nrepresentative sample of all possible cases, and due to the tetrahedral\nsymmetry only one mid-face, mid-edge and vertex needs to be considered.\nFor each of the four cases, $10^{5}$ events were generated within\nthe defined spherical sub-region of the $BRT$ tetrahedron. These\nwere selected by rejection from an intial uniform sample over amplitude\nspace (the 7-sphere). In order to simulate possible measurements of\n$F$, $G$ and $L_{x}$, a $\\pm$0.1 cut on the generated points around\na central value of each observable was imposed. The central values\nare shown for each case in table \\ref{tab:Results-of-simulated}.\nRelatively large values were chosen for clarity of illustration, and\nnote that the same values of $F$, $G$ and $L_{x}$ could not be\nused for each case because of the interdependency of these observables\nwith the chosen $BRT$ values.\n\nFor each set of $BRT$ values, three cases were studied for combinations\nof further measurements: $T_{x}$ only (choice of Ref. \\cite{Chiang:1996em}),\n$E$ and $L_{z}$ (choice of Ref. \\cite{Barker1975}) and $T_{x}$,\n$E$ and $L_{z}$. Again, a $\\pm$0.1 cut on the generated points\naround a central value of each observable is applied. The results\nfor information gain are shown in the penultimate column of table\n\\ref{tab:Results-of-simulated}.\n\n\\begin{table}\n\n\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline \n$BRT$ position & $F$ & $G$ & $L_{x}$ & $E$ & $L_{z}$ & $T_{x}$ & Information (bits) & Ambiguity?\\tabularnewline\n\\hline\n & 0.4 & 0.4 & 0.3 & - & - & 0.7 & 11.4 $\\pm$ 0.2 & Y\\tabularnewline\nCenter & 0.4 & 0.4 & 0.3 & 0.3 & 0.3 & - & 12.7 $\\pm$ 0.3 & N\\tabularnewline\n & 0.4 & 0.4 & 0.3 & 0.3 & 0.3 & 0.7 & 13.2 $\\pm$ 0.3 & N\\tabularnewline\n\\hline \n & 0.4 & -0.4 & 0.4 & - & - & 0.4 & 11.1 $\\pm$ 0.1 & Y\\tabularnewline\nMid-Face & 0.4 & -0.4 & 0.4 & 0.7 & -0.7 & - & 12.0 $\\pm$ 0.2 & N\\tabularnewline\n & 0.4 & -0.4 & 0.4 & 0.7 & -0.7 & 0.4 & 12.7 $\\pm$ 0.3 & N\\tabularnewline\n\\hline \n & 0.4 & 0.4 & 0.4 & - & - & -0.7 & 12.4 $\\pm$ 0.2 & N\\tabularnewline\nMid-Edge & 0.4 & 0.4 & 0.4 & 0.2 & -0.7 & - & 13.6 $\\pm$ 0.4 & N\\tabularnewline\n & 0.4 & 0.4 & 0.4 & 0.2 & -0.7 & -0.7 & 13.6 $\\pm$ 0.4 & N\\tabularnewline\n\\hline \n & 0.4 & 0.4 & 0.4 & - & - & 0.3 & 8.8 $\\pm$ 0.1 & Y\\tabularnewline\nVertex & 0.4 & 0.4 & 0.4 & 0.3 & 0.2 & - & 11.1 $\\pm$ 0.1 & N{*}\\tabularnewline\n & 0.4 & 0.4 & 0.4 & 0.3 & 0.2 & 0.3 & 11.5 $\\pm$ 0.2 & N{*}\\tabularnewline\n\\hline\n\\end{tabular}\\caption{\\label{tab:Results-of-simulated}Results of simulated measurements\nfor different combinations of observables. The values of each observable\nare all defined with a $\\pm$0.1 cut.}\n\n\\end{table}\n\n\nSeveral points are apparent from the results displayed. It is clear\nthat the more measurements that are made, the more information that\nis gained. It is also clear that the information gain is dependent\non the assumed measured $BRT$ values. Recall that the information\ngain obtained when measuring \\emph{only} $BRT$ values was independent\nof position in the $BRT$ tetrahedron; the difference is again due\nto the interdependency among observables. When the information gain\nis greater that 13, the number of points surviving the cuts is 10\nor less, so the estimates are of limited accuracy. \n\nAll the cases of combinations of observables that are listed in table\n\\ref{tab:Results-of-simulated} have previously been proved to result\nin mathematically complete sets. With the introduction of simulated\nexperimental uncertainty, however, this can no longer be taken to\nbe adequate. The last column of the table (headed {}``Ambiguity?'')\nindicates whether there are identifiable, unambiguous values of both\nmagnitudes and relative phases of the amplitudes. The mid-face case,\nwhere $T_{x}$ only has been measured in addition to the common set\nof observables, is illustrated in Fig. \\ref{fig:Example-of-residual}.\nDespite the few surviving points, it is fairly clear that there are\nno three relative phase angles that have a single cluster of points,\nand so an unambiguous extraction of amplitudes would not be possible.\n\nFor the cases listed in table \\ref{tab:Results-of-simulated} with\nN{*} for ambiguity, this indicates that while there is just one identifiable\ncluster of points in the distributions of relative phases, the spread\nin possible points is greater than 10\\% of the full angular range;\ni.e. there may be no quadrant ambiguity, but there remains a considerable\nuncertainty. \n\nIt appears, from this very small sample of possible outcomes, that\nfor measurement of double polarization observables an information\ngain of about 12 bits is required. Combining this number with that\nfrom the measurement of $BRT$ ($\\sim$10 bits), this leads us to\na crude, but very helpful conclusion: only when the total information\ngain from polarization observables is greater than about 21 bits should\nit be possible to extract amplitudes from experimental measurements.\nThis condition will apply irrespective of which particular set of\nobservables have been measured, since information gain is simply a\nmeasure how of much one has compressed the original uniform PDF in\namplitude space. This number is also in line with the crude calculation\ngiven in Eq. (\\ref{eq:info-gain}), where the real and imaginary parts\nof the amplitudes were assumed to be extracted to an accuracy of 0.05.\n\n\\begin{figure}\n\n\n\\includegraphics[width=0.8\\textwidth]{FGLx}\\caption{\\label{fig:Example-of-residual}\n(Color online) Example of residual ambiguity in relative phases after measurement of set $F$, $G$, $L_{x}$ and $T_{x}$. Light shade - uniform sample of amplitude space; Dark shade - region within BRT tetrahedron; Light shade - points surviving cuts in $F$, $G$, $L_{x}$ and $T_{x}$. Labels (a)-(k) are as in Fig. \\ref{fig:Bottom-left-panel}.}\n\n\n\n\\end{figure}\n\n\nThe scheme outlined above uses {}``cuts'' in the space of possible\nobservables to simulate the act of measurement, and the reduced observable\nspace is projected back into amplitude space to calculate the associated\nentropy. This is a crude, but effective, means of relating the observable-space\nPDF to the amplitude-space PDF. To apply the idea of information gain\nto the results of actual experiments, this scheme will have to be\nmodified. When the measurement of a set of observables is made, the\nresult will be an approximately multi-dimensional Gaussian PDF over\nthe range of those observables. A PDF in amplitude space can be constructed\nby sampling uniformly over all amplitude space, calculating the value\nof the observables for each sample point then weighting them with\nthe values of the experimentally determined observable PDF. The resulting\namplitude PDF can be made arbitrarily accurate, depending on the number\nof sample points, but for calculating information gains of 21 bits,\n${\\cal O}\\left(10^{7}\\right)$ points may be needed. \n\nOne final comment related to practical experiments is in order. It\nis clear that for extraction of amplitudes, it is essential to be\nable to polarize photon beams and targets, and to detect recoil polarization.\nGiven that all three components of the reaction require this technological\neffort, the most obvious strategy is to worry less about which combination\nof observables to measure, and more about trying to measure as many\nas possible, with as great an accuracy as possible. The theoretical\nwork in Refs. \\cite{Barker1975,Chiang:1996em} is, however, still\na useful guide to selecting the combinations of observables that will\nmost efficiently lead to an information gain of 21 bits. The information\nmeasure (\\ref{eq:nd-entropy}) can be used in the design of experiments\nto provide an estimate of the degree of accuracy (and hence the integrated\nluminosity) required for amplitude extraction. \n\n\n\\section{\\label{sec:Comparison-of-Models}Comparison of Models with Data}\n\nHaving established how to estimate the quantity of information contained\nin measured data, can the measured data be used to extract information\nabout the parameters of an individual model?\n\n\nAn individual model will depend on some input parameters, $\\xi$,\nsay (e.g. coupling constants). Quite often, the comparison of model\ncalculations to data is used to extract {}``best fit'' values for\nthe input parameters, $\\xi^{\\star}$. We can use the information gain\nfor measured data to tell whether a fit to the new data will yield\nan improved knowledge of input parameters, compared to prior information.\nPrior to a fit procedure, knowledge about the possible values $\\xi$\nwill be encoded in a PDF $p\\left(\\xi,M\\right)$, where $M$ is there\nas a reminder that this quantity depends on a model. The amplitude\nPDF of the model, given a specified set of input parameters is $p\\left(x_{i}\\mid\\xi,M\\right)$,\nwhere as before $x_{i}$ represents the real and imaginary parts of\nthe amplitudes. The total prior PDF of the model is an integral over\nthe range of input parameters\\[\np\\left(x_{i}\\mid M\\right)=\\int p\\left(x_{i}\\mid\\xi,M\\right)p\\left(\\xi,M\\right)d\\xi,\\]\nso a model entropy $H_{model}$ can be evaluated by plugging the model\nprior PDF into Eq. (\\ref{eq:nd-entropy}). Then only if $H_{measured}\\ll H_{model}$\nis there likely to be a significant improvement in the knowledge of\nthe input parameters when the model is fitted to the data.\n\n\n\\section{Conclusions}\n\nIn this article a measure of information content, based on Shannon\nentropy, was introduced and has been applied to measurement of polarization\nobservables in pseudo-scalar meson photoproduction. Using the uncertainties\nin the measurements, the information entropy of the four amplitudes\ncan be calculated. It is assumed that a suitably accurate determination\nof the cross-section has been made, which gives an overall scale factor\nto the amplitudes. \n\nAn important finding is that, when allowing for a realistic but small\nmeasurement uncertainty, measuring only a mathematically complete\nset of observables is not enough to guarantee the extraction of amplitudes.\nInstead, a rule of thumb, based on quantifying information gain of\nabout 21 bits for each point in $\\left\\{ W,\\theta_{CM}\\right\\} $,\nis likely to be a more robust guide.\n\nAn extension of the work presented here is likely to be applicable\nto other reactions in which information content could determine whether\nmeasurements will be adequate to extract physically meaningful results.\nExamples such as the extraction of generalized parton distributions\nfrom DVCS-like asymmetries, or inferring the details of the nucleon-nucleon\ninteraction from the database of scattering observables may be fruitful\nareas of investigation.\n\\begin{acknowledgments}\nThis work was supported by the United Kingdom's Science and Technology\nFacilities Council.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}  \nWe study varieties of complete flags in nilpotent representations associated to an oriented cycle. In this situation the filtrations by radicals and socles play a special role (for an indecomposable module they are the functorial filtrations for a string module). Restriction to the one-dimensional subspace and relative position with respect to these flags gives us (recursively) a cell decomposition into affine cells. This method goes back to a work of Spaltenstein \\cite{Sp}, compare section 3.  \nFor the loop (or Jordan) quiver with the zero endomorphism we get the Schubert cell decomposition for $\\mathbf{Gl}_n$ and for more general representations we reobtain the cell decomposition for Springer fibres in type $A$ studied by L. Fresse  in \\cite{Fr}.  \nAn efficient way to parametrize the cells is book-keeping of the relative position in terms of multi-tableaux (see section 4) and this gives a combinatorial tool to describe the Betti numbers of these quiver flag varieties. \n\nWe also observe that the torus action which is given by scalar multiplication on each indecomposable summand operates on the quiver flag variety with finitely many fixed points and one-dimensional orbits. In fact, every cell is a limit cell of a unique torus fix point in it and the cell decomposition is an instance of a Bialynicki-Birula decomposition (of a non-smooth variety, see e.g. \\cite{Car}). The theory developped by Goresky, Kottwitz and MacPherson (see \\cite{GKM}) gives a description of the torus equivariant cohomology of these varieties. \n\nAs an application, the Borel-Moore homology groups appear as modules for quiver Hecke algebras of nilpotent representations of the oriented cycle, \nsee \\cite{Ka3} for the definition in a more restrictive situation and a general study of their properties (Kato's situation does not apply because it requires all multiplicites $L_{\\lambda}$ to be non-zero).\nQuiver Hecke algebras are known to be graded Morita equivalent to (positively graded) standardly stratified algebras in the sense of Mazorchuk \\cite{Ma} and under this equivalence Kato's standard modules correspond to (Mazorchuk's) two types of standard modules. \n\n\n \n\n\\section{Nilpotent representations of the oriented cycle and quiver flag varieties}\n\nLet $K$ be an algebraically closed field. Let $A=KQ$ where $Q$ is the oriented cycle with vertices $Q_0=\\{1,\\ldots , n\\}$ identified with their residue classes in $\\mathbb{Z}\/n\\mathbb{Z}$ and  $Q_1=\\{a_i\\colon i\\to i+1 \\mid i\\in \\mathbb{Z}\/n\\mathbb{Z}\\}$. An $A$-module $M$ is given by vector spaces $V_1, \\ldots , V_n$ and linear maps $f_i\\colon V_i\\to V_{i+1}$. Its $Q_0$-graded dimension is given by $\\Dim M=(\\dim V_1, \\ldots , \\dim V_n)$.  We will only look at finite-dimensional nilpotent representations, that means $\\dim V_i<\\infty $ and \n$f_{i+n-1} \\circ \\cdots \\circ f_{i+1}\\circ f_i\\colon V_i\\to V_i$ is nilpotent for every $i\\in Q_0$.  \n\n\\paragraph{Indecomposable nilpotent $A$-modules} For $1\\leq i\\leq n$ we write $S_i=E_i[1]$ for the simple left $A$-module supported in the vertex $i$. For $j\\in Q_0, \\ell\\in \\mathbb{N} $ we write $E_{j}[\\ell]$ for the unique indecomposable $A$-module with socle $S_j$ and $K$-vector space dimension $\\ell $.\nWe have  $\\soc E_{j}[\\ell ] = S_j\\subset E_{j}[\\ell ]$ the inculsion of the socle, $\\rad E_{j}[\\ell ]= E_{j}[\\ell-1] \\subset E_{j}[\\ell ]$ (with $E_j[0]:=0$) the inclusion of the radical and $E_j[\\ell] \\surj \\Top E_j[\\ell] = S_{j-\\ell +1}$ the quotient map to the top. \n\n\n\\begin{defi}\nLet $M$ be a nilpotent $A$-module of dimension $\\Dim M =:\\underline{d} \\in \\mathbb{N}_0^{Q_0}$ and \n$\\underline{\\mathbf{d}} := (0=\\underline{d}^0 , \\underline{d}^1 ,\\ldots , \\underline{d}^{r}=\\underline{d})$ with $\\underline{d}^k \\in \\mathbb{N}_0^{Q_0}$ be a sequence  We define \n\\[\n{\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} := \\{ U=(U^0\\subset \\cdots \\subset U^{r}=M) \\mid U^k \\; A \n\\text{-submodule}, \\; \\Dim U^k =\\underline{d}^k \\}\n\\]\n\nThis defines a projective $K$-variety, we call it the \\textbf{quiver flag variety} for $(M, \\underline{\\mathbf{d}} )$. We will always assume that the flags are complete i.e. $\\left| \\underline{d}^{t+1}\\right|- \\left|\\underline{d}^t \\right| =1 , \\; 1\\leq t \\leq r -1$ (where $\\left| v\\right| =\\sum_{i=1}^nv_i$ for $v\\in \\mathbb{N}_0^{Q_0}$), with one exception: If $\\underline{\\mathbf{d}}=(0, \\underline{e} , \\underline{d})$ we denote the quiver flag variety by ${\\rm Gr}\\binom{M}{\\ee}$. \n\\end{defi} \n\n\nFor $d\\in \\mathbb{N}$ we denote $\\mathrm{Fl} (d)$ the variety of complete flags in $K^d$ and $\\mathrm{Fl}(0):=pt$.   \n\\paragraph{Relative position}\nLet $\\mathrm{Fl} (\\underline{\\mathbf{d}} ):=\\prod_{i\\in Q_0} \\mathrm{Fl} (\\underline{d}_i), \\mathrm{Fl} (\\underline{\\mathbf{e}} ) :=\\prod_{i\\in Q_0} \\mathrm{Fl} (\\underline{e}_i )$ with $\\underline{d}_i=(0=d^0_i \\leq d_i^1 \\leq \\cdots \\leq d_i^{r}),  \\underline{e}_i=(0=e^0_i \\leq e_i^1 \\leq \\cdots \\leq e_i^{\\mu})$, $d_i^{r}=e_i^{\\mu}$ for all $i\\in Q_0$. The relative position map  \n$\\rm rp \\colon \\mathrm{Fl}(\\underline{\\mathbf{d}} )\\times \\mathrm{Fl} (\\underline{\\mathbf{e}} ) \\to \\prod_{i\\in Q_0} \\Mat \\bigl((r +1)\\times (\\mu +1) , \\mathbb{N}_0 \\bigr)$ is defined as  \n$\n\\rm rp (U_i^{\\bullet}, W_j^{\\bullet})_{i,j\\in Q_0} := \\bigl((\\dim U_i^k \\cap W_i^\\ell)_{k,\\ell}\\bigr)_{i\\in Q_0}. \n$ \nNow fix $W\\in \\mathrm{Fl} (\\underline{\\mathbf{e}} )$ and given $w\\in \\prod_{i\\in Q_0} \\Mat \\bigl((r +1)\\times (\\mu +1) , \\mathbb{N}_0 \\bigr)$ we set \n\\[ \\mathrm{Fl} (\\underline{\\mathbf{d}} )_{w} :=\\{ U\\in \\mathrm{Fl} (\\underline{\\mathbf{d}} )\\mid \\rm rp (U,W)=w\\}. \\]\nFor our fixed representation $M$ we will assume from now on $V_i=K^{d_i}$, $i\\in Q_0$ and for the quiver flag variety from before, we set \n\\[ \n{\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}_{w} := {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\cap \\mathrm{Fl} (\\underline{\\mathbf{d}} )_{w} \n\\]\nThis gives the \\textbf{stratification by relative position (with the fixed flag $W$) }in ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$. \n\n\n\\section{Spaltenstein's fibration and the cell decomposition}\n\nLet $M$ be an $A$-module.\nIf $L$ is $1$-dimensional $Q_0$-graded subvector space of $M$, the inclusion \n$j\\colon L\\inj M$ is $A$-linear if and only if $L\\subset \\soc (M)$. Thus for a dimension vector $\\underline{e}$ with $\\underline{e} =e_i$ for some $i\\in Q_0$,  we have an isomorphism ${\\rm Gr} \\binom{M}{\\underline{e}} \\cong {\\rm Gr} \\binom{s_i  }{1 }=\\mathbb{P}^{s_i -1}$ where $\\underline{s}:= \\Dim \\soc (M)$. \n\nWe need the following preparation. \nLet $i\\in Q_0$ and denote by $M_{(i)}$ the maximal subrepresentation of a representation $M$ such that $\\soc (M_{(i)})$ is a direct sum of copies of $S_i$. We get $M=\\bigoplus_{i\\in Q_0}M_{(i)}$ and we can see $\\Aut (M_{(i)})\\subset \\Aut (M)$ as a subgroup.  \nWe denote by $F=F_M$ the underlying $Q_0$-graded flag of the following flag of submodules\n$ 0\\subset \\soc (M) \\cap \\rad^m (M) \\subset \\cdots \\subset \\soc (M)\\cap \\rad^2 (M) \\subset  \\soc (M) \\cap \\rad (M) \\subset \\soc (M) $. \n\nNow fix $i\\in Q_0$ and let $s_i:= \\dim \\soc(M_{(i)})$. We can consider $F_i$ as a flag in the vector space $\\soc M_{(i)}$. We denote by  $P=P_i \\subset \\mathbf{Gl}_{s_i}$ the stabilizer of this flag.  \nThe restriction to the socle gives a group homomorphism \n\\[ \n\\varphi\\colon \\Aut(M_{(i)} ) \\to \\mathbf{Gl}_{s_i}.\n\\]\nWe fix a vector space basis for $M_{(i)}$ which is compatible with the Krull-Schmidt decomposition and a complete flag refining $F_i$ such that its stabilizer $B\\subset P$ is a lower triangular standard Borel.  \n\n\n\\begin{lemma} The image of $\\varphi $ is $P$ and \nthere is a group morphism $\\theta \\colon P\\to \\Aut (M_{(i)}) $ such that $\\varphi\\circ \\theta =\\id_P$. \n\\end{lemma}\n\n\\begin{proof} Clearly the image is contained in $P$ since any $A$-linear map maps radicals to radicals. We look at the $K$-algebra homomorphism $\\Phi$ given by taking socle \nwhose restriction to the units gives the map   \n$\\varphi$. \n\\[ \\Phi \\colon \\End_A (M_{(i)}) \\to \\mathfrak{p}:=\\{ f\\in \\End_K (\\soc M_{(i)})\\mid f (F_i^k)\\subset F_i^k, k\\leq m\\} .\\]\nIf we fix a vector space basis for $M$ compatible with the direct sum decomposition, we can define a $K$-algebra homomorphism $\\Theta \\colon \\mathfrak{p}\\to \\End_A(M_{(i)})$ such that \n$\\Phi\\circ \\Theta =\\id_{\\mathfrak{p}}$ as follows. \nAny coordinate $(s,t)$ in $\\End_K(\\soc M_{(i)})$ corresponds (by our choice of a  basis) to a pair of socles of direct summands $E_i[\\ell]$ and $E_i[k]$ of $M_{(i)}$. If $k<\\ell$, then there is no nonzero $A$-linear map $E_i[\\ell]\\to E_i[k]$, we set $\\theta_{s,t} =0$. If $k\\geq \\ell $ we fix the (unique) inclusion $\\theta_{s,t}\\colon E_i[\\ell] \\subset E_i[k]$ of the submodule. Then, $\\Theta ((a_{s,t})) := \\sum_{s,t} a_{s,t}\\theta_{s,t}$ defines the desired map. \n\\end{proof}\nWe call two points $U^\\bullet, W^\\bullet \\in {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ equivalent if \nthe quotients of $M\/U^k$ and $M\/V^k$ are isomorphic $A$-modules for $1\\leq k\\leq r$. There are only finitely many equivalence classes, we denote by $sp \\colon {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\to \\mathcal{S}, U^\\bullet \\mapsto ([M\/U^k])_{1\\leq k\\leq r}$ the map to the equivalence class.  \n\nNow we can prove the following analogue of a result of Spaltenstein \\cite[Lemma, p. 453]{Sp}.\n\n\\begin{theorem}[Spaltenstein's fibration] Let $Q$ be the oriented cycle with $n$ vertices, and $\\underline{\\mathbf{d}}=(\\underline{d}^0,\\underline{d}^1,\\ldots , \\underline{d}^{r}=\\underline{d})$ a complete dimension filtration. Let $M$ be a $\\underline{d}$-dimensional nilpotent $A$-module, for $w\\in \\prod_{i\\in Q_0} \\Mat (1 \\times (s_i +1), \\mathbb{N}_0)$ we write $()_w$ for the relative position with respect to a complete flag refining $\\soc(M) \\cap \\rad^{\\bullet}(M)$. \nThen, there is an isomorphism of algebraic varieties \n\\[ \n f\\colon p^{-1} ({\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w}) \\to\n{\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w} \\times {\\rm Fl}\\binom{N}{\\underline{\\mathbf{e}}} \n\\]  \nwith $\\underline{\\mathbf{e}}:= (\\underline{d}^1-\\underline{d}^1,\\underline{d}^2-\\underline{d}^1,\\ldots , \\underline{d}^{r }-\\underline{d}^1)$, $p\\colon {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\to {{\\rm Gr} \\binom{M}{\\underline{d}^{1}}}$ is the map forgetting all but the first subspace, and $N = M\/U_0$ with $(U_0\\subset M) \\in  {{\\rm Gr} \\binom{M}{\\underline{d}^{1}}}_w $ arbitrary, such that the following \ndiagram is commutative \n\\[ \n\\xymatrix{ &{\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w} & \\\\ \np^{-1} ({\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w})\\ar[ur]^{p} \\ar[rr]^f \\ar[dr]_{sp} &&{\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w} \\times {\\rm Fl}\\binom{N}{\\underline{\\mathbf{e}}}  \\ar[ul]_{pr_1}\\ar[dl]^{([N], sp)\\circ pr_2} \\\\ &\\mathcal{S} .&\n}\n\\]\n\\end{theorem}\n  \n\\begin{proof}\nLet $L\\in {\\rm Gr} \\binom{M}{\\underline{d}^{1}}_w=\\{[0:\\cdots :0:1:x_1:\\ldots :x_s]\\in \\mathbb{P}^{s_j -1}\\mid x_i \\in K, 1\\leq i\\leq s\\}$. Observe, that by definition $B$ operates transitive on ${\\rm Gr} \\binom{M}{\\underline{d}^{1}}_w$ since it is a $B$-orbit. \nUsing the previous lemma we can find an algebraic map $\\phi \\colon  \n{\\rm Gr} \\binom{M}{\\underline{d}^{1}}_w \\to \\Aut (M), \\; L\\mapsto \\phi_L$ with $\\phi_L (L)=U_0$. More precisely, if $L$ corresponds to the column $(0,\\ldots ,0, 1, x_1, \\ldots , x_s)^t$ and $U_0$ to $(0,\\ldots ,0 ,1, 0,\\ldots ,0)^t$, then we \ntake the image of \n\\[\n\\begin{tikzpicture}[baseline=(current bounding box.center)]\n\\matrix (m) [matrix of math nodes,nodes in empty cells,right delimiter={]},left delimiter={[} ]{\n1  &    &      &  &  &   \\\\\n   &    &      &  &  &   \\\\\n   &    &      &  &  &   \\\\\n   &    & 1    &  &  &   \\\\\n   &    & -x_1 &  &  &   \\\\\n   &    &      &  &  &   \\\\\n   &    & -x_s &  &  & 1 \\\\ \n} ;\n\\draw[dotted] (m-1-1)-- (m-4-3);\n\\draw[dotted] (m-4-3)-- (m-7-6);\n\\draw[dotted] (m-5-3)-- (m-7-3);\n\\end{tikzpicture}\n\\]\nunder $B\\subset P\\subset \\Aut(M_{(i)} \\subset \\Aut (M)$. \nWe define \n\\[\n\\begin{aligned}\nf\\colon p^{-1} ({\\rm Gr} \\binom{M}{\\underline{d}^{1}})_{w} &\\to {\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w} \\times {\\rm Fl}\\binom{N}{\\underline{\\mathbf{e}}} \\\\\nU=(U^1\\subset \\cdots \\subset U^{r}=M) &\\mapsto (U^1, \\phi_{U^1}(U)\/U_0 ) .\n\\end{aligned}\n\\]\nThis is a morphism of algebraic varieties. To find the inverse, we consider $\\pi\\colon M\\to M\/U_0 =N$ the canonical projection and define \n\\[\n\\begin{aligned}\n{\\rm Gr} &\\binom{M}{\\underline{d}^{1}}_{w} \\times {\\rm Fl}\\binom{N}{\\underline{\\mathbf{e}}} \\to p^{-1} ({\\rm Gr} \\binom{M}{\\underline{d}^{1}}_{w})\\\\\n&\\left( L, V=(V^1\\subset \\cdots \\subset V^{r -1}=N)\\right) \\\\\n&\\quad \\mapsto (L \\subset \\phi_L^{-1}\\pi^{-1}(V^1)\\subset \\phi_L^{-1}\\pi^{-1}V^2 \\subset \\cdots \n\\subset \\phi_L^{-1}\\pi^{-1}V^{r -1 }=M).\n\\end{aligned}\n\\]\n\\end{proof}\n\n\\begin{defi} Let $X$ be a scheme. An \\text{affine cell decomposition} is a filtration \n\\[X=X_m \\supset X_{m-1}\\supset \\cdots \\supset X_0\\supset X_{-1}=\\emptyset \\]\nby closed subschemes, with each $X_{i}\\setminus X_{i-1}$ is a disjoint union of finitely many schemes $U_{ij}$ isomorphic to affine spaces $\\mathbb{A}^{n_{ij}}$. \n\\end{defi}\n\n\\begin{coro} \nLet ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ be a complete quiver flag variety for the oriented circle and $M$ be a nilpotent representation. Then, it has an affine cell decomposition. If $K=\\mathbb{C}$ or $\\overline{\\mathbb{Q}_{\\ell}}$ then it is pure.   \n\\end{coro}\n\n\nNote that there is a closed embedding by forgetting the $A$-module structure \n$\\kappa \\colon {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\to \\mathrm{Fl} (\\underline{\\mathbf{d}} ) =:\\mathcal{F}$. Forgetting all other then the first subspace gives a commutative square \n\\[ \n\\xymatrix{ \\mathcal{F} \\ar[r]^q & \\mathbb{P}^{d_i -1} \\\\\n{\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\ar[r]^p\\ar[u]^{\\kappa} & \\mathbb{P}^{s_i-1}\\ar[u]^{\\rho} \n}\n\\] \nwhere the vertical maps are closed immersions. By choosing appropiate cells in $\\mathcal{F}$, there is an affine cell decomposition in $\\mathcal{F}$ such that \nthe intersection with ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ is a union of cells in $\\mathcal{F}$. This implies that the open complement of ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ in $\\mathcal{F}$ also has an affine cell decomposition, this is the main ingredient to prove the following theorem, for $n=1$ compare \\cite[section 4.4, 4.5]{DP}. \n\n\n\\begin{satz}\nLet be $K=\\mathbb{C}$. Let $Q$ be the oriented cycle, $M$ be a nilpotent representation and $\\underline{d}$ a complete dimension filtration. \nThe pullback along $\\kappa \\colon {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} \\to \\mathcal{F}$ induces a surjective ring homomorphism \non singular cohomology \n\\[ \n\\kappa^* \\colon H^*(\\mathcal{F}) \\to H^*({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}).  \n\\]\n\\end{satz}\n\n\\paragraph{proof:} \nBy the universal coefficient theorem for projective varieties, it suffices to show that $\\kappa_* \\colon H_*^{BM}({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}) \\to H_*^{BM}(\\mathcal{F} )$ is injective. Let $U$ be the open complement of ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ in $\\mathcal{F}$. By the construction of the Spaltenstein fibration, we get that $\\mathcal{F}$ has a cell decomposition continuing the one of ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$, therefore $U$ also has also a cell decomposition. Since we have odd-degree vanishing also for $U$ the localization sequence gives a short exact sequence \n\\[\n0\\to H_*^{BM} ({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} ) \\to H_*^{BM} (\\mathcal{F} ) \\to H_*^{BM} (U) \\to 0 \n\\]\n\\hfill $\\Box $\n\n\n\\section{Parametrizing cells by multi-tableaux} \\label{PS}\n\n\nTo the nilpotent indecomposable $A$-module $E_i[\\ell]$ we associate a row of $\\ell $ boxes, indexing the columns by the elements $i-\\ell +1,i-\\ell, \\ldots , i$  in $\\mathbb{Z}\/n$ from left to right. \nTo any nilpotent $A$-module we associate a multipartition $Y=(Y_j)_{j\\in \\mathbb{Z}\/n}$ by taking the young diagram $Y_j$ corresponding to all indecomposable summands which have top supported on $j\\in \\mathbb{Z}\/n$. For example for $n=3$ and \\[M=(E_3[3]^2\\oplus E_2[2]) \\oplus (E_2[4] \\oplus E_3[2]) \\] we visualize the multipartition $Y_M= ((3,3,2), (4,2), \\emptyset )$ as follows, see the figure on the left.  \n\\[ \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & & \\none & \\none  \\\\\n& & & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\quad \\quad \\quad  \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & *(gray) &  \\none & \\none  \\\\\n& & *(gray) & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & *(gray) & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\quad \\quad \\quad \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & & \\none & \\none  \\\\\n& & & \\none & \\none \\\\\n& *(gray) &\\none &\\none & \\none  \\\\\n\\none & & & & *(gray) \\\\\n\\none & & & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\]\nIn the middle and the right hand side figure we shaded the socle at $3$ and the socle at $2$ respectively, the socle at $1$ is zero. \nFrom now on we choose a basis of $M$ such that each box corresponds to a basis vector and we order the basis vectors starting at the first row from left to right and then the second row from left to right and so on.  \nWhen we include a line into the vector space given by the socle at $3$, we find a first \ndirect summand (with respect to the fixed basis from before) where the inclusion is nonzero, we call the corresponding box the \\textbf{pivot box}. Looking at lines with the same pivot box defines a Schubert cell, \ne.g. we indicate the Schubert cells as follows, the pivot box gets a $1$, the number of stars indicates the dimension of the cell\n\\[ \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & 1 &  \\none & \\none  \\\\\n& & \\star & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & \\star & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\quad \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & 0 &  \\none & \\none  \\\\\n& & 1 & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & \\star & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\quad \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & & 0 &  \\none & \\none  \\\\\n& &  0 & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & 1 & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\]\nLet us look at the first cell. The quotients $M\/S$ with $S\\subset \\soc (M_{(3)})$ can be visualized by the left figure below.\n\\[ \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & &\\none  & \\none & \\none  \\\\\n& & & \\none & \\none \\\\\n& &\\none &\\none & \\none  \\\\\n\\none & & & & \\\\\n\\none & & & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\quad \\quad \\quad \\quad \\quad \\quad  \n \\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n{} & 0& \\none  & \\none & \\none  \\\\\n& & & \\none & \\none \\\\\n& 1 &\\none &\\none & \\none  \\\\\n\\none & & & & \\star  \\\\\n\\none & &  & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\]\nThe right hand side figure above shows a cell in the socle at $2$ with pivot box in row $3$. \nIn the figure above on the left each box corresponds to the residue class of a basis vector of $M$ corresponding to a box at the same place. This gives an ordered basis of $M\/S$.\nTo parametrize the cells in the complete flag variety, we need to parametrize the cells in the socles of the successive quotients. Now, to parameterize the cells in ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ with $\\underline{\\mathbf{d}}$ complete, we put an $r=\\dim M$ into an end box in the column specified by $\\underline{d}^1$. Then, we put an $r-1$ into an end or not filled box in the column specified by $\\underline{d}^2-\\underline{d}^1$, etc. We obtain a filling of $Y_M$ with numbers \n$1,\\ldots , r$ which is increasing from left to right in the rows, we call such a filling a \\textbf{row multi-tableau} of dimension $\\underline{\\mathbf{d}} $. Let $\\tau $ be a row multi-tableau, then we write $C_{\\tau} \\subset {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ for the corresponding cell. We can recover a dimension filtration $\\underline{\\mathbf{d}}=: \\Dim \\tau$ and an isomorphism class of a module $Y_{\\tau}$ from $\\tau$ by counting boxes in the columns and by looking at the shape of $\\tau$.   \nFor $k\\in \\{1, \\ldots , r\\}$ we write $c_k\\in \\mathbb{Z}\/n$ for the column containing \n$k$ and $r_k\\in \\mathbb{N}$ for the row containing $k$. We define  \n\\[ d_{\\tau }(k) := \\# \\{ s\\in \\{1, \\ldots , k-1\\}\\mid c_s=c_k, r_s>r_k, r_s\\neq r_t, \\; s<t<k\\}\n\\]\nOne has by construction\n\\[ \\dim C_{\\tau } =\\sum_{k=1}^r d_{\\tau}(k) .\\]  \n\nFor example, for $\\tau $ as follows, $d_{\\tau}(-)$ is as in the table below and therefore $\\dim C_{\\tau}=9$.\n\\[\n\\tau= \n\\begin{array}{c}\n\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\makebox[9pt][c]{\\text{\\tiny{3}}}\\makebox[9pt][c]{\\text{\\tiny{1}}}\\makebox[9pt][c]{\\text{\\tiny{2}}}\\\\\n\\ytableausetup{smalltableaux}\n\\begin{ytableau}\n5 & 11 & 14 &\\none & \\none  \\\\\n2 & 6 &  8& \\none & \\none \\\\\n3 & 13 &\\none &\\none & \\none  \\\\\n\\none & 1 & 9  &  10 & 12 \\\\\n\\none & 4 & 7 & \\none & \\none \n\\end{ytableau}\n\\end{array}\n\\]\n\n\\begin{tabular}[c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n$k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ \\\\\n\\hline \n$d_{\\tau }(k)$ & $0$ & $0$ & $0$ & $0$ & $2$ & $2$ & $0$ & $1$ & $1$ & $0$ & $0$ & $0$ & $1$ & $2$ \n\\end{tabular}\n\n\\noindent\n\\hspace{1cm}\n\nNow let $Y$ be a tuple of $n$ finite sequences of positive numbers. \nFor the $i$-th sequence we write a skew-diagram starting at column $i\\in \\mathbb{Z}\/n$ with row lengths specified by the sequence of positive numbers. \nAny (complete) numbering of the boxes of $Y$ which is increasing in the length is called a row multi-tableau. From the columns of the \ntableau, we can read of the dimension filtration denoted by $\\Dim \\tau$, we set $\\dim \\tau := \\sum_{k=1}^r d_{\\tau}(k)$ and we write \n$Y_{\\tau}$ for the $Y$ underlying $\\tau$. \n\n\n\\[\n\\begin{aligned}\n{\\rm f}\\binom{Y}{\\underline{\\mathbf{d}}} &:= \\#\\{ \\tau \\mid Y_{\\tau }=Y, \\Dim \\tau =\\underline{\\mathbf{d}} \\}\\\\\n{\\rm f}\\left( \\binom{Y}{\\ddd} ; k\\right)  &:= \\#\\{\\tau \\mid  Y_{\\tau}=Y, \\Dim \\tau =\\underline{\\mathbf{d}}, \\dim \\tau =k\\}\n\\end{aligned}\n\\]\nand for $M$ of shape $Y$ one has $\\dim H^*({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} ) ={\\rm f}\\binom{Y}{\\underline{\\mathbf{d}}}$. Moreover $H^{odd} ({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})=0$ and $\\dim H^{2k} ({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})={\\rm f}\\left( \\binom{Y}{\\ddd} ; k\\right) $. \n\n\nWe can use the following recursion to \ncalculate ${\\rm f}\\binom{Y}{\\underline{\\mathbf{d}}}$ and ${\\rm f}\\left( \\binom{Y}{\\ddd} ; k\\right) $. For $i\\in \\mathbb{Z}\/n$ we denote by $E(Y,i)=(b_1,\\ldots , b_s)$ the tuple of right end boxes in $Y$ in column $i$ ordered from top to bottom. For $b \\in E(Y,i)$ we write $Y\\setminus b$ for the skew-diagram obtained from $Y$ by removing the box $b$. We write the dimension filtration as the sequence of the supports of the subquotients $\\underline{\\mathbf{d}}= (i_1, \\ldots , i_r) \\in (Q_0)^r$, i.e. $\\underline{d}^1=e_{i_1}, \\underline{d}^2=e_{i_1}+e_{i_2}$, etc. and let $E(Y,i_1)=(b_1, \\ldots , b_s)$ \n\\[ \n\\begin{aligned}\n{\\rm f}\\binom{Y}{\\underline{\\mathbf{d}}} &=  \\sum_{b\\in E(Y, i_1)} {\\rm f}\\binom{Y\\setminus b}{\\underline{\\mathbf{d}}-\\dd^1} \\\\\n{\\rm f}\\left( \\binom{Y}{\\ddd} ; k\\right)  &= \\sum_{m=1}^s {\\rm f}\\left( \\binom{Y\\setminus{b_m}}{\\underline{\\mathbf{d}}-\\dd^1} ; k+m-s\\right) \n\\end{aligned}\n\\]\n\n\n\n \n\\section{Torus fixed points and GKM-theory}\n\n\nLet $Q$ be an arbitrary quiver and $\\mathbb{A}_{r+1}$ the quiver $0\\to 1 \\to \\cdots \\to r$.   \nLet $\\Lambda =KQ\\otimes_{K}K\\mathbb{A}_{r +1}$. Then a $\\Lambda$-module is given by a sequence of $KQ$-module homomorphisms $U_0\\xrightarrow{f_0}U_1\\xrightarrow{f_1} \\cdots \\xrightarrow{f_{r -1}} U_{r}$ and let $\\mathcal{M} $ be the full subcategory of $\\Lambda$-modules with $U_0 =0$ and every morphism $f_k$ is a set-theoretic inclusion. \n \n\n\\begin{defi} An object $0\\subset U_1\\subset \\cdots \\subset U_r =M$ in $\\mathcal{M} $ is called \\textbf{split} with respect to a direct sum decomposition $M=M_1\\oplus \\cdots \\oplus M_t$ in indecomposable submodules if \n\\[ U_s =(U_s\\cap M_1)\\oplus \\cdots \\oplus (U_s\\cap M_t), \\quad  1\\leq s\\leq r-1 .\\]\nWe call a $\\Lambda$-module $U$ in $\\mathcal{M} $ \\textbf{split} if there exists a direct sum decomposition of $U_r$ such that it is split with respect to this direct sum decomposition. \nFix an additive family of modules with for each isomorphism type precisely one module. The full subcategory of $\\mathcal{M} $ of split modules with respect to this family defines an additive subcategory $S\\mathcal{M} $ of $\\mathcal{M} $.  \n\\end{defi}\n\nLet $U:=(0\\subset U_1\\subset \\cdots \\subset U_r=M)$ be a split module with respect to $M=M_1\\oplus \\cdots \\oplus M_t$, we set \n$(U\\cap M_i):= \\bigl(0\\subset (U_1\\cap M_i) \\subset \\cdots \\subset (U_n \\cap M_i) =M_i\\bigr) $, then $U=(U\\cap M_1)\\oplus \\cdots \\oplus (U\\cap M_t)$ is a direct sum decomposition into indecomposable $\\Lambda$-submodules. \n\nLet $M= \\bigoplus_{j =1}^t M_i$ be a decomposition into indecomposables.  \n\\[T:=\\{ \\bigoplus_{j} \\lambda_{j} \\id_{M_j} \\mid \\lambda_{j} \\in K^*\\} \\subset  \\prod_{j}\\Aut (M_j) \\subset \\Aut (M)\\]\n\n\\begin{lemma} Let $U\\in{\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$. Then, $U$ is split with respect to the decomposition of $M$ if and only if it is a $T$-fixed point. \n\\end{lemma}\n\n\\begin{proof} Clearly, if $U$ is split, it is fixed under $T$. \nConversely, let $U\\in ({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})^T$. Take $x\\in U_s$, we need to see that it can be written as $\\sum_j x_j$ with $x_j \\in U_s \\cap M_j$. Write $x=x_j+x^\\prime$ with $x^\\prime \\in M^\\prime = M_1\\oplus \\cdots \\oplus M_{j-1} \\oplus M_{j+1} \\oplus \\cdots \\oplus M_t$, $x_j\\in M_j$ and we claim $x_j\\in U_s $. Let $\\lambda \\neq 1$ and apply \n$f=(\\lambda -1)\\id_{M_j} \\oplus (-1)\\id_{M^\\prime} \\in T$, then by assumption $f(x)\\in U_s$ and therefore $x+f(x)=\\lambda x_j\\in U_s$ which implies $x_j\\in U_s$.  \n\\end{proof}\n\nWe remark that for $M$ indecomposable all points in ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ are $T$-fixed.  \nAssume now $M$ is uniserial. Then quiver flag varieties ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ are either empty or a point, given by an indecomposable $\\Lambda$-module $M[\\underline{\\mathbf{d}}]$. \n\n\\begin{rem} Let $M=M_1\\oplus \\cdots \\oplus M_t$ be a module such that each $M_i$ is uniserial. For $U\\in {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ one has $U$ is a $T$-fixed point if and only if one of the following equivalent conditions is fulfilled   \\\\\n(*) $U=\\bigoplus_j M_j[\\underline{\\mathbf{d}}^j]$ as $\\Lambda$-module with $\\sum_{j=1}^t \\underline{\\mathbf{d}}^j=\\underline{\\mathbf{d}}$. \\\\\n(*) $U$ is the image of a direct sum map $\\oplus \\colon {\\rm Fl}\\binom{M_1}{{\\underline{\\mathbf{d}}}^1} \\times \\cdots \\times {\\rm Fl}\\binom{M_t}{{\\underline{\\mathbf{d}}^t}} \\to {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ with $\\sum_{j=1}^t \\underline{\\mathbf{d}}^j=\\underline{\\mathbf{d}}$. \\\\\nIn particular, $({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})^T$ is a finite set. \n\\end{rem}\n\nFrom now on we investigate the case of nilpotent representation of the oriented cycle. \nWe define the modules $M[\\underline{\\mathbf{d}}]$ for $M=E_i[\\ell]$ as follows. \nLet $\\lambda =(\\ell_1\\geq \\ell_2\\geq \\cdots \\geq \\ell_r)$ be a partition with at most $r$-parts, $\\ell_1=\\ell, \\ell_j\\geq 0$. We define an element in $\\mathcal{M}$ as follows \n\\[ \nE_i[\\lambda ] := (0\\subset E_i[\\ell_r]\\subset \\cdots \\subset E_i[\\ell_1] )\n\\]\nwhere $E_i[0]:=0$. The category $S\\mathcal{M} $ is Krull-Schmidt. \nThe isomorphism classes of indecomposable objects in \n$S\\mathcal{M}$ are given by the $E_i[\\lambda], i\\in \\mathbb{Z}\/n$ and $\\lambda$ a partition. One has $\\End_{\\Lambda}(E_{i}[\\lambda])=\\End_{A} (E_i[\\ell_1 ])$. \n\n\nNow, if $\\underline{\\mathbf{d}}$ is a complete dimension filtration, then the split modules $U\\in {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ correspond to the points in the cells $C_{\\tau}$ which are given by the inclusion of the pivot box in each successive step (compare previous section). To each row in a row-multi-tableau $\\tau$ we assign a split module as follows. Assume the row ends at $i$ and has a filling with positive numbers $a_1<a_2<\\cdots <a_{\\ell}$, then take \n$E_i[\\lambda]$ with $\\lambda = (0^{m_0}, 1^{m_1} , \\ldots , \\ell^{m_{\\ell}}) $ where $\\sum_j m_j =r$ such that $m_j$ is the number of occurences of $j$ given as follows \n\\[m_0=r-a_{\\ell}, m_1=a_{\\ell}-a_{\\ell-1}, m_2=a_{\\ell -1}- a_{\\ell-2}, \\ldots , m_{\\ell-1} = a_{2}-a_1, m_{\\ell}=a_1. \\] \n\n\\begin{lemma} Assume $\\underline{\\mathbf{d}}$ is a complete dimension filtration. \nThen, every cell $C_{\\tau}$ contains a unique $T$-fixed point. \nIn particular, there is a bijection between isomorphism classes of split modules and row root tableau. \n\\end{lemma}\n\nIt is easy to see that the cells $C_{\\tau}$ are $T$-equivariant. In fact, our cell decomposition coincides for $K=\\mathbb{C}$ with a Bialynicki-Birula cell decomposition, see \\cite{Car}. \n\nIf we choose in the Spaltenstein fibration the split inclusions as reference points $U_0$ each factor in the splitting is $T$-invariant. We conclude that there are $\\dim C_{\\tau}$ one-dimensional $T$-orbits in $C_{\\tau}$. \n\nEach one-dimensional $T$-orbit closure contains precisely two $T$-fixed points, the one corresponding to the split module in $C_{\\tau}$ and the other \none obtained by an \\textbf{admissible} swap of box entries from $\\tau$, defined as follows: Another row multi-tableau $\\tau^\\prime$ is obtained by an admissible swap \nfrom $\\tau$ if there is a connected interval of box-entries in two rows let's say $r_k$ and $r_m$ (where $k$ and $m$ are the largest entries in these intervals) which are swapped to obtain $\\tau^\\prime$ and $\\left|\\dim C_{\\tau}- \\dim C_{\\tau^\\prime}\\right| =1$. In this case we write $\\tau^\\prime := \\sigma_{k,m}(\\tau)$. \n\nFrom the $T$-fixed points $F:=\\{\\tau\\mid C_{\\tau}\\subset {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}\\}$ and the one-dimensional $T$-orbit closures we obtain a graph $\\Gamma$ which determines a ring \n \\[\n S(\\Gamma ) := \\left\\lbrace (f_{\\tau})_{\\tau \\in F} \\in \\mathbb{C}[x_1, \\ldots , x_t]^{\\oplus F}\\mid \n\\begin{aligned} \n&  \\\\\n  &f_{\\tau} - f_{\\sigma_{k,m}(\\tau )} \\in (x_k-x_m) \\mathbb{C}[x_1, \\ldots , x_t]\\\\\n  &\\text{ for all admissible swaps}\n  \\end{aligned}\n  \\right\\rbrace\n \\] \nwith $t=\\rk T$. \n\n\nLet us denote the inclusion of the torus fixed points by $i\\colon ({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})^T \\to {\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$. The $T$-equivariant cohomology with complex coefficients of ${\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}$ can be described directly by applying GKM-theory, see \\cite[Thm 1.2.2]{GKM}. \n\\begin{coro}  Let $K=\\mathbb{C}$ and $\\underline{\\mathbf{d}}$ a complete dimension filtration. Then, the map $i^*$ induces an injective ring homomorphism \n\\[i^*\\colon H_T^*({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}) \\to H_T^*(({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}})^T)\\]\nwith image isomorphic to $S(\\Gamma )$ described before.  \n\\end{coro}\n\n\\section{Kato's standard modules}\nLet $K=\\mathbb{C}$, $\\underline{d}\\in \\mathbb{N}_0^{Q_0}$, $G:= \\prod_{i\\in Q_0} \\mathbf{Gl}_{d_i}(K)$ and one has the $G$-representation $\\mathrm{R}(\\dd):= \\bigoplus_{i\\to i+1} \\Hom (K^{d_i} , K^{d_{i+1}})$ given by \n$(g_i)\\cdot (m_{i\\to i+1}) := (g_{i+1} m_{i\\to i+1} g_i^{-1})$. We set \n$I_{\\underline{d}}:= \\{ \\underline{\\mathbf{d}}:=(i_1, \\ldots , i_r )\\mid \\sum_{j=1}^r i_j =\\underline{d} \\}$. \nFor each $\\underline{\\mathbf{d}} \\in I_{\\underline{d}}$ we fix a $Q_0$-graded flag $(V^{k})_{0\\leq k \\leq r}$ in \n$\\bigoplus_{i\\in Q_0}K^{d_i}$ of dimension vector $\\Dim V^k\/V^{k-1}= i_k$. We define a parabolic group $P(\\underline{\\mathbf{d}} )\\subset G$ as the stabilizer of $V^\\bullet$ and a $P(\\underline{\\mathbf{d}} )$-subrepresentation of $\\mathrm{R}(\\dd)$ by $F(\\underline{\\mathbf{d}} ):=\\{ (m_{i\\to i+1}) \\in \\mathrm{R}(\\dd) \\mid m_{i\\to i+1} (V^k_i)\\subset V_{i+1}^{k-1}\\}$. \nWe set $\\pi_{\\underline{\\mathbf{d}}} \\colon E_{\\underline{\\mathbf{d}} }:= G\\times^{P(\\underline{\\mathbf{d}} )} F(\\underline{\\mathbf{d}} )\\to \\mathrm{R}(\\dd), \\quad \\overline{(g, f)} \\mapsto gf$ for the collapsing of the associated vector bundle\n$E_{\\underline{\\mathbf{d}}}$ over $G\/P(\\underline{\\mathbf{d}} )$ and $e_{\\underline{\\mathbf{d}}} :=\\dim_{\\mathbb{C}}E_{\\underline{\\mathbf{d}}}$. Then we define \n\\[ \\mathcal{L}:= \\bigoplus_{\\underline{\\mathbf{d}} \\in I_{\\underline{d}}} (\\pi_{\\underline{\\mathbf{d}} })_* \\underline{\\mathbb{C}}[e_{\\underline{\\mathbf{d}}}] \\quad \\in D^b_G(\\mathrm{R}(\\dd))\n\\]\nwhere $D^b_G(\\mathrm{R}(\\dd))$ is the equivariant derived category of Bernstein and Lunts \\cite{BL}. \nLet $\\Lambda$ be the set of multipartitions $\\lambda=(\\lambda^1, \\ldots , \\lambda^n)$ corresponding to the $G$-orbits $\\mathcal{O}_{\\lambda}$ in $\\mathrm{R}(\\dd)$. We fix a point $i_\\lambda\\colon \\{M\\} \\inj \\mathcal{O}_{\\lambda}$.    \nThe module corresponding in our situation to Kato's standard module $K_{\\lambda}$ shifted by $(-\\dim \\mathcal{O}_{\\lambda})$ is given by \n\\[ \nK_{\\lambda}\\langle -\\dim \\mathcal{O}_{\\lambda} \\rangle := H^*(i_{\\lambda}^{!} \\mathcal{L} ) = \\bigoplus_{\\underline{\\mathbf{d}}\\in I_{\\underline{d}}} H_{e_{\\underline{\\mathbf{d}}} -*}({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}} )\n\\]\nwhere $H_*()$ refers to Borel-Moore homology with complex coefficients. \nBy the local to global sequence in Borel-Moore homology we get $\\dim H_k({\\rm Fl}\\binom{M}{\\underline{\\mathbf{d}}}) = {\\rm f}\\left( \\binom{\\lambda }{\\ddd } ; k\\right) $.  \nFor any $\\mathbb{Z}$-graded vector space $V$ with $V_j=0$ for $j <<0$ we define \nthe graded dimension to by \n$ \\gdim V := \\sum_{j\\in \\mathbb{Z}} (\\dim V_j) t^j \\; \\; \\in (\\mathbb{Z}\\left[\\left[ t \\right]\\right])\\left[t^{-1}\\right].  $\nUsing our cell decomposition we have  \n\\[\nt^{-\\dim \\mathcal{O}_{\\lambda}} \\gdim K_{\\lambda} = \\sum_{k\\in \\mathbb{Z}} \\left( \\sum_{\\underline{\\mathbf{d}} \\in I_{\\underline{d}}} {\\rm f}\\left( \\binom{Y}{\\ddd } ; e_{\\ddd} - k\\right)  \\right) t^k. \n\\] \n\\begin{rem}\nConsider the BBD-decomposition of $\\mathcal{L}$, apply  $H^*(i_{\\lambda}^!(-))$ and then apply $\\gdim$, the resulting equation in the ring of Laurent series expresses\\\\ $t^{-\\dim \\mathcal{O}_{\\lambda}}\\gdim K_{\\lambda}$ as a sum over products given by $\\gdim L_{\\mu}$ with \n $\\gdim \\mathcal{H}^*_{\\mathcal{O}_{\\lambda}}(IC_{\\mu})$ where $L_{\\mu}$ runs through the simple modules (over the quiver Hecke algebra of the nilpotent $\\underline{d}$-dimensional representations of the quiver $Q$) and $IC_{\\mu}$ is the simple perverse sheaf on $\\mathrm{R}(\\dd)$ build from the trivial local system on $\\mathcal{O}_{\\mu}$. It is an observation of Schiffmann (see \\cite[Corollary 6.2]{Sch}, generalizing from Lusztig for $n=1$) that \n$\\gdim \\mathcal{H}^*_{\\mathcal{O}_{\\lambda}}(IC_{\\mu})$ is an affine Kazhdan Lusztig polynomial of type $A$. It naturally generalizes the notion of a Kostka polynomial in the sense that it describes the base change expressing a canonical basis $[K_{\\lambda}]$ in a PBW-basis $[L_{\\mu}]$ within the Grothendieck group of \nfinite-dimensional graded modules, see loc. cit.  \n\\end{rem}\n\n\\section*{Acknowledgements}\n\nI would like to thank Peter McNamara for pointing out that the multiplicity spaces can be zero and therefore, Kato's work can not directly be applied. Furthermore, I would like to mention financial support from the CRC 701 in Bielefeld and the SPP 1388 (Schwerpunktprogramm  Darstellungstheorie) by covering travel costs and research stays. \n\n\n\n\n\\bibliographystyle{alphadin}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Related work}\n\n\nDedicated machines with single-axis servo mechanisms are omnipresent in industrial production processes, especially given the tendency to evolve towards dedicated actuators for each machine movement \\cite{Berselli2016}. In this context, several approaches have been presented in the past literature to optimize the motion profiles of these one DOF systems.\n\nIn \\cite{Richiedei2016,Lee2020,Park1996,Carabin2021,Botan2010}, optimal motion profiles are obtained for mechanical systems with constant load parameters such as the inertia $J$. Nevertheless, as indicated in literature \\cite{Pellicciari2015,Berselli2016,Hsu2014}, it is essential to consider varying loads to cover the majority of machine applications.\n\nHowever, in those cases obtaining an analytic description of the position-dependent system properties is a challenging task for industrial machine designers. For instance, \\cite{Hsu2014,Hsu2016,Huang2012,Ha2006} use Hamilton's principle and Lagrange multipliers to obtain the differential-algebraic equations of the system. In \\cite{Sollmann2010}, a method of virtual work is described to obtain the system matrix while \\cite{Vanbecelaere2020} determines the inertia profile using the method of kinetic energy. Such approaches are cumbersome, complex and error-prone, and are not easily applicable. Especially given the trend indicated in \\cite{Walsch2014} that there is a demand for methods that take into account the ease of implementation.\n\nFortunately, machine builders often already design their machines in 3D CAD multibody software, which can be used to extract crucial information. Hence, in \\cite{VanOosterwyck2019} and \\cite{Berselli2016}, the authors of this paper describe a technique to derive the position dependency of critical parameters inertia $J(\\theta)$ and load torque $\\tau_l(\\theta)$, based on only three CAD motion simulations. If the load model is known, optimization algorithms can iterate on it to minimize energy usage.\n\nFor what concerns these optimization algorithms, several approaches are possible. On the one hand, \\cite{Park1996} and \\cite{Shiller1996} use an indirect approach such as Pontryagin's Maximum Principle to obtain the best possible control. However, this method tends to be abandoned recently due to the small convergence area and difficulties incorporating constraints \\cite{Chettibi2004}.\n\nOn the other hand, direct approaches recast the optimization into a nonlinear optimization problem, which can be solved with different numerical methods. In particular, \\cite{VanOosterwyck2020} and \\cite{Pellicciari2015} use gradient-based methods such as Sequential Quadratic Programming (SQP) or quasi-Newton methods that are known to have very low solve times and good scalability. However, these algorithms can only deliver local optimal solutions and are not suited for problems with multiple minima. As indicated in \\cite{Huang2017}, the optimum obtained with gradient-based methods is greatly influenced by the selected starting points, which are to be chosen arbitrarily.\n\nTo avoid this problem, heuristic optimization algorithms \\cite{VanOosterwyck2019,Huang2012,Hsu2014,Berselli2016}  such as generalized pattern search (GPS) or genetic algorithms (GA) are of interest. In contrast to gradient-based algorithms that do not search the entire design space, derivative-free algorithms like GA often sample a wide part of the design space in order to be successful \\cite{Wenzhong2005}. Nevertheless, because these heuristic solvers do not exploit gradient information, they are not computationally competitive with gradient-based methods \\cite{Betts1998}.\n\nRegarding the motion profile function, several papers rely on piecewise position functions (\\cite{Pellicciari2015,Berselli2016}), where either cubic, quintic, or trigonometric splines are used. However, the objective functions in these works, are characterized by many local minima, causing the risk of getting stuck in a suboptimal solution. For instance, in \\cite{Piazzi1998}, the usage of cubic splines resulted in a savings difference of $18\\%$ between the global and local optimum.\n\nOn the other hand, continuous motion profile functions such as classic polynomials (\\cite{VanOosterwyck2019,Huang2012,Hsu2014,Carabin2021,Lee2020}) are also popular because they do not introduce high jerk peaks into the system, which increases the wear of the components. However, the resulting optimization problem is known to be badly conditioned. For example in \\cite{VanOosterwyck2019}, the coefficients reached values up to $1.8 \\, 10^{20}$. Therefore, the authors of this paper proposed Chebyshev polynomials and rescalings in \\cite{VanOosterwyck2020} to obtain a more numerically stable problem formulation and to increase the robustness against getting stuck in local minima. Moreover, in contrast to the classical polynomial descriptions, the design variables of Chebyshev polynomials can be bounded as is shown in this paper.\n\n\\subsection{Method}\nIn light of the considerations mentioned above, a CAD-based method for computing energy-optimal PTP (Point-to-Point) motion profiles of single DOF mechanisms using Chebyshev polynomials has been previously presented by the authors in \\cite{VanOosterwyck2020}. However, due to the numerous symbolic calculations involved in constructing the objective function, solve times of almost 2 hours were reported. In addition, the solutions in \\cite{VanOosterwyck2020} were obtained using gradient-based solvers which have a high risk of getting stuck in local minima. Finally, only theoretical reductions were reported, thus, leaving the feasibility of the proposed motion profiles undetermined. This paper builds upon these previous results by providing five critical improvements:\n\n\\begin{itemize}\n\n    \\item In order to reduce the computational burden, a discrete approach is presented which eliminates the use of symbolic operations. To do so, the discrete system property data which originates from the CAD motion simulations have to be properly rescaled and interpolated.\n    \n    \\item As an accurate model of the system dynamics is crucial for a correct optimization, the dynamics of the mechanism are extended by including damping and friction into the optimization routine. In addition, a new identification procedure is described which is validated on an industrial case.\n    \n    \\item A derivation for exact bounds of the Chebyshev polynomial coefficients is introduced. This allows limiting the feasible design space. The latter is essential for heuristic optimizers to reduce their computation time and the chances that the global optimum remains unidentified.\n    \n    \\item To check the robustness of the proposed method against getting stuck in local optima, the resulting optimization problem is solved with both a fast gradient-based and a global heuristic solver (i.e. GA).\n    \n    \\item Experimental tests have been carried out on an industrial pick-and-place unit to quantify the actual measured energy savings.\n    \n   \n   \n\\end{itemize}\n\n\n\n\n\n\n\\section{System Modeling}\nThe complete mechatronic system can be divided into two subsystems (Fig. \\ref{fig:Simplified_Model}). On the one hand, there is the \\textit{mechanical subsystem} which describes the dynamics of a generic single-axis system. For high dynamical applications, these systems usually consist of slider-crank mechanisms and four-bar linkages \\cite{Berselli2016}. Nevertheless, the approach is valid to any position-controlled system where the mechanism is driven by a single actuator.\n\nOn the other hand, there is the \\textit{actuation subsystem} which converts the electrical energy into mechanical energy and drives the mechanism. For the envisaged position-controlled systems, PMSMs are becoming the industry standard for rotary applications, whereas linear motors are used for fast and precise linear movements \\cite{Kiel2008}. In Fig. \\ref{fig:Simplified_Model}, the PMSM actuator is represented by an equivalent DC model.\n\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Simplified_Model_Motor.pdf}\n  \\caption{Schematic of the q-axis of a single axis mechanism.}\n  \\label{fig:Simplified_Model}\n\\end{figure}\n\n\\subsection{Mechanical subsystem}\nThe dynamics of a single axis DOF mechanism can be described by means of the torque equation \\cite{Pellicciari2015,Dresig2010} :\n\n\\begin{equation} \\label{eq:torque_equation}\n\\tau_m(t) = \\tau_l(\\theta) + \\underbrace{J(\\theta)\\ddot{\\theta}}_{\\tau_a} + \\underbrace{\\frac{1}{2}\\frac{\\mathrm{d}J(\\theta)}{\\mathrm{d}\\theta}(\\dot{\\theta})^2}_{\\tau_{v}} +  \\tau_f(\\dot{\\theta}) \\, .\n\\end{equation}\n\nWith reference to Fig. \\ref{fig:Simplified_Model} and equation (\\ref{eq:torque_equation}), let us define $\\theta = \\theta(t)$ as the Lagrangian coordinate which describes the angular position of the main driving axis as a function of time $t$. The motor torque $\\tau_m(\\theta)$ is defined as the driving torque generated by the motor. The load torque $\\tau_l(\\theta)$ contains both gravitational forces as well as external process powers that act on the mechanism.\n\nFurthermore, all inertias of the mechanism's components are related to the main driving axis resorting to the concept of reduced moment of inertia. Therefore, the reduced inertia of the complete system $J(\\theta)$ is defined as a combination of the reduced load inertia $J_l(\\theta)$ and inertia of the motor shaft itself $J_m$. Note that the position-dependent inertia of the system $J(\\theta)$, results in two torque components when it is reduced to the motor side. The acceleration torque $\\tau_a$ represents the part of the motor torque responsible for the motor acceleration forces that arise during the movement, while the variation torque $\\tau_v$ compensates for the variation of inertia in the system.\n\nFinally, the frictional torque $\\tau_f(\\dot{\\theta})$ is defined as the result of frictional forces such as, for instance, viscous brush friction or dry bearing friction in the motor bearings and mechanical system. A commonly used model of friction shows three components of force: Coulomb (sliding) friction, viscous damping, and static friction \\cite{Ellis2012}. Regarding the PMSM, as indicated in \\cite[p.~175]{Westphal2001}, the only appreciable friction effect in operation is viscous friction. Thus, coulomb and static frictions can be neglected in the PMSM model. For what concerns the mechanical model, only the viscous damping is modeled since the other friction components are constant and will not have an effect on the optimal motion profile:\n\n\\begin{equation} \\label{eq:driction_torque}\n\\tau_f(\\dot{\\theta}) = \\mu_v\\dot{\\theta} \\, ,\n\\end{equation}\n\nwith $\\mu_v$ the equivalent viscous friction coefficient. \n\nThe key benefit of the formulation in \\eqref{eq:torque_equation} is that it permits to model every possible mechanism with a known geometry and allows to define a generic optimization approach.\n\n\\subsection{Actuation subsystem}\nConcerning the dynamics of the PMSM as depicted in Fig. \\ref{fig:Simplified_Model} (represented by an equivalent DC model), the electromechanical behavior can be easily described by the following basic laws \\cite[p.~843]{Rizzoni2003}:\n\n\\begin{equation} \\label{eq:basis_torque}\n\\tau_m = k_t i \\, ,\n\\end{equation}\n\n\n\n\\begin{equation} \\label{eq:voltage}\nu = R i + L\\frac{\\mathrm{d}i}{\\mathrm{d}t} + e = R i + L\\frac{\\mathrm{d}i}{\\mathrm{d}t} + p k_v \\dot{\\theta} \\, ,\n\\end{equation}\n\n\nwith electric back emf $e$, resistance $R$, back emf constant $k_v$, motor torque constant $k_t$, and number of pole pairs $p$, which can be found in the motor data sheet.\n\nIn equation (\\ref{eq:voltage}), the voltage drop $L\\frac{\\mathrm{d}i}{\\mathrm{d}t}$ due to the armature inductance is omitted as the mean value of its reactive power will be zero and therefore does not contribute to the system's energy need \\cite{Pellicciari2015}.\n\n\n\nDepending on whether the electric power flows from the drive unit to the PMSM's or vice versa, the PMSM operates in respectively motor or generator mode. In this latter condition, depending on the capabilities of the drive unit, the generated electric power can be either stored in a capacitor, dissipated as heat on a braking resistance, or transferred back to the energy source. Recent commercial PMSM drives are sized so that no electric power is actually dissipated during normal functioning so that the braking resistance is actually activated only under emergency conditions \\cite{Berselli2016}. Therefore, in what follows, it is assumed that all the generated energy is returned to the grid and no losses occur in the process.\n\nFor a correct model of the actuation subsystem and prediction of the energy usage, it is important to model other losses such as cooling fans and drive circuitry as well. Nevertheless, the power consumption of these devices is generally considered constant and is therefore not affected by the motion profile \\cite{Gadaleta2019}.\n\nIn order to minimize the total energy need $E$ of the application, it is crucial to quantify the input energy of the complete system. Therefore, similar to \\cite{Berselli2016}, a formulation of the input electrical energy $E$ is derived and a torque-based design objective is obtained which allows minimizing the energy solely based on the mechanical parameters.\n\nStarting from equations \\eqref{eq:basis_torque} and \\eqref{eq:voltage}, the instantaneous power $P_e$ is defined as\n\n\\begin{equation}\nP_{e} = u \\, i = \\frac{R}{k_t^2}\\tau_m^{2} + \\frac{p k_v}{k_t} \\tau_m \\dot{\\theta} \\, .\n\\end{equation}\n\nThe motion profile is defined on the time interval $t \\in [t_A,t_B]$ and must have zero initial and final speed and acceleration, i.e. $\\dot{\\theta}(t_A) = \\dot{\\theta}(t_B) = \\ddot{\\theta}(t_A) = \\ddot{\\theta}(t_B) = 0$. The total energy can be expressed as\n\n\\begin{equation}\n\\begin{aligned}\nE= \\int_{t_A}^{t_B} P_e \\, \\mathrm{d} t &= \\int_{t_A}^{t_B} \\left[ \\frac{R}{k_t^2}\\tau_m^{2} + \\frac{p k_v}{k_t} \\tau_m \\dot{\\theta} \\right] \\, \\mathrm{d} t \\, .\n\\end{aligned}\n\\end{equation}\n\nThen, by incorporating the torque equation from \\eqref{eq:torque_equation}, the total energy of the motion is given by:\n\n\\begin{equation} \\label{eq:energy}\n\\begin{aligned}\nE=\\frac{p k_v}{k_t}  \\underbrace{\\int_{t_A}^{t_B} (\\tau_a + \\tau_v) \\dot{\\theta} \\, \\mathrm{d} t}_{E_{k}} + \\frac{p k_v}{k_t} \\underbrace{\\int_{t_A}^{t_B} \\tau_l \\, \\dot{\\theta} \\, \\mathrm{d} t}_{E_{p}} \\\\\n+\\underbrace{\\int_{t_A}^{t_B}\\left[\\frac{R}{k_t^2} \\tau_m^{2}+ \\frac{p k_v}{k_t} \\tau_f \\dot{\\theta}\\right] \\mathrm{d} t}_{E_{l}} \\, .\n\\end{aligned}\n\\end{equation}\n\nHere, the first term $E_k$ represents the kinetic energy of the moving masses  in the system. Due to the rest-to-rest motion of the envisaged applications, this term reduces to zero. Further, the term $E_p$ represents the potential energy stored in the system. As this term $E_p$ only depends on the fixed start $\\theta_A$ and end position $\\theta_B$, it is disregarded in the optimization routine \\cite{Berselli2016}. The final term $E_l$ represents the energy that is lost due to the coil resistance and frictional forces and is the only term that is affected by optimizing the motion profile $\\theta(t)$. Nevertheless, in many industrial applications, the frictional forces $\\tau_f$ are negligible \\cite{Park1996}, especially if the inertial loads are predominant. Thus, the energy losses $E_l$ can be expressed as:\n\n\\begin{equation}\n    E_l = \\int_{t_A}^{t_B} \\frac{R}{k_t^2} \\tau_m^{2} \\mathrm{d} t = \\frac{R \\Delta t}{k_t^2}\\tau_{rms}^2 \\, ,\n\\end{equation}\n\nwhere $\\tau_{rms}$ is the RMS value of the motor Torque $\\tau_{m}$. This proves that the RMS torque $\\tau_{rms}$ can be effectively used as an optimization objective to minimize the total energy usage of the system. This is very useful in situations where the motor coil properties are unknown or where parameters are missing \\cite{Berselli2016}.\n\n\\section{Identification}\n\\subsection{Inertia and Load Torque}\nIdentification of all the position varying parameters in the highly nonlinear differential torque equation (\\ref{eq:torque_equation}) is not straightforward. Fortunately, machine builders design their machines in 3D CAD multibody software. For this reason, \\cite{Berselli2016} and \\cite{VanOosterwyck2019} describe a technique to derive the position dependency of critical parameters inertia $J(\\theta)$ and load torque $\\tau_l(\\theta)$, based on three CAD motion simulations. (Fig. \\ref{fig:Property_Extraction}).\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Property_Extraction.pdf}\n  \\caption{Schematic overview of the procedure for extracting position-dependent properties $J(\\theta)$ and $\\tau_l(\\theta)$ based on three different CAD motion simulations \\cite{VanOosterwyck2019}.}\n  \\label{fig:Property_Extraction}\n\\end{figure}\n\nIn this paper, the identification routine is illustrated by applying it to an industrial pick-and-place unit (Fig. \\ref{fig:Experimental_Setup}) that performs repetitive movements between start point $A$ with angular position $\\theta_A = 0$ and endpoint $B$ with angular position $\\theta_B = 173.6^\\circ$. The resulting inertia $J(\\theta)$ and load torque $\\tau_l(\\theta)$ profiles are presented in Fig. \\ref{fig:sytem_properties}. Because of the machine position limits $\\theta_A$ and $\\theta_B$, only the green shaded part of the system properties is relevant during operation.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\columnwidth]{Images\/Nedschroef_Real2.png}\n\\includegraphics[width=0.51\\columnwidth]{Images\/Nedschroef_Schematic_Simple.pdf}\n\\caption{Experimental set-up (left) and schematic overview (right) of the pick- and place unit.}\n\\label{fig:Experimental_Setup}\n\\end{figure}\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/System_Properties.pdf}\n  \\caption{Values of system properties inertia $J(\\theta)$ and load torque $\\tau_l(\\theta)$.}\n  \\label{fig:sytem_properties}\n\\end{figure}\n\n\\subsection{Viscous Friction Coefficient}\nOnce the system properties $J(\\theta)$ and $\\tau_l(\\theta)$ are determined, the only indefinite term in Eq. \\eqref{eq:torque_equation} is the friction torque $\\tau_f$, and more specifically $\\mu_k$. In the previous description of the energy flows, the friction torque $\\tau_f$ was neglected, leading to a simple objective (i.e. $\\tau_{rms}$) to quantify the energy consumption. However, it is important to verify this statement for the intended setup. Therefore this section describes a method to quantify the frictional forces $\\tau_f$.\n\nSince the viscous friction coefficient $\\mu_k$ parameter is highly dependent on the practical setup, it is often only possible to determine this parameter experimentally. Therefore, a first measurement is carried out by using an arbitrary motion profile $\\theta^{*}(t)$ as a set point and recording the resulting actual motor torque $\\tau_{m}^{e}(t)$ and position $\\theta^{e}(t)$. The arbitrary motion profile $\\theta^{*}(t)$ can be determined by using a default motion law such as a trapezoidal or s-curve profile.\n\nAfter this measurement, a least squares fit can be used to determine the experimental value of $\\mu_k$, by fitting the torque model $\\tau_{m}(\\theta^{e}(t),\\mu_k)$ with the measured torque $\\tau_{m}^{e}$.\n\nHowever, using the measured position $\\theta^{e}$ and its time derivatives $\\dot{\\theta}^{e}$, $\\ddot{\\theta}^{e}$ in the torque equation \\eqref{eq:torque_equation} leads to unfeasible results since the derivatives amplify any noise that is present in the measurement. Therefore, the measured position $\\theta^{e}(t)$ is fitted with an $n$-th degree polynomial $\\theta^p(t) = \\sum_{i=1}^{n} a_it^i$ and differentiated symbolically to smooth out any noise.\n\n\nThe friction parameter $\\mu_v$ is thus determined by comparing the measured torque $\\tau_{m}^{e}(t)$ with virtual model and fitted motion profile $\\tau_{m}(\\theta^{p}(t))$:\n\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mu_v \\, \\in \\, \\mathbb{R}}{\\text{minimize}}\n& & ||\\tau_{m}^{e}(t) - \\tau_{m}(\\theta^{p}(t),\\mu_v)||_2 \\, .\n\\end{aligned}\n\\end{equation}\n\nFor the pick-and-place unit, a viscous damping coefficient of $0.0157Nms\/rad$ was found. In Fig. \\ref{fig:Torque_Validation}, a comparison of the measured $\\tau_{m}^{e}(t)$ and virtual $\\tau_{m}(\\theta^{p}(t))$ torque is presented. The difference between the virtual torques with and without friction is minimal, which indicates that the friction can be neglected for the present case. The graph also shows a close correlation between the virtual and measured torque, which indicates that the virtual model can be effectively used to minimize the RMS torque $\\tau_{rms}$ and, by extension, the energy consumption $E$.\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Torque_Validation.pdf}\n  \\caption{Comparison of the virtual $\\tau_{m}(\\theta^{p}(t))$ and measured torque $\\tau_{m}^{e}(t)$ (with and without friction).}\n  \\label{fig:Torque_Validation}\n\\end{figure}\n\n\n\n\n\n\n\\section{Optimization Approach}\n\\subsection{Motion Profile Definition \\& Rescaling}\nIn this paper, a Chebyshev polynomial $\\sum_{i=0}^{n} p_iT_i(x)$ is used to define the position profile $\\theta(t)$, where $t \\in [t_A,t_B]$, in between the start-$\\theta(t_A) = \\theta_A$ and endpoint $\\theta(t_B) = \\theta_B$ of the motion task. The sequence of orthogonal Chebyshev polynomials $T_n(x) = T_n(\\cos(\\vartheta))$, defined on the interval $x \\in [-1,1]$, is obtained from the recurrence relation: \n\n\\begin{equation} \\label{eq:def_cheb}\n\\begin{aligned}\nT_0(x) &= 1, \\quad T_1(x) = x, \\\\\nT_{k+1}(x) &= 2xT_k(x)-T_{k-1}(x),\n\\end{aligned}\n\\end{equation}\n\nThis gives exactly the same results as from the trigonometric definition:\n\n\\begin{equation} \\label{eq:def_cheb_tri}\nT_k(\\cos(\\vartheta)) = \\cos(k\\vartheta).\n\\end{equation}\n\nTo use $T_n(x)$ as a representation for the position profile, a linear transformation from $t$ into the range $[-1,1]$ of $x$ is required \\cite{Thompson2013}:\n\n\\begin{equation} \\label{eq:rescale_tx}\n\\begin{aligned}\nt&=\\frac{1}{2}(t_B-t_A)x+\\frac{1}{2}(t_B+t_A) =a x+ b,\n\\end{aligned}\n\\end{equation}\n\nwhere scale factors $a$ and $b$ are defined for the purpose of the following paragraphs. In addition, the position $\\theta$ is also rescaled to the interval $[-1,1]$, which makes it possible to obtain strict bounds on the design space in \\eqref{boundsDS}. Thus, the rescaled motion profile description $\\phi(x)$ of degree $n$ with optimizable coefficients $\\mathbf{p} = [p_0, p_1, \\ldots ,p_n]^T$ is obtained.\n\n\\begin{equation} \\label{eq:position_function_cheb}\n\\phi(x)=\\sum_{i=0}^{n} p_iT_i(x), \\quad x\\in [-1,1].\n\\end{equation}\n\nThe output of the motion simulations in the previous section deliver $n_s$ samples of inertia $\\mathbf{J}= [J_1, \\ldots  ,J_{n_s}]^T$, load torque $\\mathbf{T}_l= [T_{l,1}, \\ldots  ,T_{l,n_s}]^T$ and corresponding angle query points $\\boldsymbol{\\uptheta} = [\\theta_1, \\ldots  ,\\theta_{n_s}]^T$. Due to the position rescaling of the motion profile $\\phi(x)$, the angle query points $\\boldsymbol{\\uptheta}$ of every system property vector $\\mathbf{Y}$ have to be rescaled accordingly:\n\n\\begin{equation} \\label{eq:rescale_prop}\n\\begin{aligned}\n    \\boldsymbol{\\upphi} &= \\frac{2}{(\\theta_B -\\theta_A)} \\, \\boldsymbol{\\uptheta} - \\frac{(\\theta_B +\\theta_A)}{(\\theta_B -\\theta_A)} = c \\, \\boldsymbol{\\uptheta} + d.\n    \\end{aligned}\n\\end{equation}\n\nMoreover, as the property description $Y(\\phi)$ is now defined on the rescaled interval $\\phi \\in [-1,1]$, the following relationship holds with regard to the derivative properties:\n\n\\begin{equation} \\label{eq:rescale_pder}\n\\frac{\\mathrm{d}J(\\phi)}{\\mathrm{d}\\phi} = \\frac{1}{2}(\\theta_B-\\theta_A) \\frac{\\mathrm{d}J(\\theta)}{\\mathrm{d}\\theta} =  e \\,  \\frac{\\mathrm{d}J(\\theta)}{\\mathrm{d}\\theta}.\n\\end{equation}\n\nWhen using the rescaled position profile $\\phi(x)$, it is important to rescale the torque equation \\eqref{eq:torque_equation} as well. Otherwise, the resulting values of the torque profile $T(x)$ are distorted which results in different objective values (i.e. $T_{rms}$) and solutions. To preserve the motor torque's absolute values, the following rescaled torque equation is used:\n\n\\begin{equation} \\label{eq:torque_equation_rescaled}\nT_m(x) = T_l(\\phi) + \\frac{1}{2}\\frac{\\mathrm{d}J(\\phi)}{\\mathrm{d}\\phi}\\frac{1}{e}\\left(\\frac{\\dot{\\phi}}{a.c}\\right)^2 + J(\\theta)\\frac{\\ddot{\\phi}}{a^2.c} + \\mu_k\\frac{\\dot{\\phi}}{a}.\n\\end{equation}\n\nAn overview of the position and torque rescalings is presented in Fig. \\ref{fig:rescaling}. The new system equation \\eqref{eq:torque_equation_rescaled} ensures the system dynamics are equally scaled and the minima are not altered.\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Rescaling2.pdf}\n  \\caption{Original $\\theta(t)$ and rescaled position profiles $\\theta(x)$, $\\phi(x)$ with their corresponding torque equations.}\n  \\label{fig:rescaling}\n\\end{figure}\n\nFor what concerns the constraints, the rest-to-rest motion required zero speed $\\dot{\\phi}$ and acceleration $\\ddot{\\phi}$ in the start and endpoint:\n\n\\begin{equation} \n\\begin{array}{ccccc}\n\\phi(-1)=-1 & , & \\dot{\\phi}(-1)=0 & , & \\ddot{\\phi}(-1)=0,\\\\\n\\phi(1)=1 & , & \\dot{\\phi}(1)=0 & , & \\ddot{\\phi}(1)=0.\n\\end{array}{}\n\\label{eq:constraints}\n\\end{equation}\n\n\nReferring to \\eqref{eq:position_function_cheb}, and by incorporating the motion profile constraints \\eqref{eq:constraints}, the lower degree coefficients $[p_0,... , p_5]^T$ can be written as a function of the remaining coefficients $[p_6,... , p_n]^T$, such that $n-5$ degrees of freedom (DOF) are kept available for the optimization algorithm \\cite{Hsu2014}. Thus, the energy optimal motion profile problem is formulated as the following minimization problem with design variable vector $\\mathbf{o}=[p_6,... , p_n]^T$:\n\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{o} \\, \\in \\, \\mathbb{R}^{n-5}}{\\text{minimize}}\n& & T_{m;rms}(x) = \\sqrt{\\frac{1}{2}\\int_{-1}^{1} {T_m(\\phi(x,\\mathbf{o}))}^2 \\, \\mathrm{d}x} .\n\\end{aligned}\n\\end{equation}\n\n\nIn some applications, an additional constraint of zero jerk in the begin and endpoint can be imposed to limit the vibrations:\n\n\\begin{equation} \\label{eq:constraint_jerk}\n    \\dddot{\\phi}(-1)=0 \\quad;\\quad \\dddot{\\phi}(1)=0.\n\\end{equation}\n\nBecause of these two extra equations, the DOF is reduced to $n-7$ and the design variable vector can be expressed as $\\mathbf{o}=[p_8,... , p_n]^T$.\n\n\\subsection{Initialization \\& Design Space}\nIn this paper, the resulting optimization problem is solved with both a fast \\textit{gradient-based} solver, the BFGS (Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno) quasi-Newton method \\cite{Nocedal2006}, and a global \\textit{heuristic} solver, the genetic algorithm \\cite{Holland1992}.\n\nFor gradient-based optimization, a starting point needs to be defined. The use of the Chebyshev basis $T_i(x)$ in representation \\eqref{eq:position_function_cheb} allows initializing the optimization parameter vector at zero, since the coefficients in a convergent Chebyshev series development of the motion profile function $\\theta(x)$ would converge to zero \\cite{Majidian2017}. Here, we can safely assume some similar behavior for the coefficients $p_i$ in \\eqref{eq:position_function_cheb}.\n\nFor what concerns the genetic algorithm, a similar approach is used for the initialization of the population. However, because a GA often samples a wide part of the design space \\cite{Wenzhong2005}, it is beneficial to determine the exact bounds on the design vector $\\mathbf{o}$. By doing so, the solver can cover a large part of the design space and reveal the global optimal solution. In the following paragraphs, thanks to the rescaled Chebyshev motion profile, strict bounds on the design vector $\\mathbf{o}$ can be derived.\n\nTo define these bounds, we first introduce the integral $I$:\n\n\\begin{equation}\\label{int}\n\tI = \\int\\limits_{0}^{2\\pi} T_k(\\cos\\theta) \\cos(\\ell\\theta) \\, \\mathrm{d}\\theta.\n\\end{equation}\n\nWith respect to Eq. \\eqref{eq:def_cheb_tri} and by using the inverse Simpson rule, the integral can be written as:\n\n\\begin{equation}\n\\begin{aligned}\n\tI &= \\int\\limits_{0}^{2\\pi} \\cos(k\\theta) \\cos(\\ell\\theta) \\, \\mathrm{d}\\theta \\\\\n\t&= \\frac{1}{2} \\int\\limits_{0}^{2\\pi} \\cos\\big((k+\\ell)\\theta\\big)\\,\\mathrm{d}\\theta\n\t+ \\frac{1}{2} \\int\\limits_{0}^{2\\pi} \\cos\\big((k-\\ell)\\theta\\big)\\,\\mathrm{d}\\theta.\n\\end{aligned}\n\\end{equation}\n\nThe problem is divided into three cases of which the results will be used in the following paragraphs:\n\\begin{enumerate}\n\n    \\item \\underline{$k+\\ell = 0$} \\\\\n    Since $k$ and $\\ell$ are both positive, this is only possible for $k=\\ell=0$. In this case, we find\n    \n    \\begin{equation}\\label{eq:case1}\n        I =\n        2\\pi.\n    \\end{equation}\n    \n    \\item \\underline{$k-\\ell = 0$} \\\\\n    In this case, we find that $k = \\ell$, which means that\n    \n    \\begin{equation}\\label{eq:case2}\n    \\begin{aligned}\n        I\n\n\t= \\pi.\n\t\\end{aligned}\n    \\end{equation}\n    \n    \\item \\underline{$k+\\ell \\neq 0$ and $k-\\ell \\neq 0$} \\\\\n    In this case, $k \\neq \\ell$ and both are not 0 at the same time. For the integral we then find\n    \n    \\begin{equation}\\label{eq:case3}\n    \\begin{aligned}\n        I\n        = 0.\n    \\end{aligned}\n    \\end{equation}\n    \n\\end{enumerate}\n \nFurther, a second integral $F$ is introduced which is used later on to obtain the strict bounds. \n\n\\begin{equation}  \\label{eq:integralF}\n    \\begin{aligned}\n    F &= \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)\\cos(\\ell\\theta)\\,\\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nBy taking into account \\eqref{eq:position_function_cheb}, we find the following result:\n\n\\begin{equation} \n    \\begin{aligned}\n    F &= \\int\\limits_{0}^{2\\pi} \\left(\\sum\\limits_{k=0}^n p_k T_k(\\cos\\theta)\\right)\\cos(\\ell\\theta)\\,\\mathrm{d}\\theta \\\\\n\t&= \\sum\\limits_{k=0}^n p_k \\int\\limits_{0}^{2\\pi} T_k(\\cos\\theta) \\cos(\\ell\\theta) \\, \\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nThen, by taking into account \\eqref{eq:case3}, only the term for which $k=l$ remains in the summation:\n\n\\begin{equation}\n    \\begin{aligned}\n    F = p_\\ell \\int\\limits_{0}^{2\\pi} T_\\ell(\\cos\\theta) \\cos(\\ell\\theta) \\, \\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nThis can be split into two cases. For $\\ell = 0$ and with respect to \\eqref{eq:case1} and \\eqref{eq:integralF} we find:\n\\begin{equation}\\label{c0}\n\tp_0 = \\dfrac{1}{2\\pi} \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)\\,\\mathrm{d}\\theta,\n\\end{equation}\n\nand for $\\ell > 0$ with respect to \\eqref{eq:case2} and \\eqref{eq:integralF}:\n\n\\begin{equation}\\label{cl}\n\tp_\\ell = \\dfrac{1}{\\pi} \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)\\cos(\\ell\\theta)\\,\\mathrm{d}\\theta.\n\\end{equation}\n\nFor $\\theta \\in [0,2\\pi]$, $\\cos\\theta$ lies in interval $[-1,1]$, so the image of $\\phi(\\cos\\theta)$ also lies in the interval $[-1,1]$. Thus, we find:\n\n\\begin{equation}\n\t|p_0| \\leq \\dfrac{1}{2\\pi}\\int\\limits_{0}^{2\\pi} |\\phi(\\cos\\theta)|\\,\\mathrm{d}\\theta \\leq \\dfrac{1}{2\\pi}\\int\\limits_{0}^{2\\pi} \\,\\mathrm{d}\\theta = 1.\n\\end{equation}\n\nand\n\n\\begin{equation}\n|p_\\ell| \\leq \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi} |\\phi(\\cos\\theta)||\\cos(\\ell\\theta)|\\,\\mathrm{d}\\theta \\leq \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi}|\\cos(\\ell\\theta)| \\,\\mathrm{d}\\theta.\n\\end{equation}\n\nTo calculate this last integral, we use the periodicity of the function $\\cos(\\ell\\theta)$. This function has a period of $2\\pi \/ \\ell$, so goes $\\ell$ times up and down on the interval $[0,2\\pi]$. So, after taking the absolute value of this function, we find $2\\ell$ times the integral over the positive part of a period, for example, the interval $[-\\pi\/2\\ell, \\pi\/2\\ell]$:\n\n\\begin{equation}\n\\begin{aligned}\n    \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi}|\\cos(\\ell\\theta)| \\,\\mathrm{d}\\theta = \\dfrac{2\\ell}{\\pi} \\int\\limits_{-\\pi\/2\\ell}^{\\pi\/2\\ell} \\cos(\\ell\\theta)\\,\\mathrm{d}\\theta = \\dfrac{4}{\\pi}.\n\\end{aligned}\n\\end{equation}\n\nThus, the following bounds were obtained:\n\n\\begin{equation} \\label{boundsDS}\n    |p_0| \\leq 1 \\hspace{0.5cm} \\text{ and } \\hspace{0.5cm} |p_\\ell| \\leq \\dfrac{4}{\\pi}, \\hspace{0.3cm} \\ell =1,\\dots,n.\n\\end{equation}\n\nThese constraints on the design space are exploited in the subsequent optimization.\n\n\n\\section{Results}\n\\subsection{Motion Profile Optimization}\nIn order to assess the performance of the proposed method, a set of optimizations has been performed on the industrial pick-and-place unit depicted in Fig. \\ref{fig:Experimental_Setup}. The mechanism is required to move between its start position $\\theta_A$ of $0^\\circ$ and end position $\\theta_B$ of $173.6^\\circ$ and has a motion time $\\Delta t$ of $73.5ms$. As for the constraint, two different cases are considered, namely\n\n\\begin{itemize}\n    \\item \\textit{Jerk Free (JF)}: Only the boundary constraints of \\eqref{eq:constraints} are taken into account. The corresponding rescaled Chebyshev position profile $\\phi(x)$ of degree $n$ is hereafter referred to as \\textit{cheb\"n\"}. A 5th-degree polynomial, hereafter indicated as \\textit{poly5}, is taken as the reference motion profile for comparison purposes. This is the smallest degree polynomial that satisfies the constraints.\n    \n    \\item \\textit{Jerk Zero (J0)}: In addition to the constraint of a jerk-free optimization, a zero-jerk constraint is added in the start, and endpoint \\eqref{eq:constraint_jerk} is added to the motion profile definition. The resulting $n$-th degree position profile $\\phi(x)$ is referred to as \\textit{cheb\"n\"J0}. The reference motion profile is in this case a 7th-degree polynomial, hereafter referred to as \\textit{poly7J0}.\n\\end{itemize}\n\nFor every case, the resulting optimization problem is solved in a MATLAB environment for degrees $n= 7, 9, 11,$ and $13$. The results are presented in Fig. \\ref{fig:Optimization_Results} and Tables \\ref{tab:results_jerkfree} \\& \\ref{tab:results_jerkzero} where for every motion profile, the corresponding RMS torque $\\tau_{rms}$ and solve time $t_{sol}$ are displayed. Savings up to $54.4\\%$ are obtained in under $0.77$ s. The results clearly converge towards a minimal value for increasing degree $n$. In general, the motion profiles which include the jerk constraint \\eqref{eq:constraint_jerk}, have slightly bigger $\\tau_{rms}$ values, which is to be expected due to the fact that this extra constraint limits the acceleration near the endpoints while it is desirable to have high accelerations here since the inertia is low.\n\nIn Table \\ref{tab:results_jerkfree}, the $\\tau_{rms}$ values of a conventional trapezoidal 1\/3 motion profile are presented as well, which accelerates during 1\/3rd of the time, moves at a constant speed during 1\/3rd, and decelerates at the last 1\/3rd \\cite{Park1996}. What is interesting in this table is that the torque demand can already be significantly reduced by selecting an adequate default motion law. Notwithstanding that the greatest savings are realized after optimization.\n\nIt is worth noting that for the jerk-free motion profiles, the same solution was found for both the genetic algorithm and gradient-based solver. However, the calculation times with GA are considerably higher. When including the jerk constraint, the GA comes close but does not completely reveal the full optimization potential. Therefore, for what concerns the present study, gradient-based optimizations algorithms are preferable. Since the GA did not obtain a better solution for any motion profile in the bounded search space, we can expect that the results obtained with the gradient-based method are global optimal solutions.\n\nAlthough only the forward motion is considered here, similar results can be obtained for the return motion by simply changing the position constraints.\n\n\n\\begin{table} \n\\caption{Results of the motion profile optimization (Jerk Free).}\n\\label{tab:results_jerkfree}\n\\centering\n\\begin{tabular}{lcccc}\n            & \\multicolumn{2}{c}{\\textbf{Gradient-Based }}                                       & \\multicolumn{2}{c}{\\textbf{Genetic Algorithm}}                                    \\\\\n\\textbf{JF} & \\textbf{$\\tau_{rms}\\,[Nm]$}                                & \\textbf{$t_{sol}\\, [s]$} & \\textbf{\\textbf{$\\tau_{rms}\\,[Nm]$}}                      & \\textbf{$t_{sol}\\,[s]$}  \\\\ \n\\hline\\hline\npoly5 (ref.)       & \\begin{tabular}[c]{@{}c@{}}22.48\\end{tabular}        & -                        & \\begin{tabular}[c]{@{}c@{}}22.48\\end{tabular}       & -                        \\\\ \n\\cmidrule(lr){1-5}\ntrap       & \\begin{tabular}[c]{@{}c@{}}17.16 \\\\-23.7\\%\\end{tabular} & -                     & \\begin{tabular}[c]{@{}c@{}}17.16\\\\-23.7\\%\\end{tabular} & -                     \\\\ \n\\cmidrule(lr){1-5}\ncheb7       & \\begin{tabular}[c]{@{}c@{}}13.78 \\\\-38.7\\%\\end{tabular} & 0.21                     & \\begin{tabular}[c]{@{}c@{}}13.78\\\\-38.7\\%\\end{tabular} & 3.28                     \\\\ \n\\cmidrule(lr){1-5}\ncheb9       & \\begin{tabular}[c]{@{}c@{}}12.47 \\\\-44.5\\%\\end{tabular} & 0.32                     & \\begin{tabular}[c]{@{}c@{}}12.47\\\\-44.5\\%\\end{tabular} & 40.33                    \\\\ \n\\cmidrule(lr){1-5}\ncheb11      & \\begin{tabular}[c]{@{}c@{}}12.33 \\\\-45.2\\%\\end{tabular} & 0.51                     & \\begin{tabular}[c]{@{}c@{}}12.33\\\\-45.2\\%\\end{tabular} & 67.05                    \\\\ \n\\cmidrule(lr){1-5}\ncheb13      & \\begin{tabular}[c]{@{}c@{}}12.29 \\\\-45.4\\%\\end{tabular} & 1.06                     & \\begin{tabular}[c]{@{}c@{}}12.29\\\\-45.4\\%\\end{tabular} & 142.34                   \\\\\n\\cmidrule(lr){1-5}\n\\end{tabular}\n\\end{table}\n\n\\begin{table} \n\\caption{Results of the motion profile optimization (Jerk 0).}\n\\label{tab:results_jerkzero}\n\\centering\n\\begin{tabular}{lcccc}\n            & \\multicolumn{2}{c}{\\textbf{Gradient-Based }}                                       & \\multicolumn{2}{c}{\\textbf{Genetic Algorithm}}                                    \\\\\n\\textbf{J0} & \\textbf{$\\tau_{rms}\\,[Nm]$}                                & \\textbf{$t_{sol}\\, [s]$} & \\textbf{\\textbf{$\\tau_{rms}\\,[Nm]$}}                      & \\textbf{$t_{sol}\\,[s]$}  \\\\ \n\\hline\\hline\npoly7J0 (ref.)     & \\begin{tabular}[c]{@{}c@{}}28.44\\end{tabular}        & -                        & \\begin{tabular}[c]{@{}c@{}}28.44\\end{tabular}       & -                        \\\\ \n\\cmidrule(r){1-5}\ncheb9J0     & \\begin{tabular}[c]{@{}c@{}}16.12 \\\\-43.3\\%\\end{tabular} & 0.27                     & \\begin{tabular}[c]{@{}c@{}}16.12\\\\-43.3\\%\\end{tabular} & 6.15                     \\\\ \n\\cmidrule(r){1-5}\ncheb11J0    & \\begin{tabular}[c]{@{}c@{}}13.61 \\\\-52.2\\%\\end{tabular} & 0.38                     & \\begin{tabular}[c]{@{}c@{}}14.11\\\\-50.4\\%\\end{tabular} & 175.23                   \\\\ \n\\cmidrule(lr){1-5}\ncheb13J0      & \\begin{tabular}[c]{@{}c@{}}12.98\\\\-54.4\\%\\end{tabular}  & 0.77                     & \\begin{tabular}[c]{@{}c@{}}13.15\\\\-53.8\\%\\end{tabular} & 195.02                   \\\\\n\\cmidrule(lr){1-5}\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Optimization_Results.pdf}\n  \\caption{Results of the motion profile optimization for different degrees $n$.}\n  \\label{fig:Optimization_Results}\n\\end{figure}\n\n\n\\subsection{Measurements}\nThe theoretical results are validated against experimental measurements on the pick-and-place unit (Fig. \\ref{fig:Experimental_Setup}). The setup comprises a Beckhoff CX5140 PLC, a Beckhoff AX5901 motor drive, and a Beckhoff AM3064 PMSM, which is connected to the shaft of the mechanism. In order to measure the input electrical energy, a Tektron PA4000 power analyzer is used to analyze the power supply (Fig. \\ref{fig:Schematic_Experimental_Setup}).\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Experimental_Setup.pdf}\n  \\caption{Schematic overview of the experimental setup.}\n  \\label{fig:Schematic_Experimental_Setup}\n\\end{figure}\n\nThe theoretical savings potential of the motion profile optimization is only fulfilled when the motor is capable of following the optimized position setpoint. Therefore, a performant motion controller needs to be designed in order to keep the tracking error as low as possible. Here, similar to \\cite{VanOosterwyck2019}, a cascade controller with torque and speed feedforward is employed as it has proven to be successful for high dynamic systems. The look-up table for the feedforward torque is determined using the torque equation \\eqref{eq:torque_equation_rescaled}.\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Motion_Controller.pdf}\n  \\caption{Schematic overview of the cascade motion controller with feedforward \\cite{VanOosterwyck2019}.}\n  \\label{fig:Motion_Controllers}\n\\end{figure}\n\nIn Tables \\ref{tab:measurement_jerkfree} and \\ref{tab:measurement_jerkzero}, the results of both the measured RMS torque $\\tau_{rms}$ and measured input electrical energy $E$ for different motion profiles are presented. As expected from the simulations, the lowest absolute energy consumption is obtained when using jerk-free motion profiles. When the jerk constraint is active, a decrease of 62.9\\% in energy consumption can be achieved by optimizing the motion profile, while a relative saving of 52.5\\% is possible if no extra constraint on the jerk is imposed.\n\nThe measured $\\tau_{rms;meas}$ and calculated RMS motor torque $\\tau_{rms}$ show a very high similarity, which confirms that the present system model is valid.\n\n\\begin{table}\n\\caption{Experimental results with energy measurement (Jerk Free).}\n\\label{tab:measurement_jerkfree}\n\\centering\n\\begin{tabular}{cccc}\n\\textbf{JF} & \\textbf{$\\tau_{rms}\\,[Nm]$}                               & \\textbf{$\\tau_{rms;meas}\\,[Nm]$}                          & \\textbf{$E_{meas} \\, [Wh] $}                                   \\\\ \n\\hline\\hline\npoly5       & 22.48                                                  & 19.59                                                  & 312.2                                                   \\\\ \n\\cmidrule(r){1-4}\ntrap       & \\begin{tabular}[c]{@{}c@{}}17.16\\\\-23.7\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}15.88\\\\-18.98\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}215.1\\\\-31.1\\%\\end{tabular}  \\\\ \n\\cmidrule(r){1-4}\ncheb7       & \\begin{tabular}[c]{@{}c@{}}13.78\\\\-38.7\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}13.40\\\\-31.6\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}181.7\\\\-41.8\\%\\end{tabular}  \\\\ \n\\cmidrule(lr){1-4}\ncheb9       & \\begin{tabular}[c]{@{}c@{}}12.47\\\\-44.5\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}12.07\\\\-38.4\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}152.3\\\\-51.2\\%\\end{tabular}  \\\\ \n\\cmidrule(lr){1-4}\ncheb11      & \\begin{tabular}[c]{@{}c@{}}12.33\\\\-45.2\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}11.93\\\\-39.1\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}150.1\\\\-51.9\\%\\end{tabular}  \\\\ \n\\cmidrule(lr){1-4}\ncheb13      & \\begin{tabular}[c]{@{}c@{}}12.29\\\\-45.4\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}11.83\\\\-39.6\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}148.2\\\\-52.5\\%\\end{tabular}  \\\\\n\\cmidrule(lr){1-4}\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Experimental results with energy measurement (Jerk Zero).}\n\\label{tab:measurement_jerkzero}\n\\centering\n\\begin{tabular}{cccc}\n\\textbf{J0} & \\textbf{$\\tau_{rms}\\,[Nm]$}                               & \\textbf{$\\tau_{rms;meas}\\,[Nm]$}                          & \\textbf{$E_{meas} \\, [Wh] $}                                   \\\\ \n\\hline\\hline\npoly7J0     & 28.44                                                  & 25.30                                                  & 458.5                                                   \\\\ \n\\cmidrule(lr){1-4}\ncheb9J0     & \\begin{tabular}[c]{@{}c@{}}16.12\\\\-43.3\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}15.81\\\\-37.5\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}222.9\\\\-51.4\\%\\end{tabular}  \\\\ \n\\cmidrule(lr){1-4}\ncheb11J0    & \\begin{tabular}[c]{@{}c@{}}13.61\\\\-52.2\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}13.08\\\\-48.3\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}170.3\\\\-62.9\\%\\end{tabular}  \\\\ \n\\cmidrule(lr){1-4}\ncheb13J0    & \\begin{tabular}[c]{@{}c@{}}12.98\\\\-54.4\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}12.72\\\\-49.7\\%\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}170.8\\\\-62.7\\%\\end{tabular}  \\\\\n\\cmidrule(lr){1-4}\n\\end{tabular}\n\\end{table}\n\\section{Conclusion}\nThis study proposes a novel approach for motion profile optimization of PTP motions with Chebyshev polynomials. At first, system properties have been extracted from both CAD motion simulations and measurements to obtain an accurate virtual twin of the system. A Chebyshev motion profile with scaling laws is presented. Especially novel in this paper is the derivation of the boundary conditions of this profile which enables to define bounds for the design variables. The latter allows to use an optimizer that is designed to obtain globally optimal solutions, i.e. Genetic Algorithm. In addition, the solutions are validated with fast gradient-based optimization algorithms. Finally, experimental optimization results have been considered to verify the feasibility of the proposed solutions.\n\nThe numerical results, achieved on an exemplary model, clearly show that large $\\tau_{rms}$ savings of up to 53.8\\% can be achieved. In addition, it is shown that by employing Chebyshev polynomials for the motion profile, a fast gradient-based optimization can be effectively employed with solve times under $0.8$s. At last, the validation measurements show that similar savings are obtained on the real machine with a maximum energy reduction of $62.9 \\%$.\n\nDue to the straightforward implementation of both the optimization itself and integration of the resulting motion profiles in the motor drive, the proposed method can be easily adopted in any existing configuration where the CAD is data available. Therefore, the proposed method is expected to have a beneficial impact on the energy usage of the envisaged PTP applications.\n\n\n\n\n\\section{Optimization Approach}\n\\subsection{Motion Profile Definition \\& Rescaling}\nIn this paper, a Chebyshev polynomial $\\sum_{i=0}^{n} p_iT_i(x)$ is used to define the position profile $\\theta(t)$, where $t \\in [t_A,t_B]$, in between the start- ($\\theta(t_A) = \\theta_A$) and endpoint ($\\theta(t_B) = \\theta_B$) of the motion task. The sequence of orthogonal Chebyshev polynomials $T_k(x) = T_k(\\cos(\\vartheta))$, defined on the interval $x \\in [-1,1]$, is obtained from the recurrence relation: \n\n\\begin{equation} \\label{eq:def_cheb}\n\\begin{aligned}\nT_0(x) &= 1, \\quad T_1(x) = x, \\\\\nT_{k+1}(x) &= 2xT_k(x)-T_{k-1}(x),\n\\end{aligned}\n\\end{equation}\n\nAlternatively, the polynomials can be derived from the trigonometric definition, which gives exactly the same results:\n\n\\begin{equation} \\label{eq:def_cheb_tri}\nT_k (x) = T_k(\\cos(\\vartheta)) = \\cos(k\\vartheta).\n\\end{equation}\n\nTo use $T_n(x)$ as a representation for the position profile, a linear transformation from $t$ into the range $[-1,1]$ of $x$ is required \\cite{Thompson2013}:\n\n\\begin{equation} \\label{eq:rescale_tx}\n\\begin{aligned}\nt&=\\frac{1}{2}(t_B-t_A)x+\\frac{1}{2}(t_B+t_A) =a x+ b,\n\\end{aligned}\n\\end{equation}\n\nwhere scale factors $a$ and $b$ are defined for the purpose of the following paragraphs. In addition, the position $\\theta \\in [\\theta_A, \\theta_B]$ is also rescaled to the interval $\\phi \\in [-1,1]$, which makes it possible to obtain strict bounds on the design space in \\eqref{boundsDS}. Thus, the rescaled motion profile description $\\phi(x)$ of degree $n$ with optimizable coefficients $\\mathbf{p} = [p_0, p_1, \\ldots ,p_n]^T$ is obtained.\n\n\\begin{equation} \\label{eq:position_function_cheb}\n\\phi(x)=\\sum_{i=0}^{n} p_iT_i(x), \\quad x\\in [-1,1].\n\\end{equation}\n\nThe output of the motion simulations in the previous section deliver $n_s$ samples of inertia $\\mathbf{J}= [J_1, \\ldots  ,J_{n_s}]^T$, load torque $\\boldsymbol{\\uptau}_l= [\\tau_{l,1}, \\ldots  ,\\tau_{l,n_s}]^T$ and corresponding angle query points $\\boldsymbol{\\uptheta} = [\\theta_1, \\ldots  ,\\theta_{n_s}]^T$. Due to the position rescaling of the motion profile $\\phi(x)$, the angle query points $\\boldsymbol{\\uptheta}$ have to be rescaled accordingly:\n\n\\begin{equation} \\label{eq:rescale_prop}\n\\begin{aligned}\n    \\boldsymbol{\\upphi} &= \\frac{2}{(\\theta_B -\\theta_A)} \\, \\boldsymbol{\\uptheta} - \\frac{(\\theta_B +\\theta_A)}{(\\theta_B -\\theta_A)} = c \\, \\boldsymbol{\\uptheta} + d.\n    \\end{aligned}\n\\end{equation}\n\nMoreover, as the property description is now defined on the rescaled interval $\\phi \\in [-1,1]$, the following relationship holds with regard to the derivative properties such inertia variation $\\frac{\\mathrm{d}J(\\phi)}{\\mathrm{d}\\phi}$:\n\n\\begin{equation} \\label{eq:rescale_pder}\n\\frac{\\mathrm{d}J(\\phi)}{\\mathrm{d}\\phi} = \\frac{1}{2}(\\theta_B-\\theta_A) \\frac{\\mathrm{d}J(\\theta)}{\\mathrm{d}\\theta} =  e \\,  \\frac{\\mathrm{d}J(\\theta)}{\\mathrm{d}\\theta}.\n\\end{equation}\n\nWhen using the rescaled position profile $\\phi(x)$, it is important to rescale the torque equation \\eqref{eq:torque_equation} as well. Otherwise, the resulting values of the torque profile $\\tau(x)$ are distorted which results in different objective values (i.e. $\\tau_{rms}$) and solutions. To preserve the motor torque's absolute values, the following rescaled torque equation is introduced:\n\n\\begin{equation} \\label{eq:torque_equation_rescaled}\n\\tau_m(x) = \\tau_l(\\phi) + \\frac{1}{2}\\frac{\\mathrm{d}J(\\phi)}{\\mathrm{d}\\phi}\\frac{1}{e}\\left(\\frac{\\dot{\\phi}}{a.c}\\right)^2 + J(\\theta)\\frac{\\ddot{\\phi}}{a^2.c} + \\mu_k\\frac{\\dot{\\phi}}{a.c}.\n\\end{equation}\n\nAn overview of the position and torque rescalings is presented in Fig. \\ref{fig:rescaling}. The new system equation \\eqref{eq:torque_equation_rescaled} ensures the system dynamics are equally scaled and the minima are not altered.\n\n\\begin{figure}[thpb]\n  \\centering\t\n  \\includegraphics[width=\\columnwidth]{Images\/Rescaling2.pdf}\n  \\caption{Original $\\theta(t)$ and rescaled position profiles $\\theta(x)$, $\\phi(x)$ with their corresponding torque equations.}\n  \\label{fig:rescaling}\n\\end{figure}\n\nFor what concerns the constraints, the rest-to-rest motion requires zero speed $\\dot{\\phi}$ and acceleration $\\ddot{\\phi}$ in the start and endpoint:\n\n\\begin{equation} \n\\begin{array}{ccccc}\n\\phi(-1)=-1 & , & \\dot{\\phi}(-1)=0 & , & \\ddot{\\phi}(-1)=0,\\\\\n\\phi(1)=1 & , & \\dot{\\phi}(1)=0 & , & \\ddot{\\phi}(1)=0.\n\\end{array}{}\n\\label{eq:constraints}\n\\end{equation}\n\n\nReferring to \\eqref{eq:position_function_cheb}, and by incorporating the motion profile constraints \\eqref{eq:constraints}, the lower degree coefficients $[p_0,... , p_5]^T$ can be written as a function of the remaining coefficients $[p_6,... , p_n]^T$, such that $n-5$ degrees of freedom (DOF) are kept available for the optimization algorithm \\cite{Hsu2014}. Thus, the energy optimal motion profile problem is formulated as the following minimization problem with design variable vector $\\mathbf{o}=[p_6,... , p_n]^T$:\n\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{o} \\, \\in \\, \\mathbb{R}^{n-5}}{\\text{minimize}}\n& & \\tau_{rms} = \\sqrt{\\frac{1}{2}\\int_{-1}^{1} {\\tau_m(\\phi(x,\\mathbf{o}))}^2 \\, \\mathrm{d}x} .\n\\end{aligned}\n\\end{equation}\n\n\nIn some applications, an additional constraint of zero jerk in the begin and endpoint can be imposed to limit the vibrations:\n\n\\begin{equation} \\label{eq:constraint_jerk}\n    \\dddot{\\phi}(-1)=0 \\quad;\\quad \\dddot{\\phi}(1)=0.\n\\end{equation}\n\nBecause of these two extra equations, the DOF is reduced to $n-7$ and the design variable vector can be expressed as $\\mathbf{o}=[p_8,... , p_n]^T$.\n\n\\subsection{Initialization \\& Design Space}\nIn this paper, the resulting optimization problem is solved with both a fast \\textit{gradient-based} solver, the BFGS (Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno) quasi-Newton method \\cite{Nocedal2006}, and a global \\textit{heuristic} solver, the genetic algorithm \\cite{Holland1992}.\n\nFor gradient-based optimization, a starting point needs to be defined. The use of the Chebyshev basis $T_i(x)$ in representation \\eqref{eq:position_function_cheb} allows initializing the optimization parameter vector at zero since the coefficients in a convergent Chebyshev series development of the motion profile function $\\phi(x)$ would converge to zero \\cite{Majidian2017}. Here, we can safely assume some similar behavior for the coefficients $p_i$ in \\eqref{eq:position_function_cheb}.\n\nFor what concerns the genetic algorithm, a similar approach is used for the initialization of the population. However, because a GA often samples a wide part of the design space \\cite{Wenzhong2005}, it is beneficial to determine the exact bounds on the design vector $\\mathbf{o}$. By doing so, the solver can cover a large part of the design space and reveal the global optimal solution. In the following paragraphs, thanks to the rescaled Chebyshev motion profile $\\phi(x)$, strict bounds on the design vector $\\mathbf{o}$ can be derived.\n\nTo define these bounds, we take a look at the projection of the position profile $\\phi(x)$ onto the orthogonal Chebyshev polynomial basis $T_l(x)$. Given that $x =\\cos(\\theta)$, we introduce the inner product $F$:\n\n\\begin{equation}  \\label{eq:integralF}\n    \\begin{aligned}\n    F = \\langle\\phi(x),T_l(x)\\rangle &= \\int\\limits_{-1}^{1} \\frac{\\phi(x)T_l(x)}{\\sqrt{1-x^2}}\\,\\mathrm{d}x\\\\\n    &= \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)T_l(\\cos\\theta)\\,\\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nThen, by taking into account the position function definition \\eqref{eq:position_function_cheb}, we find the following result:\n\n\\begin{equation} \n    \\begin{aligned}\n    F &= \\int\\limits_{0}^{2\\pi} \\left(\\sum\\limits_{k=0}^n p_k T_k(\\cos\\theta)\\right)T_l(\\cos\\theta)\\,\\mathrm{d}\\theta \\\\\n\t&= \\sum\\limits_{k=0}^n p_k \\int\\limits_{0}^{2\\pi} T_k(\\cos\\theta) T_l(\\cos\\theta) \\, \\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nHere, the integral $I= \\int\\limits_{0}^{2\\pi} T_k(\\cos\\theta) T_l(\\cos\\theta) \\, \\mathrm{d}\\theta$ can be further simplified by using the Chebyshev polynomial orthogonality properties, which are rederived here for the sake of readability. Because of Eq. \\eqref{eq:def_cheb_tri} and by using the inverse Simpson rule of trigonometry, the integral $I$ can be written as:\n\n\\begin{equation}\n\\begin{aligned}\n\tI &= \\int\\limits_{0}^{2\\pi} \\cos(k\\theta) \\cos(\\ell\\theta) \\, \\mathrm{d}\\theta \\\\\n\t&= \\frac{1}{2} \\int\\limits_{0}^{2\\pi} \\cos\\big((k+\\ell)\\theta\\big)\\,\\mathrm{d}\\theta\n\t+ \\frac{1}{2} \\int\\limits_{0}^{2\\pi} \\cos\\big((k-\\ell)\\theta\\big)\\,\\mathrm{d}\\theta.\n\\end{aligned}\n\\end{equation}\n\nThis integral can be split into three cases:\n\\begin{enumerate}\n    \\item \\underline{$k =\\ell = 0$} \\\\\n    \n    \\begin{equation}\\label{eq:case1}\n        I =\n        2\\pi,\n    \\end{equation}\n    \n    \\item \\underline{$k =\\ell \\neq 0$} \\\\\n    \n    \\begin{equation}\\label{eq:case2}\n    \\begin{aligned}\n        I\n\n\t= \\pi,\n\t\\end{aligned}\n    \\end{equation}\n    \n    \\item \\underline{$k \\neq \\ell$} \\\\\n    \n    \\begin{equation}\\label{eq:case3}\n    \\begin{aligned}\n        I\n        = 0.\n    \\end{aligned}\n    \\end{equation}\n\\end{enumerate}\n \n\nThus, by taking into account \\eqref{eq:case3}, only the term for which $k=l$ remains in the summation $F$:\n\n\\begin{equation}\n    \\begin{aligned}\n    F = p_\\ell \\int\\limits_{0}^{2\\pi} \\cos^2 (\\ell\\theta) \\, \\mathrm{d}\\theta.\n    \\end{aligned}\n\\end{equation}\n\nThis can be split into two cases. For $\\ell = 0$ and by making use of \\eqref{eq:case1} and \\eqref{eq:integralF} we find:\n\\begin{equation}\\label{c0}\n\tp_0 = \\dfrac{1}{2\\pi} \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)\\,\\mathrm{d}\\theta,\n\\end{equation}\n\nand for $\\ell > 0$, by making use of \\eqref{eq:case2} and \\eqref{eq:integralF}:\n\n\\begin{equation}\\label{cl}\n\tp_\\ell = \\dfrac{1}{\\pi} \\int\\limits_{0}^{2\\pi} \\phi(\\cos\\theta)\\cos(\\ell\\theta)\\,\\mathrm{d}\\theta.\n\\end{equation}\n\nFor $\\theta \\in [0,2\\pi]$, $\\cos\\theta$ lies in interval $[-1,1]$. Because of the position rescalings of the motion profile $\\phi(x)$, the image $\\phi(\\cos\\theta)$ also lies in the interval $[-1,1]$. Thus, we find:\n\n\\begin{equation}\n\t|p_0| \\leq \\dfrac{1}{2\\pi}\\int\\limits_{0}^{2\\pi} |\\phi(\\cos\\theta)|\\,\\mathrm{d}\\theta \\leq \\dfrac{1}{2\\pi}\\int\\limits_{0}^{2\\pi} \\,\\mathrm{d}\\theta = 1.\n\\end{equation}\n\nand\n\n\\begin{equation}\n|p_\\ell| \\leq \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi} |\\phi(\\cos\\theta)||\\cos(\\ell\\theta)|\\,\\mathrm{d}\\theta \\leq \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi}|\\cos(\\ell\\theta)| \\,\\mathrm{d}\\theta.\n\\end{equation}\n\nTo calculate this last integral, we use the periodicity of the function $\\cos(\\ell\\theta)$. This function has a period of $2\\pi \/ \\ell$, so goes $\\ell$ times up and down on the interval $[0,2\\pi]$. So, after taking the absolute value of this function, we find $2\\ell$ times the integral over the positive part of a period, for example, the interval $[-\\pi\/2\\ell, \\pi\/2\\ell]$:\n\n\\begin{equation}\n\\begin{aligned}\n    \\dfrac{1}{\\pi}\\int\\limits_{0}^{2\\pi}|\\cos(\\ell\\theta)| \\,\\mathrm{d}\\theta = \\dfrac{2\\ell}{\\pi} \\int\\limits_{-\\pi\/2\\ell}^{\\pi\/2\\ell} \\cos(\\ell\\theta)\\,\\mathrm{d}\\theta = \\dfrac{4}{\\pi}.\n\\end{aligned}\n\\end{equation}\n\nThus, the following bounds for the coefficients $p_i$ are obtained:\n\n\\begin{equation}\\label{boundsDS}\n    |p_0| \\leq 1 \\hspace{0.5cm} \\text{ and } \\hspace{0.5cm} |p_\\ell| \\leq \\dfrac{4}{\\pi}, \\hspace{0.3cm} \\ell =1,\\dots,n.\n\\end{equation}\n\nThese constraints on the design space simplify the subsequent optimization.\n\n\n\\section{Introduction}\n\\input{Sections\/01_Introduction}\n\\input{Sections\/02_SystemModelling}\n\\input{Sections\/03_Identification}\n\\input{Sections\/04_Optimization}\n\\input{Sections\/05_Results}\n\\input{Sections\/06_Conclusion}\n\n\n\\section*{Acknowledgements}\nResearch funded by a PhD grant of the Research Foundation Flanders (FWO) [1S88120N].\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nComplex networks have attracted much attention over the last decades.\nThey provide a natural setting to describe many phenomena\nin nature and society~\\cite{abrmp,doro1,doro2,blmch,cup}.\nOne of the salient features of most networks,\neither natural and artificial, is their scalefreeness.\nThis term refers to the broad degree distribution exhibited by these networks.\nThe probability that a node\nhas degree $k$ (i.e., is connected to exactly $k$ other nodes)\nis commonly observed to fall off as a power law:\n\\begin{equation}\nf_k\\sim k^{-\\gamma}.\n\\label{f}\n\\end{equation}\nThis power-law behavior,\nwhich holds in the limit of an infinitely large network,\nwill be referred to hereafter as `stationary'.\nThe exponent usually obeys $\\gamma>2$,\nso that the mean degree of the infinite network is finite.\nGrowing networks with a preferential attachment rule,\nsuch as the well-known Barab\\'asi-Albert (BA) model~\\cite{ba1,ba2},\nhave received a considerable interest,\nas they provide a natural explanation for the observed scalefreeness.\nThe observation that preferential attachment generates\na power-law degree distribution actually dates back\nto much earlier works~\\cite{simon,price}.\n\nScalefree networks, being chiefly characterized by the exponent $\\gamma$\nof their degree distribution,\nare therefore somewhat similar to equilibrium systems\nat their critical point.\nAs a consequence,\nfinite-size (i.e., finite-time) effects can be expected to yield\nimportant corrections to the asymptotic or stationary form~(\\ref{f})\nof the degree distribution.\nThese effects are one of the possible causes of the cutoff phenomenon\nwhich is often observed in the degree distribution of real networks~\\cite{bpv}.\nMore precisely,\nthe largest degree $k_\\star(n)$ of a scalefree network at time $n$\ncan be estimated by means of the following argument of extreme value statistics:\nit is such that the stationary probability\nof having $k\\ge k_\\star(n)$ is of order $1\/n$.\nThe largest degree thus grows as a power law~\\cite{bpv,dms1}:\n\\begin{equation}\nk_\\star(n)\\sim n^\\nu,\\quad\\nu=\\frac{1}{\\gamma-1}.\n\\label{kstar}\n\\end{equation}\nThis growth law is always subextensive,\nbecause one has $\\gamma>2$, so that $\\nu<1$.\nThe cases $2<\\gamma<3$ (i.e., $1\/2<\\nu<1$)\nand $\\gamma>3$ (i.e., $0<\\nu<1\/2$)\nhowever correspond to qualitative differences,\nespecially in the topology\nand in the various dimensions of the networks~\\cite{bck}.\n\nThe goal of this article is to provide a systematic analysis\nof the degree statistics\nof growing network models at a large but finite time $n$.\nBoth the age-resolved distribution $f_k(n,i)$\nof the degree of node $i$ at a later time $n$\nand the distribution $f_k(n)$ of an unspecified node at time $n$\nwill be considered throughout.\nSeveral works have already been devoted to this problem,\nboth for growing networks\nwith preferential attachment~\\cite{dms1,dms2,krl,kr1,kr2,ws,mj}\nand for related models of random graphs and other structures~\\cite{kk,cb}.\nThe present work aims at being systematic in the following three respects:\n\n\\noindent $\\bullet$ {\\it Models.}\nThis work is focussed onto growing network models where\na new node enters at each time step,\nso that nodes can be labeled by their birth date~$n$,\ni.e., the time they enter the network.\nNode $n$ attaches to a single earlier node ($i=1,\\dots,n-1$)\nwith probability $p_{n,i}$.\nThe attachment probabilities and the initial configuration\nentirely define the model.\nThe network thus obtained has the topology of a tree.\nThe degrees $k_i(n)$ of the nodes at time $n$ obey the sum rule\n\\begin{equation}\n\\sum_{i=1}^n k_i(n)=2L(n),\n\\label{sumr}\n\\end{equation}\nwhere $L(n)$ is the number of links of the network at time $n$.\n\nWe will successively consider the following models:\n\n\\noindent -- {\\it Uniform attachment} (UA) (Section~2).\nThe attachment probability is independent of the node,\ni.e., uniform over the network.\nThis model is not scalefree.\nIts analysis serves as a warming up for that of the subsequent models.\n\n\\noindent -- {\\it Barab\\'asi-Albert} (BA) {\\it model} (Section~3).\nThe attachment probability is proportional\nto the degree $k_i(n)$ of the earlier node.\nThis well-known model~\\cite{ba1,ba2} is scalefree,\nwith exponents $\\gamma=3$ and $\\nu=1\/2$.\n\n\\noindent -- {\\it General preferential attachment} (GPA) (Section~4).\nThe attachment probability is proportional\nto the sum $k_i(n)+c$ of the degree of the earlier node\nand of an additive constant $c>-1$.\nThis parameter,\nrepresenting the initial attractiveness of a node~\\cite{dms1},\nis relevant as it yields the continuously varying exponents\n$\\gamma=c+3$ and $\\nu=1\/(c+2)$.\nThe BA and UA model are respectively recovered when $c=0$ and $c\\to\\infty$.\n\n\\noindent $\\bullet$ {\\it Regimes.}\nFor each model, the following three regimes will be considered:\n\n\\noindent -- {\\it Stationary regime} ($k\\ll k_\\star(n)$).\nThe degree distribution is essentially given by its stationary form~(\\ref{f}),\nto be henceforth denoted by $f_{k,{\\mathrm{stat}}}$,\nin order to emphasize its belonging to the stationary regime.\n\n\\noindent -- {\\it Finite-size scaling regime} ($k\\sim k_\\star(n)$).\nIn the scalefree cases, the degree distribution obeys\na multiplicative finite-size scaling law of the form\n\\begin{equation}\nf_k(n)\\approx f_{k,{\\mathrm{stat}}}\\,\\Phi\\!\\left(\\frac{k}{k_\\star(n)}\\right).\n\\label{fssdef}\n\\end{equation}\n\n\\noindent -- {\\it Large-deviation regime} ($k_\\star(n)\\ll k\\sim n$).\nThe degree distribution is usually exponentially small in $n$.\n\n\\begin{table}\n\\caption{Various characteristics of the network for both initial conditions.\nThe listed results hold irrespective of the attachment rule.}\n\\label{tabledef}\n\\begin{tabular}{l|l|l}\n\\hline\\noalign{\\smallskip}\nInitial condition & Case~A & Case~B\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nTopology & tree & rooted tree\\\\\nNumber of links at time $n$ & $L^{(\\mathrm{A})}(n)=n-1$ & $L^{(\\mathrm{B})}(n)=n-1\/2$\\\\\nMean degree at time $n$ & $\\mean{k^{(\\mathrm{A})}(n)}=2-2\/n$ & $\\mean{k^{(\\mathrm{B})}(n)}=2-1\/n$\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nDegrees at time $1$ & $k^{(\\mathrm{A})}_1(1)=0$ & $k^{(\\mathrm{B})}_1(1)=1$\\\\\nand generating polynomials& $F^{(\\mathrm{A})}_1(x)=1$ & $F^{(\\mathrm{B})}_1(x)=x$\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nDegrees at time $2$ & $k^{(\\mathrm{A})}_1(2)=k^{(\\mathrm{A})}_2(2)=1$ & $k^{(\\mathrm{B})}_1(2)=2$, $k^{(\\mathrm{B})}_2(2)=1$\\\\\nand generating polynomials& $F^{(\\mathrm{A})}_2(x)=x$ & $F^{(\\mathrm{B})}_2(x)={\\textstyle\\frac{1}{2}}x(x+1)$\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\\noindent $\\bullet$ {\\it Initial conditions.}\nWe will consider the following two initial conditions:\n\n\\noindent -- {\\it Case~A.}\nThe first node appears at time $n=1$ with degree $k_1(1)=0$.\nThis prescription is natural because the first node initially has no connection.\nAll subsequent nodes appear with degree $k_n(n)=1$.\nIn particular, at time $n=2$ the second node connects to the first one,\nso that $k_1(2)=k_2(2)=1$.\nThe configuration thus obtained is the dimer configuration\nused e.g.~in~\\cite{kr1,kr2}.\nAt time $n$, the network has $L(n)=n-1$ links.\nIt has the topology of a tree.\n\n\\noindent -- {\\it Case~B.}\nThe first node now appears at time $n=1$ with degree $k_1(1)=1$.\nThis formally amounts to saying that this node is connected to a root,\nwhich does not belong to the network.\nIt is natural to associate half a link to this fictitious connection.\nAt time $n=2$ the second node connects to the first one,\nso that $k_1(2)=2$ and $k_2(2)=1$.\nAt time $n$, the network has $L(n)=n-1\/2$ links.\nIt has the topology of a rooted tree.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{dessin_CI2.eps}\n\\caption{\\label{figdef}\nFirst three steps of the construction of the network (upper panel)\nand corresponding interacting particle representation (lower panel)\nfor both initial conditions.}\n\\end{center}\n\\end{figure}\n\nTable~\\ref{tabledef} summarizes various characteristics of the network\nfor both initial conditions,\nwhereas Figure~\\ref{figdef} illustrates the first three steps of the network\nconstruction.\nThe upper panel shows the networks with their nodes and links.\nThe lower panel shows the corresponding representation\nas an interacting particle system,\nwhere each node is viewed as a site occupied by\na number of particles equal to its degree.\nThe total number of particles in the system is therefore $2L(n)$.\nThe information about the topology of the network,\nand especially about the genealogy of the nodes,\nis lost in the interacting particle representation,\nbut this information will not be used in the present study\nwhich is focussed on the statistics of degrees.\n\n\\section{The uniform attachment (UA) model}\n\\label{UA}\n\nThe uniform attachment (UA) model is the simplest of all:\nthe attachment probability is chosen to be uniform over all existing nodes.\nThis section is devoted to an analytical study of the distribution\nof the degree of a fixed node and of an unspecified node,\nexactly taking into account fluctuations, finite-time effects,\nand the influence of the initial condition.\n\n\\subsection{Degree statistics of a fixed node}\n\nWe start with the study of the distribution\nof the degree $k_i(n)$ of node $i$ at time $n$.\nThe node appearing at time $n\\ge2$ links to any of the $n-1$ earlier nodes\n($i=1,\\dots,n-1$) with uniform probability\n\\begin{equation}\np_{n,i}=\\frac{1}{n-1}.\n\\end{equation}\n\nIf we define the degree increment of node $i$ at a later time $j>i$ as\n\\begin{equation}\nI_i(j)=k_i(j)-k_i(j-1)=\\left\\{\\begin{array}{l l}1 &\\mathrm{with\\ probability\\\n}p_{j,i},\\\\0 &\\mathrm{else},\\end{array}\\right\n\\label{idef}\n\\end{equation}\nthe degree $k_i(n)$ of node $i$ at a later time $n$ is given by\n\\begin{equation}\nk_i(n)=k_i(i)+\\sum_{j=i+1}^n I_i(j),\n\\label{kinsum}\n\\end{equation}\nwith $k_i(i)=1$, except for $i=1$ in Case~A, where $k_1(1)=0$\n(see Table~\\ref{tabledef}).\n\nThe mean degree $\\mean{k_i(n)}$ therefore reads ($i\\ge2$)\n\\begin{equation}\n\\mean{k_i(n)}=1+\\sum_{j=i+1}^n\\frac{1}{j-1}\n=H_{n-1}-H_{i-1}+1\\approx\\ln\\frac{n}{i}+1,\n\\label{kave}\n\\end{equation}\nwhere the harmonic numbers $H_n$ are defined in~(\\ref{hardef}).\n\nThe distribution $f_k(n,i)=\\mathop{\\rm Prob}\\nolimits\\{k_i(n)=k\\}$\ncan be encoded in the generating polynomial\n\\begin{equation}\nF_{n,i}(x)=\\bigmean{x^{k_i(n)}}=\\sum_{k=1}^{n}f_k(n,i)x^k.\n\\end{equation}\nAs a consequence of~(\\ref{kinsum}), we have\n\\begin{equation}\nF_{n,i}(x)=x^{k_i(i)}\\prod_{j=i+1}^n\\bigmean{x^{I_i(j)}},\n\\label{fniprod}\n\\end{equation}\nwhere the characteristic function of the degree increment $I_i(j)$\nassumes the simple form\n\\begin{equation}\n\\bigmean{x^{I_i(j)}}=1+(x-1)p_{j,i}=\\frac{x+j-2}{j-1},\n\\end{equation}\nirrespective of $i$.\nWe thus get ($i\\ge2$)\n\\begin{equation}\nF_{n,i}(x)=\\frac{x(i-1)!\\Gamma(x+n-1)}{(n-1)!\\Gamma(x+i-1)},\n\\label{fnires}\n\\end{equation}\nwhereas only $F_{n,1}(x)$ depends on the initial condition according to\n\\begin{equation}\nF_{n,1}^{(\\mathrm{A})}(x)=\\frac{\\Gamma(x+n-1)}{(n-1)!\\Gamma(x)},\\quad\nF_{n,1}^{(\\mathrm{B})}(x)=\\frac{x\\Gamma(x+n-1)}{(n-1)!\\Gamma(x)}.\n\\end{equation}\nThroughout the following, the superscripts ${(\\mathrm{A})}$ and ${(\\mathrm{B})}$ mark a result\nwhich holds for a prescribed initial condition (Case~A or Case~B).\n\nThe product form~(\\ref{fniprod}) implies that the generating polynomials\nof node $i$ at times $n$ and $n+1$ obey the recursion\n\\begin{equation}\nF_{n+1,i}(x)=\\bigmean{x^{I_i(n+1)}}F_{n,i}(x)=\\frac{x+n-1}{n}\\,F_{n,i}(x).\n\\label{fniprec}\n\\end{equation}\nThe probabilities $f_k(n,i)$ therefore obey the recursion\n\\begin{equation}\nf_k(n+1,i)=\\frac{1}{n}\\,f_{k-1}(n,i)+\\left(1-\\frac{1}{n}\\right)f_k(n,i),\n\\label{fkdif}\n\\end{equation}\nwith initial conditions given in Table~\\ref{tabledef}, i.e.,\n\\begin{equation}\nf_k(i,i)=\\delta_{k,1}\\quad(i\\ge2),\\quad\nf_k^{(\\mathrm{A})}(1,1)=\\delta_{k,0},\\quad f_k^{(\\mathrm{B})}(1,1)=\\delta_{k,1}.\n\\label{f2init}\n\\end{equation}\nThe master equations~(\\ref{fkdif}) can be directly written down\nby means of a simple reasoning.\nThey provide an alternative way of describing\nthe evolution of the degree distribution of individual nodes.\n\nThe degree distribution encoded in~(\\ref{fnires})\nhas the following characteristics.\nThe degree of node $i$ at time $n$ ranges from the\nminimal value 1 to the maximal value $n+1-i$.\nThese extremal values occur with probabilities\n\\begin{equation}\nf_1(n,i)=\\frac{i-1}{n-1},\\quad f_{n+1-i}(n,i)=\\frac{(i-1)!}{(n-1)!}.\n\\end{equation}\nThe mean and the variance of the degree can be obtained\nby expanding the result~(\\ref{fnires}) around $x=1$, using\n\\begin{equation}\n\\mean{x^K}=1+(x-1)\\mean{K}+\\frac{1}{2}(x-1)^2\n{\\hskip -9pt}\\underbrace{\\mean{K^2-K}}_{\\mathop{\\rm var}\\nolimits{K}+\\mean{K}^2-\\mean{K}}+\\cdots,\n\\label{devt}\n\\end{equation}\nwhere $K$ is any random variable taking positive integer values.\nWe thus get\n\\begin{equation}\n\\matrix{\n\\mean{k_i(n)}=H_{n-1}-H_{i-1}+1,\\hfill\\cr\\cr\n\\mathop{\\rm var}\\nolimits{k_i(n)}=H_{n-1}-H^{(2)}_{n-1}-H_{i-1}+H^{(2)}_{i-1},\\hfill\n}\n\\label{kmom}\n\\end{equation}\nwhere the harmonic numbers $H_n$ and $H^{(2)}_n$ are defined in~(\\ref{hardef}).\nThe above results hold irrespective of the initial condition.\nThe first one coincides with~(\\ref{kave}).\n\nIn the scaling regime where both times $i$ and $n$ are large and comparable,\nintroducing the time ratio\n\\begin{equation}\nz=\\frac{n}{i}\\ge1,\n\\label{zdef}\n\\end{equation}\nthe expressions~(\\ref{kmom}) yield\n\\begin{equation}\n\\mean{k_i(n)}\\approx\\ln z+1,\\quad\\mathop{\\rm var}\\nolimits{k_i(n)}\\approx\\ln z.\n\\label{kinsca}\n\\end{equation}\n\nIn deriving the above results,\nwe have used the asymptotic behavior\nof the digamma function $\\Psi(x)=\\Gamma'(x)\/\\Gamma(x)$\nand of the trigamma function~$\\Psi'(x)$ as $x\\to\\infty$:\n\\begin{equation}\n\\Psi(x)=\\ln x-\\frac{1}{2x}+\\cdots,\\quad\n\\Psi'(x)=\\frac{1}{x}+\\frac{1}{2x^2}+\\cdots,\n\\end{equation}\nas well as their values at integers:\n\\begin{equation}\n\\Psi(n)=H_{n-1}-{\\gamma_{\\scriptscriptstyle{\\rm E}}},\\quad\n\\Psi'(n)=\\frac{\\pi^2}{6}-H^{(2)}_{n-1},\n\\end{equation}\nwhere\n\\begin{equation}\nH_n=\\sum_{i=1}^n\\frac{1}{i},\\quad\nH^{(2)}_n=\\sum_{i=1}^n\\frac{1}{i^2}\n\\label{hardef}\n\\end{equation}\nare the harmonic numbers of the first and second kind,\nand ${\\gamma_{\\scriptscriptstyle{\\rm E}}}$ is Euler's constant.\n\nThe entire degree distribution can be characterized in the scaling regime.\nEquation~(\\ref{fnires}) indeed yields\n\\begin{equation}\nF_{n,i}(x)\\approx x\\,{\\rm e}^{(x-1)\\ln z},\n\\end{equation}\nirrespective of the initial condition.\nWe recognize the generating function of a Poissonian distribution\nwith parameter $\\lambda=\\ln z$, up to a shift by one unit.\nWe thus obtain~\\cite{kr1,krlead}\n\\begin{equation}\nf_k(n,i)\\approx\\frac{(\\ln z)^{k-1}}{z\\,(k-1)!}.\n\\label{ufz}\n\\end{equation}\n\n\\subsection{Degree statistics of the whole network}\n\nWe now turn to the degree distribution of the whole network at time $n$,\n$f_k(n)=\\mathop{\\rm Prob}\\nolimits\\{k(n)=k\\}$,\nwhere $k(n)$ stands for the degree of an unspecified node.\nWe have\n\\begin{equation}\nf_k(n)=\\frac{1}{n}\\sum_{i=1}^n f_k(n,i).\n\\end{equation}\nThe corresponding generating polynomials,\n\\begin{equation}\nF_n(x)=\\bigmean{x^{k(n)}}=\\sum_{k=1}^nf_k(n)x^k\n=\\frac{1}{n}\\sum_{i=1}^n F_{n,i}(x),\n\\end{equation}\nobey the recursion\n\\begin{equation}\n(n+1)F_{n+1}(x)=(x+n-1)F_n(x)+x,\n\\label{fnrec}\n\\end{equation}\nor equivalently\n\\begin{equation}\n(n+1)f_k(n+1)=f_{k-1}(n)+(n-1)f_k(n)+\\delta_{k,1},\n\\label{fknrec}\n\\end{equation}\nwith initial conditions given in Table~\\ref{tabledef}, i.e.,\n\\begin{equation}\nf_k^{(\\mathrm{A})}(1)=\\delta_{k,0},\\quad\nf_k^{(\\mathrm{B})}(1)=\\delta_{k,1}.\n\\label{f1init}\n\\end{equation}\n\nThe recursion~(\\ref{fnrec}) has a non-polynomial solution, independent of $n$,\n\\begin{equation}\nF_{\\mathrm{stat}}(x)=\\frac{x}{2-x},\n\\end{equation}\ndescribing the stationary\ndegree distribution on an infinitely large network:\n\\begin{equation}\nf_{k,{\\mathrm{stat}}}=\\frac{1}{2^k}\\quad(k\\ge1).\n\\label{fstat}\n\\end{equation}\nThe solution of~(\\ref{fnrec}) reads\n\\begin{equation}\n\\matrix{\nF_n^{(\\mathrm{A})}(x)\n=\\frad{x}{2-x}+\\frad{2(1-x)}{2-x}\\,\\frad{\\Gamma(x+n-1)}{n!\\Gamma(x)},\\cr\\cr\nF_n^{(\\mathrm{B})}(x)\n=\\frad{x}{2-x}+\\frad{x(1-x)}{2-x}\\,\\frad{\\Gamma(x+n-1)}{n!\\Gamma(x)}.}\n\\label{fnres}\n\\end{equation}\n\nThe polynomials $F_n^{(\\mathrm{A})}(x)$ and $F_n^{(\\mathrm{B})}(x)$\nhave respective degrees $n-1$ and~$n$.\nThe first of them which are not listed in Table~\\ref{tabledef} read\n\\begin{equation}\n\\matrix{\nF_3^{(\\mathrm{A})}(x)=\\frac{1}{3}\\,x(x+2),\\hfill&\nF_3^{(\\mathrm{B})}(x)=\\frac{1}{6}\\,x(x^2+2x+3),\\hfill\\cr\\cr\nF_4^{(\\mathrm{A})}(x)=\\frac{1}{12}\\,x(x^2+4x+7),\\hfill&\nF_4^{(\\mathrm{B})}(x)=\\frac{1}{24}\\,x(x+3)(x^2+x+4).\\hfill}\n\\end{equation}\nThe degree $k(n)$ at time $n$ ranges from the\nminimal value 1 to the maximal value $n-1$ (Case~A) or $n$ (Case~B).\nThese extremal values occur with the following probabilities $(n\\ge2)$\n\\begin{equation}\nf_1^{(\\mathrm{A})}(n)=\\frac{1}{2}+\\frac{1}{n(n-1)},\\quad\nf_1^{(\\mathrm{B})}(n)=\\frac{1}{2},\\quad\nf_{n-1}^{(\\mathrm{A})}(n)=\\frac{2}{n!},\\quad\nf_n^{(\\mathrm{B})}(n)=\\frac{1}{n!}.\n\\label{ffac}\n\\end{equation}\n\nWe now turn to the finite-size scaling behavior of the degree distribution\nwhen both $k$ and $n$ are large.\nAs anticipated in the Introduction,\nit is to be expected that the probabilities $f_k(n)$ are close to their limits\n$(f_k(n)\\approx f_{k,{\\mathrm{stat}}})$ for $n$ large at fixed degree $k$,\nand more generally in the stationary regime\nwhere~$k$ is much smaller than some characteristic\ncrossover degree $k_\\star(n)$.\nConversely, the probabilities $f_k(n)$ are expected to be negligible\n$(f_k(n)\\ll f_{k,{\\mathrm{stat}}})$ for $k$ large enough at fixed time $n$,\nand more generally in the large-deviation regime where $k_\\star(n)\\ll k\\sim n$.\nThe crossover scale $k_\\star(n)$ can be estimated as\n$k_\\star(n)\\approx\\mean{k_1(n)}$ (see~(\\ref{kave})).\nNodes with highest degrees are indeed typically expected to be the oldest ones.\nAn alternative route consists in using the argument\nof extreme value statistics alluded to in the Introduction:\nthe largest degree $k_\\star$ at time $n$\nis such that the stationary probability of having $k\\ge k_\\star$\nis of order $1\/n$.\nBoth approaches consistently yield\n\\begin{equation}\nk_\\star(n)\\sim\\ln n.\n\\end{equation}\nFinite-size effects are best revealed by considering the ratios\n\\begin{equation}\nR_k(n)=\\frac{f_k(n)}{f_{k,{\\mathrm{stat}}}}=2^k f_k(n).\n\\label{rdef}\n\\end{equation}\nThese ratios are expected to fall off to zero\nfor $k$ of the order of $k_\\star(n)\\sim\\ln n$.\nFigure~\\ref{auscaling} shows a plot of the ratios $R_k(n)$ against $k\/\\ln n$,\nfor times $n=10^3$ and $n=10^6$ in Case~A.\nNumerically exact values of the $f_k(n)$ are obtained\nby iterating~(\\ref{fknrec}).\nA steeper and steeper crossover is clearly observed.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.7\\linewidth]{aus.eps}\n\\caption{\\label{auscaling}\nPlot of the ratios $R_k(n)$ against $k\/\\ln n$ (see~(\\ref{rdef})),\nfor the UA model with initial condition~A,\nat times $n=10^3$ (empty symbols) and $n=10^6$ (full symbols).}\n\\end{center}\n\\end{figure}\n\nIn order to get some quantitative information on the observed crossover,\nit is advantageous to introduce the differences\n$d_k(n)=R_{k-1}(n)-R_k(n)$ for $k\\ge2$,\ncompleted by $d_1(n)=1-R_1(n)$, i.e., $R_0(n)=1$.\nAlthough the $d_k(n)$ are not positive, most of them are,\nand they sum up to unity, so that it is tempting to think of them\nas a narrow probability distribution living in the crossover region.\nThe generating function of the $d_k(n)$ reads\n\\begin{equation}\nD_n(x)=\\sum_{k\\ge1}d_k(n)x^k=(x-1)F_n(2x)+x.\n\\label{phires}\n\\end{equation}\nThe above picture suggests to define the crossover scale as the first moment\n\\begin{equation}\nk_\\star=\\mu(n)=\\sum_{k\\ge1}kd_k(n)=D_n'(1),\n\\end{equation}\nand the squared width of the crossover front as the variance\n\\begin{equation}\n\\sigma^2(n)=\\sum_{k\\ge1}k^2d_k(n)-\\mu(n)^2=D_n''(1)+\\mu(n)-\\mu(n)^2.\n\\end{equation}\nEquations~(\\ref{fnres}),~(\\ref{phires}) yield\n\\begin{equation}\n\\mu^{(\\mathrm{A})}(n)=2H_n\\approx2(\\ln n+{\\gamma_{\\scriptscriptstyle{\\rm E}}}),\\quad\n\\mu^{(\\mathrm{B})}(n)=2H_n+1\\approx2(\\ln n+{\\gamma_{\\scriptscriptstyle{\\rm E}}})+1,\n\\end{equation}\nand\n\\begin{equation}\n\\sigma^2(n)=2H_n-4H^{(2)}_n\\approx2(\\ln n+{\\gamma_{\\scriptscriptstyle{\\rm E}}}-\\pi^2\/3),\n\\end{equation}\nthe latter result being independent of the initial condition.\n\nThe crossover scale is thus $k_\\star\\approx2\\ln n$,\nwhereas the width of the crossover front grows as\n$\\sigma(n)\\approx(2\\ln n)^{1\/2}$.\nThese predictions are in agreement with the observations which can be made\non Figure~\\ref{auscaling},\nnamely that the crossover takes place around $k\/\\ln n=2$,\nand that it becomes steeper at larger times,\nas its relative width falls off, albeit very slowly, as $(\\ln n)^{-1\/2}$.\n\nAnother illustration of finite-size effects is provided\nby the complex zeros of the polynomials $F_n(x)$.\nThe location of these zeros indeed shows\nhow fast the degree distribution of finite networks,\nencoded in the polynomials $F_n(x)$, converges to the stationary distribution,\nencoded in the function $F_{\\mathrm{stat}}(x)$.\nFor $n\\ge2$, $F_n^{(\\mathrm{A})}(x)$ and $F_n^{(\\mathrm{B})}(x)$ have one trivial zero at $x=0$,\nand respectively $n-2$ and $n-1$ non-trivial ones.\nThe explicit expressions~(\\ref{fnres}) allow one to find\nthe asymptotic locus of the zeros as follows.\nThe most rapidly varying part of these results is the rightmost ratio,\nso that the zeros are asymptotically located on the curve with equation\n$\\abs{\\Gamma(x+n-1)\/(n!\\Gamma(x))}=1$.\nSetting\n\\begin{equation}\nx=n\\xi,\n\\label{xidef}\n\\end{equation}\nand using Stirling's formula, we can recast the above estimate as\n\\begin{equation}\n\\Re{\\left[(1+\\xi)\\ln(1+\\xi)-\\xi\\ln\\xi\\right]}=0.\n\\label{lens}\n\\end{equation}\nThe non-trivial zeros of the polynomials $F_n(x)$\nare thus predicted to escape to infinity linearly with time $n$.\nOnce rescaled by $n$ according to~(\\ref{xidef}),\nthey accumulate onto a well-defined limiting curve in the complex $\\xi$-plane.\nThis curve, with equation~(\\ref{lens}),\nhas the shape of a lens connecting the points~$-1$ and $0$.\nFigure~\\ref{auzeros} illustrates this result with data at time $n=50$\nfor both initial conditions.\nThe polynomials $F_n(x)$ converge to the stationary function $F_{\\mathrm{stat}}(x)$\nwhenever the complex ratio $\\xi=x\/n$ lies within the lens.\nOtherwise they diverge exponentially with $n$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.7\\linewidth]{auz.eps}\n\\caption{\\label{auzeros}\nPlot of the non-trivial zeros of the polynomials $F_n(x)$ for the UA model,\nin the complex plane of the rescaled variable $\\xi=x\/n$.\nSymbols: zeros for $n=50$ in Case~A (empty symbols) and Case~B (full symbols).\nLine: limiting curve with equation~(\\ref{lens}).}\n\\end{center}\n\\end{figure}\n\nA related issue concerns the behavior of the probability $f_k(n)$\nof having a very large degree, of order $k\\sim n$,\nmuch larger than $k_\\star(n)\\sim\\ln n$.\nConsidering Case~A for definiteness,\nthe expression~(\\ref{fnres}) leads to the exact contour-integral representation\n\\begin{equation}\nf_k^{(\\mathrm{A})}(n)=\\oint\\frac{\\d x}{2\\pi\\i\\,x^{k+1}}\n\\left(\\frac{x}{2-x}+\\frac{2(1-x)}{2-x}\\,\\frac{\\Gamma(x+n-1)}{n!\\Gamma(x)}\n\\right).\n\\end{equation}\nThe presence of gamma functions suggests to look for a saddle point $x_{\\mathrm{s}}$\nproportional to $n$.\nSetting $\\zeta=k\/n$, we indeed find $x_{\\mathrm{s}}=n\/v$, where $\\zeta$ and\n$v$ are related through\n\\begin{equation}\n\\zeta=\\frac{\\ln(v+1)}{v}.\n\\end{equation}\nWe thus obtain the following large-deviation estimate\n\\begin{equation}\nf_k(n)\\sim\\exp\\Bigl(-n\\bigl(\\zeta\\ln n+S(\\zeta)\\bigr)\\Bigr),\n\\label{uent}\n\\end{equation}\nwhere the exponent has a usual contribution in $n$\nand a less usual one in $n\\ln n$.\nThe term linear in $n$ involves a large-deviation function $S(\\zeta)$,\nwhich is obtained in the following form, parametrized by $v$:\n\\begin{equation}\nS(\\zeta)=\\frac{1}{v}\\bigl(v\\ln v-\\ln v\\ln(v+1)-(v+1)\\ln(v+1)\\bigr).\n\\end{equation}\nThis function decreases from $S(0)=0$ to $S(1)=-1$.\nThe resulting behavior at $\\zeta=1$, i.e., $\\exp(-n(\\ln n-1))$,\nis in agreement with the inverse factorial expressions~(\\ref{ffac}).\n\n\\section{Linear preferential attachment: the Barab\\'asi-Albert (BA) model}\n\\label{BA}\n\nThe Barab\\'asi-Albert (BA) model is the simplest of the models\nwith preferential attachment:\neach new node connects to earlier nodes with a probability proportional\nto their degrees.\nThe probability that node $n$ connects to an earlier node $i$ thus reads\n\\begin{equation}\np_{n,i}=\\frac{k_i(n-1)}{Z(n-1)},\n\\end{equation}\nwhere $k_i(n-1)$ is the degree of node $i$ at time $n-1$,\ni.e., before node $n$ enters the network.\nThe partition function in the denominator,\n\\begin{equation}\nZ(n)=\\sum_{i=1}^nk_i(n)=2L(n)\n\\end{equation}\n(see~(\\ref{sumr})), ensures that the attachment probabilities add up to unity.\n\nIn the following we analyze the BA model\nalong the lines of the previous section,\nkeeping consistent notations as much as possible.\nThe dependence of the attachment probability $p_{n,i}$\non the degree $k_i(n-1)$\nhowever makes the problem more difficult than\nthe previous one of a uniform attachment.\n\n\\subsection{Degree statistics of a fixed node}\n\nLet us again begin with the\ndistribution $f_k(n,i)=\\mathop{\\rm Prob}\\nolimits\\{k_i(n)=k\\}$\nof the degree of node $i$ at time $n$.\n\nA first estimate of the degree $k_i(n)$ is provided by the following\nrecursion relation for the mean degree $\\mean{k_i(n)}$,\nwhich is a consequence of~(\\ref{idef}):\n\\begin{equation}\n\\mean{k_i(n)}=\\mean{k_i(n-1)}+\\mean{p_{n,i}}\n=\\left(1+\\frac{1}{Z(n-1)}\\right)\\mean{k_i(n-1)}.\n\\label{aveprod}\n\\end{equation}\nIn the scaling regime where both $i$ and $n$ are large,\nusing the expressions of the partition function given in Table~\\ref{tabledef},\ni.e.,\n\\begin{equation}\nZ^{(\\mathrm{A})}(n)=2n-2,\\quad Z^{(\\mathrm{B})}(n)=2n-1,\n\\label{zres}\n\\end{equation}\nthe above relation becomes the differential equation\n\\begin{equation}\n\\frac{\\partial\\mean{k_i(n)}}{\\partial n}\\approx\\frac{\\mean{k_i(n)}}{2n},\n\\end{equation}\nwhich yields\n\\begin{equation}\n\\mean{k_i(n)}\\approx\\left(\\frac{n}{i}\\right)^{1\/2}.\n\\label{bave}\n\\end{equation}\n\nThe generating polynomials $F_{n,i}(x)$ and $F_{n+1,i}(x)$\nwhich encode the distribution of the degree of node $i$\nat successive times $n$ and $n+1$ obey the recursion formula:\n\\begin{eqnarray}\nF_{n+1,i}(x)\n&=&\\bigmean{x^{k_i(n+1)}}=\\bigmean{x^{I_i(n+1)}x^{k_i(n)}}\\nonumber\\\\\n&=&\\bigmean{(1+(x-1)p_{n+1,i})x^{k_i(n)}}\\nonumber\\\\\n&=&\\bigmean{\\left(1+\\frac{x-1}{Z(n)}\\,k_i(n)\\right)x^{k_i(n)}},\n\\end{eqnarray}\ni.e.,\n\\begin{equation}\nF_{n+1,i}(x)=F_{n,i}(x)+\\frac{x(x-1)}{Z(n)}\\frac{\\d F_{n,i}(x)}{\\d x},\n\\label{fnirec}\n\\end{equation}\nwhere $Z(n)$ is given by~(\\ref{zres}).\nThe probabilities $f_k(n,i)$ themselves therefore obey the recursion\n\\begin{equation}\nf_k(n+1,i)\n=\\frac{k-1}{Z(n)}f_{k-1}(n,i)+\\left(1-\\frac{k}{Z(n)}\\right)f_k(n,i),\n\\end{equation}\nwith initial conditions~(\\ref{f2init}).\nThe initial condition for Case~A should be taken at time $n=2$,\nin order to avoid indeterminate expressions, as $Z^{(\\mathrm{A})}(1)=0$.\n\nIn order to solve the recursion~(\\ref{fnirec}),\nwe perform the rational change of variable from $x$ to $u$ such that\n\\begin{equation}\nu=\\frac{x}{1-x},\\quad x=\\frac{u}{u+1},\\quad\nx(x-1)\\frac{\\d}{\\d x}=-u\\frac{\\d}{\\d u}.\n\\label{xtou}\n\\end{equation}\nIntroducing the notation $\\widehat F_{n,i}(u)=F_{n,i}(x)$,\nthe recursion~(\\ref{fnirec}) reads\n\\begin{equation}\n\\widehat F_{n+1,i}(u)=\\widehat F_{n,i}(u)-\\frac{u}{Z(n)}\\frac{\\d\\widehat F_{n,i}(u)}{\\d u}.\n\\label{wfnirec}\n\\end{equation}\nIt is then advantageous to introduce the Mellin transform $M_{n,i}(s)$\nof $\\widehat F_{n,i}(u)$, defined as\n\\begin{equation}\nM_{n,i}(s)=\\int_0^\\infty\\widehat F_{n,i}(u)\\,u^{-s-1}\\,\\d u.\n\\end{equation}\nThe inverse transform reads\n\\begin{equation}\n\\widehat F_{n,i}(u)=\\int_{\\mathrm{C}}\\frac{\\d s}{2\\pi\\i}\\,M_{n,i}(s)\\,u^s,\n\\label{invmel}\n\\end{equation}\nwhere C is a vertical contour in the complex $s$-plane\nwhose position will be defined in a while.\nThe virtue of the Mellin transformation\nis that the recursion~(\\ref{wfnirec}) simplifies to\n\\begin{equation}\nM_{n+1,i}(s)=\\left(1-\\frac{s}{Z(n)}\\right)M_{n,i}(s),\n\\label{mfnirec}\n\\end{equation}\nwith initial condition $M_{i,i}(s)=X_0(s)$ for $i\\ge2$, with\n\\begin{equation}\nX_0(s)=\\int_0^\\infty x(u)\\,u^{-s-1}\\,\\d u=\\int_0^1 x^{-s}(1-x)^{s-1}\\,\\d x\n=\\frac{\\pi}{\\sin\\pi s}\n\\label{xdef}\n\\end{equation}\nfor $0<\\Re s<1$.\nHereafter the contour C is assumed to be in that strip.\nWe thus get $(i\\ge2)$\n\\begin{equation}\n\\matrix{\nM_{n,i}^{(\\mathrm{A})}(s)=\\frad{\\Gamma\\!\\left(n-\\frac{s}{2}-1\\right)\\Gamma(i-1)}\n{\\Gamma\\!\\left(i-\\frac{s}{2}-1\\right)\\Gamma(n-1)}\\,X_0(s),\\hfill\\cr\\cr\nM_{n,i}^{(\\mathrm{B})}(s)=\\frad{\\Gamma\\!\\left(n-\\frac{s}{2}-\\frac{1}{2}\\right)\n\\,\\Gamma\\!\\left(i-\\frac{1}{2}\\right)}\n{\\Gamma\\!\\left(i-\\frac{s}{2}-\\frac{1}{2}\\right)\n\\,\\Gamma\\!\\left(n-\\frac{1}{2}\\right)}\\,X_0(s).\\hfill\n}\n\\label{bfnires}\n\\end{equation}\nThese product formulas in the Mellin variable $s$\nare reminiscent of~(\\ref{fnires}).\n\nThe mean and the variance of the degree of node $i$ at time~$n$\ncan be extracted from these results as follows.\nThe identity~(\\ref{devt}) yields\n\\begin{equation}\n\\widehat F_{n,i}(u)=1-\\frac{\\mean{k_i(n)}}{u}\n+\\frac{\\mean{k_i(n)^2}+\\mean{k_i(n)}}{2u^2}+\\cdots\n\\end{equation}\nas $u\\to+\\infty$.\nFurthermore the coefficients of $1\/u$ and $1\/u^2$\nare respectively the residues of $M_{n,i}(s)$ at $s=-1$ and $s=-2$.\nWe thus obtain\n\\begin{equation}\n\\matrix{\n\\mean{k^{(\\mathrm{A})}_i(n)}=\\frad{\\Gamma\\!\\left(n-\\frac{1}{2}\\right)\\Gamma(i-1)}\n{\\Gamma\\!\\left(i-\\frac{1}{2}\\right)\\Gamma(n-1)},\\quad\n\\mean{k^{(\\mathrm{B})}_i(n)}=\\frad{\\Gamma(n)\\,\\Gamma\\!\\left(i-\\frac{1}{2}\\right)}\n{\\Gamma(i)\\,\\Gamma\\!\\left(n-\\frac{1}{2}\\right)}\n}\n\\end{equation}\nand\n\\begin{equation}\n\\matrix{\n\\mathop{\\rm var}\\nolimits{k^{(\\mathrm{A})}_i(n)}=2\\,\\frad{n-1}{i-1}-\\mean{k^{(\\mathrm{A})}_i(n)}^2-\\mean{k^{(\\mathrm{A})}_i(n)},\\hfill\n\\cr\\cr\n\\mathop{\\rm var}\\nolimits{k^{(\\mathrm{B})}_i(n)}=2\\,\\frad{2n-1}{2i-1}-\\mean{k^{(\\mathrm{B})}_i(n)}^2-\\mean{k^{(\\mathrm{B})}_i(n)}.\\hfill\n}\n\\end{equation}\n\nIn the scaling regime where both times $i$ and $n$ are large and comparable,\nintroducing the time ratio $z=n\/i$ (see~(\\ref{zdef})),\nthe above results yield\n\\begin{equation}\n\\mean{k_i(n)}\\approx z^{1\/2},\\quad\\mathop{\\rm var}\\nolimits{k_i(n)}\\approx z^{1\/2}(z^{1\/2}-1),\n\\label{bkin}\n\\end{equation}\nirrespective of the initial condition.\nThe mean degree is in agreement with the estimate~(\\ref{bave}).\nThe entire degree distribution can actually be derived in the scaling regime.\nEquation~(\\ref{bfnires}) indeed yields\n\\begin{equation}\nM_{n,i}(s)\\approx z^{-s\/2}\\,\\frac{\\pi}{\\sin\\pi s}.\n\\end{equation}\nWe thus obtain\n\\begin{equation}\nF_{n,i}(x)\\approx\\frac{x}{x+z^{1\/2}(1-x)}\n\\end{equation}\nand finally\n\\begin{equation}\nf_k(n,i)\\approx z^{-1\/2}\\bigl(1-z^{-1\/2}\\bigr)^{k-1}.\n\\label{bfz}\n\\end{equation}\nThe degree distribution is therefore found to be asymptotically geometric,\nirrespective of the initial condition~\\cite{kr1,krlead}.\n\n\\subsection{Degree statistics of the whole network}\n\nWe now turn to the degree distribution $f_k(n)=\\mathop{\\rm Prob}\\nolimits\\{k(n)=k\\}$,\nwhere $k(n)$ stands for the degree of an unspecified node.\n\nThe generating polynomials $F_n(x)$ obey the recursion\n\\begin{equation}\n(n+1)F_{n+1}(x)=nF_n(x)+n\\frac{x(x-1)}{Z(n)}\\frac{\\d F_n(x)}{\\d x}+x,\n\\label{bfnrec}\n\\end{equation}\nwhere $Z(n)$ is again given by~(\\ref{zres}), and\nwith initial conditions given in Table~\\ref{tabledef}.\nThe probabilities $f_k(n)$ themselves obey the recursion\n\\begin{equation}\n(n+1)f_k(n+1)\n=\\frac{k-1}{Z(n)}nf_{k-1}(n)+\\left(1-\\frac{k}{Z(n)}\\right)nf_k(n)+\\delta_{k,1}.\n\\label{bfknrec}\n\\end{equation}\n\nThe first generating polynomials which depend on the attachment rule read\n\\begin{equation}\n\\matrix{\nF_3^{(\\mathrm{A})}(x)=\\frac{1}{3}\\,x(x+2),\\hfill&\nF_3^{(\\mathrm{B})}(x)=\\frac{1}{9}\\,x(2x^2+2x+5),\\hfill\\cr\\cr\nF_4^{(\\mathrm{A})}(x)=\\frac{1}{8}\\,x(x^2+2x+5),\\hfill&\nF_4^{(\\mathrm{B})}(x)=\\frac{1}{60}\\,x(6x^3+8x^2+11x+35).\\hfill}\n\\end{equation}\n\nThe stationary degree distribution $f_{k,{\\mathrm{stat}}}$\ncan be determined as the solution of~(\\ref{bfknrec})\nwhich becomes independent of $n$ for large $n$.\nWe thus get\n\\begin{equation}\n(k+2)f_{k,{\\mathrm{stat}}}=(k-1)f_{k-1,{\\mathrm{stat}}}+2\\delta_{k,1},\n\\end{equation}\nhence~\\cite{dms1,krl,kr1}\n\\begin{equation}\nf_{k,{\\mathrm{stat}}}=\\frac{4}{k(k+1)(k+2)}.\n\\label{bfstat}\n\\end{equation}\nAn alternative approach consists in looking for the\nasymptotic generating function $F_{\\mathrm{stat}}(x)$\nas the solution of~(\\ref{bfnrec}) which becomes independent of $n$\nfor large~$n$.\nWe thus obtain the differential equation\n\\begin{equation}\nx(1-x)F'_{\\mathrm{stat}}(x)+2F_{\\mathrm{stat}}(x)=2x,\n\\end{equation}\nwhich has for solution\n\\begin{equation}\nF_{\\mathrm{stat}}(x)=3-\\frac{2}{x}-\\frac{2(1-x)^2}{x^2}\\ln(1-x).\n\\label{fst}\n\\end{equation}\nExpanding this result as a power series in $x$\nallows one to recover~(\\ref{bfstat}).\n\nThe recursion~(\\ref{bfnrec}) for the generating polynomials $F_n(x)$\ncan be solved along the lines of the above solution\nof the recursion~(\\ref{fnirec}).\nThe Mellin transforms $M_n(s)$ of the functions $\\widehat F_n(u)=F_n(x)$\nobey the recursion\n\\begin{equation}\n(n+1)M_{n+1}(s)=\\left(1-\\frac{s}{Z(n)}\\right)nM_n(s)+X_0(s),\n\\label{mfnrec}\n\\end{equation}\nwith initial condition $M_2^{(\\mathrm{A})}(s)=M_1^{(\\mathrm{B})}(s)=X_0(s)$.\nEquation~(\\ref{mfnrec}) has a special solution\n\\begin{equation}\nM_n(s)=\\frac{Z(n)X_0(s)}{(s+2)n},\n\\label{mspec}\n\\end{equation}\nwhereas the general solution of the homogeneous equation\nshares the $n$-de\\-pen\\-den\\-ce of the expressions~(\\ref{bfnires}).\nWe thus obtain\n\\begin{equation}\n\\matrix{\nM_n^{(\\mathrm{A})}(s)=\\frad{2X_0(s)}{(s+2)n}\n\\left(n-1+(s+1)\\frad{\\Gamma\\!\\left(n-\\frac{s}{2}-1\\right)}\n{\\Gamma\\!\\left(1-\\frac{s}{2}\\right)\\,\\Gamma\\!\\left(n-1\\right)}\\right),\\hfill\\cr\\cr\nM_n^{(\\mathrm{B})}(s)=\\frad{X_0(s)}{(s+2)n}\\left(2n-1\n+(s+1)\\frad{\\sqrt{\\pi}\\,\\Gamma\\!\\left(n-\\frac{s}{2}-\\frac{1}{2}\\right)}\n{\\Gamma\\!\\left(\\frac{1}{2}-\\frac{s}{2}\\right)\\,\\Gamma\\!\\left(n-\\frac{1}{2}\\right)}\n\\right).\\hfill\n}\n\\label{bfnres}\n\\end{equation}\nThe common stationary limit of both expressions,\n\\begin{equation}\nM_{\\mathrm{stat}}(s)=\\frac{2X_0(s)}{s+2},\n\\end{equation}\nis proportional to the special solution~(\\ref{mspec}).\nRecalling~(\\ref{xdef}), the inverse Mellin transform of the above result,\n\\begin{equation}\n\\widehat F_{\\mathrm{stat}}(u)=1-\\frac{2}{u}+\\frac{2}{u^2}\\ln(u+1),\n\\end{equation}\nis equivalent to~(\\ref{fst}).\n\nThe results~(\\ref{bfnres}) allow one to investigate,\nat least in principle, every feature of the degree distribution $f_k(n)$.\nLet us take the example of the probability $f_1(n)$\nfor a node to have degree one.\nThe inverse formula~(\\ref{invmel}) shows that this probability\nis equal to minus the residue of $M_n(s)$ at $s=1$.\nThe nature of the subleading corrections to the stationary value $f_{1,{\\mathrm{stat}}}=2\/3$\ndepends on the initial condition.\nFor Case~A we obtain ($n\\ge2$)\n\\begin{equation}\nf_1^{(\\mathrm{A})}(n)=\\frad{2(n-1)}{3n}\n+\\frad{4\\,\\Gamma\\!\\left(n-\\frac{3}{2}\\right)}{3\\sqrt{\\pi}\\,n\\Gamma(n-1)}\n=\\frad{2}{3}-\\frad{2}{3n}+\\frad{4}{3\\sqrt{\\pi}\\,n^{3\/2}}+\\cdots\n\\label{af1}\n\\end{equation}\nMore generally, all the probabilities $f_k(n)$ exhibit\na singular correction in $n^{-3\/2}$.\nCase~B has the remarkable property that all the probabilities $f_k(n)$\nare rational functions of time $n$.\nTheir expansion at large $n$ therefore only involves integer powers of $1\/n$.\nWe have e.g.\n\\begin{equation}\n\\matrix{\nf_1^{(\\mathrm{B})}(n)=\\frad{2n-1}{3n}=\\frad{2}{3}-\\frad{1}{3n},\\hfill\\cr\\cr\nf_2^{(\\mathrm{B})}(n)=\\frad{n^2-2n+3}{3n(2n-3)}\n=\\frad{1}{6}-\\frad{1}{12n}+\\frad{3}{8n^2}+\\cdots\\hfill\n}\n\\label{bf1}\n\\end{equation}\n\nWe now turn to the finite-size scaling behavior\nof the degree distribution when both $k$ and $n$ are large.\nThe crossover scale $k_\\star(n)$ can again be estimated\neither using~(\\ref{bave}) or by the argument of extreme value statistics.\nBoth approaches consistently yield\n\\begin{equation}\nk_\\star(n)\\sim n^{1\/2}.\n\\end{equation}\nWe will now show that the degree distribution\nobeys the multiplicative finite-size scaling law\n\\begin{equation}\nf_k(n)\\approx f_{k,{\\mathrm{stat}}}\\,\\Phi(y),\\quad y=\\frac{k}{n^{1\/2}},\n\\label{fss}\n\\end{equation}\nwhere the scaling function $\\Phi(y)$ is non-universal,\nin the sense that it depends on the initial condition~\\cite{kr2,ws}.\nThe proof of the scaling behavior~(\\ref{fss})\nand the determination of the scaling functions $\\Phi^{(\\mathrm{A})}(y)$\nand $\\Phi^{(\\mathrm{B})}(y)$ go as follows.\nLet us start with Case~A.\nThe second term of the expression~(\\ref{bfnres}) for $M_n^{(\\mathrm{A})}(s)$ scales as\na power law for large $n$:\n\\begin{equation}\nM_{n,{\\mathrm{scal}}}^{(\\mathrm{A})}(s)\\approx\\frac{2(s+1)X_0(s)}\n{(s+2)\\,\\Gamma\\!\\left(1-\\frac{s}{2}\\right)}\\,n^{-s\/2-1}.\n\\end{equation}\nThe inverse Mellin transform of the latter formula,\n\\begin{equation}\n\\widehat F_{n,{\\mathrm{scal}}}^{(\\mathrm{A})}(u)\\approx\\frac{1}{n}\\int_{\\mathrm{C}}\\frac{\\d s}{2\\pi\\i}\n\\,\\frac{2(s+1)X_0(s)}{(s+2)\\,\\Gamma\\!\\left(1-\\frac{s}{2}\\right)}\n\\left(u\/n^{1\/2}\\right)^s,\n\\end{equation}\ndescribes the scaling behavior of $\\widehat F_n^{(\\mathrm{A})}(u)$\nin the regime where $u$ and $n$ are simultaneously large,\nwith $u\/n^{1\/2}$ fixed.\nFinally, by inserting the above scaling estimate\ninto the contour-integral representation\n\\begin{equation}\nf_k^{(\\mathrm{A})}(n)=\\oint\\frac{\\d x}{2\\pi\\i}\\,\\frac{F_n^{(\\mathrm{A})}(x)}{x^{k+1}}\n=\\oint\\frac{\\d u}{2\\pi\\i}\\,\\frac{\\widehat F_n^{(\\mathrm{A})}(u)(u+1)^{k-1}}{u^{k+1}},\n\\label{bcontour}\n\\end{equation}\npermuting the order of integrals, opening up the $u$-contour and using\n\\begin{equation}\n\\int_C\\frac{\\d u}{2\\pi\\i}\\,\\frac{(u+1)^{k-1}}{u^{k-s+1}}\n=\\frac{\\Gamma(k)}{\\Gamma(s)\\Gamma(k-s+1)},\n\\end{equation}\nwe obtain after some algebra the scaling form~(\\ref{fss}), with\n\\begin{equation}\n\\Phi^{(\\mathrm{A})}(y)=1+\\frac{2}{\\sqrt{\\pi}}\n\\int_{\\mathrm{C}}\\frac{\\d s}{2\\pi\\i}\\,\\frac{s+1}{s+2}\n\\,\\Gamma\\!\\left(\\frac{1-s}{2}\\right)\\left(\\frac{y}{2}\\right)^{s+2}.\n\\end{equation}\nCase~B can be dealt with along the same lines.\nWe thus get the similar expression\n\\begin{equation}\n\\Phi^{(\\mathrm{B})}(y)=1+\\int_{\\mathrm{C}}\\frac{\\d s}{2\\pi\\i}\\,\\frac{s+1}{s+2}\n\\,\\Gamma\\!\\left(1-\\frac{s}{2}\\right)\\left(\\frac{y}{2}\\right)^{s+2}.\n\\end{equation}\nThe above expressions can be evaluated by closing the contour to the right\nand summing the residues at the poles of the gamma functions.\nWe thus get\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=1+\\frad{8}{\\sqrt{\\pi}}\n\\displaystyle\\sum_{m\\ge0}\\frad{(-1)^m(m+1)}{(2m+3)m!}\\left(\\frac{y}{2}\\right)^{2m+3},\n\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{B})}(y)=1+\n\\displaystyle\\sum_{m\\ge0}\\frad{(-1)^m(m+1)(2m+3)}{(m+2)!}\\left(\\frac{y}{2}\\right)^{2m+4},\n\\hfill\n}\n\\end{equation}\ni.e., finally\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=\\mathop{\\rm erfc}\\nolimits\\left(\\frad{y}{2}\\right)\n+\\frad{y}{\\sqrt{\\pi}}\\left(1+\\frad{y^2}{2}\\right){\\rm e}^{-y^2\/4},\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{B})}(y)=\\left(1+\\frad{y^2}{4}+\\frad{y^4}{8}\\right){\\rm e}^{-y^2\/4},\\hfill\n}\n\\label{bfss}\n\\end{equation}\nwhere erfc denotes the complementary error function.\nThe above expression for $\\Phi^{(\\mathrm{A})}$ can be found in~\\cite{kr2,ws},\nwhereas that for $\\Phi^{(\\mathrm{B})}$ has been shown in~\\cite{dms2} to hold for a slightly\ndifferent attachment rule and initial condition.\n\nFigure~\\ref{bscaling} shows a plot of the ratios $f_k(n)\/f_{k,{\\mathrm{stat}}}$,\nagainst the scaling variable $y=k\/n^{1\/2}$,\nat time $n=10^3$ for both initial conditions.\nExact values for the $f_k(n)$ are obtained by iterating~(\\ref{bfknrec}).\nThe data are well described by the predicted finite-size scaling functions\n$\\Phi^{(\\mathrm{A})}(y)$ and $\\Phi^{(\\mathrm{B})}(y)$, shown as full lines.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.7\\linewidth]{bas.eps}\n\\caption{\\label{bscaling}\nPlot of the ratios $f_k(n)\/f_{k,{\\mathrm{stat}}}$\nagainst the scaling variable $y=k\/n^{1\/2}$,\nfor the BA model at time $n=10^3$ (symbols) for both initial conditions.\nFull lines: asymptotic scaling functions $\\Phi^{(\\mathrm{A})}(y)$ and $\\Phi^{(\\mathrm{B})}(y)$.}\n\\end{center}\n\\end{figure}\n\nBoth scaling functions share similar qualitative features.\nThey start from the value 1 at $y=0$,\nincrease to a maximum, which is reached for $y=2$ in Case~A\nand for $y=\\sqrt{6}$ in Case~B, and fall off as $\\exp(-y^2\/4)$.\nThey however differ at the quantitative level,\nboth at small and large values of $y$:\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=1+\\frad{y^3}{3\\sqrt{\\pi}}+\\cdots,\\hfill&\n\\Phi^{(\\mathrm{B})}(y)=1+\\frad{3y^4}{32}+\\cdots,\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{A})}(y)\\approx\\frad{y^3}{2\\sqrt{\\pi}}\\,{\\rm e}^{-y^2\/4},\\hfill&\n\\Phi^{(\\mathrm{B})}(y)\\approx\\frad{y^4}{8}\\,{\\rm e}^{-y^2\/4}.\\hfill\n}\n\\label{phiasy}\n\\end{equation}\nApart from the additive constant 1,\nthe scaling functions $\\Phi^{(\\mathrm{A})}$ and $\\Phi^{(\\mathrm{B})}$ are respectively\nan odd and an even function of $y$.\nThis is the transcription in the finite-size scaling regime\nof the phenomenon underlined when discussing~(\\ref{af1}) and~(\\ref{bf1}).\nIn particular, the first correction term at small $y$ is in $y^3$ for $\\Phi^{(\\mathrm{A})}$,\nand in $y^4$ for $\\Phi^{(\\mathrm{B})}$.\n\nLet us again close up with the location\nof the complex zeros of the polynomials $F_n(x)$.\nConsidering Case~A for definiteness,\nthe result~(\\ref{bfnres}) can be recast as the exact formula\n\\begin{equation}\n\\widehat F_n^{(\\mathrm{A})}(u)-\\frac{n-1}{n}\\,\\widehat F_{\\mathrm{stat}}(u)\n=\\frac{1}{n}\\int_{\\mathrm{C}}\\frac{\\d s}{\\i}\\,\\frac{s+1}{s+2}\n\\,\\frac{\\Gamma\\!\\left(n-\\frac{s}{2}-1\\right)}\n{\\Gamma\\!\\left(1-\\frac{s}{2}\\right)\\Gamma(n-1)}\\,\\frac{u^s}{\\sin\\pi s}.\n\\label{bfnc}\n\\end{equation}\nThe growth of this expression with $n$\nfor a fixed value of the complex variable $u$\ncan be investigated by means of the saddle-point approximation.\nThe presence of gamma functions again suggest to look for a saddle point\n$s_{\\mathrm{s}}$ proportional to~$n$.\nSkipping details, let us mention that we find $s_{\\mathrm{s}}\\approx2n\/(1-u^2)$,\nso that the right-hand side of~(\\ref{bfnc}) can be estimated as\n\\begin{equation}\n\\widehat F_{n,{\\mathrm{sing}}}(u)\\sim\\left(1-\\frac{1}{u^2}\\right)^{-n},\n\\label{bexpo}\n\\end{equation}\nwith exponential accuracy.\nThe asymptotic locus of the complex zeros is then naturally given\nby the condition that the above estimate neither falls off\nnor grows exponentially.\nWe thus obtain $\\abs{1-1\/u^2}=1$.\nThe relevant part of this locus\ncan be parametrized by an angle $0\\le\\theta\\le2\\pi$~as\n\\begin{equation}\nu=\\left(1-{\\rm e}^{-\\i\\theta}\\right)^{-1\/2},\\quad\nx=\\frac{1}{1-\\left(1-{\\rm e}^{-\\i\\theta}\\right)^{1\/2}}.\n\\label{bacurve}\n\\end{equation}\nThis closed curve in the $x$-plane has a cusp at the point $x=1$,\ncorresponding to the scaling regime, with a right opening angle.\nWe have indeed $x-1\\approx({\\rm e}^{\\i\\pi\/2}\\theta)^{1\/2}$ as $\\theta\\to0$.\nFigure~\\ref{bzeros} illustrates this result with data at time $n=50$\nfor both initial conditions.\nThe polynomials $F_n(x)$ converge to the stationary series $F_{\\mathrm{stat}}(x)$\nwhenever the complex variable $x$ lies\nwithin the closed curve shown on the figure.\nOtherwise they diverge exponentially with $n$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.6\\linewidth]{baz.eps}\n\\caption{\\label{bzeros}\nPlot of the non-trivial zeros of the polynomials $F_n(x)$\nfor the BA model in the complex $x$-plane.\nSymbols: zeros for $n=50$ in Case~A (empty symbols) and Case~B (full symbols).\nLine: limiting curve with equation~(\\ref{bacurve}).}\n\\end{center}\n\\end{figure}\n\nThe exponential estimate~(\\ref{bexpo}) has another virtue.\nBy inserting it\ninto the contour-integral representation~(\\ref{bcontour}), we obtain\n\\begin{equation}\nf_k(n)\\sim\\oint\\frac{\\d u}{2\\pi\\i}\\left(\\frac{u+1}{u}\\right)^k\n\\left(1-\\frac{1}{u^2}\\right)^{-n}.\n\\label{fksaddle}\n\\end{equation}\nThis integral can in turn be investigated\nby means of the saddle-point approximation.\nThe result is the following large-deviation estimate\n\\begin{equation}\nf_k(n)\\sim\\exp(-n\\,S(\\zeta)),\n\\label{bent}\n\\end{equation}\nwhere $\\zeta=k\/n$, and where the large-deviation function $S(\\zeta)$ reads\n\\begin{equation}\nS(\\zeta)=(1-\\zeta)\\ln(1-\\zeta)-(2-\\zeta)\\ln\\frac{2-\\zeta}{2}.\n\\end{equation}\nThe formula~(\\ref{bent}) describes, with exponential accuracy,\nthe degree distribution in the whole large-deviation regime\nwhere $k$ and $n$ are comparable.\nThe quadratic growth $S(\\zeta)\\approx\\zeta^2\/4$\nat small $\\zeta$ matches the fall-off of the finite-size scaling functions\n$\\Phi^{(\\mathrm{A})}(y)\\sim\\Phi^{(\\mathrm{B})}(y)\\sim\\exp(-y^2\/4)$ (see~(\\ref{phiasy})).\nThe maximal value $S(1)=\\ln 2$ describes the fall-off $f_k(n)\\sim2^{-n}$\nof the probability of having\na degree $k$ equal to its maximal value ($k=n$ or $k=n-1$).\n\n\\section{The general preferential attachment (GPA) model}\n\nWe now consider the general preferential attachment (GPA) rule,\nwhere the attachment probability to a node is proportional\nto the sum $k_i(n)+c$ of the degree of the earlier node\nand of an additive constant $c$,\nrepresenting the initial attractiveness of the node~\\cite{dms1}.\nThis attachment rule interpolates between the uniform attachment rule,\nwhich is recovered in the $c\\to\\infty$ limit,\nand the BA model, which corresponds to $c=0$.\nIt can actually be continued on the other side of the BA model,\nas $c$ can be chosen in the range $-1<c<\\infty$.\nThe GPA model thus defined is scalefree for any finite value of $c$,\nwith the continuously varying exponents $\\gamma=c+3$ and $\\nu=1\/(c+2)$.\n\nThe probability that node $n$ connects to an earlier node $i$ thus reads\n\\begin{equation}\np_{n,i}=\\frac{k_i(n-1)+c}{Z(n-1)},\n\\label{gp}\n\\end{equation}\nwhere $k_i(n-1)$ is the degree of node $i$ at time $n-1$,\nand the partition function in the denominator,\n\\begin{equation}\nZ(n)=\\sum_{i=1}^n(k_i(n)+c)=2L(n)+cn\n\\label{zp}\n\\end{equation}\n(see~(\\ref{sumr})), ensures that the attachment probabilities add up to unity.\n\nIn the following we analyze the GPA model\nalong the very lines of the previous section.\n\n\\subsection{Degree statistics of a fixed node}\n\nLet us again begin with the\ndistribution $f_k(n,i)=\\mathop{\\rm Prob}\\nolimits\\{k_i(n)=k\\}$\nof the degree of node $i$ at time $n$.\n\nA first estimate of the degree $k_i(n)$ is provided by\nthe product formula~(\\ref{aveprod}) for the mean degree $\\mean{k_i(n)}$,\nwhich still holds in the present case.\nIn the scaling regime where both $i$ and $n$ are large,\nthe latter relation becomes the differential equation\n\\begin{equation}\n\\frac{\\partial\\mean{k_i(n)}}{\\partial n}\\approx\\frac{\\mean{k_i(n)}+c}{(c+2)n},\n\\end{equation}\nwhich yields\n\\begin{equation}\n\\mean{k_i(n)}\\approx(c+1)\\left(\\frac{n}{i}\\right)^{1\/(c+2)}-c.\n\\label{gave}\n\\end{equation}\nAs anticipated, this expressions exhibits a power-law growth\nwith exponent $\\nu=1\/(c+2)$ in the range $0<\\nu<1$.\n\nThe generating polynomials $F_{n,i}(x)$ and $F_{n+1,i}(x)$\nassociated with the degree of node $i$ at successive times\n$n$ and $n+1$ obey the recursion formula:\n\\begin{eqnarray}\nF_{n+1,i}(x)\n&=&\\bigmean{x^{k_i(n+1)}}=\\bigmean{x^{I_i(n+1)}x^{k_i(n)}}\\nonumber\\\\\n&=&\\bigmean{(1+(x-1)p_{n+1,i})x^{k_i(n)}}\\nonumber\\\\\n&=&\\bigmean{\\left(1+\\frac{x-1}{Z(n)}\\,(k_i(n)+c)\\right)x^{k_i(n)}},\n\\end{eqnarray}\ni.e.,\n\\begin{equation}\nF_{n+1,i}(x)=F_{n,i}(x)+\\frac{x-1}{Z(n)}\n\\left(c F_{n,i}(x)+x\\frac{\\d F_{n,i}(x)}{\\d x}\\right),\n\\label{gfnirec}\n\\end{equation}\nwhere\n\\begin{equation}\nZ^{(\\mathrm{A})}(n)=(c+2)n-2,\\quad Z^{(\\mathrm{B})}(n)=(c+2)n-1.\n\\label{gzres}\n\\end{equation}\nThe probabilities $f_k(n,i)$ themselves therefore obey the recursion\n\\begin{equation}\nf_k(n+1,i)\n=\\frac{k+c-1}{Z(n)}f_{k-1}(n,i)+\\left(1-\\frac{k+c}{Z(n)}\\right)f_k(n,i),\n\\end{equation}\nwith initial conditions~(\\ref{f2init}).\n\nIn order to solve the recursion~(\\ref{gfnirec}),\nwe again perform the change of variable~(\\ref{xtou}) from $x$ to $u$, and set\n\\begin{equation}\nF_{n,i}(x)=(1-x)^{-c}\\widehat F_{n,i}(u).\n\\end{equation}\nThe recursion~(\\ref{gfnirec}) then reads\n\\begin{equation}\n\\widehat F_{n+1,i}(u)=\\widehat F_{n,i}(u)-\\frac{1}{Z(n)}\n\\left(c\\widehat F_{n,i}(u)+u\\frac{\\d\\widehat F_{n,i}(u)}{\\d u}\\right).\n\\label{gwfnirec}\n\\end{equation}\nWe then again introduce the Mellin transform $M_{n,i}(s)$ of $\\widehat F_{n,i}(u)$,\nso that the recursion~(\\ref{gwfnirec}) simplifies to\n\\begin{equation}\nM_{n+1,i}(s)=\\left(1-\\frac{s+c}{Z(n)}\\right)M_{n,i}(s),\n\\label{gmfnirec}\n\\end{equation}\nwith initial condition $M_{i,i}(s)=X_c(s)$ for $i\\ge2$, with\n\\begin{equation}\nX_c(s)=\\int_0^1 x^{-s}(1-x)^{s+c-1}\\,\\d x\n=\\frac{\\Gamma(1-s)\\Gamma(s+c)}{\\Gamma(c+1)}\n\\label{gxdef}\n\\end{equation}\nfor $-c<\\Re s<1$.\nHereafter the contour C is assumed to be in that strip.\nWe thus get $(i\\ge2)$\n\\begin{equation}\n\\matrix{\nM_{n,i}^{(\\mathrm{A})}(s)=\\frad{\\Gamma\\!\\left(n-\\frac{s+c+2}{c+2}\\right)\n\\,\\Gamma\\!\\left(i-\\frac{2}{c+2}\\right)}\n{\\Gamma\\!\\left(i-\\frac{s+c+2}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{2}{c+2}\\right)}\n\\,X_c(s),\\hfill\\cr\\cr\nM_{n,i}^{(\\mathrm{B})}(s)=\\frad{\\Gamma\\!\\left(n-\\frac{s+c+1}{c+2}\\right)\n\\,\\Gamma\\!\\left(i-\\frac{1}{c+2}\\right)}\n{\\Gamma\\!\\left(i-\\frac{s+c+1}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{1}{c+2}\\right)}\n\\,X_c(s),\\hfill\n}\n\\label{gfnires}\n\\end{equation}\nThese product formulas are a generalization of~(\\ref{bfnires}).\nThe mean and the variance of the degree of node $i$ at time~$n$\ncan be extracted from these results as follows.\nThe identity~(\\ref{devt}) now yields\n\\begin{equation}\n\\widehat F_{n,i}(u)=\\frac{1}{u^c}-\\frac{\\mean{k_i(n)}+c}{u^{c+1}}\n+\\frac{\\mean{k_i(n)^2}+(2c+1)\\mean{k_i(n)}+c(c+1)}{2u^{c+2}}+\\cdots\n\\end{equation}\nas $u\\to+\\infty$.\nFurthermore the coefficients of this expansion\nare respectively the residues of $M_{n,i}(s)$\nat $s=-c$, $s=-c-1$ and $s=-c-2$.\nWe thus obtain\n\\begin{equation}\n\\matrix{\n\\mean{k^{(\\mathrm{A})}_i(n)}=(c+1)\\frad\n{\\Gamma\\!\\left(n-\\frac{1}{c+2}\\right)\\,\\Gamma\\!\\left(i-\\frac{2}{c+2}\\right)}\n{\\Gamma\\!\\left(i-\\frac{1}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{2}{c+2}\\right)}-c,\n\\hfill\\cr\\cr\n\\mean{k^{(\\mathrm{B})}_i(n)}=(c+1)\\frad{\\Gamma(n)\\,\\Gamma\\!\\left(i-\\frac{1}{c+2}\\right)}\n{\\Gamma(i)\\,\\Gamma\\!\\left(n-\\frac{1}{c+2}\\right)}-c\\hfill\n}\n\\end{equation}\nand\n\\begin{equation}\n\\matrix{\n\\mathop{\\rm var}\\nolimits{k^{(\\mathrm{A})}_i(n)}=(c+1)(c+2)\\frad\n{\\Gamma(n)\\,\\Gamma\\!\\left(i-\\frac{2}{c+2}\\right)}\n{\\Gamma(i)\\,\\Gamma\\!\\left(n-\\frac{2}{c+2}\\right)}\\hfill\\cr\\cr\n{\\hskip 47.5pt}-\\mean{k^{(\\mathrm{A})}_i(n)}^2-(2c+1)\\mean{k^{(\\mathrm{A})}_i(n)}-c(c+1),\\hfill\\cr\\cr\n\\mathop{\\rm var}\\nolimits{k^{(\\mathrm{B})}_i(n)}=(c+1)(c+2)\\frad\n{\\Gamma\\!\\left(n+\\frac{1}{c+2}\\right)\\,\\Gamma\\!\\left(i-\\frac{1}{c+2}\\right)}\n{\\Gamma\\!\\left(i+\\frac{1}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{1}{c+2}\\right)}\n\\hfill\\cr\\cr\n{\\hskip 47.5pt}-\\mean{k^{(\\mathrm{B})}_i(n)}^2-(2c+1)\\mean{k^{(\\mathrm{B})}_i(n)}-c(c+1).\\hfill\n}\n\\end{equation}\n\nIn the scaling regime where both times $i$ and $n$ are large and comparable,\nintroducing the time ratio $z=n\/i$ (see~(\\ref{zdef})),\nthe above results yield\n\\begin{equation}\n\\mean{k_i(n)}\\approx(c+1)z^{1\/(c+2)}-c,\\quad\n\\mathop{\\rm var}\\nolimits{k_i(n)}\\approx(c+1)z^{1\/(c+2)}(z^{1\/(c+2)}-1),\n\\label{gkin}\n\\end{equation}\nirrespective of the initial condition.\nThe mean degree is in agreement with the estimate~(\\ref{gave}).\nThe entire degree distribution can actually be derived in the scaling regime.\nEquation~(\\ref{gfnires}) indeed yields\n\\begin{equation}\nM_{n,i}(s)\\approx z^{-(s+c)\/(c+2)}\\,X_c(s).\n\\end{equation}\nWe thus obtain after some algebra\n\\begin{equation}\nF_{n,i}(x)\\approx\\frac{x}{\\left(x+z^{1\/(c+2)}(1-x)\\right)^{c+1}}\n\\end{equation}\nand finally\n\\begin{equation}\nf_k(n,i)\\approx z^{-(c+1)\/(c+2)}\\bigl(1-z^{-1\/(c+2)}\\bigr)^{k-1}\n\\frac{\\Gamma(k+c)}{\\Gamma(k)\\Gamma(c+1)}.\n\\label{gfz}\n\\end{equation}\nThis result allows one to recover both the Poissonian law~(\\ref{ufz})\nin the $c\\to\\infty$ limit and the geometric one~(\\ref{bfz}) as $c=0$.\n\n\\subsection{Degree statistics of the whole network}\n\nWe now turn to the degree distribution of the whole network at time $n$,\n$f_k(n)=\\mathop{\\rm Prob}\\nolimits\\{k(n)=k\\}$,\nwhere $k(n)$ stands for the degree of an unspecified node.\n\nThe generating polynomials $F_n(x)$ obey the recursion\n\\begin{equation}\n(n+1)F_{n+1}(x)=nF_n(x)+\\frac{n(x-1)}{Z(n)}\n\\left(c F_n(x)+x\\frac{\\d F_n(x)}{\\d x}\\right)+x,\n\\label{gfnrec}\n\\end{equation}\nwhere $Z(n)$ is given by~(\\ref{gzres}), and\nwith initial conditions given in Table~\\ref{tabledef}.\nThe probabilities $f_k(n)$ themselves obey the recursion\n\\begin{equation}\n(n+1)f_k(n+1)\n=\\frac{k+c-1}{Z(n)}nf_{k-1}(n)\n+\\left(1-\\frac{k+c}{Z(n)}\\right)nf_k(n)+\\delta_{k,1}.\n\\label{gfknrec}\n\\end{equation}\n\nThe first generating polynomials which depend on the attachment rule read\n\\begin{equation}\n\\matrix{\nF_3^{(\\mathrm{A})}(x)=\\frac{1}{3}\\,x(x+2),\\hfill\\cr\\cr\nF_3^{(\\mathrm{B})}(x)=\\frac{1}{3(2c+3)}\\,x\\left((c+2)x^2+2(c+1)x+3c+5\\right),\\hfill\\cr\\cr\nF_4^{(\\mathrm{A})}(x)=\\frac{1}{4(3c+4)}\\,x\\left((c+2)x^2+4(c+1)x+7c+10\\right),\\hfill\\cr\\cr\nF_4^{(\\mathrm{B})}(x)=\\frac{1}{4(2c+3)(3c+5)}\\,x\n\\bigl((c+2)(c+3)x^3+4(c+1)(c+2)x^2\\hfill\\cr\n{\\hskip 110pt}+(c+1)(7c+11)x+(3c+5)(4c+7)\\bigr).\\hfill\n}\n\\end{equation}\n\nThe stationary degree distribution $f_{k,{\\mathrm{stat}}}$\ncan be determined as the solution of~(\\ref{gfknrec})\nwhich becomes independent of $n$ for large $n$.\nWe thus get\n\\begin{equation}\n(k+2c+2)f_{k,{\\mathrm{stat}}}=(k+c-1)f_{k-1,{\\mathrm{stat}}}+(c+2)\\delta_{k,1},\n\\label{fstrec}\n\\end{equation}\nhence~\\cite{dms1,krl}\n\\begin{equation}\nf_{k,{\\mathrm{stat}}}=\\frac{(c+2)\\Gamma(2c+3)\\Gamma(k+c)}{\\Gamma(c+1)\\Gamma(k+2c+3)}.\n\\label{gfstat}\n\\end{equation}\nThis result has a power-law decay at large $k$:\n\\begin{equation}\nf_{k,{\\mathrm{stat}}}\\approx\\frac{(c+2)\\Gamma(2c+3)}{\\Gamma(c+1)}\\,k^{-(c+3)}.\n\\label{gfkas}\n\\end{equation}\nAn alternative approach consists in looking for the\ngenerating function $F_{\\mathrm{stat}}(x)$\nas the stationary solution of~(\\ref{gfnrec}).\nWe thus obtain the differential equation\n\\begin{equation}\nx(1-x)F'_{\\mathrm{stat}}(x)+(2c+2-cx)F_{\\mathrm{stat}}(x)=(c+2)x,\n\\end{equation}\nwhich is equivalent to~(\\ref{fstrec}).\nThe solution\n\\begin{equation}\nF_{\\mathrm{stat}}(x)=\\frac{(c+2)(1-x)^{c+2}}{x^{2c+2}}\n\\int_0^x\\frac{y^{2c+2}}{(1-y)^{c+3}}\\,\\d y\n\\end{equation}\ncan be recast in terms of a hypergeometric function,\nwhich boils down to elementary functions whenever $2c$ is an integer.\n\nThroughout the regime where the degree $k$ and the parameter $c$\nare both large and comparable,\nthe expression~(\\ref{gfstat}) assumes a stationary large-de\\-vi\\-a\\-tion form,\n\\begin{equation}\nf_{k,{\\mathrm{stat}}}\\sim\\exp(-c\\,\\phi(\\kappa)),\n\\end{equation}\nwhere $\\kappa=k\/c$, and with\n\\begin{equation}\n\\phi(\\kappa)=(\\kappa+2)\\ln(\\kappa+2)-(\\kappa+1)\\ln(\\kappa+1)-2\\ln 2.\n\\end{equation}\nThe linear behavior $\\phi(\\kappa)\\approx\\kappa\\ln 2$ as $\\kappa\\to0$\nmatches the exponential decay~(\\ref{fstat})\nof the stationary distribution in the UA model,\nformally corresponding to $c\\to\\infty$,\nwhereas the logarithmic growth $\\phi(\\kappa)\\approx\\ln\\kappa+1-2\\ln 2$\nas $\\kappa\\to\\infty$ matches the power-law decay~(\\ref{gfkas}).\n\nThe moments of the stationary distribution,\n\\begin{equation}\nm_p=\\sum_{k\\ge1}k^p\\,f_{k,{\\mathrm{stat}}},\n\\end{equation}\ncan be derived from~(\\ref{fstrec}), which yields the recursion\n\\begin{equation}\n(c+2-p)m_p=c+2+pcm_{p-1}+\\sum_{q=0}^{p-2}{{p}\\choose{q}}(m_{q+1}+cm_q).\n\\end{equation}\nWe thus get\n\\begin{equation}\n\\matrix{\nm_0=1,\\quad m_1=2,\\quad m_2=\\frad{2(3c+2)}{c},\\hfill\\cr\\cr\nm_3=\\frad{2(13c^2+17c+6)}{c(c-1)},\\quad\nm_4=\\frad{2(3c+2)(25c^2+33c+14)}{c(c-1)(c-2)},\\hfill\n}\n\\end{equation}\nand so on.\nThe power-law decay~(\\ref{gfkas})\nimplies that the moment $m_p$ is convergent for $c>p-2$.\n\nThe recursion~(\\ref{gfnrec}) for the generating polynomials $F_n(x)$\ncan again be exactly solved for a finite time $n$.\nThe Mellin transforms $M_n(s)$ of the functions\n$\\widehat F_n(u)=(1-x)^c F_n(x)$ obey\n\\begin{equation}\n(n+1)M_{n+1}(s)=\\left(1-\\frac{s+c}{Z(n)}\\right)nM_n(s)+X_c(s),\n\\label{gmfnrec}\n\\end{equation}\nwith initial condition $M_2^{(\\mathrm{A})}(s)=M_1^{(\\mathrm{B})}(s)=X_c(s)$.\nEquation~(\\ref{gmfnrec}) has a special solution\n\\begin{equation}\nM_n(s)=\\frac{Z(n)X_c(s)}{(s+2c+2)n},\n\\label{gmspec}\n\\end{equation}\nwhereas the general solution of the homogeneous equation\nshares the $n$-de\\-pen\\-den\\-ce of the expressions~(\\ref{gfnires}).\nWe thus get\n\\begin{equation}\n\\matrix{\nM_n^{(\\mathrm{A})}(s)=\\frad{X_c(s)}{(s+2c+2)n}\\hfill\\cr\n{\\hskip 36pt}\\times\n\\left((c+2)n-2+2(s+c+1)\\frad{\\Gamma\\!\\left(n-\\frac{s+c+2}{c+2}\\right)\n\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}\n{\\Gamma\\!\\left(1-\\frac{s}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{2}{c+2}\\right)}\\right)\n,\\hfill\\cr\\cr\nM_n^{(\\mathrm{B})}(s)=\\frad{X_c(s)}{(s+2c+2)n}\\hfill\\cr\n{\\hskip 36pt}\\times\n\\left((c+2)n-1+(s+c+1)\\frad{\\Gamma\\!\\left(n-\\frac{s+c+1}{c+2}\\right)\n\\,\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}\n{\\Gamma\\!\\left(\\frac{1-s}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{1}{c+2}\\right)}\\right)\n.\\hfill\n}\n\\label{gfnres}\n\\end{equation}\n\nIn order to illustrate these general results,\nlet us again consider the probability $f_1(n)$ for a node to have degree one.\nThis probability is minus the residue of $M_n(s)$ at $s=1$.\nFor Case~A we obtain ($n\\ge2$)\n\\begin{eqnarray}\nf_1^{(\\mathrm{A})}(n)&=&\\frad{1}{(2c+3)n}\n\\left((c+2)n-2+2(c+2)\\frad{\\Gamma\\!\\left(n-\\frac{c+3}{c+2}\\right)\n\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}\n{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)\\,\\Gamma\\!\\left(n-\\frac{2}{c+2}\\right)}\\right)\n\\nonumber\\\\\n&=&\\frac{1}{2c+3}\n\\left(c+2-\\frac{2}{n}+\\frac{2(c+2)\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}\n{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}\n\\,n^{-\\frat{2c+3}{c+2}}+\\cdots\\right),\n\\end{eqnarray}\nwhereas for Case~B we obtain ($n\\ge2$)\n\\begin{equation}\nf_1^{(\\mathrm{B})}(n)=\\frac{(c+2)n-1}{(2c+3)n}.\n\\end{equation}\nThis rational expression for $f_1^{(\\mathrm{B})}(n)$ is however an exception.\nThe probabilities $f_k(n)$ indeed generically have\na singular correction in $n^{-(2c+3)\/(c+2)}$\nfor both initial conditions,\nwhereas only $f_1^{(\\mathrm{B})}(n)$ and $f_2^{(\\mathrm{B})}(n)$ are rational functions of time $n$.\n\nWe now turn to the finite-size scaling behavior\nof the degree distribution when both $k$ and $n$ are large.\nThe crossover scale $k_\\star(n)$ can again be estimated\neither using~(\\ref{gave}) or by the argument of extreme value statistics.\nBoth approaches consistently yield\n\\begin{equation}\nk_\\star(n)\\sim n^{1\/(c+2)}.\n\\end{equation}\nThe degree distribution\nobeys a finite-size scaling law of the form\n\\begin{equation}\nf_k(n)\\approx f_{k,{\\mathrm{stat}}}\\,\\Phi(y),\\quad y=\\frac{k}{n^{1\/(c+2)}},\n\\end{equation}\nwhere the scaling function $\\Phi(y)$ again depends\non the initial condition~\\cite{kr2,ws}.\nThe determination of the scaling functions $\\Phi^{(\\mathrm{A})}(y)$\nand $\\Phi^{(\\mathrm{B})}(y)$ closely follows the steps of Section 3.2.\nWe thus obtain\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=1+\\frad{2\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}{(c+2)\\Gamma(2c+3)}\n\\int_{\\mathrm{C}}\\frad{\\d s}{2\\pi\\i}\\,\\frad{s+c+1}{s+2c+2}\n\\frad{\\Gamma(1-s)}{\\Gamma\\!\\left(1-\\frac{s}{c+2}\\right)}\\,y^{s+2c+2},\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{B})}(y)=1+\\frad{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}{(c+2)\\Gamma(2c+3)}\n\\int_{\\mathrm{C}}\\frad{\\d s}{2\\pi\\i}\\,\\frad{s+c+1}{s+2c+2}\n\\frad{\\Gamma(1-s)}{\\Gamma\\!\\left(\\frac{1-s}{c+2}\\right)}\\,y^{s+2c+2}.\\hfill\n}\n\\label{gfss}\n\\end{equation}\nBy closing the contours to the right,\nwe can derive the following convergent series:\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=1+\\frad{2\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}{(c+2)\\Gamma(2c+3)}\ny^{2c+3}\\displaystyle\\sum_{m\\ge0}\\frad{(m+c+2)(-y)^m}\n{(m+2c+3)m!\\,\\Gamma\\!\\left(1-\\frac{m+1}{c+2}\\right)},\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{B})}(y)=1+\\frad{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}{(c+2)\\Gamma(2c+3)}\ny^{2c+3}\\displaystyle\\sum_{m\\ge1}\\frad{(m+c+2)(-y)^m}\n{(m+2c+3)m!\\,\\Gamma\\!\\left(-\\frac{m}{c+2}\\right)}.\\hfill\n}\n\\label{gfsss}\n\\end{equation}\nThe above expression for $\\Phi^{(\\mathrm{A})}$ can be found in~\\cite{ws},\nalbeit not in a fully explicit form.\nIt is also worth mentioning that the finite-size scaling function derived\nin~\\cite{cb} for asymmetric growing networks is different from the above one\nfor generic values of the exponent $\\nu=1\/(c+2)$,\nalthough it coincides for $\\nu=1\/2$ with our result~(\\ref{bfss}) for $\\Phi^{(\\mathrm{A})}$.\n\nThe expressions~(\\ref{gfsss}) suggest that the derivatives\n${\\Phi^{(\\mathrm{A})}}'(y)$ and ${\\Phi^{(\\mathrm{B})}}'(y)$ are somewhat simpler\nthan the functions themselves.\nThe factor $(m+2c+3)$ is indeed chased away from the denominators\nunder differentiation.\nThe resulting series can be resummed by means of the identities\n\\begin{equation}\n\\matrix{\n\\displaystyle\\sum_{m\\ge0}\\frad{(-y)^m}{m!\\,\\Gamma\\!\\left(1-\\frac{m+1}{c+2}\\right)}\n=(c+2)\\int_{\\mathrm{C}}\\frad{\\d z}{2\\pi\\i}\\,{\\rm e}^{-yz+z^{c+2}},\\hfill\\cr\\cr\n\\displaystyle\\sum_{m\\ge1}\\frad{(-y)^m}{m!\\,\\Gamma\\!\\left(-\\frac{m}{c+2}\\right)}\n=y\\int_{\\mathrm{C}}\\frad{\\d z}{2\\pi\\i}\\,{\\rm e}^{-yz+z^{c+2}},\\hfill\n}\n\\end{equation}\nwhich are known e.g.~in the theory of L\\'evy stable laws.\nWe are thus left with the following alternative contour-integral expressions\nfor the derivatives:\n\\begin{equation}\n\\matrix{\n{\\Phi^{(\\mathrm{A})}}'(y)=\\frad{2\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}{\\Gamma(2c+3)}\ny^{2c+2}\\int_{\\mathrm{C}}\\frad{\\d z}{2\\pi\\i}(c+2-yz)\\,{\\rm e}^{-yz+z^{c+2}},\\hfill\\cr\\cr\n{\\Phi^{(\\mathrm{B})}}'(y)=\\frad{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}{(c+2)\\Gamma(2c+3)}\ny^{2c+3}\\int_{\\mathrm{C}}\\frad{\\d z}{2\\pi\\i}(c+3-yz)\\,{\\rm e}^{-yz+z^{c+2}}.\\hfill\n}\n\\label{phiint}\n\\end{equation}\n\nBoth scaling functions start increasing from the value 1\naccording to the power laws\n\\begin{equation}\n\\matrix{\n\\Phi^{(\\mathrm{A})}(y)=1+\\frad{2\\,\\Gamma\\!\\left(\\frac{2c+2}{c+2}\\right)}\n{\\Gamma(2c+4)\\,\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)}\\,y^{2c+3}+\\cdots,\\hfill\\cr\\cr\n\\Phi^{(\\mathrm{B})}(y)=1+\\frad{(c+3)}{2(c+2)^3\\Gamma(2c+3)}\\,y^{2c+4}+\\cdots,\\hfill\n}\n\\end{equation}\ngo through a maximum, and fall off superexponentially as\n\\begin{equation}\n\\Phi^{(\\mathrm{A})}(y)\\approx\n2(c+2)C\\Gamma\\!\\left(\\frat{2c+2}{c+2}\\right)\\Psi(y),\\quad\n\\Phi^{(\\mathrm{B})}(y)\\approx\nC\\Gamma\\!\\left(\\frat{c+1}{c+2}\\right)y\\,\\Psi(y),\n\\label{phist}\n\\end{equation}\nwith\n\\begin{equation}\n\\Psi(y)=y^{2c+3-\\!\\frat{c}{2(c+1)}}\n\\,\\exp\\left(-(c+1)\\left(\\frac{y}{c+2}\\right)^{\\frat{c+2}{c+1}}\\right)\n\\label{psist}\n\\end{equation}\nand\n\\begin{equation}\nC=\\left[(2\\pi(c+1))^{1\/2}(c+2)^{\\frat{2c+3}{2(c+1)}}\\Gamma(2c+3)\\right]^{-1}.\n\\end{equation}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.7\\linewidth]{gs.eps}\n\\caption{\\label{gscaling}\nPlot of the scaling functions\n$\\Phi^{(\\mathrm{A})}(y)$ (full lines) and $\\Phi^{(\\mathrm{B})}(y)$ (dashed lines) against $y$,\nfor (1) $c=-1\/2$, i.e., $\\nu=2\/3$;\n(2) $c=0$, i.e., $\\nu=1\/2$ (the BA model);\nand (3) $c=1$, i.e., $\\nu=1\/3$.}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{gscaling} shows a plot of the scaling functions\n$\\Phi^{(\\mathrm{A})}(y)$ and $\\Phi^{(\\mathrm{B})}(y)$\nfor (1) $c=-1\/2$, i.e., $\\nu=2\/3$;\n(2) $c=0$, i.e., $\\nu=1\/2$ (the BA model);\nand (3) $c=1$, i.e., $\\nu=1\/3$.\nThe figure demonstrates that the scaling functions present\na high and narrow maximum for the smaller values of $c$,\nand a direct crossover from 1 to 0 for the larger values of $c$.\nThese observations can be made quantitative by means of the pseudo-moments\n\\begin{equation}\n\\mu_p=-\\int_0^\\infty\\Phi'(y) y^p\\,\\d y=p\\int_0^\\infty\\Phi(y) y^{p-1}\\,\\d y.\n\\end{equation}\nThe integral formulas~(\\ref{phiint}) allow one\nto evaluate these quantities explicitly:\n\\begin{equation}\n\\matrix{\n\\mu_p^{(\\mathrm{A})}=\\frad{(p+c+1)\\,\\Gamma\\!\\left(\\frac{3c+4}{c+2}\\right)\\Gamma(p+2c+3)}\n{(c+1)\\,\\Gamma\\!\\left(\\frac{p+3c+4}{c+2}\\right)\\Gamma(2c+3)},\\hfill\\cr\\cr\n\\mu_p^{(\\mathrm{B})}=\\frad{\\Gamma\\!\\left(\\frac{c+1}{c+2}\\right)\\Gamma(p+2c+3)}\n{\\Gamma\\!\\left(\\frac{p+c+1}{c+2}\\right)\\Gamma(2c+3)}.\\hfill\n}\n\\end{equation}\n\n\\noindent $\\bullet$\nFor large values of $c$ (i.e., $c\\to\\infty$),\nthe model is close to the UA model.\nThe analysis of the scaling functions\nwill follow that of the ratios $R_k(n)$ in the UA model,\nperformed in Section~2.2.\nThe crossover value of $y$,\nat which the functions exhibit a relatively sharp crossover from 1 to 0,\ncan be estimated as~$\\mu_1$, i.e.,\n\\begin{equation}\n\\mu_1^{(\\mathrm{A})}=2c+2{\\gamma_{\\scriptscriptstyle{\\rm E}}}+2+\\cdots,\\quad\\mu_1^{(\\mathrm{B})}=2c+2{\\gamma_{\\scriptscriptstyle{\\rm E}}}+3+\\cdots,\n\\end{equation}\nwhich grows as $2c$, irrespective of the initial condition.\nSimilarly, the squared width of the crossover region can be estimated as the\npseudo-variance $\\sigma^2=\\mu_2-\\mu_1^2$, i.e.,\n\\begin{equation}\n\\sigma^{2{(\\mathrm{A})}}=2c+4{\\gamma_{\\scriptscriptstyle{\\rm E}}}+2-2\\pi^2\/3+\\cdots,\\quad\n\\sigma^{2{(\\mathrm{B})}}=2c+4{\\gamma_{\\scriptscriptstyle{\\rm E}}}+3-2\\pi^2\/3+\\cdots,\n\\end{equation}\nwhich also grows as $2c$, irrespective of the initial condition.\n\n\\noindent $\\bullet$\nFor small values of $c$ (i.e., $c\\to-1$),\nthe scaling functions exhibit a high and narrow peak around $y=1$.\nThe position of the peak can be estimated as $\\mean{y}=\\mu_2\/(2\\mu_1)$,\ni.e., setting $c=-1+\\varepsilon$,\n\\begin{equation}\n\\mean{y}^{(\\mathrm{A})}=1+(3\/2-{\\gamma_{\\scriptscriptstyle{\\rm E}}})\\varepsilon+\\cdots,\\quad\n\\mean{y}^{(\\mathrm{B})}=1+(2-{\\gamma_{\\scriptscriptstyle{\\rm E}}})\\varepsilon+\\cdots,\n\\end{equation}\nwhereas the squared width of the peak can be estimated as\n$\\mathop{\\rm var}\\nolimits{y}=(4\\mu_1\\mu_3-3\\mu_2^2)\/(12\\mu_1^2)$, i.e.,\n\\begin{equation}\n\\matrix{\n\\mathop{\\rm var}\\nolimits{y}^{(\\mathrm{A})}=\\frad{5}{6}\\,\\varepsilon+\\frad{19-10{\\gamma_{\\scriptscriptstyle{\\rm E}}}-\\pi^2}{6}\\,\\varepsilon^2+\\cdots,\\hfill\\cr\n\\mathop{\\rm var}\\nolimits{y}^{(\\mathrm{B})}=\\frad{2}{3}\\,\\varepsilon+\\frad{22-8{\\gamma_{\\scriptscriptstyle{\\rm E}}}-\\pi^2}{6}\\,\\varepsilon^2+\\cdots,\\hfill}\n\\end{equation}\nand finally the area under the peak scales as $\\mu_1$, i.e.,\n\\begin{equation}\n\\mu_1^{(\\mathrm{A})}=\\frac{1}{\\varepsilon}+2-{\\gamma_{\\scriptscriptstyle{\\rm E}}}+\\cdots,\\quad\n\\mu_1^{(\\mathrm{B})}=\\frac{1}{\\varepsilon}+3-{\\gamma_{\\scriptscriptstyle{\\rm E}}}+\\cdots\n\\end{equation}\nWe are thus left with the picture of a narrow peak around $y=1$,\nwhose width shrinks as $\\varepsilon^{1\/2}$ and whose height grows as $\\varepsilon^{-3\/2}$.\n\nLet us close up this section with the location\nof the complex zeros of the polynomials $F_n(x)$.\nThe derivation of the estimate~(\\ref{bexpo}) can be generalized\nto the present situation for arbitrary values of $c$.\nWe are thus left with\n\\begin{equation}\n\\widehat F_{n,{\\mathrm{sing}}}(u)\\sim\\left(1-\\frac{1}{(-u)^{c+2}}\\right)^{-n},\n\\label{gexpo}\n\\end{equation}\nagain with exponential accuracy.\nThe asymptotic locus of the complex zeros is\ntherefore given by the condition $\\abs{1-1\/(-u)^{c+2}}=1$.\nThe relevant part of this locus\ncan be parametrized by an angle $0\\le\\theta\\le2\\pi$~as\n\\begin{equation}\nu=\\left(1-{\\rm e}^{-\\i\\theta}\\right)^{-1\/(c+2)},\\quad\nx=\\frac{1}{1-\\left(1-{\\rm e}^{-\\i\\theta}\\right)^{1\/(c+2)}}.\n\\label{gcurve}\n\\end{equation}\nThis closed curve in the $x$-plane has a cusp at the point $x=1$,\ncorresponding to the scaling regime, with an opening angle equal to $\\pi\/(c+2)$.\nWe have indeed $x-1\\approx({\\rm e}^{\\i\\pi\/2}\\theta)^{1\/(c+2)}$ as $\\theta\\to0$.\nFigure~\\ref{gzeros} illustrates this result with data at time $n=50$\nfor three values of $c$ and both initial conditions.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=.6\\linewidth]{gz.eps}\n\\caption{\\label{gzeros}\nPlot of the non-trivial zeros of the polynomials $F_n(x)$\nin the complex $x$-plane.\nSymbols: zeros for $n=50$ in Case~A (empty symbols) and Case~B (full symbols).\nLines: limiting curves with equation~(\\ref{gcurve}).\nFrom the inside to the outside: $c=0$ (the BA model,\nalready shown in Figure~\\ref{bzeros}), $c=1$ and $c=2$.}\n\\end{center}\n\\end{figure}\n\nThe exponential estimate~(\\ref{gexpo})\ncan again be recast into a large-deviation estimate\nfor the probabilities $f_k(n)$ in the regime $k\\sim n$, of the form\n\\begin{equation}\nf_k(n)\\sim\\exp(-n\\,S(\\zeta)),\n\\label{gent}\n\\end{equation}\nwith $\\zeta=k\/n$.\nThe large-deviation function $S(\\zeta)$ is obtained in parametric form:\n\\begin{equation}\n\\matrix{\n\\zeta=\\frad{(c+2)(v-1)}{v^{c+2}-1},\\hfill\\cr\\cr\nS=\\ln(v^{c+2}-1)\n-\\frad{c+2}{v^{c+2}-1}\\left((v-1)\\ln(v-1)+(v^{c+1}-1)v\\ln v\\right),\n}\n\\end{equation}\nwhere the parameter $v$ in the range $1<v<\\infty$\nis the opposite of the saddle-point value of $u$\nin the contour integral generalizing~(\\ref{fksaddle}).\n\nThe power-law behavior\n\\begin{equation}\nS(\\zeta)\\approx\n(c+1)\\left(\\frac{\\zeta}{c+2}\\right)^{\\frat{c+2}{c+1}}\n\\end{equation}\nat small $\\zeta$ (corresponding to $v\\to\\infty$)\nexactly matches the superexponential decay~(\\ref{phist}),~(\\ref{psist})\nof the finite-size scaling functions $\\Phi^{(\\mathrm{A})}(y)$ and $\\Phi^{(\\mathrm{B})}(y)$.\nThe maximal value\n\\begin{equation}\nS(1)=\\ln(c+2),\n\\end{equation}\ncorresponding to $v\\to1$, describes the exponential decay\n$f_k(n)\\sim(c+2)^{-n}$ of the probability of having\na degree $k$ equal to its maximal value ($k=n$ or $k=n-1$).\n\n\\section{Discussion}\n\nIn this paper we have presented a comprehensive study\nof finite-size (i.e., finite-time) effects\non the degree statistics in growing networks.\nWe have considered models defined by stochastic attachment rules,\nwhere nodes enter the network one at a time\nand attach to one single earlier node,\nso that the network has the topology of a tree.\nThe present study thus generalizes and extends many results\nof References~\\cite{dms1,dms2,krl,kr1,kr2,ws,mj}.\n\nWe have successively investigated the uniform attachment rule (UA),\nthe linear attachment rule of the Barab\\'asi-Albert (BA) model,\nand a general preferential attachment rule (GPA)\ncharacterized by a continuous parameter $c>-1$,\nrepresenting the initial attractiveness of a node.\nThe UA and BA models are recovered as two special cases,\nrespectively corresponding to $c\\to\\infty$ and $c=0$.\nThe model is scalefree for any finite value of $c$,\nwith the continuously varying exponents $\\gamma=c+3$ and $\\nu=1\/(c+2)$.\nThe continuous dependence of exponents on the parameter $c$,\nand the dependence of finite-size scaling functions\non the initial condition (Case~A or Case~B in the present study),\nare two illustrations of the lack of universality\nwhich altogether characterizes the scaling behavior of growing networks.\n\nThe GPA rule is actually the most general one for which\nthe partition function $Z(n)$ (see~(\\ref{zp})) is deterministic,\ni.e., independent of the history of the network.\nWhenever the attachment probability has a non-linear dependence on\nthe degree $k$,\nthe partition function becomes a history-dependent fluctuating quantity,\nso that the analysis of size effects becomes far more difficult.\nThe general case of an arbitrary attachment rule,\ngrowing either less or more rapidly than linearly with the degree,\nhas been considered in several works~\\cite{krl,kr1,mj}.\nWhenever the degree dependence of the attachment rule is asymptotically linear,\nthe resulting network is generically scalefree.\nThe determination of the degree exponent $\\gamma$\nis however a highly non-trivial\ntask in general (see~\\cite{krl,kr1} for an explicit example).\n\nThe present study has underlined the key r\\^ole\nplayed by the typical value $k_\\star(n)$ of the largest degree\nin a finite network at time $n$.\nIn the UA model, $k_\\star(n)$ grows logarithmically with time $n$.\nThe situation is more interesting in the scalefree case, i.e., for $c$ finite.\nThe largest degree $k_\\star(n)$ grows as a subextensive power law\nwith exponent $\\nu$,\nand demarcates three regimes in the size-degree plane,\nwhere finite-size (i.e., finite-time) effects\non the degree distribution $f_k(n)$ have different forms.\n\n\\noindent --\nIn the stationary regime ($k\\ll k_\\star(n)$),\nthe degree distribution is very close to the stationary one, $f_{k,{\\mathrm{stat}}}$.\n\n\\noindent --\nIn the finite-size scaling regime ($k\\sim k_\\star(n)$),\nthe degree distribution obeys\na multiplicative finite-size scaling law.\nAs already noticed in several earlier works,\nthe finite-size scaling function $\\Phi$ depends\non the initial condition imposed on the network.\nThis lack of universality holds for all finite values of\nthe parameter~$c$.\nAnother feature of the finite-size scaling function is that\nit increases from its initial value $\\Phi(0)=1$, reaches a maximum,\nand stays above unity for a range of values of its argument\n$y=k\/k_\\star(n)$, before it eventually falls off to zero.\nThis non-monotonic overshooting behavior is however not mandatory.\nIn this respect it is worth recalling\nthe example of the so-called zeta urn model~\\cite{dgc,zeta1,zetarev}.\nThis mean-field interacting particle system with multiple occupancies\npossesses a continuous condensation transition at a finite critical density.\nIts behavior right at the critical density shares\na high amount of similarity with the present problem,\nincluding a power-law stationary distribution\nwith a continuously varying exponent, and finite-time scaling.\nThe same results have been shown\nto apply to the dynamics of condensation\nin the zero-range process (ZRP)~\\cite{fsszrp}.\nIn the critical zeta urn and ZRP models, the finite-size scaling function\nis a monotonically decreasing function, so that $\\Phi(y)<1$ for all $y$.\nThis does not contradict the conservation of probability:\nthe excess probability is carried by smaller values of~$k$,\npertaining to the stationary regime.\n\n\\noindent --\nIn the large-deviation regime ($k_\\star(n)\\ll k\\sim n$),\nthe degree distribution falls off exponentially in $n$.\nAt variance with the finite-size scaling law,\nthe corresponding large-deviation function\nis independent of the initial condition.\nThe analysis of this regime has been shown to be closely related\nto the locus of the complex zeros of the generating polynomials $F_n(x)$,\nwhich have played a central r\\^ole throughout this work.\n\nTo close up, it is to be hoped that some of the concepts and methods\nused in the present work can be used to shed some new light\neither to other observables in the network models considered here,\nsuch as e.g.~the statistics of leaders and lead changes~\\cite{krlead},\nor to the degree statistics in more complex network models,\nsuch as e.g.~the Bianconi-Barab\\'asi (BB) model~\\cite{bb1,bb2},\nwhere attachment rules involve the competing effects\nof dynamical variables (the node degrees)\nand quenched disordered ones (the node fitnesses).\nDepending on the a priori distribution of the random fitnesses,\nthe BB model may possess a low-temperature condensed phase.\nSome features of the dynamics of the condensed phase have been\ninvestigated recently,\nboth at zero temperature~\\cite{usrecords},\nwhere the model is intimately related to the statistics of records,\nand at finite temperature~\\cite{fb}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{ Introduction}\n\nThroughout this paper, we always make use of the classical definition of\nquantum concepts as follows:\n\nThe $q$-shifted factorial is defined b\n\\begin{align*}\n\\left(  a;q\\right)  _{0} &  =1,\\ \\ \\ \\left(  a;q\\right)  _{n}\n{\\displaystyle\\prod\\limits_{j=0}^{n-1}}\n\\left(  1-q^{j}a\\right)  ,\\ \\ \\ n\\in\\mathbb{N},\\\\\n\\left(  a;q\\right)  _{\\infty} &  \n{\\displaystyle\\prod\\limits_{j=0}^{\\infty}}\n\\left(  1-q^{j}a\\right)  ,\\ \\ \\ \\ \\left\\vert q\\right\\vert <1,\\ \\ a\\in\n\\mathbb{C}.\n\\end{align*}\nIt is known tha\n\\[\n\\left(  a;q\\right)  _{n}=\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}q^{\\frac{1}{2}k\\left(  k-1\\right)  }\\left(  -1\\right)  ^{k}a^{k}.\n\\]\nThe $q$-numbers and $q$-numbers factorial and their improved forms are defined\nb\n\\begin{align*}\n\\left[  a\\right]  _{q} &  =\\frac{1-q^{a}}{1-q},\\ \\ \\ \\left(  q\\neq\n1,\\ a\\in\\mathbb{C}\\right)  ;\\ \\ \\ \\\\\n\\ \\ \\left[  0\\right]  _{q}! &  =1,\\left[  n\\right]  _{q}!=\\left[  n\\right]\n_{q}\\left[  n-1\\right]  _{q}!\\ \\ \\ \\ \\ ,\\ .\n\\end{align*}\nThe $q$-polynomail coefficient and improved type of them are defined b\n\\[\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}=\\frac{\\left(  q;q\\right)  _{n}}{\\left(  q;q\\right)\n_{n-k}\\left(  q;q\\right)  _{k}},\\ \\ \\ \\ \\ (k\\leqslant n,k,n\\i\n\\mathbb{N}\n)\n\\]\nIn the standard approach to the $q$-calculus two exponential function are\nused, these $q$-exponential and improved type (see \\cite{cel}) of it are\ndefined as follows\n\n\\begin{align*}\ne_{q}\\left(  z\\right)   &  =\\sum_{n=0}^{\\infty}\\frac{z^{n}}{\\left[  n\\right]\n_{q}!}=\\prod_{k=0}^{\\infty}\\frac{1}{\\left(  1-\\left(  1-q\\right)\nq^{k}z\\right)  },\\ \\ \\ 0<\\left\\vert q\\right\\vert <1,\\ \\left\\vert z\\right\\vert\n<\\frac{1}{\\left\\vert 1-q\\right\\vert },\\ \\ \\ \\ \\ \\ \\ \\\\\nE_{q}(z)  &  =e_{1\/q}\\left(  z\\right)  =\\sum_{n=0}^{\\infty}\\frac{q^{\\frac\n{1}{2}n\\left(  n-1\\right)  }z^{n}}{\\left[  n\\right]  _{q}!}=\\prod\n_{k=0}^{\\infty}\\left(  1+\\left(  1-q\\right)  q^{k}z\\right)\n,\\ \\ \\ \\ \\ \\ \\ 0<\\left\\vert q\\right\\vert <1,\\ z\\in\\mathbb{C},\\\\\n\\mathcal{E}_{q}\\left(  z\\right)   &  =e_{q}\\left(  \\frac{z}{2}\\right)\nE_{q}\\left(  \\frac{z}{2}\\right)  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{(-1,q)_{n}}{2^{n}}\\frac{z^{n}}{\\left[  n\\right]  _{q}!}\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{z^{n}}{\\left\\{  n\\right\\}  _{q}!}\\\\\n&  =\\prod_{k=0}^{\\infty}\\frac{\\left(  1+\\left(  1-q\\right)  q^{k}\\frac{z\n{2}\\right)  }{\\left(  1-\\left(  1-q\\right)  q^{k}\\frac{z}{2}\\right)\n},0<q<1,\\left\\vert z\\right\\vert <\\frac{2}{1-q}.\n\\end{align*}\nThe form of improved type of $q$-exponential function $\\mathcal{E}_{q}\\left(\nz\\right)  $, motivate us to define a new $q$-addition and $q$-substraction as\nfollows\n\\begin{align*}\n\\left(  x\\oplus_{q}y\\right)  ^{n}  &  :\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k}(-1,q)_{n-k}}{2^{n}}x^{k}y^{n-k\n,\\ \\ \\ n=0,1,2,...,\\\\\n\\left(  x\\ominus_{q}y\\right)  ^{n}  &  :\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k}(-1,q)_{n-k}}{2^{n}}x^{k}\\left(  -y\\right)\n^{n-k},\\ \\ \\ n=0,1,2,...\n\\end{align*}\nIt follows tha\n\\[\n\\mathcal{E}_{q}\\left(  tx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)\n=\\sum_{n=0}^{\\infty}\\left(  x\\oplus_{q}y\\right)  ^{n}\\frac{t^{n}}{\\left[\nn\\right]  _{q}!}.\n\\]\nThe Bernoulli numbers $\\left\\{  B_{m}\\right\\}  _{m\\geq0}$ are rational numbers\nin a sequence defined by the binomial recursion formula\n\\begin{equation}\n\\sum_{k=0}^{m}\\left(\n\\begin{array}\n[c]{c\nm\\\\\nk\n\\end{array}\n\\right)  B_{k}-B_{m}=\\left\\{\n\\begin{tabular}\n[c]{ll\n$1,$ & $m=1,$\\\\\n$0,$ & $m>1,\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\right.  \\label{b1\n\\end{equation}\nor equivalently, the generating functio\n\\[\n\\sum_{k=0}^{\\infty}B_{k}\\frac{t^{k}}{k!}=\\frac{t}{e^{t}-1}.\n\\]\n$q$-analogues of the Bernoulli numbers were first studied by Carlitz\n\\cite{calitz1} in the middle of the last century when he introduced a new\nsequence $\\left\\{  \\beta_{m}\\right\\}  _{m\\geqslant0}$\n\\begin{equation}\n\\sum_{k=0}^{m}\\left(\n\\begin{array}\n[c]{c\nm\\\\\nk\n\\end{array}\n\\right)  \\beta_{k}q^{k+1}-\\beta_{m}=\\left\\{\n\\begin{tabular}\n[c]{ll\n$1,$ & $m=1,$\\\\\n$0,$ & $m>1.\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\right.  \\label{b2\n\\end{equation}\nHere, and in the remainder of the paper, the parameter we make the assumption\nthat $\\left\\vert q\\right\\vert <1.$Clearly we recover (\\ref{b1}) if we let\n$q\\rightarrow1$ in (\\ref{b2}).The $q$-binomial formula is known a\n\\begin{align*}\n\\left(  1\\ominus_{q}x\\right)  ^{n}  &  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k}}{2^{k}}(-x)^{k}=\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(1+1)(1+q)...(1+q^{k-1})x^{k}}{2^{k}}(-1)^{k}\\\\\n\\left(  1-a\\right)  _{q}^{n}  &  =\\left(  a;q\\right)  _{n}\n{\\displaystyle\\prod\\limits_{j=0}^{n-1}}\n\\left(  1-q^{j}a\\right)  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}q^{\\frac{1}{2}k\\left(  k-1\\right)  }\\left(  -1\\right)  ^{k}a^{k}.\n\\end{align*}\nThe above $q$-standard notation can be found in \\cite{andrew}.\n\nCarlitz has introduced the $q$-Bernoulli numbers and polynomials in\n\\cite{carlitz}. Srivastava and Pint\\'{e}r proved some relations and theorems\nbetween the Bernoulli polynomials and Euler polynomials in \\cite{sri1}. They\nalso gave some generalizations of these polynomials. In \\cite{kim2\n-\\cite{kim7}, Kim et al. investigated some properties of the $q$-Euler\npolynomials and Genocchi polynomials. They gave some recurrence relations. In\n\\cite{cenkci}, Cenkci et al. gave the $q$-extension of Genocchi numbers in a\ndifferent manner. In \\cite{kim5}, Kim gave a new concept for the $q$-Genocchi\nnumbers and polynomials. In \\cite{simsek}, Simsek et al. investigated the\n$q$-Genocchi zeta function and $l$-function by using generating functions and\nMellin transformation. There are numerous recent studies on this subject by\namong many other authors: Cenkci et al. \\cite{cenkci}, \\cite{cenkci2}, Choi et\nal \\cite{choi}, Cheon \\cite{cheon}, Luo and Srivastava \\cite{luo},\n\\cite{luo2}, \\cite{luo3}, Srivastava et al.\\cite{sri1}, \\cite{sri2}, Nalci and\nPashaev \\cite{pash} Gabouary and Kurt B., \\cite{kurt1}, Kim et al.\n\\cite{kimd}, Kurt V. \\cite{kurt}.\n\nWe first give here the definitions of the $q$-numbers and $q$-polynomials as\nfollows. It should be mentioned that the definition of $q$-Bernoulli numbers\nin Definition \\ref{D:1} can br found in \\cite{pash}.\n\n\\begin{definition}\n\\label{D:1}Let $q\\in\\mathbb{C},\\ 0<\\left\\vert q\\right\\vert <1.$ The\n$q$-Bernoulli numbers $\\mathfrak{b}_{n,q}$ and polynomials $\\mathfrak{B\n_{n,q}\\left(  x,y\\right)  $ are defined by the means of the generating\nfunctions\n\\begin{align*}\n\\widehat{\\mathfrak{B}}\\left(  t\\right)   &  :=\\frac{te_{q}\\left(  -\\frac{t\n{2}\\right)  }{e_{q}\\left(  \\frac{t}{2}\\right)  -e_{q}\\left(  -\\frac{t\n{2}\\right)  }=\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}=\\sum_{n=0\n^{\\infty}\\mathfrak{b}_{n,q}\\frac{t^{n}}{\\left[  n\\right]  _{q}!\n,\\ \\ \\ \\left\\vert t\\right\\vert <2\\pi,\\\\\n\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}\\mathcal{E}_{q}\\left(  tx\\right)\n\\mathcal{E}_{q}\\left(  ty\\right)   &  =\\sum_{n=0}^{\\infty}\\mathfrak{B\n_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n,\\ \\ \\ \\left\\vert t\\right\\vert <2\\pi.\n\\end{align*}\n\n\\end{definition}\n\n\\begin{definition}\n\\label{D:2}Let $q\\in\\mathbb{C},\\ 0<\\left\\vert q\\right\\vert <1.$ The $q$-Euler\nnumbers $\\mathfrak{e}_{n,q}$ and polynomials $\\mathfrak{E}_{n,q}\\left(\nx,y\\right)  $ are defined by the means of the generating functions\n\\begin{align*}\n\\widehat{\\mathfrak{E}}\\left(  t\\right)   &  :=\\frac{2e_{q}\\left(  -\\frac{t\n{2}\\right)  }{e_{q}\\left(  \\frac{t}{2}\\right)  +e_{q}\\left(  -\\frac{t\n{2}\\right)  }=\\frac{2}{\\mathcal{E}_{q}\\left(  t\\right)  +1}=\\sum_{n=0\n^{\\infty}\\mathfrak{e}_{n,q}\\frac{t^{n}}{\\left[  n\\right]  _{q}!\n,\\ \\ \\ \\left\\vert t\\right\\vert <\\pi,\\\\\n\\frac{2}{\\mathcal{E}_{q}\\left(  t\\right)  +1}\\mathcal{E}_{q}\\left(  tx\\right)\n\\mathcal{E}_{q}\\left(  ty\\right)   &  =\\sum_{n=0}^{\\infty}\\mathfrak{E\n_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n,\\ \\ \\ \\left\\vert t\\right\\vert <\\pi.\n\\end{align*}\n\n\\end{definition}\n\n\\begin{definition}\n\\label{D:3}Let $q\\in\\mathbb{C},\\ 0<\\left\\vert q\\right\\vert <1.$ The\n$q$-Genocchi numbers $\\mathfrak{g}_{n,q}$ and polynomials $\\mathfrak{G\n_{n,q}\\left(  x,y\\right)  $ are defined by the means of the generating\nfunctions\n\\begin{align*}\n\\widehat{\\mathfrak{G}}\\left(  t\\right)   &  :=\\frac{2te_{q}\\left(  -\\frac\n{t}{2}\\right)  }{e_{q}\\left(  \\frac{t}{2}\\right)  +e_{q}\\left(  -\\frac{t\n{2}\\right)  }=\\frac{2t}{\\mathcal{E}_{q}\\left(  t\\right)  +1}=\\sum\n_{n=0}^{\\infty}\\mathfrak{g}_{n,q}\\frac{t^{n}}{\\left[  n\\right]  _{q\n!},\\ \\ \\ \\left\\vert t\\right\\vert <\\pi,\\\\\n\\frac{2t}{\\mathcal{E}_{q}\\left(  t\\right)  +1}\\mathcal{E}_{q}\\left(\ntx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)   &  =\\sum_{n=0}^{\\infty\n}\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q\n!},\\ \\ \\ \\left\\vert t\\right\\vert <\\pi.\n\\end{align*}\n\n\\end{definition}\n\n\\begin{definition}\n\\label{D:4}Let $q\\in\\mathbb{C},\\ 0<\\left\\vert q\\right\\vert <1.$ The\n$q$-tangent numbers $\\mathfrak{T}_{n,q}$ are defined by the means of the\ngenerating functions\n\\begin{align*}\n\\tanh_{q}t  &  =-i\\tan_{q}\\left(  it\\right)  =\\frac{e_{q}\\left(  t\\right)\n-e_{q}\\left(  -t\\right)  }{e_{q}\\left(  t\\right)  +e_{q}\\left(  -t\\right)\n}=\\frac{\\mathcal{E}_{q}\\left(  2t\\right)  -1}{\\mathcal{E}_{q}\\left(\n2t\\right)  +1}\\\\\n&  =\\sum_{n=1}^{\\infty}\\mathfrak{T}_{2n+1,q}\\frac{\\left(  -1\\right)\n^{k}t^{2n+1}}{\\left[  2n+1\\right]  _{q}!}.\n\\end{align*}\n\n\\end{definition}\n\nIt is obvious that by tending $q$ to 1 from the left side, we lead to the\nclassic definition of these polynomials\n\\begin{align*}\n\\mathfrak{b}_{n,q}  &  :=\\mathfrak{B}_{n,q}\\left(  0\\right)  ,\\ \\ \\ \\lim\n_{q\\rightarrow1^{-}}\\mathfrak{B}_{n,q}\\left(  x\\right)  =B_{n}\\left(\nx\\right)  ,\\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{B}_{n,q}\\left(  x,y\\right)\n=B_{n}\\left(  x+y\\right)  ,\\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{b\n_{n,q}=B_{n},\\\\\n\\mathfrak{e}_{n,q}  &  :=\\mathfrak{E}_{n,q}\\left(  0\\right)  ,\\ \\ \\ \\lim\n_{q\\rightarrow1^{-}}\\mathfrak{E}_{n,q}\\left(  x\\right)  =E_{n}\\left(\nx\\right)  ,\\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{E}_{n,q}\\left(  x,y\\right)\n=E_{n}\\left(  x+y\\right)  ,\\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{e\n_{n,q}=E_{n},\\\\\n\\mathfrak{g}_{n,q}  &  :=\\mathfrak{G}_{n,q}\\left(  0\\right)  ,\\ \\ \\ \\lim\n_{q\\rightarrow1^{-}}\\mathfrak{G}_{n,q}\\left(  x\\right)  =G_{n}\\left(\nx\\right)  ,\\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{G}_{n,q}\\left(  x,y\\right)\n=G_{n}\\left(  x+y\\right)  \\ \\ \\ \\lim_{q\\rightarrow1^{-}}\\mathfrak{g\n_{n,q}=G_{n}.\n\\end{align*}\nHere $B_{n}\\left(  x\\right)  ,$ $E_{n}\\left(  x\\right)  $ and $G_{n}\\left(\nx\\right)  $ denote the classical Bernoulli, Euler and Genocchi polynomials\nwhich are defined b\n\\[\n\\frac{t}{e^{t}-1}e^{tx}=\\sum_{n=0}^{\\infty}B_{n}\\left(  x\\right)  \\frac{t^{n\n}{n!},\\ \\ \\ \\text{\\ }\\frac{2}{e^{t}+1}e^{tx}=\\sum_{n=0}^{\\infty}E_{n}\\left(\nx\\right)  \\frac{t^{n}}{n!}\\ \\ \\text{and\\ \\ \\ \\ }\\frac{2t}{e^{t}+1}e^{tx\n=\\sum_{n=0}^{\\infty}G_{n}\\left(  x\\right)  \\frac{t^{n}}{n!}.\n\\]\n\n\nThe aim of the present paper is to obtain some results for the above newly\ndefined $q$-polynomials. It should be mentioned that $q$-Bernoulli and\n$q$-Euler polynomials in our definitions are polynomials of $x$ and $y$ and\nwhen $y=0$ they are polynomials of $x$, but in other definitions they respect\nto $q^{x}$. First advantage of this approach is that for $q\\rightarrow1^{-}$\n,$\\mathfrak{B}_{n,q}\\left(  x,y\\right)  $ ($\\mathfrak{E}_{n,q}\\left(\nx,y\\right)  ,$ $\\mathfrak{G}_{n,q}\\left(  x,y\\right)  $) becomes the classical\nBernoulli $\\mathfrak{B}_{n}\\left(  x+y\\right)  $ (Euler $\\mathfrak{E\n_{n}\\left(  x+y\\right)  ,\\ $Genocchi $\\mathfrak{G}_{n,q}\\left(  x,y\\right)  $)\npolynomial and we may obtain the $q$-analogues of well-known results, for\nexample Srivastava and Pint\\'{e}r \\cite{pinter}, Cheon \\cite{cheon}, etc.\nSecond advantage is that similar to the classical case odd numbered terms of\nthe Bernoulli numbers $\\mathfrak{b}_{k,q}$ and the Genocchi numbres\n$\\mathfrak{g}_{k,q}$are zero, and even numbered terms of the Euler numbers\n$\\mathfrak{e}_{n,q}$ are zero.\n\n\\section{Preliminary results}\n\nIn this section we shall provide some basic formulas for the $q$-Bernoulli,\n$q$-Euler and $q$-Genocchi numbers and polynomials in order to obtain the main\nresults of this paper in the next section.\n\n\\begin{lemma}\n\\label{L:11}The $q$-Bernoulli numbers $\\mathfrak{b}_{n,q}$ satisfy the\nfollowing $q$-binomial recurrence:\n\\begin{equation\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{b}_{k,q}-\\mathfrak{b\n_{n,q}=\\left\\{\n\\begin{tabular}\n[c]{ll\n$1,$ & $n=1,$\\\\\n$0,$ & $n>1.\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\right.  \\label{ber\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nBy simple multiplication on (\\ref{D:1}) we see that\n\\[\n\\widehat{\\mathfrak{B}}\\left(  t\\right)  \\mathcal{E}_{q}\\left(  t\\right)\n=t+\\widehat{\\mathfrak{B}}\\left(  t\\right)  .\n\\]\nS\n\\\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{b}_{k,q}\\frac{t^{n\n}{\\left[  n\\right]  _{q}!}=t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{b}_{n,q}\\ \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\]\nThe statement follows by comparing $t^{m}$-coefficients.\n\\end{proof}\n\nWe use this formula to calculate the first few $\\mathfrak{b}_{k,q}$\n\\\n\\begin{tabular}\n[c]{ll\n$\\mathfrak{b}_{0,q}=$ & $1,$\\\\\n$\\mathfrak{b}_{1,q}=$ & $-\\frac{1}{2}=-\\frac{1}{\\left\\{  2\\right\\}  _{q}},$\\\\\n$\\mathfrak{b}_{2,q}=$ & $\\frac{1}{4}\\dfrac{q(q+1)}{q^{2}+q+1}=\\frac{q[2]_{q\n}{4[3]_{q}},$\\\\\n$\\mathfrak{b}_{3,q}=$ & $0.\n\\end{tabular}\n\\ \\ \\ \\ \\\n\\]\n\n\nThe similar property can be proved for $q$-Euler number\n\\begin{equation\n{\\displaystyle\\sum\\limits_{k=0}^{m}}\n\\left\\{\n\\begin{array}\n[c]{c\nm\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{e}_{k,q}+\\mathfrak{e}_{m,q}=\\left\\{\n\\begin{tabular}\n[c]{ll\n$2,$ & $m=0,$\\\\\n$0,$ & $m>0.\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\right.  \\label{euler\n\\end{equation}\n\\bigskip and $q$-Genocchi number\n\\begin{equation\n{\\displaystyle\\sum\\limits_{k=0}^{m}}\n\\left\\{\n\\begin{array}\n[c]{c\nm\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{g}_{k,q}+\\mathfrak{g}_{m,q}=\\left\\{\n\\begin{tabular}\n[c]{ll\n$2,$ & $m=1,$\\\\\n$0,$ & $m>1.\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\right.  \\label{gen\n\\end{equation}\n\n\nUsing the above recurrence formulae we calculate the first few $\\mathfrak{e\n_{n,q}$ and $\\mathfrak{g}_{n,q}$as well\n\n\\\n\\begin{tabular}\n[c]{ll\n$\\mathfrak{e}_{0,q}=$ & $1,$\\\\\n$\\mathfrak{e}_{1,q}=$ & $-\\dfrac{1}{2},$\\\\\n$\\mathfrak{e}_{2,q}=$ & $0,$\\\\\n$\\mathfrak{e}_{3,q}=$ & $\\frac{[3]_{q}[2]_{q}-[4]_{q}}{8}=\\dfrac{q\\left(\n1+q\\right)  }{8},\n\\end{tabular\n\\begin{tabular}\n[c]{ll\n$\\mathfrak{g}_{0,q}=$ & $0,$\\\\\n$\\mathfrak{g}_{1,q}=$ & $1,$\\\\\n$\\mathfrak{g}_{2,q}=$ & $-\\frac{[2]_{q}}{2}=-\\dfrac{q+1}{2},$\\\\\n$\\mathfrak{g}_{3,q}=$ & $0.\n\\end{tabular}\n\\ \\ \\ \\ \\ \\ \\\n\\]\n\n\n\\begin{remark}\nThe first advantage of the new $q$-numbers $\\mathfrak{b}_{k,q},$\n$\\mathfrak{e}_{k,q}$ and $\\mathfrak{g}_{k,q}$ is that similar to classical\ncase odd numbered terms of the Bernoulli numbers $\\mathfrak{b}_{k,q}$ and the\nGenocchi numbres $\\mathfrak{g}_{k,q}$are zero, and even numbered terms of the\nEuler numbers $\\mathfrak{e}_{n,q}$ are zero.\n\\end{remark}\n\nNext lemma gives the relationsheep between $q$-Genocchi numbers and\n$q$-Tangent numbers.\n\n\\begin{lemma}\nFro any $n\\in\\mathbb{N}$ we hav\n\\[\n\\mathfrak{T}_{2n+1,q}=\\mathfrak{g}_{2n+2,q}\\frac{\\left(  -1\\right)\n^{k-1}2^{2n+1}}{\\left[  2n+2\\right]  _{q}}.\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nFirst we recall the definition of $q$-trigonometric functions\n\\begin{align*}\n\\cos_{q}t  &  =\\frac{e_{q}\\left(  it\\right)  +e_{q}\\left(  -it\\right)  \n{2},\\ \\ \\ \\ \\ \\sin_{q}t=\\frac{e_{q}\\left(  it\\right)  -e_{q}\\left(\n-it\\right)  }{2i},\\\\\ni\\tan_{q}t  &  =\\frac{e_{q}\\left(  it\\right)  -e_{q}\\left(  -it\\right)\n}{e_{q}\\left(  it\\right)  +e_{q}\\left(  -it\\right)  },\\ \\ \\ \\ \\ \\ \\ \\cot\n_{q}t=i\\frac{e_{q}\\left(  it\\right)  +e_{q}\\left(  -it\\right)  }{e_{q}\\left(\nit\\right)  -e_{q}\\left(  -it\\right)  }.\n\\end{align*}\nNow by choosing $z=2it$ in $\\widehat{\\mathfrak{B}}\\left(  z\\right)  $, we ge\n\\[\n\\widehat{\\mathfrak{B}}\\left(  2it\\right)  =\\frac{2it}{\\mathcal{E}_{q}\\left(\n2it\\right)  -1}=\\frac{te_{q}\\left(  -it\\right)  }{\\sin_{q}t}=\\sum\n_{n=0}^{\\infty}\\mathfrak{b}_{n,q}\\frac{\\left(  2it\\right)  ^{n}}{\\left[\nn\\right]  _{q}!}.\n\\]\nIt follows tha\n\\begin{align*}\n\\widehat{\\mathfrak{B}}\\left(  2it\\right)   &  =\\frac{te_{q}\\left(  -it\\right)\n}{\\sin_{q}t}=\\frac{t}{\\sin_{q}t}\\left(  \\cos_{q}t-i\\sin_{q}t\\right)\n=t\\cot_{q}t-it\\\\\n&  =\\mathfrak{b}_{0,q}+2it\\mathfrak{b}_{1,q}+\\sum_{n=2}^{\\infty\n\\mathfrak{b}_{n,q}\\frac{\\left(  2it\\right)  ^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  =1-it+\\sum_{n=2}^{\\infty}\\mathfrak{b}_{n,q}\\frac{\\left(  2it\\right)  ^{n\n}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\nSince $t\\cot_{q}t$ is even in the above sum odd coefficients $\\mathfrak{b\n_{2k+1,q}$ , $k=1,2,...$are zero we get\n\\[\nt\\cot_{q}t=1+\\sum_{n=2}^{\\infty}\\mathfrak{b}_{n,q}\\frac{\\left(  2it\\right)\n^{n}}{\\left[  n\\right]  _{q}!}=1+\\sum_{n=1}^{\\infty}\\mathfrak{b}_{n,q\n\\frac{\\left(  2it\\right)  ^{2n}}{\\left[  2n\\right]  _{q}!}.\n\\]\n\\bigskip By choosing $z=2it$ in $\\widehat{\\mathfrak{G}}\\left(  z\\right)  $, we\nge\n\\[\n\\widehat{\\mathfrak{G}}\\left(  2it\\right)  =\\frac{4it}{\\mathcal{E}_{q}\\left(\n2it\\right)  +1}=\\frac{2ite_{q}\\left(  -it\\right)  }{\\cos_{q}t}=\\sum\n_{n=0}^{\\infty}\\mathfrak{g}_{n,q}\\frac{\\left(  2it\\right)  ^{n}}{\\left[\nn\\right]  _{q}!}.\n\\\n\\begin{align*}\n\\widehat{\\mathfrak{G}}\\left(  2it\\right)   &  =\\frac{4it}{\\mathcal{E\n_{q}\\left(  2it\\right)  +1}=\\frac{2ite_{q}\\left(  -it\\right)  }{\\cos_{q\nt}=\\frac{2it}{\\cos_{q}t}\\left(  \\cos_{q}t-i\\sin_{q}t\\right)  =2it+2t\\tan\n_{q}t\\\\\n&  =\\mathfrak{g}_{0,q}+2it\\mathfrak{g}_{1,q}+\\sum_{n=2}^{\\infty\n\\mathfrak{g}_{n,q}\\frac{\\left(  2it\\right)  ^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  =2it+\\sum_{n=2}^{\\infty}\\mathfrak{g}_{n,q}\\frac{\\left(  2it\\right)  ^{n\n}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\nIt follows tha\n\\begin{align*}\n2t\\tan_{q}t  &  =\\sum_{n=1}^{\\infty}\\mathfrak{g}_{2n,q}\\frac{\\left(\n2it\\right)  ^{2n}}{\\left[  2n\\right]  _{q}!},\\ \\ \\ \\ \\ \\ \\tan_{q}t=\\sum\n_{n=1}^{\\infty}\\mathfrak{g}_{2n,q}\\frac{\\left(  -1\\right)  ^{n}\\left(\n2t\\right)  ^{2n-1}}{\\left[  2n\\right]  _{q}!}\\\\\n\\tanh_{q}t  &  =-i\\tan_{q}\\left(  it\\right)  =-i\\sum_{n=1}^{\\infty\n}\\mathfrak{g}_{2n,q}\\frac{\\left(  -1\\right)  ^{n}\\left(  2it\\right)  ^{2n-1\n}{\\left[  2n\\right]  _{q}!}\\\\\n&  =-\\sum_{n=1}^{\\infty}\\mathfrak{g}_{2n,q}\\frac{\\left(  2t\\right)  ^{2n-1\n}{\\left[  2n\\right]  _{q}!}=-\\sum_{n=1}^{\\infty}\\mathfrak{g}_{2n+2,q\n\\frac{\\left(  2t\\right)  ^{2n+1}}{\\left[  2n+2\\right]  _{q}!}.\n\\end{align*}\nThu\n\\begin{align*}\n\\tanh_{q}t  &  =-i\\tan_{q}\\left(  it\\right)  =\\frac{e_{q}\\left(  t\\right)\n-e_{q}\\left(  -t\\right)  }{e_{q}\\left(  t\\right)  +e_{q}\\left(  -t\\right)\n}=\\frac{\\mathcal{E}_{q}\\left(  2t\\right)  -1}{\\mathcal{E}_{q}\\left(\n2t\\right)  +1}\\\\\n&  =\\sum_{n=1}^{\\infty}\\mathfrak{T}_{2n+1,q}\\frac{\\left(  -1\\right)\n^{k}t^{2n+1}}{\\left[  2n+1\\right]  _{q}!},\n\\end{align*}\nan\n\\[\n\\mathfrak{T}_{2n+1,q}=\\mathfrak{g}_{2n+2,q}\\frac{\\left(  -1\\right)\n^{k-1}2^{2n+1}}{\\left[  2n+2\\right]  _{q}}.\n\\]\n\n\\end{proof}\n\nThe following result is $q$-analogue of the addition theorem for the classical\nBernoulli, Euler and Genocchi polynomials.\n\n\\begin{lemma}\n\\label{L:1}\\emph{(Addition Theorems)} For all $x,y\\in\\mathbb{C}$ we hav\n\\begin{equation\n\\begin{tabular}\n[c]{lll\n$\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{b}_{k,q}\\left(  x\\oplus_{q}y\\right)  ^{n-k},$ &  &\n$\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{B}_{k,q}\\left(  x\\right)\ny^{n-k},$\\\\\n$\\mathfrak{E}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{e}_{k,q}\\left(  x\\oplus_{q}y\\right)  ^{n-k},$ &  &\n$\\mathfrak{E}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{E}_{k,q}\\left(  x\\right)\ny^{n-k},$\\\\\n$\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{g}_{k,q}\\left(  x\\oplus_{q}y\\right)  ^{n-k},$ &  &\n$\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{G}_{k,q}\\left(  x\\right)\ny^{n-k}.\n\\end{tabular}\n\\ \\ \\label{be01\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nWe prove only the first formula. It is a consequence of the following identit\n\\begin{align*}\n\\sum_{n=0}^{\\infty}\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[\nn\\right]  _{q}!}  &  =\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1\n\\mathcal{E}_{q}\\left(  tx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)\n=\\sum_{n=0}^{\\infty}\\mathfrak{b}_{n,q}\\frac{t^{n}}{\\left[  n\\right]  _{q\n!}\\sum_{n=0}^{\\infty}\\left(  x\\oplus_{q}y\\right)  ^{n}\\frac{t^{n}}{\\left[\nn\\right]  _{q}!}\\\\\n&  =\\sum_{n=0}^{\\infty\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{b}_{k,q}\\left(  x\\oplus_{q}y\\right)  ^{n-k}\\frac{t^{n\n}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\n\n\\end{proof}\n\nIn particular, setting $y=0$ in (\\ref{be01}), we get the following formulas\nfor $q$-Bernoulli, $q$-Euler and $q$-Genocchi polynomials, respectively\n\\begin{align}\n\\mathfrak{B}_{n,q}\\left(  x\\right)   &  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{b}_{k,q}x^{n-k\n,\\ \\ \\ \\mathfrak{E}_{n,q}\\left(  x\\right)  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{e}_{k,q}x^{n-k\n,\\label{be7}\\\\\n\\mathfrak{G}_{n,q}\\left(  x\\right)   &  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{g}_{k,q}x^{n-k}.\n\\label{be8\n\\end{align}\nSetting $y=1$ in (\\ref{be01}), we ge\n\\begin{align}\n\\mathfrak{B}_{n,q}\\left(  x,1\\right)   &  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{B}_{k,q}\\left(  x\\right)\n,\\ \\ \\ \\mathfrak{E}_{n,q}\\left(  x,1\\right)  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{E}_{k,q}\\left(  x\\right)\n,\\label{be3}\\\\\n\\mathfrak{G}_{n,q}\\left(  x,1\\right)   &  =\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\dfrac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{G}_{k,q}\\left(  x\\right)\n. \\label{be4\n\\end{align}\nClearly (\\ref{be3}) and (\\ref{be4}) are $q$-analogues o\n\\[\nB_{n}\\left(  x+1\\right)  =\\sum_{k=0}^{n}\\left(\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right)  B_{k}\\left(  x\\right)  ,\\ E_{n}\\left(  x+1\\right)  =\\sum_{k=0\n^{n}\\left(\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right)  E_{k}\\left(  x\\right)  ,\\ G_{n}\\left(  x+1\\right)  =\\sum_{k=0\n^{n}\\left(\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right)  G_{k}\\left(  x\\right)  ,\n\\]\nrespectively.\n\n\\begin{lemma}\nThe odd coefficient of the $q$-Bernoulli numbers except the first one are\nzero, that means $\\mathfrak{b}_{n,q}=0$ where $n=2r+1,\\ (r\\i\n\\mathbb{N}\n)$.\n\n\\begin{proof}\nIt follows from the fact that the function\n\\[\nf(t)\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{b}_{n,q}\\frac{t^{n}}{[n]_{q}!}-\\mathfrak{b}_{1,q}t=\\frac\n{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}+\\ \\frac{t}{2}=\\ \\frac{t}{2}\\left(\n\\frac{\\mathcal{E}_{q}\\left(  t\\right)  +1}{\\mathcal{E}_{q}\\left(  t\\right)\n-1}\\right)  \\ \\\n\\]\nis even, and the coefficient of $t^{n}$ in the Taylor expansion about zero of\nany even function vanish for all odd $n$. note that this could not happen in\nthe last $q$-analogue of these numbers, because in the case of improved\nexponential function $\\mathcal{E}_{q}\\left(  -t\\right)  =\\left(\n\\mathcal{E}_{q}\\left(  t\\right)  \\right)  ^{-1}$.\n\\end{proof}\n\\end{lemma}\n\nBy using (\\ref{be07}) and $q$-derivative approaching to the next lemma.\n\n\\begin{lemma}\n\\label{L:2}We hav\n\\begin{align*}\nD_{q,x}\\mathfrak{B}_{n,q}\\left(  x\\right)    & =\\left[  n\\right]  _{q\n\\frac{\\mathfrak{B}_{n-1,q}\\left(  x\\right)  +\\mathfrak{B}_{n-1,q}\\left(\nqx\\right)  }{2},\\ \\ \\ D_{q,x}\\mathfrak{E}_{n,q}\\left(  x\\right)  =\\left[\nn\\right]  _{q}\\frac{\\mathfrak{E}_{n-1,q}\\left(  x\\right)  +\\mathfrak{E\n_{n-1,q}\\left(  qx\\right)  }{2},\\ \\ \\ \\\\\nD_{q,x}\\mathfrak{G}_{n,q}\\left(  x\\right)    & =\\left[  n\\right]  _{q\n\\frac{\\mathfrak{G}_{n-1,q}\\left(  x\\right)  +\\mathfrak{G}_{n-1,q}\\left(\nqx\\right)  }{2}.\n\\end{align*}\n\n\\end{lemma}\n\n\\begin{lemma}\n\\label{L:3}\\emph{(Difference Equations)} We hav\n\\begin{align}\n\\mathfrak{B}_{n,q}\\left(  x,1\\right)  -\\mathfrak{B}_{n,q}\\left(  x\\right)   &\n=\\frac{\\left(  -1;q\\right)  _{n-1}}{2^{n-1}}\\left[  n\\right]  _{q\nx^{n-1},\\ \\ n\\geq1,\\label{be5}\\\\\n\\mathfrak{E}_{n,q}\\left(  x,1\\right)  +\\mathfrak{E}_{n,q}\\left(  x\\right)   &\n=2\\frac{\\left(  -1;q\\right)  _{n}}{2^{n}}x^{n},\\ \\ \\ n\\geq0,\\label{be6}\\\\\n\\mathfrak{G}_{n,q}\\left(  x,1\\right)  +\\mathfrak{G}_{n,q}\\left(  x\\right)   &\n=2\\frac{\\left(  -1;q\\right)  _{n-1}}{2^{n-1}}\\left[  n\\right]  _{q\nx^{n-1},\\ \\ \\ n\\geq1. \\label{b61\n\\end{align}\n\n\\end{lemma}\n\n\\begin{proof}\nWe prove the identity for the $q$-Bernoulli polynomials. From the identit\n\\[\n\\frac{t\\mathcal{E}_{q}\\left(  t\\right)  }{\\mathcal{E}_{q}\\left(  t\\right)\n-1}\\mathcal{E}_{q}\\left(  tx\\right)  =t\\mathcal{E}_{q}\\left(  tx\\right)\n+\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}\\mathcal{E}_{q}\\left(\ntx\\right)  ,\n\\]\nit follows tha\n\\[\n\\sum_{n=0}^{\\infty\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{B}_{k,q}\\left(  x\\right)\n\\frac{t^{n}}{\\left[  n\\right]  _{q}!}=\\sum_{n=0}^{\\infty}\\frac{(-1,q)_{n\n}{2^{n}}x^{n}\\frac{t^{n+1}}{\\left[  n\\right]  _{q}!}+\\sum_{n=0}^{\\infty\n}\\mathfrak{B}_{n,q}\\left(  x\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\]\n\n\\end{proof}\n\nFrom (\\ref{be5}) and (\\ref{be7}), (\\ref{be6}) and (\\ref{be8}) we obtain the\nfollowing formulas.\n\n\\begin{lemma}\n\\label{L:4}We have\n\\begin{align}\nx^{n}  &  =\\frac{2^{n}}{\\left(  -1;q\\right)  _{n}\\left[  n\\right]  _{q}\n\\sum_{k=0}^{n}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{\\left(  -1;q\\right)  _{n+1-k}}{2^{n+1-k}}\\mathfrak{B\n_{k,q}\\left(  x\\right) \\label{be9}\\\\\nx^{n}  &  =\\frac{2^{n-1}}{\\left(  -1;q\\right)  _{n}}\\left(  \\sum_{k=0\n^{n}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{\\left(  -1;q\\right)  _{n-k}}{2^{n-k}}\\mathfrak{E\n_{k,q}\\left(  x\\right)  +\\mathfrak{E}_{n,q}\\left(  x\\right)  \\right)  ,\\\\\nx^{n}  &  =\\frac{2^{n-1}}{\\left(  -1;q\\right)  _{n}\\left[  n+1\\right]  _{q\n}\\left(  \\sum_{k=0}^{n+1}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{\\left(  -1;q\\right)  _{n+1-k}}{2^{n+1-k}}\\mathfrak{G\n_{k,q}\\left(  x\\right)  +\\mathfrak{G}_{n+1,q}\\left(  x\\right)  \\right)  .\n\\label{be10\n\\end{align}\n\n\\end{lemma}\n\nThe above formulas are $q$-analoques of the following familiar expansion\n\\begin{align}\nx^{n}  & =\\frac{1}{n+1\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left(\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right)  B_{k}\\left(  x\\right)  ,\\ \\ \\ x^{n}=\\frac{1}{2}\\left[\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left(\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right)  E_{k}\\left(  x\\right)  +E_{n}\\left(  x\\right)  \\right]\n,\\ \\ \\ \\label{cl1}\\\\\nx^{n}  & =\\frac{1}{2\\left(  n+1\\right)  }\\left[\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\left(\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right)  E_{k}\\left(  x\\right)  +E_{n+1}\\left(  x\\right)  \\right]  ,\\nonumber\n\\end{align}\nrespectively.\n\n\\begin{lemma}\nThe following identities hold true\n\\begin{align*\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{B}_{k,q}\\left(  x,y\\right)\n-\\mathfrak{B}_{n,q}\\left(  x,y\\right)   &  =\\left[  n\\right]  _{q}\\left(\nx\\oplus_{q}y\\right)  ^{n-1},\\\\%\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{E}_{k,q}\\left(  x,y\\right)\n+\\mathfrak{E}_{n,q}\\left(  x,y\\right)   &  =2\\left(  x\\oplus_{q}y\\right)\n^{n},\\\\%\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{G}_{k,q}\\left(  x,y\\right)\n+\\mathfrak{G}_{n,q}\\left(  x,y\\right)   &  =2\\left[  n\\right]  _{q}\\left(\nx\\oplus_{q}y\\right)  ^{n-1}.\n\\end{align*}\n\n\\end{lemma}\n\n\\begin{proof}\nWe the identity for the $q$-Bernoulli polynomials. From the identit\n\\[\n\\frac{t\\mathcal{E}_{q}\\left(  t\\right)  }{\\mathcal{E}_{q}\\left(  t\\right)\n-1}\\mathcal{E}_{q}\\left(  tx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)\n=t\\mathcal{E}_{q}\\left(  tx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)\n+\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}\\mathcal{E}_{q}\\left(\ntx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)  ,\n\\]\nit follows tha\n\\[\n\\sum_{n=0}^{\\infty\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{2^{n-k}}\\mathfrak{B}_{k,q}\\left(  x,y\\right)\n\\frac{t^{n}}{\\left[  n\\right]  _{q}!}=\\sum_{n=0}^{\\infty\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k}(-1,q)_{n-k}}{2^{n}}x^{k}y^{n-k}\\frac{t^{n+1\n}{\\left[  n\\right]  _{q}!}+\\sum_{n=0}^{\\infty}\\mathfrak{B}_{n,q}\\left(\nx,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\]\n\n\\end{proof}\n\n\\section{Some new formulas}\n\nThe classical Cayley transformation $z\\rightarrow$Cay$(z,a):=\\frac{1+az\n{1-az}$ motivates us to approaching to the formula for $\\mathcal{E}_{q}\\left(\nqt\\right)  $, In addition by substitute it in the generating formula we have\n\n\\[\n\\widehat{\\mathfrak{B}}_{q}\\left(  qt\\right)  \\widehat{\\mathfrak{B}}_{q}\\left(\nt\\right)  =\\left(  \\widehat{\\mathfrak{B}}_{q}\\left(  qt\\right)  -q\\widehat\n{\\mathfrak{B}}_{q}\\left(  t\\right)  (1+(1-q)\\frac{t}{2})\\right)  \\frac{1\n{1-q}\\times\\frac{2}{\\mathcal{E}_{q}\\left(  t\\right)  +1\n\\]\nThe right hand side can be presented by improved $q$-Euler numbers .Now the\nequating coefficients of $t^{n}$ we get the following identity.In the case\nthat $n=0$, we find the first improved $q$-Euler number which is exactly 1.\n\n\\begin{proposition}\nFor all $n\\geq1$\n\n\\\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{k,q}\\mathfrak{B}_{n-k,q}q^{k}=-\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{k,q}\\mathfrak{E}_{n-k,q}[k-1]_{q}-\\frac{q}{2\n{\\displaystyle\\sum\\limits_{k=0}^{n-1}}\n\\left[\n\\begin{array}\n[c]{c\nn-1\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{k,q}\\mathfrak{E}_{n-k-1,q}[n]_{q\n\\]\n\n\\end{proposition}\n\nLet take a $q$-derivative from generating function, after simplifying the\nequation and by knowing the quotient rule for quantum derivative , also using\nthat\n\\[\n\\mathcal{E}_{q}\\left(  qt\\right)  =\\frac{1-(1-q)\\frac{t}{2}}{1+(1-q)\\frac\n{t}{2}}\\mathcal{E}_{q}\\left(  t\\right)  ,D_{q}(\\mathcal{E}_{q}\\left(\nt\\right)  )=\\frac{\\mathcal{E}_{q}\\left(  qt\\right)  +\\mathcal{E}_{q}\\left(\nt\\right)  }{2},\n\\]\nwe have\n\\[\n\\widehat{B}_{q}\\left(  qt\\right)  \\widehat{B}_{q}\\left(  t\\right)\n=\\frac{2+(1-q)t}{2\\mathcal{E}_{q}\\left(  t\\right)  (q-1)}\\left(  q\\widehat\n{B}_{q}\\left(  t\\right)  -\\widehat{B}_{q}\\left(  qt\\right)  \\right)\n\\]\n\n\nIt is clear that $\\mathcal{E}_{q}^{-1}\\left(  t\\right)  =\\mathcal{E\n_{q}\\left(  -t\\right)  $. Now the equating coefficient of $t^{n}$ we lead to\nthe following identity.\n\n\\begin{proposition}\nFor all $n\\geq1$\n\n\\\n{\\displaystyle\\sum\\limits_{k=0}^{2n}}\n\\left[\n\\begin{array}\n[c]{c\n2n\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{k,q}\\mathfrak{B}_{2n-k,q}q^{k}=-\n{\\displaystyle\\sum\\limits_{k=0}^{2n}}\n\\left\\{\n\\begin{array}\n[c]{c\n2n\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{B}_{k,q}[k-1]_{q}(-1)^{k}+\\frac{q(1-q)}{2\n{\\displaystyle\\sum\\limits_{k=0}^{2n-1}}\n\\left\\{\n\\begin{array}\n[c]{c\n2n-1\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{B}_{k,q}[k-1]_{q}(-1)^{k\n\\]\n\n\\\n{\\displaystyle\\sum\\limits_{k=0}^{2n+1}}\n\\left[\n\\begin{array}\n[c]{c\n2n+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{k,q}\\mathfrak{B}_{2n-k+1,q}q^{k}=\n{\\displaystyle\\sum\\limits_{k=0}^{2n+1}}\n\\left\\{\n\\begin{array}\n[c]{c\n2n+1\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{B}_{k,q}[k-1]_{q}(-1)^{k}-\\frac{q(1-q)}{2\n{\\displaystyle\\sum\\limits_{k=0}^{2n}}\n\\left\\{\n\\begin{array}\n[c]{c\n2n\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\mathfrak{B}_{k,q}[k-1]_{q}(-1)^{k\n\\]\n\n\\end{proposition}\n\nWe may also derive a differential equation for $\\widehat{B}_{q}\\left(\nt\\right)  .$If we differentiate both sides of generating function with respect\nto $t$, after a little calculation we find tha\n\n\\[\n\\frac{\\partial}{\\partial t}\\widehat{B}_{q}\\left(  t\\right)  =\\widehat{B\n_{q}\\left(  t\\right)  \\left(  \\frac{1}{t}-\\frac{(1-q)\\mathcal{E}_{q}\\left(\nt\\right)  }{\\mathcal{E}_{q}\\left(  t\\right)  -1}\\left(\n{\\displaystyle\\sum\\limits_{k=0}^{\\infty}}\n\\frac{4q^{k}}{4-(1-q)^{2}q^{2k}}\\right)  \\right)\n\\]\n\n\nIf we differentiate with respect to $q$, we instead obtai\n\n\\[\n\\frac{\\partial}{\\partial q}\\widehat{B}_{q}\\left(  t\\right)  =-\\widehat{B\n_{q}^{2}\\left(  t\\right)  \\mathcal{E}_{q}\\left(  t\\right)\n{\\displaystyle\\sum\\limits_{k=0}^{\\infty}}\n\\frac{4t(kq^{k-1}-(k+1)q^{k})}{4-(1-q)^{2}q^{2k}\n\\]\n\n\nAgain using generating function and combining this with the t derivative we\nget the partial differential equation\n\n\\begin{proposition}\n\n\\[\n\\frac{\\partial}{\\partial t}\\widehat{B}_{q}\\left(  t\\right)  -\\frac{\\partial\n}{\\partial q}\\widehat{B}_{q}\\left(  t\\right)  =\\frac{\\widehat{B}_{q}\\left(\nt\\right)  }{t}+\\frac{\\widehat{B}_{q}^{2}\\left(  t\\right)  \\mathcal{E\n_{q}\\left(  t\\right)  }{t\n{\\displaystyle\\sum\\limits_{k=0}^{\\infty}}\n\\frac{4t(kq^{k-1}-(k+1)q^{k})-q^{k}(1-q)}{4-(1-q)^{2}q^{2k}\n\\]\n\n\\end{proposition}\n\n\\section{Explicit relationship between the $q$-Bernoulli and $q$-Euler\npolynomials}\n\nIn this section we will give some explicit relationships between the\n$q$-Bernoulli and $q$-Euler polynomials. Here some $q$-analogues of known\nresults will be given. We also obtain new formulas and their some special\ncases below. These formulas are some extensions of the formulas of Srivastava\nand Pint\\'{e}r, Cheon and others.\n\nWe present natural $q$-extensions of the main results in the papers\n\\cite{pinter} and \\cite{luo2}, see Theorems \\ref{S-P1} and \\ref{S-P2}.\n\n\\begin{theorem}\n\\label{S-P1}For $n\\in\\mathbb{N}_{0}$, the following relationships hold true\n\\begin{align*}\n\\mathfrak{B}_{n,q}\\left(  x,y\\right)   &  =\\frac{1}{2\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}m^{k-n}\\left[  \\mathfrak{B}_{k,q}\\left(  x\\right)  \n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left\\{\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{B}_{j,q}\\left(  x\\right)  }{m^{k-j}}\\right]\n\\mathfrak{E}_{n-k,q}\\left(  my\\right) \\\\\n&  =\\frac{1}{2\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}m^{k-n}\\left[  \\mathfrak{B}_{k,q}\\left(  x\\right)  +\\mathfrak{B\n_{k,q}\\left(  x,\\frac{1}{m}\\right)  \\right]  \\mathfrak{E}_{n-k,q}\\left(\nmy\\right)  .\n\\end{align*}\n\n\\end{theorem}\n\n\\begin{proof}\nUsing the following identit\n\\[\n\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)  -1}\\mathcal{E}_{q}\\left(  tx\\right)\n\\mathcal{E}_{q}\\left(  ty\\right)  =\\frac{t}{\\mathcal{E}_{q}\\left(  t\\right)\n-1}\\mathcal{E}_{q}\\left(  tx\\right)  \\cdot\\frac{\\mathcal{E}_{q}\\left(\n\\frac{t}{m}\\right)  +1}{2}\\cdot\\frac{2}{\\mathcal{E}_{q}\\left(  \\frac{t\n{m}\\right)  +1}\\mathcal{E}_{q}\\left(  \\frac{t}{m}my\\right)\n\\]\nwe hav\n\\begin{align*\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\n&  =\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{(-1,q)_{n}}{m^{n}2^{n}}\\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  x\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  +\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  x\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  =:I_{1}+I_{2}.\n\\end{align*}\nIt is clear tha\n\\[\nI_{2}=\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  x\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q\n!}=\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nj\n\\end{array}\n\\right]  _{q}m^{k-n}\\mathfrak{B}_{k,q}\\left(  x\\right)  \\mathfrak{E\n_{n-k,q}\\left(  my\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\]\nOn the other han\n\\begin{align*}\nI_{1}  &  =\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{j=0}^{n}}\n\\left\\{\n\\begin{array}\n[c]{c\nn\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\mathfrak{B}_{j,q}\\left(  x\\right)  \\frac{t^{n}}{m^{n-j}\\left[\nn\\right]  _{q}!}\\\\\n&  =\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{E}_{n-k,q}\\left(  my\\right)\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left\\{\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{B}_{j,q}\\left(  x\\right)  }{m^{n-k}m^{k-j}\n\\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\nTherefor\n\\\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q\n!}=\\frac{1}{2\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}m^{k-n}\\left[  \\mathfrak{B}_{k,q}\\left(  x\\right)  \n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left\\{\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{B}_{j,q}\\left(  x\\right)  }{m^{k-j}}\\right]\n\\mathfrak{E}_{n-k,q}\\left(  my\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\]\nIt remains to equate coefficient of $t^{n}.$\n\\end{proof}\n\nNext we discuss some special cases of Theorem \\ref{S-P1}.\n\n\\begin{corollary}\n\\label{C:3}For $n\\in\\mathbb{N}_{0}$ the following relationship holds true\n\\begin{equation}\n\\mathfrak{B}_{n,q}\\left(  x,y\\right)  \n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(  \\mathfrak{B}_{k,q}\\left(  x\\right)  +\\frac{\\left(\n-1;q\\right)  _{k-1}}{2^{k}}\\left[  k\\right]  _{q}x^{k-1}\\right)\n\\mathfrak{E}_{n-k,q}\\left(  y\\right)  . \\label{cw1\n\\end{equation}\n\n\\end{corollary}\n\nThe formula (\\ref{cw1}) ia a $q$-extension of the Cheon's main result\n\\cite{cheon}.\n\n\\begin{theorem}\n\\label{S-P2} For $n\\in\\mathbb{N}_{0}$, the following relationship\n\\[\n\\mathfrak{E}_{n,q}\\left(  x,y\\right)  =\\frac{1}{\\left[  n+1\\right]  _{q}\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\frac{1}{m^{n+1-k}}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left\\{\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{E}_{j,q}\\left(  x\\right)  }{m^{k-j\n}-\\mathfrak{E}_{k,q}\\left(  y\\right)  \\right)  \\mathfrak{B}_{n+1-k,q}\\left(\nmx\\right)\n\\]\nhold true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is based on the following identit\n\\[\n\\frac{2}{\\mathcal{E}_{q}\\left(  t\\right)  +1}\\mathcal{E}_{q}\\left(  tx\\right)\n\\mathcal{E}_{q}\\left(  ty\\right)  =\\frac{2}{\\mathcal{E}_{q}\\left(  t\\right)\n+1}\\mathcal{E}_{q}\\left(  ty\\right)  \\cdot\\frac{\\mathcal{E}_{q}\\left(\n\\frac{t}{m}\\right)  -1}{t}\\cdot\\frac{t}{\\mathcal{E}_{q}\\left(  \\frac{t\n{m}\\right)  -1}\\mathcal{E}_{q}\\left(  \\frac{t}{m}mx\\right)  .\n\\]\nIndee\n\\begin{align*\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{t^{n-1}}{m^{n}\\left\\{  n\\right\\}  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  mx\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}\\\\\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  y\\right)  \\frac{t^{n-1}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  mx\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}\\\\\n&  =:I_{1}-I_{2}.\n\\end{align*}\nIt follows tha\n\\begin{align*}\nI_{2}  &  =\\dfrac{1}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{E}_{n,q}\\left(  y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  mx\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}=\\dfrac{1}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{1}{m^{n-k}}\\mathfrak{E}_{k,q}\\left(  y\\right)\n\\mathfrak{B}_{n-k,q}\\left(  mx\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{1}{\\left[  n+1\\right]  _{q}\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{1}{m^{n+1-k}}\\mathfrak{E}_{k,q}\\left(  y\\right)\n\\mathfrak{B}_{n+1-k,q}\\left(  mx\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q\n!},\n\\end{align*}\nan\n\\begin{align*}\nI_{1}  &  =\\dfrac{1}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  mx\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left\\{\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{E}_{k,q}\\left(  y\\right)  }{m^{n-k}}\\frac{t^{n\n}{\\left[  n\\right]  _{q}!}\\\\\n&  =\\dfrac{1}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{n-k,q}\\left(  mx\\right)\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left\\{\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{E}_{j,q}\\left(  y\\right)  }{m^{n-k}m^{k-j}\n\\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nj\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{n-j,q}\\left(  mx\\right)\n{\\displaystyle\\sum\\limits_{k=0}^{j}}\n\\left\\{\n\\begin{array}\n[c]{c\nj\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{E}_{k,q}\\left(  x\\right)  }{m^{n-k}\n\\frac{t^{n-1}}{\\left[  n\\right]  _{q}!}\\\\\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{1}{\\left[  n+1\\right]  _{q}\n{\\displaystyle\\sum\\limits_{j=0}^{n+1}}\n\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nj\n\\end{array}\n\\right]  _{q}\\mathfrak{B}_{n+1-j,q}\\left(  mx\\right)\n{\\displaystyle\\sum\\limits_{k=0}^{j}}\n\\left\\{\n\\begin{array}\n[c]{c\nj\\\\\nk\n\\end{array}\n\\right\\}  _{q}\\frac{\\mathfrak{E}_{k,q}\\left(  x\\right)  }{m^{n+1-k}\n\\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\n\n\\end{proof}\n\nNext we give an interesting relationship between the $q$-Genocchi polynomials\nand the $q$-Bernoulli polynomials.\n\n\\begin{theorem}\n\\label{S-P3}For $n\\in\\mathbb{N}_{0}$, the following relationshi\n\\begin{align*}\n\\mathfrak{G}_{n,q}\\left(  x,y\\right)   &  =\\dfrac{1}{\\left[  n+1\\right]  _{q}\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\frac{1}{m^{n-k}}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left[\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k-j}}{m^{k-j}2^{k-j}}\\mathfrak{G}_{j,q}\\left(\nx\\right)  -\\mathfrak{G}_{k,q}\\left(  x\\right)  \\right)  \\mathfrak{B\n_{n+1-k,q}\\left(  my\\right)  ,\\\\\n\\mathfrak{B}_{n,q}\\left(  x,y\\right)   &  =\\dfrac{1}{2\\left[  n+1\\right]\n_{q}\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\frac{1}{m^{n-k}}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left[\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k-j}}{m^{k-j}2^{k-j}}\\mathfrak{B}_{j,q}\\left(\nx\\right)  +\\mathfrak{B}_{k,q}\\left(  x\\right)  \\right)  \\mathfrak{G\n_{n+1-k,q}\\left(  my\\right)\n\\end{align*}\nholds true between the $q$-Genocchi and the $q$-Bernoulli polynomials.\n\\end{theorem}\n\n\\begin{proof}\nUsing the following identit\n\\begin{align*}\n&  \\frac{2t}{\\mathcal{E}_{q}\\left(  t\\right)  +1}\\mathcal{E}_{q}\\left(\ntx\\right)  \\mathcal{E}_{q}\\left(  ty\\right)  \\\\\n&  =\\frac{2t}{\\mathcal{E}_{q}\\left(  t\\right)  +1}\\mathcal{E}_{q}\\left(\ntx\\right)  \\cdot\\left(  \\mathcal{E}_{q}\\left(  \\frac{t}{m}\\right)  -1\\right)\n\\frac{m}{t}\\cdot\\frac{\\frac{t}{m}}{\\mathcal{E}_{q}\\left(  \\frac{t}{m}\\right)\n-1}\\cdot\\mathcal{E}_{q}\\left(  \\frac{t}{m}my\\right)\n\\end{align*}\nwe hav\n\\begin{align*}\n&\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  =\\frac{m}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\frac{(-1,q)_{n}}{m^{n}2^{n}}\\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}\\\\\n&  -\\frac{m}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{G}_{n,q}\\left(  x,y\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}\\\\\n&  =\\frac{m}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\left(\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{n-k}}{m^{n-k}2^{n-k}}\\mathfrak{G}_{k,q}\\left(\nx\\right)  -\\mathfrak{G}_{n,q}\\left(  x\\right)  \\right)  \\frac{t^{n}}{\\left[\nn\\right]  _{q}!\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\mathfrak{B}_{n,q}\\left(  my\\right)  \\frac{t^{n}}{m^{n}\\left[  n\\right]\n_{q}!}\\\\\n&  =\\frac{m}{t\n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n{\\displaystyle\\sum\\limits_{k=0}^{n}}\n\\frac{1}{m^{n-k}}\\left[\n\\begin{array}\n[c]{c\nn\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left[\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k-j}}{m^{k-j}2^{k-j}}\\mathfrak{G}_{j,q}\\left(\nx\\right)  -\\mathfrak{G}_{k,q}\\left(  x\\right)  \\right)  \\mathfrak{B\n_{n-k,q}\\left(  my\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}\\\\\n&  \n{\\displaystyle\\sum\\limits_{n=0}^{\\infty}}\n\\dfrac{1}{\\left[  n+1\\right]  _{q}\n{\\displaystyle\\sum\\limits_{k=0}^{n+1}}\n\\frac{1}{m^{n-k}}\\left[\n\\begin{array}\n[c]{c\nn+1\\\\\nk\n\\end{array}\n\\right]  _{q}\\left(\n{\\displaystyle\\sum\\limits_{j=0}^{k}}\n\\left[\n\\begin{array}\n[c]{c\nk\\\\\nj\n\\end{array}\n\\right]  _{q}\\frac{(-1,q)_{k-j}}{m^{k-j}2^{k-j}}\\mathfrak{G}_{j,q}\\left(\nx\\right)  -\\mathfrak{G}_{k,q}\\left(  x\\right)  \\right)  \\mathfrak{B\n_{n+1-k,q}\\left(  my\\right)  \\frac{t^{n}}{\\left[  n\\right]  _{q}!}.\n\\end{align*}\nThe second identity can be proved in a like manner.\n\\end{proof}\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the first discovery of late-time cosmic acceleration \nby supernovae type Ia (SN Ia) in 1998 \\cite{SN1,SN2}, \nthe origin of this phenomenon has not been identified yet.\nA scalar field $\\phi$ is one of the simplest candidates \nfor dark energy, whose potential energy \\cite{quin} or \nnonlinear kinetic energy \\cite{kes} can drive the acceleration. \nIf we allow for the coupling between $\\phi$ and the \ngravity sector, Horndeski theories \\cite{Horndeski} are \nknown as the most general scalar-tensor theories \nwith second-order equations of motion \\cite{Horn1,Horn2,Horn3}. \n\nDark energy models based on Horndeski theories can be constrained \nnot only by the observational data of SNIa, Cosmic Microwave \nBackground (CMB) temperature anisotropies, Baryon Acoustic \nOscillations (BAO) but also by the measurements of gravitational waves (GWs). \nThe bound on the speed of GWs from gravitational Cherenkov \nradiation \\cite{Moore} was used in Ref.~\\cite{Kimura:2011qn} \nto place constraints on the Lagrangian of Horndeski theories. \nAfter the first discovery of the GW event GW150914 \\cite{GW15}, \nthe possibility for constraining modified gravity models from the \nmeasurements of GWs along with gamma-ray bursts was pointed \nout in Ref.~\\cite{Lombriser:2015sxa}. \n{}From the Hulse-Taylor pulsar data, the speed of GWs $c_{t}$ \nwas also constrained to be close to that of light $c$ \nat the level of $10^{-2}$ \\cite{Jime}.\n\nThe GW170817 event from a neutron star merger \\cite{GW170817} \ntogether with electromagnetic counterparts \\cite{Goldstein} \nshowed that the relative difference between $c_t$ and $c$\nis less than the order \n$10^{-15}$. If we strictly demand that $c_t=c$ on the isotropic cosmological \nbackground, the allowed Horndeski Lagrangian is of the form \n$L=G_2(\\phi, X)+G_3(\\phi,X) \\square \\phi+G_4(\\phi)R$, \nwhere $G_2, G_3$ are functions of $\\phi$ and \n$X=-\\partial_{\\mu} \\phi \\partial^{\\mu} \\phi\/2$, while\n$G_4$ is a function of $\\phi$ \nalone \\cite{GW1,GW2,GW3,GW4,GW5}.\nThis includes the theories like \nquintessence \\cite{quin}, \nk-essence \\cite{kes}, \ncubic Galileons \\cite{Nicolis,Galileon1,Galileon2,braiding}, \nBrans-Dicke (BD) theory \\cite{Brans}, \n$f(R)$ gravity \\cite{Bergmann,Ruz,Staro}, \nand nonminimally coupled theories with general functions \n$G_4(\\phi)$ \\cite{Damour1,Damour2,Amen99,Uzan,Chiba99,Bartolo99,Perrotta,Boi00,Gille,BD1}.\n\nThe original massless BD theory \\cite{Brans} is equivalent \nto the Lagrangian $L=\\left( 1-6Q^2 \\right) F(\\phi) X+\n(M_{\\rm pl}^2\/2) F(\\phi)R$ with \n$F(\\phi)=e^{-2Q (\\phi-\\phi_0)\/M_{\\rm pl}}$, where \nthe constant $Q$ is related to the so-called BD \nparameter $\\omega_{\\rm BD}$, as \n$2Q^2=1\/(3+2\\omega_{\\rm BD})$ \\cite{BD1}.\nGeneral Relativity (GR) is recovered in the limit \n$\\omega_{\\rm BD} \\to \\infty$, i.e., $Q \\to 0$.\nIf we transform the action of BD theory to that in the Einstein \nframe, the constant $Q$ has a meaning of coupling between \nthe scalar field and nonrelativistic \nmatter \\cite{DT10}.\n\nThe parametrized post-Newtonian (PPN) \nformalism \\cite{Nord,Will71}\non the weak gravitational background shows that, in  \nmassless BD theory, one of the PPN parameters \nis given by \n$\\gamma=(1+\\omega_{\\rm BD})\/(2+\\omega_{\\rm BD})$ \\cite{Will05}. \nThe Cassini experiment measuring the time delay of light \nin the solar system placed the constraint \n$|\\gamma-1| \\le 2.3 \\times 10^{-5}$ \\cite{Will14}. \nThis translates to the bound $\\omega_{\\rm BD} \\ge 4.3 \\times 10^4$, \nor equivalently, $|Q| \\le 2.4 \\times 10^{-3}$. \nFor the coupling $|Q|>2.4 \\times 10^{-3}$, one needs to \nresort to some mechanism for screening fifth forces mediated \nby the BD scalar field.\n\nIf the BD scalar has a massive potential in over-density \nregions of the Universe, the propagation of fifth forces \ncan be suppressed under the chameleon mechanism \\cite{chame1,chame2}.  \nFor example, metric $f(R)$ gravity corresponds to BD theory \nwith $Q=-1\/\\sqrt{6}$ in the presence of a scalar potential \nof gravitational origin \\cite{DT10,APT07}.\nIt is possible to design the form of $f(R)$ such that the scalar \ndegree of freedom (scalaron) has a heavy mass in \nover-density regions, while realizing cosmic acceleration \nby a light scalar on Hubble scales \\cite{fR1,fR2,fR3,fR4}.\nHowever, this amounts to a fine-tuning of initial conditions of \nscalaron perturbations in the early Universe \\cite{fR2,fR4,fR5}.\nMoreover, unless the scalaron is nearly frozen until \nrecently, the large coupling $|Q| \\simeq 0.4$ leads to the \nsignificant enhancement of matter perturbations \nin the late Universe \\cite{fR1,fR2,fR4,Tsuji07,Tsuji09}. \nFor the compatibility of $f(R)$ models of late-time \ncosmic accelerationwith with observations,  \nthe deviation from GR is required to be \nvery small and hence they are hardly distinguishable from the \n$\\Lambda$-Cold-Dark-Matter ($\\Lambda$CDM) model \\cite{Lomb,Battye}.\n\nThere is yet another mechanism for screening fifth forces \nin local regions of the Universe based on nonlinear derivative \nself-interactions \\cite{Vain}. \nA representative example is the cubic Galileon Lagrangian \n$X \\square \\phi$  \\cite{Nicolis,Galileon1,Galileon2,braiding}, with which \nthe Newtonian behavior is recovered inside the so-called Vainshtein radius \n$r_V$  \\cite{Cede,Luty,Babichev,Burrage,Brax,Babi11,DKT12,KKY12,Kase13}\neven with the coupling $|Q|>2.4 \\times 10^{-3}$.\nFor uncoupled Galileons ($Q=0$) without the scalar potential, \nit is known that there exists a cosmological tracker solution finally \napproaching a de Sitter attractor \\cite{DTGa,DTGa2} \n(see also Refs.~\\cite{GS,Ali}).\nUnfortunately, this dark energy model is in tension with \nthe observational data of supernovae type Ia, CMB, \nBAO, and redshift-space \ndistortions \\cite{NDT10,AppleLin,Neveu,Barreira1,Barreira2,Renk,Peirone2}.  \nFor the nonminimally coupled light mass or massless Galileon with \na potential, e.g., the linear potential $V(\\phi)=m^3 \\phi$, it is possible \nto realize the viable cosmic expansion history, while \nrecovering the Newtonian behavior in the solar system \\cite{Ali2,KTD}.\n\nWhile the Vainshtein mechanism suppresses the scalar-matter interaction for the \ndistance $r \\ll r_V$, the gravitational coupling $G_{\\rm N}$ in over-density regions contains \ntime dependence of the dark energy field $\\phi$ through \nthe nonminimal coupling $F(\\phi)$ \\cite{Babi11,KKY12}. \nThen, $G_{\\rm N}$ varies in time even inside the solar system. \nThe LLR experiments of the earth-moon system measure \nthe time variation $\\dot{G}_{\\rm N}\/G_{\\rm N}$, so it can \nbe used to constrain nonminimally coupled dark energy models. \n\n{}From the LLR bound of $\\dot{G}_{\\rm N}\/G_{\\rm N}$ \nin 2004 \\cite{Williams}, the time variation \n$\\alpha_{\\rM} \\equiv \\dot{F}\/(HF)$ \n(where $H$ is the Hubble expansion rate) is in the range\n$|\\alpha_{\\rM} (t_0)| \\le 0.02$ today.\nIn 2011, Babichev {\\it et al.} \\cite{Babi11} used this bound\nfor nonminimally coupled cubic Galileons without the potential\nand claimed that the time variation of the field is tightly constrained at low redshifts. \nWe note that, besides this fact, the cubic Galileon without the potential \nis in tension with the observational data.  \nOn the other hand, the presence of potentials for nonminimally coupled \nGalileons allows the possibility for realizing viable cosmic \nexpansion and growth histories, even with the LLR bound in 2004, \nsee Figs.~4 and 5 in Ref.~\\cite{KT18}.\n\nThe recent LLR experiments \\cite{Hofmann} constrain the time variation \n$\\dot{G}_{\\rm N}\/G_{\\rm N}$ with the upper limit more stringent \nthan before \\cite{Williams}.\nIn particular, for $\\alpha_{\\rm M}>0$, the upper bound of  \n$\\dot{G}_{\\rm N}\/G_{\\rm N}$ translates to \n$\\alpha_{\\rm M}(t_0) \\le 7 \\times 10^{-5}$ today, which is \ntighter than the bound $\\alpha_{\\rM} (t_0) \\le 0.02$ \nby more than two orders of magnitude. \nThis LLR bound in 2018 was used to constrain dark energy \nmodels based on nonlocal gravity \\cite{Belga}. \nIt remains to be seen how nonminimally coupled Galileons \nwith the potential can be constrained with this new bound \nof $\\alpha_{\\rm M}(t_0)$.\n\nIn this paper, we exploit the new LLR bound to constrain \nnonminimally coupled dark energy models with \nthe cubic self-interaction $\\beta_3 M^{-3} X \\square \\phi$ \nand the potential $V(\\phi)$ of light mass Galileons, where \n$\\beta_3$ is dimensionless coupling constant and $M$ is \na mass scale defined later in Eq.~(\\ref{Massdef}).\nWe stress that our model is different from the nonminimally coupled \ncubic Galileon without the potential studied in Ref.~\\cite{Babi11}, \nin that the scalar potential is the dominant source for late-time cosmic \nacceleration. The Galileon term can still play an important role \nfor the scalar field dynamics in the early Universe. \nMoreover, we require that the propagation \nof fifth forces is suppressed in over-density regions. \nWe perform a detailed analysis for the cosmological dynamics from \nthe radiation era to today and put bounds on the coupling $Q$ \nby using the new LLR data.\n\nFor $|\\beta_3| \\ll 1$, there exists a so-called \n$\\phi$-matter-dominated epoch ($\\phi$MDE) \\cite{Amenco} \nin the Jordan frame followed by the stage of cosmic acceleration. \nFor the exponential potential $V(\\phi)=V_0 e^{\\lambda \\phi\/M_{\\rm pl}}$, \nwe place constraints on the allowed parameter space in \nthe $(\\lambda,Q)$ plane and derive the stringent limit  \n$Q \\le 3.4 \\times 10^{-3}$ from the LLR constraint. \nThis is almost close to the Cassini bound $Q \\le 2.4 \\times 10^{-3}$ \nobtained for massless BD theories without \nthe Vainshtein screening. \nFor $|\\beta_3| \\gg 1$, the coupling $Q$ is not particularly bounded from \nabove due to the suppression of field kinetic energy under the \ncosmological Vainshtein screening. \nIn this case, we show a new possibility for realizing \nthe dark energy equation of state $w_{\\rm DE}$ \nclose to $-1$ from high redshifts to today even for the steep \npotential satisfying $\\lambda>\\sqrt{2}$.\n\nIn our dark energy theory the speed of GWs is equivalent to that \nof light, but the existence of nonminimal coupling $F(\\phi)$ \nleads to the modified GW propagation through the existence \nof a nonvanishing term $\\alpha_{\\rM}$.\nThe possibility of using the difference between GW and luminosity \ndistances to test for the running Planck mass \nwas first pointed out in Ref.~\\cite{Saltas:2014dha}.\nThe first forecasts of such constraints were \ngiven in Ref.~\\cite{Lombriser:2015sxa}, \nwhich were followed by a sequence of papers after the \ndirect detection of GWs \\cite{Nishi17,Arai,Amendola17,Zhao,Belga17,Belga17d,Ezqu,Lagos}. \n\nIn Ref.~\\cite{Lombriser:2015sxa}, it was anticipated that the LLR bound \non the running Planck mass may be beyond the reach of the constraint \narising from standard sirens. This generally depends on the models of dark energy. \nFor example, in nonlocal gravity models studied recently in Ref.~\\cite{Belga},  \nthe difference between the GW distance  $d_{\\rm GW}$ and luminosity distance $d_L$\nis typically more than a few percent, which may be probed in future high-precision measurements. \nThis reflects the fact that, in nonlocal gravity, the gravitational coupling deep inside the Hubble \nradius (wavelength $a\/k \\ll H^{-1}$) is very close to the Newton gravitational constant $G$, \nas $G_{\\rm N}\/G=1+{\\cal O}((aH\/k)^2)$ \\cite{Nesseris,Belga}. \nHence the nonlocal gravity models can pass\nthe new LLR bound in 2018, while leaving the sizable difference between \n$d_{\\rm GW}$ and $d_L$. \n\nThe nonminimal coupling $G_4(\\phi)R$ gives rise to the effective \ngravitational coupling \n$G_{\\rm N}$ different from that in nonlocal gravity.\nHence it deserves for studying whether the new LLR data leads to the constraint on the nonminimal coupling beyond or within the reach of future \nstandard siren measurements.  \nIn this paper, we will compute the relative ratio between \n$d_{\\rm GW}$ and $d_L$ for the aforementioned nonminimally \ncoupled dark energy model. Under the LLR bound on the variation of $F(\\phi)$, \nwe show that the relative difference $d_{\\rm GW}\/d_L-1$ does not exceed \nthe order $10^{-5}$ in the redshift range \n$0<z<100$. Thus, unlike nonlocal gravity, \nthe LLR data allow only tiny deviations of $d_{\\rm GW}$ from $d_L$ \nin nonminimally coupled theories, so \nit will be difficult to detect such difference without \nvery high-precision distance measurements in future.\n\nThis paper is organized as follows.\nIn Sec.~\\ref{sec:syspf}, we present our nonminimally coupled \ndark energy model and revisit how the cubic Galileon interaction screens the scalar-matter coupling under the\nVainshtein mechanism.\nWe then interpret the recent LLR bound in terms of today's \nvalue of $\\alpha_{\\rM}$. \nIn Sec.~\\ref{darksec}, we derive the background equations \nof motion on the flat Friedmann-Lema\\^{i}tre-Robertson-Walker (FLRW) background and express them in \nautonomous forms. \nIn Sec.~\\ref{numesec}, we study the cosmological dynamics \nin the presence of exponential potential $V(\\phi)=V_0 \ne^{\\lambda \\phi\/M_{\\rm pl}}$ for unscreened \n($|\\beta_3| \\ll 1$) and screened ($|\\beta_3| \\gg 1$) cases \nafter the radiation domination. \nWe put constraints on the allowed parameter space from \nthe recent LLR bound and discuss the evolution of \n$w_{\\rm DE}$ and field density parameters. \nIn Sec.~\\ref{GWsec}, we investigate how much difference \narises between $d_{\\rm GW}(z)$ and $d_L(z)$ for the two \ndifferent background cosmologies discussed in Sec.~\\ref{numesec}.\nSec.~\\ref{consec} is devoted to conclusions.\n\nUnless otherwise stated, we use the natural unit where \nthe speed of light $c$, the reduced Planck constant $\\hbar$, \nand the Boltzmann constant $k_B$ \nare equivalent to 1.\n \n\\section{Nonminimally coupled theories and \nLLR constraints}\n\\label{sec:syspf}\n\nWe begin with a subclass of Horndeski theories \ngiven by the action  \n\\ba\n\\mL{S} \n&=& \\int \\mr{d}^4 x \\sqrt{-g}  \n\\biggl[  \\frac{M_{\\rm{pl}}^2}{2} F(\\phi) R\n+\\left( 1-6Q^2 \\right) F(\\phi) X \\nonumber \\\\\n& &\\qquad \\qquad \\quad -V(\\phi)\n+\\beta_3 M^{-3} X \\square \\phi\n\\biggr]+\\mL{S}_m \\,,\n\\label{action}\n\\ea\nwhere $g$ is the determinant of metric tensor $g_{\\mu \\nu}$, \n$X=-\\partial_{\\mu} \\phi \\partial^{\\mu} \\phi\/2$ is the kinetic  \nenergy of a scalar field $\\phi$, and $F(\\phi)$, $V(\\phi)$ \nare functions of $\\phi$. \nThe couplings $Q$ and $\\beta_3$ are dimensionless constants, \nwhile $M$ is a constant having a dimension of mass \nrelated to today's Hubble constant $H_0$ as\n\\be\nM=\\left( M_{\\rm pl} H_0^2 \\right)^{1\/3}\\,,\n\\label{Massdef}\n\\ee\nwhich is of order $10^{-22}$~GeV. \nThe mass scale (\\ref{Massdef}), which translates to the frequency \n$f \\sim 100$~Hz, corresponds to the typical strong coupling scale of \ntheories containing derivative self-interactions \nlike $X \\square \\phi$ \\cite{Cede,Luty}.\n\nWe assume that the effective field theory of dark energy is valid up \nto the mass scale $M \\sim 10^{-22}$~GeV. \nIn other words, we resort to the action (\\ref{action}) \nfor studying the physics on scales larger than $\\sim 10^6$~m. \nThis includes the dynamics of late-time cosmic acceleration \n($\\sim 10^{26}$~m) and the earth-moon local \nsystem of LLR experiments ($\\sim 10^8$~m).\nBelow the length scale $10^6$~m, some ultraviolet effects may \ncome into play to approach the General Relativistic behavior.\nIndeed, the possibility of recovering the GW value $c_t=1$ above \nthe frequency $f \\sim 100$~Hz was discussed in Ref.~\\cite{deRham:2018red}.\nThis critical frequency is of the same order as the GW frequency \nobserved by LIGO\/Virgo \\cite{GW170817}, \nso there may be more general Horndeski theories realizing \n$c_t$ very close to 1 for the frequency $f\\geq 100$~Hz even if \nthe deviation of $c_t$ from 1 is large on cosmological scales.\nIn this paper we do not pursue such a possibility, but we focus \non the theory given by the action (\\ref{action}) in which $c_t=1$ \nfor any scales of interest.\n\nThe nonminimal coupling $F(\\phi)$ is chosen to be \nof the form:\n\\be\nF(\\phi)=e^{-2Q (\\phi-\\phi_0)\/M_{\\rm pl}}\\,,\n\\label{Fphi}\n\\ee\nwhere $\\phi_0$ is today's value of $\\phi$ and hence\n$F(\\phi_0)=1$. \nWe assume that the matter sector, which is described \nby the action $\\mL{S}_m$ with the density $\\rho_m$, \nis minimally coupled to gravity. \nThe scalar field mediates fifth forces with the matter sector \nthrough the direct gravitational interaction characterized by \nthe coupling $Q$.\n\nIf $\\beta_3=0$, then the theories given by the action (\\ref{action}) \nare equivalent to BD theories \\cite{Brans} \nwith the scalar potential $V(\\phi)$.\nIndeed, by setting $\\chi=F(\\phi)$, the Lagrangian \nin the action (\\ref{action}) reduces to \n$L=\\chi R\/2-\\omega_{\\rm BD} \n\\partial_{\\mu} \\chi \\partial^{\\mu} \\chi\/(2\\chi)-V(\\phi (\\chi))$ \nin the unit $M_{\\rm pl}=1$, where $\\omega_{\\rm BD}$ \nis the BD parameter related to $Q$ according to \n$3+2\\omega_{\\rm BD}=1\/(2Q^2)$ \\cite{BD1}. \nIn the original massless BD theories with $V(\\phi)=0$, \nthe coupling strength is constrained to be \n$|Q| \\le 2.4 \\times 10^{-3}$ from the Cassini experiment \\cite{Will14}.\n\nFor the coupling $|Q|>2.4 \\times 10^{-3}$, we require the \nexistence of scalar potential $V(\\phi)$ or field derivative interaction \n$X \\square \\phi$ to screen fifth forces in the solar system.\nIn the former case, the chameleon mechanism \\cite{chame1,chame2} \ncan be at work for the potential having a large mass in regions of the high density. \nOne of such examples is $f(R)$ gravity, in which the scalar \npotential of gravitational origin arises with the coupling \n$Q=-1\/\\sqrt{6}$ \\cite{DT10}. \nIn $f(R)$ models of late-time cosmic acceleration accommodating \nthe chameleon mechanism in over-density regions, \nthe functional form of $f(R)$ needs to be designed such that\nthe scalaron mass $M_{\\phi}$ grows very rapidly\ntoward the asymptotic past \\cite{fR1,fR2,fR3,fR4}.\nThis causes the fine-tuning problem of initial conditions of perturbations \nassociated with the oscillating mode induced \nby the heavy mass \\cite{fR2,fR4,fR5}.\n\nInstead of resorting to the chameleon mechanism with \na very massive scalar in over-density regions, \nwe consider the Galileon self-interaction $X \\square \\phi$ \nto suppress fifth forces under the Vainshtein \nmechanism \\cite{Vain}. \nThe scalar potential $V(\\phi)$ of a light scalar is also \ntaken into account as a source for the cosmic acceleration. \nDefining the dimensionless quantity\n\\be\n\\lambda \\equiv \\frac{M_{\\rm pl}}{V} \\frac{{\\rm d}V}{{\\rm d}\\phi}\\,,\n\\ee\nthe condition for cosmic acceleration in the absence of Galileon \ninteractions and matter is given by \n$|\\lambda|<\\sqrt{2}$ \\cite{CLW,CST}.\nThe existence of Galileons can modify this structure, \nbut we focus on the case in which the condition\n\\be\n\\left| \\lambda \\right| \\le {\\cal O}(1)\n\\label{lamra}\n\\ee\nis satisfied during the cosmic expansion history from the past \nto today. The coupling strength $|Q|$ exceeding the order 1 leads to the \nstrong enhancement of matter density perturbations \nincompatible with observations in large-scale structures \\cite{BD1}, \nso we consider the coupling \n\\be\n|Q| \\le {\\cal O}(0.1)\\,,\n\\label{Qra}\n\\ee\nin the following discussion.\n\nThe original Galileon theory \\cite{Nicolis} has the linear \npotential $V(\\phi)=m^3 \\phi$ with $Q=0$, in which case \nthe resulting field equation of motion\nrespects the Galilean symmetry in Minkowski space-time. \nThis potential corresponds to a massless scalar \nwith $\\lambda=M_{\\rm pl}\/\\phi$, \nso the condition (\\ref{lamra}) translates to \n$\\phi \\ge M_{\\rm pl}$. \nFor $Q \\neq 0$, the cosmological dynamics with \nthe linear potential was studied in \nRef.~\\cite{KTD}. In this case, today's cosmic acceleration \nis followed by the collapsing Universe after the field enters \nthe region $V(\\phi)<0$. \n\nThe constant $\\lambda$ corresponds to the exponential potential \n$V(\\phi)=V_0 e^{\\lambda \\phi\/M_{\\rm pl}}$. \nIn this case, the scalar mass squared $M_{\\phi}^2 \\equiv {\\rm d}^2 V\/{\\rm d}\\phi^2$ \nis given by $M_{\\phi}^2=\\lambda^2 V\/M_{\\rm pl}^2$. \nSince the potential energy $V$ is the dominant contribution to \ntoday's energy density of the Universe, we have \n$V \\lesssim M_{\\rm pl}^2 H^2$, where $H$ is the Hubble expansion rate in the past (redshift $z \\geq 0$).\nThen, under the condition (\\ref{lamra}), it follows that $M_{\\phi}^2 \\lesssim \\lambda^2 H^2 \\lesssim H^2$.\nThis property also holds for the potential with \na time-varying $\\lambda$ in the range (\\ref{lamra}). \nFor the light scalar whose today's mass $M_{\\phi}$ is smaller \nthan $H_0$, the effect of $M_{\\phi}$ on the scalar-field \nequation can be ignored to study the Vainshtein mechanism \nin regions of the high density.\nIn other words, the chameleon mechanism does not \ncome into play for screening fifth forces.\n\n\\subsection{Vainshtein screening}\n\nThe behavior of scalar and gravitational fields around \na spherically symmetric over-density on a cosmological background was already studied in \nRefs.~\\cite{KKY12,Babi11}, so we briefly review it in the following.\nLet us consider the following perturbed metric in the Newtonian gauge: \n\\be\n{\\rm d}s^2=- \\left( 1+2\\Psi \\right) {\\rm d}t^2\n+\\left( 1+2\\Phi \\right) a^2(t) \\delta_{ij}{\\rm d}x^i {\\rm d}x^j\\,,\n\\ee\nwhere $a(t)$ is the time-dependent scale factor, $\\Psi$ and $\\Phi$ \nare gravitational potentials depending $t$ and \nthe radial coordinate $r=a(t)\\sqrt{ \\delta_{ij} x^i x^j}$. \nThe scalar field and matter density on the homogenous cosmological \nbackground are given by $\\bar{\\phi}(t)$ and $\\bar{\\rho}_m(t)$, respectively.\nThe existence of a compact object gives rise to the perturbations \n$\\chi (t, r)$ and $\\delta \\rho_m (t, r)$ in $\\phi$ and $\\rho_m$, \nsuch that $\\phi=\\bar{\\phi}(t)+\\chi (t, r)$ and \n$\\rho_m=\\bar{\\rho}_m(t)+\\delta \\rho_m (t, r)$. \n\nWe are interested in solutions deep inside today's \nHubble radius, $r \\ll H_0^{-1}$. \nHence we neglect time derivatives of perturbed quantities, \nwhile keeping spatial derivatives. \nThe radial dependence of the derivative \n$\\partial \\chi\/\\partial r$ changes around the Vainshtein radius \n$r_V$, which is estimated as \\cite{KKY12,DKT12}\n\\be\nr_V \\simeq \\left( \\frac{|\\beta_3 Q| M_{\\rm pl} r_g}{M^3}\n\\right)^{1\/3}= \\left( |\\beta_3 Q| r_g H_0^{-2} \\right)^{1\/3}\\,,\n\\label{rV}\n\\ee\nwhere \n\\be\nr_g=M_{\\rm pl}^{-2} \\int_0^r \\delta \\rho_m\\,\n\\tilde{r}^2 {\\rm d}\\tilde{r}\n\\label{rg}\n\\ee\nis the Schwarzschild radius of the source. \nFor $r \\gg r_V$ the field derivative has the dependence \n$\\partial \\chi\/\\partial r \\propto r^{-2}$, while, for $r \\ll r_V$, \n$\\partial \\chi\/\\partial r \\propto r^{-1\/2}$.\nIn the latter regime, the nonlinear effect arising from the cubic \nGalileon self-interaction suppresses the propagation of fifth forces \ninduced by the coupling $Q$.\nIndeed, for $r \\ll r_V$, the gravitational potentials \nare given by \\cite{KKY12,DKT12}\n\\ba\n\\Psi &\\simeq& -\\frac{r_g}{2rF}\n\\left[ 1+{\\cal O}(1)\\,Q^2 \\left( \\frac{r}{r_V} \n\\right)^{3\/2} \\right]\\,,\\label{Psilo}\\\\\n\\Phi &\\simeq& \\frac{r_g}{2rF}\n\\left[ 1+{\\cal O}(1)\\,Q^2 \\left( \\frac{r}{r_V} \n\\right)^{3\/2} \\right]\\,.\\label{Philo}\n\\ea\nSince the value of $F$ today (cosmic time $t_0$) is \nequivalent to 1 in our theory, the Newtonian \nbehavior ($-\\Psi=\\Phi=r_g\/(2r)$) is \nrecovered for $r \\ll r_V$.\nAs long as $r_V$ is much larger than \nthe solar-system scale ($\\sim 10^{15}$\\,cm), the model is \nconsistent with solar-system tests of gravity. \nSince $(r_gH_0^{-2})^{1\/3} \\simeq 3 \\times 10^{20}$~cm for \nthe Sun, this condition translates to \n\\be\n\\left| \\beta_3 Q \\right| \\gg 10^{-17}\\,.\n\\label{local}\n\\ee\nWhen $|Q|$ is of order $10^{-2}$, for example, \nthe coupling $\\beta_3$ needs to \nbe in the range $|\\beta_3| \\gg 10^{-15}$. \n\n\n\\subsection{LLR constraints}\n\n{}From Eq.~(\\ref{Psilo})-(\\ref{Philo}) with Eq.~(\\ref{rg}), the leading-order gravitational potentials deep inside the Vainshtein radius can be expressed as \n\\be\n-\\Psi \\simeq \\Phi \\simeq  \\frac{G_{\\rm N} \\delta {\\cal M}}{r}\\,,\n\\ee\nwhere $ \\delta {\\cal M}=4\\pi \\int_0^r \\delta \\rho_m \\tilde{r}^2 {\\rm d} \\tilde{r}$, and \n$G_{\\rm N}$ is the measured gravitational coupling given by \n\\be\nG_{\\rm N}=\\frac{1}{8 \\pi M_{\\rm pl}^2 F(\\phi(t))}\\,,\n\\label{GN}\n\\ee\nwhere we omitted the bar from the background value of $\\phi$.\nHere the background field $\\phi(t)$ is a cosmological scalar \ndriving the late-time cosmic acceleration.\nSince we are considering over-density regions on the cosmological \nbackground, the homogenous value $\\phi(t)$ survives even in \nthe local Universe.\nThe dark energy scalar field $\\phi(t)$ changes in time, so \nthis leads to the time variation of $G_{\\rm N}$. \nThis fact was first recognized in Ref.~\\cite{Babi11} and it was \nproved in Ref.~\\cite{KKY12} in full Horndeski theories.\n\nThe effective gravitational coupling (\\ref{GN}) \nis valid for a light scalar field operated by the Vainshtein \nmechanism in over-density regions. \nHere, the light scalar means that the slope of field potential \n$V(\\phi)$ satisfies the condition (\\ref{lamra}). \nFor the potential of a massive scalar violating this condition in regions \nof the high density (as in $f(R)$ dark energy models), \nthe chameleon mechanism can be at work \nto suppress the gravitational coupling with matter\nin a way different from Eqs.~(\\ref{Psilo})-(\\ref{Philo}).\nAs we already mentioned, we do not consider such a massive \nscalar field in this paper.\n\nFor the cubic derivative self-interaction we chose the Galileon \ncoupling $X \\square \\phi$, but this can be generalized to the \nderivative coupling $X^n \\square \\phi$ with $n>1$. \nIn such cases, the second terms on the right hand sides \nof (\\ref{Psilo}) and (\\ref{Philo}) are modified to \n${\\cal O}(1) Q^2 (r\/r_V)^{2-1\/(2n)}$, which is much smaller \nthan 1 deep inside the Vainshtein radius. \nThen the local gravitational coupling reduces to \nthe form (\\ref{GN}), so the property of $G_{\\rm N}$ \ninduced by the time-dependent background scalar field \n$\\phi(t)$ is similar to that of cubic Galileons. \nFor the models in which derivative field self-interactions \nare not employed to screen fifth forces in over-density regions, \ne.g., chameleons and nonlocal gravity,\nthe expression of $G_{\\rm N}$ is generally different from \nthat discussed above.\n\n{}From the recent LLR experiment, the variation of $G_{\\rm N}$ \nis constrained to be  \\cite{Hofmann}\n\\be\n\\frac{\\dot{G}_{\\rm N}}{G_{\\rm N}}=\\left( 7.1 \\pm 7.6 \\right) \n\\times 10^{-14}~{\\rm yr}^{-1}\\,,\n\\label{Gbou}\n\\ee\nwhere a dot represents the derivative with respect to $t$.\nThis improves the previous bound \n$\\dot{G}_{\\rm N}\/G_{\\rm N}=(4 \\pm 9) \\times 10^{-13}$~yr$^{-1}$ \n\\cite{Williams}. \nUsing the value $H_0=100~h$~km s$^{-1}$ Mpc$^{-1}\n=(9.77775~{\\rm Gyr})^{-1}h$, \nthe bound (\\ref{Gbou}) translates to \\cite{Belga}\n\\be\n\\frac{\\dot{G}_{\\rm N}}{H_0G_{\\rm N}}=\n\\left( 0.99 \\pm 1.06 \\right) \\times 10^{-3}  \n\\left( \\frac{0.7}{h} \\right)\\,.\n\\label{Gbou2}\n\\ee\nWe define the following quantity,\n\\be\n\\alpha_{\\rM} \\equiv \\frac{\\dot{F}}{HF}\n=-\\frac{2Q \\dot{\\phi}}{M_{\\rm pl}H}\\,,\n\\label{alM}\n\\ee\nwhich was used in the context of effective field theory of \ndark energy \\cite{Bellini}. \nSince $\\alpha_{\\rM}$ is related to the variation \nof $G_{\\rm N}$, as \n$\\alpha_{\\rM}=-\\dot{G}_{\\rm N}\/(H G_{\\rm N})$, \nthe bound (\\ref{Gbou2}) can be expressed as\n\\be\n-2.05 \\times 10^{-3} \\left( \\frac{0.7}{h} \\right) \\le\n\\alpha_{\\rM} (t_0) \\le 0.07 \\times 10^{-3} \\left( \\frac{0.7}{h} \\right)\\,.\n\\label{aMcon}\n\\ee\nIf $\\alpha_{\\rM}>0$, i.e., for decreasing $G_{\\rm N}$ in time, \nthe upper bound is especially stringent: \n$\\alpha_{\\rm M}(t_0) \\le 7 \\times 10^{-5}$ for $h=0.7$.\nEven when $\\alpha_{\\rM}<0$, the upper limit of \n$|\\alpha_{\\rM} (t_0)|$ is of the order $10^{-3}$. \nThey are smaller than the previous bound\n$|\\alpha_{\\rM} (t_0)| \\le 0.02$ \\cite{Babi11}\nby more than one order of magnitude.\n\n\\section{Dynamical system}\n\\label{darksec}\n\nWe study the background cosmology for theories given \nby the action (\\ref{action}) and discuss how the coupling \n$Q$ is constrained from the LLR bound (\\ref{aMcon}). \nWe consider the flat FLRW background described by the line element \n${\\rm d}s^2=-{\\rm d}t^2+a^2(t) \\delta_{ij}{\\rm d} x^i {\\rm d}x^j$.\nFor the matter action $\\mL{S}_m$, we take nonrelativistic \nmatter (density $\\rho_m$ with vanishing pressure) and \nradiation (density $\\rho_r$ and pressure $P_r=\\rho_r\/3$) \ninto account. \nThen, the Hamiltonian and momentum constraints \nlead to \\cite{Horn2,KT18}:\n\\ba\n& &\n3M_{\\rm pl}^2 H^2=\\rho_{\\rm DE}+\\rho_m+\\rho_r\\,,\n\\label{back1} \\\\\n& &\n2M_{\\rm pl}^2 \\dot{H}=-\\rho_{\\rm DE}-P_{\\rm DE}\n-\\rho_m-\\frac{4}{3} \\rho_r\\,,\n\\label{back2} \n\\ea\nwhere $H=\\dot{a}\/a$, and \n$\\rho_{\\rm DE}$ and $P_{\\rm DE}$ are the density \nand pressure of dark energy, defined, \nrespectively, by \n\\ba\n\\rho_{\\rm DE} \n&=& 3M_{\\rm pl}^2 H^2 \\left( 1-F \\right)\n+\\frac{F}{2} (1-6Q^2) \\dot{\\phi}^2 \\nonumber \\\\\n& &+6FQ H M_{\\rm pl} \\dot{\\phi}+V\n-3 \\beta_3 M^{-3} H \\dot{\\phi}^3\\,,\n\\label{rhode} \\\\\nP_{\\rm DE} \n&=& -M_{\\rm pl}^2\\left( 2\\dot{H}+3H^2 \\right) \n\\left( 1-F \\right)+\\frac{F}{2} (1+2Q^2) \\dot{\\phi}^2 \n\\nonumber \\\\\n& &-2FQM_{\\rm pl} \\left( \\ddot{\\phi}+2H \\dot{\\phi} \n\\right)-V+\\beta_3 M^{-3} \\dot{\\phi}^2 \\ddot{\\phi}\\,.\n\\label{Pde}\n\\ea\nBesides the matter continuity equations $\\dot{\\rho}_m+3H \\rho_m=0$ \nand $\\dot{\\rho}_r+4H \\rho_r=0$, the dark sector obeys \n\\be\n\\dot{\\rho}_{\\rm DE}+3H \\left( \\rho_{\\rm DE}\n+P_{\\rm DE} \\right)=0\\,.\n\\label{back3}\n\\ee\nThe dark energy equation of state is defined by \n\\be\nw_{\\rm DE} \\equiv \\frac{P_{\\rm DE}}\n{\\rho_{\\rm DE}}\\,.\n\\label{wdedef}\n\\ee\nIn nonminimally coupled theories the first terms on the right \nhand sides of Eqs.~(\\ref{rhode}) and (\\ref{Pde}) are different from \n0 in the past due to the property $F \\neq 1$.\n\nTo study the background cosmological dynamics, \nwe introduce the following density parameters,\n\\ba\n& &\n\\Omega_K \\equiv \\frac{\\dot{\\phi}^2}{6M_{\\rm pl}^2H^2},\n\\qquad\n\\Omega_{V} \\equiv \n\\frac{V(\\phi)}{3M_{\\rm pl}^2 H^2 F}, \\nonumber \\\\\n& &\n\\Omega_{G_3} \\equiv \n-\\frac{\\beta_3 \\dot{\\phi}^3}{M_{\\rm pl}^2 M^3 H F},\n\\qquad\n\\Omega_r \\equiv \\frac{\\rho_r}{3M_{\\rm pl}^2 H^2 F}\\,.\n\\label{moDdi}\n\\ea\nWe consider the case in which $\\Omega_{G_3}$ is positive\nin the expanding Universe ($H>0$),  \nwhich amounts to the condition\n\\be\n\\beta_3 \\dot{\\phi}<0\\,.\n\\ee\nWe also define the quantity \n\\be\nx \\equiv \\frac{\\dot{\\phi}}{\\sqrt{6}M_{\\rm pl}H}\\,,\n\\ee\nwhich is related to $\\Omega_K$ and $\\alpha_{\\rM}$, as\n\\be\n\\Omega_K=x^2\\,,\\qquad\n\\alpha_{\\rM}=-2\\sqrt{6} Qx\\,.\n\\ee\nWe can express Eq.~(\\ref{back1}) in the form:\n\\be\n\\Omega_m \\equiv \\frac{\\rho_m}{3M_{\\rm pl}^2 H^2 F}=\n1-\\Omega_{\\rm DE}-\\Omega_r\\,,\n\\label{Omem2}\n\\ee\nwhere $\\Omega_{\\rm DE}$ is defined by \n\\be\n\\Omega_{\\rm DE} \\equiv \\left(1-6Q^2 \\right) \\Omega_K\n-\\alpha_{\\rM}+\\Omega_V+\\Omega_{G_3}\\,.\n\\ee\n\n{}From Eqs.~(\\ref{back2}) and (\\ref{back3}), it follows that \n\\begin{widetext}\n\\ba\nh \\equiv \\frac{\\dot{H}}{H^2}\n&=&\n-\\frac{1}{{\\cal D}}\n\\left[ \\Omega_{G_3} ( 6+2\\Omega_r\n-6\\Omega_{V}+3\\Omega_{G_3}\n-\\alpha_{\\rM}+\\sqrt{6}\\Omega_V \\lambda x )\n+2\\Omega_K \\{ 3+\\Omega_r-3\\Omega_{V}+6\\Omega_{G_3}\n+6\\lambda Q \\Omega_{V} \\right. \\nonumber \\\\\n& &\\qquad \\left. +6Q^2 (1-\\Omega_r+3\\Omega_{V}\n-2\\Omega_{G_3}) \\} -\\alpha_{\\rM} \\Omega_K (1-6Q^2) \n(2-\\Omega_{G_3})+6\\Omega_K^2(1-8Q^2+12Q^4) \\right]\\,,\\label{hD}\\\\\n\\epsilon_{\\phi} \\equiv \\frac{\\ddot{\\phi}}{H \\dot{\\phi}}\n&=& \\frac{1}{{\\cal D}}\n[\\Omega_{G_3} (\\Omega_r-3-3\\Omega_{V} )\n-\\alpha_{\\rM} (\\Omega_r-1-3\\Omega_{V}\n-2\\Omega_{G_3})-2\\sqrt{6} \\Omega_V \\lambda x \\nonumber \\\\\n& &\\quad\n-3 \\Omega_K \\{ 4(1-2Q^2)\n-\\Omega_{G_3}(1+2Q^2) \\}\n-\\alpha_{\\rM} \\Omega_K (5-6Q^2)]\\,,\n\\label{epD}\n\\ea\n\\end{widetext}\nwhere\n\\be\n{\\cal D}=\\Omega_{G_3} \n\\left( 4-2\\alpha_{\\rM}+\\Omega_{G_3} \\right)\n+4\\Omega_K\\,.\n\\ee\nThe condition for cosmic acceleration to occur is that \nthe effective equation of state,\n\\be\nw_{\\rm eff} \\equiv -1-\\frac{2}{3}h\\,,\n\\ee\nis smaller than $-1\/3$.\n\nThe dimensionless variables $x$, $\\Omega_{V}$, $\\Omega_{G_3}$, and $\\Omega_r$ obey the differential equations, \n\\ba\nx' &=& x \\left( \\epsilon_{\\phi}- h \\right)\\,,\n\\label{auto1} \\\\\n\\Omega_{V}' &=& -\\Omega_{V} \n\\left( \\alpha_{\\rM}-\\sqrt{6}\\lambda x+2h \\right)\\,,\n\\label{auto2} \\\\ \n\\Omega_{G_3}' &=& -\\Omega_{G_3} \\left( \n\\alpha_{\\rM}-3\\epsilon_{\\phi}+h \\right)\\,, \n\\label{auto3} \\\\ \n\\Omega_r' &=& -\\Omega_r \n\\left( \\alpha_{\\rM}+4+2h \\right)\\,,\n\\label{auto4}\n\\ea\nrespectively, where a prime represents a derivative with respect to ${\\cal N}=\\ln a$.\nThe dark energy equation of state (\\ref{wdedef}) \nis expressed as\n\\begin{widetext}\n\\be\nw_{\\rm DE}=-\\frac{3+2h-[3+2h+3(1+2Q^2)\\Omega_K\n-3\\Omega_{V}\n+\\alpha_{\\rM} (2+\\epsilon_{\\phi})\n-\\epsilon_{\\phi}\\Omega_{G_3}]F}\n{3-3[1+(6Q^2-1)\\Omega_K\n-\\Omega_{V}+\\alpha_{\\rM}-\\Omega_{G_3}]F}\\,.\n\\label{wdeD}\n\\ee\n\\end{widetext}\nThe dimensionless field $y \\equiv \\phi\/M_{\\rm pl}$\nobeys \n\\be\ny'=\\sqrt{6}x\\,.\n\\label{dy}\n\\ee\nOnce the potential $V(\\phi)$ is specified, \nthe cosmological dynamics is known by solving \nEqs.~(\\ref{auto1})-(\\ref{auto4}) and (\\ref{dy}) \nfor given initial conditions of $x$, $\\Omega_V$, \n$\\Omega_{G_3}$, $\\Omega_r$, and $y$.\n\nFor the theory (\\ref{action}), the propagation \nspeed squared of GWs is equivalent to 1 \\cite{Horn2,DT12}. \nThe tensor ghost is absent for $F(\\phi)>0$, \nwhich is satisfied for the choice (\\ref{Fphi}). \nFor scalar perturbations, the conditions for avoiding \nghosts and Laplacian instabilities are given, respectively, by \n\\ba\n\\hspace{-0.8cm}\nq_s &\\equiv& \\Omega_{G_3} \\left( 4+ \\Omega_{G_3}\n-2\\alpha_{\\rM} \\right)+4\\Omega_K >0,\\label{Qs}\\\\\n\\hspace{-0.8cm}\nc_s^2 &\\equiv& \\frac{ \\Omega_{G_3}[4(2+\\epsilon_{\\phi})\n-\\Omega_{G_3}-2\\alpha_{\\rM} ]+12\\Omega_K}\n{3\\Omega_{G_3} \\left( 4+ \\Omega_{G_3}\n-2\\alpha_{\\rM} \\right)+12\\Omega_K}>0\\,.\n\\label{cs}\n\\ea\nIn Sec.~\\ref{numesec}, we will discuss whether these conditions are satisfied during the cosmological evolution from the radiation-dominated epoch to today.\n\n\\section{Cosmological dynamics}\n\\label{numesec}\n\nIn this section, we study the cosmological dynamics \nfor constant $\\lambda$, i.e., the exponential potential, \n\\be\nV(\\phi)=V_0 e^{\\lambda \\phi\/M_{\\rm pl}}\\,.\n\\label{exp}\n\\ee\nIn this case, the dynamical system given by \nEqs.~(\\ref{auto1})-(\\ref{auto4}) is closed. \nAs long as $\\lambda$ slowly varies in time in the range \n(\\ref{lamra}), the cosmological evolution is similar to \nthat discussed below. \n\nIn over-density regions of the Universe, the operation of Vainshtein mechanism means that the cubic Galileon term \n$X \\square \\phi$ dominates over other field Lagrangians. \nIn the cosmological context, this amounts to the dominance of \n$\\Omega_{G_3}$ over $\\Omega_K$ and $\\Omega_V$ \nin the early epoch.\nLet us consider the case in which the conditions \n\\be\n\\{ \\Omega_K, \\Omega_V \\} \\ll \\Omega_{G_3} \\ll 1\\,,\n\\qquad \\left| \\alpha_{\\rM} \\right| \\ll 1\n\\label{radcon}\n\\ee\nare satisfied during the radiation-dominated epoch\n(in which $\\Omega_r$ is close to 1). {}From Eqs.~(\\ref{hD}) \nand (\\ref{epD}), we then have $h \\simeq -2$ and \n\\be\n\\epsilon_{\\phi} \\simeq \n-\\frac{1}{2}+\\epsilon_{\\alpha}\\,,\\qquad \n\\epsilon_{\\alpha} \\equiv \\frac{\\alpha_{\\rM}}{4 \\Omega_{G_3}}\n \\left( 1-\\Omega_r \\right)\\,. \n\\label{hepes}\n\\ee\nSince $\\Omega_r$ starts to deviate from 1 in the \nlate radiation era, the term $\\epsilon_{\\alpha}$ is \nnot necessarily negligible \nrelative to $-1\/2$ for $|\\alpha_{\\rM}| \\gg \\Omega_{G_3}$.\nOn using Eqs.~(\\ref{auto1}), (\\ref{auto3}), and (\\ref{auto4}), the quantity $\\epsilon_{\\alpha}$ \nobeys the differential equation, \n\\be\n\\epsilon_{\\alpha}' \\simeq 6Q^2 \n\\frac{\\Omega_K}{\\Omega_{G_3}}\n+2 \\epsilon_{\\alpha}\n \\left(  1- \\epsilon_{\\alpha} \\right)\\,.\n\\label{epp}\n\\ee\nUnder the condition $\\Omega_{G_3} \\gg \\Omega_K$, \nthe first term on the right hand side of Eq.~(\\ref{epp}) is much smaller than 1. \nIgnoring this term and solving the \ndifferential equation $\\epsilon_{\\alpha}' \\simeq \n2 \\epsilon_{\\alpha} \\left(  1- \\epsilon_{\\alpha} \\right)$\nfor $\\epsilon_{\\alpha}$, it follows that \n\\be\n\\epsilon_{\\alpha}=\\left[1+\\frac{a_i^2}{a^2}\n\\frac{1-\\epsilon_{\\alpha}^{(i)}}{\\epsilon_{\\alpha}^{(i)}} \\right]^{-1}\\,,\n\\label{epso}\n\\ee\nwhere $\\epsilon_{\\alpha}^{(i)}$ is the initial value of \n$\\epsilon_{\\alpha}$ at $a=a_i$. \nIn the limit $a \\to \\infty$, $\\epsilon_{\\alpha}$ \nasymptotically approaches 1.\n\nIf the condition $|\\alpha_{\\rM}| \\gg \\Omega_{G_3}$ is \ninitially satisfied, $|\\epsilon_{\\alpha}^{(i)}|$ \ncan be as large as the order 1. \nThen, $\\epsilon_{\\phi}$ soon approaches \nthe asymptotic value\n\\be\n\\epsilon_{\\phi} \\to \\frac{1}{2}\\,,\n\\label{epphies}\n\\ee\nduring the radiation era. In this regime, \nthe field density parameters and $|\\alpha_{\\rM}|$ \ngrow as\n\\be\n\\Omega_K \\propto a^5\\,,\\quad \n\\Omega_V \\propto a^4\\,,\\quad \n\\Omega_{G_3} \\propto a^{7\/2}\\,,\\quad \n|\\alpha_{\\rM}| \\propto a^{5\/2}\\,.\n\\label{Omeevo}\n\\ee\nThis shows that, even if $\\Omega_{G_3} \\gg \\Omega_K$ initially, it is possible for $\\Omega_K$ to \ncatch up with $\\Omega_{G_3}$. \nIf this catch up occurs by the end of radiation era, \nwe have $\\Omega_{G_3}<\\Omega_{K}$ \nat the onset of matter dominance.\n\nIf $|\\alpha_{\\rM}| \\ll \\Omega_{G_3}$ initially, \ni.e., $|\\epsilon_{\\alpha}^{(i)}| \\ll 1$, \nthere is the stage of radiation era in which \nthe quantity $\\epsilon_{\\phi}$ is close to $-1\/2$. \nOn using Eqs.~(\\ref{auto1})-(\\ref{auto3}) in this epoch, \nthe field density parameters and $|\\alpha_{\\rM}|$ \nevolve as\n\\be\n\\Omega_K \\propto a^3\\,,\\quad \n\\Omega_V \\propto a^4\\,,\\quad \n\\Omega_{G_3} \\propto a^{1\/2}\\,,\\quad \n|\\alpha_{\\rM}| \\propto a^{3\/2}\\,,\n\\label{den1}\n\\ee\nso that $|\\alpha_{\\rM}|$ grows faster than $\\Omega_{G_3}$. \nIf $|\\alpha_{\\rM}|$ exceeds $\\Omega_{G_3}$ \nduring the radiation era, the solutions enter the regime characterized by Eqs.~(\\ref{epphies}) and (\\ref{Omeevo}). \nAlthough $\\Omega_K$ grows faster than \n$\\Omega_{G_3}$ in the two regimes explained above, \nit can happen that the inequality \n$\\Omega_{G_3}>\\Omega_K$ still holds at the beginning \nof matter era for $\\Omega_{G_3}$ initially much larger \nthan $|\\alpha_{\\rM}|$ and $\\Omega_K$.\n\nThe above discussion shows that there are two qualitatively \ndifferent cases depending on the values of \n$\\Omega_{G_3}$ and $\\Omega_K$ \nat the onset of matter dominance.\nThe first is the case in which $\\Omega_K$\ndominates over $\\Omega_{G_3}$, i.e., \n\\be\n{\\rm (i)}~~\\Omega_{G_3} \\ll \\Omega_{K}\n\\quad ({\\rm unscreened})\\,.\n\\ee\nUnder this condition, there exists \nthe $\\phi$MDE in which \nthe field kinetic energy is not screened by the  \nGalileon term.\n\nThe second is the case in which the condition \n\\be\n{\\rm (ii)}~~\\Omega_{G_3} \\gg \\Omega_{K}\n\\quad ({\\rm screened})\n\\ee\nis satisfied after the end of radiation era.\nThis corresponds to the situation in which  \nthe cosmological Vainshtein screening \nis sufficiently efficient to suppress the time variation \nof $\\phi$ throughout the evolution \nfrom the radiation era to today.\nIn the following, we study these two different cases \nin turn.\n\nWe note that, under the conditions (\\ref{radcon}),\nthe dark energy equation of state (\\ref{wdeD})\nduring the radiation dominance can be estimated as\n\\be\nw_{\\rm DE} \\simeq w_{\\rm eff} \\simeq \n\\frac{1}{3}\\,,\n\\label{wdera}\n\\ee\nirrespective of the two asymptotic values of  \n$\\epsilon_{\\phi}~(=\\pm 1\/2)$ explained above.\n\n\\subsection{Unscreened late-time cosmology with \nthe $\\phi$MDE}\n\\label{unsec}\n\nLet us first study the cosmological dynamics for the case \n(i), i.e., $\\Omega_{G_3} \\ll \\Omega_K$ after the onset \nof matte era. In this case, the coupling $\\beta_3$ \nis in the range\n\\be\n\\left| \\beta_3 \\right| \\ll 1\\,.\n\\ee\nTo derive fixed points of the dynamical system, \nwe take the limit \n$\\Omega_{G_3} \\to 0$ in the autonomous \nEqs.~(\\ref{auto1})-(\\ref{auto4}). \nFor $Q \\neq 0$, the standard matter era is replaced \nby the $\\phi$MDE characterized by the fixed point  \n\\be\n({\\rm a})~\\left( x, \\Omega_{V}, \\Omega_{G_3}, \\Omega_{r} \n\\right) =\\left( -\\frac{\\sqrt{6}Q}{3(1-2Q^2)},0,0,0 \\right)\\,,\n\\label{phiMDE}\n\\ee\nwith \n\\ba\n& &\n\\Omega_m=\\frac{3-2Q^2}{3(1-2Q^2)^2}\\,,\\quad\nw_{\\rm eff}=\\frac{4Q^2}{3(1-2Q^2)}\\,, \\nonumber \\\\\n& &\nw_{\\rm DE}=\\frac{4Q^2(1-2Q^2)}{3(1-F)-2(6-F)Q^2+12Q^4}\\,.\n\\label{phiMDE2}\n\\ea\nThe $\\phi$MDE was originally found for coupled quintessence in the \nEinstein frame \\cite{Amenco}. \nThis corresponds to the kinetically driven stage \nin which $\\Omega_K=2Q^2\/[3(1-2Q^2)^2]$ dominates \nover $\\Omega_{G_3}$. \nOn the fixed point (a), the parameter $\\alpha_{\\rM}$ is \ngiven by \n\\be\n\\alpha_{\\rM}^{(\\rm a)}=\\frac{4Q^2}{1-2Q^2}\\,,\n\\label{aMa}\n\\ee\nand hence $\\alpha_{\\rM}^{(\\rm a)}>0$ for $Q^2<1\/2$. \nThe positivity of $\\alpha_{\\rM}^{(\\rm a)}$ means that \n\\be\nQx_{(\\rm a)}<0\\,,\n\\label{Qx}\n\\ee\nwhere $x_{(\\rm a)}$ is the value of $x$ on the $\\phi$MDE.\n\nAfter $\\Omega_K$ exceeds $\\Omega_{G_3}$ by the end \nof radiation era, the solutions are naturally followed \nby the $\\phi$MDE \nin which the cosmological Vainshtein screening is no \nlonger effective.\nWhile $\\Omega_K$ is constant during the $\\phi$MDE, \nthe other field density parameters evolve as\n\\be\n\\Omega_V \\propto a^{\\frac{3-2Q \\lambda-6Q^2}{1-2Q^2}}\\,,\\quad \n\\Omega_{G_3} \\propto \na^{-\\frac{3+2Q^2}{1-2Q^2}}\\,.\n\\label{OmeVG}\n\\ee\nFor $|Q \\lambda| \\ll 1$ and $Q^2 \\ll 1$, $\\Omega_V$ grows \nin proportion to $a^3$, whereas $\\Omega_{G_3}$ \ndecreases as $\\propto a^{-3}$. \nHence the contribution of cubic Galileons to $\\Omega_{\\rm DE}$ \nbecomes negligibly small in the late matter era.\n\nThe stability of point (a) is known by linearly perturbing \nEqs.~(\\ref{auto1})-(\\ref{auto4}) with homogenous perturbations $\\delta x$, $\\delta \\Omega_V$, $\\delta \\Omega_{G_3}$, and $\\delta \\Omega_r$ \\cite{CLW,CST}. \nThe eigenvalues of Jacobian matrix associated with these perturbations are given by \n$-1$, $-(3-2Q^2)\/(2-4Q^2)$, \n$-(3+2Q^2)\/(1-2Q^2)$, and \n$(3-2Q\\lambda-6Q^2)\/(1-2Q^2)$. \nThe first three eigenvalues are negative for $\\lambda$ and $Q$ \nin the ranges (\\ref{lamra}) and (\\ref{Qra}), \nwhile the last one is positive. \nHence the $\\phi$MDE corresponds to a saddle point.\nThis shows that, as long as $\\Omega_K$ catches up with\n$\\Omega_{G_3}$ by the end of radiation era, the solutions \ntemporally approach the $\\phi$MDE with \n$\\Omega_{G_3} \\ll  \\Omega_K \\simeq {\\rm constant}$.\n\nThere are other kinetically driven fixed points \ncharacterized by $(x,\\Omega_V,\\Omega_{G_3},\\Omega_r)\n=(1\/(\\sqrt{6}Q \\pm 1),0,0,0)$. \nSince $\\Omega_m=0$, this point cannot \nbe responsible for the matter era. \nThe scaling fixed point \n$(x,\\Omega_V,\\Omega_{G_3},\\Omega_r)\n=(-\\sqrt{6}\/(2\\lambda), (3-2Q \\lambda-6Q^2)\/(2\\lambda^2),0,0)$ \nis also present, but $\\Omega_{\\rm DE}=(3-7Q \\lambda-12Q^2)\/\\lambda^2$ \nis larger than the order 1 under the conditions \n(\\ref{lamra}) and (\\ref{Qra}). \nHence this scaling solution is irrelevant to the \nmatter-dominated epoch. \nThis is also the case for the radiation scaling solution \n$(x,\\Omega_V,\\Omega_{G_3},\\Omega_r)\n=(-2\\sqrt{6}\/(3\\lambda), 4\/(3\\lambda^2),0,\n1-4(1-2Q \\lambda-4Q^2)\/\\lambda^2)$, \nwhere $\\Omega_{\\rm DE}=\n4(1-2Q \\lambda-4Q^2)\/\\lambda^2$ exceeds the order 1.  \n\nThe fixed point relevant to the dark energy \ndomination is given by \n\\ba\n\\hspace{-0.95cm}\n& &\n({\\rm b})~\\left( x, \\Omega_{V}, \\Omega_{G_3}, \\Omega_{r} \n\\right) \\nonumber \\\\\n\\hspace{-0.95cm}\n& &\n=\\left( \\frac{-\\sqrt{6}(\\lambda+4Q)}{6(1-Q\\lambda-4Q^2)},\n\\frac{6-\\lambda^2-8Q (\\lambda+2Q)}{6(1-Q\\lambda-4Q^2)^2},0,0 \\right),\n\\label{bpoint}\n\\ea\nwith \n\\be\n\\Omega_m=0\\,,\\quad \nw_{\\rm eff}=w_{\\rm DE}=\n-1+\\frac{\\lambda^2+6Q \\lambda+8Q^2}\n{3(1-Q \\lambda-4Q^2)},\n\\label{weffc}\n\\ee\nand $\\Omega_{\\rm DE}=1$. \nOn this fixed point, the quantity $\\alpha_{\\rM}$ yields \n\\be\n\\alpha_{\\rM}^{(\\rm b)}=\\frac{2Q (\\lambda+4Q)}\n{1-Q \\lambda -4Q^2}\\,.\n\\ee\n\nThe point (b) can drive the cosmic acceleration for \n$w_{\\rm eff}<-1\/3$, which translates to\n\\be\n\\lambda^2<2(1-4Q \\lambda-8Q^2)\\,.\n\\label{lamcon}\n\\ee\nUnder this bound, the four eigenvalues of \nJacobian matrix of homogeneous perturbations around \npoint (b) are all negative. \nThen, after the $\\phi$MDE, the solutions finally \napproach the stable point (b) with cosmic acceleration.  \nOn using the values of $x$ and $\\Omega_{V}$ in \nEq.~(\\ref{bpoint}), Eq.~(\\ref{auto3}) reduces to\n\\be\n\\Omega_{G_3}'=-p\\,\\Omega_{G_3}\\,,\\qquad \np=\\frac{(\\lambda+4Q)^2}{1-Q \\lambda -4Q^2}\\,.\n\\ee\nThe Galileon density parameter decreases as \n$\\Omega_{G_3} \\propto a^{-p}$ around point (b).\n\nIn the following, we focus on the couplings satisfying\n\\be\n\\lambda>0\\,,\\qquad Q>0\\,.\n\\label{lQcon}\n\\ee\nDuring the $\\phi$MDE, we showed that $\\alpha_{\\rM}>0$ \nfor $Q^2<1\/2$. \nProvided $x$ does not change the sign during the \ncosmological evolution from the radiation era to fixed \npoint (b), the parameter $\\alpha_{\\rM}$ is in the range\n\\be\n\\alpha_{\\rM}=-2\\sqrt{6} Qx>0\\,,\n\\label{aMre}\n\\ee\nand hence $x<0$.\nThe negative value of $x$ is consistent with the fact \nthat $\\dot{\\phi}<0$ when the scalar field rolls down the potential \nwith $\\lambda>0$. Alternatively, we can consider negative \nvalues of $\\lambda$ and $Q$, in which case $x>0$.\nUnder the condition (\\ref{aMre}), we have \n$Q \\dot{\\phi}<0$ for $H>0$ and \nhence the quantity $Q \\phi$ decreases in time. \nThis means that the field $\\phi$\nsatisfies the inequality $Q (\\phi-\\phi_0)>0$ in the past.\nThen, irrespective of the sign of $Q$, the quantity \n$F=e^{-2Q (\\phi-\\phi_0)\/M_{\\rm pl}}$ is \nsmaller than 1 during the past cosmic expansion history.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.4in,width=3.4in]{fig1.pdf}\n\\end{center}\n\\caption{\\label{fig1}\nEvolution of $\\Omega_K$, $\\Omega_V$, $\\Omega_{G_3}$, \n$\\Omega_m$, $\\Omega_r$, and $\\alpha_{\\rM}$ versus \n$z+1$ for $Q=5.0 \\times 10^{-4}$ and \n$\\lambda=0.1$ with the initial conditions \n$x=-1.0 \\times 10^{-15}$, $\\Omega_V=1.0 \\times 10^{-29}$, \n$\\Omega_{G_3}=1.0 \\times 10^{-23}$, \n$\\Omega_r=0.99998$, and $y=1.0$ at the redshift \n$z=1.62 \\times 10^8$. The present epoch ($z=0$) is \nidentified by the condition $\\Omega_{\\rm DE}=0.68$.\n}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.4in,width=3.4in]{fig2.pdf}\n\\end{center}\n\\caption{\\label{fig2}\nEvolution of $w_{\\rm DE}$, $w_{\\rm eff}$, and \n$c_s^2$ versus $z+1$ for the same model parameters \nand initial conditions as those given in the caption \nof Fig.~\\ref{fig1}.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig1}, we exemplify the evolution of \n$\\Omega_K$, $\\Omega_V$, $\\Omega_{G_3}$, \n$\\Omega_r$, $\\Omega_m$, and $\\alpha_{\\rM}$ \nversus $z+1~(=a(t_0)\/a(t))$\nfor $Q=5.0 \\times 10^{-4}$ and $\\lambda=0.1$.\nIn this case, the initial value of $\\epsilon_{\\alpha}$ \nin Eq.~(\\ref{hepes}) is $\\epsilon_{\\alpha}^{(i)}=1.22$, \nso $\\epsilon_{\\phi}$ starts from the value around $0.72$. \nAs estimated from Eq.~(\\ref{epphies}), $\\epsilon_{\\phi}$ \nsoon approaches the value $1\/2$ during the radiation era.\nIn Fig.~\\ref{fig1}, we can confirm that the evolution of \n$\\Omega_K$, $\\Omega_V$, $\\Omega_{G_3}$, \n$\\alpha_{\\rM}$ around \nthe redshift $10^4 \\lesssim z \\lesssim 10^8$ \nis approximately given by Eq.~(\\ref{Omeevo}). \nIn Fig.~\\ref{fig2}, we plot the evolution of $w_{\\rm DE}$ \nand $w_{\\rm eff}$ for the same model parameters and \ninitial conditions as those used in Fig.~\\ref{fig1}. \nAs the analytic estimation (\\ref{wdera}) shows, both \n$w_{\\rm DE}$ and $w_{\\rm eff}$ are close to $1\/3$ during the \ndeep radiation-dominated epoch. \n\nIn the numerical simulation of Fig.~\\ref{fig1}, $\\Omega_K$\ncatches up with $\\Omega_{G_3}$ around the \nredshift $z=4.6 \\times 10^3$.\nThen, the solutions approach the $\\phi$MDE \nwith the constant kinetic density parameter \n$\\Omega_K=2Q^2\/[3(1-2Q^2)^2] \\simeq 1.7 \\times 10^{-7}$ \nwith $\\alpha_{\\rM}=6(1-2Q^2)\\Omega_K \\simeq \n1.0 \\times 10^{-6}$.\nAs we estimated in Eq.~(\\ref{OmeVG}), $\\Omega_V$ \nincreases during the $\\phi$MDE, while $\\Omega_{G_3}$ \ndecreases. In Fig.~\\ref{fig1}, we observe that \n$\\Omega_V$ exceeds $\\alpha_{\\rM}$ around the redshift \n$z=130$. After this moment, $\\Omega_V$ becomes \nthe dominant contribution to $\\Omega_{\\rm DE}$. \nAs long as $\\Omega_V \\ll 1$, the terms containing \n$\\Omega_V$ in Eqs.~(\\ref{hD}) and \n(\\ref{epD}) hardly modify the values of $h$ and \n$\\epsilon_{\\phi}$ during the $\\phi$MDE.\nIn Fig.~\\ref{fig1}, we find that the $\\phi$MDE with \nnearly constant $\\Omega_K$ continues up to the \nredshift $z \\approx 10$.\n\nThe dark energy equation of state is more sensitive to the dominance \nof $\\Omega_V$ over other field density parameters. \nIn the regime where the condition \n$\\Omega_V \\gg \\{ \\alpha_{\\rM}, \\Omega_K, \n\\Omega_{G_3} \\}$ is satisfied, Eq.~(\\ref{wdeD}) approximately \nreduces to \n\\be\nw_{\\rm DE} \\simeq -1-\\frac{2h}{3} \n\\frac{1-F}{1-F+\\Omega_V F}\\,.\n\\label{wdetra}\n\\ee\nProvided the inequality $\\Omega_V F \\ll 1-F$ holds \nduring the early stage of matter era, it follows that \n$w_{\\rm DE} \\simeq w_{\\rm eff}\n=-1-2h\/3 \\simeq 4Q^2\/[3(1-2Q^2)]$. \nAfter $\\Omega_V F$ grows to be larger than $1-F$, \n$w_{\\rm DE}$ starts to approach $-1$. \nIn Fig.~\\ref{fig2}, we can confirm that $w_{\\rm DE}$ \ndeviates from $w_{\\rm eff}$ around the \nsame moment at which $\\Omega_V$ becomes \nthe dominant contribution to $\\Omega_{\\rm DE}$ \nand that $w_{\\rm DE}$ temporally approaches the value close to $-1$.\n\nAfter the Universe enters the stage of cosmic acceleration, \nthe solutions finally reach the fixed point (b). \nFor $Q=5.0 \\times 10^{-4}$ and $\\lambda=0.1$, \nthe analytic estimation (\\ref{bpoint}) gives the values \n$x=-0.04164$, $\\Omega_V=0.9984$, and \n$w_{\\rm DE}=w_{\\rm eff}=-0.9966$, \nwhich are in good agreement with the \nnumerical results of Figs.~\\ref{fig1} and \\ref{fig2}. \nIn this case, the future asymptotic value of \n$\\alpha_{\\rM}$ is $1.02 \\times 10^{-4}$, \nwhile its today's value is \n$\\alpha_{\\rM}(t_0)=5.61 \\times 10^{-5}$. \nTaking $h=0.7$ in Eq.~(\\ref{aMcon}), this case is within \nthe LLR bound of $\\alpha_{\\rM} (t_0)$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.2in,width=3.5in]{fig3.pdf}\n\\end{center}\n\\caption{\\label{fig3}\nParameter space in the $(\\lambda, Q)$ plane \n(colored region) consistent with the bound \n(i) $\\alpha_{\\rM}(t_0) \\le 7 \\times 10^{-5}$ and \n(ii) the condition for cosmic acceleration of point (b). \nWe also show the bound $Q \\le 4.2 \\times 10^{-3}$ \narising from the condition \n$\\alpha_{\\rM}^{({\\rm a})} \\le 7 \\times 10^{-5}$ \non the $\\phi$MDE.}\n\\end{figure}\n\n{}From Eqs.~(\\ref{phiMDE}) and (\\ref{bpoint}) we find that \nthe inequality $0>x_{(\\rm a)}>x_{(\\rm b)}$ holds, where \n$x_{(\\rm a)}$ and $x_{(\\rm b)}$ are the values of \n$x$ on points (a) and (b) respectively.\nThen, the quantity $\\alpha_{\\rM}$ on point (b) is larger than \nthat on point (a), such that \n$\\alpha_{{\\rM}}^{{(\\rm b)}}>\\alpha_{{\\rM}}^{{(\\rm a)}}>0$. \nSince $\\alpha_{\\rM}$ increases from $\\alpha_{{\\rM}}^{{(\\rm a)}}$ \nduring the $\\phi$MDE to the asymptotic value $\\alpha_{{\\rM}}^{{(\\rm b)}}$ \nin the future, the necessary condition for satisfying \nthe LLR bound (\\ref{aMcon}) for $h=0.7$ is \n$\\alpha_{{\\rM}}^{{(\\rm a)}} \\le 7 \\times 10^{-5}$, i.e., \n\\be\nQ \\le 4.2 \\times 10^{-3}\\,.\n\\label{Qup}\n\\ee\nSince today's value $\\alpha_{{\\rM}}(t_0)$ is between \n$\\alpha_{{\\rM}}^{{(\\rm b)}}$ and $\\alpha_{{\\rM}}^{{(\\rm a)}}$, \nthe condition (\\ref{Qup}) is not sufficient for the compatibility \nwith the bound (\\ref{aMcon}).\n\nIn Fig.~\\ref{fig3}, we plot the parameter space \nin the $(\\lambda, Q)$ plane constrained from the bound \n$\\alpha_{\\rM}(t_0) \\le 7 \\times 10^{-5}$, \nwhose border is denoted as the line (i). \nWe also depict the region in which the condition \n(\\ref{lamcon}) for cosmic acceleration of point (b)\nis satisfied, whose border is shown as the line (ii). \nThis condition gives the upper limit $\\lambda<\\sqrt{2}$.\nThe coupling $Q$ is constrained to be \n\\be\nQ \\le 3.4 \\times 10^{-3}\\,,\n\\label{Qup2}\n\\ee\nwhich is tighter than (\\ref{Qup}). \nThis significantly improves the upper limit $Q \\le 2.6 \\times 10^{-2}$ \nfollowing from  the LLR bound $|\\alpha_{\\rM} (t_0)| \\le 0.02$ \nin 2004 \\cite{Babi11}.\nWe note that the bound (\\ref{Qup2}) corresponds to the limit \n$\\lambda \\to 0$. For increasing $\\lambda$ from 0, \nthe constraint on $Q$ is more stringent than (\\ref{Qup2}), e.g., $Q \\le 6.2 \\times 10^{-4}$ for $\\lambda=0.1$ \nand $Q \\le 6.3 \\times 10^{-5}$ \nfor $\\lambda=1$. If $\\lambda>0.013$, then the recent LLR data give the upper limit of $Q$ tighter than the Cassini \nbound $Q \\le 2.4 \\times 10^{-3}$ derived for the massless \nscalar field without the Vainshtein screening.\n\nCosmologically, today's value of $\\Omega_{G_3}$ is \nrelated to the dimensionless coupling $\\beta_3$, as\n\\be\n\\Omega_{G_3} (t_0)=-6 \\sqrt{6} \\beta_3\\,x(t_0)^3\\,.\n\\ee\nThe numerical simulation of Fig.~\\ref{fig1} corresponds to \n$\\Omega_{G_3}(t_0)=1.76 \\times 10^{-12}$, \n$x(t_0)=-2.29 \\times 10^{-2}$, and \n$\\beta_3=9.97 \\times 10^{-9}$, with $Q=5.0 \\times 10^{-4}$. \nThese couplings satisfy the condition (\\ref{local}), so the \nVainshtein mechanism is at work in the solar system.\nThe existence of $\\phi$MDE generally requires that \n$\\beta_3 \\ll 1$, but still the fifth force can be screened around \nlocal sources for the product $\\beta_3 Q$ \nin the range (\\ref{local}).\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.2in,width=3.5in]{fig4.pdf}\n\\end{center}\n\\caption{\\label{fig4}\nEvolution of $w_{\\rm DE}$ versus $z+1$ for \n(A) $Q=6.20 \\times 10^{-4}$, $\\lambda=0.1$, \n(B) $Q=2.57 \\times 10^{-4}$, $\\lambda=0.25$, \n(C)  $Q=1.27 \\times 10^{-4}$, $\\lambda=0.5$, and \n(D)  $Q=6.32 \\times 10^{-5}$, $\\lambda=1$. \nThe initial conditions of $x$, $\\Omega_V$, \n$\\Omega_{G_3}$, $\\Omega_r$, and $y$ are \nthe same as those used in Fig.~\\ref{fig1}.}\n\\end{figure}\n\nIn Fig.~\\ref{fig4}, we show the evolution of $w_{\\rm DE}$ \nfor four different combinations of $Q$ and $\\lambda$.\nIn all these cases, $\\alpha_{\\rM}(t_0)$ is close to the LLR \nupper limit $7 \\times 10^{-5}$, with $\\beta_3$ \nof order $10^{-8}$.\nAs we estimated in Eq.~(\\ref{wdetra}), $w_{\\rm DE}$ \ntemporally approaches the value close to $-1$ after \n$\\Omega_V$ dominates over other field density \nparameters in the matter era. \nIn all the cases plotted in Fig.~\\ref{fig4}, \nthe minimum values of $w_{\\rm DE}$ are close to $-1$.\nEven for the case (D), i.e., $\\lambda=1$, \n$w_{\\rm DE}$ reaches the minimum value \n$-0.9952$ at $z=4.5$. The solutions finally approach \nthe fixed point (b), with $w_{\\rm DE}$ given by \nEq.~(\\ref{weffc}).\nFor larger $\\lambda$ closer to the border line (ii) \nin Fig.~\\ref{fig3}, the deviation of $w_{\\rm DE}$ from \n$-1$ at low redshifts is more significant.\nThis property can be used to distinguish between \nthe models with different values of $\\lambda$ from \nobservations.\n\nSince $\\Omega_{G_3}$ and $\\Omega_K$ are positive  \nwith $0<\\alpha_{\\rM} \\ll 1$ from the radiation era to \nthe accelerated point (b), the no-ghost condition  \n(\\ref{Qs}) of scalar perturbations is always satisfied. \nProvided that $1 \\gg \\Omega_{G_3} \\gg \\Omega_{K}$ in \nthe deep radiation era, the scalar propagation \nspeed squared (\\ref{cs}) reduces to \n$c_s^2 \\simeq (2+\\epsilon_{\\phi})\/3$.\nIn the numerical simulation of Fig.~\\ref{fig2}, the quantity\n$\\epsilon_{\\phi}$ approaches the value $1\/2$ around\nthe redshift $z \\approx 10^7$, and hence $c_s^2 \\simeq 5\/6$ \nfor $10^5 \\lesssim z \\lesssim 10^7$. \nDuring the late radiation era ($3000 \\lesssim z \\lesssim 10^5$) \nin which $\\Omega_r$ starts to deviate from 1, \n$c_s^2$ temporally decreases due to the decrease \nof $\\epsilon_{\\phi}$. \nFor $\\Omega_K \\gg \\Omega_{G_3}$ we have \n$c_s^2 \\simeq 1$ from Eq.~(\\ref{cs}). \nIndeed, the approach to this value can be confirmed \nin Fig.~\\ref{fig2} \nafter the onset of matter era.\nSince $c_s^2$ remains positive from the radiation era to \nthe asymptotic future, the Laplacian instability of \nscalar perturbations is absent.\nWe note that the property $c_s^2>0$ also \nholds for the four cases shown in Fig.~\\ref{fig4}.\n\n\\subsection{Screened cosmology}\n\nWe proceed to the case (ii) in which the cubic coupling $\\beta_3$ \nis in the range \n\\be\n|\\beta_3| \\gg 1\\,,\n\\ee\nwith positive values of $\\lambda$ and $Q$.\nAs we will see below, the field kinetic energy can be \nsuppressed even in the late epoch \nthrough the cosmological Vainshtein mechanism.\n\nDuring the radiation dominance the condition \n(\\ref{radcon}) holds, so the quantity $\\epsilon_{\\phi}$ can be estimated as Eq.~(\\ref{hepes}).\nThe difference from the case discussed in Sec.~\\ref{unsec} is \nthat $\\epsilon_{\\alpha}$ is much smaller than 1 due to the largeness \nof $\\Omega_{G_3}$ relative to $\\alpha_{\\rM}$.\nSince $\\epsilon_{\\phi} \\simeq -1\/2$ during most \nstage of the radiation era, the field density parameters \nand $\\alpha_{\\rM}$ evolve according to \nEq.~(\\ref{den1}). Indeed, we can confirm this behavior in Fig.~\\ref{fig5}, \nwhere the cubic coupling is $\\beta_3=1.0 \\times 10^7$. \nAlthough $\\Omega_K$ grows faster than $\\Omega_{G_3}$, \nthe inequality $\\Omega_{G_3} \\gg \\Omega_K$  holds \neven after the end of radiation era. \nHence the solutions do not reach the \n$\\phi$MDE charactrized by constant $\\Omega_K$ \nlarger than $\\Omega_{G_3}$. \nIn Fig.~\\ref{fig6}, we observe that both $w_{\\rm DE}$ and $w_{\\rm eff}$ \nare close to $1\/3$ during the radiation dominance.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.4in,width=3.4in]{fig5.pdf}\n\\end{center}\n\\caption{\\label{fig5}\nEvolution of $\\Omega_K$, $\\Omega_V$, $\\Omega_{G_3}$, \n$\\Omega_m$, $\\Omega_r$, and $\\alpha_{\\rM}$ versus \n$z+1$ for $Q=0.1$ and $\\lambda=1$ with the initial conditions \n$x=-1.0 \\times 10^{-15}$, $\\Omega_V=1.0 \\times 10^{-29}$, \n$\\Omega_{G_3}=1.0 \\times 10^{-8}$, \n$\\Omega_r=0.99998$, and $y=1.0$ at the redshift \n$z=1.62 \\times 10^8$. \n}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.4in,width=3.4in]{fig6.pdf}\n\\end{center}\n\\caption{\\label{fig6}\nEvolution of $w_{\\rm DE}$, $w_{\\rm eff}$, and \n$c_s^2$ versus $z+1$ for the \nsame model parameters and initial conditions \nas those used in Fig.~\\ref{fig5}.\n}\n\\end{figure}\n\nDuring the matter-dominated epoch, we study the \ncosmological evolution \nunder the conditions:\n\\ba\n& &\n\\Omega_K \\ll \\Omega_{G_3} \\ll 1\\,,\\quad \n\\alpha_{\\rM} \\ll 1\\,,\\quad \n\\Omega_V \\ll 1\\,, \\nonumber \\\\\n& &  \\Omega_r \\ll 1\\,,\\quad (\\lambda\/Q)\\Omega_V \\ll 1\\,.\n\\label{sccon}\n\\ea\nThen,  the quantities defined in Eqs.~(\\ref{hD}) and \n(\\ref{epD}) reduce to $h \\simeq -3\/2$ \nand $\\epsilon_{\\phi} \\simeq -3\/4+\\alpha_{\\rM}\n\/(4\\Omega_{G_3})$, respectively.\n{}From Eqs.~(\\ref{auto1})-(\\ref{auto3}), we obtain the \ndifferential equations for\n$\\alpha_{\\rM}$, $\\Omega_V$, and $\\Omega_{G_3}$, as \n\\ba\n\\alpha_{\\rM}' &\\simeq &\\frac{\\alpha_{\\rM}}{4} \n\\left( 3+\\frac{\\alpha_{\\rM}}{\\Omega_{G_3}} \\right)\\,,\\\\\n\\Omega_{V}' &\\simeq & 3\\Omega_V\\,,\\\\\n\\Omega_{G_3}' &\\simeq& -\\frac{3}{4} \n\\left( \\Omega_{G_3}-\\alpha_{\\rM} \\right)\\,.\n\\ea\nThis means that, provided $x<0$, $\\alpha_{\\rM}$ \nincreases during the matter era. \nThe density parameter associated with the field potential \nalso grows as $\\Omega_{V} \\propto a^3$. \nOn the other hand, $\\Omega_{G_3}$ decreases for \n$\\Omega_{G_3}>\\alpha_{\\rM}$, whereas it increases \nfor $\\Omega_{G_3}<\\alpha_{\\rM}$. \nIn the numerical simulation of Fig.~\\ref{fig5}, \n$\\Omega_{G_3}$ is larger than $\\alpha_{\\rM}$ \nat the onset of matter era and hence $\\Omega_{G_3}$ decreases \nby the moment at which $\\alpha_{\\rM}$ catches \nup with $\\Omega_{G_3}$. \nAfter this catch up, $\\Omega_{G_3}$ starts to grow. \nThe field kinetic density parameter increases as \n$\\Omega_K \\propto \\alpha_{\\rM}^2$, but still \n$\\Omega_K$ is smaller than $\\Omega_{G_3}$ \naround the end of matter era.\n\nIn Fig.~\\ref{fig5}, we find that $\\Omega_V$ dominates over $\\Omega_{G_3}$, \n$\\Omega_K$, and $\\alpha_{\\rM}$ for the redshift $z \\lesssim 200$.\nThen, the dark energy equation of state after the dominance of $\\Omega_V$ \nis given by Eq.~(\\ref{wdetra}). The numerical simulation of Fig.~\\ref{fig6} shows \nthat $w_{\\rm DE}$ starts to deviate from $w_{\\rm eff} \\simeq 0$ \naround $z=200$ and then $w_{\\rm DE}$ approaches the value close to $-1$ \nfor $z \\lesssim 10$. \n{}From the radiation dominance to the deep matter era, \nwe have\n$\\epsilon_{\\phi} \\simeq [\\Omega_r-3+(1-\\Omega_r)(\\alpha_{\\rM}\/\\Omega_{G_3})]\/4$ \nunder the condition (\\ref{sccon}).\nThen, the sound speed squared $c_s^2 \\simeq (2+\\epsilon_{\\phi})\/3$ can be \nestimated as\n\\be\nc_s^2 \\simeq \\frac{1}{12} \\left[ 5+\\Omega_r+\\frac{\\alpha_{\\rM}}{\\Omega_{G_3}}\n(1-\\Omega_r) \\right]\\,,\n\\label{csfd}\n\\ee\nwhich is valid for $z \\gg 10$. As $\\Omega_r$ starts to deviate from 1 \nin the late radiation era, \n$c_s^2$ decreases from the initial value close to $1\/2$. \nSince the ratio $\\alpha_{\\rM}\/\\Omega_{G_3}$ grows \nin the deep matter era, the term $(\\alpha_{\\rM}\/\\Omega_{G_3})(1-\\Omega_r)$ in \nEq.~(\\ref{csfd}) starts to increase the value of $c_s^2$. \nIndeed, in the numerical simulation of Fig.~\\ref{fig6}, $c_s^2$ \nreaches the minimum value $0.430$ around $z=365$.\n\nIn Fig.~\\ref{fig5}, we observe that $\\Omega_V$, \n$\\Omega_{G_3}$, and $\\Omega_K$ asymptotically \napproach constants with $\\Omega_V ={\\cal O}(1) \\gg \\Omega_{G_3} \\gg \\Omega_K$. \nIn the regime where \n$\\Omega_V$ dominates over $\\Omega_{G_3}$, \n$\\Omega_K$, and $\\Omega_r$, \nEq.~(\\ref{auto1}) approximately reduces to  \n\\be\nx' \\simeq \\frac{x}{4} \\left[ 3(1-3\\Omega_V) \n-\\frac{2\\sqrt{6} x}{\\Omega_{G_3}} \\{ Q+(3Q+\\lambda)\\Omega_V \\} \n\\right]\\,.\n\\label{dxeq}\n\\ee\nThen, the solutions approaching a nonvanishing constant  $x$ is given by \n\\be\nx \\simeq -\\frac{\\sqrt{6} (3\\Omega_V-1)}\n{4[Q+(3Q+\\lambda)\\Omega_V]} \\Omega_{G_3}\\,.\n\\label{xest}\n\\ee\nSubstituting this relation into Eqs.~(\\ref{auto2}) and \n(\\ref{auto3}), it follows that \n\\ba\n\\Omega_V' &\\simeq& 3\\Omega_V \n\\left( 1-\\Omega_V \\right)\\,,\\label{OmeVe} \\\\\n\\Omega_{G_3}' &\\simeq& \n3 \\left( \\Omega_V-1 \\right)\\Omega_{G_3}\\,,\n\\label{OmeG3es}\n\\ea\nwhich can be integrated to give\n\\ba\n\\Omega_V &\\simeq& \n\\left( 1+c_1 a^{-3} \\right)^{-1}\\,, \\label{Ome1late} \\\\\n\\Omega_{G_3} &\\simeq& \nc_2 \\left( 1+c_1 a^{-3} \\right)\\,,\n\\label{OmeG3late}\n\\ea\nwhere $c_1$ and $c_2$ are constants.\nThese solutions are valid only at the very late \ncosmological epoch in which $x$ starts to approach a constant.\n{}From Eqs.~(\\ref{Ome1late}) and (\\ref{OmeG3late}), $\\Omega_V$ and \n$\\Omega_{G_3}$ approach \nthe values 1 and $c_2$, respectively. \nTaking the limit $\\Omega_{V} \\to 1$ in Eq.~(\\ref{xest}), \nwe can estimate the asymptotic values of $\\alpha_{\\rM}$ \nand the ratio $\\Omega_K\/\\Omega_{G_3}$, as \n\\ba\n\\alpha_{\\rM} &=& \\frac{6Q}\n{4Q+\\lambda} \\Omega_{G_3}\\,,\\label{aMas}\\\\\n\\frac{\\Omega_K}{\\Omega_{G_3}} &=& \n\\frac{3}{2(4Q+\\lambda)^2} \\Omega_{G_3}\\,.\n\\label{aMas2}\n\\ea\nThey are in good agreement with the numerical values in \nFig.~\\ref{fig5}, i.e., $\\alpha_{\\rM}=5.24 \\times 10^{-5}$ and  \n$\\Omega_K\/\\Omega_{G_3}=9.37 \\times 10^{-5}$ with \n$\\Omega_{G_3}=1.22 \\times 10^{-4}$, so the condition \n$\\Omega_{G_3} \\gg \\Omega_K$ is satisfied. \nWe note that, for the other solution $x=0$ in Eq.~(\\ref{dxeq}), \n$\\Omega_{G_3}$ approaches 0, so this does not lead to the solution \nwith $\\Omega_{G_3} \\gg \\Omega_K>0$.\n\nIn the numerical simulation of Fig.~\\ref{fig5}, today's value of \n$\\alpha_{\\rM}$ is $3.38 \\times 10^{-5}$ and hence this case \nis within the LLR bound (\\ref{aMcon}). \nOn using Eq.~(\\ref{aMas}), the criterion for consistency \nwith the LLR experiment is that the asymptotic value of \n$\\Omega_{G_3}$ is in the range, \n\\be\n\\frac{6Q}{4Q+\\lambda}\n\\Omega_{G_3} \\le 7 \\times 10^{-5}\\,.\n\\ee\nThis is a sufficient condition, so the actual upper bound on \n$\\Omega_{G_3}$ is slightly tighter. \nUnlike the case discussed in Sec.~\\ref{unsec}, \nthe coupling $Q$ is not particularly bounded from above.\nIndeed, the numerical simulation of Fig.~\\ref{fig5} \ncorresponds to $Q=0.1$, but the LLR bound is satisfied. \nThis property comes from the fact that the \ncubic Galileon term suppresses the field kinetic energy \nthrough the cosmological Vainshtein screening, so that the variable \n$x$ in $\\alpha_{\\rM}=-2\\sqrt{6}Qx$ is restricted to be small.\nWe note that, even though $\\Omega_K \\ll \\Omega_{G_3}$, \n$\\Omega_{G_3}$ is much smaller than $\\Omega_V$, so \nthe cubic Galileon is sub-dominant as the dark energy density. \n\nThe asymptotic value of $\\epsilon_{\\phi}$ in the future \nis close to $h~(\\simeq 0)$ to realize $x'=0$ with $x \\neq 0$ in Eq.~(\\ref{auto1}). \nThen, the scalar propagation speed squared \nshould approach the value \n$c_s^2 \\simeq (2+\\epsilon_{\\phi})\/3 \\simeq 2\/3$, \nwhich is indeed the case for the numerical simulation \nin Fig.~\\ref{fig6}. Since the condition $c_s^2>0$ \nis satisfied from the radiation dominance to the future, \nthere is no Laplacian instability of scalar perturbations. \n\nThe numerical simulation of Fig.~\\ref{fig6} corresponds to $\\lambda=1$, but \n$w_{\\rm DE}$ is very close to $-1$ even in the asymptotic \nfuture. This behavior is different from the case (D) in \nFig.~\\ref{fig4} where the solutions finally \nreach the fixed point (b) with the large deviation \nof $w_{\\rm DE}$ from $-1$. \nIn the screened cosmology discussed in this section, \nthe future asymptotic solution is characterized by \nEqs.~(\\ref{aMas}) and (\\ref{aMas2}) with the strongly \nsuppressed kinetic energy ($\\Omega_K \\ll \n\\Omega_{G_3} \\ll \\Omega_V \\simeq 1$). \nIn this case, the dark energy equation of state is \ngiven by Eq.~(\\ref{wdetra}) with $h \\simeq 0$ \nin the asymptotic future and hence \n$w_{\\rm DE} \\simeq -1$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.2in,width=3.5in]{fig7.pdf}\n\\end{center}\n\\caption{\\label{fig7}\nEvolution of $w_{\\rm DE}$ versus $z+1$ for $\\lambda=2$ \nfor the same initial conditions of $x$, $\\Omega_V$, \n$\\Omega_{G_3}$, $\\Omega_r$, and $y$ as those \nused in Fig.~\\ref{fig5}.\nEach case correspond to (A) $Q=0.153$, \n(B) $Q=0.010$, and (C) $Q=0.001$.}\n\\end{figure}\n\nSince the cosmological Vainshtein screening for the \nfield kinetic energy efficiently works for $\\beta_3 \\gg 1$, \nit is possible to realize $w_{\\rm DE}$ close to $-1$ \nat low redshifts even for $\\lambda>\\sqrt{2}$. \nIn Fig.~\\ref{fig7}, we plot the evolution of $w_{\\rm DE}$ for \n$\\lambda=2$ with three different values of $Q$, all of which \ncorrespond to $\\beta_3 \\simeq 1.0 \\times 10^7$. \nEven with $\\lambda$ larger than $\\sqrt{2}$, $w_{\\rm DE}$ is\nvery close to $-1$ from the redshift $z \\approx {\\cal O}(10)$ toward the asymptotic future. \nFor decreasing $Q$, the deviation of \n$F=e^{-2Q (\\phi-\\phi_0)\/M_{\\rm pl}}$ from 1 \ntends to be smaller in the past and hence the solutions \nenter the regime $\\Omega_V F> 1-F$ at earlier time.\nThen, from Eq.~(\\ref{wdetra}), the approach of $w_{\\rm DE}$ \nto $-1$ occurs at higher redshifts. \nIn case (A) of Fig.~\\ref{fig7} we have \n$\\alpha_{\\rM}(t_0)=6.98 \\times 10^{-5}$, so this is \nclose to the LLR upper limit (\\ref{aMcon}). \nFor decreasing $Q$ with given values of $\\beta_3$ and $\\lambda$, \n$\\alpha_{\\rM}(t_0)$ gets smaller, \ne.g., $\\alpha_{\\rM}(t_0)=3.87 \\times 10^{-6}$ and $\\alpha_{\\rM}(t_0)=3.82 \\times 10^{-7}$ in cases (B) and (C) of Fig.~\\ref{fig7}, respectively. \nFor smaller $\\alpha_{\\rM}(t_0)$, the models mimic the \n$\\Lambda$CDM behavior ($w_{\\rm DE}=-1$) from earlier \ncosmological epochs to today.\n\n\\section{Modified gravitational wave propagation}\n\\label{GWsec}\n\nIn this section, we study the modified GW propagation induced \nby the nonminimal coupling $F(\\phi)R$ and compute the difference between \nGW and luminosity distances for the dark energy cosmology \ndiscussed in Sec.~\\ref{numesec}.\nThe perturbed line element containing tensor perturbations $h_{ij}$ \non the flat FLRW background is given by \n\\be\n{\\rm d} s^2=-{\\rm d}t^2+a^2(t) \n\\left( \\delta_{ij}+h_{ij} \\right) \n{\\rm d} x^i {\\rm d}x^j\\,.\n\\ee\nTo satisfy the transverse and traceless conditions \n$\\partial^j h_{ij}=0$ and ${h_i}^i=0$, we choose the nonvanishing \ncomponents of $h_{ij}$, as $h_{11}=h_1(t,z)$, $h_{22}=-h_1(t,z)$ \nand $h_{12}=h_{21}=h_2(t,z)$.\nExpanding the action (\\ref{action}) up to quadratic order \nin $h_{ij}$ and integrating it by parts, the resulting \nsecond-order action of tensor perturbations yields \\cite{Horn2,DT12,KT18}\n\\be\n{\\cal S}_t^{(2)}=\\int {\\rm d}t\\,{\\rm d}^3x \\sum_{i=1}^2\n\\frac{M_{\\rm pl}^2}{4} F(\\phi) a^3 \n\\left[ \\dot{h}_i^2- \\frac{1}{a^2} (\\partial h_i)^2 \n\\right]\\,.\n\\label{St2}\n\\ee\nIn general, the speed $c_t$ of tensor perturbations appears \nas the spatial derivative term \n$-(c_t^2\/a^2)(\\partial h_i)^2$ in the square bracket \nof Eq.~(\\ref{St2}). In our theory $c_t^2$ is equivalent to 1, so \nit automatically satisfies the observational bound of \nGW propagation speed \\cite{GW170817}.\n\nIn Fourier space with the coming wavenumber $k$, \nthe two polarization modes $h_i$ (where $i=1, 2$) \nobey the wave equation,\n\\be\n\\ddot{h}_i+H \\left( 3+\\alpha_{\\rM} \\right)\\dot{h}_i\n+\\frac{k^2}{a^2}h_i=0\\,.\n\\label{hieq}\n\\ee\nBy defining \n\\be\n\\hat{h}_i  \\equiv a_{{\\rm GW}} h_i\\,,\\qquad  \na_{\\rm GW} \\equiv \\sqrt{F}a\\,,\n\\ee\nEq.~(\\ref{hieq}) can be expressed in the form \n\\be\n\\frac{{\\rm d}^2\\hat{h}_i}{{\\rm d}\\eta^2}\n+\\left( k^2 -\\frac{1}{a_{\\rm GW}} \n\\frac{{\\rm d}^2 a_{\\rm GW}}{{\\rm d} \\eta^2}\n\\right) \\hat{h}_i=0\\,,\n\\label{hieq2}\n\\ee\nwhere $\\eta=\\int a^{-1} dt$ is the\nconformal time.\n\nFor the physical wavelength much smaller than the Hubble radius \n($k\/a \\gg H$), the second term in the parenthesis of \nEq.~(\\ref{hieq2}) can be ignored relative to $k^2$.\nThen, the solution to Eq.~(\\ref{hieq2}) \nis simply given by a plane wave with a constant \namplitude ($\\hat{h}_i \\simeq e^{\\pm i k\\eta}$).\nThe amplitude of $h_i=\\hat{h}_i\/a_{\\rm GW}$ decreases \nin proportion to $1\/a_{\\rm GW}$. \nThe GW produced by a binary inspiral (point particles \nwith two masses $m_1$ and $m_2$) at redshift \n$z$ with the comoving distance $r$ from \nan observer has the amplitude \\cite{Michele}:\n\\be\nh_A(z)=\\frac{4}{a(t_s)r} \n\\left( \\frac{G_{\\rm N}(t_s) M_c}{c^2} \\right)^{5\/3} \n\\left( \\frac{\\pi f_s}{c} \\right)^{2\/3}\\,,\n\\label{hA}\n\\ee\nwhere $t_s$ is the time at emission, $G_{\\rm N}(t_s)=G\/F(t_s)$ \nis the screened gravitational coupling at $t=t_s$\nwith $G=1\/(8 \\pi M_{\\rm pl}^2)$, \n$M_c=(m_1 m_2)^{3\/5}\/(m_1+m_2)^{1\/5}$ \nis the chirp mass, and $f_s$ is the \nfrequency measured by the clock of source. \nWe note that the speed of light $c$ is explicitly written \nin Eq.~(\\ref{hA}).\nToday's GW amplitude $h_A(0)$ observed at time $t_0$ \nis related to $h_A(z)$, as \n$h_A(0)=[a_{\\rm GW}(t_s)\/a_{\\rm GW}(t_0)]h_A(z)$.  \nOn using the property $a_{\\rm GW}(t_0)=a (t_0)$, \nit follows that \n\\be\nh_A(0)=\\frac{a_{\\rm GW}(t_s)}{a(t_s)} \n\\frac{1}{F(t_s)^{5\/3}}\nh_{A,{\\rm GR}}(0)\\,,\n\\label{hA0}\n\\ee\nwhere \n\\be\nh_{A,{\\rm GR}} (0)=\\frac{4}{a(t_0)r} \n\\left( \\frac{G M_c}{c^2} \\right)^{5\/3} \n\\left( \\frac{\\pi f_s}{c} \\right)^{2\/3}\n\\label{hGR}\n\\ee\nis the observed GW amplitude in GR.\nOn the flat FLRW background, the luminosity distance \nfrom the observer to the source is given by \n$d_L(z)=(1+z)a(t_0)r$.\nBy using $d_L(z)$ and the observed GW frequency \n$f_{\\rm obs}=f_s\/(1+z)$, one can write Eq.~(\\ref{hGR}) in the form \n\\be\nh_{A,{\\rm GR}} (0)=\\frac{4}{d_L(z)} \n\\left( \\frac{G {\\cal M}_c}{c^2} \\right)^{5\/3} \n\\left( \\frac{\\pi f_{\\rm obs}}{c} \\right)^{2\/3}\\,,\n\\label{hGR2}\n\\ee\nwhere ${\\cal M}_c \\equiv (1+z)M_c$. \nSubstituting Eq.~(\\ref{hGR2}) into Eq.~(\\ref{hA0}), \nthe observed GW amplitude is expressed as\n\\be\nh_{A} (0)=\\frac{4}{d_{\\rm GW}(z)} \n\\left( \\frac{G_{\\rm N}(t_s) {\\cal M}_c}{c^2} \\right)^{5\/3} \n\\left( \\frac{\\pi f_{\\rm obs}}{c} \\right)^{2\/3}\\,,\n\\label{hGR3}\n\\ee\nwhere \n\\be\nd_{\\rm GW}(z)=d_L(z) \\frac{a(t_s)}{a_{\\rm GW}(t_s)}\n=\\frac{d_L(z)}{\\sqrt{F(t_s)}}\\,.\n\\label{dGWL}\n\\ee\nOn using Eq.~(\\ref{alM}), the quantity $F$ at redshift \n$z$ is generally expressed as   \n\\be\nF(z)=\\exp \\left[ -\\int_0^z \\frac{\\alpha_{\\rM} (\\tilde{z})}\n{1+\\tilde{z}} {\\rm d} \\tilde{z} \\right]\\,.\n\\ee\nThen, the relative ratio between $d_{\\rm GW}(z)$ and \n$d_L(z)$ yields \n\\be\n\\frac{d_{\\rm GW}(z)}{d_L(z)}=\n\\exp \\left[ \\int_0^z \\frac{\\alpha_{\\rM} (\\tilde{z})}\n{2(1+\\tilde{z})} {\\rm d} \\tilde{z} \\right]\\,.\n\\label{dgLra}\n\\ee\nIf $\\alpha_{\\rM}(z)>0$, then $d_{\\rm GW}(z)>d_L(z)$ for $z>0$. \nFor positive $\\alpha_{\\rM}(z)$, which is the case for our \nnonminimally coupled dark energy scenario, there is the \nLLR bound $\\alpha_{\\rM}(0) \\le \\alpha_{\\rm max}$, \nwhere $\\alpha_{\\rm max}=7 \\times 10^{-5}$.\nProvided that the past value of $\\alpha_{\\rM}(z)$ \nis smaller than $\\alpha_{\\rM}(0)$, the ratio (\\ref{dgLra}) \nis in the range \n\\be\n\\frac{d_{\\rm GW}(z)}{d_L(z)} \\le \n\\left( 1+z \\right)^{\\alpha_{\\rm max}\/2}\\,.\n\\label{dgLra2}\n\\ee\nExpanding the term \n$\\left( 1+z \\right)^{\\alpha_{\\rm max}\/2}$ around \n$\\alpha_{\\rm max}=0$, it follows that \n\\be\n\\mu_d(z) \\equiv\n\\frac{d_{\\rm GW}(z)}{d_L(z)}-1 \\lesssim \n\\frac{\\alpha_{\\rm max}}{2}  \\ln \n\\left( 1+z \\right)\\,,\n\\label{dgLra2}\n\\ee\nwhere we ignored the terms \nhigher than the order $\\alpha_{\\rm max}$.\nSubstituting $\\alpha_{\\rm max}=7 \\times 10^{-5}$ into \nthe right hand side of Eq.~(\\ref{dgLra2}), we have \n$(\\alpha_{\\rm max}\/2) \\ln \\left( 1+z \\right)=1.6 \\times 10^{-4}$ \nat $z=100$.\nThen, the quantity $\\mu_d (z)$ is constrained to be\n\\be\n\\mu_d(z)\n\\lesssim 10^{-4}\\,,\\qquad \n({\\rm for}~0<z<100)\\,.\n\\label{dgLra3}\n\\ee\nThis is the maximum allowed difference between \n$d_{\\rm GW}(z)$ and $d_L(z)$ constrained from \nthe LLR data. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.2in,width=3.5in]{fig8.pdf}\n\\end{center}\n\\caption{\\label{fig8}\nThe relative difference $d_{\\rm GW}(z)\/d_L(z)-1$ versus $z$ \ncorresponding to the cases (A), (B), (C), (D) shown  \nin Fig.~\\ref{fig4}.}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[height=3.2in,width=3.5in]{fig9.pdf}\n\\end{center}\n\\caption{\\label{fig9}\nThe relative difference $d_{\\rm GW}(z)\/d_L(z)-1$ versus $z$ \ncorresponding to the cases (A), (B), (C) shown  \nin Fig.~\\ref{fig7}.}\n\\end{figure}\n\nFor concreteness, let us consider the nonminimally \ncoupled dark energy scenario given by the action (\\ref{action}).\n{}From Eq.~(\\ref{dGWL}), we have \n\\be\n\\frac{d_{\\rm GW}(z)}{d_L(z)}\n=e^{Q[\\phi (z)-\\phi_0]\/M_{\\rm pl}}\\,.\n\\label{dra}\n\\ee\nThe change of $\\phi$ from the redshift $z$ to \ntoday leads to the difference between \n$d_{\\rm GW}(z)$ and $d_L(z)$. \nAs we studied in Sec.~\\ref{numesec},  there are two \nqualitative different cases: (i) $|\\beta_3| \\ll 1$ \nwith the $\\phi$MDE, and (ii) $|\\beta_3| \\gg 1$ \nwithout the $\\phi$MDE. \n\nIn case (i), the LLR data place the tight upper \nlimit (\\ref{Qup2}) on the coupling constant $Q$. \nIn Fig.~\\ref{fig8}, we plot $\\mu_d(z)=d_{\\rm GW}(z)\/d_L(z)-1$ in the redshift range $0<z<100$ for four different combinations of $Q$ \nand $\\lambda$. Each plot corresponds to cases (A), (B), (C), (D) \nshown in Fig.~\\ref{fig4}. In all these cases, \nthe LLR bound is marginally satisfied, i.e., \n$\\alpha_{\\rM}(0) \\simeq 7 \\times 10^{-5}$.\nFor the redshift $z<1$, the values of $\\mu_d(z)$ are \nsimilar to each other between the four cases,  \nwith  $\\mu_d \\simeq 1.5 \\times 10^{-5}$ at $z=1$.\nThe difference starts to appear for $z>1$, but the orders \nof $\\mu_d(z)$ at $z=100$ are still $10^{-5}$. \nAs we estimated in Eq.~(\\ref{aMa}), the value of \n$\\alpha_{\\rM}$ during the $\\phi$MDE \nis of order $4Q^2$ and hence \n$\\alpha_{\\rM}^{({\\rm a})} \\le 4.6 \\times 10^{-5}$ \nunder the bound (\\ref{Qup2}).\nSince $\\alpha_{\\rM}^{({\\rm a})}$ is\nsmaller than today's value $\\alpha_{\\rM}(0)$, \nthe main contribution to the ratio (\\ref{dgLra}) \ncomes from $\\alpha_{\\rM}(z)$ at low redshifts.\nSince $\\alpha_{\\rM}(z)$ at $z \\le 1$ is not much \ndifferent from today's value \n$\\alpha_{\\rM}(0) \\simeq 7 \\times 10^{-5}$ in \nthe numerical simulation of Fig.~\\ref{fig8}, \nthe maximum value of $\\mu_d$ for $z \\gg 1$\ncan be estimated by substituting $z=1$ \ninto Eq.~(\\ref{dgLra2}), i.e., \n$\\mu_d \\lesssim {\\cal O}(10^{-5})$.\nIndeed, this crude estimation is consistent with \nthe numerical values of $\\mu_d$ at $z \\gg 1$ \nin Fig.~\\ref{fig8}.\nIf $\\alpha_{\\rM}(0)$ is smaller than $7 \\times 10^{-5}$, \nthe resulting values of $\\mu_d$ at high redshifts are \nless than the order $10^{-5}$.\n\nIn case (ii), the upper limit of $Q$ is not particularly constrained from the LLR experiment, but the cosmological Vainshtein screening leads to the strong suppression of $\\dot{\\phi}$. \nThe case (A) in Fig.~\\ref{fig9}, which corresponds to \n$Q=0.153$ and $\\lambda=2$, is marginally within the\nLLR bound. In this case, the value of $\\alpha_{\\rM}$ \nfor $z \\gg 1$ is of order $10^{-5}$. \nAs we see in Fig.~\\ref{fig5}, $\\alpha_{\\rM}$ \nrapidly decreases toward the asymptotic past  \nand hence the main contribution to $\\mu_d(z)$ again \ncomes from $\\alpha_{\\rM}(z)$ \nat $z \\le {\\cal O}(1)$.\nIn cases (B) and (C) of Fig.~\\ref{fig9}, which \ncorrespond to the couplings $Q=0.01$ and $Q=0.001$, \ntoday's values of $\\alpha_{\\rM}$ are smaller than\nthat in case (A) by one and two orders of magnitude, \nrespectively.\nIn cases (B) and (C), the numerical values of $\\mu_d(z)$ \nat $z=100$ are $1.1 \\times 10^{-6}$ and $1.1 \\times 10^{-7}$, \nrespectively, so the order difference of $\\alpha_{\\rM}(0)$ directly affects $\\mu_d$ at high redshifts. \n\n{}From the above discussion, we have $\\mu_d(z) \\le {\\cal O}(10^{-5})$ \nfor $0<z<100$ in both unscreened and screened cosmological backgrounds. \nThis property is mostly attributed to the fact that \nthe value of $\\alpha_{\\rM}$ at low redshifts is tightly \nlimited by the LLR bound.\nUnless the ratio $d_{\\rm GW}(z)\/d_L(z)$ is measured in \nhigh accuracy, it is challenging to observationally \ndistinguish nonminimally coupled theories from minimally coupled theories.\n \n\\section{Conclusions}\n\\label{consec}\n\nWe studied how the recent LLR measurement constrains\nnonminimally coupled dark energy models given by the action (\\ref{action}). \nThe existence of nonminimal coupling of the form $F(\\phi)R$, where \n$F(\\phi)=e^{-2Q(\\phi-\\phi_0)\/M_{\\rm pl}}$, gives rise to the \npropagation of fifth forces characterized by the coupling constant \n$Q$ with nonrelativistic matter. For a massless scalar field without \nderivative interactions, the coupling is constrained to be \nin the range $|Q| \\le 2.4 \\times 10^{-3}$ from the Cassini experiment. \nThe cubic Galileon coupling $\\beta_3 M^{-3} X \\square \\phi$ \nallows one to recover the Newtonian behavior in over-density regions \neven for $|Q|>2.4 \\times 10^{-3}$. \nSince the late-time dominance of Galileons as the dark energy density\ngenerally leads to the incompatibility with observations, \nwe considered the potential $V(\\phi)$ of a light scalar field.\n\nIn local regions of the Universe, the Galileon self-interaction screens \nfifth forces within the Vainshtein radius (\\ref{rV}).\nThe Vainshtein mechanism is at work within the solar system \nfor the cubic coupling in the range $|\\beta_3 Q| \\gg 10^{-17}$.\nIn spite of the screened scalar-matter interaction, \nthe time variation of $\\phi$ associated with the dynamics \nof dark energy survives in the expression of gravitational \ncoupling $G_{\\rm N}$ in over-density regions, with the form \n$G_{\\rm N}=1\/[8\\pi M_{\\rm pl}^2F(\\phi)]$. \nThe recent LLR data placed the tight constraint \n(\\ref{Gbou}) on the time variation of $G_{\\rm N}$, \nwhich translates to the bound (\\ref{aMcon}) on today's \nvalue of $\\alpha_{\\rM}=\\dot{F}\/(HF)$.\n\nTo investigate the evolution of $\\alpha_{\\rM}$ as well as field density parameters $\\Omega_K, \\Omega_V, \\Omega_{G_3}$, we expressed dynamical equations of motion on the flat FLRW background in the \nautonomous form given by (\\ref{auto1})-(\\ref{auto4}).\nIn addition to the dark energy equation of state $w_{\\rm DE}$, \nwe also considered the quantities $q_s$ and $c_s^2$ to ensure \nthe absence of ghosts and Laplacian instabilities.\nTogether with Eq.~(\\ref{dy}), the dynamical \nbackground equations of motion can be applied to any scalar \npotential $V(\\phi)$.\n\nIn Sec.~\\ref{numesec}, we studied the cosmological dynamics \nin details for the exponential potential (\\ref{exp}). \nFor the cubic coupling satisfying the condition (\\ref{local}), \n$\\Omega_{G_3}$ can dominate over $\\Omega_K$ in the \nradiation-dominated epoch. \nWe showed that, under the conditions \n$|\\alpha_{\\rM}| \\gg \\Omega_{G_3}$ and \n$|\\alpha_{\\rM}| \\ll \\Omega_{G_3}$, the field density parameters \nand $ |\\alpha_{\\rM}|$ evolve as Eqs.~(\\ref{Omeevo}) and (\\ref{den1}), \nrespectively, during the radiation era. \nAfter the onset of matter dominance, there are two qualitatively \ndifferent cases: (i) unscreened cosmology with \n$|\\beta_3| \\ll 1$, and (ii) screened cosmology \nwith $|\\beta_3| \\gg 1$.\n\nIn case (i), there is the kinetically driven $\\phi$MDE \nin which $\\alpha_{\\rM}$ is given by \n$\\alpha_{\\rM}^{({\\rm a})}=4Q^2\/(1-2Q^2)$. \nThe solutions finally approach the fixed point (b) with \ncosmic acceleration at which $\\alpha_{\\rM}$ is \nequivalent to $\\alpha_{\\rM}^{({\\rm b})}\n=2Q(\\lambda+4Q)\/(1-Q \\lambda-4Q^2)$. \nFor positive $\\lambda$ and $Q$ the inequality\n$\\alpha_{\\rM}^{({\\rm b})}>\\alpha_{\\rM}^{({\\rm a})}>0$ holds, \nso the necessary condition for consistency with \nthe LLR bound (\\ref{aMcon}) corresponds to \n$\\alpha_{\\rM}^{({\\rm a})} \\le 7 \\times 10^{-5}$, i.e., \n$Q \\le 4.2 \\times 10^{-3}$. Applying today's bound \n$\\alpha_{\\rM}(t_0) \\le 7 \\times 10^{-5}$ to case (i), \nthe coupling is constrained to be \n$Q \\le 3.4 \\times 10^{-3}$ in the limit $\\lambda \\to 0$. \nAs we see in Fig.~\\ref{fig3}, for increasing $\\lambda$, \nthe upper bound on $Q$ is tighter than the bound \n$Q \\le 3.4 \\times 10^{-3}$.\nWe also showed that $w_{\\rm DE}$ temporally approaches the \nvalue close to $-1$ during the matter era after the dominance \nof the term $\\Omega_V F$ over $1-F$. \nFor larger $\\lambda$, the deviation of $w_{\\rm DE}$ from \n$-1$ on the attractor point (b) tends to be larger, see \nFig.~\\ref{fig4}.\n\nIn case (ii), the cosmological Vainshtein screening of field kinetic energy \nis at work, so the condition $\\Omega_K \\ll \\Omega_{G_3}$ \nis satisfied even after the end of radiation dominance.\nAs we observe in Fig.~\\ref{fig5}, $\\alpha_{\\rM}$ grows \nduring the matter era and finally approaches a constant \nrelated to $\\Omega_{G_3}$, as \n$\\alpha_{\\rM}=6Q \\Omega_{G_3}\/(4Q+\\lambda)$. \nProvided that this asymptotic value of $\\alpha_{\\rM}$ is \nsmaller than the order $10^{-4}$, the case (ii) can be \nconsistent with today's LLR bound (\\ref{aMcon}). \nSince $\\Omega_{G_3}$ is much smaller than\n$\\Omega_V$ today, the coupling $Q$ is not \nparticularly bounded from above. \nThe field kinetic energy is strongly suppressed by \nthe cosmological Vainshtein screening, i.e., \n$\\Omega_K \\ll \\Omega_{G_3} \\ll \\Omega_V$, so \nit is possible to realize $w_{\\rm DE}$ very close to $-1$  \nat low redshifts even for $\\lambda>\\sqrt{2}$, see \nFig.~\\ref{fig7}.\nThis behavior is different from that in case (i) where\n$w_{\\rm DE}$ deviates from $-1$ in the asymptotic future \nfor increasing $\\lambda$ in the range $\\lambda<\\sqrt{2}$.\n\nIn Sec.~\\ref{GWsec}, we derived the relation between the GW and luminosity distances in the form (\\ref{dGWL}). \nIn terms of the parameter $\\alpha_{\\rM}$, the ratio between \n$d_{\\rm GW}(z)$ and $d_L(z)$ is given by Eq.~(\\ref{dgLra}). \nProvided that $\\alpha_{\\rM}(z)$ in the past is smaller than \ntoday's value $\\alpha_{\\rM}(0)$, the LLR experiment gives the upper limit on \nthe relative difference $\\mu_d(z)=d_{\\rm GW}(z)\/d_L(z)-1$ \nas Eq.~(\\ref{dgLra2}). We computed the quantity $\\mu_d(z)$ \nfor the nonminimally coupled dark energy scenario discussed\nin Sec.~\\ref{numesec} and showed that $\\mu_d(z)$ for \n$z \\geq {\\cal O}(1)$ is mostly determined by today's value \nof $\\alpha_{\\rM}$. For $\\alpha_{\\rM}(0)$ \nclose to the LLR upper limit $7 \\times 10^{-5}$,  \n$\\mu_d(z)$ is of order $10^{-5}$ in the redshift range $1<z<100$. \nThis property is independent of the unscreened and \nscreened cosmological backgrounds, so the LLR constraint \ngives a tight restriction on the deviation of $d_{\\rm GW}(z)$ \nfrom $d_{L}(z)$ in nonminimally coupled theories.\n\nIn this paper we did not study the evolution of scalar cosmological \nperturbations relevant to the observations of large-scale structures and \nweak lensing, but it is straightforward to do so by using the linear \nperturbation equations of motion derived in Refs.~\\cite{Horn2,KT18,DKT}. \nIn the unscreened cosmological background the upper limit of \n$Q$ is tightly constrained from the  LLR experiment, \nso the effective gravitational \ncouplings felt by matter and light are close to $G_{\\rm N}$ \\cite{KT18}. \nIn the screened background not only $\\Omega_K$ but also \n$\\Omega_{G_3}$ is generally much smaller than 1 at low redshifts, so it is expected that the gravitational \ninteraction is not substantially modified from that in GR.\nAt the background level, the dark energy equations of state in the \nunscreened and screened cases exhibit some difference especially \nin the late cosmological epoch.\nIt will be of interest to place further constraints \non the allowed parameter space of our theory \nby exploiting the observational \ndata of cosmic expansion and growth histories.\n\n\n\\section*{Acknowledgments}\nThe author thanks Michele Maggiore for useful discussions.\nThe author is supported by the Grant-in-Aid for Scientific Research \nFund of the JSPS No.~16K05359 and MEXT KAKENHI Grant-in-Aid \nfor Scientific Research \non Innovative Areas ``Cosmic Acceleration'' (No.\\,15H05890). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Sec0}\nA pseudorandom number generator (PRNG) which is also known as a deterministic random number generator (DRNG) is an algorithm for producing a string of random numbers whose characteristics could represent same properties of strings of random numbers produced from a truly random number generator (TRNG). The obtained number series from this PRNG is not considered truly random, because it is estimated by a comparatively set of known initial values, called the PRNG's \\textbf{seed} \\cite{c1}. Although PRNG strings are very close to TRNG sequences, thus could be generate by the use of hardware random number generators. Therefore, PRNGs are fundamental in real world applications for the speed in generate the numbers and also for their reproducibility. PRNGs are central in many recent applications such as electronic games, numerical simulations, statistical research, randomized algorithms, cryptography and lottery.\n\nIn the year 1985, elliptic curve public-key cryptosystems (also known as elliptic curve cryptography (ECC)), were introduced by Neal Koblitz \\cite{c2} and Victor Miller \\cite{c3}. By substituting the subgroup of the multiplicative group $\\mathbb{Z}_{p}^{*}$ with the group of points on an elliptic curve (EC) over a finite field $\\mathbb{F}$, these cryptosystems could be considered as EC analogues of the traditional discrete logarithm ones. The security beyond these EC cryptosystems is proven by the computational intractability support of the elliptic curve discrete logarithm problem (ECDLP). Since the ECDLP appears to be significantly harder than the ordinary discrete logarithm problem, the strength-for-key-bit is ultimately greater in elliptic curve based systems than in classical discrete logarithm based systems. Thus, parameters with small sizes is used in the case of ECC morethan with discrete logarithm systems with almost equivalent levels of security \\cite{c4}. A worthy number of merits that could be acquired from the use of such small parameters include smaller key sizes, speed (faster computations) and secured certificates. Such features are important mostly in environments where bandwidth, storage space,  power of processing, or power consumption is bounded.\n\nEC pseudorandom number generator (EC-PRNG) is considered to be the EC analogue of the previously mentioned PRNG. EC-PRNG is based mathematically on the ECDLP and is counted a cryptographically secure pseudorandom number generator (CS-PRNG). The idea of making use of ECs to be source of randomness dates back to early work introduced by Kaliski \\cite{c5}. Since then, the use of elliptic curves for generating pseudorandom numbers has been\n studied for many years \\cite{c6,c7,c8}. EC-PRNG logically can be separated into two main parts: one part that generates a sequence of EC points and one that extracts a bit string from that point sequence as a source of pseudorandom bits of uniform distribution.\n\nIn this study, we proposed a method for generating pseudorandom bit strings based on an elliptic curve points operations (Add, Double, Multiply) over finite fields ($\\mathbb{F}_{p}$), which is secured by a hash function. The obtained pseudorandom bit string constructions are based on EC pseudorandom bit generator (EC-PRBG), which supported by a hash function to reinforce the security of an EC-PRNG. It is found that the statistical properties of the obtained bit strings can be greatly improved by applying this newly designed mechanism. Also, EC-PRBG mechanism is more suitable in producing pseudorandom bits in the EC-based digital signature algorithm (ECDSA) and session keys in related encryption stages.\n\nThis work is formulated as next. The preliminaries of EC are introduced in Section \\ref{Sec1}. An overview of elliptic curves over $\\mathbb{F}_{p}$ are mentioned in Section \\ref{Sec1.1}. In Section \\ref{Sec1.2}, the related EC point operations are presented. The cryptographic hash function is discussed in Section \\ref{Sec1.3}. In Section \\ref{Sec2}, the pseudorandom bit generator mechanism is proposed. Experimental and NIST test results are presented in Section \\ref{Sec3}. In Section \\ref{Sec4}, an image encryption  application with various security analysis is given and finally conclusions are given in Section \\ref{Sec5}. \n\n\n\\section{Preliminaries} \\label{Sec1}\n\n\\subsection{Elliptic Curves over $\\mathbb{F}_{p}$} \\label{Sec1.1}\n\nLets consider an EC to be $E$ over $\\mathbb{F}_{p}$, $p>3$, given by the following affine \\textit{Weierstrass} equation in the form \n\n\\begin{equation} \\label{EQ1.1}\nE: y^{2} =x^{3} + ax + b,  \n\\end{equation}\n\n\\noindent where $a$ and $b$ are coefficients belonging to $\\mathbb{F}_{p}$ such that $4a^{3} +27b^{2} \\ne 0$. The set of $\\mathbb{F}_{p}$ consists of all the points $(x,y), x \\in \\mathbb{F}_{p}, y \\in \\mathbb{F}_{p}$, which fulfill the defining equation (\\ref{EQ1.1}), together with a special\npoint $O$ called the point at infinity \\cite{c2}.\n\n\\textbf{Example 1.} Let a prime $p = 29$ and consider the elliptic curve $E: y^{2} =x^{3} + x + 4$ defined over $\\mathbb{F}_{29}$. This curve has order $33$ and is cyclic. Note that parameters values are $a = 1$ and $b = 4$ and $4a^{3} +27b^{2} = 4+ 432 = 436 \\equiv 1 \\: (\\text{mod} \\, 29)$, in this case $E$ is considered as an EC. Also points of $\\mathbb{F}_{29}$ are point $O$ and the other points which listed in Table \\ref{tab3.1}.\n\n\\begin{table}\n\\caption{Elliptic Curve Points in $\\mathbb{F}_{29}$}\n\\label{tab3.1}      \n\\begin{center} \n\\begin{tabular}{l|l|l|l|l|l|l|l}\n\\hline\\noalign{\\smallskip}\n$(0, 2)$   &  $(0, 27)$  &  $(1, 8)$   &  $(1, 21)$   &  $(3, 11)$  &  $(3, 18)$    \\\\             \n$(6, 9)$   &  $(6, 20)$  &  $(7, 8)$   &  $(7, 21)$   &  $(10, 12)$ &  $(10, 17)$   \\\\\n$(12, 2)$  &  $(12, 27)$ &  $(14, 6)$  &  $(14, 23)$  &  $(15, 1)$  &  $(15, 28)$   \\\\\n$(17, 2)$  &  $(17, 27)$ &  $(18, 5)$  &  $(18, 24)$  &  $(19, 3)$  &  $(19, 26)$  \\\\\n$(20, 7)$  &  $(20, 22)$ &  $(21, 8)$  &  $(21, 21)$  &  $(25, 9)$  &  $(25, 20)$  \\\\\n$(27, 9)$  &  $(27, 20)$  \\\\  \n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center} \n\\end{table}\n\n\\subsection{Elliptic Curve Point Operations} \\label{Sec1.2}\nThe addition operation of two points on an elliptic curve $E(\\mathbb{F}_{p})$ to result in a third point on same curve, the chord-and-tangent rule is used. With this addition operation, the set of points $E(\\mathbb{F}_{p})$ forms a group with $O$ serving as its identity. The formed group is used in the elliptic curve cryptosystem structure. Let $P=(x_{1} ,y_{1})$ and $Q=(x_{2} ,y_{2})$ to be two featured points on an elliptic curve $E$. The sum of such points $P$ and $Q$, denoted $P + Q = R =(x_{3} ,y_{3})$, is defined as next. First, a line is drawn through points $P$ and $Q$; this line intersects the same EC in a third point as drawn in Figure \\ref{Fig121}. As a result, point $R$ is the reflection of that process in the $x$-axis \\cite{c9}. \n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.43 \\textwidth]{Fig1.eps}\n\\caption{Description of two EC-points addition: $P + Q = R$}\n\\label{Fig121}       \n\\end{figure}\n\nConsider point $P=(x_{1} ,y_{1})$, the double of $P$ which is denoted $2P = R =(x_{3} ,y_{3})$, is obtained as follows. The tangent line with EC at point $P$ is drawn first. This line intersects the EC in a second point. As a result, point $R$ is the reflection of this process in the $x$-axis as shown in Figure \\ref{Fig122}. The sum of two points, also the double of one point is deduced from algebraic description as follows:\n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.43 \\textwidth]{Fig2.eps}\n\\caption{Description of one EC-point doubling: $P + P = 2P = R$}\n\\label{Fig122}       \n\\end{figure}\n\n\\begin{enumerate}\n\\item Note that $P + O = O + P$ \\quad all of $P \\in E(\\mathbb{F}_{p})$. \n\\item For point $P = (x, y) \\in E(\\mathbb{F}_{p})$, so that $(x, y)+ (x, -y) = O$, the point $(x, -y)$ is indicate as $-P$, so it is called the negative of $P$; notice that $-P$ is in fact a point on the same curve $E$. \n\\item Addition of points $P= (x_{1}, y_{1}) \\in E(\\mathbb{F}_{p})$ and $Q= (x_{2}, y_{2}) \\in E(\\mathbb{F}_{p})$, with $P \\neq \\pm Q$. The operation $P + Q = R = (x_{3} ,y_{3})$, where \n\n\\begin{equation} \\label{EQ1.2}\n\\begin{array}{c c c}x_{3} &=\\left(\\frac{y_{2} -y_{1} }{x_{2} -x_{1} } \\right)^{2} -x_{1} -x_{2}   \\\\\n\\text{and} \\\\\ny_{3} &=\\left(\\frac{y_{2} -y_{1} }{x_{2} -x_{1} } \\right)(x_{1} -x_{3} )-y_{1} \\\\ \\end{array} .\n\\end{equation}\n\n\\item Doubling of point $P= (x_{1}, y_{1}) \\in E(\\mathbb{F}_{p})$, with $P \\neq -P$. The operation $P + P = 2P = (x_{3} ,y_{3})$, where \n\n\\begin{equation} \\label{EQ1.3}\n\\begin{array}{c c c} x_{3} &=\\left(\\frac{3x_{1}^{2} +a}{2y_{1} } \\right)^{2} -2x_{1} \\\\\n \\text{and} \\\\\n y_{3} &=\\left(\\frac{3x_{1}^{2} +a}{2y_{1} } \\right)(x_{1} -x_{3} )-y_{1} \\\\  \\end{array} .\n\\end{equation}\n \n\\end{enumerate}\n\nNote that, the addition of these two points $P$ and $Q$ in $E(\\mathbb{F}_{p})$ needs some arithmetic operations such as (addition, subtraction, multiplication, and inversion) in the $\\mathbb{F}_{p}$ field .\n\n\\textbf{Example 2.} Let EC defined in Example $1$ used again.\n\\begin{enumerate}\n\\item Points $P = (3, 11)$ and $Q = (14, 23)$, so that $P + Q = (x_{3}, y_{3})$ can be computed as: \n\n\\begin{equation} \\label{EQ1.4}\n\\begin{array}{c c c} x_{3} =\\left(\\frac{23 - 11}{14 - 3} \\right)^{2} - 3- 14 = 64 \\equiv 6 (\\text{mod} \\; 29) \\end{array} \n\\end{equation}\n\n\\noindent and \n\n\\begin{equation} \\label{EQ1.5}\n\\begin{array}{c c c} y_{3} = 9(3- 6)- 11= -38 \\equiv 20 (\\text{mod} \\; 29) \\end{array} .\n\\end{equation}\n\nResulting $P+ Q= (6, 20)$\n\n\\item Point $P = (3, 11)$ which doubled as $2P = P+ P =(x_{3}, y_{3})$ can be computed as: \n\n\\begin{equation} \\label{EQ1.6}\n\\begin{array}{c c c}x_{3} =\\left(\\frac{3(3^2) + 1}{22} \\right)^{2} - 6 = 619 \\equiv 10 (\\text{mod} \\; 29) \\end{array}  \n\\end{equation}\n\n\\noindent and\n\n\\begin{equation} \\label{EQ1.7}\n\\begin{array}{c c c} y_{3} = 25(3- 10)- 11= -186 \\equiv 17 (\\text{mod} \\; 29) \\end{array} .\n\\end{equation}\n\nHence, $2P= (10, 17)$.\n\\end{enumerate}\n\n\\subsection{Secure Hash Function} \\label{Sec1.3}\nThe PRBG mechanism can be based on a non-invertible or one-way hash function \\cite{c1}. The hash-based EC-PRBG mechanism which used here is designed to make use of any suitable secure hash which is used by exhaustion applications that need different security strengths, providing that a suitable hash function is utilized and adequate entropy is gained for the seed value.\n\nThereafter, $H$ hash function is defined as:\n\n\\begin{equation} \\label{EQ1.3.1}\nH_i = f(X_i, H_{i-1}) \\quad i = 1,2, \\cdots, t\n\\end{equation}\n\nFrom the equation, $f$ be the round function, $H_t$ is the hashcode and $H_0$ is equal to an initial value (IV). For hash function safely usage ought to concur or exceed the in demand security strength of the exhaustion applications. \n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.48 \\textwidth]{fig1n.eps}\n\\caption{Pseudorandom Bit Generator Mechanism}\n\\label{Fig1N}       \n\\end{figure}\n\n\\section{The Proposed Random Bit Generator Mechanism} \\label{Sec2}\nThe proposed EC-PRBG is based on the hardness of ECDLP which can be described as: given two points $P$ and $Q$ on EC of order $n$, how to get $a$ such that $Q = aP$?. The EC-PRBG mechanism is depicted in Figure \\ref{Fig1N}.\n\nThe instantiation stage of the EC-PRBG mechanism requires choosing an appropriate EC and points on that curve for the required security strength. The seed value which is used to locate the initial value $(s)$ of the EC-PRBG must have enough bits of entropy with sufficient security strength. The value of $t$ is accounted for the seedlen-bit number in the initial state, so we can consider that $t = s_{0}$ in this case. The EC-PRBG can offer security strength as long as the security strength of the used curve. The main reason of using the hash function $H$ is to ensure that the entropy is distributed throughout the extracted bits, provided that they are verifiably random.\n\nBacktracking resistance in this mechanism is deep-seated, even in the case that the internal state is vulnerable to exposure. As shown in Figure \\ref{Fig2N}, EC-PRBG generates a seed value for each step $i = 1 ,2 ,3 ,\\cdots, $ as follows:\n\n\\begin{align*} \\label{EQ2.1}\ns_{i} &= \\varphi (x([s_{i-1}])P) \\quad , i=1,2,..., \\\\ \nH_{i} &= \\varphi (x(s_{i}),H_{i-1}) \\quad , i=1,2,...,  \n\\end{align*}\n\n\\noindent where $s_{0} \\in E(\\mathbb{F}_{p})$ is the \"initial value\". The EC-PRBG method represents an EC scalar multiplication with the extraction of the $x-$coordinate from the resulting points $s_{i}$ and from the random hash output $H_{i}$ with truncation operation to obtain the output pseudorandom bits. Following a line in the same direction of the arrow is the normal operation; inverting that direction reveals the ability to solve the ECDLP for that specific curve. The ability of an adversary to invert the arrow in Figure \\ref{Fig2N}, implies that the adversary has solved the ECDLP for that specific elliptic curve. Backtracking resistance is built into the mechanism design, as knowledge of $s_1$ does not allow an adversary to determine $s_0$ (and so forth) unless the adversary is able to solve the ECDLP for that specific curve. Furthermore, knowledge of $H_1$ does not allow an adversary to determine $s_1$ (and so forth) unless the adversary is able to solve the ECDLP for that specific curve.\n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.26 \\textwidth]{mad2.eps}\n\\caption{EC-PRBG Backtracking Resistance}\n\\label{Fig2N}       \n\\end{figure}\n\nThe EC-PRBG generates pseudorandom bit strings by extracting bits from an EC points. The internal state of the EC-PRBG is a secret value $s$ that represents the $x$-coordinate of a point on an EC. Output bits are produced by first computing $H$ to be the $x$-coordinate of the point $[s]P$, and then extracting low order bits from the $x$-coordinate of the hashcode output $H$.\n\n\\section{Experimental Results} \\label{Sec3}\nThe implementation of the presented EC-PRBG mechanism need to include an approved curve. Once the designer chooses the security level  required by a given application, he can then start the implementation of an EC that most NIST SP 800-90A \\cite{c1} appropriately meets this requirement. \n\n\\subsection{Implementation Example} \\label{Sec3.1}\nThe EC-PRBG algorithm allows an exhaustion application to instantiate using a prime curve. In accordance security key strengths of 112, 128, 192 and 256 bits may then be supported. The secure hash algorithm (SHA)-256 is chosen as the hash function. SHA-256 function generates an almost-unique, fixed size 256-bit hash. In this experiment, the implementation process used the following EC equation:\n\n \\begin{equation} \\label{EQ3.1.1} \n E: y^{2} =x^{3} + 4 x + 1               \n \\end{equation}\n\n \\noindent where $a = 4, b = 1$ are $E$ parameters over $\\mathbb{F}_{p}, p = 503$ and the cardinality of $E$ is $N = \\#E(\\mathbb{F}_{503}) = 516$. Also the generator point $G = (283, 315)$ of order $\\ell = 129$ is selected for our EC-PRBG mechanism.\n \n\n\\subsection{The NIST Randomness Tests} \\label{Sec3.2}\nThe NIST \\cite{c11} test suite is a statistical package that consisting of up to $15$ tests. It is developed for testing the randomness of binary strings obtained by either hardware or software based cryptographic pseudorandom and random number generators. The $15$ tests concentrate on a variety of different non-randomness types that could exist in a bit strings. The proposed mechanism produces a very random bit strings as reflected by the high p-values as shown in Table \\ref{NIST}. \n\n\n{\\begin{table} [htbp]\n\\caption{Test results for 1048576 bit strings}\n\\label{NIST}\n\\begin{center}\n\\begin{tabular}{| l | c | c |} \\hline\n\\textbf{Test-name}        & \\textbf{P-value} & \\textbf{Result} \\\\ \\hline\nBlock Frequency (m = 100)        & 0.046169  & Succeed     \\\\ \\hline\nFrequency                        & 0.681211  & Succeed     \\\\ \\hline\nCusum (Forward)                    & 0.878529  & Succeed     \\\\ \\hline\nCusum (Reverse)                    & 0.674391  & Succeed    \\\\ \\hline\nLong Runs of Ones                & 0.128851  & Succeed     \\\\ \\hline\nSpectral DFT                     & 0.149590  & Succeed     \\\\ \\hline\nRank                             & 0.638151  & Succeed     \\\\ \\hline\nLempel Ziv Complexity            & 1.000000  & Succeed      \\\\ \\hline\nOverlapping Templates (m = 9)    & 0.120402  & Succeed     \\\\ \\hline\nNonOverlapping Templates (m = 9) & 0.197506  & Succeed \\\\ \\hline\nApproximate Entropy (m = 10)     & 0.681211  & Succeed     \\\\ \\hline\nUniversal (L = 7, Q = 1280)      & 0.051599  & Succeed     \\\\ \\hline\nRandom Excursions (x = +1)       & 0.297235  & Succeed     \\\\ \\hline\nSerial (m = 16)                  & 0.343750  & Succeed     \\\\ \\hline\nRandom Excursions Variant (x = +1)& 0.050388 & Succeed     \\\\ \\hline\nRuns                             & 0.499889  & Succeed     \\\\ \\hline\nLinear Complexity (M = 500)      & 0.703017  & Succeed    \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}}\n\n\\section{Application In Image Encryption} \\label{Sec4}\nWith the huge amount of the data and high correlation between the adjacent pixels in images, stream ciphers are highly preferred over block ciphers in image encryption applications. The security of digital images need pseudorandom bit strings that have pretty good randomness properties and also high periodicity. Recently, several works that using ECs for digital and medical images encryption has been presented in literature such as \\cite{c12,c13,c14,c14a,c15}. In this work, the EC-PRBG is used for encrypting a $256 \\times 256$ grayscale of Lena image pixels as shown in Figure \\ref{Figmain}. Each image pixel has a $8$-bit value of between $0$ and $255$, so the pseudorandom bit strings in turn divided into blocks of $8$-bit each. Next, bitwise XOR operation is carried on every bit of the $8$-bit block. The resulted bits then grouped together to obtain the cipherimage. The decryption process is done vice-versa and the following security analysis is carried out.\n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.45 \\textwidth]{flowchart.eps}\n\\caption{EC-PRBG Based Image Encryption Schema}\n\\label{Figmain}       \n\\end{figure}\n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.44 \\textwidth]{enclenanew.eps}\n\\caption{Lena plainimage and the corresponding cipherimage with the EC-PRBG mechanism}\n\\label{Fig3}      \n\\end{figure}\n\n\\subsection{Entropy Analysis} \\label{Sec4.1}\nThe entropy $H(m)$ of a message source $m$ is calculated from the equation:\n\n\\begin{equation}\n\\label{EQ4.1.1}\nH{(m)} = - \\sum\\limits_{k=0}^{255} {P{(m_{k})}log_{2}}{P(m_{k})}\n\\end{equation}\n\n\\noindent where $P(m_{k})$ represents the probability of message $m_{k}$ \\cite{c16}. The various entropy values for Lena plain and encrypted image which shown in Figure \\ref{Fig3} are indicated in Table \\ref{tab3.2}. It is remarkable that the entropy of the encrypted image is too close to the theoretical value of $8$ which elucidate that all of the pixels in the encrypted image occur with almost equal probability. Therefore, the proposed EC-PRBG is secure against the entropy-based attack and the information leakage is negligible. \n\n{\\begin{table}[h!]\n\\caption{Entropy and basic parameters for Lena image}\n\\label{tab3.2}      \n\\begin{center}\n\\begin{tabular}{ l | c | l | l | l } \\hline\nScheme           & Entropy & PSNR     & MAE    & MSE      \\\\  \\hline\nProposed EC-PRBG & 7.9971  & 8.5631   & 79.52  & 9305.32  \\\\  \\hline\nRef\\cite{c18}    & 7.9898  & 8.5838   & -----   & 9009.33  \\\\  \\hline\nRef\\cite{c19}    & 7.9968  & 11.30    & 79.22  & 4859.03  \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}} \n\n\n\\subsection{Mean Absolute Error and Mean Square Error} \\label{Sec4.2}\nThe cipherimage must demonstrates a significant difference with it's corresponding plainimage. This difference can be measured by two major techniques, Mean Absolute Error (MAE) and Mean Square Error (MSE) \\cite{c17}. MAE and MSE values are calculated by using the following equations:\n\n\\begin{equation}\n\\label{EQ4.2.1}\nMAE = \\frac{1}{W*H} \\sum\\limits_{j=1}^{H} {\\sum\\limits_{i=1}^{W} {|(P_{ij}-C_{ij})|}}\n\\end{equation}\n\n\\begin{equation}\n\\label{EQ4.2.2}\nMSE = \\frac{1}{W*H} \\sum\\limits_{j=1}^{H} {\\sum\\limits_{i=1}^{W} {(P_{ij}-C_{ij})^2}}\n\\end{equation}\n\n\nIn equations (\\ref{EQ4.2.1}) and (\\ref{EQ4.2.2}), parameters $W$ and $H$ are the width and height of that image. Also $P_{ij}$ is the gray level of the pixel in the plainimage and $C_{i,j}$ is the gray level of the pixel in the cipherimage. MAE and MSE values of the cipherimage are reported in Table \\ref{tab3.2}. As shown from the table, MAE and MSE tests have produced  high values which then guarantee the resistance of the EC-PRBG mechanism against differential attacks.\n\n\\subsection{Peak Signal-to-noise Ratio (PSNR)} \\label{Sec4.3}\nPSNR is mainly used in image processing area as a consistent image quality metric \\cite{c20} and the greater PSNR, the better the output image quality. The performance of the proposed EC-PRBG method is estimated on the basis of PSNR and the measure values obtained are shown in Table \\ref{tab3.2}. The obtained results clearly illustrated that the EC-PRBG mechanism is well suited for many types of image encryption operations.\n\n\\subsection{Correlation Analysis} \\label{Sec4.4}\nFor any common image, two neighboring pixels in a plainimage are strongly correlated vertically, horizontally and diagonally. The maximum value of correlation coefficient is $1$ and the minimum value is $0$ \\cite{c21}. Horizontal, vertical and diagonal directions results are obtained as shown in Table \\ref{tab3.3} for plainimage of Lena and for it's ciphered image by the EC-PRBG method respectively. The obtained results elucidate that there is negligible correlation between the two adjacent pixels in the cipherimage, even when this two adjacent pixels in the plainimage are highly correlated as shown in Figure \\ref{Fig5}.\n\n\\subsection{Sensitivity Analysis} \\label{Sec4.5}\nIf one small change in a plainimage able to cause a significant change in the corresponding cipherimage, with respect to diffusion and confusion properties, then the known-plaintext attack actually loses its efficiency and becomes practically useless. To quantify that demand, two joint measures are used: Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) \\cite{c22}. The test results shown that the average values of the percentage of pixels changed in cipherimage is greater than 99.47\\% for NPCR and 30.48\\% for UACI for the pseudorandom bits. This means that the EC-PRBG method works perfectly and precisely with respect to small changes in the plainimage pixels.\n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.50 \\textwidth]{histlena.eps}\n\\caption{Histogram of Lena cipherimage with the EC-PRBG operation}\n\\label{Fig4}      \n\\end{figure}\n\n\\subsection{Histogram Analysis} \\label{Sec4.6}\nTo block the information leak to an adversary, an image encryption schema should always produce a cipherimage of the uniform histogram for all of the corresponding plainimage \\cite{c17}. The histograms for Lena plainimage and cipherimage are estimated. Lena plainimage histogram contains large spikes while the histogram of it's cipherimage is almost flat and uniform as depicted in Figure \\ref{Fig4} which denotes equal probability of occurrence of each pixel. Histogram of Lena cipherimage is remarkably different from the respective plainimage and consequently does not provide any evidence to appoint known statistical attacks on the image encryption application. \n\n\\begin{figure} [h!]\n\\centering\n\\includegraphics[width= 0.53 \\textwidth]{corrlena.eps}\n\\caption{Correlation of Lena plainimage and cipherimage with the EC-PRBG}\n\\label{Fig5}      \n\\end{figure}\n\n{\\begin{table}[h!]\n\\caption{Correlation coefficients for Lena image}\n\\label{tab3.3}      \n\\begin{center}\n\\begin{tabular}{ l | l | l | l } \\hline\nScheme           & Horizontal & Vertical   & Diagonal  \\\\ \\hline\nPlainimage       & 0.93915    & 0.96890    & 0.91686   \\\\ \\hline\nProposed EC-PRBG & -0.00287   & 0.03450    & 0.00196   \\\\ \\hline\nRef\\cite{c18}    & -0.00041   & -0.00025   & -0.000027   \\\\ \\hline\nRef\\cite{c19}    & -0.0043    & -0.0090    & -0.0031    \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}} \n\n\n\\section{Conclusion} \\label{Sec5}\nThis paper presented a new mechanism for generating pseudorandom bit strings based on elliptic curve group over finite fields ($\\mathbb{F}_{p}$) and hash derivation function to achieve high security levels. The performance analysis and security results showed that the obtained bit strings have high periodicity and good randomness properties. Moreover, an application in image encryption based on cipher bit strings stream was examined and various security analysis of the cipherimage is reported.\n\n\\section*{Acknowledgment}\nThe prepared work has been supported financially by Shaqra University, Saudi Arabia and Sohag University, Egypt.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nZipf's law constitutes a striking quantitative regularity \nin the usage of language \\cite{Baayen,Baroni2009,Zanette_book,Piantadosi}.\nIt states that, for a large enough piece of text, \nthe frequency of use $n$ of any word decreases with its\nrareness $r$ in the text in an approximately hyperbolic way, i.e., $n \\propto 1\/r$,\nwhere the symbol ``$\\propto$'' denotes proportionality. \nTechnically, $r$ is called the rank, and the most common (i.e., less rare) word is assigned $r=1$, \nthe second most common, $r=2$, and so on.\nA slightly more general formulation includes a parameter in the form of an exponent $\\alpha$; \nthen, the rank-frequency relation takes the form of a power law,\n\\begin{equation}\nn \\propto \\frac 1 {r^\\alpha}.\n\\label{laprimera}\n\\end{equation}\nwith the value of $\\alpha$ close to one. \n\n\nThis pattern (\\ref{laprimera})\nhas been found across different languages, literary styles, time periods, and\nlevels of morphological abstraction \\cite{Baroni2009,Zanette_2005,Corral_Boleda}.\nMore fascinatingly, the same law has been claimed in other codes of communication, as in \nmusic \\cite{Serra_scirep}\nor for the timbres of sounds \\cite{Haro},\nand also in disparate discrete systems where individual units or agents\ngather into different classes \\cite{Li02}, for example, \nemployees into firms \\cite{Axtell},\nbelievers into religions \\cite{Clauset},\ninsects into plants \\cite{Pueyo},\nunits of mass into animals present in ecosystems \\cite{Camacho_sole},\nvisitors or links into web pages \\cite{Adamic_Huberman},\ntelephone calls to users \\cite{Newman_05},\nor abundance of proteins (in a single cell) \\cite{Furusawa2003}.\nThe attempts to find an explanation have been diverse \\cite{Simon,Miller_monkey,Ferrer2002a,Mitz,Newman_05,Saichev_Sornette_Zipf,Zanette_book,Corominas-Murtra_2011,Peterson_Dill,Corominas_dice},\nbut no solution has raised consensus \\cite{Mitz,Cancho_monkey,Dickman_Moloney_Altmann}. \n\n\nDespite its quantitative character, \nZipf's law has been usually checked for in a qualitative way, \nplotting the logarithm of the frequency $n$\nversus the logarithm of the rank $r$\nand looking for some domain with a roughly linear behavior, \nwith slope more or less close to $-1$. \nA more refined approach consists in fitting a straight line\nto the double-logarithmic plot by linear regression\n\\cite{Li2010a}.\nBut several authors have recently pointed out the limitations\nof this method when applied to probability distributions\n\\cite{Goldstein,Bauke,White,Clauset}, \nand the advantages of using an asymptotically unbiased and minimum-variance procedure \nsuch as maximum likelihood (ML) estimation \\cite{White}, \nwhose solutions, moreover, are invariant under reparameterizations \\cite{Casella,Corral_Deluca}.\nOne should consider then ML estimation as the most reliable procedure\nof estimation for parametric models \n(when a maximum of the likelihood does exist \nand the number of data is large).\n\nFurthermore, for the particular case of linguistics, \nthe search for Zipf's law has been traditionally performed in very limited sets\nof texts (less than a dozen in a typical research article \\cite{FontClos_Corral,Corral_Boleda}). \nMore recently, however, large corpora have been considered\n--these are representative collections of different texts aggregated together\ninto a single bag, so, instead of many separated texts one deals with one enormous mixed text.\nWhen ``rare'' words are not considered\n(i.e., for high frequencies), it seems that \nZipf's law still holds in these large collections\n\\cite{Baroni2009,Ferrer2000a,Petersen_scirep,Gerlach_Altmann,Williams}.\nAt present, there is agreement that Zipf's law is a rough approximation\nin lexical statistics, but its range of validity is totally unknown, \ni.e., we ignore how good Zipf's law is in order to account for the appearance of words, \nand for which texts it should work --and with which level of precision-- \nand for which texts it should fail.\n\nAn extra difficulty emerges when one recognizes the ill-defined nature of Zipf's law.\nIn fact, the law has two formulations, with the first one being Eq. (\\ref{laprimera}), \nwhich just counts the frequency of words.\nFor the sake of clarity, the words that are counted are referred to as word types,\nin order to distinguish them from  each repetition, \nwhich is called a token.\nThe second formulation of Zipf's law arises when, after counting the frequency of word types, \none performs a second statistics and counts \nhow many values of the frequency are repeated,\nthat is, \nhow many word types have the same frequency.\nThis means that the frequency $n$ is considered the random variable.\nOne can realize that the rank, when normalized by its maximum value in text, \nis just the empirical estimation of the complementary cumulative distribution \nfunction of $n$, \nand then, the derivative of the expression for $r(n)$ (the inverse of Eq. (\\ref{laprimera}))\nyields a continuous approximation\nfor the probability mass function $f(n)$ of the frequency $n$.\nFrom here one obtains another power law,\n\\begin{equation}\nf(n) \\propto \\frac 1 {n^\\beta},\n\\label{lasegunda}\n\\end{equation}\nwith the new exponent $\\beta$ fulfilling $\\beta=1+1\/\\alpha$, \nwhich yields values of $\\beta$ close to 2.\nThe expression given by Eq. (\\ref{lasegunda})\nwas in fact the first approach followed by Zipf's himself\n\\cite{Zanette_book}, and\nis usually considered as equivalent \nto Eq. (\\ref{laprimera})\n\\cite{Adamic_Huberman,Newman_05,Zanette_book,Li02,FontClos_Corral};\nhowever, as it is derived in the continuum limit, \nboth expressions can only be \nequivalent asymptotically, for large $n$ \\cite{Mandelbrot61}.\nConsequently, if one wants to be precise, a natural question follows:\nwhich one is the ``true'' Zipf's law (if any)?\n\n\nWe cannot know a priori which of the two Zipf's laws better describes real texts, \nbut we can argue which of the two representations (that of $n(r)$, Eq.\\eqref{laprimera}, or that of $f(n)$, Eq.\\eqref{lasegunda})\nis better for statistical purposes,\nindependently of the functional dependency they provide.\nIt is clear that the rank-frequency representation, given by $n(r)$,\npresents several difficulties, due to the peculiar nature of the rank variable.\nFirst, in Ref. \\cite{Corral_Cancho}, Zipf-like texts were randomly generated\nfollowing Eq. (\\ref{laprimera}), keeping the original ranks ``hidden''\n(as it happens in the real situation), and it was found that the rank reconstructed \nfrom the sample deviated considerably from the original ranks when \nthese were taking large values\n(which for a power law happens with a high probability). \nThe resulting ML estimations of the exponent\n$\\alpha$ were highly biased and \nthe Kolmogorov-Smirnov test rejected the power-law hypothesis, \nalthough the original ranks were power-law indeed.\n\nOne might argue that the problem could be escaped by using an upper truncated power-law distribution \n(introducing then an additional parameter for the truncation), \nin order to avoid the inconsistency of the rank representation for high values.\nBut a second problem is that\nthe rank is not a true random variable \\cite{Kolmogorov}, \nas its values are assigned a posteriori,\nonce the sample (i.e., the text) is analyzed.\nThis means that the rank does not show ``enough'' statistical fluctuations, \nthat is, if $r_a < r_b$, then the frequency of $a$ is always larger, by construction, than the frequency of $b$. \nThis does not necessarily happen for a true random variable.\nThe negative correlation between the variable and its frequency of occurrence makes the \n{power-law hypothesis} harder to reject. \nIn fact, inflated $p$-values (not uniformly distributed between 0 and 1) \nhave been found when fitting truncated power laws to simulated power-law\nrank-frequency representations \\cite{Corral_Cancho}.\nThis problem could still be avoided by choosing a low enough upper truncation parameter\n(yielding a very short range of ranks, for which the fluctuations would be very little) \nbut at the expense of disregarding an important part of the data.\n\nA third inconvenience is the impossibility, due to normalization, that a non-truncated power law\ncomprises values of the $\\alpha-$exponent smaller than 1.\nThis yields the necessity of introducing a truncation parameter\nthat may be artifactual, i.e., not present in the real system.\nAll this leads to the conclusion that the most reliable method of parameter estimation \n(ML, in a frequentist framework) \ncannot be directly applied to the rank-frequency representation.\nIn contrast, the representation in terms of the distribution of frequencies\nis devoid of these problems \\cite{Corral_Cancho}, \nas $n$ is a well-defined random variable,\nand this will be the representation used in this paper for statistical inference.\nNevertheless, for alternative arguments, see Ref. \\cite{Altmann_Gerlach}.\n\n\n\n\nThe purpose of this paper is to quantify, at a large, big-data scale, \ndifferent versions of Zipf's law and their ranges of validity. \nIn the next section, we present and justify the three Zipf-like distributions\nwe are going to fit, \nand we briefly explain the selected fitting method and the goodness-of-fit test.\nThe corpus of texts under consideration is also detailed. \nThe following section\npresents the results, with special attention to their statistical significance\nand their dependence with text length.\nFinally, we end with the conclusions and some technical appendices.\n\n\n\\section{Zipf-like distributions}\n\\label{secdos}\n\nAs implicit in the introduction, and in contrast with continuous random variables, \nin the discrete case a power law in the probability mass function $f(n)$ does not\nlead to a power law in the complementary cumulative distribution or survival function $S(n)$, \nand vice-versa.\nLet us specify our definition for both functions,\n$f(n)=\\mbox{Prob}[\\mbox{frequency} =n]$ (as usual),\nand $S(n)=\\mbox{Prob}[\\mbox{frequency} \\ge n]$\n(changing, for convenience, the usual strict inequality sign by the non-strict inequality).\nThen, the relation between both is $f(n)=S(n)-S(n+1)$ and\n$S(n)=\\sum_{n'=n}^\\infty f(n')$.\n\nWe consider that the values the random variable takes, given by $n$, are discrete,\nstarting at the integer value $a$, \ntaking values then \n$\nn=a, a+1,\\dots\n$\nup to infinity.\nIn this study we will fix the parameter $a$ to $a=1$,\nin order to fit the whole distribution and not just the tail.\nThen, although \nfor large $n$ and smooth $S(n)$ we may approximate\n$f(n) \\simeq - dS(n) \/ dn$, \nthis simplification, which lies \nin the equivalence between Eqs. (\\ref{laprimera}) and (\\ref{lasegunda}),\nis clearly wrong for small $n$.\n\nFor the first distribution that we consider, the power-law form is in $f(n)$, then,\n\\begin{equation}\n\\label{distro1}\nf_1(n)=\\frac{1}{\\zeta(\\beta,a) n^{\\beta}}.\n\\end{equation}\n This is just the normalized version of Eq. (\\ref{lasegunda}),  and then,\n$$\nS_1(n)=\\frac{\\zeta(\\beta,n)}{\\zeta(\\beta,a)}\n$$\nwith $\\beta > 1$ and\n$\\zeta(\\beta,a)=\\sum_{k=0}^{\\infty}{(a+k)^{-\\beta}}$ denotes the Hurwitz zeta function, \nwhich ensures normalization of both expressions.\nA preliminary analysis in terms of this distribution was done\nin Ref. \\cite{Font_Clos_Corral}.\nIn contrast, when the power law is in $S(n)$, this leads to our second case,\n\\begin{equation}\n\\label{distro2}\nf_2(n)=\\left(\\frac{a}{n}\\right)^{\\beta-1}-\\left(\\frac{a}{n+1}\\right)^{\\beta-1}\n\\end{equation}\nand\n$$\nS_2(n)=\\left(\\frac{a}{n}\\right)^{\\beta-1} \n$$\nwith $\\beta > 1$ again.\nNote that this corresponds to a power law in the empirical rank-frequency relation.\n\nFinally, it is interesting to consider also the frequency distribution \nderived by Mandelbrot \\cite{Mandelbrot61} when ranks are generated independently from a power law in \nEq. (\\ref{laprimera}), which is,\n\\begin{equation}\n\\label{distro3}\nf_3(n)=\n\\frac{(\\beta-1)\\Gamma(a)}{\\Gamma(a+1-\\beta)}\n\\frac{\\Gamma(n+1-\\beta)}{\\Gamma(n+1)}\n\\end{equation}\nand\n$$\nS_3(n)=\\frac{\\Gamma(a)\\Gamma(n+1-\\beta)}{\\Gamma(n)\\Gamma(a+1-\\beta)},\n$$\nwith $1 < \\beta < 2$ and $\\Gamma(\\gamma)=\\int_0^\\infty x^{\\gamma-1} e^{-x}dx$ denotes\nthe gamma function \\cite{Abramowitz}.\nIn this case the power law is the underlying theoretical rank-frequency relation $n(r)$.\nNote that $f_3(n)$ can be written as\n$$\nf_3(n) = \\frac{B(n+1-\\beta,\\beta)}{B(a+1-\\beta,\\beta-1)}\n$$\nusing the beta function \\cite{Abramowitz}, $B(x,y)=\\Gamma(x)\\Gamma(y)\/\\Gamma(x+y)$,\nwith an analogous expression for $S_3(n)$,\nbut do not confuse this distribution with the beta distribution.\n\nIn all three cases it is easy to show that we have well-defined, normalized probability distributions, when $n$ takes values $n=a, a+1, \\dots $, with $a$ being a positive integer.\nMoreover, in the limit $n\\rightarrow \\infty$ all of them yield a power-law tail,\n$f(n) \\propto 1\/n^\\beta$,\n so $\\beta$ will be referred to as the power-law exponent.\n Indeed, it is easy to show that\n$$f_2(n) \\xrightarrow{n\\to\\infty} (\\beta-1)\n\\frac{a^{\\beta-1}}{n^\\beta},$$ \nwhereas \n$$\nf_3(n) \\xrightarrow{n\\to\\infty} \\frac{(\\beta-1)\\Gamma(a)}{\\Gamma(a+1-\\beta) n^\\beta}\n$$\nusing Stirling's formula \\cite{Abramowitz}.\nThe main difference between the three distributions is \nin the smaller values of $n$, taking $f_2(n)$ a convex shape\nin log-scale (as seen ``from above'');\n$f_3(n)$ a concave one;\nand $f_1(n)$ being somehow in between, as it is neither concave nor convex.\n\n\\section{Methodology and data}\nIn order to fit these three distributions to the different texts, \nand test the goodness of such fits, we use maximum likelihood estimation \\cite{Pawitan2001}\nfollowed by the Kolmogorov-Smirnov (KS) test \\cite{Press_C}.\nThe procedure seems \nsimilar to the one proposed in Ref. \\cite{Clauset},\nbut as $a$ is fixed here, the problems resulting from the search of the optimum $a$ \\cite{Corral_nuclear,Corral_Deluca} do not arise in this case.\n\n\n\n\nThe method of ML estimation proceeds in the following simple way.\nGiven a set of data $\\{n_i\\}$ with $i=1,2, \\dots N$, \nand a probability mass function parameterized by $\\beta$, denoted as \n$f(n;\\beta)=f(n)$,\nthe log-likelihood function is obtained as\n\\begin{equation}\nl(\\beta) = \\sum_{i=1}^N \\ln f(n_i;\\beta).\n\\label{loglikelihood}\n\\end{equation}\nWe are assuming that the data points $n_i$ are independent from each other, \nin other words, we are calculating the likelihood that the data are\ngenerated independently from $f(n;\\beta)$.\nThe ML estimation of $\\beta$ is obtained as the value of $\\beta$\nwhich maximizes $l(\\beta)$; we undertake this numerically, \nusing Brent's method \\cite{Press_C}.  \nIn the case of the distribution $f_1$ the log-likelihood function takes the simple form \n$l_1(\\beta)\/N=-\\ln(\\zeta(\\beta,a)) -\\beta \\ln G$, with $G$ the geometric mean\nof the set $\\{n_i\\}$.\nFor the other distributions no closed-form expression\nis possible and we use Eq. (\\ref{loglikelihood}) directly.\n\nAs mentioned, the goodness-of-fit test is done through the Kolmogorov-Smirnov\nstatistic \\cite{Press_C,Clauset}, in the discrete case \\cite{Corral_Deluca_arxiv}, \nfor which the $p$-value is calculated from Monte Carlo simulations\n(due to the fact that, as the value of the exponent \nis calculated from the same data is going to be tested, \nthe theoretically computed $p$-value \\cite{Press_C}\nwould be positively biased).\nIn this paper we use 100 Monte Carlo simulations for each test.\nThe proper simulation of the 3 distributions is explained in the Appendix.\n Remember that a small enough $p$-value\nleads to the rejection of the fit.\nAlthough we perform multiple testing, we do not incorporate any Bonferroni-like correction\n\\cite{Abdi_Bonferroni,Bland_Altman,Benjamini}, due to the fact that \nthese corrections increase the number of non-rejected null hypothesis\n(that is, decrease the number of type I errors),\ninflating the performance of the fits, in the case of goodness-of-fit tests. \nWithout Bonferroni-like corrections, our acceptance (i.e., non-rejection) \nof the fits is more strict.\n\nIn order to apply this methodology we consider a set of 37\\,041 texts \nstored in UTF-8 encoding in the Project Gutenberg database (accessed July 2014 \\cite{Gutenberg}).\nThese texts correspond to different languages, styles, \nand time periods, although\nmost of them are works of literature from the Western cultural tradition \\cite{EWikip}.\nFirst of all, parts of text that do not pertain to the piece under consideration \n(copyright notes, headers,...) are removed by an automatized process. \nBooks that have not been filtered in this step (mainly because they do not have standard delimiters) are discarded. \nAfter this, we still keep 97.5\\,\\% of the total (i.e., 36\\,108). \nTo perform our study, we restrict ourselves to the subset of texts in English, \nwhich represent the 86\\,\\% of these 36\\,108 (i.e., 31\\,102).\n \nAn important characteristic of each text  \nis its length, $L$, counting the number of word tokens it contains.\nIt turns out to be that in the database $L$ expands from very small values up to 4\\,659\\,068 tokens, \n with a distribution that is shown in\nFig.~\\ref{Ldens}. Observe the roughly uniform distribution up to about $L=10^5$, \nand the decay afterwards.\nWe consider only the 31\\,075 English texts that \nconsist of more than 100 word tokens, \n as smaller texts would not have statistical value.\nFor each of these texts \nwe select then actual word types (punctuation signs, numbers and any character different from letters are not considered) to count their frequencies $n$,\nwhich will be our primary object of study. \n\nThe values of these frequencies, for each text, are available\non  http:\/\/dx.doi.org\/10.6084\/m9.figshare.1515919, in order to facilitate the reproducibility\nof our results.\n \nIn summary, we apply the above described fitting and goodness-of-fit procedure\n--using ML estimation and the Kolmogorov-Smirnov test-- to a total of 31\\,075\ntexts from the English Project Gutenberg database, using three different possibilities\nfor the distribution of frequencies: $f_1$ (Eq.~\\eqref{distro1}), $f_2$ (Eq.~\\eqref{distro2}),\nand $f_3$ (Eq.~\\eqref{distro3}). This yields a total of $3\\times31\\,075$ \nfits and associated $p$-values, which we analyze and interpret in what follows.\n\n\\begin{figure} [!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_hist_long_pg}\n\\end{center}\n\\caption{\nEstimation of the probability density function of text length $L$ \nin the English Project Gutenberg database,\nusing logarithmic binning (5 bins per decade).\nTexts with less than 100 tokens are not considered.\nA power-law fit of the tail \\cite{Corral_Deluca} yields an exponent $2.92 \\pm 0.15$. \n}\n\\label{Ldens}\n\\end{figure}\n\n\n\\section{Results}\\label{sectionresults}\nContrary to previous studies where the number of texts considered was, at most, in the order of tens, the large-scale approach taken in this work requires a statistical analysis of the fitting results, as a case-by-case interpretation is out of hand.\nWe first focus on the distribution of $p$-values,\nsee Fig.~\\ref{pvfits_en} and Fig.~\\ref{Nvspv1y2}.\nIf \\emph{all} texts were truly\ngenerated by a mechanism following a given distribution, \nthe corresponding $p$-values for that distribution \nwould be uniformly distributed between zero and one. \nAs seen in Fig. \\ref{pvfits_en}, this is not the case and, furthermore, \nmost texts have rather small $p$-values for the three fits;\nnevertheless, for distributions $f_1$ and $f_2$\nthere are still many texts that yield high enough $p$-values. \nThis implies that, although we cannot conclude that the whole database is  generated by any of these distributions, \nthese cannot be rejected as good \n descriptions for large subsets of the database. \nRegarding distribution $f_3$,\nit is clear from the histogram of $p$-values that it can be discarded as a good description of \nthe distribution of frequencies in any non-negligible subset of texts.\nSo, from now on, we will concentrate on the remaining options, $f_1$ and $f_2$,\nto eventually quantify which of these better describes\nour corpus.\nIn essence, what we are interested in is which version of Zipf's law, \neither distribution $f_1$ or $f_2$, fits better a reasonable number of texts, \nand which range of validity these simple one-parameter distributions have.\n\nThe outcome is that, independently of the significance level\n(as long as this \nis not below our resolution of 0.01 given the number of Monte Carlo simulations),\nthe ratio between the number of texts fitted by distribution $f_2$\nand those fitted by $f_1$ is nearly constant, taking a value around 2.6.\nFor example, considering significance level (i.e., minimum $p$-value)\nequal to 0.05,\nFig. \\ref{Nvspv1y2} shows that \ndistribution $f_2$ fits about 40\\,\\% of all texts, \nwhereas distribution $f_1$ fits just 15\\,\\%.\nBoth percentages include a 2.7\\,\\%\nof texts that are fitted by both distributions simultaneously, although \nthis number does not keep a constant ratio with the other two, rather, it\ndecreases when the significance level is increased \n(as it is implicit in the values of Fig. \\ref{Nvspv1y2}).  \nGiven that the aforementioned ratio of 2.6 is independent of the significance level,\nit is fair to say that distribution $f_2$ provides, compared to $f_1$, a better description of our database.\nAs a visual illustration of the performance of the fits we\ndisplay in Fig.~\\ref{visual} the word frequency distribution of\nthe longest texts that have $p > 0.5$, for distributions  $f_1$, $f_2$ and $f_3$. \n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_hist_pv_2}\n\\end{center}\n\\caption{\nHistograms of $p$-values obtained\nwhen the Zipf-like distributions $f_1$, $f_2$, and $f_3$\nare fitted to the texts of the English Project Gutenberg.\nThe histograms just count the number of texts in each bin of width 0.01.\nNote the poor performance of distribution 3\nand the best performance of 2.\nPower-law approximations to the histograms for $f_1$ and $f_2$,\nwith respective exponents 0.74 and 0.78,\nare shown as a guide to the eye.\n}\n\\label{pvfits_en}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_acum_pv}\n\\end{center}\n\\caption{\nComplementary cumulative distributions \n(i.e., survival functions) of $p$-values\nobtained when our three distributions\nare fitted to the texts of the English Project Gutenberg. \n This corresponds, except for normalization, to the integral of the previous figure,\nbut we have included a fourth curve for the fraction of texts whose $p$-values for fits $1$ and $2$\nare both higher than the value marked in the abscissa.\nNote that the values of $p$ can play the role of the significance level.\nThe value for $p=0$ is not shown, in order to have higher resolution. \n}\n\\label{Nvspv1y2}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_ejemplo1}\n\\includegraphics[width=90mm]{fig_ejemplo2}\n\\includegraphics[width=90mm]{fig_ejemplo3}\n\\end{center}\n\\caption{\nComplementary cumulative distribution and probability mass function of texts:\n(a) \\textit{A Chronicle of London, from 1089 to 1483};\n(b) \\textit{The letters of Charles and Mary Lamb}, edited by E. V. Lucas;\n(c) \\textit{A Popular History of France from the Earliest Times, Vol. I,} by F. Guizot.\nThese texts are the ones with the largest length $L$ (83\\,720, 239\\,018 and 2\\,081  respectively)\nof those that fulfill  $p> 1\/2$, for fits 1, 2 and 3 \nrespectively.\nThe exponent $\\beta$ takes values 1.96, 1.89, and 1.82,\nin each case. \n}\n\\label{visual}\n\\end{figure}\n\n       \nThe next question we address is the dependence of the performance of  fits on text length $L$.\nIn order to  asses this,\nnote that from the shape of the histograms in Fig.~\\ref{pvfits_en} we can distinguish two groups of texts:\nthose that lie in the zero bin (whose $p$-value is strictly less than 0.01), and the rest. \nTaking the last group, i.e., texts with $p\\geq 0.01$,\nand partitioning it into different subsets according to text length\n(i.e., looking at the distribution of $p$ conditioned to $p\\ge 0.01$ for different ranges of $L$), \nit holds that the shape of the resulting distribution of $p$ does not strongly depend on $L$, as shown in Fig.~\\ref{pv_Ls_fit2}.\nIn contrast, the number of texts that yield $p$-value near zero certainly varies with $L$, see Fig.~\\ref{pv0}.\nTherefore, in order to compare the performances of $f_1$ and $f_2$ as a function of the text length $L$,\nit is enough to consider a single value of the significance level (greater than zero)\nas the results for any other significance level \nwill be  the same, in relative terms.\n\nIndeed, Fig.~\\ref{perc_L}(a) shows how distribution $f_1$ fits some more texts than\ndistribution $f_2$ for small values of $L$, up to about $13\\,000$ tokens.\nBut for larger texts, distribution $f_2$ clearly outperforms distribution $f_1$,\nwhich becomes irrelevant for $L$ beyond $100\\,000$ (at 0.05 significance level), \nwhereas distribution $f_2$ is able to fit many texts with $L$ larger than $200\\,000$.\nThe figure shows that this is the case no matter if the significance level is 0.05, 0.20, or 0.50;\nthe collapse of the curves in  Fig.~\\ref{perc_L}(b) confirms this fact.\nFrom Fig.~\\ref{pv0} one could infer the same for significance level equal to 0.01.\nThis stability of the performance of the fits for different significance levels arises from \nthe observed fact that the distributions of $p$-values (conditioned to $p \\ge 0.01$)\nare nearly identical for different $L$,\nas shown in Fig.~\\ref{pv_Ls_fit2}.\n      \n\\begin{figure} [!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{pv_collapse_fit1}\n\\includegraphics[width=90mm]{pv_collapse_fit2}\n\\end{center}\n\\caption{\nEstimated probability  density functions \nof $p$-values conditioned to $p\\ge 0.01$ \nseparating for different ranges of text length $L$. $p$-values correspond\nto the fitting of  word frequencies to (a) distribution $f_1$ and (b) distribution $f_2$.\nWe divide the distribution of text length into 15 intervals of 2\\,000 texts each. \nFor distribution $f_1$ only the first seven groups (up to length 34\\,400) are displayed (beyond this value we do not have enough statistics to see the distribution of  $p$-values greater than 0.01, as displayed in Fig.~\\ref{pv0};\nfor distribution 2  this happens only in the last two groups). The intervals $L_i$ range from $L_{1}=[115, 5\\,291]$ to $L_{13}=[89\\,476, 103\\,767]$.\n}\n\\label{pv_Ls_fit2}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_pv0_fit.pdf} \n\\end{center}\n\\caption{Number of texts with $p$-value near zero ($p<0.01$) \nin different ranges of $L$ divided by the number of texts in the same ranges,\nfor the fits of distributions $f_1$ and $f_2$. \nValues of $L$ denote the geometric mean of ranges containing 1000 texts each.\nThe higher value for fit 1 (except for $L$ below about 13000 tokens) denotes its worst performance. \n}\n\\label{pv0}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{L_accepted}\n\\includegraphics[width=90mm]{L_accepted2}\n\\end{center}\n\\caption{\n(a)  Histograms showing the fraction of accepted texts by the three distributions\nas a function of their text length, for three different significance levels $p_0$: \n0.05 (upper curves), 0.20 (middle), 0.50 (lower).\nTo be concrete, for each range of $L$, \nthe ratio between the number of texts with $p\\ge p_0$\nand the number of texts in that range is calculated.\n(b) Same curves (removing those for distribution 3) under rescaling.\nWe rescale  the $y-$axis by the number of $p\\ge p_0$, in each case,\nshowing that the relative performance of each fit\nwith regard $L$ is independent on the significance level.\nBins are selected to contain 1000 texts each.\n}\n\\label{perc_L}\n\\end{figure}\n          \nIn order to  check the consistency of our fitting procedure, we also perform a direct comparison of models through the likelihood ratio (LR) test \\cite{Vuong,Clauset}.\nWe apply this test to all texts that have been fitted, \nconsidering 0.05 as significance level,\nby at least one of the two distributions $f_1$ and $f_2$.\nThen the log-likelihood-ratio between distributions $f_1$ and $f_2$ is \n\\begin{equation}\nR_{1,2}= \\sum_{i=1}^{N}\\left(\\ln f_1(n_i)-\\ln f_2(n_i)\\right), \\nonumber\n\\end{equation}\nand, under the null hypothesis that both models are equally good to describe the data,\n$R_{1,2}$ should be normally distributed with zero mean and a variance that can be estimated as $N \\sigma^2$,\nwith $\\sigma^2$ the variance of the random variable\n$\\ln f_1(n)-\\ln f_2(n)$.\nLarge absolute values of $R_{1,2}$ will lead to the rejection of the null hypothesis.\n\nTable \\ref{tabla_lkh} merges the results of the LR test and our previous procedure (based on \nML estimation plus the KS test). \nThe total number of texts previously fitted by $f_1$ or\/and $f_2$ is displayed depending on the sign of the corresponding log-ratio $R_{1,2}$.\nHowever we must take into account that the sign of the obtained value of $R_{1,2}$ could be a product of just statistical fluctuations if the true value were zero and thus, the sign of $R_{1,2}$ \ncannot be trusted in order to discriminate between two models.\nThe probability under the null hypothesis, of obtaining an absolute value of the log-ratio greater than the empirical value\n$|R_{1,2}|$ is computed through:  \n\\begin{equation}\np_{LR}=\n\\mbox{erfc}\\left(\\frac{|R_{1,2}|}{\\sqrt{2N\\sigma^2}}\\right)\\nonumber\n\\end{equation}\nwhere \nerfc  is the complementary error function \\cite{Abramowitz}.\nWe will take as statistically significant those $R_{1,2}$ that yield $p_{LR}<0.05$.\nEquivalently, at 0.05 significance level,\n$R_{1,2}$ is significant if its absolute value is greater than \n$R_c=1.96 \\sqrt{N \\sigma^2}$. The results are shown in Table \\ref{tabla_lkh_pv}\n\nNote that the LR test cannot conclude if a fit is good or bad, \nas it only compares the relative performance of two fits;\nin other words, if the LR test selects a particular distribution, \nthat distribution can still yield a bad fit, in absolute terms.\nAnyway, there is no mismatch between the results of both tests: \nany time the ML-KS method selects one distribution over the other,\nthe LR test either supports the selection or does not give significant results, \nbut it never selects the other option (as shown in Table~\\ref{tabla_lkh_pv}).\n\n\\begin{center}\n\\begin{table}[!htbp]  \n\\begin{tabular}{| l | r | r | r |}\\hline\n                    & $R_{1,2}>0$ & $R_{1,2}<0$ & Total ML-KS \\\\\\hline\n$f_1$ (exclusively) & 3614        & 81          & 3695        \\\\\\hline\n$f_2$ (exclusively) & 120         & 11366       & 11486       \\\\\\hline \n$f_1$ and $f_2$     & 431         & 398         & 829         \\\\\\hline \nTotal LR            & 4165        & 11845       & 16010       \\\\\\hline \n\\end{tabular}\n\\caption[]{\nThe number of texts that are fitted by $f_1$ or $f_2$ or both\nat 0.05 significance level of the ML-KS procedure,\nseparated into two columns according to the sign of $R_{1,2}$. \nPositive $R_{1,2}$ means that the likelihood \nfor $f_1$ is greater than that for $f_2$,\nand conversely for negative $R_{1,2}$.\nNevertheless, the sign of $R_{1,2}$ is not an indication of significance,\nfor significant LR tests see\nTable~\\ref{tabla_lkh_pv}.\n}\n\\label{tabla_lkh}\n\\end{table}\n\\end{center}\n\n\\begin{center}\n\\begin{table}[!htbp] \n\\begin{tabular}{| l | r | r |}\\hline\n                                & $R_{1,2}>R_c$ & $R_{1,2}<-R_c$  \\\\\\hline\n$f_1$ (exclusively)             & 1666          & 0               \\\\\\hline\n$f_2$ (exclusively)             & 0             & 9423            \\\\\\hline \n$f_1$ and $f_2$                 & 0             & 3               \\\\\\hline \nTotal LR test                   & 1666          & 9426            \\\\\\hline \nNone (neither $f_1$ nor $f_2$)  & 510           & 11431           \\\\\\hline \n\\end{tabular}\n\\caption[]{\nNumber of texts with a significant LR test, \nat the 0.05 level, either favouring distribution $f_1$ \n($R_{1,2}>R_c$)\nor distribution $f_2$ $(R_{1,2}<-R_c)$,\nfor different outcomes of the ML-KS procedure (at the 0.05 level also). \nNote that these cases correspond to a subset of the previous table.\nAn additional row shows the number of texts that are\nfitted neither by distribution $f_1$ nor $f_2$;\nnotice that in this case a significant LR test does not guarantee a good fit.\n}\n\\label{tabla_lkh_pv}\n\\end{table}\n\\end{center}\n\n  \n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_hist_expo}\n\\end{center}\n\\caption{\nEstimation of the probability density  of the exponent $\\beta$ \nfor texts yielding $p\\ge0.05$ in fit 1 and fit 2.\nCurves have been calculated from the histograms via normal kernel smoothing method\nas implemented in MatLab (\\textit{ksdensity} function).\nEstimated mean and standard deviation are 2.03 and 0.15 respectively for fit 1, and 2.02 and 0.17 for fit 2.}\n\\label{betadist}\n\\end{figure}\n      \nTaking now those texts whose frequency distributions could be approximated by $f_1$ or $f_2$, we draw attention to the distribution of the estimated exponents (i.e., the parameter $\\beta$). \nThe original formulation of Zipf's law implies $\\beta=2$ and \nFig.~\\ref{betadist} shows that $\\beta$ is certainly distributed around 2, \nwith a  bell-like shape.\nIf we check the effect of the text length $L$ in the distribution of $\\beta$, \nwe find a decreasing trend of $\\beta$ with $L$, as can be seen in Fig.~\\ref{betadist_L}.\nWe have tested that this observation is not an artifact of the fitting method, as synthetic texts generated with fixed $\\beta$ do not show this behavior.\nWe have no theoretical explanation for this fact, but notice that this trend\nis not in disagreement with the claims of Ref. \\cite{FontClos_Corral}, \nwhere the stability of the exponent $\\beta$ was demonstrated\nfor a single growing text (i.e., comparing small parts of a text with the whole).\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=90mm]{fig_expodist_L_fit1}\n\\includegraphics[width=90mm]{fig_expodist_L_fit2}\n\\end{center}\n\\caption{\nEstimated probability density  of $\\beta$ for fits with $p\\ge 0.05$, in different length ranges.  \nWe have divided both groups of accepted texts into 4 percentiles according to $L$.\nAs in the previous figure, the normal kernel smoothing method is applied.\n(a) For distribution $f_1$.\n(b) For distribution $f_2$.\n}\n\\label{betadist_L}\n\\end{figure}\n\n\\section{Conclusions}\nZipf's law is probably the most intriguing and at the same time well-studied experimental law of quantitative linguistics, and extremely popular in its wider sense in the science of complex systems.\nAlthough the previous literature is vast, as far as we know our work constitutes the first large-scale\nanalysis of Zipf's law in single (non-aggregated) texts. \nThus, we are in a position to make a well-grounded statement about the validity of Zipf's law in such texts.\n\nLet us first briefly summarize, however, some key technical points of our study.\nFirst, we have analyzed a total of 31\\,075 English texts from the Project Gutenberg database using rigorous fitting procedures, \nand have tested how well they are described by three Zipf-like distributions.\nOur choice of distributions has not been exhaustive;\nrather, we have limited ourselves to different interpretations \nof what can be understood as ``Zipf's law'', \nin the sense of having a perfect power law either in the probability mass function\nof word frequencies, \nor in the complementary cumulative distribution function\n(whose empirical estimation leads to the rank-frequency relation of the sample),\nor in the rank-frequency relation of an underlying population.\nRemarkably, the resulting distributions have a unique parameter, $\\beta$, \nwhich in all cases is the exponent of an asymptotic power law \nin the probability mass function of the frequency.\nIt is left to explore how other, more complicated extensions of Zipf's law perform on this large corpus, but it is obvious that, by including additional parameters, one might provide good fits to a larger number of texts\n(although in this case, proper model selection will require to balance number of parameters and parsimony).\n\nOur aim in this paper has not been to fit as many texts as possible, but to test the\nperformance of the simplest Zipf-like distributions within a very strict, conservative framework.\nIndeed, by requiring the three versions of Zipf's law to hold\non the full range of frequencies $n=1,2,\\dots$ (and not only on the tail of the distribution)\nwe put ourselves in the strictest range of demands.\nIt is hence remarkable that, e.g., at the standard significance level of $0.05$, \nand for text lengths between $10^4$ and $10^5$\nword tokens, more than 40\\,\\% of the considered texts are statistically compatible \nwith the pure power law in the \ncomplementary cumulative distribution function represented\nby distribution $f_2$ (see Fig.~\\ref{perc_L}).\nSo, we can state that, for the corpus under consideration, \nthe most appropriate version of Zipf's law\nis given by a probability mass function\n$$\nf(n) = \\mbox{Prob}[\\mbox{frequency} = n] = \n\\frac 1 {n^{\\beta-1}} - \\frac 1 {(n+1)^{\\beta-1}},\n$$\nor, equivalently, by a complementary cumulative distribution function\n$$\nS(n) = \\mbox{Prob}[\\mbox{frequency} \\ge n]=\n\\frac 1 {n^{\\beta-1}}.\n$$\nDue to the broad coverage of the Project Gutenberg corpus\nwe speculate that this distribution should fit\na large fraction of generic (non-technical) English texts. \nOf course, testing this speculation in front of all possible corpora\nis an impossible task.\n\nWe have also shown that our conclusions regarding the \\emph{relative} performance of \na pure power law in the probability mass function, given by\ndistribution $f_1$, versus distribution $f_2$ are robust with respect to changes in the significance level: \nabout twice as many texts are statistically compatible with distribution $f_2$ than\nthose compatible with $f_1$, at any significance level  \n(obviously, in absolute terms, the number of accepted texts varies with the significance level).\nHence we can conclude that\ndistribution $f_2$ gives a better description of English texts than distribution $f_1$,\nat least for the corpus considered in this work.\nWe also conclude that distribution $f_3$, first derived by Mandelbrot \\cite{Mandelbrot61},\nis irrelevant for the description of texts in this corpus.\nFinally, we have corroborated that the exponent $\\beta$ of Zipf's law certainly varies from text to text, as had been previously claimed using other approaches for defining \nwhat Zipf's law is \\cite{Zanette_2005,Corral_Boleda}.\nInterestingly, the value $\\beta=2$ originally proposed by Zipf himself is among the most frequent ones.\n\nWe believe that our analysis constitutes a major advancement in the understanding of Zipf's law.\nIt is astonishing how good the simplest one-parameter Zipf-like distributions perform on such a large set of texts, particularly with the strict set of requirements we have imposed.\nThis is in sharp contrast for instance with Zipf's law in demography \\cite{Malevergne_Sornette_umpu} \nand in the distribution of income \\cite{Drgulescu2001}, \nwhere the power law seems to be valid only for the tail corresponding to the largest sizes,\nas it happens also for the distribution of word frequency in large text corpora,\nas mentioned above \\cite{Baroni2009,Ferrer2000a,Petersen_scirep,Gerlach_Altmann,Williams}.\n\nZipf's law has been subject to much debate, and will probably continue to be so for many years.\nIndeed, one can always cast doubt on its validity on the basis of some particular examples.\nYet it seems clear to us that, in our modern times of big data and large computational capabilities, more efforts should be put towards large-scale analysis of Zipf's law.\nWe hope this paper constitutes a first step in this direction.\n\n\\section{Appendix: Simulation of discrete Zipf-like distributions}\nAs part of the testing procedure, we need simulated samples from $f_1$, $f_2$, and $f_3$,\nwhich are discrete distributions defined for $n=a, a+1, \\dots$. \nWe will give the recipe of simulation for an arbitrary positive integer value of the lower cut-off $a$. \nIt is simpler to start with $f_2$, as this is used as an auxiliary distribution\nin the simulation of the other two. \n\\\\\n{\\it Simulation of $f_2$.}\nFixed $a$ and given the parameter $\\beta$,  we want a set of random numbers  whose cumulative distribution function is a discrete power law: \n$S_2(n)=\\left({a}\/{n}\\right)^{\\beta -1}$. \nFor that, we first generate a random number $u$ from a uniform distribution in the interval \n$(0, u_{max})$, with $u_{max}={1}\/{a^{\\beta-1}}$.\nIf we take $x={1}\/{u^{1\/(\\beta -1)}}$, it turns out to be that the values of  $x$ yield a continuous power law with $S^c_2(x)=\\left({a}\/{x}\\right)^{\\beta -1}$, for $x\\ge a$, where the superscript $c$ distinguishes the continuous distribution from its discrete analogous one.\nSo, taking $n$ equal to the integer part of $x$, i.e., \n$n=\\text{int}(x)$, yields a discrete distribution with $S_2(n)=\\left({a}\/{n}\\right)^{\\beta -1}$, as desired.\nThis is so because, for any $X$, $\\text{int}(X)\\ge n$ is equivalent to $X\\ge n$ for $n$ integer.\nIn a recipe:\n\\begin{itemize}\n\\item generate $u$ from a uniform distribution in $(0,1\/a^{\\beta-1}]$,\n\\item calculate $x= {1}\/{u^{1\/(\\beta -1)}}$,\n\\item take $n =\\text{int}(x)$.\n\\end{itemize}\n\nBy means of the so-called rejection method  \\cite{Devroye},\nsimulated integers distributed following $f_2$ can be used for the simulation of\nintegers  following $f_1$ or $f_3$.\nThe key point to achieve a high performance in the rejection method is to use a ``good'' auxiliary function,\ni.e., one that leads to a low rejection rate.\nThis is certainly the case in our framework, as explained below.\n\\\\\n{\\it Simulation of $f_1$.}\nIn this case, the steps are:\n\\begin{itemize}\n\\item generate $n$ from $f_2(n)$,\n\\item  generate  $v$ from a uniform distribution in the unit interval,\n\\item  $n$ is accepted if \n$$v\\le\\frac{f_1(n)}{f_2(n)C},$$\nand rejected otherwise,\nwhere $C$ is the rejection constant given by \n$C=\\max_{n\\ge a}\\left\\{{f_1(n)}\/{f_2(n)}\\right\\}$.\n\\end{itemize}\n\nIt is easy to check that the maximum of $f_1\/f_2$ is reached at $n=a$ as this is a decreasing function\n\\cite{Corral_Cancho}.\nThe acceptance condition above can be simplified by\ntaking $\\tau=(1+n^{-1})^{\\beta-1}$, and $b=(1+a^{-1})^{\\beta-1}$, \nthen, the condition becomes: \n$$\nb v n (\\tau-1) \\le a (b-1) \\tau,\n$$\nwhich is devoid of the calculation of the Hurwitz-zeta function.\nThis is a generalization for $a>1$ of the method of Ref. \\cite{Devroye}.\nThe choice of $f_2$ as the auxiliary distribution function is justified by the small value that $C$ takes, as this is the expected number of  generated values of $n$ until we accept one. \nFor instance, for $\\beta=2$ and $a=1$ we get $C=1.2158$.\n\\\\\n{\\it Simulation of $f_3$.}\nProceeding similarly, we get in this case low values of \n$C=\\max_{n\\ge a}\\left\\{{f_3(n)}\/{f_2(n)}\\right\\}$ as well \n(we get $C=2$ in the limit $\\beta\\rightarrow 2$ for $a=1$).\nThe maximum of ${f_3(n)}\/{f_2(n)}$ is numerically seen to be reached at $n=a$.\nIn summary, the steps are:\n\\begin{itemize}\n\\item generate $n$ from $f_2(n)$\n\\item  generate  $v$ from a uniform distribution in the unit interval\n \\item $n$ is accepted if \\begin{equation}\n vf_2(n)\\le a\\left(1-\\left(\\frac{a}{a+1}\\right)^{\\beta-1}\\right)\\frac{\\Gamma(n-(\\beta-1))}{\\Gamma(n+1)}\\frac{\\Gamma(a)}{\\Gamma(1+a-\\beta)}\\nonumber\n\\end{equation}\nand rejected otherwise.\n\\end{itemize}\n\n\n\\section{Acknowledgments}\nWe are grateful to the Project Gutenberg initiative, \nand to those who help maintain it alive.\nS. Pueyo provided bibliographic wisdom for Zipf's law in ecology,\nand I. Serra assistance for ML estimation.\nI. M.-S. enjoys a contract from  \nthe Collaborative Mathematics\nProject of La Caixa Foundation.\nResearch projects in which this work is included are\nFIS2012-31324, from Spanish MINECO, \nand  2014SGR-1307, from AGAUR.\n\n\n\\addcontentsline{toc}{chapter}{Bibliography}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnmanned Aerial Vehicles (UAVs) equipped with cameras have grown into an important asset in a wide range of fields, such as agriculture, delivery, surveillance, and search and rescue (SAR) missions \\cite{adao2017hyperspectral,san2018uav,geraldes2019uav}. In particular, UAVs are capable of assisting in SAR missions due to their fast and versatile applicability while providing an overview over the scene \\cite{mishra2020drone,karaca2018potential,albanese2020sardo}. Especially in maritime scenarios, where wide areas need to be quickly overseen and searched, the efficient use of autonomous UAVs is crucial \\cite{yeong2015review}. Among the most challenging issues in this application scenario is the detection, localization, and tracking of people in open water \\cite{gallego2019detection,nasr2019shipwrecked}. The small size of people relative to search radii and the variability in viewing angles and altitudes require robust vision-based systems. \n\n\\begin{figure}[t]\n\t\n\t\\centering\n\n{\\includegraphics[scale=0.1,trim=100 290 0 0,clip]{199.png}}\n{\\includegraphics[scale=0.12]{Screenshot_from_DJI_0001_resized}}\\\\ \n\\vspace{0.5mm}\n{\\includegraphics[scale=0.12,trim=22 0 0 0,clip]{Screenshot_from_DJI_0058_resized.png}}\n{\\includegraphics[scale=0.12]{Screenshot_from_DJI_0068_resized.png}}\\\\\n(a)\n\n\n{\\includegraphics[scale=0.09,trim=21 200 0 0,clip]{IMG_0218_4.png}}\n{\\includegraphics[scale=0.09,trim=0 200 0 0,clip]{IMG_0218_5.png}}\\\\\n(b)\n\n\n\n\t\\caption{(a) Typical image examples with varying altitudes and angles of view: 250 m, $90^\\circ$; 50 m, $30^\\circ$; 10 m, $0^\\circ$ and 20 m, $90^\\circ$ (from top left to bottom right). (b) Examples of the Red Edge (717 nm, left) and Near Infrared (842 nm, right) light spectra of an image captured by the MicaSense RedEdge-MX. Note the glowing appearance of the swimmers.}\n\t\\label{fig:front_image}\n\t\\vspace{-0.4cm}\n\t%\n\\end{figure}\n\n\nCurrently, these systems are implemented via data-driven methods such as deep neural networks. These methods depend on large-scale data sets portraying real-case scenarios to obtain realistic imagery statistics. However, there is a great lack of large-scale data sets in maritime environments. Most data sets captured from UAVs are land-based, often focusing on traffic environments, such as VisDrone \\cite{zhu2018vision} and UAVDT \\cite{du2018unmanned}. Many of the few data sets that are captured in maritime environments fall in the category of remote sensing, often leveraging satellite-based synthetic aperture radar \\cite{crisp2004state}. All of these are only valuable for ship detection \\cite{corbane2010complete} as they don't provide the resolution needed for SAR missions. Furthermore, satellite-based imagery is susceptible to clouds and only provides top-down views. Finally, many current approaches in the maritime setting rely on classical machine learning methods, incapable of dealing with the large number of influencing variables and calling for more elaborate models \\cite{prasad2019object}.\n\nThis work aims to close the gap between large-scale land-based data sets captured from UAVs to maritime-based data sets. We introduce a large-scale data set of people in open water, called SeaDronesSee. We captured videos and images of swimming probands in open water with various UAVs and cameras. As it is especially critical in SAR missions to detect and track objects from a large distance, we captured the RGB footage with 3840$\\times$2160 px to 5456$\\times$3632 px resolution. We carefully annotated ground-truth bounding box labels for objects of interest including swimmer, floater (swimmer with life jacket), life jacket, swimmer$^\\dagger$ (person on boat not wearing a life jacket), floater$^\\dagger$ (person on boat wearing a life jacket), and boat.\n\nMoreover, we note that current data sets captured from UAVs only provide very coarse or no meta information at all. We argue that this is a major impediment in the development of multi-modal systems, which take these additional information into account to improve accuracy or speed. Recently, methods that rely on these meta data were proposed. However, they note the lack of large-scaled publicly available data set in that regime  (see \\eg \\cite{kiefer2021leveraging,wu2019delving,messmer2021gaining}). Therefore, we provide precise meta information for every frame and image including altitude, camera angle, speed, time, and others.\n\nIn maritime settings, the use of multi-spectral cameras with Near Infrared channels to detect humans can be advantageous \\cite{gallego2019detection}. For that reason, we also captured multi-spectral images using a MicaSense RedEdge. This enables the development of detectors taking into account the non-visible light spectra Near Infrared (842 nm) and Red Edge (717 nm).\n\nFinally, we provide detailed statistics of the data set and conduct extensive experiments using state-of-the-art models and hereby establish baseline models. These serve as a starting point for our SeaDronesSee benchmark. We release the training and validation sets with complete bounding box ground truth but only the test set's videos\/images. The ground truth of the test set is used by the benchmark server to calculate the generalization power of the models. We set up an evaluation web page, where researchers can upload their predictions and opt to publish their results on a central leader board such that transparent comparisons are possible. The benchmark focuses on three tasks: (i) object detection, (ii) single-object tracking and (iii) multi-object tracking, which will be explained in more detail in the subsequent sections. Our main contributions are as follows:\n\\begin{itemize}\n\t\\item To the best of our knowledge, SeaDronesSee is the first large annotated UAV-based data set of swimmers in open water. It can be used to further develop detectors and trackers for SAR missions.\n\t\n\t\\item We provide full environmental meta information for every frame making SeaDroneSee the first UAV-based data set of that nature.\n\t\n\t\\item We provide an evaluation server to prevent researches from overfitting and allow for fair comparisons.\n\t\n\t\\item We perform extensive experiments on state-of-the-art object detectors and trackers on our data set.\n\t\n\\end{itemize}\n\n\n\\begin{table*}\n\t\n\t\\begin{center}\n\t\t\n\t\t\\begin{tabular}{c|c|c|c|cc|cc|c}\n\t\t\tObject detection  & Env. &  Platform  & Image widths   & Altitude & Range & Angle & Range &  Other meta\\\\\n\t\t\t\\hline\n\t\t\tDOTA \\cite{xia2018dota} & cities & satellite & 800-20,000 & -- & -- & \\ding{53} & $90^\\circ$ & \\ding{53} \\\\\n\t\t\t\n\t\t\tUAVDT \\cite{du2018unmanned} & traffic & UAV & 1,024 & \\ding{53} & 5-200 m* & \\ding{53} & $0-90^\\circ$* & \\ding{53}\\\\\n\t\t\n\t\t\tVisDrone \\cite{zhu2018vision} & traffic & UAV & 960-2,000 & \\ding{53} & 5-200 m* & \\ding{53} & $0-90^\\circ$* & \\ding{53}\\\\\n\t\t\t\n\t\t\tAirbus Ship \\cite{airbus-ship} & maritime & satellite & 768 & -- & -- & \\ding{53} & $90^\\circ$\\hphantom{*} & \\ding{53}\\\\\n\t\t\t\n\t\t\tAU-AIR \\cite{bozcan2020air} & traffic & UAV & 1,920 & \\checkmark & 5-30 m\\hphantom{*} & \\ding{53} & $45-90^\\circ$\\hphantom{*} & \\checkmark \\\\\n\t\t\t\n\t\t\t\\bf SeaDronesSee & maritime & UAV  & 3,840-5,456 & \\checkmark & 5-260 m\\hphantom{*} & \\checkmark & $\\ 0-90^\\circ$\\hphantom{*} & \\checkmark  \\\\\t\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\n\t\\begin{center}\n\t\t\n\t\t\\begin{tabular}{c|c|c|c|cc|cc|c}\n\t\t\tSingle-object tracking  & Env.  & \\#Clips\n\t\t\n\t\t\t& Frame widths   & Altitude & Range & Angle & Range & Other meta\\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\tUAV123 \\cite{mueller2016benchmark} & traffic & 123 &\n\t\t\n\t\t\t1,280 & \\ding{53} & 5-50 m* & \\ding{53} &  $0-90^\\circ$* & \\checkmark \\\\\n\t\t\t\n\t\t\tDTB70 \\cite{li2017visual} & sports & 70 & 1,280 & \\ding{53} & 0-10 m* & \\ding{53} & $0-90^\\circ$* & \\ding{53}\\\\\n\t\t\t\n\t\t\tUAVDT-SOT \\cite{du2018unmanned} & traffic & 50 &\n\t\t\n\t\t\t1,024 & \\ding{53} & 5-200 m* & \\ding{53} &  $0-90^\\circ$* & \\checkmark \\\\\n\t\t\t\n\t\t\tVisDrone \\cite{zhu2018vision} & traffic & 167 &\n\t\t\n\t\t\t960-2,000   &\\ding{53} & 5-200 m* & \\ding{53} & $0-90^\\circ$* & \\checkmark  \\\\\n\t\t\t\n\t\t\t\\bf SeaDronesSee & maritime & 208 &\n\t\t\n\t\t\t3,840 & \\checkmark & 5-150 m\\hphantom{*} & \\checkmark & $0-90^\\circ$\\hphantom{*} & \\checkmark  \\\\\n\t\t\t\n\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\n\t\\begin{center}\n\t\t\n\t\t\\begin{tabular}{c|c|c|c|cc|cc|c}\n\t\t\tMulti-object tracking  & Env. & \\#Frames\n\t\t\n\t\t\t& Frame widths   & Altitude & Range & Angle & Range & Other meta\\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\tUAVDT-MOT \\cite{du2018unmanned} & traffic & 40.7 k &\n\t\t\n\t\t\t1,024 & \\ding{53} & 5-200 m* & \\ding{53} &  $0-90^\\circ$* & \\checkmark \\\\\n\t\t\t\n\t\t\tVisDrone \\cite{zhu2018vision} & traffic & 40 k &\n\t\t\n\t\t\t960-2,000   &\\ding{53} & 5-200 m* & \\ding{53} & $0-90^\\circ$* & \\checkmark  \\\\\n\t\t\t\n\t\t\t\\bf SeaDronesSee & maritime & 54 k &\n\t\t\n\t\t\t3,840 & \\checkmark & 5-150 m\\hphantom{*} & \\checkmark & $0-90^\\circ$\\hphantom{*} & \\checkmark  \\\\\n\t\t\t\n\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\n\t\n\t\\caption{Comparison with the most prominent annotated aerial data sets. 'Altitude' and 'Angle' indicate whether or not there are precise altitude and angle view information available. 'Other meta' refers to time stamps, GPS, and IMU data and in the case of object tracking can also mean attribute information about the sequences. The values with stars have been estimated based on ground truth bounding box sizes and corresponding real world object sizes (for altitude) and qualitative estimation of sample images (for angle). For DOTA and Airbus Ship the range of altitudes is not available because these are satellite-based data sets.}\n\t\n\t\\label{table:comparison_datasets}\n\\end{table*}\n\n\\section{Related Work}\nIn this section, we review major labeled data sets in the field of computer vision from UAVs and in maritime scenarios which are usable for supervised learning models.\n\n\\subsection{Labeled Data Sets Captured from UAVs}\nOver the last few years, quite a few data sets captured from UAVs have been published. The most prominent are these that depict traffic situations, such as VisDrone \\cite{zhu2018vision} and UAVDT \\cite{du2018unmanned}. Both data sets focus on object detection and object tracking in unconstrained environments. Pei \\etal \\cite{pei2019human}  collect videos (Stanford Drone Dataset) showing traffic participants on campuses (mostly people) for human trajectory prediction usable for object detection. UAV123 \\cite{mueller2016benchmark} is a single-object tracking data set consisting of 123 video sequences with corresponding labels. The clips mainly show traffic scenarios and common objects. Both, Hsieh \\etal \\cite{hsieh2017drone} and Mundhenk \\etal \\cite{mundhenk2016large} capture a data set showing parking lots for car counting tasks and constrained object detection. Li \\etal \\cite{li2017visual} provide a single-object tracking data set showing traffic, wild life and sports scenarios. \nCollins \\etal capture a single-object tracking data set showing vehicles on streets in rural areas. Krajewski \\etal \\cite{krajewski2018highd} show vehicles on freeways.\n\nAnother active area of research focuses on drone-based wildlife detection. Van \\etal \\cite{van2014nature} release a data set for the tasks of low-altitude detection and counting of cattle. Ofli \\etal \\cite{ofli2016combining} release the African Savanna data set as part of their crowd-sourced disaster response project.\n\n\n\n\n\\subsection{Labeled Data Sets in Maritime Environments}\nMany data sets in maritime environments are captured from satellite-based synthetic aperture radar and therefore fall into the remote sensing category. In this category, the airbus ship data set \\cite{airbus-ship} is prominent, featuring 40k images from synthetic aperture radars with instance segmentation labels. Li \\etal \\cite{li2018hsf} provide a data set of ships with images mainly taken from Google Earth, but also a few UAV-based images. In \\cite{xia2018dota}, the authors provide satellite-based images from natural scenes, mainly land-based but also harbors.\nThe most similar to our work is \\cite{lygouras2019unsupervised}. They also consider the problem of human detection in open water. However, their data mostly contains images close to shores and of swimming pools. Furthermore, it is not publicly available. \n\n\n\\subsection{Multi-Modal Data Sets Captured from UAVs}\n\nUAVDT \\cite{du2018unmanned} provides coarse meta data for their object detection and tracking data: every frame is labeled with altitude information (low, medium, high), angle of view (front-view, side-view, bird-view) and light conditions (day, night, foggy). Wu \\etal \\cite{wu2019delving} manually label VisDrone after its release with the same annotation information for the object detection track. Mid-Air \\cite{Fonder2019MidAir} is a synthetic multi-modal data set with images in nature containing precise altitude, GPS, time, and velocity data but without annotated objects. Blackbird \\cite{antonini2018blackbird} is a real-data indoor data set for agile perception also featuring these meta information. In \\cite{majdik2017zurich}, street-view images with the same meta data are captured to benchmark appearance-based localization. Bozcan \\etal \\cite{bozcan2020air} release a low-altitude ($<30$ m) object detection data set containing images showing a traffic circle and provide meta data such as altitude, GPS, and velocity but exclude the import camera angle information.\n\nTracking data sets often provide meta data (or attribute information) for the clips. However, in many cases these do not refer to the environmental state in which the image was captured. Instead, they abstractly describe the way in which a clip was captured: UAV123 \\cite{mueller2016benchmark} label their clips with information such as aspect ratio change, background clutter, and fast motion, but do not provide frame-by-frame meta data. The same observation can be made for the tracking track of VisDrone \\cite{fan2020visdrone}. See Table \\ref{table:comparison_datasets} for an overview of annotated aerial data sets.\n\n\n\\section{Data Set Generation}\n\n\\label{sec:imagedatacollection}\n\n\nWe gathered the footage on several days to obtain variance in light conditions. Taking into account safety and environmental regulations, we asked over 20 test subjects to be recorded in open water. Boats transported the subjects to the area of interest, where quadcopters were launched at a safe distance from the swimmers. At the same time, the fixed-wing UAV Trinity F90+ was launched from the shore. We used waypoints to ensure a strict flight schedule to maximize data collection efficiency. Care was taken to maintain a strict vertical separation at all times. Subjects were free to wear life jackets, of which we provided several differently colored pieces (see also Figure \\ref{fig:objects_examples}).\n\nTo diminish the effect of camera biases within the data set, we used multiple cameras, as listed in Table \\ref{table:cameras}, mounted to the following drones: DJI Matrice 100, DJI Matrice 210, DJI Mavic 2 Pro, and a Quantum Systems Trinity F90+.\n\\begin{table}\t\n\t\\begin{center}\t\t\n\t\t\\begin{tabular}{ccc}\n\t\t\tCamera  & Resolution & Video   \\\\\n\t\t\t\\hline\t\t\t\n\t\t\tHasselblad L1D-20c & 3,840$\\times$2,160 & 30 fps \\\\\n\t\t\tMicaSense RedEdge-MX & 1,280$\\times$ 960 \\ \\  & \\ding{53} \\\\\n\t\t\tSony UMC-R10C & 5,456$\\times$3,632 & \\ding{53} \\\\\n\t\t\tZenmuse X5 & 3,840$\\times$2,160 & 30 fps \\\\\n\t\t\tZenmuse XT2 & 3,840$\\times$2,160 & 30 fps\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Overview of used cameras.}\n\t\\label{table:cameras}\n\\end{table}\n With the video cameras, we captured videos at 30 fps. For the object detection task, we extract at most three frames per second of these videos to avoid having redundant occurrences of frames.\n See Section \\ref{sec:datasetstatistics} for information on the distribution of images with respect to different cameras.\n \n Lastly, we captured top-down looking multi-spectral imagery at 1 fps. We used a MicaSense RedEdge-MX, which records five wavelengths (475 nm, 560 nm, 668 nm, 717 nm, 842 nm). Therefore, in addition to the RGB channels, the recordings also contain a RedEdge and a Near Infrared channel. The camera was referenced with a white reference before each flight. As the RedEdge-MX captures every band individually, we merge the bands using the development kit provided by MicaSense. \n\n\n\\subsection{Meta Data Collection}\n\n\\begin{table}\t\n\t\\begin{center}\t\t\n\t\t\\begin{tabular}{cccc}\n\t\t\tData  & Unit & Min. value & Max.value  \\\\\n\t\t\t\\hline\n\t\t\n\t\t\tTime since start & ms & 0 & $\\infty$ \\\\\n\t\t\tDate and Time & ISO 8601 & -- & -- \\\\\n\t\t\tLatitude & degrees & $-90$ & $+90$ \\\\\n\t\t\tLongitude & degrees & $-90$ & $+90$ \\\\\n\t\t\tAltitude & meters & $0$ & $\\infty$ \\\\\n\t\t\tGimbal pitch & degrees & $0$ & 90 \\\\\n\t\t\tUAV roll & degrees & $-90$ & $+90$ \\\\\n\t\t\tUAV pitch & degrees & $-90$ & $+90$ \\\\\n\t\t\tUAV yaw & degrees & $-180$ & $+180$ \\\\\n\t\t\t$x$-axis speed & m\/s & $0$ & $\\infty$ \\\\\n\t\t\t$y$-axis speed & m\/s & $0$ & $\\infty$ \\\\\n\t\t\t$z$-axis speed& m\/s & $0$ & $\\infty$\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Meta data that comes with every image\/frame.}\n\t\\label{table:meta_data}\n\\end{table}\n\nAccompanied with every frame there is a meta stamp, that is logged at 10 hertz. To align the video data (30 fps) and the time stamps, a nearest neighbor method was performed. The data in Table \\ref{table:meta_data} is logged and provided for every image\/frame read from the onboard clock, barometer, IMU and GPS sensor, and the gimbal, respectively.\n\\iffalse\n\\begin{itemize}\n\t\\itemsep-0.5em\n\t\\item $d$: current date and time of capture\n\t\\item $t$: relative time stamp since beginning of capture\n\t\\item $la$: latitude of the UAV\n\t\\item $lo$: longitude of the UAV\n\t\\item $a$: altitude of the UAV\n\t\\item $\\alpha$: camera pitch angle (viewing angle)\n\t\\item $\\phi$: UAV roll angle\n\t\\item $\\theta$: UAV pitch angle\n\t\\item $\\psi$: UAV yaw angle \n\t\\item $V_x$: speed along the $x$-axis\n\t\\item $V_y$: speed along the $y$-axis\n\t\\item $V_z$: speed along the $z$-axis\n\\end{itemize}\n\\fi\n\n Note that $\\alpha=90^\\circ$ corresponds to a top-down view, and $\\alpha=0^\\circ$ to a horizontally facing camera. The date format is given in the extended form of ISO 8601.\nFurthermore, note that the UAV roll\/pitch\/yaw-angles are of minor importance for meta-data-aware vision-based methods as the onboard gimbal filters out movement by the drone such that the camera pitch angle is roughly constant if it is not intentionally changed  \\cite{jkedrasiak2013prototype}. Note that the gimbal yaw angle is not included, as we fix it to coincide with the UAV's yaw angle.\n\nWe need to emphasize that the meta values lie within the error thresholds introduced by the different sensors, but an extended analysis is beyond the scope of this paper (see \\eg \\cite{zimmermann2017precise,webinar2020gps,kulhavy2017accuracy} for an overview).\n\n\n\n\\subsection{Annotation Method}\n\nUsing the non-commercial labeling tool DarkLabel \\cite{darklabel}, we manually and carefully annotated all provided images and frames with the categories swimmer (person in water without life jacket), floater (person in water with life jacket), life jacket, swimmer$^\\dagger$ (person on boat without life jacket), floater$^\\dagger$ (person on boat with life jacket), and boats. We note that it is not sufficient to infer the class floater by the location from swimmer and life jacket as this can be highly ambiguous. Subsequently, all annotations were checked by experts in aerial vision. We choose these classes as they are the hardest and most critical to detect in SAR missions. Furthermore, we annotated regions with other objects as ignored regions, such as boats on land. Moreover, the  data  set  also  covers unlabeled objects, which may not be of interest, like driftwood, birds or the coast such that detectors can be robust to distinguish from those objects. Our guidelines for the annotation are described in the appendix. See Figure \\ref{fig:objects_examples} for examples of objects.\n\n\n\n\\begin{figure*}\n\t\\centering\n\n\t\\setlength{\\tabcolsep}{2pt}\n\t\t\t\\begin{tabular}{ccccc}\n\t            \n\t\t\t\t\n\t\t\t\n\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{floater_example1_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Floater}}\\end{overpic}\t\t\t\t\n\t\t\t\n\t\t\t\t&\n\t\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{floater_example2_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Floater}}\\end{overpic}\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\hphantom{abc}\n\t\t\t\t&\n\t\t\t\n\n\t\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{floater_example3_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Swimmer}}\\end{overpic}\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{floater_example4_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Swimmer}}\\end{overpic}\n\t\t\t\t\n\t\t\t\t\\hphantom{abc} &\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{human_on_boat_with_jacket_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Floater$^\\dagger$}}\\end{overpic}\n\t\t\t\t\n\t\t\t\t\\\\\t\t\t\n\t\t\n\t\t\t\n\t\t\t\t\t\t\n\t\t\n\t\t\t\n\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{floater_example5_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Floater}}\\end{overpic}\t\t\t\n\t\t\t\n\t\t\t&\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{life_jacket_l.png} \\put (0,6) {\\large \\colorbox{blue!30}{Life jacket}}\\end{overpic}\n\t\t\t\n\t\t\t\t\n\t\t\t\\hphantom{abc}\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t&\n\t\t\t\n\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{swimmer_example4_l.png} \\put (0,3) {\\large \\colorbox{blue!30}{Swimmer}}\\end{overpic}\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{swimmer_example2_l.png} \\put (0,4) {\\large \\colorbox{blue!30}{Swimmer}}\\end{overpic}\n\t\t\t\t\n\t\t\t\\hphantom{abc} &\n\t\t\n\t\t\t\\begin{overpic}[width=31mm,height=30mm,tics=10]{human_on_boat_better_l.png} \\put (0,3) {\\large \\colorbox{blue!30}{Swimmers$^\\dagger$}}\\end{overpic}\n\t\t\t\\end{tabular}\n\n\n\t\n\t\\caption{Examples of objects. Note that these examples are crops from high-resolution images. However, as the objects are small and the images taken from high altitudes, they appear blurry.\\protect\\\\\n}\n\t\\label{fig:objects_examples}\n\t\\vspace{-0.6cm}\n\n\\end{figure*}\n\n\n\n\n\\subsection{Data Set Split}\n\\label{sec:datasetsplit}\n\n\\subsubsection*{Object Detection}\nTo ensure that the training, validation, and testing set have similar statistics, we roughly balance them such that the respective subsets have similar distributions with respect to altitude and angle of view, two of the most important factors of appearance changes. Of the individual images, we randomly select $\\nicefrac{4}{7}$ and add it to the training set, add $\\nicefrac{1}{7}$ to the validation set and another $\\nicefrac{2}{7}$ to the testing set. In addition to the individual images, we randomly cut every video into three parts of length $\\nicefrac{4}{7}$, $\\nicefrac{1}{7}$, and $\\nicefrac{2}{7}$ of the original length and add every 10-th frame of the respective parts to the training, validation, and testing set. This is done to avoid having subsequent frames in the training and testing set such that a realistic evaluation is possible. We release the training and validation set with all annotations and the testing set's images, but withhold its annotations. Evaluation will be available via an evaluation server, where the predictions on the test set can be uploaded.\n\\vspace*{-3mm}\n\n\n\\subsubsection*{Object Tracking}\n\nSimilarly, we take $\\nicefrac{4}{7}$ of our recorded clips as the training clips, $\\nicefrac{1}{7}$ as the validation clips and $\\nicefrac{2}{7}$ as the testing clips. As for the object detection task, we withhold the annotations for the testing set and provide an evaluation server.\n\n\n\n\\section{Data Set Tasks}\n\\label{sec:datasetstatistics}\n\n\nThere are many works on UAV-based maritime SAR missions, focusing on unified frameworks describing the process of how to search and rescue people \\cite{mishra2020drone,gallego2019detection,lvsouras2020new,lygouras2019unsupervised,queralta2020autosos,roberts2016unmanned,ghazali2016determining}. These works answer questions corresponding to path planning, autonomous navigation and efficient signal transmission. Most of them rely on RGB sensors and detection and tracking algorithms to actually find people of interest. This commonality motivates us to extract the specific tasks of object detection and tracking, which pose some of the most challenging issues in this application scenario.\n\nMaritime environments from a UAV's perspective are difficult for a variety of reasons: Reflective regions and shadows resulting from different cardinal points (such as in Fig. \\ref{fig:front_image}) that could lead to false positives or negatives; people may be hardly visible or occluded by waves or sea foam (see Supplementary material); typically large areas are overseen such that objects are particularly small \\cite{mishra2020drone}. We note that these factors are on top of general UAV-related detection difficulties.\n\nNow, we proceed to describe the specific tasks.\n\n\n\n\n\n\n\n\n\\subsection{Object Detection}\n\\label{sec:od_statistics}\n\nThere are 5,630 images (training: 2,975; validation: 859; testing: 1,796). See Figure \\ref{fig:camera_and_class_distribution} for the distribution of images\/frames with respect to cameras and the class distribution. We recorded most of the images with the L1D-20c and UMC-R10C, having the highest resolution. Having the lowest resolution, we recorded only 432 images with the RedEdge-MX. Note, for the Object Detection Task only the RGB-channels of the multi-spectral images are used to support a uniform data structure.\n\n Furthermore, the class distribution is slightly skewed towards the class 'boat', since safety precautions require boats to be nearby. We emphasize that this bias can easily be diminished by blackening the respective regions, as is common for areas which are not of interest or undesired (such as boats here; see \\eg \\cite{du2018unmanned}). Right after that, swimmers with life jacket are the most common objects. We argue that this scenario is very often encountered in SAR missions. This type of class often is easier to detect than just swimmer as life jackets mostly are of contrasting color, such as red or orange (see Fig. \\ref{fig:objects_examples} and Table \\ref{table:od_results}). However, as it is also a likely scenario to search for swimmers without life jacket, we included a considerable amount. There are also several different manifestations\/visual appearances of that class which is why we recorded and annotated swimmers with and without adequate swimwear (such as wet suit). To be able to discriminate between humans in water and humans on boats, we also annotated humans on boats (with and without life jackets). Lastly, we annotated a small amount of life jackets only. However, we note that the discrimination between life jackets and humans in life jackets can become visually ambiguous, especially in higher altitudes. See also Fig. \\ref{fig:objects_examples}.  \n\n\n\\begin{figure}\n\t\n\t\\centering\n\t\n\t\\includegraphics[scale=0.5,trim=0 0 0 8,clip]{camera_image_distribution.pdf}\t\n\t\\includegraphics[scale=0.5]{class_distribution.pdf}\n\t\n\n\n\t\n\t\\caption{Distribution of training images over camera types (left) and distribution of objects over classes (right).}\n\t\\label{fig:camera_and_class_distribution}\n\t%\n\\end{figure}\n\nFigure \\ref{fig:meta_distribution} shows the distribution of images with respect to the altitude and viewing angle they were captured at. Roughly 50\\% of the images were recorded below 50 m because lower altitudes allow for the whole range of available viewing angles ($0-90^\\circ$). That is, to cover all viewing angles, more images at these altitudes had to be taken. On the other hand, there are many images facing downwards ($90^\\circ$), because images taken at greater altitudes tend to face downwards since acute angles yield image areas with tiny pixel density, which is unsuitable for object detection. Nevertheless, every altitude and angle interval is sufficiently represented.\n\n\n\n\\begin{figure}\n\t\\vspace*{-3mm}\n\t\\centering\n\t\\includegraphics[scale=0.5]{altitude_angle_distribution.pdf}\t\n\n\t\n\t\n\t\\caption{Distribution of images over altitudes (left) and angles (right), respectively.}\n\t\\label{fig:meta_distribution}\n\t\\vspace*{-5mm}\n\t%\n\\end{figure}\n\n\n\n\n\\subsection{Single-Object Tracking}\n\nWe provide 208 short clips ($>$4 seconds) with a total of 393,295 frames (counting the duplicates), including all available objects labeled. We randomly split the sequences into 58 training, 70 validation and 80 testing sequences. We do not support long-term tracking. The altitude and angle distributions are similar to these in the object detection section since the origin of the images of the object detection task is the same.\n\n\\subsection{Multi-Object Tracking}\n\nWe provide 22 clips with a total of 54,105 frames and 403,192 annotated instances, the average consists of 2,460 frames. We differentiate between two use-cases. In the first task, only the persons in water (floaters and swimmers) are tracked, it is called MOT-Swimmer. In the second task, all objects in water are tracked (also the boats, but not people on boats), called MOT-All-Objects-In-Water. In both tasks, all objects are grouped into one class. The data set split is performed as described in section \\ref{sec:datasetsplit}.\n\n\\subsection{Multi-Spectral Footage}\n\\label{sec:multi_spectral}\n\nAlong with the data for the three tasks, we provide multi-spectral images. We supply annotations for all channels of these recordings, but only the RGB-channels are currently part of the Object Detection Task. \nThere are 432 images with 1,901 instances. See Figure \\ref{fig:front_image} for an example of the individual bands.\n\n\\section{Evaluations}\n\\label{sec:evaluations}\n\nWe evaluate current state-of-the-art object detectors and object trackers on SeaDronesSee. All experiments can be reproduced by using our provided code available on the evaluation server. Furthermore, we refer the reader to the Supplementary Material for the exact form and uploading requirements.\n\n\n\\subsection{Object Detection}\n\\label{sec:evaluation_od}\n\n\n\n\n\n\\begin{table*}[h!]\t\n\t\\begin{center}\t\t\n\t\t\\begin{tabular}{c|ccccc|cccccc|c}\n\t\t\t\\multicolumn{1}{c!{\\vline height 1.1\\ht\\strutbox}}{Model}  & AP & AP$_{50}$ & AP$_{75}$ & AR$_{1}$ & {AR$_{10}$}  & S & F & S$^\\dagger$ & F$^\\dagger$ & B & LJ & FPS \\\\\n\t\t\t\\hline\n\t\t\tF. ResNeXt-101-FPN \\cite{xie2017aggregated}  & 30.4 & 54.7 & 29.7 & 18.6 & 42.6 & 78.1 & 82.4 & 25.9 & 44.3 & 96.7 & 0.6 & 2\\\\\n\t\t\tF. ResNet-50-FPN \\cite{girshick2015fast} & 14.2 & 30.1 & 7.2 & 6.4 & 17.7 & 24.6 & 54.1 & 4.9 & 7.5 & 89.2 & 0.3 & 14 \\\\ \\hline\n\t\t\tCenterNet-Hourglass104 \\cite{zhou2019objects} & 25.6 & 50.3 & 22.2 & 17.7 & 40.1 & 65.1 & 73.6 & 19.1 & 48.1 & 95.8 & 0.3 & 6 \\\\\n\t\t\tCenterNet-ResNet101 \\cite{zhou2019objects} & 15.1 & 36.4 & 10.8 & 9.6 & 21.4 & 16.8 & 39.8 & 0.8 & 1.7 & 74.3 & 0 &  22 \\\\\n\t\t\n\t\t\tCenterNet-ResNet18 \\cite{zhou2019objects} & 9.9 & 21.8 & 9.0 & 7.2 & 19.7 & 20.9 & 21.9 & 2.6 & 3.3 & 81.9 & 0.4 & 78 \\\\ \n\t\t\t\\hline\n\t\t\tEfficientDet--$D0$ \\cite{tan2020efficientdet} & 20.8 & 37.1 & 20.6 & 11.5 & 29.1 & 65.3 & 55.1 & 3.1 & 3.3 & 95.5 & 0.1 & 26\n\t\t\n\t\t\t\t\n\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Average precision results for several baseline models. The right part contains AP$_{50}$--values for each class individually. All reported FPS numbers are obtained on a single Nvidia RTX 2080 Ti. The abbreviation 'F.' stands for Faster R-CNN. For visualization purposes, the classes are abbreviated as swimmer($^\\dagger$) $\\rightarrow$ S($^\\dagger$), floater($^\\dagger$) $\\rightarrow$ F($^\\dagger$), boat $\\rightarrow$ B, life jacket $\\rightarrow$ LJ. }\n\t\\label{table:od_results}\n\t\\vspace{-0.4cm}\n\n\\end{table*}\n\n\n\nThe used detectors can be split into two groups. The first group consists of two-stage detectors, which are mainly built on Faster R-CNN \\cite{girshick2015fast} and its improvements. Built for optimal accuracy, these models often lack the inference speed needed for real-time employment, especially on embedded hardware, which can be a vital use-case in UAV-based SAR missions. For that reason, we also evaluate on one-stage detectors. In particular, we perform experiments with the best performing single-model (no ensemble) from the workshop report \\cite{zhu2018visdrone}: a Faster R-CNN with a ResNeXt-101 64-4d \\cite{xie2017aggregated} backbone with P6 removed. For large one-stage detectors, we take the recent CenterNet \\cite{zhou2019objects}. To further test an object detector in real-time scenarios, we choose the current best model family on the COCO test-dev according to \\cite{paperswithcode}, i.e. EfficientDet \\cite{tan2020efficientdet}, and take the smallest model, $D0$, which can run in real-time on embedded hardware, such as the Nvidia Xavier \\cite{kiefer2021leveraging}. We refer the reader to the appendix for the exact parameter configurations and training configurations of the individual models.\n\nSimilar to the VisDrone benchmark \\cite{zhu2018vision}, we evaluate detectors according to the COCO json-format \\cite{lin2014microsoft}, i.e. average precision at certain intersection-over-union-thresholds. More specifically, we use AP$=$AP$^{\\text{IoU=0.5:0.05:0.95}}$, AP$_{50}=$AP$^{\\text{IoU=0.5}}$ and AP$_{75}=$AP$^{\\text{IoU=0.75}}$. Furthermore, we evaluate the maximum recalls for at most 1 and 10 given detections, respectively, denoted AR$_1=$AR$^{\\text{max=1}}$, and AR$_{10}=$AR$^{\\text{max=10}}$. All these metrics are averaged over all categories (except for \"ignored region\"). We furthermore provide the class-wise average precisions. \nMoreover, similar to \\cite{kiefer2021leveraging}, we report AP$_{50}$-results on different equidistant levels of altitudes 'low' = 5-56 m (L), 'low-medium' = 55-106 m (LM), 'medium' = 106-157 m (M), 'medium-high' = 157-208 m (MH), and 'high' = 208-259 m (H). To measure the universal cross-domain performance, we report the average over these domains, denoted AP$_{50}^\\text{avg}$. Similarly, we report AP$_{50}$-results for different angles of view: 'acute' = 7-23$^\\circ$ (A), 'acute-medium' = 23-40$^\\circ$ (AM), 'medium' = 40-56$^\\circ$ (M), 'medium-right' = 56-73$^\\circ$ (MR), and 'right' = 73-90$^\\circ$ (R). Ultimately, it is the goal to have robust detectors across all domains uniformly, which is better measured by the latter metrics.\n\nTable \\ref{table:od_results} shows the results for all object detection models. As expected, the large Faster R-CNN with ResNeXt-101 64-4d backbone performs best, closely followed by CenterNet-Hourglass104. Medium-sized networks, such as the ResNet-50-FPN, and fast networks, such as CenterNet-ResNet18 and EfficientDet-$D0$, expectedly perform worse. However, the latter can run in real-time on an Nvidia Xavier \\cite{kiefer2021leveraging}. Swimmers are detected significantly worse than floaters by most detectors. Notably, life jackets are very hard to detect since from a far distance these are easily confused with swimmers$^\\dagger$ (see Fig. \\ref{fig:objects_examples}). Since there is a heavy class imbalance with many fewer life jackets, detectors are biased towards floaters.\n\nTable \\ref{table:domain_aps_altitude} and \\ref{table:domain_aps_angle} show the performances for different altitudes and angles, respectively. These evaluations help assess the strength and weaknesses of individual models. For example, although ResNeXt-101-FPN performs overall better than Hourglass104 in AP$_{50}$ (54.7 vs. 50.3), the latter is better in the domain of medium angles (45.2 vs. 49.7). Furthermore, the great performance discrepancy between CenterNet-ResNet101 and CenterNet-ResNet18 in AP$_{50}$ (36.4 vs. 21.8) vanishes when averaged over angle domains  (23.8 vs. 23.1 AP$_{50}^{\\text{avg}}$) possibly indicating ResNet101's bias towards specific angle domains.  \n\n\n\\begin{table}\n\t\n\n\t\t\\resizebox{240pt}{!}{%\n\t\t\t\\begin{tabular}{c|ccccc|c}\n\t\t\t\t\t\t\t\tModel &  L & LM & M & MH & H & AP$_{50}^{\\text{avg}}$ \\\\ \\hline \n\t\t\t\t\n\t\t\t\t\n\t\t\t\tResNeXt-101-FPN & 56.8 & 54.6 & 49.2 & 65 & 78.3 & 60.8\\\\ \n\t\t\t\tResNet-50-FPN & 32.8 & 29.8 & 23.5 & 40.5 & 48.9 & 35.1 \\\\ \\hline\n\t\t\t\tHourglass104 & 50.6 & 52.0 & 47.5 & 64.9 & 73.2 & 57.6\\\\\n\t\t\t\tResNet101 & 20.2 & 30.4 & 24.1 & 35.1 & 38.0 & 29.6\\\\\n\t\t\t\tResNet18 & 23.8 & 20.3 & 19.2 & 29.3 & 31.9 & 24.9\\\\ \\hline\n\t\t\t\t$D0$ & 39.6 & 38.0 & 30.4 & 42.5 & 54.5 & 41.0\n\t\t\t\t\n\t\t\\end{tabular}}\n\n\t\\caption{Results on different altitude-domains. E.g. ResNeXt's AP$_{50}$ performance in low-medium (LM) altitudes is 54.6 AP$_{50}$.}\n\t\\label{table:domain_aps_altitude}\n\n\\vspace*{3mm}\n\n\t\n\n\t\t\\resizebox{240pt}{!}{%\n\t\t\t\\begin{tabular}{c|ccccc|c}\n\t\t\t\t\t\t\t\tModel &  A & AM & M & MR & R & AP$_{50}^{\\text{avg}}$ \\\\ \\hline \n\t\t\t\t\n\t\t\t\t\n\t\t\t\tResNeXt101-FPN & 68.3 & 55.1 & 45.2 & 63.6 & 51.5 & 56.7\\\\ \n\t\t\t\tResNet50-FPN & 32.8 & 35.5 & 32.7 & 35.7 & 27.6 & 32.9 \\\\ \\hline\n\t\t\t\tHourglass104 & 66.4 & 42.1 & 49.7 & 58.7 & 46.9 & 52.76\\\\\n\t\t\t\tResNet101 & 7.4 & 35.8 & 20.5 & 33.6 & 21.7 & 23.8\\\\\n\t\t\t\tResNet18 & 9.6 & 29.5 & 26.3 & 27.9 & 22.1 & 23.1\\\\ \\hline\n\t\t\t\t$D0$ & 26.9 & 47.0 & 40.5 & 40.3 & 36.8 & 38.3\n\t\t\t\t\n\t\t\\end{tabular}}\n\n\t\\caption{Results on different angle-domains. For example, ResNeXt's AP$_{50}$ performance in medium-right (MR) angles (57-73$^\\circ$) is 63.6 AP$_{50}$.}\n\t\\label{table:domain_aps_angle}\n\t\\vspace*{-3mm}\n\\end{table}\n\n\n\\subsection{Single-Object Tracking}\n\n\\begin{table*}\n\t\\begin{center}\t\t\n\t\t\\begin{tabular}{c|ccccccccccccccc}\n\t\t\tModel & MOTA & IDF1 & MOTP & MT & ML & FP & FN & Recall & Prcn & ID Sw. & Frag \\\\\n\t\t\t\\hline\n\t\t\n\t\t\tFairMOT-D34 \\cite{zhang2020fairmot} & 39.0 & 44.8 & 23.6 & 17 & 17 & 3,604 & 9,445 & 57.2 & 77.8 & 307 & 1,687 \\\\\n\t\t\tFairMOT-R34 \\cite{zhang2020fairmot} & 15.2 & 27.6 & 33.7 & 6 & 37 & 2,502 & 12,592 & 30.1 & 68.4 & 181 & 807 \\\\\n\t\t\tTracktor++ \\cite{tracktor_2019_ICCV} & 55.0 & 69.6 & 25.6 & 62 & 4 & 7,271 & 3,550 & 85.5 & 74.2 & 165 & 347\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Multi-Object Tracking evaluation results for the \\textbf{Swimmer} task.}\n\n\n\n\n\t\\begin{center}\t\t\n\t\t\\begin{tabular}{c|ccccccccccccccc}\n\t\t\tModel & MOTA & IDF1 & MOTP & MT & ML & FP & FN & Recall & Prcn & ID Sw. & Frag \\\\\n\t\t\t\\hline\n\t\t\tFairMOT-D34 \\cite{zhang2020fairmot} & 36.5 & 43.8 & 20.9 & 28 & 49 & 3,788 & 20,867 & 47.2 & 83.1 & 447 & 1,599 \\\\\n\t\t\tFairMOT-R34 \\cite{zhang2020fairmot} & 30.5 & 40.8 & 27.3 & 29 & 127 & 4,401 & 28,999 & 40.2 & 81.6 & 285 & 1,588 \\\\\n            Tracktor++ \\cite{tracktor_2019_ICCV} & 71.9 & 80.5 & 20.1 & 123 & 5 & 7,741 & 5,496 & 88.5 & 84.5 & 192 & 438\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Multi-Object Tracking evaluation results for the \\textbf{All-Objects-In-Water} task.}\n\t\\label{table:mot_results}\n\t\\vspace{-0.4cm}\n\n\\end{table*}\n\n\nLike VisDrone \\cite{zhu2020vision}, we provide the success and precision curves for single-object tracking and compare models based on a single number, the success score. As comparison trackers, we choose the DiMP family (DiMP50, DiMP18, PrDiMP50, PrDiMP18) \\cite{bhat2019learning,danelljan2020probabilistic} and Atom \\cite{danelljan2019atom} because they were the foundation of many of the submitted trackers to the last VisDrone workshop \\cite{fan2020visdrone}.\n\n\n\n\n\n\\begin{figure}\n\t\n\t\\centering\n\t\\includegraphics[scale=0.16]{success_plot.pdf}\t\n\t\\includegraphics[scale=0.16]{precision_plot.pdf}\t\n\t\n\t\\caption{Success and precision plots for single-object tracking task (best viewed in color).}\n\t\\label{fig:sot_results}\n\t\\vspace*{-0.4cm}\n\t%\n\\end{figure}\n\n\n\nFigure \\ref{fig:sot_results} shows that the PrDiMP- and DiMP-family expectedly outperform the older Atom tracker in both, success and precision. Surprisingly, PrDiMP50 slightly trails the accuracy of its predecessor DiMP50. Furthermore, all trackers' performances on SeaDronesSee are similar or worse than on UAV123 (\\eg Atom with 65.0 success) \\cite{bhat2019learning,danelljan2020probabilistic,danelljan2019atom}, for which they were heavily optimized. We argue that in SeaDronesSee there is still room for improvement, especially considering that the clips feature precise meta information that may be helpful for tracking. Furthermore, in our experiments, the faster trackers DiMP18 and Atom run at approximately 27.1 fps on an Nvidia RTX 2080 Ti. However, we note that they are not capable of running in real-time on embedded hardware, a use-case especially important for UAV-based SAR missions. \n\n\n\\begin{table}\n\t\n\n\t\t\\resizebox{240pt}{!}{%\n\t\t\t\\begin{tabular}{c|ccccc|c}\n\t\t\t\t\t\t\t\tModel &  L & LM & M & MH & H & AP$_{50}^{\\text{avg}}$ \\\\ \\hline \n\t\t\t\t\n\t\t\t\t\n\n\n\t\t\t\t\\hline\n\t\t\t\tF. ResNet-50-FPN &\\bf 32.8 & 29.8 & 23.5 & 40.5 &\\bf 48.9 & 35.1\\\\ \n\t\t        5$\\times$Altitude@3\\cite{kiefer2021leveraging} &\\bf 32.8 &\\bf 29.9 &\\bf 26.2 &\\bf 41.5 &\\bf 48.9 &\\bf 35.9 \\\\\n\n\t\t\t\t\n\t\t\\end{tabular}}\n\n\n\t\t\t\\resizebox{240pt}{!}{%\n\t\t\t\\begin{tabular}{c|ccccc|cc}\n\t\t\t\t\t\t\t\tModel &  A & AM & M & MR & R & AP$_{50}^{\\text{avg}}$ \\\\ \\hline \n\t\t\t\t\n\t\t\t\t\n\t\t\n\t\t\t\t\\hline\n\t\t\t\tF. ResNet-50-FPN & 32.8 &\\bf 35.5 & 32.7 &\\bf 35.7 & 27.6 & 32.9\\\\ \n\t\t        5$\\times$Angle@3\\cite{kiefer2021leveraging}& \\bf 42.0 &\\bf 35.5 &\\bf 39.3 &\\bf 35.7 &\\bf 27.7 &\\bf 36.0   \\\\\n\n\t\t\t\t\n\t\t\\end{tabular}}\n\n\t\\caption{Results on different altitude- and angle-domains.\n\t\n\t\\label{table:domain_aps_leveraging}\n\t\\vspace{-0.4cm}\n\\end{table}\n\n\n\n\n\\subsection{Multi-Object Tracking}\n\n\nWe use a similar evaluation  protocol as the MOT benchmark \\cite{milan2016mot16}. That is, we report results for Multiple Object Tracking Accuracy (MOTA), Identification F1 Score (IDF1), Multiple Object Tracking Precision (MOTP), number of false positives (FP), number of false negatives (FN), recall (R), precision (P), ID switches (ID sw.), fragmentation occurrences (Frag). We refer the reader to \\cite{ristani2016performance} or the appendix for a thorough description of the metrics. \\\\\nWe train and evaluate FairMOT \\cite{zhang2020fairmot}, a popular tracker, which is the base of many trackers submitted to the challenge \\cite{fan2020visdrone2}. FairMOT-D34 employs a DLA34 \\cite{yu2018deep} as its backbone while FairMOT-R34 makes use of a ResNet34. Another SOTA tracker is Tracktor++ \\cite{tracktor_2019_ICCV}, which we also use for our experiments. It performed well on the MOT20 \\cite{Dendorfer} challenge and is conceptually simple. \\\\\nSurprisingly, Tracktor++ was better than FairMOT in both tasks. One reason for this may be the used detector. Tracktor++ utilizes a Faster-R-CNN with a ResNet50 backbone. In contrast, FairMOT is using a CenterNet with a DLA34 and a ResNet34 backbone, respectively.\n\n\n\\iffalse\n\\begin{figure}\n\t\n\t\\centering\n\n\t\\includegraphics[scale=0.135,trim=0 600 500 0,clip]{2760_pred.png}\n\t\n\t\n\t\n\t\\caption{Example predictions. Unseen wave patterns result in false positive detections: The detector falsely detects swimmers (light green bounding boxes) in the right part of the image.}\n\t\\label{fig:false_positives}\n\t%\n\\end{figure}\n\\fi\n\n\n\n\\subsection{Meta-Data-Aware Object Detector}\n\nDeveloping meta-data-aware object detectors is difficult since there are no large-scale data sets to evaluate their performances. However, some works provide promising preliminary results using this metadata \\cite{wu2019delving,messmer2021gaining,kiefer2021leveraging}. We provide an initial baseline from \\cite{kiefer2021leveraging} incorporating the meta data. We evaluate the performances of 5$\\times$Altitude@3- and 5$\\times$Angle@3-experts, which are constructed on top of a Faster R-CNN with ResNet-50-FPN, respectively. Essentially, these experts make use of meta-data by allowing the features to adapt to their responsible specific environmental domains.\n\nAs Table \\ref{table:domain_aps_leveraging} shows, meta data can enhance the accuracy of an object detector considerably. For example, 5$\\times$Angle@3 outperforms its ResNet-50-FPN baseline by $3.1$ AP$^{\\text{avg}}_{50}$ while running at the same inference speed. The improvements are especially significant for underrepresented domains, such as $+9.2$ and $+6.4$ AP$^{\\text{avg}}_{50}$ for the acute angle (A) and the medium angle (M), respectively, which are underrepresented as can be seen from Fig. \\ref{fig:meta_distribution}.\n\n\n\n\\section{Conclusions}\n\nThis work serves as an introductory benchmark in UAV-based computer vision problems in maritime scenarios. We build the first large scaled-data set for detecting and tracking humans in open water. Furthermore, it is the first large-scaled benchmark providing full environmental information for every frame, offering great opportunities in the so-far restricted area of multi-modal object detection and tracking. \nWe offer three challenges, object detection, single-object tracking, and multi-object tracking by providing an evaluation server. We hope that the development of meta-data-aware object detectors and trackers can be accelerated by means of this benchmark. \nMoreover, we provide multi-spectral imagery for detecting humans in open water. These images are very promising in maritime scenarios, having the ability to capture wavelengths, which set apart objects from the water background.\n\n\n\n\n\\section*{Acknowledgment}\nWe would like to thank Sebastian Koch, Hannes Leier and Aydeniz Soezbilir, without whose contribution this work would not have been possible.\\\\\nThis work has been supported by the German Ministry for Economic\nAffairs and Energy, Project Avalon, FKZ: 03SX481B.\n\n\\newpage\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}\nWith the increasing requirements for real-time transaction and analysis processing, traditional disk-based data management techniques are no longer applicable to these scenarios. \nIn order to meet the requirements for real-time performance of critical business, in-memory database (IMDB) came into being. \nMore and more flexible and efficient in-memory data storage and access methods are used to increase system throughput and reduce response time.\n\nAs an important component of the database system, indexing technology has always been a research hotspot in the field of IMDB. \nA large number of index structures for memory data are proposed.\nCompared with early work such as T tree \\cite{LehmanC1986:T-tree}, CSB+ tree \\cite{Rao2000:csb+-tree}, and CSS tree \\cite{RaoR1999:css-tree}, state-of-the-art in-memory indexes like FAST \\cite{KimCSSNKLBD2010:fast-index}, Masstree \\cite{MaoKM2012:mass-tree}, BwTree \\cite{LevandoskiLS2013a:BwTree}, PSL \\cite{XieCJOW2017:PSL} and ART \\cite{LeisKN2013:ART-index} achieve better performance by making good use of concurrent synchronization and new hardware technologies \\cite{XieCCMZ2018:evaluation-imdb-index}.\nThe ART (Adaptive Radix Tree) is a representative one of these indexes which shows a competitive small memory footprint and overall performance especially for dense dataset by building a trie structure tree with adaptive variable length type internal nodes and supporting efficient SIMD processing \\cite{XieCCMZ2018:evaluation-imdb-index}. \n\nAlthough indexes can improve query efficiency, for IMDB, there are more details to be considered, as it is inherently designed to pursue more rapid system response. \nFirstly, the time overhead of index construction cannot be ignored. \nObviously, a one-time construction of the entire index may cause the system to become unavailable for a short period of time. \nSecondly, the effectiveness of an index is largely determined by the actual query workload. \nIndexes built in the absence of query workload information are likely to be time-consuming, space-intensive, and what's more may even be of no use. \nThirdly, index reconstruction due to data updates can also increase system response time.\n\n\nThe database cracking technology \\cite{IdreosKM2007:DB-cracking} provides a way to solve these problems. \nInstead of building a complete index in the preprocessing stage, it builds and refines the index along with the query processing. \nThus, the cracker index also enables query workload aware, and hence avoids the case of ineffective indexing. \nAt the same time, such dynamic index construction mechanism can better adapt to data updates. \nThe philosophy behind the database cracking is to delay unavoidable changes as far as possible \\cite{IdreosKM2007:updating-cracked}.\nHowever, as traditional database cracking technology was originally designed towards simple array alike column data structures, it alone is uncompetitive compared to modern IMDB indexes. \nRecent work \\cite{SchuhknechtJD2016:evaluation-cracking} shows that the data lookup speed of ART index is 3.6 times faster than the traditional database cracking method after 1M queries. \nTherefore, an ideal solution might be applying database cracking technology to modern IMDB index construction to further improve the overall system responsiveness (considering the overhead of index construction and update, and the index effectiveness). \nTo the best of our knowledge, there is currently no specific research on database cracking for complex index structures. \nThus, it is necessary to study the feasibility and relevant general techniques of the complex index cracking.\nFor this reason, a preliminary case study is conducted on the ART index in this paper, which will be representative for in-memory index.\n\nWe studied the construction overhead of ART index and proposed an algorithm that cracks ART with the help of auxiliary data structures. The main contributions of this paper are as follows.\n\\begin{enumerate}\n    \\item We investigated the impact of data ordering on the construction of ART index and the range lookup performance of ART index. \n    We found that cracking technology can improve the construction of ART because ordered data come into being gradually during the cracking process; \n   \n    \\item We  proposed a cracking algorithm for the Adaptive Radix Tree with the help of auxiliary data structures which has a low index initialization overhead and guarantees to eventually form a complete ART index in the process of constant queries. \n    The algorithm is easy to implement and is applicable to other complex index structures that support range query. \n    Furthermore, in order to improve the update performance, caching and shuffling techniques are introduced. \n    \\item Extensive experiments were conducted under different workloads on two datasets, \\textit{i}.\\textit{e}.,  a synthetic dataset and the YCSB \\cite{CooperSTRS2010:YCSB} benchmark. \n    The experimental results were analyzed to show the feasibility of the proposed cracking algorithm from the convergence speed, response time, and selectivity.\n\\end{enumerate}\n\nThe organization of this paper is as follows: \nSection 2 introduces related work; \nSection 3 discusses the feasibility of cracking ART index by analyzing the construction process of the index and the characteristics of query processing on it; \nSection 4 describes the design and implementation of the ART cracker in detail;\nSection 5 shows and analyzes the experimental results; \nFinally, conclusions are given in Section 6.\n\n\\section{Related Work}\\label{sec:relate-work}\nMore and more index structures are proposed for in-memory database systems.\nRao \\textit{et al}.  \\cite{Rao2000:csb+-tree} proposed the Cache-Sensitive B+-Tree (CSB+-Tree) that retains the good cache behavior of CSS-Trees while at the same time being able to support incremental updates.\nHankins \\textit{et al}.  \\cite{Hankins2003:csb-tree} explored the effect of node size on the performance of CSB+-trees and found that using node sizes larger than acache line size (i.e., larger than 512 bytes) produces better searchperformance. While trees with nodes that are of the same size as a cache line have the minimum number of cache misses, they found that TLB misses are much higher than on trees with large node sizes, thus favoring large node sizes.\nKim \\textit{et al}.  \\cite{KimCSSNKLBD2010:fast-index} proposed the Fast Architecture Sensitive Tree (FAST),a binary tree logically organized to optimize for architecture features like page size, cache\nline size, and SIMD width of the underlying hardware.Additionally, they proposed to interleave the stages of multiple queries in order to increase the throughput of their search algorithm.\nHowever, both  \\cite{LeisKN2013:ART-index} and \\cite{FelixMartin2013:uncracked} indicate that ART outperforms other main-memory optimised search trees such as CSB+-Tree  and FAST.\n\n\nAdaptive Radix Tree (ART) index was first proposed in \\cite{LeisKN2013:ART-index}. \nAs the example shown in Figure \\ref{fig:ART}, ART index has two types of nodes where the internal nodes provide mappings from partial key to other nodes, and the leaf node stores the value corresponding to the keyword. \nThe height of the tree is only determined by the maximum length of the indexed keyword rather than the number of indexed keywords, and all keys are sorted lexicographically. \nThe capacity of an internal node changes adaptively with the inserted keywords, which makes ART cache aware, and query efficient. \nIn addition, by employing the path compression and lazy expansion techniques, ART further reduces the space consumption. \nART index has been integrated in an in-memory database system, HyPer \\cite{KemperN2011:Hyper}, which shows better query performance than other in-memory database index struc-tures. \nRecently, Leis V \\textit{et al}.  proposed the Optimistic Lock Coupling and Read-Optimized Write EXclusion (ROWEX) protocols to deal with the synchronization problem of ART index \\cite{LeisSKN2016:ART-sync}, which further improve the performance of ART index and expand its application scope.\n\n\\begin{figure}[htp!]\n   \\centering\n   \\includegraphics[scale=0.35]{art}\n   \\caption{An example ART index structure \\cite{LeisKN2013:ART-index}.}\n   \\label{fig:ART}\n\\end{figure}\n\nThe idea of database cracking was first proposed in \\cite{KerstenM2005:cracking-DS} and simulated with SQL statements, which showed the application prospect of the algorithm. \nFor an attribute to be cracked, the database cracking technology first makes a copy (called the cracker column) of the corresponding column, and then partially records the tuples in the cracker column into tuple clusters (called the column slices) according to the continuously arrived range queries on the attribute until the column is completely sorted. \nFigure \\ref{fig:cracking} depicts the standard Database Cracking when executing two queries.\nThe tuples are clustered in three pieces from the range predicate of Q1.The result of Q1 is then retrieved as a view on Piece 2(\\textit{i}.\\textit{e}.,  indexing 10 < A < 14).\nLater, query Q2 requires a refinement of Pieces 1 and 3(\\textit{i}.\\textit{e}.,  respectively indexing A > 7 and A < 16) and splitting each in two new pieces.More database cracking implementation details are discussed in \\cite{IdreosKM2007:DB-cracking}. \n\n\n\\begin{figure*}[htp!]\n   \\includegraphics[scale= 0.75]{cracking}\n   \\caption{Database Cracking when executing two queries \\cite{IdreosKM2007:DB-cracking}.}\n   \\label{fig:cracking}\n\\end{figure*}\n\n\nIt shows good performance when the cracking algorithm is embedded in MonetDB \\cite{IdreosGNMMK2012:MonetDB}. \nRecently, research on database cracking technology has become more extensive and in-depth.\nIn \\cite{IdreosKM2007:updating-cracked}, Idreos \\textit{et al}.  discussed the problem of database cracking update. \nIn \\cite{Halim:stochastic},Halim  \\textit{et al}.  proposed the Stochastic Database Cracking,it alleviates the sensitivity of the cracking process to the kind of queries by introducing random physical reorganization steps for efficient incremental index-building, while also taking the actual queries into account.\nIn \\cite{IdreosMKG2011:merging-cracking},Idreos  \\textit{et al}.  proposes a series of Hybrid Cracking algorithms based on the combination of Database Cracking and Adaptive Merging \\cite{Graefe2010:self-tuning,Graefe2010:adaptive}.\nThese hybrids are intended to meet both the database cracking design goal of minimizing initial per query overhead and also the adaptive merging design goal of exploiting the concept of runs and merges to converge quickly.\nIn \\cite{IdreosKM2009:self-organizing-cs}, a number of technologies such as Sideways Cracking, Multi-projection Queries, and Tuple Reconstruction were proposed to solve the problem of cross-column query in column databases. \nThe concept of tuple reorganization is put forward in this paper as well. \nMoreover, the authors summarized existing database cracking algorithms comprehensively in \\cite{SchuhknechtJD2016:evaluation-cracking}, and further improved the original algorithm in terms of the convergence speed, robustness and parallelism, and suggested that the future development direction of database cracking is to adapt to the changes of in-memory index structures.\n\n\\section{Feasibility Analysis of ART Index Cracking} \\label{sec:feasibility}\nIn this section, we will present the necessity and possibility of applying cracking technologies on ART index.\n\n\\subsection{ART Index Construction Overhead} \\label{subsec:art-overhead}\nDuring the construction of the index, ART recursively maps each part of the key to a child internal node along a path of the tree from the root. \nIn order to efficiently manage the size of the internal node, four internal node types with different capacities are provided. \nAn appropriate internal node type is selected according to the number of its children, \\textit{i}.\\textit{e}.,  its fanout.\n\nAs we know, ART is a trie structure tree. \nIt means that the tree structure remains the same regardless of the node insertion order. \nIn other words, there will be no rebalancing of the tree. \nAlthough different key insertion orders lead to the same ART structure, we found that the order actually has a significant impact on the time overhead of index construction.\n\nFigure \\ref{fig:InsertOrderImapct} shows our experimental results of the impact of key insertion order on ART index construction. \nThe experiment was conducted on a commodity server with an Intel(R) Xeon(R) CPU E7-4820 v4 @ 2.00GHz, 25600K L3 Cache, and 1TB RAM. \nWe prepared three sequences of integer data. \nThey have the same length $N>0$, but the elements are distributed in different orders. \nFor the \\emph{Ordered} sequence, all elements are arranged in ascending order within the range [1, $N$]; for the \\emph{Disordered} sequence, all elements are randomly arranged within the range [1, $N$]; and for the \\emph{Even order} sequence, all even elements are arranged in ascending order within the range [1, 2$N$]. \nThe index construction time increases approximately linearly with the amount of data inserted for all three cases. For the same amount of data, the comparison of the time overhead of the three is Disordered > Even order > Ordered. \nAnd their gaps also increase with the amount of data. \nThe index construction time for Disordered even becomes twice of that for the Ordered when data amount to reaching 90 million. \nMoreover, the experiment also shows that larger data intervals (the case of Even order here) require more index construction time.\n\\begin{figure}[h]\n   \\includegraphics[width=\\linewidth]{InsertOrderImapct}\n   \\caption{The impact of key insertion order on ART.}\n   \\label{fig:InsertOrderImapct}\n\\end{figure}\n\nThe experiment is a good illustration of the fact that the construction of the ART index is sensitive to the key insert order. \nAs we know, the data exchange between main memory and CPU cache is through cache lines, \\textit{i}.\\textit{e}.,  fixed size (usually 32 or 64 bytes) of blocks. \nSuccessively inserted disordered keys or evenly ordered keys may cause them to span more ART tree nodes, which make more different nodes to be visited in an insertion, and hence cache misses occur more frequently. \nSince accessing the cache is much faster than accessing the main memory, the cases of disordered insertion and large interval insertion inevitably need more index construction time. \n\nSince keys are sorted gradually according to the continuously arrived queries during the process of cracking, it is worthwhile applying the cracking technology to the construction of ART index.\n\n\\subsection{ART Range Query} \\label{subsec:art-range}\nThe process of database cracking is driven by continuously issued range queries.\nTherefore, it should be ensured that any data structure to be cracked will support range queries. \nAccording to \\cite{LeisKN2013:ART-index}, the ART index supports not only point queries but also range scans. \nAs child pointers are sorted in an internal node, the range scan can be performed efficiently by returning all leaf nodes in a subtree between lower and upper bounds of a range.\n\nFor conventional database cracking technology, the overall query performance is bounded by the computational complexity of the binary search on an ordered array, \\textit{i}.\\textit{e}.,  O(log$N$) where $N$ is the amount of data in the array. \nFortunately, the number of comparisons is only related to the path length of the key in the ART index tree rather than the number of keys indexed, and the path length is usually much smaller than log$N$. \nFrom this point of view, to index a column, it seems faster to start with cracking an ART index on the column than to crack the column directly.\n\n\\section{ART Index Cracking} \\label{sec:art-cracking}\nIn this section, we propose an ART index cracking method which is based on the assumption that the time overhead of index construction and maintenance cannot be ignored for large scale IMDB where real-time query response is critical in the meanwhile. \nAlthough a completed ART index has excellent query response performance, the initialization phase to construct the index from scratch makes it unavailable for the scenarios that require instant query response at any time. \nThe ART index cracking algorithm contributes a possible solution to the performance trade-off between instant query and overall query. \nIt can also ensure the index effectiveness under unknown query workload. \nThe basic idea behind the ART index cracking algorithm is to use auxiliary data structures to gradually construct an ART index rather than a sorted column along with the range queries issued on the column. \n\n\\subsection{Components for ART Index Cracking} \\label{subsec:components}\nAccording to \\cite{IdreosKM2007:DB-cracking}, \\emph{Cracker Column} and \\emph{Cracker Index} are the basic components of the conventional database cracking algorithm. \nSince our aim is to obtain a completed ART index rather than a sorted column, we assemble a cracker index with an ART index and an auxiliary data structure for maintaining the information of column data organization.\nThe components of our ART index cracking algorithm are illustrated in Figure \\ref{fig:RangeQuery2}.\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{RangeQuery2}\n   \\caption{The process of ART index cracking with three range queries.}\n   \\label{fig:RangeQuery2}\n\\end{figure}\n\n\\begin{itemize}\n\\item[-] \\emph{Cracker Column} is a copy of the database column from which an ART index is constructed. \nFrom the perspective of the ART index, it holds an array of keys to be indexed.\n\\item[-] \\emph{Cracker Index} is an auxiliary index structure to locate and organize data in the cracker column into column slices which are the clusters of data on the column identified by range intervals, and to sort the column slices in a sorted order. \nIn conventional implementation, any structure that supports search in sorted order (at least a total preorder) is available, such as the red-black tree and the AVL tree.\nAs analyzed in \\ref{subsec:art-range}, ART is available this case.\nIn order to provide a sound ART cracking function, the cracker index here consists of two parts.\n    \\begin{itemize}\n        \\item[-] \\emph{ART Index} is what to be gradually constructed from scratch according to the continuously arrived range queries on the column. \n        Internal nodes collapsing techniques are welcome. \n        \\item[-] \\emph{Range Lookup Table} is used to record those column slices which have already been indexed in the ART so that the cracking algorithm can decide whether to return the query results directly through the ART index or to perform the standard cracking operation through the cracker index. \n        To achieve this goal, merged historical query ranges and the corresponding covered column slices are recorded as the keys and values in the range lookup table. \n    \\end{itemize}\n\\end{itemize}\n\nThe following points should be noted. \nFirstly, the column slices are dynamic during the entire cracking process. \nA range query on the cracker column may cause a column slice to be created by splitting or removed by merging existing slices. \nSecondly, any column slices covered by previous range queries has already been kept in the range lookup table. \nThis ensures the improvement of overall query performance based on the strategy that uses the ART index for related query evaluation as much as possible.\n\n\\subsection{ART Index Cracking with Range Queries} \\label{subsec:cracking}\nFigure 3 illustrates the case of ART index cracking with three consecutive range query processing.\nFor the convenience of discussion, suppose queries occur on column $A$ of table $R$, and the data type of values in column A is Integer.\n\nSince the range lookup table is empty when the first range query Q1 (80 $\\leq$ A $\\leq$ 110) is issued, a cracker column will be created by making a copy of the original column A. \nThen an ART index is constructed to index all data in the cracker column according to the lower bound and the upper bound of the range query, \\textit{i}.\\textit{e}.,  80 and 110 respectively. \nHence, the cracker column is split and rearranged into three column slices, P1 (A < 80), P2 (80 $\\leq$ A $\\leq$ 110), and P3 (A > 110), which guarantees a sorted order in the cracker column between the column slices, \\textit{i}.\\textit{e}.,  P1 $<$ P2 $<$ P3, \\textit{i}.\\textit{e}.,  the lower bound index of P2 larger than the upper bound index of P1 and the upper bound index of P2 lower than the lower bound index of P3. \nAll data in P2 satisfy the query, hence are returned as the result set.\nFinally, the queried range [80, 110] and the covered column slice P2 are recorded in the range lookup table as a key-value pair.\n\n\nSince there is no intersection between the query ranges of Q2 (220 $\\leq$ A $\\leq$ 300) and any queried ranges recorded in the range lookup table ([80,110] currently), a similar cracking process was conducted as that of Q1. \nFirstly, it can be infer that the result set must be included in the column slices larger than P2, \\textit{i}.\\textit{e}.,  P3 according to the key-value pair recorded in the range lookup table, because the lower bound of Q2 is larger than the upper bound of the only key, \\textit{i}.\\textit{e}.,  $110 < 220$. \nThis results in two more column slices separated from P3, \\textit{i}.\\textit{e}.,  P4 (220 $\\leq$ A $\\leq$ 300) and P5 (A > 300).\nThe result set consisting of all data from P4 is returned and further inserted into the ART index. \nThe range lookup table is then updated with [220, 300] as a key and the corresponding covered column slice P4 as its value. \n\nFor the range query Q3 (80 $\\leq$ A $\\leq$ 350) whose range stretches over P2, P4, P3 and part of P5, with the former two column slices having been recorded in the range lookup table, while P3 and partial P5 not been visited and sorted yet, the algorithm will form a union result set comprised of both the search results that are obtained directly from the ART and the search results that are obtained from P3 and P5 on cracker column. \nNote that a new column slice P6 will be split from P5 with respect to the upper bound value 350. \nAt the end of the query process of Q3, those data in P3 and new P5 are sorted and inserted into the ART index, and the ranges in the range lookup table are replaced by a new key-value pair, \\textit{i}.\\textit{e}.,  an interval [80, 350] and a merged column slice across P2, P3, P4, and P5, because both [80,110] and [220, 300] can be covered by [80, 350].\n\nIn this way, the complete ART index is gradually constructed by continuously arrived range queries. \nThrough the analysis of the above algorithm execution process, it can be seen that the selectivity of range queries is an important factor to the performance.\nOn the one hand, the fewer tuples each query selects, the faster the query response will be, but the longer it will take to construct the complete ART index. \nOn the other hand, the more tuples each query selects, the slower the query response will be, but the shorter time it will cost to construct the complete ART index.\nAlthough it seems impractical that the build speed of ART index depends on the range query selectivity, we should be aware that it coincides with the philosophy of database cracking, \\textit{i}.\\textit{e}.,  delaying unavoidable changes as far as possible.\nIt means that the ART index creation of cracking ART reflects the practical workload.\nThe analysis of the impact of selectivity will be discussed in detail in the experimental section.\n\n\\begin{algorithm}[htbp]\n\t\\caption{ART Index Cracking with Range Queries}\n\t\\label{alg:ARTCracking}\n\n\t\\begin{algorithmic}[1]\n\t\\Require \\qquad CrackerColumn $col$, RangeLookupTable $tbl$, \\qquad \\qquad \\qquad ARTIndex $art\\_idx$\n\t\\Function {art$\\_$cracking}{[$lb$, $ub$]}\n\t    \\If{$tbl$.empty()} \n\t        \\State $art\\_idx$.init() \n\t        \\State $col$.init()\n\t    \\EndIf\n\t    \\State $hit\\_range \\gets \\emptyset, hit\\_slices \\gets \\emptyset, rs\\_art \\gets \\emptyset$\n\t    \\ForAll {$entry \\in tbl$}\n\t        \\State $hit\\_range \\gets entry.key \\cap [lb, ub]$\n\t        \\If {$hit\\_range \\not= \\emptyset$ }\n\t            \\State $rs \\gets rs \\cup art\\_idx.$scan($hit\\_range$)\n\t            \\State $hit\\_slices \\gets hit\\_slices \\cup entry.values$\n\t        \\EndIf\n\t    \\EndFor\n\t    \\State $new\\_slices \\gets \\emptyset$\n\t    \\ForAll {$slice \\in (col.\\text{slices()} - hit\\_slices)$}\n\t        \\If{$slice \\cap [lb, ub] \\not= \\emptyset$}\n    \t        \\State $new\\_slice = slice \\cap [lb, ub]$\n    \t        \\State $new\\_slices \\gets new\\_slices \\cup \\{new\\_slice\\}$\n    \t        \\State $rs\\_db \\gets$ \\Call{db$\\_$cracking}{$new\\_slice$}\n    \t        \\State $art\\_idx$.insert($rs\\_db$)\n    \t        \\State $rs \\gets rs \\cup rs\\_db$\n\t        \\EndIf\n        \\EndFor\n        \\State $key \\gets [lb,ub]$, $value \\gets new\\_slices$\n        \\State $tbl.\\text{insert\\_and\\_merge}(key, value)$\n        \\State \\textbf{return} $rs$\n    \\EndFunction\n\t    \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\t\\end{algorithmic}\n\\end{algorithm}\n\nAs shown in Algorithm \\ref{alg:ARTCracking}, given a range query $[lb, ub]$ on a column as the input, the algorithm outputs a set of values $rs$ as the range query result on the column $col$ while cracking the ART index $art\\_idx$.\nThe algorithm of ART index cracking consists of four phases, \\textit{i}.\\textit{e}.,  the initialization phase (line $2\\sim 5$), the range search in the ART index (line $6\\sim 13$), \nthe cracking phase (line $14\\sim 23$), and the finishing phase (line $24\\sim 26$).\n\nIf the the range lookup table $tbl$ is empty, \\textit{i}.\\textit{e}.,  there is not any range recorded in the table, the algorithm will first initialize the ART index $art\\_idx$ and the cracker column $col$ (line 3 and 4). \n\nAfter the inialization, the algorithm performs a range search in the ART index guided by the range lookup table.\nIt first finds out all entries recorded in $tbl$ whose key has non-empty intersection $hit\\_range$ with $[lb, ub]$ (line 8).\nFor each found entry, perform the range search $hit\\_range$ in the ART index and make $rs$ union with current results (line 10).\nCorresponding column slices in each found entry are collected in a set $hit\\_slices$ for use in the next phase (line 11).\n\nEntering the cracking phase, the algorithm iterates through all the column slices in the cracker column $col$ that are not indexed in $art\\_idx$ but intersected with $[lb, ub]$ (line 15 and 16).\nNote that there is only one complete slice across the entire cracker column initially.\nAll such intersections are collected in a set $new\\_slices$ for updating the range lookup table $tbl$ in the last phase (line 17 and 18).\nThe traditional database cracking function \\textbf{DB\\_CRACKING()} is called for cracking each such intersection (line 19).\nAfter calling \\textbf{DB\\_CRACKING()}, new column slices are sperated from the original ones, all values in the intersection are sorted and returned in $rs\\_db$.\nFurthermore, values in $rs\\_db$ are inserted into the ART index $art\\_idx$ (line 20), and union is made between $rs$ and $rs\\_db$ (line 21).\n\nIn the finishing phase, a key-value pair is constructed by taking $[lb, ub]$ as key and all new column slices $new\\_slices$ as value (line 24).\nThe key-value pair is inserted into the range lookup table $tbl$ and merged with the existing entries (line 25).\nTaking the entry in the lookup table in Figure \\ref{fig:RangeQuery2} after executing Q3 for example, the previous two entries with keys [80, 110] and [220, 300] are merged with the new key-value pair [80, 350]:\\{P3, P5\\} to be a new and only one entry [80, 350]:\\{P2, P3, P4, P5\\}.\nFinally, $rs$ is returned as the result set. \n\n\\subsection{ART Index Cracking with Updates} \\label{subsec:updats}\nThe algorithm of ART index cracking with range query simply considers the case where the column to be queried and indexed is read-only. However, updates (insertions and deletions) to the columns are inevitable in the actual database application scenario. Index update overhead brought by this may result in a decrease in database system response performance. Preliminary idea to this problem is discussed below.\n\n\\subsubsection{Caching}\\label{subsubsec:caching}\nThe basic idea is to delay the update operations by caching them until the arrived range query has an intersection with any cached update operations or until the cache is full. An exception is that the updated data fall within any range in the range lookup table, which will be handled immediately by ART index. In this way, write lock contention is effectively reduced. \n\nFor this purpose, a sorted list is used to cache the updated data and their corresponding operation type (either insertion or deletion) according to the data value. \nAssume that this simple cache structure is rather small compared with the entire column, so the cost of sorting is negligible. \nFor the same data value, a deletion after an insertion will both be removed from the list, while an insertion after a deletion will cover the previous deletion. \nOnce the arrived range query has an intersection with any cached update operations, all data in the intersection will be inserted or deleted from the ART index, while all data in the query range which are not in the sorted list cache will be cracked as stated in Section \\ref{subsec:cracking}. \nFinally, those update operations that have been performed in the ART index are removed from the sorted list.\n\nFor the case when the sorted list (cache) is full, a range query between the lower and upper bound of the data in the sorted list is constructed and issued actively by the algorithm to trigger an update process on the ART index. \nThe process after that is just the same as the previous case.\n\nIt should be noted that the data to be deleted in the sorted list cache should be carefully examined and not to be returned in the result set of a range query.\n\nAlthough there is a more straightforward approach to perform updates directly on the cracker column, the overhead of updating the cracker column includes not only the maintenance of the cracker column but also the maintenance of the ART index and the range lookup table.\n\n\n\\begin{algorithm}[htbp]\n\t\\caption{ART Index Cracking with Insert}\n\t\\label{alg:ARTCracking-insert}\n\n\t\\begin{algorithmic}[1]\n\t\\Require \\qquad CrackerColumn $col$, RangeLookupTable $tbl$, \\qquad \\qquad \\qquad ARTIndex $art\\_idx$, Cache $cache$ \n\t\\Function {insert$\\_$art$\\_$cracking}{$lb$, $ub$,$value$}\n\t    \\State $next$ = point at the last position of $col$  \t\n\t    \\State $cur$ = point at the first position of last $slice$\n\t    \\If{$value \\in tbl$} \n\t        \\State $art\\_idx$.insert() \n\t    \\Else \n\t        \\State $cache$.insert()\n\t    \\EndIf\n\t    \\If{$cache \\text{ is full}$}\n\t        \\State $result \\gets cache$\n\t    \\Else\n\t        \\State $result \\gets cache \\cap [lb, ub]$\n\t    \\EndIf\n\t     \\If{$result \\not= \\emptyset$ }\n\t        \\ForAll{$entry \\in  result$}\n\t            \\While{$entry \\notin cur\\_slice(cur)$}  \n\t                \\State $col[next] \\gets col[cur]$\n\t                \\State $next \\gets cur$\n\t                \\State $cur \\gets \\text{point at previous slice of $cur$}$\n\t            \\EndWhile\n\t            \\State $col.\\text{insert}(entry,cur)$\n\t            \\State $tbl.\\text{update}()$\n\t            \\State $cache.\\text{update}()$\n\t            \\State $art\\_idx.\\text{updare}()$\n\t        \\EndFor\n\t    \\EndIf\n    \\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\\subsubsection{Shuffling}\\label{subsubsec:shuffling}\nObviously, the maintenance overhead of the cracker column triggered by the update is not negligible.\nIn order to ensure the existing sorted order in the slices of the cracker column, a naive method is to move all the data behind the position to be inserted or deleted backward or forward respectively.\nA more practical technology, \\emph{shuffling}, has been introduced in \\cite{IdreosKM2007:updating-cracked}.\n\n\\begin{figure}[htp!]\n   \\includegraphics[scale=1.3]{Insert}\n   \\caption{Shuffling of the cracker column after the insertion of 45.}\n   \\label{fig:Insert}\n\\end{figure}\n\nPseudo code for ART cracking with insert is shown in  Algorithm \\ref{alg:ARTCracking-insert}. The algorithm of ART index cracking with insert consists of two phases, \\textit{i}.\\textit{e}.,  the cache insert phase (line $4\\sim 8$) and the shuffing a cracker column in range search (line $9\\sim 25$), the necessary data is initialized in lines 2 and 3.\n\nIf the value to be inserted is in the range lookup table  $tbl$ , it can be inserted directly into the ART index  $art\\_idx$ (line 5), otherwise it is inserted into the cache $cache$ to delay the update (line 7). \n\nThe data in the cache $cache$ is  handled in (line $9\\sim 13$).\nIf the cache is full, all data needs to be inserted(line 10), otherwise the intersection of the cache and the reached range query $[lb, ub]$ is processed(line 12). For all results $result$ that satisfy the condition,  move the element through shuffling a cracker cloumn(line $16\\sim 20$ and get the position to insert.Then the value is inserted into the cracker cloumn and the range table $tbl$ (line 21), cache $cache$(line 23),the ART index $art\\_idx$ (line 24) are updated.\n\n\nThe Algorithm \\ref{alg:ARTCracking-insert} is explained below by an example as shown in Figure~\\ref{fig:Insert}.When 45 in the cache is to be inserted, it first searches in the range lookup table.\nSince 45 is smaller than the lower bound of the key ([80, 350]) of the only entry, it can be inferred that 45 should be inserted in the column slice P1.\nHence, according to the shuffling algorithm \\cite{IdreosKM2007:updating-cracked}, for each lower bound value of the column slices that higher than P1, it is moved to the position of the lower bound of the next column slice.\nAs column slices P2, P3, P4, and P5 are all indexed in the ART index, and they are merged as one complete interval in the range lookup table, the shuffling process is actually only happens in P6 and P2.\nFinally, 45 is inserted in the absent position that originally was owned by 100.\nAlthough 100 is moved to the position behind 320 during shuffling, the sorted order of data in [80,350] is still maintained through the ART index.\nIn this way, the shuffling technology only needs two movements compared to the naive method requiring 7 movements.\n\nApparently, the shuffling technology can be adapted to deletion operations in a similar way.\n\n\\section{Experiments}\\label{sec:experiments}\nIn this section, the advantages and disadvantages of standard cracking, standard ART, and ART cracking algorithms in various situations are compared through experiments.\n\nThe experimental settings are as follows. \nAll algorithms were implemented in C++ language. \nAll experiments were conducted on an Intel(R) Xeon(R) CPU E7-4820 v4 @ 2.00GHz server. \nThe server has 512GB RAM and 25600K L3 Cache. \nThe operating system is Ubuntu 18.04.4 LTS (GNU\/Linux hp50 4.15.0-38-generic x86\\_64).\nAll experiments were performed in memory to simulate the in-memory database.\n\nIn order to examine the performance of algorithms under different scenarios, we designed two sets of experiments.\n\nThe first set of experiments follow the experimental design and setting from \\cite{IdreosMKG2011:merging-cracking} where the dataset is synthetic and the range query pattern is as follows. \nIf there is no special statement, the experiment selects $N$ non-repetitive integers randomly distributed between $[1, N]$ where $N$ equals $10$M, the query selectivity $S$ is $0.0001N$, and the workload type is random.\n\n\\textbf{SELECT} $A$ \\textbf{FROM} $R$\n\n\\textbf{WHERE} $A \\geq low$ \\textbf{AND} $A \\leq high;$\n\nThe second set of experiments were conducted on YCSB (Yahoo! Cloud Serving Benchmark) which is a popular benchmark for evaluating different key-value and cloud database \\cite{CooperSTRS2010:YCSB}.\n\n\\subsection{Space overhead}\\label{subsec:space}\nIn order to compare the space overhead of the standard ART, the standard database cracking, and the ART cracking.\nWe did two experiments on the synthetic dataset with the default setting, \\textit{i}.\\textit{e}.,  $N=10$M and non-repetitive integers randomly distributed between $[1, N]$.\nFor the first experiment, as shown in Figure~\\ref{fig:space-fixed}, the lower bound and upper bound of each query range were selected with a fixed selectivity $S=0.005$.\nFor the second experiment, as shown in Figure~\\ref{fig:space-random}, the lower bound and upper bound of each query range were generated randomly between $[1, N]$.\n\nAs demonstrated in both experiments, the standard ART and our ART cracking consume more space than the standard database cracking, because the index structure of ART is more complicated than that of database cracking (usually AVL or RBTree).\nAccording to Figure~\\ref{fig:space-fixed}, the space overhead of ART cracking increases with the number of queries and has a trend over that of ART.\nDifferent from cracking methods, the space overhead of ART keeps unchanged.\nComparing Figure~\\ref{fig:space-random} and Figure~\\ref{fig:space-fixed}, we can also find that the space overhead grows faster in a random query range mode.\nThe reason is that there is a higher probability of generating a large query range for the random mode than that for the fixed selectivity mode.\nAs stated in Section~\\ref{subsec:cracking}, a larger query range means more data to be indexed, and hence a larger space consumption. \n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{space-fixed}\n   \\caption{Space overhead comparisons when selectivity $S$= 0.005}\n   \\label{fig:space-fixed}\n\\end{figure}\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{space-random}\n   \\caption{Space overhead comparisons for random query ranges}\n   \\label{fig:space-random}\n\\end{figure}\n\n\n\\subsection{Response Time}\\label{subsec:response-time}\nIn Figure~\\ref{fig:response-time}, we compare the response time of the standard ART, the ART cracking algorithm and the binary search algorithm in the synthetic dataset and random query mode.\nThe binary search algorithm applies the quick sort to sort the array first, and then use the binary search to find the result.\nWe have the following observations.\n\nFirstly, compared with the binary search and the standard ART algorithm, the initialization time of the binary search algorithm and the standard ART algorithm is longer than that of the ART cracking algorithm.\nThis is due to the fact that the binary search algorithm needs to sort the data first, and the standard ART takes a lot of time to initialize the ART index as well. \nWith the increase of the number of queries, the response time of standard ART and ART cracking algorithm increases linearly, while the increase of the response time of the binary algorithm is not significant.\n\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{response-time}\n   \\caption{Response time comparisons for three algorithms}\n   \\label{fig:response-time}\n\\end{figure}\n\n\nSecondly, when the number of queries exceeds 5000, the response time of the ART cracking algorithm exceeds the binary search algorithm because of the good performance advantage of using binary search to perform range queries on ordered arrays.\nAs can be seen from the figure, when the number of queries exceeds 8000, the response time of the ART cracking algorithm exceeds the standard ART.\nThe reason is that ART cracking algorithm still needs a part of time to build a complete ART index, but the maximum difference between ART cracking algorithm and standard ART in response time is less than 1 second, and this difference is gradually reduced as ART is built up.\n\nIn summary, ART cracking algorithm can avoid building a complete ART index at one time and has a relatively low initialization cost.\nAt the same time, the hot data query is real-time, \\textit{i}.\\textit{e}.,  when the query arrives, it avoids the waiting overhead due to the long time to initialize the ART.\n\n\\subsection{Selectivity}\\label{subsec:selectivity}\nThe above experiments merely show the excellent performance of the ART cracking algorithm under a fixed selectivity. \nHowever, the impact of change to the selectivity cannot be ignored. \nOn the one hand, when the selectivity is high, the algorithm converges quickly, while the initialization time is long, and the complete index structure is established promptly which fails to show the advantages of the ART cracking algorithm. \nOn the other hand, when the degree of selection is low, the algorithm takes a long time to converge, while the initialization time is short. \nEspecially for the case of querying hot data, though the query is increased, the cost of subsequent index maintenance is rather small.\n\n\\begin{figure}[htp!]\n   \\includegraphics[scale=0.31]{Selectivity}\n   \\caption{The effect of selectivity on the response time}\n   \\label{fig:selectivity}\n\\end{figure}\n\nIn order to show the effect of selectivity on the performance of the algorithm, we conducted an experiments with the synthetic dataset and a random query mode. \nThe selectivity is manipulated constantly in accordance with the number of the ART query, and the response time of the algorithm for different number of queries is recorded. \nThe experimental results are shown in Figure~\\ref{fig:selectivity}. \n\nFirstly, the greater the degree of selection, the greater the initialization time for the first execution of the algorithm will be. \nAnd the response time increases with the increase of the number of queries. \nSecondly, when the number of queries is constant, the response time increases slowly at first, if the query selectivity is small. \nIt increased significantly as the selectivity doubled.\nAs can be seen from the figure, when the selectivity is 0.001 and 0.01, the response time changes significantly as the number of queries increases.\nWith the increase of selectivity, the interval between range queries is increased by a large amount compared with the smaller selectivity, and the overhead of ART index initialization and ART range queries for ART cracking algorithm is increased significantly.\nTherefore, the efficiency of the ART cracking algorithm is influenced by the selectivity. \nThe appropriate selectivity will result in better performance for the ART carcking algorithm, which is beneficial to the construction of the ART index and the query response time.\n\n\\subsection{Convergence Speed}\\label{subsec:convergence}\nAs stated in Section~\\ref{subsec:art-range}, there are four phases in the ART cracking, \\textit{i}.\\textit{e}.,  initialization, search ART, cracking, and finishing.\nApparently, the overall query efficiency will be greatly improved if the ART index is completely constructed (or converged) as soon as possible.\nTo measure the convergence degree of the algorithm, the definition of the ART building rate $R$ is given as follows.\n\\[R = key.size \/ N * 100\\%\\]\n\nThe tree building rate $R$ is defined to be the ratio of the number of keys indexed in the ART to the amount of data $N$ after certain times of range queries.\nThe change in the rate of establishment in a unit of time reflects the convergence speed of the algorithm.\n\nThe tree building rate $R$ is an important indicator to measure the performance of the algorithm. \nThe more proximate the $R$ value is to 1, the closer the generated ART is to the complete index. \nThe degree of convergence of the algorithm depends on the query execution. \nOn the one hand, whatever the query pattern is, with the increasing number of queries, the algorithm will gradually converge to the complete index. \nSince the hot data will be queried repeatedly, it has a good query performance even though the tree building rate $R$ is low. \nOn the other hand, for the queries with higher selectivity, they usually converge faster, \\textit{i}.\\textit{e}.,  the tree building rate increase faster. \nIn the extreme case, if the first query selects all tuples, the ART cracking process will degenerate into a standard ART query process, and hence the selectivity will become the primary factor for algorithm convergence.\n\nIn this experiment, we utilized the synthetic dataset with default setting and the random query range mode, and observe the convergence degree of the algorithm under different selection degrees. \nThe experimental results are shown in Figure~\\ref{fig:BuildRate}. \nWe choose three different selectivity. \nWhen the selectivity is fixed, as the number of queries increases, the tree building rate is gradually approaching 1, and the rate of growth tends to be flat. \nThis is mainly because that as the number of queries increases, the overlapping scope of the query range also increases. \nWith a fixed selectivity, the number of updated data each time is reduced, so the change rate of the ART building also declines accordingly. \nAt the same time, it is noted that as the number of queries increases, the algorithm converges gradually, and the larger the selection rate, the faster the convergence of the algorithm will be. \nWhen the number of queries reaches 10,000, the selectivity doubles itself and the tree building rate increases by more than 30\\%.\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{BuildRate}\n   \\caption{The effect of selectivity on the ART building rate}\n   \\label{fig:BuildRate}\n\\end{figure}\n\n\\subsection{Range Query Workload Modes}\\label{subsec:workload}\nThe impact of range query workload mode on the standard ART is almost negligible, but it has an important impact on the ART cracking algorithm, which will be shown in this section.\nFor this experiment, we consider the following range query workloads:\n\\begin{enumerate}\n    \\item \\emph{Random mode}: the minimum of the range is randomly generated, and the selectivi-ty is fixed.\n    \\item \\emph{Sequential mode}: the minimum of the range increases as the number of queries increases, the maximum value is randomly determined, and the query range may be overlapped.\n    \\item \\emph{Distorted mode}: The top 80\\% of the queries are concentrated on 20\\% of the data, and the hot data may be queried multiple times.\n    \\item \\emph{Two-way incremental mode}: The query range expands on both the left and right ends on the basis of the previous query.\n    \\item \\emph{One-way incremental mode}: The minimum value of the current query`s range is the maximum value of last query`s range, and the selectivity is fixed.\n\\end{enumerate}\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{workload}\n   \\caption{Response time comparisons for different workload modes}\n   \\label{fig:workload}\n\\end{figure}\n\nThe experimental results are shown in Figure~\\ref{fig:workload}. \nFor sequential search, the algorithm converges quickly due to the large degree of randomness of the range selection, thus maintaining a high initialization cost. \nFor distorted query, the response time is relatively low at the first 80\\% of the time, and then increases sharply in the latter part of the query. \nFor random queries, as the number of queries increases, the ART cracking algorithm converges slowly and the overall response time is minimal.\n\n\\begin{figure}[htp!]\n   \\includegraphics[scale=0.53]{incremental}\n   \\caption{Response time comparisons for two incremental workload modes}\n   \\label{fig:incremental}\n\\end{figure}\n\nThe experimental results for the two incremental modes are shown in Figure~\\ref{fig:incremental}. \nIt can be seen that as the range of queries increases, the response time of the two incremental query modes increases almost linearly. \nThis is mainly because there is no overlap for the query range of this mode, hence the algorithm degenerates into a standard cracking algorithm which updates the ART each time and never use it in subsequent queries. \nFor the two-way incremental mode, the two ends are continuously expanded, and the historical data is stored in the ART, however, each query is inserted at both ends of the range, hence in addition to the algorithm convergence, the overhead of the maintenance algorithm is huge, so the response time increases.\n\nFrom the two groups of experiments it can be concluded that ART cracking algorithm depends on the change of the workload. \nWhen the query is in incremental mode, the algorithm has a lower initialization cost, but the subsequent response time is far beyond the ART standard query. \nFor other modes, the ART cracking algorithm has both a lower initialization cost and a lower query response time.\n\n\\subsection{Per-query response times}\\label{subsec:p-query}\nFigure ~\\ref{fig:pre-response} shows the per-query response times for different methods in the synthetic dataset and random mode.\nSince the available ART implementations do not support bulk loading, the ART cracking algorithm greatly reduces the time required to build full indexes before data is accessed for the first time compared to standard ART.\nAt that same time, the response time of each query in the ART cracking algorithm is very close to the standard cracking algorithm.\n\n\\begin{figure}[htp!]\n   \\includegraphics[scale=0.33]{pre-response}\n   \\caption{Per-Query Response Time of Different Algorithms}\n   \\label{fig:pre-response}\n\\end{figure}\n\n\\subsection{Updates}\\label{subsec:updates}\nThe above experiments exhibit good performance of the ART cracking algorithm without considering the update.\nHowever, the update of the index in the actual scenario is inevitable.\nFor this reason, the YCSB benchmark is used to compare the impact of the update on the ART cracking algorithm and the standard ART respectively.\nYCSB workload is workload mode e,The experimental results are shown in Figure~\\ref{fig:insert-update} and Figure~\\ref{fig:delete-update}.\n\nFigure~\\ref{fig:insert-update} shows the impact of the insert operation on the standard ART and ART cracking algorithms.\nThe throughput of the standard ART is significantly higher than that of the ART cracking algorithm because the ART cracking has the overhead of maintaining cracker columns and the range lookup table in addition to maintaining the ART index compared to the standard ART.\nAt the same time, by using the general insertion method for ART cracking comparison, we can see that using Shuffling method can greatly reduce the impact of insertion on ART cracking algorithm.\nThe main reason is that the Shuffling method greatly reduces the overhead of inserting cracker columns and the range lookup table maintenance, so that it can achieve almost the same impact as the ART cracking algorithm without insertion.\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{insert-update}\n   \\caption{The effect of insertion on two algorithms for the YCSB workload}\n   \\label{fig:insert-update}\n\\end{figure}\n\n\\begin{figure}[htp!]\n   \\includegraphics[width=\\linewidth]{delete-update}\n   \\caption{The effect of deletion on two algorithms for the YCSB workload}\n   \\label{fig:delete-update}\n\\end{figure}\n\n\n\nFigure~\\ref{fig:delete-update} shows the impact of the delete operation on the standard ART and ART cracking algorithms. \nThe deletion has the most obvious impact on the standard ART, even its throughput is lower than that of the ART cracking with general deletion. \nWhen the deletion operation is performed, the maximum overhead of both standard ART and ART cracking is to maintain the cracker column.\nHowever, the column slices information is maintained in the ART cracking, the position of the element to be deleted can be quickly located in the cracker column.\nTherefore, the throughput of the ART cracking is slightly higher than that of the standard ART. \nSimilar to the insertion, using the Shuffling method reduces such overhead.\n\n\nIn summary, by using Shuffing technology, the impact of update on ART cracking algorithm is greatly reduced.\n\n\\section{Conclusions and Future work}\\label{sec:conclusion}\nIn order to cope with the huge overhead of in-memory database index creation, this paper proposes an Adaptive Radix Tree (ART) index cracking algorithm based on ART index structure, which improves the instant query response speed by distributing the complete index creation cost to each range query.\nThe algorithm adapts the conventional database cracking technique to the ART index which enables the query to continuously establish the complete ART index structure during the process to avoid unnecessary initialization overhead and to have a high query efficiency.\nThe experimental results show the effectiveness of the algorithm.\n\nUndoubtedly, as a case study of the in-memory database index cracking techniques, the experience and lessons learned from the proposed ART index cracking approach are still very preliminary.\nFirstly, cracking in-memory index is a simple yet effective technique.\nIn contrast to building a complete index in the preprocessing stage, cracking the in-memory index is more lightweight.\nIt does not penalise the first query heavily and also reduces unnecessary initialization time for those cold data seldom visited.\nIt exhibits better performance when the selectivity is between 0.01\\% and 1\\% and the workload is random workloads.\nSecondly, cracking in-memory index still has the same performance bottlenecks as the standard database cracking, \\textit{i}.\\textit{e}.,  being sensitive to the access patterns when the selectivity is too large or too small, and the workload is too harsh.\nThis remains a challenge for cracking in-memory index.\n\nThe future research will focus on improving the convergence and robustness of cracking in-memory index and the combination of the range lookup table and the ART index to eliminate the access overhead accross different data structure. \nMoreover, a concurrent version of ART cracking is another direction of our efforts. \nBoth  \\cite{GraefeHIKM2012:concurrent-cracking} and \\cite{LeisSKN2016:ART-sync} are good references.\n\n\\begin{acks}\nGang Wu is supported by the NSFC (Grant No. 61872072) and the State\nKey Laboratory of Computer Software New Technology Open Project Fund (Grant\nNo. KFKT2018B05).\nBaiyou Qiao is supported by the National Key R\\&D Program of China (No. 2016YFC1401900).\nDonghong Han is supported by the NSFC (Grant No. 61672144).\nGuoren Wang is supported by the NSFC (Grant No. U1401256, 61732003, 61332006 and 61729201).\nYe Yuan is supported by the NSFC (Grant No. 61572119 and 61622202) andthe Fundamental Research Funds for the Central Universities (Grant No. N150402005).\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMesoscopic systems such as nanocapsules, viruses, and ribosomes evolve through the \ncoupling of processes across multiple scales in space and time. \nTherefore, a theory of the dynamics of\nthese systems must somehow account for the coevolution of coarse-grained (CG)\nand microscropic variables. Methods based on deductive multiscale analysis (DMA)\nhave shown great promise in achieving efficiency \n\\cite{Ortoleva2005,Ortoleva2008,Ortoleva2009,Cheluvaraja2010,Singharoy2011,Singharoy2012}\nover conventional molecular dynamics (MD) via coevolving the micro and CG states. \nSome of these methods have used\nthe space-warping approach \\cite{Khuloud2002} to coarse grain the atomistic configuration,\nevolve the CG variables, and then reconstruct the microstate. The drawback with this\napproach is that bonds tend to be highly strained when overall deformations implied \nby the CG state are expressed. This leads to computational difficulties because most of these\nconfigurations are typically unlikely because of their negligible Boltzmann\nweight. Therefore, to construct thermal averages, these configurations must be\nmodified using energy minimization and thermalization \\cite{Long2010}\n(see Figures \\ref{fig:EM} and \\ref{fig:therm}), \nand similarily for single evolution scenarios developed using Trotter factorization \n\\cite{AbiMansour2013}. \nThis creates a computational burdern and thereby detracts from\nmuch of the efficiency that would have otherwise been afforded by the\nmultiscale approach. Therefore, while there is much discussion about the microstate to CG state mapping \n\\cite{Khuloud2002,Murtola2007,Schulten2006,Schulten2007,Bahar1997,Haliloglu1997}, \nthe information gap between these levels  has been a challenge for understanding multiscale systems. \n\nIn this paper, we extend the space-warping method introduced earlier \\cite{Khuloud2002}\nby accounting for CG to microstate transformation in a way that respects short scale effects\nsuch as the constraints imposed by stiff bonds. This\nis achieved by constraining the fine-scaling algorithm to a set of equations\nthat enforce constants bond lengths and angles. Constraint algorithms\nare often applied to MD simulations inorder to increase the time step a \nsolver can take. This is because at the atomic scale, Newton's equations of motions \nare highly stiff, and MD integrators use explicit methods \\cite{Iserles2008}\nto numerically integrate these equations. Therefore, the time step is restricted \nto the timescale of bond fluctuations. Constrained MD attempts to resolve this\nproblem by neglecting the bond fluctuations. This is achieved by adding constraint forces \nto Newton's equations of motion such that these forces prevent any two bonded atoms from changing\ntheir bond lengths, and any three consecutively bonded atoms from changing their\nangle (See Figure \\ref{fig:tri}). Constraint algorithms are usually based on some form of the \nmethod of undetermined Lagrange multipliers \\cite{SETTLE1992,SHAKE1977,SHAKE1994,MSHAKE1998}, \nand they perform well for specific systems and when \nthe time step of evolution  is quite small. The first algorithm to satisfy \ngeometrical bond constraints was SHAKE \\cite{SHAKE1977,SHAKE1994}. This algorithm \nwas limited to mechanical systems with a tree structure. A later extension of \nthe method, QSHAKE (Quaternion SHAKE) was developed to amend this \\cite{QSHAKE1998}. It works \nsatisfactorily for rigid loops such as aromatic ring systems but fails for \nflexible loops \\cite{McBride1998} (e.g. proteins having a disulfide bridge). \nFurther extensions include RATTLE \\cite{RATTLE1983}, WIGGLE \\cite{WIGGLE2005}. and MSHAKE \\cite{MSHAKE1998}. \nRATTLE is similar to SHAKE in that it uses the Velocity Verlet time integration scheme, but\nit offers higher precision. WIGGLE extends SHAKE and RATTLE\nby using an initial estimate for the Lagrange multipliers based on the\nparticle velocities. MSHAKE computes corrections on the\nconstraint forces to achieve better convergence. Another modification is the\nP-SHAKE algorithm \\cite{PSHAKE2007} for rigid or semi-rigid molecules. P-SHAKE computes and\nupdates a pre-conditioner which is applied to the constraint gradients before\nthe SHAKE iteration, causing the Jacobian to become diagonal or strongly\ndiagonally dominant. The de-coupled constraints converge much faster\n(quadratically as opposed to linearly) at a cost of $O(N^{2})$ flops. An alternative\nconstraint method called LINCS (Linear Constraint Solver) was developed in 1997 by\nHess, Bekker, Berendsen and Fraaije \\cite{LINCS1997}. LINCS approximates the inverse of the\nJacobian of constraints with a power series using geometric progression for\neach Newton step. This approximation works only for molecules with low connectivity\nsince the eigenvalues of the Jacobian have to be smaller than $1$.\n\nThe objective of the present study is to develop a robust and scalable\nalgorithm that computes the all-atom configuration from a given CG state\nwithout any restrictions on the connectivity of the macromolecule. Rather than modifying\nthe CG dynamical equations, here a CG-to-atom resolved state map that\ntakes into account the information needed to avoid uphysical bond\nstrains is presented. The algorithm exploits the connectivity of atomic bonds \nand angles in a given macromolecule. This is based on the fact that atoms that \nare spatially far from each other are independent and therefore do not contribute\nto the same constraint equation. Therefore, all\nthe assembled matrices are naturally sparse, which we exploit in solving the\nlinearized systems. This is the source of the efficiency and scalability of\nthe constrained mapping algorithm, which allows efficient parallelization of\nthe code on distributed and shared memory systems. The study is based on implementing the\nproposed algorithm in the framework of Deductive Multiscale Simulator (DMS)\npackage \\cite{Joshi2011,Singharoy2011,Ortoleva2012}. The formulation is provided, as are the\nspecific algorithm implementation and parallelization, results and\ndiscussion are given, and conclusions are drawn.\n\n\\section{Formulation}\n\nThe development starts with a set of space warping CG variables (denoted $\\mathbf{\\phi}$)\nthat were used to simulate large-scale macromolecular conformational changes \\cite{Khuloud2002}. \nThe fine graining relationship, i.e. obtaining the atom-resolved\nconfiguration consistent with the CG variables, takes the form\n\\begin{equation}\n\\mathbf{r}_{i}=\\sum_{k}\\mathbf{U}_{k}(\\mathbf{r}_{i}^{0})\\mathbf{\\phi}\n_{k}+\\mathbf{\\sigma}_{i}.\\label{Unconstrained-Mapping}\n\\end{equation}\nHere $\\mathbf{r}$ is a vector of all $3N$ atomic coordinates,\n$\\mathbf{\\sigma}$ is a vector of $3N$ residual displacements from \ncoherent deformation generated by the first term of the RHS of Eq. (\\ref{Unconstrained-Mapping}), and\n$\\mathbf{U}$is a matrix (of size $N \\times N_{CG}$) \nof mass-weighted orthonormalized legendre functions\n\\cite{Ortoleva2012}. These CG variables \ndescribe the overall features of the system whereas $\\mathbf{r}$ describes \nthe atomic configuration directly. The basis functions depend on the reference\nall-atom configuration $\\mathbf{r}^{0}$. The reference structure introduces\na configuration as determined by X-ray and Cryo-EM data. Thus, the CG variables\nspecify how the structure is deformed from this reference configuration (i.e. due to\nthe temperature and fluid condition of interest). This\nmethod was extended \\cite{Pankavich2008} later on by maximizing the amount of information\ncontained in $\\phi$, and in particular by minimizing the mass-weighted square residual. \nThe result is a reversible $\\mathbf{r} \\rightleftharpoons \\mathbf{\\phi}$ map, which provies a unique\nCG state given $\\mathbf{r}$, but an ensemble of $\\mathbf{r}$ given $\\mathbf{\\phi}$. \n\nFor conciseness, Eq. (\\ref{Unconstrained-Mapping}) is recast in matrix form:\n\\begin{equation}\n\\mathbf{r=K\\phi+\\sigma,}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{K} =\n\\begin{pmatrix}\n\\mathbf{U} & \\mathbf{0} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{U} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{0} & \\mathbf{U}\n\\end{pmatrix}\n_{3N \\times N_{CG}}\n\\end{equation}\nThe constraint mapping approach begins by defining the objective function $f$ via\n\\begin{equation}\nf\\left(  \\mathbf{r}\\right)  =\\frac{1}{2}\\left(  \\mathbf{r}-\\mathbf{K\\phi\n-\\sigma}\\right)  ^{t}\\left(  \\mathbf{r}-\\mathbf{K\\phi-\\sigma}\\right)  .\n\\end{equation}\nThis function is convex and therefore admits a single unique minimum at\n$\\mathbf{r}_{u}\\mathbf{=K\\phi}+\\mathbf{\\sigma}$, the unconstrained map defined in\nEq. (\\ref{Unconstrained-Mapping}). Here we intoduce two sets of constraints that enforce\nconstant bond lengths and angles, respectively. Thus, as $\\mathbf{\\phi}$\nchanges, the minimization of $f$ with repect to $\\mathbf{r}$ subject to these\nconstraints yields atom-resolved configurations consistent with the CG state.\nThe first set of constraints is written in the form:\n\\[\n\\left\\vert \\mathbf{r}_{i}-\\mathbf{r}_{j}\\right\\vert =d_{ij},\n\\]\nsuch that $d_{ij}$ is the equilibrium bond length between atoms $i$ and $j$. The\nsecond set of constraints ensures the three-body angle, defined between\ntwo consecutive bonds (see Figure(\\ref{fig:tri})), is respected\n\\begin{equation}\n\\theta_{ijk}=\\theta_{ijk}^{eq},\n\\end{equation}\nsuch that the indices $i,j,k$ span every two consecutive bonds and\n$\\theta_{ijk}^{eq}$ is the value of the angle between these two bonds \nat equilibrium. This constraint can be satisfied by\nsetting $\\left\\vert \\mathbf{r}_{i}-\\mathbf{r}_{k}\\right\\vert $ equal to\n$d_{ik}$. For convenience, the inter-atomic distance squared is used in\nall constraint equations. The\noptimization problem can then be stated as follows:\n\\begin{equation}\n\\min_{\\mathbf{r}}f\\left(  \\mathbf{r}\\right),\n\\end{equation}\nsubject to%\n\\begin{align}\n\\left\\vert \\mathbf{r}_{i}-\\mathbf{r}_{j}\\right\\vert ^{2} &  =d_{ij}%\n^{2},\\label{Cons1}\\\\\n\\left\\vert \\mathbf{r}_{i}-\\mathbf{r}_{k}\\right\\vert ^{2} &  =d_{ik}%\n^{2}.\\label{Cons2}\n\\end{align}\nFor efficient parallelization, these equations are\nformulated in matrix form. To achieve that, the entire marcmolecule is represented \nas a Graph. In Graph theory \\cite{Chartrand1985}, a set of vertices (atoms) can link in pairs to form\nedges (bonds). To take the\nnon-bonded angle-determining distance ($d_{ik}$) into account, we introduce an adjacency-like\nmatrix $\\mathbf{A}$ that captures the location of each\natomic index in every constraint equation, i.e., for the $l^{th}$ constraint spanning\natoms $i$ and $j$, row $l$ in $\\mathbf{A}$ has $+1$ entry at column $i$, and $-1$\nat column $j$, while all remaining columns have zero entries.\nThe Lagrangian $\\mathcal{L}$ of the optimization problem can be written as\n\\begin{equation}\n\\mathcal{L}\n=f(\\mathbf{r})+\\mathbf{\\lambda}^t \\left(\\sum_{d=1}^{3}\\mathbf{D}\\left(\\mathbf{Ar}\n_{d}\\right)  \\mathbf{Ar}_{d}-\\mathbf{l}_{eq}\\right) ,\n\\end{equation}\nfor a vector of \\ Lagrange multipliers $\\mathbf{\\lambda}$, and vector\n$\\mathbf{l}_{eq}=\\sum_{d=1}^{3}\\mathbf{D}\\left(\\mathbf{A}\\mathbf{r}_d^{eq}\\right)\n\\mathbf{A}\\mathbf{r}_d^{eq}$ that represents the equilibrium inter-atomic distances squared. \nTo simplify the notation, we defined $\\mathbf{r} = (\\mathbf{r}_1,\\mathbf{r}_2,\\mathbf{r}_3)$.\nHere $\\mathbf{D(v)}$ represents the diagonal matrix form\nof vector $\\mathbf{v}$.\nBy using the Lagrange method for equality constraints\n\\cite{Optimization2006}, the Lagrangian with respect to $\\mathbf{r}$ and $\\mathbf{\\lambda}$ was\nminimized i.e. the equations $\\nabla_{\\mathbf{r}}\n\\mathcal{L} =0$ and $\\nabla_{\\mathbf{\\lambda}}\\mathcal{L}=0$ were solved such \nthat at the optimum solution $\\left(  \\mathbf{r}^{\\ast\n}\\mathbf{,\\lambda}^{\\ast}\\right)$ the Hessian of $\\mathcal{L}$\nis positive definite. This leads to the following non-linear coupled\nequations:\n\\begin{align}\n\\mathbf{r-r}_{u}-\\mathbf{\\mathbf{J}}_{\\mathbf{r}}^{t}\\mathbf{\\mathbf{\\lambda}} &\n\\mathbf{\\mathbf{=0},}\\label{mod_coord}\\\\\n\\sum_{d=1}^{3}\\mathbf{D}\\left(  \\mathbf{A}\\mathbf{r}_{d}\\right)  \\mathbf{A} \\mathbf{r}_{d}\n-\\mathbf{l}_{eq} &=\\mathbf{0}. \\label{lagrange}\n\\end{align}\nwhere $\\mathbf{J}_{\\mathbf{r}}$ is the Jacobian of Eqs. (\\ref{Cons1}\n,\\ref{Cons2}). When $\\mathbf{\\lambda}$ is zero, the unconstrained solution is\nrecovered. Unlike constrained MD \\cite{Barth2004}, this procedure does not does not modify \nthe ensuing Newtonian Physics, and furthermore yields the same coarse-graining\nmap $\\mathbf{r} \\rightleftharpoons \\mathbf{\\phi}$ noted above, i.e., the definition\nof $\\mathbf{\\phi}$ is unaltered (see Appendix).\n\n\\section{Methodology}\n\nFinding an analytical solution to the optimization problem presented in the\nprevious section is not feasible. Therefore, the following numerical approach was\nimplemented. First the constraints equations were recast in the form\n\\begin{equation}\n\\mathbf{c}=\\sum_{d=1}^{3}\\mathbf{D}\\left(\\mathbf{Ar}_{d}\\right)\n\\mathbf{Ar}_{d}-\\mathbf{l}_{eq}.\n\\end{equation}\nThe vector function $\\mathbf{c}$ depends on $\\mathbf{\\lambda}$, in a way\nthat can be made explicit by substituting $\\mathbf{r}$ from Eq. (\\ref{mod_coord}) \nto give\n\\begin{align}\n\\mathbf{c(\\lambda)}  & =\\sum_{d=1}^{3}\\mathbf{D}\\left(  \\mathbf{Ar}_{d}\n^{u}\\right)  \\mathbf{Ar}_{d}^{u}+\\mathbf{D}\\left(  \\mathbf{Ar}_{d}^{u}\\right)\n\\mathbf{A\\mathbf{J}}_{\\mathbf{r},d}^{t}\\mathbf{\\mathbf{\\lambda}}\\nonumber\\\\\n& +\\mathbf{D}\\left(  \\mathbf{A\\mathbf{J}}_{\\mathbf{r},d}^{t}\\mathbf{\\mathbf{\\lambda}\n}\\right)  \\mathbf{Ar}_{d}^{u}+\\mathbf{D}\\left(  \\mathbf{A\\mathbf{J}}_{\\mathbf{r}}\n^{t}\\mathbf{\\mathbf{\\lambda}}\\right)  \\mathbf{A\\mathbf{J}}_{\\mathbf{r},d}\n^{t}\\mathbf{\\mathbf{\\lambda}}-\\mathbf{l}_{eq}\\label{non-linear-cons}\n\\end{align}\nHere $\\mathbf{J}_{\\mathbf{r},0}^{t},~\\mathbf{J}_{\\mathbf{r},1}^{t}\n,\\mathbf{J}_{\\mathbf{r},2}^{t}$ are the $x,$ $y,$ and $z$ component block\nmatrices of $\\mathbf{J}_{\\mathbf{r}}^{t}$, i.e.\n\\begin{equation}\n\\mathbf{J}_{\\mathbf{r}}^{t}=\n\\begin{pmatrix}\n\\mathbf{J}_{\\mathbf{r},0}^{t}\\\\\n\\mathbf{J}_{\\mathbf{r},1}^{t}\\\\\n\\mathbf{J}_{\\mathbf{r},2}^{t}\n\\end{pmatrix}\n_{3N\\times N_{c}}.\n\\end{equation}\nThus, the optimization problem is reduced to solving a system of\nnon-linear equations for $\\mathbf{\\lambda}$. Newton's\nmethod can therefore be used to solve Eqs. (\\ref{non-linear-cons}) by setting\n$\\mathbf{\\lambda}$ equal to $\\mathbf{0}$, updating $\\mathbf{r}$ through Eqs.\n(\\ref{mod_coord}), and then by equating $\\mathbf{r}_{u}$ to $\\mathbf{r}$. This\nprocedure is iterative i.e. it is repeated until the change from iteration\nto iteration falls below a specified tolerance.\nThe Jacobian $\\mathbf{J}_{\\lambda}$ of Eq.\n(\\ref{non-linear-cons}) is found to be\n\\begin{equation}\n\\left.  \\mathbf{J}_{\\lambda}\\right\\vert _{\\mathbf{\\lambda}=\\mathbf{0}%\n}=2\\mathbf{D}\\left(  \\mathbf{Ar}_{d}^{u}\\right)  \\mathbf{A\\mathbf{J}}_{\\mathbf{r}}^{t}.\n\\end{equation}\nSolving the linear system, however, proves to be impractical due to the high\ncondition number of $\\mathbf{J}_{\\lambda}$. Shifting the matrix by a factor of\n$\\alpha$ significantly reduces the condition number and makes the problem\ntracable. Newton's iteration becomes\n\\begin{equation}\n\\left(  \\mathbf{J}_{\\lambda}+\\alpha\\mathbf{I}\\right)  \\mathbf{\\lambda\n}=-\\mathbf{c(0),} \\label{newton_iter}\n\\end{equation}\nwith $\\mathbf{I}$ being the identity matrix. Given that $\\lambda$ will\neventually vanish if the system converges to the correct solution, the shift\nterm $\\alpha\\mathbf{I}$ does not alter the numerical solution but makes the\nprocedure tracable. With this scheme, only\none linear system is solved per Newton iteration. A Krylov subspace\niterative solver based on the Conjugate Gradient method \\cite{Optimization2006} \nis used to solve Eq. (\\ref{newton_iter}).\nSince the Jacobian matrices ($\\mathbf{J}_{\\mathbf{r}}$ and\n$\\mathbf{J}_{\\mathbf{\\lambda}}$) contain a large number of zero entries (see\nFigure \\ref{fig:sparse}), they were stored in compressed sparse row  format \\cite{Timothy2006}. \nIn sparse storage, a low density matrix is compressed into a set of three vectors. \nThe first stores the non-zero values, while the second and third store the \nrow and column indices, respectively. \n\n\\section{Implementation}\n\\subsection{Parallelization}\nThe matrix form of Eqs. (\\ref{mod_coord}, \\ref{lagrange}, \\ref{newton_iter}) and the\nsize of the biological systems of interest (virus capsid, ribosomes, etc.) make \nthe algorithm a good candidate for parallelization.\nIn the current implementation, the algorithm is parallelized for distributed memory\nsystems. Since the most expensive part of the algorithm is the numerical solution of the linear \nsystem (Eq. (\\ref{newton_iter})), the sparse matrices $\\mathbf{A}$, $\\mathbf{J}_{\\mathbf{r}}$, and \n$\\mathbf{J}_{\\mathbf{\\lambda}}$ were constructed or assembled in parallel. This was done\nwith the aid of PETSC (portable extensible scientific toolkit) library \n\\cite{petsc-efficient,petsc-user-ref,petsc-web-page}, which uses message\npassing interface (MPI) to perform linear algebra subroutines in parallel. The library\nsupports sparse storage of matrices and vectors distributed on multiple nodes. In particular,\nan MPI matrix is partitioned into a set of sequential matrices, each of which is stored \nlocally on one specific node. The partitioning scheme used is shown in Figure \\ref{fig:storage}. \nMPI vectors such as \nthe coordinates $\\mathbf{r}$ and the lagrange multipliers $\\mathbf{\\lambda}$ were\nstored distributively on all processors. With the input read on node 0 and parallelized\nacross all available processors (see Figure \\ref{fig:diag}), the algorithm proceeds by constructing\nthe RHS of Eq. (\\ref{newton_iter}) and the Jacobian $\\mathbf{J}_{\\mathbf{r}}$, \nassembling the Jacobian $\\mathbf{J}_{\\mathbf{\\lambda}}$, and then \nsolving the linear system with a scalable iterative solver.\nSince $\\mathbf{J}_{\\mathbf{\\lambda}}$ is symmetric, and the shifting factor makes it positive\ndefinite, the Conjugate Gradient method (KSPCG in PETSC) was used to solve\nthe linear system of Eq. (\\ref{newton_iter}).\n\\subsection{Results and discussion}\n\nIn order to assess the accuracy, efficiency, and scalability of the constrained mapping\nalgorithm, three systems were simulated in vacuum. The first is Lactoferrin\n(ID 1LFG) undergoing a structural transition \\cite{AbiMansour2013}.\nIt was experimentally shown that this iron binding protein has two free energy \nminimizing conformations \\cite{Norris1991}: diferric with closed proximal lobes (PDB code\n1LFG), and apo with open ones (PDB code 1LFH). Here, we start \nwith an open lactoferrin structure and simulate its closing in vacuum. The \nsecond system is N$\\omega$v (ID 1OHF) triangular structure \\cite {Johnson2003} shrinking after equilibration in an explicit \nsolvent at pH = 5.4, and the third is the native CCMV viral capsid (ID 1CWP) shrinking due to \nstrong protein-protein interactions. All three systems were previously simulated \\cite{AbiMansour2013} using\nDMS \\cite{Pankavich2008,Cheluvaraja2010,Ortoleva2012,AbiMansour2013} and NAMD \\cite{NAMD2005} under NVT conditions. \nHere we show that the constrained map captures the system dynamics accurately \n(Figures \\ref{fig:LFG}-\\ref{fig:CCMV}) and with a greater efficiency (Figure \\ref{fig:speedup}) over\nboth DMS and conventional MD. \n\n\\section{Conclusion}\n\nWe have shown that the coarse-grained to all-atom information gap\ncan be decreased by taking bond lengths and angles into\naccount when constructing atom-resolved states from CG variables. The mapping\nalgorithm proposed in this paper provided efficiencies not only over traditional MD \nbut also over the multiscale approach without constraints. \nThese efficiencies will enable the simulation of mesoscopic systems\nsuch as viruses over longer periods of time. Furthermore, the mapping algorithm is\nflexible so that future implementations can also account for experimental\nnanocharacterization data \\cite{Long2006} (e.g., atomic force microscopy, \ntime of flight techniques, and micro-fluidics).\n\n\\section{Appendix}\n\n\\subsection{Construction of CG variables from atomic positions}\n\nIn the Formulation section, we modified Eq. (\\ref{Unconstrained-Mapping}) by\nimposing a set of constraints on the bond lengths and angles. We showed\nthat the solution to the optimization problem can be obtained in terms of a vector of Lagrange \nmultipliers (Eq. (\\ref{mod_coord})). Here we show that this approach does not modify the \nCG variables derived from the unconstrained map \\cite{Ortoleva2012}. We begin by constructing $\\mathbf{\\phi}$ \nfrom $\\mathbf{r}$ by doing a mass-weighted least-squares minimization of $\\mathbf{\\sigma}$ \nwith respect to $\\mathbf{\\phi}$, i.e.\n\\begin{equation}\n\\min_{\\mathbf{\\phi}}\\frac{1}{2}\\left(\\mathbf{M\\sigma}\\right)^{t}\\sigma.\n\\label{phi_min}\n\\end{equation}\nHere $\\mathbf{M}$ is a diagonal matrix of all atomic masses. \nSince $\\mathbf{\\sigma}=\\mathbf{r}-\\mathbf{U\\phi}-\\mathbf{J}_{\\mathbf{r}}^{t}\n\\mathbf{\\lambda}$, the solution to Eq. (\\ref{phi_min}) is\n\\begin{equation}\n\\mathbf{MU}^{t}\\mathbf{U\\phi=MU}^{t}\\left(\\mathbf{r}-\\mathbf{J}_{\\mathbf{r}}\n^{t}\\mathbf{\\lambda}\\right)\n\\end{equation}\nHowever, in constructing $\\mathbf{\\phi}$, the all-atom configuration used is\nfrom MD, which implies that the constraints are satisfied. The Jacobian\n$\\mathbf{J}_{\\mathbf{r}}$ vanishes, and the CG variables are therefore unaltered.\n\n\\subsection{The adjacency-like matrix}\n\nFor a simple triatomic non-linearly bonded molecule (see Figure \\ref{fig:tri}), \nthe adjacency-like matrix $\\mathbf{A}$ defined in the Formulation section takes the form\nshown in Figure \\ref{fig:adj}. Here a total of three constraints must be taken into account\nfor three bonded atoms.\n\n\n\\acknowledgement\nThis project was supported in part by the National Science Foundation\n(Collaborative Research in Chemistry Program), National Institutes of Health\n(NIBIB), METAcyt through the Center of Cell and Virus Theory, Indiana\nUniversity College of Arts and Sciences, and the Indiana University\ninformation technology services (UITS) for high performance computing resources.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n\n\n\\begin{document}\n\n\\title{Recent results on 4-body, charm semileptonic decays}\n\n\n\\author{Jim Wiss}\n\\affiliation{University of Illinois, 1110 W. Green, Urbana IL , 61801}\n\n\\begin{abstract}\nWe summarize recent data on 4-body charm semileptonic decay concentrating on \\ensuremath{D_s^+ \\rightarrow K^+ K^- e^+ \\nu }{} and \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}.\nWe begin with giving some motivation for the study of these decays. We discuss several of \nthe models traditionally used to describe these decays and conclude by presenting a non-parametric\nanalysis of \\ensuremath{D^+ \\rightarrow K^- \\pi^+ e^+ \\nu }{} and its possible extension into non-parametric studies of \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\mu^+ \\nu }{}.\n\\end{abstract}\n\\maketitle\n\n\\thispagestyle{fancy}\n\n\n\\section{Introduction}\n\nFigure \\ref{cartoon} shows a cartoon of the $D^0 \\rightarrow K^- \\ell^+ \\nu$ decay process.  \nAll of the  hadronic complications for this process is contained\nin $\\ensuremath{q^2}{}$ dependent form factors that are computable \nusing non-perturbative methods such as LQCD.  Although semi-leptonic process can in principle provide a  determination of charm CKM elements, one frequently uses the (unitarity constrained) CKM measurements, lifetime, and branching\nfraction to measure the scale of charm semileptonic decay constants and compare them to \nLQCD predictions. The $\\ensuremath{q^2}{}$ dependence of the semileptonic form factor can also be directly measured\nand compared to theoretical predictions.  \n\nThe hope is that charm semileptonic decays can provide high statistics, precise tests of LQCD calculations and thus\nvalidate the computational techniques for charm.  \nOnce validated, the same LQCD techniques can be used in related calculations for $B$-decay\nand thus produce CKM parameters with significantly reduced theory systematics.  \n\n  \\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.5in]{slcartoon.eps}\n  \\caption{Diagrams for the semileptonic decay of charmed mesons. The\nhadronic,QCD complications are contained in \\ensuremath{q^2}{} dependent form factors. \n\\label{cartoon}} \\end{center}\n\\end{figure}\nAlthough recent, unquenched LQCD calculations are unavailable for \\ensuremath{D \\rightarrow {\\rm vector}~ \\ell^+ \\nu }{} processes, owing to the instability \nof the vector parent, I hope that the 4-body will provide additional tests of LQCD for a variety of spin\nstates which will further help calibrate the lattice, and provide confidence in analogous decays for the beauty sector.\n\nI find it remarkable that 4-body semileptonic decays such as \\ensuremath{D_s^+ \\rightarrow K^+ K^- \\ell^+ \\nu }{} and \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\mu^+ \\nu }{} are so heavily\ndominated by the vector decays \\ensuremath{D_s^+ \\rightarrow \\phi\\; e^+ \\nu_e }{} and \\ensuremath{D^+ \\rightarrow \\krzb \\mu^+ \\nu}{}. Figure \\ref{VecDom} illustrates this dominance by showing\ndata from FOCUS\\cite{swave} and recent data from BaBar\\cite{BBPhilnu}.\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=2.in]{kpi_lineshape.ps}\n  \\includegraphics[width=2.in]{mkkBaBar.eps}\n  \\caption{   We show the $m(K^+K^-)$ spectra obtained in \\ensuremath{D_s^+ \\rightarrow K^+ K^- \\ell^+ \\nu }{} by BaBar\\cite{BBPhilnu} and \n$m(K^-\\pi^+)$ by FOCUS\\cite{swave}. The curve on the FOCUS $m(K^-\\pi^+)$ spectra is a \\ensuremath{\\overline{K}^{*0}}{} line shape\nboth with (A = 0.36) and without (A = 0) a small s-wave, non-resonant component which was found through an interference\nin the decay intensity and is described later. The $m(K^+K^-)$ spectra obtained by BaBar is very strongly dominated by the $\\phi$\nresonance along with a few known backgrounds. \n\\label{VecDom}} \\end{center}\n\\end{figure}\nThe absence of a substantial non-resonant, or higher spin resonance component to these decays means\nthe decay angular distribution can be described in terms of three, \\ensuremath{q^2}{}-dependent helicity\nbasis form factors that describe the coupling of the lepton system to the three helicity states\nof the vector meson according to Eq. (\\ref{dkelect}) :\n\n\\begin{eqnarray}\n\\left| \\cal{A} \\right|^2  \\approx \\frac{{q^2 }}{8}\\left| \\begin{array}{l}\n (1 + \\cos \\theta _l )\\sin \\theta _V e^{i\\chi } H_ + (\\ensuremath{q^2}) \\\\ \n  - (1 - \\cos \\theta _l )\\sin \\theta _V e^{ - i\\chi } H_ - (\\ensuremath{q^2})  \\\\ \n  - 2\\sin \\theta _l \\cos \\theta _V H_0 (\\ensuremath{q^2}) \\\\ \n \\end{array} \\right|^2\n\\label{dkelect}\n\\end{eqnarray}\n\nThe three decay angles describing the \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}\ndecay, referenced in Eq.(\\ref{dkelect}), are illustrated by Fig. \\ref{anglesA}. \n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=2.in]{angles.eps}\n    \\caption{Definition of kinematic variables.\n \\label{anglesA}}\n \\end{center}\n\\end{figure}\n\\mysection{Analytic models for form factors}\n\nWe begin by describing the three form factors relevant to \\ensuremath{D \\rightarrow {\\rm vector}~ \\ell^+ \\nu }{} although there is strong evidence \\cite{swave}\\cite{cleo-ff} for a non-resonant, s-wave component to \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}. A new, fifth form factor $H_T(\\ensuremath{q^2})$ is also required for \\ensuremath{D^+ \\rightarrow \\krzb \\mu^+ \\nu}{} to describe\nthe suppressed coupling of the \\ensuremath{\\overline{K}^{*0}}{} to a left-handed $\\mu^+$.\n\nThe   \\ensuremath{H_+(\\ensuremath{q^2})}{} , \\ensuremath{H_-(\\ensuremath{q^2})}{}, \\ensuremath{H_0(\\ensuremath{q^2})}{} form factors are linear combinations of two axial and one vector form factor~\\cite{KS} according to Eq. ~(\\ref{helicity}): \n\\begin{widetext}\n\\begin{eqnarray}\nH_\\pm(\\ensuremath{q^2}) &=&\n   (M_D+\\ensuremath{m_{K\\pi}})A_1(\\ensuremath{q^2})\\mp 2{M_D K\\over M_D+m_{K\\pi}}V(\\ensuremath{q^2}) \\,,\n                               \\nonumber \\\\\nH_0(\\ensuremath{q^2}) &=&\n   {1\\over 2\\ensuremath{m_{K\\pi}}\\sqrt{\\ensuremath{q^2}}}\n   \\left[\n    (M^2_D -m^2_{K\\pi}-\\ensuremath{q^2})(M_D+\\ensuremath{m_{K\\pi}})A_1(\\ensuremath{q^2}) \n    -4{M^2_D K^2\\over M_D+\\ensuremath{m_{K\\pi}}}A_2(\\ensuremath{q^2}) \\right] \\label{helicity} \\nonumber \\\\ \n\\label{helform}\n\\end{eqnarray}\n\\end{widetext}\nwhere $K$ is the momentum of the $K^- \\pi^+$ system and \\ensuremath{m_{K\\pi}}{} is its mass.\n\nEq.(\\ref{analytic}) provides considerable insight into the expected analytic form for semileptonic form factors.\nIt uses a dispersion relation obtained using Cauchy's Theorem under the assumption that a form factor is an analytic, complex function apart from some known singularities. Fig. \\ref{cut} illustrates the Cauchy's Theorem contour for the case for the $f_+(\\ensuremath{q^2}{})$ form factor\ndescribing \\ensuremath{D^0 \\rightarrow K^- \\ell^+ \\nu }{}. \n\n\\begin{figure}[tbph!]\n\\begin{center}\n  \\includegraphics[width=2.5in]{cut.eps}\n  \\caption{ Each form factor is assumed to be an analytic function with pole singularities at the \nmasses of bound states, and cuts that start at the start of the continuum. We illustrate\nthe case of \\ensuremath{D^0 \\rightarrow K^- \\ell^+ \\nu }{}. One can use Cauchy's\ntheorem with the indicated contour to write an dispersion expression for each form factor in the physical range \n$0 < \\ensuremath{q^2}{} < \\left(m_D - m_K \\right)^2$ \n\\label{cut}} \n\\end{center}\n\\end{figure}\nThe form factor singularities will consist of a sum of simple poles at the D meson -kaon vector bound states (e.g. $D_s^{*+}$) plus\na cut beginning at the $D-{\\rm kaon}$ continuum in the cross process: $\\nu \\ell^+ \\rightarrow D~{\\rm kaon}$. The dispersion relation\ngives the form factor ($\\rm{F}(\\ensuremath{q^2})$) as a sum over the spectroscopic poles plus an integral over the cut.\n\\begin{eqnarray}\n\\rm{F}(q^2 ) = \\frac{\\mathcal{R}}{{m_{D_s^*}^2  - q^2 }} + \\frac{1}{\\pi }\\int_{\\left( {m_D  + K} \n\\right)^2 }^\\infty  {\\frac{{{\\mathop{\\rm Im}\\nolimits} \\left\\{ {f_ +  (s)} \\right\\}}}{{s - q^2  - \ni\\varepsilon }}ds} \n\\label{analytic}\n\\end{eqnarray}\n\nBoth the cuts and poles are generally beyond the physical $\\ensuremath{q^2}{}_{\\rm max}$ and thus can never be actually realized. \n\nSpectroscopic pole dominance (SPD) was an early parameterization for the form factors relevant to both \\ensuremath{D \\rightarrow {\\rm vector}~ \\ell^+ \\nu }{} and \\ensuremath{D \\rightarrow {\\rm pseudoscalar}~ \\ell^+ \\nu }{}.\nSPD ignores the cut integral entirely and approximates $\\rm{F}(q^2 )$ using just the first term of Eq.(\\ref{analytic}). \nThe advantage of SPD approach is that it requires only a single unknown fitting parameter $\\mathcal{R}$ to describe  \neach $\\rm{F}(\\ensuremath{q^2})$ since the positions of the bound states are well known. SPD entirely predicts {\\it shape} of \\ensuremath{D \\rightarrow {\\rm pseudoscalar}~ \\ell^+ \\nu }{} decay intensity\nand predicts that the {\\it shape} for the \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} can be fit by just two parameters\nwhich are traditionally taken to be the axial and vector form factor ratios at $\\ensuremath{q^2} = 0$: $\\ensuremath{r_v} = V(0)\/A_1(0)$ and $\\ensuremath{r_2} = A_2(0)\/A_1(0)$.\n\nBaBar \\cite{BBPhilnu} has recently published an interesting SU(3) test based on SPD applied to \\ensuremath{D_s^+ \\rightarrow \\phi\\; e^+ \\nu_e }{}. \nFigure \\ref{phi-ff} compares the \\ensuremath{r_v}{} and \\ensuremath{r_2}{} parameters measured for \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} to those previously measured for\n\\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{}. By SU(3) symmetry and explicit calculation, the \\ensuremath{r_v}{} and \\ensuremath{r_2}{} form factor ratios for \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} and \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} decays are\nexpected to be very close to each other. This is true for \\ensuremath{r_v}{}, but previous to the recent measurement by the FOCUS Collaboration\\cite{focusPhi}, \\ensuremath{r_2}{} for  \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} was measured to be roughly a factor of two larger than that\nfor \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}. BaBar\\cite{BBPhilnu} has confirmed the expected consistency between the form factor ratios obtained for \n\\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} and \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} with unparalleled statistics.\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=2.75in]{rvBaBar.eps}\n\\includegraphics[width=2.75in]{r2BaBar.eps}\n  \\caption{   \nThe \\ensuremath{r_v}{} and \\ensuremath{r_2}{} form factor ratios measured for \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} by various experiments. The blue\nlines show $\\pm 1 \\sigma$ bands for the weighted average of the \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} form factor ratios compiled in Reference \\cite{fpcp}. It is\nexpected from SU(3) symmetry that the \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} form factors should be very close to those for\n\\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{}\n \\label{phi-ff}} \\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.3in]{mpole.eps}\n  \\caption{ Effective pole mass measurement in $D^0 \\rightarrow K^- e^+ \\nu$ over the years.\nThe green line is the $m_{D_S^*}$ sectroscopic pole mass and is inconsistent with the \naverage of the displayed data by 5.1 $\\sigma$. \\label{mpole}} \\end{center}\n\\end{figure}\n\nSeveral experiments have tested SPD by measuring an ``effective\" pole mass ($m_{pole}$)in $D^0 \\rightarrow K^- e^+ \\nu$ decay \nwhere the pole mass is defined using ${\\rm{f}}_{\\rm{ + }} (q^2 ) \\propto 1\/{(m_{pole}^2  - q^2)}$.  As Fig. \\ref{mpole} from Reference \\cite{fpcp} shows, as errors have\nimproved over the years, it becomes clear that effective pole is significantly lower than the spectroscopic \npole, underscoring the importance of the cut integral contribution for this decay.\n\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.2in]{hillTrans.eps}\n  \\caption{Illustration of Hill transformation approach. \\label{hillTrans}} \\end{center} \\end{figure}\n\n\nSeveral parameterizations have been proposed to include the cut integral in Eq. (\\ref{analytic}) as well as the spectroscopic poles. \nBecirevic and Kaidalov (1999) \\cite{BK} proposed a new parameterization for the \\ensuremath{D \\rightarrow {\\rm pseudoscalar}~ \\ell^+ \\nu }{} for factor $f_+(\\ensuremath{q^2}{})$ that replaces the cut integral by an effective pole where the heavy quark symmetry and other theoretical ideas are used to relate the residue and effective pole position. These constraints\nleads to a modified pole form with a single additional parameter $\\alpha$ that describes the degree to which\nthe single spectroscopic pole fails to match $f_+(\\ensuremath{q^2}{})$ for a given process. \n\n\\begin{eqnarray}\nf_ +  (q^2 ) = \\frac{{f_ +  (0)}}{{\\left( {1 - q^2 \/m_{D*}^2 } \\right)\\left( {1 - \\alpha q^2 \/m_{D*}^2 } \\right)}}\n\\label{modpole}\n\\end{eqnarray}\nS. Fajfer and J. Kamenik \\cite{FK} have recently extended the effective pole approach to the three helicity form \nfactors relevant to \\ensuremath{D \\rightarrow {\\rm vector}~ \\ell^+ \\nu }{} decays.\n\nR.J. Hill\\cite{Hill2}\\cite{Hill} has proposed an alternative way of viewing form factors which is illustrated in Fig. \\ref{hillTrans}.\nThe basic idea is to devise a transformation of a form factor from the complex \\ensuremath{q^2}{} plane to a complex $z$ plane. This transformation is devised to (1) remove the spectroscopic poles and (2) put the cuts far away from the physical $z$ region. After the \ntransformation, since the singularities\nhave been removed or diminished, each form factor can be well represented by a low order Taylor series in $z$.  The transformation approach is known\\cite{Hill} to work very well in $B$-decays where the physical \\ensuremath{q^2}{} region gets very close to the singularities for pseudo-scalar $B$ semileptonic decay. It also works well for pseudoscalar charm pseudoscalar semileptonic decay\\cite{Hill2}. \n\\mysection{ \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} Decays}\nAlthough historically \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} have been the most accessible semileptonic\ndecays in fixed target experiments owing to their ease of isolating a signal, they are significantly more complicated\nto analyze than \\ensuremath{D \\rightarrow {\\rm pseudoscalar}~ \\ell^+ \\nu }{}. One problem is that a separate helicity form factor is required\nfor each of the three helicity states of vector meson. The \\ensuremath{q^2}{} dependence of these form factors cannot\nbe simply measured from the \\ensuremath{q^2}{} dependence of the decay rate as is the case in \\ensuremath{D \\rightarrow {\\rm pseudoscalar}~ \\ell^+ \\nu }{} but rather\nmust be entangled from the \\ensuremath{q^2}{} dependence of the angular distribution such as that given by Eq. (\\ref{dkelect}).\n\nAnother complication is that since \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} states result in a multihadronic\nfinal state, the \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} final states can potentially interfere with \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} processes\nwith the $K^- \\pi^+$ in various angular momentum waves with each wave requiring its own form factor.\nBecause the \\ensuremath{m_{K\\pi}}{} distribution in \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} was an excellent fit to the \\ensuremath{\\overline{K}^{*0}}{} Breit-Wigner as shown in Fig. \\ref{VecDom},\nit was assumed for many years that any non-resonant component to \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} must be negligible.\nIn 2002, FOCUS observed a strong, forward-backward asymmetry in \\ensuremath{\\cos\\thv}{} for events with \\ensuremath{m_{K\\pi}}\nbelow the \\ensuremath{\\overline{K}^{*0}}{} pole with essentially no asymmetry above the pole as shown in Figure \\ref{asym}.\nThe simplest explanation for this asymmetry is the presence of a linear \\ensuremath{\\cos\\thv}{} term in the decay intensity due to interference\nbetween the \\ensuremath{D^+ \\rightarrow \\krzb \\mu^+ \\nu}{} and a non-resonant, s-wave amplitude. This interference is the second-to-last term in  Eq. (\\ref{KSE}), which is basically an expanded\nout version of Eq. (\\ref{dkelect}), integrated over acoplanarity $\\chi$. We also explicitly include the \\ensuremath{\\overline{K}^{*0}}{} Breit-Wigner amplitude ($BW$). \nNote that all other interference terms (such as a possible $H_+(\\ensuremath{q^2}) \\times H_-(\\ensuremath{q^2})$ contribution) vanish because of the \n$\\int_0^{2\\pi} d \\chi \\exp(i \\Delta \\chi)$ integration.  Only ``same\" helicity contributions can interfer in the acoplanarity averaged intensity.  We will argue shortly that an appropriate $\\delta$ can create the asymmetry pattern shown in Fig. \\ref{asym}  \n\n\n\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.25in]{asym.ps}\n  \\caption{Evidence for s-wave interference in \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}.\n \\label{asym}}\n \\end{center}\n\\end{figure}\nFinally we introduce an additional form factor  (\\ensuremath{h_0(\\ensuremath{q^2})}{}) in\nEq. (\\ref{KSE}) to describe the coupling to the s-wave amplitude.  \n\\begin{widetext}\n\\begin{eqnarray}\n\\int {\\left| {\\rm{A}} \\right|^2 d\\chi }  = \\frac{1}{8}q^2 \\left\\{ \\begin{array}{l}\n \\left( {(1 + \\cos \\theta _l )\\sin \\theta _V } \\right)^2 \\left| {H_ +  (q^2 )} \\right|^2 \\left| {BW} \\right|^2  \\\\ \n  + \\left( {(1 - \\cos \\theta _l )\\sin \\theta _V } \\right)^2 \\left| {H_ -  (q^2 )} \\right|^2 \\left| {BW} \\right|^2  \\\\ \n  + \\left( {2\\sin \\theta _l \\cos \\theta _V } \\right)^2 \\left| {H_0 (q^2 )} \\right|^2 \\left| {BW} \\right|^2  \\\\ \n  + 8\\left( {\\sin ^2 \\theta _l \\cos \\theta _V } \\right)H_0 (q^2 )h_o (q^2 ){\\mathop{\\rm Re}\\nolimits} \\left\\{ {Ae^{ - i\\delta } BW} \\right\\} \\\\ \n  + O(A^2 ) \\\\ \n \\end{array} \\right\\}\n\\label{KSE}\n\\end{eqnarray}\n\\end{widetext}\n\\subsection{\\label{asymtopia} Asymtotic Forms}\nAssuming that $A_{1,2}(\\ensuremath{q^2})$ and $V(\\ensuremath{q^2})$ approach a constant in the low \\ensuremath{q^2}{} limit, as expected in \nspectroscopic pole dominance, Eq. (\\ref{helform}) shows $\\ensuremath{q^2}{} \\rightarrow 0$, both \\ensuremath{H_+(\\ensuremath{q^2})}{} and \\ensuremath{H_-(\\ensuremath{q^2})}{} approach a constant as well. \nBy way of constrast, \\ensuremath{H_0(\\ensuremath{q^2})}{} will diverge in the low \\ensuremath{q^2}{} limit according to Eq. (\\ref{helform}) owing to the $1\/\\sqrt{\\ensuremath{q^2}}$ \nprefactor. \nSince the helicity intensity contributions are proportional to $\\ensuremath{q^2}{} H^2_\\pm(\\ensuremath{q^2}{})$, according to Eq.(\\ref{KSE}),\nthe $H_\\pm$ intensity contributions vanish in this limit, while $\\ensuremath{q^2}{} H^2_0(\\ensuremath{q^2}{})$ will approach a constant.\n\nFigure \\ref{hcartoon}\nexpains why this is true. As $\\ensuremath{q^2}{} \\rightarrow 0$, the $e^+$ and $\\nu$ become collinear with the virtual $W^+$.\nFor \\ensuremath{H_+(\\ensuremath{q^2})}{} and \\ensuremath{H_-(\\ensuremath{q^2})}{}, the virtual $W^+$ must be in the $| 1 , \\pm 1 \\rangle$ state which means\nthat the  $e^+$ and $\\nu$ must both appear as either right-handed or left-handed thus violating the charged current helicity rules. \nHence $\\ensuremath{q^2}{}H_\\pm (\\ensuremath{q^2}{})$ vanishes at low \\ensuremath{q^2}{}.  For \\ensuremath{H_0(\\ensuremath{q^2})}{}, the $W^+$ is in $| 1 , 0 \\rangle$ state thus allowing the \n$e^+$ and $\\nu$ to be in their (opposite) natural helicity state.  Hence at low \\ensuremath{q^2}{},  $\\ensuremath{q^2}{}\\ensuremath{H_0(\\ensuremath{q^2})}{} \\rightarrow {\\rm constant}$\nwhich allows for  \\ensuremath{D^+ \\rightarrow \\krzb \\mu^+ \\nu}{} decays as $\\ensuremath{q^2}{}\\rightarrow 0$. Presumably $\\ensuremath{h_0(\\ensuremath{q^2})}{} \\rightarrow 1\/\\sqrt{\\ensuremath{q^2}{}}$ as well since\nit also describes a process with $W^+$ in the $| 1 , 0 \\rangle$ state \n  \n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.0in]{hcartoon.eps}\n  \\caption{The electron helicity state in the low \\ensuremath{q^2}{} limit. When the virtual $W^+$ is in the zero\nhelicity state, the $e^+$ and $\\nu$ have the opposite helicity and can be in their charged-current\nhelicity states. When the virtual $W^+$ is in the $| 1 , \\pm 1 \\rangle$ state the\n$e^+$ and $\\nu$ must be in the same helicity states and violate the weak helicity rules. \n \\label{hcartoon}}\n \\end{center}\n\\end{figure}\nHere is a final observation on the expected asymtotic behavior of the helicity form factors. As $\\ensuremath{q^2}{} \\rightarrow q^2_{\\rm max}$, the\nmomenta of the virtual $W^+$ and \\ensuremath{\\overline{K}^{*0}}{} approaches zero and \\ensuremath{\\theta_\\textrm{v}}{} and \\ensuremath{\\theta_\\ell}{} can no longer be defined.  This means the \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} decay\nmust be isotropic and Eq. (\\ref{KSE}) implies that $|H_\\pm|^2 \\rightarrow |H_0|^2$ as $\\ensuremath{q^2}{} \\rightarrow q^2_{\\rm max}$. \nA spectroscopic pole dominance model for the axial and vector form factors will automatically satisfies these asymtotic limits\naccording to Eq.(\\ref{helform}). \n\n\\mysection{Projection weighting technique} \n\nWe next describe the projective weighting technique that we use to extract\nthe helicity basis form factors.  This technique was initially developed by the FOCUS Collaboration\\cite{focus-helicity}\nand applied to CLEO\\cite{cleo-ff} data.  As shown in Eq. (\\ref{KSE}), after integrating over acoplanarity, the decay intensity\nis just a sum over four terms that consist of a form factor product times a characteristic angular distribution in \\ensuremath{\\theta_\\textrm{v}}{} and \n\\ensuremath{\\theta_\\ell}{}. The acoplanarity integration has significantly simplified the problem by eliminating the five of the possible six interference terms between the four form factor amplitudes with different helicities.\nWe begin by making a binned version of Eq. (\\ref{KSE}) given by Eq. (\\ref{series}), where for simplicity we only write three of the terms.\n\n\\begin{equation}\n  \\vec D_i = f_+(q_i^2)\\:\\vec m_+ + f_-(q_i^2)\\:\\vec m_- \n            + f_0(q_i^2)\\:\\vec m_0\n\\label{series}\n\\end{equation}\n\nWe use 25 joint  $\\Delta \\ensuremath{\\cos\\thv}  \\times \\Delta \\ensuremath{\\cos\\thl}$ angular bins:  \n5 evenly spaced bins in \\ensuremath{\\cos\\thv}{} times 5 bins in \\ensuremath{\\cos\\thl}{} and 6 bins in \\ensuremath{q^2}{} ($i = 0 \\rightarrow 6$).\nThe number of \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} events observed in each of the 25 angular bins is packed into a twenty-five component $\\vec D_i$ ``data\" vector. \n\n\nThe $f_\\pm(q_i^2)$ and $f_0(q_i^2)$ are proportional to \\ensuremath{H^2_\\pm(\\ensuremath{q^2})}{}, \\ensuremath{H^2_0(\\ensuremath{q^2})}{} averaged over the $q^2_i$ bin along with all phase space and efficiency factors. The $\\vec m_\\pm$ and $\\vec m_0$ are the angular distributions due to each individual form factor\nproduct packed into a 25-vector for each of the six \\ensuremath{q^2}{} bins. The acceptance and phase space corrected $m$-vectors\nare obtained directly from a Monte Carlo simulation where a given form factor product is turned on and all others are turned off.\nWe can write Eq.~(\\ref{series}) as the ``component equation\" shown in  \nEq.~(\\ref{system}) by forming the dot product with each of the three $m$-vectors:\n\\begin{widetext}\n\\begin{equation}\n\\left( {\\begin{array}{*{20}c}\n   {\\vec m_+ \\cdot \\vec D_i }  \\\\\n   {\\vec m_- \\cdot \\vec D_i }  \\\\\n   {\\vec m_0 \\cdot \\vec D_i }  \\\\\n\\end{array}} \\right) = \\left( {\\begin{array}{*{20}c}\n   {\\vec m_+ \\cdot \\vec m_+} & {\\vec m_+ \\cdot \\vec m_-} \n      & {\\vec m_+ \\cdot \\vec m_0 }  \\\\\n   {\\vec m_- \\cdot \\vec m_+} & {\\vec m_- \\cdot \\vec m_-} \n      & {\\vec m_- \\cdot \\vec m_0 }  \\\\\n   {\\vec m_0 \\cdot \\vec m_+} & {\\vec m_0 \\cdot \\vec m_-} \n      & {\\vec m_0 \\cdot \\vec m_0 }  \\\\\n\\end{array}} \\right) \\left( {\\begin{array}{*{20}c}\n   {f_+(q_i^2)}  \\\\\n   {f_-(q_i^2)}  \\\\\n   {f_0(q_i^2)}  \\\\\n\\end{array}} \\right)\n\\label{system}\n\\end{equation}\nThe solution to Eq.~(\\ref{system}) can be written as:\n\n\\begin{equation}\nf_+ (q_i^2) = {}^i\\vec P_+ \\cdot \\vec D_i\\:,\\\nf_- (q_i^2) = {}^i\\vec P_- \\cdot \\vec D_i\\:,\\\nf_0 (q_i^2) = {}^i\\vec P_0 \\cdot \\vec D_i\n\\label{solution}\n\\end{equation}\nwhere ${}^i\\vec P_\\alpha$ vectors are given by Eq.~(\\ref{projectors}).\n\n\\begin{equation}\n\\left( {\\begin{array}{*{20}c}\n   {{}^i\\vec P_+}  \\\\\n   {{}^i\\vec P_-}  \\\\\n   {{}^i\\vec P_0}  \\\\\n\\end{array}} \\right) = \\left( {\\begin{array}{*{20}c}\n   {\\vec m_+ \\cdot \\vec m_+} & {\\vec m_+ \\cdot \\vec m_-}\n       & {\\vec m_+ \\cdot \\vec m_0}  \\\\\n   {\\vec m_- \\cdot \\vec m_+} & {\\vec m_- \\cdot \\vec m_-}\n       & {\\vec m_- \\cdot \\vec m_0}  \\\\\n   {\\vec m_0 \\cdot \\vec m_+} & {\\vec m_0 \\cdot \\vec m_-}\n       & {\\vec m_0 \\cdot \\vec m_0}  \\\\\n\\end{array}} \\right)^{- 1} \\left( {\\begin{array}{*{20}c}\n   {\\vec m_+}  \\\\\n   {\\vec m_-}  \\\\\n   {\\vec m_0}  \\\\\n\\end{array}} \\right)\n\\label{projectors}\n\\end{equation}\n\\end{widetext}\nIt is useful to think of forming the dot products in Eq. (\\ref{solution})\nby making a weighted histogram:\n\\begin{equation}\n\\vec P_+ \\cdot \\vec D = \n    \\left[\\vec P_+\\right]_1 n_1 + \\left[\\vec P_+\\right]_2 n_2 \n       + \\cdots \\left[\\vec P_+\\right]_{25} n_{25}  \n\\label{likeweighting}\n\\end{equation}\nEq.~(\\ref{likeweighting}) demonstrates the product $\\vec P_ + \\cdot \\vec D$ is\nequivalent to weighting the $n_1$ events in angular bin 1 by\n$\\left[\\vec P_+\\right]_1$, weighting the $n_2$ events in angular bin 2 by\n$\\left[\\vec P_+\\right]_2$, etc.\nHence each form factor product such as $f_+(q^2_i)$ can be obtained by simply weighting\nthe data by $\\left[\\vec P_+\\right]_i$ where $i$ is the angular bin of the given datum.\nThe acceptance and phase space factors can be easily included the projective weights as well in order to directly\nproduce each form factor product.\nHence the (arbitrarily normalized)\nform factor products $H_+^2(\\ensuremath{q^2})$, $H_-^2(\\ensuremath{q^2})$, and $H_0^2(\\ensuremath{q^2})$ can then be obtained by making three weighted histograms using the efficiency rescaled ${{}^i\\vec P}_+$, \n${{}^i\\vec P}_-$, and ${{}^i\\vec P}_0$ weights respectively.\n\nThe same, basic projective weighting approach has been recently applied by the FOCUS Collaboration\\cite{kkpi} for a non-parametric\nanalysis of the $K^- \\pi^+$ amplitudes in the hadronic decay $D^+ \\rightarrow K^- K^+ \\pi^+$. To whet the appetite,\nFig.\\ref{sysall2} shows the $K^- \\pi+$ amplitudes obtained in that analysis. The s-wave amplitude shown \nin Fig. \\ref{sysall2} (a) and\nbegs comparison with the s-wave amplitude obtained in a K-matrix analysis\\cite{kpipi} of  $D^+ \\rightarrow K^- \\pi^+ \\pi^+$ described by S. Malvezzi in these proceedings.\n\n\\begin{figure}[htp]\n\\begin{center}\n\\includegraphics[width=3.2in]{Psysall_shft.ps}\n\\end{center}\n\\caption{ \nResults of a non-parametric analysis\\cite{kkpi} of $D^+ \\rightarrow K^- K^+ \\pi^+$ using a variant of the projective\nweighting technique described here.\nThe plots are:\n(a)~\\ensuremath{S^2~(\\ensuremath{m_{K\\pi}})}{} direct term,\n(b)~\\ensuremath{2~\\ess \\times \\pee}{} interference term,\n(c)~\\ensuremath{P^2~(\\ensuremath{m_{K\\pi}})}{} direct term,\n(d)~\\ensuremath{2~\\pee \\times \\dee}{} interference term and\n(e)~\\ensuremath{D^2~(\\ensuremath{m_{K\\pi}})}{} direct term.\nThe overlay is a model including the \\ensuremath{\\overline{K}^{*0}}{} which dominates \\ensuremath{P^2~(\\ensuremath{m_{K\\pi}})}{}, and a wider\n\\ensuremath{\\overline{K}_0^{*0}(1430)}{} which dominates \\ensuremath{S^2~(\\ensuremath{m_{K\\pi}})}{}.\n\\label{sysall2}}\n\\end{figure}\n\n\\section{\\label{results} A non-parametric analysis of the helicity form factors in \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}}\n\n\nFigure \\ref{hsq} shows the four weighted histograms from an analysis of 281 $pb^{-1}$ $\\psi(3770)$ \nCLEO data\\cite{cleo-ff}.   \n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.in]{fpcp06-hel.ps}\n  \\caption{The four helicity form factor products obtained using the 281 $pb^{-1}$ data set\nfrom CLEO\\cite{cleo-ff}.  The curves represent the model of Reference \\cite{formfactor}.\n \\label{hsq}}\n \\end{center}\n\\end{figure}\nFigure \\ref{hsq} shows the expected behavior discussed in Section \\ref{asymtopia}. In particular, $H_\\pm {\\ensuremath{q^2}} \\rightarrow {\\rm constant}$ as $\\ensuremath{q^2}{} \\rightarrow 0$ while the zero-helicity form factors, \\ensuremath{H_0(\\ensuremath{q^2})}{} and \\ensuremath{h_0(\\ensuremath{q^2})}{}, diverge as $1\/\\sqrt{\\ensuremath{q^2}{}}$.  It is interesting to note that although the non-resonant, s-wave amplitude is too small to see in \nthe $K^- \\pi^+$ mass spectrum (Fig. \\ref{VecDom}), its form factor is measured with roughly the same precision\nas \\ensuremath{H^2_+(\\ensuremath{q^2})}{} or \\ensuremath{H^2_-(\\ensuremath{q^2})}{}. The curves give the\nhelicity form factors according to Eq. (\\ref{KSE}) , using spectrocopic pole dominance and \nthe \\ensuremath{r_v}{}, \\ensuremath{r_2}{}, and s-wave parameters measured by FOCUS\\cite{formfactor}. Apart from the \\ensuremath{\\hzer\\,\\Hzer}{} interference form \nfactor product, the spectroscopic pole dominance model is a fairly good match to the \nCLEO non-parametric analysis.  This suggests that the ad-hoc assumption, used by FOCUS, that \\ensuremath{h_0(\\ensuremath{q^2})}{}=\\ensuremath{H_0(\\ensuremath{q^2})}{} \nis questionable but it will probably take more data, and some theoretical guidance, to gain insight into the nature of the \ndiscrepancy.\n\nFigure \\ref{qsqhsq} gives a different insight into the helicity basis form factors by\nplotting the intensity contributions of each of the form factor products. This is the\nform factor product multiplied by \\ensuremath{q^2}{}. \nSince \\ensuremath{q^2}{}\\ensuremath{H^2_0(\\ensuremath{q^2})}{} dominates, we normalized form factors such that $\\ensuremath{q^2}{}\\ensuremath{H^2_0(\\ensuremath{q^2})}{} = 1$ at \\ensuremath{q^2}{} = 0 but use the\nsame scale factor for the other three form factors.\nAs expected, both \\ensuremath{q^2}{} \\ensuremath{H^2_+(\\ensuremath{q^2})}{} and \\ensuremath{q^2}{}\\ensuremath{H^2_-(\\ensuremath{q^2})}{} rise from zero with increasing \\ensuremath{q^2}{} and they both appear to approach \\ensuremath{q^2}{}\\ensuremath{H^2_0(\\ensuremath{q^2})}{} at  \\ensuremath{q^2}{}$_{max}$ -- although \\ensuremath{q^2}{} \\ensuremath{H^2_+(\\ensuremath{q^2})}{} seems slightly lower than\n\\ensuremath{q^2}{} \\ensuremath{H^2_0(\\ensuremath{q^2})}{} at \\ensuremath{q^2}{}$_{max}$.\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.in]{pretty_SRedo6-25VGPB_NICGZ.ps}\n  \\caption{Non-parametric form factor products obtained for the data sample\n(multiplied by \\ensuremath{q^2}{}) The reconstructed form factor products are shown as the points with error bars,\nwhere the error bars represent the statistical uncertainties.\nThe solid curves in the histograms represent a form factor model described\nin Ref.~\\cite{formfactor}. \nThe histogram plots are:\n(a)~$\\ensuremath{q^2}{} H_+^2(\\ensuremath{q^2})$,\n(b)~$\\ensuremath{q^2}{} H_-^2(\\ensuremath{q^2})$,\n(c)~$\\ensuremath{q^2}{} H_0^2(\\ensuremath{q^2})$, and\n(d)~$\\ensuremath{q^2}{} h_0(\\ensuremath{q^2}) H_0(\\ensuremath{q^2})$.  The form factors are normalized such that $\\ensuremath{q^2}{} \\ensuremath{H^2_0(\\ensuremath{q^2})} \\rightarrow 1$\nas $\\ensuremath{q^2}{} \\rightarrow 0$.\n \\label{qsqhsq}}\n \\end{center}\n\\end{figure}\n\nWhat can we learn about the pole masses?  Unfortunately Fig. \\ref{infpole} shows that the present\ndata is insufficient to learn anything useful about the pole masses.  On the \nleft of Figure \\ref{infpole}, the helicity form factors are compared to \na model generated with the FOCUS form factor ratios\\cite{formfactor} and the standard pole masses\nof 2.1 GeV for the vector pole and 2.5 GeV for the two axial poles. On the right side\nof Fig. \\ref{infpole}, the form factors are compared to a model where the pole masses\nare set to infinity meaning that the axial and vector form factors are constant.  Both models fit the data equally well.\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.in]{infpole.ps}\n  \\caption{Non-parametric form factor products obtained for data (multiplied by \\ensuremath{q^2}{})  \nThe solid curves are based on the $s$-wave model and measurements described in Reference~\\cite{formfactor}. \nThe reconstructed form factor products are the points with error bars. The three plots on the right are the usual model with the \nspectroscopic pole masses; while the three plots on the right are run with the axial and vector\npole masses taken to infinity. \\label{infpole}}\n \\end{center}\n\\end{figure}\n\nThe data of Fig. \\ref{infpole} is consistent with the spectroscopic pole dominance albeit with essentially no \nsensitivity to the pole masses.  Fig. \\ref{hillH0} shows that it is also consistent with the expected behavior\nunder a Hill transformation, illustrated earlier in Fig. \\ref{hillTrans}. \nFig. \\ref{hillH0} shows the result of transforming from \\ensuremath{q^2}{} to $z$ according to the Hill prescription\\cite{Hill2}.\nOver the very narrow $-z$ range accessible for \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} , it is not surprising that that the transformed form factor is\nessentially constant.\n\nIt is interesting to note that the FOCUS analysis was based on a sample of 11400 \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\mu^+ \\nu }{} events, while the CLEO\nanalysis was based on a sample of only 2470 \\ensuremath{D^+ \\rightarrow K^- \\pi^+ e^+ \\nu }{} events. The error bars in Fig. \\ref{hillH0} for FOCUS data are much larger than those for the much smaller CLEO data set and only four FOCUS \\ensuremath{q^2}{} bins are reported on. This is because of the much poorer \\ensuremath{q^2}{} resolution in fixed target semileptonic decay compared to the order-of-magnitude better \\ensuremath{q^2}{} resolution obtainable for semileptonic analyses in charm threshold data from $e^+ e^-$ colliders where the neutrino can be reconstructed using energy-momentum balance. This is especially relevant for \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}\nsince the 1 GeV$^2$ \\ensuremath{q^2}{} range for \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} is a factor of two smaller than that in \\ensuremath{D^0 \\rightarrow K^- \\ell^+ \\nu }{}. Error inflation due to \ndeconvolution grows dramatically once the bin-to-bin separation, $\\Delta \\ensuremath{q^2}{}$, approaches the r.m.s. resolution, $\\sigma(\\ensuremath{q^2})$, which was\ntypically 0.18 GeV$^2$ in the four bins reported on by FOCUS\\cite{focus-helicity}.  \n\n\n \\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=2.7in]{plot_DKstar_z.eps}\n  \\caption{Transformation of \\ensuremath{H_0(\\ensuremath{q^2})}{} into $H_0(z)$ by R.J. Hill\\cite{Hill2}.  Here $t \\equiv \\ensuremath{q^2}$ and $P$ and $\\phi$ are functions of \\ensuremath{q^2}{} designed\nto remove the simple poles.\nThe FOCUS data is from Reference\n\\cite{focus-helicity} and the CLEO data is from Reference \\cite{cleo-ff}.\n\\label{hillH0}} \\end{center} \\end{figure}\n\n\nWhat can we learn about the phase of the s-wave contribution?  Recall in \nFigure \\ref{asym} the asymmetry created by the interference between the\ns-wave and \\ensuremath{D^+ \\rightarrow \\krzb \\ell^+ \\nu_\\ell}{} only appeared below the \\ensuremath{\\overline{K}^{*0}}{} pole in FOCUS data and thus\nthe s-wave phase was such that it was orthogonal with the $\\ensuremath{m_{K\\pi}} > m(\\ensuremath{\\overline{K}^{*0}})$ half\nof the  Breit-Wigner amplitude or $\\langle BW_+ \\rangle$. Since the asymmetry is ``negative\" according to the \nconvention of Eq. (\\ref{KSE}), in that favors the backward over the forward \\ensuremath{\\cos\\thv}{}\ndirection, it must be anti-collinear to $\\langle BW_- \\rangle$ as well. Hence it must have roughly the phase of 40$^0$ as illustrated by Fig. \\ref{swave-cartoon}. FOCUS\\cite{formfactor} measured the s-wave phase to be $\\delta = (39 \\pm 4 \\pm 3)^0$.\n\n\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=2.3in]{swave-phs.eps}\n  \\caption{Illustration of s-wave phase\n\\label{swave-cartoon}} \\end{center} \\end{figure}\n\nAs Figure \\ref{split} shows, the same thing happens in CLEO data.\nThe effective \\ensuremath{\\hzer\\,\\Hzer}{} disappears above the \\ensuremath{\\overline{K}^{*0}} pole and is very strong\nbelow the pole.  The amplitude $A$ of the s-wave piece is arbitrary since\nusing interference we can only observe the product $A~H_0 (\\ensuremath{q^2}{})~h_0 (\\ensuremath{q^2}{})$.  This\nmeans any change in $A$ scale can be compensated by a change of scale in $h_0 (\\ensuremath{q^2}{})$.\nThe fact that the \\ensuremath{\\hzer\\,\\Hzer}{} data was a tolerable match (at least in the low \\ensuremath{q^2}{} region) to the FOCUS curve in Figure \\ref{hsq} \ndoes imply, however, that the s-wave amplitude observed in CLEO is consistent with that of FOCUS.  A more formal\nfit of the s-wave parameters in CLEO data is in progress.\n\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.in]{split_arb.eps}\n  \\caption{The s-wave interference term for events below the \\ensuremath{\\overline{K}^{*0}} pole (left) and above the pole (right).\nThe interference term depends on the s-wave phase relative to the phase average phase of each \nhalf of the Breit-Wigner.  All of the \\ensuremath{\\cos\\thv}{} interference observed by FOCUS was also below the \n\\ensuremath{\\overline{K}^{*0}} pole as shown in Fig. \\ref{asym}\n \\label{split}}\n \\end{center}\n\\end{figure}\n\n\n\nFinally, is there evidence for higher $K^- \\pi^+$ angular momentum amplitudes in \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{}? \nWe searched for possible additional interference terms such as a (zero helicity) d-wave contribution:\n$4\\,\\ensuremath{\\sin^2\\thl} (3\\,\\cos^2 \\ensuremath{\\theta_\\textrm{v}} - 1)\\,H_0(q^2)\\,h^{(d)}_0(q^2)\\,\n{\\mathop{\\mathrm{Re}}\\nolimits}\\{\\mathrm{A}e^{-i\\delta} \\ensuremath{\\textrm{BW}}\\}$  or\nan f-wave contribution: \n$4\\,\\ensuremath{\\sin^2\\thl} \\cos \\ensuremath{\\theta_\\textrm{v}} (5\\,\\cos^2 \\ensuremath{\\theta_\\textrm{v}} - 3)\\,H_0(q^2)\\,h^{(f)}_0(q^2)\\,\n{\\mathop{\\mathrm{Re}}\\nolimits}\\{Ae^{-i\\delta} \\ensuremath{\\textrm{BW}} \\}$. \nAs shown in Figure \\ref{fdwave}, there is no evidence for such additional contributions which should diverge\nas $1\/\\ensuremath{q^2}{}$ at low \\ensuremath{q^2}{}.\n\\begin{figure}[tbph!]\n \\begin{center}\n  \\includegraphics[width=3.in]{wavelimits.ps}\n  \\caption{Search for (a) $d$-wave and (b) $f$-wave interference effects as described in the text.\n \\label{fdwave}}\n \\end{center}\n\\end{figure}\n\n\\section{Future Directions}\nIt will be interesting to pursue the non-parametric \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} analysis with more data.  One motivation is \nwill be to further study the $h_0(\\ensuremath{q^2})$ form factor which appears to be somewhat different than\n$H_0(\\ensuremath{q^2})$. It would also be interesting to pursue tighter limits on possible d-wave and f-wave non-resonant contributions\nto \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} and make more stringent tests of SPD.\nCLEO is slated to increase their luminosity at the $\\psi(3770)$ from the 280 pb$^{-1}$ reported here to 750 pb$^{-1}$.\nIn addition Surik Mehrabyan and I, are studying \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\mu^+ \\nu }{} as well as \\ensuremath{D^+ \\rightarrow K^- \\pi^+ e^+ \\nu }{} in CLEO data. This is\na somewhat challenging project since the CLEO muon detector was designed for higher energy B-meson running and the muons\nfrom charm semileptonic decay tend to range out before being identified. Hence special care must be exercised to \nreduce backgrounds. Besides increasing our statistics, the \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\mu^+ \\nu }{} should allow us to make the first measurements\nof the $H_T(\\ensuremath{q^2})$ form factor which is suppressed by a factor of $m^2_\\ell\/\\ensuremath{q^2}$. Since this is a zero helicity factor,  it can interfere\nwith \\ensuremath{H_0(\\ensuremath{q^2})}{} and hence two new projectors will be required: one for the $H_T^2(\\ensuremath{q^2})$ term and one for $H_0(\\ensuremath{q^2}) \\times H_T(\\ensuremath{q^2})$ interference. At present the prognosis for making these measurements looks good.\n\n\\mysection{Summary}\n\n\nProgress in understanding \\ensuremath{D \\rightarrow {\\rm vector}~ \\ell^+ \\nu }{} decays was reviewed.  These have\nhistorically been analyzed under the assumption of spectroscopic pole dominance (SPD).  \nA recent result from BaBar was reviewed that used SPD to show that the form factors for \\ensuremath{D_s^+ \\rightarrow \\phi\\; \\ell^+ \\nu_\\ell}{} are\nconsistent with those from \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} as expected from SU(3) symmetry.\nExperiments\nhave obtained consistent results with the SPD assumption, but as of yet there have been\nno incisive tests of spectroscopic pole dominance.   We concluded by describing a first non-parametric look at the \\ensuremath{D^+ \\rightarrow K^- \\pi^+ \\ell^+ \\nu }{} form factors.  Although the results were very consistent\nwith the traditional pole dominance fits, the data was not precise enough to incisively\nmeasure \\ensuremath{q^2}{} dependence of the axial and vector form factors and thus test SPD.  This\npreliminary analysis did confirm the existence of an $s$-wave effect first observed by FOCUS \\cite{swave},\nbut was unable to obtain evidence for $d$ and $f$-waves. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpectral line series of alkali metal atoms display remarkable features with\nprominent minima in the transition probabilities, emission oscillator\nstrengths or photoionization cross-sections~\\cite{Bates, Aymar1976, Cooper1962, FanoCooper, Ditchburn1943, TheodosiouMin1980,  HudsonCarter1967,  Aymar1978,\n Manson, Hoogenraad1995, HoogenraadDipole, Duncan,\nPetrov}. These minima arise from the cancellation of the radial integral for\nsome transitions, depending on the overlap between the wavefunctions of the\ninitial and final quantum states of the atoms~\\cite{Bates, Aymar1976}, and are\nwell known as Cooper minima~\\cite{Cooper1962, FanoCooper}. \n\nObservation of the Cooper minima  in the photoionization cross-sections and\nin the transition probabilities of the discrete spectrum provides valuable\ninformation about the electronic structure of the atoms. Minima in\nphotoionization cross sections were first found experimentally in Ref.~\\cite{Ditchburn1943} and explained 20 years later by Cooper~\\cite{Cooper1962}. The\nsharp minima in the emission probabilities for some Rydberg states of\nalkali-metal atoms were first discussed by Theodosiou~\\cite{TheodosiouMin1980}.\n\nThe experimental investigation of Cooper minima can be used for the\nverification of theoretical calculations of spectroscopic properties of\natoms and molecules. For example, experimentally measured photoionization\ncross-sections for sodium ground state showing the Cooper minimum~\\cite{HudsonCarter1967}, were in good agreement with the theoretical calculations\nof Aymar~\\cite{Aymar1978}, which confirmed the accuracy of the theoretical model.\n\nCooper minima in the discrete spectrum are revealed as a suppression of the\ntwo-photon photoionization~\\cite{Hoogenraad1995} or sharp decrease of the\nemission oscillator strengths~\\cite{TheodosiouMin1980}. The map of these minima\ncould be valuable for systematic studies of the processes which involve a\nlarge number of transitions, such as calculation of lifetimes of Rydberg\natoms~\\cite{BeterovLifetimes}, blackbody-radiation-induced photoionization rates~\\cite{BBRIonization, BeterovNJP} or collisional ionization cross-sections of cold\natoms~\\cite{Amthor2009}.\n\nPhotoionization of alkali-metal atoms recently attracted a lot of interest,\nas it has taken a central role in experiments with cold atoms in\nfar-off-resonance traps~\\cite{AdamsPhoto, ShafferCs}, in photoionization\nspectroscopy~\\cite{RbDroplets, Haq2010, Nadeem2011},   measurement of oscillator\nstrengths~\\cite{Haq2010, Nadeem2011,Piotrowicz2011}, photoionization cross-sections and lifetimes of excited atoms~\\cite{Haq2010, Nadeem2011, Piotrowicz2011, Fabry, LiOscStr, BaigNa, Gabbanini2006,BeterovLifetimes, Feng, Tate}, \nand photoionization of the Bose-Einstein condensate~\\cite{BECion,AlkaliLasers}.\n\nRydberg atoms with large principal quantum numbers \\textit{n}$\\sim $50-100\nrecently received attention due to the progress achieved in experiments with\ncold atomic samples. These samples are often prepared in optical\ntraps whose intense laser field could make the ionization lifetimes of\nRydberg atoms extremely short. However, for certain wavelengths, ionization\nrates could be significantly reduced due to Cooper minima in the\nphotoionization cross-sections~\\cite{Cooper1962}. Therefore it would be most\nuseful to exploit trapping schemes operated at wavelengths displaying Cooper\nminima to avoid photoionization~\\cite{AdamsPhoto, SaffmanFORT}.\n\nIn this paper we have calculated the radial matrix elements of arbitrary bound-bound,\nbound-free and free-free transitions between \\textit{S, P, D} and \\textit{F}\nstates of alkali-metal atoms using the quasiclassical model by Dyachkov and\nPankratov (DP model)~\\cite{Dyachkov1994, DyachkovFree}. In section II we present\nexamples of the Cooper minima for bound-free and bound-bound transitions and discuss the\naccuracy of the theoretical method. Numerical\nresults are presented in Section III as density plots, revealing the Cooper\nminima in bound-bound, bound-free and free-free transitions.\n\n\n\\section{The quasiclassical model and its applicability}\n\n\nRadial matrix elements of the electric dipole transitions between arbitrary\natomic states (e.g., bound-bound or bound-free transitions) are required to\ncalculate the spectroscopic properties of atoms, including oscillator\nstrengths, lifetimes, photoionization cross-sections, and rates of\ncollisional ionization.\n\nAlthough alkali-metal atoms have a single valence electron, only states with\nsmall quantum defects exhibit truly hydrogen-like behavior. Due to\nnon-hydrogenic character of alkali-metal atoms, the calculation of radial\nmatrix elements remains a challenging task, since no exact analytical\nsolution for arbitrary transitions is available yet~\\cite{BetheSalpeter}. The\noscillator strengths for alkali-metal atoms can strongly deviate from the\nvalues for hydrogen. Accurate calculation of the radial integral for\ntransitions between states with small angular momentum are difficult because\nof the need to take into account the interaction of the valence electron\nwith the atomic core.\n\nA method based on the Coulomb approximation relies on the idea that the\nRydberg electron is localized mostly outside the atomic core, where the\npotential is Coulombic. In the Numeric Coulomb Approximation~\\cite{NCA} the\nradial wavefunctions are obtained by solution of the Shrodinger equation\nwith the exact energies of the alkali-metal quantum states, expressed\nthrough the quantum defect (Rydberg-Ritz formula, atomic units are used in this paper):\n\n\\begin{equation}\n\\label{eq1}\nE_{n} = - \\frac{{1}}{{2n_{eff}^{2}} }.\n\\end{equation}\n\n\n\n\n\\noindent Here $n_{eff} = n - \\mu _{L} $ is an effective quantum number, $\\mu _{L} $\nis the quantum defect. Quantum defect accounts for the penetration of the\nvalence electron into the ionic core of a Rydberg atom. The quantum defects\nare used as input parameter for the calculations and the integration is\ntruncated at the inner core radius.\n\nAlternative forms of the Coulomb approximation were developed in~\\cite{MCA}. The\nModified Coulomb Approximation (MCA) is a generalization of the analytical\nexpression for the hydrogen radial integral for non-integer quantum numbers.\nIt allows direct calculation of the radial matrix elements without numeric\nintegration.\n\nFurther simplification of the calculations in the Coulomb approximation is\nachieved by extension of the quasiclassical approximation to the states with\nlow principal quantum numbers~\\cite{DyachkovFree, Dyachkov1994}. The\nradial matrix elements are expressed through transcendental functions, which\nhelp to avoid inaccurateness of the direct numerical integration. This\nmethod can substantially improve both the speed and reliability of the\ncalculations. However, the validity of most quasiclassical models was\nrestricted by transitions between neighboring excited states~\\cite{Bureeva,\nNaccache, Edmonds, DavydkinZon, GDK1994}.\n\nIn our previous works~\\cite{BeterovLifetimes, BeterovNJP} we used the\nquasiclassical model developed by Dyachkov and Pankratov~\\cite{Dyachkov1994,DyachkovFree}. Their original approach provides more precise\nvalues for the wave functions of the Rydberg and continuum states, compared\nto the other quasiclassical models. Good agreement with numeric results\nbased on NCA model~\\cite{NCA} or various model potential methods~\\cite{\nOvsiannikov2, Marinescu} is observed.\n\n\nRadial matrix elements for transitions between excited states of alkali-metal atoms display numerous features in their depenencies on $n$.\nFor example, minima were revealed in the transition probabilities for $nD \\to {n}'F$ in\ncesium~\\cite{TheodosiouMin1980} and for $nD \\to {n}'P$ in potassium. Hoogenraad\net~al.~\\cite{Hoogenraad1995} observed theoretically and experimentally a Cooper\nminimum for $nS \\to {n}'P$ transitions in lithium.\nTo study the validity of the quasiclassical model for bound-bound transitions, we have calculated the transition probabilities for $nL \\to {n}'{L}'$\ntransitions in alkali-metal atoms with $n < 100$ and $L,{L}' = 1,2,3$ using\nthe DP quasiclassical model, which is depicted in Appendix~B. Figure~\\ref{BoundMin} shows the dependencies of the radial\nmatrix elements on the principal quantum number \\textit{n} for $60F \\to nD$\ntransitions in rubidium and cesium [Fig.~\\ref{BoundMin}(a) and Fig.~\\ref{BoundMin}(b), respectively], $60D\n\\to nP$ transitions in potassium [Fig.~\\ref{BoundMin}(c)] and $60P \\to nS$ transitions in\nlithium [Fig.~\\ref{BoundMin}(d)]. The observed minima for lithium, potassium and cesium are in\nagreement with the results of Refs.~\\cite{TheodosiouMin1980, HoogenraadDipole}.\nThe minima for $nF \\to {n}'D$ transitions in rubidium appear only for high\n\\textit{n}, and have not been located yet, to the best of our knowledge.\nThese minima lie in the microwave region of about 150 GHz and could be\nstudied using microwave spectroscopy~\\cite{Gallagher}.\n\n\\begin{figure}\n\\includegraphics[scale=0.38]{Fig1.eps}\n\\caption{\n\\label{BoundMin}(Color online).\nThe calculated radial matrix elements for bound-bound  (a) $60F \\to nD$ transitions in rubidium; (b) $60F \\to nD$ \ntransitions in cesium; (c) $60D \\to nP$ \ntransitions in potassium; (d) $60P \\to nS$ \ntransitions in lithium.}\n\\end{figure}\n\n\nDirect measurement of the radial matrix elements is of great importance for verification of the theory. However, due to lack of available experimental data for transitions between excited states of alkali-metal atoms, new measurements are required. In order to benchmark the model, we have earlier measured  the reduced matrix element for the diffuse series of rubidium~\\cite{Piotrowicz2011}. \n\nWe observed the Autler Townes splitting in a sample of ultra-cold Rb atoms using a 3-level ladder system. Briefly, we monitored the absorption of a weak probe laser scanned over the $5S_{1\/2} \\to 5P_{3\/2}$ whilst simultaneously a strong coupling laser, locked to the $ 5P_{3\/2}\\to nD_{5\/2}$ transition illuminated the atoms. The strong coupling laser gave rise to two absorption peaks separated by the Rabi frequency of the atom-coupling laser interaction. Knowledge of the laser intensity allowed us to measure the dipole matrix elements of the $5P_{3\/2}\\to nD_{5\/2}$ transitions to within $7\\%$ accuracy.\n\nWe have also compared the reduced dipole moments calculated using DP model with\nother available  experimental data on diffuse series of rubidium~\\cite{Nadeem2011,Piotrowicz2011} and cesium~\\cite{Haq2010,Fabry}. Good agreement with experiments of Refs.~\\cite{Piotrowicz2011, Nadeem2011} is confirmed in Fig.~\\ref{CooperPhoto}(a) for rubidium $5P\n\\to nD$ transitions. For cesium the theoretical values in Fig.~\\ref{CooperPhoto}(c) are in\nagreement with the experiment only for the lowest \\textit{nD} states and\ndiffer from the experimental values for higher $n$ by a factor of two. Our\ntheoretical results for cesium are, however, in excellent agreement with the\nprevious calculations of Ref.~\\cite{Fabry}. The observed disagreement between experiment and theory can result from improper\naccount for core polarization for heavy cesium atoms~\\cite{Migdalek}. However, we\nexpect that for transitions between the states with larger principal quantum\nnumbers the accuracy of the semiclassical approximation will be\nsignificantly increased due to smaller interaction of the Rydberg electron with the atomic core.\n\\begin{figure}\n\\includegraphics[scale=0.4]{Fig2.eps}\n\\caption{\n\\label{CooperPhoto}(Color online).\n(a), (c) Comparison of the calculated reduced dipole moments for (a) rubidium atoms with \nexperiment~\\cite{Nadeem2011, Piotrowicz2011} and (c) cesium atoms with experiment~\\cite{Haq2010, Fabry}.\n(b), (d) Cooper minima in photoionization cross-sections of $nS$ (b)~rubidium and (d)~sodium atoms. Solid\ncurves - this work. Broken curves - theoretical calculations from\nRef.~\\cite{Aymar1978}; \n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{Fig3.eps}\n\\caption{\n\\label{Lithium}(Color online).\nLithium 2S photoionization cross-section. Comparison of the quasiclassical calculations with the \nexperiment~\\cite{HudsonCarter1967} and theory~\\cite{LiNaPlasmaSahoo, Peach}.}\n\\end{figure}\n\n\n\nCalculated Cooper minima in the photoionization cross-sections of the \\textit{nS}\nstates of rubidium and sodium are shown in Fig.~\\ref{CooperPhoto}(b) and (d), respectively.\nThe solid curves in Fig.~\\ref{CooperPhoto}(b) and (d) represent the photoionization cross-sections\ncalculated using our model based on the DP method. In the case of sodium,\nthese are compared with the quantum-mechanical calculations of\nRef.~\\cite{Aymar1978}, shown as broken curves in Fig.~\\ref{CooperPhoto}(d). Good agreement is\nobserved for sodium \\textit{nS} states with $n < 5$. For higher states\nthe positions of the Cooper minimum are significantly shifted. In Ref.~\\cite{Bezuglov1999} it has been argued that the discrepancy in the quasiclassical\ncalculations could be corrected by adjusting the phase in the radial\nintegral in order to compensate the phase shift of the radial wavefunction\nfrom the value given by the quantum defect. At the same time, the\nreliability of the quasiclassical approximation is also expected to improve\nwith the increase of the principal quantum number, and only experimental\ndata could confirm the validity of the theory.\n\n\nIn figure~\\ref{Lithium} we have compared the calculated photoionization cross-section\nfor lithium 2S state with the experiment of Ref.~\\cite{HudsonCarter1967} and theory\nof Refs.~\\cite{LiNaPlasmaSahoo, Peach}. It is seen that our approach provides a better agreement with the experiment.\n\nWe conclude that the DP model is suitable for calculation of the radial\nmatrix elements and photoionization cross-sections with an accuracy better\nthan a factor of two for low \\textit{n} and much enhanced for higher excited\nstates, as confirmed by the good agreement between experiment and theory in\nrecent lifetime measurements~\\cite{Tate, Feng}.\n\n\n\\section{Maps of the cooper minima}\n\nUsing the quasiclassical model of Delone et al.~\\cite{GDK1994}, in Ref.~\\cite{HoogenraadDipole} it has been shown that the radial matrix elements for\nbound-bound, bound-free and free-free transitions could be expressed in a\nuniversal way through the numerically calculated relative matrix elements\n$R_{rel} $, multiplied by the appropriate normalization factors:\n\n\\begin{align}\n\\label{eq2}\n&R\\left( {nL \\to {n}'{L}'} \\right) = \\frac{{0.4108 \\times R_{rel} \\left(\n{E_{n} L \\to E_{n'}{L}'} \\right)}}{{n_{eff}^{'3\/2} \\times n_{eff}^{3\/2} \\times\n\\left| {E_{n'} - E_{n}}  \\right|^{5\/3}}} = &\\nonumber\\\\\n &= \\frac{{0.4108 \\times R_{rel} \\left( {E_{n} L \\to E_{n'}{L}'}\n\\right)}}{{\\left( { - 2E_{n'}}  \\right)^{- 3\/4} \\times \\left( {- 2E_{n}}\n\\right)^{ - 3\/4} \\times \\left| {E_{n'}- E_{n}}  \\right|^{5\/3}}} &\\nonumber\\\\\n &R\\left( {nL \\to {E}'{L}'} \\right) = \\frac{{0.4108 \\times R_{rel} \\left(\n{E_{n} L \\to {E}'{L}'} \\right)}}{{n_{eff}^{3\/2} \\times \\left| {{E}' - E_{n}\n} \\right|^{5\/3}}} =&\\nonumber \\\\\n &= \\frac{{0.4108 \\times R_{rel} \\left( {E_{n} L \\to {E}'{L}'}\n\\right)}}{{\\left( { - 2E_{n}}  \\right)^{ - 3\/4} \\times \\left| {{E}' - E_{n}\n} \\right|^{5\/3}}} &\\nonumber\\\\\n&R\\left( {EL \\to {E}'{L}'} \\right) = \\frac{{0.4108 \\times R_{rel} \\left( {EL\n\\to E'L'} \\right)}}{{\\left| {{E}' - E} \\right|^{5\/3}}} &\n \\end{align}\n\n\n\n\\noindent The prefactor $\\left( {4\/3} \\right)^{1\/3}\/\\Gamma \\left( {1\/3} \\right) =\n0.4108$ results from the asymptotic expression for the quasiclassical matrix\nelements for $n \\to n + 1$ transitions \\cite{HoogenraadDipole}. Relative matrix\nelements $R_{rel} \\left( {EL \\to {E}'{L}'} \\right)$ introduced in Eq.(\\ref{eq2}) are\nconvenient as these are slowly varying functions of $E$ and ${E}'$. The\ndependence of $R_{rel} \\left( {EL \\to {E}'{L}'} \\right)$ on the energy\n${E}'$ of the final state passes smoothly through the ionization threshold\n\\cite{HoogenraadDipole}. The asymptotic $\\left| {{E}' - E} \\right|^{ - 5\/3}$\ndependence of the radial matrix elements is incorrect for transitions\nbetween neighboring states with $E \\approx {E}'$, where the dipole matrix\nelements rapidly increase~\\cite{HoogenraadDipole}. In this case the radial matrix\nelements can be calculated numerically using a DP model~\\cite{Dyachkov1994,\nDyachkovFree}, or NCA~\\cite{NCA}.\n\nWe have calculated the relative matrix elements $R_{rel} \\left( {E_{n} L \\to\nE_{{n}'} {L}'} \\right)$, $R_{rel} \\left( {E_{n} L \\to {E}'{L}'} \\right)$,\n$R_{rel} \\left( {EL \\to {E}'L} \\right)$ for transitions between \\textit{S},\n\\textit{P}, \\textit{D} and \\textit{F} states of alkali-metal atoms, starting\nfrom the ground state. The energies of the continuum states were taken\nwithin $0 < E < 0.5$ (atomic units) to extend the results of Ref.\n\\cite{HoogenraadDipole} to the area where Cooper minima are expected for\nalkali-metal atoms.\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig4.eps}\n\\caption{\n\\label{RbSP}(Color online).\nDensity plots of the relative matrix elements for (a) Rb bound-free $nS_{1\/2} \\to\n{E}'P_{1\/2} $ transitions; (b) arbitrary Rb $ES_{1\/2} \\to {E}'P_{1\/2}$\ntransitions including discrete and continuum spectra; (c) Rb bound-bound\n$nS_{1\/2} \\to {n}'P_{1\/2}$  transitions; (d) Rb bound-free\n$nP_{1\/2} \\to {E}'S_{1\/2}$ transitions.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig5.eps}\n\\caption{\n\\label{RbPDF}(Color online).\nDensity plots of the relative matrix elements in Rb atoms for (a) bound-free $nP_{3\/2} \\to\n{E}'D_{5\/2}$ transitions; (b) all $EP_{3\/2} \\to {E}'D_{5\/2}$ transitions;\n(c) bound-bound $nP_{3\/2} \\to {n}'D_{5\/2}$ transitions; (d) bound-free\n$nD_{5\/2} \\to {E}'P_{3\/2}$ transitions; (e) bound-free $nD_{3\/2} \\to\n{E}'F_{5\/2}$ transitions; (f) all $ED_{3\/2} \\to {E}'F_{5\/2}$ transitions;\n(g) bound-bound $nD_{3\/2} \\to {n}'F_{5\/2}$ transitions; (h) bound-free\n$nF_{5\/2} \\to {E}'D_{3\/2}$ transitions. }\n\\end{figure}\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig6.eps}\n\\caption{\n\\label{CsSPD}(Color online).\nDensity plots of the relative matrix elements in Cs atoms for (a) bound-free\n$nS_{1\/2} \\to {E}'P_{1\/2} $ transitions; (b) all $ES_{1\/2} \\to {E}'P_{1\/2} $\ntransitions; (c) bound-bound $nS_{1\/2} \\to {n}'P_{1\/2}$ transitions; (d)\nbound-free $nP_{1\/2} \\to {E}'S_{1\/2}$ transitions; (e) bound-free $nP_{3\/2}\n\\to {E}'D_{5\/2}$ transitions; (f) all $EP_{3\/2} \\to {E}'D_{5\/2} $\ntransitions; (g) bound-bound $nP_{3\/2} \\to {n}'D_{5\/2}$ transitions; (h)\nbound-free $nD_{5\/2} \\to {E}'P_{3\/2}$ transitions.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig7.eps}\n\\caption{\n\\label{CsDF}(Color online).\nDensity plots of the relative matrix elements in Cs atoms for (a) bound-free $nD_{3\/2} \\to\n{E}'F_{5\/2} $ transitions; (b) all $ED_{3\/2} \\to {E}'F_{5\/2} $ transitions;\n(c) bound-bound $nD_{3\/2} \\to {n}'F_{5\/2}$ transitions; (d) bound-free\n$nF_{5\/2} \\to {E}'D_{3\/2}$ transitions.}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig8.eps}\n\\caption{\n\\label{LiSPD}(Color online).\nDensity plots of the relative matrix elements in Li atoms for (a) bound-free $nS \\to {E}'P$\ntransitions; (b) all $ES \\to {E}'P$ transitions; (c) bound-bound $nS \\to\n{n}'P$ transitions; (d) bound-free $nP \\to {E}'S$ transitions; (e) bound-free\n$nP \\to {E}'D$ transitions; (f) all $EP \\to {E}'D$ transitions; (g)\nbound-bound $nP \\to {n}'D$ transitions; (h) bound-free $nD \\to {E}'P$\ntransitions.}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig9.eps}\n\\caption{\n\\label{LiDF}(Color online).\nDensity plots of the relative matrix elements in Li atoms for (a) bound-free $nD \\to\n{E}'F$ transitions; (b) all $ED \\to {E}'F$ transitions; (c) bound-bound $nD\n\\to {n}'F$ transitions; (d) bound-free $nF \\to {E}'D$ transitions.}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig10.eps}\n\\caption{\n\\label{NaSPD}(Color online).\nDensity plots of the relative matrix elements in Na atoms for (a) bound-free $nS_{1\/2} \\to\n{E}'P_{1\/2}$ transitions; (b) all $ES_{1\/2} \\to {E}'P_{1\/2}$ transitions;\n(c) bound-bound $nS_{1\/2} \\to {n}'P_{1\/2}$ transitions; (d) bound-free\n$nP_{1\/2} \\to {E}'S_{1\/2}$ transitions; (e) bound-free $nP_{3\/2} \\to\n{E}'D_{5\/2}$ transitions; (f) all $EP_{3\/2} \\to {E}'D_{5\/2}$ transitions;\n(g) bound-bound $nP_{3\/2} \\to {n}'D_{5\/2}$ transitions; (h) bound-free\n$nD_{5\/2} \\to {E}'P_{3\/2} $ transitions.}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig11.eps}\n\\caption{\n\\label{NaDF}(Color online).\nDensity plots of the relative matrix elements in Na atoms for (a) bound-free $nD_{3\/2} \\to\n{E}'F_{5\/2} $ transitions; (b) all $ED_{3\/2} \\to {E}'F_{5\/2} $ transitions;\n(c) bound-bound $nD_{3\/2} \\to {n}'F_{5\/2}$ transitions; (d) bound-free\n$nF_{5\/2} \\to {E}'D_{3\/2} $ transitions;}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig12.eps}\n\\caption{\n\\label{KSPD}(Color online).\nDensity plots of the relative matrix elements in K atoms for (a) bound-free $nS_{1\/2} \\to\n{E}'P_{1\/2} $ transitions; (b) all $ES_{1\/2} \\to {E}'P_{1\/2} $ transitions;\n(c) bound-bound $nS_{1\/2} \\to {n}'P_{1\/2}$ transitions; (d) bound-free\n$nP_{1\/2} \\to {E}'S_{1\/2} $ transitions; (e) bound-free $nP_{3\/2} \\to\n{E}'D_{5\/2} $ transitions; (f) all $EP_{3\/2} \\to {E}'D_{5\/2}$ transitions;\n(g) bound-bound $nP_{3\/2} \\to {n}'D_{5\/2}$ transitions; (h) bound-free\n$nD_{5\/2} \\to {E}'P_{3\/2}$ transitions.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{Fig13.eps}\n\\caption{\n\\label{KDF}(Color online).\nDensity plots of the relative matrix elements in K atoms for (a) bound-free\n$nD_{3\/2} \\to {E}'F_{5\/2} $ transitions; (b) all $ED_{3\/2} \\to {E}'F_{5\/2} $\ntransitions; (c) bound-bound $nD_{3\/2} \\to {n}'F_{5\/2}$ transitions; (d)\nbound-free $nF_{5\/2} \\to {E}'D_{3\/2}$ transitions.}\n\\end{figure}\n\n\\subsection{Rubidium}\n\n\nRubidium and cesium atoms are widely employed  in laser cooling experiments, and\nwe shall discuss them in more detail. The relative matrix elements $R_{rel}\n\\left( {S_{1\/2} \\to P_{1\/2}}  \\right)$ for rubidium are shown in Fig.~\\ref{RbSP}.\nFollowing  Ref.~\\cite{HoogenraadDipole}, we present our numerical\nresults as density plots. We use both \\textit{E}-scaled and\n\\textit{n}-scaled plots, since the latter are more appropriate to the\nrelative matrix elements for bound-bound and bound-free transitions from\nstates with large principal quantum numbers \\textit{n.} The signs of\nthe radial matrix elements are unimportant in the calculation of transition\nprobabilities and photoionization cross-sections, therefore we present only their\nabsolute values.\n\n\n\n\nFigure~\\ref{RbSP}(a) shows the relative matrix elements $R_{rel} \\left( {nS_{1\/2} \\to\n{E}'P_{1\/2}}  \\right)$ for bound-free transitions in rubidium atoms. The\nhorizontal axis is the principal quantum number of the \\textit{nS} states,\nwhile the vertical axis is the binding energy ${E}'$ of the continuum\n\\textit{P} states (in atomic units). From Fig.~\\ref{RbSP}(a) one finds that for a\nbound-free transition between the Rydberg 40\\textit{S}$_{1\/2}$ state and the\ncontinuum ${E}'P_{1\/2} $ state with ${E}' = 0.18$ the relative matrix\nelement is $R_{rel} \\left( {nS_{1\/2} \\to {E}'P_{1\/2}}  \\right) = 0.02$.\nAccording to Eq.(\\ref{eq1}) and Table~1 in the appendix~A the energy of the 40S$_{1\/2\n}$ state is $E_{n} = - 3.68 \\times 10^{-4}$, and the energy difference is\n$\\left| {{E}' - E_{n}}  \\right| = 0.18$. Then the absolute radial matrix\nelement can be found from Eq.(\\ref{eq2}): $R\\left( {40S_{1\/2} \\to {E}'P_{1\/2}}\n\\right) = 6.1 \\times 10^{ - 4}$. To highlight Cooper minima, the regions where the relative matrix\nelement $R_{rel} \\left( {nS_{1\/2} \\to {E}'P_{1\/2}}  \\right)$ falls below\n0.02 are filled by black. The interesting feature of Fig.~\\ref{RbSP}(a) is the\npresence of the two sharp Cooper minima at ${E}' = 0.09$ and ${E}' = 0.21$\nfor $n >10$.\n\n\nFigure~\\ref{RbSP}(b) shows the relative matrix elements for all possible bound-bound,\nbound-free and free-free $R_{rel} \\left( {S_{1\/2} \\to P_{1\/2}}  \\right)$\ntransitions in rubidium atoms, plotted in the energy scale. The horizontal axis carries the binding\nenergy of \\textit{S} states while the\nbinding energy of \\textit{P} states is given in   the vertical axis.\nFor a bound-free transition between the\n6\\textit{P}$_{1\/2}$ state with $E = - 0.045$ and the continuum\n\\textit{S}$_{1\/2}$ state with ${E}' = 0.15$, the relative matrix element is\n$R_{rel} \\left( {E_{n} P_{1\/2} \\to {E}'S_{1\/2}}  \\right) = 0.8$. Then Eq.~(\\ref{eq2})\ngives the absolute radial matrix element $R\\left( {E_{n} P_{1\/2} \\to\n{E}'S_{1\/2}}  \\right) = 0.82$ (the energy difference is $\\left| {{E}' - E}\n\\right| = 0.195$). The same procedure can be applied to calculate radial\nmatrix elements for all bound-bound, bound-free and free-free transitions.\n\nA prominent Cooper minimum is observed in Fig.~\\ref{RbSP}(a),~(b) for the bound-free\n$nS_{1\/2} \\to {E}'P_{1\/2}$ and free-free $ES_{1\/2} \\to {E}'P_{1\/2} $\ntransitions in rubidium. For the \\textit{nS}$_{1\/2}$ states with $E_{n} > -\n0.03$ (corresponding to $n>7$) the relative matrix elements fall down\nbelow 0.02, while for the lower \\textit{nS}$_{1\/2}$ states the minimum is\nnot so sharp.\n\nThe relative matrix elements $R_{rel} \\left( {E_{n} P_{1\/2} \\to E_{{n}'} S_{1\/2}\n} \\right)$ for bound-bound transitions in rubidium  are presented in\nFig.~\\ref{RbSP}(c).  Since the\nrelative matrix elements slowly vary with $n$ and ${n}'$, accurate\ncalculation of the radial matrix elements from the data of Fig.~\\ref{RbSP}(c) is\npossible as described earlier. For example, $R_{rel} \\left( {27S_{1\/2} \\to\n80P_{1\/2}}  \\right) = 0.45$, $E_{27S} = - 8.776 \\times 10^{ - 4}$, $E_{80P}\n= - 8.36 \\times 10^{ - 5}$ and the energy difference is $\\left| {E_{{n}'} -\nE_{n}}  \\right| = 7.94 \\times 10^{ - 4}$. From Eq.~(\\ref{eq2}) one finds $R\\left(\n{27S_{1\/2} \\to 80P_{1\/2}}  \\right) = 0.342$.\n\nRelative matrix elements $R_{rel} \\left( {nP_{1\/2} \\to {E}'S_{1\/2}}\n\\right)$ for bound-free transitions in rubidium atoms are shown in Fig.~\\ref{RbSP}(d).\n As an example, the relative\nmatrix element is $R_{rel} \\left( {60P_{1\/2} \\to {E}'S_{1\/2}}  \\right) =\n0.84$ for ${E}' = 0.2$; then the radial matrix element is $R\\left(\n{60P_{1\/2} \\to {E}'S_{1\/2}}  \\right) = 0.012$.\n\nFigure~\\ref{RbPDF} displays the relative matrix elements $R_{rel} \\left( {P_{3\/2} \\to\nD_{5\/2}}  \\right)$ and $R_{rel} \\left( {D_{3\/2} \\to F_{5\/2}}  \\right)$ for\nrubidium atoms in the same way as in Fig.~\\ref{RbSP}. Relative matrix elements for\nother fine-structure components of the rubidium \\textit{P} and \\textit{D}\nstates are not presented, since the difference between them is too small to\nbe distinguishable on our density plots. Cooper minima are observed for\nbound-free $nP_{3\/2} \\to {E}'D_{5\/2} $ transitions with ${E}' \\approx 0.37$\n[Fig.~\\ref{RbPDF}(a)], free-free $EP_{3\/2} \\to {E}'D_{5\/2} $ and $ED_{3\/2} \\to F_{5\/2}$ transitions [Fig.~\\ref{RbPDF}(b)],\nand bound-free $nF_{5\/2} \\to {E}'D_{3\/2} $ transitions with ${E}' \\approx\n0.17$ [Fig.~\\ref{RbPDF}(h)].\n\n\n\n\n\n\\subsection{Cesium}\n\nFigure~\\ref{CsSPD} shows relative matrix elements $R_{rel} \\left( {S_{1\/2} \\to P_{3\/2}\n} \\right)$ and $R_{rel} \\left( {P_{3\/2} \\to D_{5\/2}}  \\right)$ for cesium\natoms. Relative matrix elements $R_{rel} \\left( {D_{3\/2} \\to F_{5\/2}}\n\\right)$ for cesium atoms are presented in Fig.~\\ref{CsDF}. Two Cooper minima are\nobserved for bound-free $nS_{1\/2} \\to {E}'P_{1\/2} $ transitions with ${E}'\n\\approx 0.06$ and ${E}' \\approx 0.17$ in Fig.~\\ref{CsSPD}(a). The Cooper minima are also\nnoticed for the bound-free $n D_{5\/2}\\to EP_{3\/2} $ transitions with ${E}'\n\\approx 0.28$ [Fig.~\\ref{CsSPD}(h)], bound-free $nF_{5\/2} \\to {E}'D_{3\/2} $ transitions\nwith ${E}' = 0.19$[Fig.~\\ref{CsDF}(h)], and bound-bound $nD_{3\/2} \\to {n}'F_{5\/2}\n$ transitions with $n<23$ [Fig.~\\ref{CsDF}(g)].\n\nCooper minimum in the discrete spectrum was first discussed in Ref.~\\cite{TheodosiouMin1980}. In Ref.~\\cite{HoogenraadDipole}  a\ncontinuation of this minimum in the bound-free $nD_{3\/2} \\to {E}'F_{5\/2} $\ntransitions was found,  with the energy ${E}' \\approx 10^{ - 3}$ being close to the\nionization threshold. These features were reproduced in our calculations,\nbut they are not shown due to the large energy scale of our density plots.\nThe near-threshold Cooper minima for Rydberg states of alkali-metal atoms\nwere also discussed in Ref.~\\cite{HoogenraadDipole}.\n\n\\subsection{Lithium}\n\n\nFor lithium atoms the relative matrix elements obtained for $R_{rel} \\left(\n{S \\to P} \\right)$ and $R_{rel} \\left( {P \\to D} \\right)$ transitions are\nshown in Fig.~\\ref{LiSPD}, and for $R_{rel} \\left( {D \\to F} \\right)$ transitions in\nFig.~\\ref{LiDF}. Fine structure is neglected due to the small fine splitting. One may see\nthat the matrix elements of the bound-free $S \\to P$ transitions slowly decrease\nas the energy of the continuum state grows. A Cooper minimum is observed for\nthe bound-bound $nS_{1\/2} \\to {n}'P_{1\/2} $ transitions [Fig.~\\ref{LiSPD}(a)]. This\nminimum has been found earlier in Ref.~\\cite{Hoogenraad1995} in an experimental\nstudy of the far-infrared transitions between Rydberg states. Later on we\nhave shown that such minimum can also appear in the BBR-induced transitions\n\\cite{BeterovJETP2008}.\n\n\\subsection{Sodium}\n\nFor sodium atoms the relative matrix elements obtained for $R_{rel} \\left(\n{S_{1\/2} \\to P_{1\/2}}  \\right)$ and $R_{rel} \\left( {P_{3\/2} \\to D_{5\/2}}\n\\right)$ transitions are shown in Fig.~\\ref{NaSPD}, and for $R_{rel} \\left( {D_{3\/2}\n\\to F_{5\/2}}  \\right)$ transitions in Fig.~\\ref{NaDF}. A Cooper minimum is observed\nfor the bound-free $nS_{1\/2} \\to {E}'P_{1\/2} $ transitions at ${E}' \\approx\n0.06$ [Fig.~\\ref{NaSPD}(a)], bound-free $nD_{5\/2} \\to {E}'P_{3\/2} $ transitions at\n${E}' \\approx 0.05$ [Fig.~\\ref{NaSPD}(h)], bound-free $nP_{3\/2} \\to {E}'D_{5\/2} $\ntransitions at ${E}' \\approx 0.25$ [Fig.~\\ref{NaSPD}(e)], and bound-free $nF_{5\/2} \\to\n{E}'D_{3\/2} $ transitions at ${E}' \\approx 0.29$ [Fig.~\\ref{NaDF}(d)]. For $nS_{1\/2}\n\\to {E}'P_{1\/2} $ and $nD_{5\/2} \\to {E}'P_{3\/2} $ transitions in sodium a\nCooper minimum is found to be close to the ionization threshold.\n\n\\subsection{Potassium}\n\nFor potassium atoms the relative matrix elements obtained for $R_{rel}\n\\left( {S_{1\/2} \\to P_{1\/2}}  \\right)$ and $R_{rel} \\left( {P_{3\/2} \\to\nD_{5\/2}}  \\right)$ transitions are shown in Fig.~\\ref{KSPD}, and for $R_{rel} \\left(\n{D_{3\/2} \\to F_{5\/2}}  \\right)$ transitions in Fig.~\\ref{KDF}. Interesting features\nare observed in the radial matrix elements of bound-bound and bound-free\ntransitions. The two minima have been located in the relative matrix elements\nof the bound-free $nS_{1\/2} \\to {E}'P_{1\/2} $ transitions at ${E}' \\approx\n0.05$ and ${E}' \\approx 0.45$ [Fig.~\\ref{KSPD}(a)]. A Cooper minimum is also observed\nfor $nP_{3\/2} \\to {E}'D_{5\/2} $ transitions at ${E}' \\approx 0.29$ [Fig.~\\ref{KSPD}(g)].\n A minimum in the discrete spectrum has been found [Fig.~\\ref{KSPD}(c)],\nwhich is similar to the lithium $S \\to P$ and cesium $D \\to F$ transitions.\nThis minimum in potassium atoms was first discussed by Theodosiou\n\\cite{TheodosiouMin1980}. Finally, for the bound-free $nF_{5\/2} \\to {E}'D_{3\/2} $\ntransitions in potassium, a Cooper minimum is registered at ${E}' \\approx\n0.18$ and ${E}' \\approx 0.38$ [Fig.~\\ref{KDF}(d)].\n\n\n\\section{Conclusion}\n\n\n\n\n\nThe quasiclassical model developed by Dyachkov and Pankratov~\\cite{Dyachkov1994,\nDyachkovFree} can be used for fast and reliable calculations of the radial\nmatrix elements for bound-bound, bound-free and free-free transitions\nbetween arbitrary states of alkali-metal atoms. We have demonstrated this by\nperfoming the numerical calculations of the radial matrix elements for\ntransitions between \\textit{S}, \\textit{P}, \\textit{D} and \\textit{F} states\nwith the energies $E < 0.5$ (in atomic units) in all alkali-metal atoms. Our\nresults on radial matrix elements are in good agreement with  numerical calculations in the Coulomb\napproximations~\\cite{HoogenraadDipole}. Our theoretical results~\\cite{BeterovLifetimes} are also\nconsistent with the experimental measurements of the effective lifetimes of\nRydberg states~\\cite{Tate, Feng}, oscillator strengths~\\cite{Piotrowicz2011,\nNadeem2011}, and photoionization cross-sections~\\cite{HudsonCarter1967}. Our\napproach allowed us to reveal several unknown Cooper minima, both in the discrete\nand continuum spectra, which would be interesting to confirm experimentally. Reliability of the \nquasiclassical model for study of the Cooper minima is verified by good agreement with calculations of Ref.~\\cite{TheodosiouMin1980} for bound-bound transitions\nand satisfactory agreement with calculations of Ref.~\\cite{Aymar1978} for bound-free transitions.\nWe conclude that the quasiclassical model of Dyachkov and Pankratov is a\nuniversal method for systematic calculation of the radial matrix elements for transitions\nbetween excited states of alkali-metal atoms.\n\n\n\n\\section{Acknowledgments}\n\n\n\n\nThis work was supported by Grants of the President of Russia MK.7060.2012.2, MK.3727.2011.2, RFBR Grant No.~10-02-00133, Russian Academy of Sciences\nand the Dynasty foundation. SB , CM, AK and CMC aknowledge support from\nEPSRC grant No.~EP\/F031130\/1.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:introduction}Introduction}\n\nThe formation of contacts is an important technological aspect of device fabrication. Ohmic contacts on SiC require a post deposition anneal (PDA) of the deposited metal film. PDA induces a chemical reaction between the metal and SiC. Metal silicides are frequently observed (see refs. \\cite{crofton1997a,tanimoto2003a} and references therein) after PDA. For Ni \\cite{crofton1995a} and Co \\cite{lundberg1993a} graphite was observed as a byproduct of the silicide formation. In order to elucidate the impact of graphite on the electrical properties of metal contacts we have determined the Schottky barriers between graphite and 6H-SiC\\{0001\\} surfaces. Both, (0001) and ($000\\overline{1}$) oriented surfaces have to be considered for devices such as Schottky diodes, pn-diodes, MOSFETs and JFETs. \n\n\\section{\\label{sec:experimental}Experimental}\n\nThe samples used in this study were on-axis oriented, n-type 6H-SiC(0001) with a doping level of around $1\\times10^{18}$~cm$^{-3}$, 3.5 degree off axis oriented, p-type 6H-SiC(0001) with a p-type epilayer with a doping concentration of $1\\times10^{16}$~cm$^{-3}$, and  on-axis oriented, n-type 6H-SiC$(000\\overline{1})$ with a doping level of approximately $1\\times10^{18}$~cm$^{-3}$.\n\nGraphite layers were grown by solid state graphitization \\cite{forbeaux1998a,forbeaux1999a}, i.e. by sublimation of Si from the surface at elevated temperatures ($\\geq1150^{\\circ}$C) as described elsewhere \\cite{seyller2006a,seyller2006b,emtsev2006b}. First the samples were annealed in a Si flux at 950$^{\\circ}$C. Thereafter, Si was gradually removed from the surface by annealing steps at increasing temperatures until graphitization occured. The sequence of surface reconstructions observed during the preparation is face-dependent \\cite{starke2003a}. On SiC(0001) it is $(3\\times3)$, $(\\sqrt3\\times\\sqrt3)R30^\\circ$, $(6\\sqrt3\\times6\\sqrt3)R30^\\circ$, and $(1\\times1)_{\\mathrm{graph}}$. On SiC($000\\overline{1}$) different reconstructions occur: $(2\\times2)_{\\mathrm{Si}}$, $(3\\times3)$, $(2\\times2)_{\\mathrm{C}}$, $(1\\times1)_{\\mathrm{graph}}$. In agreement with previous results \\cite{forbeaux1998a,forbeaux1999a} the graphite layers obtained on 6H-SiC(0001) showed a sharp LEED pattern which indicated that the graphite lattice was rotated against the substrate by 30 degrees \\cite{seyller2006a,seyller2006b}. In contrast to that, graphite layers grown on 6H-SiC($000\\overline{1}$) led to a ring-like LEED pattern indicative of rotationally disordered domains. The rings had some intensity modulation with maxima in positions 30 degrees from the substrate spots suggesting a preference for an alignment of the graphite overlayer.\n\nPhotoelectron spectroscopy measurements were carried out using the end station MUSTANG operated either at beam line U49\/2-PGM1 or at U49\/2-PGM2 of the synchrotron radiation source BESSY II. The end station is equipped with a Specs Phoibos 150 electron analyzer. The combined energy resolution was estimated from the width of the Fermi level measured of the Mo-made sample holder and ranged from 60~meV at $\\hbar\\omega=130$~eV to 150~meV at $\\hbar\\omega=700$~eV.\n \n\\section{\\label{sec:results}Results}\n\nFig. \\ref{fig:C1sspectra}(a) displays C~1s core level spectra measured for 2~nm thick graphite films on the three different samples mentioned above. The overlayer leads to a sharp, asymmetric line at 284.41$\\pm$0.06 eV, which is typical for graphite. In addition, weak signals of the underlying SiC substrate are visible as well. For the Si-face samples the binding energy of the SiC substrate was practically the same $E_b^{C1s}=283.7\\pm0.06$~eV \\cite{seyller2006b} independent of the doping type (see  table \\ref{tab:properties}). On the n-type C-face sample the binding energy was $E_b^{C1s}=282.6\\pm0.06$~eV.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics [width=6cm]{Fig1a.EPS}\n    \\includegraphics [width=6cm]{Fig1b.EPS}\n    \\includegraphics [width=4cm]{Fig1c.EPS}\n    \\caption{(a) C~1s core level spectra of 2~nm graphite grown on n-type 6H-SiC(0001), p-type 6H-SiC(0001), and n-type 6H-SiC($000\\overline{1}$) by solid state graphitization. (b) Determination of the energy difference between the C~1s core level and the valence band maximum with XPS.(c) Band diagram of the n-type SiC\/graphite contact.}\n    \\label{fig:C1sspectra}\n\\end{figure}\n\nIn order to determine the Schottky barrier from the measured core levels we first determine the difference between binding energy $E_b^{C1s}$ of the bulk C1s core level and the valence band maximum $E_v$ for SiC as shown in figure \\ref{fig:C1sspectra}(b). This quantity amounts to 281.1$\\pm$0.1~eV. Note, that the binding energies determined by photoelectron spectroscopy are referenced to the Fermi level (see figure \\ref{fig:C1sspectra}(c)). The inelastic mean free path of photoelectrons is short (typically $\\leq$~2~nm for the present experiments) compared to the width of the depletion layer $w$ (see below), i.e. we are probing only a small region close to the surface in which the band bending can be considered constant. The position of the Fermi level at the surface $E_F^s$ with respect to the valence band maximum is thus given by $E_F^s-E_v=E_b^{C1s}-281.0$~eV. The extracted numerical values are collected in table \\ref{tab:properties}. The position of the bulk Fermi level with respect to the valence band maximum $E_F^{bulk}-E_v$ estimated \\cite{sze1981} for our n-type and p-type 6H-SiC is also listed. The band bending $V_b$ present at the surface of SiC is simply the difference between the bulk and the surface Fermi level, i.e. $e_0V_b=E_F^s-E_F^{bulk}$. From the band bending we have also calculated the surface charge $Q_{sc}$ and the width of the depletion layer $w$ (see table \\ref{tab:properties}). In all cases studied here, the latter is much larger than the inelastic mean free path of the electrons, indeed. Finally table \\ref{tab:properties} also lists the derived Schottky barriers which are $\\phi_{b,p}=E_F^s-E_v$ and $\\phi_{b,n}=E_g-(E_F^s-E_v)$ for the case of p-type and n-type SiC, respectively. For the Si-face the Schottky barriers are dramatically different: quite small ($\\phi_{b,n}^{Si}$=0.3$\\pm$0.1~eV) for electrons and huge ($\\phi_{b,p}^{Si}$=2.7$\\pm$0.1~eV) for holes \\cite{seyller2006b}. On the other hand, on the C-face a sizable barrier of $\\phi_{b,n}^{C}$=1.4$\\pm$0.1~eV for electrons was derived. In addition, taking into account that $\\phi_{b,n}+\\phi_{b,p}=E_g$ we predict that $\\phi_{b,p}^{C}$=1.6$\\pm$0.1~eV. \n \n\\begin{table}\n\\begin{center}\n\\vspace{4mm}\n\\small\n\\begin{tabular}{c|c|c|c|c|c|c|c|c} \n  Surface & Doping & $E_b^{C1s}$ & $E_F^s-E_v$ & $E_F^{bulk}-E_v$ & $V_b$ & $Q_{sc}$ & $w$ & $\\phi$ \\\\\n   & (cm$^{-3}$) & (eV) & (eV) & (eV) & (eV) & ($e_0$) & (nm) & (eV) \\\\  \n  \\hline\\hline\n  (0001) & N, $1\\times10^{18}$ & 283.7 & 2.7 & 2.9 & 0.2 & $-9.5\\times10^{11}$ & 23 & 0.3$\\pm$0.1\\\\\n  (0001) & Al, $1\\times10^{16}$ & 283.7 & 2.7 & 0.2 & -2.5 & $2.9\\times10^{11}$ & 958 & 2.7$\\pm$0.1 \\\\\n  \\hline\n  $(000\\overline{1})$ & N, $1\\times10^{18}$ & 282.6 & 1.6 & 2.9 & 1.3 & $-2.5\\times10^{12}$ & 58 & 1.4$\\pm$0.1 \\\\\n\\end{tabular}\\\\\n\\normalsize\n\\caption{\\label{tab:properties}C~1s binding energy of SiC ($E_b^{C1s}$), position of the Fermi level above the valence band maximum at the surface ($E_F^s-E_v$) and in the bulk ($E_F^{bulk}-E_v$), band bending ($V_b$), surface charge ($Q_{sc}$), width of depletion region ($w$), and barrier ($\\phi$) of the graphite\/SiC interfaces for graphite grown on n-type and p-type 6H-SiC(0001), and n-type 6H-SiC($000\\overline{1}$), respectively.}\n\\end{center}\n\\end{table}\n\n\n\\section{\\label{sec:discussion}Discussion}\n\nOn the Si-face we observe a dramatic difference between the barriers for holes and electrons. For n-type 6H-SiC(0001) the barrier is extremely low. Thus, Schottky contacts on n-type SiC(0001) are expected to degrade if graphite is formed at the interface. Indeed this is observed experimentally: Co\/SiC Schottky contacts transform into ohmic contacts after annealing at 900~$^{\\circ}$C \\cite{lundberg1993a}. The transformation is accompanied by the formation of Co$_2$Si and a buildup of carbon in the contact layer. Consequently, for Schottky contacts to n-type SiC(0001) we recommend metals for which the phase diagram prohibits the formation of graphite.\n\nOn the other hand, ohmic contacts on this surface orientation could benefit from interfacial graphite. In fact, Lu et al. \\cite{lu2003a} observed an ohmic behavior of pure carbon contacts on n-type 4H-SiC(0001) annealed at 1150-1300~$^{\\circ}$C which is consistent with a small Schottky barrier between n-type SiC(0001) and graphite. In another work, Lu et al. \\cite{lu2003b} observed that Ni\/carbon\/SiC(0001) and Co\/carbon\/SiC(0001) stacks annealed at 800~$^{\\circ}$C developed ohmic behavior but similar stacks with W, Mo, Au, or Al remained rectifying. They also reported that the ohmic contacts contained graphite while the rectifying did not. According to our results, the observed contact behavior is readily explained by the small value for $\\phi_{b,n}^{Si}$ for the graphite\/SiC(0001).\n\nOur results suggest that graphite will be detrimental for ohmic contacts to p-type SiC(0001) and that on p-type SiC(0001) metals should be preferred which form stable carbides or which have a high solubility for carbon. In agreement with that, Al\/Ti contacts have shown low contact resistivities (see ref. \\cite{tanimoto2003a} and references therein). Furthermore, metal\/Si co-deposition is expected to lead to better ohmic contacts than simple deposition of metals. This is so because in a metal\/Si\/SiC stack, silicide formation can occur without SiC decomposition and graphite formation. Accordingly, Lundberg and \\\"Ostling \\cite{lundberg1996a} reported that Si\/Co\/SiC stacks annealed at 900~$^{\\circ}$C showed an ohmic behavior superior to Co\/SiC contacts treated in the same way. \n\nOur experiments also show a clear-cut face dependence of the barriers. The microscopic origin of the different barriers observed on Si-face and C-face is unclear as yet. Considering that different reconstructions occur on SiC(0001) and ($000\\overline{1}$), it is very likely that the atomic arrangement of the interface between graphite and these two polar surfaces is not the same. On SiC(0001) the interface consists of the $(6\\sqrt3\\times6\\sqrt3)R30^\\circ$ structure \\cite{emtsev2006b}. The atomic arrangement at the interface between SiC($000\\overline{1}$) and graphite is currently unknown. A different interface structure on the atomic scale is likely to produce different interface dipole moments as well as different interface states which may pin the Fermi level. In terms of contact properties we note that the large value of $\\phi_{b,n}^{C}$=1.4$\\pm$0.1~eV observed here supports the conclusions of Nikitina et al. \\cite{nikitina2005a}, namely, that graphite is not responsible for ohmic contact behavior of Ni\/SiC($000\\overline{1}$) after PDA.\n\n\\section{\\label{sec:conclusion}Conclusion}\n\nWe have determined the Schottky barriers between graphite and n- and p-type 6H-SiC(0001) as well as n-type 6H-SiC($000\\overline{1}$). While we find a rather small value of $\\phi_{b,n}^{Si}$=0.3$\\pm$0.1~eV on n-type 6H-SiC(0001), the Schottky barrier on p-type SiC(0001) is extremely large $\\phi_{b,p}^{Si}$=2.7$\\pm$0.1~eV. The barrier is strongly face dependent: for n-type 6H-SiC($000\\overline{1}$) we determined $\\phi_{b,n}^{C}$=1.4$\\pm$0.1~eV.  The origin for the face dependence is the subject of ongoing investigations. Our results give a consistent interpretation of numerous reports on the contact behavior of metals on SiC\\{0001\\} after PDA. \n\n\n\\section{Acknowledgments}\n\nThe authors are grateful for support by the BESSY staff and by G. Gavrila from TU Chemnitz. Traveling costs for beam times were provided by the BMBF through Grant No. 05 ES3XBA\/5. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{section1}\n\\footnote{This version corrects an error in the published version.}Deep neural networks (DNNs) have been demonstrated to be vulnerable to adversarial examples \\cite{szegedy2013intriguing, goodfellow2014explaining,liao2018defense,shen2017ape,ma2018characterizing,wu2020adversarial, pmlr-v139-zhou21e}. The adversarial samples are typically generated by adding imperceptible but adversarial noise to natural samples. The vulnerability of DNNs seriously threatens many decision-critical deep learning applications, such as image processing \\cite{lecun1998gradient, Zagoruyko2016WRN, 2017Mask, xia2020part,  ma2021understanding, xia2021robust}, natural language processing \\cite{sutskever2014sequence} and speech recognition \\cite{sak2015fast}. The urgent demand to reduce these security risks prompts the development of adversarial defenses.\n\nMany researchers have made extensive efforts to improve the adversarial robustness of DNN. A major class of adversarial defense focuses on exploiting adversarial samples to help train the target model to achieve adversarially robust performance \\cite{madry2017towards,ding2019sensitivity,zhang2019theoretically,wang2019improving,zhou2021modeling,zheng2021regularizing,yang2021class}. However, the dependence between the output of the target model and the input adversarial sample has not been well studied yet. Explicitly measuring this dependence could help train the target model to make predictions that are closely relevant to the given objectives \\cite{belghazi2018mutual,sanchez2020learning}.\n\nIn this paper, we investigate the dependence from the perspective of information theory. Specifically, we exploit the mutual information (MI) to explicitly measure the dependence of the output on the adversarial sample. MI is an entropy-based measure that can reflect the dependence degree between variables. A larger MI typically indicates stronger dependence between the two variables. However, directly exploiting MI between the input and its corresponding output (called \\textit{standard MI}) to measure the dependence has limitations in improving classification accuracy for adversarial samples.\n\nNote that adversarial samples contain natural and adversarial patterns. As shown in \\cref{fig1}, given a target model and an adversarial sample, the natural pattern is derived from the corresponding natural sample, and the adversarial pattern is derived from the adversarial noise in the adversarial sample. The adversarial pattern controls the flip of the prediction from the correct label to the wrong label \\cite{ilyas2019adversarial}. The standard MI of the adversarial sample reflects the confused dependence of the output on the natural pattern and the adversarial pattern. Maximizing the standard MI of the adversarial sample to guide the target model may result in a larger dependence between the output and the adversarial pattern. This may cause more interference with the prediction of the target model. Therefore, directly maximizing the standard MI to help train the target model cannot surely promote the adversarial robustness.\n\nTo handle this issue, we propose to disentangle the standard MI to explicitly measure the dependence of the output on the natural pattern and the adversarial pattern, respectively. As shown in \\cref{fig1}, we disentangle the standard MI into the \\textit{natural MI} (i.e., the MI between the output and the natural pattern) and the \\textit{adversarial MI} (i.e., the MI between the output and the adversarial pattern). To present the reasonability of the disentanglement, we theoretically demonstrate that standard MI is closely related to the linear sum of natural MI and adversarial MI. In addition, how to effectively estimate the natural MI and the adversarial MI is a crucial problem. Inspired by the \\textit{mutual information maximization} in \\citet{hjelm2018learning,zhu2020learning}, we design a neural network-based method to train two MI estimators to estimate the natural MI and adversarial MI respectively. The detailed discussion can be found in \\cref{section3.2}. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=3.0in]{fig1.pdf}}\n\\caption{A visual illustration of disentangling the standard MI into the natural MI and the adversarial MI. The \\textit{longdash} lines show that the adversarial sample is disassembled into the natural pattern (derived from the natural sample) and the adversarial pattern (derived from the adversarial noise). The \\textit{dotted} lines denote the operation of disentangling the standard MI into the natural MI and the adversarial MI.}\n\\label{fig1}\n\\end{center}\n\\vskip -0.3in\n\\end{figure}\n\nBased on the above MI estimation, we develop an adversarial defense algorithm called \\textit{natural-adversarial mutual information-based defense} (NAMID) to enhance the adversarial robustness. Specifically, we introduce an optimization strategy using the natural MI and the adversarial MI on the basis of the adversarial training manner. The optimization strategy is to maximize the natural MI of the input adversarial sample and minimize its adversarial MI simultaneously. By iteratively executing the procedures of generating adversarial samples and optimizing the model parameters, we can learn an adversarially robust target model. \n\nThe main contributions in this paper are as follows:\n\\begin{itemize}\n    \\item Considering the adversarial sample has the natural pattern and the adversarial pattern, we propose the natural MI and the adversarial MI to explicitly measure the dependence of the output on the different patterns.\n    \\item We design a neural network-based method to effectively estimate the natural MI and the adversarial MI. By exploiting the MI estimation networks, we develop a defense algorithm to train a robust target model. \n    \\item We empirically demonstrate the effectiveness of the proposed defense algorithm on improving the classification accuracy. The evaluation experiments are conducted against multiple adversarial attacks.\n\\end{itemize}\n\nThe rest of this paper is organized as follows. In Section~\\ref{section2}, we introduce some preliminary information and briefly review related works. In Section~\\ref{section3}, we propose the natural MI and the adversarial MI, and develop an adversarial defense method. Experimental results are provided in Section~\\ref{section4}. Finally, we conclude this paper in Section~\\ref{section5}.\n\n\\section{Preliminaries}\n\\label{section2}\nIn this section, we introduce some preliminary about notation, the problem setting and mutual information (MI). We also review some representative literature on adversarial attacks and defenses.\n\n\\noindent\\textbf{Notation.} \nWe use \\textit{capital} letters such as $X$ and $Y$ to represent random variables, and \\textit{lower-case} letters such as $x$ and $y$ to represent realizations of random variables $X$ and $Y$, respectively. For norms, we denote by $\\|x\\|$ a generic norm, by $\\|x\\|_{\\infty}$ the $L_{\\infty}$ norm of $x$, and by $\\|x\\|_{2}$ the $L_{2}$ norm of $x$. In addition, let $\\mathbb{B}(x, \\epsilon)$ represent the neighborhood of $x$: $\\{\\tilde{x}:\\|\\tilde{x}-x\\| \\leq \\epsilon$\\}, where $\\epsilon$ is the perturbation budget. We define the \\textit{classification function} as $f: \\mathcal{X} \\rightarrow \\{1,2,\\ldots,C\\}$. It can be parameterized, e.g., by a deep neural network $h_{\\theta}$ with model parameter $\\theta$.\n\n\n\\noindent\\textbf{Problem setting.} \nIn this paper, we focuses on a classification task under the adversarial environment. Let $X$ and $Y$ be the variables for natural instances and ground-truth labels respectively. We sample natural data $\\{(x_i, y_i)\\}_{i=1}^{n}$ according to the distribution of variables $(X,Y)$. Given a pair of natural data $(x,y)$, the adversarial instance $\\tilde{x}$ satisfies the following constraint:\n\\begin{equation}\n\\label{eq1}\nf\\left(\\tilde{x}\\right) \\neq y \\quad \\text { s.t.} \\quad\\left\\|x-\\tilde{x}\\right\\| \\leq \\epsilon \\text{,}\n\\end{equation}\nwhere $\\tilde{x}=x+n$, $n$ denotes the adversarial noise. Our aim is to develop an adversarial defense method to help train the classification model $h_{\\theta}$ to make normal predictions.\n\n\\noindent\\textbf{Mutual information.}\nMI is an entropy-based measure that can reflect the dependence degree between variables. A larger MI typically indicates a stronger dependence between the two variables. Various methods have been proposed for estimating MI \\cite{moon1995estimation,darbellay1999estimation,kandasamy2015nonparametric,moon2017ensemble}. A representative and efficient estimator is the mutual information neural estimator (MINE) \\cite{belghazi2018mutual}. MINE is empirically demonstrated its superiority in estimation accuracy and efficiency, and proved that it is strongly consistent with the true MI. Besides, the work in \\citet{hjelm2018learning} points that using the complete input to estimate MI is often insufficient for classification task. Instead, estimating MI between the high-level feature and local patches of the input is more suitable. Therefore, in this paper, we refer the local DIM estimator \\cite{hjelm2018learning} to estimate MI. The definition of MI and other details are presented in \\cref{appendix_1}\n\n\\noindent\\textbf{Adversarial attacks.} \nAdversarial noise can be crafted by optimization-based attacks, such as PGD \\cite{madry2017towards}, AA \\cite{croce2020reliable}, CW \\cite{carlini2017towards} and DDN \\cite{rony2019decoupling}. Besides, some attacks such as FWA \\cite{wu2020stronger} and STA \\cite{xiao2018spatially} focus on mimicking non-suspicious vandalism by exploiting the geometry and spatial information. These attacks constrain the perturbation boundary by a small norm-ball $\\|\\cdot\\|_{p} \\leq \\epsilon$, so that their adversarial instances can be perceptually similar to natural instances. \n\n\\noindent\\textbf{Adversarial defenses.} The issue of adversarial attacks promotes the development of adversarial defenses. A major class of adversarial defense methods is devoted to enhance the adversarial robustness in an adversarial training manner \\cite{madry2017towards,ding2019sensitivity,zhang2019theoretically,wang2019improving}. They augment training data with adversarial samples and use a min-max formulation to train the target model \\cite{madry2017towards}. However, these methods do not explicitly measure the dependence between the adversarial sample and its corresponding output. In addition, some data pre-processing based methods try to remove adversarial noise by learning denoising functions \\cite{liao2018defense,naseer2020self,zhou2021removing} or feature-squeezing functions \\cite{guo2017countering}. However, these methods may suffer from the problems of human-observable loss \\cite{xu2017feature} and residual adversarial noise \\cite{liao2018defense}, which would affect the final prediction. To avoid the above problems, we propose to exploit the natural MI and the adversarial MI to learn an adversarially robust classification model in the adversarial training manner.\n\\vskip -0.1in\n\n\\section{Methodology}\n\\label{section3}\nIn this section, we first illustrate the motivation for using mutual information (MI) and disentangling the standard mutual information into the \\textit{natural MI} and the \\textit{adversarial MI} (\\cref{section3.1}). Next, we theoretically prove the reasonability of the disentanglement and introduce how to effectively estimate the natural MI and the adversarial MI (\\cref{section3.2}). Finally, we propose an adversarial defense algorithm which contains an MI-based optimization strategy (\\cref{section3.3}). The code is available at \\url{https:\/\/github.com\/dwDavidxd\/MIAT}.\n\n\\subsection{Motivation}\n\\label{section3.1}\nFor adversarial samples, The predictions of the target model are typically significantly irrelevant to the given objectives in the inputs. Studying the dependence between the adversarial sample and its corresponding output is considered to be beneficial for improving the adversarial robustness. The dependence could be exploited as supervision information to help train the target model to make correct predictions.\n\nEstimating the standard MI of the adversarial sample (i.e., the MI between the adversarial sample and its corresponding output in a target model) is a simple strategy to measure the dependence. However, different from natural samples, adversarial samples have two patterns, i.e., the natural pattern and the adversarial pattern. The standard MI cannot respectively consider the dependence of the output on the different patterns, which may limit its performance in helping the target model improve the adversarial robustness. \n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.8in]{fig2.pdf}}\n\\caption{The visualization of the proof-of-concept experiment. Given the natural instance $x$, its adversarial instance $\\tilde{x}$ and a target model $h$, the logit output of $h$ for $\\tilde{x}$ is denoted by $h(\\tilde{x})$. We respectively estimate the standard MI of the natural instance $\\widehat{I}(x, h(x))$, the standard MI of the adversarial instance $\\widehat{I}(\\tilde{x}, h(\\tilde{x}))$ and the MI between $h(\\tilde{x})$ and $x$. The \\textit{shaded} area represents the difference between $\\widehat{I}(\\tilde{x}, h(\\tilde{x}))$ and $\\widehat{I}(x, h(\\tilde{x}))$.}\n\\label{fig2}\n\\vskip -0.3in\n\\end{center}\n\\end{figure}\n\nSpecifically, the natural pattern is derived from the original natural sample. It provides available information for the target model to produce the right output. The adversarial pattern is derived from the adversarial noise. It controls the flip of the prediction from the correct label to the wrong label \\cite{ilyas2019adversarial}. Both the natural and adversarial patterns cause important impacts on the output, but they are mutually exclusive. Therefore, the standard MI actually measures a confused dependence.\n\nTo clearly illustrate the confused dependence, we conduct a proof-of-concept experiment. We randomly select a set of natural instances and use five attacks to generative adversarial instances. We use a classification model as the target model. The MI estimator is trained on natural instances and their outputs via the MI maximization \\cite{hjelm2018learning}. By exploiting the estimator, we respectively estimate the standard MI of the natural instance, the standard MI of the adversarial instance and the MI between the natural instance and the output for the corresponding adversarial instance. The details of the experiment are presented in \\cref{appendix_2}\n\nAs shown in \\cref{fig2}, the result shows that the standard MI of the adversarial sample is indeed smaller than that of the natural sample. However, it is still significantly larger than the MI between the output and natural pattern only (see the pink line). This shows that the standard MI of the adversarial sample contains the dependence of the output on the adversarial pattern. Thus, maximizing the standard MI may increase the dependence of the output on the adversarial pattern and cause more disturbance to the prediction. Directly maximizing the standard MI cannot comprehensively promote the target model to make more accurate predictions for adversarial samples. \n\nTo solve this problem, in this paper, we propose to disentangle the standard MI into two parts related to the natural and adversarial patterns respectively.\n\n\\subsection{Natural MI and adversarial MI}\n\\label{section3.2}\nWe define two new concepts: \\textit{natural mutual information} (natural MI) and \\textit{adversarial mutual information} (adversarial MI). The natural MI is MI between the output and the natural pattern of the input. The adversarial MI is MI between the output and the adversarial pattern of the input. To explicitly measure the dependence of the output on different patterns of the input, we need to disentangle the standard MI into the natural MI and the adversarial MI. \n\n\\subsubsection{Disentangling the standard MI}\n\\label{section3.2.1}\nIn this section, we introduce how to disentangle the standard MI and describe the reasonability of the disentanglement. We first provide Theorem 1 to illustrate the transformation relationship of MI among four variables.\n\n\\noindent \\textbf{Theorem 1.} \nLet $X, \\widetilde{X}, N, Z$ denote four random variables respectively, where $\\widetilde{X}=X+N$. Let $\\widetilde{\\mathcal{X}}$ be the feature space of $\\widetilde{X}$ and $\\mathcal{Z}$ be the feature space of $Z$. \nThen, for any function $h: \\widetilde{\\mathcal{X}} \\rightarrow \\mathcal{Z}$, we have \n\\begin{equation}\n\\label{eq2}\n\\begin{aligned}\n I(\\widetilde{X};Z) &= I(X;Z) + I(N;Z) - I(X;N;Z) \\\\ &+ H(Z|X,N) - H(Z|\\widetilde{X}) \\text{,}\n\\end{aligned}\n\\end{equation}\nwhere $I(\\cdot;\\cdot)$ denotes the MI between two variables and $I(\\cdot;\\cdot;\\cdot)$ denotes the MI between two three variables. A detailed proof is provided in \\cref{appendix_3}. \n\nThen, we apply Theorem 1 to the adversarial learning setting and obtain Corollary 1.\n\n\\noindent \\textbf{Corollary 1.}\nLet $X, \\widetilde{X}, N$ denote the random variables for the natural instance, adversarial instance and adversarial noise respectively, where $\\widetilde{X}=X+N$. Given a function parameterized by a target model $h_{\\theta}$ with model parameter $\\theta$, the logit output of $h_{\\theta}$ for $\\widetilde{X}$ is denoted by $h_{\\theta}(\\widetilde{X})$. Considering that the effects of the natural instance and the adversarial noise on the output are mutually exclusive, we assume that the MI between $X, N$ and $h_{\\theta}(\\widetilde{X})$ (i.e., $I(X;N;h_{\\theta}(\\widetilde{X}))$) is small. We also assume that the difference between $H(Z|X,N)$ and $H(Z|\\widetilde{X})$ is small (see \\cref{appendix_4} for more details), then we have:\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq3}\nI(\\widetilde{X};h_{\\theta}(\\widetilde{X})) \\approx \\underbrace{I(X;h_{\\theta}(\\widetilde{X}))}_{I_{N}}+\\underbrace{I(N;h_{\\theta}(\\widetilde{X}))}_{I_{A}} \\text{,}\n\\end{equation}\n\\vskip -0.15in\nwhere $I_{N}$ denote the MI between the output and the natural instance and $I_{A}$ denote the MI between the output and the adversarial noise. \n\nActually, we could use the original natural instance and the adversarial noise to represent the natural pattern and the adversarial pattern of the input respectively. In this way, $I_{N}$ and $I_{A}$ could denote the natural MI and the adversarial MI respectively.\n\nAccording to \\cref{eq3}, we can approximately disentangle the standard MI into the natural MI and the adversarial MI. The latter two can not only reflect the dependency between input and output as the former, but also provides independent measurements for different patterns. This is more conducive to designing specific optimization strategies for the two patterns to better alleviate the negative effects of the adversarial noise.\n\n\\subsubsection{Estimating the natural MI and the adversarial MI}\n\\label{section3.2.2}\nThe local DIM estimation method has been demonstrated to be efficient for estimating MI \\cite{hjelm2018learning,zhu2020learning}. We thus first use this method to train a estimation network for the natural MI and the adversarial MI respectively. Let $E_{\\phi_{n}}$ denote the estimation network for the natural MI and $E_{\\phi_{a}}$ denote the estimation network for the adversarial MI. Considering the inherent close relevance between the natural\/adversarial pattern and the output for the natural\/adversarial sample, we naturally use the natural\/adversarial samples to train the $E_{\\phi_{n}}$\/$E_{\\phi_{a}}$. The optimization goals for $\\phi_{n}$ and $\\phi_{a}$ are as follows:\n\n\\vskip -0.1in\n\\begin{equation}\n\\label{eq4}\n\\begin{aligned}\n&\\widehat{\\phi}_{n}=\\underset{\\phi_{n} \\in \\Phi_{N}}{\\arg \\max } \\,E_{\\phi_{n}}(h_{\\theta_{0}}(X)) \\text{,} \\\\\n&\\widehat{\\phi}_{a}=\\underset{\\phi_{a} \\in \\Phi_{A}}{\\arg \\max } \\, E_{\\phi_{a}}(h_{\\theta_{0}}(\\widetilde{X})) \\text{,}\n\\end{aligned}\n\\end{equation}\n\\vskip -0.1in\n\nwhere $\\Phi_{N}$ and $\\Phi_{A}$ denote the sets of model parameters, $h_{\\theta_0}$ denotes the pre-trained target model. It can be naturally trained or be adversarially trained. We use a ResNet-18 optimized by standard AT \\cite{madry2017towards} as the pre-trained model $h_{\\theta_0}$.  $E_{\\widehat{\\phi}_{n}}\\left(\\cdot \\right)$ is the estimated natural MI and $E_{\\widehat{\\phi}_{a}}\\left(\\cdot \\right)$ is the estimated adversarial MI, i.e.,  $E_{\\widehat{\\phi}_{n}}\\left(h_{\\theta}(X)\\right)=\\widehat{I}_{N}(X; h_{\\theta}(X))$, $E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\widetilde{X}))=\\widehat{I}_{N}(X; h_{\\theta}(\\widetilde{X}))$ and $E_{\\widehat{\\phi}_{a}}\\left(h_{\\theta}(X)\\right)=\\widehat{I}_{A}(N; h_{\\theta}(X))$,  $E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\widetilde{X}))=\\widehat{I}_{A}(N; h_{\\theta}(\\widetilde{X}))$.\n\nBy exploiting the two MI estimation network, we estimate the natural MI of the natural sample and the adversarial sample (i.e., $\\widehat{I}_{N}(X, h_{\\theta_{0}})$ and $\\widehat{I}_{N}(\\widetilde{X}, h_{\\theta_{0}})$), and the adversarial MI of the natural sample and the adversarial sample (i.e., $\\widehat{I}_{A}(X, h_{\\theta_{0}})$ and $\\widehat{I}_{N}(\\widetilde{X}, h_{\\theta_{0}})$), respectively. We find that adversarial samples usually have larger adversarial MI and smaller natural MI compared with those \\textit{w.r.t.} natural samples, which is consistent with our intuitive cognition. However, the change is relatively insignificant, and thus has limitations in reflecting the difference between the adversarial sample and the natural sample in the natural MI and the adversarial MI. We will show this observation later in \\cref{section4.2}. \n\nTo adequately represent the inherent difference in the natural MI and the adversarial MI between the natural sample and the adversarial sample, we design an optimization mechanism for the MI estimation. This is, we minimize the natural MI of the adversarial sample and minimize the adversarial MI of the natural sample during training the estimators. In addition, to estimate the two MI more accurately, we select samples that are correctly predicted by the target model and the corresponding adversarial samples are wrongly predicted, to train the estimation networks. The reformulated optimization goals are as follows:\n\\begin{equation}\n\\label{eq5}\n\\begin{aligned}\n&\\widehat{\\phi}_{n}=\\underset{\\phi_{n} \\in \\Phi_{N}}{\\arg \\max } \\,[E_{\\phi_{n}}( h_{\\theta_{0}}(X^{\\prime})) - E_{\\phi_{n}}(h_{\\theta_{0}}(\\widetilde{X}^{\\prime}))] \\text{,} \\\\\n&\\widehat{\\phi}_{a}=\\underset{\\phi_{a} \\in \\Phi_{A}}{\\arg \\max } \\, [E_{\\phi_{a}}(h_{\\theta_{0}}(\\widetilde{X}^{\\prime})) - E_{\\phi_{a}}\\left(h_{\\theta_{0}}(X^{\\prime}\\right))] \\text{,}\n\\end{aligned}\n\\end{equation}\n\nwhere $X^{\\prime}$ is the selected data:\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq6}\n    X^{\\prime}=\\arg _{X}\\left[\\delta\\left(h_{{\\theta}_0}(X)\\right)=Y \\& \\delta(h_{{\\theta}_0}(\\widetilde{X})) \\neq Y\\right]  \\text{,} \n\\end{equation}\n\nwhere $\\delta$ is the operation that transforms the logit output into the prediction label.\n\n\\subsection{Adversarial defense algorithm}\n\\label{section3.3}\nBased on the above two MI estimation networks, we develop an adversarial defense algorithm called \\textit{natural-adversarial mutual information-based defense} (NAMID) to enhance the adversarial robustness. In this section, we first introduce the natural-adversarial MI-based optimization strategy and then illustrate the training algorithm.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=5.6in]{fig3.pdf}}\n\\caption{The overview of our proposed NAMID adversarial defense algorithm.}\n\\label{fig3}\n\\vskip -0.2in\n\\end{center}\n\\end{figure*}\n\n\\subsubsection{natural-adversarial MI-based defense}\n\\label{section3.3.1}\nAccording to the observations and analysis in \\cref{section3.1} and \\cref{section3.2}, we plan to use the natural MI and the adversarial MI to help train the target model. We aim to guide the target model to increase the attention to the natural pattern while reducing the attention to the adversarial pattern. \n\nSpecifically, The optimization strategy is to maximize the natural MI of the input adversarial sample and minimize its adversarial MI simultaneously. The optimization goal for the target model is as follows:\n\n\\begin{equation}\n\\label{eq7}\n\\widehat{\\theta}=\\underset{\\theta \\in \\Theta}{\\arg \\max }\\left[E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\widetilde{X}))-E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\widetilde{X}))\\right] \\text{.}\n\\end{equation}\n\nTo achieve the optimization goal, we can directly utilize the natural and adversarial MI of the adversarial sample to construct a loss function. However, this loss function does not consider the difference in natural\/adversarial MI between the natural sample and the adversarial sample. Thus, we transform this absolute metric-based loss to a relative metric-based loss. In addition, as described in \\cref{section3.2.2}, we use the selected samples to compute the loss function. The loss function is formulated as:\n\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq8}\n\\begin{gathered}\n\\mathcal{L}_{m i}(\\theta)=\\frac{1}{m} \\sum_{i=1}^{m}\\{\\mathcal{L}_{\\cos }(E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime})), E_{\\widehat{\\phi}_{n}}(h_{\\theta}(x_{i}^{\\prime}))) \\\\ + \\mathcal{L}_{\\cos }(E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}), E_{\\widehat{\\phi}_{a}}(h_{\\theta}(x_{i}^{\\prime}))) \\\\ + \\lambda \\cdot [E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}))-E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}))]\\} \\text{,}\n\\end{gathered}\n\\end{equation}\n\nwhere $m$ is the number of the selected data $X^{\\prime}$ and $\\lambda$ is a hyperparameter. $\\mathcal{L}_{\\cos}(\\cdot, \\cdot)$ is the cosine similarity-based loss function, i.e., $\\mathcal{L}_{\\cos}(a, b) = ||1 - sim (a,b)||_1$, $sim(\\cdot,\\cdot)$ denotes the cosine similarity measure.\n\nThe MI-based optimization strategy can exploited together with the adversarial training manner. The overall loss function for training the target model is as follows:\n\n\\begin{equation}\n\\label{eq9}\n    \\mathcal{L}_{all}(\\theta)= \\mathcal{L}_{adv}(\\theta) + \\alpha \\cdot \\mathcal{L}_{mi}(\\theta) \\text{,}\n\\end{equation}\n\\vskip 0.1in\n\nwhere $\\mathcal{L}_{adv}(\\theta)$ is the loss function of the adversarial training method, which is typically the cross-entropy loss between the adversarial outputs and the ground-truth labels: $\\mathcal{L}_{adv}(\\theta)= -\\frac{1}{n} \\sum_{i=1}^{n} [\\boldsymbol{y_i} \\cdot \\log(\\sigma(h_{\\theta}(\\tilde{x}_{i})))]$. $n$ is the number of training samples and $\\sigma$ denotes the softmax function. $\\alpha$ is a trade-off hyperparameter. We provide the overview of the adversarial defense method in \\cref{fig3}.\n\n\\subsubsection{Training algorithm} \n\\label{section3.3.2}\nWe conduct the adversarial training on the procedures of generating adversarial samples and optimizing the target model parameter. The details of the overall procedure are presented in \\cref{alg1}.\n\n Specifically, the procedure requires the target model $h_{\\theta}$ with parameter $\\theta$, the natural MI estimation network with parameter $\\widehat{\\phi}_{n}$, the adversarial MI estimation network with parameter $\\widehat{\\phi}_{a}$ and perturbation budget $\\epsilon$. For the natural instance $x$ in mini-batch $\\mathcal{B}=\\{x_i\\}_{i=1}^{n}$ sampled from natural training set, we first craft adversarial noise $n$ and generate adversarial instance $\\tilde{x}$ via the powerful PGD adversarial attack \\cite{madry2017towards}. Then, we input the natural and adversarial training data into the target model $h_{\\theta}$ and obtain the selected instance $x^{\\prime}$, $\\tilde{x}^{\\prime}$ according to \\cref{eq6}. Next, we estimate the natural MI $E_{\\widehat{\\phi}_{n}}(h_{\\theta}(x^{\\prime})), E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}^{\\prime}))$ and the adversarial MI $ E_{\\widehat{\\phi}_{a}}(h_{\\theta}(x^{\\prime})), E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}^{\\prime})$. Finally, we optimize the parameter $\\theta$ according to \\cref{eq9}. By iteratively conduct the adversarial training, $\\theta$ is expected to be optimized well.\n\n\\begin{algorithm}[t]\n   \\caption{\\small Natural-adversarial mutual information-based defense (NAMID) algorithm}\n   \\label{alg1}\n\\begin{algorithmic}[1]\n   \\begin{small}\n   \\REQUIRE Target model $h_{\\theta}(\\cdot)$ parameterized by $\\theta$, natural MI estimation network $E_{\\widehat{\\phi}_{n}}$, adversarial MI estimation network $E_{\\widehat{\\phi}_{a}}$, batch size $n$, and the perturbation budget $\\epsilon$;\n   \\REPEAT\n   \\STATE Read mini-batch $\\mathcal{B}=\\{x_i\\}_{i=1}^{n}$ from training set;\n   \\FOR{$i=1$ to $n$ (in parallel)}\n   \\STATE Craft adversarial noise $n_{i}$ and generate adversarial instance $\\tilde{x}_i$ at the given perturbation budget $\\epsilon$ for $x_i$;\n   \\STATE Forward-pass $x_i$, $\\tilde{x}_i$ through $h_{\\theta}(\\cdot)$ and obtain $h_{\\theta}(x_i)$, $h_{\\theta}(\\tilde{x}_i)$;\n   \\STATE Select samples according to \\cref{eq6};\n   \\ENDFOR\n   \\STATE Calculate $\\mathcal{L}_{all}$ using \\cref{eq9} and optimize $\\theta$;\n   \\UNTIL training converged.\n   \\end{small}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\\label{section4}\nIn this section, we first introduce the experiment setups including datasets, attack setting and defense setting in \\cref{section4.1}. Then, we show the effectiveness of our optimization mechanism for evaluating MI in \\cref{section4.2}. Next, we evaluate the performances of the proposed adversarial defense algorithm in \\cref{section4.3}. Finally, we conduct ablation studies in \\cref{section4.4}.\n\n\\subsection{Experiment setups}\n\\label{section4.1}\n\\noindent\\textbf{Datasets.}\nWe verify the effective of our defense algorithm on two popular benchmark datasets, i.e., \\textit{CIFAR-10} \\cite{krizhevsky2009learning} and \\textit{Tiny-ImageNet} \\cite{wu2017tiny}. \\textit{CIFAR-10} has 10 classes of images including 50,000 training images and 10,000 test images. \\textit{Tiny-ImageNet} has 200 classes of images including 100,000 training images, 10,000 validation images and 10,000 test images. Images in the two datasets are all regarded as natural instances. All images are normalized into [0,1], and are performed simple data augmentations in the training process, including random crop and random horizontal flip. \n\n\\noindent\\textbf{Model architectures.} We use a ResNet-18 \\cite{he2016deep} as the target model for both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}. For the MI estimation network, we utilize the same neural network as in \\citep{zhu2020learning}. The estimation networks for the natural MI and the adversarial MI have same model architectures.\n\n\\noindent\\textbf{Baselines.} (1) \\textit{Standard AT} \\cite{madry2017towards}; (2) TRADES \\cite{zhang2019theoretically}; (3) MART \\cite{wang2019improving}; and (4)\\textit{WMIM}: A defense that refers to \\citet{zhu2020learning}. The first three are representative adversarial training methods, and the last one combines adversarial training with standard MI maximization (on adversarial samples).\n\n\\noindent\\textbf{Attack settings.}\nAdversarial data for evaluating defense models are crafted by applying state-of-the-art attacks. These attacks are divided into two categories: $L_{\\infty}$-norm attacks and $L_{2}$-norm attacks. The $L_{\\infty}$-norm attacks include PGD \\cite{madry2017towards}, AA \\cite{croce2020reliable}, TI-DIM \\cite{dong2019evading,xie2019improving}, and FWA \\cite{wu2020stronger}. The $L_{2}$-norm attacks include PGD, CW$_2$ \\cite{carlini2017towards} and DDN \\cite{rony2019decoupling}. Among them, the AA attack algorithm integrates three non-target attacks and a target attack. Other attack algorithms are utilized as non-target attacks. The iteration number of PGD and FWA is set to 40 with step size $\\epsilon\/4$. The iteration number of CW$_2$ and DDN are set to 20 respectively with step size 0.01. For \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}, the perturbation budgets for $L_{2}$-norm attacks and $L_{\\infty}$-norm attacks are $\\epsilon=0.5$ and $8\/255$ respectively.  \n\n\\noindent\\textbf{Defense settings.} For both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}, the adversarial training data for $L_{\\infty}$-norm and $L_{2}$-norm is generated by using $L_{\\infty}$-norm PGD-10 and $L_{2}$-norm PGD-10 respectively. The step size is $\\epsilon\/4$ and the perturbation budget is $8\/255$ and $0.5$ respectively. The epoch number is set to 100. For fair comparisons, all the methods are trained using SGD with momentum 0.9, weight decay $2 \\times 10^{-4}$, batch-size 1024 and an initial learning rate of 0.1, which is divided by 10 at the 75-th and 90-th epoch. In addition, we adjust the hyperparameter settings of the defense methods so that the natural accuracy is not severely compromised and then compare the adversarial accuracy. We set $\\alpha=5, \\lambda= 0.1$ for our algorithm. \n\n\\subsection{Effectiveness of MI estimation networks}\n\\label{section4.2}\nIn \\cref{section3.2.2}, we point out that training the MI estimation network directly by MI maximization may not clearly reflect the difference between the adversarial sample and the natural sample in natural MI and adversarial MI. We thus design an optimization mechanism for training the MI estimation network. To demonstrate the effectiveness of the optimization mechanism, we compare the performance of the estimation networks trained by \\cref{eq4} and \\cref{eq5} in \\cref{fig4}. The performances of the defenses based on the two different estimators are shown in \\cref{appendix_5_1}\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.5 in]{fig4.pdf}}\n\\caption{The performances of MI estimation networks trained by \\cref{eq4} (MIM) and \\cref{eq5} (Our). The left half is the estimated natural MI, and the right half is the adversarial MI.}\n\\label{fig4}\n\\vskip -0.35in\n\\end{center}\n\\end{figure}\n\n\\begin{table*}[hbtp]\n\\caption{Adversarial accuracy (percentage) of defense methods against white-box attacks on \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}. The target model is ResNet-18.}\n\\label{tab1}\n\\renewcommand\\tabcolsep{6pt}\n\\renewcommand\\arraystretch{1.1}\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{l|l|ccccc|cccc}\n\\hline\n\\multirow{2}{*}{Dataset} &\\multirow{2}{*}{Defense}  &\\multicolumn{5}{c|}{$L_{\\infty}$-norm} &\\multicolumn{4}{c}{$L_{2}$-norm} \\\\\n& & None & PGD-40 & AA & FWA-40 &TI-DIM & None &PGD-40& CW & DDN \\\\ \\hline\n\\multirow{7}{*}{CIFAR-10} &Standard & 83.39 & 42.38 & 39.01 & 15.44 &55.63 &83.97 &61.69 & 30.96 &29.34 \\\\\n&WMIM & 80.32 & 40.76 & 36.05 & 12.14 &53.10 &81.29 &58.36 & 28.41 &27.13 \\\\\n&NAMID & \\textbf{83.41} & \\textbf{44.79} & \\textbf{39.26} & \\textbf{15.67} &\\textbf{58.23} &\\textbf{84.35} &\\textbf{62.38} & \\textbf{34.48} &\\textbf{32.41}\\\\ \\cdashline{2-11}[3pt\/5pt]\n&TRADES & \\textbf{80.70} & 46.29 & 42.71 & 20.54 &57.06 &83.72 &63.17 & 33.81 &32.06 \\\\\n&NAMID\\_T &80.67  &\\textbf{47.53}  &\\textbf{43.39}  &\\textbf{21.17} &\\textbf{59.13} &\\textbf{84.19} &\\textbf{64.75} &\\textbf{35.41} &\\textbf{34.27} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&MART & 78.21 & 50.23 & 43.96 & 25.56 &58.62  &83.36 &65.38 & 35.57 &34.69  \\\\\n&NAMID\\_M &\\textbf{78.38} &\\textbf{51.69}  &\\textbf{44.42}  &\\textbf{25.64} &\\textbf{61.26} &\\textbf{84.07} &\\textbf{66.03} &\\textbf{36.19} &\\textbf{35.76} \\\\ \\hline\n\\multirow{7}{*}{Tiny-ImageNet} &Standard & 48.40 & 17.35 & 11.27 & 10.29 &27.84 &49.57 &26.19 &12.73 &11.25 \\\\\n&WMIM & 47.43 & 16.50 & 9.87 & 9.25 & 25.19 &48.16 &24.10 &11.35 &10.16  \\\\\n&NAMID & \\textbf{48.41} & \\textbf{18.67} & \\textbf{12.29} & \\textbf{11.32} &\\textbf{29.37}  &\\textbf{49.65} &\\textbf{28.13} & \\textbf{14.29} &\\textbf{12.57} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&TRADES & \\textbf{48.25} & 19.17 & 12.36 & 10.67 &29.64  & 48.83 & 27.16 & 13.28 &12.34 \\\\\n&NAMID\\_T &48.21  &\\textbf{20.12}  &\\textbf{12.86}  &\\textbf{14.91} &\\textbf{30.81}  &\\textbf{49.07} &\\textbf{28.83} &\\textbf{14.47} &\\textbf{13.91} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&MART & \\textbf{47.83} & 20.90 & 15.57 & 12.95 &30.71 &48.56 &27.98 & 14.36 &13.79  \\\\\n&NAMID\\_M &47.80  &\\textbf{21.23}  &\\textbf{15.83}  &\\textbf{15.09} & \\textbf{31.59} &\\textbf{48.72} &\\textbf{29.14} &\\textbf{15.06} &\\textbf{14.23} \\\\ \\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\vskip -0.15in\n\\end{table*}\n\nWe use the test data from \\textit{CIFAR-10} to evaluate the performance. For the natural MI, we offset the estimated MI so that the worst-case of the natural MI equals 0, and calculate the average of all samples. Similarly, we offset the estimated MI so that the worst-case of the adversarial MI equals 0. Note that for a fair comparison, we use selected samples to train the estimation networks for both methods. As shown in \\cref{fig4}, the results demonstrate that the optimization mechanism could help adequately represent the inherent difference in the natural MI and the adversarial MI between the natural sample and the adversarial sample.\n\n\\subsection{Robustness evaluation and analysis}\n\\label{section4.3}\n\nTo demonstrate the effectiveness of our adversarial defense algorithm, we evaluate the adversarial accuracy using white-box and black-box adversarial attacks, respectively.\n\n\\noindent\\textbf{White-box attacks.} \nIn the white-box setting, all attacks can access the architectures and parameters of target models. We evaluate the robustness by exploiting six types of adversarial attacks for both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}: $L_{\\infty}$-norm PGD, FWA, AA, TI-DIM attacks and $L_{2}$-norm PGD, DDN, CW attacks. The average natural accuracy (i.e., the results in the third column) and the average adversarial accuracy of defenses are shown in \\cref{tab1}.\n\nThe results show that our method (i.e., NAMID) can achieve better robustness compared with \\textit{Standard AT}. The performance of our method on the natural accuracy is competitive (83.39\\% vs. 83.41\\%), and it provides more gains on adversarial accuracy (e.g., 5.69\\% against PGD-40). Compared with \\textit{WMIM}, the results show that our proposed strategy of disentangling the standard MI into the natural MI and the adversarial MI is effective. The standard deviation is shown in \\cref{appendix_5_2}.\n\nNote that the default adversarial training loss in our method (i.e., $\\mathcal{L}_{adv}$ in \\cref{eq9}) is the same as \\textit{Standard AT}. To avoid the bias caused by different adversarial training methods, we apply the adversarial training losses of \\textit{TRADES} and \\textit{MART} to our method respectively (i.e., NAMID\\_T and NAMID\\_M). As shown in \\cref{tab1}, the results show that our method can improve the adversarial accuracy (e.g., the accuracy against PGD is improved by 2.68\\% and 2.91\\% compared with \\textit{TRADES} and \\textit{MART} on \\textit{CIFAR-10}).\n\n\\noindent\\textbf{Black-box attacks.}\nBlock-box adversarial instances are crafted by attacking a surrogate model. We use a VggNet-19 \\cite{he2016deep} as the surrogate model. The surrogate models and the defense models are trained separately. We use \\textit{Standard AT} method to train the surrogate model and use PGD, AA and FWA to generate adversarial test data. The performances of our defense method is reported in \\cref{tab2}. The results show that our method is a practical strategy for real scenarios, which can protect the target model from black-box attacks by malicious adversaries.\n\n\\begin{table}[t]\n\\caption{Adversarial accuracy (percentage) of defense methods against black-box attacks on \\textit{CIFAR-10}. The target model is ResNet-18 and the surrogate model is adversarially trained VggNet-19. We show the most successful defense with \\textbf{bold}.}\n\\label{tab2}\n\\renewcommand\\tabcolsep{8pt}\n\\renewcommand\\arraystretch{1.05}\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{l|cccc}\n\\hline\nDefense  & None & PGD-40 & AA & FWA-40 \\\\ \\hline\nStandard & 83.39    & 65.88    & 60.93   & 56.42       \\\\\nWMIM     & 80.32     & 62.79       & 57.86   &  53.05      \\\\\nNAMID      & \\textbf{83.41}     & \\textbf{69.57}       & \\textbf{63.72}   & \\textbf{59.30}       \\\\ \\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\vskip -0.25in\n\\end{table}\n\n\n\\subsection{Ablation study}\n\\label{section4.4}\nTo clearly elucidate the role of each component of our method in improving adversarial robustness, we conduct ablation studies in three different settings: (i) removing the adversarial MI; (ii) removing the natural MI; and (iii) setting the hyperparameter $\\lambda$ (in \\cref{eq8}) to 0. We use $L_{\\infty}$-norm PGD and FWA attacks to evaluate the performances of these variants. As shown in \\cref{fig5}, the results demonstrate that each component of our method contributes positively to improving adversarial accuracy.\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.5 in]{fig5.pdf}}\n\\caption{The ablation study.The bars with different colors represent the performance under different settings. Among them, 'Ori' denotes our method NAMID and 'Standard' denotes \\textit{Standard AT}.}\n\\label{fig5}\n\\vskip -0.35in\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{section5}\nTo the best of our knowledge, the dependence between the output of the target model and input adversarial samples have not been well studied. In this paper, we investigate the dependence from the perspective of information theory. Considering that adversarial samples contain natural and adversarial patterns, we propose to disentangle the standard MI into the natural MI and the adversarial MI to explicitly measure the dependence of the output on the different patterns. We design a neural network-based method to train two MI estimation networks to estimate the natural MI and the adversarial MI. Based on the above MI estimation, we develop an adversarial defense algorithm called natural-adversarial mutual information-based defense (NAMID) to enhance the adversarial robustness. The empirical results demonstrate that our defense method can provide effective protection against multiple adversarial attacks. Our work provides a new adversarial defense strategy for the community of adversarial learning. In future, we will design more efficient mechanisms for training MI estimators and further optimize the natural-adversarial MI-based defense to improve the performance against stronger attacks. In addition, \n\n\\section{Acknowledgements}\nThis work was supported in part by the National Key Research and Development Program of China under Grant 2018AAA0103202, in part by the National Natural Science Foundation of China under Grant 61922066, 61876142, 62036007, 62006202, 61922066, 61876142, 62036007, and 62002090, in part by the Technology Innovation Leading Program of Shaanxi under Grant 2022QFY01-15, in part by Open Research Projects of Zhejiang Lab under Grant 2021KG0AB01, in part by the RGC Early Career Scheme No. 22200720, in part by Guangdong Basic and Applied Basic Research Foundation No. 2022A1515011652, in part by Australian Research Council Projects DE-190101473, IC-190100031, and DP-220102121, in part by the Fundamental Research Funds for the Central Universities, and in part by the Innovation Fund of Xidian University. The authors thank the reviewers and the meta-reviewer for their helpful and constructive comments on this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a neutrinoless double-beta ($0\\nu\\beta\\beta$) decay an atomic nucleus decays into another one with two more protons and two fewer neutrons, emitting two electrons. In other words, two leptons are created. Such violation of the lepton number conservation is only possible if neutrinos---unlike any other fundamental particle---are its own antiparticle, a possibility first suggested by Ettore Majorana in the 1930's. In spite of the challenges associated with the detection of a process that involves new physics, $0\\nu\\beta\\beta$ decay is being pursued by several experimental collaborations~\\cite{KamLAND-Zen16,EXO18,GERDA18,MAJORANA18,CUORE18,NEXT18,CUPID18}. A major advantage is that the parameter $m_{\\beta\\beta}$ that controls the $0\\nu\\beta\\beta$ decay half-life is fully fixed by the the known neutrino mass differences and mixing angles, in such a way that $m_{\\beta\\beta}$ only depends on the ordering ---``normal\" or ``inverted\"---of the neutrinos masses~\\cite{engelmen_review}:\n\\begin{equation}\n\\label{eq:half-life}\n[T^{0\\nu}_{1\/2}]^{-1}=G^{0\\nu}\n\\left|M^{0\\nu\\beta\\beta}\\right|^2  m_{\\beta\\beta}^2\\,.\n\\end{equation}\nThere is, however, a catch. The $0\\nu\\beta\\beta$ decay half-life also depends on the value of an associated nuclear matrix element (NME), $M^{0\\nu\\beta\\beta}$, like any other {\\it nuclear} decay---$G^{0\\nu}$ is a known phase-space factor. NMEs need to be calculated theoretically, and their value is key to assess the prospects to observe $0\\nu\\beta\\beta$ decay in present and next-generation experiments.\n\nAt present, predicted NME values vary by a factor two or three depending on the many-body method used to calculate them. In addition, the results may need to be ``quenched\" as is common for $\\beta$ decays, but since the momentum transfer in $0\\nu\\beta\\beta$ decay is much larger, the necessity of such ``quenching\" is unclear~\\cite{engelmen_review}. These proceedings discuss recent ideas towards a more reliable determination of the \nNMEs, with focus on improved many-body calculations, and on the relation between $0\\nu\\beta\\beta$ decay and double Gamow-Teller (GT) transitions.\n\n\n\\section{Shell model nuclear matrix elements in two harmonic oscillator shells}\n\nAmong the nuclear many-body methods used to study $0\\nu\\beta\\beta$ decay, the nuclear shell model plays a prominent role, as one of the most successful approaches to nuclear structure~\\cite{cau05}. Nonetheless the main drawback of shell-model NMEs is that they are typically calculated limiting the configuration space to one harmonic oscillator shell. While, in general, such restriction works very well to describe the nuclear structure and spectroscopy of stable nuclei, it has been claimed that such a configuration space may not be large enough to obtain converged $0\\nu\\beta\\beta$ decay NMEs~\\cite{vogel12}.\n\nThe lightest $\\beta\\beta$ emitter is $^{48}$Ca. This is therefore the nucleus for which shell model calculations beyond one harmonic oscillator shell are  less demanding computationally. Ref.~\\cite{iwata16} calculated the NME for the $0\\nu\\beta\\beta$ decay in a configuration space consisting of two harmonic oscillator shells, the $sd$ and $pf$ shells. Previous shell model calculations were restricted to the $pf$ shell, while Ref.~\\cite{iwata16} was able to include up to $2\\hbar\\omega$ $sd$-$pf$ excitations. The calculation was validated by reproducing the excitation spectra of the initial and final nuclei of the decay, $^{48}$Ca and $^{48}$Ti~\\cite{iwata16}. In addition, the shell model calculation of Ref.~\\cite{iwata16} showed a good description of the GT strength, including the GT giant resonance (GR), of $^{48}$Ca and $^{48}$Ti into $^{48}$Sc~\\cite{iwata15}---these GT strengths had been measured in charge-exchange experiments~\\cite{yako-ca48}---, and reproduced the two-neutrino $\\beta\\beta$ decay matrix element of $^{48}$Ca as well. For the transition operators, the agreement to experiment was only possible after a ``renormalization\", or ``quenching\", of the theoretical predictions by a factor $q=0.71$ for each spin-isospin ${\\bm \\sigma}\\tau$ term present in the corresponding operator. This is, once for the GT strength and twice for two-neutrino $\\beta\\beta$ decay matrix element.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=.82\\textwidth]{nme_shells.pdf}\n\\end{center}\t\n\\caption{\\label{fig:ca48_nme}\n$^{48}\\textrm{Ca}$ and $^{76}$Ge $0\\nu\\beta\\beta$ decay\nNME calculated in a one-shell (shaded) and two-shell (solid) configuration space. One-shell~\\cite{menendez09} and $^{48}\\textrm{Ca}$~\\cite{iwata16} calculations are performed with the full shell model, while the $^{76}$Ge two shell calculation~\\cite{jiao17} use the approximate generator-coordinate method.}\n\\end{figure}\n\nThe result of Ref.~\\cite{iwata16} is shown in Fig.~\\ref{fig:ca48_nme}. Note that the NME does not include a possible ``renormalization\" of the NMEs, even though such ``renormalization\" is required to reproduce the $^{48}$Ca two-neutrino $\\beta\\beta$ decay half-life. The main conclusion found in Ref.~\\cite{iwata16} is that, in spite of performing the calculation in a significantly larger configuration space, the $^{48}$Ca NMEs was enhanced by only $\\sim30\\%$ in the two-shell calculation compared to the one-shell one. The main reason for such a relatively small effect is the competition between two type of contributions: on the one hand, pair-type two-particle--two-hole excitations into the additional harmonic oscillator shell tend to enhance the value of the NMEs~\\cite{caurier08}; on the other hand, one-particle--one-hole type excitations---which are generally related to two decaying nucleons coupled to angular momentum $J>0$---tend to reduce the value of the NME~\\cite{caurier08}. Overall, the competition between these two kinds of contributions results in a moderate enhancement of the NME in the expanded configuration space. Since the competition is expected to be general, similar effects are expected for extending the configuration space of shell model NME calculations in heavier $\\beta\\beta$ emitters.\n\nThis expectation is consistent with the recent result of Ref.~\\cite{jiao17}, which calculated the nuclear matrix element of $^{76}$Ge in a configuration space consisting of two harmonic oscillator shells. \nThis calculation is based on the generator-coordinated method, which does not include all the many-body correlations present in the shell model, because a full shell model diagonalization of the two-shell configuration space is beyond present computing capabilities. Fig.~\\ref{fig:ca48_nme} shows the result of Ref.~\\cite{jiao17} in comparison with a shell model calculation in one oscillator shell. Similarly to the findings in $^{48}$Ca, the impact of increasing the size of the configuration space is small, with the NME even slightly reducing its value in the larger space.\n\n\n\n\\section{$\\beta\\beta$ decay and double Gamow-Teller transitions}\n\nIn the absence of a $0\\nu\\beta\\beta$ detection, theoretical calculations of the NMEs have to be tested against different nuclear structure data. First, all calculations compare their predictions to the nuclear structure of the initial and final states of the decay. In addition, an obvious observable to test calculations is the two-neutrino $\\beta\\beta$ decay, which shares initial and final states with $0\\nu\\beta\\beta$ decay and has similar spin-isospin structure. However, the momentum transfers in the two $\\beta\\beta$ modes are very different: while in the two-neutrino case the momentum transfer is limited by the $Q$-value---a couple of MeV---in the neutrinoless case momentum is transferred via the not-to-be-emitted virtual neutrinos. A test of the relevant momentum-transfer regime---about $q\\sim100$~MeV---would involve a comparison to muon capture or neutrino scattering. Unfortunately, data on these observables is limited.\nGT transition strengths measured in charge-exchange experiments are also typically used to test calculations. Described by the same operator as GT $\\beta$ decays, GT transitions are not limited by the $Q$-value, and can be studied to energies even past the GT GR at $E\\sim10-15$~MeV.\n\nA closer connection to $0\\nu\\beta\\beta$ decay can be expected to come from double GT (DGT) transitions that are being looked for in double charge-exchange experiments~\\cite{takaki-aris,uesaka-nppac,cappuzzello,takahisa-17}. The operator structure $0\\nu\\beta\\beta$ decay of DGT transitions is very similar, with the corresponding matrix elements given by\n\\begin{eqnarray}\n&M^{0\\nu\\beta\\beta}(i\\rightarrow f)&\n=M_{GT}^{0\\nu}+\\frac{M_F^{0\\nu}}{g_A^2}+M_T^{0\\nu}= \\sum_{X=GT,F,T}\n\\bra{f} \\sum_{a,b} H_X(r_{ab})\\,S_X\\, \\tau^+_a \\tau^+_b \\ket{i}\\,,\\\\\n\\label{eq:2GT}\n&M^{DGT}(i\\rightarrow f)&\n=\\bra{f} \\sum_{a,b}\n[{\\bm \\sigma}_a \\tau^+_a \\times {\\bm \\sigma}_b \\tau^+_b]^{\\lambda} \\ket{i}\\,,\n\\end{eqnarray}\nwhere $\\bm \\sigma$, $\\tau$ denote spin and isospin, respectively, and $g_A$ is the axial coupling. The labels $F$ and $T$ stand for the subleading Fermi and tensor parts of the $0\\nu\\beta\\beta$ NME, much smaller---less than 20\\%---than the dominant GT piece associated to the spin structure $S_{GT}={\\bm\\sigma}_1\\cdot{\\bm\\sigma}_2$. Therefore besides the small effect of the $F$ and $T$ terms, for DGT transitions to the ground state of the final nucleus---where the DGT operator can only couple to $\\lambda=0$---the $0\\nu\\beta\\beta$ and DGT operators only differ by the presence of the neutrino potential $H$, which depends on the internucleon distance $r_{ab}$. The form of the neutrino potentials is given in detail in Ref.~\\cite{menendez_heavy}.\n\nReference~\\cite{2GT0nbb} predicted the DGT strength of $^{48}$Ca, including the DGT GR using large-scale shell model calculations up to two oscillator shells. Interestingly, the energy of the resonance was found to be correlated---in the shell model calculation---to the value of the $0\\nu\\beta\\beta$ decay NME. This relation is due to the dependence of the two observables to particle-like pairing correlations. As a consequence, a measurement of the DGT GR in $^{48}$Ca could provide an indication of the value of the NME of the same nucleus.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{nme_linear_ge-xe.pdf}\t\n\t\\end{center}\t\n\t\\caption{\\label{fig:nme_linear}\t\n\t\tCorrelation between\n\t\t$0\\nu\\beta\\beta$ decay NMEs\n\t\t$M^{0\\nu\\beta\\beta}(0^+_{gs,i} \\rightarrow 0^+_{gs,f})$\n\t\tand the DGT matrix elements\n\t\t$M^{\\rm DGT}(0^+_{gs,i} \\rightarrow 0^+_{gs,f})$.\n\t\tShell model results for germanium, tellurium, tin, tellurium, and xenon isotopes (black) including the $\\beta\\beta$ emitters $^{76}$Ge, $^{82}$Se, $^{124}$Sn, $^{130}$Te and $^{136}$Xe (blue) are compared to EDF theory~\\cite{rodriguez} (green) and QRPA predictions~\\cite{simkovic-11} (open red symbols).\n\t\tThe calculations use several shell model interactions for each isotope~\\cite{menendez09,JUN45,qi-12}.  Adapted from Ref.~\\cite{2GT0nbb}.\n\t}\n\\end{figure}\n\nIn addition, Ref.~\\cite{2GT0nbb} studied DGT transitions to the ground state, and compared the results to the $0\\nu\\beta\\beta$ decay NMEs. Note that the initial and final states of both processes are the same, and also the transition operator is very similar, as discussed above. Instead of limiting to one particular case---as in the study of the DGT GR---the calculations included a set of nuclei ranging from calcium to xenon isotopes, with nuclear mass number $42\\leq A\\leq 136$. Therefore several $\\beta\\beta$ emitters but also many isotopes not relevant for $0\\nu\\beta\\beta$ decay searches were studied. Nonetheless, these additional calculations are very useful to illuminate systematic effects.\n\nFigure~\\ref{fig:nme_linear} summarizes the results of Ref.~\\cite{2GT0nbb}. A good linear correlation is found between the DGT transitions to the ground state and $0\\nu\\beta\\beta$ NMEs. The linear correlation does not depend on the details of the shell-model interaction used, or in the correlations included in the shell-model initial and final states---as long as particle-like pairing correlations are present.  Furthermore, the correlation between $0\\nu\\beta\\beta$ and DGT matrix elements is valid for $\\beta\\beta$ emitters, shown in blue in Fig.~\\ref{fig:nme_linear}, and for all the other nuclei---seventeen isotopes in total. Moreover, the correlation observed in the shell model is consistent with the results of energy-density functional (EDF) theory~\\cite{rodriguez}---also shown in Fig.~\\ref{fig:nme_linear}---even if for the latter approach $0\\nu\\beta\\beta$ and DGT matrix elements are much larger than the shell model ones. In contrast, quasiparticle random-phase approximation (QRPA) results~\\cite{simkovic-11}---shown in Fig.~\\ref{fig:nme_linear} as well---do not support the linear correlation found for the shell model.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=.95\\textwidth]{density_se_te_DGT.pdf}\t\n\t\\end{center}\t\n\t\\caption{\\label{fig:radial_density}\t\n$^{82}$Se (left panel) and $^{130}$Te (right) normalized radial density distributions $C(r)$ of the GT $0\\nu\\beta\\beta$ (red) and DGT (orange) matrix elements. Shell model interactions from Ref.~\\cite{menendez09} are used.\n\t}\n\\end{figure}\n\nThe linear correlation shown in Fig.~\\ref{fig:nme_linear} relates the $0\\nu\\beta\\beta$ decay NME, driven by the weak interaction, and the DGT matrix element, a result of the strong interaction. It therefore opens the door to exploring $0\\nu\\beta\\beta$ decay NMEs in nuclear double charge-exchange experiments~\\cite{takaki-aris,uesaka-nppac,cappuzzello,takahisa-17}. This is, however, a formidable challenge at the experimental and theoretical level. First, the DGT transition is a tiny---0.03 per mil---piece of the DGT sum rule. In addition, dedicated reaction theory efforts are needed to establish the relation between double charge-exchange cross-sections and DGT matrix elements.\n\nWhat is the origin of the linear correlation between $0\\nu\\beta\\beta$ decay and DGT transitions? To address this question, Fig.~\\ref{fig:radial_density} shows the normalized radial densities of the $0\\nu\\beta\\beta$ and DGT matrix elements, defined as\n\\begin{eqnarray}\nC_{GT}^{0\\nu}(r)= \\langle f | \\sum_{ab} \\delta(r-r_{ab}) \\,H_{GT}(r_{ab})\\,\n{\\bm \\sigma}_a\\cdot{\\bm \\sigma}_b \\,\\tau_a \\tau_b | i \\rangle \/ M_{GT}^{0\\nu}\\,, \\\\\nC^{DGT}(r)= \\langle f | \\sum_{ab} \\delta(r-r_{ab}) \\,\n[{\\bm \\sigma}_a\\times{\\bm \\sigma}_b]^0 \\,\\tau_a \\tau_b | i \\rangle \/ M^{DGT}\\,.\n\\label{eq:density_r}\n\\end{eqnarray}\nFigure~\\ref{fig:radial_density} shows that the two matrix elements are dominated by the contribution of nucleons that are relatively close to each other, $r_{ab}\\lesssim3$~fm. In the case of DGT transitions this is the result of the partial cancellation of the longer-range contributions. This short-range dominance is non trivial, as Fig.~\\ref{fig:radial_density} shows that the shell model calculation naturally probes internucleon distances up to twice the nuclear radius.\n\nThe short-range character provides an explanation for the existence of the linear correlation between the two matrix elements. The work of Bogner et al.~\\cite{bogner-10,bogner-12} shows that when an operator probes only the short-range physics of low-energy states, the corresponding matrix elements factorize into a universal operator-dependent constant times a state-dependent number which is common to all short-range operators. Since both $0\\nu\\beta\\beta$ decay and DGT shell-model matrix elements fulfill these conditions, a linear relation between them is predicted. In contrast, the QRPA DGT matrix elements receive contributions from longer range, so that the correlation is not predicted in their case, in agreement with Fig.~\\ref{fig:nme_linear}.\n\n\n\\section{Conclusions}\nWe have summarized two advances that improve our understanding of $0\\nu\\beta\\beta$ decay. On the one hand, shell model calculations in a configuration space comprising two oscillator shells suggest that the NME obtained in standard shell model calculations are reasonably converged. On the other hand, the finding of a good linear correlation between the NMEs and DGT transitions, valid across the nuclear chart, brings the opportunity to obtain precious information on $0\\nu\\beta\\beta$ decay in double charge-exchange nuclear reactions. These advances pave the way towards a more reliable determination of the $0\\nu\\beta\\beta$ NMEs in the mid-term future.\n\n\n\\section*{Acknowledgments}\nI would especially like to thank Prof. T. Otsuka for many stimulating discussions and for his support, as well as for his kind introduction to research in Tokyo and Japanese culture. I am grateful to my co-authors T. Abe, M. Honma, Y. Iwata, T. Otsuka, N. Shimizu, Y.~Utsuno, and K. Yako for using in these proceedings results of our common research.\nThis work was supported by the\nCNS-RIKEN joint project for large-scale nuclear structure calculations, and by MEXT and JICFuS as a priority issue \n(Elucidation of the fundamental laws and evolution of the universe, hp170230) \nto be tackled by using Post K Computer.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the early days of the holographic correspondence \\cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}, $(2+1)$-dimensional gravitational systems have played a central role in testing and exploring the ideas behind the duality. In fact, with the benefit of hindsight, one can see that the work of Brown and Henneaux on the asymptotic symmetries of three-dimensional spacetimes with a negative cosmological constant \\cite{Brown:1986nw} displayed some basic features of the correspondence as early as a decade before it was proposed. The 3$d$ gravitational systems of interest in the framework of holography are special in that their field theory duals enjoy an infinite-dimensional (local) conformal symmetry; via the powerful techniques of conformal field theory (CFT), one then has a better grasp of the boundary theory structure which is often lacking in higher-dimensional examples. A beautiful example of this fact is the precise connection between the CFT spectrum and retarded Green's functions, and black hole quasinormal modes in the bulk \\cite{Birmingham:2001pj}.\n\nThe feature that makes the gauge\/gravity correspondence outstanding is that it postulates the equivalence between gravitational weakly-coupled degrees of freedom propagating in the bulk spacetime, and strongly-coupled degrees of freedom in a dual quantum field theory in one less dimension (``the boundary\"). A pivotal ingredient in the proposal is the dilatation symmetry of the boundary CFT; broadly speaking, using this symmetry one can relate the masses of the bulk fields to the conformal dimensions of operators in the quantum theory on the boundary, as first established in \\cite{Witten:1998qj}. However, for a given bulk field, the spectrum of conformal dimensions in the dual quantum field theory is not entirely determined by the masses of the fields. This is intimately related to the fact that the boundary conditions that yield well-defined dynamics are not unique. In fact, in the so-called ``bottom-up\" holography where the bulk theory is phenomenologically devised, the operator content of the possible dual theories is completely specified only after the boundary conditions for bulk fields have been properly chosen.\n\nIn the present article, we will focus on the study of the Abelian Maxwell-Chern-Simons (MCS) theory, frequently referred to as ``Topologically Massive Electrodynamics\" \\cite{Deser:1981wh,Deser:1982vy}, in three-dimensional asymptotically-AdS spacetimes. The emphasis will be on determining a set of ``admissible\" boundary conditions, in a sense that will be made precise below; this is crucial to the dictionary problem in the context of the AdS\/CFT correspondence, as discussed above. One of the motivations to study Chern-Simons terms in the bulk is that these arise naturally in the context of string theory compactifications, and endow the bulk black hole solutions with $U(1)$ charge (see \\cite{Kraus:2006nb,Kraus:2006wn}, for example). It is worth mentioning, however, that the MCS system plays a central role in condensed matter physics as well, in particular in the study of fermionic systems in two spatial dimensions, where it describes the low-energy effective theory of the Fractional Quantum Hall Effect (FQHE). Furthermore, even in flat space the MCS theory is often said to be holographic, albeit in a different sense from the above: in the topological limit (where the bulk quasiparticles become infinitely massive), the degrees of freedom are effectively localized on the boundary \\cite{Wen:1991mw}.\\footnote{The key difference being that the AdS\/CFT correspondence is an \\textit{equivalence} between bulk and boundary degrees of freedom, while in the topological theory the bulk degrees of freedom are gapped, and the low-energy excitations propagate exclusively on the boundary.} From a mathematical point of view, this is the well-known correspondence between three-dimensional Chern-Simons gauge theory and a chiral rational CFT \\cite{Witten:1988hf, Moore:1989yh, Elitzur:1989nr,Balachandran:1991dw, Bos:1989kn, Bos:1989wa,Schwarz:1979ae}. More recently, the latter correspondence has been refined to reconcile the modular transformation properties of the string theory partition function on $AdS_3$ and those of the Chern-Simons theory which dominates its infrared dynamics \\cite{Gukov:2004id}, and the potential relevance of these observations for condensed matter physics was also pointed out. This provides yet another motivation to carefully study the holographic dictionary of the full (finite coupling) MCS theory; here we will do so from a bottom-up perspective, in the hope that our results could be useful in the study of other models which might be interesting for applications of holography to condensed matter physics.\n\nOur analysis starts by determining a broad set of boundary conditions under which the bulk theory is expected to have well-posed dynamics. We find it convenient to approach this problem using the covariant phase space formalism, along the lines of \\cite{Marolf:2006nd,Amsel:2006uf,Amsel:2008iz}. Within this framework, the first requirement on the boundary conditions is that they lead to a conserved symplectic structure (in the sense of timelike evolution). In the context of holography, this condition can be conveniently rephrased as the vanishing of the symplectic flux on the (radial) boundary. Roughly speaking, the bulk gauge field splits into a ``massive\", gauge-invariant piece, and the flat connections. Accordingly, the boundary theory operators organize themselves into two sectors: a vector operator dual to the massive part of the connection, and the well-known $U(1)$ chiral currents (which also arise in the pure Chern-Simons theory). We will obtain a variety of boundary conditions that correspond to double-trace deformations from the dual field theory perspective. In particular, we shall note the possibility of coupling the vector operator and the chiral currents via this mechanism. To our knowledge, these ``hybrid\" boundary conditions intertwining the massive and topological sectors have not been discussed in the literature; their existence was anticipated in \\cite{Gukov:2004id}, however, where the topology of the spacetime manifold was chosen in such a way that the two sectors effectively decouple.\n\nIt is worth emphasizing that all of this physics occurs at finite Maxwell coupling. It is often argued in the literature that the Maxwell coupling should be irrelevant in the infrared; this is certainly true from the bulk perspective. However, the Maxwell coupling is not irrelevant in the UV, and so is an important parameter holographically. One also notes in parallel that in condensed matter systems such as quantum Hall, the Maxwell coupling sets the cut-off scale where quasi-particle excitations live, going away only in the topological limit. It seems quite plausible that such excitations will exist in the holographic theory as well, a subject that we will explore elsewhere.\n\nHaving obtained the class of boundary conditions that lead to a conserved symplectic structure, one can examine in detail which of these are consistent with unitarity. Our motivation to consider this restriction comes primarily from the existence of the unitarity bound in conformal field theories (see \\cite{Mack:1975je,Minwalla:1997ka}, for example), which dictates that the presence of operators whose dimension is ``too low\" leads to negative norm states (ghosts). Via the holographic correspondence, this fact should manifest in the bulk physics as well, which is the question we address. As first noted in \\cite{Balasubramanian:1998sn,Witten:2001ua}, a closely analogous concern arises when considering bulk scalar fields with sufficiently high masses if one imposes boundary conditions that allow the slow-decaying branch to fluctuate. As a result of this choice, the conformal dimension of the dual operator lies below the unitarity bound and one expects the bulk theories to be ill-defined. Recently, these setups were considered in \\cite{Andrade:2011dg}, which confirmed that such bulk theories are indeed pathological and that, generically, they suffer from ghosts.\n\n We will address the question of unitarity by studying the dynamics of the MCS system in $AdS_{3}$ in both global and Poincar\\'e coordinates; in particular, we will discuss the resulting spectrum and symplectic products for the various boundary conditions for which the symplectic structure is conserved. Our main result is that the only boundary conditions consistent with unitarity do not mix the massive and topological sectors, and in particular they require to hold fixed the slower fall-off of the massive mode (i.e. they are of Dirichlet type). In short, the class of permissible boundary conditions is severely restricted by unitarity considerations. Interestingly, we will also find additional ghosts in the flat sector whose presence cannot be linked to unitarity bounds in an obvious way. We will also include an analysis of the symmetries we expect to be present in the dual theory as a result of various choices of boundary conditions.\n\nThe three-dimensional MCS theory has been previously considered in the context of AdS\/CFT. We refer the reader to \\cite{Gukov:2004id,Kraus:2006wn, Kraus:2006nb,Jensen:2010em} for work which focuses on the flat (topological) sector of the theory. The massive sector has also received some attention and the holographic dictionary problem has been studied to some extent \\cite{Minces:1999tp,DHoker:2010hr,Yee:2011yn}. Our results agree with the references above as far as the operator content is concerned. The novelty of our analysis lies in the fact that we have considered a wider class of boundary conditions, including ``hybrid\" boundary conditions that mix the massive and topological sectors, and analyzed their consistency with unitarity in detail. Additional related work includes \\cite{Carlip:2008jk,Fujita:2009kw,Balasubramanian:2010sc,Fosco:2011ra}.\n\nThis paper is organized as follows. In section \\ref{sec: prelim} we review the MCS system, the solution of the asymptotic equations of motion on asymptotically $AdS_3$ backgrounds, and the corresponding conformal dimensions of dual operators. In section \\ref{section:symplectic} we briefly describe the covariant phase space formalism, and use the conservation of the symplectic structure as a criterion to determine a wide class of \\textit{a priori} admissible boundary conditions in the holographic setup. For all the boundary conditions of interest, we construct the appropriate action principles and compute the one-point functions of the dual operators holographically. We also review the notion of symplectic product, which will play a central role in the analysis of unitarity. In section \\ref{section:spectrum} we discuss the spectrum of excitations in the dual field theory for the class of boundary conditions previously found, and discuss the normalizability of the various bulk modes. In section \\ref{section:norms} we present the calculation of the symplectic product for the various normalizable modes, focusing on the existence of ghosts; the requirement of unitarity in the dual theory then leads to a restricted class of permissible boundary conditions, which constitutes our main result. We conclude in section \\ref{section:discussion} with a discussion of our findings, along with possible extensions and applications. Some useful results used in the body of the paper have been collected in the appendices, as well as a brief discussion of the $U(1)$ symmetries in the dual field theory for the different boundary conditions under consideration.\n\n\\section{The Maxwell-Chern-Simons system}\\label{sec: prelim}\nWe consider the Maxwell-Chern-Simons (MCS) system in $(2+1)$ spacetime dimensions,\n\\begin{align}\\label{TME bulk action}\n I\n={}&\n- \\frac{1}{4\\gc^2}\\int_{M}d^{3}x\\,\\sqrt{|g|}F_{\\mu\\nu}F^{\\mu\\nu} -\\frac{\\hat{\\alpha}}{4} \\int_{M}d^{3}x\\, \\varepsilon^{\\mu\\nu\\rho}A_{\\mu}F_{\\nu\\rho}\n \\, ,\n\\end{align}\n\\noindent where $\\gc^2$ is the gauge coupling (with units of $[length^{-1}]$) and $\\hat{\\alpha}$ is the (dimensionless) Chern-Simons (CS) coupling. Throughout this paper we work in a fixed background in which we neglect the backreaction of the gauge field on the metric, {\\it i.e.} $G_N\/\\gc^2 \\to 0$, where  $G_N$ is the three dimensional gravitational coupling (which has units of $[length]$). Where appropriate, we will occasionally comment on issues of backreaction, and will consider them in a subsequent publication. The background metrics we consider are solutions of the Einstein equations in the presence of a negative cosmological constant $\\Lambda = -1\/L^2$, and the normalization is chosen such that pure $AdS_3$ space is a vacuum solution of the decoupled gravitational sector with radius $L$ and scalar curvature $R=-6\/L^2$. As usual, in a holographic context the action \\eqref{TME bulk action} must be supplemented by a collection of boundary terms that render the variational problem well defined and remove divergent contributions; these will be fully specified later on in the paper.\n\nThe equations of motion that follow from \\eqref{TME bulk action} are\\footnote{Our convention for the Levi-Civita tensor is $\\epsilon^{\\mu\\nu\\rho} = -\\frac{1}{\\sqrt{|g|}}\\varepsilon^{\\mu\\nu\\rho}$, where $\\varepsilon^{\\mu\\nu\\rho}$ is the Levi-Civita symbol.}\n\\begin{align}\n\\nabla_{\\nu}F^{\\nu\\mu} + \\frac{\\alpha}{2L}\\epsilon^{\\mu\\nu\\rho}F_{\\nu\\rho}=0\\, ,\\label{MaxCS equation}\n\\end{align}\n\\noindent where we have defined the rescaled CS coupling $\\alpha$ as\n\\begin{equation}\\label{rescaled CS coupling}\n\\alpha = \\gc^2 L\\hat{\\alpha}\\, ,\n\\end{equation}\n\\noindent which is also dimensionless. Without loss of generality, we will assume $\\alpha > 0$. When taking backreaction on the metric into account, asymptotically AdS solutions exist only for $\\alpha<1$, and we will restrict our discussions in the present paper to that range.\n\nIn form language, the Maxwell-CS equation \\eqref{MaxCS equation} can be written as\\footnote{On a $D$-dimensional spacetime, our convention for the Hodge dual is $*(dx^{\\nu_{1}}\\wedge\\cdots \\wedge dx^{\\nu_{r}}) = \\frac{1}{(D-r)!}\\epsilon^{\\nu_{1}\\ldots \\nu_{r}}_{\\phantom{\\nu_{1}\\ldots \\nu_{r}}\\mu_{1}\\ldots \\mu_{D-r}}dx^{\\mu_{1}}\\wedge\\cdots \\wedge dx^{\\mu_{D-r}}$.}\n\\begin{equation}\\label{Maxwell-CS eom}\nd^{\\dagger}F = \\frac{\\alpha}{L}*F\n\\end{equation}\n\\noindent where $d^\\dagger$ is the adjoint exterior derivative, which in our conventions acts on $F$ as $d^\\dagger F = -*d(*F) = -\\nabla_{\\mu}F^{\\mu}_{\\phantom{\\mu}\\nu}\\, dx^{\\nu}$. Hence, the equation of motion implies\n\\begin{equation}\\label{gauge field splitting}\n    A = A^{(0)} + B\\, ,\n\\end{equation}\n\\noindent  where $A^{(0)}$ is a flat connection and we have defined\n\\begin{equation}\\label{definition B}\nB \\equiv  -\\frac{L}{\\alpha}*F\\,.\n\\end{equation}\n\\noindent We note that $B$ is, by definition, invariant under the $U(1)$ gauge symmetry of the theory. In a later section we will study the consequences of the splitting \\eqref{gauge field splitting} at the level of the symplectic structure and the boundary conditions in a holographic context.\n\nSince $dB=dA=F$, the equation of motion \\eqref{Maxwell-CS eom} becomes a first order equation for $B$:\n\\begin{equation}\\label{equation of motion for B}\n*dB + \\frac{\\alpha}{L}B=0\\, ,\n\\end{equation}\n\\noindent which is the familiar equation for a massive vector field. In components, this equation reads\n\\begin{equation}\\label{first order equation for B}\n \\epsilon^{\\mu\\nu\\rho}\\partial_{\\nu}B_{\\rho} + \\frac{\\alpha}{L}B^{\\mu}=0\\, .\n\\end{equation}\n\\noindent Notice also that the definition \\eqref{definition B} implies a consistency condition:\n\\begin{equation}\nd^\\dagger B = 0\\, ,\n\\end{equation}\n\\noindent i.e. $B$ is a co-closed form ($\\nabla^{\\mu}B_{\\mu}=0$); naturally, this also follows from the equation of motion \\eqref{equation of motion for B}.  Acting on \\eqref{equation of motion for B} with $*d$ we can write a second-order equation for $B$,\n\\begin{equation}\n0 = d^{\\dagger}d B +\\frac{\\alpha^2}{L^2}B=\\Delta B + \\frac{\\alpha^2}{L^2}B\\, ,\n\\end{equation}\n\\noindent where $\\Delta = d^\\dagger d + d d^\\dagger$ is the Laplacian.\n\n\\subsection{Asymptotic solutions}\\label{subsection:asymptotic sols}\nFor the sake of concreteness,  we will write the metric of the asymptotically AdS spacetimes we are interested in  as\n\\begin{equation}\\label{asympt expansion of metric}\nds^{2} \\xrightarrow[r\\to \\infty]{} L^{2}\\frac{dr^{2}}{r^{2}} + \\frac{r^{2}}{L^{2}}g^{(0)}_{ij}(x)dx^{i}dx^{j} + \\ldots\n\\end{equation}\n\\noindent Restricting ourselves to flat connections which are finite at the conformal boundary, the asymptotic form of the solution for the gauge field is then of the form\\footnote{As we will see in appendix \\ref{section:sym bndy}, any finite $r$-dependent piece in the near-boundary behavior of the flat connection can be removed with the appropriate gauge transformation.}\n\\begin{equation}\\label{gen asympt}\nA(r,x) \\xrightarrow[r\\to \\infty]{} A^{(0)}(x) + r^{\\alpha}\\left(B^{(+)}(x) +\\mathcal{O}(r^{-2})\\right)+ r^{-\\alpha}\\left(B^{(-)}(x)+\\mathcal{O}(r^{-2})\\right)  \\,,\n\\end{equation}\nwhere $A^{(0)}$ is flat, i.e. $F^{(0)}=dA^{(0)}=0$. Similarly, solving the equations of motion asymptotically  one finds that the radial component $B_{r}$ of the gauge-invariant mode is subleading with respect to the $B_{i}$ components, which are moreover constrained by\n\\begin{equation}\\label{asymptotic constraint}\nP_{\\pm}^{ij}B_{j}^{\\left(\\mp\\right)} = 0\\, ,\\qquad \\mbox{where } \\quad  P_{\\pm}^{ij} = \\frac{1}{2}\\left(g^{(0) ij} \\pm  \\epsilon^{ij}\\right) .\n\\end{equation}\n\\noindent We have adopted the convention that $ \\epsilon^{ij} = -\\varepsilon^{ij}\/\\sqrt{|g^{(0)}|}$, where $\\varepsilon^{ij}$ is the two-dimensional Levi-Civita symbol, related to its three-dimensional counterpart by $\\varepsilon^{ij} = \\varepsilon^{rij}$. Notice that the projectors $P_{\\pm}^{ij}$ satisfy the usual properties: $\\left(P_{+}P_{-}\\right)^{ij} = P_{+}^{ik}g^{(0)}_{kl}P_{-}^{lj}=0$, $\\left(P_{\\pm}^{2}\\right)^{ij} = P_{\\pm}^{ik}g^{(0)}_{kl}P_{\\pm}^{lj} = P_{\\pm}^{ij}\\,$.\n\n\\subsection{Conformal dimensions}\\label{subsection:conf dims}\nGiven the asymptotic expansion \\eqref{gen asympt} and noting that the pullback to the boundary of the bulk vector field is simply a boundary vector, we conclude that the standard AdS\/CFT dictionary relates $B^{(+)}$ and $B^{(-)}$ with vector operators of dimensions $\\Delta_- = 1 - \\alpha$ and $\\Delta_+ = 1 + \\alpha$, respectively. On the other hand, the components of $A^{(0)}$ have scaling dimension one. As we shall review below, the components of $A^{(0)}$ along the boundary directions correspond to chiral currents that live on the boundary theory, \\cite{Witten:1988hf, Moore:1989yh, Elitzur:1989nr,Wen:1991mw,Balachandran:1991dw, Bos:1989kn, Bos:1989wa,Schwarz:1979ae}. We note that the lower scaling dimension is positive as long as $\\alpha < 1$, which implies that we can allow both fall-offs to fluctuate while preserving locally AdS asymptotics\\footnote{Here we use the terminology of \\cite{Skenderis:2002wp}, i.e., we mean that the curvature near the conformal boundary is that of AdS plus subleading corrections.} if $\\alpha < 1$. We have verified this statement explicitly by studying the effect of backreaction on a general asymptotically locally AdS metric of the form \\eqref{asympt expansion of metric}.\n\nIt should be noted that the operator of dimension $\\Delta_-$ violates the unitarity bound $\\Delta_V = 1$ for vector operators in two dimensions for all $\\alpha>0$\n\\cite{Mack:1975je,Minwalla:1997ka}, see also \\cite{ElShowk:2011gz} for the explicit expression. This suggests that boundary conditions that allow this degree of freedom to fluctuate should yield pathologies in the bulk; in subsequent sections we shall verify that this is indeed the case.\n\n\n\\section{Symplectic structure and boundary conditions}\\label{section:symplectic}\nIn the present section we study the issue of boundary conditions in the holographic description of the MCS system. We find it convenient to work within the covariant phase space formalism, which we will review shortly. The motivation for employing this formalism is two-fold: first, the classification of the allowed boundary conditions is nicely encoded in a simple vanishing-flux condition; and second, it allows us to classify the spectrum of excitations in a clean way. We emphasize however that this decision is just a matter of personal preference, and the results obtained within this framework should indeed be equivalent to the ones arrived at by more familiar, say canonical, methods.\n\nWe now proceed to briefly review the covariant phase space techniques; more detailed discussions can be found in \\cite{Lee:1990nz, Wald:1995yp,Wald:1999wa,Iyer:1994ys,Ashtekar:1990gc}. First, we stress that the construction is inherently Lorentzian, so we shall assume that the spacetime is endowed with a Lorentzian metric. Now, the ingredient that lies at the heart of this construction is the identification of the phase space with the space of solutions of the equations of motion which satisfy certain boundary conditions. This is possible since in any well-defined setup the specification of a point in canonical phase-space, i.e. of initial data, completely determines the subsequent evolution of the system. The other main ingredient is an algebraic structure that determines the dynamics once a Hamiltonian function is given, or crudely speaking, something that contains information about the Poisson brackets. This is nothing but the pre-symplectic structure of the theory, $\\Omega$, which can be thought of as a (possibly degenerate) two-form in the tangent space of (linearized) solutions. In other words, $\\Omega$ maps a pair of tangent vectors in the space of solutions to the real numbers. Given a background solution $\\bar{s}$ and two linearized solutions $\\delta_1 s$ and $\\delta_2 s$, we denote the symplectic product of $\\delta_1 s$ with $\\delta_2 s$ by $\\Omega(\\delta_1 s, \\delta_2 s; \\bar{s})$. Quite conveniently, this object can be constructed algorithmically given a Lagrangian \\cite{Lee:1990nz}, and we will illustrate this below.\n\nFrom the discussion above, it follows that the pre-symplectic structure must indeed be conserved in order for the identification of the initial data with the space of solutions to be independent of the surface on which the initial data is specified. This conservation condition is what we shall take as a guiding principle to classify the allowed boundary conditions for the MCS system. It is worth emphasizing here that the boundary conditions are in fact a crucial part of the definition of the phase space of a given theory. As pointed out above, the covariant phase space formalism also provides a useful way to classify the spectrum of excitations of the theory. In particular, we mention that in the presence of gauge symmetries the pre-symplectic structure is degenerate, the gauge orbits being precisely its null directions. Thus, we shall refer to any solution of the equations of motion whose symplectic product with an arbitrary solution vanishes as ``pure gauge\".\\footnote{We mention that the prefix ``pre\" makes reference to the degeneracy of $\\Omega$: by definition, a symplectic structure is non-degenerate. In a slight  abuse of notation we drop the prefix from now on, even when the kernel of $\\Omega$ is non-empty.} Further nomenclature will be discussed in section \\ref{sec: symp prod}.\n\nAfter taking the quotient by the gauge directions, the symplectic structure has a unique inverse and this corresponds to the Poisson bracket defined for gauge-invariant quantities. As discussed in detail in \\cite{Wald:1995yp}, this relation can be written succinctly as\n\\begin{equation}\\label{PB omega}\n    \\{ \\Omega(\\delta_1 s, \\cdot; \\bar{s}) , \\Omega(\\delta_2 s, \\cdot; \\bar{s}) \\}_{PB} = - \\Omega(\\delta_1 s, \\delta_2 s; \\bar{s}) \\, .\n\\end{equation}\n\\noindent Here $\\Omega(\\delta_1 s, \\cdot; \\bar{s})$ is to be understood as a linear function in covariant phase space. Then, the fact that the Poisson bracket and $\\Omega$ are the inverse of each other follows trivially by writing \\eqref{PB omega} in component notation. Finally, we mention that, at the classical level, one can construct conserved charges directly in terms of $\\Omega$. More precisely, given an infinitesimal transformation $\\delta_\\lambda s$ and an arbitrary linearized solution $\\delta s$, the infinitesimal variation of the generator $Q_\\lambda$ along $\\delta s$ is given by\n\\begin{equation}\\label{d Q}\n    \\delta Q_\\lambda  = \\Omega(\\delta_\\lambda s, \\delta s;\\bar{s}) \\, ,\n\\end{equation}\n\\noindent which, once again, is most easily visualized by translating \\eqref{d Q} into component notation. We stress that the charge $Q_\\lambda$ is only defined if \\eqref{d Q} is finite and satisfies the appropriate integrability conditions, see e.g. \\cite{Wald:1993nt}. Expression \\eqref{d Q} also makes it clear that gauge transformations, i.e. null directions of $\\Omega$, have a vanishing generator. This is just the familiar statement that the generators of gauge symmetries are constraints, and as such vanish on-shell.  On the other hand, global symmetries are associated to a non-zero charge.\n\n\\subsection{The symplectic flux}\\label{subsection:symplectic flux}\nIn this section we apply the method of \\cite{Lee:1990nz} to construct the symplectic structure of the MCS theory and determine the expression for the symplectic flux, which serves as a first step in classifying the allowed boundary conditions. Under an infinitesimal variation $\\delta A_{\\mu}$ of the gauge field  (and assuming a fixed background metric), the first order variation of the bulk action\\footnote{Note that we have not included the boundary terms in the action here. We will come back to them later, and confirm that they do not contribute to the symplectic structure.} is\n\\begin{equation}\\label{first variation action}\n\\delta I = \\int _{M}d^{3}x\\sqrt{|g|}\\, \\mbox{EOM}(A)^{\\mu}\\delta A_{\\mu} - \\int_{\\partial M}d^{2}x\\,\\sqrt{|\\gamma|}\\, \\rho_{\\mu}\\left(\\frac{1}{q^2}F^{\\mu\\nu} + \\frac{\\hat{\\alpha}}{2}\\epsilon^{\\mu\\rho\\nu}A_{\\rho}\\right)\\delta A_{\\nu}\\, ,\n\\end{equation}\n\\noindent where $\\mbox{EOM}(A)^{\\mu}=0$ is the equation of motion of the background gauge field, $\\gamma$ is the determinant of the induced metric on the timelike boundary (a ``constant radius\" slice), and $\\rho^{\\mu}$ denotes the corresponding unit normal.  From the above variation we read off the symplectic 1-form (see \\cite{Lee:1990nz,Compere:2008us})\n\\begin{align}\n\\theta^{\\mu}\n={}&\n - \\left(\\frac{1}{q^2}F^{\\mu\\nu} + \\frac{\\hat{\\alpha}}{2}\\epsilon^{\\mu\\rho\\nu}A_{\\rho}\\right)\\delta A_{\\nu}\\, .\n\\end{align}\n\\noindent Next, denoting by $\\delta_{1}A$ and $\\delta_{2}A$ two independent solutions of the linearized equations of motion\\footnote{We note that in the probe approximation the equations of motion for the background gauge field and its fluctuation have the same form, because the MCS system is linear.} we define the symplectic 2-form\n\\begin{align}\n\\omega^{\\mu}(\\delta_1 A, \\delta_2 A; \\bar{A})\n\\equiv{}&\n\t\\delta_{1}\\theta[\\delta_{2}A] - \\delta_{2}\\theta[\\delta_{1}A]\n\t\\nonumber\\\\\n={}&\n\t-\\frac{1}{q^2}\\Bigl(\\delta_{1}F^{\\mu\\nu}\\delta_{2} A_{\\nu}-\\delta_{2}F^{\\mu\\nu}\\delta_{1} A_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_{1}A_{\\rho}\\delta_{2} A_{\\nu}\\, .\n\\end{align}\n\\noindent Using the equation of motion for $\\delta F_{\\mu\\nu}$ (which is the same as \\eqref{MaxCS equation}, because we are ignoring backreaction on the metric) one can then show the crucial property\n\\begin{equation}\\label{symplectic continuity eq}\n\\nabla_{\\mu}\\omega^{\\mu} = 0\\, .\n\\end{equation}\n\nAs stated above, we assume that the $(2+1)$ manifold is Lorentzian, with the topology $X\\times \\mathds{R}$, where the $\\mathds{R}$ factor is parameterized by the timelike coordinate ($t$, say). The boundary $\\partial M$ is a surface of constant $r$. We now define the symplectic structure by\n\\begin{equation}\\label{OM bulk}\n\\Omega(\\delta_1 A, \\delta_2 A; \\bar{A}) = \\int_{\\Sigma}d^{2}x\\sqrt{h }\\, n_{\\mu}\\omega^{\\mu}\\, ,\n\\end{equation}\n\\noindent where $\\Sigma$ is a spacelike hypersurface (a $t={\\rm constant}$ slice, for example) with unit normal $n^{\\mu}$ and induced metric determinant $h$. Since the theory under consideration is linear, we can take the background to be the trivial configuration, i.e. $\\bar{A} = 0$, without loss of generality. We shall do so henceforth and omit the explicit reference to the background as an argument of the symplectic structure. We mention that, in principle, the bulk expression \\eqref{OM bulk} may require renormalization; the appropriate counterterms can be read off from a well-defined action principle as explained in \\cite{Compere:2008us}. However, working in the range $0 < \\alpha < 1$, no (UV) divergences arise in \\eqref{OM bulk} even if we allow the slow fall-off of the field to fluctuate, as we will verify by explicit computation in section \\ref{section:norms}. This is intimately related to the fact that, for $0 < \\alpha < 1$, the counterterms that render the variational principle well-defined do not include derivatives along the timelike direction, see section \\ref{sec:1pt}.\n\nAs discussed above, in order to obtain a well-defined phase space it is necessary to impose boundary conditions on our solutions in such a way that the symplectic structure is \\textit{conserved} (i.e. independent of $\\Sigma$). Integrating equation \\eqref{symplectic continuity eq} over a ``pillbox\" bounded by two spacelike hypersurfaces $\\Sigma_{1}$ and $\\Sigma_{2}$ and a region $R \\subset \\partial M$ (i.e. $R$ is an open subset of the boundary slice at constant $r$, see figure \\ref{pillbox}), one learns that the symplectic structure is independent of $\\Sigma$ provided the symplectic flux $\\Phi$ through $R$ vanishes, i.e.\n\\begin{figure}[htb]\n\\center\n\\includegraphics[width=0.65\\linewidth]{pillbox.pdf}\n\\label{pillbox}\n\\caption{The symplectic structure is conserved, i.e. $\\Omega(\\Sigma_{1})=\\Omega(\\Sigma_{2})$, when the symplectic flux through the region $R \\subset \\partial M$ vanishes.}\n\\end{figure}\n\\begin{equation}\\label{vanishing of flux}\n\\Phi = \\int_{R}d^{2}x\\sqrt{|\\gamma|}\\,\\rho_{\\mu}\\omega^{\\mu} =0\\, ,\n\\end{equation}\n\\noindent where, as before, $\\rho^{\\mu}$ and $\\gamma$ are the unit normal and the determinant of the induced metric on $R$, respectively. We suppose that this is attained {\\it locally}, so that the flux through the boundary vanishes through any open subset $R$. We mention that, from the point of view of the dual theory, these local boundary conditions correspond to the insertion of local operators. In the presence of additional boundaries, e.g. the Poincar\\'e horizon, one must also require the flux to vanish there. Given our assumption of locality, the boundary conditions at the extra boundaries are of course independent of the ones at the conformal boundary. It is worth noting that, for black hole spacetimes, the phase space is typically defined including the interior of the black hole, so a non-vanishing flux through the horizon is not in conflict with conservation of $\\Omega$.\n\nIn the coordinates introduced in \\eqref{asympt expansion of metric} the only non-vanishing component of $\\rho$ is $\\rho_{r} = \\sqrt{g_{rr}} = N_{r}$, where $N_{r}$ is the lapse in a radial foliation. Since $\\sqrt{|g|} = N_{r}\\sqrt{|\\gamma|}$, we have\n\\begin{equation}\n\\Phi  = \\int_{R}d^{2}x\\sqrt{|g|}\\,\\bar{\\rho}_{\\mu}\\omega^{\\mu}\\, ,\n\\end{equation}\n\\noindent where $\\bar{\\rho}_{\\mu}\\, dx^{\\mu}=dr$ and $g$ is the determinant of the full $(2+1)$ metric, as before. If we now split the connection as in \\eqref{definition B}, so that in an obvious notation the gauge field fluctuation is $\\delta A  = \\delta B + \\delta A^{(0)}$, we find\n\\begin{equation}\n\\omega^{\\mu} = \\omega^{\\mu}_{B} + \\omega^{\\mu}_{0} + \\omega^{\\mu}_{mix}\\, ,\n\\end{equation}\n\\noindent where we have defined\n\\begin{align}\n\\omega^{\\mu}_{B}\n\\equiv{}&\n\t-\\frac{1}{q^2}\\Bigl(f_{1}^{\\mu\\nu}\\delta_2 B_{\\nu}-f_{2}^{\\mu\\nu}\\delta_1 B_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_1 B_{\\rho} \\delta_2 B_{\\nu}\n\\\\\n\\omega^{\\mu}_{0}\n\\equiv {}&\n\t-\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_1 A^{(0)}_{\\rho} \\delta_2 A^{(0)}_{\\nu}\n\\\\\n\\omega^{\\mu}_{mix}\n\\equiv {}&\n\t-\\frac{1}{q^2}\\Bigl(f_{1}^{\\mu\\nu}\\delta_2 A^{(0)}_{\\nu}-f_{2}^{\\mu\\nu}\\delta_1 A^{(0)}_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\bigl(\\delta_1 A^{(0)}_{\\rho} \\delta_2 B_{\\nu} + \\delta_1 B_{\\rho} \\delta_2 A^{(0)}_{\\nu}\\bigr)\n\\end{align}\n\\noindent with $f$ the field strength of $\\delta B$. We now notice that contracting equation \\eqref{first order equation for B} with the Levi-Civita tensor results in $0 = F_{\\mu\\nu} - q^{2}\\hat{\\alpha}\\,\\epsilon_{\\mu\\nu\\rho}B^{\\rho}$. Consequently, the fluctuations of the gauge-invariant mode satisfy\n\\begin{equation}\\label{massive mode fluctuation eq}\nf^{\\mu\\nu} =   q^{2}\\hat{\\alpha}\\,\\epsilon^{\\mu\\nu\\rho}\\delta B_{\\rho}\\, .\n\\end{equation}\n\n\\noindent Using this on-shell condition in the above expression for $\\omega^{\\mu}$ we find\n\\begin{align}\\label{omegas flat and non-flat}\n\\omega^{\\mu}_{B}\n=\n\t\\hat{\\alpha}\\,\\epsilon^{\\mu\\nu\\rho}\\delta_1 B_{\\nu}\\delta_2 B_{\\rho}\\, ,\\qquad\n\\omega^{\\mu}_{0}\n=\n\t-\\hat{\\alpha}\\, \\epsilon^{\\mu\\nu\\rho}\\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\rho}\n\\,, \\qquad\n\\omega^{\\mu}_{mix}\n=\n\t0\\, .\n\\end{align}\n\nAs a result of the splitting \\eqref{omegas flat and non-flat}, the symplectic structure can be written as\n\\begin{equation}\\label{omega split}\n    \\Omega = \\int_\\Sigma d^2 x \\sqrt{h}\\, n_\\mu \\omega^{\\mu}_{B} + \\int_\\Sigma d^2 x \\sqrt{h}\\, n_\\mu \\omega^{\\mu}_{0} \\, .\n\\end{equation}\n\\noindent This suggests that the space of solutions is a direct product of the flat and non-flat sectors. However, a more detailed analysis reveals that this is only true if the boundary conditions do not mix modes in the various sectors, see section \\ref{sec:bc}.\n\nLet us now find an expression for the symplectic flux that will allow us to determine the allowed boundary conditions. In order to do so, it is important to keep in mind that the modes $\\delta B^{(\\pm)}$ are constrained by the asymptotic equations of motion, and therefore obey \\eqref{asymptotic constraint}. For example, in light-cone coordinates $(u,v)$ in which the boundary metric takes the form\n\\begin{equation}\\label{bndy light cone}\n    g^{(0)}_{ij} = \\left(\n                     \\begin{array}{cc}\n                       0 & 2 \\\\\n                       2 & 0 \\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\n\\noindent these lead to\n\\begin{equation}\n\\delta B_{v}^{(+)} = \\delta B_{u}^{(-)}=0\\,.\n\\end{equation}\n\\noindent Taking the asymptotic constraints \\eqref{asymptotic constraint} into account then, we find that the symplectic flux through $R$ is given by\n\\begin{align}\\label{flux coeff}\n\\Phi\n  ={}&\n \\hat{\\alpha}\\int_{R}d^{2}x\\,\n\t\\varepsilon^{r\\nu\\lambda}\\left( \\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\lambda}-\\delta_1 B_{\\nu}^{(+)}\\delta_2 B_{\\lambda}^{(-)} + \\delta_2 B_{\\nu}^{(+)} \\delta_1 B_{\\lambda}^{(-)}  \\right)\n\t\\nonumber\\\\\n\t={}&\n\t\t\\hat{\\alpha}\\int_{R}d^{2}x\\,\\varepsilon^{ij}\\left( \\delta_1 A^{(0)}_{i} \\delta_2 A^{(0)}_{j}-\\delta_1B_{i}^{(+)}\\delta_2 B_{j}^{(-)} + \\delta_2 B_{i}^{(+)} \\delta_1 B_{j}^{(-)}  \\right).\n\\end{align}\n\n\\subsection{Boundary conditions}\\label{sec:bc}\nAs discussed above, demanding the vanishing of the symplectic flux gives us a useful way of classifying the boundary conditions. Momentarily giving up covariance in the boundary directions, in light-cone coordinates \\eqref{bndy light cone} we find that possible local boundary conditions include\n\\begin{eqnarray}\\label{bcs flat sector super}\nA^{(0)}_{u} = W\\bigl[A^{(0)}_{v}\\bigr],\\qquad  B_{u}^{(+)} = V\\bigl[ B_{v}^{(-)}\\bigr].\n\\end{eqnarray}\n\n\\noindent For general ``potentials\" $W$ and $V$, such boundary conditions would correspond to multi-trace deformations in the dual CFT. For simplicity, let us focus on the linear case\n\\begin{eqnarray}\\label{bcs flat sector}\n\\delta A^{(0)}_{u} = \\bar{\\beta}\\, \\delta A^{(0)}_{v},\\qquad \\delta B_{u}^{(+)} = \\beta\\, \\delta B_{v}^{(-)},\n\\end{eqnarray}\n\\noindent for any constants $\\beta,\\bar\\beta$. Note that $\\beta=0,\\infty$ correspond to chiral boundary conditions, while other values mix the modes and break covariance. We will refer to $\\delta B_{u}^{(+)} = 0$ as Dirichlet and to $\\delta B_{v}^{(-)} = 0$ as Neumann boundary conditions, in close analogy to the terminology commonly used for scalar fields in AdS. We will term the boundary condition $\\delta B_{u}^{(+)} = \\beta\\, \\delta B_{v}^{(-)}$ as ``mixed\" when $\\beta$ is finite. As usual, the boundary conditions with finite $\\beta$ and $\\bar\\beta$ are related to double-trace deformations of the boundary theory \\cite{Witten:2001ua,Berkooz:2002ug}, as we will review later on. Furthermore, we notice that, because $B_{u}^{(+)}$ and $B_{v}^{(-)}$ have scaling dimensions $\\Delta_- = 1- \\alpha$ and $\\Delta_+ = 1 + \\alpha$, respectively, the constant $\\beta$ has dimension $\\Delta_\\beta = - 2\\alpha$. The RG flow interpretation of double-trace deformations has been discussed in, for example, \\cite{Witten:2001ua,Gubser:2002zh,Gubser:2002vv,Leigh:2003gk,Hartman:2006dy}.\\footnote{This interpretation requires both end points of the RG flow to be well-defined, e.g. as in the case of scalar fields with masses close to Breitenlohner-Freedman bound in AdS. We shall see below that in the present case the Neumann theories are ill-defined so this picture does not strictly hold.} On the other hand, since $A^{(0)}_u$ and $A^{(0)}_v$ both have dimension one, the constant $\\bar{\\beta}$ is dimensionless.\nInterestingly, we also note the possibility of a ``hybrid\" boundary condition\n\\begin{equation}\\label{gen hyb}\n\\delta A^{(0)}_{u} = \\kappa\\, \\delta B_{u}^{(+)} \\qquad \\mbox{\\textbf{and}}\\qquad \\delta A^{(0)}_{v} = \\frac{1}{\\kappa}\\delta B_{v}^{(-)}\\, ,\n\\end{equation}\n\\noindent that mixes the flat connections with the massive sector. Here, $\\kappa$ is a constant of scaling dimension $\\Delta_\\kappa = \\alpha$. Notice that, in view of the flatness condition on $\\delta A^{(0)}$, \\eqref{gen hyb} implies\n\\begin{equation}\\label{hyb on b}\n    \\kappa^2\\, \\partial_v \\delta B_u^{(+)} = \\partial_u \\delta B_v^{(-)}\\, .\n\\end{equation}\n\\noindent In analogy with the linear boundary conditions discussed above, this hybrid boundary condition has the interpretation of a double-trace deformation. To our knowledge, the possibility of such boundary conditions has not been explicitly discussed in the literature.\n\nIt is now clear from the decomposition \\eqref{omega split} and the analysis of the boundary conditions above that, as anticipated in \\cite{Gukov:2004id}, the flat and massive sectors do not always decouple. In fact, for our hybrid boundary conditions \\eqref{gen hyb} both sectors indeed interact with one another. The decoupling only occurs if one imposes boundary conditions which do not mix both sectors, i.e. if we impose  boundary conditions like those in \\eqref{bcs flat sector}. This is because it is only in this case that the symplectic structure effectively splits as a direct sum of two independent pieces.\n\n\\subsection{One-point functions}\n\\label{sec:1pt}\n\nAs usual in the context of holography, the Maxwell-Chern Simons action \\eqref{TME bulk action} must be supplemented by a series of boundary terms that serve two purposes: achieving a well-defined variational principle for a chosen set of boundary conditions, and removing divergences. We will refer to the latter as counterterms. We recall now that the first variation of the bulk action is given by \\eqref{first variation action}. Evaluating this expression on-shell we find\n\\begin{equation}\n\\left.\\delta I \\right|_{os} = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\sqrt{|\\gamma|}\\, \\rho_{\\mu}\\epsilon^{\\mu\\rho\\nu}\\left(B_{\\rho}- A^{(0)}_{\\rho}\\right)\\delta A_{\\nu}\\, ,\n\\end{equation}\n\n\\noindent where the gauge field fluctuations are understood to be evaluated on the solution of the linearized equations of motion.\\footnote{Since we are ignoring backreaction, the various metric quantities are always understood to be evaluated on their (fixed) background values.}  Employing the notation established above, we find\n\\begin{align}\n\\left.\\delta I \\right|_{os}\n={}&\n -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B_{i}- A^{(0)}_{i}\\right)\\left( \\delta A^{(0)}_{j} + \\delta B_{j}\\right)\n  \\nonumber\\\\\n ={}&\n  -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B^{(+)}_{i}\\delta B_{j}^{(-)} + B_{i}^{(-)}\\delta B_{j}^{(+)}- A^{(0)}_{i}\\delta A^{(0)}_{j}  \\right)\n  \\nonumber\\\\\n &\n  -\\frac{\\hat{\\alpha}}{2}\\lim_{r\\to \\infty} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij} r^{\\alpha}\\left(B_{i}^{(+)}\\delta A^{(0)}_{j} + A_{j}^{(0)}\\delta B_{i}^{(+)}\\right)\\, ,\n\\end{align}\n\\noindent where in the last equality we used the restrictions placed by the asymptotic equations of motion on the $B^{(\\pm)},\\delta B^{(\\pm)}$ modes (c.f. section \\ref{subsection:asymptotic sols}). We note the presence (for any finite Maxwell coupling $q^2$) of the divergent term, which we cancel by the addition of a counterterm. Noticing that $\\varepsilon^{ij} r^{\\alpha}\\left(B_{i}^{(+)}\\delta A^{(0)}_{j} + A_{j}^{(0)}\\delta B_{i}^{(+)}\\right) = \\delta \\left(r^{\\alpha}\\varepsilon^{ij}A_{j}^{(0)}B_{i}^{(+)}\\right)$ it is easy to check that the desired counterterm is given by the covariant expression\n\\begin{align}\nI_{ct} = \\frac{1}{2q^2} \\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}F^{i}A_{i}\\, ,\n\\end{align}\n\\noindent where, as before, $\\gamma$ is the determinant of the induced metric on the $r={\\rm constant}$ surface, and we have defined\n\\begin{equation}\nF^{i}\\equiv \\rho_{\\mu}F^{\\mu i}\\, ,\n\\end{equation}\n\\noindent with $\\rho_{\\mu}$ the unit normal 1-form on the radial slices. Therefore, we have that\n\\begin{align}\n\\left.\\delta\\left( I + I_{ct}\\right) \\right|_{os}\n ={}&\n  -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B^{(+)}_{i}\\delta B_{j}^{(-)} + B_{i}^{(-)}\\delta B_{j}^{(+)}- A^{(0)}_{i}\\delta A^{(0)}_{j}  \\right)\n \\end{align}\n\\noindent is finite as $r\\to\\infty$.\n\\subsubsection{Covariant boundary conditions}\nIn order to proceed further we need to discuss the additional \\textit{finite} boundary terms needed in order to enforce different boundary conditions of interest. Confining ourselves to covariant terms for the moment, we consider the following quantities:\n\\begin{align}\nB_{\\pm}\n ={}&\n  \\mp \\frac{1}{4\\gc^{4}\\hat{\\alpha}}\\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}F^{i}\\gamma_{ij}F^{j}\\, ,\n\\\\\nB_{(0)} ={}&\n  \\frac{1}{2q^{2}}\\int_{\\partial M}d^{2}x\\, \\varepsilon^{ij}F_{i}A_{j}+\\frac{1}{4}\\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}\\gamma^{ij}\\left(\\frac{1}{q^4\\hat\\alpha}F_{i}F_{j}-\\hat\\alpha A_{i}A_{j}\\right)\\, .\n\\end{align}\n\n\\noindent Evaluating on-shell we find\n\\begin{align}\\label{on shell covariant counter-terms}\n\\left. B_{\\pm}\\right|_{os}\n ={}&\n \\pm  \\frac{\\hat\\alpha}{2}\\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}g^{(0)ij}B_{i}^{(-)}B_{j}^{(+)}\n\\nonumber\\\\\n={}&\n\\pm  \\frac{\\hat\\alpha}{2}\\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}B_{i}^{(+)}B_{j}^{(-)}\n\\\\\n \\left. B_{(0)} \\right|_{os}\n ={}&\n-\\frac{\\hat\\alpha}{4}\\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}g^{(0)ij}A_{i}^{(0)}A_{j}^{(0)}\\, .\n\\end{align}\n\\noindent By taking linear combinations of these finite boundary terms we can achieve a variational principle well-suited for the various boundary conditions \\eqref{bcs flat sector} of interest in the flat and gauge-invariant (massive) sectors. For example, we find\n\\begin{align}\\label{variation with Bplus fixed}\n\\delta \\left. \\left(I + I_{ct}\\pm B_{(0)} + B_{+}\\right) \\right|_{os}\n={}&\n     \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\left[\\epsilon^{ij}B_{i}^{(-)}\\delta B_{j}^{(+)} \\mp A_{i}^{(0)}P_{\\pm}^{ij}\\delta A^{(0)}_{j}\\right]\\, ,\n\\end{align}\n\\noindent and\n\\begin{align}\\label{variation with Bminus fixed}\n\\delta \\left. \\left(I + I_{ct}\\pm B_{(0)} + B_{-}\\right) \\right|_{os}\n={}&\n     \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\left[\\epsilon^{ij}B^{(+)}_{i}\\delta B_{j}^{(-)}\\mp A_{i}^{(0)}P_{\\pm}^{ij}\\delta A^{(0)}_{j}\\right].\n\\end{align}\n\\noindent Now that we have identified the sources for the covariant boundary conditions, i.e. $\\delta B^{(\\pm)}_{i}$ and $\\left(P_{\\pm}\\delta A^{(0)}\\right)_{i} = g^{(0)}_{ij}P_{\\pm}^{jk}\\delta A^{(0)}_{k}$, we write the variation of the renormalized action $I_{ren}$ generically as\n\\begin{equation}\\label{I ren gen}\n\\left.\\delta I_{ren}\\right|_{os} =  \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\biggl[\\langle \\mathcal{O}^{(\\pm)i}\\rangle \\delta B^{(\\pm)}_{i} + \\langle\\mathcal{O}_{\\pm}^{(0)\\, i}\\rangle \\left(P_{\\pm}\\delta A^{(0)}\\right)_{i}\\biggr].\n\\end{equation}\n\\noindent Comparing with \\eqref{variation with Bplus fixed} and \\eqref{variation with Bminus fixed} and using the properties of $P_{\\pm}$ we can read-off the one-point functions of the dual operators, and we obtain\n\\begin{align}\\label{O b}\n\\langle \\mathcal{O}^{(\\pm)i}\\rangle\n ={}&\n  \\pm \\hat{\\alpha}\\,g^{(0)ij}B_{j}^{(\\mp)}\\, ,\n  \\\\ \\label{O A0}\n \\langle\\mathcal{O}_{\\pm}^{(0)\\, i}\\rangle\n ={}&\n \\mp \\hat{\\alpha} P_{\\mp}^{ij}A_{j}^{(0)}\\, .\n\\end{align}\n\nSince $A_{i}^{(0)}$ is constrained by the flatness condition, the variational derivatives with respect to its components are ill-defined, and, as a consequence, the one-point functions \\eqref{O A0} suffer from an ambiguity. However, this ambiguity is nothing but the one associated to the $U(1)$ gauge transformations. In other words, \\eqref{O A0} are only defined up to the transformations $\\delta A_i  = \\partial_i \\lambda$ that preserve the boundary conditions in the variational principle. See \\cite{Marolf:2006nd} for a related discussion in the context of (pure) Maxwell fields.\n\\subsubsection{Symmetry-breaking boundary conditions}\nLet us now turn to the less symmetric scenarios. First, we consider the case of ``mixed\" boundary conditions, i.e. $B^{(+)}_u - \\beta B^{(-)}_v = 0$, where $\\beta$ is a finite dimensionful constant. It is clear that this requirement breaks both conformal and Poincar\\'e symmetry, so we are allowed to write down the appropriate boundary terms simply in terms of the coefficients of the asymptotic expansion. As we shall see shortly, it is useful to generalize the above boundary condition and consider instead\n\\begin{equation}\\label{mix bc J}\n    B^{(+)}_u - \\beta B^{(-)}_v = J_\\beta\\, ,\n\\end{equation}\n\\noindent where $J_\\beta$ is an arbitrary fixed function of the boundary coordinates.  We ignore the contribution from the flat sector momentarily. Starting from the Neumann theory, i.e. the theory in which $B^{(-)}_i$ is fixed and whose action we denote by $I_N$, the boundary term we need to add in order to attain the mixed boundary condition is\n\\begin{equation}\\label{I def mix}\n    I_{def,\\beta} = - \\frac{\\hat{\\alpha}}{4 \\beta} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left(B^{(+)}_u \\right)^2\\, .\n\\end{equation}\n\\noindent In fact, using the variation of the Neumann action (\\ref{variation with Bminus fixed}) and the explicit boundary term \\eqref{I def mix}, we obtain\n\\begin{equation}\\label{dI mixed}\n    \\delta\\left(I_N + I_{def,\\beta}\\right) = \\frac{\\hat{\\alpha}}{2 \\beta} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} B^{(+)}_u\\left( \\beta\\, \\delta B^{(-)}_v - \\delta B^{(+)}_u \\right),\n\\end{equation}\n\\noindent which is finite and stationary when the boundary condition \\eqref{mix bc J} holds. Comparing \\eqref{dI mixed} with \\eqref{mix bc J}, we note that the quantity that is being held fixed in the variational principle is in fact $J_\\beta$. This means that $J_\\beta$ is to be interpreted as the source for the dual operator in the boundary theory. Given this, it follows from \\eqref{dI mixed} that the one-point function in the presence of sources for the dual operator in the deformed theory is given by\n\\begin{equation}\\label{1pt mix}\n    \\langle {\\cal O}^{(-)}_u \\rangle_{\\beta} = - \\frac{\\hat{\\alpha}}{2 \\beta} B^{(+)}_u,\n\\end{equation}\n\\noindent where, as usual, $B^{(+)}_u$ must be thought of as a function of the source $J_\\beta$ defined in \\eqref{mix bc J}. Before constructing variational principles suitable for the remaining boundary conditions, we comment that the computation above provides a simple illustration of the well-known fact that linear boundary conditions of the form \\eqref{mix bc J} correspond to double-trace deformations in the dual theory. The argument is as follows. First, we recall that in AdS\/CFT the Neumann action $I_N$ is interpreted as the generating function for the operator associated to $B^{(+)}_u$ in the dual CFT. Then, the boundary term \\eqref{I def mix} is transparently identified with a double-trace deformation for this operator. Moreover, the inclusion of \\eqref{I def mix} implies that the original Neumann boundary condition needs to be shifted in such a way that the the modified action has an extremum. As noted above, the new boundary condition is nothing but the linear relation \\eqref{mix bc J}, which completes the argument. It is worth commenting on the possibility of thinking of the \\eqref{mix bc J} as a deformation of the Dirichlet theory. In such case, the boundary term that implements the shift in the boundary condition is quadratic in $B^{(-)}_v$, so it has dimension $2(1+\\alpha)$. We see that the deformation is then irrelevant.\n\nWe now construct an appropriate action for the boundary condition\n\\begin{equation}\\label{mix bc A0 J}\n    A^{(0)}_u - \\bar{\\beta} A^{(0)}_v = J_{\\bar \\beta}\n\\end{equation}\n\\noindent where $\\bar\\beta$ is a non-zero dimensionless constant and $J_{\\bar \\beta}$ is a fixed arbitrary function of the boundary coordinates. In analogy with the previous case, $J_{\\bar \\beta}$ corresponds to the source of the dual operator. Note that since $\\bar \\beta$ is dimensionless the boundary condition \\eqref{mix bc A0 J} for $J_{\\bar \\beta} = 0$ preserves scale invariance, yet it breaks Lorentz invariance. Once again, as a consequence of this, it is licit to write extra boundary terms which are not Lorentz densities. Moreover, because this boundary condition does not mix the flat and massive sectors, we concentrate on the flat connections and temporarily drop the contribution from the massive modes. Now, assuming that we start with an action $I^{(1)}_{ren}$ which attains an extremum when $P_- A^{(0)}$ is fixed, we find\n\\begin{equation}\\label{dI1}\n    \\delta I^{(1)}_{ren} = \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\langle\\mathcal{O}_{-}^{(0)\\, i}\\rangle (P_- \\delta A^{(0)})_i = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\delta A^{(0)}_v\n\\end{equation}\n\\noindent as follows from \\eqref{I ren gen} and \\eqref{O A0}.  In this case, the boundary term that we need to add to $I^{(1)}_{ren}$ in order for \\eqref{mix bc A0 J} to hold can be written as\n\\begin{equation}\\label{I def bar beta}\n    I_{def, \\bar \\beta} = - \\frac{\\hat{\\alpha}}{4 \\bar{\\beta}} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left( A^{(0)}_u \\right)^2\\, .\n\\end{equation}\n\\noindent In fact, with this choice the on-shell variation of the action reads\n\\begin{equation}\\label{d I mix A0}\n    \\delta \\left(I^{(1)}_{ren} + I_{def, \\bar \\beta}\\right) = \\frac{\\hat{\\alpha}}{2 \\bar{\\beta}} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\left( \\bar{\\beta} \\delta A^{(0)}_v - \\delta A^{(0)}_u \\right),\n\\end{equation}\n\\noindent as desired. As discussed above, the boundary condition \\eqref{mix bc A0 J} is in one-to-one correspondence with the inclusion of the double-trace deformation \\eqref{I def bar beta} in the dual theory. The relevant one-point function is given by\n\\begin{equation}\\label{1pt mix A0}\n    \\langle {\\cal O}^{(0)}_{+ u} \\rangle_{\\bar \\beta} = - \\frac{\\hat{\\alpha}}{2 \\bar{\\beta}} A^{(0)}_u\\, .\n\\end{equation}\n\n\\noindent Once again, we mention that the one-point function \\eqref{1pt mix A0} is only defined up to the appropriate $U(1)$ transformation.\n\nFinally, we consider the ``hybrid\" boundary conditions defined in \\eqref{gen hyb}, which admit the obvious generalization\n\\begin{equation}\\label{hyb bc J}\n    A^{(0)}_u - \\kappa B^{(+)}_u = J_\\kappa\\,, \\qquad A^{(0)}_v - \\kappa^{-1} B^{(-)}_v = \\tilde{J}_\\kappa \\, ,\n\\end{equation}\n\\noindent where we take $J_\\kappa$, $\\tilde{J}_\\kappa $ to be the sources of the dual operators. It is convenient to start with a renormalized action $I^{(2)}_{ren}$ such that\n\\begin{equation}\\label{I ren for hyb}\n    \\delta I^{(2)}_{ren} = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left( A^{(0)}_u \\delta A^{(0)}_v + B^{(-)}_v \\delta B^{(+)}_u \\right).\n\\end{equation}\n\\noindent With $I^{(2)}_{ren}$ as a starting point, the boundary term that implements hybrid boundary conditions is given by\n\\begin{equation}\\label{I def hyb}\n    I_{def, \\kappa} = - \\frac{\\hat{\\alpha}}{2 \\kappa} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} B^{(-)}_v A^{(0)}_u,\n\\end{equation}\n\\noindent as it follows from\n\\begin{align}\\label{dI hyb}\n    \\delta\\left(I^{(2)}_{ren} + I_{def, \\kappa}\\right)\n    ={}&\n     \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\left( \\delta A^{(0)}_v - \\kappa^{-1} \\delta B^{(-)}_v \\right)\n     \\nonumber \\\\\n    &-\n     \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\kappa^{-1} B^{(-)}_v \\left( \\delta A^{(0)}_u - \\kappa\\, \\delta B^{(+)}_u \\right).\n\\end{align}\n\n\\noindent As pointed out before, the hybrid boundary conditions correspond to a double-trace deformation in the dual theory. Note that, in this case, the deformation \\eqref{I def hyb} explicitly mixes the flat and massive sectors, so indeed these do not decouple in the theory defined by the hybrid boundary conditions. It follows from \\eqref{dI hyb} that the one-point functions in the dual theory are given by\n\\begin{align}\\label{O kappa 1}\n    \\langle {\\cal O}^{(0)}_{+ u} \\rangle_\\kappa\n    =\n     \\frac{\\hat{\\alpha}}{2} A^{(0)}_u \\qquad \\mbox{and}\\qquad\n    \\langle {\\cal O}^{(0)}_{- v} \\rangle_\\kappa\n    =\n     - \\frac{\\hat{\\alpha}}{2 \\kappa} B^{(-)}_v\\, .\n\\end{align}\n\\noindent It is worthwhile noting that, since the $U(1)$ transformations do not preserve the boundary conditions \\eqref{gen hyb}, the one-point functions \\eqref{O kappa 1} are unambiguously defined.\n\nBefore closing this section, we emphasize that, provided $0 < \\alpha < 1$, the boundary terms involved do not contain derivatives along the timelike direction. Given the results of \\cite{Compere:2008us}, this strongly suggests that the bulk symplectic structure does not need to be supplemented by additional boundary contributions. This is indeed the case, as we will explicitly verify below. Specifically, we will check that, for $0 < \\alpha < 1$, the bulk symplectic structure is finite and conserved for all the boundary conditions under scrutiny.\n\n\\subsection{The symplectic product}\\label{sec: symp prod}\nRecall that the symplectic structure is given by \\eqref{omega split} with \\eqref{omegas flat and non-flat}, i.e.\n\\begin{equation}\\label{omega recap}\n    \\Omega = \\hat{\\alpha} \\int_\\Sigma d^2 x \\sqrt{h} n_\\mu \\epsilon^{\\mu\\nu\\rho}\\delta_1 B_{\\nu}\\delta_2 B_{\\rho} - \\hat{\\alpha}  \\int_\\Sigma d^2 x \\sqrt{h} n_\\mu \\epsilon^{\\mu\\nu\\rho}\\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\rho}\\, .\n\\end{equation}\n\\noindent As mentioned above, it turns out that the restriction $0 < \\alpha < 1$ ensures that \\eqref{omega recap} is finite for all the boundary conditions of interest, provided one imposes additional requirements on the solutions in the deep interior.\n\nQuite generally, given a symplectic structure it is possible to endow the space of solutions with an inner product defined in terms of $\\Omega$, as we now review briefly. A more detailed discussion can be found in \\cite{Wald:1995yp}, for example. We start by complexifying the space of solutions and consider\\footnote{The reader uneasy with the use of the complex conjugates in \\eqref{ip} can think of using a basis of solutions in momentum space in which the modes are generically complex despite the fact that the field is real.}\n\\begin{equation}\\label{ip}\n    (A_1, A_2) = - i \\Omega(A_1^*, A_2)\\, .\n\\end{equation}\n\\noindent We will refer to \\eqref{ip} as the symplectic product of the theory. One can verify that \\eqref{ip} satisfies the expected properties of bi-linearity and Hermiticity, although in general it fails to be positive definite.\n\nThe inner product \\eqref{ip} allows us to introduce some useful terminology. First, we shall term a given solution $A_0$ as \\textit{normalizable} if $(A_0, A)$ is finite for all $A$. As stated above, in our particular setup this translates into a requirement on the fields in the deep interior. Second, we define a \\textit{ghost} to be an excitation of definite positive(negative) frequency with negative(positive) norm. Here, we will use the definition of positive frequency associated to the timelike Killing vector of the relevant background geometry under consideration. For example, if $\\partial_t$ is a vector field which is timelike everywhere, the solution $A$ is said to be a positive frequency solution if $\\partial_t A = - i \\omega A $ with $\\omega > 0$. Third, we will refer to a solution $A_{gauge}$ as \\textit{pure gauge} if $(A_{gauge}, A) = 0$ for all $A$.\n\nIt should be noted that the presence of ghosts in a given system is correlated with the lack of unitarity in the associated quantum theory. At the classical level, the presence of ghosts also signals pathologies since these give negative contributions to the energy.\n\n\n\\section{The dual field theory spectrum}\\label{section:spectrum}\nIn this section we determine the spectrum of normalizable solutions of the MCS system in $AdS_3$, in both global and Poincar\\'e coordinates, for the various boundary conditions of interest. As explained in section \\ref{sec: symp prod}, by ``normalizable\" we mean excitations that have finite symplectic product with all the modes. We mention that, while normalizability at the conformal boundary is guaranteed by restricting the coupling $\\alpha$ defined in \\eqref{rescaled CS coupling} to satisfy $0 < \\alpha < 1$, normalizability at the interior is achieved by restricting the wave functions appropriately. More precisely, when the geometry is global AdS we shall require the wave functions to be smooth at the origin, as is customary. In the Poincar\\'e AdS case, in addition to smoothness in the interior, we restrict the wave functions in such a way that no symplectic flux can leak through the Poincar\\'e horizon.\n\nAs explained in section \\ref{sec: prelim}, the connection splits into flat and ``massive\" pieces, and we can solve the bulk equations of motion separately for each sector. Moreover, as discussed in section \\ref{sec:bc}, these sectors decouple unless we impose the ``hybrid\" boundary condition \\eqref{gen hyb}. Our strategy to find the spectrum will be to focus on the massive and flat sectors separately, and incorporate the effects of the mixing only when we discuss the hybrid boundary conditions. For the sake of simplifying the exposition, we display the general solution to the equations of motion of the massive mode in appendix \\ref{section:solutions}, while here we focus exclusively on imposing the appropriate boundary conditions.\n\n\\subsection{Global $AdS_{3}$} \\label{sec: spec global adS}\nWe first consider the MCS theory in global $AdS_{3}$, whose line element is given by \\eqref{Global AdS3}. Since the spacetime is topologically trivial, there is no room for holonomies and the connection must be smooth at the origin, where the vector field $\\partial_x =(1\/L)\\partial \\varphi$ becomes singular. As a result, in addition to normalizability we must impose $A[\\partial_x] = A_x = 0$ at $\\rho = 0$. It should be stressed that setting $A^{(0)}_\\rho = 0$ \\textit{everywhere} in the bulk is generically in conflict with smoothness. To see why, we note that this implies that the components of $A^{(0)}$ along the boundary directions are independent of $\\rho$ everywhere, so any boundary condition other than $A^{(0)}_x |_{\\partial M} = 0$ would yield singular configurations. Having said this, we initiate the study of the spectrum for all the boundary conditions of interest.\n\\subsubsection{Flat sector}\n\\label{spec flat sector global}\n\nWe first consider the flat sector. Since there are no holonomies, the flat connections can be written as\n\\begin{equation}\\label{soln flat global}\n    \\delta A^{(0)}_\\mu = \\partial_\\mu \\lambda\\, ,\n\\end{equation}\n\\noindent where $\\lambda$ is smooth everywhere, with  $\\partial_x \\lambda = 0 $ at $\\rho=0$. Recall that in our analysis of the symplectic flux, we encountered the allowed boundary condition  \\eqref{bcs flat sector}, which in terms of the $(t,x)$ coordinates defined in appendix \\ref{section:solutions} takes the form\n\\begin{equation}\\label{bc flat t x}\n   \\left.\\left( \\delta A^{(0)}_t - \\hat{\\beta}\\, \\delta A^{(0)}_x \\right)\\right|_{\\partial M} = 0\\, ,\n\\end{equation}\n\\noindent where $\\hat{\\beta} = (\\bar{\\beta}-1)\/(\\bar{\\beta} +1)$ is a (possibly vanishing or infinite) constant. Fourier-decomposing $\\lambda$ as $\\lambda = e^{-i \\omega t + i k x} \\hat \\lambda(k,\\omega)$ with $k\\in\\mathds{Z}$, and using \\eqref{soln flat global} in \\eqref{bc flat t x}, we learn that the frequencies must satisfy\n\\begin{equation}\\label{freq global}\n    \\omega = - \\hat{\\beta} k\\, ,\n\\end{equation}\n\\noindent which determines the spectrum of the flat sector. As is well-known \\cite{Witten:1988hf, Moore:1989yh,Elitzur:1989nr,Wen:1991mw,Balachandran:1991dw,Bos:1989kn,Bos:1989wa,Schwarz:1979ae}, the degrees of freedom of the flat sector reside exclusively on the spacetime boundary,\\footnote{In particular, if $\\lambda$ goes to zero at the boundary the flat connections are pure gauge.} a fact that we will briefly review in appendix \\ref{section:sym bndy}.   We will consider the flat solutions for hybrid boundary conditions in the next subsection.\n\n\\subsubsection{Massive sector}\nFocusing now on the massive sector we use the ansatz \\eqref{ansatz global AdS}, in which case the solution is given by \\eqref{solution b in global AdS3}-\\eqref{the cs 2} in terms of functions $F(\\omega,\\pm k,\\pm\\alpha;\\rho)$. We observe that only the $F(\\omega,|k|,\\alpha ;\\rho)$ profiles are regular in the interior ($\\rho \\to 0$). Hence, for $k<0$, we take the $F(\\omega,-k,\\alpha ;\\rho)$ solution. Consequently, we will write the general solution which is smooth in the interior of $AdS_3$ as\n\\begin{align}\\label{regular solution b in global AdS3}\nb_{u} ={}&\nC_{u}F( \\omega, |k|,\\alpha;  \\rho)\n\\\\\nb_{v} ={}&\nC_{v}F(  \\omega, |k|, -\\alpha;  \\rho)\\, ,\n\\end{align}\n\\noindent where $(u,v)$ are the light-cone coordinates defined in \\eqref{nc}, $F( \\omega, k,\\alpha;  \\rho)$ is defined as in \\eqref{F hyp}, and\n\\begin{equation}\\label{relation coeff global AdS}\n   \\frac{C_{v}}{C_{u}} = \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha}\\, .\n\\end{equation}\n\n\\noindent Here, $s$ denotes the sign function, i.e. $s(k) = 1 $ for $k \\geq 0$ and $s(k) = - 1$ for $k < 0$. The $b_{\\rho}$ component is obtained from $b_{u}$ and $b_{v}$ via \\eqref{radial constraint global ads} and it is subleading with respect to them near the conformal boundary of $AdS_{3}$. Expanding $F(\\omega,k,\\alpha ;\\rho)$ near $\\rho = \\infty$ and using \\eqref{regular solution b in global AdS3}-\\eqref{relation coeff global AdS}, we learn that the relevant coefficients in the asymptotic expansion are\n\\begin{align}\\label{explicit coeffs b+}\n    b^{(+)}_u\n     ={}&\n      C_u C(\\alpha, |k|, \\omega)\\,,\n      &    b^{(+)}_v ={}&0 \\, ,\n\\\\\n\\label{explicit coeffs b-}\n    b^{(-)}_v\n    ={}&\n      C_u \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega)\\,,\n     &   b^{(-)}_u ={}&0\\, ,\n\\end{align}\n\\noindent where\n\\begin{equation}\\label{C alpha}\n    C(\\alpha, k, \\omega) = \\frac{\\Gamma(k+1) \\Gamma(1+\\alpha)}{\\Gamma\\left(\\displaystyle{1+\\frac{ k + \\alpha - \\omega}{2}}\\right) \\Gamma\\left(\\displaystyle{1+\\frac{ k + \\alpha + \\omega}{2}}\\right)}\\, .\n\\end{equation}\n\nA choice of asymptotic boundary conditions will constrain the allowed values of $(\\omega,k)$, corresponding to the normal modes of the system. In the Dirichlet case $(\\beta=0)$ the source is identified with $b^{(+)}_{u}$ and the normal modes are given by the zeros of $C(\\alpha, |k|, \\omega)$, located at\n\\begin{equation}\\label{omega D2}\n    \\omega^{\\pm}_{nk} = \\pm \\left(2 n + |k|  +\\alpha \\right)\\, ,\\qquad   n = 1,2,\\ldots\n\\end{equation}\n\\noindent and the zeros of the denominator in \\eqref{relation coeff global AdS}, located at\n\\begin{alignat}{2}\\label{omega D zero modes}\n    \\omega^{+}_{0k}\n    ={}&\n   +(  |k| + \\alpha)    &\\phantom{aaaa}&{\\rm for} \\ k > 0\n     \\\\\n     \\omega^{-}_{0k}\n     ={}&\n      -(|k| + \\alpha)  &\\phantom{aaaa}&{\\rm for} \\ k < 0\\, .\n      \\label{omega D zero modes 2}\n\\end{alignat}\n\\noindent We notice that the normal modes with $n \\geq 1$ are doubly degenerate, with each frequency attained for both $k$ and $-k$, while $\\omega^\\pm_{0k}$ occur only once.\n\n\\noindent Similarly, for Neumann boundary condition $(b^{(-)}_v = 0)$ we find the eigenfrequencies\n\\begin{equation}\\label{omega N2}\n    \\omega^{\\pm}_{nk} = \\pm \\left(2 n + |k|  - \\alpha \\right)\\, ,\\qquad   n = 1,2,\\ldots\n\\end{equation}\n\\noindent in addition to\n\\begin{alignat}{2}\n    \\omega^{-}_{0k}\n    ={}& \\alpha - |k|  &\\phantom{aaaa}&{\\rm for} \\ k > 0\n     \\\\\n     \\label{omega N zero modes k <0}\n     \\omega^{+}_{0k}\n     ={}&\n      |k| - \\alpha &\\phantom{aaaa}&{\\rm for} \\ k < 0\\, .\n\\end{alignat}\n\\noindent More generally, the boundary condition $b^{(+)}_u = \\beta\\, b^{(-)}_v $ for finite $\\beta$ gives\n\\begin{equation}\\label{bc eq for w}\n    C(\\alpha, |k|, \\omega) - \\beta \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega) = 0\\, .\n\\end{equation}\n\\noindent For generic $\\beta$, we will proceed numerically, examining the structure of the solutions of \\eqref{bc eq for w} in the complex-$\\omega$ plane as a function of $k$, $\\beta$ and $\\alpha$. For $\\beta > 0$ and all values of $k$, we find an infinite discrete set of real frequency solutions, in analogy to the Dirichlet and Neumann cases. Now, while for $\\beta < 0$ and $k>0$ all frequencies are real, for $\\beta < 0$ and $k<0$ a pair of complex solutions occurs in addition to the series of real solutions. Notice that, with the exception of $\\omega$, all the parameters in \\eqref{bc eq for w} are real, which implies that complex solutions must appear in complex conjugate pairs. These complex solutions go off to $\\pm i \\infty$ as $\\beta \\rightarrow 0$, in agreement with our analysis for Dirichlet boundary conditions.  See figures \\ref{w mix bc} and \\ref{w mix bc k neg}. The complex frequency solutions signal an instability of the system, since some perturbations can grow exponentially with time. This instability is associated with ghosts, as we will see in section \\ref{ip massive global}. We stress that, aside from the existence of complex frequencies, there is nothing particularly special about $\\beta < 0$. In fact, we will see below that all values of $\\beta \\neq 0$ are qualitatively equivalent, since they all yield ghosts.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{w mix bc}\n\\includegraphics[width=0.4\\linewidth]{wmixbcGkpos.pdf}\n}\\qquad\\qquad\n\\subfigure[][]{\n\\label{w mix bc k neg}\n\\includegraphics[width=0.4\\linewidth]{wmixbcGkneg.pdf}}\n\\caption{\\ref{w mix bc}:  We plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{bc eq for w} in the complex-$\\omega$ plane for $\\{k =2,\\,\\alpha = 0.2,\\,\\beta = 1.7\\}$. We observe that these solutions only intersect for $\\mbox{Im}(\\omega) = 0$, which illustrates the fact  that \\eqref{bc eq for w} has only real solutions for $\\beta > 0$. \\ref{w mix bc k neg}: For $\\{k =-2,\\,\\alpha = 0.2,\\, \\beta = -1.7\\}$, we plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{bc eq for w} in the complex-$\\omega$ plane. We note that in this case there are complex frequency solutions. }\\end{figure}\n\nFinally, we consider the hybrid boundary conditions \\eqref{gen hyb}. As noted in section \\ref{sec:bc}, the condition \\eqref{gen hyb} along with the flatness of $\\delta A^{(0)}$ imply the extra requirement \\eqref{hyb on b}, which in view of our mode decomposition translates into\n\\begin{equation}\\label{hyb bc coeff}\n    \\kappa^2 \\frac{k + \\omega}{k -\\omega} C(\\alpha, |k|, \\omega) - \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega) = 0\\, .\n\\end{equation}\nThus, the spectrum of frequencies is given by the solutions of \\eqref{hyb bc coeff} provided the flat components of the connection are related to the massive ones by \\eqref{gen hyb}. Lacking an analytic solution of \\eqref{hyb bc coeff} for finite $\\kappa$, we proceed numerically. Studying \\eqref{hyb bc coeff} for various values of the parameters, we find that generically there is an infinite set of real solutions. Additionally, a pair of complex solutions occurs when $k > 0$ and $|\\kappa| > |\\kappa_c|$, where $\\kappa_c$ is an increasing function of $\\alpha$ and $k$. See figures \\ref{tach hyb real}, \\ref{tach hyb cplex} for an illustration of this fact. We have also verified numerically that the complex solutions go off to $\\pm i \\infty$ as $|\\kappa|$ approaches infinity, consistent with the Dirichlet result. As in the case of mixed boundary conditions, the complex frequency solutions correspond to a dynamical instability of the system that is associated to ghosts. We shall also find that the all finite values of $\\kappa$ yield ghosts, in agreement with the CFT unitarity bound.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{tach hyb real}\n\\includegraphics[width=0.4\\linewidth]{tachhybcplex.pdf}\n}\\qquad\\qquad\n\\subfigure[][]{\n\\label{tach hyb cplex}\n\\includegraphics[width=0.4\\linewidth]{tachhybreal.pdf}}\n\\caption{\\ref{tach hyb real}: We plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{hyb bc coeff} in the complex-$\\omega$ plane for $\\{k =1,\\,\\alpha = 0.8,\\,\\kappa = 1.0\\}$. We note that there are real solutions but also a pair of complex solutions near $|\\omega| = 0$.\n\\ref{tach hyb cplex}: Solutions for $\\{k =1,\\,\\alpha = 0.8,\\, \\kappa = 0.9\\}$. We observe that the complex solutions become real, which shows that for $\\alpha = 0.8$ the critical value of $\\kappa$ is near $|\\kappa_c| = 0.95\\,$.}\n\\end{figure}\n\nAs mentioned above, given a solution of \\eqref{hyb bc coeff} the components of the flat connection are uniquely determined by \\eqref{gen hyb}.\nIt is worth mentioning that with these boundary conditions the chiral currents acquire a non-vanishing expectation value. See section \\ref{sec:1pt}.\n\n\\subsection{Poincar\\'e patch of  $AdS_{3}$}\\label{sec: spec PP}\nWe now carry out the study of the spectrum of normalizable excitations for the boundary conditions of interest in the Poincar\\'e patch of $AdS_{3}$. As in the global AdS case, normalizability at the conformal boundary is guaranteed by the restriction $0 < \\alpha < 1$. On the other hand, the treatment of the Poincar\\'e horizon turns out to be more delicate as we will discuss in detail below.\n\\subsubsection{Flat sector}\nLet us first consider the flat sector. As mentioned in section \\ref{subsection:symplectic flux}, when the geometry is the Poincar\\'e patch of $AdS_3$, symplectic flux can generically leak through the Poincar\\'e horizon. In the flat sector, the easiest way to see this is to note that in this sector the theory is actually topological, so there is no difference between the Poincar\\'e horizon and the conformal boundary. From our experience with the latter, we conclude that good boundary conditions in the flat sector correspond to fixing half of the connection on the Poincar\\'e horizon. We will impose the condition\n\\begin{equation}\\label{bc A0 at P hor}\n    \\left. \\delta A^{(0)}_x \\right|_{z = \\infty} = 0\\, .\n\\end{equation}\nAs reviewed in appendix \\ref{section:sym bndy}, when fixing the spatial part of $A^{(0)}_i$, the degrees of freedom that reside at the Poincar\\'e horizon become pure gauge, which allows us to focus on the physics at the boundary. Note however that \\eqref{bc A0 at P hor} can be generalized in the same way as the boundary conditions discussed in section \\ref{sec:bc}. Also, in analogy with the global case, we see that $U(1)$ transformations that set $A_z = 0$ everywhere in the bulk generically do not preserve the boundary condition \\eqref{bc A0 at P hor}, so they are not allowed symmetries of the system.\n\nFrom the above discussion, it is clear that the spectrum of the flat connections in the Poincar\\'e case is analogous to the one in global AdS discussed in section \\ref{sec: spec global adS}. In particular, the frequencies are fixed as \\eqref{freq global} as a consequence of the boundary conditions at the conformal boundary, which are identical to the ones we consider in the Poincar\\'e patch. Note however that in the present case the spatial momentum $k$ is not quantized, so the spectrum of eigenfrequencies is continuous.\n\\subsubsection{Massive sector}\nLet us now focus on the massive sector. In order to solve the equations of motion, we use the mode decomposition $\\delta B_\\mu =  e^{i( k_u u + k_v v )} b_\\mu\\, $; see appendix \\ref{sec: soln PP} for the explicit solutions. We classify the modes according to the value of $m^2:= -k_u k_v = \\omega^2 - k^2$ as: timelike ($m^2 > 0$), lightlike ($m^2 = 0$), and spacelike $(m^2 < 0)$.\n\nFrom \\eqref{gen asympt} it follows that the asymptotic expansion of the solution for the massive mode reads (after noting that near the boundary we have $r = 1\/z$)\n\\begin{equation}\\label{asympt PP}\n    b_\\mu = z^{-\\alpha} b^{(+)}_\\mu + z^{\\alpha} b^{(-)}_\\mu + \\mathcal{O}\\left(z^{1-\\alpha}\\right).\n\\end{equation}\nHere $z$ is the radial variable defined in \\eqref{ds2}. Note that under the isometry \\eqref{dilation PP}, the coefficients in \\eqref{asympt PP} scale as\n\\begin{equation}\\label{scaling}\n    b^{(+)}_\\mu \\rightarrow c^{\\alpha-1} b^{(+)}_\\mu~,  \\qquad b^{(-)}_\\mu \\rightarrow c^{-\\alpha-1} b^{(-)}_\\mu~,\n\\end{equation}\n\\noindent in agreement with our discussion of section \\ref{subsection:conf dims} regarding the conformal dimensions of the dual operators.\n\nHaving said this, let us consider the spectrum of timelike modes, whose radial profile is given by \\eqref{soln tl PP}. Comparing \\eqref{soln tl PP} with \\eqref{asympt PP}, we read-off\n\\begin{align}\\label{}\n    b^{(+)}_{u}\n     ={}&\n      k_u C(\\vec{k}) \\frac{2^{1+\\alpha} m^{-(\\alpha+1)}}{\\Gamma(-\\alpha)}\\,,&\n       b^{(+)}_v ={}&\n        b^{(+)}_z = 0\\, ,\n\\\\\n    b^{(-)}_v\n    ={}&\n     k_v A (\\vec{k}) \\frac{2^{1-\\alpha} m^{\\alpha-1}}{\\Gamma(\\alpha)}\\,,&\n      b^{(-)}_u ={}&\n       b^{(-)}_z = 0\\, .\n\\end{align}\n\\noindent Thus, $C(\\vec{k}) = 0$ corresponds to Dirichlet and $A(\\vec{k})= 0$ to Neumann boundary conditions. We also find that mixed boundary conditions imply\n\\begin{equation}\\label{mix bc}\n    C(\\vec{k}) = \\beta \\frac{k_v}{k_u} \\frac{\\Gamma(-\\alpha)}{4^\\alpha \\Gamma(\\alpha)} m^{2\\alpha} A(\\vec{k})\\,,\n\\end{equation}\n\\noindent while hybrid boundary condition translate into\n\\begin{equation}\\label{hybrid bc}\n    C(\\vec{k}) = \\kappa^{-2} \\frac{\\Gamma(-\\alpha)}{4^\\alpha \\Gamma(\\alpha)} m^{2\\alpha} A(\\vec{k})\\, .\n\\end{equation}\n\nWe stress that the timelike modes above oscillate rapidly near $z= \\infty$. As a result, one can construct wave packets that behave smoothly near the Poincar\\'e horizon. Alternatively, one can work with the modes as they stand and treat their symplectic products in the appropriate distributional sense, and this is the strategy we adopt below. More precisely, in section \\ref{sec ip PP} we find that the timelike modes are in fact (plane wave-)normalizable for all the boundary conditions of interest.\n\nWe now study the existence of spacelike solutions, whose profiles are given by \\eqref{soln sl PP}. Taking $\\mbox{Re}( p) >0$ by convention, we see that unless we set $C(\\vec{k}) = 0$ in \\eqref{soln sl PP}, the solutions blow up exponentially at the horizon ($z=\\infty$) and are thus non-normalizable. Therefore, we set  $C(\\vec{k}) = 0$ which implies that the coefficients of the asymptotic expansion for the spacelike solution can be written as\n\\begin{align}\\label{bplus tach}\n    b^{(+)}_u\n     ={}&\n       A(\\vec{k}) k_u 2^{\\alpha} p^{-\\alpha-1} \\Gamma(1+\\alpha)\\,,& b^{(+)}_v ={}& b^{(+)}_z = 0\\, ,\n    \\\\\n\\label{bminus tach}\n    b^{(-)}_v ={}&\n     A(\\vec{k}) k_v  2^{-\\alpha} p^{\\alpha-1} \\Gamma(1-\\alpha)\\,,& b^{(-)}_u ={}& b^{(-)}_z = 0\\, .\n\\end{align}\n\\noindent Both Dirichlet and Neumann boundary conditions require $A(\\vec{k}) = 0$, so in these cases spacelike solutions do not exist. Mixed boundary conditions $b^{(+)}_u = \\beta b^{(-)}_v$, in turn, imply the relation\n\\begin{equation}\\label{mix bc tach PP}\n    \\tilde{\\beta} = \\frac{(k-\\omega)^{1-\\alpha}}{(k+\\omega)^{1+\\alpha}}\\, ,\n\\end{equation}\n\\noindent where we have defined $\\tilde{\\beta} = 4^{-\\alpha} \\frac{\\Gamma(1-\\alpha)}{\\Gamma(1+\\alpha)} \\beta$. Spacelike solutions are then in one-to-one correspondence with the solutions of \\eqref{mix bc tach PP}, which we now study. First, we observe that regularity at transverse infinity, $|x| \\rightarrow \\infty$, requires $k \\in \\mathbb{R}$. On the other hand, recall that we derived \\eqref{mix bc tach PP} only under the assumption $\\mbox{Re}(p) >0$, so in principle complex frequency solutions are allowed and their existence is exclusively dictated by \\eqref{mix bc tach PP}. Examining \\eqref{mix bc tach PP} it is not hard to conclude that for all $\\beta >0$ there are real solutions in the region $k-\\omega>0$, $k+\\omega >0$; see figure \\ref{w real tach}. On the other hand, if $\\beta < 0$ real solutions are ruled out, but we find instead a pair of complex-frequency solutions that are conjugate to each other, see figure \\ref{w cplex tach}. The fact that our results depend on the sign of $\\beta$ only can be easily understood in terms of the scaling symmetry \\eqref{dilation PP}, which acts non-trivially on $\\beta$.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{w real tach}\n\\includegraphics[width=0.42\\linewidth]{tachrealw.pdf}\n} \\qquad \\qquad\n\\subfigure[][]{\n\\label{w cplex tach}\n\\includegraphics[width=0.4\\linewidth]{complexwtach.pdf}}\n\\caption{\\ref{w real tach}: We plot in red the real solutions of \\eqref{mix bc tach PP} in the $(k,\\omega)$ plane for $\\{\\tilde{\\beta} = 0.5\\,,\\alpha = 0.6\\}$. The dashed line corresponds to the light-cone in momentum space. \\ref{w cplex tach}: For $\\{\\tilde{\\beta} = -0.5,\\,\\alpha = 0.6,\\, k = -1\\}$, we plot in red\/blue the solutions to the real\/imaginary part of \\eqref{mix bc tach PP} in the complex-$\\omega$ plane. Complex solutions are given by the intersection of both lines at $\\omega \\approx 1.45 \\pm i 2.01\\,$. This implies $p \\approx 2.18 - i 1.34$ so $\\mbox{Re}(p) > 0$, consistent with the assumption under which the solution is regular at the Poincar\\'e horizon. }\n\\end{figure}\n\nSimilarly, for spacelike solutions satisfying the ``hybrid'' boundary condition \\eqref{hyb on b} we have\n\\begin{equation}\\label{hyb bc tach PP}\n    \\tilde{\\kappa}^2 = (k^2 - \\omega^2)^\\alpha\\, ,\n\\end{equation}\n\\noindent where we defined $\\tilde{\\kappa}^2 = 4^\\alpha \\frac{\\Gamma(1+\\alpha)}{\\Gamma(1-\\alpha)} \\kappa^2 $. Since $\\tilde{\\kappa}^2 > 0$, it follows that $ \\omega^2 = k^2 - \\tilde{\\kappa}^{2\/\\alpha}\\,$. Now, because $k$ can be arbitrarily small, we find real as well as imaginary frequency solutions for all values of $\\kappa$. In analogy to the mixed boundary conditions studied above, we can use the scaling symmetry \\eqref{dilation PP} to set $\\kappa$ to any desired value. Furthermore, in this case the spectrum is insensitive to the sign of $\\kappa$ due to the structure of the boundary condition \\eqref{hyb on b}.\n\nFinally, we discuss the lightlike modes. For the right-moving modes, i.e. those with $k_{v}=0$, the general solution is given in \\eqref{soln PP kv=0}. Examining the expression for the inner product, we conclude that the norm of the right-moving modes diverges if $b_z \\neq 0$. Therefore, we find that right-moving modes are only allowed for Neumann boundary conditions. In this case, they read\n\\begin{equation}\\label{RM Neumann}\n    \\delta B = A(k_u) z^{-\\alpha} e^{i k_u u} du\\, .\n\\end{equation}\n\n\\noindent We emphasize that the solution \\eqref{RM Neumann} is smooth at the Poincar\\'e horizon. Similarly, the left-moving modes \\eqref{soln PP ku=0} are only normalizable only for Dirichlet boundary conditions, in which case they can be written as\n\\begin{equation}\\label{LM Dirichlet}\n    \\delta B = C(k_v) z^{\\alpha} e^{i k_v v} dv\\,.\n\\end{equation}\n\\noindent Note however that in this case they fail to be smooth at $z = \\infty$, which removes them from the spectrum.\n\n\n\\section{Evaluating the symplectic product}\\label{section:norms}\nNext, we compute the symplectic product of the various solutions found in section \\ref{section:spectrum}. The emphasis will be on determining the existence of ghosts, which, as stated above, correspond to positive (resp. negative) frequency modes having negative (resp. positive) norm. According to CFT considerations regarding unitarity bounds for vector operators, we expect the theories in which $B^{(+)}$ fluctuates to contain ghosts. Up to certain subtleties present in the Poincar\\'e patch, we will verify that the expected ghosts arise in the bulk, consistent with the field theory result. In addition, we will also find ghosts in the flat sector for a certain class of double-trace boundary conditions; the latter are not related to unitarity bounds of the kind mentioned above. The presence of these ghosts should not be at all surprising, however, since the symplectic structure restricted to the flat sector is not manifestly positive definite, see e.g. \\eqref{omega split}. We find it convenient to study first the flat sector separately, assuming that we have chosen boundary conditions which decouple this sector from the massive one. The results for the massive sector and the mixed hybrid case will be presented later in this section.\n\n\\subsection{Flat sector}\\label{ip flat sector}\nWe start by discussing the case of global AdS. The symplectic product is evaluated on a slice of constant $t$, so we have $\\sqrt{h} n_\\mu = \\sqrt{g} \\delta_\\mu^t$. Then, using \\eqref{omega recap} and \\eqref{ip} the symplectic product reads\n\\begin{equation}\\label{ip flat 1}\n    (A_1,A_2) = - i \\hat{\\alpha} \\int d^2x\\, \\varepsilon^{t \\lambda \\nu} \\left(\\delta_1 A^{(0)}_{\\lambda}\\right)^* \\delta_2 A^{(0)}_{\\nu}\\,.\n\\end{equation}\n\\noindent Using the solution \\eqref{soln flat global} and the mode decomposition $\\lambda = e^{-i \\omega t + i k x} \\hat \\lambda$, it is straightforward to arrive at\n\\begin{equation}\\label{ip flat 2}\n    (A_1,A_2) = - 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2} k_1 e^{i t (\\omega_1 - \\omega_2)} \\int_0^\\infty d\\rho \\left(\\hat{\\lambda}_2\\partial_\\rho \\hat{\\lambda}_1^*  -  \\hat{\\lambda}_1^* \\partial_\\rho \\hat{\\lambda}_2\\right).\n\\end{equation}\n\\noindent \nUpon using \\eqref{freq global} in \\eqref{ip flat 2} we see that the time dependence in the symplectic product cancels out, as required by conservation of the symplectic structure. Finally, integrating by parts the first term in \\eqref{ip flat 2} and using the smoothness condition $\\delta A_x^{(0)} = 0 $ at $\\rho = 0\\,$, we conclude\n\\begin{equation}\\label{ip flat 3}\n    (A_1,A_2) = 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2} \\frac{\\omega_1}{\\hat{\\beta}} \\bigl| \\hat{\\lambda}_{1, \\partial}  \\bigr|^2\\, .\n\\end{equation}\n\\noindent where $\\hat{\\lambda}_{\\partial} = \\hat{\\lambda} \\bigr|_ {\\partial M}$ are the (finite) boundary values of the Fourier components of $\\lambda$. Note that \\eqref{ip flat 3} is manifestly finite and conserved, as promised.\nWe observe that the symplectic product is local on the boundary values of $\\lambda$, as expected in a topological theory with a boundary. In other words, flat connections for which $\\lambda$ vanishes on the boundary are pure gauge degrees of freedom. Moreover, for the boundary condition $\\bigl.\\delta A^{(0)}_x \\bigr|_{\\partial M} = 0$, i.e. $k=0$, we also find that the flat sector becomes pure gauge. We refer the reader to appendix \\ref{section:sym bndy} for a discussion on gauge symmetries.\nRecall that ghost excitations are defined as positive(negative) frequency solutions with negative(positive) norm. Thus, with the assumption that $\\hat \\alpha > 0$, we conclude that there are ghosts in the flat sector for $\\hat\\beta < 0$. Although in this case there is no obvious violation of unitarity bounds (recall that $A^{(0)}$ has scaling dimension one), the mere fact that the symplectic product \\eqref{ip flat 1} is not positive definite is an indication that such ghosts might occur.\n\nLet us now focus on the case of Poincar\\'e coordinates. As discussed in section \\ref{sec: spec PP}, with our choice of boundary conditions at the Poincar\\'e horizon, the flat sector largely resembles that of global $AdS_3$. Carrying out a calculation analogous to the one above we find that the symplectic product for flat modes in the Poincar\\'e patch is given by (\\ref{ip flat 3}), with the replacement of the Kronecker-$\\delta$ by a Dirac $\\delta$-function since $k$ is no longer quantized.\n\n\n\\subsection{Massive sector in global AdS}\n\\label{ip massive global}\n\nNext we  evaluate the symplectic products \\eqref{ip} for the positive frequency modes found in section \\ref{sec: spec global adS}. We first focus on the non-flat sector, and at the end of this section we consider the hybrid boundary conditions which introduce a mixing with the flat sector. We choose to evaluate the symplectic product on a surface $\\Sigma$ in which $t = {\\rm const}$, in which case we obtain\n\\begin{equation}\\label{ip global nf1}\n    (A_1, A_2) = i \\hat{\\alpha} \\int dz dx\\, \\varepsilon^{t \\lambda \\nu} \\delta_1 B^*_{ \\lambda} \\delta_2 B_{ \\nu}\\, .\n\\end{equation}\n\\noindent Using the mode decomposition $\\delta B_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} b_\\mu (k) $ in \\eqref{ip global nf1} and computing the integral over $x$, we get\n\\begin{equation}\\label{ip global}\n    (A_1,A_2) = - 2 \\pi i\\hat{\\alpha}\\, \\delta_{k_1, k_2} e^{i\\frac{t}{L} (\\omega_1- \\omega_2)} \\int_0^\\infty \\left( b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\right) d \\rho\\, .\n\\end{equation}\n\\noindent It will prove convenient to express \\eqref{ip global} in terms of $b_u$ and $b_v$. To do so, we recall that $b_x = \\frac{1}{2}(b_u + b_v)$ along with the fact that the first order equation for $b$ yields\n\\begin{equation}\\label{elim brho}\n    b_\\rho = \\frac{i \\alpha \\rho}{ 2 k(1+\\rho^2)} (b_u - b_v) - \\frac{i}{2 k} (b_u + b_v)' .\n\\end{equation}\n\\noindent Therefore, we have\n\\begin{align}\\label{brho bx}\n\\nonumber\n    - i \\int_0^\\infty d \\rho \\left(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\right)\n     ={}&\n      \\frac{\\alpha}{2 k_1}\\Bigl(\\langle b_{1v}, b_{2v} \\rangle - \\langle b_{1u}, b_{2u} \\rangle\\Bigr)\n      + \\frac{1}{4 k_1}\\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\bigg| ^\\infty_0\n       \\\\\n\\nonumber\n    ={}&\n     \\frac{\\alpha \\rho(1+ \\rho^2)}{2 k_1(\\omega_1 - \\omega_2)} \\Bigl[\\bigl(b_{1v} b'_{2v} - b_{2v} b'_{1v}\\bigr) - \\bigl(b_{1u} b'_{2u} - b_{2u} b'_{1u}\\bigr)  \\Bigr] \\bigg| ^\\infty_0\n      \\\\\n    &\n    + \\frac{1}{4 k_1}\\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\bigg| ^\\infty_0\\, .\n\\end{align}\n\\noindent Here, $\\langle \\cdot, \\cdot \\rangle$ is the Sturm-Liouville (SL) product defined in appendix \\ref{SL gen}. It is straightforward to verify that regularity of the modes at the origin guarantees that the contribution to \\eqref{brho bx} from $\\rho = 0$ vanishes, so the solutions found in \\ref{sec: spec global adS} are indeed normalizable, as promised. For generic frequencies $\\omega_1$ and $\\omega_2\\,$, the contribution from $\\rho = \\infty$ is finite and it evaluates to\n\\begin{equation}\\label{IP gen omega}\n    - i \\int_0^\\infty d \\rho \\bigl(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\bigr) = \\frac{b^{(+)}_{2 u} b^{(-)}_{1 v} - b^{(+)}_{1 u} b^{(-)}_{2 v} }{2(\\omega_1 - \\omega_2)}\\, .\n\\end{equation}\n\\noindent It is not hard to see that \\eqref{IP gen omega} vanishes for Dirichlet, Neumann and mixed boundary conditions if $\\omega_1 \\neq \\omega_2$. Using this fact in \\eqref{ip global} we conclude that the inner product is conserved (i.e. independent of $t$) for all of the above boundary conditions, in agreement with our analysis of the symplectic flux. In order to calculate the norms, we take the limit $\\omega_2 = \\omega_1$ in \\eqref{IP gen omega} and set $\\omega_1$ to its quantized value at the end of the calculation. Since \\eqref{IP gen omega} vanishes for $\\omega_1 \\neq \\omega_2$, we can write\n\\begin{equation}\\label{IP gen omega2}\n    - i \\int_0^\\infty d \\rho \\bigl(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\bigr) = \\delta_{\\omega_1, \\omega_2}\n    \\frac{1}{2} \\left(b^{(+)}_{1 u} \\partial_{\\omega_1} b^{(-)}_{1 v} - b^{(-)}_{1 v} \\partial_{\\omega_1} b^{(+)}_{1 u} \\right)\\, .\n\\end{equation}\n\\noindent Plugging \\eqref{IP gen omega2} into \\eqref{ip global} we find the following general expression for the symplectic products:\n\\begin{equation}\\label{ip global formal}\n    (A_1,A_2) = \\pi \\hat{\\alpha} \\delta_{\\vec{k}_1, \\vec{k}_2} \\left(b^{(+)}_{1 u} \\partial_{\\omega_1} b^{(-)}_{1 v} - b^{(-)}_{1 v} \\partial_{\\omega_1} b^{(+)}_{1 u} \\right).\n\\end{equation}\n\\noindent We now specialize to the various boundary conditions of interest. For Dirichlet boundary conditions, the spectrum of eigenfrequencies is given by \\eqref{omega D2}, \\eqref{omega D zero modes}. The positive frequency solutions can be expressed more succinctly as\n\\begin{equation}\\label{omega D positive}\n   \\omega^{+}_{n,k} =  2 \\bigl[n + \\theta(-k)\\bigr] + |k| + \\alpha \\qquad  n = 0,1,2,\\ldots\\,,\n\\end{equation}\n\\noindent where $\\theta(x)$ is the Heaviside function, defined as $\\theta(x) = 1 $ for $x \\geq 0$ and $\\theta(x) = 0$ for $x < 0$. Using \\eqref{omega D positive} in \\eqref{ip global formal} we find\n\\begin{equation}\\label{ip D global}\n    (A_1,A_2) = \\pi \\hat{\\alpha}  \\frac{(-1)^n \\pi  \\alpha  \\csc(\\pi  \\alpha) n!\\, \\Gamma \\bigl(2\\theta(-k)+|k|+n \\bigr) \\Gamma\\bigl(- s(-k) -n-\\alpha\\bigr)}{4 \\Gamma\\bigl(1-\\alpha\\bigr)^2\\, \\Gamma\\bigl(1+|k|+n+\\alpha\\bigr)}\\, ,\n\\end{equation}\n\\noindent where $n$ is a non-negative integer. Here we have normalized the modes in such a way that the leading term has coefficient $1$. We shall continue to do so henceforth, unless explicitly otherwise stated. Similarly, the spectrum of positive frequency solutions for Neumann boundary conditions can be written as\n\\begin{equation}\\label{omega N positive}\n    \\omega_N =  2 \\bigl[n + \\theta(k)\\bigr] + |k| - \\alpha \\qquad  n = 0,1,2,\\ldots\n\\end{equation}\n\\noindent as it follows from \\eqref{omega N2} and \\eqref{omega N zero modes k <0}. Inserting \\eqref{omega N positive} in \\eqref{ip global formal} we conclude that the Neumann norms read\n\\begin{equation}\\label{ip N global}\n    (A_1,A_2) = \\pi \\hat{\\alpha}  \\frac{(-1)^{n+1} \\pi  \\alpha  \\csc(\\pi  \\alpha) n! \\Gamma \\bigl(2\\theta(k)+|k|+n \\bigr) \\Gamma\\bigl(-s(k) -n+\\alpha\\bigr)}{4 \\Gamma\\bigl(1+\\alpha\\bigr)^2\\, \\Gamma\\bigl(1+|k|+n-\\alpha\\bigr)}\\, ,\n\\end{equation}\n\\noindent where $n$ is a non-negative integer. Inspecting \\eqref{ip D global} and \\eqref{ip N global}, we note that all the modes have positive norm with the exception of the Neumann modes characterized by $n=0$, $k<0$, whose frequencies are given by \\eqref{omega N zero modes k <0}. Therefore, we conclude that the theory contains ghosts for Neumann boundary conditions, as expected.\n\nLet us now focus on mixed boundary conditions. In this case, the lack of a closed expression for the frequencies prevents us from displaying the norm explicitly. However, we find substantial evidence that ghosts must be present in the system for generic values of the deformation parameter $\\beta$. For $\\beta < 0$, the existence of ghosts follows immediately from the existence of complex frequency solutions, see for example \\cite{Andrade:2011dg}. The argument is as follows. First we recall that, as all the parameters are real, the complex frequencies always occur in pairs of complex conjugate values;  c.f. figure \\ref{w mix bc k neg}. Second, denoting the aforementioned solutions by $\\psi_1$, $\\psi_2$, we can verify that $(\\psi_1, \\psi_1) = (\\psi_2, \\psi_2) = 0$. A simple way to see this is to note that the norms have the overall time-dependent factor $\\exp(- 2 t\\, \\mbox{Im}(w))$. Since this is in conflict with conservation, the norms must vanish. Third, the definition of the norm guarantees that cross-terms satisfy $(\\psi_1, \\psi_2) = (\\psi_2, \\psi_1)^*$, and we can explicitly verify that they are non-vanishing. Finally, diagonalizing the symplectic structure we find that one of the excitations has negative norm, signaling the presence of ghosts. We turn now to the case $\\beta > 0$. In this situation we did not find evidence of complex frequency solutions, so we need to examine the norms in more detail. Indeed, we found numerical evidence that ghosts should be present for this case as well, c.f. figure \\ref{ghostsmixbc}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.75]{ghostmixbckappapos.pdf}\n\\caption{For $\\{\\alpha = 0.7,\\, k = -2,\\, \\beta = 1.1\\}$, we plot the formal expression for the norm \\eqref{ip global formal} with a red dashed line and the left hand side of \\eqref{bc eq for w}, whose zeroes correspond to the allowed frequencies, with a red solid line. Note that the smallest frequency corresponds to a ghost. Exploring the norm numerically for various values of the parameters, we find that this behavior is generic.}\n\\label{ghostsmixbc}\n\\end{center}\n\\end{figure}\n\nFinally, we consider the hybrid boundary conditions. Recall that in section \\ref{sec: spec global adS} we found that there is a pair of complex frequency solutions if the absolute value of the deformation parameter $\\kappa$ is large enough. Following the conventions in section \\ref{sec: spec global adS}, we denote the critical value by $|\\kappa_c|$. As mentioned above, these complex frequency solutions correspond to ghost\/anti-ghost pairs, so we do not discuss this case any further. Let us now examine the norms of the real frequency solutions. We expect the pair of complex frequency solutions that occur for $|\\kappa| > |\\kappa_c|$ to remain as a ghost\/anti-ghost pair when we tune $|\\kappa|$ below $|\\kappa_c|$. We will exhibit ample numerical evidence that this is indeed the case, and thus conclude that ghosts are present for all values of $\\kappa$.\n\nRecall that the symplectic structure splits into the contributions from the flat and non-flat sectors as in \\eqref{omega recap}, and that the symplectic product is given in terms of the symplectic structure by \\eqref{ip}. Choosing the Cauchy slice $\\Sigma$ on which we evaluate the product to be a surface of constant $t$, we can write the inner product as\n\\begin{equation}\\label{ip hyb bc}\n    (A_1, A_2) = (A_1, A_2)_{nf} + (A_1, A_2)_{f}\\, ,\n\\end{equation}\n\\noindent where\n\\begin{align}\\label{ip nf exp}\n    (A_1, A_2)_{nf}\n    ={}&\n     - i \\hat{\\alpha} \\int_{\\Sigma} d^{2}x\\Bigl(\\delta B^*_{1 \\rho} \\delta B_{2x} - \\delta B^*_{1x} \\delta B_{2 \\rho}\\Bigr)\n\\\\\n\\label{ip f exp}\n    (A_1, A_2)_{f}\n    ={}&\n     i \\hat{\\alpha} \\int_{\\Sigma}d^{2}x \\Bigl(\\delta_1 A^{(0) *}_{ \\rho} \\delta_2 A^{(0)}_{ x} - \\delta_1 A^{(0) *}_{ x} \\delta_2 A^{(0)}_{\\rho}\\Bigr).\n\\end{align}\n\\noindent Next, we introduce Fourier decompositions as $\\delta B_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} b_\\mu (k) $, $\\delta A^{(0)}_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} a^{(0)}_\\mu$ and proceed as above by computing \\eqref{ip nf exp} and \\eqref{ip f exp} for generic frequencies $\\omega_1$ and $\\omega_2$. The expression \\eqref{ip nf exp} is then given by \\eqref{IP gen omega} and it only remains to compute \\eqref{ip f exp}. We manipulate \\eqref{ip f exp} noting that the flatness condition implies that the modes satisfy\n\\begin{equation}\\label{flat fourier 1}\n    a^{(0)}_\\rho = - \\frac{i}{k}\\left( a^{(0)}_x\\right)'\\, ,\n\\end{equation}\n\\noindent where the prime denotes a radial derivative. Using \\eqref{flat fourier 1} in \\eqref{ip f exp} we thus find\n\\begin{equation}\\label{ip flat fourier}\n    (A_1, A_2)_f = \\left. - 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2}\\, k^{-1} a^{(0)}_{1 x} a^{(0)}_{2 x} \\right|^{\\rho = \\infty}_{\\rho = 0}\\, .\n\\end{equation}\n\\noindent From the regularity condition $a^{(0)}_x =0$ at $\\rho  = 0$ we conclude that only the term at $\\rho = \\infty$ contributes to \\eqref{ip flat fourier}. Thus, gathering the results \\eqref{IP gen omega} and \\eqref{ip flat fourier} we find for generic frequencies\n\\begin{equation}\\label{IP hyb generic}\n    (A_1, A_2) = 2 \\pi \\hat{\\alpha}\\, e^{it(\\omega_1 - \\omega_2)} \\delta_{k_1, k_2} \\left [ \\frac{b^{(+)}_{2 u} b^{(-)}_{1 v} - b^{(+)}_{1 u} b^{(-)}_{2 v}  }{2(\\omega_1 - \\omega_2)} - \\frac{\\bigl(a^{(0)}_{1 u} + a^{(0)}_{1 v}\\bigr) \\bigl(a^{(0)}_{2 u} +  a^{(0)}_{2 v}\\bigr) }{4 k} \\right].\n\\end{equation}\n\\noindent We can readily verify that the symplectic structure is conserved by noting that \\eqref{IP hyb generic} vanishes for $\\omega_1 \\neq \\omega_2$ when the boundary conditions \\eqref{gen hyb}, \\eqref{hyb on b}, hold. Therefore, the symplectic product can be written in terms of the coefficients of the asymptotic expansion as\n\\begin{equation}\\label{IP hyb final}\n    (A_1, A_2) = \\pi \\hat{\\alpha}\\, \\delta_{\\vec{k}_1, \\vec{k}_2} \\biggl[ b^{(+)}_u \\partial_{\\omega_1} b^{(-)}_{v} - \\kappa^2 \\left( \\frac{k+\\omega}{k-\\omega} b^{(+)}_u \\partial_{\\omega_1} b^{(+)}_u  + \\frac{2 k}{(k-\\omega)^2} (b^{(+)}_u)^2 \\right) \\biggr],\n\\end{equation}\n\\noindent where $\\omega$ is implicitly given by the solutions of \\eqref{hyb bc coeff}. Studying \\eqref{IP hyb final} numerically, we find that there is always a ghost among the lowest real frequency modes that occur for $|\\kappa| < |\\kappa_c|$; see figures \\ref{ghost hyb bc 1}, \\ref{ghost hyb bc 2}. Furthermore, we find that this picture continues to hold true for all $|\\kappa|$ in the range $0 \\leq |\\kappa| < |\\kappa_c|$. This must indeed be the case since $\\kappa = 0$ corresponds to Neumann boundary conditions, which were found above to induce violations of unitarity in the bulk.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{ghost hyb bc 1}\n\\includegraphics[width=0.42\\linewidth]{ghosthybbc1.pdf}\n} \\qquad \\qquad\n\\subfigure[][]{\n\\label{ghost hyb bc 2}\n\\includegraphics[width=0.42\\linewidth]{ghosthybbc2.pdf}}\n\\caption{\\ref{ghost hyb bc 1}: For $\\{\\alpha = 0.8,\\,  k = 1,\\, \\kappa = 0.9\\}$ we plot the left hand side of \\eqref{hyb bc coeff} (solid line), whose zeros correspond to the allowed frequencies, and the expression for the norm \\eqref{IP hyb final} (dashed line). We notice that the second to lowest frequency solution is a ghost. By slightly increasing the value of $\\kappa$ the solutions move to the complex plane, as seen in figures \\ref{tach hyb real} and \\ref{tach hyb cplex}. \\ref{ghost hyb bc 2}: For $\\{\\alpha = 0.8$,  $k = 1$, $\\kappa = 0.1\\}$, we plot the left hand side of \\eqref{hyb bc coeff} (solid line) and the expression for the norm \\eqref{IP hyb final} (dashed line). We observe that the lowest frequency mode found for higher values of $|\\kappa|$ disappears, but there is still a ghost in the system. }\n\\end{figure}\n\n\\subsection{Massive sector in Poincar\\'e AdS}\\label{sec ip PP}\nWe now proceed to compute the symplectic product for the Poincar\\'e AdS modes, focusing on the non-flat piece of the connection. We start with the timelike modes. It should be noted that, since the spectrum is continuous, the products should be understood in the sense of distributions. As usual, in order to evaluate \\eqref{ip} we choose $\\Sigma$ to be a surface of $t = {\\rm const}$, so we have\n\\begin{equation}\\label{ip nf1}\n    (A_1, A_2) = i \\hat{\\alpha} \\int dz dx\\, \\varepsilon^{t \\lambda \\nu} \\delta_1 B^*_{ \\lambda} \\delta_2 B_{ \\nu}\\,.\n\\end{equation}\nWe find it convenient to use the mode decomposition $    \\delta B_\\mu = e^{i(-\\omega t + k x)} b_\\mu (k)$ in \\eqref{ip nf1} and computing the integral over $x$ we get\n\\begin{equation}\\label{ip nf2}\n    (A_1, A_2) = 2 \\pi i \\hat{\\alpha}\\, \\delta(k_{1} - k_{2}) e^{i t (\\omega_{1} - \\omega_{2})} \\int_0^\\infty dz \\bigl(b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z} \\bigr),\n\\end{equation}\n\\noindent where we have used $\\varepsilon^{ztx} = - 1$, in consistency with the convention $\\varepsilon^{zuv} = - 1$ employed in appendix \\ref{sec: soln PP}.\nIt is convenient to express \\eqref{ip nf2} in terms of the components $b_u$ and $b_v$. To this end we note that the first order equation for $b$ implies\n\\begin{equation}\\label{bz in bu bv}\n    b_z = -\\frac{i \\alpha}{2 k z} (b_u - b_v) - \\frac{i}{2 k} \\left(b_u + b_v\\right)'\\, ,\n\\end{equation}\n\\noindent and we recall $ b_x = \\frac{1}{2}(b_u + b_v)$, $b_t = \\frac{1}{2}(b_u - b_v)$. It follows that we can write\n\\begin{equation}\\label{rad int PP}\n    i \\int_0^\\infty dz (b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z} ) = \\frac{\\alpha}{2 k} \\int dz\\, z^{-1}\\bigl(b^*_{1 v} b_{2 v} - b^*_{1 u} b_{2 u}\\bigr) - \\frac{1}{4 k} \\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)^*\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\Big|_{0}^\\infty\\, .\n\\end{equation}\n\\noindent The first two terms in \\eqref{rad int PP} correspond to the SL inner product associated to \\eqref{decoupled v} and \\eqref{decoupled u}, respectively. Thus, from the results of appendix \\ref{SL gen} it follows that\n\\begin{align}\\label{rad int PP Wrons}\n\\nonumber\n    i \\int_0^\\infty dz \\bigl(b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z}\\bigr)\n     ={}&\n      \\frac{\\alpha z^{-1}}{2 k (m_1^2 - m_2^2)} \\Bigl[ \\bigl(b_{1 v} b'_{2 v}- b_{2 v} b'_{1 v}\\bigr)  - \\bigl( b_{1 u} b'_{2 u} - b_{2 u} b'_{1 u} \\bigr) \\Bigr] \\Big|_{0}^\\infty\n       \\\\\n     &\n     - \\frac{1}{4 k} \\Bigl[ \\bigl(b_{1 u} + b_{1 v} \\bigr)^*\\bigl(b_{2 u} + b_{2 v}\\Bigr)] \\Big|_{0}^\\infty\\, ,\n\\end{align}\n\\noindent where we have used the explicit form of the SL coefficients \\eqref{SL bu} and \\eqref{SL bv}. Next, using the near-boundary expansion one can readily verify that the contribution from $z = 0$ to \\eqref{rad int PP Wrons} vanishes for Dirichlet, Neumann and mixed boundary conditions. The contribution at the Poincar\\'e horizon can be evaluated by introducing a regulator $z_\\infty$ at large $z$ and using\n\\begin{equation}\\label{J large x}\n    J_\\nu(x) \\to \\sqrt{\\frac{2}{\\pi x}} \\cos \\left( x - \\frac{\\nu  \\pi }{2} - \\frac{\\pi }{4} \\right) \\quad\\mbox{for }\\,\\, x \\gg 1 \\, .\n\\end{equation}\n\\noindent From this point on the calculation proceeds in close analogy to that for a scalar field in Poincar\\'e AdS. We refer the reader to \\cite{Andrade:2011dg} for details.\\footnote{The present calculation exhibits one additional complication: roughly speaking, the third term in \\eqref{rad int PP Wrons} has the structure $\\sim (m_1 m_2)^{-1\/2} z_\\infty \\cos[(m_1 - m_2) z_\\infty]$, so it is indeed power-counting divergent as $z_\\infty \\rightarrow \\infty\\,$. However, integrating this against test functions $f(m_1)$ and $f(m_2)$ of compact support, one can show that the contribution from this type of terms vanishes as we remove the regulator.} Up to terms that vanish in the distributional sense, we find that the general expression for the inner product \\eqref{ip nf2} reads\n\\begin{equation}\\label{ip PP gen}\n    (A_1,A_2) = 2 \\pi \\hat{\\alpha}\\, \\delta^{(2)}(k_1^i - k_2^i) {\\cal Q}(\\alpha, k)\\, ,\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{Q PP gen}\n   {\\cal Q}(\\alpha, k_i) =  2 \\alpha \\left|A(\\vec{k}) + e^{i \\pi \\alpha} C(\\vec{k}) \\right|^2\\, .\n\\end{equation}\n\\noindent Here we have used that $C$ and $A$ satisfy the relation \\eqref{mix bc}. Clearly, the norm \\eqref{ip PP gen} is manifestly positive definite for Dirichlet, Neumann and mixed boundary conditions.\n\nLet us now calculate the products of the spacelike excitations that are present for mixed boundary conditions. As discussed in section \\ref{sec: spec PP}, their radial profile is given by \\eqref{soln sl PP} with $C(\\vec{k}) = 0$, which ensures normalizability since they vanish exponentially at the horizon. Furthermore, the mixed boundary condition holds provided the frequencies satisfy \\eqref{mix bc tach PP}. Recall also that for all $\\beta < 0$ the spectrum contains a pair of solutions $\\psi_1$, $\\psi_2$ whose frequencies are complex conjugate to each other. As argued above, there is always a ghost among these degrees of freedom, so we do not consider this case any further. For $\\beta >0$ we found real frequency spacelike solutions, whose norm we now compute.\n\nSince the real frequency spacelike solutions form a discrete set, we compute their norms in analogy to the calculation of the inner product in global coordinates. Our starting point is the general expression \\eqref{rad int PP Wrons}. Because the radial profiles decay exponentially at the horizon, the only non-vanishing contribution to \\eqref{rad int PP Wrons} comes from the boundary asymptotics. A simple computation reveals that the norm of the real frequency spacelike solutions can be written in terms of the coefficients of the asymptotic expansion as\n\\begin{equation}\\label{norm tach1}\n    (A_1, A_2)_{T} = \\pi \\hat{\\alpha}\\, \\delta(k_1 - k_2) \\Bigl(b^{(+)}_u \\partial_\\omega b^{(-)}_v - b^{(-)}_v \\partial_\\omega b^{(+)}_u\\Bigr)\\, ,\n\\end{equation}\n\\noindent where $\\omega$ satisfies \\eqref{mix bc tach PP}. Plugging in \\eqref{norm tach1} the explicit expressions for the coefficients $b^{(\\pm)}$ found previously in \\eqref{bplus tach} and \\eqref{bminus tach}, we find that the norm of the spacelike solution is\n\\begin{equation}\\label{norm tach2}\n    (A_1, A_2)_{T} =  \\pi \\hat{\\alpha}\\, \\delta(k_1 - k_2)  |A|^2 \\frac{\\pi \\alpha (k- \\alpha \\omega) \\csc (\\pi  \\alpha)}{p^2}\\, .\n\\end{equation}\n\\noindent Note that the sign of the norm \\eqref{norm tach2} is controlled by the factor $(k - \\alpha \\omega)$. Now, it follows from \\eqref{mix bc tach PP} that positive norm solutions occur for positive and negative frequencies, see also figure \\ref{w real tach}. Thus, we conclude that there are ghosts in the theory for mixed boundary conditions and $\\beta > 0$.\n\nIt only remains to discuss hybrid boundary conditions. In this case, the spectrum consists of both real and imaginary frequencies, regardless of the value of the deformation parameter $\\kappa$. As argued above, the existence of non-real frequencies is associated with ghosts on general grounds. Therefore, no detailed calculation of the norms is required to show that this class of theories violate unitarity in the bulk.\n\nAs pointed out in \\cite{Andrade:2011dg}, bulk theories dual to CFT's in which the unitarity bound is violated do not necessarily contain ghosts when the geometry is that of Poincar\\'e AdS. Alternatively, the two-point function suffers from a divergence near the light-cone, which implies that the theory does not exist. This motivates us to inspect the near light-cone structure of the Neumann correlators.\n\nThe boundary (Wightman) two-point function can be easily computed given the matrix of symplectic products, see e.g. \\cite{Andrade:2011dg}. For the timelike modes in the Neumann theory we find\n\\begin{equation}\\label{2pt N}\n    \\langle 0| b^{(+)}_u (-k_i) b^{(+)}_u (k_i) |0 \\rangle = (A_1,A_2)^{-1} |_{Neumann} = \\frac{4^\\alpha q^2 L}{\\pi \\alpha^2 \\Gamma(-\\alpha)^2}  \\frac{(\\omega-k)^{1-\\alpha}}{(\\omega+k)^{1+\\alpha}}\\, .\n\\end{equation}\n\\noindent In order to obtain \\eqref{2pt N} we have normalized the radial profiles such that the leading term is $1$. As expected, the Fourier transform does not converge due to the behavior near $\\omega = -k$; this behavior is to be contrasted with the Dirichlet case, in which we find\n\\begin{equation}\\label{2pt D}\n    \\langle 0| b^{(-)}_v (-k_i) b^{(-)}_v (k_i) |0 \\rangle = (A_1,A_2)^{-1} |_{Dirichlet} =   \\frac{4^{-\\alpha} q^2 L}{\\pi \\alpha^2 \\Gamma(\\alpha)^2}\\frac{(\\omega+k)^{1+\\alpha}}{(\\omega-k)^{1-\\alpha}}\\, .\n\\end{equation}\n\\noindent This is clearly finite as we approach $\\omega = - k$. In the parameter range of interest, namely $0 < \\alpha < 1$, the divergence near $\\omega = k$ is mild enough so that the Fourier transform of \\eqref{2pt D} converges.\n\nThe parallel with \\cite{Andrade:2011dg} extends beyond the existence of the light-cone divergence discussed above, in that this divergence can be related to the appearance of lightlike gauge modes. In fact, recall that in section \\ref{sec: spec PP} we found that the Neumann theory admits the lightlike solution \\eqref{RM Neumann}, which in position space can be written as\n\\begin{equation}\\label{RM Neumann gen}\n    \\delta B = f(u) z^{-\\alpha}  du\\, ,\n\\end{equation}\n\\noindent where $f$ is an arbitrary function of $u$. From \\eqref{ip nf2}, it is clear that the aforementioned solution has zero norm since its $z$-component vanishes. Moreover, it is straightforward to verify that the inner product of \\eqref{RM Neumann gen} with the timelike modes vanishes in the distributional sense. Thus, assuming that the spectrum of the Neumann theory we found in \\ref{sec: spec PP} is complete,\\footnote{In principle, there could be solutions with anharmonic time dependence, which lie outside of the class of modes we consider here. Although we have not studied this possibility in detail, the present setup is self-consistent in that it provides the correct physical results, namely that the Neumann theory is sick.} i.e. that there are only timelike and lightlike modes, we conclude that the lightlike solution \\eqref{RM Neumann gen} is a null direction of the symplectic structure and is thus pure gauge. The reader might be somewhat puzzled by the fact that there is a gauge mode which is not flat. However, one can argue that this must be the case by noting the large arbitrariness in \\eqref{RM Neumann gen} parametrized by the function $f(u)$, which is unconstrained by the equations of motion.\n\n\n\\section{Discussion}\\label{section:discussion}\nBy studying the bulk symplectic structure, we have obtained a class of admissible boundary conditions for the MCS system in asymptotically-AdS spaces. According to the holographic dictionary, these boundary conditions determine the operator content in the possible dual theories. In agreement with the existing literature, we find that there is a vector operator of conformal dimension $1 \\pm \\alpha\\,$, in addition to the well-known chiral currents which are also present in the pure Chern-Simons theory. The vector operator is associated with the Hodge dual of the bulk field strength, which behaves as a massive vector with a mass proportional to the Chern-Simons coupling. It is worth mentioning that the components of these operators satisfy a constraint, so they have less degrees of freedom than naively expected. This feature is reminiscent of the situation in topologically massive gravity, where similar constraints exist \\cite{Skenderis:2009nt}. The chiral currents, on the other hand, are associated to the flat piece of the connection, and are in that sense topological.\n\nOur analysis reveals that, whereas it is possible to impose boundary conditions such that the topological and massive sectors decouple, it is also in principle valid to introduce a mixing between them. In particular, we studied a class of boundary conditions that corresponds to double-trace deformations that couple the chiral currents with the vector operators. Regarding the boundary conditions in which both sectors decouple, we have also considered boundary conditions that yield double-trace deformations within each sector. In this case, it is even possible to generalize these to incorporate multi-trace deformations in the usual way. Our main result is that this apparently large freedom in the choice of boundary conditions is severely restricted once we impose unitarity as an extra requirement.\n\nWe have addressed the issue of unitarity by studying the MCS theory both in Poincar\\'e and global AdS. In these setups, the violations of unitarity generically manifest themselves as ghost excitations in the spectrum of the theories defined with given boundary conditions. The boundary conditions that pass the test of unitarity correspond to fixing the leading behavior of the massive sector (Dirichlet boundary conditions), while separately specifying a linear relation between the two components of the flat connection along the boundary directions. It is worth mentioning that the latter also requires a specific choice of sign in the proportionality constant. Furthermore, we mention that for boundary conditions that fix the spatial part of the boundary connection, the topological degrees of freedom become pure gauge (in the absence of holonomies).\n\nFor the boundary conditions corresponding to double-trace deformations which involve the massive sector, we contented ourselves with numerical results and the reader may wonder whether our analysis was exhaustive enough to rule out the existence of a non-trivial phase diagram. In particular, since the Dirichlet theory is well defined and we can in principle approach it by continuously tuning the deformation parameters, it is reasonable to ask whether there is an open set of unitarity-preserving values near the Dirichlet point. The answer to this question is negative, as it is most easily seen when the geometry is taken to be the Poincar\\'e patch of $AdS_{3}$. In this case, the presence of a scaling symmetry dictates that, up to sign changes, all non-zero values of the coupling constants are equivalent. One can use this fact to draw conclusions regarding the mixed boundary condition $\\delta B^{(+)}_u = \\beta\\, \\delta B^{(-)}_v$. For the reason we just mentioned, it suffices to study the cases $\\beta = 0,\\, \\infty,\\, \\pm 1$, where $\\beta = 0 $ corresponds to Dirichlet boundary conditions. Then, finding ghosts for $\\beta = \\pm 1$ implies that these remain for all non-zero $\\beta$, forbidding a non-zero critical value. Moreover, noting that the Poincar\\'e patch theory captures the high-momentum dynamics of the theory in global AdS, \\footnote{This is most easily seen by noting that, for short characteristic lengths, a cylinder is equivalent to a plane.} one can extend this result to the global case. Clearly, the analogous statement holds true for our hybrid boundary conditions parametrized by the constant $\\kappa$.\n\nIn many scenarios, the presence of the ghosts is in one-to-one correspondence with violations of the unitarity bound in the dual theory, which establishes that the scaling dimension of vector operators must be greater than one. In fact, the operator of dimension $1-\\alpha$ violates the bound for all $\\alpha$ and, accordingly, we find ghosts whenever the corresponding slow-decaying branch fluctuates. The only exception are the conformally invariant Neumann boundary conditions in the Poincar\\'e patch, which set to zero the faster fall-off. In analogy with the scalar case discussed in \\cite{Andrade:2011dg}, we have found in this case that the spectrum is ghost-free, and that the expected pathologies arise instead in the 2-point function, which is ill-defined even at large distances. Interestingly, we also found ghosts in the flat sector, which occur for some choices of the parameter that controls the double-trace deformation. Since the chiral currents have dimension one, these unitarity violations cannot be linked to the bound on the scaling dimension. These pathologies are indeed to be expected, however, because the expression for the symplectic product restricted to the flat sector is not positive definite in any obvious way.\n\nIt is worth commenting in more detail on the mixed boundary conditions in relation to the unitarity bound. Above, we obtained these as a deformation of the Neumann theory, as it is customary for bulk scalars whose mass lies in the Breitenlohner-Freedman window. Had the Neumann theory been well defined, the inclusion of the relevant double-trace operator could have been thought of as triggering an RG flow towards the Dirichlet theory. However, as we have seen, the Neumann theory is sick, and inclusion of the double-trace operator does not cure its pathologies. Thus, the aforementioned flow is not well defined. Attempting to remedy this, one might try to understand the mixed boundary conditions as triggering a flow from the Dirichlet theory. In this case, unfortunately, the deformation term one needs to add is of the form $\\sim \\Bigl(B^{(-)}_v\\Bigr)^2$, so it corresponds to an irrelevant operator of dimension $2(1+\\alpha)$. It follows that the resulting theory is non-renormalizable and the ghosts that arise can be understood as being the result of our loss of control of the theory in the UV.\n\nIt is interesting to contrast our results with those of \\cite{DHoker:2010hr}, in which the authors found an effective three-dimensional MCS theory in the context of holographic RG flows. More precisely, they constructed five-dimensional solutions in the Einstein-Maxwell-Chern-Simons theory which have the interpretation of magnetic branes. Perturbations around these backgrounds turn out to describe RG flows from four-dimensional field theories in the UV to two-dimensional ones in the IR, and the dynamics of the latter are captured by a 3$d$ MCS theory. Requiring the usual Dirichlet boundary conditions in the UV and imposing matching conditions in the bulk, the effective IR theory contains a double-trace for the vector operators which we denoted by $B^{(\\pm)}$. Our analysis reveals that this theory must contain ghosts, and indeed, the results of \\cite{DHoker:2010hr} indicate that violations of unitarity are present if one tries to extend the domain of validity of the IR description to the entire bulk. Then, what saves the theory is the existence of an effective cut-off associated to the domain wall solution, whose presence implies that the IR description breaks down at some intermediate value of the radial coordinate. This is to be expected since Dirichlet boundary conditions were imposed in the UV, and these respect the dual unitarity bounds. The issue of removing the bulk ghosts by introducing the appropriate cut-offs will be discussed in an upcoming publication\n\\cite{T.Andrade}.\n\nWe now briefly comment on the implications of our results in the context of potential condensed matter applications. For illustrative purposes, we first review the relevant results of the pure Maxwell theory and then move on to describe how the addition of the Chern-Simons term changes the picture. In terms of the radial variable of \\eqref{Global AdS3}, the asymptotics of the gauge field in the pure Maxwell theory are of the form\n\\begin{equation}\\label{maxwell asympt}\n    A_i = \\log r A^{(1)}_i + A^{(0)}_i + \\ldots \\qquad {\\rm with} \\quad \\nabla^{(0)}_i A^{(1) i} = 0\\, ,\n\\end{equation}\n\\noindent where $i$ is a boundary index and $\\nabla^{(0)}$ is the covariant derivative associated with the conformal boundary metric. The conservation equation satisfied by the coefficient $A^{(1)}_i$ indicates that it should be interpreted as the $U(1)$ current. This fact was overlooked in\n\\cite{Ren:2010ha,Nurmagambetov:2011yt,Liu:2011fy,Lashkari:2010ak}, in which the authors discussed the construction of a holographic $1+1$ dimensional superfluid\/superconductor incorrectly interpreting $A^{(0)}$ as the boundary current. We mention that this confusion was resolved in \\cite{Jensen:2010em} using the conservation argument given above. However, there is still an obstruction to the study of such holographic theory, since the boundary conditions that allow for a fluctuating current yield ghosts \\cite{Andrade:2011dg}. Thus, the applicability of the by now standard procedure \\cite{Hartnoll:2008vx,Hartnoll:2008kx} to the study of holographic $1+1$ superconductors remains, at least, unclear. Given this, it is compelling to ask ourselves what are the implications of adding the Chern-Simons term to the Maxwell theory and the possible AdS\/CMT applications of the resulting setup.\\footnote{This possibility was suggested in \\cite{Lashkari:2010ak}, with a different motivation.} As we have seen, the inclusion of the Chern-Simons term drastically modifies the scenario, as the $U(1)$ vector current is replaced by the topological chiral currents associated to the flat connections, and one can imagine introducing an order parameter (dual to a minimally coupled bulk charged scalar, say) which could potentially break the associated symmetry spontaneously.\\footnote{We thank Per Kraus for pointing out this possibility.} We leave the exploration of this line of research for future work.\n\n\\vskip 1cm\n\\centerline{\\bf Acknowledgments}\n\nWe are grateful to Geoffrey Comp\\`ere, Eduardo Fradkin, Gary Horowitz, Per Kraus, Mukund Rangamani, Simon Ross and Jorge Santos for helpful conversations, and specially to Don Marolf for many useful discussions on these and related topics. T.A. was partly supported by a Fulbright-CONICYT fellowship, by the US National Science Foundation under grant PHY08-55415 and by funds from the University of California. T.A. is also pleased to thank the Department of Physics of the University of Illinois at Urbana-Champaign and the Department of Mathematics of the University of California, Davis, for their hospitality during the completion of this work. RGL is partially supported by the US Department of Energy under contract FG02-91-ER40709. The work of J.I.J. is supported by the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe main theorem of the Bass-Serre theory of groups acting on\ntrees states that a group $G$ acting on a tree $T$ is the\nfundamental group of a graph of groups whose vertex and edge\ngroups are the stabilizers of certain vertices and edges of $T$.\nThis tells that $G$ can be obtained by successively forming\namalgamated free products and ${\\rm HNN}$-extensions.\nThe pro-$p$ version of this theorem does not hold in general\n({\\it cf.} Example \\ref{ex-4.3}),\nnamely a pro-$p$ group acting on a pro-$p$ tree does not have\nto be isomorphic to the fundamental pro-$p$ group of\na graph of finite $p$-groups (coming from the stabilizers).\nMoreover, the fundamental pro-$p$ group of a profinite graph\nof pro-$p$ groups does not have to split as\nan amalgamated free pro-$p$ product or as\na pro-$p$ ${\\rm HNN}$-extension over some edge stabilizer\n(the reason is that by deleting an edge\nof the profinite graph one may destroy its compactness).\nThese two facts are usually the main obstacles for proving\nsubgroup theorems of free constructions in the category of\npro-$p$ groups.\n\nWe show that the two Bass-Serre theory principal results\nmentioned above hold for finitely generated infinite\npro-$p$ groups acting {\\it virtually freely} on pro-$p$ trees, \n{\\it i.e.} such that the restriction of the action \non some open subgroup is free.\nSuch a group is then virtually free pro-$p$. \n\n\\begin{thm}\\label{t-treeacting_intro}\nLet $G$ be a finitely generated infinite pro-$p$ group acting\nvirtually freely on a pro-$p$ tree $T$.\nThen\n\n\\begin{enumerate}\n  \\item[\\rm (a)] $G$ splits either as\n  an amalgamated free pro-$p$ product or as\n  a pro-$p$ ${\\rm HNN}$-extension over some edge stabilizer;\n\n \\item[\\rm (b)] $G$ is isomorphic to\n the fundamental pro-$p$ group $\\Pi_1(\\cG,\\Gamma)$ of\n a finite graph of finite $p$-groups.\n\\end{enumerate}\n\\end{thm}\n\n\nOne should say that in contrast to the classical theorem from\nBass-Serre theory our $\\Gamma$ in item (b) is not $T\/G$.\nThe graph $\\Gamma$ is constructed in a special way by first modifying $T$\nwithout loosing the essential information of the action.\n\nAs a corollary we deduce the following subgroup theorem.\n\n\\begin{thm}\\label{t-subgrouptheorem_intro}\nLet $H$ be a finitely generated subgroup of a\nfundamental pro-$p$ group $G$ of a finite graph of\nfinite $p$-groups.\nThen $H$ is the fundamental pro-$p$ group of a finite graph of\nfinite $p$-groups which are intersections of $H$\nwith some conjugates of vertex and edge groups of $G$.\n\\end{thm}\n\n\nMoreover, as an application of Theorem \\ref{t-treeacting_intro},\nwe obtain the following result.\nIt is a pro-$p$ analogue of a classical result of\nG. Baumslag \\cite[Thm. 2]{Baumslag:62}\nthat gave an impulse to the theory known now as the theory of limit groups. Note that our theorem also generalizes \nthe pro-$p$ {\\it ipsis litteris} version of\n\\cite{BBaumslag:68}, as well as \\cite[Thm. 7.3]{KZ:11}.\n\n\\begin{thm} \\label{t:freeorabelian_intro}\n  Let $G=A\\amalg_{C} B$ be a free pro-$p$ product of\n  $A$ and $B$ with procyclic amalgamating subgroup $C$.\n  Suppose that\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n  If each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group\n  then so is each $2$-generated pro-$p$ subgroup of $G$.\n\\end{thm}\n\nThe method of proof is to consider the standard pro-$p$ tree $T$\non which $G$ acts naturally; so $A$ and $B$ are stabilizers of\nvertices $v$ and $w$, and $C$ is the stabilizer of the edge\nconnecting $v$ and $w$.\nThen we decompose the pair $(G, T)$ as a\ninverse limit of $(G_U,T_U)$ satisfying the hypothesis of\nTheorem \\ref{t-treeacting_intro}.\n\n\\\n\n\\noindent {\\bf Notation.}\nThroughout this paper, $p$ is a fixed but arbitrary prime number.\nThe additive group of the ring of $p$-adic integers is ${\\mathbb Z}_p$;\nthe natural numbers, ${\\mathbb N}$.\nFor $x$, $y$ in a group we shall write $y^x \\!:= x\\inv y x$.\nAll groups are pro-$p$, subgroups are closed and\nhomomorphisms are continuous.\nFor $A\\subseteq G$ we denote by $\\gp A$ the subgroup of $G$\n(topologically) generated by $A$\nand by $A^G$ the normal closure of $A$ in $G$, {\\it i.e.},\nthe smallest closed normal subgroup of $G$ containing $A$.\nBy $d(G)$ we denote the smallest cardinality of a\ngenerating subset of $G$.\nRecall that a cyclic profinite group is always finite.\nThe Frattini subgroup of $G$ will be denoted by $\\Phi(G)$.\nBy $\\tor G$ we mean the set of all torsion elements of $G$.\n\nFor a pro-$p$ group $G$ acting continuously\non a space $X$ we denote the set of fixed points of $G$ by $X^G$\nand for each $x\\in X$ the {\\em point stabilizer} by $G_x$.\nWe define $\\widetilde{G}=\\gp{ G_x\\,|\\, x\\in X}$.\n\nThe rest of our notation is very standard and basically follows\n\\cite{RZ:00a}\nand\n\\cite{RZ:00b}.\n\n\n\n\\section{Preliminary Results}\n\nIn this section we collect properties of amalgamated free pro-$p$\nproducts, pro-$p$ {\\rm HNN}-extensions and pro-$p$ groups acting\non pro-$p$ trees to be used in the paper.\nFurther information on this subject can be found in\n\\cite{RZ:00a} and \\cite{RZ:00b}.\nRecall the following two notions.\nFirst, an amalgamated free pro-$p$ product\n$G\\!:=A\\amalg_CB$ is {\\em non-fictitious}\nif $C$ is a proper subgroup of both, $A$ and $B$.\nUnless differently stated we shall consider exclusively\nnon-fictitious free amalgamated products\nand we shall make use of the fact from\n\\cite{Ribes:71} that a free pro-$p$ product\nwith either procyclic or finite amalgamating subgroup\nis always {\\em proper}, {\\it i.e.},\nthe factors $A$ and $B$ embed in $G$ via the natural maps.\n\nSecond,\na pro-$p$ {\\rm HNN}-extension $G={\\rm HNN}(H,A,f,t)$\nis {\\em proper} if the natural map from $H$ to $G$ is injective.\nOnly such free pro-$p$ products and pro-$p$ {\\rm HNN}-extensions\nwill be used in this paper and they are therefore always proper.\n\nWe start with a simple general lemma.\n\n\\begin{lemma}\\label{l:invsys}\nLet $G\\!:=\\varprojlim G_i$ be the inverse limit of\nan inverse system $\\{G_i,\\varphi_{ij},I\\}$ of\npro-$p$ groups and $H_i\\le G_i$ so that $\\varphi_{ij}(H_i)\\le H_j$ holds whenever $j\\le i$.\nSuppose that there is a constant $d$ with $d(G_i)=d$ for all $i\\in I$.\nThe following statements hold:\n\\begin{itemize}\n\n\\item[{\\rm (a)}]\n If $d(G)=d$,\n then there exists $j\\in I$ such that\n the projection $G\\to G_j$ is surjective.\n\\item[{\\rm (b)}]\nFor the induced inverse limit $H\\!:=\\varprojlim H_i\\le G$,\nwe have equality $H^G=\\varprojlim H_i^{G_i}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFor each $i\\in I$, let $\\varphi_i\\colon G\\to G_i$ be the\nprojection.\n\n\\noindent (a)\nThere is an induced inverse system of Frattini\nquotients with $G\/\\Phi(G)=\\varprojlim_iG_i\/\\Phi(G_i)$.\nIf $\\varphi_{ij}(G_i)\\Phi(G_j)\/\\Phi(G_j)$ is a proper subgroup of\n$G_j\/\\Phi(G_j)$, for all $i,j$ belonging to a cofinal subset  of $I$,\nthen  $G\/\\Phi(G)=\\varprojlim_i \\varphi_{j}(G)\\Phi(G_j)\/\\Phi(G_j)$\nand so $G$ can be generated by $d-1$ elements.\nOtherwise $\\varphi_{ij}$ must be surjective for $i,j$\nbelonging to a cofinal subset of $I$, and so is $\\varphi_j$\n({\\it cf.} \\cite[Prop. 1.1.10]{RZ:00b}).\n\n\\medskip\n\n\\noindent (b)\nSet, for the moment, $K\\!:=\\varprojlim H_i^{G_i}$.\nSince $H\\le K$, we have $H^G\\le K$, as $K\\triangleleft G$.\nSo, it suffices to establish $K\\le H$.\nThis is certainly true when there is a bound on the orders of the $G_i$.\nFix $n\\in\\mathbb N$.\nThen, as $d(G_i)=d$, there is a bound on the\norders of all $G_i\/\\Phi^n(G_i)$.\nThen the statement reads\n$\\varprojlim H_i^{G_i}\\Phi^n(G_i)\\le H^G\\Phi^n(G)$\nand therefore $K\\le H^G\\Phi^n(G)$.\nSince, $d(G)\\le d$, $G$ is finitely generated, and so the set $(\\Phi^n(G))_{n\\ge1}$\nis a fundamental system of neighbourhoods of the identity in $G$\n({\\it cf.} \\cite[Prop. 2.8.13]{RZ:00b}).\nHence $K\\le H$, as needed.\n\\end{proof}\n\nWe recollect the following fundamental results\nfrom the theory of pro-$p$ groups acting on pro-$p$ trees\nand their consequences for\nan amalgamated free pro-$p$ product or\na pro-$p$ {\\rm HNN}-extension.\n\nRecall that for a pro-$p$ group $G$ acting on a pro-$p$ tree $T$,  the closed subgroup generated by all vertex stabilizers\nis denoted by $\\widetilde{G}$; also, \nthe (unique) smallest pro-$p$ subtree of $T$ containing\ntwo vertices $v$ and $w$ of $T$ is denoted by $[v,w]$\nand called the geodesic connecting $v$ to $w$ in $T$\n({\\it cf.} \\cite[p. 83]{RZ:00a}).\n\n\\begin{thm}\n\\label{t:trees1}\nLet $G$ be a pro-$p$ group acting on a pro-$p$ tree $T$.\n\\begin{itemize}\n\n \\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Prop. 3.5]{RZ:00a}})}\n  $T\/\\widetilde{G}$ is a pro-$p$ tree.\n\n \\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Cor. 3.6]{RZ:00a}})}\n  $G\/\\widetilde{G}$ is a free pro-$p$ group.\n\n \\item[{\\rm (c)}] {\\rm (\\protect{\\cite[Cor. 3.8]{RZ:00a}})}\n If $v$ and $w$ are two different vertices of $T$,\n then $E([v,w])\\neq\\emptyset$ and\n $(G_v\\cap G_w)\\leq G_e$ for every $e\\in E([v,w])$.\n\n \\item[{\\rm (d)}] {\\rm (\\protect{\\cite[Thm. 3.9]{RZ:00a}})}\n If $G$ is finite, then $G=G_v$, for some $v\\in V(T)$.\n \\end{itemize}\n\\end{thm}\n\n\\begin{thm}\n\\label{t:afpproperties}\nLet $G=G_1\\coprod_H G_2$ be a proper amalgamated free pro-$p$\n product of pro-$p$ groups.\n\\begin{itemize}\n\\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Thm. 4.2(b)]{RZ:00a}})}\n Let $K$ be a finite subgroup of $G$.\n Then  $K\\leq G_i^{g}$ for some $g\\in G$ and for some $i=1$ or $2$.\n\n\\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Thm. 4.3(b)]{RZ:00a}})}\n Let $g\\in G$. Then\n\n $G_i\\cap G_j^{g}\\leq H^{b}$\n\n for some $b\\in G_i$, whenever $1\\leq i\\neq j\\leq 2$ or $g\\not\\in G_i$.\n\\end{itemize}\n\\end{thm}\n\n\\begin{thm}\n\\label{t:hnnproperties}\n Let $G={\\rm HNN}(H,A,f)$ be a proper pro-$p$ {\\rm HNN}-extension.\n\\begin{itemize}\n\\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Thm. 4.2(c)]{RZ:00a}})}\n Let $K$ be a finite subgroup of $G$.\n Then $K\\leq H^{g}$ for some $g\\in G$.\n\n\n\\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Thm. 4.3(c)]{RZ:00a}})}\n Let  $g\\in G $. Then\n \\begin{equation*}\n H\\cap H^{g}\\leq A^{b}\n \\end{equation*}\n for some $b\\in H\\cup tH$, whenever $g\\not\\in H$.\n\\end{itemize}\n\\end{thm}\n\n\nThe next  technical results concern inverse systems that will play\nan essential role during the proof of\nTheorem \\ref{t:freeorabelian_intro} in section \\ref{s:2-generated}.\nUntil the end of this section the directed set $I$\nwill be assumed to be order isomorphic to $\\mathbb N$.\n\n\\begin{prop} \\label{p:decomphnn}\n Let $G$ be the inverse limit of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of pro-$p$ groups.\n Suppose that each $G_i={\\rm HNN}(H_i,A_i,t_i)$\n is an\n {\\rm HNN}-extension with\n $H_i$ finite and $\\varphi_{ij}(H_i)\\cong H_j$.\n Then\n there are inverse systems of groups\n $\\{H_i',\\varphi_{ij},I\\}$ and\n $\\{A_i'',\\varphi_{ij},I\\}$ such that\n $G=\\text{\\rm HNN}(H,A,t)$ with\n $H\\!:=\\varprojlim H_i'$, $A\\!:=\\varprojlim A_i''$\n where each $H_i'$ (resp. $A_i''$)\n is a conjugate of $H_i$ (resp. $A_i$)\n by an element of $G_i$.\n\\end{prop}\n\n\\begin{proof} \\\nFix $k\\in I$.\nBy Theorem \\ref{t:hnnproperties}(a)\nthere are $g_k\\in G_k$ with $\\varphi_{jk}(H_j)= H_k^{g_k}$\n(remember they are isomorphic by the hypothesis).\nPick $g_j\\in\\varphi_{jk}^{-1}(g_k)$ and define\n$H_j' \\!:= H_j^{g_j\\inv}$, $A_j' \\!:= A_j^{g_j\\inv}$, and,\n$t_j' \\!:= t_j^{g_j\\inv}$; clearly $\\varphi_{jk}(H_j')= H_k$.\nSince $A_j = H_j \\cap H_j^{t_j}$ and\n$\\varphi_{jk}$ is surjective, we have\n\\begin{equation*}\n\\varphi_{jk}(A_j')\n\\le H_k \\cap H_k^{\\varphi_{jk}(t_j')}\n\\le A_k^{t_k^{\\epsilon_k}h_k} \\, ,\n\\end{equation*}\nfor suitable $h_k\\in H_k$ and $\\epsilon_k=0$ or $1$,\nby Theorem \\ref{t:hnnproperties}(b).\nChoose $h_j\\in\\varphi_{jk}^{-1}(h_k)\\cap H_j'$ and\n$x_j\\in\\varphi_{jk}^{-1}(t_k)$.\nDefining\n$A_j'' \\!:= {A_j'}^{(x_j^{\\epsilon_k}h_j)\\inv}$\nwe obtain\n$\\varphi_{jk}(A_j'') \\le A_k$\nin both cases.\nContinuing inductively \nwe obtain the desired inverse systems\n $\\{H_i',\\varphi_{ij},I\\}$ and\n $\\{A_i'',\\varphi_{ij},I\\}$.\n\nIt is straightforward to check that the other associated subgroup\nalso ``fits'' into the inverse system, that is\n$\\varphi_{jk}({A_j''}^{t_j''}) \\le A_k ^{t_k}$ where $t_j'' \\!:=\n{t_j'}^{(x_j^{\\epsilon_k}h_j)\\inv}$.\n\nNow, let $H\\!:=\\varprojlim H_i'$,\n$A\\!:=\\varprojlim A_i''$ and $B\\!:=\\varprojlim {A_i''}^{t_i''}$.\nFor each $i\\in I$ let us consider the subset\n\\begin{equation*}\nX_i\\!:=\\{\\tau_i\\in G\\,|\\, A^{\\tau_i}\\!=\\!B\n{ \\, \\ and \\, \\ }\nG_i\\!:=\\langle H_i, \\tau_i\\rangle \\}\\,.\n\\end{equation*}\nClearly every $X_i$ is compact, and\nsince $X_{i+1}\\subseteq X_i$ for all $i\\in I$,\nthere exists $t\\in \\bigcap_i X_i$ so that $B=A^t$.\n\nThe desired isomorphism from\n$\\text{\\rm HNN}(H,A,t)$ onto $G$ follows now from\nthe universal property of {\\rm HNN}-extensions.\n\\end{proof}\n\n\\begin{prop} \\label{p:decompfp}\n Let $G$ be the inverse limit\n of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of pro-$p$ groups\n $G_i$ each a free pro-$p$ product\n $G_i=A_i\\amalg B_i$ whith $A_i$ cyclic and $B_{i}$ procyclic.\n\n Then\n there are inverse systems of pro-$p$ groups,\n $\\{A_{i}',\\varphi_{ij},I\\}$ and\n $\\{B_{i}',\\varphi_{ij},I\\}$,\n where each $A_{i}'$\n is a conjugate of $A_{i}$\n by an element of $G_i$, and $B_{i}'\\le G_i$, and,\n $G\\cong \\left(\\varprojlim A_{i}'\\right)\n \\amalg \\left(\\varprojlim B_{i}'\\right)$.\n\\end{prop}\n\n\\begin{proof}\nSuppose first that there exists $i_0\\in I$ such that\n$B_{i_0}\\cong {\\mathbb Z}_p$.\nThen, since each $\\varphi_{ij}$ is surjective,\nthe induced homomorphism between the continuous abelianizations\n$A_i\\times {\\mathbb Z}_p\\cong\nG_i\/[G_i,G_i]\\to G_{j}\/[G_{j},G_{j}]\\cong A_j\\times {\\mathbb Z}_p$\nis surjective for $i_0\\le j\\le i$.\nTherefore $\\varphi_{ij}(A_i)\\le A_j\\Phi(G_j)$ and\nby Theorem \\ref{t:afpproperties}(a)\nthere is $g_{j}\\in G_j$ with $\\varphi_{ij}(A_i)\\le A_j^{g_j}$\nshowing that $A_i$ maps onto a conjugate of $A_j$.\nNow, observing that $G=\\varprojlim G_i$ with\n$G_i\\cong {\\rm HNN}(A_{i},1,t_i)$,\nwhere $t_i$ generates $B_{i}$, we can apply\nProposition \\ref{p:decomphnn}\nto obtain the result.\n\nSuppose that each $B_{i}$ is finite.\nSince $\\varphi_{ij}$ are surjective, from\nTheorem \\ref{t:afpproperties}(a),\nwe obtain that distinct free factors of $G_i$ are mapped,\nup to conjugation, to distinct free factors of $G_j$.\nSo, there is $k_0$ in $I$ so that\nfor all $i,j$ we have\n\\begin{equation*}\n\\varphi_{ij}(A_{i})= A_{j}^{x_{j}}\n{ \\ \\, and \\, \\ }\n\\varphi_{ij}(B_{i})= B_{j}^{y_{j}} \\, ,\n\\end{equation*}\nfor some $x_{j},y_{j}\\in G_j$.\nThen inductively the desired inverse systems $\\{A_{i}',\\varphi_{ij},I\\}$ and\n $\\{B_{i}',\\varphi_{ij},I\\}$,\ncan be exhibited.\nThe result follows now from\n\\cite[Lemma 9.1.5]{RZ:00b}.\n\\end{proof}\n\n\n\\begin{lemma} \\label{l:2gen}\n Let $G$ be a $2$-generated pro-$p$ group.\n \\begin{itemize}\n \\item[{\\rm (a)}]\n If $G$ is a free pro-$p$ product with\n procyclic amalgamation,\n then one of its free factors is procyclic.\n \\item[{\\rm (b)}]\n If $G$ is a proper {\\rm HNN}-extension with\n procyclic associated subgroups,\n then its base subgroup $H$ is at most $2$-generated.\n Moreover, if $d(H)=2$ then $H$ is generated by the associated subgroups.\n \\item[{\\rm (c)}]\n If $G$ is the fundamental pro-$p$ group of a finite tree of finite  groups\n such that all edge groups are cyclic,\n then either $|G|< \\infty$ or $G=K\\amalg_{C} R$ with $K$ cyclic and $R$ finite,\n or $G=K\\amalg_CM\\amalg_DN$,\n with $K$ and $N$ cyclic and $M\\le \\Phi(G)$.\n \\end{itemize}\n\\end{lemma}\n\n\\begin{proof} \\\n\n\\noindent (a) Suppose that $G=A\\amalg_{C} B$ and let ``bar''\nindicate passing to the Frattini quotient. We have an obvious\nepimorphism from $G$ to the induced pushout \n$P\\!:= \\bar A\\amalg_{\\bar C} \\bar B$. Let\n$n\\!:=d(A)+d(B)$. Since $C$ is procyclic, the image $M$ of the\nkernel of the canonical map $\\bar A\\amalg \\bar B \\to \\bar G$ via\nthe cartesian map $\\bar A\\amalg\\bar B\\to \\bar A\\times \\bar B$ is\nalso procyclic. The latter map induces an epimorphism from $\\bar\nG$ to the at least $(n-1)$-generated elementary abelian pro-$p$\ngroup $(\\bar A\\times \\bar B)\/M$. Therefore, $n\\leq 3$ and the\nresult follows.\n\n\\noindent (b) Suppose that $G={\\rm HNN}(H,C,f,t)$ with $C=\\gp c$.\nIf $d(H)\\ge3$ then $d(G)\\ge3$ as can be seen by using the obvious\nepimorphism $G\\to (H\\times {\\langle t\\rangle})\/{\\langle tct^\n{-1}f(c)^ {-1}\\rangle}$. Thus $d(H)\\le2$.\n\nFinally suppose that $d(H)=2$. Now $G$ is the quotient of \n$Q\\!:=H\\amalg \\gp t$  modulo the relation\n$f(c)\\inv c^{t}$. Since $d(Q)=3$ we can conclude that $c\\not\\in\n\\Phi(G)$ and $f(c)\\not\\in\\Phi(G)$. Therefore neither $c\\in\\Phi(H)$\nnor $f(c)\\in\\Phi(H)$. So we cannot have $f(c)\\inv c\\in\\Phi(H)$\nelse $d(G\/\\Phi(G))$ turns out to be 3. Hence $H=\\langle C,\nC^{t}\\rangle$.\n\n\\noindent (c) Let $G=\\Pi_1({\\mathcal G},\\Gamma)$ with finite\nvertex groups ${\\mathcal G}(v)$ and cyclic edge groups ${\\mathcal\nG}(e)$. We claim that $|V(\\Gamma)|\\le 3$. By assumption\n$|V(\\Gamma)|\\ge2$, and therefore it has an edge $e$. Splitting $G$\nover $e$, we can assume that ${\\mathcal G}(d_0(e))$ is procyclic\nby (a); hence $d_0(e)$ is a pending vertex of $\\Gamma$. Suppose\nnow that $\\Gamma$ has at least $3$ vertices, and let $a$ be an\narbitrary edge $\\neq e$ of $\\Gamma$ having initial or terminal\nvertex $v=d_1(e)$. Without loss of generality, suppose that\n$d_0(a)=v$. Then $d_1(a)$ is a pending vertex with procyclic\nvertex group ${\\mathcal G}(d_1(a))$; for, otherwise, by splitting\n$G$ over the edge $a$ we would obtain that $d(G)>2$, a\ncontradiction. Now, if we have $r\\geq 2$ edges with initial or\nterminal vertex $v$ then it follows from the pro-$p$ presentation\nof $G$ that it has a free pro-$p$ abelian group ${\\mathbb Z}_p^r$\nas a quotient; this implies $r=2$, whence $|V(\\Gamma)|\\le 3$.\n\nIf $|V(\\Gamma)|=2$ then $G=K\\amalg_CM$ with $K$ and $M$ finite,\nand, by (a), we can assume that $K$ is cyclic.\n\nSuppose now that $|V(\\Gamma)|=3$. Then $G=K\\amalg_CM\\amalg_DN$\nwith $C$ and $D$ cyclic and $K$, $M$, and $N$ finite. By the\nproperness of our decomposition we have\n$d(K\\amalg_CM)=d(M\\amalg_DN)=2$ and, making use of (a), we can\nconclude that $K$ and $N$ must both be cyclic. Since $d(G)=2$ then\n$M\\le\\Phi(G)$ follows.\n\\ignor{\n\n}\n\\end{proof}\n\n\n\\begin{prop} \\label{p:treeprod}\n Let $G$ be the inverse limit of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of  pro-$p$ groups\n $G_i$. Suppose $G_i$ decomposes as\n an amalgamated free pro-$p$ product\n $G_i=K_i\\amalg_{C_i} R_i$ with $K_i$ cyclic and $R_i$ finite or\n $G_i=K_i\\amalg_{C_i}M_i\\amalg_{D_i}N_i$,\n with $K_i$ and $N_i$ cyclic and $M_i\\le \\Phi(G_i)$. Then,\n passing to a cofinal subset of $I$, if necessary,\n there are inverse systems\n $\\{K_i',\\varphi_{ij},I\\}$ and $\\{C_i'',\\varphi_{ij},I\\}$ such\n that $C_i''\\le K_i'$,\n $\\varphi_{ij}(K_i')= K_j'$ and $\\varphi_{ij}(C_i'')\\le C_j''$\n where each $K_i'$ (resp. $C_i''$) is a conjugate of $K_i$ (resp. $C_i$)\n by an element of $G_i$.\n\\end{prop}\n\\begin{proof}\n\nUsing Theorem \\ref{t:afpproperties}(a) in both cases we can pass to a cofinal\nsubset $J$ of $I$ such that for all $i\\geq j$ in $J$ we have\n$\\varphi_{ij}(K_i)\\leq K_j^{g_j}$, for some $g_j\\in G_j$.\nIndeed, in the first case $\\varphi_{ij}$ sends factors to the\nfactors up to conjugation and in the second case $\\varphi_{ij}$\nsends cyclic factors to cyclic factors up to conjugation. Then in\nfact, since $K_j$ is cyclic $\\varphi_{ij}(K_i)=K_j^{g_j}$ (indeed,\notherwise $\\varphi_{ij}(K_i)^{G_j}\\neq K_j^{G_j}$ contradicting\nthe surjectivity of $\\varphi_{ij}$). Now  selecting\n$g_i\\in\\varphi_{ij}^{-1}(g_j)$ and letting\n$K_i'\\!:=K_i^{g_i\\inv}$, and using an induction argument, we\nobtain the desired inverse system $\\{K_i',\\varphi_{ij},J\\}$. Next,\nletting $C_i'\\!:=C_i^{g_i\\inv}$ we have $C_i'\\le K_i'\\cap\nM_i^{g_i}$; then, by Theorem \\ref{t:afpproperties}(b),\n$\\varphi_{ij}(C_i')\\le K_j\\cap \\varphi_{ij}(M_i)\\le {C_j'}^{b_j}$,\nfor some $b_j\\in K_j'$. Choosing $b_i\\in\\varphi_{ij}^{-1}(b_j)\\cap\nK_i'$ and letting $C_i''\\!:={C_i'}^{b_i\\inv}$ we obtain the other\ninverse system $\\{C_i'',\\varphi_{ij},I\\}$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:fp}\nLet $X$ be a $G$-space and $(\\widetilde U_n)_{n\\ge1}$ be a subset\nof normal subgroups of $G_n$ with $\\bigcap \\widetilde U_n=1$.\nWrite $X_n\\!:=X\/\\widetilde U_n$ and $G_n\\!:=G\/\\widetilde U_n$. Let\nthere be subgroups $S_n\\le G_n$ so that $\\varphi_{nm}(S_n)\\le S_m$\nand $S\\!:=\\varprojlim S_n$ be the inverse limit. If\n$X_n^{S_n}\\neq\\emptyset$ for all $n\\in\\mathbb N$ then\n$X^S\\neq\\emptyset$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi_n$ denote the canonical projection from $X\\cup G$\nonto $X_n\\cup G_n$. Then $X_n^{\\varphi_n(S)}\\supseteq\nX_n^{S_n}\\neq\\emptyset$. Therefore\n$Y_n:=\\varphi_n\\inv(X_n^{\\varphi_n(S)})\\neq\\emptyset$. Now\n$Y_n=\\{x\\in X\\mid xS\\subseteq x\\widetilde U_n\\}$ so that\n$Y_{n+1}\\subseteq Y_n$; by the compactness of $X$ we can deduce\nthat $\\emptyset\\neq \\bigcap Y_n\\subseteq X^S$.\n\\end{proof}\n\n\n\nWe end this section by quoting results to be used\nin the next section.\n\n\\begin{prop}[\\protect{\\cite[Thm. 1.1]{Scheiderer:99}}]\n\\label{p:scheiderer}\nLet $G$ be a finitely generated pro-$p$ group which\ncontains an open free pro-$p$ subgroup of index $p$.\nThen $G$ is isomorphic to a free pro-$p$ product\n\\begin{equation*}\nH_0 \\amalg (S_1\\times H_1 )\\amalg\\cdots\\amalg (S_m\\times H_m)\n\\end{equation*}\nwhere $m\\geq 0$, the $S_i$ are cyclic groups of order $p$\nand the $H_i$ are free pro-$p$ groups of finite rank.\n\\end{prop}\n\n\\begin{cor}[\\protect{\\cite[Cor. 1.3(a)]{Scheiderer:99}}]\n\\label{c:finiteconjugacy}\nEvery pro-$p$ group which contains an open\nfree pro-$p$ subgroup of finite rank\nhas, up to conjugation, only a finite number\nof finite subgroups.\n\\end{cor}\n\n\n\\begin{prop}\\label{p-freeedgeaction}\nA pro-$p$ group $G$ acting on a pro-$p$ tree $T$\nwith trivial edge stabilizers such that there\nexists a continuous section $\\sigma:V(T)\/G\\longrightarrow V(T)$ is\nisomorphic to a free pro-$p$ product\n\\begin{equation*}\n\\left(\\coprod_{\\dot v\\in V(T)\/G} G_{\\sigma(\\dot v)}\\right)\\amalg \\left(G\/\\gp{G_w\\mid w\\in V(T)}\\right) \\, .\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nThis follows from the proof of \\protect{\\cite[Thm. 3.6]{Z:96}}.\nSee also the last section of \\cite{Melnikov:90}.\n\\end{proof}\n\n\n\n\n\\section{Groups acting virtually freely on trees}\n\\label{s-proofs}\n\nIf a pro-$p$ group $G$ acts on a profinite graph $\\Gamma$ we shall\ncall sometimes $\\Gamma$ a $G$-graph.\n\n  \\begin{lemma}\\label{l-actionfree}\n  Let $G$ be a non-trivial finitely generated pro-$p$ group,\n  and let $\\Gamma$ be a connected $G$-graph.\n  Suppose that $\\Delta$ is a connected subgraph of $\\Gamma$\n  such that $\\Delta G=\\Gamma$.\n  Then there exists a minimal set of generators $X$ of $G$\n  such that $\\Delta\\cap \\Delta x\\neq \\emptyset$ holds for each $x\\in X$.\n  \\end{lemma}\n\n  \\begin{proof}\n  It is enough to prove the lemma under the\n  additional assumption that $G$ is elementary abelian.\n  Indeed, using ``bar'' to denote passing to the quotient modulo $\\Phi(G)$,\n  making $Z$  a minimal generating set of $G$, suppose that for each\n  $z\\in Z$ there exists a vertex $v_z\\in \\Delta$ with\n  $\\overline{v_z}\\in\\ol \\Delta\\cap \\ol \\Delta\\ol z\\neq\\emptyset$.\n  Then there exists $f_z\\in\\Phi(G)$ with $v_zz\\in \\Delta f_z$, so that\n  the set  $X\\!:=\\{zf_z\\inv\\mid z\\in Z\\}$ is a minimal set of\n  generators of $G$ for  which the assertion of the Lemma holds.\n\n  Suppose that the lemma is false for an elementary abelian group.\n  Then there is a counterexample $G$ with minimal $d(G)$.\n  Select a minimal generating subset $X$ of $G$.\n  If $d(G)=1$ then,\n  due to the connectedness  of $\\Gamma$,\n  there are $g_1,g_2\\in G$ with $g_1\\neq g_2$ such that\n  $\\Delta g_1\\cap \\Delta g_2\\neq\\emptyset$.\n  Replacing $X$ by $\\{g_1g_2\\inv\\}$ shows that\n  the conclusion of the lemma holds, a contradiction.\n  Hence $d(G)\\ge2$.\n  Select an element $z\\in X$ and let ``bar'' denote passing\n  to the quotient modulo $\\gp z$.\n  Since $d(\\ol G)=d(G)-1$, by the minimality assumption there is a\n  subset $\\ol Y$ of $\\ol G$ which is\n  a minimal set of generators of $\\ol G$ such that\n  $\\ol\\Delta\\cap \\ol\\Delta \\ol y\\neq\\emptyset$ for all $\\ol y\\in \\ol Y$.\n  Let $Y$ denote a transversal of $\\ol Y$ in $G$.\n  Then there are elements $z_y\\in \\gp z$ such that\n  $\\Delta\\cap \\Delta yz_y\\neq\\emptyset$ for all $y\\in Y$.\n  Set $W=\\{yz_y\\mid y\\in Y\\}$. Since\n  $\\Gamma$ can be viewed as a $\\gp z$-graph,\n  $\\Delta W \\cap \\Delta W z\\neq\\emptyset$ can be assumed by  the\n  minimality assumption.\n  This means that there exist $w_1,w_2\\in W$ such that\n  $\\Delta w_1\\cap \\Delta w_2z\\neq \\emptyset$, so\n  $X=W\\cup\\{w_1^{-1}w_2z\\}$ would satisfy the assertion\n  of the lemma, a contradiction.\n  \\end{proof}\n\n \\begin{lemma}\\label{l-action}\n Let $G$ be a finitely generated infinite pro-$p$ group.\n Suppose that $G$ acts on a pro-$p$ tree $T$ containing a\n pro-$p$ subtree $D$ such that $DG=T$.\n Then there exists a minimal set of generators $X$ of a retract $H$\n of $G$ such that $D\\cap Dx\\neq \\emptyset$ for each $x$ in $X$.\n \\end{lemma}\n\n  \\begin{proof}\n  Let ``bar'' denote passing to the quotient modulo\n  $\\widetilde G=\\gp{G_v\\mid v\\in V(T)}$.\n  By Theorem \\ref{t:trees1}(a)\n  the quotient graph $\\ol T\\!:=T\/\\widetilde G$ is a pro-$p$ tree.\n  Applying Lemma \\ref{l-actionfree} to $\\ol G$ acting on $\\ol T$\n  yields a subset $Z$ of $G$ such that $\\ol Z$ is\n  a minimal set of generators of $\\ol G$ and\n  for each $z$ there exists a vertex $v_z\\in D$ such that\n  $\\overline{v_z}\\ol z\\in\\ol D\\cap \\ol D\\ol z\\neq\\emptyset$ holds.\n  Hence there  exists $k_z\\in \\widetilde G$\n  with $v_zzk_z\\in Dk_z\\cap Dz$ and so $v_zz\\in D\\cap Dzk_z^{-1}$.\n  Now set $X\\!:=\\{zk_z^{-1}\\mid z\\in Z\\}$ and $H\\!:=\\gp X$.\n  Finally observe that by\n  Theorem \\ref{t:trees1}(b),\n  $\\ol G$ is a free pro-$p$ group, so that $H$ is indeed a retract.\n  \\end{proof}\n\n\n\n\\begin{lemma}\\label{l-rankformula}\nLet $G$ be a finitely generated pro-$p$ group\nacting on a pro-$p$ tree $T$.\nSuppose that all vertex stabilizers are finite and\nall edge stabilizers are pairwise conjugated.\nAssume further that there exist an edge $e\\in T$ and\na finite subset $V\\subseteq T^{G_e}$ such that:\n\\begin{itemize}\n\\item[\\rm (i)] for every $v_1, v_2\\in V$,\n$v_1G=v_2G$ implies $v_1=v_2$,\n\\item[\\rm (ii)]  $G$ is generated by the $G_v$, $v\\in V$.\n\\end{itemize}\nIf $F$ is a free pro-$p$ open normal subgroup of $G$, then \n\\begin{equation*}\n{\\rm rank}(F)-1=\n[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V} \\frac{1}{|G_v|}\\right) \\, .\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof} \nWe use induction on the index $[G:F]$.\nObviously $F\\neq G$, from hypothesis (ii); so,\nlet us consider the preimage $N$ in $G$ of a central subgroup\nof order $p$ of $G\/F$.\n\n\\medskip\n\\noindent {\\it Case 1}.  $N\\cap G_e=1$.\n\nIt follows that each non-trivial torsion element\n$t$ of $N$ generates a self-centralized subgroup.\nIndeed, by\nTheorem \\ref{t:trees1}(d)\n$t$ stabilizes some vertex $w$, so if $g$ centralizes\n$t$, the element $t$ also stabilizes $wg$.\nBut then by\nTheorem \\ref{t:trees1}(c)\n$t$ stabilizes\nthe geodesic $[wg,w]$.\nSince, however, $G_e\\cap N=1$,\nthe element $t$ cannot stabilize any edge,\nso $wg=w$, and therefore $g$ is a power of $t$.\n\nThus the decomposition of $N$ according to\nProposition \\ref{p:scheiderer}\nbecomes $N=\\left(\\coprod_{i\\in I}C_i\\right)\\amalg F_1$,\nwith $F_1$ a free pro-$p$ subgroup of $F$.\nTaking into account that $G$ acts upon\nthe conjugacy classes of subgroups of order $p$ we have\n\\begin{equation}\n\\label{e-scheiderer}\nN=\\coprod_{v\\in V}\\left(\\coprod_{r_v\\in G\/NG_v}\n(N\\cap G_{v})^{r_v}\\right) \\amalg F_1 \\, .\n\\end{equation}\nSet $V_1=\\{v\\in V\\mid N\\cap G_{v}\\neq 1\\}$.\nSince $NG_v=FG_v$ for every $v\\in V_1$ we can rewrite\nEq.~(\\ref{e-scheiderer}) as\n\\begin{equation*}\nN=\\coprod_{v\\in V_1}\\left(\\coprod_{r_v\\in G\/FG_v}\n(N\\cap G_{v})^{r_v}\\right) \\amalg F_1\\, .\n\\end{equation*}\nUsing this free decomposition and comparing it with\nProposition \\ref{p:scheiderer}\nwe find\n\\begin{equation}\n|I|=\\sum_{v\\in V_1}|G\/FG_v|=[G:F]\\sum_{v\\in V_1}\\frac{1}{|G_v|} \\, ,\\label{e-AinLemma}\n\\end{equation}\nand\n\\begin{equation}\n{\\rm rank}(F)-1=p\\ {\\rm rank}(F_1)+(p-1)(|I|-1)-1 \\, .\\label{e-rankF}\n\\end{equation}\n\nIf $N=G$ then $F_1=1$, since $G_v, v\\in V$ generate $G$.\nThen $G_e=1$ since otherwise $G$ is finite.\nSo $|V|=|I|$ and the last equation becomes exactly the needed one.\nThis gives the base of induction.\n\nSuppose now that $N\\not= G$.\nThen the product $p\\ {\\rm rank}\\of{F_1}$ can be computed\nby observing that passing to the quotient modulo $\\torgp{N}$\nand indicating it by ``bar'' we have\n${\\rm rank}\\of{\\ol F}={\\rm rank}\\of{F_1}$,\nso that using $[G:F]=p[\\ol G :\\ol F]$\nthe induction hypothesis yields\n\\begin{eqnarray*}\np\\  {\\rm rank}\\of{F_1}&=&p\\ {\\rm rank}\\of{\\ol F}\\\\\n&=&p[{\\ol G}:{\\ol F}]\\left(\\frac{|V|-1}{|\\ol G_e|}-\\sum_{v\\in V} \\frac{1}{|\\ol G_v|}\\right)+p\\\\\n&=&[G:F] \\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V_1} \\frac{1}{|\\ol G_v|}-\\sum_{v\\in V\\,-\\, V_1} \\frac{1}{|G_v|}\\right)+p\\\\\n&=&[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V_1} \\frac{p}{|G_v|}-\\sum_{v\\in V\\,-\\, V_1} \\frac{1}{|G_v|}\\right)+p\n\\end{eqnarray*}\n(we used $G_e\\cap N=1=G_v\\cap N$\nfor all $v\\in V\\,-\\,V_1$ and $|G_v\\cap N|=p$ for all $v\\in V_1$\nto obtain the last equality).\nInserting this expression and the expression for $|I|$\nfrom Eq.~(\\ref{e-AinLemma}) into Eq.~(\\ref{e-rankF})\nyields the claimed formula for ${\\rm rank}\\of F$.\n\n\\medskip\n\\noindent {\\it Case 2.} $N\\cap G_e \\not= 1$.\n\nSince for all $v\\in V$ the edge group $G_e$ is contained in $G_v$\nby the hypothesis, $G_v$ centralizes $N\\cap G_e$.\nBut $G=\\gp{G_{v} \\mid v\\in V}$\nso $N\\cap G_e$ is  a central subgroup of $G$ of order $p$.\nThen, using ``bar'' to pass to the quotient modulo $N\\cap G_e$\nand the inductive hypothesis, for $\\ol G$ we have\n\\begin{eqnarray*}\n{\\rm rank}{(F)}-1&=& {\\rm rank}{(\\ol F)}-1 \\\\\n&=&[\\ol G:\\ol F]\\left(\\frac{|V|-1}{|\\ol G_e|}-\\sum_{v\\in V} \\frac{1}{|\\ol G_v|}\\right) \\\\\n&=&\\frac{1}{p}[G:F]\\left(\\frac{p(|V|-1)}{|G_e|}-\\sum_{v\\in V} \\frac{p}{|G_v|}\\right) \\\\\n&=&[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V} \\frac{1}{|G_v|}\\right)\n\\end{eqnarray*}\nas needed.\n\\end{proof}\n\n\nRecall that a pro-$p$ group $G$ acts\n{\\it faithfully} on a pro-$p$ tree $T$ if the kernel of the action is trivial;\nand $G$ acts {\\it irreducibly} on $T$\nif $T$ does not contain a proper $G$-invariant pro-$p$ subtree.\n\n\\begin{lemma}\\label{l-exchange1\nLet a pro-$p$ group $G$ act faithfully and irreducibly\non a pro-$p$ tree $T$.\nSuppose that $G_e$ is a minimal edge stabilizer and\nthe set of edges $E(T^{G_e})G$ is open in $T$.\nThen $T'\\!:=T\\,-\\, E(T^{G_e})G$ is a subgraph having each connected component a pro-$p$ tree.\n\nLet $\\ol T$ be the quotient graph obtained by collapsing distinct\nconnected components of $T'$ to distinct points.\nThen $\\ol T$ is a pro-$p$ tree on which\n$G$ acts faithfully and irreducibly, and $\\ol T=\\ol T^{G_e}G$.\n\\end{lemma}\n\n\\begin{proof}\nSince  $T'$ is closed and contains $V(T)$, it is a subgraph of $T$;\nhence its connected components \nare pro-$p$ trees.\nMoreover, $\\ol T$ is a $G$-graph, and by\n\\cite[Proposition p. 486]{Zalesskii:89},\nit is simply connected and hence a pro-$p$ tree.\n\nNow, we have $m\\in T'$ if and only if there exists a\nsubgroup $L\\le G_m$, an edge stabilizer, so that $L^g$ is not\ncontained in $G_e$ for every $g\\in G$.\nTherefore, since $G_e$ is a minimal edge stabilizer,\nwe conclude that all edge stabilizers\nof edges in $T^{G_e}G$ are conjugates of $G_e$.\n\nLet us show that $G_{\\ol e}$ is a conjugate of $G_e$ for every\n$\\ol e\\in E(\\ol T)$.\nLet $f\\in E(T)$ and $u,v$ be its end points which, \nby construction, belong to $T'$.\nFix $g\\in G_{\\ol e}=G_{\\ol u}\\cap G_{\\ol v}$.\nThen $ug$ and $vg$ belong\nrespectively to the same connected components as $u$ and $v$.\nThe collapsing procedure induces a\ncanonical epimorphism which is injective when restricted to\n$E(T^{G_e})G$.\nSince $\\ol T$ is a pro-$p$ tree we find that\nafter collapsing $e$ and $eg$ both map to $\\ol e$, and\nas edges of $eG$ under the collapsing procedure\nare not identified, $e=eg$ must follow.\nHence $G_{\\ol e}$ is a conjugate of $G_e$ indeed.\n\nSuppose that $G$ does not act irreducibly on $\\ol T$.\nSince $\\ol T$ is obtained by collapsing pro-$p$ subtrees,\nthe preimage of a proper $G$-invariant pro-$p$ subtree of $\\ol T$\nis a proper $G$-invariant pro-$p$ subtree of $T$; a contradiction.\n\nSuppose that $g\\in G$ acts trivially upon all of $\\ol T$.\nThen, in particular, $\\ol eg=\\ol e$ and,\nas edges of $eG$ under the collapsing procedure are not identified,\nwe must have $eg=e$, {\\it i.e.}, $g\\in G_e$.\nTherefore the kernel of the action of $G$ upon\n$\\ol T$ is contained in $G_e$ and so $G_e$ contains a normal\nsubgroup of $G$ which, by\n\\cite[Thm. 3.12]{RZ:00a}\nmust act trivially on $T$.\nHence $G$ acts faithfully on $\\ol T$.\n\\end{proof}\n\n\nRecall from the introduction that $G$ acts virtually freely\non a space $X$ if some open subgroup $H$ of $G$ acts freely on $X$.\n\n\\begin{lemma}\\label{l-treetech\nLet $G$ be a finitely generated pro-$p$ group acting\nfaithfully, irreducibly and virtually freely on a pro-$p$ tree $T$.\nThen there are a pro-$p$ tree $D$, an edge $e\\in E(D)$,\na finite subset $V\\subseteq D^{G_e}$ and\na finite subset $X\\subseteq G$ such that\n\\begin{itemize}\n\\item[{\\rm (a)}]\n$G$ acts faithfully upon $D$.\n\\item[{\\rm (b)}]\nAll edge stabilizers are pairwise conjugate;\nin particular, $D=D^{G_e}G$.\n\\item[{\\rm (c)}]\nfor every $v_1,v_2\\in V$, $v_1G=v_2G$ implies $v_1=v_2$.\n\\item[{\\rm (d)}]\n$X$ freely generates a free pro-$p$ subgroup $H$\nsuch that for $G_0\\!:=\\gp{G_v\\mid v\\in V}$\nwe have $G=\\gp{G_0, H}$ and $H\\cap G_0^G=1$.\n\\item[{\\rm (e)}] For each $x\\in X$, we have\n\\begin{equation*}\nG_e^x\\subseteq\\bigcup_{v\\in V} G_v \\, .\n\\end{equation*}\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof}\nLet $e\\in T$ be an edge with the stabilizer $G_e$ \nof minimal order.\nLet $\\Sigma$ denote the set of all non-trivial finite subgroups\n$L$ of $G$ that are not conjugate to a subgroup of $G_e$.\nSince $G$ is finitely generated,\nCorollary \\ref{c:finiteconjugacy}\nsays that there exist up to conjugation\nonly finitely many finite subgroups in $G$;\nin particular there is a finite subset $S$ of\n$\\Sigma$ such that $\\Sigma=\\{L^g\\mid L\\in S, \\ g\\in G\\}$.\nTherefore the subset\n$T_\\Sigma\\!:=\\{m\\in T\\mid \\exists L\\in\\Sigma, m\\in T^L\\}$,\nwhich is the union of all subtrees of fixed points $T^L$ for\nsubgroups $L\\in\\Sigma$ can be represented in the form\n$T_\\Sigma=\\bigcup_{L\\in S}T^LG$ and is hence a closed\n$G$-invariant subgraph of $T$.\nTherefore\n$E(T^{G_e})G=T\\,-\\, T_\\Sigma$ is open and we can apply\nLemma \\ref{l-exchange1}\nto obtain, by collapsing the connected components of $T_\\Sigma$,\na pro-$p$ tree $D$ on which $G$ acts irreducibly and faithfully with\nall edge stabilizers conjugate to $G_e$.\nThus $D$ satisfies (a) and (b).\n\nWe come to proving (c),(d) and (e).\nSet $N\\!:=\\gp{G_v\\mid v\\in V(D)}$.\nBy Lemma \\ref{l-action}\nthere is finite minimal subset $X$ of generators of\na retract $H$ in $G$ of $G\/N$ such that\n$D^{G_e}\\cap D^{G_e}x\\neq\\emptyset$ for every $x$ in $X$;\nin fact, as $G\/N$ is free pro-$p$ by\nTheorem \\ref{t:trees1}(b),\n$X$ freely generates $H$.\nMoreover, by the construction of $D$, there is only a finite subset\n$V$ of vertices up to translation with stabilizers that\nare not conjugates of $G_e$;\nto see this we observe that if vertices $v,w$ are both\nstabilized by $L\\in\\Sigma$, then $L$ stabilizes the geodesic $[v,w]$\n(see Theorem \\ref{t:trees1}(c))\nand so $v,w$ belong to the same connected component of $T_\\Sigma$.\n\nIt follows that $G=\\gp{G_v, H\\mid v\\in V}$ and\n$H\\cap \\gp{G_v\\mid v\\in V}=1$.\nMoreover, since $G$ is pro-$p$ we can reduce $V$ such that\nno two distinct vertices of it lie in the same orbit.\n\nBy construction, for every group element\n$x\\in X$ there is\na vertex $v_x\\in D^{G_e}$ with $v_xx\\inv\\in D^{G_e}$.\nWhen $f$ is any edge in the geodesic $[v_x,v_xx\\inv]$ then\n$G_e=G_f=G_{v_x}\\cap G_{v_x}^{x\\inv}$\n(see Theorem \\ref{t:trees1}(c)),\nso that, in particular, $G_e^{x}\\le G_{v_x}$.\nFinally modify $V$ by replacing for every $x\\in X$\na vertex $v\\in V$ by the vertex $v_x$ whenever $vG=v_xG$.\nThen we see that (c), (d), and (e) all hold.\n\\end{proof}\n\nIt is now convenient to introduce a notion of\n{\\em pro-$p$ {\\rm HNN}-extension} as a\ngeneralization of the construction described in\n\\cite[Sec. 4, p. 97]{RZ:00a}.\n\n\\begin{definition}\\label{d:HNN-ext}\n\\rm Suppose that $G$ is a pro-$p$ group, and for a finite set\n$X$, there are given monomorphisms $\\varphi_x:A_x\\to G$ for subgroups\n$A_x$ of $G$.\nThe {\\em {\\rm HNN}-extension}\n$\\tilde G\\!:={\\rm HNN}(G,A_x,\\varphi_x,x\\in X)$\nis defined to be the quotient of $G\\amalg F(X)$ modulo the\nequations $\\varphi_x(a_x)=a_x^x$ for all $x\\in X$.\nWe call $\\tilde G$ an {\\em {\\rm HNN}-extension} and\nterm $G$ the {\\em base group}, $X$ the set of {\\em stable letters},\nand the subgroups $A_x$ and $B_x\\!:=\\varphi_x(A_x)$\n{\\em associated}.\n\\end{definition}\n\nOne can see that every {\\rm HNN}-extension in the sense of the\npresent definition can be obtained\nby successively forming {\\rm HNN}-extensions, as defined\nin \\cite{RZ:00a}, each time defining the base group\nto be the just constructed group and then\nselecting a pair of associated subgroups\nand adding a new stable letter.\n\nThe {\\rm HNN}-extension $\\tilde G\\!:={\\rm HNN}(G,A_x,\\varphi_x,X)$\ncan also be defined by a {\\em universal property} as follows.\nThere are canonical maps\n$\\tilde f:G\\to\\tilde G$, $\\tilde f_x:A_x\\to\\tilde G$, $\\tilde g:X\\to \\tilde G$,\nwith $\\tilde f_x(a_x)^{\\tilde g(x)}=\\tilde f\\varphi_x(a_x)$\nfor all $a_x\\in A_x$,\nso that, given any pro-$p$ group $H$, any homomorphisms $f:G\\to H$, $f_x:A_x\\to H$ and a map $g:X\\to H$\nsuch that for all $x\\in X$ and all $a_x\\in A_x$\nwe have $f(\\varphi_i(a_x))=f_x(a_x)^{g(x)}$ then there is a unique\nhomomorphism $\\omega:\\tilde G\\to H$ with $f=\\omega\\tilde f$, $g=\\omega\\tilde g$, and, for all $x\\in X$,\n$f_x=\\omega\\tilde f_x$.\n\n\n\n\\begin{thm}\\label{t-treeacting\n Let $G$ be a finitely generated infinite pro-$p$ group acting\n virtually freely on a pro-$p$ tree $T$.\n Then $G$ splits either as an amalgamated free pro-$p$\n product or as a proper pro-$p$ ${\\rm HNN}$-extension\n over some edge stabilizer.\n\\end{thm}\n\n\\begin{proof}\\setcounter{claims}0\nWe consider $G$ to be a counterexample to the theorem\nwith minimal index $[G:F]$\nwhere $F$ is an open subgroup of $G$ acting freely on $T$.\n\n\\bcl\n$G$ does not have a non-trivial finite normal subgroup.\nIn particular, we can assume that\n$G$ acts on $T$ faithfully and irreducibly.\n\\ecl\n\nBy \\cite[Lemma 3.11]{RZ:00a} there\nexists a unique minimal $G$-invariant subtree in $T$.\nReplacing $T$ by this subtree we may assume that\nthe action of $G$ is irreducible.\n\nNow, if $G$ contains a non-trivial finite normal subgroup,\nit contains a central subgroup $C$ of order $p$.\nBy the minimality assumption on $[G:F]$\nand as $[{G\/C}:{FC\/C}]<[G:F]$\nthe quotient group $\\ol G\\!:=G\/C$ satisfies\nthe conclusion of the theorem, {\\it i.e.} $\\ol G$ is either\nan amalgamated free pro-$p$ product\n$\\ol G=\\ol G_1\\amalg_{\\ol H} \\ol G_2$ with finite\namalgamating subgroup or it is\nan ${\\rm HNN}$-extension $\\ol G={\\rm HNN}(\\ol G_1, \\ol H, t)$\nwith finite associated subgroups.\nThen $G$ is either a non-trivial amalgamated free pro-$p$ product\n$G=G_1\\amalg_H G_2$ or ${\\rm HNN}(G_1,  H, t)$\nwith $G_1,G_2, H$ being preimages of $\\ol G_1,\\ol G_2, \\ol H$ in $G$,\nrespectively, as needed.\nHence $G$ does not possess a non-trivial\nfinite normal subgroup.\nSince the vertex stabilizers are finite, the kernel of the\naction of $G$ upon $T$ is finite, hence it is trivial.\n\n\\medskip\n{\\em Thus, there exist $D$, $e$, $V$, $X$ and $G_0$\nhaving the properties {\\rm (a)}-{\\rm (e)} of\nLemma {\\rm \\ref{l-treetech}}.\nNote that the stabilizers of vertices in $D$ may well be infinite.}\n\n\\bcl\nThe pro-$p$ subgroup $H$ of $G$\nfreely generated by $X$ must be trivial.\n\\ecl\n\nSuppose that $H\\neq1$.\nLet $\\tilde G={\\rm HNN}(G_0, G_e, X)$ and\n$\\lambda:\\tilde G\\longrightarrow G$ be the epimorphism\ngiven by the universal property.\nBy induction on ${\\rm rank}\\of F$ we show that\n$\\lambda$ is an isomorphism.\n\nIt suffices to show that the rank of $F$ equals the rank of\n$\\tilde F\\!:=\\lambda^{-1}(F)$.\nIf $F_0\\!:=G_0\\cap F\\neq 1$ we can factor out\nthe normal closure of $F_0$ in $G$\n(and, if necessary, the kernel of the action as well)\nin order to obtain the quotient group $\\ol G$\nwhich acts on $D\/F_0^G$ and satisfies\n${\\rm rank}\\of{\\ol F}<{\\rm rank}\\of F$.\nTherefore the induced epimorphism\n$\\ol\\lambda:{\\rm HNN}(\\ol G_0,\\ol G_{\\ol e},X)\\to\\ol G$\nis an isomorphism, and it is not hard to see that\n${\\rm HNN}(\\ol G_0,\\ol G_{\\ol e},X)$ is isomorphic to\n$\\tilde G\/\\tilde F_0^{\\tilde G}$,\nwhere $\\tilde F_0\\!:=G_0\\cap \\tilde F$.\nThis shows that the image $\\ol F$ of $F$ in $\\ol G$ is\nisomorphic to  $\\tilde F\/\\tilde F_0^{\\tilde G}$.\nBy Proposition \\ref{p-freeedgeaction}\n$F$ is a free pro-$p$ product $F\\cong F_0\\amalg \\ol F$ and\n$\\tilde F\\cong  \\tilde F_0\\amalg \\tilde F\/\\tilde F_0^{\\tilde G}$, so $F\\cong \\tilde F$ and we are done.\nThus we may assume that $G_0$ is finite.\nNow applying\nLemma \\ref{l-rankformula}\nto $\\tilde G$ and $G$ we deduce that\n${\\rm rank}(\\tilde F)={\\rm rank}(F)$, so\n$\\lambda\\restr{\\tilde F}$ turns out to be an isomorphism,\ncontradicting $G$ being a counterexample.\nThis finishes the proof of the claim.\n\n\\bcl\nThe natural epimorphism\n\\begin{equation*}\n\\lambda:{\\coprod_{v\\in V}}\\raisebox{-1.2ex}{\\small $G_e$}\nG_{v}\\longrightarrow G\n\\end{equation*}\nfrom the free pro-$p$ product of vertex stabilizers $G_v$\namalgamated along the single edge group $G_e$\nonto $G$ is an isomorphism.\n\\ecl\n\nLet us use induction on ${\\rm rank}\\of F$.\nSince $F$ acts freely on $E(D)$\nby Proposition \\ref{p-freeedgeaction},\nfor each $v\\in V$ the intersection $F\\cap G_{v}$\nis a free factor of $F$, so similarly as in the proof of Claim 2\nwe can use the induction hypothesis,\nin order to achieve all $G_{v}$ to be finite.\n\nPut $\\tilde F=\\lambda^{-1}(F)$.\nSince $\\lambda$ restricted to all $G_v$ is injective,\nit suffices to prove that $\\lambda\\restr{\\tilde F}$ is an isomorphism.\nBut by applying\nLemma \\ref{l-rankformula} to $\\tilde G$ and $G$\nwe get that $F$ and $\\tilde F$ have the same rank\nand therefore $\\lambda$ is an isomorphism.\nThe result follows.\n\n\\medskip\n\nClaim 3 shows that $G$ is not a counterexample, a final contradiction.\n\\end{proof}\n\n\n\\begin{thm}\\label{t-fund\nA finitely generated pro-$p$ group $G$ acting\nvirtually freely on a pro-$p$ tree $T$\nis isomorphic to the\nfundamental pro-$p$ group $\\Pi_1(\\cG,\\Gamma)$ of a finite graph of\nfinite $p$-groups whose edge and vertex groups are isomorphic to\nthe stabilizers of some edges and vertices of $T$.\n\\end{thm}\n\n\\begin{proof}\nBy induction on the rank of a maximal normal free\npro-$p$ subgroup $F$ of $G$.\nIf ${\\rm rank}(F)=0$, that is $G$ is finite,\ntake as graph of groups the single vertex $G$.\nIn the general case, we apply\nTheorem \\ref{t-treeacting}\nto split $G$ as an amalgamated free pro-$p$ product\n$G=G_1\\amalg_K G_2$ or as a pro-$p$ {\\rm HNN}-extension\n$G={\\rm HNN}(G_1,K,t)$ over a finite subgroup $K$.\nMoreover, we are free to choose $K$ up to\nconjugation in $G_1$.\nThen every free factor, or the base group,\nsatisfies the induction hypothesis and\nso exists the fundamental group of a finite graph of finite $p$-groups.\nBy \\cite[Thm. 3.10]{ZM:89}\n$K$ is conjugate to some vertex group of $G_1$ and so\nwe may assume that\n$K$ is contained in a vertex group of $G_1$.\nNow in the case of an amalgamated product\nthere is $g_2\\in G_2$ such that $K^{g_2}$ is contained in\na vertex group of $G_2$, so $G$ admits a decomposition\n$G=G_1^{g_2}\\amalg_{K^{g_2}} G_2$.\nThus in both cases $G$ becomes\nthe fundamental group of a finite graph of finite $p$-groups.\n\\end{proof}\n\n\\begin{thm}\\label{t:subgrouptheorem\nLet $H$ be a finitely generated subgroup of the\nfundamental pro-$p$ group $G$ of a finite graph of finite $p$-groups.\nThen $H$ is the fundamental pro-$p$ group of\na finite graph of finite $p$-groups which are\nintersections of $H$ with some conjugates of vertex and edge\ngroups of $G$.\n\\end{thm}\n\n\\begin{proof}\nThe fundamental pro-$p$ group $G=\\Pi_1(\\cG,\\Gamma)$\nacts naturally on the standard pro-$p$ tree $T$\n({\\it cf.} \\cite[Sec. 3]{ZM:89})\nand therefore so does $H$.\nMoreover, since the graph $\\Gamma$ is finite,\nthere exists an open normal subgroup $U$ of $G$\nthat intersects all vertex groups trivially and so acts freely on $T$.\nThus Theorem \\ref{t-fund} can be applied.\n\\end{proof}\n\n\\begin{example}\\label{ex-4.3}\\rm\nLet $A$ and $B$ be groups of order $2$ and\n$G_0=\\gp{ A\\times B, t\\mid A^{t}=B}$ be a\npro-$2$ {\\rm HNN}-extension of $A\\times B$ with\nassociated subgroups $A$ and $B$.\nNote that $G_0$ admits an automorphism of order $2$ that\nswaps $A$ and $B$ and inverts $t$.\nLet $G=G_0\\rtimes C$ be the holomorph.\nSet $H_0=\\torgp{G_0}$ and $H=H_0\\rtimes C$.\nSince $G_0$ acts on its standard pro-$2$ tree\nwith finite vertex stabilizers, so does $H$.\nThe main result in\n\\cite{HZ:10}\nshows that $H$ does not decompose as the\nfundamental group of a profinite graph of finite $2$-groups.\nIts proof also shows that $H$ does not decompose as\nan amalgamated free pro-$p$ product or\nas a pro-$p$ {\\rm HNN}-extension over a finite group.\n\\end{example}\n\n\n\n\\section{\\texorpdfstring{$2$}{2}-generated subgroups}\n\\label{s:2-generated}\n\nThe final section is devoted to the proof of\nTheorem \\ref{t:freeorabelian_intro}.\nSo, henceforth, $G:=A\\amalg_{C} B$ is a free pro-$p$ product of\n$A$ and $B$ with procyclic amalgamating subgroup $C$\nsatisfying the following assumptions:\n\\begin{itemize}\n \\item[{\\rm (i)}]\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n\n \\item[{\\rm (ii)}]\n  each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group.\n\\end{itemize}\n\n\\begin{lemma}\\label{l:cent}\nFor every subgroup $D\\le C$ we have $N_G(D)=C_G(D)$.\n\\end{lemma}\n\n\\begin{proof}\nBy the pro-$p$ version of \\cite[Cor. 2.7(ii)]{RZ:96},\n\\begin{equation*}\nN_{G}(D)=N_{A}(D)\\amalg_{C}N_{B}(D)\\, .\n\\end{equation*}\nSince solvable $2$-generated subgroups of $A$ and $B$ are abelian,\n$N_{A}(D)=C_{A}(D)$ and $N_{B}(D)=C_{B}(D)$;\nhence $N_G(D)=\\gp{C_{A}(D),C_{B}(D)}\\subseteq C_G(D)$,\nas needed.\n\\end{proof}\n\n\\begin{thm}\\label{t:freeorabelian}\n  Let $G=A\\amalg_{C} B$ be a free pro-$p$ product of\n  $A$ and $B$ with procyclic amalgamating subgroup $C$.\n  Suppose that\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n  If each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group\n  then so is each $2$-generated pro-$p$ subgroup of $G$.\n\\end{thm}\n\n\\begin{proof}\nLet $T$ be the standard pro-$p$ tree on which $G$ acts\n({\\it cf.} \\cite[Sec. 4]{RZ:00a})\nand let $L$ be a $2$-generated pro-$p$ subgroup of $G$.\nIt follows from the definition of $T$ that if\n$L$ stabilizes a vertex of $T$, then $L$ is up to conjugation\nin one of the free factors of $G$; hence $L$ is either free pro-$p$ or\nfree abelian pro-$p$, by hypothesis (ii).\n\nLet us assume that $L$ fixes no vertex of $T$.\nSince $L$ is finitely generated, we have \n$L\\cong \\varprojlim L\/U_n$ where\n$\\{U_n\\mid n\\in{\\mathbb N}\\}$ is a set of open normal subgroups of\n$L$ with $\\bigcap U_n =1$.\nRecall our notation $\\widetilde{U_n}$\nfor the closed subgroup of $U_n$ generated by all vertex\nstabilizers with respect to the action of $U_n$ on $T$.\nWe consider the infinite set $I$ of integers $n$ such that\n$U_n\/\\widetilde{U_n}$ is an infinite free pro-$p$ group\n({\\it cf.} Theorem \\ref{t:trees1}(b)).\nSo, defining $L_n\\!:=L\/\\widetilde{U_n}$ we see that\n$L_n$ acts virtually freely on a pro-$p$ tree $T\/\\widetilde{U_n}$\n({\\it cf.} Theorem \\ref{t:trees1}(a))\nand so we are in position to apply\nTheorem \\ref{t-fund}\nto each of them.\nThus $L_n=\\Pi_1(\\cL_n,\\Gamma_n)$ is the fundamental pro-$p$ group of\na finite graph of finite $p$-groups whose edge and vertex groups\nare stabilizers of certain edges and vertices of $T\/\\tilde U_n$.\nClearly we have $L\\cong \\varprojlim \\{L_n, \\varphi_{nm}, I\\}$\nwhere each $\\varphi_{nm}$ is the canonical map.\n\nNow, since $L\/\\widetilde{L}$ is a free pro-$p$ group\nof rank at most $2$,\nwe need to consider only the two cases\n$L=\\widetilde{L}$ and $L\/\\widetilde{L}\\cong {\\mathbb Z}_p$;\nin the remaining case, when $d(L\/\\widetilde L)=2$,\n$L$ is itself free pro-$p$ of rank 2 -- by the Hopfian property.\nWe can assume that $\\widetilde{L}\\neq 1$, otherwise\nthere is nothing to prove.\n\n\\medskip\n\\noindent {\\it Case 1}.  $L=\\widetilde{L}$.\n\n\\medskip\n\nWe claim that $\\Gamma_n$ is a tree.\nIf not then there is an edge $e\\in \\Gamma_n$\nso that $L_n={\\rm HNN}(P_n,G(e),t)$ for $G(e)$ finite.\nBut then there is a homomorphism from $L_n$ onto ${\\mathbb Z}_p$\ncontradicting $L_n=\\torgp{L_n}$.\n\nThen in light of\nLemma \\ref{l:2gen}(c) and of Proposition \\ref{p:treeprod},\nwe have inverse systems of conjugates of $K_n$ and $D_n$; \nfollowing the notation of the referred Proposition, we define\ntwo procyclic groups $K\\!:=\\varprojlim K_n'$ and $D \\!:=\n\\varprojlim D_n''$.\n\nWe claim that $D=1$.\nNote that since each $D_n$ is an edge stabilizer\nwith respect to the $L_n$-action, we have\n$D=L\\cap C^g$, for some $g\\in G$. \nSince $C_L(D) = L\\cap C_G(D)$, it follows from (i) that \n$C^g$ is a direct factor of $C_G(D)$,\nhence $D$ is a direct factor of $C_L(D)$.\nSuppose on the contrary that $D\\neq 1$. \nSince the procyclic group $K$ contains $D$,\nit follows that $D=K$. \nNow, the projection $K\\to K_{n_0}'$ is\nsurjective for some sufficiently large $n_0$, by\nLemma \\ref{l:invsys}(a).\nHence $D_{n_0}=K_{n_0}$; a contradiction to the\nnon-fictitious decomposition of $L_{n_0}$.\n\nThus $D=1$, and so\n$\\varprojlim {D_n''}^{L_n} =1$, by\nLemma \\ref{l:invsys}(b).\nThen, writing $L_n\\cong K_n'\\amalg_{D_n''} R_n'$\nwe have\n$L\\cong \\varprojlim L_{n}\/{D_n''}^{L_n}\n\\cong \\varprojlim (K_n'\/D_n'' \\amalg R_n'\/{D_n''}^{R_n'})$.\nNow, if $d(L_n\/D_n''^{L_n})=1$ for every $n$, then\n$L$ is procyclic; thus\nwithout loss of generality we may and do assume that each\n$L_{n}\/{D_n''}^{L_n}$ is $2$-generated.\nSince $K_n'\/D_n''$ is $1$-generated,\nso is $R_n'\/{D_n''}^{R_n'}$.\nTherefore $L\\cong \\mathbb{Z}_p\\amalg \\mathbb{Z}_p\\,$,\nby Proposition \\ref{p:decompfp}.\nOur proof is finished for {\\it Case 1}.\n\n\n\\medskip\n\\noindent {\\it Case 2}. $L\/\\widetilde{L}\\cong {\\mathbb Z}_p$.\n\n\\medskip\n\nFor $n\\in \\mathbb N$ we have ${\\mathbb Z}_p\\cong\nL\/\\widetilde L\\cong L_n\/(\\widetilde L\/\\widetilde U_n)$\nand therefore $\\Gamma_n$ cannot be a tree.\nThen we can select a suitable edge $e_n$ of $\\Gamma_n$,\nset $\\Delta_n:=\\Gamma_n\\,-\\,\\{e_n\\}$, and present\n$L_n={\\rm HNN}(K_n,D_n,t_n)$\nwith cyclic edge group $D_n$ of $e_n$ and\n$K_n=\\Pi_1({{\\mathcal G}_n}\\restr{\\Delta_n},\\Delta_n)$.\n\nSince $\\widetilde{L}\/\\widetilde{U_n}$ is generated by torsion,\nas a consequence of\nTheorem \\ref{t:hnnproperties}(a), it is contained in ${K_n}^{L_n}$;\nso, ${\\langle {\\rm tor}(L_n) \\rangle}={K_n}^{L_n}$.\nBy \\cite[Prop. 1.7(ii)]{Zalesskii:04},\n$K_n\/{\\langle {\\rm tor}(K_n)\\rangle}$ is a free pro-$p$ group,\nwhence ${\\langle {\\rm tor}(L_n) \\rangle}$ has trivial image\nin the quotient ${\\rm HNN}(K_n\/{\\langle {\\rm tor}(K_n)\\rangle},1, t_n)$\nof $L_n$.\nThus $K_n={\\langle {\\rm tor}(K_n)\\rangle}$.\nSince $K_n$ acts on the pro-$p$ tree $T\/\\widetilde{U_n}$\nwe have $K_n=\\widetilde{K_n}$\n({\\it cf.} Theorem \\ref{t:trees1}(d)),\nso in particular, $\\Delta_n$ must be a tree.\nPassing now to a cofinal subset of $\\mathbb N$,\nif necessary, we may assume that for all $n$ either\n$\\Delta_n$ is a single vertex or $\\Delta_n$ contains an edge.\nWe discuss the two subcases.\n\n\\medskip\n\\noindent {\\it Subcase 2($\\alpha$).\nFor each $n\\in\\mathbb N$ the tree $\\Delta_n$ is a single vertex.}\n\n\\medskip\n\nThen $K_n$ is finite.\nPassing again to a cofinal subset of $\\mathbb N$, if necessary,\nwe can, making use of\nTheorem \\ref{t:hnnproperties}(a)\nand a projective limit argument, arrange that\n$\\varphi_{n+1n}(K_{n+1})\\le K_n$ holds for all $n$.\nPassing again to a cofinal subset of $\\mathbb N$, if necessary,\nand making use of\nLemma \\ref{l:2gen}(b)\nwe can arrange that for all $n$ either\n$K_n$ is cyclic or that $d(K_n)=2$.\nWe shall discuss the situations when $K\\!:=\\varprojlim K_n$\nis procyclic and when $d(K)=2$.\n\nIf $K$ is procyclic, then for every $m$ there exists $n>m$\nsuch that the $\\varphi_{nm}(K_n)$ is cyclic and so\n$\\varphi_{nm}(D_n)=\\varphi_{nm}(D_n^{t_n})$.\nHence $\\varphi_{nm}(t_n)$ normalizes $\\varphi_{nm}(D_n)$\nand so $L_m=N_{L_m}(\\varphi_{nm}(D_n))$.\nSince $L=\\varprojlim L_m$ it follows that\n$D\\!:=\\varprojlim D_m$ is normal in $L$.\nSince $E(T)$ is a compact $L$-space, setting in\nLemma \\ref{l:fp}\n$X\\!:=E(T)$, $G\\!:=L$, and $S_n\\!:=D_n$,\nwe find $e\\in E(T)$ with $D\\le G_e$.\nTherefore $D^g\\le C$ for some $g\\in G$ and,\nif $D\\neq 1$, making use of \nLemma \\ref{l:cent},\nwe find that $L\\cong {\\mathbb Z}_p\\times {\\mathbb Z}_p$\nby hypothesis (i), as needed.\n\nNext assume that $D=1$.\nIt follows from\nLemma \\ref{l:invsys}(b)\nthat $\\varprojlim D_m^{L_m}=1$ and\nso $L=\\varprojlim L_m\/D_m^{L_m}$.\nObserving that\n$L_m\/D_m^{L_m}\n=(K_m\/K_m\\cap D_m^{L_m})\\amalg \\langle t_m\\rangle$\nProposition \\ref{p:decompfp}\nimplies that $L\\cong {\\mathbb Z}_p\\amalg {\\mathbb Z}_p$,\nwhence the result when $K$ is procyclic.\n\nFor finishing Subcase 2($\\alpha)$\nwe can now assume that $d(K)=2$.\nThen\nLemma \\ref{l:invsys}(a)\nin conjunction with a projective limit argument implies that\n$\\varphi_{n+1 n}(K_{n+1})\\cong K_n$ for every $n$.\nBy virtue of\nProposition \\ref{p:decomphnn},\nwe have inverse systems of conjugates $K_n'$ and $D_n''$\nof the finite $p$-groups $K_n$ and\n$D_n$, and $L ={\\rm HNN}(K,D,t)$ where $K\\!:=\\varprojlim K_n'$ and\n$D\\!:=\\varprojlim D_n''$ is procyclic.\nWe must have $D\\neq 1$,\nelse $L\\cong K\\amalg \\gp t$, and so $2=d(K)=d(L)-1=1$;\na contradiction.\n\nAn application of Lemma \\ref{l:fp} shows that\n$K$ stabilizes a vertex in $T$; it is therefore, up to conjugation,\ncontained in either $A$ or $B$ and so by hypothesis (ii) is either\nfree pro-$p$ or free abelian pro-$p$.\nIn the first case we observe that\nLemma \\ref{l:2gen}(b)\nimplies that $K=D\\amalg D^t$ and so\n$L=D\\amalg \\langle t\\rangle$ is a free pro-$p$ group.\n\nSo assume in the sequel that\n$K$ is a free pro-$p$ abelian group.\nNote that $L={\\rm HNN}(K,D,t)$ contains\n$H\\!:=K\\amalg_DK^t$ which is not abelian.\nOn the other hand since $E(T)$ is a compact $L$-space, setting in\nLemma \\ref{l:fp}\n$X\\!:=E(T)$, $G\\!:=L$, and $S_n\\!:=D_n$\nwe find $e\\in E(T)$ with $D\\le G_e$.\nHence $D\\le C^g$ for suitable $g\\in G$.\nSince $D\\le C^g$, by hypothesis (i) $C_G(D)$ is abelian,\nand it contains $H$; a contradiction.\nHence we are done with Subcase 2($\\alpha$).\n\n\n\\medskip\n\n\\noindent {\\it Subcase 2($\\beta$).\nFor each $n\\in\\mathbb N$ the tree $\\Delta_n$ contains an edge.}\n\n\\medskip\n\nLemma \\ref{l:2gen}(c)  and Proposition \\ref{p:treeprod}\nimply that $K_n$ can be written as\n$K_n=X_n\\amalg_{Z_n}W_n$, with cyclic $p$-groups $X_n$,\nand there are inverse systems $\\{X'_n\\}$ and $\\{Z''_n\\}$\nwith $Z''_n\\le X'_n$ of conjugates of $X_n$ and $Z_n$\nrespectively.\nDefine procyclic groups $X=\\varprojlim X_n'$ and\n$Z=\\varprojlim Z_n''$.\nWe must have $Z\\neq X$ else by\nLemma \\ref{l:invsys}(a) we could find $n$ with $Z_n=X_n$\nand so the decomposition $K_n=X_n\\amalg_{Z_n}W_n$\nwould be fictitious; a contradiction.\nSetting in\nLemma \\ref{l:fp}\n$X:=E(T)$, $G:=L$, and $S_n:=Z_n''$\nwe find $e\\in E(T)$ with $Z\\le G_e$.\nHence there is $g\\in G$ with $Z\\le C^g$.\nNow, since $Z\\neq X$, \nhypothesis (i) implies $Z=1$.\nLet $\\bar K_n=K_n\/{Z_n}^{K_n}$\nand let $\\bar D_n$ be the canonical image of $D_n$ in $\\bar K_n$.\nThen, we consider\n\\begin{equation*}\n\\bar L_n \\!=\\! L_n\/{Z_n}^{L_n} \\!=\\!\n\\text{\\rm HNN}(\\bar K_n, \\bar D_n, \\bar t_n) \\!=\\! \\text{\\rm HNN}\n(X_n\/{Z_n}^{X_n}\\amalg W_n\/{Z_n}^{W_n}, \\bar D_n, \\bar t_n)\n\\, .\n\\end{equation*}\nBy Lemma \\ref{l:2gen}(b),\neach pro-$p$ group $K_n$ is at most $2$-generated,\nhence considering $\\bar L_n$ modulo its Frattini subgroup,\nwe can conclude that $d(W_n\/{Z_n}^{W_n})=1$.\nSo, taking into account\nLemma \\ref{l:2gen}(b) we conclude that\n$X_n\/{Z_n}^{X_n}$ and $W_n\/{Z_n}^{W_n}$ are isomorphic\ncyclic $p$-groups.\nThus $\\bar L_n\\cong X_n\/{Z_n}^{X_n} \\amalg\n\\langle\\bar t_n\\rangle$.\nBy Lemma \\ref{l:invsys}(b) and Proposition \\ref{p:decompfp}\nwe obtain that $L\\cong \\varprojlim\n\\bar L_n \\cong {\\mathbb Z}_p\\amalg {\\mathbb Z}_p$.\nThis concludes the proof of the theorem.\n\\end{proof}\n\n\\begin{cor}\\label{c:2-free}\nSuppose that neither $A$ nor $B$ contains a\n$2$-generated non-procyclic abelian subgroup.\nThen any $2$-generated subgroup $L$ of $G$\nis a free pro-$p$ group.\n\\end{cor}\n\n\\begin{proof}\nSuppose that $L$ is a free abelian pro-$p$ group of rank $2$.\n\nLet $T$ be the standard pro-$p$ tree on which $G$ acts.\nThen by\n\\cite[Thm. 3.18]{RZ:00a}\neither $L$ stabilizes a vertex or there\nis an edge $e$ of $T$ such that $L\/L_e\\cong{\\mathbb Z}_p$.\nBut $L$ cannot stabilize a vertex; else it is conjugate to \na subgroup of one of the free factors of $G$,\ncontradicting the supposition.\n\nTherefore $L\/L_e\\cong {\\mathbb Z}_p$ for some edge $e$.\nSince $d(L)=2$ we must have $L_e \\not= 1$.\nConjugating $L$ by some element of $G$\nwe may assume that $L_e$ is contained in $C$.\nThen, $L=N_G(L_e)=C_G(L_e)$, by\nLemma \\ref{l:cent}(a), and,\nby the centralizer condition of the theorem,\n$L=C\\cong {\\mathbb Z}_p$, a contradiction.\nThus, by\nTheorem \\ref{t:freeorabelian},\n$L$ must be free pro-$p$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe central prediction of the Weakly Interacting Massive Particles (WIMP)  paradigm is that Dark Matter (DM) particles should have a thermally averaged \nannihilation cross section of $\\langle \\sigma v \\rangle \\sim 10^{-26} \\, \\mathrm{cm^3 \\, s^{-1}}$ during freeze-out. \nIn many DM models, the present-day annihilation cross section in astrophysical systems is predicted to be of a similar magnitude, \nproviding a clear target for indirect detection experiments searching for the products of DM annihilation processes.\n\nWhile the most robust constraints on the DM annihilation cross section stem from observations of the {CMB~\\cite{Aghanim:2018eyx} and of the} $\\gamma$-ray sky, \nin particular from Fermi-LAT~\\cite{Ackermann:2013yva,Ackermann:2015zua,Ackermann:2015lka}, highly complementary information\ncan be obtained by precisely measuring the flux \nof charged anti-particles arriving on Earth. Very recently, AMS-02 has released results \nfrom the first seven years of data taking~\\cite{Aguilar:2021tos}, which include in particular the flux of antiprotons with unprecedented precision. \nTheoretical predictions for this flux however require detailed modelling of the production and propagation of \ncharged cosmic rays (CRs) in the Galaxy, which are subject to significant uncertainties {and are currently constrained using CR data (see e.g.\\ Ref.~\\cite{Korsmeier:2021brc}), as well as their non-thermal emissions (see e.g.\\ Ref.~\\cite{Orlando:2017mvd}).} \n\nWhile various numerical codes, such as \\textsc{Galprop}~\\cite{Strong:1998fr} and \\textsc{Dragon}~\\cite{Evoli:2008dv}, exist to address this challenge and simulate the propagation of CRs, they require as input a large number of parameters that need to be varied to assess their impact on the predictions. \nAs a result these simulations are typically computationally so expensive that they become prohibitive in the context \nof a global analysis of DM models, where also the fundamental model parameters need to be varied~\\cite{Ambrogi:2018jqj}. Recent analyses of the AMS-02 antiproton data have therefore typically focused on simplified DM models with only a single annihilation channel, see e.g.\\ Ref.~\\cite{Cuoco:2016eej, Cui:2016ppb, Reinert:2017aga,Cholis:2019ejx}. \n\n\nIn the present work we explore the potential of artificial neural networks (ANNs) to solve this issue and substantially speed up \nthe calculation of predictions for the primary antiproton flux for a very broad range of DM models.\\footnote{For other recent works on the use of machine learning for cosmic ray propagation in the context of DM we refer to Refs.~\\cite{Lin:2019ljc,Tsai:2020vcx}.} Specifically, we employ recurrent neural networks (RNNs), which are particularly well suited for the prediction of continuous spectra. \nThe network is trained on a large sample of antiproton fluxes based on propagation parameters that are chosen to broadly agree with recent AMS-02 data, and a general parametrisation of the properties of the DM particle in terms of its mass and the branching fractions for various different final states. Using the same approach we have also developed and trained ANNs to accurately predict further CR species, like secondary antiprotons, protons or helium.\n\nThe predictions of the network can then be used to calculate the likelihood of the AMS-02 data for a given DM model and varying propagation parameters in order to calculate exclusion limits using a range of frequentist or Bayesian methods.  However, it is important to ensure that the resulting constraints are not biased by regions of parameter space for which the ANN has not been sufficiently trained. In the Bayesian approach this potential pitfall is avoided by evaluating the likelihood for a fixed sample of propagation parameter points drawn from the posterior probability distribution in the absence of a DM signal. The marginalisation over propagation uncertainties can then be performed via importance sampling, i.e.\\ by appropriately reweighing and combining the points in the sample. This approach is particularly well suited for the analysis of antiproton data, since the propagation parameters are rather \ntightly constrained by the proton flux and the secondary antiproton flux, so that the presence of a DM signal \ndoes not dramatically shift the relevant regions of parameter space. \n\nWe emphasise that, while the initial generation of a sample from the posterior is computationally expensive, it does not require specific assumptions on the properties of DM and therefore only needs to be carried out once in advance. Moreover, the same simulation step can be used to set up the training data for the ANN, ensuring that the network is trained specifically on the most interesting regions of parameter space. Once training is completed, the remaining steps are computationally cheap and can be performed for a large number of DM parameters. Indeed, the full marginalisation over propagation parameters can be performed in a similar amount of time as it would take to \nsimulate a single parameter point in the conventional approach.\n\nWe apply our fully trained ANN to a number of cases of particular interest. For the case of DM annihilations exclusively into bottom quarks we show that the most recent AMS-02 data leads to results that are compatible with previous studies. \nIn particular, we recover a notable excess for DM masses around 100 GeV in the case that no correlations in the AMS-02 data are considered. \nWe also present new constraints on the well-studied model of scalar singlet DM and find that antiproton data places competitive constraints on this model. \nHowever, we emphasise  that the ANN is not limited to these cases and can be applied to a wide variety of DM models. \nMoreover, the general approach that we present can be extended to consider different propagation models (provided a suitable simulator exists), systematic uncertainties (such as correlations in the AMS-02 data) or cross-section uncertainties, enabling the community to fully explore the wealth of the available CR data.\n\nThe remainder of this work is structured as follows. In section~\\ref{sec:cr} we briefly review the fundamental concepts of CR production and propagation and present the specific implementation that we adopt in the present work. We also carry out a first analysis of the most recent AMS-02 data and perform a parameter scan to identify the most interesting regions of parameter space. In section~\\ref{sec:ANN} we introduce our machine learning approach to simulating CRs and discuss how we train and validate our ANNs. Finally, in section~\\ref{sec:constraints} we apply the fully trained ANNs to constrain DM models. We present the relevant statistical methods and discuss the resulting exclusion limits.\n\n\\section{Cosmic-ray antiprotons in the Galaxy}\\label{sec:cr}\n\nFor the following discussion it is useful to distinguish between primary and secondary CRs. Primary CRs are directly accelerated and emitted by standard astrophysical sources \nlike supernova remnants or pulsars. But also more exotic scenarios such as the production of (anti)particles by DM annihilation or decay are considered as primary origin. \nProtons provide the dominant contribution to primary CRs (about 90\\%) while helium (He) makes up about 10\\%. Heavier nuclei only contribute at the percent level.\nOn the other hand, secondary CRs are produced during the propagation of primary CRs by fragmentation or decay.\nMore specifically, when the primary CRs interact with the gas in the Galactic disc, commonly called interstellar medium (ISM), secondary particles are produced.\nBecause of the different production mechanism, secondaries are suppressed with respect to primary CRs.\nIt is commonly believed that CR antiprotons do not have standard astrophysical sources%\n\\footnote{\n\tWe note that the possibility of primary antiprotons that are directly produced and accelerated at supernova remnants\n\t\\cite{Blasi:2009bd,Mertsch:2014poa,Mertsch:2020dcy, Kohri:2015mga} is also discussed in literature. \n}\nsuch that their dominant contribution comes from secondary production. As a consequence, antiprotons are \nsuppressed by 4--5 orders of magnitude with respect to protons, which makes them (together with other antimatter CRs, e.g.\\ antideuterons \\cite{Aramaki:2015pii,vonDoetinchem:2020vbj}) \na promising channel for constraining DM signals. \n\nIn this section we first discuss the production of antiprotons in the annihilation of dark matter particles in our Galaxy, followed by a discussion of backgrounds from secondary antiprotons. We then present the framework that we use to simulate CR propagation and the strategy to fit the resulting spectra to data. Finally, we perform a scan over the propagation parameters in order to create the training set for the machine learning approach introduced in section~\\ref{sec:ANN}.\n\n\\subsection{Antiprotons from dark matter annihilation}\\label{sec::antipdm}\n\nCR antiprotons are a long standing target used to search for signals of WIMP DM in our Galaxy\n\\cite{Bergstrom:1999jc,Donato:2003xg,Bringmann:2006im,Donato:2008jk,Fornengo:2013xda,Evoli:2011id,Bringmann:2014lpa,Pettorino:2014sua,Cirelli:2014lwa,Cembranos:2014wza,Hooper:2014ysa,Boudaud:2014qra,Giesen:2015ufa,Evoli:2015vaa,Johannesson:2016rlh,Luque:2021ddh,DiMauro:2021qcf}.\nMore recently, there has been a discussion of an antiproton excess at about 20~GeV, which could be fitted with a \nprimary DM source~\\cite{Cuoco:2016eej, Cui:2016ppb, Reinert:2017aga, Cuoco:2019kuu, Cholis:2019ejx}. \nHowever, the excess might also be accounted for by a combination of systematic effects~\\cite{Boudaud:2019efq, Heisig:2020nse, Heisig:2020jvs}.\nIf DM particles annihilate into standard model particle final states $f$ within the diffusion halo of our Galaxy as ${\\rm DM}\\!+\\!{\\rm DM} \\to f\\!+\\!\\bar{f}$, \nwe expect a corresponding flux contribution to antiprotons in CRs, coming from the subsequent decay of for example $q \\!+\\!\\bar{q}$ modes (see e.g.\\ \\cite{Cirelli:2010xx}). \nThe source term of this primary antiproton component,  $q_{\\bar{p}}^{(\\mathrm{DM})}$, is a function of the Galactic coordinates $\\bm{x}$ and the antiproton kinetic energy $E_\\mathrm{kin}$. For a generic combination of standard model final states $f$ it reads: \n\\begin{eqnarray}\n  \\label{eqn::pbar_DM_source_term}\n  q_{\\bar{p}}^{(\\mathrm{DM})}(\\bm{x}, E_\\mathrm{kin}) =\n      \\frac{1}{2} \\left( \\frac{\\rho(\\bm{x})}{m_\\mathrm{DM}}\\right)^2  \n      \\sum_f \\left\\langle \\sigma v \\right\\rangle_f \\frac{\\mathrm{d} N^f_{\\bar{p}}}{\\mathrm{d} E_\\mathrm{kin}} \\; .\n\\end{eqnarray}\nThe factor $1\/2$ in eq.~\\eqref{eqn::pbar_DM_source_term} corresponds to Majorana fermion DM.\nFurthermore, $m_\\mathrm{DM}$ is the DM mass, $\\rho(\\bm{x})$ the DM halo energy density profile, and $\\langle \\sigma v \\rangle_f$ is the thermally \naveraged annihilation cross section for the individual final states $f$. In the following, we fix $\\langle \\sigma v \\rangle$ independent of $f$ and account for this by assigning branching fractions into the relevant final states.\nFinally, $\\mathrm{d} N^f_{\\bar{p}}\/\\mathrm{d} E_\\mathrm{kin}$ denotes the energy spectrum of antiprotons for a single DM annihilation. \nThis quantity depends on the DM mass and the standard model final state. Here we implement the  widely used tabulated results for the antiproton energy spectrum \npresented in Ref.~\\cite{Cirelli:2010xx} which include electroweak corrections.%\n\\footnote{\n\tIf DM annihilates into a pair of $W$ or $Z$ bosons it is possible to produce one of them off-shell. This possibility is not taken \n\tinto account in the original tables. We extend the tables of $W$ and $Z$ bosons to lower DM masses using the tables from Ref.~\\cite{Cuoco:2017rxb}.\n}\n\n\nWe assume that the DM density in our Galaxy follows an NFW profile~\\cite{Navarro:1995iw}  \n$\\rho_{\\mathrm{NFW}}(r) = \\rho_h \\, r_h\/r\\, \\left( 1 + r\/r_h \\right)^{-2}$, \nwith a scale radius of $r_h=20\\;$kpc and a characteristic halo density, $\\rho_h$, which is normalised such that the local \nDM density at the solar position of $8.5\\;$kpc is fixed to $0.43\\;$GeV\/cm$^3$~\\cite{Salucci:2010qr}, compatible also with more recent estimates \\cite{deSalas:2020hbh}.\nWe note that the NFW profile is only one of many viable DM profiles currently investigated. Other profiles \ncan have a significantly different behavior towards the Galactic center, see e.g.\\ the discussion in Ref.~\\cite{Benito:2019ngh}.\nHowever, we stress that choosing a different DM density profile only has a small impact on the results presented in this paper  \nsince CR antiprotons from DM annihilation dominantly arrive from the local environment. \nTherefore they are mostly sensitive to the local DM density and the resulting flux depends only weakly on the shape of the DM density profile at \nthe Galactic center. \nMore specifically, the impact of changing the cuspy NFW profile to the cored Burkert profile~\\cite{Burkert:1995yz} has been quantified \nin Ref.~\\cite{Cuoco:2017iax}; it was found that a core radius of $5\\;$kpc only weakens DM limits by about 20\\%. \n\n\n\n\\subsection{Secondary antiprotons}\\label{sec::sec}\n\nThe ISM consists of roughly  90\\% hydrogen (H) and 10\\% He. Thus secondary antiprotons are mostly produced by the \ninteraction of $p$ and He CRs with the H and He components of the ISM. \nThe source term for the secondary antiprotons $q_{\\bar p}^{(\\mathrm{sec})}$ is thus given by the convolution of the primary CR fluxes $\\phi$ of isotope $i$, \nthe ISM density $n_{\\mathrm{ISM}}$ of component $j \\in \\lbrace \\mathrm{H}, \\mathrm{He} \\rbrace$, and the energy-differential \nproduction cross section $\\mathrm{d}\\sigma_{ij\\rightarrow\\bar p}\/\\mathrm{d}  E_{\\mathrm{kin},\\bar{p}}$:\n\\begin{eqnarray}\n\t\\label{eqn::pbar_sec_source_term}\n\tq_{\\bar p}^{(\\mathrm{sec})}({\\bm x},E_{\\mathrm{kin},\\bar{p}}) &=&  \n\t                                \\!\\!\\!\\!\\sum\\limits_{j \\in \\lbrace \\mathrm{H}, \\mathrm{He} \\rbrace} \\!\\!\\!\\! 4\\pi \\,n_{\\mathrm{ISM},j}({\\bm x}) \n\t                                \\sum\\limits_{i} \n\t                                \\int\n\t                                \\mathrm{d} E_{\\mathrm{kin},i} \\,\n                                    \\phi_i  ( E_{\\mathrm{kin},i}) \\, \n                                    \\frac{\\mathrm{d}\\sigma_{ij\\rightarrow\\bar p}}{\\mathrm{d}  E_{\\mathrm{kin},\\bar{p}} }(E_{\\mathrm{kin},i} ,  E_{\\mathrm{kin},\\bar{p}} )\\,.\n\\end{eqnarray}\nBy construction, secondaries are suppressed with respect to primary CRs. In the case of antiprotons, the experimentally \nobserved suppression compared to protons is {5 orders of magnitude at 1~GV and increases to about 4 orders of magnitude above 10~GV}.\nSince secondary CRs constitute the dominant contribution of the measured antiproton flux, \nconsidering standard astrophysical sources only already results in a good fit to the data \\cite{Korsmeier:2016kha, Cuoco:2016eej, Boudaud:2019efq}, \nsee also  discussion in section~\\ref{sec:fitams}.\n\nThe cross section of secondary antiproton production is a very important ingredient of eq.~\\eqref{eqn::pbar_sec_source_term}, which has been discussed \nby various groups recently~\\cite{diMauro:2014zea,Winkler:2017xor,Korsmeier:2018gcy,Kachelriess:2019ifk}. \nIn general there are two different strategies to determine this cross section. On the one hand, Monte Carlo generators, \nwhich are tuned to the relevant cross section data~\\cite{Kachelriess:2019ifk}, can be used to infer the relevant cross section.\nOn the other hand, a parametrisation of the Lorentz invariant cross section can be fitted to all available cross section data. \nThen the required energy-differential cross section is obtained by an angular integration~\\cite{diMauro:2014zea,Winkler:2017xor,Korsmeier:2018gcy}. \nWe follow the second approach and use the analytic cross section parametrisation from Ref.~\\cite{Winkler:2017xor} with the updated parameters from Ref.~\\cite{Korsmeier:2018gcy}. \nAn important advantage of the analytic cross section parametrisation is that it is explicitly tuned to cross-section data at low energies, and therefore more reliable \nbelow antiproton energies of $\\sim 10$~GeV as discussed in Ref.~\\cite{Donato:2017ywo}.\n\nFinally, we consider that secondary antiprotons may scatter inelastically with the ISM and lose energy. This antiproton contribution, commonly referred to as tertiary \\cite{Moskalenko:2001ya},\nis suppressed with respect to the secondaries.\n\n\n\\subsection{Propagation in the Galaxy and solar modulation}\\label{sec::prop}\nThe sources, acceleration and propagation of Galactic CRs are research topics by themselves \\cite{Amato:2017dbs, Gabici:2019jvz}. \nFast evolution and progresses has been driven in the last years by newly available and very precise data by \nAMS-02 \\cite{Aguilar:2021tos}, PAMELA \\cite{Adriani:2014xoa} and Voyager \\cite{2013Sci...341..150S}. \nSome recent developments include the studies of systematic uncertainties from solar modulation, correlated experimental data points, \nsecondary production\/fragmentation cross sections as well as detailed studies of propagation phenomena below a rigidity of 10 GV to disentangle \ndiffusion and reacceleration \\cite{Genolini:2019ewc,Evoli:2019wwu,Evoli:2019iih,Boschini:2018baj,Boschini:2019gow,Weinrich:2020cmw,Weinrich:2020ftb,Luque:2021nxb,Luque:2021joz,Schroer:2021ojh, Korsmeier:2021brc}, {where the rigidity $R$ of a CR particle \nis defined as its momentum divided by the absolute value of its charge}.\nHere we will not explore these exciting directions and instead focus on one standard setup of CR propagation, which was already studied in the context DM searches with antiprotons in Ref.~\\cite{Cuoco:2019kuu}. The machine learning approach and the statistical methods introduced below can however be readily applied also to alternative assumptions and more refined descriptions. We briefly summarise below the main ingredients  of this specific approach and refer to Ref.~\\cite{Cuoco:2019kuu} for a more detailed discussion.\n\n\\medskip\n\nCharged CRs propagate within a diffusion halo assumed to be cylindrically symmetric, which extends a few kpc above and below the Galactic plane. \nIn particular, it has a fixed radial extent of 20~kpc, while the \nhalf height of the diffusion halo is denoted by $z_\\mathrm{h}$ and typically enters CRs fits as a free parameters (see section~\\ref{sec:fitams}).\nWhen CRs cross the boundary of the diffusion halo they escape from the Galaxy, while the propagation within the halo is described by a chain of coupled diffusion equations.\n\nThe diffusion equation for the {CR number density per volume and absolute momentum} \n$\\psi_i (\\bm{x}, p, t)$ of CR species $i$ at position $\\bm{x}$ and momentum $p$  is given by~\\cite{StrongMoskalenko_CR_rewview_2007}:\n\\begin{eqnarray}\n  \\label{eqn::propagationEquation}\n  \\frac{\\partial \\psi_i (\\bm{x}, p, t)}{\\partial t} = \n    q_i(\\bm{x}, p) &+&  \n    \\bm{\\nabla} \\cdot \\left(  D_{xx} \\bm{\\nabla} \\psi_i - \\bm{V} \\psi_i \\right)  \\\\ \\nonumber\n     &+&  \\frac{\\partial}{\\partial p} p^2 D_{pp} \\frac{\\partial}{\\partial p} \\frac{1}{p^2} \\psi_i - \n    \\frac{\\partial}{\\partial p} \\left( \\frac{\\mathrm{d} p}{\\mathrm{d} t} \\psi_i  \n    - \\frac{p}{3} (\\bm{\\nabla \\cdot V}) \\psi_i \\right) -\n    \\frac{1}{\\tau_{f,i}} \\psi_i - \\frac{1}{\\tau_{r,i}} \\psi_i \\; .\n\\end{eqnarray}\nWe briefly describe each of the terms in eq.~\\eqref{eqn::propagationEquation} below. \nTo solve these equations numerically we employ \\textsc{Galprop}~56.0.2870~\\cite{Strong:1998fr,Strong:2015zva} and \\textsc{Galtoollibs}~855\\footnote{https:\/\/galprop.stanford.edu\/download.php} \n\twith a few custom modification as described in Ref.~\\cite{Cuoco:2019kuu}. Alternatively, solutions might be obtained analytically, utilizing various simplifying assumption \\cite{Putze:2010zn,Maurin:2018rmm}, or using other fully numerically codes like \\textsc{Dragon}~\\cite{Evoli:2008dv,Evoli:2017vim} or \\textsc{Picard}~\\cite{Kissmann:2014sia}.\n\\textsc{Galprop} assumes that CRs are in a steady state and solves the diffusion equations on a 3-dimensional grid.\nTwo dimensions describe the spatial distribution of CRs, the radial distance $r$ from the Galactic center and distance $z$ perpendicular to the plane, and \none dimension contains the CR's kinetic energy. The grid points of the spatial dimensions are spaced linearly with step size of $\\Delta r = 1$\\;kpc \nand $\\Delta z = 0.1$\\;kpc, respectively, while the grid is spaced logarithmically in kinetic energy with a ratio between successive grid points of 1.4.\n\nThe source term $q_i$ in eq.~(\\ref{eqn::propagationEquation}) depends on the CR species. For secondary antiprotons and antiprotons from DM annihilation it takes the form of \neq.~\\eqref{eqn::pbar_sec_source_term} and eq.~\\eqref{eqn::pbar_DM_source_term}, respectively. \nFor primary CRs the source term factorizes into a spatial and a rigidity-dependent term. \nThe spatial term traces the distribution of supernova remnants.%\n\\footnote{\n    We use the default prescription of \\textsc{Galprop} where the parameters of the\n    source term distribution are fixed to $\\alpha = 0.5$, $\\beta=2.2$, $r_s=8.5$~kpc, and $z_0=0.2$~kpc.\n    This is slightly different from recent values in the literature \\cite{Green:2015isa}.\n    We note, however, that nuclei are only very weakly sensitive to the chosen distribution as discussed in Ref.~\\cite{Korsmeier:2016kha}.\n} \nOn the other hand, the rigidity dependence is modeled as a smoothly broken power-law:\n\\begin{eqnarray}\\label{eq::psp}\n    \\label{eqn::SourceTerm_2}\n    q_R(R)     &=&   \\left( \\frac{R}{R_0} \\right)^{-\\gamma_1}\n                     \\left( \\frac{R_0^{1\/s}+R^{1\/s} }\n                                 {2\\,R_0^{1\/s}      } \\right)^{-s (\\gamma_2-\\gamma_1)},\n\\end{eqnarray}\nwhere $R_0$ is the break position and  $\\gamma_{1,i}$ and $\\gamma_{2,i}$ are the\nspectral indices above and below the break for the CR species $i$, respectively.\nThe parameter $s$ regulates the amount of smoothing at the break.\n{\n     In the following analysis we will assume that all primary nuclei except for protons have a \n     universal injection spectrum such that we adopt $\\gamma_{1,i}=\\gamma_{1}$ and $\\gamma_{2,i}=\\gamma_{2}$. \n     For protons we allow different spectral behaviour and keep the subscript $i=p$.}\nThe broken power-law spectrum in eq.~(\\ref{eq::psp}) is a widely used  phenomenological approximation which describes well the data in the considered rigidity range. \nAll CR species are affected by several processes that contribute to CR propagation, which are diffusion, reacceleration, convection, and energy losses.\nWe assume that diffusion is spatially homogeneous and isotropic. In this case, the diffusion coefficient, $D_{xx}$, \ncan be modeled as a broken power-law in rigidity\n\\begin{eqnarray}\n    \\label{eqn::diffusionConstant}\n    D_{xx} &=&\n        \\begin{cases} \n       \t    \\beta D_{0} \\left( \\frac{R}{4 \\, \\mathrm{GV}} \\right)^{\\delta} &\\text{if}\\; R<R_1\\\\\n       \t    \\beta D_{0} \\left( \\frac{R_1}{4 \\, \\mathrm{GV}} \\right)^{\\delta} \\left( \\frac{R}{R_1} \\right)^{\\delta_h} &\\text{otherwise}\\,,\n        \\end{cases}\n\\end{eqnarray}\nwhere $D_0$ is an overall normalisation and $\\delta$ and $\\delta_h$ are the power-law indices below and above the break at position $R_1$.\nAt low energies the diffusion coefficient is proportional to the velocity $\\beta=v\/c$ of the CRs.\nWe allow for a diffusive reacceleration of CRs by scattering off Alfv\\`en magnetic waves. The amount of reacceleration \nis then determined by the velocity $v_\\mathrm{Alfven}$  of the waves  \\cite{Ginzburg:1990sk,1994ApJ...431..705S}:\n\\begin{eqnarray}\n    \\label{eqn::DiffusivReaccelerationConstant}\n    D_{pp} = \\frac{4 \\left(p \\, v_\\mathrm{Alfven} \\right)^2 }{3(2-\\delta)(2+\\delta)(4-\\delta)\\, \\delta \\, D_{xx}} \\; .\n\\end{eqnarray}\n\nThe terms proportional $\\bm{V}(\\bm{x})$ in eq.~(\\ref{eqn::propagationEquation}) describe  convective winds which drive the CRs away from the Galactic plane. They are taken constant and orthogonal \nto the Galactic plane, such that $\\bm{V}(\\bm{x})= {\\rm sign}(z)\\, v_{0,{\\rm c}}\\,{\\bm e}_z$.\nThe remaining terms describe different contributions of energy losses, for which we adopt the \ndefault \\textsc{Galprop} implementation. In particular, continuous energy losses like ionisation or bremsstrahlung are \nincluded in the term $\\mathrm{d} p\/\\mathrm{d} t$, while catastrophic energy losses by fragmentation or decay \nare modeled by the last two terms. The parameters $\\tau_f$ and $\\tau_r$ are the corresponding lifetimes. \n\nWe emphasise again that the setup of CR propagation described above reflects one specific choice. The available measurements of \nCR nuclei are also described well by other setups. In particular, a model without diffusive re-acceleration,\nbut with an additional break in the diffusion coefficient between 5 and 10\\;GeV \nis currently discussed in the literature \\cite{Genolini:2019ewc,Weinrich:2020cmw,Korsmeier:2021brc}. \n\n\\medskip\n\nCRs measured at the top of the atmosphere  have to traverse a significant part of the heliosphere where they\nare deflected and decelerated by solar winds. The strength of this effect varies in a 22-year cycle and is commonly \nknown as solar modulation. It mostly affects low-energetic CRs; in practice the impact on the spectra above a few tens of \nGV is negligible. We use the common force-field approximation~\\cite{Fisk:1976aw} to model the impact on the CR spectra:\n\\begin{eqnarray}\n    \\label{eqn::solarModulation}\n    \\phi_{\\oplus,i}(E_{\\oplus,i}) &=& \\frac{E_{\\oplus,i}^2 - m_i^2}{E_{\\text{LIS}, i}^2 - m_i^2} \\phi_{\\text{LIS}, i}(E_{\\text{LIS}, i}) \\,,\\\\\n    E_{\\oplus,i} &=& E_{\\text{LIS}, i} - e|Z_i|\\varphi_{i}\\,.\n\\end{eqnarray} \n{Here $\\phi$ and $E$ label the energy-differential flux and the kinetic energy, respectively. \nThe subscripts on the energy or flux denote the position which can either be local interstellar (LIS) or top of the atmosphere ($\\oplus$). \nFurthermore, }$Z_i$ is the charge number, $e$ is the elementary charge, and $\\varphi_{i}$ is the solar modulation potential. \nThe potential is known to be time and charge-sign dependent.\nWe note that the force-field approximation is probably an oversimplified treatment of solar modulation.%\n\\footnote{\n    In more sophisticated models solar modulation is described by propagation equations similar to eq.~\\eqref{eqn::propagationEquation} but tuned to the \n    environment of the heliosphere. These are typically solved numerically~\\cite{Kappl:2015hxv,Vittino:2017fuh,Boschini:2017gic,Fiandrini:2020puf,Ngobeni:2020quz}.\n}\nTo minimise systematic impacts from solar modulation on our results we will exclude data below 5~GV from our analysis. Furthermore, \nwe allow a different solar modulation potential for antiprotons  to account for a possible charge-sign dependence.  \n\n\\subsection{Fit to AMS-02 data}\\label{sec:fitams}\n\nIn the following we summarise very briefly the considered data sets and the fit strategies, where the latter are directly adopted from Ref.~\\cite{Cuoco:2019kuu}. \nThe most precise measurement of CR antiprotons above 1~GV is currently provided by the AMS-02 experiment \\cite{Aguilar:2021tos}. \nWe consider the data sets of proton, helium, and {the antiproton-to-proton ratio} from AMS-02~\\cite{Aguilar:2021tos} collected over 7 years from 2011 to \n2018 and complement with low-energy data for protons and helium from Voyager~\\cite{Cummings:2016pdr}.\nWhen fitting the CR data with the model outlined below, the CR likelihood is defined by \n\\begin{eqnarray}\n  \t\\label{eqn::likelihood_CR}\n\t-2\\,\\log{{\\cal L}_{{\\rm CR}} }(\\bm\\theta) =  \\chi^2_{\\rm CR}(\\bm\\theta) =  \\sum\\limits_{e,s,i} \t\n\t\t\t\t\t\t\t\\left(\\frac{\\phi^{(e)}_{{ s},i}- \\phi^{(\\text{m})}_{s,i,e}(\\bm \\theta)}\n\t\t\t\t\t\t\t           {\\sigma^{\\left(e\\right)}_{s,i}}\\right)^2 \\,,\n\\end{eqnarray}\nwhere $\\phi^{(e)}_{{s},i}$ denotes the flux of the CR species $s$ that was measured by the experiment $e$\nat the rigidity $R_i$ or energy $E_{\\mathrm{kin},i}$, \nwhile $\\phi^{(\\text{m})}_{s,i,e}(\\theta)$ is the flux computed with \\textsc{Galprop}  for the corresponding species and energy. \nFinally, $\\sigma^{\\left(e\\right)}_{s,i}$ is the experimental uncertainty of the flux measurement.\nThe AMS-02 experiment provides separate statistical and systematic uncertainties. Here we assume that \nthe systematic uncertainties are uncorrelated and add the two contribution in quadrature. This \nis certainly a simplified treatment.\nIn particular, it was shown that the significance of a potential excess can depend critically on the assumptions made for  the correlations of this\nuncertainty~\\cite{Boudaud:2019efq,Heisig:2020nse}. \nHowever, we expect that the impact on DM limits is less severe,\n{\n    because of two opposing effects: The covariance matrices as modeled in Refs.~\\cite{Boudaud:2019efq,Heisig:2020nse}\n    contain contributions with both large and small correlation lengths. A large correlation length corresponds \n    to a change of the overall normalisation which looks very different from a peaked signature \n    such as expected from DM annihilation. This potentially leads to \n    a stronger DM limit. On the other hand, a small correlation length allows for signatures similar to those from DM and, therefore, potentially weakens the limit.\n    Overall, we expect these two effects to partly cancel each other. \n}\nWe leave the study of different systematics and correlations within the new methods introduced in this paper to future investigation.\n\n\nWithin the phenomenological description of CR injection and propagation outlined in section~\\ref{sec::prop}, the parameters of \neq.~\\eqref{eqn::propagationEquation} are largely unconstrained a priori and are directly inferred from CR data. \nWe allow for a total of 15 parameters to describe CR injection and propagation. To sample this large parameter space efficiently we use the Nested Sampling algorithm implemented in the \\textsc{MultiNest} code~\\cite{Feroz:2008xx}. The computing efficiency is increased even further by exploiting a hybrid strategy where only a subset of parameters \tis sampled by \\textsc{MultiNest} (``slow parameters'') and the remaining parameters are profiled on-the-fly (``fast parameters''). \tThe slow parameters are the ones that are needed as input for \\textsc{Galprop} and thus changing them is time consuming. More specifically, these are the following eleven parameters: the slopes of the primary injection spectra $\\gamma_{1,p}$, $\\gamma_{1}$, $\\gamma_{2,p}$, \nand $\\gamma_{2}$, the break position $R_0$ and its smoothing $s$, the normalisation $D_0$ and slope\n$\\delta$ of the diffusion coefficient, the half-height of the diffusion halo $z_\\mathrm{h}$, and the velocities of\nconvection $v_{0c}$ and Alfv{\\`e}n magnetic waves  $v_\\text{Alfv{\\`e}n}$.\nThe scan ranges for all of these parameters are summarised in table~\\ref{tab:param_ranges_v2}.\n{In the following we will give results in the frequentist and the Bayesian interpretation. \nFor the Bayesian interpretation we assume flat priors in the scan ranges.}\n\nThe four remaining parameters describe the normalisation of the proton ($A_p$) and helium ($A_\\mathrm{He}$) fluxes and the solar modulation potentials ($\\varphi_{\\text{AMS-02},p,{\\rm He}}$ for $p$ and He and $\\varphi_{\\text{AMS-02},{\\bar{p}}}$ for  ${\\bar{p}}$). These are the fast parameters, which are treated in a simplified way in our analysis and therefore can be varied much more easily.\nInstead of explicitly including them in the \\textsc{MultiNest} parameter scans, we profile over them on-the-fly at each likelihood evaluation of \\textsc{MultiNest}, i.e.\\ we maximise the likelihood over the fast parameters using \\textsc{Minuit}~\\cite{James:1975dr}. A very weak Gaussian prior is applied to $\\varphi_{\\text{AMS-02},{\\bar{p}}}$ by adding to the main likelihood the term\n$-2\\,\\log({{\\cal L}_{{\\rm SM}} }) = (\\varphi_{\\text{AMS-02},p,{\\rm He}}-\\varphi_{\\text{AMS-02},{\\bar{p}}})^2 \/ \\sigma_\\varphi^2$\nwhere $\\sigma_\\varphi = 100$ MV,%\n\\footnote{\n\t {The prior expresses that the solar modulation potential of antiprotons \n     and the one of protons and helium are related, even if they are not forced to be the same.\n     In Ref.~\\cite{Vittino:2019yme} the average difference between the potential of electrons and positrons \n     was found to be around 100 MV.}\n}\nwhile no priors are applied on $\\varphi_{\\text{AMS-02},p,{\\rm He}}$. \n\nWe truncate the rigidity range of the fit to the range between 5 to 300\\;GV. \nAs mentioned above, data below 5\\;GV is \nexcluded to avoid a strong bias from our modeling of solar modulation.%\n\\footnote{\n    It was shown in Ref.~\\cite{Cuoco:2019kuu} that the cut at $R=5$\\;GV does not artificially enhance the significance of a potential DM signal.  \n}\nAt high energies, the spectra of CR nuclei show a break at $R\\sim300$\\;GV, which is more pronounced \nin secondaries with respect to primaries~\\cite{Aguilar:2018njt,Aguilar:2017hno}.\nWhile in general it would be possible to introduce spectral breaks in the injection spectrum or in the diffusion coefficient, \nonly the latter naturally explains the different behavior of the primaries and secondaries~\\cite{Genolini:2017dfb}. We therefore \nfix the parameters of eq.~\\eqref{eqn::diffusionConstant} to $R_1=300$ GV and $\\delta_h=\\delta - 0.12$. The proton and helium data \nof AMS-02 is described well by this choice. Truncating our fit at $R\\sim300$\\;GV avoids unnecessary bias.%\n\\footnote{\n    As an alternative, $R_1$ and $\\delta_h-\\delta$ could be treated at free parameters in the fit which would, however, increase the \n    complexity of the already high-dimensional parameter fit. \n}\n\n\\begin{table}[t]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\caption{\n        Results of the CR fits to AMS-02 and Voyager data of protons, helium, and antiprotons. The parameter ranges for the \\textsc{MultiNest} scan are stated in column 2. \n        In the remaining columns we state the best-fit parameter values and their uncertainty at the 68\\% C.L. for a fit with and without a DM signal. \n        Results are given both in the frequentist and the Bayesian interpretation. \n        \\label{tab:param_ranges_v2}\n\t }\n\t\\begin{tabular}{lccccc}\n\t\t\\hline\n\t\t\\hline\n\t\t                                                                    &               &  \\multicolumn{2}{c}{Frequentist}                                                        & \\multicolumn{2}{c}{Bayesian}                                                            \\\\\n\t\tParameter                                                           &  Scan ranges  & w\/o DM                                     & w\/ DM                                      & w\/o DM                                     & w\/ DM                                      \\\\\n\t\t\\hline\n\t\t$\\gamma_{1,p}$                                                      & $[1.2, 2]   $ & $                  {1.80}^{+0.04}_{-0.03}$ & $                  {1.79}^{+0.07}_{-0.06}$ & $                  {1.77}^{+0.07}_{-0.04}$ & $                  {1.68}^{+0.14}_{-0.07}$ \\\\\n\t\t$\\gamma_{1}$                                                        & $[1.2, 2]   $ & $                  {1.79}^{+0.04}_{-0.04}$ & $                  {1.74}^{+0.08}_{-0.06}$ & $                  {1.75}^{+0.07}_{-0.04}$ & $                  {1.63}^{+0.15}_{-0.07}$ \\\\\n\t\t$\\gamma_{2,p}$                                                      & $[2.3, 2.6] $ & $               {2.405}^{+0.013}_{-0.007}$ & $                  {2.48}^{+0.02}_{-0.03}$ & $                  {2.41}^{+0.01}_{-0.01}$ & $                  {2.48}^{+0.02}_{-0.03}$ \\\\\n\t\t$\\gamma_{2}$                                                        & $[2.3, 2.6] $ & $               {2.357}^{+0.014}_{-0.005}$ & $                  {2.42}^{+0.02}_{-0.03}$ & $               {2.366}^{+0.009}_{-0.012}$ & $                  {2.42}^{+0.02}_{-0.02}$ \\\\\n\t\t$R_{0}\\,\\mathrm{[ 10^{3}\\;MV]}$                                     & $[1, 20]    $ & $                  {7.92}^{+0.82}_{-0.80}$ & $                  {7.32}^{+1.16}_{-0.83}$ & $                  {7.06}^{+0.93}_{-1.04}$ & $                  {6.42}^{+0.97}_{-1.13}$ \\\\\n\t\t$s_{}$                                                              & $[0.1, 0.9] $ & $                  {0.37}^{+0.03}_{-0.03}$ & $                  {0.40}^{+0.03}_{-0.04}$ & $                  {0.38}^{+0.04}_{-0.04}$ & $                  {0.44}^{+0.04}_{-0.06}$ \\\\\n\t\t$D_{0}\\,\\mathrm{[ 10^{28}\\;cm^2\/s]}$                                & $[0.5, 10]  $ & $                  {2.05}^{+1.48}_{-0.39}$ & $                  {2.92}^{+2.09}_{-0.96}$ & $                  {3.58}^{+1.30}_{-0.73}$ & $                  {5.37}^{+1.52}_{-1.78}$ \\\\\n\t\t$\\delta$                                                      & $[0.2, 0.6] $ & $               {0.419}^{+0.009}_{-0.012}$ & $                  {0.35}^{+0.03}_{-0.02}$ & $                  {0.42}^{+0.01}_{-0.01}$ & $                  {0.33}^{+0.03}_{-0.03}$ \\\\\n\t\t$v_\\mathrm{Alfven}\\,\\mathrm{[km\/s]}$                                & $[0, 30]    $ & $                  {8.84}^{+1.45}_{-2.58}$ & $                 {10.25}^{+2.12}_{-2.06}$ & $                  {6.02}^{+3.57}_{-2.51}$ & $                  {7.70}^{+4.15}_{-3.10}$ \\\\\n\t\t$v_{0,\\mathrm{c}}\\,\\mathrm{[km\/s]}$                                 & $[0, 60]    $ & $                  {0.09}^{+1.08}_{-0.08}$ & $                  {0.90}^{+6.77}_{-0.78}$ & $                  {2.48}^{+0.32}_{-2.48}$ & $                {13.36}^{+2.44}_{-13.36}$ \\\\\n\t\t$z_\\mathrm{h}\\,\\mathrm{[kpc]}$                                      & $[2, 7]     $ & $                  {2.60}^{+2.25}_{-0.48}$ & $                  {2.79}^{+2.87}_{-0.75}$ & $                  {4.70}^{+2.30}_{-0.86}$ & $                  {4.84}^{+2.13}_{-0.75}$ \\\\\n\t\t$\\log_{10}(m_\\mathrm{DM}\/\\mathrm{MeV})$                          & $[4, 7]     $ & -                                          & $                  {5.07}^{+0.03}_{-0.05}$ & -                                          & $                  {5.08}^{+0.04}_{-0.05}$ \\\\\n\t\t$\\log_{10}( {\\langle} \\sigma v {\\rangle} \\mathrm{s\/cm^{3}})$     & $[-27,-22]  $ & -                                          & $                {-25.42}^{+0.22}_{-0.48}$ & -                                          & $                {-25.76}^{+0.13}_{-0.26}$ \\\\\n\t\t$\\varphi_{\\mathrm{AMS-02,pHe}}\\,\\mathrm{[GV]}$                      &               & $                  {0.26}^{+0.04}_{-0.03}$ & $                  {0.25}^{+0.05}_{-0.03}$ & $                  {0.30}^{+0.04}_{-0.05}$ & $                  {0.28}^{+0.04}_{-0.06}$ \\\\\n\t\t$(\\varphi_{\\bar p} - \\varphi_{p})_\\mathrm{AMS-02} \\,\\mathrm{[GV]}$  &               & $               {0.200}^{+0.000}_{-0.036}$ & $                  {0.13}^{+0.07}_{-0.12}$ & $               {0.177}^{+0.023}_{-0.001}$ & $                  {0.09}^{+0.11}_{-0.03}$ \\\\\n\t\t$A_{\\mathrm{p,AMS-02}}$                                             &               & $               {1.173}^{+0.004}_{-0.003}$ & $               {1.173}^{+0.003}_{-0.004}$ & $               {1.178}^{+0.004}_{-0.004}$ & $               {1.177}^{+0.004}_{-0.004}$ \\\\\n\t\t$A_{\\mathrm{He,AMS-02}}$                                            &               & $               {1.257}^{+0.006}_{-0.014}$ & $                  {1.20}^{+0.02}_{-0.01}$ & $               {1.253}^{+0.010}_{-0.010}$ & $                  {1.20}^{+0.02}_{-0.02}$ \\\\\n\t\t$\\chi^2_{\\mathrm{p,AMS-02}}$                                        &               & $                                     7.2$ & $                                     6.2$ \\\\\n\t\t$\\chi^2_{\\mathrm{He,AMS-02}}$                                       &               & $                                     3.2$ & $                                     2.1$ \\\\\n\t\t$\\chi^2_{\\mathrm{pbar\/p,AMS-02}}$                                   &               & $                                    35.0$ & $                                    21.5$ \\\\\n\t\t$\\chi^2_{\\mathrm{p,Voyager}}$                                       &               & $                                     7.9$ & $                                     4.1$ \\\\\n\t\t$\\chi^2_{\\mathrm{He,Voyager}}$                                      &               & $                                     3.9$ & $                                     3.2$ \\\\\n\t\t$\\chi^2$                                                            &               & $                                    57.2$ & $                                    37.1$ \\\\\n\t\t\\hline \\hline\n\t\\end{tabular}\n\t\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nTo gain a first understanding of the allowed regions of parameter space, we perform a CR fit as detailed above. The scan is conducted using 1000 live points, a stopping criterion of \\textsc{tol=0.1} and an enlargement factor \\textsc{efr=0.7}. \nThe final efficiency of the scan is found to be around 9\\%, \nwith about 350\\,000\nlikelihood evaluations in total.\n     The fit is heavily parallelised using 96 cores at the same time. The \\textsc{Galprop} code is parallelised using \\textsc{openMP} while the nested sampling algorithm of\n     \\textsc{MultiNest} can be expanded to multiple \\textsc{MPI} tasks. We follow a hybrid strategy with 24 \\textsc{MPI} tasks using 4 cores for each task. \n     We have verified that the parallelisation efficiency lies above 70\\%. \n     In total the fit requires about 5.5 days to converge and consumes 12500 cpu hours, which means that a single likelihood evaluation requires on average about 130 cpu seconds. \nTo perform this fit, \\textsc{MultiNest} starts by broadly sampling the entire parameter space and then continuously shrinks to the allowed parameters. The result is an ensemble of parameter points which is denser in the most interesting parameter region. We will make use of this property in the following section, where our goal is to train the ANNs in such a way that they perform particularly well in the parameter range preferred by data. Thus, we save all the sample points during the fit and then use them as a starting point for the training in section~\\ref{sec:ANN}.\n\nIn table~\\ref{tab:param_ranges_v2} we summarise the best-fit (most probable) values of the various parameters based on a frequentist (Bayesian) interpretation as well as their 68\\% confidence intervals (credible intervals).\nThe best-fit point corresponds to $\\chi^2 = 57.2$ for the AMS-02 data. \nThese results broadly agree with the ones from Ref.~\\cite{Cuoco:2019kuu} even though we use the more recent 7-year AMS-02 data for $p$, He, and $\\bar p$.\nAs expected, for the parameters that are well-constrained by data, there is good agreement between the frequentist and the Bayesian approach. For less constrained parameters (such as for example $z_\\mathrm{h}$) there can be sizeable differences between the best-fit point (obtained by maximizing the profile likelihood) and the most probable point (obtained by maximizing the marginalised likelihood). We will return to this issue in section~\\ref{sec:constraints}.\n\nPrevious analysis have discussed a potential DM signal that could be accommodated at antiproton energies between 10 and 20 GeV where the antiproton flux shows a small anomaly at the level of a few percent. This potential signal corresponds, for example, to DM particles with a mass of about 80~GeV that self-annihilate into $b\\bar{b}$ final states at a thermal cross section. However, the significance of this potential signal has been discussed controversially in the literature. The most recent works suggest that the anomaly is well explained by the combination of several systematic uncertainties, namely uncertainties in the secondary antiproton production cross section, correlated systematics in the AMS-02 data, and some additional \nfreedom in the CR propagation model \\cite{Boudaud:2019efq, Heisig:2020nse, Heisig:2020jvs}, which we do not include here. \n\nThe focus of this work lies instead on developing new methods for exploiting ANNs and importance sampling to derive DM limits. In contrast to a DM signal, we expect the limits to be only weakly dependent on those systematics and leave their investigation to future studies. Nevertheless, for comparison we also perform one fit where antiprotons from DM annihilation are included. We choose DM annihilation into a pair of $b\\bar b$ quarks as our benchmark. In this case,\ntwo further parameters are considered, the mass of the DM particle $m_{\\rm DM}$ and the thermally averaged annihilation cross section $\\langle \\sigma v \\rangle$ (see eq.~\\eqref{eqn::pbar_sec_source_term}). We explore values of $m_{\\rm DM}$ from 10\\;GeV to 10\\;TeV and values of $\\langle \\sigma v \\rangle$ between $10^{-22}$ and $10^{-27}\\;\\mathrm{cm^3\/s}$, with our results being independent of the precise choice of these ranges.\nTheses two additional parameters are sampled with \\textsc{MultiNest}. \n\nThe results of the additional fit are also shown in table~\\ref{tab:param_ranges_v2}.\nIncluding a DM signal formally improves the $\\chi^2$ by 20.1 which, however, \ngiven the discussion above should not be interpreted as significant. Nonetheless, we can take this value as a point of \ncomparison for the performance of the ANN in section~\\ref{sec:constraints}. \n{We furthermore observe that, while the additional DM signal affects most CR propagation parameters only marginally, there is a sizeable shift of the preferred parameter regions for $\\gamma_{2}$, $\\gamma_{2,p}$ and $\\delta$. While this shift is likely overestimated in our analysis for the reasons mentioned earlier, it highlights the challenges for the training of the ANN (see section~\\ref{sec:ANN}) and for the statistical inference via importance sampling (see section~\\ref{sec:constraints}).}\n\nIn the previous paragraph, we focused on a specific case of DM annihilation into a pair of bottom quarks which serves as an example and a point of comparison. In general,\nmuch more complex scenarios with a range of different final states and combinations at different branching fractions are possible. The naive approach to obtain\nresults would be to perform an entirely new parameter scan for each case of interest which obviously requires a substantial amount of computational resources. \nInstead, in the following we will discuss methods to speed up the calculation of CR spectra in a model-independent fashion to quickly obtain constraints \nfor any given DM model. \n\n\n\\section{Deep neural network setup and training}\n\\label{sec:ANN}\n\nOur aim is to predict the output of \\textsc{Galprop} for a wide range of input parameters representing both uncertainties in the propagation model and the unknown properties of DM. This output can then be used to calculate experimental likelihoods as described in section~\\ref{sec:fitams} without computationally expensive simulations. To achieve this goal, we build and train suitable ANNs and validate their performance. Considering the two different contributions to the antiproton flux (i.e.\\ primary and secondary CRs), we construct two separate ANNs to provide fast predictions of each component based on the relevant physical parameters. \nWe will refer to the networks for the DM component and the secondary component as DMNet and sNet, respectively.\nAs the underlying method in the development of the neural networks is the same, both ANNs will be presented in parallel in this section. \n\n\\subsection{Training Set}\n\\label{sec:training_set}\n\nThe information that a neural network should be able to learn, in general, has to be represented in the data that is used to train the network. This allows for the interpolation of data within the parameter space that would, in a conventional approach, require new simulations. To remain impartial on the specific parameters of the DM model, we consider a wide range in the mass of the DM particle from 5 GeV to 5 TeV and randomly sample from a logarithmic distribution in this range. A similar approach is taken for the branching fractions, where we consider all SM final states that give a non-negligible contribution to a CR antiproton flux~\\cite{Cirelli:2010xx}: $q\\bar{q}$, $c\\bar{c}$, $b\\bar{b}$, $t\\bar{t}$, $W^+W^-$, $ZZ$, $hh$ and $gg$. We logarithmically sample each branching fractions in the range $[10^{-5}, 1]$ and then normalise the result in such a way that the sum of all  branching fractions equals one. The DM annihilation cross section is fixed to $\\langle \\sigma v \\rangle = 3 \\times 10^{-26} \\, \\text{cm}^3 \\, \\text{s}^{-1}$ in the complete training set, as variations in this parameter can be included at a later stage by an appropriate rescaling of the flux. \nThese DM parameters, which we will collectively denote by $\\mathbf{x}_\\text{DM}$, are only relevant to the DM component of the antiproton flux and the corresponding neural network, while the secondary flux is independent of $\\mathbf{x}_\\text{DM}$ and hence these parameters will not be used as inputs to the sNet. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = \\textwidth,clip,trim=5 0 5 0]{.\/figures\/2dhist_training_data_v2.png}\n\t\\caption{\\textit{Triangle plot:} One and two dimensional histograms {showing the frequency} of propagation parameters used in the training set, constructed in such a way that the highest density is achieved in the regions most favoured by the combination of AMS-02 proton, antiproton and helium data without DM signal. \\textit{Top right:} One dimensional histogram of the training set for each of the branching fractions {$\\chi \\chi \\rightarrow$ SM SM}.}\n\t\\label{img:training_data_distibution}\n\\end{figure}\n\nFor the propagation parameters we face the significant challenge that the parameter space introduced in section~\\ref{sec:cr} is 11-dimensional and that only a very small volume of this parameter space gives an acceptable fit to AMS-02 data. If we were to simply perform a grid scan or draw random samples from this parameter space, we would include large regions of parameter space for which accurate predictions are unnecessary, as they will anyways be strongly excluded. Conversely, in the preferred regions of parameter space, we want to achieve an accuracy that is significantly better than the typical relative errors of about 5\\% in AMS-02 data, which requires large amounts of training data.\n\nTo obtain sufficiently accurate network predictions in the most interesting regions of parameter space without spending unnecessary time on the simulation and training of less interesting regions, we want to make use of the AMS-02 data already for the creation of the training set. Indeed, we can directly use the \\textsc{MultiNest} scan described in section~\\ref{sec:fitams} to obtain a sample of propagation parameters (denoted by $\\bm{\\theta}_\\text{prop}$ in the following) that is focused on the regions of parameter space with the highest likelihood (see also Ref.~\\cite{Coccaro:2019lgs}). {Since in the following we will be most interested in the calculation of exclusion limits, we will base our training on the \\textsc{MultiNest} scan without DM signal. For a detailed investigation of the excess the same procedure outlined below could be applied to the sample of propagation parameters from the \\textsc{MultiNest} scan with DM signal.}\n\nFor the creation of the training set we exclude any parameter point in the \\textsc{MultiNest} sample that gives a poor fit to AMS-02 data, specifically with $\\Delta \\chi^2 \\ge 30$ compared to the best-fit point. This results in a total of 117335 remaining parameter points which we show in figure~\\ref{img:training_data_distibution}. We emphasise that for each parameter the training data extends well beyond the 68\\% confidence\/credible intervals {without DM annihilations quoted in table~\\ref{tab:param_ranges_v2}}. To ensure a sufficiently good coverage also of the DM parameter space, we sample 8 combinations of DM parameters for each propagation parameter point, leading to a very large simulation set of $\\mathcal{O} (10^6)$ parameter points. \n\n\\subsection{Neural Network Architectures}\n\\label{sec:architectures}\n\nAlthough the two networks that we use to predict the two components of the antiproton flux can be set up and trained in a similar way, we face distinct challenges in each component. For the DMNet the key challenge is the very large number of input parameters, namely the DM mass plus 8 branching fractions in $\\mathbf{x}_\\text{DM}$ and a total of 11 propagation parameters in $\\bm{\\theta}_\\text{prop}$, each with a different physical effect on the output, i.e.\\ the antiproton flux. As we want to have accurate predictions for variations in each of the parameters, we treat the DM mass, the branching fractions, and the propagation parameters as three distinct sets of inputs, which are first processed by three independent dense networks before combining the outputs (see below).\n\nFor the sNet the key challenge is to achieve sufficient accuracy in the prediction of the secondary antiprotons flux, which is tightly constrained by AMS-02 data. Given these constraints, the secondary antiproton flux only exhibits relatively small variations across the training set, which nevertheless need to be accurately captured using the sNet. To achieve the desired accuracy, we provide an increased number of trainable parameters that define the network.\nAs we will show within the following sections, the training duration consequently increases with respect to the DMNet but a very good accuracy is achieved.\n\nRather than directly feeding the physical parameters as inputs to the network, we map the logarithm of $\\mathbf{x}_\\text{DM}$ to values in the range $\\left[0, 1\\right]$ and the remaining parameters $\\bm{\\theta}_\\text{prop}$ to a distribution with a mean of 0 and a standard deviation of 1. Each of the networks is then trained in a supervised approach. The simulated fluxes serve as training labels or `true' fluxes to which the network output can be compared.\n\nGiven the large variations in the CR fluxes that are desired as the output of the ANNs, here we choose a natural scaling of the original (simulated) flux $\\Phi ( E )$ for the sNet outputs, \n\\begin{equation}\n\t\\tilde{\\Phi}_\\text{s} (E) = \\log_{10} \\left( \\Phi(E) \\, E^{2.7} \\right) \\, .\n\t\\label{eq:trafo}\n\\end{equation}\nThe $\\log_{10}$ further decreases the variations in the flux values, which would otherwise cover several orders of magnitude.\nThe energies and their respective fluxes are binned values, identical to the output from the simulations, which extend over the energy range of the AMS-02 antiproton measurement. Consequently, we have sequences of distinct values in the scaled flux as training labels. The transformation in eq.~(\\ref{eq:trafo}) is easily invertible and thus allows for direct comparison of the network output to the simulated spectra.\n\nAs the DM component of the flux predominantly scales with the DM mass, we choose a different scaling for that flux component, \n\\begin{equation}\n\t\\tilde{\\Phi}_\\text{DM} (x) = \\log_{10} \\left( m_\\text{DM}^3 \\, x \\, \\Phi(E) \\right) \\, ,\n\t\\label{eq:trafo_DM}\n\\end{equation}\nwhere $x = E\/m_\\text{DM}$ is a dimensionless quantity. We use a grid in $x$ with 40 points logarithmically spaced in the interval $[10^{-3.7},1]$, on which we evaluate the training labels and DMNet output. The advantage of this scaling compared to eq.~\\eqref{eq:trafo} is that it substantially reduces the impact of changing the DM mass and therefore leads to much less variation across the training set.\\footnote{To first approximation the DM component of the antiproton flux follows the source term $\n\\Phi_{\\overline{p}, \\text{DM}} \\left(E \\right) \\propto q_\\text{DM} \\propto m_\\text{DM}^{-2} \\, \\mathrm{d} N\/\\mathrm{d} E \\propto m_\\text{DM}^{-3} x^{-1} \\, \\mathrm{d} N\/ \\mathrm{d} \\log_{10} x$, where $\\mathrm{d} N \/ \\mathrm{d} \\log_{10} x$ depends only very mildly on $m_\\text{DM}$.\n}\nThis is illustrated in figure~\\ref{img:training_outputs}, which shows the resulting DM antiproton fluxes $\\tilde{\\Phi}_\\text{DM}$ as a function of $x$ for a representative set of final state combinations and DM masses in the training set. We find that for each combination of input parameters we obtain a slowly-varying function of $x$ that reaches a maximum and then drops towards $x \\to 1$. The general trend is similar across the entire range of DM masses that we consider, but some information on the DM mass is retained. We find that this approach significantly improves the training of the DMNet compared to the scaling in eq.~\\eqref{eq:trafo}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.7\\textwidth]{.\/figures\/Output_trafo.png}\n\t\\caption{Transformed DM antiproton fluxes following eq.~\\eqref{eq:trafo_DM} for our training set, which varies propagation parameters, branching fractions and DM masses as discussed in sec.~\\ref{sec:training_set}. The modest amount of variation across different parameter points results in a more easily processable version of the input \\textsc{GALPROP} simulated flux for the DMNet.} \n\t\\label{img:training_outputs}\n\\end{figure}\n\nSubsequent to the pre-processing of the input, the ANNs contain densely connected ('dense') layers, that process the information from the inputs. To address the individual challenges for the networks we set up the architecture as depicted in figure~\\ref{img:architecture}. We provide dense layers for each of the different inputs in the DMNet which are concatenated in the next step and followed by large dense layers. In the sNet the pre-processed input is fed through a more intricate set of dense layer, specifically (56, 28, 14, 28, 56) nodes in the set of layers. We use ReLU activations and add a small dropout probability of 0.1\\% between the layers. The precise values of these hyperparameters do not significantly affect the training performance.\n\nThe main feature of each of the networks is a recurrent layer. The choice to work with a recurrent setup instead of other network architecture types has lead to significant improvements in the architecture development process. Even though the typical application for RNNs is time-series data, we find our spectra as functions of energy to be handled just as well by this network type. In particular, it can be reasoned that the information on the flux that is contained in a specific energy bin is highly correlated with the prior and subsequent energy bins and a network architecture that is able to propagate the information of neighbouring units is very beneficial for the task at hand.\nWe chose a GRU layer as proposed in Ref.~\\cite{DNN:gru} in the DMNet and a LSTM layer following \\cite{DNN:lstm} in the sNet. Each of these layer types is useful for long data sequences and far-reaching information propagation without leading to vanishing or exploding gradients during training. While both methods can in principle be used for either networks, the final implementations that achieved the best results was based on different layer types.\nAs network output a final dense layer is set up. We build the networks using the deep learning API \\textsc{Keras} \\cite{DNN:keras} which uses \\textsc{Tensorflow} \\cite{DNN:tensorflow} as backend. \n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 1\\textwidth]{.\/figures\/Archtecture_sketch_both_simple.pdf}\n\t\t\\begin{tabular}{ll}\n            \\hline \\hline\n\t\t\t\\multicolumn{2}{l}{\\textbf{Hyperparameters}} \\\\\n\t\t\t\\hline\n\t\t\tActivation & ReLU \\\\ \n\t\t\tDropout fraction & 0.1 \\% \\\\ \n\t\t\tOptimizer & Adam, learning rate scheduling $l \\in [10^{-2}, 10^{-5}]$, patience 10 epochs \\\\ \n\t\t\tLoss & Mean squared error (MSE) \\\\\n\t\t\tBatch size & 500 \\\\ \n\t\t\tValidation split & 20~\\% \\\\\n\t\t\tEarly stopping & Monitor val. loss, patience = 40 \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\caption{Schematic of the network structure. \\textit{Top left:} Architecture set up for handling the complete set of inputs. This network type can be used to be trained on the DM component of the $\\overline{p}$ flux (DMNet). \\textit{Top right:} Simplified architecture for networks that require only the CR propagation parameters as input. This network architecture is designed for learning the $\\overline{p}_\\text{secondary}$ fluxes (sNet) and can be employed to train on proton and helium spectra as well (see appendix~\\ref{app:p_and_He}). \\textit{Bottom:} The hyperparameters used during the training process for each of these networks.}\n\t\\label{img:architecture}\n\\end{figure}\n\n\\subsection{Training process}\n\\label{sec:train_process}\n\nWe use approximately $75$\\% of the previously described simulation set for the network training.\\footnote{Note that the sNet has a smaller training set compared to the DMNet, as here we have fewer unique spectra following from our parameter sampling for simulating the training set.} The remainder is used as a test set on which network performance evaluations can be conducted. Within the training set, a validation split of $20$\\% is used during training to monitor the generalisation capabilities of the network. Unlike the training loss, the loss calculated on the validation set is not used to update the model parameters during the optimisation process. \n\nThe network training was conducted using the ADAM optimizer~\\cite{adam}  and a mean squared error (MSE) loss. The initial learning rate of $10^{-2}$ is decreased during the training process, based on the behaviour of the validation loss, for an optimal convergence to a minimal loss. After the learning rate reaches its predefined minimum (lr = $10^{-5}$) the training process is terminated after 40 epochs without improvement of the validation loss, using an early stopping mechanism. This process helps ensure the convergence of the network optimisation. The MSE loss for both the training and validation loss over the training epochs is shown in figure \\ref{img:loss_curves} for both ANNs. \n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/loss_full_DM.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/loss_secondary.pdf}\n\t\\end{minipage}\n\t\\caption{Evolution of the MSE loss for the DMNet \\textit{(left)} and sNet \\textit{(right)} over the training epochs.}\n\t\\label{img:loss_curves}\n\\end{figure}\n\nWe performed the training on a V100-SXM2 GPU. Given the depth of the individual networks, this resulted in training durations of about $4$~minutes per epoch of the DMNet and about $12$~minutes per epoch for the sNet. \n\n\\subsection{Validation of the Network Performance}\n\nTraining performance measures, such as the loss based on the training set, can be helpful while adjusting the architecture and hyperparameters of the networks. The usage of the networks however, requires an evaluation of their ability to replace the simulations.\nUsing the fully trained networks we can compare the simulated spectra from \\textsc{Galprop} within the test set to the network predictions based on the same parameter point. An example for such a comparison is shown in figure~\\ref{img:example_fluxes}. We show the simulated spectra and the output of the respective ANNs for both the secondary and DM component of the antiproton flux (as well as for their sum). In the top panel we depict the fluxes in physical space alongside the AMS-02 antiproton data, demonstrating that the network provides fluxes that are extremely similar to the corresponding simulations. This is illustrated even more clearly in the bottom panel, which shows the relative differences between the ANN and the simulation with respect to the simulated total antiproton flux, compared to the relative uncertainties of the AMS-02 antiproton data. Prior to plotting each CR flux we infer the solar modulation and overall normalization by maximizing the likelihood for the AMS-02 data, as outlined in section~\\ref{sec:fitams}. This enables a fit to the data measured within the heliosphere and is automatically applied to each CR flux evaluated in the following. As this is not computationally expensive, it is not necessary to already include this step in the training process for the ANN. The parameters inferred for the \\textsc{Galprop} and ANN fluxes respectively are in agreement with each other.\nThe parameter point for the specific example presented in figure~\\ref{img:example_fluxes} was randomly selected from the extensive test set.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Antiprotons.pdf}\n\t\\caption{Exemplary comparison of the ANN versus \\textsc{Galprop} antiprotons flux of only the DM component and the combination of secondary and DM component where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. The lower panel depicts the relative difference between the \\textsc{Galprop} (`true') and ANN (`predicted') fluxes with respect to the \\textsc{Galprop} flux compared to the relative AMS-02 uncertainty. The listed solar modulation potential and overall normalization were inferred based on the AMS-02 data for each combined antiproton flux as described in section~\\ref{sec:fitams}.}\n\t\\label{img:example_fluxes}\n\\end{figure}\n\nWe conclude that the accuracy of the sNet is fully sufficient: the relative difference between the fluxes predicted by the ANN and the simulated \\textsc{Galprop} fluxes are always well below the relative uncertainty of the AMS-02 measurements. The architecture and training process used for the sNet can analogously be applied to train an ANN on proton and Helium spectra based on the same \\textsc{Galprop} simulation set, achieving a comparable accuracy. We provide additional details on these networks in appendix~\\ref{app:p_and_He}.\n\nGiven that the DMNet is trained on a parameter space of much higher dimensionality, it is unsurprising that its predictions are on average less accurate than the ones of the sNet. Indeed when calculating the relative differences between simulations and network predictions for the DM component only, we find that only 72\\% of samples lie on average within the AMS-02 relative uncertainties. However, it is essential to realise that in any realistic setting the DM component will only constitute a subdominant contribution to the antiproton flux. Indeed, if the DM contribution in a given bin significantly exceeds the uncertainty of the AMS-02 data (which is typically at the level of 5\\%) the model is expected to be excluded. \n\nIn order to provide more realistic estimates of the general accuracy and stability of the DMNet performance within the test set, we therefore focus on DM signals that contribute 5\\% to the total antiproton flux in the bin where the relative DM contribution is largest. We then calculate the differences between simulations and network output relative to the total antiproton flux. This approach shows that even if the DMNet itself is only accurate at the level of 10\\%, the total antiproton flux can still be predicted with an accuracy at the sub-percent level for allowed DM models.\n\nIn figure~\\ref{img:residual_bands}, we show this accuracy estimate for a total of 3000 DM component samples from the test set (1000 samples each for three different mass bins corresponding to the three different rows). Here we compute the deviation between DMNet prediction and \\textsc{Galprop} simulation to the corresponding total antiproton flux, as in the lower panel of figure~\\ref{img:example_fluxes}. Since the deviations are found to be miniscule when compared to the AMS-02 relative uncertainties, we provide in the right column of figure~\\ref{img:residual_bands} a zoomed-in version. It can be seen that the uncertainty bands (containing the central 68\\% of the network predictions) are typically at the level of 0.1\\% and do not exhibit any systematic shifts nor any significant dependence on the DM mass.  \n\nIn the following we will be interested in comparing the total antiproton flux to data in order to determine which DM signals are allowed by observations. The comparison between the network accuracy and the AMS-02 uncertainties in figure~\\ref{img:residual_bands} clearly shows that it is fully sufficient for this purpose to use the ANNs instead of running \\textsc{Galprop}. Indeed, we will show explicitly in the next section that both approaches lead to very similar values for the $\\chi^2$ statistic described in section~\\ref{sec:fitams}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/residual_bands_tot.png}\n\t\\caption{Relative deviations between predicted and simulated fluxes for the DM component binned into three mass bins and their 68 percentile. Each of the panel shows 1000 samples from the test set. As in the lower panels in figure~\\ref{img:example_fluxes}, in the left panel we again compare this to the benchmark of the relative AMS-02 uncertainties. The right panel shows a zoomed-in version of the left panel in the interval [-0.010, 0.010].}\n\t\\label{img:residual_bands}\n\\end{figure}\n\nThe fully trained networks, as described in this section and appendix~\\ref{app:p_and_He}, are publicly available as \\textsc{DarkRayNet} at \\url{https:\/\/github.com\/kathrinnp\/DarkRayNet}. In this repository, we provide an interface to easily access flux predictions for the corresponding CR species. This tool can for example be used for indirect DM searches as we outline in our analysis in the subsequent section.  \n\n\\section{Constraining the dark matter annihilation cross section}\n\\label{sec:constraints}\n\n\\subsection{Statistical method}\n\\label{sec:marg_importance}\n\nThe ANNs described in the previous section enable us to obtain predictions of the primary and secondary antiproton flux as a function of the DM parameters $\\mathbf{x}_\\text{DM}$ and the propagation parameters $\\bm{\\theta}_\\text{prop}$. Given data from observations we can then construct a likelihood function $\\mathcal{L}(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop})$. We emphasise that, given suitable predictions for the CR fluxes, this likelihood is quick to evaluate and therefore does not need to be predicted by the ANN. This has the significant advantage that the ANN does not need to learn the various fluctuations that may be present in the data.\n\nThe likelihood function can then be used to constrain both $\\mathbf{x}_\\text{DM}$ and $\\bm{\\theta}_\\text{prop}$. In the present work we primarily focus on the constraints on the DM parameter space, meaning that we will treat the propagation parameters simply as nuisance parameters that need to be varied in order to draw robust conclusions. The two main ways to achieve this is to either calculate the profile likelihood\n\\begin{equation}\n \\hat{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) = \\mathcal{L}(\\mathbf{x}_\\text{DM},\\hat{\\bm{\\theta}}_\\text{prop}(\\mathbf{x}_\\text{DM})) \\; ,\n\\end{equation}\nwhere $\\hat{\\bm{\\theta}}_\\text{prop}(\\mathbf{x}_\\text{DM})$ denote the propagation parameters that maximise the likelihood for given DM parameters $\\mathbf{x}_\\text{DM}$, or to calculate the marginalised likelihood\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) = \\int \\mathcal{L}(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop}) p(\\bm{\\theta}_\\text{prop}) \\mathrm{d}\\bm{\\theta}_\\text{prop} \\; ,\n\\end{equation}\nwhere $p(\\bm{\\theta}_\\text{prop})$ denotes the prior probability for the propagation parameters. Given sufficiently constraining data, the profile likelihood and the marginalised likelihood are expected to be similar and the dependence of the result on the chosen priors should be small. We find that this is largely true for the case considered here, with some notable exceptions to be discussed below.\n\nFrom the point of view of our machine learning approach, however, the two ways of varying the nuisance parameters are very different. The profile likelihood depends only on the antiproton flux for a single value of $\\bm{\\theta}_\\text{prop}$, meaning that highly accurate predictions are needed close to the maximum of the likelihood. For extreme choices of the DM parameters this maximum may be pushed to corners of parameter space where the network has not been sufficiently trained. A single outlier in the prediction will then completely bias the result and lead to numerical instabilities {when sampling the parameter space}. This makes accurate calculations of the profile likelihood a highly challenging task.\n\nThe marginalised likelihood, on the other hand, depends on the likelihood across a range of propagation parameters, which should have substantial overlap with the parameter regions seen during training. The impact of individual outliers in the predictions is also reduced significantly compared to the case of the profile likelihood, making the calculation of marginalised likelihoods based on ANN predictions more robust. Nevertheless, the challenge remains to ensure that results are not biased by regions of parameter space where only little training has been performed. In the present work we address this challenge using the technique of importance sampling~\\cite{mcbook}, which we describe in the following.\\footnote{For a different approach to Bayesian analyses of cosmic ray propagation with the help of neural networks we refer to Ref.~\\cite{Johannesson:2016rlh}.}\n\nFirst of all, we note that an approximate marginalisation can be performed by drawing a random sample of parameter points $\\bm{\\theta}_i$ from the prior probability $p(\\bm{\\theta}_\\text{prop})$ and calculating the sum\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i) \\; .\n\\end{equation}\nIn fact, the same can be done by drawing a random sample from any probability distribution function $q(\\bm{\\theta}_\\text{prop})$ provided the individual points are reweighted accordingly (so-called importance sampling):\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{\\sum_{i=1}^N \\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i) \\frac{p(\\bm{\\theta}_i)}{q(\\bm{\\theta}_i)}}{\\sum_{i=1}^N \\frac{p(\\bm{\\theta}_i)}{q(\\bm{\\theta}_i)}} \\; .\n \\label{eq:sample}\n\\end{equation}\nA particularly interesting case is that $q(\\bm{\\theta}_\\text{prop})$ is taken to be the posterior probability for the propagation parameters in the absence of a DM signal, i.e.\n\\begin{equation}\n\tq(\\bm{\\theta}_\\text{prop}) \\propto \\mathcal{L}(\\mathbf{x}_\\text{DM} = 0, \\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\equiv \\mathcal{L}_0(\\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\, .\n\\end{equation}\nIn this case $p(\\bm{\\theta}_i) \/ q(\\bm{\\theta}_i) \\propto 1\/\\mathcal{L}_0(\\theta_i)$ and hence\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{\\sum_{i=1}^N  \\frac{\\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i)}{\\mathcal{L}_0(\\bm{\\theta}_i)}}{\\sum_{i=1}^N \\frac{1}{\\mathcal{L}_0(\\bm{\\theta}_i)}} \\; .\n\\end{equation}\n\nThe great advantage of this approach is that the likelihood is only evaluated for plausible values of the propagation parameters, meaning for values that give a large posterior probability in the absence of a DM signal. These are exactly the same parameter regions on which we have focused for the training of the ANNs described above. Indeed, it is possible to generate the training data and sample the posterior probability using the same prior probabilities and the same \\textsc{MultiNest} runs such that a large overlap between the two is ensured.\\footnote{We emphasise that the posterior sample is not part of the training data, i.e.\\ the ANN is never evaluated on the exact same values seen during training.} Another significant advantage is that it is straight-forward to include additional constraints on the propagation parameters that are independent of the DM parameters and therefore not part of the ANN training. For example, to also include likelihoods for proton data $\\mathcal{L}_p$ and He data $\\mathcal{L}_\\text{He}$, it is sufficient to draw a sample from the joint posterior\n\\begin{equation}\n q(\\bm{\\theta}_\\text{prop}) \\propto \\mathcal{L}_0(\\bm{\\theta}_\\text{prop}) \\, \\mathcal{L}_p(\\bm{\\theta}_\\text{prop}) \\, \\mathcal{L}_\\text{He}(\\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\, .\n\\end{equation}\n\nTo conclude this discussion, we note that in the case that the likelihood can be written in terms of a $\\chi^2$ function, $\\mathcal{L} \\propto e^{- \\chi^2\/2}$, we can define a marginalised $\\chi^2$ function as $\\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) \\equiv -2 \\log \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM})$. Importance sampling then yields\n\\begin{equation}\n\t\\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) = - 2 \\log {  \\frac{\\sum_{i=1}^N  \\exp{ \\left(- \\frac{\\Delta\\chi^2(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_\\text{prop})}{2} \\right)}} {\\sum_{i=1}^N  \\exp{ \\left(\\frac{\\chi^2_0(\\bm{\\theta}_\\text{prop})}{2} \\right)} }} \\, ,\n\t\\label{eq:marg_chi}\n\\end{equation}\nwhere $\\chi^2_0(\\bm{\\theta}_\\text{prop}) = \\chi^2(\\mathbf{x}_\\text{DM} = 0, \\bm{\\theta}_\\text{prop})$ and $\\Delta\\chi^2(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_\\text{prop}) = \\chi^2(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop}) - \\chi^2_0(\\bm{\\theta}_\\text{prop})$.\nTo calculate confidence intervals and exclusion limits for the DM parameters, we then define\n\\begin{equation}\n\t\\Delta \\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) = \\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) - \\bar{\\chi}^2_0\\, .\n\t\\label{eq:upper_bound}\n\\end{equation}\nHence, $\\Delta \\bar{\\chi}^2 < 0$ corresponds to a preference for a DM signal, while parameter points with $\\Delta \\bar{\\chi}^2 > 3.84$ can be excluded at 95\\% confidence level.\\footnote{Note that although our treatment of nuisance parameters is motivated by Bayesian statistics, we still interpret the resulting marginalised likelihood using frequentist methods, such that there is no need to choose priors for the DM parameters.}\n\n\\subsection{Example A: Single Dark Matter Annihilation Channel}\n\\label{sec:example_A}\n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/MN_sample_galprop_vs_NN_pbar_delta_100GeV.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/MN_sample_galprop_vs_NN_pbar_delta_1TeV.pdf}\n\t\\end{minipage}\n\t\\caption{One and two dimensional histograms of $\\Delta \\chi^2$ for the AMS-02 antiproton measurement based on the antiproton fluxes provided by the Neural Network and \\textsc{Galprop} for different combinations of propagation parameters. \n\tWe consider the annihilations of DM particles with $m_\\text{DM} = 100$ GeV \\textit{(left)} and 1 TeV \\textit{(right)} into $b \\overline{b}$ with a cross section of $\\langle \\sigma v \\rangle = 10^{-26}$ cm$^3$ s$^{-1}$. The values for $\\Delta \\hat{\\chi}^2$ indicated by the black dashed lines represent the marginalised values obtained by the importance sampling technique described in section~\\ref{sec:marg_importance}. } \n\t\\label{img:chi_comp}\n\\end{figure}\n\nLet us first consider a frequently-used benchmark scenario and assume that the DM particles annihilate exclusively into pairs of bottom quarks, such that the injection spectrum is fully characterised by the (velocity-independent) annihilation cross section $\\langle \\sigma v \\rangle$ and the DM mass $m_\\text{DM}$. As a first step, we can then calculate $\\Delta \\chi^2(m_\\text{DM}, \\langle \\sigma v \\rangle, \\bm{\\theta}_\\text{prop})$ for different values of the propagation parameters. Figure~\\ref{img:chi_comp} compares the results that we obtain when using the ANN predictions of the antiproton flux and when employing \\textsc{Galprop}. The two panels correspond to different values of the DM mass and use the same 10122 sets of propagation parameters drawn randomly from the posterior distribution $q(\\bm{\\theta}_\\text{prop})$ as discussed above. In both cases we find a very strong correlation between the two ways of calculating $\\Delta \\chi^2$ ($r > 0.98$). Indeed, for 95\\% of parameter points the absolute difference in $\\Delta \\chi^2$ is smaller than $2.1$ ($0.9$) for $m_\\text{DM} = 100\\,\\mathrm{GeV}$ ($m_\\text{DM} = 1\\,\\mathrm{TeV}$), confirming the excellent performance of our ANN.\n\nIn each case we use a dashed line to indicate $\\Delta \\bar{\\chi}^2$ as defined in eq.~(\\ref{eq:marg_chi}). We emphasise that, since we average over $\\exp(-\\Delta \\chi^2 \/ 2)$, the final result is dominated by the points with the smallest $\\Delta \\chi^2$. Again, we find very good agreement between the marginalised $\\Delta \\chi^2$ obtained from the ANN and from \\textsc{Galprop}. The values obtained in the left panel correspond to a substantial preference for a DM signal, while the parameter point considered in the right panel is slightly disfavoured by data. Although the value $\\Delta \\bar{\\chi}^2 = -31.5$ ($-32.7$) that we obtain for $m_\\text{DM} = 100 \\, \\mathrm {GeV}$ from the ANN (\\textsc{Galprop}) would at face value correspond to quite a significant excess, we {emphasize that our set-up is not designed to provide an accurate characterisation of this excess. In particular we} caution the reader that due to our simplified implementation of AMS-02 data (in particular neglecting correlations) this number should be interpreted with care. We expect that a more detailed analysis of AMS-02 data would lead to a much lower significance.\n\nComparing the evaluations of the marginalised $\\Delta \\chi^2$ with the ANN and \\textsc{Galprop} respectively, the reduction of the computational cost achieved with our neural network method becomes apparent. For the ANN the prediction of the set of CR fluxes for each of the specific DM parameter points only takes $\\mathcal{O}(1)$ cpu second in total for the 10122 parameter points, but the calculation of the respective $\\chi^2$ while inferring the solar modulation potential takes up the majority of the computation time ($\\mathcal{O}(10)$ cpu seconds in total). This time is however negligible compared to the \\textsc{Galprop} simulations which take $\\mathcal{O}(10)$ cpu hours to obtain the same number of CR fluxes.\n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/chi_dist_zoom.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/chi_dist_marg_zoom.pdf}\n\t\\end{minipage}\n\t\\caption{$\\Delta \\chi^2$ for the AMS-02 antiproton measurement based on the antiproton fluxes provided by the Neural Network and \\textsc{Galprop} as a function of $\\langle \\sigma v \\rangle$ and for different values of $m_\\text{DM}$. We assume a dominant DM DM $\\rightarrow \\, b \\overline{b}$ annihilation in each case. \\textit{Left:} Propagation parameters are fixed to the best-fit values in a frequentist setup when only secondary antiprotons are considered (see table~\\ref{tab:param_ranges_v2}). \\textit{Right:} Propagation parameters are marginalised over using importance sampling. We also include the 95 \\% upper bound values of the annihilation cross section following eq.~(\\ref{eq:upper_bound}).}\n\t\\label{img:chi_dist_zoom}\n\\end{figure}\n\nA complementary perspective to the results in figure~\\ref{img:chi_comp} is provided in figure~\\ref{img:chi_dist_zoom}, which shows $\\Delta \\chi^2$ as a function of $\\langle \\sigma v \\rangle$ for different values of the DM mass. In the left panel we fix the propagation parameters to their best-fit values in the absence of a DM signal (see table~\\ref{tab:param_ranges_v2}), while in the right panel we marginalise over all propagation parameters using importance sampling. Solid (dotted) curves correspond to the ANN (\\textsc{Galprop}) predictions and again show excellent agreement. The horizontal dashed lines indicate the 95\\% confidence level upper bound on $\\langle \\sigma v \\rangle$ obtained following eq.~(\\ref{eq:upper_bound}).\n\nAs expected, allowing variations in the propagation parameters generally leads to smaller values of $\\Delta \\chi^2$ and hence relaxes the upper bounds on the annihilation cross section. This effect is most dramatic for the case $m_\\text{DM} = 100 \\, \\mathrm{GeV}$ (blue line), where there is a preference for a DM signal in the data and hence the exclusion limit is relaxed by about an order of magnitude. The small bumps in the blue curve in the right panel are a result of the finite size of the sample of propagation parameters used for the marginalisation and result from the approximation made in eq.~(\\ref{eq:sample}).\n\nRepeating this procedure for different values of the DM mass, we can obtain exclusion limits on $\\langle \\sigma v \\rangle$ as a function of $m_\\text{DM}$. These are shown in figure~\\ref{img:bounds_bb} for the case of fixed propagation parameters (left) and when marginalising over propagation parameters (right). The colour shading indicates parameter regions where $\\Delta \\chi^2 > 0$, such that a DM signal is disfavoured, while greyscale is used to indicate parameter regions where $\\Delta \\chi^2 < 0$ such that a DM signal is preferred. We find that this is the case for DM masses in the range $50\\text{--}250\\,\\mathrm{GeV}$. Again, marginalisation leads to relaxed exclusion bounds  and an increased preference for a DM signal. We reiterate however that the magnitude of this preference is likely overestimated in our analysis.\n\n\\begin{figure}[t]\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/bb_both.pdf}\n\t\\caption{$\\Delta \\chi^2$ for the AMS-02 antiproton measurement as a function of $\\langle \\sigma v \\rangle$ and $m_\\text{DM}$ using {the fixed propagation parameters specified in table~\\ref{tab:param_ranges_v2}} (\\emph{left}) and performing the marginalisation via importance sampling (\\emph{right}). The dashed lines represent the 95~\\% CL upper bounds on the annihilation cross section. The white regions in the upper part of each panel correspond to $\\Delta \\chi^2 > 1000$ and are excluded to improve numerical stability.}\n\t\\label{img:bounds_bb}\n\\end{figure}\n\nTo assess the impact of marginalisation let us finally compare our results with those obtained using a profile likelihood. As discussed in section~\\ref{sec:marg_importance}, special care needs to be taken when using the ANN predictions to calculate a profile likelihood in order to ensure that the result is not dominated by regions of parameter space with insufficient training data. We achieve this goal by restricting the allowed parameter regions as follows: $0.1 < s < 0.6$, $1 \\, \\mathrm{GV} < R_0 < 10 \\, \\mathrm{GV}$, $0.35 < \\delta < 0.6$ and $2.3 < \\gamma_{2,(p)} < 2.5$. We then use \\textsc{MultiNest} to explore the remaining parameter space for fixed values of the DM mass and varying annihilation cross section in order to find the largest value of $\\langle \\sigma v \\rangle$ such that $\\Delta \\hat{\\chi}^2(m_\\text{DM}, \\langle \\sigma v \\rangle) \\equiv  -2 \\Delta \\log \\hat{\\mathcal{L}}((m_\\text{DM}, \\langle \\sigma v \\rangle)) < 3.84$. Repeating this procedure for different values of $m_\\text{DM}$ then yields the exclusion limit.\n\nThe results are shown in figure~\\ref{img:bounds_bb_compare} together with the exclusion limits obtained for fixed propagation parameters and when marginalising over propagation parameters as shown in figure~\\ref{img:bounds_bb}. We find that in most regions of parameter space the profile likelihood approach yields somewhat weaker exclusion limits than the marginalisation. Such a difference is to be expected whenever substantial tuning in the propagation parameters is required in order to accommodate a DM signal. For example, for $m_\\text{DM} = 1 \\, \\mathrm{TeV}$ and $\\langle \\sigma v \\rangle = 5 \\times 10^{-26} \\, \\mathrm{cm^3 \\, s^{-1}}$ we find that $\\Delta \\hat{\\chi}^2 < 3.84$ can be achieved only if $D_0$, $v_{0,c}$ and $z_\\mathrm{h}$ all take values close to their lower bounds. Such a tuning is not penalised in the profile likelihood, but the contribution of these solutions to the marginalised likelihood will be suppressed according to the small volume of the viable parameter space. The same conclusion can be reached from the right panel of figure~\\ref{img:chi_comp}: Although there are sets of propagation parameters that yield $\\Delta \\chi^2 \\approx 0$, most parameter combinations give significantly larger $\\Delta \\chi^2$, such that marginalisation leads to $\\Delta \\hat{\\chi}^2 \\approx 2.6$, close to the 95\\% confidence level upper bound. In other words, the difference between the two approaches is a direct consequence of the different statistical methods and not an artefact of the ANN predictions.\n\nIn general the dependence of the DM limit on the chosen value for the halo height is very well known. To first order the normalisation of the \nDM flux is proportional to $z_\\mathrm{h}$ and thus the DM limit is anti-proportional to $z_\\mathrm{h}$ as again nicely demonstrated in \na very recent analysis \\cite{Genolini:2021doh}.\nThe CR fit conducted in section \\ref{sec:cr} varies $z_\\mathrm{h}$ between 2 and 7 kpc. Because of the well-known $z_\\mathrm{h}$-$D_0$ \ndegeneracy the resulting posterior of $z_\\mathrm{h}$ is almost flat in the entire fit range. \nThe DM limit derived from the marginalisation of the $\\Delta \\hat{\\chi}^2$ should be understood to refer to 4.8 kpc, \nnamely the average value of $z_\\mathrm{h}$ in the posterior. This is in perfect agreement with recent analyses \nof secondary fluxes by AMS-02~\\cite{Evoli:2019iih,Weinrich:2020ftb,Luque:2021joz,Korsmeier:2021brc}.\nOn the other hand, when limits are derived in a frequentist approach and in the absence of a DM preference, \n$z_\\mathrm{h}$ values are pushed towards the lower boundary of the fit range at 2 kpc. \nThis again explains the difference between the marginalised and profiled limit in the figure~\\ref{img:bounds_bb_compare}.\nOne possible way to study the $z_\\mathrm{h}$ dependence explicitly in the marginalisation framework is to \nfurther restrict the range of $z_\\mathrm{h}$. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.7\\textwidth]{.\/figures\/Compare_bounds_part.pdf}\n\t\\caption{A comparison of the 95~\\% CL exclusion bounds in figure~\\ref{img:bounds_bb} (blue and light blue) with the bounds obtained when profiling over the propagation parameters using the CR spectra provided by our ANNs (green). The black dashed line indicates the thermal annihilation cross section for WIMPs from \\cite{Steigman2012}.}\n\t\\label{img:bounds_bb_compare}\n\\end{figure}\n\nThe differences between marginalised and profiled limits are particularly relevant given how they affect the conclusions drawn from figure~\\ref{img:bounds_bb_compare}. When using the marginalised likelihood we find that the thermal cross section (indicated by the black dashed line) can be excluded for DM masses in the range $300\\text{--}2000\\,\\mathrm{GeV}$, implying that WIMP models in this mass range can only be viable if the injection of antiprotons are suppressed. When using the profile likelihood, on the other hand, almost the entire mass range above $70 \\, \\mathrm{GeV}$ is found to be viable. \tWe note that the agreement between the frequentist and Bayesian approach will improve with a better determination of $z_\\mathrm{h}$ as expected from the analysis of the forthcoming Be isotope measurements by AMS-02 \\cite{Derome:ICRC2021}.\n\nIn addition to the reduction in computing time achieved when using the ANN instead of \\textsc{Galprop}, we find that the use of importance sampling leads to another improvement compared to the more conventional profiling approach.\nCrucially, our marginalisation using importance sampling is based on a fixed set of 10122 data points in the propagation model, which can be evaluated in parallel. The ANN therefore gives a negligible contribution to the time needed to calculate the upper bound on the annihilation cross section for each of the 100 mass bins shown in figures~\\ref{img:bounds_bb} and \\ref{img:bounds_bb_compare}. \nFor the profiling approach on the other hand the evaluation of the data points cannot be performed in parallel by the ANN due to their sampling. This leads to an increase in computation time, such that  the speed-up of the runtime when using the ANN instead of \\textsc{Galprop} is reduced to two orders of magnitude (rather than three orders of magnitude for importance sampling).\n\n\\subsection{Example B: Scalar Singlet Dark Matter}\n\nWe now illustrate the use of the ANN for the analysis of a specific model of DM with a singlet scalar field $S$.  Imposing a $Z_2$ symmetry, $S \\to -S$, the scalar particle is stable and thus a DM candidate. The Lagrangian of this scalar singlet DM (SSDM) model reads~\\cite{Silveira:1985rk,McDonald:1993ex,Burgess:2000yq} \n\\begin{equation}\n{\\cal L} = {\\cal L}_\\text{SM} + \\frac 12 \\partial_\\mu  S  \\partial^\\mu  S  - \\frac12 m_{S,0}^2S^2- \\frac 14 \\lambda_S  S^4- \\frac 12 \\lambda_{H\\!S}\\, S^2 H^\\dagger H\\,, \n\\label{eq:lagr}\n\\end{equation}\nwhere ${\\cal L}_\\text{SM}$ is the Standard Model Lagrangian and $H$ is the Standard Model Higgs field. After electroweak symmetry breaking, the last three terms of the Lagrangian become\n\\begin{equation}\n{\\cal L} \\supset  - \\frac12 m_{S}^2\\, S^2- \\frac 14 \\lambda_S\\,  S^4 - \\frac 14 \\lambda_{H\\!S}\\, h^2 S^2 - \\frac {1}{2} \\lambda_{H\\!S}\\, v h S^2\\,,\n\\label{eq:ewbr}\n\\end{equation}\nwith $H = (h+v, 0)\/\\sqrt{2}\\,$, $v = 246\\,$GeV, and where we introduced the physical mass of the singlet \nfield, $m_S^2  = m_{S,0}^2 + \\lambda_{H\\!S} \\,v^2 \/ 2$. The DM phenomenology of the SSDM has been extensively studied in the literature, see e.g.\\ \\cite{Cline:2013gha,Beniwal:2015sdl,Cuoco:2016jqt,Cuoco:2017rxb,GAMBIT:2017gge,Athron:2018ipf} and references therein. \n\nThe DM phenomenology of the SSDM is fully specified by the mass of the DM particle, $m_S=m_{\\rm DM}$, and the strength of the coupling between the DM and Higgs particle, $\\lambda_{H\\!S}$. Below the Higgs-pair threshold, $m_S < m_h$, DM annihilation proceeds through $s$-channel Higgs exchange only, and the relative weight of the different SM final states is determined by the SM Higgs branching ratios, independent of the Higgs-scalar coupling $\\lambda_{H\\!S}$. Above the Higgs-pair threshold, $m_S \\ge m_h$, the $hh$ final state opens up. The strength of the annihilation into Higgs pairs, as compared to $W$, $Z$ or top-quark pairs, depends on the size of the Higgs-scalar coupling. For our specific analysis we require that the SSDM provide the correct DM relic density, $\\Omega h^2 = 0.1198\\pm 0.0015$~\\cite{Ade:2015xua}, which in turn determines the size of $\\lambda_{H\\!S}$ for any given DM mass $m_S$. The corresponding branching fractions for DM annihilation within the SSDM are shown in figure~\\ref{img:bounds_SSDM} (left panel) as a function of the DM mass. \n\nUsing the ANN we analyse the $\\Delta\\chi^2$ distribution of the model, marginalising over propagation uncertainties as described in section~\\ref{sec:marg_importance}. The result is shown in figure~\\ref{img:bounds_SSDM} (right panel). Comparing figure~\\ref{img:bounds_SSDM} with the analogous result for the single annihilation channel into $b\\bar{b}$, figure~\\ref{img:bounds_bb} (right panel), we observe a similar overall shape of the $\\Delta\\chi^2$ distribution. \n\nFor light DM the SSDM annihilates dominantly into bottom final states, so one expects results that are very similar to the case of the single $b\\bar{b}$ channel. However, for the smallest DM masses that we consider ($m_\\chi \\approx 10 \\, \\mathrm{GeV}$) we find that the constraints become considerably stronger when including even a sub-dominant contribution from $c\\bar{c}$. The reason is that in this mass range, most antiprotons resulting from annihilation into bottom quarks have energies below $5\\,\\mathrm{GeV}$ and do therefore not give a contribution in our fits. Annihilation into charm quarks, on the other hand, can give rise to more energetic antiprotons, leading to stronger constraints. For DM masses above about $50 \\, \\mathrm{GeV}$, a variety of SM final states contributes in the SSDM, including in particular $WW$, $hh$ and $ZZ$.  However, as shown in Ref.~\\cite{Cuoco:2017iax}, the limits for heavy DM are similar for these final states and for annihilation into bottom quarks, so that the overall constraints for the SSDM are comparable to those for annihilation into bottom quarks only. \n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.435\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/SSDM_fractions.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.565\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/SSDM_marginalized.pdf}\n\t\\end{minipage}\n\t\\caption{\\textit{Left:} Mass dependence of branching fractions of $S S \\rightarrow \\text{SM} \\text{ SM}$ in the SSDM model for $\\lambda_\\mathrm{HS}$ fixed by the relic density requirement. \\textit{Right:} Marginal $\\chi^2$ distribution of the $\\langle \\sigma v \\rangle - m_\\text{DM}$ parameter space in the SSDM model.}\n\t\\label{img:bounds_SSDM}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThe analysis of cosmic ray (CR) antiprotons is a powerful method for the indirect detection of dark matter (DM). The accurate experimental measurements, in particular from AMS-02, allow to probe DM annihilation cross sections close to the value predicted by thermal freeze-out for a wide range of DM masses. However, a precise description of CR propagation through the Galaxy is required to exploit the potential of the experimental data. The propagation models depend on a large number of parameters, and the  standard numerical simulation tools, such as \\textsc{Galprop}, are computationally expensive. Therefore, global analyses of generic models of DM can only be carried out with an immense computational effort, if at all. \n\nIn this work we have developed an artificial neural network (ANN) that allows extremely fast and accurate predictions of the cosmic ray flux for generic DM models. Specifically, we have employed recurrent neural networks (RNNs) to predict the CR energy spectrum. RNNs are particularly well suited to learn the correlations between the fluxes contained in neighbouring energy bins. Additional improvements in performance are achieved by grouping input parameters that have similar physical origin and by performing a suitable rescaling of the output spectra. \n\nWe have trained the ANN with a large set of antiproton fluxes simulated with \\textsc{Galprop}, where the propagation parameters have been chosen to be broadly compatible with the most recent AMS-02 data, and a generic parametrisation of the dark matter model in terms of the DM mass and the branching fractions for the annihilation into various \nStandard Model final states. We emphasise that the contribution of different DM models to the antiproton flux only has a marginal impact on the preferred range of the propagation parameters. It is therefore possible to focus the training of the ANN on the relevant range of propagation parameters without specifying the details of the DM model in advance. We have validated the performance and accuracy of the network by comparing both the predicted antiproton fluxes and the resulting AMS-02 likelihoods to the ones obtained from explicit \\textsc{Galprop} simulations for a range of different propagation and DM model parameters.\n\nWe have then used the neural network predictions to test specific DM models against current AMS-02 data. We have focused on the DM parameter space and treated the propagation parameters as nuisance parameters by calculating both the corresponding profile and marginalised likelihoods. While the former approach requires an explicit restriction of the parameter space to the regions where the ANN has been sufficiently trained, this requirement can be automatically fulfilled in the latter case by employing importance sampling. Comparing the ANN to \\textsc{Galprop} we find a speed-up in runtime of about two (three) orders of magnitude when using profiling (importance sampling).\n\nFor DM annihilation into bottom quarks we have obtained results that are consistent with previous studies based on simulations and a profile likelihood approach. We find more stringent bounds on the DM parameter space when using the marginalised likelihood; here a thermal cross section can be excluded for DM annihilating fully into bottom quarks for DM  masses in the range between approximately 300~GeV and 2~TeV. \nTo illustrate the flexibility of our approach, we have also used the ANN to derive constraints on scalar singlet DM, for which DM annihilation results in a variety of Standard Model final states with branching fractions that depend strongly on the DM mass. \n\nThe ANN developed in this work, and the corresponding method for efficient training, can {also be used to study more closely the potential DM interpretation of the antiproton excess around 20 GeV, for example regarding the impact of correlations in AMS-02 data. Moreover, it can} be easily extended to alternative propagation models and can be applied to a wide class of DM scenarios.\nIt will thus be possible to fully exploit the potential of current and future cosmic-ray data in global analyses of general DM models. {In future work a transformation of the ANNs into Bayesian neural networks can be incorporated in the analysis. With this step, additional more in-depth studies of the uncertainties of the network predictions will be possible.} The fully trained networks together with a suitable user interface are publicly available as \\textsc{DarkRayNet} at \\url{https:\/\/github.com\/kathrinnp\/DarkRayNet}.\n\n\\acknowledgments\n\nWe thank Thorben Finke and Christoph Weniger for discussions, Alessandro Cuoco and Jan Heisig for helpful comments on the manuscript and Sven Guenther for testing \\textsc{DarkRayNet}. F.K.\\ is  supported by  the  Deutsche Forschungsgemeinschaft (DFG) through the Emmy Noether Grant No.\\ KA 4662\/1-1.\nM.Ko.\\ is partially supported by the Swedish National Space Agency under contract 117\/19 and the European Research Council under grant 742104.\nSimulations and ANN training were performed with computing resources granted by RWTH Aachen University under project jara0184 and rwth0754.\n\n\n\\begin{appendix}\n\n\n\\section{Predicting proton and helium spectra}\n\\label{app:p_and_He}\n\nWhen simulating the antiproton fluxes as described in section~\\ref{sec:training_set} we can  also obtain the CR spectra of protons, deuterium, and helium ($^3$He and $^4$He) without significant additional computation costs due to the setup of \\textsc{Galprop}. The task of modelling these spectra using an ANN is very comparable with the task fulfilled by the sNet. We have thus examined the ability of the sNet architecture (as described in sec.~\\ref{sec:architectures}) to also accurately predict proton and helium spectra. The inputs of the sNet remain the same, but we have extended the length of the final output layer, to accommodate a wider energy range, appropriate for the proton and Helium AMS-02 and Voyager data. Using also the same training process (see sec.~\\ref{sec:train_process}) we achieve a similar accuracy as for the secondary antiprotons, as each of the predictions deviates from the simulations only marginally with respect to the experimental uncertainties. In figures~\\ref{img:example_fluxes_p} and \\ref{img:example_fluxes_He} we show exemplary results for protons, resp. helium, and their individual components analogous to figure~\\ref{img:example_fluxes}. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Protons.pdf}\n\t\\caption{Exemplary comparison of the simulated versus predicted protons flux of the individual components protons and Deuterium and the combination of both where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. Lower panel as figure~\\ref{img:example_fluxes}.}\n\t\\label{img:example_fluxes_p}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Helium.pdf}\n\t\\caption{Exemplary comparison of the simulated versus predicted He flux of the individual components $^3$He and $^4$He and the combination of both where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. Lower panel as figure~\\ref{img:example_fluxes}.}\n\t\\label{img:example_fluxes_He}\n\\end{figure}\n\n\\end{appendix}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\\label{introd}\n\nNuclear collective motion, such as fission, or multipole\nvibration and rotation excitation modes,\nwas successfully studied by using several\nmicroscopic-macroscopic approximations to\nthe description of the finite Fermi systems of the strongly interacting\nnucleons \\cite{migdal,myswann69,fuhi,bohrmot,mix,belyaevzel}.\nMany significant phenomena deduced from experimental data on nuclear\nfission, vibrations and rotations were explained within the\ntheoretical approaches based\n mainly on the cranking model \\cite{inglis,bohrmotpr,valat,bohrmot}\nand its extensions to the pairing correlations\n\\cite{belyaevfirst,belyaev61,belzel,belsmitolfay87},\nincluding shell and temperature effects \\cite{brackquenNPA1981},\nand to non-adiabatic effects\n\\cite{fraupash,zelev,mix,zshimansky,marshalek,belyaevhighspin,afanas,belyaevbif},\nwhich were originally applied for the\nrotational modes (see also \\cite{sfraurev} for the review paper\nand references therein).\n\n\nFor the nuclear collective excitations\nwithin the general response-function theory\n\\cite{bohrmot,siemjen,hofbook}, the basic idea is to parametrize\nthe complex dynamical problem of the collective motion of many\nstrongly interacting particles in terms of a few collective\nvariables found from the physical\nmeaning of the considered dynamical problem, for example\nthe nuclear surface itself \\cite{strtyap,strutmagbr,strutmagden}\nor its multipole deformations \\cite{bohrmot}.\nWe can then study the response to an external field of the dynamical\nquantities describing the nuclear collective motion in terms of\nthese variables. Thus, we get important information on the transport\nproperties of nuclei. For such a theoretical description of the\ncollective motion it is very important to take into account the\ntemperature dependence of the\ndissipative nuclear characteristics as the friction coefficient, as\nshown in \\cite{hofivyam,ivhofpasyam,hofmann,hofbook}.\nThe friction depends strongly on the temperature and its\ntemperature dependence can therefore not be ignored\nin the description of the collective excitations in nuclei.\nConcerning the temperature dependence of\nthe nuclear friction, one of the most important problems is\nrelated to the properties of the static susceptibilities and\nergodicity of the Fermi systems like nuclei.\n\nHowever, the quantum description of dissipative phenomena in\nnuclei is rather complicated because we have to take into account\nthe residual interactions beyond the mean-field approximation.\nTherefore, more simple models\n\\cite{strutmagbr,kolmagpl,magkohofsh,kolmagsh}\naccounting for some\nmacroscopic properties of the many-body Fermi-system are helpful\nto understand the global average properties of the collective\nmotion.\nSuch a model is based on the Landau Fermi-liquid theory\n\\cite{landau,abrikha,pinenoz},  applied for the nuclear interior\nand simple\nmacroscopic boundary conditions on the nuclear surface\n\\cite{strutmagbr,strutmagden,magstrut,magboundcond,kolmagsh,magsangzh,BMRV}\n(see\nalso macroscopic approaches with different boundary conditions\n\\cite{bekhal,ivanov,abrditstrut,komagstrvv,abrdavkolsh}).\nIn \\cite{magkohofsh}, the response-function theory can be applied to\ndescribe collective nuclear excitations as the isoscalar quadrupole\nmode. The\ntransport coefficients, such as friction and inertia, are simply\ncalculated within\nthe macroscopic Fermi-liquid droplet model (FLDM)\n\\cite{kolmagpl,magkohofsh,kolmagsh} \nand\ntheir temperature dependence  can be clearly discussed\n(see also earlier works\n\\cite{strutmagden0,galiqmodgen,galiqmod,strutmagden,magstrut,denisov}). \nThe asymmetry of\nheavy nuclei near their stability line\nand the structure of the isovector dipole resonances\nare studied in \\cite{kolmag,kolmagsh,BMV,BMR}\n(see also \\cite{abrIVGDR,abrdavpl}). In this way,\nthe giant multipole resonances were described, and, with\nincreasing temperature \\cite{kolmagpl,magkohofsh}, a transition\nfrom zero sound modes to the hydrodynamic first sound.\nThe friction in \\cite{kolmagpl,magkohofsh}\nis due to the collisions of particles, which were taken into\naccount in the relaxation-time approximation\n\\cite{abrikha,pinenoz,sykbrook,brooksyk,heipethrev,baympeth} with a\ntemperature and\nfrequency dependence (retardation effects)\n\\cite{landau,kolmagpl}.\n\n\nThe most important\nresults obtained in \\cite{magkohofsh,hofivmag} are related to the\noverdamped surface excitation mode for the low energy region and\nits dissipative characteristics as friction.\nFor the low excitation energy region these investigations can be\ncompleted by the additional sources of the friction related to a\nmore precise description of the heated Fermi liquids presented in\n\\cite{heipethrev,baympeth} for the infinite matter. Following\n\\cite{heipethrev}, we should take into account the thermodynamic\nrelations along with the dynamical Landau--Vlasov equation and\nintroduce the local equilibrium distribution instead of the one of global\nstatics, used earlier in \\cite{magkohofsh,hofivmag} for the\nlinearization procedure of this equation. These new developments\nof the Landau theory are especially important for the further\ninvestigations of the temperature dependence of the friction.\nFor the first step we have to work out in more details the theory\n\\cite{heipethrev} of the heated Fermi liquids for nuclear matter\nto apply then it for the dynamical description of the collective\nmotion in the interior of nuclei in the macroscopic FLDM\n \\cite{kolmagpl,magkohofsh}. Our purpose is also to find the\nrelations to some general points of the response function theory\nand clarify them taking the example of the analytically solved\nmodel based on the non-trivial temperature-dependent Fermi-liquid\ntheory. One of the most important questions which would be better\nto clarify is the above mentioned ergodicity property, temperature\ndependence of the friction and coupling constant.\n\nAnother important\nextension of this macroscopic theory is to study the structure\nof the isovector giant\ndipole resonance (IVGDR) as a splitting phenomenon due to the nuclear\nsymmetry interaction between neutrons and protons\n near the stability line \\cite{kolmag,abrIVGDR,abrdavpl,kolmagsh,BMV,BMRV,BMR}.\nThe neutron skin of exotic nuclei with a large excess of neutrons\nis also still one of the exciting subjects of\n    nuclear physics and nuclear astrophysics\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85,danielewicz1,pearson,danielewicz2,vinas1,vinas2,vinas3,vinas4}.\nSimple and accurate solutions for the isovector  particle density\ndistributions were obtained within the nuclear effective surface (ES)\napproximation\n\\cite{strtyap,strutmagbr,strutmagden,magsangzh,BMRV}.\nIt exploits the saturation of nuclear matter and a\nnarrow diffuse-edge region in finite heavy nuclei. The ES is defined as\nthe location of points of the maximum density gradient. The coordinate\nsystem, connected locally with\nthe ES, is specified by the distance  from the\ngiven point to the surface and by tangent coordinates  at the ES.\nThe variational condition for the nuclear energy with some\nadditional fixed integrals of motion in the local energy-density theory\n\\cite{rungegrossPRL1984,marquesgrossARPC2004} is\nsignificantly\nsimplified in these coordinates. In particular,  in the extended\nThomas--Fermi (ETF) approach \\cite{brguehak,sclbook}\n(with Skyrme forces\n\\cite{chaban,reinhard,bender,revstonerein,ehnazarrrein,pastore})\n this can be done for any deformations by using an expansion  in a small\nleptodermic parameter. The latter\nis of the order of the diffuse edge\nthickness of heavy enough nucleus over its mean curvature radius, or\nthe number of nucleons in power one third\nunder the distortion constraint in the case of\ndeformed nuclei. The accuracy of the ES  approximation in the ETF\napproach without spin-orbit (SO) and asymmetry terms was checked\n\\cite{strutmagden} by comparing results of  Hartree--Fock (HF)\n\\cite{brink,ringschuck} and ETF calculations  \\cite{brguehak,sclbook}  for some\nSkyrme forces. The ES approach (ESA) \\cite{strtyap,strutmagbr,strutmagden}\nwas then extended by taking SO and asymmetry effects into account\n\\cite{magsangzh,BMRV}.\nSolutions for the\nisoscalar and isovector particle densities and\nenergies at the quasi-equilibrium\nin the ESA of the ETF approach were applied to\nanalytical calculations of the neutron skin and isovector stiffness\ncoefficients in the leading order of the leptodermic parameter\nand to the derivations of the macroscopic boundary conditions \\cite{BMRV}.\nOur results are compared with the fundamental researches\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85} in the\nliquid droplet model (LDM). These analytical expressions for the\nenergy surface constants can be used\nfor IVGDR calculations within the FLDM\n\\cite{denisov,kolmag,kolmagsh,BMV,BMR}.\n\nA further interesting application\nof the semiclassical response theory would consist in the study of the\nproperties of collective rotation bands in heavy deformed nuclei.\nOne may consider nuclear collective rotations\nwithin the cranking model\nas a response to the Coriolis external-field perturbation.\nThe moment of inertia (MI)\ncan be calculated as a susceptibility with respect to this external field.\nThe rotation frequency of the rotating Fermi system in the cranking model\nis determined for a given nuclear\nangular momentum through a constraint, as\nfor any other integral of motion, as in particular the particle number\nconservation \\cite{ringschuck}. In order to simplify\nsuch a rather complicated problem, the Strutinsky shell correction method (SCM)\n\\cite{strut,fuhi} was adjusted to the collective nuclear rotations in\n\\cite{fraupash,mix}. The collective MI\nis expressed as function of the particle number and temperature\nin terms of a smooth part\nand an oscillating shell correction. The smooth component can be described\nby a suitable macroscopic model, like the dynamical ETF approach\n\\cite{bloch,amadobruekner,rockmore,jenningsbhbr,sclbook,brguehak,bartelnp,bartelpl}\n similar to the FLDM,\nwhich has proven to be both simple and precise.\nFor the definition of the MI shell correction, one can apply the Strutinsky\naveraging procedure to the single-particle (s.p.) MI, in the same way as for\nthe well-known free-energy shell correction.\n\nFor a deeper understanding of the quantum results and the correspondence\nbetween classical and quantum physics of the MI shell components, it\nis worth to analyze these shell components in terms of periodic orbits\n(POs), what is now well established as the semiclassical periodic-orbit\ntheory (POT) \\cite{gutz,bablo,strutmag,bt,creglitl,sclbook,migdalrev}\n(see also its\nextension to a given angular momentum projection along with the\nenergy of the particle \\cite{magkolstr} and to the particle densities\n\\cite{strutmagvvizv1986,brackrocciaIJMPE2010} and\npairing correlations\n\\cite{brackrocciaIJMPE2010}).\nGutzwiller was the first who developed the POT for\ncompletely chaotic Hamiltonians with only one integral of motion\n (the particle energy) \\cite{gutz}.\nThe Gutzwiller approach of the POT extended\nto potentials with continuous symmetries for the description of the\nnuclear shell structure can be found in \\cite{strutmag,smod,creglitl,sclbook}.\nThe semiclassical shell-structure corrections to the level density\nand energy have been tested for a large number of s.p.\\ Hamiltonians\nin two and three dimensions (see, for instance,\n\\cite{sclbook,magosc,ellipseptp,spheroidpre,spheroidptp,maf,magNPAE2010,magvlasar}).\nFor the Fermi gas the\n entropy shell corrections of the POT as a sum of periodic orbits were\nderived in \\cite{strutmag}, and with its help,\nsimple analytical expressions for the\nshell-structure energies in cold nuclei were obtained there following\na general semiclassical theory \\cite{sclbook}.\nThese energy shell corrections are in good agreement with the\nquantum SCM results, for\ninstance for elliptic and spheroidal cavities, including the\n superdeformed bifurcation region\n\\cite{ellipseptp,spheroidptp}.\nIn particular in three dimensions, the superdeformed bifurcation\nnanostructure leads as function of deformation to the\ndouble-humped shell-structure energy with the first and second\npotential wells in heavy enough nuclei\n\\cite{smod,migdalrev,spheroidptp,sclbook,magNPAE2010},\nwhich is well known as the double-humped fission barriers in the region\nof actinide nuclei.\nAt large deformations the second well can be understood semiclassically,\nfor spheroidal type shapes, through the bifurcation of equatorial orbits\ninto equatorial and the shortest 3-dimensional periodic orbits, because of\nthe enhancement of the POT amplitudes of the shell correction to the level\ndensity near the Fermi surface at these bifurcation deformations.\n\nFor finite heated fermionic systems, it was also shown\n\\cite{strutmag,kolmagstr,magkolstrutizv1979,richter,sclbook,brackrocciaIJMPE2010}\n within the POT that the\nshell-structure of the entropy, the thermodynamical (grand-canonical) potential\n and the free-energy shell\ncorrections can be obtained by multiplying the terms of the POT expansion\nby a temperature-dependent factor, which is exponentially decreasing\nwith temperature. For the case of the so called\n``{\\it classical rotations}$\\,$'' around the symmetry $z$ axis of the nucleus,\nthe MI shell correction is obtained, for any rotational frequency and at finite\ntemperature, within the extended Gutzwiller POT through the averaging of the\nindividual angular momenta aligned along this symmetry axis\n\\cite{magkolstr,kolmagstr,magkolstrutizv1979}.\nA similar POT problem, dealing with the magnetic susceptibility of fermionic\nsystems like metallic clusters and quantum dots, was worked out in\n\\cite{richter,fraukolmagsan}.\n\n\n\nIt was suggested in \\cite{dfcpuprc2004} to use the spheroidal cavity and\nthe classical perturbation approach to the POT by Creagh \\cite{creagh,sclbook}\nto describe the collective rotation of deformed nuclei around an axis ($x$\naxis) perpendicular to the symmetry $z$ axis.\nThe small parameter of the POT perturbation approximation turns out to be\nproportional to the rotational frequency, but also to the classical action\n(in units of $\\hbar$), which causes an additional restriction to Fermi\nsystems (or particle numbers) of small enough size, in contrast to the usual\nsemiclassical POT approach.\n\nIn \\cite{mskbg,mskbPRC2010}, the nonperturbative extended Gutzwiller POT\nwas used for the calculation of the MI shell corrections within the\nmean-field cranking model for both the collective and the alignment\nrotations.\nIn these works, for the statistical equilibrium nuclear rotations,\nthe semiclassical MI shell corrections were obtained in good agreement with the\nquantum results\nin the case of the harmonic-oscillator potential.\nWe extend this approach for collective rotations\nperpendicular to symmetry axis to the analytical calculations of\nthe MI shell corrections for the  case of different mean fields,\nin particular with spheroidal\nshapes and sharp edges. The main purpose is to study semiclassically\nthe enhancement effects in the MI within the improved stationary\nphase method (improved SPM or shortly ISPM)\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar},\ndue to the bifurcations of the\nperiodic orbits in the superdeformed region.\n\n\nIn the present review in Section \\ref{eqmotion} we present some\nbasic formulas of the temperature-dependent Fermi-liquid theory\n\\cite{heipethrev}. We consider in Sec.\\ \\ref{conserveqs} the\nparticle number and momentum conservation equations and derive\nfrom them the energy conservation and general transport equations,\nin particular,\nthe expressions for the viscosity, shear modulus and thermal conductivity\ncoefficients. In Sec.\\ \\ref{respfunsec} we determine the\ndensity-density and density-temperature response functions with\nthe low temperature corrections. Section\n\\ref{longwavlim} shows the long wave-length (LWL, or hydrodynamic) limit\nfor the response functions, and the specific expressions for the\ntransport coefficients.\nIn Sec.\\ \\ref{suscept}, one obtains the static\nisolated, isothermal, and adiabatic susceptibilities to clarify\nsome important points of the general response function theory,\nmainly, the ergodicity property of the Fermi systems\n\\cite{hofmann,kubo}. We study the relaxation and correlation\nfunctions on the basis of the fluctuation-dissipation theorem and\nestablish their relations to the ergodicity of the Fermi-liquid\nsystem in section \\ref{relaxcorr}.\nGeneral aspects of the response function theory for the collective\nmotion in nuclei are presented in\nSec.\\ \\ref{basdef}\nin line with \\cite{hofmann,hofbook}. Section\n\\ref{fldm} shows the basic ingredients and\nthe collective response function of the nuclear FLDM.\nSection \\ref{transprop} is devoted to the\nderivation of the temperature dependence of the\ntransport coefficients, such as friction, inertia,\nand stiffness for the density modes\nfor slow collective motion. The numerical illustrations are given in Sec.\\\n\\ref{discuss}.  In Sec.\\ \\ref{npcorivgdr}, the semiclassical theory is extended\nto neutron-proton asymmetric nuclei and applied for the\ncalculations of IVGDRs.\nIn Sec.\\ \\ref{semshellmi}, the smooth\nETF and fluctuating shell-structure components of the moments of inertia\nare derived for collective rotations of heavy nuclei.\nThe MI shell component is analytically presented in terms of the\nperiodic orbits and their bifurcations within the POT. This component\nis compared with the quantum results for the simplest case of the deformed\nharmonic oscillator Hamiltonian.\nComments and conclusions are finally given in Sec.\\ \\ref{concl}.\nSome details of the thermodynamical, FLDM\n(in the LWL limit) and POT calculations, such as\nthe analytical derivations\nof the in-compressibility, viscosity, thermoconductivity,\ncoupling, and surface symmetry-energy constants, as well as the\nsemiclassical MI\nare presented in Appendices A-E.\n\n\n\n\n\\section{The quasiparticle kinetic theory}\n\\label{kinapp}\n\n\\subsection{Equations of motion for the heated Fermi liquid}\n\n\\label{eqmotion}\n\nIn the semiclassical approximation the dynamics of a Fermi liquid\nmay be described by the distribution function $f({\\bf r},{\\bf p},t)$ in the\none body phase-space. Restricting to small deviations of particle\ndensity $\\rho({\\bf r},t)$ and temperature $T$, from their values in a\nthermodynamic  equilibrium one may apply the linearized\nLandau--Vlasov equation \\cite{abrikha,heipethrev}:\n\\begin{eqnarray}\\label{landvlas}\n\\frac{\\partial}{\\partial t}\n    \\delta f({\\bf r},{\\bf p},t) &+&\n    \\frac{\\partial \\varepsilon_{{\\bf p}}^{\\rm g.e.}}{\\partial {\\bf p}}\n    {\\bf \\nabla}_r \\delta f({\\bf r},{\\bf p},t) - \n    - {\\bf \\nabla}_r \\left[\\delta \\varepsilon({\\bf r},{\\bf p},t) \\right.\\nonumber\\\\ \n    &+&\\left.V_{\\rm ext}\\right]\n    {\\bf \\nabla}_p f_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.}) =\n    \\delta St. \n\\end{eqnarray}\nThe right hand side (r.h.s.) represents the dynamic component\nof the integral collision term $\\delta St$, and\n$V_{\\rm ext}$ stands for an external field. We introduce here\nthe Fermi distribution\n\\bel{fgeq}\nf_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.})  =\n    \\left[1 + \\hbox{exp} \\left(\\frac{\\varepsilon_{\\bf p}^{\\rm g.e.}-\\mu}{T}\\right)\\right]^{-1}\n\\end{equation}\nof the {\\it global equilibrium} (g.e.), with  $\\mu$ being the chemical\npotential,  the temperature $T$ is given, as usually\nin nuclear physics, in the energy\n(MeV) units (without Boltzmann's constant), and\n$\\delta f({\\bf r},{\\bf p},t)$ measures the deviation\n\\bel{dfgeqrpt} \\delta f({\\bf r},{\\bf p},t) =  f({\\bf r},{\\bf p},t) -\nf_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.}).\n\\end{equation}\nFor the sake of simplicity, the s.p.\\ energy $\\varepsilon_{\\bf p}^{\\rm g.e.}$\nwill be assumed to be of the form\n$~\\varepsilon_{\\bf p}^{\\rm g.e.} =\np^2\/2m^*~$ with $m^*$ being the effective nucleonic\nmass. In (\\ref{landvlas}),\n$\\delta \\varepsilon({\\bf r},{\\bf p},t)$ stands for the variation of the\nquasiparticle energy $\\varepsilon({\\bf r},{\\bf p},t)$,\n\\begin{eqnarray}\\label{deleps}\n&&\\delta\n\\varepsilon({\\bf r},{\\bf p},t) = \\varepsilon({\\bf r},{\\bf p},t)- \\varepsilon_{\\bf p}^{\\rm g.e.}  \\nonumber\\\\\n&=&    \\frac{1}{\\mathcal{N}(T)} \\int\n\\frac{2 d{\\bf p}^\\prime}{(2 \\pi {\\hbar})^3}\\;\n    \\mathcal{F}({\\bf p}, {\\bf p}^\\prime)\\;\n    \\delta f({\\bf r}, {\\bf p}^\\prime, t).\n\\end{eqnarray}\nThe quasiparticles' density of\nstates $\\mathcal{N}(T)$ at the chemical potential $\\mu$ is given\nby\n \\bel{enerdensnt}\n \\mathcal{N}(T)=\\int\n  \\frac{2 d{\\bf p}^\\prime}{(2 \\pi {\\hbar})^3}\n    \\left(\n-\\frac{\\partial f_{{\\bf p}^{\\prime}}} {\\partial \\varepsilon_{{\\bf p}^{\\prime}}} \\right)_{g.e}.\n\\end{equation}\nEvidently, because of our linearization the density\n$\\mathcal{N}(T)$ here is the one of equilibrium. In the sequel such a\nconvention will inherently be applied to any coefficient of\nquantities of order $\\delta f$.\nThe factor 2 accounts for the spin degeneracy.\nThe amplitude of the\nquasiparticle interaction, $\\mathcal{F}({\\bf p},{\\bf p}^\\prime)$,\ncommonly is written in terms of\nthe Landau parameters $\\mathcal{F}_0$ and $\\mathcal{F}_1$, according to\n\\bel{intampfpp}\n\\mathcal{F}({\\bf p}, {\\bf p}^\\prime) =\n    \\mathcal{F}_0 + \\mathcal{F}_1 {\\hat p}\n    \\cdot {\\hat p}^\\prime\\;, \\qquad   {\\hat p} =\n    {\\bf p}\/p.\n\\end{equation}\nThese two constants may be related to the two properties of\nnuclear matter, namely the isothermal in-compressibility $K^{T}$\n(see Appendix A.1),\n\\bel{isotherk}\nK^{T} =\n9 \\rho\\mathcal{G}_0\/\\mathcal{N}(T),\n\\end{equation}\nand the effective mass $m^*$,\n\\bel{effmass}\nm^*=  \\mathcal{G}_1 m, \\quad\n\\mathcal{G}_n=\\left(1+\\frac{\\mathcal{F}_n}{2n+1}\\right)\n\\end{equation}\n($n=0, 1$). The equation for the effective\nmass $m^*$ is known \\cite{abrikha,heipethrev} to be valid\nfor systems obeying Galileo invariance, which shall be assumed\nhere.\n\nIn principle, the Landau parameters $\\mathcal{F}_0$ and $\\mathcal{F}_1$\nmight vary with the momenta $p$ and $p'$. Such a dependence will\nbe neglected henceforth.  This approximation appears to be\nreasonable as we are going to stick to small excitations near the\nFermi surface and to temperatures $T$, which are small as compared to\nthe chemical potential $\\mu$. Likewise, we shall discard any\ntemperature dependence of the effective mass. Notice that\nin addition to the ratio $({T \/\\mu})^2$, this dependence would be\ngoverned by the additional factor $|{m^* \/ m}-1|$ which is small\nfor nuclear matter. These assumptions will allow us to simplify\nfurther the theory \\cite{heipethrev} and to get\nmore explicit results by making use of the temperature expansion\nfor the response functions in the small parameter $T\/\\mu$, as well\nas of the standard perturbation approach to eigenvalue problems\nneeded later for the hydrodynamic (long-wave\nlength) limit. We will follow \\cite{heipethrev} in neglecting\nhigher order terms in the expansion (\\ref{intampfpp}) in Legendre\npolynomials.\n\n\nLater on we want to study motion of the system which can be\nclassified as an excitation on top of the {\\it local equilibrium}.\nFollowing  \\cite{abrikha,heipethrev}, the\ncollision term $\\delta St$ can  be considered in the relaxation\ntime approximation,\n\\begin{eqnarray}\\label{intcoll}\n&&\\delta St =\n-\\frac{\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)}{\\tau}\\;, \\qquad\n    f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{l.e.}\\right) =  \\nonumber\\\\\n    &=&\\left[1 + \\hbox{exp}\\left(\\frac{\\varepsilon_{\\bf p}^{\\rm l.e.}-\\mu({\\bf r},t)\n    -{\\bf p}{\\bf u}({\\bf r},t)}{T({\\bf r},t)}\\right)\\right]^{-1}.\n\\end{eqnarray}\nHere, $f_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$  is the distribution function\nof a {\\it local equilibrium} (l.e.), and $\\varepsilon_{\\bf p}^{\\rm l.e.}$ is the associated\nquasiparticle energy. $\\mu({\\bf r},t)$ represents the chemical\npotential, ${\\bf u}({\\bf r},t)$ the mean velocity field, and $T({\\bf r},t)$\nthe temperature, all defined in the local sense.  Like in\n\\cite{magkohofsh}, the relaxation time $\\tau$ is assumed to be\nindependent of the quasiparticle momentum ${\\bf p}$. However, it  will be\nallowed to depend $\\tau$ on $T$  as well as on the frequency of the\nmotion (thus, accounting for retardation effects in collision\nprocesses).   In\n(\\ref{intcoll}), $\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$\nis defined as\n\\bel{dfleqrpt} \\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},t) =  f({\\bf r},{\\bf p},t) - f_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.}).\n\\end{equation}\nIt differs from $\\delta f({\\bf r},{\\bf p},t)$ of (\\ref{dfgeqrpt}) by the\nvariations of local quantities.  For the latter, we may write\n\\bel{dfgeqdfleq} \\delta f({\\bf r},{\\bf p},t) =\n    \\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t) +\n    \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{l.e.}\\right)\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{dfleq}\n&&\\delta f_{\\rm l.e.} \\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right)\n    = f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) -\n    f_{\\rm g.e.}\\left(\\varepsilon_{\\bf p}^{\\rm g.e.}\\right)  \\qquad \\nonumber\\\\\n&=&\n   \\left(\\frac{\\partial f_{\\bf p}}{\\partial \\varepsilon_{\\bf p}}\\right)_{\\rm g.e.}\n    \\left(\\delta \\varepsilon_{\\bf p}^{\\rm l.e.} - \\delta \\mu -{\\bf p} {\\bf u} -\n    \\frac{\\varepsilon_{\\bf p}^{\\rm g.e.}-\\mu}{T}\\delta T\\right).\\qquad\n\\end{eqnarray}\nFor the l.e. quasiparticle energy $\\varepsilon_{\\bf p}^{\\rm l.e.}$, one has\n\\bel{delelocglo} \\varepsilon_{\\bf p}^{\\rm l.e.}=\\varepsilon_{\\bf p}^{\\rm g.e.}+\n    \\delta \\varepsilon_{\\bf p}^{\\rm l.e.},\n\\end{equation}\nwhere $\\delta \\varepsilon_{\\bf p}^{\\rm l.e.}$ is defined like in\n(\\ref{deleps}) with only $\\delta f({\\bf r},{\\bf p},t)$ replaced by\n$\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$. According to (\\ref{dfgeqdfleq}) and\n(\\ref{dfleq}), for the simplified interaction\n(\\ref{intampfpp}), one gets\n\\bel{delepsleq} \\delta \\varepsilon_{\\bf p}^{\\rm l.e.}=\n    \\delta \\varepsilon ({\\bf r},{\\bf p},t)=\n    \\frac{\\mathcal{F}_0}{\\mathcal{N}(T)} \\delta \\rho({\\bf r},t)\n    + \\frac{\\mathcal{F}_1 m \\rho}{\\mathcal{N}(T) p_{{}_{\\! {\\rm F}}}^2} {\\bf p}{\\bf u},\n\\end{equation}\n where $\\delta \\rho$ is the dynamical component of the particle\ndensity\n \\bel{densit} \\rho({\\bf r}, t) =\n    \\int \\frac{2 d{\\bf p}}{(2 \\pi \\hbar)^3} \\;\n    f({\\bf r}, {\\bf p}, t) = \\rho_\\infty + \\delta \\rho({\\bf r},t)\n\\end{equation}\n with $\\rho_\\infty$ being its g.e. value\nassociated to $f_{\\rm g.e.}(\\varepsilon_{\\bf p}^{\\rm g.e.})$ for the infinite Fermi liquid.\nThe vector of the mean\nvelocity ${\\bf u}$ can be expressed in terms of the first moment of the\ndistribution function (current density) and the particle density\n(\\ref{densit}),\n\\bel{veloc} {\\bf u}({\\bf r},t) =    \\frac{1}{\\rho}\n    \\int \\frac{2  d {\\bf p}}{(2 \\pi \\hbar)^3} \\>\n\\frac{{\\bf p}}{m} \\delta f({\\bf r}, {\\bf p}, t).\n\\end{equation}\n\nThe definition of the collision term in the form (\\ref{intcoll})\nis incomplete without posing conditions for the conservation of\nthe particle number, momentum, and energy (for simplicity\nof notations, we shall omit index $\\infty$ in\nthe static nuclear-matter density component $\\rho$ at\nsecond order terms in the energy density variations).\nNotice that to the order considered, in the equation\nfor energy conservation, $\\varepsilon$ may be replaced by\n$\\varepsilon_{\\bf p}^{\\rm g.e.}$ (see also \\cite{baympeth}).\nIncidentally, for the\nquasiparticle interaction (\\ref{intampfpp}), this substitution\neven becomes {\\it exact}, as the dynamical part $\\delta \\varepsilon$\nwould drop out of the last integral (as follows from\n(\\ref{delepsleq}), (\\ref{delelocglo}), and two first equations\nin the following set of conditions\n\\cite{heipethrev},\n\\begin{eqnarray}\n    \\int {\\rm d}{\\bf p}\\; \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t)\\; = \\;0\\;, \\qquad\n    \\int {\\rm d}{\\bf p}\\;{\\bf p}\\;\\delta f_{\\rm l.e.}({\\bf r},{\\bf p}, t)\n    \\; = \\;0\\;, \\nonumber\\\\\n    \\int {\\rm d}{\\bf p}\\;\n    \\varepsilon\\;\n    \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t) \\; = 0. \\qquad\\qquad\\qquad\\qquad\n\\label{consereq}\n\\end{eqnarray}\nThese equations\nmimics conservation of the\ncorresponding quantities in each collision of quasiparticles and\nensures that of the same quantities calculated for the total\nsystem (without external fields). Together with the basic equation\n(\\ref{landvlas}), one thus has 6 equations for the 6 unknown\nquantities $\\delta \\rho({\\bf r},t)$, $\\delta \\mu({\\bf r},t)$, ${\\bf u}({\\bf r},t)$ and\n$\\delta T({\\bf r},t)$. They allow one to find unique solutions as\nfunctionals of the external field $V_{\\rm ext}(t)$. Below we shall\nsolve these equations in terms of response functions.\nIt may be noted that, due\nto the conditions (\\ref{consereq}), the first variation of the\ndistribution function $\\delta f({\\bf r},{\\bf p},t)$\n(\\ref{dfgeqdfleq}) disappears from the dynamical component\n$\\delta \\rho({\\bf r},t)$ of the density $\\rho({\\bf r},t)$ and of the\nvelocity field ${\\bf u}({\\bf r},t)$. As one knows (see, e.g.,\n\\cite{heipethrev,abrikha,baympeth}), the equation for the velocity field\nreduces to an identity if one takes into account the definition of\nthe effective mass $m^*$ given by (\\ref{effmass}).\n\n\\subsection{The conserving equations}\n\\label{conserveqs}\n\nIn this section, we like to deduce conserving equations for the\nparticle number, momentum, and energy, which later on will turn out\nhelpful to find appropriate solutions of the Landau--Vlasov\nequation (\\ref{landvlas}). The procedure, which basis on a moment\nexpansion, is well known from textbooks\n\\cite{pinenoz,baympeth,forster}.\nWe will follow more\nclosely the version of \\cite{kolmagpl,magkohofsh} (see also \n\\cite{galiqmod}).\n\n\\subsubsection{THE MOMENT EXPANSION}\nWhereas particle number conservation implies to have\n \\bel{conteq}\n\\frac{\\partial \\rho}{\\partial t} +\n    {\\bf \\nabla} \\left(\\rho {\\bf u}\\right) =  0,\n\\end{equation}\nthe  momentum conservation is reflected by the following set of\nequations\n \\bel{momenteq}\nm \\rho\n\\frac{\\partial u_\\alpha}{\\partial t} +\n    \\sum_{\\beta}\n\\frac{\\partial \\Pi_{\\alpha \\beta}}{\\partial r_{\\beta}}  =\n    -\\frac{\\partial V_{\\rm ext}}{\\partial r_\\alpha}.\n\\end{equation}\n Besides quantities introduced before, they involve\n\\bel{momentflux}\n\\Pi_{\\alpha\\beta}=\n    \\int \\frac{2  d {\\bf p}}{(2 \\pi \\hbar)^3 } \\>\n    \\frac{p_\\alpha p_\\beta}{m^*} \\delta f({\\bf r}, {\\bf p}, t)\n    + \\frac{\\mathcal{F}_0}{\\mathcal{N}(T)} \\delta \\rho\\left({\\bf r},t\\right)\n    \\;\\delta_{\\alpha\\beta}.\n\\end{equation}\n Substituting for  $\\delta f({\\bf r}, {\\bf p}, t)$\n(\\ref{dfgeqdfleq})  into (\\ref{momentflux}), one gets\n\\bel{momentflux1} \\Pi_{\\alpha\\beta} = - \\sigma_{\\alpha\\beta} +\n\\delta {\\cal P} ~\\delta_{\\alpha\\beta}.\n\\end{equation}\nThe first component $\\sigma_{\\alpha\\beta}$,\nwhich results from the first term $\\delta\nf_{\\rm l.e.}({\\bf r}, {\\bf p}, t)$ on the right of (\\ref{dfgeqdfleq}),\ndetermines the dynamic shear stress tensor,\n\\bel{presstens}\n\\sigma_{\\alpha\\beta}({\\bf r}, t)=\n   - \\int \\frac{2  d {\\bf p}}{(2 \\pi \\hbar)^3}\\;\n    \\frac{p_{\\alpha} p_{\\beta}}{m^*}\n    \\delta f_{\\rm l.e.}({\\bf r}, {\\bf p}, t)\\;,\n\\end{equation}\nwhose trace vanishes.\nFor a linearized dynamics, the non-diagonal components of the\nmomentum flux tensor $\\Pi_{\\alpha\\beta}$ equal the corresponding\nstress tensor (but with the opposite sign), with correction terms\nbeing proportional to $u_\\alpha u_\\beta$ in $\\delta f$, and thus,\nof higher order, see (\\ref{veloc}) for $u_{\\alpha}$.\n\nThe second component of the momentum flux tensor of\n(\\ref{momentflux1}) can be derived from the variation  $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$ as given by (\\ref{dfleq}). It\nrepresents the compressional part of the momentum flux tensor,\n\\begin{eqnarray}\\label{pressdef}\n&&\\int \\frac{2  \\hbox{d} {\\bf p} }{(2 \\pi \\hbar)^3}~\n    \\frac{p_\\alpha p_\\beta }{m^*}~\n    \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) +\n\\frac{\\mathcal{F}_0}{\\mathcal{N}(T)}~\\delta \\rho~\n\\delta_{\\alpha\\beta}= \\delta {\\cal P} \\delta_{\\alpha\\beta}\n\\nonumber\\\\\n&&{\\rm with}\\quad \\delta \\rho \\equiv \\delta \\rho({\\bf r},t)= \\int\n\\frac{2\\hbox{d} {\\bf p}}{(2 \\pi \\hbar)^3}~\n    \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right)\n\\end{eqnarray}\n[mind (\\ref{dfgeqdfleq}) and (\\ref{consereq})]. Notice, that here\nonly the diagonal parts survive. The only non-diagonal ones could come\nfrom the terms in (\\ref{dfleq}) involving  ${\\bf p}{\\bf u}$; but they vanish\nwhen integrating over angles in momentum space. Traditionally,\n$\\delta {\\cal P}$ in (\\ref{pressdef}) is referred to as the\nscalar pressure, see \\cite{brenig}.\nUsing\n(\\ref{dfleq}) for the distribution  $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\bf p}^{\\rm l.e.})$ and its properties mentioned above,\nafter some simple algebraic transformations, one\ngets\n \\begin{eqnarray}\\label{pressureq}\n\\delta {\\cal P} &=&\\frac{2}{3}~ \\int \\frac{2 \\hbox{d} {\\bf p}}{(2 \\pi \\hbar)^3}\n    \\frac{p^2}{2 m^*}\n    \\delta f_{\\rm l.e.}\\left(\\varepsilon_{\\bf p}^{\\rm l.e.}\\right) +\n\\frac{\\mathcal{F}_0 }{\\mathcal{N}(T)}~\\delta \\rho \\nonumber\\\\\n&=&{K^T \\over 9}\n\\delta \\rho + \\rho\\left(\\varsigma-\n\\frac{\\mathcal{M} }{\\mathcal{N}}\\right)~\\delta T,\n\\end{eqnarray}\n with $K^T$ being the isothermal\nin-compressibility (\\ref{isotherk}). For the derivation of the\nsecond equation in (\\ref{pressureq}),  one can use (i)\nthe transformation of $\\delta \\mu$ to the\nvariations of $\\delta \\rho$ and $\\delta T$\n[see (\\ref{dmu})], and (ii) the relations\n (\\ref{entropydef}), (\\ref{densstat}), and (\\ref{mcapt}) for\nthe entropy per particle $\\varsigma$, the particle density $\\rho$\nas well as for the quantity $\\mathcal{M}$ (\\ref{mcapt}),\nrespectively. Inspecting (\\ref{dpdtr}) and\n(\\ref{incomprTdef}), it becomes apparent that the expression on\nthe very right of (\\ref{pressureq}) may indeed be interpreted as\nan  expansion of the static pressure to the first order in\n$\\delta \\rho$ and $\\delta T$. It is thus seen that the truly\nnon-equilibrium component $\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$ only appears\nin the shear stress tensor $ \\sigma_{\\alpha\\beta} $ given in\n(\\ref{presstens}).\n\n\nNote, here and below within\nSec.\\ \\ref{conserveqs}, we omit immaterial constants related to the\nglobal equilibrium (static) components of the moments to simplify\nthe notations and adopt them to the ones of the standard textbooks\nwhen it will not lead to misunderstanding. We should\nemphasize that the Landau quasiparticle theory which is a basis of\nour derivations is working in a self-consistence way with small\ndeviations from (small excitations near) the Fermi surface\nwhich are denoted by symbol \"$\\delta$\" and takes\nabove mentioned static components as those of the external phenomenological\n(experimental) data. Therefore, all relations discussed below in\nthis section should be understood as the ones between such\nclose-to-Fermi-surface quantities within our linearized\nLandau--Vlasov phase space dynamics after exclusion of all above\nmentioned immaterial constants. Nevertheless, we keep the symbol\n$\\delta$ with the scalar pressure $\\delta \\mathcal{P}$ to avoid\npossible misunderstanding related to the linearization procedure,\nsee more comments below after (\\ref{deformtens}).\n\n\n\\subsubsection{THE STRESS TENSOR}\n\nIt may be worthwhile to relate the stress tensor  $\n\\sigma_{\\alpha \\beta}$ given in (\\ref{presstens}) to the standard\nform  in terms of the\ncoefficients of the shear modulus $\\lambda$  and the viscosity\n$\\nu$,\n \\bel{prestensone}\n\\sigma_{\\alpha\\beta}=\n    \\sigma_{\\alpha\\beta}^{(\\lambda)}\n    + \\sigma_{\\alpha\\beta}^{(\\nu)}.\n\\end{equation}\n Here, the first term $ \\sigma_{\\alpha\\beta}^{(\\lambda)}$\nis the conservative part of the stress tensor $\n\\sigma_{\\alpha\\beta}$,\n\\bel{presslamb}\n\\sigma_{\\alpha\\beta}^{(\\lambda)}=\n    \\lambda\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n    \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha \\beta}\\right)\n\\end{equation}\nwith $~{\\bf u} = \\partial {\\bf w}\/\\partial t$\nand ${\\bf w}$ being the displacement field.\nThe second term in (\\ref{prestensone}) can be written as\n\\bel{pressnu}\n\\sigma_{\\alpha\\beta}^{(\\nu)}=\n    \\nu\\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n     \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right),\n\\end{equation}\n where $\\nu$ is the coefficient of the shear viscosity (or the\nfirst viscosity). For more details see Appendix A.2,\nin particular for expressions of the coefficients $\\lambda$\n(\\ref{shearmod}) and $\\nu$ (\\ref{viscos}) in terms of\nFermi liquid interaction parameters.\n\nTo obtain microscopic expressions for the shear modulus $\\lambda$\nand the viscosity $\\nu$, one needs to exploit the solution $\\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},t)$ of the Landau--Vlasov equation (\\ref{landvlas})\nfor the stress tensor $\\sigma_{\\alpha\\beta}({\\bf r},t)$ (\\ref{presstens}),\nreducing the latter to the form (\\ref{prestensone}). Such a\ncalculation of $\\lambda$ and $\\mu$ in terms of the\nLandau Fermi-liquid parameters is discussed in Appendix A.2,\nin which Fourier transforms are exploited \\cite{kolmagpl}.\nEquivalently, one may\nexpress functions of space and time by plane waves, which for the\ndistribution function reads \\cite{strutmagden0,kolmagpl}\n\\bel{planewave}\n\\delta f({\\bf r}, {\\bf p}, t)=\n\\delta {\\tilde f}\\left({\\bf q},{\\bf p},\\omega \\right)\n  \\hbox{exp}\\left[{i({\\bf q}{\\bf r}-\\omega  t)}\\right]\n\\end{equation}\n with ${\\bf q}$ being the wave vector and $\\omega $ the frequency of the\nvibrational modes of nuclear matter. Such a plane-wave\nrepresentation is to be applied to both sides of (\\ref{presstens})\nand (\\ref{prestensone}). The amplitudes for the velocity\n${\\bf u}$ and the displacement ${\\bf w}$ field then satisfy ${\\bf\n{\\tilde w}}={\\tilde{\\bf u}}\/(-i\\omega )$.\n\nUsing (\\ref{presstens}),\n(\\ref{prestensone}), (\\ref{presslamb}) and (\\ref{pressnu}) for\nthe stress tensor $\\sigma_{\\alpha\\beta}$\nand (\\ref{pressureq}) for the scalar pressure $\\delta \\mathcal{P}$,\n  \\bel{incomprtotdef}\n  \\delta {\\cal P}=\\frac{K_{\\rm tot}}{9}~\\delta \\rho ,\n\\end{equation}\none\nfinally may write down a general expression for the momentum flux\ntensor $\\Pi_{\\alpha\\beta}({\\bf r},t)$ (\\ref{momentflux1}),\n\\begin{eqnarray}\\label{momentfluxtot}\n&&\\Pi_{\\alpha\\beta}=\n    -\\lambda\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n    \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha \\beta}\\right) ~~\\nonumber\\\\\n    &-&\\nu\\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n    \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right)\n    + \\frac{K_{\\rm tot}}{9}~\\delta \\rho~\\delta_{\\alpha\\beta}.~~~\n\\end{eqnarray}\nA total effective in-compressibility $K_{\\rm tot}$ includes the change of\nthe pressure due to\nvariations of the temperature with density.\nWith the help of (\\ref{dpdtr}), the in-compressibility $K_{\\rm tot}$\ncan be expressed through the specific\nheat per particle  ${\\tt C}_{\\mathcal{V}}$\n(\\ref{specifheatpdef}),\n \\bel{incomprtot}\nK_{\\rm tot}=K^{T}+ 6 {\\tt C}_{\\mathcal{V}}\\; \\rho\n    ~\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}.\n\\end{equation}\nAgain, $\\delta {\\tilde T}$ and $\\delta {\\tilde \\rho}$ are the\nFourier components of $\\delta T({\\bf r},t)$ and $\\delta \\rho({\\bf r},t)$.\nLike all other kinetic coefficients, such as\n$\\lambda$ and $\\nu$ given in (\\ref{shearmod}) and (\\ref{viscos}),\nrespectively, this effective, total in-compressibility modulus\n$K_{\\rm tot}$, too, depends on $\\omega $ and $q$.  Later on we shall\ndiscuss in more detail these quantities in the LWL\nlimit. In this limit, the total in-compressibility $K_{\\rm tot}$ will\nbe seen to become identical to the adiabatic one $K^{\\varsigma}$\ngiven in (\\ref{incompradexp}).\n\n\\subsubsection{ENERGY CONSERVATION AND THE GENERAL TRANSPORT\\\\ EQUATION}\n\nSo far we have not looked at the energy conservation. For this purpose,\none needs to consider thermal aspects as they appear in equations\nfor the change of entropy and temperature. To do this we will\nfollow standard procedures. We first built the\nscalar product of the mean velocity ${\\bf u}$ with the vector equation,\nwhose component $\\alpha$ is given by (\\ref{momenteq}). Making\nuse of the continuity equation (\\ref{conteq}), after some\nmanipulations, one gets\n\\begin{eqnarray}\\label{enerconstwo}\n&&\\frac{\\partial}{\\partial t}\n    \\left(\\frac{1}{2} m\\rho u^2+\\rho {\\cal E}\\right) = \\nonumber\\\\\n    &=& -\\sum_{\\alpha\\beta}\\frac{\n\\partial}{\\partial r_{\\beta}} \\left[u_{\\alpha}\n    \\left(\\frac{1}{2} m \\rho u^2 \\delta_{\\alpha\\beta}\n    +\\rho W_{\\alpha\\beta}-\n    \\sigma_{\\alpha\\beta}^{(\\nu)}\n    -\\kappa \\frac{\\partial T}{\\partial r_{\\alpha}}\n    \\delta_{\\alpha\\beta}\\right)\\right] \\nonumber\\\\\n    &+& \\rho T \\left(\\frac{\\partial \\varsigma}{\\partial t}\n    + {\\bf u}{\\bf \\nabla} \\varsigma\\right)\n   -{\\bf \\nabla}\\left(\\kappa {\\bf \\nabla}T\\right) \\nonumber\\\\\n    &-&\\frac{\\nu}{2}\\sum_{\\alpha\\beta}\n    \\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta} }+\n    \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha\\beta}\\right)^2\n    -\\rho {\\bf u}{\\bf \\nabla} V_{\\rm ext}.\n\\end{eqnarray}\nOn the left hand side, there appears the mean kinetic energy\ndensity and the internal energy density $\\rho \\mathcal{E}$ per unit\nvolume (defined again up to an immaterial constant). The density\n${\\cal E}$ itself may be split in three different components,\n\\bel{enerintr}\n{\\cal E}= {\\cal E}^{(\\lambda)}+{\\cal E}_{\\rm tot}^{(K)}\n+ T \\delta \\varsigma\\;.\n\\end{equation}\n  The first one,\n\\bel{shearen}\n{\\cal E}^{(\\lambda)}\n =\\frac{\\lambda}{4 \\rho}\n    \\sum_{\\alpha \\beta}\\left(\\frac{\n\\partial w_{\\alpha}}{\\partial r_{\\beta}} +\n    \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf w}\\;\\delta_{\\alpha\n    \\beta}\\right)^2\\;,\n\\end{equation}\n is related to shear deformations, which is known from the solid state\nphysics and for Fermi liquids as coming from\ndistortions of the Fermi surface \\cite{landau}.\nThe second one may\nbe written as\n\\bel{enerintrk}\n {\\cal E}_{\\rm tot}^{(K)}= \\frac{K_{\\rm tot}}{18} \\left(\\delta \\rho\\right)^2 ;\n\\end{equation}\n it represents the\ncompressional component, associated to the effective total\nin-compressibility $K_{\\rm tot}$,\nwhich is\nin line of the known thermodynamic relations.\nEquation (\\ref{enerintrk}) resembles the expression\nfound in \\cite{strutmagbr,strutmagden}, except for a generalization\nof the physical meaning of the in-compressibility modulus\n$K_{\\rm tot}$ as function of $\\omega $ and $q$\ngiven in (\\ref{incomprtot}), as compared to the quasistatic adiabatic\ncase. The third one in (\\ref{enerintr}) represents the\nchange of heat part resulting from a change of entropy.\nWe keep here the dynamical variation symbol $\\delta$ for the\nentropy $\\varsigma$ (also for the pressure $\\mathcal{P}$ here and\nbelow) to remember that all quantities of the Landau Fermi-liquid\ntheory are presented for small dynamical deviations near the\nFermi surface in the linear (or quadratic after multiplying\n(\\ref{momenteq}) by ${\\bf u}$) form in $\\delta f$. We avoid here a\nmisunderstanding with following transformations of the energy\n${\\cal E}$ (\\ref{enerintr}), say Legendre ones, to the\ndifferential form  in line of a\ngeneral comment at the beginning of this section.\nOn the r.h.s. of (\\ref{enerconstwo})\nthe enthalpy $W_{\\alpha\\beta}$ per particle has been introduced,\n\\bel{enthalp} W_{\\alpha\\beta}=\n    {\\cal E} \\delta_{\\alpha\\beta}\n    + \\frac{1}{\\rho}\\left(\\sigma_{\\alpha\\beta}^{(\\lambda)} +\n    \\delta {\\cal P}~ \\delta_{\\alpha\\beta} \\right)\n\\end{equation}\n (see the comment above concerning $\\delta {\\cal P}$).\nFurthermore, the thermodynamic relation for the dynamical\nvariations of the internal energy ${\\cal E}$ in\nterms of those $\\varsigma$ for the entropy per particle $\n\\varsigma$, the density $\\rho$ and the displacement tensor\n$w_{\\alpha\\beta}$, is given by\n\\bel{thermener}\n{\\rm d} {\\cal E} =T {\\rm d}\n\\varsigma+ \\frac{\\delta {\\cal P}}{\\rho^2} {\\rm d} \\rho +\n\\frac{1}{\\rho}\n    \\sum_{\\alpha\\beta}\\sigma_{\\alpha\\beta}^{\\lambda}\n    {\\rm d} w_{\\alpha\\beta}.\n\\end{equation}\nThe displacement tensor $w_{\\alpha\\beta}$ is defined as\n\\bel{deformtens}\nw_{\\alpha\\beta}=\n    \\frac{1}{2}\\left(\\frac{\\partial w_{\\alpha}}{\\partial r_{\\beta}}\n    + \\frac{\\partial w_{\\beta}}{\\partial r_{\\alpha}}\\right)\\;.\n\\end{equation}\nNote that equation (\\ref{incomprtotdef}) for the pressure $\\delta\n{\\cal P}$ is important to get\n(\\ref{enerintr}) by integration of (\\ref{thermener}).\nAccording to (\\ref{enthalp}), we get the standard relation of\n{\\it linearized} thermodynamics  of\n\\cite{brenig}, for instance,\nbetween the enthalpy (\\ref{enthalp}) and entropy $\\varsigma$ up to the\nsecond order term in $\\delta {\\cal P}$.\n\nIn (\\ref{enerconstwo}), we also added and subtracted the term\n${\\bf \\nabla} {\\bf j}_T$ containing the heat current,\n\\bel{currheat}\n{\\bf j}_{{}_{\\! T}}= -\\kappa {\\bf \\nabla} T\\;,\n\\end{equation}\nwith the coefficient $\\kappa$ for the thermal conductivity.\nWe may now write the\nequation for energy conservation as\n\\begin{eqnarray}\\label{enerconserv}\n&&\\frac{\\partial}{\\partial t}\n    \\left(\\frac{1}{2} m\\rho u^2+\\rho {\\cal E} \\right)=\n    -\\sum_{\\alpha\\beta}\\frac{\n\\partial}{\\partial r_\\beta} \\left[u_\\alpha\n    \\left(\n   \\frac{1}{2} m \\rho u^2 \\delta_{\\alpha\\beta} \\right.\\right.\\nonumber\\\\\n    &+&\\left.\\left.\\rho W_{\\alpha\\beta}\n       - \\sigma_{\\alpha\\beta}^{(\\nu)}\n    -\\kappa \\frac{\n\\partial T}{\\partial r_{\\alpha}} \\delta_{\\alpha\\beta}\\right)\\right]\n    -\\rho {\\bf u}{\\bf \\nabla} V_{\\rm ext}.\n\\end{eqnarray}\nIn this way, it is seen that from (\\ref{enerconstwo}) and\n(\\ref{enerconserv}), together with the continuity equation\n(\\ref{conteq}) and the definition of the heat current ${\\bf j}_T$\n(\\ref{currheat}), one\ngets for the change of entropy:\n\\begin{eqnarray}\\label{entropyeq}\n    \\frac{\\partial (\\rho \\delta \\varsigma)}{\\partial t}\n    &=&-{\\bf \\nabla}\\left(\\rho \\delta \\varsigma\\; {\\bf u}\n    + \\frac{1}{T} {\\bf j}_{{}_{\\! T}}\\right)\n    + \\frac{\\kappa}{T^2}\\left({\\bf \\nabla}T\\right)^2 \\nonumber\\\\\n    &+&\\frac{\\nu}{2T} \\sum_{\\alpha\\beta}\n    \\left(\\frac{\\partial u_{\\alpha}}{\\partial r_{\\beta}} +\n    \\frac{\\partial u_{\\beta}}{\\partial r_{\\alpha}} -\n    \\frac{2}{3}{\\bf \\nabla}{\\bf u}\\;\\delta_{\\alpha \\beta}\\right)^2\n\\end{eqnarray}\n[again the variation $\\delta$ in $\\delta \\varsigma$ is not\nomitted because of the following derivations of the Fourier\nequation (\\ref{fouriereq}) and thermal conductivity\n(\\ref{kappadef}) in Appendix A.2].\nThese two equations have a very clear physical meaning\nfor normal liquids and amorphous solids (a very\nviscose liquids are associated to the amorphous\nsolids with some shear modulus $\\lambda$ in our notations, i.e.,\nsolids without any crystal structure).\nThe first equation (\\ref{enerconserv}) claims that the change of the\ncollective and internal energy, concentrated in unit volume per\nunit of time and presented as the sum of the collective kinetic\nand internal parts, equals the corresponding energy flux through\nits surface and work of the external field. The second equation\n(\\ref{entropyeq}) is usually called as a general\nheat transport equation.\nThis\nequation states that the change of entropy in the unit volume\nper unit of time equals the entropy flux through its surface\n(heat energy flux). Two other terms\nshow the entropy increase related to the gradient of the\ntemperature and dissipation due to the shear ($\\nu$) viscosity.\nNote that there is no explicit dependence on the\nexternal field in (\\ref{entropyeq}). This dependence is\nmanifested only through the solutions of the dynamical equations\nin terms of the moments. For zero external field (closed system)\nthe entropy is increasing  because of the basic thermodynamic law.\nTherefore, according to (\\ref{entropyeq}), the\nshear viscosity $\\nu$ (\\ref{viscos}) and the thermal conductivity\n$\\kappa$ should be positive. The energy conservation equation for\nthe Fermi liquid (\\ref{enerconserv}) differs from the one for\nclassical hydrodynamics  by the same\nFermi-surface distortions related to the shear modulus $\\lambda$\n(\\ref{shearmod}) as discussed above. That is similar to the\namorphous solids  (in the above mentioned sense\nof very viscose liquids). However, in\ncontrast to the latter, one obtains the energy conservation\ncondition (\\ref{enerconserv}) for the dynamical variations of the\nFermi-liquid collective and internal energy with the specific\nconstants $\\lambda$ (\\ref{shearmod}), $\\nu$ (\\ref{viscos}) and\n$K_{\\rm tot}$ (\\ref{incomprtot}) found from the relation to the\nLandau--Vlasov equation (\\ref{landvlas}). Our way of the\nderivation of the energy conservation equation\n(\\ref{enerconserv}) for the Fermi liquids within the {\\it\nlinearized} Landau--Vlasov dynamics (\\ref{landvlas}),\nas for normal liquids and\nsolids, leads to a more explicit form of the energy conservation\nequation than that suggested in \\cite{heipethrev,baympeth}. In this way,\nwe get rather simple expressions for the collective and internal\ncomponents of the energy out the hydrodynamical limit.\n\n\\subsubsection{POTENTIAL FLOW: FERMI LIQUID VERSUS HYDRODYNAMICS}\n\\label{potenflow}\n\nBelow, we shall be interested in the case of a viscous potential\nflow, for which one has\n\\bel{velpoten}\n{\\bf u}={\\bf\n\\nabla}\\varphi,\\quad {\\bf w}={\\bf \\nabla}\\varphi_w\\quad\n{\\rm with} \\quad \\varphi=\\dot{\\varphi}_w\n\\end{equation}\n [cf. the second\nequation with (\\ref{presslamb})]. With the help of these\ndefinitions, the momentum equation (\\ref{momenteq}) and the flux\ntensor (\\ref{momentfluxtot}) can be brought to the following\nforms:\n\\bel{navstokeq0}\nm \\rho \\frac{\\partial \\varphi}{\\partial\nt} - \\frac{4}{3} \\nu~ \\Delta \\varphi\n - \\frac{4}{3} \\lambda~ \\Delta \\varphi_w\n+ \\frac{K_{\\rm tot}}{9} \\delta \\rho=-V_{\\rm ext}\n\\end{equation}\nand\n\\begin{eqnarray}\\label{momentfluxpot}\n\\Pi_{\\alpha\\beta}&=& -2 \\left(\n\\frac{\\lambda}{-i\\omega } +\\nu \\right)\n\\left(\\frac{\n\\partial^2\\varphi}{\\partial r_\\alpha \\partial r_\\beta}- \\triangle \\varphi \\;\n\\delta_{\\alpha\\beta} \\right) \\nonumber\\\\\n&-&\\left(m \\rho \\frac{\n\\partial \\varphi}{\\partial t}+ V_{\\rm ext}\\right) \\delta_{\\alpha\\beta}.\n\\end{eqnarray}\nThe diagonal term given on the very right of\n(\\ref{momentfluxpot}) had been used to remove $\\delta \\rho$\nwhich still appears in (\\ref{momentfluxtot}). With the continuity\nequation (\\ref{conteq}) for the plane wave solutions\n(\\ref{planewave}), one has from (\\ref{navstokeq0}) the equation\nfor the velocity potential $\\varphi$ \\cite{magkohofsh}:\n\\bel{navstokeq}\nm \\rho \\frac{\\partial^2 \\varphi}{\\partial t^2}\n-\\frac{\\rho}{9}\\left(K_{\\rm tot} +12 \\lambda\/\\rho \\right) \\Delta\n\\varphi\n - \\frac{4}{3} \\nu~ \\Delta \\frac{\\partial \\varphi}{\\partial t}\n = - \\frac{\\partial V_{\\rm ext}}{\\partial t}.\n\\end{equation}\nThe structure of\n(\\ref{navstokeq}) for the potential flow is similar to that of\nthe Navier-Stokes equation for the velocity potential $\\varphi$.\nThe difference to the case of the common classical liquid, is seen in\nthe terms proportional to $\\lambda$, viz in the presence of the\nanisotropy term (\\ref{presslamb}), which actually represents a\n{\\it reversible} motion. Such a term is known from the dynamics of\namorphous solids. We emphasize that for Fermi liquids, this term\narises only in the presence of the Fermi surface distortions, which\nsurvive even in the non-viscous limit; they will turn out\nimportant for our applications below. The shear modulus $\\lambda$\nmay be interpreted as a measure of those distortions which are\nrelated to a reversible anisotropy of the momentum flux tensor.\nThey disappear in the hydrodynamic limit, and so does\n$\\lambda$, in which case all formulas of this section turn into\nthose for normal liquids; for more details see section\n\\ref{longwavlim}.\n\nAt this place an important remark is in order. It should be noted\nthat in contrast to classical hydrodynamics\nour system of equations for the moments is {\\it not closed} to the\nfirst few ones, namely particle density $\\delta \\rho$ and velocity\nfield ${\\bf u}$. This is true in particular for (\\ref{navstokeq})\nfor the  potential flow $\\varphi$. Indeed, the coefficients\n$\\lambda$, $\\nu$ and $K_{\\rm tot}$ depend on the variable $\\omega \/q$\nwhich yet is unknown. The latter is determined from a\ndispersion relation, which in turn has to be derived from the\nLandau--Vlasov equation (\\ref{landvlas}). Such a procedure goes\nback to \\cite{landau} where the dispersion relation was exploited\nfor the collisionless case at $T=0$. A collision term in the\nrelaxation approximation has been taken into account in\n\\cite{abrikha}. The extension to heated Fermi liquids and low\nexcitations, in the way, which we are going to use later on, has been\ndeveloped in \\cite{heipethrev}. It may be noted that this version\nof the dispersion relation, which we are aiming at, differs\nessentially from the one obtained in the \"truncated\" (scaling model)\nversions of\nthe Fermi-liquid theory of \\cite{holzwarth,nixsierk}, where the\nmomentum flux tensor is not influenced by higher moments of the\ndistribution function.\nWe take into account all other  multipolarities (larger the quadrupole\none) of the Fermi-surface distortions when there is no\nconvergence in multipolarity expansion of the distribution\nfunction for finite and large $\\omega \\tau$ or for finite\n$K_{\\rm tot}$, for instance for nuclear matter with small $\\mathcal{F}_0$,\nin contrast to the Fermi liquid $^3$He.\n\n\n\\subsection{Response functions}\n\\label{respfunsec}\n\n\\subsubsection{DYNAMIC RESPONSE}\n\\label{dynresp}\n\nAs mentioned earlier, we want to solve the linearized equations of\nmotion in terms of response functions. We concentrate on two\nquantities, namely particle density $\\rho({\\bf r},t)$ and temperature\n$T({\\bf r},t)$ and examine how they react to the external field\n$V_{\\rm ext}({\\bf r},t)$ introduced earlier. This may be quantified by the\nfollowing two response functions: The density-density response\n$\\chi_{DD}^{\\rm coll}$ and the temperature-density response\n$\\chi_{TD}^{\\rm coll}$ defined as\n\\bel{ddrespdef}\n\\chi_{DD}^{\\rm coll}(q,\\omega )=\n    - \\frac{\\delta \\rho(q,\\omega )}{V_{\\rm ext}(q,\\omega )}\n\\end{equation}\n  and\n \\bel{dtrespdef} \\chi_{TD}^{\\rm coll}(q,\\omega )=\n    - \\frac{\\delta T(q,\\omega )}{V_{\\rm ext}(q,\\omega )},\n\\end{equation}\nrespectively. To keep the notation simple, we will omit the\ntilde characterizing the Fourier transform of the distribution\nfunction (\\ref{planewave}) (it should suffice to only show the\narguments $q,\\omega $). The definition of the response functions is\nidentical to the one of \\cite{heipethrev}, except that we have\nintroduced the suffix \"coll\". This was done adopting a notation\nused in the literature of nuclear physics when the dynamics of a\nfinite nucleus is expressed in terms of shape variables, to which\nwe will come below. Notice, however, that $V_{\\rm ext}(q,\\omega )$ is only\nproportional to the density,\n$V_{\\rm ext}(q,\\omega )=q_{\\rm ext}(\\omega )\\rho(q,\\omega )$, with $q_{\\rm ext}(\\omega )$\nbeing some externally determined function. Often, one therefore\ndefines response functions in a slightly modified way, in that the\nfunctional derivatives are performed with respect to\n$q_{\\rm ext}(\\omega )$ instead of $V_{\\rm ext}(q,\\omega )$ (see,\ne.g., \\cite{pinenoz}).\n\nAs will be seen below, these functions only depend on the wave\nnumber $q$ but not on the angles of the wave vector ${\\bf q}$. For this\nreason, it is convenient to introduce the dimensionless quantities\n$s$ and $\\tau_q$ (with $v_{{}_{\\! {\\rm F}}}=p_{{}_{\\! {\\rm F}}}\/m^*$)\n\\bel{som}\ns= \\frac{\\omega }{v_{{}_{\\! {\\rm F}}} q},\\quad\n\\tau_q =\\tau v_{{}_{\\! {\\rm F}}} q,\n    \\quad{\\rm implying}\\quad \\omega  \\tau = s  \\tau_q,\n\\end{equation}\n  instead of the frequency $\\omega $ and the wave number $q$.\n\nTo calculate the response functions (\\ref{ddrespdef}) and\n(\\ref{dtrespdef}) we follow the procedure of \\cite{heipethrev}. As\nany further details may be found there, it may suffice to outline\nbriefly the main features. In short, the strategy is as follows.\nFirstly, one rewrites the Landau--Vlasov equation (\\ref{landvlas})\nin terms of the Fourier coefficients introduced in\n(\\ref{planewave}). Evidently, in the spirit of the separation\nspecified in (\\ref{dfgeqdfleq}), we need to evaluate explicitly\nonly the first component\n$\\delta f_{\\rm l.e.}({\\bf r},{\\bf p},t)$  which\nenters the conditions (\\ref{consereq}). By a\nstraightforward calculation, one may then express $\\delta\nf_{\\rm l.e.}({\\bf r},{\\bf p},\\omega )$ in terms of the unknown quantities $\\delta\n\\rho$, ${\\bf u}$, $\\delta \\mu$ and $\\delta T$ for any given external\nfield $V_{\\rm ext}$. The form is given in (\\ref{basiceq}).\nThe continuity equation (\\ref{conteq})\nin the Fourier representation through\n (\\ref{planewave}),\n${\\bf q}{\\bf u}=\\omega  \\delta\n    \\rho \/ \\rho$,\nmay be used to eliminate the velocity field ${\\bf u}$.\nFurthermore, the thermodynamic relation [see (\\ref{drhomt}),\n(\\ref{incomprTdef}) and (\\ref{isotherk})]\n\\bel{dmu} \\delta \\mu =\n    \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T \\delta \\rho +\n    \\left(\\frac{\\partial \\mu }{\\partial T}\\right)_{\\rho} \\delta T =\n    \\frac{K^{T} }{9\\rho}\\;\n    \\delta \\rho - \\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\;\\delta T\n\\end{equation}\nallows one to express the chemical potential $\\delta \\mu$ in\nterms of the two unknown variables $\\delta \\rho$ and $\\delta T$.\nNext, one may exploit the conditions (\\ref{consereq}). As the\nsecond (set of) equation(s) is just an identity, provided one\nuses the appropriate definition of the effective mass\n(\\ref{effmass}), it is only\nthe first and the third equation which matter. They\nmay determine the remaining two\nvariables $\\delta \\rho$ and $\\delta T$ in terms of the external\nfield,\n\\bel{drhodteqone}\n    \\left(\\frac{i s \\tau_q }{i s \\tau_q -1}\n    -\\wp(s) \\chi _0 \\right) \\delta \\rho\n    + \\frac{1}{1-i s \\tau_q } \\chi _1 \\delta T\n    =-\\chi _0 \\delta V_{\\rm eff}\n\\end{equation}\n and\n\\bel{drhodteqtwo} -\\wp(s) \\chi _1 \\delta \\rho\n    + \\frac{1}{1-i s \\tau_q } \\left(\\chi _2\n    -i  s \\tau_q \\rho \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right) \\delta\nT\n    = -\\chi _1 \\delta V_{\\rm eff}.\n\\end{equation}\n Here, the quantity\n\\bel{alphas}\n   \\wp(s) = \\frac{1}{\\mathcal{N}(T)}\\frac{1}{i s \\tau_q -1}\n    - \\frac{3is}{\\tau_q \\mathcal{N}(0)}\n\\end{equation}\nhas been introduced with $\\mathcal{N}(0)$ being the level density\n(\\ref{enerdensnt}) of the quasiparticles at $T=0$,\n\\bel{nzero} \\mathcal{N}(0)= \\frac{p_{{}_{\\! {\\rm F}}} m^*}{\\pi^2 \\hbar^3}=\n    \\frac{3}{2} \\frac{\\rho _0}{\\varepsilon_{{}_{\\! {\\rm F}}}},\n\\end{equation}\nand $\\varepsilon_{{}_{\\! {\\rm F}}}=p_{\\rm F}^2\/2m^*$.\nThe functions $\\chi _n$ are given by\n\\bel{chinfun}\n\\chi _n = -\\mathcal{N}(T) \\left\\langle\n\\frac{{\\bf q}{{\\bf v}_{\\bf p}}}{\\mathcal{D}_{\\bf p}}\n\\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu}{T} -\n\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right)^n\\; \\right\\rangle\n\\end{equation}\n with $n=0,1,2,...\\;$,\n\\begin{equation}\\label{domindp}\n\\mathcal{D}_{\\bf p}=\n    \\omega -{\\bf q}{\\bf v}_{\\bf p}+ i\/\\tau, \\qquad{\\bf v}_{\\bf p}={\\bf p}\/m^*.\n\\end{equation}\n Furthermore, in (\\ref{drhodteqone}) and\n(\\ref{drhodteqtwo}) a short hand notation $\\delta V_{\\rm eff}$ has\nbeen used for the sum of two terms, namely\n\\bel{delueff} \\delta\nV_{\\rm eff}=\n    V_{\\rm ext} + k(\\omega ,T) \\delta \\rho\n\\end{equation}\n with\n\\bel{couplconst} k(\\omega ,T) =\\frac{1}{\\mathcal{N}(T)}\n    \\left[\\mathcal{F}_0 + \\frac{\\mathcal{F}_1}{\\mathcal{G}_1}\n    \\left(\\frac{\\omega }{v_{{}_{\\!{\\rm F}}} q}\\right)^2\\right].\n\\end{equation}\nIn (\\ref{delueff}), $\\delta V_{\\rm eff}$ may be considered as an\n{\\it effective} field which includes the true external field\n$V_{\\rm ext}$ and the \"screened\" field $k\\delta \\rho$\n\\cite{heipethrev}. Our notation follows the one often used for\nfinite nuclei: The second term in (\\ref{delueff}) plays the\nrole of the collective variable and $k$ of (\\ref{couplconst})\nrepresents the \"coupling\" constant (see, e.g.,\n\\cite{bohrmot,hofbook,hofmann}).\n\nThe response function $\\chi_{DD}^{\\rm coll}$ of (\\ref{ddrespdef})\ncan be now obtained\nfrom (\\ref{drhodteqone}) and (\\ref{drhodteqtwo}),\n\\bel{ddresp} \\chi_{DD}^{\\rm coll}( \\tau_q ,s)=\n    \\frac{\\aleph( \\tau_q ,s)}{D( \\tau_q ,s)},\n\\end{equation}\nwhere\n\\bel{despfunc} D( \\tau_q ,s)=\n    D_0( \\tau_q ,s)+ k( \\tau_q ,s) \\aleph( \\tau_q ,s)\n\\end{equation}\n with\n\\bel{chidomin} D_0( \\tau_q ,s)=\n    \\left(\\frac{i s \\tau_q }{i s \\tau_q-1}\n    -\\wp(s)\\chi _0 \\right)\n    \\left(\\chi _2 - i s \\tau_q \\rho\n\\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right)\n    + \\wp \\chi _1 ^2.\n\\end{equation}\nIn (\\ref{ddresp}), $\\aleph(\\tau_q ,s)$ finally is given by\n\\bel{chinumer}\n\\aleph(\\tau_q ,s)=\n    \\chi _0 \\left(\\chi _2\n    -i s \\tau_q \\rho\n\\frac{{\\tt C}_{\\mathcal{V}}}{T} \\right)- \\chi _1 ^2.\n\\end{equation}\n\nIt is worth noticing that the collective response function for the\ndensity-density mode, as given by (\\ref{ddrespdef}) or\n(\\ref{ddresp}), can be expressed as\n\\bel{ddrespchiin}\n\\chi^{\\rm coll}(q,\\omega )=\n    \\frac{\\chi (q,\\omega )}{1+k(\\omega ,T) \\chi (q,\\omega )}.\n\\end{equation}\nThis form is analogous to the form used to describe the\ndynamics of shape variables \\cite{bohrmot,hofbook,hofmann}. We omit\nhere the suffix $DD$ because the $TD$ response function takes on a\nsimilar form\n(with some modification of the numerator). It is here where the\n\"coupling constant\" $k$ appears, as defined in\n(\\ref{couplconst}), together with the \"intrinsic\" (or\n\"un-screened\" (see \\cite{pinenoz,heipethrev}) response\nfunction $\\chi $,\n\\bel{chiindef} \\chi ( \\tau_q ,s)=\n    - \\frac{\\delta \\rho}{\\delta V_{\\rm eff}}\n    = \\frac{\\aleph( \\tau_q ,s)}{D_0( \\tau_q ,s)}.\n\\end{equation}\n Both expressions can be found already in \\cite{heipethrev}.\nHowever, later we will find the form (\\ref{ddresp}) more\nconvenient for our applications, in particular for the discussion\nof the low frequency limit  $\\omega  \\tau \\ll 1$.\nWhen we shall expand first $\\chi $ (\\ref{chiindef}) in\n(\\ref{ddrespchiin}) in small $\\omega  \\tau$ near the poles of\n$\\chi ^{\\rm coll}$ (\\ref{ddresp}) (see next section), one should\nassume that the singularities of $\\chi$ related to zeros of $D_0$\nin (\\ref{chiindef}) are far away from zeros of $D$ in\n(\\ref{ddresp}), i.e., a smoothness of $\\chi$ as function of\n$\\omega \\tau$ near these poles. After the cancellation of a\npossible singularity source $D_0$ in (\\ref{ddresp}) we are free\nfrom such an assumption.\n\n\nFinally, let us turn to the temperature-density response function\n$\\chi_{TD}^{\\rm coll}$ (\\ref{dtrespdef}). It is determined by the same\nsystem of equations (\\ref{drhodteqone}) and (\\ref{drhodteqtwo})\nand can be written in the form (\\ref{ddrespchiin}) but with\nanother ``intrinsic'' response function $\\chi_{{}_{\\! TD}}$ appearing in the\nnumerator,\n\\bel{chitindef} \\chi_{{}_{\\! TD}}( \\tau_q ,s)=\n    - \\frac{\\delta T}{\\delta V_{\\rm eff}}.\n\\end{equation}\n From (\\ref{drhodteqone}), (\\ref{drhodteqtwo}) and\n(\\ref{chitindef}), one obtains\n\\bel{chitbar}\n\\chi_{{}_{\\! TD}}(\\tau_q,s)=\n    -\\frac{is\\tau_q~\\chi _1}{D_0(\\tau_q ,s)},\n\\end{equation}\nwhere $D_0( \\tau_q ,s)$ is given by (\\ref{chidomin}).\n As compared to the one printed in \\cite{heipethrev},\nthis expression contains an additional factor $i s\\tau_q\/\\chi_1$,\nwhich later on will turn out important, for instance, when\ncalculating susceptibilities and the in-compressibility $K_{\\rm tot}$\n(\\ref{incomprtot}).  (We are grateful to H.\nHeiselberg for confirming this misprint.)\nSubstituting (\\ref{chitbar}) into the\nnumerator of (\\ref{ddrespchiin})\ninstead of $\\chi$, one gets the temperature-density response\nfunction (\\ref{dtrespdef}) in the form similar to (\\ref{ddresp}),\n\\bel{dtresp} \\chi_{TD}^{\\rm coll}( \\tau_q,s)=\n    -\\frac{is\\tau_q~\\chi _1}{D( \\tau_q ,s)}.\n\\end{equation}\n Notice that according to (\\ref{ddresp}) and\n(\\ref{dtresp}), both response functions (\\ref{ddrespdef}) and\n(\\ref{dtrespdef}) have the same set of poles, which lie at the\nroots of the equation\n\\bel{despeq} D( \\tau_q ,s)= 0.\n\\end{equation}\nThis is identical to the condition of zero determinant\nfor the system of the linear equations (\\ref{drhodteqone})\nand (\\ref{drhodteqtwo}).\n\n\n\\bigskip\n\n\\subsubsection{LOW TEMPERATURE LIMIT}\n\\label{lowtemlim}\n\nThe expressions for the collective response functions become much\nsimpler at low temperatures $T \\ll \\mu$. In\nthis case, one may calculate $\\chi _n$ of (\\ref{chinfun}) by\nexpanding in powers of ${T \/ \\mu}$. For those applications to\nnuclear physics we have in mind the temperature is sufficiently\nsmall such that it suffices to mainly stick to order two. Fourth\norder terms shall be shown only when necessary.\n\nA basic element for the quantities which we need to evaluate is\nthe derivative $\\partial f_{{\\bf p}} \/ {\\partial \\varepsilon_{{\\bf p}}}$ taken at\nglobal equilibrium:\n\\bel{dfpdep} \\left(\\frac{\n\\partial f_{{\\bf p}}}{\\partial \\varepsilon_{{\\bf p}}}\\right)_{\\rm g.e.}=\n    -\\left[\n4 T \\hbox{cosh}^2\\left(\\frac{\\varepsilon_{\\bf p}-\\mu}{2T}\\right)\\right]^{-1}_{\\rm g.e.}.\n\\end{equation}\n It appears in $\\mathcal{N}(T)$ of (\\ref{enerdensnt}) [see\nalso (\\ref{averag})], which in turn is needed for $\\chi _n$\nof (\\ref{chinfun}). For small $T$, this derivative is a sharp\nbell-shaped function of $\\varepsilon_{\\bf p}^{\\rm g.e.}$, such that one may evaluate\nthe averaging integrals (\\ref{chinfun}) and (\\ref{averag}) by\nexpanding the smooth functions in terms of $\\varepsilon_{\\bf p}^{\\rm g.e.}$ near\n$\\varepsilon_{\\bf p}^{\\rm g.e.}=\\mu_{\\rm g.e.}$. In this way, the Fourier-Bernoulli\nintegrals over the dimensionless variable\n$\\left[(\\varepsilon_{\\bf p}-\\mu)\/T\\right]_{\\rm g.e.}$ appear (see, e.g.,\n\\cite{heipethrev})\nwhich lead to\n\\begin{eqnarray}\\label{chitemzero}\n\\chi_0&=&\n     \\left[-Q_1(\\zeta)+ \\frac{\\pi^2 {\\bar T}^2}{12}\n     \\left(Q_1(\\zeta)-\\zeta Q_1^{\\prime}(\\zeta) \\right.\\right. \\nonumber\\\\\n   &-& \\left.\\left. \\frac{1}{2} \\zeta^2 Q_1^{\\prime \\prime}(\\zeta)\\right)\n     + \\mathcal{O}\\left({\\bar T}^4\n\\right)\\right]\n     \\mathcal{N}(0),\n\\end{eqnarray}\n \\bel{chitemone} \\chi_1=\n     \\left[\\frac{\\pi^2 {\\bar T}}{6} \\zeta Q_1^{\\prime}(\\zeta)\n     + \\mathcal{O}\\left({\\bar T}^3\n\\right)\\right]\n     \\mathcal{N}(0),\n\\end{equation}\n and\n\\begin{eqnarray}\\label{chitemtwo}\n\\chi_2 &=&\n    \\left\\{-\\frac{\\pi^2}{3} \\left[Q_1(\\zeta)+\n\\frac{\\pi^2 {\\bar T}^2}{120}\n    \\left(36 Q_1(\\zeta)\n - 46 \\zeta Q_1^{\\prime}(\\zeta) \\right.\\right.\\right. \\nonumber\\\\\n &-&\\left.\\left.\\left.\n    21 \\zeta^2 Q_1^{\\prime \\prime}(\\zeta)\\right)\\right]\n    + \\mathcal{O}\\left({\\bar T}^4\n\\right) \\right\\}\n    \\mathcal{N}(0).\n\\end{eqnarray}\n Here $Q_1(\\zeta)$ is the Legendre function of second kind with\n$\\zeta= s+i\/\\tau_q$,\n and\n${\\bar T}=T\/\\varepsilon_{{}_{\\! {\\rm F}}}$ is used also in Appendix A.3.\nThese quantities may now be used to calculate the response\nfunctions (\\ref{ddrespdef}) and (\\ref{dtrespdef}), [or more\nspecifically (\\ref{ddresp}) and (\\ref{dtresp})].  For zero\ntemperature, one gets the standard solutions\n\\cite{abrikha,heipethrev}. So far no assumption has been made\nconcerning the parameter $\\omega  \\tau$ which specifies the importance\nof  collision in various regimes of the collective motion\n\\cite{abrikha}. In particular, the formulas obtained in this\nsection are valid both for the regimes of zero sound ($\\omega  \\tau\n\\gg 1$) and  hydrodynamics ($\\omega  \\tau \\ll 1$). For $\\omega  \\tau \\gg\n1$ our solutions agree with those of \\cite{abrikha,heipethrev}.\nHowever, below we shall be interested mainly in  collective\nexcitations of low frequencies. The notion \"low frequencies\" is\nmeant to indicate that the corresponding excitation energies are\nsmaller than those of the\ngiant resonances. Next we will turn to the hydrodynamic regime\nwhere $\\omega  \\tau \\to 0$. As we shall see, at low temperatures our\nsolutions approach the ones of normal classical liquids, in\nagreement with \\cite{forster,brenig}.\n\n\\bigskip\n\n\\subsection{Hydrodynamic regime}\n\\label{longwavlim}\n\n\\medskip\n\n\\subsubsection{DISPERSION RELATION}\n\\label{disrel}\n\nThe response functions can be\nsimplified significantly in the long-wave length\nlimit. Using\n$\\tau_q$ introduced in (\\ref{som}), this (LWL) limit may be defined as\n$\\tau_q \\ll 1$. It can be reached in two ways, namely for small\nwave numbers $q$ and finite collision time $\\tau$ or for small\n$\\tau$ but finite $q$. Both cases imply that the dimensionless\nparameter $\\omega  \\tau=s \\tau_q$, which determines the collision\nrate in comparison to the frequency of the modes, becomes small\nfor any finite value $s$ of (\\ref{som}) ($|s| \\siml 1$).\nAs will be shown below for nuclear matter at low temperatures, this\nquantity $s$ is not enough large, in distinction to the\ncase of liquid $^3He$. Therefore, a small $\\tau_q$ implies\nhydrodynamic behavior, in contrast to the zero sound regime; where\n$\\tau_q \\gg 1$, or $\\omega  \\tau \\gg 1$.\n\nThe Landau--Vlasov equation (\\ref{landvlas}) is an integral\nequation. Its solution may be sought for in terms of an eigenvalue\nproblem with the distribution function $\\delta f$ being the\neigenfunctions and the sound velocity $s$ (\\ref{som}) being the\neigenvalues, see also \\cite{abrikha,pinenoz}. This eigenvalue\nproblem may be solved perturbatively with $\\tau_q$ being the\nsmallness parameter\n\\cite{sykbrook,brooksyk}.\nIt may be noted in passing that this method may be applied to some extent\nas well\nto the eigenvalue problem of the Schr\\\"odinger equation.\nWe shall use it to get the hydrodynamic sound\nvelocities from the kinetic equation, see\n\\cite{sykbrook,brooksyk}. To this end, we expand the solutions for\n$s$ and $\\delta f$ into  power series with respect to $\\tau_q$,\nbut restricted to linear order. Thus, we may write\n\\bel{somexp} s =\ns_0 + i s_1 \\tau_q,\n\\end{equation}\n where $s_0$ and $s_1$ are independent of\nthe expansion parameter $\\tau_q$. In Appendix A.2,\nit is\nshown how the density-density response function may be calculated\nin the LWL limit.\nThere two non-linear equations for the\ncoefficients $s_0$ and $s_1$ are obtained from the dispersion\nrelation (\\ref{despeq}), namely (\\ref{eqzero}) and (\\ref{eqone})).\nThe first equation [see (\\ref{eqzero})] has one obvious solution\n$s_0=s_0^{(0)}=0$ and two others $s_0=\\pm s_0^{(1)}$ with the\nsame modulus,\n\\begin{eqnarray}\\label{sfirst0}\n    s_0^{(1)} &=& \\sqrt{\\frac{\\mathcal{G}_0 \\mathcal{G}_1\n\\mathcal{N}(0)}{3 \\mathcal{N}(T)}\n    \\left(1 + \\frac{\\pi^2 \\tbar^2}{3 \\mathcal{G}_0}\\right)} \\nonumber\\\\\n&\\approx&\n    \\sqrt{\\frac{\\mathcal{G}_0 \\mathcal{G}_1}{3}\n    \\left[1 +\n    \\frac{\\pi^2 \\tbar^2 \\left(4 + \\mathcal{G}_0\\mathcal{G}_1\\right)}{12 \\mathcal{G}_0}\n\\right]}.\n\\end{eqnarray}\nSubstituting $s_0^{(0)}$ and $s_0^{(1)}$ into the second equation (\\ref{eqone}),\none finds the two solutions for $s_1$, $s_1^{(0)}$ and $s_1^{(1)}$,\nrespectively. These solutions for $s$ (\\ref{somexp})\ncan be written in terms of the dimensional\nfrequency $\\omega $ by means of (\\ref{som})\nin the following form:\n\\begin{eqnarray}\\label{shp}\n\\omega ^{(0)} &=& -i\n\\frac{{\\Gamma}^{(0)}}{2}\\;, \\qquad {\\Gamma}^{(0)} = s_1^{(0)}v_{{}_{\\! {\\rm F}}} q \\qquad\\nonumber\\\\\n&=&\n\\frac{2 \\tau_q v_{{}_{\\! {\\rm F}}} q}{3}\n\\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right]\\qquad\n\\end{eqnarray}\nand\n\\bel{sfirst}\n\\omega _{\\pm}^{(1)} =\n            \\pm \\omega _0^{(1)} -i \\frac{{\\Gamma}^{(1)}}{2}\n\\quad {\\rm with}\\quad\n            \\omega _0^{(1)}=s_0^{(1)} v_{{}_{\\! {\\rm F}}} q ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{gambarone}\n{\\Gamma}^{(1)} &=&\n     s_1v_{{}_{\\! {\\rm F}}} q =\n     \\frac{4}{15} \\tau_q v_{{}_{\\! {\\rm F}}} q \\mathcal{G}_1\n    \\left[1 \\!+\\! \\frac{5 \\pi^2 \\tbar^2}{6}\n    \\left(\\frac{1}{\\mathcal{G}_0 \\mathcal{G}_1}\\right)\\right] \\nonumber\\\\\n   &\\approx&  \\frac{4}{15} \\tau_q v_{{}_{\\! {\\rm F}}} q \\mathcal{G}_1\n    \\left[1 \\! +\\! \\frac{5 \\pi^2 \\tbar^2}{12}\n    \\left(1\\!+\\! \\frac{1}{\\mathcal{G}_0 \\mathcal{G}_1}\\right)\\right].\n\\end{eqnarray}\nThe first root $\\omega ^{(0)}$ given in (\\ref{shp}) is purely\nimaginary and corresponds to the overdamped excitations of the\nhydrodynamic Raleigh mode \\cite{kubo,forster}. The second\nand third ones $\\omega _{\\pm}^{(1)}$ correspond to the usual first\nsound mode, expressed in terms of the (macroscopic) parameters of\nviscosity and thermal conductivity of normal liquids\n \\cite{forster,brenig}.\nIn (\\ref{sfirst0}) and\n(\\ref{gambarone}), small corrections  of the order of the product\nof the two small quantities ${\\bar T}^2$ and $\\mathcal{F}_0$ have been\nneglected, along with ${\\bar T}^2 |(m^*-m)\/m|={\\bar T}^2|\\mathcal{F}_1|\/3$.\nThis procedure should be valid for nuclear matter; where\nthe relevant parameters are small, both $\\mid\\mathcal{F}_0\\mid$ and\n$|(m^*-m)\/m|$ being of order $ \\approx 0.2$. Discarding such\nsmall corrections, our results for the sound frequencies\n$\\omega _\\pm^{(1)}$ (\\ref{sfirst}) are in agreement with\n\\cite{heipethrev}. In particular, up to these small corrections,\nthe volume (or second) viscosity disappears, as it is the case in\n\\cite{heipethrev}. In the expressions (\\ref{shp}) to\n(\\ref{gambarone}) more explicit temperature corrections are given\nfor $\\omega ^{(0)}$ and $\\omega ^{(1)}$ than those discussed in\n\\cite{heipethrev}. This will turn out important for the thermal\nconductivity $\\kappa$, which we shall address in\nSec.\\ \\ref{viscosthermcond} [see (\\ref{kappaexp})]. The \"widths\"\n${\\Gamma}^{(0)}$ and ${\\Gamma}^{(1)}$ are proportional to $\\tau_q$, and thus, to\nthe relaxation time $\\tau$ which represents the effects of\ntwo-body collisions. For nuclear matter, the Landau parameters\n$\\mathcal{F}_0$ and $\\mathcal{F}_1$ are small [$\\mathcal{G}_0$ and\n$\\mathcal{G}_1$ are close to unity, see (\\ref{effmass})]. For this reason,\naccording to\nthe last equation in (\\ref{som}), the sound velocities cannot be\nlarge [see the approximation\n(\\ref{sfirst0})]. So, the LWL limit\n($\\tau_q \\ll 1$)\nmay be identified with the hydrodynamic collision regime\n$\\omega  \\tau =s \\tau_q \\siml \\tau_q \\ll 1$.\nNote that for the Fermi liquid $^3He$, for instance, the parameters\n$\\mathcal{F}_0$ and $\\mathcal{F}_1$ are large and second order equation of\n(\\ref{sfirst0}) can not be applied. Moreover, according to the\nfirst line in (\\ref{sfirst0}), the sound velocity is large.\nTherefore, in this case a smallness $\\tau_q$ does not mean yet that\n$\\omega  \\tau$ is also small, i.e., the LWL\ncondition is not\nenough for the hydrodynamical collision regime.\n\n\n\\bigskip\n\n\\subsubsection{RESPONSE FOR INDIVIDUAL MODES}\n\\label{respmodes}\n\n\nIn the following, we are going to examine the collective response\nfunction $\\chi_{DD}^{\\rm coll}$ (\\ref{ddresp}), in particular its\nbehavior in the neighborhood of the individual modes given by\n(\\ref{shp}) and (\\ref{sfirst}). To simplify the notation, we shall\nat times omit the lower index \"DD\" and move down the upper index\n\"coll\". Near any of the sound poles $\\omega _{\\pm}^{(1)}$ given in\n(\\ref{sfirst}), the collective response function $\\chi_{\\rm coll}$\n(\\ref{ddresp}) may be written as\n\\begin{eqnarray}\\label{chicollone}\n\\chi_{\\rm coll}^{(1)}\\left(q,\\omega \\right) &=&\n    a^{(1)} \\left( \\frac{1}{\\omega -\\omega _{-}^{(1)}} -\n    \\frac{1}{\\omega -\\omega _{+}^{(1)}} \\right)\\quad {\\rm with} \\nonumber\\\\\na^{(1)} &=&\n    \\frac{\n\\omega _0^{(1)} \\mathcal{N}(T)}{2 \\mathcal{G}_0\n\\left[1 + \\pi^2 \\tbar^2 \/(3 \\mathcal{G}_0) \\right]}.\n\\end{eqnarray}\n Here, we have made use of (\\ref{ampsexp}), (\\ref{dszexpzero}),\n(\\ref{couplconst}) as well as of (\\ref{somexp}).\nIt will turn out convenient to present separately the dissipative\nand reactive parts, $\\chi_{\\rm coll}^{(1)\\;\\prime\\prime}$ and\n$\\chi_{\\rm coll}^{(1)\\;\\prime}$, respectively,\n\\begin{eqnarray}\\label{chiconeqompp}\n\\chi_{\\rm coll}^{(1)\\;\\prime\\prime}(q,\\omega ) &=&\n    \\frac{1}{2} a^{(1)} \\left[\n\\frac{\\Gamma^{(1)}}{\\left(\\omega -\\omega _0^{(1)} \\right)^2 +\n    \\left(\\Gamma^{(1)}\\right)^2 \/ 4 } \\; \\right.\\nonumber\\\\\n   &-&\\left. \\frac{\\Gamma^{(1)}}{\\left(\\omega  + \\omega _0^{(1)} \\right)^2 +\n    \\left(\\Gamma^{(1)}\\right)^2 \/ 4} \\right]\n\\end{eqnarray}\n and\n\\begin{eqnarray}\\label{chiconeqomp}\n\\chi_{\\rm coll}^{(1)\\;\\prime}(q,\\omega )&=&\n    a^{(1)} \\left[\\frac{\n\\omega  + \\omega _0^{(1)}}{\\left(\\omega  + \\omega _0^{(1)}\\right)^2 +\n    \\left(\\Gamma^{(1)}\\right)^2 \/ 4} \\right.\\nonumber\\\\\n   &-&\\left.\\frac{\\omega -\\omega _0^{(1)}}{\\left(\\omega -\\omega _0^{(1)}\\right)^2 +\n    \\left(\\Gamma^{(1)}\\right)^2 \/ 4}\\right].\n\\end{eqnarray}\n Notice that for $\\tau_q=+0$ the Lorentzians in (\\ref{chiconeqompp}) turn into\n$\\delta$-functions.\n\nThe relaxation time $\\tau$, which determines the dimensionless\nquantity $\\tau_q =\\tau \\, v_{{}_{\\! {\\rm F}}} q$, might depend on temperature and\nfrequency. A useful form is found in\n\\bel{tautom} \\tau = \\frac{\n\\tau_o}{T^2 + c_o \\left(\\hbar \\omega  \\right)^2} \\quad\n    \\approx \\quad \\frac{\\tau_o}{T^2} \\quad {\\rm for} \\quad\nc_o(\\hbar \\omega )^2 \\ll T^2,\n\\end{equation}\n with some parameters $\\tau_o$ and\n$c_o$ independent of $T$ and $\\omega $; see, e.g., \\cite{magkohofsh}. As\nindicated on the very right, for our present purpose we may\nneglect the frequency dependence, simply because we are interested\nin describing low frequency modes at larger temperatures (with\nrespect to $\\hbar \\omega $). Indeed,\nit is such a condition which\nhelps justifying the assumption of local equilibrium. We shall\nreturn to this question later, when we are going to apply the Landau\ntheory to a finite Fermi-liquid drop. Substituting (\\ref{tautom})\ninto the damping coefficient ${\\Gamma}^{(1)}$ (\\ref{gambarone}),\none has\n \\bel{gambaronet} {\\Gamma}^{(1)} =\n     \\frac{4 \\tau_o v_{{}_{\\! {\\rm F}}}^2 q^2 \\mathcal{G}_1}{15 T^2}\n    \\left[1 +\n    \\frac{5 \\pi^2 \\tbar^2}{12}\\left(1+ \\frac{1}{\\mathcal{G}_0\n\\mathcal{G}_1}\\right)\\right].\n\\end{equation}\n To leading order, this gives the expected\ndependence on temperature commonly associated to hydrodynamics,\nnamely ${\\Gamma}^{(1)} \\propto 1\/T^2$.\n\nFinally, we may note that in the long-wave limit the effective,\ntotal in-compressibility $K_{\\rm tot}$ (\\ref{incomprtot}) becomes\nidentical to the adiabatic in-compressibility $K^{\\varsigma}$\n(\\ref{incompraddef}) specified in Appendix A,\n\\bel{Kadiabat}\nK^{\\varsigma}=\nK^{T}\\left(1 + \\frac{4T{\\tt C}_{\\mathcal{V}}}{K^{T}}\\right),\n\\end{equation}\nsee (\\ref{isotherk}) for the isothermal in-compressibility $K^{T}$\nand (\\ref{incompradexp}) at small temperatures.\nFor the derivation\nof this identity, it is more easy to\nconsider variations $\\delta \\tilde{T}$ and $\\delta \\tilde{\\rho}$\nas caused formally by some external field $V_{\\rm ext}$. Then,\none can represent $\\delta {\\tilde T}\/\\delta {\\tilde \\rho}$ in\n(\\ref{incomprtot}) in terms of the ratio of the\ntemperature-density $\\chi_{{}_{\\! TD}}(\\tau_q,s)$ (\\ref{chitbar}) to\ndensity-density $\\chi_{{}_{\\! DD}}(\\tau_q,s)$ (\\ref{chiindef}),\n\\bel{dtdrhoexp0}\n\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}\n\\equiv \\frac{\\delta {\\tilde T}\/V_{\\rm ext}}{\\delta {\\tilde\n\\rho}\/V_{\\rm ext}}= \\frac{\n\\chi_{{}_{\\! TD}}(\\tau_q,s)}{\\chi_{{}_{\\! DD}}(\\tau_q,s)}\\;.\n\\end{equation}\n Using then the LWL\nexpansions\n(\\ref{chiexpone}) and (\\ref{ampsexp}) up to the third order\nterms in $\\tau_q$, from (\\ref{dtdrhoexp0}) we get\n\\bel{dtdrhoexp}\n\\frac{\\delta {\\tilde T}}{\\delta {\\tilde \\rho}}\n\\approx \\frac{{\\bar T}}{\\mathcal{N}(0)}\\;\\left(1-\n\\frac{i \\tau_q}{3\ns_0^{(1)}}\\right) \\quad {\\rm for}\\quad \\tau_q \\rightarrow 0,\n\\end{equation}\n where $s_0^{(1)}$ was defined in (\\ref{sfirst0}). As the\nspecific heat ${\\tt C}_{\\mathcal{V}}$ (\\ref{specifheatv}) is\nproportional to ${\\bar T}$, we need in (\\ref{dtdrhoexp}) only\nlinear terms to get the temperature correction of the second\norder in ${\\bar T}$ in the total in-compressibility $K_{\\rm tot}$\n(\\ref{incomprtot}). Substituting (\\ref{dtdrhoexp}),\n(\\ref{specifheatv}) and (\\ref{isotherkexp}) into\n(\\ref{incomprtot}) for the total in-compressibility $K_{\\rm tot}$,\none obtains identically the same as in (\\ref{incompradexp}) for\nthe adiabatic in-compressibility $K^\\varsigma$. The same result\n(\\ref{dtdrhoexp}) in the LWL limit can be obtained\nalso from (\\ref{dtdrho}).\n\n\nLet us address now the pole at $\\om^{(0)}$ [see (\\ref{shp})]. Near\nthe latter, the collective response function $\\chi_{\\rm coll}$\n(\\ref{ddresp}) becomes [as may be checked with the help of\n(\\ref{ampsexp}), (\\ref{dszexpzero}), (\\ref{couplconst}) and\n(\\ref{somexp})]\n\\begin{eqnarray}\\label{chicollhp} \\chi_{\\rm coll} ^{(0)}\n&=&\n    \\frac{i a^{(0)}}{\\omega -\\omega ^{(0)}}=\n    \\frac{i a^{(0)}}{\\omega  + \\frac{i {\\Gamma}^{(0)}}{2}}\n\\qquad {\\rm with} \\nonumber\\\\\n a^{(0)} &=&\n    \\frac{\n\\pi^4 {\\bar T}^2 \\tau_q (8-3 \\mathcal{G}_0)}{108\n\\mathcal{G}_0^2}~v_{{}_{\\! {\\rm F}}} q ~\\mathcal{N}(0),\n\\end{eqnarray}\nand  ${\\Gamma}^{(0)}$ being defined in (\\ref{shp}). It may\nbe rewritten in a more traditional form, see \\cite{kubo},\n\\cite{forster} and \\cite{brenig}. Introducing the \"diffusion\ncoefficient\"\n\\bel{diffusd}\n{\\tt D}_T = \\frac{\\kappa}{{\\tt C}_{\\cal P} \\rho}\n    =\\frac{\\tau v_{{}_{\\! {\\rm F}}}^2}{3}\n\\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right]\n\\end{equation}\n$[{\\tt D}_T =\n\\Gamma^{(0)}\/(2 q^2)]$,  one gets\n\\bel{chichpqom} \\chi_{\\rm coll}^{(0)} =\n\\frac{i {\\tt D}_T q^2}{\\omega  + i {\\tt D}_T q^2} ~\\chi_{\\rm coll}^{(0)} (q, \\omega  = 0).\n\\end{equation}\nNote that according to (\\ref{shp}) and (\\ref{tautom}), the\ntemperature dependence of $\\Gamma^{(0)}$ becomes similar to the\none found in (\\ref{gambaronet}),\n\\bel{gambarhpt} {\\Gamma}^{(0)}\n= \\frac{2 \\tau_o v_{\\rm F}^2 q^2}{3 T^2} \\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29\n\\mathcal{G}_0\\right)}{120 \\mathcal{G}_0}\\right].\n\\end{equation}\n For the dissipative and reactive parts of the response function\n$\\chi_{\\rm coll}^{(0)}$ (\\ref{chichpqom}), from (\\ref{chichpqom}) one gets\n\\begin{eqnarray}\\label{chichpqompp}\n\\chi_{\\rm coll}^{(0)\\;\\prime\\prime}(q,\\omega )&=&\n     a^{(0)} \\frac{\\omega }{\\omega ^2 + \\left(\\Gamma^{(0)}\\right)^2 \/ 4},\\nonumber\\\\\n\\chi_{\\rm coll}^{(0)\\;\\prime}(q,\\omega )&=&\n     a^{(0)} \\frac{\n\\Gamma^{(0)}\/2}{\\omega ^2 + \\left(\\Gamma^{(0)} \\right)^2 \/ 4}.\n\\end{eqnarray}\n The strength distribution $\\chi_{\\rm coll}^{(0)\\;\\prime\\prime}$\nhas a maximum at $\\omega =\\Gamma^{(0)}\/2$ and a width $\\Gamma^{(0)}\/2\n\\propto \\tau_q$. In the LWL\nlimit $\\tau_q \\ll 1$\nthis distribution becomes quite sharp with the maximum lying close\nto $\\omega =0$. As may be inferred with the help of (\\ref{chicollhp})\nand (\\ref{shp}), the maximal value does not depend on $\\tau_q$\nand is proportional to ${\\bar T}^2$.  It will be demonstrated\nshortly that the pole at $\\om^{(0)}$ (\\ref{shp}) is related to the heat\nconduction, for which reason it sometimes is called \"heat pole\".\nNotice that the reactive response function\n$\\chi_{\\rm coll}^{(0) \\; \\prime}$ is finite at $\\omega =0$, with a value\nindependent of $\\tau_q$.\n\n\nIn the hydrodynamic regime with $\\tau_q \\ll 1$, the response\nfunction $\\chi_{\\rm coll}$ found for the {\\it Fermi liquid} becomes\nidentical to the one for {\\it normal liquids}\n\\cite{forster,brenig}. This can be made more apparent after\nintroducing the dimensional sound velocity $c$, a width parameter\n$\\Gamma$, determined as\n\\bel{liqparam} c=v_{{}_{\\! {\\rm F}}} s_0^{(1)}\\;, \\qquad\\qquad\n\\Gamma=\\Gamma^{(1)}\/q^2 \\;,\n\\end{equation}\n  as well as the diffusion\ncoefficient ${\\tt D}$ (\\ref{diffusd}) and the specific heats. The\nsum of the two contributions discussed above may then be written\nas\n\\begin{eqnarray}\\label{forstereq} \\chi_{\\rm coll}^{\\prime\\prime} &=&\n    \\rho \\left(\\frac{\\partial \\rho}{\\partial {\\cal P}}\\right)_T\n    \\left[\\frac{\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\n    c^2 q^4 \\Gamma\\; \\omega }{\\left(\\omega ^2-c^2q^2\\right)^2\n+ \\left(\\omega  q^2\n\\Gamma\\right)^2} \\right.\\nonumber\\\\\n    &+& \\left.\\frac{\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T\\; \\omega }{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2}\\right].\n\\end{eqnarray}\n Traditionally, the peaks related to the first and second terms\nare called Brillouin and Rayleigh (or Landau--Placzek) peak,\nrespectively. The ratio of the specific heats ${\\tt C}_{\\cal P}$ and\n${\\tt C}_{\\mathcal{V}}$ per particle is discussed in Appendix A.1,\nsee (\\ref{cpcvkakt}) and (\\ref{cpcvexp}). Note\nthat the sound speed $s_0^{(1)}$, see (\\ref{sfirst0}), is identical\nto the adiabatic sound velocity found in Appendix A.1,\nsee\n(\\ref{velocad}) ($c$ in dimensional units for normal liquids), as\nit should be for normal liquids\n\\cite{brenig,forster}.\nThe structure of (\\ref{forstereq}) is identical to that discussed\nin the literature (see, e.g., (4.44a) of \\cite{forster}), if one\nonly expresses the quantities appearing here in terms of viscosity\nand thermal conductivity. As a matter of fact, the alert reader\nmight expect a third term (as in (4.44a) of \\cite{forster}), but\nthis one is of the order of $\\tau_q^2$ and thus is neglected here.\nThe specific temperature dependence of these parameters (in the\nLWL limit) will be discussed in the next subsection,\nwith respect to the specific heats, see also Appendix A.1.\n\n\nNote that in the derivation of the both amplitudes $a^{(0)}$\n(\\ref{chicollhp}) and $a^{(1)}$ (\\ref{chicollone}) we took\n$D(s)$ (\\ref{despfunc}) at low temperatures using\n(\\ref{chitemzero}) to (\\ref{chitemtwo}); and then, expand first\nit near the poles (\\ref{shp}) and (\\ref{sfirst0}), respectively;\nand second, in small $\\tau_q$ of the LWL\nlimit. This way\nof the calculation is much more simpler because the two last\noperations can be exchanged only when we shall take into account\nnext order terms in $\\tau_q$, that takes much hard work. If we\nexchange the last two operations, expanding first in $\\tau_q$ in\nthe {\\it linear} LWL\napproximation (\\ref{somexp}), and then,\ndoing expansion near the poles, some important terms will be lost.\n\n\n\\subsubsection{SHEAR MODULUS, VISCOSITY AND THERMAL CONDUCTIVITY}\n\\label{viscosthermcond}\n\nAs explained in Appendix A.2,\nthese coefficients may be\nobtained by applying expansions to $\\chi _n$ within the\nperturbation theory mentioned above, for low temperatures (with\n${\\bar T} \\ll 1$); see  in particular (\\ref{chiexpzero}), (\\ref{chiexpone}) and\n(\\ref{chiexptwo}).\nThey specify the stress tensor $\\sigma_{\\alpha\\beta}$\n(\\ref{prestensone})-(\\ref{pressnu}) and the heat current ${\\bf\nj}_T$ (\\ref{currheat}).\n\nThe shear\nmodulus $\\lambda $ (\\ref{shearmod}) in the time-reversible part $\n\\sigma_{\\alpha\\beta}^{(\\lambda)}$ (\\ref{presslamb}) of the\nstress tensor $ \\sigma_{\\alpha\\beta}$ (\\ref{prestensone})\nturns into zero in the long wave-length approximation linear in\n$\\tau_q$ as in \\cite{heipethrev}\nup to immaterial corrections of the order of ${\\bar T}^4$. By\nanother words, in this case, $\\lambda $ is a small quantity of\nthe order of $\\tau_q^2$ because such corrections were neglected\neverywhere. It means a disappearance of the Fermi-surface\ndistortions in our linear approach (\\ref{somexp}) which are the\nmain peculiarity of Fermi liquids compared to the normal ones.\n\n\nFor the shear viscosity $\\nu $\n(\\ref{viscos}) taken at the first sound frequency\n$\\omega =\\omega _0^{(1)}$ (\\ref{sfirst}), one obtains\n\\bel{shearvisonehp} \\nu = \\nu^{(1)}+\\nu^{(2)}\\;,\n\\end{equation}\n where\n\\bel{shearvisone} \\nu^{(1)} = \\frac{2}{5} \\rho \\varepsilon_{{}_{\\! {\\rm F}}} \\tau\n    \\left(1 + \\frac{5 \\pi^2 \\tbar^2}{12} \\right)~\n\\end{equation}\n and\n\\bel{shearvishp} \\nu^{(2)}=\\frac{13 \\pi^4}{720} \\frac{\\rho\n\\varepsilon_{{}_{\\! {\\rm F}}} {\\bar T}^4 }{v_{{}_{\\! {\\rm F}}}^2 q^2 \\tau}.\n\\end{equation}\n The first term\n$\\nu^{(1)}$ (\\ref{shearvisone}) in (\\ref{shearvisonehp}) is proportional to the\nrelaxation time $\\tau$ and coincides mainly with that obtained\nearlier for mono-atomic gases and\nfor a Fermi liquid by using another method\n\\cite{heipethrev},\nexcept for the specific explicit dependence on temperature\npresented here. The temperature dependence of the shear viscosity\n$\\nu^{(1)}$ (\\ref{shearvisone}) is mainly the same as for the\nrate of the sound damping $\\Gamma ^{(1)}$ (\\ref{gambaronet}),\n$\\nu^{(1)} \\propto {1\/T^2}$, with the temperature dependence of the\nrelaxation time $\\tau$ (\\ref{tautom}). Although the viscosity component\n $\\nu^{(2)}$,\ntoo, is related to the first sound solution $\\omega _0^{(1)}$, it is\nproportional to $1\/\\tau$, similar to the viscosity of zero sound\nbut in contrast to the standard first sound viscosity\n(\\ref{shearvisone}). The $\\nu^{(2)}$ component (\\ref{shearvishp})\nof the viscosity (\\ref{shearvisonehp}) increases with temperature\nas $T^6$, see also (\\ref{tautom}) for the relaxation time $\\tau$.\nAlthough the second component $\\nu^{(2)}$ of the shear viscosity\nis proportional to ${\\bar T}^4$, and thus may be considered small\nunder usual conditions, it may become important for small wave\nnumbers $q$ (or frequencies $\\omega $) [for more details, see the\ndiscussion to come below in Sec.\\ \\ref{heatcorrfun}].\nThis component of the viscosity was not discussed\nin \\cite{heipethrev}.\n\n\nLet us finally turn to the thermal conductivity $\\kappa$ which\nshows up in the equation for variations of temperature $T({\\bf r},t)$\nwith ${\\bf r}$ and $t$ (see Appendix A.2).\nThe form (\\ref{kappadef}) [for\nthe heat mode $\\omega =\\omega ^{(0)}$ of (\\ref{shp})] may be rewritten as\n\\bel{kappaexp} \\kappa =\n    \\rho \\frac{{\\tt C}_{\\cal P} \\Gamma^{(0)}}{2 q^2} \\approx\n   \\frac{1}{3} \\rho {\\tt C}_{\\cal P} v_{{}_{\\! {\\rm F}}}^2 \\tau\n   \\left[1 - \\frac{\\pi^2 \\tbar^2 \\left(80-29 \\mathcal{G}_0\\right)}{120\n\\mathcal{G}_0}\\right].\n\\end{equation}\n We present here also explicitly the temperature\ncorrections up to the terms of the order of ${\\bar T}^2$. Our\nexpression for the thermal conductivity $\\kappa$ (\\ref{kappaexp})\ndiffers from the one found in \\cite{forster} and\n\\cite{heipethrev} by small ${\\bar T}^2$ corrections. However, they are not\nimportant in the calculations of the damping coefficient\n$\\Gamma^{(1)}$ for the first sound mode defined in\n\\cite{forster}, and \\cite{brenig},\nsee also the comment before (\\ref{forstereq}),\n\\bel{gammaland}\n\\Gamma ^{(1)}=\n    \\frac{q^2}{m \\rho }\n    \\left[\n    \\frac{4}{3} \\nu^{(1)}\n    + \\frac{m \\kappa}{{\\tt C}_{\\cal P}}\n    \\left(\\frac{{\\tt C}_{\\cal P}}{{\\tt C}_{\\mathcal{V}}} -1\\right)\n     \\right],\n\\end{equation}\n Here, $\\nu^{(1)}$ is the part of the shear viscosity\ncoefficient related to the first sound mode, see\n(\\ref{shearvisone}); ${\\tt C}_{\\cal P}\/ {\\tt C}_{\\mathcal{V}}$ is the\nadiabatic ratio of the specific heats, see\n(\\ref{specifheatpdef}) and (\\ref{cvcpkSkT}). We omitted here\ncorrections related to the second viscosity in line of the second\napproximation in (\\ref{gambarone}).  In\n(\\ref{gammaland}),  $\\kappa $ is multiplied by a small quantity of the\norder of the ${\\bar T}^2$ as follows from (\\ref{cpcvexp}) and the\ntemperature corrections to $\\kappa $ written explicitly in\n(\\ref{kappaexp}) can be neglected in (\\ref{gammaland}) . The\nexpression for the damping coefficient $\\Gamma^{(1)}$\n(\\ref{gammaland}) with the viscosity coefficient $\\nu^{(1)}$\n(\\ref{shearvisone}), thermal conductivity $\\kappa$\n(\\ref{kappaexp}) and specific heats from (\\ref{cpcvexp}) and\n(\\ref{specifheatp}) for viscose normal liquids is in agreement\nwith our result for $\\Gamma^{(1)}$ (\\ref{gambarone}) including the\ntemperature corrections.\n\n\nThus, up to\nthe temperature corrections discussed above, we have agreement\nwith the results of\n\\cite{heipethrev} for the dispersion equation, viscosity and\nthermal conductivity coefficients in the hydrodynamic limit. Our\nderivations are more strict and direct within the perturbation\ntheory for the eigenvalue problem. We have the transition to the\nhydrodynamics of normal liquids discussed in\n\\cite{forster,brenig,kubo}\nin terms of\nthe  macroscopic parameters mentioned above.\n\n\\subsection{Susceptibilities}\n\\label{suscept}\n\nIn this section, we want to address the calculation of the static\nsusceptibilities, for which one distinguishes isolated, isothermal\nand adiabatic ones \\cite{kubo,forster,brenig}. Their comparison is\nrelevant for ergodicity properties, see \\cite{kubo,brenig}. Here\nwe will concentrate on the density mode of nuclear matter\nconsidered as an infinite Fermi-liquid system.\n\n\\subsubsection{ADIABATIC AND ISOTHERMAL SUSCEPTIBILITIES}\n\\label{susceptadt}\n\nThe isolated susceptibility $\\chi_{{}_{\\! DD}}(0)$ is defined as the\nstatic limit of the response function $\\chi_{{}_{\\! DD}}(q,\\omega )$ (or\n$\\chi_{{}_{\\! DD}}(\\tau_q,s)$ of (\\ref{chiindef}) in dimensionless\nvariables), for which one first has to take the limit $q \\to 0$\n(or $\\tau_q \\to 0$), and then, $\\omega  \\to 0$ (or $s \\to 0$) (see, e.g.,\n\\cite{forster})\n\\bel{statresponse} \\chi_{{}_{\\! DD}}(0)=\n \\lim_{\\omega  \\rightarrow 0}\n    \\left[\\lim_{q \\rightarrow 0}\\;\\chi_{{}_{\\! DD}} \\left(q,\\omega \\right)\\right]=\n \\lim_{s \\rightarrow 0}\n    \\left[\\lim_{\\tau_q \\rightarrow 0}\\;\\chi_{DD} \\left(\\tau_q,s\\right)\\right].\n\\end{equation}\n Apparently, $\\chi_{{}_{\\! DD}}(0)$ satisfies the relation\n\\bel{vardenschi0}\n\\delta \\rho \\equiv -\\chi_{{}_{\\! DD}}(0) \\delta V_{\\rm eff},\n\\end{equation}\n where $\\delta V_{\\rm eff}$ and $\\delta \\rho$ are quasistatic\nvariations. They can be considered as independent of time, in\ncontrast to the ones discussed in Sec.\\ \\ref{dynresp}, see\n(\\ref{delueff}).\n\n\nThe isothermal susceptibility $\\chi_{DD}^{T}$ is defined as the\ndensity-density response at constant temperature $T$, and the\nadiabatic one, $\\chi_{DD}^{\\varsigma}$, as that at constant\nentropy (per particle $\\varsigma$). Suitable variables for\nstudying the variations of the density $\\rho$ are therefore\npressure ${\\cal P}$ and temperature $T$ in the first case, and\npressure ${\\cal P}$ and entropy per particle $\\varsigma$ in the\nsecond one. These two representations of $\\delta \\rho$ can be\nwritten as\n\\bel{vardens} \\delta \\rho \\equiv\n    \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_T \\delta {\\cal P}\n    + \\left(\\frac{\\partial \\rho }{\\partial T} \\right)_{\\cal P} \\delta T \\equiv\n    \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_\\varsigma\n    \\delta {\\cal P}\n    + \\left(\\frac{\\partial \\rho }{\\partial \\varsigma} \\right)_{\\cal P}\n    \\delta \\varsigma.\n\\end{equation}\n For the isothermal and adiabatic susceptibilities\n$\\chi_{DD}^{T}$ and  $\\chi_{DD}^{\\varsigma}$, one thus gets the\nfollowing two relations:\n\\bel{vardenschiT} \\delta \\rho \\equiv\n    \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_T \\delta {\\cal P}\n    =-\\chi_{DD}^T \\delta V_{\\rm eff}\n\\end{equation}\n and\n\\bel{vardenschiS} \\delta \\rho \\equiv\n    \\left(\\frac{\\partial \\rho }{\\partial {\\cal P}} \\right)_\\varsigma\n    \\delta {\\cal P}\n    =-\\chi_{DD}^{\\varsigma} \\delta V_{\\rm eff}.\n\\end{equation}\n The variations of the density with pressure are related to the\n(in-)compressibilities, see (\\ref{incompraddef}) and\n(\\ref{incomprTdef}). As shown in Appendix A.1,\ntheir ratio\ncan be expressed by that of the corresponding specific heats, see\n(\\ref{cvcpkSkT}). Building the ratio, one therefore gets\nfrom (\\ref{vardenschiT}) and (\\ref{vardenschiS})\n\\bel{chitchiskSkT}\n\\frac{\\chi_{DD}^{T}}{\\chi_{DD}^{\\varsigma}}=\n\\frac{K^{\\varsigma}}{K^{T}}=\n\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}.\n\\end{equation}\n This is a\ngeneral relation from thermodynamics where we only have replaced\nthe system's total entropy \\cite{brenig}\nby the entropy per particle $\\varsigma$ applied\nfor the intensive systems as normal and Fermi liquids.\n\n\nWe are interested more in the calculation of the differences\nbetween the isothermal susceptibility $\\chi_{DD}^{T}$ defined by\nthe relations in (\\ref{vardenschiT}) (or adiabatic one\n$\\chi_{DD}^{\\varsigma}$, see (\\ref{vardenschiS})) and isolated\n(static) susceptibility $\\chi_{{}_{\\! DD}}(0)$ presented by\n(\\ref{vardenschi0}) \\cite{hofmann}. For this purpose, we\nfind first the ratio of the isothermal-to-isolated\nsusceptibilities $\\chi_{DD}^{T}\/\\chi_{{}_{\\! DD}}(0)$ in terms of the\nratio of the static \"intrinsic\" temperature-density response\nfunction to the isolated one $\\chi_{{}_{\\! DD}}(0)$ (\\ref{chiindef}). The\nstatic temperature-density susceptibility $\\chi_{{}_{\\! TD}}(0)$ is\ndefined in the same way (\\ref{statresponse}) as the static limit\nof the \"intrinsic\" temperature-density response function\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ given by (\\ref{chitindef}). Note that\nthe limits $\\omega  \\rightarrow 0$ (or $s \\rightarrow 0$) and $q\n\\rightarrow 0$ (or $\\tau_q \\rightarrow 0$)  which we consider to\nget the static response functions are not commutative \\cite{forster}.\nTaking the second equations in (\\ref{vardenschiT})  and\n(\\ref{vardenschi0}) for the intensive systems as liquids, one\ngets\n\\bel{chitchi0}\n\\frac{\\chi_{DD}^{T}}{\\chi_{{}_{\\! DD}}(0)}= 1 +\n\\left[\\varsigma \\left(\n\\frac{\\partial \\rho }{\\partial \\mu}\\right)_T\n- \\left(\\frac{\\partial \\rho }{\\partial T}\\right)_\\mu\\right]\n\\frac{\\chi_{{}_{\\! TD}}(0)}{\\chi_{{}_{\\! DD}}(0)}.\n\\end{equation}\n We used here\nthe definitions (\\ref{chiindef}) and (\\ref{chitindef}) for the\ndensity-density and temperature-density response functions and\n(\\ref{statresponse}) for their static limits $\\chi_{{}_{\\! DD}}(0)$\nand $\\chi_{{}_{\\! TD}}(0)$. We then applied the thermodynamic relations of\nAppendix A.1  for the transformations\nof the derivative\n$\\left(\\partial \\rho \/ \\partial {\\cal P}\\right)_T$. This derivative\n appears from the\ndefinition of the isothermal susceptibility $\\chi_{DD}^{T}$ in\n(\\ref{vardenschiT}) to another simpler thermodynamic\nderivatives for the application to Fermi liquids, see below.\nFor this aim, we transform the variables $(T,{\\cal P})$ to the new\nones $(T,\\mu)$. The derivatives of pressure ${\\cal P}$ over\nthese two new variables can be then reduced to the ones of the\ndensity $\\rho$ shown in the r.h.s. of (\\ref{chitchi0})\nwith the help of (\\ref{gibbsduh}).\n\nSo, the calculations of the susceptibilities are resulted\nin the\nderivation of the static limits defined by\n(\\ref{statresponse}) for the temperature-density\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ and density-density $\\chi_{{}_{\\! DD}}(\\tau_q,s)$\nresponse functions, see (\\ref{chiindef}) and\n(\\ref{chitindef}), and their ratio $\\chi_{{}_{\\! TD}}(0)\/\\chi_{{}_{\\! DD}}(0)$ for\nthe case of a heated Fermi liquid. We can then calculate the two\nratios of the susceptibilities (\\ref{chitchi0}) and\n(\\ref{chitchiskSkT}) which both determine separately each\nconsidered susceptibilities.\n\n\\subsubsection{FERMI-LIQUID SUSCEPTIBILITIES}\n\\label{flsuscept}\n\nThe expression for the ratio of the isothermal-to-static\nsusceptibilities (\\ref{chitchi0}) can be simplified my making use\nof the specific properties of Fermi liquids given by\n(\\ref{drhomt}) and second equation in (\\ref{dpdtr}),\n\\bel{chitchi0fl}\n\\frac{\\chi_{DD}^{T} }{\\chi_{{}_{\\! DD}}(0)} = 1\n+\\left(\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}-1\\right)\n\\frac{\\chi_{{}_{\\! TD}}(0) }{{\\bar T} \\chi_{{}_{\\! DD}}(0)}\\;\\mathcal{N}(0).\n\\end{equation}\nAccording to the definition (\\ref{statresponse}) of the static\nresponse functions applied to the ones in the ratio\n$\\chi_{{}_{\\! TD}}(0)\/\\chi_{{}_{\\! DD}}(0)$ of (\\ref{chitchi0fl}), we shall\nuse (\\ref{chitbar}) and (\\ref{chiindef}) for the corresponding\nintrinsic susceptibilities ($\\mathcal{F}_{0}=\\mathcal{F}_{1}=0$ there).\nThe static limit (\\ref{statresponse}) of the\nresponse functions $\\chi_{{}_{\\! DD}}(\\tau_q,s)$ (\\ref{chiindef}) and\n$\\chi_{{}_{\\! TD}}(\\tau_q,s)$ (\\ref{chitbar}) in (\\ref{chitchi0fl})\ncan be found by using the LWL\nexpansions over a\nsmall parameter $\\tau_q \\ll 1$ at low temperatures, see\nSec.\\ \\ref{lowtemlim} and Appendix A.2 for the\nfirst limit ($\\tau_q \\rightarrow 0$)\nin (\\ref{statresponse}). We substitute now the\nperturbation theory expansions for small $\\tau_q$ for the\nquantities $s$ (\\ref{somexp}), $\\chi_1$\n(\\ref{chiexpone}),\n$\\aleph$ (\\ref{ampsexp}), and $D_0$ (\\ref{dszexpzero}) into\n(\\ref{chiindef}) and (\\ref{chitbar}). We get this limit as\nfunctions of $s_0$ and $s_1$, and then, we shall take the second limit of\n$s_{0} \\to 0$ and $s_{1} \\to 0$ [ $s \\rightarrow 0$ in (\\ref{statresponse})].\nFinally, we\narrive then at the very simple result\n \\bel{chitchid}\n\\frac{\\chi_{{}_{\\! TD}}(0) }{\\chi_{{}_{\\! DD}}(0)}=\n\\frac{{\\bar T} }{\\mathcal{N}(0)}\n\\end{equation}\nneglecting small cubic terms in ${\\bar T}$, which correspond to ${\\bar T}^4$\ncorrections in susceptibilities and do not matter in this section.\nNote that the sequence of the limit transitions defined in\n(\\ref{statresponse}) and recommended in \\cite{forster} is\nimportant for the calculation of this ratio: We get zero for this\nratio if we take first $s \\rightarrow 0$, and then, $\\tau_q\n\\rightarrow 0$.\n\nSubstituting now the ratio (\\ref{chitchid}) of the\nsusceptibilities into (\\ref{chitchi0fl}),\none obtains\n\\bel{chitchi0s}\n\\frac{\\chi_{DD}^{T} }{\\chi_{{}_{\\! DD}}(0)} =\n\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}\n = 1 + \\frac{\\pi^2 \\tbar^2 }{3 \\mathcal{G}_0},\n\\end{equation}\nsee also (\\ref{cpcvexp}) for the second equation. We compare\nnow this result with (\\ref{chitchiskSkT}) and get that our\nFermi-liquid system satisfies the ergodicity property:\n\\bel{ergodicity}\n\\chi_{DD}^{(\\varsigma)}=\\chi_{{}_{\\! DD}}(0) \\;.\n\\end{equation}\n This ergodicity property was proved at low temperatures, for which the\nLandau Fermi-liquid theory  can be applied. It is related to\nthe adiabaticity of the velocity of the sound mode $s_0^{(1)}$,\nsee (\\ref{sfirst0}) and discussion\nafter (\\ref{forstereq}). Moreover, we got the normal liquid\n(hydrodynamic) limit from the Fermi-liquid dynamics, and  therefore, the\nergodicity property is general for  heated Fermi liquids and\nnormal (classical) ones.\n\nAnother aspect of the discussed ergodicity property might be the\nrelation to the non-degeneracy of the excitation spectrum in the\ninfinite Fermi liquids beside of the spin degeneracy. We have only\nthe two-fold degenerate quasiparticle states, due to the spin\ndegeneracy. However, it does not influence on our results concerning the\nergodicity relations because we consider the density-density\nexcitations, which do not disturb  the spin degree of freedom. We\nhave only the multiplication factor two in all susceptibilities,\ndue to the spin degeneracy, that does not change the ratios of the\nsusceptibilities which are only important for the ergodicity\ndiscussed here.\n\nOur susceptibilities obtained above satisfy the Kubo relations,\nsee (4.2.32) of \\cite{kubo}:\n\\bel{kuborel} \\chi ^{T} \\geq \\chi\n^{\\varsigma} \\geq \\chi(0)\n\\end{equation}\n with the equal sign for the second\nrelation because of the ergodicity property. To realize this, we\nshould take into account that\n${\\tt C}_{\\cal P} > {\\tt C}_{\\mathcal{V}}$ (or $K^{\\varsigma}>K^{T}$),\naccording to (\\ref{cpcvkakt}),\nbecause all quantities on the r.h.s. of this equation are positive\nfor the stable modes $\\mathcal{G}_0=1+\\mathcal{F}_0>0$. The equal sign\nfor the first relation in (\\ref{kuborel}) becomes true in the\ntwo limit cases: For the temperature $T$ going to 0 or for the\nin-compressible matter when the interaction constant $\\mathcal{F}_0$\ntends to $\\infty$. In both limit cases we made obvious equality\n${\\tt C}_{\\cal P}={\\tt C}_{\\mathcal{V}}$ and all susceptibilities are\nidentical [equal signs in the both relations of\n(\\ref{kuborel})].\n\nNote now that namely the specific Fermi-liquid expression of the\nstatic susceptibility $\\chi_{DD}(0)$, see (\\ref{chiindef}) with\n$\\mathcal{F}_0=\\mathcal{F}_1=0$ for the case of the intrinsic response\nfunctions, depends on the sequence of the limit transitions\ndiscussed near\n(\\ref{statresponse}), (\\ref{chitchi0}), (\\ref{chitchid}) above\nand in \\cite{forster}.\nFor the definition\n(\\ref{statresponse}) of \\cite{forster}, one gets\n\\bel{statrespfl}\n\\chi_{{}_{\\! DD}}(0)=\\left(1 - \\frac{5 \\pi^2 \\tbar^2 }{12}\\right)\n\\mathcal{N}(0),\\quad \\chi_{DD}^{T}=\\mathcal{N}(T).\n\\end{equation}\n In the last\nequation, we used also (\\ref{chitchi0s}). Taking the opposite\nsequence of the limit transitions, first $s \\rightarrow 0$, one\nhas the result $\\mathcal{N}(T)$ (\\ref{enerdensnt})\nfor the isolated susceptibility\n$\\chi_{{}_{\\! DD}}(0)$ like for the isothermal one $\\chi_{DD}^{T}$. The\ndifference is in ${\\bar T}^2$ corrections. Ignoring them, the both\nversions of the limit transitions coincide, and we come to the\nresult independent on temperature discussed in \\cite{pinenoz}~.\nThe ergodicity property (\\ref{ergodicity}), Kubo's relations\n(\\ref{kuborel}) and relation of the isothermal susceptibility to\nadiabatic one (\\ref{chitchiskSkT}) do not depend on the specific\npeculiarities of the static limit of the response function\ndiscussed here in connection to Fermi liquids.\n\n\n\\subsection{Relaxation and correlation functions}\n\\label{relaxcorr}\n\n\\medskip\n\n\\subsubsection{RELAXATION FUNCTION}\n\\label{relaxfun}\n\nComing back to the dynamical problem, we note that\nthe dissipative part of the response function\n$\\chi^{\\prime\\prime}(\\omega )$ is related to the relaxation function\n$\\Phi^{\\prime\\prime} (\\omega )$ \\cite{kubo} by\n\\bel{chiimpsi}\n\\chi\n^{\\prime\\prime} =\n    \\omega  \\Phi ^{\\prime\\prime} (\\omega  )\\;.\n\\end{equation}\nWe follow the notations of \\cite{hofmann,hofbook} and omit the\nindex \"coll\" in this section: For the comparison with the\nmicroscopic results of \\cite{hofmann} we need really the\nrelaxation and correlation functions related to the {\\it\nintrinsic} response functions. According to\n(\\ref{ddrespchiin}) and (\\ref{couplconst}), all these intrinsic\nfunctions can be formally obtained from the collective ones at the\nzero Landau constants $\\mathcal{F}_0$ and $\\mathcal{F}_1$. Taking into\naccount also (\\ref{chiconeqompp}) and (\\ref{chichpqompp}), one has\n\\begin{eqnarray}\\label{relaxcom}\n    \\Phi^{\\prime\\prime}(\\omega ) &=&\n   \\frac{a^{(1)} }{2 \\omega _0^{(1)}}\n    \\left[\\frac{\n\\Gamma^{(1)} }{\\left(\\omega -\\omega _0^{(1)} \\right)^2\n    + \\left(\\Gamma^{(1)}\\right)^2\/4} \\right. \\nonumber\\\\\n    &+&\\left.\\frac{\\Gamma^{(1)} }{\\left(\\omega  + \\omega _0^{(1)}\\right)^2\n    + \\left(\\Gamma^{(1)}\\right)^2 \/4}\\right] \\nonumber\\\\\n    &+&\n    \\frac{a^{(0)} }{{\\Gamma}^{(0)}}\\;\n    \\frac{\\Gamma^{(0)} }{\\omega ^2 + \\left(\\Gamma^{(0)} \\right)^2 \/ 4}.\n\\end{eqnarray}\n This equation can be re-written in the same way like to\n(\\ref{forstereq}) in terms of the parameters $c$, $\\Gamma$ and\n${\\tt D}_T$, see (\\ref{liqparam}) and (\\ref{diffusd}),\n\\begin{eqnarray}\\label{relaxbrenig}\n\\Phi^{\\prime\\prime}(\\omega ) &=&\n    \\chi^T\n    \\left[\\frac{\n\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right) {\\tt C}^2 q^4\n\\Gamma}{\\left(\\omega ^2-{\\tt C}^2q^2\\right)^2 +\\left(\\omega  q^2 \\Gamma\\right)^2}\n\\right.\\nonumber\\\\\n    &+&\\left. \\frac{\n\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T}{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2}\\right].\n\\end{eqnarray}\n We used here the Jacobian relations\nand (\\ref{incomprTdef}), (\\ref{isotherk})\nfor the transformation of the coefficient in front of the square\nbrackets in (\\ref{forstereq}) to the one, the {\\it intrinsic}\nisothermal susceptibility $\\chi^T$ (\\ref{statrespfl})\n($\\mathcal{F}_0=0$). We also neglected\nterms of the order of $\\tau_q^2$ as in the derivation of\n(\\ref{forstereq}). Equation (\\ref{relaxbrenig}) for the relaxation\nfunction $\\Phi^{\\prime\\prime}(\\omega )$ is identical to the imaginary\npart of the r.h.s. of (28.29) in \\cite{brenig}\nwith the transparent physical meaning as (\\ref{forstereq}). The\nfirst term in the square brackets of (\\ref{relaxcom}) and\n(\\ref{relaxbrenig}) is the\nfirst sound Brillouin component with the poles (\\ref{sfirst})\nassociated to the finite frequencies $\\pm\\omega _0^{(1)}$  of the\ntime-dependent relaxation-function oscillations and their damping\nrate $1\/\\Gamma^{(1)}$ ($\\pm\\omega _s$ and $1\/\\gamma_s$ in the notation\nof \\cite{brenig}, respectively, see more complete discussion of\nproperties of the time-dependent relaxation function as a Fourier\ntransform of the relaxation function $\\Phi(\\omega )$ in\n\\cite{brenig}). The second term in (\\ref{relaxcom}) and\n(\\ref{relaxbrenig}) describes the pure\ndamped Raleigh mode corresponding to the overdamped pole\n$\\omega ^{(0)}$ (\\ref{shp}) defined by the diffuseness coefficient\n${\\tt D}_T \\propto \\Gamma^{(0)}$  (or  $ \\propto\\gamma_{{}_{\\! T}}$ in the\nnotation of \\cite{brenig}). As noted in \\cite{brenig}, the\nstrength of this peak is a factor $1-{\\tt C}_{\\mathcal{V}}\/{\\tt\nC}_{\\cal P}$ smaller than for the two first sound peaks. According\nto (\\ref{cpcvexp}), in the zero temperature limit $T\n\\rightarrow 0$, the Raleigh peak disappears but the\nBrillouin ones become dominating because of $\\Gamma \\propto\n\\Gamma^{(1)} \\propto 1\/T^2$; see the second equation of\n(\\ref{liqparam}) for the relation of $\\Gamma$ to\n$\\Gamma^{(1)}$ and (\\ref{gambaronet}). Note also that the\ncoefficient in front of the square brackets in\n(\\ref{relaxcom}) is finite in the limit $T \\rightarrow 0$.\n\n\n\\subsubsection{CORRELATION FUNCTION}\n\\label{corfun}\n\nWe like to present also the correlation function, partly for the\nsake of completeness and partly to allow for comparisons with\ncalculations of the function in the nuclear SM approach of\n\\cite{hofivyam,ivhofpasyam}, see also\n\\cite{hofmann,hofbook}, to the collective motion of finite nuclei. Let us\nuse now the fluctuation-dissipation theorem \\cite{kubo} to get the\ncorrelation function $\\psi^{\\prime\\prime}(\\omega )$,\n\\bel{fludiptheor}\n\\psi^{\\prime\\prime}(\\omega ) \\rightarrow\n    \\hbar \\omega  \\coth \\left(\\frac{\\hbar \\omega  }{2T} \\right)\n    \\Phi ^{\\prime\\prime}(\\omega )=\n     \\hbar \\coth \\left(\\frac{\\hbar \\omega  }{2T}\\right)\n     \\chi ^{\\prime\\prime}(\\omega )\\;.\n\\end{equation}\n In the semiclassical limit $\\hbar \\rightarrow 0$ considered\nhere, one has\n\\bel{corrfun}\n\\psi^{\\prime\\prime}(\\omega ) =\n    ~\\frac{2 T }{\\omega }~ \\chi ^{\\prime\\prime}(\\omega )=~\n    2T~ \\Phi^{\\prime\\prime}\\left(\\omega \\right).\n\\end{equation}\n According to (\\ref{forstereq}),(\\ref{relaxbrenig}), this\ncorrelation function can be split into the two components as in\n\\cite{hofivyam,hofmann},\n\\bel{corrfunhof}\n\\psi^{\\prime\\prime}(\\omega ) =\n    \\psi_0^{\\prime\\prime}(\\omega ) + \\psi_R^{\\prime\\prime}(\\omega )\\;.\n\\end{equation}\n Here, $\\psi_0^{\\prime\\prime}$ is the heat pole part,\n\\begin{eqnarray}\\label{corrfunhp}\n\\psi_0^{\\prime\\prime}(\\omega ) &=&\n    \\frac{2 T }{\\omega } \\chi ^{(0)\\;\\prime\\prime}(\\omega ) \\nonumber\\\\\n  &=& 2T \\chi^T~\n\\frac{\\left(1-{\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\nq^2 {\\tt D}_T}{\\omega ^2 +\\left(q^2 {\\tt D}_T\\right)^2},\n\\end{eqnarray}\n $\\chi^{(0)\\;\\prime\\prime}$ is given by the first equation in\n(\\ref{chichpqompp}) and is related to the second heat pole\nterms in the square brackets of (\\ref{forstereq}) and\n(\\ref{relaxbrenig}) [through (\\ref{chiimpsi})]. This part is\nsingular at the zero frequency point $\\omega  = 0$ for $\\tau_q\n\\rightarrow 0$, see (\\ref{diffusd}) and (\\ref{som}). The\nother term $\\psi_R^{\\prime\\prime}$ in (\\ref{corrfunhof}) is\nassociated with the first sound component in the square brackets\nof (\\ref{forstereq}), (\\ref{relaxbrenig}),\n\\begin{eqnarray}\\label{corrfun1}\n\\psi_R^{\\prime\\prime}(\\omega ) &=&\n    ~\\frac{2 T }{\\omega }~ \\chi ^{(1)\\;\\prime\\prime}(\\omega ) \\nonumber\\\\\n   &=& ~2T~ \\chi^{T}~\\frac{\n\\left({\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\cal P}\\right)\n   c^2 q^4 \\Gamma}{\\left(\\omega ^2-c^2q^2\\right)^2 +\\left(\\omega  q^2 \\Gamma\\right)^2}.\n\\end{eqnarray}\n This component has no such singularity at $\\omega  = 0$ for $\\tau_q \\to 0$, as\nseen from (\\ref{liqparam}), (\\ref{sfirst}) and\n(\\ref{gambaronet}) [see (\\ref{chiconeqompp}) for\n$\\chi^{(1)\\;\\prime\\prime}(\\omega )$ in the middle of\n(\\ref{corrfun1})]. According to the second equation in\n(\\ref{corrfunhp}), the heat pole part\n$\\psi_0^{\\prime\\prime}(\\omega )$ of (\\ref{corrfunhof}) for the {\\it\nintrinsic} correlation function can be written as in\n\\cite{hofivyam,hofmann},\n\\bel{corrfunhphof}\n\\psi_0^{\\prime\\prime}(\\omega )= \\psi^{(0)}\n     \\frac{\\hbar {\\it \\Gamma}_T }{(\\hbar \\omega )^2 + {\\it \\Gamma}_T^2 \/ 4},\n\\end{equation}\n where\n\\bel{GammaT}\n{\\it \\Gamma}_T=\n2 \\hbar  q^2 {\\tt D}_T= \\hbar\n\\Gamma^{(0)},\n\\end{equation}\nand\n\\bel{psi0}\n(1 \/ T) \\psi^{(0)}=\\chi^T-\\chi^\\varsigma\n = \\chi^T-\\chi(0).\n\\end{equation}\n We applied here (\\ref{chitchiskSkT}) in the first equation\nof (\\ref{psi0}) and ergodicity condition (\\ref{ergodicity})\nfor the second one. The specific expressions for the quantities\n$\\Gamma^{(0)}$, $\\chi^T$ and $\\chi(0)$ in the last two equations\n(\\ref{GammaT}) and (\\ref{psi0}) can be found in (\\ref{shp}),\n(\\ref{gambarhpt}) and (\\ref{statrespfl}). Note that the\ncorrelation function (\\ref{corrfunhphof}), corresponding to the\nheat pole, has the Lorentzian multiplier. This multiplier  approaches the\n$\\delta(\\omega )$ function in the hydrodynamic limit $\\tau_q\n\\rightarrow 0$ because of $\\Gamma_T \\rightarrow 0$, according to\n(\\ref{GammaT}) and (\\ref{shp}) ($\\Gamma^{(0)} \\rightarrow\n0$), i.e.,\n \\bel{corrdeltafunlim}\n    \\psi^{(0)}(\\omega )\n     \\rightarrow 2 \\pi \\psi^{(0)} \\delta (\\omega )\\qquad\\mbox{for}\\qquad\n     \\tau_q \\rightarrow 0.\n\\end{equation}\n The relations (\\ref{corrfunhphof}), (\\ref{psi0}) and\n(\\ref{corrdeltafunlim}) confirm the discussion in\n\\cite{hofmann} concerning the heat pole contribution to the\ncorrelation function. The specific property of the Fermi liquid is\nthat this system is exactly ergodic, see (\\ref{ergodicity}), as\nused in the second equation of (\\ref{psi0}).\n\n\n\\section{Nuclear response within the Fermi-liquid droplet model}\n\\label{respfuntheor}\n\n\\subsection{Basic definitions}\n\\label{basdef}\n\nSo far we considered the Fermi-liquid theory  for study of the\ncollective excitations at finite temperatures much smaller than\nthe Fermi energy $\\varepsilon_{{}_{\\! {\\rm F}}}$ in the {\\it infinite} nuclear matter.\nThis theory can be also helpful for investigation of the\ncollective modes and transport properties of heavy heated nucleus\nconsidered as a {\\it finite} Fermi system within the macroscopic FLDM\n\\cite{strutmagbr,strutmagden0,galiqmodgen,galiqmod,magstrut,denisov,magboundcond,kolmagpl,magkohofsh,kolmagsh}.\nSuch a semiclassical nuclear model applied\nearlier successfully to the giant multipole resonance description\n\\cite{strutmagden0,galiqmod,strutmagden,denisov,magpl,kolmagpl,kolmagsh,magkohofsh} is expected\nto be also\nincorporated in practice as an asymptotic high temperature limit\nof the quantum transport theory \\cite{hofmann} based on the shell model.\nThis theory takes into account the residue\ninteractions like particle collisions for study of the low energy\nexcitations in nuclei. The latter application  of the FLDM is very\nimportant for understanding itself the dissipative processes\nlike nuclear fission at finite temperatures\n(see, g.e., \\cite{hofivyam,hofmann,hofbook,hofivmag,hofmag})\n\n\nFollowing \\cite{hofmann,hofbook}, let us describe the many-body\nexcitations of nuclei in terms of the response to an external\nperturbation\n\\bel{extfield}\nV_{\\rm ext}=q_{\\rm ext}(t)~{\\hat F},\n\\end{equation}\nwhere\n${\\hat F}$ is some one-body operator,\n\\bel{qext}\nq_{\\rm ext}(t)=q_{\\rm ext}^{\\omega }\\hbox{exp}[-i(\\omega  + i\\epsilon)],\\qquad (\\epsilon=+0)\n\\end{equation}\n The linear\nresponse function can be determined through the Fourier transform\n$\\langle {\\hat F} \\rangle_\\omega $ of the time-dependent quantum\naverage $\\langle {\\hat F} \\rangle_t$ by\n\\bel{defresp}\n\\langle\n{\\hat F} \\rangle_\\omega = -\\chi_{FF}^{\\rm coll}(\\omega )~q_{\\rm ext}^\\omega .\n\\end{equation}\n Here\nand below we omit an unperturbed average value $\\langle {\\hat F}\n\\rangle_0$ and use the same notation as in \\cite{hofmann}. In the\nfollowing, we shall consider the operators ${\\hat F}$ neglecting the\nmomentum dependence in a phase space representation in the linear\napproximation for an external field $V_{\\rm ext}$ and writing\n${\\hat F}={\\hat F}({\\bf r})$. According to (\\ref{defresp}), one can\nthen express explicitly $\\chi_{FF}^{\\rm coll}(\\omega )$ in terms of the\nFourier transform $\\delta \\rho_\\omega ({\\bf r})$ of the transition density\n$\\delta \\rho({\\bf r},t)$ \\cite{magkohofsh} as\n\\bel{chicollrho}\n\\chi_{FF}^{\\rm coll}(\\omega )= -\\frac{1 }{q_{\\rm ext}^\\omega } \\int {\\rm\nd}{\\bf r}~{\\hat F}({\\bf r})~\\delta \\rho_\\omega ({\\bf r}).\n\\end{equation}\n Note that in a\nmacroscopic picture the transition density is the dynamical part\n$\\delta \\rho({\\bf r},t)$ of the particle density,\n\\bel{partdens}\n\\rho({\\bf r},t)=\\rho_{\\rm qs}+ \\delta \\rho({\\bf r},t).\n\\end{equation}\n Here, $\\rho_{\\rm qs}$ is\nthe quasistatic equilibrium particle density. We define now\n${\\hat F}$ as related to the variation of the self-consistent mean\nfield $V$ in the nuclear Hamiltonian:\n\\bel{hamil}\n {\\hat H}={\\hat H}_0 + V\n= {\\hat H}_0 + Q {\\hat F} + \\frac{1 }{2} Q^2 \\left\\langle \\left(\\frac{\\partial^2\n{\\hat H}}{\\partial Q^2}\\right)_{Q=0}\\right\\rangle_0+...,\n\\end{equation}\n where\n$H_0$ is an unperturbed Hamiltonian. Introducing the\n collective variable $Q$  ($Q=0$ in equilibrium), one may write\n \\bel{foper}\n{\\hat F}=\\left(\\frac{\\partial {\\hat H}}{\\partial Q}\\right)_{Q=0}=\n\\left(\\frac{\\partial V }{\\partial Q}\\right)_{Q=0}.\n\\end{equation}\n The total Hamiltonian ${\\hat H}_{\\rm tot}$ is given by\n\\bel{hamiltot}\n{\\hat H}_{\\rm tot}={\\hat H}+q_{\\rm ext}(t){\\hat F}.\n\\end{equation}\n\nAs shown in \\cite{hofmann,hofbook}, a\nconservation of the nuclear energy\n$\\langle {\\hat H} \\rangle$ for the Hamiltonian ${\\hat H}$\n(\\ref{hamil}) leads to the equation of motion which is the\nsecular equation in the Fourier representation,\n\\bel{seculareq}\nk^{-1} +\\chi(\\omega )=0 .\n\\end{equation}\n The coupling constant $k$ is given by\n\\bel{kstiffC0chi0}\n -k^{-1} = C(0) + \\chi_{{}_{\\! FF}}(0),\n\\end{equation}\n$C(0)=\\left(\\partial^2 E(Q,S)\/ \\partial Q^2\\right)_{Q=0}$ is the stiffness\ncoefficient of the internal energy\n$E(Q,S)$ for the constant nuclear entropy, $S_0$,\nand $\\chi_{FF}(0)$ is the static (isolated) susceptibility.\n$\\langle {\\hat F} \\rangle_\\omega $ and $Q_\\omega $ are related then each\nother by the self-consistency condition\n \\bel{selfconsist}\nk \\langle {\\hat F} \\rangle_\\omega  = Q_\\omega \n\\end{equation}\n with $Q_\\omega $ being the\nFourier component of the collective variable $Q(t)$.\nThe ergodicity condition,\n\\bel{ergodicity1}\n\\chi_{{}_{\\! FF}}(0)=\\chi_{FF}^{\\rm ad},\n\\end{equation}\n with $\\chi_{FF}^{\\rm ad}$ being the\nadiabatic susceptibility was not used really in the derivation of\nthe self-consistency condition (\\ref{selfconsist}) with the\ncoupling constant $k$ from (\\ref{kstiffC0chi0}) in\n\\cite{hofmann} if the definition of slow variation of the\ntime-dependent $\\langle {\\hat F} \\rangle$ is employed under the\ncertain physical conditions, see (3.7-14),(3.3-15) in\n\\cite{hofmann} and discussion there. The isolated susceptibility\n$\\chi_{{}_{\\! FF}}(0)$ is the static limit $\\omega  \\rightarrow 0$ of the\nintrinsic response function $\\chi_{{}_{\\! FF}}(\\omega )$ defined by\n\\bel{defrespintr}\n\\langle {\\hat F}\\rangle_\\omega  = -\\left(Q_\\omega  +\nq_{\\rm ext}^\\omega \\right) \\chi_{{}_{\\! FF}}(\\omega ) .\n\\end{equation}\n Thus, the intrinsic response\nfunction $\\chi_{{}_{\\! FF}}(\\omega )$ is related to the collective response\nfunction $\\chi_{FF}^{\\rm coll}(\\omega )$ through the relation\n(\\ref{ddrespchiin}) \\cite{bohrmot,hofmann}.  Within the FLDM\nformulated below, it is simpler to derive first the collective\nresponse function $\\chi_{FF}^{\\rm coll}(\\omega )$ by making directly\nuse of the\ndefinition (\\ref{chicollrho}). For comparison with\nthe microscopic quantum theory \\cite{hofmann} and for study of\nthe susceptibilities and of the ergodicity property, it is helpful\nto present the intrinsic response function $\\chi(\\omega )$ in terms\nof the collective response function $\\chi_{FF}^{\\rm coll}(\\omega )$ found\nfrom (\\ref{ddrespchiin}) as\n\\bel{respintr}\n\\chi_{{}_{\\! FF}}(\\omega )=\n\\frac{\\chi_{FF}^{\\rm coll}(\\omega ) }{1 - k \\chi_{FF}^{\\rm coll}(\\omega )}.\n\\end{equation}\n\n\n\\subsection{Fermi-Liquid Droplet Model}\n\\label{fldm}\n\nIn this section we follow \\cite{magkohofsh} for the basic grounds\nof the FLDM \\cite{magstrut,kolmagpl} for heavy nuclei taking into account\nthe quasiparticle Landau--Vlasov theory for the collective\ndynamics of the {\\it heated} Fermi liquids described in\n\\cite{heipethrev} and developed in the previous sections in more\ndetails for nuclear matter. The main idea is to apply this\nsemiclassical theory for the distribution function {\\it inside the\nnucleus} with the macroscopic {\\it boundary conditions}\n\\cite{strutmagbr,magstrut} like for normal liquids {\\it at its\nmoving surface}. These boundary conditions are used for the\nsolutions of the dynamical collisional Landau--Vlasov equation\n(\\ref{landvlas}) {\\it coupled with the thermodynamic relations}\nfor motion in the Fermi-liquid-drop interior. Our derivations are\nbased on the conception of the linearized dynamics near the {\\it\nlocal} equilibrium instead of the global one considered earlier in\n\\cite{kolmagpl,magkohofsh}. This is important for a {\\it low} frequency\nregion of the nuclear excitations which we are interested in this\nreview.\n\nWe shall consider below small isoscalar vibrations of the nuclear surface\nnear a spherical shape, which are induced by the external field\n$V_{\\rm ext}(t)$ (\\ref{extfield}). To this end, we define\na collective variable $Q(t)$ in the usual way:\n\\bel{surface}\nR = R_0\n\\left[1 + Q(t)Y_{L0}({\\hat r})\\right],\n\\end{equation}\nwhere $R_0$ is the\nequilibrium radius of nucleus, and $Y_{L0}({\\hat r})$ is the spherical\nharmonics which represent the axially symmetric shapes as\nfunctions of the radius vector angles ${\\hat r}$. For $Q(t)$\nwe expect the form\n\\bel{collvarq}\nQ(t) = Q_\\omega  \\hbox{exp}\\left(-i\\omega \nt\\right)\n\\end{equation}\n with the same frequency $\\omega $ as for the external\nfield (\\ref{extfield}).\n\n\n\\subsubsection{EQUATIONS OF MOTION INSIDE THE NUCLEUS}\n\n Quasiparticle conceptions\nof the Landau Fermi-liquid theory  can be justified\nin the nuclear volume, where variations of the density\n$\\rho({\\bf r},t)$ (\\ref{densit}) are small.  Therefore, in\nthe interior of sufficiently heavy nuclei, one may describe the\nsemiclassical phase-space dynamics\nin terms of the distribution function $\\delta f({\\bf r},{\\bf p},t)$\n(\\ref{dfgeqdfleq}) which satisfies the\ncollisional Landau--Vlasov equation (\\ref{landvlas}). We recall now\nthe equations of Sec.\\ \\ref{eqmotion} which present the\ncollective dynamics linearized with respect to the local\nequilibrium (\\ref{intcoll}). Our interior nuclear collective\ndynamics is then described by 6 equations, see (\\ref{landvlas}) and\n(\\ref{consereq}), for the 6 local quantities $\\delta \\rho({\\bf r},t)$,\n$\\delta \\mu({\\bf r},t)$, ${\\bf u}({\\bf r},t)$ and $\\delta T({\\bf r},t)$ defined inside\nof the nucleus as for the nuclear matter. The conserving\nequations (\\ref{conteq}), (\\ref{momenteq}) [or (\\ref{navstokeq})\nfor a potential flow], (\\ref{enerconserv}) and (\\ref{entropyeq})\nare helpful to find them in the semiclassical approximation.\n\nFor the isoscalar multipole vibrations of the Fermi-liquid drop\nsurface (\\ref{surface}), we  shall look for the solutions of\n(\\ref{landvlas}), (\\ref{consereq})\nin terms of  a superposition\nof the plane sound waves (\\ref{planewave}) over all  angles\n${\\hat q}$ of the unit wave vector ${\\bf q}$ with the amplitude\n$\\mathcal{A}_L({\\hat q})$,\n\\begin{eqnarray}\\label{superpos}\n \\delta f({\\bf r},{\\bf p},t) &=& \\int {\\rm d} \\Omega_{\\bf q}\n\\mathcal{A}_L({\\hat q})\n~\\delta {\\tilde f} ({\\bf q},{\\bf p},\\omega )\\; \\hbox{exp} [i({\\bf q}{\\bf r}-\\omega  t)]\n\\nonumber\\\\\n&&{\\rm with} \\qquad \\mathcal{A}_L({\\hat q})=\nY_{L0} ({\\hat q}_z).\n\\end{eqnarray}\n Here $L$ is the multipolarity of collective vibrations,\n${\\hat q}_z$ is the projection of the unit\nvector ${\\hat q}={\\bf q}\/q$ on the symmetry $z$-axis.\nThe Fourier amplitudes\n$\\delta f({\\bf q},{\\bf p},\\omega )$ are presented as a spherical harmonic\nexpansion in momentum space,\n\\bel{tildef}\n\\delta {\\tilde\nf}({\\bf q},{\\bf p},\\omega ) = \\left(\\frac{\\partial f_{\\rm g.e.} (\\varepsilon_{\\bf p})}{\\partial\n\\varepsilon_{\\bf p}} \\right)_{\\rm g.e.} \\sum_{l^\\prime} \\mathcal{A}_{l^\\prime}(\\omega ,q)\nY_{l^\\prime 0} ({\\hat p} \\cdot {\\hat q}),\n\\end{equation}\n where $\\mathcal{A}_{l^\\prime}$ are small vibration amplitudes.\nFor such\nsolutions, the velocity field ${\\bf u}$ corresponds to the\npotential flow (\\ref{velpoten}).\n\nThe relaxation time $\\tau$ in (\\ref{intcoll}) is assumed to be\nfrequency and temperature dependent as in (\\ref{tautom}).\nFollowing \\cite{hofmann,kolmagpl,magkohofsh}, we take the form:\n\\bel{relaxtime}\n \\tau(\\omega ,T)=\\frac{\\hbar}{{\\it \\Gamma}(\\omega ,T)},\n\\end{equation}\nwhere\n\\bel{widthG}\n {\\it \\Gamma}(\\omega ,T)=\\frac{\\pi^2 }{{\\it\n\\Gamma}_0}~ \\frac{T^2 + c_o (\\hbar \\omega )^2 }{\n1+\\frac{\\pi^2}{c^2}\\left[T^2 + c_o (\\hbar \\omega )^2\\right]}.\n\\end{equation}\n For $c_o$ one has several values. For instance, $c_o = 1 \/ 4\\pi^2$,\naccording to \\cite{landau,ayik}, $c_o = 1 \/\\pi^2$ follows\nfrom \\cite{pinenoz,hofmann}, $3 \/\n4\\pi^2$ from \\cite{kolmagpl} and several numbers near these\nconstants were suggested in \\cite{sykbrook,brooksyk}. Formula\n(\\ref{widthG}) with the $c_o = 1\/\\pi^2$ and finite cut-off\nconstant $c$ which weakens the dependence on both\nfrequency $\\omega $ and\ntemperature $T$ at large values of these quantities may in some\nsense be compared with the expressions suggested in \\cite{hofmann} for\nthe imaginary part of the self-energy to be used in microscopic\ncomputations \\cite{hofmann,hofbook}\n[$c$ in (\\ref{widthG}) should not be\nconfused with the sound velocity $c$ used for the description of\nnormal liquids [see, g.e., (\\ref{liqparam}) and (\\ref{forstereq})].\nIn line\nof these computations, we shall use ${\\it \\Gamma}_o=33.3$ MeV and\n$c=20$ MeV in our FLDM calculations.\nThe value of the parameter $c_o=3\/4 \\pi^2$ is taken as in\n\\cite{kolmagpl,magkohofsh}. The specific value of this\nparameter is not important for the following derivations and\nresults in this section because we shall apply the temperature-dependent\nFermi-liquid theory for low frequencies and large temperatures.\nNote that for $c \\to \\infty$ the expression (\\ref{widthG}) was derived in\n\\cite{landau,pinenoz,sykbrook,brooksyk,kolmagpl}.\n\n\n\n\\subsubsection{BOUNDARY CONDITIONS AND COUPLING CONSTANT}\n\\label{boundary}\n\nThe dynamics in the surface layer of nucleus can be described by\nmeans of the macroscopic boundary conditions as in\n\\cite{magstrut} by using the effective surface approximation\n\\cite{strutmagbr,strutmagden,magboundcond}. For small vibration\namplitudes, they read:\n\\bel{bound1}\nu_r \\Big\\vert_{r=R_0} = {\\dot\nR}(t) \\equiv R_0 {\\dot Q}(t) Y_{L0}({\\hat r}),\n\\end{equation}\n \\bel{bound2}\n\\Pi_{rr} \\Big\\vert_{r=R_0} = P_{S} + P_{\\rm ext},\n\\end{equation}\n where $u_r$\nand $\\Pi_{rr}$ are the radial components of the velocity field\n${\\bf u}$ (\\ref{veloc}) and the momentum flux tensor\n$\\Pi_{\\alpha\\beta}$ (\\ref{momentflux}) which are determined in\nthe nuclear volume, see\n\\cite{bekhal,ivanov,magboundcond,abrditstrut,komagstrvv,abrdavkolsh}\nfor other (mirror and diffused) boundary conditions used directly\nfor the distribution function as a solution of the Landau--Vlasov equation.\nIn the case of the potential flow (\\ref{velpoten}),\n we shall use the specific expression for the\nmomentum flux tensor (\\ref{momentfluxpot}) with the shear modulus\n($\\lambda$) and viscosity ($\\nu$) coefficients given by\n(\\ref{shearmod}) and (\\ref{viscos}), respectively. The\nsurface pressure $P_{S}$, which is due to the tension forces\nfor the isoscalar motion in symmetric nuclei, is\ngiven by\n\\bel{surfpress}\nP_{S} = \\frac{\\alpha }{R_0} (L-1)\n(L+2)\\; Q(t) Y_{L0} ({\\hat r}),\n\\end{equation}\n where $\\alpha$ is the surface\ntension coefficient, see Sec.\\ \\ref{npcorivgdr} and\nAppendix D\nfor the isovector asymmetric modes. For the\ntension coefficient $\\alpha$, we used an expression found in\n\\cite{strutmagden} within  the  ESA.\nThis approximation is based on expansion of the nuclear\ncharacteristics, such as the particle density and the total energy in\nsmall parameter $a\/R_{0} \\sim A^{-1\/3}$, where $a$ is the diffuseness\nparameter and $R_{0}$ is the mean curvature radius of the nuclear surface\n\\cite{strutmagbr,strutmagden}, see also \\cite{magsangzh,BMRV} and\nAppendix D.\nIn this way, one derives the\nnuclear energy expansion [Wiezs\\\"acker formula (\\ref{EvEs}),\n(\\ref{Espm})], $E=E_{\\mathcal{V}} +\nE_{S} +...$,\nwith the volume part of the energy $E_{\\mathcal{V}}$\nproportional to the particle number $A$,\nand the surface energy $E_{S}$, $E_{S}=b_{{}_{\\! S}}A^{2\/3}$\n($b_{{}_{\\! S}}=4\\pi r_0^2 \\alpha$ corresponds to the surface tension\nconstant $\\alpha$, $b_{{}_{\\! S}}\\approx 20~\\mbox{MeV}$,\n$r_{{}_{\\! 0}}=R_0\/A^{1\/3} \\approx 1.1-1.2$ fm) and so on, see\n\\cite{strutmagbr,strutmagden,magsangzh,BMRV} and Appendices A.4\n(symmetrical nuclei) and D (asymmetrical ones) for more details\n(the suffix ``$+$'' is omitted here).\nAccording to (\\ref{sigma}) of \\cite{strutmagden,BMRV},\n\\bel{tensionconst}\n\\alpha \\approx 2\n\\mathcal{C}\n\\int\\limits^\\infty_0 {\\rm d} r \\left(\\frac{\n\\partial \\rho_{\\rm qs}}{\\partial r}\\right)^2.\n\\end{equation}\nHere and below we neglect the relatively small\ncorrections of the order of $A^{-1\/3}$ of the\nESA, which are in\nparticular related to the semiclassical $\\hbar$ corrections and\nexternal field. The coefficient $\\mathcal C$\nappears earlier in\nfront of the term which is proportional to $\\left({{\\bf \\nabla}}\n\\rho_{\\rm qs}(r)\\right)^2$\nin the nuclear energy-density formula [see (\\ref{enerden})],\n$\\mathcal{C}=40-60~ {\\rm MeV} \\cdot {\\rm fm}^5$ \\cite{BMRV}.\n\n\nAn external pressure\n$P_{\\rm ext}$ appears in (\\ref{bound2}),\nwhere we make connection to the external\npotential $V_{\\rm ext}$ (\\ref{extfield})\n\\cite{komagstrvv,magkohofsh},\n\\bel{extpress}\nP_{\\rm ext} =\n-\\int\\limits_0^\\infty dr \\> \\> \\rho_{\\rm qs} (r)\n\\frac{\\partial V_{\\rm ext}}{\\partial r}.\n\\end{equation}\n For the density in equilibrium,\none has\n\\bel{denseq}\n\\rho_{\\rm qs} (r) =\\rho_{{}_{\\! 0}} w(\\xi),\\quad \\xi=\\frac{r-R}{a},\\quad\na=\\sqrt{\\frac{\\mathcal{C}_{+}\\; \\rho_\\infty\\; K}{30\\;\nb_{\\mathcal{V}}^2}}.\n\\end{equation}\nThis density is expressed in terms of the\nprofile function $w(\\xi)$ with a sharp decrease from one to zero in\nthe narrow region of the order of the diffuseness parameter\n$a$\nnear $\\xi=0$ as in a step function ($w(\\xi) \\rightarrow \\theta(R-r)$\nfor $a \\rightarrow 0$), $b_{\\mathcal{V}} \\approx 16$ MeV\nis the separation energy per\nnucleon\n\\cite{strutmagbr,magstrut,strutmagden,BMRV}. The value\nof equilibrium density inside the nucleus $\\rho_{{}_{\\! 0}}$\n\\cite{strutmagbr} is given by\n\\bel{rho0}\n\\rho_{{}_{\\! 0}}=\\rho_\\infty\\left(1+\\frac{6\nb_{{}_{\\! S}} r_0}{K R_0}\\right),\n\\end{equation}\n where\n$\\rho_\\infty$ is the particle density of the infinite nuclear\nmatter, $\\rho_\\infty=3\/(4\\pi r_0^3)$. The surface energy constant,\n$b_{{}_{\\! S}}$,  and in-compressibility modulus, $K$, in\n(\\ref{rho0}) depend on the condition of the constant temperature,\nentropy and of the static limit, as shown in Appendix C.\nIn (\\ref{rho0}) and below, we omit the index ${\\tt X}$ of these\nquantities which specifies one of these conditions, see Appendix C.\nFor instance, the in-compressibility in\n(\\ref{rho0})  is denoted simply as\n$K=K_{\\rm tot}(\\omega =0)=K^{\\varsigma}$, as shown above through\n(\\ref{incomprtot}), (\\ref{dtdrhoexp}) and (\\ref{incompradexp}).\nThe surface energy constant $b_{{}_{\\! S}}$ in (\\ref{rho0}) is also\nidentical to the adiabatic one as the in-compressibility\n(see Appendix C). The second term in the circle brackets of\n(\\ref{rho0}) is a small correction proportional to $A^{-1\/3}$,\ndue to the surface tension.\n\n\nBoundary conditions (\\ref{bound1}) and (\\ref{bound2}) were\nre-derived here from (\\ref{conteq}) and\n(\\ref{momenteq}) where all quantities are now extended to the\nsurface region with a sharp coordinate dependence of the particle\ndensity as in the approach \\cite{magstrut}.\nHowever, we used the specific properties of the {\\it\nheated} Fermi-liquid drop following the same ESA\n\\cite{strutmagbr,magstrut,strutmagden}. For the derivation of (\\ref{bound2}),\ng.e.,\nthe key equation\n(\\ref{keybound}) for the Gibbs thermodynamic potential per\nparticle $g$, which satisfies the thermodynamic relations\n(\\ref{thermrelmu}), was applied instead of the energy per particle\n$\\varepsilon$ of \\cite{magstrut}. The result\n(\\ref{bound2}) has the same form as in\n\\cite{magstrut,magkohofsh} because in its derivation we have\nsimultaneously to use  (\\ref{gradrelener}) of the\ntemperature-dependent Fermi-liquid theory (with the entropy term\n$T d \\varsigma$), in contrast to the adiabatic equation (17) of\n\\cite{magstrut}, see Appendix A.4 for details.\n\n\nThe external field $V_{\\rm ext}$ (\\ref{extfield}) in\n(\\ref{extpress}) is determined by the operator ${\\hat F}({\\bf r})$\n(\\ref{foper}), and hence, $V_{\\rm ext}$ is concentrated in the surface\nregion of the nucleus. Indeed, for the operator ${\\hat F}({{\\bf r}})$\n(\\ref{foper}) in the FLDM, one gets the form\n\\bel{foperl}\n{\\hat F}({{\\bf r}}) = \\left(\\frac{\\delta V }{\\delta \\rho}~ \\frac{\\partial\n\\rho}{\\partial Q}\\right)_{Q=0}^{\\rm qs}\n=-R_0\\left(\\frac{\\partial V }{\\partial r}\\right)_{R=R_0}^{\\rm qs}\nY_{L0}({\\hat r}),\n\\end{equation}\n see (\\ref{dhdq}). After substitution of\n(\\ref{foperl}) into (\\ref{extpress}) we have\n\\bel{extpressl}\nP_{\\rm ext} = - \\frac{1 }{kR_0^3}\\; q_{\\rm ext}(t) Y_{L0}\n({\\hat r}),\n\\end{equation}\nwhere\n\\bel{kfld}\nk^{-1} = \\frac{K \\alpha R_0^4 }{18\n\\mathcal{C} \\rho_\\infty}\n       \\left[1 + \\mathcal{O}\\left(A^{-1\/3}\\right)\\right]\n       \\approx \\frac{K b_{{}_{\\! S}} r_0^5 }{54 \\mathcal{C}} A^{4\/3}.\n\\end{equation}\n The integration by parts in (\\ref{extpress}) and the\nequation (\\ref{couplfld}) for the quasistatic coupling constant $k^{-1}$\nwere used in the derivation of (\\ref{extpressl}), (\\ref{kfld}), see\nthe second equation of (\\ref{couplxchix}), and also applications to\ncalculations of the collective vibration modes in \\cite{yaf,gzhmagfed}.\n\n\n\n\\subsubsection{COLLECTIVE RESPONSE FUNCTION}\n\\label{collresponse}\n\nAs shown in \\cite{strutmagbr,strutmagden,galiqmod}, the \nlinearized dynamic part of the\nnucleonic density $\\delta \\rho ({\\bf r},t)$ for the isoscalar modes\ncan be represented as a sum\nof the \"volume\" and the \"surface\" term,\n\\bel{densvolsurf}\n\\delta\n\\rho({{\\bf r}},t) = \\delta \\rho^{\\rm vol}({{\\bf r}},t) w(\\xi)\n- \\frac{\\partial\nw}{\\partial r} \\rho_{{}_{\\! 0}} \\delta R,\n\\end{equation}\n where $\\delta R$ is the\nvariation of nuclear radius (\\ref{surface}), $\\delta\nR = R_0 Q(t) Y_{L0} ({\\hat r})$, $w$ is defined around\n(\\ref{denseq})\nand in Appendix D. For\nisovector vibration modes of the odd multipolarity (dipole), one has to\naccount for the mass center conservation \\cite{kolmagsh,BMR}\n[see (\\ref{trandenscl})].\nThe upper index \"vol\" in $\\delta\n\\rho^{\\rm vol}({{\\bf r}},t)$ of (\\ref{densvolsurf}) denotes that\nthe dynamical particle-density variation\nis determined by the equations of motion in the nuclear\nvolume and is given in terms of the local part $\\delta\nf_{\\rm l.e.}(\\varepsilon_{\\rm l.e.})$ (\\ref{dfleq}) of the distribution function\n$\\delta f({{\\bf r}}, {{\\bf p}},t)$ (\\ref{dfgeqdfleq}) through\n(\\ref{densit}).\n\n\nSolving (\\ref{navstokeq}) with the first\nboundary condition (\\ref{bound1}), one gets the potential\n$\\varphi$ in the form\n\\bel{potensolut}\n\\varphi({{\\bf r}},t) = \\frac{1 }{q^2}\n\\frac{qR_0}{ j^\\prime_L (qR_0)} {\\dot Q}(t) j_{{}_{\\! L}}(qr)  Y_{L0}\n({\\hat r}),\n\\end{equation}\n where $j_{{}_{\\! L}}(x)$ is the spherical Bessel function\nand $j^\\prime_L(x) = dj_{{}_{\\! L}}(x)\/dx$.  From the continuity equation\n(\\ref{conteq}) with (\\ref{potensolut}), one has\n\\bel{densolut}\n\\delta \\rho^{\\rm vol}({{\\bf r}},t) = \\rho_{{}_{\\! 0}} \\frac{qR_0 }{ j^\\prime_L(qR_0)}\nQ(t) j_{{}_{\\! L}}(qr) Y_{L0} ({\\hat r}).\n\\end{equation}\n Therefore, according to\n(\\ref{densvolsurf}) and (\\ref{densolut}), one finds\n\\bel{trandensolut}\n\\delta \\rho({{\\bf r}},t) = \\rho_{{}_{\\! 0}} Q(t) Y_{L0}({\\hat\nr}) \\left[\\frac{qR_0 }{ j^\\prime_L(qR_0)}  j_{{}_{\\! L}}(qr) w(\\xi)\n- \\frac{\\partial w}{\\partial r} R_0\\right].\n\\end{equation}\n\nWith this solution, we may now proceed to calculate the response\nfunction $\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollrho})\nby expressing the integral over the coordinates ${\\bf r}$ for the average\n$\\langle {\\hat F}\\rangle_\\omega $ (\\ref{defresp}) in\nthe numerator of (\\ref{chicollrho})\nin terms of our collective variable $Q_\\omega $ given by (\\ref{collvarq}).\nIndeed, substituting the Fourier transform of\n(\\ref{trandensolut}) together with ${\\hat F}$ from\n(\\ref{foperl}) into (\\ref{chicollrho}), we obtain\n\\bel{chicollQq}\n \\chi_{FF}^{\\rm coll}(\\omega ) = - \\frac{Q_\\omega  }{k q^\\omega _{\\rm ext}}.\n\\end{equation}\n\n\nUsing  (\\ref{momentfluxpot}),\n(\\ref{surfpress}), (\\ref{extpressl}) and (\\ref{potensolut}),  one may\nwrite the second boundary condition (\\ref{bound2}) in terms of the\ncollective variable $Q(t)$ and periodic time dependence of the\nexternal field $V_{\\rm ext}$ in the form of the equation of motion\n\\bel{eqmotionQ}\n\\mathcal{B}_L(x) {\\ddot Q} + \\mathcal{C}_L(x)Q +\n\\mathcal{Z}_L(x) {\\dot Q} = - q_{\\rm ext}.\n\\end{equation}\n We have introduced various new\nquantities,\n\\bel{x}\nx = \\frac{\\omega }{\\Omega}=\\frac{\\omega  R_0 }{ v_{{}_{\\! {\\rm F}}} s}, \\quad \\mbox{with}\n\\quad\n\\Omega=\\frac{v_{{}_{\\! {\\rm F}}}}{R_0}\\sim \\frac{\\varepsilon_{{}_{\\! {\\rm F}}} }{A^{1\/3} \\hbar},\n\\end{equation}\n which  is a complex function of $\\omega $ by\nmeans of (\\ref{despeq}) for the sound velocity $s$ with\n(\\ref{despfunc})-(\\ref{chinumer}),\n(\\ref{chitemzero})-(\\ref{chitemtwo}).\nIn (\\ref{x}), $\\Omega$ is the characteristic frequency of the classical\nparticle rotation\nin a mean potential well of the radius $R_0$ with the\nenergy near $\\varepsilon_{{}_{\\! {\\rm F}}}$, as a convenient frequency unit.\nOther quantities are defined as\n\\bel{blx}\n\\mathcal{B}_L(x) = m \\rho_{{}_{\\! 0}} R_0^5 \\frac{j_{{}_{\\! L}}(x)}{x\nj^\\prime_L(x)},\n\\end{equation}\n \\bel{clx} \\mathcal{C}_L(x) = C_L^{(S)} +\n\\mathcal{C}_L^{(\\lambda)} (x),\n\\end{equation}\n \\bel{LDstiff} C_L^{(S)} =\n{\\alpha} R^2_0 (L-1) (L+2)= \\frac{b_{{}_{\\! S}} }{4\\pi} A^{2\/3} (L-1)\n(L+2),\n\\end{equation}\n \\bel{clmu} \\mathcal{C}_L^{(\\lambda)}(x) = 2 \\lambda R^3_0\n\\frac{x }{j^\\prime_L(x)} (j^{\\prime\\prime}_L (x) + j_{{}_{\\! L}}(x))\\;,\n \\end{equation}\nand\n \\bel{zlx} \\mathcal{Z}_L(x) = 2 \\nu R^3_0 \\frac{x }{ j^\\prime_L(x)}\n(j^{\\prime \\prime}_L (x)+ j_{{}_{\\! L}}(x)).\n\\end{equation}\n\nFrom (\\ref{eqmotionQ}), one has\n\\bel{Qqext}\n\\frac{Q_\\omega  }{q^\\omega _{\\rm ext}} = - \\frac{1 }{k D_L(\\omega )}\n\\end{equation}\n with\n\\begin{eqnarray}\\label{glx}\n&&\\mathcal{D}_L(\\omega ) \\equiv - \\left[\\mathcal{B}_L(x) \\omega ^2 -\n\\mathcal{C}_L(x) + i \\omega \n\\mathcal{Z}_L(x)\\right] ~~~\\nonumber\\\\\n&=& \\frac{C_L^{(S)} }{j^\\prime_{{}_{\\! L}} (x)}\n\\left\\{ j^\\prime_L(x) - \\frac{6 A^{1\/3}\\lambda\\;\nx }{b_{{}_{\\! S}}(L-1) (L+2) \\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}}} \\left[\n\\left(i \\nu \\omega  -\\lambda\\right) \\right.\\right.~~~\\nonumber\\\\\n&\\times& \\left.\\left.  j^{\\prime \\prime}_L (x) +\n\\left(\\frac{s^2\n\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{\\mathcal{G}_1} - \\lambda + i \\nu \\omega  \\right)\nj_{{}_{\\! L}}(x)\\right] \\right\\}.~~~\n\\end{eqnarray}\nIn (\\ref{Qqext}),\n$k$ is the coupling constant (\\ref{kfld}), see (\\ref{blx})-(\\ref{zlx}).\nThe kinetic\ncoefficients $\\lambda$ and $\\nu$ are the shear modulus $\\lambda$\nand viscosity $\\nu$ given by (\\ref{shearmod}) and\n(\\ref{viscos}), respectively. The two latter quantities enter\n(\\ref{glx}) in the following combination\n\\bel{lambvischixz}\n\\lambda -i \\nu \\omega = s \\chi _{xz} \\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}}\n\\end{equation}\nthrough a function $\\chi _{xz}$ defined by\n(\\ref{chixz}). Finally, the\nresponse function $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollQq}) with\n(\\ref{Qqext}) writes\n\\bel{chicollfldm}\n\\chi_{FF}^{\\rm coll}(\\omega ) =  \\frac{1 }{k^2 \\mathcal{D}_L(\\omega )}.\n\\end{equation}\nThe poles of this collective response function\nare determined by the following equation,\nsee (\\ref{glx}) for $\\mathcal{D}_L(\\omega )$,\n\\bel{poleseq}\n- \\mathcal{B}_L(x) \\omega ^2 + \\mathcal{C}_L(x) -\ni \\omega  \\mathcal{Z}_L(x) = 0,\n\\end{equation}\n with $\\,x\\,$ defined by (\\ref{x}).\nThe complex\nsolution of the dispersion equation (\\ref{despeq}) for $s$\nhas two branches of the solutions. They are related\nasymptotically to the Landau--Placzek heat $s^{(0)}$ and the sound\n$s^{(1)}$ solutions considered all in the hydrodynamic limit for\nthe infinite nuclear matter in Sec.\\ \\ref{longwavlim}, see\n(\\ref{shp})-(\\ref{gambarone}) for the corresponding\nfrequencies $\\omega ^{(0)}$ and $\\omega ^{(1)}$. For each branch denoted\nbelow by the same upper index\n$n=0,1$ as well in\nSec.\\ \\ref{longwavlim}, we have the roots of the secular equation\n(\\ref{poleseq}) written as\n\\bel{rootomega}\n \\omega ^{(n)} =\n\\omega _i^{(n)} - i \\Gamma_i^{(n)}\/2, \\> \\> \\> i = 0,1,...,\n\\end{equation}\n where\n$i$ numbers $\\omega _i^{(n)}$ in order of their increasing magnitude.\nWe shall consider these roots with $\\omega _i^{(n)}$ in the frequency\nregion of about $\\hbar \\omega  \\siml \\hbar \\Omega$ which overlaps the\nlow frequency energy region discussed below. We shall consider\nenough large\ntemperatures $T \\simg \\sqrt{c_o} \\hbar \\Omega\\; \\simg \\sqrt{c_o}\n\\hbar \\omega $ but smaller than the Fermi energy. This approximately\nmeans $2 MeV \\siml T \\siml 10 MeV$ for $c_o=3\/4\\pi^2$ of\n\\cite{kolmagpl} ($A \\sim 200$). (Low temperature limit is about\n$1$ MeV for $c_o=1\/4\\pi^2$ of\n\\cite{landau,ayik}.)\nFor above mentioned frequencies $\\omega $ and temperatures $T$, for which\nthe quasiparticle and local-equilibrium conceptions of the theory\nfor the heated Fermi liquids can be applied, the only lowest\nsolutions have been found in the infinite sequence\n(\\ref{rootomega}). They are\nassociated with $i=0$, $1$ and $2$ for the \"first sound\"\nbranch $n=1$, and\nthat with $i=0$ for the \"Landau--Placzek\"\nbranch ($n=0$). (Quote marks show that the corresponding names\nare realized in fact only asymptotically in the hydrodynamical\nlimit.) The total response function is the sum of the two\nbranches mentioned above. The response function\n(\\ref{chicollfldm}) contains all important information concerning\nthe excitation modes of the Fermi-liquid drop. One of the ways of\nthe receipt of this information is to analyze the response\nfunction poles  (\\ref{rootomega}) and their residua. However, this\nway is often not so convenient and too complicate in the case when\na few poles are close to each other or they belong (or are close)\nto the imaginary axis of the complex plane $\\omega $. More transparent\nway which is free from such disadvantages is to describe the\nresponse function in terms of the transport coefficients\n\\cite{hofmann}.\n\n\n\\subsection{Transport properties for a slow collective motion}\n\\label{transprop}\n\nThe macroscopic response of nucleus to an external field\nis a good tool for calculations of\nthe transport coefficients. To achieve this goal we follow\nthe lines of \\cite{hofmann,hofbook}. For instance, in cranking model type\napproximations, one assumes the collective motion to be\nsufficiently slow such that the transport coefficients can be\nevaluated simply in the \"zero-frequency\" limit. For a such slow\ncollective motion we shall study here the transport coefficients\nwithin the FLDM having a look at excitation energies smaller than the\ndistance between gross shells \\cite{strutmag},\n\\begin{equation}\\label{hom}\n\\hbar \\Omega = \\frac{\\hbar v_{{}_{\\! F}}}{R} \\approx\n\\frac{\\varepsilon_{{}_{\\! F}}}{A^{1\/3}} = 7-10\\, {\\rm MeV}\n\\end{equation}\n in heavy nuclei [$\\Omega$  is the particle rotation frequency\n(\\ref{x})] $\\hbar\\omega \n\\siml \\hbar\\Omega$, i.e., less than or of the order of the giant\nmultipole resonance\nenergies. Within the low collective motion ($\\omega  \\siml \\Omega$),\nwe shall deal with first more simple\ncase of the hydrodynamic approximation which can be applied for\nfrequencies much smaller than the characteristic \"collisional\nfrequency\" $1\/\\tau$ related to the relaxation time $\\tau$\n(\\ref{relaxtime}), $\\omega  \\tau \\ll 1$. Using this hydrodynamic\nexpansion of the macroscopic response function (\\ref{chicollfldm})\nin small parameter $\\omega  \\tau$, we shall look for in\nSec.\\ \\ref{zerofreqlim} the relation to the \"zero frequency limit\"\ndiscussed in \\cite{hofmann}. Another problem of our interest in\nthis section is related to the correlation functions, \"heat pole\nfriction\" and ergodicity property, see \\cite{hofmann,hofbook}. We shall\nconsider in the next Sec.\\ \\ref{heatcorrfun} a smaller\nfrequency region where the nuclear heat pole like the Landau--Placzek mode\nfor the infinite matter appears within the hydrodynamic\napproximation. This subsection will be ended by a more general\ntreatment of the transport coefficients in terms of the parameters\nof the oscillator response function.\nThe method of \\cite{hofmann,hofbook} can be applied for the low frequency\nexcitations, $\\omega  \\siml \\Omega$, but also in the case when the\nhydrodynamic approach fails, i.e. for $\\omega  \\tau \\simg  1$.\n\n\n\nFollowing \\cite{hofmann,hofbook}, we shall study the \"intrinsic\"\nresponse function $\\chi_{{}_{\\! FF}}(\\omega )$ related to the collective\none $\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollrho}) by\nthe relation (\\ref{respintr}). The collective response function\n$\\chi_{FF}^{\\rm coll}(\\omega )$ (\\ref{chicollfldm}) in the FLDM was\nderived straightly from (\\ref{chicollrho}) in terms of the\nsolution for the transition density $\\delta\\rho$\n(\\ref{densvolsurf}). The \"intrinsic\" response function can be\nthen got with help of (\\ref{respintr}). This way is more\nconvenient in the FLDM with respect to the opposite one used\nusually in the microscopic quantum calculations based on the shell\nmodel \\cite{hofmann}.\n\n\nBy making use of expansion of the denominator $\\mathcal{D}_L(\\omega )$\n(\\ref{glx}) of the response function $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollfldm}) up to fourth order terms in small parameter\n$\\omega  \\tau$ in the low frequency region $\\omega \/\\Omega \\ll 1$, and then,\nof (\\ref{respintr}), one gets the response function in the $F$ mode\nin the form:\n\\begin{eqnarray}\\label{respfunas}\n\\chi(\\omega )&=& k^{-2}\\left(-M\\omega ^2- i\n\\gamma \\omega  + C_{\\rm in} - \\frac{i \\Upsilon C } {2 \\omega }\\right)^{-1},\\nonumber\\\\\nC_{\\rm in}&=&C - k^{-1},\n\\end{eqnarray}\n $M$, $C$ and $\\gamma$ can be\ndefined as the $Q$-~mode mass, the stiffness and the friction coefficients\nwhich are the values of $B_L(x)$ (\\ref{blx}), $C_L(x)$\n(\\ref{clx}) and $\\mathcal{Z}_L(x)$ (\\ref{zlx}) for $x=0$ ($\\omega =0$).\nHere and below we omit the low index \"FF\" in the\n\"FF\"-response functions everywhere when it will not lead to\nmisunderstanding. Note that the formulas which we derive here and\nbelow for the \"intrinsic\" response function $\\chi(\\omega )$ can be\napplied also to the collective response function\n$\\chi^{\\rm coll}(\\omega )$ if we only omit the index \"in\" in $C_{\\rm in}$\nand in functions of $C_{\\rm in}$ denoted by the same index (except\nfor some approximations based on the specific properties of\n$C_{\\rm in}$ compared to $C$ and noted below if necessary).\nAnother argument of the presentation of our results in terms of\nthe \"intrinsic\" response functions  is to compare them more\nstraightly with the discussed ones in \\cite{hofmann} in connection\nto correlation functions and ergodicity. For the\ninertia  $M$ and stiffness $C$,  we obtain the parameters of the\nclassic hydrodynamic model, namely:\n\\bel{mass0}\nM=\\mathcal{B}_L(0) =\n\\frac{1 }{L} m \\rho_{{}_{\\! 0}} R^5_0 \\equiv M_{\\rm LD},\n\\end{equation}\n the inertia of\nirrotational flow, and\n\\bel{stiffness0}\n C=\\mathcal{C}_L(0) =\nC_L^{(S)} \\equiv  C_{\\rm LD}\n\\end{equation}\n with $C_L^{(S)}$ being the\nstiffness coefficient of the surface energy (\\ref{LDstiff}). (We\nintroduced here more traditional notations labeled by index \"LD\"\nwhich means the relation to the usual liquid-drop model of\nirrotational flow). For friction $\\gamma$, we arrive at\nthe temperature dependence typical for hydrodynamics,\n\\begin{eqnarray}\\label{friction0}\n\\gamma &=&\\mathcal{Z}_L(0)=2 \\nu_{{}_{\\! {\\rm LD}}} R_0^3 (L-1) \\nonumber\\\\\n&=& \\frac{3 A (L-1)\\varepsilon_{{}_{\\! {\\rm F}}}\n}{5 \\pi}~ \\frac{\\nu_{{}_{\\! {\\rm LD}}}(T) }{\\nu_{{}_{\\! {\\rm LD}}}(0)}~  \\tau \\equiv\n\\gamma_{{}_{\\! {\\rm LD}}}.\n\\end{eqnarray}\n Here, $\\nu_{{}_{\\! {\\rm LD}}}$ is the classical hydrodynamic\nlimit $\\nu^{(1)}$ (\\ref{shearvisone}) for the viscosity\ncoefficient, $\\tau$ the relaxation time\n(\\ref{relaxtime}), (\\ref{widthG}) for $\\omega =0$,\n\\bel{wdthGT}\n\\tau\\equiv\\tau(0,T)=\\frac{\\hbar }{{\\it \\Gamma}(0,T)},\\quad\n{\\it \\Gamma}(0,T)= \\frac{\\pi^2 T^2 }{{\\it \\Gamma}_0 \\left(1+\n\\pi^2 T^2 \/ c^2\\right)}.\n\\end{equation}\n However, our result\n(\\ref{friction0}) for the\nclassical liquid-drop model of irrotational flow, if only extended\nto include the two-body viscosity, differs from the one found in\n\\cite{nixsierk,davinixsierk}\nby an additional factor of\n$(2L+1)\/L$, see \\cite{magkohofsh,ikmPRC}. We neglected the fourth order\nterms in ${\\bar T}$ (Sec.\\ \\ref{lowtemlim}) in\n(\\ref{mass0}), (\\ref{stiffness0})\nand (\\ref{friction0}) because of the presence of more important\nlower order terms there. For the coefficient $\\Upsilon$ in the term\nproportional to $1\/\\omega $ in (\\ref{respfunas}), one obtains\n\\bel{gamma0hp}\n\\Upsilon=\\frac{13A^{1\/3}\\varepsilon_{{}_{\\! {\\rm F}}} \\pi^4 {\\bar T}^4 s_0^2\n}{60 b_{{}_{\\! S}}(L-1)(L+2) \\tau} =\n\\frac{24 \\mathcal{G}_0 \\varepsilon_{{}_{\\! {\\rm F}}} A^{1\/3}\n}{b_{{}_{\\! S}}(L-1)(L+2)}~ \\nu^{(2)}.\n\\end{equation}\n The expression in the middle\nof these equations turns into zero for the Landau--Placzek kind\n($n=0$) of the solutions (\\ref{shp}) of dispersion equation\n(\\ref{despeq}) for the velocity $s$. It is, however, finite for the first\nsound mode $n=1$ presented in (\\ref{sfirst}) for\n$s_0=s_0^{(1)}$. The second equation being true only for the first\nsound mode was obtained by making use of (\\ref{sfirst}) for\nthe first sound velocity $s_0^{(1)}$ and (\\ref{shearvishp}) for\nthe viscosity component $\\nu^{(2)}$  up to small temperature\ncorrections of the next order. The both equations (\\ref{gamma0hp}) show\nthe main term in the temperature expansion of the coefficient\n$\\Upsilon$ in front of $1\/\\omega $ in (\\ref{respfunas}). Note that it\nappears in the order ${\\bar T}^4$ and can not be neglected for enough\nsmall frequencies $\\omega $. As seen from (\\ref{respfunas})\nconsidered for the case of the collective response, i.e., with\nomitted index \"in\" in $C_{\\rm in}$ of (\\ref{respfunas}), for\nenough small frequencies $\\omega $, there is the pole which equals\napproximately $i\\Upsilon\/2$. Therefore, the physical meaning of the parameter\n$\\Upsilon$ (\\ref{gamma0hp}) is a \"width\"  of the overdamped pole in\nthe asymptotic collective response function $\\chi^{\\rm coll}(\\omega )$ for\nenough low frequencies. As shown below, this pole and\ncorresponding pole of the intrinsic response function\n(\\ref{respfunas}) is overdamped. It is similar to the Landau--Placzek\npole in the infinite nuclear matter and to the nuclear heat pole found in\n\\cite{hofmann}, see more detailed discussion in\nSec.\\ \\ref{heatcorrfun}. The \"width\" $\\Upsilon$ (\\ref{gamma0hp}) of such\n\"heat pole\" is inversely proportional to the relaxation time\n$\\tau$ and increases with temperature and particle number.\nNote also that this \"width\" is proportional to the component\n$\\nu^{(2)}$ (\\ref{shearvishp}) of the viscosity discussed in\nSec.\\ \\ref{viscosthermcond}. It is somewhat similar to the\nviscose part of the standard expression for the first sound\n\"width\" $\\Gamma$ (\\ref{gammaland}) in terms of the first component\n$\\nu^{(1)}$ of the viscosity coefficient (\\ref{shearvisonehp}),\n$\\Gamma \\propto \\nu^{(1)}$. However, there is in\n(\\ref{gamma0hp}) the surface energy constant $b_{{}_{\\! S}}$ and\nparticle number factor $A^{1\/3}$ which are both the specific\nparameters of a {\\it finite} Fermi-liquid drop.\n\nThus, the denominator of the hydrodynamical response function\n(\\ref{respfunas}) contains the two friction terms. One of them is\nproportional to the friction coefficient $\\gamma$, $\\gamma\n\\propto \\nu^{(1)}$, and another one which is proportional to\n$\\Upsilon$ ($\\Upsilon \\propto \\nu^{(2)}$). We shall consider in\nthe next two Secs. \\ref{zerofreqlim} and \\ref{heatcorrfun}\nthe two limit cases neglecting first the heat pole  $\\Upsilon$\nterm for enough large frequencies $\\omega $ within the hydrodynamic\napproximation $\\omega \\tau \\ll 1$, and then, the $\\gamma$ friction one\nfor smaller frequencies with the dominating heat pole, respectively.\n\n\n\\subsubsection{HYDRODYNAMIC SOUND RESPONSE}\n\\label{zerofreqlim}\n\nFor enough large frequencies  $\\omega $ within the frequent\ncollisional (hydrodynamic) regime,\n\\begin{eqnarray}\\label{hydpolcond}\n&&\\omega _{\\rm crit} \\tau \\ll \\omega  \\tau \\ll 1 \\;, \\qquad \\omega _{\\rm crit}\\tau\n=\\tau~\\sqrt{C\\Upsilon \/2\\gamma}  \\nonumber\\\\\n&=&\\frac{\\pi^2\\sqrt{13 \\mathcal{G}_0 \\mathcal{G}_1}\n~s_0~\\nu_{{}_{\\! {\\rm LD}}}(0) }{36(L-1)~\\nu_{{}_{\\! {\\rm LD}}}(T)}~{\\bar T}^2 ,\n\\end{eqnarray}\n one finds the\n first sound ($i=1$) solution $s$ (\\ref{sfirst0}).\nIn this case, one can neglect\nthe last term proportional to $1\/\\omega $ compared to the friction\nterm in the denominator of the asymptotic expression\n(\\ref{respfunas}). The critical frequency $\\omega _{\\rm crit}$ is defined\nin the second equation of (\\ref{hydpolcond}) as a frequency\nfor which these two compared terms coincide,\n$\\omega _{\\rm crit}=\\sqrt{C\\Upsilon\/2\\gamma}=\\omega _{{}_{\\! {\\rm LD}}}\\sqrt{M\\Upsilon\/2\\gamma}~$\n($\\omega _{{}_{\\! {\\rm LD}}}=\\sqrt{C\/M}$ is the frequency of the  surface\nliquid-drop\nvibrations). The critical value $\\omega _{\\rm crit}\\tau$ increases with\nincreasing temperature as ${\\bar T}^2$ and does not depend on\nparticle number for the first sound mode $n=1$.  It equals zero\nfor the Landau--Placzek branch $n=0$, according to\n(\\ref{gamma0hp}) for $\\Upsilon$. For the $n=1$ mode,\n$\\omega _{\\rm crit}\\tau$ is small for all temperatures $T ~\\siml~\n10~\\mbox{MeV}$, $\\omega _{\\rm crit}\\tau \\approx 0.6~ {\\bar T}^2 ~\\ll~ 1$ at\ntypical values of the parameters, $\\varepsilon_{{}_{\\! {\\rm F}}}=40~\\mbox{MeV}$ and\n$r_0=1.2~\\mbox{fm}$, and\nfor a value $\\mathcal{C}$ of the Skyrme forces considered in\n\\cite{strutmagden},\n$\\mathcal{C} =80~\\mbox{MeV} \\cdot \\mbox{fm}^5$,\nwhich is somewhat larger than those of \\cite{BMRV}\n(Sec.\\ \\ref{npcorivgdr} and Appendix D)\nin the ESA, where $A^{-1\/3}$ is assumed to be\nsmall.  We took here and below $L=2$ for the quadrupole\nvibrations, $\\mathcal{F}_0=-0.2$, $\\mathcal{F}_1=-0.6$ for the Landau\nconstants which are close to the values common used for the\ncalculations of the nuclear giant multipole resonances\n\\cite{spethwoud,hasse}, a little more \"realistic\" than in\n\\cite{kolmagpl,magkohofsh}. For frequencies $\\omega $ within the\ncondition (\\ref{hydpolcond}), we arrive at the\noscillator-like response function,\n\\bel{oscresponse}\n\\chi(\\omega )\\equiv k^{-2}\\chi_{\\rm osc}(\\omega ) =k^{-2}\\left(-M \\omega ^2 -i\n\\gamma \\omega  +C_{\\rm in}\\right)^{-1},\n\\end{equation}\n with all hydrodynamic\ntransport coefficients presented in (\\ref{mass0}),\n(\\ref{stiffness0}) and (\\ref{friction0}). In the middle of (\\ref{hydpolcond}),\n$\\chi_{\\rm osc}(\\omega )$ is\nthe \"intrinsic\" oscillator response function which describes the\ndynamics in terms of the $Q(t)$ variable for the collective\nharmonic oscillator potential. As seen now, the constants $M$, $C$\nand $\\gamma$ were naturally called above as the transport\ncoefficients: The collective response function $\\chi^{\\rm coll}(\\omega )$\nwithin the approximation (\\ref{hydpolcond}) is the same\n(\\ref{oscresponse}) but with omitted index \"in\" in the\nstiffness coefficient, as noted above. This remark is related also\nto the oscillator $QQ$- response function $\\chi_{\\rm osc}^{\\rm coll}(\\omega )$\nuseful for the following analysis of the response functions in\nterms of the transport coefficients,\n\\bel{oscrespcoll}\n\\chi_{\\rm osc}^{\\rm coll}(\\omega ) =\\left(-M \\omega ^2 -i \\gamma \\omega \n+C\\right)^{-1}.\n\\end{equation}\n We obtain the $QQ$- response functions from\nthe $FF$-ones, for instance, from $\\chi(\\omega )$ (\\ref{oscresponse}),\nsimply multiplying by the constant $k^2$ because of the\nself-consistency condition (\\ref{selfconsist}). Note also that\nthe condition (\\ref{hydpolcond}) for the Landau--Placzek branch of\nthe solutions for the sound velocity $s$ [see (\\ref{shp})], is\nalways fulfilled for $\\omega \\tau \\ll 1$.\n\n\nIn order to compare our results with those of previous\ncalculations \\cite{hofmann},\nwe introduce the dimensionless quantity\n\\begin{eqnarray}\\label{eta}\n\\eta &=& \\gamma \/ \\left(2 \\sqrt{M |C|}\\right)  \\nonumber\\\\\n&=& \\frac{2 \\varepsilon_{{}_{\\! {\\rm F}}}}{5 p_{{}_{\\! {\\rm F}}} r_0 A^{1\/6}} ~\n\\sqrt{\\frac{6 L (L-1)\n\\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_1 }{(L+2) b_{{}_{\\! S}}}}\n~\\frac{~\\nu_{{}_{\\! {\\rm LD}}}(T)~\\tau\n}{~\\nu_{{}_{\\! {\\rm LD}}}(0)},\n\\end{eqnarray}\n see\n(\\ref{friction0}), (\\ref{mass0}) and (\\ref{stiffness0}) for\n$\\gamma$, $M$ and $C$, respectively. The quantity $\\eta$ in\n(\\ref{eta}) characterizes the effective damping rate of the\ncollective motion. Neglecting small temperature corrections of the\nviscosity coefficient $\\nu_{{}_{\\! {\\rm LD}}}=\\nu^{(1)}$ in (\\ref{eta}), see\n(\\ref{shearvisone}), and substituting\n(\\ref{relaxtime}), (\\ref{widthG}) for the relaxation time\n$\\tau$ at $\\omega =0$, one writes\n\\bel{eta0}\n\\eta \\approx \\frac{2 \\hbar\n\\varepsilon_{{}_{\\! {\\rm F}}} {\\it \\Gamma}_0 }{5 \\pi^2 p_{{}_{\\! {\\rm F}}} r_0 A^{1\/6}} ~\\sqrt{{6 L\n(L-1) \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_1 \\over {(L+2)b_{{}_{\\! S}}}}} ~\n\\frac{1 + \\pi^2 T^2 \/\nc^2}{T^2}.\n\\end{equation}\n This hydrodynamic effective friction $\\eta$\nmainly decreases with temperature as $1\/T^2$. For large\ntemperatures and finite cut-off parameter $c$,\nthe dimensionless friction parameter\n$~\\eta$ (\\ref{eta0}) approaches the\nconstant.\n\nWe have the two kind of poles of the response function\n(\\ref{oscresponse}) as roots of the quadratic polynomial in the\ndenominator, the overdamped poles, see\n\\cite{hofmann},\n\\begin{eqnarray}\\label{overdamp}\n\\omega _{\\pm}^{\\rm over}&=&-i \\Gamma_{\\pm}^{\\rm in}\/2,\\nonumber\\\\\n\\Gamma_{\\pm}^{\\rm in}&=&2\\varpi_{\\rm in}\\left(\\eta_{\\rm in}\\pm\n\\sqrt{\\eta_{\\rm in}^2-1}\\right),\\quad \\eta_{\\rm in}>1,\n\\end{eqnarray}\n and the\nunderdamped ones,\n\\bel{underdamp}\n\\omega _{\\pm}^{\\rm under}=\n\\varpi_{\\rm in}\\left(\\pm \\sqrt{1-\\eta_{\\rm in}^2} - i\n{\\eta_{\\rm in}}~\\right), \\quad \\eta_{\\rm in}<1\\;.\n\\end{equation}\n These solutions\ndepend on the two parameters,\n\\bel{varpietain}\n\\varpi_{\\rm in}=\\sqrt{\\frac{|C_{\\rm in}|}{M}}~ \\qquad \\mbox{and} \\qquad\n\\eta_{\\rm in}= \\frac{\\gamma }{2 \\sqrt{M|C_{\\rm in}|}} .\n\\end{equation}\n Note also that\nthe two hydrodynamic poles in (\\ref{oscresponse}) coincide\napproximately for the both branches $n=0$ and $1$ of solutions to\nthe dispersion equation (\\ref{despeq}) for the velocity $s$. The\ndifference between these two modes is related only to the last\nterm proportional to $\\Upsilon$ in the brackets of r.h.s. of\n(\\ref{respfunas}), and it was neglected under the condition\n(\\ref{hydpolcond}).\n\nFor the real and imaginary parts of the response function\n$\\chi(\\omega )$ (\\ref{oscresponse}) with the help of\n(\\ref{varpietain}) for the overdamped case (\\ref{overdamp}),\nfor instance, one gets for completeness\n[see (\\ref{overdamp})-(\\ref{varpietain})]\n\\begin{eqnarray}\\label{oscrespre}\n\\chi^{\\prime}(\\omega )&=& \\frac{1 }{4 M k^2\n\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}}\n\\left(\\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. {\\Gamma_{+}^{\\rm in} \\over\n{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}}\\right),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{oscrespim}\n\\chi^{\\prime\\prime}(\\omega )&=& \\frac{\\omega  }{4 M\nk^2\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}} \\left(\\frac{1\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. \\frac{1\n}{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right).\n\\end{eqnarray}\n\n\nFor a more simple case of the collective response in the FLDM, we\nomit index \"in\" in formulas of this section [see the comment\nafter (\\ref{respfunas})]. From (\\ref{eta}) for $\\eta$ with\nthe parameters used above for the estimate of $\\omega _{\\rm crit} \\tau$\nof (\\ref{hydpolcond}), and the \"standard\"\n${\\it \\Gamma}_0=33.3$ MeV \\cite{hofmann}; one has an overdamped motion, $\\eta > 1$,\nfor all temperatures $T \\siml 10~ {\\rm MeV}$ and particle numbers\n$A \\siml 230$, as seen from Fig.\\ \\ref{fig1}. Moreover, for such\ntemperatures and particle numbers, one can expand the \"widths\"\n$\\Gamma_{\\pm}$ in small parameter $MC\/\\gamma^2=(4\\eta^2)^{-1}$,\nsee (\\ref{overdamp}) omitting index \"in\". From\n(\\ref{overdamp}) (without index \"in\") one gets approximately\n\\bel{overdamp1}\n\\Gamma_{\\pm}=4\\varpi \\eta \\left\\{{1-\\left(4\n\\eta^{2}\\right)^{-1} \\atop {\\left(4 \\eta^2\\right)^{-1}}}\\right\\}\n= 2\\left\\{{\\gamma\/M \\atop {C\/\\gamma}}\\right\\},\\,\\,\n\\eta^2~ \\gg~ 1.\n\\end{equation}\n Fig.\\ \\ref{fig1} shows that the above\nmentioned parameter $1\/(4 \\eta^2)$ for the expansion in\n(\\ref{overdamp1}) is really small for all considered\ntemperatures. Using (\\ref{mass0}),\n(\\ref{stiffness0}), (\\ref{friction0}) for the transport\ncoefficients and  the definition of $\\tau$ (\\ref{wdthGT}) as in\nthe derivation of (\\ref{eta}), (\\ref{eta0}), one obtains from\n(\\ref{overdamp1})\n\\begin{eqnarray}\\label{Gammaplus}\n\\Gamma_{+}&=& \\frac{16  \\mathcal{G}_1\nL(L-1)\\varepsilon_{\\rm F}^2 }{5 \\left(p_{{}_{\\! {\\rm F}}} r_0\\right)^2~A^{2\/3}}~\n\\frac{~\\nu_{{}_{\\! {\\rm LD}}}(T) ~\\tau }{ ~\\nu_{{}_{\\! {\\rm LD}}}(0)} \\nonumber\\\\\n&\\approx& \\frac{16 \\hbar^2\n\\mathcal{G}_1 L(L-1)\\Gamma_0 \\varepsilon_{\\rm F}^2 }{5\n\\pi^2\\left(p_{{}_{\\! {\\rm F}}}r_0\\right)^2~A^{2\/3}}~ ~\n\\frac{1 + \\pi^2 T^2 \/ c^2\n}{T^2},\n\\end{eqnarray}\n \\begin{eqnarray}\\label{Gammaminus}\n\\Gamma_{-}&=&\\frac{5 b_{{}_{\\! S}} (L+2)\n}{6 \\varepsilon_{{}_{\\! {\\rm F}}}~A^{1\/3}}~ \\frac{~\\nu_{{}_{\\! {\\rm LD}}}(0)\n}{~\\nu_{{}_{\\! {\\rm LD}}}(T) ~\\tau} \\nonumber\\\\\n&\\approx& \\frac{5 \\pi^2 b_{{}_{\\! S}} (L+2) }{6 \\hbar {\\it \\Gamma}_0\n\\varepsilon_{{}_{\\!{\\rm F}}}~A^{1\/3}}~ ~\\frac{T^2 }{1 + \\pi^2 T^2\/c^2 }.\n\\end{eqnarray}\n One of\nthe \"widths\" specified by  $\\Gamma_{+}$ (\\ref{Gammaplus}) is\nmainly the decreasing function of temperature, $\\Gamma_{+}\n\\propto \\tau\\propto 1\/T^2$ at low temperatures. It is typical for\nthe hydrodynamic modes as the first sound vibrations in normal\nliquids; in contrast to another \"width\" $\\Gamma_{-}$\n(\\ref{Gammaminus}), $\\Gamma_{-} \\propto 1\/\\tau \\propto T^2$,\nsimilar to the zero sound damping in relation to the $\\tau $-\ndependence. They both become about a constant for high\ntemperatures, due to the cut-off factor $c$.\n\nNote, $\\Gamma_{+}$ (\\ref{Gammaplus}) decreases with particle\nnumber as $A^{-2\/3}$ while $\\Gamma_{-} \\propto A^{-1\/3}$, see\n(\\ref{Gammaminus}). The different $A$-dependence of the \"widths\"\n$\\Gamma_{-}$ (\\ref{Gammaminus}) and $\\Gamma_{+}$\n(\\ref{Gammaplus}) can not be nevertheless\nreferred even formally to the\nso-called  \"one- and two-body\ndissipation\", respectively.  (Collisions with potential walls\nwithout the integral collision term in the Landau--Vlasov equation\nbut with the mirror or diffused boundary conditions might lead to\nthe \"widths\" proportional to $\\Omega$\nin (\\ref{x}), $\\Omega\n\\propto A^{-1\/3}$, as in equation (49) of \\cite{komagstrvv} or\nthrough the wall formula \\cite{wall,blocki,BMY}.)\nThey both depend on the collisional\nrelaxation time $\\tau$ and correspond to the \"two-body\"\ndissipation. The latter means here the collisional damping of the\nviscose Fermi liquid as in \\cite{kolmagpl,magkohofsh}. The\nphysical source of the damping in the both cases is the same\ncollisions of particles in the nuclear volume, due to the integral\ncollision term (\\ref{intcoll}) with the relaxation time $\\tau$.\nWe would like to emphasize, however, that the collisional\n$\\Gamma_{-}$ (\\ref{Gammaminus}) depends on the surface energy\nconstant $b_{{}_{\\! S}}$ and disappears proportionally to $A^{-1\/3}$\nwith increasing particle number $A$ like $\\Omega$\nof (\\ref{x}) because we took into account a {\\it finite} size of\nthe system through the boundary conditions\n(\\ref{bound1}), (\\ref{bound2}). An additional overdamped pole with\nthe \"width\" $\\Gamma_{-}$ (\\ref{Gammaminus}) appears because of the\n{\\it finiteness of the system and collisions inside the nucleus.}\nThis looks rather in contrast to the wall friction \\cite{blocki,BMY}\ncoming from\nthe collisions with the only walls of the potential well.\n\n\nWe shall come back now to the {\\it intrinsic}\nresponse function $\\chi(\\omega )$ (\\ref{oscresponse}).\nFor the \"intrinsic stiffness\" $C_{\\rm in}$, one has\n\\bel{Cintr}\nC_{\\rm in}=-\\left(1-kC\\right)\/k\\approx -1\/k.\n\\end{equation}\n In the last equation,\nwe neglected a small parameter $kC$,\n\\begin{eqnarray}\\label{smallpar}\nkC&=& \\frac{54\n(L-1)(L+2) \\mathcal{C} }{4 \\pi K r_0^5 A^{2\/3}} \\nonumber\\\\\n&\\approx&\n\\frac{9(L-1)(L+2)\n\\mathcal{C} }{4 \\pi \\mathcal{G}_0 \\varepsilon_{{}_{\\! {\\rm F}}} r_0^5 A^{2\/3}}\n \\approx \\frac{3 }{A^{2\/3}},\n\\end{eqnarray}\n for the typical values of the\nparameters mentioned above before (\\ref{oscresponse}). We\nneglected also small temperature corrections of\n(\\ref{incompradexp}) for the in-compressibility modulus\n$K$, $K=K^{\\varsigma}$, in the second equation of (\\ref{smallpar}).\n\n\nUsing a smallness of the parameter $kC$ (\\ref{smallpar}), we shall\nget now the relation of the coupling constant $k^{-1}$ with the\nisolated susceptibility $\\chi(0)$ and stiffness $C$ as in\nequation (3.1.26) of \\cite{hofmann}. For this purpose, we take the limit\n$\\omega  \\rightarrow 0$ in (\\ref{oscresponse}) for the \"intrinsic\"\nresponse function $\\chi(\\omega )$ and  expand then the obtained\nexpression for $\\chi(0)$,\n$\\chi(0)=k^{-2}C_{\\rm in}^{-1}=-k^{-2}(1-kC)^{-1}$, in powers of the\nsmall parameter $kC$ (\\ref{smallpar}) up to {\\it second} order\nterms. As result, we arrive at the relation\n\\bel{kstiffCchi0}\n-k^{-1}=\\chi(0) + C.\n\\end{equation}\n\nThe liquid-drop transport coefficients $M$ (\\ref{mass0}), $C$\n(\\ref{stiffness0}) and $\\gamma$ (\\ref{friction0}) can be now\ncompared with the ones in the \"zero frequency limit\" $M(0)$,\n$C(0)$ and $\\gamma(0)$, respectively, defined by\nequations (3.1.84)-(3.1.86) in \\cite{hofmann}:\n\\bel{C0}\nC(0)=-\\left(1\/k\n+ \\chi(0)\\right)=C,\n\\end{equation}\n\\bel{gamma0} \\gamma(0)=-i\\left(\\partial\n\\chi(\\omega )\/\\partial \\omega \\right)_{\\omega =0}= \\gamma,\n\\end{equation}\n \\begin{eqnarray}\\label{M0}\nM(0)&=&\\left(\\frac{1}{2}\\partial^2\n\\chi(\\omega )\/\\partial\\omega ^2\\right)_{\\omega =0}= M\\left(1+\\gamma^2\nk\/M\\right)  \\nonumber\\\\\n&=&M\\left(1+4\\eta_{\\rm in}^2\\right).\n\\end{eqnarray}\n Expanding $\\chi(\\omega )$\nnear the zero frequency $\\omega =0$ in the secular equation\n(\\ref{seculareq}), see \\cite{hofmann}, we assumed here and will\nshow below that the \"intrinsic\" response function $\\chi(\\omega )$ is\na smooth function of $\\omega $ for small frequencies $\\omega $ within the\nhydrodynamic condition (\\ref{hydpolcond}). The second and third\nequations in (\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0}) were\ngot approximately in the ESA from (\\ref{oscresponse}) up to\nsmall corrections in the parameter $kC$ with help of\n(\\ref{kstiffCchi0}), second equation in\n(\\ref{varpietain}) and (\\ref{Cintr}). As the liquid drop\nstiffness $C$ equals approximately the stiffness in the \"zero\nfrequency limit\" $C(0)$, according to (\\ref{C0}), the\nequation (\\ref{kstiffCchi0}) is identical to the relation\n(\\ref{kstiffC0chi0}) of the general response-function theory\n\\cite{hofmann} within the same ESA. As seen from\n(\\ref{C0})-(\\ref{M0}), the stiffness $C(0)$ and friction\n$\\gamma(0)$  equal to the liquid drop parameters, but the inertia\n$M(0)$ differs from the liquid-drop mass value $M$ by a positive\ncorrection.\n\nFor the definition of transport coefficients in \"the zero\nfrequency limit\" (\\ref{C0})-(\\ref{M0}), we needed to know also\nthe properties of the \"intrinsic\" response function in the secular\nequation (\\ref{seculareq}),\nconcerning its pole\nstructure. For the \"intrinsic\" case the quantity\n$\\eta_{\\rm in}$, see (\\ref{varpietain}), plays a\nrole similar to the effective damping $\\eta$ (\\ref{eta}) for the\ncollective motion.\nMoreover, $\\eta_{\\rm in}$  determines the correction to the liquid\ndrop mass parameter $M$ in (\\ref{M0}) for the inertia $M(0)$\nin \"the zero frequency limit\". Due to a smallness of the parameter\n$kC$ (\\ref{smallpar}), $\\eta_{\\rm in}$ is much smaller than $\\eta$\n(\\ref{eta0}) for large particle numbers $A \\approx 200-230$, as\nseen from Fig.\\ \\ref{fig1},\n\\begin{eqnarray}\\label{etaintr}\n &&\\eta_{\\rm in} = \\frac{\\gamma\n}{ 2 \\sqrt{M |C_{\\rm in}|}} \\approx \\eta^2 kC ~~~ \\nonumber\\\\\n&\\approx& \\frac{3 (L-1) \\hbar\n{\\it \\Gamma}_0 \\varepsilon_{{}_{\\! {\\rm F}}} }{5 \\pi^2 p_{{}_{\\! {\\rm F}}} r_0}~ \\sqrt{\n\\frac{6 l \\mathcal{C}\n\\mathcal{G}_1 }{\\pi \\mathcal{G}_0 b_{{}_{\\! S}} r_0^5~A}} ~\\frac{1 + \\pi^2\nT^2 \/ c^2 }{T^2},~~~\n\\end{eqnarray}\n see (\\ref{friction0}), (\\ref{mass0}), (\\ref{stiffness0}),\n(\\ref{Cintr}), and (\\ref{smallpar}).\n\n\nFor such heavy nuclei ($A \\approx 200-230$) and enough large\ntemperatures, $T \\simg 5 ~\\mbox{MeV} $, one has formally the\n\"underdamped\" pole structure (\\ref{underdamp}) ($\\eta_{\\rm in} < 1$)\nfor the parameters selected above. Using the expansion of the\npoles $\\omega _{\\pm}^{\\rm in}$ (\\ref{underdamp}) of the intrinsic\nresponse function in powers of small $\\eta_{\\rm in}^2$\n(\\ref{etaintr}) up to terms of the order of $\\eta_{\\rm in}^4$,  one\nwrites\n\\begin{eqnarray}\\label{underdamp1}\n&&\\omega _{\\rm in}^{\\pm} = \\omega _{\\rm in}\\left[\\pm\n\\left(1-\\frac{1}{2} \\eta_{\\rm in}^2\\right) -i \\eta_{\\rm in}\\right] \\nonumber\\\\\n&\\approx& \\pm \\varpi_{{}_{\\! {\\rm LD}}}\/\\sqrt{kC}-i\\Gamma_{+}\/4 \\quad{\\rm\nfor}\\quad \\eta_{\\rm in}^2 \\ll 1;\n\\end{eqnarray}\n see (\\ref{varpietain}), (\\ref{overdamp1}) (for $\\Gamma_{+}$ on the\nvery r.h.s.), and (\\ref{Cintr}) (for $kC$ there) in the\nderivation of the second equation. The \"underdamped\" poles\n$\\omega _{\\rm in}^{\\pm}$ approach the real axis on a large distance from\nthe imaginary one as compared to the liquid drop frequency\n$\\varpi_{{}_{\\! {\\rm LD}}}=\\sqrt{C\/M}$, $|\\omega _{\\rm in}^{\\pm}| \\gg\n\\omega _{{}_{\\! {\\rm LD}}}$. They\nhave a small \"width\" $2\\omega _{\\rm in}\\eta_{\\rm in}=\\gamma\/M\\propto 1\/T^2$\nfor our choice of large temperatures ($T \\simg 5 {\\rm MeV}$); see\n(\\ref{varpietain}), (\\ref{friction0}), (\\ref{mass0}), and\n(\\ref{wdthGT}). By this reason, for the \"underdamped\" case of\nsmall $\\eta_{\\rm in}^2$ and low frequencies\n$\\omega  \\siml \\omega _{{}_{\\! {\\rm LD}}}$, the\nintrinsic response function $\\chi(\\omega )$ is a smooth function of\n$\\omega $.\n\nFor smaller temperatures $T ~\\siml~ 4~ \\mbox{MeV}$ and for our\nparameters used in (\\ref{etaintr}), one has the \"overdamped\" poles\n(\\ref{overdamp}) of the intrinsic response function $\\chi(\\omega )$,\n$\\eta_{\\rm in} > 1$. For such temperatures, $\\eta_{\\rm in}$\n(\\ref{etaintr}) is enough large. We can use therefore the\nexpansion of the \"widths\" $\\Gamma_{\\pm}^{\\rm in}$ of\n(\\ref{overdamp}) in a small parameter $(M\n\\omega _{\\rm in}\/\\gamma)^2=(4 \\eta_{\\rm in}^2)^{-1}$ (see Fig.\\ \\ref{fig1}),\n\\begin{eqnarray}\\label{overdampintr}\n&&\\Gamma_{\\pm}^{\\rm in}= 4\\omega _{\\rm in}\\eta_{\\rm in}\n\\left\\{{1-\\left(4 \\eta_{\\rm in}^{2}\\right)^{-1} \\atop {\\left(4\n\\eta_{\\rm in}^2\\right)^{-1}}}\\right\\} \\nonumber\\\\\n&\\approx& 2~\\left\\{{\\gamma\/M\n\\atop {1\/(k\\gamma)}}\\right\\} \\quad {\\rm for} \\quad\n\\left(4\\eta_{\\rm in}^2\\right)^{-1} \\ll 1,\n\\end{eqnarray}\nsee\n(\\ref{varpietain}), (\\ref{Cintr}) and (\\ref{smallpar}). The\n\"intrinsic width\" $\\Gamma_{+}^{\\rm in}$ in the upper row of\n(\\ref{overdampintr}) and the \"collective width\" $\\Gamma_{+}$\n(\\ref{Gammaplus}) [see (\\ref{overdamp1}] are the same.\n$\\Gamma_{-}^{\\rm in}$ in the low row has the temperature dependence\nas for $\\Gamma_{-}$ in (\\ref{Gammaminus}) but a different\n$A$-dependence, $\\Gamma_{-}^{\\rm in} \\propto A^{1\/3}$ [see\n(\\ref{kfld}) and (\\ref{friction0})]. Moreover,\n$\\Gamma_{-}^{\\rm in} \\gg \\Gamma_{-}$ because of smallness of the\nparameter $kC$ (\\ref{smallpar}). It becomes clear after dividing\nand multiplying the last expression for the $\\Gamma_{-}^{\\rm in}$ in\n(\\ref{overdampintr}) by the factor $C$ and using\n(\\ref{stiffness0}) and (\\ref{friction0}).\n\n\nThe \"intrinsic width\" $\\Gamma_{+}^{\\rm in}$, see\n(\\ref{overdampintr}), is mainly larger than $\\Gamma_{-}^{\\rm in}$.\nThey become comparable when increasing temperature, i.e.,\n$\\Gamma_{+}^{\\rm in}\\simg \\Gamma_{-}^{\\rm in}$. As $\\Gamma_{-}^{\\rm in}$\nfrom (\\ref{overdampintr}),\n\\bel{Gammaminintr}\n\\Gamma_{-}^{\\rm in} =\\frac{10 \\pi \\mathcal{G}_0 b_{{}_{\\! S}}\nr_0^5 ~A^{1\/3}}{27 (L-1) \\mathcal{C} ~\\tau},\n\\end{equation}\n is large compared to the\ncharacteristic collisional frequency $1\/\\tau$\n(for the same choice of the parameters) the both poles are far\naway from the zero, see more discussions of the \"intrinsic widths\"\nbelow in connection with the heat pole in the next section\n\\ref{heatcorrfun}. Therefore, the intrinsic response function\n$\\chi(\\omega )$ (\\ref{oscresponse}) is a smooth function of $\\omega $ for\nthe \"overdamped\" case of enough large $\\eta_{\\rm in}$ used in\nthe derivations of (\\ref{overdampintr}) as for the\n\"underdamped\" one discussed above. Thus, we expect that the \"zero\nfrequency limit\" based on the expansion of the intrinsic response\nfunction $\\chi(\\omega )$ is a good approximation for low frequencies\nlarger the critical value $\\omega _{\\rm crit}$ within the\nhydrodynamic condition (\\ref{hydpolcond}) for all considered\ntemperatures.\nIt means that the definition of\nthe transport coefficients in this limit\n(\\ref{C0})-(\\ref{M0}) is justified within the hydrodynamic\napproximation (\\ref{hydpolcond}).\n\nThe correction to the liquid drop mass parameter in the inertia\n$M(0)$ (\\ref{M0}) is always positive. This correction is the\ndecreasing function of the temperature and particle number which\ncan be presented approximately as\n\\begin{eqnarray}\\label{masscorr}\n&&(M(0)-M)\/M =\nk\\gamma^2\/M =4\\eta_{\\rm in}^2 \\nonumber\\\\\n&\\propto& ~\\left(1 + \\pi^2 T^2 \/\nc^2\\right)^2 \/ \\left(A~ T^4\\right),\n\\end{eqnarray}\n see (\\ref{etaintr}).\nFor smaller temperatures when the expansion in\n(\\ref{overdampintr}) is justified, this correction is equal\napproximately to the ratio of the \"intrinsic widths\"\n$\\Gamma_{+}^{\\rm in}\/\\Gamma_{-}^{\\rm in}$ taken from (\\ref{overdampintr}).\nThe relative mass correction\n(\\ref{masscorr}) and the \"intrinsic width\" ratio\n$\\Gamma_{+}^{\\rm in}\/\\Gamma_{-}^{\\rm in}$, see (\\ref{overdampintr}),\ndecreases with temperature $T$\nmainly as\n$1\/T^4$ if $T$ is not too big, as shown in Fig.\\ \\ref{fig1}.\nThe dimensionless inertia correction (\\ref{masscorr}) is proportional\napproximately to $1\/A$. The zero frequency mass\n$M(0)$ exceeds much the liquid drop inertia and turns asymptotically to\nthe latter for high temperatures, see Fig.\\ \\ref{fig1}.\nNote, for enough large temperatures $T \\simg 5 {\\rm MeV}$ and\nparticle numbers $A \\sim 200$ when $\\eta_{\\rm in}^4$ terms can be\nneglected in accordance with  (\\ref{etaintr}), all zero\nfrequency transport coefficients $C(0)$, $\\gamma(0)$ and $M(0)$\n[see (\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0})] approach the\ncorresponding liquid drop parameters.\n\n\n\nIt would be interesting now to get the \"overdamped\" correlation\nfunction $\\psi^{\\prime\\prime}(\\omega )$ determined by the imaginary\npart of the corresponding response function\n$\\chi^{\\prime\\prime}(\\omega )$ (\\ref{oscrespim}) through the\nfluctuation-dissipation theorem, see (\\ref{fludiptheor}) and\n (\\ref{corrfun}). In the semiclassical approximation\n(\\ref{corrfun}) and (\\ref{oscrespim})\nfor the first sound mode, one writes\n\\begin{eqnarray}\\label{corroscim}\n &&\\frac{1}{T}\\psi^{s \\; \\prime\\prime}(\\omega )=\\frac{2}{\\omega }\\chi^{\\prime\\prime}(\\omega )\n= \\frac{2 }{4 M\nk^2\\varpi_{\\rm in}\\sqrt{\\eta_{\\rm in}^2+1}} \\nonumber\\\\\n&\\times&\\left(\\frac{1\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4}- \\frac{1\n}{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right).\n\\end{eqnarray}\n Using the approximations\nas in (\\ref{overdampintr}) and (\\ref{Cintr}), one gets from\n(\\ref{corroscim})\n\\begin{eqnarray}\\label{corroscim1}\n&&\\frac{1}{T}\\psi^{s\n\\;\\prime\\prime}(\\omega )= \\frac{1 }{k} \\left(\\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4} \\right.\\nonumber\\\\\n&-& \\left. \\frac{1 }{4 \\eta_{\\rm in}^2}\\;\n\\frac{\\Gamma_{+}^{\\rm in} }{\\omega ^2+(\\Gamma_{+}^{\\rm in})^2\/4}\\right)\n\\approx \\frac{1}{k}\\; \\frac{\\Gamma_{-}^{\\rm in}\n}{\\omega ^2+(\\Gamma_{-}^{\\rm in})^2\/4}.\n\\end{eqnarray}\n The second Lorentzian in the\nmiddle is negligibly small compared to the first one because\n\\bel{gammapmincond}\n\\Gamma_{+}^{\\rm in}\\simg \\Gamma_{-}^{\\rm in} \\gg\n1\/\\tau \\gg \\omega ,\n\\end{equation}\n and $4\\eta_{\\rm in}^2$ is large in these\nderivations, see the discussions in between\n(\\ref{overdampintr}) and (\\ref{Gammaminintr}). It seems that\nwe are left with the Lorentzian term of this correlation function\non very right of (\\ref{corroscim1}) which looks as the\nLandau--Placzek heat-pole correlation function\n(\\ref{corrfunhphof}) and equation (4.3.30) of \\cite{hofmann} with\nobvious constants $\\psi^{(0)}$ and ${\\it \\Gamma}_T$. However, we can\nnot refer the found correlation function\n$\\psi^{s\\;\\prime\\prime}(\\omega )$ (\\ref{corroscim1}) to the heat pole\none. The \"width\" $\\Gamma_{-}^{\\rm in}$ of (\\ref{overdampintr}) in\n(\\ref{corroscim1}) is finite and large compared to  the\ncharacteristic collision frequency $1\/\\tau$ which, in turn, is\nmuch larger considered frequencies $\\omega $, as shown above, see\n(\\ref{gammapmincond}). The limit $\\Gamma_{-}^{\\rm in} \\to 0$ for a\nfixed finite $\\omega$  and the corresponding\n$\\delta(\\omega )$-function which would show the relation to the heat\npole correlation function do not make sense within the\napproximation (\\ref{hydpolcond}) used in (\\ref{corroscim1}).\nIn particular, the response (\\ref{oscresponse}) and the correlation\n(\\ref{corroscim1}) functions were derived for\nenough large frequencies $\\omega  \\gg \\omega _{\\rm crit}$ due to the\ncondition (\\ref{hydpolcond}). Note also that the inertia\nparameter $M$ (\\ref{M0}) is not zero, as it should be for the heat\npole.\n\n\n\\subsubsection{HYDRODYNAMIC CORRELATIONS AND HEAT POLE}\n\\label{heatcorrfun}\n\nFor lower frequencies $\\omega $, which are smaller the critical value\n$\\omega _{\\rm crit}$, we should take into account the last additional term\nin the denominator of (\\ref{respfunas}) for the response\nfunction. For such small frequencies, this friction term being\nproportional to $\\Upsilon$ (\\ref{gamma0hp}) becomes dominating as\ncompared to the liquid-drop one $\\gamma=\\gamma_{{}_{\\! {\\rm LD}}}$. Within this\napproximation, we shall derive the heat-pole response and\ncorrelation functions, and relate $\\Upsilon$\n(\\ref{gamma0hp}) of (\\ref{respfunas}) with the corresponding heat\npole friction. This subsection will be ended by\ndiscussions of the nuclear ergodicity.\n\n\nFor smaller frequencies,\n\\bel{heatpolcond}\n\\omega  \\tau \\ll\n\\omega _{\\rm crit}\\tau \\ll 1,\n\\end{equation}\n (see the second equation in\n(\\ref{hydpolcond}) for the critical frequency $\\omega _{\\rm crit}$),\none can neglect the friction $i\\gamma \\omega $ term in the\ndenominator of the asymptotic response function (\\ref{respfunas})\nas compared to the last one, $\\gamma \\omega  \\ll \\Upsilon C\/ 2\\omega $. The\nmass term there is even smaller than the friction one for\nfrequencies $\\omega  \\siml \\omega _{\\rm crit}$ for  the considered parameters\nand will be neglected too, $M \\omega ^2 \\ll \\gamma\\omega $. In this\napproximation, from (\\ref{respfunas}) one\nobtains the heat pole response function $\\chi(\\omega ) \\approx\n\\chi^{\\rm hp}(\\omega )$, which is similar to\n(\\ref{chicollhp}), (\\ref{chichpqom}) for the infinite nuclear\nmatter,\n\\bel{hprespfunas}\n\\chi ^{\\rm hp}(\\omega )= \\frac{\\omega  }{k^2 C_{\\rm in}\n\\left(\\omega  +i \\Gamma^{\\rm hp}\/2\\right)} \\approx - \\frac{\\omega  }{k\n\\left(\\omega  +i \\Gamma^{\\rm hp}\/2\\right)},\n\\end{equation}\n where\n\\bel{heatpole}\n\\Gamma^{\\rm hp}= -C \\Upsilon\/C_{\\rm in} \\approx kC\\Upsilon.\n\\end{equation}\n In these derivations, we used the\nspecific properties of the {\\it intrinsic} response functions\nwhich we now are interested in for analysis of the correlation\nfunctions and ergodicity conditions \\cite{hofmann}. In\n(\\ref{hprespfunas}) and in all approximate equations below in this subsection,\nwe\napplied also the expansion in small parameter $kC$\n(\\ref{smallpar}) as in (\\ref{Cintr}).\n\n\nThe real and imaginary parts of the response function\n$\\chi ^{\\rm hp}(\\omega )$ (\\ref{hprespfunas}) are, respectively,\n\\begin{eqnarray}\\label{rehprespas}\n\\chi ^{\\rm hp\\;\\prime}(\\omega )&=& \\frac{\\omega ^2 }{k^2 C_{\\rm in} \\left[\\omega ^2\n+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx& - \\frac{\\omega ^2 }{k \\left[\\omega ^2\n+(\\Gamma^{\\rm hp})^2\/4\\right]},\n\\end{eqnarray}\n\\begin{eqnarray}\\label{imhprespas}\n\\chi^{\\rm hp\\;\\prime\\prime}(\\omega )&=& - \\frac{\\omega  \\Gamma^{\\rm hp} }{2 k^2 C_{\\rm in}\n\\left[\\omega  ^2+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx& \\frac{\\omega \n\\Gamma^{\\rm hp} }{2 k \\left[\\omega  ^2+(\\Gamma^{\\rm hp})^2\/4\\right]}\n\\end{eqnarray}\nup to small $kC$ corrections, see (\\ref{smallpar}).\n\nWe shall derive now the correlation function\n$\\psi^{{\\rm hp}\\,\\prime\\prime}(\\omega )$  applying the\nfluctuation-dissipation theorem (\\ref{corrfun}) to the \"intrinsic\" response\nfunction $\\chi ^{{\\rm hp}\\;\\prime\\prime}(\\omega )$ (\\ref{imhprespas})\nobtained in the asymptotic limit (\\ref{heatpolcond}). From\n(\\ref{corrfun}) and (\\ref{imhprespas}) one gets\n\\begin{eqnarray}\\label{corrfunashp}\n\\frac{1}{T}\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )&=& \\frac{2}{\\omega }\n\\chi ^{{\\rm hp} \\;\\prime\\prime}(\\omega ) = -\n\\frac{\\Gamma^{\\rm hp}}{k^2 C_{\\rm in}\\;\n\\left[\\omega  ^2+(\\Gamma^{\\rm hp})^2\/4\\right]} \\nonumber\\\\\n&\\approx&\n\\frac{\\Gamma^{\\rm hp}}{k \\left[\\omega  ^2+(\\Gamma^{\\rm hp})^2\/4\\right]}.\n\\end{eqnarray}\n This correlation\nfunction looks as the Landau--Placzek peak for the infinite Fermi\nliquid, see (\\ref{corrfunhphof}),\n\\bel{psifunhp}\n\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )= \\psi_{\\rm hp}^{(0)} \\frac{\\hbar\n{\\it \\Gamma}_T^{\\rm hp} }{(\\hbar \\omega )^2 + ({\\it\n\\Gamma}_T^{\\rm hp})^2\/4}.\n\\end{equation}\n It is identical to the r.h.s. of equation\n(4.3.30) in \\cite{hofmann}, but with the specific parameters\n$\\psi^{(0)}=\\psi_{\\rm hp}^{(0)}$ and\n${\\it \\Gamma}_T={\\it \\Gamma}_T^{\\rm hp}$,\n\\bel{psi0GammaT}\n\\frac{1}{T} \\psi_{\\rm hp}^{(0)}=-\\frac{1}{k^2 C_{\\rm in}}~\n\\approx \\frac{1}{k}, \\quad {\\it \\Gamma}_T^{\\rm hp} =\\hbar\n\\Gamma^{\\rm hp}\\approx \\hbar kC\\Upsilon.\n\\end{equation}\n The \"width\" $\\Gamma^{\\rm hp}$ in\n(\\ref{psi0GammaT}) is much smaller than the characteristic\ncollision frequency $1\/\\tau$,\n\\bel{Gammahptau}\n\\Gamma^{\\rm hp} = \\frac{13\n\\pi^4 \\mathcal{G}_1 \\mathcal{C}~ T^4 }{20\\varepsilon_{\\rm F}^4b_{{}_{\\! S}}r_0^5 {\\it\n\\Gamma}_0~A^{1\/3}~\\tau} \\ll \\frac{1}{\\tau},\n\\end{equation}\nsee (\\ref{gamma0hp})\nand (\\ref{smallpar}). The relationship (\\ref{Gammahptau}) for\n$\\Gamma^{\\rm hp}$ is in contrast to the one (\\ref{gammapmincond}) for\nthe \"intrinsic overdamped widths\" $\\Gamma_{\\pm}^{\\rm in}$\n(\\ref{overdampintr}) which are much larger the collision\nfrequency $1\/\\tau$ for the same selected parameters at all\ntemperatures $T \\siml 10$ MeV and particle numbers\n$A=200-230$.\n\nFor the following discussion of the friction coefficients,\nwe compare now the\n\"width\" $\\Gamma^{\\rm hp}$ (\\ref{heatpole}), (\\ref{Gammahptau}) with\n$\\Gamma_{-}^{\\rm in}$ in (\\ref{overdampintr})\n[see (\\ref{friction0}), (\\ref{stiffness0}), (\\ref{gamma0hp}),\n(\\ref{sfirst}), (\\ref{Cintr}) and (\\ref{kfld})],\n\\bel{zetaintr}\n\\frac{\\Gamma^{\\rm hp}}{\\Gamma_{-}^{\\rm in}} =\n\\frac{\\gamma C \\Upsilon }{2C_{\\rm in}^2} \\approx\n\\frac{1}{2}\\gamma C k^2 \\Upsilon.\n\\end{equation}\n For all temperatures and particle numbers\nwhich we discuss here, this ratio is small,\n\\bel{zetapar}\n\\frac{\\Gamma^{\\rm hp}}{\\Gamma_{-}^{\\rm in}}\n = \\frac{351 \\pi^2 (L-1) \\mathcal{C}^2 }{800\nb_{{}_{\\! S}}^2 r_0^{10}~A^{2\/3}} ~{\\bar T}^4 \\approx\n\\frac{15{\\bar T}^4 }{A^{2\/3}}.\n\\end{equation}\nIn the second\nequation of (\\ref{zetapar}) we used the same values of the\nparameters as in (\\ref{smallpar}). Note that the \"width\" of the\nLandau--Placzek peak $\\Gamma^{(0)}$, $\\Gamma^{(0)} \\sim\n\\tau_q^2\/\\tau \\ll 1\/\\tau$ for $\\tau_q \\ll 1$, is similar to\n$\\Gamma^{\\rm hp}$ and is unlike $\\Gamma_{\\pm}^{\\rm in}$\n(\\ref{overdampintr}) in (\\ref{corroscim1}) for the\nhydrodynamical sound correlation function. In contrast to the\nhydrodynamical sound case, see (\\ref{corroscim1}), we can consider\n(\\ref{corrfunashp}) for the correlation function approximation\nin the zero width limit $\\Gamma^{\\rm hp} \\to 0$ (or in the zero\ntemperature limit $T \\to 0$) taking any small but finite frequency\n$\\omega $ under the condition (\\ref{heatpolcond}). Therefore, for such\nfrequencies $\\omega $, the correlation function (\\ref{corrfunashp}) can\nbe approximated by $\\delta(\\omega )$-like function as in\n(\\ref{corrdeltafunlim}) for the correlation function of the\ninfinite Fermi liquid (\\ref{corrfunhphof}). Because of a very\nclose analogy of equation for the correlation function\n$\\psi^{{\\rm hp}\\,\\prime\\prime}(\\omega )$ (\\ref{psifunhp}) to the\nLandau--Placzek peak for the infinite Fermi-liquids in the\nhydrodynamic limit, see (\\ref{corrfunhphof}), and to\nequation (4.3.30) of \\cite{hofmann}, we associate the pole\n(\\ref{heatpole}) and corresponding asymptotics of the response\n (\\ref{hprespfunas})\nand correlation (\\ref{corrfunashp}) functions with the \"heat pole\". As in the case of the\ninfinite nuclear matter, this pole for the finite Fermi-liquid\ndrop is situated at zero frequency $\\omega =0$. Moreover, they  are\nboth called as the \"heat poles\" because they disappear in the\nzero temperature limit $T \\to 0$ in line of the discussions near\nequation (4.3.30) of \\cite{hofmann} and after. In the case of the\ninfinite matter, we can see this property from\n(\\ref{corrfunhp}) because\n${\\tt C}_{\\mathcal{V}}\/{\\tt C}_{\\mathcal{P}}\n\\to 1 $) [or due to (\\ref{psi0}) for $\\psi^{(0)}$ in\n(\\ref{corrfunhphof})]. For the finite Fermi-liquid drop, the\nreason is that $\\Upsilon \\to 0$ in the zero temperature limit $T \\to 0\n$, see (\\ref{gamma0hp}), and the only hydrodynamical sound\ncondition (\\ref{hydpolcond}) is then satisfied with the response\nfunction (\\ref{oscresponse}) and correlation function\n(\\ref{corroscim1}) where the heat pole is absent, see the\ndiscussion after (\\ref{corroscim1}).\n\n\nTo get more explicit expressions for $\\psi^{(0)}$ and ${\\it\n\\Gamma}_T$ of (\\ref{psi0GammaT}) we use now (\\ref{kfld}),\n(\\ref{gamma0hp}), (\\ref{friction0}) and (\\ref{stiffness0}) for the\ncoupling constant $k^{-1}$, parameter $\\Upsilon$, friction $\\gamma$\nand stiffness $C(0)$, respectively. With these expressions, one\nobtains approximately from (\\ref{psi0GammaT})\n\\bel{psi0GammaT1}\n\\frac{1}{T} \\psi_{\\rm hp}^{(0)} = \\frac{\\mathcal{G}_0\\varepsilon_{{}_{\\! {\\rm F}}} b_{{}_{\\! S}}\nr_0^5 A^{4\/3} }{9 \\mathcal{C}} \\approx\n2 A^{4\/3},\n\\end{equation}\n \\begin{equation}\\label{GammaT2m}\n{\\it \\Gamma}_{T}^{\\rm hp} = \\hbar \\Gamma^{\\rm hp}\n=\\frac{13 \\pi^6 \\mathcal{G}_1\n\\mathcal{C} }{20 \\varepsilon_{\\rm F}^4 b_{{}_{\\! S}} r_0^5 {\\it \\Gamma}_0 ~\nA^{1\/3}}~ \\frac{T^6 }{\\left(1+\\pi^2T^2\/c^2\\right)}.\n\\end{equation}\n  In the derivation of\n(\\ref{GammaT2m}), we used (\\ref{sfirst}) for the first\nsound solution $s_0=s_0^{(1)}$  ($n=1$) in (\\ref{gamma0hp})\nfor $\\Upsilon$ and (\\ref{wdthGT}) for the relaxation time $\\tau$.\nFor simplicity, we neglected small\ntemperature corrections in the viscosity coefficient $\\nu^{(1)}$\n(\\ref{shearvisone}) and in the first sound velocity $s_0^{(1)}$\n(\\ref{sfirst}). Other approximations are the same as well in the\nderivation of (\\ref{smallpar}) for $kC$ used in\n(\\ref{GammaT2m}) through (\\ref{heatpole}). The\ntemperature dependences of the \"intrinsic overdamped width\"\n$\\Gamma_{-}^{\\rm in}$ (\\ref{Gammaminintr}) and \"heat pole one\"\n$\\Gamma^{\\rm hp}$ (\\ref{GammaT2m}), (\\ref{Gammahptau}) are different,\nnamely $\\Gamma^{\\rm hp} \\propto {\\bar T}^4\/\\tau(0,T)$ and $\\Gamma_{-}^{\\rm in}\n\\propto 1\/\\tau(0,T)$ where the temperature dependence of the\nrelaxation time $\\tau(0,T)$ can be found in (\\ref{wdthGT}). The\nboth \"widths\" are the growing function of temperature as in\n\\cite{hofmann} but with a different power. The dependence on\nparticle number $A$ completely differs for these compared poles\nbeing the growing function of $A$ for the \"width\"\n$\\Gamma_{-}^{\\rm in}$, $\\Gamma_{-}^{\\rm in} \\propto A^{1\/3}$, and\ndecreasing function of $A$ for the $\\Gamma^{\\rm hp}$, $\\Gamma^{\\rm hp}\n\\propto A^{-1\/3}$.  As noted above, like for the Landau--Placzek\npeak [see (\\ref{corrfunhphof}), (\\ref{GammaT}) and (\\ref{psi0})],\nthe heat pole with the \"width\" $\\Gamma^{\\rm hp}$ (\\ref{GammaT2m})\nexists only in heated systems with a temperature $T \\neq 0$.\nHowever, in contrast to the result (\\ref{GammaT}),\n(\\ref{gambarhpt}) for the ${\\it \\Gamma}_T$ of the heat pole in the\ninfinite Fermi liquid, the heat pole \"width\" ${\\it \\Gamma}_T$\n(\\ref{GammaT2m}) disappears with increasing particle number $A$,\ni.e., ${\\it \\Gamma}_T \\rightarrow 0$ for $A \\rightarrow \\infty$. It\nallows us to emphasize also that this kind of the heat pole\nappears only in a {\\it finite} Fermi system.\n\n\nThe correlation function $\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$\n(\\ref{psifunhp}) was obtained approximately\nnear the pole $-i\\Gamma^{\\rm hp}\/2$, see (\\ref{heatpole}). The\ncorresponding $QQ$- correlation function\n$\\psi_{QQ}^{{\\rm hp}\\;\\prime\\prime}(\\omega )=k^2\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$\nis identical to the oscillator correlation function\n$\\psi_{\\rm osc}^{{\\rm hp}\\;\\prime\\prime}(\\omega )$ defined through the imaginary part of\n$\\chi_{\\rm osc}(\\omega )$ from the second equation of\n(\\ref{oscresponse}) at the zero mass parameter $M$, $M=0$, see\n\\cite{hofmann},\n\\begin{eqnarray}\\label{oscresphp}\n\\frac{1}{T}\\psi_{\\rm osc}^{{\\rm hp}\\,\\prime\\prime}(\\omega )&=&\n\\frac{2}{\\omega }\\chi_{\\rm osc}^{{\\rm hp}\\;\\prime\\prime}(\\omega )=\n\\frac{2}{|C_{\\rm in}|}~ \\frac{\n|C_{\\rm in}|\/\\gamma ^{\\rm hp} }{\\omega ^2 + \\left(C_{\\rm in}\/\\gamma\n^{\\rm hp}\\right)^2} \\nonumber\\\\\n&\\approx& 2 k ~ \\frac{ 1\/\\left(k\\gamma ^{\\rm hp}\\right)\n}{\\omega ^2+ 1\/\\left(k\\gamma ^{\\rm hp}\\right)^2},\n\\end{eqnarray}\n see again\n(\\ref{Cintr}) for the last approximation. The\nresponse (\\ref{hprespfunas}) and\ncorrelation (\\ref{corrfunashp}) functions are\nidentical to the corresponding oscillator ones (\\ref{oscresphp})\nwith a friction coefficient $\\gamma ^{\\rm hp}$,\n\\bel{frictionhp}\n\\gamma^{\\rm hp} =2|C_{\\rm in}|\/\\Gamma^{\\rm hp} \\approx 2k^{-1}\n\\left(\\Gamma^{\\rm hp}\\right)^{-1} \\approx 2\/(k^2C\\Upsilon).\n\\end{equation}\n Here,\nthe same equation (\\ref{Cintr}) was used, $\\Gamma^{\\rm hp}$ is given by\n(\\ref{heatpole}), (\\ref{GammaT2m}), $k^{-1}$ is the coupling\nconstant (\\ref{kfld}). [For $C$ and $\\Upsilon$ in\n(\\ref{frictionhp}), one has (\\ref{stiffness0}) and\n(\\ref{gamma0hp}), respectively.] According to\n(\\ref{corroscim1}) and (\\ref{corrfunashp}) for the correlation\nfunctions $\\psi ^{s\\;\\prime\\prime}(\\omega )$ and\n$\\psi^{{\\rm hp}\\;\\prime\\prime}(\\omega )$, with the help of (\\ref{heatpole}),\n(\\ref{frictionhp}) and (\\ref{gamma0}), one gets\n\\begin{eqnarray}\\label{corrfriction}\n(1\/2T)\\psi^{s\\;\\prime\\prime}(0)&=&\\gamma=\\gamma(0),\\nonumber\\\\\n(1\/2T)\\psi^{{\\rm hp}\\,\\prime\\prime}(0)&=& 2\/(k^2C\\Upsilon)=\\gamma^{\\rm hp},\n\\end{eqnarray}\n in\nline of the last right equation in (3.1.85) of \\cite{hofmann}.\n\nFor the friction $\\gamma ^{\\rm hp}$ (\\ref{frictionhp}) related to the\n\"heat pole width\" $\\Gamma^{\\rm hp}$ (\\ref{heatpole}), one\napproximately writes\n\\begin{equation}\\label{frictionhpm}\n\\gamma ^{\\rm hp} =\\frac{40 \\mathcal{G}_0 {\\it \\Gamma}_0\n\\varepsilon_{\\rm F}^5b_{{}_{\\! S}}^2 r_0^{10} A}{117 \\pi^6\n\\mathcal{G}_1 \\mathcal{C}^2 T^6 }\\;\\left(1+\\frac{\\pi^2T^2}{c^2}\\right),\n\\end{equation}\n see\n(\\ref{GammaT2m}) for $\\Gamma^{\\rm hp}$ and (\\ref{kfld}) for\n$k^{-1}$ in (\\ref{frictionhp}). We neglected here small\ntemperature corrections in the adiabatic in-compressibility\nmodulus $K=K^{\\varsigma}$ (\\ref{incompradexp}). The heat pole\nfriction $\\gamma ^{\\rm hp}$ (\\ref{frictionhpm}) is proportional to\n$1\/T^6$ for smaller temperatures and $1\/T^4$ for larger ones (due\nto the cut-off parameter $c$). This decreasing temperature\ndependence is much more sharp compared to the liquid drop one\n$\\gamma$ (\\ref{friction0}), (\\ref{wdthGT}); $\\gamma \\propto\n1\/T^2$ for smaller temperatures, and $\\gamma$ is a constant for\nlarge ones.\nNotice, according to (\\ref{frictionhp}), the \"width\" ratio\n$\\Gamma^{\\rm hp}\/\\Gamma_{-}^{\\rm in}$ (\\ref{zetaintr}),(\\ref{zetapar})\nhas a clear physical meaning as the ratio of the hydrodynamic\nfriction coefficient $\\gamma$ (\\ref{friction0}) to the heat pole\none $\\gamma^{\\rm hp}$ (\\ref{frictionhpm}),\n\\bel{zintr1}\n\\Gamma^{\\rm hp}\/\\Gamma_{-}^{\\rm in} \\approx \\gamma\/\\gamma ^{\\rm hp} \\approx\n\\gamma(0)\/\\gamma ^{\\rm hp}.\n\\end{equation}\n A smallness of this ratio shown above\nclaims that the heat pole friction $\\gamma ^{\\rm hp}$ is much larger\nthan the typical hydrodynamic one $\\gamma$, see more discussions\nconcerning this comparison of different friction coefficients\nbelow.\n\n\nAs seen from the inequalities (\\ref{heatpolcond}) with the\ndefinition of $\\omega _{\\rm crit}$ from (\\ref{hydpolcond}), the heat\npole appears only in the \"sound\" branch $n=1$ and does not exist\nfor the Landau--Placzek branch of the solutions of\n(\\ref{despeq}) for $s$. We realize it immediately noting that\nthe width parameter $\\Upsilon$ (\\ref{gamma0hp}) is proportional to\n$s_0$ which is finite for $n=1$ and zero for $n=0$ case, see\n(\\ref{shp}) and (\\ref{sfirst}), respectively.\n\n\nAs shown in \\cite{hofmann}, for enough small ${\\it \\Gamma}_T$, the\ncoefficient $\\psi^{(0)}$ in front of the Lorentzian-like correlation\nfunction, see (\\ref{corrfunhphof}) and (\\ref{psifunhp}),\nis related to the difference of\nsusceptibilities,\n\\bel{psichi}\n(1\/T) \\psi^{(0)}\n= \\chi^T-\\chi(0)=\\chi^T-\\chi^{\\rm ad} +\\chi^{\\rm ad}-\\chi(0).\n\\end{equation}\nNeglecting a small difference $\\chi^T-\\chi^{\\rm ad}$ according to\n(\\ref{chiTchiad1}), (\\ref{chiTchiad}), see Appendix C\nand \\cite{hofmann} for details, one notes that the ergodicity\ncondition (\\ref{ergodicity1}) means smallness of the $(1\/T)\n\\psi^{(0)}$ compared to the stiffness $C$.\n\nHowever, from (\\ref{psi0GammaT}), (\\ref{psi0GammaT1}) one gets\na large quantity $(1\/T) \\psi_{\\rm hp}^{(0)}\/C \\approx 1\/(kC) \\gg 1$.\nNote, in the derivations of\n(\\ref{corrfunashp}), (\\ref{psi0GammaT}) we took first $\\omega \n\\rightarrow 0$ (small $\\omega \\tau$) for the finite ${\\it\n\\Gamma}_T^{\\rm hp}$, see also (\\ref{gamma0hp}) for $\\Upsilon$ in the\nsecond equation of (\\ref{psi0GammaT}), and then, considered\n${\\it \\Gamma}_T^{\\rm hp} \\rightarrow 0$ (small temperature limit $T\n\\to 0$ ).\nWe emphasize that the limits $\\omega  \\rightarrow 0$ and ${\\it\n\\Gamma}_T^{\\rm hp} \\rightarrow 0$ are not commutative, i. e., the\nresult of the correlation function calculations depends on the\norder of executing of these two operations like for the infinite\nFermi-liquid matter \\cite{forster}. This\nis obvious if we take into account that the \"heat pole\" last term\nin the denominator of (\\ref{respfunas}) appears in the next\n(${\\bar T}^4$) order in ${\\bar T}$  and is proportional to $1\/(\\omega \n\\tau)$ in contrast to the other classical (sound) hydrodynamic\nterms, i.e., this $\\Upsilon$-term turns into zero for ${\\it \\Gamma}_T\n\\to 0 $ ($T \\to 0$).\n\nThe relation (\\ref{psichi}) was derived in \\cite{hofmann} using\nthe opposite sequence of the above mentioned limits, namely, first\n${\\it \\Gamma}_T \\rightarrow 0$ and then $\\omega  \\rightarrow 0$ in\nline of the recommendations of Forster \\cite{forster} [first ${\\it\n\\Gamma}_T \\propto q^2 \\rightarrow 0$ [or $\\tau_q \\to 0$, see\n(\\ref{GammaT}), (\\ref{shp})], and then, $\\omega  \\rightarrow 0$ ($s\n\\rightarrow 0$) for the infinite Fermi-liquid]. In this case\nthere is no contradiction with the ergodicity for the finite\nFermi-liquid drop. In the limit ${\\it \\Gamma}_T^{\\rm hp} \\rightarrow\n0$ ($T \\to 0$) for {\\it a finite} value of $\\omega $, the condition\n(\\ref{hydpolcond}) is fulfilled instead of (\\ref{heatpolcond}),\nand the \"heat pole\" term proportional to $1\/\\omega $ in the\ndenominator of the response function (\\ref{respfunas}) disappears\nwithin the ESA used in the FLDM, as noted above. It means\nformally that one can neglect $\\psi_{\\rm hp}^{(0)} $ in (\\ref{psifunhp}),\nand we have small quantities on the both sides of\n(\\ref{psichi}) taking into account the ergodicity condition\n(\\ref{ergodicity1}) derived in Appendix C.\nIt is not\nobvious that the relation (\\ref{psichi}) can be also derived for\nthe opposite consequence of the above mentioned limit transitions\nunlike the Forster recommendations, i.e., taking first limit $\\omega \n\\rightarrow 0$ for a finite ${\\it \\Gamma}_T$, and then, considering\nthe limit ${\\it \\Gamma}_T \\rightarrow 0$. In particular,\n(\\ref{corroscim1}) for the overdamped correlation function was\nobtained for the last choice of the limit sequences.\nEquation (\\ref{corroscim1}) does have also the Lorentzian-like shape\nbut it is not related to the \"heat pole\" because the coefficient\nin front of the Lorentzian is {\\it not} equal to\n$\\chi^T-\\chi(0)$.\nThis equation was derived only for\nlarge $\\Gamma_{-}^{\\rm in}$ compared to the $1\/\\tau$, see\n(\\ref{gammapmincond}),\nand is true {\\it only} under these conditions and within\ninequalities (\\ref{hydpolcond}). There is no a $\\delta(\\omega )$\nfunction-like peak in (\\ref{corroscim1}) for all possible\nvariations of the parameters for which this equation was derived.\nThe overdamped shape of the correlation function like\n(\\ref{corrfunhphof}) does not mean yet that this function is\nthe \"heat pole\" one though the opposite statement is true. We\npoint out again that\n $(1\/T)\\psi^{(0)}$ (\\ref{corroscim1}) is really\nlarge compared to the stiffness $C$, $(1\/T)\\psi^{(0)}=1\/k$, and the\nergodicity condition (\\ref{ergodicity1}) is fulfilled rather than\nthe relation (\\ref{psichi}) between $(1\/T)\\psi^{(0)}$ and\n$\\chi^T-\\chi(0)$ within the hydrodynamic conditions\n(\\ref{hydpolcond}).\n\n\n\nFollowing\nthe Forster's recommendations \\cite{forster},\ni.e., take first the limit of small ${\\it \\Gamma}_T$  (${\\it\n\\Gamma}_T \\rightarrow 0$) or small temperature ($T \\rightarrow 0$),\none gets the typical hydrodynamic response function\n(\\ref{oscresponse}) without \"heat pole\" terms. The next limit $\\omega \n\\rightarrow 0$ ($\\omega \\tau \\rightarrow 0$) in (\\ref{oscresponse})\nleads to the finite value,\n\\bel{isorespk}\n\\chi(0)=\\frac{1\n}{k^2C_{\\rm in}} \\approx - \\frac{1}{k} -C,\n\\end{equation}\n up to the\nrelatively small corrections of higher order in parameter $kC$\n(\\ref{smallpar}). This is in line of Appendix C,\nand the\nergodicity condition (\\ref{ergodicity1}) is fulfilled for the\nfinite Fermi-liquid drop within the ESA. Note that we accounted\nabove for the $kC$ correction at the second order\nin (\\ref{isorespk}). In this way, we got the relation\n(\\ref{kstiffCchi0}) between the coupling constant $k^{-1}$,\nisolated susceptibility $\\chi(0)$ and stiffness $C$ provided that\nthe condition (\\ref{hydpolcond}) is true, see also\n(\\ref{kstiffC0chi0}) with the stiffness $C(0)=C$ of the \"zero\nfrequency limit\". Note also that the \"heat pole\" response\nfunction $\\chi^{\\rm hp}(\\omega )$ (\\ref{hprespfunas}) has a sharp peak\nnear the zero frequency, and hence, is not smooth, i.e., \"the zero\nfrequency limit\" for the transport coefficients can not be\napplied in the case (\\ref{heatpolcond}).\n\nThus, all properties of the finite Fermi liquids within the ESA\nconcerning the ergodicity relation (\\ref{ergodicity1}), as applied\nto (\\ref{psichi}), are quite similar to the ones for the\ninfinite nuclear matter [besides the expressions (\\ref{Gammaminus}),\n$\\Gamma_{-} \\propto b_{{}_{\\! S}}\/A^{1\/3}$, and\n (\\ref{Gammahptau}),  $\\Gamma^{\\rm hp} \\propto 1\/( b_{{}_{\\! S}} A^{1\/3})$,\nthemselves depending on  $b_{{}_{\\! S}}$]. Our study of these properties is helpful\nfor understanding the microscopic shell-model approach\n\\cite{hofmann,hofivmag,hofbook}. We point out\nthat the strength function corresponding to the asymptotics\n(\\ref{respfunas}) is the curve with the two maxima which are\nrelated to the \"heat pole\" and standard (sound) hydrodynamic\nmodes. However, for intermediate frequencies $\\omega  $ of the order\nof $\\omega _{\\rm crit}$ in the low frequency region, see\n(\\ref{hydpolcond}) and (\\ref{heatpolcond}), the asymptotic\nresponse function (\\ref{respfunas}) can not be presented exactly\nin terms of a sum of the two  oscillator response functions like\n(\\ref{oscrespcoll}). For instance, in this case we have the\ntransition from the \"heat pole\" mode to the sound hydrodynamic\npeak, and the response function (\\ref{respfunas}) is more\ncomplex. We have a similar problem when the hydrodynamic\ncondition $\\omega  \\tau \\ll 1$ becomes not valid. However, as shown in\nthe next subsection, such problems can be overcome approximately\nusing an alternative definition for the transport coefficients\nsuggested in \\cite{hofmann}.\n\n\nFor larger frequencies, i.e., for $\\omega  \\tau$ larger or of the order\nof 1, but within the low frequencies $\\omega $ smaller than $\\Omega$,\nsee (\\ref{x}), the equation for the collective motion becomes more\ncomplicate. It is not reduced generally speaking to the second\norder differential equation with the constant coefficients as in\nthe zero frequency limit of the hydrodynamic approach\n(\\ref{hydpolcond}). As shown and applied in\n\\cite{hofivyam,hofmann} (see also \\cite{magkohofsh} in\nconnection to the FLDM), the problem of the definition of transport\ncoefficients can be nevertheless overcome by  defining them\nthrough a procedure of fitting an oscillator response function\n(\\ref{oscrespcoll})  to selected peaks of the collective\nresponse function $\\chi_{QQ}^{\\rm coll}(\\omega )$ of\n(\\ref{chicollfldm}) with respect to the parameters $M$, $C$ and\n$\\gamma$. Here such a fitting procedure would also be adequate for\ntemperatures mentioned above, especially because our response\nfunction (\\ref{chicollfldm}) has several poles (\\ref{rootomega}),\nfor instance, with $i = 0,1,2; n=1$ and $i=0;n=0$. Some of them\nare the overdamped poles close to the imaginary axis in the\n$\\omega $-complex plane. This procedure can be done analytically in\nthe zero frequency limit provided that the response function\n(\\ref{chicollfldm}) can be approximated\nby the oscillator\nresponse functions as in (\\ref{oscrespcoll}) or by\n$\\chi_{\\rm osc}^{\\rm hp}(\\omega )$ in (\\ref{oscresphp}). In this case, we\nhave analytical fitting of the collective response function\n(\\ref{chicollfldm}) by these oscillator response functions and\nget the expressions for the transport coefficients\n(\\ref{C0}) -\n(\\ref{M0}) in the zero frequency limit\n(\\ref{hydpolcond}) [or (\\ref{frictionhp}) for the heat pole\nfriction in a smaller frequency region (\\ref{heatpolcond})]. For\nlarger frequencies, we need to carry out the fitting procedure\nnumerically.\n\nWe should also comment a little more the definition of the\ntransport coefficients in the zero frequency limit in connection\nto the one through the fitting procedure to avoid some possible\nmisunderstanding. The transport coefficients in the zero frequency\nlimit can be related to the \"intrinsic\" response function and its\nderivatives taken at $\\omega  \\rightarrow 0$ \\cite{hofmann,hofbook};\nsee (\\ref{C0}),\n (\\ref{gamma0}), and (\\ref{M0}). For\napplication of this method of the transport coefficient\ncalculations, we should be carefully in the case when we have\nseveral peaks in the strength function but we need to get the\ntransport coefficient, for instance, for the second or more high\npeaks. In these cases the zero frequency limit might be applied\nalso, but we have first to remove all lower peaks in the collective\nresponse function and take then the corresponding \"intrinsic\"\nresponse function and its derivatives without these lower peaks.\nIn practical applications, this limit for the transport\ncoefficients obtained in a such way is close to the same limit for\nthe oscillator response function which fits the selected peak. The\nlatter could be also the second or more high one.\n\n\nWe shall consider now the hydrodynamical approximation\n$\\omega  \\tau \\ll 1$ for the response function, see\n(\\ref{respfunas}), for the two cases:  The sound response\nfunction (\\ref{oscresponse}) for the sound condition\n(\\ref{hydpolcond}) and the heat-pole response function\n(\\ref{hprespfunas}) for the heat pole condition\n(\\ref{heatpolcond}). The corresponding correlation functions are\nthe sound correlation function (\\ref{corroscim1}) and the\nheat-pole correlation one (\\ref{corrfunashp}). These two different\napproximations are realized for different consequence of the\nlimit transitions, i.e., the approximate result depends on the\nconsequence of their applying. The heat pole case\n(\\ref{hydpolcond}) is realized when we take first the limit $\\omega \n\\to 0$ for a finite width ${\\it \\Gamma}_T$, and then, ${\\it\n\\Gamma}_T \\to 0$ (or zero temperature limit $T \\to 0$). This leads\napproximately to the $\\delta(\\omega )$-like function for the\ncorrelation function. In contrast to this, the sound pole case\n(\\ref{heatpolcond}) is realized when we take first ${\\it\n\\Gamma}_T \\to 0$ (or $T \\to 0$) to remove the last heat pole term\nproportional to $\\Upsilon$ in the hydrodynamical response\n(\\ref{respfunas}), and then, $\\omega  \\to 0$. We like to follow this\nlast consequence of the limit transition in line of the Forster\nrecommendations \\cite{forster} when we have the response\n(\\ref{oscresponse}) and correlation (\\ref{corroscim1}) functions\nwithout heat pole. In this case the transport coefficients for\n$\\omega  \\tau \\ll 1$ are the standard hydrodynamical ones\n(\\ref{C0}), (\\ref{gamma0}) and (\\ref{M0}) related to the parameters\nof the standard hydrodynamical model (\\ref{stiffness0}),\n(\\ref{friction0}) and (\\ref{mass0}), respectively. Exception\nshould be done for the\nmodified mass parameter in (\\ref{M0}) which turns into the\nirrotational flow inertia (\\ref{mass0}) for high temperatures.\n\n\n\\subsection{Discussion of the results}\n\\label{discuss}\n\nIn this subsection, we discuss the\nresults of the FLDM calculations for the collective response\nfunction and transport coefficients. We shall explain now in more details\nthe application of the general fitting procedure for the definition\nof the transport coefficients. We discuss also the\nstiffness and inertia parameters found within the FLDM.\nThis subsection will be ended by\nthe discussion of the friction versus temperature. One\nof the important points of this discussion is the \"heat pole\"\nfriction and comparison with the quantum shell-model calculations\n\\cite{hofivyam,hofmann,hofbook}.\n\n\n\nWe show first the imaginary part of the response function\n$\\chi_{QQ}^{\\rm coll}(\\omega )$ (\\ref{chicollfldm}) (its strength)  for\ndifferent temperatures in Fig.\\ \\ref{fig2}.  The total collective\nresponse function $\\chi_{QQ}^{\\rm coll}$ is presented in\nFig.\\ \\ref{fig2} as a sum of the two branches $n=0$ and $1$ of\neigen-frequencies $\\omega ^{(n)}$, see (\\ref{rootomega}), in the\nimaginary part (strength) of the response function\n(\\ref{chicollfldm}). They are related to the two different\nsolutions of the dispersion equation (\\ref{despeq}) for the sound\nvelocity $s^{(n)}$. These solutions are similar to the\nLandau--Placzek (Raleigh) and the sound (Brillouin) ones in normal\nliquids. The latter are approached exactly by $s^{(0)}$ and\n$s^{(1)}$ solutions for sound velocity $s$ in the\nhydrodynamic limit $\\omega \\tau \\rightarrow 0$, which are related to\nthe eigen-frequencies of the infinite-matter vibrations\n$\\omega ^{(0)}$ (\\ref{shp}) and $\\omega ^{(1)}$ (\\ref{sfirst}),\nrespectively. The integral collision term is parametrized in\nterms of the relaxation time $\\tau(\\omega ,T)$ (\\ref{relaxtime}),\n(\\ref{widthG}) with $c=20~\\mbox{MeV}$. We took the nucleus Pu-230\nwith particle numbers $A=230$ as an example of enough heavy\nnucleus.\n\nFor the intermediate temperatures $4~\\mbox{MeV} ~\\siml~ T~ \\siml~\n6~\\mbox{MeV}$ we have the three peak structure. More detailed\nplots for smaller frequencies are shown in Fig.\\ \\ref{fig3} for the\ntemperature $T=6~\\mbox{MeV}$ for which the first two peaks (\"heat\npole\" and usual hydrodynamic ones) are seen better in a normal\nscale. In Fig.\\ \\ref{fig3}, we show also the separate contributions\nof the two branches $n=0$ (dotted line) and $n=1$\n(dashed one) for the eigen-frequencies $\\omega ^{(n)}$\n(\\ref{rootomega}) calculated from the secular equation\n(\\ref{poleseq}) at each $s^{(n)}$ ($n=0,1$) as in\nFig.\\ \\ref{fig2}. We present also the imaginary part of the\nasymptotic response function (\\ref{respfunas}) obtained\nanalytically above in the hydrodynamic frequent-collision limit.\nAs seen from Fig.\\ \\ref{fig3}, we found from (\\ref{chicollfldm})\nthe $n=1$ mode with the two ($i=0, 1$) peaks  and the $n=0$ mode\nwith one peak ($i=0$) for small frequencies $\\omega $ and small\nparameter $\\omega \\tau$ in agreement with asymptotics\n(\\ref{respfunas}). The heat pole contribution is shown separately\nby the dotted curve.\nNote that the two curves for $i=0 $ and $1$ at $n=1$\nin Fig.\\ \\ref{fig3} coincide because they both\nwere calculated without the last $\\Upsilon$ term in\n(\\ref{respfunas}). For the dotted curve,\none has $\\Upsilon \\propto\ns_0^{(0)}=0$, and for the dashed one,\nthe last $\\Upsilon$ term in\n(\\ref{respfunas}) is omitted  under the asymptotical sound\ncondition (\\ref{hydpolcond}). Therefore, the upper asymptotical\ndata (thin solid) marked also by the condition (\\ref{hydpolcond})\nare in factor about two larger than the dotted,\nor dashed, or asymptotical (\\ref{respfunas}) ones.\n\n\nThe third peak in Fig.\\ \\ref{fig2} appears for intermediate\ntemperatures and larger frequencies. This peak is coming from the\nthird pole $i=2$ which belongs to the branch $n=1$ in\n(\\ref{rootomega}). This is the essentially Fermi-liquid\nunderdamped mode due to the Fermi-surface distortions related to\nthe shear modulus $\\lambda$ given by (\\ref{shearmod}). Such a peak is\nmoving from a large zero-sound-frequency region of the giant\nresonances  to smaller frequencies with increasing temperature.\nThe second ($i=1$) peak in the $n=1$ branch and first ($i=0$)\npeak in the $n=0$ one in the low frequency region ($\\omega \\tau \\ll\n1$) are related to the overdamped motion described approximately\nby the overdamped oscillator response function like\n(\\ref{oscrespcoll}) for the same cut-off parameter $c=20~\\mbox{MeV}$.\nFor $c=\\infty$ the overdamped motion turns into the underdamped\none for large temperatures $T \\simg 7~\\mbox{MeV}$. The next\n(third) peak in a more high frequency region ($\\omega \\tau ~\\simg~ 1$)\ncorresponds to the underdamped mode for the both $c$ values. The\nfirst lowest peak in Figs.\\ \\ref{fig2} and \\ref{fig3}, which is not seen\nin Fig.\\ \\ref{fig2} being too close to the ordinate axis and\nstudied separately in Fig.\\ \\ref{fig3}, is due to the overdamped\n\"heat pole\" $i\\Upsilon\/2$ in the collective response function, see\n(\\ref{gamma0hp}) for $\\Upsilon$. The most remarkable property of\nthis \"heat pole\" peak for smaller temperatures is that it has\nmainly a very narrow width (\\ref{gamma0hp}) which increases with\nthe temperature as $T^6$, see the comments concerning the heat\npole \"width\" after (\\ref{gamma0hp}) and (\\ref{GammaT2m}).\nThis is in\ncontrast to the temperature behavior of the width $\\Gamma_{-}$\n(\\ref{Gammaminus}) like $T^2$ for the hydrodynamic sound peak at\nlarge temperatures.\nFig.\\ \\ref{fig2}  shows the three  peaks only for the intermediate\ntemperatures $4~ \\siml~ T ~\\siml~ 6~\\mbox{MeV}$ because for\nsmaller temperatures the third peak moves to the high frequency\nregion larger $\\Omega$ corresponding to the giant resonances and\nfirst peak is very close to the ordinate axis.\n\n\nThe transport coefficients for such two- or three resonance\nstructure were calculated by a fitting procedure of the oscillator\nresponse functions to the selected peaks. We subtract first the\n\"heat pole\" peak known analytically, see (\\ref{hprespfunas}), from\nthe total response function (\\ref{chicollfldm}). We are left then\nwith the two-humped curve and fit then it by the sum of the two\noscillator response functions as (\\ref{oscrespcoll}). One of\nthem which fits the first (hydrodynamic) peak in the curve with\nthe remaining two maxima is the overdamped oscillator response\nfunction ($\\eta ~>~ 1$) and other one (more high in the low energy\nregion) corresponds to the underdamped motion ($\\eta ~<~ 1$). In\nthis way, we get the two consequences of the transport coefficients\npresented in Figs.\\ \\ref{fig4}-\\ref{fig7}. In these Figures,\nthe heavy squares\nare related to the second, hydrodynamic-sound peak of\nFigs.\\ \\ref{fig2},\\ref{fig3} for the mostly overdamped modes with\nthe effective friction $\\eta~\n>~1$. The open squares show the third Fermi-liquid peak (see\nFig.\\ \\ref{fig2}) related to the underdamped motion ($\\eta~ < ~1$)\nand Fermi-surface distortions, very specific for the\nFermi liquids, in contrast to the normal liquids.\n\n\nFor the temperatures smaller about $6~\\mbox{MeV}$ the second peak\n$i=1$ in the total response function is {\\it overdamped} and is\ncoming from the two poles $(i=1,n=1)$ and $(i=0,n=0)$ which are\nclose to the standard hydrodynamic approach. The third peak, due\nto the Fermi-surface distortions as noted above, can not be found\nin principle in the hydrodynamic limit. The main difference\nbetween the second and third peaks can be found in the comparison\nof the stiffness coefficient $C$ with the liquid-drop value\n$C_{\\rm LD}$ obtained both from the fitting procedure mentioned above.\nFor the third (\"Fermi liquid\" in sense of the relation to the\nFermi surface distortions specific for the Fermi liquids, in\ncontrast to normal ones) peak the stiffness $C$ is much high\nthan the liquid drop value $C_{\\rm LD}$ in contrast to the second\n(typical hydrodynamical) one for which the stiffness $C$ is very\nclose to $C_{\\rm LD}$ almost for all temperatures, see\nFig.\\ \\ref{fig4}. It means that the third peak is essentially of\ndifferent nature than the second one because exists only due to\nthe Fermi-surface distortions. A measure of these distortions is\nthe anisotropy (or shear modulus) coefficient $\\lambda$, see\n(\\ref{shearmod}), which disappears in the hydrodynamic limit.\n\nFor enough large temperature (larger than or of the order of $7~\n\\mbox{MeV}$~) all three peaks are not distinguished in\nFig.\\ \\ref{fig2}. For such large temperatures the fitting procedure\nis a little modified to select these three peaks which are close\nto each other. For the finite $c=20~\\mbox{MeV}$ and all large\ntemperatures presented in Fig.\\ \\ref{fig2} nearly $7-10~\\mbox{MeV}$,\nwe have one wide peak which can be analyzed as the superposition\nof the three peaks, namely the \"heat-pole\", usual overdamped\nhydrodynamic and underdamped \"Fermi-liquid\" ones. Subtracting the\nfirst \"heat pole\" peak [see (\\ref{hprespfunas})] as for lower\ntemperatures, we fit then the remaining curve by the only one\noverdamped oscillator function like (\\ref{oscrespcoll}) for $\\eta\n> 1$. We subtract then again this overdamped fitted oscillator function\nfrom the response function (\\ref{chicollfldm}) without the heat\npole one (\\ref{hprespfunas}) and fit the rest by the single\nunderdamped oscillator. The found parameters of the two last\noscillator response functions are used as initial values for the\niteration fitting procedure of the sum of the two oscillator\nresponse functions of the same types to the response function\n(\\ref{chicollfldm}) (without the heat pole). The found transport\ncoefficients are presented in Figs.\\ \\ref{fig4}-\\ref{fig7}. For\nenough large temperature nearly $10~\\mbox{MeV}$ in the case\n$c=\\infty$ the only one underdamped oscillator can be used for\nfitting procedure of one peak [after an exclusion of the heat\npole from (\\ref{chicollfldm})].\n\nWe show also the mass parameters found from the above described\nfitting procedure for several selected peaks in Fig.\\ \\ref{fig5}.\nFor the third \"Fermi-liquid\" peaks the mass parameter $M$ is\nclose to the liquid drop values $M_{\\rm LD}$ related to the\nirrotational flow. The mass parameter of the second\n\"hydrodynamic\" peak, due to the mixture of the identical\n($i=1,n=1$) and ($i=0, n=0$) poles, is significantly smaller than\nthe liquid drop value $M_{\\rm LD}$ but finite. For the first \"heat\npole\" ($i=0,n=1$) peak the mass parameter can be approximated\nonly by zero. As noted above,\nthe stiffness parameter for the\nthird peak is much larger than the one for other (hydrodynamic)\npoles which is mainly close to the liquid drop value (see\nFig.\\ \\ref{fig4}).\nAs shown in Figures \\ref{fig4} and \\ref{fig5},\nfor enough large temperatures the temperature\ndependences of the stiffness ($C$) and mass ($M$) parameters are\nclose to their zero frequency limit, see (\\ref{C0}) for $C(0)$\nand (\\ref{M0}) for $M(0)$. For\nsmaller temperatures, the inertia $M(0)$ [Fig.\\ \\ref{fig5}]\nbecomes essentially larger than that found from the response function\n(\\ref{chicollfldm}). It is in contrast to the stiffness $C(0)$\nwhich is identical to the liquid-drop quantity in the\nsemiclassical limit $\\hbar \\rightarrow 0$ when $C(0)$ does not contain\nquantum shell corrections.\n\n\nFigs. \\ref{fig6} and \\ref{fig7} show the results for the friction\ncoefficient $\\gamma\/\\hbar$ versus the temperature for the\ncollective response function $\\chi_{QQ}^{\\rm coll}(\\omega ) =k^2(T)\n\\chi_{FF}^{\\rm coll}(\\omega )$ related to the $\\chi_{FF}^{\\rm coll}(\\omega )$\n(\\ref{chicollfldm}). We used here the same parameters as well\nin Figs.\\ \\ref{fig2} and \\ref{fig3} for the response function.\nThe solid line for the friction\n$\\gamma$ (\\ref{friction0}) corresponds to the response function\n(\\ref{oscrespcoll}) in the hydrodynamic limit (\\ref{hydpolcond}),\nthe same as for the zero frequency approach (\\ref{gamma0}). The\nheavy squares show the result of the fit of (\\ref{chicollfldm})\nto the oscillator response function (\\ref{oscrespcoll}).\nWe presented also the \"heat\npole\" contribution to the friction obtained from the fitting\nprocedure by one \"heat pole\" (overdamped) oscillator response\nfunction (\\ref{hprespfunas}), see circles in Fig. \\ref{fig7}. We\nmight compare the results of this fit to the friction\nanalytically found in terms of the heat pole asymptotics\n(\\ref{frictionhp}) valid for smaller temperatures and shown by\nsolid thin lines in Figs.\\ \\ref{fig6} and \\ref{fig7}. They are in a\ngood agreement for smaller temperatures where the overdamped\n\"heat pole\" with the \"width\" $\\Upsilon$ (\\ref{gamma0hp}) is more\nimportant. This \"heat pole\" friction is too big as compared to\nother friction components related to the hydrodynamical-sound\n(full squares) and \"Fermi-liquid\" poles in  the usual scale of\nFig.\\ \\ref{fig6}. Therefore, we use the logarithmic scale in\nFig.\\ \\ref{fig7}.\n\n\nOur FLDM friction, except for the \"heat pole\" one, is similar to\nthe corresponding result of SM calculations\n\\cite{hofivyam,hofmann}, see Fig.\\ \\ref{fig8}. A large SM\nfriction coming from the diagonal matrix elements in\nFig.\\ \\ref{fig8} and standard hydrodynamic friction\n(\\ref{friction0}) as well as heavy squares shown in\nFigs.\\ \\ref{fig6} and \\ref{fig7} are obviously similar. All these curves\nfor temperatures $T \\simg 2~\\mbox{MeV}$ show the mainly\ndiminishing friction, $\\gamma \\propto \\tau \\propto 1\/T^2$ roughly\nlike in hydrodynamics, see (\\ref{friction0}). Some deflection of\nthe friction temperature dependence in Fig.\\ \\ref{fig6} for large\ntemperatures $T$ from usual hydrodynamic one $1\/T^2$, i.e., a\nconstant asymptotics is related to a different temperature\nbehavior of the ${\\it \\Gamma}(0,T)$ (\\ref{widthG}) for a finite\nand infinite cut-off parameter $c$: This ${\\it \\Gamma}(0,T)$ goes\nto a constant for large temperatures if $c$ is finite and to zero\nfor $c=\\infty$, see the solid and dashed lines in Fig.\\ \\ref{fig6}.\n\n\nIt is noted also a similarity concerning the third\n(\"Fermi-liquid\") peak presented by the lower open squares with\nmainly increasing friction in Figs.\\ \\ref{fig6}, \\ref{fig7} and by\njoint full squares in Fig.\\ \\ref{fig8}. For $c=20~\\mbox{MeV}$ and\ntemperatures smaller about $10~\\mbox{MeV}$ the friction  of this\nmode increases, see Figs.\\ \\ref{fig6}, \\ref{fig7}, in contrast to\nthe standard hydrodynamic behavior (for $c=\\infty$ this friction\nincreases first up to about $6-7~\\mbox{MeV}$, and then, decreases\nat larger temperatures). In Fig.\\ \\ref{fig8} the lower curve with\ngrowing dependence on the temperature for $c= 20~\\mbox{MeV}$ was\nobtained by excluding the contribution of the diagonal terms in\nthe response function within the quantum approach based on the\nSM, see \\cite{hofmann,hofbook} for the detailed explanations.\nWithin the conceptions of the FLDM and classical hydrodynamics of\nthe normal liquid drops the first \"heat pole\" friction obtained\nfor enough small frequencies (\\ref{heatpolcond}) within the\nhydrodynamic collision regime $\\omega  \\tau \\ll 1$ at finite\ntemperature is the physical mode which can be excited when this\nregime might be realized like the Landau--Placzek pole for normal\nliquids. However, the hydrodynamic collision regime  being still\nwithin a low frequency region (enough small collision frequency\n$1\/\\tau$) is expected to be not achieved in fission experiments.\nTherefore, the friction is related mainly to another Fermi-liquid mode\ncorresponding to the only third peak owing to the Fermi-surface\ndistortions. The friction of this mode is much smaller than the\nhydrodynamic one for small temperatures, and they become\ncomparable for high ones. The Fermi-surface distortion friction\ncan be characterized by completely other, mainly growing\ntemperature behaviour, see the lower curve marked by open squares\nin Figs.\\ \\ref{fig6} and \\ref{fig7}. Concerning the SM calculations, it\nseems that we should omit the diagonal matrix elements, see\n\\cite{hofmann}, because of similar arguments: The hydrodynamic\ncollision regime seems to be not realized for nuclear fission\nprocesses. (These diagonal matrix elements might correspond to the\nphysical hydrodynamic mode if it is excited, say in another\nsystems like a normal liquid drop). The quantum shell-model\nfriction without contributions of diagonal matrix elements is\nrelated probably to another non-hydrodynamic mode, such as the third peak\nfor a Fermi-liquid drop, and this might be the physical reason\nfor an exclusion of these matrix elements.\n\nNote that in the SM response-function derivations the diagonal\nmatrix elements mentioned above do not contribute in the Forster's\nsequence of the limit transitions discussed at the end of the\nprevious section, first ${\\it \\Gamma}_T \\to 0$, for exclusion of\nthe diagonal matrix elements at finite $\\omega$, and then, $\\omega  \\to\n0$ limit. In this case, we have not contribution of the diagonal\nmatrix elements in the friction, and we are left with the low\nfriction curves with increasing temperature dependence shown in\nFigs.\\ \\ref{fig6}-\\ref{fig8}. For the opposite limit sequence if we\nconsider first the small frequency limit $\\omega  \\to 0$ for the\nfinite (large) ${\\it \\Gamma}_T$ we have the contribution of\ndiagonal matrix elements to the friction shown by the curves\ndecreasing with temperature which correspond to the hydrodynamic\nlimit here. As noted above, the exclusion of diagonal matrix\nelements for this last case could be justified because the\nphysical condition of the hydrodynamic limit $\\omega  \\tau \\ll 1$ is\nnot probably realized in fission processes. In that case, we\nexpect the increasing friction; which has essentially other,\nnon-hydrodynamic nature. We might interpret it within the FLDM as\nrelated to the third peak, due to the Fermi-surface distortions.\n\n\n\n\\section{NEUTRON-PROTON CORRELATIONS AND IVGDR}\n\\label{npcorivgdr}\n\n\\subsection{Extensions to the asymmetric nuclei}\n\\label{extensiontoas}\n\nThe FLDM was successfully applied for studying the global properties\nof the isoscalar multipole\ngiant resonances having nice agreement of their basic characteristics,\nsuch as the energies and sum rules, with experimental data\nfor collective excitations of heavy nuclei\n\\cite{strutmagden0,kolmagpl}. For the collective excitation modes in\nasymmetric neutron-proton nuclei, the FLDM was straightly extended\n in particular for calculations of the IVGDR structure\n\\cite{denisov,kolmagsh,BMV,BMR}.\nIn this case,\none has the two coupled (isoscalar and isovector) Landau--Vlasov equations\nfor the dynamical variations of distribution functions,\n$\\delta f^{}_{\\pm}({\\bf r},{\\bf p},t)$, in the nuclear phase-space volume \\cite{kolmagsh},\n\\begin{eqnarray}\\label{LVeq}\n\\frac{\\partial }{\\partial t}\n    \\delta f^{}_{\\pm}({\\bf r},{\\bf p},t) &+&\n    \\frac{{\\bf p} }{m_\\pm^*}\n   {\\bf \\nabla}_r  \\left[ \\delta f^{}_{\\pm}({\\bf r},{\\bf p},t) ~~\\right.\\nonumber\\\\\n  &+&\\left.\n    \\delta \\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right)\n    \\delta \\varepsilon_{\\pm}\n    +V_{\\rm ext}^{\\pm}\\right]\n    =\\delta St^{}_{\\pm}.~~\n\\end{eqnarray}\nHere $m_\\pm^*$ are the isoscalar (+) and isovector (-) effective masses,\n$\\varepsilon=p^2\/(2 m_\\pm^*)$ , $\\varepsilon_{{}_{\\! {\\rm F}}}=\n(p_{\\rm F}^{\\pm})^2\/(2 m_\\pm^*)$  is the Fermi energy.\nThe splitting between\nthe Fermi momenta $p_{\\rm F}^{\\pm}$ is originated by the difference of the\nneutron and proton potential well depths, due to the Coulomb interaction\n\\cite{migdal,kolmagsh},\n\\bel{momentumdef}\np_{\\rm F}^\\pm =p_{{}_{\\! {\\rm F}}}(1\\mp \\Delta,\\quad \\Delta=2(1+\\mathcal{F}_0')\n\\mathcal{I}\/3,\n\\end{equation}\n where\n$\\mathcal{F}_0'=3J\/\\varepsilon_{{}_{\\! {\\rm F}}}-1$  is the isotropic isovector\nLandau constant of the quasiparticle interaction (\\ref{fasymint}),\n$J$ is the volume symmetry energy constant \\cite{myswann69}.\nThe asymmetry parameter\n$~\\mathcal{I}=(N-Z)\/A~$ is assumed to be small\nnear the nuclear stability line, $N$ and $Z$ are the neutron and proton\nnumbers in the nucleus ($A=N+Z$).\nIn (\\ref{LVeq}), for the dynamical variations of the\nself-consistent quasiparticle\n(mean-field) interaction\n$\\delta \\varepsilon_\\pm ({\\bf r},{\\bf p},t)$, one has\n\\bel{interaction}\n\\delta \\varepsilon_{\\sigma}=\\pi^2\\hbar^3\n\\sum_{\\sigma'}\\left[\\frac{F_{0,\\sigma \\sigma'}\n}{p_{\\rm F}^{\\sigma'} m_{\\sigma'}^*}\n~\\delta \\rho_{\\sigma'} +\n\\frac{m F_{1,\\sigma \\sigma'} }{m_{\\sigma'}^* p_{\\rm F}^\\sigma\\left(\np_{\\rm F}^{\\sigma'}\\right)^2}\n~{\\bf p} \\cdot {\\bf j}_{\\sigma'}\\right].\n\\end{equation}\nThe sum is taken over the sign index $\\sigma=\\pm$.\nThe dynamical variations\nof the quasiparticle interaction\n$\\delta \\varepsilon_\\pm$  at the first order with respect to the equilibrium energy\n$p^2\/(2 m_\\pm^*)$ is defined through those of the\nparticle density,\n\\bel{densitydef}\n\\delta \\rho_\\pm({\\bf r},t)=\n\\int \\frac{2{\\rm d}{\\bf p} }{(2 \\pi\\hbar)^3}\\;\\delta f_\\pm({\\bf r},{\\bf p},t)\n\\end{equation}\n[zero ${\\bf p}$-moments\nof the dynamical distribution functions $\\delta f_{\\sigma}({\\bf r},{\\bf p},t)$\n(\\ref{planewave})], and the current density,\n\\bel{currentdef}\n{\\bf j}_\\pm({\\bf r},t)=\\int \\frac{2{\\rm d}{\\bf p} }{(2 \\pi\\hbar)^3}\n~\\frac{{\\bf p}}{m}~ \\delta f_\\pm({\\bf r},{\\bf p},t)\n\\end{equation}\n(their first ${\\bf p}$-moments).\nThe Landau interaction constants $F_{l,\\sigma \\sigma'}$ in\n(\\ref{interaction}) are defined by\nexpansion of the scattering quasiparticle's interaction amplitude\n$F_{\\sigma \\sigma'}({\\bf p},{\\bf p}')$\nin the Legendre polynomial series,\n\\bel{fasymint}\nF_{\\sigma \\sigma'}({\\bf p},{\\bf p}')=F_{0,\\sigma \\sigma'} + F_{1,\\sigma \\sigma'}\n{\\hat p} \\cdot {\\hat p}'+ ..., \\quad {\\hat p}={\\bf p}\/p.\n\\end{equation}\nFor the sake of simplicity,\nwe assume that $F_{l,\\sigma \\sigma'}$ is a symmetrical matrix\n($l \\leq 1)$ and\n$F_{l,pp}-F_{l,nn}$\nis of the second order in parameter $\\Delta$ [see below (\\ref{LVeq})], and\ncan be neglected in the linear approximation with respect to $\\Delta$,\n\\bel{symmetry}\nF_{l,pp}=F_{l,nn},\\quad F_{l,pn}=F_{l,np}.\n\\end{equation}\nThus, we arrive at usual simple definitions for the isoscalar\n$F_0$ and $F_1$ and isovector $F_0'$ and $F_1'$ Landau interaction\nconstants \\cite{migdal,kolmagsh},\n\\begin{eqnarray}\\label{landauconst}\nF_{l}&=&(F_{l,pp}+F_{l,pn})\/2,\n\\nonumber\\\\\n F_{l}^\\prime&=&(F_{l,pp}-F_{l,pn})\/2,\n\\qquad l=0,1.\n\\end{eqnarray}\nThese constants are related to the\nSkyrme interaction constants\nin the usual way \\cite{liu}.\nThe isoscalar ($\\mathcal{F}_0$)\nand isovector ($\\mathcal{F}_0'$) isotropic interaction constants are\nassociated with\nthe volume in-compressibility modulus $K$ and symmetry energy constant  $J$,\nrespectively.\nThe anisotropic interaction constants $\\mathcal{F}_1$ and\n$\\mathcal{F}_1'$ correspond to\nthe effective masses by equations $m_{+}^*=m(1+\\mathcal{F}_1\/3)$ and\n$m_{-}^*=m(1+\\mathcal{F}_1'\/3)$.\nThe periodic time-dependent external\nfield in (\\ref{LVeq}) is given by  $V_{\\rm ext} \\propto \\hbox{exp}(-i \\omega  t)$\nas in (\\ref{extfield}). The collision term $\\delta St^{}_{\\pm}$ is\ntaken in the simplest $\\tau_\\pm$-relaxation time approximation\n(\\ref{intcoll}).\nFor simplicity, we consider in this section the low temperature\nlimit $T \\rightarrow 0$ neglecting the difference between the local and\nglobal equilibrium for the quasistatic distribution function.\n\n\nSolutions of these equations (\\ref{LVeq}) associated with the dynamic\nmultipole particle-density variations,\n$\\delta \\rho_\\pm({\\bf r},t) \\propto Y_{L0}({\\hat r})$ in\nthe spherical coordinates $r$, $\\theta$ , $\\varphi$,  can be found\nin terms of a superposition of the plane waves (\\ref{planewave})\nover angles of the wave vector ${\\bf q}$  as\n\\begin{eqnarray}\\label{planewaves}\n&& \\delta f_{\\pm}=\\delta\\left(\\varepsilon-\n(p_{\\rm F}^{\\pm})^2 \/2m_{\\pm}^*\\right) ~~\\nonumber\\\\\n&\\times&\\int {\\hbox{d}}\\Omega_{\\bf q} \\mathcal{A}_{\\pm} Y_{L0}\\left({\\hat q}\\right)~\n\\hbox{exp}\\left[i\\left({\\bf q}{{\\bf r}}-\\omega  t\\right)\\right] ,  \\quad {\\hat q}={\\bf q}\/q,~~\n\\end{eqnarray}\n$\\omega =p_{\\rm F}^{\\pm}s q \\sqrt{NZ\/A^2}\/m_\\pm^*$,  $q=|{\\bf q}|$.\nThe factor $~\\sqrt{NZ\/A^2}~$ ensures the conservation of the center-of-mass\n position for the odd vibration multipolarities $~L~$\n\\cite{eisgrei}), in particular, for the dipole modes ($L=1$).\nThe amplitudes of the Fermi surface\ndistortions $\\mathcal{A}_\\pm~$ are determined by (\\ref{LVeq}).\nFor the simplest case of the zero anisotropic interaction  ($F_1=F_1'=0$)\nin the collisionless limit $\\omega \\tau \\to \\infty$,\nthe dispersion equation for the sound velocity $s$\ntakes the form:\n\\begin{eqnarray}\\label{dispeq}\n&&4F_0F_0'\\left(F_0 Q_1(s)-1\\right) \\nonumber\\\\\n&-&\\frac14\\Delta^2 F_0^2{F_0'}^2\n\\left(\\frac{s^2}{s^2-1} + Q_1(s)\\right)^2=0,\n\\end{eqnarray}\n(We accounted for a small $\\Delta$\nand large $\\omega \\tau$ at the zero temperature.)\nThis equation has the two solutions $s=s_n$ related to the main peak $n=1$\nand $2$ for its satellite, see (26) of \\cite{kolmagsh}\nfor the finite $\\omega \\tau$ and nonzero $F_1$ and $F_1'$.\nIn the limit $\\Delta \\to 0$, the dispersion equations\ngiven by (25) of \\cite{kolmagsh}\nwith our definitions for $s_1$ and $s_2$ modes $n=1$ and $2$\nare resulted in the two (isovector and isoscalar) equations\nfor the equations for the separated zero sounds,\nrespectively,\n\\bel{splitdispeq}\nQ_1(s)= 1\/F_0', \\qquad \\mbox{and}\\qquad\nQ_1(s)= 1\/F_0.\n\\end{equation}\n\n        For the finite Fermi-liquid drop with a sharp ES\n\\cite{strutmagden,magstrut,magboundcond},\nthe macroscopic boundary conditions for the pressures\nand those for the velocities  were derived in\n\\cite{kolmagsh,magsangzh,BMRV}. For small isovector vibrations near\nspherical shape, the radial mean-velocity $u_{r}$   and\nmomentum-flux-tensor $\\Pi_{rr}$ components, defined through\nthe moments of the distribution function\n$\\delta f_{-}$ as solutions of the kinetic equation (\\ref{LVeq}) [see\n(\\ref{veloc}) and (\\ref{momentflux})]\nare given by (\\ref{bound1}) and  (\\ref{bound2}) with\n$u_{r}=u_{r}^{+}-u_{r}^{-}$ and\n$\\Pi_{rr}=\\Pi_{rr}^{+}-\\Pi_{rr}^{-}$.\nThe r.h.s.s of these boundary conditions are the isovector\nES velocity  $u_{{}_{\\! S}}=R \\dot{Q}_S Y_{L0}({\\hat r})$   and  capillary\npressure exceed\n\\bel{boundcondiv}\n\\delta P_S=2 Q_S b_S^{(-)} \\rho_{{}_{\\! 0}} A^{1\/3}\nY_{10}(\\hat{r})\/3,\n\\end{equation}\ngiven through the isovector\nsurface energy constant $b_S^{(-)} \\propto \\alpha_{-}$ [see (\\ref{sigma}) and\n(\\ref{bsplusminus})],  where $Q_S$ is\nthe dynamical isovector-dipole ($L=1$)\namplitude of the motion of the neutron drop\nES against the proton one (\\ref{surface}),\nkeeping also the volume and the position of the center of mass conserved).\nNote that another interpretation of the surface symmetry-energy\nconstant $b_{\\rm S}^{(-)}$ in (\\ref{boundcondiv}) is considered  in\n\\cite{denisov,abrIVGDR}.\nThis constant essentially differs from the isovector stiffness introduced\nin \\cite{myswann69} for the description of the neutron skin as a\ncollective variable, see more detailed discussions in \\cite{BMV,BMRV}.\n\n       The energy constant,  $~D=\\hbar \\omega  A^{1\/3}~$,\nand energy weighted sum\nrules (EWSR),\n\\begin{equation}\\label{strength}\n{\\tt S}_{1}=\\frac{\\hbar^2}{\\pi} \\int \\hbox{d} \\omega \\; \\omega \\; {\\mbox {\\rm Im}} \\chi^{\\rm coll}(\\omega ),\n\\end{equation}\nfor the IVGDR can be found from the collective response function\n$\\chi^{\\rm coll}(\\omega )$ .\nThe response function (\\ref{chicollrho}) is determined by the\ntransition density (\\ref{densvolsurf}) generalized\nto the dynamic isoscalar and isovector components \\cite{BMR}:\n\\begin{eqnarray}\\label{trandenscl}\n\\delta \\rho_{\\pm}({{\\bf r}},t) &=&\n\\delta \\rho_{\\pm}^{\\rm vol}({{\\bf r}},t)\\; w_{\\pm}(\\xi) \\nonumber\\\\\n&-&\n\\frac{1}{a}\\frac{\\hbox{d} w_{\\pm}(\\xi)}{ \\hbox{d} \\xi}\\; \\overline{\\rho}\\;\n\\left[\\delta R_{\\pm}- \\delta \\aleph_{L}^{\\pm} \\;\nY_{L0} ({\\hat r})\\right],\n\\end{eqnarray}\nwhere\n$\\delta \\aleph_{L}^{\\pm}$ is defined by the mass center conservation\n($\\int \\hbox{d} {\\bf r} \\;{\\bf r}\\; \\delta \\rho_{\\pm}=0$),\n$w_\\pm(\\xi)$ is given by (\\ref{ysolplus}) and (\\ref{ysolminus}).\nIn Fig.\\ \\ref{fig9etf}, a strong SO dependence of the isovector\ndensity $w_{-}(\\xi)$\nis compared with\nthat of the isoscalar one $w_{+}(\\xi)$\n(low index ``+'' is omitted here and below)\nfor the SLy7 force as a typical example\n\\cite{magsangzh,BMRV}.\nAs  shown in \\cite{BMRV}, the isoscalar $w(\\xi)$, and therefore, the isovector\n$w_{-}(\\xi)$ densities depend rather strongly on the\nmost of the Skyrme forces \\cite{chaban,reinhard} near the ES.\nIn  Fig.\\ \\ref{fig10etf} (in logarithmic scale),\none observes notable differences in the isovector densities $w_{-}$ derived\nfrom different Skyrme forces\nwithin the edge diffuseness. In particular,\nthis is important for the calculations of the neutron skins of nuclei\n\\cite{BMRV}.\n\nWe emphasize that the dimensionless densities, $w(x)$\n(\\ref{ysolplus}) and $w_{-}(x)$\n(\\ref{ysolminus}), shown in Figs. \\ref{fig9etf} and \\ref{fig10etf} were\nobtained in\nthe leading ES approximation ($a\/R \\ll 1$) as functions of the\nspecific combinations\nof the Skyrme force parameters, such as $\\beta$ and $ c_{sym}$ of\n(\\ref{defpar}). Therefore, they are the universal distributions\nindependent of the specific properties of the nucleus such as the neutron and\nproton numbers, and the deformation and curvature of the nuclear ES;\nsee also \\cite{strtyap,strutmagden,magsangzh}.\nThese distributions yield approximately the spatial coordinate dependence\nof local densities in the normal-to-ES direction $\\xi$.\nWith the correct asymptotical behavior outside of the ES layer for any\nES deformation, they satisfy the leptodermic\ncondition $a\/R \\ll 1$, in particular,\nfor the semi-infinite nuclear matter.\n\n\nThe universal functions $w(\\xi)$ (\\ref{ysolplus})  and $w_{-}(x)$\n(\\ref{ysolminus}) of the leading order in the ESA\ncan be used [explicitly analytically in the quadratic\napproximation for $\\epsilon(w)$] for the calculations of the surface\nenergy coefficients\n$b_S^{(\\pm)}$ (\\ref{sigma}), the neutron skin  and isovector stiffness\n(see \\cite{BMRV}). As shown in Appendices B and C of \\cite{BMRV},\nonly these particle-density distributions  $w_{\\pm}(\\xi)$\nwithin the surface layer\nare needed through their derivatives [the lower limit of the integration\nover $\\xi$ in (\\ref{sigma}) can be approximately extended to\n$-\\infty$ because of no contributions from the internal volume region\nin the evaluation of the main surface terms of the pressure and energy].\nTherefore,  the surface symmetry-energy coefficient\n$k_{{}_{\\! S}}$ in (\\ref{bsplusminus}) and (\\ref{Jm}) (also the neutron skin\nand the isovector stiffness \\cite{BMRV})\ncan be approximated analytically in terms of the functions\nof the definite critical combinations of the Skyrme parameters such as $\\beta$,\n$c_{sym}$, $a$ [see (\\ref{defpar})],\nand the parameters of the infinite nuclear matter\n($b_{\\mathcal{V}}, \\rho_\\infty, K$). Thus,  they are independent of the\nspecific properties\nof the nucleus (for instance, the neutron and proton numbers), and\nthe curvature and deformation of the nuclear surface in the\nconsidered ESA.\n\n\nSolving the Landau--Vlasov equations (\\ref{LVeq})  in terms of the\nzero sound plane waves\n(\\ref{planewaves}) with using the dispersion equations (26)\nin \\cite{kolmagsh}  for the sound velocities $s_n$ and\nmacroscopic boundary conditions\n(\\ref{bound1}) and (\\ref{bound2}) with (\\ref{boundcondiv}) on the\nnuclear  ES,  from (\\ref{chicollrho}) and (\\ref{trandenscl}) one obtains\n\\begin{eqnarray}\\label{respfun}\n &&\\chi_{L}^{\\rm coll}(\\omega )=\\sum_n \\frac{\\mathcal{A}_{L}^{(n)}(q)\n}{\\mathcal{D}_{L}^{(n)}(\\omega -i\\Gamma\/2)},\\quad{\\rm with}\\quad\n\\mathcal{D}_{L}^{(n)}(\\omega ) \\nonumber\\\\\n&=&\nj_1'(qR)+\\frac{3 \\varepsilon_{{}_{\\! {\\rm F}}} qR}{2b_S^{(-)}A^{1\/3}}\\;\n\\left[c_n j_1''(qR)+d_nj_1(qR)\\right].\n\\end{eqnarray}\nHere,\n$~c_1\\approx 1-s_1^2+\\mathcal{F}_0'~$,\n$d_1\\approx 1-s_1^2+\\mathcal{F}_0'$   for the main ($n=1$)  IVGDR\npeak. Small anisotropic $\\mathcal{F}_1$ and $\\mathcal{F}_1'$\ncorrections\n and more bulky expressions for  $s_2$  of the satellite ($n=2$) peak\nof a smaller ($\\propto I$)\nstrength were omitted (see (D11) in \\cite{kolmagsh} for more precise\nexpressions). We present  here also the simplest expressions for the amplitudes,\n$\\mathcal{A}_1(q)\\approx -\\rho_\\infty R^3 j_1(qR)\/(m \\omega ^2)$  and\n$\\mathcal{A}_1(q)\\propto \\Delta \\propto I$\nfor the $n=1$ and $2$ modes [see a more complete equation\n(60) in \\cite{kolmagsh}]. The Bessel functions  $j_1(z)$\nand its derivative  $j_1'$ were defined after (\\ref{potensolut}) ($L=1$).\nThe poles of the response function\n$\\chi^{\\rm coll}(\\omega )$\n(\\ref{respfun}) (roots $\\omega _n$ of\nthe equation $D^{(n)}(\\omega -i \\Gamma\/2)=0$ or $q_n$ )\ndetermine the IVGDR energies $\\hbar \\omega $ as their\nreal part (the IVGDR width $\\Gamma$\nis determined  by their imaginary part).\nThe residue $\\mathcal{A}_n$   is important for the calculations\nof the EWSR (\\ref{strength})\nat a small width of the IVGDR $\\Gamma$.\nNote that the expression like  (\\ref{respfun})\nfor the only one  main peak (without the IVGDR structure)\nin symmetrical nuclei ($N=Z$)\nwith using the phenomenological boundary conditions\nwhich have the same form as (\\ref{bound1}) and (\\ref{bound2}),\nwhere however the isovector neutron-skin stiffness was applied\ninstead of the surface symmetry-energy constant $b_{{}_{\\! S}}^{(-)}$ in the\ncapillary pressure exceed (\\ref{boundcondiv})\nwas obtained earlier  in \\cite{denisov}.\n\n\n\\subsection{ Discussions of the asymmetry effects}\n\\label{discaseff}\n\n      The isovector surface energy constants $k_{{}_{\\! S}}$ (\\ref{bsplusminus})\nin the ESA using the simplest quadratic approximation\nfor $\\epsilon(w)$ of\nthe energy density (\\ref{enerden})\nare shown in Table 1 for several critical Skyrme forces\n\\cite{chaban,reinhard}. These constants are rather sensitive to the choice of\nthe Skyrme forces. The modulus of  $k_{{}_{\\! S}}$\nfor the Lyon Skyrme forces SLy4-7\n\\cite{chaban} is significantly larger than for other forces,\nall of them much smaller than those related to\n\\cite{myswann69,myswnp80pr96,myswprc77,myswiat85}.\nFor T6 \\cite{chaban}, one has $\\mathcal{C}_{-}=0$,\nand therefore,  $k_S=0$,\nin contrast to all of other forces shown in Table 1. Notice that the\nisovector gradient terms  which are important for the consistent\nderivations within the ESA\n\\cite{BMRV} are not also included  ($\\mathcal{C}_{-}=0$)  into the energy density in \\cite{danielewicz1,danielewicz2}. For RATP \\cite{chaban},\nthe isovector stiffness ($\\propto -1\/k_{{}_{\\! S}}$),\ncorresponding inversed $k_{{}_{\\! S}}$ but with the opposite sign\n\\cite{BMRV}, is even negative as $\\mathcal{C}_{-}>0$\n($k_{{}_{\\! S}}>0$). The reason of significant differences in these  values\nmight be related to those of the critical  isovector\nSkyrme parameter $\\mathcal{C}_{-}$\n  in the gradient terms of the energy density (\\ref{enerden}).\nDifferent experiments used for fitting this parameter were found to be\nalmost insensitive in determining uniquely its value, and hence, $k_S$\n[or $b_S^{(-)}$, see (\\ref{bsplusminus})],\nas compared to the well-known isoscalar surface-energy constant $b_S^{(+)}$.\nThe isovector  surface-energy constant $k_{{}_{\\! S}}$  (\\ref{bsplusminus})\nand the corresponding stiffness\ndepend much on the SO $\\beta$ parameter through the constant\n$\\mathcal{J}_{-}$ (\\ref{Jm}).\n\n\nThe IVGDR energy constants $D=\\hbar \\omega ^{(-)} A^{1\/3}$\nof the hydrodynamic model (HDM)\nare roughly in good agreement with the well-known experimental value\n$D_{\\rm exp}\\approx 80$ MeV for heavy nuclei within a precision better\nor of the order of 10\\%,\nas shown in \\cite{BMV,BMRV} (see also\n\\cite{denisov,kolmagsh,plujko}).\nMore precise $A^{-1\/3}$ dependence of $D$\nseems to be beyond the accuracy of these HDM calculations. This takes place\n even accounting more consistently for the ES motion because of several\nother reasons (the macroscopic Fermi-surface distortions\n\\cite{denisov}, also including  structure of the IVGDR\n\\cite{kolmag,kolmagsh,BMR,abrIVGDR,abrdavpl,plujko},\ncurvature, Coulomb, quantum-shell,\nand pairing \\cite{belyaevzel} effects\ntowards the realistic\nself-consistent calculations\nbased on the Skyrme HF approach\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2}.\nLarger values 30-80 MeV of the isovector stiffness\n\\cite{myswann69}\n(smaller $k_{{}_{\\! S}}$ ) were found in\n\\cite{myswnp80pr96,myswiat85,vinas2,brguehak}.\nWith smaller $|k_{{}_{\\! S}}|$ (see Table 1, or larger the isovector\nstiffness) the fundamental parameter of the LDM expansion in\n\\cite{myswann69,myswnp80pr96}\nis really small for $A \\simg 40$, and therefore, the results\nobtained by using this expansion are justified \\cite{BMRV}.\n\n\n        Table 1 shows also the mean IVGDR energies $D$\nobtained \\cite{BMV,BMRV} within a more precised FLDM \\cite{kolmagsh}.\nThe IVGDRs even for the spherical nuclei have a double-resonance structure,\nthe main peak $n=1$ which exhausts mainly the EWSR\nfor almost all Skyrme forces and the satellite one $n=2$ with\nthe significantly smaller EWSR contributions proportional to\nthe asymmetry parameter $I$, typical for heavy nuclei. The\nlast row shows the average\n$D(A)$ weighted by their EWSR\ndistribution in rather good agreement with the experimental data\nwithin the same accuracy about 10 \\%, and in agreement with the results\nof different other macroscopic IVGDR models\n\\cite{denisov,abrIVGDR,abrdavpl,plujko}.\nExclusion can be done (see Table 1) for  the\nSkyrme forces SIII \\cite{chaban} and SkL3 \\cite{reinhard} where we\nobtained a little larger IVGDR energies.\nNote that the main characteristics of the\n IVGDR described by mean $D$\nare almost insensitive to the isovector surface-energy constant\n$k_{{}_{\\! S}}$ \\cite{BMRV,BMV}. Therefore, we suggested \\cite{BMRV,BMR}\nto study the IVGDR\ntwo-peak (main and satellite) structure in order to fix the ESA value of\n$k_{{}_{\\! S}}$ \\cite{BMRV} from comparison with the experimental data\n\\cite{adrich,wieland,kievPygmy} and theoretical results\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2,ditoro}.\n\n\n \\section{NUCLEAR COLLECTIVE ROTATIONS}\n\\label{semshellmi}\n\n\\subsection{General ingradients of the cranking model}\n\\label{cranmod}\n\nWithin the cranking model, the nuclear collective\nrotation of the Fermi independent-particle system\nassociated with a many-body Hamiltonian,\n$H^{\\boldsymbol\\omega }=H+H_{\\rm CF}^{\\boldsymbol\\omega }$,\ncan be described, to a good approximation \\cite{eisgrei},\nin the restricted subspace of Slater determinants, by\nthe eigenvalue problem for\n a s.p.\\ Hamiltonian, usually called the {\\it Routhian}.\nFor this Routhian, in the body-fixed rotating frame \\cite{bohrmot,mix,fraupash},\none has\n\\bel{raussian}\nh^{\\boldsymbol\\omega }=h + h_{\\rm CF}^{\\boldsymbol\\omega },\\qquad\nh_{\\rm CF}^{\\boldsymbol\\omega }=-\\boldsymbol\\omega  \\cdot\n\\left(\\boldsymbol\\ell + {\\bf s}\\right),\n\\end{equation}\nwhere $h_{\\rm CF}^{\\boldsymbol\\omega }$ is the s.p.\\ cranking field which is\napproximately equal to the Coriolis interaction\n(neglecting a smaller centrifugal term, $\\propto \\omega ^2$).\nThe Lagrange multiplier $\\boldsymbol\\omega $\n(rotation frequency of the body-fixed coordinate system) is defined\nthrough the constraint on the nuclear angular momentum ${\\bf I}$,\nevaluated through the quantum average\n$\\langle \\boldsymbol\\ell +{\\bf s} \\rangle^{\\boldsymbol\\omega }={\\bf I} $, of the\ntotal s.p.\\ operator, $\\boldsymbol\\ell + {\\bf s}$,\nwhere $\\boldsymbol\\ell$\nis the orbital angular momentum and ${\\bf s}$ is the spin of the quasiparticle,\nthus defining a function\n$\\boldsymbol\\omega =\\boldsymbol\\omega ({\\bf I})$.\nThe quantum average of the total s.p.\\ operator $\\boldsymbol\\ell\n+ {\\bf s}$ is obtained by evaluating expectation values of the\nmany-body Routhian\n$H_{\\rm CF}^{\\boldsymbol\\omega }$ in the subspace of Slater determinants.\nFor the specific case of a rotation around the $x$ axis ($\\omega =\\omega _x$)\nwhich is perpendicular to the symmetry $z$ axis of the axially-symmetric mean\nfield $V$, one has (dismissing for simplicity spin (spin-isospin) variables),\n\\bel{constraint0}\n\\langle \\ell_x \\rangle^{\\omega } \\equiv\nd_s \\sum_i n_i^{\\omega } \\int \\hbox{d} {\\bf r}\n\\;\\psi_i^{\\omega }\\; \\left({\\bf r}\\right) \\; \\ell_x\\;\n\\overline{\\psi}_i^{\\omega }\\left({\\bf r}\\right)=I_x,\n\\end{equation}\nwhere $d_s$ as the spin (spin-isospin) degeneracy in the case of the\ncorresponding symmetry of the mean potential $V$.\nThe occupation numbers $n_i^{\\omega }$\nfor the Fermi system of independent nucleons are given by\n\\bel{ocupnumbi}\nn_i^{\\omega }\\equiv n\\left(\\varepsilon_i^{\\omega }\\right)\n=\\{1+ \\hbox{exp}\\left[(\\varepsilon_i^{\\omega } - \\mu^{\\omega })\/T\\right]\\}^{-1}.\n\\end{equation}\nIn (\\ref{constraint0}),  $\\psi_i^{\\omega }({\\bf r})$ are the eigenfunctions\nand\n$\\overline{\\psi}_i^{\\omega }({\\bf r})$ their complex conjugate,\n$\\varepsilon_i^{\\omega }$\nthe eigenvalues of the Routhian $h^{\\omega }$ (\\ref{raussian}),\n$\\mu^{\\omega }$ is the chemical potential. For relatively small frequencies\n$\\omega $ and temperatures $T$, $\\mu^{\\omega }$ is to a good approximation\nequal to the Fermi energy,\n$\\mu^{\\omega } \\approx \\varepsilon_{{}_{\\! {\\rm F}}} =\\hbar^2 k_{\\rm F}^2\/2 m^*$,\nwhere $k_{{}_{\\! {\\rm F}}}$ is the Fermi momentum in units of $\\hbar$.\nFrom (\\ref{constraint0}), the rotation frequency $\\omega $ can be\nspecifically expressed in terms of a given angular momentum of nucleus\n$I_x$, $\\omega =\\omega \\left(I_x \\right)$.\nWithin the same approach,\none approximately has for the particle number\n\\bel{partconspert}\nA= d_s\\sum_i n_i^{\\omega }\n\\int \\hbox{d} {\\bf r}\\; \\psi_i^{\\omega }({\\bf r})\\;\\overline{\\psi}_i^{\\omega }({\\bf r})\n\\approx d_s \\int \\hbox{d} \\varepsilon \\; n(\\varepsilon),\n\\end{equation}\nwhich determines the chemical potential $\\mu^{\\omega }$ for a given number of\nnucleons $A$. As we introduce the continuous\nparameter $\\omega $ and ignore the uncertainty\nrelation between the angular\n momentum and angles of the body-fixed coordinate system,\nthe cranking model is semiclassical\nin nature \\cite{ringschuck}.\nThus, we may consider the collective MI $\\Theta_x$ (for a rotation\naround the $x$ axis, and omitting, to simplify the notation, spin and\nisospin variables) as a response of the quantum average\n$\\delta \\langle \\ell_x \\rangle^{\\omega }$ (\\ref{constraint0}), to the\nexternal cranking field $h_{\\rm CF}^{\\omega }$ in (\\ref{raussian}).\nSimilarly to the magnetic or isolated\nsusceptibilities\n\\cite{richter,fraukolmagsan,magvvhof,yaf},\none can write\n\\bel{response}\n\\delta \\langle \\ell_x \\rangle^{\\omega }=\n\\Theta_x(\\omega ) \\delta\\omega ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{thetaxdef}\n&&\\Theta_{x}(\\omega )=\n\\partial \\langle \\ell_x\\rangle^\\omega \/\\partial \\omega =\n\\partial^2 E(\\omega )\/\\partial \\omega ^2,\n\\quad\\;\\;\\;\n\\nonumber\\\\\n&& E(\\omega )=\\langle h \\rangle^\\omega \n\\equiv d_s \\sum_i n_i^{\\omega }\\int \\hbox{d} {\\bf r}\n\\;\\psi_i^{\\omega }\\; \\left({\\bf r}\\right) \\;h \\;\n\\overline{\\psi}_i^{\\;\\omega }\\left({\\bf r}\\right).\\;\\;\\;\n\\end{eqnarray}\nTraditionally \\cite{mix,dfcpuprc2004,mskbPRC2010},\nanother parallel (alignment)\nrotation with respect to the\nsymmetry $z$ axis can be also considered as presented\nin Appendix A of \\cite{mskbPRC2010}.\n\nAs was shown in\n\\cite{inglis,bohrmotpr,belyaevzel,valat,bohrmot,fraupash,mix}, one can treat\nthe term\n$-\\boldsymbol\\omega  \\cdot \\boldsymbol\\ell = -\\omega  \\, \\ell^{}_x$ as\na perturbation for a nuclear rotation around the $x$ axis.\nWith the constraint (\\ref{constraint0})\nand the MI (\\ref{thetaxdef}) treated in second order perturbation theory,\none obtains the well known Inglis cranking formula.\nInstead of carrying out the rather involved calculations\npresented above, one could, to obtain the yrast line energies $E(I_x)$\nfor small enough temperatures $T$ and frequencies $\\omega $, approximate\nthe angular frequency by $\\omega =I_x\/\\Theta_x$ and write the energy in the form\n\\bel{yrast}\nE(I_x)= E(0) + \\frac{I^{2}_x}{2 \\Theta_x}.\n\\end{equation}\nAs usually done, the rotation term above needs to be quantized\nthrough $I^{2}_x \\rightarrow I_x (I_x+1)$ in order to study the\nrotation bands.\n\n\\subsection{Self-consistent ETF description of nuclear rotations}\n\\label{etfmi}\n\nFollowing\nreference \\cite{bartelnp}, a microscopic description of rotating nuclei\nwas obtained in the Skyrme Hartree--Fock formalism, within the Extended\nThomas--Fermi density-functional theory up to order $\\hbar^2$.\nWithin a variational space restricted to Slater determinant,\n the minimization of the\nexpectation value of\nthe nuclear Hamiltonian\nlead to the s.p. Routhian $h^{\\boldsymbol\\omega}_{\\rm q}$ (\\ref{raussian})\nthat is determined by a one-body potential $V_{\\rm q}({\\bf r})$,\na spin-orbit field ${\\bf W}_{\\rm q}({\\bf r})$ and an effective mass\nform factor $f^{\\rm eff}_{\\rm q}({\\bf r})=m\/m^*_{\\rm q}$ (see\nalso \\cite{brguehak}). In addition,\nin the case when the time\nreversal symmetry is broken, a cranking field form factor\n$\\boldsymbol\\alpha_{\\rm q}({\\bf r})$ and a spin field form factor\n${\\bf S}_{\\rm q}({\\bf r})$\nalso appear. In this subsection the (roman) subscript $\\rm q$\nrefers to the nucleon isospin\n($\\rm q = \\{n,p\\}$) and should not be confused\nwith the wave number $q$ in other sections.\nAll these fields can be written\nas functions of local densities and their derivatives, like the\nneutron-proton particle\ndensities $\\rho_{\\rm q}({\\bf r})$, the kinetic energy densities\n$\\tau_{\\rm q}({\\bf r})$,\nthe spin densities (also referred to\nas spin-orbit densities)\n${\\bf J}_{\\rm q}({\\bf r})$, the current densities\n${\\bf j}_{\\rm q}({\\bf r})$, and\nthe spin-vector densities\n$\\boldsymbol\\rho_{\\rm q}({\\bf r})$.\nNote that in the present subsection, $\\tau_{\\rm q}({\\bf r})$\nstands for the kinetic energy\ndensity which should not be confused with the relaxation time in\nprevious sections (here, however, with a different subscript ${\\rm q}$\nas compared to $q$ in \nsections \\ref{kinapp},\\ref{respfuntheor},\\ref{npcorivgdr} and Appendices\nA,B). In principle, two\nadditional densities appear, a spin-vector kinetic energy density\n$\\boldsymbol\\tau_{\\rm q}({\\bf r})$ and a tensor coupling\n$J^{}_{\\alpha \\beta}({\\bf r})$\nbetween spin and gradient vectors, which have, however, been neglected\nsince their contribution should be small, as suggested by \\cite{BFH87}.\n\n\n\nThe cranking-field form factor\n$\\boldsymbol\\alpha_{\\rm q}({\\bf r})$ contains two\ncontributions.\nOne of them is coming from the orbital part of the constraint,\n$-\\boldsymbol\\omega  \\, \\boldsymbol\\ell$,\nwhich has been shown in \\cite{BV78} to correspond to the Inglis cranking\nformula \\cite{inglis}. The other,\na Thouless--Valatin self-consistency\ncontribution \\cite{TV62} has its origin in the self-consistent response\nof the mean field to the time-odd part of the density matrix generated by\nthe cranking term of the Hamiltonian.\nThe aim is now to find functional relations for the local densities\n$\\tau_{\\rm q}({\\bf r})$, ${\\bf J}_{\\rm q}({\\bf r})$,\n${\\bf j}_{\\rm q}({\\bf r})$ and $\\boldsymbol\\rho_{\\rm q}({\\bf r})$ in\nterms of the particle densities $\\rho_{\\rm q}({\\bf r})$,\nin contrast to those given by\nGrammaticos and Voros \\cite{GV79} in terms of the form factors\n$V_{\\rm q}$,\n$f^{\\rm eff}_{\\rm q}$, ${\\bf W}_{\\rm q}$,\n$\\boldsymbol\\alpha_{\\rm q}$ and\n${\\bf S}_{\\rm q}$.\nTaking advantage of the fact that, at the leading Thomas--Fermi order,\nthe cranking field form factor is given by \\cite{bartelnp}\n\\bel{ETF02}\n  \\boldsymbol\\alpha^{({\\rm TF})}_{\\rm q}\n                        = f^{\\rm eff}_{\\rm q}\\left( {\\bf r}\n    \\times \\boldsymbol\\omega \\right),\n\\end{equation}\none simply obtains the rigid-body value for the Thomas--Fermi current density\n\\bel{ETF03}\n   {\\bf j}^{({\\rm TF})}_{\\rm q} = \\displaystyle \\frac{m}{\\hbar}\n   \\left( \\boldsymbol\\omega  \\times {\\bf r} \\right) \\, \\rho_{\\rm q}.\n\\end{equation}\nThis result is not that trivial, since it is only through the effect\nof the Thouless--Valatin self-consistency terms that such a simple\nresult is obtained. Notice also that (\\ref{ETF03}) corresponds to a\ngeneralization to the case $f^{\\rm eff}_{\\rm q} \\neq 1$ of a result\nalready found by Bloch \\cite{Bl54}.\nEquation  (\\ref{ETF03})\ncan be also considered as an extension of\nthe Landau quasiparticle (generalized TF) theory\n\\cite{landau,abrikha} presented in\nSecs. \\ref{etfmi}, \\ref{cranmod} to the case of rotating Fermi-liquid systems,\ncf. (\\ref{ETF03}) with (\\ref{currentdef})\nfor the current density as an average of the \\textit{particle}\nvelocity, ${\\bf p}_{\\rm rot}\/m=\\boldsymbol\\omega  \\times {\\bf r}$,\nrotating with the frequency\n$\\boldsymbol\\omega $.\nIn particular, the re-normalization of the cranking field form factor\n$\\boldsymbol\\alpha^{({\\rm TF})}_{\\rm q}=\nf_{\\rm q}^{\\rm eff} \\boldsymbol\\alpha_{\\rm o}$ with\n$\\boldsymbol\\alpha_{\\rm o} =\n\\left( {\\bf r} \\times \\boldsymbol\\omega \\right)$,\nby (\\ref{ETF02}) can be also explained\nas related to the effective mass corrections,\n$f_{\\rm q}^{\\rm eff} \\neq 1$, obtained\nby Landau \\cite{landau} with using\nboth the Galileo principle and the Thouless--Valatin self-consistency\ncorrections to a particle mass $m$ due to the quasiparticles'\n(self-consistent) interaction\nthrough a mean field. They lead in \\cite{bartelnp}\nto  the self-consistent TF angular momentum\nof the \\textit{quasiparticle}\n$\\boldsymbol\\ell_{\\rm q} =f_{\\rm q}^{\\rm eff} \\boldsymbol\\ell_{\\rm o}$ with the\nclassical angular momentum $\\boldsymbol\\ell_{\\rm o}= {\\bf r} \\times {\\bf p} $\nof the particle,\nso that $-\\boldsymbol\\omega \\cdot \\boldsymbol\\ell_{\\rm q}=\n\\boldsymbol\\alpha_{\\rm q} \\cdot {\\bf p}$. This effect is similar to that\nfor the kinetic energies of the \\textit{quasiparticles},\n$\\varepsilon_{\\rm q}=p^2\/(2 m^*_{\\rm q})=f_{\\rm q}^{\\rm eff}\\varepsilon_{\\rm o}$\nwhere $\\varepsilon_{\\rm o}=p^2\/(2m)$, see after (\\ref{LVeq}).\nWith this transparent connection to the Landau quasiparticle\ntheory, it is clear that\nthere is no contradictions with the TF limit\nof the current densities (\\ref{currentdef}), $\\hbar \\rightarrow 0$,\naccounting for the particle\ndensities (\\ref{densitydef}), as well as\nwith the definitions in subsections \\ref{extensiontoas} and\n\\ref{semshell}, because  $\\hbar$ in (\\ref{ETF03})\n appears formally due to a traditional\n use of the dimensionless units for the angular momenta in the\nquantum-mechanical picture to compare with experimental\nnuclear data. Another reason is related to\na consistent treatment of the essentially quantum\nspin degrees of freedom, beyond the Landau quasiparticle approach\nto the description of Fermi liquids,\nwhich have no \\textit{straight} classical limit,\nin contrast to the orbital angular momentum\n$\\boldsymbol\\ell$.\nThe convergence in the TF limit $\\hbar \\rightarrow 0$ can be realized\nfor smooth already quantities after the statistical (macroscopic)\naveraging over many\ns.p.\\ (more generally speaking, many-body) quantum states\nto remove the fluctuating (shell) effects  which\nappear in the denominators of the exponents within the POT (see \nSec.\\ \\ref{semshell} for\nmore detailed discussions). Finally, the spin paramagnetic  effect\n can be considered as a macroscopic one in the MI\nlike the orbital diamagnetic contribution.\nFor instance, the spin-vector density does not\nhave a \\textit{straight} classical analogue,\nsuch as the orbital angular momentum,\nand is considered as the object of leading order  $\\hbar$.\n\n\nStarting from these results and taking advantage of the fact that in\nthe functional ETF expressions up to the order\n$\\hbar^2$, it is sufficient to replace quantities, such as\nthe cranking field form factor $\\boldsymbol\\alpha_{\\rm q}$, by\ntheir Thomas--Fermi expressions\n(after the statistical averaging mentioned above).\nIn order to obtain a\nsemiclassical expression,\nthat is correct to that order in $\\hbar$, one obtains for the spin-vector\ndensities $\\boldsymbol\\rho_n$ and $\\boldsymbol\\rho_p$, which are of order\n$\\hbar$ in the considered ETF expansion, a system of linear\nequations. They  can be easily resolved \\cite{bartelnp}.\nOne also notices from this system of\nequations that the spin-vector densities are proportional to the\nangular velocity $\\omega$.\nExploiting the well known analogy of the microscopic Routhian problem with\nelectromagnetism, one may then define {\\it spin susceptibilities}\n$\\chi_{\\rm q}$,\n\\bel{ETF04}\n  \\boldsymbol\\rho_{\\rm q} = \\hbar \\,\n\\chi_{\\rm q} \\, \\boldsymbol\\omega \\; .\n\\end{equation}\n\n\nThe key question now is to assess the sign of\nthese susceptibilities and to\ndecide whether or not the corresponding alignment is of a\n``Pauli paramagnetic''\ncharacter. The study of \\cite{bartelnp} shows that this is the case, i.e., that\nthe spin polarization is, indeed, of paramagnetic character, thus confirming\nthe conclusions of the work performed\nby Dabrowski \\cite{Da75} in a simple model of non-interacting\nnucleons.\n\n\n\nSince the cranking field factor\n$\\boldsymbol\\alpha_{\\rm q}$ is, appart from that of the\nconstraining field\n$\\boldsymbol\\alpha_{\\rm o}$ determined\nonly by the current densities ${\\bf j}_{\\rm q}$\nand the spin-vector densities\n$\\boldsymbol\\rho_{\\rm q}$, one can then write down \\cite{bartelnp}\nthe contributions to the\ncurrent densities ${\\bf j}_{\\rm q}$\ngoing beyond the Thomas--Fermi approach.\nThe semiclassical corrections\nof order $\\hbar^2$ can be split into\ncontributions\n$(\\delta {\\bf j}_{\\rm q})_{\\ell}$ and\n$(\\delta {\\bf j}_{\\rm q})_{s}$ coming\nrespectively from the orbital motion and the spin degree of freedom. It is\nfound \\cite{bartelnp} that the orbital correction\n$(\\delta {\\bf j}_{\\rm q})_{\\ell}$\ncorresponds to a surface-peaked {\\it counter-rotation} with respect to the\nrigid-body current proportional\nto $\\left( \\boldsymbol\\omega \\times {\\bf r} \\right)$,\nthus recovering the Landau diamagnetism characteristic of a finite Fermi gas.\nWith the expressions of the current densities ${\\bf j}_{\\rm q}$\nand the spin-vector\ndensities $\\boldsymbol\\rho_{\\rm q}$ up to order $\\hbar^2$,\none can write down the\ncorresponding ETF expressions for the kinetic energy density\n$\\tau_{\\rm q}({\\bf r})$ and\nspin-orbit density ${\\bf J}_{\\rm q}({\\bf r})$.\n\n\nHaving now at hand the ETF functional expressions up to order $\\hbar^2$ of all\nthe densities entering our problem, one is able to write down the energy of the\nnucleus in the laboratory frame as a functional of these local densities,\n\\bel{endentau}\nE = \\int \\hbox{d} {\\bf r} \\; \\rho \\; \\mathcal{E}[\\rho_{\\rm q},\n\\tau_{\\rm q}, {\\bf J}_{\\rm q}, {\\bf j}_{\\rm q},\n\\boldsymbol\\rho_{\\rm q}],\n\\end{equation}\nwhere $\\rho=\\rho_n+\\rho_p$ as in Appendix D,\n$\\rho \\approx \\rho_\\infty w_{+}$.\nUpon some integration by parts, one finds that\n$\\mathcal{E}$ can be written as a sum of\n the energy density per particle of\nthe non-rotating system\n$\\mathcal{E}(0)$ and its rotational part,\nin line of (\\ref{yrast}).\nWithin the ETF approach, one has from (\\ref{endentau})\n\\bel{endentauSCM}\nE_{\\rm ETF}= \\int \\hbox{d}{\\bf r} \\rho \\mathcal{E}(0) +\n\\frac12 \\Theta_{\\rm ETF}^{({\\rm dyn})}\\,\\omega ^2,\n\\end{equation}\nwhere $\\Theta_{\\rm TF}^{({\\rm dyn})}$ is the ETF dynamical moment of\ninertia for the\nnuclear rotation with\nthe frequency $\\boldsymbol\\omega $. This MI is given in the form:\n\\begin{eqnarray}\\label{ETF05}\n  \\Theta_{\\rm ETF}^{({\\rm dyn})} &=& m \\sum_{\\rm q} \n\\int \\hbox{d} {\\bf r} \\; \\left\\{ r_\\perp^2 \\,\n\\rho_{\\rm q}\n  - \\left(3 \\pi^2 \\right)^{-2\/3} f^{\\rm eff}_{\\rm q} \\; \\rho^{1\/3}_{\\rm q} \n\\right.\\nonumber\\\\\n&+&  \\left.\n       \\left[\\frac{\\hbar^2}{2m} + W^{}_0 \\left(\\rho + \\rho_{\\rm q}\\right)\n\\right] \\, \\chi_{\\rm q}   \\right\\},\n\\end{eqnarray}\nwhere\n$ r_\\perp$ is the distance of a given point to the rotation axis and\n$W^{}_0$ is the Skyrme-force strength parameter of the spin-orbit\ninteraction \\cite{brguehak}.\n\n\n\n\nOne notices that the Thomas--Fermi term which comes from the orbital motion\nturns out to be the rigid-body moment of inertia. Semiclassical corrections\nof order $\\hbar^2$ come from both the orbital motion\n($\\Theta_{\\rm orb.}^{({\\rm dyn})}$) and from the spin degrees of freedom\n($\\Theta_{\\rm spin}^{({\\rm dyn})}$). The contribution $\\Theta_{\\rm orb.}^{({\\rm dyn})}$\nis\nnegative corresponding to a surface-peaked counter rotation in the rotating\nframe. Such a behavior is to be expected\nfor a N-particle system bound by\nattractive short-range forces (see \\cite{BJ76}).\nThe spin contribution\n$\\Theta_{\\rm spin}^{({\\rm dyn})}$ turns out to be of the {\\it paramagnetic} type,\nthus leading to a positive contribution which corresponds to an alignment of\nthe nuclear spins along the rotation axis. It can\nalso be shown (see \\cite{BBQ93})\nthat the ETF kinematic moment of inertia,\n\\bel{mikin}\n  \\Theta_{\\rm ETF}^{({\\rm kin})} = \\displaystyle\n\\frac{\\langle{\\boldsymbol\\ell}+{\\bf s}\\rangle^{\\omega }}{\\omega},\n\\end{equation}\nis identical\nto the ETF dynamical moment of\ninertia presented above.\n\n\n\n\n\nIt is now interesting to study the importance of the Thouless--Valatin\nself-consistency terms.  This has accomplished by calculating\nthe moment of inertia\nin the Thomas--Fermi approximation but omitting, this time, the\nThouless--Valatin terms. One then finds \\cite{bartelnp} the following\nexpressions for the dynamical moment of inertia, in what is simply the\nInglis cranking (IC) limit\n\\begin{eqnarray}\\label{ETF06}\n  \\Theta_{\\rm IC}^{({\\rm dyn})} &=& m \\sum_{\\rm q} \\int \\hbox{d} {\\bf r} \\;\n\\left[ \\frac{\\rho_{\\rm q}}{\\left(f^{\\rm eff}_{\\rm q}\\right)^2} \\right.\n\\nonumber\\\\\n  &+& \\left.\\frac{m B_3}{\\hbar^2} \\, \\rho_{\\rm q} \\, \\rho_{\\bar {\\rm q}}\n          \\left(\\frac{1}{f^{\\rm eff}_{\\rm q}} -\n\\frac{1}{f^{\\rm eff}_{\\bar {\\rm q}}} \\right)^{\\!\\!2} \\,\n                                        \\right] r_\\perp^2,\n\\end{eqnarray}\nwhere ${\\bar \\rm q}$ is the {\\it other}\ncharge state (${\\bar \\rm q} \\!\\!=\\!\\! p$ when\n$\\rm q \\!\\!=\\!\\! n$ and vice-versa)\nand $B_3$ is defined through the Skyrme force parameters\n$t^{}_1, t^{}_2, x^{}_1$ and $x^{}_2$ (see \\cite{bartelnp}).\nApart from the corrective term in $\\rho_{\\rm q} \\, \\rho_{\\bar {\\rm q}}$,\none notices\nthat the first term in the expression above, which is the leading term,\n yields, at least for a standard HF-Skyrme\n    force where $f^{\\rm eff}_{\\rm q} \\geq 1$, to a smaller moment of\n    inertia than the corresponding term in (\\ref{ETF05})\ncontaining the Thouless--Valatin corrections.\nIt is also worth noting that in\nthis approximate case, the kinematic moment of inertia is given by\n\\bel{ETF07}\n  \\Theta_{\\rm IC}^{({\\rm kin})}\n        = m \\sum_{\\rm q} \\int \\hbox{d} {\\bf r} \\;\n\\frac{\\rho_{\\rm q}}{f^{\\rm eff}_{\\rm q}} \\;\nr_\\perp^2,\n\\end{equation}\nwhich turns out to be quite different from the above given\ndynamical moment\nof inertia, (\\ref{ETF06}),\nobtained in the same limit (Thomas--Fermi limit, omitting the\nThouless--Valatin self-consistency terms).\n\n\nTo investigate the importance of the different contributions to the total\nmoment of inertia, we have performed\nself-consistent ETF calculations up to\norder $\\hbar^4$ for 31 non-rotating nuclei,\nimposing a spherical symmetry,\nand\nusing\nthe SkM$^*$ Skyrme effective nucleon-nucleon interaction \\cite{BQB82}. Such\ncalculations yield variational semiclassical density profiles for neutrons\nand protons \\cite{brguehak} which are then used to calculate the above given\nmoments of inertia. The nuclei included in our calculations are $^{16}$O,\n$^{56}$Ni, $^{90}$Zr, $^{140}$Ce, $^{240}$Pu and three isotopic chains for\nCa ($A \\!=\\! 36-50$), Sn ($A \\!=\\! 100-132$) and Pb ($A \\!=\\! 186-216$).\nThe results of these calculations\nare displayed in figure \\ref{fig11} taken from\n\\cite{bartelnp}.\n\nOne immediately notices the absence of any significant isovector dependence.\nThe good reproduction of the total ETF moment of inertia obtained by the\nThomas--Fermi (rigid-body) value is also quite striking. One finds that the\norbital and spin semiclassical corrections are not small individually but\ncancel each other to a large extent. To illustrate this fact the ETF moments\nobtained by omitting only the spin contribution are also shown on the figure.\nOne thus obtains a reduction of the Thomas--Fermi result that is about 6\\% in\n$^{240}$Pu but as large as 43\\% in $^{16}$O.\n\n\nThe Inglis cranking approach performed at the Thomas--Fermi level underestimates\nthe kinematic moment of inertia by as much as 25\\% and the dynamical moment of\ninertia by about 50\\% in heavy nuclei, demonstrating in this way the importance\nof the Thouless--Valatin self-consistency terms.\n\n\n\nIn \\cite{bartelnp}, a crude estimate of the semiclassical corrections due to orbital and spin degrees of freedom has been made by considering the nucleus as a piece of\nsymmetric nuclear matter (no isovector dependence as already indicated by the\nself-consistent results shown in figure \\ref{fig11} above).\nIt turns out that these\nsemiclassical corrections have an identical $A$ dependence\n($A_{}^{-2\/3}$ relative to the leading order Thomas--Fermi,\ni.e.\\ rigid-body, term)\n\\bel{ETF08}\n  \\Theta_{\\rm  ETF} = \\Theta^{\\rm (RB)}\n     \\left[ 1 + \\left( \\eta_{\\ell} +\n\\eta_s \\right) A^{-2\/3} \\right] \\; .\n\\end{equation}\nA fit of the parameters $\\eta_{\\ell}$ and $\\eta_s$ to\nthe numerical\nresults displayed\nin Fig.\\ \\ref{fig11} yields $\\eta_{\\ell} = -1.94$ and $\\eta_s = 2.63$\ngiving a total (orbital\n+ spin) corrective term of $0.69 \\, A^{-2\/3}$. For a typical rare-earth\nnucleus (A = 170) all this would correspond to a total corrective term equal to\n2.2\\% of the rigid-body value, resulting from a -6.3\\% correction\nfor the orbital\nmotion and a 8.5\\% correction for the spin degree of freedom.\n\n\n\nWhereas in the calculations that lead to figure \\ref{fig11} above,\nspherical symmetry\nwas imposed, fully variational calculations have been performed in\n\\cite{bartelpl}, imposing however the nuclear shapes to be of spheroidal form.\nIn this way, the\nnuclear rotation clearly impacts on the specific form of the\nmatter densities $\\rho^{}_n$ and $\\rho^{}_p$ which, in\nturn, in the framework\nof the ETF approach determine all the other local densities, as explained\nabove.\n\n\nTrying to keep contact with usual shape parametrizations,\nby the standard quadrupole\nparameters $\\beta$ and $\\gamma$ equating the semi-axis lengths of the spheroids\nwith the lengths of a standard quadrupole drop.\n\n\nAs a result,\nfigure \\ref{fig12} shows the evolution of the equilibrium\nsolutions (the ones that minimize the energy for given angular momentum I)\nas a function of I. One clearly observes that at low\nvalues of\nthe angular momentum\n(I in the range between 0 and 50 $\\hbar$) the nuclear drop takes on\nan oblate\nshape, corresponding to increasing values of the quadrupole parameter $\\beta$\nwith increasing I values, but keeping the non-axiality parameter fixed at\n$\\gamma = 60^{\\circ}$.  For larger values of the total angular momentum\n(I beyond 55 $\\hbar$), one observes a transition into triaxial shapes, where\nthe nucleus evolves rapidly to more and more elongated shapes. For even\nhigher values of I (I beyond 70 $\\hbar$) the nucleus approaches\nthe fission\ninstability. These results are in excellent qualitative agreement with those\nobtained by Cohen, Plasil and Swiatecki \\cite{CPS74} in a rotating LDM.\n\n\nIt is\namusing to observe\nhere\na {\\it backbending} phenomena at the semiclassical\nlevel when one is plotting,\nas usual, the moment of inertia $\\Theta_{\\rm ETF}$\nvs the rotational angular momentum,\nsee Fig.\\ \\ref{fig13}.\nOne should, however,\ninsist on the fact that this {\\it backbending} has strictly nothing to do with\nthe breaking of a Cooper pair. The rapid increase\nof the moment of inertia\nat about I$ \\,= 60 \\hbar$\nwith a practically constant (or even slightly\ndecreasing)\nrotational frequency $\\omega $ comes simply\nfrom the fact that at such a value of I\n(between I $\\approx$ 60 and I $\\approx$ 70) the nucleus elongates\nsubstantially\nincreasing in this way its deformation and at the same time its\nmoment of inertia.\n\nIt is therefore interesting to notice that the semiclassical ETF\napproach leads to a moment of inertia that is very well approximated\nby its Thomas--Fermi, i.e. rigid-body value. Thouless--Valatin terms which\narise from the self-consistent response of the mean field to the time-odd\npart of the density matrix generated by the cranking piece of the\nHamiltonian are naturally taken care of in this approach. Semiclassical\ncorrections of order $\\hbar^2$ coming from the orbital motion and the spin\ndegree of freedom are not small individually, but compensate each other\nto a large extent. One has, however, to keep in mind  that the\nshell and pairing effects, that go beyond the ETF approach, are not included\nin this description. These effects are not only both present, but influence\neach other to a large extent, especially for collective high-spin\nrotations of strongly deformed nuclei, as shown\nin \\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev}.\n\n\n\n\\subsection{MI shell structure and periodic orbits}\n\\label{semshell}\n\nWe shall outlook first the basic points of the POT for the semiclassical\nlevel-density and free-energy shell corrections\n\\cite{strut,fuhi,migdalrev}. We apply then the POT\nfor the derivation of the MI through the rigid-body MI\n(with the shell corrections, see Appendix E)\nin the NLLLA\nrelated to the equilibrium collective rotation\nwith a given frequency $\\omega$ \\cite{mskbPRC2010}. For simplicity,\nwe shall discard the spin and isospin degrees of freedom, in particular,\nthe spin-orbit and asymmetry interaction.\n\nNotice also that from the results presented in\nFigs. \\ref{fig11} and \\ref{fig13} (with the help of\nFig. \\ref{fig12}), one may conclude that the main contribution to\nthe moment of inertia\nof the strongly deformed heavy nuclei can be found within the\nETF approach to the rotational problems as a smooth\nrigid body MI.\n\n\\medskip\n\n\\subsubsection{\nGREEN'S FUNCTION TRAJECTORY EXPANSION}\n\\label{greenfun}\n\nFor the derivations of shell effects \\cite{strut} within the POT\n\\cite{gutz,strutmag,bt,creglitl,sclbook,migdalrev},\nit turns out to be helpful\nto use the coordinate representation of the MI\nthrough the Green's functions $G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon \\right)$\n\\cite{magvvhof,yaf,gzhmagsit,mskbg,mskbPRC2010},\n\\begin{eqnarray}\\label{thetaxdefG}\n\\Theta_{x}&=&\\frac{2 d_s}{\\pi}\\;\\int^{\\infty}_0 \\hbox{d}\\varepsilon\\;\nn({\\varepsilon}) \\int \\hbox{d} {\\bf r}_1 \\int \\hbox{d} {\\bf r}_2 \\; \\ell_{x}({\\bf r}_1)\\; \\ell_{x}({\\bf r}_2)\n\\nonumber\\\\\n&\\times&\n {\\mbox {\\rm Re}} \\left[ G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon \\right) \\right]\n{\\mbox {\\rm Im}} \\left[ G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right) \\right].\n\\end{eqnarray}\nThe Fermi occupation numbers $n(\\varepsilon)$ (\\ref{ocupnumbi})\nare approximately considered at $\\omega =0$ ($\\varepsilon=\\varepsilon_i$).\nIn (\\ref{thetaxdefG}), $\\ell_x({\\bf r}_1)$ and $\\ell_x({\\bf r}_2)$\nare the s.p.\\ angular-momentum projections onto the perpendicular rotation\n$x$ axis at the spatial points ${\\bf r}_1$ and ${\\bf r}_2$, respectively.\nWith the usual energy-spectral representation for the one-body\nGreen's function $G$ in the mean-field approximation, one finds\nthe standard cranking model expression, which however includes\nthe diagonal matrix elements of the operator $\\ell_x$.\nIn this sense, equation (\\ref{thetaxdefG}) looks more general beyond the\nstandard perturbation approximation,\nsee \\cite{mskbPRC2010}.\nMoreover, the quantum criterion of the application\nof this standard cranking model approximation, which is a smallness\nof the cranking field perturbation $h^\\omega _{\\rm CF}$ in\n(\\ref{raussian}) as\ncompared to the distance between\nthe neighboring states of the non-perturbative spectrum, becomes weaker\nin the semiclassical approach, see more comments below in relation to\n\\cite{belyaevfirst,belyaevbif}.\n\n\nFor the MI calculations by (\\ref{thetaxdefG}),\nthrough the Green's function $G$, one may use the\nsemiclassical Gutzwiller trajectory expansion \\cite{gutz} extended to\ncontinuous symmetry \\cite{strutmag,smod,magosc,creglitl,magkolstr,sclbook}\nand symmetry breaking \\cite{spheroidptp,maf,sclbook,migdalrev} problems,\n\\bel{GRdefsem}\nG\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)=\n\\sum_{\\rm CT} G_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right),\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{Gct}\n&&G_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)=\n\\mathcal{A}_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right) \\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar}{\\rm S}_{\\rm CT}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right)\n - \\frac{i \\pi}{2}\\sigma_{{}_{\\! {\\rm CT}}}-\ni\\phi_{\\rm d}\\right].\n\\end{eqnarray}\nThe sum runs over all isolated\n classical\ntrajectories (CTs) or their families inside the\npotential well $V({\\bf r})$ which, for a given energy $\\varepsilon$, connect the two\nspatial points ${\\bf r}_1$ and ${\\bf r}_2$.\nHere ${\\rm S}_{\\rm CT}$ is the classical action along such a CT,\nand $\\sigma_{{}_{\\! {\\rm CT}}}$ denotes the phase associated with the Maslov index\nthrough the number of caustic and turning points along the path CT,\n$\\phi_{\\rm d}$\nis the constant phase depending on the dimension\nof the problem \\cite{sclbook,strutmag,maf,migdalrev}.\nThe amplitudes $\\mathcal{A}_{\\rm CT}$ of the Green's function depend on the\nclassical stability factors and trajectory degeneracy, due to the symmetries\nof that potential \\cite{sclbook,strutmag,smod,spheroidptp,maf}.\nFor the case of the isolated CTs\n\\cite{gutz,sclbook},\n one has the explicit semiclassical expression for the amplitudes\nthrough the stability characteristics of classical dynamics,\n\\bel{Agutz}\n\\mathcal{A}_{\\rm CT}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)\n=-\\frac{1}{2\\pi\\hbar^2}\\,\\sqrt{\\Big|\\mathcal{J}_{\\rm CT}({\\bf\np}_{1},t_{{}_{\\! {\\rm CT}}};{\\bf r}_{2},\\varepsilon)\\Big|}.\n\\end{equation}\nHere, $\\mathcal{J}_{\\rm CT}({\\bf p}_{1},t_{{}_{\\! {\\rm CT}}};{\\bf r}_{2},\\varepsilon)$\nis the Jacobian for the transformation between the two sets of variables\n${\\bf p}_{1},t_{{}_{\\! {\\rm CT}}}$ and ${\\bf r}_{2},\\varepsilon$; ${\\bf p}_1$ and $t_{{}_{\\! CT}}$ are the\ninitial momentum and time of motion of the particle along a\nCT, $t_{{}_{\\! {\\rm CT}}}=\\partial S_{\\rm CT}\/\\partial\\varepsilon$ ,\n${\\bf r}_2$ and $\\varepsilon$ are its final coordinate and energy.\nIn more general case, if the mean field Hamiltonian $h$ obeys a\nhigher symmetry like that of spherical or harmonic-oscillator\npotentials with rational ratios of frequencies,\none has to use other expressions for the amplitude\n$\\mathcal{A}_{\\rm CT}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)$ for close trajectories\nof a finite action (with reflection from the potential boundary),\ntaking into account such symmetries.\nThey account for an enhancement in $\\hbar$ owing to their classical\ndegeneracy (see \\cite{strutmag,sclbook,maf,migdalrev} and the discussion\nin subsection  below). In the case of\nthe bifurcation of POs, generated by a symmetry-breaking, one\nmay use the ISPM\n\\cite{spheroidptp,maf}, especially\nfor superdeformed shapes of the potential. Some examples\nof the specific amplitudes for the degenerate families of\nclosed POs in the harmonic oscillator (HO) potential are given in \nAppendix E of\n\\cite{mskbPRC2010}.\nNote that (\\ref{Agutz}) can be applied for any potential wells for the\ncontributions of closed and\nnon-closed trajectories which can be considered as isolated\n(no PO families) ones for the given end points ${\\bf r}_1$ and ${\\bf r}_2$.\n\n\nAmong all of CTs in (\\ref{GRdefsem}),\nwe may single out ${\\rm CT}_{0} $\nwhich connects directly ${\\bf r}_1 $ and ${\\bf r}_2 $ without\nintermediate turning points, see Fig.\\ \\ref{fig14}.\nIt is associated with the component\n$G_{{\\rm CT}_0}$ of the sum (\\ref{GRdefsem}) for the semiclassical\nGreen's function.\nTherefore, for the Green's function $G({\\bf r}_1,{\\bf r}_2;\\varepsilon)$ (\\ref{GRdefsem}),\none has then a separation,\n\\bel{Ggsplit}\nG = G_{CT_{0}} + G_{1} \\approx  G_0 + G_1.\n\\end{equation}\nIn the NLLLA \\cite{gzhmagsit,gzhmagsit,mskbPRC2010},\n \\begin{equation}\\label{nllla}\ns_{{}_{\\! 12}}  \\ll \\hbar\/p_{{}_{\\! {\\rm F}}},\n\\end{equation}\nthe first term\n$G_{CT_{0}}$ of the splitting in the middle of (\\ref{Ggsplit}) is given\nby\n\\begin{equation}\\label{G0}\nG_{CT_{0}} \\approx G_0(s_{{}_{\\! 12}},p)=\n-\\frac{m}{2 \\pi \\hbar^2  s_{{}_{\\! 12}}} \\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} s_{{}_{\\! 12}} \\, p\\left({\\bf r}\\right) \\right],\n\\end{equation}\nwhere $p\\left({\\bf r}\\right) = \\sqrt{2 m [\\varepsilon - V({\\bf r})]\\,}~$, $~V({\\bf r})~$\nis a mean nuclear potential,\n\\bel{srvar}\ns_{{}_{\\! 12}}=\\left|{\\bf r}_2-{\\bf r}_1\\right|,\\qquad {\\bf r}=\\left({\\bf r}_1+{\\bf r}_2\\right)\/2,\n\\end{equation}\n$p=|{\\bf p}|$, ${\\bf p}=({\\bf p}_1+{\\bf p}_2)\/2$.\nThe second term $G_1$ in (\\ref{Ggsplit})\nis the fluctuating part of the Green's function\n(\\ref{GRdefsem}) determined  by all other trajectories\n${\\rm CT}_1 \\neq {\\rm CT}_{0}$ in the sum\n(\\ref{GRdefsem}) with reflection points at the potential surface\n(see one of such trajectories ${\\rm CT}_1$ in Fig.\\ \\ref{fig14}),\n\\bel{Gosc}\nG_{1}({\\bf r}_{1},{\\bf r}_{2};\\varepsilon)=\n\\sum_{{\\rm CT}_1}\nG_{{\\rm CT}_1}\\left({\\bf r}_{1},{\\bf r}_{2};\\varepsilon\\right),\n\\end{equation}\nwhere $G_{{\\rm CT}_1}$ is the Green's function component (\\ref{Gct}) taken at\nthe ${\\rm CT} \\neq {\\rm CT}_0$, i.e., ${\\rm CT}_1$.\n\n\n\\medskip\n\n\\subsubsection{\nLEVEL-DENSITY AND ENERGY SHELL CORRECTIONS}\n\\label{general}\n\nThe\nlevel density, $g(\\varepsilon)=\\sum_i\\delta(\\varepsilon-\\varepsilon_i)$, where\n$\\varepsilon_i$ is the quantum spectrum, is identically\nexpressed in terms of the Green's\nfunction $G$ as\n\\bel{totdensgen}\ng(\\varepsilon)=- \\frac{1}{\\pi}\\;{\\mbox {\\rm Im}}\n\\int \\hbox{d} {\\bf r}\\; \\left[G({{\\bf r}}_1,{{\\bf r}}_2;\\varepsilon)\n\\right]_{{\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}}.\n\\end{equation}\nAccording to (\\ref{Ggsplit}), this level density\ncan be presented semiclassically\nas a sum of the smooth and oscillating components\n\\cite{gutz,strutmag,sclbook,migdalrev},\n\\bel{totdensscl}\ng_{\\rm scl}(\\varepsilon)=g_{{}_{\\! {\\rm ETF}}}(\\varepsilon) + \\delta g_{\\rm scl}(\\varepsilon),\n\\end{equation}\nwhere\n$g_{{}_{\\! {\\rm ETF}}}(\\varepsilon)$ is given by the ETF approach related to the\ncomponent $G_0$ in (\\ref{Ggsplit}) in the NLLLA (\\ref{nllla})\n${\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}$\n\\cite{brguehak,sclbook,gzhmagfed,migdalrev}. The local part of\n$g_{{}_{\\! {\\rm ETF}}}(\\varepsilon)$ is the main simplest Thomas--Fermi (TF) level density\n$g_{{}_{\\! {\\rm TF}}}(\\varepsilon)$ \\cite{sclbook}.\nThe second oscillating term $\\delta g_{\\rm scl}(\\varepsilon)$\nof the level density (\\ref{totdensscl}) corresponds to the fluctuating\n$G_1$ in the sum\n(\\ref{Ggsplit}) for the Green's function $G$ near\nthe Fermi surface. The stationary phase conditions for the\n(standard or improved)\nSPM evaluation\nof the integral taken from $G_1$ over the spatial coordinates ${\\bf r}$\nare the PO equations.\nAs the result, one arrives at the sum over PO sum for this\noscillating level density\n\\cite{gutz,strutmag,bt,sclbook},\n\\begin{eqnarray}\\label{dlevdenscl}\n&&\\delta g_{\\rm scl}(\\varepsilon)\n= {\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)\\quad \\mbox{with}\\qquad\n\\nonumber\\\\\n&&\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)= \\mathcal{B}_{\\rm PO}\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar}\\; S_{\\rm PO}(\\varepsilon) -\ni \\frac{\\pi}{2}\\;\\sigma_{{}_{\\! {\\rm PO}}} -i \\phi_{d}\\right],\\qquad\n\\end{eqnarray}\nwhere $B_{\\rm PO}$ is an amplitude of the oscillating PO terms, see\n\\cite{gutz,strutmag,bt,creglitl,sclbook,migdalrev,spheroidptp}.\nThe above sum runs over the isolated POs and, in the case of\ndegeneracies owing to the symmetries of\n  a given potential well, over all families of POs.\n  $\\mathcal{B}_{\\rm PO}$ is the oscillation amplitude depending on the\n  stability factors, $S_{\\rm PO}(\\varepsilon)$ the action integral along a given\n  PO, and $\\sigma_{{}_{\\! {\\rm PO}}}$ is\nthe Maslov phase associated with the turning and\n  caustic points along the PO, see\n\\cite{sclbook,maf,migdalrev} for the detailed explanations.\n\nThe semiclassical free-energy shell corrections,\n$\\delta F_{\\rm scl}$ at finite temperature\n($T \\siml \\hbar \\Omega \\ll \\varepsilon_{{}_{\\! {\\rm {\\rm F}}}}$),\ncan be expressed through the PO components of the energy shell corrections\n$\\delta U_{\\rm scl}$ \\cite{strutmag,sclbook,migdalrev}\n(see Appendix E.1),\n\\begin{eqnarray}\\label{descl}\n\\delta U_{\\rm scl} &=& {\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta U_{\\rm PO},\n\\nonumber\\\\\n\\delta U_{\\rm PO}&=&d_s \\; \\frac{\\hbar^2}{t^{2}_{\\rm PO}} \\;\n\\delta g_{{}_{\\! {\\rm PO}}}(\\mu),\n\\end{eqnarray}\nwith the exponentially decreasing temperature-dependent factor\n\\cite{strutmag,kolmagstr,richter,fraukolmagsan,sclbook,mskbPRC2010},\n\\begin{eqnarray}\\label{fpotau}\n\\delta F_{\\rm scl}&=&{\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta F_{\\rm PO} \\nonumber\\\\\n&=&\n{\\mbox {\\rm Re}} \\sum_{\\rm PO}\\frac{\\pi t_{{}_{\\! {\\rm PO}}}T\/\\hbar}{\n\\hbox{sinh}\\left(\\pi t_{{}_{\\! {\\rm PO}}}T\/\\hbar\\right)}\\;\\delta U_{\\rm PO}.\n\\end{eqnarray}\nFinally through (\\ref{descl}), the shell corrections\n$\\delta F_{\\rm scl}$ and $\\delta U_{\\rm scl}$ are determined by\nthe PO level-density shell-correction components\n$\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ of (\\ref{dlevdenscl}) at the chemical potential,\n$\\varepsilon=\\mu \\approx \\varepsilon_{{}_{\\! {\\rm F}}}$.\nIn (\\ref{descl}), one has the additional factor,\n$\\propto 1\/t^2_{\\rm PO}$,\nwhich yields the convergence of the PO sum (without averaging of\n$\\delta g(\\varepsilon)$\nover the s.p.\\ spectrum), $t_{{}_{\\! {\\rm PO}}}$ is the time of motion along\nthe PO (PO period). Another (exponential) convergence of\n$\\delta F_{\\rm scl}$\n(\\ref{fpotau}) with increasing the period $t_{{}_{\\! {\\rm PO}}}$ and temperature\n$T$ is giving by the temperature-dependent factor in front of\n$\\delta U_{\\rm PO}$.\n\n\n\\subsubsection{FROM CRANKING MODEL TO THE RIGID BODY ROTATION}\n\\label{relperprigrot}\n\nSubstituting (\\ref{Ggsplit}) into (\\ref{thetaxdefG}), one has a sum\nof several terms,\n\\bel{thetaxsum}\n\\Theta_{x\\; {\\rm scl}} \\approx \\Theta_x^{00} + \\Theta_x^{01} +\n\\Theta_x^{10} + \\Theta_x^{11},\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{thetaxnnp}\n\\Theta_x^{n n'}&=&\\frac{2 d_s}{\\pi}\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r}_1 \\int \\hbox{d} {\\bf r}_2\\;\n\\ell_x\\left({\\bf r}_1\\right)\\;\\ell_x\\left({\\bf r}_2\\right)\n\\nonumber\\\\\n&\\times&\n{\\mbox {\\rm Re}} \\left[G_{n}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right]\\;\n{\\mbox {\\rm Im}} \\left[G_{n'}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right].\n\\end{eqnarray}\nIndexes $n $ and $n'$ run independently\nthe two integers 0 and 1. As shown in Appendix E.2a,\nthe main smooth part of the semiclassical\nMI $\\Theta_{x\\; {\\rm scl}}$ (\\ref{thetaxsum})\nis associated with the TF (ETF)\nrigid-body component through the first term $\\Theta_x^{00}$\naveraged over the phase-space variables; see section \\ref{etfmi}, also\n\\cite{bartelpl,gzhmagsit,mskbPRC2010}, and previous publications\n\\cite{bloch,amadobruekner,rockmore,jenningsbhbr}.\nThe statistical averaging over phase space coordinates removes\nthe non-local long-length correlations.\nThe $\\hbar$ corrections of the smooth ETF approach to the TF\napproximation were obtained in\n\\cite{jenningsbhbr,bartelnp,bartelpl},\nsee Sect. \\ref{etfmi} for the review of these\nworks.\n\n\n Using the transformation of the coordinates\n${\\bf r}_1$ and ${\\bf r}_2$ to the center-of-mass and relative ones\n${\\bf r}$ and ${\\bf s}_{{}_{\\! 12}}$,\n\\bel{newcoord}\n{\\bf r}=({\\bf r}_1+{\\bf r}_2)\/2 \\qquad {\\rm and}\\qquad {\\bf s}_{{}_{\\! 12}}={\\bf r}_2-{\\bf r}_1,\n\\end{equation}\nin (\\ref{thetaxnnp}), respectively, one simplifies much the calculations of\nthe oscillating terms,\n$\n\\Theta_x^{01} +\n\\Theta_x^{10} + \\Theta_x^{11}~$.\nIn this way, one finds that the shell\ncomponent $\\delta \\Theta_x^{01}$ of $\\Theta_x^{01}$\n[see (\\ref{thetaxnnp}) at $n=0$\nand $n'=1$]\nis dominating in the MI shell correction\n$\\delta \\Theta_{x\\; {\\rm scl}}$ within the NLLLA (\\ref{nllla}),\nsee Appendix E.2b.\nIndeed, in this approximation, substituting the\ncomponents, $G_0$ and $G_1$, of the Green's function (\\ref{Ggsplit})\n[see (\\ref{G0}) for $G_0$] into\n(\\ref{thetaxnnp}) for $\\Theta_x^{01}$,\nand  using the averaging over the phase-space\nvariables in the fluctuating (shell) part $\\delta\n\\Theta_{x}$ of $\\Theta_{x}$,\none results in the relationship for the\ncorresponding shell corrections (see Appendix E.2b):\n\\bel{dmilocapproach}\n\\delta \\Theta_{x\\; {\\rm scl}} \\approx\n\\delta \\Theta_{x}^{01} \\approx \\delta \\Theta_{x}^{\\rm (RB)}.\n\\end{equation}\nHere, $\\delta \\Theta_x^{\\rm (RB)}$ is\nthe shell correction to the rigid-body\nMI $\\Theta_x^{\\rm (RB)}$,\n which is related to the semiclassical particle-density\n$\\rho({\\bf r})$ through\n\\bel{rigbodmomgen}\n\\Theta_x^{\\rm (RB)} =\nm \\int \\hbox{d} {\\bf r}\\;r_{\\perp x}^2\\; \\rho\\left({\\bf r}\\right),\n\\end{equation}\nwith\n\\bel{rperpcoord}\nr_{\\perp x}^2=y^2+z^2.\n\\end{equation}\nThe particle density $\\rho({\\bf r})$, and therefore, the MI\n(\\ref{rigbodmomgen}),  can be expressed in terms of the Green's\nfunction $G$,\n\\bel{denpartgen}\n\\rho({\\bf r}) = -\\frac{d_s}{\\pi}\\; {\\mbox {\\rm Im}} \\int \\hbox{d} \\varepsilon\\; n\\left(\\varepsilon\\right)\\;\n\\left[G\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)\\right]_{{\\bf r}_1 \\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}}.\n\\end{equation}\nWith the splitting of the Green's function (\\ref{Ggsplit}),\none obtains the semiclassical sum of the smooth and oscillating (shell)\ncomponents \\cite{strutmagvvizv1986,brackrocciaIJMPE2010}:\n\\bel{denpartscl}\n\\rho({\\bf r})\\approx  \\rho_{\\rm scl}({\\bf r})=\\rho_{{}_{\\! {\\rm ETF}}}({\\bf r}) +\n\\delta \\rho_{\\rm scl}({\\bf r}).\n\\end{equation}\nThe integration over $\\varepsilon$ in (\\ref{denpartgen}) is performed\nover the whole s.p.\\ energy spectrum.\nFor the Green's function $G$, we applied the semiclassical expansion\n(\\ref{GRdefsem}) in terms of the sum (\\ref{Ggsplit}) of CTs in the\nlast equation for the semiclassical particle density $\\rho_{\\rm scl}({\\bf r})$.\nThe first term in (\\ref{denpartscl}) is the (extended) Thomas--Fermi component\n(see Appendix E.2a).\nSubstituting the particle density splitting (\\ref{denpartscl}) into\n(\\ref{rigbodmomgen}), one has the corresponding semiclassical\nexpression\nof the rigid-body MI,\n\\bel{rigmomsplit}\n\\Theta_x^{\\rm (RB)} \\approx \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)} =\n\\Theta_{x\\; {\\rm ETF}}^{\\rm (RB)} +\n\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)}.\n\\end{equation}\nWe introduced the shell corrections $\\delta \\rho$\n(see \\cite{brackrocciaIJMPE2010}) to the particle\ndensity $\\rho$ and $\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}$ to the rigid-body MI\n$\\Theta_x^{\\rm (RB)}$, and their semiclassical counterparts,\n\\bel{drigbodmomgen}\n\\delta \\Theta_{x}^{\\rm (RB)} \\approx\n\\delta \\Theta_{x \\; {\\rm scl}}^{\\rm (RB)} =\nm \\int \\hbox{d} {\\bf r}\\;r_{\\perp x}^2\\; \\delta \\rho_{\\rm scl}\\left({\\bf r}\\right),\n\\end{equation}\nwhere\n\\bel{ddenpart}\n\\delta \\rho_{\\rm scl}\\left({\\bf r}\\right)=-\\frac{d_s}{\\pi}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1}\\int \\hbox{d} \\varepsilon\\; n\\left(\\varepsilon\\right)\\;\nG_{{\\rm CCT}_1}\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right),\n\\end{equation}\nwhere $G_{{\\rm CCT}_1}$ is given by (\\ref{Gct}) with ${\\rm CT}$ being the\nclosed ${\\rm CT}_1$, i.e.,\n${\\rm CCT}_1$ (${\\bf r}_1\\rightarrow {\\bf r}_2 \\rightarrow {\\bf r}$).\nWith the smooth (extended) TF MI component (\\ref{TFrig}),\nsee also the section \\ref{etfmi}, the equation (\\ref{dmilocapproach})\nyields semiclassically\n\\bel{milocapproach}\n\\Theta_{x\\; {\\rm scl}} \\approx \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)},\n\\end{equation}\nthat is in agreement with the adiabatic picture\nof the statistically equilibrium rotation \\cite{mskbPRC2010}.\nNote that the non-adiabatic MI at arbitrary rotation\nfrequencies for the HO mean field by Zelevinsky \\cite{zelev} was extended\nto the finite\ntemperatures in \\cite{mskbPRC2010}.\n\n\nWe emphasize that due to\nan averaging over the phase space variables, one survives with\nthe NLLLA.\nNote also that the classical angular-momentum projection (\\ref{l2})  in the\nrotating body-fixed coordinate system is caused by the global rotation\nwith a given frequency $\\omega$ rather than by\nthe motion of particles along the trajectories  inside the nucleus\nwith respect to this system,\nconsidered usually in the cranking model.\nAccording to the time-reversible symmetry of the Routhian,\nthe particles are, indeed, moving in the non-rotating coordinate system\nalong these trajectories in both opposite directions.\nTheir contributions to the total angular momentum of the nucleus turns out to\nbe zero. Performing then\nthe integration over ${\\bf s}$ in (\\ref{dthetax01new})\n in the spherical coordinate system,\none obtains the rigid-body shell correction\n$\\delta \\Theta_x^{\\rm (RB)}$ in the NLLLA\nas explained in Appendix E.2.\nNote that\nthe cranking model for the nuclear rotation\nimplies that the correlation (non-local) corrections to\n(\\ref{dthetax01new}) and (\\ref{rigbodmomgen}) should be\nsmall enough with respect to the main rigid-body shell component\n$\\delta \\Theta_x^{\\rm (RB)}$ to be neglected within the adiabatic picture\nof separation of the global rotation of the Fermi system\nfrom its vibration and then,\nboth from the internal motion of particles.\nOther contributions,\nexcept for a smooth rigid-body part coming from $\\Theta_x^{00}$, like\n$\\Theta^{10}_{x}$ and $\\Theta^{11}_{x}$, as\nreferred to the fluctuation (non-local) correction\nto the rigid body MI are found semiclassically to be negligibly small\nin the NLLLA\ndue to the averaging\nover phase-space variables, see Appendix E.2b.\nIn particular, for the HO Hamiltonian, it was shown that there is almost\nno contribution of the\n$\\delta \\Theta_x^{11}$ at leading order in $\\hbar$ in \\cite{mskbPRC2010}.\nThus, with the semiclassical precision, from the\nadiabatic cranking model expression (\\ref{thetaxdefG}) we come to the\nMI of the statistically equilibrium rotation (\\ref{milocapproach}),\nwhich must be the rigid-body MI, according to\nthe general theorem of the statistical physics. This is in agreement\nwith the ETF approach of section \\ref{etfmi}.\nOur semiclassical derivations, valid for the rotation frequencies\n$\\hbar \\omega  \\ll \\hbar \\Omega$, are beyond the quantum criterion\nof the application of the standard\n2nd order perturbation approach within the cranking model where\n$\\hbar \\omega $ is small as compared to the distance between the\nneighboring levels of quantum spectra. We point out that this weakness\nof the perturbation theory criterion is similar to that with the statistical\naveraging in the heated Fermi systems and with accounting for the pairing\ncorrelations \\cite{belyaevhighspin,belyaevbif}, where the role of the\ndistance between the quantum neighboring energy levels plays the\ntemperature and the pairing gap, as distance between gross shells\n$\\hbar \\Omega$ (\\ref{hom}) in the POT \\cite{strutmag}, respectively.\n\n\n\\medskip\n\n\\subsubsection{\nSHELL CORRECTIONS TO THE RIGID-BODY MI}\n\\label{SCperprigrot}\n\nUsing (\\ref{ddenpart}) for calculations of\nthe MI rigid-body shell correction\n$\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)}$ (\\ref{drigbodmomgen}),\none may exchange the order of integrations over the coordinate ${\\bf r}$\nand energy $\\varepsilon$.\nBy making use also of the semiclassical\ntrajectory expansion (\\ref{GRdefsem}) for the oscillating Green's\nfunction component\n$G_1\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)$ of the sum (\\ref{Ggsplit}), one finds\n\\begin{eqnarray}\\label{dTxrigSCL}\n&&\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm (RB)} =\n-\\frac{m d_s}{\\pi}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} \\varepsilon\\;\nn(\\varepsilon) \\qquad \\nonumber\\\\\n&\\times& \\int \\hbox{d} {\\bf r}\\;\\left\\{r_{\\perp x }^2 \\mathcal{A}\\left({\\bf r},{\\bf r};\\varepsilon\\right)\n\\right.\\qquad\\nonumber\\\\\n&\\times& \\left.\n\\hbox{exp}\\left[\\frac{i}{\\hbar}\\;\n S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right) -\n\\frac{i \\pi}{2}\\sigma - i\\phi_{d}\\right]\n\\right\\}_{{\\rm CCT}_1}.\\qquad\n\\end{eqnarray}\nAs usually, with the  semiclassical precision, we evaluate\nthe spatial integral by the SPM extended to continuous symmetries\n\\cite{strutmag,sclbook,migdalrev} and the\nbifurcation phenomena (ISPM)\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar}.\nThe SPM (ISPM) condition writes\n\\begin{eqnarray}\\label{spmcond}\n&&\\left[\\frac{\\partial S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)}{\\partial {\\bf r}_1}\n+ \\frac{\\partial S\\left({\\bf r}_1,{\\bf r}_2;\\varepsilon\\right)}{\\partial {\\bf r}_2}\n \\right]^{\\ast}_{{\\rm CCT}_1}\n\\nonumber\\\\\n &\\equiv&\n\\left(-{\\bf p}_1+{\\bf p}_2\\right)^{\\ast}_{{\\rm CCT}_1},\n\\end{eqnarray}\nwhere the asterisk means the SPM value of the spatial coordinates\nand momenta,\n$~{\\bf r}_j={\\bf r}_j^{\\ast}~$ and $~{\\bf p}_j={\\bf p}_j^{\\ast}~$ ($~j=1,2$)\nat the closed ${\\rm CT}_1$s\nin the phase space,\n${\\bf r}^{\\ast}_1={\\bf r}^{\\ast}_2$ and ${\\bf p}^{\\ast}_1={\\bf p}^{\\ast}_2$. Thus, with the\nstandard relations for the canonical variables by using the action as a\ngenerating function, one arrives\nat the PO condition  on right of (\\ref{spmcond}).\nWithin  the simplest ISPM\n\\cite{spheroidptp,maf,migdalrev,magvlasar},\nthe other smooth factors $r_{\\perp x}^2$ and\n$\\mathcal{A}_{{\\rm CT}_1}\\left({\\bf r},{\\bf r},\\varepsilon\\right)$\nof the integrand in (\\ref{dTxrigSCL})\n can be taken off the\nintegral over ${\\bf r}$ at these stationary points. Assuming that the\nquantum averages\n$\\langle (y^2+z^2)^2 \\rangle\/\\varepsilon$ are smooth enough functions of $\\varepsilon$\nas compared to other factors, for instance, $\\delta n$,\none may take them approximately also off the integral\nover $\\varepsilon$ at the chemical potential, $\\varepsilon=\\mu$. For example,\nfor the HO potential (see \\cite{mskbPRC2010}),\nthey are simply exact constants.\nTherefore, the main contribution into the integral\nin (\\ref{dTxrigSCL}) is coming from\nthe PO stationary-phase points, determined by (\\ref{spmcond}),\n as for calculations of the\nlevel-density shell corrections $\\delta g_{\\rm scl}$ (\\ref{dlevdenscl})\n\\cite{sclbook,strutmag,migdalrev,mskbPRC2010}.\nThe SPM condition (\\ref{spmcond}) is\n identity for any stationary point of the classically accessible\nspatial region for a particle motion filled by PO families\nin the case of their high\ndegeneracy\n$\\mathcal{K}\\geq 3$.\nFor instance, it is the case for the contribution  of\nthe three dimensional (3D) orbits in the axially symmetric HO-potential well\nwith commensurable frequencies, $\\omega _x=\\omega _y=\\omega _\\perp$ and $\\omega _z$\n\\cite{magosc,mskbPRC2010}.\nThe stationary points occupy\nsome spatial subspace for a smaller degeneracy $\\mathcal{K}$.\nIn the latter case of the equatorial orbits (EQs) ($\\mathcal{K}=2$)\nin this HO potential well,\nthe SPM condition is\nidentity in the equatorial plane $z=0$.\nFollowing\nsimilar derivations of  the oscillating component $\\delta g_{\\rm scl}$\n(\\ref{dlevdenscl})\nof the level density $g_{\\rm scl}(\\varepsilon)$ (\\ref{totdensscl})\nand free-energy shell correction $\\delta F_{\\rm scl}$ (\\ref{fpotau}),\none expands the\nsmooth amplitudes and action phases of the\nMI shell corrections\n$\\delta \\Theta_{\\kappa {\\rm scl}}^{\\rm rig}$ (\\ref{dTxrigSCL}) up to the\nfirst nonzero terms\n(see Appendix C of \\cite{mskbPRC2010} and Appendix E.2\nhere).\nFinally, from (\\ref{dTxrigSCL}), one obtains \\cite{mskbPRC2010}\n\\bel{dTxrigSCLgen}\n\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}=\\frac{m}{\\mu}\\;{\\mbox {\\rm Re}} \\sum_{\\rm PO}\n\\langle r_{\\perp x}^2 \\rangle_{{}_{\\! {\\rm PO},\\mu}}\\;\\delta F_{\\rm PO},\n\\end{equation}\nwhere $\\langle r_{\\perp x}^2 \\rangle_{{\\rm PO},\\mu}$\nis the average given by\n\\bel{avsqcoor}\n\\langle r_{\\perp x}^2 \\rangle_{{\\rm PO},\\varepsilon} =\n\\frac{\\int \\hbox{d} {\\bf r}\\;\\mathcal{A}_{\\rm PO}\\left({\\bf r},{\\bf r};\\varepsilon\\right)\\;r_{\\perp x}^2\n }{\\int \\hbox{d} {\\bf r}\\;\\mathcal{A}_{\\rm PO}\\left({\\bf r},{\\bf r};\\varepsilon\\right)}\n\\end{equation}\nat $\\varepsilon=\\mu$, $\\mathcal{A}_{\\rm PO}({\\bf r},{\\bf r};\\varepsilon)$ are the\nGreen's function amplitudes\nfor a closed ${\\rm CT}_1$ in the phase space, i.e., PO.\nIntegration over ${\\bf r}$ is performed  over the classically\naccessible region of the spatial coordinates.\nSemiclassical expression (\\ref{dTxrigSCLgen})\nis general for any potential well.\nShorter POs are\ndominating in the PO sum (\\ref{dTxrigSCLgen})\n\\cite{sclbook,strutmag,kolmagstr,mskbPRC2010,migdalrev},\nsee\n(\\ref{fpotau}), (\\ref{descl}). Therefore,\naccording to (\\ref{fpotau}) for $\\delta F_{\\rm scl}$,\nwe obtain approximately the relation\n\\bel{dTxrigSCLgen1}\n\\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}\\approx\n\\frac{m}{\\mu}\\;\n\\langle r_{\\perp x}^{2}\\rangle_{\\mu}\\;\n\\delta F_{\\rm scl},\n\\end{equation}\nwhere $\\langle r_{\\perp x}^2 \\rangle_{\\mu}$ is an\naverage value of the quantity (\\ref{avsqcoor}),\nindependent of the specific PO, at $\\varepsilon=\\mu$ over short\ndominating POs.\n\n\n\nFor the axially symmetric\n HO potential well with the commensurable frequencies $\\omega _\\perp$\nand $\\omega _z$, as the simplest example,\nthe integration in (\\ref{avsqcoor}) over ${\\bf r}$\nfor the 3D contribution means over the 3D volume occupied by\nthe 3D families of orbits.\nFor the EQ component\nthe integral is taken\nover the 2D spatial region filled by the EQ families\nin the equatorial ($z=0$) plane \\cite{mskbPRC2010}.\nIn the incommensurable-frequency\ncase (irrational $\\omega _\\perp\/\\omega _z$), one has the only EQ-orbit contributions.\nThe average (\\ref{avsqcoor}) can be easily calculated\nby using the Green's  function amplitudes $\\mathcal{A}_{\\rm PO}$\nfor 3D and\nfor EQ orbits, which are given in \\cite{magosc,mskbPRC2010}.\nFinally, for the considered HO potential, one may arrive at\n\\bel{dthetaxdehoscl}\n\\delta \\Theta_{x\\; {\\rm scl}} \\approx \\delta \\Theta_{x {\\rm scl}}^{\\rm (RB)}=\\frac{1 +\n\\eta_{\\rm HO}^2}{3 \\omega _\\perp^2}\\;\\delta F_{\\rm scl},\n\\end{equation}\nwhere $\\delta F_{\\rm scl}$ is the semiclassical PO sum\n(\\ref{fpotau}), (\\ref{descl})\nfor the semiclassical free-energy shell-corrections,\n$\\eta_{{}_{\\!{\\rm HO}}}=\\omega _\\perp\/\\omega _z$ is the deformation parameter.\nFor the parallel (alignment) rotations around the symmetry axis,\n one finds similar relations of the MI through the rigid-body MI to\nthe free-energy shell corrections.\nMoreover, one has such relations\nfor the smooth TF parts, in particular for the HO case, see Appendices E.2.1\nhere and D1 in\n\\cite{mskbPRC2010}.\nThus, for the total\nmoment $\\Theta_x$ [see (\\ref{thetaxsum})], one may prove semiclassically\nwithin the POT, up to the same $\\hbar$ corrections in a\nsmooth TF part, that\nthe shell MI and free-energy shell corrections are approximately\nproportional, in particular exactly\n for that\nHO Hamiltonian \\cite{mskbPRC2010}:\n\\bel{thetaxhoscl}\n\\Theta_{x\\; {\\rm scl}}=\\frac{1 + \\eta^2}{3 \\omega _\\perp^2}\\;F_{\\rm scl},\\qquad\nF_{\\rm scl}=F_{{\\rm ETF}}+\\delta F_{\\rm scl}.\n\\end{equation}\nWe emphasize that the POT expressions (\\ref{dthetaxdehoscl})\nfor $\\delta \\Theta_{x\\; {\\rm scl}}$ and\n(\\ref{thetaxhoscl})\nof $\\Theta_{x\\; {\\rm scl}}$  were derived\nwithout a direct use of the statistically equilibrium rotation\ncondition \\cite{bohrmot,mskbPRC2010}.\n\n\nSubstituting the semiclassical PO expansion\n(\\ref{descl}) for the free-energy shell correction\n$\\delta F_{{\\rm scl}}$ (\\ref{fpotau}) (after \\cite{spheroidptp})\nfor 3D orbit families and\nfor EQ POs into (\\ref{dthetaxdehoscl}),\none arrives  finally at the explicit POT expressions for the MI\nshell corrections $\\delta \\Theta_x$\nin terms of the characteristics of the classical\nPOs. For the mean field with the spheroidal shapes and sharp edges\n(spheroid cavity), these derivations can be performed similarly\nas for the HO Hamiltonian in \\cite{mskbPRC2010}\nbut with accounting\nfor the specific PO degeneracies.\nNote that\nthe parallel, $\\delta \\Theta_z$,\nand perpendicular, $\\delta \\Theta_x$,\nMI shell components  are expressed through the 3D and EQ POs\nthrough the free-energy shell correction which contains generally\nspeaking both them for the deformations larger the bifurcation ones.\nThe dominating\n contributions\nof one of these families or coexistence of both together\ndepend on the surface deformation parameter (semi-axis ratio\nof spheroid). For the critical deformations and on right of them, one observes\nthe significant enhancement of the MI shell corrections through the PO\nlevel-density amplitudes $\\mathcal{B}_{\\rm POT}$\n[see (\\ref{dlevdenscl})] of the free-energy shell corrections\n(\\ref{fpotau}), (\\ref{descl}).\n\n\n\\subsubsection{COMPARISON OF SHELL STRUCTURE CORRECTIONS WITH\\\\ QUANTUM RESULTS}\n\\label{qmsclcomparison}\n \n\nFig.\\ \\ref{fig15} shows the\nsemiclassical free-energy shell correction $\\delta F_{\\rm scl}$,\n[(\\ref{fpotau}), (\\ref{descl}), see also \\cite{magosc,mskbPRC2010}]\nvs the particle-number variable, $A^{1\/3}$,\nat a small temperature of $T=0.1\\; \\hbar \\omega_{{}_{\\! 0}}$ for different critical\nsymmetry-breaking and bifurcation deformations\n$\\eta_{{}_{\\! HO}}=1$, $6\/5$, and $2$ of the HO potential\n\\cite{sclbook,mskbPRC2010} with\nthe corresponding\nquantum SCM results for the same\ndeformations.\nThis comparison also shows practically a perfect agreement\nbetween the\nsemiclassical,\n(\\ref{fpotau}) and (\\ref{descl}),\nand quantum results.\nFor the spherical case ($\\eta_{{}_{\\! HO}}=1$),\none has only contributions of the families\nof 3D orbits with the highest degeneracy $\\mathcal{K}=4$.\nAt the bifurcation points $\\eta_{{}_{\\! HO}}=6\/5$ and $2$ the relatively simple\nfamilies of these 3D POs appear\nalong with EQ orbits of smaller degeneracy.\nFor $\\eta_{{}_{\\! HO}}=6\/5$, one mainly has the contributions from EQ POs\nbecause the 3D\norbits are generally too long in this case.\nFor the bifurcation point $\\eta_{{}_{\\! HO}}=2$,\n one finds an interference of the two\ncomparably large contributions of EQ and 3D\norbits\nwith  essentially the different time periods $t_{{}_{\\! EQ}}$ and\n$t_{{}_{\\! 3D}}$, respectively.\n\n\nThe quantum (QM) and semiclassical\n(SCL) shell corrections to\nthe MI $\\delta \\Theta^{}_x$ of (\\ref{dthetaxdehoscl})\nare compared in Fig.\\ \\ref{fig16}.\nAn excellent agreement is observed\nbetween the semiclassical and quantum results as for the free-energy shell\ncorrections $\\delta F$.\n It is not really astonishing\nbecause of the proportionality of the $\\delta \\Theta^{}_x$ to\n$\\delta F$ [see (\\ref{dthetaxdehoscl})].\nOne finds in particular the same clear interference of contributions of\n3D and EQ POs in the shell corrections to the MI at $\\eta_{{}_{\\! HO}}=2$.\nThe exponential decrease of shell oscillations with increasing temperature,\ndue to the temperature factor\nin front of the PO energy-shell correction components $\\delta E_{\\rm PO}$\nin (\\ref{fpotau})\nis clearly seen in Fig.\\ \\ref{fig16}. As the MI and free-energy\nshell corrections are basically proportional [see (\\ref{dTxrigSCLgen})]\nfor any mean potential well, we may emphasize\nthe amplitude enhancement of the MI near the bifurcation\ndeformations due to that\nfor the energy-shell corrections found in\n\\cite{ellipseptp,spheroidptp,maf,migdalrev,magvlasar}.\nThe\ncritical temperature for a disappearance of shell effects in the MI\nis found for prolate deformations ($\\eta>1$)\nand particle numbers\n$A \\sim 100-200$, approximately at $T^{}_{cr}=\n\\hbar \\omega^{}_{EQ}\/\\pi \\sim \\hbar \\omega^{}_{0}\/\\pi \\approx 2-3$ MeV\njust as for $\\delta F$, see \\cite{sclbook,strutmag,mskbPRC2010}. This effect\nis also general for any potentials.\nThe particle-number dependence of the shell corrections\n $\\delta \\Theta^{}_z$  to the total MI $\\Theta^{}_z$\n(alignment)\nis not shown because it is similar to that of $\\delta \\Theta^{}_x$\nthrough their approximate relations,\n$\\delta \\Theta^{}_z \\propto \\delta \\Theta^{}_x \\propto \\delta F$.\n\n\n\n\\section{Conclusions}\n\\label{concl}\n\nWe derived the dynamical equations of motion, such as the conservation\nof the particle number, momentum and energy as well as the\ngeneral transport equation for the entropy for \\textit{low}\nfrequency excitations in nuclear matter within the Landau\nquasiparticle theory of \\textit{heated} Fermi-liquids.\nOur approach is based essentially on the\nLandau--Vlasov equation for the distribution function, and it\nincludes all its moments in phase space, in contrast to several\ntruncated versions of fluid dynamics\nsimilar to the hydrodynamic description\nin terms of a few first moments. From the dynamics of the\nLandau--Vlasov equation for the distribution function, linearized\nnear the \\textit{local} equilibrium, we obtained the momentum flux\ntensor and heat current in terms of the shear modulus, viscosity,\nin-compressibility and thermal conductivity coefficients as for\nvery viscose liquids called sometimes \\textit{amorphous solids}.\nWe obtain the dependence of these coefficients on the\n         temperature, the frequency and the Landau interaction parameters.\nWe derived the \\textit{temperature\nexpansions} of the density-density and\ntemperature-density response functions for nuclear matter and got\ntheir \\textit{specific expressions for small temperatures} as\ncompared to the chemical potential. The \\textit{hydrodynamic limit}\nof normal liquids for these response functions \\textit{within the\nperturbation theory} was obtained from the Landau--Vlasov equation for\nboth distribution function and sound\n     velocity, as for an eigenvalue problem.\nIn this way we found the Landau--Placzek and first sound peaks in\nthe corresponding strength functions as the hydrodynamic limit of\nthe Fermi-liquid theory for heated Fermi-systems. The former (heat\npole) peak was obtained only because of the use of the local\nequilibrium in the Landau--Vlasov linearized dynamics instead of\nthe global static Fermi-distribution of the giant multipole-resonance\nphysics. This is very important for the dispersion equation and\nits wave velocity solutions.\n\nWe got the \\textit{isolated, isothermal and adiabatic\nsusceptibilities for the Fermi-liquids} and showed that they\nsatisfy the \\textit{ergodicity} condition of equivalence of the\nisolated and adiabatic susceptibilities as well as the general\nKubo inequality relations. We found the \\textit{correlation function}\nusing the fluctuation-dissipation theorem and discussed its\nrelation to the susceptibilities and Landau--Placzek \"heat pole\" in\nthe hydrodynamic limit.\n\nWe applied the theory of heated Fermi-liquids to the Fermi-liquid\ndrop model of finite nuclei within the Landau--Vlasov dynamics in the\nnuclear interior and macroscopical boundary conditions in the\neffective sharp surface approximation. Solutions of this\nproblem in terms of the response functions and transport\ncoefficients were obtained. We considered the hydrodynamic limit\nof these solutions and found the ``heat pole'' correlation function\nfor frequencies smaller than some critical frequency. The latter\nwas realized only because of using the local equilibrium for the\ndistribution function. The isolated, isothermal and adiabatic\nsusceptibilities for finite nuclei within the FLDM in the ESA\nwere derived. We showed that the ergodicity condition is satisfied\nalso for finite Fermi-systems as for infinite nuclear\nmatter in the same ESA.\n\nWe found a three-peak structure of the collective strength\nfunction: the \"heat\", standard hydrodynamic and essentially\nFermi-liquid peaks. The conditions for the existence of such modes\nwere analyzed and the temperature dependence of their transport\ncoefficients such as friction, stiffness and inertia were\n obtained\nin particular, in the hydrodynamic limit.\nWe arrived at the increasing temperature dependence of the\nfriction coefficient for the specific Fermi-liquid mode which\nexist due to the Fermi-surface distortions. At enough large\ntemperatures, we showed a nice agreement with the results for the\nfriction which were obtained earlier within the microscopic\nshell-model approach of \\cite{hofbook}. The correlation functions\nfound in the FLDM and quantum shell models were discussed in\nrelation to the susceptibilities and ergodicity properties of\nfinite nuclei.\n\nThe expression for the surface symmetry-energy constant $k_{{}_{\\! S}}$\nwas derived from simple isovector solutions\nof the particle density and energies in the leading ES\napproximation. We used them for the calculations\nof the energies, sum rules of the  IVGDR strength and the transition\ndensities\nwithin the HDM and FLDM  \\cite{kolmagsh} for several Skyrme-force parameters.\nThe surface symmetry-energy constant\ndepends much on the fundamental\nwell-known parameters of the Skyrme forces, mainly through the\ncoefficient in the  density gradient terms of the isovector part of the\nenergy density.\nThe value of this isovector constant\n is rather sensitive  also\non the SO interaction.\nThe IVGDR strength is split into the two main and satellite peaks.\nThe mean energies and EWSRs within both HDM and FLDM are in fairly\ngood agreement with the\nexperimental data.\n\nSemiclassical functional expressions were derived\n in the framework of the Extended Thomas--Fermi approach. We used\n these analytical expressions to obtain a self-consistent description\n of rotating nuclei where the rotation velocity impacts on the structure\n of the nucleus. It has been shown that such a treatment leads, indeed,\n to the Jacobi phase transition to triaxial shapes as already predicted\n in \\cite{CPS74} within the rotating LDM. We emphasize\n that the rigid-body moment of inertia gives a quite accurate\n approximation for the full ETF value. Being aware of the mutual influence\n between rotation and pairing correlations\n \\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev}, it would be\n especially  interesting to work on an approach that is able to\ndetermine  the nuclear structure depending on its angular velocity,\nas we have  done here in the ETF approach, but taking pairing correlations\nand their rotational quenching into account.\n\nWe derived also the shell corrections\nof the MI\nin terms of free-energy shell corrections\nwithin the nonperturbative extended POT through those of the rigid-body MI of\nthe equilibrium rotations,\nwhich is exact for the HO potential.\nFor the HO, we extended\nto the finite temperature case the Zelevinsky\nderivation of the non-adiabatic\nMI at any rotation frequency.\nFor the deformed HO potential, one finds a perfect agreement\nbetween the semiclassical POT and quantum\nresults for the free-energy\nand the MI shell corrections\nat several critical deformations and temperatures.\nFor larger temperatures, we show that the short EQ orbits are\nmostly dominant.\nFor small temperatures, one observes a remarkable interference of the\nshort 3D and EQ orbits in the superdeformed region.\nAn exponential decrease of all shell corrections\nwith increasing temperature is observed, as expected. We point out\nalso the amplitude enhancement of the MI shell corrections due to\nthe bifurcation catastrophe phenomenon.\n\nAs further perspectives, it would be worth to apply our results to\ncalculations of the IVGDR structure within the Fermi-liquid droplet\nmodel to determine the value of the fundamental surface symmetry-energy\nconstant from comparison with experimental data for the pygmy resonance\n\\cite{adrich,wieland} and theoretical calculations\n\\cite{vretenar1,vretenar2,ponomarev,nester1,nester2,BMR}.\nFor further extensions to\nthe description of  the isovector low-lying collective states, one has first to\nuse the POT for including semiclassically the shell effects\n\\cite{strutmag,sclbook,gzhmagfed,blmagyas,BM}.\nIt would be also worth to apply this semiclassical theory\nto the shell corrections of the MI for the spheroid cavity\nand for the inertia parameter of the\nlow-lying collective excitations in nuclear dynamics involving\nmagic nuclei\n\\cite{dfcpuprc2004,magvvhof,yaf,gzhmagfed}. One of the most attractive\nsubject of the semiclassical periodic orbit theory, in line of the main\nworks of S.T. Belyaev\n\\cite{belyaevfirst,belyaevzel,belzel,belsmitolfay87},\nis its extension\nto the {\\it pairing} correlations\n\\cite{brackquenNPA1981,brackrocciaIJMPE2010},\nand their influence on the collective\nvibrational and rotational excitations in heavy deformed neutron-rich\nnuclei\n\\cite{belyaevhighspin,pomorbartelPRC2011,sfraurev} (see also\n\\cite{abrpairing} for the semiclassical phase-space dynamical approach\nto the Hartree--Fock--Bogoliubov theory).\n\n\\bigskip\n\n\\section*{Acknowledgement}\n\n\\label{aknow}\n\n\\medskip\n\n\nAuthors gratefully acknowledge \nH.\\ Hofmann for many suggestions and fruitful discussions, also \n S.\\ Aberg, V.\\ I.\\ Abrosimov, J.\\ Blocki, R.\\ K.\\ Bhaduri, M.\\ Brack, \nV.\\ Yu.\\ Denisov, S.\\ N.\\ Fedotkin,\nH.\\ Heisenberg, F.\\ A.\\ Ivanyuk,\nV.\\ M.\\ Kolomietz, M.\\ Kowal, J.\\ Meyer, V.\\ O.\\ Nesterenko, V.\\ V.\\ Pashkevich,\nM.\\ Pearson, V.\\ A.\\ Plujko, P.\\ Ring, V.\\ G.\\ Zelevinsky, \nA.\\ I.\\ Sanzhur, S.\\ Siem, J.\\ Skalski,\nand X.\\ Vinas for many useful discussions.  One of us (A.G.M.) is\nalso very gratitude for a nice hospitality during his working visits of the\nTechnical Munich University in Garching and the University \nof Regensburg in Germany, the Interdisciplinary \nHubert Curien Institute  of the Louis Pasteur University \nin Strassburg of France,  and \nNational Centre for Nuclear Research in Otwock-Swierk of Poland.\n\n\n\\vspace{1cm}\n\n\\noindent\n\\begin{appendix}\n\\setcounter{equation}{0}\n\\renewcommand{\\thesubsection}{\\Alph{section}.\\arabic{subsection}}\n\\renewcommand{\\theequation}{\\mbox{\\Alph{section}.\\arabic{equation}}}\n\\noindent\n\\section{Elements of Landau theory for equilibrized systems}\n\\label{app1}\n\\subsection{Thermodynamic relations}\n\\label{app1thermrel} Let us begin recalling the fundamental\nequations $TdS = dE-\\mu dN +{\\cal P} \\hbox{d} \\mathcal{V}$ and $-S\\hbox{d} T = \\hbox{d} F-\\mu\n\\hbox{d} N +{\\cal P} \\hbox{d} \\mathcal{V}$, which are related to each other by the\nLegendre transformation $F=E-TS$. They imply the following\nrelations for the chemical potential $\\mu$ and pressure\n${\\cal P}$: \n\\bel{chempot} \n\\mu =-T\\left(\\frac{\\partial S }{\\partial\nN}\\right)_{E,{\\mathcal{V}}}= \\left(\\frac{\\partial E }{\\partial\nN}\\right)_{S,{\\mathcal{V}}}= \\left(\\frac{\\partial F }{\\partial\nN}\\right)_{T,{\\mathcal{V}}}, \n\\end{equation} \n\\bel{pressure} {\\cal P} =\n-\\left(\\frac{\\partial E }{\\partial \\mathcal{V}}\\right)_{S,N}=\n-\\left(\\frac{\\partial F }{\\partial \\mathcal{V}}\\right)_{T,N}. \n\\end{equation} \nFor\nhomogeneous systems the {\\it intensive} quantities depend only on\ntwo independent variables. For instance, the entropy per particle\n$S\/N=\\varsigma(E\/N,\\mathcal{V}\/N)$ only depends on the energy and\nvolume per particle, $E\/N $ and $\\mathcal{V}\/N$ respectively. For\nsuch systems, the adiabadicity condition may simply be expressed as\n$\\varsigma = {\\rm const}$. \nCommonly in\nnuclear physics, one uses  the particle density $\\rho=N\/\\mathcal{V}$,\nin which case the chemical potential can be expressed as\n\\bel{chempothom} \n\\mu = \\left({\\partial \\phi \\over\n\\partial \\rho}\\right)_T \n\\end{equation}\n with $\\phi = F\/\\mathcal{V}$ being the\nfree internal energy per unit volume.\n\nFor differential quantities there exist various variants of the  \nGibbs-Duheim relation \n\\bel{giduvar} \n\\hbox{d} \\phi = -\\varsigma \\rho \\hbox{d} T + \\mu \\hbox{d} \\rho \\qquad \\mbox{or} \n\\end{equation}\n\\begin{eqnarray}\\label{gibbsduh} \n&&\\hbox{d}{\\cal P} = \\varsigma \\rho \\hbox{d} T + \\rho \\hbox{d} \\mu~\n\\quad{\\rm implying} \\quad \\left({\\partial \\varsigma \\over\n\\partial \\mu}\\right)_T \\nonumber\\\\\n&=& \\frac{1}{ \\rho} \\left[\\left(\\frac{\\partial \\rho\n}{\\partial T}\\right)_\\mu- \\varsigma \\left(\\frac{\\partial \\rho\n}{\\partial \\mu}\\right)_T \\right], \n\\end{eqnarray}\n  as follows from\nLegendre transformations. Thus for the derivatives of the pressure\n$\\mathcal{P}$, considered as functions of $T$ and $\\rho$, one gets from\n(\\ref{gibbsduh}):\n\\begin{eqnarray}\\label{dpdmt}\n &&\\left(\\frac{\\partial {\\cal P} }{\\partial \\rho} \\right)_T =\n    \\rho \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T,\n    \\qquad\n    \\left(\\frac{\\partial {\\cal P} }{\\partial T} \\right)_\\rho =\\nonumber\\\\\n  &=&\\rho\\left(\\varsigma-\n    \\left(\\frac{\\partial \\rho } {\\partial T}\\right)_{\\mu}\n    \\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T\\right).\n\\end{eqnarray}\n In deriving such relations, it is useful to employ special\nproperties of the Jacobian,  which allows\none to perform transformations between different variables (see\ne.g.,\n\\cite{brenig}).\nThese relations will be used below to get the specific heats as\nwell as the isothermal and adiabatic compressibilities, together\nwith the corresponding susceptibilities. At first, we shall look at\n{\\it in}-compressibilities  defined by the\nderivative of the pressure over the particle density\n (multiplied by a factor of 9). At {\\it\nconstant entropy per particle} $\\varsigma$, the {\\it adiabatic}\nin-compressibility $K^{\\varsigma}$ writes \n\\bel{incompraddef}\nK^{\\varsigma} \\equiv 9 \\left(\\frac{\\partial {\\cal P} }{\\partial\n\\rho}\\right)_\\varsigma = 9 \\left(\\frac{\\partial {\\cal P} }{\\partial\n\\mu}\\right)_\\varsigma \\left(\\frac{\\partial \\mu }{\\partial\n\\rho}\\right)_\\varsigma. \n\\end{equation} \nTo get the corresponding quantity\nat constant temperature $K^{T}$, one only needs to replace\n$\\varsigma$ by $T$. According to (\\ref{dpdmt}) and\n(\\ref{chempothom}), one obtains \n\\bel{incomprTdef} \nK^{T} \\equiv 9\n\\left(\\frac{\\partial {\\cal P} }{\\partial \\rho}\\right)_T = 9 \\rho\n\\left(\\frac{\\partial \\mu }{\\partial \\rho}\\right)_T = 9 \\rho\n    \\left(\\frac{\\partial^2 \\phi }{\\partial \\rho^2}\\right)_T.\n\\end{equation}\n\nNext we turn to the specific heats at constant volume and constant\npressure. If measured per particle, they can be defined in terms\nof the entropy per particle $\\varsigma$  as \n\\bel{specifheatpdef}\n{\\tt C}_{\\mathcal{V}} =\n    T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_{\\mathcal{V}}=\n    T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_\\rho,\n\\quad {\\tt C}_{\\cal P} =\n    T \\left(\\frac{\\partial \\varsigma }{\\partial T} \\right)_{\\cal P}.\n\\end{equation} \nThey obey the following, well known relation to the\nin-compressibilities \\cite{forster,brenig} \n\\bel{cvcpkSkT} \n\\left(\\frac{{\\tt C}_{\\cal P} \n}{{\\tt C}_{\\mathcal{V}}}\\right)= \n\\frac{\\left(\\partial {\\cal P} \/\\partial\n\\rho\\right)_\\varsigma }{\\left(\\partial {\\cal P} \/\\partial \\rho \\right)_T} = \n\\frac{K^{\\varsigma} }{K^{T}}. \n\\end{equation}\n\nFor the variation of the entropy $\\varsigma$ per particle,  one\nfinds %\n\\bel{entropypart}\n \\hbox{d} \\varsigma = - \\frac{1}{\\rho}\n\\left[\\varsigma + \\left(\\frac{\\partial \\mu }{\\partial\nT}\\right)_\\rho\\right] \\hbox{d} \\rho + \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\hbox{d} T,\n\\end{equation} \nafter using (\\ref{giduvar}) and the  specific heat ${\\tt\nC}_{\\mathcal{V}}$ of (\\ref{specifheatpdef}). To get the first term we\napplied \n\\bel{derphi} -\\left(\\frac{\\partial (\\varsigma\n\\rho) }{\\partial \\rho}\\right)_T= \\left(\\frac{\\partial \\mu \n}{\\partial T}\\right)_\\rho \\equiv \\frac{\\partial^2 \\phi }{\\partial\n\\rho \\partial T}, \n\\end{equation} \nwhich is a consequence of (\\ref{giduvar}).\n\n\n\\subsection{Landau theory proper}\n\\label{landtheorprop}\n\nIn the following, we will repeat some important relations discussed\nin \\cite{heipethrev} without arguing much about their proofs.\nThese relations will be needed to derive some specific\nthermodynamic properties for quantities, as the entropy or the\nspecific heats.\nA basic element in Landau theory is the microscopic\nexpression for the entropy per particle,  \n\\begin{eqnarray}\\label{entropydef} \n\\varsigma &=&\n    - \\frac{1}{\\rho} \\int \n\\frac{2 d {\\bf p} }{\\left(2 \\pi \\hbar \\right)^3}\n    \\left[f_{\\bf p} \\ln f_{\\bf p} + \\left(1-f_{\\bf p}\\right)\n    \\ln \\left(1-f_{\\bf p}\\right) \\right]\\nonumber\\\\\n   &=& \\left\\langle \\frac{p^2 }{3 m^*}\n    \\left(\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\\right) \\right\\rangle \/\n\\left\\langle \\frac{p^2 }{3 m^*} \\right\\rangle. \n\\end{eqnarray}\n in terms of\nthe Fermi distribution  $f_{{\\bf p}}$ [c.f. (\\ref{fgeq})]. The (static)\nquasiparticle density $\\rho$ in (\\ref{entropydef}) may be\nexpressed as \n\\bel{densstat} \n\\rho  =\\frac{N}{\\mathcal{V}} =  \\frac{p_{\\rm F}^3\n}{3 \\pi^2 \\hbar^3}=\n     \\int \\frac{2 \\hbox{d} {\\bf p} }{\\left(2 \\pi \\hbar \\right)^3} f_{\\bf p}\n = \\mathcal{N}(T) \\left\\langle \\frac{p^2 }{3 m^*} \\right\\rangle,\n\\end{equation} \nwith the density of states $\\mathcal{N}(T)$ \n(\\ref{enerdensnt}). The additional factor 2 in the integration\nmeasure accounts for the spin degeneracy. The expressions on the\nright in both (\\ref{entropydef}) and (\\ref{densstat}) are obtained\nafter integrating by parts. The brackets $<\\cdots>$ denote some\nkind of average, which if written for any quantity $A({\\bf r},{\\bf p},t)$ is\ndefined as \n\\bel{averag} \n\\langle A({\\bf r},{\\bf p},t) \\rangle\n    =\\frac{1}{\\mathcal{N}(T)} \\int \\frac{2 \\hbox{d} {\\bf p}^\\prime\n    }{(2 \\pi \\hbar)^3}\n    \\left(-\\frac{\\partial f_{{\\bf p}^\\prime}\n    }{\\partial \\varepsilon_{{\\bf p}^\\prime}}\\right)\n    A({\\bf r},{\\bf p}^\\prime,t).\n\\end{equation}\n In addition to the $\\mathcal{N}(T)$, one needs \n\\bel{mcapt} \n\\mathcal{M}(T) = \\mathcal{N}(T) \\left\\langle\n\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n    \\right\\rangle.\n\\end{equation}\n From (\\ref{densstat}) one may derive (see (2.9) and\n(2.11) of \\cite{heipethrev}) \n\\bel{drhomt} \n\\hbox{d} \\rho =\n    \\frac{\\mathcal{N}(T) }{\\mathcal{G}_0}~ \\hbox{d} \\mu +\n\\frac{\\mathcal{M}(T)}{\\mathcal{G}_0}~\\hbox{d} T,\n\\end{equation}\n[see also (\\ref{effmass}) for $\\mathcal{G}_0$] which allows\none to express the isothermal in-compressibility $K^{T}$ \n(\\ref{incomprTdef}) by (\\ref{isotherk}).\nFor the variation of the pressure with temperature, one gets\nfrom (\\ref{dpdmt}) and (\\ref{drhomt}) \n\\bel{dpdtr}\n    \\left(\\frac{\\partial {\\cal P} }{\\partial T} \\right)_{\\rho} =\n    \\left(\\varsigma- \\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right) \\rho =\n    \\frac{2}{3} \\rho {\\tt C}_{\\mathcal{V}}.\n\\end{equation} \nFor the proof of the second equation, we refer to (3.35) of\n\\cite{heipethrev} (mind however a difference in the notations for\nthe specific heat: Our $\\rho {\\tt C}_{\\mathcal{V}}$ is identical to the\n$ {\\tt C}_{\\mathcal{V}}$ of \\cite{heipethrev}). For our \n$ {\\tt C}_{\\mathcal{V}}$, one may derive the formula (see (3.34) of\n\\cite{heipethrev}) \n\\bel{specheatveq} \n{\\tt C}_{\\mathcal{V}} = \n\\frac{T \\mathcal{N}(T)}{\\rho} \\left\\langle \n\\left[\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n   -\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)} \\right]^2 \\right\\rangle.\n\\end{equation} \nCollecting (\\ref{incomprTdef}), (\\ref{isotherk}) and\n(\\ref{dpdtr}), one can write the variation of the pressure \nas\n\\bel{dpress} \n{\\rm d}{\\cal P} = \\rho \n\\left(\\frac{\\mathcal{G}_0}{\\mathcal{N}(T)} \n{\\rm d} \\rho + \\frac{2}{3} {\\tt C}_{\\mathcal{V}} {\\rm d} T\\right). \n\\end{equation}\n\n \nThermodynamic quantities such as\nin-compressibilities and susceptibilities are calculated under\ndifferent conditions as fixed temperature or entropy. As one\nknows (see, e.g., \n\\cite{forster}), these\n(in)-compressibilities may be associated to different sound\nvelocities. To make use of the adiabaticity condition mentioned\nearlier, we need the derivatives of the entropy per particle\n$\\varsigma(\\rho,T)$. The ones arising in (\\ref{entropypart}) can\nbe simplified by exploiting the specific Fermi-liquid expressions\ngiven in (\\ref{drhomt}) and the second relation of (\\ref{dpdtr})\nbetween the entropy per particle $\\varsigma$ and the specific heat\n${\\tt C}_{\\mathcal{V}}$,\n\\bel{dspdmt}\n  \\hbox{d} \\varsigma =\n   - \\frac{2}{3 \\rho} {\\tt C}_{\\mathcal{V}} \\hbox{d} \\rho +\n    \\frac{{\\tt C}_{\\mathcal{V}}}{T} \\hbox{d} T.\n\\end{equation} \nNext we turn to the  adiabatic in-compressibility\n$K^{\\varsigma}$ (\\ref{incompraddef}).  It may be expressed by\nthe isothermal one $K^{T}$ given in (\\ref{isotherk}), \nsee (\\ref{Kadiabat}).\nTo derive this relation, \nthe Jacobian transformation from $(\\rho,\\varsigma)$ to $(\\rho,T)$ for\nthe derivatives of the pressure in (\\ref{incompraddef}) has been applied\n[mind also\n(\\ref{dpdtr}), (\\ref{incomprTdef}) and (\\ref{dspdmt})]. Finally,\nfor the ratio of the specific heats, we find from\n(\\ref{cvcpkSkT}), (\\ref{isotherk}) and (\\ref{Kadiabat})\n\\bel{cpcvkakt} \n\\left(\\frac{{\\tt C}_{\\cal P} }{{\\tt C}_{\\mathcal{V}}}\\right)= \n1 + \\frac{4 T {\\tt C}_{\\mathcal{V}} \\mathcal{N}(T)}{9 \\rho\n\\mathcal{G}_0}. \n\\end{equation}\n\n\\subsection{Low temperature expansion}\n\\label{lowtempexp}\n\nIn this subsection, we address the temperature dependence of the\nquantities introduced above. It may be derived as discussed in\n\\cite{heipethrev} and conveniently\nexpressed by expansions in terms of ${\\bar T} = T\/\\varepsilon_{{}_{\\! {\\rm F}}}$; with\n$\\varepsilon_{{}_{\\! {\\rm F}}}$ being the Fermi energy at zero temperature, \n$\\varepsilon_{{}_{\\! {\\rm F}}}=p_{\\rm F}^2\n\/ (2m^*)=(3\\pi^2\\hbar^3 \\rho)^{2\/3}\/(2m^*)$.  For some of the\nquantities discussed below we shall include terms of third order\nin ${\\bar T} = T\/\\varepsilon_{{}_{\\! {\\rm F}}}$, which are not considered in \\cite{heipethrev}.\n\nFrom (\\ref{densstat}) one gets for the particle density\n$\\rho(\\mu,T)$ \n\\bel{densexp} \n\\rho =\n    \\frac{\\left(2 m^* \\mu\\right)^{3\/2} }{3 \\pi^2 \\hbar^3}\n    \\left(1 + \\frac{\\pi^2 \\tbar^2 }{8} \\right)\n\\end{equation} \nas function of the chemical potential $\\mu$ and the\ntemperature  $T$.  For the chemical potential $\\mu$, one obtains\n\\bel{chemicpot} \\mu = \\varepsilon_{{}_{\\! {\\rm F}}} \\left(1-\\frac{\\pi^2 \\tbar^2 }{12}\n\\right),\n\\end{equation} \nwhich is typical for a system of independent fermions.\n At this stage it may be worth while to mention\nthat the formulas presented here remain largely unchanged in case\nof the presence of a density dependent potential $V(\\rho)$. As\nlong as such a potential does not depend on the momentum, we may\njust change our s.p.\\ energy $\\varepsilon_{\\bf p}^{\\rm g.e.}$ to $\np^2\/(2 m^*) + V(\\rho)$, and the chemical potential $\\mu$ to the\n$\\mu'=\\mu-V(\\rho)$ of \\cite{heipethrev}.\n\n\nFor the density of states $\\mathcal{N}(T)$ of the quasiparticles, one\nfinds from (\\ref{enerdensnt}) \n\\bel{capnexp} \n\\mathcal{N}(T) = \n\\mathcal{N}(0)\\left(1-\\frac{\\pi^2 \\tbar^2 }{12} \\right), \n\\end{equation} \nwhere $\\mathcal{N}(0)$ is given by (\\ref{nzero}).\nSimilarly, for \n$\\mathcal{M}(T)$ defined in (\\ref{mcapt}),  one gets \n\\bel{mcapexp} \n\\mathcal{M}(T) = \n\\frac{\\pi^2}{6}\\;\\mathcal{N}(0)\\; {\\bar T}\n    \\left(1 + \\frac{13\\pi^2 \\tbar^2 }{60}\\right).\n\\end{equation} \nAs different to \\cite{heipethrev}, we include a temperature\ncorrection here, which is of interest for some of the quantities\ndescribed in the text. The specific heat \n${\\tt C}_{\\mathcal{V}}$ (\\ref{specheatveq}) per particle for the constant volume\nbecomes \n\\bel{specifheatv} \n{\\tt C}_{\\mathcal{V}}=\n   \\frac{\\pi^2{\\bar T}}{2}\n    \\left(1-\\frac{3 \\pi^2 \\tbar^2 }{10} \\right).\n\\end{equation} \nFor the isothermal in-compressibility $K^{T}$, one\ngets from (\\ref{isotherk}) and (\\ref{capnexp}) \n\\bel{isotherkexp}\nK^{T} = 6 \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_0\n    \\left(1 + \\frac{\\pi^2 \\tbar^2 }{12} \\right).\n\\end{equation} \nLikewise, for the in-compressibility modulus $K^{\\varsigma}$\n(\\ref{Kadiabat}) at constant entropy $\\varsigma$ per particle, one\nobtains \n\\bel{incompradexp} \nK^{\\varsigma} =\n    6 \\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{G}_0 \\left[1 + \\frac{\\pi^2 \\tbar^2 }{12}\n    \\left(1 + \\frac{4 }{\\mathcal{G}_0}\\right) \\right]\\;.\n\\end{equation} \nUsing (\\ref{incompradexp}), the adiabatic sound velocity\n$v^{(\\varsigma)}$ (cf. \n\\cite{forster,brenig}) \ncan be expressed as\n\\bel{speedad} \nv^{(\\varsigma)} =\n    \\sqrt{\\frac{K^{\\varsigma} }{9 m}} = v_{{}_{\\! {\\rm F}}} s^{\\varsigma},\n\\end{equation} \nwhere \n\\bel{velocad} \ns^{\\varsigma} = \\sqrt{{\\mathcal{G}_0 \n\\mathcal{G}_1 \\over 3}\n    \\left[1 + \\frac{\\pi^2 \\tbar^2 }{12} \\left(1 + \n\\frac{4 }{\\mathcal{G}_0}\\right)\\right]}.\n\\end{equation} \nThe ratio of the specific heats (\\ref{cvcpkSkT}) may be\ncalculated using either the expansions of the in-compressibilities\n(\\ref{incompradexp}) and (\\ref{isotherkexp}) or\n(\\ref{cpcvkakt}) together with (\\ref{capnexp}) and\n(\\ref{specifheatv}). Finally, one gets \n\\bel{cpcvexp} \n\\left(\\frac{{\\tt C}_{\\cal P}\n}{{\\tt C}_{\\mathcal{V}}}\\right) =\n    1 + \\frac{\\pi^2 \\tbar^2 }{3 \\mathcal{G}_0}.\n\\end{equation} \nThus, from (\\ref{specifheatv}) and (\\ref{cpcvexp}),\n\\bel{specifheatp} C_{\\cal P} =\n   \\frac{\\pi^2 {\\bar T}}{2}\n    \\left[1-\\frac{3 \\pi^2 \\tbar^2 }{10}\n    \\left(1-\\frac{10 }{9 \\mathcal{G}_0} \\right)\\right]\n\\end{equation} \nis the specific heat at the fixed pressure.\n\n\n\\subsection{Thermodynamic relations for a finite Fermi-liquid drop}\n\\label{thermgibbs}\n\nIn this subsection, we apply the formulas derived above to extend\nthe derivations of the boundary conditions in\n\\cite{strutmagbr,magstrut,magboundcond} to the case of equilibrium\nat  a finite $T$. Like in these papers, the finite Fermi-liquid drop\nis treated in the effective sharp surface approximation, see\nsubsection \\ref{boundary} and Appendix D.\nApplying to the standard thermodynamic relations $~{\\rm d}E=T{\\rm d}S\n-{\\cal P} {\\rm d}\\mathcal{V}-{\\cal P}_Q{\\rm d}Q~$ and $~{\\rm d}G=-S{\\rm\nd}T + \\mathcal{V} {\\rm d}{\\cal P}-{\\cal P}_Q{\\rm d}Q~$, we include the\nchange of the collective variable $Q$\n(see, e.g., \\cite{hofmann,hofbook}). $G$ is the Gibbs free\nenergy $G=F+{\\cal P} \\mathcal{V}=E+TS+{\\cal P} \\mathcal{V}$, defined\nsimilarly to the free energy $F$ with only the volume \n$\\mathcal{V}$\nreplaced by the pressure ${\\cal P}$.\nFor the FLDM \nit is more convenient to use $G$\nrather than $F$, simply because in general volume may not be\nconserved but the pressure has to be fixed by the boundary\ncondition (\\ref{bound2}). The Gibbs free energy is used for\nderiving these boundary conditions as well as for the\ncalculations of the coupling constants and susceptibilities\nassociated to the operator ${\\hat F}({\\bf r})$ (\\ref{foperl}).\n\nFor the following derivations, we need the relations for the\nthermodynamical potentials per particle. The Gibbs free energy per\nparticle $G\/N$ which is identical to the chemical potential\n$\\mu$  \nis related to the corresponding\nfree energy $F\/N$ by the relation $G\/N \\equiv\n\\mu=F\/N+{\\cal P}\/\\rho$~. For a finite Fermi-liquid drop where the\nparticle density $\\rho$ is function of the coordinates (smooth\ninside and sharp decreasing in the surface region) they are\nwritten as in \\cite{strutmagbr,magstrut,magboundcond} through the\nvariational derivatives $\\delta g\/\\delta \\rho$ and $\\delta\n\\phi\/\\delta \\rho$ with the thermodynamical potential densities\n$g$ and $\\phi$ per unit of volume, respectively, and this\nrelation reads now \n\\bel{gibbspart} \n\\frac{\\delta g }{\\delta \\rho}\n\\equiv \\mu= \\frac{\\delta \\phi }{\\delta \\rho} +\n\\frac{{\\cal P} }{\\rho}.\n\\end{equation}\n These densities depend on the coordinates through \n$\\rho$ and its gradients. Their calculation is carried out\nfrom the variations of the corresponding total integral\nquantities $G$ and $F$ with the following integration by parts,\nsee \\cite{strutmagbr,magstrut,magboundcond} for details. Taking\ninto account also that the particles in the Fermi-liquid drop\nmove in a mean field $V$ with the coordinate dependence similar to\nthe density $\\rho$, one gets from (\\ref{gibbspart})\n\\begin{eqnarray}\\label{thermrelmu} \n{\\rm d} \\frac{\\delta g }{\\delta \\rho} &\\equiv&\n{\\rm d} \\mu= -\\varsigma {\\rm d} T + \\frac{1}{\\rho} {\\rm d}{\\cal P}\n+{\\rm d} V\\quad{\\rm with}\\nonumber\\\\ {\\rm d}\nV&=&-\\left({\\cal P}_Q\/N\\right) {\\rm d} Q. \n\\end{eqnarray}\nFrom (\\ref{thermrelmu}) one has \n\\bel{gradrelmu} \n{\\bf \\nabla} \\mu\n=-\\varsigma {\\bf \\nabla} T + \\frac{1}{\\rho} {\\bf \\nabla}{\\cal P}\n+{\\bf \\nabla} V. \n\\end{equation}\n\nFor the derivation of the boundary condition (\\ref{bound2}), we\nused (\\ref{gradrelmu}) for the transformations of\n(\\ref{momenteq}) instead of (17) of \\cite{magstrut}. The\none-to-one correspondence of this derivation with that explained in\n\\cite{strutmagbr,magstrut,magboundcond} becomes obvious if we note\nthat equation (17) \nwas found from \n\\bel{gradrelener} \n{\\bf \\nabla}\n\\varepsilon =T{\\bf \\nabla} \\varsigma + \\frac{1}{\\rho} {\\bf\n\\nabla}{\\cal P} +{\\bf \\nabla} V, \n\\end{equation}\n for the adiabatic condition of\na constant entropy per particle ($\\varepsilon$ here is the same as\n$\\delta \\varepsilon \/ {\\delta \\rho}$ in the notation of\n\\cite{magstrut}). The variational derivative $\\delta g \/ {\\delta\n\\rho}$ (\\ref{gibbspart}) (or the chemical potential $\\mu$)\nappears now in the following key equation for the derivation of\nthe surface condition (\\ref{bound2}): \n\\bel{keybound}\n\\rho_\\infty\\left(\\frac{\\delta g }{\\delta\n\\rho}\\right)_{S}^{\\rm vol}= -b_{\\mathcal{V}}\\rho_\\infty +2 \\alpha \\mathcal{H},\n\\end{equation} \nwhere $b_{\\mathcal{V}}$ is the nucleon binding energy in the infinite\nnuclear matter, $\\mathcal{H}$ the mean curvature of the nuclear\nsurface, $\\mathcal{H}=1\/R_0$ for the spherical shape at equilibrium.\nIndex \"vol\" means that the Gibbs free energy per particle is\nconsidered as that found in the nuclear \n interior. Hence, is a\nsmooth quantity taken at the nuclear surface as the quantities\nin the l.h.s. of the boundary conditions (\\ref{bound1}) and\n(\\ref{bound2}) within the precision of the ESA. \n\n\nThe temperature $T$ and chemical potential $\\mu$ in\n(\\ref{thermrelmu}) and (\\ref{keybound}) are constants as\nfunction of the coordinates ${\\bf r}$ within our Fermi-liquid-drop\ninterior at equilibrium. \nWith these properties, one gets \n\\bel{derVderP}\n{\\bf \\nabla} V =-\\frac{1 }{\\rho} {\\bf \\nabla}{\\cal P} =\n\\frac{K }{9\n\\rho_{{}_{\\! 0}}} {\\bf \\nabla} \\rho.  \n\\end{equation} \nIn the second equation, we applied\n(\\ref{dpress}) which shows that the expression in the middle\nof (\\ref{derVderP}) is proportional to the gradient of the\nparticle density with some smooth coefficient related to the\nin-compressibility  $K$. The relation (\\ref{derVderP}) will be\nused in the Appendix C  \nfor the calculation of several coupling\nconstants and susceptibilities for the constant temperature and\nentropy, as well as for the static limit $\\omega  \\to 0$, with the\ncorresponding in-compressibility modulus and particle density in\nthe last equation (\\ref{derVderP}).\n\nFor the derivations of the susceptibilities in Appendix C \nand ratio of the surface energy constants (\\ref{bstbs}), we need  \nhere also the following thermodynamic relation:\n\\begin{eqnarray}\\label{ctcaddif}\n\\left(\\frac{\\partial^2 G}{\\partial Q^2}\\right)_T &-& \n\\left(\\frac{\\partial^2 E}{\\partial Q^2}\\right)_S=\n\\left[\\left(\\left(\\frac{\\partial^2 G}{\\partial T^2}\\right)_Q\\right)^{-1}\n\\!\\!\\left(\\frac{\\partial^2 G}{\\partial T\\partial Q}\\right)^2\\right]_{Q=0}\n\\nonumber\\\\ \n&=&-\\left[\\left(\\left(\\frac{\\partial S}{\\partial\nT}\\right)_Q\\right)^{-1} \\left(\\frac{\\partial S}{\\partial\nQ}\\right)^2\\right]_{Q=0}. \n\\end{eqnarray}\n\nWe obtained this relation as explained in Appendix A1 in\n\\cite{hofmann} with the only one change of the free energy $F$ to\nthe Gibbs free energy $G$.  \nThe derivatives in\nthese equations should be considered for the constant pressure\ninstead of the volume of the Fermi-liquid drop.\n\n\n\n\\setcounter{equation}{0}\n\\section{Stress tensor and heat current}\n\\label{app2}\nWe shall derive\nthe specific expressions for the shear modulus and viscosity\nin the stress tensor\n$\\sigma_{\\alpha\\beta}$ (\\ref{presstens}) representing it in the form\n(\\ref{prestensone}) with (\\ref{presslamb}) and (\\ref{pressnu})\nin subsection B.2. \nWe are going also to obtain the expression for\nthe thermal conductivity in the heat current in subsection B.1. \nThe next subsection B.3  \nis devoted to the long wave approximation\nfor the above mentioned coefficients.  \nIn the latter subsection,\nwe derive some basic formulas for this approximation which are\nused for the response function in whole section \\ref{longwavlim}, \nbeside\nthe above mentioned coefficients, in particular equations\nfor poles of the response function.\n\n\\subsection{Stress tensor, shear modulus and viscosity}\n\\label{app2stress}\n\nFor the calculation of the stress tensor $\\sigma_{\\alpha\\beta}$\n(\\ref{presstens}), we shall show first that it really has\nthe form given in (\\ref{prestensone}), (\\ref{presslamb}), (\\ref{pressnu})\nwith some coefficients $\\lambda$ and $\\nu$ \nin Appendix B.1a, \nand then, find their specific expressions in B.1b. \n\n\\subsubsection{STRESS TENSOR FOR  FERMI LIQUIDS}\n\\label{stresscontfliq}\n\nFirst, after a short calculation of the r.h.s. of (\\ref{presslamb})\nand (\\ref{pressnu}), one simply gets \n\\bel{prestensone1}\n{\\tilde \\sigma}_{\\alpha\\beta}= -\\left(\\frac{\\lambda \n}{\\omega } -i \\nu\\right)~ \\left(q_\\beta {\\tilde u}_\\alpha + q_\\alpha\n{\\tilde u}_\\beta -\\frac{2}{3}{\\bf q}{\\tilde {\\bf u}}\n\\delta_{\\alpha\\beta}\\right) .\n\\end{equation}\nTo simplify more these expressions we\nnote now, that the Fourier components\n${\\tilde \\sigma}_{\\alpha\\beta}$ (\\ref{prestensone1}) of the stress tensor\n$\\sigma_{\\alpha\\beta}$ (\\ref{presstens}) is a symmetric\ntensor with the two independent components ${\\tilde\n\\sigma}_{zz}$ and ${\\tilde \\sigma}_{xz}$ in the Cartesian\ncoordinate system $(x,y,z)$ with the axis $z$ directed to the wave\nvector ${\\bf q}$ because of axial symmetry. The tensor (\\ref{prestensone1})\nhas also zero trace. Hence, from the set of\nequations (\\ref{prestensone1}) only two independent ones \nsurvive, namely, \n\\bel{prestensone2}  \n{\\tilde \\sigma}_{zz}=\n-\\frac{4}{3} \\left(\\frac{\\lambda }{\\omega } - i \\nu\\right)~ q\n{\\tilde u}_z,\\quad {\\tilde \\sigma}_{xz}= \n-\\left(\\frac{\\lambda }{\\omega } - i \\nu\\right)~q{\\tilde u}_x, \n\\end{equation}\n with\n\\bel{prestensone2b} \n{\\tilde \\sigma}_{xx}= {\\tilde\n\\sigma}_{yy}= -\\frac{1}{2} {\\tilde \\sigma}_{zz}, \\quad\n {\\tilde \\sigma}_{yz}= {\\tilde \\sigma}_{xz}\n \\quad {\\rm and} \\quad  {\\tilde \\sigma}_{xy}=0.\n\\end{equation}\n\n\nOn the other hand, the stress tensor $\\sigma_{\\alpha \\beta}$ (\\ref{presstens})\nin the l.h.s. of (\\ref{prestensone2})\nis determined by the distribution function \n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ in the\nplane-wave representation, see\n(3.10) from \\cite{heipethrev},\n\\begin{eqnarray}\\label{basiceq}\n&&\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )= \n\\left(\\frac{\\partial f_{{\\bf p}} \n}{\\partial \\varepsilon_{{\\bf p}}}\\right)_{\\rm g.e.} \n\\left\\{\\frac{\\omega  }{\\mathcal{D}_{{\\bf p}}}\n\\left[\\delta {\\tilde \\mu} +\\frac{m }{m^*} {\\bf p}{\\tilde\n{\\bf u}}\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left. \\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu }{T}\\right)_{\\rm g.e.} \n\\delta {\\tilde T}\n- \\frac{\\mathcal{F}_0 }{\\mathcal{N}(T)} \\delta {\\tilde\n\\rho}\\right]\\right.\\nonumber\\\\ \n&-&\\left.\n\\frac{{\\bf q} {\\bf v}_{{\\bf p}} }{\\mathcal{D}_{{\\bf p}}} \n\\left[\\delta {\\tilde \\mu}+{\\bf p}{\\tilde {\\bf u}}+\n\\left(\\frac{\\varepsilon_{{\\bf p}}-\\mu }{T}\\right)_{\\rm g.e.} \\delta {\\tilde\nT}\\right]\\right\\}.\n\\end{eqnarray}\nThis expression can be easy derived from (\\ref{landvlas})\nafter not too lengthy and simple transformations \n\\cite{heipethrev}, besides of the adaptation to our notations.\nWe substitute then the distribution function\n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ given by (\\ref{basiceq})\nto the l.h.s. of (\\ref{prestensone2}) through\n(\\ref{presstens}) in the considered representation.\nIn this way, we easy realize that the stress tensor (\\ref{presstens})\nhas the above mentioned symmetry properties, and\nits components ${\\tilde \\sigma}_{zz}$, and $\n{\\tilde \\sigma}_{xz}$ of l.h.s. of (\\ref{prestensone2})\nwith some shear modulus $\\lambda$ and viscosity $\\nu$\nare indeed proportional to $q {\\tilde\nu}_z$ and $q {\\tilde u}_x$, respectively.\nAs result, these stress tensor components can be represented for convenience\nin terms of the\ntwo dimensionless quantity $\\chi _{zz}$ and $\\chi _{xz}$ independent of\nthe mean velocity ${\\bf u}$, \n\\bel{dpresszz}\n {\\tilde \\sigma}_{zz}=\n    -\\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}} \\chi _{zz} {\\tilde u}_z , \\qquad\n     {\\tilde \\sigma}_{xz}=\n    -\\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}} \\chi _{xz} {\\tilde u}_x,\n\\end{equation} \nwhere \n\\bel{chizzfun} \n\\chi _{zz} = \\mathcal{J}_1 \\frac{\\rho_{{}_{\\! 0}} }{\\mu}\n    \\frac{\\delta {\\tilde T} }{\\delta {\\tilde \\rho}} + \\mathcal{J}_2,\n\\end{equation}\n \\bel{J1def} \n\\mathcal{J}_1=\\frac{2i \\mathcal{N}(T) }{s \\tau \\varepsilon_{{}_{\\! {\\rm F}}}\n\\mathcal{N}(0)} \\left\\langle P_2\\left({\\hat p}_z\\right)~\n\\frac{\\varepsilon_{\\bf p} \n}{\\mathcal{D}_{\\bf p}} \\left(\\frac{\\varepsilon_{\\bf p}-\\mu }{T}\n-\\frac{\\mathcal{M}(T)}{\\mathcal{N}(T)}\\right)\\right\\rangle_{\\rm g.e.}, \n\\end{equation}\n\\begin{eqnarray}\\label{J2def}\n\\mathcal{J}_2&=&-\\frac{4 \\mathcal{N}^2(T) }{3 s \n\\varepsilon_{{}_{\\! {\\rm F}}} \\mathcal{N}^2(0)} \n\\left\\langle \\varepsilon_{\\bf p} P_2\\left({\\hat p}_z\\right) \n\\left[\\frac{\\omega  }{\\mathcal{D}_{\\bf p}}~\\left(\n\\frac{\\mathcal{N}(0) }{\\mathcal{N}(T)} + \n\\frac{3 p \\omega  }{2 \\mathcal{G}_1 q \\varepsilon_{{}_{\\! {\\rm F}}}}~ {\\hat p}_z \\right) \n\\right.\\right. \\nonumber\\\\\n&-&\\left.\\left. \\frac{{\\bf q} {\\bf v}_{\\bf p} }{\\mathcal{D}_{\\bf p}}~ \n\\left(\\frac{\\mathcal{G}_0 \n\\mathcal{N}(0) }{\\mathcal{N}(T)} + \\frac{3 p\\omega  \n}{2q\\varepsilon_{{}_{\\! {\\rm F}}}}~{\\hat p}_z\\right)\\right] \\right\\rangle_{\\rm g.e.},\n\\end{eqnarray}\n$P_2(x)$ is the Legendre polynomial,  ${\\hat p}_\\alpha$ is the\n$\\alpha$ component of the unit vector ${\\hat p}$ defined in\n(\\ref{intampfpp}). Other quantities were defined in Secs.\\\n\\ref{eqmotion}, \\ref{respfunsec} and Appendix D,  \nsee also (\\ref{dfgeqrpt}),\n(\\ref{enerdensnt}), (\\ref{mcapt}), (\\ref{nzero}),\n(\\ref{domindp}) and (\\ref{som}). $\\chi _{xz}$ in\n(\\ref{dpresszz}) is given by \n\\begin{eqnarray}\\label{chixzdef}\n&&\\chi _{xz}=-\\frac{3 \\omega  \\mathcal{N}(T) }{s p_{\\rm F}^3 \\mathcal{N}(0)} \n\\left\\langle p^3\n\\frac{{\\hat p}_x^2{\\hat p}_z }{\\mathcal{D}_{\\bf p}} \n\\left(\\frac{s }{\\mathcal{G}_1 } -\n\\frac{p }{ p_{{}_{\\! {\\rm F}}}} {\\hat p}_z\\right)\\right\\rangle\\quad\n\\nonumber\\\\\n&=&-\\frac{3 \\omega  \\mathcal{N}(T) }{2 s p_{\\rm F}^3 \\mathcal{N}(0)} \n\\left\\langle p^3\n\\frac{(1-{\\hat p}_z^2){\\hat p}_z }{\\mathcal{D}_{\\bf p}} \\left(\n\\frac{s }{\\mathcal{G}_1 }-\\frac{p}{p_{{}_{\\! {\\rm F}}}} {\\hat p}_z\\right)\n\\right\\rangle. \\quad\n\\end{eqnarray} \nIn the second equation, we used the invariance of the average\nin the first equation with respect to the replace ${\\hat p}_x\n\\rightarrow {\\hat p}_y$, due to the axial symmetry and the equation\n$\\sum_\\alpha {\\hat p}_\\alpha^2=1$ for the unit vector ${\\hat p}$. We\napplied also the thermodynamic relation (\\ref{dmu}) for \n$\\delta {\\tilde \\mu}$ in the distribution function (\\ref{basiceq})\nin these derivations.\n\n\\subsubsection{THE SHEAR MODULUS AND VISCOSITY}\n\\label{calclambnu}\n\nThe shear modulus $\\lambda$ and viscosity $\\nu$ can be now found\nfrom the comparison of (\\ref{prestensone2}) for continuous matter\nand explicit expressions\n(\\ref{dpresszz}) obtained above from the Fermi-liquid distribution function\n$\\delta {\\tilde f}_{\\rm l.e.}({\\bf q},{\\bf p},\\omega )$ (\\ref{basiceq}) for the same\nstress tensor components ${\\tilde \\sigma}_{zz}$ and ${\\tilde \\sigma}_{xz}$.\nIndeed, substituting (\\ref{dpresszz}) to the l.h.s. of\n(\\ref{prestensone2}) and canceling the velocity field\ncomponents from their both sides,  one finds \n\\bel{lamviseq} \n\\mathcal{J}_1~\\frac{\\rho_{{}_{\\! 0}}}{\\mu_{\\rm g.e.}}~ \n\\frac{\\delta {\\tilde T} }{\\delta\n{\\tilde \\rho}} +\\mathcal{J}_2\n    =\\frac{4}{3} \\chi _{xz},\\quad\n\\lambda-iv_{{}_{\\! {\\rm F}}} qs\\nu=\\rho_{{}_{\\! 0}}\\varepsilon_{{}_{\\! {\\rm F}}} s~\\chi _{xz}\\;. \n\\end{equation}\nFrom the first equation one has the ratio \n\\bel{dtdrho}\n\\frac{\\delta {\\tilde T} }{\\delta {\\tilde \\rho}} = \\frac{\\mu_{\\rm g.e.}\n}{\\rho_{{}_{\\! 0}} \\mathcal{J}_1}~ \\left(\\frac{4}{3} \\chi _{xz}-\n\\mathcal{J}_2\\right). \n\\end{equation}\nSeparating real and imaginary parts in the second\nequation, one obtains the shear modulus $\\lambda$ and viscosity\n$\\nu$: \n\\bel{shearmod} \n\\lambda=\n      \\frac{\\mid s \\mid ^2 \\chi _{xz} ^\\prime }{s ^\\prime}\\rho_{{}_{\\! 0}}\n\\varepsilon_{{}_{\\! {\\rm F}}}q\n\\end{equation}\n and \n\\bel{viscos} \\nu =\n    -\\frac{s ^{\\prime\\prime} \\chi _{xz} ^\\prime +\n    s ^\\prime \\chi _{xz} ^{\\prime\\prime} }{s^\\prime }~\n    \\frac{\\rho_{{}_{\\! 0}} \\varepsilon_{{}_{\\! {\\rm F}}} }{v_{{}_{\\! {\\rm F}}}}.\n\\end{equation} \nWith these constants $\\lambda$ and $\\nu$,  the equations\n(\\ref{prestensone}), (\\ref{presslamb}), (\\ref{pressnu}) and  \n(\\ref{presstens}) are identities.\n\nThe aim of the following derivations of the shear modulus and the viscosity\nis to simplify $\\mathcal{J}_1$\n(\\ref{J1def}), $\\mathcal{J}_2$ (\\ref{J2def}) and $\\chi _{xz}$\n(\\ref{chixzdef}). For this aim, we make use of\ntransformations of the averages like $\\langle p^k{\\hat p}_z^l\n\\varepsilon_{\\bf p}^m ({\\bf q}{\\bf v}_{\\bf p})^n\/\\mathcal{D}_{\\bf p} \\rangle_{\\rm g.e.}$ with\nsome integer numbers $0 \\leq k \\leq 4$, $0 \\leq l \\leq 4$, $m=0,1$\nand $n=0,1$ in terms of more simpler functions $\\chi _n$ \n($n=0,1,2$) introduced\nin \\cite{heipethrev} for the response functions, see\n(\\ref{chinfun}). For these functions, one has simple temperature and\nhydrodynamic expansions presented below at the end subsection\nof this Appendix B. Using\nsuch enough lengthy and simple algebraic derivations, one\nfinally gets\n\\begin{eqnarray}\\label{J1}\n\\mathcal{J}_1&=&\n    \\frac{1}{s (1-i s \\tau_q)}\n    \\left[\\left(1+ {\\bar T} \\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)}\n    +\\frac{3}{\\tau_q ^2}\n    (1-i s \\tau_q)^2\\right)\\frac{\n\\chi _1 }{\\mathcal{N}(0)}\\right.\\nonumber\\\\\n    &-&\\left.{\\bar T}\\left(\\frac{\\pi^2 }{3} \n{\\bar C}_V-\\frac{\\chi _2 }{\\mathcal{N}(0)}\n    \\right)\\right],\n\\end{eqnarray}\n\\begin{eqnarray}\\label{J2}\n   && \\mathcal{J}_2=\n    \\frac{2i \\mathcal{N}^2(T) }{3s \\tau_q (1-is \\tau_q) \\mathcal{N}^2(0)}\n    \\left[3s\\left(1-i s \\tau_q\\right)^2\\right.\\nonumber\\\\\n&+& \\left.{3i \\tau_q \\over {\\mathcal{G}_1}}\n    \\left(1-i s \\tau_q\\right)\n\\left(s^2-\\frac{\\mathcal{G}_0\n    \\mathcal{G}_1 \\mathcal{N}(0) }{3 \\mathcal{N}(T) }\\right)\n    +\\frac{s \\mathcal{N}(0)\\tau_q ^2 }{\\mathcal{N}(T)}\\right]\\nonumber\\\\\n    &\\times& \\left[ \\left(\\frac{3 \\mathcal{N}(0)\n    }{ \\mathcal{N}(T)\\tau_q ^2}(1-i s \\tau_q)^2 \n    + 1 + {\\bar T}\\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)} \\right)\n    \\frac{\\chi _0 }{\\mathcal{N}(T)} \\right.\\nonumber\\\\\n    &-&\\left. 1\n    -{\\bar T}\\frac{\\mathcal{M}(T) }{ \\mathcal{N}(T)}\n    +{\\bar T}\\frac{\\chi _1 }{\\mathcal{N}(T)} \\right],\n\\end{eqnarray}\n\\begin{eqnarray}\\label{chixz}\n&&\\chi _{xz} = -\\frac{3i}{\\tau_q}\n    \\left(1-\\frac{i \\mathcal{F}_1 s \\tau_q }{3 \\mathcal{G}_1}\\right)\n    \\left[\\left(\\frac{(1-i s \\tau_q)^2\\mathcal{N}(0)\n    }{ \\tau_q ^2 \\mathcal{N}(T) } +1\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.{\\bar T}\n    \\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)} \\right)\n    \\frac{\\chi _0 }{\\mathcal{N}(0)} \n    +  {\\bar T} \\frac{\\chi _1 }{\\mathcal{N}(0)}\n    -\\frac{1}{3}\\left(1+{\\bar T}\n\\frac{\\mathcal{M}(T) }{\\mathcal{N}(T)}\\right)\n    \\frac{\\mathcal{N}(T) }{\\mathcal{N}(0)} \\right]\\nonumber\\\\\n    &=& \\chi _{xz} ^\\prime + i\\chi _{xz} ^{\\prime\\prime}.\n\\end{eqnarray} \nNote that the shear\nmodulus $\\lambda$ (\\ref{shearmod}) and viscosity\n$\\nu$ (\\ref{viscos})\ndepend on the sound velocity $s$, and hence,\non the solution of the Landau-Vlasov\nequation (\\ref{landvlas}) for $s$ [(\\ref{J1}),(\\ref{J2})\nand (\\ref{chixz})].\n\n\\subsection{Heat current}\n\\label{app2heat}\n\nFor the following derivations of\nthe thermal conductivity $\\kappa$ in Fermi liquids, we need to\nderive the equation for the temperature $T$ from the general\ntransport equation (\\ref{entropyeq}). The latter equation\n(\\ref{entropyeq}) in the linear approximation with respect to the\ndynamical variations $\\delta f$ in terms of the moments, such as the\nvelocity field ${\\bf u}$ (\\ref{veloc}), particle density $\\delta \\rho$,\nentropy density per particle $\\delta \\varsigma$ and so on, writes\n\\bel{entropeqlin} \n\\rho T {\\partial \\varsigma \\over\n{\\partial t}}\n    +{\\bf \\nabla} \\cdot {\\bf j}_T = 0,\n\\end{equation} \nwhere ${\\bf j}_T$ is the heat current given in terms of the\nthermal conductivity $\\kappa$ and temperature gradient by\n(\\ref{currheat}). By making use of the thermodynamic relation\nfor the entropy $\\varsigma$ per particle, \n\\bel{thermeqspt} \n\\hbox{d} \\varsigma=\n    \\left(\\frac{\\partial \\varsigma }{\\partial {\\cal P}}\\right)_T\n    \\hbox{d} {\\cal P}\n    +\\left(\\frac{\\partial \\varsigma }{\\partial T}\\right)_{\\cal P}\n    \\hbox{d} T,\n\\end{equation}  \nand  the well known arguments\nto get the\nthermal conductivity equation, we consider the process with the\n{\\it constant pressure} rather than the constant of particle density.\n(We again omitted the symbol variation $\\delta$ as in Sec.\\ \\ref{conserveqs}).\nWith the help of (\\ref{thermeqspt}), one then results in the\nFourier thermal conductivity equation \n\\bel{fouriereq}\n   \\rho {\\tt C}_{\\cal P} \\frac{\\partial T }{\\partial t}-\n   \\kappa \\triangle T = 0,\n\\end{equation} \nwhere ${\\tt C}_{\\cal P}$ is the specific heat for the\nconstant pressure per particle, see (\\ref{specifheatp}).\n(Equation (\\ref{currheat}) was also used in (\\ref{fouriereq}) for the\nheat current ${\\bf j}_T$). Solving equation (\\ref{fouriereq}) for the\ntemperature $T({\\bf r},t)=T_{\\rm g.e.}+\\delta T$ in terms of the plane\nwaves for the dynamical part of the temperature $\\delta T({\\bf r},t)$\nas in (\\ref{planewave}) and using the relations\n(\\ref{som}), one gets \n\\bel{kappadef} \n\\kappa=i \\rho {\\tt C}_{\\mathcal{P}} v_{{}_{\\! {\\rm F}}}s\/q. \n\\end{equation} \nNotice, the thermal conductivity $\\kappa$\n(\\ref{kappadef}) depends on the sound velocity $s$ as the shear\nmodulus $\\lambda$ (\\ref{shearmod}) and viscosity\n$\\nu$ (\\ref{viscos}), and therefore, on the solution of the Landau-Vlasov\nequation (\\ref{landvlas}) for $s$.\n\n\n\\subsection{Long wave-length limit}\n\\label{app2exp}\n\nAs shown in section \\ref{respfunsec} and subsections \nB.1a and B.1b, \nmany physical quantities, such as the response\nfunctions, see (\\ref{ddresp}), the shear modulus (\\ref{shearmod})\nand viscosity (\\ref{viscos}) can be expressed in terms of the\nsame helpful functions $\\chi _n$ (\\ref{chinfun}). By this reason,\nit is easy to get their \nLWL limit \nby expanding the only $\\chi _n$ in small parameter $\\tau_q$.\n\nFor small $\\tau_q$, one can use asymptotic expansions \nfor the Legendre function of second kind\n$Q_1(\\zeta)$ \nand its derivatives\nwhich\nenter $\\chi _n$ with its derivatives, according to (\\ref{chitemzero}),\n(\\ref{chitemone}) and (\\ref{chitemtwo}). This approximation is\nvalid for large arguments $\\zeta$. Substituting these expansions\ninto the functions $\\chi _n$ (\\ref{chinfun}) and $\\wp$\n(\\ref{alphas}), one gets to fourth order in $\\tau_q$:\n\\begin{eqnarray}\\label{chiexpzero} \n\\chi _0 &=&\n    \\frac{\\tau_q^2 }{3} \\left[1 + 2 i s_0 \\tau_q\n    -\\left(2 s_1 + 3 s_0^2 +\\frac{3}{5}\\right) \\tau_q^2\\right.\n\\nonumber\\\\\n    &-& \\left. \\frac{\\pi^2 \\tbar^2 }{4} \\tau_q^2 \\right] \\mathcal{N}(0),\n\\end{eqnarray} \n\\bel{chiexpone} \n\\chi _1 =\n    \\frac{\\pi^2 {\\bar T} \\tau_q^2 }{9}\n    \\left[1+2 i s_0 \\tau_q\n    -\\left(2 s_1 + 3 s_0^2 +\\frac{6}{5}\\right) \\tau_q^2\\right] \n\\mathcal{N}(0),\n\\end{equation} \n\\begin{eqnarray}\\label{chiexptwo}\n&&\\chi _2 =\n    \\frac{\\pi^2 \\tau_q^2 }{9}\n    \\left[1+2 i s_0 \\tau_q\n    -\\left(2 s_1 + 3 s_0^2 +\\frac{3}{5}\\right) \\tau_q^2 \n    -\\frac{\\pi^2 \\tbar^2 }{60}\\right.\\nonumber\\\\\n    &\\times&\\left.\\left(1+2 i s_0 \\tau_q\n    -\\left(2 s_1 + 3 s_0^2 - 60 \\right) \\tau_q^2 \\right) \\right] \n\\mathcal{N}(0).\n\\end{eqnarray}\nWith these expressions, \none obtains the collective response\nfunction $\\chi _{DD}^{\\rm coll}$ of (\\ref{ddresp}), (\\ref{despfunc}) \nthrough\n\\begin{eqnarray}\\label{ampsexp}\n&&\\aleph(s) \\equiv \\aleph(\\tau_q,s_0,s_1) = \\frac{\\pi^2 \\taubar^3 }{27}\n   \\left\\{-3 i s_0 + \\left(1 + 6 s_0^2 +3 s_1\\right) \\tau_q\n\\right.\\nonumber\\\\\n   &+& \\left.\\frac{\\pi^2 \\tbar^2 }{120}\n   \\left[93 i -\\left(2+186 s_0^2 + 93 s_1\\right)\n   \\tau_q \\right]\\right\\} \\mathcal{N}^2(0),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{dszexpzero}\n&&D_0(s) \\equiv D_0(\\tau_q,s_0,s_1) =\n    \\frac{\\pi^2 \\taubar^3 }{9}\n    \\left\\{-i s_0 \\left(1 - 3 s_0^2\\right)\\right.\\nonumber\\\\\n   &+& \\left. \\frac{1}{15}\n    \\left[5 + 3 s_0^2 \\left(1-30 s_0^2\\right)+\n    15 s_1\\left(1 -9 s_0^2 \\right)\\right] \\tau_q ~~~\\right.\\nonumber\\\\\n    &+& \\left. \\frac{\\pi^2 \\tbar^2 }{120}\n    \\left[-i s_0 \\left(19+93 s_0^2\\right)\n+\\frac{1}{15}\n    \\left(-160 +18 s_0^2\\right.\\right.\\right.\\nonumber\\\\\n&\\times&\\left.\\left.\\left.\\left(54  +155 s_0^2\\right) \n+ \n    5 s_1 \\left(57 +837 s_0^2 \\right)\\right)\\tau_q\n    \\right]\\right\\} \\mathcal{N}(0)~~~\n\\end{eqnarray}\n[also for the temperature-density response function (\\ref{dtresp})].\nThese two quantities determine the expansion of the function $D(s)\n\\equiv D(\\tau_q,s_0,s_1)$ (\\ref{despfunc}) in powers of $\\tau_q$,\nand  then approximately, the excitation modes given by the\ndispersion relation (\\ref{despeq}). Indeed, equaling zero the\ncoefficients which appear in front of the each power of $\\tau_q $\nin this expansion of $D(\\tau_q,s_0,s_1)$, we get  \nequations for the unknown quantities $s_0$ and $s_1$ of\n(\\ref{somexp}), \n\\begin{eqnarray}\\label{eqzero} \n&&\\frac{i s_0 }{\\mathcal{G}_1}\n\\left[s_0^2 -\\frac{\\mathcal{G}_0 \\mathcal{G}_1 }{3}\n + \\frac{\\pi^2 \\tbar^2 }{120}\n \\left(-40 \\mathcal{G}_1 + 21 \\mathcal{G}_0 \\mathcal{G}_1\n\\right.\\right.\\nonumber\\\\ \n&-&\\left.\\left. 63 s_0^2\n - 30 \\mathcal{G}_1 s_0^2\\right)\\right] = 0,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eqone}\n&&\\frac{1}{45 \\mathcal{G}_1}\\left\\{5 \\mathcal{G}_0 \\mathcal{G}_1 - 3 s_0^2\n\\left[5 +\n        2 \\mathcal{G}_1 \\left(2 - 5 \\mathcal{G}_0 \\right) + 30 s_0^2 \\right] +\n        15 s_1 \\right.\\nonumber\\\\\n&\\times&\\left.\n\\left(\\mathcal{G}_0 \\mathcal{G}_1 -\n        9 s_0^2\\right) \n  +\\frac{\\pi^2 \\tbar^2 }{120}\n   \\left[40 \\mathcal{G}_1\\left( \\mathcal{G}_0 -5\\right)\n   - 6s_0^2 \\left(60 - 287 \\mathcal{G}_1 \\right.\\right.\\right.\\nonumber\\\\\n&+& \\left.\\left.\\left.\n   105 \\mathcal{G}_0 \\mathcal{G}_1 +\n           15 s_0^2 \\left(21 + 10 \\mathcal{G}_1 \\right)\\right)\n        + 15 s_1 \\right.\\right.\\nonumber\\\\\n&\\times&\\left.\\left.\n\\left(40 \\mathcal{G}_1 - \n21 \\mathcal{G}_0 \\mathcal{G}_1 +\n           9 s_0^2\\left(21 + 10 \\mathcal{G}_1 \\right) \\right)\\right]\\right\\} \n    = 0.\n\\end{eqnarray}\nSolving these equations, one obtains the position of the poles as\ngiven in (\\ref{shp}) and (\\ref{sfirst}).\n\nThe shear modulus ($\\lambda$) and viscosity ($\\nu$) coefficients\nenter the response function $\\chi_{FF}^{\\rm coll}$ (\\ref{chicollfldm})\nand (\\ref{glx}) in terms of the sum $(\\lambda - i \\nu \\omega )\/\\rho_{{}_{\\! 0}}\n\\varepsilon_{{}_{\\! {\\rm F}}}$. The  \nLWL expansion of this sum can be\nobtained with help of (\\ref{chiexpzero}), (\\ref{chiexpone})\nand (\\ref{somexp}) and expansions of all static quantities in\n${\\bar T}$ there, see Appendix A, but with taking into account fourth\norder terms,\n\\begin{eqnarray}\\label{chixzexp}\n&&(\\lambda - i \\nu \\omega )\/\\rho_{{}_{\\! 0}} \\varepsilon_{\\rm F}= s \\chi_{xz}({\\bar T},\\omega \\tau)=\n-i \\frac{2}{5}\\left(1 + \\frac{5 \\pi^2 \\tbar^2 }{12} \\right) \\omega \\tau +\n\\nonumber\\\\\n&&  \\frac{\\pi^4 {\\bar T}^4 }{2160}\n  \\left[13 \\left(\\mathcal{G}_0 - \\frac{4}{5} \\mathcal{G}_1\n  - \\mathcal{G}_0 \\mathcal{G}_1\\right) -\n     \\frac{13 i  \\mathcal{G}_0 \\mathcal{G}_1 }{\\omega \\tau} + \n\\right.\\nonumber\\\\\n   &&  \\left. \\frac{i \\omega \\tau }{25 \\mathcal{G}_0}\n     \\left(780 + 2815 \\mathcal{G}_0 + 52 \\mathcal{G}_1\n     + 260 \\mathcal{G}_0 \\mathcal{G}_1\\right)\\right].\n\\end{eqnarray}\nSeparating the real and imaginary parts in these equations, one\ngets the LWL approximation of the both {\\it real}\ncoefficients $\\lambda$ and $\\nu$.  \nThe terms linear in\n$\\omega  \\tau$ determine the hydrodynamic viscosity $\\nu^{(1)}$\n(\\ref{shearvisone}), and the terms proportional to $1\/\\omega \\tau$ are\nrelated to $\\nu^{(2)}$ (\\ref{shearvishp}),  see \ndiscussions of the \"heat pole\" for the FLDM transport coefficients \nin Sec.\\ \\ref{transprop}.\n\nThe  LWL approximation for\nthe thermal conductivity $\\kappa$ is determined by \nequation (\\ref{kappadef})\nand solutions (\\ref{shp}) for the heat pole \nand (\\ref{sfirst0}) for the sound velocity\n$s$ of the\ndispersion equations (\\ref{eqzero}) and\n(\\ref{eqone}).\n\n\nThe explicit final expressions for the viscosity $\\nu$ and the\nthermal conductivity $\\kappa$ are presented and discussed in\nsubsection \nII D in the LWL limit in\nconnection with the first sound and overdamped (heat pole) modes,\nsee (\\ref{shearvisonehp}) and (\\ref{kappaexp}). As seen\nimmediately from (\\ref{chixzexp}), the linear terms in\n$\\tau_q$ for the shear modulus $\\lambda$ appear as high\ntemperature corrections proportional to ${\\bar T}^4$. They are\nregular in $\\omega \\tau$, and therefore, are totally immaterial, see\nmore discussions in the subsection mentioned above.\nIn the\nlinear approximation in $\\tau_q$ of the  \nLWL limit,\nit is easy to check that the\nderivative $\\delta {\\tilde T} \/ \\delta {\\tilde \\rho}$ \n(\\ref{dtdrho}) is the same as obtained in terms of the response\nfunctions in (\\ref{dtdrhoexp}), and therefore, the in-compressibility\n$K_{\\rm tot}$ (\\ref{incomprtot}) turns into the adiabatic one.\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Coupling constants and\nsusceptibilities} \\label{app4}\n\nLet us consider the change of average $\\langle {\\hat F}({\\bf r})\n\\rangle$ of the operator ${\\hat F}({\\bf r})$ (\\ref{foper}) due to a\nquasistatic variation of the particle density $\\rho_{\\rm qs}({\\bf r},Q,T)$,\n\\bel{quasavf} \n\\delta \\langle {\\hat F}({\\bf r}) \\rangle_{\\rm qs}^{\\tt X} =\n\\int~{\\rm d} {\\bf r} ~{\\hat F}^{\\tt X}({\\bf r})~ \\delta \\rho_{\\rm qs}^{\\tt X}\n=-\\chi_{FF}^{\\tt X}\\; \\delta Q\\;, \n\\end{equation}\nwhere \n\\bel{quasdens} \n\\delta \\rho_{\\rm qs}^{\\tt X}\n=\\left[\\left(\\frac{\\partial \\rho_{\\rm qs} }{\\partial Q}\\right)_T\n+\\left(\\frac{\\partial \\rho_{\\rm qs} }{\\partial T}\\right)_Q \n\\frac{\\delta T}{\\delta Q}\\right]^{\\tt X} \\delta Q \\;.\n\\end{equation} \nThe index \"${\\tt X}$\" shows one of the conditions of the constant temperature \n(${\\tt X}=T$), entropy \n(${\\tt X}$ is \"ad\") or static limit $\\omega  \\to 0$ (${\\tt X}$ is \"$\\omega =0$\").\nWe shall follow the notations of \\cite{hofmann,hofbook} omitting the\nindex $\\omega =0$ for the coupling constant ($k_{\\omega =0} \\equiv k$),\nsurface energy constant ($b_{{}_{\\! S}}^{\\omega =0} \\equiv b_{{}_{\\! S}}$), and\nin-compressibility [$K^{\\omega =0} \\equiv K=K_{\\rm tot}$ for $\\omega =0$, see\n(\\ref{incomprtot})]. We write it as the zero argument for the\nisolated susceptibility, $\\chi^{\\omega =0} \\equiv \\chi(0)$, and\nstiffness coefficient, $C^{\\omega =0}=C(0)$. $f^{\\tt X}$ and\n$\\delta f^{\\tt X}$ denote the quantity $f$ and its variation\nprovided that the  \n${\\tt X}$ condition is\ncarried out. The index \"qs\" stands for the quasistatic quantities\nas in \\cite{hofmann} and will be omitted within this Appendix. \nNote that the\noperator ${\\hat F}({\\bf r})$ (\\ref{foperl}) depends in the FLDM on \n${\\tt X}$ through the derivatives of the particle density, and by\nthis reason, the upper index ${\\tt X}$ appears in \n${\\hat F}^{\\tt X}({\\bf r})$ of (\\ref{quasavf}).\nFrom the first of (\\ref{quasavf}) with (\\ref{foperl}) and\n(\\ref{quasdens}), one gets the self-consistency condition\n(\\ref{selfconsist}),\n\\bel{Favfld} \n\\delta \\left\\langle {\\hat F}({\\bf r})\n\\right\\rangle^{\\tt X} =k_{\\tt X}^{-1} \\delta Q,\n\\end{equation}\n with the following expression for the\ncoupling constant, \n\\begin{eqnarray}\\label{couplfld} \nk_{\\tt X}^{-1}\n&=& - R_0 \\int~{\\rm d} {\\bf r} \\left\\{\\frac{\\partial V }{\\partial r}\nY_{L0}({\\hat r}) \\left[\\left(\\frac{\\partial \\rho }{\\partial\nQ}\\right)_T \\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.\\left(\\frac{\\partial \\rho }{\\partial T}\\right)_Q\n\\frac{\\delta T }{\\delta Q}\\right]\\right\\}_{Q=0}^{\\tt X}\\mathcal{O}_1, \n\\end{eqnarray} \nwithin the ESA parameter of smalleness\n$a\/R \\sim A^{-1\/3}\\ll 1$,\n$\\mathcal{O}_1=1+\\mathcal{O}\\left(a\/R\\right)=\n1+\\mathcal{O}\\left(A^{-1\/3}\\right)$.\nFor the susceptibilities $\\chi_{FF}^{\\tt X}$ defined by (\\ref{quasavf}) \nup to small corrections of the\norder of $A^{-1\/3}$ in the same approximation,  from\n(\\ref{couplfld}) for $k_{\\tt X}^{-1}$ one gets \n\\bel{chix} \n\\chi^{\\tt X}=-k_{\\tt X}^{-1}\\mathcal{O}_1.\n\\end{equation}\nWe omit also the low indexes \"FF\"\nfor the susceptibilities. Note that we have not identities of\n$-k_{\\tt X}^{-1}$ to $\\chi^{\\tt X}$ because we neglected earlier\nhigh order $A^{-1\/3}$ corrections in the derivation of the\noperator ${\\hat F}$ (\\ref{foperl}), in particular, in the FLDM\napproximation (\\ref{denseq}) for the quasistatic particle density\n$\\rho_{\\rm qs}$.  \nThe equation (\\ref{chix}) is in\nagreement with (\\ref{kstiffC0chi0}) (identical to equation (3.1.26)\nof \\cite{hofmann}), see also\n(\\ref{smallpar}), (\\ref{isorespk}), for the specific relation\nbetween the coupling constant $k^{-1}$ and isolated susceptibility\n$\\chi(0)$ {\\it with presence of the stiffness term} $C(0)$ in \"the\nzero frequency limit\" within the FLDM. As shown in \nSec.\\ \\ref{zerofreqlim} through (\\ref{chicollfldm}) by using the\nexpansion in small parameter $kC$ (\\ref{smallpar}) up to the\nsecond order terms in $kC$, the isolated susceptibility\n$\\chi(0)$, see (\\ref{respintr}) at $\\omega  =0$, is related to the\ncoupling constant $k^{-1}$ by (\\ref{kstiffC0chi0}) with the\nstiffness term $C(0)$. The correction related to the stiffness\n$C(0)$ appears in (\\ref{chix}) in a \nhigher order than\n$A^{-1\/3}$ because it is of the order of the small parameter $kC\n\\sim A^{-2\/3}$, see (\\ref{smallpar}) and discussion near this\nequation. The zero frequency stiffness $C(0)$ is equal\napproximately to the liquid drop one $C$ (\\ref{stiffness0}) in the\nFLDM for the considered enough large temperatures for which the\nquantum shell effects can be neglected.\n\n\nThe derivatives of the quasistatic particle density in\n(\\ref{quasdens}), (\\ref{Favfld}) and (\\ref{couplfld}) can be\nfound from (\\ref{denseq}), \n\\begin{eqnarray}\\label{drdqdt} \n\\left(\\frac{\\partial \\rho }{\\partial Q}\\right)_T^{\\tt X}\n&=& -\\left(\\frac{\\partial \\rho \n}{\\partial r}\\right)^{\\tt X} R_0 Y_{L0}({\\hat r}),\n\\nonumber\\\\\n\\left(\\frac{\\partial \\rho }{\\partial T}\\right)_Q^{\\tt X} &=& \n\\frac{1}{\\rho_0^{\\tt X}}\\frac{\\partial \\rho_0^{\\tt X} }{\\partial T}\n\\rho^{\\tt X}+\\frac{R_0 }{3\\rho_\\infty} \\frac{\\partial \\rho_\\infty\n}{\\partial T} \\left(\\frac{\\partial \\rho }{\\partial\nr}\\right)^{\\tt X}, \n\\end{eqnarray}\n for $Q=0$ with \n\\bel{rho0x} \n\\rho_0^{\\tt X}=\n\\rho_\\infty \\left(1+\\frac{6 b_{{}_{\\! S}}^{\\tt X} r_{{}_{\\! 0}} }{K^{\\tt X} R_0\n}\\right), \n\\end{equation}\n as in (\\ref{rho0}). We emphasize\nthat the surface energy constant $b_{{}_{\\! S}}^{\\tt X}$ (or the\nsurface tension coefficient $\\alpha^{\\tt X}$) depends also on the\ntype of the process specified by index ${\\tt X}$ as the\nin-compressibility $K^{\\tt X}$ because of the ${\\tt X}$-dependence\nof the particle density derivative in the integrand of\n(\\ref{tensionconst}) for the tension coefficient. The total\nquasistatic energy is the sum of the volume  and surface parts\ndetermined by the in-compressibility $K^{\\tt X}$ and surface\n$b_{{}_{\\! S}}^{\\tt X}$ constants, respectively. The in-compressibility\nmodulus $K^{\\tt X}$ (responsible for the change of the volume\nenergy) is given by (\\ref{incomprTdef}) for ${\\tt X}=T$, and\n(\\ref{incompraddef}) for ${\\tt X}$ is \"ad\", see also\n(\\ref{isotherk}), (\\ref{Kadiabat}) or \n(\\ref{isotherkexp}), (\\ref{incompradexp}) of their more \nspecific expressions for\nnuclear matter.\nThe in-compressibility $K$ equals the adiabatic one\n$K^{\\varsigma}$ as shown through (\\ref{dtdrhoexp}) and\n(\\ref{incompradexp}), $K=K_{\\rm tot}(\\omega =0)=K^{\\varsigma}$. In the\nderivations of (\\ref{drdqdt}), we took into account that \n$\\rho_0^{\\tt X}$ (\\ref{rho0x}) does not depend on $Q$, and the\ndensity $\\rho_\\infty$ (or $r_0$) is assumed to be approximately\nindependent of index \"${\\tt X}$\" in (\\ref{drdqdt}).\n\n\nSubstituting (\\ref{drdqdt}) into (\\ref{couplfld}) for the\ncoupling constant $k_{\\tt X}^{-1}$, one writes \n\\begin{eqnarray}\\label{couplfld1}\n&&k_{\\tt X}^{-1} = R_0^2 \\int~{\\rm d} {\\bf r} \\left\\{{\\partial V \\over\n{\\partial r}} \\frac{\\partial \\rho }{\\partial r} Y_{L0}({\\hat r})~\\qquad\n\\right.\\nonumber\\\\\n&\\times & \\left.\\left[Y_{L0}({\\hat r}) -\\frac{1}{3 \\rho_\\infty} \n\\frac{\\partial\n\\rho_\\infty }{\\partial T} \\frac{\\delta T }{\\delta\nQ}\\right]\\right\\}_{Q=0}^{\\tt X} \\mathcal{O}_1.\n\\end{eqnarray}\nThe\nfirst term proportional to the density in $\\partial \\rho\/\n\\partial T$ of (\\ref{drdqdt}) leads to small $A^{-1\/3}$\ncorrections to the coupling constant $k_{\\tt X}^{-1}$\n(\\ref{couplfld1}) with respect to the second component depending\non the coordinate derivative $\\partial \\rho\/\\partial r$. However,\nall terms including these corrections related to the variation of\nthe temperature $\\delta T$ in (\\ref{couplfld}) [or\n(\\ref{couplfld1})] can be neglected as compared to the first\nterm in the square brackets. (It comes from the variation of the\ncollective variable $\\delta Q$ up to the same relatively small\ncorrections of the order of $A^{-1\/3}$.) Indeed, for the\nisothermal case \"${\\tt X}=T$\" one has it exactly by its\ndefinition. For other \"${\\tt X}$\" the quantity $\\delta T\/\\delta\nQ$ in (\\ref{Favfld}) and (\\ref{couplfld1}) with the density\n(\\ref{denseq}) can be transformed within the ES precision\n\\bel{dtdqx} \n\\left(\\frac{\\delta\nT }{\\delta Q}\\right)^{\\tt X} =\\left(\\frac{\\delta T }{\\delta\n\\rho} \\frac{\\partial \\rho }{\\partial Q}\\right)^{\\tt X}\n=\\left(\\frac{\\partial T }{\\partial r}\\right)^{\\tt\nX}R_0Y_{L0}({\\hat r}).\n\\end{equation}\n For\ninstance, for the constant entropy (adiabatic) condition $S=\\int\n{\\rm d}r \\rho \\varsigma=S(\\rho,T)={\\rm const.}$, see\n(\\ref{entropydef}) with the quasistatic particle density\n$\\rho$ (\\ref{denseq}), the derivative $\\delta T\/\\delta Q$ can be\ncalculated through a variation of this density $\\rho$ as shown in\nthe middle of (\\ref{dtdqx}). In the quasistatic limit $\\omega \n\\to 0$ all quantities of the equilibrium state can be considered\nalso as a functional of the only density $\\rho$ (\\ref{denseq})\nin the ESA \nand one has again\n(\\ref{dtdqx}). We have used already this property in the\nderivation of the operator ${\\hat F}({\\bf r})$ for transformations of\nthe derivatives of a mean field $V$ in (\\ref{foperl}). As noted\nand used for the derivations in Appendix A.1,  \nthe\ntemperature $T({\\bf r})$ is approximately independent of the spatial\ncoordinates ${\\bf r}$ at equilibrium.  \nTherefore, according to (\\ref{dtdqx}),\nthe second terms in (\\ref{Favfld}) and (\\ref{couplfld1}),\nwhich appear due to the temperature variation $\\delta T$, turn\ninto zero with the FLDM precision. \n\nAfter the simple integration over the angles ${\\hat r}$ in\n(\\ref{couplfld1}) for the coupling constant $k_{\\tt X}^{-1}$,\none then arrives at\n\\bel{couplfld2}\nk_{\\tt X}^{-1}\n= R_0^4 \\int_0^\\infty~{\\rm d} r \\left[ \\frac{\\delta V }{\\delta\n\\rho} \\left(\\frac{\\partial \\rho }{\\partial r}\\right)^2\n\\right]_{Q=0}^{\\tt X} \\mathcal{O}_1.\n\\end{equation}\nAccording to (\\ref{denseq}),\nthe integrands in (\\ref{couplfld2}) contains the sharp bell\nfunction ${\\partial \\rho\/\\partial r}$ of $r$. Therefore, the\nintegrals converges there to a small spatial region near the\neffective nuclear surface defined as the positions of maxima of\nthis derivative at $r=R_0$ (Appendix D). We use these properties of the\nintegrand in the derivation of (\\ref{couplfld2}) taking\nsmooth quantities as $r^2$ at the nuclear surface point $r=R_0$\noff the integrals up to small corrections of the order of\n$A^{-1\/3}$ within the same ESA. [This is like for the\nderivations of the boundary conditions\n(\\ref{bound1}), (\\ref{bound2}), see\n\\cite{strutmagbr,magstrut,strutmagden}, and of (\\ref{foperl})\nfor the operator ${\\hat F}({\\bf r})$.] In this way, we get the expansion\nof the coupling constant $k_{\\tt X}^{-1}$ (\\ref{couplfld}), in\npowers of the $A^{-1\/3}$ with the leading term shown in the second\nequation there, see (\\ref{couplfld2}).\n\nFor the following derivations, we specify now the quasistatic\nderivative $\\left(\\delta V \/ \\delta \\rho\\right)^{\\tt X}$ at $Q=0$\ntaken it from (\\ref{gradrelmu}), \n\\bel{derVderP1} \n{\\bf \\nabla}\nV^{\\tt X}~ =~ \\frac{K^{\\tt X} }{9 \\rho_0^{\\tt X}}~ {\\bf \\nabla}\n\\rho^{\\tt X}, \n\\end{equation}\n where index ${\\tt X}$ in ${\\bf \\nabla} f^{\\tt\nX}$ means the gradient of the quantity $f$ taken for the condition\nmarked by ${\\tt X}$ as in the variation $\\delta f^{\\tt X}$. The\nproportionality of the gradients in (\\ref{derVderP1}) shows the\nself-consistency within the ESA precision, see\n\\cite{magstrut} for more general relations of the self-consistency\nin the FLDM.\n\nUsing (\\ref{derVderP}) and (\\ref{tensionconst}) in \n(\\ref{couplfld2}) for the coupling\nconstants and (\\ref{chix}) for the susceptibilities, one obtains\nthe identical results for these quantities with \nsmall corrections\nof the order of $A^{-1\/3}$, \n\\bel{couplxchix} \n\\chi^{\\tt X}= -k_{\\tt X}^{-1}\\mathcal{O}_1=\n\\frac{K^{\\tt X} b_{{}_{\\! S}}^{\\tt X} R_0^4 }{72 \\pi\nr_0^2 \\rho_\\infty^{\\tt X} \\mathcal{C}} \\mathcal{O}_1.\n\\end{equation}\n\nWe shall show now from (\\ref{couplxchix}) that the adiabatic\nsusceptibility $\\chi ^{\\rm ad}$ and coupling constant $k_{\\rm ad}^{-1}$\nare equal to the isolated ($\\chi(0)$) and quasistatic ($k^{-1}$)\nones, respectively, up to small $A^{-1\/3}$ corrections within the\nESA.\n\nAs noted above, for the adiabatic ($K^{\\varsigma}$) and\nquasistatic ($K$) in-compressibility modula,  we got\n$K^{\\varsigma}=K$, see after (\\ref{dtdrhoexp}).  \nThe surface energy constant $b_{{}_{\\! S}}$ equals\nthe adiabatic one $b_{S}^{\\rm ad}$, according to\n(\\ref{keybound}). Indeed, the volume energy per particle  is\nalso approximately the same for these cases, \n$b_{\\mathcal{V}}^{\\rm ad} = b_{{}_{\\! \\mathcal{V}}}$,\nbecause of its relation  $b_{{}_{\\! \\mathcal{V}}} \\approx K\/18$ to the\nin-compressibility modulus ($b_{\\mathcal{V}}^{\\rm ad}=K^{\\varsigma}\/18$)\nwithin the ESA \\cite{strutmagden}\nand equivalence of the corresponding in-compressibility modula. The\nfunctional derivative in (\\ref{keybound}) is the quasistatic\nchemical potential $\\mu$ which does not depend obviously on the\ntype of the process ${\\it X}$.\nFrom (\\ref{keybound}), one gets now $\\alpha^{\\rm ad}=\\alpha$ for\nthe surface tension coefficient or\n$b_{{}_{\\! S}}=b_{S}^{\\rm ad}$ for the surface energy constant. Namely,\nthis quantity should be identified with the experimental value\n$b_{{}_{\\! S}}=17-19~\\mbox{MeV}$ in the FLDM computations. \nThus, from (\\ref{couplxchix}) one obtains the ergodicity condition\n(\\ref{ergodicity1}) for the susceptibilities within the ESA precision,\n\\bel{chi0adfld} \n\\chi(0)=\\chi^{\\rm ad}=\\frac{K b_{{}_{\\! S}} r_0^5\n}{54 \\mathcal{C}} A^{4\/3}, \n\\end{equation}\n up to small $A^{-1\/3}$ corrections.\nAs seen from  (\\ref{couplxchix}), one gets  also\n$k^{-1}=k_{\\rm ad}^{-1}$ for the coupling constants within the same\napproximation , see (\\ref{kfld}) for $k^{-1}$. The index \"ad\"\nfor the coupling constant will be omitted below in line of\n\\cite{hofmann}.\n\nWe are interested also in the discussion of the difference between\nthe susceptibilities $\\chi^T$ and $\\chi^{\\rm ad}$. From\n(\\ref{chix}), one has \n\\bel{chiTchiad}\n\\chi^T-\\chi^{\\rm ad}=\\left(k_T^{-1}-k^{-1}\\right) \\mathcal{O}_1.  \n \\end{equation} \nIt is useful to re-derive this\nrelation by applying Appendix A1 of \\cite{hofmann} for the\nspecific Fermi-liquid drop thermodynamics, see\n(\\ref{ctcaddif}). As noted in Appendix A.4,  \nit is\nmore convenient to use the Gibbs free energy $G$ instead of the\nfree energy $F$. As the derivations of the $\\chi^T-\\chi^{\\rm ad}$ in\nAppendix A1 of \\cite{hofmann} do not contain any change in the\nvolume and pressure variables, we can use all formulas in A1 of\n\\cite{hofmann} here with the replace of the free energy $F$ by\nthe Gibbs free energy $G$, in particular (\\ref{ctcaddif}).\nThere is a specific property of the FLDM with respect to the\nmicroscopic shell model with the residue interaction of\n\\cite{hofmann}. The second derivatives of the Hamiltonian\n$\\langle\\left(\\partial ^2 {\\hat H} \/ {\\partial\nQ^2}\\right)_{Q=0}\\rangle_0$ in equations (A.1.6) and (A.1.7) of\n\\cite{hofmann} \ndepend in the FLDM on the type of the\nprocess ${\\tt X}$, isothermal and adiabatic one, relatively. The\nfirst derivative of the Hamiltonian ${\\partial H\/\\partial Q}$ in\nequations (A.1.2) and (A.1.3), see \\cite{hofmann}, is proportional to\nthe derivatives of the density ${\\partial \\rho\/\\partial Q}$ like\nin the susceptibilities (A.1.8), \n\\bel{dhdq} \n\\left(\\frac{\\partial H\n}{\\partial Q}\\right)^{\\tt X} \n=\\left(\\frac{\\delta V }{\\delta\n\\rho} \\frac{\\partial \\rho }{\\partial Q}\\right)^{\\tt X}\n=-\\left(\\frac{\\partial V }{\\partial r}\\right)^{\\tt\nX}R_0Y_{L0}({\\hat r}). \n\\end{equation}\nWe used also this self-consistent\ndependence of mean potential $V$ through the particle density\n$\\rho$ in the derivations of the operator ${\\hat F}({\\bf r})$\n(\\ref{foperl}) in the FLDM: The derivatives of $V$ are\nproportional to the ones of the density $\\rho$ [see (\\ref{derVderP1})], \nwhich both depend\non ${\\tt X}$, i.e., whether we consider the latter for the fixed temperature\nor entropy. Therefore, (A.1.6) and (A.1.7) with the definitions\n(A.1.8) of \\cite{hofmann} as applied to the FLDM, see\n(\\ref{ctcaddif}), should be a little modified to \n\\begin{eqnarray}\\label{d2GEdq2}\n\\left(\\frac{\\partial^2 G }{\\partial Q^2}\\right)_T&=&\n\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^T -\\chi^T ,\\nonumber\\\\\n\\left(\\frac{\\partial^2 E }{\\partial Q^2}\\right)_S&=&\n\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^{\\rm ad} -\\chi^{\\rm ad} .\n\\end{eqnarray} \nThe derivatives of\nthe thermodynamic potential G are considered for the constant\npressure instead of the constant volume as used for the free\nenergy case. Similar calculations of the average value of the\nsecond derivative of the Hamiltonian $\\langle\\partial ^2 {\\hat H}\/\n\\partial Q^2\\rangle^{\\tt X}$ as for the coupling constants lead\nto\n\\begin{eqnarray}\\label{secderiv}\n&&\\left\\langle\\left(\\frac{\\partial ^2 {\\hat H} }{\\partial\nQ^2}\\right)_{Q=0}\\right\\rangle^{\\tt X}= \\int {\\rm d} {\\bf r} \n\\left(\\frac{\\partial\n^2 V }{\\partial Q^2}\\right)_{Q=0}^{\\tt X} \\rho \\qquad\\nonumber\\\\\n&=& -R_0^4\\int_0^{\\infty}\n{\\rm d} r \\left(\\frac{\\partial V }{\\partial r}~ \\frac{\\partial \\rho\n}{\\partial r}\\right)_{Q=0}^{\\tt X} \\mathcal{O}_1. \n\\end{eqnarray}\nWe integrated in the second equation of (\\ref{secderiv}) by\nparts. Taking then (\\ref{denseq}) for the quasistatic density\n$\\rho_{\\rm qs}$, one gets \n\\bel{d2hdq2} \n\\left\\langle\\left(\\frac{\\partial ^2\n{\\hat H} }{\\partial Q^2}\\right)_{Q=0}\\right\\rangle_0^{\\rm qs}=\n-k^{-1} \\mathcal{O}_1\n\\end{equation} \nwith the\ncoupling constant $k$ (\\ref{kfld}). Using the same transformations\nof the thermodynamic derivatives as in Appendix A1 of\n\\cite{hofmann}, see (\\ref{ctcaddif}),\nfrom (\\ref{d2GEdq2}) and (\\ref{d2hdq2}) up\nto relatively small $A^{-1\/3}$ corrections of the ESA, one gets\n\\begin{eqnarray}\\label{chiTchiad1}\n&&\\chi^T-\\chi^{\\rm ad}= \n-\\left[\\left(\\left(\\frac{\\partial^2 G}{\\partial\nT^2}\\right)_Q\\right)^{-1} \\left(\\frac{\\partial^2 G}{\\partial T\\partial\nQ}\\right)^2\\right]_{Q=0}\\nonumber\\\\\n&+& k_T^{-1}-k^{-1} \\;.\n\\end{eqnarray}\nApplying then the relation of the Gibbs free energy $G=A \\mu$ to\nthe chemical potential $\\mu$, we note that there is the factor $A^{-1}$ which\nsuppress much the contribution of the first term compared to\nsecond one, $k_T^{-1}-k^{-1}$, see (\\ref{couplxchix}),\n\\bel{ktk} \nk_T^{-1}-k^{-1}= \\frac{r_0^5 A^{4\/3} K b_{{}_{\\! S}}  }{56\n\\mathcal{C}} \\left(1-\\frac{K^{T} }{K} \\frac{b_{{}_{\\! S}}^T \n}{b_{{}_{\\! S}}}\\right) \\mathcal{O}_1.\n\\qquad\\qquad \n\\end{equation} \nMoreover, the terms in\nthe square brackets of\n(\\ref{chiTchiad1}) are zero because the second derivative\n$~\\left(\\partial \\mu\/\\partial Q\\right)_T~$ is zero within the\nprecision of the FLDM. To show this, let us take the equation as\nfor the temperature (\\ref{dtdqx}) with the only replace of the\ntemperature $T$ by the chemical potential $\\mu$. \nThe above mentioned statement becomes\nnow obvious because the chemical\npotential $\\mu$ is a constant as function of the spatial\ncoordinates at the equilibrium as the temperature $T$\nindependently on the type of the process ${\\tt X}$\nwithin the FLDM.\nAs the result, we obtain the same\nrelation (\\ref{chiTchiad}) with the difference of the coupling\nconstants shown in (\\ref{ktk}).\n\nWe can evaluate the ratio of the surface energy coefficients\n$b_{S}^T\/b_{{}_{\\! S}}$ of (\\ref{couplxchix}) using in\n(\\ref{chiTchiad}) the fundamental relation\n(\\ref{chitchiskSkT}) for the ratio of the susceptibilities\n$\\chi^T\/\\chi^{\\rm ad}$ in terms of the in-compressibility modula\n$K\/K^T$ ($K=K^{\\varsigma}=K^{\\rm ad}$), \n\\bel{bstbs} \n\\frac{b_{S}^T }{b_{{}_{\\! S}}}= \n\\frac{K }{K^T}~\\left(2-\\frac{K }{K^T}\\right)\n\\approx 1+\\frac{2 \\pi^2 {\\bar T}^2 }{3 \\mathcal{G}_0}. \n\\end{equation} \nIn the last\nequation, we used the temperature expansions for the\nin-compressibilities $K$ (\\ref{incompradexp}) and $K^T$\n(\\ref{isotherkexp}). Thus, the surface energy constant $b_{S}^T$\nfor the constant temperature is larger than adiabatic (or\nquasistatic) $b_{{}_{\\! S}}$ and their difference is small as\n${\\bar T}^2$.\n\n\n\\setcounter{equation}{0}\n\n\\section{Symmetry-energy density functional and boundary conditions}\n\\label{endenfun}\n\nThe nuclear energy, \n$E=\\int \\hbox{d} {\\bf r}\\; \\rho_{+}\\;{\\cal E}\\left(\\rho_{+},\\rho_{-}\\right)\\;$, \nin the local density approach\n\\cite{brguehak,chaban,reinhard,bender,revstonerein,ehnazarrrein,pastore}\ncan be calculated through the energy density \n${\\cal E}\\left(\\rho_{+},\\rho_{-}\\right)$ per particle, \n\\begin{eqnarray}\\label{enerden}\n&&{\\cal E}\\left(\\rho_{+},\\rho_{-}\\right) =\n- b_{{}_{\\! \\mathcal{V}}} \n+ J \\mathcal{I}^2 \n+  \n\\frac{K}{18}\\epsilon_{+}(\\rho_{+}) -\nJ \\mathcal{I}^2\\epsilon_{-}(\\rho_{+},\\rho_{-}) +\n\\quad\\nonumber\\\\\n&& \n\\left(\\frac{\\mathcal{C}_{+}}{\\rho_{+}} +\\mathcal{D}_{+} \n\\right) \n\\left(\\nabla \\rho_{+}\\right)^2  \n+ \\left(\\frac{\\mathcal{C}_{-}}{\\rho_{+}} + \n\\mathcal{D}_{-} \n\\right) \n\\left(\\nabla \\rho_{-}\\right)^2.\\quad\n\\end{eqnarray}\nHere, $\\rho_{\\pm}=\\rho_n \\pm \\rho_p$ are the isoscalar, $\\rho_{+}$, and \nisovector, $\\rho_{-}$, particle densities, $\\mathcal{I}=(N-Z)\/A$ \nis the asymmetry parameter, \n$N$ and $Z$ are the neutron and proton numbers in the nucleus, $A=N+Z$. \nThe particle separation energy  \n$b_{{}_{\\! \\mathcal{V}}} \\approx$ 16 MeV  \nand  the symmetry energy constant of nuclear matter $~J \\approx$ \n30 MeV are introduced also in (\\ref{enerden}). \nThe in-compressibility modulus of \nthe symmetric nuclear matter \n $K \\approx 220-260$ MeV is shown in Table I of \n\\cite{chaban,reinhard,BMRV}). \nEquation (\\ref{enerden})\ncan be applied approximately to the\nrealistic Skyrme forces \\cite{chaban,reinhard}, in particular\nby neglecting small semiclassical $\\hbar$ corrections \nand Coulomb terms \\cite{brguehak,strtyap,strutmagbr,strutmagden,magsangzh}. \n$\\mathcal{C}_{\\pm}$ and $\\mathcal{D}_{\\pm}$ are constants defined by the \nbasic Skyrme force parameters. The isoscalar \nsurface energy-density part, independent explicitly \nof the density gradient terms, is determined by the dimensionless function  \n$\\epsilon_{+}(\\rho_{+})$ satisfying\nthe standard saturation conditions \\cite{strutmagden,magsangzh,BMRV}.\nFor the derivation of the explicitly analytical results, we \nuse the quadratic approximation $\\epsilon_{+}(\\rho_{+}) = \n(1-\\rho_{+}\/\\rho_\\infty)^2=(1-w_{+})^2$, where $\\rho_\\infty\\approx$ 0.16 fm$^{-3}$ \nis the density of infinite nuclear matter [see around \n(\\ref{denseq})].\nThe isovector component can be simply evaluated as   \n$\\epsilon_{-}=1 - \\rho_{-}^2\/(\\mathcal{I}\\rho_{+})^2=1-w_{-}^2\/w_{+}^2$.\nIn both these energies $\\epsilon_{\\pm}$, \n$w_{\\pm}=\\rho_{\\pm}\/(\\mathcal{I}_{\\pm} \\rho_\\infty)$ are the dimensionless\nparticle densities, $\\mathcal{I}_{+}=1$ and \n$\\mathcal{I}_{-}= \\mathcal{I}$.\nThe isoscalar SO gradient terms in (\\ref{enerden}) are defined with a\nconstant: \n$\\mathcal{D}_{+} = -9m W_0^2\/16 \\hbar^2$, where\n$W_0 \\approx$100 - 130~ MeV$\\cdot$fm$^{5}$ \n and $\\mathcal{D}_{-}$ is relatively small \n\\cite{brguehak,chaban,reinhard}.\n\n\nFrom the condition of the minimum energy $E$   \nunder the constraints\nof the fixed particle number $A=\\int \\hbox{d} {\\bf r}\\; \\rho_{+}({\\bf r})$ and\n neutron excess $N-Z= \\int \\hbox{d} {\\bf r}\\; \\rho_{-}({\\bf r})$,\none arrives at the Lagrange equations for $\\rho_{\\pm}$ with the corresponding \nmultipliers being the \nisoscalar and isovector  chemical potentials\nwith the surface corrections at the first order, \n$\\Lambda_\\pm \\propto \\mathcal{I}_\\pm a\/R \\sim A^{-1\/3}$ \n\\cite{strutmagbr,strutmagden,magsangzh,BMRV}.  \n\n\nThe isoscalar  and isovector  \nparticle densities $w_\\pm$  \ncan be derived from (\\ref{enerden})\nfirst at the leading approximation in a small parameter \n$a\/R$. For the isoscalar particle density $w_{+}=w_{+}(\\xi)$ \n[$\\xi$ is the distance\nof the given point ${\\bf r}$ from the ES in units of the diffuseness\nparameter $a$ in the local ES coordinates, see (\\ref{denseq}), \n$\\xi = (r-R)\/a$ for the spherical nuclei], \none finds (Appendix B of \\cite{BMRV} and\n\\cite{strutmagden,magsangzh}),\n\\bel{ysolplus}\n\\xi=-\\int_{w_r}^{w}\\hbox{d} y\\; \\sqrt{\\frac{1 +\\beta y}{y\\epsilon(y)}}\\;,\\qquad\n\\end{equation}\nbelow the turning point $~\\xi(w=0)~$ and $~w=0~$ for $~\\xi \\geq \\xi(w=0)~$,\n$\\beta=\\mathcal{D}_{+}\\rho_\\infty\/\\mathcal{C}_{+}$ is the dimensionless SO\nparameter (for simplicity of the notations, we omit the low index\n``+'' in $w_{+}$).  \nFor $w_r=w(\\xi=0)$,\none has the boundary condition,\n$\\hbox{d}^2 w(\\xi)\/\\hbox{d} \\xi^2=0$\nat the ES ($\\xi=0$):\n\\bel{boundeq}\n\\epsilon(w_r)+w_r(1 +\\beta w_r) \\left[\\hbox{d} \\epsilon(w_r)\/\\hbox{d} w\\right]=0.\n\\end{equation}\n(see Appendix B of \\cite{BMRV} where the specific solutions for $\\xi(w)$\nin the quadratic approximation for $\\epsilon_{+}(w)$ in terms of \nelementary functions were derived).\nFor $\\beta=0$ (i.e. without SO terms), it simplifies to\nthe solution $w(\\xi)=\\tanh^2\\left[(\\xi-\\xi_0)\/2\\right]$ for\n$\\xi\\leq \\xi_0=2{\\rm arctanh}(1\/\\sqrt{3})$\nand zero for $\\xi$ outside the nucleus ($\\xi>\\xi_0$).\nFor the same leading term of the isovector density, $w_{-}(w)$,\none approximately (for large enough constants $c_{sym}$ \nof all desired Skyrme forces \n\\cite{chaban,reinhard})  \nfinds (Appendix A of \\cite{BMRV}) \n\\bel{ysolminus}\nw_{-} =  w\\;\\left(1-\\frac{\\widetilde{w}^2(w) \\left[1\n+ \\tilde{c}\\widetilde{w}(w)\\right]^2}{2\\left(1+\\beta\\right)}\n\\right).\n\\end{equation}\nwhere\n\\bel{defpar}\n\\widetilde{w}=\\frac{1-w}{c_{sym}}, \\quad \nc_{sym}=a \\sqrt{\\frac{J}{\\rho_\\infty\\vert\\mathcal{C}_{-}\\vert}},\\quad\n\\widetilde{c}=\\frac{\\beta c_{sym}\/2-1}{1+\\beta},\n\\end{equation}\nand $a \\approx 0.5 - 0.6$ fm is the \ndiffuseness parameter [see (\\ref{denseq})]. \n\n\nSimple expressions for the constants \n$b_S^{(\\pm)}$ \n(\\ref{bsplusminus}) can be easily\nderived  in terms of\nthe elementary functions \nin the quadratic approximation to $\\epsilon_{+}(w)$,\ngiven explicitly in Appendix A \\cite{BMRV}.\nNote that in these derivations we neglected curvature terms \nand being of the same order shell corrections. \nThe isovector energy terms were obtained within the ES \napproximation with high accuracy up to the product of two\nsmall quantities, $\\mathcal{I}^2$ and $(a\/R)^2$.\n\n\n\nWithin the improved ES approximation accounting also \nfor next order corrections in a small parameter \n$a\/R$, we derived \nthe macroscopic boundary conditions \n(Appendix B of \\cite{BMRV})\n\\begin{eqnarray}\\label{macboundcond}\n&&\\delta \\mathcal{P}_{\\pm}\\Big|_{ES} \n\\equiv \\left(\\rho_\\infty\\;\\mathcal{I}_{\\pm}\\;\n\\Lambda_{\\pm}\\right)_{ES} =  \\delta P_{S}^{\\pm},\\nonumber\\\\ \n&&\\mbox{where}\\quad \\delta P_{S}^{\\pm} \\equiv 2 \\alpha_{\\pm} \n\\delta \\mathcal{H} \n\\end{eqnarray}\n are the\nisovector and isoscalar surface-tension (capillary) pressures, \n$\\delta \\mathcal{H} \\approx -\\delta R_\\pm\/R_\\pm^2$ are small \nvariations of \nmean ES curvatures $\\mathcal{H}$ (\\ref{keybound}), \n$\\delta R_\\pm $\nare radius variations (\\ref{surface}),\nand $\\alpha_{\\pm}$ are the tension coefficients, respectively,\n\\begin{eqnarray}\\label{sigma}\n\\alpha_{\\pm}&=&b_S^{(\\pm)}\/4 \\pi r_0^2,\\quad b_{S}^{(\\pm)} \\approx\n\\frac{8 \\pi}{a}\\left(\\rho_\\infty \\mathcal{I}_\\pm\\right)^2 \n\\mathcal{C}_{\\pm}\\nonumber\\\\\n&\\times&\\int_{-\\infty}^{\\infty} \\hbox{d} \\xi\\;\n\\left(1 + \\frac{\\mathcal{D}_{\\pm}\\rho_\\infty}{\\mathcal{C}_{\\pm}} w_{+}\\right)\n\\left(\\frac{\\partial w_{\\pm}}{\\partial \\xi}\\right)^2.\\quad\n\\end{eqnarray}\nThe conditions (\\ref{macboundcond}) ensure \nthe equilibrium through the equivalence\nof the volume and surface pressure \n(isoscalar or isovector) variations, see detailed derivations \nin Appendix B of \\cite{BMRV}.\nAs shown in Sec.\\ III \\cite{strutmagbr,strutmagden,kolmagsh}, \nthe pressures \n$\\delta \\mathcal{P}_{\\pm}$\ncan be obtained through moments of dynamical variations of the\ncorresponding phase-space distribution functions \n$\\delta f_\\pm({\\bf r},{\\bf p},t)$ (\\ref{planewave}) in the nuclear volume. \n\n\nFor the nuclear energy $E$  \nin this improved ESA (Appendix C of \\cite{BMRV}), \none obtains\n\\bel{EvEs}\nE \\approx -b^{}_V\\; A + J (N-Z)^2\/A + E_S^{(+)} + E_S^{(-)},\n\\end{equation}\nwith the following \nisoscalar (+) and isovector (-) surface energy components,\n\\bel{Espm}\n E_S^{(\\pm)}= \\alpha_{\\pm}\\mathcal{S}=b_{S}^{(\\pm)} \\mathcal{S}\/(4\\pi r_0^2),\n\\end{equation}\nand the ES area $\\mathcal{S}$. \nThese energies  \nare determined by the\nisoscalar and isovector \nsurface energy constants $b_{S}^{(\\pm)} \\propto \\alpha_{\\pm}$ (\\ref{sigma}) \nthrough the solutions \nfor $w_{\\pm}(\\xi)$ taken \n at the leading order in $a\/R$. \n\n\nFor the energy surface coefficients\n $b_{S}^{(\\pm)}$ (\\ref{sigma}) with $\\mathcal{D}_{-} \\approx 0$] \nin the quadratic approximation  $\\epsilon_{+}(w)=(1-w)^2$,\nwe finally arrived at the following explicit analytical expressions\nin terms of\nthe Skyrme force parameters \n(Appendix C of \\cite{BMRV})\n\\bel{bsplusminus} \nb_{S}^{(+)}=\\frac{6 \\mathcal{C}_{+}\n\\rho_\\infty \\mathcal{J}_{+}}{ r_0 a},\\quad\nb_{S}^{(-)}=k^{}_S \\mathcal{I}^2,\\quad\nk_{{}_{\\! S}}= 6 \\rho_\\infty \\mathcal{C}_{-}\\mathcal{J}_{-}\/(r_0 a), \n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{Jp}\n&&\\mathcal{J}_{+}=\\int_0^1 \\hbox{d} w\\; \\sqrt{w(1+\\beta w)}\\;(1-w)\\nonumber\\\\\n&=&\\frac{1}{24}\\;(-\\beta)^{-5\/2}\\;\n\\left[\\mathcal{J}_{+}^{(1)}\\; \\sqrt{-\\beta(1+\\beta)}\n+\\mathcal{J}_{+}^{(2)}\\; \\arcsin\\sqrt{-\\beta}\\right],\\nonumber\\\\\n&&\\mathcal{J}_{+}^{(1)}=3 + 4 \\beta(1+\\beta),\\quad\n\\mathcal{J}_{+}^{(2)}=-3-6\\beta,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{Jm}\n\\mathcal{J}_{-}&=&-\\frac{1}{1+\\beta}\\;\n\\int_0^1 \\hbox{d} w\\; \\sqrt{\nw(1+\\beta w)}\\;(1-w)(1+\\widetilde{c} \\widetilde{w})^2\n\\nonumber\\\\\n&=&\\frac{\\widetilde{c}^2}{1920 (1+\\beta) (-\\beta)^{9\/2}}\\;\n\\left[\\mathcal{J}_{-}^{(1)}\\left(c_{sym}\/\\widetilde{c}\\right)\\;\n\\sqrt{-\\beta(1+\\beta)}\\right.\\nonumber\\\\ \n&+& \\left.\n\\mathcal{J}_{-}^{(2)}\\left(c_{sym}\/\\widetilde{c}\\right)\\;\n\\arcsin\\sqrt{-\\beta}\\right],\n\\nonumber\\\\\n\\mathcal{J}_{-}^{(1)}(z)&=& 105- 4 \\beta \\left\\{95 +75 z +\n\\beta \\left[119+10z (19+6z) \\right.\\right.\\nonumber\\\\\n&+& \\left.\\left. 8 \\beta^2\n\\left(1+ 10z(1+z)\\right)\n+ 8 z \\left(5 z (3 +2 z) -6\\right)\\right]\\right\\},\n\\nonumber\\\\\n\\mathcal{J}_{-}^{(2)}(z)&=&15 \\left\\{7+2\\beta \\left[\n5 (3 + 2 z)\\right.\\right.\\nonumber\\\\ \n&+& \\left.\\left. 8 \\beta (1+z)\n\\left(3 +z +2 \\beta (1+z)\\right)\\right]\\right\\},\n\\end{eqnarray}\nsee also (\\ref{defpar})\nfor $\\widetilde{c}$, $c_{sym}$ and $\\widetilde{w}$.\nFor the limit $\\beta \\rightarrow 0$ from (\\ref{Jp}) and (\\ref{Jm}),\none has\n$\\mathcal{J}_{\\pm} \\rightarrow 4\/15$. In the limit\n$\\mathcal{C}_{-} \\rightarrow 0$,  one obtains $k^{}_S \\rightarrow 0$. \n\n\n\n\\setcounter{equation}{0}\n\\section{POT calculations of the MI}\n\\label{semcalmi}\n\n\\subsection{Energy shell corrections}\n\\label{enshcor}\n\nThe energy shell corrections $\\delta E$ can be expressed approximately  \nthrough the oscillating level density component \n$\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)$, \naveraged locally by using the convolution (folding) integral \nwith a small averaging parameter $\\Gamma$ of the \nGaussian weight function\n \\cite{strut,fuhi}.\nAs shown in  \\cite{fuhi}, neglecting small corrections of the order of \nthe squares of the Fermi energy shell fluctuations  \n$(\\delta \\varepsilon_{{}_{\\! {\\rm F}}})^2$ at \n$\\Gamma \\ll \\hbar \\Omega \\sim \\varepsilon_{{}_{\\! {\\rm F}}}\/A^{1\/3}$  (see (\\ref{hom}), \nalso \\cite{strutmag}), one has\n\\begin{eqnarray}\\label{dedge}\n&&\\delta E = \\int \\hbox{d} \\varepsilon\\; n(\\varepsilon) \n\\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right)\\; \\delta g_{{}_{\\! \\Gamma}}(\\varepsilon),\\nonumber\\\\\n&& {\\rm with} \\quad  N=\\int \\hbox{d} \\varepsilon\\; n(\\varepsilon),\\quad \nn(\\varepsilon)=\\theta(\\varepsilon_{{}_{\\! {\\rm F}}}-\\varepsilon). \n\\end{eqnarray}\nSubstituting \n\\bel{avdden}\n\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)=\n{\\mbox {\\rm Re}} \\sum_{\\rm PO} \\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)\\;\n\\hbox{exp}\\left[-\\left(\\frac{t_{{}_{\\! {\\rm PO}}} \\Gamma}{\\hbar}\\right)^2\\right]\n\\end{equation}\nwith (\\ref{dlevdenscl}) for $\\delta g_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ into (\\ref{dedge}),\none can expand a smooth action in exponent at the linear order,\n\\bel{actexpef}\nS_{\\rm PO}(\\varepsilon) \\approx S_{\\rm PO}(\\varepsilon_{{}_{\\! {\\rm F}}}) + t_{{}_{\\! {\\rm PO}}} \n\\left(\\varepsilon-\\varepsilon_{{}_{\\! {\\rm F}}}\\right),\n\\end{equation} \nand pre-exponent \namplitude at zero order  over $\\varepsilon$ near the Fermi energy\n $\\varepsilon_{{}_{\\! {\\rm F}}}$\n[$t_{{}_{\\! {\\rm PO}}}= \\partial S_{\\rm PO}(\\varepsilon_{{}_{\\! {\\rm F}}})\/\\varepsilon$]\n(see a similar derivation of the averaged density $\\delta g_{{}_{\\! \\Gamma}}(\\varepsilon)$\nin \\cite{strutmag,sclbook,migdalrev}). \nThese expansions are valid for a small enough\nwidth $\\Gamma$ mentioned above to get a sharped bell-shaped Gaussian \naveraging function near $\\varepsilon_{{}_{\\! {\\rm F}}}$.  \nCalculating then\nsimple Gaussian integrals over the energy $\\varepsilon$ by integration\nby parts, one arrives at \n(\\ref{descl}). In these derivations at the leading order in expansion in\n$(\\delta \\varepsilon_{{}_{\\! {\\rm F}}})^2$, we accounted for the zero value\noriginated by the lower limit $\\varepsilon=0$ in (\\ref{dedge}) \nby using that\n$t_{{}_{\\! {\\rm PO}}}(\\varepsilon)$ is relatively large at small but finite $\\Gamma$.\nThus, one stays with the only contribution (independent of $\\Gamma$)\n at the upper limit\n$\\varepsilon=\\varepsilon_{{}_{\\! {\\rm F}}}$, in line of the basic concepts that\nthe energy shell correction is determined by the quantum s.p.\\ states\nnear the Fermi surface \\cite{fuhi,strut}. \nSimilarly, the same result can be obtained\nby using the Lorentzian weight function [the summand in \n(\\ref{avdden}) is proportional to \n$\\hbox{exp}\\left(-t_{{}_{\\! {\\rm PO}}} \\Gamma\/\\hbar\\right)$ \nin the Lorenzian\ncase, instead of the Gaussian exponent]. In this case, the \nlocal convolution averaging\nof the oscillation level density component with the Lorentzian width\nparameter $\\Gamma$ \nis resulted in a formal shift of the energy \n$\\varepsilon \\to \\varepsilon +i\\Gamma$ ($\\Gamma \\ll \\hbar \\Omega$). Thus, the \nstrightforward\ncalculations by the residue method also gives (\\ref{descl}).   \n\n\n\\subsection{Derivation of the rigid-body MI }\n\\label{derrigmi}\n\n\n\\subsubsection{TF COMPONENT }\n\\label{tfmi}\n\nWe substitute approximately the Green's function\n$\\langle G_{0}\\rangle_{\\Gamma_p}$, locally averaged over the momentum $p$\nby using the Gaussian weight function with a finite small width \n$\\Gamma_p$, into $\\Theta^{00}$ [see (\\ref{thetaxnnp}) at $\\nu=\\nu'=0$] \ninstead of $G_0$,\n\\begin{eqnarray}\\label{G0av}\n\\left\\langle G_{0}\\right\\rangle_{\\Gamma_p} &=& \n\\frac{1}{\\Gamma_p\\sqrt{\\pi}}\\int_{-\\infty}^\\infty \\hbox{d} p'\\; G_0(s,p')\\;\n\\hbox{exp}\\left[-\\left(\\frac{p'-p}{\\Gamma_p}\\right)^2\\right]\n\\nonumber\\\\\n&\\approx&\nG_0(s,p)\\; \\hbox{exp}\\left[\n-\\frac{s^2 \\Gamma_p^2}{4\\hbar^2}\\right],\n\\end{eqnarray}\nas in \\cite{strutmag} for the level density.\nTransforming then\nthe integration variables ${\\bf r}_1$ and ${\\bf r}_2$ to the canonical \naverage ${\\bf r}$ and\ndifference ${\\bf s}={\\bf s}_{{}_{\\! 12}}$ ones (\\ref{newcoord}), \nfor the corresponding locally averaged MI component\n$\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}$, one approximately gets  \n\\begin{eqnarray}\\label{thetaxnnp00}\n&&\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}\n\\approx\\frac{d_sm^2}{4\\pi^3}\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r} \\int \\frac{\\hbox{d} {\\bf s}}{s^2}\\;\n\\ell_x\\left({\\bf r}+\\frac{{\\bf s}}{2}\\right)\\quad\n\\nonumber\\\\\n&\\times&\n\\ell_x\\left({\\bf r}-\\frac{{\\bf s}}{2}\\right)\\;\\hbox{sin}\\left(\\frac{2 s p}{\\hbar}\\right)\\;\n\\hbox{exp}\\left[- \\frac{s^2 \\Gamma_p^2}{2\\hbar^2}\\right].\\quad\n\\end{eqnarray}\nFor simplicity, we omit here and below the \nindex in $s_{{}_{\\! 12}}$ within this \nAppendix E.2  because it will not interfer with\ndifferent notations. As shown below (at the end of this Appendix  E.2a), \nthe final result for $\\langle \\Theta_x^{00} \\rangle_{\\Gamma_p}$\n(\\ref{thetaxnnp00}) does not depend approximately on $\\Gamma_p $,\nthat looks as a plateau of the SCM (without correction polynomials).\nWithin the NLLLA (\\ref{nllla}), \nused already in  (\\ref{G0})\n\\cite{gzhmagsit,gzhmagsit,mskbPRC2010} after the averaging over \nthe phase-space variables, the main contribution is given\nby small distance $s_{{}_{\\! 12}}$ with respect to\nthe wave length $\\hbar\/p_{{}_{\\! {\\rm F}}}$ of the particle near the Fermi surface.\nIn this approximation at the leading zero order, due to the \nexponential cut-off factor decreasing with $s$ and $\\Gamma_p$,\none may expand smooth classical quantities in $sp\/\\hbar$ in the argument\nof exponent and pre-exponent amplitude factors in (\\ref{thetaxnnp00})\n at the leading \norder, in particular, applying \n\\bel{l2}\n\\ell_x\\left({\\bf r}+s\/2\\right)\\;\\ell_x\\left({\\bf r}-{\\bf s}\/2\\right) \\approx\n\\ell_x^2({\\bf r}) \\approx p^2 r_{\\perp x}^2.\n\\end{equation}\nIn (\\ref{thetaxnnp00}), we integrate over ${\\bf s}$ in the spherical \ncoordinates, $\\hbox{d} {\\bf s}=s^2\\hbox{d} s\\; \\hbox{sin}\\theta_s\\hbox{d} \\theta_s\\;\\hbox{d} \\varphi_s$,\nwith the polar axis $z_s$\ndirected along ${\\bf p}({\\bf r})$ (see Fig.\\ \\ref{fig14}). Then, \nfor the ${\\rm CT}_0$ momentum ${\\bf p}({\\bf r})$, i.e., \nalong the ${\\bf s}_{{}_{\\! 12}}$, \none takes into account that the integrand and \nlimits of the integration  \nover angles $\\theta_s$,\nand $\\varphi_s$ are constants independent of \nother variables. \nTherefore, this integration over all angles\ngives simply $4 \\pi$, and we arrive at \n\\begin{eqnarray}\\label{thetaxnnpint00}\n\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma_p}\n&\\approx&\\frac{d_sm^2}{\\pi^2}\\int \\hbox{d} {\\bf r} \\;r_\\perp^2\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \n\\hbox{d} \\varepsilon \\; \\left[{\\tt I}_{00}(s_{\\rm max},\\varepsilon,\\Gamma_p)\\right.\n\\nonumber\\\\\n&-& \\left.{\\tt I}_{00}(0,\\varepsilon,\\Gamma_p)\\right].\n\\end{eqnarray}\nHere, we exchanged the order of integrations over $\\varepsilon$\nand ${\\bf r}$. The remaining indefinite integral ${\\tt I}_{00}(s,\\varepsilon,\\Gamma_p)$ \nover $s$ as function of $s$, $\\varepsilon$ and $\\Gamma_p$\ncan be approximately (within the NLLLA) taken\nanalytically, \n\\begin{eqnarray}\\label{sinttf}\n&&{\\tt I}_{00}(s,\\varepsilon,\\Gamma)=\\int \\hbox{d} s\\; \\hbox{sin}\\left(\\frac{2 s p}{\\hbar}\\right)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma^2}{2\\hbar^2}\\right] \n\\nonumber\\\\\n&=& \n\\sqrt{\\frac{\\pi}{2}}\\;\\frac{i \\hbar}{2 \\Gamma}\\;\n\\hbox{exp}\\left[-2 \\left(\\frac{p}{\\Gamma}\\right)^2\\right]\\;\n\\left[{\\rm erf}\\left(\\frac{ip \\sqrt{2}}{\\Gamma}+\n\\frac{\\Gamma s}{\\hbar\\sqrt{2}}\\right)\\right.\n\\nonumber\\\\ \n&+&\\left.\n{\\rm erf}\\left(\\frac{ip \\sqrt{2}}{ \\Gamma}-\n\\frac{\\Gamma s}{\\hbar \\sqrt{2}}\\right)\\right],\n\\end{eqnarray}\nwhere \n${\\rm erf}(z)$\nis the standard error function, ${\\rm erf}(z)=\n(2\/\\sqrt{\\pi})\\int_0^z \\hbox{d} t\\; \\hbox{exp}(-t^2)$.\nThis integral, taken at the upper limit\n$s=s_{max}$, is rather a complicated function\nof ${\\bf r}$, especially near the ES of the potential well. \nHowever, the Gaussian factor in the integrand \nwith any small but \na finite Gaussian parameter $\\Gamma_p$,\n\\bel{avcondgp}\n\\hbar\/R \\ll \\Gamma_p \\ll p_{{}_{\\! {\\rm F}}},\n\\end{equation}\nremoves the oscillating contribution\narising from the upper limit $s_{\\rm max}$ ($R$ is the mean \nnuclear radius). The reason is due to the exponential \nasymptotics at a large argument\n$s$, such as\n\\begin{eqnarray}\\label{int00as}\n&&\\hbox{exp}\\left[-\\frac{\\Gamma_p^2 s_{\\rm max}^2}{2 \\hbar^2}\\right], \\qquad\n{\\rm or} \n\\nonumber\\\\\n&&\\hbox{exp}\\left[-\\frac{2 p^2}{\\Gamma_p^2}\\right],\\quad \n{\\rm at}\\quad p\\sim p_{{}_{\\! {\\rm F}}}, \n\\end{eqnarray}\nor even strongly as the product of these exponents.\nThen, according to another asymptotics for small $s \\rightarrow 0$,\n\\begin{eqnarray}\\label{int00as0}\n{\\tt I}_{00}(s,\\varepsilon,\\Gamma) &=& -\\frac{\\hbar}{2p} + \n\\frac{i\\; \\sqrt{2 \\pi}\\;\\hbar}{\\Gamma}\\;\n\\hbox{exp}\\left(-\\frac{p^2}{2\\Gamma^2}\\right)\\nonumber\\\\\n&+&\\frac{p s^2}{\\hbar}\\left\\{1 + \n\\mathcal{O}\\left[\\left(\\frac{ps}{\\hbar}\\right)^2\\right]\\right\\},\n\\end{eqnarray}\nwe are left with the only constant contribution from the lower\nlimit $s=0$, independent of $s$ and of\nthe Gaussian averaging parameter $\\Gamma_p$ satisfying the conditions \n(\\ref{avcondgp}),\n \\bel{sinttffin}\n{\\tt I}_{00}(s,\\varepsilon,\\Gamma) \\approx -\\hbar\/(2 p).\n\\end{equation}\nFinally, from (\\ref{thetaxnnpint00}) and (\\ref{sinttffin})\none obtains \n\\begin{eqnarray}\\label{TFrig}\n\\left\\langle\\Theta_x^{00}\\right\\rangle&=&\n\\frac{d_sm^2}{2\\pi^2 \\hbar^3}\\left\\langle\\int \\hbox{d} \\varepsilon \\;n(\\varepsilon)\n\\int \\hbox{d} {\\bf r} \\; r_{\\perp x}^2 \\; p({\\bf r})\\right\\rangle\\nonumber\\\\\n&=& d_s m \\int \\hbox{d} {\\bf r}\\; r_{\\perp x}^2\\;\\rho_{{}_{\\! {\\rm TF}}}({\\bf r})=\n\\Theta_{x,{\\rm TF}}^{\\rm (RB)}.\n\\end{eqnarray}\nWe used also the expression for the TF particle density through\n$G_0$ (\\ref{G0}),\n\\bel{tfpartden}\n\\rho_{{}_{\\! {\\rm TF}}}({\\bf r}) = \n-\\frac{1}{\\pi} {\\mbox {\\rm Im}} \\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\; G_0\\Big|_{s\\rightarrow 0}=\n\\frac{m}{2 \\pi^2\\hbar^3}\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\; p({\\bf r}).\n\\end{equation}\nSimilarly, using the Lorentzian weight function for the averaging in \n(\\ref{G0av}) instead of the Gaussian one,\n\\bel{G0avlor}\n\\left\\langle G_{0}\\right\\rangle_{\\Gamma} =\n\\frac{1}{\\pi}\\int_{-\\infty}^\\infty \\hbox{d} p'\\; \n\\frac{\\Gamma G_0(s,p')}{\\left(p'-p\\right)^2 + \\Gamma^2}\n=G_0(s,p+i\\Gamma),\n\\end{equation}\none obtains the same result (\\ref{TFrig}) independently of the choice of\nthe averaging function ($\\Gamma=\\Gamma_p$ in this Appendix E.2a). \nIn these derivations, we used\nthe residue technics for the analytical evaluations of the integrals,\nthat means formally the replace of such a local averaging by the shift of the\nmomentum, $p \\rightarrow p +i \\Gamma_p$ [see (\\ref{thetaxnnp}) \nat $\\nu=\\nu'=0$ and (\\ref{l2})],\n\\begin{eqnarray}\\label{theta00avlor}\n&&\\left\\langle \\Theta_x^{00} \\right\\rangle_{\\Gamma}\\approx\n\\frac{d_sm^2}{\\pi^2} {\\mbox {\\rm Im}} \\int \\hbox{d} {\\bf r}\\; \\left(y^2+z^2\\right)\n\\nonumber\\\\ \n&\\times&\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \n\\hbox{d} \\varepsilon \\int_{0}^{s_{\\rm max}} \\hbox{d} s\\;\n\\hbox{exp}\\left[ \\frac{2 i s \\left(p+i \\Gamma\\right)}{\\hbar}\\right]\n\\approx \\frac{d_s m \\hbar}{2 \\pi^2}\n\\nonumber\\\\\n&\\times& \\int \\hbox{d} {\\bf r} \\left(y^2+z^2\\right)\\;\\int_0^{p_{{}_{\\! {\\rm F}}}}\n\\hbox{d} p\\; p^2 \\left\\{1 - \\hbox{exp}\\left[-\\frac{2 \\Gamma s_{\\rm max}}{\\hbar}\\right]\\right.\n\\nonumber\\\\\n&\\times& \\left. \n\\left[\\frac{\\Gamma}{p}\\; \n\\hbox{sin}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right) +\n\\hbox{cos}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\\right]\\right\\}. \n\\end{eqnarray}\nAgain, according to (\\ref{avcondgp}), the second strongly \noscillating term of the integrand coming\nfrom the upper limit $s=s_{\\rm max}$ in the last\nline can be neglected as exponentially small, instead of \nthe Gaussian behavior above. Then, \nwe are left with the main first TF term [see (\\ref{TFrig})] \nindependent of $\\Gamma_p$, as in the case of the Gaussian averaging.  \n\n\\bigskip\n\\subsubsection{MI SHELL CORRECTIONS}\n\\label{mishcor}\n\nTo average the oscillating component $\\delta \\Theta_x^{01}$ of the sum \n(\\ref{thetaxsum})  (see \n(\\ref{thetaxnnp}) at $n=0$ and $n'=1$) over the phase space variables,\none may use the Green's function \n$\\left\\langle G_0\\right\\rangle_\\Gamma$ (\\ref{G0av}), locally averaged\nwith a Gaussian weight \ninstead of $G_0$, and similarly, instead of $G_1$ \\cite{strutmag},\n\\begin{eqnarray}\\label{G1av}\n&&\\left\\langle G_1\\right\\rangle_\\Gamma=\n\\frac{1}{\\Gamma \\sqrt{\\pi}} \\int \\hbox{d} \\varepsilon'\\; G_1\\left({\\bf r}_1,{\\bf r}_2,\\varepsilon'\\right)\\;\n\\hbox{exp}\\left[-\\frac{(\\varepsilon'-\\varepsilon)^2}{\\Gamma^2}\\right]=\\qquad\n\\nonumber\\\\\n&=&\n\\sum_{CT_1}\\mathcal{A}_{CT_1}\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{CT_1}\\! -\\! \\frac{i\\pi}{2}\\sigma_{{}_{\\! CT_1}}\n\\!- \\! i \\phi_d \n\\! -\\! \\frac{t_{{}_{\\! CT_1}}^2 \\Gamma^2}{4\\hbar^2}\\right].\\,\\,\\,\n\\end{eqnarray}\nTransforming also the integration variables ${\\bf r}_1$ and ${\\bf r}_2$\nto the canonical ones (\\ref{newcoord}) at zero temperature, one finds\n\\begin{eqnarray}\\label{dthetax01new}\n&&\\left\\langle\\delta \\Theta_x^{01}\\right\\rangle=-\\frac{d_s m}{\\pi^2 \\hbar^2}\n\\sum_{{\\rm CT}_1} {\\mbox {\\rm Im}}\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\int \\hbox{d} {\\bf r} \\int \\frac{\\hbox{d} {\\bf s}}{s} \\;\n\\ell_x\\left({\\bf r}-\\frac{{\\bf s}}{2}\\right)\n\\nonumber\\\\\n&\\times& \\ell_x\\left({\\bf r}+\\frac{{\\bf s}}{2}\\right)\n\\;\\hbox{cos}\\left[\\frac{s}{\\hbar} p({\\bf r})\\right]\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\times& \\mathcal{A}_{{\\rm CT}_1}\\left(\n{\\bf r}-\\frac{{\\bf s}}{2},{\\bf r}+\\frac{{\\bf s}}{2};\\varepsilon\\right)\\;\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CT}_1}\\left(\n{\\bf r}-\\frac{{\\bf s}}{2},{\\bf r}+\\frac{{\\bf s}}{2};\\varepsilon\\right)\\right. \n\\nonumber\\\\\n&-&\\left.\ni \\frac{\\pi}{2} \\sigma_{{}_{\\! CT_1}}- i \\phi_{\\rm d}\n- \\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right].\n\\end{eqnarray}\nWe shall put $\\Gamma$ and $\\Gamma_p$ to be zero in the final \nexpressions\nfor this average $\\left\\langle\\delta \\Theta_x^{01}\\right\\rangle$, as far as\n$\\Gamma$ is much smaller than the distance between gross shells \n$\\hbar \\Omega$ (\\ref{hom}) and $\\Gamma_p$ satisfies inequalities\n (\\ref{nllla}).\nExpanding then the action phase of the second exponent and its \npre-exponent factors \nin small ${\\bf s} p\/\\hbar$ up to first nonzero terms \n(i.e., \nup to the first and zeroth order ones, respectively), due to \nthe first sharp-peaked exponential Gaussian factor in the second line of \n(\\ref{dthetax01new}), \none applies (\\ref{l2}) and \n\\begin{eqnarray}\\label{ampactoc}\n\\mathcal{A}_{{\\rm CT}_1}\\left({\\bf r} - \\frac{{\\bf s}}{2},{\\bf r} + \\frac{{\\bf s}}{2}; \\varepsilon\\right)\n&\\approx& \\mathcal{A}_{{\\rm CCT}_1}\\left({\\bf r},{\\bf r}; \\varepsilon\\right),\\qquad\n\\nonumber\\\\\nS_{{\\rm CT}_1}\\left({\\bf r} - \\frac{{\\bf s}}{2},{\\bf r} + \\frac{{\\bf s}}{2}; \\varepsilon\\right)\n&\\approx& S_{{\\rm CCT}_1}\\left({\\bf r},{\\bf r}; \\varepsilon\\right) +{\\bf p} {\\bf s}.\\qquad\n\\end{eqnarray}\nWith these expansions in (\\ref{dthetax01new}), \nfor the integration over $d{\\bf s}=s^2 \\hbox{d} s\\; \\hbox{d} x_s\\;\\hbox{d} \\varphi_s$ \nin (\\ref{dthetax01new}), we use the \nsame spherical coordinate system ($s, \\theta_s, \\varphi_s$)\nwith the polar axis $z_s$ directed again along the momentum vector \n${\\bf p}({\\bf r})=({\\bf p}_1 +{\\bf p}_2)\/2$, \n$x_s=\\hbox{cos} \\theta_s$ (Fig.\\ \\ref{fig14}). The integral over \nthe azimuthal angle $\\varphi_s$ gives simply $2 \\pi$ due to the azimuthal\nsymmetry. The integration limits  over $x_s$ \ncan be considered as from -1 to 1 within the NLLLA (\\ref{nllla})\n(neglecting thus the dependence of limits \nfor the integration over angles $\\theta_s$ on $s_{\\rm max}$ \nand ${\\bf r}$),\none approximately finds from (\\ref{dthetax01new}) \n\\begin{eqnarray}\\label{dthetax01new1}\n\\left\\langle \\delta \\Theta_{x}^{01} \\right\\rangle\n&\\approx&  \\frac{2d_s m}{\\pi \\hbar}\\;\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} {\\bf r} \\;r_{\\perp x}^2\n\\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\mathcal{A}_{{\\rm CCT}_1}\n\\quad\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CCT}_1} - \\frac{i \\pi}{2}\\sigma_{{}_{\\! {{\\rm CCT}_1}}} - i \\phi_{d}\n\\right]\\;I_{01},  \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\\label{int01}\nI_{01}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\; s\\; \\int_{-1}^{1} \\hbox{d} x_s\\; \n\\hbox{cos}\\left[s p({\\bf r})\/\\hbar\\right]\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right]\\nonumber\\\\\n&\\times&\\hbox{exp}\\left[\\frac{i}{\\hbar} s p({\\bf r})\\; x_s\\right]\\;\n\\hbox{exp}\\left[-\\frac{t_{{}_{\\! CT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n& \\approx&  \\left[{\\tt I}_{00}\\left(s_{\\rm max},\\varepsilon,\n\\frac{\\Gamma_p}{\\sqrt{2}}\\right)\n-{\\tt I}_{00}\\left(0,\n\\varepsilon,\\frac{\\Gamma_p}{\\sqrt{2}}\\right)\\right]\n\\nonumber\\\\\n&\\times& \\hbox{exp}\\left[-\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\approx \\frac{\\hbar}{2p}\\;\n\\hbox{exp}\\left[-\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right].\n\\end{eqnarray}\nThe sum runs all of ${\\rm CCT}_1$s (closed ${\\rm CT}_1$s).\nTaking then the integral over the angle variable $x_s$ \nin the NLLLA (\\ref{nllla}), one then integrate\nover the\nmodulus $s$ within integration limits\nfrom $0$ to $s_{\\rm max}$.  Note that with the approximation \n$t_{{}_{\\! CT_1}} \\approx t_{{}_{\\! CCT_1}}$, due to $\\Gamma \\ll \\hbar \\Omega$\n(but significantly larger\nthan a distance between neighboring energy levels), this integral is \nreduced to $s=0$ and $s_{\\rm max}$ boundaries of \n${\\tt I}_{00}(s,\\varepsilon,\\Gamma_p\/\\sqrt{2})$ (\\ref{sinttf}), \nsee the third line in (\\ref{int01}).\nCalculating approximately the integral over $s$ as \nin the subsection E.2a of this Appendix, and using the \nsame asymptotics (\\ref{int00as})\nat large upper integration limit $s=s_{\\rm max}$ \nand\n(\\ref{int00as0}) at small lower one $s=0$, one obtains\nthe nonzero contribution only from the lower integration limit $s=0$\nas in the previous subsection E.2a.\nOther contributions of the upper limit $s_{\\rm max}$ can be neglected \nbecause the integrand over $s$ contains rapidly\noscillating functions, and after a local averaging in \nthe phase space variables (even with a small but finite Gaussian\naveraging parameter), they  exponentially disappear under the \ncondition (\\ref{avcondgp})\nfor $\\Gamma_p$ as in the calculations of \nthe Thomas-Fermi MI component\n(Appendix E.2a). Finally, by making use of (\\ref{int01}) \nin (\\ref{dthetax01new1}), \none obtains\n\\begin{eqnarray}\\label{dthetax01new2}\n&&\\left\\langle \\delta \\Theta_{x}^{01}\\right\\rangle \\approx \n-\\frac{d_s m}{\\pi}\n{\\mbox {\\rm Im}} \\sum_{{\\rm CCT}_1} \\int \\hbox{d} {\\bf r}\\; r_{\\perp x}^2 \\int_0^{\\varepsilon_{{}_{\\! {\\rm F}}}} \\hbox{d} \\varepsilon\\;\n\\mathcal{A}_{{\\rm CCT}_1}\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[\\frac{i}{\\hbar} S_{{\\rm CCT}_1} - \n\\frac{i \\pi}{2}\\sigma_{{}_{\\! {{\\rm CCT}_1}}} - i \\phi_{d}\n-\\frac{t_{{\\rm CCT}_1}^2\\Gamma^2}{4\\hbar^2}\\right]\n\\nonumber\\\\\n&=&d_s m\\int \\hbox{d} {\\bf r} \\;\\left(z^2+y^2\\right)\\; \\delta \\rho_{\\rm scl}({\\bf r})=\n\\delta \\Theta_{x,{\\rm scl}}^{\\rm (RB)}.\n\\end{eqnarray}\nIn these derivations, we used\n(\\ref{rperpcoord}) for \nthe perpendicular coordinate $r_{\\perp x}$,\nand (\\ref{dTxrigSCL}) for the oscillating shell component \n$\\delta \\Theta_{x\\; {\\rm scl}}^{\\rm RB}$ of the semiclassical MI \n(\\ref{rigmomsplit}). This component is \nrelated to the oscillating shell part $\\delta \\rho_{\\rm scl}({\\bf r})$\n[see (\\ref{ddenpart}) and (\\ref{Gct}) with a closed ${\\rm CT}_1$] \nin the semiclassical particle\ndensity (\\ref{denpartscl}). \nLike in the previous subsection of Appendix E.2, we obtain the \nsame result (\\ref{dthetax01new2}) \nby using the Lorentzian weight function for the local average\n($\\varepsilon \\rightarrow \\varepsilon+i \\Gamma$). Indeed, using its definition\n(\\ref{G0avlor}) for both Green function components $G_0$ and $G_1$,\nand performing the same integrations in the NLLLA (\\ref{nllla}),\none gets\n\\begin{eqnarray}\\label{dthetax01new2lor}\n&&\\langle \\delta \\Theta_x^{01}\\rangle \n=-\\frac{m d_s}{\\pi}\\int \\hbox{d} {\\bf r} \\;\\left(y^2 +z^2\\right){\\mbox {\\rm Im}}\n\\sum_{{\\rm CCT}_1}\\int \\hbox{d} \\varepsilon \n\\mathcal{A}_{{\\rm CCT}_1} \n\\nonumber\\\\\n&\\times&\\hbox{exp}\\left[\\frac{i}{\\hbar}\\; S_{{\\rm CCT}_1}\n-\\frac{i \\pi}{2}\\sigma_{{}_{\\! {\\rm CCT}_1}} -i \\phi_{\\rm d} -\n\\frac{\\Gamma t_{{}_{\\! {\\rm CCT}_1}}}{\\hbar}\\right]\n\\nonumber\\\\\n&\\times& \\left\\{1 + \\hbox{exp}\\left(-\\frac{\\Gamma_ps_{\\rm max}}{\\hbar}\\right)\n\\left[\\frac{\\Gamma_p}{2p}\\; \\hbox{sin}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\n\\right.\\right.\n\\nonumber\\\\\n&-&\\left.\\left.\\hbox{cos}\\left(\\frac{2 p s_{\\rm max}}{\\hbar}\\right)\\right]\\right\\}.\n\\end{eqnarray}\nAs transparently seen from this explicit expression,  \none has exponential disappearance of the oscillating contributions\non the upper integration limit $s_{\\rm max}$ \nunder the conditions (\\ref{avcondgp})\nfor $\\Gamma_p$, see the second term in figure brackets of the last two lines\nof (\\ref{dthetax01new2lor}). Therefore, \nthe first constant term \nin these brackets (coming from the lower integration limit $s=0$) \nyields immediately the finite $\\Gamma \\rightarrow 0$ rigid-body limit \n(\\ref{dthetax01new2}) for $\\langle \\delta \\Theta_x^{01}\\rangle$.  \n \n\nUsing analogous analytical calculations of the other terms \n$\\langle\\delta \\Theta_{x}^{10}\\rangle$ and \n$\\langle\\delta \\Theta_{x}^{11}\\rangle$ [see (\\ref{thetaxnnp})], \none finds the essentially different integrals over $s$, such as\n\\begin{eqnarray}\\label{int10} \nI_{10}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\n\\hbox{sin}^2 (ps\/\\hbar)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4  \\hbar^2}-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\approx& \\hbox{exp}\\left[-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\\;\n\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\n\\left[1-\\hbox{cos}\\left(\\frac{2ps}{\\hbar}\\right)\\right]\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right],\n\\end{eqnarray}\n and \n\\begin{eqnarray}\\label{int11} \nI_{11}&=&\\int_0^{s_{\\rm max}} \\hbox{d} s\\; s\\; \\hbox{sin}(2 p s\/\\hbar)\\;\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}-\n\\frac{t_{{\\rm CT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\n\\nonumber\\\\\n&\\approx& \\hbox{exp}\\left[-\n\\frac{t_{{\\rm CCT}_1}^2\\Gamma^2}{4 \\hbar^2}\\right]\\;\n\\int_0^{s_{\\rm max}} \\hbox{d} s\\;\ns\\;\\hbox{sin}\\left(\\frac{2ps}{\\hbar}\\right)\n\\nonumber\\\\\n&\\times&\n\\hbox{exp}\\left[-\\frac{s^2 \\Gamma_p^2}{4 \\hbar^2}\\right],\n\\end{eqnarray}\nrespectively.   \nIntegrating analytically in (\\ref{int10}) and (\\ref{int11}),\none can see that any contributions coming from the upper limit \n$s_{\\rm max}$ exponentially disappear as shown in (\\ref{int00as}) \n(with the formal replace $\\Gamma_p$ by $\\Gamma_p\/\\sqrt{2}$) as above.\nHowever, in contrast to the calculations \nof $\\left\\langle \\Theta_x^{00}\\right\\rangle$ \nand $\\left\\langle \\Theta_x^{01}\\right\\rangle$, the\n contributions from the lower \nintegration limit\n$s=0$ turn into zero too, according to the asymptotics at the 4th order in \ndistance $s$ in units of the wave-length $\\hbar\/p$:\n\\begin{eqnarray}\\label{intas1011}\nI_{10} &=& \\frac{2 p^2 s^3}{3 \\hbar^2} \\hbox{exp}\\left[-\n\\frac{t_{{}_{\\! CCT_1}}^2\\Gamma^2}{4 \\hbar^2}\\right]\\left[1 +\n\\mathcal{O}\\left(\\frac{ps}{\\hbar}\\right)\\right],\\qquad\n\\nonumber\\\\\nI_{11} &=& \\frac{2 p s^3}{3 \\hbar} \\hbox{exp}\\left[-\n\\frac{t_{{}_{\\! CCT_1}}^2 \\Gamma^2}{4 \\hbar^2}\\right]\\left[1 +\n\\mathcal{O}\\left(\\frac{ps}{\\hbar}\\right)\\right].\\qquad \n\\end{eqnarray}\nTherefore, \nin addition to (\\ref{dthetax01new2}), independently of the\nweight function for averaging, the two components associated with \nintegrals (\\ref{int10}) and (\\ref{int11})\ndo not contribute at both integration limits within \nthe NLLLA, as explained above. Thus, for  \nall nonzero terms of the oscillating part of the MI,\n$\\left\\langle \\delta \\Theta_x\\right\\rangle$,\none finally approximately arrives\nat the same rigid-body MI shell component \nin the NLLLA (\\ref{nllla}). \nThis result does not depend on the choice of the weight (Gaussian\nand Lorentzian) functions for the local averaging over the phase space.\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe spectroscopic observations of the \\textit{Infrared Space Observatory}\n(ISO) (Kessler et al. \\cite{kes96}) opened a new window for the study of the physical and\nchemical properties of IR-bright, ultraluminous infrared galaxies\n(ULIRG) and active galactic nuclei (AGN). Most of the pioneering work on the mid-to-far infrared spectra of active and starburst galaxies is derived from ISO spectroscopy (Sturm {\\em et al.\\\/} \\cite{stu02}; Spinoglio {\\em et al.\\\/} \\cite{spi05}; Verma {\\em et al.\\\/} \\cite{ver03}; Verma {\\em et al.\\\/} \\cite{ver05}).\n\n\\begin{figure}\n\\includegraphics[width=11cm]{fig1.eps}\n\\caption{Far infrared spectra observed with the LWS onboard of ISO of a sample of ultraluminous infrared galaxies (Fischer {\\em et al.\\\/} \\cite{Fis99})}\n\\end{figure}\n\nThe mid-IR spectral range includes most of the fine-structure lines excited by the hard radiation produced\nby black hole accretion as well as those mainly excited by stellar\nionization (Spinoglio \\& Malkan \\cite{sm92}) and thus represents an essential tool\nto distinguish between the two processes, especially in obscured\nnuclei suffering severe dust extinction. With the advent of the \\textit{Spitzer Space Telescope}, \nwith its powerful mid infrared instrument, the IRS (Houck {\\em et al.\\\/} \\cite{hou04}), systematic spectroscopic \nstudies of samples of galaxies have stated to appear (Dale {\\em et al.\\\/} \\cite{dal06}; Higdon {\\em et al.\\\/} \\cite{hig06}; Brandl {\\em et al.\\\/} \\cite{bra06}; Armus {\\em et al.\\\/} \\cite{arm07}; Farrah {\\em et al.\\\/} \\cite{far07}; Buchanan {\\em et al.\\\/} \\cite{buc06}; Tommasin {\\em et al.\\\/} \\cite{tom07}).\n\nIn a complementary way, the far-IR range contains a large variety of molecular (OH,\nH$_{2}$O, high-J CO) and low excitation ionic\/atomic transitions,\nin emission or in absorption, that can reveal the geometry and\nmorphology of the circumnuclear and nuclear regions in galaxies.\nIn particular far-infrared molecular lines could trace the expected conditions of\nX-UV illuminated dusty tori predicted from the unified models (Antonucci \\cite{ant93} and\nwhose presence in type 2 active galaxies is\nforeseen to reconcile the type1\/type2 dichotomy.\n \n \\begin{figure}\n\\includegraphics[width=12cm]{fig2.eps}\n\\caption{Fine-structure lines in the 3-200 $\\mu$m range. \nThe lines are plotted  as a function of their\nionization potential and critical density. Different symbols are\nused for lines from photodissociation regions (squares),\nstellar\/HII region lines (triangles), AGN lines (circles) and\ncoronal lines (stars). The two [OI] lines have been plotted - for\ngraphical reasons - at a ionization potential higher than their\neffective value.}\\end{figure}\n\nThe far-IR spectra of local IR-bright and ULIRG galaxies, as measured by\nISO-LWS (Fischer {\\em et al.\\\/} \\cite{Fis99}), showed an unexpected sequence of features,\nas can be seen in Fig. 1, from strong [OIII]52, 88 $\\mu$m and\n[NIII]57 $\\mu$m line emission to detection of only faint\n[CII]157$\\mu$m line emission and [OI]63 $\\mu$m in absorption. The\n[CII]157 $\\mu$m line in 15 ULIRGs (L$_{IR}\\geq10^{12}L_{\\odot}$)\nrevealed an order of magnitude deficit compared to normal and\nstarburst galaxies relative to the FIR continuum. Non-PDR\ncomponents, such as dust-bounded photoionization regions,\ngenerating much of the FIR continuum but not contributing\nsignificant [CII] emission, can explain the [CII] deficiency. Such\nenvironments may also explain the suppression of FIR\nfine-structure emission from ionized gas and PAHs, and the warmer\nFIR colors of ULIRGs. (Luhman {\\em et al.\\\/} \\cite{Lum03}).\n\n\nLWS observations of Arp 220 show absorption in molecular lines of OH, H$_2$O, CH,\nNH, and NH$_3$, as well as in the [OI]63$\\mu$m line and faint\nemission in the [CII]158$\\mu$m line. The molecular absorption in\nthe nuclear region is characterized by high\nexcitation due to high infrared radiation density (Gonz\\'alez-Alfonso {\\em et al.\\\/} \\cite{go04}).\nNotably, the LWS spectrum of the prototype Seyfert 2 galaxy NGC\n1068, beside the expected ionic fine structure emission lines,\nshows the 79, 119 and 163$\\mu$m OH rotational lines in emission,\nnot in absorption as in every other galaxy yet observed.\nModeling the three FIR lines of OH suggests the gas lies in small\n(0.1pc) and dense clouds ($\\sim 10^4 {\\rm cm^{-3}}$) in\nthe nuclear region (potentially a signature of the torus) with a minor contribution from\nthe circumnuclear starburst ring at 3kpc (Spinoglio {\\em et al.\\\/} \\cite{spi05}).\n\n\\section{Fine-structure emission lines}\n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig3.eps}\n\\caption{[NeVI]7.6$\\mu$m\/[OIV]26$\\mu$m ratio as a function of the \n[NeV]14.3$\\mu$m\/24.3$\\mu$m ratio. The line rations of Seyfert 1's (NGC1365, NGC4151, Tol0109), Seyfert 2's (MKN3, CenA, Circinus, NGC1068, PKS2048) and NLXR galaxies (NGC5503, NGC7582) are presented, from ISO SWS observations (Spinoglio {\\em et al.\\\/} \\cite{spi00}; Sturm {\\em et al.\\\/} \\cite{stu02}). }\n\n\\end{figure}\n\nMid-IR and far-IR spectroscopy of fine-structure emission lines\nare powerful tools to understand the physical conditions in\ngalaxies from the local universe to distant cosmological objects.\nFig. 2 shows the critical density (i.e. the density for which the\nrates of collisional and radiative de-excitation are equal) of\neach line as a function of the ionization potential of its ionic\nspecies. This diagram shows how these lines can measure\ntwo fundamental physical quantities (density and ionization) of\nthe gas. Lines from different astrophysical emission regions in\ngalaxies are shown in the figure with different symbols. The ratio\nof two lines with similar critical density, but different\nionization potential,  gives a good estimate of the ionization,\nwhile the ratio of lines with similar ionization potential, but\nwith diffrerent critical density, can measure the density of the\ngas in the region (see, e.g., Spinoglio \\& Malkan \\cite{sm92}).\nFrom Fig. 2 it is clear that infrared spectroscopy has a thorough\ndiagnostic power for gas with densities from about 10$^2$ cm$^{-3}$ \nto as high as 10$^8$ cm$^{-3}$ and ionization potentials up to 350 eV, \nusing, to trace these extreme conditions, the so called coronal lines. \nMoreover, increasing the wavelength \nof the transition used, the spectroscopic diagnostics become more and \nmore insensitive to dust extinction, and can therefore probe regions \nhighly obscured at optical or even near-to-mid infrared wavelengths.\n\n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig4.eps}\n\\caption{[CII]157$\\mu$m\/[OI]63$\\mu$m ratio as a function of the \n[OIII]88$\\mu$m\/[OI]63$\\mu$m ratio. The grid represents starburst photoionization models computed using the CLOUDY code. At the right bottom are shown the gas density values as derived from AGN photoionization models with log U = -2.5, while at the left are given the densities derived from photodissociation region models  (Spinoglio {\\em et al.\\\/} \\cite{spi03}).}\n\n\\end{figure}\n\nTo give an example of the diagnostic power of infrared spectroscopy, we show in Fig. 3 \na diagram showing lines excited only in the highly energetic environments of AGN and not from stellar ionization. In this diagram the [NeVI]7.6$\\mu$m\/[OIV]26$\\mu$m ratio is shown as a function of the \n[NeV]14.3$\\mu$m\/24.3$\\mu$m ratio (Spinoglio {\\em et al.\\\/} \\cite{spi00}). The photoionization models using the CLOUDY code (Ferland \\cite{fer00}) have been computed and shown as a grid in the diagram. As expected, the former ratio is sensitive to  ionization, \nwhile the latter is sensitive to density. In the figure are also shown measurements on a small sample of Seyfert galaxies for which we can determine the ionization potential (basically the ratio of ionizing photons over the number of hydrogen atoms) and the gas density. We note that the observed galaxies have average densities ranging from  less than 10$^2$ cm$^{-3}$ to less than 10$^4$ cm$^{-3}$ and ionization potential of \n10$^{-2.0}$ $<$ log~U $<$ 10$^{-1.5}$. This is in agreement, for the lower density objects,  with conditions of \"coronal emission regions\" in AGNs  (Spinoglio \\& Malkan \\cite{sm92}).\n\nAnother example, presented in Fig. 4,  is given by the [CII]157$\\mu$m\/[OI]63$\\mu$m ratio as a function of the \n[OIII]88$\\mu$m\/[OI]63$\\mu$m ratio. These low ionization lines are copiously produced in the ISM of galaxies (in photodissociation regions) and the [OIII] line is excited also in HII regions. However we can see from this diagram that while normal galaxies are clustering in a central region that can easily be explained by starburst models, most of the Seyfert galaxies are far from this locus and their rations cannot be reproduced by starburst models. They have much stronger emission of [OI]63$\\mu$m than it would be expected from stellar emission only. To obtain better statistics on this diagram, we will have to wait for the \\textit{Herschel} satellite, that will have the sensitivity to collect far-infrared spectroscopic observations of large samples of local galaxies.\n\n \n\\section{From the local to the distant Universe} \n\n\\begin{figure}\n\\includegraphics[width=12cm]{fig5.eps}\n\\caption{Same as fig.1, but for differerent redshift intervals.\n Different symbols are used for lines from photodissociation regions (squares),\nstellar\/HII region lines (triangles), AGN lines (circles) and\ncoronal lines (stars). The two [OI] lines have been plotted - for\ngraphical reasons - at a ionization potential higher than their\neffective value.}\\end{figure}\n\nMid-IR and far-IR spectroscopy of fine-structure emission lines\ncan be used not only in the local universe but also to measure \nthe excitation conditions in distant cosmological objects.\nFig. 5 shows again, as in Fig.2,  the critical density of the lines as a function \nof their ionization potential, not only for the local universe, but \nin three different redshift ranges, one for each frame,\nfor which the rest-frame wavelength of the line is shifted in the\nfar-infrared range. \nIt appears from the figure that although the photodissociation (PDR) \nregime can be probed only in the relatively local universe, because of the long \nwavelengths of the lines tracing this regime, however, the\nstellar emission (e.g. in starburst galaxies) can be probed up to\nhigh z, using many lines in the rest-frame spectral range of 3\n$\\leq$ $\\lambda (\\mu m)$ $\\leq$ 30. Moreover, the ionization from AGN can be\nprobed from the local universe up to redshifs of z$\\leq$5 and the extremely high \nexcitation coronal emission regions are probed by near-IR lines shifted into the\nfar-IR at a redshift of z $\\sim$ 5.\n\nOnce we have proved that we have in the mid-to-far infrared the adequate diagnostic lines, we need\nstill to understand if these lines could be detected by the future space facilities under development or study \nin the future years. Do do so, we will use the observed infrared spectra in local galaxies and let them evolve backwards at earlier cosmological times.\n\n \n\\begin{figure}\n\\includegraphics[width=12cm]{fig6.eps}\n\\caption{Predicted line fluxes as a function of redshift, using as a \nlocal template the prototypical Seyfert type 2 galaxy NGC1068. \nSquares indicate the fluxes of HII region lines, triangles the fluxes \nof lines emitted by AGN, filled circles the fluxes of the \n[OI]$\\mu$m line and open circles the three OH lines \ndetected in emission in NGC1068. The solid lines show how \nthe line fluxes change with redshift adopting luminosity evolution, \nwhile dotted lines without any evolution.}\n\\end{figure}\n\nFor simplicity, and to cover as many transitions possible, we use a \nlocal object which contains both an active nucleus and a starburst, \nthe bright prototype Seyfert 2 galaxy NGC1068.\nThe ISO spectrometers detected in this galaxy many of the lines plotted in Fig. 2 at flux levels\nof 5-200 $\\times$ 10$^{-16}$ W m$^{-2}$ (Alexander {\\em et al.\\\/} \\cite{Ale00}; Spinoglio et al. {\\em et al.\\\/} \\cite{spi05}). \nConsidering this galaxy as a template object, we computed the line\nintensities expected at redshifts ranging from 0.1 to 5. For\nsimplicity, we adopted an Einstein-De Sitter model Universe, with\n$\\Omega_{\\Lambda}$ = $\\Omega_{vac}=0$ and $\\Omega_{M}$= 1,\nH$_{0}$=75 km s$^{-1}$ Mpc$^{-1}$. The luminosity distances have been derived using:\n\\begin{equation}\nd_{L} (z)= (2c\/H_{0})\\cdot [1+z - (1+z)^{1\/2}]\n\\end{equation}\n\nThe results are reported in Fig. 6, where the line intensities are given in W m$^{-2}$, and the expected sensitivities of spectrometers, such ESI (European SPICA Instrument, Swinyard \\cite{swi06}), onboard of the future space observatories, SPICA (Space Infrared Telescope for Cosmology and\nAstrophysics, ISAS-JAXA) (Nakagawa \\cite{Nag04}; Onaka \\& Nakagawa \\cite{ON05}) and FIRI (Far-Infrared Interferometer) are also shown.\n\nWe have assumed that the line luminosities scale as the bolometric luminosity and we have chosen two cases: \\\\\nA) a luminosity evolution proportional to the (z+1)$^{2}$, consistent with the {\\it Spitzer}\nresults at least up to redshift z=2 (P\\'{e}rez-Gonz\\'{a}lez {\\em et al.\\\/} \\cite{Per05}) ;\\\\\nB) no luminosity evolution.\\\\\n\nBecause the star formation process in galaxies was much more enhanced at z=1-2 than today, we consider reasonable to adopt the model with strong evolution at least for the stellar\/HII region lines and, to be conservative, the \"no\nevolution\" one  for the AGN lines.\\\\\nWe note that the dependence on different cosmological models is not very strong.\nThe popular model with $\\Omega_{M}$= 0.27, $\\Omega_{vac}$=0.73,\nH$_{0}$=71 km s$^{-1}$ Mpc$^{-1}$ shows greater dilutions,\nincreasing with z, by factors of 1.5 for z=0.5 to 2.5 for z=5. In\nthis case the line intensities of Fig. 6 would decrease by these factors.\n \nWe conclude that a relatively low luminosity object like NGC1068, with an infrared luminosity of \n2 $\\times$ 10$^{11}$ L$_{\\odot}$, \nwill be detected up to a redshift of z=5 by the cooled 3.5m mirror of the SPICA satellite in a few bright and important diagnostic lines, such as the [OI]63$\\mu$m, the   [OIII]52$\\mu$m,  [OIV]26$\\mu$m, assuming luminosity evolution in the lines.\n\nThe fainter AGN lines (like [NeVI]7.6$\\mu$m) and the molecular lines of OH, will be detected by SPICA in such an object at z $\\sim$ 0.5. For detecting the fainter lines up to z $\\sim$ 5, we will have to wait for larger collecting area space telescopes, as the FIRI project, foreseen beyond the next decade.\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Maximizing $p$-Mean Welfare}\n\nAddressing fair-division instances with identical subadditive valuations, this section presents an efficient algorithm for computing a constant-factor approximation to the $p$-mean welfare objective, uniformly for all $p \\in (-\\infty, 1]$. \n\nThe algorithm consists of two phases, Algorithm~\\ref{Alg} (\\textsc{Alg}) and Algorithm~\\ref{AlgSub} (\\textsc{AlgLow}). In the first phase, ``high-value'' goods are assigned as singletons--we use the approximation algorithm of Feige~\\cite{feige2009maximizing} to obtain an estimate of the optimal  $1$-mean welfare and deem a good to be of high value if its valuation is at least a constant (specifically, $\\nicefrac{1}{3.53}$) times this estimate. Intuitively, the estimate provides a useful benchmark, since the optimal $1$-mean welfare upper bounds the optimal $p$-mean welfare for all $p \\in (-\\infty, 1]$ (Proposition~\\ref{Monotonicity}); this bound essentially follows from the generalized mean inequality~\\cite{bullen1988mathematics} which asserts that, for all $p \\in (-\\infty, 1]$, the $p$-mean welfare of any allocation $\\mathcal{A}$ is at most its $1$-mean welfare, ${\\textrm M}_p(\\mathcal{A}) \\leq {\\textrm M}_1(\\mathcal{A})$. \n\nTherefore, in phase one of the algorithm, we sort the goods in non-increasing order by value and iteratively select goods, which by themselves provide a value comparable to that of the optimal $p$-mean welfare. In each iteration, the selected good is assigned as a singleton to an agent and this agent-good pair is removed from consideration. Note that such an update leads to a new fair-division instance with one less good and one less agent, as well as a potentially different optimal $1$-mean welfare. The key technical issue here is that the change in the optimal $1$-mean welfare (and, hence, its estimate obtained via Feige's  algorithm) can be non-monotonic. Nonetheless, via an inductive argument, we show that the welfare contribution of the goods assigned (as singletons) in the first phase is sufficiently large (Lemma~\\ref{Induction_argument}). \n\nThe first phase terminates when we obtain an instance $\\mathcal{J}$ wherein each good is of value no more than a constant times its optimal $1$-mean welfare. The second phase (\\textsc{AlgLow}) is designed to address such a fair-division instance. In particular, we show that, in the absence of high-value goods, we can efficiently find an allocation $\\mathcal{B}=(B_i)_i$ such that each bundle $B_i$ is of value at least constant times the optimal $p$-mean welfare of $\\mathcal{J}$. To obtain the allocation $\\mathcal{B}$, we first compute (via Feige's approximation algorithm) an allocation $\\mathcal{S}=(S_j)_j$ that provides a $2$-approximation to the optimal $1$-mean welfare of $\\mathcal{J}$. Subsequently, we show that the subsets $S_j$s, that have appropriately high value, can be  partitioned to form the desired bundles $B_i$s, which constitute the allocation $\\mathcal{B}$. \n\nMultiple technical lemmas (in Sections~\\ref{Supporting_Lemmas} and~\\ref{section:stitching-lemma}) are required to show that  the two phases in combination lead to the desired $p$-welfare bound. It is also relevant to note that, while the above-mentioned ideas hold at a high level, the formal guarantees are obtained by separately analyzing different ranges of the exponent parameter $p$.\n\n\n\n\n\\floatname{algorithm}{Algorithm}\n\\begin{algorithm}[ht]\n  \\caption{\\textsc{Alg}} \\label{Alg\n  \\textbf{Input:} A fair-division instance $\\mathcal{I}= \\langle [m],[n],v \\rangle$ with demand oracle access to the subadditive valuation function $v$. \\\\\n  \\textbf{Output:} An allocation $\\mathcal{A} = (A_1, A_2, \\ldots, A_n)$ \n  \\begin{algorithmic}[1]\n        \\STATE Initialize the set of agents ${U}=[n]$, the set of goods $G=[m]$, and bundle $A_i= \\emptyset $ for all $i\\in {U}$\n        \\STATE Index all the goods in non-increasing order of value $v(g_1) \\geq v(g_2) \\geq  \\ldots \\geq v(g_m)$\\label{Ordered_Goods}\n        \\STATE Set $\\mathcal{I}^0= \\I{G}{{U}}$ and initialize $t=1$    \\COMMENT{Recall that ${\\rm F}(\\mathcal{I}) = {\\rm M}_1 (\\mathcal{S})$, where $\\mathcal{S}$ denotes the allocation obtained by executing Feige's algorithm~\\cite{feige2009maximizing} on instance $\\mathcal{I}$.}\n        \\WHILE {  $v(g_t) \\geq \\frac{1}{3.53} \\ {\\rm F}(\\mathcal{I}^{t-1})$ } \\label{Threshold}\n         \\STATE Allocate $A_t \\leftarrow  \\{ g_t \\}$ and update $G \\leftarrow G\\setminus \\{g_t\\}$ along with ${U} \\leftarrow {U} \\setminus \\{ t \\}$\n                   \\STATE Set $\\mathcal{I}^{t}=\\I{G}{{U}}$ and update $t \\leftarrow t+1$\n        \\ENDWHILE \\label{step:end-while}\n        \\STATE Set $(A_{t}, A_{t+1},\\ldots, A_n) = \\textsc{AlgLow}(G, {U} ,v)$ \\COMMENT{This step corresponds to the second phase of the algorithm which assigns bundles to the remaining $| {U}| = n-t +1$ agents. Also, note that, in the current instance $\\mathcal{J} \\coloneqq \\langle G, {U}, v \\rangle$, for every good $g \\in G$ we have $v(g) < \\frac{1}{3.53} {\\rm F} (\\mathcal{J})$.}\n        \\RETURN allocation $\\mathcal{A} = (A_1,A_2,...,A_n).$\n     \n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\floatname{algorithm}{Algorithm}\n\\begin{algorithm}[h]\n  \\caption{\\textsc{AlgLow} } \\label{AlgSub}\n   \\textbf{Input:} A fair-division instance $\\mathcal{J} = \\langle G, U,v \\rangle$ with demand oracle access to the subadditive valuation function $v$. \\\\\\textbf{Output:} An allocation $\\mathcal{B} = (B_1, B_2, \\ldots, B_{|U|})$ \n  \\footnotesize\n  \\begin{algorithmic} [1]\n        \\STATE Execute Feige's approximation algorithm~\\cite{feige2009maximizing} on the given instance $\\mathcal{J}$ to compute allocation $\\mathcal{S} = (S_1, S_2, \\ldots, S_{|U|})$.  \n        \\COMMENT{Note that allocation $\\mathcal{S}$ provides a $2$-approximation to the optimal $1$-mean welfare of $\\mathcal{J}$, ${\\rm M}_1 (\\mathcal{S}) = {\\rm F}(\\mathcal{J}) \\geq \\frac{1}{2} {\\rm M}_1 \\left(\\mathcal{A}^*(\\mathcal{J}, 1) \\right)$}\n        \\STATE Index the bundles such that $v(S_1) \\geq \\ldots \\geq v(S_{|U|})$ and initialize $i=a = 1$ along with $B_\\ell = \\emptyset$ for $1 \\leq \\ell \\leq |U|$ \\\\ \\COMMENT{Lemma~\\ref{Low_valued} shows that the following loop runs to completion} \n         \\WHILE {agent index $a < |U|$}\n         \\STATE Consider an arbitrary good $g \\in S_i$   \n         \\IF {$v( B_a \\cup \\{ g\\}) < \\frac{1}{3} {\\rm F} (\\mathcal{J}) $ } \n            \\STATE Update $B_a \\leftarrow B_a \\cup \\{ g \\}$ and $S_i \\leftarrow S_i \\setminus \\{ g\\}$ \\COMMENT{Here good $g$ is assigned to bundle $B_a$ to  increase its value}\n        \\ELSE\n        \\STATE Update $a \\leftarrow a + 1$ \\COMMENT{This update is performed when sufficient value has been accumulated in a bundle}\n        \\ENDIF\n        \\IF{$v(S_i) < \\frac{1}{3} {\\rm F} (\\mathcal{J}) $}\n        \\STATE Update $i \\leftarrow i + 1$ \\COMMENT{Once the value of $S_i$ drops below $\\frac{1}{3} {\\rm F}(\\mathcal{J})$ we consider the next bundle in $\\mathcal{S}$}\n        \\ENDIF\n\\ENDWHILE\n\\STATE $B_{|U|} \\leftarrow B_{|U|} \\cup \\left( G \\setminus (\\bigcup \\limits _{a=1}^{|U|-1} B_a )\\right)$ \\COMMENT{Assign the remaining elements to $B_{|U|}$} \n   \n      \n       \n       \n        \n        \\RETURN partition $\\mathcal{B} = (B_1,\\ldots ,B_{|U|}).$\n     \n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\nThe following theorem constitutes the main result of the current work. It asserts that Algorithm~\\ref{Alg} (\\textsc{Alg}) achieves a constant-factor  approximation ratio for the $p$-mean welfare maximization problem.\n\n\n\\newtheorem{thm}{Theorem}[section]\n\\newcommand{Main Theorem }{Main Theorem }\n\n\\newtheorem*{genericthm*}{Main Theorem }\n\\newenvironment{namedthm*}[1]\n  {\\renewcommand{Main Theorem }{#1}%\n   \\begin{genericthm*}}\n  {\\end{genericthm*}}\n  \n\n  \n\\begin{theorem} [Main Result] \\label{MainTheorem}\nLet $\\mathcal{I} = \\langle [m], [n], v \\rangle$ be a fair-division instance wherein all the agents have an identical, subadditive valuation function $v$. Given  demand oracle access to $v$, \\textsc{Alg} computes in polynomial time an allocation $\\mathcal{A}$ that, for all $p \\in (-\\infty, 1]$, provides a $40$-approximation to the optimal $p$-mean welfare, i.e., ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} \\ {\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p))$, for all $p \\in (-\\infty, 1]$; here, $\\mathcal{A}^*(\\mathcal{I},p)$ is the $p$-optimal allocation in $\\mathcal{I}$. \n\\end{theorem}\n\n \n We first consider an instance $\\mathcal{J}$ wherein all the goods are of value a constant times less than ${\\rm F}(\\mathcal{J})$ and prove that, for such an instance, \\textsc{AlgLow} finds an allocation in which the value of every bundle is comparable to the optimal average social welfare of $\\mathcal{J}$. In Section \\ref{subsection:p-inf-half}, we use this fact and supporting lemmas from Sections~\\ref{Supporting_Lemmas} and~\\ref{section:stitching-lemma} to prove Theorem~\\ref{MainTheorem} for $p\\in (-\\infty,0.4)$. Finally, in Section \\ref{subsection:p-half-one}, we prove the main result for $p\\in [0.4,1].$ \n \nWe start with the following observation to upper bound the optimal $p$-mean welfare in terms of the optimal $1$-mean welfare. \n\\begin{Proposition}\\label{Monotonicity}\nLet $\\mathcal{I}$ be a fair-division instance in which all the agents have an identical, subadditive valuation $v$. Then, for each $p\\in(-\\infty,1],$ the optimal $1$-mean welfare is at least as large as the optimal $p$-mean welfare:\n\\begin{align*}\n{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{I},1)) \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)) \\  \\ \\text{for every } \\; p \\in \\left(- \\infty , 1\\right]\n\\end{align*}  \n\\end{Proposition}\n\\begin{proof}\nThe generalized mean inequality (see, e.g., ~\\cite{bullen1988mathematics}) applied to the allocation $\\mathcal{A}^{*}(\\mathcal{I},p)$, gives us ${\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)) \\leq {\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},p))$, for all $p \\in (-\\infty, 1]$. By definition, the allocation $\\mathcal{A}^*(\\mathcal{I}, 1)$ maximizes the $1$-mean welfare, ${\\rm M}_1(\\cdot)$, and, hence, the claim follows ${\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},1)) \\geq {\\rm M}_1 (\\mathcal{A}^{*}(\\mathcal{I},p))  \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$. \n\\end{proof}\n\n\n\\section{Approximation Guarantee for \\textsc{AlgLow}}\n\\label{Approx_for_AlgSub}\nThis section addresses the second phase of the algorithm (\\textsc{AlgLow}) that---by the processing performed in the while-loop of \\textsc{Alg}---solely needs to  consider fair-division instances $ \\mathcal{J}=\\I{G}{U}$ wherein all the goods $g \\in G$ satisfy $v(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})$, i.e., the goods are of ``low value.''  The following lemma establishes that, for such instances, \\textsc{AlgLow} finds bundles each with value comparable to the optimal $1$-mean welfare (and, hence, comparable to the optimal $p$-mean welfare) of $\\mathcal{J}$. \n\nRecall that here $G$ is a subset of the original set of goods $[m]$, $U$ is a subset of the $[n]$ agents, and ${\\rm F}(\\mathcal{J})$ denotes the $1$-mean welfare (average social welfare) of the allocation computed by Feige's approximation algorithm for instance $\\mathcal{J}$.\n\n\n\n \n\\begin{Lemma}\\label{Low_valued}\nLet $\\mathcal{J} = \\I{G}{U}$ be a fair-division instance in which all the agents have an identical subadditive valuation function $v$, and every good $g\\in G$ satisfies   \n$v(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})$. Then, in the demand oracle model, the algorithm \\textsc{AlgLow} efficiently computes an allocation  $\\mathcal{B}=(B_1,\\ldots , B_{|U|})$ with the property that, for all $i \\in \\{1, \\ldots, |U|\\}$,\n\\begin{align*}\nv(B_i)\\geq \\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1)) \\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p)).\n\\end{align*}\n\\end{Lemma}\n\\begin{proof}\nFor input instance $\\mathcal{J}$, Feige's algorithm returns an allocation $\\mathcal{S}=(S_1,S_2,\\ldots,S_{|U|})$ with near-optimal average social welfare:\n\\begin{align*}\n\\frac{1}{|U|}\\summi{1}{|U|}{S} & = {\\rm F}(\\mathcal{J}) \\geq \\frac{1}{2}{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))\n\\end{align*}\n\nGiven allocation $\\mathcal{S}=(S_1,S_2,\\ldots,S_{|U|})$, we show that one can partition $S_i$s to form $|U|$ bundles such that the value of each  bundle is at least $\\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))$. Note that, by the assumption in the lemma, for each $g \\in G$ we have \n\\begin{align}\\label{2}\nv(g)\\leq \\frac{1}{3.53}{\\rm F}(\\mathcal{J})\n\\end{align} \n\nLet $H \\coloneqq \\left\\{ i \\in \\{1, 2, \\ldots, |U|\\} \\mid v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\right\\}$ denote the subsets in $\\mathcal{S}$ with value at least $\\frac{1}{3} {\\rm F}(\\mathcal{J})$. These are the subsets we split to form bundles $B_i's$ that form the output allocation $\\mathcal{B}$.\n\nSince $v$ is subadditive, we have the following lower bound on the cumulative value of the subsets in $H$\n\\begin{align}\\label{3}\n\\sum \\limits _{i \\in H} v(S_i) \\geq \\sum \\limits _{j=1}^{|U|} v(S_j) - \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\cdot |U| = \\frac{2}{3}\\ |U|  \\cdot  {\\rm F}(\\mathcal{J})\n\\end{align}\n\\newtheorem{Claim}{Claim}\n\\begin{Claim}\nLet $\\mathcal{S} = (S_1,\\ldots,S_{|U|})$ be the allocation computed by Feige's algorithm for input instance $\\mathcal{J}.$ Then, every $S_i$, with the property that $v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J})$, can be partitioned into at least $k=\\left ( \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})}-1\\right )$ subsets $T^1_i,\\ldots , T^k_i$ such that $v(T^j_i)\\geq \\frac{1}{20}{\\rm F}(\\mathcal{J})$ for each $T^j_i$.\n\\end{Claim}\n\\begin{proof}\nInitialize $T_i^1$ to be the empty set. Then, we keep transferring goods---in any order and one at a time---from $S_i$ to $T_i^1$ till the value of $T_i^1$ goes over $\\frac{1}{3}{\\rm F}(\\mathcal{J}).$ Returning the last such good back into $S_i$, the populated set $T_i^1$ satisfies \n\\begin{align*}\nv(T_i^1) & \\geq \\frac{1}{3}{\\rm F}(\\mathcal{J}) - \\frac{1}{3.53}{\\rm F}(\\mathcal{J}) \\tag{using (\\ref{2}) and the subadditivity of $v$} \\nonumber \\\\ & \\geq \\frac{1}{20}{\\rm F}(\\mathcal{J}) \n\\end{align*}\nNote that, by construction, $v(T_i^1) \\leq \\frac{1}{3}{\\rm F}(\\mathcal{J})$. Therefore, using the subadditivity of $v$, we get $v(S_i\\setminus T_i^1) \\geq v(S_i)-\\frac{1}{3}{\\rm F}(\\mathcal{J})$.\n\nWe can repeat the above process to obtain subsets $T_i^1,\\ldots T_i^k$ and stop when $v\\left(S_i\\setminus  ( T_i^1 \\cup \\ldots \\cup T_i^k) \\right) \\leq \\frac{1}{3}{\\rm F}(\\mathcal{J}).$ Given that, for each $T_i^j$, we remove a subset of value atmost $\\frac{1}{3} {\\rm F}(\\mathcal{I})$ from $S_i$, the subadditivity of $v$ gives us $v \\left(S_i\\setminus ( T_i^1 \\cup \\ldots \\cup T_i^k) \\right) \\geq v(S_i) - \\frac{k{\\rm F}(\\mathcal{J})}{3}$. Hence, the following lower bound holds $k\\geq \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1$.\nIn other words, we can extract at least $\\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1$ bundles, each of value no less than $\\frac{1}{20}{\\rm F}(\\mathcal{J})$, from $S_i$.\n\\end{proof}\nWe now apply the same procedure to every subset in $H \\coloneqq \\left\\{ i \\in \\{1, 2, \\ldots, |U| \\} \\mid v(S_i) \\geq \\frac{1}{3} {\\rm F}(\\mathcal{J}) \\right\\}$ to obtain $k^{'} = \\sum \\limits _{i=1}^{|H|} \\left ( \\frac{3v(S_i)}{{\\rm F}(\\mathcal{J})} -1 \\right ) $ bundles, each of value at least $\\frac{1}{20}{\\rm F}(\\mathcal{J})$. Using this equation and inequality (\\ref{3}), we get\n\\begin{align*}\nk^{'} & \\geq \\frac{2|U|{\\rm F}(\\mathcal{J})}{{\\rm F}(\\mathcal{J})} - |H| \\geq 2|U| - |U|  \\tag{Since $|H|<|U|$} \\\\\n& = |U| \\hspace{3.3cm}\n\\end{align*}\n\nIn conclusion, one can construct at least $|U|$ bundles of value at least $\\frac{1}{20}{\\rm F}(\\mathcal{J}) \\geq \\frac{1}{40}{\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1))$. Note that this observation implies that \\textsc{AlgLow} successfully finds $|U|$ bundles each of value at least $\\frac{1}{40} {\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1))$. \n\nProposition~\\ref{Monotonicity} gives us ${\\rm M}_1(\\mathcal{A}^*(\\mathcal{J},1)) \\geq {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p))$ and, hence, the stated claim follows. \n\\end{proof}\n\n\\section{Proof of Numeric Inequality from Lemma \\ref{Good_Transfer}} \\label{app 3}\nThis section establishes the numeric inequalities used in Sections \\ref{subsection:p-infty-zero-good-transfer} and \\ref{subsection:p-zero-half-good-transfer}. Specifically, for $p\\in (-\\infty,0)$\n\\begin{align}\n \\left(  \\frac{1}{2} - \\frac{1}{40}\\right )^p + \\left ( \\frac{1}{2}\\right )^p &\\leq 1 + \\left ( \\frac{2}{11.33} \\right )^p \\label{ineq_neg}\n\\end{align}\nAlso, for $p\\in (0,0.4)$, we have \n\\begin{align}\n\\left(  \\frac{1}{2} - \\frac{1}{40}\\right )^p + \\left ( \\frac{1}{2}\\right )^p &\\geq 1 + \\left ( \\frac{2}{11.33} \\right )^p \\label{ineq_pos} \n\\end{align}\n\n\nWrite $a$ to denote $\\left(  \\frac{1}{2} - \\frac{1}{40}\\right ) $, $b$ to denote $\\left ( \\frac{1}{2}\\right )$ and $c$ to denote $\\left ( \\frac{2}{11.33} \\right ).$ Consider the following differentiable function $f(p)\\coloneqq a^p+b^p-1-c^p$. In order to obtain bounds (\\ref{ineq_neg}) and (\\ref{ineq_pos}), it suffices to prove that $f(p)>0$ for all $p\\in (0,0.4)$ and $f(p)\\leq 0$ for all $p\\in(-\\infty,0)$. We first show a useful property of the function $f(p)$ that will help us in deriving these inequalities.\n\\begin{center}\n    \\includegraphics[scale=0.3]{graph1.png}\n    \\captionof{figure}{Plot of the function  $f(p)$ }\n    \\label{fig:sample_figure}\n\\end{center}\n\n\n\n\\begin{Claim}\\label{extreme_points_are_maxima}\nEvery extreme point of the function $f(p)=a^p+b^p-1-c^p$ is a local maximum; here, $a=\\left(  \\frac{1}{2} - \\frac{1}{40}\\right ) $, $b=\\left ( \\frac{1}{2}\\right )$ and $c=\\left ( \\frac{2}{11.33} \\right ).$\n\\end{Claim}\n\n\\begin{proof}\nWe consider the first and second derivatives of the smooth function $f$:\n\\begin{align}\nf'(p)= a^p\\log  (a) + b^p\\log  (b) - c^p\\log  (c) \\label{$f'$}\\\\  \\text{and } f''(p) = a^p(\\log (a))^2 + b^p(\\log  (b))^2 - c^p(\\log  (c))^2  \\label{$f''$}\n\\end{align} \nLet $\\overline{p}$ be any point that satisfies $f'(\\overline{p})=0. $ That is, $c^{\\overline{p}}\\log (c)= a^{\\overline{p}}\\log (a) + b^{\\overline{p}}\\log  (b) $. Instantiating equation (\\ref{$f''$}) with $p=\\overline{p}$ and substituting the previous expression for $c^{\\overline{p}}\\log (c)$, we get \n\\begin{center}\n$f''(\\overline{p})=a^{\\overline{p}}\\log  (a)(\\log  (a)-\\log  (c))+b^{\\overline{p}}\\log  (b)(\\log (b)-\\log (c))$\n\\end{center}\nIn the above equation, note that the terms $(\\log  (a)-\\log  (c))$ and $(\\log  (b)-\\log  (c))$ are positive, since $\\frac{a}{c} >1$ and $\\frac{b}{c} >1$, while the terms $\\log (a)$ and $\\log (b)$ are negative since $a,b<1.$ Therefore, $f''(\\overline{p})<0$, which means $\\overline{p}$ is a local maximum. We have now shown that every extreme point of $f$ is a local maximum.\n\n\\end{proof}\nNote that $f(0.4)>0$ and $f(0.41)<0$, hence, the intermediate value theorem implies that there exists a point $r \\in (0.4, 0.41)$ such that $f(r) = 0$. We will show that $f$ is nonnegative over the interval $[0,r]$ and, hence, $f(p) \\geq 0$ for all $p \\in (0, 0.4)$. \n\nGiven that $f(0)=0$ and $f(r) = 0$, we can apply the mean value theorem to $f$ in the range $[0,r]$ to conclude that there is a point $p_0 \\in [0,r]$ satisfying $f'(p_0)=0$. Using Claim \\ref{extreme_points_are_maxima}, we will next show that $f(p)$ has a unique extreme point, which is a local maximum, in the range $[0,r]$. \n\n\\begin{Claim}\\label{unique_maximum}\nLet $p^*  \\in [0,r]$ be a point that satisfies $f'(p^*)=0$, then it is the unique extreme point of $f(\\cdot)$ and is a local maximum.\n\\end{Claim}\n\\begin{proof}\nSince $f'(p^*)=0$, $p^*$ is an extreme point. Additionally, from Claim \\ref{extreme_points_are_maxima}, $p^*$ is a local maximum. \n\nAssume, towards a contradiction, that there is another local maximum $p_1 \\in \\mathbb{R}$ and, without loss of generality, assume that $p_1>p^*.$ We will show that this implies the existence of a local minimum between $p^*$ and $p_1$, hence contradicting Claim \\ref{extreme_points_are_maxima}.\n\nConsider the point $p_{min}$ defined as $p_{min}=\\text{arginf} \\left\\{ \\ f(p) \\mid p \\in [p^*, p_1] \\right\\}$. Note that $p_{min}\\neq p^*,p_1$ because $p^*$ and $p_1$ are local maxima, which means their function values are greater than those of the points in an $\\varepsilon$-neighborhood around them. Therefore, applying Fermat's theorem for stationary points, we get $f'(p_{min})=0$. Since $p_{min}$ is also the infimum of $p\\in[p^*,p_1],$ it is a local minimum  between $p^*$ and $p_1.$ Therefore, by way of contradiction, we obtain the stated claim. \n\\end{proof}\n\\noindent \\textit{Proof of inequality (\\ref{ineq_pos}):} We now prove that $f(p)\\geq0$ for $p \\in [0, 0.4].$\nAssume, for the sake of contradiction, that there exists is a point $q \\in [0, 0.4]$ such that $f(q)<0$.\n\nThe facts that $f(0) = 0$, $f'(0) >0$ and $f(q)<0$, along with the mean value theorem, imply that there exists a point $\\widehat{p} \\in [0, q)$ such that $f'(\\widehat{p})=0$. Additionally, applying mean value theorem with inequalities $f(0.4)>0$, $f'(0.4)<0$, and $f(q)<0$, we get a different point $\\overline{p}\\in (q, 0.4]$ such that $f'(\\overline{p})=0$. Existence of two distinct extreme points contradicts Claim \\ref{unique_maximum} and, hence, establishes that $f(p)\\geq 0$ for all $p \\in [0,0.4]$. \\\\\n\n\\noindent \\textit{Proof of inequality (\\ref{ineq_neg}):} We split the proof of this inequality depending on the range of $p$:\\\\\n\n\\noindent\nCase 1: $p\\in (-\\infty,-1]$. Note that $a>\\frac{1}{2.2}$, $b=\\frac{1}{2}$, $c<\\frac{1}{5}$, and $p$ is negative.\nSubstituting these bounds for $a,b$ and $c$ in the expression for $f(p)$, we get $f(p)<\\left(\\frac{1}{2.2}\\right)^p+\\left(\\frac{1}{2}\\right)^p-1-\\left(\\frac{1}{5}\\right)^p.$  Equivalently, we have $f(p)<(2.2)^{|p|}+2^{|p|}-1-5^{|p|}.$  Since $5^{|p|}\\geq(2.2)^{|p|}+2^{|p|}$ for $p\\leq-1$, we conclude that $f(p)<0$ for $p\\in(-\\infty,-1].$\\\\\n\\noindent\nCase 2: $p\\in (-1,0)$. Assume, for the sake of contradiction, that there is a $\\widetilde{p} \\in(-1,0)$ such that $f(\\widetilde{p})>0$.\nNote that $f(-1) <0$. Hence, by applying intermediate value theorem to $-1$ and $\\widetilde{p}$, we have a point $\\widehat{p} \\in (-1, \\widetilde{p})$ such that $f(\\widehat{p})=0$. Additionally, mean value theorem with $0$ and $\\widehat{p}$ implies that there exists a point $\\overline{p}\\in [\\widehat{p},0)$ such that $f'(\\overline{p})=0$. Since this contradicts Claim \\ref{unique_maximum}, we have $f(p) \\leq 0$ for all $p\\in(-1,0)$.\n\nThese two cases imply that inequality (\\ref{ineq_neg}) holds for all $p\\in (-\\infty,0).$\n\n\n\\section{APX-Hardness of Maximizing $p$-Mean Welfare} \\label{APX_Hardness}\nIn this section, we prove that the problem of computing a $p$-optimal allocation is {\\rm APX}-hard, for all $p\\in (-\\infty,1]$, in the demand oracle model. \nThat is, we show that there exists a constant $c>1$ such that it is {\\rm NP}-hard to approximate the optimal $p$-mean welfare within a factor of $c$, even if we are given access to demand queries.\n\n\\begin{theorem}\\label{APX_hardness_theorem}\nGiven a fair division instance $\\mathcal{I}=\\I{[m]}{[n]}$, wherein the agents have identical, subadditive valuations, the $p$-mean welfare maximization problem is {\\rm APX}-Hard for all $p\\in (-\\infty,1]$, in the demand oracle model.\n \\end{theorem}\n\nWe prove this hardness result by developing a gap-preserving reduction from the Gap-3DM problem. An instance $\\mathcal{C}$ of this problem consists of three disjoint sets $X$, $Y$, and $Z$, of cardinality $q$ each, along with a collection of $3$-uniform hyperedges $E \\subset X \\times Y \\times Z$. \nThe goal is to find a matching (i.e., a subset of pairwise disjoint hyperedges) $M \\subset E$ of maximum cardinality. \n\nFormally, Gap-3DM is the gap version of $3$-dimensional matching and it entails distinguishing between the following types of instances: \\\\\n(i) {\\rm YES}-instance: There is a perfect matching (a matching of size $q$) in the given instance $\\mathcal{C}$.\\\\\n(ii) {\\rm NO}-instance: All matchings in $\\mathcal{C}$ are of size at most $\\alpha q$, with $\\alpha <1$.\\\\\nThe Gap-3DM problem is known to be {\\rm NP}-hard, for an absolute constant $\\alpha <1$~\\cite{ostrovsky2014s}.\\footnote{Note that the given instances in this  problem are promised to be either {\\rm YES} or {\\rm NO} instances.} \n\nTheorem \\ref{APX_hardness_theorem} follows from a gap-preserving reduction. In particular, we will prove that in the {\\rm YES} case (i.e., when \nthe given instance of Gap-3DM has a perfect matching) the reduced instance of the welfare problem admits an allocation with $p$-mean welfare at least $3$. In the {\\rm NO} case, the optimal $p$-mean welfare is less than $3 \\ c (\\alpha)$, where $c (\\alpha) <1$ is a constant (that depends of $\\alpha$). \n\nTherefore, given a $\\frac{1}{c(\\alpha)}$-approximation algorithm for $p$-mean welfare maximization, one could distinguish between the {\\rm YES} and {\\rm NO} instances of Gap-3DM. Hence, the hardness of Gap-3DM implies that it is {\\rm NP}-hard to approximate the optimal $p$-mean welfare with a constant factor of $\\nicefrac{1}{c(\\alpha)}$. \n\n\\subsection{Proof of Theorem \\ref{APX_hardness_theorem}} \n\nGiven an instance $\\mathcal{C}$ of Gap-3DM with $3$-uniform hyperedges $E=\\{E_1, \\ldots, E_T\\}$ over size-$q$ sets $X$, $Y$, and $Z$, we construct an instance $\\mathcal{I}$ of $p$-mean welfare maximization with $q$ agents and $3q$ goods, one for each vertex $X \\cup Y \\cup Z$. All the agents share a common XOS valuation function $v$: define $v(S) \\coloneqq \\max \\limits _{1\\leq i\\leq T} \\left\\{|S \\cap E_i| \\right\\}$ for each subset of goods $S\\subset X\\cup Y\\cup Z$.\n\nNote that the value of any subset $S$ is upper bounded by 3, since each $E_i$ is a 3-uniform hyperedge. We now prove that this reduction is gap preserving. \n\nWhen the input instance $\\mathcal{C}$ is a {\\rm YES} instance, there is a matching of size $q$. We assign the three goods corresponding to each edge in the matching to a distinct agent. In this allocation each agent gets a bundle of value $3$. Therefore, the optimal $p$-mean welfare in this case is at least $3$. Conversely, when $\\mathcal{C}$ is a {\\rm NO} instance, every matching is of size at most $\\alpha q$. We claim that in this case, the optimal $p$-mean welfare in the reduced instance $\\mathcal{I}$ is upper bounded by $3 \\ c(\\alpha)$, where $c(\\alpha) =\\frac{2+\\alpha}{3} <1$. \n\nLet $\\mathcal{A}^*(\\mathcal{I},p)=(A^*_1,\\ldots, A^*_q)$ denote a $p$-optimal allocation in instance $\\mathcal{I}$. Here, in any allocation, an agent's value for her bundle is either $0$, $1$, $2$, or $3$. We partition the bundle in the optimal allocation into two collections\n$\\mathcal{H} \\coloneqq \\{A^*_i \\mid v(A^*_i)=3\\}  \\text{ and } \\overline{\\mathcal{H}} \\coloneqq \\{A^*_i \\mid v(A^*_i)\\leq 2\\}$.\nEvery bundle $A^*_i$ in $\\mathcal{H}$ contains at least one hyperedge $E_j$. Since the bundles in $\\mathcal{H}$ are disjoint, the hyperedges contained in different bundles are nonintersecting. Hence, such hyperedges form a matching in $\\mathcal{C}$ of size at least $|\\mathcal{H}|.$ Recall that $\\mathcal{C}$ is a {\\rm NO} instance, i.e., every matching in $\\mathcal{C}$ of size at most $\\alpha q.$ Therefore,  there are at most $\\alpha q$ bundles in $\\mathcal{A}^*(\\mathcal{I},p)$ of value $3$, $|\\mathcal{H}| \\leq \\alpha q$. Using this inequality we can upper bound the optimal $p$-mean welfare in instance $\\mathcal{I}$ as follows. \n\\begin{Claim}\n${\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p))\\leq 2+ \\alpha = 3 \\left( \\frac{2+\\alpha}{3}\\right)  $\n\\end{Claim}\n\\begin{proof}\nRecall that $\\mathcal{H}$ and $\\overline{\\mathcal{H}}$ form a partition of the $q$ bundles in $\\mathcal{A}^*(\\mathcal{I},p)$. Let $|\\mathcal{H}|=\\overline{\\alpha} q$ for some $\\overline{\\alpha}\\leq \\alpha$. Then, $\\sum \\limits _{A^*_i \\in \\overline{\\mathcal{H}}} v(A^*_i)\\leq 2(1-\\overline{\\alpha})q$. Therefore, \n\\begin{align*}\n{\\rm M}_p(\\mathcal{A}^*(\\mathcal{I},p)) &\\leq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{I},p)) \\tag{via the generalized mean inequality}\\\\\n& \\leq \\frac{1}{q}\\left( 3\\overline{\\alpha} q + 2(1-\\overline{\\alpha})q \\right) \\tag{averaging over the values of bundles in $\\mathcal{H}$ and $\\overline{\\mathcal{H}}$}\\\\\n&\\leq 2+\\alpha \\tag{since $\\overline{\\alpha}\\leq \\alpha$}\n\\end{align*}\n\\end{proof}\nHence, we have a polynomial-time reduction from the Gap-3DM to the $p$-mean welfare maximization problem such that:\\\\\n(i) When Gap-3DM instance $\\mathcal{C}$ is a {\\rm YES} instance, the optimal $p$-mean welfare is at least $3$.\\\\\n(ii) When $\\mathcal{C}$ is a {\\rm NO} instance, the optimal $p$-mean welfare is at most $3\\ c(\\alpha)$, for a constant $c(\\alpha) <1$.  \n\nAs mentioned previously, such a gap-preserving reduction establishes the {\\rm APX}-hardness of $p$-mean welfare maximization. \n\nFinally, note that we can efficiently simulate the demand oracle for the valuation function $v$ in the constructed instance $\\mathcal{I}$. For any additive function $f$, the response to a demand query---with prices $p_j$s associated with the goods---is simply the subset of goods $g$ that satisfy $f(g) - p_g \\geq 0$. The XOS valuation function $v$ in the reduction is obtained by considering a maximum over $T$ additive functions. The parameter $T$ is the number of edges in the given Gap-3DM instance and, hence, is polynomially bounded. Therefore, we can efficiently simulate the demand oracle for $v$ by explicitly optimizing over all the $T$ additive functions.  Overall, we get that the {\\rm APX}-hardness holds in the demand oracle model and the theorem follows. \n\\section{Conclusion and Future Work}\nThis work studies the problem of allocating indivisible goods among agents that share a common subadditive valuation. We show that, for such settings, one can always (and in polynomial-time) find a single allocation that simultaneously approximates a range of generalized-mean welfares, to within a constant factor of the optimal. \n\n\nFor ease of presentation, we focussed on the case in which the agents' valuations are exactly identical. Nonetheless, it can be shown that the developed results are somewhat robust: if, say, the agents' valuations are point-wise and multiplicatively close to each other, then again one can obtain meaningful approximation guarantees. Here, an interesting direction of future work is to address settings in which we have a fixed number of distinct valuation functions across all the agents. A nontrivial improvement on the developed approximation guarantee will also be interesting. \n\n\n\n\n\n\\section{Introduction}\n\nA significant body of recent work, in algorithmic game theory, has been directed towards the study of fair and efficient allocation of indivisible goods among agents; see, e.g.,~\\cite{endriss2017trends} and~\\cite{brandt2016handbook}. This thread of research has led to the development of multiple algorithms and platforms (e.g., {Spliddit}~\\cite{goldman2015spliddit}) which, in particular, address settings wherein discrete resources (that cannot be fractionally allocated) need to be partitioned among multiple agents. Contributing to this line of work, the current paper studies discrete fair division from a welfarist perspective. \n\nWe specifically address the problem of finding allocations (of indivisible goods) that (approximately) maximize the \\emph{generalized means} of the agents' valuations. Formally, for exponent parameter $p \\in \\mathbb{R}$, the $p${th} generalized mean, of $n$ nonnegative values $\\{v_i\\}_{i=1}^n$, is defined as ${\\rm M}_p(v_1, \\ldots, v_n) \\coloneqq \\left( \\frac{1}{n} \\sum_i v_i^p \\right)^{\\frac{1}{p}}$. Parameterized by $p$, this family of functions includes well-studied fairness and efficiency objectives, such as average social welfare ($p=1$), Nash social welfare ($p \\to 0$), and egalitarian welfare ($p \\to -\\infty$). In fact, generalized means---with the exponent parameter $p$ in the range $(-\\infty, 1]$---admit a fundamental axiomatic characterization: up to monotonic transformations, generalized means (with $p \\in (-\\infty, 1]$) exactly constitute \\emph{the} family of welfare functions that satisfy the \\emph{Pigou-Dalton transfer principle} and a few other key axioms~\\cite{moulin2004fair}.\\footnote{Note that generalized means are ordinally equivalent to CES (constant elasticity of substitution) functions.} Hence, by way of developing a single approximation algorithm for maximizing generalized means, the current work provides a unified treatment of multiple fairness and efficiency measures. \n\nWith generalized mean as our objective, we focus on fair-division instances in which the agents have a common \\emph{subadditive} (i.e., complement free) valuation. Formally, a set function $v$, defined over a set of indivisible goods $[m]$, is a said to be subadditive iff, for all subsets $A$ and $B$ of $[m]$, we have $v(A \\cup B) \\leq v(A) + v(B)$. This class of functions includes many other well-studied valuation families, namely \\emph{XOS}, \\emph{submodular}, and \\emph{additive} valuations.\\footnote{Recall that a submodular function $f$ is defined by a diminishing returns property: $f(A + e) - f(A) \\geq f(B + e) - f(B)$, for all subsets $A \\subseteq B$ and $e \\notin B$.} These function classes have been used extensively in computer science and mathematical economics to represent agents' valuations. Of particular relevance here are results that (in the context of combinatorial auctions) address the problem of maximizing social welfare under submodular, XOS, and, more generally,  subadditive valuations~\\cite{nisan2007algorithmic}. \n\n\nThe focus on a common valuation function across the agents provides a technically interesting and applicable subclass of fair-division problems--as a stylized application, consider a setting in which the agents' values represent money, i.e., for every agent, the value of each subset (of the goods) is equal to the subset's monetary worth. Here, one encounters subadditivity when considering goods that are substitutes of each other. Also, from a technical standpoint, we note that the problem of maximizing social welfare is {\\rm APX}-hard even under identical submodular~\\cite{khot2008inapproximability} and subadditive valuations~\\cite{dobzinski2005approximation}. Appendix \\ref{APX_Hardness} extends this hardness result to all $p \\in (-\\infty, 1]$.\\\\\n\n\n\\noindent\n{\\bf Our Results:} Addressing fair-division instances with identical subadditive valuations, we develop an efficient constant-factor approximation algorithm for the generalized-mean objective (Theorem~\\ref{MainTheorem}). Specifically, our algorithm computes an allocation (of the indivisible goods among the agents), $\\mathcal{A}$, with the property that its generalized-mean welfare, ${\\rm M}_p (\\mathcal{A})$, is at least ${1}\/{40}$ times the optimal $p$-mean welfare, for all $p \\in (-\\infty, 1]$. This result in fact implies an interesting existential guarantee as well: if in a fair-division instance the agents' valuations are identical and subadditive, then there exists a single allocation that uniformly approximates the optimal $p$-mean welfare for all $p \\in (-\\infty, 1]$.  \n\nThe tradeoff between fairness and economic efficiency is an important consideration in fair division literature.\\footnote{For example, consider the work on price of fairness~\\cite{bertsimas2011price,bei2019price}} The relevance of the above-mentioned existential guarantee is substantiated by the fact that this result reasonably mitigates the fairness-efficiency tradeoff in the current context; it shows that for identical subadditive valuations there exists a single allocation which is near optimal with respect to efficiency objectives (in particular, social welfare) as well as fairness measures (e.g., egalitarian welfare). Note that such an allocation cannot be simply obtained by selecting an arbitrary partition that (approximately) maximizes social welfare: under identical additive valuations, all the allocations have the same social welfare, even the ones with egalitarian welfare equal to zero. One can also construct instances, with identical subadditive valuations, wherein particular allocations have optimal egalitarian welfare, but subpar social welfare. \n\nEven specific instantiations of our algorithmic guarantee provide novel results: while the problem of maximizing Nash social welfare, among $n$ agents, admits an $\\mathcal{O}(n \\log n)$-approximation under nonidentical submodular valuations~\\cite{garg2020approximating}, the current work provides a novel (constant-factor) approximation guarantee for maximizing Nash social welfare when the agents share a common subadditive (and, hence, submodular) valuation.\\footnote{Under nonidentical additive valuations, there exists a polynomial-time $1.45$-approximation algorithm for maximizing Nash social welfare~\\cite{barman2018finding}. Furthermore, under identical additive valuations, maximizing Nash social welfare admits a polynomial-time approximation scheme~\\cite{nguyen2014minimizing,barman2018greedy}.}  Analogously, the instantiation of our result for egalitarian welfare is interesting in and of itself.\n\nGiven that the valuations considered in this work express combinatorial preferences, a naive representation of such set functions would require exponential (in the number of goods) values, one for each subset of the goods. Hence, to primarily focus on the underlying computational aspects and not on the representation details, much of prior work assumes that the valuations are provided via oracles that can only answer  particular type of queries. The most basic oracle considered in literature answers \\emph{value queries}: given a subset of the indivisible goods, the value oracle returns the value of this subset. In this value oracle model, the work of Vondr\\'{a}k~\\cite{vondrak2008optimal} considers submodular valuations and provides an efficient $\\frac{e}{e-1}$-approximation algorithm for maximizing social welfare. Using this method as a subroutine and, hence, completely in the value oracle model, our algorithm achieves the above-mentioned approximation guarantee for identical submodular valuations. \n\n\nAnother well-studied oracle addresses \\emph{demand queries}. Specifically, such an oracle, when queried with an assignment of prices $p_1, \\ldots, p_m \\in \\mathbb{R}$ to the $m$ goods, returns $\\max_{S \\subseteq [m]} \\left( v(S) - \\sum_{j \\in S} p_j \\right)$, for the underlying valuation function $v$.\\footnote{Observe that a value query can be simulated via polynomially many demand queries. Though, the converse is not true~\\cite{nisan2007algorithmic}.} Demand oracles have been often utilized in prior work for addressing social welfare maximization in the context of subadditive and XOS valuations~\\cite{nisan2007algorithmic}. In particular, the work of Fiege~\\cite{feige2009maximizing} shows that, under subadditive valuations and assuming oracle access to {demand queries},\\footnote{This result holds even if the agents have distinct, but subadditive, valuations.} the social welfare maximization problem admits an efficient $2$-approximation algorithm. Demand queries are unavoidable in the subadditive case: one can directly extend the result of Dobzinski et al.~\\cite{DobzinskiNS10} to show that, even under identical (subadditive) valuations, any sub-linear (in $n$) approximation of the optimal social welfare requires exponentially many value queries. At the same time, we note that our algorithm requires demand oracle access \\emph{only} to implement the $2$-approximation algorithm of Fiege~\\cite{feige2009maximizing} as a subroutine. Beyond this, we can work with the value oracle.  \\\\\n\n\n\n\\noindent\n{\\bf Related Work:} Multiple algorithmic and hardness results have been developed to address welfare maximization in the context of indivisible goods\/discrete resources. Though, in contrast to the present paper, prior work in this direction has primarily addressed one welfare function at a time. \n\nAs mentioned previously, maximizing social welfare and Nash social welfare (see, e.g., \\cite{cole2018approximating} and references therein) has been actively studied in algorithmic game theory. Egalitarian welfare has also been addressed in prior work--this welfare maximization problem is also referred to as the  max-min allocation problem (or the Santa Claus problem); see, e.g.,~\\cite{DBLP:journals\/corr\/AnnamalaiKS14}. Specifically, for maximizing egalitarian welfare under additive and nonidentical valuations, the result of Chakrabarty et al.~\\cite{chakrabarty2009allocating} provides an $\\widetilde{\\mathcal{O}}(n^{\\varepsilon})$-approximation algorithm that runs in time $\\mathcal{O}(n^\\frac{1}{\\varepsilon})$; here $n$ denotes the number of agents and $\\varepsilon >0$.  Furthermore, under nonidentical submodular valuations, the problem of maximizing egalitarian welfare is known to admit a polynomial-time $\\widetilde{\\mathcal{O}}(n^{1\/4} m^{1\/2})$-approximation algorithm~\\cite{goemans2009approximating}; here $m$ is the number of goods. In contrast to these sublinear approximations, this paper shows that, if the agents' valuations are identical, then even under subadditive valuations the problem of maximizing egalitarian welfare admits a constant-factor approximation guarantee. \n\n\n\n\n\\section{Notation and Preliminaries}\n\nAn instance of a fair-division problem corresponds to a tuple $\\langle [m], [n], v \\rangle$, where $[m]= \\left\\{1,2,\\ldots, m \\right\\}$ denotes the set of $m \\in \\mathbb{N}$ indivisible {goods} that have to be allocated (partitioned) among the set of $n \\in \\mathbb{N}$ agents, $[n]=\\{1, 2, \\ldots, n\\}$. \nHere, $v: 2^{[m]} \\mapsto \\mathbb{R}_+$ represents the (identical) valuation function of the agents;\\footnote{Recall that this work addresses fair-division instances in which all the agents have a common valuation function.} specifically, $v(S) \\in \\mathbb{R}_+$ is the value that each agent $i \\in [n]$ has for a subset of goods $S \\subseteq [m]$. \n\nWe will assume throughout that the valuation function $v$ is (i) normalized: $v(\\emptyset) = 0$, (ii) monotone:  $v(A) \\leq v(B)$ for all $A \\subseteq B \\subseteq [m]$, and (iii) {subadditive}: $v(A \\cup B) \\leq v(A) + v(B)$ for all subsets $A, B \\subseteq [m]$. \n\n\nWrite $\\Pi_n([m])$ to denote the collection of all $n$ partitions of the indivisible goods $[m]$. We use the term \\textit{allocation} to refer to an $n$-partition $\\mathcal{A} = \\allo{A}{}{1} \\in \\Pi_n([m])$ of the $m$ goods. Here, $A_i$ denotes the subset of goods allocated to agent $i \\in [n]$ and will be referred to as a \\emph{bundle}.\n\n\nGeneralized (H\\\"{o}lder) means, ${\\rm M}_p$, constitute a family of functions that capture multiple fairness and efficiency measures. Formally, for an exponent parameter $p \\in \\mathbb{R}$, the $p${th} generalized mean of $n$ nonnegative numbers $x_1,\\ldots , x_n \\in \\mathbb{R}_+$ is defined as $\\Mp{x}{1} \\coloneqq \\left( \\frac{1}{n} \\sum \\limits _{i=1}^n x_i^p \\right )^\\frac{1}{p}$.\n\nNote that, when $p=1$, ${\\rm M}_p$ reduces to the arithmetic mean. Also, as $p$ tends to zero, ${\\rm M}_p$, in the limit, is equal to the geometric mean and $\\lim_{p \\rightarrow -\\infty}  \\Mp{x}{1} = \\min\\{x_1, x_2, \\ldots, x_n\\}$. Hence, following standard convention, we will write ${\\rm M}_0(x_1, \\ldots, x_n) = \\left(\\prod_{i=1}^n x_i \\right)^{1\/n}$ and ${\\rm M}_{-\\infty}(x_1, \\ldots, x_n) = \\min_i x_i $.\n\nConsidering generalized means as a parameterized collection of welfare objectives, we define the \\emph{$p$-mean welfare}, ${\\rm M}_p(\\mathcal{A})$, of an allocation $\\mathcal{A}=(A_1, A_2, \\ldots, A_n)$ as  \\begin{align}\n{\\rm M}_p(\\mathcal{A}) & \\coloneqq  {\\rm M}_p\\left( v(A_1), \\ldots, v(A_n) \\right) = \\left( \\frac{1}{n} \\sum_{i=1}^n v (A_i)^p \\right)^{1\/p} \\label{eq:generalized-mean}\n\\end{align}\nHere, $v$ is the (common) valuation function of the agents. Indeed, with $p$ equal to one, zero, and $-\\infty$, the $p$-mean welfare, respectively,  corresponds to (average) social welfare, Nash social welfare, and egalitarian welfare.  \n\n\nGiven a fair-division instance $\\mathcal{I}=\\langle [m], [n], v \\rangle$ and $p \\in (-\\infty, 1]$, ideally, we would like to find an allocation $\\mathcal{A} = (A_1, \\ldots, A_n)$ with as large an ${\\rm M}_p(\\mathcal{A})$ value as possible, i.e., maximize the $p$-mean welfare. An allocation that achieves this goal will be referred to as a \\emph{$p$-optimal allocation} and denoted by $\\mathcal{A}^*(\\mathcal{I}, p)=(A^*_1(\\mathcal{I}, p), A^*_2(\\mathcal{I}, p), \\ldots, A^*_n(\\mathcal{I}, p))$. \n\n\n\nWe note that, under identical, subadditive valuations, finding a $p$-optimal allocation is {\\rm APX}-hard, for any $p \\in (-\\infty, 1]$ (Appendix~\\ref{APX_Hardness}). Hence, the current work considers approximation guarantees. In particular, for fair-division instances $\\mathcal{I}$ in which the agents have a common subadditive valuation, we develop a polynomial-time algorithm that computes a single allocation $\\mathcal{A}$ with the property that ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p({\\mathcal{A}^{*}}(\\mathcal{I}, p))$ for all $p \\in (-\\infty, 1]$. That is, the developed algorithm achieves an approximation ratio of $40$ uniformly for all $p \\in (-\\infty, 1]$. \n\n\nThe work of Fiege~\\cite{feige2009maximizing} shows that, for subadditive valuations, the social-welfare maximization problem (equivalently, the problem of maximizing ${\\rm M}_1(\\cdot)$) admits an efficient $2$-approximation algorithm, assuming oracle access to {demand queries}. In particular, such an oracle, when queried with an assignment of prices $p_1, \\ldots, p_m \\in \\mathbb{R}$ to the $m$ goods, returns $\\max_{S \\subseteq [m]} \\left( v(S) - \\sum_{j \\in S} p_j \\right)$. Our algorithm requires demand oracle access {only} to implement the $2$-approximation algorithm of Fiege~\\cite{feige2009maximizing} as a subroutine. Beyond this, we can work with the basic value oracle, which when queried with a subset of goods $S \\subseteq [m]$, returns $v(S)$.\n\nIn fact, if the underlying valuation is submodular, then one can invoke the result of Vondr\\'{a}k~\\cite{vondrak2008optimal} (instead of using the approximation algorithm by Feige~\\cite{feige2009maximizing}) and efficiently obtain a $\\frac{e}{e-1}$-approximation for the social-welfare maximization problem in the value oracle model. Hence, under a submodular valuation, our algorithm can be implemented entirely in the standard value oracle model. \n\nFor a fair-division instance $\\mathcal{I}$, write ${\\rm F}(\\mathcal{I})$ to denote the $1$-mean welfare ${\\rm M}_1$ (i.e., the average social welfare) of the allocation computed by the approximation algorithm of Feige~\\cite{feige2009maximizing}. The approximation guarantee established in~\\cite{feige2009maximizing} ensures that---for any instance $\\mathcal{I}$ with a subadditive valuation---we have ${\\rm F}(\\mathcal{I}) \\geq \\frac{1}{2} {\\textrm M}_1(\\mathcal{A}^{*}(\\mathcal{I}, 1))$. Here, $\\mathcal{A}^{*}(\\mathcal{I}, 1)$ denotes a $1$-optimal allocation, i.e., it maximizes the (average) social welfare in $\\mathcal{I}$. \n\n\\section{Proof of Theorem \\ref{MainTheorem} for $p \\in [0.4,1]$} \\label{subsection:p-half-one}\n\n\n\nFor instance $\\mathcal{I}$, let \\textsc{Alg} assign the $k$ highest-valued goods as singletons in its while-loop. Specifically, write $\\widehat{G}=\\left\\{g_1,\\ldots, g_k\\right\\}$ to denote the $k$ goods that are assigned in {while}-loop of \\textsc{Alg}. Instance $\\mathcal{J} = \\langle [m] \\setminus \\{g_1, \\ldots, g_k\\}, [n] \\setminus [k], v \\rangle$ is passed as input to \\textsc{AlgLow}, which returns allocation $\\mathcal{B}=(B_{k+1},\\ldots,B_n)$. Recall that $\\mathcal{B}$  satisfies Lemma~\\ref{Low_valued}. Finally, let $\\mathcal{A}=(\\{g_1\\},\\ldots,\\{g_k\\},B_{k+1},\\ldots , B_{n})$ denote the allocation returned by \\textsc{Alg}.\nAlso, as before, let $\\mathcal{A}^*(\\mathcal{I},p) = (A^*_1(\\mathcal{I},p),\\ldots, A^*_n(\\mathcal{I},p))$ denote the $p$-optimal allocation of $\\mathcal{I}$. \n\n \nWe will prove the following bound for $p \\in  [0.4,1]$ and, hence, establish the stated approximation guarantee \n\\begin{align}\n\\left( \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p +  \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p \\right)^{\\frac{1}{p}} & \\geq \\frac{1}{40} \\left(\\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p))^p \\right)^{\\frac{1}{p}}\n\\end{align}\n\nWrite $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ to denote the allocation (specifically, an $n$-partition) obtained by removing the goods $\\widehat{G} =\\{g_1,\\ldots,g_k\\}$ from the bundles in $\\mathcal{A}^*(\\mathcal{I},p)$, i.e., $ \\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G} \\coloneqq \\left( A^*_i(\\mathcal{I},p) \\setminus \\widehat{G}  \\right)_{i=1}^n$.\nSubadditivity of $v$ ensures that, for all $i \\in [n]$, the bundle $A^{*}_i(\\mathcal{I},p)$ satisfies \n\\begin{align*}\nv(A^{*}_i(\\mathcal{I},p))  & \\leq v \\left( A^{*}_i (\\mathcal{I},p)  \\setminus \\widehat{G}\\right)  + \\sum \\limits_{ g \\in \\widehat{G} \\cap A^{*}_i(\\mathcal{I},p)} v(g).  \n\\end{align*}\n\nSince $p>0$, exponentiating the previous inequality by $p$ gives us   \n\\begin{align}\n  \\left(v(A^{*}_i(\\mathcal{I},p)) \\right)^p & \\leq \\left(v \\left( A^{*}_i (\\mathcal{I},p)  \\setminus \\widehat{G}\\right)  + \\sum \\limits_{ g \\in \\widehat{G} \\cap A^{*}_i(\\mathcal{I},p)} v(g) \\right)^p \\nonumber \\\\\n  & \\leq \\left(v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G}) \\right)^p + \\sum \\limits_{g\\in \\widehat{G}\\cap A^{*}_i(\\mathcal{I},p)} v(g)^p \\label{ineq:p-exp}\n\\end{align}\n\nThe last inequality follows from the fact that $(x+y)^p \\le x^p+y^p$, for all $p \\in [0.4, 1]$ and $x, y \\in \\mathbb{R}_+$.\n\nAveraging equation (\\ref{ineq:p-exp}) over $i \\in [n]$ leads to  \n \\begin{align}\\label{17}\n  \\frac{1}{n} \\summi{1}{k}{g}^p + \\frac{1}{n}\\sum\\limits_{i=1}^{n}{v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G}))^p} \n  \\geq \\frac{1}{n}\\sum\\limits_{i=1}^{n}{v(A^{*}_i(\\mathcal{I},p))^p}\n \\end{align}\n\nTo show that the $k$ goods in $\\widehat{G}$ (which are allocated as singletons) substantially contribute towards $p$-mean welfare of the computed allocation $\\mathcal{A}$, we will next establish the following lower bound for all $1\\leq t \\leq k$\n\\begin{align*}\nv(g_t)^p & \\geq \\frac{1}{(7.06)^p}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) ^p \\right)\n\\end{align*}\n\nRecall that $\\mathcal{I}^t$ denotes the fair-division instance $\\I{[m]\\setminus \\{ g_1,\\ldots, g_t\\}}{[n]\\setminus \\{1,\\ldots t\\}}$, for $1\\leq t\\leq k$. Write $\\mathcal{A}^*(\\mathcal{I}^t,p)=(A^*_{t+1} (\\mathcal{I}^t,p),\\ldots,A^*_{n}(\\mathcal{I}^t,p))$ to denote a $p$-optimal allocation of instance $\\mathcal{I}^t$.\n\nThe selection criterion of the while-loop in \\textsc{Alg} and the fact that Feige's algorithm achieves an approximation ratio of $2$ ensure \n\\begin{align}\n  v(g_t) &\\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{7.06}\\left ( \\frac{1}{n-t+1} \\sum\\limits_{j=t}^{n}v(A^{*}_j(\\mathcal{I}^{t-1},1))  \\right) \\label{ineq:bound-g-t-low}\n\\end{align}\n\nIndex the $n$ bundles in allocation $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ in non-increasing order of value $  v(A^*_1(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq  v(A^*_2(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq \\ldots \\geq v(A^*_n(\\mathcal{I},p)\\setminus \\widehat{G})$ and note that the arithmetic mean of the values of the first $n-t+1$ bundles is at least as large as the overall arithmetic mean\n\\begin{align}\n\\frac{1}{n-t+1}\\sum \\limits _{j=1}^{n-t+1}v(A^*_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\geq \\frac{1}{n}\\sum \\limits _{j=1}^{n}v(A^*_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\label{ineq:top-mean}\n\\end{align}\n\nGiven that allocation $\\mathcal{A}^*(\\mathcal{I},p)\\setminus \\widehat{G}$ constitutes an $n$-partition of the set of goods $[m] \\setminus \\widehat{G}$ and allocation $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 1) = (A^*_{t} (\\mathcal{I}^{t-1},1),\\ldots, A^*_{n}(\\mathcal{I}^{t-1},1))$ is an $(n-t+1)$-partition of $[m]\\setminus \\{g_1, \\ldots, g_{t-1}\\} \\supseteq [m] \\setminus \\widehat{G}$, we have the following containment  $\\bigcup \\limits _{j=1}^{n-t+1}\\left(A_j^*(\\mathcal{I},p)\\setminus \\widehat{G}\\right ) \\subseteq \\bigcup \\limits _{j=t}^{n} \\left(\\ A^*_j(\\mathcal{I}^{t-1},1)\\right)$. Furthermore, by definition, allocation $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 1)$ achieves the maximum possible average social welfare among all $(n-t+1)$ partitions of $[m] \\setminus \\{g_1, \\ldots, g_t\\}$. Therefore,  we have \n\\begin{align}\n \\frac{1}{n-t+1}\\sum\\limits_{j=t}^{n} v(A^{*}_j(\\mathcal{I}^{t-1},1)) & \\geq   \\frac{1}{n-t+1}\\sum\\limits_{j=1}^{n-t+1} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) \\label{ineq:opt-sub-problem}\n\\end{align}\n\nEquations (\\ref{ineq:bound-g-t-low}) and (\\ref{ineq:opt-sub-problem}) lead to \n \\begin{align*}\n  v(g_t) & \\geq \\frac{1}{7.06}\\left( \\frac{1}{n-t+1}\\sum\\limits_{j=1}^{n-t+1} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})  \\right)\\\\\n  &\\geq \\frac{1}{7.06}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})  \\right) \\tag{using inequality (\\ref{ineq:top-mean})} \\\\ \n  &\\geq \\frac{1}{7.06}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})^p  \\right) ^{\\frac{1}{p}} \\tag{via the generalized mean inequality}\n\\end{align*}\nExponentiating both sides of the previous inequality by $p$ gives us the desired lower bound\n\\begin{align}\\label{19}\n v(g_t)^p \\ge \\frac{1}{(7.06)^p}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G}) ^p \\right)\n \\end{align} \n \nEquation (\\ref{19}) enables us to bound the $p$-welfare contribution of the goods $\\widehat{G}=\\{g_1, \\ldots, g_k\\}$ assigned as singletons \n\\begin{align}\\label{20}\n \\frac{2}{n} \\sum\\limits_{i=1}^kv(g_i)^p \\geq  \\frac{1}{n} \\sum\\limits_{i=1}^kv(g_i)^p\n + \\frac{k}{n} \\ \\frac{1}{(7.06)^p} \\left(\\frac{1}{n} \\sum \\limits_{i=1}^n v ( A^{*}_i (\\mathcal{I},p)\\setminus \\widehat{G})^p\\right)\n\\end{align}\n\nRecall that \\textsc{AlgLow}---with input instance $\\mathcal{J}=\\mathcal{I}^k$---returns allocation $\\mathcal{B}=(B_{k+1}, \\ldots B_n)$. Next we lower bound the values of these bundles $B_{k+1}, \\ldots, B_n$. \n\\begin{align*}\nv(B_j) & \\geq  \\frac{1}{40} {\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1)) \\tag{via Lemma~\\ref{Low_valued}} \\\\\n& = \\frac{1}{40 }\\left( \\frac{1}{n-k} \\sum\\limits_{i=k+1}^{n} v(A^{*}_i(\\mathcal{J},1))  \\right) \\tag{by defintion, $\\mathcal{A}^{*}(\\mathcal{J},1) = \\left( \\ A^{*}_i(\\mathcal{J},1) \\ \\right)_{i=k+1}^n $} \\\\\n& \\geq \\frac{1}{40 }\\left(  \\frac{1}{n-k}\\sum\\limits_{j=1}^{n-k} v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})  \\right)\\tag{instantiating inequality (\\ref{ineq:opt-sub-problem}) for $\\mathcal{J} = \\mathcal{I}^k$} \\\\\n&\\geq \\frac{1}{40}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})  \\right) \\tag{using inequality (\\ref{ineq:top-mean})} \\\\ \n  &\\geq \\frac{1}{40}\\left( \\frac{1}{n}\\sum\\limits_{j=1}^n v(A^{*}_j(\\mathcal{I},p)\\setminus \\widehat{G})^p  \\right) ^{\\frac{1}{p}} \\tag{via the generalized mean inequality}\n\\end{align*}\n\n\n\n\nExponentiating by $p$ and summing over all $j \\in \\{k+1,\\ldots n\\}$, we have\n\\begin{align} \\label{21}\n \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p \\ge \\frac{n-k}{n} \\frac{1}{(40) ^p}\\left(\\frac{1}{n}\n \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right)\n\\end{align}\n\nCombining inequalities (\\ref{20}) and (\\ref{21}) gives us\n\\begin{align*}\n \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p +  \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p & \\geq\n   \\frac{1}{2n}\\ \\sum\\limits_{i=1}^k v(g_i)^p +\n  \\frac{k}{2n} \\cdot \\frac{1}{(7.06)^p} \\left (\\frac{1}{n}\\sum\\limits_{j=1}^{n}v ( A^{*}_i (\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \\\\ & \\ \\ \\ \\ \\   +\n  \\frac{n-k}{n}\\frac{1}{(40) ^p}\\left(\\frac{1}{n}\\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \n  \\end{align*}\n\nNote that $2 \\times (7.06)^p \\leq (40) ^p$ for all $p  \\in [0.4,1]$, hence, the previous inequality simplifies to \n  \\begin{align*}\n    \\frac{1}{n}\\sum\\limits_{i=1}^k v(g_i)^p +  \\frac{1}{n}\\sum\\limits_{j=k+1}^{n}v(B_j)^p & \\geq\n    \\frac{1}{2n}\\sum\\limits_{i=1}^{k}v(g_i)^p +\n  \\frac{1}{(40) ^p}\\left(\\frac{1}{n}\\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p\\right) \\\\\n  & \\geq \\frac{1}{(40) ^p} \\left(\\frac{1}{n} \\sum\\limits_{i=1}^{k}v(g_i)^p +\n \\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p)\\setminus \\widehat{G})^p \\right) \\tag{since $(40)^p > 2$ for $p \\in [0.4,1]$} \\\\\n & \\geq \\frac{1}{(40) ^p}\\left(\\frac{1}{n} \\sum\\limits_{i=1}^{n}v(A^{*}_i(\\mathcal{I},p))^p \\right) \\tag{using inequality (\\ref{17})}\n  \\end{align*}\n\nTaking the $p${th} root (with $p>0$) on both sides of the last inequality gives us the desired result for the computed allocation $\\mathcal{A} = (\\{g_1\\},\\ldots,\\{g_k\\},B_{k+1},\\ldots , B_{n})$\n\\begin{align*}\n{\\rm M}_p(\\mathcal{A})\\ge \\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p)).\n\\end{align*}\n\nThis completes the proof of Theorem \\ref{MainTheorem} for all $p \\in (-\\infty,1]$.\n\n\\section{Proof of Theorem \\ref{MainTheorem} for $p \\in (-\\infty,0.4)$}\n\\label{subsection:p-inf-half}\n\n\nAs before, we will write $\\mathcal{I}=\\I{[m]}{[n]}$ to denote the given fair-division instance. Also, write $\\mathcal{J} = \\mathcal{I}^k = \\I{[m]\\setminus \\{g_1,\\ldots,g_k\\}}{[n]\\setminus [k]}$ to denote the instance obtained at the termination of the while loop in \\textsc{Alg}. That is, instance $\\mathcal{J}$ is obtained after allocating the $k$ highest-valued goods as singletons to different agents and $\\mathcal{J}$ is passed on as an input to \\textsc{AlgLow}. \n\nAlso recall that $\\mathcal{A}^*(\\mathcal{I},p)=(A^*_1(\\mathcal{I},p),\\ldots, A^*_n(\\mathcal{I},p))$ and $\\mathcal{A}^*(\\mathcal{J}, p)= \\left( A^*_{k+1}(\\mathcal{J},p),\\ldots, A^*_n(\\mathcal{J},p) \\right)$ denote the $p$-mean optimal allocations of instances $\\mathcal{I}$ and $\\mathcal{J}$, respectively.\n\nSo far, we have established two results \\\\\n\\noindent\n(i) Lemma~\\ref{Induction_argument}: The allocation $(\\{g_1\\},\\ldots ,\\{g_k\\}, A^*_{k+1}(\\mathcal{J},p), \\ldots, A^*_{n}(\\mathcal{J},p) )$ achieves welfare comparable to the optimal $p$-mean welfare (i.e., comparable to ${\\rm M}_p \\left(\\mathcal{A}^*(\\mathcal{I},p) \\right)$), for $p \\in (-\\infty, 0.4)$. \\\\\n\\noindent\n(ii) Lemma~\\ref{Low_valued}: For the instance $\\mathcal{J}$ and any $p\\in (-\\infty, 0.4]$, \\textsc{AlgLow} computes an allocation $\\mathcal{B}= (B_{k+1},\\ldots,B_{n})$ such that, for all $j \\in \\{k+1, \\ldots, n\\}$,\\footnote{For notational convenience, we index the bundles in allocation $\\mathcal{B}$ from $k+1$ to $n$.}\n\\begin{align}\\label{16}\nv(B_j)\\geq\\frac{1}{40}{\\rm M}_1(\\mathcal{A}^{*}(\\mathcal{J},1))\\geq\\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{J},p)) \n\\end{align}\n\nThe allocation returned by \\textsc{Alg} for input instance $\\mathcal{I}$ is $\\mathcal{A}=(\\{g_1\\},\\ldots,\\{g_k\\}, B_{k+1}, \\ldots ,B_n)$. We will prove Theorem~\\ref{MainTheorem}, for $p \\in (-\\infty,0.4)$, by showing that Lemma~\\ref{Induction_argument} and Lemma~\\ref{Low_valued}, in conjunction, imply that the $p$-mean welfare of $\\mathcal{A}$ is a constant times the optimal $p$-mean optimal; specifically, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40}{\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, for $p\\in(-\\infty,0.4)$.\n\nWe split the proof of this inequality into three parts, depending on the range of the exponent parameter $p$.\n\n\\noindent\n\\textbf{Case 1:} $p \\in (-\\infty, 0)$. Since in this case $p$ is negative, to obtain the desired inequality, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, it suffices to show that\n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(B_j\\right)^p\\leq \\frac{1}{(40) ^p}\\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align*}\n\nExponentiating both sides of equation (\\ref{16}) by $p <0$ and summing over $j \\in \\{k+1, \\ldots, n\\}$ lead to \n\\begin{align}\n\\sum \\limits _{j=k+1}^nv(B_j)^p\\leq\\frac{1}{(40) ^p}\\sum \\limits _{j=k+1}^{n}v(A^{*}_j(\\mathcal{J},p))^p \n\\end{align}\n\nWe add $\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p$ to both sides of the previous equation and apply Lemma~\\ref{Induction_argument} (with $p \\in (-\\infty, 0)$)  to obtain the desired inequality \n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits _{j=k+1}^n v(B_i)^p \\leq \\frac{1}{\\left( 40 \\right)^p}  \\left( (40)^p\\sum \\limits _{i=1}^k v(g_i)^p + \\sum \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},p))^p \\right)\n\\leq \\frac{1}{40 ^p}\\left( \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\\right)\\hspace{3.6cm}\n\\end{align*}\n\n\n\\noindent {\\bf Case 2:} $p  \\in (0,0.4)$.\nNote that in this case $p>0$. Hence, to obtain the desired inequality, ${\\rm M}_p(\\mathcal{A})\\geq \\frac{1}{40} {\\rm M}_p(\\mathcal{A}^{*}(\\mathcal{I},p))$, it suffices to show that\n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(B_j\\right)^p\\geq \\frac{1}{(40) ^p}\\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align*}\n\nExponentiating both sides of equation (\\ref{16}) by $p >0$ and summing over $j \\in \\{k+1, \\ldots, n\\}$ lead to \n\n\\begin{align*}\n\\sum \\limits _{j=k+1}^nv(B_i)^p & \\geq \\frac{1}{(40) ^p}\\sum \\limits _{j=k+1}^{n}v(A^{*}_j(\\mathcal{J},p))^p.\n\\end{align*}\n\nWe add $\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p$ to both sides of the previous equation and apply Lemma~\\ref{Induction_argument} (with $p \\in (0,0.4)$)  to obtain the desired inequality \n\\begin{align*}\n\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits _{j=k+1}^n v(B_j)^p &\\geq  \\frac{1}{\\left( 40\\right)^p}  \\left( (40)^p \\sum \\limits _{i=1}^k v(g_i)^p + \\sum \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},p))^p \\right)\n\\geq \\frac{1}{(40) ^p}\\left( \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\\right)\\hspace{3.6cm}\n\\end{align*}\n\n\n\n\n\\noindent {\\bf Case 3:} $p=0$. In this case, ${\\rm M}_0(\\mathcal{A}) = \\left( \\prod \\limits _{i=1}^kv(g_i) \\prod \\limits _{j=k+1}^{n} v(B_j)\\right)^\\frac{1}{n}.$\nMultiplying inequality (\\ref{16}) over all $j \\in \\{k+1,\\ldots, n\\}$ gives us \n\\begin{align*}\n\\prod \\limits _{j=k+1}^{n} v(B_i)  \\geq \\frac{1}{(40)^{n-k}} \\left( {\\rm M}_0 \\left(\\mathcal{A}^{*}(\\mathcal{J}, 0) \\right) \\right)^{n-k} & =  \\frac{1}{(40)^{n-k}}\\prod \\limits _{j=k+1}^{n} v(A^{*}_j(\\mathcal{J},0))\n\\end{align*}\n\nNext we multiply both sides of this inequality by  $\\prod_{i=1}^k v\\left(g_i\\right)$ and obtain \n\\begin{align*} \n\\prod \\limits _{i=1}^k v(g_i) \\prod \\limits _{j=k+1}^n v(B_i)\\geq \\frac{1}{(40) ^{n-k}}\\prod \\limits_{i=1}^k v(g_i) &\\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{J},0) ) \\geq \\frac{1}{(40) ^n}({\\rm M}_0(\\mathcal{A}^{*}(\\mathcal{I}, 0)))^n   \\tag{via Lemma~\\ref{Induction_argument} with $p=0$}\n\\end{align*}\nTaking the $n${th} root on both sides, we obtain the desired result ${\\rm M}_0(\\mathcal{A}) \\geq \\frac{1}{40} {\\rm M}_0(\\mathcal{A}^{*}(\\mathcal{I},0))$. \\\\\n\n\nTheorem \\ref{MainTheorem} now stands proved for $p \\in (-\\infty,0.4)$. \n\n\\section*{Acknowledgements}\nSiddharth Barman gratefully acknowledges the support of a Ramanujan Fellowship (SERB - {SB\/S2\/RJN-128\/2015}) and a Pratiksha Trust Young Investigator Award.\n\n\\bibliographystyle{alpha}\n\n\\section{Structural Lemma} \\label{Supporting_Lemmas}\nThe following lemma provides a structural property of $p$-optimal allocations $\\mathcal{A}^*(\\mathcal{I}, p)$, for $p \\in (-\\infty,0.4)$. It states that the only way allocation $\\mathcal{A}^*(\\mathcal{I}, p)$ has a bundle $A^*_i(\\mathcal{I}, p)$ of notably high value is through a single good $g \\in A^*_i(\\mathcal{I}, p)$ that by itself has high value. \n\n\n\\begin{Lemma}\\label{Good_Transfer}\nLet $\\mathcal{L}= \\I{[M]}{[N]}$ be a fair-division instance wherein all the $N \\in \\mathbb{N}$ agents have an identical, subadditive valuation $v$ over the set of $M\\in\\mathbb{N}$ goods. In addition, let $\\mathcal{A}^{*}(\\mathcal{L},p) = \\{ A^{*}_{1}(\\mathcal{L},p), \\cdots , A^{*}_{N}(\\mathcal{L},p)\\}$ be a $p$-mean optimal allocation in $\\mathcal{L}$,  for any  $p \\in \\left( -\\infty ,0.4 \\right)$. \n\nIf for any bundle $A^{*}_i(\\mathcal{L},p)$, with $i \\in [N]$, we have $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L})$, then there exists a good $g\\in A_i^*(\\mathcal{L},p)$ with the property that that $v(g)\\geq \\frac{1}{40} v(A_i^*(\\mathcal{L},p)).$\n\\end{Lemma}\n\nThe proof of the above lemma is divided into  three parts (Sections~\\ref{subsection:p-infty-zero-good-transfer}, \\ref{subsection:p-zero-half-good-transfer}, and~\\ref{subsection:p-zero-good-transfer}) depending on the range of the exponent parameter $p.$ \n\\subsection{Proof of Lemma \\ref{Good_Transfer} for $p \\in (-\\infty,0)$}\n\\label{subsection:p-infty-zero-good-transfer}\n\nAssume, towards a contradiction, that $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L}) $, for some $i \\in [N] $, and $v(g) \\leq  \\frac{1}{40} v(A^{*}_i(\\mathcal{L},p))$ for all $g \\in A^{*}_i(\\mathcal{L},p).$ Recall that the $1$-mean welfare of the allocation returned by Feige's algorithm satisfies ${\\rm F}(\\mathcal{L}) \\geq \\frac{1}{2}{\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1))$. Pick a bundle $A^{*}_j(\\mathcal{L},p)$ (in $\\mathcal{A}^*(\\mathcal{L}, p)$) with the property that $v(A^{*}_j(\\mathcal{L},p)) \\leq 2{\\rm F}(\\mathcal{L})$. Such a bundle exists, since  $2{\\rm F}(\\mathcal{L})\\geq {\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1)) \\geq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{L},p))$. \n\nDefine a partition---$A'_i$ and $A'_j$---of $A^*_i$ as follows: \\\\\n\\noindent\n(i) Initialize $A'_j$ to be the empty set. Then, we keep transferring goods from $A_i^*(\\mathcal{L},p)$ to $A'_j$ (one at a time and in an arbitrary order) and stop as soon as the value of $A'_j$ exceeds $\\frac{v(A^*_i(\\mathcal{L},p))}{2}$.\\\\\n(ii) Denote the remaining set of goods as $A'_i \\coloneqq A^*_i(\\mathcal{L},p) \\setminus A'_j$. Note that, by construction, $v(A'_j)\\geq \\frac{v(A^{*}_i(\\mathcal{L},p))}{2}$ and \n$v(A'_i)\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right ) v(A^*_i(\\mathcal{L},p))$. The last inequality follows from the fact that $v$ is subadditive and the assumption that goods in $A^*_i(\\mathcal{L},p)$ are of value at most $\\frac{1}{40}v(A^*_i(\\mathcal{L},p))$. \\\\\n(iii) Write $\\mathcal{B}= \\{B_1, \\ldots, B_N\\}$ to denote the allocation obtained by replacing the bundles $A^{*}_i(\\mathcal{L},p)$ and $A^{*}_j(\\mathcal{L},p)$ in $\\mathcal{A}^*(\\mathcal{L},p)$ by $A'_i$ and $A^{*}_j(\\mathcal{L},p) \\cup A'_j$, respectively: $B_i = A'_i$ and $B_j = A^*_j(\\mathcal{L},p) \\cup A'_j$ along with $B_\\ell = A^*_\\ell (\\mathcal{L},p)$ for all $\\ell \\in [N] \\setminus \\{i, j \\}$. \n\nWe will show that $\\mathcal{B}$ has $p$-mean welfare strictly greater than that of the $p$-optimal allocation ${\\mathcal{A}^{*}}(\\mathcal{L},p)$. Hence, by way of contradiction, the desired result follows.\n\nRecall that the current case addresses exponent parameters that are negative, $p \\in (-\\infty, 0)$. Hence, the following  inequality implies that the $p$-mean welfare of $\\mathcal{B}$ is strictly greater than that of $\\mathcal{A}^*(\\mathcal{L},p)$:\n\\begin{align}\nv(A'_i)^p+v(A'_j)^p< v(A^{*}_i(\\mathcal{L},p))^p+v(A^{*}_j(\\mathcal{L},p))^p \\label{ineq:desired-neg-p}\n\\end{align}\n\nHowever, for negative $p$, the lower bounds on the values of $A'_i$ and $A'_j$ gives us\n\\begin{align*}\nv(A'_i)^p+v(A'_j)^p\\leq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p v(A^{*}_i(\\mathcal{L},p))^p + \\left (\\frac{1}{2} \\right )^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nIn addition, using the bounds $v(A^*_j(\\mathcal{L},p)) \\leq 2 {\\rm F}( \\mathcal{L}) < \\frac{2}{11.33} v(A^*_i(\\mathcal{L},p))$, we get  \n\\begin{align*}\nv(A^{*}_i(\\mathcal{L},p))^p+ v(A^{*}_j(\\mathcal{L},p))^p & > v(A^{*}_i(\\mathcal{L},p))^p+ \\left( \\frac{2}{11.33} \\right)^p v(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nTherefore, the desired equation (\\ref{ineq:desired-neg-p}) follows from the following numeric inequality, which is established in Appendix \\ref{app 3}.\n\\begin{align*}\n \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p + \\left (\\frac{1}{2} \\right )^p &\\leq 1+ \\left( \\frac{2}{11.33} \\right)^p \\quad \\text{ for all } p\\in (-\\infty,0).\n\\end{align*}\nThis establishes Lemma \\ref{Good_Transfer} for $p \\in (-\\infty,0)$.\n\n\\subsection{Proof of Lemma \\ref{Good_Transfer} for $p \\in (0, 0.4)$}\n\\label{subsection:p-zero-half-good-transfer}\nAssume, towards a contradiction, that $v(A^{*}_i(\\mathcal{L},p)) > 11.33\\ {\\rm F}(\\mathcal{L}) $, for some $i \\in [N] $, and $v(g) \\leq  \\frac{1}{40} v(A^{*}_i(\\mathcal{L},p))$ for all $g \\in A^{*}_i(\\mathcal{L},p).$ Recall that the $1$-mean welfare of the allocation returned by Feige's algorithm satisfies ${\\rm F}(\\mathcal{L}) \\geq \\frac{1}{2}{\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1))$. Pick a bundle $A^{*}_j(\\mathcal{L},p)$ (in $\\mathcal{A}^*(\\mathcal{L}, p)$) with the property that $v(A^{*}_j(\\mathcal{L},p)) \\leq 2{\\rm F}(\\mathcal{L})$. Such a bundle exists, since  $2{\\rm F}(\\mathcal{L})\\geq {\\rm M}_1({\\mathcal{A}^{*}}(\\mathcal{L},1)) \\geq {\\rm M}_1(\\mathcal{A}^*(\\mathcal{L},p))$. \n\nDefine a partition---$A'_i$ and $A'_j$---of $A^*_i(\\mathcal{L},p)$ as follows: \\\\\n\\noindent\n(i) Initialize $A'_j$ to be the empty set. Then, we keep transferring goods from $A_i^*(\\mathcal{L},p)$ to $A'_j$ (one at a time and in an arbitrary order) and stop as soon as the value of $A'_j$ exceeds $\\frac{v(A^*_i(\\mathcal{L},p))}{2}$.\\\\\n(ii) Denote the remaining set of goods as $A'_i \\coloneqq A^*_i(\\mathcal{L},p) \\setminus A'_j$. Note that, by construction, $v(A'_j)\\geq \\frac{v(A^{*}_i(\\mathcal{L},p))}{2}$ and \n$v(A'_i)\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right ) v(A^*_i(\\mathcal{L},p))$. The last inequality follows from the fact that $v$ is subadditive and the assumption that goods in $A^*_i(\\mathcal{L},p)$ are of value at most $\\frac{1}{40}v(A^*_i(\\mathcal{L},p))$. \\\\\n(iii) Write $\\mathcal{B}= \\{B_1, \\ldots, B_N\\}$ to denote the allocation obtained by replacing the  bundles $A^{*}_i(\\mathcal{L},p)$ and $A^{*}_j(\\mathcal{L},p)$ in $\\mathcal{A}^*(\\mathcal{L},p)$ by $A'_i$ and $A^{*}_j(\\mathcal{L},p) \\cup A'_j$, respectively: $B_i = A'_i$ and $B_j = A^*_j(\\mathcal{L},p) \\cup A'_j$ along with $B_\\ell = A^*_\\ell (\\mathcal{L},p)$ for all $\\ell \\in [N] \\setminus \\{i, j \\}$. \n\nWe will show that $\\mathcal{B}$ has $p$-mean welfare strictly greater than that of the $p$-optimal allocation ${\\mathcal{A}^{*}}(\\mathcal{L},p)$. Hence, by way of contradiction, the desired result follows.\n\nNotice that the construction of $A'_i$, $A'_j$ and $\\mathcal{B}$ hold for $p=0$ as well. We will use these sets in the next section to prove an analogous result for Nash social welfare. \n\nThe current case addresses exponent parameters that are positive $p \\in (0, 0.4)$. Hence, the following  inequality implies that the $p$-mean welfare of $\\mathcal{B}$ is strictly greater than that of $\\mathcal{A}^*(\\mathcal{L},p)$\n\\begin{align}\nv(A'_i)^p+v(A'_j)^p & > v(A^{*}_i(\\mathcal{L},p))^p+v(A^{*}_j(\\mathcal{L},p))^p \\label{ineq:desired-pos-p}\n\\end{align}\n\nHowever, for positive $p$, the lower bounds on the values of $A'_i$ and $A'_j$ gives us\n\\begin{align*}\nv(A'_i)^p+v(A'_j)^p & \\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p v(A^{*}_i(\\mathcal{L},p))^p + \\left (\\frac{1}{2} \\right )^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nIn addition, using the bounds $v(A^*_j(\\mathcal{L},p)) \\leq 2 {\\rm F}( \\mathcal{L}) < \\frac{2}{11.33} v(A^*_i(\\mathcal{L},p))$, we get  \n\\begin{align*}\nv(A^{*}_i(\\mathcal{L},p))^p+ v(A^{*}_j(\\mathcal{L},p))^p & < v(A^{*}_i(\\mathcal{L},p))^p + \\left( \\frac{2}{11.33} \\right)^pv(A^{*}_i(\\mathcal{L},p))^p.\n\\end{align*}\n\nTherefore, the desired equation (\\ref{ineq:desired-pos-p}) follows from the following numeric inequality, which is established in Appendix \\ref{app 3}.\n\\begin{align*}\n \\left (\\frac{1}{2}-\\frac{1}{40} \\right )^p + \\left (\\frac{1}{2} \\right )^p &\\geq 1+ \\left( \\frac{2}{11.33} \\right)^p \\hbox{ for } p\\in (0,0.4).\n\\end{align*}\n\nThis establishes Lemma \\ref{Good_Transfer} for $p \\in (0, 0.4)$.\n\n\n\\subsection{Proof of Lemma \\ref{Good_Transfer} for Nash Social Welfare ($p=0$)}\n\\label{subsection:p-zero-good-transfer}\n\n\nRecall the sets $A'_i$, $A'_j$, and the allocation $\\mathcal{B}$ defined in Section \\ref{subsection:p-zero-half-good-transfer}. We will show that $\\mathcal{B}$ has Nash welfare strictly greater than that of ${\\mathcal{A}^{*}}(\\mathcal{L}, 0)$, and hence, by way of contradiction, establish the desired result.\n\n In order to obtain ${\\rm M}_0(\\mathcal{B})>{\\rm M}_0(\\mathcal{A}^*(\\mathcal{L},0))$, it suffices to prove that $v(A'_i)v(A'_j)>v\\left( A^{*}_i(\\mathcal{L},0)\\right)v( A^{*}_j(\\mathcal{L},0)).$\nThe lower bounds we obtained on the values of $A'_i$ and $A'_j$ gives us \n\\begin{align*}\nv(A'_i)v(A'_j)&\\geq \\left (\\frac{1}{2}-\\frac{1}{40} \\right )v\\left( A^{*}_i (\\mathcal{L},0)\\right) \\frac{v\\left( A^{*}_i (\\mathcal{L},0)\\right)}{2}\n\\geq  0.2 \\ v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2 \n\\end{align*}\n\nIn addition, we have \n\\begin{align*}\nv\\left( A^{*}_i (\\mathcal{L},0)\\right) v\\left( A^{*}_j (\\mathcal{L},0)\\right)  & <\\frac{2}{11.33} \\  v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2\n<0.18 \\ v\\left( A^{*}_i (\\mathcal{L},0)\\right)^2 \n\\end{align*}\nTherefore, the lemma holds for $p=0$ as well. \n\n\n\n\\section{Combination Lemma} \\label{section:stitching-lemma}\nThe following lemma shows that the goods assigned as singletons in the while-loop of $\\textsc{Alg}$ (Algorithm~\\ref{Alg}), along with a $p$-optimal allocation of instance $\\mathcal{J}$ that remains at the termination of the loop, lead to a $p$-mean welfare that is comparable to the optimal, for all $p \\in (-\\infty, 0.4)$.\n\nAs shown previously in Lemma~\\ref{Low_valued}, $\\textsc{AlgLow}$---with instance $\\mathcal{J}$ as input---achieves a constant-factor approximation for the $p$-mean welfare objective. Hence, the lemma established in this section will enable us to combine the welfare guarantees of the goods assigned in the while-loop of \\textsc{Alg} and the allocation computed by \\textsc{AlgLow} to obtain the desired approximation result for $p \\in (-\\infty, 0.4)$. \n\nSpecifically, given a fair-division instance $\\mathcal{I} = \\langle [m], [n], v \\rangle$ as input, let $\\{g_1, \\ldots, g_k\\}$ denote the set of goods that get assigned as singletons in the while-loop of $\\textsc{Alg}$ (Algorithm~\\ref{Alg}). Furthermore, for $1 \\leq t \\leq k$, let $\\mathcal{I}^t$ denote the instance obtained at the end of the $t$th iteration of this while-loop. Since in the first $t$ iterations $\\textsc{Alg}$ assigns goods $\\{g_1, \\ldots, g_t\\}$ to the first $t$ agents as singletons, we have $\\mathcal{I}^t = \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$. In particular, $\\mathcal{J} \\coloneqq \\mathcal{I}^k$ is the instance that remains after the termination of the while-loop in $\\textsc{Alg}$ and this instance is passed on to $\\textsc{AlgLow}$ as input.\n\nInstance $\\mathcal{J}$ consists of $n-k$ agents and, hence, in $\\mathcal{J}$, any $p$-optimal allocation $\\mathcal{A}^*(\\mathcal{J},p)$ contains $(n-k)$ bundles. For notational convenience, we will index these bundles from $k+1$ to $n$, i.e., $\\mathcal{A}^*(\\mathcal{J},p)= \\left( A^*_{k+1} (\\mathcal{J},p), \\ldots, A^*_{n} (\\mathcal{J},p) \\right)$.\n\n\n\\begin{Lemma}\\label{Induction_argument}\nGiven a fair-division instance $\\mathcal{I} = \\langle [m], [n], v \\rangle$ with an identical subadditive valuation $v$, let $\\{g_1, \\ldots, g_k\\}$ denote the set of goods that get assigned as singletons in the while-loop of $\\textsc{Alg}$ and let $\\mathcal{J}=\\langle [m] \\setminus \\{g_1, \\ldots, g_k \\}, [n] \\setminus [k], v \\rangle$ be the instance that remains after the termination of this loop. In addition, let $\\mathcal{A}^*(\\mathcal{I},p)=\\left(A^*_1 (\\mathcal{I},p), \\ldots, A^*_n(\\mathcal{I},p) \\right)$ and $\\mathcal{A}^*(\\mathcal{J},p) = \\left( A^*_{k+1} (\\mathcal{J},p), \\ldots, A^*_{n} (\\mathcal{J},p) \\right)$ denote $p$-optimal allocations of instances $\\mathcal{I}$ and $\\mathcal{J}$, respectively. Then, with constant $\\alpha = 40$,   \n\\begin{itemize}\n\\item For $p \\in (-\\infty, 0)$, we have $\\alpha^p  \\sum \\limits_{i=1}^k v(g_i)^p \\ + \\ \\sum \\limits_{j=k+1}^{n} v (A^{*}_j(\\mathcal{J},p) )^p \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p ))^p$.  \n\n\\item For $p \\in (0,0.4)$, we have $\\alpha^p \\sum \\limits_{i=1}^k v \\left( g_i\\right)^p  \\ +  \\ \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{J},p)\\right)^p \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p ))^p$.\n\\item  For $p=0$, we have $\\alpha^k \\prod \\limits_{i=1}^k v(g_i) \\  \\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{J},p) ) \\geq  \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},p))$. \n \n\\end{itemize}\n\\end{Lemma}\nWe will prove Lemma~\\ref{Induction_argument} by considering different ranges of the exponent parameter $p$ separately. However, in all of the ranges, the desired inequality is obtained by inducting on the number of iterations of the while-loop in \\textsc{Alg}. \n\n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for $p\\in(-\\infty,0)$}\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $p$-mean optimal, $\\mathcal{A}^*(\\mathcal{I}^t,p)=\\left( A^*_{t+1}(\\mathcal{I}^{t},p),\\ldots , A^*_n(\\mathcal{I}^t,p)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\\begin{align}\\label{7}\n(40) ^p\\sum \\limits_{i=1}^t v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^t,p)\\right)^p & \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align}\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{7}) are equal to each other. Therefore, the base case holds.\n\n\\noindent\n\\emph{Induction Step:} We establish inequality  (\\ref{7}) for $t$, assuming that it holds for $t-1$. \n\n Consider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},p)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is a $p$-optimal allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p) =\\left( A^*_{t}(\\mathcal{I}^{t-1},p),\\ldots , A^*_n(\\mathcal{I}^{t-1},p) \\right)$. \n \n We lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},p)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},p)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},p))$; since, $3.53 \\times 11.33 \\leq 40$. \\\\\n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},p)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the highest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},p )) \\label{8}\n\\end{align}\nNote that inequality (\\ref{8}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. Furthermore, since the current case addresses negative $p \\in (-\\infty, 0)$, we have $ (40)^p \\ v(g_{t})^p\\leq v(A^*_{t}(\\mathcal{I}^{t-1},p  ))^p$. \n\nWe add $(40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p $ to both sides of the previous inequality to obtain\n\n\\begin{align}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\leq (40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p +v(A^*_{t}(\\mathcal{I}^{t-1},p  ))^p +\\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p \\label{ineq:interim}\n\\end{align}\n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},p )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1},p)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1},p)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t},p)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},p) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},p)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is the $p$-optimal allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align*}\n\\left(\\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p\\right)^\\frac{1}{p} & \\leq \\left( \\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j( \\mathcal{I}^{t},p)\\right)^p \\right)^\\frac{1}{p}.\n\\end{align*}\n\nExponentiating both sides by $p$ (which in the current case is negative) and multiplying by $n-t$, gives us $ \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\geq \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p .$ Therefore, inequality (\\ref{ineq:interim}) extends to\n\n\\begin{align*}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\leq 40 ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\\\\n& \\leq \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align*}\nThe last inequality follows from the induction hypothesis. Setting $t=k$ gives us the desired inequality for $p\\in(-\\infty,0)$\n\\begin{align}\\label{9}\n( 40 )^p\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^k,p)\\right)^p & \\leq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align}\nTherefore, Lemma \\ref{Induction_argument} holds for $p\\in (-\\infty,0)$.\n\n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for $p\\in (0,0.4)$}\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $p$-mean optimal, $\\mathcal{A}^*(\\mathcal{I}^t,p)=\\left( A^*_{t+1}(\\mathcal{I}^{t},p),\\ldots , A^*_n(\\mathcal{I}^t,p)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\\begin{align} \n(40) ^p\\sum \\limits_{i=1}^t v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^t,p)\\right)^p & \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \\label{ineq:ind}\n\\end{align}\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{7}) are equal to each other. Therefore, the base case holds.\n\n\\noindent\n\\emph{Induction Step:} We establish inequality  (\\ref{ineq:ind}) for $t$, assuming that it holds for $t-1$. \n\nConsider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},p)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is a $p$-optimal allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p) =\\left( A^*_{t}(\\mathcal{I}^{t-1},p),\\ldots , A^*_n(\\mathcal{I}^{t-1},p) \\right)$. \\\\\n\nWe lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},p)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},p)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},p))$; since, $3.53 \\times 11.33 \\leq 40$.  \n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},p)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the highest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},p )) \\label{ineq:g-t-val}\n\\end{align}\n\nNote that inequality (\\ref{ineq:g-t-val}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. Furthermore, since the current case addresses positive $p\\in (0,0.4)$, we have $ (40)^p \\ v(g_{t})^p \\geq v(A^*_{t}(\\mathcal{I}^{t-1},p  ))^p$. \n\nWe add $(40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p $ to both sides of the previous inequality to obtain\n\n\\begin{align}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\geq (40) ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p +v(A^*_{t}(\\mathcal{I}^{t-1},p  ))^p +\\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p \\label{ineq:interim-positive}\n\\end{align}\n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},p)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},p )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1},p)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1},p)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t},p)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},p) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},p)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t},p)$ is the $p$-optimal allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align*}\n\\left(\\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p\\right)^\\frac{1}{p} & \\leq \\left( \\frac{1}{n-t} \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j( \\mathcal{I}^{t},p)\\right)^p \\right)^\\frac{1}{p}.\n\\end{align*}\n\nExponentiating both sides by $p$ (which in the current case is positive) and multiplying by $n-t$, gives us $ \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\leq \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p .$ Therefore, via inequality (\\ref{ineq:interim-positive}), we obtain\n\\begin{align*}\n(40) ^p \\sum \\limits_{i=1}^{t} v \\left( g_i\\right)^p + \\sum \\limits_{j=t+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t},p)\\right)^p & \\geq 40 ^p \\sum \\limits_{i=1}^{t-1} v \\left( g_i\\right)^p + \\sum \\limits_{j=t}^{n} v\\left(A^{*}_j(\\mathcal{I}^{t-1},p)\\right)^p \\\\\n& \\geq \\sum \\limits _{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p\n\\end{align*}\nThe last inequality follows from the induction hypothesis. Setting $t=k$ gives us the desired inequality for $p \\in (0, 0.4)$\n\\begin{align} \n( 40 )^p\\sum \\limits_{i=1}^k v \\left( g_i\\right)^p + \\sum \\limits_{j=k+1}^{n} v\\left(A^{*}_j(\\mathcal{I}^k,p)\\right)^p & \\geq \\sum \\limits_{i=1}^n v(A^{*}_i(\\mathcal{I},p))^p \n\\end{align}\nTherefore, Lemma \\ref{Induction_argument} holds for $p \\in (0, 0.4)$. \n\n\\subsection{Proof of Lemma \\ref{Induction_argument} for Nash Social Welfare ($p=0$)}\n\\label{subsection:p-zero}\n\nFor $0 \\leq t \\leq k$, recall that instance $\\mathcal{I}^t \\coloneqq \\langle [m]\\setminus \\{g_1, \\ldots, g_t \\}, [n]\\setminus \\{1, \\ldots, t\\}, v \\rangle$ and its corresponding $0$-mean optimal (i.e., Nash optimal), $\\mathcal{A}^*(\\mathcal{I}^t, 0)=\\left( A^*_{t+1}(\\mathcal{I}^{t},0),\\ldots , A^*_n(\\mathcal{I}^t,0)\\right)$. \n\nWe prove by induction over all $0\\leq t \\leq k, $ that\n\n\\begin{align}\\label{13}\n \\prod \\limits_{i=1}^t v(g_i) \\  \\prod \\limits_{j=t+1}^{n} v(A^*_j (\\mathcal{I}^t,0) ) \\geq  \\left( \\frac{1}{40}\\right)^t \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0)).\n\\end{align}\n\n\\noindent\n\\emph{Base Case:} When $t=0,$ we have $\\mathcal{I}^t = \\mathcal{I}$ and, hence, both sides of equation (\\ref{13}) are equal to each other. Therefore, the base case holds. \\\\\n\n\n\\noindent\n\\emph{Induction Step:} We establish inequality  (\\ref{13}) for $t$, assuming that it holds for $t-1$.  \n\n Consider the good $g_{t}$ that was assigned in the $t${th} iteration of the while-loop in \\textsc{Alg}. Note that $v(g_{t}) \\geq \\frac{1}{3.53}{\\rm F}(\\mathcal{I}^{t-1})$ (see Step \\ref{Threshold}). Without loss of generality, we may assume that $g_{t} \\in A^{*}_{t}(\\mathcal{I}^{t-1},0)$. This assumption is justified since ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0)$ is a $0$-optimal (Nash optimal) allocation of the instance $\\mathcal{I}^{t-1}$ and, hence, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0)$ is an $(n-t+1)$-partition of the goods $[m] \\setminus \\{g_1, g_2, \\ldots, g_{t-1}\\}$, i.e., $g_t$ belongs to one of bundles in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1}, 0) =\\left( A^*_{t}(\\mathcal{I}^{t-1},0),\\ldots , A^*_n(\\mathcal{I}^{t-1},0) \\right)$. \\\\\n \n We lower bound the value of $g_t$ in terms of the value of the bundle $A^*_t(\\mathcal{I}^{t-1},0)$.\n\n\\noindent {Case {\\rm I}:} $v(A^*_{t} (\\mathcal{I}^{t-1},0)) \\leq 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. In this case, we have $v(g_{t}) \\geq \\frac{1}{3.53} {\\rm F}(\\mathcal{I}^{t-1}) \\geq \\frac{1}{40} v({A_{t}^{*}}(\\mathcal{I}^{t-1},0))$; since, $3.53 \\times 11.33 \\leq 40$. \n\n\\noindent {Case {\\rm II}:} $v(A^*_{t}(\\mathcal{I}^{t-1},0)) > 11.33 \\ {\\rm F}(\\mathcal{I}^{t-1})$. Recall that the goods are indexed in non-increasing order of value (see Step \\ref{Ordered_Goods} of \\textsc{Alg}) and, hence, $g_t$ is the largest valued good in the instance $\\mathcal{I}^{t-1}$. Therefore, Lemma \\ref{Good_Transfer} gives us \n\\begin{align}\nv(g_{t})\\geq \\frac{1}{40} v(A^*_{t}(\\mathcal{I}^{t-1},0 )) \\label{ineq:val-g-t-0}\n\\end{align}\n\n\nHere, inequality (\\ref{ineq:val-g-t-0}) holds in both Cases {\\rm I} and {\\rm II} mentioned above. \n\nNote that the allocation ${\\mathcal{A}^{*}}(\\mathcal{I}^{t-1},0)$ is defined over the goods $[m]\\setminus \\{g_1,\\ldots , g_{t-1}\\}$ and the bundle $A^{*}_{t}(\\mathcal{I}^{t-1},0 )$ contains $g_{t}$. On the other hand, ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$ is defined over $[m]\\setminus \\{g_1,\\ldots , g_{t}\\}$. Hence, all the goods in ${\\mathcal{A}^*}(\\mathcal{I}^{t-1}, 0)$, with the exception of $g_{t}$, appear in ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$. \n\nIn other words, the last $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t-1}, 0)$ and all the $n-t$ bundles of $\\mathcal{A}^*(\\mathcal{I}^{t}, 0)$ satisfy $\\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t-1},0) \\subseteq \\bigcup \\limits _{j=t+1}^n A^{*}_j(\\mathcal{I}^{t},0)$. Using this containment and the fact that ${\\mathcal{A}^{*}}(\\mathcal{I}^{t}, 0)$ is the $0$-optimal (Nash optimal) allocation for the instance $\\mathcal{I}^{t}$, we have\n\\begin{align}\n\\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right ) > \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t-1},0)\\right ) \\label{ineq:prod-induction}\n\\end{align}\n\nMultiplying by $v(g_1) \\ v(g_2) \\ldots v(g_{t-1}) \\ v(g_{t})$ on both sides of the previous inequality and using equation (\\ref{ineq:val-g-t-0}), we get\n\\begin{align*}\nv(g_1) \\ v(g_2) \\ldots v(g_{t-1}) \\ v(g_{t}) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right) & \\geq \\frac{1}{40}v\\left(g_1\\right)\\ldots v\\left(g_{t-1}\\right) \\ v\\left(A^{*}_{t}(\\mathcal{I}^{t-1},0)\\right) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t},0)\\right) \\\\\n& \\geq \\frac{1}{40}v\\left(g_1\\right)\\ldots v\\left(g_{t-1}\\right) \\ v\\left(A^{*}_{t}(\\mathcal{I}^{t-1},0)\\right) \\prod\\limits_{j=t+1}^n v\\left(A^{*}_j(\\mathcal{I}^{t-1},0)\\right) \\tag{via inequality (\\ref{ineq:prod-induction})}\\\\\n  &\\geq \\left(\\frac{1}{40}\\right)^{t} \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0))  \\tag{using the induction hypothesis}\n\\end{align*}\n\n\nSetting $t=k$ gives us the desired inequality for $p =0$\n\\begin{align*}\n(40)^k \\prod \\limits_{i=1}^k v(g_i) \\  \\prod \\limits_{j=k+1}^{n} v(A^*_j (\\mathcal{I}^k,0) ) \\geq  \\prod \\limits_{i=1}^{n} v(A^*_i(\\mathcal{I},0)).\n\\end{align*}\nTherefore, Lemma \\ref{Induction_argument} holds for $p =0$ as well. \\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn \\cite{FFT16}, a special class of group actions on CAT(0) cube complexes is defined by the behaviour of halfspaces under the action (see Definition \\ref{RAAGlike} in this paper). Since they generalise right-angled Artin groups (RAAGs) acting on the associated RAAG-complex (the universal cover of the associated Salvetti complex), they are called \\emph{RAAG-like} actions.\\par\nThe authors define quasimorphisms on all groups acting non-transversally \\footnote{A group acts \\emph{non-transversally}, if orbits of halfspaces are nested. In particular, RAAG-like actions are non-transverse.} on CAT(0) cube complexes, generalising Fujiwara--Epstein counting quasimorphisms on free groups (introduced in \\cite{EF97}). The quasimorphisms in \\cite{FFT16} are shown to have defect of at most $6$ and, hence, their homogenisation at most $12$. This is done using the median property of CAT(0) spaces.\\par\nThe authors then prove \\emph{effectiveness} of these quasimorphisms, that is, for every element $g$ acting hyperbolically on the CAT(0) cube complex, one of their homogeneous quasimorphisms $\\overline{\\phi}$ satisfies $\\overline{\\phi}(g)\\geq 1$. For this, using Haglunds combinatorial axis (\\cite{Hag07}), they isolate a subcomplex with respect to $g$ called `essential set' and construct a $g$-equivariant embedding of this into a Euclidean space. They then use a rather intricate series of technical lemmas to prove that $\\overline{\\phi}(g)\\geq 1$.\\par\nThe purpose of this paper is to show that these technical arguments can be avoided and replaced by a short proof that also uses the full properties of RAAG-like actions.\\par\nFinally, we conclude as in \\cite{FFT16} that hyperbolic elements of RAAG-like actions have stable commutator length at least $1\/24$ using the Bavard Duality. In particular, RAAGs have a stable commutator length gap of $1\/24$. Note that, using a different class of quasimorphisms, Heuer \\cite{Heu19} has already proved a better (and optimal) bound of $1\/2$ for the stable commutator length in RAAGs.\n\n\\section{Premliminaries}\n\\subsection{Halfspaces in CAT(0) cube complexes}\nLet $X$ be a CAT(0) cube complex. Denote by $\\mathcal{H}(X)$ the set of half-spaces and by $\\overline{\\Phi}$ the complement of a halfspace $\\Phi$. Two halfspaces $\\Phi,\\Psi \\in \\mathcal{H}$ are said to be \\emph{nested}, if either $\\Phi\\subseteq \\Psi$, $\\overline{\\Phi} \\subseteq \\Psi$, $\\Phi\\subseteq \\overline{\\Psi}$ or $\\overline{\\Phi}\\subseteq \\overline\\Psi$. Otherwise, they are \\emph{transverse}. Two distinct $\\Phi,\\Psi \\in \\mathcal{H}$ are \\emph{tightly nested}, if they are nested and there is no $\\Phi'\\in \\mathcal{H}$ with $\\Phi\\subsetneq \\Phi' \\subsetneq \\Psi$ or $\\overline{\\Phi}\\subsetneq \\Phi' \\subsetneq \\Phi$ etc..\\par\nFor every oriented edge $E$ in $X$, there is exactly one $\\Phi\\in\\mathcal{H}$ such that the beginning vertex of $E$ is in $\\Phi$ and the end vertex is in $\\overline{\\Phi}$. We say that $\\Phi$ and $E$ are \\emph{dual} to each other.\\par\n\nGiven $x,y\\in \\mathcal{H}$ the \\emph{interval} between $x$ and $y$ is \n$$[x,y]:= \\left\\{\\Phi\\in\\mathcal{H}: x\\notin\\Phi \\textrm{ and } y\\in\\Phi \\right\\}.$$\nGiven vertices $x,y,z\\in X$, there is a unique vertex $m(x,y,z)$ called \\emph{median} with the property $[a,b]=[a,m]\\cup[m,b]$ for any distinct pair $a,b$ in ${x,y,z}$ (see Preliminaries in \\cite{FFT16} for a simple proof).\n\n\n\\subsection{Haglund's combinatorial axis}\nLet $X$ be a CAT(0) cube complex. We can introduce a metric called \\emph{combinatorial distance} $d^c$ on the set of vertices $X^{(0)}$, by defining $d^c(x,y)$ as the minimal number of edges in an edge path from $x$ to $y$. A \\emph{combinatorial geodesic} is an optimal (with respect to $d^c$) oriented edge path.\\par\n\nThe \\emph{translation distance} of an automorphism $g$ of $X$ is the natural number\n$$\\delta(g)=\\min_{x\\in X^{(0)}} d^c(x,gx).$$\nAn automorphism $g$ of a CAT(0) cube complex $X$ is \\emph{hyperbolic}, if it fixes no vertex of $X$, i.e. $\\delta(g)>0$. Otherwise, $g$ is called \\emph{elliptic}. A combinatorial axis is an infinite combinatorial geodesic on which $g$ acts as a shift. According to Haglund in \\cite{Hag07}, if $g$ is a hyperbolic automorphism all of whose powers act without inversion (that is, there are no $\\Phi\\in\\mathcal{H}$ and $n\\in\\mathbb{Z}$ with $g^n\\Phi=\\overline{\\Phi}$), then every vertex in $X^{(0)}$ on which $g$ attains its translation distance is contained in some combinatorial axis.\\par\n\nAs in \\cite{FFT16}, let $A^+_g$ denote all halfspaces dual to an oriented edge in some combinatorial axis (indeed, a halfspace dual to some combinatorial axis is also dual to all other ones according to \\cite{Hag07}). Clearly, $A^+_g = \\bigcup_{n\\in \\mathbb{N}} [g^no,g^{n+1}o]$ for any vertex $o$ where $g$ attains its translation distance. An important fact is that for $\\Phi,\\Psi \\in A^+_g$ either $\\Phi\\subseteq \\Psi$, $\\Phi\\supseteq \\Psi$ or $\\Phi$ and $\\Psi$ are transverse, which follows because a combinatorial geodesic may never leave a halfspace after entering.\n\n\n\\subsection{The Bavard Duality}\nGiven a group $G$, the \\emph{commutator length} $\\textrm{cl}$ is a function $\\textrm{cl}: [G,G] \\rightarrow \\mathbb{N}$, where $[G,G]$ is the commutator subgroup. For $g\\in[G,G]$, it is defined as the minimal number of commutators whose product is $g$. The \\emph{stable commutator length} is the well defined limit \n$$\\textrm{scl}(g)= \\lim_{n\\to \\infty} \\frac{\\textrm{cl}(g^n)}{n}.$$\\par\nBounds of $\\textrm{scl}$ can be estimated using \\emph{homogeneous quasimorphisms} and the \\emph{Bavard Duality}. A quasimorphism is a function $\\phi: G\\rightarrow \\mathbb{R}$ with a bounded \\emph{defect} \n$$D(\\phi):= \\sup_{g,h\\in G} |\\phi(gh)-\\phi(g)-\\phi(h)|.$$ \nThe quasimorphism $\\phi$ is called homogeneous, if $\\phi(g^n)= n\\phi(g)$ for all $g$ and $n\\in \\mathbb{Z}$. Denote by $Q(G)$ the space of homogeneous quasimorphisms on $G$. Every quasimorphism yields a homogeneous quasimorphism called \\emph{homogenisation} $\\overline{\\phi}(g):= \\lim_{n\\to\\infty} \\frac{\\phi(g^n)}{n}$ with the following property:\n\\begin{mylem}\nLet $\\phi$ be a quasimorphism. Then its homogenisation satisfies $D(\\overline{\\phi})\\leq 2D(\\phi)$.\n\\end{mylem}\nThe Bavard Duality states that:\n\\begin{mythm}[\\cite{Bav91}]\\label{bavard}\nFor any $g\\in[G,G]$\n$$\\mathrm{scl}(g)= \\sup_{\\overline{\\phi}\\in Q(G)} \\frac{\\overline{\\phi}(g)}{2D(\\overline{\\phi}))}.$$\n\\end{mythm}\nTherefore, to prove that $\\textrm{scl}(g)$ is bounded from below by some constant, it suffices to find a homogeneous quasimorphism which has low enough defect (we say it is \\emph{efficient}) and at the same time does not vanish on $g$ (that is, it is \\emph{effective}).\\footnote{See \\cite[Chapter 2]{Cal09} for more detail and proofs on quasimorphisms and scl.}\n\n\\section{RAAG-like actions}\nLet us reproduce the definition of RAAG-like actions given in \\cite[Chapter 7]{FFT16}:\n\\begin{mydef}\\label{RAAGlike}\nLet $G$ be a group acting on a CAT(0) cube complex $X$ with halfspaces $\\mathcal{H}(X)$. The action is called \\emph{RAAG-like} if the following are satisfied:\n\\begin{enumerate}[(i)]\n\\item There are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $h\\overline{\\Phi} = \\Phi$ (``no inversions'')\n\\item there are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi$ and $h\\Phi$ transverse (``non-transverse''),\n\\item there are no tightly nested $\\Phi,\\Phi'\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi$ and $h\\Phi'$ transverse,\n\\item there are no $\\Phi\\in\\mathcal{H}(X)$ and $h\\in G$ with $\\Phi\\subset h\\overline{\\Phi}$ tightly.\n\\end{enumerate}\nA group is called RAAG-like, if it has a faithful RAAG-like action on some CAT(0) cube complex.\n\\end{mydef}\n\\begin{myrem}\nIf $G$ acts on $X$ freely, then RAAG-likeness of the action is equivalent to $X\/G$ being a \\emph{A-special} (often simply called \\emph{special}) cube complex in the sense of Haglund and Wise \\cite[Definition 3.2]{HW08}. In particular, we have the following correspondences: \n\\begin{enumerate}[(i)]\n\\item corresponds to all hyperplanes in $X\/G$ being two-sided, \n\\item corresponds to no hyperplane in $X\/G$ intersecting itself,\n\\item corresponds to no pair of hyperplanes in $X\/G$ inter-osculating and \n\\item corresponds to no pair of hyperplanes in $X\/G$ directly self-osculating.\n\\end{enumerate}\nHence $G$ is the fundamental group of an A-special cube complex and conversely the fundamental group of an A-special cube complex acts RAAG-like and freely on its universal cover. In particular, RAAGs are RAAG-like, as they are the fundamental group of an A-special cube complex (their Salvetti complex).\n\\end{myrem}\n\\begin{mylem}\\label{RAAGhyperbolic}\nEvery non-trivial element of a RAAG-like action is hyperbolic.\n\\end{mylem}\n\\begin{proof}\nSuppose $h\\in G$ is elliptic, i.e. $h$ has at least one fixed vertex, and acts non-trivially. If for some fixed vertex of $h$ in $X$, every incident edge is fixed, then all neighbouring vertices of $v$ are also fixed. Therefore, there must be some fixed vertex $v$ with an incident edge which is not fixed, or else every single vertex of $X$ would be fixed. Let $E$ be adjacent to $v$ and mapped to some other edge $F$ adjacent to $v$. If $E$ and $F$ bound a square, then $h$ is transverse, as the halfspace $\\Phi$ dual to $E$ is transverse to $h\\Phi$, the halfspace dual to $F$. If they do not bound a square, then $\\Phi$ and $h\\Phi$ are tightly nested if they are not transverse.\n\\end{proof}\n\n\\section{The quasimorphisms and their defect}\nFrom now on, let $G$ be a group with a non-transverse (not necessarily RAAG-like) action on a CAT(0) cube complex $X$.\\par\nWe recall the quasimorphisms defined in \\cite[Chapter 4]{FFT16} and, for completeness, the proof that their defect is bounded by $12$.\n\n\\begin{mydef}\nA \\emph{segment} is a series of half-spaces $\\gamma= \\left\\{\\Phi_0,..., \\Phi_r\\right\\}$ such that $\\Phi_i\\supsetneq \\Phi_{i+1}$ \\emph{tightly} for $0\\leq i<r$.\\\\\nThe \\emph{reverse} segment of $\\gamma$ is $\\overline{\\gamma}= \\left\\{\\overline{\\Phi}_r,..., \\overline{\\Phi}_0\\right\\}$.\n\\end{mydef}\n\\begin{mydef}\nTwo segments $\\gamma_1, \\gamma_2\\in \\mathcal{H}(X)$ are said to \\emph{overlap}, if there is $\\Phi_1\\in \\gamma_1$ and $\\Phi_2\\in \\gamma_2$ that are equal or transverse. A set of segments is called \\emph{non-overlapping}, if no two of its segments overlap.\n\\end{mydef}\n\\begin{myrem}\nIf $\\gamma= \\left\\{\\Phi_0,..., \\Phi_r\\right\\}$ and $\\gamma'= \\left\\{\\Psi_0,..., \\Psi_s\\right\\}$ are non-overlapping segments, then one of the following holds:\n\\begin{itemize}\n\\item[] $\\Phi_0\\supset...\\supset \\Phi_r\\supset \\Psi_0\\supset... \\supset\\Psi_s$\n\\item[] $\\Psi_0\\supset...\\supset \\Psi_s\\supset \\Phi_0\\supset... \\supset\\Phi_r$\n\\item[] $\\overline{\\Phi}_r\\supset...\\supset \\overline{\\Phi}_0\\supset \\Psi_0\\supset... \\supset\\Psi_s$\n\\item[] $\\Psi_0\\supset...\\supset \\Psi_s\\supset \\overline{\\Phi}_r\\supset...\\supset \\overline{\\Phi}_0$\n\\end{itemize}\nOtherwise, it can easily be seen, that tight nestedness of the segments is contradicted. We write $\\gamma>\\gamma'$, $\\gamma'>\\gamma$, $\\overline{\\gamma}> \\gamma'$ and $\\gamma'> \\overline{\\gamma}$ respectively in these cases.\n\\end{myrem}\n\\begin{myrem}\nIf $S$ is a set of non-overlapping segments, then for any $\\gamma_1, \\gamma_2\\in S$ either $\\gamma_1> \\gamma_2$ or $\\gamma_2> \\gamma_1$. Thus, if $S$ is finite, it must contain a maximal segment that contains every other segment in $S$, and a minimal segment that is contained by every other segment in $S$, respectively.\n\\end{myrem}\n\\begin{mydef}\nGiven a segment $\\gamma$, let $G\\gamma = \\{g\\gamma: g\\in G\\}$ denote the set of \\emph{copies of $\\gamma$}. The function $c_{\\gamma}: X^2\\rightarrow \\mathbb{R}$ is defined sucht that $c_{\\gamma}(x,y)$ is the cardinality of the largest non-overlapping subset of $G\\gamma$ in $[x,y]$.\\par\nFurthermore, define $\\omega_{\\gamma}:X^2\\rightarrow \\mathbb{R}$ by $\\omega_{\\gamma}(x,y):= c_{\\gamma}(x,y) - c_{\\bar{\\gamma}}(x,y)$.\n\\end{mydef}\n\\begin{myrem}\n$\\omega_{\\gamma}(\\cdot,\\cdot)$ is $G$-invariant, i.e. $\\omega_{\\gamma}(x,y)= \\omega_{\\gamma}(gx,gy)$ for any $x,y\\in X$ and $g\\in G$, since any non-overlapping subset of $G\\gamma$ in $[x,y]$ can be pushed by $g$ to one in $[gx,gy]$, and vice versa.\\par\nFurthermore, $\\omega_{\\gamma}(\\cdot,\\cdot)$ is antisymmetric, since if $g\\gamma\\in [x,y]$, then $g\\overline{\\gamma}\\in [y,x]$, and vice versa.\n\\end{myrem}\nThe following lemmas show that $\\omega_{\\gamma}(o,go)$ as a function of $g$ (where $o$ is any vertex of $X$) is a quasimorphism.\n\\begin{mylem}\nFor $x,m,y\\in X$ with $m=m(x,m,z)$,\n$$\\left| \\omega_{\\gamma}(x,y)-\\omega_{\\gamma}(x,m) -\\omega_{\\gamma}(m,y)\\right| < 2$$\nholds.\n\\end{mylem}\n\\begin{proof}\nLet us first prove $c_{\\gamma}(x,y)\\geq c_{\\gamma}(x,m)+ c_{\\gamma}(m,y)-1$. Let $S_1$ and $S_2$ be maximal non-overlapping sets of copies of $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r\\right\\}$ in $[x,m]$ and $[m,y]$, respectively. Let $g\\gamma$ be the minimal element of $S_1$. We have $a\\gamma> b\\gamma$ for any $a\\gamma\\in S_1\\setminus\\{g\\gamma\\}$ and $b\\gamma\\in S_2$, because for any $a\\Phi_k \\in a\\gamma$ and $b\\Phi_l \\in b\\gamma$ we have $a\\Phi_k\\subsetneq g\\Phi_0 \\subsetneq b\\Phi_0 \\subsetneq b\\Phi_l$. Thus, $S_1\\setminus\\{g\\gamma\\}$ and $S_2$ do not overlap, whence the inequality follows.\\par\nLet us now prove $c_{\\gamma}(x,z)\\leq c_{\\gamma}(x,m)+ c_{\\gamma}(m,y) +1$. Let $S$ be a maximal set of copies of $\\gamma$ in $[x,m]$. There can be at most one copy $g\\gamma\\in S$ containing halfspaces $\\Phi$ and $\\Psi$ such that $y\\in\\Phi$ and $y\\notin \\Psi$ since all other copies of $\\gamma$ in $S$ either elementwise contain $\\Phi$ or are contained elementwise in $\\Psi$. The remaining $|S|-1$ copies can be assigned to sets $S_1$ and $S_2$ contained in $[x,y]$ and $[m,y]$, respectively, which proves the inequality.\\par\nThis proves $|c_{\\gamma}(x,y)- c_{\\gamma}(x,m)- c_{\\gamma}(m,y)| <1$ and therefore the lemma, as\n\\begin{equation*}\n\\begin{split}\n&\\left|\\omega_{\\gamma}(x,y)-\\omega_{\\gamma}(x,m) -\\omega_{\\gamma}(m,y)\\right|\\\\\n &\\leq \\left|c_{\\gamma}(x,y)-c_{\\gamma}(x,m) -c_{\\gamma}(m,y)\\right|\n+ \\left|c_{\\bar{\\gamma}}(x,y)-c_{\\bar{\\gamma}}(x,m) -c_{\\bar{\\gamma}}(m,y) \\right|\\\\\n&\\leq 2\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\begin{mylem}\\label{omega}\nFor any $x,y,z\\in X$\n$$\\left|\\omega_{\\gamma}(x,y)+\\omega_{\\gamma}(y,z) + \\omega_{\\gamma}(z,x)\\right| \\leq 6$$\nholds.\n\\end{mylem}\n\\begin{proof}\nLet $m$ be the median of $x,y,z$. By the last lemma, $\\left| \\omega_{\\gamma}(x,y)- \\omega_{\\gamma}(x,m)- \\omega_{\\gamma}(m,y)\\right|< 2$ holds, and analogous inequalities after replacing $x$ or $y$ by $z$. Therefore, \n\\begin{equation*}\n\\begin{split}\n\\left|\\omega_{\\gamma}(x,y)+\\omega_{\\gamma}(y,z) + \\omega_{\\gamma}(z,x)\\right| \\leq& |\\omega_{\\gamma}(x,m)+\\omega_{\\gamma}(m,y)\n\t\t\t+\\omega_{\\gamma}(y,m)\\\\\n\t\t&+\\omega_{\\gamma}(m,z)\n\t\t\t+\\omega_{\\gamma}(z,m)+\\omega_{\\gamma}(m,x)| + 6\\\\\n=& 6,\n\\end{split}\n\\end{equation*}\nwhere antisymmetry of $\\omega_{\\gamma}$ was used on the last line.\n\\end{proof}\n\n\\begin{mylem}\\label{defect}\nGiven a segment $\\gamma$ and a vertex $x_0\\in X$, the map $\\phi_{\\gamma}:G\\rightarrow \\mathbb{R}$ given by $\\phi_{\\gamma}(g)= \\omega_{\\gamma}(x_0,gx_0)$ is a quasimorphism with defect bounded by $6$.\\par\nAs a consequence, its homogenisation $\\overline{\\phi}_{\\gamma}$ has defect bounded by $12$.\n\\end{mylem}\n\\begin{proof}\n\\begin{equation*}\n\\begin{split}\n|\\delta\\phi_{\\gamma}(g,h)| &= |\\phi_{\\gamma}(gh)-\\phi_{\\gamma}(g)-\t\\phi_{\\gamma}(h)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)-\\omega_{\\gamma}(x_0,gx_0)-\t\\omega_{\\gamma}(x_0,hx_0)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)-\\omega_{\\gamma}(x_0,gx_0)-\t\\omega_{\\gamma}(gx_0,ghx_0)|\\\\\n\t&= |\\omega_{\\gamma}(x_0,ghx_0)+\\omega_{\\gamma}(gx_0,x_0)+\t\\omega_{\\gamma}(ghx_0,gx_0)|\\\\\n\t&\\leq 6,\n\\end{split}\n\\end{equation*}\nwhere $\\omega_{\\gamma}(x_0,hx_0)= \\omega_{\\gamma}(gx_0, ghx_0)$, antisymmetry of $\\omega_{\\gamma}$ and Lemma \\ref{omega} were used in this order.\n\\end{proof}\n\n\n\n\\section{Effectiveness}\nFrom now on let $G$ be a group with a RAAG-like action on $X$. Let $g\\in G$ be a hyperbolic element and let $o\\in X$ denote a vertex where $g$ attains its translation distance.\\par\nThe aim is to find a segment $\\gamma$ in $[o,go]$ such that for $m\\in\\mathbb{N}$ the interval $[g^mo,g^{m+1}o]$ contains at least one copy of $\\gamma$ and no copies of $\\overline{\\gamma}$. This will guarantee $\\overline{\\phi_{\\gamma}}(g)\\geq 1$.\\par\nThe following are the segments we need:\n\\begin{mydef}\nA segment $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ in $[o,go]$ is called $g$-nested if $\\gamma> g\\gamma$. It is a maximal $g$-nested segment, if it is not contained in any other $g$-nested segment in $[o,go]$\n\\end{mydef}\n\\begin{myrem}\nA maximal $g$-nested segment always exists, since a single halfspace inn $[o,go]$ is a $g$-nested segment by non-transversality.\n\\end{myrem}\nThe $g$-nestedness is to guarantee, that $[g^mo,g^{m+1}o]$ contains a copy of $\\gamma$ for every $m\\in\\mathbb{N}$, while the maximality will be crucial to ensure that no copies of $\\overline{\\gamma}$ occur in these intervals.\\par\nHere is a useful characterisation of maximality:\n\\begin{mylem}\\label{maxnestchar}\nLet $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ be maximal $g$-nested in $[o,go]$. Then:\n\\begin{enumerate}[(i)]\n\\item Any $\\Psi\\in [o,go]$ with $\\Psi \\supsetneq \\Phi_0$ is transverse to $g^{-1}\\Phi_r$.\n\\item Any $\\Psi\\in [o,go]$ with $\\Phi_r \\supsetneq \\Psi$ is transverse to $g\\Phi_0$.\n\\end{enumerate}\n\\end{mylem}\n\\begin{proof}\nLet $\\Psi\\in [o,go]$ with $\\Psi \\supsetneq \\Phi_0$ and suppose by contradiction that $\\Psi$ is not transverse to $g^{-1}\\Phi_r$. Since $o\\notin\\Psi$, we have $g^{-1}\\Phi_r \\supsetneq \\Psi$ as $\\Psi\\supseteq g^{-1}\\Phi_r$ would imply $o\\notin g^{-1}\\Phi_r$. Let $\\Psi'$ be a halfspace with $\\Psi'\\supsetneq \\Phi_0$ tightly and $\\Psi\\supseteq\\Psi'$. Clearly, $g^{-1}\\Phi_r \\supsetneq \\Psi'$. Therefore, $\\{g^{-1}\\Psi'\\}\\cup g^{-1}\\gamma > \\{\\Psi'\\}\\cup \\gamma$. Applying $g$ yields $\\{\\Psi'\\}\\cup \\gamma> \\{g\\Psi'\\}\\cup g\\gamma$ which means $\\{\\Psi'\\}\\cup \\gamma$ is $g$-nested and thereby $\\gamma$ not maximal $g$-nested.\\par\nThe proof of the second part is symmetric.\n\\end{proof}\nThe following lemma, overlooked in \\cite{FFT16}, will be the key:\n\\begin{mylem}\\label{lesserorgreater}\nLet $\\alpha = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ be a segment in $A^+_g$ and $h\\in G$ such that $h\\overline{\\alpha}\\subset A^+_g$. Then either $h\\overline{\\alpha}> \\alpha$ or $\\alpha> h\\overline{\\alpha}$.\n\\end{mylem}\n\\begin{proof}\nNote that for $0\\leq k\\leq r$, exactly one of $\\Phi_k\\supsetneq h\\overline{\\Phi}_k$ or $h\\overline{\\Phi}_k  \\supsetneq \\Phi_k$ must hold, because they are nested ($h$ is non-transverse) and equality would amount to an inversion ($\\Phi_k\\supsetneq h\\Phi_k$ and $h\\Phi_k  \\supsetneq \\Phi_k$ are impossible, as $\\alpha,h\\overline{\\alpha}\\subset A^+_g$).\\par\nWe may assume $0<r$ as the case $0=r$ is trivial.\nIf the Lemma were false, then $\\Phi_0\\supsetneq h\\overline{\\Phi}_0$ and $h\\overline{\\Phi}_r \\supsetneq \\Phi_r$. There must be some $0\\leq i< r$ such that $\\Phi_i\\supsetneq h\\overline{\\Phi}_i$ and $h\\overline{\\Phi}_{i+1} \\supsetneq \\Phi_{i+1}$. \nLet us show that all four possible relations between $h\\overline{\\Phi}_{i+1}$ and $\\Phi_{i}$ are impossible, leading to a contradiction:\n\\begin{enumerate}[(a)]\n\\item $h\\overline{\\Phi}_{i+1} = \\Phi_{i}$: By RAAG-like (iv) (see Definition \\ref{RAAGlike}), this is impossible.\n\\item $h\\overline{\\Phi}_{i+1} \\supsetneq \\Phi_{i}$: Then $h\\overline{\\Phi}_{i+1} \\supsetneq \\Phi_{i} \\supsetneq h\\overline{\\Phi}_{i}$, which contradicts that $h\\overline{\\Phi}_{i+1}$ and $h\\overline{\\Phi}_{i}$ are tightly nested.\n\\item $\\Phi_{i} \\supsetneq  h\\overline{\\Phi}_{i+1}$: Then $\\Phi_{i} \\supsetneq  h\\overline{\\Phi}_{i+1} \\supsetneq \\Phi_{i+1}$, which contradicts that $\\Phi_{i}$ and $\\Phi_{i+1}$ are tightly nested.\n\\item $h\\overline{\\Phi}_{i+1}$ and $\\Phi_{i}$ are transverse: Then RAAG-like (iii) is violated.\n\\end{enumerate}\n\\end{proof}\nAnd we are ready to show, that no reverse copies of maximal $g$-nested segments occur:\n\\begin{mylem}\\label{almostdone}\nLet $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$ in $A^+_g$ maximal $g$-nested. Then there is no $h\\in G$ such that $h\\overline{\\gamma} \\subset A^+_g$ and $h\\overline{\\Phi}_r \\in [o,go]$. \n\\end{mylem}\n\\begin{proof}\nBy Lemma \\ref{lesserorgreater} there are two cases: either (1) $h\\overline{\\gamma}> \\gamma$ or (2) $\\gamma> h\\overline{\\gamma}$.\\par\nCase (1): If $h\\overline{\\gamma}>g^{-1}\\gamma$, then $o\\notin h\\overline{\\Phi}_r$ clearly cannot hold. If $g^{-1}\\gamma> h\\overline{\\gamma}$, then $h\\overline{\\Phi}_r$ is in $[o,go]$ and contains $\\Phi_0$, but is not transverse to $g^{-1}\\Phi_r$, in contradiction to Lemma \\ref{maxnestchar}.\\par\t\nCase (2): If $g\\gamma> h\\overline{\\gamma}$, then $go\\in h\\overline{\\Phi}_r$ clearly cannot hold. If $h\\overline{\\gamma}>g\\gamma$, then $h\\overline{\\Phi}_r$ is in $[o,go]$ and contained in $\\Phi_r$, but is not transverse to $g\\Phi_0$, in contradiction to Lemma \\ref{maxnestchar}.\n\\end{proof}\nTying things together gives us the following:\n\\begin{mythm}\\label{effective}\nLet $\\gamma$ be maximal $g$-nested in $[o,go]$. Then $\\overline{\\phi}_{\\gamma}(g) \\geq 1$.\n\\end{mythm}\n\\begin{proof}\nLet $\\gamma = \\left\\{\\Phi_0,..., \\Phi_r \\right\\}$. For $n>0$, $\\left\\{ g^0\\gamma,...,g^n\\gamma \\right\\}$ is non-overlapping and in $[o,go]$. Therefore, $c_{\\gamma}(g^n)\\geq n$.\\par\nOn the other hand, if $h\\overline{\\gamma}\\subset [o,g^no]$, then $g^{-m}h\\overline{\\Phi}_r\\in [o,go]$ (and $g^{-m}h\\overline{\\gamma}\\subset A^+$) for some $m\\in\\mathbb{N}$. But this contradicts Lemma \\ref{almostdone}. Hence, $c_{\\overline{\\gamma}}(g^n)= 0$.\\par\nNow \n\\begin{equation*}\n\\overline{\\phi}_{\\gamma}(g)= \\lim_{n\\rightarrow \\infty} \\frac{\\phi_{\\gamma}(g^n)}{n}= \\lim_{n\\rightarrow \\infty} \\frac{\\omega_{\\gamma}(o,g^no)}{n}\\geq \\frac{n}{n} = 1\n\\end{equation*}\n\\end{proof}\nAn application of the Bavard Duality yields the main result: \n\\begin{mycor}\nLet $G$ be a group with a RAAG-like action on a CAT(0) cube complex. Then any element acting non-trivially has $\\mathrm{scl}(g)\\geq \\frac{1}{24}$. In particular RAAG-like groups have a stable commutator length gap of $1\/24$.\n\\end{mycor}\n\\begin{proof}\nBy Lemma \\ref{RAAGhyperbolic} every element of $G$ is hyperbolic. By the Theorem \\ref{effective} there is a quasimorphism $\\overline{\\phi}_{\\gamma}$ with $\\overline{\\phi}_{\\gamma}(g) \\geq 1$ that has defect $\\leq 12$ by Lemma \\ref{defect}. By the Bavard Duality (Theorem \\ref{bavard}) the corollary follows.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDeep discriminative models (\\emph{e.g.}~deep regression forests, deep neural decision forests) have recently been applied to many computer vision problems with remarkable success.\nThey compute the input to output mapping for regression or classification by virtue of deep neural networks~\\cite{Kontschieder_2015_ICCV,he2016deep,Simonyan2015,shen_deep_2018,chendeepage,chen_using_2017}.\nIn general, DDMs probably perform better when large amounts of effective training data (less noisy and balanced) is available.\nHowever, such ideal data is hard to collect, especially when large amounts of labels are required.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{Figure1.pdf}\n\t\\caption{The motivation of considering underrepresented examples in DRFs. \\textbf{(a):} The histogram shows the number of face images at different ages, and the average entropy curve represents the predictive uncertainty. We observe the high entropy values correspond to \\emph{underrepresented samples}. \\textbf{(b):} The histogram of the selected face images at pace 1 in SPL. \\textbf{(c):} The proposed new self-paced learning paradigm: easy and underrepresented samples first.}\n\t\\label{Figure1}\n\\end{figure}\n\nComputer vision literatures are filled with scenarios in which we are required to learn DDMs, not only robust to confusing and noisy examples, but also capable to conquer imbalanced data problem~\\cite{zeng2019soft,ren2018learning,cui2019class,khan2019striking,Kortylewski_2019_CVPR_Workshops}.\nOne typical approach is to learn discriminative features through rather deep neural networks, and feed them into a \\emph{cost-sensitive} discriminative function, often with regularization~\\cite{Kai2018Deep}.\nThe other typical approach reweights training samples according to their cost values~\\cite{cui2019class,khan2019striking} or gradient directions~\\cite{ren2018learning} (\\emph{i.e.}~meta learning).\nThese strategies are unlike our human beings, who lean things gradually---start with easy concepts and build up to complex ones, and can exclude extremely hard ones.\nMore importantly, we have a sense of \\emph{uncertainty} for some samples (\\emph{e.g.}~seldom seen) and progressively improve our capability to recognize them.\nThus, the main challenge towards realistic discrimination lies how to mimic our human discrimination system might work.\n\n\n\nThis line of thinking makes us resort to self-paced learning---a gradual learning regime inspired by the manner of humans~\\cite{Kumar2010Self}.\nIn fact, there are rare studies on the problem of self-paced DDMs.\nThen, a natural question arises: \\emph{can the self-paced regime lead DDMs to achieve more robust and less biased solutions? }\n\nMotivated by this, we propose a new self-paced learning paradigm for DDMs, which tackles the fundamental ranking and selecting problem in SPL from a new perspective: fairness.\nTo the best of our knowledge, this is the first work considering \\emph{ranking fairness} in SPL.\nSpecifically, we focus on deep regression forests (DRFs), a typical discriminative method, and propose self-paced deep regression forests with consideration on underrepresented examples (SPUDRFs).\nFirst, by virtue of SPL, our model distinguishes confusing and noisy examples from regular ones, and emphasizes more on ``good'' examples to obtain robust solutions.\nSecond, our method considers underrepresented examples, which may incur neglect in SPL since visual data is often imbalanced, renderring less bisaed solutions.\nThird, we build up a new self-paced learning paradigm: ranking samples on the basis of both likelihood and entropy (predictive uncertainty), as shown in Fig.~\\ref{Figure1}, which could be easily combined with a variety of DDMs.\n\nFor verification, we apply the SPUDRFs framework on two computer vision problems: (\\romannumeral1)  facial age estimation, and (\\romannumeral2) head pose estimation.\nExtensive experimental results demonstrate the efficacy of our proposed new self-paced paradigm for DDMs.\nMoreover, on both aforementioned problems, SPUDRFs almost achieve the state-of-the-art performances.\n\n\n\\section{Related Work}\n\nThis section reviews the deep discriminative methods for facial age estimation and head pose estimation, and SPL methods.\n\n\\noindent\\textbf{Facial Age Estimation.}\nDDM based facial age estimation methods, for example~\\cite{niu_ordinal_2016,chen_using_2017,gao_age_2018,shen_deep_2018,li2019bridgenet}, employ DNNs to precisely model the mapping from image to age.\nOrdinal-based approaches~\\cite{niu_ordinal_2016,chen_using_2017} resort to a set of sequential binary queries---each query refers to a comparison with a predefined age, to exploit the inter-relationship (ordinal information) among age labels.\nImproved deep label distribution learning (DLDL-v2)~\\cite{gao_age_2018} explores the underlying age distribution patterns to effectively accommodates age ambiguity.\nBesides, deep regression forests (DRFs)~\\cite{shen_deep_2018} connect random forests to deep neural networks and achieve promising results.\nBridgeNet~\\cite{li2019bridgenet} uses local regressors to partition the data space and gating networks to provide continuity-aware weights.\nThe final age estimation result is the mixture of the weighted regression results.\nOverall, these DDM based approaches have enhanced age estimation performance largely; however, they plausibly ignore one problem: the interference arising from confusing and noisy examples---facial images with PIE (\\emph{i.e.}~pose, illumination and expression) variation, occlusion, misalignment and so forth.\n\n\n\\noindent\\textbf{Head Pose Estimation.}\nFor head pose estimation, Riegler~\\cite{riegler2013hough} \\emph{et al.}~utilized convolutional neural networks (CNNs) to learn patch features of facial images and achieved better performance.\nIn~\\cite{huang2018mixture}, Huang \\emph{et al.}~adopted multi-layer perceptron (MLP) networks for head pose estimation and proposed multi-modal deep regression networks to fuse RGB and depth information.\nIn~\\cite{wang2019deep}, Wang \\emph{et al.}~proposed a deep coarse-to-fine network for head pose estimation.\nIn~\\cite{ruiz2018fine}, Ruiz \\emph{et al.}~used a large synthetically expanded head pose dataset to train rather deep multi-loss CNNs for head pose estimation and gained satisfied accuracy.\nIn~\\cite{kuhnke2019deep}, Kuhnke \\emph{et al.}~proposed domain adaptation for head pose estimation, assuming shared and continuous label spaces.\nDespite seeing much success, these methods seldom consider the potential problems caused by imbalanced and noisy training data, which may exactly exist in visual problems.\n\n\\noindent\\textbf{Self-Paced Learning.}\nThe SPL is a gradual learning paradigm, which builds on the intuition that, rather than considering all training samples simultaneously, the algorithm should be presented with the training data from easy to difficult, which facilitates learning~\\cite{Kumar2010Self,meng_theoretical_2017}.\nVariants of SPL methods have been proposed recently with varying degrees of success.\nFor example, in~\\cite{jiang2015self}, Zhao \\emph{et al.}~generalized the conventional binary (hard) weighting scheme for SPL to a more effective real valued (soft) weighting manner.\nIn~\\cite{ma2017self}, Ma \\emph{et al.}~proposed self-paced co-training which applies self-paced learning to multi-view or multi-modality problems.\nIn~\\cite{han2017self}, Han \\emph{et al.}~made some efforts on mixture of regressions with SPL strategy, to avoid poorly conditioned linear sub-regressors.\nIn~\\cite{Ren2017RoSR,Ren2020SAMVC}, Ren \\emph{et al.}~introduced soft weighting schemes of SPL to reduce the negative influence of outliers and noisy samples.\nIn fact, the majority of these mentioned methods can be cast as the combination of SPL and shallow classifiers, where SVM and logistic regressors are usually involved.\nIn computer vision, due to the remarkable performance of DNNs, some authors have realized SPL may guide DDMs to achieve more robust solutions recently.\nIn~\\cite{ijcai2017}, Li \\emph{et al.}~sought to enhance the learning robustness of CNNs with SPL, and proposed SP-CNNs.\nHowever, \\cite{ijcai2017} omits one important problem in the discriminative model: the imbalance of training data.\nIn contrast to SP-CNNs, our SPUDRFs model has three advantages: (i) it emphasizes ranking fairness (\\emph{i.e.}~considering underrepresented examples) in SPL, and hence tends to achieve less biased solutions; (ii) its learning regime is fundamental and can be easily combined with other DDMs, especially the ones with predictive uncertainty; (iii) it creatively explores how SPL can integrate with DMMs with a probabilistic interpretation.\n\n\n\nOur work is inspired by the existing works~\\cite{jiang2014self,yang2019self} which take the class diversity in the sample selection of SPL into consideration.\nJiang \\emph{et al.}~\\cite{jiang2014self} encouraged the class diversity in sample selection at the early paces of self-paced training.\nYang \\emph{et al.}~\\cite{yang2019self} defined a metric, named complexity of image category, to measure sample number and recognition difficult jointly, and adopted this measure for sample selection in SPL.\nIn fact, the aforementioned two methods realize the lack of class diversity in SPL's sample selection may achieve biased solutions since visual data is often imbalanced.\nBut what causes lack of class diversity is exactly the ranking unfairness as underrepresented examples may often have large loss (particular in DDMs).\nNot only that, \\cite{yang2019self,jiang2014self} are only suitable for classification, but not regression (with continuous and high dimensional output).\nIn this paper, we will go further along this direction, aiming to tackle the fundamental problem in SPL: ranking unfairness.\n\n\n\\section{Preliminaries}\n\nIn this section, we review the basic concepts of deep regression forests (DRFs)~\\cite{shen_deep_2018}.\n\n\n\n\\noindent \\textbf{Deep Regression Tree.} DRFs usually consist of a number of deep regression trees.\nA deep regression tree, given input-output pairs $\\left\\{\\mathbf{x}_i, y_i\\right\\}_{n=1}^N$, where $\\mathbf{x}_i\\in\\mathbb{R}^{D_x}$ and $y_i\\in\\mathbb{R}$, models the mapping from input to output through DNNs coupled with a regression tree.\nA regression tree $\\mathcal{T}$ consists of split nodes $\\mathcal{N}$ and leaf nodes $\\mathcal{L}$~\\cite{shen_deep_2018}.\nMore specifically, each split node $n \\in \\mathcal{N}$ possesses a split to determine whether input $\\mathbf{x}_i$ goes to the left or right subtree; each leaf node $\\ell \\in \\mathcal{L}$ corresponds to a Gaussian distribution $p_{\\ell}(y_i)$ with mean $\\mu_l$ and variance $\\sigma^2_l$.\n\n\n\n\\noindent \\textbf{Split Node.}\nSplit node has a split function, $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta}) : \\mathbf{x}_i \\rightarrow[0,1]$, which is parameterized by $\\bm{\\Theta}$---the parameters of DNNs.\nConventionally, the split function is formulated as $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})=\\sigma\\left(\\mathbf{f}_{\\varphi(n)}(\\mathbf{x}_i ; \\bm{\\Theta})\\right)$, where $\\sigma(\\cdot)$ is the sigmoid function, $\\varphi(\\cdot)$ is an index function to specify the $\\varphi(n)$-th element of $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ in correspondence with a split node $n$, and $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ denotes the learned deep features.\nAn example to illustrate the sketch chart of the DRFs is shown in Fig.~\\ref{Figure1}, where $\\varphi_1$ and $\\varphi_2$ are two index functions for two trees.\nThe probability that $\\mathbf{x}_i$ falls into the leaf node $\\ell$ is given by:\n\n\\begin{equation}\n\\label{Eq.1}\n\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}=\\prod_{n \\in \\mathcal{N}} s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})^{[\\ell \\in \\mathcal{L}_{n_l}]}\\left(1-s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})\\right)^{\\left[\\ell \\in \\mathcal{L}_{n_r}\\right]},\n\\end{equation}\nwhere $[\\mathcal{H}]$ denotes an indicator function conditioned on the argument $\\mathcal{H}$. In addition, $\\mathcal{L}_{n_l}$ and $\\mathcal{L}_{n_r}$ correspond to the sets of leaf nodes owned by the subtrees $\\mathcal{T}_{n_l}$ and $\\mathcal{T}_{n_r}$ rooted at the left and right children ${n}_{l}$ and ${n}_{r}$ of node $n$, respectively.\n\n\n\n\\noindent \\textbf{Leaf Node.} For tree $\\mathcal{T}$, given $\\mathbf{x}_i$,  each leaf node $\\ell \\in \\mathcal{L}$ defines a predictive distribution over $y_i$, denoted by $p_{\\ell}(y_i)$.\nTo be specific, $p_{\\ell}(y_i)$ is assumed to be a Gaussian distribution: $\\mathcal{N}\\left(y_i|\\mu_l, \\sigma^2_l\\right)$.\nThus, considering all leaf nodes, the final distribution of $y_i$ conditioned on $\\mathbf{x}_i$ is averaged by the probability of reaching each leaf:\n\\begin{equation}\n\\label{Eq.2}\np_{\\mathcal{T}}(y_i | \\mathbf{x}_i ; \\bm{\\Theta}, \\bm{\\pi})=\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i),\n\\end{equation}\nwhere $\\bm{\\Theta}$ and $\\bm{\\pi}$ represent the parameters of DNNs and the distribution parameters $\\left\\{\\mu_l,\\sigma^2_l\\right\\}$, respectively.\nIt can be viewed as a mixture distribution, where $\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}$ denotes mixing coefficients and $ p_{\\ell}(y_i)$ denotes the Gaussian distributions associated with the $\\ell^{th}$ leaf node.\nNote that $\\bm{\\pi}$ varies along with tree $\\mathcal{T}_k$, and thus we rewrite it as $\\bm{\\pi}_k$ below.\n\n\n\n\\noindent \\textbf{Forests of Regression Trees.}\nSince a forest comprises a set of deep regression trees $\\mathcal{F}=\\left\\{\\mathcal{T}_1,...,\\mathcal{T}_k\\right\\}$, the predictive output distribution, given $\\mathbf{x}_i$, is obtained by averaging over all trees:\n\\begin{equation}\n\\label{Eq.3}\np_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\n=\n\\frac{1}{K}\\sum_{k=1}^K p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i, \\bm{\\Theta}, \\bm{\\pi}_k\\right),\n\\end{equation}\nwhere $K$ is the number of trees and $\\bm{\\Pi}=\\left\\{\\bm{\\pi}_1,...,\\bm{\\pi}_K\\right\\}$.\n$p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ can be viewed as the likelihood that the $i^{th}$ sample has output $y_i$.\n\n\n\n\n\\section{Self-Paced DRFs with Consideration on Underrepresented Examples}\nThe problems in training DDMs for visual tasks arise from: (\\romannumeral1) the noisy and confusing examples, and (\\romannumeral2) the imbalance of training data.\nIntuitively inspired by the gradual learning manner of humans, we resort to self-paced learning and explore whether the DDMs, by virtue of SPL, tend to achieve more robust solutions.\nPerhaps not easily, in existing SPL, we observe ranking unfairness, as shown in Fig.~\\ref{Figure1}.\nMotivated by this observation, we propose SPUDRFs, which starts learning with easy yet underrepresented examples, and build up to complex ones.\nSuch a paradigm avoids overlooking the ``minority'' of training samples, leading to less biased solutions.\n\n\n\n\\subsection{Underrepresented Examples}\n\\label{Uncertainty}\nUnderrepresented examples mean ``minority'', as which the examples with similar or the same labels are scarce.\nUnsurprisingly, we observe that they may incur unfairness treatment in the early paces of SPL (see Fig.~\\ref{Figure1}(b)), due to imbalanced data distribution.\nThe underrepresented level could be measured by predictive uncertainty.\nGiven the sample $\\mathbf{x}_i$, its predictive uncertainty is formulated as the entropy of its predictive output distribution $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$:\n\\begin{equation}\n\\label{Eq.4}\nH\\left [p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\\right] = \\frac{1}{K}\\sum^K_{k=1}H\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right],\n\\end{equation}\nwhere $H\\left[ \\cdot\\right ]$ denotes entropy, and the entropy corresponds to the $k^{th}$ tree is:\n\\begin{equation}\n\\label{Eq.5}\nH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right] = -\\int p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\ln p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right) dy_i,\n\\end{equation}\nThe large the entropy is, the more uncertain the prediction should be, \\emph{i.e.}, the more underrepresented the sample is.\nConsidering underrepresented samples can be interpreted as adequately utilizing the ``information'' inherent in such examples in SPL training.\n\n\nAs previously discussed, $p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i; \\bm{\\Theta}, \\bm{\\pi}_k\\right)$ is a mixture distribution, taking the form $\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i)$, where $\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}$ denotes mixing coefficients and $p_{\\ell}(y_i)$ denotes the Gaussian distribution associated with the $\\ell\n^{th}$ leaf node.\nIn Eq.~(\\ref{Eq.5}), the integral of mixture of Gaussians is non-trivial. Monte Carlo sampling provides a way to calculate it, but incurs large computational cost~\\cite{huber2008entropy}.\nHere, we use the lower bound of this integral to approximate its true value:\n\\begin{equation}\n\\label{Eq.6}\nH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right]\\approx\\frac{1}{2}\\sum_{\\ell\\in\\mathcal{L}}\\omega_\\ell(\\mathbf{x}_i|\\mathbf{\\Theta})\\left[\\ln \\left(2\\pi \\sigma_\\ell^2\\right)+1\\right].\n\\end{equation}\nThe underrepresented examples are often scarce, and have not been treated fairly, resulting in large prediction uncertainty (\\emph{i.e.}~entropy).\n\n\\subsection{ Objective Function}\n\nRather than considering all the samples simultaneously, our proposed SPUDRFs are presented with the training data in a meaningful order, that is, easy and underrepresented examples first.\nSpecifically, we define a latent variable $v_i$ that indicates whether the $i^{th}$ sample is selected $(v_i = 1)$ or not $(v_i = 0)$ depending on how easy and underrepresented it is for training.\nOur objective is to jointly maximize the log likelihood with\nrespect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$, and learn the latent selecting variables $\\mathbf{v}=\\left(v_1,...,v_N\\right)^T$.\nWe prefer to select the underrepresented examples, which probably have higher predictive uncertainty (\\emph{i.e.}~entropy), particularly in the early paces.\nIt builds on the intuition that the underrepresented examples may incur neglect since they are the ``minority'' in training data.\nTherefore, we maximize a self-paced term regularized likelihood function, meanwhile considering predictive uncertainty,\n\\begin{equation}\n\\label{Eq.7}\n\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i \\right \\}  + \\lambda\\sum_{i=1}^N v_i ,\n\\end{equation}\nwhere $\\lambda$ is a parameter controlling the learning pace, $\\lambda>0$, $\\gamma$ is the parameter imposing on entropy, and $H_i$ denotes the predictive uncertainty of the $i^{th}$ sample, as previously discussed in Sec.~\\ref{Uncertainty}.\nWhen $\\gamma$ decays to 0, the objective function is equivalent to the log likelihood function with respect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nEq.~(\\ref{Eq.7}) indicates each sample is weighted by $v_i$, and whether $\\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i>-\\lambda$ determines\nthe $i^{th}$ sample is selected or not.\nThat is, the sample with high likelihood value or high predictive uncertainty may be selected.\nThe optimal $v_i^*$ is:\n\\begin{align}\n\\label{Eq.8}\nv_i^* = \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i > -\\lambda$}\\\\\n0 & \\textrm{otherwise}\n\\end{array} \\right.,\n\\end{align}\nwhere $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ is written as $ p_{\\mathcal{F}i}$ for simplicity.\n\nOne might argue the noisy and hard examples tend to have high predictive uncertainty also, rendering being selected in the early paces.\nIn fact, from Eq.~(\\ref{Eq.8}), we observe whether one sample is selected is determined by both its predictive uncertainty and the log likelihood of being predicted correctly.\nThe noisy and hard examples probably have relatively large loss \\emph{i.e.}~low log likelihood, avoiding being selected at the very start.\n\nIteratively increasing $\\lambda$ and decreasing $\\gamma$, samples are dynamically involved in the training of DRFs, starting with easy and underrepresented examples and ending up with all samples.\nNote every time we retrain DRFs, that is, maximizing Eq.~(\\ref{Eq.7}), our model is initialized to the result of the last iteration.\nAs such, our model is initialized progressively by the result of the previous pace---adaptively calibrated by ``good'' examples.\nThis also means we place more emphasis on easy and underrepresented examples rather than confusing and noisy ones.\nThus, SPUDRFs are prone to have more robust and less biased solutions since we adequately consider the underrepresented examples.\n\n\\noindent \\textbf{Mixture Weighting.}\nIn the previous section, we adopt a hard weighting scheme to assign data points to paces, in which one sample is either selected $(v_i=1)$ or not $(v_i=0)$.\nSuch a weighting scheme appears to be less accurate as it omits the importance of samples.\nHence, we adopt a mixture weighting scheme~\\cite{jiang2014easy}, where the selected samples are weighted by its importance, ling in the range $0\\leq v_i \\leq 1$.\nThe objective function with mixture weighting is defined as:\n\\begin{equation}\n\\label{Eq.9}\n\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i\\right \\}  + \\zeta \\sum_{i=1}^N \\log\\left(v_i + \\zeta\/\\lambda\\right) ,\n\\end{equation}\nwhere $\\zeta$ is a parameter controlling the learning pace.\nWe set $\\zeta=\\left(\\frac{1}{\\lambda'}-\\frac{1}{\\lambda}\\right)^{-1}$, and $\\lambda>\\lambda'>0$ to construct a reasonable soft weighting formulation.\nThe self-paced regularizer in Eq.~(\\ref{Eq.9}) is convex with respect to $v\\in\\left[0,1\\right]$.\nThen, setting the partial gradient of Eq.~(\\ref{Eq.9}) with respect to $v_i$ as zero will lead to the following:\n\\begin{equation}\n\\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i + \\frac{\\zeta}{v_i + \\zeta\/\\lambda} = 0.\n\\end{equation}\nThen, the optimal solution of $v_i$ is given by:\n\\begin{align}\nv_i^* = \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\geq -\\lambda'  $}\\\\\n0 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\leq -\\lambda $}\\\\\n\\frac{-\\zeta}{\\log p_{\\mathcal{F}i} + \\gamma H_i} - \\zeta\/\\lambda & \\textrm{otherwise}\n\\end{array} \\right.\n\\label{Eq.11}\n\\end{align}\nIf either the log likelihood or the predictive uncertainty is too large,  $v^*_i$ equals to 1.\nIn addition, if the likelihood and the predictive uncertainty are both too small, $v^*_i$ equals to 0.\nExcept the above two situations, the soft weighting calculation (\\emph{i.e.}, the last line of Eq.~(\\ref{Eq.11})) is adopted.\n\n\\noindent \\textbf{Curriculum Reconstruction.}\nThe underrepresented examples play an important role in our SPUDRFs algorithm.\nAs previously mentioned, the proposed new self-paced regime coupled with a mixture weighting scheme emphasizes more on underrepresented examples, rendering better solutions.\nSince the intrinsic reason that causes predictive uncertainty is plausibly the imbalanced training data, we further re-balance data distribution via a curriculum reconstruction strategy.\nMore specifically, we distinguish the underrepresented examples (whose $H_i$ is lager than $\\beta$) from regular ones at each pace, and augment them into the training data.\n\n\n\n\\subsection{Optimization}\n\\label{Learning}\nWe propose a two-step alternative search strategy (ASS) algorithm to solve SPUDRFs: (\\romannumeral1) update $\\mathbf{v}$ for sample selection with fixed $\\bm{\\Theta}$ and $\\bm{\\Pi}$, and (\\romannumeral2) update $\\bm{\\Theta}$ and $\\bm{\\Pi}$ with current fixed sample weights $\\mathbf{v}$.\n\n\\noindent\\textbf{Optimizing $\\bm{\\Theta}$ and $\\bm{\\Pi}$.}\nThe parameters $\\left\\{\\bm{\\Theta},\\bm{\\Pi}\\right\\}$ and weights $\\mathbf{v}$ are optimized alternatively.\nWith fixed $\\mathbf{v}$,  our DRFs is learned by alternatively updating $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nIn \\cite{shen_deep_2018}, the parameters $\\bm{\\Theta}$ for split nodes (\\emph{i.e.}~parameters for VGG) are updated through gradient descent since the loss is differentiable with respect to $\\bm{\\Theta}$.\nWhile the parameters $\\bm{\\Pi}$ for leaf nodes are updated by virtue of variational bounding~\\cite{shen_deep_2018} when fixing $\\bm{\\Theta}$.\n\n\\noindent\\textbf{Optimizing $\\mathbf{v}$.}\nAs previously discussed, $v_i$ is a binary variable or real variable ranged in $\\left[0, 1\\right]$.\nIt indicates how to weight the $i^{th}$ sample during training.\nThe parameter $\\lambda$ could be initialized to obtain 50\\% samples to train the model, and is then progressively increased to involve 10\\% more data in each pace.\nThe parameter $\\gamma$ could be initialized empirically and is progressively decayed to zero.\nThe training stops when all the samples are selected, at $\\gamma=0$.\nAlong with increasing $\\lambda$ and decreasing $\\gamma$, DRFs are trained to be more ``mature''.\nThis learning process is like how our human beings learn one thing from easy and uncertain to complex.\n\n\n\n\n\\section{Experimental Results}\n\\subsection{Tasks and Benchmark Datasets}\n\\noindent \\textbf{Age Estimation.}\nThe Morph \\uppercase\\expandafter{\\romannumeral2~\\cite{ricanek2006morph}} dataset contains 55,134 unique face images of 13618 individuals with unbalanced gender and ethnicity distributions, and is the most popular publicly available real age dataset.\nThe FG-NET~\\cite{panis2016overview} dataset includes 1,002 color or gray images of 82 people with each subject almost accompanied by more than 10 photos at different ages.\nSince all images were taken in a totally uncontrolled environment, there exists a large deviation on lighting, pose and expression (\\emph{i.e.}~PIE) of faces inside the dataset.\n\n\n\\noindent \\textbf{Head Pose Estimation.}\nThe BIWI dataset~\\cite{fanelli2013random} contains 20 subjects, of which 10 are male and 6 are female, besides, 4 males have been chosen twice with wearing glasses or not.\nIt includes 15678 images collected by a Kinect sensor device for different persons and head poses\nwith pitch, yaw and roll angles mainly ranging within $\\pm 60^{\\circ}$, $\\pm 75^{\\circ}$ and $\\pm 50^{\\circ}$.\n\n\\subsection{Experimental Setup}\n\n\\noindent\\textbf{Dataset Setting.}\nThe settings of different datasets are given below.\n\\begin{itemize}\n\t\\item \\textbf{Morph \\uppercase\\expandafter{\\romannumeral2}.} Following the recent relevant work~\\cite{shen_deep_2018}, the images in Morph \\uppercase\\expandafter{\\romannumeral2} were divided into two sets: 80\\% for training and the rest 20\\% for testing. The random division was repeated\n\t5 times and the reported performance was averaged over these 5 times. The VGG-Face~\\cite{parkhi2015deep} networks were chosen as the pre-trained model.\n\t\\item \\textbf{FG-NET.} The leave-one-person-out scheme~\\cite{shen_deep_2018} was adopted, where the images of one person were selected for testing and the remains for training. The VGG-16 networks were pre-trained on the IMDB-WIKI~\\cite{rothe2018deep} dataset.\n\t\\item \\textbf{BIWI.} Similarly, 80\\% of the whole data was randomly chosen for training and the rest 20\\% for testing, and this operation was repeated 5 times. Moreover, the VGG-FACE networks were the pre-trained model.\n\\end{itemize}\n\n\n\\noindent\\textbf{Evaluation Metrics.}\nThe first evaluation metric is the mean absolute error (MAE), which is defined as the average absolute error between the ground truth and the predicted output: $\\sum_{i=1}^{N}\\left|\\hat{y}_{i}-y_{i}\\right|\/N$, $\\hat{y_{i}}$ represents the estimated output of the $i^{th}$ sample, and $N$ is the total number of testing images.\nThe other evaluation metric is cumulative score (CS), which denotes the percentage of images sorted in the range of $\\left[y_{i}-L, y_{i}+L\\right]$: $CS(L)=\\sum_{i=1}^{N}\\left[\\vert\\hat{y}_{i}-y_{i}\\vert \\leq L\\right]\/N \\cdot 100 \\%$, where $[ \\cdot ]$ denotes an indicator function and $L$ is the error range.\n\n\\noindent\\textbf{Preprocessing and Data Augmentation.}\nOn the Morph \\uppercase\\expandafter{\\romannumeral2} and FG-NET datasets, MTCNN~\\cite{zhang_joint_2016} was used for joint face detection and alignment.\nFurthermore, following~\\cite{shen_deep_2018}, we augmented training images in three ways: (\\romannumeral1) random cropping (5 times); (\\romannumeral2) adding Gaussian white noise with variance of 0.0001 (2 times); (\\romannumeral3) random horizontal flipping (2 times). The whole number of samples was increased by 20 times after augmentation.\nOn the BIWI dataset, we utilized the depth images for training and did not augment training images.\n\n\n\\noindent\\textbf{Parameters Setting.}\nThe VGG-16~\\cite{Simonyan2015} was employed as the fundamental backbone networks of SPUDRFs.\nThe hyper-parameters of VGG-16 were: training batch size (32 on Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI, 8 on FG-NET), drop out ratio (0.5), max iterations of each pace ($80k$ on Morph \\uppercase\\expandafter{\\romannumeral2},  $20k$ on FG-NET, and $40k$ on BIWI), stochastic gradient descent (SGD), initial learning rate (0.2 on Morph \\uppercase\\expandafter{\\romannumeral2}, 0.1 on BIWI, 0.02 on FG-NET) by reducing the learning rate ($\\times$0.5) per $10k$ iterations. The hyper-parameters of SPUDRFs were: tree number (5), tree depth (6), output unit number of feature learning (128), iterations to update leaf node predictions (20), number of mini-batches used to update leaf node predictions (50).\nIn the first pace, 50\\% samples which are easy or underrepresented were selected for training.\nHere, $\\lambda$ was set to guarantee the first 50\\% samples with large $\\log p_{\\mathcal{F}i} + \\gamma H_i$ values  involved.\n$\\lambda'$ was set to ensure 10\\% of selected samples with soft weighting.\n$\\gamma$ was initialized to be 15 on the Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI datasets, and 5 on the FG-NET dataset.\n$\\beta$ was set to select 1180 and 2000 samples as the ones needed to be augmented twice at each pace on the Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI datasets.\nThe number of paces was empirically set to be 10, 3 and 6 on the Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET, and BIWI datasets, and except the first pace, an equal proportion of the rest data was gradually involved at each pace.\n\n\n\n\n\n\\subsection{Validity of Our Proposed Method}\n\\label{valid}\n\\noindent \\textbf{Self-paced Learning Strategy.}\nThe validity of self-paced strategy in training DDMs is mainly demonstrated by the following experiments on the MorphII dataset.\nWe first used all training images in the Morph \\uppercase\\expandafter{\\romannumeral2} datasets to train DRFs so as to rank samples at the beginning pace.\nRetraining proceeded with progressively increasing $\\lambda$ such that every 1\/9 of the rest data was gradually involved at each pace, where $\\gamma$ was decreased to the half of its previous value every time.\nIn the last pace, the value $\\gamma$ was constrainedly set to be 0.\nThe visualization of this process can be found in Fig.~\\ref{SPUDRFs_validation}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{SPUDRFs_validation.pdf}\n\t\\caption{The gradual learning process of SP-DRFs and SPUDRFs. \\textbf{Left:} The typical worst cases at each iteration become more confusing and noisy along\n\t\twith iteratively increasing $\\lambda$ and decreasing $\\gamma$. The two numbers below each image are the real age (left) and predicted age (right). \\textbf{Right:} The MAEs of SP-DRFs and SPUDRFs at each pace descend gradually. The SPUDRFs show its superiority of taking predictive uncertainty into consideration, when compared with SP-DRFs.}\n\t\\label{SPUDRFs_validation}\n\\end{figure}\nFig.~\\ref{SPUDRFs_validation} illustrates the representative face images in each learning pace of SPUDRFs, along with increasing $\\lambda$ and decreasing $\\gamma$.\nThe two numbers below each image are the real age (left) and predicted age (right).\nWe observe that the training images in the latter paces are obviously more confusing and noisy than the ones in the early paces.\nSince our model is initialized by the results of the previous retraining pace, meaning adaptively calibrated by ``good'' examples.\nAs a result, it has improved performance than DRFs, where the MAE is improved from 2.17 to 1.91, and the CS is promoted from 92.79\\% to 93.31\\% (see Fig.~\\ref{morph_experiment}(a)).\n\n\nFig.~\\ref{SPUDRFs_validation} also shows the comparison between SP-DRFs and SPUDRFs on the Morph \\uppercase\\expandafter{\\romannumeral2} datasets.\nThe yellow bar denotes the MAE of SP-DRFs, while the orange bar denotes for SPUDRFs.\nWe find the MAE of SPUDRFs is lower than SP-DRFs at each pace, particularly the last pace ($1.91$ against $2.02$).\nAs we discussed previously, as in Fig.~\\ref{Figure1}, SPUDRFs are prone to reach less biased solutions due to the wider covering range of leaf nodes, owing to considering underrepresented examples.\nThis experiment could be regarded as an ablation study of considering ranking fairness in SPL.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{Uncertainty_efficacy.pdf}\n\t\\caption{The leaf node distribution of SP-DRFs and SPUDRFs in gradual learning process. Three paces, \\emph{i.e.}~pace 1, 3, and 6, are randomly chosen for visualization. For SP-DRFs, the Gaussian means of leaf nodes (the red points in the second row) are concentrated in a small range, incurring seriously biased solutions. For SPUDRFs, the Gaussian means of leaf nodes (the orange points in the third row) distribute widely, leading to much better MAE performance.}\n\t\\label{Uncertainty_efficacy}\n\\end{figure}\n\\noindent\\textbf{Considering Underrepresented Examples.}\n\\label{ExpUnderSamples}\nOn the BIWI dataset, the necessity of considering ranking fairness in SPUDRFs is further demonstrated.\nIn SP-DRFs, DRFs was first trained on the basis of all data, and the samples were ranked and selected for the first pace according to this result.\nSubsequently, every $10\\%$ of the rest samples were progressively involved for retraining.\n$\\lambda$ was progressively increased while $\\gamma$ was progressively decreased until zero.\nIn SP-DRFs, the same self-paced strategy was adopted as in SPUDRFs, but without considering ranking fairness (\\emph{i.e.}~underrepresented examples).\n\n\nFig.~\\ref{Uncertainty_efficacy} visualizes the leaf node distributions of SP-DRFs and SPUDRFs in the progressive learning process.\nThe Gaussian means $\\mu_l$ associated with the 160 leaf nodes, where each 32 leaf nodes are defined for 5 trees, are plotted in each sub-figures.\nThree paces, \\emph{i.e.}~pace 1, 3, and 6, are randomly chosen for visualization.\nOnly pitch and yaw angles are shown for clarity.\nBesides, the distribution of angle labels (\\emph{i.e.}~pitch and yaw) are also shown, where the imbalance problem of data distribution is obvious.\n\nIn Fig.~\\ref{Uncertainty_efficacy}, the comparison results between SP-DRFs and SPUDRFs demonstrate the efficacy of considering ranking fairness in SPL.\nFor SP-DRFs, the Gaussian means of leaf nodes (red points in the second row) are concentrated in a small range, incurring seriously biased solutions.\nThat means the underrepresented examples have been neglected in SPL training.\nThe poor MAEs are the evidence for this, which are even inferior to DRFs (see Fig.~\\ref{biwi_experiment}(a)).\nSPUDRFs rank samples by log likelihood coupled with entropy, and are prone to achieve less biased solutions,  as shown in the third rows of Fig.~\\ref{Uncertainty_efficacy}.\nSuch an experiment could be also regard as an ablation study of the proposed ranking algorithm.\n\n\\subsection{Comparison with State-of-the-art Methods}\n\\label{sec:blind}\n\nWe compared our SPUDRFs with other state-of-the-art methods on the Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET and BIWI datasets.\n\\begin{figure}[t]\n\t\\centering                                                                        \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.82}{\n\t\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tLSVR \\cite{guo_human_2009}     & 4.31 & 66.2\\% \\\\\n\t\t\t\tRCCA \\cite{Huerta2014Facial}   & 4.25 & 71.2\\% \\\\\n\t\t\t\tOHRank \\cite{Chang2011Ordinal} & 3.82 & N\/A \\\\\n\t\t\t\tOR-CNN \\cite{niu_ordinal_2016} & 3.27 & 73.0\\% \\\\\n\t\t\t\tRanking-CNN \\cite{chen_using_2017} & 2.96 & 85.0\\% \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 2.17 & 91.3\\% \\\\\n\t\t\t\tDLDL-v2 \\cite{gao_age_2018}& 1.97 & N\/A \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.02} & \\textbf{92.79\\%} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{1.91} & \\textbf{93.31\\%} \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t& \\raisebox{-0.83in}{\\includegraphics[width=0.555\\textwidth]{morph_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves of the comparison methods.}\n\t\\label{morph_experiment}\n\\end{figure}\n\n\\noindent\\textbf{Results on Morph \\uppercase\\expandafter{\\romannumeral2}.}\nFig. \\ref{morph_experiment}(a) compares SPUDRFs with other baseline methods: LSVR~\\cite{guo_human_2009}, RCCA \\cite{Huerta2014Facial}, OHRank~\\cite{Chang2011Ordinal}, OR-CNN \\cite{niu_ordinal_2016}, Ranking-CNN \\cite{chen_using_2017}, DRFs~\\cite{shen_deep_2018}, and DLDL-v2~\\cite{gao_age_2018}.\nFirstly, owing to the effective feature learning ability of DNNs, the SPUDRFs method is much superior to the shallow model based approaches, such as LSVR~\\cite{guo_human_2009} and OHRank~\\cite{Chang2011Ordinal}.\nSecondly, duing to the valid self-paced regime, our SPUDRFs outperform other DDMs, and lead to more robust and less biased solutions.\nThirdly, SPUDRFs outperform SP-DRFs on both MAE and CS, and achieve state-of-the-art performance.\nFig.~\\ref{morph_experiment}(b) shows the CS comparison on this dataset.\nWe observe that the CS of SPUDRFs reachs 93.31\\% at error level $L=5$, which is significantly better than DRFs and obtained 2.01\\% increment.\n\\begin{figure}\n\t\\centering                                                                        \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.82}{\n\t\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tIIS-LDL \\cite{xin_geng_facial_2013} & 5.77 & N\/A \\\\\n\t\t\t\tLARR \\cite{guodong_guo_image-based_2008} & 5.07 & 68.9\\% \\\\\n\t\t\t\tMTWGP \\cite{Yu2010Multi} & 4.83 & 72.3\\% \\\\\n\t\t\t\tDIF \\cite{han_demographic_2015} & 4.80 & 74.3\\% \\\\\n\t\t\t\tOHRank \\cite{Chang2011Ordinal} & 4.48 & 74.4\\% \\\\\n\t\t\t\tCAM \\cite{Luu2013Contourlet} & 4.12 & 73.5\\% \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 3.06 & 83.33\\% \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.84} & \\textbf{84.73\\%} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{2.77} & \\textbf{85.53\\%}\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\n\t\t& \\raisebox{-0.83in}{\\includegraphics[width=0.555\\textwidth]{fgnet_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the FGNET dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves of the comparison methods.}\n\t\\label{fgnet_experiment}\n\\end{figure}\n\n\n\\noindent\\textbf{Results on FG-NET.} Fig.~\\ref{fgnet_experiment}(a) shows the comparison results of SPUDRFs with the state-of-the-art approaches on FG-NET dataset.\nAs can be seen, SPUDRFs reach an MAE of 2.77 years, which reduces the MAE of DRFs by 0.29 years.\nBesides, the CS comparison is shown in Fig.~\\ref{fgnet_experiment}(b), SPUDRFs consistently outperform other recent proposed methods at different error levels, proving that our method is effective in enhancing the robustness of facial age estimation.\n\n\n\\begin{figure}\n\t\\centering                                                                        \n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{0.78}{\n\t\t\t\\begin{tabular}{@{}l|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tHF \\cite{riegler2013hough} & 4.95 \\\\\n\t\t\t\tSVR \\cite{drucker1997support} & 3.14 \\\\\n\t\t\t\tRRF \\cite{liaw2002classification} & 3.06 \\\\\n\t\t\t\tKPLS \\cite{al2012partial} & 2.88 \\\\\n\t\t\t\tSAE \\cite{hinton2006reducing} & 1.94 \\\\\n\t\t\t\tMoDRN \\cite{huang2018mixture} & 1.62 \\\\\n\t\t\t\tDRFs \\cite{shen_deep_2018} & 1.44 \\\\\n\t\t\t\t\\textbf{SP-DRFs} & \\textbf{2.08} \\\\\n\t\t\t\t\\textbf{SPUDRFs} & \\textbf{1.18} \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t& \\raisebox{-0.8in}{\\includegraphics[width=0.53\\textwidth]{biwi_cs5.pdf}} \\\\\n\t\t{\\small (a) } & {\\small (b)}\n\t\\end{tabular}\n\t\\caption{The comparison results on the BIWI dataset. (a) The MAE comparison with the state-of-the-art methods, (b) the CS curves the comparison methods.}\n\t\\label{biwi_experiment}\n\\end{figure}\n\n\\noindent\\textbf{Results on BIWI.}\nFig.~\\ref{biwi_experiment}(a) shows the comparison results of our method with several state-of-the-art approaches.\nThe experimental results reveal the proposed SPUDRFs method achieves the best performance with an MAE of 1.18, which is state-of-the-art.\n\\emph{Besides, we observe one important phenomenon: the MAE of SP-DRFs is even much worse than DRFs.\n\tThis further demonstrates the obvious drawback of the ranking and selecting algorithm in original SPL---incurring seriously biased solutions.}\nIn the first pace of the original SPL, as illustrated in Fig.~\\ref{Uncertainty_efficacy}, the Gaussian means of leaf nodes are concentrated in a small range, leading biased solutions.\nIncorporating underrepresented examples in the early pace of SPUDRFs renders to more reasonable distributions of the leaf nodes.\nFig.~\\ref{biwi_experiment}(b) plots only three CS curves for brevity, \\emph{i.e.}, DRFs, SP-DRFs and SPUDRFs, which is the average of the three angles.\nSPUDRFs also outperform DRFs and SP-DRFs at different error levels.\n\n\n\\section{Conclusion and Future Work}\nThis paper explored how self-paced regime leads deep discriminative models (DDMs) to achieve more robust and less biased solutions on different computer vision tasks (\\emph{e.g.}~facial age estimation and head pose estimation).\nSpecifically, a novel self-paced paradigm, which considers ranking fairness, was proposed.\nThe new ranking scheme jointly considers loss and predictive uncertainty.\nSuch a paradigm was combined with deep regression forests (DRFs), and led to a new model, namely self-paced deep regression forests with consideration on underrepresented examples (SPUDRFs).\nExtensive experiments on two well-known computer vision tasks demonstrated the efficacy of the proposed paradigm.\n\n\n\nWe are currently applying self-paced DDMs for other computer vision tasks, \\emph{e.g.}~viewpoint estimation, indoor scene classification, where the ability to handle ranking unfairness is fundamental to the success.\nThus, investigating the causes of algorithm unfairness in DDMs is a worthy direction.\nObviously, except data imbalance, there exist some other causing factors.\nIn addition to this, exploring how to combine the new self-paced paradigm with other DDMs, including deep regressors and  classifiers, will also be our future work.\n\n\n\\noindent \\textbf{Acknowledgement.} The authors gratefully acknowledge the support of China Postdoctoral Science Foundation No.2017M623007.\n\n\\clearpage\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Derivation of Eq.~\\eqref{eq:Ea}}\n\\label{app:chambers}\n\nIn this section we derive~\\eqref{eq:Ea}, that the energetic dependence of a single band of the HH model for rational flux $\\alpha = p\/q$ is given by\n\\begin{equation}\n    E_j(x_0,y_0) =  E_{j}^* - 2 V_x' \\cos q y_0 - 2 V_y' \\cos q x_0 + O(W_j^2\/\\Delta_j).\n\\end{equation}\nIn the related, AA model the quantities $(x_0,y_0)$ have a straightforward interpretation: $x_0$ acts as the phase of the potential, and $y_0$ is a crystal momentum, and \\emph{vice versa} in the dual model obtained by writing in the $y$-basis. The results of this section are obtained using well known properties of the characteristic equation and spectrum of the HH model~\\cite{chambers1965linear,bellissard1982cantor,thouless1983bandwidths,thouless1990scaling,thouless1990scaling,last1994zero,last1992sum}.\n\nOur first step is to obtain a useful form for the characteristic equation, the roots of which are the bands $E_j(x_0,y_0)$. We begin by noting that, per~\\eqref{eq:AA}, when written in the $x$-basis $\\hat{x} \\ket{x} = x \\ket{x}$ the HH hamiltonian takes the form $H  = \\int_0^{2\\pi\\alpha} d x_0 H_\\mathrm{AA}(x_0)$ with\n\\begin{equation}\n    \\begin{aligned}\n    H_\\mathrm{AA}(x_0) & =  \\sum_{n \\in \\mathbb{Z}}   \\Big[ V_y \\left(  \\ket{x_0 + 2 \\pi \\alpha (n + 1) }\\bra{x_0 + 2 \\pi \\alpha n} +\\mathrm{h.c} \\right) + 2 V_x \\cos x_n \\ket{x_0 + 2 \\pi \\alpha n}\\bra{x_0 + 2 \\pi \\alpha n} \\Big].\n    \\end{aligned}\n\\end{equation}\nThis Hamiltonian is manifestly periodic under a shift $x \\to x + 2 \\pi \\alpha q$, and so we may project in a momentum sector, in which we obtain the Bloch Hamiltonian\n\\begin{equation}\n    \\begin{aligned}\n    H_\\mathrm{B}(x_0,y_0) & = -  \\sum_{n \\in \\mathbb{Z}}   \\Big[ V_y \\left( \\mathrm{e}^{ i y_0} \\ket{n+1 }\\bra{n} +\\mathrm{h.c} \\right) + 2 V_x \\cos x_n \\ket{n}\\bra{n} \\Big]\n    \\end{aligned}\n\\end{equation}\nwhere we identify $\\ket{n} \\equiv \\ket{n+q}$. The spectrum of $H$ is thus made up of bands $E_j(x_0,y_0)$ determined by the solutions of the characteristic equation \n\\begin{equation}\n    C(E_j(x_0,y_0),x_0,y_0) = 0\n\\end{equation}\nwhere $C$ is the characteristic polynomial given by\n\\begin{equation}\n    C(E,x_0,y_0) := \\det ( H_\\mathrm{B}(x_0,y_0) - E).\n\\end{equation}\nRemarkably, the characteristic polynomial has a very simple dependence on $x_0,y_0$\n\\begin{equation}\n    C(E,x_0,y_0) = P(E) + C_0(x_0,y_0)\n\\end{equation}\nwhere $P(E)$ is a $q$th order polynomial independent of $x_0$ and $y_0$, and $C_0$ is an energy independent constant\n\\begin{equation}\n    \\begin{aligned}\n       P(E) & = \\det ( H_\\mathrm{B}(\\pi\/2 q,\\pi \/ 2 q) - E)\n       \\\\\n       C_0(x_0,y_0) &= - 2 V_x^q \\cos q x_0 - 2 V_y^q \\cos q y_0.\n    \\end{aligned}\n    \\label{eq:PC}\n\\end{equation}\nHence the roots of the characteristic polynomial are the solutions to the equation\n\\begin{equation}\n    P(E) = - C_0(x_0,y_0).\n    \\label{eq:PC}\n\\end{equation}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{Fig_chambers.pdf}\n    \\caption{\n    \\emph{Graphical illustration of solutions to~\\eqref{eq:PC} for $V_x = V_y = V$ and $\\alpha = p\/q = 1\/7$}: $P(E)$ is shown in blue, the range of values swept out by $C_0$ is demarcated by the dashed green lines, and the corresponding range of values swept out by the roots of~\\eqref{eq:PC} is marked in red on the horizontal axis.\n    }\n    \\label{Fig:Chambers}\n\\end{figure}\n\n\nThe solutions to~\\eqref{eq:PC} are plotted in Fig~\\ref{Fig:Chambers} for $V_x = V_y = V$ and $\\alpha = 1\/7$. In this figure $P(E)$ is shown in blue, the range of values swept out by $C_0$ is demarcated by the dashed green lines, and the corresponding range of values swept out by the solutions to~\\eqref{eq:PC} is marked in red on the horizontal axis. Each red interval corresponds to one band $E_j(x_0,y_0)$. Intuitively, it follows from eyeballing Fig.~\\ref{Fig:Chambers} that each band takes the form\n\\begin{equation}\n    E_j(x_0,y_0) \\approx E_j^* + \\frac{C_0(x_0,y_0)}{P'(E_j^*)}\n\\end{equation}\nwhich may be obtained by linearizing $P(E)$ about its roots $E_j^*$. We expect the corrections to this to be small if the variation of the gradient $P'(E)$ is small over the interval in which $E_j(x_0,y_0)$ varies, i.e. to leading order, that $W_j P''(E_j^*)  \\ll P'(E_j^*)$ where $W_j$ is the bandwidth. There is reason to expect this leading order analysis of the error should be expected to provide an accurate answer: $P''(E)$ varies only on the scale of the separation between successive roots, and thus we generically expect $P''(E_j^*)$ to provide a good order of magnitude estimate for $P''(E)$ for $E$ in the range $E_{j-1}^* \\leq E \\leq E_{j+1}^*$. In the remainder of this section, we perform such a leading order analysis to make intuitive statement more precise.\n\nHaving obtained a form for the characteristic equation, we see that linearizing $P(E)$ about its roots $E_j^*$, yields\n\\begin{equation}\n    0 = P'(E_j^*) (E_j(x_0,y_0) - E_j^*) + C_0(x_0,y_0) + O(P''(E_j^*)(E_j(x_0,y_0) - E_j^*)^2).\n\\end{equation}\nThe solutions to this equation give the band structure up to an error which must be estimated\n\\begin{equation}\n    \\begin{aligned}\n    E_j(x_0,y_0) & = E_j^* + \\frac{C_0(x_0,y_0)}{P'(E_j^*)} - O\\left( \\frac{C_0^2(x_0,y_0) P''(E_j^*)}{(P'(E_j^*))^3} \\right)\n    \\\\\n    &\n    = E_j^* - 2 V_x' \\cos q x_0 - 2 V_y' \\cos q y_0 - O\\left( \\frac{W_j^2 P''(E_j^*)}{P'(E_j^*)} \\right)\n    \\end{aligned}\n    \\label{eq:ejsupp}\n\\end{equation}\nwhere in the second line we have substituted~\\eqref{eq:PC} and defined $V_x' = V_x^q \/ P'(E_j^*)$, $V_y' = V_y^q \/ P'(E_j^*)$ and $W_j = 4 V_x' + 4 V_y'$\n\nFinally, we note that the ratio $P''(E_j^*)\/P'(E_j^*)$ may be related to the band spacing. Specifically, for a quadratic expansion of the characteristic polynomial\n\\begin{equation}\n    P(E) = P'(E_j^*) (E_j(x_0,y_0) - E_j^*) - C_0(x_0,y_0) + \\tfrac12 P''(E_j^*)(E_j(x_0,y_0) - E_j^*)^2 + O(P'''(E_j^*)(E_j(x_0,y_0) - E_j^*)^3)\n\\end{equation}\nthis equation has roots at\n\\begin{equation}\n    E = E_j^*, \\qquad \\text{and} \\qquad E = E_j^* - \\frac{2 P'(E_j^*)}{ P''(E_j^*) } + O \\left(\\frac{P'(E_j^*)^2 P'''(E_j^*)}{ P''(E_j^*)^2}\\right)\n\\end{equation}\nthis yields and distance between the roots of\n\\begin{equation}\n    \\Delta_j = \\frac{2 P'(E_j^*)}{ P''(E_j^*) } + O \\left(\\frac{P'(E_j^*)^2 P'''(E_j^*)}{ P''(E_j^*)^3}\\right)\n\\end{equation}\nwhich, combined with~\\eqref{eq:ejsupp}, yields~\\eqref{eq:Ea} in the main text.\n\n\n\n\n\\section{Numerical evidence of Eq.~\\eqref{eq:Wgap_size}}\n\\label{app:89}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{Fig_supp.pdf}\n    \\caption{\n    \\emph{Limiting forms for the bandwidths $W_j$ and band gaps $\\Delta_j$}: (a,b) the bandwidths and band gaps are analysed for $\\alpha = 1\/q$ in the limit $q \\to \\infty$ (a) $q^{-1} \\log W_j$ is plotted versus $E_j^*$ for various values of $q$ (legend inset), in (b) $( \\Delta_j q)^{-1}$ is plotted versus $E_j^{(\\mathrm{gap})}$, the inset shows the same data on a logarithmic horizontal scale. In (c,d) analogous plots are shown for $\\alpha = p\/q$ with $q = 2 p + 1$.\n    }\n    \\label{Fig:supp}\n\\end{figure}\n\nEq.~\\eqref{eq:Wgap_size} is obtained via anayltic arguments by Wilkinson in Refs.~\\cite{wilkinson1984critical,wilkinson1987exact}. Nevertheless, here we provide some numerical evidence of this result.\n\nConsider the HH Hamiltonian~\\eqref{eq:H1} with $\\alpha = 1\/q$ tuned to the critical point $V_x = V_y = V$. Per the previous Appendix, this Hamiltonian has $q$ bands $E_j(x_0,y_0)$ for $j = 1 \\cdots q$. We denote the extrema of each band by $E_j^{( \\min )}$ and $E_j^{( \\max )}$. We further denote the band centers, bandwidths, and gap centers respectively by\n\\begin{equation}\n    E_j^* = \\tfrac12 \\left( E_j^{( \\max )} + E_j^{( \\min )} \\right),\n    \\qquad \n    \\Delta_j = \\rho_j^{-1} =  E_{j+1}^{( \\min )} - E_j^{( \\max )} ,\n    \\qquad\n    E_j^{(\\mathrm{gap})} = \\tfrac12 \\left( E_{j+1}^{( \\min )} + E_j^{( \\max )} \\right)\n\\end{equation}\nwhere $\\rho_j = \\Delta_j^{-1}$ is the density of states at the gap center.\n\nIn Fig.~\\ref{Fig:supp}a we show the bandwidths are decaying exponentially in $q$, specifically we plot $q^{-1} \\log W_j$ versus $E_j^*$ for various values of $q$ (legend inset). The different $q$ series approaches a limiting form at large $q$\n\\begin{equation}\n    \\log W_j \\sim - q \\ell(E_j^* \/ V).\n\\end{equation}\nIn Fig.~\\ref{Fig:supp}b we plot $\\rho_j \/ q$ as a function of $E_j^{(\\mathrm{gap})}$, showing that in the same limit the density of states has the limiting form\n\\begin{equation}\n    \\rho_j = \\Delta_j^{-1} \\sim q \\varrho(E_j^* \/ V) \/ V.\n\\end{equation}\nIn the inset of Fig.~\\ref{Fig:supp}b the same data is shown on a log scale, showing that $\\varrho(z)$ has an integrable (specficially logarithmic) divergence at $z=0$. Plots Fig.~\\ref{Fig:supp}c-d show the equivalent plots in the limit of large $q$ with $q = 2 p+1$, illustrating that analgous limits occur for $\\alpha \\to 1\/2$. Indeed similar limits apply for $\\alpha$ approaching any rational. \n\n\n\\section{The ergodic map $B(\\bar{\\alpha})$}\n\\label{app:T}\n\nIn this section we show the ergodicity of the map $B:[0,1\/2]\\to[0,1\/2]$ corresponding to an RG where we project into the lower band at each step. That, for $B$ given by\n\\begin{equation}\n    B(\\bar{\\alpha}) = \\min\\left[ b(\\bar{\\alpha}) , 1 - b(\\bar{\\alpha})\\right], \\qquad b(\\bar{\\alpha}) = \\frac{1}{\\bar{\\alpha}} - \\left\\lfloor \\frac{1}{\\bar{\\alpha}} \\right\\rfloor,\n\\end{equation}\nas defined in the main text in Eq.~\\eqref{eq:beta_scaling}. Results for an RG projecting into the middle band at each step, as used in the latter part of the paper, follow by the same methods.\n\nWe employ a numerical approach previously used in Ref.~\\cite{briggs2003precise,flajolet1995gauss} to calculate the spectral gap of the Gauss map. Consider an initial set values $\\bar{\\alpha}_0^{(i)} \\in [0,1\/2]$ characterized by a smooth distributed $g_0(\\bar{\\alpha})$. Each of these values can be renormalized to yield $\\bar{\\alpha}_n^{(i)} = B^n(\\bar{\\alpha}_0^{(i)})$, which is also characterised by a smooth distribution function $g_n(\\bar{\\alpha})$. As $n$ is taken large $g_n$ converges to the unique steady state distribution\n\\begin{equation}\n    g_n(\\bar{\\alpha}) \\to f(\\bar{\\alpha}) = \\frac{1}{\\log \\varphi} \\cdot \\frac{\\varphi^3}{\\varphi^3 + \\bar{\\alpha}-\\bar{\\alpha}^2}\n    \\label{eq:f_app}\n\\end{equation}\nwhere $\\varphi = (1 + \\sqrt{5})\/2$ is the golden ratio. Moreover, the deviation from the limiting distribution is exponentially small in $n$\n\\begin{equation}\n    \\log |g_n(\\bar{\\alpha}) - f(\\bar{\\alpha})| \\sim  - n\\Delta,\n   \n    \\label{eq:Delta}\n\\end{equation}\nwhere\n\\begin{equation}\n    \\Delta = 3.7856665519818449128 \\ldots\n\\end{equation}\nis the spectral gap of $B$. The statement~\\eqref{eq:Delta} together with $\\Delta>0$ demonstrates the ergodicity of the map $B$. Moreover, the large value $\\Delta \\gg 1$, indicates that the convergence of $g_n(x)$ to $f(x)$ occurs rapidly over an $O(1)$ number of steps. Eq.~\\eqref{eq:Delta} is the main result of this section. Prior to the main result, we arrive at two further results: (i) we show that $f(\\bar{\\alpha})$ in~\\eqref{eq:f_app} is the steady state, and (ii) we show that the map $B$ is chaotic with maximal Lyapunov exponent\n\\begin{equation}\n    \\Lambda = \\frac{\\pi^2}{6 \\log \\varphi}.\n\\end{equation}\n\n\\subsection{Steady state distribution of $B$}\n\nThe sequence of distributions $g_n$ are defined by recursive application of the map $B$, i.e. $g_{n+1}(\\bar{\\alpha}) = [B g_n](\\bar{\\alpha})$, where the action of $B$ on $g$ is given explicitly by\n\\begin{equation}\n    \\begin{aligned}\n     [B g](\\bar{\\alpha}) & : = \\int_0^{1\/2} d\\bar{\\alpha}' \\delta(\\bar{\\alpha} - B(\\bar{\\alpha}')) g(\\bar{\\alpha}')  = \\sum_{q=2}^\\infty \\left[ \\frac{g\\left(\\frac{1}{q+\\bar{\\alpha}}\\right)}{(q+\\bar{\\alpha})^2} + \\frac{g\\left(\\frac{1}{q+1-\\bar{\\alpha}}\\right)}{(q+1-\\bar{\\alpha})^2}\\right].\n     \\end{aligned}\n\\end{equation}\nNote that the action of $B$ on the space of distributions $g$ is linear, $[B(g+h)] = [B g] + [B h]$, and thus the steady state distribution $f$ is obtained as the leading eigenfunction of $B$, which has a corresponding eigenvalue of unity\n\\begin{equation}\n    [B f](\\bar{\\alpha}) = f(\\bar{\\alpha}).\n\\end{equation}\nIt is then straightforward to verify that~\\eqref{eq:f_app} satisfies this relation. The uniqueness of this solution is verified numerically in App.~\\ref{app:ergB}.\n\n\\subsection{Chaoticity of $B$}\n\nThe Lyapunov exponent of a discrete map is given by\n\\begin{equation}\n    \\Lambda = \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{m = 1}^n \\log |B'(\\bar{\\alpha}_n)|\n\\end{equation}\nwhere $\\bar{\\alpha}_n = B^n(\\bar{\\alpha}_0)$, $B'(\\bar{\\alpha})$ is the derivative of $B(\\bar{\\alpha})$ and the Lyapunov exponent $\\Lambda$ is independent of $\\bar{\\alpha}_0$ due to the ergodicity of $B$.\n\nMoreover, as $B$ is ergodic, $\\Lambda$ may be straightforwardly evaluated using the steady state distribution $f(\\bar{\\alpha})$\n\\begin{equation}\n\\begin{aligned}\n    \\Lambda & = \\int_0^{1\/2}d \\bar{\\alpha} \\, f(\\bar{\\alpha}) \\, \\log| B(\\bar{\\alpha})|\n    = - 2 \\int_0^{1\/2}d \\bar{\\alpha} \\, f(\\bar{\\alpha}) \\, \\log \\bar{\\alpha} \n    = \\frac{\\pi^2}{6 \\log \\varphi}\n    \\end{aligned}\n    \\label{eq:lyapunov_app}\n\\end{equation}\nIn Eq.~\\eqref{eq:lyapunov_app} we have used that $| B(\\bar{\\alpha})| = \\bar{\\alpha}^{-2}$ except at a measure zero set of points, where the derivative is undefined. As $\\Lambda > 0$, $B$ is chaotic.\n\n\\subsection{Ergodicity of $B$}\n\\label{app:ergB}\n\n\n\n\nAs $B$ is a linear operator, it has a spectrum of eigenvalues $\\beta_k \\geq 0$ with associated eigenfunctions $v_k(\\bar{\\alpha})$ which form a complete basis\n\\begin{equation}\n    [B v_k](\\bar{\\alpha}) = \\beta_k v_k(\\bar{\\alpha}).\n    \\label{eq:B_eigs}\n\\end{equation}\nIn principle the spectrum of eigenvalues may have discrete and continuous components, though in the present case we find only a discrete spectrum allowing us to index them in descending order $1 = |\\beta_0| \\geq |\\beta_1| \\geq |\\beta_2| \\geq \\cdots$. The distribution at late times is found by projecting onto the subspace of eigenfunctions with eigenvalues $|\\beta_k| =1$. If there is exactly one such eigenvalue, which we denote as $\\beta_0 = 1$ (the eigenvalue cannot have a phase as $g_n(\\bar{\\alpha})$ is strictly non-negative), then the steady state distribution $f(\\bar{\\alpha})$ is unique, independent of $g_0$, and given by the corresponding eigenfunction $f = v_0$. The deviation of $g_n$ from $v_0$ is then determined by the first sub-leading eigenvalue: $|f - g_n| = O(\\beta_1^n) = O(\\mathrm{e}^{- \\Delta n})$, where we have defined\n\\begin{equation}\n    \\Delta = - \\log |\\beta_1|\n\\end{equation}\nas the spectral gap of $B$.\n\nThe eigenfucntion(s) $v_k(\\bar{\\alpha})$ may be obtained as the solutions to the eigenvalue equation~\\eqref{eq:B_eigs}, however in the absence of an analytic technique to solve this equation, we resort to numerics. To numerically tackle this problem we first re-write in terms of the coordinate $y = 1\/2 - \\bar{\\alpha} \\in [0,1\/2]$. In this coordinate $B$ has the action\n\\begin{equation}\n     [B g](y) = \\sum_{h} \\left[ \\frac{g\\left(\\tfrac12 - \\frac{1}{h - y}\\right)}{(h- y)^2} + \\frac{g\\left(\\tfrac12 - \\frac{1}{h+y }\\right)}{(h+y)^2}\\right]\n\\end{equation}\nwhere the sum is taken over the half-integers $h = \\tfrac52, \\tfrac72, \\tfrac92, \\tfrac{11}{2} \\ldots$. To make the problem numerically tractable we subsequently write $B$ in a basis spanned by a countable set of basis elements. For simplicity we choose the basis monomials\n\\begin{equation}\n    u_p(y) = y^p = (1\/2 - x)^p\n\\end{equation}\nupon which $B$ acts as\n\\begin{equation}\n    \\begin{aligned}\n     [B u_p](y) &= \\sum_{h} \\frac{1}{2^p}\\left[ \\frac{\\left(1 - \\frac{2}{h - y}\\right)^p}{(h- y)^2} + \\frac{\\left(1 - \\frac{2}{h+y }\\right)^p}{(h+y)^2}\\right]\n     \\\\\n    \n    \n     & = \\sum_{h} \\frac{1}{2^{p+2}} \\sum_{k = 0}^p \\binom{p}{k}\\left( -\\frac{2}{h}\\right)^{k+2}\\left[ \\left( \\frac{1}{1 - y\/h}\\right)^{k+2} + \\left( \\frac{1}{1+y\/h }\\right)^{k+2}\\right]\n     \\\\\n     & = \\sum_{h} \\frac{1}{2^{p+2}} \\sum_{k = 0}^p \\binom{p}{k}\\left( -\\frac{2}{h}\\right)^{k+2}\\left[ 2 \\sum_{n = 0}^\\infty \\binom{2n + k + 1 }{2n} \\left(\\frac{y}{h}\\right)^{2n}\n     \\right]\n     \\end{aligned}\n\\end{equation}\nRecalling the definition of the Hurwitz zeta function $\\zeta(s,a) = \\sum_{n=0}^\\infty (n+a)^{-s}$, and rerranging we find\n\\begin{equation}\n    [B u_p](y) =  \\sum_{k = 0}^p \\sum_{n = 0}^\\infty  \\binom{p}{k} \\binom{2n + k + 1 }{2n}   \n    \\frac{(-1)^k}{2^{p - k - 1}} \\zeta(2n + k + 2,5\/2) u_{2n}(y).\n    \\label{eq:Tup}\n\\end{equation}\nWe re-write~\\eqref{eq:Tup} to define $M_{pq}$, the transfer matrix on the basis on monomials $u_p$ we \n\\begin{equation}\n    [B u_p](y) = \\sum_{q = 0}^\\infty M_{pq} u_q(y)\n\\end{equation}\nwhere the matrix elements are given by \n\\begin{equation}\n    M_{pq} = \\begin{cases}\n    \\displaystyle\n    \\frac{1}{2^{p-1}}\\sum_{k = 0}^p  \\binom{p}{k} \\binom{q + k + 1 }{q}  \n    (-2)^k\\zeta(q + k + 2,5\/2) & \\qquad q  \\text{\\,\\, even},\n    \\\\[15pt]\n    0 & \\qquad q  \\text{\\,\\, odd}.\n    \\end{cases}\n\\end{equation}\n\nThe spectrum of $M$, and hence $B$, may then be numerically estimated by evaluating $M_{pq}$ up to a cutoff $p,q \\leq p_{\\max}$ and diagonalising. The eigenvalues $\\beta_k$ are found to be discrete, non-degenerate and exponentially decaying in $k$. As a result the values of low order eigenvalues converge exponentially as a function of $p_{\\max}$, allowing them to be accurately numerically estimated. The numerical limitation is the evaluation of the matrix elements, which require high precision numerics for even moderately large $p_{\\max}$. Numerically extracted values for the magnitudes of the first five sub-leading eigenvalues are given below (to 20 significant figures)\n\\begin{equation}\n    \\begin{aligned}\n       - \\log |\\beta_0| &= 0\n     \\\\\n     \\Delta = \\Delta_1 = - \\log |\\beta_1| & = 3.7856665519818449128\n     \\\\\n     \\Delta_2 = - \\log |\\beta_2| & = 6.7251453074741971174\n     \\\\\n     \\Delta_3 = - \\log |\\beta_3| & = 11.339665867968595165\n     \\\\\n     \\Delta_4 = - \\log |\\beta_4| & = 12.043871233196576668\n     \\\\\n     \\Delta_5 = - \\log |\\beta_5| & = 16.966376007200018885\n    \\end{aligned}\n\\end{equation}\n\n\nIndeed, as expected, the associated leading eigenfunction is found to be\n\\begin{equation}\n    f(\\bar{\\alpha}) \\propto \\sum_{q = 0}^\\infty \\left( \\frac{ 1 - 2 \\bar{\\alpha}}{\\varphi^3} \\right)^{2q}\n    \\label{eq:f_sum_app}\n\\end{equation}\nwhere $\\varphi = (1 + \\sqrt{5})\/2$ is the Golden Ratio. Performing the sum in~\\eqref{eq:f_sum_app} and normalising yields~\\eqref{eq:f_app}.\n\n\n\n\n\\end{widetext}\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor $\\sigma\\dvtx \\mathbb{R}\\to\\mathbb{R}$ and $b\\dvtx\n\\mathbb{R}\\to\\mathbb {R}$, we are interested in the simulation of the\nstochastic differential equation\n\\begin{equation}\ndX_t=\\sigma(X_t)\\,dW_t+b(X_t)\\,dt,\n\\label{sde}\n\\end{equation}\nwhere $X_0=x_0\\in\\mathbb{R}$ and $W=(W_t)_{t\\geq0}$ is a standard Brownian\nmotion. We make the standard Lipschitz assumptions on the coefficients,\n\\[\n\\exists K\\in(0,+\\infty), \\forall x,y\\in\\mathbb{R}\\qquad\n\\bigl|\\sigma(x)-\\sigma(y)\\bigr|+\\bigl|b(x)-b(y)\\bigr|\\leq K|x-y|.\n\\]\n\n\nFor $T>0$, we are interested in the approximation of\n$X=(X_t)_{t\\in[0,T]}$ by its Euler scheme\n$\\bar{X}=(\\bar{X}_t)_{t\\in[0,T]}$ with $N\\geq1$ time-steps. We consider\nthe regular grid $\\{0=t_0<t_1<t_2<\\cdots<t_N=T\\}$ of the interval\n$[0,T]$ with $t_k=\\frac{kT}{N}$ and define inductively $\\bar{X}_0=x_0$\nand\n\\begin{equation}\n\\bar{X}_t=\\bar{X}_{t_k}+\\sigma(\\bar{X}_{t_k})\n(W_t-W_{t_k})+b(\\bar{X}_{t_k})\n(t-t_k)\\qquad\\mbox{for }t\\in[t_k,t_{k+1}].\n\\label{eul}\n\\end{equation}\nIt is well known that the order of convergence of the strong error of\ndiscretization is $N^{-1\/2}$. Indeed, we have (see~\\cite{Ka})\n\\begin{equation}\n\\forall p\\geq1, \\exists C<+\\infty, \\forall N\\geq1\\qquad\n\\mathbb{E}^{1\/p}\n\\Bigl[\\sup_{t\\leq T}|X_t-\\bar{X}_t|^p\n\\Bigr]\\leq\\frac{C}{\\sqrt{N}}.\n\\end{equation}\nSee Section~\\ref{Sec_res_std} for a more precise statement. This\nupper-bound gives the correct order of convergence since according to\nRemark 3.6 \\cite{kurtz5}, when $\\sigma$ and $b$ are continuously\ndifferentiable, $(\\sqrt{N}(X_t-\\bar{X}_t))_{t\\leq T}$ converges in law\nas $N$ goes to $\\infty$ to some diffusion limit which is nonzero as\nsoon as $\\sigma$ is positive and nonconstant (see also \\cite\n{kurtz6} and \\cite{jac1} where stable convergence is also proved). When\n$\\sigma$ is constant, then the Euler scheme coincides with the Milstein\nscheme, and the strong order of convergence is $N^{-1}$.\n\nOn the other hand, the order of convergence of the weak error of\ndiscretization is always $N^{-1}$. For example, according to \\cite{tt},\nwhen $\\sigma$ and $b$ are $C^\\infty$ with bounded derivatives of all\norders and $f\\dvtx \\mathbb{R}\\to\\mathbb{R}$ is $C^\\infty$ with\npolynomial growth together with its derivatives, then for each integer\n$L\\geq1$, the expansion\n\\begin{equation}\n\\mathbb{E} \\bigl[f(X_T) \\bigr]-\\mathbb{E} \\bigl[f(\n\\bar{X}_T) \\bigr]=\\sum_{l=1}^L\n\\frac\n{a_l}{N^l}+{\\mathcal O} \\bigl(N^{-(L+1)} \\bigr)\\label{deverrfaib}\n\\end{equation}\nin powers of $N^{-1}$ holds for the weak error. The bound\n$|\\mathbb{E}[f(\\bar{X}_T)]-\\mathbb{E}[f(X_T)]|\\leq\\frac{C}{N}$ holds\nwhen $\\sigma,b$ and $f$ are $C^4$ with the same growth assumptions.\nWhen $f$ is only assumed to be measurable and bounded, it is proved in\n\\cite{bt1,bt2} that the expansion (\\ref{deverrfaib}) still holds for\n$L=1$ if $b$ and $\\sigma$ are smooth functions satisfying an\nhypoellipticity condition. Under uniform ellipticity, Guyon\n\\cite{gu} even extends this expansion by only assuming that $f$ is a\ntempered distribution acting on the densities of both $X_T$ and\n$\\bar{X}_T$.\n\nIn view of financial applications, the weak error analysis gives the\nconvergence rate to $0$ of the discretization bias introduced when\nreplacing $X$ by its Euler scheme $\\bar{X}$ for the computation of the\nprice $\\mathbb{E}[f(X_T)]$ of a vanilla European option with payoff $f$\nand maturity $T$ written on $X$. Let $\\mathcal{C}$ denote the space\n$C([0,T],\\mathbb{R})$ of continuous paths endowed with the sup norm.\nWhen dealing with exotic options with payoff $F\\dvtx\n\\mathcal{C}\\to\\mathbb{R}$ Lipschitz continuous,\n\\[\n\\bigl\\llvert\\mathbb{E} \\bigl[F(X) \\bigr]-\\mathbb{E} \\bigl[F(\\bar{X})\n\\bigr]\n\\bigr\\rrvert\\leq\\mathbb{E}\\bigl|F(X)-F(\\bar{X})\\bigr|\\leq\\frac{C}{\\sqrt{N}},\n\\]\nwhere the second inequality follows from the strong error estimate. But\nthe first inequality is very rough and prevents us from taking\nadvantage of the cancellations in the mean which occur and permit us to\nobtain the upper-bound $\\frac{C}{N}$ for vanilla options. The weak\nerror analysis has been performed for specific path-dependent payoffs,\ntypically when $F(X)=f(X_T,Y_T)$ with $Y_t$ a function of $(X_s)_{0\\leq\ns\\leq t}$ such that $((X_t,Y_t))_{0\\leq t\\leq T}$ is a Markov process.\nThe cases $Y_t=\\int_0^tX_s\\,ds$ and $Y_t=\\max_{0\\leq s\\leq t}X_s$,\nrespectively, correspond to Asian \\cite{thesestemam} and barrier \\cite\n{gob1,gob2,gobmen} or lookback options \\cite{thess}. But no general\ntheory has been developped so far to analyze the weak trajectorial\nerror. The Wasserstein distance between the laws $\\mathcal{L}(X)$ and\n$\\mathcal{L}(\\bar{X})$ of $X$ and $\\bar{X}$ defined by\n\\[\n\\mathcal{W}_1 \\bigl(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}) \\bigr)=\\sup\n_{F\\dvtx \\mathcal{C}\\to\\mathbb\n{R}\\dvtx \\mathrm{Lip}(F)\\leq1}\\bigl|\\mathbb{E} \\bigl[F(\\bar{X}) \\bigr]-\\mathbb{E} \\bigl[F(X)\n\\bigr]\\bigr|,\n\\]\nwhere $\\mathrm{Lip}(F)$ denotes the Lipschitz constant of $F$ is the\nappropriate measure to deal with the whole class of exotic Lipschitz\npayoffs. Notice that this distance has already been used in the context\nof discretization schemes for SDEs: in the multidimensional setting, by\na clever rotation of the driving Brownian motion, Cruzeiro, Malliavin\nand Thalmaier~\\cite{cruz} construct a modified Milstein scheme which\ndoes not involve the simulation of iterated Brownian integrals and with\norder of convergence $N^{-1}$ for the Wasserstein distance. A simpler\nscheme with the same convergence properties is exhibited in\n\\cite{js} for usual stochastic volatility models.\n\nThe weak and strong error estimations recalled above imply that\n\\begin{equation}\n\\exists c, C<+\\infty, \\forall N\\geq1\\qquad \\frac{c}{N}\\leq\\mathcal{W}_1\n\\bigl(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}) \\bigr)\\leq\\frac{C}{\\sqrt{N}}.\n\\label{minmajw1}\n\\end{equation}\nA very nice feature of the Wasserstein distance is its primal\nrepresentation in the Kantorovitch duality theory. This representation\nis obtained by choosing $p=1$, $E=\\mathcal{C}$ and\n$(\\mu,\\nu)=(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}))$ in the general\ndefinition\n\\begin{equation}\n\\mathcal{W}_p(\\mu,\\nu)= \\biggl(\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int\n_{E\\times E}|x-y|^p\\pi(dx,dy) \\biggr)^{1\/p},\n\\label{defwas}\n\\end{equation}\nwhere $p\\in[1,+\\infty)$, $(E,|~|)$ is a normed vector space, $\\mu$ and\n$\\nu$ are two probability measures on $E$ endowed with its Borel\nsigma-field and the infimum is computed on the set $\\Pi(\\mu,\\nu)$ of\nprobability measures on $E\\times E$ with respective marginals $\\mu$ and\n$\\nu$; see, for instance, Remark 6.5 page 95 \\cite{villani}. When one\nis able to exhibit some coupling $(Y,\\bar{Y})$ with\n$Y\\stackrel{\\mathcal{L}}{=}X$ and\n$\\bar{Y}\\stackrel{\\mathcal{L}}{=}\\bar{X}$, then the law of\n$(Y,\\bar{Y})$ belongs\\vspace*{1pt} to\n$\\Pi(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}))$ and necessarily\n$\\mathcal{W}_p(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}))\\leq\n\\mathbb{E}^{1\/p} [\\sup_{t\\in[0,T]}|Y_t-\\bar{Y}_t|^p ]$. For the obvious\ncoupling $(Y,\\bar{Y})=(X,\\bar{X})$ obtained by choosing the same\ndriving Brownian motion for the diffusion and its Euler scheme, one\nrecovers the upper bound in (\\ref{minmajw1}) from the strong error\nanalysis. The main result of the present paper is the construction of a\nbetter coupling which leads to the upper bound\n\\[\n\\forall p\\geq1, \\forall\\varepsilon>0, \\exists C<+\\infty, \\forall N\\geq1\n\\qquad\n\\mathcal{W}_p \\bigl(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}) \\bigr)\\leq\n\\frac{C}{N^{2\/3-\\varepsilon}}\n\\]\nproved in Section~\\ref{sec_pathwise} under additional regularity\nassumptions on the coefficients and uniform ellipticity. To construct\nthis coupling, we first obtain in Section~\\ref{sec_marginal} a\ntime-uniform estimation of the Wasserstein distance between the\nrespective laws $\\mathcal{L}(X_t)$ and $\\mathcal{L}(\\bar{X}_t)$ of\n$X_t$ and $\\bar{X}_t$,\n\\[\n\\forall p\\geq1, \\exists C<+\\infty, \\forall N\\geq1\\qquad\n\\sup_{t\\in[0,T]} \\mathcal{W}_p \\bigl(\\mathcal{L}(X_t),\\mathcal{L}(\n\\bar{X}_t) \\bigr)\\leq\\frac{C\\sqrt{\\log(N)}}{N}.\n\\]\nPreviously, in Section~\\ref{Sec_res_std}, we recalled well-known\nresults concerning the moments and the dependence on the initial\ncondition of the solution to the SDE~(\\ref{sde}) and its Euler scheme.\nAlso, we make explicit the dependence of the strong error estimations\n$\\mathbb{E} [\\sup_{s\\le t}|\\bar{X}_s-X_s|^p]$ with respect to\n$t\\in[0,T]$, which will play a key role in our analysis.\n\n\n\n\\section{Basic estimates on the SDE and its Euler scheme}\\label{Sec_res_std}\n\nWe recall some well-known results concerning the flow defined\nby~(\\ref{sde}) (see, e.g., Karatzas and Shreve~\\cite{KS}, page 306) and\nits Euler approximation.\n\n\n\n\\begin{aprop}\nLet us denote by $(X^{x}_t)_{t\\in[0,T]}$ the solution of (\\ref{sde}),\nstarting from $x\\in\\mathbb{R}$. One has that for any $p\\geq1$, the existence\nof a positive constant $C\\equiv C(p,T)$ such that\n\\begin{eqnarray}\n\\forall x \\in\\mathbb{R}\\qquad\n\\mathbb{E} \\Bigl[\\sup_{t\\in[0,T]}\\bigl|X^{x}_t\\bigr|^p\\Bigr]&\\leq& C\\bigl(1+|x|\\bigr)^p, \\label{momenteds}\n\\\\\n\\qquad\\quad \\forall x\\in\\mathbb{R}, \\forall s \\leq t \\leq T\n\\qquad \\mathbb{E} \\Bigl[\\sup _{u\\in[s,t]}\\bigl|X^{x}_{u}-X^{x}_{s}\\bigr|^p\n\\Bigr] &\\leq& C\\bigl(1+|x|\\bigr)^p(t-s)^{p\/2},\\label{accroisseds}\n\\\\\n\\forall x, y\\in\\mathbb{R}\\qquad \\mathbb{E} \\Bigl[\\sup_{t\\in\n[0,T]}\\bigl|X^{x}_t-X^{y}_t\\bigr|^p\n\\Bigr] &\\leq& C|y-x|^p.\\label{cieds}\n\\end{eqnarray}\n\\end{aprop}\n\n\\begin{aprop}\\label{vitfort_prop}\nLet $(\\bar{X}^{x}_t)_{t\\in[0,T]}$\ndenote the\nEuler scheme~(\\ref{eul}) starting from~$x$.\nFor any $p\\in[1,\\infty)$, there exists a positive constant $C\\equiv\nC(p,T)$ such that\n\\begin{eqnarray}\n\\forall N\\geq1, \\forall x\\in\\mathbb{R}\\qquad \\mathbb{E} \\Bigl[\\sup\n_{t\\in[0,T]}\\bigl|\\bar{X}^{x}_t\\bigr|^p\n\\Bigr]&\\leq& C\\bigl(1+|x|\\bigr)^p,\\label{momenteul}\n\\\\\n\\hspace*{30pt}\\forall N\\geq1, \\forall x\\in\\mathbb{R}, \\forall t\\in[0,T]\\qquad \\mathbb{E} \\Bigl[\\sup_{r\\in[0,t]}\\bigl|\\bar{X}^{x}_r-X^{x}_r\\bigr|^p\n\\Bigr]&\\leq&\\frac{C t^{p\/2}(1+|x|)^p}{N^{p\/2}}.\\label{vitfort}\n\\end{eqnarray}\n\\end{aprop}\n\nThe moment bound~(\\ref{momenteul}) for the Euler scheme holds in fact\nas soon as the drift and the diffusion coefficients have a sublinear\ngrowth. The strong convergence order is established in\nKanagawa~\\cite{Ka} for Lipschitz and bounded coefficients. In fact, it\nis straightforward to extend Kanagawa's proof to merely Lipschitz\ncoefficients by using the estimates~(\\ref{momenteds})\nand~(\\ref{momenteul}) and obtain\n\\begin{equation}\n\\label{vitfort2}\n\\qquad\\forall N\\geq1, \\forall x\\in\\mathbb{R}, \\forall t\\in[0,T]\\qquad \\mathbb{E} \\Bigl[\\sup_{r\\in[0,t]}\\bigl|\\bar{X}^{x}_r-X^{x}_r\\bigr|^p\n\\Bigr]\\leq\\frac{C (1+|x|)^p}{N^{p\/2}}.\n\\end{equation}\nThe estimate~(\\ref{vitfort}) precises the dependence on $t$. This\nslight improvement will in fact play a crucial role in constructing the\ncoupling between the diffusion and the Euler scheme. We prove it for\nthe sake of completeness, even though the arguments are standard.\n\n\n\\begin{pf*}{Proof of~(\\ref{vitfort})}\nLet $\\tau_s=\\sup\\{t_i, t_i\\le s\\}$ denote the last discretization time\nbefore~$s$. We have $\\bar{X}^{x}_t-X^x_t=\\int_0^t\nb(\\bar{X}^{x}_{\\tau_s})-b(X^x_s) \\,ds + \\int_0^t\n\\sigma(\\bar{X}^{x}_{\\tau_s})-\\sigma(X^x_s) \\,dW_s$. By the Jensen and\nBurkholder--Davis--Gundy inequalities,\n\\begin{eqnarray*}\n&& \\mathbb{E} \\Bigl[ \\sup_{r \\in[0,t]} \\bigl|\\bar{X}^{x}_r-X^x_r\\bigr|^p\n\\Bigr]\n\\\\\n&&\\qquad \\le 2^p \\biggl( \\mathbb{E} \\biggl[ \\biggl(\\int\n_0^t \\bigl|b \\bigl(\\bar{X}^{x}_{\\tau_s}\n\\bigr)-b \\bigl(X^x_s \\bigr)\\bigr| \\,ds \\biggr)^p\n\\biggr]\n\\\\\n&&\\hspace*{48pt}{} + C_p\\mathbb{E} \\biggl[ \\biggl(\\int_0^t\n\\bigl(\\sigma\\bigl(\\bar{X}^{x}_{\\tau_s} \\bigr)-\\sigma\n\\bigl(X^x_s \\bigr) \\bigr)^2 \\,ds\n\\biggr)^{p\/2} \\biggr] \\biggr)\n\\\\\n&&\\qquad \\le 2^p \\biggl( t^{p-1}\\int_0^t\n\\mathbb{E} \\bigl[ \\bigl|b \\bigl(\\bar{X}^{x}_{\\tau\n_s} \\bigr)-b\n\\bigl(X^x_s \\bigr)\\bigr|^p \\bigr] \\,ds\n\\\\\n&&\\hspace*{48pt}{}+ C_pt^{p\/2-1}\\int_0^t \\mathbb{E} \\bigl[\n\\bigl|\\sigma\\bigl(\\bar{X}^{x}_{\\tau_s} \\bigr)- \\sigma\\bigl(X^x_s\n\\bigr)\\bigr|^p \\bigr]\\,ds \\biggr).\n\\end{eqnarray*}\n\nDenoting by $\\mathrm{Lip}(\\sigma)$ the finite Lipschitz constant of\n$\\sigma$, we have $|\\sigma(\\bar{X}^{x}_{\\tau_s})-\\sigma(X^x_s)|\\le\n\\mathrm{Lip}(\\sigma)(|\\bar{X}^{x}_{\\tau_s}-X^x_{\\tau_s}|+|X^x_{\\tau_s}-X^x_s|)$.\nThus, (\\ref{accroisseds}) and~(\\ref{vitfort2}) yield\\break\n$\\mathbb{E}[|\\sigma(\\bar{X}^{x}_{\\tau_s})-\\sigma(X^x_s)|^p]\\le \\frac{C\n(1+|x|)^p}{N^{p\/2}}, $ and the same bound holds for $b$\nreplacing~$\\sigma$. Since $t^p\\leq T^{p\/2}t^{p\/2}$, we easily conclude.\n\\end{pf*}\n\n\\section{The Wasserstein distance between the marginal laws}\\label{sec_marginal}\n\nIn this section, we are interested in finding an upper bound for the\nWasserstein distance between the marginal laws of the SDE~(\\ref{sde})\nand its Euler scheme. It is well known that the optimal coupling\nbetween two one-dimensional random variables is obtained by the inverse\ntransform sampling. Thus, let $F_t$ and $\\bar{F}_t$ denote the\nrespective cumulative distribution functions of $X_t$ and $\\bar{X}_t$.\nThe $p$-Wasserstein distance between the time-marginals of the\nsolution\\vadjust{\\goodbreak}\nto the SDE and its Euler scheme is given by (see Theorem~3.1.2\nin~\\cite{raru})\n\\begin{equation}\n\\mathcal{W}_p \\bigl(\\mathcal{L}(X_t),\\mathcal{L}(\n\\bar{X}_t) \\bigr)= \\biggl(\\int_0^1\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^p\\,du \\biggr)^{1\/p}.\n\\label{wpinv}\n\\end{equation}\nLet us state now the main result of this section. We set\n\\begin{eqnarray*}\nC^k_b&=& \\bigl\\{f\\dvtx \\mathbb{R} \\rightarrow\\mathbb{R}\\ k \\mbox{\ntimes continuously differentiable  s.t. }\n\\\\\n&&\\hspace*{115pt}\\bigl\\|f^{(i)} \\bigr\\|_\\infty<\n\\infty, 0\\le i\\le k \\bigr\\}.\n\\end{eqnarray*}\n\n\n\n\\begin{ahyp}\\label{hyp_wass_marginal}\nLet $a=\\sigma^2$. We assume that $a, b \\in C^2_b$, $a''$\nis globally \\mbox{$\\gamma$-}H\\\"older continuous with $\\gamma>0$ and\n\\[\n\\exists\\underline{a}>0, \\forall x\\in\\mathbb{R}, a(x)\\geq\\underline\n{a} \\mbox{ (uniform ellipticity)}.\n\\]\n\\end{ahyp}\n\n\nSince $\\sigma$ is Lipschitz continuous, under\nHypothesis~\\ref{hyp_wass_marginal}, we have either\n$\\sigma\\equiv\\sqrt{a}$ or $\\sigma\\equiv-\\sqrt{a}$. From now on, we\nassume without loss of generality that $\\sigma\\equiv\\sqrt{a}$ which is\na $C^2_b$ function bounded from below by the positive constant\n$\\underline{\\sigma}=\\sqrt{\\underline{a}}$.\n\n\n\n\\begin{theorem}\\label{wasun}\nUnder Hypothesis~\\ref{hyp_wass_marginal}, we have for\nany $p\\ge1$,\n\\[\n\\forall N\\geq1\\qquad \\sup_{t\\in[0,T]}\\mathcal{W}_p \\bigl(\n\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t) \\bigr)\\leq\n\\frac{C\\sqrt{\\log(N)}}{N},\n\\]\nwhere $C$ is a positive constant that only depends on $p$, $T$,\n$\\underline{a}$ and ($\\|a^{(i)}\\|_\\infty$, $\\|b^{(i)}\\|_\\infty$, $0\\le\ni\\le2$) and does not depend on the initial condition~$x\\in\\mathbb{R}$.\n\\end{theorem}\n\n\n\\begin{arem}\\label{w1unif}\nWhen $p=1$, the slightly better bound\n$\\sup_{t\\in[0,T]}\\mathcal{W}_1(\\mathcal{L}(X_t),\\allowbreak \\mathcal{L}(\\bar\n{X}_t))\\leq\\frac{C}{N}$ holds if $\\sigma$ is uniformly elliptic,\naccording to~\\cite{thesesbai}, Chapter~3. This is proved in a\nmultidimensional setting for $C^\\infty$ coefficients $\\sigma$ and $b$\nwith bounded derivatives by extending the results of \\cite{gu} but can\nalso be derived from a result of Gobet and Labart~\\cite{goblab} only\nsupposing that $b,\\sigma\\in C^{3}_b$. Let $p_t(x,y)$ and\n$\\bar{p}_t(x,y)$ denote, respectively, the density of~$X^{0,x}_t$ and\n$\\bar{X}^{0,x}_t$. Then Theorem~2.3 in~\\cite{goblab} gives the\nexistence of a constant $c>0$ and a finite nondecreasing function $K$\n(depending on the upper bounds of $\\sigma$ and $b$ and their\nderivatives) such that\n\\[\n\\forall(t,x,y) \\in(0,T]\\times\\mathbb{R}^2\\qquad\n\\bigl|p_t(x,y)-\\bar{p}_t(x,y)\\bigr|\\leq\\frac{TK(T)}{N t}\\exp\\biggl(-\\frac\n{c|x-y|^2}{t}\\biggr).\n\\]\nAs remarked in~\\cite{thesesbai}, Chapter~3, for $f\\dvtx \\mathbb{R}\\to\n\\mathbb{R}$ a Lipschitz continuous function with Lipschitz constant not\ngreater than one, one deduces that\n\\begin{eqnarray*}\n\\bigl|\\mathbb{E} \\bigl[f(X_t) \\bigr]-\\mathbb{E} \\bigl[f(\n\\bar{X}_t) \\bigr]\\bigr|&=& \\biggl\\llvert\\int_{\\mathbb\n{R}}\n\\bigl(f(y)-f(x) \\bigr) \\bigl(p_t(x,y)-\\bar{p}_t(x,y)\n\\bigr)\\,dy \\biggr\\rrvert\n\\\\\n&\\leq&\\frac\n{K(T)T}{N t}\\int_{\\mathbb{R}}|y-x|\\exp\\biggl(-\n\\frac{c|x-y|^2}{t} \\biggr)\\,dy\n\\\\\n&=&\\frac{K(T)T}{cN},\n\\end{eqnarray*}\nwhich gives $\\sup_{t\\leq\nT}\\mathcal{W}_1(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t))\\leq\n\\frac{CK(T)T}{N}$ by the dual formulation of the $1$-Wasserstein\ndistance.\n\\end{arem}\n\nOur approach consists of controlling the time evolution of the\nWasserstein distance. To do so, we need to compute the evolution of\nboth $F_t^{-1}(u)$ and $\\bar{F}_t^{-1}(u)$. In the two next\npropositions, we derive partial differential equations satisfied by\nthese functions by integrating in space the Fokker--Planck equations\nand then applying the implicit function theorem.\n\n\\begin{aprop}\\label{propevolftm1}\nAssume that\nHypothesis~\\ref{hyp_wass_marginal} holds\nThen for any $t\\in(0,T]$, the cumulative distribution function\n$x\\mapsto F_t(x)$ is invertible with inverse denoted by $F_t^{-1}(u)$.\nMoreover, the function $(t,u)\\mapsto F_t^{-1}(u)$ is $C^{1,2}$ on\n$(0,T]\\times(0,1)$ and satisfies\n\\begin{equation}\n\\partial_t F_t^{-1}(u)=-\\frac{1}{2}\n\\partial_u \\biggl(\\frac\n{a(F_t^{-1}(u))}{\\partial_u F_t^{-1}(u)} \\biggr)+b \\bigl(F_t^{-1}(u)\n\\bigr).\\label{fpinvfr}\n\\end{equation}\n\\end{aprop}\n\n\\begin{aprop}\\label{propevolbarftm1}\nAssume that $\\sigma$ and $b$ have linear growth $\\exists C>0$, $\\forall\nx\\in\\mathbb{R}$, $|\\sigma(x)|+|b(x)|\\leq C(1+|x|)$ and that uniform\nellipticity holds, $\\exists\\underline{a}>0$, $\\forall x\\in\\mathbb{R}$,\n$a(x)\\geq\\underline{a}$. Then for any $t\\in(0,T]$, $\\bar{X}_t$ admits a\ndensity $\\bar{p}_t(x)$ with respect to the Lebesgue measure and its\ncumulative distribution function $x\\mapsto\\bar{F}_t(x)$ is invertible\nwith inverse denoted by $\\bar{F}_t^{-1}(u)$. Moreover, for each\n$k\\in\\{0,\\ldots,N-1\\}$, the function $(t,u)\\mapsto\\bar{F}^{-1}_t(u)$ is\n$C^{1,2}$ on $(t_k,t_{k+1}]\\times(0,1)$ and, on this set, it is a\nclassical solution of\n\\begin{eqnarray}\n\\label{eqevolbarftm1}\n\\partial_t \\bar{F}_t^{-1}(u)&=&-\n\\frac{1}{2}\\partial_u \\biggl(\\frac\n{\\alpha_t(u)}{\\partial_u \\bar{F}_t^{-1}(u)} \\biggr)+\n\\beta_t(u),\n\\end{eqnarray}\nwhere $\\alpha_t(u)=\\mathbb{E}[a(\\bar{X}_{t_k})|\\bar{X}_t=\\bar\n{F}_t^{-1}(u)]$ and\n$\\beta_t(u)=\\mathbb{E}[b(\\bar{X}_{t_k})|\\bar{X}_t=\\bar{F}_t^{-1}(u)]$.\n\\end{aprop}\n\n\nThe proofs of these two propositions are postponed to\nAppendix~\\ref{App_Sec1}. Let us mention here that\nProposition~\\ref{propevolftm1} also holds when $b'$ is only H\\\"older\ncontinuous: the Lipschitz assumption on~$b'$ is needed later to prove\nTheorem~\\ref{wasun}. The PDEs (\\ref{fpinvfr})~and~(\\ref{eqevolbarftm1})\nenable us to compute the time derivative of the $p$th power of the\nWasserstein distance (\\ref{wpinv}) and prove, again in\nAppendix~\\ref{App_Sec1} the following key lemma.\n\n\\begin{alem}\\label{lemmajoderwp}\nUnder Hypothesis~\\ref{hyp_wass_marginal}, for $p\\geq2$, the function\n$t\\mapsto\\mathcal{W}_p^p(\\mathcal{L}(X_t),\\allowbreak\n\\mathcal{L}(\\bar{X}_t))$ is continuous on $[0,T]$, and its first order\ndistribution\\vadjust{\\goodbreak} derivative\\break\n$\\partial_t\\mathcal{W}_p^p(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t))$ is\nan integrable function on $[0,T]$. Moreover, $dt$ a.e.,\n\\begin{eqnarray}\\label{majoderwp}\n&& \\partial_t\\mathcal{W}_p^p \\bigl(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t) \\bigr)\\nonumber\n\\\\[-1pt]\n&&\\qquad \\leq C \\biggl(\\mathcal{W}_p^p \\bigl(\\mathcal{L}(X_t),\n\\mathcal{L}(\\bar{X}_t) \\bigr)\n\\nonumber\\\\[-9pt]\\\\[-9pt]\n&&\\hspace*{45pt}{} +\\int_0^1\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^{p-1}\\bigl|b \\bigl(\n\\bar{F}_t^{-1}(u) \\bigr)-\\beta_t(u)\\bigr|\\,du\\nonumber\n\\\\[-1pt]\n&&\\hspace*{45pt}{}+\\int_0^1\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^{p-2} \\bigl(a \\bigl(\n\\bar{F}_t^{-1}(u) \\bigr)-\\alpha_t(u)\n\\bigr)^2\\,du \\biggr),\\nonumber\n\\end{eqnarray}\nwhere $C$ is a positive constant that only depends on $p$,\n$\\underline{a}$, $\\|a'\\|_\\infty$ and $\\|b'\\|_\\infty$.\n\\end{alem}\n\nThe last ingredient of the proof of Theorem~\\ref{wasun} is the next\nlemma, the proof of which is also postponed in Appendix~\\ref{App_Sec1}.\n\n\n\\begin{alem}\\label{malcal} Let $\\tau_t=\\sup\\{t_i, t_i\\le t\\}$ denote\nthe last discretization time before~$t$. Under\nHypothesis~\\ref{hyp_wass_marginal}, we have for all $p\\geq1$,\n\\[\n\\exists C<+\\infty, \\forall N\\geq1, \\forall t\\in[0,T]\\qquad \\mathbb{E} \\bigl[\n\\bigl\\llvert\\mathbb{E} [ W_{t}-W_{\\tau_{t\n}|\\bar{X}_{t} ]\n\\bigr\\rrvert^{p} \\bigr] \\leq C \\biggl(\\frac{1}{N\\vee(N^2t)}\n\\biggr)^{p\/2}.\n\\]\n\\end{alem}\n\n\\begin{pf*}{Proof of Theorem~\\ref{wasun}}\nSince\n$\\mathcal{W}_p(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t))\\le\\mathcal\n{W}_{p'}(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t)) $ for $p\\le p'$, it\nis enough to prove the estimation for $p\\geq2$. Therefore we suppose\nwithout loss of generality that $p\\ge2$. Let\n$\\psi_p(t)=\\mathcal{W}^2_p(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t))$\nand\\looseness=-1\n\\begin{eqnarray}\n\\mbox{for any integer } k\\geq1\\qquad h_k(x)=k^{-2\/p}h(kx)\\nonumber\n\\\\\n\\eqntext{\\mbox{where }h(x)= \\cases{x^{2\/p}, &\\quad if $x\\geq1$,\n\\vspace*{2pt}\\cr\n1+\\dfrac{2}{p}(x-1), &\\quad otherwise.}}\n\\end{eqnarray}\\looseness=0\nSince $h_k$ is $C^1$ and nondecreasing, Lemma~\\ref{lemmajoderwp} and\nH\\\"older's inequality imply that\n\\begin{eqnarray*}\n&& h_k \\bigl(\\psi^{p\/2}_p(t) \\bigr)\n\\\\[-2pt]\n&&\\qquad = h_k\n\\bigl(\\mathcal{W}_p^p \\bigl(\\mathcal{L}(X_0),\n\\mathcal{L}(\\bar{X}_0) \\bigr) \\bigr)+\\int_0^th_k'\n\\bigl(\\psi^{p\/2}_p(s) \\bigr)\\partial_s\n\\mathcal{W}_p^p \\bigl(\\mathcal{L}(X_s),\n\\mathcal{L}(\\bar{X}_s) \\bigr)\\,ds\n\\\\[-2pt]\n&&\\qquad \\leq h_k(0)\n+C\\int_0^th_k'\n\\bigl(\\psi^{p\/2}_p(s) \\bigr)\n\\\\[-2pt]\n&&\\hspace*{91pt} {}\\times\\biggl[\\psi ^{p\/2}_p(s)\n\\\\[-2pt]\n&&\\hspace*{107pt}{}\n +\\psi^{(p-1)\/2}_p(s)\n\\biggl(\\int _0^1\\bigl|b \\bigl(\\bar{F}_s^{-1}(u)\n\\bigr)-\\beta_s(u)\\bigr|^p\\,du \\biggr)^{1\/p} \\nonumber\n\\\\\n&&\\hspace*{107pt}\n{}+\\psi^{(p-2)\/2}_p(s) \\biggl(\\int_0^1\\bigl|a\n\\bigl(\\bar{F}_s^{-1}(u) \\bigr)-\\alpha\n_s(u)\\bigr|^p\\,du \\biggr)^{2\/p} \\biggr]\\,ds.\n\\end{eqnarray*}\nSince for fixed $x\\geq0$, the sequence $(h'_k(x))_k$ is nondecreasing\nand converges to $\\frac{2}{p}x^{(2\/p)-1}$ as $k\\to\\infty$, one may take\nthe limit in this inequality thanks to the monotone convergence theorem\nand remark that the image of the Lebesgue measure on $[0,1]$ by\n$\\bar{F}_s^{-1}$ is the distribution of $\\bar{X}_s$ to deduce\n\\begin{eqnarray}\\label{pregron}\n\\psi_p(t)&\\leq&\\frac{2C}{p}\\int_0^t\n\\psi_p(s)+\\psi^{1\/2}_p(s)\\mathbb{E}\n^{1\/p} \\bigl(\\bigl|b(\\bar{X}_s)-\\mathbb{E} \\bigl(b(\n\\bar{X}_{\\tau_s})|\\bar{X}_s \\bigr)\\bigr|^p \\bigr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{} + \\mathbb{E}^{2\/p} \\bigl(\\bigl|a(\\bar{X}_s)-\\mathbb{E} \\bigl(a(\n\\bar{X}_{\\tau _s})|\\bar{X}_s \\bigr)\\bigr|^p \\bigr) \\,ds.\\nonumber\n\\end{eqnarray}\nOne has\n\\begin{eqnarray*}\na(\\bar{X}_{\\tau_s})-a(\\bar{X}_s)&=&a'(\n\\bar{X}_s)\\sigma(\\bar{X}_{s}) (W_{\\tau_s}-W_s)\n\\\\[-1pt]\n&&{} -a'(\\bar{X}_s) \\bigl[ \\bigl(\\sigma(\\bar{X}_{\\tau\n_s})-\\sigma(\n\\bar{X}_{s}) \\bigr) (W_s-W_{\\tau_s})+b(\n\\bar{X}_{\\tau\n_s}) (s-\\tau_s) \\bigr]\n\\\\[-2pt]\n&&{}+(\\bar{X}_{\\tau_s}-\\bar{X}_s)\\int_0^1a'\n\\bigl(v \\bar{X}_{\\tau_s}+(1-v)\\bar{X}_s\n\\bigr)-a'( \\bar{X}_s)\\,dv.\n\\end{eqnarray*}\nUsing Jensen's inequality, the boundedness assumptions on $a,b$ and\ntheir derivatives and Lemma \\ref{malcal}, one gets\n\\begin{eqnarray*}\n&& \\mathbb{E} \\bigl(\\bigl|a(\\bar{X}_s)-\n\\mathbb{E} \\bigl(a( \\bar{X}_{\\tau_s})|\\bar{X}_s\n\\bigr)\\bigr|^p \\bigr)\n\\\\\n&&\\qquad \\leq C\\mathbb{E} \\bigl(\\bigl|\\sigma a'(\n\\bar{X}_s)\\bigr|^p\\bigl| \\mathbb{E}\\bigl((W_s-W_{\\tau_s})|\n\\bar{X}_s \\bigr)\\bigr|^p \\bigr)\n\\\\\n&&\\quad\\qquad{}+C\\mathbb{E} \\bigl((s-\\tau_s)^p+\\bigl| \\bigl(\\sigma(\\bar\n{X}_{\\tau\n_s})-\\sigma(\\bar{X}_{s}) \\bigr) (W_s-W_{\\tau_s})\\bigr|^p+|\n\\bar{X}_{\\tau\n_s}-\\bar{X}_s|^{2p} \\bigr)\n\\\\\n&&\\qquad \\leq\\frac{C}{N^{p\/2}\\vee(N^ps^{p\/2})}.\n\\end{eqnarray*}\nThe same bound holds with $a$ replaced by $b$. With (\\ref{pregron}) and\nYoung's inequality, one deduces\n\\begin{eqnarray*}\n\\psi_p(t)&\\leq& C\\int_0^t\n\\psi_p(s)+\\frac{\\psi^{1\/2}_p(s)}{\\sqrt{N}\\vee(N\\sqrt{s})}+\\frac\n{1}{N\\vee(N^2s)}\\,ds\n\\\\\n&\\leq& C\\int _0^t\\psi_p(s)+\n\\frac{1}{N\\vee(N^2s)}\\,ds.\n\\end{eqnarray*}\nOne concludes by Gronwall's lemma.\n\\end{pf*}\n\n\\begin{arem} When $a(x)\\equiv a$ is constant, the term $\\mathbb\n{E}^{2\/p}\n(|a(\\bar{X}_s)-\\break\\mathbb{E}(a(\\bar{X}_{\\tau_s})|\\bar{X}_s)|^p )$ in\n(\\ref{pregron}) vanishes and the above reasoning ensures that $\\bar\n{\\psi}_p(t)$ defined as $\\sup_{s\\in[0,T]}\\psi_p(s)$ satisfies\n\\begin{eqnarray*}\n\\bar{\\psi}_p(t)&\\leq& C\\int_0^t \\bar{\\psi}_p(s) \\,ds\n+C\\bar{\\psi}^{1\/2}_p(t) \\int\n_0^t\\frac{1}{\\sqrt{N}\\vee(N\\sqrt{s})}\\,ds\n\\\\\n&\\le& C\\int _0^t\\bar{\\psi}_p(s)\n\\,ds + \\frac{1}{2}\\bar{\\psi}_p(t) +\\frac{C^2 (T+1)^2}{2N}.\n\\end{eqnarray*}\nBy Gronwall's lemma, we recover the estimation\n$\\sup_{t\\in[0,T]}\\mathcal{W}_p(\\mathcal{L}(X_t),\\break \\mathcal{L}(\\bar\n{X}_t))\\leq\\frac{C}{N}$\nwhich is also a consequence of the strong order of convergence of the\nEuler scheme when the diffusion coefficient is constant.\n\\end{arem}\n\n\\section{The Wasserstein distance between the pathwise laws}\\label{sec_pathwise}\nWe now state the main result of the paper.\n\\begin{ahyp}\\label{hyp_wass_pathwise}\nWe assume that $a \\in C^4_b, b \\in C^3_b$ and\n\\[\n\\exists\\underline{a}>0, \\forall x\\in\\mathbb{R}\\qquad a(x)\\geq\\underline{a}\n\\mbox{ (uniform ellipticity)}.\n\\]\n\\end{ahyp}\nClearly, Hypothesis~\\ref{hyp_wass_pathwise} implies\nHypothesis~\\ref{hyp_wass_marginal}.\n\n\n\\begin{theorem}\\label{main_thm}\nUnder Hypothesis~\\ref{hyp_wass_pathwise}, we have\n\\[\n\\forall p\\geq1, \\forall\\varepsilon>0, \\exists C<+\\infty, \\forall N\\geq1\n\\qquad \\mathcal{W}_p \\bigl(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}) \\bigr)\\leq\n\\frac{C}{N^{2\/3-\\varepsilon}}.\n\\]\n\\end{theorem}\n\nBefore proving the theorem, let us state some of its consequences for\nthe pricing of lookback options. It is well known (see, e.g.,\n\\cite{glas} page 367) that if $(U_k)_{0\\leq k\\leq N-1}$ are independent\nrandom variables uniformly distributed on $[0,1]$ and independent from\nthe Brownian increments $(W_{t_{k+1}}-W_{t_{k}})_{0\\leq k\\leq N-1}$\nthen\n$\\bar{\\hspace*{-1.2pt}\\bar{X}}\\stackrel{\\mathrm{def}}{=}\\frac{1}{2}\\max_{0\\leq\nk\\leq N-1} (\\bar{X}_{t_k}+\\bar{X}_{t_{k+1}}+\\sqrt{(\\bar\n{X}_{t_{k+1}}\\,{-}\\,\\bar{X}_{t_k})^2-2\\sigma^2(\\bar{X}_{t_k})t_1\\ln(U_k)} ) $\nis such\\vspace*{-2pt} that $\n(\\bar{X}_0,\\bar{X}_{t_1},\\ldots,\\bar{X}_{T},\\bar{\\hspace*{-1.2pt}\\bar{X}}\n)\\stackrel{\\mathcal{L}}{=}(\\bar{X}_0,\\bar{X}_{t_1},\\ldots,\\bar\n{X}_{T},\\max_{t\\in [0,T]}\\bar{X}_t)$.\n\\begin{acor}\nIf $f\\dvtx \\mathbb{R}^2\\to\\mathbb{R}$ is Lipschitz continuous, then,\nunder Hypothesis~\\ref{hyp_wass_pathwise},\n\\begin{eqnarray}\\label{vitfaiblook}\n&& \\forall\\varepsilon>0, \\exists C<+\\infty, \\forall N\\geq1\n\\nonumber\\\\[-6pt]\\\\[-12pt]\n&&\\qquad \\Bigl\\llvert \\mathbb{E} \\Bigl[f \\Bigl(X_T,\\max_{t\\in[0,T]}X_t\n\\Bigr) \\Bigr]-\\mathbb{E}\n\\bigl[f(\\bar{X}_T,\\bar{\\hspace*{-1.2pt}\\bar{X}}) \\bigr]\n\\Bigr\\rrvert\\leq\\frac{C}{N^{2\/3-\\varepsilon}}.\\nonumber\n\\end{eqnarray}\n\\end{acor}\n\nTo our knowledge, this result appears to be new. Of course, when $f$ is\nalso differentiable with respect to its second variable, one has\n\\begin{eqnarray*}\n&& \\mathbb{E} \\Bigl[f \\Bigl(X_T,\\max_{t\\in[0,T]}X_t \\Bigr)\n\\Bigr]\n\\\\\n&&\\qquad =\\mathbb{E} \\bigl[f (X_T,x_0 ) \\bigr]+\n\\int_{x_0}^{+\\infty}\\mathbb{E} \\bigl[\n\\partial_2f(X_T,x)1_{\\{\\max_{t\\in[0,T]}X_t\\geq x\\}} \\bigr]\\,dx.\n\\end{eqnarray*}\nOne could try to combine the weak error analysis for the first term on\nthe right-hand side with Theorem 2.3 \\cite{gob2} devoted to barrier\noptions to obtain the order $N^{-1}$ instead on $N^{-2\/3+\\varepsilon}$\nin (\\ref{vitfaiblook}). Unfortunately, one cannot succeed for two main\nreasons. First, it is not clear whether the estimation in Theorem 2.3\n\\cite{gob2} is preserved by integration over $[x_0,+\\infty)$. More\nimportantly, for this estimation to hold, a structure condition on the\npayoff function implying that $\\partial_2f(x,x)=0$ for all $x\\geq x_0$\nis needed.\n\n\\begin{pf*}{Proof of Theorem \\ref{main_thm}}\nWe first deduce from Theorem \\ref{wasun} some bound on the Wasserstein\ndistance between the finite-dimensional marginals of the diffusion $X$\nand its Euler scheme $\\bar{X}$ on a coarse time-grid. For\n$m\\in\\{1,\\ldots,N-1\\}$, we set $n=\\lfloor N\/m\\rfloor$ and define\n\\[\ns_l=\\frac{lmT}{N}\\qquad\\mbox{for } l\\in\\{0,\\ldots,n-1\\}\\mbox{ and } s_n=T.\n\\]\nWe will use this coarse time-grid $(s_l)_{1\\leq l\\leq n}$ to\napproximate the supremum norm on $\\mathcal{C}$ and therefore we endow\nconsistently $\\mathbb{R}^n$ with the norm $|(x_1,\\ldots,\\break x_n)|=\\max\n_{1\\leq l\\leq n}|x_l|$. Combining the next proposition, the proof of\nwhich is postponed in Appendix~\\ref{App_Sec2} with Theorem \\ref{wasun},\none obtains that\n\\begin{equation}\n\\mathcal{W}_p \\bigl(\\mathcal{L}(X_{s_1},\\ldots,X_{s_n}),\\mathcal{L}(\\bar{X}_{s_1},\\ldots,\n\\bar{X}_{s_n}) \\bigr)\\leq\\frac{C\\sqrt{\\log\nN}}{m},\\label{wasfindim}\n\\end{equation}\nwhere the constant $C$ does not depend on $(m,N)$.\n\\begin{aprop}\\label{prop_wass_multi}\nLet $\\mathbb{R}^n$ be endowed\nwith the norm $|(x_1,\\ldots,x_n)|=\\max_{1\\leq l\\leq n}|x_l|$. For any\n$p\\geq1$, there is a constant $C$ not depending on $n$ such that\n\\[\n\\mathcal{W}_p \\bigl(\\mathcal{L}(X_{s_1},\\ldots,X_{s_n}),\\mathcal{L}(\\bar{X}_{s_1},\\ldots,\n\\bar{X}_{s_n}) \\bigr)\\leq Cn\\sup_{0\\leq t\\leq T,x\\in\n\\mathbb{R}}\n\\mathcal{W}_p \\bigl(\\mathcal{L} \\bigl(\\bar{X}^{x}_t\n\\bigr),\\mathcal{L} \\bigl(X^{x}_t \\bigr) \\bigr).\n\\]\n\\end{aprop}\nThere is a probability measure\n$\\pi(dx_1,\\ldots,dx_n,d\\bar{x}_1,\\ldots,d\\bar{x}_n)$ in $\\Pi\n(\\mathcal{L}(X_{s_1},\\allowbreak\\ldots,\nX_{s_n}),\\mathcal{L}(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_n}))$ which\nattains the Wasserstein distance in the left-hand side of\n(\\ref{wasfindim}); see, for instance, Theorem 3.3.11 \\cite{raru},\naccording to which $\\pi$ is the law of\n$(X_{s_1},\\ldots,X_{s_n},\\xi_{s_1},\\ldots,\\xi_{s_n})$ with\n$(\\xi_{s_1},\\ldots,\\xi_{s_n})\\in\\partial_{|~|}\\varphi\n(X_{s_1},\\ldots,\\break X_{s_n})$ where $\\partial_{|~|}\\varphi$ is the\nsubdifferential, for the above defined norm $|~|$ on $\\mathbb{R}^n$, of\nsome \\mbox{$|~|$-}convex function $\\varphi$. Let\n$\\tilde{\\pi}(x_1,\\ldots,x_n,d\\bar{x}_1,\\ldots,d\\bar{x}_n)$ denote a\nregular conditional probability of $(\\bar{x}_1,\\ldots,\\bar{x}_n)$ given\n$(x_1,\\ldots,x_n)$ when $\\mathbb{R}^{2n}$ is endowed with $\\pi$ and\n$(\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n})$ be distributed according to\n$\\tilde{\\pi}(X_{s_1},\\ldots,X_{s_n},\\break d\\bar{x}_1,\\ldots,d\\bar{x}_n)$. The\nvector $(X_{s_1},\\ldots,X_{s_n},\\bar{Y}_{s_1},\\ldots,\\bar {Y}_{s_n})$\nis distributed according to $\\pi$ so that\n\\begin{eqnarray}\\label{xybar}\n(\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n})&\\stackrel{\n\\mathcal{L}} {=}&(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_n})\\quad\\mbox{and}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n\\mathbb{E}^{1\/p} \\Bigl[\\max_{1\\leq l\\leq n}|X_{s_l}-\n\\bar{Y}_{s_l}|^p \\Bigr]&\\leq&\\frac{C\\sqrt{\\log N}}{m}.\\nonumber\n\\end{eqnarray}\nLet $p_t(x,y)$ denote the transition density of the SDE~(\\ref{sde}) and\n$\\ell_t(x,y)=\\log(p_t(x,y))$. According to Appendix \\ref{diff_bridge}\ndevoted to diffusion bridges, the processes\n\\[\n\\biggl(W^l_t=\\int\n_{s_l}^t \\bigl(dW_s-\n\\sigma(X_s)\\partial_x\\ell_{s_{l+1}-s}(X_s,X_{s_{l+1}})\n\\,ds \\bigr),t \\in[s_l,s_{l+1}) \\biggr) _{0\\leq l\\leq n-1}\n\\]\nare independent Brownian motions independent from\n$(X_{s_1},\\ldots,X_{s_n})$. We suppose from now on that the vector\n$(\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n})$ has been generated independently\nfrom these processes and so will be all the random variables and\nprocesses needed in the remaining of the proof (see in particular the\nconstruction of $\\beta$ below). Moreover, again by Appendix\n\\ref{diff_bridge}, the solution of\n\\begin{equation}\\label{defz}\n\\cases{\n\\displaystyle Z^{x,y}_t=x+\n\\int_{s_l}^t\\sigma\\bigl(Z^{x,y}_s\n\\bigr)\\,dW^l_s\n\\vspace*{2pt}\\cr\n\\displaystyle\\hspace*{31pt}{} +\\int_{s_l}^t \\bigl[b \\bigl(Z^{x,y}_s \\bigr)+\\sigma^2 \\bigl(Z^{x,y}_s\n\\bigr)\\partial_x\\ell _{s_{l+1}-s} \\bigl(Z^{x,y}_s,y \\bigr) \\bigr]\\,ds,\n\\vspace*{2pt}\\cr\n\\hspace*{200pt} t\\in[s_l,s_{l+1}),\n\\vspace*{4pt}\\cr Z^{x,y}_{s_{l+1}}=y}\n\\end{equation}\nis distributed according to the conditional law of\n$(X_t)_{t\\in[s_l,s_{l+1}]}$ given $(X_{s_l},\\break X_{s_{l+1}})=(x,y)$ and for\neach $l\\in\\{0,\\ldots,n-1\\}$, one has $(Z^{X_{s_l},X_{s_{l+1}}}_t)_{t\\in\n[s_l,s_{l+1}]}=(X_t)_{t\\in[s_l,s_{l+1}]}$.\n\n\nIn order to construct a good coupling between $\\mathcal{L}(X)$ and\n$\\mathcal{L}(\\bar{X})$, a natural idea would be to extend\n$(\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n})$ to a process\n$(\\bar{Y}_t)_{t\\in[0,T]}$ with law $\\mathcal{L}(\\bar{X})$ by defining\nfor each $l\\in\\{0,\\ldots,n-1\\}$, $(\\bar{Y}_t)_{t\\in[s_l,s_{l+1}]}$ as\nthe process obtained by inserting the Brownian motion $W^l$, the\nstarting point $\\bar{Y}_{s_l}$ and the ending point $\\bar{Y}_{s_{l+1}}$\nin the It\\^{o}'s decomposition of the conditional dynamics of\n$(\\bar{X}_t)_{t\\in[s_l,s_{l+1}]}$ given $\\bar{X}_{s_l}=x$ and\n$\\bar{X}_{s_{l+1}}=y$. Unfortunately, even if this Euler scheme bridge\nis deduced by a simple transformation of the Brownian bridge on a\nsingle time-step, it becomes a complicated process\\vspace*{1pt} when\nthe difference between the starting and ending times is larger than\n$\\frac{T}{N}$ because of the lack of Markov property. At the end of the\nproof, we will choose the difference $s_{l+1}-s_l$ of order\n$\\frac{T}{N^{1\/3}}$ and, therefore,\\vspace*{-2pt} much larger than the\ntime-step $\\frac{T}{N}$. In addition, it is not clear how to compare\nthe paths of the diffusion bridge and the Euler scheme bridge driven by\nthe same Brownian motion. That is, why we are going to introduce some\nnew process $(\\tilde{\\chi}_t)_{t\\in[0,T]}$ such that the comparison\nwill be performed at the diffusion bridge level, which is not so\neasy~yet.\n\nTo construct $\\tilde{\\chi}$, we are going to exhibit a Brownian motion\n$(\\beta_t)_{t\\in[0,T]}$ such that $\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n}$\nare the values on the coarse time-grid $(s_l)_{1\\leq l\\leq n}$ of the\nEuler scheme~(\\ref{eul}) driven by $\\beta$ instead of $W$. The\nextension $(\\bar{Y}_t)_{t\\in[0,T]}$ with law $\\mathcal{L}(\\bar{X})$ is\nthen simply defined as the whole Euler scheme driven by $\\beta$:\n\\begin{eqnarray}\n\\bar{Y}_t=\\bar{Y}_{t_k}+\\sigma(\\bar{Y}_{t_k}) (\n\\beta_t-\\beta_{t_k})+b(\\bar{Y}_{t_k})\n(t-t_k),\\nonumber\n\\\\\n\\eqntext{ t\\in[t_{k},t_{k+1}], 0\\le k\\le N-1.}\n\\end{eqnarray}\nThe construction of $\\beta$ is postponed at the end of the\npresent proof. One then defines\n\\[\n\\chi_t=\\bar{Y}_{s_l}+\\int_{s_{l}}^t\n\\sigma(\\chi_{s})\\,d\\beta_s+\\int_{s_{l}}^tb(\n\\chi_s)\\,ds,\\qquad t\\in[s_{l},s_{l+1}), 0\\le l\\le\nn-1.\n\\]\n\nNotice that the process $\\chi=(\\chi_t)_{t\\in[0,T]}$ which evolves\naccording to the SDE~(\\ref{sde}) with $\\beta$ replacing $W$ on each\ntime-interval $[s_l,s_{l+1})$ is c\\`adl\\`ag: discontinuities may arise\nat the points $\\{s_{l+1}, 0\\leq l\\leq n-1\\}$. We denote by\n$\\chi_{s_{l+1}-}$ its left-hand limit at time $s_{l+1}$ and set\n$\\chi_{T}=\\chi_{s_n-}$. The strong error estimation (\\ref{vitfort})\nwill permit us to estimate the difference between the processes\n$\\bar{Y}$ and $\\chi$. For the subsequent choice of $\\beta$, we do not\nexpect the processes $\\chi$ and $X$ to be close. Nevertheless, the\nprocess $\\tilde{\\chi}$ obtained by setting\n\\[\n\\forall l\\in\\{0,\\ldots,n-1\\}, \\forall t\\in[s_l,s_{l+1})\n\\qquad\n\\tilde{\\chi}_t=Z^{\\chi_{s_l},\\chi\n_{s_{l+1}-}}_t\\quad\\mbox{and}\\quad\\tilde{\\chi}_T=\\chi_T,\n\\]\nwhere $Z^{x,y}$ is defined in (\\ref{defz}) is such that\n$\\mathcal{L}(\\tilde{\\chi})=\\mathcal{L}(\\chi)$ by\nPropositions~\\ref{prop_bridge} and~\\ref{prop_bridge2}. On each coarse\ntime-interval $[s_l,s_{l+1})$ the diffusion bridges associated with $X$\nand $\\tilde{\\chi}$ are driven by the same Brownian motion $W^l$.\nMoreover the differences $|X_{s_l}-Y_{s_l}|$ between the starting\npoints and $|X_{s_{l+1}}-\\chi_{s_{l+1}-}|\\leq\n|X_{s_{l+1}}-Y_{s_{l+1}}|+| Y_{s_{l+1}}-\\chi_{s_{l+1}-}|$ between the\nending points is controlled by (\\ref{xybar}) and the above mentionned\nstrong error estimation. That is, why one may expect to obtain a good\nestimation of the difference between the processes $X$ and\n$\\tilde{\\chi}$. By the triangle inequality and since\n$\\mathcal{L}(\\bar{X})=\\mathcal{L}(\\bar{Y})$ and\n$\\mathcal{L}(\\tilde{\\chi})=\\mathcal{L}(\\chi)$,\n\\begin{eqnarray}\\label{triangle}\n\\qquad\\mathcal{W}_p \\bigl(\\mathcal{L}(\\bar{X}),\\mathcal{L}(X) \\bigr)&\\leq&\n\\mathcal{W}_p \\bigl(\\mathcal{L}(\\bar{X}),\\mathcal{L}(\\chi) \\bigr)+\n\\mathcal{W}_p \\bigl(\\mathcal{L}(\\chi),\\mathcal{L}(X) \\bigr)\n\\nonumber\n\\\\\n&\\leq&\\mathbb{E}^{1\/p} \\Bigl[\\sup_{t\\in[0,T]}|\n\\bar{Y}_t-\\chi_t|^p \\Bigr]+\\mathbb{E}\n^{1\/p} \\Bigl[\\sup_{t\\in[0,T]}|X_t-\\tilde{\n\\chi}_t|^p \\Bigr],\n\\end{eqnarray}\nwhere, for the definition of\n$\\mathcal{W}_p(\\mathcal{L}(\\bar{X}),\\mathcal{L}(\\chi))$ and\n$\\mathcal{W}_p(\\mathcal{L}(\\chi),\\mathcal{L}(X))$, the space of\nc\\`adl\\`ag sample-paths from $[0,T]$ to $\\mathbb{R}$ is endowed with\nthe supremum norm. Let us first estimate the first term on the\nright-hand side. From~(\\ref{vitfort}), we get\n\\[\n\\mathbb{E} \\Bigl[\\sup_{t\\in[s_l,s_{l+1})}|\\bar{Y}_t-\\chi\n_t|^{p} \\big|\\bar{Y}_{s_l} \\Bigr]\\leq C\n\\frac{m^{p\/2}(1+|\\bar{Y}_{s_l}|)^{p}}{N^{p}},\n\\]\nwhere the constant $C$ does not depend on $(N,m)$. We deduce that\n\\begin{eqnarray*}\n\\mathbb{E} \\Bigl[\\sup_{t\\in[0,T]}|\\bar{Y}_t-\n\\chi_t|^{p} \\Bigr]\n&=&\\mathbb{E} \\Bigl[\\max _{0\\leq l\\leq n-1}\\sup_{t\\in[s_l,s_{l+1})}|\\bar{Y}_t-\\chi\n_t|^{p} \\Bigr]\n\\\\\n&\\leq& \\sum_{l=0}^{n-1}\n\\mathbb{E} \\Bigl[\\mathbb{E} \\Bigl[\\sup_{t\\in\n[s_l,s_{l+1})}|\n\\bar{Y}_t-\\chi_t|^{p} \\big|\\bar{Y}_{s_l}\n\\Bigr] \\Bigr]\n\\\\\n&\\leq& C\\frac{m^{p\/2}}{N^{p}}\\sum_{l=0}^{n-1}\n\\mathbb{E} \\bigl[\\bigl(1+|\\bar{Y}_{s_l}|\\bigr)^{p} \\bigr]\n\\\\\n&\\leq& C\n\\frac{m^{p\/2-1}}{N^{p-1}},\n\\end{eqnarray*}\nwhere we used (\\ref{momenteul}) for the last inequality. As a\nconsequence,\n\\begin{equation}\n\\label{Wass_chi_se} \\mathbb{E}^{1\/p} \\Bigl[\n\\sup_{t\\in\n[0,T]}|\\bar{Y}_t-\\chi_t|^p\n\\Bigr] \\le C\\frac{m^{1\/2-1\/p}}{N^{1-1\/p}}.\n\\end{equation}\n\nLet us now estimate the second term on the right-hand side of\n(\\ref{triangle}). By Proposition~\\ref{prop_bridge2} and since for\n$l\\in\\{0,\\ldots,n-1\\}$, $\\chi_{s_l}=\\bar{Y}_{s_l}$,\n\\begin{eqnarray*}\n\\sup_{t\\leq T}|X_t-\\tilde{\\chi}_t|&=&\\max _{0\\leq l\\leq\nn-1}\\sup_{t\\in\n[s_l,s_{l+1})}\\bigl|Z^{X_{s_l},X_{s_{l+1}}}_t-Z^{\\chi_{s_l},\\chi\n_{s_{l+1}-}}_t\\bigr|\n\\\\\n&\\leq& C\\max_{0\\leq l\\leq n-1}|X_{s_l}-\\bar{Y}_{s_l}| \\vee|X_{s_{l+1}}-\\chi_{s_{l+1}-}|.\n\\end{eqnarray*}\nSince, by the triangle inequality and the continuity of $\\bar{Y}$,\n\\begin{eqnarray*}\n|X_{s_{l+1}}-\\chi_{s_{l+1}-}|&\\leq&|X_{s_{l+1}}-\\bar\n{Y}_{s_{l+1}}|+|\\bar{Y}_{s_{l+1}}-\\chi_{s_{l+1}-}|\n\\\\\n&\\leq&\n|X_{s_{l+1}}-\\bar{Y}_{s_{l+1}}|+\\sup_{t\\in[0,T]}|\n\\bar{Y}_{t}-\\chi_{t}|,\n\\end{eqnarray*}\none deduces that\n\\[\n\\sup_{t\\leq T}|X_t-\\tilde{\\chi}_t|\\leq C\n\\Bigl(\\max_{1\\leq l\\leq\nn}|X_{s_l}-\\bar{Y}_{s_l}|+\n\\sup_{t\\in[0,T]}|\\bar{Y}_{t}-\\chi_{t}| \\Bigr).\n\\]\nCombined with (\\ref{xybar}) and (\\ref{Wass_chi_se}), this implies\n\\begin{eqnarray*}\n\\mathbb{E}^{1\/p} \\Bigl[\\sup_{t\\leq T}|X_t- \\tilde{\\chi}_t|^p\n\\Bigr]&\\leq& C\\mathbb{E} ^{1\/p} \\Bigl[\\max_{1\\leq l\\leq\nn}|X_{s_l}-\\bar{Y}_{s_l}|^p \\Bigr]+C\\mathbb{E}^{1\/p}\n\\Bigl[\\sup_{t\\in[0,T]}| \\bar{Y}_{t}-\\chi_{t}|^p \\Bigr]\n\\\\\n&\\leq& C \\biggl(\n\\frac{\\sqrt{\\log N}}{m}+\\frac{m^{1\/2-1\/p}}{N^{1-1\/p}} \\biggr).\n\\end{eqnarray*}\nPlugging this inequality together with (\\ref{Wass_chi_se}) in\n(\\ref{triangle}), we deduce that\n\\[\n\\mathcal{W}_p \\bigl(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}) \\bigr) \\leq\nC \\biggl(\\frac{\\sqrt{\\log N}}{m}+\\frac{m^{1\/2-1\/p}}{N^{1-1\/p}} \\biggr)\n\\]\nand conclude by choosing $m=\\lfloor N^{2\/3} \\rfloor$ that for\n$p\\geq\\frac{1}{3\\varepsilon}$,\n$\\mathcal{W}_p(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}))\\leq\\frac{C}{N^{2\/3-\\varepsilon}}$.\nWhen $\\frac{1}{3\\varepsilon}>1$, the conclusion follows for\n$p\\in[1,\\frac{1}{3\\varepsilon})$ since\n$\\mathcal{W}_p(\\mathcal{L}(X),\\allowbreak\\mathcal{L}(\\bar{X}))\\leq\\mathcal\n{W}_{1\/3\\varepsilon}(\\mathcal{L}(X),\\mathcal{L}(\\bar{X}))$.\n\nTo complete the proof, we still have to construct the Brownian motion\n$\\beta$. We first reconstruct on the fine time grid $(t_k)_{1\\leq k\\leq\nN}$ an Euler scheme $(\\bar{Y}_{t_k},0\\le k\\le N)$ interpolating the\nvalues on the coarse grid $(s_l)_{1\\leq l\\leq n}$. Let us denote by\n$\\bar{p}(x,y)$ the density of the law\n$\\mathcal{N}(x+b(x)T\/N,\\sigma(x)^2T\/N)$ of the Euler scheme starting\nfrom~$x$ after one time step~$T\/N$. Thanks to the ellipticity\nassumption, we have $\\bar{p}(x,y)>0$ for any $x,y\\in\\mathbb{R}$.\nConditionally on $(\\bar{Y}_{s_1},\\ldots,\\bar{Y}_{s_n})$, we generate\nindependent random vectors\n\\[\n(\\bar{Y}_{s_{l-1}+t_1},\\ldots,\\bar{Y}_{s_{l-1}+t_{m-1}})_{1\\leq\nl\\leq n-1}\\quad\\mbox{and}\\quad (\\bar{Y}_{s_{n-1}+t_1},\\ldots,\\bar{Y}_{t_{N-1}})\n\\]\nwith respective densities\n\\[\n\\frac{\\bar{p}(\\bar{Y}_{s_{l-1}},x_1)\\bar{p}(x_1,x_2)\\cdots\\bar\n{p}(x_{n-1},\\bar{Y}_{s_{l}})\n}{\\int_{\\mathbb{R}^{n-1}}\\bar{p}(\\bar{Y}_{s_{l-1}},y_1)\\bar\n{p}(y_1,y_2)\\cdots\\bar{p}(y_{n-1},\\bar{Y}_{s_{l}})\\,dy_1\\cdots\ndy_{n-1}}\n\\]\nand\n\\[\n\\frac{\\bar{p}(\\bar{Y}_{s_{n-1}},x_1)\\bar{p}(x_1,x_2)\\cdots \\bar\n{p}(x_{N-1-m(n-1)},\\bar{Y}_{s_{n}}) }{\\int_{\\mathbb\n{R}^{N-1-m(n-1)}}\\bar{p}(\\bar{Y}_{s_{n-1}},y_1)\\bar{p}(y_1,y_2)\\cdots\n\\bar{p}(y_{N-1-m(n-1)},\\bar{Y}_{s_{n}})\\,dy_1\\cdots dy_{N-1-m(n-1)}}.\n\\]\nThis ensures that $(\\bar{Y}_{t_k})_{0\\leq k\\leq\nn}\\stackrel{\\mathcal{L}}{=}(\\bar{X}_{t_k})_{0\\leq k\\leq n}$. Then we\nget, thanks to the ellipticity condition, that\n$ (\\frac{1}{\\sigma(\\bar{Y}_{t_{k-1}})}(\\bar{Y}_{t_k}-\\bar\n{Y}_{t_{k-1}}-b(\\bar{Y}_{t_{k-1}})) )_{1\\le k\\le N}$~are\nindependent centered Gaussian variables with variance~$T\/N$. By using\nindependent Brownian bridges, we can then construct a Brownian motion\n$(\\beta_t)_{t\\in[0,T]}$ such that\n\\[\n\\beta_{t_k}-\\beta_{t_{k-1}}= \\frac{1}{\\sigma(\\bar{Y}_{t_{k-1}})} \\bigl(\n\\bar{Y}_{t_k}-\\bar{Y}_{t_{k-1}}-b(\\bar{Y}_{t_{k-1}})\n\\bigr),\n\\]\nwhich completes the construction.\n\\end{pf*}\n\n\\section*{Conclusion}\nIn this paper, we prove that the order of convergence of the\nWasserstein distance $\\mathcal{W}_p$ on the space of continuous paths\nbetween the laws of a uniformly elliptic one-dimensional diffusion and\nits Euler scheme with $N$-steps is not worse that\n$N^{-2\/3+\\varepsilon}$. In view of a possible extension to\nmultidimensional settings, two main difficulties have to be resolved.\nFirst, we take advantage of the optimality of the inverse transform\ncoupling in dimension one to obtain a uniform bound on the Wasserstein\ndistance between the marginal laws with optimal rate $N^{-1}$ up to a\nlogarithmic factor. In dimension $d>1$, the optimal coupling between\ntwo probability measures on $\\mathbb{R}^d$ is not available, which\nmakes the estimation of the Wasserstein distance between the marginal\nlaws much more complicated even if, for $\\mathcal{W}_1$, the order\n$N^{-1}$ may be deduced from the results of \\cite{goblab}; see Remark\n\\ref{w1unif}. Next, one has to generalize the estimation on diffusion\nbridges given by Proposition~\\ref{prop_bridge2} which we deduce from\nthe Lamperti transform in dimension $d=1$.\n\nIn the perspective of the multi-level Monte Carlo method introduced by\nGiles~\\cite{giles}, coupling with order of convergence\n$N^{-2\/3+\\varepsilon}$ the Euler schemes with $N$ and $2N$ steps would\nalso be of great interest for variance reduction, especially in\nmultidimensional situations where the Milstein scheme is not feasible;\nsee \\cite{js} for the implementation of this idea in the example of a\ndiscretization scheme devoted to usual stochastic volatility models.\nBut this does not seem obvious from our nonconstructive coupling\nbetween the Euler scheme and its diffusion limit. For both the\nderivation of the order of convergence of the Wasserstein distance on\nthe path space and the explicitation of the coupling, the limiting step\nin our approach is Proposition \\ref{prop_wass_multi}. In this\nproposition, we bound the dual formulation of the Wasserstein distance\nbetween $n$-dimensional marginals by the Wasserstein distance between\none-dimensional marginals multiplied by $n$.\n\nEven if the order of convergence of the Wasserstein distance on the\npath space obtained in the present paper may not be optimal, it\nprovides the first significant step from the order $N^{1\/2}$ obtained\nwith the trivial coupling where the diffusion and the Euler scheme are\ndriven by the same Brownian motion.\n\\begin{appendix}\n\\section{Proofs of Section~\\lowercase{\\protect\\texorpdfstring{\\ref{sec_marginal}}{2}}}\\label{App_Sec1}\n\n\\begin{pf*}{Proof of Proposition \\ref{propevolftm1}}\nAccording to \\cite{friedman}, Theorems 5.4 and 4.7, for any\n$t\\in(0,T]$, the solution $X_t$ of (\\ref{sde}) starting from $X_0=x_0$\nadmits a density $p_t(x)$ w.r.t. the Lebesgue measure on the real line,\nthe function $(t,x)\\mapsto p_t(x)$ is $C^{1,2}$ on\n$(0,T]\\times\\mathbb{R}$, and on this set,\\vspace*{-1pt} it is a\nclassical solution of the Fokker--Planck equation\n\\begin{equation}\n\\partial_t p_t(x)=\\tfrac{1}{2}\\partial\n_{xx} \\bigl(a(x)p_t(x) \\bigr)-\\partial_x\n\\bigl(b(x)p_t(x) \\bigr). \\label{fp}\n\\end{equation}\nMoreover, the following Gaussian bounds hold:\n\\begin{eqnarray}\\label{gb}\n\\bigl|p_t(x)\\bigr|+\\sqrt{t}\\bigl|\\partial_x p_t(x)\\bigr|\\leq\\frac{C}{\\sqrt{t}}e^{-(x-x_0)^2\/Ct}\n\\nonumber\\\\[-10pt]\\\\[-10pt]\n\\eqntext{\\exists C>0, \\forall t\\in(0,T], \\forall x\\in\\mathbb{R}.}\n\\end{eqnarray}\nThe partial derivatives $\\partial_x F_t(x)=p_t(x)$ and\n$\\partial_{xx}F_t(x)=\\partial_xp_t(x)$ exist and are continuous on\n$(0,T]\\times\\mathbb{R}$. For $0<s<t\\leq T$ and $y\\leq x$, integrating\n(\\ref{fp}) over $[s,t]\\times[y,x]$, then letting $y\\to-\\infty$ thanks\nto (\\ref{gb}), one obtains\n$F_t(x)-F_s(x)=\\int_{s}^t\\frac{1}{2}\\partial_{x}(a(x)p_r(x))-b(x)p_r(x)\\,dr$.\nBy continuity of the integrand w.r.t. $(r,x)$ one deduces that the\npartial derivative $\\partial_tF_t(x)$ exists and is continuous on\n$(0,T]\\times\\mathbb{R}$. So, $(t,x)\\mapsto F_t(x)$ is $C^{1,2}$ on\n$(0,T]\\times\\mathbb{R}$ and solves\n\\begin{equation}\n\\label{fpfr}\\partial_t F_t(x)=\\tfrac{1}{2}\n\\partial_{x} \\bigl(a(x)\\partial_x F_t(x)\n\\bigr)-b(x)\\partial_x F_t(x).\n\\end{equation}\n\nAccording to Aronson \\cite{aron}, the density is also bounded from\nbelow by some Gaussian kernel $\\exists c>0, \\forall\n(t,x)\\in(0,T]\\times\\mathbb{R}, |p_t(x)|\\geq\n\\frac{c}{\\sqrt{t}}e^{-(x-x_0)^2\/ct}$. This enables us to apply the\nimplicit function theorem to $(t,x,u)\\mapsto F_t(x)-u$ to deduce that\nthe inverse $u\\mapsto F_t^{-1}(u)$ of $x\\mapsto F_t(x)$ is $C^{1,2}$ in\nthe variables $(t,u)\\in(0,T]\\times(0,1)$ and solves\n\\begin{eqnarray*}\n\\partial_t F_t^{-1}(u)&=&-\\frac{\\partial_t F_t}{\\partial_x\nF_t}\n\\bigl(F_t^{-1}(u) \\bigr)\n\\\\\n&=&-\\frac{1}{2}\\partial_{x} \\bigl(a(x)\\partial_x\nF_t(x) \\bigr)\\big|_{x=F_t^{-1}(u)}\\partial_u\nF_t^{-1}(u)+b \\bigl(F_t^{-1}(u)\n\\bigr)\n\\\\\n&=&-\\frac{1}{2}\\partial_{u} \\biggl(\\frac{a(F_t^{-1}(u))}{\\partial_u\nF_t^{-1}(u)}\n\\biggr)+b \\bigl(F_t^{-1}(u) \\bigr),\n\\end{eqnarray*}\nwhere we used\\vspace*{-1pt} (\\ref{fpfr}) for the second equality and\n$\\partial_u F_t^{-1}(u)=\\frac{1}{\\partial_xF_t( F_t^{-1}(u))}$ for both\nthe second and the third equalities.\n\\end{pf*}\n\n\\begin{pf*}{Proof of Proposition \\ref{propevolbarftm1}}\nFor $t\\in(0,t_1]$,\n$\\bar{X_t}$ admits the Gaussian density with mean $x_0+b(x_0)t$ and\nvariance $a(x_0)t$. By induction on $k$ and independence of\n$W_t-W_{t_k}$ and $\\bar{X}_{t_k}$ in (\\ref{eul}), one checks that for\n$k\\in\\{1,\\ldots,\\allowbreak n-1\\}$, $\\bar{X}_{t_k}$ admits a density\n$\\bar{p}_{t_k}(x)$ and that for $t\\in(t_{k},t_{k+1}]$, $(\\bar\n{X}_{t_k},\\bar{X_t})$ admits the density\n\\[\n\\rho(t_k,t,y,x)=\\bar{p}_{t_k}(y)\\frac{\\exp({-\n{(x-y-b(y)(t-t_k))^2}\/{2a(y)(t-t_k)}})}{\\sqrt{2\\pi a(y)(t-t_k)}}.\n\\]\nThe marginal density $\\bar{p}_t(x)=\\int_\\mathbb{R}\\bar\n{p}_{t_k}(y)\\frac{\\exp({-({x-y-b(y)(t-t_k)^2})\/{2a(y)(t-t_k)}})}{\\sqrt{2\\pi\na(y)(t-t_k)}}\\,dy$ of $\\bar{X}_t$ is continuous on\n$(t_k,t_{k+1}]\\times\\mathbb{R}$ by Lebesgue's theorem and positive.\n\nLet $N(x)=\\int_{-\\infty}^xe^{-y^2\/2}\\frac{dy}{\\sqrt{2\\pi}}$ denote the\ncumulative distribution function of the standard Gaussian law and\n$k\\in\\{0,\\ldots,N-1\\}$. Again by the independence structure in\n(\\ref{eul}), for $(t,x)\\in(t_k,t_{k+1}]\\times\\mathbb{R}$,\n$\\bar{F}_t(x)=\\break \\mathbb{E} (N (\\frac{x-\\bar{X}_{t_k}-b(\\bar\n{X}_{t_k})(t-t_k)}{\\sqrt{a(\\bar{X}_{t_k})(t-t_k)}} ) )$. One~has\n\\begin{eqnarray*}\n\\partial_t N \\biggl(\\frac{x-y-b(y)(t-t_k)}{\\sqrt{a(y)(t-t_k)}} \\biggr\n)&=& - \\biggl(\\frac{x-y-b(y)(t-t_k)}{2\\sqrt{2\\pi a(y)(t-t_k)^3}}+\\frac\n{b(y)}{\\sqrt{2\\pi a(y)(t-t_k)}} \\biggr)\n\\\\\n&&{}\\times \\exp\\biggl(-\\frac\n{(x-y-b(y)(t-t_k))^2}{2a(y)(t-t_k)}\\biggr).\n\\end{eqnarray*}\nBy the growth assumption on $\\sigma$ and $b$, one easily checks that\n$\\forall k\\in\\{0,\\ldots,N\\}$, $\\mathbb{E}(\\bar{X}^2_{t_k})<+\\infty$.\nWith the uniform ellipticity assumption, one deduces by a standard\nuniform integrability argument that $\\bar{F}_t(x)$ is differentiable\nw.r.t. $t$ with partial\\vadjust{\\goodbreak} derivative\n\\begin{eqnarray}\\label{evolfbart}\n\\qquad \\partial_t\\bar{F}_t(x)&=&-\\mathbb{E} \\biggl[ \\biggl(\n\\frac{x-\\bar{X}_{t_k}-b(\\bar{X}_{t_k})(t-t_k)}{2\\sqrt{2\\pi a(\\bar\n{X}_{t_k})(t-t_k)^3}}+\\frac{b(\\bar{X}_{t_k})}{\\sqrt{2\\pi a(\\bar\n{X}_{t_k})(t-t_k)}} \\biggr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\hspace*{70pt}\n{}\\times\n\\exp \\biggl(-\\frac{(x-\\bar{X}_{t_k}-b(\\bar\n{X}_{t_k})(t-t_k))^2}{2a(\\bar{X}_{t_k})(t-t_k)}\\biggr) \\biggr]\\nonumber\n\\end{eqnarray}\ncontinuous in $(t,x)\\in(t_k,t_{k+1}]\\times\\mathbb{R}$. In the same way,\none checks smoothness of $\\bar{F}_t(x)$ in the spatial variable $x$ and\nobtains that this function is $C^{1,2}$ on \\mbox{$(t_k,t_{k+1}]\\times\n\\mathbb{R}$.}\n\nWhen $k\\geq1$,\n\\begin{eqnarray*}\n&& \\mathbb{E} \\biggl[b(\\bar{X}_{t_k})\\frac{\\exp (-{(x-\\bar\n{X}_{t_k}-b(\\bar{X}_{t_k})(t-t_k))^2}\/{(2a(\\bar{X}_{t_k})(t-t_k)}))}{\\sqrt{2\\pi a(\\bar\n{X}_{t_k})(t-t_k)}} \\biggr]\n\\\\\n&&\\qquad =\\int_{\\mathbb{R}}b(y)\\rho(t_k,t,y,x)\\,dy\n\\\\\n&&\\qquad =\\mathbb{E} \\bigl[b(\n\\bar{X}_{t_k})|\\bar{X}_t=x \\bigr]\\bar{p}_t(x).\n\\end{eqnarray*}\nFor $k=0$, even if $(\\bar{X}_0,\\bar{X}_t)$ has no density, the\nequality between the opposite sides of this equation remains true.\n\nCombining Lebesgue's theorem and a similar reasoning, one checks that\n\\begin{eqnarray*}\n&& -\\mathbb{E} \\biggl[\\frac{x-\\bar{X}_{t_k}-b(\\bar\n{X}_{t_k})(t-t_k)}{\\sqrt{2\\pi a(\\bar{X}_{t_k})(t-t_k)^3}}\\exp\\biggl({-\\frac\n{(x-\\bar{X}_{t_k}-b(\\bar\n{X}_{t_k})(t-t_k))^2}{2a(\\bar{X}_{t_k})(t-t_k)}}\\biggr) \\biggr]\n\\\\\n&&\\qquad =\\partial\n_x\\mathbb{E} \\biggl[a(\\bar{X}_{t_k})\\frac{\\exp({-{(x-\\bar\n{X}_{t_k}-b(\\bar\n{X}_{t_k})(t-t_k))^2}\/({2a(\\bar{X}_{t_k})(t-t_k)})})}{\\sqrt{2\\pi a(\\bar\n{X}_{t_k})(t-t_k)}}\n\\biggr]\n\\\\\n&&\\qquad =\\partial_x \\bigl[\\mathbb{E} \\bigl(a(\\bar{X}_{t_k})|\n\\bar{X}_t=x \\bigr)\\bar{p}_t(x) \\bigr].\n\\end{eqnarray*}\nWith (\\ref{evolfbart}), one deduces that\n\\begin{equation}\n\\label{gyongyfbart}\n\\qquad\\partial_t\\bar{F}_t(x)=\n\\tfrac{1}{2}\\partial_x \\bigl(\\mathbb{E} \\bigl[a(\\bar\n{X}_{t_k})|\\bar{X}_t=x \\bigr]\\partial_x\n\\bar{F}_t(x) \\bigr)-\\mathbb{E} \\bigl[b(\\bar{X}_{t_k})|\n\\bar{X}_t=x \\bigr]\\partial_x\\bar{F}_t(x).\n\\end{equation}\nOne checks that the function $(t,u)\\mapsto\\bar{F}_t^{-1}(u)$ is smooth\nand satisfies the partial differential equation (\\ref{eqevolbarftm1})\nby arguments similar to the ones given at the end of the proof of\nProposition \\ref{propevolftm1}.\n\\end{pf*}\n\n\n\\begin{arem}\nIn the same way, for $k\\in\\{0,\\ldots,N-1\\}$, one could prove that on\n$(t_k,t_{k+1}]\\times\\mathbb{R}$, $(t,x)\\mapsto\\bar{p}_t(x)$ is\n$C^{1,2}$ and satisfies the partial differential\n\\[\n\\partial_t\\bar{p}_t(x)=\\tfrac{1}{2}\n\\partial_{xx} \\bigl(\\mathbb{E} \\bigl[a(\\bar{X}_{t_k})|\n\\bar{X}_t=x \\bigr]\\bar{p}_t(x) \\bigr)-\n\\partial_x \\bigl(\\mathbb{E} \\bigl[b(\\bar{X}_{t_k})|\n\\bar{X}_t=x \\bigr] \\bar{p}_t(x) \\bigr)\\vadjust{\\goodbreak}\n\\]\nobtained by spatial derivation of (\\ref{gyongyfbart}). This shows that\n$(\\bar{X}_t)_{t\\in[0,T]}$ has the same marginal distributions as the\ndiffusion process with coefficients given by the above conditional\nexpectations, which is also a consequence of \\cite{gyongy}.\n\\end{arem}\n\n\n\\begin{pf*}{Proof of Lemma \\ref{lemmajoderwp}}\nBy the continuity of\nthe paths of $X$ and $\\bar{X}$ and the finiteness of\n$\\mathbb{E} [\\sup_{t\\leq T}(|X_t|^{p+1}+|\\bar\n{X}_t|^{p+1}) ]$, one easily checks that\n$t\\mapsto\\mathcal{W}_p^p(\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t))$ is\ncontinuous.\n\nLet $k\\in\\{0,\\ldots,N-1\\}$ and $s,t\\in(t_k,t_{k+1}]$ with $s\\leq t$.\nCombining Propositions \\ref{propevolftm1}~and~\\ref{propevolbarftm1}\nwith a spatial integration by parts, one obtains for $\\varepsilon\\in\n(0,1\/2)$,\n\\begin{eqnarray}\\label{preipp}\n\\hspace*{12pt}&& \\int_\\varepsilon^{1-\\varepsilon}\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^p\\,du\\nonumber\n\\\\[-2pt]\n&&\\qquad =\\int_\\varepsilon^{1-\\varepsilon}\\bigl|F_s^{-1}(u)-\\bar\n{F}_s^{-1}(u)\\bigr|^p\\,du\\nonumber\n\\\\[-2pt]\n&&\\quad\\qquad{} +p\\int_s^t\\int_\\varepsilon\n^{1-\\varepsilon}\\bigl|F_r^{-1}(u)-\\bar{F}_r^{-1}(u)\\bigr|^{p-2}\n\\bigl(F_r^{-1}(u)-\\bar{F}_r^{-1}(u)\\bigr)\\nonumber\n\\\\[-2pt]\n&&\\hspace*{92pt}\n{}\\times\\bigl(b \\bigl(F_r^{-1}(u) \\bigr)-\n\\beta_r(u) \\bigr)\\,du\\,dr\\nonumber\n\\\\[-2pt]\n&&\\quad\\qquad{} +\\frac{p(p-1)}{2}\\int_s^t\\int\n_\\varepsilon^{1-\\varepsilon\n}\\bigl|F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u)\\bigr|^{p-2} \\bigl(\n\\partial_u F_r^{-1}(u)-\\partial_u\n\\bar{F}_r^{-1}(u) \\bigr)\n\\nonumber\\\\[-9pt]\\\\[-9pt]\n&&\\hspace*{128pt}\n{}\\times \\biggl(\\frac\n{a(F_r^{-1}(u))}{\\partial_u F_r^{-1}(u)}-\n\\frac{\\alpha_r(u)}{\\partial\n_u \\bar{F}_r^{-1}(u)} \\biggr)\\,du\\,dr\\nonumber\n\\\\[-2pt]\n&&\\quad\\qquad{}+\\frac{p}{2}\\int_s^t\\bigl|F_r^{-1}(1-\n\\varepsilon)-\\bar{F}_r^{-1}(1-\\varepsilon)\\bigr|^{p-2}\n\\bigl(F_r^{-1}(1-\\varepsilon)-\\bar{F}_r^{-1}(1-\n\\varepsilon) \\bigr)\\nonumber\n\\\\[-2pt]\n&&\\hspace*{68pt}\n{}\\times\\biggl(\\frac{\\alpha_r(1-\\varepsilon\n)}{\\partial_u \\bar{F}_r^{-1}(1-\\varepsilon)}-\\frac\n{a(F_r^{-1}(1-\\varepsilon))}{\\partial_u F_r^{-1}(1-\\varepsilon\n)} \\biggr)\\,dr\\nonumber\n\\\\[-2pt]\n&&\\quad\\qquad{} -\\frac{p}{2}\\int_s^t\\bigl|F_r^{-1}(\n\\varepsilon)-\\bar{F}_r^{-1}(\\varepsilon)\\bigr|^{p-2}\n\\bigl(F_r^{-1}(\\varepsilon)-\\bar{F}_r^{-1}(\n\\varepsilon) \\bigr)\\nonumber\n\\\\[-2pt]\n&&\\hspace*{68pt}\n{}\\times \\biggl(\\frac{\\alpha_r(\\varepsilon\n)}{\\partial_u \\bar{F}_r^{-1}(\\varepsilon)}-\\frac\n{a(F_r^{-1}(\\varepsilon))}{\\partial_u F_r^{-1}(\\varepsilon)} \\biggr)\\,dr.\\nonumber\n\\end{eqnarray}\n\nWe are now going to take the limit as $\\varepsilon\\to0$. We will check\nat the end of the proof that\n\\begin{eqnarray}\\label{IPP_terms}\n&& \\lim_{u\\rightarrow0^+\\ \\mathrm{or}\\ 1^-}\\ \\sup\n_{r\\in[s,t]}\\frac{a(F_t^{-1}(u))}{\\partial_u\nF_t^{-1}(u) } \\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u) \\bigr|^{p-1}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\quad{} + \\sup _{r\\in [s,t]}\\frac{\\alpha_t(u) }{\\partial_u \\bar{F}_t^{-1}(u) }\n\\bigl|F_t^{-1}(u)- \\bar{F}_t^{-1}(u) \\bigr|^{p-1}=0,\\nonumber\n\\end{eqnarray}\nwhich enables us to get rid of the two last boundary terms.\\vadjust{\\goodbreak}\n\nCombining Young's inequality with the uniform ellipticity assumption\nand the positivity of $\\partial_uF_t^{-1}(u)$ and $\\partial_u\n\\bar{F}_t^{-1}(u)$, one obtains\n\\begin{eqnarray*}\n&& \\bigl(\\partial_u F_r^{-1}(u)-\n\\partial_u \\bar{F}_r^{-1}(u) \\bigr) \\biggl(\n\\frac{a(F_r^{-1}(u))}{\\partial_u F_r^{-1}(u)}-\\frac{\\alpha_r(u)}{\\partial\n_u \\bar{F}_r^{-1}(u)} \\biggr)\n\\\\\n&&\\qquad = \\bigl(a \\bigl(F_r^{-1}(u) \\bigr)-\n\\alpha_r(u) \\bigr)\\frac{\\partial_u\nF_r^{-1}(u)-\\partial_u \\bar{F}_r^{-1}(u)}{\\partial_u F_r^{-1}(u)\\vee\n\\partial_u \\bar{F}_r^{-1}(u)}\n\\\\\n&&\\quad\\qquad{} -a \\bigl(F_r^{-1}(u) \\bigr)\\frac{((\\partial_u\n\\bar{F}_r^{-1}(u)-\\partial_u F_r^{-1}(u))^+)^2}{\\partial_u\nF_r^{-1}(u)\\partial_u \\bar{F}_r^{-1}(u)}\n\\\\\n&&\\quad\\qquad{}-\\alpha_r(u)\\frac\n{((\\partial_u F_r^{-1}(u)-\\partial_u \\bar\n{F}_r^{-1}(u))^+)^2}{\\partial_u F_r^{-1}(u)\\partial_u \\bar\n{F}_r^{-1}(u)}\n\\\\\n&&\\qquad\\leq\\frac{1}{4\\underline{a}} \\bigl(a \\bigl(F_r^{-1}(u)\n\\bigr)- \\alpha_r(u) \\bigr)^2+\\underline{a}\n\\frac{(\\partial_u F_r^{-1}(u)-\\partial\n_u \\bar{F}_r^{-1}(u))^2}{(\\partial_u F_r^{-1}(u)\\vee\\partial_u \\bar\n{F}_r^{-1}(u))^2}\n\\\\\n&&\\quad\\qquad{} - \\bigl(a \\bigl(F_r^{-1}(u) \\bigr)\\wedge\n\\alpha_r(u) \\bigr)\\frac{(\\partial\n_u \\bar{F}_r^{-1}(u)-\\partial_u F_r^{-1}(u))^2}{\\partial_u\nF_r^{-1}(u)\\partial_u \\bar{F}_r^{-1}(u)}\n\\\\\n&&\\qquad\\leq\\frac{1}{4\\underline{a}} \\bigl(a \\bigl(F_r^{-1}(u)\n\\bigr)- \\alpha_r(u) \\bigr)^2.\n\\end{eqnarray*}\nHence, up to the factor $\\frac{p(p-1)}{2}$, the third term on the\nright-hand side of (\\ref{preipp}) is equal to\n\\begin{eqnarray*}\n&&\\int_s^t\\int_\\varepsilon^{1-\\varepsilon}\\bigl|F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u)\\bigr|^{p-2}\n\\\\[-2pt]\n&&\\hspace*{38pt}{}\\times \\biggl[ \\bigl(\n\\partial_u F_r^{-1}(u)-\\partial_u\n\\bar{F}_r^{-1}(u) \\bigr) \\biggl(\\frac{a(F_r^{-1}(u))}{\\partial_u\nF_r^{-1}(u)}-\n\\frac{\\alpha_r(u)}{\\partial_u \\bar{F}_r^{-1}(u)} \\biggr)\n\\\\\n&&\\hspace*{174pt}{} -\\frac{ (a(F_r^{-1}(u))-\\alpha_r(u) )^2}{4\\underline\n{a}} \\biggr]\\,du\\,dr\n\\\\[-2pt]\n&&\\quad{} +\\frac{1}{4\\underline{a}}\\int_s^t\n\\int_\\varepsilon^{1-\\varepsilon}\\bigl|F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u)\\bigr|^{p-2} \\bigl(a\n\\bigl(F_r^{-1}(u) \\bigr)-\\alpha_r(u)\n\\bigr)^2\\,du\\,dr,\n\\end{eqnarray*}\nwhere the integrand in the first integral is nonpositive. Since\n\\begin{eqnarray*}\n\\hspace*{-5pt}&&\\int_s^t\\hspace*{-1pt}\\int_0^1\\bigl|F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u)\\bigr|^{p-2}\n\\\\[-1pt]\n\\hspace*{-5pt}&&\\hspace*{25pt}{} \\times\\hspace*{-0.3pt}  \\bigl(\\bigl|F_r^{-1\\hspace*{-0.3pt}}(u)-\\hspace*{-0.2pt}\n\\bar{F}_r^{-1\\hspace*{-0.3pt}}(u)\\bigr|\\bigl|b \\bigl(F_r^{-1\\hspace*{-0.3pt}}(u)\n\\bigr)-\\beta_r(u)\\bigr|\n+ \\bigl(a \\bigl(F_r^{-1\\hspace*{-0.3pt}}(u)\n\\bigr)-\\alpha_r(u) \\bigr)^2 \\bigr)\\,du\\,dr\n\\\\[-1pt]\n\\hspace*{-5pt}&&\\qquad \\leq2\\|b\\|_\\infty\\int_s^t\n\\mathcal{W}_{p}^{p-1} \\bigl(\\mathcal{L}(X_r),\n\\mathcal{L}(\\bar{X}_r) \\bigr)\\,dr\n\\\\[-1pt]\n\\hspace*{-5pt}&&\\quad\\qquad{}+4\\|a\\|^2_\\infty\n\\int_s^t\\mathcal{W}_{p}^{p-2}\n\\bigl(\\mathcal{L}(X_r),\\mathcal{L}(\\bar{X}_r)\n\\bigr)\\,dr<+\\infty,\n\\end{eqnarray*}\none can take the\nlimit $\\varepsilon\\to0$ in (\\ref{preipp}) using Lebesgue's theorem for\nthe second term on the right-hand side and combining Lebesgue's theorem\nwith monotone convergence for the third term to obtain\n\\begin{eqnarray}\\label{vipp}\n&& \\mathcal{W}_{p}^{p} \\bigl(\\mathcal{L}(X_t),\n\\mathcal{L}(\\bar{X}_t) \\bigr)\\nonumber\n\\\\\n&&\\qquad =\\mathcal{W}_{p}^{p}\n\\bigl(\\mathcal{L}(X_s),\\mathcal{L}(\\bar{X}_s) \\bigr)\n\\nonumber\n\\\\\n&&\\quad\\qquad{}+p\\int_s^t\\int_0^{1}\\bigl|F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u)\\bigr|^{p-2} \\bigl(F_r^{-1}(u)-\n\\bar{F}_r^{-1}(u) \\bigr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\hspace*{82pt}{}\\times \\bigl(b \\bigl(F_r^{-1}(u)\n\\bigr)-\\beta_r(u) \\bigr)\\,du\\,dr\n\\nonumber\n\\\\\n&&\\quad\\qquad{}+\\frac{p(p-1)}{2}\\int_s^t\\int\n_0^{1}\\bigl|F_r^{-1}(u)-\\bar\n{F}_r^{-1}(u)\\bigr|^{p-2} \\bigl(\\partial_u\nF_r^{-1}(u)-\\partial_u \\bar\n{F}_r^{-1}(u) \\bigr)\\nonumber\n\\\\\n&&\\hspace*{118pt}\n{}\\times \\biggl(\\frac{a(F_r^{-1}(u))}{\\partial_u\nF_r^{-1}(u)}-\n\\frac{\\alpha_r(u)}{\\partial_u \\bar{F}_r^{-1}(u)} \\biggr)\\,du\\,dr.\\nonumber\n\\end{eqnarray}\nThe last term which belongs to $[-\\infty,+\\infty)$ is finite since so\nare all the other terms. We deduce the integrability of\n\\begin{eqnarray*}\n(r,u)&\\mapsto&\\bigl|F_r^{-1}(u)-\\bar{F}_r^{-1}(u)\\bigr|^{p-2}\n\\bigl(\\partial_u F_r^{-1}(u)-\n\\partial_u \\bar{F}_r^{-1}(u) \\bigr)\n\\\\\n&&\\times{} \\biggl(\\frac{a(F_r^{-1}(u))}{\\partial_u\nF_r^{-1}(u)}-\\frac{\\alpha_r(u)}{\\partial_u \\bar{F}_r^{-1}(u)} \\biggr)\n\\end{eqnarray*}\non $[s,t]\\times(0,1)$. Similar arguments show that the integrability\nproperty and (\\ref{vipp}) remain true for $s=t_k$. By summation, they\nremain true for $0\\leq s\\leq t\\leq T$. So the integrability holds on\n$[0,T]$ for the derivative in the distributional sense\n\\begin{eqnarray*}\n&& \\partial_t\\mathcal{W}_{p}^{p} \\bigl(\n\\mathcal{L}(X_t),\\mathcal{L}(\\bar{X}_t) \\bigr)\n\\\\\n&&\\qquad =p\\int_0^{1}\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^{p-2} \\bigl(F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u) \\bigr) \\bigl(b \\bigl(F_t^{-1}(u)\n\\bigr)-\\beta_t(u) \\bigr)\\,du\n\\\\\n&&\\quad\\qquad{}+\\frac{p(p-1)}{2}\\int_0^{1}\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^{p-2} \\bigl(\n\\partial_u F_t^{-1}(u)-\\partial_u\n\\bar{F}_t^{-1}(u) \\bigr)\n\\\\\n&&\\hspace*{103pt}\n{}\\times \\biggl(\\frac\n{a(F_t^{-1}(u))}{\\partial_u F_t^{-1}(u)}-\n\\frac{\\alpha_t(u)}{\\partial\n_u \\bar{F}_t^{-1}(u)} \\biggr)\\,du\n\\\\\n&&\\qquad \\leq p\\int_0^{1}\\bigl|F_t^{-1}(u)-\n\\bar{F}_t^{-1}(u)\\bigr|^{p-2}\n\\\\\n&&\\hspace*{53pt}\n{}\\times \\biggl[\n\\bigl(F_t^{-1}(u)-\\bar{F}_t^{-1}(u)\n\\bigr) \\bigl(b \\bigl(F_t^{-1}(u) \\bigr)-\n\\beta_t(u) \\bigr)\n\\\\\n&&\\hspace*{106pt}{} +\\frac{(p-1) (a(F_t^{-1}(u))-\\alpha_t(u) )^2}{8\\underline {a}} \\biggr]\\,du.\n\\end{eqnarray*}\nEquation (\\ref{majoderwp}) follows by remarking that\n\\begin{eqnarray*}\n&& \\bigl(a \\bigl(F_t^{-1}(u) \\bigr)-\\alpha_t(u)\n\\bigr)^2\n\\\\\n&&\\qquad \\leq2 \\bigl(\\bigl\\|a'\\bigr\\|_\\infty\n^2\\bigl|F_t^{-1}(u)-\\bar{F}_t^{-1}(u)\\bigr|^2+\n\\bigl(a \\bigl(\\bar{F}_t^{-1}(u) \\bigr)-\n\\alpha_t(u) \\bigr)^2 \\bigr)\n\\end{eqnarray*}\nand using a similar idea for $|b(F_t^{-1}(u))-\\beta_t(u)|$.\n\nTo prove (\\ref{IPP_terms}) for $0<s\\leq t\\leq T$, we use the Aronson\nestimates recalled in the proof of Proposition \\ref{propevolftm1} for\n$X_t$ and deduced from Theorem 2.1 \\cite{lemairemenozzi}, for the Euler\nscheme\n\\begin{eqnarray}\\label{aronson}\n&& \\frac{c}{\\sqrt{r}}\\exp\\biggl(-\\frac{(x-x_0)^2}{cr} \\biggr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad \\le\np_r(x)\\wedge\\bar{p}_r(x)\\le p_r(x)\\vee\n\\bar{p}_r(x)\n\\le \\frac{C}{\\sqrt{r}}\\exp\\biggl(-\\frac{(x-x_0)^2}{C r}\\biggr).\\nonumber\n\\end{eqnarray}\n\nSetting $K_1=\\frac{c}{\\sqrt{t}}$, $c_1=cs\/2$, $K_2=\\frac{C}{\\sqrt{s}}$\nand $c_2=Ct\/2$, one has\n\\begin{eqnarray}\\label{aronson2}\nK_1 \\exp\\biggl(-\\frac{(x-x_0)^2}{2c_1 } \\biggr) \\le\\rho_r(x) \\le\nK_2 \\exp\\biggl(-\\frac{(x-x_0)^2}{2c_2 } \\biggr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n\\eqntext{\\forall r\\in[s,t], \\forall x \\in\\mathbb{R},}\n\\end{eqnarray}\nwhere $\\rho_r$ denotes either $p_r$ or $\\bar{p}_r$. The four limits in\n(\\ref{IPP_terms}) can be obtained similarly, and we focus on the one of\n$\\sup_{r\\in[s,t]}\\frac{a(F_r^{-1}(u))}{\\partial_u\nF_r^{-1}(u) } |F_r^{-1}(u)-\\bar{F}_r^{-1}(u) |^{p-1}$.\nUp to modifying $K_1>0$ and decreasing $c_1>0$, we get\nfrom~(\\ref{aronson2}) that\n\\begin{eqnarray}\nK_1(x_0-x)\\exp\\biggl(-\\frac{(x-x_0)^2}{2c_1 } \\biggr) \\le\\rho_r(x) \\le\nK_2 (x_0-x) \\exp\\biggl(-\\frac\n{(x-x_0)^2}{2c_2 } \\biggr)\\nonumber\n\\\\\n\\eqntext{\\forall r\\in[s,t], \\forall x \\le x_0-1,}\n\\end{eqnarray}\nwhich leads to\n\\[\n\\forall x \\le x_0-1\\qquad K_1c_1\\exp\\biggl(-\n\\frac{(x-x_0)^2}{2c_1 } \\biggr) \\le G_r(x) \\le K_2c_2\n\\exp\\biggl(-\\frac{(x-x_0)^2}{2c_2 } \\biggr),\n\\]\nwhere $G_r$ denotes either $F_r$ or $\\bar{F}_r$. Thus, the inverse\nfunction satisfies\n\\begin{equation}\nx_0- \\sqrt{-2c_2 \\log\\biggl(\\frac{u}{K_2c_2}\n\\biggr)} \\le\\bar{F}_r^{-1}(u)\\le x_0-\n\\sqrt{-2c_1 \\log\\biggl(\\frac{u}{K_1c_1} \\biggr)}\\label{continvrep}\n\\end{equation}\nfor $u$ small\nenough. The two last inequalities imply that when $x\\rightarrow-\n\\infty$,\n\\[\n\\forall r\\in[s,t]\\qquad \\bar{F}_r^{-1} \\bigl(F_r(x)\n\\bigr) \\ge x_0 - \\sqrt{-2 c_2 \\biggl[ \\log\\biggl(\n\\frac{K_1c_1}{K_2c_2} \\biggr) -\\frac{(x-x_0)^2}{2c_1} \\biggr]}\n\\]\nand $\\sup_{r\\in[s,t]}|x-\\bar{F}_r^{-1}(F_r(x))\n|\\mathop{=}\\limits_{x\\rightarrow- \\infty}O(x)$. With the boundedness\nof $a$\\vspace*{-1pt} and (\\ref{aronson2}), we easily deduce that\n\\[\n\\lim_{x\\rightarrow- \\infty} \\sup_{r\\in[s,t]}a(x)\np_r(x) \\bigl|x-\\bar{F}_r^{-1}\n\\bigl(F_r(x) \\bigr) \\bigr|^{p-1}=0.\n\\]\nSince, by (\\ref{continvrep}), $\\bar{F}_r^{-1}(u)$\nconverges to $-\\infty$ uniformly in $r\\in[s,t]$ as $u$ tends to $0$,\nwe conclude that\n\\[\n\\lim_{u\\rightarrow0^+}\\sup_{r\\in[s,t]}\\frac\n{a(F_r^{-1}(u))}{\\partial_u\nF_r^{-1}(u) }\n\\bigl|F_r^{-1}(u)-\\bar{F}_r^{-1}(u)\n\\bigr|^{p-1}=0.\n\\]\\upqed\n\\end{pf*}\n\n\\begin{pf*}{Proof of Lemma \\ref{malcal}}\nBy Jensen's inequality,\n\\begin{eqnarray*}\n\\mathbb{E} \\bigl[\\bigl|\\mathbb{E}(W_t-W_{\\tau_t}|\n\\bar{X}_t)\\bigr|^p \\bigr]&\\leq&\\mathbb{E} \\bigl[|W_t-W_{\\tau_t}|^{p}\n\\bigr]\\leq\\frac{C}{N^{p\/2}}.\n\\end{eqnarray*}\nLet us now check that the left-hand side is also smaller than\n$\\frac{C}{t^{p\/2}N^{p}}$. To do this, we will study\n\\[\n{\\mathbb{E}} \\bigl[ (W_{t}-W_{\\tau_{t}})g(\\bar{X}_{t})\n\\bigr],\n\\]\nwhere $g$ is any smooth real valued function.\n\nIn order to continue, we need to do various estimations on the Euler\nscheme and its Malliavin derivative, which we denote by $D_u\\bar{X}_t$.\nLet $\\eta_{t}=\\min\\{t_{i};t\\leq t_{i}\\}$ denote the discretization\ntime just after $t$. We have $D_u\\bar{X}_t=0$ for $u>t$, and\n\\begin{eqnarray}\nD_{u}\\bar{X}_{t}&=&1_{\\{t\\leq\n\\eta_u\\}}\\sigma(\\bar{X}_{\\tau_{t}})\\nonumber\n\\\\\n&&{} +1_{\\{t>\\eta_u\\}} \\bigl(1+\\sigma'(\n\\bar{X}_{\\tau_{t}}) (W_t-W_{\\tau_t})+b'(\\bar{X}_{\\tau_{t}}) (t-\\tau_t) \\bigr)D_u\n\\bar{X}_{\\tau_{t}}\\nonumber\n\\\\\n\\eqntext{\\mbox{for }u\\leq t.}\n\\end{eqnarray}\n\nThen by induction, one clearly obtains that for $u\\le t$,\n\\begin{eqnarray*}\nD_{u}\\bar{X}_{t} & =&\\sigma(\\bar{X}_{\\tau_{u}})\n\\bar{\\mathcal{E}}_{u,t},\n\\\\\n\\bar{\\mathcal{E}}_{u,t} & =& \\cases{ 1, &\\quad if $\\tau_{t}\n\\leq\\eta_{u}$,\n\\vspace*{3pt}\n\\cr\n\\bigl( 1+b^{\\prime}(\n\\bar{X}_{\\tau_{t}}) (t-\\tau_{t})+\\sigma^{\\prime}(\n\\bar{X}_{\\tau_{t}}) (W_{t}-W_{\\tau_{t}}) \\bigr), &\\quad if $\n\\eta_{u}=\\tau_{t}$,\n\\vspace*{4pt}\n\\cr\n\\displaystyle\\prod\n_{i=N\\eta_{u}\/T}^{N\\tau_{t}\/T-1} \\bigl( 1+b^{\\prime}(\n\\bar{X}_{t_{i}}) (t_{i+1}-t_{i})+\\sigma^{\\prime}(\\bar{X}_{t_{i}}) (W_{t_{i+1}}-W_{t_{i}})\n\\bigr)\n\\cr\n\\hspace*{33pt}{}\\times\\bigl( 1+b^{\\prime}(\\bar{X}_{\\tau_{t}}) (t-\\tau_{t})+\n\\sigma^{\\prime}(\\bar{X}_{\\tau_{t}}) (W_{t}-W_{\\tau_{t}})\n\\bigr),&\\quad if $\\eta_{u}<\\tau_{t}$.}\n\\end{eqnarray*}\n\nNote that $\\bar{\\mathcal{E}}$ satisfies the following properties: (1)\n$\\bar{\\mathcal{E}}_{u,t}=\\bar{\\mathcal{E}}_{\\eta(u),t}$ and\\break (2)~$\\bar{\\mathcal{E}}_{t_i,t_j}\\bar{\\mathcal{E}}_{t_j,t}=\\bar{\\mathcal\n{E}}_{t_i,t}$ for $t_i\\le t_j\\le t$. We also introduce the process\n$\\mathcal{E}$ defined by\n\\[\n\\mathcal{E}_{u,t}=\\exp\\biggl( \\int_{u}^{t}b^{\\prime}(X_{s})-\n\\frac\n{1}{2\n\\sigma^{\\prime}(X_{s})^{2}\\,ds+\n\\int_{u}^{t}\\sigma^{\\prime\n}(X_{s})\\,dW_{s}\n\\biggr).\n\\]\nThe next lemma, the proof of which is postponed at the end of the\npresent proof states some useful properties of the processes\n$\\mathcal{E}$ and $\\bar{\\mathcal{E}}$.\n\n\n\n\\begin{alem}\\label{lemme_majorations} Let us assume that $b,\\sigma\\in\nC^2_b$. Then we have\n\\begin{eqnarray}\n\\sup_{0\\leq s\\leq t \\le T}{\\mathbb{E}} \\bigl[\n\\mathcal{E}_{s,t}^{-p} \\bigr]+{\\mathbb{E}} \\bigl[\n\\mathcal{E}_{s,t}^{p} \\bigr] &\\leq& C, \\label{eq:propE}\n\\\\\n\\sup_{0\\leq s\\leq t \\le T}{\\mathbb{E}} \\bigl[ \\bar{\\mathcal{E}}_{s,t}^{p}\n\\bigr] &\\leq& C,\\label{eqA.13}\n\\\\\n\\sup_{0\\leq s,u\\leq t \\le T}{\\mathbb{E}} \\bigl[ |\nD_u\\bar{\\mathcal{E}}_{s,t}|^p+|\nD_u \\mathcal{E}_{s,t}|^p \\bigr] &\\leq& C,\n\\label{eq:A13}\n\\\\\n\\sup_{0\\leq t \\le T}{\\mathbb{E}} \\bigl[ \\llvert\\mathcal\n{E}_{0,t}-\\bar{\\mathcal{E}}_{0,t}\\rrvert\n^{p} \\bigr] &\\leq&\\frac\n{C}{N^{p\/2}}, \\label{vitesse_forte}\n\\end{eqnarray}\nwhere $C$ is a positive constant depending only on $p$ and $T$.\n\\end{alem}\n\nWe next define the localization given by\n\\[\n\\psi=\\varphi\\bigl( \\mathcal{E}_{0,t}^{-1} (\n\\mathcal{E}_{0,t\n-\\bar{\\mathcal{E}}_{0,t} ) \\bigr).\n\\]\nHere $\\varphi\\dvtx \\mathbb{R\\rightarrow}[0,1]$ is a~$C^\\infty$\nsymmetric function so that\n\\[\n\\varphi(x)=\\cases{ 0, &\\quad if $|x|>\\frac{1}{2}$, \\vspace*{2pt}\n\\cr\n1, &\n\\quad if $|x|<\\frac{1}{4}$.}\n\\]\nOne has\n\\begin{eqnarray*}\n\\mathbb{E} \\bigl[ (W_{t}-W_{\\tau_{t}})g(\\bar{X}_{t})\n\\bigr] &=&\\mathbb{E} \\bigl[ (W_{t}-W_{\\tau_{t}})g(\n\\bar{X}_{t})\\psi\\bigr]+\\mathbb{E} \\bigl[ (W_{t}-W_{\\tau_{t}})g(\n\\bar{X}_{t}) (1-\\psi) \\bigr]\n\\\\\n&=&\\int_{\\tau_t}^t\\mathbb{E} \\bigl[\\psi\ng'(\\bar{X}_{t})D_u\\bar{X}_{t}\n\\bigr] \\,du+\\mathbb{E} \\biggl[g(\\bar{X}_{t})\\int_{\\tau_t}^tD_u\n\\psi \\,du \\biggr]\n\\\\\n&&{}+\\mathbb{E} \\bigl[ (W_{t}-W_{\\tau_{t}})g(\n\\bar{X}_{t}) (1-\\psi) \\bigr],\n\\end{eqnarray*}\nwhere the second equality follows from the duality formula; see, for\nexample, Definition 1.3.1 in \\cite{N}. Since for $\\tau_{t}\\leq u\\leq t$\n\\begin{eqnarray*}\n{\\mathbb{E}} \\bigl[ \\psi g^{\\prime}(\\bar{X}_{t})D_{u}\n\\bar{X}_{t} \\bigr] & =& {\\mathbb{E}} \\bigl[ \\psi g^{\\prime}(\n\\bar{X}_{t})\\sigma(\\bar{X}_{\\tau_{t\n}) \\bigr]\n\\\\\n&=&t^{-1}\n\\mathbb{E} \\biggl[\\int_0^t D_sg(\n\\bar{X}_{t})\\frac\n{\\psi\n\\sigma(\\bar{X}_{\\tau_{t\n})}{D_s\\bar{X}_t}\\,ds \\biggr]\n\\\\\n& =&t^{-1}{\\mathbb{E}} \\biggl[ g(\\bar{X}_{t})\\int\n_{0}^{t}\\psi\\sigma(\\bar{X}_{\\tau_{t}})\n\\sigma^{-1} ( \\bar{X}_{\\tau_{s}} ) \\bar{\\mathcal{E\n}_{s,t}^{-1}\\delta W_{s} \\biggr],\n\\end{eqnarray*}\none deduces\n\\begin{eqnarray}\n\\qquad\\qquad\\mathbb{E} [ W_{t}-W_{\\tau_{t}}\\rrvert\\bar{X}_{t} ]\n&=&t^{-1}\\int_{\\tau_{t}}^{t} \\mathbb{E}\n\\biggl[ \\int_{0}^{t}\\psi\\sigma(\n\\bar{X}_{\\tau_{t}})\\sigma^{-1} ( \\bar{X}_{\\tau\n_{s}} )\\bar{\\mathcal{E}}_{s,t}^{-1}\\delta W_{s}\\big|\n\\bar{X}_{t} \\biggr] \\,du\n\\nonumber\\\\[-8pt]\\label{espcond}\\\\[-8pt]\n&&{}+ \\mathbb{E} \\biggl[\\int_{\\tau_t}^t\nD_u\\psi \\,du \\big| \\bar{X}_{t} \\biggr]+\\mathbb{E} \\bigl[ (\nW_{t}-W_{\\tau_{t}} ) (1-\\psi)\\rrvert\\bar{X}_{t}\n\\bigr].\\nonumber\n\\end{eqnarray}\nHere $\\delta W$ denotes the Skorohod integral. In order to obtain the\nconclusion of the lemma, we need to bound the $L^p$-norm of each term\non the right-hand side of~(\\ref{espcond}). In particular, we will use\nthe following estimate (which also proves the existence of the Skorohod\nintegral on the left-hand side below) which can be found in Proposition\n1.5.4 in \\cite{N}:\n\\begin{equation}\n\\label{controle_normep} \\biggl\\llVert\\int_{0}^{t}\n\\psi\\sigma(\\bar{X}_{\\tau_{t}})\\sigma^{-1} ( \\bar{X}_{\\tau_{s}}\n) \\bar{\\mathcal{E}}_{s,t}^{-1}\\delta W_{s} \\biggr\n\\rrVert_{p}\\leq C(p) \\bigl\\llVert\\psi\\sigma(\\bar{X}_{\\tau_{t}})\n\\sigma^{-1} (\\bar{X}_{\\tau_{\\cdot}} )\\bar{\\mathcal{E}}_{\\cdot,t}^{-1}\n\\bigr\\rrVert_{1,p},\\hspace*{-35pt}\n\\end{equation}\nwhere $\\|F_\\cdot\\|_{1,p}^p =\\mathbb{E} [ (\\int_0^t F_s^2 \\,ds )^{p\/2}+\n(\\int_0^t \\int_0^t (D_uF_s)^2 \\,ds\\,du )^{p\/2} ]$. By Jensen's\ninequality for $p\\ge2$, we have\n\\begin{equation}\n\\label{upper_bound_1p} \\qquad\\|F_\\cdot\n\\|_{1,p}^p \\le t^{p\/2-1} \\int_0^t\n\\mathbb{E} \\bigl[|F_s|^p \\bigr] \\,ds + t^{p-2}\n\\int_0^t\\int_0^t\n\\mathbb{E} \\bigl[|D_uF_s|^p \\bigr]\\,ds\\,du\n\\end{equation}\nand we will use this inequality to upper bound~(\\ref{controle_normep}).\nWhen $1\\le p\\le2$, we will use alternatively the following upper bound\n$\\|F_\\cdot\\|_{1,p}^p \\le( \\int_0^t \\mathbb{E}[F_s^2] \\,ds )^{p\/2}+\n(\\int_0^t \\int_0^t \\mathbb{E}[(D_uF_s)^2] \\,ds\\,du )^{p\/2}$ that comes\nfrom H\\\"older's inequality.\n\nFor $\\psi>0$, we have $\\mathcal{E}_{0,t}^{-1} ( \\mathcal\n{E}_{0,t\n-\\bar{\\mathcal{E}}_{0,t} )\\leq\\frac{1}{2}$ so that\n$\\bar{\\mathcal{E}}_{0,t}\\geq\\frac{1}{2}\\mathcal{{E}}_{0,t}>0$. From\nHypothesis~\\ref{hyp_wass_pathwise}, there are constants\n$0<\\underline{\\sigma}\\le\\bar{\\sigma}<\\infty$ such that\n$0<\\underline{\\sigma}\\leq\\sigma\\leq\\bar{\\sigma}$, and one has\n\\begin{eqnarray*}\n&& \\int_{0}^{t}{\\mathbb{E}} \\bigl[ \\bigl( \\psi\n\\sigma(\\bar{X}_{\\tau_{t}} )\\sigma^{-1} ( \\bar{X}_{\\tau_{s}} )\n\\bar{\\mathcal{E}}_{s,t} ^{-1} \\bigr) ^{p} \\bigr]\\,ds\n\\\\\n&&\\qquad \\leq \\biggl(\\frac{\\bar{\\sigma\n}}{\\underline{\\sigma}} \\biggr)^p\\int_{0}^{t}{\n\\mathbb{E}} \\bigl[ \\psi^{p}\\bar{\\mathcal{E}}_{0,t}^{-p}\n\\bar{\\mathcal{E}}_{0,\\eta\n(s)}^{p} \\bigr]\\,ds\n\\\\\n&&\\qquad \\leq \\biggl(\\frac{2\\bar{\\sigma}}{\\underline{\\sigma}} \\biggr)^p\\sqrt\n{\\mathbb{E}\n\\bigl[\\mathcal{{E}}_{0,t}^{-2p} \\bigr]} \\int\n_0^t \\sqrt{\\mathbb{E} \\bigl[|\\bar{\\mathcal{\nE}}_{0,\\eta\n(s)}|^{2p} \\bigr]}\\,ds \\leq C t,\n\\end{eqnarray*}\nby using the estimates~(\\ref{eq:propE}) and (\\ref{eqA.13}).\n\nNext, we focus on getting an upper bound for\n\\begin{equation}\n\\int_0^t \\int_{0}^{t}{\n\\mathbb{E}} \\bigl[ \\bigl\\llvert D_{u} \\bigl( \\psi\\sigma(\n\\bar{X\n_{\\tau_{t}})\\sigma^{-1} ( \\bar{X}_{\\tau_{s}}\n) \\bar{\\mathcal{E\n}_{s,t}^{-1} \\bigr) \\bigr\n\\rrvert^{p} \\bigr] \\,ds \\,du.\\label{eq:Dloc\n\\end{equation}\nTo do so, we compute the derivative using basic derivation rules, which\ngives\n\\begin{eqnarray}\\label{eq:ft}\n&& D_{u} \\bigl( \\psi\\sigma(\\bar{X}_{\\tau_{t}})\n\\sigma^{-1} ( \\bar{X\n_{\\tau_{s}} ) \\bar{\\mathcal{\nE}}_{s,t}^{-1} \\bigr)\\nonumber\n\\\\\n&&\\qquad =D_u\\psi\\sigma(\n\\bar{X}_{\\tau_{t}})\\sigma^{-1} ( \\bar{X}_{\\tau\n_{s}} )\\bar{\\mathcal{E}}_{s,t}^{-1}+\\psi\\sigma^{\\prime}(\n\\bar{X}_{\\tau_{t}})D_{u}\\bar{X}_{\\tau_{t}}\n\\sigma^{-1} ( \\bar{X}_{\\tau_{s}} ) \\bar\n{\\mathcal{E}}_{s,t}^{-1}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\quad\\qquad{}-\\psi\\sigma(\\bar{X}_{\\tau_{t}}) \\sigma^{-2}\\sigma'\n( \\bar{X}_{\\tau_{s}} )\\sigma(\\bar{X}_{\\tau_u}){\\mathcal{\n\\bar E}} _{u,\\tau_s}\\bar{\\mathcal{E}}_{s,t}^{-1}\n\\mathbf{1}_{u\\le\\tau_s}\\nonumber\n\\\\\n&&\\quad\\qquad{}-\\psi\\sigma(\\bar{X}_{\\tau_{u}})\\sigma^{-1} (\n\\bar{X}_{\\tau_{s}} ) \\bar{\\mathcal{E}}_{s,t}^{-2}D_{u}\n\\bar{\\mathcal{E}}_{s,t}.\\nonumber\n\\end{eqnarray}\nOne has then to get an upper bound for the $L^p$-norm of each term. As\nmany of the arguments are repetitive, we show the reader only some of\nthe arguments that are involved. Let us start with the first term. We\nhave\n\\begin{eqnarray*}\nD_u\\psi&=&\\varphi^{\\prime} \\bigl( \\mathcal{E}_{0,t}^{-1}\n( \\mathcal{E\n_{0,t}-\\bar{\\mathcal{E}}_{0,t} )\n\\bigr) D_{u} \\bigl[ \\mathcal{E\n_{0,t}^{-1}\n( \\mathcal{E}_{0,t}-\\bar{\\mathcal{E}}_{0,t} ) \\bigr]\n\\end{eqnarray*}\nand $D_{u} [ \\mathcal{E\n_{0,t}^{-1} ( \\mathcal{E}_{0,t}-\\bar{\\mathcal{E}}_{0,t} ) ]=\n\\mathcal{E\n_{0,t}^{-2}D_{u}\\mathcal{E\n_{0,t}\\bar{\\mathcal{E}}_{0,t}\n-\\mathcal{E\n_{0,t}^{-1}D_u\\bar{\\mathcal{E}}_{0,t} $. From the estimates\nin~(\\ref{eq:propE}), (\\ref{eqA.13}) and~(\\ref{eq:A13}), we obtain\n\\begin{equation}\n\\label{eq:Dest} \\sup_{u\\in[0,t]}\\llVert D_u\\psi\n\\rrVert_{p}\\leq\\bigl\\|\\varphi^{\\prime}\\bigr\\|_\\infty C(p).\n\\end{equation}\nSince $\\bar{\\mathcal{E}}_{s,t}^{-1}=\\bar{\\mathcal{E}}_{0,\\eta(s)}\n\\bar{\\mathcal{E}}_{0,t}^{-1}$ and $\\bar{\\mathcal{E}}_{0,t}\\geq\n\\frac{1}{2}\\mathcal{{E}}_{0,t}>0$ if $\\varphi^{\\prime} (\n\\mathcal{E}_{0,t}^{-1} ( \\mathcal{E\n_{0,t}-\\bar{\\mathcal{E}}_{0,t} ) ) \\neq0$, we have\n\\[\n \\mathbb{E} \\bigl[ \\bigl\\llvert D_u\\psi\\sigma(\n\\bar{X}_{\\tau_{t}})\\sigma^{-1} ( \\bar{X}_{\\tau\n_{s}} )\n\\bar{\\mathcal{E}}_{s,t}^{-1} \\bigr\\rrvert^p\n\\bigr] \\le\\biggl( \\frac{2\\bar{\\sigma}}{\\underline{\\sigma}} \\biggr)^p\n\\llVert\nD_u\\psi\\rrVert_{2p}^p \\mathbb{E} \\bigl[\n\\bigl\\llvert\\mathcal{{E}}_{0,t}^{-1}\\bar{\\mathcal{\nE}}_{0,\\eta(s)} \\bigr\\rrvert^{2p} \\bigr]^{1\/2}.\n\\]\nSimilar bounds hold for the three other terms. Note that the highest\nre\\-quirements on the derivatives of~$b$ and~$\\sigma$ will come from the\nterms involving $D_u\\bar{\\mathcal{E}}$ in~(\\ref{eq:ft}). Gathering all\nthe upper bounds,\\vspace*{-2pt} we get that\\break  $\\llVert\n\\psi\\sigma(\\bar{X}_{\\tau_{t}})\\sigma^{-1} (\\bar {X}_{\\tau_{\\cdot}}\n)\\bar{\\mathcal{E}}_{\\cdot,t}^{-1}\\rrVert _{1,p}^p \\le C(t^{p\/2}+t^p)\n\\le Ct^{p\/2}$ since $0\\le t\\le T$. From~(\\ref{controle_normep}), we\nfinally obtain\n\\[\n\\biggl\\llVert\\int_{0}^{t}\\psi\\sigma(\n\\bar{X}_{\\tau_{u}})\\sigma^{-1} ( \\bar{X}_{\\tau_{s}} )\n\\bar{\\mathcal{E}}_{s,t}^{-1}\\delta W_{s} \\biggr\n\\rrVert_{p}\\leq C(p)t^{1\/2}.\n\\]\n\nWe are now in position to conclude. Using Jensen's inequality, the\nresults (\\ref{eq:propE}), (\\ref{vitesse_forte}), (\\ref{espcond}),\n(\\ref{eq:Dest}) and the definition of $\\varphi$ together with\nChebyshev's inequality, we have for any $k>0$ that\n\\begin{eqnarray*}\n&& \\mathbb{E} \\bigl[ \\bigl\\llvert\\mathbb{E} \\bigl[ W_{t}-W_{\\tau_{t}}\n|\\bar{X}_{t} \\bigr] \\bigr|^{p} \\bigr]\n\\\\[-2pt]\n&&\\qquad \\leq C \\biggl( t^{-p}(t-\\tau_{t})^{p\n\\biggl\\llVert\\int_{0}^{t}\\psi\\sigma(\n\\bar{X}_{\\tau_{t}\n)\\sigma^{-1} ( \\bar{X}_{\\tau_{s}}\n) \\bar{\\mathcal{E}}_{s,t}^{-1}\\delta\nW_{s} \\biggr\\rrVert_{p}^p\n\\\\[-2pt]\n&&\\hspace*{45pt}{} +(t-\\tau_{t})^{p-1}\\int\n_{\\tau_{t}}^{t}\\llVert D_u\\psi\\rrVert\n_{p}^p\\,du\n\\\\[-2pt]\n&&\\hspace*{45pt}{}+\\sqrt{\\mathbb{E} \\bigl(|W_t-W_{\\tau_t}|^{2p}\n\\bigr)} 4^{k\/2} \\bigl(\\mathbb{E} \\bigl(|\\mathcal{{E}}_{0,t}-\n\\bar{\\mathcal{E}}_{0,t}|^{2k} \\bigr)\\mathbb{E} \\bigl(\\mathcal\n{{E}}_{0,t}^{-2k} \\bigr) \\bigr)^{1\/4} \\biggr)\n\\\\[-2pt]\n&&\\qquad \\leq C \\biggl( t^{-p\/2}(t-\\tau_{t})^{p}+(t-\\tau\n_{t})^{p}+ \\biggl(\\frac{1}{N} \\biggr)^{ ( 2p+k ) \/4}\n\\biggr)\n\\\\[-2pt]\n&&\\qquad \\leq C \\biggl( \\frac{1}{t^{p\/2}N^{p}}+\\frac{1}{N^{p\/2+k\/4}} \\biggr).\n\\end{eqnarray*}\\upqed\n\\end{pf*}\\eject\n\n\\begin{pf*}{Proof of Lemma~\\ref{lemme_majorations}}\nThe upper bounds~(\\ref{eq:propE}) and (\\ref{eqA.13}) on $\\mathcal{{E}}$ and $\\bar{\\mathcal\n{E}}$ are obvious since $b'$ and $\\sigma'$ are bounded. Now, let us\nremark that $\\bar{\\mathcal{E}}$ and $\\mathcal{{E}}$ satisfy\n\\begin{eqnarray*}\n\\mathcal{{E}}_{u,t}&=&1+\\int_u^t\n\\sigma' ({X}_{s})\\mathcal{{E}}_{u,s}\n\\,dW_s+\\int_u^tb'\n({X}_{s})\\mathcal{{E}}_{u,s} \\,ds,\n\\\\\n\\bar{\\mathcal{E}}_{\\eta_u,t}&=&1+\\int_{\\eta_u}^t\n\\sigma' (\\bar{X}_{\\tau_s})\\bar{\\mathcal{E}}_{\\eta_u,\\tau_s}\n\\,dW_s+\\int_{\\eta_u}^tb' (\n\\bar{X}_{\\tau_s})\\bar{\\mathcal{E}}_{\\eta\n_u,\\tau_s} \\,ds.\n\\end{eqnarray*}\nThus, (\\ref{vitesse_forte}) can be easily obtained by noticing that\n$(\\bar{X}_t,\\bar{\\mathcal{E}}_{0,t})$ is the Euler scheme for the SDE\n$(X_t,\\mathcal{E}_{0,t})$ which has Lipschitz coefficients, and by\nusing the strong convergence order of $1\/2$; see, for\nexample,~\\cite{Ka}.\\vadjust{\\goodbreak}\n\nThe estimate (\\ref{eq:A13}) on $D_u{\\mathcal{E}}$ is given, for\nexample, by Theorem 2.2.1 in \\cite{N}. On the other hand, we have for\n$\\eta(s)\\le u\\le t$\n\\begin{eqnarray*}\nD_u\\bar{\\mathcal{E}}_{\\eta_s,t}&=&\\sigma'(\n\\bar{X}_{\\tau_u}) \\bar{\\mathcal{E}}_{\\eta_s,\\tau_u}\n\\\\\n&&{} +\\int _{\\eta_u}^t \\bigl[ \\sigma''(\n\\bar{X}_{\\tau_r}) \\sigma(\\bar{X}_{\\tau_u}) \\bar{\n\\mathcal{E}}_{\\eta_u,\\tau_r} \\bar{\\mathcal{E}}_{\\eta_s,\\tau_r} +\n\\sigma'(\\bar{X}_{\\tau_r}) D_u\\bar{\n\\mathcal{E}}_{\\eta_s,\\tau_r} \\bigr] \\,dW_r\n\\\\\n&&{}+\\int_{\\eta_u}^t \\bigl[ b''(\n\\bar{X}_{\\tau_r}) \\sigma(\\bar{X}_{\\tau_u}) \\bar{\n\\mathcal{E}}_{\\eta_u,\\tau_r} \\bar{\\mathcal{E}}_{\\eta_s,\\tau_r} +b'(\n\\bar{X}_{\\tau_r}) D_u\\bar{\\mathcal{E}}_{\\eta_s,\\tau_r}\n\\bigr]\\,dr.\n\\end{eqnarray*}\nIn order to obtain a $L^p(\\Omega)$ estimate, we then\nuse~(\\ref{eqA.13}), $b,\\sigma\\in C^2_b$ and Gronwall's lemma.\n\\end{pf*}\\vspace*{-15pt}\n\n\\section{Proofs of Section~\\lowercase{\\protect\\texorpdfstring{\\ref{sec_pathwise}}{3}}}\\vspace*{-5pt}\\label{App_Sec2}\n\n\\begin{pf*}{Proof of Proposition \\ref{prop_wass_multi}}\nWe use the dual representation of the Wasserstein distance\n(\\ref{defwas}) deduced from Kantorovitch duality theorem (see, e.g.,\nTheorem 5.10, page 58 \\cite{villani}),\n\\[\n\\mathcal{W}^p_p(\\mu,\\nu)=\\sup_{\\phi\\in L^1(\\nu)}\n\\biggl(\\int_E\\tilde{\\phi}(x)\\mu(dx)-\\int\n_E\\phi(x)\\nu(dx) \\biggr),\n\\]\nwhere $\\tilde{\\phi}(x)=\\inf_{y\\in E} (\\phi(y)+|y-x|^p )$.\n\nWe also denote by $(X^{s,x}_t)_{t\\in[s,T]}$ the solution to (\\ref{sde})\nstarting from $x\\in\\mathbb{R}$ at time $s\\in[0,T]$ and by\n$(\\bar{X}^{t_j,x}_t)_{t\\in[t_j,T]}$ the Euler scheme starting from $x$\nat time $t_j$ with $j\\in\\{0,\\ldots,N\\}$. It is enough to check that\n\\begin{eqnarray*}\nw_k&\\stackrel{\\mathrm{def}}{=}&\\mathcal{W}_p \\bigl(\\mathcal{L}\n\\bigl(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_k},X^{s_k,\\bar\n{X}_{s_k}}_{s_{k+1}},\\ldots,X^{s_k,\\bar{X}_{s_k}}_{s_{n}} \\bigr),\n\\\\\n&&\\hspace*{21pt}\n\\mathcal{L} \\bigl(\n\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},X^{s_{k-1},\\bar\n{X}_{s_{k-1}}}_{s_{k}},\\ldots,X^{s_{k-1},\\bar{X}_{s_{k-1}}}_{s_{n}} \\bigr) \\bigr)\n\\end{eqnarray*}\nis smaller\\vspace*{1pt} than $C\\sup_{0\\leq t\\leq\nT,x\\in\\mathbb{R}}\\mathcal{W}_p(\\mathcal{L}(\\bar{X}^{x}_t),\\mathcal{L}(X^{x}_t))$\nsince $\\mathcal{W}_p(\\mathcal{L}(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_n}),\\allowbreak \\mathcal\n{L}(X_{s_1},\\ldots,X_{s_n}))\\leq\\sum_{k=1}^nw_k$. For $f\\dvtx\n\\mathbb{R}^n\\rightarrow\\mathbb{R}$ a bounded measurable function and\n\\[\n\\tilde{f}(x_1,\\ldots,x_n)=\\inf_{(y_1,\\ldots,y_n)\\in\\mathbb{R}^n}\n\\Bigl\\{ f(y_1,\\ldots,y_n)+\\max_{1\\leq\nj\\leq n}|y_j-x_j|^p\n\\Bigr\\},\n\\]\nwe set\n$f_k(x_1,\\ldots,x_k)=\\mathbb{E}(f(x_1,\\ldots,x_k,X^{s_k,x_k}_{s_{k+1}},\\ldots,X^{s_k,x_k}_{s_n}))$.\nFirst choosing\n\\[\n(y_1,\\ldots,y_{k-1},y_{k+1},\\ldots,y_n)= \\bigl(\\bar{X}_{s_1},\\ldots,\\bar\n{X}_{s_{k-1}},X^{s_k,y_k}_{s_{k+1}},\\ldots,X^{s_k,y_k}_{s_{n}}\n\\bigr),\n\\]\nthen conditioning to $\\sigma(W_s,s\\leq s_{k})$ and using (\\ref{cieds}),\nnext conditioning to $\\sigma(W_s,s\\leq s_{k-1})$ and using the dual\nformulation of the Wasserstein distance, one gets\n\\begin{eqnarray*}\n&&\\mathbb{E} \\bigl(\\tilde{f} \\bigl(\\bar{X}_{s_1},\\ldots,\\bar\n{X}_{s_k},X^{s_k,\\bar\n{X}_{s_k}}_{s_{k+1}},\\ldots,X^{s_k,\\bar{X}_{s_k}}_{s_{n}}\n\\bigr)\n\\\\[-2pt]\n&&\\hspace*{9pt}\n{}-f \\bigl(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},X^{s_{k-1},\\bar\n{X}_{s_{k-1}}}_{s_{k}},\\ldots,X^{s_{k-1},\\bar\n{X}_{s_{k-1}}}_{s_{n}} \\bigr) \\bigr)\n\\\\[-2pt]\n&&\\qquad \\leq\\mathbb{E} \\Bigl(\\inf_{y_k\\in\\mathbb{R}} \\Bigl\\{f \\bigl(\\bar\n{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},y_k,X^{s_k,y_k}_{s_{k+1}},\\ldots,X^{s_k,y_k}_{s_{n}} \\bigr)\n\\\\[-2pt]\n&&\\hspace*{125pt}{}+\\max_{k\\leq j\\leq\nn}\\bigl|X^{s_k,y_k}_{s_{j}}-X^{s_k,\\bar{X}_{s_k}}_{s_{j}}\\bigr|^p\n\\Bigr\\}\n\\\\[-2pt]\n&&\\hspace*{10pt}\\quad\\qquad{}-f \\bigl(\\bar{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},X^{s_{k-1},\\bar\n{X}_{s_{k-1}}}_{s_{k}},\\ldots,X^{s_{k-1},\\bar{X}_{s_{k-1}}}_{s_{n}} \\bigr) \\Bigr)\n\\\\[-2pt]\n&&\\qquad\\leq\\mathbb{E} \\Bigl(\\inf_{y_k\\in\\mathbb{R}} \\bigl\\{f_k(\\bar\n{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},y_k)+C|y_k-\n\\bar{X}_{s_k}|^p \\bigr\\}\n\\\\[-2pt]\n&&\\hspace*{95pt}{}\n-f_k \\bigl(\\bar\n{X}_{s_1},\\ldots,\\bar{X}_{s_{k-1}},X^{s_{k-1},\\bar\n{X}_{s_{k-1}}}_{s_{k}}\n\\bigr) \\Bigr)\n\\\\[-2pt]\n&&\\qquad\\leq C\\mathbb{E} \\bigl(\\mathcal{W}_p^p \\bigl(\\mathcal{L}\n\\bigl(X^{s_{k-1},x}_{s_k} \\bigr),\\mathcal{L} \\bigl(\\bar\n{X}^{s_{k-1},x}_{s_k} \\bigr) \\bigr)\\big|_{x=\\bar{X}_{s_{k-1}}} \\bigr)\n\\\\[-2pt]\n&&\\qquad \\leq C\\sup_{x\\in\\mathbb{R}}\\mathcal{W}^p_p \\bigl(\n\\mathcal{L} \\bigl(\\bar{X}^{x}_{s_k-s_{k-1}} \\bigr),\\mathcal{L}\n\\bigl(X^{x}_{s_k-s_{k-1}} \\bigr) \\bigr)\n\\\\[-2pt]\n&&\\qquad\\leq C\\sup_{0\\leq t\\leq T,x\\in\\mathbb{R}}\\mathcal{W}^p_p\n\\bigl( \\mathcal{L} \\bigl(\\bar{X}^{x}_t \\bigr),\\mathcal{L}\n\\bigl(X^{x}_t \\bigr) \\bigr).\n\\end{eqnarray*}\\upqed\n\\end{pf*}\n\n\\section{Some properties of diffusion bridges}\\label{diff_bridge}\nLet us suppose that the SDE $dX_t=b(X_t)\\,dt+\\sigma(X_t)\\,dW_t$,\n$X_0=x$ has a~transition density $p_t(x,y)$ which is positive and of\nclass $\\mathcal{C}^{1,2}$ with respect to $(t,x)\\in\\mathbb{R}_+^*\n\\times \\mathbb{R}$. We check later in this section that this holds\nunder Hypothesis~\\ref{hyp_wass_pathwise}. Then, the law of the\ndiffusion bridge with deterministic time horizon~$\\mathcal{T}$ is given\nby (see, e.g., Fitzsimmons, Pitman and Yor~\\cite{FPY})\n\\begin{eqnarray}\n\\mathbb{E} \\bigl[F(X_u,0\\le u\\le t)|X_\\mathcal{T}=y \\bigr]=\n\\mathbb{E} \\biggl[F(X_u,0\\le u\\le t)\\frac{\np_{\\mathcal{T}-t}(X_t,y)}{p_\\mathcal{T}(x,y)} \\biggr],\\nonumber\n\\\\\n\\eqntext{0\\le t<\\mathcal{T},}\n\\end{eqnarray}\nwhere $F\\dvtx C([0,t],\\mathbb{R})\\rightarrow\\mathbb{R}$ is a bounded\nmeasurable function. Indeed for\\allowbreak $g\\dvtx \\mathbb{R}\\to\\mathbb{R}$\nmeasurable and bounded, using that $X_\\mathcal{T}$ has the density\n$p_\\mathcal{T}(x,y)$, then the Markov property at time $t$, one checks\nthat\n\\begin{eqnarray*}\n&& \\mathbb{E} \\biggl[\\mathbb{E} \\biggl[F(X_u,0\\le u\\le t)\n\\frac{\np_{\\mathcal{T}-t}(X_t,y)}{p_\\mathcal{T}(x,y)} \\biggr] \\bigg|_{y=X_\\mathcal{T}\n}g(X_\\mathcal{T}) \\biggr]\n\\\\\n&&\\qquad =\n\\mathbb{E} \\biggl[F(X_u,0\\le u\\le t)\\int_\\mathbb\n{R}g(y)p_{\\mathcal{T}-t}(X_t,y)\\,dy \\biggr]\n\\\\\n&&\\qquad =\\mathbb{E} \\bigl[F(X_u,0\\le u\\le t)\\mathbb{E}\n\\bigl[g(X_\\mathcal{T})|X_t \\bigr] \\bigr]\n\\\\\n&&\\qquad =\\mathbb{E}\n\\bigl[F(X_u,0\\le u\\le t)g(X_\\mathcal{T}) \\bigr].\n\\end{eqnarray*}\n\n\nWe thus focus on the change of probability measure\n\\[\n\\frac{d\\mathbb{P}^y}{d\\mathbb{P}} \\bigg|_{\\mathcal{F}_t}=\\frac\n{p_{\\mathcal{T}-t}(X_t,y)}{p_\\mathcal{T}(x,y)}=:M_t,\n\\]\nso that $\\mathbb{E}[F(X_u,0\\le u\\le t)|X_\\mathcal{T}=y]=\\mathbb\n{E}^y[F(X_u,0\\le u\\le t)]$ where $\\mathbb{E}^y$ denotes the expectation\nwith respect to $\\mathbb{P}^y$. We define $\\ell_t(x,y)=\\log p_t(x,y)$.\nThe process $(M_t)_{t\\in[0,\\mathcal {T})}$ is a martingale, and by\nIt\\^o's formula, we get $dM_t=M_t \\partial_x\n\\ell_{\\mathcal{T}-t}(X_t,y) \\sigma(X_t)\\,dW_t$, which gives\n\\[\nM_t=\\exp\\biggl( \\int_0^t\n\\partial_x\\ell_{\\mathcal{T}-s}(X_s,y) \\sigma\n(X_s)\\,dW_s -\\frac{1}{2} \\int\n_0^t \\partial_x\n\\ell_{\\mathcal{T}\n-s}(X_s,y)^2 \\sigma(X_s)^2\\,ds\n\\biggr).\n\\]\nGirsanov's theorem then gives that for all $y\\in\\mathbb{R}$,\n$(W^y_t=W_t-\\int_0^t \\partial_x\\ell_{\\mathcal{T}-s}(X_s,\\break y)\n\\*\\sigma(X_s)\\,ds)_{t\\in [0, \\mathcal{T})}$ is a Brownian motion\nunder~$\\mathbb{P}^y$, so that $(W^{X_\\mathcal{T} }_t)_{t\\in [0,\n\\mathcal{T})}$ is a Brownian motion independent of~$X_\\mathcal{T}$.\nMoreover, we have\n\\begin{equation}\n\\label{bridge_dyn} dX_t= \\bigl[b(X_t)+\n\\partial_x\\ell_{\\mathcal{T}-t}(X_t,y)\n\\sigma(X_t)^2 \\bigr]\\,dt +\\sigma(X_t)\\,dW_t^y,\n\\end{equation}\nwhich gives precisely the diffusion bridge dynamics.\n\nConversely, we would like now to reconstruct the diffusion from the\ninitial and the final value by using diffusion bridges. The following\nresult, stated in dimension one, may be generalized to higher\ndimensions.\n\n\n\\begin{aprop}\\label{prop_bridge} We consider an SDE\n$dX_t=b(X_t)\\,dt+\\sigma(X_t)\\,dW_t$, $X_0=x$ with a transition density\n$p_t(x,y)$ positive and of class $\\mathcal{C}^{1,2}$\nwith respect to $(t,x)\\in\\mathbb{R}_+^* \\times\\mathbb{R}$. Let $(B_t,t\\ge0)$ be a\nstandard Brownian motion and $Z_\\mathcal{T}$ be a~random variable with\ndensity $p_\\mathcal{T}(x,y)$ drawn independently from~$B$. We assume\nthat pathwise uniqueness holds for the SDE\n\\begin{eqnarray}\\label{eds_pont}\ndZ^{x,y}_t &=& \\bigl[b\n\\bigl(Z^{x,y}_t \\bigr)+\\partial_x\n\\ell_{\\mathcal{T}-t} \\bigl(Z^{x,y}_t,y \\bigr) \\sigma\n\\bigl(Z^{x,y}_t \\bigr)^2 \\bigr]\\,dt +\\sigma\n\\bigl(Z^{x,y}_t \\bigr)\\,dB_t,\\quad t\\in[0,\\mathcal{T}),\\hspace*{-25pt}\n\\nonumber\\\\[-4pt]\\\\[-4pt]\nZ^{x,y}_0 &=& x,\\nonumber\\hspace*{-25pt}\n\\end{eqnarray}\nfor any $x,y\\in\\mathbb{R}$, and set $Z_t= Z^{x,Z_\\mathcal{T}}_t$ for $t\n\\in\n[0,\\mathcal{T})$. Then, $(Z_t)_{t \\in[0,\\mathcal{T}]}$ and\n$(X_t)_{t\\in[0,\\mathcal{T}]}$ have the same law.\n\\end{aprop}\n\nA consequence of this result is that $(Z_t,t \\in[0,\\mathcal{T}])$ has\ncontinuous paths, which gives that $\\lim_{t\\rightarrow\n\\mathcal{T}-}Z^{x,y}_t = y$ a.s., $dy$-a.e.\n\n\\begin{pf}\nLet $t\\in[0,\\mathcal{T})$ and $F\\dvtx C([0,t],\\mathbb\n{R})\\to\\mathbb{R}$ and $g:\\mathbb{R}\\to\\mathbb{R}$ be bounded and\nmeasurable functions. Since pathwise uniqueness for the\nSDE~(\\ref{eds_pont}) implies weak uniqueness, we get\n\\begin{eqnarray*}\n\\mathbb{E} \\bigl[F \\bigl(Z_{u}^{x,y},0\\leq u\\leq t \\bigr)\n\\bigr]&=&\\mathbb{E}^y \\bigl[F(X_u,0\\leq u\\leq t) \\bigr]\n\\\\\n&=& \\mathbb{E} \\biggl[F(X_u,0\\leq u\\leq t)\\frac\n{p_{\\mathcal{T}\n-t}(X_t,y)}{p_\\mathcal{T}(x,y)} \\biggr].\n\\end{eqnarray*}\nThus we have\n\\begin{eqnarray*}\n\\mathbb{E} \\bigl[F(Z_{u},0\\leq u\\leq t)g(Z_\\mathcal{T})\n\\bigr]&=&\\mathbb{E} \\biggl[F(X_u,0\\leq u\\leq t)\\int\n_\\mathbb{R}p_{\\mathcal{T}\n-t}(X_t,y)g(y)\\,dy \\biggr]\n\\\\\n&=&\n\\mathbb{E} \\bigl[F(X_u,0\\leq u\\leq t)g(X_\\mathcal{T} )\n\\bigr].\n\\end{eqnarray*}\nHence the finite-dimensional marginals of the two processes are equal.\nSince $(X_t)_{t\\in[0,\\mathcal{T}]}$ has continuous paths and\n$(Z_t)_{t\\in[0,\\mathcal{T}]}$ has c\\`adl\\`ag paths (continuous on\n$[0,\\mathcal{T})$ with a possible jump at $\\mathcal{T}$), this\ncompletes the proof.\n\\end{pf}\n\n\nFrom now on, we assume that Hypothesis~\\ref{hyp_wass_pathwise} holds.\nWe introduce the Lamperti transformation of the stochastic process\n$(X_t,t\\ge0)$. We\\vspace*{-4pt} define $\\varphi(x)=\\int_0^x\\frac{dy}{\\sigma(y)}$ and\n$\\alpha(y)= (\\frac{b}{\\sigma}-\\frac{\\sigma'}{2} )\\circ\n\\varphi^{-1}(y)$, $\\hat{X}_t\\stackrel{\\mathrm{def}}{=}\\varphi(X_t)$ so\nthat we have\n\\begin{equation}\n\\label{sde_lamperti} d\\hat{X}_t= \\alpha(\n\\hat{X}_t)\\,dt + dW_t,\\qquad t\\in[0,T].\n\\end{equation}\nBy\\vspace*{-2pt} Hypothesis~\\ref{hyp_wass_pathwise}, $\\varphi$ is a $C^5$ bijection,\n$\\alpha\\in C^3_b$ and both $\\varphi$ and $\\varphi^{-1}$ are Lipschitz\ncontinuous. We denote by $\\hat{p}_t(\\hat{x},\\hat{y})$ the transition density\nof~$\\hat{X}$ and $\\hat{\\ell}_t(\\hat{x},\\hat{y})=\\log(\\hat\n{p}_t(\\hat{x},\\hat{y}))$.\n\n\n\\begin{alem}\\label{lem_densite} The density $\\hat{p}_t(\\hat{x},\\hat\n{y})$ is\n$C^{1,2}$ with respect to $(t,\\hat{x})\\in\\mathbb{R}_+^*\\times\n\\mathbb{R}$. Besides, we have\n\\[\n\\partial_{\\hat{x}} \\hat{\\ell}_t(\\hat{x},\\hat{y})=\n\\frac{\\hat\n{y}-\\hat{x}}{t}-\\alpha(\\hat{x})+ g_t(\\hat{x},\\hat{y}),\n\\]\nwhere $g_t(\\hat{x},\\hat{y})$ is a continuous function on $\\mathbb\n{R}_+\\times\n\\mathbb{R}^2$ such that\n$\\partial_{\\hat{x}} g_t(\\hat{x},\\hat{y})$ and $\\partial_{\\hat{y}}\ng_t(\\hat{x},\\hat{y})$\nexist and\n\\[\n\\forall T>0\\qquad\n\\sup_{t\\in[0,T], \\hat{x},\\hat{y}\\in\\mathbb\n{R}}\\bigl|\\partial_{\\hat{x}}\ng_t(\\hat{x},\\hat{y})\\bigr|+\\bigl|\\partial_{\\hat{y}} g_t(\\hat\n{x},\\hat{y})\\bigr|<\\infty.\n\\]\n\\end{alem}\n\n\n\\begin{pf}\nIt is well known that we can express the transition\ndensity~$\\hat{p}_t(\\hat{x},\\hat{y})$ by using Girsanov's theorem as an\nexpectation on a Brownian bridge between $\\hat{x}$~and~$\\hat{y}$.\nNamely, since $\\alpha$ and its derivatives are bounded, we can apply a\nresult stated in Gihman and Skorohod~\\cite{gs} (Theorem~1, Chapter~3,\nSection~13) or in Rogers \\cite{rog} to get that $\\hat{p}_t(\\hat{x},\\hat{y})$ is positive and\n\\begin{eqnarray*}\n\\hat{\\ell}_t(\\hat{x},\\hat{y})&=&-\\frac{(\\hat{x}-\\hat\n{y})^2}{2t}+\\int\n_{\\hat{x}}^{\\hat{y}\n}\\alpha(z)\\,dz\n\\\\\n&&{} +\\log\\mathbb{E}\n\\biggl(\\exp\\biggl({-\\frac{1}{2}\\int_0^t\\bigl(\\alpha\n'+\\alpha\n^2\\bigr)\\biggl(\\hat{x}+W_s+\\frac{s}{t}(\\hat{y}-\\hat{x}-W_t)\\biggr)\\,ds} \\biggr)\\biggr)\n\\\\\n&&{} -\\frac{1}{2}\\log(2\\pi t).\n\\end{eqnarray*}\nClearly, $\\hat{\\ell}_t(\\hat{x},\\hat{y})$ is $C^{1,2}$ in\n$(t,\\hat{x})\\in\\mathbb{R}_+^*\\times\n\\mathbb{R}$ (we can use carefree the dominated convergence theorem for\nthe third\nterm since $\\alpha\\in C^3_b$), and we have\n\\begin{eqnarray*}\ng_t(\\hat{x},\\hat{y})\n&=& -\\frac{1}{2}\n\\biggl(\\mathbb{E} \\biggl[\\exp\\biggl({-\\frac\n{1}{2}\\int_0^t\\bigl(\\alpha'+\\alpha^2\\bigr)\\biggl(\\hat{x}+W_s+\\frac{s}{t}(\\hat{y}-\\hat\n{x}-W_t)\\biggr)\\,ds}\\biggr)\n\\\\\n&&\\hspace*{33pt}{}\\times \\int_0^t\\frac{t-s}{t}\\bigl(\\alpha''+2\\alpha\\alpha'\\bigr)\\biggl(\\hat\n{x}+W_s+\\frac\n{s}{t}(\\hat{y}-\\hat{x}-W_t)\\biggr)\\,ds \\biggr]\\biggr)\n\\\\\n&&\\hspace*{9pt}{}\\bigg\/\\biggl({\\mathbb{E} \\biggl[\\exp\\biggl({-\\frac\n{1}{2}\\int\n_0^t\\bigl(\\alpha'+\\alpha^2\\bigr)\\biggl(\\hat{x}+W_s+\\frac{s}{t}(\\hat{y}-\\hat\n{x}-W_t)\\biggr)\\,ds}\\biggr) \\biggr]}\\biggr).\n\\end{eqnarray*}\nThis is a continuous function on~$\\mathbb{R}_+\\times\\mathbb{R}^2$,\nand we easily\nconclude by using the dominated convergence theorem and~$\\alpha\\in\nC^3_b$.\n\\end{pf}\n\nBy straightforward calculations, we have\n\\[\np_t(x,y)=\\frac{1}{\\sigma(y)}\\hat{p}_t \\bigl(\\varphi(x),\n\\varphi(y) \\bigr)\n\\]\nand $p_t(x,y)$ is thus positive and $C^{1,2}$ with respect to $(t,x)$.\nThe diffusion bridge~(\\ref{bridge_dyn}) is thus well defined. Since\n$\\partial_x \\ell_t(x,y) =\\frac{1}{\\sigma(x)}\\partial_{\\hat x}\n\\hat{\\ell}_t(\\varphi(x),\\varphi(y))$, we get by It\\^{o} formula\nfrom~(\\ref{bridge_dyn})\n\\begin{eqnarray*}\nd \\hat{X}_t&=& \\bigl[\\alpha(\\hat{X}_t)+\n\\partial_{\\hat{x}} \\hat{\\ell}_{\\mathcal{T}-t} \\bigl(\\hat{X}_t,\n\\varphi(y) \\bigr) \\bigr]\\,dt+dW^y_t,\n\\\\\ndW^y_t &=&dW_t-\n\\partial_{\\hat{x}} \\hat{\\ell}_{\\mathcal{T}-t} \\bigl(\\hat{X}_t,\n\\varphi(y) \\bigr)\\,dt.\n\\end{eqnarray*}\nTherefore, as one could expect, the Lamperti transform on the diffusion\nbridge coincides with the diffusion bridge on the Lamperti transform.\n\n\\begin{aprop}\\label{prop_bridge2}\nLet Hypothesis~\\ref{hyp_wass_pathwise} hold. There exists a\ndeterministic constant~$C$ such that\n\\[\n\\forall\\mathcal{T}\\in(0,T], x,x',y,y'\\in\\mathbb{R}\\qquad\n\\sup_{t\\in[0,\\mathcal{T})} \\bigl|Z^{x,y}_t-Z^{x',y'}_t\\bigr|\n\\le C \\bigl(\\bigl|x-x'\\bigr|\\vee\\bigl|y-y'\\bigr| \\bigr)\n\\]\nand in particular, pathwise uniqueness holds for~(\\ref{eds_pont}).\n\\end{aprop}\n\n\n\\begin{pf}\nFor $\\hat{x},\\hat{y}\\in\\mathbb{R}$, we consider the following SDE:\n\\begin{eqnarray}\\label{pont_Z2}\nd \\hat{Z}^{\\hat{x},\\hat\n{y}}_t&=&dB_t+\n\\biggl[\\frac{\\hat{y}-\\hat{Z}^{\\hat{x},\\hat{y}}_t}{\\mathcal\n{T}-t}+g_{\\mathcal{T}-t} \\bigl(\\hat{Z}^{\\hat{x},\\hat{y}}_t,\n\\hat{y} \\bigr) \\biggr]\\,dt,\\qquad t\\in[0, \\mathcal{T}),\n\\nonumber\\\\[-20pt]\\\\\n\\hat{Z}^{\\hat{x},\\hat{y}}_{0}&=&\\hat{x},\\nonumber\n\\end{eqnarray}\nwhich corresponds to the diffusion bridge on the Lamperti\ntransform~$\\hat{X}$. We set\n$\\Delta_t=\\hat{Z}^{\\hat{x},\\hat{y}}_t-\\hat{Z}^{\\hat{x}',\\hat\n{y}'}_t$ for\n$t\\in[0,\\mathcal{T})$ and $\\hat{x}',\\hat{y}' \\in\\mathbb{R}$. We have\n\\[\nd \\Delta_t = \\biggl[\\frac{\\hat{y}-\\hat{y}'-\\Delta_t}{\\mathcal{T}-t}\n+g_{\\mathcal{T}-t} \\bigl(\n\\hat{Z}^{\\hat{x},\\hat{y}}_t,\\hat{y} \\bigr)-g_{\\mathcal{T}-t} \\bigl(\\hat\n{Z}^{\\hat{x}',\\hat{y}'}_t,\\hat{y}' \\bigr) \\biggr]\\,dt\n\\]\nand thus $d(|\\Delta_t| \\vee|\\hat{y}-\\hat{y}'|)=\n\\operatorname{sign}(\\Delta_t)\\mathbf{1}_{|\\Delta_t|\\ge|\\hat{y}-\\hat{y}'|}\\,d\\Delta_t$.\nOn the one hand, we observe that\n$\\mathbf{1}_{|\\Delta_t|\\ge|\\hat{y}-\\hat{y}'|}[\\operatorname{sign}(\\Delta_t)\n(\\hat{y}-\\hat{y}')-|\\Delta_t|]\\le0$. On the other hand, $g_t$ is\nuniformly Lipschitz w.r.t. $(\\hat{x},\\hat{y})$ on $t\\in[0,T]$ by\nLemma~\\ref{lem_densite}, which leads to\n\\[\nd \\bigl(|\\Delta_t| \\vee\\bigl|\\hat{y}-\\hat{y}'\\bigr| \\bigr)\\le C\n\\bigl(|\\Delta_t| \\vee\\bigl|\\hat{y}-\\hat{y}'\\bigr| \\bigr)\n\\]\nfor some positive constant~$C$. Gronwall's lemma gives then\n$|\\Delta_t|\\le e^{CT}(|\\hat{x}-\\hat{x}'|\\vee|\\hat{y}-\\hat{y}'|)$.\nThis gives in\nparticular pathwise uniqueness for~(\\ref{pont_Z2}).\n\nNow, let us\\vspace*{-1pt} assume that $(Z^{x,y}_t)_{t\\in[0,\\mathcal{T})}$\nsolves~(\\ref{eds_pont}). Then\n$\\varphi(Z^{x,y}_t)$ solves~(\\ref{pont_Z2}) with $\\hat{x}=\\varphi\n(x)$ and\n$\\hat{y}=\\varphi(y)$, and we necessarily have\n$Z^{x,y}_t=\\varphi^{-1}(\\hat{Z}_t^{\\varphi(x),\\varphi(y)})$ by pathwise\nuniqueness. Both $\\varphi$ and $\\varphi^{-1}$ are Lipschitz, and we\ndenote by $K$ a~common Lipschitz constant. Then we get\n\\begin{eqnarray*}\n\\bigl|Z^{x,y}_t-Z^{x',y'}_t\\bigr| &=&\\bigl|\n\\varphi^{-1} \\bigl(\\hat{Z}_t^{\\varphi(x),\\varphi(y)} \\bigr) -\n\\varphi^{-1} \\bigl(\\hat{Z}_t^{\\varphi(x'),\\varphi(y')} \\bigr) \\bigr|\n\\\\\n&\\le&\nK^2 e^{CT} \\bigl(\\bigl|x-x'\\bigr|\\vee\\bigl|y-y'\\bigr|\n\\bigr),\n\\end{eqnarray*}\nwhich gives the desired result.\n\\end{pf}\n\\end{appendix}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnderstanding the evolution of neutral hydrogen (HI{}) in dark matter\nhaloes is important for models of galaxy formation\n\\citep{somerville2015, blanton2009, barkana2016}.  The HI{} content\nof dark matter haloes forms an intermediate state in the baryon cycle\nthat connects the hot shock-heated gas and star-forming molecular gas\nin haloes \\citep{2010ApJ...718.1001B, 2010MNRAS.409..515F,\n  2012ApJ...753...16K}.  Constraints on HI{} in galaxies therefore\nreveal the role of gas dynamics, cooling, and regulatory processes\nsuch as stellar feedback and gas inflow and outflow in galaxy\nformation \\citep{prochaska09, 2011MNRAS.414.2458V,\n  2015MNRAS.447.1834B, 2015MNRAS.451..878K, 2016MNRAS.456.1115B}.\nHI{} also traces environmental processes like satellite quenching,\ntidal interactions and ram-pressure stripping\n\\citep{2012MNRAS.427.2841F, 2012MNRAS.424.1471L, 2013MNRAS.429.2191Z,\n  lagos2014}.  The average HI mass content of dark matter haloes can\nbe expressed as an HI-mass halo-mass (HIHM) relation.\n\nAt low redshifts ($z \\sim 0$), constraints on HI{} in galaxies are\nderived from the observations of the 21~cm emission line of hydrogen\nin large-area blind galaxy surveys like the HI{} Parkes All Sky\nSurvey \\citep[HIPASS,][]{meyer2004} and the Arecibo Fast Legacy ALFA\nsurvey \\citep[ALFALFA,][]{giovanelli2005}, which provide measurements\nof the mass function and clustering of HI{}-selected galaxies. There\nare also targeted surveys such as The HI{} Nearby Galaxy Survey\n\\citep[THINGS,][]{walter2008}, the Galex Arecibo SDSS Survey\n\\citep[GASS,][]{catinella2010}, and the Westerbork HI{} survey of\nSpiral and Irregular Galaxies \\citep[WHISP,][]{vanderhulst2001}, which\nfocus on a smaller number of resolved galaxies. Efforts are also\ncurrently underway to constrain the density and clustering of HI{}\nusing intensity mapping without resolving individual galaxies\n\\citep{chang10, masui13, switzer13}. In the future, current and\nupcoming facilities such as MeerKAT \\citep{jonas2009}, the Square\nKilometre Array \\citep[SKA,][]{2015aska.confE..19S} and its\npathfinders, and the Canadian Hydrogen Intensity Mapping Experiment\n\\citep[CHIME,][]{2014SPIE.9145E..22B}, will provide unprecedented\ninsight into the evolution of the cosmic neutral hydrogen content\nacross redshifts.\n\n\nUnfortunately, the intrinsic faintness of the 21~cm line and the\nlimits of current radio facilities hamper direct detection of HI{}\nfrom individual galaxies at redshifts above $z \\sim 0.1$. Spectral\nstacking has been used to probe the HI{} content of undetected\nsources out to redshifts $z \\sim 0.24$ \\citep{lah07, lah2009, rhee13,\n  delhaize13}. At higher redshifts, therefore, constraints on the\ndistribution and evolution of HI in galaxies come chiefly from high\ncolumn density Lyman-$\\alpha$ absorption systems (Damped\nLyman-$\\alpha$ Absorbers; DLAs) with column density\n$N_\\mathrm{HI}>10^{20.2}$~cm$^{-2}$ in the spectra of bright\nbackground sources such as quasars.  DLAs are the main reservoir of HI\nbetween redshifts $z\\sim 2$--$5$, containing $> 80 \\%$ of the cosmic\nHI content \\citep{wolfe1986, lanzetta1991, gardner1997, prochaska09,\n  rao06, noterdaeme12, zafar2013}.  At low redshift, DLAs have been\nfound to be associated with galaxies \\citep{lanzetta1991} and to\ncontain the vast majority ($\\sim 81\\%$) of the HI{} gas in the local\nuniverse \\citep{zwaan2005a}.  At high redshift, the kinematics of DLAs may\nsupport the hypothesis that they probe HI in large rotating disks\n\\citep{1997ApJ...487...73P, 2001MNRAS.326.1475M, 2015MNRAS.447.1834B} or proto-galactic clumps \\citep{haehnelt1998}.\nThe three-dimensional clustering of DLAs \\citep{fontribera2012} points\nto DLAs being preferentially hosted by dark matter haloes with mass $M\n\\sim 10^{11} M_{\\odot}$ at redshift $z \\sim 3$.\n\nSemi-analytical models and hydrodynamical simulations have provided\nclues towards the evolution of HI{} in galaxies and its relation to\nstar-formation, feedback and galaxy evolution \\citep{dave2013,\n  duffy2012, lagos2011, obreschkow2009a, nagamine2007, pontzen2008,\n  tescari2009, hong2010, cen2012, fu2012, kim2013, bird2014,\n  popping2009, popping2014, eagle2016, kim2016,\n  martindale2016}. {Semi-analytical methods \\citep[e.g.,][]{berry2014, popping2014, somerville15} typically reproduce the\nHI{} mass functions and the HI{}-to-stellar-mass scaling relations\nfound in low-redshift HI{} observations and DLA observables.}\nSimulation techniques have also been used to model DLA populations at\nhigher redshifts \\citep{pontzen2008} and their relation to galaxy\nformation and feedback processes \\citep{bird2014, rahmati2013,\n  rahmati2014}.  Hydrodynamical simulations suggest that DLAs are\nhosted in haloes with mass $10^{10}$--$10^{11} h^{-1}$ M$_\\odot$\n\\citep[e.g.,][]{bird2014}.  In the presence of strong stellar\nfeedback, these simulations can reproduce the observed abundance and\nclustering of DLAs but end up having an excess of HI{} at low\nredshifts ($z<3$).\n\nAnalytical techniques offer complementary insight into the processes\ngoverning the HI{} content of dark matter halos. Analytical methods\nhave been used for modelling 21~cm intensity mapping observables,\nparticularly the HI{} bias and power spectrum\n\\citep{marin2010,wyithe2010, sarkar2016} as well as DLAs \\citep{haehnelt1996,\n  haehnelt1998,barnes2009, barnes2010, 2013ApJ...772...93K,\n  barnes2014}. These models use prescriptions for assigning HI{} mass\nto dark matter halos as inputs to the model, either directly or in\nconjunction with cosmological simulations \\citep{bagla2010, marin2010,\n  gong2011, guhasarkar2012}. In \\citet{hptrcar2016}, the 21-cm- and\nDLA-based analytical approaches are combined towards a consistent\nmodel of HI{} evolution across redshifts. It is found that a model\nthat is consistent with low-redshift radio as well as high-redshift\noptical\/UV observations requires a fairly rapid transition of HI{}\nfrom low-mass to higher-mass haloes at high redshifts. A more complete\nstatistical {data-driven} approach \\citep{2016arXiv160701021P}\nconstrains the HIHM relation using low- and high-redshift\nobservations {in a halo model framework}.\n\nAn essential ingredient in analytical techniques is therefore the HIHM\nrelation.  In this paper, we employ the technique of abundance\nmatching to quantify the observational constraints on the HIHM\nrelation in the post-reionization Universe. Abundance matching has\nbeen widely used to describe the relation between the stellar mass of\ngalaxies and the mass of their host dark matter halos \\citep{vale2004,\n  vale2006, conroy2006, behroozi2010, guo2010, shankar2006,\n  moster2010, moster2013}.  The basic assumption involved is that\nthere is a monotonic relationship between a galaxy property (say,\nstellar mass or galaxy luminosity) and the host dark matter halo\nproperty (say, the host halo mass). In its simplest form, abundance\nmatching involves matching the cumulative abundance of galaxies to\nthat of their (sub)haloes, thereby assigning the most luminous\ngalaxies to the most massive haloes. The mapping between the\nunderlying galaxy property and the host halo mass can be derived from\nthis. {A key feature of this approach is that being completely empirical\\footnote{{A caveat is that the halo mass function being used is theoretical, and the assumption of matching the most massive haloes is involved.}}, it is free from the uncertainties involved in physical models of HI{} and galaxy evolution. {It is therefore a complementary analysis to forward modelling techniques, including semi-analytical models  and hydrodynamical simulations.}\n\nThe HI{} mass function \\citep{rao1993} is the radio equivalent of the\noptical luminosity function in galaxies and is an important\nstatistical quantity in the observations of gas-rich galaxies. It\nmeasures the volume density of HI{}-selected galaxies as a function\nof the HI{} mass and simulations suggest that its shape is a more\nsensitive probe of some aspects of galaxy formation physics than the\ngalaxy luminosity function \\citep{kim2013}. At low redshifts, the\nHI{} mass function is fairly well-constrained over four decades in HI\nmass \\citep{zwaan05, martin10}. \\citet{papastergis2013} constrained\nthe HIHM relation at low redshift using ALFALFA data and found that\nthe observed clustering of HI was reproduced well by this approach.  In\nthis work, we describe the results of abundance matching HI{} mass to\ndark matter halo mass using the low-redshift radio observations of the\nHI{} mass function \\citep{zwaan05, martin10} and then evolve the\nrelation using the complementary information available through DLA\nmeasurements at high redshift. The combination of the radio data at\nlow redshifts and DLA observations at higher redshifts constrains a\nmulti-epoch HI{}-halo mass relation with the available data. We also compare how the results from this approach are consistent with those from studies in previous literature.\n\nThe paper is organized as follows. In Section~\\ref{sec:abmatch}, we\ndetail the abundance matching technique and apply it to three\nlow-redshift HI mass function measurements.  We also combine the\nresultant HIHM relation with the stellar-mass halo-mass (SHM) relation\nto discuss the HI-to-stellar-mass ratio in low-redshift galaxies. In\nSection~\\ref{sec:observations}, we extend the low-redshift HIHM\nrelation to higher redshifts using measurements of DLA column density\ndistribution and clustering.  We compare the relation so derived with\nother HI{} models in the literature, and conclude in\nSection~\\ref{sec:conc}.\n\n\\section{HIHM relation at low redshift}\n\\label{sec:abmatch}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{.\/mhiandm.pdf}\n  \\end{center}\n  \\caption{The blue and red curves show the HI mass functions derived\n    from the HIPASS \\citep{zwaan05} and ALFALFA data \\citep{martin10},\n    respectively.  The shaded region shows the combined uncertainty.\n    The black curve shows the halo mass function.}\n  \\label{fig:mhiandm}\n\\end{figure} \n\nWe derive the HIHM relation at $z\\sim 0$ by abundance matching dark\nmatter haloes with HI-selected galaxies.  We use the HI mass function\nfrom the HIPASS \\citep{meyer2004} and ALFALFA \\citep{martin10} datasets,\nthe latter derived using the $1\/V_{\\rm max}$ as well as the 2DSWML\n(2-Dimensional StepWise Maximum Likelihood) methods:\n\n\\begin{itemize}\n\\item HIPASS: This complete catalogue of HI sources contains 4,315\n  galaxies \\citep{meyer2004}. The HI mass function $\\phi(M_{\\rm HI})$ is fitted by a  Schechter function    using the the 2-Dimensional StepWise Maximum  Likelihood (2DSWML) method, with a total of 4010 galaxies.  The\n  effective volume $V_{\\rm eff}$ is calculated for each galaxy\n  individually and the values of $1\/V_{\\rm eff}$ are summed in bins of\n  HI mass to obtain the 2DSWML mass function.  The resultant best-fit\n  parameters are $\\alpha = -1.37 \\pm 0.03 \\pm 0.05$,\n  $\\log(M_{*}\/M_{\\odot}) = 9.80 \\pm 0.03 \\pm 0.03 h_{75}^{-2}$ and\n  $\\phi^* = (6.0 \\pm 0.8 \\pm 0.6) \\times 10^{-3} h_{75}^3$ Mpc$^{-3}$\n  (the two error values show statistical and systematic errors,\n  respectively; \\citealt{zwaan05}).  The distribution of HI masses is\n  calculated using 30 equal-sized mass bins spanning $6.4 <\n  \\log_{10}M_{\\rm HI} < 10.8$ (in $M_{\\odot}$).\n\n\n\\item ALFALFA: This catalogue contains 10,119 sources to form the\n  largest available sample of HI-selected galaxies \\citep{martin10}.\n  The ALFALFA survey measures the HI mass function by using both the\n  2DSWML as well as the $1\/V_{\\rm max}$ methods.  The HI mass function\n  is fitted with the Schechter form, with the best-fitting parameters\n  $\\phi^* = (4.8 \\pm 0.3) \\times 10^{-3} h_{70}^3$ Mpc$^{-3}$, $\\log (M_*\/M_{\\odot}) + 2\n  \\log(h_{70}) = 9.95 \\pm 0.04$, and $\\alpha = -1.33 \\pm 0.03$ with\n  the $1\/V_{\\rm max}$ method, and $\\phi^* = (4.8 \\pm 0.3) \\times\n  10^{-3} h_{70}^3$ Mpc$^{-3}$, $\\log (M_*\/M_{\\odot}) + 2 \\log(h_{70}) = 9.96 \\pm 0.2$,\n  and $\\alpha = -1.33 \\pm 0.02$ with the 2DSWML method.  The two\n  determinations of the HI mass function are in good agreement.\\footnote{In the figures, we only indicate the ALFALFA 2DSWML mass function fit for clarity.}\n\\end{itemize}\n\nTo match HI-selected galaxies to dark matter haloes, we use the\nSheth-Tormen \\citep{sheth2002} form of the dark matter halo mass\nfunction.  Figure~\\ref{fig:mhiandm} shows the comparison of the three\nHI mass functions mentioned above with the halo mass function, which\nis shown by the solid black curve.  This corresponds to the assumption\nthat each dark matter halo hosts one HI galaxy with its HI mass\nproportional to the host dark matter halo mass.  The shaded region in\nFigure~\\ref{fig:mhiandm} shows the combined uncertainty in the\nobserved HI mass functions.  \nMatching the abundance of the halo mass function and the fitted HI\nmass function then leads to the relation between the HI mass and the\nhalo mass \\citep[e.g.,][]{vale2004}:\n\\begin{equation}\n  \\int_{M (M_{\\rm HI})}^{\\infty} \\frac{dn}{ d \\log_{10} M'} \\ d \\log_{10} M' = \\int_{M_{\\rm HI}}^{\\infty} \\phi(M_{\\rm HI}') \\ d \\log_{10} M_{\\rm HI}'\n  \\label{eqn:abmatch}\n\\end{equation}\nwhere $dn \/ d \\log_{10} M$ is the number density of dark matter haloes with logarithmic\nmasses between $\\log_{10} M$ and $\\log_{10} (M$ + $dM)$, and $\\phi(M_{\\rm HI})$ is the\ncorresponding number density of HI galaxies in logarithmic mass bins.  Solving\nEquation~(\\ref{eqn:abmatch}) gives a relation between the HI-mass\n$M_{\\rm HI}$ and the halo mass $M$.  Note that this approach assumes\nthat there is a monotonic relationship between $M_{\\rm HI}$ and $M$.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{.\/abmatch_coldgasfrac.pdf}\n  \\end{center}\n  \\caption{\\textit{Top panel}: The HIHM relation at $z=0$ derived from HIPASS\n    (blue curve) and ALFALFA (red curve) HI mass functions.  The black\n    curve shows a combined fit to the mass functions using the parametric form of\n    Equation~(\\ref{moster12}).  The shaded region shows the error in\n    the fit.  \\textit{Lower panel}: The HI mass fraction, $M_{\\rm\n      HI}\/M$ as a function of halo mass $M$ at $z=0$.  Also shown for\n    comparison in both panels is the SHM relation \\citep{moster2013}.}\n\\label{fig:coldgasfrac}\n\\end{figure} \n\nSolving Equation~(\\ref{eqn:abmatch}) in the mass range\n$10^6$~M$_{\\odot} < M_{\\rm HI} < 10^{11}$~M$_{\\odot}$, we show the\nresultant HIHM relation in the top panel of\nFigure~\\ref{fig:coldgasfrac}.  The red curve shows the HIHM relation\nobtained from the ALFALFA data, while the blue curve shown the same\nfor the HIPASS data.   We find that the HI mass monotonically increases\nas a function of the halo mass and changes slope at a characteristic\nvalue of the halo mass.  This behaviour is qualitatively similar to\nthe SHM relation \\citep{moster2013}, which is shown by the dashed red\ncurve in the top panel of Figure~\\ref{fig:coldgasfrac}.  For small\nmass haloes, the HI mass is nearly equal to the stellar mass.  But the\nHI mass decreases more rapidly than the stellar mass as a function of\nhalo mass, and for high mass haloes the HI mass is down to almost a tenth of\nthe stellar mass.  The characteristic mass for the HIHM relation is also slightly smaller ($10^{11.7} M_{\\odot}$) than that for the SHM relation ($\\sim 10^{12} M_{\\odot}$).\nThe HIHM relation is shown as the ratio of the HI and halo masses in\nthe lower panel of Figure~\\ref{fig:coldgasfrac}.  The peak HI mass\nfraction is about 1\\%, and  this reduces down to 0.01\\% at both high and low\nmasses.  The peak HI mass fraction is in good agreement with the\nabundance matching estimates of \\citet{puebla2011, evoli2011,\n  baldry2008} and the direct estimate of \\citet{papastergis2012} for\nthe baryonic mass fraction. It had been found that the clustering of\nthe HI selected galaxies in ALFALFA \\citep{papastergis2013} was also\nwell-matched by abundance matching at $z \\sim 0$, and the cold gas\nfraction showed a maximum at halo masses close to $10^{11.1 - 11.3}\nM_{\\odot}$, which was lower than the corresponding peak for the stellar mass\nfraction ($10^{11.8} M_{\\odot}$).\n\nWe parameterise the HIHM relation by a function of the form introduced\nfor the SHM relation by \\citet{moster2013},\n\\begin{equation}\nM_{\\rm HI} = 2 N_{10} M \\left[\\left(\\frac{M}{M_{10}}\\right)^{-b_{10}} + \\left(\\frac{M}{M_{10}}\\right)^{y_{10}}\\right]^{-1}.\n\\label{moster12}\n\\end{equation}\nWe fit the HIHM relation by the function of this form using non-linear\nleast squares.  The best-fitting values of the free parameters are\n$M_{10}=(4.58 \\pm 0.19)\\times 10^{11}$~M$_\\odot$, $N_{10}=(9.89\\pm\n4.89)\\times 10^{-3}$, $b_{10}=0.90 \\pm 0.39$ and $y_{10}=0.74 \\pm\n0.03$.\nThe errors here are estimated by propagating the uncertainties in\nFigure~\\ref{fig:mhiandm}.  The best-fit HIHM relations are shown in\nFigure~\\ref{fig:coldgasfrac} (black curves), with the corresponding\nerror indicated by the shaded region.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{.\/mhimstar.pdf} \n  \\end{center}\n  \\caption{The HI-mass stellar-mass relation obtained by abundance\n    matching combined with the SHM relation determined by\n    \\citet{moster2013}, are shown by the solid curves. {The 68\\%\n      scatter in the relation is indicated by the blue band.} The\n    green band shows the region around the median in which 68\\% of the\n    galaxies in the EAGLE reference simulation lie on this plane\n    \\citep{eagle2016}. Also shown are the data from individual objects\n    detected in the GASS and COLD GASS surveys, and the nearby\n    galaxies in HERACLES and THINGS \\citep{leroy2008}.}\n  \\label{fig:mhimstar}\n\\end{figure} \n\n\\subsection{The HI-mass stellar-mass relation}\n\nWe can combine our derived HIHM relation with known SHM relations to\nunderstand the relationship between the HI mass and stellar mass in\ndark matter haloes.  \\citet{moster2013} use a multi-epoch abundance\nmatching method with observed stellar mass functions (SMFs) to\ndescribe the evolution of the SHM relation across redshifts.  At each\nredshift, they parameterise the SHM relation using the functional form\nin Equation~(\\ref{moster12}).  At low redshifts, the SMFs of\n\\citet{li2009} based on the Sloan Digital Sky Survey (SDSS) DR7\n\\citep{york2000, abazajian2009} are used, along with the observations\nof \\citet{baldry2008}. At higher redshifts, the SMFs by\n\\citet{gonzalez2008} are used for massive galaxies, and those by\n\\citet{santini2012} for the low mass galaxies. From the results of\nabundance matching, the mean SHM relation is obtained, which is then\nused to populate haloes in the Millennium\n\\citep[MS-I;][]{millenium2005} and the Millennium - II\n\\citep[MS-II;][]{boylan2009} simulations with galaxies. From this, the\nmodel stellar mass functions are derived and directly compared to\nobservations to constrain the free parameters in the SHM relation. The\nresulting mean stellar mass fraction at $z \\sim 0$ is shown by the\ndashed line in Figure~\\ref{fig:coldgasfrac}.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{.\/mhimstarm.pdf} \n  \\end{center}\n  \\caption{The HI-mass to stellar-mass ratio as a function of the halo\n    mass at $z\\sim 0$.  The blue and red curves combine our results\n    for HIPASS and ALFALFA data, respectively, with the SHM relation\n    from \\citet{moster2013}. The parametrized fit is indicated by the black curve. The shaded region shows the uncertainty in the HI-mass to\n    stellar-mass ratio obtained by propagating errors from\n    Figure~\\ref{fig:coldgasfrac}.}\n  \\label{fig:mhimstarm}\n\\end{figure} \n\nWe use the \\citet{moster2013} results for the SHM relation, coupled to\nour abundance matching results for HIHM to arrive at a HI-mass\nstellar-mass relation.  This is shown by the solid red and blue curves\nin Figure~\\ref{fig:mhimstar} for HIPASS and ALFALFA\nrespectively. {The 68\\% scatter in the relation is indicated by\n  the blue band.} For comparison, we also show the measurements from\n750 galaxies in the redshift range $0.025 < z < 0.05$ and $M_{*} >\n10^{10}$~M$_{\\odot}$ from the GALEX Arecibo SDSS survey\n\\citep[GASS;][]{catinella2010, catinella2013}, and 366 galaxies from\nthe COLD GASS survey \\citep{saintonge2011, saintonge2011a,\n  catinella2012}.  We also show results from \\citet{leroy2008}, which\nis a compilation of individual galaxies detected in the HERA CO Line\nExtragalactic Survey \\citep[HERACLES;][]{leroy2009} that are part of\nThe HI Nearby Galaxy Survey \\citep[THINGS;][]{walter2008}, which\ncovers HI{} masses in the range $(0.01$--$14) \\times\n10^9$~M$_{\\odot}$.  These measurements are consistent with our result,\nalthough the observational data exhibit a somewhat large\nscatter. {We note that the HI-stellar mass relation\n    from the ALFALFA data and the THINGS data show some discrepancy at\n    low stellar masses (also seen in \\citet{popping2015}, which\n    matches the data in \\citet{leroy2008}, but has difficulty matching\n    the ALFALFA data mass function at low HI\n    masses).  However, the main\n    aim of the present work is to provide an understanding of the\n    HI-mass halo-mass relation, and as such, we do not conjecture on\n    the observed discrepancy of the \\citet{leroy2008} results with the\n    ALFALFA data.} We also compare our HI-mass stellar-mass relation\nwith that found in the EAGLE hydrodynamical simulations\n\\citep{schaye2015, crain2015}.  The EAGLE simulations model the\nformation and evolution of galaxies in the presence of various\nfeedback processes.  They also model the HI content of galaxies by\nusing calibrated fitting functions from radiative transfer simulations\nto estimate self-shielding, and also employing empirical relations to\ncorrect for molecular gas formation \\citep{eagle2016}. The green band\nin Figure~\\ref{fig:mhimstar} shows the region around the median on the\nHI-mass stellar-mass diagram occupied by 68\\% of galaxies in the\nreference EAGLE simulation (labelled ``L100N1504'' in\n\\citealt{schaye2015}).  Our results are in good agreement with the\nEAGLE predictions, except possibly at the highest stellar masses\n($M_*>10^{10}$~M$_\\odot$) where the HI mass in EAGLE galaxies starts\nto decrease.  This is likely a reflection of the AGN feedback in\nEAGLE, that heats and expunges cold gas from high mass galaxies by\ntheir massive central black holes \\citep{eagle2016}.\n\nFigure~\\ref{fig:mhimstarm} shows the HI-mass to stellar-mass ratio as\na function of the halo mass.  The blue and red curves show the results\nfor HIPASS and ALFALFA respectively, and the black curve shows the\nparametrized fit.  In each case, we obtain the HI-mass to stellar-mass\nratio by combining our HIHM relation with the SHM relation of\n\\citet{moster2013}.  The HI-mass to stellar-mass ratio is about 25\\%\nin a rather broad range of halo masses from $10^{11}$ to\n$10^{13}$~M$_\\odot$.  {The ratio decreases to about 10\\% at halo masses\nabove this range, and is more uncertain below this range, due to the uncertainty in the data and the fitting (Fig. \\ref{fig:coldgasfrac} lower panel) at lower masses.}  The shaded regions show the\nuncertainty in the HI-mass to stellar-mass ratio, obtained by\npropagating the errors from Figure~\\ref{fig:coldgasfrac}. \n\n\\section{HIHM relation at high redshift}\n\\label{sec:observations}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{cll}\n\\hline\n$z$   & Observable               & Source                  \\\\\n\\hline\n$\\sim$ 1    & $\\Omega_{\\rm HI}b_{\\rm HI}$ & \\citet{switzer13}       \\\\\n     & $f_{\\rm HI}$               & \\citet{rao06}           \\\\\n     & $dN\/dX$                   & \\citet{rao06}           \\\\\n2.3  & $\\Omega_{\\rm DLA}$         & \\citet{zafar2013}       \\\\\n     & $f_{\\rm HI}$               & \\citet{noterdaeme12}    \\\\\n     & $b_{\\rm DLA}$              & \\citet{fontribera2012}  \\\\\n     & $dN\/dX$                   & \\citet{zafar2013}       \\\\\n > 3 & $dN\/dX$                   & \\citet{zafar2013}       \\\\\n \\hline\n\\end{tabular}\n\\caption{High-redshift data used in this paper.  The measurement of\n  $\\Omega_{\\rm HI}b_{\\rm HI}$ comes from HI intensity mapping at\n  $z\\sim 0.8$ by \\citet{switzer13}. \\citet{rao06} use measurements of\n  absorption systems at median redshifts $z \\sim 0.609$ and $z \\sim\n  1.219$ to derive the DLA parameters. All other data come from\n  Lyman-$\\alpha$ absorption measurements using high-redshift quasar\n  spectra.}\n\\label{table:data}\n\\end{table}\n\nDue to the intrinsic faintness of the 21~cm line, the direct detection\nof HI from resolved galaxies is difficult at redshifts above $z \\sim\n0.1$.  At higher redshifts ($z<5$), therefore, constraints on the\ndistribution and evolution of HI in galaxies mainly come from high\ncolumn density Lyman-$\\alpha$ absorption systems (Damped\nLyman-$\\alpha$ Absorbers; DLAs) with column densities\n$N_\\mathrm{HI}>10^{20.3}$~cm$^{-2}$ in the spectra of bright\nbackground sources such as quasars.  The relevant observables at these\nredshifts are the incidence rate $dN\/dX$ of DLAs, the column density\ndistribution $f_\\mathrm{HI}(N_\\mathrm{HI},z)$ of DLAs at high column\ndensities, the three-dimensional clustering of DLAs as quantified by\ntheir clustering bias relative to the underlying dark matter, and the\ntotal amount of neutral hydrogen in DLAs \\citep{wolfe1986,\n  lanzetta1991, gardner1997, prochaska09, rao06, noterdaeme12,\n  zafar2013}.  A detailed summary of the low- and high-redshift HI\nobservables is provided in \\citet{hptrcar2015}. We now extend the HIHM\nrelation obtained at $z=0$ to higher redshifts by using these\nobservables. Throughout the analysis, we use the cosmological\nparameters $h = 0.71$, $\\Omega_m = 0.281$, $\\Omega_{\\Lambda} = 0.719$,\n$\\sigma_8 = 0.8$, $n_s = 0.964$. \n\n\\subsection{Modelling the HI observables}\n\nTo model the distribution of HI density within individual dark matter\nhaloes, we use the redshift- and mass-dependent modified\nNavarro-Frenk-White (NFW; \\citealt{1996ApJ...462..563N}) profile\nintroduced by \\citet{barnes2014}:\n\\begin{equation}\n  \\rho_{\\rm HI}(r) = \\frac{\\rho_0 r_s^3}{(r + 0.75 r_s) (r+r_s)^2},\n  \\label{rhodef}\n\\end{equation} \nwhere $r_s$ is the scale radius defined as $r_s=R_v(M)\/c(M,z)$, with\n$R_v(M)$ being the virial radius of the halo.  The halo concentration\nparameter, $c(M,z)$ is approximated by:\n\\begin{equation}\n  c(M,z) = c_{\\rm HI} \\left(\\frac{M}{10^{11} M_{\\odot}} \\right)^{-0.109} \\left(\\frac{4}{1+z} \\right).\n\\end{equation} \nThe profile in Equation~(\\ref{rhodef}) is motivated by {the analytical modelling of} cooling in multiphase halo gas by \\cite{maller2004}.  In the above\nequation, $c_{\\rm HI}$ is a free parameter, the concentration\nparameter for the HI, analogous to the dark matter halo concentration\n$c_0 = 3.4$ \\citep{maccio2007}. The value of this parameter can be\nconstrained by fitting to the observations.  The \n$\\rho_0$ in Equation~(\\ref{rhodef}) is determined by normalization to\nthe total HI mass:\n\\begin{equation}\n \\int_0^{R_v(M)} 4 \\pi r^2 \\rho_{\\rm HI}(r) dr = M_{\\rm HI} (M)\n\\end{equation} \nThus, both the HI-halo mass relation as well as the radial\ndistribution of HI are required for constraining the HI profile.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width = \\columnwidth, scale=0.45]{.\/mostercompare.pdf} \n  \\end{center}\n  \\caption{The evolution of the parameters of the HIHM relation\n    (Equation~\\ref{mosterredshiftevol}).  The green curves show our\n    best-fit parameter inferences with 68\\% confidence intervals shown\n    by the orange shaded region. For comparison, the evolution of the\n    corresponding quantities for the SHM relation of\n    \\citet{moster2013} is shown in blue.}\n  \\label{fig:evolution}\n\\end{figure} \n\n\n \\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{.\/columndensity.pdf} \n  \\end{center}\n  \\caption{The best-fit column density distribution (red curves) in\n    our model at redshifts 0, 1 and 2.3, compared to the\n    observations. The blue shaded regions show the 68\\% confidence\n    limits.  The model fits the high redshift column density\n    distributions quite well but has difficulty in fitting the column\n    density distribution at $z=0$, especially at low column\n    densities.}\n  \\label{fig:fhibias0}\n\\end{figure*}  \n\nThe DLA based quantities at different redshifts can now be computed by\ndefining the column density of a halo at impact parameter $s$ as\n\\citep{barnes2014, hptrcar2016}:\n\\begin{equation}\n N_{\\rm HI}(s) = \\frac{2}{m_H} \\int_0^{\\sqrt{R_v(M)^2 - s^2}}dl \\ \\rho_{\\rm HI}\\left(\\sqrt{s^2 + l^2}\\right)\n \\label{coldenss}\n\\end{equation} \nwhere $m_H$ is the hydrogen atom mass and $R_v(M)$ is the virial radius\nassociated with a dark matter halo of mass $M$.  We define the DLA\ncross-section of the halo as $\\sigma_{\\rm DLA} = \\pi s_*^2$, where\n$s_*$ is defined such that $N_{\\rm HI}(s_*) = 10^{20.3}$ cm$^{-2}$.  The\nclustering bias of DLAs, $b_{\\rm DLA}$, can then be written as \n\\begin{equation}\n b_{\\rm DLA} (z) =  \\frac{\\int_{0}^{\\infty} dM n (M,z) b(M,z) \\sigma_{\\rm DLA} (M,z)}{\\int_{0}^{\\infty} dM n (M,z) \\sigma_{\\rm DLA} (M,z)},\n\\end{equation}\nwhere $n(M,z)$ is the comoving halo mass function and $b(M,z)$ is the\nclustering bias factor of haloes \\citet{scoccimarro2001}.  The DLA\nincidence $dN\/dX$ can be calculated as\n\\begin{equation}\n \\frac{dN}{dX} = \\frac{c}{H_0} \\int_0^{\\infty} n(M,z) \\sigma_{\\rm DLA}(M,z) \\ dM,\n \\label{dndxdef}\n\\end{equation} \nand the column density distribution $f_{\\rm HI}(N_{\\rm HI}, z)$ is given by\n\\begin{multline}\n  f(N_{\\rm HI}, z) \\equiv \\frac{d^2 n}{dX d N_{\\rm HI}} \\\\\n  = \\frac{c}{H_0} \\int_0^{\\infty} n(M,z) \\left|\\frac{d \\sigma}{d N_{\\rm HI}} (M,z) \\right| \\ dM\n  \\label{coldensdef}\n\\end{multline}\nwhere\n\\begin{equation}\n  \\frac{d\\sigma}{dN_{\\rm HI}}=2\\pi s\\frac{ds}{dN_{\\rm HI}},\n\\end{equation}\nwith $N_{\\rm HI}(s)$ defined by Equation~(\\ref{coldenss}).  The\ndensity parameter for DLAs, $\\Omega_{\\rm DLA}$ is obtained by\nintegrating the column density distribution \n\\begin{equation}\n \\Omega_{\\rm DLA}(N_{\\rm HI}, z)  = \\frac{m_H H_0}{c \\rho_{c,0}} \\int_{10^{20.3}}^{\\infty} f_{\\rm HI}(N_{\\rm HI}, z) \\ N_{\\rm HI} \\ d N_{\\rm HI},\n\\end{equation}\nwhere $\\rho_{c,0}$ is the present-day critical density.\n\nAt high redshifts, we also use the measurement of $\\Omega_{\\rm HI}b_{\\rm\n  HI}$ from HI intensity mapping at $z\\sim 0.8$ by \\citet{switzer13}.\nTo calculate this quantity in our model, the HI density parameter is\ngiven by\n\\begin{equation}\n \\Omega_{\\rm HI} (z) = \\frac{1}{\\rho_{c,0}} \\int_0^{\\infty} n(M, z) M_{\\rm HI} (M,z) dM \\ .\n \\label{omegaHI}\n\\end{equation} \nThe bias of HI is given by\n\\begin{equation}\nb_{\\rm HI} (z) = \\frac{\\int_{0}^{\\infty} dM n(M,z) b (M,z) M_{\\rm HI} (M,z)}{\\int_{0}^{\\infty} dM n(M,z) M_{\\rm HI} (M,z)}\n\\label{biasHI}\n\\end{equation}\nwhere $b(M,z)$ is the dark matter halo bias. We fit the HI density\nprofiles of haloes at $z=0$ by using the  column\ndensity distribution at $z=0$ for $N_\\mathrm{HI}>10^{20.3}$~cm$^{-2}$,\nderived from the WHISP data by \\citet{zwaan2005a}. \n\n\n\n\\subsection{Extending the HIHM relation to high redshifts}\n\\label{sec:extending}\nWe can now extend the HIHM relation developed in\nSection~\\ref{sec:abmatch} to higher redshifts.  We do this by\nparameterising the HIHM relation evolution in a manner similar to the\nparameterisation of the SHM relation evolution by \\citet{moster2013}.\nWe write the HIHM relation at higher redshifts as\n\\begin{equation}\nM_{\\rm HI} = 2 N_{1} M \\left[\\left(\\frac{M}{M_{1}}\\right)^{-b_{1}} + \\left(\\frac{M}{M_{1}}\\right)^{y_{1}}\\right]^{-1},\n\\label{mosterredshiftevol}\n\\end{equation}\nwhich has the same form as Equation~(\\ref{moster12}).  The parameters\nin Equation~(\\ref{mosterredshiftevol}) are written as:\n\\begin{align}\n& \\log_{10} M_{1} = \\log_{10}  M_{10} + \\frac{z}{z + 1}  M_{11}, \\nonumber \\\\\n& N_{1}  = N_{10} + \\frac{z}{z + 1} N_{11}, \\nonumber \\\\ \n& b_{1} = b_{10} + \\frac{z}{z + 1} b_{11},\\ \\mathrm{and}\\nonumber \\\\\n& y_{1}  = y_{10} + \\frac{z}{z + 1} y_{11}. \n\\label{eq:evol}\n\\end{align}\n\nThe parameters $M_{10}$, $N_{10}$, $b_{10}$ and $y_{10}$ are defined\nin Equation~(\\ref{moster12}) for $z=0$.  The four additional\nparameters, $M_{11}$, $N_{11}$, $b_{11}$ and $y_{11}$, introduced by\nEquations~(\\ref{eq:evol}) govern the evolution of the HIHM at high\nredshift.  These four parameters together with the HI density profile\nparameter $c_{\\rm HI}$ are to be constrained from the high redshift\nobservations.  This is done by using the data available from $z=0$ to\n$5$ as summarised in Table~\\ref{table:data}.  We use the measurements  of the incidence rate $dN\/dX$ of DLAs, the column\ndensity distribution $f_\\mathrm{HI}(N_\\mathrm{HI},z)$ of DLAs at high\ncolumn densities, the three-dimensional clustering of DLAs as\nquantified by their clustering bias relative to the dark matter,\nand the total amount of neutral hydrogen in DLAs \\citep{wolfe1986,\n  lanzetta1991, gardner1997, prochaska09, rao06, noterdaeme12,\n  zafar2013}, as well as the measurements of the HI column density\ndistribution and clustering from radio data at $z<1$ \\citep{zwaan2005a,\n  switzer13}.\n  \n\n\\begin{figure}\n  \\begin{center}\n    \\hskip-0.2in \\includegraphics[width = \\columnwidth, scale=0.6]{.\/omegabiasdndx.pdf} \n  \\end{center}\n\\caption{Our model predictions for the density parameter, clustering\n  bias, and DLA incidence rate (red, with 68\\% confidence intervals\n  indicated by the error bars) compared to the observations.  Note\n  that at redshift $z \\sim 1$, \\citet{switzer13} constrain the product\n  $\\Omega_{\\rm HI} b_{\\rm HI}$.  Shown here is the observed\n  $\\Omega_{\\rm HI} b_{\\rm HI}$ divided by the model value of $b_{\\rm\n    HI}$ (top panel) and $\\Omega_{\\rm HI}$ (second panel).  The model\n  successfully matches these observations, including the bias at high\n  redshifts.}\n\\label{fig:panels123}\n\\end{figure} \n\nThe best-fitting values for the five parameters $M_{11}, N_{11},\nb_{11}$, $y_{11}$ and $c_{\\rm HI}$, and their errors are now estimated\nby a Bayesian Markov Chain Monte Carlo (MCMC) analysis using the\n\\textsc{CosmoHammer} package \\citep{akeret2013}. The likelihood,\n\\begin{equation}\n\\mathcal{L} = \\exp\\left(-\\frac{\\chi^2}{2}\\right) \n\\end{equation}\nis maximized with respect to the five free parameters, with:\n\\begin{equation}\n\\chi^2 = \\sum_i\\frac{(f_{\\rm i} - f_{\\rm obs,i})^2}{\\sigma^2_{\\rm obs,i}}\n\\end{equation}\nwhere the $f_{\\rm i}$ are the model predictions, $f_{\\rm obs,i}$ are\nthe observational data and $\\sigma^2_{\\rm obs,i}$ are the squares of\nthe associated uncertainties (here assumed independent).\n\nThe best fitting parameters and their 68\\% errors are\n$M_{11}=1.56^{+0.53}_{-2.70}$,\n$N_{11}=0.009^{+0.06}_{-0.001}$, $b_{11}=-1.08^{+1.52}_{-0.08}$,\n$y_{11}=4.07^{+0.39}_{-2.49}$, and $c_{\\rm\n  HI}=133.66^{+81.39}_{-56.23}$.  The inferred evolution of the four\nparameters of the HIHM relation in Equation~(\\ref{mosterredshiftevol})\nis shown in Figure~\\ref{fig:evolution} together with the 68\\%\nerrors.  For comparison, the evolution of the corresponding parameters\nin the SHM relation parametrization of \\citep{moster2013} are also\nshown.  {The model allows for a wide range of parameters in the HIHM relation at high redshifts.}  The increase in the {best-fitting} characteristic mass follows the\nincrease in the characteristic halo mass of the SHM relation.  The\nevolution of the high mass slope $y_1$ is much more rapid for the HIHM\nrelation than the SHM relation.  As we will see below, the high value\nof the clustering bias factor for DLAs at high redshifts forces the\nincrease in the characteristic halo mass of the HIHM relation but the\nmore gradual increase observed in the DLA incidence rate prevents us\nfrom putting too much HI in high mass halos, which constrains the high\nmass slope to very steep values. \n\nFigure~\\ref{fig:fhibias0} shows the column density distribution\nderived from our model at $z\\sim 0, 1,$ and $2.3$ together with the\nassociated 68\\% statistical error. \n\n{At $z \\sim 0$, only the concentration parameter of the profile is used to obtain the column density distribution, since the  HIHM relation has been directly fixed by the results of abundance matching. The concentration parameter is assumed to be equal to that obtained from the fitting of higher redshifts, which is done using the analysis outlined in Sec. \\ref{sec:extending}.} The relation fits the available\ndata reasonably well, but leads to an underprediction of the observed\ncolumn density distribution at $z \\sim 0$ at low column densities\n($N_\\mathrm{HI}<10^{21.4}$~cm$^{-2}$).  \\footnote{{The two datasets for the column density distribution at $z \\sim 0$ (which indicate a systematic offset) are shown only for comparison, and not directly fitted. The parameters involved in the HIHM are obtained from the abundance matching fits, and the concentration parameter is obtained from the results of the higher-redshift column density fitting. \n The steep\nslope of the HIHM relation  for $z=0$ leads to a lower column\ndensity distribution than observed, suggesting that {the\n  altered NFW profile may not fully describe} the HI density profiles\nof halos at $z=0$, or that there {may be} a possible tension between\nthe HI mass function and the column density distribution at $z=0$. We\nexplore this issue in further detail in future work.}} Figure~\\ref{fig:panels123}\ncompares other quantities in our model to their observed values.  The\nincidence rate of DLAs is fit very well by the model throughout the\nredshift range considered here.  {The measurements of the density\nparameters of HI and DLAs, and the clustering bias of $z \\sim 2.3$ DLAs are also fit well.}  The fit to the measured HI bias at $z=0$ is also good,\nalthough it is somewhat poor at $z=1$.\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[scale=0.6, width = \\textwidth]{.\/allz.pdf} \n  \\end{center}\n  \\caption{\\textit{Left panel}: The HIHM relation inferred at redshifts $z=0$,\n    $1$, $2$, $3$ and $4$ from the present work. \\textit{Right panels}: The\n    HIHM relation relation in the present work compared to the results\n    of other approaches in the literature at redshifts $z=0$, $1$, $2$\n    and $3$.}\n  \\label{fig:massevolall}\n\\end{figure*} \n \n\\subsection{Comparison to other models of HI at high redshift}\n\n\nFigure~\\ref{fig:massevolall} shows the inferred best-fitting HIHM at $z=0$, $1$,\n$2$, $3$ and $4$ in the present model, together with their associated uncertainties. In each case, the black curve shows the best-fit HIHM relation\n  and the grey band shows the 68\\% scatter around it. The figure also presents a\ncomparison of the HIHM obtained from hydrodynamical simulations and\nother approaches in the literature at $z=0$, $1$, $2$ and $3$. {{These are briefly described below:\n\\begin{enumerate}\n\n\\item  At $z=0$, the model that comes closest to the present work is the\nnon-parametric HIHM relation of \\citet{marin2010}, although their\nlow-mass slope is shallower.  \n\n\\item The hydrodynamical simulations of\n\\citet{dave2013} produce an HIHM relation that has very similar\nhigh-mass and low-mass slopes as the present HIHM relation.  The high characteristic mass of\nthe {average best-fitting} HIHM relation in the present work is a natural consequence of\nmatching the abundance of haloes with HI-selected galaxies, under the\nassumption that HI-mass of dark matter haloes scales monotonically\nwith their virial mass.\n\n\\item  \\citet{bagla2010} used a set of analytical\nprescriptions to populate HI{} in dark matter haloes.  In their\nsimplest model, HI{} was assigned to dark matter haloes with a\nconstant fraction $f$ by mass, within a mass range. The maximum and\nminimum masses of haloes that host HI{} were assumed to be\nredshift-dependent. It was also assumed that haloes with virial\nvelocities of greater than 200 km\/s and less than 30 km\/s do not host\nany HI.  \n\n\\item \\citet{gong2011} provide nonlinear analytical forms of the\nHIHM relation at $z=1$, $2$ and $3$, derived from the results of the\nsimulations of \\citet{obreschkow2009a}. These predict a\n  slightly different form for the HIHM relation. \n  \n  \\item  The model of\n\\citet{barnes2014} uses an HIHM relation that reproduces the observed\nbias of DLA systems at $z \\sim 2.3$, and constrains stellar feedback\nin shallow potential wells. \n\n\\item \\citet{2016arXiv160701021P} used a {statistical data-driven} approach to\nderive the best-fitting HIHM relation and radial distribution profile\n$\\rho_{\\rm HI}(r)$ for $z=0$--$4$, from a joint analysis combining the\ndata from the radio observations at low redshifts and the Damped\nLyman-Alpha (DLA) system observables at high redshifts, along the\nlines of the present work. This approach also produces results\n  consistent with the present work, although the present best-fit HIHM\n  relation at high redshifts may prefer a higher characteristic halo mass.\n  \n  \\end{enumerate}\n  \n  \n  It can be seen that all these\n  models are consistent with each other and with the data at the 68\\% confidence level. Tighter constraints on the HIHM relation at high redshifts may be achieved with the availability of better quality data with upcoming radio telescopes.\n\n}}\n\n\n\n\\section{Conclusions}\n\\label{sec:conc}\n\nIn this paper, we have explored the evolution of the neutral hydrogen\ncontent of galaxies in the last 12 Gyr (redshifts $z=0$--$4$).  At\nredshift $z=0$, this work follows the approach of abundance matching,\nwhich has been widely used for the stellar mass content of galaxies to\nmodel galaxy luminosity functions \\citep{vale2004,\n  vale2006, conroy2006, shankar2006,\n  guo2010, behroozi2010, moster2010, moster2013}.\nA parameterised functional form for a monotonic relationship between\nthe HI{} and halo mass is assumed to obtain the HI- Halo Mass (HIHM)\nrelation.  The best fit values of the parameters that fit the observed\nHI{} mass function from radio data are then obtained.  This approach\nof modelling the HIHM relation at $z=0$ from the radio data at low\nredshifts has been followed previously by \\citet{papastergis2013}.\nOur abundance matched HIHM agrees with that derived by these authors.\n\nWe further explore how well the abundance matching approach at $z = 0$\ncan be constrained by fitting to the high redshift data. We extend the\nlow redshift determination of the HIHM relation by postulating that\nthe evolution of the HIHM relation is similar to the\nstellar-to-halo-mass (SHM) relation.  We parameterize this evolution\nanalogously to the evolution of the SHM relation by\n\\citet{moster2013}. {The physical motivation for the\n  parametrization is that the HI-follows-stars functional form works\n  well at low redshifts, which is in turn a consequence of the fact\n  that the underlying mass\/luminosity functions can both be described\n  by the Schechter form.} Observational measurements of the HI{} mass\nfunction are not {yet} available at these redshifts.  Hence, we use\nmeasurements of the HI{} column density distribution function and the\nHI{} clustering from UV\/optical observations of quasar absorption\nspectra.  We assume that high column density systems (DLAs;\n$N_\\mathrm{HI}>10^{20.3}$~cm$^{-2}$) probe systems are high-redshift\nanalogs of HI{} in galaxies detected in radio surveys at low\nredshifts \\citep{zwaan2005a}.\n\nOur procedure allows a modeling of low and high redshift measurements\nof the HI{} content of galaxies to obtain the evolution of the HIHM\nrelation from $z=0$ to $2.3$ with the associated uncertainty.  This\ntechnique is complementary to the forward modelling approach which\naims to characterize HI{} using a halo model framework similar to\nthat of the underlying dark matter \\citep{2016arXiv160701021P}.\n{However, the present work represents a first attempt to characterize\n  the HIHM relation empirically, directly from the data. Due to the\n  sparse nature of the high-redshift data at present, there is\n  considerable scatter in the high-redshift HIHM relation. {As\n    a result,  other apparently dissimilar models from the\n    literature are also consistent with the data and the allowed range of the present work. The scatter in the HIHM relation at higher redshifts can be reduced with tighter constraints on the HI mass functions from upcoming and future radio surveys.} \n\nOur results provide a useful benchmark to calibrate the HI{} physics\nin hydrodynamical simulations, especially at low redshifts where\ncorrect treatment of star formation and feedback as well as cooling\nand formation of molecular hydrogen are critical. They also provide an\nestimate of the uncertainty in the HIHM relation coming from the\nhigh-redshift data, and motivate further work towards possibly tighter\nconstraints on the HIHM relation.\n\n\n\\section*{Acknowledgements}\n\nWe thank Alireza Rahmati, Alexandre Refregier and Sergey Koposov for\nuseful discussions, Daniel Lenz for pointing out a minor typo and Robert Crain for kindly providing the data\nfrom the EAGLE simulations. This work has made use of the VizieR\ncatalogue access tool, CDS, Strasbourg, France. The original\ndescription of the VizieR service was published in the A\\&AS 143, 23.\nHP's research is supported by the Tomalla Foundation.  GK gratefully\nacknowledges support from the ERC Advanced Grant 320596 `The\nEmergence of Structure During the Epoch of Reionization'.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional neural network (CNN) architectures exhibit state-of-the-art performance on a variety of learning tasks dealing with 1D, 2D, and 3D grid-structured data such as acoustic signals, images, and videos, in which\nconvolution   serves as a feature extractor~\\cite{lecun2015deep}. However, the (usual) convolution operation is not applicable when  applying CNN  to data that is  supported on an arbitrary graph rather than on a regular grid structure, since the number and topology of neighbors of each vertex on the graph varies, and  it is difficult to design a fixed-size filter scanning over the graph-structured data for feature extraction.\n\nRecently, there has been an increasing interest in graph CNNs~\\cite{bruna2013spectral, defferrard2016convolutional, thomas2017semi,monti2017geometric, levie2017cayleynets}, attempting to generalize deep learning methods to graph-structured data, specifically focusing on the design of graph convolution.\nIn this paper, we propose the topology adaptive graph convolutional network (TAGCN), a unified convolutional neural network to learn nonlinear representations for the graph-structured data. It slides a set of fixed-size learnable filters on the graph simultaneously, and the output is the weighted sum of these filters' outputs, which extract both vertex features and strength of correlation between vertices. Each filter is adaptive to the topology of the local region on the graph where it is applied. TAGCN unifies filtering in both the spectrum and vertex domains; and applies to both directed and undirected graphs.\n\n\nIn general, the existing graph CNNs can be grouped into two types: spectral domain techniques and vertex domain techniques. In \\citet{bruna2013spectral}, CNNs have been generalized to   graph-structured data, where convolution is achieved by a pointwise product in the spectrum domain according to the convolution theorem.\nLater, \\citet{defferrard2016convolutional} and \\citet{levie2017cayleynets} proposed   spectrum filtering based methods that utilize  Chebyshev polynomials and Cayley polynomials, respectively. \n\\citet{thomas2017semi} simplified this spectrum method and obtained a filter in the vertex domain, which achieves   state-of-the-art performance.\nOther researchers~\\citep{diffusion, monti2017geometric} worked on designing feature propagation models in the vertex domain for graph CNNs.\n\\citet{yang2016revisiting, dai2016discriminative, grover2016node2vec, du2016convergence}  study transforming graph-structured  data to embedding vectors for  learning problems.\nMore recently, \\citep{Bengio18} proposed graph attention networks leveraging masked self-attentional layers to address the approximation of exiting graph convolutions networks.\nNevertheless, it still remains open how to extend CNNs from grid-structured data to arbitrary graph-structured data with local feature extraction capability.\n\nWe define rigorously the graph convolution operation on the vertex domain as multiplication by polynomials of the graph adjacency matrix, which is consistent with the notion of convolution in graph signal processing.\nIn graph signal processing \\cite{sandryhaila2013discrete}, polynomials of the adjacency matrix are graph filters, extending to graph based data from the usual concept of filters in traditional time or image based signal processing. Thus, comparing ours with existing work on graph CNNs, our paper provides a solid theoretical foundation for our proposed convolution step instead of an ad-hoc approach to convolution in CNNs for graph structured data. \n\n{Further, our method avoids computing the spectrum of the graph Laplacian as in \\citet{bruna2013spectral}, or approximating the spectrum using high degree Chebyshev polynomials of the graph Laplacian matrix (in \\citet{defferrard2016convolutional}, it is suggested that one needs a $25^{\\textrm{th}}$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum) or using high degree Cayley polynomials of the graph Laplacian matrix (in \\citet{levie2017cayleynets}, $12^{\\textrm{th}}$ degree Cayley polynomials are needed). We also clarify that the GCN method in \\citet{thomas2017semi} is a first order approximation of the Chebyshev polynomials approximation in \\citet{defferrard2016convolutional}, which is very different from our method. Our method has a much lower computational complexity than the spectrum based methods, since our method only uses polynomials of the adjacency matrix with maximum degree $2$ as shown in our experiments. Finally, the method that we propose exhibits better performance than existing methods as no approximation is required.}\nOur contributions are summarized as follows:\n\\begin{itemize}\n\t\\item\n\tThe proposed TAGCN explores a general $K$-localized filter for graph convolution in the vertex domain to extract  local features on a set of size-$1$ up to size-$K$ receptive fields.\n\tThe topologies of these filters are adaptive to the topology of the graph  as they scan  the graph to perform convolution.\n\tIt replaces the fixed   square filters in traditional CNNs for the  grid-structured input data volumes in  traditional CNNs.\n\tThus, the convolution  that we define in the convolution step for the vertex domain is consistent with convolution in traditional  CNNs.\n\t\\item\n\t{We analyze the mechanisms of the graph convolutional layers and prove that if only a size-k filter is used, as the convolutional layers go deeper under certain condition, the output of the last convolutional layer is the projection of the output of the first convolutional layer  along the eigenvector corresponding to the  eigenvalue of the graph adjacency matrix with the largest amplitude.\n\t\tThis linear approximation leads to information loss and classification accuracy degradation.\n\t\tIn contrast, using a set of size-1 up to size-K filters (as in our TAGCN) can avoid the linear approximation and increases the representation capability. Therefore, it leads to improved classification accuracy.}\n\t\\item\n\t{TAGCN  is  consistent with  the convolution  in graph signal processing.\n\t\tIt  applies to both directed and undirected graphs.\n\t\tMoreover, it has a much lower computational complexity compared with recent methods  since it only needs polynomials of the adjacency matrix with maximum degree $2$ compared with the  $25^{\\textrm{th}}$ and $12^{\\textrm{th}}$ degree Laplacian matrix polynomials  in \\citet{defferrard2016convolutional} and \\citet{levie2017cayleynets}.  }\n\t\n\t\\item\n\t{As no approximation to the convolution is needed in TAGCN, it achieves better performance compared with existing methods.}\n\tWe contrast TAGCN with  recently proposed  graph CNN  including both   spectrum filtering methods \\citep{bruna2013spectral,defferrard2016convolutional} and vertex domain propagation methods \\citep{thomas2017semi,monti2017geometric,diffusion},\n\tevaluating their performances on three commonly used graph-structured data sets.\n\tOur experimental tests show that   TAGCN consistently achieves superior performance on  all these data sets.\n\\end{itemize}\n\n\\section{Convolution on Graph}\nWe use boldface uppercase and lowercase letters to represent matrices and vectors, respectively.\nThe information and their relationship on a graph $\\mathcal{G}$ can be represented by\n$\\mathcal{G} = (\\mathcal V, \\mathcal {E}, \\bar{\\textbf A})$, where $\\mathcal V$ is the set of vertices, $\\mathcal E$ is the set of edges, and $\\bar{\\textbf A}$ is the weighted  adjacency matrix of the graph; the graph can be weighted or unweighted, directed or undirected.\nWe assume there is no isolated vertex in $\\mathcal G$.\nIf $\\mathcal{G}$ is a \\emph{directed weighted} graph, the weight $\\bar{\\textbf A}_{n,m}$ is on the directed edge from vertex $m$ to $n$.\nThe entry  $\\bar{\\textbf A}_{n,m}$ reveals the dependency between node $n$ and $m$ and  can take arbitrary real or  complex values.\nThe  graph convolution  is general and can be adapted to graph CNNs  for particular tasks.\nIn this paper, we focus on  the vertex semisupervised learning problem,\nwhere we have access to very limited labeled vertices, and the task is to classify the remaining unlabeled vertices by feeding the output of the last convolutional layer to a fully connected layer.\n\n\n\\subsection{Graph Convolutional Layer for TAGCN}\nWithout loss of generality,  we  demonstrate   graph convolution on the $\\ell$-th hidden layer. The results apply to any other hidden layers.\nSuppose on the $\\ell$-th hidden layer, the input feature map for each vertex of the graph has $C_{\\ell}$ features.\nWe collect the $\\ell$-th hidden layer\ninput data on all vertices\nfor  the $c$-th feature by the vector  $\\textbf x^{(\\ell)}_{c}\\in \\mathbb R^{N_{\\ell}}$, where\n$c = 1, 2,\\ldots C_{\\ell}$ and $N_{\\ell}$ is the number of vertices\\footnote{Graph coarsening could be used and the number of vertices may vary for different layers.}.\nThe components of $\\textbf x^{(\\ell)}_{c}$ are indexed by vertices of the data graph representation\n$\\mathcal G=(\\mathcal V, \\mathcal {E}, \\bar{\\textbf A})$\\footnote{We use superscript $(\\ell)$ to denote data on the $\\ell$th layer and superscript $ {\\ell}$ to denote the $\\ell$-th power of a matrix.}.\nLet $\\textbf G^{(\\ell)}_{c,f}\\in \\mathbb R^{N_{\\ell}\\times N_{\\ell}}$ denote the $f$-th graph filter.\nThe graph convolution is the matrix-vector product, i.e., $\\textbf G^{(\\ell)}_{c,f}\\textbf x^{(\\ell)}_{c}$.\nThen  the $f$-th output feature map followed by a ReLU function is given by\n\\begin{equation}\\label{out_f}\n\\textbf y_f^{(\\ell)} = \\sum_{c=1}^{C_{\\ell}}\\textbf G^{(\\ell)}_{c,f}\\textbf x^{(\\ell)}_{c} +  b_f\\textbf 1_{N_{\\ell}},\n\\end{equation}\nwhere $ b_f^{(\\ell)}$ is a learnable bias, and\n$\\textbf 1_{N_{\\ell}}$ is the $N_{\\ell}$ dimension  vector of all ones.\nWe   design  $\\textbf G^{(\\ell)}_{c,f}$ such that $\\textbf G^{(\\ell)}_{c,f}\\textbf x^{(\\ell)}_{c}$ is a meaningful convolution on a graph with arbitrary topology.\n\nIn the recent theory on graph signal processing~\\citep{sandryhaila2013discrete}, the \\emph{graph shift} is defined as a local operation that replaces a graph signal at a graph vertex by a linear weighted combination of the values of the graph signal at the neighboring vertices:\n$$\\tilde{\\textbf x}^{(\\ell)}_{c} = \\bar{\\textbf A} \\textbf x^{(\\ell)}_{c}.$$\nThe graph shift $\\bar{\\textbf A}$ extends the time shift in traditional signal processing to graph-structured data.\nFollowing \\citet{sandryhaila2013discrete},\na graph filter $\\textbf G^{(\\ell)}_{c,f}$ is shift-invariant, i.e.,\nthe shift $\\bar{\\textbf A}$ and the filter\n$\\textbf G^{(\\ell)}_{c,f}$ commute,\n$\\bar{\\textbf A}(\\textbf G^{(\\ell)}_{c,f}\\textbf x_c^{(\\ell)}) =\n\\textbf G^{(\\ell)}_{c,f} (\\bar{\\textbf A} \\textbf x_c^{(\\ell)})$,  if under appropriate assumption $\\textbf G^{(\\ell)}_{c,f}$ is a polynomial in $\\textbf A$,\n\\begin{equation}\\label{con}\n\\textbf G^{(\\ell)}_{c,f} =  \\sum_{k = 0}^{K} g^{(\\ell)}_{c,f,k}\\textbf A^{k}.\n\\end{equation}\nIn (\\ref{con}),\nthe $g^{(\\ell)}_{c,f,k}$ are the graph filter polynomial coefficients; the quantity\n$\\textbf A = \\textbf D^{-\\frac{1}{2}}\\bar{\\textbf A}\\textbf D^{-\\frac{1}{2}}$ is the normalized adjacency matrix\nof the graph, and $\\textbf D = \\text{diag}[\\textbf d]$ with the $i$th component being $\\textbf d(i) = \\sum_j \\textbf A_{i,j}$.\\footnote{There is freedom to  normalize $\\textbf A$ in different ways; here it is assumed that $\\bar{\\textbf A}_{m,n} $ is nonnegative and the above normalization is well defined.}\nWe adopt the normalized  adjacency matrix  to guarantee that\nall the eigenvalues of $\\textbf A$  are inside  the unit circle, and therefore $\\textbf G^{(\\ell)}_{c,f}$ is computationally stable.\nThe next subsection shows we will adopt   $  1\\times C_{\\ell},   2\\times C_{\\ell},\\ldots,$ and $  K\\times C_{\\ell}$ filters sliding on the graph-structured data.\nThis fact coincides with  GoogLeNet \\citep{googlenet}, in  which a set of  filters with different sizes are used in each convolutional layer.\nFurther, It is shown in the appendix that the convolution operator defined in (\\ref{con}) is consistent with  that in\nclassical  signal processing when the graph is in the 1D cyclic  form, as shown in Fig.~\\ref{f1}.\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.4\\columnwidth, height = 1.6cm]\n\t\t{.\/1D.PNG}\n\t\\end{center}\n\t\\caption{Graph topology of a 1-D cyclic graph.}\n\t\\label{f1}\n\\end{figure}\n\n\nFollowing the CNN architecture, an additional  nonlinear operation, e.g,   rectified linear unit (ReLU) is used after every graph convolution operation:\n$$\\textbf x_{f}^{(\\ell+1)} = \\sigma\\left(\\textbf y_{f}^{(\\ell)}\\right), $$\nwhere $\\sigma(\\cdot)$ denotes the ReLU activation function  applied to the vertex values.\n\n\\subsection{Analysis of Graph Convolutional Layers}\n\n{In the following, we analyze the mechanisms of the graph convolutional layers. \n\tWe start from the graph filter with the form of a {monomial} $g_{\\ell}\\textbf A^{k_{\\ell}}$ for the $\\ell$-th layer and $C_{\\ell} = F_{\\ell} = 1$ with $\\ell =1,\\ldots L$.\n\tIn the following, we show that when the graph convolutional layers go deeper, the output of the last graph convolutional layer is proportional to the projection of the output data of the first convolutional layer along the  eigenvector corresponding to the eigenvalue of the graph adjacency matrix with the largest amplitude.\n\t\n\t\\begin{theorem}\n\t\tFor any filters $\\textbf A^{k_{\\ell}}$, with $k_{\\ell}\\in \\{1,2,3,\\ldots\\}$, \n\t\t$$\\lim_{L\\to +\\infty} \\underbrace{\\sigma\\left( \\sigma\\cdots \\sigma \\left(g_2 \\textbf A^{k_2} \\sigma\\left(g_1 \\textbf A^{k_1} \\textbf x \\right)\\right)\\right)}_{\\textrm{$L$ times $\\sigma(\\cdot)$}}\n\t\t= m\n\t\t\\\n\t\t\\langle \\textbf y_1^{(1)}, \\textbf v_1 \\rangle \\textbf v_1.\n\t\t$$\n\t\twith $m=\\prod_{\\ell=1}^{\\ell = L}g_{\\ell}$ and $\\textbf y_1^{(1)} =\\sigma\\left(g_1 \\textbf A^{k_1} \\textbf x\\right)$.\n\t\\end{theorem}\t\n\t\n\t\\noindent {\\textbf {Proof}} \\quad For any input  data  $\\textbf x\\in \\mathbb R^N$ on the graph with $N$ vertices, the output of the first graph convolutional layer is \n\t$\\textbf y_1^{(1)} = \\sigma\\left(g_1 \\textbf A^{k_1} \\textbf x\\right).$\n\tAccording to the definition of ReLU function, we know that each component in $\\textbf y_1^{(1)}$ is nonnegative. \n\tThe data feeded to the fully connected layer for classification, which is the output of  the $L$-th graph convolutional layer, is \n\t$  \\sigma\\left( \\sigma\\cdots\\left( \\sigma\\left(g_2\\textbf A^{k_2}\\textbf y_1^{(1)} \\right)\\right)\\right).  $\n\tIt can be observed that all the learned $g_i$ with $i\\geq 2$ should be positive, otherwise, the output would be $\\textbf 0$ resulting in an all-zero vector feeded to the fully connected layer for classification.\n\tFurther, since all the components of $\\textbf A^{k_{\\ell}}$ are nonnegative and $\\textbf y_1^{(1)}$ is nonnegative, by conduction,    the input of ReLU function in each layer is nonnegative. \n\tTherefore, the output of the $L$-th graph convolutional layer   can be  written equivalently as \n\t\\begin{equation}\n\t\\begin{split}\n\t&\\underbrace{\\sigma\\left( \\sigma\\cdots \\sigma \\left(g_2 \\textbf A^{k_2} \\sigma\\left(g_1 \\textbf A^{k_1} \\textbf x \\right)\\right)\\right)}_{\\textrm{$L$ times $\\sigma(\\cdot)$}}\\\\\n\t=& m\n\t\\textbf A^{\\sum_{\\ell = 2}^{L} k_{\\ell}}\\textbf y_1^{(1)}\\\\\n\t{\\overset{(a)}{=}} & m\n\t\\textbf V\\textbf J^{\\sum_{\\ell = 2}^{L} k_{\\ell}}\\textbf V^{-1} \\textbf y_1^{(1)} \\\\\n\t{\\overset{(b)}{=}}&m\n\t\\textbf V\\textbf J^{\\sum_{\\ell = 2}^{L} k_{\\ell}}\\textbf V^{-1}\\left( c_1\\textbf v_1 + c_2\\textbf v_2\\ldots c_N\\textbf v_N\\right)\\\\\n\t{\\overset{(c)}{=}}& m\n\t\\textbf V\\textbf J^{\\sum_{\\ell = 2}^{L} k_{\\ell}}\\left(c_1\\textbf e_1+c_2\\textbf e_2\\ldots+c_N\\textbf e_N\\right).\n\t\\end{split}\n\t\\end{equation}\n\tIn (a), we use eigendecomposition of $\\textbf A$, where $\\textbf V=[\\textbf v_1, \\ldots, \\textbf v_N]$ with $\\textbf v_i$ the eigenvector of $\\textbf A$, and $\\textbf J$ is a diagonal matrix with diaonal elements being the eigenvalues of $\\textbf A$.\\footnote{When $\\textbf A$ is asymmetric and  rank deficient, Jordan decomposition is adopted to replace eigendecomposition, and then $\\textbf J$ is a block diagonal matrix. The remaining analysis also applies for the Jordan decomposition.}\n\tEquation (b) is due to the fact that \n\tthe set of  eigenvectors $\\left\\{\\textbf v_i\\right\\}_{i=1}^N$ form an orthogonal basis, and one can express $\\textbf  y_1^{(1)}\\in \\mathbb R^N$ by a linear combination of    those vectors, with $c_i = \\langle \\textbf y_1^{(1)}, \\textbf v_i \\rangle$.\n\tIn (c), $\\left\\{\\textbf e_i\\right\\}_{i=1}^{N}$ is the standard basis.\n\t\n\tWithout loss of generality,  the graph is assumed to be strongly connected, and  we have the unique largest eigenvalue. Then,  following the definition of   $\\textbf A$, we have \n\t$\\textbf J =\\textrm{diag}\\left([1, \\lambda_2, \\ldots, \\lambda_N] \\right)$  with $|\\lambda_k|<1$ for all $k>2$. Then we obtain \n\t\\begin{equation}\n\t\\begin{split}\n\t&\\lim_{L\\to +\\infty}\\textbf V\\textbf J^{\\sum_{\\ell = 2}^{L} k_{\\ell}}\\left(c_1\\textbf e_1+c_2\\textbf e_2\\ldots+c_N\\textbf e_N\\right)\\\\\n\t&=\n\tc_1\\textbf v_1\n\t= \\langle \\textbf y_1^{(1)}, \\textbf v_1 \\rangle \\textbf v_1.\n\t\\end{split}\n\t\\end{equation} \\QEDA\n\t\n\tNote that  when $k_{\\ell}=1$ for all $\\ell\\in \\{1,2,\\ldots L\\}$, the graph convolutional filter reduces to $g_{\\ell}\\textbf A$ which is used in \\cite{thomas2017semi}.\n\tDue to the linear approximation (projection along the  eigenvector corresponding to the largest eigenvalue amplitude), the information loss would degrade the classification accuracy.\n\tHowever, if we choose the graph filter as a set of filters from size-1 to size-$K$, it is not a projection anymore, and \n\tthe representation capability of graph convolutional layers is improved. }\n\n\n\n\\subsection{Filter Design for TAGCN Convolutional Layers}\n{{ In this section, we would like to understand the proposed convolution\n\t\tas a feature extraction operator in traditional CNN rather than as embedding propagation. Taking this point of view helps us to profit from the design knowledge\/experience from traditional CNN and apply it to grid structured data. Our definition of weight of a path and the following filter size for graph convolution in this section make it possible to design a graph CNN architecture similar to GoogLeNet \\citep{googlenet}, in which a set of filters with different sizes are used in each convolutional layer. In fact, we found that a combination of size 1 and size 2 filters gives the best performance in all three data sets studied, which is a polynomial with maximum order 2. }\n\t\n\t\n\tIn traditional CNN,  a $K\\times K\\times C_{\\ell}$  filter scans over the input grid-structured data  for feature extraction.\n\tFor image classification problems, the value  $K$ varies for different  CNN architectures and tasks to achieve  better performance.\n\tFor example, in VGG-Verydeep-16 CNN model \\citep{vgg}, only $3\\times 3\\times C_{\\ell}$  filters are used; in ImageNet CNN model \\citep{imagenet}, $11\\times 11\\times C_{\\ell}$ filters are adopted; and in GoogLeNet \\citep{googlenet}, rather than using the same size  filter in each convolutional layer,  different size filters, for example, $1\\times 1\\times C_{\\ell}$, $3\\times 3\\times C_{\\ell}$ and $5\\times 5\\times C_{\\ell}$ filters, are concatenated in each convolution layer.\n\tSimilarly,  we propose a general $K$-localized filter for graph CNN.\n\t\n\tFor a graph-structured data, we cannot\n\tuse a square filter window since the graph topology is no longer a grid.\n\tIn the following, we demonstrate that the convolution operation $ \\textbf G^{(\\ell)}_{c,f}\\textbf x^{(\\ell)}_{c}$ with $\\textbf G^{(\\ell)}_{c,f}$ a polynomial filter\n\t$\\textbf G^{(\\ell)}_{c,f} =  \\sum_{k = 0}^{K} g^{(\\ell)}_{c,f,k}\\textbf A^{k}$   is equivalent to using a  set of filters with filter size  from $1$ up to $K$.\n\tEach $k$-size filter, which is used for local feature extraction on the graph, is\n\t$k$-localized in the vertex domain.\n\t\n\tDefine a \\emph{path} of length $m$ on a graph $\\mathcal G$ as a sequence $v = (v_0,v_1,...,v_m)$ of vertices $v_k \\in \\mathcal V $  such that\n\teach step of the path $(v_k,v_{k+1})$ corresponds to an (directed) edge of the graph, i.e., $(v_k,v_{k+1}) \\in \\mathcal {E}$.\n\tHere one path may\n\tvisit the same vertex or cross the same edge multiple times.\n\tThe following adjacency matrix $\\textbf A $ is one such   example:\n\t\\[\\textbf A =\n\t\\begin{bmatrix}\n\t\\begin{smallmatrix}\n\t0      & 1   & 0 & 2&3&0&0&\\cdots \\\\\n\t1     & 0   & 4 & 5&0&0&0&\\cdots   \\\\\n\t0     & 1   & 0 & 0&0&0&1&\\cdots   \\\\\n\t1     & 1   & 0 & 0&6&0&0&\\cdots   \\\\\n\t1     &0   & 0 & 1&0&1&0&\\cdots\\\\\n\t0     &0   & 0 & 0&1&0&0&\\cdots\\\\\n\t0     &0   & 0 & 1&0&0&0&\\cdots\\\\\n\t\\vdots     &\\vdots    & \\vdots  & \\vdots &\\vdots &\\vdots &\\vdots &\\ddots\n\t\\end{smallmatrix}\n\t\\end{bmatrix}.\\]\n\tSince $\\textbf A$ is asymmetric, it represents a directed graph, given  in  Fig. \\ref{f2}.\n\tIn this example, there are $6$ different length $3$-paths on the graph from vertex $2$ to vertex $1$, namely, $(2,1,4,1)$, $(2,1,2,1)$,  $(2,1,5,1)$, $(2,3,2,1)$, $(2,4,2,1)$, and $(2,4,5,1)$.\n\t\n\tWe further define the\n\t\\emph{weight of a path} to be the product of the edge weights along the path, i.e.,\n\t$\\phi( p_{0,m}) = \\prod_{k=1}^{m}\\textbf A_{v_{k-1},v_k}$, where $p_{0,m}=(v_0, v_1,\\ldots v_m)$.\n\tFor example,\n\tthe weight of the path $(2,1,4,1)$ is $1\\times 1\\times 2=2$.\n\tThen, the $(i,j)$th entry of $\\textbf A^k$ in (\\ref{con}), denoted by $\\omega (p_{j,i}^k) $, can be interpreted as\n\tthe sum of the weights of  all the length-$k$ paths from $j$ to $i$, which is $$\\omega (p_{j,i}^k) = \\sum_{j\\in \\left\\{\\tilde j|\\tilde j   \\textrm{ is $k$ paths to $i$}\\right\\} }\\phi( p_{j,i}).$$\n\tIn the above example, it can be easily verified that\n\t$\\textbf A^3_{1,2} = 18$ by summing up\n\tthe weights of all the above six paths  from vertex $2$ to vertex $1$ with length $3$.\n\tThen, the $i$th component of\n\t$\\textbf A^k\\textbf x^{(\\ell)}_{c}$\n\tis the weighted sum of the input features of each vertex  $\\textbf x^{(\\ell)}_{c}$ that are length-$k$ paths  away to vertex $i$.\n\tHere,\n\t$k$ is defined as the\n\t\\emph{filter size}.\n\tThe output feature map is a vector with each component  given by the size-$k$ filter sliding on the graph following a fixed order of the vertex indices.\n\t\n\tThe output at the $i$-th component\n\tcan be written explicitly as\n\t$\n\t\\sum_{c=1}^{C_{\\ell}}\\sum_{j}\n\tg^{(\\ell)}_{c,f,k}\\omega (p_{j,i}^k) \\textbf x^{(\\ell)}_{c}(j)$.\n\tThis weighted sum is similar to the dot product for convolution for a grid-structured data in traditional CNNs.\n\tFinnaly, the output feature map is a weighted sum of convolution results from filters with different sizes, which is\n\t\\begin{equation}\\label{tagcn}\n\t\\textbf y_f^{(\\ell)}(i) = \\sum_{k=1}^{K_{\\ell}} \\sum_{c=1}^{C_{\\ell}}\\sum_{j\\in \\{\\tilde j|\\tilde j   \\textrm{ is $k$ paths to $i$}\\} }\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\tg^{(\\ell)}_{c,f,k}\\omega (p_{j,i}^k) \\textbf x^{(\\ell)}_{c}(j) +  b_f\\textbf 1_{N_{\\ell}}.\n\t\\end{equation}\n\tThe above equation shows that\n\teach neuron in the graph convolutional layer is connected only to a local region (local vertices and edges) in the vertex domain of the input data volume, which is adaptive to the graph topology.\n\tThe strength of correlation   is explicitly utilized in  $\\omega (p_{j,i}^k)$.\n\tWe refer to this method as topology adaptive graph convolutional network (TAGCN).\n\t\n\t\n\tIn Fig.~\\ref{f2}, we show TAGCN with an example of $2$-size filter sliding from vertex $1$ (figure on the left-hand-side) to vertex $2$ (figure on the right-hand-side).\n\tThe filter is first placed at vertex $1$. Since paths  $(1,2,1)$ $(5,4,1)$ and so on (paths with red glow) are all $2$-length paths to vertex $1$, they are covered by this $2$-size filter. Since paths $(2,3)$ and  $(7,3)$ are not on any $2$-length path to vertex $1$, they are not covered by this filter.\n\tFurther, when this 2-size filter moves to vertex $2$, paths $(1,5)$, $(4,5)$ and $(6,5)$ are no longer covered, but paths $(2,3)$ and $(7,3)$ are first time covered and contribute to the convolution with output at vertex $2$.\n\t\n\tFurther, $\\textbf y_f^{(\\ell)}(i)$\n\tis the weighted sum of the input features of vertices in  $\\textbf x^{(\\ell)}_{c}$ that are within $k$-paths  away to vertex $i$, for $k=0, 1,\\ldots K$, with weights given by the products of components of $\\textbf A^k$ and $g^{(\\ell)}_{c,f,k}$.\n\tThus the output is the weighted sum of the feature map given by the filtered results from $1$-size up to $K$-size filters.\n\tIt is evident  that the vertex convolution on the graph using $K$th order polynomials is $K$-paths localized.\n\tMoreover, different vertices on the graph share $g^{(\\ell)}_{c,f,k}$.\n\tThe above local convolution and weight sharing properties  of the convolution  (\\ref{tagcn}) on a graph  are very similar to those in traditional CNN.\n\t\n\t{Though the convolution operator defined in (\\ref{con}) is  defined on the vertex domain, it can also be understood as a filter  in the spectrum domain, and it is consistent with the definition of convolution in graph signal processing. We provide detailed discussion in the appendix.}\n\n\n\n\n\n\n\n\n\t\n\t\\begin{figure}[t]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=0.8\\columnwidth, height =5.5cm]\n\t\t\t{.\/Conv.PNG}\n\t\t\\end{center}\n\t\t\\caption{An example of a directed graph with weights along directed edges corresponding to $\\textbf A$.\t\n\t\t\tThe parts with  glow on left(right)-hand-side represent filters at different locations.\n\t\t\tThe figure on the left-hand-side denotes the filtering\/convolution starting from vertex $1$, then the filter slides  to vertex $2$  as shown on the right-hand-side with filter topology adaptive to the new local region.\n\t\t\n\t\t\n\t\t}\n\t\t\\label{f2}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\section{Relation with Other Existing Formulations}\n\n\tIn general, there are two types of graph convolution operators for the CNN architecture.\n\tOne   defines the convolution  in the spectrum domain, whose output feature map is the multiplication of the inverse Fourier transform matrix with the filtered results in the spectrum domain \\citep{bruna2013spectral, defferrard2016convolutional, levie2017cayleynets}.\n\tBy doing further approximations based on this spectrum domain operator, a simplified convolution was obtained in~\\citet{thomas2017semi}.\n\tThe other defines convolution by a feature propagation model   in the vertex domain such as MoNet in ~\\citet{monti2017geometric} and the diffusion CNN (DCNN) in~\\cite{diffusion}.\n\tWe investigate in detail  each alternative.\n\t\n\t\n\tIn \\citet{bruna2013spectral,defferrard2016convolutional, levie2017cayleynets}, the convolution operation was defined using the convolution theorem and  filtering operation in the spectrum domain by computing the eigendecomposition of the normalized Laplacian matrix of the graph.\n\tThe Laplacian matrix $\\textbf L$ is defined as\n\t$\\textbf L = \\textbf D - \\textbf A$ with the further assumption that $\\textbf A$ is symmetric to guarantee that $\\textbf L$ is positive semi-definite.\n\tThe convolution defined by the multiplication in the spectrum domain is approximated by \\citet{defferrard2016convolutional} by\n\t\\begin{equation}\\label{Cheby}\n\t\\textbf U g_{\\theta} \\textbf U^T\\textbf x\n\t\\approx \\sum_{k=0}^{K}\n\t\\theta_kT_k\\left[\\frac{2}{\\lambda_{\\textrm{max}}}\\textbf L - \\textbf I\\right]\n\t\\textbf x^{(\\ell)}_{c},\n\t\\end{equation}\n\twhere $T_k\\left[\\cdot\\right]$ is the $k$th order matrix Chebyshev polynomial~\\citep{shuman2013emerging} where\n\t\\begin{equation}\\label{ChePoly}\n\tT_k(\\textbf L) = 2\\textbf L T_{k-1}[\\textbf L] - T_{k-2}[\\textbf L],\n\t\\end{equation}\n\twith the initial values defined as $T_0[\\textbf L]=\\textbf I$ and $T_1\\left[\\textbf L\\right] = \\textbf L.$\n\tWe refer later to this method as ChebNet for performance comparison.\n\tNote that the Laplacian matrix can be seen as\n\ta differentiator operator.\n\tThe assumption of  symmetric  $\\textbf A$ restricts the application to undirected graphs.\n\tNote that in \\citet{defferrard2016convolutional}, Laplacian matrix polynomials with maximum order $K =25$  is needed to approximate the convolution operation on the left-hand side in (\\ref{Cheby}), which imposes the computational burden.\n\tWhile TAGCN only needs an adjacency matrix polynomials with maximum order $2$ to achieve better performance as shown in the experiment part.\n\t\n\tIn \\citet{thomas2017semi},   a graph convolutional network (GCN) was obtained by { a first order approximation of (\\ref{Cheby}).}\n\tIn particular, let $K=1$   and  make the further assumptions that  $\\lambda_{\\textrm{max}} =2$ and $\\theta_0 = \\theta_1=\\theta$.\n\tThen a simpler convolution operator that does not depend on the spectrum  knowledge is obtained as\n\t\\begin{equation}\\label{Cheby2}\n\t\\textbf U g_{\\theta} \\textbf U^T\\textbf x\n\t\\approx \\sum_{k=0}^{1}\n\t\\theta_kT_k\\!\\!\\left[\\frac{2\\textbf L}{\\lambda_{\\textrm{max}}} \\!- \\textbf I\\right]\\!\\!\n\t\\textbf x^{(\\ell)}_{c}\n\t\\!\\approx\n\t\\theta(\\textbf I + \\textbf D^{-\\frac{1}{2}}\n\t\\textbf A \\textbf D^{-\\frac{1}{2}} )\\textbf x_c^{(\\ell)}.\\nonumber\n\t\\end{equation}\n\tNote that $\\textbf I + \\textbf D^{-\\frac{1}{2}}\n\t\\textbf A \\textbf D^{-\\frac{1}{2}}$ is a matrix with eigenvalues in $[0,2]$.\n\tA renormalization trick is adopted here by letting\n\t$\\widetilde{\\textbf A} = \\textbf A + \\textbf I$ and $\\widetilde{\\textbf D}_{i,i} =  \\sum_{j}\\widetilde{\\textbf A}_{i,j}$.\n\tFinally, the convolutional operator is approximated by\n\t\\begin{equation}\\label{Cheby22}\n\t\\textbf U g_{\\theta} \\textbf U^T\\textbf x\n\t\\approx\n\t\\theta \\widetilde{\\textbf D}^{-\\frac{1}{2}}\n\t\\widetilde{\\textbf A} \\widetilde{\\textbf D}^{-\\frac{1}{2}}\n\t\\textbf x_c^{(\\ell)}= \\theta \\widehat{\\textbf A},\n\t\\end{equation}\n\twhere $ \\widehat{\\textbf A} =  \\widetilde{\\textbf D}^{-\\frac{1}{2}}\n\t\\widetilde{\\textbf A} \\widetilde{\\textbf D}^{-\\frac{1}{2}} $.\n\tIt is interesting to observe that this method though obtained by simplifying  the spectrum method has a better performance than  the spectrum method \\citep{defferrard2016convolutional}.\n\tThe reason may be because the simplified form is equivalent to propagating vertex features on the graph, which can be seen as a special case of our TAGCN method, though there are other important  differences.\n\t\n\tAs we analyzed in Section 2.2,  GCN  as in (\\ref{Cheby22}) or even extending it to higher order, i.e., $\\theta \\widehat{\\textbf A}^k$ only project the input data to the graph eigenvector of the largest eigenvalue when the convolutional layers go deeper.\n\t\n\t\n\t\n\t\n\t\n\n\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\t\n\t\n\t\n\t{ Our TAGCN is able to leverage information at a farther distance, but it is not a simple extension of GCN \\citet{thomas2017semi}.\n\t\tFirst,  the graph convolution in GCN is defined as a first order Chebyshev polynomial of the graph Laplacian matrix, which is an approximation to the graph convolution defined in the spectrum domain in \\citet{defferrard2016convolutional}.\n\t\tIn contrast, our graph convolution is rigorously defined as multiplication by polynomials of the graph adjacency matrix; this is not an approximation, rather, it simply is filtering with graph filters as defined and as being consistent with graph signal processing.}\t\n\t\n\t{Next, we show the difference between our work and the GCN method in \\citet{thomas2017semi} when using 2nd order (K=2, 2 steps away from the central node) Chebyshev polynomials of Laplacian matrix.\n\t\tIn the GCN paper \\citet{thomas2017semi}, it has been shown that $\\sum_{k=0}^{1} \\theta_k T_k(\\textbf L) \\approx \\widehat{\\textbf A}$ as repeated in (\\ref{Cheby22}), and $T_2[\\textbf L] =2\\textbf L^2$\n\t\tby the definition of Chebyshev polynomial. Then, extending GCN to the second order Chebyshev polynomials (two steps away from a central node) can be obtained from the original definition in T. Kipf's GCN \\citep[eqn (5)]{thomas2017semi} as $\\sum_{k=0}^{2} \\theta T_k(L)= \\widehat{\\textbf A}   + 2\\textbf L^2 -\\textbf I$, which is different from our definition as in (\\ref{con}). Thus, it is evident that our method is not a simple extension of GCN. We apply graph convolution as proposed from basic principles in the graph signal processing, with no approximations involved, while  both GCN in \\citet{thomas2017semi} and \\citet{defferrard2016convolutional} \\citet{levie2017cayleynets} are based on  approximating the convolution defined  in the spectrum domain.\n\t\tIn our approach, the degree of freedom is the design of the graph filter-its degree and its coefficients. Ours is a principled approach and provides a generic methodology. The performance gains we obtain are the result of capturing the underlying graph structure with no approximation in the convolution operation.   }\n\t\n\t\n\t\n\t\\citet{simonovsky2017dynamic} proposed the edge convolution network (ECC)   to extend the convolution operator from regular grids to arbitrary graphs.\n\tThe convolution operator is defined similarly to (\\ref{Cheby2}) as\n\t\\begin{equation}\n\t\\textbf y_f^{(\\ell)}(i)\n\t=\n\t\\sum_{j\\in \\mathcal N(i)}  \\frac{1}{\\left|\\mathcal {N}(i)\\right|} \\bm\\Theta^{(\\ell)}_{j,i}\\textbf x^{(\\ell)}_{c}(j) + b_{f}^{(\\ell)},\\nonumber\n\t\\end{equation}\n\twith $\\bm\\Theta^{(\\ell)}_{j,i}$ is the weight matrix that needs to be learned.\n\t\n\t\n\t\n\tA mixture model network (MoNet) was proposed in  \\cite{monti2017geometric}, with convolution  defined as\n\t\\begin{equation}\n\t\\textbf y^{(\\ell)}_{f} (i)\n\t=\n\t\\sum_{f=1}^F\\sum_{j\\in \\mathcal N(i)}\n\tg_f \\kappa_f\n\t\\textbf x^{(\\ell)}_{c}(j),\\nonumber\n\t\\end{equation}\n\t\\vspace{-0.1cm}where\n\t$\\kappa_f$ is a Gaussian kernel with\n\t$\\kappa_f = \\exp\\left\\{-\\frac{1}{2}(\\textbf u - \\bm\\mu_f)^T\\mathbf\\Sigma_f^{-1}(\\textbf  u - \\bm\\mu_f)\\right\\}$\n\tand\n\t$g_f$ is the weight  coefficient for each Gaussian kernel $\\kappa_f$.\n\tIt is further assumed that $\\mathbf\\Sigma_f$ is a $2\\times 2$ diagonal matrix.\n\t\n\tGCN, ECC, and MoNet  all  design a propagation model on the graph; their\n\tdifferences are on the weightings used by each model.\n\t\n\t\n\n\t\n\t\n\t\n\t\\cite{diffusion} proposed a diffusion CNN (DCNN) method that considers a diffusion process over the graph.\n\tThe transition probability of a random walk on a graph is given by $\\textbf P =\\textbf D^{-1}\\textbf A$, which is equivalent to the normalized  adjacency matrix.\n\t\\begin{equation}\n\t\\textbf y_{c,f}^{(\\ell)} = \\textbf g^{(\\ell)}_{c,f}\\textbf P^f\\textbf x^{(\\ell)}_{c}.\\nonumber\n\t\\end{equation}\n\n\t\n\tBy comparing the above methods with TAGCN in (\\ref{tagcn}), it can be concluded that\n\tGCN, ECC and MoNet can be seen as a special case of TAGCN because in (\\ref{tagcn}) the item with $k=1$  can be seen as an  information propagation term.\n\tHowever, as shown in Section~2.2, when the convolutional layers go deeper, the output of the last convolutional layer  of GCN, ECC and MoNet are all   linear approximations of the output of the corresponding first convolutional layer which degradates the representation capability. TAGCN overcomes this  by\n\tdesigning a set of fixed size  filters that is adaptive  to the input graph topology when performing convolution on the graph.\n\tFurther,\tcompared with  existing spectrum methods \\citep{bruna2013spectral, defferrard2016convolutional, levie2017cayleynets}, TAGCN satisfies the convolution theorem as shown in the previous subsection and  implemented in the vertex domain, which avoids performing costly and practically numerical unstable eigendecompositions.\n\t\n\tWe further compare the number of weights that need to be  learned in each hidden layer for these different methods in Table 1.\n\tAs we show later,  $K=2$ is selected in our experiments\n\tusing cross validation.\n\tHowever, for ChebNet in \\citep{defferrard2016convolutional}, it is suggested that one needs a $25^{\\textrm{th}}$ degree Chebyshev polynomial to provide a good approximation to the graph Laplacian spectrum.\n\tThus we have a moderate number of weights to be learned.\n\tIn the following, we show that our method achieves the best performance for each of those commonly used graph-structured data sets.\n\t\n\t\n\t\\section{Experiments}\n\t\\label{others}\\begin{table}[t]\n\t\t\\caption{Number of weights need to be learned for the $\\ell$-th layer. }\n\t\t\\label{sample-table}\n\t\t\\begin{center} \\fontsize{9}{9}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t\\multicolumn{1}{c}{\\bf DCNN}  &\\multicolumn{1}{c}{\\bf ECC}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf ChebNet}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf GCN}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf MoNet}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf TAGCN}\n\t\t\t\t\\\\ \\hline \\\\\n\t\t\t\t$F_{\\ell}C_{\\ell}$         &$F_{\\ell}C_{\\ell}$   & $25F_{\\ell}C_{\\ell}$  &\n\t\t\t\t$ F_{\\ell}C_{\\ell}$  &$4F_{\\ell}C_{\\ell}$  &$2F_{\\ell}C_{\\ell}$\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\end{table}\n\tThe proposed TAGCN is general and can be fit to the general graph CNN architectures for different tasks. In the experiments, we focus on the vertex semisupervised learning problem, where we have access to only a few labeled vertices, and the task is to classify the remaining unlabeled vertices. To compare the performance of TAGCN with that of existing methods, we extensively evaluate TAGCN on three graph-structured datasets, including the Cora, Citesser and Pubmed datasets. The datasets split and experiments setting closely follow the standard criteria in \\cite{yang2016revisiting}.\n\t\n\t\n\n\n\tIn each data set, the vertices are the documents and the edges are the citation links. Each document is represented by sparse bag-of-words feature vectors, and the citation links between documents are provided. Detailed statistics of these three data sets are summarized in Table \\ref{data}. It shows the number of nodes and edges that corresponding to documents and citation links, and  the number of document classes in each data set. Also, the number of features at each vertex is given. Label rate denotes the number of labeled documents that are used for training divided by the total number of documents in each data set.\n\t\n\t\n\t\\label{others}\\begin{table}[t]\n\t\t\\caption{Dataset statistics, following \\cite{yang2016revisiting}}\n\t\t\\label{data}\n\t\t\\begin{center}\\fontsize{9}{9}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t\\multicolumn{1}{c}{\\bf Dataset}  &\\multicolumn{1}{c}{\\bf Nodes}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Edges}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Classes}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Features}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Label Rate}\n\t\t\t\t\\\\ \\hline \\\\\n\t\t\t\tPubmed         &19,717 & 44,338 & 3&500&0.003\\\\\n\t\t\t\tCiteseer         &3,327 & 4,732 &6 & 3,703&0.036\\\\\n\t\t\t\tCora         &2,708 & 5,429 & 7 & 1,433&0.052\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\end{table}\n\t\\begin{table}\n\t\t\\caption{Summary of results in terms of percentage classification accuracy with standard variance}\n\t\t\\label{result}\n\t\t\\begin{center}\\fontsize{9}{9}\n\t\t\t\\begin{tabular}{lllll}\n\t\t\t\t\\multicolumn{1}{c}{\\bf Dataset}  &\\multicolumn{1}{c}{\\bf Pubmed }\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Citeseer}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Cora}\n\t\t\t\n\t\t\t\t\\\\ \\hline \\\\\n\t\t\t\tDeepWalk &65.3 & 43.2 &67.2\\\\\n\t\t\t\tPlanetoid     & 77.2   &64.7 & 75.7  \\\\\n\t\t\t\tDCNN     & 73.0$\\pm$0.5    &- & 76.8$\\pm$0.6  \\\\\n\t\t\t\n\t\t\t\tChebNet    & 74.4     &69.8 & 79.5  \\\\\n\t\t\t\tGCN    & \n\t\t\t\t{{79.0}}     &{{70.3} }& 81.5 \\\\\n\t\t\t\n\t\t\t\tMoNet        & 78.81$\\pm$0.4 &-  & {{81.69$\\pm$0.5}}  \\\\\n\t\t\t\tGAT  &{79.0$\\pm$0.3}      &\\textbf{72.5$\\pm$ 0.7}   & {83.0$\\pm$0.7}      \\\\\n\t\t\t\tTAGCN (\\textrm{ours})  &\\textbf{81.1$\\pm$0.4}      &{71.4$\\pm$ 0.5}   & \\textbf{83.3$\\pm$0.7}      \\\\\n\t\t\t\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\end{table}\n\t\n\t\n\t\n\t\\subsection{Experimental Settings}\n\tWe construct a graph for each data set with nodes representing documents and undirected edges\\footnote{We use undirected graphs here as citation relationship gives positive correlation between two documents. However, in contrast with the other approaches surveyed here, the TAGCN method is not limited to  undirected graphs if directed graphs are better suited to the applications. } linking two papers if there is a citation relationship. We obtain the adjacency matrix $\\bar{\\textbf A}$ with $0$ and $1$ components and further obtain the normalized  matrix $\\textbf A$.\n\t\n\tIn the following experiments, we design a TAGCN with two hidden layers (obtained from cross validation) for the semi-supervised node classification. In each hidden layer, the proposed TAGCN is applied for convolution, followed by a ReLU activation. $16$ hidden units (filters) are designed for each hidden layer, and dropout is applied after each hidden layer. The softmax activation function is applied to the output of the second hidden layer for the final classification. For ablation study, we evaluate the performance of TAGCN with different filter sizes from 1 to 4. To investigate the performance for different number of parameters, we also design a TAGCN with $8$ filters for each hidden layer and compare its classification accuracy with all the baselines and TAGCN with $16$ filters. We train our model using Adam \\citep{Adam} with a learning rate of $0.01$ and early stopping with a window size\n\tof $45$. Hyperparameters of the networks (filter size, dropout rate, and number of hidden layers) are selected by cross validation.\n\t\n\tTo make a fair comparison, we closely follow the same split of training, validation, and testing sets as in \\cite{thomas2017semi, yang2016revisiting}, i.e.,\n\t$500$ labeled examples for hyperparameters  (filter size, dropout rate, and number of hidden layers) optimization  and cross-entropy error is used for  classification accuracy   evaluation.\n\tThe performance results of the proposed TAGCN method are an  average over $100$ runs.\n\t\n\t\\subsection{Quantitative Evaluations}\n\t\n\tWe compare the classification accuracy with other recently proposed graph CNN methods as well as a graph embedding methods known as DeepWalk and Planetoid \\cite{perozzi2014deepwalk, yang2016revisiting}. \n\tThe  recent published graph attention networks (GAT) \\cite{Bengio18} leveraging masked self-attentional layers is also compared.\n\t\n\tThe quantitative results are summarized in Table~\\ref{result}. Reported numbers denote classification accuracy in percentage. Results for DeepWalk, Planetoid, GCN, and ChebNet are taken from \\citet{thomas2017semi}, and results for DCNN and MoNet are taken from \\citet{monti2017geometric}. All the experiments for different methods are based on the same data statistics shown in Table 2. The datasets split and experiments settings closely follow the standard criteria in \\cite{yang2016revisiting,thomas2017semi}. Table~\\ref{result} shows that our method outperforms all the recent state-of-the-art graph CNN methods (DCNN, ChebNet, GCN, MoNet) by obvious margins for all the three datasets. \n\t\n\t{These experiment results \n\t\tcorroborate our analyses in Section~2.2 and Section~3 that as no approximation to the convolution is needed in TAGCN, it achieves better performance compared with spectrum approximation method such as ChebNet and GCN.\n\t\tFurther, using a set of size-1 up to size-2 filters avoids the linear approximation by the simply size-1 filter (\\textbf A in GCN \\cite{thomas2017semi}), which further verify the efficacy of the proposed TAGCN.\n\t\tCompared with the most recent GAT method \\cite{Bengio18}, our method exhibit obvious advantage for the largest dataset Pubmed.\n\t\tIt should be note that GAT suffers from storage limitation in their model and not scale well for large-scale graph as explained by the authors. }\n\t\n\tFor ablation study, we further compare the performance of different filter sizes from $K=1$ to $K=4$ in Table \\ref{order}. It shows that the performances for filter size $K=2$ are always better than that for other filter sizes. The value $K=1$ gives the worst classification accuracy. As $K=1$ the filter is a monomial, this further validates the analysis in Section 2.2 that monomial filter results in a very rough approximation.  In Table \\ref{order}, we also compare the performance of different number of filters, which reflects different number of network parameters.\n\tNote, we also choose filer size $K=2$ and filter number $F_{\\ell} =8$ that results in the same number of network parameters as that in GCN, MoNet, ECC and DCNN according to Table~1.\n\tIt shows that the classification accuracy using $8$ filters is comparable with that using  $16$ filters in each hidden layer for TAGCN. Moreover,  TAGCN with $8$ filters can still achieve higher accuracy than GCN, MoNet, ECC and DCNN methods.\n\t{ This proves that, even with a similar number of parameters or architecture, our method still exhibits superior performance than GCN. }\n\t\n\t\n\t{ As we analyzed and explained  in Section 2.2 and Section 3, TAGCN in our paper is not simply extending GCN \\citet{thomas2017semi} to $k$-th order. Nevertheless, we implement $\\textbf A^2$ and compare its performance with ours. For the data sets Pubmed, Cora, and Citeseer, the classification accuracies are $79.1 (80.8)$, $81.7(83.0)$ and $70.8 (71.2)$, where the numbers in parentheses are the results obtained with our method. Our method still achieves a noticeable performance advantage over $\\textbf A^2$; in particular, we note the significant performance gain with the Pubmed database that has the largest number of nodes among these three data sets. }\n\t\n\t\n\t\\begin{table}[t]\n\t\t\\caption{ TAGCN  classification accuracy (ACC) with different parameters}\n\t\t\\label{order}\n\t\t\\begin{center} \\fontsize{9}{9}\n\t\t\t\\begin{tabular}{llll}\n\t\t\t\t\\multicolumn{1}{c}{\\bf Data Set}  &\\multicolumn{1}{c}{\\bf Filter Size}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf Filter Number}\n\t\t\t\t&\\multicolumn{1}{c}{\\bf ACC}\n\t\t\t\n\t\t\t\t\\\\ \\hline  \\\\[-1.5ex]\n\t\t\t\tCiteseer\n\t\t\t\t&1 & 16&68.9 \\\\\n\t\t\t\t&2 & 16&\\textbf{71.4} \\\\\n\t\t\t\t&3 & 16&70.0 \\\\\n\t\t\t\t&4 & 16&69.8 \\\\\n\t\t\t\t&2 & \\textbf{8}&\\textbf{71.2} \\\\\n\t\t\t\t\\hline  \\\\[-1.5ex]\n\t\t\t\tCora\n\t\t\t\t&1 & 16&81.4 \\\\\n\t\t\t\t&2 & 16&\\textbf{83.3} \\\\\n\t\t\t\t&3 & 16&82.1 \\\\\n\t\t\t\t&4 & 16&81.8 \\\\\n\t\t\t\t&2 & \\textbf{8}&\\textbf{83.0} \\\\\n\t\t\t\t\\hline  \\\\[-1.5ex]\n\t\t\t\tPubmed\n\t\t\t\t&1& 16&79.4 \\\\\n\t\t\t\t&2 & 16&\\textbf{81.1} \\\\\n\t\t\t\t&3 & 16&80.9 \\\\\n\t\t\t\t&4 & 16&79.5 \\\\\n\t\t\t\t&2 & \\textbf{8}&\\textbf{80.8} \\\\\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\end{table}\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\n\n\n\n\n\n\t\n\t\n\t\\section{Conclusions}\n\tWe have defined a novel graph convolutional network that rearchitects the  CNN architecture for graph-structured data.\n\tThe proposed   method, known as TAGCN,  is adaptive to the graph topology as   the filter scans the   graph.\n\tFurther, TAGCN inherits properties of the convolutional layer in classical CNN, i.e., local  feature extraction  and weight sharing.\n\tOn the other hand, by the convolution theorem, TAGCN that implements in the vertex domain offers implement in the spectrum domain unifying graph CNN in both the spectrum domain and the vertex domain.\n\tTAGCN is  consistent with  convolution in graph  signal processing.\n\tThese nice properties  lead to a noticeable performance advantage  in  classification accuracy on\n\tsemi-supervised graph vertex classification problems with low computational complexity.\n\t\t\t\n\t\t\t\n\t\t\t\\section{Appendix: Spectrum response of TAGCN}\n\t\t\tIn  classical signal processing \\citep{oppenheim1999discrete}, the convolution in the time domain is equivalent to  multiplication in the spectrum domain.\n\t\t\tThis relationship is known as the convolution theorem.\n\t\t\t\\citet{sandryhaila2013discrete}  showed\n\t\t\tthat the  graph filtering defined in the vertex domain satisfies the generalized convolution theorem naturally and can also interpret spectrum filtering for both directed and undirected graphs.\n\t\t\tRecent work \\citep{bruna2013spectral, defferrard2016convolutional} used the convolution theorem for undirected graph-structured data and designed a spectrum graph filtering.\n\t\t\t\n\t\t\t\\begin{figure}[h]\n\t\t\t\t\\begin{center}\n\t\t\t\t\t\\includegraphics[width=0.4\\columnwidth, height = 1.8cm]\n\t\t\t\t\t{.\/1D.PNG}\n\t\t\t\t\\end{center}\n\t\t\t\t\\caption{Graph topology of a 1-D cyclic graph.}\n\t\t\t\t\\label{f1}\n\t\t\t\\end{figure}\n\t\t\t\n\t\t\tAssume that the  adjacency matrix $\\textbf A$ for a  graph is diagonalizable, i.e.,\n\t\t\t$\\textbf A = \\textbf F^{-1}\\textbf J \\textbf F$ with $\\textbf J$  a  diagonal matrix.\n\t\t\tThe components on the diagonal of $\\textbf J$ are eigenvalues of $\\textbf A$, and the column vectors of $\\textbf F^{-1}$ are the  right   eigenvectors of $\\textbf A$; the  row vectors of $\\textbf F$ are the  left   eigenvectors of $\\textbf A$ \\footnote{\n\t\t\t\tWhen $\\textbf A$ is not diagonalizable,\n\t\t\t\tthe columns of $\\textbf F^{-1}$ and the rows  of $\\textbf F$ are the generalized right and left eigenvectors of $\\textbf A$, respectively.\n\t\t\t\tIn this case, $\\textbf F$ is no longer a unitary matrix. Also, $\\textbf J$ is a block diagonal matrix; it is then in Jordan form. Interested readers may refer to \\citet{sandryhaila2013discrete} or \\citet{Deri} and references therein.}.\n\t\t\tBy diagonalizing $\\textbf A$ in (\\ref{con}) for TAGCN, we obtain\n\t\t\t\\begin{equation} \\label{generalization}\n\t\t\t\\textbf G^{(\\ell)}_{c,f} \\textbf x_c^{(\\ell)}=\n\t\t\t\\textbf F^{-1} \\left( \\sum_{k = 0}^{K} g^{(\\ell)}_{c,f,k} \\textbf J^k \\right)\\textbf F \\textbf x_c^{(\\ell)}.\n\t\t\t\\end{equation}\n\t\t\tThe expression on the left-hand-side of the above equation represents the filtering\/convolution on the vertex domain.\n\t\t\tMatrix $\\textbf F$   defines the graph Fourier transform \\citep{sandryhaila2013discrete, SPM}, and $ \\textbf F \\textbf x_c^{(\\ell)}$\n\t\t\tis the  input feature spectrum map, which is a linear mapping from the input feature on the vertex domain to the spectrum domain.\n\t\t\tThe polynomial $ \\sum_{k = 0}^{K} g^{(\\ell)}_{c,f,k} \\textbf J^k$ is the spectrum of the\n\t\t\tgraph filter.\n\t\t\tRelation   (\\ref{generalization}), which is equation (27) in \\citet{sandryhaila2013discrete} generalizes   the classical convolution\n\t\t\ttheorem to graph-structured data: convolution\/filtering on the vertex domain becomes multiplication in the spectrum domain.\n\t\t\tWhen the graph is in the 1D cyclic  form, as shown in Fig. \\ref{f1}, the corresponding adjacency matrix is of the form\n\t\t\t\\[\\textbf A =\n\t\t\t\\begin{bmatrix}\n\t\t\t&    &  & 1 \\\\\n\t\t\t1        &     &   &   \\\\\n\t\t\t&\\ddots &  &  \\\\\n\t\t\t&    &  1 &\n\t\t\t\\end{bmatrix}.\\]\n\t\t\tThe eigendecomposition of $\\textbf A$\n\t\t\tis\n\t\t\t$$\\textbf A =\n\t\t\t\\frac{1}{N} \\textrm{DFT}^{-1}\n\t\t\t\\begin{bmatrix}\n\t\t\t& e^{-j\\frac{2\\pi 0}{N}}     &  \\\\\n\t\t\t&\\ddots     &      \\\\\n\t\t\t&    &   e^{-j\\frac{2\\pi (N-1)}{N}}\n\t\t\t\\end{bmatrix} \\textrm{DFT},\n\t\t\t$$\n\t\t\twhere $\\textrm{DFT}$ is the discrete Fourier transform matrix.\n\t\t\tThe convolution operator defined in (\\ref{con}) is consistent with  that in\n\t\t\tclassical  signal processing.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Physics context and motivation for quantitative analysis}\n\n\nUnderstanding the dependence of material properties of continuous media on frequency is a natural and practically relevant task, stemming from the theoretical and experimental studies of ``metamaterials'', {\\it e.g.} materials that exhibit negative refraction of propagating wave packets. Indeed, it was noted as early as in the pioneering work \\cite{Veselago}, that negative refraction is only possible under the assumption of frequency dispersion, {\\it i.e.} when the material parameters (permittivity and permeability in electromagnetism, elastic moduli and mass density in acoustics) are not only frequency-dependent, but also become negative in certain frequency bands.\n\nIndependently of the search for metamaterials, in the course of the development of the theory of electromagnetism, it has transpired in modern physics that the Maxwell equations need to be considered with time-nonlocal ``memory'' terms, see {\\it e.g.} \\cite[Section 7.10]{Jackson} and also \\cite{Cessenat}, \\cite{Tip_1998}. The related generalised system (in the absence of charges and currents in the domain of interest) has the form\n\\begin{equation}\n\\rho\\partial_tu+\\int_{-\\infty}^ta(t-\\tau)u(\\tau)d\\tau+{\\rm i}Au=0,\\qquad A=\\left(\\begin{array}{cc}0 & {\\rm i}\\,{\\rm curl}\\\\[0.3em] {\\rm -i}\\,{\\rm curl}& 0\\end{array}\\right),\n\\label{gen_Maxwell}\n\\end{equation}\n\\noindent where $u$ represents the (time-dependent) electromagnetic field $(H, E)^\\top$, the matrix $\\rho$ depends on the electric permittivity and magnetic permeability,\nand $a$ is a matrix-valued ``susceptibility\" operator, set to zero in the more basic form of the system.\\footnote{From the rigorous operator-theoretic point of view, $A$ in (\\ref{gen_Maxwell}) is treated as a self-adjoint operator in a Hilbert space $\\mathbb H$ of functions of $x\\in\\Omega,$ for example $\\mathbb H=L^2(\\Omega; {\\mathbb R}^6),$ where $\\Omega$ is the part of the space occupied by the medium.}\n\n\nApplying the Fourier transform in time $t$ to (\\ref{gen_Maxwell}), an equation in the frequency domain is obtained:\n\\begin{equation}\n\\bigl(i\\omega\\rho+\\widehat{a}(\\omega)\\bigr)\\widehat{u}(\\cdot,\\omega)+iA\\widehat{u}(\\cdot, \\omega)=0,\n\\label{frequency_dep}\n\\end{equation}\n\\noindent where $\\widehat{u}$ is the Fourier transform of $u,$ and $\\omega$ is the frequency.\nEquation (\\ref{frequency_dep}) is often interpreted as a ``non-classical'' version of Maxwell's system of equations, where the permittivity and\/or permeability are frequency-dependent. The existence of such media (commonly known as Lorentz materials) and the analysis of their properties go back a few decades in time and has also attracted considerable interest quite recently, {\\it e.g.} in the study of plasma in tokamaks, see \\cite{Weder} and references therein.\n\n\n\nSimultaneously with the above developments in the physics literature, recent mathematical evidence, see \\cite{Jikov}, \\cite{Bullshitte}, suggests that  such novel material behaviour, which is incompatible (see \\cite{Birman,ChKisYe_PDE,ChKisYe}) with the mathematical assumption of uniform ellipticity of the corresponding differential operators\n(such as $A$ in (\\ref{gen_Maxwell})), may be explained by means of the asymptotic analysis (``homogenisation'') of operator families with rapidly oscillating, and non-uniformly elliptic, coefficients.\n\nIt is therefore reasonable to ask the question of whether frequency dispersion laws such as pertaining to (\\ref{frequency_dep}), which in turn may provide one with metamaterial behaviour in appropriate frequency intervals \\cite{Veselago}, can be derived by some process of homogenisation of composite media with contrast (or, as we shall suggest below, any other miscroscopic degeneracies resonating with the macroscopic wavefields).\n\n\\subsection{Basis for the mathematical framework} If one were to look for an asymptotic expansion of eigenmodes of a high-contrast composite,  {\\it restricted} to the soft component of the medium, one would notice (see, e.g., \\cite{CherKis}) that their leading order terms can be understood as the eigenmodes of boundary-value problems with impedance (i.e., frequency dependent) boundary conditions. Such problems have been considered in the past (see, e.g., \\cite{PavlovFaddeev}), motivated by the analysis of the wave equation. On the other hand, by the celebrated analysis of the so-called generalised resolvents of \\cite{Naimark,Naimark1943} one knows, that a problem of this type admits a conservative dilation, which is constructed by adding the hidden degrees of freedom. In fact, precisely this latter observation has been used in \\cite{Figotin_Schenker_2005,Figotin_Schenker_2007b} in devising a conservative ``extension'' of a time-dispersive system of the type \\eqref{gen_Maxwell}. The substance of the argument that is proposed in the present paper is that the aforementioned conservative dilation is in fact precisely the asymptotic model of the original high-contrast composite. Furthermore, the leading order terms of its eigenmodes restricted to the {\\it stiff} component are solutions to a problem of the type \\eqref{frequency_dep} with frequency dispersion. They can be easily expressed in terms of the above impedance boundary value problems, thus yielding an explicit description of the link between the resonant soft inclusions and the macroscopic time-dispersive properties.\n\nTherefore, models of continuous media with frequency-dependent effective boundary conditions can be seen as natural building blocks for media with frequency dispersion.\n\nIt is of a considerable value to relate these ideas to the earlier works \\cite{KuchmentZeng, KuchmentZeng2004, Exner}, where similar limiting impedance-type problems are obtained in the spectral analysis of ``thin\" periodic structures, converging to metric graphs. Here, one obtains the aforementioned impedance setup (see Fig. \\ref{fig:exner}) on the limiting graph as the asymptotics of the eigenmodes of a Neumann Laplacian, when the ``thickness'' of the structure vanishes for one particular (resonant) scaling between the ``edge'' and ``vertex''  volumes of the structure.\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1.0]{hren3.pdf}\n\\end{center}\n\\caption{{\\scshape An example of a resonant thin network} {\\small Edge volumes are asymptotically of the same order as vertex volumes. The stiffness of the material of the structure is of the order period-squared. }}\n\\label{fig:exner}\n\\end{figure}\n\n\n\n\nIt is instructive to point out that the results of \\cite{CherKis} establish a thrilling relationship between the analysis of thin structures and the homogenisation theory of high-contrast composites.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1.0]{hren2.pdf}\n\\end{center}\n\\caption{{\\scshape High-contrast superlattice} {\\small The problem for a superlattice is reduced to a one-dimensional high-contrast problem. This is asymptotically equivalent to an impedance-type problem on the soft component.}}\n\\label{chain}\n\\end{figure}\nNamely, the paper \\cite{CherKis} deals with the case of the so-called superlattices \\cite{Tsu} with high contrast, see Fig. \\ref{chain}.\n While simple to set up, the related system of ordinary differential equations (subject to the  appropriate conditions of continuity of fields and fluxes) is nontrivial from the point of view of quantitative analysis, see also \\cite{ChCG}. It is shown that the asymptotic model for this system is precisely the one derived in \\cite{KuchmentZeng, KuchmentZeng2004, Exner} in the case of a resonant thin structure converging to a chain-graph, see Fig. \\ref{fig:exner}.\nAs we shall argue in the present article, such superlattices (and the corresponding chain-graphs) offer a simple prototype for a metamaterial, via the mathematical approach outlined above.\n\nThe result described above suggests, that thin networks might acquire the same asymptotic properties as those of the corresponding high-contrast composites. It is therefore a viable conjecture, that the metamaterial properties of a medium  can be attained via a version of geometric contrast instead of relying upon the contrast between material components. This is especially promising when the required material contrast cannot be guaranteed, as is commonly the case in elasticity and electromagnetism. The corresponding thin networks on the other hand have been made available in the study of graphenes and related areas. This subject will be further pursued in a forthcoming publication.\n\nThe above exposition vindicates the value of quantum graph models in the analysis of high-contrast composites, where we follow the well-established convention, see \\cite{Kuchment2}, to use the term {\\it quantum graph} for an ordinary differential operator of second order defined on a metric graph. These graph-based models are seen as natural limits of composite thin networks consisting of a large number of channels (for, say, acoustic or electromagnetic waves), where a combination of high-contrast and rapid oscillations becomes increasingly taxing at small scales and often leads to impractical numerical costs.\nFor channels with low cross-section-to-length ratios,\nthe material response of such a system, see Fig.\\,\\ref{thick_chain}, is closely approximated by a quantum graph as described above.\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.7]{hren1.pdf}\n\\end{center}\n\\caption{{\\scshape Thin network} {\\small An example of a high-contrast periodic network. Stiff channels are in grey, soft channels are in blue.}}\n\\label{thick_chain}\n\\end{figure}\nSystems of this type are a particular example of high-contrast composites and thus, as explained above, they possess resonant properties at the miscroscale, which leads to macroscopic dispersion by the above argument. At a very crude level, this  is similar to the way in which particle motion on the atomic scale leads to Lorentz-type electromagnetism, see {\\it e.g.} \\cite[Chapter 1]{Nussenzweig} for the analysis of a related model of damped harmonic oscillator.\n\nFurthermore, periodic quantum graphs with vanishing period can serve as realistic explicitly solvable ODE models for multidimensional continuous media, as demonstrated\\footnote{We remark, that it was Professor Pavlov who had pioneered the mathematical study of quantum graphs, see \\cite{Pavlov_old}.}, e.g., in \\cite{MelnikovPavlov}, where an $h-$periodic cubic lattice is shown to be close (up to and including the scattering properties) to the Laplacian in $\\mathbb R^d$. More involved periodic graphs can be used to model non-trivial media, including anisotropic ones.\n\nAs a particular realistic example of a thin network with high contrast, consider the problem of modelling acoustic wave propagation in a  system of channels $\\Omega^{\\varepsilon, \\delta}$,  $\\varepsilon$-periodic in one direction, of thickness $\\delta\\ll\\varepsilon$, and with contrasting material properties (cf. Fig. \\ref{thick_chain}). To simplify the presentation, we assume the antiplane shear wave polarisation (the so called S-waves), which leads to a scalar wave equation for the only non-vanishing component $W,$ of the form\n\\[\nW_{tt}-\\nabla_x\\cdot (a^\\varepsilon(x)\\nabla_xW)=0,\\qquad u=W(x,t),\\quad x, t\\in{\\mathbb R},\n\\]\nwhere the coefficient $a^\\varepsilon$  takes values one and $\\varepsilon^2$ in different channels of the $\\varepsilon$-periodic structure.\nLooking for time-harmonic solutions $W(x,t)=U(x)\\exp({\\rm i}\\omega t),$ $\\omega>0,$ one arrives at the spectral problem\n\\begin{equation}\n-\\nabla\\cdot (a^\\varepsilon\\nabla U)=\\omega^2U.\n\\label{spectral}\n\\end{equation}\nAs we argue below, the behaviour of (\\ref{spectral}) is close, in a quantitatively controlled way as $\\varepsilon\\to0,$ to that of an ``effective medium'' on ${\\mathbb R}$ described by an equation of the form\n\\begin{equation}\n-U''=\\beta(\\omega)U,\n\\label{dispersive}\n\\end{equation}\nfor an appropriate function $\\beta=\\beta(\\omega)$, explicitly given in terms of the material parameters $a^\\varepsilon$ and the topology of the original system of channels.\n\nThe goal of the present paper is to derive an explicit general formula for the function $\\beta$ in (\\ref{dispersive}), in terms of the topology of the graph representing the original domain of wave propagation, which is no longer restricted to the example shown in Fig.\\,\\ref{thick_chain}. As noted above, the presence of both rapid oscillations and high contrast make the task mathematically nontrivial. In our approach, which is new, we call upon some recently developed machinery in the operator-theoretic analysis of abstract boundary-value problems (which in our case take the form of boundary-value problems for differential operators of interest). In the subsequent work \\cite{ChKisYe_PDE} we develop the corresponding analysis for the multidimensional case, which is neither included nor an extension of the analysis for graphs presented in this article. However, it is based on the same set of mathematical ideas, which makes us hope that the foundations for (\\ref{dispersive}) in the case of PDEs is clear from what follows.\n\nUnlike the approach aimed at derivation of norm-resolvent convergence, which we adopt in \\cite{ChKisYe,ChKisYe_PDE}, in the present paper, having the convenience of the more physically inclined reader in mind, we systematically treat the subject from the point of view of spectral problems and in particular of the asymptotic analysis of eigenmodes. We refer the interested reader to the aforementioned papers, where further mathematical details, which we think are out of scope here, are contained.\n\n\n\nThe present paper can be viewed as following in the footsteps of \\cite{CherKis} in that it relies upon the analysis of the fibre representations (obtained via the Floquet-Gelfand transform) of the original periodic operator. This is carried out using the boundary triples  theory (see, e.g., \\cite{Gor,DM}), which generalises the classical methods based on the Weyl-Titchmarsh $m-$coefficient, applied to self-adjoint extensions of symmetric operators. This allows us to develop a novel approach to the homogenisation of a class of periodic high-contrast problems on ``weighted quantum graphs'', {\\it i.e.} one-dimensional versions of thin composite media where the material parameters on one of the components are much lower than on the others and scaled in a ``critical'' way with respect to the period of the composite. We reiterate  that the idea that such media can be viewed as idealised models of thin periodic critical-contrast networks has been explored in the mathematics literature, see \\cite{KuchmentZeng2004}, \\cite{Exner}, \\cite{Zhikov_singular_structures} and elsewhere. The backbone of our approach is, as explained above, the study of eigenfunctions of the problem restricted to one (``soft'') component of the composite only. After the asymptotics for these is obtained, it proves possible to reconstruct the ``complete'' eigenfunctions, where we implicitly rely upon the classical results of operator theory, in particular dealing with out-of-space self-adjoint extensions of symmetric operators and associated generalised resolvents.\n\n\\subsection {Physics interpretation and relevance to metamaterials}\n\nOur argument leads to the understanding of the phenomenon of critical-contrast homogenisation limit as a manifestation of a frequency-converting device: if one restricts the eigenfunctions to the ``stiff'' component, they prove to be close to those of the medium where the soft component has been replaced with voids, \\emph{but} correspond to non-trivially shifted eigenfrequencies. This is precisely what one would expect in the setting of time-dispersive media after the passage to the frequency domain, {\\it cf.} (\\ref{frequency_dep}).\n\nFrom the physics perspective, this link between homogenisation and frequency conversion can be viewed as a justification of an ``asymptotic equivalence'' between eigenvalue problems for periodic composites with high contrast and problems with nonlinear dependence on the spectral parameter, which in the frequency domain characterise ``time-dispersive media'', as in (\\ref{gen_Maxwell}), see also \\cite{Tip_1998, Tip_2006, Figotin_Schenker_2005,\nFigotin_Schenker_2007b}.\n\nAs we mentioned above, the phenomenon of frequency dispersion emerging as a result of homogenisation has been observed in the two-scale formulation applied to critical-contrast PDEs in, {\\it e.g.}, \\cite{Jikov, Bullshitte}. Our approach goes beyond the results of \\cite{Jikov, Bullshitte} in several ways. First, being based on an explicit asymptotic analysis of operators, using the recent developments in the theory of abstract boundary-value problems (see {\\it e.g.} \\cite{Ryzh_spec}), it  provides an explicit procedure for recovering the dispersion relation and does not draw upon the well-known two-scale asymptotic techniques.\n\nThe approach we develop in the present paper thus offers a new perspective on frequency-dispersive (time non-local) continuous media in the sense that it provides a recipe for the construction of such media with prescribed dispersive properties from periodic composites whose individual components are non-dispersive.\nIt has been known that time-dispersive media \\cite{Figotin_Schenker_2005} in the frequency domain can be realised as a ``restriction'' of a conservative Hamiltonian defined on a space which adds the ``hidden'' degrees of freedom.\\footnote{\nThis is based on the observation that the equation (\\ref{frequency_dep}) can be written in the form of an eigenvalue problem ${\\mathcal A}U=\\omega U,$ $U\\in{\\mathcal H},$ for a suitable  self-adjoint ``dilation\" ${\\mathcal A}$ of the operator $A,$ so that ${\\mathcal A}$ acts in a space ${\\mathcal H}\\supset{\\mathbb H}.$ The vector field  $U$ has a natural physical interpretation in terms of additional electromagnetic field variables, the so-called polarisation $P$ and magnetisation $M,$ so that the full (12-dimensional) field vector is $(H, E, P, M)^\\top.$ }\n\nIn summary, the existing belief in the engineering and physics literature that time-dispersive properties often arise as the result of complex microstructure of composites suggests to look for a rather concrete class of such conservative Hamiltonian dilations, namely, those pertaining to differential operators on composites with critical contrast. Our results can be viewed as laying foundations for rigorously solving this problem.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Infinite-graph setup}\n\nConsider a graph ${\\mathbb G}_\\infty,$ periodic in one direction, so that ${\\mathbb G}_\\infty+\\ell={\\mathbb G}_\\infty,$ where $\\ell$ is a fixed vector, which defines the graph axis.\nLet the periodicity cell  ${\\mathbb G}$ be a finite compact graph of total length $\\varepsilon\\in(0,1),$ and denote by\n$e_j,$ $j=1,2,\\dots n,$ $n\\in{\\mathbb N}$ its edges. For each $j=1,2,\\dots, n,$ we identify $e_j$ with the interval $[0,\\varepsilon l_j],$ where $\\varepsilon l_j$ is the length of $e_j.$  We associate with the graph ${\\mathbb G}_\\infty$ the Hilbert space\n$$\nL_2({\\mathbb G}_\\infty):=\\bigoplus\\limits_{{\\mathbb Z}}\\bigoplus\\limits_{j=1}^n L_2(0, \\varepsilon l_j).\n$$\n\nConsider a sequence of operators $A^\\varepsilon,$ $\\varepsilon>0,$ in $L_2({\\mathbb G}_\\infty),$ generated by second-order differential expressions\n\\begin{equation}\n-\\frac{d}{dx}\\left(\\bigl(a^\\varepsilon\\bigr)^2\\frac{d}{dx}\\right),\n\\label{diff_operator}\n\\end{equation}\nwith positive ${\\mathbb G}$-periodic coefficients $(a^\\varepsilon)^2$ defined on ${\\mathbb G}_\\infty,$\nwith the domain ${\\rm dom}(A^\\varepsilon)$ that describes the coupling conditions at the vertices of\n${\\mathbb G}_\\infty:$\n\\begin{equation}\n{\\rm dom}(A^\\varepsilon)=\n\\left\\{\nu\\in\\bigoplus\\limits_{e\\in{\\mathbb G}_\\infty}W^{2,2}\\bigl(e)\\Big|\\ u\n\\text{\\ continuous,}\\ \\sum_{e\\ni V}\\sigma_e(a^\\varepsilon)^2u'(V)=0\\\n\\forall\\ V\\in{\\mathbb G}_\\infty\\right\\},\n\\label{Atau}\n\\end{equation}\nIn the formula (\\ref{Atau}) the summation is carried out over the edges $e$\nsharing the vertex $V,$ the coefficient $(a^\\varepsilon)^2$ in the vertex condition is calculated on the edge $e,$ and $\\sigma_{ e}=-1$ or $\\sigma_{e}=1$ for $e$  incoming or outgoing  for $V,$ respectively.\nThe matching conditions \\eqref{Atau} represent the so-called standard, or Kirchhoff, conditions of combined continuity of the function and equality to zero of sums of co-normal derivatives at all vertices.\n\n\\section{Gelfand transform}\n\\label{Gelfand_section}\nWe seek to apply the one-dimensional Gelfand transform\n\\begin{equation}\\label{1dGelfand}\nv(x)=\\sqrt{\\frac{\\varepsilon}{2\\pi}}\\sum\\limits_{n\\in \\mathbb{Z}}u(x+\\varepsilon n) {\\rm e}^{-it(x+\\varepsilon n)}.\n\\end{equation}\nto the operator $A^\\varepsilon$ defined on ${\\mathbb G}_\\infty$ in order to obtain the direct fibre integral for the operator $A^\\varepsilon:$\n\\begin{equation}\\label{vonNeumann}\nA^{\\varepsilon}=\\int_{\\oplus}A^\\varepsilon_t dt.\n\\end{equation}\nIn order to do achieve this goal, we first note that the geometry of ${\\mathbb G}_\\infty$ is encoded in the matching conditions \\eqref{Atau} \\emph{only}. This opens up a possibility to embed the graph ${\\mathbb G}_\\infty$ into $\\mathbb R^1$ by rearranging it edges\nas consecutive segments of the real line (leading to a one-dimensional $\\varepsilon$-periodic chain graph). In doing so we drop the customary practice of drawing graphs in a way reflecting matching conditions ({\\it i.e.}, so that these are local relative to graph vertices). The above embedding leads to rather complex non-local matching conditions, but, on the positive side, allows us to use the Gelfand transform as required by \\eqref{1dGelfand}, \\eqref{vonNeumann}.\n\nThe Gelfand transform leads to periodic conditions on the boundary of the cell $\\mathbb G$ and thus in our case identifies the ``left\" boundary vertices of the graph $\\mathbb G$ with their translations by $\\ell$, which results in a modified graph $\\widehat{\\mathbb G}$. Apart from this, the matching conditions for the internal vertices of $\\mathbb G$ admit the same form as for $A^\\varepsilon$, except for the fact that the Kirchhoff matching is replaced by a Datta-Das Sarma one (the latter can be viewed as a weighted Kirchhoff), see below in \\eqref{Atau1}. Unimodular weights appearing in Datta-Das Sarma conditions are precisely due to the non-locality of matching conditions mentioned above for the embedding of $\\mathbb G_\\infty$ into $\\mathbb R^1$.\n\nThe image of the Gelfand transform is described as follows. There exists a unimodular list $\\{w_V(e)\\}_{e\\ni V},$ {\\it cf.} \\cite{ChKisYe}, defined at each vertex $V$ of $\\widehat{\\mathbb G}$ as a finite collection of values corresponding to the edges adjacent to $V$.\nFor each $t\\in[-\\pi\/\\varepsilon,\\pi\/\\varepsilon)$, the fibre operator $A^\\varepsilon_t$  is generated by the differential expression\n\\begin{equation}\n\\left(\\frac 1i \\frac d{dx}+t\\right)(a^\\varepsilon)^2\\left(\\frac 1i \\frac d{dx}+t\\right)\n\\label{diff_expr}\n\\end{equation}\non the domain\n\\begin{multline}\n{\\rm dom}(A^\\varepsilon_t)=\n\\Bigg\\{\nv\\in\\bigoplus\\limits_{e\\in {\\mathbb G}}W^{2,2}\\bigl(e)\\ \\Big|\\ w_V(e)v|_e(V)=w_V(e')v|_{e'}(V)\n\\text{\\ for all } e,e' \\\\ \\text{ adjacent to } V, \\ \\sum_{e\\ni V}\\partial^{(t)}v(V)=0\\ \\ \\\n{\\rm for\\ each\\ vertex}\\ V\\Bigg\\},\n\\label{Atau1}\n\\end{multline}\nwhere $\\partial^{(t)}v(V)$ is the weighted ``co-derivative''  $\\sigma_{e}w_V(e)(a^\\varepsilon)^2(v'+{\\rm i}t v)$  of the function $v$ on the edge $e,$ calculated at $V.$\n\n\n\n\\section{Boundary triples for extensions of symmetric operators}\n\\label{triples_section}\n\nIn the analysis of the asymptotic behaviour of the fibres of the original operator representing the quantum graph, we employ the framework of boundary triples for a symmetric operator with equal deficiency indices for the description of a class of its extensions. Part of the toolbox of the theory of boundary triples is the generalisation of the classical Weyl-Titchmarsh $m$-function to the case of a matrix (finite deficiency indices) and operators (infinite deficiency indices).\n\nThe boundary triples theory is a very convenient toolbox for dealing with extensions of linear operators, originating in the works of M.\\,G. Kre\\u\\i n. In essence, it is an operator-theoretic interpretation of the second Green's identity. As such, it allows one to pass over from the consideration of functions in Hilbert spaces to a formulation in which one deals with objects in the boundary spaces (such as traces of functions and traces of their normal derivatives), which in the context of quantum graphs are finite-dimensional. Furthermore, it allows one to use explicit concise formulae for the resolvents of operators under scrutiny and for other related objects. Thus it facilitates the analysis by expressing the familiar, commonly used in this area, objects in a concise way.\n\n\n\\begin{definition}[\\cite{Gor,Ko1,DM}]\nSuppose that $A_{\\rm max}$ is the adjoint to a densely defined symmetric operator on a separable Hilbert space $H$ and let $\\Gamma_0,$ $\\Gamma_1$ be linear mappings of ${\\rm dom}(A_{\\max})\\subset H$\nto a separable Hilbert space $\\mathcal{H}.$\n\nA. The triple\n$(\\mathcal{H}, \\Gamma_0,\\Gamma_1)$ is called \\emph{a boundary\ntriple} for the operator $A_{\\max}$ \nif the following two conditions hold:\n\\begin{enumerate}\n\\item For all $u,v\\in {\\rm dom}(A_{\\max})$ one has the second Green's identity\n\\begin{equation}\n\\langle A_{\\max} u,v \\rangle_H -\\langle u, A_{\\max} v \\rangle_H = \\langle \\Gamma_1 u, \\Gamma_0\nv\\rangle_{\\mathcal{H}}-\\langle\\Gamma_0 u, \\Gamma_1 v\\rangle_{\\mathcal{H}}.\n\\label{Green_identity}\n\\end{equation}\n\\item The mapping\n${\\rm dom}(A_{\\max})\\ni u\\longmapsto (\\Gamma_0 u,\n\\Gamma_1 u)\\in{\\mathcal H}\\oplus{\\mathcal H}$\nis onto\n\\end{enumerate}\n\nB. A restriction ${A}_B$ of the operator $A_{\\rm max}$ such\nthat $A_{\\rm max}^*=:A_{\\min}\\subset  A_B\\subset A_{\\max}$  is called\nalmost solvable if there exists a boundary triple\n$(\\mathcal{H}, \\Gamma_0,\\Gamma_1)$ for $A_{\\max}$ and a bounded\nlinear operator $B$ defined on $\\mathcal{H}$ such that\n\\[\n{\\rm dom}({A_B})=\\bigl\\{u\\in{\\rm dom}(A_{\\rm max}):\\ \\Gamma_1u=B\\Gamma_0u\\bigr\\}.\n\\]\n\n\nC. The operator-valued Herglotz\\footnote{For a definition and properties of Herglotz functions, see {\\it e.g.} \\cite{Nussenzweig}.} function $M=M(z),$ defined by\n\\begin{equation}\n\\label{Eq_Func_Weyl}\nM(z)\\Gamma_0 u_{z}=\\Gamma_1 u_{z}, \\ \\\nu_{z}\\in \\ker (A_{\\max}-z),\\  \\ z\\in\n\\mathbb{C}_+\\cup{\\mathbb C}_-,\n\\end{equation}\nis called the Weyl-Titchmarsh $M$-function of the operator\n$A_{\\max}$ with respect to the corresponding boundary triple.\n\\end{definition}\n\n\nSuppose $A_B$ be a self-adjoint almost solvable restriction of $A_{\\rm max}$ with compact resolvent. Then $M(z)$ is analytic on the real line away from the eigenvalues of $A_\\infty,$ where $A_\\infty$ is the restriction of $A_{\\rm max}$ to domain $\\dom(A_\\infty)=\\dom(A_{\\rm max})\\cap\\ker(\\Gamma_0).$ It is a key observation for what follows that $u\\in{\\rm dom}(A_B)$ is an eigenvector of $A_B$ with eigenvalue $z_0\\in{\\mathbb C}\\setminus{\\rm spec}(A_\\infty)$ if and only if\n\\begin{equation}\n\\bigl(M(z_0)-B\\bigr)\\Gamma_0u=0.\n\\label{eigeneq}\n\\end{equation}\n\n\nIn the next section we utilise a particular operator $A_{\\rm max}$ and a boundary triple $({\\mathcal H}, \\Gamma_0, \\Gamma_1),$ which we use to analyse the resolvents of the operators on quantum graphs\nintroduced earlier.\n\n\n\n\n\n\n\n\\section{Graph with high contrast: prototype for time-dispersive media}\n\\label{our_graph}\nIn what follows we develop a general approach to the analysis of weighted quantum graphs with critical contrast. We demonstrate it on one particular example, which, as we show in Appendix A, exhibits all the properties of the  generic case. We have therefore chosen to present the analysis in the terms that are immediately applicable to the general case and, whenever advisable, we provide statements that carry over without modifications. Speaking of a ``general'' case, we imply an operator of the class introduced in Section 2, where some of the edges $e_{\\text{soft}}$ of the cell graph $\\mathbb G$ carry the weight $a^\\varepsilon=\\varepsilon$, with the remaining edges carrying weights of order 1 uniformly in $\\varepsilon$.\n\nThe rationale of the present section is in fact extendable to an even  more general setup (including the one of periodic high-contrast PDEs), which we treat in the paper \\cite{ChKisYe_PDE}. However, in the present paper we consider a rather simplified model, in view of keeping technicalities to a bare minimum and thus hopefully making the matter transparent to the reader.\n\nConsider the graph ${\\mathbb G}_\\infty$ with the periodicity cell ${\\mathbb G}$ shown in Figure  \\ref{infinite_graph_figure}.\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1.5]{gebra0.png}\n\\end{center}\n\\caption{{\\scshape Periodicity cell $\\mathbb G.$} {\\small The intervals of lengths $\\varepsilon l_1$ and $\\varepsilon l_3$ are ``stiff\", {\\it i.e.} they carry the weights $a_1^2$ and $a_3^2$, respectively, whereas the interval of length $\\varepsilon l_2$ is ``soft\", with weight $\\varepsilon^2.$}}\n\\label{infinite_graph_figure}\n\\end{figure}\nThe Gelfand transform, see Section \\ref{Gelfand_section}, applied to this graph, yields the graph $\\widehat{\\mathbb G}$ of Figure \\ref{compact_fig}. In the present section we show that there exists a boundary triple such that $A^\\varepsilon_t$ is an almost solvable extension of the corresponding $A_{\\min}$, and the\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.7]{gebra00.png}\n\\end{center}\n\\caption{{\\scshape The graph $\\widehat {\\mathbb G}.$} {\\small The left and right boundary vertices have been identified.}}\n\\label{compact_fig}\n\\end{figure}\n$M$-function (which is in our case a matrix-valued function; for convenience, it is written as a function of $k:=\\sqrt{z}$, with the branch chosen so that $\\Im k>0$) of $A_{\\max}$ is given by\n\\begin{equation}\n\\label{Msplit}\nM(k,\\varepsilon, t)=k\\widetilde{M}^{\\rm stiff}(\\varkappa,\\tau)+\\varepsilon\\widetilde{M}^{\\rm soft}(k, \\tau),\\quad \\varkappa:=\\varepsilon k,\\quad\\tau:=\\varepsilon t,\n\\end{equation}\nwhere\n\\[\n\\widetilde{M}^{\\rm stiff}(\\varkappa,\\tau):=\\left(\\begin{array}{cc}\n-a_1\\cot\\dfrac{\\varkappa l_1}{a_1} -a_3\\cot\\dfrac{\\varkappa l_3}{a_3}\\ &\\\na_1\\dfrac{{\\rm e}^{-i(l_1+l_3)\\tau}}{\\sin\\dfrac{\\varkappa l_1}{a_1}}+a_3 \\dfrac{{\\rm e}^{il_2\\tau}}{\\sin\\dfrac{\\varkappa l_3}{a_3}}\\\\[3.3em]\na_1\\dfrac{{\\rm e}^{i(l_1+l_3)\\tau}}{\\sin\\dfrac{\\varkappa l_1}{a_1}}+a_3 \\dfrac{{\\rm e}^{-il_2\\tau}}{\\sin\\dfrac{\\varkappa l_3}{a_3}}\\ &\\\n-a_1\\cot\\dfrac{\\varkappa l_1}{a_1}-a_3\\cot\\dfrac{\\varkappa l_3}{a_3}\n\\end{array}\\right),\n\\]\n\\begin{equation}\n\\widetilde{M}^{\\rm soft}(k,\\tau):=k\\left(\\begin{array}{cc}-\\cot k l_2\\ &\\ \\dfrac{{\\rm e}^{il_2\\tau}}{\\sin k l_2}\\\\[1.6em]\n\\dfrac{{\\rm e}^{-il_2\\tau}}{\\sin k l_2}\\ &\\ -\\cot k l_2\n\\end{array}\\right),\n\\label{M_soft_tilde}\n\\end{equation}\n\nNote that  for all $\\tau\\in[-\\pi, \\pi),$ the function $\\widetilde{M}^{\\rm soft}(\\cdot,\\tau)$ is meromorphic and regular at zero.\n\n\nEssentially, the claim made is a straightforward consequence of the double integration by parts, followed by a simple rearrangement of terms.\nIn the rest of this section we sketch the construction applicable in the general case, which in particular yields the above claim for the model graph considered. Under the definitions of Section \\ref{triples_section}, the maximal operator\n$A_{\\rm max}=A_{\\rm min}^*$ is defined by the same differential expression (\\ref{diff_expr}) on the domain\n\\begin{multline}\\label{domAmax}\n{\\rm dom}(A_{\\rm max})=\n\\biggl\\{\nv\\in\\bigoplus\\limits_{e\\in \\widehat{\\mathbb G}}W^{2,2}\\bigl(e)\\ \\Big|\\ w_V(e)v|_e(V)=w_V(e')v|_{e'}(V)\n\\\\\n\\text{\\ for all } e,e' \\text{ adjacent to } V,\\ \\\n\\forall\\,V \\in\\widehat{\\mathbb G}\n\\biggr\\}.\n\n\\end{multline}\nIn what follows we use the triple $({\\mathbb C}^m, \\Gamma_0, \\Gamma_1),$ where $m$ is the number of vertices in the graph $\\widehat{\\mathbb G}$, and\n\\begin{equation}\n\\Gamma_0v=\\bigl\\{v(V)\\bigr\\}_V,\n\\qquad \\Gamma_1v=\\Bigl\\{\\sum_{e\\ni V}\\partial^{(t)}v(V)\\Bigr\\}_V,\\qquad v\\in{\\rm dom}(A_{\\rm max}),\n\\label{boundary_operators}\n\\end{equation}\nwhere $u(V)$ is defined as the common value of  $w_V(e)v|_e(V)$ for all edges $e$ adjacent to $V$.\n\n\nBy definition of the $M$-matrix one has\n$\n\\Gamma_1v=M\\Gamma_0v,$\n$\nv\\in\\ker (A_{\\rm max} - z).\n$\nFunctions $v\\in\\ker (A_{\\rm max} - z)$\nhave the form\n$$\nv(x)=\\exp(-{\\rm i}xt)\\biggl\\{A_e\\exp\\biggl(-\\frac{{\\rm i}kx}{a^\\varepsilon}\\biggr)+B_e\\exp\\biggl(\\frac{{\\rm i}kx}{a^\\varepsilon}\\biggr)\\biggr\\},\\quad x\\in e,\\quad A_e, B_e\\in{\\mathbb C},\n$$\nwhere $k:=\\sqrt{z}$, and the co-derivative is given by\n$$\na_\\varepsilon^2(v'(x)+it v(x)\n= {\\rm i}ka^\\varepsilon\\exp(-{\\rm i}xt)\n\\biggl\\{-A_e\\exp\\biggl(-\\frac{{\\rm i}kx}{a^\\varepsilon}\\biggr)+B_e\\exp\\biggl(\\frac{{\\rm i}kx}{a^\\varepsilon}\\biggr)\\biggr\\}, \\qquad x\\in e,\n$$\nFor the vertex $V$\nand for every ``Dirichlet data\" vector $\\Gamma_0v$ one of whose entries is unity and the other entries vanish,\nthe ``Neumann data'' vector $\\Gamma_1v$ gives the column of the $M$-matrix corresponding to $V.$\nThe corresponding $\\Gamma_1v$ has diagonal and off-diagonal entries of the form, respectively,\n$$\n-\\sum_{e\\in V} ka^\\varepsilon\\cot\\left(\\dfrac{k \\varepsilon l_e}{a^\\varepsilon}\\right),\\qquad\n\\qquad \\sum_{e\\in V} ka^\\varepsilon \\widetilde w_V(e)\\biggl(\\sin \\dfrac{k \\varepsilon l_e}{a^\\varepsilon}\\biggr)^{-1},\n$$\nwhere $\\{\\widetilde w_V(e)\\}_{e\\ni V}$ is a unimodular list uniquely determined by the list $\\{w_V(e)\\}_{e\\ni V}$.\nThe resulting $M$-matrix is constructed from these columns over all vertices $V.$\n\nIn particular, for the example of Fig. \\ref{infinite_graph_figure} -- \\ref{compact_fig}, we have the following:\nthe boundary space $\\mathcal H$ pertaining to the graph $\\widehat{\\mathbb G}$ is chosen as $\\mathcal H=\\mathbb C^2$. The unimodular list functions $w_{V_1}$ and $w_{V_2}$ are chosen as follows:\n\\begin{equation*}\\label{eq:1-weights}\n\\begin{gathered}\n\\{w_{V_1}(e^{(j)})\\}_{j=1}^3=\\{1,1,e^{i\\tau(l^{(2)}+l^{(3)})}\\},\\quad \\{w_{V_2}(e^{(j)})\\}_{j=1}^3=\\{e^{i\\tau l^{(3)}},1,1\\}\n\\end{gathered}\n\\end{equation*}\nand\n\\begin{equation*\n\\begin{gathered}\n\\{\\widetilde w_{V_1}(e^{(j)})\\}_{j=1}^3=\\{e^{-i\\tau(l^{(1)}+l^{(3)})},e^{i\\tau l^{(2)}},e^{i\\tau l^{(2)}}\\},\\\\ \\{\\widetilde w_{V_2}(e^{(j)})\\}_{j=1}^3=\\{e^{i\\tau(l^{(1)}+l^{(3)})},e^{-i\\tau l^{(2)}},e^{-i\\tau l^{(2)}}\\},\n\\end{gathered}\n\\end{equation*}\nyielding the formula \\eqref{M_soft_tilde}.\n\n\n\n\n\n\n\n\n\\section{Asymptotic diagonalisation of the $M$-matrix and the eigenvector asymptotics}\\label{sect:asymp_diag}\n\nThe present section is the centrepiece of our approach. The major difficulty to overcome is of course the fact that the operator $A^\\varepsilon_t$ entangles in a non-trivial way the stiff and soft components of the medium. On the level of the analysis of the operator itself this problem admits no obvious solution, unless one is prepared to introduce a two-scale asymptotic ansatz. On the other hand, the $M$-matrix calculated above will be shown to be additive with respect to the decomposition of the medium (hence the notation $M^{\\rm soft}$ and $M^{\\rm stiff}$). Thus, via the representation \\eqref{eigeneq}, it proves possible to use the asymptotic expansion of $M^{\\rm stiff}$, which is readily available, to recover the asymptotics of eigenmodes, restricted to the soft component. This way, the homogenisation task at hand can be viewed as a version of the perturbation analysis in the boundary space pertaining to the problem.\n\n\n\n\n\n\nIn the example considered (and in the general case in view of Appendix A)\nit follows from (\\ref{eigeneq}), (\\ref{Msplit})\nthat $u_\\varepsilon$ is an eigenfunction of the operator $A^\\varepsilon_t,$ see (\\ref{diff_expr})--(\\ref{Atau1}), if and only if\n\\begin{equation}\nM^{\\rm soft}\\Gamma_0u_\\varepsilon=-M^{\\rm stiff}\\Gamma_0u_\\varepsilon,\\qquad M^{\\rm soft}:=\\varepsilon\\widetilde{M}^{\\rm soft},\\quad\nM^{\\rm stiff}:=k\\widetilde{M}^{\\rm stiff}.\n\\label{bc1}\n\\end{equation}\nIn writing (\\ref{bc1}), we assume, without loss of generality, that the eigenvalue $z_\\varepsilon=k^2$ corresponding to the eigenfunction $u_\\varepsilon$ does not belong to the spectrum of the Dirichlet decoupling $A_\\infty^t,$ defined according to the general theory of Section \\ref{triples_section} for the operators we introduce in Section \\ref{Gelfand_section}. Indeed, in any compact subset of $\\mathbb C,$ for small enough $\\varepsilon,$ this spectrum coincides with the $\\varepsilon$-independent set of poles of the matrix\n$\\widetilde{M}^{\\rm soft},$ see (\\ref{M_soft_tilde}). For the same reason, we can safely assume that the eigenvalues $z_\\varepsilon$ do not belong to the spectrum of the Dirichlet operator on the soft inclusion. These assumptions ensure that that the condition $z_0\\in{\\mathbb C}\\setminus{\\rm spec}(A_\\infty)$ for the validity of (\\ref{eigeneq}) is satisfied in both cases: for the $M$-matrix of the operator $A^\\varepsilon_t,$ where $B=0,$ and for the $M$-matrix of the operator on the soft component represented by (\\ref{bc1}), where the role of $B$ is played by the matrix $-M^{\\rm stiff}.$\n\nWe proceed by observing that the matrices $M^{\\text{soft}}$ and $M^{\\text{stiff}}$ in (\\ref{bc1}) can be treated as $M$-matrices of certain triples on their own. In particular, it will be instrumental in what follows to attribute this meaning to $M^{\\text{soft}}$. To this end, consider the decomposition of the graph $\\widehat{\\mathbb G}$ into its ``soft'' $\\mathbb G^{\\text{soft}}$ and ``stiff'' $\\mathbb G^{\\text{stiff}}$ components (each of these is treated as a graph, so that $\\widehat{\\mathbb G}=\\mathbb G^{\\text{soft}}\\cup \\mathbb G^{\\text{stiff}}$) and the operator $A_{\\max}^{\\text{soft}}$ defined by \\eqref{diff_expr}, \\eqref{domAmax}, with $\\widehat{\\mathbb G}$ replaced by $\\mathbb G^{\\text{soft}}$. The boundary space for $A_{\\max}^{{\\rm soft}}$ can be defined as $\\mathcal H$, the same as the boundary space for the operator $A_{\\max}$ (again by Appendix A in the general case). The boundary operators $\\Gamma_j^{\\text{soft}}$, $j=0,1,$ are defined as in \\eqref{boundary_operators} for the graph $\\mathbb G^{\\text{soft}}$. Then, by inspection, the $M$-matrix for the operator $A_{\\max}^{\\text{soft}}$ is nothing but $M^{\\text{soft}}$ (see \\cite{CherKisSilva} for further details).\n\nFor each $v\\in{\\rm dom}(A_{\\rm max}),$ define $\\widetilde{v}$ to be the restriction of $v$ to the soft component $\\mathbb G^{\\text{soft}}$. It is obvious that $\\widetilde{v}\\in\\dom(A_{\\max}^{{\\rm soft}}).$\n\n\n\n\n\nWe notice that (\\ref{bc1}) implies, in particular, that\n\\begin{equation}\nM^{\\rm soft}\\Gamma_0^{\\rm soft}\\widetilde{u}_\\varepsilon=B^\\varepsilon\\Gamma_0^{\\rm soft}\\widetilde{u}_\\varepsilon,\\qquad\\qquad B^\\varepsilon:=-M^{\\rm stiff}.\n\\label{suggested_bound_cond}\n\\end{equation}\nFurthermore, since $M^{\\rm soft}$ is the $M$-matrix for the pair $(\\Gamma^{\\rm soft}_0, \\Gamma^{\\rm soft}_1),$ one has\n\\[\nM^{\\rm soft}\\Gamma_0^{\\rm soft}\\widetilde{u}_\\varepsilon=\\Gamma_1^{\\rm soft}\\widetilde{u}_\\varepsilon,\n\\]\nso the condition (\\ref{suggested_bound_cond}) takes a form similar to (\\ref{Eq_Func_Weyl}):\n\\begin{equation}\n\\Gamma_1^{\\rm soft}\\widetilde{u}_\\varepsilon=B^\\varepsilon\\Gamma_0^{\\rm soft}\\widetilde{u}_\\varepsilon.\n\\label{eq_eigenvector3}\n\\end{equation}\n\nThis condition involves the Dirichlet data of the solution to the spectral equation for $A_{\\max}^{\\text{soft}}$ which is an ODE on the graph $\\mathbb G^{\\text{soft}}$ with a constant coefficient. The Dirichlet data $\\Gamma_0^{\\rm soft}\\widetilde{u}_\\varepsilon$ determine the vector $\\widetilde{u}_\\varepsilon$ uniquely. The named vector is interpreted as a solution to the spectral equation on the soft component of the graph $\\widehat{\\mathbb G}$ subject to\n$z$-dependent boundary conditions, encoded in \\eqref{eq_eigenvector3}. On the other hand, this vector can also be used to reconstruct the vector $u_\\varepsilon$: indeed, from $\\Gamma_0 u_\\varepsilon=\\Gamma_0^{\\text{soft}}\\widetilde{u}_\\varepsilon$ it follows, that $u_\\varepsilon$, which is by assumption an eigenvector to $A^\\varepsilon_t$ at the point $z$, is nothing but a continuation of $\\widetilde{u}_\\varepsilon$ to the rest of the graph $\\widehat{\\mathbb G}$ based on its Dirichlet data at the boundary of the soft component. It follows, cf. \\eqref{eq_eigenvector3}, that the asymptotic analysis can be reduced to the soft component, with the information about the presence of the stiff component fed into the related asymptotic procedure by means of the stiff-soft interface.\n\nBefore we proceed further, let us take another look at the equation $M\\Gamma_0u_\\varepsilon=0,$ {\\it cf.} (\\ref{bc1}), which is equivalent to $u_\\varepsilon$ being an eigenvector of $A^\\varepsilon_t$ at the value of spectral parameter $z$. Using the fact that $M=M^{\\text{soft}}+M^{\\text{stiff}}$ as well as the explicit expressions for the matrices $M^{\\text{soft}},$ $M^{\\text{stiff}},$ {\\it cf.} \\eqref{Msplit}, it is easily seen that the leading-order term of $\\Gamma_0 u_\\varepsilon$, and thus of $u_\\varepsilon$, does not depend on the soft component of the medium, since the elements of $M^{\\text{soft}}$ are $\\varepsilon$-small. On the other hand, the situation is drastically different from the viewpoint of the associated dispersion relation, which must be guaranteed for the \\emph{solvability} of $M \\Gamma_0 u_\\varepsilon=0$. The dispersion relation follows from the condition $\\det M=0$, and it is \\emph{here, and here only}, that the soft component of the medium makes its presence felt in the problem. Due to the fact that $M^{\\text{stiff}}$ is rank one at $\\tau=0$, it transpires that the leading-order term of the equation $\\det M=0$ \\emph{in the case of critical contrast only} blends together in a non-trivial way the stiff and soft components of the medium. Bearing this in mind, the phenomenon of critical-contrast homogenisation can be seen as a manifestation of a frequency-converting device: if one restricts the eigenfunctions to the stiff component, they are $\\varepsilon$-close to those of the medium where the soft component has been replaced with voids, \\emph{but} correspond to non-trivially shifted eigenfrequencies. This is precisely what one would expect in the setting of time-dispersive media after the passage to the frequency domain, {\\it cf.} (\\ref{gen_Maxwell}), (\\ref{frequency_dep}). We will come back to this discussion in Section 8.\n\nLet us return to the analysis of \\eqref{eq_eigenvector3}, which, as explained above, contains all the information on the asymptotic behaviour of $A^\\varepsilon_t$. We notice that the named equation corresponds to a homogeneous ODE; the non-trivial dependence on $\\varepsilon$ is concealed in the right-hand side, which describes $\\varepsilon$- \\emph{and} frequency-dependent boundary conditions. The problem of asymptotic analysis of eigenfunctions of $A^\\varepsilon_t$ is thus effectively reduced to the analysis of the asymptotic behaviour of these boundary conditions. This analysis however is greatly simplified by the fact that $B^\\varepsilon$ is equal to $-M^{\\text{stiff}}$, where $M^{\\text{stiff}}$ is shown to be the $M$-matrix of $A_{\\max}^{\\text{stiff}}$ (see Appendix A) by a similar argument to that applied above to $M^{{\\rm soft}}$. Hence, the asymptotics sought for $M^{{\\rm stiff}}$ is simply the asymptotics of the Dirichlet-to-Neumann map of a uniformly elliptic problem at zero frequency, which allows to use well-known elliptic techniques.\n\nFirstly, we notice that the results of Section 5 combined with the asymptotic formulae\n\n\\begin{equation*}\na_e \\cot \\frac{\\varkappa l_e}{a_e} = \\frac{a_e^2}{\\varkappa l_e}-\\frac 13\\varkappa l_e+ O(\\varkappa^3),\\quad\\quad\\quad a_e\\biggl(\\sin\\dfrac{\\varkappa l_e}{a_e}\\biggr)^{-1} = \\frac{a_e^2}{\\varkappa l_e}+\\frac{1}{6}\\varkappa l_e+ O(\\varkappa^3),\n \\end{equation*}\nyield the following statement.\n\n\n\\begin{lemma}\n\\label{M_expansion}\nSuppose that $K\\subset{\\mathbb C}$ is compact.\nOne has\n\\begin{equation*}\n\\widetilde{M}^{\\rm stiff}(\\varkappa, \\tau)\n=\\varkappa^{-1}M_0(\\tau)+\\varkappa M_1(\\tau)+O(\\varkappa^3),\\quad \\tau\\in[-\\pi, \\pi),\\ \\varkappa=\\varepsilon k,\\ \\varepsilon\\in(0,1),\\ k\\in K,\n\\end{equation*}\nwhere $M_0$ and $M_1$ are analytic matrix functions of $\\tau$.\n\\end{lemma}\n\n\nIt follows from Lemma \\ref{M_expansion} that, for all  $\\tau\\in[-\\pi,\\pi),$\n\\begin{equation}\nB^\\varepsilon(z)=\\varepsilon^{-1}B_0\n+\\varepsilon zB_1+O(\\varepsilon^3z^2),\\qquad\\varepsilon\\in(0,1),\\ \\sqrt{z}\\in K,\n\\label{asympt}\n\\end{equation}\nwhere $B_0,$ $B_1$ are Hermitian matrices that depend on $\\tau$ only.\nThe following two lemmata carry over to the general case with minor modifications, since they only pertain to the stiff component of the medium and therefore rely upon the general uniformly elliptic properties of the latter.\n\\begin{lemma}\n\\label{mu_lemma}\nThere exist $\\gamma\\geq0$ (where $\\gamma=0$ if and only if the graph $\\mathbb{G}^{\\rm stiff}$  is a tree\\footnote{Recall that a tree is a connected forest \\cite{Cvetkovich}.}) and an eigenvalue branch $\\mu^{(\\tau)}$ for the matrix $B_0,$ such that\n$\\dim\\Ker\\bigl(B_0-\\mu^{(\\tau)}\\bigr)=1,$ $\\tau\\in[-\\pi,\\pi),$ and\n\\begin{equation}\n\\mu^{(\\tau)}=\\gamma\\tau^2 + O(\\tau^4).\n\\label{mu_asymptotics}\n\\end{equation}\n\\end{lemma}\n\n\nWe denote by $\\psi^{(\\tau)}$ the normalised eigenvector for the eigenvalue $\\mu^{(\\tau)},$ so that\n\n $\\psi^{(0)}=(1\/\\sqrt{2})(1,1)^\\top,$ {\\it i.e.} the trace of the first eigenvector of the Neumann problem on the stiff component at zero quiasimomentum, which is clearly constant.\n  Let $\\mathcal P:=\\langle \\cdot, \\psi^{(\\tau)}\\rangle \\psi^{(\\tau)}$ and $\\mathcal P_{\\bot}$ be the orthogonal projections in the boundary space onto $\\psi^{(\\tau)}$ and its orthogonal complement, respectively.\n\n\\begin{lemma}\n\\label{bound_below_lemma}\nThere exists $C_\\perp>0$ such that\n\\begin{equation}\n\\mathcal P_{\\bot} B_0 \\mathcal P_{\\bot}\\ge C_\\perp\\mathcal P_{\\bot},\n\\label{bound_below}\n\\end{equation}\nin the sense that the operator $\\mathcal P_{\\bot}(B_0-C_\\perp)\\mathcal P_{\\bot}$ is non-negative.\n\\end{lemma}\n\n\n\nWe use Lemma \\ref{bound_below_lemma} to solve \\eqref{eq_eigenvector3} asymptotically. The overall idea is to diagonalise the leading order term $\\varepsilon^{-1} B_0$ of the asymptotic expansion of $B^\\varepsilon$ in \\eqref{eq_eigenvector3}. From Lemma \\ref{mu_lemma} we infer that $B_0$ has precisely one eigenvalue quadratic in $\\tau$ (which thus gets close to zero), while Lemma \\ref{bound_below_lemma} provides us with a bound below on the remaining eigenvalue. The fact that the eigenvalue $\\mu^{(\\tau)}$ degenerates requires that the next term in the asymptotics of $B^\\varepsilon$ be taken into account in the related eigenspace. This additional term is easily seen to be $z-$dependent (in fact, linear in $z$).\n\nWe start with an auxiliary rescaling of the soft component. Namely, we introduce the unitary operator\n$\\Phi_\\varepsilon$ mapping $v\\mapsto \\widehat{v}$ according to the formula $\\widehat{v}(\\cdot)=\\sqrt{\\varepsilon}{v}(\\varepsilon\\cdot)$. Under this mapping, the length of the soft component loses its dependence on $\\varepsilon$. The operator $\\widehat{A}_{\\rm max}^{\\text{soft}}$ is defined as the unitary image of $A_{\\rm max}^{\\text{soft}}$ under the mapping $\\Phi_\\varepsilon$, and $\\widehat{\\Gamma}_0^{\\rm soft},$ $\\widehat{\\Gamma}_1^{\\rm soft}$ are the boundary operators for the rescaled soft component:\n\\[\n\\widehat{\\Gamma}_0^{\\rm soft}\\widehat{v}:=\\bigl\\{\\widehat{v}(V)\\bigr\\}_V,\n\\qquad \\widehat{\\Gamma}_1^{\\rm soft}\\widehat{v}:=\\biggl\\{\\sum_{e\\ni V}\\widehat{\\partial}^{(\\tau)}\\widehat{v}(V)\\biggr\\}_V,\\qquad \\widehat{v}\\in {\\rm dom}\\bigl(\\widehat{A}_{\\rm max}^{\\text{soft}}\\bigr),\n\\]\nwhere we set $\\widehat{v}(V)$ as the common value of  $w_V(e)\\widehat{v}|_e(V)$ for all $e$ adjacent to $V,$ and\n$\\widehat{\\partial}^{(\\tau)} \\widehat{v}(V)$ is the expression  $\\sigma_{e}w_V(e)(\\widehat{v}'+{\\rm i}\\tau \\widehat{v})$  on the edge $e,$ calculated at $V.$ Note that $\\widehat{\\Gamma}_1^{\\rm soft}$ does not depend on $\\varepsilon$.\n\nUnder the rescaling $\\Phi_\\varepsilon$ the equation \\eqref{eq_eigenvector3} becomes\n\\begin{equation}\n\\widehat{\\Gamma}_1^{\\rm soft}\\widehat{u}_\\varepsilon=\\varepsilon^{-1} B^\\varepsilon\\widehat{\\Gamma}_0^{\\rm soft}\\widehat{u}_\\varepsilon,\n\\label{eq_eigenvector_resc}\n\\end{equation}\nwhere in accordance with the above convention $\\widehat{u}_\\varepsilon=\\Phi_\\varepsilon \\widetilde{u}_\\varepsilon$.\n\n\nWe start our diagonalisation procedure by considering the non-degenerate ei\\-gen\\-spa\\-ce of $B^\\varepsilon$.\nApplying $\\mathcal P_{\\bot}$ to both sides of \\eqref{eq_eigenvector_resc}, replacing $B^\\varepsilon$ by its asymptotics (\\ref{asympt}) and using (\\ref{bound_below}) yields\n\\begin{equation}\n\\mathcal P_{\\bot} \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}_\\varepsilon =\\varepsilon^{-2}\\mathcal P_{\\bot} B_0 \\mathcal P_{\\bot} \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon + O(1)\\ge \\varepsilon^{-2}C_\\perp\\mathcal P_{\\bot} \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon + O(1),\n\\label{asym_rel}\n\\end{equation}\nwhere we assume that $u_\\varepsilon$ is $L^2$-normalised.\nMultiplying by $\\varepsilon^2$ both sides of (\\ref{asym_rel}) and applying the Sobolev embedding theorem to the left-hand side of (\\ref{asym_rel}),\nwe infer\n\\begin{equation}\\label{eq_part-solution}\n\\mathcal P_{\\bot} \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon = O(\\varepsilon^2).\n\\end{equation}\nPlugging this partial solution back into \\eqref{eq_eigenvector_resc}, to which $\\mathcal P$ is applied on both sides, we obtain\n\\begin{align*}\n\\mathcal P \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}_\\varepsilon\n&= \\varepsilon^{-2}\\mathcal P B_0 \\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon +  z\\mathcal P B_1\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon + O(\\varepsilon^2)\\\\[0.4em]\n&=\\varepsilon^{-2}\\mu^{(\\tau)}\\mathcal P\\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon +  z\\mathcal P B_1\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon + O(\\varepsilon^2).\n\\end{align*}\n\nWe have proved that up to an error term admitting a uniform estimate $O(\\varepsilon^2)$ one has the following asymptotically equivalent problem for the eigenvector $\\widehat{v}_\\varepsilon$:\n\\begin{equation}\\label{eq:eigenvector_asymp}\n  \\mathcal P_{\\bot} \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon =0,\\quad\n  \\mathcal P \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}_\\varepsilon = \\varepsilon^{-2}\\mu^{(\\tau)}\\mathcal P\\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon +  z\\mathcal P B_1\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon.\n\\end{equation}\n\nWe use Lemma \\ref{mu_lemma} and expand $\\mathcal P B_1 \\mathcal P$ in powers of $\\tau=\\varepsilon t$ as follows\\footnote{In the example considered in the present paper, as opposed to the general case, one can prove that $\\mathcal P B_1 \\mathcal P= \\mathcal P B_1^{(0)}\\mathcal P+O(\\tau^2)$, see the calculation in \\cite[Appendix B]{ChKisYe} for details. This yields the error bound $O(\\varepsilon^2)$ in the statement of Theorem \\ref{eff_thm} below.}: $\\mathcal P B_1 \\mathcal P= \\mathcal P B_1^{(0)}\\mathcal P+O(\\tau)$.\nThe second equation in \\eqref{eq:eigenvector_asymp} admits the form\n\\begin{equation}\n\\label{eq_eigenvector4}\n\\mathcal P \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}_\\varepsilon= \\gamma t^2 \\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon+z \\mathcal P B_1^{(0)}\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon +\n(O(\\tau)+O(\\tau^4\/\\varepsilon^2))\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon.\n\\end{equation}\nExpressing $\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon$ from the latter equation, it is easily seen based on embedding theorems that \\eqref{eq_eigenvector4} is asymptotically equivalent, up to an error uniformly estimated as $O(\\varepsilon)$, to the following equation:\n\\begin{equation}\n\\label{eq_eigenvector5}\n\\mathcal P \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}_\\varepsilon= \\gamma t^2 \\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon+z \\mathcal P B_1^{(0)}\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}_\\varepsilon.\n\\end{equation}\n\n\n   We formulate the above result as the following theorem.\n\n\\begin{theorem}\n\\label{eff_thm}\nLet $\\widehat{u}$ solve the  following equation on the re-scaled soft component:\n\\begin{equation*}\\label{eq:eq}\n\\begin{aligned}\n\\widehat{A}_{\\rm max}^{\\rm soft}\\widehat{u}(x)&=z\\widehat{u}(x),\\\\[0.3em]\n\\mathcal P_{\\bot} \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u} &= 0,\\\\[0.3em]\n\\mathcal P \\widehat{\\Gamma}_1^{\\rm soft} \\widehat{u}&= \\gamma t^2 \\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}+z \\mathcal P B_1^{(0)}\\mathcal P \\widehat{\\Gamma}_0^{\\rm soft} \\widehat{u}.\n\\end{aligned}\n\\end{equation*}\n\nThen the eigenvalues $z_\\varepsilon$ and their corresponding eigenfunctions $u_\\varepsilon$ of the operators $A^\\varepsilon_t$\nare $O(\\varepsilon)$-close uniformly in $t\\in[-\\pi\/\\varepsilon, \\pi\/\\varepsilon)$, in the sense of $\\mathbb C$ and in the sense of the $L^2$ norm,  respectively, to the values $z$ as above and functions ${u}_{\\rm eff}$ defined as follows. On the soft component ${\\mathbb G}^{\\rm soft}$ we set $u_{\\rm eff}(\\cdot):=(1\/\\sqrt{\\varepsilon})\\widehat{u}({\\varepsilon}^{-1}\\cdot)$. On the stiff component ${\\mathbb G}^{\\rm stiff}$ the function $u_{\\rm eff}$ is obtained as the extension by $(1\/\\sqrt{\\varepsilon})v,$\nwhere $v$ is the solution of the operator equation\n\\[\nA_{\\rm max}^{\\rm stiff} v=0,\n\\]\ndetermined by the Dirichlet data of $\\widehat{u}(\\varepsilon^{-1}\\cdot),$ where $A_{\\rm max}^{\\rm stiff}$ is defined by (\\ref{maternoe_slovo}), Appendix A.\n\n\n\n\n\n\n\n\\begin{remark}\nIt is straightforward to see that the eigenvalue $\\mu^{(\\tau)}$ in Lemma \\ref{mu_lemma} is the least, by absolute value, Steklov eigenvalue of $A_{\\max}^{\\rm stiff}$, {\\it i.e.} the least $\\kappa$ such that the problem\n$$\n\\begin{aligned}\nA_{\\max}^{{\\rm stiff}}\\breve v&=0, \\quad \\breve v\\in W^{2,2}(\\mathbb G^{\\rm stiff}),\\\\[0.3em]\n\\Gamma_1^{\\rm stiff}\\breve v&=\\kappa \\Gamma_0^{\\rm stiff}\\breve v.\n\\end{aligned}\n$$\nadmits a non-trivial solution $\\breve v.$  Note that for this solution $\\breve v$ one has $\\Gamma_0^{\\rm stiff}\\breve v=\\psi^{(\\tau)}.$ It follows that for the function $v$ of Theorem \\ref{eff_thm} one has $v=c\\breve v,$ where $c$ is a constant determined by $\\widehat{u}.$\n\\end{remark}\n\n\n\n\\end{theorem}\n\n\n\n\n\\section{Eigenvalue and eigenvector asymptotics in the example of Section \\ref{our_graph}}\n\nHere we provide the result of an explicit calculation applying the general procedure described in the previous section to the specific example of Section \\ref{our_graph} (see \\cite{ChKisYe} for details).\nWe start by expanding the matrix $B^\\varepsilon$\nas a series in powers of $\\varepsilon$:\n$$\n\\widehat{B}:=\\varepsilon^{-1}B^\\varepsilon=\n\\widehat{B}_0+z\\widehat{B}_1+O(\\varepsilon^2z^2),\\\n\\widehat{B}_0:= { \\frac{1}{\\varepsilon^2 }\\begin{pmatrix} D&\\overline{\\xi}\\\\[0.7em]\n                                                     \\xi& D\n                                                     \\end{pmatrix}},\\ \\widehat{B}_1:=\n                                                     {\\begin{pmatrix}  E & \\overline{\\eta}\\\\[0.7em]\n                                                    \\eta& E\n                                                     \\end{pmatrix}},\n$$\nwhere\n\\begin{align}\n\\xi:&=-\\frac{a_1^2}{l_1}\\exp\\bigl({\\rm i}\\tau(l_1+l_3)\\bigr)-\\frac{a_3^2}{l_3}\\exp(-{\\rm i}\\tau l_2),\\quad\\quad\\quad D:=\\frac{a_1^2}{l_1}+\\frac{a_3^2}{l_3},\n\\label{xi_def}\\\\[0.75em]\n\\eta:&=\\dfrac{1}{6}\\Bigl(l_1\\exp\\bigl({\\rm i}\\tau(l_1+l_3)\\bigr)+l_3\\exp(-{\\rm i}\\tau l_2)\\Bigr),\\quad\\quad\\quad E:=\\dfrac{1}{3}(l_1+l_3).\\nonumber\n\\end{align}\n\n The matrix $\\varepsilon^2 \\widehat{B}_0$ is Hermitian and has two distinct eigenvalues, $\\mu=D-|\\xi|$ and $\\mu_\\bot=D+|\\xi|$. The eigenvalue branch $\\mu$ is singled out by the condition $\\mu\\vert_{\\tau=0}=0$. In order to diagonalise the matrix $\\widehat{B}_0$, consider the normalised eigenvectors $\\psi^{(\\tau)}=(1\/\\sqrt{2})(1,-\\xi\/|\\xi|)^\\top$ and $\\psi^{(\\tau)}_\\bot=(1\/\\sqrt{2})(1,\\xi\/|\\xi|)^\\top$ corresponding to the eigenvalues $\\mu$ and $\\mu_\\bot$, respectively,  and the matrix $X:=\\bigl(\\psi^{(\\tau)}, \\psi^{(\\tau)}_\\bot\\bigr).$\nThe projections ${\\mathcal P},$ ${\\mathcal P}_\\bot$ introduced in the previous section are as follows:\n\\[\n{\\mathcal P}=\\frac{1}{2}\\left(\\begin{array}{cc}1&\\dfrac{\\overline{\\xi}}{\\vert\\xi\\vert}\\\\[1.3em] \\dfrac{\\xi}{\\vert\\xi\\vert} &1\\end{array}\\right),\\quad\\quad {\\mathcal P}_\\bot=\\frac{1}{2}\\left(\\begin{array}{cc}1 &-\\dfrac{\\overline{\\xi}}{\\vert\\xi\\vert}\\\\[1.3em] -\\dfrac{\\xi}{\\vert\\xi\\vert} & 1\\end{array}\\right).\n\\]\n\nIt follows by a straightforward calculation that the effective spectral problem\nis given by\n\\begin{equation}\n-\\biggl(\\frac d{dx}+{\\rm i}\\tau\\biggr)^2u=zu\n\\label{res_eq}\n\\end{equation}\n\\begin{multline}\nu(0)=-\\frac {\\overline{\\xi}}{|\\xi|}u(l_2),\\\\\n(u'+{\\rm i}\\tau u)(0)+\\frac {\\overline{\\xi}}{|\\xi|}(u'+{\\rm i}\\tau u)(l_2)=\\Biggl(\\biggl(\\dfrac{l_1}{a_1^2}+\\dfrac{l_3}{a_3^2}\\biggr)^{-1}\\biggl(\\frac{\\tau}{\\varepsilon}\\biggr)^2-(l_1+l_3)z\\Biggr)u(0),\n\\label{res_bc}\n\\end{multline}\n\nBy invoking Theorem \\ref{eff_thm}, the problem (\\ref{res_eq})--(\\ref{res_bc}) on the scaled soft component provides the asymptotics, as $\\varepsilon\\to0,$ of the eigenvalue problems for the family $A^\\varepsilon_t,$ $t=\\tau\/\\varepsilon\\in[-\\pi\/\\varepsilon,\\pi\/\\varepsilon).$  Its spectrum, {\\it i.e.} the set of values $z$ for which (\\ref{res_eq})--(\\ref{res_bc}) has a non-trivial solution, as well as the corresponding eigenfunctions approximate, up to terms of order $O(\\varepsilon^2),$ the corresponding spectral information for the family $A^\\varepsilon_t,$ and consequently, $A^\\varepsilon.$ Notice that the stiff component of the original graph (where the eigenfunctions converge to a constant, in a suitable scaled sense), appears in this limit problem through the boundary datum $u(0).$ In the next section we show that an appropriate extension of the function space for (\\ref{res_eq})--(\\ref{res_bc}) by the (one-dimensional) complementary space of constants leads to an eigenvalue problem for a self-adjoint operator, describing a conservative system. Solving this latter eigenvalue problem for the element in the complementary space yields a frequency-dispersive formulation we announced in the introduction.\n\n\n\n\n\n\n\n\\section{Frequency dispersion in a ``complementary\" medium}\n\n\\subsection{Self-adjoint out-of-space extension}\n\n\\label{Ahom}\n\nFollowing the strategy outlined at the end of the last section, we treat $u(0)$ in (\\ref{res_bc}) as an additional field variable, and reformulate (\\ref{res_eq})--(\\ref{res_bc}) as an eigenvalue problem in a space of pairs $(u, u(0)),$ see (\\ref{spectral_eq}).\n\nMore precisely, for all values $\\tau\\in[-\\pi, \\pi),$ consider an operator $A^{\\rm hom}_\\tau$ in the space $L^2(0, l_2)\\oplus \\mathbb{C}$\ndefined as follows. The domain $\\text{\\rm dom}\\bigl(A^{\\rm hom}_\\tau\\bigr)$ consist of all pairs $(u,\\beta)$ such that $u\\in W^{2,2}(0, l_2)$ and the quasiperiodicity condition\n\\begin{equation}\nu(0)=\\overline{w_\\tau}u(l_2)=:\\frac{\\beta}{\\sqrt{l_1+l_3}},\\qquad w_\\tau\\in{\\mathbb C},\n\\label{quasi_cond}\n\\end{equation}\nis satisfied.  On $\\text{\\rm dom}\\bigl(A^{\\rm hom}_\\tau\\bigr)$\nthe action of the operator is set by\n\\begin{equation}\nA^{\\rm hom}_\\tau\\left(\\begin{matrix}u\\\\[0.3em] \\beta\\end{matrix}\\right)=\n\\left(\\begin{array}{c}\\biggl(\\dfrac{1}{\\rm i}\\dfrac{d}{dx}+\\tau\\biggr)^2u\\\\[1.1em]\n\\dfrac{1}{\\sqrt{l_1+l_3}}\\Gamma_\\tau\\left(\\begin{matrix}u\\\\[0.3em] \\beta\\end{matrix}\\right)\n\\end{array}\\right),\n\\label{lim_form}\n\\end{equation}\nwhere $\\Gamma_\\tau: W^{2,2}(0, l_2)\\oplus{\\mathbb C}\\to{\\mathbb C}$ is bounded. We set\n\\begin{equation}\n\\Gamma_\\tau\\left(\\begin{matrix}u\\\\[0.3em] \\beta\\end{matrix}\\right)=-(u'+{\\rm i}\\tau u)(0)+\\overline{w_\\tau}\n(u'+{\\rm i}\\tau u)(l_2)+\n\\frac{(\\sigma t)^2}{\\sqrt{l_1+l_3}}\\beta,  \\quad\\sigma^2:=\\biggl(\\dfrac{l_1}{a_1^2}+\\dfrac{l_3}{a_3^2}\\biggr)^{-1},\n\\label{Gamma_part}\n\\end{equation}\nwhere $w_\\tau=-{\\xi}\/{|\\xi|}$ (see \\eqref{xi_def} for the definition of $\\xi$), in which case $A^{\\rm hom}_\\tau$ is a self-adjoint operator on the domain described by (\\ref{quasi_cond}). Moreover, (\\ref{res_eq})--(\\ref{res_bc}) is the problem on the first component of spectral problem for the operator $A^{\\rm hom}_\\tau:$\n\\begin{equation}\nA^{\\rm hom}_\\tau\\left(\\begin{matrix} u\\\\[0.3em] \\beta\\end{matrix}\\right)=z\\left(\\begin{matrix} u\\\\[0.3em] \\beta\\end{matrix}\\right).\n\\label{spectral_eq}\n\\end{equation}\n\nWe now re-write this spectral problem in terms of the complementary component $\\beta\\in{\\mathbb C}.$ In order to do this, we represent the function\n$u$ in (\\ref{spectral_eq}) as a sum of two: one of them is a solution to the related inhomogeneous Dirichlet problem, while the other takes care of the boundary condition. More precisely, consider the solution $v$ to the problem\n\\begin{equation*}\n-\\biggl(\\frac{d}{dx}+{\\rm i}\\tau\\biggr)^2v=0,\\qquad\\qquad\nv(0)=1,\\ \\ \\ \\ \\ v(l_2)=w_\\tau,\n\\end{equation*}\n{\\it i.e.}\n\\begin{equation}\nv(x)=\\Bigl\\{1+l_2^{-1}\\Bigl(w_\\tau\\exp({\\rm i}\\tau l_2)-1\\Bigr)x\\Bigr\\}\\exp(-{\\rm i}\\tau x),\\quad\\quad x\\in(0, l_2).\n\\label{function_v}\n\\end{equation}\nThe function\n\\[\n\\widetilde{u}:=u-\\frac{\\beta}{\\sqrt{l_1+l_3}}v\n\\]\nsatisfies\n\\begin{equation*}\n-\\biggl(\\frac{d}{dx}+{\\rm i}\\tau\\biggr)^2\\widetilde{u}-z\\widetilde{u}=\\frac{z\\beta}{\\sqrt{l_1+l_3}}v,\\quad\\qquad\n \\widetilde{u}(0)=\\widetilde{u}(l_2)=0.\n\n\\end{equation*}\nIn other words, one has\n\\begin{equation*}\n\\widetilde{u}=\\frac{z\\beta}{\\sqrt{l_1+l_3}}(A_{\\rm D}-zI)^{-1}v,\n\\end{equation*}\nwhere $A_{\\rm D}$ is the Dirichlet operator in $L^2(0, l_2)$ associated with the differential expression\n\\[\n-\\biggl(\\frac{d}{dx}+{\\rm i}\\tau\\biggr)^2.\n\\]\nWe can now write the ``boundary''  part of the spectral equation (\\ref{spectral_eq}) as\n\\begin{equation}\nK(\\tau, z)\\beta=z\\beta,\\quad K(\\tau, z):=\\dfrac{1}{l_1+l_3}\\left\\{z\\Gamma_\\tau\\left(\\begin{matrix\n(A_{\\rm D}-zI)^{-1}v\\\\[0.3em] 0\\end{matrix}\\right)+\n\\Gamma_\\tau\\left(\\begin{matrix}v\\\\[0.3em] \\sqrt{l_1+l_3}\\end{matrix}\\right)\\right\\}.\n\\label{K_expr}\n\\end{equation}\nIn accordance with the rationale for introducing the component $\\beta,$ the effective dispersion relation for the operator $A_{\\tau\/\\varepsilon}^\\varepsilon,$\n$\\tau\\in[-\\pi,\\pi),$ is given by\n\\[\nK(\\tau, z)=z.\n\\]\nThe explicit expression for this relation that we have obtained, see (\\ref{K_expr}), is new, and it quantifies explicitly the r\\^ole of the soft component of the composite in the macroscopic frequency-dispersive properties. In particular, the expression (\\ref{K_expr}) shows that the soft inclusions enter the macroscopic equations via a Dirichlet-to-Neumann map on the boundary of the inclusions.\n\n\\subsection{Explicit formula for the time-dispersion kernel}\n\nHere we compute explicitly the kernel $K(\\tau, z)$ entering the effective dispersion relation for $A_\\tau^\\varepsilon.$ In view of possible generalisations, and recalling the pioneering formula in \\cite[Section 8]{Jikov} for effective dispersion in double-porosity media, we represent the action of the resolvent $(A_{\\rm D}-zI)^{-1}$ as a series in terms of the normalised eigenfunctions\n\\begin{equation}\n\\phi_j(x)=\\sqrt{\\frac{2}{l_2}}\\exp(-{\\rm i}\\tau x)\\sin\\frac{\\pi jx}{l_2},\\qquad x\\in(0,l_2),\\qquad\\qquad j=1,2,3,\\dots,\n\\label{function_phi}\n\\end{equation}\nof the operator $A_{\\rm D}.$ This yields\n\\begin{equation}\nK(\\tau, z):=\\dfrac{1}{l_1+l_3}\\left\\{z\\sum_{j=1}^\\infty\\dfrac{\\langle v, \\phi_j\\rangle}{\\mu_j-z}\\Gamma_\\tau\\left(\\begin{matrix\n\\varphi_j\\\\[0.3em] 0\\end{matrix}\\right)+\n\\Gamma_\\tau\\left(\\begin{matrix}v\\\\[0.3em] \\sqrt{l_1+l_3}\\end{matrix}\\right)\\right\\}.\n\\label{K_general1}\n\\end{equation}\nwhere $\\mu_j=(\\pi j\/l_2)^2,$ $j=1,2,3,\\dots,$ are the eigenvalues corresponding to (\\ref{function_phi}). For the choice (\\ref{Gamma_part}) of $\\Gamma_\\tau$ we obtain (see (\\ref{function_v}), (\\ref{function_phi}))\n\\[\n\\Gamma_\\tau\\left(\\begin{matrix}v\\\\[0.3em] \\sqrt{l_1+l_3}\\end{matrix}\\right)\n=\\frac{2}{l_2}\\bigl(1-\\Re\\theta(\\tau)\\bigr)+\\biggl(\\frac{\\sigma\\tau}{\\varepsilon}\\biggr)^2,\\qquad \\theta(\\tau):=\\frac{\\dfrac{a_1^2}{l_1}{\\rm e}^{-{\\rm i}\\tau}+\\dfrac{a_3^2}{l_3}}{\\biggl|\\dfrac{a_1^2}{l_1}{\\rm e}^{-{\\rm i}\\tau}+\\dfrac{a_3^2}{l_3}\\biggr|},\n\\]\n\\[\n\\Gamma_\\tau\\left(\\begin{matrix\n\\varphi_j\\\\[0.3em] 0\\end{matrix}\\right)=-\\sqrt{\\frac{2}{l_2}}\\frac{\\pi j}{l_2}\\bigl((-1)^{j+1}\\overline{\\theta(\\tau)}+1\\bigr),\\\n\\langle v, \\phi_j\\rangle=\\frac{\\sqrt{2l_2}}{\\pi j}\\bigl((-1)^{j+1}\\theta(\\tau)+1\\bigr),\\ j=1,2,\\dots\n\\]\nSubstituting the above expressions into (\\ref{K_general1}) and making use of  the formulae, see {\\it e.g.} \\cite[p.\\,48]{Gradshteyn_Ryzhik},\n\\begin{equation*}\n\\sum_{j=1}^\\infty\\frac{1}{(\\pi j)^2-x^2}=\\frac{1}{2}\\biggl(\\frac{1}{x^2}-\\frac{\\cos x}{x\\sin x}\\biggr),\\quad\n\\sum_{j=1}^\\infty\\frac{(-1)^j}{(\\pi j)^2-x^2}=\\frac{1}{2}\\biggl(\\frac{1}{x^2}-\\frac{1}{x\\sin x}\\biggr),\\quad x\\notin\\pi{\\mathbb Z},\n\\end{equation*}\nwe obtain\n\n\\begin{equation}\nK(\\tau, z)=\\frac{1}{l_1+l_3}\\biggl\\{\n\\frac{2\\sqrt{z}\\cos(l_2\\sqrt{z})}{\\sin(l_2\\sqrt{z})}\n-\\frac{2\\sqrt{z}}{\\sin(l_2\\sqrt{z})}\\Re\\theta(\\tau)+\\biggl(\\frac{\\sigma\\tau}{\\varepsilon}\\biggr)^2\\biggr\\}.\n\\label{K_example}\n\\end{equation}\n\n\n\\subsection{Asymptotically equivalent model on the real line}\n\nIn this section we are going to treat (\\ref{K_expr}), (\\ref{K_example}) as a nonlinear eigenvalue problem in the space of second components of pairs\n$(u, \\beta)= L^2(0, l_2)\\oplus{\\mathbb C}.$ As is evident from above, this problem is closely related to (\\ref{res_eq})--(\\ref{res_bc}), via the construction presented in Section \\ref{Ahom}.\nWe show next that the aforementioned macroscopic field is governed by a certain frequency-dispersive formulation. In order to obtain the latter,\nwe will use a suitable inverse Gelfand transform.\n\nOur strategy can be seen as motivated by the following elementary observation, closely linked with the Birman-Suslina study of homogenisation in the moderate contrast case, albeit understood in terms of spectral equations. Starting with the spectral problem\n\\begin{equation}\\label{eq:b-s-problem}\n-\\frac {d^2u}{dx^2}=zu \\text{\\ \\ on\\ \\ } L_2(\\mathbb R),\n\\end{equation}\none applies the Gelfand transform\\footnote{Recall, {\\it cf.} Section \\ref{Gelfand_section}, that the Gelfand transform is a map\n$L^2({\\mathbb R})\\to L^2\\bigl((0, \\varepsilon)\\times(-\\pi\/\\varepsilon,\\pi\/\\varepsilon)\\bigr)$ given by\n\\begin{equation*}\n{\\mathcal G}u(y, t)=\\sqrt{\\frac{\\varepsilon}{2\\pi}}\\sum_{n\\in{\\mathbb Z}}u(x+\\varepsilon n)\\exp\\bigl(-{\\rm i}t(x+\\varepsilon n)\\bigr),\\qquad t\\in\\bigl[-\\pi\/\\varepsilon, \\pi\/\\varepsilon\\bigr),\\qquad x\\in(0,\\varepsilon).\n\\end{equation*}} (well-defined on generalised eigenvectors due to the rigging procedure, see, {\\it e.g.,} \\cite{Berezansky,BS}) to obtain for $\\widetilde u:=\\mathcal G u$\n$$\n-\\biggl(\\frac d {dx}+i t\\biggr)^2 \\widetilde u(x,t) = z \\widetilde u(x,t), \\quad x\\in(0,\\varepsilon),\\quad t\\in[-\\pi\/\\varepsilon,\\pi\/\\varepsilon).\n$$\nWe compute the inner products of both sides in $L_2(0,\\varepsilon)$ with the normalised constant function $(1\/\\sqrt{\\varepsilon})\\mathbbm 1$, which yields the dispersion relation of the original problem via the equation\n$$\nt^2 \\widehat u (t)=z \\widehat u(t),\n$$\nwhere $\\widehat u$ is the Fourier transform of the function $u\\in L_2(\\mathbb R)$. The latter equation is then solved in the distributional sense,\n\\begin{equation}\\label{eq:beta}\n\\beta (t)=\\sum_{m} c_m \\delta(t-t_m),\n\\end{equation}\nwhere $\\beta (t):=\\widehat u(t)$ and the sum in \\eqref{eq:beta} is taken over $m=1,2$,  $t_1, t_2$ being the zeroes of the equation $t^2=z$ and $c_m$ are arbitrary constants. Ultimately, one applies the inverse Gelfand transform\n\\[\n({\\mathcal G}^*f)(x)=\\sqrt{\\frac{\\varepsilon}{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}f(t)\\exp({\\rm i}tx)dt,\\quad f\\in L^2\\biggl(-\\frac{\\pi}{\\varepsilon}, \\frac{\\pi}{\\varepsilon}\\biggr),\\qquad x\\in{\\mathbb R},\n\\]\nto the function $\\mathfrak B (x,t):=(1\/\\sqrt{\\varepsilon})\\beta(t)\\mathbbm 1(x),$ {\\it i.e.}\n$$\nv(x):=\\sqrt{\\frac{\\varepsilon}{2\\pi}}\\int_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon} \\mathfrak B(x,t) \\exp(i t x) dt, \\qquad x\\in{\\mathbb R}.\n$$\nIt is easily seen that this function is precisely the solution to \\eqref{eq:b-s-problem}.\n\nWe emulate the above argument for the case of interest to us, starting from the eigenvalue problem\n$K(\\tau, z)\\beta=z\\beta,$ which we now treat as an equation in the distributional sense with $K$ given by (\\ref{K_example}). It admits  the form\n\\begin{equation}\n(\\sigma t)^2\\beta=\\biggl\\{(l_1+l_3)z-\\frac{2\\sqrt{z}\\cos(l_2\\sqrt{z})}{\\sin(l_2\\sqrt{z})}+\\frac{2\\sqrt{z}}{\\sin(l_2\\sqrt{z})}\\Re \\theta(\\varepsilon t)\\biggr\\}\\beta,\\qquad t=\\frac{\\tau}{\\varepsilon},\n\\label{spectral_final}\n\\end{equation}\nThe solution is defined by \\eqref{eq:beta}, where $\\{t_m\\}$ is the set of zeroes of the equation $K(\\varepsilon t,z)=z$.\n\nSecond, we argue that the function $\\mathfrak B(x,t)$ as  defined above\nis the $\\varepsilon$-periodic Gelfand transform of the solution to a spectral equation on ${\\mathbb R}$ for a differential operator with constant coefficients, where the conventional spectral parameter $z$ is replaced by a nonlinear in $z$ expression,  as on the right-hand side of (\\ref{spectral_final}).\n\nIndeed, expand the function $\\Re\\theta(\\tau)$ into Fourier series\n\\[\n\\Re\\theta(\\tau)=\\frac{1}{\\sqrt{2\\pi}}\\sum_{n=-\\infty}^\\infty c_n\\exp({\\rm i}n\\tau),\\qquad\nc_n:=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\pi}^{\\pi}\\Re\\theta(\\tau)\\exp(-{\\rm i}n\\tau)d\\tau,\\qquad n\\in{\\mathbb Z}.\n\\]\nand apply to $\\mathfrak B(x,t)$  the inverse\nGelfand transform ${\\mathcal G}^*:$\n\n\\[\n({\\mathcal G}^*f)(x)=\\sqrt{\\frac{\\varepsilon  }{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}f(t)\\exp({\\rm i}tx)dt,\\quad f\\in L^2\\biggl(-\\frac{\\pi}{\\varepsilon  }, \\frac{\\pi}{\\varepsilon  }\\biggr),\\qquad x\\in{\\mathbb R}.\n\\]\nWe denote $U:={\\mathcal G}^*\\mathfrak B$ and notice that\n\\[\n\\sqrt{\\frac{\\varepsilon  }{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}t^2\\mathfrak B(x,t)\\exp({\\rm i}tx)dt=-\\frac{d^2}{dx^2}\\Biggl(\\sqrt{\\frac{\\varepsilon  }{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}\\mathfrak B(x,t)\\exp({\\rm i}tx)dt\\Biggr)=-U''(x)\n\\]\nand\n\\begin{align*}\n&\\sqrt{\\frac{\\varepsilon  }{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}\\Re\\theta(\\varepsilon t)\\mathfrak B(x,t)\\exp({\\rm i}tx)dt=\\sum_{n=-\\infty}^\\infty c_n{\\frac{\\sqrt{\\varepsilon}  }{2\\pi}}\\int\\limits_{-\\pi\/\\varepsilon}^{\\pi\/\\varepsilon}\\mathfrak B(x,t)\\exp\\bigl({\\rm i}t(x+\\varepsilon   n)\\bigr)dt\\\\\n\\\\\n&=\\frac{1}{\\sqrt{2\\pi}}\\sum_{n=-\\infty}^\\infty c_n U(x+\\varepsilon   n)\n\\sim\n\\frac{1}{\\sqrt{2\\pi}}\\sum_{n=-\\infty}^\\infty c_n U(x)\n=\\Re\\theta(0) U(x)=U(x),\\qquad \\varepsilon\\to0.\n\\end{align*}\n\nThe above asymptotics as $\\varepsilon\\to0$ is understood in the sense of $W^{-2,2}(\\mathbb R).$ It can be demonstrated, see \\cite{ChKisYe}, that the order of convergence is $O(\\varepsilon^{2})$ (and $O(\\varepsilon)$ in the general case), however we do not dwell on the complete proof here. The idea of the proof, which is standard, can be, for example, the following. Instead of the function $\\beta,$ define $\\beta^0$ by the expression  (\\ref{eq:beta}), where the sequence $\\{t_m\\}$ is replaced by the sequence $\\{t_m^0\\}$ of zeros of the equation $K^0(\\tau, z)=z.$ Here $K^0$ is defined by (\\ref{K_example}) with $\\Re\\theta(\\tau)$ replaced by $\\Re\\theta(0)=1.$ It is then shown that $\\beta$ is $O(\\varepsilon^{2})$-close, in the sense of distributions, to $\\beta^0,$ from where one obtains the claim by taking the inverse Gelfand transform of the function ${\\mathfrak B}^0(x, t)=(1\/\\sqrt{\\varepsilon})\\beta^0(t){\\mathbbm 1}(x).$\n\nIt follows that the limit equation on the  function $U$ takes the form\n\\begin{equation}\n-\\sigma^2 \\,U''(x)\n=\\biggl\\{(l_1+l_3)z+2\\sqrt{z}\\tan\\biggl(\\frac{l_2\\sqrt{z}}{2}\\biggr)\\biggr\\}U(x),\\qquad x\\in{\\mathbb R}.\n\\label{limit_spectral}\n\\end{equation}\nIn particular, the limit spectrum is given by the set of $z\\in{\\mathbb R}$ for which the expression in brackets on the right-hand side of (\\ref{limit_spectral}) is non-negative, see Fig.\\,\\ref{fig:tangens}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1]{tangens.pdf}\n\\end{center}\n\\caption{{\\scshape Dispersion function.} {\\small The plot of the dispersion function on the right-hand side of (\\ref{limit_spectral}), for $L=0.2.$ The spectral gaps are highlighted in bold.}}\n\\label{fig:tangens}\n\\end{figure}\n\n\n\n\n\n\n\\section*{Appendix A: The reduction of the general case to the one treated in Section \\ref{sect:asymp_diag}}\\label{App_Kis}\n\nWe proceed as follows. First, we decompose the graph $\\widehat{\\mathbb G}$ into the union of its stiff and soft components, $\\widehat{\\mathbb G}=\\mathbb G^{\\text{soft}}\\cup \\mathbb G^{\\text{stiff}}$, each of these being a graph on its own. The common boundary of them is $\\partial \\mathbb G:=\\mathbb G^{\\text{soft}}\\cap \\mathbb G^{\\text{stiff}},$ and it is treated as a set of vertices. Second, we consider two maximal operators $\\breve A_{\\max}^{\\text{soft}}$ and $\\breve A_{\\max}^{\\text{stiff}},$ which are densely defined in $L_2(\\mathbb G^{\\text{soft}})$ and $L_2(\\mathbb G^{\\text{stiff}})$, respectively, by \\eqref{diff_expr}, \\eqref{domAmax} applied to $\\mathbb G^{\\text{soft}}$ and $\\mathbb G^{\\text{stiff}}$. Furthermore, we introduce the orthogonal projections $P^{\\text{soft}}, P^{\\text{stiff}}$ in the boundary space $\\mathcal H$ onto the subspaces pertaining to vertices of $\\mathbb G^{\\text{soft}}$ and $\\mathbb G^{\\text{stiff}}$, respectively. Finally, we construct boundary triples for $\\breve A_{\\max}^{\\text{soft (stiff)}}$  with boundary spaces $P^{\\text{soft (stiff)}}\\mathcal H$ and boundary operators $\\breve\\Gamma_j^{\\text{soft (stiff)}}$, $j=0,1$ ({\\it cf.} \\eqref{boundary_operators}), respectively.\n\nNow consider the restrictions\n\\begin{equation}\n\\begin{aligned}\n&A_{\\max}^{\\text{soft (stiff)}}=\\breve A_{\\max}^{\\text{soft (stiff)}}\\big|_{\\dom(A_{\\max}^{\\text{soft (stiff)}})},\\\\[0.4em]\n&\\dom\\bigl(A_{\\max}^{\\text{soft (stiff)}}\\bigr):=\\Bigl\\{u\\in \\dom\\bigl(\\breve A_{\\max}^{\\text{soft (stiff)}}\\bigr)\\Big| (1-P_{\\partial \\mathbb G})\\breve\\Gamma_1^{\\text{soft (stiff)}}u=0\\Bigr\\},\n\\end{aligned}\n\\label{maternoe_slovo}\n\\end{equation}\nwhere $P_{\\partial \\mathbb G}$ is defined as an orthogonal projection in $\\mathcal H$ onto the subspace pertaining to the vertices belonging to $\\partial \\mathbb G$. For these two maximal operators, one has the common boundary space $P_{\\partial\\mathbb G}\\mathcal H$ and boundary operators defined by\n$$\n\\Gamma_j^{\\text{soft (stiff)}}:=P_{\\partial\\mathbb G} \\breve\\Gamma_j^{\\text{soft (stiff)}},\\quad j=0,1.\n$$\nThe corresponding $M$-matrices $M^{\\text{soft (stiff)}}$ are computed as inverses of the matrices $$P_{\\partial \\mathbb G}\\bigl(\\breve M^{\\text{soft (stiff)}}\\bigr)^{-1}P_{\\partial \\mathbb G},$$ where the latter are considered in the reduced space $P_{\\partial \\mathbb G} \\mathcal H$ and $\\breve M^{\\text{soft (stiff)}}$ are $M$-matrices of $\\breve A_{\\max}^{\\text{soft (stiff)}}$ relative to the boundary triples\n$\\bigl(P^{\\text{soft (stiff)}}\\mathcal H, \\breve\\Gamma_0^{\\text{soft (stiff)}}, \\breve\\Gamma_1^{\\text{soft (stiff)}}\\bigr)$.\n\nIt is easily shown that the operator $A^\\varepsilon_t$ is expressed as an almost solvable extension parameterised by the matrix $B=0$ relative to a triple which has the $M$-matrix $M=M^{\\text{soft}}+M^{{\\rm stiff}}$. It follows that all the prerequisites of the analysis carried out in Section \\ref{sect:asymp_diag} are met.\n\n\n\n\n\n\\section*{Appendix B: Proof of Lemma \\ref{mu_lemma}}\n\n\nThe proof could be carried out on the basis of \\cite{Yorzh3}, \\cite{Yorzh4} and is rather elementary. Nevertheless, in the present paper we have elected to follow an alternative approach to this proof, which has an advantage of carrying over to the PDE case with minor modifications.\n\nFor simplicity we set $w_V(e)=1$ for all $e, V$ in (\\ref{Atau1}), as the argument below is unaffected by the concrete choice of the list $\\{w_V(e)\\}_{e\\ni V},$ $V\\in\\widehat{\\mathbb G},$ in the construction of Section \\ref{Gelfand_section}. For convenience, we also imply that the unitary rescaling to a graph of length one has been applied to the  operator family $A_t^\\varepsilon$. For brevity, we keep the same notation for the unitary images of graphs $\\widehat{\\mathbb{G}}$, $\\mathbb{G}^{\\rm stiff}$ and $\\partial \\mathbb G$ under this transform.\n\nFor each $\\tau\\in[-\\pi, \\pi),$ the eigenvalues of $B_0(\\tau)$ are those $\\mu\\in{\\mathbb C}$ for which there exists $u\\neq0$ satisfying\n\\begin{equation}\n\\left\\{\\begin{array}{ll}\\biggl(\\dfrac{d}{dx}+{\\rm i}\\tau\\biggr)^2u=0\\quad{\\rm in}\\ {\\mathbb G}^{\\rm stiff}, \\\\[1.2em]\n-\\sum_{e\\ni V}\\sigma_e\\bigl(u'_{e}(V)+{\\rm i}\\tau u(V)\\bigr)=\\mu u(V),\\quad V\\in\\partial{\\mathbb G},\\\\[1em]\nu\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff},\n\\end{array}\\right.\n\\label{problem}\n\\end{equation}\nwhere $u'_{e}(V)$ is the derivative of $u$ along the edge $e$ of ${\\mathbb G}^{\\rm stiff}$ evaluated at $V\\in\\partial{\\mathbb G},$ and, as before,\n$\\sigma_{e}=-1$ or $\\sigma_{e}=1,$ depending on whether $e$ is  incoming or outgoing for $V,$ respectively.\nIt is known that the spectrum of (\\ref{problem}) is discrete and the least eigenvalue, which clearly coincides with $\\mu^{(\\tau)},$ is simple.\n\n{\\it Formal series.}  In order to show (\\ref{mu_asymptotics}), we first consider  series in powers of ${\\rm i}\\tau:$\n\\begin{equation}\n\\mu=\\sum_{k=1}^\\infty\\alpha_j({\\rm i}\\tau)^{2k},\\qquad\n  u=\\sum_{j=0}^\\infty u_j({\\rm i}\\tau)^j,\n  \\label{expansion}\n\\end{equation}\nwhere $u_j,$ $j=1,2,\\dots$ are continuous on ${\\mathbb G}^{\\rm stiff}.$\n\n\nNote that the expansion for $\\mu$ contains only even powers of the parameter $\\tau,$ as it is an even function of $\\tau.$ Indeed, the function obtained from the eigenfunction $u$ in (\\ref{problem}) by changing the directions of all edges of the graph is clearly an eigenfunction for (\\ref{problem}) with $\\tau$  replaced by $-\\tau.$ (On such a change of edge direction, the weights $w_e(V),$ ${e\\ni V},$ $V\\in\\widehat{\\mathbb G},$ are replaced by their complex conjugates.) In view of the fact that for all $\\tau\\in(-\\pi,\\pi]$ the eigenvalue $\\mu^{(\\tau)}$ is simple, we obtain $\\mu^{(-\\tau)}=\\mu^{(\\tau)}.$\n\nSubstituting the expansion (\\ref{expansion}) into (\\ref{problem}) and equating the coefficients on different powers of\n$\\tau,$ we obtain a sequence of recurrence relations for $u_j,$ $j=0,1,\\dots$  In particular, the problem for $u_0$ is obtained by comparing the coefficients on $\\tau^0:$\n\\[\n\\left\\{\\begin{array}{lll}u_0''=0\\ \\ \\ {\\rm on}\\ \\ {\\mathbb G}^{\\rm stiff}, \\\\[0.5em]\n\\sum_{e\\ni V}\\sigma_e(u_0)_e'(V)=0,\\quad V\\in\\partial{\\mathbb G},\\\\[0.6em]\nu_0\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff}.\n\n\\end{array}\\right.\n\\]\nAssuming that ${\\mathbb G}^{\\rm stiff}$ contains a loop, it follows that\n$u_0$ is a constant, which we set to be unity. In the case opposite, i.e., when  ${\\mathbb G}^{\\rm stiff}$ is a tree, $\\mu^{(\\tau)}\\equiv 0$  for all $\\tau$,  and the claim of Lemma follows trivially.\n\nWe impose the condition of vanishing mean of $u_j,$ $j=1,2,\\dots$ over ${\\mathbb G}^{\\rm stiff}.$ This is justified by the convergence estimates below as well as the fact that the eigenvalue $\\mu$ is simple. The choice $u_0=1$ thus corresponds to the ``normalisation\" condition that the mean over ${\\mathbb G}^{\\rm stiff}$ of the eigenfunction $u$ for (\\ref{problem}) is close to unity\\footnote{The eigenfunction $u$ clearly does not vanish identically, at least for small values of $\\tau.$} for small values of $\\tau.$\n\n\nProceeding with the asymptotic procedure, the problem for $u_1$ is obtained by comparing the coefficients on $\\tau^1:$\n\\[\n\\left\\{\\begin{array}{ll}u''_1=0\\ \\ {\\rm on}\\ \\ {\\mathbb G}^{\\rm stiff}, \\\\[0.7em]\n\\sum_{e\\ni V}\\sigma_e\\bigl((u_1)_e'(V)+1\\bigr)=0,\\quad V\\in\\partial{\\mathbb G},\\\\[0.8em]\nu_1\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff},\\\\[0.8em]\n\\int_{{\\mathbb G}^{\\rm stiff}}u_1=0.\n\\end{array}\\right.\n\\]\nFurther, the equation for $u_2$ is obtained by comparing the coefficients on  $\\tau^2:$\n\\begin{equation}\n\\left\\{\\begin{array}{ll}u''_2=-2u'_1-1 \\ \\ {\\rm on}\\ \\ {\\mathbb G}^{\\rm stiff}, \\\\[1.1em]\n-\\sum_{e\\ni V}\\sigma_e\\bigl((u_2)_e'(V)+u_1(V)\\bigr)=\\alpha_2,\\quad V\\in\\partial{\\mathbb G},\\\\[1.2em]\nu_2\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff},\\\\[0.8em]\n\\int_{{\\mathbb G}^{\\rm stiff}}u_2=0.\n\\end{array}\\right.\n\\label{u_2}\n\\end{equation}\n The condition for solvability of the problem (\\ref{u_2}) yields the expression for $\\alpha_2,$ as follows:\n\\[\n\\int_{{\\mathbb G}^{\\rm stiff}}(-2u'_1-1)=\\int_{{\\mathbb G}^{\\rm stiff}}u''_2=-\\sum_{V\\in\\partial{\\mathbb G}}\\ \\sum_{e\\ni V}\\sigma_e(u_2)_e'(V)\n=\\sum_{V\\in\\partial{\\mathbb G}}\\Bigl(\\sum_{e\\ni V}\\sigma_e u_1(V)+\\alpha_2\\Bigr).\n\\]\nRe-arranging the terms in the last equation, we obtain\n\\[\n\\alpha_2=-\\bigl\\vert\\partial{\\mathbb G}\\bigr\\vert^{-1}\\int_{{\\mathbb G}^{\\rm stiff}}(u'_1+1).\n\\]\nThe above asymptotic procedure is continued, to obtain the terms of all orders in (\\ref{expansion}). In particular, for the term $u_3$ in the expansion for $u$ we obtain\n\\begin{equation*}\n\\left\\{\\begin{array}{ll}u''_3=-2u'_2-u_1 \\ \\ {\\rm on}\\ \\ {\\mathbb G}^{\\rm stiff}, \\\\[1.1em]\n-\\sum_{e\\ni V}\\sigma_e\\bigl((u_3)_e'(V)+u_2(V)\\bigr)=\\alpha_2u_1,\\quad V\\in\\partial{\\mathbb G},\\\\[1.2em]\nu_3\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff},\\\\[0.8em]\n\\int_{{\\mathbb G}^{\\rm stiff}}u_3=0.\n\\end{array}\\right.\n\\label{u_3}\n\\end{equation*}\n\n\n{\\it Error estimates.}\nWe write\n\\[\nu=1+{\\rm i}\\tau u_1+({\\rm i}\\tau)^2u_2+({\\rm i}\\tau)^3u_3+R,\\qquad \\mu^{(\\tau)}=\\alpha_2({\\rm i}\\tau)^2+r,\n\\]\nso that $R,$ $r$ satisfy\n\\begin{empheq}[right=\\empheqrbrace]{align}\n&\\biggl(\\dfrac{d}{dx}+{\\rm i}\\tau\\biggr)^2R=-({\\rm i}\\tau)^4(2u_3'+u_2)-({\\rm i}\\tau)^5u_3\\quad \\text{ on } \\mathbb{G}^{\\rm stiff},\\label{R_equation}\n\\\\[0.4em]\n&-\\sum_{e\\ni V}\\sigma_e (R'_{e}(V)+{\\rm i}\\tau R(V))=\\label{bc}\\\\\n&=\\bigl(r+\\alpha_2({\\rm i}\\tau)^2\\bigr)\n\\bigl(1+{\\rm i}\\tau u_1+({\\rm i}\\tau)^2u_2+({\\rm i}\\tau)^3u_3+R\\bigr)\\nonumber\\\\[0.5em]\n&-\\alpha_2({\\rm i}\\tau)^2(1+{\\rm i}\\tau u_1),\\quad V\\in\\partial \\mathbb{G}\\nonumber\\\\[0.4em]\n&R\\ {\\rm continuous\\ on\\ }{\\mathbb G}^{\\rm stiff},\\nonumber\\\\[0.4em]\n&\\int_{{\\mathbb G}^{\\rm stiff}}R=0.\\ \\ \\ \\ \\ \\ &\\nonumber\n\\end{empheq}\n\nNotice first that\n\\begin{multline}\nr+\\alpha_2({\\rm i}\\tau)^2=\\mu^{(\\tau)}=\\min_{u\\in W^{2,2}({\\mathbb G}^{\\rm stiff})}\\biggl(\\sum_{\\partial{\\mathbb G}}\\vert u\\vert^2\\biggr)^{-1}\\int_{{\\mathbb G}^{\\rm stiff}}\\Biggl\\vert\\biggl(\\dfrac{d}{dx}+{\\rm i}\\tau\\biggr)u\\Biggr\\vert^2 \\le\\bigl\\vert\\partial{\\mathbb G}\\bigr\\vert^{-1}\\bigl\\vert {\\mathbb G}^{\\rm stiff}\\bigr\\vert\\tau^2.\n\\label{mu_est}\n\\end{multline}\nMultiplying (\\ref{R_equation}) by $R$, integrating by parts, and using (\\ref{bc}), we obtain the estimate\n\\begin{equation}\n\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}^2\\le C\\bigl(\\vert\\tau\\vert\\vert r\\vert\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}+\\vert\\tau\\vert^4\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}+\\vert r\\vert^2\\bigr),\\qquad C>0,\n\\label{R_est}\n\\end{equation}\nand hence, by virtue of (\\ref{mu_est}), we obtain\n\\begin{equation}\n\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}\\le C\\tau^2.\n\\label{first_R_estimate}\n\\end{equation}\n\nNext, we re-arrange the right-hand side of (\\ref{bc}):\n\\begin{multline*}\n\\bigl(r+\\alpha_2({\\rm i}\\tau)^2\\bigr)\n\\bigl(1+{\\rm i}\\tau u_1+({\\rm i}\\tau)^2u_2+({\\rm i}\\tau)^3u_3+R\\bigr)-\\alpha_2({\\rm i}\\tau)^2(1+{\\rm i}\\tau u_1)\\\\[0.5em]=r\\bigl(1+{\\rm i}\\tau u_1+({\\rm i}\\tau)^2u_2+({\\rm i}\\tau)^3u_3+R\\bigr)+\\alpha_2({\\rm i}\\tau)^2\\bigl(({\\rm i}\\tau)^2u_2+({\\rm i}\\tau)^3u_3+R\\bigr).\n\\end{multline*}\nMultiplying  (\\ref{R_equation}) by $1$, integrating by parts, and using (\\ref{bc}) once again yields the existence of  $C>0$ such that\n\\begin{equation}\n\\vert r\\vert\\le C\\bigl(\\vert\\tau\\vert\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}+\\vert\\tau\\vert^4\\bigr).\n\\label{r_estimate}\n\\end{equation}\nCombining this with (\\ref{first_R_estimate}) yields $\\vert r\\vert\\le C\\tau^3,$ which, by virtue of (\\ref{R_est}) again, implies\n\\begin{equation}\n\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}\\le C\\vert\\tau\\vert^3.\n\\label{second_R_estimate}\n\\end{equation}\nFinally, the inequalities (\\ref{r_estimate}) and (\\ref{second_R_estimate}) together yield\n\\begin{equation}\n\\vert r\\vert\\le C|\\tau|^4,\n\\label{r_final}\n\\end{equation}\nas claimed.\\footnote{Combining (\\ref{r_final}) with (\\ref{mu_est}), we also obtain the estimate $\\Vert R\\Vert_{L^2({\\mathbb G}^{\\rm stiff})}\\le C\\tau^4.$}\n\n\n\n\n\\section*{Appendix C: Proof of Lemma \\ref{bound_below_lemma}}\n\n\nFor all $\\tau\\in[-\\pi,\\pi),$ using the formula for the second eigenvalue $\\mu_2^{(\\tau)}$ of the problem (\\ref{problem}) via the Rayleigh quotient, we obtain\n\\begin{align*}\n\\mu_2^{(\\tau)}&=\\min\\Biggl\\{\\biggl(\\sum_{\\partial{\\mathbb G}}\\vert u\\vert^2\\biggr)^{-1}\\int_{{\\mathbb G}^{\\rm stiff}}\\Biggl\\vert\\biggl(\\dfrac{d}{dx}+{\\rm i}\\tau\\biggr)u\\Biggr\\vert^2: u\\in W^{2,2}({\\mathbb G}^{\\rm stiff}), \\int_{{\\mathbb G}^{\\rm stiff}}u=0\\Biggr\\}\\\\[0.9em]\n&\\ge \\min\\Biggl\\{\\biggl(\\sum_{\\partial{\\mathbb G}}\\vert u\\vert^2\\biggr)^{-1}\\int_{{\\mathbb G}^{\\rm stiff}}\\vert u'\\vert^2: u\\in W^{2,2}({\\mathbb G}^{\\rm stiff}), \\int_{{\\mathbb G}^{\\rm stiff}}u=0\\Biggr\\}=\\mu_2^{(0)}>0,\n\\end{align*}\nfrom which the claim follows by setting $C_\\perp=\\mu_2^{(0)}.$\n\n\n\\section*{Acknowledgements}\n\nWe are grateful to Professor S. Naboko for suggesting a calculation in Section 8.\n\n\\bibliographystyle{siamplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{introduction}\n\n\n\nLet $f = \\sum_{j = 0}^d a_j X^j\\in \\mathbb{R}[X]$ be a univariate polynomial\nof degree $d \\in \\mathbb{Z}^+$.  It is a classical result due to\nNewton (see \\cite{HLP}, \\S2.22 and \\S4.3 for two proofs) that\nwhenever all the roots of $f$ are real, then the coefficients of $f$\nsatisfy the following log-concavity condition:\n\n\\begin{equation} a_i^2 \\geq \\frac{d-i+1}{d-i} \\frac{i+1}{i}\\, a_{i-1} a_{i+1}\n{\\rm \\ for \\ all \\ } i \\in\n\\{1,\\ldots,d-1\\}.\\label{newton}\\end{equation} Moreover, if the roots\nof $f$ are not all equal, these inequalities are strict. When $d =\n2$, condition (\\ref{newton}) becomes $a_1 \\geq 4 a_0 a_2$, which is\nwell known to be a necessary and sufficient condition for all the\nroots of $f$ to be real. Nevertheless, for $d \\geq 3$, the converse\nof Newton's result does not hold any more~\\cite{Kurtz}.\n\n\\medskip\n\nWhen $f \\in \\mathbb{R}^+[X]$, i.e., when $f = \\sum_{j = 0}^d a_j X^j$ with\n$a_j \\geq 0$ for all $j \\in \\{0,\\ldots,d\\}$, a weak converse of\nNewton's result holds true. Namely, a sufficient condition for $f$\nto only have real (and distinct) roots is that\n$$a_i^2\n> 4 a_{i-1} a_{i+1} {\\rm \\ for\\  all}\\ i \\in \\{1,\\ldots,d-1\\}.$$\nWhenever a polynomial fulfills this condition, we\nsay that it satisfies the {\\it Kurtz condition} since this converse\nresult is often attributed to Kurtz~\\cite{Kurtz}.\nNote however that it was obtained some 70 years earlier by Hutchinson~\\cite{Hutchinson}.\n\\medskip\n\nIf $f$ satisfies the Kurtz condition, all of its $d+1$ coefficients\nare nonzero except possibly the constant term. Such a polynomial is\ntherefore very far from being sparse (recall that a polynomial is\ninformally called {\\em sparse} if the number of its nonzero coefficients\nis small compared to its degree).\nOne question that we investigate in this paper is: how can we\nconstruct polynomials satisfying the Kurtz condition using sparse\npolynomials as building blocks?\nMore precisely, consider $f$ a polynomial of the form\n\\begin{equation}\\label{sumprod} f = \\sum_{i = 1}^k \\prod_{j = 1}^m\nf_{i,j}\\end{equation} where $f_{i,j}$ are polynomials with at most\n$t$ monomials each. By expanding the products in~(\\ref{sumprod}) we\nsee that $f$ has at most $k t^m$ monomials.\nAs a result, $d \\leq k t^m$ if $f$ satisfies the Kurtz condition.\nOur goal is to improve this very coarse bound. For the case of\npolynomials $f_{i,j}$ with nonnegative coefficients, we obtain the\nfollowing result.\n\n\\begin{theorem}\\label{bound}\nConsider a polynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f\n= \\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j},$$ where $m \\geq 2$ and the\n$f_{i,j} \\in \\mathbb{R}^+[X]$ have at most $t$ monomials. If $f$ satisfies\nthe Kurtz condition, then $d = \\mathcal O(k m^{2\/3} t^{2m\/3} {\\rm\nlog^{2\/3}}(kt))$.\n\\end{theorem}\nWe prove this result in Section \\ref{kurtzsection}. After that, in\nSection \\ref{strongsection}, we study the following stronger\nlog-concavity condition\n\\begin{equation} \\label{stronglogconcave}a_i^2\n> d^{2d} a_{i-1} a_{i+1} {\\rm \\ for\\  all}\\ i \\in \\{1,\\ldots,d-1\\}.\\end{equation} In\nthis setting we prove the following improved analogue of Theorem\n\\ref{bound}.\n\n\\begin{theorem}\\label{bound2} Consider a\npolynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f = \\sum_{i =\n1}^k \\prod_{j = 1}^m f_{i,j},$$ where $m \\geq 2$ and the $f_{i,j}\n\\in \\mathbb{R}^+[X]$ have at most $t$ monomials. If $f$\nsatisfies~$(\\ref{stronglogconcave})$, then $d \\leq k m t$.\n\\end{theorem}\n\n This\ninvestigation has a complexity-theoretic motivation: we show in\nSection~\\ref{complexity} that a suitable\nextension of\nTheorem~\\ref{bound2} (allowing negative coefficients for the\npolynomials $f_{ij}$) would imply\na separation of the algebraic complexity classes $\\vp$ and $\\vnp$.\nThe classes $\\vp$ of ``easily computable polynomial families'' and\n$\\vnp$ of ``easily definable polynomial families'' were proposed by\nValiant~\\cite{Val79} as algebraic analogues of $\\p$ and $\\np$. As\nshown in Theorem~\\ref{monotone}, Theorem~\\ref{bound2} as it now\nstands is strong enough to provide a new example of a family of\npolynomials in $\\vnp$ which cannot be computed by monotone\narithmetic circuits of polynomial size.\n\n\\section{The Kurtz log-concavity condition}\\label{kurtzsection}\n\nOur main tool in this section is a result of convex geometry\n\\cite{EPRS}. To state this result, we need to introduce some\ndefinitions and notations. For a pair of planar finite sets $R, S\n\\subset \\mathbb{R}^2$, the {\\it Minkowski sum} of $R$ and $S$ is the set $R\n+ S := \\{y + z \\, \\vert \\, y \\in R, z \\in S\\} \\subset \\mathbb{R}^2$. A\nfinite set $C \\subset \\mathbb{R}^2$ is {\\it convexly independent} if and\nonly if its elements are vertices of a convex polygon. The following\nresult provides an upper bound for the number of elements of a\nconvexly independent set contained in the Minkowski sum of two other\nsets.\n\n\\begin{theorem}\\cite[Theorem 1]{EPRS}\\label{convex} Let $R$ and $S$ be two planar point sets with $\\vert\nR \\vert = r$ and $\\vert S \\vert = s$. Let $C$ be a subset of the\nMinkowski sum $R + S$. If $C$ is convexly independent we have that\n$\\vert C \\vert = \\mathcal O(r^{2\/3} s^{2\/3} + r + s)$.\n\\end{theorem}\n\n\n\\medskip From this result the following corollary follows easily.\n\n\\begin{corollary}\\label{maxconvex}Let $R_1,\\ldots,R_k,S_1,\\ldots,S_k,Q_1,Q_2$ be planar point sets with $\\vert\nR_i \\vert = r, \\ \\vert S_i \\vert = s$ for all $i \\in \\{1,\\ldots,k\\}$,\n$\\lvert Q_1\\rvert = q_1$\nand $\\lvert Q_2 \\rvert = q_2$. Let $C$ be a subset of $\\cup_{i = 1}^k (R_i\n+ S_i) + Q_1+Q_2$. If $C$ is convexly independent, then $\\vert C \\vert =\n\\mathcal O(k r^{2\/3} s^{2\/3} q_1^{2\/3}q_2^{2\/3} + k r q_1 + k s q_2)$.\n\\end{corollary}\n\\begin{proof}We observe that $\\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2 =\n\\cup_{i = 1}^k ((R_i+Q_1) + (S_i + Q_2))$. Therefore, we partition $C$ into\n$k$ convexly independent disjoint sets $C_1,\\ldots,C_k$ such that\n$C_i \\subset (R_i+Q_1) + (S_i + Q_2)$ for all $i \\in \\{1,\\ldots,k\\}$. Since\n$\\vert R_i +Q_1\\vert = rq_1$ and $\\vert S_i + Q_2 \\vert \\leq sq_2$, by Theorem\n\\ref{convex}, we get that $\\vert C_i \\vert = \\mathcal O(r^{2\/3}\ns^{2\/3} q_1^{2\/3}q_2^{2\/3}+ rq_1 + sq_2)$ and the result follows.\n\\end{proof}\n\n\\medskip\n\n\n\n\\begin{theorem}\\label{bound2summands}Consider a polynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f =\n\\sum_{i = 1}^k g_i h_i,$$ where $g_i,h_i \\in \\mathbb{R}^+[X]$, the $g_i$\nhave at most $r$ monomials and the $h_i$ have at most $s$ monomials.\nIf $f$ satisfies the Kurtz condition, then $d = \\mathcal O(k\nr^{2\/3}s^{2\/3}\\,{\\rm log}^{2\/3}(k r)+ k(r+s)\\log^{1\/2}(kr))$.\n\\end{theorem}\n\\begin{proof}We write $f = \\sum_{i = 0}^d c_i X^i$, where $c_i > 0$ for all $i \\in \\{1,\\ldots,d\\}$ and $c_0 \\geq 0$.\nSince $f$ satisfies the Kurtz condition, setting $\\epsilon := {\\rm\nlog}(4)\/2$ we get that\n\\begin{equation}\\label{ineq} 2 {\\rm log}(c_i) > {\\rm log}(c_{i-1}) + {\\rm log}(c_{i+1})\n+ 2 \\epsilon. \\end{equation} for every $i \\geq 2$. For every\n$\\delta_1,\\ldots,\\delta_{d} \\in \\mathbb{R}$, we set\n$C_{(\\delta_1,\\ldots,\\delta_d)} := \\{(i,{\\rm log}(c_i) + \\delta_i)\n\\, \\vert \\, 1 \\leq i \\leq d\\}$. We observe that (\\ref{ineq}) implies\nthat $C_{(\\delta_1,\\ldots,\\delta_d)}$ is convexly independent\nwhenever $0 \\leq \\delta_i < \\epsilon$ for all $i \\in\n\\{1,\\ldots,d\\}$.\n\n\\medskip\nWe write $g_i = \\sum_{j = 1}^{r_i} a_{i,j} X^{\\alpha_{i,j}}$ and\n$h_i = \\sum_{j = 1}^{s_i} b_{i,j} X^{\\beta_{i,j}}$, with $r_i \\leq\nr$, $s_i \\leq s$ and $a_{i,j}, b_{i,j}\n> 0$ for all $i,j$. Then, $c_l = \\sum_{i = 1}^k (\\sum_{\\alpha_{i,j_1} + \\beta_{i,j_2} =\nl} a_{i,j_1} b_{i,j_2})$. So, setting $M_l := {\\rm max} \\{a_{i,j_1}\nb_{i,j_2} \\, \\vert \\, i \\in \\{1,\\ldots,k\\}, \\alpha_{i,j_1} +\n\\beta_{i,j_2} = l\\}$ for all $l \\in \\{1,\\ldots,d\\}$, we have that\n$M_l \\leq c_l \\leq k r M_l$, so ${\\rm log}(M_l) \\leq {\\rm log}(c_l)\n\\leq {\\rm log}(M_l) + {\\rm log}(k r)$.\n\n\\medskip For every $l \\in \\{1,\\ldots,d\\}$, we set\n\\begin{equation}\\label{lambdaepsilon} \\lambda_l := \\left\\lceil \\frac{{\\rm log}(c_l) - {\\rm\nlog}(M_l)}{\\epsilon} \\right\\rceil {\\rm \\ and \\ } \\delta_l := {\\rm\nlog}(M_l) + \\lambda_l \\epsilon - {\\rm log}(c_l),\\end{equation} and\nhave that $0 \\leq \\lambda_l \\leq \\lceil ({\\rm log}(k r))\/\\epsilon\n\\rceil$ and that $0 \\leq \\delta_l < \\epsilon$.\n\n\\medskip\nNow, we consider the sets\n\\begin{itemize}\n\\item $R_i :=\n  \\{(\\alpha_{i,j}, {\\rm log}(a_{i,j}))\\, \\vert \\, 1 \\leq j \\leq r_i\\}$\n  for $i = 1,\\ldots,k$,\n\\item $S_i := \\{(\\beta_{i,j}, {\\rm\n    log}(b_{i,j}))\\, \\vert \\, 1 \\leq j \\leq s_i\\}$ for $i = 1,\\ldots,k$,\n\\item $Q := \\{(0, \\lambda \\epsilon) \\, \\vert \\, 0 \\leq \\lambda\n  \\leq \\lceil {\\rm log}(k r) \/ \\epsilon \\rceil \\}$,\n\\item $Q_1 := \\{(0, \\mu \\epsilon) \\, \\vert \\, 0 \\leq \\mu\n  \\leq \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\}$, and\n\\item $Q_2 := \\{(0, \\nu \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\epsilon) \\, \\vert \\, 0 \\leq \\nu\n  \\leq \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\}$.\n\\end{itemize}\nIf $(0,\\lambda\\epsilon)\\in Q$, then there exist $\\mu$ and $\\nu$ such that\n$\\lambda=\\nu\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil + \\mu$ where\n$\\mu,\\nu\\leq\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil$. We have,\n\\begin{align*}\n  (0,\\lambda\\epsilon)=\n  (0,\\nu\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil\\epsilon)+(0,\\mu\\epsilon)\\in Q_1+Q_2,\n\\end{align*}\nso $Q\\subset Q_1+Q_2$.\nThen, we claim that $C_{(\\delta_1,\\ldots,\\delta_d)} \\subset \\cup_{i = 1}^k\n(R_i + S_i) + Q$. Indeed, for all $l \\in \\{1,\\ldots,d\\}$, by\n(\\ref{lambdaepsilon}),  $${\\rm log}(c_l) + \\delta_l = {\\rm log}(M_l)\n+ \\lambda_l \\epsilon = {\\rm log}(a_{i,j_1}) + {\\rm log}(b_{i,j_2}) +\n\\lambda_l \\epsilon$$ for some $i \\in \\{1,\\ldots,k\\}$ and some\n$j_1,j_2$ such that $\\alpha_{i,j_1} + \\beta_{i,j_2} = l$; thus\n$$(l,{\\rm log}(c_l) + \\delta_l) = (\\alpha_{i,j_1}, {\\rm log}(a_{i,j_1})) +\n(\\beta_{i,j_2}, {\\rm log}(b_{i,j_1})) + (0, \\lambda_l \\epsilon) \\in\n\\cup_{i = 1}^k (R_i + S_i) + Q.$$ Since\n$C_{(\\delta_1,\\ldots,\\delta_d)}$ is a convexly independent set of\n$d$ elements contained in $\\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2$, a direct\napplication of Corollary \\ref{maxconvex} yields the result.\n\\end{proof}\n\n\\medskip\n\n From this result it is easy to derive an upper bound for the\ngeneral case, where we have the products of $m \\geq 2$ polynomials.\nIf suffices to divide the $m$ factors into two groups of\napproximately $m\/2$ factors, and in each group we expand the product\nby brute force.\n\n\\medskip\n\n\\begin{proof}[Proof of Theorem~\\ref{bound}]\nWe write each of the $k$ products as a product of two\npolynomials $G_i := \\prod_{j = 1}^{\\lfloor m\/2 \\rfloor} f_{i,j}$ and\n$H_i := \\prod_{j = \\lfloor m\/2 \\rfloor + 1}^{m} f_{i,j}$. We can now\napply Theorem \\ref{bound2summands} to $f = \\sum_{i = 1}^k G_i H_i$\nwith $r = t^{\\lfloor m\/2 \\rfloor}$ and $s = t^{m - \\lfloor m\/2\n\\rfloor}$ and we get the result.\n\\end{proof}\n\n\\bigskip\n\n\\begin{remark}We observe that the role of the constant $4$ in the\nKurtz condition can be played by any other constant $\\tau > 1$ in\norder to obtain the conclusion of Theorem \\ref{bound}, i.e., we\nobtain the same result for $f = \\sum_{i = 0}^d a_i X^i$\nsatisfying that $a_i^2 > \\tau a_{i-1} a_{i+1}$ for all $i \\in\n\\{1,\\ldots,d-1\\}$. For proving this it suffices to replace the value\n$\\epsilon = {\\rm log}(4) \/ 2$ by $\\epsilon = {\\rm log}(\\tau) \/ 2$ in\nthe proof of Theorem \\ref{bound2summands} to conclude this more\ngeneral result.\n\\end{remark}\n\n\\bigskip\n\nFor $f = g h$ with $g,h \\in \\mathbb{R}^+[X]$ with at most $t$ monomials,\nwhenever $f$ satisfies the Kurtz condition, then $f$ has only real\n(and distinct) roots and so do $g$ and $h$. As a consequence, both\n$g$ and $h$ satisfy (\\ref{newton}) with strict inequalities and we\nderive that $d \\leq 2t$. Nevertheless, in the similar setting where\n$f = g h + x^i$ for some $i > 0$, the same argument does not apply\nand a direct application of Theorem \\ref{bound} yields $d = \\mathcal\nO(t^{4\/3}\\, {\\rm log^{2\/3}}(t))$, a bound\nwhich seems to be very far from optimal.\n\n\\subsection*{Comparison with the setting of Newton polygons}\n\n\nA result similar to Theorem~\\ref{bound} was obtained in~\\cite{KPTT}\nfor the Newton polygons\nof bivariate polynomials. Recall that the Newton polygon of a\npolynomial $f(X,Y)$ is the convex hull of the points $(i,j)$ such\nthat the monomial $X^iY^j$ appears in $f$ with a nonzero coefficient.\n\\begin{theorem}[Koiran-Portier-Tavenas-Thomass\\'e] \\label{mpolys}\nConsider   a bivariate polynomial\n of the form\n\\begin{equation} \\label{bivariateSPS}\nf(X,Y)=\\sum_{i=1}^k \\prod_{j=1}^m f_{i,j}(X,Y)\n\\end{equation}\nwhere $m \\geq 2$ and the $f_{i,j}$ have at most $t$ monomials. The\nNewton polygon  of  $f$ has $O(k t^{2m\/3})$ edges.\n\\end{theorem}\nIn the setting of Newton polygons, the main issue is how to deal with\nthe cancellations arising from the addition of the $k$ products\nin~(\\ref{bivariateSPS}).\nTwo monomials of the form $cX^iY^j$ with the\nsame pair $(i,j)$ of exponents but oppositive values of the\ncoefficient $c$ will cancel, thereby deleting the point $(i,j)$\nfrom the Newton polygon.\n\nIn the present paper we associate to the monomial $cX^i$ with $c>0$\nthe point $(i,\\log c)$. There are no cancellations since we only\nconsider polynomials $f_{i,j}$ with nonnegative coefficients in\nTheorems~\\ref{bound} and~\\ref{bound2summands}. However, the addition\nof two monomials $cX^i, c'X^i$ with the same exponent will ``move''\nthe corresponding point along the coefficient axis. By contrast, in\nthe setting of Newton polygons points can be deleted but cannot\nmove. In the proof of Theorem~\\ref{bound2summands} we deal with the\nissue of ``movable points'' by an approximation argument, using the\nfact that the constant $\\epsilon=\\log(4)\/2>0$ gives us a little bit\nof slack.\n\n\\section{A stronger log-concavity condition}\\label{strongsection}\n\nThe objective of this section is to improve the bound provided in\nTheorem \\ref{bound} when $f = \\sum_{i = 0}^d a_i X^i \\in \\mathbb{R}^+[x]$\nsatisfies a stronger log-concavity condition, namely, when\n$a_i^2 > d^{2d} a_{i-1} a_{i+1}$ for all $i \\in \\{1,\\ldots,d-1\\}$.\n\nTo prove this bound, we make use of the following\nwell-known lemma (a reference and similar results for polytopes in\nhigher dimension can be found in~\\cite{karavelas2012}).\nFor completeness, we provide a short proof.\n\\begin{lemma}\\label{convexhull} If $R_1,\\ldots,R_s$ are planar sets and $\\vert R_i\n\\vert = r_i$ for all $i \\in \\{1,\\ldots,s\\}$, then the convex hull of\n$R_1 + \\cdots + R_s$ has at most $r_1 + \\cdots + r_s$ vertices.\n\\end{lemma}\n\\begin{proof}We denote by $k_i$ the number of vertices of the\nconvex hull of $R_i$. Clearly $k_i \\leq r_i$. Let us prove that the\nconvex hull of $R_1 + \\cdots + R_s$ has at most $k_1 + \\cdots + k_s$\nvertices. Assume that $s = 2$. We write $R_1 =\n\\{a_1,\\ldots,a_{r_1}\\}$, then $a_i \\in R_1$ is a vertex of the\nconvex hull of $R_1$ if and only if there exists $w \\in S^1$ (the\nunit Euclidean sphere) such that $w \\cdot a_i\n> w \\cdot a_j$ for all $j \\in \\{1,\\ldots,r_1\\} \\setminus \\{i\\}$.\nThus, $R_1$ induces a partition of $S^1$ into $k_1$ half-closed\nintervals. Similarly, $R_2$ induces a partition of $S^1$ into $k_2$\nhalf-closed intervals. Moreover, these two partitions induce a new\none on $S^1$ with at most $k_1 + k_2$ half-closed intervals; these\nintervals correspond to the vertices of $R_1 + R_2$ and; thus, there\nare at most $k_1 + k_2$. By induction we get the result for any\nvalue of $s$.\n\\end{proof}\n\n\\begin{proposition}\\label{SPS}\n  Consider a polynomial $f=\\sum_{i=0}^d a_i X^i \\in \\mathbb{R}^+[X]$ of the form\n  \\begin{align*}\n    f=\\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j}\n  \\end{align*}\n  where the $f_{i,j} \\in \\mathbb{R}^+[x]$. If $f$ satisfies the\n  condition\n  \\begin{align*}\n    a_i^2 > k^2 d^{2m} a_{i-1}  a_{i+1} ,\n  \\end{align*}\n  then there exists a polynomial $f_{i,j}$ with at least $d \/ km$ monomials.\n\\end{proposition}\n\n\\begin{proof}\nEvery polynomial $f_{i,j} := \\sum_{l = 0}^{d_{i,j}} c_{i,j,l}\\,\nX^l$, where $d_{i,j}$ is the degree of $f_{i,j}$, corresponds to a\nplanar set\n\\begin{align*} R_{i,j} := \\{(l, {\\rm log}(c_{i,j,l}))\\, \\vert \\,\nc_{i,j,l} > 0\\} \\subset \\mathbb{R}^2.\n\\end{align*} We set,\n$C_{i,l}  :=  {\\rm max} \\{0, \\prod_{r = 1}^m c_{i,r,l_r} \\, \\vert \\,\nl_1 + \\cdots + l_m = l \\},$ for all $i \\in \\{1,\\ldots,k\\}$, $l \\in\n\\{0,\\ldots,d\\}$, and $ C_l  :=  {\\rm max}\\{C_{i,l} \\, \\vert \\, 1\n\\leq i \\leq k\\}$ for all $l \\in \\{0,\\ldots,d\\}$. Since the\npolynomials $f_{i,j} \\in \\mathbb{R}^+[X]$  and\n$$a_l = \\sum_{i = 1}^k \\left(\\sum_{l_1 + \\cdots + l_m = l}\\\n\\prod_{r= 1}^m c_{i,r,l_r}\\right)$$ for all $l \\in \\{0,\\ldots,d\\}$,\nwe derive the following two properties:\n\\begin{itemize}\n\\item $C_l \\leq a_l \\leq k d^m C_l$ for all $l \\in \\{0,\\ldots,d\\}$,\n\\item either $C_{i,l} = 0$ or $(l,\n{\\rm log}(C_{i,l})) \\in R_{i,1} + \\cdots + R_{i,m}$ for all $i \\in\n\\{1,\\ldots,k\\}, \\, l \\in \\{0,\\ldots,d\\}$. Since $a_l > 0$ for all $l\n\\in \\{1,\\ldots,d\\}$, we have that $C_l > 0$ and $(l, {\\rm log}(C_l))\n\\in \\bigcup_{i = 1}^k \\left(R_{i,1} + \\cdots + R_{i,m}\\right)$\n\\end{itemize}\n\n\nWe claim that the points in the set $\\{(l, {\\rm log}(C_l)) \\, \\vert\n\\, 1 \\leq l \\leq d\\}$ belong to the upper convex envelope of\n$\\bigcup_{i = 1}^k (R_{i,1} + \\cdots + R_{i,m})$. Indeed, if\n$(a,\\log(b)) \\in \\bigcup_{i = 1}^k (R_{i,1} + \\cdots + R_{i,m})$,\nthen $a \\in \\{0,\\ldots,d\\}$ and $b \\leq C_{a}$; moreover, for all $l\n\\in \\{1,\\ldots,d-1\\}$, we have that $$C_l^2 \\geq a_l^2 \/ (k^2\nd^{2m}) > a_{l-1} \\, a_{l+1} \\geq C_{l-1} C_{l+1}.$$\n\nHence, there exist $i_0 \\in \\{1,\\ldots,k\\}$ and $L \\subset\n\\{1,\\ldots,d\\}$ such that $\\vert L\\vert \\geq d\/k$ and $C_l =\nC_{i_0,l}$ for all $l \\in L$. Since the points in $\\{(l,{\\rm\nlog}(C_{l}))\\, \\vert \\, 1 \\leq l \\leq d\\}$ belong to the upper\nconvex envelope of $\\bigcup_{i = 1}^k (R_{i,1} + \\cdots +\n R_{i,m})$ we easily get that the set $\\{(l, {\\rm\n log}(C_{i_0,l}))  \\, \\vert \\, l \\in L\\}$ is a subset of the vertices in the convex hull\nof $R_{i_0,1} + \\cdots  + R_{i_0,m}$. By Lemma \\ref{convexhull}, we\nget that there exists $j_0$ such that $\\vert R_{i_0,j_0} \\vert \\geq\n\\lvert L \\rvert \/ m \\geq d \/ km$ points. Finally, we conclude that\n$f_{i_0,j_0}$ involves at least $d \/ km$ monomials.\n\\end{proof}\n\n\\bigskip\n\n\\begin{proof}[Proof of Theorem \\ref{bound2}]If $d \\leq k$ or $d \\leq m$, then $d \\leq kmt$.\nOtherwise, $d^{2d} > k^2 d^{2(d-1)} \\geq k^2 d^{2m}$ and, thus, $f$\nsatisfies (\\ref{stronglogconcave}). A direct application of\nProposition \\ref{SPS} yields the result.\n\\end{proof}\n\n\n\n\\section{Applications to Complexity Theory}\\label{complexity}\n\n\nWe first recall some standard definitions from algebraic complexity\ntheory (see e.g.~\\cite{Burgi} or~\\cite{Val79} for more\ndetails). Fix a field $K$.\nThe elements of the complexity class $\\vp$ are sequences $(f_n)$ of\nmultivariate polynomials with coefficients from $K$. By definition,\nsuch a sequence  belongs to $\\vp$ if the degree of $f_n$ is bounded by\na polynomial function of $n$ and if $f_n$ can be evaluated in a\npolynomial number of arithmetic operations (additions and\nmultiplications) starting from variables and from constants in $K$.\nThis can be formalized with the familiar model of {\\em arithmetic\n  circuits}.\nIn such a circuit, input gates are labeled by a constant or a\nvariable and the other gates are labeled by an arithmetic operation\n(addition or multiplication). In this paper we take $K = \\mathbb{R}$ since\nthere is a focus on polynomials with nonnegative coefficients. An\narithmetic circuit is {\\em monotone} if input gates are labeled by\nnonnegative constants only.\n\nA family of polynomials\nbelongs to the complexity class $\\vnp$ if it can be\nobtained  by summation from a family in $\\vp$.\nMore precisely, $f_n(\\overline{x})$ belongs to $\\vnp$ if there exists\na family $(g_n(\\overline{x},\\overline{y}))$ in $\\vp$ and a polynomial $p$\nsuch that the tuple of variables\n$\\overline{y}$ is of length $l(n) \\leq p(n)$ and\n$$f_n(\\overline{x})=\\sum_{\\overline{y} \\in \\{0,1\\}^{l(n)}} g_n(\\overline{x},\\overline{y}).$$\nNote that this summation over all boolean values of $\\overline{y}$\nmay be of exponential size.\nWhether the inclusion $\\vp \\subseteq \\vnp$ is strict is a major open\nproblem in algebraic complexity.\n\nValiant's criterion~\\cite{Burgi,Val79} shows that ``explicit''\npolynomial families belong to $\\vnp$. One version of it is as follows.\n\\begin{lemma}\nSuppose that the function $\\phi:\\{0,1\\}^* \\rightarrow \\{0,1\\}$ is\ncomputable in polynomial time. Then the family $(f_n)$ of multilinear\npolynomials defined by\n$$f_n=\\sum_{e \\in \\{0,1\\}^n} \\phi(e)x_1^{e_1} \\cdots x_n^{e_n}$$\nbelongs to $\\vnp$.\n\\end{lemma}\nNote that more general versions of Valiant's criterion are know. One\nmay allow polynomials with integer rather than\n0\/1 coefficients~\\cite{Burgi}, but in Theorem~\\ref{monotone}\nbelow we will only have to deal with 0\/1 coefficients.\nAlso, one may allow $f_n$ to depend on any (polynomially bounded)\nnumber of variables rather than exactly $n$ variables and in this case, one may\nallow the algorithm for computing the coefficients of $f_n$ to take as\ninput the index $n$ in addition to the tuple $e$ of exponents\n(see~\\cite{Koi04}, Theorem~2.3).\n\n\n\nReduction of arithmetic circuits to depth~4 is an important\ningredient in the proof of the forthcoming results. This phenomenon\nwas discovered by Agrawal and Vinay \\cite{AV}. Here we will use it\nunder the form of \\cite{Tavenas}, which is an improvement of\n\\cite{Koiran2012}. We will also need the fact that if the original\ncircuit is monotone, then the resulting depth~4 circuit is also\nmonotone (this is clear by inspection of the proof\nin~\\cite{Tavenas}). Recall that a depth~4 circuit is a sum of\nproducts of sums of products of inputs; sum gates appear on layers\n 2 and 4 and product gates on layers 1\nand 3. All gates may have arbitrary fan-in.\n\n\\begin{lemma}\\label{redprof4}Let $C$ be an arithmetic circuit of size $s > 1$\ncomputing a $v$-variate polynomial of degree $d$. Then, there is an\nequivalent depth $4$ circuit $\\Gamma$ of size $2^{\\, \\mathcal\nO\\left(\\sqrt{d \\log (ds) \\log (v)} \\right)}$ with multiplication\ngates at layer $3$ of fan-in $\\mathcal O(\\sqrt{d})$. Moreover, if\n$C$ is monotone, then $\\Gamma$ can also be chosen to be monotone.\n\\end{lemma}\n\nWe will use this result under the additional hypothesis that $d$ is\npolynomially bounded by the number of variables $v$. In this\nsetting, since $v \\leq s$, we get that the resulting depth $4$\ncircuit $\\Gamma$ provided by Lemma \\ref{redprof4} has size\n$s^{\\mathcal O(\\sqrt{d})}$.\n\n\n\\medskip\nBefore stating the main results of this section, we construct an explicit family of log-concave polynomials.\n\\begin{lemma}\\label{lem_concavVn}Let $n, s \\in \\mathbb{Z}^+$ and\nconsider $g_{n,s}(X) := \\sum_{i=0}^{2^n-1} a_i X^i$, with\n\\begin{align*} a_i := 2^{si(2^n-i-1)} {\\text \\ for \\ all \\ } i \\in \\{0,\\ldots,2^n\n- 1\\}. \\end{align*}\n Then, $a_i^2\n> 2^s \\, a_{i-1} \\, a_{i+1}$.\n\\end{lemma}\n\\begin{proof}Take $i \\in \\{1,\\ldots,2^n - 2\\}$, we have that\n  \\begin{align*}\n    \\log\\left(2^s a_{i-1}a_{i+1}\\right) & = s + s2^n(i-1) - s(i-1)i\n    +s2^n(i+1) - s(i+1)(i+2) \\\\\n    & = 2s2^n i - 2s i(i+1)- s  \\\\\n    & < 2s 2^n i -2s i(i+1) \\\\\n    & = \\log(a_i^2).\n  \\end{align*}\n\\end{proof}\n\nIn the next theorem we start from the family $g_{n,s}$\nof Lemma~\\ref{lem_concavVn} and we set $s=n2^{n+1}$.\n\n\n\\begin{theorem}\\label{ifvpvnp}\nLet $(f_n) \\in \\mathbb{N}[X]$ be the family of polynomials $f_n(x)=g_{n,n2^{n+1}}(x)$.\n\\begin{itemize}\n\\item[(i)] $f_n$ has degree $2^n-1$ and satisfies the log-concavity condition {\\rm (\\ref{stronglogconcave})}.\n\\item[(ii)] If $\\vp=\\vnp$, $f_n$ can be written under form~{\\rm\n    (\\ref{sumprod})} with $k=n^{O(\\sqrt{n})}$,\n $m=O(\\sqrt{n})$ and $t=n^{O(\\sqrt{n})}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nIt is clear that $f_n \\in\n\\mathbb{N}[X]$ has degree $2^n - 1$ and, by Lemma \\ref{lem_concavVn}, $f_n$\nsatisfies (\\ref{stronglogconcave}).\n\nConsider now the related family of bivariate polynomials\n$g_n(X,Y)=\\sum_{i=0}^{2^n-1}X^i Y^{e(n,i)},$ where $e(n,i) = s i (2^n - i - 1)$. One can check in time polynomial in $n$ whether\na given monomial $X^iY^j$ occurs in $g_n$: we just need to check\nthat $i<2^n$ and that $j=e(n,i)$. By mimicking the proof of Theorem\n1 in \\cite{KPTT} and taking into account Lemma \\ref{redprof4} we get\nthat, if $\\vp = \\vnp$, one can write\n\\begin{equation} \\label{sumprod2}\ng_n(X,Y)=\\sum_{i=1}^k\\prod_{j=1}^m g_{i,j,n}(X,Y)\n\\end{equation}\nwhere the bivariate polynomials $g_{i,j,n}$ have $n^{O(\\sqrt{n})}$\nmonomials, $k=n^{O(\\sqrt{n})}$ and $m=O(\\sqrt{n})$. Performing the\nsubstitution $Y=2$ in~(\\ref{sumprod2}) yields the required\nexpression for $f_n$.\n\\end{proof}\n\nWe believe that there is in fact no way to write $f_n$ under\nform~(\\ref{sumprod}) so that the parameters $k,m,t$ satisfy the\nconstraints $k=n^{O(\\sqrt{n})}$,\n $m=O(\\sqrt{n})$ and $t=n^{O(\\sqrt{n})}$.\nBy part (ii) of Theorem~\\ref{ifvpvnp}, a proof of this would\nseparate $\\vp$ from $\\vnp$. The proof of Theorem~\\ref{monotone} below\nshows that our belief is actually correct in the special case where\nthe polynomials $f_{i,j}$ in~(\\ref{sumprod}) have nonnegative\ncoefficients.\n\n\\medskip\nThe main point of Theorem \\ref{monotone} is to present an\nunconditional lower bound for a polynomial family $(h_n)$ in $\\vnp$\nderived from $(f_n)$. Note that $(f_n)$ itself is not in $\\vnp$\nsince its degree is too high. Recall that\n\\begin{equation} \\label{fneq}\n f_n(X) := \\sum_{i=0}^{2^n-1} 2^{2n2^ni(2^n-i-1)} X^i.\n\\end{equation}\nTo construct $h_n$ we write down in base 2 the exponents of ``2'' and\n``$X$'' in~(\\ref{fneq}).\nMore precisely, we take $h_n$ of the form:\n\\begin{equation} \\label{hneq}\n h_n :=\\sum_{\\alpha \\in \\{0,1\\}^{n} \\atop \\beta \\in \\{0,1\\}^{4n}}\n\\lambda(n, \\alpha, \\beta)\\, X_0^{\\alpha_0} \\cdots\nX_{n-1}^{\\alpha_{n-1}} Y_0^{\\beta_0} \\cdots\nY_{4n-1}^{\\beta_{4n-1}},\n\\end{equation}\n where $\\alpha =\n(\\alpha_0,\\ldots,\\alpha_{n-1}),\\, \\beta =\n(\\beta_0,\\ldots,\\beta_{4n-1})$ and $\\lambda(n,\\alpha,\\beta) \\in\n\\{0,1\\}$; we set $\\lambda(n,\\alpha,\\beta) = 1$ if and only\nif $\\sum_{j = 0}^{4n-1} \\beta_j 2^j = 2n2^n i (2^n - i - 1) <\n2^{4n},$ where $i := \\sum_{k = 0}^{n-1} \\alpha_{i,k} 2^k$.\nBy construction, we have:\n\\begin{equation} \\label{transfor} f_n(X) = h_n(X^{2^0}, X^{2^1},\\ldots,X^{2^{n-1}},2^{2^0},\n2^{2^1},\\ldots,2^{2^{4n-1}}). \\end{equation}\nThis relation will be useful in the proof of the following lower bound theorem.\n\\begin{theorem}\\label{monotone}\nThe family $(h_n)$ in~{\\rm(\\ref{hneq})} is in $\\vnp$. If $(h_n)$ is\ncomputed by depth $4$ monotone arithmetic circuits of size $s(n)$,\nthen $s(n) = 2^{\\,\\Omega(n)}$. If $(h_n)$ is computed by monotone\narithmetic circuits of size $s(n)$, then $s(n) =\n2^{\\,\\Omega(\\sqrt{n})}$. In particular, $(h_n)$ cannot be computed\nby monotone arithmetic circuits of polynomial size.\n\\end{theorem}\n\\begin{proof}\nNote that $h_n$ is a polynomial in $5n$ variables, of degree at most\n$5n$, and its coefficients $\\lambda(n, \\alpha, \\beta)$ can be\ncomputed in polynomial time. Thus, by Valiant's criterion\nwe conclude that $(h_n)\\in \\vnp$.\n\nAssume that $(h_n)$ can be computed by depth $4$ monotone arithmetic\ncircuits of size $s(n)$. Using (\\ref{transfor}), we get that $f_n =\n\\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j}$ where $f_{i,j} \\in \\mathbb{R}^+[X]$\n have at most $t$ monomials and $k,m,t$ are $\\mathcal\nO(s(n))$. Since the degree of $f_n$ is $2^n - 1$, by Theorem\n\\ref{bound2}, we get that $2^n - 1 \\leq kmt$. We conclude that $s(n)\n= 2^{\\,\\Omega(n)}$.\n\nTo complete the proof of the theorem, assume that $(h_n)$ can be\ncomputed by monotone arithmetic circuits of size $s(n)$. By\nLemma~\\ref{redprof4}, it follows that the polynomials $h_n$ are\ncomputable by depth~4 monotone circuits of size $s'(n) :=\ns(n)^{\\mathcal O(\\sqrt{n})}$. Therefore $s'(n) = 2^{\\,\\Omega(n)}$\nand we finally get that $s(n) = 2^{\\,\\Omega(\\sqrt{n})}$.\n\\end{proof}\n\n\nLower bounds for monotone arithmetic circuits have been known for a\nlong time (see for instance~\\cite{jerrum82,valiant79negation}).\nTheorem~\\ref{monotone} provides yet another example of a polynomial\nfamily which is hard for monotone arithmetic circuits, with an\napparently new proof method.\n\n\n\\section{Discussion}\n\nAs explained in the introduction, log-concavity plays a role\nin the study of real roots of polynomials.\nIn~\\cite{Koi10a} bounding the number of real roots of sums of products\nof sparse polynomials was suggested as an approach for separating\n$\\vp$ from $\\vnp$. Hrube\\v{s}~\\cite{Hrubes13} suggested to bound the\nmultiplicities of roots, and~\\cite{KPTT} to bound the number of edges\nof Newton polygons of bivariate polynomials.\n\nTheorem~\\ref{ifvpvnp} provides another\n plausible approach to $\\vp \\neq \\vnp$: it\nsuffices to show that if a polynomial $f \\in \\mathbb{R}^+[X]$  under\nform~(\\ref{sumprod}) satisfies the Kurtz condition or the stronger\nlog-concavity condition (\\ref{stronglogconcave}) then its degree is\nbounded by a ``small'' function of the parameters $k,m,t$.  A degree\nbound which is polynomial bound in $k,t$ and $2^m$ would be good\nenough to separate $\\vp$ from $\\vnp$. Theorem~\\ref{bound} improves\non the trivial $kt^m$ upper bound when $f$ satisfies the Kurtz\ncondition, but certainly falls short of this goal: not only is the\nbound on $\\deg(f)$ too coarse, but we would also need to allow\nnegative coefficients in the polynomials $f_{i,j}$.\nTheorem~\\ref{bound2} provides a polynomial bound on $k,m$ and $t$\nunder a stronger log-concavity condition, but still needs the extra\nassumption that the coefficients in the polynomials $f_{i,j}$ are\nnonnegative. The unconditional lower bound in Theorem~\\ref{monotone}\nprovides a ``proof of concept'' of this approach for the easier\nsetting of monotone arithmetic circuits.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe intersection cohomology \nof a complex projective variety\nenjoys many of the good properties of the\nordinary cohomology of a smooth variety,\ncollectively known as the \\textit{K\\\"{a}hler package} \n(Poincar\\'{e} duality, weak and hard Lefschetz,\nHodge decomposition and Hodge signature theorem).\nThese properties deal primarily with the\nintersection cohomology group \nthat has attracted most of the attention from algebraic topologists and geometers:\nthe middle-perversity intersection cohomology group.\nHowever, there is additional geometric information carried by other\nintersection cohomology groups, as well as\nby cohomological operations that are defined when allowing other perversities than the middle one\n(such as cup products or Steenrod squares).\nIt is in this context, that Goresky raised the following question in the introduction of \\cite{Goresky}:\n\\begin{quotation}\n{\\small ``It remains as open question whether there is an intersection \nhomology-analogue\nto the rational homotopy theory of Sullivan. For example,\none would like to know when Massey triple products are defined\nin intersection homology\nand whether they always vanish on a (singular) projective algebraic variety''.}\n\\end{quotation}\nThe first part of Goresky's question has been answered by Chataur-Saralegi-Tanr\\'{e}\nin the foundational work \\cite{CST} on rational intersection homotopy theory,\nwhere the \\textit{perverse algebraic model} \nof a topological pseudomanifold is introduced. This is \na perverse commutative differential graded algebra (perverse cdga for short) defined over the rationals,\nwhose cohomology is isomorphic to the rational\nintersection cohomology with all perversities\nand is, when forgetting its multiplicative structure, quasi-isomorphic to the intersection cochains\noriginally introduced by\nGoresky and MacPherson \\cite{GMP1,GMP2}. In general, the perverse algebraic model \ncontains more information than the intersection cohomology ring (for instance, it contains the Massey products) and\ngives rise to a well-defined notion of intersection-formality for \ntopological pseudomanifolds, analogously to the notion of formality \nappearing in the classical rational homotopy theory of Sullivan \\cite{Su}.\n\nOther significant contributions in this direction are the homotopy theory of perverse cdga's\ndeveloped by Hovey \\cite{Hov2} within the context of Quillen model categories,\nthe works of Friedman \\cite{Friedman} and Friedman-McClure \\cite{FMC} \non intersection pairings and cup products in intersection cohomology respectively\nand Banagl's theory of intersection spaces \\cite{Banagl}.\n\n\nThe present work draws its main motivation from the second\npart of Goresky's question, which is almost equivalent to asking whether\nsingular complex projective varieties are intersection-formal.\nThis question is legitimated by a well-known application of the Hodge decomposition to topology: the Formality Theorem \nof Deligne-Griffiths-Morgan-Sullivan \\cite{DGMS}, which states that\nthe rational homotopy type of a compact K\\\"{a}hler manifold\nis entirely determined by its cohomology ring.\n\nIn general, the Hodge decomposition on the intersection cohomology of a singular projective variety\nfails for perversities other than the middle one. Instead, \neach intersection cohomology group carries a mixed Hodge structure. Since the perverse algebraic model\ndepends on all perversities, we do not\nexpect an intersection-analogous statement of the Formality Theorem for singular projective varieties,\nbut of a generalization of this statement involving the weight spectral sequence.\n\nIn this paper, we study the rational intersection homotopy type\nof complex projective varieties with only isolated singularities,\nvia mixed Hodge theory.\n\\\\\n\nWe next explain the contents and main results of this paper. For the rest of this introduction, let $X$ be a complex\nprojective variety with only isolated singularities.\n\nIn Section $\\ref{Section_RIHT}$, we\ncollect preliminary definitions and results\non intersection cohomology and on the homotopy theory of perverse cdga's.\nFollowing \\cite{CST}, we\ndescribe the perverse algebraic \nmodel $I\\mathcal{A}_{\\ov\\bullet}(X)$ of $X$.\nThis can be computed from the morphism\nof rational algebras of piece-wise linear forms $\\mathcal{A}_{pl}(X_{reg})\\to \\mathcal{A}_{pl}(L)$\ninduced by the inclusion $L\\hookrightarrow X_{reg}$\nof the link $L$ of the singularities into the regular part of $X$.\n\nSection $\\ref{Section_MHS}$ is the core of this paper. \nIn this section, we endow the perverse algebraic model $I\\mathcal{A}_{\\ov\\bullet}(X)$\nof $X$ with natural mixed Hodge structures \n(this result is stated in a more technical form in Theorem $\\ref{MHSmodel}$). Our proof relies, first, on the existence of \nmixed Hodge structures on the rational homotopy types of $X_{reg}$ and $L$ due to Morgan \\cite{Mo} and \nDurfee-Hain \\cite{DH} respectively, and second, on the existence of relative models of mixed Hodge diagrams\nproven by Cirici-Guill\\'{e}n in \\cite{CG1}. \nAs an important application of the existence of mixed Hodge structures on the perverse algebraic model,\nwe study the \\textit{perverse weight spectral sequence} \n$IE_{1,\\ov \\bullet}^{*,*}(X)$,\na perverse differential bigraded algebra\nwhose cohomology computes the weight filtration on the intersection cohomology:\n$IE_{2,\\ov \\bullet}^{*,*}(X):=H^{*,*}(IE_{1,\\ov \\bullet}(X))\\cong Gr_{\\bullet}^WIH^*_{\\ov\\bullet}(X;\\mathbb{Q})$.\nIn Theorem $\\ref{IE1formality}$, we prove that \nthe complex intersection homotopy type of $X$ is a direct consequence of \nits perverse weight spectral sequence. In other words: there is a string of quasi-isomorphisms\nof perverse cdga's from\n$I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes\\mathbb{C}$ to \n$IE_{1,\\ov \\bullet}(X)\\otimes\\mathbb{C}$. This result descends to the rationals for perverse cdga's of finite type \nand is the intersection-analogue of the main result of \\cite{CG1},\nwhich in turn is the generalization to singular varieties,\nof the Formality Theorem of \\cite{DGMS}. \nAs in the classical setting,\nthe perverse weight spectral sequence can be described \nin terms of the cohomologies of varieties associated with a resolution of singularities of $X$.\nHence Theorem $\\ref{IE1formality}$ implies that the complex intersection homotopy type of\n$X$ has a finite-dimensional model, determined by\ncohomologies of smooth projective varieties.\n\nThe last two sections contain applications of Theorem $\\ref{IE1formality}$.\nIn Section $\\ref{Section_OIS}$, we prove\nthat if $X$\nadmits a resolution of singularities \nin such a way that the exceptional divisor is smooth, and if the link\nof each singular point is $(n-2)$-connected, where $n$ is the complex dimension of $X$, then $X$ is GM-intersection-formal\nover $\\mathbb{C}$ (the prefix GM accounts for Goresky-MacPherson,\nsince we consider finite perversities only).\nThe main class of examples to which this result applies are varieties with \nordinary multiple points, but it also applies to a large family of\nhypersurfaces and more generally, to complete\nintersections\nadmitting  a resolution of singularities with smooth exceptional divisor. \nThis extends a result of \\cite{CST}, where it is shown that any\nnodal hypersurface of $\\mathbb{C}\\mathbb{P}^4$ is intersection-formal.\nLikewise, in Section $\\ref{Section_OIS}$ we prove GM-intersection-formality over $\\mathbb{C}$\nfor every isolated surface singularity.\nIf a variety is (GM)-intersection-formal,\nthen its normalization is formal in the classical sense.\nWe remark that these results generalize our previous work\n\\cite{ChCi1}, where we study the (classical) rational homotopy type of complex \nprojective varieties with normal isolated singularities,\nusing the multiplicative weight spectral sequence.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Rational intersection homotopy types}\\label{Section_RIHT}\nIn this preliminary section, we recall the description \nof the intersection cohomology of a complex projective variety with only isolated singularities\nappearing in \\cite{GMP1}, as well as its main properties. Then, we\nintroduce the notion of rational intersection homotopy equivalence and study its relation with\nthe classical notion of rational homotopy equivalence.\nLastly, we collect the necessary definitions and results\non the homotopy theory of perverse differential graded algebras,\nsuch as the intersection-analogous notions of quasi-isomorphism and formality,\nand describe the perverse algebraic model of a complex\nprojective variety with only isolated singularities, following \\cite{CST}.\n\n\n\\subsection{Intersection cohomology}\\label{sectionintersectioncohomology}\nIntersection cohomology is defined for any topological pseudomanifold and\ndepends on the choice of a multi-index called \\textit{perversity},\nwhich measures how far cycles are allowed to deviate from transversality.\nFor a complex projective variety of dimension $n$ having only isolated singularities,\na perversity $\\ov p$ is determined by a single integer $p$ such that\n$0\\leq p\\leq 2n-2$. We will denote by $\\mathcal{P}$ the totally ordered set of such perversities.\nThere are three distinguished elements in $\\mathcal{P}$ that we shall refer to:\nthe $\\ov{0}$-perversity $\\ov{0}=0$, the middle perversity $\\ov{m}=n-1$\nand the top perversity $\\ov{t}=2n-2$. The complementary perversity of $\\ov p\\in\\mathcal{P}$ is given by $\\ov{t}-\\ov{p}:=2n-2-p$.\nNote that the middle perversity is complementary to itself.\nWe enlarge the set of perversities $\\widehat{\\mathcal{P}}=\\mathcal{P}\\cup \\{\\ov{\\infty}\\}$ \nby adjoining the $\\ov{\\infty}$-perversity. We define\nthe sum of two perversities $\\ov p$ and $\\ov q$ in $\\widehat{\\mathcal{P}}$ by letting\n$\\ov p+\\ov q:=\\ov{p+q}$ if $p+q\\leq 2n-2$ and $\\ov p+\\ov q:=\\ov\\infty$ otherwise.\n\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities. Denote by\n$\\Sigma$ the singular locus of $X$ and by $X_{reg}:=X-\\Sigma$ its regular part.\nThe intersection cohomology of $X$ with perversity $\\ov p$ (and coefficients in a commutative ring $R$) is given by\n(see $\\S$6.1 of \\cite{GMP1})\n$$\nIH^k_{\\ov{p}}(X;R)=\\left\\{\n\\begin{array}{ll}\nH^k(X_{reg};R)&\\text{ if }k\\leq p\\\\\n\\mathrm{Im }\\left(H^k(X;R)\\longrightarrow H^k(X_{reg};R)\\right)&\\text{ if }k=p+1\\\\\nH^k(X;R)&\\text{ if }k>p+1\n\\end{array}\n\\right..\n$$\n\nFor the $\\ov{0}$-perversity we have an isomorphism of graded algebras $IH^*_{\\ov0}(X;R)\\cong H^*(\\overline{X};R)$, \nwhere $\\overline{X}\\to X$ is a normalization of $X$\n(see $\\S$4 of \\cite{GMP1}). \nFor the $\\ov\\infty$-perversity we recover the cohomology ring $IH^*_{\\ov{\\infty}}(X;R)\\cong H^*({X}_{reg};R)$ \nof the regular part of $X$ (see \\cite{CST}).\nA main feature of intersection cohomology is that, when $R=\\mathbb{Q}$,\nfor every finite perversity $\\ov p\\in \\mathcal{P}$ we have a Poincar\\'{e} duality isomorphism (see $\\S$3.3 of \\cite{GMP1})\n$$IH^k_{\\ov{p}}(X;\\mathbb{Q})\\cong (IH^{2n-k}_{\\ov{t}-\\ov{p}}(X;\\mathbb{Q}))^\\vee:=\\mathrm{Hom}(IH^{2n-k}_{\\ov{t}-\\ov{p}}(X;\\mathbb{Q}),\\mathbb{Q}).$$\n\nThe graded objects $IH_{\\ov p}^*(X;R)$ together with the morphisms\n$IH_{\\ov p}^*(X;R)\\longrightarrow IH_{\\ov q}^*(X;R)$ for every pair $\\ov p\\leq \\ov q$,\nand the products\n$IH_{\\ov p}(X;R)\\otimes IH_{\\ov q}(X;R)\\longrightarrow IH_{\\ov p+\\ov q}(X;R)$ \ninduced by the cup products of $H^*(X;R)$ and $H^*(X_{reg};R)$\nfor any pair $\\ov p,\\ov q\\in \\widehat \\mathcal{P}$,\nconstitute the prototypical example of a \\textit{perverse commutative graded $R$-algebra}:\nthis is a commutative monoid in the category\nof functors from $\\widehat\\mathcal{P}$ to the category of graded $R$-modules.\n\nDenote by $\\mathcal{V}_\\mathbb{C}$ the category whose objects are complex projective varieties with only isolated singularities \nand whose morphisms $f:X\\longrightarrow Y$ satisfy $f(X_{reg})\\subset Y_{reg}$.\nThe above formula defines a contravariant functor $IH_{\\ov\\bullet}^*(-;R):\\mathcal{V}_\\mathbb{C}\\longrightarrow \\pga{R}$\nwith values in the category of perverse commutative graded $R$-algebras.\n\n\n\n\n\\subsection{Intersection homotopy equivalence}\nThe consideration of the intersection cohomology ring with all perversities\nleads to a natural notion of rational intersection homotopy equivalence.\n\n\n\\begin{defi}Let\n$f:X\\longrightarrow Y$ be a morphism between simply connected topological pseudomanifolds, such that $f(X_{reg})\\subset Y_{reg}$.\nThen $f$ is said to be a \\textit{rational intersection homotopy equivalence}\nif it induces an isomorphism of perverse graded algebras $f^*:IH^*_{\\ov \\bullet}(Y;\\mathbb{Q})\\longrightarrow IH^*_{\\ov \\bullet}(X;\\mathbb{Q})$.\n\\end{defi}\n\nIf $f:X\\to Y$ is a rational intersection homotopy equivalence then the morphism induced on\nthe normalizations $\\overline f:\\overline X\\longrightarrow \\overline Y$ is a rational\nhomotopy equivalence. The following result exhibits how the notion of rational\nintersection homotopy equivalence is stronger than the classical notion of rational homotopy equivalence.\n\n\\begin{prop}\\label{proj_cones_htp_equiv}\nLet $S$ and $S'$ be two simply connected smooth projective surfaces of $\\mathbb{C}\\mathbb{P}^n$. \nDenote by $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ the projective cones of $S$ and $S'$ respectively.\nThen:\n\\begin{enumerate}[(1)]\n \\item   $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally homotopy equivalent if and only if $\\mathcal{X}(S)=\\mathcal{X}(S')$.\n \\item  $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally intersection homotopy equivalent if and only if $S$ and $S'$ \n are rationally homotopy equivalent.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nLet $w\\in H^2(S;\\mathbb{Q})$ denote the Poincar\\'{e} dual of the\nhyperplane section of $S\\subset\\mathbb{C}\\mathbb{P}^n$. Since $w^2\\neq 0$, using\nPoincar\\'{e} duality we obtain an orthogonal decomposition \n$H^2(S;\\mathbb{Q})\\cong\\mathbb{Q}\\langle w\\rangle \\oplus^\\bot V.$\nThe projective cone $\\mathbb{P}_cS$ of $S$ is isomorphic to\nthe Thom space of the restriction $S(1)$ of the hyperplane bundle on $\\mathbb{C}\\mathbb{P}^n$ to $S$.\nThe rational cohomology algebra of $\\mathbb{P}_cS$ \ncan be written as $H^*(\\mathbb{P}_cS;\\mathbb{Q})\\cong \\mathbb{Q}\\langle Th\\rangle \\oplus V'$,\nwhere $Th$ has degree $2$ and satisfies $Th^4=0$ and $V'$ is a vector space of degree 4.\nThom's isomorphism $\\cup Th: H^*(S;\\mathbb{Q})\\to \\widetilde H^*(\\mathbb{P}_cS;\\mathbb{Q})$ identifies $w$ with $Th^2$\nand $V$ with $V'$. Furthermore, $Th\\cup V'=0$. This proves (1).\nThe intersection cohomology of $\\mathbb{P}_cS$ can be written as:\n\\begin{equation*}\nIH^s_{\\ov p}(\\mathbb{P}_cS;\\mathbb{Q})\\cong\n\\def1.4{1.4}\n\\begin{array}{| c | c | c | }\n\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov m$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov t$}}}\\\\\n\\hline\n\\mathbb{Q}\\langle Th^3\\rangle   &\\mathbb{Q}\\langle Th^3\\rangle&\\mathbb{Q}\\langle Th^3\\rangle  \\\\ \\hline\n0&0&0\\\\ \\hline\n\\mathbb{Q}\\langle Th^2\\rangle\\oplus V'  &\\mathbb{Q}\\langle Th^2\\rangle\\oplus V'  & H^4(S;\\mathbb{Q})   \\\\ \\hline\n0&0&0  \\\\ \\hline\n\\mathbb{Q}\\langle Th\\rangle  &\\mathbb{Q}\\langle w\\rangle \\oplus V&   \\mathbb{Q}\\langle w\\rangle \\oplus V                   \\\\\\hline\n0  &0&0                     \\\\\\hline\n\\mathbb{Q}  &\\mathbb{Q}&\\mathbb{Q}                         \\\\\\hline\n\\end{array}\n\\def1.4{1.4}\n\\begin{array}{ l }\n\\\\\n\\text{\\tiny{$s=6$}}\\\\\n\\text{\\tiny{$s=5$}}\\\\\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\n\\end{array}\n\\end{equation*}\n\nwhere the product $IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Q})\\otimes IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Q})\\longrightarrow IH_{\\ov t}^4(\\mathbb{P}_cS;\\mathbb{Q})\\cong H^4(S;\\mathbb{Q})=\\mathbb{Q}$\ncorresponds to the product on $H^2(S;\\mathbb{Q})$ and determines the signature of $S$.\nThis proves (2).\n\\end{proof}\n\n\\begin{example}\nLet $S$ be a K3-surface and let $S'$ be the projective plane blown-up at 19 \npoints. Then $\\mathcal{X}(S)=\\mathcal{X}(S')=24$, $Sign(S)=(3,19)$ and $Sign(S')=(1,21)$.\nTherefore $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally homotopy equivalent, but not rationally \nintersection homotopy equivalent.\n\\end{example}\n\n\n\\subsection{Integral intersection cohomology}\nWe prove an analogous statement of Proposition $\\ref{proj_cones_htp_equiv}$\nfor intersection cohomology with integer coefficients.\n\n\\begin{prop}\\label{proj_cone_homeo}\nLet $S$ and $S'$ be two simply connected smooth projective surfaces of $\\mathbb{C}\\mathbb{P}^n$. \nThen their projective cones $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are homeomorphic if and only if\n$IH^*_{\\ov \\bullet}(\\mathbb{P}_cS;\\mathbb{Z})$ and $IH^*_{\\ov \\bullet}(\\mathbb{P}_cS';\\mathbb{Z})$ are isomorphic as perverse graded algebras.\n\\end{prop}\n\\begin{proof}\nWe follow the notation of the proof of Proposition $\\ref{proj_cones_htp_equiv}$.\nThe intersection cohomology algebra of $\\mathbb{P}_cS$ is given by:\n\\begin{equation*}\nIH^s_{\\ov p}(\\mathbb{P}_cS;\\mathbb{Z})\\cong\n\\def1.4{1.4}\n\\begin{array}{| c | c | c | }\n\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov m$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov t$}}}\\\\\n\\hline\n\\mathbb{Z}\\langle T\\rangle , Th^3=\\deg(S)\\cdot T&\\mathbb{Z}\\langle T\\rangle&\\mathbb{Z}\\langle T\\rangle  \\\\ \\hline\n0&0&0\\\\ \\hline\nH^4(\\mathbb{P}_cS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z}) &H^4(\\mathbb{P}_cS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z}) & H^4(S;\\mathbb{Z})\\cong \\mathbb{Z}   \\\\ \\hline\n0&0&0  \\\\ \\hline\n\\mathbb{Z}\\langle Th\\rangle  &H^2(S;\\mathbb{Z})&     H^2(S;\\mathbb{Z})    \\\\\\hline\n0  &0&0                     \\\\\\hline\n\\mathbb{Z}  &\\mathbb{Z}&\\mathbb{Z}                         \\\\\\hline\n\\end{array}\n\\def1.4{1.4}\n\\begin{array}{ l }\n\\\\\n\\text{\\tiny{$s=6$}}\\\\\n\\text{\\tiny{$s=5$}}\\\\\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\n\\end{array}\n\\end{equation*}\nThe morphism \n$$H^2(\\mathbb{P}_sS;\\mathbb{Z})\\cong IH_{\\ov 0}^2(\\mathbb{P}_sS;\\mathbb{Z})\\longrightarrow IH_{\\ov m}^2(\\mathbb{P}_sS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z})$$\ndetermines up to sign a class $\\pm w\\in H^2(S;\\mathbb{Z})$ given by the image of a generator of\n$H^2(\\mathbb{P}_sS;\\mathbb{Z})$. We get line bundles $L_s^+$ and $L_s^-$ over $S$\nsatisfying $c_1(L_S^\\pm)=\\pm w$. Since these two bundles are isomorphic as rank 2 vector bundles,\ntheir Thom spaces $Th(L_S^\\pm)\\cong \\mathbb{P}_cS$ are homeomorphic.\n\nAssume that we have an isomorphism $\\Psi:IH_{\\ov\\bullet}(\\mathbb{P}_cS;\\mathbb{Z})\\longrightarrow IH_{\\ov\\bullet}(\\mathbb{P}_cS';\\mathbb{Z})$.\nThen the intersection forms of $S$ and $S'$ are equivalent, and it follows form Freedman's \nTheorem that $S$ and $S'$ are homeomorphic.\nFrom the commutative diagram\n$$\n\\xymatrix{\n\\ar[d]_\\Psi IH_{\\ov 0}^2(\\mathbb{P}_cS;\\mathbb{Z})\\ar[r]&IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Z})\\ar[d]^\\Psi\\\\\nIH_{\\ov 0}^2(\\mathbb{P}_cS';\\mathbb{Z})\\ar[r]&IH_{\\ov m}^2(\\mathbb{P}_cS';\\mathbb{Z})\n}\n$$\nwe deduce that $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are homeomorphic.\n\\end{proof}\n\n\n\\begin{example}\nLet $S$ be a surface of degree $4$ in $\\mathbb{C}\\mathbb{P}^3$, let $S'$ be the intersection of a quadric\n and a cubic in $\\mathbb{C}\\mathbb{P}^4$, and let $S''$ be the intersection of three quadrics in $\\mathbb{C}\\mathbb{P}^5$.\nAll three surfaces are examples of K3-surfaces with different intersection cohomology algebras.\nHence their projective cones are non-homeomorphic.\n\\end{example}\n\n\nLet $S$ be a simply-connected 4-dimensional\nsmooth manifold and let $w\\in H^2(S;\\mathbb{Z})$.\nTo such a pair $(S,\\pm w)$  one can associate two homeomorphic Thom spaces\n$Th(L_w^\\pm)$.\nThe proof of Proposition $\\ref{proj_cone_homeo}$ is easily generalized \nto this setting. We have:\n\n\\begin{prop}\\label{equiv_pairs}\n Let $(S,\\pm w)$ and $(S',\\pm w')$ be two pairs. The following are equivalent:\n \\begin{enumerate}[(1)]\n  \\item   The pairs are topologically equivalent: there is a homeomorphism $\\phi:S\\to S'$ \nsuch that $\\phi^*(w)=w'$. \n  \\item  The line bundles $L_w^\\pm$ and $L_{w'}^\\pm$ are isomorphic as real vector bundles.\n  \\item The Thom spaces $Th(L_w^\\pm)$ and $Th(L_{w'}^\\pm)$ are homeomorphic.\n  \\item  The integral intersection cohomologies $IH^*_{\\ov\\bullet}(Th(L_w^\\pm);\\mathbb{Z})$ and\n  $IH^*_{\\ov\\bullet}(Th(L_{w'}^\\pm);\\mathbb{Z})$ are isomorphic as perverse graded algebras.\n \\end{enumerate}\n\\end{prop}\n\n\n\n\\subsection{Perverse differential graded algebras}\\label{perversecdgas}\nAs in the classical rational homotopy theory of Sullivan \\cite{Su}, the study of rational intersection homotopy \ntypes is closely related to the homotopy theory of\nperverse differential graded algebras. We next recall\nthe main definitions. Given our interest in varieties with only isolated singularities, we \nrestrict to the particular case where perversities \nare given by a single integer, and refer \\cite{Hov2} and \\cite{CST} for the general definitions, \nin which perversities are given by multi-indexes.\nFor the rest of this section we let $\\mathbf{k}$ be a field of characteristic 0.\n\n\\begin{defi}\\label{pcdga_def}\nA \\textit{perverse commutative differential graded algebra} \\textit{over $\\mathbf{k}$}\n is a commutative monoid in the category of functors from $\\widehat\\mathcal{P}$\nto the category $C^+(\\mathbf{Vect_\\mathbf{k}})$ of cochain complexes of $\\mathbf{k}$-vector spaces:\nthis is a bigraded $\\mathbf{k}$-vector \nspace $A_{\\ov\\bullet}^*=\\{A^i_{\\ov{p}}\\}$, with $i\\geq 0$ and $\\ov{p}\\in \\widehat\\mathcal{P}$,\ntogether with a linear differential $d:A^i_{\\ov{p}}\\to A^{i+1}_{\\ov{p}}$,\nan associative product $\\mu:A^i_{\\ov{p}}\\otimes A^j_{\\ov{q}}\\to A^{i+j}_{\\ov{p}+\\ov{q}}$ with unit $\\eta:\\mathbf{k}\\to A^0_{\\ov{0}}$ \nand a poset map $A^i_{\\ov{q}}\\to A^i_{\\ov{p}}$ for every $\\ov{q}\\leq \\ov{p}$. \nProducts and differentials satisfy the usual graded commutativity and graded Leibnitz rules, \nand are compatible with poset maps:\nfor all $\\ov{p}\\leq \\ov{p}'$ and $\\ov{q}\\leq \\ov{q}'$ the following diagrams commute:\n$$\\xymatrix{\nA_{\\ov p}\\otimes A_{\\ov q}\\ar[d]\\ar[r]^\\mu& A_{\\ov {p}+\\ov{q}}\\ar[d]\\\\\nA_{\\ov p'}\\otimes A_{\\ov q'}\\ar[r]^\\mu& A_{\\ov{p}'+ \\ov{q}'}\n}\\,\\,\\,\\,\\,\\,;\\,\\,\\,\\,\\,\\,\n\\xymatrix{\nA_{\\ov p}\\ar[d]\\ar[r]^d& A_{\\ov {p}}\\ar[d]\\\\\nA_{\\ov p'}\\ar[r]^d& A_{\\ov {p'}}\n}.\n$$\n\\end{defi}\nThe cohomology of a perverse cdga naturally inherits the structure of a perverse commutative graded algebra.\nDenote by $\\pdga{\\mathbf{k}}$ the category of perverse cdga's over $\\mathbf{k}$.\n\n\\begin{defi}\nA morphism of perverse cdga's $f:A_{\\ov\\bullet}\\to B_{\\ov\\bullet}$ is called \\textit{quasi-isomorphism}\nif for every perversity $\\ov p\\in\\widehat\\mathcal{P}$\nthe induced map $H^*(A_{\\ov p})\\to H^*(B_{\\ov q})$ is an isomorphism.\n\\end{defi}\n\nThe category $\\pdga{\\mathbf{k}}$ admits a Quillen model structure with quasi-isomorphisms as weak equivalences and\nsurjections as fibrations (see \\cite{Hov2}). The existence and uniqueness of minimal models of perverse cdga's\n\\`{a} la Sullivan\nis proven in \\cite{CST}.\nDenote by $\\mathrm{Ho}(\\pdga{\\mathbf{k}})$ the homotopy category of perverse cdga's, defined by inverting quasi-isomorphisms.\n\n\\begin{defi}\n A perverse cdga $A_{\\ov\\bullet}$ is said to be \\textit{intersection-formal} if there is an isomorphism in $\\mathrm{Ho}(\\pdga{\\mathbf{k}})$\nfrom $A_{\\ov\\bullet}$ to $H^*(A_{\\ov \\bullet})$.\n\\end{defi}\nNote that if a perverse cdga $A_{\\ov{\\bullet}}$ is intersection-formal, then both $A_{\\ov{0}}$ and $A_{\\ov{\\infty}}$\nare formal cdga's.\n\nWe shall consider the following weaker notion of intersection-formality, which excludes the infinite perversity.\nDenote by $\\GMpdga{\\mathbf{k}}$ the category of \\textit{GM-perverse cdga's} defined by replacing $\\widehat\\mathcal{P}$ by $\\mathcal{P}$ \nin Definition $\\ref{pcdga_def}$.\nNote that for a GM-perverse cdga $A_{\\ov\\bullet}$ the products\n$A_{\\ov p}\\otimes A_{\\ov q}\\longrightarrow A_{\\ov p + \\ov q}$ need only be defined whenever $\\ov p+\\ov q<\\ov \\infty$.\nThe prefix ``GM'' accounts for Goresky-MacPherson, since only finite perversities are involved.\nDenote by $\\mathrm{U}:\\pdga{\\mathbf{k}}\\longrightarrow \\GMpdga{\\mathbf{k}}$ the forgetful functor.\n\n\\begin{defi}\nA perverse cdga $A_{\\ov\\bullet}$ is said to be \\textit{GM-intersection-formal} if\nthere is an isomorphism in $\\mathrm{Ho}(\\GMpdga{\\mathbf{k}})$ from $A_{\\ov\\bullet}$ to $H^*(A_{\\ov \\bullet})$.\n\\end{defi}\nNote that if a $A_{\\ov{\\bullet}}$ is GM-intersection-formal, then $A_{\\ov{0}}$ is formal, but $A_{\\ov{\\infty}}$ \nneed not be formal.\nWe remark that intersection-formality implies the vanishing of Massey products in intersection cohomology,\nwhile GM-intersection-formality implies the vanishing of Massey products in $\\mathrm{U}(IH^*_{\\ov\\bullet}(A))$.\nWe refer to $\\S$3 of \\cite{CST} for a proof of these statements and further discussion on (GM)-intersection-formality.\n\n\n\n\\subsection{Perverse algebraic model}\nWe next describe the perverse algebraic model of a complex projective variety with only isolated singularities,\nas introduced in $\\S$3.2 of \\cite{CST}.\n\n\nLet us first fix some notation.\nDenote by $\\Lambda(t,dt)=\\mathbf{k}(t,dt)$ the free cdga over $\\mathbf{k}$ generated by $t$ in degree 0 and $dt$ in degree 1.\nFor $\\lambda\\in\\mathbf{k}$ denote by $\\delta_\\lambda:\\Lambda(t,dt)\\to \\mathbf{k}$ the evaluation map defined \nby $t\\mapsto \\lambda$ and $dt\\mapsto 0$.\nGiven a perversity $\\ov p\\in \\widehat{\\mathcal{P}}$, we will denote by\n$\\xi_{\\leq \\ov p}A(t,dt)$\nthe truncation \nof $A(t,dt)=A\\otimes \\Lambda(t,dt)$\nby perverse degree $\\ov p$, given in degree $k$ by:\n$$\\xi_{\\leq \\ov p}A(t,dt)^k=\\left\\{\n\\begin{array}{ll}\nA^k\\otimes\\Lambda(t)\\oplus A^{k-1}\\otimes\\Lambda(t)\\otimes dt&,\\text{ if } k<p\\\\\n\\mathrm{Ker } (d^k)\\oplus A^k\\otimes\\Lambda(t)\\otimes t\\oplus A^{k-1}\\otimes\\Lambda(t)\\otimes dt&,\\text{ if } k=p\\\\\nA^k\\otimes\\Lambda(t)\\otimes t\\oplus A^{k-1}\\otimes\\Lambda(t)\\otimes dt&,\\text{ if } k>p\n\\end{array}\n\\right..\n$$\nThis truncation is compatible with differentials, products and poset maps: \n$$d(\\xi_{\\leq \\ov p})\\subseteq \\xi_{\\leq \\ov {p}}\\text{ and }\\xi_{\\leq \\ov p}\\times \\xi_{\\leq \\ov q}\\subseteq \\xi_{\\leq \\ov {p}+\\ov{q}}\n\\text{ for all }\\ov p,\\ov q\\in\\widehat\\mathcal{P}, \\text{ and }\n\\xi_{\\leq \\ov {q}}\\subseteq \\xi_{\\leq \\ov p}\\text{ for all }\\ov q\\leq \\ov p.$$\n\n\\begin{defi}\\label{Pullbackpervers}\nLet $f:A\\to B$ be a morphism of cdga's over $\\mathbf{k}$. Given a perversity $\\ov p\\in\\widehat\\mathcal{P}$, consider the pull-back\nin the category of complexes of $\\mathbf{k}$-vector spaces:\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n\\mathcal{I}_{\\ov{p}}(f)\\ar[r]&\\xi_{\\leq \\ov p}B(t,dt)\\ar[d]^{\\delta_1}\\\\\nA\\ar[r]^{f}&B\n}.\n$$\nSince $\\xi_{\\leq\\ov p}$ is compatible with differentials, products and poset maps, $\\mathcal{I}_{\\ov{\\bullet}}(f)$ with the products\nand differentials\ndefined component-wise, is a perverse cdga, called the \\textit{perverse cdga associated with $f$}.\n\\end{defi}\n\n\nLet $X$ be a complex projective variety with only isolated singularities.\nLet $T$ be a closed algebraic neighborhood of the singular locus $\\Sigma$ in $X$\n(in such a way that the inclusion $\\Sigma\\subset T$ is a homotopy equivalence, see \\cite{Durfee2}).\nThen the link of $\\Sigma$ in $X$ is $L:=\\partial T\\simeq T^*:=T-\\Sigma$.\nThe inclusion $\\iota:L\\hookrightarrow X_{reg}$ of the link into\nthe regular part of $X$ induces a morphism $\\iota^*:\\mathcal{A}_{pl}(X_{reg})\\to \\mathcal{A}_{pl}(L)$\nof cdga's over $\\mathbb{Q}$, between the rational algebras of piecewise linear forms of $X_{reg}$ and $L$. \n\n\\begin{defi}\nThe \\textit{perverse algebraic model for $X$} is the rational perverse cdga\n$I\\mathcal{A}_{\\ov{\\bullet}}(X):=\\mathcal{I}_{\\ov{\\bullet}}(\\iota^*)$ associated with the morphism $\\iota^*$. It is\ngiven by the pull-back diagrams\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\nI\\mathcal{A}_{\\ov{p}}(X)\\ar[r]&\\xi_{\\leq \\ov p}\\mathcal{A}_{pl}(L)(t,dt)\\ar[d]^{\\delta_1}\\\\\n\\mathcal{A}_{pl}(X_{reg})\\ar[r]^{\\iota^*}&\\mathcal{A}_{pl}(L)\n}.\n$$\n\\end{defi}\nWe have an isomorphism of perverse commutative graded algebras $H^*(I\\mathcal{A}_{\\ov \\bullet}(X))\\cong IH^*_{\\ov\\bullet}(X;\\mathbb{Q})$.\nFor the $\\ov{0}$-perversity\nwe have a quasi-isomorphism of cdga's $I\\mathcal{A}_{\\ov0}(X)\\simeq \\mathcal{A}_{pl}(\\overline{X})$, where $\\overline{X}\\to X$ is a normalization of $X$.\nFor the $\\ov\\infty$-perversity we recover the rational homotopy type\n$I\\mathcal{A}_{\\ov{\\infty}}(X)\\simeq \\mathcal{A}_{pl}({X}_{reg})$ of the regular part of $X$.\n\n\nThe above construction defines a contravariant functor $I\\mathcal{A}_{\\ov\\bullet}:\\mathcal{V}_\\mathbb{C}\\longrightarrow \\mathrm{Ho}(\\pdga{\\mathbb{Q}})$\nfrom the category $\\mathcal{V}_\\mathbb{C}$ of complex projective varieties with only isolated singularities \nand stratified morphisms, to the the homotopy category of perverse cdga's over $\\mathbb{Q}$.\n\n\\begin{defi}Let $\\mathbb{Q}\\subset \\mathbf{K}$ be a field.\nA complex projective variety $X$ with isolated singularities is called \\textit{(GM)-intersection-formal\nover $\\mathbf{K}$} if and only if $I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes \\mathbf{K}$ is (GM)-intersection-formal.\n\\end{defi}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Mixed Hodge Structures and Perverse Weight Spectral Sequence}\\label{Section_MHS}\nIn this section, we endow the perverse algebraic model of a complex projective variety $X$ with only isolated singularities,\nwith natural mixed Hodge structures.\nWe then study the perverse weight spectral sequence of $X$\nand prove that\nthe complex intersection homotopy type of $X$ is a direct consequence of \nits perverse weight spectral sequence.\nLastly, we describe the perverse weight spectral sequence in terms of the cohomologies\nof the varieties associated with a resolution of $X$.\n\n\\subsection{Mixed Hodge structures on intersection cohomology}\nDeligne showed that the rational cohomology ring\nof every complex algebraic variety $X$ is endowed with \\textit{mixed Hodge structures}:\nfor every $k\\geq 0$, there is an increasing filtration\n$W$ of the rational cohomology $H^k(X;\\mathbb{Q})$,\ncalled the \\textit{weight filtration}, together with a decreasing filtration \n$F$ of the complex cohomology $H^k(X;\\mathbb{C})$, called the \\textit{Hodge filtration}, \nin such a way that the filtration induced by $F$ and its complex conjugate $\\overline{F}$ on\nthe graded objects\n$Gr_m^WH^k(X;\\mathbb{C})\\cong Gr_m^WH^k(X;\\mathbb{Q})\\otimes\\mathbb{C}$\ndefine a Hodge decomposition of pure weight $m$.\nFurthermore, these filtrations are functorial and\ncompatible with products of varieties (we refer to \\cite{DeHII}, \\cite{DeHIII} or the book \\cite{PS} for details).\n\nIf $X$ is a complex projective variety with only isolated singularities,\nthe compatible mixed Hodge structures on the cohomologies\nof $X$ and $X_{reg}$ define canonical mixed Hodge structures on\n $IH^k_{\\ov{p}}(X;\\mathbb{Q})$, which are compatible with products and\nposet maps. In particular, for every $k\\geq 0$ the morphism\n $IH^k_{\\ov{0}}(X;\\mathbb{Q})\\to IH^k_{\\ov{\\infty}}(X;\\mathbb{Q})$ induced by the inclusion\n$X_{reg}\\hookrightarrow \\ov{X}$ preserves mixed Hodge structures.\n\nA well-known result on the mixed Hodge theory of projective varieties with isolated singularities\nis that for the middle perversity, \nthe weight filtration $W$ on $IH^k_{\\ov{m}}(X;\\mathbb{Q})$ is pure of weight $k$, for all $k\\geq 0$, that is:\n$0=W_{k-1}\\subset W_k=IH^k_{\\ov{m}}(X;\\mathbb{Q}).$\nThis is a consequence of Gabber's purity theorem and the decomposition theorem of\nintersection homology (see \\cite{Ste}. A direct proof using Hodge theory appears in \\cite{Na2}).\nWe next give the bounds on the weight filtration $W$ for an arbitrary perversity.\n\\begin{lem}\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities.\n\\begin{enumerate}[(1)]\n\\item  If $\\ov p<n-1$ then $0=W_{k-1}\\subset W_k\\subset \\cdots \\subset W_{2k}=IH^k_{{\\ov p}}(X;\\mathbb{Q})$. If in addition,\n $k\\leq \\ov p+1$ or $k>n$, then the weight filtration $W$ on $IH^k_{{\\ov p}}(X;\\mathbb{Q})$ is pure of weight $k$.\n \\item  If $\\ov p=n-1$ then the weight filtration $W$ on $IH^k_{{\\ov p}}(X;\\mathbb{Q})$ is pure of weight $k$ for, all $k\\geq 0$.\n\\item If $\\ov p>n-1$ then $0=W_{-1}\\subset W_0\\subset \\cdots \\subset W_{k}=IH^k_{{\\ov p}}(X;\\mathbb{Q})$.\nIf in addition, $k<n$ or $k\\geq \\ov p+1$, then the weight filtration $W$ on $IH^k_{{\\ov p}}(X;\\mathbb{Q})$ is pure of weight $k$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe weight filtration on the cohomologies of $X$ and $X_{reg}$ is bounded respectively by:\n$$0=W_{-1}\\subset W_0\\subset\\cdots\\subset W_k=H^k(X;\\mathbb{Q})\\text{ and }0=W_{k-1}\\subset W_k\\subset\\cdots\\subset W_{2k}=H^k(X_{reg};\\mathbb{Q}).$$\nBy Theorem 2.3.5 of \\cite{DeHII} the category of mixed Hodge structures is abelian and morphisms of mixed Hodge\nstructures are strictly compatible with filtrations.\nHence $\\mathrm{Im }(H^k(X;\\mathbb{Q})\\longrightarrow H^k(X_{reg};\\mathbb{Q}))$ carries a pure Hodge structure of weight $k$, for all $k\\geq 0$.\nSince $X$ has only isolated singularities, for $k>n$, the filtration $W$ on $H^k(X;\\mathbb{Q})$ is pure of weight $k$,\nwhile for $k<n$, the filtration $W$ on $H^k(X_{reg};\\mathbb{Q})$ is pure of weight $k$ (see Theorem 1.13 of \\cite{Ste}).\nThe result is now a matter of verification.\n\\end{proof}\n\n\\subsection{Mixed Hodge perverse cdga's}\nWe next introduce mixed Hodge perverse cdga's. These are perverse cdga's carrying\ncompatible mixed Hodge structures in each perversity and degree.\nUsing Deligne's splitting of mixed Hodge structures we show that, over the\ncomplex numbers, certain mixed Hodge perverse cdga's are isomorphic\nto the perverse cdga defined by the first term\nof the spectral sequence associated with the weight filtration.\n\\begin{defi}\nA \\textit{filtered perverse cdga} $(A_{\\ov\\bullet},W)$ is a perverse cdga $A_{\\ov\\bullet}$ together with a filtration\n$\\{W_{m}A_{\\ov\\bullet}\\}$ indexed by the integers and satisfying: \n\\begin{enumerate}[(i)]\n \\item  $W_{m-1}A_{\\ov p}\\subset W_mA_{\\ov p}$ and\n$d(W_{m}A_{\\ov p})\\subset W_mA_{\\ov p}$ for all $m\\in\\mathbb{Z}$ and all $\\ov{p}\\in \\widehat\\mathcal{P}$,\n\\item $W_mA_{\\ov p}\\cdot W_nA_{\\ov q}\\subset W_{m+n}A_{\\ov p+\\ov q}$ for all $m,n\\in\\mathbb{Z}$ and all $\\ov{p},\\ov{q}\\in\\widehat\\mathcal{P}$,\n\\item $W_mA_{\\ov q}\\subset W_mA_{\\ov p}$, for all $m\\in\\mathbb{Z}$ and all $\\ov q\\leq \\ov p$,\n\\item For all $n\\geq 0$ and all $\\ov{p}\\in\\widehat \\mathcal{P}$ there exists integers $m$ and $l$ such that\n$W_mA^n_{\\ov p}=0$ and $W_lA^n_{\\ov p}=A^n_{\\ov p}$.\n\\end{enumerate}\n\\end{defi}\n\nThe spectral sequence associated with a filtered perverse cdga $(A_{\\ov\\bullet},W)$ is compatible with the multiplicative\nstructure. Hence for all $r\\geq 0$, the term $E_r(A_{\\ov\\bullet},W)$ is a perverse differential \nbigraded algebra with differential $d_r$ of bidegree $(r,1-r)$.\n\n\n\n\\begin{defi}\nA \\textit{mixed Hodge perverse cdga} is a filtered perverse cdga $(A_{\\ov\\bullet},W)$ over $\\mathbb{Q}$,\ntogether with a filtration $F$ on $A_{\\ov\\bullet}\\otimes\\mathbb{C}$, such that for each $n\\geq 0$ and each $\\ov p\\in\\widehat \\mathcal{P}$ the triple\n$(A^n_{\\ov p},\\mathrm{Dec} W,F)$\nis a mixed Hodge structure and the differentials $d:A^n_{\\ov p}\\to A^{n+1}_{\\ov p}$, products\n$A^n_{\\ov p}\\times A^m_{\\ov q}\\to A^{n+m}_{\\ov p+\\ov q}$ and poset maps $A^n_{\\ov {q}}\\to A^n_{\\ov p}$\nare compatible with $W$ and $F$.\n\\end{defi}\nBy an abuse of notation, we shall denote such a mixed Hodge perverse cdga by a triple $(A_{\\ov\\bullet},W,F)$, noting that the second filtration $F$\nis not defined over $A_{\\ov\\bullet}$, but on its complexification $A_{\\ov\\bullet}\\otimes\\mathbb{C}$.\nThe filtration $\\mathrm{Dec} W$ denotes Deligne's d\\'{e}calage of the weight filtration $W$ (see Definition 1.3.3 of \\cite{DeHII}),\ngiven by $\\mathrm{Dec} W_p A_{\\ov\\bullet}^n:= W_{p-n}A^{n}_{\\ov\\bullet} \\cap d^{-1}(W_{p-n-1}A^{n+1}_{\\ov\\bullet})$.\nBy forgetting the perversities we recover the notion of mixed Hodge cdga appearing in \\cite{CG1}.\nThis differs form Morgan's original definition (see \\cite{Mo}) by a shift, which we introduce to make it compatible with \nDeligne's mixed Hodge complexes.\n\n\\begin{lem}\\label{splittingQ}\nLet $(A_{\\ov{\\bullet}},W,F)$ be a mixed Hodge perverse cdga such that $d(W_pA_{\\ov{\\bullet}})\\subset W_{p-1}A_{\\ov{\\bullet}}$.\nThere is an isomorphism of complex perverse cdga's\n$A_{\\ov{\\bullet}}\\otimes\\mathbb{C}\\cong E_1(A_{\\ov\\bullet}\\otimes\\mathbb{C},W)$.\n\\end{lem}\n\\begin{proof}\nThe proof is an adaptation to the perverse setting of Lemma 3.20 of \\cite{CG1} for mixed Hodge cdga's, see also \\cite{Mo}. We indicate the main steps.\nSince the triple $(A_{\\ov{p}}^n,\\mathrm{Dec} W,F)$ is a mixed Hodge structure, by Lemma\n1.2.11 of \\cite{DeHII}, there are functorial decompositions\n$$A^n_{\\ov{p}}\\otimes\\mathbb{C}=\\bigoplus V^{i,j}_{\\ov{p},n},\\text{ with }\\mathrm{Dec} W_m(A^n_{\\ov p}\\otimes\\mathbb{C})=\\bigoplus_{i+j\\leq m} V^{i,j}_{\\ov{p},n}.$$\nSince the differentials, products and poset maps of $A_{\\ov\\bullet}$ are morphisms of mixed Hodge structures,\nthese decompositions are compatible with the perverse cdga structure.\nDefine complex vector spaces $$A^{i,n-i}_{\\ov p}:=\\bigoplus_{r} V_{\\ov p,n}^{n-r-i,r}.$$\nThis gives a decomposition $A_{\\ov \\bullet}^n\\otimes\\mathbb{C}=\\bigoplus A^{i,n-i}_{\\ov p}$ compatible with products and poset maps, and such that\n$d A^{i,n-i}_{\\ov{p}}\\subset A^{i+1,n-i}_{\\ov{p}}$. Since\n$d(W_pA_{\\ov{\\bullet}})\\subset W_{p-1}A_{\\ov{\\bullet}}$, it follows that\n$W_pA_{\\ov{\\bullet}}^n=\\mathrm{Dec} W_{p+n}A_{\\ov{\\bullet}}^n$. Then\n$$W_m(A_{\\ov p}^n\\otimes \\mathbb{C})=\\bigoplus_{i\\leq m} A^{-i,n+i}_{\\ov p}.$$\nSince $Gr^W_m(A_{\\ov p}^n\\otimes \\mathbb{C})=A^{-m,n+m}_{\\ov p}$ and $d A^{m,n-m}_{\\ov{p}}\\subset A^{m+1,n-m}_{\\ov{p}}$,\nthe differential of $E_r(A_{\\ov\\bullet},W)$ is \nis trivial for all $r\\neq 1$. This gives an isomorphism\n$\\pi:A_{\\ov \\bullet}\\otimes\\mathbb{C}\\to E_1(A_{\\ov \\bullet}\\otimes\\mathbb{C},W)$\nof perverse cdga's such that\n$\\pi(A^{-m,n+m}_{\\ov p})\\cong Gr_m^WA_{\\ov p}^n=E_1^{-m,n+m}(A_{\\ov p}\\otimes\\mathbb{C},W).$\n\\end{proof}\n\n\\begin{rmk}\nLet $(A,W)$ be a filtered cdga over a field $\\mathbf{k}$ and let $\\mathbf{k}\\subset\\mathbf{K}$ be a field extension. If $A$ has finite type\nthen by Theorem 2.26 of \\cite{CG1} we have that\n$A\\cong E_r(A,W)$ if and only \nif $A\\otimes_\\mathbf{k}\\mathbf{K}\\cong E_r(A\\otimes_\\mathbf{k}\\mathbf{K},W)$.\nThe same proof is valid for perverse cdga's of finite type. \nHence in this case, the isomorphism of Lemma $\\ref{splittingQ}$ descends to an\nisomorphism over $\\mathbb{Q}$.\n\\end{rmk}\n\nConsider on $\\mathbb{Q}(t,dt)$ the b\\^{e}te filtration $\\sigma$.\nThis is the multiplicative filtration defined by setting\n$t$ of weight $0$ and $dt$ of weight $-1$.\nEndow $\\mathbb{C}(t,dt)=\\mathbb{Q}(t,dt)\\otimes\\mathbb{C}$ with the b\\^{e}te filtration $\\sigma$ and the trivial filtration $t$.\nSince $\\mathrm{Dec} \\sigma=t$, the triple $(\\mathbb{Q}(t,dt), \\sigma, t)$ is a mixed Hodge cdga,\nwhich induces on $H^0(\\mathbb{Q}(t,dt))\\cong \\mathbb{Q}$ the trivial mixed Hodge structure.\n\n\\begin{prop}\\label{pbMHdga}\nLet $f:(A,W,F)\\longrightarrow (B,W,F)$ be a morphism of mixed Hodge cdga's.\nThen the perverse cdga $\\mathcal{I}_{\\ov\\bullet}(f)$ of Definition $\\ref{Pullbackpervers}$\nwith the filtrations $W$ and $F$ defined via the pull-backs\n $$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n(\\mathcal{I}_{\\ov{p}}(f),W,F)\\ar[r]&(\\xi_{\\leq \\ov p}B(t,dt),W*\\sigma,F*t)\\ar[d]^{\\delta_1}\\\\\n(A,W,F)\\ar[r]^{f}&(B,W,F)\n}.\n$$\nis a mixed Hodge perverse cdga, where the filtrations $W*\\sigma$ and $F*t$ are defined by convolution:\n$$(W*\\sigma)_pB(t,dt)=\nW_pB\\otimes\\Lambda(t)+W_{p+1}B\\otimes\\Lambda(t)\\otimes dt\n\\text{ and }(F*t)^pB(t,dt)=F^pB\\otimes\\Lambda(t,dt).$$\n\n\\end{prop}\n\\begin{proof}\nIt suffices to verify that for all $n\\geq 0$ and all $\\ov p\\in\\widehat \\mathcal{P}$, the triple\n$(\\mathcal{I}_{\\ov{p}}(f)^n,\\mathrm{Dec} W,F)$ is a mixed Hodge structure, and that the products\nand poset maps of $\\mathcal{I}_{\\ov{\\bullet}}(f)$ are compatible with the filtrations $W$ and $F$.\nSince d\\'{e}calage commutes with pull-backs and $\\mathrm{Dec}(W*\\sigma)=\\mathrm{Dec} W* t$, we have\n $$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n(\\mathcal{I}_{\\ov{p}}(f)^n,\\mathrm{Dec} W,F)\\ar[r]&(\\xi_{\\leq \\ov p}B(t,dt)^n,\\mathrm{Dec} W*t,F*t)\\ar[d]^{\\delta_1}\\\\\n(A^n,\\mathrm{Dec} W,F)\\ar[r]^{f}&(B^n,\\mathrm{Dec} W,F)\n}.\n$$\nSince the category of mixed Hodge structures is abelian, the truncations\n$(\\xi_{\\leq \\ov p}B(t,dt)^n,\\mathrm{Dec} W,F)$\nare mixed Hodge structures, and hence the above pull-back gives a mixed Hodge structure.\nIt is straightforward to verify that the products\n$\\xi_{\\leq \\ov q}B(t,dt)\\times \\xi_{\\leq \\ov p}B(t,dt)\\to \\xi_{\\leq \\ov q+\\ov p}B(t,dt)$\nand poset maps\n$\\xi_{\\leq \\ov q}B(t,dt)\\longrightarrow \\xi_{\\leq \\ov p}B(t,dt)$ for $\\ov q\\leq \\ov p$,\nare compatible with filtrations, so that the perverse algebra structure of \n$\\mathcal{I}_{\\ov{\\bullet}}(f)$ is also compatible with $W$ and $F$.\n\\end{proof}\n\n\n\\begin{lem}\\label{commutepb}\nLet $f:(A,W,F)\\to (B,W,F)$ be a morphism of mixed Hodge cdga's. There is an isomorphism of perverse differential bigraded algebras \n$E_1(\\mathcal{I}_{\\ov{\\bullet}}(f),W)\\cong \\mathcal{I}_{\\ov{\\bullet}}(E_1(f,W)).$\n\\end{lem}\n\\begin{proof}\nThe evaluation map\n$\\delta_1:(\\xi_{\\leq \\ov p}B(t,dt),W*\\sigma)\\longrightarrow (B,W)$ induces a surjective quasi-isomorphism at the level of $E_1$,\nfor any perversity $\\ov p$.\nTherefore we have\n$E_1(\\mathrm{Ker }(f-\\delta_1))=\\mathrm{Ker }(E_1(f-\\delta_1)$\n(c.f. Proposition\n3.9 of \\cite{Cirici}). \nIt remains to observe that $E_1$ commutes with the truncations $\\xi_{\\leq \\ov p}$ and that we have\na canonical isomorphism of differential bigraded algebras\n$E_1(B(t,dt), W*\\sigma)\\cong E_1(B,W)(t,dt).$\n\\end{proof}\n\n\n\\subsection{Mixed Hodge structures on the perverse algebraic model}\nWe next show that the perverse algebraic model of a complex projective variety $X$ with only isolated singularities\ncarries well-defined mixed Hodge structures (in the homotopy category) which are compatible\nwith the mixed Hodge structures on the rational homotopy types of\n$\\ov{X}$ and $X_{reg}$, and are functorial for stratified morphisms.\n\n\nWe first recall some basic definitions for the theory of mixed Hodge structures \nin rational homotopy. We refer to \\cite{Mo, Na, CG1} for further details.\n\\begin{defi}\nA \\textit{mixed Hodge diagram (of cdga's over $\\mathbb{Q}$)} consists of\na filtered cdga $(A_{\\mathbb{Q}},W)$ over $\\mathbb{Q}$, a bifiltered cdga $(A_{\\mathbb{C}},W,F)$ over $\\mathbb{C}$, together with\na string of filtered quasi-isomorphisms\nfrom $(A_\\mathbb{Q},W)\\otimes \\mathbb{C}$ to $(A_\\mathbb{C},W).$\nIn addition, the following axioms are satisfied:\n\\begin{enumerate}\n\\item[($\\mathrm{MH}_0$)] The weight filtrations $W$ are regular and exhaustive. The Hodge filtration $F$ is biregular.\nThe cohomology $H(A_\\mathbb{Q})$ has finite type.\n\\item[($\\mathrm{MH}_1$)] For all $p\\in\\mathbb{Z}$, the differential of $Gr_p^WA_\\mathbb{C}$ is strictly compatible with $F$.\n\\item[($\\mathrm{MH}_2$)] For all $n\\geq 0$ and all $p\\in\\mathbb{Z}$, the filtration $F$ induced on $H^n(Gr^W_pA_{\\mathbb{C}})$ defines a pure Hodge structure of\nweight $p+n$ on $H^n(Gr^W_pA_\\mathbb{Q})$.\n\\end{enumerate}\n\\end{defi}\n\nMorphisms of mixed Hodge diagrams are given by level-wise morphisms of (bi)filtered cdga's making the corresponding diagrams strictly commutative.\nNote that by forgetting the multiplicative structures we recover the original notion of mixed Hodge complex introduced by Deligne\n(see 8.1 of \\cite{DeHIII}).\n\nAxiom ($\\mathrm{MH}_2$) implies that for all $n\\geq 0$ the triple $(H^n(A_\\mathbb{Q}),\\mathrm{Dec} W,F)$ is a mixed Hodge structure over $\\mathbb{Q}$.\nIn particular, the cohomology of every mixed Hodge diagram is a mixed Hodge cdga with trivial\ndifferential. Since the category of mixed Hodge structures is abelian, every mixed Hodge cdga is a mixed Hodge diagram in which the\ncomparison quasi-isomorphisms are identities.\n\n\\begin{defi}\nLet $X$ be a topological space. A \\textit{mixed Hodge diagram for $X$} is a mixed Hodge diagram \n$\\mathcal{A}(X)$ whose rational component $\\mathcal{A}(X)_\\mathbb{Q}\\simeq A_{pl}(X)$ is quasi-isomorphic to the rational algebra of piecewise linear forms of $X$.\n\\end{defi}\nNote that when such a mixed Hodge diagram $\\mathcal{A}(X)$ exists, the above quasi-isomorphism induces an isomorphism\n$H^*(\\mathcal{A}(X)_\\mathbb{Q})\\cong H^*(X;\\mathbb{Q})$ endowing the cohomology of $X$ with mixed Hodge structures.\n\nWe now prove the main result of this section, endowing the perverse algebraic model of a projective variety with only isolated singularities,\nwith mixed Hodge structures. The proof relies on the existence of compatible mixed Hodge structures on the rational\nhomotopy types of the link of the singularities and the regular part of the variety respectively (\\cite{Mo, Na, DH}),\ntogether with the existence of minimal models \\`{a} la Sullivan of morphisms of cohomologically \nconnected mixed Hodge diagrams, which is proven in \\cite{CG1} (see also \\cite{Cirici} for a homotopical framework of such models).\n\n\\begin{teo}\\label{MHSmodel}\nLet $X$ be a complex projective variety with only isolated singularities and let $I\\mathcal{A}_{\\ov\\bullet}(X)$ denote the perverse algebraic model of $X$.\nThere is a mixed Hodge perverse cdga $(IM_{\\ov\\bullet}(X),W,F)$ and a string\n$IM_{\\ov\\bullet}(X)\\leftarrow \\ast \\to I\\mathcal{A}_{\\ov\\bullet}(X)$ of quasi-isomorphisms of perverse cdga's such that:\n\\begin{enumerate}[(1)]\n\\item $IM_{\\ov\\bullet}(X)=\\mathcal{I}_{\\ov\\bullet}(\\widetilde \\iota)$, where $\\widetilde \\iota:M(X_{reg})\\to M(L)$ is a \nmodel of mixed Hodge cdga's for the rational homotopy type of the inclusion $\\iota:L\\hookrightarrow X_{reg}$.\n\\item There is an isomorphism of mixed Hodge perverse graded algebras $H^*(IM_{\\ov\\bullet}(X))\\cong IH^*_{\\ov\\bullet}(X;\\mathbb{Q})$.\n\\item The mixed Hodge cdga's $IM_{\\ov{0}}(X)$ and  $IM_{\\ov{\\infty}}(X)$ \ndefine the mixed Hodge structures on the rational homotopy type of a normalization $\\ov{X}$ of $X$ and the regular part $X_{reg}$ of $X$\nrespectively.\n\\item The differential of $IM_{\\ov\\bullet}(X)$ satisfies $d(W_pIM_{\\ov\\bullet}(X))\\subset W_{p-1}IM_{\\ov\\bullet}(X)$.\n\\end{enumerate} \nThis construction defines a functor $IM_{\\ov \\bullet}:\\mathcal{V}_\\mathbb{C}\\longrightarrow \\mathrm{Ho}(\\widehat{\\mathcal{P}}\\mathbf{MH}\\mathsf{CDGA})$\nwith values in the homotopy category of mixed Hodge perverse cdga's.\n\\end{teo}\n\\begin{proof}\nBy Theorem 3.2.1 of \\cite{DH} (see also $\\S$13 of \\cite{Na}) there are mixed Hodge diagrams $\\mathcal{A}(X_{reg})$ and $\\mathcal{A}(L)$ \nfor $X_{reg}$ and $L$ respectively, together with a morphism\n$\\mathcal{A}(X_{reg})\\longrightarrow \\mathcal{A}(L)$ \nwhose rational component is the morphism\n$\\iota^*:\\mathcal{A}_{pl}(X_{reg})\\longrightarrow \\mathcal{A}_{pl}(L)$ of rational piecewise linear forms induced by the inclusion $L\\hookrightarrow X_{reg}$.\nBy Theorem 3.19 of \\cite{CG1} on the existence of relative minimal models for mixed Hodge diagrams,\nwe can construct a commutative diagram of mixed Hodge diagrams\n$$\n\\xymatrix{\n\\mathcal{A}(X_{reg})\\ar[r]^{\\iota^*}&\\mathcal{A}(L)\\\\\n\\ar[u]\\ar[d]\\ast\\ar[r]&\\ast\\ar[u]\\ar[d]\\\\\nM(X_{reg})\\ar[r]^{\\widetilde \\iota}&M(L)\n}\n$$\nwhere the vertical maps are quasi-isomorphisms and $\\widetilde \\iota$ is a morphism of mixed Hodge cdga's\nwhose differential satisfies $d(W_p)\\subset W_{p-1}$.\nBy Proposition $\\ref{pbMHdga}$, the perverse cdga $IM_{\\ov{\\bullet}}(X):=\\mathcal{I}_{\\ov\\bullet}(\\widetilde i)$\nassociated with $\\widetilde \\iota$ is a mixed Hodge cdga.\nFurthermore, the above commutative\ndiagram gives a string of quasi-isomorphisms of perverse cdga's from \n$IM_{\\ov\\bullet}(X)$ to $I\\mathcal{A}_{\\ov\\bullet}(X)$.\nThis proves (1).\nThe induced map $\\widetilde \\iota:M(X_{reg})\\longrightarrow M(L)$ induces \na morphism of mixed Hodge structures in cohomology $H^*(X_{reg})\\longrightarrow H^*(L)$.\nHence (2) follows from the isomorphism $\\mathrm{Ker }(H^*(X_{reg})\\longrightarrow H^*(L))\\cong \\mathrm{Im }(H^*(X)\\longrightarrow H^*(X_{reg})$.\nAssertion (3) is easily verified. Lastly, (4) follows from the fact that\nthe differential on \n$IM_{\\ov p}(X)$ is defined component-wise by differentials satisfying $d(W_p)\\subset W_{p-1}$.\nThe construction of $IM_{\\ov \\bullet}(X)$ is functorial (in the homotopy category) for morphisms $f:X\\to Y$ such that $f(X_{reg})\\subset Y_{reg}$.\n\\end{proof}\n\n\n\\subsection{Perverse weight spectral sequence}\nLet $X$ be a complex projective variety with only isolated singularities.\nThe inclusion $\\iota:L\\hookrightarrow X_{reg}=X-\\Sigma$ of the link into the regular part of $X$\ninduces a morphism of (multiplicative) weight spectral sequences $E_1(\\iota^*):E_1(X_{reg})\\longrightarrow E_1(L)$,\nwhere $E_1(X_{reg})$ and $E_1(L)$\nare the spectral sequences associated with the weight filtration of a mixed Hodge diagram\nfor $X_{reg}$ and $L$ respectively.\n\n\\begin{defi}\nThe \\textit{perverse weight spectral sequence of $X$} is the perverse differential bigraded algebra\n$IE_{1,\\ov\\bullet}(X):=\\mathcal{I}_{\\ov\\bullet}(E_1(\\iota^*))$\nassociated with the morphism $E_1(\\iota^*)$,\ngiven by the pull-back diagrams\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\nIE_{1,\\ov p}(X)\\ar[r]&\\xi_{\\leq \\ov p}E_1(L)(t,dt)\\ar[d]^{\\delta_1}\\\\\nE_1(X_{reg})\\ar[r]^{E_1(\\iota^*)}&E_1(L)\n}.\n$$\n\n\n\n\\end{defi}\nWe remark that $IE_{1,\\ov\\bullet}(X)$ is only well-defined in the homotopy category of perverse differential bigraded algebras.\nIts cohomology is a well-defined algebraic invariant of $X$ and satisfies\n$$IE^{r,s}_{2,\\ov p}(X):=H^{r,s}(IE_{1,\\ov p}(X))\\cong Gr_{s}^WIH_{\\ov p}^{r+s}(X;\\mathbb{Q}).$$\n\nThe following result states that the complex intersection homotopy type of a projective variety with isolated singularities is\ndetermined by its perverse weight spectral sequence.\n\n\\begin{teo}\\label{IE1formality}Let $X$ be a complex projective variety with only isolated singularities. There is an isomorphism\nfrom $I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes\\mathbb{C}$ to $IE_{1,\\ov\\bullet}(X)\\otimes\\mathbb{C}$\nin the homotopy category $\\mathrm{Ho}(\\pdga{\\mathbb{C}})$.\n\\end{teo}\n\\begin{proof}\nLet $(IM_{\\ov\\bullet}(X),W,F)$ be the mixed Hodge perverse cdga given by Theorem \n$\\ref{MHSmodel}$.\nSince the differential satisfies $d(W_pIM_{\\ov\\bullet}(X))\\subset W_{p-1}IM_{\\ov\\bullet}(X)$,\nby Lemma $\\ref{splittingQ}$, we have an isomorphism of complex perverse cdga's\n$IM_{\\ov{\\bullet}}(X)\\otimes\\mathbb{C}\\cong E_1(IM_{\\ov\\bullet}(X)\\otimes\\mathbb{C},W).$\n\nBy construction, we have that $IM_{\\ov\\bullet}(X)=\\mathcal{I}_{\\ov\\bullet}(\\widetilde \\iota)$,\nwhere $\\widetilde \\iota:(M(X_{reg}),W,F)\\to (M(L),W,F)$ is a morphism of mixed Hodge cdga's computing the rational homotopy type of\n$\\iota:L\\hookrightarrow X_{reg}$. Therefore by Lemma $\\ref{commutepb}$, we have an isomorphism of perverse cdga's\n$E_1(IM_{\\ov\\bullet}(X),W)\\cong \\mathcal{I}_{\\ov{\\bullet}}(E_1(\\widetilde \\iota,W)).$\nIt only remains to note that there is a string of quasi-isomorphisms from\n$\\mathcal{I}_{\\ov{\\bullet}}(E_1(\\widetilde \\iota))$ to $IE_{1\\ov\\bullet}(X)=\\mathcal{I}_{\\ov{\\bullet}}(E_1(\\iota^*))$.\n\\end{proof}\n\nAs a direct application we have the following \"purity implies formality\" result in the context of rational intersection homotopy:\n\\begin{cor}\nLet $X$ be a complex projective variety with only isolated singularities.\nIf the weight filtration on $IH^{k}_{\\ov p}(X)$ is pure of weight $k$,\nfor all $k\\geq 0$ and every finite perversity $\\ov p\\in \\mathcal{P}$,\nthen $X$ is GM-intersection-formal over $\\mathbb{C}$.\n\\end{cor}\n\n\n\\begin{example}\n[$\\mathbb{Q}$-homology manifolds]\nLet $X$ be a complex projective variety which is also a\n$\\mathbb{Q}$-homology manifold. Then \n$IH^k_{\\ov p}(X;\\mathbb{Q})\\cong IH^k_{\\ov 0}(X;\\mathbb{Q})$ for every finite perversity $\\ov p\\in\\mathcal{P}$ (see 6.4 of \\cite{GMP1}).\nIn particular, the weight filtration on $IH^k_{\\ov p}(X;\\mathbb{Q})$ is pure of weight $k$, for all $k\\geq 0$\nand every $\\ov p\\in \\mathcal{P}$. Hence $X$ is GM-intersection-formal over $\\mathbb{C}$.\nExamples of such varieties are given by weighted projective spaces or more generally $V$-manifolds\n(see Appendix B of \\cite{Di}),\nsurfaces with $A_1$-singularities, the Cayley cubic or the Kummer surface.\n\\end{example}\n\n\n\\subsection{Resolution of singularities and weight spectral sequence}\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities.\nDenote by $\\Sigma$ the singular locus of $X$ and by\n$L:=L(\\Sigma,X)$ the link of $\\Sigma$ in $X$. \n\nLet $f:\\widetilde X\\longrightarrow X$ be a resolution of singularities of $X$ such that $D:=f^{-1}(\\Sigma)$ is a simple normal crossings divisor.\nWe may write\n$D=D_1\\cup\\cdots \\cup D_N$ as the union\nof irreducible smooth varieties meeting transversally. Let $D^{(0)}=\\widetilde X$ and for all $r>0$, \ndenote by $D^{(r)}=\\bigsqcup_{|I|=r}D_I$ the disjoint union of all \n$r$-fold intersections \n$D_I:=D_{i_1}\\cap\\cdots \\cap D_{i_r}$ where $I=\\{i_1,\\cdots,i_r\\}$ denotes an ordered subset of $\\{1,\\cdots,N\\}$.\nSince $D$ has simple normal crossings, it follows that $D^{(r)}$ is a smooth projective variety of dimension $n-r$. \nFor $1\\leq k\\leq r$, denote by $j_{I,k}:D_I\\hookrightarrow D_{I\\setminus \\{i_k\\}}$ \nthe inclusion and let $j_{r,k}:=\\bigoplus_{|I|=r} j_{I,k}:D^{(r)}\\hookrightarrow D^{(r-1)}$.\nThese maps define a simplicial resolution\n$D_\\bullet=\\{D^{(r)}, j_{r,k}\\}$.\n\nLet $r\\geq 1$. For every $1\\leq k\\leq r$ we\nwill denote by $j_{r,k}^*:=(j_{r,k})^*:H^*(D^{(r-1)})\\to H^*(D^{(r)})$ the restriction morphism induced by the inclusion $j_{r,k}$ and\nby $\\gamma_{r,k}:=(j_{r,k})_!:H^{*-2}(D^{(r)})\\to H^*(D^{(r-1)})$ the corresponding Gysin map.\nWe have combinatorial restriction morphisms\n$$j_{(r)}^s:=\\sum_{k=1}^r (-1)^{k-1} (j_{r,k})^{*}:H^{s}(D^{(r-1)};\\mathbb{Q})\\longrightarrow H^s(D^{(r)};\\mathbb{Q})$$\nand combinatorial Gysin maps\n$$\\gamma_{(r)}^s:=\\sum_{k=1}^r (-1)^{k-1} (j_{r,k})_{!}:H^{s-2r}(D^{(r)};\\mathbb{Q})\\longrightarrow H^{s-2(r-1)}(D^{(r-1)};\\mathbb{Q}).$$\n\nWith this notation, the weight spectral sequence for $X_{reg}$ can be written as:\n$$\nE_1^{r,s}(X_{reg})=\n\\def1.4{1.6}\n\\begin{array}{c c c c c c c c c}\n\\multicolumn{1}{c}{}\\\\\n\\cdots&\\longrightarrow &H^{s-4}(D^{(2)};\\mathbb{Q})&\\xra{\\gamma^s_{(2)}}&H^{s-2}(D^{(1)};\\mathbb{Q})&\\xra{\\gamma^s_{(1)}}&H^s(\\widetilde X;\\mathbb{Q})&\\longrightarrow&0\\\\ \n\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}.\n$$\nIts algebra structure \nis given by the maps\n$H^{m}(D^{(p)};\\mathbb{Q})\\otimes H^{l}(D^{(q)};\\mathbb{Q})\\longrightarrow H^{m+l}(D^{(p+q)};\\mathbb{Q})$\ninduced by combinatorial restriction morphisms, for $p+q\\leq n$ (see \\cite{Mo}).\n\nWe next describe the multiplicative weight spectral sequence of the link $L\\simeq L(D,\\widetilde X)$.\nIn \\cite{Durfee}, Durfee endows the cohomology of the link of an\nisolated singularity with mixed Hodge structures, and describes\nits weight spectral sequence in terms of a resolution of singularities.\nHowever, such spectral sequence is not multiplicative,\nsince it is the spectral sequence associated with a mixed Hodge complex for $L$.\nTo describe the multiplicative weight spectral sequence of the link we\nanalyze the construction due to Durfee-Hain \\cite{DH} \nof a mixed Hodge diagram of cdga's for $L$.\n\nFor all $i\\in \\{1,\\cdots,N\\}$ define \n$$L_i:=L(D_i-\\bigsqcup_{i\\neq j} D_i\\cap D_j,\\widetilde X).$$\nFor all $r>0$ denote by $L^{(r)}=\\bigsqcup_{|I|=r}L_I$ the disjoint union of all \n$r$-fold intersections \n$L_I:=L_{i_1}\\cap\\cdots \\cap L_{i_r}$ where $I=\\{i_1,\\cdots,i_r\\}$ denotes an ordered subset of $\\{1,\\cdots,N\\}$.\nWe have\n$$L^{(1)}:=\\bigsqcup_i L_i,\\,\\, L^{(2)}:=\\bigsqcup_{i\\neq j}L_i\\cap L_j,\\cdots$$\nWe obtain a\nsimplicial manifold\n$L_\\bullet=\\{L^{(r)}, i_{r,k}\\}$,\nwhere $i_{r,k}:L^{(r)}\\hookrightarrow L^{(r-1)}$, for\n$1\\leq k\\leq r$, denote the natural inclusions.\n\nThe multiplicative weight spectral sequence for $L^{(r)}$ is given by:\n$$E_1^{*,*}(L^{(r)})=\\bigoplus_{I=\\{i_1,\\cdots,i_r\\}} E_1(D_I-\\mathrm{Sing}(D_I))\\widetilde\\otimes \\Lambda(\\theta_{i_1},\\cdots,\\theta_{i_r}).$$\nwhere $\\theta_k$ are generators of bidegree $(-1,2)$ \nand $\\widetilde \\otimes$ accounts for the fact that the differential of $\\theta_k$ is given by\n$d(\\theta_{k})=c_k$, where $c_k\\in H^2(D_{k};\\mathbb{Q})$ is the Chern class of $D_{k}$.\n\nThe multiplicative weight spectral sequence for $L$ is then given by the end\n$$E_1^{p,q}(L):=\\int_\\alpha \\bigoplus_mE_1^{p-m,q}(L^{(\\alpha)})\\otimes \\Omega_\\alpha^m,$$\nwhere $\\Omega_\\alpha$\nis the simplicial cdga given by\n$\\Omega_\\alpha:={{\\Lambda(t_0,\\cdots,t_\\alpha,dt_0,\\cdots,dt_\\alpha)}\/{\\sum t_i-1,\\sum dt_i}}$,\nwith $t_i$ of degree 0 and $dt_i$ of degree 1.\n\nIn Sections $\\ref{perverseordinary}$ and $\\ref{perversesurface}$ we provide a description\nof the morphism $E_1(X_{reg})\\longrightarrow E_1(L)$ in the particular cases of\nordinary isolated singularities and isolated surface singularities respectively,\nthus giving an explicit description of the perverse weight spectral sequence in these cases.\n\n\n\n\n\n\n\n\n\n\n\\section{Ordinary Isolated Singularities}\\label{Section_OIS}\nFor the rest of this section, let $X$ be a complex projective variety of dimension $n$ with isolated singularities. \nWe will show that if $X$ admits a resolution of singularities \nin such a way that the exceptional divisor is smooth, and if the link\nof each singular point is  $(n-2)$-connected, then $X$ is GM-intersection-formal over $\\mathbb{C}$.\nThe main class of examples to which this result applies are varieties with \nordinary multiple points, but it also applies to a large family of\nhypersurfaces with isolated singularities and more generally, to complete\nintersections with isolated singularities\nadmitting  a resolution of singularities with smooth exceptional divisor.\n\n\\subsection{Notation}\nDenote by $\\Sigma$ the singular locus of $X$ and by $X_{reg}=X-\\Sigma$ its regular part.\nDenote by $L:=L(\\Sigma,X)$ the link of $\\Sigma$ in $X$, and by $\\iota:L\\hookrightarrow X_{reg}$ the natural inclusion.\nSince $\\Sigma$ is discrete, the link $L$ can be\nwritten as a disjoint union $L=\\sqcup L_\\sigma$, where $L_\\sigma=L(\\sigma,X)$ is the link of $\\sigma\\in\\Sigma$\nin $X$.\n\nAssume that $X$ admits a resolution of singularities \n$f:\\widetilde X\\longrightarrow X$ of $X$ such that the exceptional\ndivisor $D:=f^{-1}(X)$ is smooth.\nDenote by\n$$j^k:H^k(\\widetilde X)\\longrightarrow H^k(D)\\text{ and }\\gamma^k:H^{k-2}(D)\\longrightarrow H^k(\\widetilde X)$$\nthe restriction morphisms and the Gysin maps \ninduced by the inclusion $j:D\\hookrightarrow \\widetilde X$.\nFor all $k\\geq 2$, define $j_{\\#}^k:=j^k\\circ \\gamma^k:H^{k-2}(D)\\longrightarrow H^{k}(D)$.\n\nUnless stated otherwise, all cohomologies are taking with rational coefficients.\n\n\\subsection{Perverse weight spectral sequence}\\label{perverseordinary}\nThe morphism $E_1(\\iota^*):E_1^{*,*}(X_{reg})\\longrightarrow E_1^{*,*}(L)$ of weight spectral sequences\ninduced by the inclusion $\\iota:L\\hookrightarrow X_{reg}$ can be written as:\n\n$$\n\\xymatrix@R=8pt@C=36pt{\nE_1^{r,s}(X_{reg})=\\ar[ddd]&\\ar[ddd]^{Id} H^{s-2}(D)\\ar[r]^{\\gamma^s}&\\ar[ddd]^{j^s} H^{s}(\\widetilde X)\\\\\n\\\\\n\\\\\nE_1^{r,s}(L)=& H^{s-2}(D)\\ar[r]^{j_{\\#}^s}& H^{s}(D)\\\\\n&\\text{\\tiny{$r=-1$}}&\\text{\\tiny{$r=0$}}&\n}\n$$\nThe algebra structure of $E_1^{*,*}(X_{reg})$ is induced by the \ncup product of $H^*(\\widetilde X)$, together with the maps\n$H^s(\\widetilde X)\\times H^{s'}(D)\\longrightarrow H^{s+s'}(D)$\ngiven by $(x,a)\\mapsto j^*(x)\\cdot a$.\nSince $\\gamma(a\\cdot j^*(x))=\\gamma(a)\\cdot x$,\nthis algebra structure is compatible with the differential $\\gamma$.\nThe non-trivial products of $E_1^{*,*}(L)$ are the maps $E_1^{0,s}(L)\\times E_1^{r,s'}(L)\\longrightarrow E_1^{r,s+s'}(L)$,\nwith $r\\in\\{0,1\\}$ and $s,s'\\geq 0$, induced by the cup product of $H^*(D)$.\n\n\nThe perverse weight spectral sequence $IE^{*,*}_{1,\\ov \\bullet}(X):=\\mathcal{I}_{\\ov{\\bullet}}(E_1(i^*))$ for $X$ \n can be written as:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{p}}(X)=\n\\def1.4{1.6}\n\\begin{array}{| c c c c c |}\n\\hline\nH^{s-2}(D)\\otimes\\Lambda(t)\\otimes t&\\longrightarrow&\\mathcal{J}^s_{1}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline\n\\mathrm{Ker }(j_{\\#}^s)\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes t&\\longrightarrow&\\mathcal{J}^s_{1}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline\nH^{s-2}(D)\\otimes\\Lambda(t)&\\longrightarrow&\\mathcal{J}^s_{0}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}\n\\end{array}\n\\begin{array}{ l }\n\\text{\\tiny{$s>p+1$}}\\\\\n\\text{\\tiny{$s=p+1$}}\\\\\n\\text{\\tiny{$s<p+1$}}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{array}\n$}\\end{equation*}\n\nwhere $\\mathcal{J}^s_{\\alpha}$,with $\\alpha\\in\\{0,1\\}$ and $0\\leq s\\leq n$, is the vector space defined via the pull-back:\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n\\mathcal{J}^s_{\\alpha}\\ar[r]&H^{s}(D)\\otimes\\Lambda(t)\\otimes t^{\\alpha}\\ar[d]^{\\delta_1}\\\\\nH^{s}(\\widetilde X)\\ar[r]^{j^s}&H^{s}(D)\n}.\n$$\n\nThe differential $d_1^{-1,s}:IE^{-1,s}_{1,\\ov{p}}(X)\\longrightarrow IE^{0,s}_{1,\\ov{p}}(X)$ is given by\n$$\\sum a_it^i\\mapsto \\left((\\sum \\gamma^s(a_i), \\sum j_{\\#}^s(a_i)t^i), \\sum i a_i t^{i-1}dt)\\right); \\, a_i\\in H^{s-2}(D).$$\nThe differential $d_1^{0,s}:IE^{0,s}_{1,\\ov{p}}(X)\\longrightarrow IE^{-1,s}_{0,\\ov{p}}(X)$ is given by\n$$\\left( (x,\\sum a_i t^i), \\sum b_it^i dt)\\right)\\mapsto \\sum i a_i t^{i-1} dt+\\sum j_{\\#}^s(b_i)t^i dt\\,; \n\\left\\{\\begin{array}{l}\na_i \\in H^{s}(D), b_i\\in H^{s-2}(D)\\\\\nx\\in H^{s}(X), j^s(x)=\\sum a_i\n\\end{array}\\right..$$\n\nThe algebra structure of $IE^{*,*}_{1,\\ov \\bullet}(X)$ is given by the following maps:\n$$\n\\def1.4{1.6}\n\\begin{array}{llll}\n\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}\\,& IE^{0,*}_{1,\\ov p}(X)\\times IE^{0,*}_{1,\\ov q}(X)&\\longrightarrow &IE^{0,*}_{1,\\ov p+\\ov q}(X)\\\\\n& ((x,a+b\\cdot dt), (y,c+e\\cdot dt))&\\mapsto &(x y, a c+(a e+c b)\\cdot dt)\\\\\n\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}\\,&IE^{0,*}_{1,\\ov p}(X)\\times IE^{-1,*}_{1,\\ov q}(X)&\\longrightarrow &IE^{-1,*}_{1,\\ov p+\\ov q}(X)\\\\ \n & ((x,a+b\\cdot dt), c)&\\mapsto &a c\\\\\n\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}\\,& IE^{0,*}_{1,\\ov p}(X)\\times IE^{1,*}_{1,\\ov q}(X)&\\longrightarrow &IE^{1,*}_{1,\\ov p+\\ov q}(X)\\\\ \n& ((x,a+b\\cdot dt), c\\cdot dt)&\\mapsto &a c\\cdot dt\\\\\n\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}\\,&  IE^{1,*}_{1,\\ov p}(X)\\times IE^{-1,*}_{1,\\ov q}(X)&\\longrightarrow &IE^{0,*}_{1,\\ov p+\\ov q}(X)\\\\ \n& (a,c\\cdot dt)&\\mapsto &a c \\cdot dt\n\\end{array}\n$$\nwhere $x,y\\in H^*(\\widetilde X)$ and $a,b,c,e\\in H^*(D)\\otimes\\Lambda(t)$.\n\nBy computing the cohomology of $IE_{1,\\ov \\bullet}(X)$ we find:\n\n$$\nIE^{r,s}_{2,\\ov{p}}(X)\\cong\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\hline\n0&\\mathrm{Ker }(j^s)&\\mathrm{Coker }(j^s)\\\\ \\hline\n\\mathrm{Ker }(\\gamma^s)&\\mathrm{Ker }(j^s)\/(\\mathrm{Ker }(j^s)\\cap\\mathrm{Im }(\\gamma^s))&\\mathrm{Coker }(j^s)\\\\ \\hline\n\\mathrm{Ker }(\\gamma^s)&\\mathrm{Coker }(\\gamma^s)&0\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}\n\\end{array}\n\\begin{array}{ l }\n\\text{\\tiny{$s>p+1$}}\\\\\n\\text{\\tiny{$s=p+1$}}\\\\\n\\text{\\tiny{$s<p+1$}}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{array}\n$$\n\nThis gives the following formula for the intersection cohomology of $X$:\n$$IH^s_{\\ov p}(X;\\mathbb{Q})=\\left\\{\n\\begin{array}{ll}\nH^s(X_{reg})\\cong \\mathrm{Ker }(\\gamma^{s+1})\\oplus  \\mathrm{Coker }(\\gamma^s) &\\text{ if }s<p+1\\\\\n\\mathrm{Im }(H^s(\\widetilde X)\\longrightarrow H^s(X_{reg}))\\cong \\dfrac{\\mathrm{Ker }(j^s)}{\\mathrm{Im }(\\gamma^s)\\cap \\mathrm{Ker }(j^s)}&\\text{ if }s=p+1\\\\\nH^s(X)\\cong \\mathrm{Coker }(j^{s-1})\\oplus \\mathrm{Ker }(j^{s})&\\text{ if }s>p+1\n\\end{array}\n\\right..\n$$\n\n\nThe following is straightforward.\n\\begin{lem}\\label{PD}\nFor all $0\\leq s\\leq n$ have  Poincar\\'{e} duality isomorphisms\n$$\\mathrm{Coker }(\\gamma^{n+s})\\cong \\mathrm{Ker }(j^{n-s})^\\vee\\text{ and }\\mathrm{Ker }(\\gamma^{n+s})\\cong \\mathrm{Coker }(j^{n-s})^\\vee.$$\n\\end{lem}\n\n\n\\subsection{Conditions on the cohomology of the link}\nSince $\\dim(\\Sigma)=0$, the weight filtration on the cohomology of the link\nis semi-pure: the weights on $H^k(L)$ are less than or equal to $k$ for $k<n$,\nand greater or equal to $k+1$ for $k\\geq n$.\nThis is a consequence of Gabber's purity theorem and the decomposition theorem of\nintersection homology (see \\cite{Ste}, see also \\cite{Na2} for a direct proof).\n\n\\begin{lem}\\label{semipure0}With the above notation we have:\n\\begin{enumerate}[(1)]\n\\item The map $j_{\\#}^k:H^{k-2}(D)\\to H^k(D)$ is injective for $k\\leq n$ and surjective for $k\\geq n$.\n\\item The Gysin map $\\gamma^k:H^{k-2}(D)\\to H^k(\\widetilde X)$ is injective for $k\\leq n$ and $\\gamma^{2n}$ is surjective.\n\\item The restriction morphism $j^k:H^{k}(\\widetilde X)\\to H^k(D)$ is surjective for $k\\geq n$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe first assertion follows from the semi-purity of the link and the isomorphisms\n$$Gr_{k+1}^WH^{k}(L)\\cong \\mathrm{Ker }(j_{\\#}^{k+1})\\text{ and } Gr_{k}^WH^{k}(L)\\cong \\mathrm{Coker }(j_{\\#}^{k}).$$\nSince $j_{\\#}^k=j^k\\circ \\gamma^k$, it follows from (1) that $\\gamma^k$ is injective for $k\\leq n$ and\n$j^k$ is surjective for $k\\geq n$. Since $H^{2n}(X_{reg})\\cong \\mathrm{Coker }(\\gamma^{2n})$ and $X_{reg}$ is non-compact of real dimension $2n$,\n$\\gamma^{2n}$ is surjective.\n\\end{proof}\n\nAssume that the rational cohomology of the link $L$ of $\\Sigma$ in $X$ satisfies\n$H^i(L;\\mathbb{Q})=0$ for all $0<i\\leq n-2$.\nFor instance, this is the case when $X$ has only ordinary isolated singularities, as we shall later see.\n\n\\begin{lem}\\label{ordinary0}With the above assumption:\n\\begin{enumerate}[(1)]\n \\item   The map $j_{\\#}^{n-1}:H^{n-3}(D)\\to H^{n-1}(D)$ is injective, the map $j_{\\#}^{n+1}:H^{n-1}(D)\\to H^{n+1}(D)$ is surjective \n and $j_{\\#}^s:H^{s-2}(D)\\to H^{s}(D)$ is an isomorphism for all $s\\neq 0,n-1,n+1,2n$.\n \\item  The map $\\gamma^s:H^{s-2}(D)\\to H^{s}(X)$ is injective for all $s\\neq n+1,2n$, and $j^s:H^{s}(X)\\to H^{s}(D)$ is surjective for all $s\\neq 0,n-1$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nIt follows from the isomorphisms $H^s(L)\\cong \\mathrm{Ker }(j_{\\#}^{s+1})\\oplus \\mathrm{Coker } (j_{\\#}^s)$ together with Lemma $\\ref{semipure0}$.\n\\end{proof}\n\n\\begin{lem}\\label{ordinary1}With the above assumption we have the following isomorphisms:\n\\begin{enumerate}[(1)]\n \\item  $H^s(X)\\cong \\mathrm{Ker }(j^s)\\oplus \\mathrm{Im }(\\gamma^s)$ for all $s\\neq 0,n-1,n+1,2n$, \n \\item  $\\mathrm{Ker }(j^{n-1})\\cap \\mathrm{Im }(\\gamma^{n-1})=\\{0\\}$ and $\\mathrm{Ker }(j^{n+1})\/(\\mathrm{Ker }(j^{n+1})\\cap \\mathrm{Im }(\\gamma^{n+1}))\\cong \\mathrm{Coker }(\\gamma^{n+1})$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nAssume that $s\\neq 0,n-1,n+1,2n$. From (1) and (2) of Lemma $\\ref{ordinary0}$, $j_{\\#}^s$ is an isomorphism and $j^s$ is surjective.\nThen the composition $\\gamma^s\\circ (j_{\\#}^s)^{-1}$ defines a splitting for the short exact sequence\n$$0\\longrightarrow \\mathrm{Ker }(j^{s})\\longrightarrow H^s(X)\\stackrel{j^{s}}{\\longrightarrow}H^s(D)\\longrightarrow 0$$\nHence (1) follows. The isomorphisms in (2) follow from the injectivity of $j_{\\#}^{n-1}$ and the surjectivity of $j_{\\#}^{n+1}$ respectively.\n\\end{proof}\n\n\\subsection{Intersection cohomology}\nBy Lemma $\\ref{ordinary1}$ the second term of the weight spectral sequences for $X_{reg}$ and $L$ are given by:\n$$\nE_2^{r,s}(X_{reg})\\cong\n\\def1.4{1.6}\n\\begin{tabular}{| c | c | }\n\\hline\n$\\mathrm{Ker }(\\gamma^{2n})$&0\\\\ \\hline\n0&$\\mathrm{Coker }(\\gamma^s)$\\\\ \\hline\n$\\mathrm{Ker }(\\gamma^{n+1})$&$\\mathrm{Coker }(\\gamma^{n+1})$\\\\ \\hline\n& \\\\\n0&$\\mathrm{Coker }(\\gamma^s)$\\\\\n& \\\\\\hline\n0&$H^0(\\widetilde X)$\\\\ \\hline\\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,;\\,\\,\\,\\,\\,\nE_2^{r,s}(L)\\cong\n\\begin{tabular}{| c | c | }\n\\hline\n$H^{2n-2}(D)$&0\\\\ \\hline\n0&0\\\\  \\hline\n$\\mathrm{Ker }(j_{\\#}^{n+1})$&0\\\\ \\hline\n0&0\\\\ \\hline\n0&$\\mathrm{Coker }(j_{\\#}^{n-1})$\\\\ \\hline\n0&0\\\\ \\hline\n0&$H^0(D)$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}\n\\end{tabular}\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\ \\\\\n\\end{tabular}\n$$\n\nBy Lemmas $\\ref{ordinary0}$ and $\\ref{ordinary1}$ the second term $IE^{r,s}_{2,\\ov{p}}(X)\\cong Gr_{s}^WIH_{\\ov p}^{r+s}(X;\\mathbb{Q})$\nof the perverse weight spectral sequence for $X$ is\ngiven in each perversity by the following tables:\n\n$\n\\def1.4{1.6}\n\\begin{tabular}{| c | c | c | }\n\\multicolumn{1}{l}{\\tiny{$\\ov{0}\\leq \\ov{p}<\\ov{m}$}}\\\\\n\\hline\n0&$H^{2n}(X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n-1})$&$\\mathrm{Coker }(j^{n-1})$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\  \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=-1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=0$}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | c | c |}\n\\multicolumn{1}{l}{\\tiny{$\\ov{p}=\\ov{m}$}}\\\\\n\\hline\n0&$H^{2n}(X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\ \\hline\n0&$\\mathrm{Coker }(\\gamma^{n-1})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=-1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=0$}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}\n\\end{tabular}\n\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s\\geq n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\n\\end{tabular}\n$\n\n\n\n$\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\multicolumn{1}{l}{\\tiny{$\\ov{m}<\\ov{p}\\leq \\ov{t}$}}\\\\\n\\hline\n0&$H^{2n}(X)$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&$0$\\\\ \\hline\n$\\mathrm{Ker }(\\gamma^{n+1})$&$\\mathrm{Coker }(\\gamma^{n+1})$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&$0$\\\\ \\hline\n0&$\\mathrm{Coker }(\\gamma^{n-1})$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&$0$\\\\ \\hline\n0&$H^0(\\widetilde X)$&$0$ \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=-1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=0$}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | c | c |}\n\\multicolumn{1}{l}{\\tiny{$\\ov{p}=\\ov{\\infty}$}}\\\\\n\\hline\n$\\mathrm{Ker }(\\gamma^{2n})$&$0$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&$0$\\\\ \\hline\n$\\mathrm{Ker }(\\gamma^{n+1})$&$\\mathrm{Coker }(\\gamma^{n+1})$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&$0$\\\\ \\hline\n0&$\\mathrm{Coker }(\\gamma^{n-1})$&$0$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&$0$\\\\ \\hline\n0&$H^0(\\widetilde X)$ &$0$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=-1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=0$}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}&\n\\multicolumn{1}{c}{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tiny{$r=1$}\\,\\,\\,\\,\\,\\,\\,\\,\\,}\n\\end{tabular}\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\n\\end{tabular}\n$\n\n\n\\subsection{Intersection-formality}\nWe now prove the main result of this section.\n\\begin{teo}\\label{intersformal}\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities. \nDenote by $\\Sigma$ the singular locus of $X$.\nAssume that there is a resolution of singularities $f:\\widetilde X\\longrightarrow X$ of $X$ such that $D=f^{-1}(\\Sigma)$ is smooth, \nand that the link $L_\\sigma$ of $\\sigma$ in $X$,\nfor all $\\sigma\\in\\Sigma$ is $(n-2)$-connected. Then $X$ is GM-intersection-formal over $\\mathbb{C}$. \nFurthermore, if $\\Sigma=\\{\\sigma\\}$ is given by a single point then $X$ is intersection-formal over $\\mathbb{C}$.\n\\end{teo}\n\n\\begin{proof}\nBy Theorem $\\ref{IE1formality}$ there is a string of quasi-isomorphisms of \nperverse cdga's from $I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes\\mathbb{C}$ to\n$IE_{1,\\ov\\bullet}(X)\\otimes\\mathbb{C}$. Furthermore, we have $IE_{2,\\ov \\bullet}(X)\\cong IH_{\\ov\\bullet}(X;\\mathbb{Q})$.\nWe next define a perverse cdga $M_{\\ov\\bullet}$ together with quasi-isomorphisms\n$IE_{1,\\ov{\\bullet}}(X){\\longleftarrow} M_{\\ov{\\bullet}}{\\longrightarrow} IE_{2,\\ov{\\bullet}}(X).$\n\nFor projective curves, the Theorem is trivially satisfied. Hence we may assume that $n>1$.\nWe will define $M_{\\ov\\bullet}$ step by step, for the perversities $\\ov 0$, $\\ov m$, $\\ov n$, $\\ov t$ and $\\ov \\infty$.\nWe begin with the $\\ov 0$-perversity. Let\n$M_{\\ov{0}}$ be the bigraded complex with trivial differential given by\n$$\nM_{\\ov{0}}^{r,s}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c | }\n\\hline\n\\,\\,\\,\\,\\,\\,0\\,\\,\\,\\,\\,\\,&$H^{2n}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\  \\hline\n0&$\\mathrm{Ker }(j^{n-1})$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s\\geq n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n$$\nBy Lemma $\\ref{PD}$ we have \n$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\cong \\mathrm{Coker }(j^{n-1})$. The assignation $\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  dt\\mapsto \\mathrm{Coker }(j^{n-1})$\ndefines an isomorphism of bigraded complexes\n$\\psi_{\\ov 0}:M_{\\ov 0}\\to IE_{2,\\ov 0}(X)$.\nWe next define an inclusion of bigraded complexes $\\varphi_{\\ov 0}:M_{\\ov 0}\\to IE_{1,\\ov 0}(X)$.\nRecall that\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{0}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{s-2}(D)\\Lambda(t)\\otimes t$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\Lambda(t)\\otimes t\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline\n$0$&$H^0(\\widetilde X)\\oplus_{H^0(D)}H^0(D)\\otimes \\Lambda(t)$&$H^{0}(D)\\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\begin{tabular}{ l }\n\\tiny{$s\\geq 1$}\\\\\n\\tiny{$s=0$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nFor $s\\geq 0$ define $M^{0,s}_{\\ov{0}}\\longrightarrow IE_{1,\\ov 0}^{0,s}(X)$\nby $x\\mapsto (x,j^{s}(x))$.\nSince $\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\subset H^{n+1}(D)^\\vee\\cong H^{n-1}(D)$, we have an injective map\n$\\mathrm{Ker }(\\gamma^{n+1})^\\vee dt \\longrightarrow IE_{1,\\ov{0}}^{1,n-1}(X)=H^{n-1}(D)\\otimes\\Lambda(t)\\otimes dt$.\nThen the diagram\n$$\n\\xymatrix@R=5pt{\nM^{*,n-1}_{\\ov{0}}=&\\ar[ddd]\\mathrm{Ker }(j^{n-1})\\ar[r]^{0}&\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes dt\\ar[ddd]& \\\\\n\\\\\n\\\\\nIE_{1,\\ov{0}}^{*,n-1}(X)=&IE_{1,\\ov{0}}^{0,n-1}(X)\\ar[r]^-{d}&IE_{1,\\ov{0}}^{1,n-1}(X)&\n}\n$$\ncommutes. Hence we have quasi-isomorphisms of complexes\n$IE_{1,\\ov{0}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{0}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{0}}(X).$\n\nThe $\\ov m$-perversity weight spectral sequence for $X$ is given by:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{m}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)\\otimes t$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\otimes t\\right)\\oplus H^{s-2}(D)\\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\begin{tabular}{ l }\n\\tiny{$s\\geq n$}\\\\\n\\tiny{$s<n$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine $M_{\\ov{m}}$ as the bigraded sub-complex of $IE_{1,\\ov{m}}(X)$ given by\n$$\nM^{r,s}_{\\ov{m}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n\\,\\,\\,\\,\\,0\\,\\,\\,\\,\\,&$H^{2n}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\  \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes(t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\\\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s\\geq n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n$$\nThe only non-trivial differential of $M_{\\ov m}$ is given by differentiation with respect to $t$.\nNote that by Lemma $\\ref{PD}$ we have \n$\\mathrm{Coker }(\\gamma^{n-1})\\cong \\mathrm{Ker }(j^{n+1})^\\vee\\subset H^{n+1}(\\widetilde X)\\cong H^{n-1}(\\widetilde X).$\nThe assignations\n$\\mathrm{Ker }(j^{n+1})^\\vee\\mapsto \\mathrm{Coker }(\\gamma^{n-1})$\nand $\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\mapsto 0$ give a commutative\ndiagram\n$$\n\\xymatrix@R=5pt{\nM^{*,n-1}_{\\ov{m}}=&\\ar[ddd]_{}\\mathrm{Ker }(j^{n-1})^\\vee \\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes(t-1)\\ar[r]^-{d}&\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes dt\\ar[ddd]& \\\\\n\\\\\n\\\\\nIE_{2,\\ov{m}}^{*,n-1}(X)=&\\mathrm{Coker }(\\gamma^{n-1})\\ar[r]&0&\n}.\n$$\nHence we have quasi-isomorphisms of complexes\n$IE_{1,\\ov{m}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{m}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{m}}(X).$\n\n\nBy Lemma $\\ref{ordinary1}$ we have $\\mathrm{Ker }(j^{n-1})\\cap \\mathrm{Im }(\\gamma^{n-1})=\\{0\\}$. Hence\n$\\mathrm{Ker }(j^{n-1})\\subset \\mathrm{Ker }(j^{n+1})^\\vee$ and we have an\ninjective morphism of complexes $M_{\\ov 0}\\longrightarrow M_{\\ov m}$ making the following diagram commute\n$$\n\\xymatrix{\n\\ar[d]IE_{1,\\ov 0}(X) & \\ar[d]\\ar[l]_-{\\sim}M_{\\ov 0}\\ar[r]^-{\\sim} & \\ar[d]IE_{2,\\ov 0}(X)\\\\\nIE_{1,\\ov m}(X) & \\ar[l]_-{\\sim}M_{\\ov m}\\ar[r]^-{\\sim} & IE_{2,\\ov m}(X)\n}.\n$$\n\nWe next study the case $\\ov m+\\ov 1=\\ov n$.\nThe $\\ov n$-perversity weight spectral sequence for $X$ is given by:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{n}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)\\otimes t$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\otimes t\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline\n$\\mathrm{Ker }(j_{\\#}^{n+1})\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes t$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\otimes t\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\begin{tabular}{ l }\n\\tiny{$s> n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s\\leq n$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine\n$M_{\\ov{n}}$ as the bigraded sub-complex of $IE_{1,\\ov{n}}(X)$ given by\n$$\nM^{r,s}_{\\ov{n}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes t$&$H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\  \\hline\n$\\mathrm{Ker }(j_{\\#}^{n+1})$&$\\mathrm{Ker }(j^{n+1})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes (t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n$$\n\nThe non-trivial differentials of $M_{\\ov n}$ are given by the map\n$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  (t-1)\\longrightarrow  \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  dt$ in degree $s=n-1$ defined by\ndifferentiation with respect to $t$,\nthe map $\\gamma^{n+1}:\\mathrm{Ker }(j_{\\#}^{n+1})\\to \\mathrm{Ker }(j^{n+1})$ in degree $s=n+1$ and the map\n$H^{2n-2}(D)\\otimes t\\longrightarrow H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$ in degree $s=2n$\ndefined by $a\\cdot t\\mapsto (\\gamma^{2n}(a),a\\cdot dt)$.\nTo define a morphism of complexes $M_{\\ov{n}}\\longrightarrow IE_{2,\\ov{n}}(X)$ it suffices to\ndefine the maps in degrees $s=n+1$ and $s=2n$.\n\nBy Lemma $\\ref{ordinary1}$ there is a projection map\n$$\\mathrm{Ker }(j^{n+1}) \\twoheadrightarrow \\mathrm{Ker }(j^{n+1})\/(\\mathrm{Ker }(j^{n+1})\\cap \\mathrm{Im }(\\gamma^{n+1}))\\cong \\mathrm{Coker }(\\gamma^{n+1}).$$\nSince $\\mathrm{Ker }(\\gamma^{n+1})\\subset \\mathrm{Ker }(j_{\\#}^{n+1})$ we may find a direct sum decomposition\n$\\mathrm{Ker }(j_{\\#}^{n+1})=\\mathrm{Ker }(\\gamma^{n+1})\\oplus C$. Define a map $\\mathrm{Ker }(j_{\\#}^{n+1})\\longrightarrow \\mathrm{Ker }(\\gamma^{n+1})$ by projection to the first component.\nSince $\\gamma^{n+1}(\\mathrm{Ker }(j_{\\#}^{n+1})\\subset \\mathrm{Im }(\\gamma^{n+1})\\cap \\mathrm{Ker }(j^{n+1})$ we\nhave a commutative diagram\n$$\n\\xymatrix@R=5pt{\nM^{*,n+1}_{\\ov{n}}=&\\ar[ddd]\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[r]^-{\\gamma^{n+1}}&\\mathrm{Ker }(j^{n+1})\\ar@{->>}[ddd]& \\\\\n\\\\\n\\\\\nIE_{2,\\ov{n}}^{*,n+1}(X)=&\\mathrm{Ker }(\\gamma^{n+1})\\ar[r]^{0}&\\mathrm{Coker }(\\gamma^{n+1})&\n}.\n$$\nIn degree $2n$ we have a commutative diagram\n$$\n\\xymatrix@R=5pt{\nM^{*,2n}_{\\ov{n}}=&\\ar@{->>}[ddd]_{}H^{2n-2}(D)\\otimes t\\ar[r]^-{d}&H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt\\ar@{->>}[ddd]^\\pi& \\\\\n\\\\\n\\\\\nIE_{2,\\ov{n}}^{*,2n}(X)=&0\\ar[r]^{0}&H^{2n}(\\widetilde X)&\n}\n$$\nwhere \n$d(a\\cdot t)=(\\gamma^{2n}(a), a\\cdot dt)$ and \n$\\pi(x,a\\cdot dt)=\\gamma^{2n}(a)-x$.\nThis gives quasi-isomorphisms of complexes \n$IE_{1,\\ov{n}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{n}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{n}}(X)$\ncompatible with the inclusion $M_{\\ov m}\\to M_{\\ov n}$.\n\nThe $\\ov t$-perversity weight spectral sequence for $X$ is given by:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{t}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes \\Lambda(t)\\otimes t$&$H^{2n}(X)\\oplus H^{2n-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$0$\\\\ \\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\Lambda(t)\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s<2n$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine\n$M_{\\ov{t}}$ as the bigraded sub-complex of $IE_{1,\\ov{t}}(X)$ given by\n$$\nM^{r,s}_{\\ov{t}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes t$&$H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\  \\hline\n$H^{n-1}(D)$&$H^{n+1}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  (t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n$$\n\nNote that $M_{\\ov n}$ to $M_{\\ov t}$ we only replaced the complex $\\gamma^{n+1}:\\mathrm{Ker }(j_{\\#}^{n+1})\\to \\mathrm{Ker }(j^{n+1})$\nby the quasi-isomorphic complex $\\gamma^{n+1}:H^{n-1}(D)\\to H^{n+1}(\\widetilde X)$, in degree $s=n+1$.\nAnalogously to the previous case, choose a direct sum decomposition $H^{n-1}(D)\\cong \\mathrm{Ker }(\\gamma^{n+1})\\oplus C\\oplus C'$ and define a map\n$H^{n-1}(D)\\to \\mathrm{Ker }(\\gamma^{n+1})$ by projecting to the first component.\nWe have quasi-isomorphisms of complexes\n$IE_{1,\\ov{t}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{t}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{t}}(X)$\ncompatible with the inclusion $M_{\\ov n}\\to M_{\\ov t}$.\n\nLastly, the $\\ov \\infty$-perversity weight spectral sequence for $X$ is given by:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{\\infty}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\multicolumn{1}{c}{}\\\\\n\\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\otimes \\Lambda(t)\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine $M_{\\ov{\\infty}}$ as the bigraded sub-complex of $IE_{1,\\ov{\\infty}}(X)$ given by\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nM^{r,s}_{\\ov{\\infty}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\oplus H^{2n-2}(D)\\otimes t$&$H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\  \\hline\n$H^{n-1}(D)$&$H^{n+1}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes (t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes  dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s<n-1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n$}\\end{equation*}\nNote that from $M_{\\ov t}$ to $M_{\\ov \\infty}$ we only added $H^{2n-2}(D)$ in bidegree $(-1,2n)$.\nThe differential in degree $s=2n$ is given by $(a+b\\cdot t)\\mapsto (\\gamma^{2n}(a)+\\gamma^{2n}(b),b\\cdot dt)$.\nIt is straightforward to define quasi-isomorphisms of complexes\n$IE_{1,\\ov{\\infty}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov \\infty}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{\\infty}}(X)$\ncompatible with the inclusion $M_{\\ov t}\\longrightarrow M_{\\ov \\infty}$.\n\nLet $M_{\\ov{\\bullet}}$ be the perverse complex given by\n$$\nM_{\\ov{p}}=\\left\\{\\begin{array}{ll}\nM_{\\ov{0}}&\\text{ if }0\\leq p<n-1\\\\\nM_{\\ov{m}}&\\text{ if }p=n-1\\\\\nM_{\\ov{n}}&\\text{ if }p=n\\\\\nM_{\\ov{t}}&\\text{ if }n< p\\leq 2n-2\\\\\nM_{\\ov{\\infty}}&\\text{ if }p=\\infty\\\\\n\\end{array}\\right..\n$$\nWe have quasi-isomorphisms of perverse complexes\n$$IE_{1,\\ov{\\bullet}}(X)\\stackrel{\\varphi_{\\ov \\bullet}}{\\longleftarrow} M_{\\ov{\\bullet}}\\stackrel{\\psi_{\\ov \\bullet}}{\\longrightarrow} IE_{2,\\ov{\\bullet}}(X),$$\nwhere $\\varphi_{\\ov \\bullet}$ is injective and $\\psi_{\\ov \\bullet}$ is surjective.\n\n\nConsider on $M_{\\ov\\bullet}$ the multiplicative structure induced by the inclusion $\\varphi_{\\ov\\bullet}$.\nWe next show that $M_{\\ov p}\\times M_{\\ov q}\\subset M_{\\ov p+ \\ov q}$ for all $\\ov p,\\ov q\\in\\widehat \\mathcal{P}$,\nthe map $\\varphi_{\\ov\\bullet}$ is a quasi-isomorphism of perverse cdga's.\n\nThe multiplicative structure of $M_{\\ov0}$ is given by the maps $M_{\\ov 0}^{0,*}\\times M_{\\ov 0}^{0,*}\\longrightarrow M_{\\ov 0}^{0,*}$\ninduced by the cup product of $H^*(\\widetilde X)$\nand the map $M_{\\ov 0}^{0,*}\\times M_{\\ov 0}^{1,*}\\longrightarrow M_{\\ov 0}^{1,*}$ given by\n$(a,x)\\mapsto a\\cdot j^*(x)$. Since $M_{\\ov 0}^{0,s}=\\mathrm{Ker }(j^s)$ for all $s>0$, the only \nnon-trivial product in $M_{\\ov 0}^{1,*}$ is\n$H^{0}(\\widetilde X)\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\longrightarrow \\mathrm{Ker }(\\gamma^{n+1})^\\vee$.\nAlso, from the multiplicative structure of $IE_{2,\\ov 0}(X)$ we have\n$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee=0.$\nThis proves that $M_{\\ov 0}\\times M_{\\ov 0}\\subseteq M_{\\ov 0}$ and that the map $\\varphi_{\\ov 0}:M_{\\ov 0}\\longrightarrow IE_{1,\\ov 0}(X)$ is a morphism of cdga's.\nTo see that $M_{\\ov 0}\\times M_{\\ov m}\\subseteq M_{\\ov m}$ it suffices to prove that\n$\\mathrm{Ker }(j^{n+1})^\\vee\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee=0.$\nThis follows from the algebra structure of $IE_{2,\\ov \\bullet}(X)$ together with the\ncorresponding Poincar\\'{e} duality isomorphisms.\nWe now show that $M_{\\ov 0}\\times M_{\\ov n}\\subseteq M_{\\ov n}$.\nNote that for all $s>0$ we have\n$\\mathrm{Ker }(j^s)\\cdot \\mathrm{Ker }(j_{\\#}^{n+1})=0.$\nThe remaining inclusions are trivial.\nTherefore $M_{\\ov\\bullet}$ is a perverse cdga and the inclusion $\\varphi_{\\ov \\bullet}:M_{\\ov \\bullet}\\longrightarrow IE_{1,\\ov \\bullet}(X)$ \nis a quasi-isomorphism of perverse cdga's.\n\nLastly, we show that for\nevery pair of perversitites $\\ov p,\\ov q\\in\\mathcal{P}$  such that $\\ov{p}+\\ov{q}<\\ov{\\infty}$, the diagram\n$$\n\\xymatrix{\n\\ar[d]^{\\psi_{\\ov p}\\otimes \\psi_{\\ov q}}M_{\\ov p}\\otimes M_{\\ov p}\\ar[r]&M_{\\ov p+\\ov q}\\ar[d]^{\\psi_{\\ov p+\\ov q}}\\\\\nIE_{2,\\ov p}(X)\\otimes IE_{2,\\ov q}(X)\\ar[r]&IE_{2,\\ov p+\\ov q}(X)\n}\n$$\ncommutes.\nThe only non-trivial cases are when $\\ov p=\\ov 0$ and $\\ov q=\\ov n$ or $q=\\ov t$.\nWe show that the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times H^0(\\widetilde X)\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov 0}}\\ar[r]^-{\\mu}&\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]_{\\psi_{\\ov n}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times H^0(\\widetilde X)\\ar[r]^-{\\mu}&\\mathrm{Ker }(\\gamma^{n+1})\n}\n$$\ncommutes, where $\\mu(a,x)=a\\cdot j^*(x)$. Recall that the morphism $\\psi_{\\ov n}:\\mathrm{Ker }(j_{\\#}^{n+1})\\longrightarrow \\mathrm{Ker }(\\gamma^{n+1})$ is defined by\ntaking a direct sum decomposition $\\mathrm{Ker }(j_{\\#}^{n+1})=\\mathrm{Ker }(\\gamma^{n+1})\\oplus C$ and choosing the projection to the first component.\nLet $(a,x)\\in \\mathrm{Ker }(j_{\\#}^{n+1})\\times H^0(\\widetilde X)$, and decompose $a=\\overline{a}+c$ with $\\overline{a}\\in \\mathrm{Ker }(\\gamma^{n+1})$ and $c\\in C$.\nThen $\\mu(a,x)=(\\overline{a}+c)\\cdot j^*(x)$. Since $\\gamma(\\overline{a}\\cdot j^*(x))=\\gamma(\\overline{a})\\cdot x=0$,\nit suffices to show that $c\\cdot j^*(x)\\in C$. Since $x=1\\in H^0(\\widetilde X)$ and $\\gamma(c)\\neq 0$, it follows that $\\gamma(c\\cdot j^*(x))=\\gamma(c)\\cdot x\\neq 0$.\nHence  $c\\cdot j^*(x)\\in C$, and the above diagram commutes.\nWe next show that the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov 0}}\\ar[r]^-{\\mu}&H^{2n-2}(D)dt\\ar[d]_{\\psi_{\\ov n}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times \\mathrm{Coker }(j^{n-1})\\ar[r]^-{\\mu}&H^{2n}(\\widetilde X)\n}\n$$\ncommutes.\nLet $(a,b)\\in \\mathrm{Ker }(j_{\\#}^{n+1})\\times \\mathrm{Ker }(\\gamma^{n+1})^\\vee$. Then\n$\\psi_{\\ov t}(\\mu(a,b))=\\gamma^{2n}(a\\cdot b).$\nOn the other hand we have \n$\\mu(\\psi_{\\ov t}(a),\\psi_{\\ov 0} b)=\\gamma^{2n}(\\overline{a}\\cdot b),$\nwhere $a=\\overline{a}+c$ is a decomposition such that $\\ov{a}\\in \\mathrm{Ker }(\\gamma^{n+1})$ and $c\\in C$.\nHence to prove that the above diagram\ncommutes, it suffices to see that $c\\cdot b=0$. This follows from the fact that $C\\cap \\mathrm{Ker }(\\gamma^{n+1})=\\{0\\}$ and $b\\in \\mathrm{Ker }(\\gamma^{n+1})^\\vee$.\nThis proves that \n$\\psi_{\\ov{0}}\\cdot \\psi_{\\ov{n}}=\\psi_{\\ov n}$.\nThe same arguments allow us to prove that $\\psi_{\\ov{0}}\\cdot \\psi_{\\ov{t}}=\\psi_{\\ov t}$.\nTherefore \n$\\psi_{\\ov \\bullet}$ is multiplicative for finite perversities, and \n$X$ is GM-intersection-formal over $\\mathbb{C}$.\n\nAssume now that $X$ has only one isolated singularity. Then $\\mathrm{Ker }(\\gamma^{2n})=0$ and the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times \\mathrm{Ker }(j^{n+1})^\\vee\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov m}}\\ar[r]^-{\\mu}&H^{2n-2}(D)dt\\ar[d]_{\\psi_{\\ov \\infty}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times \\mathrm{Coker }(j^{n-1})\\ar[r]^-{\\mu}&\\mathrm{Ker }(\\gamma^{2n})\n}\n$$\ncommutes. This proves that\n$\\psi_{\\ov{p}}\\cdot \\psi_{\\ov{q}}=\\psi_{\\ov p + \\ov q}$ for all $p,q\\in\\widehat\\mathcal{P}$. Hence in this case, $X$ is intersection-formal over $\\mathbb{C}$.\n\\end{proof}\n\n\\subsection{Applications}\nA singular point $\\sigma\\in X$ is called \\textit{ordinary} if there exists a neighborhood of $\\sigma$\nisomorphic to an affine cone $C_\\sigma$ with vertex $\\sigma$, over a\nsmooth hypersurface $S_\\sigma$ of $\\mathbb{C}\\mathbb{P}^n$.\nIn such case, the link $L_\\sigma$ of $\\sigma$ in $X$\nis a smooth real manifold of dimension $(2n-1)$ which is $(n-2)$-connected\nHence we have:\n\n\\begin{cor}\n Let $X$ be a complex projective variety with only ordinary isolated singularities.\n Then $X$ is GM-intersection-formal over $\\mathbb{C}$.\n Furthermore, if $X$ has only one singular point, then $X$ is intersection-formal over $\\mathbb{C}$.\n\\end{cor}\n\n\\begin{example}[Segre cubic]\nLet $S$ denote the set of points $(x_0:x_1:x_2:x_3:x_4:x_5)$ of $\\mathbb{C}\\mathbb{P}^5$\nsatisfying\n$x_0+x_1+x_2+x_3+x_4+x_5=0$ and $x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=0$.\nThis is a normal projective threefold with 10 isolated ordinary singular points,\nknown as the \\textit{Segre cubic}.\nA resolution of $S$ is given by the moduli space $f:\\ov{\\mathcal{M}}_{0,6}\\longrightarrow S$ of stable rational curves with 6 marked points,\nand $D:=f^{-1}(\\Sigma)=\\bigsqcup_{i=1}^{10} \\mathbb{C}\\mathbb{P}^1\\times\\mathbb{C}\\mathbb{P}^1$, where $\\Sigma=\\{\\sigma_1,\\cdots,\\sigma_{10}\\}$\ndenotes the singular locus of $S$.\nFor each $0\\leq i\\leq 10$ the link of $\\sigma_i$ in $S$ is homeomorphic to a product\nof spheres $L_i\\simeq S^2\\times S^3$. In particular, $L_i$ is simply connected.\nHence $S$ is GM-intersection-formal over $\\mathbb{C}$.\nThe intersection homotopy type of $S$ is determined\nby the perverse graded algebra $IH_{\\ov\\bullet}^*(S;\\mathbb{Q})$, which we next describe.\nThe rational cohomology of $\\ov{\\mathcal{M}}_{0,6}$ is well-known, with non-trivial Betti numbers:\n$b_0(\\ov{\\mathcal{M}}_{0,6})=b_6(\\ov{\\mathcal{M}}_{0,6})=1$ and \n$b_2(\\ov{\\mathcal{M}}_{0,6})=b_4(\\ov{\\mathcal{M}}_{0,6})=16$.\nLet $j^s:H^s(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})\\to H^s(D;\\mathbb{Q})$ denote the restriction map induced by the inclusion\n$j:D\\hookrightarrow \\ov{\\mathcal{M}}_{0,6}$, and $\\gamma^s:H^{s-2}(D;\\mathbb{Q})\\to H^s(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})$\nthe corresponding Gysin map.\nThe rational cohomology of $S$ is:\n$$\nH^*(S;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^4)\\cong \\mathbb{Q}^6$\\\\ \\hline\n$\\mathrm{Coker }(j^2)\\cong \\mathbb{Q}^5$\\\\ \\hline\n$\\mathrm{Ker }(j^2)\\cong \\mathbb{Q}$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline\n\\end{tabular}\n$$\nNote that for $k\\neq 3$, the weight filtration on $H^k(S;\\mathbb{Q})$ is pure of weight $k$,\nwhile for $k=3$ we have a non-trivial weight filtration, with $Gr^W_3H^3(S;\\mathbb{Q})\\cong \\mathrm{Ker }(j^3)=0$ and\n$Gr^W_2H^3(S;\\mathbb{Q})\\cong H^3(S;\\mathbb{Q})\\cong \\mathbb{Q}^5.$\n\nDenote by $Van:=\\mathrm{Coker }(j^2)\\cong \\mathbb{Q}^5$ and let $Exc\\cong \\mathbb{Q}^5$ be defined via the direct sum decomposition\n$H^2(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})\\cong \\mathrm{Ker }(j^2)\\oplus \\mathrm{Coker }(\\gamma^2)\\oplus Exc$.\nThe rational intersection cohomology of $S$ is given by:\n$$\nIH^*_{\\ov p}(S;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{0}\\leq \\ov{p}\\leq \\ov{1}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee\\oplus Exc^\\vee$\\\\ \\hline\n$Van$\\\\ \\hline\n$H^2(S;\\mathbb{Q})$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c |}\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov 2$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee \\oplus Exc^\\vee$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})\\oplus Exc$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{3}\\leq \\ov{p}\\leq \\ov{4}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee$\\\\ \\hline\n$Van^\\vee$\\\\ \\hline\n$H^2(S;\\mathbb{Q})\\oplus Exc$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n$$\nNote that the weight filtration of $IH^*_{\\ov \\bullet}(S;\\mathbb{Q})$ is non-trivial, since\n$Gr^W_2IH^3_{\\ov 0}(S;\\mathbb{Q})\\cong Van\\neq 0$.\n\nSince $S$ is simply connected, and $IH_{\\ov 0}(S;\\mathbb{Q})\\cong H^*(S;\\mathbb{Q})$,\none may compute the rational homotopy groups $\\pi_*(S)\\otimes\\mathbb{Q}$ from a minimal model of\n$IH_{\\ov 0}^*(S;\\mathbb{Q})$, as done in Example 4.7 of \\cite{ChCi1}.\nLikewise, a perverse minimal model (in the sense of \\cite{CST}) of the perverse cdga $IH_{\\ov \\bullet}^*(S;\\mathbb{Q})$ would give\nthe ``rational intersection homotopy groups'' of $S$.\n\\end{example}\n\n\n\nFor a complete intersection $X$ with isolated singularities, the link of each singular point in $X$ is $(n-2)$-connected\n(this result is due to Milnor \\cite{Milnor} in the case of hypersurfaces\nand to Hamm \\cite{Hamm} for general complete intersections).\nAs another direct consequence of Theorem $\\ref{intersformal}$ we have:\n\n\\begin{cor}\nLet $X$ be a complete intersection with singular locus $\\Sigma$ of dimension 0.\nAssume that there exists a resolution of singularities $f:\\widetilde X\\to X$ such that\n$D=f^{-1}(\\Sigma)$ is smooth. Then $X$ is GM-intersection-formal over $\\mathbb{C}$.\n\\end{cor}\n\n\n\n\n\n\n\n\n\\section{Isolated Surface Singularities}\\label{Section_ISS}\nIn this last section we prove that isolated surface singularities are GM-intersection-formal over $\\mathbb{C}$.\n\n\\subsection{Notation}\nLet $X$ be a complex projective surface with only isolated singularities and denote by $\\Sigma$ the singular locus of $X$.\nLet $f:\\widetilde X\\longrightarrow X$ be a resolution of singularities of $X$ such that\n$D:=f^{-1}(\\Sigma)=D_1\\cup\\cdots\\cup D_N$ is a\nsimple normal crossings divisor.\nLet $\\widetilde D:=D^{(1)}=\\sqcup_i D_i$ and $Z:=D^{(2)}=\\sqcup_{i\\neq j} D_i\\cap D_j$. Then\n$\\widetilde D$ is a disjoint union of smooth projective curves and $Z$ is a finite collection of points.\nDenote by $j:\\widetilde D\\longrightarrow \\widetilde X$ the natural inclusion.\nLet $i_1:Z\\to \\widetilde D$ be the inclusion defined by $D_i\\cap D_j\\mapsto D_i$, for every $i<j$,\nand $i_2:Z\\to \\widetilde D$ be defined by $D_i\\cap D_j\\mapsto D_j$, for every $i<j$.\nThen\n$j\\circ i_1=j\\circ i_2$.\n\nDenote by\n$j^*:H^*(\\widetilde X)\\to H^*(\\widetilde D)$ and $i_k^*:H^0(\\widetilde D)\\to H^s(Z)$ the induced restriction morphisms\nand by \n$\\gamma:H^{*}(\\widetilde D)\\to H^{*+2}(\\widetilde X)$ \nand \n$\\eta_k:H^{0}(Z)\\to H^{2}(\\widetilde D)$\nthe corresponding Gysin maps.\nLet $i^*:=i_{1}^*-i_{2}^*$ and $\\eta:=\\eta_{1}-\\eta_{2}$ denote the combinatorial restriction and Gysin maps respectively.\nWe have $i^*\\circ j^*=0$ and $\\gamma\\circ \\eta=0$.\nLastly, define $j_{\\#}:=j^*\\circ \\gamma:H^0(\\widetilde D)\\longrightarrow H^2(\\widetilde D)$.\n\n\n\n\\subsection{Perverse weight spectral sequence}\\label{perversesurface}\nThe weight spectral sequence for $X_{reg}$ can be written as\n$$\nE_1^{r,s}(X_{reg})=\n\\arraycolsep=1pt\n\\def1.4{1.6}\n\\begin{array}{|cccccc|}\n\\hline\nH^0(Z)&&\\xra{\\,\\,\\,\\eta\\,\\,\\,}&H^2(\\widetilde D)&\\xra{\\,\\,\\,\\gamma^4\\,\\,\\,}&H^4(\\widetilde X)\\\\ \\hline\n&&&H^1(\\widetilde D)&\\xra{\\,\\,\\,\\gamma^3\\,\\,\\,}&H^3(\\widetilde X)\\\\ \\hline\n&&&H^0(\\widetilde D)&\\xra{\\,\\,\\,\\gamma^2\\,\\,\\,}&H^2(\\widetilde X)\\\\ \\hline\n&&&&&H^1(\\widetilde X)\\\\ \\hline\n&&&&&H^0(\\widetilde X)\\\\ \\hline\n\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&&&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}\n\\def1.4{1.6}\n\\begin{array}{c}\n\\text{ \\tiny{$s=4$}}\\\\\n\\text{ \\tiny{$s=3$}}\\\\\n\\text{ \\tiny{$s=2$}}\\\\\n\\text{ \\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\\\\\\\\\n\\end{array}\n$$\nIts algebra structure is\ninduced by the combinatorial restriction morphisms together with the cup products of $H^*(\\widetilde X)$, $H^*(\\widetilde D)$ and $H^0(Z)$.\n\nThe multiplicative weight spectral sequence for the link $L=L(\\Sigma,X)$ of $\\Sigma$ in $X$ can be written as:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\nE_1^{r,s}(L)\\cong\n\\arraycolsep=1pt\n\\def1.4{1.6}\n\\begin{array}{|cccccccc|}\n\\hline\nH^0(Z)\\Lambda(l)&&\\xra{\\,\\,\\,d^{-2,4}\\,\\,\\,}&H^2(\\widetilde D)\\oplus H^0(Z)\\Lambda(l)dl&&&&\\\\ \\hline\n&&&H^1(\\widetilde D)&&&&\\\\ \\hline\n&&&\\mathcal{P}(i_1^*,i_2^*)\\oplus H^0(Z)\\Lambda(l)&\\xra{\\,\\,\\,d^{-1,2}\\,\\,\\,}&H^2(\\widetilde D)\\oplus H^0(Z)\\Lambda(l)dl\\oplus H^0(Z)\\Lambda(l)dl&&\\\\ \\hline\n&&&&&H^1(\\widetilde D)&&\\\\ \\hline\n&&&&&\\mathcal{P}(i_1^*,i_2^*)&\\xra{\\,\\,\\,d^{0,0}\\,\\,\\,}&H^0(Z)\\Lambda(l)dl\\\\ \\hline\n\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&&&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}\n\\end{array}\n\\def1.4{1.6}\n\\begin{array}{c}\n\\text{ \\tiny{$s=4\\,\\,$}}\\\\\n\\text{ \\tiny{$s=3\\,\\,$}}\\\\\n\\text{ \\tiny{$s=2\\,\\,$}}\\\\\n\\text{ \\tiny{$s=1\\,\\,$}}\\\\\n\\text{ \\tiny{$s=0\\,\\,$}}\\\\\\\\\n\\end{array}\n$}\n\\end{equation*}\nwhere $\\mathcal{P}(i_1^*,i_2^*)$ is the vector space given by the pull-back\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n\\mathcal{P}(i_1^*,i_2^*)\\ar[r]&H^0(Z)\\Lambda(l)\\ar[d]^{(\\delta_0,\\delta_1)}\\\\\nH^{0}(\\widetilde D)\\ar[r]^-{(i_1^*,i_2^*)}&H^{0}(Z)\\times H^0(Z)}\n$$\nand the differential is given by\n$$\\left\\{\n\\def1.4{1.6}\n\\begin{array}{l}\nd^{0,0}(a,z(l))= z'(l)dl,\\\\\nd^{-1,2}((a,z(l)),w(l))=(j_{\\#}(a)+\\eta_1(w(0))-\\eta_2(w(1)),z'(l)dl,w'(l)dl),\\\\\nd^{-2,4}(z(l))=(\\eta_1(z(0))-\\eta_2(z(1)),z'(l)dl).\n\\end{array}\\right.$$\n\n\nWe describe the morphism $\\iota^*:E_1(X_{reg})\\longrightarrow E_1(L)$ induced by the inclusion $\\iota:L\\hookrightarrow X_{reg}$.\n\\begin{itemize}\n \\item [\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}]  In degree $s=0$, the map $H^0(\\widetilde X)\\longrightarrow \\mathcal{P}(i_1^*,i_2^*)$ is given by $x\\mapsto (j^*(x),i_1^*\\circ j^*(x))=(j^*(x),i_2^*\\circ j^*(x))$.\n \\item [\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}]  In degree $s=1$ we have the restriction morphism $j^*:H^1(\\widetilde X)\\longrightarrow H^1(\\widetilde D)$.\n\n \\item [\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}]  In degree $s=2$, the map $H^0(\\widetilde D)\\longrightarrow \\mathcal{P}(i_1^*,i_2^*)\\oplus H^0(Z)\\Lambda(l)$ is given by\n$a\\mapsto ((a,i_1^*(a)(1-t)+i_2^*(a)t), 0)$ and\n$H^2(\\widetilde X)\\longrightarrow H^2(\\widetilde D)\\oplus H^0(Z)\\Lambda(l)dl\\oplus H^0(Z)\\Lambda(l)dl$\nis given by $b\\mapsto (j^*(b),-i^*(b), 0)$.\n \\item [\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}]  In degree $s=3$ we have the identity of $H^1(\\widetilde D)$.\n \\item [\\begin{picture}(-1,1)(1,-3)\\circle*{2.5}\\end{picture}]  In degree $s=4$ the maps $H^0(Z)\\to H^0(Z)\\Lambda(l)$\nand $H^2(\\widetilde D)\\to H^2(\\widetilde D)\\oplus H^0(Z)\\Lambda(l)dl$\nare given by the natural inclusions $z\\mapsto z$ and $a\\mapsto (a,0)$ respectively. \n\\end{itemize}\n\n\n\n\n\\subsection{Intersection cohomology}\nBy semi-purity, the weight filtration on $H^k(X_{reg})$ is pure of weight $k$, for $k=0,1$.\nAlso, the weights on $H^k(L)$ are less than or equal to $k$ for $k=0,1$,\nand greater or equal to $k+1$ for $k\\geq 2$. The \nsecond terms $E^{r,s}_{2}(X_{reg})\\cong Gr_s^WH^{r+s}(X_{reg})$\nand  $E^{r,s}_{2}(L)\\cong Gr_s^WH^{r+s}(L)$\nof the weight spectral sequences for $X_{reg}$ and $L$ respectively are:\n\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\nE_2^{r,s}(X_{reg})\\cong\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\hline\n\\mathrm{Ker }(\\eta)&\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta)&0\\\\\\hline\n\\cellcolor{black!8}&\\mathrm{Ker }(\\gamma^3)&\\mathrm{Coker }(\\gamma^3)\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Coker }(\\gamma^2)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^0(\\widetilde X)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}\\,\\,\\,;\\,\\,\\,\nE_2^{r,s}(L)\\cong\n\\def1.4{1.6}\n\\begin{array}{| c | c | c | c |}\n\\hline\n\\mathrm{Ker }(\\eta)&\\mathrm{Coker }(\\eta)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&H^1(\\widetilde D)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&0&0&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde D)&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\\mathrm{Ker }(i^*)&\\mathrm{Coker }(i^*)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}\n\\end{array} \n\\def1.4{1.6}\n\\begin{array}{c}\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\\\\\n\\\\\n\\end{array} \n$}\\end{equation*}\n\n\nThe second term $IE^{r,s}_{2,\\ov p}(X)\\cong Gr_s^WIH^{r+s}_{\\ov p}(X)$ of the perverse weight spectral sequence for $X$ is:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 0$}}}&\\multicolumn{1}{c}{}\\\\\n\\hline\nH^4(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\nH^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\n\\mathrm{Ker }(j^1)&\\mathrm{Coker }(j^1)&\\cellcolor{black!8}\\\\\\hline\nH^0(\\widetilde X)&0&\\mathrm{Coker }(i^*)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\\,\\,\\,;\\,\\,\\,\n\\def1.4{1.6}\n\\begin{array}{| c   |}\n\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 1$}}}\\\\\n\\hline\nH^4(\\widetilde X)\\\\\\hline\nH^3(\\widetilde X)\\\\\\hline\n\\mathrm{Ker }(j^2)\\\\\\hline\nH^1(\\widetilde X)\\\\\\hline\nH^0(\\widetilde X)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}\n\\,\\,\\,;\\,\\,\\,\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 2$}}}&\\multicolumn{1}{c}{}\\\\\n\\hline\n\\mathrm{Ker }(\\eta)&0&H^4(\\widetilde X)\\\\\\hline\n\\cellcolor{black!8}&\\mathrm{Ker }(\\gamma^3)&\\mathrm{Coker }(\\gamma^3)\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Coker }(\\gamma^2)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^0(\\widetilde X)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}\n\\,\\,\\,;\\,\\,\\,\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov \\infty$}}}&\\multicolumn{1}{c}{}\\\\\n\\hline\n\\mathrm{Ker }(\\eta)&\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta)&0\\\\\\hline\n\\cellcolor{black!8}&\\mathrm{Ker }(\\gamma^3)&\\mathrm{Coker }(\\gamma^3)\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Coker }(\\gamma^2)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^0(\\widetilde X)\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}\n\\def1.4{1.6}\n\\begin{array}{c}\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\\\\\n\\end{array} \n$}\n\\end{equation*}\nNote that except for $IH_{\\ov 0}^2(X)$ and $IH_{\\ov 2}^2(X)$,\nwhich may have weights $(0,1,2)$ and $(2,3,4)$ respectively,\nthe weight filtration on $IH_{\\ov p}^s(X)$ is pure of weight $s$.\n\n\n\n\\subsection{Intersection formality} We prove the main result of this section.\n\\begin{teo}\nLet $X$ be a projective surface with only isolated singularities. Then $X$ is GM-intersection-formal over $\\mathbb{C}$.\nFurthermore, if $X$ has only one singular point then $X$ is\nintersection-formal over $\\mathbb{C}$.\n\\end{teo}\n\\begin{proof}\nBy Theorem $\\ref{IE1formality}$ there is a string of quasi-isomorphisms of \nperverse cdga's from $I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes\\mathbb{C}$ to\n$IE_{1,\\ov\\bullet}(X)\\otimes\\mathbb{C}$. Furthermore, we have $IE_{2,\\ov \\bullet}(X)\\cong IH_{\\ov\\bullet}(X;\\mathbb{Q})$.\nWe will next define a perverse cdga $M_{\\ov\\bullet}$ together with quasi-isomorphisms\n$IE_{1,\\ov{\\bullet}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{\\bullet}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{\\bullet}}(X).$\n\nLet $M_{\\ov 0}^{*,*}$ be the bigraded complex with trivial differential given by\n\\begin{equation*}\nM_{\\ov 0}^{r,s}=\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\hline\nH^4(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\nH^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\n\\mathrm{Ker }(j^1)&\\mathrm{Ker }(\\gamma^3)^\\vee \\otimes dt&\\cellcolor{black!8}\\\\\\hline\nH^0(\\widetilde X)&0&\\mathrm{Ker }(\\eta)^\\vee \\otimes  dt\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\n\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=4$}\\\\\n\\tiny{$s=3$}\\\\\n\\tiny{$s=2$}\\\\\n\\tiny{$s=1$}\\\\\n\\tiny{$s=0$}\\\\\\\\\n\\end{tabular}\n\\end{equation*}\nBy Poincar\\'{e} duality we have $\\mathrm{Ker }(\\eta)^\\vee\\cong \\mathrm{Coker }(i^*)$ and $\\mathrm{Ker }(\\gamma^3)^\\vee\\cong \\mathrm{Coker }(j^1)$. The assignation $dt\\mapsto 1$\ndefines an isomorphism of bigraded complexes\n$\\psi_{\\ov 0}:M_{\\ov 0}\\to IE_{2,\\ov 0}(X)$.\nWe next define an injective morphism\n$\\varphi_{\\ov 0}:M_{\\ov 0}\\longrightarrow IE_{1,\\ov 0}(X)$. \nTo ease notation we will let $E^{r,s}:=E_1^{r,s}(L)$ and $d^{r,s}:=d_1^{r,s}(L)$.\nThe $\\ov 0$-perversity weight spectral sequence $IE_{1,\\ov 0}(X)$ for $X$ is:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\n\\def1.4{2}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)\\widehat\\oplus E^{-2,4}\\Lambda(t)\\otimes t&\n\\left( H^2(\\widetilde D)\\widehat\\oplus E^{-1,4}\\Lambda(t)\\otimes t\\right) \\oplus E^{-2,4}\\Lambda(t)\\otimes dt&\nH^4(\\widetilde X)\\oplus E^{-1,4}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^1(\\widetilde D)\\widehat\\oplus E^{-1,3}\\Lambda(t)\\otimes t&\nH^3(\\widetilde X)\\oplus E^{-1,3}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^0(\\widetilde D)\\widehat\\oplus E^{-1,2}\\Lambda(t)\\otimes t&\n\\left(H^2(\\widetilde X)\\widehat\\oplus E^{0,2}\\Lambda(t)\\otimes t\\right) \\oplus E^{-1,2}\\Lambda(t)\\otimes dt&\nE^{0,2}\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^1(\\widetilde X)\\widehat\\oplus E^{0,1}\\Lambda(t)\\otimes t&\nE^{0,1}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^0(\\widetilde X)\\widehat\\oplus \\left(\\mathrm{Ker }(d^{0,0})\\oplus E^{0,0}\\Lambda(t)\\otimes t\\right)&\nE^{1,0}\\Lambda(t)\\otimes t\\otimes (t-1)\\oplus E^{0,0}\\Lambda(t)\\otimes dt&\nE^{1,0}\\Lambda(t)\\otimes dt\\\\\\hline\n\\multicolumn{1}{c}{r=-2}&\\multicolumn{1}{c}{r=-1}&\\multicolumn{1}{c}{r=0}&\\multicolumn{1}{c}{r=1}&\\multicolumn{1}{c}{r=2}\\end{array}\n$}\\end{equation*}\n\nFor $s\\geq 0$, define an injective morphism $M_{\\ov 0}^{0,s}\\longrightarrow IE_{1,\\ov 0}^{0,s}(X)$\nby letting $x\\mapsto (x,\\iota^*(x))$. This is well-defined, since $\\iota^*(H^0(\\widetilde X))\\subset \\mathrm{Ker }(d^{0,0})$\nand $\\iota^*(\\mathrm{Ker }(j^s))=0$. Since $\\mathrm{Ker }(\\eta)^\\vee \\subset H^0(Z)^\\vee\\cong H^0(Z)$\nand $\\mathrm{Ker }(\\gamma^3)^\\vee\\subset H^1(\\widetilde D)^\\vee\\cong H^1(\\widetilde D)$ we have injective morphisms \n$\\mathrm{Ker }(\\eta)^\\vee\\longrightarrow E^{1,0}\\cong H^0(Z)\\otimes\\Lambda(l)$ and $\\mathrm{Ker }(\\gamma^3)^\\vee\\longrightarrow E^{0,1}\\cong H^0(\\widetilde D)$.\nWe have quasi-isomorphisms\n$IE_{1,\\ov0}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov0}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov0}(X).$\n\nThe middle-perversity $\\ov m=\\ov 1$ weight spectral sequence $IE_{1,\\ov 1}(X)$ for $X$ is:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\n\\def1.4{2}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)\\widehat\\oplus E^{-2,4}\\Lambda(t)\\otimes t&\n\\left( H^2(\\widetilde D)\\widehat\\oplus E^{-1,4}\\Lambda(t)\\otimes t\\right) \\oplus E^{-2,4}\\Lambda(t)\\otimes dt&\nH^4(\\widetilde X)\\oplus E^{-1,4}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^1(\\widetilde D)\\widehat\\oplus E^{-1,3}\\Lambda(t)\\otimes t&\nH^3(\\widetilde X)\\oplus E^{-1,3}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^0(\\widetilde D)\\widehat\\oplus E^{-1,2}\\Lambda(t)\\otimes t&\n\\left(H^2(\\widetilde X)\\widehat\\oplus E^{0,2}\\Lambda(t)\\otimes t\\right) \\oplus E^{-1,2}\\Lambda(t)\\otimes dt&\nE^{0,2}\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^1(\\widetilde X)\\widehat\\oplus E^{0,1}\\Lambda(t)&\nE^{0,1}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^0(\\widetilde X)\\widehat\\oplus E^{0,0}\\Lambda(t)&\nE^{1,0}\\Lambda(t)\\otimes (t-1)\\oplus E^{0,0}\\Lambda(t)\\otimes dt&\nE^{1,0}\\Lambda(t)\\otimes dt\\\\\\hline\n\\multicolumn{1}{c}{r=-2}&\\multicolumn{1}{c}{r=-1}&\\multicolumn{1}{c}{r=0}&\\multicolumn{1}{c}{r=1}&\\multicolumn{1}{c}{r=2}\n\\end{array}\n$}\\end{equation*}\n\nLet $M_{\\ov 1}$ be the bigraded sub-complex of $IE_{1,\\ov 1}(X)$ given by:\n\\begin{equation*}\nM_{\\ov 1}=\n\\def1.4{1.6}\n\\begin{array}{| c | c | c |}\n\\hline\nH^4(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\nH^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\nH^1(\\widetilde X)\\oplus \\mathrm{Ker }(\\gamma^3)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\gamma^3)^\\vee\\otimes dt&\\cellcolor{black!8}\\\\\\hline\nH^0(\\widetilde X)&\\mathrm{Ker }(\\eta)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\eta)^\\vee\\otimes  dt\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\n\\end{equation*}\nThe non-trivial differential of $M_{\\ov 1}$ is given by $(t-1)\\mapsto dt$.\nThe assignations $\\mathrm{Ker }(\\gamma^3)^\\vee\\mapsto 0$ and $\\mathrm{Ker }(\\eta)^\\vee\\mapsto 0$ define a surjective morphism of complexes $\\psi_{\\ov 1}:M_{\\ov 1}\\to IE_{2,\\ov 1}(X)$.\nHence we have quasi-isomorphisms\n$IE_{1,\\ov 1}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov1}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov1}(X)$\ncompatible with the inclusion $M_{\\ov 0}\\longrightarrow M_{\\ov 1}$.\n\nThe top-perversity $\\ov t=\\ov 2$ weight spectral sequence $IE_{1,\\ov 2}(X)$ for $X$ is:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\n\\def1.4{2}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)\\widehat\\oplus E^{-2,4}\\Lambda(t)\\otimes t&\n\\left( H^2(\\widetilde D)\\widehat\\oplus E^{-1,4}\\Lambda(t)\\otimes t\\right) \\oplus E^{-2,4}\\Lambda(t)\\otimes dt&\nH^4(\\widetilde X)\\oplus E^{-1,4}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^1(\\widetilde D)\\widehat\\oplus E^{-1,3}\\Lambda(t)&\nH^3(\\widetilde X)\\oplus E^{-1,3}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^0(\\widetilde D)\\widehat\\oplus E^{-1,2}\\Lambda(t)&\n\\left(H^2(\\widetilde X)\\widehat\\oplus E^{0,2}\\Lambda(t)\\right) \\oplus E^{-1,2}\\Lambda(t)\\otimes dt&\nE^{0,2}\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^1(\\widetilde X)\\widehat\\oplus E^{0,1}\\Lambda(t)&\nE^{0,1}\\Lambda(t)\\otimes dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^0(\\widetilde X)\\widehat\\oplus E^{0,0}\\Lambda(t)&\nE^{1,0}\\Lambda(t)\\otimes (t-1)\\oplus E^{0,0}\\Lambda(t)\\otimes dt&\nE^{1,0}\\Lambda(t)\\otimes dt\\\\\\hline\n\\multicolumn{1}{c}{r=-2}&\\multicolumn{1}{c}{r=-1}&\\multicolumn{1}{c}{r=0}&\\multicolumn{1}{c}{r=1}&\\multicolumn{1}{c}{r=2}\n\\end{array}\n$}\\end{equation*}\n\nLet $M_{\\ov 2}$ be the bigraded sub-complex of $IE_{1,\\ov 2}(X)$ given by:\n\\begin{equation*}\nM_{\\ov 2}=\n\\def1.4{1.6}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\n\\mathrm{Ker }(\\eta)^\\vee&H^2(\\widetilde D)\\otimes t&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&H^1(\\widetilde D)&H^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\oplus \\mathrm{Ker }(\\gamma^3)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\gamma^3)^\\vee\\otimes  dt&\\cellcolor{black!8}\\\\\\hline\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&H^0(\\widetilde X)&\\mathrm{Ker }(\\eta)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\eta)^\\vee\\otimes dt\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\n\\end{equation*}\nThe non-trivial differentials of $M_{\\ov 2}$ are given by the map\n$H^2(\\widetilde D)\\otimes t\\longrightarrow H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt$ in degree $s=4$ defined by $a\\cdot t\\mapsto (\\gamma^4(a),a\\cdot dt)$,\nthe Gysin map\n$\\gamma^3:H^1(\\widetilde D)\\to H^3(\\widetilde X)$ in degree $s=3$ and the map $(t-1)\\mapsto dt$ defined by differentiation with respect to $t$.\nWe next define a morphism of complexes $\\psi_{\\ov 2}:M_{\\ov 2}\\to IE_{2,\\ov 2}(X)$.\nIn degrees $s=0$ and $s=1$ we let $\\psi_{\\ov 2}=\\psi_{\\ov 1}$.\nIn degree $s=2$ we have a Poincar\\'{e} duality isomorphism $\\mathrm{Ker }(j^2)\\cong \\mathrm{Coker }(\\gamma^2)$.\nFor $s=3$ choose a direct sum decomposition $H^1(\\widetilde D)\\cong \\mathrm{Ker }(\\gamma^3)\\oplus C_1$\nand consider the projection $H^1(\\widetilde D)\\to \\mathrm{Ker }(\\gamma^3)$ to the first component.\nThis gives a commutative diagram\n$$\n\\xymatrix{\nM_{\\ov 2}^{*,3}=&H^1(\\widetilde D)\\ar[r]^-{\\gamma^3}\\ar@{->>}[d]&\\ar@{->>}[d]H^3(\\widetilde X)&\\\\\nIE_{2,\\ov 2}^{*,3}(X)=&\\mathrm{Ker }(\\gamma^3)\\ar[r]^-0&\\mathrm{Coker }(\\gamma^3)\n}\n$$\nFor $s=4$ we have a commutative diagram\n$$\n\\xymatrix{\nM_{\\ov 2}^{*,4}=&\\mathrm{Ker }(\\eta)\\ar[r]^-{0}\\ar[d]^{Id}&\\ar[d]H^2(\\widetilde D)\\otimes t\\ar[r]^-{d}&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt\\ar@{->>}[d]^\\pi\\\\\nIE_{2,\\ov 2}^{*,4}(X)=&\\mathrm{Ker }(\\eta)\\ar[r]&0\\ar[r]&T\n}\n$$\nwhere $d(a\\cdot t)=(\\gamma^4(a),a\\cdot dt)$ and $\\pi(x,a\\cdot dt)=\\gamma^4(a)-x$.\nHence we have quasi-isomorphisms of complexes\n$IE_{1,\\ov 2}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov2}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov2}(X)$\ncompatible with the inclusion $M_{\\ov 1}\\longrightarrow M_{\\ov 2}$.\n\n\nThe $\\ov \\infty$-perversity weight spectral sequence $IE_{1,\\ov \\infty}(X)$ for $X$ is:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\n\\def1.4{2}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)\\Lambda(t)&\nH^2(\\widetilde D)\\Lambda(t)\\oplus H^0(Z)\\Lambda(t)dt&\nH^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^1(\\widetilde D)\\Lambda(t)&\nH^3(\\widetilde X)\\oplus H^1(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^0(\\widetilde D)\\Lambda(t)&\nH^2(\\widetilde X)\\oplus_{H^2(\\widetilde D)}\\left(H^2(\\widetilde D)\\Lambda(t)\\oplus H^0(\\widetilde D)\\Lambda(t)dt\\right)&\nH^2(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^1(\\widetilde X)\\oplus_{H^1(\\widetilde D)}H^1(\\widetilde D)\\Lambda(t)&\nH^1(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^0(\\widetilde X)\\oplus_{H^0(\\widetilde D)}H^0(\\widetilde D)\\Lambda(t)t&\nH^0(Z)\\Lambda(t)(t-1)\\oplus H^0(\\widetilde D)\\Lambda(t)dt&\nH^0(Z)\\Lambda(t)dt\\\\\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{-2}}}&\\multicolumn{1}{c}{\\text{\\tiny{-1}}}&\\multicolumn{1}{c}{\\text{\\tiny{0}}}&\\multicolumn{1}{c}{\\text{\\tiny{1}}}&\\multicolumn{1}{c}{\\text{\\tiny{2}}}\n\\end{array}$}\n\\end{equation*}\n\nLet $M_{\\ov \\infty}$ be the bigraded sub-complex of $IE_{1,\\ov \\infty}(X)$ given by:\n\\begin{equation*}\nM_{\\ov \\infty}=\n\\def1.4{1.6}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)&H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&H^1(\\widetilde D)&H^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\oplus \\mathrm{Ker }(\\gamma^3)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\gamma^3)^\\vee\\otimes  dt&\\cellcolor{black!8}\\\\\\hline\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&H^0(\\widetilde X)&\\mathrm{Ker }(\\eta)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\eta)^\\vee\\otimes dt\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\n\\end{equation*}\nNote that for $s<4$ we have $M_{\\ov\\infty}^{*,s}=M_{\\ov t}^{*,s}$.\nIn degree $s=4$, the differential of $M_{\\ov\\infty}$ is given by the map\n$H^0(Z)\\to H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t$ defined by\n$z\\mapsto (\\eta(z),0)$ and the map $H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t\\to H^4(\\widetilde X)\\to H^2(\\widetilde D)\\otimes dt$\ndefined by $(a,b\\cdot t)\\mapsto (\\gamma^4(a)+\\gamma^4(b),b\\cdot dt)$.\nTo define a surjective morphism of complexes $\\psi_{\\ov \\infty}:M_{\\ov \\infty}\\to IE_{2,\\ov \\infty}(X)$ it suffices to define\n$\\psi_{\\ov \\infty}:M^{*,4}_{\\ov \\infty}\\to IE^{*,4}_{2,\\ov \\infty}(X)$.\nChoose a decomposition $H^0(Z)\\cong \\mathrm{Ker }(\\eta)\\oplus C_0$ and\nconsider the projection $H^0(Z)\\to \\mathrm{Ker }(\\eta)$ to the first component. Also,\nchoose a decomposition\n$H^2(\\widetilde D)\\cong \\mathrm{Ker }(\\gamma^4)\\oplus C_2$ and consider the composition\n$\\rho:H^2(\\widetilde D)\\twoheadrightarrow \\mathrm{Ker }(\\gamma^4)\\twoheadrightarrow \\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)$.\nThis gives a commutative diagram\n$$\n\\xymatrix{\nM_{\\ov \\infty}^{*,4}=&H^0(Z)\\ar[r]^-{d}\n\\ar@{->>}[d]&\\ar[d]^{(\\rho,0)}H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t\\ar[r]^-{d}&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt\\ar@{->>}[d]\\\\\nIE_{2,\\ov \\infty}^{*,4}(X)=&\\mathrm{Ker }(\\eta)\\ar[r]^-0&\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)\\ar[r]&0\n}.\n$$\nHence we have\nquasi-isomorphisms of complexes\n$IE_{1,\\ov \\infty}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov\\infty}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov\\infty}(X)$\ncompatible with the inclusion $M_{\\ov 2}\\longrightarrow M_{\\ov \\infty}$.\n\nConsider on $M_{\\ov\\bullet}$ the multiplicative structure induced by the inclusion \ninclusion $\\varphi_{\\ov\\bullet}:M_{\\ov\\bullet}\\to IE_{1,\\ov\\bullet}(X)$. It is a matter of verification to\nsee that this structure is closed in $M_{\\ov\\bullet}$, so that $\\varphi_{\\ov\\bullet}$\n is a morphism of perverse cdga's, which is a quasi-isomorphism.\n\nWe next show that for every pair of perversities $\\ov p$ and $\\ov q$ such that $\\ov p+\\ov q<\\ov \\infty$, the diagram\n$$\n\\xymatrix{\nM_{\\ov p}\\times M_{\\ov q}\\ar[d]_{(\\psi_{\\ov p}, \\psi_{\\ov q})}\\ar[r]^-{\\mu}&M_{\\ov p+\\ov q}\\ar[d]^-{\\psi_{\\ov p+\\ov q}}\\\\\nIE_{2,\\ov p}(X)\\times IE_{2,\\ov q}(X)\\ar[r]^-{\\ov \\mu}&IE_{2,\\ov p+\\ov q}(X)\n}\n$$\ncommutes, so that $\\psi_{\\ov\\bullet}$ is multiplicative for finite perversities. The only non-trivial case is\n$$\n\\xymatrix{\nH^1(\\widetilde D)\\times \\mathrm{Ker }(\\gamma^3)^\\vee\\ar[d]_-{(\\psi_{\\ov 2},\\psi_{\\ov 0})}\\ar[r]^-{\\mu}&H^2(\\widetilde D)\\ar[d]^-{\\psi_{\\ov 2}}\\\\\n\\mathrm{Ker }(\\gamma^3)\\times \\mathrm{Ker }(\\gamma^3)^\\vee\\ar[r]^-{\\ov \\mu}&H^4(\\widetilde X)\n}.\n$$\nLet $(a,b)\\in H^1(\\widetilde D)\\times \\mathrm{Ker }(\\gamma^3)^\\vee$. Then $\\psi_{\\ov 2}\\mu(a,b)=\\gamma^4(a\\cdot b)$.\nLet $a=\\ov a+c$ be a decomposition of $a$ such that $a\\in \\mathrm{Ker }(\\gamma^3)$ and $c\\in C_1$.\nThen $\\ov \\mu(\\psi_{\\ov 2}(a),\\psi_{\\ov 0}(b))=\\gamma^4(\\ov a\\cdot b)$. Hence to prove that the above diagram commutes it suffices to show that $\\gamma^4(c\\cdot b)=0$.\nThis follows from the fact that $\\mathrm{Ker }(\\gamma^3)\\cap C_1=\\{0\\}$.\nThis proves that $X$ is GM-intersection-formal over $\\mathbb{C}$.\n\nAssume now that $X$ has only one isolated singularity. Then $\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)=0$ and $X$ is\nintersection-formal over $\\mathbb{C}$.\n\\end{proof}\n\n\n\\subsection{An example}\nWe end with an example of a projective surface with an \nisolated singularity and non-trivial weight filtration on its intersection cohomology.\n\n\\begin{example}[Cusp singularity]\nLet $C$ be a nodal cubic curve in $\\mathbb{C}\\mathbb{P}^2$. Choose a smooth plane quartic $C'$\nintersecting $C$ transversally, so that $|C\\cap C'|=12$.\nConsider the blow-up $\\widetilde X=Bl_{C\\cap C'}\\mathbb{C}\\mathbb{P}^2$ of $\\mathbb{C}\\mathbb{P}^2$ at the $12$ points of $C\\cap C'$.\nThen the proper transform $\\widetilde C$ of $C$ has negative self-intersection, and we may consider the blow-down $X$ of $\\widetilde C$ to a point. \nThen $X$ is a projective surface with a normal isolated singularity (see $\\S7$ of \\cite{ToChow},\nsee also Example 4.2 of \\cite{ChCi1} for a more general construction).\nTo make $\\widetilde C$ into a simple normal crossings divisor\nwe blow-up $2$ further times at the node of $\\widetilde C$.\nThis gives a resolution $f:Y\\to X$ where $Y\\simeq \\#_{15} \\mathbb{C}\\mathbb{P}^2$\nand the exceptional divisor $D$ is a cycle of three rational curves, so that\n$D^{(1)}=\\sqcup_{i=1}^3\\mathbb{C}\\mathbb{P}^1$ and $D^{(2)}=\\sqcup_{i=1}^3p_i$.\nLet $j^s:H^s(Y;\\mathbb{Q})\\longrightarrow H^s(D^{(1)};\\mathbb{Q})$ and $i^*:H^0(D^{(1)};\\mathbb{Q})\\to H^2(D^{(2)};\\mathbb{Q})$ denote the restriction morphisms.\nThe rational intersection cohomology of $X$ is given by:\n$$\nIH^*_{\\ov p}(X;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov{0}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)\\oplus Van$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline \\hline\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c |}\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov 1$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline \\hline\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov{2}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)^\\vee\\oplus Van^\\vee$\\\\ \\hline\n0\\\\  \\hline\n$\\mathbb{Q}$ \\\\ \\hline\n\\end{tabular}\n$$\nwhere $Van:=\\mathrm{Coker }(i^*)\\cong \\mathbb{Q}$ and $\\mathrm{Ker }(j^2)\\cong \\mathrm{Ker }(j^2)^\\vee\\cong \\mathbb{Q}^{12}$.\nThe weight filtration on $IH^*_{\\ov \\bullet}(X;\\mathbb{Q})$ is non-trivial, with\n$Gr^W_2IH^2_{\\ov 0}(X;\\mathbb{Q})\\cong \\mathbb{Q}^{12}$, $Gr^W_1IH^2_{\\ov 0}(X;\\mathbb{Q})=0$ and \n$Gr^W_0IH^2_{\\ov 0}(X;\\mathbb{Q})\\cong \\mathbb{Q}$.\n\\end{example}\n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{results -- to dos}\n\n\\begin{comment}\n\\section*{Carlos notes (to address)}\n\n\\begin{itemize}\n\t\\item talk about reservoir papers in the introduction\n\t\\item  in the trajectory difference part want to include 50\/50 autoencoders, have to also show Power diss and Re Im plot in some section, also explain why I'm using these, what to they show us, should consider changing terms from hibernating to quiescent. \n\t\\item  in the hibernating and bursting time section I think we might\/should also be able to take the initial peak out, by setting a threshold of the data we want to include in this PDF\n\t\\item maybe talk about neural ODEs in the conclusion\n\\end{itemize}\n\\end{comment}\n\\section{Introduction} \\label{sec:Intro}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\nDevelopment of reduced order dynamical models for complex flows is an issue of long-standing interest, with applications in improved understanding, as well as control, of flow phenomena. The classical approach for dimension reduction of these systems consists of extracting dominant modes from data via principal component analysis (PCA), also known as proper orthogonal decomposition (POD) and Karhunen-Lo\\'{e}ve decomposition \\cite{holmes2012turbulence}. PCA determines a set of basis vectors ordered by their contribution to the total variance (fluctuating kinetic energy) of the flow. Given $N_s$ data vectors (``snapshots\") $x_i\\in \\reals^N$, one can obtain these basis vectors by performing singular value decomposition (SVD) on the data matrix $X=[ x_1,x_2,\\cdots ] \\in \\mathbb{R}^{N \\times N_s}$ such that $X=U \\Sigma V^T$. Projecting the data onto the first $d_h$ basis vectors (columns of $U$) then gives a low-dimensional representation -- a projection onto a linear subspace of the full state space. To find a reduced order model (ROM), a Galerkin approximation of the Navier-Stokes Equations (NSE) using this basis can be implemented; these have shown some success in capturing the dynamics of coherent structures \\cite{noack1994low, aubry1988dynamics}. Previous research has also used POD as well as a filtered version thereof  \\cite{sieber2016spectral}, which are linear reduction techniques, to reduce dimensions and learn a time evolution map from data with the use of neural networks (NNs) \\cite{lui2019construction}.\n\nAlthough PCA provides the best linear representation of a data set in $d_h$ dimensions, in general the long-time dynamics of a general nonlinear dynamical systems are not expected to lie on a linear subspace of the state space. For a primer and more details on data-driven dimension reduction methods for dynamical systems refer to Linot \\& Graham \\cite{linot2022data}. For dissipative systems, such as the NSE, it is expected that the long-time dynamics will lie on an invariant manifold $\\mathcal{M}$, which can be represented \\emph{locally} with Cartesian coordinates, but may have a complex global topology. In fluid mechanics, this manifold is often called an \\emph{inertial manifold}\n\\cite{foias1988modelling, temam1989inertial, zelik2022attractors}. Hence to find a high-fidelity low-dimensional model one desires to find $\\mathcal{M}$ and the dynamical system on it. The present work will consider discrete-time models, though differential equation models could be found as well \\cite{linot2022data}. \n\nIn general one can think of breaking up $\\mathcal{M}$ into overlapping regions that cover the domain, to find the local representation. These are called charts and are equipped with a coordinate domain and a coordinate map \\cite{lee2013smooth}. The strong Whitney's embedding theorem states that any smooth manifold of dimension $d_\\mathcal{M}$ can be embedded into a Euclidean space of so-called \\emph{embedding} dimension $2d_\\mathcal{M}$ \\cite{lee2013smooth, whitney1944self}. This means that in the worst case we can expect in principle to be able to find a $2d_\\mathcal{M}$-dimensional Euclidean space in which the dynamics lie. To find a $d_\\mathcal{M}$-dimensional Euclidean space one would in general need to develop overlapping local representations and evolution equations -- this avenue is not pursued in the present work but has been done elsewhere  \\cite{floryan2021charts}.  In this work we aim to find a high-fidelity low-dimensional dynamical model using data from simulations of two-dimensional Kolmogorov flow. In this work, the governing Navier-Stokes Equations will only be used to generate the data -- the models will only use this data, not the equations that generated it.  Neural networks (NNs) will be used to map between the full state space and the manifold, as well as for the dynamical system model on the manifold. \n\n\n\n\nA number of previous studies have focused on finding {data-driven} models for fluid flow problems with the use of NNs. Srinivasan \\textit{et al.} \\cite{srinivasan2019predictions} developed NN models to attempt to predict the time evolution of the Moehlis-Faisst-Eckhardt (MFE) model \\cite{moehlis2004low}, which is a nine-dimensional  model for turbulent shear flows. They used two approaches to finding discrete-time dynamical systems. The first is to simply use a neural netowrk as a discrete-time map, yielding a Markovian representation of the time evolution. The second is to use a long short-term memory (LSTM) network, which yields a non-Markovian evolution equation. Despite the fact that the dynamics are in fact Markovian, the LSTM approach worked better, yielding reasonable agreement with the Reynolds stress profiles. Page \\textit{et al.} used deep convolutional autoencoders (CAEs) to learn low-dimensional representations for two-dimensional (in physical space) Kolmogorov flow, showing that these networks retain a wide spectrum of lengthscales and capture meaningful patterns related to the embedded invariant solutions \\cite{page2021revealing}. They considered the case where bursting dynamics is obtained at a Reynolds number of $\\text{Re}=40$ and $n=4$ wavelengths in the periodic domain. Nakamura \\textit{et al.} used CAEs for dimension reduction combined with LSTMs and applied it to minimal turbulent channel flow for $\\text{Re}_\\tau=110$ where they showed to capture velocity and Reynolds stress statistics \\cite{nakamura2021convolutional}. They studied various degrees of dimension reduction, showing good performance in terms of capturing the statistics; however for drastic dimension reduction they showed how only large vortical structures were captured. Hence, the selection of the minimal dimension to accurately represent the state becomes a challenging task. Reservoir networks  have also shown great potential in learning nonlinear models for time evolution. For example, Doan \\textit{et al.} trained what they call an Auto-Encoded Reservoir-Computing (AE-RC) framework where the latent space is fed into an Echo State Network (ESN) to model evolution in discrete time \\cite{doan2021auto}. By considering the two-dimensional Kolmogorov flow for $\\text{Re}=30$ and $n=4$ good performance was obtained when comparing the kinetic energy and dissipation evolution in time. They also showed how the model captures the velocity statistics. However, the nature of the reservoir in the ESN stores past history, making the model non-Markovian. \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\\begin{comment}\nThe Navier-Stokes Equations (NSE) are known to be infinite dimensional partial differential equations and when solved numerically many dimensions are usually needed to capture the correct dynamics. This mainly arises due to the need of capturing the smallest scales that are present in the system, which are responsible for dissipating turbulent kinetic energy \\cite{moin1998direct}. Attempts to reduce the domain in turbulent channel flow have been accomplished by finding the smallest domain in which turbulence is sustained in a channel flow \\cite{jimenez1991minimal}. Other attempts to find low-dimensional representations of the state consist in extracting dominant patterns by the use of principal component analysis (PCA), also known as POD and Karhunen\u2013Lo\\'{e}ve. This state can subsequently be evolved in this basis using, for example, Galerkin methods \\cite{noack1994low} which have shown success in capturing the dynamics of coherent structures \\cite{aubry1988dynamics}. However dynamics on a linear transformation require many modes, or dimensions, to resolve all of the relevant scales. Recent studies have focused on finding nonlinear projections with the use of Neural Networks (NNs)  \\cite{murata2020nonlinear}. By learning a nonlinear reduced order model (ROM) a more natural representation of the state can be found due to the nonlinearities that appear in the governing equations, which combined with spatio-temporal evolution has shown success in prediction of turbulent flows \\cite{san2018extreme, srinivasan2019predictions}. \n\\end{comment}\n\n\n\n\n Although previous research has found data-driven ROMs for fluid flow problems, the focus on these has not been to find the minimal dimension required to capture the data manifold and dynamics. Linot \\& Graham have addressed this issue for the Kuramoto-Sivashinsky equation (KSE) \\cite{linot2020deep, linot2022data}. They showed that the mean squared error (MSE) of the reconstruction of the snapshots using an AE for the domain size of $L=22$ exhibited an orders-of-magnitude drop when the dimension of the inertial manifold is reached. Furthermore, modeling the dynamics with a dense NN at this dimension either with a discrete time map \\cite{linot2020deep} or a system of ordinary differential equations (ODE) \\cite{linot2022data} yields excellent trajectory predictions and long-time statistics. Increasing domain size to $L=44$ and $L=66$, which makes the system more chaotic, affects the drops of MSE significantly. However a drop is still seen, and when obtaining the dynamics and calculating long time statistics, good agreement with the true data is obtained. This work, denoted ``Data-driven manifold dynamics\" (DManD) has been extended to incorporate reinforcement learning control for reduction of dissipation in the KSE, yielding a very effective control policy \\cite{Zeng.2022}.\n \n\n\nWe aim to extend this approach to the NSE, specifically to the two-dimensional Kolmogorov flow, where an external forcing drives the dynamics. As $\\text{Re}$ increases, the trivial state becomes unstable, giving rise to periodic orbits (POs), relative periodic orbits (RPOs) and eventually chaos. Relative periodic orbits correspond to periodic orbits in in a moving reference frame, such that in a fixed frame, the pattern at time $t+T$ is a phase-shifted replica of the pattern at time $t$. The nature of the weakly turbulent dynamics at a Reynolds number of $\\operatorname{Re} = 14.4$, and connections with  RPO solutions are the focus of this study. Due to the symmetries of the system the chaotic dynamics travels between unstable RPOs \\cite{crowley2022turbulence} through bursting events \\cite{armbruster1996symmetries} that shadow heteroclinic orbits connecting the RPOs. A past study \\cite{armbruster1992phase} shows that low-dimensional representations can be found with PCA for  two-dimensional Kolmogorov flow where in the case of weakly turbulent data, the first two PCA basis in the streamfunction formulation  capture most of the energetic content when filtering out the bursting events before the analysis, and including a third basis function captures the bursting information. This point hints at the low-dimensional nature of this system, where a low number of PCA basis functions can energetically represent the data. However, even though the energy can be contained in a low number of basis functions, this does not imply that these will properly capture the dynamics  \\cite{rowley2017model}. In \\cite{armbruster1992phase}, development of a model of time-evolution was not considered.  \n\n\n\nReturning to the aims of the present work, our focus is twofold. We aim to learn a minimal-dimensional high fidelity data-driven model for the long-time dynamics of two-dimensional Kolmogorov flow with the use of an autoencoder (AE), and a discrete-time map, in the form a dense NN, of the dynamics on the manifold. In this map, the future time prediction only depends on the present state (on the manifold), in keeping with the Markovian nature of the dynamics on the manifold. We will evaluate model predictions as a function of dimension, considering short-time trajectories, long-time statistics, quiescent and bursting time distributions, and predictions of bursting events.  This paper is structured as follows: in Section \\ref{sec:Framework} we present the governing equations together with the symmetries of the system. We also present the dynamics at the two values of $\\operatorname{Re}$ considered and the connections of the RPOs with the chaotic regime. In Section \\ref{sec:AEs} we show the methodology for data-driven dimension reduction and dynamic modeling, which includes the AE architecture and the time map NN. Section \\ref{sec:Results} shows the results, and concluding remarks are given in Section \\ref{sec:Conclusion}.\n\n\\newpage\n\n\\section{Kolmogorov flow formulation and dynamics} \\label{sec:Framework}\n\nThe two-dimensional Navier-Stokes equations (NSE) with Kolmogorov forcing are\n\\begin{gather}\n\\frac{\\partial \\boldsymbol{u}}{\\partial t}+\\boldsymbol{u} \\cdot \\nabla \\boldsymbol{u}+\\nabla p=\\frac{1}{\\operatorname{Re}} \\nabla^{2} \\boldsymbol{u}+\\sin (n y) \\hat{\\boldsymbol{x}}  \\\\\n\\nabla \\cdot \\boldsymbol{u}=0\n\\end{gather}\nwhere $\\boldsymbol{u}=[u,v]$ is the velocity vector, $p$ is the pressure, $n$ is the wavenumber of the forcing, and $\\hat{\\boldsymbol{x}}$ is the unit vector in the $x$ direction. Here $\\operatorname{Re}=\\frac{\\sqrt{\\chi}}{v}\\left(\\frac{L_{y}}{2 \\pi}\\right)^{3 \/ 2}$ where $\\chi$ is the dimensional forcing amplitude, $\\nu$ is the kinematic viscosity, and $L_y$ is the size of the domain in the $y$ direction. We consider the periodic domain $[0,2 \\pi \/ \\alpha] \\times[0,2 \\pi]$ with $\\alpha=1$. Vorticity is defined as $\\omega = \\nabla \\times \\boldsymbol{u}$. The equations are invariant under several symmetry operations \\cite{chandler2013invariant}, namely a shift (in $y$)-reflect (in $x$), a rotation through $\\pi$, and a continuous translation in $x$:\n\\begin{gather}\n\\mathscr{S}:[u, v, \\omega](x, y) \\rightarrow[-u, v,-\\omega]\\left(-x, y+\\frac{\\pi}{n}\\right), \\\\\n\\mathscr{R}:[u, v, \\omega](x, y) \\rightarrow[-u,-v, \\omega](-x,-y), \\\\\n\\mathscr{T}_{l}:[u, v, \\omega](x, y) \\rightarrow[u, v, \\omega](x+l, y) \\quad \\text { for } 0 \\leqslant l<\\frac{2 \\pi}{\\alpha}.\n\\end{gather}\n\\begin{figure}\n\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KE_po_v2.pdf}\n\t\n\t\t\\caption{}\n\t\t\\label{fig:sub1}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KE_bur_hib_norm_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub2}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/I_D_Re14d4_v6.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subID}\n\t\\end{subfigure}\n\t\\caption{(a) Time evolution of $KE$ at $\\operatorname{Re}=13.5$. (b) Time evolution of $KE$ at $\\operatorname{Re}=14.4$. (c) Time evolution of $D$ and $I$ at $\\operatorname{Re}=14.4$.}\n\t\\label{fig:test}\n\\end{figure}\nThe total kinetic energy for this system ($KE$), dissipation rate ($D$) and power input ($I$) are \n\\begin{equation}\nKE =\\frac{1}{2}\\left\\langle\\boldsymbol{u}^{2}\\right\\rangle_{V}, D=\\frac{1}{\\operatorname{Re}}\\left\\langle|\\nabla \\boldsymbol{u}|^{2}\\right\\rangle_{V}, \\quad I=\\langle u \\sin (n y)\\rangle_{V}\n\\end{equation}\nwhere subscript $V$ corresponds to the average taken over the domain. For the case of $n=1$ the trivial solution is linearly stable at all $\\text{Re}$ \\cite{iudovich1965example}. It is not until $n=2$ that the laminar state becomes unstable, with a critical value of $\\operatorname{Re}_c=n^{3\/2}2^{1\/4}$\\cite{meshalkin1961investigation, green1974two, thess1992instabilities}.\n\n\nThe NSE are evolved numerically in time in the vorticity representation on a $[d_x \\times d_y]=[32 \\times 32]$ grid following the pseudo-spectral scheme given by Chandler \\& Kerswell \\cite{chandler2013invariant}, which is based on the code by Bartello \\& Warn \\cite{bartello1996self}. We show here time series results for the two dynamical regimes considered in this work, an RPO regime at $\\mathrm{Re}=13.5$ and a chaotic regime at $\\mathrm{Re}=14.4$. Figure \\ref{fig:sub1} shows the $KE$ evolution for an RPO obtained at $\\operatorname{Re} = 13.5$.  Due to the discrete symmetries of the system, there are several RPOs \\cite{armbruster1996symmetries}, as we further discuss below. Figure \\ref{fig:sub2} shows the $KE$ evolution for a trajectory at  $\\operatorname{Re} = 14.4$. The dynamics are characterized by quiescent intervals where the trajectories are close to RPOs (which are now unstable), punctuated by heteroclinic-like excursions between the RPOs, which are indicated by the intermittent increases of the $KE$. The RPOs are all related by the symmetries $\\mathscr{S}$ and $\\mathscr{R}$ \\cite{armbruster1996symmetries, platt1991investigation, nicolaenko1990symmetry}. This behavior can also be seen in Figure \\ref{fig:subID}, where the black curve corresponds to the time evolution of $D$ and the blue curve to the time evolution of $I$. Figure \\ref{re14d4re13d5}, shows a state-space projection of a trajectory onto the plane $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ where $a(k_x,k_y,t) = a_{k_x,k_y}(t)=\\mathcal{F}\\{\\omega(x,y,t)\\}$ is the discrete Fourier transform in $x$ and $y$. The grey curve corresponds to $\\operatorname{Re} = 14.4$ and the different blue curves show four different RPOs related by the shift-reflect symmetry $\\mathscr{S}$ at $\\operatorname{Re} = 13.5$. \n\n\n\n\n \\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/Re13d5_Re14d4_v3.pdf}\n\t\\caption{Evolution of the real and imaginary components corresponding to the  $a_{0,1}(t)$ Fourier mode for $\\operatorname{Re}=13.5$ and $\\operatorname{Re}=14.4$.}\n\t\\label{re14d4re13d5}\n\\end{figure}\n \n\n\n\\section{Data-driven dimension reduction and dynamic modeling} \\label{sec:AEs}\n\n\\subsection{Dimension reduction with autoencoders}\\label{sec:aesub}\n\n\n\nTo learn a minimal dimension high-fidelity model for the two-dimensional Kolmogorov flow we first have to find a low-dimensional nonlinear mapping from the full state to the reduced representation.  For this purpose we consider a common machine learning architecture known as an undercomplete autoencoder (AE), whose purpose is to learn a reduced representation of the state such that the reconstruction error with respect to the true data is minimized. The AE consists of an encoder, $\\mathcal{E}(\\cdot)$, that maps from the full space $\\mathbb{R}^{N}$ to the lower dimensional latent space $h(t) \\in \\mathbb{R}^{d_h}$ (i.e., coordinates on the manifold $\\mathcal{M}$), and a decoder, $\\mathcal{D}(\\cdot)$, that maps back to the full space. Flattened versions of $\\omega (x,y,t)$ are used, which we refer from this point on as $\\omega(t)$, so $N = 32 \\times 32 = 1024$. We shall see that the latent space dimension $d_h$ will be much smaller than the dimension $N$ of the full spatially-resolved state. The encoder $\\mathcal{E}(\\omega(t))$ is a coordinate mapping from $\\mathbb{R}^{N}$ to $\\mathcal{M}$, and the decoder $\\mathcal{D}(h(t))$ is the mapping back from $\\mathcal{M}$ to $\\mathbb{R}^{N}$.\n\n\n\n\nWe train the AEs with $\\omega(t)$ obtained from the evolution of NSE for the original data as well as accounting for the discrete and continuous symmetries. By accounting for the symmetries it is expected that the networks will perform better, by not having to learn the symmetries in the latent space mapping. We account for the continuous symmetry in $x$, $\\mathscr{T}_{l}$, with the method of slices \\cite{budanur2015periodic, budanur2015reduction}. The $k_x=1, k_y=0$ Fourier mode is used to find the spatial phase: $\\phi_x(t)=\\operatorname{atan} 2\\left\\{\\operatorname{Im}\\left[a_{1,0}(t)\\right], \\operatorname{Re}\\left[a_{1,0}(t)\\right]\\right\\}$. This can then be used to phase-align the vorticity snapshots such that this mode is a pure cosine: $\\hat{\\omega}(x,y,t)=\\mathcal{F}^{-1}\\left\\{\\mathcal{F}\\{\\omega(x,y,t)\\} e^{-i k \\phi_x(t)}\\right\\}$. Doing this ensures that the snapshots lie in a reference frame were no translation happens in the $x$ direction. We will learn evolution equations for both $\\hat{\\omega}(t)$ and $\\phi_x(t)$, which we will denote as the pattern dynamics and phase dynamics, respectively. We also consider the shift-reflect (SR) symmetry, $\\mathscr{S}$, as well as the rotation through $\\pi$, $\\mathscr{R}$. To account for the SR symmetry the goal is to collapse the phase-aligned snapshots to the same common state. We can define two indicator functions such that the SR subspace is specified. The first one, $I_{Even}=\\operatorname{sgn}(\\phi_y)$, where $\\phi_y(t)=\\operatorname{atan} 2\\left\\{\\operatorname{Im}\\left[a_{0,1}(t)\\right], \\operatorname{Re}\\left[a_{0,1}(t)\\right]\\right\\}$ is the spatial phase in $y$. The second indicator function is $I_{odd}=\\operatorname{sgn}(\\operatorname{Re}[a_{2,0} (t)])$, the sign of the real part of the second Fourier mode in $x$. We can then map the vorticity snapshots in such a way that $I_{Even},I_{Odd}>0$ by applying SR operations to the state. The rotation symmetry is accounted for, on top of the SR symmetry, by minimizing the $L_2$ norm of the data with respect to a template snapshot. This is done by applying the discrete operation that rotates and shift-reflects the vorticity snapshots and selecting the snapshot that minimizes the norm. We note that we take a different approach for reducing the symmetries compared to previous research on symmetry-aware AEs \\cite{kneer2021symmetry}.\n\n\n\n\n\nPrevious work \\cite{linot2020deep} has shown that training a NN to learn the difference between the data and the projection onto the leading PCA basis vectors improved reconstruction performance compared to learning a latent space directly from the full data. To present the framework, we will use the phase-aligned and flattened vorticity $\\hat{\\omega} (t)$, since that is what we use for the time-evolution. Below, however, we will present some results where other versions of the data are used -- e.g.~the data with phase-shifting. The autoencoder aspect of the analysis is identical.  \n\nWe begin he process by computing the projection of the data onto the first $d_h$ basis vectors, $P_{d_{h}} U^{T}\\hat{\\omega}(t)$. We then seek to learn a $d_h$-dimensional correction to that projection, $E\\left(U^{T} \\hat{\\omega}(t)\\right)$ -- the sum of these is the latent-space representation $h(t)$. In other words, the encoding step learns the deviation from PCA\n\\begin{equation}\nE\\left(U^{T} \\hat{\\omega}(t)\\right)=h(t)-P_{d_{h}} U^{T} \\hat{\\omega}(t).\n\\label{encoder}\n\\end{equation}\nWe emphasize that this step \\emph{is not} simply a projection onto a linear subspace defined by $d_h$ PCA modes-- rather it is an approach that learns the deviation of the data from that projection.\nSimilarly the decoding section learns the difference \n\\begin{equation}\nD(h(t))=U^{T} \\tilde{\\hat{\\omega}}(t)-\\left[\\begin{array}{c}h(t) \\\\ 0\\end{array}\\right],\n\\label{decoder}\n\\end{equation}\nwhere $\\tilde{\\hat{\\omega}}(t)$ corresponds to the reconstruction of $\\hat{\\omega} (t)$. Inserting Equation \\ref{encoder} into Equation \\ref{decoder} and noting that by definition $\\tilde{\\hat{\\omega}}(t) = U[P_{d_{h}} U^{T} \\hat{\\omega}(t), P_{d-d_{h}} U^{T} \\hat{\\omega}(t)]^{T}$  we get that the exact solution satisfies $E\\left(U^{T} \\hat{\\omega}(t)\\right)+D_{d_{h}}((h(t))=0$. To satisfy this constraint we add it to the loss function as a penalty to obtain\n\\begin{equation}\nL=\\|\\hat{\\omega}(t)-\\tilde{\\hat{\\omega}}(t)\\|^{2}+\\alpha_L \\left\\|E(U^{T}\\hat{\\omega}(t))+D_{d_{h}}(h(t))\\right\\|^{2}\n\\end{equation}\nwhere $\\| \\cdot \\|$ is the $l^2$-norm and we select $\\alpha_L=1$.  We can now train the AEs by minimizing $L$ via stochastic gradient descent. We train 4 AEs at each of several values of $d_h$ to study the MSE of the reconstruction of $\\hat{\\omega} (t)$.   All models were trained for 300 epochs with an Adam optimizer using Keras. The training data consists of long time series from the direct simulations, with initial transients removed. We use a total of $10^5$ snapshots separated by $\\tau=5$ time units for $\\text{Re}=14.4$, and $10^4$ snapshots separated by $\\tau=5$ for $\\text{Re}=13.5$. We do an $80\\%\/20\\%$ split for training and testing respectively.  Figure \\ref{framework1}\\textcolor{blue}{a} shows a summary of the AE and Table \\ref{tablenn} gives information on the layer dimensions, and activations used in each layer of the encoder and decoder. At each value of $d_h$, the model with the smallest MSE over a test data set from the phase-aligned data is then selected for the discrete time map. We will show in  Section \\ref{sec:autoencoders} that factoring out the phase dramatically increases AE performance.\n\n\n\n\n\\subsection{Time evolution via a dense NN}\\label{sec:aetime}\n\n\nAfter finding $h(t)$ from the AEs, we seek a discrete-time map\n\\begin{equation}\n\th(t+\\tau)=F(h(t))\n\\end{equation}\nthat evolves $h(t)$ from time $t$ to $t+\\tau$. We fix $\\tau=5$. The function $F$ is also expressed as a dense NN. Here we train 5 NNs for the different $d_h$ cases with the following loss\n\\begin{equation}\nL_t=\\|\\tilde{h}(t+\\tau)-h(t+\\tau)\\|^{2},\n\\end{equation}\nwhere $h(t+\\tau)$ comes from true data and $\\tilde{h}(t+\\tau) = F(h(t))$ from the prediction, and select the one with the best performance. For the discrete time map we trained for 600 epochs with the use of a learning rate scheduler. In this case we noticed an increase in performance when dropping the learning rate hyperparameter by an order of magnitude after 300 epochs. Figure \\ref{framework1}\\textcolor{blue}{b}  shows a summary the framework just described, and Table \\ref{tablenn} gives information on the layer dimensions and activations used in each layer.\n\n\n\nAs discussed previously, the time evolution is done in the phase-aligned space. To complete the dynamical picture we seek a discrete-time map for the phase evolution \n\\begin{equation}\n\t\\Delta \\tilde{\\phi}_x(t + \\tau) = G(h(t)),\\label{eq:phaseevolution}\n\\end{equation}\nwhere $\\Delta \\phi_x(t + \\tau)=\\phi_x(t + \\tau)-\\phi_x(t)$.  Because of translation equivariance, the actual phase is only unique to within a constant. We train 5 NNs for the the different $d_h$ cases with the following loss\n\\begin{equation}\nL_p=\\|\\Delta \\tilde{\\phi}_x(t + \\tau)-\\Delta \\phi_x(t + \\tau)\\|^{2},\n\\end{equation}\nsuch that $\\Delta \\tilde{\\phi}_x(t + \\tau) = G(h(t))$. Figure \\ref{framework1}\\textcolor{blue}{c}  shows a summary of the framework we have described, and Table \\ref{tablenn} gives information on the layer dimensions and activations used in each layer.\n\n\\begin{table}[h]\n\\caption{Neural network layer dimensions and activations used in each layer. Sigmoid function are denoted 'S'.}\n$$\n\\begin{array}{lccc}\\hline \\hline & \\text { Function } & \\text { Shape } & \\text { Activation } \\\\ \\hline \\text { Encoder } & E & 1024: 5000:1000:d_{h} & \\text { S:S:S } \\\\ \\text { Decoder } & D & d_{h}: 1000: 5000: 1024 & \\text { S:S:linear } \\\\ \\text { Evolution } & F & d_{h}: 500: 500: d_{h} & \\text { S:S:linear } \\\\ \\text { Phase Prediction } & G & d_{h}: 500: 500:500: 1 & \\text { S:S:S:linear } \\\\ \\hline \\hline\\end{array}\n$$\n\\label{tablenn}\n\\end{table}\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/NNarch10.pdf}\n\t\\caption{Neural network frameworks for (a) autoencoder (b) discrete-time map for pattern prediction and (c) discrete-time for phase prediction.}\n\t\\label{framework1}\n\\end{figure}\n\n\n\n\n\\section{Results} \\label{sec:Results}\nWe present results as follows. First we will show the AE performance for the various $d_h$ and symmetries considered. We then report results for time evolution models, again studying performance as a function of the number of dimensions. Both evolution of the pattern and phase dynamics are considered. {We wrap up the results by predicting bursting events based on  the low-dimensional representation.} \n\n\n\\subsection{Dimension reduction with autoencoders}\\label{sec:autoencoders}\n\n\n\n\nWe begin by showing results for $\\operatorname{Re} = 13.5$. In Figure \\ref{fig:subMSEre13d5} we see the MSE versus $d_h$ trend where the grey curve corresponds to the PCA reconstruction for the original data ($\\tilde{\\omega}(t)=U_{d_h}U^T_{d_h} \\omega (t)$), the black curve to the AE with the original data, and the blue curve to the AE with the phase factored out before training. The MSE is calculated over the test data set. Notice that, as expected, the AEs perform better than PCA. This is because of the nonlinearities that are added to the linear optimal latent space found in PCA in combination with the nonlinear decoder. The blue curve exhibits a sharp drop in the MSE at a dimension of $d_h=2$, which is the correct embedding dimension for a limit cycle. This happens because the phase is accounted for; the dynamics of the system in the phase-aligned reference frame corresponds to a PO and the autoencoder does not have to learn all the possible phases due to the continuous translation in $x$. The overall embedding dimension is $d_h+1 = 3$, where 1 corresponds to the phase. Hence we are able to estimate the dimension for this system by looking at the drop in the MSE curve. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.7\\linewidth]{figures\/MSE_Re13d5.pdf}\n\t\\caption{MSE for test data corresponding to $\\operatorname{Re}=13.5$}\n\t\\label{re13d5intro}\n\\end{figure}\n\\end{comment}\n\nWe now consider the $\\operatorname{Re} = 14.4$ case, where the dynamics are chaotic, moving between the regions near the now unstable RPOs. In Figure \\ref{fig:subMSEre14d4} we show the same curves as in Figure \\ref{fig:subMSEre13d5} but we also include the green and magenta curves, which in addition factor out the SR and the SR-Rotation symmetries respectively before training the AEs. These are included due to the added complexity of $\\operatorname{Re}=14.4$, where the chaotic trajectory travels in the vicinity of the RPOs related by the symmetry groups previously discussed. A monotonic decrease in MSE can be seen for the different symmetries considered in the blue, green, and magenta curves, but no sharp drop is apparent. Instead we notice that the MSE drops at different rates in different regions. For example, in the blue curve corresponding to the phase aligned data, we see a sharp drop from $d_h=1-6$ followed by a more gradual drop from $d_h=6-13$. In the following sections we couple the dimension-reduction analysis with models for prediction of time evolution for the phase aligned data. We expect that this combination will help us determine how many dimensions are needed to correctly represent the state. \n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.7\\linewidth]{figures\/MSE_Re14d4.pdf}\n\t\\caption{MSE for test data corresponding to $\\operatorname{Re}=14.4$}\n\t\\label{re14d4auto}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/MSE_Re13d5_leg3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subMSEre13d5}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/MSE_Re14d4_leg4.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subMSEre14d4}\n\t\\end{subfigure}\n\n\n\n\n\n\t\\caption{MSE versus dimension $d_h$ over the test data corresponding to (a) $\\operatorname{Re}=13.5$ and (b) $\\operatorname{Re}=14.4$. The PCA curve corresponds to the MSE of the reconstruction for the test data set with respect to the true data $\\omega (t)$, with no symmetries factored out, using the truncated $U$ into $d_h$ dimensions such that $\\tilde{\\omega}(t)=U_{d_h}U^T_{d_h} \\omega (t)$ ; the `Original', `Phase', `Phase-SR', and `Phase-SR-Rotation' curves correspond to the MSEs of the reconstruction for the test data set with respect to the true data using AEs. In the curve labeled `Original', no symmetries are factored out and in the other curves the corresponding symmetries in the labels are factored out. }\n\t\\label{re13d5_14d4_auto}\t\n\n\\end{figure}\n\n\n\\subsection{Time evolution as a function of dimension  - Short time predictions}\\label{sec:dimredevshort}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/traj_true_re13d5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipdtruere13d5}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/traj_dh2_re13d5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipd2_re13d5}\n\t\\end{subfigure}\n\t\\caption{Trajectory of $I(t)$ vs $D(t)$ corresponding to $\\operatorname{Re}=13.5$ for (a) true and (b) predicted data  corresponding to dimensions $d_h=2$.}\n\t\\label{re13d5PD}\t\n\\end{figure}\n\nThe focus of this work is the chaotic dynamics at $\\mathrm{Re}=14.4$.  Before considering  that case, for completeness we briefly present results for $\\operatorname{Re}=13.5$. In Figure \\ref{re13d5PD} we see $D(t)$ versus $I(t)$ for the true and predicted dynamics at $d_h=2$; they are indistinguishable. At $d_h=1$, which is not shown, the model fails and the dynamics can not be captured. The reason for this is simple -- the embedding dimension for a limit cycle is two.\n\nNow we return to the case of $\\mathrm{Re}=14.4$, focusing first on short-time trajectory predictions. The Lyapunov time $t_L$ for this system is approximately $t_L \\approx 20$ \\cite{inubushi2012covariant}, hence $t_L \\approx 4\\tau$. We take initial conditions $h(t) \\in \\mathbb{R}^{d_{h}}$ to evolve recurrently with the discrete time map $F(\\cdot)$, such that $\\tilde{h}(t+\\tau) = F(h(t))$, $\\tilde{h}(t+2\\tau) = F(\\tilde{h}(t+\\tau))$, $\\tilde{h}(t+3\\tau) = F(\\tilde{h}(t+2\\tau))$ and so on. After evolving in time the data is then decoded to get $\\tilde{\\hat{\\omega}}_h (t)$ and compared with $\\hat{\\omega} (t)$. \nWe consider trajectories with ICs starting in the quiescent as well as in the bursting regions. The nature of the intermittency of the data makes it challenging to assign either bursting or quiescent labels. We consider a window of past and future snapshots and a criterion on $ \\| \\hat{\\omega} (t) \\| $ to make this decision, using the algorithm described in Algorithm \\ref{alg:bursting}. \n\n\\begin{algorithm}[H]\n\t\\caption{Quiescent\/Bursting labeling of vorticity snapshots}\\label{alg:cap}\n\t\\begin{algorithmic}\n\t\t\\State $W \\gets [\\hat{\\omega}(t_1),\\hat{\\omega}(t_2) \\cdots]$ \\Comment{Matrix with $N_s$ vorticity snapshots, $W \\in \\mathbb{R}^{N \\times N_s}$}\n\t\t\\State $S$ \\Comment{Initialize label array $S$}\n\t\t\\State $W_{l2} \\gets \\| W \\|$ \\Comment{Calculate $l^2$-norm of snapshots, $W_{l2} \\in \\mathbb{R}^{ N_s}$ }\n\t\t\\State $b \\gets 10$ \\Comment{Number of past snapshots in time to consider }\n\t\t\\State $f \\gets 10$  \\Comment{Number of future snapshots in time to consider }\n\t\t\\For {$i=b$, $b+1,\\ldots N_s-f$} \\Comment{$i$ is snapshot I.D.}\n\t\t\\If{$W_{l2}[i]<60$ }\n\t\t\\State $d_p \\gets \\operatorname{abs}(W_{l2}[i-b:i]-W_{l2}[i])$ \\Comment{Difference between current and past snapshots}\n\t\t\\State $b_p \\gets \\operatorname{sum}(d_p>5)$ \\Comment{Sums values that exceed a threshold of 5 (user defined)}\n\t\t\\State $d_f \\gets \\operatorname{abs}(W_{l2}[i:i+f]-W_{l2}[i])$ \\Comment{Difference between current and future snapshots}\n\t\t\\State $b_f \\gets \\operatorname{sum}(d_f>5)$ \\Comment{Sums values that exceed a threshold of 5 (user defined)}\n\t\t\\If{$b_p = 0$ or $b_f = 0$ }\n\t\t\\State $S[i-b] \\gets 0$\n\t\t\\Else \n\t\t\\State $S[i-b] \\gets 1$\n\t\t\\EndIf\n\t\t\\Else\n\t\t\\State $S[i-b] \\gets 1$\n\t\t\\EndIf\n\t\t\\EndFor\n\t\\end{algorithmic}\\label{alg:bursting}\n\\end{algorithm}\n\\noindent Doing this we ensure that snapshots that are contained in the bursting events and have a value of $ \\| \\hat{\\omega} (t) \\| $ similar to quiescent snapshots are correctly classified. We use a threshold on $ \\| \\hat{\\omega} (t) \\| $ to determine if a check is needed. For the classification strategy any snapshot above a threshold of 60 is classified as bursting with a label of 1, below 60 we enter a loop as shown in Algorithm \\ref{alg:cap} to determine if it should be classified as bursting or quiescent, where quiescent corresponds to a label of 0. This check is needed to correctly label snapshots that have comparable $ \\| \\hat{\\omega} (t) \\| $ but are still in the bursting regime. Figure \\ref{re14d4labels} shows a short time trajectory where the black line corresponds to $ \\| \\hat{\\omega} (t) \\| $ and the red to the 0\/1 labels. Notice that, as shown in Algorithm \\ref{alg:cap}, some of the data at the beginning and at the end of the time series will not be labeled, there are no past or future snapshots to compare to, and can be removed.\n\nAfter labeling the data as quiescent or bursting, we then consider the time evolution from ICs of $h(t)$ using the models of various dimensions. We will first show sample trajectories from ICs starting in the two regions, then show the ensemble-averaged prediction error as a function of time. Figure \\ref{KEhib} shows the KE evolution for an IC starting in the quiescent region. The black curve corresponds to the true data and the colored curves to the different $d_h$ models. At a dimension of $d_h=3$ we see that the predicted $KE$ diverges quickly with respect to the true $KE$. In the case of $d_h=4$ the $KE$ seems to fall on top of the unstable RPO with good agreement in the oscillatory behavior; however the bursting that occurs at $t \\approx 180$ is not captured. It is not until we reach $d_h=5$ that we see that the bursting event is correctly captured. In the case of $d_h=6$ we see that the bursting event is captured but with a time delay, and $d_h=7$ does not capture the bursting in this time frame considered. Figure \\ref{KEbur} shows the KE evolution for an IC starting in the bursting region. The black curve corresponds to the true data and the colored curves to the different $d_h$ models. At a dimension of $d_h=3$ the $KE$ stays bursting and does not show agreement with the true $KE$. However the cases $d_h=4$ and $d_h=5$ show better agreement and are also capable of closely predicting the end of the bursting event.  In this case $d_h=6$ and $d_h=7$  agree closely with the $KE$ evolution, and specifically $d_h=7$ seems to track better before traveling to the quiescent region. \n\n\n\nTurning from examples of individual trajectories to ensemble averages, Figure \\ref{re14d4ICs} shows ensemble averages of the difference between the true and predicted trajectories, separately considering ICs in the bursting and quiescent regions. Blue curves correspond to quiescent ICs and red curves to bursting ICs. Starting from $d_h=3$ (lightest curve) we increase up to $d_h=7$ (darkest). We selected $10^4$ ICs in total where approximately 1\/3 of the ICs correspond to bursting. As expected, predictions at $d_h=3$ diverge quickly from the true dynamics in both quiescent and bursting IC scenarios. With increasing $d_h$, trajectories track better for both types of ICs. We can also notice that the three darkest curves, corresponding to $d_h=5,6,7$, perform best, and in the case of the quiescent ICs there is not much increase in performance between the three. We also notice that the trajectories for the quiescent ICs track almost perfectly for approximately two Lyapunov times. In Figure \\ref{re14d4ICstotal} we show ensemble averages of the difference between the true and predicted dynamics based on all ICs. The same trend is obtained as discussed for Figure \\ref{re14d4ICs}, and as expected the errors increase for all of the curves due to the divergence of the bursting ICs. We can conclude that models $d_h=5,6,7$ are very good at capturing trajectories the quiescent regions, which happens through the accurate prediction of the oscillatory behavior of the unstable RPO right before a bursting occurs. Prediction from bursting ICs is harder, due to the complex dynamics involved in this region. \n\n\n\n\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.55\\linewidth]{figures\/hibbur_labels_v3.pdf}\n\t\\caption{Labeling of $\\hat{\\omega} (t)$ snapshots in a short time series where 1 corresponds to bursting and 0 to quiescent.}\n\t\\label{re14d4labels}\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KEshorthib_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{KEhib}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KEshortbur_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{KEbur}\n\t\\end{subfigure}\n\t\n\t\\caption{Trajectory of $KE$ at different $d_h$ for (a) quiescent and (b) bursting ICs.}\n\t\\label{KEtots}\n\t\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/hibbur_ICs_v5diffcolors.pdf}\n\t\t\\caption{}\n\t\t\\label{re14d4ICs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/total_ICs_v5diffcolors.pdf}\n\t\t\\caption{}\n\t\t\\label{re14d4ICstotal}\n\t\\end{subfigure}\n\t\n\t\\caption{Difference between true vorticity evolution and vorticity evolution obtained from the time map $F$ from $h(t)$ where (a) correspond to averages taken over bursting  and quiescent ICs and (b) averages over all the data. Lightest color curve corresponds to $d_h=3$ and darkness increases until $d_h=7$.}\n\t\\label{totalsplitICs}\n\t\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure\n\\centering\n\\includegraphics[width=.6\\linewidth]{figures\/hibbur_ICs.pdf}\n\\caption{Difference between predicted and real vorticity evolution. Green curves correspond to averages taken over bursting ICs and black over hibernating ICs. Lightest curve corresponds to $d_h=3$, $d_h$ is increased until reaching the darkest curve corresponding to $d_h=7$}\n\\label{re14d4ICs}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\n\\subsection{Time evolution as a function of dimension  - Long time predictions}\\label{sec:dimredev}\n\nIn this section we present long time statistics for the models and  true data at $\\mathrm{Re}=14.4$.  From ICs on the attractor, we evolve for $2 \\times 10^5$ time units, yielding to get $4 \\times 10^4$ snapshots of data. This duration is sufficient to densely sample the quiescent and bursting regions.  We note that long time statistics did not change if the IC was in a bursting or quiescent region.\n\n\n\nFigure \\ref{re14d4PD} shows the joint probability density function (PDF) of $I$ and $D$ for true and predicted data from models with $d_h$ from 3 to 7 -- note the logarithmic scale, here and below. We notice that at $d_h=3$ the different areas corresponding to quiescent and bursting regions are populated similarly in terms of the probability intensity compared with the true PDF shown, but the shape of the predicted PDF takes a curved form that is not seen in the true PDF. When we get to $d_h=4$ the shape of $D$ and $I$ event region approximates better the true data. However scattered points of high $D$ do not compare with the true data. It is not until $d_h=5$ is reached that the $D$ and $I$ events are better captured, and similarly for $d_h=6,7$. We also compute the joint PDF of $\\operatorname{Re}\\left[a_{0,1}\\right]$ and $\\operatorname{Im}\\left[a_{0,1}\\right]$, shown in Figure \\ref{re14d4PD01}. From this quantity we can observe the heteroclinic-like connections between the unstable RPOs, which correspond to the four ribbon-like regions of high probability. Here we see similar trends as in the joint PDF for $I$-$D$: $d_h=3,4$ show poor qualitative reconstruction compared with higher dimensions, and once $d_h\\geq 5$, the joint PDFs from the model prediction are virtually indistinguishable from the true PDFs. To further quantify the relationship of  the PDFs from the models to the true data, we calculate the Kullback-Leibler (KL) divergence,  \n\\begin{equation}\nD_{KL}(\\tilde{P}||P)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\tilde{P}\\{a,b\\} \\text{ln}\\dfrac{\\tilde{P}\\{a,b\\} }{P\\{a,b\\}}da \\; db,\n\\end{equation}\nwhere $\\tilde{P}$ corresponds to the predicted PDF and $P$ to the true PDF. Due to the approximation of the integral to discrete data we ignore areas where either the true or predicted PDFs are zero. Let us first consider the case $a=I$ and $b=D$. Figure \\ref{fig:subKLIPD} shows $D_{KL}$ calculated with varying $d_h$. The dashed grey line corresponds to $D_{KL}$ calculated over different true data sets. This serves as a baseline for comparison to the predicted PDFs.  A significant decrease happens at $d_h=4$ followed by small decreases at higher dimensions. We see that after $d_h=5$ no significant information is gained. We can also look at the case where $a= \\;$Re $\\left[a_{0,1}\\right]$ and $b= \\;$Im $\\left[a_{0,1}\\right]$ in Figure \\ref{fig:subKLF01}. We notice that errors of the joint PDF in Figure \\ref{fig:subKLF01} show the same trend as Figure \\ref{fig:subKLIPD}. We can infer from these results that the embedding dimension of this system lies in the range $d_h=5-7$, and furthermore that the data-driven model can reproduce the long-time statistics with very high fidelity.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=0.85\\linewidth]{figures\/ip_diss_true_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipdtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh3_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh4_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh5_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh6_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd6}\n\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh7_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd7}\n\\end{subfigure}\n\t\\caption{$\\operatorname{Re}=14.4$: Joint PDFs of $I$-$D$ corresponding to $\\operatorname{Re}=14.4$ for (a) true and  (b)-(f) predicted data  corresponding to dimensions $d_h=3-7$.}\n\t\\label{re14d4PD}\t\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/ipdpdferror.pdf}\n\t\\caption{Joint PDFs difference between true and predicted data corresponding to $Re=14.4$ of power input and dissipation.}\n\t\\label{re14d4pdferror}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.85\\linewidth]{figures\/01_true_pdf_bin01_notime.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01dtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh3_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh4_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh5_pdf_bin01_notime.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01d5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh6_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d6}\n\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh7_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d7}\n\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: Joint PDFs of $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ corresponding to $\\operatorname{Re}=14.4$ for (a) true and  (b)-(f) predicted data  corresponding to dimensions $d_h=3-7$.}\n\t\\label{re14d4PD01}\n\t\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KLDiv_IP_D_Re14d4_noerror_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLIPD}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KLDiv_F01_Re14d4_noerror_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLF01}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: $D_{KL}$ vs dimension $d_h$ for (a) $I$-$D$ and (b) $\\operatorname{Re}\\left[a_{0,1}\\right]-\\operatorname{Im}\\left[a_{0,1}\\right]$ predicted vs true joint PDFs. Dashed grey line corresponds to $D_{KL}$ calculated over true data sets.}\n\t\\label{re14d4KLDiv}\n\t\n\\end{figure}\n\n\n\\begin{comment}\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/01pdferror.pdf}\n\t\\caption{Joint PDFs difference between true and predicted data corresponding to $Re=14.4$ of real and imaginary part for Fourier coefficient (1,0)}\n\t\\label{re14d4PD01error}\n\\end{figure}\n\\end{comment}\n\n\n\\begin{comment}\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/ip_diss_true_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipdtrueRe20}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/ip_diss_dh10_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipd10Re20}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/01_true_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01dtruRe20e}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/01_dh10_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01d10Re20}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/KLDiv_IP_D_Re20_werror.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLIPDRe20}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/KLDiv_F01_Re20_werror.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLF01Re20}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=20$: Joint PDFs of $I(t)$-$D(t)$ for (a) True and  (b) Predicted data  corresponding to dimension $d_h=10$. Joint PDFs of $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ for (c) True and  (d) Predicted data  corresponding to dimension $d_h=10$. $D_{KL}$ vs dimension $d_h$ corresponding to the difference between (e) $I(t)$-$D(t)$ and (f) $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ predicted vs true joint PDFs. }\n\t\\label{re20PD}\t\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/MSE_Re20.pdf}\n\t\\caption{MSE of HNN-AE vs dimension $d_h$ over the test data corresponding to $\\operatorname{Re}=20$}\n\t\\label{re20}\n\\end{figure}\n\\end{comment}\n\n\n\nThe above PDFs yield no information about the temporal behavior of the system. One temporal feature of significant interest in problems with intermittency is the probability density of the durations of time intervals with different behavior.  To address this, we consider the PDFs of time spent in bursting ($t_b$) and in quiescent ($t_q$) regions. The labeling method  discussed in the previous section is used. For this calculation we take a trajectory of $10^5$ snapshots from an arbitrary IC. The PDF for the true data is shown in Figure \\ref{fig:subhibburtrue} followed by the PDFs that come from the $d_h=3-7$ models in Figures \\ref{fig:subhibburd3} - \\ref{fig:subhibburd7}. The true data shows that $t_q$ is mostly concentrated between $t \\approx 200-300$ with a high intensity peak shown at $t=5$. We attribute this peak to a small fraction of snapshots in the bursting region that get mislabeled as quiescent due to the weakly chaotic nature of the data. We do not expect for this to drastically change our conclusions because the same labeling system is used for the true data and the models. In the case of $t_b$ we notice that these are mostly concentrated between $t \\approx 0-200$. We also report the means  of these time durations as well as the standard error of the mean (SE) in Table \\ref{tablehibburtime}. Looking at both the PDFs and averages of the times we see that $d_h=3$ fails to correctly capture the shape of the PDF and also underpredicts $ \\langle t_q \\rangle$ and $ \\langle t_b \\rangle$. At $d_h=4$,  $ \\langle t_q \\rangle$ and $ \\langle t_b \\rangle$ get closer to the true values, but the shape of the PDF still looks different from the true data. At $d_h=5$ we start getting better agreement where we see that the PDFs clearly show the two regions where $t_b$ and $t_q$ are concentrated. Performance is similar at $d_h=6$, however at $d_h=7$  we can see that the quiescent PDF spreads into regions with higher $t_q$. Figure \\ref{re14d4tqtb} shows $D_{KL}$ with varying $d_h$ for these PDFs. As expected from observing the PDFs we see that $D_{KL}$ decreases up until $d_h=5$ for both cases. In the case of $t_q$ we see an increase in the error after $d_h=5$ which agrees with the above observation of the PDFs. For $t_q$, $D_{KL}$ seems to keep slightly decreasing after $d_h=5$. In short, while the low-dimensional models do not achieve the same agreement with the true results for these duration statistics as we do for the static quantities considered above, they nevertheless capture the key features of the distributions and capture their means with reasonable accuracy, within about $20\\%$. \n\n\n \n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.85\\linewidth]{figures\/hibburtime_true_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh3_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh4_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh5_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh6_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd6}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh7_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd7}\n\t\\end{subfigure}\n\t\n\t\\caption{PDFs of $t_q$ and $t_b$ at $\\operatorname{Re}=14.4$ for (a) true and  (b)-(f) predicted data  for dimensions $d_h=3-7$.}\n\t\\label{re14d4PDhibbur}\n\t\n\\end{figure}\n\n\\begin{table}[h]\n\t\\caption{Average and standard error of the mean of $t_q$ and $t_b$ for true and dimensions $d_h=3-7$.}\n\t$$\n\t\\begin{array}{lcccc}\\hline \\hline & \\text { $ \\langle t_q \\rangle$ } \\; \\; \\; & \\text { $\\langle t_b \\rangle$ } \\; \\; \\; & \\text { SE($ t_q$) } \\; \\; \\; & \\text { SE($ t_b$)  } \\; \\; \\; \\\\ \\hline \\text { True } & 176& 97 & 3 & 2 \\\\ \\text { $d_h=3$ } & 37 & 73 & 1 & 2 \\\\ \\text { $d_h=4$ } & 174 & 85 & 5 & 3 \\\\ \\text { $d_h=5$ } & 160 & 105 & 3 & 3\\\\ \\text { $d_h=6$ } & 185 & 106 & 3 & 3\\\\ \\text { $d_h=7$ } & 202 & 101 & 4 & 3 \\\\ \\hline \\hline\\end{array}\n\t$$\n\t\\label{tablehibburtime}\n\\end{table}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/tqKL_nostdev_v2.pdf\n\t\t\\caption{}\n\t\t\\label{fig:subtq}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/tbKL_nostdev_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subtb}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: $D_{KL}$ vs dimension $d_h$ corresponding to PDFS for (a) $t_q$ (b) $t_b$. Dashed grey line corresponds to $D_{KL}$ calculated over different true data sets.}\n\t\\label{re14d4tqtb}\n\t\n\\end{figure}\n\n\n\\subsection{Phase prediction}\\label{phasepred}\n\n\n\n\n\nRecall that we gain substantial accuracy in dimension reduction by factoring out the spatial phase $\\phi_x(t)$ of the data. Here we complete the dynamical picture of the model predictions at $\\operatorname{Re}=14.4$ by illustrating the predictions of phase evolution, as given by the learned phase evolution equation \\eqref{eq:phaseevolution}. Figure \\ref{re14d4phasecompare} shows a short time evolution of $\\phi_x(t)$ corresponding to the true and predicted data for the $d_h=5$ model. We select $d_h=5$ because previous short time tracking and long time statistics show agreement with respect to the true data. The smooth increases and decreases in Figure \\ref{re14d4phasecompare} correspond to trajectories during time intervals where they are near an RPO and thus are traveling in the $x$-direction. The intervals where the phase flucuates rapidly are the bursts during which the trajectories are moving between the RPO regions. This behavior is well-captured at $d_h=5$. Notice that although the trajectories diverge, for short times we get around two $t_L$ of prediction horizon where the model still captures the correct dynamics, and Figure \\ref{re14d4phasecompare} provides a clear visual indications that the loss of predictability occurs during the bursts. \n\nWe now take an approach to quantify how well the model performs with respect to the true data. Taking a look at the drops and increases for $\\phi_x(t)$ we can observe that after every burst the trajectory will either travel, essentially randomly, in the positive (increasing $\\phi_x$) or negative (decreasing $\\phi_x$) $x$ direction. This behavior is essentially a run and tumble or random walk behavior in the sense that the long periods of positive or negative phase drift correspond to ``runs\" that are separated by ``tumbles\" that correspond to the bursts, in which the direction of phase motion is reset. Hence, a natural analysis of quantification for this type of dynamics consists of calculating the mean squared displacement (MSD) of the phase: \n\\begin{equation}\n\\mbox{MSD} (t)= \\langle (\\phi_x (t) - \\phi_x (0))^2 \\rangle.\n\\end{equation}\nFigure \\ref{re14d4phaseMSD} shows the time evolution of MSD of true and predicted data. The black line corresponds to the true data and the black and blue dashed lines serve as references with slopes of 1 and 1.5, respectively. The colored lines correspond to models with various dimensions. Looking at the true curve we notice a change from superdiffusive (slope = 1.5) to diffusive (slope = 1) scaling that happens around $t \\approx 200$, which corresponds to the mean duration of the quiescent intervals, as discussed above: i.e., to the average time the trajectories travel along the RPOs before bursting. The trajectory then bursts and reorients which is captured by the long time diffusive trend. Looking at the performance of the models we observe that $d_h=3,4$ do a good job at capturing the short time scaling, however these are not to able capture the change in slope that is observed in the true data.  It is not until $d_h=5,6,7$ that the correct behavior at long times is observed -- indeed the predictions agree very well with the data, the slight upward shift upward at long times corresponding to the slight overprediction of the mean duration of the quiescent periods.\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/phasecorr_withdh5_moretime_v2.pdf}\n\t\\caption{Time evolution of $\\phi_x$ corresponding to the true data and $d_h=5$.}\n\t\\label{re14d4phasecompare}\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/phasecorr_withdh5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subphase}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/deltaphase_withdh5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subdeltaphase}\n\t\\end{subfigure}\n\t\n\t\\caption{(a) Time evolution of $\\phi_x$ corresponding to the True data and $d_h=5$. (b) Time evolution of $\\Delta \\phi_x(t + \\tau)$ corresponding to the True data and $d_h=5$.\\MDG{Aren't these just short- and long-time trajectories of the same thing? You don't seem to say so anywhere. Why are you labeling one $\\phi_x$ and the other $\\Delta\\phi_x$?  Rethink this plot -- do you really need two separate ones?}  }\n\t\\label{re14d4phasevis}\n\t\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/MSD_phase_v3.pdf}\n\t\\caption{MSD of $\\phi_x (t)$ corresponding to true data and models with dimensions $d_h=3-7$.}\n\t\\label{re14d4phaseMSD}\n\\end{figure}\n\n\\begin{comment}\n\\subsection{Short and long -time vorticity prediction}\\label{sec:dataspacing}\n\n\\MDG{Show some time evolution predictions.  I think the presentation order here is backwards.  You have long-time statistics before short-time trajectory predictions. And for the trajectory predictions, you only show averages (figure 10), not any examples of actual predictions as a function of time. I know you have these figures -- you have shown them to me.  Ask yourself what the reviewer\/reader is going to want to see. } \n\n\\MDG{Observation that linear encoder (PCA) gives good representation of dynamics as long as nonlinear decoder is used}\n\n\\MDG{Anything at higher Reynolds number? Look back at older results -- let's revisit.}\n\\end{comment}\n\n\n\n\n\n\n\n\n\\subsection{Bursting prediction}\\label{burpred}\n\n\n\n\nPrevious research has focused on finding indicators that guide predictions of when a burst will occur. It has been shown for the Kolmogorov flow that before a burst there is a depletion of the content in the $(1,0)$ Fourier mode, which then feeds into the forcing mode $(0,n)$ \\cite{nicolaenko1990symmetry}. Figure \\ref{indandKE} shows how this looks for $\\operatorname{Re} = 14.4$, $n=2$.  By considering a variational framework and finding solutions to a constrained optimization problem it was also found that examination of these modes can lead to predictions of when a burst will occur \\cite{farazmand2017variational}.\n\n\n\n\\begin{figure\n\n\n\n\n\t\\includegraphics[width=0.6\\linewidth]{figures\/KE_02_10_v4.pdf\n\n\n\n\t\\begin{comment}\n\t\\begin{subfigure}{0.5\\textwidth}\n\n\t\\includegraphics[width=.9\\linewidth]{figures\/Indicator_v2.pdf\n\t\\caption{}\n\t\\label{Indicators}\n\t\\end{subfigure}\n\t\\end{comment}\n\t\n\t\\caption{Time evolution of $KE$ and amplitudes corresponding to $(1,0)$ and $(0,2)$ Fourier mode for $\\operatorname{Re}=14.4$. }\n\t\\label{indandKE}\n\t\n\\end{figure}\n\nWith our framework,  natural indicators are the latent variables $h$, which we will consider here along with some variations, including the indicators used in previous work.   To predict bursting events based on a given indicator, we will use a simple binary classifier in the form of a support vector machine (SVM) with a radial basis function kernel \\cite{boser1992training}. These have shown success in predicting extreme events for problems such as extreme rainfall  \\cite{nayak2013prediction}. With this approach, data at time $t$ is used to learn a function that outputs a binary label of bursting\/not bursting at time $t+\\tau_b$. For all of the cases considered we use the $d_h=5$ model, taking a dataset of $5 \\times 10^4$ snapshots to train the SVM and another $5 \\times 10^4$ as a test set. \n\n\nFigure \\ref{Pburs} shows the percent correct classification of bursting events with varying time $\\tau_b$ in the future. The black curve corresponds to predicting the events based on the PCA projection of the data, $P_{d_{h}} U^{T} \\omega$, into the first $d_h=5$ coefficients and the cyan curve corresponds to $h$ of dimension $d_h=5$.  We notice that the PCA and $h$ curve fall on top of another and have a high probability of correct classification when considering prediction horizons less than one $t_L$. For this purpose we see that PCA is enough to predict bursting events. Figure \\ref{Pbursind} shows the percent correct classification of bursting at time $\\tau_b$ in the future for the previous discussed indicators. None of these work nearly as well as $P_{d_{h}} U^{T} \\omega$ or $h$. The blue curve corresponds to $(1,0)$ amplitude of the original true data, the green curve to the forcing $(0,2)$ amplitude, and we also consider $\\Delta \\phi$ in the purple curve. In the case of $\\Delta \\phi$ we see some predictability at times longer than one $t_L$ and less than two.  This also happens for the case of $(1,0)$, however there seems to be no decrease or increase in the probability of correct classification. We can see from Figure \\ref{indandKE} that even though a depletion from $(1,0)$ mode is seen, the part corresponding to the bursting in the amplitude oscillates closely in terms of magnitude to the values corresponding to the quiescent region, which might be the reason of the poor prediction. The amplitude $(0,2)$ shows to be the better predictor for bursting events. At small $\\tau_b$ its predictions outperform $(1,0)$ and $\\Delta \\phi$, however at times larger than one $t_L$, $\\Delta \\phi$ performs better. \n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/Pburs_model_v7.pdf}\n\t\t\\caption{}\n\t\t\\label{Pburs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/Pburs_ind_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{Pbursind}\n\t\\end{subfigure}\n\n\t\n\n\n\n\n\n\t\n\n\n\n\n\t\\caption{Percent of correctly classified bursting events at $\\tau_b$ forward in time for: (a) $P_{d_{h}} U^{T} \\omega$ and $h$ at $d_h=5$, (b) and indicators $\\Delta \\phi$, $(1,0)$, and $(0,2)$. Note that the vertical scales on (a) and (b) are very different.}\n\t\\label{Pbursall}\n\t\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{figures\/KE_02_10_v2.pdf}\n\t\\caption{Time evolution of $KE$ and amplitudes corresponding to $(1,0)$ and $(0,2)$ Fourier mode for $\\operatorname{Re}=14.4$}\n\t\\label{KE0210}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{figures\/Pburs_v2.pdf}\n\t\\caption{Probability of correctly predicting a burst at $\\tau_b$ forward in time}\n\t\\label{Pburs}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion} \\label{sec:Conclusion}\n\n\n\nThe nonlinearity of the NSE poses challenges when using ROMs, where the dynamics are expected to evolve on an invariant manifold that will not lie in a linear subspace. Neural networks have proven to be powerful tools for learning efficient ROMs solely from data, however finding and exploiting a \\emph{minimal}-dimensional model has not been emphasized. We present a data-driven methodology to learn an estimate of the embedding dimension of the manifold for chaotic Kolmogorov flow and the time evolution on it. An autoencoder is used to find a nonlinear low-dimensional subspace and a dense neural network to evolve it in time.\n\n\nOur autoencoders are trained on vorticity data from two cases: a case where the dynamics show a relative periodic orbit solution ($\\operatorname{Re}=13.5$), and a case with chaotic dynamics ($\\operatorname{Re}=14.4$). The chaotic regime we consider comes with challenges due to the intermittent behavior observed where the trajectory travels in between quiescent intervals and bursting events. We factor out the rich symmetries of Kolmogorov flow before training of the autoencoders, which dramatically improves reconstruction error of the snapshots. This improves training efficiency by not having to learn a compression of the full state. Specifically, factoring out the translation symmetry decreases the mean-squared reconstruction error by an order of magnitude compared to the case where phase is not factored out, and several orders of magnitude compared to PCA. The phase-aligned low-dimensional subspace is then used for time evolution where the RPO dynamics is learned essentially perfectly at $d_h=2$ for $\\operatorname{Re}=13.5$ and very good agreement for short and long time statistics is obtained at $d_h=5$ for $\\operatorname{Re}=14.4$. For comparison, the full state space of the numerical simulation data is $N=1024$.\n\n\nWe also show phase prediction evolution results based on the low-dimensional subspace learned. The time evolution of the true phase exhibits a superdiffusive scaling at short times and a diffusive scaling at long times which we attribute to the traveling near an RPO and the reorientation due to bursting. \nFinally, using the low-dimensional representation enables accurate prediction of bursting events based on conditions about a Lyapunov time ahead of the event.\n This work opens new avenues for data-driven reduced order models with applications such as control for drag reduction. One important challenge that remains is more effective treatment of systems with intermittent dynamics like those described here. A recent study \\cite{floryan2021charts} has introduced a method that uses the differential topology formalism of charts and atlases to develop \\emph{local} manifold representations and dynamical model that can be stitched together to form a global dynamical model. One attractive feature of that formalism is that it enables use of separate representations for regions of state space with very different dynamics, and has already shown in specific cases to provide dramatically improved results for dynamics with intermittency.   \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\\begin{comment}\n\n\n\n\nchallenges are higher Re, symmetries, \n\n\\MDG{Rewrite to be half to two-thirds as long.  Pick out the important points rather than trying to summarize everything (this isn't a summary, it's a conclusion). You have missed half the point of the modeling, which is not just dimension reduction but also time evolution. You didn't say anything about data-driven, or how K-flow is challenging because of its intermittency.  Once the summary is finished, draw some conclusions.  What works about the approach so far and what doesn't?  What future work might be done to improve the predictions?}    \n\nWe present an AE architecture and apply it to two-dimensional Kolmogorov flow. Specifically for the onset on which chaotic dynamics is sustained which happens for $n=2$, $\\operatorname{Re}=14.4$. We combine results from MSE, short time predictions, as well as PDFs of $I(t)$-$D(t)$ and Re $\\left[a_{0,1}(t)\\right]$ - Im $\\left[a_{0,1}(t)\\right]$ obtained from time maps at $d_h \\ll N$ to guide us in the estimation of the embedding dimension. The difference between the PDFs is calculated using KL divergence which guides us to selecting $d_h=5$. After this dimension no significant information is gained. This is further reassured with the short time tracking with respect to the true data as well as with the calculation of the quiescent and bursting fractions. We then complete the picture by predicting $\\Delta \\phi_x (t)$ from $h(t)$. Two behaviors are observed when calculating the MSD of $\\phi_x (t)$. At short times MSD scales in a super diffusive fashion while at long times diffusive scaling is obtained. The change to super diffusive happens at approximately $t \\approx 200$ which agrees with the quiescent time scale. Here we see that starting at $d_h=5$ the change to diffusive scaling is obtained. We further ask the question: can we predict bursting events from our model at $d_h=5$? To this end we use SVM to predict if a bursting will occur. We find that predicting with the first 5 PCA modes gives similar results as predicting with $h$, which we attribute to the similarity of these two projections. Using the mode $(1,0)$ shows no predictive capabilities which is not the case at higher Re and $n$. However $\\Delta \\phi_x$ and $(0,2)$ amplitude show to better predict bursting events. With varying dimensions we see that after $d_h=5$ is reached, no further improvement is obtained in the bursting prediction and it outperforms the different indicators considered. The selection of $d_h=5$ arises naturally in this case. Furthermore we consider the encoder section of the AE to predict bursting events. Using the full state PCA$-h$ (not shown) did not improve the event prediction, however predicting based on individual units gives some interesting outcomes. As shown previously, we see that PCA$_2-h_2$ does a better job at predicting bursting events than PCA$_2$ or $h_2$. The correction of the encoder shows to come into play in the bursting events where the difference increases in magnitude as seen in Figure \\ref{fig:diffh2dh6}. When taking the discrete Fourier transform of PCA$_2$ we see that most of its content lies in the $(0,1)$ Fourier mode. Recall that this mode corresponds to the projection in which the traveling between the different symmetric subspaces is observed. We find that using the amplitude of $(0,1)$ alone as an indicator does not provide good bursting prediction compared to the other indicators considered in this manuscript. The nonlinearity introduced by the encoder in PCA$_2$ seems to split quiescent and bursting events in such a way that prediction is improved. This work motivates new avenues such as flow control based on minimal model. Finding these models in experiments could also be powerful tools for controls. Extensions of this work also include charting the manifold such that the inertial manifold dimension can be estimated.\\MDG{what would be the advantage of this? Not just determining manifold dimension!!!! Cite Daniel's paper here!}\n\n\n\\end{comment}\n\n\n\n\\begin{comment}\n\nWe can argue that features extracted by the NNs are the necessary ones to correctly evolve the state in time and capture long time statistics. All models evolved in such a way that the turbulence was sustained, with no apparent blow up of the trajectory evolution for the models showed. \n\nFor bursting at lower n and Re delta phi is a better predictor\n\n\\end{comment}\n\n\\begin{comment}\n\n\\subsection{Trajectory Difference}\\label{sec:dataspacing}\n\nShort and long-time prediction of the trajectory is compared with the true data. We first take a look at the calculation of the KE for $d_h = 5$. Figure  \\ref{re14d4ke} shows the trajectory comparison with respect to the true data. We notice that predicted trajectory travels close to the true trajectory for about a Lyapunov time which in this case is $\\approx 93$ time units \\hl{[cite]}. \n\n\\begin{figure}[H]\n\n\t\\centering\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/kecomparisonshort.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub7}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/kecomparisonlong.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub8}\n\t\\end{subfigure}\n\n\t\\caption{(a) Short-time prediction comparison of KE with true data (b) Long-time prediction comparison of KE with true data. Notice that model is able to capture quiescent and bursting events.}\n\t\\label{re14d4ke}\n\t\n\\end{figure}\n\nWe also compare the vorticity snapshots obtained from the model with the true data. To this end we calculate the MSE between the two trajectories. Similar to the KE results we see that the trajectory travels well for about a Lyapunov time. \n\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/wdifference.pdf}\n\t\\caption{Vorticity difference between predicted data from the model and true data.}\n\t\\label{re14d4wdiff}\n\\end{figure}\n\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\\begin{acknowledgments}\nThis work was supported by AFOSR  FA9550-18-1-0174 and ONR N00014-18-1-2865 (Vannevar Bush Faculty Fellowship). We also want to thank the Graduate Engineering Research Scholars (GERS) program and funding through the Advanced Opportunity Fellowship (AOF) as well as the PPG Fellowship.\n\\end{acknowledgments}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAnti-de Sitter (AdS) gravity coupled to scalar fields only, has received recently considerable \nattention.\nThis simple model admits both soliton \n\\cite{Hertog:2004rz}, \n\\cite{Hertog:2004ns}, \n\\cite{Radu:2005bp},\n\\cite{Faulkner:2010fh},\nand black hole solutions (see $e.g.$\n\\cite{Torii:2001pg},\n\\cite{Sudarsky:2002mk},\n\\cite{Winstanley:2002jt},\n\\cite{Hertog:2004dr},\n\\cite{Martinez:2004nb},\n\\cite{Park:2008zzb})\nwith interesting properties, the resulting picture being in strong contrast\nwith the one found \nin the absence of a cosmological constant.\nMoreover, since scalar fields generically enter the gauge supergravity\nmodels, the study of such solutions is also relevant\nto the AdS\/CFT conjecture.\n \nHowever, most of the studies in the literature assume that the scalar fields \nare real and have the same symmetries as the underlying spacetime. \nIt is interesting to ask if the known solutions in \\cite{Hertog:2004rz}-\\cite{Martinez:2004nb} \ncan be generalized \nfor a  complex scalar field. \nIn formulating a  scalar field ansatz in this case, it is natural to take \na separation of variables \n\\begin{equation}\n\\label{Phi}\n\\Phi(\\vec x,t)=e^{-i  \\omega t}\\phi(\\vec x),\n\\end{equation}\n(with $t,\\vec x$ the time and space coordinates, respectively) \nsuch that the energy-momentum tensor is time independent\n(note that in general, $\\phi(\\vec x)$ is a complex function as well).\nThis model  possesses a conserved current \n\\begin{eqnarray}\nj^{\\mu}= - i \\left( \\Phi^* \\partial^{\\mu} \\Phi \n - \\Phi \\partial^{\\mu}\\Phi ^* \\right)  ,~~~\nj^{\\mu} _{\\ ; \\, \\mu}  =  0 \\ ,\n\\end{eqnarray}\nand a conserved Noether charge $Q$, which is the integral over a $t=const.$ hypersurface\nof the $j^t$ component of the current.\n \nScalar field configurations carrying a global U(1) Noether charge have been extensively studied in the literature, \nfor a Minkowski spacetime background and four spacetime dimensions.\nWhile black hole solutions are rather difficult to find in this case \n\\cite{Pena:1997cy},\n\\cite{Kleihaus:2010ep},\nthe spectrum of  smooth horizonless solutions with harmonic \n time dependence is very rich.\nFor example, one finds (non-topological) soliton solutions even\nin the absence of gravity--the so-called $Q-$balls \n\\cite{Friedberg:1976me},\n\\cite{Coleman:1985ki}, \n\\cite{Lee:1991ax}. \nThere, the scalar field possesses a potential which\nis necessarily non-renormalizable\\footnote{It is interesting to note that $Q-$ball solutions appear in supersymmetric \ngeneralizations of the standard model \n\\cite{Kusenko:1997zq}.\nThey may be responsible for the generation of\nbaryon number or may be regarded as candidates for dark matter  \n\\cite{Kusenko:1997si}. \n}.\nWhen  gravity is coupled to $Q-$balls, boson stars arise \n(see the review work \\cite{Jetzer:1991jr}).\nMoreover,  in this case, solutions with rather similar properties are found even for a\npotential consisting in a mass term only\n\\cite{Kaup:1968zz}, \n\\cite{Ruffini:1969qy}, \n\\cite{Mielke:1980sa}.\nThe self-gravity of such objects is balanced by the dispersive effect\ndue to the wave character of the complex scalar field.\n\n The study of boson stars and Q-balls in \n AdS spacetime has received relatively little attention, only spherically symmetric solutions\n being discussed so far \n \\cite{Astefanesei:2003qy}, \n \\cite{Prikas:2004yw}, \n \\cite{Hartmann:2012wa} \n (see, however, the planar solitons with a complex scalar field in \\cite{Horowitz:2010jq}).\n While the results in \\cite{Astefanesei:2003qy} for a gravitating \n massive scalar field without self-interaction are rather similar to those valid in \n the asymptotically flat limit, \n the recent study \n \\cite{Hartmann:2012wa}\n  has shown the existence of some new features\n in the case of Q-ball solutions. \n\n\nFor a vanishing cosmological constant, $\\Lambda=0$, the scalar solitons admit also \nrotating generalizations \n\\cite{Schunck:1996wa}, \n\\cite{Yoshida:1997qf}, \n\\cite{Volkov:2002aj},\n\\cite{Kleihaus:2005me}, \n\\cite{Kleihaus:2007vk}.\nThese are stationary localized solutions possessing a finite mass and angular momentum.\nInterestingly, their angular momentum is quantized\n$J=n Q$ (with $n$ an integer), \nand the energy density exhibits a toroidal distribution.\n \nThe main purpose of this work is to investigate the existence of spinning scalar solitons \nfor the case of a four dimensional AdS background\\footnote{Rotating AdS boson stars were found \nhowever in $d=3$ \\cite{Astefanesei:2003rw} and also in $d=5$ \\cite{Dias:2011at}\ndimensions, where a special ansatz proposed in \\cite{Hartmann:2010pm} allows to deal with ODEs.}.\nThese solutions are found by solving numerically\na set of partial differential equations with suitable boundary conditions. \nIrrespective of the scalar field potential, they exhibit  the same quantization of the angular momentum\nas for $\\Lambda=0$.\nWe find that the spinning \n  solutions emerge as perturbations of the AdS spacetime\nfor a critical value of the frequency  which is fixed by the scalar field mass and the cosmological constant.\nIn the absence of a scalar field self-interaction, the basic\nproperties of the solutions\nare rather similar to those of the asymptotically flat counterparts.\nNew features are found once we allow for self-interaction terms in the potential\nleading to a violation of the positive energy condition.\nFor example, we find a class of solutions with a smooth $\\omega=0$\nlimit, describing static axially symmetric solitons. \n \n\n\\section{The  model}\n\\subsection{The action and field equations}\n We consider the action of a self-interacting complex scalar field \n$\\Phi$ coupled to Einstein gravity with a negative cosmological constant\n$\\Lambda=-3\/\\ell^2$,\n\\begin{equation}\n\\label{action}\nS=\\int  d^4x \\sqrt{-g}\\left[ \\frac{1}{16\\pi G}(R-2 \\Lambda)\n   -\\frac{1}{2} g^{\\mu\\nu}\\left( \\Phi_{, \\, \\mu}^* \\Phi_{, \\, \\nu} + \\Phi _\n{, \\, \\nu}^* \\Phi _{, \\, \\mu} \\right) - U( \\left| \\Phi \\right|) \n \\right] , \n\\end{equation}\nwhere $R$ is the curvature scalar,\n$G$ is Newton's constant,\nthe asterisk denotes complex conjugation,\nand $U$ denotes the scalar field potential.\n\nVariation of the action with respect to the metric\nleads to the Einstein equations\n\\begin{equation}\n\\label{Einstein-eqs}\nE_{\\mu\\nu}= R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R+\\Lambda g_{\\mu\\nu} - 8 \\pi G T_{\\mu\\nu}=0\\ , \n\\end{equation}  \nwhere $T_{\\mu\\nu}$ is the\n stress-energy tensor  of the scalar field\n\\begin{eqnarray}\n\\label{tmunu} \nT_{\\mu \\nu}  \n&=&\n\\left(\n \\Phi_{, \\, \\mu}^*\\Phi_{, \\, \\nu}\n+\\Phi_{, \\, \\nu}^*\\Phi_{, \\, \\mu} \n\\right )\n-g_{\\mu\\nu} \\left[ \\frac{1}{2} g^{\\alpha\\beta} \n\\left( \\Phi_{, \\, \\alpha}^*\\Phi_{, \\, \\beta}+\n\\Phi_{, \\, \\beta}^*\\Phi_{, \\, \\alpha} \\right)+U(\\left|\\Phi\\right|)\\right]\n \\ .\n\\end{eqnarray}\nVariation with respect to the scalar field\nleads to the matter field equation,\n\\begin{eqnarray}\n\\label{scalar-eq}\n\\frac{1}{\\sqrt{-g}} \\partial_\\mu \\big(\\sqrt{-g} \\partial^\\mu\\Phi \\big)=\\frac{\\partial U}{\\partial\\left|\\Phi\\right|^2} \\Phi.\n\\end{eqnarray} \nThe potential $U$ can be decomposed according to\n\\begin{eqnarray}\n\\label{pot1}\nU(|\\Phi|)= \\mu^2 |\\Phi|^2 +U_{int}(|\\Phi|),\n\\end{eqnarray}\n where $\\mu$ is the mass of the field, and $U_{int}$ is a self-interaction potential.\nAs discussed in \\cite{Astefanesei:2003qy} for $\\mu^2>0$, this model possesses finite mass solutions  \neven in the absence of self-interaction,\n$U_{int}=0$, the so-called 'mini-boson stars'.\n\nHowever, the inclusion\nof an interaction potential may lead\nto a more complex picture (see $e.g.$ \\cite{Colpi:1986ye} for the $\\Lambda=0$ case). \nOur choice of  $U_{int}$ was guided by the requirement that nontopological solitons\nexist also in a fixed AdS background.\nMoreover, we are interested in the case when the solutions would possess a nontrivial static limit. \nAs discussed in \\cite{Hertog:2004rz}, this \nrequires the occurrence of negative energy densities,\n $i.e.$ $U(|\\Phi|)<0$\nin some region.\nSince, in order \nto make contact with the previous work on mini-boson stars, we restrict the numerical part of our\nstudy to the case $\\mu^2>0$, this implies that  $U_{int}$ is not strictly positive. \n\nWe have found that the simplest choice of the interaction potential \nsatisfying these conditions is $U_{int}=-\\lambda|\\Phi|^{2k}$, with $k>1$ and $\\lambda>0$.\nMost of the  results in this work are found\\footnote{Note that the  exact solution with spherical symmetry\n in Section 3 covers a more general range of $k$.} for $k=2$,\nsuch that \n\\begin{eqnarray}\n\\label{U}\n U(|\\Phi|)=\\mu^2 |\\Phi|^2  -\\lambda |\\Phi|^{4}.\n\\end{eqnarray} \nAlthough the action (\\ref{action}) together with (\\ref{U}) does not seem to correspond to any supergravity \nmodel, it is likely that some features of its solutions are generic.\n In particular, we have found the same general picture \n for a more general potential (which was used in the previous studies \n \\cite{Volkov:2002aj},\n\\cite{Kleihaus:2005me}, \n\\cite{Kleihaus:2007vk}\n on $\\Lambda=0$ spinning Q-balls and boson stars)\n \\begin{eqnarray}\n\\label{Un}\n U(|\\Phi|)=\\mu^2 |\\Phi|^2 -\\lambda |\\Phi|^{4}+\\nu |\\Phi|^{6},\n\\end{eqnarray} \n\n provided that\n the new coupling constant $\\nu>0$ is  small enough.\n \n \n\n\\subsection{The Ansatz }\n\nWe are interested in stationary axially symmetric configurations, \nwith a spacetime geometry admiting two Killing vectors \n$\\partial_t$\nand \n$\\partial_{\\varphi}$,\nin a system of adapted coordinates $\\{t, r, \\theta, \\varphi\\}$.\nThus the line element can be written as\n\\begin{eqnarray}\n\\label{ansatzg}\nds^2 =- F_0 N dt^2 \n+ F_1 \\left( \\frac{dr^2}{N} + r^2 \\, d\\t^2 \\right) \n  + F_2 r^2 \\sin^2 \\t  \\left( d \\varphi\n- \\frac{W}{r}   dt \\right)^2   . \n\\end{eqnarray}\nThe  metric functions $F_0$, $F_1$, $F_2$ and $W$\ndepend on the variables $r$ and $\\theta$ only, while\n\\begin{eqnarray}\nN=1+\\frac{r^2}{\\ell^2}\n\\end{eqnarray}\nis a suitable 'background' function.\n \n\nFor the scalar field $\\Phi$ we adopt the stationary ansatz \n\\begin{eqnarray}\n\\label{ansatzp}\n\\Phi (t,r,\\t, \\varphi)= \\phi (r, \\t)\n e^{ i( n \\varphi-\\omega  t )}   , \n\\end{eqnarray}\nwhere $\\phi (r, \\theta)$ is a real function,\nand $\\omega $ and $n$ are real constants.\nSingle-valuedness of the scalar field requires\n$\\Phi(\\varphi)=\\Phi(2\\pi + \\varphi)$;\nthus the constant $n$ must be an integer,\n$i.e.$, $n \\, = \\, 0, \\, \\pm 1, \\, \\pm 2, \\, \\dots$~.\nIn what follows, we shall take $n\\geq 0$ and $\\omega \\geq 0$,\nwithout any loss of generality. \n\n\n The spherically symmetric limit is found for $n=0$, in which case the\n functions $F_0,F_1,F_2$ and $\\phi$\n depend  only on $r$, with $F_1=F_2$ and $W=0$. \n \n\\subsection{The boundary conditions}\nThe solutions in this work describe horizonless, particle-like configurations. \nA study of an approximate form of the solutions as a power series around $r=0$ leads to the following\n boundary conditions at the origin\\footnote{For spherically\nsymmetric solutions, the scalar field is nonvanishing ar $r=0$.}:\n\\begin{eqnarray}\n\\label{bc0} \n\\partial_r F_i|_{r=0}=0, ~~\nW|_{r=0}=0,~~\n\\phi| _{r =0}=0~,\n\\end{eqnarray}\n (with $i=0,1,2$).\nAt infinity, the AdS background is approached, while the scalar field vanishes.\nWithout any loss of generality,\nwe are choosing a frame in which the solutions do not rotate at infinity,\nthe conformal boundary being a static Einstein universe $R\\times S^2$.\nThe boundary conditions compatible with these requirement are\n\\begin{eqnarray}\n\\label{bcinf} \nF_i|_{r \\rightarrow \\infty} =1,~~\nW|_{r \\rightarrow \\infty} =0, ~~\n\\phi| _{r \\rightarrow \\infty}=0 \\ .\n\\end{eqnarray}\nFor $\\t=0,\\pi$  \nwe require the boundary conditions\n\\begin{eqnarray}\n\\label{bct0} \n\\partial_{\\t} F_i|_{\\t=0,\\pi}=0, ~~\n\\partial_{\\t} W |_{\\t=0,\\pi}=0,~~\n\\f |_{\\t=0,\\pi}=0.\n\\end{eqnarray}\n The absence of conical singularities\n  imposes on the symmetry axis the supplementary condition \n$F_1|_{\\theta=0,\\pi}=F_2|_{\\theta=0,\\pi},$\nwhich is  used to verify the accuracy of the solutions.\n\nAlso, all solutions in this work \nare invariant under the parity transformation $\\theta \\to\\pi-\\theta $.\nWe make use of this symmetry to integrate the equations for $0\\leq \\theta\\leq \\pi\/2$ only, the\nfollowing boundary conditions being imposed in the equatorial plane\n\\begin{eqnarray}\n\\label{bctpi2} \n\\partial_{\\t} F_i|_{\\t=\\pi\/2}=0 ,~~\n\\partial_{\\t} W |_{\\t=\\pi\/2}=0 \\ ,~~\n\\partial_{\\t} \\phi |_{\\t=\\pi\/2}=0 \\ .\n \\end{eqnarray} \n\n\n\\subsection{The far field asymptotics and global charges}\nFor solutions\nwith $\\mu^2>0$ (the only case considered in the numerics),\nthe scalar field decays asymptotically as \n\\begin{eqnarray}\n\\label{asym-scalar0}\n\\phi\\sim \n\\frac{c_1(\\theta)}{r^{\\frac{3}{2}\\left( 1+\\sqrt{1+\\frac{4}{9}\\mu^2 \\ell^2} \\right)}}+\\dots~.\n\\end{eqnarray}\nThe physical interpretation of $c_1(\\theta)$\nis that it corresponds, up to a normalization,\nto the expectation value of some scalar operator in the dual theory.\n\nWithout entering into details, \nwe mention that the picture is more complicated \\cite{Henneaux:2006hk} if one allows for a tachyonic\nmass of the scalar field, $\\mu^2<0$.\nFor $-9\/4< \\mu^2\\ell^2<-5\/4$,\nthe general asymptotic behaviour of the scalar field \nis more complicated\\footnote{For\na field which saturates the Breitenlohner-Freedman bound $\\mu^2\\ell^2=-9\/4$,\none finds $\\Delta_+=\\Delta_-=\\Delta$, and the second solution asymptotically behaves \nlike $\\log r\/r^\\Delta$.\n},  with the existence of a second mode apart from (\\ref{asym-scalar0}): \n\\begin{eqnarray}\n\\label{asym-scalar}\n\\phi\\sim \n\\frac{c_1(\\theta)}{r^{\\Delta_+}}+\n\\frac{c_2(\\theta)}{r^{\\Delta_-}},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Deltapm}\n\\Delta_{\\pm}=\\frac{3}{2}\\left( 1\\pm \\sqrt{1+\\frac{4}{9}\\mu^2 \\ell^2} \\right). \n\\end{eqnarray}\nFor this range of $\\mu^2<0$, both modes above are normalizable in the sense that the spatial integral of $j^t$ is finite, $i.e.$\nthe scalar field possesses a finite Noether charge.\nTo have a well defined theory,\none must specify a boundary condition at infinity, $i.e.$\nto choose a relation between $c_1$ and $c_2$, the standard choice being $c_2=0$. \nHowever, as discussed $e.g.$ in  \\cite{Hertog:2004dr}, \\cite{Henneaux:2004zi},\nthe solutions with a slower decay at infinity, $c_2 \\neq 0$,\nare also physically acceptable, the AdS charges involving in this\ncase a scalar field contribution (thus the \nexpression (\\ref{EJ}) below would not be valid).\n\n \nRestricting to the case $\\mu^2>0$, the scalar field decays asymptotically faster than $1\/r^3$ and\nthus the Einstein equations (\\ref{Einstein-eqs}) imply the following form of the metric functions\nas $r\\to \\infty$\n \\begin{eqnarray}\n\\nonumber\n&&F_0=1+ \\frac{f_{03}(\\theta)}{r^3}+O(1\/r^5),\n~~~F_1=1+ \\frac{f_{13}(\\theta)}{r^3}+O(1\/r^5), \n\\\\\n\\label{asym1}\n&&F_2=1+ \\frac{f_{23}(\\theta)}{r^3}+O(1\/r^5),\n~~W=\\frac{w_2(\\theta)}{r^2}+O(1\/r^4),\n\\end{eqnarray}\nin terms of two functions $f_{13}(\\theta)$ and $w_2(\\theta)$ which result from the numerics, \nwith \n\\begin{eqnarray}\nf_{03}(\\theta)=-3 f_{13}(\\theta)-\\frac{4}{3}\\tan\\theta f_{13}'(\\theta),~~{\\rm and}~~\nf_{23}(\\theta)=  f_{13}(\\theta)+\\frac{4}{3}\\tan\\theta f_{13}'(\\theta).\n\\end{eqnarray} \n\nA straightforward computation based on the formalism in \\cite{Balasubramanian:1999re} \nleads to the following expression for \nthe mass-energy $E$ and angular momentum $J$ of the  configurations\n\\begin{eqnarray}\n\\label{EJ}\nE= \\frac{1}{8 G \\ell^2}\\int_{0}^\\pi d\\theta \\sin\\theta \\bigg(5 f_{13}(\\theta)+3 f_{23}(\\theta)\\bigg),~~\nJ=-\\frac{3}{8 G }\\int_{0}^\\pi d\\theta \\sin^3\\theta ~w_2(\\theta) .\n\\end{eqnarray} \n(Note that the same result can be derived by using\n the Ashtekar-Magnon-Das conformal mass definition \\cite{Ash}).\n Moreover, \nthe same expression for the angular momentum is found from the Komar integral:\n\\begin{eqnarray}\n\\label{j1}\n &J=\\frac{1}{8 \\pi G }\\int R_{\\varphi}^t \\sqrt{-g} dr  d\\theta d\\varphi\n =\n\n\n\n  -\\frac{1}{8 G }\\int_{0}^{\\infty}dr \\int_0^\\pi d \\theta\n \\bigg [\n \\bigg(r^4 \\sqrt{\\frac{F_2^4}{F_0}}\\sin^3 \\theta   \\big(\\frac{W}{r} \\big)_{,r} \\bigg)_{,r}\n + \\bigg(\\frac{r}{N} \\sqrt{\\frac{F_2^3}{F_0}}\\sin^3 \\theta   W_{,\\theta} \\bigg)_{,\\theta}\n \\bigg ].~{~~}\n\\end{eqnarray} \nThe solutions possess also a conserved Noether charge ($i.e.$ the total\nparticle number)\n\\begin{eqnarray}\n\\label{Q1}\nQ= \\int j^t \\sqrt{-g} dr  d\\theta d\\varphi=\n2 \\pi \\int_0^\\infty dr \\int_0^\\pi d\\theta~\n2r \\sin \\theta F_1\\sqrt{\\frac{F_2}{F_0}}\n\\frac{ \\phi^2}{N}(\\omega r-n W).\n\\end{eqnarray} \nSince $T_\\varphi^t=n j^t$, from (\\ref{j1}) \nand the Einstein equation $R_{\\varphi}^t=8\\pi G T_{\\varphi}^t$\nwe find that the generic relation\n\\begin{eqnarray}\n\\label{JQ}\nJ= n Q~,\n\\end{eqnarray} \n(which was proven in \\cite{Schunck:1996wa}\n for asymptotically flat spinning solutions)\nholds also in the AdS case.\nAs a result, the spinning solitons\ndo not emerge as perturbations\nof the spherically symmetric\nconfigurations, $i.e.$\nthere are no slowly rotating\nsolutions in this model (note also that the expression of the scalar field\npotential was not used in the derivation of (\\ref{JQ})). \n\nThese scalar solitons have no horizon and therefore they are zero entropy objects,\nwithout an intrinsic temperature.\nThe first law of thermodynamics\nreads   in this case  \\cite{Lee:1991ax}\n$dE= \\omega dQ=\\frac{\\omega }{n} dJ$.\n\n\\subsection{The numerical scheme}\nThere are not  many studies on numerical \nsolutions of Einstein gravity with negative cosmological constant\ncoupled with  matter fields,\n describing stationary, axially symmetric configurations.\nThe model in this work provides perhaps \nthe simplest test ground for \ninvestigating various approaches and developing numerical techniques for elliptic \nproblems with AdS asymptotics\\footnote{Because the ansatz (\\ref{ansatzp}) has an explicit dependence \n on both $\\varphi$ and $t$,\n the scalar field is neither static nor axisymmetric.\n However, all physical quantities, such as the current $j^\\mu$\n and the energy-momentum tensor $T_{\\mu\\nu}$,\n will exhibit no dependence on  $\\varphi$ and $t$.}.\n\n\n\nThe solutions of the field equations (\\ref{Einstein-eqs}), (\\ref{scalar-eq}) \nare found  by using an approach originally proposed in \\cite{Kleihaus:1996vi}\nfor $d=4$ asymptotically flat solutions of Einstein \ngravity coupled with Yang-Mills gauge fields. \nThe equations for the metric functions  $ \\mathcal F=(F_0,F_1,F_2,W)$ \n employed in the numerics,\nare found by using a  suitable combination of the Einstein equations (\\ref{Einstein-eqs}),\n$E_t^t =0,~E_r^r+E_\\theta^\\theta =0$,  $E_\\varphi^\\varphi=0$\nand  $E_{\\varphi}^{t} =0$,\n which diagonalizes them $w.r.t.$ $\\nabla^2 \\mathcal F$ \n (where $\\nabla^2=\\partial_{rr}+\\frac{1}{r}\\partial_{r}+\\frac{1}{r^2 N}\\partial_{\\theta\\theta}$).\n An important issue here concerns the status of the \nremaining equations $E_\\theta^r =0,~E_r^r-E_\\theta^\\theta  =0$,\n which\nyield two constraints. \nFollowing \\cite{Wiseman:2002zc}, one can show that\nthe identities $\\nabla_\\mu E^{\\mu r} =0$ and $\\nabla_\\mu E^{\\mu \\theta}=0$, \nimply the Cauchy-Riemann relations\n$\n\\partial_{\\bar r} {\\cal P}_2  +\n\\partial_\\theta {\\cal P}_1  \n= 0 ,~~\n \\partial_{\\bar r} {\\cal P}_1  \n-\\partial_{\\theta} {\\cal P}_2\n~= 0 ,\n$\nwith ${\\cal P}_1=\\sqrt{-g} E^r_\\theta$, ${\\cal P}_2=\\sqrt{-g}r \\sqrt{N}(E^r_r-E^\\theta_\\theta)\/2$\nand $d\\bar r=\\frac{dr}{r \\sqrt{N}}$.\nTherefore the weighted constraints still satisfy Laplace equations, and the constraints \nare fulfilled, when one of them is satisfied on the boundary and the other \nat a single point\n\\cite{Wiseman:2002zc}. \nFrom the boundary  conditions (\\ref{bc0})-(\\ref{bct0}) we are imposing,\nit turns out that this is the case for our solutions,\n $i.e.$ the numerical scheme is consistent.\n \nTo obtain spinning boson star solutions,\nwe solve  numerically the set of five coupled non-linear\nelliptic partial differential equations for $(\\mathcal F,\\phi)$,\nsubject to the  boundary conditions (\\ref{bc0})-(\\ref{bct0}).\nWe employ a compactified radial coordinate $\\bar r=  r\/(1+ r)$\n which maps spatial infinity to the finite value $\\bar r=1$.\n Then the equations are discretized on a non-equidistant grid in\n$\\bar r$ and $\\theta$.\nTypical grids used have sizes $250 \\times 30$,\ncovering the integration region\n$0\\leq \\bar r \\leq 1$ and $0\\leq \\bar \\theta \\leq \\pi\/2$.\n(See \\cite{Kleihaus:1996vi} and \\cite{schoen}  \nfor further details and examples for the numerical procedure.) \nThe numerical calculations are based on the Newton-Raphson method\nand are performed with help of the  software package FIDISOL \\cite{schoen},\nwhich provides also an error estimate for each unknown function.\nThe typical relative error for the solutions \nin this work is smaller that\n$10^{-3}$.\n \n\\section{The solutions in the probe limit}\n\nWe shall start with a discussion of the solutions in a fixed AdS\nbackground, $i.e.$ \nwithout backreaction, $F_0=F_1=F_2=1,~W=0$ in the metric ansatz (\\ref{ansatzg}).\nThe problem is much easier to study in this limit and\nthe solutions exhibit already some basic features of\nthe gravitating configurations.\n\nThe usual Derick-type scaling argument (see $e.g.$\nthe discussion in \\cite{Radu:2008pp} \nfor the $\\Lambda=0$ limit) implies that\nthe Q-balls in a fixed AdS background satisfy\nthe following virial identity\n\\begin{eqnarray}\n\\label{virial}\n\\int_0^\\infty dr \\int_0^\\pi  d\\theta \\sin \\theta\n\\bigg [\nN\\phi_{,r}^2\n+\\frac{\\phi_{,\\theta}^2}{r^2}\n+\\frac{n^2\\phi^2}{r^2\\sin^2\\theta}\n+3\\left(U(\\phi)-\\frac{\\omega^2\\phi^2}{N}\\right)\n+\\frac{2r}{\\ell^2}\\left(\\phi_{,r}^2+\\frac{ \\omega^2 \\phi^2}{N^2}\\right)\n\\bigg]=0,\n\\end{eqnarray}\nwhich was used as a further test of the numerical accuracy.\nIt is clear that the \nsolutions with a strictly positive potential, $U(\\phi)>0$,\nowe their existence to the harmonic time dependence of the scalar field.\nAlso, the solutions may exist in the $ \\omega\\to 0$ limit \nas long as $U(\\phi)$ is allowed to take negative values.\n\nAs usual in the absence of back reaction, \nthe total mass-energy $E$ and angular momentum of the configurations\nare found by integrating over the entire space\nthe components $-T_t^t$ and $T_\\varphi^t$ of the energy \nmomentum tensor.\n\n\n\n\\subsection{Spherically symmetric configurations.\nAn exact solution}\n\nIn the spherically symmetric limit, the equation (\\ref{scalar-eq}) with a self-interaction\npotential $U_{int}=-\\lambda \\phi^{2k}$\nadmits the following simple exact solution, which to our knowledge,\nwas not yet discussed in the literature: \n\\begin{eqnarray}\n\\label{ex-sol}\n\\Phi(r,t)=\\left(\n\\frac{\\mu^2}{\\lambda}\\frac{\\Delta^2-\\ell^2  \\omega^2}{(\\Delta-3)(\\Delta+1))}\n\\right)^{\\frac{\\Delta}{2}} {\\left(1+\\frac{r^2}{\\ell^2} \\right)^{-\\frac{\\Delta}{2}}} e^{-i  \\omega t}, \n\\end{eqnarray}\nwith $\\Delta=\\Delta_{\\pm}$, as  given by (\\ref{Deltapm}).\n \n \nFor this exact solution, \nthe coefficient $k$ of the self-interaction term \nin the scalar field potential (\\ref{U}) is fixed by $\\Delta$,\n\\begin{eqnarray}\n\\label{k}\nk=1+\\frac{1}{\\Delta}.\n\\end{eqnarray}\nThis is a one parameter family of solutions which, for given input parameters $\\mu^2,\\lambda$ and $\\ell$,\ncan be parametrized by the frequency of the field.\nChoosing $\\Delta=\\Delta_{+}$ leads to a range $1<k<5\/3$ for the parameter $k$ in the self-interaction potential, \nthe solutions possessing in this case a finite mass-energy\nand Noether charge:\n\\begin{eqnarray}\n&&\nE= V_2 \\sqrt{\\pi} \\ell \\left(\\frac{\\mu^2}{\\lambda} \\right)^\\Delta \n\\frac{\\Gamma(\\Delta-\\frac{1}{2})}{\\Gamma(\\Delta+2)}\n\\left( \\frac{\\Delta^2-\\ell^2 \\omega^2}{(\\Delta-3)(\\Delta+1)} \\right)^\\Delta\n\\left( \\Delta^2+ \\omega^2\\ell^2(2\\Delta+1) \\right),\n\\\\\n\\nonumber\n&&\nQ=V_2\\sqrt{\\pi}\\frac{\\ell^3  \\omega}{2}\\left(\\frac{\\mu^2}{\\lambda} \\right)^\\Delta \n\\frac{\\Gamma(\\Delta-\\frac{1}{2})}{\\Gamma(\\Delta+1)}\n\\left( \\frac{\\Delta^2-\\ell^2  \\omega^2}{(\\Delta-3)(\\Delta+1)} \\right)^\\Delta~, \n\\end{eqnarray}\n with $V_2=4\\pi$ the area of two-sphere. \n\n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=ex-sol1.eps,width=8cm}}\n\\put(8,0.0){\\epsfig{file=ex-sol2.eps,width=8cm}}\n\\end{picture}\n\\\\\n\\\\\n{\\small {\\bf Figure 1.} The mass-energy and Noether charge of the \nexact solution (\\ref{ex-sol}) with a $\\phi^3$-self interaction potential are shown as a function of\nfrequency.\n   }\n\\vspace{0.5cm}\n\n\n \\\\\n \n  An interesting case here corresponds to $k=3\/2$, $i.e.$ \nan $(\\Phi^*\\Phi)^{3\/2}$ interaction term\\footnote{A $U(1)$-symmetric\ncubic interaction $(\\Phi^*\\Phi)^{3\/2}$ in the potential\nis not unusual in the Q-ball literature, see $e.g$\n\\cite{Kusenko:1997ad}, \\cite{Tamaki:2010zz}, such terms \noccurring in the minimal supersymmetric standard model.}\nin the  potential (\\ref{U}).\nThis case is realized for a scalar field mass $\\mu^2=-{2}\/{\\ell^2}$  and a choice $\\Delta=\\Delta_+=2$.\nThe total mass-energy and Noether charge of this solution\nare given by \n\\begin{eqnarray}\nE= V_2 \n\\frac{\\pi}{108\\lambda^2\\ell^3}(4-\\ell^2  \\omega^2)^2\n(4+5\\ell^2  \\omega^2),~~\nQ= V_2 \\frac{\\pi  \\omega (4-\\ell^2  \\omega^2)^2}{18 \\lambda^2 \\ell},\n\\end{eqnarray}\nbeing shown in Figure 1 as a function of the frequency.\n \n Remarkably, the  picture in Figure 1a \n has been recovered for most of the numerical solutions with a self-interaction potential and $\\mu^2>0$\nin this work (including the spinning ones with $n\\neq 0$).\n The \noccurrence of an extra-branch of solutions with $ \\omega> \\omega_c$ (see Figure 1b)\nis a feature of the $\\phi^3$ potential.\n(Note that this branch does not appear for the exact solution with a $\\phi^4$ potential.) \n\n\nFor the same value of the scalar field mass, when choosing instead\n$\\Delta=\\Delta_-=1$ in (\\ref{ex-sol}), (\\ref{k}), one recovers the model with\na $\\phi^4$ potential.   \nUnfortunately, the total mass-energy of these solutions, as defined\nin the usual way as the integral of $T_t^t$,\ndiverges linearly\\footnote{However, the mass can be regularized\nby supplementing the action with a suitable scalar field boundary counterterm.},\n$E=E_{div}+E_0$, with $E_{div}=- V_2\\frac{ (1-\\ell^2  \\omega^2) }{2\\lambda \\ell}r_c$ \n(where $r_c\\to \\infty$), \nwhile the charge is finite:\n\\begin{eqnarray}\n\\label{phi4}\nE_0=V_2\\frac{\\pi (1-\\ell^2  \\omega^2)(1+3 \\ell^2  \\omega^2)}{16\\lambda \\ell},~~\nQ=V_2\\frac{\\pi \\ell  \\omega(1- \\omega^2 \\ell^2)}{4 \\lambda}.\n\\end{eqnarray} \n \n\nSolutions beyond the framework (\\ref{ex-sol}), (\\ref{k})\nare found by using a numerical approach,\nfor a generic ansatz $\\Phi = \\phi (r) e^{-i  \\omega t} $. \nHere,  for  simplicity we restrict ourselves to the case of nodeless solutions.\nThen, for $\\mu^2>0$ and a self-interaction\npotential $U_{int}=-\\lambda \\phi^{4}$,\nit turns out that the picture in Figure 1a is generic.\nFor any $\\Lambda$, the solutions exist for  a limited range of frequencies,\n$0\\leq  \\omega< \\omega_c= \\Delta_+\/\\ell$.\nThe mass-energy and Noether charge are bounded and approach a maximum\nfor some $ \\omega$ around $ \\omega_c\/2$.\nThe same pattern is recovered  \n in the presence of an extra $\\nu \\phi^6$ self-interaction term (with $\\nu>0$), provided that\n the new coupling constant is small enough.\n \n \n \n\n\\subsection{Spinning scalar solitons in an AdS background}\n\nThe spinning generalizations of these solutions are found by taking $n\\neq 0$\nin the general ansatz (\\ref{ansatzp}).\nFor a $\\phi^4$ self-interaction potential,\napart from the winding number $n$,\nthe input parameters are $\\ell$, $ \\omega$, $\\mu$ and $\\lambda$.\n\\\\\n \n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=var-cosm.eps,width=8cm}}\n\\put(8,0.0){\\epsfig{file=fi6.eps,width=8cm}}\n\\end{picture}\n\\\\\n\\\\\n{\\small {\\bf Figure 2.} {\\it Left:} The mass-energy $E$ and angular momentum $J$\n of $n=1$ spinning  nongravitating Q-ball solutions \n are shown as a function of the cosmological constant $\\Lambda$ for two values of\n the frequency $ \\omega$.\n  {\\it Right:}\n  The mass-energy $E$  \n of $n=1$ spinning  nongravitating Q-ball solutions with a potential \n $U(\\phi)=\\mu^2 \\phi^2-\\lambda \\phi^4+\\nu \\phi^6$\n are shown as a function of the frequency for several values of the coefficient of the\n $\\phi^6$ term.\n   }\n\\vspace{0.5cm}\n\n \\\\\nHowever, the model admits the scaling symmetry\n$r \\to r\/\\mu$, $ \\omega\\to  \\omega \\mu$, $\\ell \\to \\ell\/\\mu$,\ntogether with the scalar field redefinition $\\phi\\to \\phi\/c$, $\\lambda\\to \\lambda c^2$,\nwhich allows us to set $\\mu=\\lambda=1$ without any loss of generality.\n\nThe following picture appears to be generic for the numerical solutions in this work:\nfirst, no solutions exist \nfor frequencies above a critical value $ \\omega= \\omega_c$.\nAs $\\omega\\to \\omega_c$, the solution emerges \n as a perturbation around the ground state $\\phi=0$, with\n\\begin{eqnarray}\n\\label{pert2}\n\\delta \\phi(r,\\theta) \\simeq \\frac{(r \\sin\\theta)^n}{\\left(1+\\frac{r^2}{\\ell^2}\\right)^{\\frac{1}{2} \\omega_c \\ell}} \n\\end{eqnarray}\nbeing the regular solution of the linearized Klein-Gordon equation in a fixed AdS background\n(here we restrict our discussion to nodeless configurations).\nThe critical frequency is given by\n\\begin{eqnarray}\n\\label{pert1}\n  \\omega_{c}=\\frac{n+\\Delta_{+}}{\\ell},\n\\end{eqnarray}\nbeing found by requiring the perturbation $\\delta \\phi(r,\\theta)$\nto be regular at both $r\\to 0$ and $r\\to \\infty$. \nThis upper bound on the frequency\nis universal, \nand holds also in the presence of gravity. \n\n\n\nWhen decreasing the frequency, the nonlinear term in $\\phi$\nstarts to contribute\n and the mass-energy and the angular momentum of the \nsolitons start to increase.\nBoth $E$ and $J$ approach a maximum at some intermediate value of $ \\omega$,\ndecreasing afterwards.\nAs $ \\omega\\to 0$, \nthe solutions describe finite mass, static, axially symmetric (for $n\\neq 0$) solitons, \nthough\nwith a vanishing Noether charge.\n\n\nThe dependence of the solutions on the cosmological constant is shown in Figure 2 (left).\nOne can see that the pattern depends on the value of the frequency.\nFor $\\omega>\\mu$, the solutions stop to exist for a maximal value of the cosmological constant,\n $\\Lambda=-\\frac{4(\\mu^2-\\omega^2)^2}{ 3( \\omega(1+2n\/3)+\\sqrt{ \\omega^2+4\\mu^2n(n+3)\/9})^2}$ \n (which results from (\\ref{pert1})),\n where both $E$ and $Q$ vanish  in that limit.\nFor $\\omega<\\mu$,\n one finds solutions for all values of cosmological constant, including $\\Lambda=0$.\n \n \n \n  We have also constructed AdS generalizations of the flat spacetime Q-balls\n with the usual potential choice (\\ref{Un}).\nAs one can see in Figure 2 (right), the picture found\nfor the $\\phi^4$-model is recovered for small enough values of  $\\nu$.\nHowever, for $\\nu$ above a critical value (around $0.67$ for those parameters),  the solutions exist\nonly for $  \\omega_{min}< \\omega< \\omega_c$ (with $\\omega_{min}>0$).\nSimilar to the flat space solutions \\cite{Kleihaus:2005me}, both the \nmass-energy and Noether charge diverge\nas $ \\omega\\to  \\omega_{min}$.\n\n \n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=w-BS.eps,width=8cm}}\n\\put(8,0.0){\\epsfig{file=w-Q.eps,width=8cm}}\n\\end{picture}\n\\\\\n\\\\\n{\\small {\\bf Figure 3.} \nThe mass-energy $E$ and the angular momentum $J$ are shown as a function of the frequency\n$\\omega$\nfor (mini-)boson star solutions without a self-interaction potential (left)\nand for gravitating Q-balls with a $\\phi^4$ interaction potential (right).\n   }\n\\vspace{0.1cm}\n\n \n \n\\section{Gravitating spinning scalar soliton}\n\nWe have found that all Q-balls in a fixed AdS background \nallow for gravitating generalizations.\nMoreover, nontrivial solutions are found in this case\neven in the absence of a self-interaction term in the scalar potential (\\ref{U}),\ngeneralizing for $J\\neq 0$ the static AdS\nboson stars in \\cite{Astefanesei:2003qy}.\n\nLet us start with a brief discussion of those configurations with  \n  $U=\\mu^2 \\phi^2$ \n(usually called\nmini-boson stars in the literature), \nwhich play an important role in the limiting behaviour of \nthe configurations with a $\\phi^4$ term in the potential.\nRestricting again to a real mass of the scalar field, $\\mu^2>0$, the usual \nrescaling $r\\to r\\mu$ implies that the model depends on two\ndimensionless parameters, $\\ell \\mu$ and $ \\omega\/\\mu$, only, the factor $1\/\\sqrt{8\\pi G}$\nbeing absorbed in $\\phi$.  \nOur results show that similar to the spherically symmetric case,\nthe spinning solutions exist for a limited range of frequencies\n$0< \\omega_{min}< \\omega< \\omega_c$,  emerging\nas a perturbation of AdS spacetime \nfor a critical frequency $ \\omega_c$ as given by (\\ref{pert1}). \nThus, as expected, the value $ \\omega=0$ is not approached in the absence of a \nscalar field self-interaction\\footnote{Finding\nAdS\nstatic solitons requires a violation of the energy conditions \\cite{Hertog:2004rz}, which is not the case for\nthe boson stars  with $U(|\\Phi|)=\\mu^2 |\\Phi|^2>0$.}.\nAs $ \\omega\\to  \\omega_{min}$,\na backbending towards larger values of $ \\omega$ is observed, see Figure 3 (left).\nWe conjecture that, similar to the spherically symmetric case, \nthis backbending  would lead to an inspiraling of the solutions\ntowards a limiting configuration with $ \\omega_0> \\omega_{min}$.\nNote also that the mass-energy and the angular momentum of these mini-boson star solutions stay finite\nin the allowed range of frequencies.\n\n\nThe picture is more complicated for solitons\nwith a self-interaction term in the potential.\nFor simplicity, we shall restrict\\footnote{However, we have also constructed  \n gravitating solutions with a $\\phi^6$-term in the potential,\nin which case we did not find new qualitative features.} our discussion to the case of \na $\\phi^4$ potential (\\ref{U}).\nWithout any loss of generality,\n one can set  $\\mu=\\lambda=1$ for the two parameters in (\\ref{U}).\nThis choice is achieved by using the rescaling \n$r \\to  r\/\\mu$, $\\ell \\to  \\ell\/\\mu  $, $\\omega \\to \\omega \\mu$, \ntogether with a redefinition of the scalar field\n$\\phi \\to \\phi \\mu\/\\sqrt{\\lambda}$.\nThis  reveals the existence of a dimensionless parameter $\\alpha^2=4 \\pi G \\mu^2\/\\lambda$,\nsuch that the\nEinstein equations read $R_{ij}-\\frac{1}{2}g_{ij}R+\\Lambda g_{ij}=2\\alpha^2 T_{ij}$,\nwith $\\alpha=0$ corresponding \nto the probe limit discussed in Section 3.\n \n  Starting with the dependence of the solutions on the frequency, we have found \nthat for a given $\\Lambda$, this is fixed by the parameter $\\alpha$.\n The spinning solutions with large  enough values of $\\alpha$  exhibit the same pattern as in the \nabsence of a self-interaction term \n(although with a smaller value of  $ \\omega_{min}$),\nand the picture in Figure 3 (left) is recovered \n(the same result  was found also for $\\Lambda=0$ solutions \\cite{Kleihaus:2005me}).\n However, for values of $\\alpha$ below a critical value ($i.e.$ for large enough $\\lambda$)\nthe picture for $Q-$balls in an AdS background is recovered. \n  \n \n \\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=Ttt.eps,width=8cm}}\n\\put(8.1,0.0){\\epsfig{file=var-alfa.eps,width=8cm}}\n\\end{picture}\n\\\\\n{\\small {\\bf Figure 4.} \n {\\it Left:}\n The energy density of a static axially symmetric gravitating soliton\n with $ \\omega=0$, $n=2$\n  is shown for several different angles as a function of the radial coordinate.\n{\\it Right:} \n The mass-energy $E$ and the angular momentum $J$ are shown as a function of the  coupling constant\n$\\alpha=\\sqrt{4\\pi G \\mu^2\/\\lambda}$\nfor boson star solutions \nwith a $\\phi^4$ self interaction potential for two different frequencies.\n}\n\\vspace{0.5cm}\n\\\\\n  Thus, for any $n$, the solutions exist for a  range of frequencies $0\\leq \\omega< \\omega_c$, see Figure 3 (right).\nIn both cases, the configurations with  small $E$, $J$ are just \nperturbations of AdS.\nThese solitons branch off from the\nAdS spacetime for the specific value of the frequency given by (\\ref{pert1}).\n\n\n\nHowever, different from the case discussed above with $U(|\\Phi|)=\\mu^2 |\\Phi|^2$,\na self-interaction term in the potential leading to negative energy densities,\nallows for a nontrivial limiting solution with $\\omega=0$.\nThis is a new type of soliton, which is different from other solutions with\ngravitating scalar fields in the literature \n\\cite{Hertog:2004rz}, \n\\cite{Hertog:2004ns}, \n\\cite{Radu:2005bp}.\nAlthough it is static ($\\partial\/\\partial t$ being a Killing vector of the configuration), \nthe geometry has axial symmetry only, being\n regular everywhere, in particular at $r=0$ and on the symmetry axis.\nAlso, this solution has a \nvanishing Noether charge; however, its mass-energy is finite and nonzero\n(see Figure 4 (left) for a plot of the energy density for a typical configuration\nwith $n=2$; one notices the existence of a region \nwith negative energy density, $\\rho=-T_t^t<0$).\n\n\nConcerning the dependence on $\\alpha^2=4 \\pi G \\mu^2\/\\lambda$, \na central role is played here by the solutions with a  $|\\Phi|^2$ potential only.\nWe have found that for a range of the frequency  $ \\omega_{min}< \\omega< \\omega_c$ \n(with $ \\omega_{min}$ the minimal allowed value\nof the frequency for the boson stars without a self-interaction term),\nthe solutions exist for arbitrarily large values of $\\alpha$.\nSimilar to the asymptotically flat  case \\cite{Kleihaus:2005me},\nthe limit $\\alpha \\to \\infty$\ncorresponds (after a rescaling) to the solution of the $|\\Phi|^2$-model.\nThe picture is different for smaller frequencies, $ \\omega< \\omega_{min}$,\nin which case there are no solutions in the $|\\Phi|^2$ model.\nThe numerical calculations indicate that, in this case, the range of $\\alpha$ is bounded from above,\nand a critical configuration is approached for $\\alpha \\to \\alpha_c$,\nwith $\\alpha_c$ depending on $ \\omega$ and $\\ell$.\nThis limiting soliton has a finite mass-energy and Noether charge.\n(These two cases are illustrated in Figure 4 (right).)\n Unfortunately, \n the numerical accuracy does not allow  to clarify the  limiting behaviour \nat the critical value of $\\alpha$.\nWe notice only that, as  $\\alpha \\to \\alpha_c$, \nthe metric function $F_0$  almost reaches zero at $r=0$,\nwhile the other functions remain finite and nonzero \n(although $F_1$ and $F_2$ take  large values at the origin). \n\n\n\n We remark also that for all spinning solutions,\nthe distributions of the mass-energy density $-T_t^t$\nare very different from those of the spherically symmetric configurations,\n$i.e.$ the typical energy density isosurfaces have a toroidal shape.\nHowever, although\n the violation of the positive energy condition is a generic feature of \n the solutions with $U_{int}=-\\lambda|\\Phi|^4<0$ (at least for small enough values of $\\omega$),\n the mass-energy of all our solutions as given by (\\ref{EJ}) is strictly positive.  \n \n We close this Section by noticing that, similar to the $\\Lambda=0$ \n case \\cite{Kleihaus:2007vk}, the AdS rotating boson stars possess ergoregions \nin a large part of their domain of existence.\n The ergoregion resides inside the ergosurface  \n defined by the condition $g_{tt}=-F_0N+F_2\\sin^2\\theta W^2=0$,\n in the metric parametrization (\\ref{ansatzg}). \nThis type of configurations are typically found for large enough values of $\\omega$, $n$ and $\\alpha$.\n \n\\section{Further remarks}\nIn this work we have initiated a preliminary\ninvestigation of  spinning Q-balls and boson stars in four dimensional AdS spacetime. \nThis study was partially motivated by the recent interest in \nsolutions of AdS gravity coupled to scalar fields only. \nThe picture we have found has some interesting new features as compared to the \nwell-known case of solutions with a vanishing cosmological constant.\nPerhaps the most interesting new result is the existence of \naxially symmetric solitons with a smooth static limit possessing a vanishing Noether charge.\n Also, all solutions have an upper bound on frequencies,\n which is fixed by the scalar field mass and the cosmological constant.\n\nMoreover,\nwe expect the existence of a  much richer set of spinning scalar solitons apart from the \nsolutions reported in this work.\nFor example,  the $\\Lambda=0$\nQ-balls and boson stars with odd parity with respect to \na reflection in the equatorial plane reported in \\cite{Kleihaus:2007vk},\nshould allow for AdS generalizations.\nIn particular, it would be interesting \nto construct AdS `{\\it twisted}' Q-balls and boson stars,\nwhich combine features of both even and odd parity solutions \\cite{Radu:2008pp}.\nFor such configurations, the scalar field is endowed with a $(r,\\theta)$-dependent phase,\n$\\Phi=\\phi (r,\\theta)e^{i(n\\varphi- \\omega t+\\Psi(r,\\theta))}=(X(r,\\theta)+i Y(r,\\theta))e^{i(n\\varphi- \\omega t)}$,\nsuch that the amplitude of the scalar field\nvanishes in the equatorial plane.\nThis would lead to a 'topological charge' of the solutions \n(see \\cite{Radu:2008pp} for the details of this\nconstruction in the flat spacetime case).\nAlso, the issue of AdS vortons, $i.e.$\nspinning\nvortex loops stabilized by the centrifugal force,\nstill remains to be investigated.\nThe results in this work suggest that the  \nAdS picture may be very different as compared to \nthe one found in the flat spacetime limit \\cite{Radu:2008pp}, \\cite{Battye:2008mm}. \n\n\nA natural question which arises concerns the issue \nof higher dimensional counterparts of the solutions discussed in this paper.\nWorking in the probe limit,\nwe have found that the general picture we have presented for $d=4$\nremains valid \nfor spinning solitons \nwith a single angular momentum in $d=5,6$ dimensions.\nTherefore we expect it to be generic for any $d\\geq 4$.\nMoreover, for the same self-interaction potential $U_{int}=-\\lambda \\phi^{2k}$ \n(with $k$ still given by (\\ref{k})), the exact Q-ball solution (\\ref{ex-sol})\nadmits a straightforward generalization\\footnote{Here we consider a fixed AdS background,\nwith $ds^2=\\frac{dr^2}{1+r^2\/\\ell^2}+r^2d\\Omega_{d-2}^2-(1+r^2\/\\ell^2)dt^2$.}\n for any $d\\geq 3$, with \n$\n\\Phi(r,t)= (\n\\frac{\\mu^2}{\\lambda}\\frac{\\Delta^2-\\ell^2  \\omega^2}{(\\Delta-(d-1))(\\Delta+1))}\n )^{\\frac{\\Delta}{2}} {\\left(1+\\frac{r^2}{\\ell^2} \\right)^{-\\frac{\\Delta}{2}}} e^{-i  \\omega t}, \n$\n and $\\Delta=\\frac{1}{2}\\left( (d-1)\\pm \\sqrt{(d-1)^2+4\\mu^2 \\ell^2} \\right)$.\n\nAlso, based on some preliminary results, \nwe conjecture that it is possible to add\na small black hole in the center of the  $d=4$ solitons  with a  harmonic time dependence studied in this work.\nTherefore the inclusion of rotation would allow  to circumvent the\nno-hair results in \\cite{Pena:1997cy}, \\cite{Astefanesei:2003qy}.\nIndeed,  such solutions \nwere constructed recently in \\cite{Dias:2011at} for $d=5$\nand a complex doublet scalar field,\nin which case a special ansatz \\cite{Hartmann:2010pm} allows to deal with ODEs.\n\n \n It would be desirable to study all these solutions\n also\nfrom an AdS\/CFT perspective and\nto see what they correspond to\nin the dual theory.\n\nWe close by remarking that\nthe study of Q-balls and boson stars\nis interesting from yet another point of view.\nThis type of relatively simple \nconfigurations provide an ideal ground for \ninvestigating various numerical techniques\non axially symmetric problems with AdS asymptotics,\nwhich thereafter can be applied to more complex models.\n\\\\\n\\\\\n\\noindent{\\textbf{~~~Acknowledgements.--~}} \nWe are grateful to Jutta Kunz\nfor her careful reading of the manuscript and many helpful comments.\n We also thank Burkhard Kleihaus for collaboration in the initial stages of this work. \nWe gratefully acknowledge support by the DFG,\nin particular, also within the DFG Research\nTraining Group 1620 ''Models of Gravity''. \n\n \\begin{small}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Article}\n\nThe Kondo effect~\\cite{DeHaas36,Kondo64} has captured the attention of experimentalists and theorists alike for decades because of its complex many-body physics. In metals with dilute magnetic impurities, the experimental signature of Kondo effect is the low temperature increase of resistivity, which is attributed to the many-body antiferromagnetic s-d exchange interaction between the impurity spin and the conduction electron spins of the host metal. The Kondo effect has also been observed in semiconductor quantum dot (QD) systems where an unpaired spin in a QD is coupled to the surrounding electron reservoirs.~\\cite{Goldhaber98} A popular but controversial physical picture of the Kondo effect is the Kondo screening cloud, which is an electron cloud surrounding the impurity site with an overall spin polarization opposite to the impurity spin. At temperatures well below the Kondo temperature $ T_K $, the net spin of the Kondo cloud completely screens the impurity spin forming a Kondo singlet state. The spatial extent $ \\xi_K $ of the Kondo cloud is given by $ \\hbar{v_F}\/k_BT_K $  in ballistic transport regime and  $ \\sqrt{{\\hbar}D\/k_{B}T_{K}} $ in diffusive regime,~\\cite{Chandr00book,Affleck09} where $ v_F $ is the Fermi velocity, $ k_B $ is the Boltzmann constant, and $ D $ is the diffusion constant. Experimental evidence for the screening cloud is scarce and therefore its physical existence has been questioned.~\\cite{Boyce74} Recently Borzenets $et~al.$~\\cite{Borzenets20} found convincing evidence for micrometer-sized Kondo clouds in a QD system. In diffusive metals, $ \\xi_K $ is expected to be $\\sim 100$ nm, but has not yet been experimentally confirmed.   \n\n\nIn recent years, the Kondo effect crosses paths with spintronics. In the Cu channels of nonlocal spin valves (NSLVs)~\\cite{Johnson85,Jedema01} with dilute Fe impurities, the spin relaxation rate $ {\\tau_{s}}^{-1} $ is found to increase at low temperatures complementing Kondo effect's low temperature increase of the momentum relaxation rate $ {\\tau_{e}}^{-1} $.~\\cite{Obrien14,Batley15,Hamaya16,Watts19} Here $ \\tau_s $ and $ \\tau_e $ are the spin relaxation time and momentum relaxation time, respectively. For spin relaxation in general, Elliott-Yafet (EY)~\\cite{Elliott54,Yafet83} and Dyakonov-Perel (DP)~\\cite{Dyakonov72} models give explicit relations between $ {\\tau_s}^{-1} $  and $ {\\tau_e}^{-1} $. The EY spin relaxation is caused by weak spin-orbit coupling between energy bands and $ {\\tau_s}^{-1} $  is proportional to  $ {\\tau_e}^{-1} $. The ratio $ \\tau_e\/\\tau_s $ is the spin flip probability $ \\alpha $.  The DP spin relaxation originates from spin-orbit coupling, caused by inversion symmetry breaking, between two spin subbands within the same energy band and the  $ {\\tau_s}^{-1} $ is inversely proportional to $ {\\tau_e}^{-1} $. The Kondo spin relaxation, however, is caused by s-d exchange interaction instead of spin orbit effects. The relation between the Kondo spin relaxation rate    $ {\\tau_{sK}}^{-1} $ and Kondo momentum relaxation rate  $ {\\tau_{eK}}^{-1} $, to the best of our knowledge,  has not yet been explored.\n\n\nIn this work, we extract values of $ {\\tau_{sK}}^{-1} $ and $ {\\tau_{eK}}^{-1} $ from Cu-based NLSVs fabricated by 2-step electron beam lithography. Each NLSV includes a spin injector $\\mathrm{F_1}$, a spin detector $\\mathrm{F_2}$, and a Cu channel, as shown in Figure 1 (a). Magnetic electrodes $\\mathrm{F_1}$ and $\\mathrm{F_2}$, made of $\\mathrm{Ni_{81}Fe_{19}}$ alloy (permalloy or Py), are patterned in the first step and Cu channels are patterned in the second step. The materials are deposited by electron beam evaporation. Before the deposition of Cu, low energy ion milling is performed to clean the surface of Py and a 3 nm $\\mathrm{AlO_x}$ layer is deposited. The Py\/$\\mathrm{AlO_x}$\/Cu interface has been shown to provide a higher effective spin polarization than the ohmic Py\/Cu interfaces.~\\cite{Wang09,Cai16} The distance $L$ between $\\mathrm{F_1}$ and $\\mathrm{F_2}$ varies from 1 to 5  $\\mu$m with 1 $ \\mu $m increment. All Cu channels are 500 nm wide and 300 nm thick to prevent the suppression of Kondo clouds.~\\cite{Chen91,Blachly95} This work involves data from two sample substrates (chip 11 and chip 12) with 10 devices on each. Devices on the same substrate undergo identical fabrication conditions. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure1.eps}\n\t\\caption{\\label{fig1}(a) SEM image of a NLSV. Plots of (b) $R_s$ versus $B$, (c) $ \\Delta{R}_s $ versus $T$, and (d) $ \\rho_{cu} $ versus $T$ for device 11-43 ($ L=3.0\\,\\mu $m).}\n\\end{figure}\n\n\nThe measurement configuration is shown in Figure 1 (a). A low frequency excitation alternating current (AC) $ I_e $ is driven from $ \\mathrm{F_1} $ to the upper end of the Cu channel, and the spin accumulation is detected by measuring the nonlocal voltage $V_{nl}$ between $ \\mathrm{F_2} $ and the lower end of the channel. Figure 1 (b) shows the nonlocal resistance $ R_s={V_{nl}}\/{I_e} $ as a function of magnetic field $B$ applied parallel to  $\\mathrm{F_1}$ and $\\mathrm{F_2}$ stripes. The high and low states of $R_s$ correspond to the parallel and antiparallel states of $\\mathrm{F_1}$ and $\\mathrm{F_2}$ magnetizations, respectively. The difference is the spin signal~\\cite{Johnson93} \n\\begin{equation} \n\\Delta{R_s}=\\frac{{P_e}^2\\rho_{cu}\\lambda_{cu}}{A_{cu}}e^{-\\frac{L}{\\lambda_{cu}}}, \\label{Rs} \n\\end{equation}\nwhere $ P_e $ is the effective spin polarization of $\\mathrm{F_1}$ and $\\mathrm{F_2}$, $ \\rho_{cu} $ the Cu resistivity, $ \\lambda_{cu} $ the Cu spin relaxation length, and $ A_{cu} $ the Cu channel cross sectional area. $ \\Delta{R_s}(T) $ of each NLSV is measured from 5 K to 100 K and Figure 1 (c) shows the data of device 11-43 (device 43 on chip 11). As $ T $ decreases, $ \\Delta{R_s} $ initially increases, reaching its maximum at 30 K, and then decreases. This feature is well documented~\\cite{Kimura08,Mihajlovic10,Zou12APL,Villamor13} for NLSVs and convincingly attributed to the Kondo effect.~\\cite{Obrien14,Batley15,Hamaya16,Watts19,Obrien16,Kim17}\n\n\n\nThe resistivity $ \\rho_{cu} $ of a given NLSV is deduced from its Cu channel resistance $ R_{cu} $, which is obtained by sending in a current through the channel and measuring the voltage difference between $ \\mathrm{F_1} $ and $\\mathrm{F_2}$. The $ \\rho_{cu}(T) $ for device 11-43 is shown in Figure 1 (d) with $ \\rho_{cu}=0.43 \\,\\mu\\Omega\\cdot $cm at 5 K and $ \\rho_{cu}=2.60 \\,\\mu\\Omega\\cdot $cm at 295K. The ratio of the two values (6.1) is the residual resistivity ratio (RRR). The inset of Figure 1 (d) shows the low temperature portion of $ \\rho_{cu}(T) $. The low $T$ increase of $ \\rho_{cu} $ indicates Kondo effect from dilute magnetic impurities in Cu.  \n\nNext, we extract the average $ P_e $ and $ \\lambda_{cu} $ values of devices on the same substrate. $ \\Delta{R_s} $ versus $ L $ is plotted for 10 devices on chip 11 at 30 K in Figure 2 (a). Fitting Eq.~(\\ref{Rs}) to the plot yields $ \\lambda_{cu}=2.6\\pm0.1 \\,\\mu$m and $ P_e=0.066\\pm0.003 $. The average  $ \\rho_{cu} $ used in this process is deduced from the linear fitting of the $R_{cu}$ versus $L$ data in Figure 2 (b). In this manner, the average $P_e$ and $ \\lambda_{cu} $ are obtained between 5 K and 100 K and shown in Figure 2 (c) and its inset, respectively. $ \\lambda_{cu}(T) $ resembles $ \\Delta{R_s}(T) $  in Figure 1(c) and reaches its maximum of 2.6 $ \\mu $m at 30 K. $ \\lambda_{cu} $ decreases to 2.2 $ \\mu $m at 5 K because of the enhanced Kondo spin relaxation. The plot of $ P_e(T) $ shows a rather flat trend around 0.07 within the temperature range of our measurements. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure2.eps}\n\t\\caption{\\label{fig2}(a) Spin signal $\\Delta{R}_s$ and (b) Cu resistance $R_{cu}$ versus channel length $L$ for NLSVs on chip 11 at 30 K. (c) Fitted average $P_e$ and $ \\lambda_{cu} $ (inset) as a function of $T$. (d) $ \\lambda_{cu} $ versus $T$ for device 11-43. }\n\\end{figure}\n\nAs suggested by previous works on Py\/Cu NLSVs, the Kondo effect originates from Fe impurities.~\\cite{Obrien14,Batley15,Hamaya16,Obrien16,Kim17} The maximum $ \\lambda_{cu} $ occurs at 30 K, which is the Kondo temperature $T_K$ for Fe impurities in Cu host. Data analysis of $ {\\tau_s}^{-1}(T) $  and $ {\\tau_e}^{-1}(T) $ later in the text is also consistent with $T_K = 30$ K. The Fe impurities are likely introduced in the fabrication processes. When the Py surface is ion milled, Fe atoms are removed and deposited on the side walls of the resist. When Cu is evaporated, the vapor flux of Cu transfers momentum to the Fe atoms on the side walls and redeposits them into the Cu channel. In some of the previous works,~\\cite{Obrien14,Watts19,Obrien16,Kim17} Fe impurities are concentrated near the ohmic Py\/Cu interfaces, and as a result the spin polarization $ P_e(T) $ is suppressed at low $T$. In our devices, the Fe impurities are located throughout the Cu channel. This is evident from the low $T$ upturn of $ \\rho_{cu}(T) $, the low $T$ downturn of $ \\lambda_{cu}(T) $, and the flat trend of $ P_e(T) $.\n\n\n\nIt is noticeable that data points disperse around the fitted lines in Figure 2 (a) and (b). For the two devices with $ L=3\\,\\mu $m, for example, data points of $ \\Delta{R_s} $ are above the fitted line and those of $ R_{cu} $ are below. The two devices with $ L=4\\,\\mu $m have $ \\Delta{R_s} $ below the fitted line and $ R_{cu} $ above. These indicate variations of $ \\lambda_{cu} $ and $ \\rho_{cu} $ between devices. Assuming a common $P_e$ (the fitted $ P_e $) for all devices on the same substrate at a specific $T$, we deduce $ \\lambda_{cu} $ for each individual NLSV from its $ \\Delta{R_s} $ and $ \\rho_{cu} $ by using Eq.~(\\ref{Rs}). $ \\lambda_{cu}(T) $ for device 11-43 is shown in Figure 2 (d) with a maximum  $ \\lambda_{cu}=3.0\\pm0.1 \\,\\mu $m at 30 K. In this manner $ \\lambda_{cu}(T) $ are obtained for all 20 NLSVs. The spin relaxation rate $ {\\tau_s}^{-1}(T) $ is then calculated from $ \\lambda_{cu}(T) $ by using the relation $ \\lambda_{cu}=\\sqrt{D\\tau_s} $ and shown in Figure 3 (a) and (b) for devices 11-33 and 12-32, respectively. $ D=\\frac{1}{3}{v_F}^2\\tau_e $ is the diffusion constant and $ v_F=1.57\\times10^6 $ m\/s is the Fermi velocity of Cu. $ \\tau_e $ can be derived from $ \\rho_{cu} $ by using the Drude model $ \\rho_{cu}=m\/(\\tau_ene^2) $, where $ n=8.47\\times10^{28}\\:\\mathrm{m^{-3}} $ is the Cu electron density and $m$ and $e$ are electron mass and charge, respectively. With a decreasing $T$, $ {\\tau_s}^{-1} $ initially decreases, reaches its minimum around 30 K, and then increases upon further cooling. This resembles Kondo effect's low temperature increase of $ \\rho_{cu} $ as shown in the insets of Figure 3 (a) and (b). The low $T$ increase of $ \\rho_{cu} $ of 11-33 is much smaller than that of 12-32, indicating a lower impurity concentration in 11-33. However, the low $T$ increase of  $ {\\tau_s}^{-1}$ of the two devices are surprisingly comparable. This provides the first hint for an unusual relation between Kondo momentum relaxation and Kondo spin relaxation. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure3.eps}\n\t\\caption{\\label{fig3}Spin relaxation rate $ {\\tau_s}^{-1} $  versus $T$ for (a) device 11-33 and (b) device 12-32. $ \\rho_{cu}(T) $ plots are shown in the insets. (c) $ {\\tau_s}^{-1} $  versus $ {\\tau_{eK}}^{-1} $  for $ T \\le 30 $ K for the two devices. The slopes of the linear fittings are compared with $ \\alpha_K $ values obtained from fittings with Eq.~(\\ref{taus}). (d) $ {\\tau_{s,ph}}^{-1} $  versus $ {\\tau_{e,ph}}^{-1} $  plots. }\n\\end{figure}\n\n\nApplying Matthiessen's rule to spin relaxation, the total $ {\\tau_s}^{-1} $ is given by $ {\\tau_s}^{-1}={\\tau_{s,def}}^{-1}+{\\tau_{s,ph}}^{-1}+{\\tau_{sK}}^{-1} $, where $ {\\tau_{s,def}}^{-1} $, $ {\\tau_{s,ph}}^{-1} $ , and $ {\\tau_{sK}}^{-1} $  are the spin relaxation rates attributed to defects, phonon, and Kondo effects, respectively. Defining $ {\\tau_{e,def}}^{-1} $, $ {\\tau_{e,ph}}^{-1} $, and $ {\\tau_{eK}}^{-1} $  as the corresponding momentum relaxation rates and $ \\alpha_{def} $, $ \\alpha_{ph} $, and $ \\alpha_K $ as the associated spin flip probabilities, we have\n\\begin{equation}\n\t\\frac{1}{\\tau_s(T)}=\\alpha_{def}\\frac{1}{\\tau_{e,def}}+\\alpha_{ph}\\frac{1}{\\tau_{e,ph}(T)}+\\alpha_{K}\\frac{1}{\\tau_{eK}(T)}. \\label{taus}   \n\\end{equation}   \nIt is well justified to assume a linear relation between $ {\\tau_s}^{-1} $  and  $ {\\tau_e}^{-1} $ for defects and phonons, because EY mechanism is dominant in these processes. We will show later that $ {\\tau_{sK}}^{-1} $  is also proportional to $ {\\tau_{eK}}^{-1} $  under varying $T$.\n\n\n\nThe  $ {\\tau_e}^{-1} $ of each type (total, defect, phonon, or Kondo) is linked to the corresponding $ \\rho $ by the Drude model $ \\rho=m\/(\\tau_ene^2) $. The defect resistivity $ \\rho_{def} $ is $T$ independent and the phonon resistivity can be described as $ \\rho_{ph}(T)=AT^5 $ at low $T$, where $A$ is constant related to the Debye temperature.~\\cite{Zimanbook} The Kondo resistivity can be described by a phenomenological formula~\\cite{Goldhaber98} \n\\begin{equation}\n\t\\rho_K(T)=\\rho_{K0}{\\left({\\frac{{T_K'}^2}{T^2+{T_K'}^2}}\\right)}^s, \\label{Kondo}\n\\end{equation}\nwhere $ T_K'=T_K\/\\sqrt{2^{1\/s}-1} $, $s=0.225$ and $T_K = 30$ K. From $ {\\tau_e}^{-1}={\\tau_{e,def}}^{-1}+{\\tau_{e,ph}}^{-1}+{\\tau_{sK}}^{-1} $, the total resistivity is \n\\begin{equation}\n\t\\rho_{cu}(T)=\\rho_{def}+AT^5+\\rho_K(T). \\label{rho}\n\\end{equation}\nFitting Eq.~(\\ref{rho}) along with Eq.~(\\ref{Kondo}) to the measured $ \\rho_{cu}(T) $ data below 20 K yields $ \\rho_{def} $, $A$, and $ \\rho_{K0} $. Note that the fitting does not work well for $ T> $ 20 K, because $ \\rho_{ph}(T)=AT^5 $ is an approximation valid at low $T$. For the data of 11-33 and 12-32 in the insets of Figure 3 (a) and (b), the fitted values of $ \\rho_{K0} $ are 0.0013 $ \\mu\\Omega\\cdot $cm and 0.0067 $ \\mu\\Omega\\cdot $cm, respectively. $ \\rho_{K0} $ or $ {\\tau_{eK0}}^{-1} $  represents the $ \\rho_K $ or $ {\\tau_{eK}}^{-1} $ value at $T << T_K$.  \n\nTo extract $ \\alpha_{def} $, $ \\alpha_{ph} $, and $ \\alpha_K $, we fit Eq.~(\\ref{taus}) to the $ {\\tau_s}^{-1}(T) $ data by using the empirical data of $ {\\tau_{e,def}}^{-1} $, $ {\\tau_{e,ph}}^{-1}(T) $, and $ {\\tau_{eK}}^{-1}(T) $ obtained from the measured $ \\rho_{cu}(T) $ and fitting. More specifically, $  {\\tau_{e,def}}^{-1} $  can be obtained from the fitted $ \\rho_{def} $ and $ {\\tau_{eK}}^{-1}(T) $ from the fitted $ \\rho_{K0} $ and Eq.~(\\ref{Kondo}). For $ {\\tau_{e,ph}}^{-1}(T) $  we use the relation $ \\rho_{ph}(T)=\\rho_{cu}(T)-\\rho_{def}-\\rho_K(T) $. We do not use $ \\rho_{ph}(T)=AT^5 $ because it significantly deviates from experimental data when $ T>20 $ K. The best fits for $ \\alpha_K $ are 0.30 $\\pm$ 0.03 and 0.066 $\\pm$ 0.006 and the best fits for $ \\alpha_{ph} $ are $(8.4 \\pm 0.3) \\times 10^{-4}$ and $ (9.3\\pm0.4) \\times 10^{-4}$ for devices 11-33 and 12-32, respectively. While $ \\alpha_{ph} $ values are comparable, $ \\alpha_K $ values are quite different. Again, the results point to the unusual scaling for Kondo spin relaxation. \n\n\n\nWe should justify the assumed linear relation $ {\\tau_{sK}}^{-1}(T)=\\alpha_K\\cdot{{\\tau_{eK}}^{-1}(T)} $ under varying $T$ in Eq.~(\\ref{taus}). In Figure 3 (c), $ {\\tau_s}^{-1} $  is plotted versus $ {\\tau_{eK}}^{-1} $   between 5 K and 30 K for the two NLSVs and we observe clear linear dependences. At $T\\leq$ 30 K, the variation of $ {\\tau_s}^{-1} $  should be dominated by $ {\\tau_{sK}}^{-1} $ , because $ {\\tau_{s,def}}^{-1} $  is $T$ independent and $ {\\tau_{s,ph}}^{-1} $  is negligible compared to $ {\\tau_{sK}}^{-1} $. Therefore, Figure 3 (c) confirms the linear relation between $ {\\tau_{sK}}^{-1}(T) $ and $ {\\tau_{eK}}^{-1}(T) $ under varying $T$. In addition, the slopes of the linear fittings to the $ {\\tau_s}^{-1} $  versus $ {\\tau_{eK}}^{-1} $  data are very close to the fitted $ \\alpha_K $ values using Eq.~(\\ref{taus}). Similarly, linear relation for phonons between $ {\\tau_{s,ph}}^{-1}(T) $ and $ {\\tau_{e,ph}}^{-1}(T) $ is also verified in Figure 3 (d). The data of $ {\\tau_{s,ph}}^{-1} $  is obtained by subtracting $ \\alpha_{def}\\cdot{\\tau_{e,def}}^{-1} $ and $ \\alpha_{K}\\cdot{\\tau_{eK}}^{-1} $  from the total $ {\\tau_s}^{-1} $. The slopes of the fitted lines are the same as the fitted $ \\alpha_{ph} $ values by using Eq.~(\\ref{taus}). \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure4.eps}\n\t\\caption{\\label{fig4}(a) Kondo spin flip probability $ \\alpha_K $ versus Kondo resistivity $ \\rho_{K0} $. (b) Phonon spin flip probability $ \\alpha_{ph} $ versus 100 K phonon resistivity $ \\rho_{ph,100K} $.}\n\\end{figure}\n\n\nNext, we demonstrate the unusual relation between $ {\\tau_{sK}}^{-1} $  and $ {\\tau_{eK}}^{-1} $ under a varying impurity concentration $ C_{Fe} $ which is approximately proportional to $ \\rho_{K0} $ or $ {\\tau_{eK0}}^{-1} $. Figure 4 (a) shows $ \\alpha_K $ versus $ \\rho_{K0} $ extracted from all NLSVs. Strikingly, $ \\alpha_K $ decreases drastically from $ 0.44 \\pm 0.05 $ to $ 0.045 \\pm 0.004 $ as $ \\rho_{K0} $ ($ \\propto{\\tau_{eK0}}^{-1} $) increases from $ < 0.001 \\,\\mu\\Omega\\cdot $cm to $> 0.009 \\,\\mu\\Omega\\cdot $cm. As a comparison, Figure 4 (b) shows $ \\alpha_{ph} $ versus $ \\rho_{ph,100K} $, which is the $ \\rho_{ph} $ at 100K, for all NLSVs. $ \\alpha_{ph} $ remains nearly a constant and independent of $ \\rho_{ph,100K} $ as expected for processes governed by EY mechanism. The average value of $ \\alpha_{ph} $ ($\\sim 8.5\\times10^{-4} $) is in good agreement with previous works.~\\cite{Watts19,Villamor13,Monod79} The average value of $ \\alpha_{def} $ is $ 3.2\\times10^{-4} $ and the data are shown in the Supplementary Materials (Note S1). The decreasing trend in Figure 4 (a) suggests that the relation between $ {\\tau_{sK0}}^{-1} $  and $ {\\tau_{eK0}}^{-1} $  is not linear, where $ {\\tau_{sK0}}^{-1} $ is the value of $ {\\tau_{sK}}^{-1} $  at $T \\ll T_K$. Figure 5 (a) shows $ {\\tau_{sK0}}^{-1} $ , obtained by using the definition $ {\\tau_{sK0}}^{-1}=\\alpha_K\\cdot{\\tau_{eK0}}^{-1} $, versus $ {\\tau_{eK0}}^{-1} $. While $ {\\tau_{eK0}}^{-1} $  varies by a factor of 10,  $ {\\tau_{sK0}}^{-1} $  stays nearly constant clearly defying a linear dependence. In contrast, the few previous theoretical treatments of Kondo spin relaxation assume a linear relation and yield a constant $ \\alpha_K $ of $2\/3$.~\\cite{Kondo64,Kim17} The dependences shown in Figure 4 (a) and 5 (a) have been neither anticipated nor addressed previously. These plots with horizontal error bars are available in the Supplementary Materails (Note S2). The $ C_{Fe} $ for each NLSV can be extracted from the temperature $ T_{min} $ that corresponds to the minimum of the fitted $ \\rho_{cu}(T) $ curve.~\\cite{Obrien16,Franck61} Figure 5 (b) shows the extracted $ C_{Fe} $ versus $ \\rho_{K0} $ for all NLSVs. \n\n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure5.eps}\n\t\\caption{\\label{fig5}(a) Kondo spin relaxation rate $ {\\tau_{sK0}}^{-1} $  versus Kondo momentum relaxation rate $ {\\tau_{eK0}}^{-1} $  from 20 NLSVs. (b) Fe impurity concentration $ C_{Fe} $ versus $ \\rho_{K0} $. (c) Illustration of the Kondo medium. The gray scale indicates the spin density, and the white arrows indicate the polarization directions of the domains. }\n\\end{figure}\n\nTo address this unusual scaling between the Kondo momentum and spin relaxation, the physical picture of the Kondo cloud becomes appealing. If Kondo clouds exist, it is valid to consider them as momentum scattering barriers as well as spin scattering barriers for conduction electrons passing through them.~\\cite{Simon03} The $ {\\tau_{eK}}^{-1} $ should be proportional to the average charge density of the cloud. The $ {\\tau_{sK}}^{-1} $ should be proportional to the average spin density of the cloud. It may also be related to the relative orientation between the conduction electron spin and the polarization direction of the cloud. The observed unusual scaling arises when the Kondo clouds of adjacent impurities overlap. \n\nTwo relevant length scales are the size of a single Kondo cloud $ \\xi_K $ and the average distance $ d_{Fe} $ between Fe impurities. The former is estimated to be $ \\xi_K=\\sqrt{{\\hbar}D\/k_BT_K}\\approx 100 $ nm for diffusive Cu channels. The latter is 10 nm $ < d_{Fe} <$ 20 nm, estimated from the $ C_{Fe} $ of our NLSVs, and obviously $ \\xi_K > d_{Fe} $. Therefore, the Kondo clouds from adjacent impurities overlap and the conduction electrons associated with the clouds form a continuous medium in the Cu channel. The medium can be characterized by its local charge density, spin density, and polarization direction with some spatial variations. The charge density of overlapping clouds should simply add up. However, the spin density of overlapping clouds may cancel out each other. Because impurity spin directions are random and so are the polarization directions of the clouds. Such cancellation effect of spin density has important implications on the $ {\\tau_{sK}}^{-1} $. Figure 5 (c) is a qualitative illustration of the spin density distribution and polarization directions of the Kondo medium. Domains with random polarization directions are formed in the medium around impurity sites. \n\n\nWhen a conduction electron traverses through the medium, the spin and momentum relaxation occur through the interaction between the electron and the Kondo medium. The $ {\\tau_{eK0}}^{-1} $  or $ {\\tau_{sK0}}^{-1} $  should be proportional to the average charge density or the average spin density of the medium, respectively, along the electron's path. The influence of the polarization directions on $ {\\tau_{sK0}}^{-1} $ can be neglected, because the traversing electron passes through many ($ \\approx 10^4 $) randomly oriented Kondo domains within the time of $ \\tau_{sK0} $. As $ C_{Fe} $ increases, more electrons are added to the Kondo medium, leading to a higher charge density and a higher $ {\\tau_{eK0}}^{-1} $. However, the spin density may not increase, because a higher $ C_{Fe} $ enhances cloud overlapping and the cancellation effect. The exact trend is challenging to predict, because it requires precise knowledge of the spatial distributions of spin and charge densities of Kondo clouds and how overlapping clouds interact.  From experimental results in Figure 5 (a), we infer that the average spin density of the medium maintains a nearly constant value within the range of 1 ppm $ < C_{Fe} < $ 12 ppm, corresponding to 10 nm $ < d_{Fe} < $ 20 nm. The red curve in Figure 5 (a) is a guide to the eye with a reasonable assumption that $ {\\tau_{sK0}}^{-1}\\rightarrow 0 $ as $ {\\tau_{eK0}}^{-1}\\rightarrow 0 $. We  speculate that the initial slope of the curve, representing $ \\alpha_K $ in the limit of $ {\\tau_{eK0}}^{-1}\\rightarrow 0 $, should be the theoretically predicted $ 2\/3 $.~\\cite{Kondo64,Kim17} \n\nIn conclusion, we extract the Kondo momentum relaxation rate $ {\\tau_{eK0}}^{-1} $  and the Kondo spin relaxation rate $ {\\tau_{sK0}}^{-1} $  from Cu-based nonlocal spin valves with Fe impurities. While $ {\\tau_{eK0}}^{-1} $  is tuned by a factor of 10 by varying Fe concentrations, $ {\\tau_{sK0}}^{-1} $  remains nearly constant and defies a more intuitive linear dependence on $ {\\tau_{eK0}}^{-1} $. Such a relation can be understood by considering a continuous Kondo medium formed by overlapping Kondo clouds. Spin relaxation occurs through interaction between a conduction electron spin and the medium. As the impurity concentration increases, the polarized spins of overlapping Kondo clouds partially cancel each other, and the average spin density of the Kondo medium reaches a stable value giving rise to a nearly constant $ {\\tau_{sK0}}^{-1} $. Our experimental results provide evidence for the physical existence of the elusive Kondo screening clouds.\n\n\\subsection{}\n\\subsubsection{}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn supervised learning, ensemble approaches are popular techniques for the prediction of tabular data. The most prominent ensemble methods include averaging ensembles like Random Forest \\cite{breiman-2001-randforests} or Random Subspace \\cite{ho98}, as well as gradient boosting techniques \\cite{friedman01}. While Random Subspace is a model-agnostic approach, i.e. an approach that be can combined with any type of base model, Random Forest is designed specifically for the aggregation of randomized decision trees (Table~\\ref{tab:ensembles}). One advantage of decision trees is their interpretability, as feature importance scores can be easily derived from a trained tree-based model. On the other hand, while model-agnostic approaches are more general and flexible, they cannot be used, at least in a straightforward way, to derive feature importances, as soon as they are combined with models other than trees. Note that while gradient boosting is a model-agnostic approach, it is designed to aggregate weak models, and is hence typically used with shallow decision trees.  \n\nIn this paper, we propose a model-agnostic ensemble approach for supervised learning (Figure~\\ref{fig:PRSB}). The proposed approach is based on a parametric version of Random Subspace, in which each base model is learned from a feature subset sampled according to a Bernoulli distribution. We formulate the training procedure as an optimization problem where the goal is to identify the parameters of the Bernoulli distribution that minimize the generalization error of the ensemble model, and we show that this optimization problem can be solved using gradient descent even when the base models are not differentiable. The optimization of the Bernoulli distribution is however intractable, as the computation of the exact output of the full ensemble model would require the training of one model for each possible feature subset. To render the parameter optimization tractable, we use Monte Carlo sampling to approximate the ensemble model output. We further use an importance sampling approach that circumvents frequent re-training of the base models after each update of the gradient descent. \n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{Figures\/PRSB} \n\\caption{{\\bf Parametric Random Subspace.} A model-agnostic ensemble is constructed, in which each base model is learned from a feature subset sampled according to a Bernoulli distribution with parameters $\\boldsymbol\\alpha$. The training procedure consists in identifying the parameters $\\boldsymbol\\alpha$ that minimize the generalization error of the ensemble model. This optimization problem is solved using gradient descent and importance sampling.}\n\\label{fig:PRSB}\n\\end{figure}\n\n\\begin{table}[t]\n\\caption{{\\bf Ensemble Methods.} The proposed approach, called parametric Random Subspace, is model-agnostic and provides feature importance scores.}\n\\label{tab:ensembles}\n\\small\n\\begin{center}\n\\begin{tabular}{lcc}\n\\toprule\n& Model-agnostic & Importances\\\\\n\\midrule\nRandom Forest & $\\times$ & $\\surd$\\\\\nGradient Boosting & $\\surd$ & $\\times^*$ \\\\\nRandom Subspace & $\\surd$ & $\\times^*$ \\\\\nParametric Random Subspace & $\\surd$ & $\\surd$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n{\\scriptsize $^*$when used with models other than decision trees.}\n\\end{table}\n\nWhile the degree of randomization is controlled by a hyper-parameter in standard Random Subspace, it has the advantage to be automatically tuned in our parametric version. Furthermore, model-agnostic feature importance scores can be easily derived from the trained ensemble model. We show the good performance of the proposed approach, both in terms of prediction and feature ranking, on simulated and real-world datasets. We also show that our approach can be successfully used for the reconstruction of gene regulatory networks. \n\n\\section{Methods}\n\nWe assume a supervised setting, where we have at our disposal a learning set containing $N$ input-output pairs $\\{(\\mathbf{x}_i, y_i)\\}_{i=1}^N$ drawn from an unknown probability distribution. Let us denote by $M$ the number of input variables. The output $y$ can be either continuous (regression problem) or discrete (classification problem). Our goal is to train a model-agnostic predictive model, while deriving for each input variable a score that measures its importance for the output prediction.\n\nTo achieve this goal, we propose a parametric version of a supervised learning method that combines Random Subspace \\cite{ho98} with Bagging \\cite{breiman-1996-bagging}, which we call RSB. The original, non-parametric RSB method consists in learning an ensemble of predictive models, where each model is built from a bootstrap sample of the original dataset and a randomly chosen subset of $K$ input variables (with $K < M$), sampled according to a uniform distribution. This method has been shown to yield competitive predictive performance with respect to Random Forest, with the advantage of being applicable with any type of base model \\cite{panov07,louppe12}. Here, instead of using a uniform distribution, we adopt a parametric distribution for the selection of the input features, and feature importance scores are derived through the identification of the distribution parameters that yield the lowest generalization error. In the following, after introducing the parametric RSB model (Section~\\ref{sec:PRSB_model}), we show how this model can be trained in a tractable way (Section~\\ref{sec:PRSB_training}) and we discuss our approach with respect to related works (Section~\\ref{sec:PRSB_discussion}).\n\n\\subsection{The Parametric RSB Approach}\n\\label{sec:PRSB_model}\n\nLet us denote by $\\mathbf{z} = (z_1, \\ldots, z_M)^\\top\\in \\{0,1\\}^M$ a binary vector of length $M$ encoding a subset of selected input variables: $z_j=1$ if the $j$-th variable is selected and $z_j=0$ otherwise, $\\forall j \\in \\{1,\\ldots,M\\}$. In the approach that we propose, each indicator variable $z_j$ is assumed to follow a Bernoulli distribution with parameter $\\alpha_j$. The probability mass function for $\\mathbf{z}$ is then given by:\n\\begin{equation}\n\\label{eq:bernoulli}\np(\\mathbf{z} \\vert \\boldsymbol\\alpha) = \\prod_{j=1}^M \\alpha_j^{z_j} (1-\\alpha_j)^{(1-z_j)},\n\\end{equation}\nwhere $\\alpha_j \\in [0,1]$ is the probability of selecting the $j$-th variable and $\\boldsymbol\\alpha = (\\alpha_1, \\ldots, \\alpha_M)^\\top$. Let $\\mathcal{Z} = \\{\\mathbf{z}^1, \\mathbf{z}^2, \\ldots, \\mathbf{z}^{\\vert \\mathcal{Z} \\vert}\\}$ be the set of all the possible feature subsets, where $\\vert \\mathcal{Z} \\vert = 2^M$ is the cardinality of $\\mathcal{Z}$.\n\nWe assume an ensemble method that consists in averaging base models trained independently of each other using subsets of features drawn from $p(\\mathbf{z}|\\boldsymbol\\alpha)$. Let us denote by $\\mathcal{F}$ some functional space corresponding to a given learning algorithm and by $\\mathcal{F}_\\mathbf{z}\\subseteq \\mathcal{F}$ the subset of functions from $\\mathcal{F}$ that only depend on the variables indicated by $\\mathbf{z}$. Let $f_{\\mathbf{z}^t}\\in \\mathcal{F}_{\\mathbf{z}^t}$ be the base model learned by this learning algorithm from the feature subset $\\mathbf{z}^t$ ($\\forall t \\in \\{1,\\ldots,|\\mathcal{Z}|\\}$). Asymptotically, the prediction of the ensemble model for a given input $\\mathbf{x}$ is given by:\n\\begin{equation}\n\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x})]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} = \\sum_{t=1}^{\\vert \\mathcal{Z} \\vert} p(\\mathbf{z}^t \\vert \\boldsymbol\\alpha) f_{\\mathbf{z}^t}(\\mathbf{x}).\n\\end{equation}\n\nFor a fixed $\\boldsymbol\\alpha$, a practical approximation of\n$\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x})]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$ can be obtained by Monte-Carlo sampling, i.e. by drawing $T$\nfeature subsets from $p(\\mathbf{z}|\\boldsymbol\\alpha)$ and then training a model from\neach of these subsets, using the chosen learning algorithm (Figure~\\ref{fig:PRSB}). If all the $\\alpha_j$'s are equal, the resulting ensemble method is very close to the Random Subspace approach \\cite{ho98}, the only difference being that the number of selected features will be slightly randomized from one model to the next. In this work, we would like however to identify the parameters $\\boldsymbol\\alpha$ that\nyield the most accurate expected predictions $\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x})]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$ over our training set. Given a loss function $L$, the corresponding optimization problem can be formulated as follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:global_optim}\n& \\min_{\\boldsymbol\\alpha \\in [0,1]^M} F(\\boldsymbol\\alpha),\\\\\n& \\mathrm{where~} F(\\boldsymbol\\alpha) = \\frac{1}{N} \\sum_{i=1}^N L  \\left( y_i,  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} \\right).\n\\end{aligned}\n\\end{equation}\nA nice advantage is that the selection probabilities $\\boldsymbol\\alpha$ after optimization can be interpreted as measures of variable importances: useless variables are expected to get low selection probabilities, while the most important ones are expected to get selection probabilities close to 1.\n\n\n\\subsection{Training the Parametric RSB Model}\n\\label{sec:PRSB_training}\n\nWe propose to solve the optimization problem in Eq~\\eqref{eq:global_optim} using gradient descent. More specifically, since  $\\alpha_j$ must be between 0 and 1, $\\forall j$, we use the projected gradient descent technique, where $\\boldsymbol\\alpha$ is projected into the space $[0, 1]^M$ after each step of the gradient descent.  In the following, we first derive the analytical formulation of the gradient of the objective function. We then explain how to estimate this gradient by using Monte Carlo sampling and show how to incrementally update this gradient estimate using importance sampling (Figure~\\ref{fig:PRSB_maths}). Precise pseudo-code of the algorithm is given in Section~\\ref{sec:pseudocode} and our Python implementation is available is available at \\url{https:\/\/github.com\/vahuynh\/PRSB}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{Figures\/PRSB_maths} \n\\caption{{\\bf Gradient Estimation.} The score function is used to compute the gradient of the expectation of the non-differentiable function $f_\\mathbf{z}$ (Section~\\ref{sec:gradient_computation}). When $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$ is the Bernoulli distribution, the $j$-th component of the gradient amounts to the difference between the expected output of a model that does take as input the $j$-th variable (in blue) and the expected output of a model that does not use it (in orange) (Section~\\ref{sec:gradient_computation}). The expectations are estimated with Monte Carlo sampling (Section~\\ref{sec:gradient_estimation}). Importance sampling approximation allows to update the gradient without the need to train new models (Section~\\ref{sec:gradient_update}).}\n\\label{fig:PRSB_maths}\n\\end{figure}\n\n\n\\subsubsection{Computing the Gradient}\n\\label{sec:gradient_computation}\n\nAssuming that the loss function $L$ is differentiable, the gradient of the objective function $F(\\boldsymbol\\alpha)$ w.r.t. $\\boldsymbol\\alpha$ is:\n\\begin{equation}\n\\nabla_{\\boldsymbol\\alpha} F(\\boldsymbol\\alpha) = \\frac{1}{N} \\sum_{i=1}^N \\frac{\\mathrm{d}L}{\\mathrm{d} \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}} \\nabla_{\\boldsymbol\\alpha}  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}.\n\\end{equation}\nTo compute the gradient $\\nabla_{\\boldsymbol\\alpha}  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$, we resort to the {\\it score function} approach \\cite{rubinstein93}, also known as the {\\it REINFORCE} method \\cite{williams92} or the {\\it likelihood-ratio} method \\cite{glynn90}, which allows us to express the gradient of an expectation as an expectation itself (see Appendix~\\ref{sec:gradient_derivation}):\n\\begin{equation}\n\\label{eq:score_function}\n\\nabla_{\\boldsymbol\\alpha}  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} = \\mathbb{E} \\left[ f_\\mathbf{z}(\\mathbf{x}_i) \\nabla_{\\boldsymbol\\alpha} \\log p(\\mathbf{z} \\vert \\boldsymbol\\alpha)  \\right]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}.\n\\end{equation}\nA major advantage of the score function approach is that, in order to compute the gradient in Eq.~\\eqref{eq:score_function},  only the distribution $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$ needs to be differentiable, and not the base model $f_\\mathbf{z}$. By using the score function method with the Bernoulli distribution in Eq.~\\eqref{eq:bernoulli}, the $j$-th component of the gradient is given by (see Appendix~\\ref{sec:gradient_derivation}):\n\\begin{equation}\n\\frac{ \\partial  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}}{\\partial \\alpha_j} = f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i) - f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i)\n\\end{equation}\nwhere $\\mathbf{z}_{-j} = \\mathbf{z} \\backslash z_j$, $\\boldsymbol\\alpha_{-j} = \\boldsymbol\\alpha \\backslash \\alpha_j$ and where, for the simplicity of notations, we have defined:\n\\begin{align}\n f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i) & = \\mathbb{E} [ f_\\mathbf{z}(\\mathbf{x}_i) \\vert z_j=0 ]_{p(\\mathbf{z}_{-j} \\vert \\boldsymbol\\alpha_{-j})},\\\\\n f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i) & =\\mathbb{E} [ f_\\mathbf{z}(\\mathbf{x}_i) \\vert z_j=1 ]_{p(\\mathbf{z}_{-j} \\vert \\boldsymbol\\alpha_{-j})}.\n\\end{align}\n$f^{\\boldsymbol\\alpha}_{j,0}$ (resp. $f^{\\boldsymbol\\alpha}_{j,1}$) is thus the expected output of a model that does not take (resp. takes) as input the $j$-th variable. We thus finally have:\n\\begin{equation}\n\\label{eq:gradient}\n\\frac{\\partial F}{\\partial \\alpha_j} = \\frac{1}{N} \\sum_{i=1}^N \\frac{\\mathrm{d} L}{\\mathrm{d} \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}} \\left( f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i) - f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i)\\right).\n\\end{equation}\nThe above derivative can be easily interpreted in the context of a gradient descent approach. For example, when $ \\frac{\\mathrm{d} L}{\\mathrm{d} \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}}$ is positive, the loss $L$ decreases with a lower model prediction $\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$. This means that if $f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i) < f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i) $, the model without variable $j$ will give a lower loss than the model with variable $j$. In that case, the derivative $\\frac{\\partial F}{\\partial \\alpha_j}$ is positive and a gradient descent step  (i.e. $\\alpha_j \\leftarrow \\alpha_j - \\eta  \\frac{\\partial F}{\\partial \\alpha_j}$, where $\\eta$ is the learning rate) will decrease the value of $\\alpha_j$. \n\n\n\\subsubsection{Estimating the Gradient}\n\\label{sec:gradient_estimation}\n\nGiven the current selection probabilities $\\boldsymbol\\alpha$, the exact computation of $\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x})]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$, $f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x})$ and $f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x})$, as required to compute the gradient, is obviously intractable as it implies training $|\\mathcal{Z}|$ models. An unbiased estimation of $\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x})]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$ can be obtained by Monte Carlo sampling, i.e. by averaging the output over an ensemble of $T$ models (with $T\\ll |\\mathcal{Z}|$), where each model $f_{\\mathbf{z}^{(t)}}, t=1, \\ldots, T,$ is trained using a subset of features $\\mathbf{z}^{(t)}$ sampled from $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$. From this ensemble of $T$ models, one can then build similarly Monte Carlo approximations of $f^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i)$ and $f^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i)$:\n\\begin{align}\n\\label{eq:cond_expec1}\nf^{\\boldsymbol\\alpha}_{j,0}(\\mathbf{x}_i) & \\simeq  \\frac{1}{T_{j,0}} \\sum_{t:z_j^{(t)}=0} f_{\\mathbf{z}^{(t)}}(\\mathbf{x}_i),\\\\\n\\label{eq:cond_expec2}\nf^{\\boldsymbol\\alpha}_{j,1}(\\mathbf{x}_i)  & \\simeq  \\frac{1}{T_{j,1}} \\sum_{t:z_j^{(t)}=1} f_{\\mathbf{z}^{(t)}}(\\mathbf{x}_i),\n\\end{align}\nwhere $T_{j,0}$ (resp. $T_{j,1}$) is the number of models where $z_j^{(t)}=0$ (resp. $z_j^{(t)}=1$), with $T_{j,0}+T_{j,1}=T$.\n\nIt remains to be explained on which data the models $f_{\\mathbf{z}^{(t)}}$ are\ntrained. Using the same $N$ samples as the ones used to compute the gradient in Eq.~\\eqref{eq:gradient} would lead to biased predictions $f_\\mathbf{z}(\\mathbf{x}_i)$ and hence to overfitting. We thus use a mini-batch gradient descent approach, in which a small subset of the training dataset (e.g. 10\\% of the samples) are used for computing the gradient, while the remaining samples (or a subset of those) are used for training the base models. We furthermore build each base model from a bootstrap sample following the original RSB method. Note that in the case\nwhere $\\mathbf{z}^{(t)}$ is the empty set, which can happen when all the $\\alpha_j$ parameters are very low, we set $f_{\\mathbf{z}^{(t)}}$ to a constant model that\nalways returns the mean value of the output in the bootstrap sample (for\nregression problems) or the majority class (for classification problems). \n\n\\subsubsection{Updating the Gradient}\n\\label{sec:gradient_update}\n\nThe above procedure allows us to estimate the gradient and to perform one\ngradient descent step.  However, after this step, the\ndistribution parameters $\\boldsymbol\\alpha$ are updated to $\\boldsymbol\\beta=\\boldsymbol\\alpha - \\eta \\nabla\nF$ and a new gradient estimation requires to compute $f^{\\boldsymbol\\beta}_{j,0}$\nand $f^{\\boldsymbol\\beta}_{j,1}, \\forall j$. To be able to compute the approximations in\nEqs~\\eqref{eq:cond_expec1} and \\eqref{eq:cond_expec2}, new models $\\{f_{\\mathbf{z}^{(t)}}\\}_{t=1}^T$ must thus in principle be learned by\nsampling each $\\mathbf{z}^{(t)}$ from the new distribution $p(\\mathbf{z} \\vert \\boldsymbol\\beta)$. This\nwould result in a very computationally expensive algorithm where new models are\nlearned after each parameter update.\n\n\nIn order to estimate the effect of a change in the feature selection probabilities $\\boldsymbol\\alpha$  without learning new models, we use the {\\it importance sampling} approximations of the expectations. Given a new vector of feature selection probabilities $\\boldsymbol\\beta \\neq \\boldsymbol\\alpha$, any expectation under $p(\\mathbf{z} \\vert \\boldsymbol\\beta)$ can be approximated through $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$. We have, for any input $\\mathbf{x}_i$:\n\\begin{align}\n\\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\beta)}  & =  \\sum_{t=1}^{\\vert \\mathcal{Z} \\vert}  \\frac{p(\\mathbf{z} \\vert \\boldsymbol\\beta)}{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} p(\\mathbf{z} \\vert \\boldsymbol\\alpha) f_{\\mathbf{z}^{t}}(\\mathbf{x}_i)\\\\\n& = \\mathbb{E} \\left[\\frac{p(\\mathbf{z} \\vert \\boldsymbol\\beta)}{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} f_{\\mathbf{z}}(\\mathbf{x}_i) \\right]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}\\\\\n\\label{eq:cond_expec_IS}\n& \\simeq  \\frac{1}{T} \\sum_{t=1}^T \\frac{p(\\mathbf{z}^{(t)} \\vert \\boldsymbol\\beta)}{p(\\mathbf{z}^{(t)} \\vert \\boldsymbol\\alpha)} f_{\\mathbf{z}^{(t)}}(\\mathbf{x}_i),\n\\end{align} \nwhere the feature subsets $\\{\\mathbf{z}^{(t)}\\}_{t=1}^T$ in Eq~\\eqref{eq:cond_expec_IS} have been sampled from $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$. Similarly, we have:\n\\begin{align}\n\\label{eq:f0}\nf^{\\boldsymbol\\beta}_{j,0}(\\mathbf{x}_i) & \\simeq \\frac{1}{T_{j,0}} \\sum_{t:z_j^{(t)}=0} \\frac{p(\\mathbf{z}^{(t)}_{-j} \\vert \\boldsymbol\\beta_{-j})}{p(\\mathbf{z}^{(t)}_{-j} \\vert \\boldsymbol\\alpha_{-j})} f_{\\mathbf{z}^{(t)}}(\\mathbf{x}_i),\\\\\n\\label{eq:f1}\nf^{\\boldsymbol\\beta}_{j,1}(\\mathbf{x}_i) & \\simeq \\frac{1}{T_{j,1}} \\sum_{t:z_j^{(t)}=1} \\frac{p(\\mathbf{z}^{(t)}_{-j} \\vert \\boldsymbol\\beta_{-j})}{p(\\mathbf{z}^{(t)}_{-j} \\vert \\boldsymbol\\alpha_{-j})} f_{\\mathbf{z}^{(t)}}(\\mathbf{x}_i).\n\\end{align}\nUsing these approximations, the expectations $\\mathbb{E}[f_\\mathbf{z}]_{p(\\mathbf{z} \\vert \\boldsymbol\\beta)} $, $f^{\\boldsymbol\\beta}_{j,0}$ and $f^{\\boldsymbol\\beta}_{j,1}$ can be estimated for any $\\boldsymbol\\beta$ by using the ensemble of models $\\{f_{\\mathbf{z}^{(t)}}\\}_{t=1}^T$ obtained when the $\\mathbf{z}^{(t)}$ were sampled from $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$.  \n\n\nAs shown by Eq.\\eqref{eq:cond_expec_IS}, the importance sampling approximation consists of a weighted average of the functions $f_{\\mathbf{z}^{(t)}}$, using weights $w_t = \\frac{p(\\mathbf{z}^{(t)} \\vert \\boldsymbol\\beta)}{ p(\\mathbf{z}^{(t)} \\vert \\boldsymbol\\alpha)}$. When $\\boldsymbol\\beta$ becomes very different from $\\boldsymbol\\alpha$, some models $f_{\\mathbf{z}^{(t)}}$ might be hardly used for the importance sampling approximation because they have a very low weight $w_t$. The effective number of used models can be computed as \\cite{doucet01}:\n\\begin{equation}\nT_{eff} = \\frac{\\left( \\sum_{t=1}^T w_t \\right)^2}{\\sum_{t=1}^T w_t^2}.\n\\end{equation}\nWith imbalanced weights, the importance sampling approximation is equivalent to averaging over $T_{eff}$ models. When $T_{eff}$ is too low, the gradient estimation thus becomes unreliable. When this happens, we train $T$ new models $f_{\\mathbf{z}^{(t)}}$ by sampling the feature subsets $\\mathbf{z}^{(t)}$ from the current distribution $p(\\mathbf{z} \\vert \\boldsymbol\\beta)$. In practice, new models are trained as soon as $T_{eff}$ drops below $\\frac{T}{2}$, with a maximum of 100 gradient descent steps between two training phases. \n\n\n\\subsection{Discussion}\n\\label{sec:PRSB_discussion}\n\nThe proposed algorithm has the advantage of being model-agnostic in\nthat any supervised learning method can be used to fit the $f_{\\mathbf{z}^{(t)}}$\nmodels. Despite the use of gradient descent, no hypothesis of\ndifferentiability is required for the model family. The framework can\nalso be easily adapted to any differentiable loss. \n\n\\paragraph{Computational Complexity}\n\nOnce the models are trained, the computation of the gradient is linear with respect to the number $N$ of samples, the number $M$ of features and the number $T$ of base models in the ensemble. The costliest step of the algorithm is the construction of the base models. The complexity of the construction of the models depends on the type of model, but note that each model is grown only from a potentially small subset of features.\n\n\n\n\\paragraph{Regularization}\n\nWhile we have not\nused any regularization term in (\\ref{eq:global_optim}), incorporating\none is straightforward (and this will be exploited for image classification and the inference of gene regulatory networks). A natural regularization term to enforce sparsity\ncould be simply the sum $\\sum_{j=1}^M \\alpha_j$, which can be nicely\ninterpreted as $\\mathbb{E}[||\\mathbf{z}||_0]_{p(\\mathbf{z}|\\boldsymbol\\alpha)}$, i.e., the\naverage size of the subsets drawn from $p(\\mathbf{z}|\\boldsymbol\\alpha)$. Adding this\nterm to (\\ref{eq:global_optim}) with a regularization coefficient\n$\\lambda$ would simply consists in adding $\\lambda$ to the gradient in\n(\\ref{eq:gradient}). We did not systematically include such regularization in\nour experiments below to reduce the number of hyper-parameters. Despite\nthe lack of regularization, the algorithm has a natural propensity for\nselecting few features. Incorporating a useless\nfeature $j$ will indeed often deteriorates the quality of the\npredictions and lead to a decrease of the corresponding $\\alpha_j$. \nNote however that the sparsity of the resulting selection weights will depend on the robustness of the learning algorithm to the presence of irrelevant features. This will be illustrated in our experiments.\n\n\\paragraph{Related Works}\n\nOur method has direct connections with Random Subspace ensemble\nmethods \\cite{ho98,panov07,louppe12}. In addition to providing a feature\nranking, it has the obvious added flexibility w.r.t. Random Subspace that the feature\nsampling distribution (and thus also the subspace size) is\nautomatically adapted to improve performance. Our approach also falls\ninto the general family of wrapper feature selection methods. Among this family, the closest works are the\napproaches based on Estimation of Distribution Algorithms (EDA,\n\\cite{armananzas08}). EDA are generic optimization techniques that\nbelong to the family of evolutionary algorithms. When applied to\nfeature selection, they iteratively generate and evaluate a population\nof feature subsets $\\{\\mathbf{z}^{(t)}\\}_{t=1}^T$. In particular, EDA\nidentify a probability distribution $p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$ from the\nbest solutions found at the previous iteration and use this\nprobability distribution to generate new solutions. EDA and our parametric RSB\napproach are thus similar in the sense that they both iteratively\nsample feature subsets from an explicit probability distribution\n$p(\\mathbf{z} \\vert \\boldsymbol\\alpha)$ and update the\nlatter. However, the goal of EDA is to minimize the expected value of\nthe loss function:\n\\begin{equation}\n\\label{eq:optim_eda}\n  \\min_{\\boldsymbol\\alpha} E \\left[\\frac{1}{N}\\sum_{i=1}^N\n    L(y_i, f_\\mathbf{z}(\\mathbf{x}_i))\\right]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)},\n\\end{equation}\nwhile we are trying to minimize the loss of the {\\it ensemble} model\n$\\mathbb{E}[f_{\\mathbf{z}}(\\mathbf{x})]_{p(\\mathbf{z}|\\boldsymbol\\alpha)}$ (see\nEq.(\\ref{eq:global_optim})). Both approaches also greatly differ in\nthe optimization technique: EDA iteratively update $p(\\mathbf{z}|\\boldsymbol\\alpha)$\nfrom the best solutions in the current population, while our approach\nis based on gradient descent and importance sampling. Finally and most importantly, contrary to our approach, EDA focuses exclusively on the identification of important features and does not train any predictive model.\n\nOur optimization procedure has also some links with variational\noptimization (VO, \\cite{staines13}).  VO is a general technique for\nminimizing a function $F(x)$ that is non-differentiable or combinatorial. It is based\non the bound:\n\\begin{equation}\n\\label{eq:VO}\n\\min_x F(x) \\leq E[F(x)]_{p(x \\vert \\theta)} = U(\\theta).\n\\end{equation}\nInstead\nof minimizing $F$ with respect to $x$, one can thus minimize the upper\nbound $U$ with respect to $\\theta$. Provided the distribution $p(x\n\\vert \\theta)$ is rich enough, this is equivalent to minimizing\n$F(x)$. In the same spirit as VO, the introduction of the distribution\n$p(\\mathbf{z}|\\boldsymbol\\alpha)$ and the use of an ensemble model in our problem\nformulation in (\\ref{eq:global_optim}) can be interpreted as a way to\ndefine a more tractable proxy to the original combinatorial feature selection\nproblem, i.e.:\n\\begin{equation}\\label{eq:feat_sel}\n\\min_{\\mathbf{z}\\in \\mathcal{Z}} \\frac{1}{N} \\sum_{i=1}^N L(y_i,f_{\\mathbf{z}}(\\mathbf{x}_i)).\n\\end{equation}\nLike in VO, the formulation in (\\ref{eq:global_optim}) allows us to\nuse gradient descent optimization despite the fact that the models\n$f_{\\mathbf{z}}$ are not necessarily differentiable. Note however that VO uses the expectation $\\mathbb{E} [F(x)]$ as an upper bound of the function $F$ that it seeks to minimize, while in our case the expectation is exactly what we want to minimize. Note also that our objective\nfunction is not an upper bound of (\\ref{eq:feat_sel}), as an ensemble is more flexible than its individual\nconstituents. \n\nSeveral works have used gradient descent to solve the feature selection problem in Eq.~\\eqref{eq:feat_sel}, by using a continuous relaxation of the discrete variables $\\mathbf{z}$ \\cite{sheth20,yamada20,dona21}. However, these methods are designed to be used with differentiable models (neural networks, polynomial models), so that both the feature selection and the model parameters can be updated in a single gradient descent step, while our approach is model-agnostic.\n\n\n\\section{Results}\n\n As base model $f_\\mathbf{z}$, we used either a CART decision tree \\cite{cartbook}, a $k$-nearest neighbors (kNN) model \\cite{altman92} with $k=5$ or a support vector machine (SVM)  \\cite{boser92} with a radial basis function kernel and the (inverse) regularization parameter $C$ set to 1.\n\nWe report the predictive performance with the mean square error for regression problems and the misclassification rate for classification problems. A ranking of features can be obtained by sorting them by decreasing value of optimized importances $\\boldsymbol\\alpha$. If the relevant variables are known, the feature ranking can be evaluated using the area under the precision-recall curve (AUPR). A perfect ranking (i.e. all the relevant features have a higher importance than the irrelevant ones) yields an AUPR of 1, while a random ranking has an AUPR close to the proportion of relevant features. \n\nWe compare our approach to three baselines: the standard RSB method, Random Forest (RF) and Gradient Boosting with Decision Trees (GBDT). When evaluating feature rankings, we also compare ourselves to Estimation of Distribution Algorithms (EDA). Implementation details for all the methods are provided in Appendix~\\ref{sec:implementation}. \n\n\\subsection{Simulated Problems}\n\\label{sec:simulated}\n\n\n\n\nWe simulated four problems, for which the relevant feature are known (see Appendix~\\ref{sec:simulation_protocol} for the detailed simulation protocol). Compared to single base models and standard RSB, our parametric RSB method (PRSB) yields lower errors for all the base models (Table~\\ref{tab:sim_base_models}, Figure~\\ref{fig:sim_TS_errors}). The improvement of performance over standard RSB is larger in the case of kNN and SVM, compared to decision trees. This can be explained by the fact a decision tree, contrary to kNN and SVM, has an inner feature selection mechanism and is hence able to maintain a good performance even in the presence of irrelevant features. Therefore, for a given irrelevant feature $j$, the difference between $f^{\\boldsymbol\\beta}_{j,0}$ and $f^{\\boldsymbol\\beta}_{j,1}$ (Eq~\\eqref{eq:f0} and Eq~\\eqref{eq:f1} respectively) will be lower in the case of trees, which can prevent the corresponding $\\alpha_j$ to decrease towards zero during the gradient descent. \n\n\\begin{table}[t]\n\\caption{{\\bf Results on the Hypercube Problem.} The table shows the misclassification rates on the test set (TS error) obtained with three model types: the single base model built using all the features, the standard RSB ensemble model, where the number $K$ of randomly sampled features is tuned by 10-fold cross-validation, and our parametric RSB method (PRSB). Lowest errors are indicated in bold type. The last column indicates for RSB the tuned value of $K$ and for PRSB the sum of optimized $\\boldsymbol\\alpha$ (i.e. the average number of selected variables per base model). Values are mean and standard deviation over 10 datasets.}\n\\label{tab:sim_base_models}\n\\begin{center}\n\\begin{tabular}{llcc}\n\\toprule\n \\multicolumn{2}{c}{Model} & TS error & $K$ \/ $\\sum \\boldsymbol\\alpha$ \\\\ \n\\midrule\ntree & Single & 0.24 $\\pm$ 0.09 &\\\\ \n& RSB & 0.18 $\\pm$ 0.12 & 259.10 $\\pm$ 70.11\\\\ \n& PRSB & {\\bf 0.12} $\\pm$ {\\bf 0.03} & 14.41 $\\pm$ 10.01\\\\ \n\\midrule\nkNN & Single & 0.44 $\\pm$ 0.05 &\\\\ \n& RSB & 0.41 $\\pm$ 0.07 & 64.70 $\\pm$ 39.88\\\\ \n& PRSB & {\\bf 0.09} $\\pm$ {\\bf 0.03} & 7.49 $\\pm$ 5.31\\\\ \n\\midrule\nSVM & Single & 0.34 $\\pm$ 0.10 &\\\\ \n& RSB & 0.34 $\\pm$ 0.09 & 187.90 $\\pm$ 78.95\\\\ \n& PRSB & {\\bf 0.12} $\\pm$ {\\bf 0.04} & 18.82 $\\pm$ 9.22\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[t]\n\\centering\n\\caption{{\\bf Results on the Simulated Problems.} We report here the prediction error on the test set (TS) and the feature ranking quality (AUPR). Values are mean and standard deviation over 10 datasets. Lowest errors and highest AUPRs are indicated in bold type.}\n\\label{tab:sim_comp}\n\\begin{tabular}{llccccc}\n\\toprule\n& & RF & GBDT & PRSB - tree & PRSB - kNN & PRSB - SVM \\\\ \n\\midrule\nCheckerboard & TS error & 8.91 $\\pm$ 0.87 & 9.05 $\\pm$ 1.01 & 5.26 $\\pm$ 1.15 & {\\bf 3.68} $\\pm$ {\\bf 0.54} & 3.85 $\\pm$ 0.54\\\\ \n& AUPR & 0.37 $\\pm$ 0.13 & 0.40 $\\pm$ 0.14 & 0.67 $\\pm$ 0.31 & {\\bf 0.71} $\\pm$ {\\bf 0.23} & 0.57 $\\pm$ 0.22\\\\ \n\\midrule\nFriedman & TS error & 2.78 $\\pm$ 0.27 & 1.26 $\\pm$ 0.13 & 1.61 $\\pm$ 0.21 & 1.26 $\\pm$ 0.11 & {\\bf 1.12} $\\pm$ {\\bf 0.21}\\\\ \n& AUPR & 0.68 $\\pm$ 0.06 & 0.89 $\\pm$ 0.04 & 0.99 $\\pm$ 0.03 & 0.98 $\\pm$ 0.06 & {\\bf 1.00} $\\pm$ {\\bf 0.01}\\\\ \n\\midrule\nHypercube & TS error & 0.18 $\\pm$ 0.11 & 0.17 $\\pm$ 0.13 & 0.12 $\\pm$ 0.03 & {\\bf 0.09} $\\pm$ {\\bf 0.03} & 0.12 $\\pm$ 0.04\\\\ \n& AUPR & 0.81 $\\pm$ 0.25 & 0.78 $\\pm$ 0.27 & 0.89 $\\pm$ 0.11 & {\\bf 0.96} $\\pm$ {\\bf 0.08} & 0.91 $\\pm$ 0.11\\\\ \n\\midrule\nLinear & TS error & 0.21 $\\pm$ 0.03 & 0.17 $\\pm$ 0.03 & 0.19 $\\pm$ 0.04 & 0.15 $\\pm$ 0.04 & {\\bf 0.08} $\\pm$ {\\bf 0.01}\\\\ \n& AUPR & 0.66 $\\pm$ 0.12 & 0.69 $\\pm$ 0.13 & 0.57 $\\pm$ 0.09 & 0.60 $\\pm$ 0.13 & {\\bf 0.84} $\\pm$ {\\bf 0.11}\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\nNote also that the standard RSB model greatly improves over the single model only in the case of decision trees. The decision tree being a model that is prone to high variance, its prediction performance is indeed usually improved by using ensemble methods \\cite{breiman-1996-bagging,breiman-2001-randforests,geurts06-ml}. On the other hand, since kNN and SVM have a sufficiently low variance, their performance is not improved with a standard ensemble. \n\nWhile the degree of randomization is controlled by the parameter $K$ (i.e. the number of randomly sampled features for each base model) in standard RSB, it has the advantage to be automatically tuned in parametric RSB. Tables~\\ref{tab:sim_base_models} and \\ref{tab:sim_alphas_sum} indicate the sum of optimized parameters $\\boldsymbol\\alpha$ for each PRSB model, which is equivalent to $\\mathbb{E}[\\vert \\vert \\mathbf{z} \\vert \\vert_0]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)}$, i.e. the average number of selected variables per base model.  By comparing this average number to the parameter value $K$ of RSB, we can see that PRSB effectively selects a much lower number of features, while no explicit constraint on sparsity is used during model training. The average number of selected variables remains however slightly higher than the actual number of relevant features, indicating that a certain degree of randomization is introduced during model construction.\n\nOverall, PRSB outperforms RF and GBDT both in terms of predictive performance and feature ranking (Table \\ref{tab:sim_comp}), the best performance being obtained with kNN and SVM. Overall, our method also returns better feature rankings than EDA (Figure~\\ref{fig:sim_EDA}).\n\n\n\n\\subsection{Real-World Problems}\n\n\\begin{table*}[t]\n\\caption{{\\bf Average Misclassification Rates on the Real Datasets.} For each model type, lowest errors are indicated in bold type.}\n\\label{tab:scikit_RSB_PRSB}\n\\begin{center}\n\\begin{tabular}{lccccccccc}\n\\toprule\n & \\multicolumn{3}{c}{Base model = tree} & \\multicolumn{3}{c}{Base model = kNN} & \\multicolumn{3}{c}{Base model = SVM}\\\\ \n & Single & RSB & PRSB & Single & RSB & PRSB & Single & RSB & PRSB\\\\ \n\\midrule\narcene& 0.30& 0.59& {\\bf 0.23}& {\\bf 0.20}& 0.36& 0.21& 0.44& 0.44& {\\bf 0.35}\\\\ \nCLL\\_SUB\\_111& 0.40& {\\bf 0.21}& 0.22& 0.45& 0.51& {\\bf 0.43}& {\\bf 0.49}& {\\bf 0.49}& {\\bf 0.49}\\\\ \nProstate\\_GE& 0.19& {\\bf 0.08}& 0.09& 0.16& {\\bf 0.11}& 0.12& 0.16& {\\bf 0.13}& 0.33\\\\ \nSMK\\_CAN\\_187& 0.38& {\\bf 0.33}& 0.34& 0.34& {\\bf 0.33}& 0.34& 0.32& {\\bf 0.30}& 0.36\\\\ \nTOX\\_171& 0.49& 0.46& {\\bf 0.19}& 0.23& 0.27& {\\bf 0.16}& 0.64& 0.74& {\\bf 0.53}\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nWe retrieved biological, classification datasets from the {\\it scikit-feature} repository \\cite{scikit-feature}. These datasets have the particularity to have very few ($\\sim 100$) samples for several thousands features. Among the biological datasets available in the repository, we filtered out datasets and classes in order to have only datasets with at least 30 samples per class. The final dataset sizes are indicated in Table~\\ref{tab:scikit_feature_datasets}. \n\nTable~\\ref{tab:scikit_RSB_PRSB} shows the average misclassification rates, estimated with 5-fold cross-validation. PRSB either outperforms RSB or has an equivalent misclassification rate (the exceptions being on the Prostate\\_GE and SMK\\_CAN\\_187 datasets with SVM). PRSB models also tend to be much sparser than RSB ensembles, when comparing the (expected) number of selected features per base model (Table~\\ref{tab:scikit_feature_RSB_PRSB_K}). PRSB combined with trees or kNNs tends to be on par with RF and GBDT, while PRSB-SVM yields the worse performance  (Table~\\ref{tab:scikit_feature_RSB_PRSB_comp}). Note that a single SVM already tends to be the worse performer compared to a single decision tree or kNN (Table~\\ref{tab:scikit_RSB_PRSB}).\n\n\n\\clearpage\n\\subsection{MNIST}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{Figures\/mnist} \n\\caption{{\\bf PRSB Feature Selection Probabilities on MNIST.} The first row shows three exemples of (noisy) images from the dataset.  The second and third rows show the optimized values of the parameters $\\boldsymbol\\alpha$, obtained with kNN and for different values of the regularization coefficients $\\lambda_1$ and $\\lambda_2$. Increasing $\\lambda_1$ enforces sparsity, while increasing $\\lambda_2$ forces the parameters of neighbouring pixels to be similar.}\n\\label{fig:mnist}\n\\end{figure}\n\nWe applied our method to classify images of handwritten digits 5's and 6's. The images were taken from the MNIST dataset \\cite{mnist} and random noise was added to them to make the task more challenging (Figure~\\ref{fig:mnist}). We used the image pixels as individual features and we combined PRSB with the following objective function:\n\n\\begin{multline}\nF(\\boldsymbol\\alpha) = \\frac{1}{N} \\sum_{i=1}^N L  \\left( y_i,  \\mathbb{E}[f_\\mathbf{z}(\\mathbf{x}_i)]_{p(\\mathbf{z} \\vert \\boldsymbol\\alpha)} \\right) + \\lambda_1 \\sum_{j=1}^W \\sum_{k=1}^H \\alpha_{j,k} \\\\\n+ \\lambda_2 \\left( \\sum_{j=2}^H \\vert \\alpha_{j,k} - \\alpha_{j-1, k} \\vert + \\sum_{k=2}^W \\vert \\alpha_{j, k} - \\alpha_{j, k-1} \\vert \\right),\n\\end{multline}\nwhere $W$ and $H$ are respectively the width and height of the image, and $\\alpha_{j,k}$ is the selection probability for the pixel in the $j$-row and $k$-th column. The second term is a regularization term that enforces sparsity, while the last term penalizes large differences between the $\\alpha_{j,k}$ parameters corresponding to neighbouring pixels (Figure~\\ref{fig:mnist}). Such regularization is known as the {\\it fused lasso} \\cite{tibshirani05} and allows us to account for the spatial structure of the features. As shown on Figure~\\ref{fig:mnist}, without any regularization ($\\lambda_1=0, \\lambda_2=0$), the pixels with the highest feature selection probabilities tend to be spread over the whole digit, while they cluster around the bottom-left of the digit as regularization is increased.\n\nWe used a grid-search strategy to tune the values of the hyper-parameters $\\lambda_1$ and $\\lambda_2$, using as validation set 20\\% of the samples of the original training set. The resulting misclassification rates of PRSB are higher than GBDT, but equivalent to RF (Table~\\ref{tab:mnist}).\n\n\n\n\\begin{table}[t]\n\\caption{{\\bf Misclassification Rates on the MNIST Test Set.}}\n\\label{tab:mnist}\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{ccccc}\n\\toprule\nRF & GBDT &  PRSB-tree & PRSB-kNN & PRSB-SVM\\\\\n\\midrule\n0.031 & 0.018 & 0.028 & 0.043 & 0.039\\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Gene Network Inference}\n\nAn open problem in computational biology is the reconstruction of gene regulatory networks, which attempt to explain the joint variability in the expression levels of a group of genes through a sparse pattern of interactions. One approach to gene network reconstruction is the application of a feature selection approach that identifies the regulators of each target gene. Such approach is used by GENIE3, one of the current state-of-the-art network inference algorithms \\cite{huynhthu10}. This method learns for each target gene a RF model predicting its expression from the expressions of all the candidate regulators, and identifies the regulators of that target gene through the RF-based feature importance scores. The PRSB and EDA approaches can be used in the same way for gene network inference, with however the advantage that the base models are not restricted to decision trees. Furthermore, while in GENIE3 the different models, corresponding to the different target genes, are learned independently of each other, PRSB can be extended to introduce a global constraint on the topology of the network. More specifically, we use a joint regularizer (with a coefficient $\\lambda$) that enforces modular networks, a property often encountered in real gene regulatory networks (see details in Section~\\ref{sec:PRSB_network_obj}).\n\nWe evaluate the ability of PRSB to reconstruct the five 100-gene networks of the DREAM4 {\\it Multifactorial Network} challenge \\cite{marbach10,marbachcostello12}, for which GENIE3 was deemed the best performer. The DREAM4 networks are artificial networks for which the true regulatory links are known and an AUPR can thus be computed given a ranking of links. To reconstruct each network, a simulated gene expression dataset with 100 samples was made available to the challenge participants. \n\nIn the absence of a validation set\\footnote{Since each dataset contains only 100 samples, we did not use a subset of the dataset as validation set, to avoid reducing the training set size.}, a criterion other than the prediction performance must be chosen for tuning the value of the regularization coefficient $\\lambda$. As $\\lambda$ determines the number of used candidate regulators (Figure~\\ref{fig:dream4_nreg}), we selected the largest value of $\\lambda$ such that the average $\\boldsymbol\\alpha_{., g}$ sum (i.e. the average number of selected candidate regulators per base model) is higher than 1. The resulting AUPRs are given in Table~\\ref{tab:dream4_aupr}. The highest AUPRs are obtained with the regularized PRSB combined with SVM and kNN, while EDA is the worse performer.\n\n\\begin{table}\n\\caption{{\\bf AUPRs Obtained on the DREAM4 Networks.}}\n\\label{tab:dream4_aupr}\n\\begin{small}\n\\begin{center}\n\\begin{tabular}{llccccc}\n\\toprule\n& & Net1 & Net2 & Net3 & Net4 & Net5 \\\\ \n\\midrule\nGENIE3 & RF & 0.18 & 0.14 & 0.26 & 0.24 & 0.23\\\\\n\\midrule\nEDA & tree & 0.05 & 0.06 & 0.10 & 0.10 & 0.10\\\\\n& kNN & 0.07 & 0.09 & 0.10 & 0.10 & 0.08\\\\\n& SVM & 0.08  & 0.07 & 0.13 & 0.10 & 0.10\\\\\n\\midrule\nPRSB & tree & 0.15 & 0.11 & 0.25 & 0.21 & 0.22\\\\\n$\\lambda=0$ & kNN & 0.17 & 0.15 & 0.24 & 0.22 & 0.24\\\\\n& SVM & {\\bf 0.19} & 0.15 & 0.25 & 0.22 & 0.24\\\\\n\\midrule\nPRSB & tree & 0.15 & 0.16 & 0.25 & 0.24 & 0.22\\\\\n$\\lambda >0$ & kNN & 0.18 & 0.20 & 0.27 & {\\bf 0.26} & 0.25\\\\\n& SVM & {\\bf 0.19} & {\\bf 0.22} & {\\bf 0.28} & {\\bf 0.26} & {\\bf 0.27}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{small}\n\\end{table}\n\n\n\\section{Conclusions}\n\nWe proposed a model-agnostic ensemble method \nbased on the idea of averaging base\nmodels independently trained on feature subsets sampled from a Bernoulli\ndistribution. We show that the parameters of the latter distribution\ncan be trained using gradient descent even if the base models are not\ndifferentiable. The required iterative gradient computations can\nfurthermore be performed efficiently by exploiting importance\nsampling. The resulting approach has interesting features: it\nis model agnostic, it can use any combination of differentiable loss\nfunction and regularization term, and it provides variable importance\nscores. Experiments show that PRSB improves over standard RSB, or is at least equivalent to it. On the simulated datasets, PRSB generally outperforms its competitors (RF, GBDT and EDA) both in terms of predictive performance and feature ranking quality. On the real problems, PRSB is competitive with RF and GBDT. We also showed that an appropriate regularization strategy allows PRSB to outperform the state-of-the-art GENIE3 in the inference of gene regulatory networks.\n\nWhile we adopted an ensemble strategy, the same\noptimization technique, combining gradient descent and importance\nsampling, can be used to solve the feature selection problem as\ndefined in (\\ref{eq:optim_eda}) and addressed also by EDA. It would be\ninteresting to investigate this approach and compare it with the\nensemble version explored in this paper. Note however that it would\nrequire to exploit a stronger learning algorithm, because it would not\nbenefit from the ensemble averaging effect. Applying this technique, and its associated derivation of feature importance scores, on\ntop of modern deep learning models would be also highly desirable\ngiven the challenge to explain these models. This would require\nhowever to develop specific strategies to reduce the non negligible\ncomputational burden of the approach. Finally, exploiting more complex\nfeature subset distributions, beyond independent Bernoulli\ndistributions, would be also very interesting  but adapting the\noptimization strategy might not be trivial.\n\n\n\\section*{Acknowledgements}\nWe thank Antoine Wehenkel for the helpful discussions on the methodology. This work was supported by Service Public de Wallonie Recherche under Grant No. 2010235 - ARIAC by DIGITALWALLONIA4.AI. Computational resources have been provided by the Consortium des \\'Equipements de Calcul Intensif (C\\'ECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region.\n\n\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA unifying idea for most modelling techniques used for multibody system (MBS) dynamics\nis to describe the equations of motion in terms of generalized\ncoordinates and generalized velocities. In classical mechanics of\nconstrained systems, a generalized velocity is taken to be an\nelement of tangential space of configuration manifold, and a\ngeneralized force is taken to be the cotangent space. However,\nneither space possesses a natural metric as the generalized\ncoordinates or the constrains may have a combination of rotational\nand translational components. As a result, the corresponding dynamic\nformulation is not invariant and a solution depends on measure units\nor a weighting matrix selected\n\\cite{Lipkin-Duffy-1988,Manes-1992,,Aghili-Piedboeuf-2003a,Angeles-2003,Aghili-2005,Luca-Manes-1994,Aghili-2015b,Aghili-2020a}. \n\nTransformation of the Lagrange dynamics into quasi-Lagrange dynamics, and with feedback force\/position control,  have been established in the literature \\cite{McClamroch-Wang-1988,Brogliato-Niculescu-1997,Brogliato-2014}. Mathematical models for constrained robot dynamics, incorporating the effects of constraint force required to maintain satisfaction of the constraints, and tracking using feedback control were presented  in \\cite{McClamroch-Wang-1988}. The problem of the control of a class of mechanical systems with a finite number of degrees-of-freedom, subject to unilateral constraints on the position is presented in \\cite{Brogliato-Niculescu-1997}. Various switching control strategies are analysed in this work based on a nonsmooth dynamics formulation.  The use of kinetic quasi-velocities was also developed for a particular case in \\cite{Brogliato-1996,Brogliato-2016}.  Alternatively, the dynamics of a multibody system can be formulated in terms of the vector of quasi-velocities, i.e., a vector whose Euclidean norm is proportional to the {\\em square root} of the system's\nkinetic energy. It is known that this formulation can lead to simplification of the equations of unconstrained MBSs\n\\cite{Koditschek-1985,Gu-Loh-1987,Spong-1992,Rodriguez-Delgado-1992,Aghili-2020a,Bahar-1994,Jain-Rodriguez-1995,Aghili-2021a,Kozlowski-1998,Papastavridis-1998,Gu-2000,Herman-2005,Herman-Kolowski-2006,Aghili-Buehler-Hollerbach-1997a,Lduha-Ravani-1995,Junkins-Schaub-1997,Aghili-Buehler-Hollerbach-2001,Sinclair-Hurtado-Junkins-2006,Aghili-2007c,Bedrossian-1992,Aghili-2021a}. \nIn short, the square-root\nfactorization of the mass matrix is used as a transformation to obtain\nthe quasi-velocities, which are a linear combination of the\nvelocity and the generalized coordinates\n\\cite{Papastavridis-1998,Herman-Kolowski-2006,Pila-2020,Aghili-2021a}.\n\nIn the literature, the concept of quasi-velocities has been used for dynamics modelling of unconstrained MBSs. The differential variational principle of Jourdain was extended in \\cite{Bahar-1994} to cover the dynamics of impulsive motion formulated in terms of quasi-velocities. It was shown by Kodistchek \\cite{Koditschek-1985} that if the\nsquare-root factorization of the inertia matrix is integrable, then\nthe dynamics can be significantly simplified. In such a case,\ntransforming the generalized coordinates to quasi-coordinates by\nmaking use of the integrable factorization modifies the dynamics to the system of a double integrator. It was later realized by Gu {\\em et al.} \\cite{Gu-Loh-1987} that\nsuch a transformation is a canonical transformation because it\nsatisfies Hamilton's equations. Rather than deriving the mass matrix\nof MBS first and then obtaining its factorization, Rodriguez {\\em et\nal.} \\cite{Rodriguez-Delgado-1992} derived the closed-form\nexpressions of the mass matrix factorization of a MBS and its\ninverse directly from the link geometric and inertial parameters.\nThis eliminates the need for the matrix inversion required to\ncompute the forward dynamics. The interesting question of when the factorization of the inertia matrix is integrable, i.e., the factorization being the Jacobian of\nsome quasi-coordinates, was addressed independently in\n\\cite{Spong-1992} and \\cite{Bedrossian-1992}.  It was shown that Riemannian manifold defined by the inertia matrix should be locally flat. The advantages of using the notion quasi-velocities for control of unconstrained manipulators have been recognized by many researchers and\nvarious setpoint PD controllers based on the quasi-velocities\nfeedbacks have been proposed\n\\cite{Jain-Rodriguez-1995,Kozolowski-Herman-2000,Herman-Kozlowski-2001,Herman-2005}.  A closed inverse dynamic formulation by the Lagrangian approach in terms of quasi-coordinates for the general Stewart platform manipulator is presented in \\cite{Chen-2003}. The quasi-Lagrange equations are derived with and without friction in \\cite{Brogliato-2014}  based on  quasi-velocities computed with the kinetic metric of a Lagrangian system. The quasi-Lagrange dynamics involves  the mass matrix inversion that allows for a clear splitting between normal and tangential dynamics. Generalization of Lagrangian dynamics equations by taking into account the unilateral and bilateral contacts as well as frication  can be also found in \\cite{Brogliato-2016}. In spite of different quasi-Lagrange dynamics formulations proposed in the literature, preservation of homogeneous units yet needs to be rendered in these  formulations.\n\n\nA problem that often arises in motion\/force control of MBSs with minimum solution to joint rate or force, is that generalized coordinate  may have a combination of rotational and\ntranslational components that can be even compounded by having\ncombination of rotational and translational constraints\n\\cite{Doty-Melchiorri-Bonivento-1993}. This may lead to inconsistent\nresults unless adequate weighting matrices are used\n\\cite{Doty-Melchiorri-Bonivento-1993,Manes-1992,Featherstone-Fijany-1999,Featherstone-Thiebaut-Khatib-1999,Aghili-2005}.\nFor example, the minimum joint rate rates\nor minimum norm force are not meaningful\nquantities if the MBS has both revolute and prismatic joints\n\\cite{Doty-Melchiorri-Bonivento-1993}. The contribution of this paper is to extend the concept of square-root factorization of inertia matrix to define homogeneous  vectors of quasi-velocities and quasi-forces\nfor dynamics formulation of constrained MBS that can be\nused for simulation, analysis, and control purposes. In this paper, we introduce new state and input variables comprising of reduced quasi-velocities, input quasi-forces, and constraint quasi-forces by making use of Cholesky decomposition and adequate orthogonal matrices in order to derive Lagrangian dynamics of constrained mechanical systems.\nThe advantages of the square--root factorization based formulation of the constrained Lagrangian dynamics is that every  vectors of quasi-velocities,  input quasi-forces, or constraint quasi-forces all have the same physical units. Therefore, unlike other approaches\n\\cite{Doty-Melchiorri-1993,Luca-Manes-1994,Schutter-Bruyinckx-1996,Aghili-2005},\nthis formulation does not require any weighting matrix when the generalized coordinates or the constraints have both translational and rotational components.\nFurthermore, the equations of motions and\nthe equation of constraints are decoupled in such a way that separate control\ninputs are associated to each set of equations, which facilitates\nmotion\/force control of constrained systems such as robotic\nmanipulators. This paper is organized as follows: Section~\\ref{sec:Modeling} presents the derivation of constrained Lagrangian dynamics formulation based on the notion of reduced quasi-velocities and decoupling quasi-forces. Properties of the dynamic formulation that could be useful for control purposes are presented in Sections~\\ref{sec:properties}. Finally, Section~\\ref{sec:control} is devoted to force\/motion control based on reduced quasi-velocities and quasi-forces.\n\n\\section{Quasi-Variables Transformation} \\label{sec:Modeling}\n\nThe kinetic energy of a MBS has the following quadratic form:\n\\begin{equation} \\label{eq:K}\nK(\\bm q, \\dot{\\bm q})=\\frac{1}{2} \\dot {\\bm q}^T \\bm M(\\bm q)\n\\dot{\\bm q},\n\\end{equation}\nwhere vectors $\\bm q\\in \\mathbb{R}^n$ and  $\\dot{\\bm q}\\in \\mathbb{R}^n$ are the  generalized\ncoordinates and generalized velocities, and $\\bm M(\\bm q)$ is the generalized inertia matrix, which is {\\em\nsymmetric} and {\\em positive definite} for all $\\bm q$.\nAccording to the {\\em Cholesky decomposition}, the symmetric and\npositive-definite matrix $\\bm M$ can be decomposed into\n\\begin{equation} \\label{eq:M_decompose}\n\\bm M = \\bm Q \\bm Q^T,\n\\end{equation}\nwhere $\\bm Q$ is a lower--triangular matrix\nwith strictly positive-diagonal elements; $\\bm Q$ is also called\nthe {\\em Cholesky triangle}. The following formula can be used\nto obtain the Cholesky triangle through some elementary operations\n\\begin{align} \\label{eq:Cholesky}\nQ_{ii} & = \\big( M_{ii} - \\sum_{k=1}^{i-1} Q_{ik}^2 \\big)^{1\/2}\n\\quad  \\quad \\quad \\forall i=1,\\cdots,n \\\\ \\nonumber Q_{ji} &= \\big(\nM_{ji} - \\sum_{k=1}^{i-1} Q_{jk}Q_{ik} \\big)\/Q_{ii} \\quad \\quad\n\\forall j=i+1,\\cdots,n\n\\end{align}\nSince $\\bm Q$ is a lower-triangular matrix, its inverse can be\nsimply computed by the back substitution technique.\n\nThe dynamics equations of a constrained MBS with kinetic energy $K$ can be derived by the {\\em Euler--Lagrange} equations\n\\begin{subequations}\n\\begin{align} \\label{eq:constraint_lagrange}\n\\frac{\\rm d}{{\\rm d}t} \\left( \\frac{\\partial K}{\\partial \\dot{\\bm\nq}} \\right) &- \\frac{\\partial K}{\\partial \\bm q} = \\bm\\tau - \\bm A^T \\bm\\lambda\\\\ \\label{eq:Adotq}\n\\bm A(\\bm q) \\dot{\\bm q} &= \\bm 0\n\\end{align}\n\\end{subequations}\nHere, $\\bm\\tau=\\bm\\tau_a + \\bm\\tau_f +\\bm\\tau_p$  is the generalized forces containing all applied loads including the actuator forces applied to the joints $\\bm\\tau_a$ and the joint friction $\\tau_f$ plus the conservative forces $\\bm\\tau_p=-\\partial P\/\\partial \\bm\nq$ owing to gravitational energy, vector $\\bm\\lambda \\in \\mathbb{R}^m$ represents the {\\em generalized Lagrangian multipliers}, and matrix $\\bm A \\in\n\\mathbb{R}^{m \\times n}$ is the corresponding constraint matrix associated wit the constraints  imposed on the generalized velocities. Equations \\eqref{eq:Adotq} can be representation of holonomic or non-holonomic constraints. Also, it should be pointed out that $\\bm A$  is not necessarily a full-rank matrix because of the possible redundant constraints. Substituting \\eqref{eq:M_decompose} into \\eqref{eq:K} and\nthen applying \\eqref{eq:constraint_lagrange} yields\n\\begin{align} \\nonumber\n\\bm\\tau - \\bm A^T \\bm\\lambda &= \\frac{\\rm d}{{\\rm d}t}\\big( \\bm Q \\bm Q^T \\dot{\\bm q} \\big)\n- \\frac{1}{2} \\Big( \\frac{\\partial}{\\partial {\\bm q}} \\| \\bm Q^T(\\bm\nq) \\dot{\\bm q} \\|^2 \\Big)^T\\\\ \\notag\n&= \\bm Q \\frac{\\rm d}{{\\rm d}t} \\big(\\bm Q^T \\dot{\\bm q} \\big) + \\dot {\\bm Q} \\bm Q^T \\dot{\\bm q} -  \\frac{\\partial \\big(\\bm Q^T \\dot{\\bm q} \\big) }{\\partial \\bm q} \\bm Q^T \\dot{\\bm q} \\\\ \\label{eq:LLddotq}\n&= \\bm Q \\frac{\\rm d}{{\\rm d}t} \\big(\\bm Q^T \\dot{\\bm q} \\big)+ \\Big( \\dot{\\bm\nQ} - \\frac{\\partial \\big( \\bm Q^T \\dot{\\bm q}\n\\big)}{\\partial \\bm q} \\Big) \\bm Q^T \\dot{\\bm q}\n\\end{align}\nDefine the vectors of {\\em quasi-velocities} and\n{\\em input quasi-forces} as follows:\n\\begin{subequations} \\label{eq:quasi}\n\\begin{align} \\label{eq:v_def}\n\\bm v & \\triangleq \\bm Q^T(\\bm q) \\dot{\\bm q} \\\\ \\label{eq:u_def}\n\\bm u & \\triangleq \\bm Q^{-1}(\\bm q) \\bm\\tau,\n\\end{align}\n\\end{subequations}\n\\begin{remark}\nSince $\\det{\\bm Q}=\\sqrt{\\det{\\bm M}}\\neq 0$, matrix $\\bm Q$ is always full rank and thus $\\bm Q^{-1}$ always exists. Therefore, \\eqref{eq:quasi} implies that there are  one-to-one  relationships between the set of generalized velocity and generalized force $\\{ \\dot{\\bm q}, \\bm\\tau \\}$ on one hand and the set of  quasi-velocities  and quasi-forces $\\{ \\bm v, \\bm u\\}$ on the other hand.\n\\end{remark}\nNow, pre-multiplying both sides of \\eqref{eq:LLddotq} by\n$\\bm Q^{-1}$ and then substituting \\eqref{eq:quasi} into the\nresultant equation, we arrive at the equations of mechanical systems\nexpressed by the quasi-variables:\n\\begin{subequations} \\label{eq:quasi_dyn}\n\\begin{align} \\label{eq:dv=f}\n\\dot {\\bm v} + \\bm\\Gamma \\bm v & = \\bm u - \\bm\\Lambda^T \\bm\\lambda \\\\ \\label{eq:Qv=0}\n\\bm\\Lambda \\bm v &= \\bm 0\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{align} \\label{eq:Omega}\n\\bm\\Gamma \\triangleq & \\bm Q^{-1} \\Big( \\dot{\\bm Q} - \\frac{\\partial\n\\bm v}{\\partial \\bm q} \\Big) \\\\ \\label{eq:Lambda}\n\\bm\\Lambda  \\triangleq & \\bm A \\bm Q^{-T}\n\\end{align}\nAlternatively, matrix $\\bm\\Gamma$ can be  described by $\\bm\\Gamma = \\bm Q^{-1}\\bm\\Psi$\nwhere the $ij$th entries of matrix $\\bm\\Psi$ can be calculated through the following partial derivative equations\n\\begin{equation} \\label{eq:Qij}\n\\Psi_{ij} = \\sum_k \\Big( \\frac{\\partial  Q_{ij}}{\\partial q_k} - \\frac{\\partial Q_{kj}}{\\partial q_i} \\Big) \\dot q_k.\n\\end{equation}\nIt is worth noting that the constraint equations \\eqref{eq:Qv=0} are imposed on the quasi-velocities analogous to constraint equations \\eqref{eq:Adotq} imposed on the generalized velocities. Since matrix $\\bm Q$ is always full-rank, we can say\n$\\mbox{rank}(\\bm\\Lambda)=\\mbox{rank}(\\bm A)=r$, where $r\\leq m$ is\nthe number of independent constraints. Then, according to the {\\em\nsingular value decomposition} (SVD) there exist unitary (orthogonal)\nmatrices $\\bm U=[\\bm U_1 \\;\\; \\bm U_2]\\in \\mathbb{R}^{m \\times m}$\nand $\\bm V=[\\bm V_1 \\;\\; \\bm V_2]\\in \\mathbb{R}^{n \\times n}$ (i.e.,\n$\\bm U^T \\bm U = \\bm U \\bm U^T = \\bm I_m$ and $\\bm V^T \\bm V = \\bm V \\bm V^T = \\bm I_n$) such that\n\\begin{equation} \\label{eq:svd}\n\\bm\\Lambda = \\bm U \\bm\\Sigma \\bm V^T \\quad \\text{where} \\quad\n\\bm\\Sigma =\n\\begin{bmatrix} \\bm S & \\bm 0 \\\\ \\bm 0 & \\bm 0 \\end{bmatrix}\n\\end{equation}\nand $\\bm S= \\mbox{diag}(\\sigma_1 , \\cdots, \\sigma_r)$ with $\\sigma_1\n\\geq \\cdots \\geq \\sigma_r > 0$ being the non-zero singular values\n\\cite{Klema-Laub-1980,Press-Flannery-1988}. The unitary matrices are\npartitioned so that the dimensions of the submatrices $\\bm U_1$ and\n$\\bm V_1 \\in \\mathbb{R}^{n \\times r}$ are consistent with those of $\\bm S$. That is the columns\nof $\\bm U_1$ and $\\bm V_2 \\in \\mathbb{R}^{n \\times (n-r)}$ are the corresponding sets of orthonormal\neigenvalues which span the range space and the null space of\n$\\bm\\Lambda$, respectively \\cite{Golub-VanLoan-1996}. Thus\n\\begin{equation}  \\label{eq:LambdaV2}\n\\bm\\Lambda \\bm V_2^T = \\bm 0\n\\end{equation}\nDefine {\\em reduced order quasi-velocities} or {\\em independent quasi-velocities}  $\\bm v_r \\in \\mathbb{R}^{n-r}$ as follow:\n\\begin{align} \\notag\n\\bm v_r &= \\bm V_2^T \\bm v \\\\ \\label{eq:vr=V2v}\n&= \\bm V_2^T \\bm Q^T \\dot{\\bm q}\n\\end{align}\nSince $\\bm V_2 \\bm V_2^T$ is a projection matrix corresponding to the kernel of matrix $\\bm\\Lambda$, we have  $\\bm V_2 \\bm V_2^T\\bm v = \\bm v$. Therefore, we can readily obtain the reciprocal of \\eqref{eq:vr=V2v} by per-multiplying both sides of the latter equation by $\\bm V_2$, i.e.,\n\\begin{equation} \\label{eq:v=V2vr}\n\\bm v= \\bm V_2 \\bm v_r\n\\end{equation}\nThe time-derivative of the reduced quasi-velocities can be expressed by\n\\begin{equation} \\label{eq:dot_vr}\n\\dot{\\bm v}_r = \\bm V_2^T \\dot{\\bm v} + \\dot{\\bm V}_2^T \\bm V_2 \\bm v_r\n\\end{equation}\nIt can be inferred from  \\eqref{eq:v_def} and \\eqref{eq:v=V2vr} that the\ngeneralized velocities can be constructed from the reduced quasi-velocities via the following mapping\n\\begin{equation} \\label{eq:dotq=Qvr}\n\\dot{\\bm q} = \\bm Q^{-T} \\bm V_2 \\bm v_r\n\\end{equation}\nFinally, pre-multiplying both sides of \\eqref{eq:dv=f} by $\\bm V_2^T$ and then using identities \\eqref{eq:LambdaV2},  \\eqref{eq:v=V2vr}, and \\eqref{eq:dot_vr}, we arrive at\n\\begin{equation} \\label{eq:dot_v_r}\n\\dot {\\bm v}_r = \\bm\\Gamma_v \\bm v_r + \\bm u_v\n\\end{equation}\nwhere\n\\begin{subequations}\n\\begin{align} \\label{eq:Gamma_v}\n\\bm\\Gamma_v &= \\bm V_2^T \\bm\\Gamma \\bm V_2 + \\dot{\\bm V}_2^T \\bm V_2 \\\\ \\label{eq:u_v}\n\\bm u_v &= \\bm V_2^T \\bm Q^{-1} \\bm\\tau\n\\end{align}\n\\end{subequations}\nNote that matrix $\\dot{\\bm V}_2^T$ in the RHS of \\eqref{eq:Gamma_v} can be obtained from time-derivative of \\eqref{eq:LambdaV2} as follows: $\\bm\\Lambda \\dot{\\bm V}_2^T = - \\dot{\\bm\\Lambda} \\bm V_2^T$, and thus\n\\begin{equation} \\notag\n\\dot{\\bm V}_2^T = - \\bm V \\bm\\Sigma^{+} \\bm U^T \\dot{\\bm\\Lambda}\\bm V_2^T\n\\end{equation}\nwhere $\\bm\\Sigma^{+}$ contains the inverse of non-zero singular values. On the other hand, using the orthogonality property $\\bm V_1^T \\bm V_2 = \\bm0$ and pre-multiplying both sides of \\eqref{eq:v=V2vr} by $\\bm V_1^T$, one can conclude that $\\bm V_1^T$ indeed acts an annihilator for the quasi-velocities, i.e., $\\bm V_1^T \\bm v = \\bm 0$. Thus, time-derivative of the latter identity gives us\n\\begin{equation} \\label{eq:V1dotv}\n\\bm V_1^T \\dot{\\bm v} + \\dot{\\bm V}_1^T \\bm v = \\bm0\n\\end{equation}\nPre-multiplying both sides of \\eqref{eq:dv=f} by $\\bm V_1^T$ and then using identity \\eqref{eq:V1dotv}, we arrive at\n\\begin{equation} \\label{eq:xi}\n\\bm\\xi  = \\bm\\Gamma_{\\xi} \\bm v_r + \\bm u_{\\xi}\n\\end{equation}\nwhere\n\\begin{subequations}\n\\begin{equation} \\label{eq:Gamma_xi}\n\\bm\\Gamma_{\\xi} = - \\bm V_1^T \\bm\\Gamma \\bm V_2 + \\dot{\\bm V}_1^T \\bm V_2 ,\n\\end{equation}\nand $\\bm\\xi $  and $\\bm u_{\\xi}$ are, respectively, the constraint quasi-forces and the corresponding input quasi-forces defined by\n\\begin{align}\n\\bm\\xi &=  \\bm V_1^T \\bm Q^{-1} \\bm A^T \\bm\\lambda\\\\ \\label{eq:u_xi}\n\\bm u_{\\xi} &= \\bm V_1^T \\bm Q^{-1} \\bm\\tau\n\\end{align}\n\\end{subequations}\nEquations \\eqref{eq:dot_v_r} and \\eqref{eq:xi} completely characterize the dynamics behaviour of a constrained MBS in terms of quasi variables. It should be noted that that the corresponding input quasi-forces for the equations of motion and the constraint forces, i.e., $\\bm u_v$ and $\\bm u_{\\xi}$ are naturally decoupled. The decoupling of the equations of motions and constraint forces allows the development of independent motion and force controllers without any need to compensate for the cross-coupling terms. It should be also pointed out that the conventional transformation used in quasi-Lagrangian dynamics  allows for splitting between the normal and tangential dynamics \\cite{Brogliato-2014}. However, the normal dynamics is eliminated from the above equations, which are expressed in terms of the reduced quasi-velocities.\n\n\\subsection{Properties of the System  in Terms of Quasi-Velocities and Quasi-Forces} \\label{sec:properties}\nIn the following analysis, we explore some properties of the quasi-variable formulation \\eqref{eq:dot_v_r} and \\eqref{eq:xi}\nthat will be useful in control design purposes.\n\\begin{enumerate} \\label{eq:kinetic_vr}\n\\item Kinetic energy\n\\begin{equation} \\label{K=vr^2}\nK = \\frac{1}{2} \\| \\bm v_r \\|^2.\n\\end{equation}\n\\item Skew-symmetric property\n\\begin{equation}  \\label{eq:skew}\n\\bm v_r^T \\bm\\Gamma_v \\bm v_r =0 \\qquad \\forall \\bm v_r \\in \\mathbb{R}^{n-r}.\n\\end{equation}\n\\item Boundedness of matrices $\\bm\\Gamma_v$ and $\\bm\\Gamma_{\\xi}$, i.e.,\n\\begin{equation} \\label{eq:boundedGam}\n\\|\\bm\\Gamma_v \\|, \\; \\|\\bm\\Gamma_{\\xi} \\| \\leq \\gamma \\| \\bm v_r \\|  \\qquad \\exists \\gamma>0.\n\\end{equation}\n\\end{enumerate}\nThe kinetic energy  is trivially given by\n\\begin{equation} \\label{eq:T_normv}\nK = \\frac{1}{2} \\| \\bm v \\|^2 = \\frac{1}{2} \\bm v_r \\bm V_2 ^T \\bm V_2 \\bm v_r = \\frac{1}{2} \\| \\bm v_r \\|^2,\n\\end{equation}\nand hence \\eqref{K=vr^2} is proven. Furthermore, in the absence of any external\nactive force, the principle of conservation of kinetic energy dictates that\nthe kinetic energy of mechanical system is bound to be constant,\ni.e., $\\bm u_v= \\bm 0 \\Longrightarrow \\dot K =0$. On the other hand,\nthe zero-input response of a mechanical system is $\\dot {\\bm v} = -\n\\bm\\Gamma_v \\bm v_r$. Substituting the latter equation in the\ntime-derivative of \\eqref{eq:T_normv} gives $\\bm v_r^T \\bm\\Gamma_r \\bm v_r =  0$, which implies \\eqref{eq:skew}.\nAssume that $c_m$ denote the minimum eigenvalue of $\\bm M$ for all\nconfigurations $\\bm q$, that is, $c_m \\bm I \\leq \\bm M(\\bm q)$.\nThen, using the norm properties leads to\n\\begin{equation} \\label{eq:norm_invW}\n\\| \\bm Q^{-1} \\| \\leq \\frac{1}{\\sqrt{c_m}}.\n\\end{equation}\nIn view of \\eqref{eq:dotq=Qvr}  and \\eqref{eq:norm_invW} and knowing\nthat $\\| \\bm V_2 \\|=1$, we can say\n\\begin{equation} \\label{eq:bound_dq}\n\\| \\dot{\\bm q} \\| \\leq \\| \\bm Q^{-T} \\| \\| \\bm V_2 \\| \\| \\bm v_r \\|\n\\leq \\frac{\\|\\bm v_r \\|}{\\sqrt{c_m}}.\n\\end{equation}\nMoreover, if the factorization $\\bm Q(\\bm q)$ is a sufficiently\nsmooth function, then all partial-derivative terms in \\eqref{eq:Qij} are bounded\nand hence there exists a finite $c_{\\psi}> 0$ such that $\\| \\bm\\Psi \\| \\leq\nc_{\\psi} \\| \\dot{\\bm q} \\|$, and hence we can say\n\\begin{equation} \\label{eq:norm_Gamma}\n\\| \\bm\\Gamma \\| \\leq \\frac{c_{\\psi}}{c_m} \\| \\bm v_r \\|.\n\\end{equation}\nAlso, the entries of the time-derivative of matrix $\\bm V$ can be written as\n\\begin{equation}\n\\dot{V}_{ij} = \\sum_k \\frac{\\partial V_{ij} }{\\partial q_k} \\dot q_k\n\\end{equation}\nConsequently, we can say $\\| \\dot {\\bm V} \\| \\leq c_V \\| \\dot{\\bm q} \\| \\leq c_V\/c_m \\| \\bm v_r \\|$. Finally, knowing that  $\\|\\bm V_1 \\| = \\| \\bm V_2 \\|=1$ and $\\| \\dot{\\bm V}_1 \\| , \\| \\dot{\\bm V}_2 \\| \\leq \\| \\dot{\\bm V} \\|$, one can infer \\eqref{eq:boundedGam} from the RHS expressions in \\eqref{eq:Gamma_v} and \\eqref{eq:Gamma_xi} where $\\gamma =(c_V + c_{\\psi})\/c_m$.\n\n\\subsection{Natural Metric}\nA problem that often arises in robotics, namely hybrid control or\nthe minimum solution to joint rate or force is that generalized\ncoordinate $\\bm q$ may have a combination of rotational and\ntranslational components that can be even compounded by having\ncombination of rotational and translational constraints\n\\cite{Doty-Melchiorri-Bonivento-1993}. This may lead to inconsistent\nresults, i.e., results that are invariant with respect to changes in\ndimensional units unless adequate weighting matrices are used\n\\cite{Doty-Melchiorri-Bonivento-1993,Manes-1992,Featherstone-Fijany-1999,Featherstone-Thiebaut-Khatib-1999,Aghili-2005}.\nFor example, the minimum joint rate rates, $\\min \\| \\dot{\\bm q} \\|$,\nor minimum norm force, $\\min \\| \\bm\\tau \\|$, are not meaningful\nquantities if the robot has both revolute and prismatic joints\n\\cite{Doty-Melchiorri-Bonivento-1993}.\n\\begin{remark}\nIt is also important to note the important property of the reduced quasi-velocities and quasi-forces is that they always have homogeneous units.\nThe expression of kinetic energy \\eqref{K=vr^2} implies  $ \\| \\bm v_r \\| = \\| \\bm v \\| = \\sqrt{2K}$ meaning that  all elements of the vector of quasi-velocities $\\bm\nv$ or $\\bm v_r$ must have a homogeneous unit $[\\sqrt{\\rm kg}{\\rm\nm}\/{\\rm s}]$. This is true even if the vector of the generalized\ncoordinate or the constraints have combinations of rotational and\ntranslational components. Similarly, one can argue from \\eqref{eq:dot_v_r} and \\eqref{eq:xi}\n that the elements\nof the quasi-forces $\\bm u_v$ and $\\bm\\xi$ have always identical unit $[\\sqrt{\\rm kg}{\\rm\nm}\/{\\rm s}^2]$, regardless of the units of the generalized force or\nthe constraint wrench.\n\\end{remark}\n\nTherefore, minimization of  $\\| {\\bm v} \\|$\nor $\\min \\| \\bm u \\|$ is legitimate because the latter vectors have\nalways homogeneous units. Moreover, the selection matrices which are\noften needed in hybrid position-force control of manipulators when\nboth translational and rotational constraints are involved between\nits end effector and its environment~\\cite{Featherstone-Thiebaut-Khatib-1999} becomes a non-issue here.\n\n\n\\subsection{Existence of Quasi-Coordinates} \\label{sec:quasi-coordinates}\n\nIt should be pointed out that despite the one-to-one\ncorrespondence between velocity coordinate $\\dot {\\bm q}$ and the\nquasi-velocities $\\bm v$, they are not synonymous. This is because\nthe integration of the former variable leads to the generalized\ncoordinate, while integration of the latter variable does not always lead\nto a meaningful vector describing the configuration of the\nmechanical system. Now let us assume that $\\dot{\\bm\\phi} = \\bm v$, where ${\\bm\\phi}$ is called {\\em\nquasi-coordinates}. For $\\bm\\phi$ to be an explicit function of $\\bm\nq$, i.e., $\\bm\\phi = \\bm\\phi(\\bm q)$, it must be the gradient of a\nscalar function meaning that $\\bm\\phi$ is a {\\em conservative vector\nfield}. In that case, \\eqref{eq:v_def} implies that $\\bm Q^T(\\bm q)$\nis actually a Jacobian as\n\\begin{equation}\nQ_{ij}= \\frac{\\partial \\phi_j}{\\partial q_i}.\n\\end{equation}\nIf $\\bm\\phi(\\bm q)$\nexists and it is a smooth function, then we can say\n\\begin{equation}\n\\frac{\\partial Q_{ij}}{\\partial q_k} - \\frac{\\partial Q_{kj}}{\\partial q_i}= \\frac{\\partial^2 \\phi_j}{\\partial q_i \\partial q_k} - \\frac{\\partial^2 \\phi_i}{\\partial q_k \\partial q_i} =0\n\\end{equation}\nUnder this circumstance, the expression in the\nparenthesis of the right-hand side of \\eqref{eq:Qij} vanishes, i.e., $\\bm\\Psi \\equiv \\bm 0$ and hence $\\bm\\Gamma \\equiv \\bm 0$. Therefore, the equations of motion become a simple\nintegrator system and hence $\\bm\\phi$ and $\\bm v$ are\nindeed alternative possibilities for generalized coordinates and\ngeneralized velocities. Technically speaking, a necessary and sufficient condition for the\nexistence of the quasi-coordinates, $\\bm\\phi$, is that the\nRiemannian manifold defined by the inertia matrix $\\bm M(\\bm\nq)$ be locally flat--by definition, a Riemannian manifold\nthat is locally isometric to Euclidean manifold is called a locally\nflat manifold \\cite{Spong-1992}. However, that has been proven to\nbe a very stringent condition \\cite{Bedrossian-1992}. Nevertheless, state variables $\\{\n\\bm q,  \\bm v_r \\}$ are sufficient to describe\ncompletely the states of MBS. Setting \\eqref{eq:dotq=Qvr} and \\eqref{eq:dot_v_r}  in state space\nform gives\n\\begin{equation} \\label{eq:ode}\n\\frac{\\rm d}{{\\rm d}t} \\begin{bmatrix} \\bm q \\\\ \\bm v_r \\end{bmatrix}\n=\n\\begin{bmatrix}  \\bm Q^{-T}\\bm V_2\n\\\\ - \\bm\\Gamma_v \\end{bmatrix} \\bm v_r  +\n\\begin{bmatrix} \\bm 0 \\\\ \\bm I \\end{bmatrix} \\bm u_r.\n\\end{equation}\nIt is interesting to note that dynamics system \\eqref{eq:ode} is in\nthe form of the so-called {\\em second-order kinematic model} of\nconstrained mechanism, which appears in kinematics of nonholonomic\nsystems. This is the manifestation of the fact that the integration\nof quasi-velocities, in general, does not lead to\nquasi-coordinates.\n\n\\section{Force\/Motion Control Based on Quasi-Velocities and Quasi-Forces} \\label{sec:control}\n\nIn this section, we use dynamics formulation  \\eqref{eq:dot_v_r} and \\eqref{eq:xi} for tracking control and regulation control of constrained MBSs. Due to presence of only $r$ independent constraints, the actual number of degrees of freedom of the system is reduced to $n-r$.\nConsequently, there must be $n-r$ independent variables\n$\\bm\\theta(\\bm q)\\in \\mathbb{R}^{n-r}$, which is also called a {\\em\nminimal set of generalized coordinates}. Thus, we can say\n\\begin{equation} \\label{eq:dot_tet}\n\\dot{\\bm\\theta} = \\left( \\frac{\\partial \\bm\\theta}{\\partial \\bm q}  \\right) \\dot{\\bm q}\n\\end{equation}\nSubstituting $\\dot{\\bm q}$ from \\eqref{eq:dotq=Qvr} into \\eqref{eq:dot_tet} yields\n\\begin{equation} \\label{eq:dtheta}\n\\dot{\\bm\\theta}  =  \\bm D(\\bm\\theta) \\bm v_r, \\qquad \\text{where}\n\\qquad \\bm D \\triangleq \\left( \\frac{\\partial \\bm\\theta}{\\partial \\bm q} \\right)\n\\bm Q^{-T} \\bm V_2.\n\\end{equation}\nSince  both variables $\\bm v_r$ and $\\dot{\\bm\\theta}$ are\nwith the same dimension, the reciprocal of mapping \\eqref{eq:dtheta}\nmust uniquely exist, i.e.,  $ \\bm D^{-1}$ is always well-defined. Now we adopt a Lyapunov-based  control scheme \\cite[p.74]{Canudas-Siciliano-Bastin-book-1996} for designing a feedback\ncontrol in terms of quasi-velocities. Define the composite error\n\\begin{equation} \\label{eq:edef}\n\\bm\\epsilon \\triangleq\n\\bm\\epsilon_{v_r} + k_p \\bm D^{-1} \\bm\\epsilon_{\\theta} ,\n\\end{equation}\nwhere $k_p>0$, $\\bm\\epsilon_{v_r}={\\bm v}_r - {\\bm v}_{r_d}$, and\n$\\bm\\epsilon_{\\theta} =\\bm\\theta - \\bm\\theta_d $. Also, define the auxiliary\nvariable $\\bm\\sigma = \\bm v_r -\\bm\\epsilon =   \\bm v_{r_d} - k_p\\bm D^{-1}\n\\bm\\epsilon_{\\theta}$, which is used in the following control law:\n\\begin{equation} \\label{eq:contrlaw_zeta}\n\\bm u_v = \\dot{\\bm\\sigma} + \\bm\\Gamma_v \\bm v_r - k_d \\bm\\epsilon,\n\\end{equation}\nwhere $k_d >0$. Applying control law \\eqref{eq:contrlaw_zeta} to\nsystem \\eqref{eq:quasi_dyn} gives the dynamics of the error $\\epsilon$\nin terms of the following first-order differential equation:\n\\begin{equation} \\label{eq:diff_eps}\n\\dot{\\bm \\epsilon} = - k_d \\bm\\epsilon.\n\\end{equation}\nIn other words,  the composite error $\\bm\\epsilon$ is exponentially stable\n\\begin{equation} \\label{eq:eps_norm}\n\\bm\\epsilon  =  \\bm\\epsilon (0)   e^{-k_d t}.\n\\end{equation}\nPre-multiplying both sides of \\eqref{eq:edef} by $\\bm D(\\bm\\theta)$,\nthe resultant equation can be rearranged to the following\ndifferential equation\n\\begin{equation} \\label{eq:dot_tildeq}\n\\dot{\\bm\\epsilon}_{\\theta}   =  - k_p \\bm\\epsilon_{\\theta} +  \\bm D\n\\bm \\epsilon.\n\\end{equation}\nNow, it remains to show that the solution of the above\nnon-autonomous system converges to zero. First, notice from definition  \\eqref{eq:dtheta} that $\\bm D$  is the product of three bounded matrix and thus it should be a bounded matrix too. That is because $\\| \\bm Q^{-T} \\|\\leq 1\/ \\sqrt{c_m}$ according to \\eqref{eq:norm_invW} and $\\bm V_2$ is a unitary matrix meaning that $\\| \\bm V_2 \\| \\leq 1$. Thus,\nthere exists scalar $c_d>0$ such that\n\\begin{equation} \\label{eq:bounded_B}\n\\bm D(\\bm\\theta)  \\leq c_d \\bm I.\n\\end{equation}\nwhere $c_d=\\| \\partial \\bm Q\/ \\partial \\bm q \\| \/ \\sqrt{c_m}$. One can show that the solution of \\eqref{eq:dot_tildeq} satisfies\n\\begin{equation} \\label{eq:solution_normdq}\n\\| \\bm\\epsilon_{\\theta}  \\| \\leq \\Big( \\| \\bm\\epsilon_{\\theta} (0) \\|  + c_d \\| \\bm\\epsilon(0) \\| \\Big) e^{-k_d t},\n\\end{equation}\nwhich implies exponential stability of the tracking error; see Appendix for details.  Tracking of the desired constraint force $ \\bm\\xi_d = \\bm\\Lambda_r^T \\bm\\lambda_d$ can be\nsimply achieved by compensating the velocity perturbation term\nin \\eqref{eq:xi}, i.e.,\n\\begin{equation}\n\\bm u_{\\xi}  = \\bm\\xi_d + \\bm\\Gamma_{\\xi}  \\bm v_r \\quad \\Longrightarrow \\quad \\bm\\xi = \\bm\\xi_d.\n\\end{equation}\nFrom definitions of $\\bm u$, $\\bm u_v$, and $\\bm u_{\\xi}$ in \\eqref{eq:u_def}, \\eqref{eq:u_v} and \\eqref{eq:u_xi}, one can write the following relationship\n\\begin{equation} \\notag\n\\begin{bmatrix} \\bm u_{\\xi} \\\\ \\bm u_{v} \\end{bmatrix} = \\bm V^T \\bm u\n\\end{equation}\nand thus $\\bm u_{\\xi}^T \\bm u_{\\xi} + \\bm u_v^T \\bm u_v = \\bm u \\bm V \\bm V^T \\bm u$. Since $\\bm V$ is a unitary matrix, the latter identity is equivalent to $\\|\\bm u_v \\|^2 + \\|\\bm u_{\\xi} \\|^2 = \\|\\bm u \\|^2$. It is worth noting that in view of the latter norm identity , we can say that\n\\begin{equation} \\notag\n\\min \\| \\bm u \\| \\quad \\leftarrow \\quad \\bm u_{\\xi} \\equiv \\bm 0\n\\end{equation}\nThat is tantamount to minimization of weighted norm of the generalized forces where the\nweight matrix is the inverse of the inertia matrix because\n\\begin{equation} \\label{eq:norm_u}\n\\|\\bm u \\|= \\sqrt{\\bm\\tau^T \\bm M^{-1} \\bm\\tau}.\n\\end{equation}\nIn other word, the kinetic metric of the generalized force in minimized.\n\n\n\\subsection{Setpoint Control}\n\nIn this section, we extend such a feedback control for hybrid\nmotion\/force control of constrained robotic systems. Consider the following control law for system \\eqref{eq:quasi_dyn}\n\\begin{equation} \\label{eq:control}\n\\bm u_v =  - k_d \\bm v_r - k_p \\bm\\epsilon_{\\theta} ,\n\\end{equation}\nwhere $\\bm\\epsilon_{\\theta} =\\bm\\theta - \\bm\\theta_d$, and  $k_d,k_p>0$. Then, the dynamics of the closed--loop system becomes\n\\begin{equation} \\label{eq:closed_reps}\n\\dot{\\bm v}_r = - \\bm\\Gamma_v \\bm v_r - k_d \\bm v_r - k_p \\bm\\epsilon_{\\theta} .\n\\end{equation}\nChoose the following standard Lyapunov function\n\\begin{equation} \\label{eq:Lyap}\nV= \\frac{1}{2} \\| \\bm\\epsilon_{v_r}  \\|^2 + \\frac{1}{2} k_p \\| \\bm\\epsilon_{\\theta} \\|^2.\n\\end{equation}\nThen, using Property~\\eqref{eq:skew} in the time-derivative of\n\\eqref{eq:Lyap} along \\eqref{eq:closed_reps} yields\n\\begin{equation} \\notag\n\\dot V = - \\bm v_r^T \\bm K_d \\bm v_r \\leq 0,\n\\end{equation}\nwhich is negative-semidefinite.  Therefore, according to LaSalle's Global Invariant Set\nTheorem \\cite{Lasalle-1960}, \\cite[p.115]{Khalil-1992}, the solution\nof system~\\eqref{eq:closed_reps} asymptotically converges to the\ninvariant set ${\\cal S}= \\{\\bm\nv_r, \\bm\\epsilon_{\\theta} : \\bm v_r= \\bm 0 ,  \\bm\\epsilon_{\\theta} =\\bm 0 \\}$, i.e., $\\bm\\theta \\rightarrow \\bm\\theta_d$ as $t$ goes to infinity. Define $\\bm\\xi_d=\\bm\\Lambda_r^T \\bm\\lambda_d$ where is the desired value of the Lagrangian multiplier. Then the force control law can be simply given by\n\\begin{equation} \\label{eq:lambda_d}\n\\bm u_{\\xi} = \\bm\\xi_d.\n\\end{equation}\nSubstituting \\eqref{eq:lambda_d} into \\eqref{eq:xi}  and using\nProperty~\\eqref{eq:boundedGam}, we get\n\\[ \\| \\bm\\xi - \\bm\\xi_d \\| \\leq \\gamma \\| \\bm v_r \\|^2. \\]\nSince $\\bm v_r~\\rightarrow~0$, then\n$\\bm\\xi~\\rightarrow~\\bm\\xi_d$ as $t$ goes to infinity.\n\n\n\n\n\\section{Conclusions}\nA consistent  formulation for modelling of constrained MBSs  using\nthe concept of quasi-velocities and quasi-forces has been presented. The main advantage of this formulation is that it does not require adequate weighting matrices when the generalized coordinates involve in both translational and rotational components and\/or the generalized force or constraint wrench involve in both force and moment components. It has been also shown that using the Cholesky decomposition of the mass matrix and the unitary transformation corresponding to the kernel of the Pfaffian constraints of the quasi-velocities led to the decoupling of motion and force control inputs. This allowed the possibility to develop a simple force control action that is totally independent of the motion control action. Some properties of the constrained Lagrangian dynamics formulation based on the quasi-variables were presented that could be useful for control purposes. It  followed by the development of the force\/motion control of constrained MBSs based on the quasi-velocities and quasi-forces.\n\n\n\\section*{Appendix}\nConsider the following positive--definite function\n\\begin{equation} \\label{eq:positive_function}\nV = \\frac{1}{2} \\| \\bm\\epsilon_{\\theta}  \\| ^2,\n\\end{equation}\nwhose time-derivative along \\eqref{eq:dot_tildeq} gives\n\\begin{equation} \\notag\n\\dot V = - k_p \\| \\bm\\epsilon_{\\theta}  \\|^2 + \\bm\\epsilon_{\\theta}^T \\bm D(\\bm\\theta) \\bm\\epsilon.\n\\end{equation}\nFrom \\eqref{eq:eps_norm} and \\eqref{eq:bounded_B}, one can find a\nbound on $\\dot V$ as\n\\begin{equation}\n\\dot V  \\leq  - 2 k_p V +  c_d  \\| \\bm\\epsilon(0) \\|  \\| \\bm\\epsilon_{\\theta}  \\| e^{-k_p t},\n\\end{equation}\nwhich is in the form of a Bernoulli differential inequality. The\nabove nonlinear inequality can be linearized by the following change\nof variable $U=\\sqrt{V}$, i.e.,\n\\begin{equation} \\label{eq:dU_inequality}\n\\dot U \\leq - k_p U + \\frac{c_d \\| \\bm\\epsilon(0) \\|}{\\sqrt{2}}\ne^{-k_p t}\n\\end{equation}\nIn view of the comparison lemma \\cite[p. 222]{Khalil-1992} and\n\\eqref{eq:eps_norm}, one can show that the solution of\n\\eqref{eq:dU_inequality} must satisfy\n\\begin{equation} \\notag\nU \\leq \\Big( U(0) + \\frac{c_d \\| \\bm\\epsilon(0)\n\\|}{\\sqrt{2}} \\Big)  e^{- k_p t},\n\\end{equation}\nwhich is equivalent to \\eqref{eq:solution_normdq}.\n\n\\bibliographystyle{IEEEtran}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn spite of the great successes of General Relativity (GR), cosmological models deriving from Einstein theory of gravity still lack a proper explanation for inflation and dark energy. Inflation and the present cosmic acceleration are indeed two of the main reasons which motivate the study of theories of gravity alternative to General Relativity, at least at large scales.\nAmong these, scalar--tensor gravitational theories are ones of the most widely investigated already for the beginning 1960s \\cite{BD}. Scalar tensor theories also arise in other contexts like the low energy limit of Kaluza-Klein gravity \\cite{Overduin:1998pn}, in quantum field theory in curved spacetimes \\cite{QFT-CS} and in the tree level action of string theory \\cite{Strings}. The basic paradigm of such theories is the non--minimal coupling of gravity to a scalar field, whose so far unknown nature is still the subject of intense scientific research. Among the others, one of the suggested hypotheses is that the scalar field is not fundamental but it is costituted by a fermion condensate. The idea that scalar fields can be composed by other fields (for example Dirac fields) is not very new; for instance, in particle physics it was already proposed  by Weinberg with specific reference to the Higgs field \\cite{W} (in this regard see also \\cite{Fabbri1,Fabbri2}). \n\nIn cosmology fermion fields have been mostly considered since the 1990s; they have been studied as possible sources of inflation and dark energy, driving the universe into accelerated expansions at both early and late time \\cite{Kovalyov,Obukhov,Saha1,Binetruy,Saha2,Picon,Saha3,Ribas1,Saha4,Boehmer1,Boehmer2,Ribas2,Rakhi,Rakhi2,Ribas4,Ribas3,Saha14}.\nIn most of the papers appeared in the literature fermions are minimally coupled to gravity; only a few and quite recent works instead investigate the effects of fermionic non--minimal couplings \\cite{Souza,Devecchi,CVC,GSK}. \n\nIn this paper, we explore some cosmological scenarios when a Dirac field is non--miminally coupled to gravity with torsion. The non--minimal interaction term we take into account is of mass dimension 5 and reduces to the product $\\bar\\psi\\psi\\\/R$ between the condensate $\\bar\\psi\\psi$ of the Dirac field and the scalar curvature $R$. In a previous paper \\cite{FVC}, we have studied the consequences of this non--minimal coupling on the renormalizability of the Dirac equations, showing that in the case torsion is not neglected fermionic non--minimal couplings are renormalizable and possess a well defined behaviour even in the ultraviolet regimes. {  In the present work, we investigate the cosmological counterpart of the theory proposed in \\cite{FVC}. Accordingly, here we work within the geometrical setting of Einstein--Cartan--Sciama--Kibble gravity (ECSK), where curvature and torsion couple to energy and spin of the Dirac field respectively. As we shall see, in this metric--affine approach we obtain a \ndynamical equation for the scale volume of the universe which is easier to handle than the analogous one in the purely metric case (compare with \\cite{CVC}). As a result, we present a simple analysis of cosmological issues such as cosmological singularity, inflation and dark energy when the above mentioned non--minimal coupling is taken into account. In particular, with the help of some illustrative examples, we discuss the role played by different self--interaction potentials in driving inflation and dark eras in connection with the non--minimal coupling}.\n \nThe layout of the paper is the following: in section II we briefly outline the theory introduced in \\cite{FVC}, recalling its main features; in section III we analyze different cosmological scenarios, first in the presence of a  Dirac field only, then when dust and radiation fluid are present too; finally, we devote section V to the conclusions. Throughout this paper natural units ($\\hbar=c=k_{B}=8\\pi G=1$) and metric signature $(+,-,-,-)$ are used.\n\n\n\\section{The $(1+\\epsilon\\bar\\psi\\psi)R$-theory with torsion}\nIn this section we briefly review the theory introduced in \\cite{FVC}. In the general framework of ECSK gravity, let us consider a Lagrangian density of the form\n\\begin{equation}\\label{3.1}\n{\\cal L}= (1+\\epsilon\\bar\\psi\\psi)\\\/eR - e\\\/L_D\n\\end{equation}\nwhere the Einstein--Hilbert term $R$ is non--minimally coupled to a Dirac Lagrangian of the form\n\\begin{equation}\\label{3.2}\nL_D = + \\frac{i}{2}\\left( \\bar{\\psi}\\Gamma^iD_i\\psi - D_i\\bar{\\psi}\\Gamma^i\\psi\\right) -m\\bar{\\psi}\\psi + V(\\bar\\psi\\psi)\n\\end{equation}\nthrough the non--minimal coupling term $\\epsilon\\bar\\psi\\psi\\\/R$, $\\epsilon$ being a suitable coupling constant. Here, we denote by $\\gamma^\\mu$ ($\\mu=0,1,2,3$) Dirac matrices and we introduce the notation $\\Gamma^i = e^i_\\mu\\gamma^\\mu$ where $e^\\mu_i$ indicate a tetrad field associated with a metric $g_{ij}=e^\\mu_{i}e^\\nu_{j}\\eta_{\\mu\\nu}$. In eq. \\eqref{3.2}, $D_i$ denote the covariant derivative of the Dirac field $\\psi$ defined as \n$D_i\\psi = \\de\\psi\/de{x^i} + \\omega_i^{\\;\\;\\mu\\nu}S_{\\mu\\nu}\\psi\\\/$ and $D_i\\bar\\psi = \\de{\\bar\\psi}\/de{x^i} - \\bar\\psi\\omega_i^{\\;\\;\\mu\\nu}S_{\\mu\\nu}\\\/$, where $\\omega_i^{\\;\\;\\mu\\nu}$ is a spin connection and $S_{\\mu\\nu}:= \\frac{1}{8}[\\gamma_\\mu,\\gamma_\\nu]$. Equivalently, we have $D_i\\psi = \\de\\psi\/de{x^i} - \\Omega_i\\psi$ and $D_i\\bar\\psi = \\de{\\bar\\psi}\/de{x^i} + \\bar{\\psi}\\Omega_i$ where\n\\begin{equation}\\label{3.3}\n\\Omega_i := - \\frac{1}{4}g_{jh}\\left\\{\\Gamma_{ik}^{\\;\\;\\;j} - e^j_\\mu\\partial_i\\\/e^\\mu_k \\right\\}\\Gamma^h\\Gamma^k\n\\end{equation}\nand $\\Gamma_{ik}^{\\;\\;\\;j}$ are the coefficients of a linear connection $\\Gamma$, associated with the spin connection through the usual relation\n\\begin{equation}\\label{3.4}\n\\Gamma_{ij}^{\\;\\;\\;h} = \\omega_{i\\;\\;\\;\\nu}^{\\;\\;\\mu}e_\\mu^h\\\/e^\\nu_j + e^{h}_{\\mu}\\partial_{i}e^{\\mu}_{j}\n\\end{equation}\nDenoting by $\\varphi:=(1+\\epsilon\\bar\\psi\\psi)$ and by $V':=\\frac{dV}{d(\\bar\\psi\\psi)}$, from \\eqref{3.1} we can derive field equations of the form\n\\begin{subequations}\\label{3.9}\n\\begin{equation}\\label{3.9a}\nR_{ij} -\\frac{1}{2}Rg_{ij}= \\frac{1}{\\varphi}\\Sigma_{ij} \n\\end{equation}\n\\begin{equation}\\label{3.9b}\nT_{ij}^{\\;\\;\\;h} = - \\frac{1}{2\\varphi}\\de{\\varphi}\/de{x^p}\\left(\\delta^p_i\\delta^h_j - \\delta^p_j\\delta^h_i\\right) + \\frac{1}{\\varphi}S^{\\;\\;\\;h}_{ij}\n\\end{equation}\n\\begin{equation}\\label{3.9c}\ni\\Gamma^{h}D_{h}\\psi + \\frac{i}{2}T_h\\Gamma^h\\psi- m\\psi + V'(\\bar\\psi\\psi)\\psi - \\epsilon\\psi\\\/R=0\n\\end{equation}\n\\end{subequations}\nwhere\n\\begin{equation}\\label{3.10}\n\\begin{split}\n\\Sigma_{ij} := \\frac{i}{4}\\\/\\left( \\bar\\psi\\Gamma_{i}{D}_{j}\\psi - {D}_{j}\\bar{\\psi}\\Gamma_{i}\\psi \\right) -\\frac{1}{2}L_D\\,g_{ij} =\\\\\n\\frac{i}{4}\\\/\\left( \\bar\\psi\\Gamma_{i}{D}_{j}\\psi - {D}_{j}\\bar{\\psi}\\Gamma_{i}\\psi \\right) - \\frac{1}{2}\\epsilon\\bar\\psi\\psi\\\/R\\,g_{ij} - \\frac{1}{2}V(\\bar\\psi\\psi)\\,g_{ij} + \\frac{1}{2}\\bar\\psi\\psi\\\/V'(\\bar\\psi\\psi)\\,g_{ij}\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\\label{3.6}\nS_{ij}^{\\;\\;\\;h}=\\frac{i}{2}\\bar\\psi\\left\\{\\Gamma^{h},S_{ij}\\right\\}\\psi\n\\end{equation}\nare respectively the energy--momentum and the spin density tensors. In eqs. \\eqref{3.9b} and \\eqref{3.9c} $T_{ij}^{\\;\\;\\;h}:=\\Gamma_{ij}^{\\;\\;\\;h} - \\Gamma_{ji}^{\\;\\;\\;h}$ denotes the torsion tensor and $T_i:=T_{ij}^{\\;\\;\\;j}$ its contraction, while in eq. \\eqref{3.6} $S_{ij}:= \\frac{1}{8}[\\Gamma_i,\\Gamma_j]$. The energy--momentum and spin tensors satisfy the conservation laws\n\\begin{subequations}\\label{c.2.6}\n\\begin{equation}\\label{c.2.6a}\n\\nabla_i\\Sigma^{ij} + T_i\\Sigma^{ij} - \\Sigma_{pq}T^{jpq} -\\frac{1}{2}S_{pqr}R^{pqrj} + \\frac{1}{2}R\\nabla^j\\\/\\varphi= 0\n\\end{equation}\n\\begin{equation}\\label{c.2.6b}\n\\nabla_h\\\/S^{ijh} + T_h\\\/S^{ijh} + \\Sigma^{ij} - \\Sigma^{ji} = 0 \n\\end{equation}\n\\end{subequations}\nautomatically ensured by the Dirac equations \\eqref{3.9c} \\cite{FVC}. It is seen that the antisymmetric part of the Einstein--like equations \\eqref{3.9a} amounts to the conservation law for the spin \\eqref{c.2.6b}. The significant part of the Einstein--like equations is then the symmetric one which, making use of the trace of \\eqref{3.9a} and separating the purely metric terms from the torsional ones through eq. \\eqref{3.9b}, can be written in the final form\n\\begin{equation}\\label{3.19}\n\\begin{split}\n\\tilde{R}_{ij} -\\frac{1}{2}\\tilde{R}g_{ij}= \\frac{1}{\\varphi}\\tilde{\\Sigma}_{ij}\n+ \\frac{1}{\\varphi^2}\\left( - \\frac{3}{2}\\de\\varphi\/de{x^i}\\de\\varphi\/de{x^j} + \\varphi\\tilde{\\nabla}_{j}\\de\\varphi\/de{x^i} + \n\\frac{3}{4}\\de\\varphi\/de{x^h}\\de\\varphi\/de{x^k}g^{hk}g_{ij} \\right. \\\\\n\\left. - \\varphi\\tilde{\\nabla}^h\\de\\varphi\/de{x^h}g_{ij}\\right) + \\frac{3}{64\\varphi^2}(\\bar{\\psi}\\gamma_5\\gamma^\\tau\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\tau\\psi)g_{ij} \\\\\n- \\frac{\\epsilon(\\bar\\psi\\psi)\\left(\\frac{m}{2}\\bar\\psi\\psi -2V + \\frac{3}{2}\\bar\\psi\\psi\\\/V'\\right)}{2\\varphi\\left(\\frac{1}{2}\\varphi - \\frac{3}{2}\\right)}\\,g_{ij} - \\frac{1}{2\\varphi}V(\\bar\\psi\\psi)\\,g_{ij} + \\frac{1}{2\\varphi}\\bar\\psi\\psi\\\/V'(\\bar\\psi\\psi)\\,g_{ij}\n\\end{split}\n\\end{equation}\nwhere $\\tilde{R}_{ij}$, $\\tilde R$ and $\\tilde{\\nabla}_i$ are respectively the Ricci tensor, the Ricci scalar curvature and the covariant derivative induced by the Levi--Civita connection and \n\\begin{equation}\\label{3.15}\n\\tilde{\\Sigma}_{ij} := \\frac{i}{4}\\\/\\left[ \\bar\\psi\\Gamma_{(i}\\tilde{D}_{j)}\\psi - \\left(\\tilde{D}_{(j}\\bar\\psi\\right)\\Gamma_{i)}\\psi \\right]\n\\end{equation}\n$\\tilde{D}_i$ denoting spinor covariant derivative with respect to the Levi--Civita connection. \nThe Dirac equations can be handled in a similar way, assuming the expression\n\\begin{equation}\\label{3.20}\ni\\Gamma^{h}\\tilde{D}_{h}\\psi\n- \\frac{1}{\\varphi}\\frac{3}{16}\\left[(\\bar{\\psi}\\psi)\n+i(i\\bar{\\psi}\\gamma_5\\psi)\\gamma_5\\right]\\psi-m\\psi + V'(\\bar\\psi\\psi)\\psi - \\epsilon\\psi\\\/R=0\n\\end{equation}\nFor further details, the reader is referred to \\cite{FVC}.\n\n\\section{Bianchi--I cosmological models}\n\\subsection{Coupling to Dirac field only}\n{  In order to investigate cosmological scenarios deriving from \\eqref{3.19} and \\eqref{3.20}}, let us consider a Bianchi type I metric of the form\n\\begin{equation}\\label{4.1}\nds^2 = dt^2 - a^2(t)\\,dx^2 - b^2(t)\\,dy^2 - c^2(t)\\,dz^2\n\\end{equation}\nDenoting by $\\tau := abc$ the scale volume, evaluating the linear and spin connection coefficients associated with the metric tensor \\eqref{4.1} and inserting the results together with \\eqref{4.1} itself in equations \\eqref{3.19}, the latter are seen to assume the form\n\\begin{subequations}\\label{4.10}\n\\begin{equation}\\label{4.10a}\n\\frac{\\dot a}{a}\\frac{\\dot b}{b} + \\frac{\\dot b}{b}\\frac{\\dot c}{c} + \\frac{\\dot a}{a}\\frac{\\dot c}{c} =\n\\frac{1}{2\\varphi}m\\bar\\psi\\psi - \\frac{3}{64\\varphi^2}(\\bar{\\psi}\\gamma_5\\gamma^\\nu\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\nu\\psi) + \n\\frac{1}{\\varphi^2}\\left[- \\frac{3}{4}{\\dot\\varphi}^2 - \\varphi\\dot\\varphi\\frac{\\dot\\tau}{\\tau}\\right] - \\frac{1}{2\\varphi}V(\\bar\\psi\\psi) \n\\end{equation}\n\\begin{equation}\\label{4.10b}\n\\begin{split}\n\\frac{\\ddot b}{b} + \\frac{\\ddot c}{c} + \\frac{\\dot b}{b}\\frac{\\dot c}{c} = \n\\frac{1}{\\varphi^2}\\left[\\varphi\\dot\\varphi\\frac{\\dot a}{a} + \\frac{3}{4}{\\dot\\varphi}^2 -\\varphi\\left( \\ddot\\varphi + \\frac{\\dot\\tau}{\\tau}\\dot\\varphi \\right)\\right] + \\frac{3}{64\\varphi^2}(\\bar{\\psi}\\gamma_5\\gamma^\\nu\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\nu\\psi)\\\\\n- \\frac{\\epsilon(\\bar\\psi\\psi)\\left(\\frac{m}{2}\\bar\\psi\\psi -2V + \\frac{3}{2}\\bar\\psi\\psi\\\/V'\\right)}{2\\varphi\\left(\\frac{1}{2}\\varphi - \\frac{3}{2}\\right)} - \\frac{1}{2\\varphi}V(\\bar\\psi\\psi) + \\frac{1}{2\\varphi}(\\bar\\psi\\psi)V'(\\bar\\psi\\psi)\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{4.10c}\n\\begin{split}\n\\frac{\\ddot a}{a} + \\frac{\\ddot c}{c} + \\frac{\\dot a}{a}\\frac{\\dot c}{c} =  \n\\frac{1}{\\varphi^2}\\left[\\varphi\\dot\\varphi\\frac{\\dot b}{b} + \\frac{3}{4}{\\dot\\varphi}^2 -\\varphi\\left( \\ddot\\varphi + \\frac{\\dot\\tau}{\\tau}\\dot\\varphi \\right) \\right] + \\frac{3}{64\\varphi^2}(\\bar{\\psi}\\gamma_5\\gamma^\\nu\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\nu\\psi)\\\\\n- \\frac{\\epsilon(\\bar\\psi\\psi)\\left(\\frac{m}{2}\\bar\\psi\\psi -2V + \\frac{3}{2}\\bar\\psi\\psi\\\/V'\\right)}{2\\varphi\\left(\\frac{1}{2}\\varphi - \\frac{3}{2}\\right)} - \\frac{1}{2\\varphi}V(\\bar\\psi\\psi) + \\frac{1}{2\\varphi}(\\bar\\psi\\psi)V'(\\bar\\psi\\psi)\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{4.10d}\n\\begin{split}\n\\frac{\\ddot a}{a} + \\frac{\\ddot b}{b} + \\frac{\\dot a}{a}\\frac{\\dot b}{b} =  \n\\frac{1}{\\varphi^2}\\left[\\varphi\\dot\\varphi\\frac{\\dot c}{c} + \\frac{3}{4}{\\dot\\varphi}^2 -\\varphi\\left( \\ddot\\varphi + \\frac{\\dot\\tau}{\\tau}\\dot\\varphi \\right) \\right] + \\frac{3}{64\\varphi^2}(\\bar{\\psi}\\gamma_5\\gamma^\\nu\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\nu\\psi)\\\\\n- \\frac{\\epsilon(\\bar\\psi\\psi)\\left(\\frac{m}{2}\\bar\\psi\\psi -2V + \\frac{3}{2}\\bar\\psi\\psi\\\/V'\\right)}{2\\varphi\\left(\\frac{1}{2}\\varphi - \\frac{3}{2}\\right)} - \\frac{1}{2\\varphi}V(\\bar\\psi\\psi) + \\frac{1}{2\\varphi}(\\bar\\psi\\psi)V'(\\bar\\psi\\psi)\n\\end{split}\n\\end{equation}\n\\end{subequations}\ntogether with the conditions\n\\begin{subequations}\\label{4.11}\n\\begin{equation}\\label{4.11a}\n\\tilde\\Sigma_{12}=0\\quad \\Rightarrow \\quad a\\\/\\dot{b} - b\\\/\\dot{a}=0 \\quad \\cup \\quad \\bar\\psi\\gamma^5\\gamma^3\\psi =0\n\\end{equation}\n\\begin{equation}\\label{4.11b}\n\\tilde\\Sigma_{23}=0\\quad \\Rightarrow \\quad c\\\/\\dot{b} - b\\\/\\dot{c}=0 \\quad \\cup \\quad \\bar\\psi\\gamma^5\\gamma^1\\psi =0\n\\end{equation}\n\\begin{equation}\\label{4.11c}\n\\tilde\\Sigma_{13}=0\\quad \\Rightarrow \\quad a\\\/\\dot{c} - c\\\/\\dot{a}=0 \\quad \\cup \\quad \\bar\\psi\\gamma^5\\gamma^2\\psi =0\n\\end{equation}\n\\end{subequations}\nThe equations $\\tilde\\Sigma_{0A}=0$ ($A=1,2,3$) result to be identities. Conditions \\eqref{4.11} are constraints imposed on the metric or on the Dirac field. There are three ways to satisfy these conditions: one is to impose constraints of purely geometrical origin by requiring that $a\\dot{b}-b\\dot{a}=0$, $a\\dot{c}-c\\dot{a}=0$, $c\\dot{b}-b\\dot{c}=0$ obtaining an isotropic universe; another is to impose constraints of purely material origin by insisting that $\\bar\\psi\\gamma^5\\gamma^1\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^2\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^3\\psi=0$ giving an anisotropic universe without spin--torsion interactions (in fact in this case necessarily we have that $\\bar\\psi\\gamma^5\\gamma^0\\psi =0$, otherwise the condition $\\bar\\psi\\gamma^0\\psi =0$ must be true, implying that the whole spinor must vanish); the last situation would be of both geometrical and material origin by insisting that for instance $a\\dot{b}-b\\dot{a}=0$ with $\\bar\\psi\\gamma^5\\gamma^1\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^2\\psi=0$ giving a partial isotropy for only two axes with the corresponding two components of the spin vector vanishing. We will be back to this issue in a moment.\n\nFollowing a useful procedure \\cite{Saha1,Saha2,Saha3,VFC,FVC}, we can suitably combine eqs. \\eqref{3.19}, obtaining the expressions of the scale factors as functions of the scale volume $\\tau$\n\\begin{subequations}\\label{4.13bis}\n\\begin{equation}\\label{4.13bisa}\na= \\tau^{\\frac{1}{3}}\\left(XY\\right)^{\\frac{1}{3}}e^{\\left(\\frac{Z+W}{3}\\int{\\frac{dt}{\\varphi\\tau}}\\right)}\n\\end{equation}\n\\begin{equation}\\label{4.13bisb}\nb=\\tau^{\\frac{1}{3}}X^{-\\frac{2}{3}}Y^{\\frac{1}{3}}e^{\\left(\\frac{-2Z+W}{3}\\int{\\frac{dt}{\\varphi\\tau}}\\right)}\n\\end{equation}\n\\begin{equation}\\label{4.13bisc}\nc=\\tau^{\\frac{1}{3}}X^{\\frac{1}{3}}Y^{-\\frac{2}{3}}e^{\\left(\\frac{Z-2W}{3}\\int{\\frac{dt}{\\varphi\\tau}}\\right)}\n\\end{equation}\n\\end{subequations}\n($X,Y,Z$ and $W$ being integration constants) and the dynamical equation \\footnote{ We notice that eq. \\eqref{4.14} differs from the analogous equation deduced in \\cite{FVC}, due to a trivial calculation error made in \\cite{FVC} and responsible for a different expression before the term $V'$; as most of the results obtained in A concern the case $V = 0$, they remain equally exact; the only case discussed in \\cite{FVC} where $V\\not =0$ remain qualitatively correct.} for $\\tau$\n\\begin{equation}\\label{4.14}\n2\\frac{\\ddot\\tau}{\\tau}  = - 3\\frac{\\ddot\\varphi}{\\varphi} - 5\\frac{\\dot\\tau}{\\tau}\\frac{\\dot\\varphi}{\\varphi} \n- \\frac{3m\\bar\\psi\\psi -3V\\left(\\varphi +1\\right) + 3\\varphi\\bar\\psi\\psi\\\/V'}{\\varphi\\left(\\varphi - 3\\right)}\n\\end{equation}\nHere, it is noteworthy\nthat equation \\eqref{4.10a} plays the role of a constraint on the initial data and thus on the integration constants. In this regard, in \\cite{FVC} it has been actually checked that if the Hamiltonian constraint \\eqref{4.10a}\nis satisfied initially, then it is preserved in time.\nAnalogously, in the metric \\eqref{4.1} the Dirac equations \\eqref{3.20} become \n\\begin{subequations}\\label{4.9}\n\\begin{equation}\\label{4.9a}\n\\dot\\psi + \\frac{\\dot\\tau}{2\\tau}\\psi + im\\gamma^0\\psi +\n\\frac{3i}{16\\varphi}\\\/\\left[ (\\bar\\psi\\psi)\\gamma^0 +i\\\/(i\\bar\\psi\\gamma^5\\psi)\\gamma^0\\gamma^5 \\right]\\psi + i\\epsilon\\\/R\\gamma^0\\psi -iV'\\gamma^0\\psi=0\n\\end{equation}\n\\begin{equation}\\label{4.9b}\n\\dot{\\bar\\psi} + \\frac{\\dot\\tau}{2\\tau}\\bar\\psi - im\\bar{\\psi}\\gamma^0 - \\frac{3i}{16\\varphi}\\bar\\psi\\\/\\left[ (\\bar\\psi\\psi)\\gamma^0 +i\\\/(i\\bar\\psi\\gamma^5\\psi)\\gamma^5\\gamma^0 \\right] - i\\epsilon\\\/R\\bar\\psi\\gamma^0 + iV'\\bar\\psi\\gamma^0=0\n\\end{equation}\n\\end{subequations}\n{  In order to solve eqs. \\eqref{4.9}, we can adapt to the present context the arguments developed in \\cite{Saha14}. First of all we combine eqs. \\eqref{4.9} obtaining the differential equations\n\\begin{equation}\\label{4.17a}\n\\frac{d}{dt}\\left(\\tau\\bar\\psi\\psi\\right) + \\frac{3\\tau}{8\\varphi}\\left(i\\bar\\psi\\gamma^5\\psi\\right)\\left(\\bar\\psi\\gamma^5\\gamma^0\\psi\\right)=0\n\\end{equation}\nWe search for solutions of \\eqref{4.9} satisfying the condition\n\\begin{equation}\\label{4.17a.1}\n\\bar\\psi\\gamma^5\\psi =0\n\\end{equation} \nUnder such a hypothesis, eq. \\eqref{4.17a} implies\n\\begin{equation}\\label{4.18}\n\\bar\\psi\\psi = \\frac{K}{\\tau}\n\\end{equation}\nwhere $K$ is an integration constant. At the same time, from eqs. \\eqref{3.9a} and \\eqref{3.9c} we can derive the expression of the scalar curvature $R$ as function of the bilinear spinor $\\bar\\psi\\psi$\n\\begin{equation}\\label{4.17a.2}\nR(\\bar\\psi\\psi)=\\frac{m\\bar\\psi\\psi -4V + 3\\bar\\psi\\psi\\\/V'}{\\varphi -3}\n\\end{equation}\nIn view of eqs. \\eqref{4.17a.1}, \\eqref{4.18} and \\eqref{4.17a.2}, the Dirac equation \\eqref{4.9a} can be rewritten as\n\\begin{equation}\\label{4.17a.3}\n\\dot\\psi + \\frac{\\dot\\tau}{2\\tau}\\psi + iG(\\tau)\\gamma^0\\psi =0\n\\end{equation} \nwhere we have set\n\\begin{equation}\\label{4.17a.4}\nG(\\tau) := \\left(m + \\frac{3}{16\\varphi}\\bar\\psi\\psi + \\epsilon\\\/R(\\bar\\psi\\psi) - V'(\\bar\\psi\\psi)\\right)_{|\\bar\\psi\\psi = \\frac{K}{\\tau}}\n\\end{equation}\nConsidering the 4-component spinor field \n\\begin{eqnarray}\\label{4.17a.5}\n\\psi =\\left(\\begin{tabular}{c}\n$\\psi_1$\\\\\n$\\psi_2$\\\\\n$\\psi_3$\\\\\n$\\psi_4$\n\\end{tabular}\\right)\n\\end{eqnarray}\neqs. \\eqref{4.17a.3} assume the explicit form\n\\begin{subequations}\\label{4.17a.6}\n\\begin{equation}\n\\dot{\\psi}_1 + \\frac{\\dot\\tau}{2\\tau}\\psi_1 + iG(\\tau)\\psi_1 =0\n\\end{equation}\n\\begin{equation}\n\\dot{\\psi}_2 + \\frac{\\dot\\tau}{2\\tau}\\psi_2 + iG(\\tau)\\psi_3 =0\n\\end{equation}\n\\begin{equation}\n\\dot{\\psi}_3 + \\frac{\\dot\\tau}{2\\tau}\\psi_3 - iG(\\tau)\\psi_3 =0\n\\end{equation}\n\\begin{equation}\n\\dot{\\psi}_4 + \\frac{\\dot\\tau}{2\\tau}\\psi_4 - iG(\\tau)\\psi_4 =0\n\\end{equation}\n\\end{subequations}\nA solution of eqs. \\eqref{4.17a.6} is then given by \n\\begin{eqnarray}\\label{4.17a.7}\n&\\psi=\\frac{1}{\\sqrt{\\tau}}\\left(\\begin{tabular}{c}\n$C_1\\exp\\left(-i\\int{Gdt}\\right)$\\\\\n$C_2\\exp\\left(-i\\int{Gdt}\\right)$\\\\\n$C_3\\exp\\left(+i\\int{Gdt}\\right)$\\\\\n$C_4\\exp\\left(+i\\int{Gdt}\\right)$\n\\end{tabular}\\right)\n\\end{eqnarray}\nwhere $C_i$ are integration constants which, because of constraints \\eqref{4.17a.1} and \\eqref{4.18}, have to satisfy the equations\n\\begin{subequations}\\label{4.17a.8}\n\\begin{equation}\nC_1^*C_1 + C_2^*C_2 - C_3^*C_3 - C_4^*C_4 = K\\label{a}\n\\end{equation}\n\\begin{equation}\nC_1^*C_3 + C_2^*C_4 =0\\label{b}\n\\end{equation}\n\\end{subequations}\nMoreover, the constants $C_i$ have to satisfy further constraints deriving from the non diagonal part of the Einstein--like equations \\eqref{3.19}. As we have discussed above, these additional conditions result in a maximum of three real equations given by\n\\begin{subequations}\\label{4.17a.9}\n\\begin{equation}\nC_1^*C_2 + C_2^*C_1 + C_3^*C_4 + C_4^*C_3 =0\n\\end{equation} \n\\begin{equation}\nC_1^*C_2 - C_2^*C_1 + C_3^*C_4 - C_4^*C_3 =0\n\\end{equation} \n\\begin{equation}\n- C_1^*C_1 + C_2^*C_2 - C_3^*C_3 + C_4^*C_4 =0\n\\end{equation} \n\\end{subequations}\nEquations \\eqref{4.17a.8} and \\eqref{4.17a.9} form a system of at most six real equations for eight real unknowns, thus in general one should expect that solutions exist. \n\nHowever, if all of the last constraints were considered then we can draw some additional conclusions. By combining the first two of \\eqref{4.17a.9} we see that \n\\begin{equation}\nC_1^*C_2+C_3^*C_4=0\\label{aux}\n\\end{equation}\nwhich can be combined together with the second of \\eqref{4.17a.8} to show that \n\\begin{equation}\nC_2(|C_1|^2-|C_4|^2)= C_2C_1^*C_1-C_2C_4^*C_4=-C_4C_3^*C_1+C_3^*C_1C_4=0\n\\end{equation} \nand so either $C_2=0$ or $|C_1|^2=|C_4|^2$ in general. If $|C_1|^2=|C_4|^2$  we would have that the third of \\eqref{4.17a.9} remains $|C_2|^2=|C_3|^2$ and so $K=0$. If $C_2=0$ we have that \\eqref{aux} and the second of \\eqref{4.17a.8}  imply that either $C_3=0$ or $C_1=C_4=0$, which by the \\eqref{4.17a.9} yields again $C_3=0$. In both cases this returns again $K=0$. Therefore, if all \\eqref{4.17a.9} are accounted for, then $K$ is necessarily zero, and there would be no condensate: this is to be expected, because these three conditions are equivalent to the requirement of total isotropy of the spinor field. Indeed, if this were to be the case, then  all of the spatial components of the spin vector would have to vanish, and because the algebraic identity $\\bar\\psi\\gamma^5\\gamma^\\mu\\psi\\bar\\psi\\gamma_\\mu\\psi=0$ is always true, then  $\\bar\\psi\\gamma^5\\gamma^0\\psi\\bar\\psi\\gamma_0\\psi=0$. Now, if  $\\psi\\bar\\psi\\gamma_0\\psi=\\psi^{\\dagger}\\psi=0$, the spinor itself would be zero.  Therefore we have to select the condition $\\bar\\psi\\gamma^5\\gamma^0\\psi =0$, which means that also the temporal component of the spin vector is zero, and therefore the entire spin vector is zero. Furthermore, since  \n\\begin{equation}\n|i\\bar\\psi\\gamma^5\\psi|^2+|\\bar\\psi\\psi|^2=-\\bar\\psi\\gamma^5\\gamma^\\mu\\psi\\bar\\psi\\gamma^5\\gamma_\\mu\\psi\n\\end{equation}\nthe reasoning above leads the conditions $i\\bar\\psi\\gamma^5 \\psi=\\bar\\psi\\psi=0$, which imply that there is no condensate, and thus no non-minimal coupling. In conclusion, we have to dismiss the case  $\\bar\\psi\\gamma^5\\gamma^1\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^2\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^3\\psi=0$ because it is not possible to have a geometrically anisotropic universe filled with isotropic matter. \n\nA second scenario is to have partial isotropy in both geometry and matter as for instance in the case $a\\dot{b}-b\\dot{a}=0$ with $\\bar\\psi\\gamma^5\\gamma^1\\psi=0$, $\\bar\\psi\\gamma^5\\gamma^2\\psi=0$. In this circumstance the last of \\eqref{4.17a.9} is lost and thus a solution can be found. For example, a solution is given by $|C_1|^2=K$ and all other constants equal to zero or by $|C_2|^2=K$ and all other constants equal to zero: they give\n\\begin{eqnarray}\\label{soluzioniesatte}\n&\\psi=\\exp\\left(-i\\int{Gdt}\\right)\\sqrt{\\frac{K}{\\tau}}\\left(\\begin{tabular}{c}\n$1$\\\\\n$0$\\\\\n$0$\\\\\n$0$\n\\end{tabular}\\right)\\ \\ \\ \\ \\ \\ \\ \\mathrm{or} \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\psi=\\exp\\left(-i\\int{Gdt}\\right)\\sqrt{\\frac{K}{\\tau}}\\left(\\begin{tabular}{c}\n$0$\\\\\n$1$\\\\\n$0$\\\\\n$0$\n\\end{tabular}\\right)\n\\end{eqnarray}\nwhich can respectively be interpreted as a spinor in the spin $\\frac{1}{2}$ or the spin $-\\frac{1}{2}$ eigenstate in non-relativistic case (that is, with the two lower components vanishing in standard representation).\n\nThe third and last case is given by a totally isotropic universe $a\\dot{b}-b\\dot{a}=0$, $a\\dot{c}-c\\dot{a}=0$, $c\\dot{b}-b\\dot{c}=0$ filled with anisotropic matter \\footnote{This might seem a contradiction, but the fact hat an anisotropic spinor could be compatible with an isotropic metric is due to the way in which the spinor gravitates. Looking at equations \\eqref{4.10} it clear that the spin vector, which would be responsible of the breaking of isotropy enters in the field equations only as $(\\bar{\\psi}\\gamma_5\\gamma^\\tau\\psi)(\\bar{\\psi}\\gamma_5\\gamma_\\tau\\psi)$. Therefore spacetime does not ``respond'' to the spin vector and can be isotropic.}.  In such a circumstance eqs. \\eqref{4.17a.9} do not apply and eqs. \\eqref{4.17a.8} certainly admit solutions, for instance still of the form \\eqref{soluzioniesatte}.\n\nWe remark that in any case, the condensate evolves as $\\bar\\psi\\psi=\\frac{K}{\\tau}$ and that is all we need to perform the analysis of the cosmological model. The fact that the entire information about the spinor is not necessary and that only the condensate is important may sound strange but it is exactly what we would expect to have in macroscopic systems (after all, also in the physics of condensates one does not need the complete dynamical behaviour of each single electron or Cooper couple to know the evolution of the condensate itself --- similar arguments can be used to justify why one does not need the motion of each single atom or molecule to know the evolution of a gas).} \n\nSo, resuming the problem of finding the dynamical equation for the scale volume $\\tau$, we may insert the relation $\\bar\\psi\\psi = \\frac{K}{\\tau}$ into \\eqref{4.14} getting the final equation\n\\begin{equation}\\label{1}\n2\\frac{\\ddot\\tau}{\\tau}\\varphi+3\\ddot\\varphi+5\\frac{\\dot\\tau}{\\tau}\\dot\\varphi\n=\\frac{3mK}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} - \\frac{3\\left(\\epsilon K + 2\\tau\\right)\\\/V}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} + \\frac{3\\left(\\epsilon\\\/K+\\tau\\right)KV'}{\\tau^2\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)}\n\\end{equation}\nThe fact that one can combine the equations in this way should not be surprising. In fact, given a time-like normalized vector field $X_a$ and the projection tensor $h_{ab}=g_{ab}-X_aX_b$, the physical properties of any anisotropic cosmology can be characterized by the expansion scalar and and the shear scalar:  \n\\begin{equation}\n\\theta=h^{ab}\\tilde{\\nabla}_{a}X_b \\qquad \\sigma=\\frac{1}{2}\\sqrt{\\sigma_{ab}\\sigma^{ab}} \\qquad \\sigma_{ab}=\\frac{1}{2}h^{c}_{a}h^{d}_{b}\\left(\\tilde{\\nabla}_{c}X_{d}+\\tilde{\\nabla}_{d}X_{c}\\right) -\\frac{1}{3}h_{ab}\\theta\n\\end{equation}\nIn the particular case of the metric \\eqref{4.1} the expansion is \n\\begin{equation}\n\\theta=\\frac{\\dot{a}}{a}+\\frac{\\dot{b}}{b}+\\frac{\\dot{c}}{c}=\\frac{\\dot{\\tau}}{\\tau}\n\\end{equation}\nso that the \\eqref{1} is an analogous of the Raychaudhuri equation. It is also useful to write the shear scalar in terms of the metric \\eqref{4.1} ad the $\\tau$\n\\begin{equation}\\label{1.sigma}\n\\sigma=\\frac{1}{2}\\left[\\left(\\frac{\\dot{a}}{a}\\right)^2+\\left(\\frac{\\dot{b}}{b}\\right)^2+\\left(\\frac{\\dot{c}}{c}\\right)^2- \\frac{1}{3}\\left(\\frac{\\dot{a}}{a}+\\frac{\\dot{b}}{b}+\\frac{\\dot{c}}{c}\\right)^2\\right]^{1\/2}=\\frac{1}{2}\\left[\\frac{2(Z^2+W^2-ZW)}{3(\\tau+\\epsilon K)^2}\\right]^{1\/2}\n\\end{equation}\nwhere in the last expression we have used the \\eqref{4.13bis}. It is immediately clear that the only way to increase the anisotropy of the system is to have a contraction, so these models, if expanding, tend to isotropize. In addition, and differently from GR, for $\\tau \\rightarrow 0$ (and for $\\epsilon >0$) the shear tends to a finite value depending on $K$ and other constants of integration; on the contrary, if $\\epsilon < 0$, the shear scalar can blow up before that $\\tau =0$.\n\nUsing the identity $2\\ddot\\tau\\varphi+3\\tau\\ddot\\varphi+5\\dot\\tau\\dot\\varphi = \\frac{d^2}{dt^2}\\left(2\\tau-\\epsilon K\\ln\\tau\\right)$, \\eqref{1}  yields\n\\begin{equation}\\label{2}\n\\frac{d}{dt}\\left[\\frac{d}{dt}\\left(2\\tau-\\epsilon K\\ln\\tau\\right)\\right]^{2}\n=6\\left[mK - \\left(\\epsilon\\\/K + 2\\tau\\right)V + \\frac{\\left(\\epsilon\\\/K+\\tau\\right)K}{\\tau}V'\\right]\\dot{\\tau}\n\\end{equation}\nIn the following, by exploiting the linear dependence on the potential and its derivative in eq. \\eqref{1} (or \\eqref{2}), we analyze different scenarios associated to various choices of the potential $V$. To do that we follow two different approaches: the first one is a reconstruction technique, where a given time evolution for the scale volume is assumed and then eq. \\eqref{1} is solved for $V$, making systematically use of the relation $\\bar\\psi\\psi = \\frac{K}{\\tau}$; the second one consists in choosing $V$ in such a way that the right--hand side of eq. \\eqref{2} becomes easily solvable (exactly or at least for some approximations) and the corresponding solutions represent interesting cosmological evolutions. The properties of these scenarios will be characterised in terms of the behavior of $\\theta$ and $\\sigma$.\n\n\n\\subsubsection{The case $V=0$.} \nTo start with, we discuss the simplest case $V=0$. In this circumstance eq. \\eqref{2} assumes the form\n\\begin{equation}\\label{3}\n\\frac{d}{dt}\\left[\\frac{d}{dt}\\left(2\\tau-\\epsilon K\\ln\\tau\\right)\\right]^{2}=6mK\\dot{\\tau}\n\\end{equation}\nwhich can be integrated as\n\\begin{equation}\\label{4}\n\\frac{d}{dt}(2\\tau-\\epsilon K\\ln{\\tau})\\!=\\!\\pm\\sqrt{6mK\\tau\\!-\\!A}\n\\end{equation}\nyielding a first--order differential equation for $\\tau$ with integration constant $A$. Assuming $A$ be negative, equation \\eqref{4} can be integrated as\n\\begin{equation}\\label{5}\nt\\!+\\!B\\!=\\!\\pm\\frac{2\\sqrt{|A|}}{3mK}\\left(\\sqrt{\\frac{6mK}{|A|}\\tau\\!+\\!1}\\right)\\!\\pm\\!\n\\frac{2\\epsilon K}{\\sqrt{|A|}}\\mathrm{arctanh}\\left(\\sqrt{\\frac{6mK}{|A|}\\tau\\!+\\!1}\\right)\n\\end{equation}\nbut as it is also clear, $A$ negative (with of course $\\tau$ positive) means that the argument of the $\\mathrm{arctanh}$ is larger than one and thus such function is ill-defined. Therefore we are forced to assume $A\\geq 0$: in the case $A >0$ the differential equation is integrated as\n\\begin{equation}\\label{6}\nt\\!+\\!B\\!=\\!\\pm\\frac{2\\sqrt{A}}{3mK}\\left(\\sqrt{\\frac{6mK}{A}\\tau\\!-\\!1}\\right)\\!\\mp\\!\n\\frac{2\\epsilon K}{\\sqrt{A}}\\arctan{\\left(\\sqrt{\\frac{6mK}{A}\\tau\\!-\\!1}\\right)}\n\\end{equation}\nwhich is well-defined whenever the volume is larger than a given lower-bound $\\tau_{0}\\!\\geqslant\\!\\frac{A}{6mK}$ and thus showing that, regardless the value of $B$, there is no way in which the minimal volume $\\tau_{0}$ can be zero; if $A=0$, we get the solution\n\\begin{equation}\nt\\!+\\!B\\!=\\!\\pm\\frac{\\sqrt{2}\\left(\\epsilon\\\/K+2\\tau\\right)}{\\sqrt{3mK\\tau}}\n\\label{zero}\n\\end{equation}\nfrom which again we cannot have zero scale volume at a finite time. In all these cases then, singularities are avoided due to the presence of the non-minimal coupling term we have here: in fact, if $\\epsilon\\!=\\!0$ then there will be nothing preventing us to have a negative $A$, so that it would be possible to have the solution \\eqref{5} which in this case would reduce to\n\\begin{equation}\nt\\!+\\!B\\!=\\!\\pm\\frac{2\\sqrt{|A|}}{3mK}\\left(\\sqrt{\\frac{6mK}{|A|}\\tau\\!+\\!1}\\right)\n\\end{equation}\nallowing zero scale volume $\\tau=0$ at the finite time $t\\!=\\!-B\\pm\\frac{2\\sqrt{|A|}}{3mK}$. These phenomena are not new in the context of ECSK theories (with minimal \\cite{Poplawski:2009su,Poplawski:2011jz,Poplawski:2011wj,Poplawski:2010jv} and non minimal couplings \\cite{Magueijo:2012ug}). However, differently form these studies, our analysis relies exclusively on the exact field equations and therefore it is of purely mathematical nature. \n\nAlso, it must be pointed out that the analysis of the Hamiltonian constraint \\eqref{4.10a} provides interesting constraints on the constants of this model. To see this point, using  eqs. \\eqref{4.13bis} and \\eqref{4}, we easily get the identities    \n\\begin{subequations}\\label{0.0.1}\n\\begin{equation}\n\\frac{\\dot a}{a} = \\frac{1}{3}\\frac{\\dot \\tau}{\\tau} + \\frac{(Z+W)}{3} \\frac{1}{\\varphi\\tau}\n\\end{equation}\n\\begin{equation}\n\\frac{\\dot b}{b} = \\frac{1}{3}\\frac{\\dot \\tau}{\\tau} + \\frac{(-2Z+W)}{3} \\frac{1}{\\varphi\\tau}\n\\end{equation}\n\\begin{equation}\n\\frac{\\dot c}{c} = \\frac{1}{3}\\frac{\\dot \\tau}{\\tau} + \\frac{(Z-2W)}{3} \\frac{1}{\\varphi\\tau}\n\\end{equation}\n\\begin{equation}\n{\\dot \\tau}^2 =\\frac{\\tau^2}{(2\\tau - \\epsilon\\\/K)^2}(6mK\\tau -A)\n\\end{equation}\n\\end{subequations}\nInserting the content of \\eqref{0.0.1} into \\eqref{4.10a}, we obtain the relation\n\\begin{equation}\\label{0.0.8}\n-\\frac{A}{12} + \\frac{1}{9}\\left[-3(Z+W)^2 + 9ZW\\right] = K^2\\left(\\frac{m}{2}\\epsilon + \\frac{3}{64}\\right)\n\\end{equation}\nBecause of the restriction imposed on A ($A\\geq 0$) found above, the left hand side of \\eqref{0.0.8} is always non--positive and so must be the right hand side: this necessarily requires \n\\begin{equation}\\label{0.0.9}\n\\epsilon \\leq -\\frac{3}{32m}\n\\end{equation}\nwhich represents an upper bound for the coupling constant $\\epsilon$ in the case the self--interaction potential $V$, or also other kinds of matter different from the only fermionic field, are absent. This fact, together with eq. \\eqref{1.sigma}, implies that the singularity on the scale factors can be replaced by a singularity in the shear that happens at finite time (if $\\frac{A}{6mK}\\leq |\\epsilon|K$). In this respect, therefore, the claim that these models are singularity free is an incomplete statement, as the model could retain a singularity (albeit of a different type) at some point in its history.\n\nAnother interesting aspect associated with the non--minimal coupling we are studying is that if there were a (cosmological) time interval in which the first term on the right hand side of equation \\eqref{6} were negligible with respect to the second one, then in such a time interval we would have an expansion of the universe according to $\\tau\\!\\sim\\!\\left(\\tan{t}\\right)^{2}$, which could account for an accelerated behaviour possibly fitting inflationary scenarios (at least for isotropic models). The above mentioned circumstance could be achieved for example by assigning initial data and then integration constants such that $\\sqrt{A}\/K$ is very small.\n\nThe model outlined above is therefore rather intriguing, because it can solve the problem of the cosmological singularity in quite elegant a way and simultaneously, by a careful fine tuning, it can address the issue of inflationary scenarios. Unfortunately, the model with $V=0$ is unable to account for cosmic acceleration at late time. This is easily seen still considering equation \\eqref{6}, this time evaluated for large values of $\\tau$ (with respect to a given reference volume of the universe), obtaining a behaviour of the scale volume as $\\tau\\!\\sim\\!t^{2}$ i.e. $\\theta=2\/t$, which at late time ensures isotropization (see eqs. \\eqref{4.13bis}) but under a decelerated expansion of the scale factors. \n\n\n\\subsubsection{The potential for a decelerated power law expansion.}\nAs a first example in which a potential is present, following a reconstruction approach we look for a potential $V$ which gives rise to an expansion law of the form $\\tau=\\tau_0\\\/t^2$ already treated in the previous section. This behaviour of $\\tau$ implies that the scale factors $a, b ,c$ have a decelerated expansion law, at least at late time.\n\nInserting $\\tau=\\tau_0\\\/t^2$ into \\eqref{1}, multiplying by $\\tau$ and expressing all in terms of $\\bar\\psi\\psi$, we get the differential equation for the unknown $V$ \n\\begin{equation}\\label{F1}\n2\\tau_0 + 2\\epsilon\\tau_0\\bar\\psi\\psi = \\frac{3mK}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)} - \\frac{3K\\left(\\epsilon\\bar\\psi\\psi + 2\\right)}{\\bar\\psi\\psi\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V + \\frac{3K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V'\n\\end{equation}\nThe solution of \\eqref{F1} is\n\\begin{equation}\\label{F2}\nV(\\bar\\psi\\psi)=\\frac{1}{\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\\left[-\\frac{2\\epsilon\\tau_0}{3K}\\left(\\bar\\psi\\psi\\right)^3 - \\frac{4\\tau_0}{3K}\\left(\\bar\\psi\\psi\\right) + \\frac{2\\epsilon\\tau_0}{3K}\\left(\\bar\\psi\\psi\\right)^2\\ln\\left(\\bar\\psi\\psi\\right) + m\\left(\\bar\\psi\\psi\\right)\\right]\n\\end{equation}\nIn addition, the cosmology isotropizes ( $\\sigma\\rightarrow 0$) in the future, since  \n\\begin{equation}\\label{s-V=0}\n\\sigma=\\frac{1}{2}\\left[\\frac{2(Z^2+W^2-ZW)}{3\\left(\\tau_0 t^2+\\epsilon K\\right)^2}\\right]^{1\/2}\n\\end{equation} \nAs above, this results could be deduced also from the \\eqref{4.13bis}, which converge to $a\\propto b\\propto c$ for this behaviour of $\\tau$.\n\n\n\n\\subsubsection{Potentials for exponential expansion.}\nAs a second example, we search for potentials inducing exponential expansion of the scale volume. We begin by a reconstruction technique considering a scale volume of the form $\\tau=\\tau_0\\exp(t)$. In this case $\\theta=1$, and\n\\begin{equation}\n\\sigma=\\frac{1}{2}\\left\\{\\frac{2(Z^2+W^2-ZW)}{3\\left[\\tau_0\\exp(t)+\\epsilon K\\right]^2}\\right\\}^{1\/2}\n\\end{equation}\nso that the anisotropy becomes quickly zero. Inserting $\\tau=\\tau_0\\exp(t)$ into \\eqref{1}, multiplying by $\\tau$ and using $\\bar\\psi\\psi$ as independent variable, we get the final equation\n\\begin{equation}\\label{E1}\n\\frac{2K}{\\bar\\psi\\psi} = \\frac{3mK}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)} - \\frac{3K\\left(\\epsilon\\bar\\psi\\psi + 2\\right)}{\\bar\\psi\\psi\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V + \\frac{3K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V'\n\\end{equation}\nThe latter admits the solution\n\\begin{equation}\\label{E2}\nV(\\bar\\psi\\psi)=\\frac{\\bar\\psi\\psi\\left(2\\epsilon+3m\\right)-2}{3\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\n\\end{equation}\nAnother potential which yields exponential expansion at least at late time is given by\n\\begin{equation}\\label{E3}\nV(\\bar\\psi\\psi) = - \\frac{1}{6\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\n\\end{equation}\nIndeed, with the choice \\eqref{E3}, equation \\eqref{2} can be integrated as\n\\begin{equation}\\label{ded11}\n\\frac{\\left(2\\tau-\\epsilon K\\right)\\dot\\tau}{\\tau} = \\sqrt{6mK\\tau + \\tau^2 + A}\n\\end{equation}\nwith $A$ denoting an integration constant. It is evident that if  $A$ is negative there exists automatically a strictly positive minimum value of the scale volume, then the singularity in the scale volume is avoided. For instance, setting $A=-1$ for simplicity, eq. \\eqref{ded11} can be integrated as\n\\begin{equation}\\label{ded11bis}\nt+C= 2\\ln\\left(\\sqrt{6mK\\tau+\\tau^2-1}+ 3mK + \\tau\\right) - \\epsilon\\\/K\\arctan{\\left(\\frac{3mK\\tau - 1}{\\sqrt{6mk\\tau + \\tau^2 -1}}\\right)}\n\\end{equation}\nwhich for large values of $\\tau$ yields exponential expansion. We discuss more in detail the case $A>0$; in such a circumstance by integrating eq. \\eqref{E3} we get\n\\begin{equation}\\label{ded13}\nt+C=2\\ln{\\left(\\sqrt{6mK\\tau + \\tau^2 + A}+3mK+\\tau\\right)}\\!+\\!\n\\frac{\\epsilon K}{\\sqrt{A}}\\ln\\left(\\frac{A+3mK\\tau+\\sqrt{A}\\sqrt{6mK\\tau + \\tau^2 + A}}{\\tau}\\right) \n\\end{equation}\nFor large values of $\\tau$ we have as above exponential expansion of the scale volume; moreover, setting $\\epsilon<0$, for very small values of $\\tau$ (with respect to a given reference volume of the universe) we can approximate the solution \\eqref{ded13} to \n\\begin{equation}\\label{ded13bis}\nt+D = \\frac{\\epsilon\\\/K}{\\sqrt{A}}\\ln{\\left(\\frac{2A}{\\tau}\\right)}\n\\end{equation}\nyielding again exponential expansion. We notice that both the potentials \\eqref{E2} and \\eqref{E3} are not trivial in view of the non--minimal coupling. Indeed, if $\\epsilon =0$ \\eqref{E3} reduces to a cosmological constant while \\eqref{E2} makes the Lagrangian \\eqref{3.2} identical to that of a massless Dirac spinor with cosmological constant. \n\n\n\\subsubsection{Potentials for transition from an early power law inflation to a decelerated power law expansion era.}\nLet us consider the potential\n\\begin{equation}\\label{IF1}\nV(\\bar\\psi\\psi)= \\frac{\\gamma\\left(\\bar\\psi\\psi\\right)^{p+1}}{6K^{p+1}\\left(p-1\\right)\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\n\\end{equation}\nwhere $\\gamma$ is a suitable constant. It is easily seen that for such choice of potential eq. \\eqref{2} becomes\n\\begin{equation}\\label{IF2}\n\\frac{d}{dt}\\left[\\frac{d}{dt}\\left(2\\tau-\\epsilon K\\ln\\tau\\right)\\right]^{2} = \\left(6mK + \\gamma\\tau^{-p}\\right)\\dot\\tau\n\\end{equation}\nFrom \\eqref{IF2}, by integrating we get\n\\begin{equation}\\label{IF3}\n\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)^2{\\dot\\tau}^2 = 6mK\\tau + \\frac{\\gamma}{-p+1}\\tau^{-p+1} + A\n\\end{equation}\n$A$ being an integration constant. Now, for large values of $\\tau$ eq. \\eqref{IF3} approximates the equation\n\\begin{equation}\\label{IF5}\n2|\\dot\\tau| = \\sqrt{6mK\\tau}\n\\end{equation}\ngiving rise to $\\tau \\approx t^2$ and then to isotropization. On the contrary, for very small values of $\\tau$ eq. \\eqref{IF3} can be approximated by\n\\begin{equation}\\label{IF6}\n|\\epsilon|\\\/K\\frac{|\\dot\\tau|}{\\tau}= \\sqrt{\\frac{\\gamma}{1-p}}\\tau^{\\frac{1-p}{2}}.\n\\end{equation}\nBy choosing $p$ such that $\\frac{1-p}{2} = -\\frac{1}{3q}$, $q\\geq 2$ being an even number, we have $\\tau \\approx t^{3q}$ which represents power law inflation  at least for isotropic models. In the case of initial anisotropy, the shear scalar \n\\begin{equation}\n\\sigma=\\frac{1}{2}\\left[\\frac{2(Z^2+W^2-ZW)}{3\\left(\\tau_0 t^{3q}+\\epsilon K\\right)^2}\\right]^{1\/2}\n\\end{equation} \nensures a quick isotropization, depending on $q$. {  It should be noted that potentials of the form \\eqref{IF1} would work equally well also in the case of minimal coupling ($\\epsilon =0$)}.\n\n\n\\subsubsection{Potentials for transition from a decelerated expansion era to dark era.}\nLet us consider the scale volume function of the form \n\\begin{equation}\\label{FD1}\n\\tau = \\tau_0\\left(\\sinh\\left(\\lambda\\\/t\\right)\\right)^2\n\\end{equation}\nfor which the expansion is $\\theta=2 \\lambda  \\coth (\\lambda  t)$ and the shear scalar is\n\\begin{equation}\n\\sigma=\\frac{1}{2}\\left\\{\\frac{2(Z^2+W^2-ZW)}{ 3\\left[\\tau_0\\sinh^2\\left(\\lambda\\\/t\\right)+\\epsilon K\\right]^2}\\right\\}^{1\/2}\n\\end{equation}\ni.e. a cosmology for which there is a transition between a power law and a de Sitter expansion and the anisotropy decreases converging eventually to zero.\nIn this case, we have the identity\n\\begin{equation}\\label{FD2}\n\\frac{d^2}{dt^2}\\left(2\\tau-\\epsilon K\\ln\\tau\\right) = 8\\lambda^2\\tau + 4\\tau_0\\lambda^2 + \\frac{2\\epsilon\\\/K\\lambda^2\\tau_0}{\\tau} = \\frac{8\\lambda^2\\\/K}{\\bar\\psi\\psi} + 4\\tau_0\\lambda^2 + 2\\epsilon\\lambda^2\\\/\\tau_0\\bar\\psi\\psi\n\\end{equation}\nIn view of \\eqref{FD2}, eq. \\eqref{1} assumes the form\n\\begin{equation}\\label{FD3}\n\\frac{8\\lambda^2\\\/K}{\\bar\\psi\\psi} + 4\\tau_0\\lambda^2 + 2\\epsilon\\lambda^2\\\/\\tau_0\\bar\\psi\\psi = \\frac{3mK}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)} - \\frac{3K\\left(\\epsilon\\bar\\psi\\psi + 2\\right)}{\\bar\\psi\\psi\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V + \\frac{3K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)}V'\n\\end{equation}\nA solution of \\eqref{FD3} is given by\n\\begin{equation}\\label{FD4}\nV(\\bar\\psi\\psi)= - \\frac{2\\lambda^2\\tau_0\\bar\\psi\\psi}{3K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\\left[\\epsilon^2\\left(\\bar\\psi\\psi\\right)^2 + 4\\right] + \\frac{m\\bar\\psi\\psi}{\\epsilon\\bar\\psi\\psi+1} - \\frac{8\\lambda^2}{3}\n\\end{equation}\n\\subsubsection{Potentials for transition power law inflation -- decelerated power law expansion -- dark era}\nNow, let us consider a suitable combination of the potentials introduced above as  \n\\begin{equation}\\label{IFD1}\nV(\\bar\\psi\\psi) = - \\frac{\\alpha}{6\\left(\\epsilon\\bar\\psi\\psi+1\\right)} - \\frac{\\beta\\bar\\psi\\psi}{6K\\left(\\epsilon\\bar\\psi\\psi+1\\right)} + \\frac{\\gamma\\left(\\bar\\psi\\psi\\right)^{p+1}}{6K^{p+1}\\left(p-1\\right)\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\n\\end{equation}\n$\\alpha,\\beta$ and $\\gamma$ being constants. This particular choice of potential gives rise to a dynamical equation for the scale volume of the form\n\\begin{equation}\\label{IFD2}\n\\frac{d}{dt}\\left[\\frac{d}{dt}\\left(2\\tau-\\epsilon K\\ln\\tau\\right)\\right]^{2} = \\left(6mK + 2\\alpha\\tau + \\beta + \\gamma\\tau^{-p}\\right)\\dot\\tau\n\\end{equation}\nwhich, integrated a first time, yields\n\\begin{equation}\\label{IFD3}\n\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)^2{\\dot\\tau}^2 = \\left(6mK+\\beta\\right)\\tau +\\alpha\\tau^2 + \\frac{\\gamma}{-p+1}\\tau^{-p+1} - A\n\\end{equation}\n$A$ being an integration constant. Choosing $p$ as above, for very small values of $\\tau$ we recover a power law inflation phase; for large values of $\\tau$ we recover exponential expansion but, by carefully choosing the values of the parameter $\\alpha$ and $\\beta$, we can have a phase where the term $\\left(6mK+\\beta\\right)\\tau$ is very dominant over the term $\\alpha\\tau^2$ and thus obtain a decelerated power law expansion.\n\nAs a side remark, it should be noted that in the presence of potentials of the form \\eqref{IFD1}, the Hamiltonian constraint \\eqref{F2} reduces to a relation identical to \\eqref{0.0.8}.  However now the integration constant $A$ does not have to satisfy the condition $A\\geq 0$. Thus in this case no restrictions are imposed on the coupling constant $\\epsilon$.\n\n\\subsubsection{A note on renormalizability in the case of a non trivial potential.}\nSo far, we have studied a list of potentials and we have given the expression of the single potential that condenses them all: altogether, they are capable of fitting within a unique scheme all expansion eras, but there is still a problem we must address about renormalizability. As it is well known, the presence of torsion renders the Dirac equation non-renormalizable; and as it is also widely recognized, non-minimal coupling do that too: one would then reasonably expect that torsion in non-minimal coupling would induce for the Dirac equation an even higher degree of non-renormalizability. But what happens is quite the contrary: opposite to our intuition, the degree of non-renormalizability is lowered. In fact, the resulting non-linear terms are even super-renormalizable \\cite{FVC}. This is a nice result, and consequently it would be desirable that it be maintained also in presence of this potential. We split tho two cases: in the ultra-violet case, we have that\n\\begin{equation}\nV(\\bar\\psi\\psi\\rightarrow\\infty)\\rightarrow-\\frac{\\beta}{6\\epsilon K}\n+\\frac{\\gamma\\left(\\bar\\psi\\psi\\right)^{p}}{6\\epsilon K^{p+1}\\left(p-1\\right)}\n\\end{equation}\nso that the potential is reduced to one term that behave as a cosmological constant, which in high-energy physics is irrelevant, plus a term that scales as $\\left(\\bar\\psi\\psi\\right)^{p}$, which therefore is renormalizable if and only if $p\\leqslant\\frac{4}{3}$, and specifically in the case of the equality the theory is renormalizable, while for the inequality the theory is super-renormalizable. In the infra-red case, it is \n\\begin{equation}\nV(\\bar\\psi\\psi\\rightarrow0)\\rightarrow-\\frac{\\alpha}{6}\n+\\frac{\\gamma\\left(\\bar\\psi\\psi\\right)^{p+1}}{6K^{p+1}\\left(p-1\\right)}\n\\end{equation}\nwith a cosmological constant that now is relevant, and it constitutes the reason why the dark energy behaviour is recovered, plus an additional term in $\\left(\\bar\\psi\\psi\\right)^{p+1}$, for which we have to require $p\\geqslant-1$ if we want the results about dark energy preserved. All in all, the constraint given by $-1\\leqslant p\\leqslant\\frac{4}{3}$ is the one that keeps the theory both in infra-red and in ultra-violet regimes completely renormalizable. And nicely, these are also the exact constraints we would need to get for $\\frac{1-p}{2}=-\\frac{1}{3q}$ the limiting condition $q\\geq 2$ needed to provide inflation and also the limiting condition $p\\geqslant-1$ needed to maintain the dark energy results. In this sense the potential we have furnished, together with the constraining conditions $-1\\leqslant p\\leqslant\\frac{4}{3}$, is such that it recovers the correct dynamics for the expansion of the universe \\emph{precisely} because it is the potential for which the theory is renormalizable. This is a \nsurprisingly good double-take of the theory, because if at first the form of the potential might have looked quite arbitrary, and some might have thought it was chosen to yield the wanted cosmology, in reality that potential could not have been any different, or else the theory would have been ill-defined in terms of particle-physics. That the expected behaviour of the standard model of cosmology be implied by constraints on the standard model of particle-physics was, in our knowledge, not known before. \n\n\n\\subsection{In the presence of dust fluid}\nIn the case of presence of dust fluid with density $\\rho$, the conservation laws for the fluid together with the relation $\\tau=\\frac{K}{\\bar\\psi\\psi}$ ensure the relation\n\\begin{equation}\\label{D1}\n\\rho = \\frac{\\rho_0}{\\tau} = \\frac{\\rho_0}{K}\\bar\\psi\\psi\n\\end{equation}\nIn such a circumstance, setting\n\\begin{equation}\\label{D2}\n\\frac{\\bar m}{2} := \\frac{\\rho_0}{K} + \\frac{m}{2}\n\\end{equation}\nit is easily seen that the dynamical equation for the scale volume $\\tau$ becomes\n\\begin{equation}\\label{D3}\n2\\frac{\\ddot\\tau}{\\tau}\\varphi+3\\ddot\\varphi+5\\frac{\\dot\\tau}{\\tau}\\dot\\varphi\n=\\frac{3\\bar{m}K}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} - \\frac{3\\left(\\epsilon K + 2\\tau\\right)\\\/V}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} + \\frac{3\\left(\\epsilon\\\/K+\\tau\\right)KV'}{\\tau^2\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)}\n\\end{equation}\nformally identical to eq. \\eqref{1}, with $\\bar m$ replacing $m$. The conclusion follows that, by substituting $m$ by $\\bar m$, all results and conclusions stated in subsection A hold also in presence of dust. \n\n\n\\subsection{In the presence of radiation fluid}\nWe consider the presence of a radiation fluid with equation of state $p=\\frac{1}{3}\\rho$. The conservation laws for the fluid provide the relation $\\rho =\\frac{\\rho_0}{\\tau^{\\frac{4}{3}}}$.\nIn this case, the dynamical equation for $\\tau$ is given by\n\\begin{equation}\\label{R1}\n2\\frac{\\ddot\\tau}{\\tau}\\varphi+3\\ddot\\varphi+5\\frac{\\dot\\tau}{\\tau}\\dot\\varphi\n=\\frac{2\\rho_0}{\\tau^{\\frac{4}{3}}} + \\frac{3mK}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} - \\frac{3\\left(\\epsilon K + 2\\tau\\right)\\\/V}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} + \\frac{3\\left(\\epsilon\\\/K+\\tau\\right)KV'}{\\tau^2\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)}\n\\end{equation}\nChoosing a potential of the form $V=\\bar{V} + \\tilde{V}$, with $\\bar V$ satisfying the equation\n\\begin{equation}\\label{R2}\n\\frac{2\\rho_0}{\\tau^{\\frac{4}{3}}} - \\frac{3\\left(\\epsilon K + 2\\tau\\right)\\\/\\bar{V}}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} + \\frac{3\\left(\\epsilon\\\/K+\\tau\\right)K\\bar{V}'}{\\tau^2\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)} = 0\n\\end{equation}\namounting to\n\\begin{equation}\\label{R3}\n\\frac{2\\rho_0}{K^{\\frac{1}{3}}}(\\bar\\psi\\psi)^{\\frac{1}{3}} - \\frac{3K\\left(\\epsilon\\bar\\psi\\psi+2\\right)}{\\bar\\psi\\psi\\left(2-\\epsilon\\bar\\psi\\psi\\right)}\\bar{V} + \\frac{3K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}{\\left(2-\\epsilon\\bar\\psi\\psi\\right)}\\bar{V}' = 0,\n\\end{equation}\neq. \\eqref{R1} reduces to \n\\begin{equation}\\label{R4}\n2\\frac{\\ddot\\tau}{\\tau}\\varphi+3\\ddot\\varphi+5\\frac{\\dot\\tau}{\\tau}\\dot\\varphi\n=\\frac{3mK}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} - \\frac{3\\left(\\epsilon K + 2\\tau\\right)\\\/\\tilde{V}}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} + \\frac{3\\left(\\epsilon\\\/K+\\tau\\right)K\\tilde{V}'}{\\tau^2\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)}\n\\end{equation}\nwhich is identical to \\eqref{1}. Again, an analysis analogous to that developed in subsection A is then applicable also in this case with identical results. A solution of \\eqref{R3} is given by\n\\begin{equation}\\label{R5}\n\\bar V = 2\\rho_0\\left(\\frac{\\bar\\psi\\psi}{K}\\right)^{\\frac{4}{3}}\n\\end{equation}\n\n\n\\subsubsection{The case $V=0$.}\nIn this case eq. \\eqref{R1} simplifies to\n\\begin{equation}\\label{R6}\n2\\frac{\\ddot\\tau}{\\tau}\\varphi+3\\ddot\\varphi+5\\frac{\\dot\\tau}{\\tau}\\dot\\varphi\n=\\frac{2\\rho_0}{\\tau^{\\frac{4}{3}}} + \\frac{3mK}{\\tau\\left(2-\\frac{\\epsilon K}{\\tau}\\right)} \n\\end{equation}\nwhich can be handled as above, giving rise to the final equation \n\\begin{equation}\\label{R7}\n\\left(2-\\frac{\\epsilon\\\/K}{\\tau}\\right)^2{\\dot\\tau}^2 = 12\\rho_0\\tau^{\\frac{2}{3}} + \\frac{12\\epsilon\\\/K\\rho_0}{\\tau^{\\frac{1}{3}}} + 6mK\\tau + A\n\\end{equation}\n$A$ being a suitable integration constant. For very small values of $\\tau$ and supposing $\\epsilon >0$, eq. \\eqref{R7} can be approximated to\n\\begin{equation}\\label{R8}\n\\epsilon\\\/K\\frac{|\\dot\\tau|}{\\tau} = \\sqrt{12\\epsilon\\\/K\\rho_0}\\tau^{-\\frac{1}{6}}\n\\end{equation}\nyielding $\\tau \\approx t^6$ which can account for an accelerated early phase of the universe (at least for isotropic models). {  As it is clear from \\eqref{R8}, we underline that this dynamics is strictly due to the non--minimal coupling. This is a remarkable difference with respect to the minimally coupled theory where the presence of a fermionic self--interacting potential is necessary to generate inflationary phases at early time \\cite{VFC}}. For very large values of $\\tau$, eq. \\eqref{R7} can be approximated to\n\\begin{equation}\\label{R9}\n2|\\dot\\tau| = \\sqrt{6mK\\tau}\n\\end{equation}\nyielding $\\tau \\approx t^2$ and thus a decelerated expansion and isotropization of universe.\n\n\n\n\n\\subsubsection{Transition power law inflation -- decelerated power law expansion -- dark era}\nFinally, taking the potential\n\\begin{equation}\nV(\\bar\\psi\\psi) = - \\frac{\\alpha}{6\\left(\\epsilon\\bar\\psi\\psi+1\\right)} - \\frac{\\beta\\bar\\psi\\psi}{6K\\left(\\epsilon\\bar\\psi\\psi+1\\right)}\n\\end{equation}\ninto account, suitably choosing the parameters $\\alpha$ and $\\beta$ and repeating the arguments as in III.A.6, we recover again a phase transition: power law inflation -- decelerated power law expansion -- exponential expansion.\n\n\n\n\n\\section{Conclusions}\nIn this paper we have considered cosmological models in the framework of Einstein--Cartan--Sciama--Kibble gravity in which a Dirac field is non--minimally coupled to gravity. {  This non--minimal coupling has been investigated in a previous paper \\cite{FVC} in connection with the renormalizability issue of Dirac equations. Here, we study some cosmological scenarios arising from such a theory}. In order to account for possible initial anisotropies of the universe, we have considered Bianchi--I models, looking at spatially flat FRW models as a particular case. We have shown that the non--minimal coupling can in general avoid the initial cosmological singularity in the scale volume (scale factors), in agreement with the results recently obtained in \\cite{Magueijo:2012ug}, where another type of fermionic non-minimal coupling was studied. However this does not necessarily imply that the model is singularity free as the Hamiltonian constraint can induce bounds on $\\epsilon$ which could cause a singularity in the \nshear at finite time. In this respect therefore, care should be taken in stating that these models are not ``singularity free''.\n\n{  Using two different approaches, we have obtained several examples of fermionic self--interaction potential which generate a number of interesting cosmological phases (power law inflation, decelerated power law expansion, dark era). In fact by an accurate fine tuning,  even a transition  power law inflation --  decelerated power law expansion -- dark era is possible. Some of the potentials we obtained have the remarkable properties to be relatively simple combination of power of $\\bar\\psi\\psi$ and to be able to lead dynamically to a dark era. The presence of cosmological fluids does not substantially modify the results achieved in the case only a Dirac field is present. We have analysed specifically the cases of dust and radiation. In this last case it became evident that the non minimal coupling alone is the origin of a power law inflation at early time.  \n\nFrom our results it emerges that a fermionic self--interaction potential is necessary in order to generate an accelerated expansion phase of the universe at late time, when the contribution of the non--minimal coupling vanishes. Conversely and differently from what happens in the minimally coupled theory, in the presence of non--minimal coupling the fermionic potential can be no longer necessary for inflation; indeed there exist cases where the non--minimal coupling alone is sufficient to generate inflationary phases at early time (small values of scale volume). }\n\n\\begin{acknowledgments}\nThe authors would like to thanks the referee for useful comments that improved the original version of the manuscript.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{I. Density functional theory calculations.}\nWe have performed electronic structure calculations of these materials within the framework of DFT and the results are shown in Fig.~1 of the main text. Our calculations were carried out by using the full-potential linearized augmented plane wave (FP-LAPW) method as implemented in the WIEN2k code~\\cite{PBlaha2001}. The generalized gradient approximation (GGA)~\\cite{JPPerdew1996} was used for the exchange-correlation functional. The spin-orbit coupling was included in a second variational way, for which relativistic $p_{1\/2}$ local orbitals were added into the basis set for the description of the $6p$ states of plutonium \\cite{Kunes2001}. The muffin-tin radii were: 2.5$a_0$ for Pu, 2.46$a_0$ (PuCoGa$_5$) and  2.5$a_0$ (PuCoIn$_5$) for Co, 2.5$a_0$ for Rh, 2.18$a_0$ (PuCoGa$_5$) and 2.23$a_0$ (PuRhGa$_5$) for Ga, and 2.39$a_0$ for In, where $a_0$ is the Bohr radius. The energy spread to separate the localized valence states was -6 Ryd. The criterion for the number of plane waves was $R_{MT}^{{\\rm min}} K^{{\\rm max}} = 8$ and the number of $\\mathbf{k}$-points was $40\\times 40 \\times 25$. The experimentally determined crystallographic structures were used~\\cite{PuCoGa5,PuRhGa5,PuCoIn5}.\n\n\n\\section{II. Point-contact spectroscopy calculation using anisotropic Fermi surfaces and anisotropic order parameter}\n\n\n\n\nWe calculate the point-contact spectrum (PCS) by taking into account the full Fermi surface (FS) anisotropy of a multiband system with anisotropic order parameter by following the formalism given in Refs.~\\cite{Kashiwaya,PCSMazin,PCSBrinkman,PCSDaghero}. For simplification, however, we only include the anisotropy in the FS of the superconducting (SC) material, while that of the normal metal tip is neglected. Let us define ${\\bf n}$ as the unit vector in the direction of the total injected current, which for simplicity we choose to be perpendicular to the contact interface between the SC (S) and normal metal (N) interface. As a consequence the components  along the direction ${\\bf n}$ of\nthe Fermi velocities at wavevector ${\\bf k}$ in the $i^{\\rm th}$ FS sheet of the superconductor are ${\\bf v}_{i{\\bf k}}\\cdot {\\bf n} = v_{i{\\bf k},n}$, where\n${\\bf v}_{i{\\bf k}}=-\\frac{1}{\\hbar} \\nabla_{\\bf k} E_{i{\\bf k}}$, and $E_{i{\\bf k}}$ is the corresponding quasiparticle band. Generalizing the Blonder, Tinkham and Klapwijk (BTK) formula \\cite{BTK} to anisotropic FSs of multiband superconductors, it was shown that the total normalized conductance seen along the direction ${\\bf n}$ can be written as \\cite{PCSDaghero}\n\\begin{eqnarray}\n\\left\\langle G(E)\\right\\rangle_{I\\parallel{\\bf n}}=\\frac{\\sum_i\\left\\langle \\sigma_{i{\\bf k}} (E) D_{i{\\bf k}} v_{i{\\bf k},n}\\right\\rangle_{{\\rm FS}_i}}{\\sum_i\\left\\langle D_{i{\\bf k}} v_{i{\\bf k},n}\\right\\rangle_{{\\rm FS}_i}} ,\n\\label{totconductance}\n\\end{eqnarray}\nwhere $D_{i{\\bf k}}=1\/v_{i{\\bf k}}$ is the density of states on the $i^{\\rm th}$ FS sheet, and $\\sigma_{i{\\bf k}} (E)$ is the BTK SC transition probability calculated as follows. Here we neglect interband interference effects and variations in the tunneling matrix elements (different weight factors) between the normal tip and different bands of the superconductor.\n\n\\begin{figure}\n\\rotatebox[origin=c]{0}{\\includegraphics[width=.95\\columnwidth]{fig8}}\n\\vspace{16pt}\n\\parbox{\\textwidth}{\n{\\bf Fig.~S1}: Calculated PCS spectra, decomposed in different bands for two parameter sets for PuCoGa$_5$. The zero-bias conductance peak is present in all bands for $d$-wave pairing and in band 2 for $s^{\\pm}$-pairing as expected. For a larger value of $\\alpha$, we find that a zero-bias conductance peak is induced in bands 3 and 4 in $s^{\\pm}$-wave pairing, in addition to band 2. Since band 3 and 4 have larger gap amplitudes which enables a broader PCS conductance peak and match the experimental data, as shown in (a)-(d). For $\\alpha=\\pi\/25$, the conductance peak only survives in band 2 for this pairing.}\n\\label{fig7}\n\\end{figure}\n\nLet us assume that $\\theta_{i{\\bf k}}$ is the transmission angle at the interface between the normal and SC materials for band $i$ at Fermi momentum ${\\bf k}$. The specific SC gap function is defined in terms of $\\theta_{i{\\bf k}}$ as $\\Delta_{i{\\bf k}}(\\theta_{i{\\bf k}})=\\Delta_{0}(\\cos{k_x^i}\\pm\\cos{k_y^i})$, for $s^{\\pm}$- and $d$-wave pairing, respectively and $k_{x,y}^i$ are the Fermi momentum for the $i^{th}$-band. Then $\\theta_{i{\\bf k}}={\\rm tan}^{-1}(k_y^i\/k_x^i)$. Lets us also define $\\alpha$ as the rotation of the crystallographic $a$-axis with respect to the normal to the interface ($x$ axis). In this circumstance, the electron-like and hole-like quasiparticle (EQs\/HQs) injected in the SC material with angles $\\pm\\theta_{i{\\bf k}}$, they access different gap values as $\\Delta^{\\pm}_{i{\\bf k}}=\\Delta_{i{\\bf k}}(\\pm\\theta_{i{\\bf k}}-\\alpha)$. In this case the SC transition probability becomes\n\\begin{eqnarray}\n\\sigma_{j{\\bf k}} (E) = \\tau_{\\rm N} \\frac{1+\\tau_{\\rm N} \\mid \\gamma^+_{j{\\bf k}}(E)\\mid^2 + (1-\\tau_{\\rm N}) \\mid\\gamma^+_{j{\\bf k}}(E)\\gamma^-_{j{\\bf k}}(E)\\mid^2}{\\mid 1+(1-\\tau_{\\rm N}) \\mid\\gamma^+_{j{\\bf k}}(E)\\gamma^-_{j{\\bf k}}(E)\\exp{(i\\phi_{i{\\bf k}})}\\mid^2},\n\\end{eqnarray}\nwhere the function\n\\begin{equation}\n\\gamma^{\\pm}_{i{\\bf k}}(E)=\\frac{\\mid E\\mid-\\sqrt{E^2-\\mid \\Delta^{\\pm}_{i{\\bf k}}\\mid^2 }}{\\mid \\Delta^{\\pm}_{i{\\bf k}}\\mid},\n\\end{equation}\nand $\\phi_{i{\\bf k}}=\\phi^-_{i{\\bf k}}-\\phi^+_{i{\\bf k}}$, with $\\phi^{\\pm}_{i{\\bf k}}$ being the phases of $\\Delta^{\\pm}_{i{\\bf k}}$. It should be noted that $\\gamma^{\\pm}_{i{\\bf k}}(E)$ are, in general, complex functions even if the momentum-dependent gap $\\Delta_{i{\\bf k}}$ is real when $E<\\Delta_{i{\\bf k}}$. For simplicity, we further assume a band-independent and momentum-independent interface barrier, the parameter $\\tau_{\\rm N}$ is the transparency factor of the barrier in the BTK approximation of current injection perpendicular to the SN interface, defined as $\\tau_{\\rm N}=1\/(1+Z^2)$. The limit $Z=0$ gives the perfectly transparent junction in the transmission limit, i.e., the ideal Andreev reflection regime.\n\n\n\\begin{figure}\n\\rotatebox[origin=c]{0}{\\includegraphics[width=.8\\columnwidth]{fig9}}\n\\vspace{16pt}\n\\parbox{\\textwidth}{\n{\\bf Fig.~S2}: Calculated PCS spectra at different parameters of the interface barrier potential $Z$ with $\\alpha=\\pi\/8$, and $\\Delta_0$=10~meV.  }\n\\label{fig9}\n\\end{figure}\n\n\nIn Fig.~6 of the main text, we presented results of the PCS conductance for the nodal $d_{x^2-y^2}$-wave, nodal $s^{\\pm}$-wave pairing symmetries, and compared these results with a nodeless and fully isotropic $s$-wave pairing symmetry. For the best fit to the available experimental data of PuCoGa$_5$,\\cite{Daghero}, we set $Z=1.55$, $\\alpha=\\pi\/8$ and a fixed SC gap amplitude of $\\Delta_0=10$~meV for all three pairing symmetries. We showed that a single and sharp zero-bias peak is reproduced for $s^{\\pm}$-pairing. Of course, even in this multiband setup, it may be possible to fit  the experimental data with $d$-wave pairing.\n\nIn Fig.~S1, we showed the band decomposed PCS spectra for two different parameter sets for three different parameter sets. The evolution of the PCS spectrum for different values of $Z$ is discussed in Fig.~S2.\n\n\n\\section{III. Multiband coulomb interactions}\nFor intermetallic actinides, the spin-orbit coupling is very strong and causes a band splitting of the 5$f$ states of about 1~eV.\nThis quenches the Hund's coupling term, because $J_H\\ll \\lambda_{SOC}$. The remaining interaction terms account for the onsite intra- and inter-band Coulomb repulsions as given by the interaction Hamiltonian\n\\begin{eqnarray}\nH_{int} = \\sum_{{\\bf k},{\\bf k}^{\\prime}}\\left[\\sum_{n} U_{n} c^{n\\dag}_{{\\bf k}\\uparrow}c^n_{{\\bf k}\\uparrow}c^{n\\dag}_{{\\bf k}^{\\prime}\\downarrow}c^n_{{\\bf k}^{\\prime}\\downarrow}\n+\\sum_{n\\ne m,\\sigma,\\sigma^{\\prime}} V_{nm} c^{n\\dag}_{{\\bf k}\\sigma}c^n_{{\\bf k}\\sigma}c^{m\\dag}_{{\\bf k}^{\\prime}\\sigma^{\\prime}}c^m_{{\\bf k}^{\\prime}\\sigma^{\\prime}}\\right].\n\\end{eqnarray}\nHere $c^{n\\dag}_{{\\bf k}\\sigma} (c^{n}_{{\\bf k}\\sigma})$ creates (annihilates) a Bloch state at momentum ${\\bf k}$, (pseudo-) spin $\\sigma=\\uparrow\/\\downarrow$ in the $n^{\\rm th}$-band. The interaction matrices $\\tilde{U}^{s\/c}$ used in the RPA formalism consist of components $U_n$ and $V_{nm}$. The different bandwidths of different bands amount to different critical values of $U$ and $V$, determined by the positive RPA denominator as $\\tilde{U}^{s\/c}_{nm}\\tilde{\\chi}_{nm}\\le 1$. Using this condition we find that the critical interaction values for bands 2 and 3 are considerably small and we fix these values to be $2U_{2}=V_{13}=V_{23}=200$ meV at all points for the calculations of $\\lambda$ in Fig.~3. The rest of the interactions are taken to be same for all bands as $U_n=U$, and $V_{nm}=V$. In drawing the parameter space in Fig.~3, we limit the values of $U$ and $V$ to below 600 meV and 700 meV, respectively, because for larger values the RPA susceptibilities for bands 3 and 4 become negative, which is above the critical values set by the RPA denominator. However the general conclusion of the dominant $s^{\\pm}$-wave pairing than $d$-wave pairing symmetry strength $\\lambda$ is consistent throughout the $U-V$ map. Since the essential physics is determined by the FS topology and nesting conditions, we anticipate that this conclusion also remains valid for higher values of $U$ and $V$.\n\n\\section{IV. BCS susceptibility and spin resonance}\n\nIn this section, we give the details of our spin susceptibility calculation presented in the main text. All ``super\" matrices with tilde are defined as ${\\tilde \\chi}_{ij}=\\chi_{nm}\\delta_{ij}$, etc., with super indices $i=4(n-1)+m$,  and $n, m=1-4$ are band indices. Thus variables with tilde have matrix dimension $16\\times 16$. We follow closely earlier work \\cite{Takimoto,Graser} and evaluate the spin-resonance susceptibility in the SC state within the random phase approximation (RPA) of the BCS formalism, which is given by the bare bubble transverse spin susceptibility\n\\begin{widetext}\n\\begin{eqnarray}\n\\chi_{nm}({\\bf q},\\omega) &=& \\int \\frac{d {\\bf k}}{\\Omega_{\\rm BZ}}~M_{nm}({\\bf k},{\\bf q}) \\left\\{\\frac{1}{2}\\left[1+\\frac{\\xi^{n}_{\\bf k}\\xi^{m}_{{\\bf k}+{\\bf q}} + \\Delta^{n}_{\\bf k}\\Delta^{m}_{{\\bf k}+{\\bf q}} }{E^{n}_{\\bf k}E^{m}_{{\\bf k}+{\\bf q}}}\\right]\\frac{f(E^{n}_{\\bf k})-f(E^{m}_{{\\bf k}+{\\bf q}})}{\\omega-E^{n}_{\\bf k}+E^{m}_{{\\bf k}+{\\bf q}}+i\\delta}\\right.\\nonumber\\\\\n&& \\hskip0.cm + \\frac{1}{4}\\left[1+\\frac{\\xi^{n}_{\\bf k}}{E^{n}_{\\bf k}}-\\frac{\\xi^{m}_{{\\bf k}+{\\bf q}}}{E^{m}_{{\\bf k}+{\\bf q}}}     - \\frac{\\xi^{n}_{\\bf k}\\xi^{m}_{{\\bf k}+{\\bf q}} + \\Delta^{n}_{\\bf k}\\Delta^{m}_{{\\bf k}+{\\bf q}} }{E^{n}_{\\bf k}E^{m}_{{\\bf k}+{\\bf q}}}\\right]\\frac{1-f(E^{n}_{\\bf k})-f(E^{m}_{{\\bf k}+{\\bf q}})}{\\omega+E^{n}_{\\bf k}+E^{m}_{{\\bf k}+{\\bf q}}+i\\delta} \\nonumber\\\\\n&&\\hskip0.cm \\left.+ \\frac{1}{4}\\left[1 - \\frac{\\xi^{n}_{\\bf k}}{E^{n}_{\\bf k}}+\\frac{\\xi^{m}_{{\\bf k}+{\\bf q}}}{E^{m}_{{\\bf k}+{\\bf q}}} -\\frac{\\xi^{n}_{\\bf k}\\xi^{m}_{{\\bf k}+{\\bf q}} + \\Delta^{n}_{\\bf k}\\Delta^{m}_{{\\bf k}+{\\bf q}} }{E^{n}_{\\bf k}E^{m}_{{\\bf k}+{\\bf q}}}\\right]\\frac{f(E^{n}_{\\bf k})+f(E^{m}_{{\\bf k}+{\\bf q}})-1}{\\omega-E^{n}_{\\bf k}-E^{m}_{{\\bf k}+{\\bf q}}+i\\delta} \\right\\}\n\\nonumber\\\\\n\\label{seq:1}\n\\end{eqnarray}\n\\end{widetext}\n\\noindent\nand the RPA susceptibility is attained with the onsite Coulomb interaction super matrix, $\\tilde U$,\n\\begin{eqnarray}\n\\tilde\\chi_{RPA}({\\bf q},\\omega) &=&  \\left[ 1 - \\tilde U \\tilde\\chi({\\bf q},\\omega) \\right]^{-1} \\tilde\\chi({\\bf q},\\omega) .\n\\label{seq:2}\n\\end{eqnarray}\nHere $E^{n}_{\\bf k}=[(\\xi_{\\bf k}^{n})^2+(\\Delta_{\\bf k}^{n})^2]^{1\/2}$ is the SC quasiparticle energy of the eigenstate $\\xi_{\\bf k}^{n}$ ($n$ is the band index) and $\\Delta_{\\bf k}^{n}$ is the SC gap function. $M_{nm}({\\bf k},{\\bf q})$ is the matrix element consisting of the eigenstates of the initial and final scattered quasiparticle states. In Eq.~\\ref{seq:1}, the first term is called particle-hole scattering term, which vanishes for $\\omega\\le 2\\Delta$, regardless of the pairing symmetry due to particle-hole symmetry. The second and third terms are for particle-particle and hole-hole scattering, respectively, which become active in the SC state. Focusing on the third term (a similar analysis applies to the second term), we find that this term contributes a non-zero value only when\nsign$[\\Delta^{n}_{\\bf k}]\\ne$sign$[\\Delta^{m}_{{\\bf k}+{\\bf q}}]$\n(since $\\xi^{n}_{\\bf k} =0$ on the Fermi surface). A pole is thus obtained in the imaginary part of $\\chi_{nm}$ at\n\\begin{eqnarray}\n\\omega^{\\rm res}_{nm}({\\bm q}) = |\\Delta^{n}_{\\bf k}|+|\\Delta^{m}_{{\\bf k}+{\\bf q}}|.\n\\end{eqnarray}\nOf course, the many-body  and  matrix-element effects can shift the energy scale as discussed in the Method section in the main text.\n\n\n\\section{III. Computed gap function and weak higher-order harmonics}\n\\begin{figure}\n\\rotatebox[origin=c]{0}{\\includegraphics[width=.95\\columnwidth]{fig10}}\n\\vspace{16pt}\n\\parbox{\\textwidth}{\n{\\bf Fig.~S3}: Top panel: Computed $k$-dependent SC gap function plotted in colormap as in the main text. Bottom panel: The same gap function of four bands plotted as a function of FS angle. The solid lines are the fit with the gap function $g(k)$ including higher harmonics as given in Eq.~(11), and the relevant parameters are given in Table~I. The FS angle is defined to be zero along the zone boundary direction (100) for band 1, and (010) for all other bands, and 45$^o$ along the diagonal direction (110) for all bands. }\n\\label{gapfnc}\n\\end{figure}\n\n\nIn the main text, we calculated the pairing eigenfunction $g({\\bf k})$ by directly solving the eigenvalue problem, which is obtained by rewriting the linearized weak-coupling multiband gap equation. We also calculated the so-called  pairing strength $\\lambda$ in two different, yet equivalent, procedures by using Eqs.~(3) and (4). We calculated the maximum eigenvalue $\\lambda$ and corresponding eigenfunction $g({\\bf k})$ by solving the eigenvalue matrix problem for the spin-fluctuation pairing vertex, in Eq.~(3). This involves summation over the 3D FSs of all 4 bands. For each band we expand the  $\\Gamma_{nm}$ matrix into discretized Fermi momenta ${\\bf k}_n$ and ${\\bf k}_m^{\\prime}$. In our case, since the $\\Gamma_{nm}$ matrix is defined in the band basis, each matrix index $n$ assumes dimension $N_n$ (number of points for the $n^{th}$-band), with a total number of $N=\\sum_{n=1}^4 N_n$ FS momenta, where $N_n$ is the number of 3D Fermi momenta for the $n^{\\rm th}$ band. Therefore the matrix we diagonalize has a dimension of $N\\times N$. \n\nThe  maximum eigenvalue is proportional to the highest superconducting transition temperature and the corresponding\neigenvector gives the leading pairing function $g({\\bf k})$, which is plotted in Fig.~3 (left-hand side) and in Fig~S3. The first point to notice in Fig.~S3 is that there is no clear fourfold symmetry breaking in $g({\\bf k})$, and that there is no gap node along the diagonal direction. This finding clearly excludes the presence of any significant $d$-wave pairing component. \nIn a final step, we analyze the pairing function in more detail by fitting it up to third harmonics of $s^{\\pm}$-pairing symmetry \\cite{Graser}:\n\\begin{equation}\ng({\\bf k}) = a_1(\\cos{k_x}+\\cos{k_y}) + 2a_2\\cos{2k_x}\\cos{2k_y} + 2a_3\\cos{4k_x}\\cos{4k_y} + a_0.\n\\label{eq:gapfnc}\n\\end{equation}\nThe corresponding fits are shown in Fig.~S3. The general conclusion drawn from these fits is that the coefficients for second and third harmonics $a_2$ and $a_3$, respectively, are almost an order of magnitude lower than the first harmonic of $s^{\\pm}$- pairing. We also notice a weak $k_z$ dependence in the fit parameters. Furthermore, the result shows that the gap anisotropy is largest in band 2. This is where the nodes are located. The gap amplitude is smallest in band 3 and then it increases from band 2 to band 1 to band 4. This is expected from the $s^{\\pm}$-pairing symmetry as the gap maxima lie at the $\\Gamma$ and M points, with opposite sign. Therefore, our conclusion about the $s^{\\pm}$ pairing symmetry in the Pu-based superconductor is a robust feature.\n\n\nSecondly, we calculated the pairing strength through the usual projection of the eigenvalue Eq.~(3) onto selected orthogonal pairing functions with characteristic symmetry. The projected pairing strength for a given pairing symmetry $g_{\\alpha}$ is calculated from Eq.~(4) of the main text, which was used earlier in Refs.~\\cite{scalapino86,Graser}. Finally, the total pairing strength is obtained by summing over all indices, $\\lambda^\\alpha = \\sum_{n,m} \\lambda_{nm}^\\alpha$, and is plotted as a function of the Coulomb potentials $U$ and $V$ in Fig.~3 (right-hand side). The line integrals over each FS sheet were performed over FS pockets in each corresponding $k_z$ plane, and then summed over $k_z$ slices. The intra- and interband pairing strength $\\lambda_{nm}^\\alpha$ is plotted in Fig.~4 as a function of the $k_z$ slices. \n\n\\begin{table}[tb]\n\\parbox{\\textwidth}{{\\bf Table.~I}: Coefficients of various harmonics of the gap function given in Eq.~(11).}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline \\hline\n&\\multicolumn{2}{|c|}{Band 1}& \\multicolumn{2}{|c|}{Band 2}&\\multicolumn{2}{|c|}{Band 3}&\\multicolumn{2}{|c|}{Band 4}\\\\\n\\hline\nPuCoIn$_5$ & $k_z=0$ & $k_z=\\pi$ & $k_z=0$ & $k_z=\\pi$ & $k_z=0$ & $k_z=\\pi$ & $k_z=0$ & $k_z=\\pi$\\\\\n\\hline\n$a_1$ & 0.35 & -- & 0.15 & 0.5 & 0.12 & 0.12 & 0.3 & 0.2 \\\\\n$a_2$ & 0.06 & -- & 0.025 & 0.05 & 0.015 & 0.015 & -0.06 & -0.02 \\\\\n$a_3$ & 0.1 & -- & 0 & 0.05 & 0 & 0 & -0.06 & -0.02 \\\\\n$a_0$ & 0.9 & -- & 0.25 & 0.21 & 0.1 & 0.1 & 0.1 & -0.08\\\\\n\\hline\nPuCoGa$_5$ & \\multicolumn{2}{|c|}{}& \\multicolumn{2}{|c|}{}&\\multicolumn{2}{|c|}{}&\\multicolumn{2}{|c|}{} \\\\\n\\hline\n$a_1$ & 0.13 & -- & 0.35 & 0.7 & 0.1 & 0.1 & 0.25 & 0.25 \\\\\n$a_2$ & -0.013 & -- & -0.02 & -0.07 & -0.01 & -0.01 & -0.025 & -0.025\\\\\n$a_3$ & 0 & -- & -0.02 & -0.07 & -0.01 & -0.01 & -0.05 & 0.05\\\\\n$a_0$ & 0.15 & -- & 0.05 & 0.1 & 0.08 & 0.08 & -0.4 & 0.1 \\\\\n\\hline\nPuRhGa$_5$ & \\multicolumn{2}{|c|}{}& \\multicolumn{2}{|c|}{}&\\multicolumn{2}{|c|}{}&\\multicolumn{2}{|c|}{} \\\\\n\\hline\n$a_1$ & 0.18 & -- & 0.35 & 0.1 & 0.1 & 0.05 & 0.2 & 0.17 \\\\\n$a_2$ & -0.01 & -- & -0.087 & 0.07 & -0.03 & 0 & -0.05 & -0.17\\\\\n$a_3$ & -0.013 & -- & -0.025 & 0.025 & 0 & 0 & 0.2 & -0.2 \\\\\n$a_0$ & 0.8 & -- & 0.27 & 0.24 & 0.11 & 0.05 & 0.05 & 0 \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\\references\n\n\\bibitem{PBlaha2001}\nBlaha P. {\\em et al.},\n{\\em An augmented plane wave + local orbitals program for calculating crystal properties},\n(K. Schwarz, Tech. Universit\\\"{a}t Wien, Austria, 2001).\n\n\\bibitem{JPPerdew1996}\nPerdew, J. P., Burke, S., \\& Ernzerhof, M.\n{\\em Generalized gradient approximation made simple},\nPhys. Rev. Lett. {\\bf 77}, 3865 (1996).\n\n\\bibitem{Kunes2001}\nKune{\\u s}, J., Nov{\\'a}k, P., Schmid, R., Blaha, P., \\& Schwarz, K.\n{\\em Electronic structure of fcc Th: Spin-orbit calculation with $6p_{1\/2}$ local orbital extension},\nPhys. Rev. B {\\bf 64}, 153102 (2001).\n\n\\bibitem{PuCoGa5}\nSarrao, J. L. {\\it et al.} \nPlutonium-based superconductivity with a transition temperature above 18 K.\n{\\it Nature} {\\bf 420}, 297 (2002).\n\n\\bibitem{PuRhGa5}\nWastin, F. {\\it et al.} \nAdvances in the preparation and characterization of transuranium systems.\n{\\it J. Phys. Condens. Matter} {\\bf 15}, S2279 (2003).\n\n\\bibitem{PuCoIn5}\nBauer, E. D. {\\it et al.}\nLocalized 5$f$ electrons in SC PuCoIn$_5$: Consequences for superconductivity in PuCoGa$_5$.\n{\\it J. Phys. Condens. 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+{"text":"\\section{Introduction}\n\\label{intro}\nThe hydrogen atom has a central position in the history of atomic physics. As it is the simplest of atoms, it has played a key role in testing fundamental theories, and hydrogen spectroscopy is associated with successive advances in the understanding of the atomic structure. Since the advent in the seventies of tunable lasers and methods of Doppler free spectroscopy, hydrogen spectroscopy has been renewed in the last decades. Consequently, several optical frequencies of hydrogen are now known with a fractional accuracy better than $10^{-11}$. In a long series of experiments, H\\\"{a}nsch and coworkers have improved the precision on the measurement of the $1\\rm S-2\\rm S$ frequency to obtain now a relative uncertainty of about $1.4\\times 10^{-14}$ \\cite{1S-2Sa}. In Paris, we have studied in the nineties the $2\\mathrm{S}-n\\mathrm{S\/D}$ two-photon transitions with $n=8~\\mathrm{and}~12$ ($n$ is the principal quantum number) \\cite{{Ryd97},{Ryd99},{EPJD00}}. For instance we have measured the frequency of the $2\\mathrm{S}_{1\/2}-8\\mathrm{D}_{5\/2}$ transition with an uncertainty of $5.9~\\mathrm{kHz}$, i.e. a relative uncertainty of $7.6\\times 10^{-12}$. The goal of these high precision measurements is to determine the Rydberg constant $R_\\infty$ and the hydrogen Lamb shifts.\n\nThe hydrogen energy levels can be conventionally expressed as the sum of three terms: the energy given by the Dirac equation for a particle with the reduced mass, the first relativistic correction due to the recoil of the proton and the Lamb shift. The first two terms have an exact expression as a function of the quantum numbers and of the fundamental constants (the Rydberg constant $R_\\infty$, the fine structure constant  $\\alpha$ and the electron-to-proton mass ratio $m_{e}\/m_{p}$). The Lamb shift takes into account all the other corrections: corrections due to quantum electrodynamics (QED), other corrections due to the recoil of the proton and the effect of the proton charge distribution. The calculation of the Lamb shift is very difficult and a review of the results obtained so far is made in the report of the CODATA (Committee on Data for Science and Technology) \\cite{codata06}. Today, for the $1\\mathrm{S}$ level, the uncertainties are due to the calculation of the two-loop and three-loop QED corrections (this uncertainty is estimated to $3.7~\\mathrm{kHz}$) and to the measurement of the proton charge distribution. Using the value of the radius $r_{\\mathrm{p}}$ of this charge distribution deduced from the electron-proton scattering experiments ($r_{\\mathrm{p}}=0.895\\,(18)~\\mathrm{fm}$) \\cite{Sick}, this uncertainty is $50~\\mathrm{kHz}$. At this level, the uncertainty of the theoretical value of the Lamb shift is not limited by the ones of $R_\\infty$, $\\alpha$  and $m_{e}\/m_{p}$. Consequently, the uncertainty in the proton radius $r_{\\mathrm{p}}$ is a strong limitation to extract the Rydberg constant from high precision measurements. For instance, using the frequency of the $1\\rm S-2\\rm S$ transition and the calculated value of the $1\\mathrm{S}$ and $2\\mathrm{S}$ Lamb shifts, the Rydberg constant can be deduced with a relative uncertainty of only  $1.8\\times 10^{-11}$ \\cite{LesHouches}.\n\nIn practice, it is possible to avoid this difficulty by using the $1\/n^3$ scaling law for the Lamb shift. Numerous terms of the Lamb shift vary with the principal quantum number exactly as $1\/n^{3}$ (for instance the effect of the charge distribution of the nucleus), and the deviation from this scaling law has been precisely calculated by Karshenboim \\cite{Karshenboim}, and more recently by Pachucki \\cite{Pachucki}. Then it is possible to eliminate the Lamb shift by forming a suitable linear combination of two frequency measurements. For instance, in the linear combination $7\\nu(2\\rm S_{1\/2}-8\\rm D_{5\/2})-\\nu(1\\rm S_{1\/2}-2\\rm S_{1\/2})$ of the  $2\\mathrm{S}_{1\/2}-8\\mathrm{D}_{5\/2}$ and $1\\mathrm{S}_{1\/2}-2\\mathrm{S}_{1\/2}$ frequencies, the quantity $L_{1\\mathrm{S}}-8L_{2\\mathrm{S}}$ appears and the effect of the proton charge distribution is eliminated ($L_{1\\mathrm{S}}$ and $L_{2\\mathrm{S}}$ are the Lamb shifts of the $1\\mathrm{S}$ and $2\\mathrm{S}$ levels). For example, from the measurements of the $1\\rm S_{1\/2}-2\\rm S_{1\/2}$, $2\\rm S_{1\/2}-8\\rm D_{5\/2}$ and $2\\rm S_{1\/2}-12\\rm D_{5\/2}$ frequencies in hydrogen and deuterium, one obtains a value of $R_{\\infty}$ with a relative uncertainty of $9 \\times 10^{-12}$ \\cite{EPJD00}. Moreover, this method gives the values of the Lamb shifts, and, taking into account the theoretical calculations of the Lamb shift, it is possible to deduce a value of the rms charge radius of the proton ($r_{\\mathrm{p}}=0.8746\\,(94)~\\mathrm{fm}$) which is more precise than the one deduced from the electron-proton scattering experiments. In this method, the accuracy is presently limited by the uncertainties of the $2\\rm S_{1\/2}-8\\rm D_{5\/2}$ and $2\\rm S_{1\/2}-12\\rm D_{5\/2}$ frequency measurements. In our experiment \\cite{EPJD00}, these uncertainties were mainly constrained by the natural width of $8\\mathrm{D}$ and $12\\mathrm{D}$ levels and by the inhomogeneous light shift experienced by the atoms passing through the gaussian profile of the laser beams.\n\nTo circumvent the limitation due to the uncertainties of the $2\\mathrm{S}-n\\mathrm{D}$ measurements, our group studies the $1\\rm S-3\\rm S$ two-photon transition, with the aim to deduce $R_\\infty$ and $r_{\\mathrm{p}}$ from the comparison between the $1\\rm S-2\\rm S$ and $1\\rm S-3\\rm S$ frequencies. Indeed, as, in an atomic beam, the number of hydrogen atoms in the $1\\mathrm{S}$ level is about eight orders of magnitude larger than the number of metastable atoms, the $1\\rm S-3\\rm S$ transition can be observed with very low light intensity so with a negligible light shift. In 1996, we have observed the $1\\rm S-3\\rm S$ transition and deduced, from the comparison with the $2\\mathrm{S}-6\\mathrm{S\/D}$ frequency intervals, a value of the $1\\mathrm{S}$ Lamb shift \\cite{Bourzeix}. Since then we have undertaken optical frequency measurements of the $1\\rm S-3\\rm S$ transition. Because there is no simple way to determine the velocity distribution of the $1\\rm S$ atomic beam, a difficulty is the determination of the second order Doppler effect, which induces a red shift of $-v^2\/2c^2$. To measure it, the atomic beam is placed in a transverse magnetic field which induces a motional electric field and so a quadratic Stark shift varying also as $v^2$. The velocity of the atoms and the second order Doppler effect are deduced from the variation of this Stark shift with the magnetic field \\cite{{Doppler},{Hagel}}. With these techniques, we have made a preliminary measurement of the $1\\rm S-3\\rm S$ optical frequency.\n\nThe aim of this paper is to relate in detail these last measurements. Section 2 describes the experimental method. Section 3 and 4 are devoted to the method used to determine the velocity distribution and to the theoretical analysis of the line shape. Finally the results are presented and analyzed in section 5.\n\n\\section{Spectroscopy of the 1S-3S transition}\n\\label{sec:2}\n\n\\subsection{Experimental setup}\n\\label{sec:21}\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure1-1S-3S.eps}\n}\n\\caption{Experimental setup for the excitation of the 1S-3S transition and the optical frequency measurement (Laser TiSa:  titanium sapphire laser, FP: Fabry-Perot cavity, DL\/Rb: laser diode stabilized to a two-photon transition of Rubidium).}\n\\label{fig:1}      \n\\end{figure}\n\nThe experimental setup (see Figure \\ref{fig:1}) has been described elsewhere \\cite{{EPJD00},{Bourzeix},{Hagel}}. The excitation wavelength of the 1S-3S two-photon transition is in the UV range, at 205 nm. This radiation is produced by quadrupling in frequency a CW titanium sapphire laser at 820~nm, with two frequency doubling stages by a lithium triborate crystal (LBO) and a beta-barium borate crystal (BBO). These crystals are placed in two successive enhancement ring cavities (LBO cavity and BBO cavity respectively). The first frequency doubling is efficient and delivers about 800~mW at 410~nm from an incident power of 2~W at 820~nm \\cite{LBO}. This first cavity is locked to the laser wavelength\nwithout any length modulation thanks to the polarization method \\cite{Hansch-Couillaud}. The second doubling step is by far more challenging \\cite{BBO}. For the second harmonic generation at 410~nm, the only possible choice is a BBO crystal used at the limit of its phase-matching angle. This results in a low conversion efficiency, with a UV power below 1~mW. The second doubling step is operated in an $\\mathrm{O}_2$ environment to slow the chemical reactions at the surface of the crystal. Furthermore, a photo-refractive effect takes place inside the crystal, which results in a reflected blue beam at 410~nm appearing from one end facet of the BBO crystal. A counter propagating wave at 410 nm develops in the ring BBO cavity. To reduce this effect, we have worked in a quasi-continuous regime where the UV intensity consists of $6~\\mu \\mathrm{s}$\npulses at a frequency of 30 kHz. This is done by overmodulating the length of the BBO cavity at the frequency $\\nu _{0}$ of 15~kHz, so as to be resonant only twice per modulation period for 6 $\\mu$s. First-order Doppler effect due to the motion of the cavity mirror results in frequency shifts, which induce a splitting of the observed 1S-3S line. This effect is described in the references \\cite{{Hagel},{DeuxBosses}}.\n\n\\begin{figure}\n\\resizebox{0.9\\columnwidth}{!}{%\n  \\includegraphics{Figure2-1S-3S-2.eps}\n}\n\\caption{Experimental setup for the frequency stabilization of the titanium sapphire laser. See the explanations in the text (TiSa: titanium sapphire laser, AOM: acousto-optic modulator, EOM: electro-optic modulator, FP and FPA:  Fabry-Perot cavities, DL\/Rb: diode laser stabilized on a two-photon transition of Rb).}\n\\label{fig:2}      \n\\end{figure}\n\nThe frequency stabilization of the titanium sapphire laser is described in reference \\cite{EPJD00}. The setup is shown in Figure \\ref{fig:2}. The short term and long term stabilities are assured by two Fabry-Perot cavities, labelled FPA (auxiliary Fabry-Perot) and FP respectively. The principle of this stabilization arrangement is to lock the titanium-sapphire laser to the FPA\ncavity, the FPA cavity to the FP cavity and, finally, the FP cavity to a diode laser stabilized on a two-photon transition of rubidium. A secondary laser beam from the titanium-sapphire laser is split after a double pass through an\nacousto-optic modulator (model 3200 from Crystal Technology at 200~MHz) and sent on the FPA and FP cavities. The FPA cavity (free spectral range 600~MHz and finesse of about 400) is placed in a robust brass vacuum box (wall thickness of 2~cm) and carefully isolated from external vibrations \\cite{LBO}. To reduce the frequency jitter, the laser is locked,\nin a first step, to the FPA~cavity with a FM sideband method \\cite{Drever-Hall}. Thanks to this servo-loop, the frequency jitter is reduced from 500~kHz (free running laser) to about 2~kHz \\cite{LBO}.\n\nThe long term stability is guaranteed by the FP cavity. This cavity is very stable. It consists of a 50~cm long zerodur spacer and two silver coated mirrors, one flat and one spherical (60~cm curvature radius). Its finesse is about 120 at 800~nm. A piezoelectric transducer moves the flat mirror thanks to a mechanical construction (made in fused silica) which avoid the rotation of the mirror (the principle is to deform a parallelogram)~\\cite{These Nez}. This cavity\nis also placed in a vacuum box with the same design than for the FPA cavity. To obtain long term stability, the FP cavity is stabilized on a standard laser, namely a laser diode at 778~nm stabilized to the 5S$_{1\/2}$-5D$_{5\/2}$ two-photon transition of rubidium (DL\/Rb laser). This standard has been described previously  \\cite{{TowardsRb},{MesRubidium97},{RbLPTF}}. With this setup, as the zerodur spacer is very stable, the servo-loop of the FP cavity on the DL\/Rb laser always uses the same fringe of the FP cavity and the length of the FP cavity is exactly known. Consequently, to excite the $1\\mathrm{S}-3\\mathrm{S}$ transition, the titanium sapphire laser is always locked on a fringe of the FP cavity which is also exactly known (fringe number 1219477 of the FP cavity) . The advantage of this method is that the 1S-3S signal always appears around the same frequency of the acousto-optic modulator. The frequency characteristics of this system will be presented in section \\ref{sec:23}. Finally, to scan the laser frequency, we sweep the frequency of the radiofrequency wave which drives the acousto-optic modulator.\n\nThe 1S-3S transition is excited in a thermal 1S atomic beam colinear with the UV laser beams. The hydrogen atoms are produced through a radio-frequency discharge from molecular hydrogen. In order to increase the UV intensity, the atomic beam is placed inside a linear build-up cavity (UV cavity) formed by two spherical mirrors (radius of curvature 25~cm, 49~cm apart). Inside the cavity, the UV beam is focused within a waist of 48~$\\mu $m. Because of the average characteristics of the UV mirrors (reflection and transmission of the input mirror 89~\\% and 8.5~\\%, reflection of the end mirror 96~\\%), the cavity finesse is about 40. The mirrors are mounted on piezoelectric translators and the cavity length is locked to the 205~nm wavelength. This locking is detailed in reference \\cite{DeuxBosses}. For this servo-loop, the amplitude of the frequency modulation due to the modulation of the BBO cavity is too small with respect to the width of the UV cavity to obtain a good error signal. Therefore the length of the UV cavity needs to be modulated at the same frequency as that of the BBO cavity (15~kHz), but in quadrature. This way, the UV pulses issued from the BBO cavity are sent to the UV cavity when the length of the UV cavity is at an extremum. If the UV cavity length is minimum for one pulse, it will be maximum for the following one and so on: the UV pulses test successively the two sides of the Airy's peak of the UV cavity. For this servo loop, the modulation amplitude is a small fraction of the width\nof an Airy's peak of the UV cavity (typically 20~\\%). Finally, the UV intensity inside the cavity is monitored by a photodiode placed after the end mirror and a phase sensitive detection at 15 kHz compares the transmission of the UV cavity for two successive pulses to obtain the servo signal.\n\nThe two-photon transition is detected by monitoring the Balmer-$\\alpha $ fluorescence due to the radiative decay 3S-2P. This fluorescence is collected with a spherical mirror and a $f\/0.5$ aspheric lens system and selected with an interference filter at 656 nm. We have used two different detectors, a photomultiplier (Hamamatsu R943-02) or a CCD camera (Princeton Instruments Spec 100B). The following section presents the data acquisition in these two cases.\n\n\n\\subsection{Experimental spectra}\n\\label{sec:22}\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure3-1S-3S.eps}\n}\n\\caption{Record of the 1S-3S two-photon transition in hydrogen with the photomultiplier (color online). This spectrum is the mean of 8 runs. For each point the acquisition time is 80 s. The two curves correspond to the signal recorded when the BBO cavity length increases (blue curve, blue shifted) or decreases (red curve, red shifted).}\n\\label{fig:3}      \n\\end{figure}\n\nFor the photomultiplier detection, the data acquisition takes benefit from the short time response of the photomultiplier. Each scan is divided in 31 frequency points. For each point, the photomultiplier signal is counted during 1~s and we make 10 scans of the line to achieve a 7~minutes run. Furthermore, acquisition electronics are designed to select only the time window during which the 205~nm UV light is resonant inside the atomic beam cavity (as a consequence of the 15 kHz over modulation of the BBO cavity). The signal of the photomultiplier is sent, after an amplifier discriminator and an electronic switch, to a multiplexer. To reduce the noise due to the dark current of the photomultiplier, the electronic switch transmits the signal only when the instantaneous UV intensity is above a reference level. Then, a multiplexer dispatches the signal to two counters in phase with the modulation at 15 kHz: when the BBO cavity length increases, the signal is sent to counter~1, and, when it decreases, to counter~2. This way, we can observe the splitting of the $1\\mathrm{S}-3\\mathrm{S}$ line due to the Doppler shift induced by the motion of the mirror (see section \\ref{sec:21}). As the signal-to-noise ratio is small, we take the mean of several runs to obtain an observable signal. Figure~\\ref{fig:3} shows the atomic signal. The two curves, corresponding to the two counters are separated by about $1.1~ \\mathrm{MHz}$ in terms of atomic frequency. Consequently we will have to take into account this effect in the line shape analysis.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure4-1S-3S.eps}\n}\n\\caption{Record of the 1S-3S two-photon transition in hydrogen with a CCD camera. The total acquisition time is about 20 minutes. In this case the signals obtained when the BBO cavity length increases or decreases are not separated.}\n\\label{fig:4}      \n\\end{figure}\n\nIn comparison with the photomultiplier, the CCD camera has two advantages: the quantum efficiency is higher (about 90 \\% for a back-illuminated camera when it is 15 \\% for the photomultiplier) and it is possible to obtain an image of the fluorescence of the atoms in the laser beam. The drawback is the longer time response of the CCD, of the order of one millisecond, which does not allow us to perform the electronic time selection described above. Moreover a readout noise, independent of exposure time, is also superimposed on each pixel (or group of pixels) every time the chip is being read. This leads to the use of long exposure times, of the order of 1 minute, to gather as many signal photons as possible before reading the chip. Nevertheless, this exposure time should remain small compared to the characteristic drifting time of the titanium sapphire laser and of the frequency doubling stages. Finally the exposure time of each frequency point of the scan is 37 s and the acquisition of the 31 frequency points lasts about twenty minutes. Then each image is carefully analyzed to reduce the parasitic signal due to the UV and to take into account the variation of the UV intensity during the scan (see the references \\cite{{Arnoult},{TheseArnoult}} for the detail of this analysis). An example of $1\\mathrm{S}-3\\mathrm{S}$ spectrum obtained with the CCD camera is shown in figure \\ref{fig:4}. For this record the UV intensity was about 70 \\% the one of the record shown in figure \\ref{fig:3}. As the two-photon excitation is a quadratic process, this corresponds to a reduction by a factor of about 2 of the excitation probability. Nevertheless the CCD camera signal is about five times larger than the photomultiplier signal. This corresponds to the gain in the quantum efficiency and to a better collection of the atomic fluorescence. For this reason, the main results presented in this article have been obtained by using the CCD camera.\n\n\\subsection{Optical frequency measurement}\n\\label{sec:23}\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure5-1S-3S-2.eps}\n}\n\\caption{Schematic diagram of the optical setup for the optical frequency measurements (Ti:Sa: titanium-sapphire laser, BS: beam splitter, DM: dichroic mirror, PD: photodiode). The repetition rate is detected with PD1, the offset $f_0$ with PD2 and the beat note between the CW Ti:Sa laser and the frequency comb with PD3 (color online).}\n\\label{fig:5}      \n\\end{figure}\n\nThe optical frequency measurements are made with an optical frequency synthesizer, following the design introduced by Hall and H\\\"{a}nsch \\cite{{Femto1},{Femto2}}. Figure \\ref{fig:5} shows the experimental arrangement. We use a second titanium sapphire laser, a six chirped mirrors mode-locked femtosecond laser with a repetition rate $f_{rep}$ of about 900~MHz (laser GigaJet 20 from Menlo Systems GmbH) with a 5~W pump laser (laser Verdi V5 from Coherent). The output of the titanium sapphire laser forms a frequency comb. The frequency $f_N$ of each line of the comb is controlled by the frequency rate $f_{rep}$ and the global shift $f_0$ of the frequency comb with respect to the zero frequency: $f_N=Nf_{rep}+f_0$. The offset frequency $f_0$ is determined thanks to the self-referencing technique. The spectrum of the femtosecond titanium sapphire laser is broadened inside a microstructure photonic crystal fiber (from CrystalFiber) as to span over more than an octave. Then the infrared part of the spectrum is frequency doubled in a KNbO$_3$ nonlinear crystal to obtain green radiation which is recombined, after a time delay, with the green part of the spectrum generated by the photonic crystal fiber. The result is a beat note detected by the photodiode PD2 (see figure \\ref{fig:5}) at the offset frequency $f_0$ or at the frequency $f_{rep}-f_0$. As the output spectrum of the comb before the fiber (ranging from 780 to 820~nm) includes the wavelength of the two-photon excitation, the CW titanium-sapphire laser is mixed with the frequency comb before the photonic crystal fiber. The advantage is that the resulting beat note (frequency $f_1$) is very stable with a good signal-to-noise ratio (40 dB in a 300 kHz bandwidth). This frequency $f_1$ corresponds to the frequency difference between the titanium sapphire laser frequency and the closest frequency of the comb, either lower (case (a)) or higher (case (b)). The spacing between the lines of the comb is fixed by phase-locking the repetition rate to a reference signal related to a Cs clock. Thanks to an optical link between our laboratory and LNE ({\\it Laboratoire National d'Essais})-SYRTE {\\it (Syst\\`{e}me de R\\'{e}f\\'{e}rence Temps Espace}) at the {\\it Observatoire de Paris} \\cite{Fibre}, we receive a radio-frequency signal at 100~MHz locked to a primary frequency standard. Then a radio frequency chain generates a signal at 11~GHz which is mixed with the twelfth harmonic of the repetition rate \\cite{FemtoSyrte}. The phase error signal is then amplified and fed back to a piezoelectric translator controlling the length of the femtosecond ring cavity. Finally the frequency of the titanium-sapphire laser is given by:\n\\begin{equation}\nf_{\\mathrm{Ti:Sa}}=N \\times f_{rep}+ f_0 \\pm f_1  \\label{Eq1}\n\\end{equation}\nwith a $+$ sign in case (a) and $-$ sign in case (b). Then, by mixing both beat notes electronically and filtering, one can record the suitable combination of $f_0$ and $f_1$ with a tracking oscillator and a frequency counter.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure6-1S-3S.eps}\n}\n\\caption{Measurements of the frequency $f(1219477)$ of the peak numbered 1219477 of the FP cavity, in 2005 (upper part) and in 2009 (lower part). The horizontal axis corresponds to the successive number of the runs (about 200 runs during 15 days in 2005 and 300 runs during 30 days in 2009). Each point corresponds to a 20 minute data acquisition run in 2005, and 7 minute in 2009. During this period, the drift of the cavity is of a few kHz.}\n\\label{fig:6}      \n\\end{figure}\n\nThanks to the optical link with the Observatoire de Paris, the DL\/Rb standard laser has been simultaneously measured during 300 s in our laboratory and in the LNE-SYRTE laboratory. The result of this test is a frequency difference of $2~(26)~\\mathrm{Hz}$, the uncertainty corresponds to one standard deviation of the mean. The absolute frequency of the DL\/Rb standard has also been measured several times. The results obtained after extrapolation to zero laser intensity (in order to eliminate the light shift of the two-photon rubidium transition) are $385~285~142~376.3~(1.0)$ $\\mathrm{kHz}$ in May 2004, $385~285~142~376.4~(1.0)~\\mathrm{kHz}$ in June 2005 and $385~285~142~378.8~(1.0)~\\mathrm{kHz}$ in February 2009. These values are close to the first measurement of the DL\/Rb standard en 1996 which was $385~285~142~376.7~(1.0)~\\mathrm{kHz}$ \\cite{{MesRubidium97},{MesCO2-3}}.\n\nDuring the recordings of the 1S-3S spectra, the frequency of the titanium-sapphire laser was continuously measured. Figure \\ref{fig:6} shows the optical frequency of the peak of the FP cavity numbered 1219477 which is used to lock the titanium-sapphire laser, measured in 2005 and in 2009. This figure illustrates the repeatability of the setup used to stabilized the titanium-sapphire laser (see figure \\ref{fig:2}). The frequency jumps are due to some degradations of the servo-loops of the lasers on the FP cavity. Usually it is due to a bad alignment of the lasers with respect to the cavity or to a diminution of the electronic gain. The shift of about 5~kHz between the 2005 and 2009 measurements can be explained by the frequency of the DL\/Rb laser or by an ageing of the silver coated mirrors of the FP cavity.\n\\section{Determination of the velocity distribution}\n\\label{sec:3}\n\n\\begin{figure}\n\\resizebox{1.1\\columnwidth}{!}{%\n  \\includegraphics{Figure7-1S-3S.eps}\n}\n\\caption{Experimental arrangement for the determination of the second-order Doppler shift. The atomic beam is horizontal. The UV light is focused in front of the detection system using a photomultiplier (PM). The two coils produce a vertical magnetic field ${\\bf B}$ perpendicular to the direction of the atomic beam.}\n\\label{fig:7}      \n\\end{figure}\n\nIn a thermal hydrogen beam at room temperature, the typical atomic velocity is 3 km\/s. For the 1S-3S transition, this velocity induces a second-order Doppler shift $-\\nu(1{\\rm S}-3{\\rm S}) \\times v^2\/2c^2$ which is about 146 kHz ($\\nu(1{\\rm S}-3{\\rm S})$ is the atomic frequency). To measure this effect, we use a method proposed in reference \\cite{Doppler} which is convenient when the 1S-$n$S (or 2S-$n$S) two-photon hydrogen transitions are produced in an atomic beam. The principle is to\napply a transverse magnetic field ${\\bf B}$ with respect to the direction of the atomic beam (see figure \\ref{fig:7}). This magnetic field produces a motional electric field ${\\bf E} = {\\bf v} \\times {\\bf B}$ which induces a quadratic Stark shift proportional, as the second-order Doppler effect, to ${v}^{2}$. From the observation of this effect, we can deduce the second order Doppler shift.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure8-1S-3S-2.eps}\n}\n\\caption{Zeeman diagram of the 1S-3S transition. The levels are labelled by the quantum numbers ($F,m_{F}$). In the intermediate regime the 1S$_{1\/2}$-3S$_{1\/2}(F=1)$ line is split in four components following the selection rule $\\Delta m_{F}=0$.\nThe motional Stark effect is important, for the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=-1$) component, around the crossing between the 3S$_{1\/2}$ ($F=1,m_{F}=-1$) and 3P$_{1\/2}$($F=1,m_{F}=0$) levels. For high magnetic field, the magnetic quantum numbers of these two levels are respectively ($m_J=-1\/2$, $m_{I}=-1\/2$) and ($m_J=1\/2$, $m_{I}=-1\/2$).}\n\\label{fig:8}      \n\\end{figure}\n\nThe first effect of the magnetic field is a Zeeman splitting (see figure \\ref{fig:8}). In low magnetic field, because of the selection rules of a two-photon transition ($\\Delta F=0$ and $\\Delta m_{F}=0$ \\cite{JPhys73}), the line between the hyperfine levels 1S$_{1\/2}(F=1$) and 3S$_{1\/2}(F=1$) is split in three lines. For a higher magnetic field, the line between the 1S$_{1\/2}(F=1, m_{F}=0$) and  3S$_{1\/2}(F=0, m_{F}=0$) Zeeman sub-levels becomes allowed, because the hyperfine structure of the 3S level is in the Paschen-Back regime when it is not the case for the 1S level. Practically, the shift of the $m_{F}=\\pm 1$ components is very small, because, in first approximation, the Land\\'{e} factors are the same for the 1S and 3S levels. Following a relativistic calculation, the Land\\'{e} factors $g_{J}$ of the 1S and 3S levels are respectively 2.002284 and 2.002315 \\cite{Paul}. This difference induces, for a field of 20 mT, a residual Zeeman shift of about $\\pm 4.3$~kHz. For this field the diamagnetic shift is about 809 Hz.  On the other hand, the $F=1, m_{F}=0$ component is shifted, because the Zeeman effect is not the same for the 1S$_{1\/2}(F=1,m_{F}=0$) and 3S$_{1\/2}(F=1,m_{F}=0$ ) sub-levels. For 20~mT, this shift is about 200~MHz.\n\nThe second effect of this magnetic field is a motional electric field ${\\bf E=v\\times B}$. For a velocity of 3 km\/s and a magnetic field of 20 mT, this electric field is 0.6 V\/cm. It induces a coupling of the 3S$_{1\/2}$ level with the 3P$_{1\/2}$ and 3P$_{3\/2}$ levels. For the 3S$_{1\/2}$ level, in the limit of high magnetic field, the electronic and nuclear spins are decoupled and the good quantum numbers are ($J$, $I$, $m_J$, $m_I$). For example, the 3S$_{1\/2}$($F=1,m_{F}=-1$) level corresponds to the quantum numbers $m_J=-1\/2$ and $m_I=-1\/2$. It is mainly coupled to the 3P$_{1\/2}$($m_J=1\/2$, $m_{I}=-1\/2$), 3P$_{3\/2}$($m_J=1\/2$, $m_{I}=-1\/2$) and 3P$_{3\/2}$($m_J=-3\/2$, $m_{I}=-1\/2$) sub-levels following the selection rules $\\Delta m_J= \\pm 1$ and $\\Delta m_I=0$. This kind of motional Stark mixing is well known and, for instance, was used by Lamb and Retherford to polarize a metastable atomic beam \\cite{Lamb}. We consider the quadratic Stark effect due to this motional electric field. This shift, proportional to ${\\bf v}^{2}$, is very small, but observable in the case of the $m_{F}=\\pm 1$ line components, because the Zeeman shift of these components is negligible. In particular, this effect is important around the level crossing between the 3S$_{1\/2}$($F=1,m_{F}=-1$) and 3P$_{1\/2}$($F=1,m_{F}=0$) Zeeman sub-levels, which appears for a magnetic field of about 18 mT (see figure \\ref{fig:8}).\n\nThe complete calculation of the line shape will be described in the section \\ref{sec:4}. We present here a simple picture. The initial state $g$ (the levels 1S$_{1\/2}$($F=1,m_{F}=\\pm 1$)) is coupled by two-photon excitation\nto an excited state $e$ (here the levels 3S$_{1\/2}$($F=1,m_{F}=\\pm 1$)). This state $e$ is mixed by the Stark hamiltonian $V_{ef}$ with several $f$ states (the Zeeman sub-levels of the 3P$_{1\/2}$ and 3P$_{3\/2}$ states). In a simple approach, the perturbed energy $E^\\prime_e$ and width $\\Gamma^\\prime_e$ of the considered $e$ level are deduced from the unperturbed energy and width $E_e$ and $\\Gamma_e$, using the perturbation theory in lowest order:\n\\begin{equation}\nE^\\prime_e-i\\frac{\\hbar}{2} \\Gamma^\\prime_e= E_e-i\\frac{\\hbar}{2} \\Gamma_e+\\sum_{f}\\frac{\\left| V_{ef}\\right| ^{2}}{( E_e-E_f)-i\\frac{\\hbar}{2}(\\Gamma_e-\\Gamma_f)} \\label{Eq2}\n\\end{equation}\nThen the Stark shift $\\delta_S $ of the level $e$ is given by:\n\\begin{equation}\n\\delta_S=\\mathop{\\rm Re}(E^\\prime_e-E_e)=\\sum_{f}\\frac{( E_e-E_f)\\left| V_{ef}\\right|^{2}} {(E_e-E_f)^2+(\\frac{\\hbar}{2})^2(\\Gamma_e-\\Gamma_f)^2} \\label{Eq3}\n\\end{equation}\nIn these equations, the energies $E_e$ and $E_f$ take into account the shifts of the $e$ and $f$ levels due to the Zeeman effect: then $E_e-E_f$ is a function of the magnetic field and the Stark shift appears as a sum of dispersion curves corresponding to the different level crossings between the 3S$_{1\/2}$ and 3P$_J$ levels. There is no divergence at the level crossing when $E_e=E_f$ because of the difference between the natural widths of the 3S$_{1\/2}$ and 3P$_{1\/2}$ levels (respectively 1 MHz and 30.6 MHz). As the effect of the electric field is negligible for the 1S$_{1\/2}$ level, the total shift $\\delta $ of the two-photon line is:\n\\begin{equation}\n\\delta = \\delta_Z+\\delta_S-\\nu(1{\\rm S}-3{\\rm S}) \\times \\frac{v^2}{2c^2} \\label{Eq4}\n\\end{equation}\nwhere $\\delta_Z $ is the shift due to the difference of the Zeeman effect for the $g$ and $e$ levels and the third term the second Doppler effect for an atom of velocity $v$.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure9-1S-3S-2.eps}\n}\n\\caption{Shift of the 1S-3S line due to the motional electric field (calculation for an atom at a velocity of 3~km\/s). Zero: line position for an atom at rest; small dashed line: shift of the line due to the second-order Doppler effect. Curves (a) and (b): line position of the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=+1$) and 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=-1$) components. In A and B, the second-order Doppler effect of the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=-1$) component is exactly compensated by the motional Stark effect. Curve (c): global motional Stark effect for the two components\n1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=\\pm 1$).}\n\\label{fig:9}      \n\\end{figure}\n\nFigure \\ref{fig:9} shows, for an atom at 3~km\/s, the shifts of the 1S$_{1\/2}-$3S$_{1\/2}(F=1,m_{F}=\\pm 1$) transitions around the level crossing at 18 mT. The large dashed line represents the line position without any shifts for the velocity $v=0$ and the small dashed line the line position shifted by the second-order Doppler effect for $v=3~\\mathrm{km\/s}$ and $B=0$. This shift is about $-146~\\mathrm{kHz}$. With respect to this reference, the shift of the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=+1$) line (curve (a)) is small (about 10~kHz), because, in this range of magnetic field, the 3S$_{1\/2}$($F=1,m_{F}=+1$) level is far from the 3P$_{1\/2}$ and 3P$_{3\/2}$ levels. On the other hand, the shift of the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=-1$) line (curve (b)) is\nimportant around the level crossing. It has a dispersion shape and is larger than the second-order Doppler effect. Moreover, there are two values of the magnetic field (points A and B) where the second-order Doppler effect is\nexactly compensated by this motional Stark effect. For these values, the position of the line is independent of the atomic velocity, because the second-order Doppler effect and the motional Stark effect both vary as $v^2$.\nNevertheless, this effect cannot be used directly, because the 1S$_{1\/2}$-3S$_{1\/2}$($F=1,m_{F}=\\pm 1$) lines are not separated (the motional Stark effect is small with respect to the 3S$_{1\/2}$ natural width (about 1~MHz). Curve (c) shows the barycenter of these two lines. Though reduced by a factor two, the dispersion amplitude is about 230~kHz, and the compensation of the second-order Doppler effect is still 73\\% for a magnetic field of 17~mT. We have observed this effect to deduce the second-order Doppler effect.\n\n\\section{Line shape}\n\\label{sec:4}\n\n\\subsection{Theoretical background }\n\\label{sec:41}\n\nThe aim of this calculation is to take into account simultaneously the natural width and the Zeeman and Stark effects. We follow the procedure described in reference \\cite{EPJD00}. The evolution of the density operator $\\rho $ is:\n\\begin{equation}\n\\frac{d\\rho }{dt}=\\frac{1}{i\\hbar }\\left[ \\left( H_{0}+V_{L}+V_{S}\\right)\n,\\rho \\right] +\\Gamma \\rho  \\label{Schrodinger}\n\\end{equation}\nwhere $V_{S}$ is the Stark Hamiltonian and the operators $V_{L}$ and $\\Gamma $ describe the two-photon excitation and the spontaneous emission. We use the notations of section \\ref{sec:3}: the states $g$, $e$ and $f$ are respectively the  1S$_{1\/2}$, 3S$_{1\/2}$ and 3P$_J$ levels. We make the rotating wave approximation and we introduce the two-photon Rabi frequency $\\Omega _{e}$:\n\\[\n\\left\\langle e\\right| V_{L}\\left| g\\right\\rangle =\\frac{\\hbar \\Omega _{e}}{2}\\exp(-2i\\omega t)\n\\]\n\\begin{equation}\n\\Omega _{e}=\\frac{8\\pi a_{0}^{2}\\left| \\langle e\\left| Q_{tp}\\right|\ng\\rangle \\right| I}{mc^{2}\\alpha }  \\label{Rabi2ph}\n\\end{equation}\nwhere $\\omega \/2\\pi$ is the laser frequency, $I$ the power density of the light, $a_{0}$ the Bohr radius, $\\alpha $ the fine structure constant and $m$ the electron mass. For a polarization along the z-axis, the two-photon operator $Q_{tp}$ is given in atomic units ($\\hbar =\\alpha c=m=1$) by:\n\\begin{equation}\nQ_{tp}=\\sum_{r} \\frac{z\\left| r\\right\\rangle \\left\\langle r\\right| z}{\\omega -\\omega _{rg}}\n\\label{Qtwo-photon}\n\\end{equation}\nwhere the sum is made on all the atomic states $r$ and $\\omega _{rg}\/2\\pi$ is the atomic frequency difference between the $g$ and $r$ states. Because of the selection rules for the two-photon excitation, the initial state $g$ is coupled to a single excited state $e$.\n\nIf we assume that $\\Omega _{e}\\ll \\Gamma _{e}$, we can neglect in a first step the populations and coherences $\\rho _{ee^{\\prime }}$, $\\rho _{ff^{\\prime }}$ or $\\rho _{ef}$ of the upper levels. In the rotating frame, we replace the density operator by an operator $\\sigma $ with $\\sigma _{gg}=\\rho _{gg}$, $\\sigma _{eg}=\\rho _{eg}\\exp (2i\\omega t)$ and $\\sigma _{ge}=\\rho _{ge}\\exp\n(-2i\\omega t)$ and we introduce the frequency detunings $\\Delta _{e}=2\\omega-(\\omega _{e}-\\omega _{g})$ and $\\Delta _{f}=2\\omega -(\\omega _{f}-\\omega _{g})$ ($\\hbar\\omega_i$ is the energy of the level $i$). In this way, we obtain from equation (\\ref{Schrodinger}) a set of equations:\n\\begin{equation}\n\\frac{d\\sigma _{gg}}{dt}=-\\frac{i}{2} \\Omega _{e}\\left( \\sigma_{eg}-\\sigma _{ge}\\right)\n\\label{Population}\n\\end{equation}\n\\begin{equation}\n\\frac{d\\sigma _{eg}}{dt}=\\left( i\\Delta _{e}-\\frac{\\Gamma _{e}}{2}\\right)\\sigma _{eg}\n-i\\frac{\\Omega _{e}}{2}\\sigma _{gg}-\\frac{i}{\\hbar }\\sum\\limits_{f}V_{ef}\\sigma _{fg}\n\\label{Coherence1}\n\\end{equation}\n\\begin{equation}\n\\frac{d\\sigma _{fg}}{dt}=\\left( i\\Delta _{f}-\\frac{\\Gamma _{f}}{2}\\right)\\sigma _{fg}\n-\\frac{i}{\\hbar }V_{fe}\\sigma _{eg}\n\\label{Coherence2}\n\\end{equation}\nwhere $V_{fe}=\\left\\langle f\\right|V_{S}\\left| e\\right\\rangle $. Then we assume that the optical coherences $\\sigma _{eg}$ and $\\sigma _{fg}$ follow adiabatically the population $\\sigma_{gg}$, {\\it i.e.} that:\n\\[\n\\frac{d\\sigma _{eg}}{dt}=0\\qquad {\\rm and\\qquad }\\frac{d\\sigma _{fg}}{dt}=0\n\\]\nWith these hypotheses, the equations (\\ref{Coherence1}, \\ref{Coherence2})\ngive:\n\\begin{equation}\n\\left( i\\Delta _{e}-\\frac{\\Gamma _{e}}{2}\\right) \\sigma_{eg}\n+\\sum\\limits_{f}\\frac{V_{ef}V_{fe}}{\\hbar^{2}\\left( i\\Delta_{f}-\\frac{\\Gamma _{f}}{2}\\right) }\\sigma _{eg}\n=i\\frac{\\Omega _{e}}{2}\\sigma _{gg}\n\\label{Coherence3}\n\\end{equation}\nThis equation gives the coherence $\\sigma_{eg}$ as a function of the population $\\sigma_{gg}$. It can be written more simply:\n\\begin{equation}\n\\sigma_{eg}=\\frac{1}{A}i\\frac{\\Omega _{e}}{2}\\sigma _{gg}\n\\label{Coherence4}\n\\end{equation}\n\nwhere:\n\\begin{equation}\nA=\\left( i\\Delta _{e}-\\frac{\\Gamma _{e}}{2}\\right)\n+\\sum\\limits_{f}\\frac{V_{ef}V_{fe}}{\\hbar^{2}\\left( i\\Delta_{f}-\\frac{\\Gamma _{f}}{2}\\right) }\n\\label{Coherence5}\n\\end{equation}\nCombining with equation (\\ref{Population}), one obtains the evolution of the population $\\sigma_{gg}$ and the probability of the two-photon excitation.\n\nTo obtain the correct line shape, we have to calculate the 656 nm fluorescence from the $n=3$ levels towards the $n=2$ levels. This fluorescence is due to the cascades 3S$_{1\/2}$ $\\rightarrow$ 2P$_{J}$ and 3P$_{J}$ $\\rightarrow$ 2S$_{1\/2}$. Because of the Stark mixing, the 3P$_{J}$ states are also populated. Then, to obtain the line shape, it is required to calculate the populations $\\sigma _{ee}$, $\\sigma _{ff}$ and $\\sigma _{ff^{\\prime }}$. They are deduced from equation \\ref{Schrodinger}:\n\\begin{eqnarray}\n\\frac{d\\sigma _{ee}}{dt}&=&-\\Gamma _{e}\\sigma _{ee}+\\frac{i}{2} \\Omega _{e}\\left( \\sigma_{eg}-\\sigma _{ge}\\right) \\nonumber \\\\\n&&-\\frac{i}{\\hbar}\\sum\\limits_{f}\\left(V_{ef} \\sigma_{fe}-\\sigma _{ef}V_{fe}\\right)\n\\label{Population1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{d\\sigma _{fe}}{dt}&=&-\\frac{\\Gamma _{e}+\\Gamma _{f}}{2}\\sigma _{fe}\n-i\\left(\\omega_f-\\omega_e\\right)\\sigma _{fe}  \\nonumber \\\\\n&&+\\frac{i}{2} \\Omega _{e} \\sigma_{fg}\n-\\frac{i}{\\hbar}V_{fe}\\left(\\sigma_{ee}-\\sigma_{ff}\\right)\n\\label{Population2}\n\\end{eqnarray}\n\\begin{equation}\n\\frac{d\\sigma _{ff}}{dt}=-\\Gamma _{f}\\sigma _{ff}-\\frac{i}{\\hbar}\\left(V_{fe} \\sigma_{ef}-\\sigma _{fe}V_{ef} \\right)\n\\label{Population3}\n\\end{equation}\nIn our experimental conditions, the two-photon excitation probability is very small, typically $10^{-3}$\/s for an atom at the center of a 1 mW UV beam. As this probability is very small with respect to the natural width of the 3S level (1 MHz), we are in a stationary regime and $d\\sigma _{ii^{\\prime}}\/dt=0$ (with $i$ and $i^{\\prime}$ are the levels $e$ or $f$). Then it is possible to calculate the populations $\\sigma_{ii^{\\prime}}$ from the set of equations (\\ref{Population1}$-$\\ref{Population3}).\n\nFirst we consider the simple case where the $e$ level is coupled with only one $f$ level by the Stark Hamiltonian. From equations (\\ref{Coherence2}), (\\ref{Coherence4}) and (\\ref{Population2}), we deduce:\n\\begin{equation}\n\\sigma _{fe}=\\frac{\\frac{1}{\\hbar}V_{fe}\\left(\\sigma_{ee}-\\sigma_{ff}\\right)-i\\frac{\\Omega_e^2}{4\\hbar}\\frac{1}{A}\n\\frac{V_{fe}}{\\Delta_f+i\\Gamma_f\/2}\\sigma_{gg}}\n{\\omega_e-\\omega_f+i\\frac{\\Gamma_e+\\Gamma_f}{2}}\n\\label{population4}\n\\end{equation}\nThe populations $\\sigma_{ee}$ and $\\sigma_{ff}$ are obtained from the equations (\\ref{Population1}) and (\\ref{Population3}):\n\\begin{eqnarray}\n\\Gamma _{e}\\sigma _{ee}&=&2\\mathop{Re}\\left(i\\frac{\\Omega _{e}}{2}\\sigma_{eg}\\right)\n-\\frac{2}{\\hbar}\\mathop{Re}\\left(iV_{ef}\\sigma_{fe}\\right)\\nonumber \\\\\n\\Gamma _{f}\\sigma _{ff}&=&\\frac{2}{\\hbar}\\mathop{Re}\\left(iV_{ef}\\sigma_{fe}\\right)\n\\label{Population5}\n\\end{eqnarray}\nUsing the expressions of $\\sigma_{eg}$ (equation (\\ref{Coherence4})) and $\\sigma_{fe}$ (equation (\\ref{population4})), we can express $\\sigma_{ee}$ and $\\sigma_{ff}$ as a  function of $\\sigma_{gg}$:\n\\begin{equation}\n\\left(\\begin{array}{c}\\Gamma _{e}\\sigma _{ee} \\\\ \\Gamma _{f}\\sigma _{ff}\\end{array}\\right)=\nB\\left(\\begin{array}{cc}1&-1 \\\\ -1&1\\end{array}\\right)\\left(\\begin{array}{c}\\sigma_{ee}\\\\\\sigma_{ff}\\end{array}\\right)\n+\\left(\\begin{array}{c}C_1 \\\\ C_2 \\end{array}\\right)\\sigma_{gg}\n\\label{Population6}\n\\end{equation}\nwhere the coefficients $B$, $C_1$ and $C_2$ are:\n\\begin{eqnarray}\nB&=&-\\frac{\\frac{1}{\\hbar^2}\\vert V_{ef}\\vert^2\\left(\\Gamma_e+\\Gamma_f\\right)}\n{\\left(\\omega_e-\\omega_f\\right)^2+\\left(\\frac{\\Gamma_e+\\Gamma_f}{2}\\right)^2}  \\nonumber \\\\\nC_1&=&-\\frac{\\Omega_e^2}{2}\\mathop{Re}\\left(\\frac{1}{A}\\left[1+\\frac{\\vert V_{ef}\\vert^2\/\\hbar^2}\n{\\left(\\Delta_f+i\\frac{\\Gamma_f}{2}\\right)\\left(\\omega_e-\\omega_f+i\\frac{\\Gamma_e+\\Gamma_f}{2}\\right)}\\right]\\right)\n\\nonumber \\\\\nC_2&=&\\frac{\\Omega_e^2}{2}\\mathop{Re}\\left(\\frac{1}{A}\\frac{\\vert V_{ef}\\vert^2\/\\hbar^2}\n{\\left(\\Delta_f+i\\frac{\\Gamma_f}{2}\\right)\\left(\\omega_e-\\omega_f+i\\frac{\\Gamma_e+\\Gamma_f}{2}\\right)}\\right)\n\\label{coeffB}\n\\end{eqnarray}\nFinally, the populations $\\sigma_{ee}$ and $\\sigma_{ff}$ are:\n\\begin{equation}\n\\left(\\begin{array}{c}\\sigma _{ee} \\\\ \\sigma _{ff}\\end{array}\\right)=\n\\left(\\begin{array}{cc}\\Gamma_e-B&B \\\\B&\\Gamma_f-B\\end{array}\\right)^{-1}\n\\left(\\begin{array}{c}C_1 \\\\ C_2 \\end{array}\\right)\\sigma_{gg}\n\\label{Population7}\n\\end{equation}\nThe calculations are similar in the case where there are several $f$ levels mixed by the electric field with the $e$ level. They are described in detail in the reference \\cite{These Hagel}.\n\n\\subsection{Theoretical line shape}\n\\label{sec:42}\n\nFor an atom in the initial state $g$ with the velocity $v$ in the magnetic field $B$, the detected fluorescence $F_g(\\omega, v, B)$ is:\n\\begin{equation}\nF_g(\\omega, v, B)=D\\left(\\Gamma_e \\sigma_{ee}+\\sum\\limits_{f}\\gamma_f \\Gamma_f \\sigma_{ff}\\right)\n\\label{Fluorescence}\n\\end{equation}\nwhere $D$ describes the efficiency of the photon detection and $\\gamma_f$ is the branching ratio of the fluorescence from the 3P levels towards the 2S levels (about 0.1183). The populations are calculated following equation (\\ref{Population7}) and, to take into account the second-order Doppler effect, the laser frequency $\\omega$ is replaced by $\\omega(1+v^2\/2c^2)$ in the expression of $\\Delta_e$ and $\\Delta_f$. Figure \\ref{fig:10} shows the line shape of the two components $m_F=1$ and $m_F=-1$ for a velocity of 3 km\/s and a magnetic field of 17.1 mT. This calculation takes into account the Stark mixing of the 3S$_{1\/2}$($F=1,m_{F}=\\pm1$) with all the 3P$_{1\/2}$ and 3P$_{3\/2}$ levels. For the $m_F=1$ component, the effect of the Stark mixing is negligible and the line is red shifted by the second-order Doppler effect. On the other hand, the $m_F=-1$ component is blue shifted and the shift due to the Stark mixing is larger than the second-order Doppler effect. The widths of the two components are also different: 1.451 MHz for the $m_F=-1$ component and 1.006 MHz for the $m_F=1$ one (the natural width of the 3S level is 1.005 MHz).\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure10-1S-3S-2.eps}\n}\n\\caption{Line shape of the $m_F=+1$ and $m_F=-1$ Zeeman components of the line for an atom at a velocity of 3~km\/s in a magnetic field of 17.1 mT. The dashed line is the line position for an atom at rest. The red shift of the $m_F=+1$ component is principally due to the second-order Doppler effect. On the contrary, because of the motional Stark effect, the $m_F=-1$ component is blue shifted and broadened and its intensity is reduced.}\n\\label{fig:10}      \n\\end{figure}\n\nIn the precedent calculation, we have considered an atom which is continuously inside the UV beam and equation (\\ref{Fluorescence}) gives the probability per unit of time that we detect a 656 nm photon. Now we have to take into account the velocity distribution of the atoms. The atomic velocity distribution of a hydrogen beam produced by a radio frequency discharge dissociator has been studied by Jaduszliwer and Chan \\cite{Jet}. In our experimental conditions (dissociator pressure in the range of 0.4~torr), we\ncan assume that this distribution $f(v, \\sigma)$ is close to the Maxwellian form:\n\\begin{equation}\nf(v,\\sigma)=v^{3}\\exp(-v^{2}\/2\\sigma ^{2})\n\\label{vitesse}\n\\end{equation}\nwhere $\\sigma =\\sqrt{kT\/M}$ ($T$ temperature, $M$ atomic mass). Then the line shape $R(\\omega, \\sigma, B)$ is:\n\\begin{equation}\nR(\\omega, \\sigma, B)=a\\int _{0}^{\\infty }\\frac{1}{v}f(v,\\sigma)\\sum\\limits_{g}F_g(\\omega, v, B)dv\n\\label{forme1}\n\\end{equation}\nwhere $g$ are the levels 3S$_{1\/2}$($F=1,m_{F}=\\pm1)$ and $a$ a normalization factor which depends of the atomic density in the atomic beam. If $L$ is the length of the part of the atomic beam in front of the detection system, the two-photon excitation probability is proportional to the  transit time $L\/v$. Consequently, we have introduced the factor $1\/v$ into equation (\\ref{forme1}). In our calculation, we have also taken into account a slight dependence of the two-photon excitation probability with the velocity $v$ which is due to the form of the UV gaussian beam and the life time $\\tau$ of the excited level: if the detection is made exactly at the waist of the UV beam, the excitation has been made at a distance $v\\tau$ from this waist. Then, in equation (\\ref{Rabi2ph}), the power density of the light $I$ depends on the velocity $v$. The calculation of this effect is described in detail in the reference \\cite{TheseArnoult}.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure11-1S-3S-2.eps}\n}\n\\caption{Theoretical line shape of the 1S$_{1\/2}$($F=1,m_F=\\pm 1$)-3S$_{1\/2}$($F=1,m_F=\\pm 1$) transition for a thermal atomic beam ($\\sigma = 1.6$ km\/s) in a magnetic field of 17.1 mT. The broadening and the shift due to the modulation of the length of the BBO cavity are taken into account with the parameter $\\Delta = 57~ {\\rm kHz}$. The full line is the sum of the two contributions due to an increase or a decrease in the cavity length (dashed lines).}\n\\label{fig:11}      \n\\end{figure}\n\nAs explained in section \\ref{sec:21}, the length of the BBO cavity is overmodulated: the UV intensity consists in a succession of pulses which corresponds to an increase or a decrease in the cavity length. A splitting and a broadening of the line consequently appear. This effect is described in reference \\cite{DeuxBosses}. Following \\cite{FORME79}, the line shape of the two-photon transition is given by:\n\\begin{equation}\n\\int_{-\\infty }^{+\\infty }\\left| F(\\Omega )\\right| ^{2}R\\left(\\frac{\\Omega}{2}, \\sigma, B\\right)d\\Omega\n\\label{convolution}\n\\end{equation}\nwhere $F(\\Omega )$ is the Fourier transform of the square of the electric field at 205~nm.\n\nIf $\\omega_L\/2\\pi$ is the laser frequency at 410 nm inside the BBO cavity, the electric field is:\n\\begin{equation}\nE_{410}(t) =\\mathop{Re}\\left[ \\frac{\\tau_C E_{0}\\exp( i\\omega _{L}t)}{1-r_C\\exp( i\\delta t)}\\right]\n\\label{E410}\n\\end{equation}\nwhere $E_{0}e^{i\\omega _{L}t}$ is the incident field on the BBO cavity, $\\tau_C $ the transmission of the input mirror, $r_C$ the reflection coefficient for a round trip in the cavity and $\\delta \/2\\pi $ the frequency shift due to the motion of the mirror ($\\delta >0$ if the cavity length decreases). The equation (\\ref{E410}) is valid because the lifetime of a photon inside the BBO cavity (about 0.2 $\\mu$s) is short in comparison with the duration of the UV pulse (6 $\\mu$s). Then the electric field at 205 nm generated by the BBO crystal is proportional to:\n\\begin{equation}\nE_{205}(t)\\sim\\mathop{Re}\\left[ \\frac{\\tau_C E_{0}\\exp( i\\omega _{L}t)}{1-r_C\\exp( i\\delta t)}\\right] ^{2}\n\\label{E205}\n\\end{equation}\nThen the function $F(\\Omega )$ is:\n\\begin{equation}\nF(\\Omega )=\\frac{1}{2\\pi }\\int_{-\\infty }^{+\\infty }\\frac{\\tau_C\n^{4}E_{0}^{4}\\exp \\left[ i\\left( \\Omega -4\\omega _{L}\\right) t\\right] }{%\n4\\left[ 1-r_C\\exp (-i\\delta t)\\right] ^{4}}dt{\\rm ,}  \\label{Fourier}\n\\end{equation}\nwhere we have only kept the term resonant with the two-photon transition. As the duration of the UV pulses (6 $\\mu$s) is short with respect to the period of the modulation (67 $\\mu$s), we can linearize the denominator in equation (\\ref{Fourier}) and obtain $\\left| F(\\Omega )\\right| ^{2}$. When the length of the cavity decreases ($\\delta >0$), we have:\n\\begin{eqnarray}\n\\left| F(\\Omega )\\right| ^{2} &=&0{\\rm \\qquad if\\qquad \\ }\\Omega <4\\omega_{L}{\\rm ,}  \\nonumber \\\\\n\\left| F(\\Omega )\\right| ^{2} &=&\\left( \\frac{\\tau_C ^{4}E_{0}^{4}}{24r_C^{4}\\delta ^{4}}\\right) ^{2}\n\\left( \\Omega -4\\omega _{L}\\right)^{6} \\nonumber \\\\\n&&\\times \\exp \\left[ -\\frac{\\Omega -4\\omega _{L}}{\\Delta }\\right] \\qquad {\\rm if}\\quad \\ \\Omega >4\\omega _{L}\n\\label{profil}\n\\end{eqnarray}\nwhere $\\Delta =r_C\\delta \/2(1-r_C)$ characterizes the shift and the broadening of the line. When the cavity length increases, the function $\\left| F(\\Omega )\\right|^{2} $ is symmetric with respect to $\\Omega =4\\omega _{L}$.\nThis symmetry supposes that the frequency shifts $\\delta_{incr}$ and $\\delta_{decr}$ are the same for the forward and backward scanning. In fact, a non-linear refractive index of the BBO crystal can produce a dissymmetry. This effect will be estimated in section \\ref{sec:5}.\nFrom equations (\\ref{profil}), a straightforward calculation shows that the mean position of each function $\\left| F(\\Omega )\\right| ^{2}$  is shifted by $\\pm 7\\Delta $ with respect to $4\\omega _{L}$ \\cite{These Hagel}.\n\n\n\nFinally, the line shape is obtained from equations (\\ref{convolution}) and (\\ref{profil}). Figure \\ref{fig:11} shows the profile calculated with a velocity distribution parameter $\\sigma=1.6$ km\/s, in a magnetic field of 17.1 mT and for a modulation $\\Delta = 57~ {\\rm kHz}$. This modulation splits the line shape in two contributions corresponding to an increase or a decrease in the cavity length (dashed lines on figure \\ref{fig:11}) with a separation of about 800 kHz.\n\n\\subsection{Fit of the experimental profiles}\n\\label{sec:43}\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure12-1S-3S.eps}\n}\n\\caption{Fit of the theoretical curve to the experimental data of figure \\ref{fig:4}. The velocity distribution width is fixed at $\\sigma=1.6$ km\/s. The fitted parameters are a modulation $\\Delta$ of 69 kHz and a frequency of the 1S-3S transition of $2~922~742~936.745(17)~{\\rm MHz}$.\n}\n\\label{fig:12}      \n\\end{figure}\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure13-1S-3S.eps}\n}\n\\caption{Fit of the theoretical curves to the experimental data of figure \\ref{fig:3}. The upper part of the figure (respectively the lower part) shows the fit of the sum (respectively the difference) of the two curves of figure \\ref{fig:3} corresponding to an increase or a decrease in the length of the BBO cavity. The velocity distribution is fixed at $\\sigma=1.6$ km\/s. The fitted parameters are a modulation $\\Delta$ of 92 kHz and a frequency of the 1S-3S transition of $2~922~742~936.722(18)~{\\rm MHz}$.\n}\n\\label{fig:13}      \n\\end{figure}\n\nThe parameters of the theoretical curves are the frequency of the 1S-3S transition $\\nu _{eg}=(\\omega _{e}-\\omega _{g})\/\\hbar$ for an atom at rest without magnetic field, the parameter $\\Delta$ which characterizes the splitting due to the modulation of the length of the BBO cavity, and $\\sigma$ which describes the velocity distribution. In a simple model the second-order Doppler shift is given by $-3\/2(\\sigma\/c)^2 \\nu _{eg}$ \\cite{EPJD00}. Consequently the parameters $\\nu _{eg}$ and $\\sigma$ are strongly correlated and it is not possible to deduce these two parameters from the fit of the line shape. Practically, $\\sigma$ is fixed and the fit gives $\\Delta$ and $\\nu_{eg}$; $\\sigma$ will be deduced in the next section from the comparison of the results for different values of the magnetic field.\n\nFigure \\ref{fig:12} shows an example of the fit of the theoretical curve to the data obtained with the CCD camera. In addition to $\\Delta$ and $\\nu_{eg}$, the other parameters of the fit are the amplitude of the signal and an offset corresponding to various detection noises (parasitic light and read-out noise of the CCD camera). The signal-to-noise ratio of this record is about 24 and the atomic frequency is determined with a statistical uncertainty of about 17 kHz.\n\nThrough detection with the photomultiplier, it is possible to fit simultaneously the two spectra corresponding to an increase or a decrease in the BBO cavity length. Nevertheless it is preferable to fit the sum and the difference of the two curves, because, in the case of the difference, a large part of the noise due to the UV parasitic light is rejected. An example is shown on figure \\ref{fig:13}. The signal-to-noise ratio is about 16 for the fit of the sum of the curves and 13 for the difference. The statistical uncertainty of the atomic frequency $\\nu_{eg}$ is 18 kHz.\n\n\\section{Results}\n\\label{sec:5}\n\n\n\\begin{table*}\n\\caption{Features of the recordings for the different values of the magnetic field.}\n\\label{tab:1}      \n\\begin{tabular}{lrrrr}\n\\hline\\noalign{\\smallskip}\n Magnetic field & 0.029 mT & 16.01 mT & 17.12 mT & 19.15 mT  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nNumber of recordings & 39 & 19 & 66 & 28 \\\\\nMean frequency $\\nu_{eg}-2~922~742~930$ (MHz) &6.724(5)  &6.728(10) &6.721(6) & 6.707(10) \\\\\n$\\chi^2\/(n-1)$ & 0.69 & 1.03 & 1.21 & 0.56\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\nThe main results have been obtained by using the CCD camera. The two-photon transition has been observed for four values of the magnetic field, 0.029 mT, 16.01 mT, 17.12 mT and 19.15 mT. The first value corresponds to the residual earth magnetic field. The magnetic field is calibrated with an accuracy of about 0.01 mT by observing the Zeeman effect of the 1S$_{1\/2}$-3S$_{1\/2}(F=1,m_F=0)$ transition. Table \\ref{tab:1} gives, for each value of the magnetic field, the number of recordings $n$, the mean $\\nu_{eg}$ of the atomic frequencies obtained by fitting each recording with $\\sigma=1.6~{\\rm km\/s}$ and the values of $\\chi^2\/(n-1)$. For $B=0.029~{\\rm mT}$, the uncertainty of the frequency measurement is 5~kHz, {\\it i.e.} a relative uncertainty of $1.7 \\times 10^{-12}$. Unfortunately it is not the final accuracy, because the parameter $\\sigma$ of the velocity distribution has been fixed to $1.6~{\\rm km\/s}$.\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{%\n  \\includegraphics{Figure14-1S-3S-3.eps}\n}\n\\caption{Frequencies $\\nu_{eg}$ as a function of the parameter $\\sigma$ for four values of the magnetic field. The calculation has been made for 6 values of $\\sigma$ spanning between 1 km\/s and 2 km\/s (small dots). The full lines are the interpolations between these points and the dashed lines the error bars. The large dot is the best value for $\\sigma$ and $\\nu_{eg}$ with corresponding uncertainties (color online).\n}\n\\label{fig:14}      \n\\end{figure}\n\nTo determine $\\sigma$ the mean frequency $\\nu_{eg}$ is calculated for different values of $\\sigma$ (in the range 1 to 2~km\/s) to obtain four curves $\\nu_{eg}\\left(\\sigma,B\\right)$ (see figure \\ref{fig:14}). The values of $\\sigma$ and $\\nu_{eg}$ are obtained from the crossing of these four curves. With a least squares method, we obtain:\n\\begin{equation}\n\\sigma=1.646~(89)~{\\rm km\/s}\n\\label{sigma}\n\\end{equation}\n\\begin{equation}\n\\nu_{eg}=2~922~742~936.7275~(120)~{\\rm MHz}\n\\label{frequence}\n\\end{equation}\nThe corresponding point is indicated in figure \\ref{fig:14} (large dot); $\\nu_{eg}$ is the frequency of the 1S$_{1\/2}$-3S$_{1\/2}(F=1)$ two-photon transition for an atom at rest. The statistical uncertainty (12 kHz) corresponds to a relative uncertainty of $4.1 \\times 10^{-12}$.\n\nFor a thermal atomic beam one has $\\sigma =\\sqrt{kT\/M}$ (see section \\ref{sec:42}). The value of $\\sigma$ (equation (\\ref{sigma})) corresponds to a temperature of 328(35)~K which is compatible with a hydrogen atomic beam produced with a radio frequency discharge. This $\\sigma$ value is also in agreement with our previous measurements \\cite{Hagel}. In this work, we did not measure the absolute frequency of the 1S-3S transition. Nevertheless, by comparing measurements with and without magnetic field, we had observed the effect of the motional electric field and deduced a value for $\\sigma$ of 1.55(11)~km\/s. For this analysis, we had approximated the theoretical line shape by a Lorentzian shape. We have remade a complete analysis of these data following the procedure described in section \\ref{sec:4}. The result is $\\sigma=1.633(180)~{\\rm km\/s}$. This value is more reliable than the result given in reference  \\cite{Hagel} and in good agreement with the new result (equation (\\ref{sigma})). It is also possible to compare this result with the measurement of the velocity distribution of the metastable 2S atomic beam used to observe the 2S-$n$S and 2S-$n$D two-photon transition in hydrogen \\cite{EPJD00}. In these experiments, the metastable atomic beam was obtained by electronic excitation of a 1S atomic beam which had the same design than the atomic beam used in the present experiment. Because of the electron impact, there was an angle $\\theta=20{^{\\circ }}$ between the 1S and the 2S atomic beams and the metastable atoms were slowed with respect to the 1S atomic beam by a factor of about $cos(\\theta)$. Using Doppler spectroscopy of the Balmer-$\\alpha$ line, the $\\sigma$ value of the metastable atomic beam was 1.525(10)~km\/s \\cite{JPhys2}. Taking into account the factor $cos(\\theta)$, this corresponds to a value of 1.623(11)~km\/s. In spite of the roughness of this model, this value is also in perfect agreement with the present result.\n\n\\begin{table}\n\\caption{Error budget.}\n\\label{tab:2}      \n\\begin{tabular}{lr}\n\\hline\\noalign{\\smallskip}\nFrequency measurements &  $8\\times 0.3~{\\rm kHz}$ \\\\\nLight shift & $0.3~{\\rm kHz}$ \\\\\nPressure shift   & $1.2~{\\rm kHz}$\\\\\nVelocity distribution & $3.0~{\\rm kHz}$\\\\\nScan of the BBO cavity&$2.6~{\\rm kHz}$\\\\\nStatistic  & $12.0~{\\rm kHz}$\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nQuadratic sum  & $13.0~{\\rm kHz}$\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\nThe error budget is summarized in Table \\ref{tab:2}. The frequency measurements are made with respect to the 100~MHz reference signal from LNE-SYRTE. During the measurements of the 1S-3S transition, this frequency reference was obtained from a hydrogen maser numbered 40805 of LNE-SYRTE. Simultaneously to the 1S-3S measurement this standard was also compared to UTC-OP (Coordinated Universal Time-Observatoire de Paris) and linked to the SI (Syst\\`{e}me International). The result was:\n\\begin{equation}\n\\nu_{SYRTE}=\\nu_{SI}\\left[1+\\left(575 \\pm 6 \\right)\\times 10^{-15}\\right]\n\\label{freqsyrte}\n\\end{equation}\nThat produces a correction of $+1680(18)~{\\rm Hz}$ of the 1S-3S frequency. On the other hand, the Allan variance of the frequency measurement of the titanium sapphire laser (about $5\\times 10^{-13}$ in 1000 s \\cite{TheseArnoult}) corresponds to a statistical uncertainty of 180~Hz for a frequency measurement during a 20 minute run, conservatively rounded to 300~Hz in table \\ref{tab:2}.\n\nFollowing the notations of the reference \\cite{JPhys2}, the light shift $\\Delta \\nu_{ls}$ is:\n\\begin{equation}\n\\Delta \\nu_{ls}=\\left(\\frac{2a_0^2}{mc^2\\alpha}\\right)\\left(\\beta_e-\\beta_g\\right){\\rm ,}\n\\label{lightshift}\n\\end{equation}\nwhere $\\beta_g$ and $\\beta_e$ are the matrix elements of the light shift operators in atomic units for the level $g$ and $e$. Their values are: $\\beta_{1S}=-6.44539$ and $\\beta_{3S}=20.9264$ \\cite{Delande}. The UV power inside the atomic beam cavity is estimated at the maximum to 2~mW. This produces a light shift of 280~Hz, rounded to 300~Hz in table \\ref{tab:2}.\n\n The pressure in the vacuum apparatus is about $6\\times 10^{-5}$ torr. The collision shift of the 1S-3S transition is not known. Nevertheless it would be similar to the ones of the 2S-3P hydrogen transition because these shifts are dominated by the two upper levels of these transitions of which the wave functions have the same spatial size. The shift of the 2S-3P transition in He buffer gas has been measured by Weber {\\it et al.} to be about $-9~{\\rm MHz\/torr}$ \\cite{Weber}. Then, to estimate the uncertainty due to the pressure shift, we suppose an upper limit of $20~{\\rm MHz\/torr}$, corresponding to the value of $1.2~{\\rm kHz}$ in table \\ref{tab:2}.\n\n Several other effects can modify the frequency determination. In the theoretical analysis, we suppose that the two Zeeman sub levels 1S$_{1\/2}(F=1,m_F=\\pm1)$ are equally populated. Actually, for a magnetic field of 20~mT, the energy splitting between these two levels is about $10^{-4}$ times the thermal energy. That produces a small population difference and a frequency shift of about 4~Hz which is negligible. To test the sensitivity of the result to the form of the velocity distribution, an analysis of the data has been made with a different velocity distribution $f(v,\\sigma)=v^{4}\\exp(-v^{2}\/2\\sigma ^{2})$. The result of equation (\\ref{frequence}) is then shifted by $-3~{\\rm kHz}$. We have adopted this value as the upper limit for the uncertainty due to the velocity distribution. A last possible effect is a dissymmetry in the modulation of the BBO cavity, if the velocity of the mirror is not the same when the cavity length decreases or increases. The two Doppler shifts $\\delta_{decr}$ and $\\delta_{incr}$ due to the motion of the mirror cavity would not be exactly opposite. The maximum relative difference between these shifts is estimated to $2\\%$. The analysis of the data with this dissymmetry shifts the frequency of the 1S-3S transition by $2.6~{\\rm kHz}$. This value corresponds to the uncertainty labeled \"Scan of the BBO cavity\" in Table \\ref{tab:2}.\n\n Taking into account the statistical uncertainty ($12~{\\rm kHz}$), the final result is:\n\\begin{equation}\n\\nu\\left[{\\rm 1S-3S}(F=1)\\right]=2~922~742~936.729~(13)~{\\rm MHz}\n\\label{frequencefinal}\n\\end{equation}\nThe total uncertainty is $13~{\\rm kHz}$, {\\it i.e.} a relative uncertainty of $4.5\\times 10^{-12}$. After the measurement of the 1S-2S transition, this result is the most precise value of an optical frequency in hydrogen. The hyperfine structure of the 1S level is well known \\cite{shf1S} and that of the 3S level is evaluated from the Fermi formula with the Breit correction to be $52~609.4~{\\rm kHz}$. After correction of the hyperfine splitting, the frequency of the 1S-3S transition is:\n\\begin{equation}\n\\nu\\left({\\rm 1S-3S}\\right)=2~922~743~278.678~(13)~{\\rm MHz}\n\\label{frequencecorr}\n\\end{equation}\n\nUnfortunately, the accuracy of this result is not sufficient to improve the determination of the Rydberg constant. Several methods can be used to extract $R_{\\infty}$. The first one is to use the measurement of the charge distribution of the proton from electron scattering to calculate the Lamb shifts of the 1S and 3S levels. One then obtains for the Rydberg constant:\n\\begin{equation}\nR_{\\infty}=10~973~731.568~75~(18)~{\\rm m}^{-1}\n\\label{rydberg1}\n\\end{equation}\nFor the calculation of the Lamb shift, we have taken into account all the terms given in the CODATA report \\cite{codata06}. The fundamental constants used are also the values of CODATA, except for the fine structure constant. For that we use the recent result of Gabrielse \\cite{Gabrielse2008}. With a relative uncertainty of $1.7\\times 10^{-11}$, this result is in acceptable agreement with the value given by the CODATA adjustment of the fundamental constants: $R_{\\infty}=10~973~731.568~527~(73)~{\\rm m}^{-1}$.\n\nAs explained in section \\ref{intro}, another method to obtain the Rydberg constant is to compare the 1S-2S and 1S-3S frequencies by using the scaling law of the Lamb shifts \\cite{{Karshenboim},{Pachucki}}. The result is:\n\\begin{equation}\nR_{\\infty}=10~973~731.568~88~(68)~{\\rm m}^{-1}\n\\label{rydberg2}\n\\end{equation}\nThe relative uncertainty of this result is $6.2 \\times 10^{-11}$ and it is about ten times less precise than the CODATA value. The reason is that the relative accuracy of the 2S-3S transition which appears in this calculation is only $2.8 \\times 10^{-11}$. With this method, one also obtains a value for the radius of the charge distribution of the proton:\n\\begin{equation}\nr_{\\rm p}=0.911~(65)~{\\rm fm}\n\\label{proton}\n\\end{equation}\nThis value is in agreement with the result deduced from the electron scattering ($r_{\\rm p}=0.895~(18)~{\\rm fm}$) and from the adjustment of the data in hydrogen and deuterium ($r_{\\rm p}=0.8760~(78)~{\\rm fm}$).\n\n\\section{Conclusion}\n\\label{sec:6}\n\nTo conclude, the optical frequency of the 1S-3S transition has been measured for the first time by using a femtosecond frequency comb with a relative accuracy of $4.5\\times 10^{-12}$. It is the best measurement of an optical frequency in hydrogen after the one of the 1S-2S transition. The second-order Doppler effect has been determined from the observation of the motional Stark effect due to a transverse magnetic field. A careful theoretical analysis has been presented to describe the main features of the line shape. In spite of this high accuracy, this result does not improve significantly the determination of the Rydberg constant. For that, an accuracy of few $10^{-13}$ would be useful.\n\n Presently the experiment is mainly limited by the low intensity of the UV source at 205~nm. To circumvent this difficulty we plan to modify the UV source by replacing the two frequency doubling stages at 820~nm and 410~nm by a frequency sum of a UV source at 266~nm and a titanium sapphire laser at 896~nm. With this scheme, similar to the one used by Bergquist to obtain 194~nm \\cite{Bergquist}, the UV power would be increased by a factor of ten and the signal-to-noise significantly improved. Moreover it would be also possible to observe the 1S-4S transition in hydrogen which lies at 194.5~nm.\n\n The authors thank G. Hagel for his essential contribution to the first step of this experiment and P. Clad\\'{e} for fruitful discussions on the manuscript. They thank also O. Acef for the frequency measurements of DL\/Rb standard and they are indebted to the {\\it Service du Temps} of LNE-SYRTE laboratory for the time reference. This work was partially supported by the {\\it Bureau National de M\\'{e}trologie} (now {\\it Laboratoire National d'Essais}).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nSHORT CUTS\n\nINTRODUCTIONS TO FILM STUDIES\nOTHER TITLES IN THE SHORT CUTS SERIES\n\nTHE HORROR GENRE: FROM BEELZEBUB TO BLAIR WITCH Paul Wells\n\nTHE STAR SYSTEM: HOLLYWOOD'S PRODUCTION OF POPULAR IDENTITIES Paul McDonald\n\nSCIENCE FICTION CINEMA: FROM OUTERSPACE TO CYBERSPACE Geoff King and Tanya Krzywinska\n\nEARLY SOVIET CINEMA: INNOVATION, IDEOLOGY AND PROPAGANDA David Gillespie\n\nREADING HOLLYWOOD: SPACES AND MEANINGS IN AMERICAN FILM Deborah Thomas\n\nDISASTER MOVIES: THE CINEMA OF CATASTROPHE Stephen Keane\n\nTHE WESTERN GENRE: FROM LORDSBURG TO BIG WHISKEY John Saunders\n\nPSYCHOANALYSIS AND CINEMA: THE PLAY OF SHADOWS Vicky Lebeau\n\nCOSTUME AND CINEMA: DRESS CODES IN POPULAR FILM Sarah Street\n\nMISE-EN-SC\u00c8NE: FILM STYLE AND INTERPRETATION John Gibbs\n\nNEW CHINESE CINEMA: CHALLENGING REPRESENTATIONS Sheila Cornelius with Ian Haydn Smith\n\nSCENARIO: THE CRAFT OF SCREENWRITING Tudor Gates\n\nANIMATION: GENRE AND AUTHORSHIP Paul Wells\n\nWOMEN'S CINEMA: THE CONTESTED SCREEN Alison Butler\n\nBRITISH SOCIAL REALISM: FROM DOCUMENTARY TO BRIT GRIT Samantha Lay\n\nFILM EDITING: THE ART OF THE EXPRESSIVE Valerie Orpen\n\nAVANT-GARDE FILM: FORMS, THEMES AND PASSIONS Michael O'Pray\n\nPRODUCTION DESIGN: ARCHITECTS OF THE SCREEN Jane Barnwell\n\nNEW GERMAN CINEMA: IMAGES OF A GENERATION Julia Knight\n\nEARLY CINEMA: FROM FACTORY GATE TO DREAM FACTORY Simon Popple and Joe Kember\n\nMUSIC IN FILM: SOUNDTRACKS AND SYNERGY Pauline Reay\n\nFEMINIST FILM STUDIES: WRITING THE WOMAN INTO CINEMA Janet McCabe\n\nMELODRAMA: GENRE STYLE SENSIBILITY John Mercer and Martin Shingler\n\nFILM PERFORMANCE: FROM ACHIEVEMENT TO APPRECITATION Andrew Klevan\n\nNEW DIGITAL CINEMA: REINVENTING THE MOVING IMAGE Holly Willis\n\nFILM AND PHILOSOPHY: IMAGES OF WISDOM Adrian Page\n\nTHE MUSICAL: RACE, GENDER AND PERFORMANCE Susan Smith\n\nTEEN MOVIES: AMERICAN YOUTH ON SCREEN Timothy Shary\n\nTHE NEW HOLLYWOOD: FROM BONNIE AND CLYDE TO STAR WARS Peter Kr\u00e4mer\n\nDOCUMENTARY: THE MARGINS OF REALITY Paul Ward\n\nITALIAN NEOREALISM: REBUILDING THE CINEMATIC CITY Mark Shiel\nFILM NOIR\n\nFROM BERLIN TO SIN CITY\n\nMARK BOULD\n\nWALLFLOWER\n\nLONDON and NEW YORK\nA Wallflower Book\n\nPublished by\n\nColumbia University Press\n\nPublishers Since 1893\n\nNew York \u2022 Chichester, West Sussex\n\ncup.columbia.edu\n\nCopyright \u00a9 Mark Bould 2005\n\nAll rights reserved.\n\nE-ISBN 978-1-904-76450-2\n\nA complete CIP record is available from the Library of Congress\n\nISBN 978-1-904764-50-2 (pbk. : alk. paper)\n\nISBN 978-0-231-50391-4 (e-book)\n\nA Columbia University Press E-book.  \nCUP would be pleased to hear about your reading experience with this e-book at cup-ebook@columbia.edu.\nCONTENTS\n\nacknowledgments\n\na note on terminology\n\n on dangerous ground: introducing film noir\n\n1 the set-up: fabricating film noir\n\n2 out of the past: the prehistory of film noir\n\n3 dark passage: the main cycle of film noir\n\n4 against all odds: neo-noir\n\n afterword: kiss tomorrow goodbye\n\nfilmography\n\nbibliography\n\nindex\nACKNOWLEDGMENTS\n\nIt started out as a kind of joke, and then it wasn't funny anymore because money became involved. Deep down, nothing about money is funny.\n\n\u2013 Charles Willeford (2000: 3)\n\nWith thanks to Susan Alexander, Caroline Bainbridge, Emma Bircham, Anita Biressi, Andrew M. Butler, Istvan Csicsery-Ronay, Jr., Graham Fraser, Carl Freedman, Gillian Glitre, Joan Gordon, Veronica Hollinger, Rob Latham, Iris Luppa, China Mi\u00e9ville, Heather Nunn, Mike Sanders, Greg Tuck, friends and colleagues at BCUC and Historical Materialism, my family and, above all, Kathrina Glitre \u2013 their friendship, solicitude and kindnesses made the year before I wrote this book easier to bear. Additional thanks to Iris for helpful comments on Weimar cinema; Carl for his Walter Neff secret; Mike Harrison for 'Fat Michael Douglas, every woman's dream'; and Kathrina for making this a better book than it would otherwise have been. Thanks also, of course, to Glenn Ford, Gloria Grahame, Ida Lupino, Robert Mitchum, Dick Powell, Robert Ryan, Barbara Stanwyck, Claire Trevor, Richard Widmark...\nA NOTE ON TERMINOLOGY\n\nDeterminism argues that the state of a system at one moment gives rise to the state of that system in the following moment. Determinism should not be confused with fate, fatalism, cause-and-effect or predictability. Fate intrudes a metaphysics in which the entire history of a system, from which it is impossible to deviate, is laid down in advance. Fatalism is a resigned belief in this inescapable fate. Cause-and-effect is a narrative technique by which we make sense of the transition of a system from moment to moment. It is always a retrospective and partial account, an abstraction which marginalises or ignores the totality of the system. (It is, nonetheless, a useful tool for modelling the world and for telling it.) Predictability is the inverse of cause-and-effect. The ability to construct retrospective cause-and-effect chains implies that it should be possible to extend their construction into the future; this is an error based on forgetting that cause-and-effect is a retrospective abstraction. A determinist system does not require fate, inevitability, predictability or cause-and-effect. In the non-linear dynamics of complex systems, there is no necessary correspondence of magnitude between a microscopic fluctuation in a system and the macroscopic divergences it can produce in that system. This is not about a small cause having a large effect, but about the initial conditions of the entire system producing unforeseeable conditions in the entire system at a subsequent moment; it is about a sensitive dependence on initial conditions; it is about determinism without predictability. In complex systems, order can emerge from chaos, and chaos often contains deeply-encoded structures of order. (See Earman 1986, Hall 1992, Hayles 1990 and 1991, and Hoefer 2003.)\n\nThe point of all this will become clear.\nON DANGEROUS GROUND: INTRODUCING FILM NOIR\n\nOne day in 1993, Emmy Award-winning filmmaker Ara Chekmayan visited a Pennsylvania fleamarket, where he discovered a statuette that looked exactly like the Maltese Falcon. Chekmayan purchased the black bird for $8 and, not long afterward, believing it to be one of two identical props that had been used in the famous 1941 Warner Bros. movie, he offered it up for auction at Christie's, who estimated its value at $50,000. Before an auction could take place, however, a Los Angeles collector pointed out that identical copies of the statuette could be purchased at $45 a piece from a book dealer in Long Beach...\n\n\u2013 James Naremore (1998: 254)\n\nJames Naremore takes this anecdote to prove that certain 1940s Hollywood thrillers have accumulated sufficient 'artistic and cultural cachet' to become 'valuable as other things besides movies' (1998: 255). Looked at differently, it provides a key to the central problematic of identifying, delineating, defining film noir: this book opens by quoting from another book on film noir which retells a story from _People_ magazine about a man who found one of many copies based on the original (two copies of the) prop of the Maltese Falcon \u2013 which, diegetically, was a fake \u2013 in the third film to be based on Dashiell Hammett's 1931 hard-boiled detective novel _The Maltese Falcon,_ itself originally serialised in _Black Mask_ pulp magazine between September 1929 and January 1930. So what exactly was it that Chekmayan thought he had found when he thought he had found (one of) the original Falcon(s)? In this welter of copies of copies in different media and adaptations from one medium to another the notion of an original evanesces. Even Hammett is no guarantor. His cynical deflation of the Grail myth in the modern urban waste land is not the first (his 1931 _The Glass Key_ alludes to T. S. Eliot's 1922 'The Waste Land'), nor is his Sam Spade the original hard-boiled detective (a more likely contender is Carroll John Daly's Terry Mack).\n\nSimilarly, when we approach film noir, we are faced with neither an objectively-existing object out there in the world nor some ideal to which particular films more or less conform. Instead, as Naremore argues, film noir 'has less to do with a group of artefacts than with a discourse \u2013 a loose evolving system of arguments and readings that helps to shape commercial strategies and aesthetic ideologies' (1998: 11). Like any genre, film noir is an intersubjective discursive phenomenon: a fabrication. But as Dudley Andrew observes, 'A fabrication... is by no means a fiction' (1995: 12).\n\nAfter indicating the number of films that have been considered film noirs, this introduction suggests how that canon might be further expanded and considers one influential attempt to more rigidly delineate the genre, detailing some of the difficulties implicit in it. Marc Vernet notes that 'the Americans made [film noir] and then the French invented it' (1993: 1). Chapter one therefore surveys some of the Francophone criticism which first identified \u2013 or fabricated the idea of \u2013 film noir before outlining some of the early Anglophone statements about film noir, showing how the idea was taken up and developed prior to the academicisation of film studies in the 1970s. The purpose of this chapter is to demonstrate the historically-contingent discursive fabrication of the genre.\n\nIf the idea of film noir originated in mid-1940s Paris, the origins of the genre are typically traced back to four sources: German expressionism, French poetic realism, American hard-boiled fiction and American crime films (see Richardson 1992 on Italian neorealism as a neglected influence). Chapter two expands the consideration of interwar German cinema beyond expressionism, noting similarities with Hollywood's 1930s fallen-women cycle before turning to the work of Robert Siodmak and Fritz Lang, German directors who made numerous film noirs in exile. It then considers 1930s French cinema, contrasting several poetic realist films with American film noirs. It examines 1930s American crime films, particularly the gangster cycle \u2013 how the anachronistic figure of the 1930s gangster proved integral to the meaning of 1940s film noirs, and how the moratorium on producing gangster pictures produced alternative kinds of crime film \u2013 and three films, two of them directed by Lang, in which the increasing psychologisation of crime already apparent in hard-boiled fiction can be traced. In focusing on these filmic sources, the treatment of literary sources will remain cursory (on hard-boiled fiction, see Forter 2000, Madden 1968, Marling 1995, McCann 2000, Nolan 1985, O'Brien 1997, O'Connell 2002).\n\nChapter three offers detailed consideration of a variety of film noirs from the main cycle, which began with _Stranger on the Third Floor_ (1940) and _The Maltese Falcon_ (1941). Production peaked in the late 1940s and early 1950s and continued into the late 1950s, petering out with _Cape Fear_ (1962), _The Manchurian Candidate_ (1962), _Shock Corridor_ (1963) and _The Naked Kiss_ (1964). The films treated in this chapter were selected in order to explore the genre's recurring concern with notions of determinism, particularly the constitution and shaping \u2013 and simultaneous shattering and dissolution \u2013 of individual subjectivity. By examining two overlapping groups of film noirs \u2013 one dominated by images and ideas of entrapment, the other by images and ideas of investigation \u2013 this chapter does not argue that these are the primary concerns of the genre or even necessarily of these particular films, but rather explores how they are articulated in a significant proportion of film noirs. The construction of gender, particularly white masculinities, is a recurring concern throughout (on film noir and race, see Diawara 1993, Kaplan 1998b, Oliver and Trigo 2003).\n\nChapter four offers a brief overview of the development of neo-noir, culminating in a discussion of _Femme Fatale_ (2002), a film whose self-conscious combination of various elements of film noir, spectacle-dominated action movie and erotic thriller is typical of many neo-noirs (as are its homophobia, misogyny and racism), while its major plot twist combines a fantastic coincidence which rearticulates noirish determinism.\n\nFinally, the afterword considers _This Is Not a Love Song_ (2002) and _Sin City_ (2005) as digital remediations of film noir.\n\nThe inclusive filmography offered by Paul Duncan (2003) lists 1,028 film noirs:\n\nGerman expressionist films, 1920\u201333 | 5  \n---|---  \nAmerican precursors, 1927\u201339 | 26  \nFrench poetic realist films, 1931\u201343 | 8  \nAmerican film noirs, 1940\u201360 | 647  \nAmerican noir westerns, 1940\u201353 | 7  \nAmerican post-noirs, 1961\u201376 | 48  \nAmerican neo-noirs, 1976\u201392 | 167  \nFrench films, 1949\u201391 | 31  \nBritish films, 1927\u201391 | 71  \nItalian films, 1943\u201371 | 5  \nMexican films, 1949\u201371 | 3  \nJapanese films, 1962\u20132000 | 10\n\nFoster Hirsch (1999) lists a further 47 American neo-noirs released between 1993 and 1997, and Robin Buss (1994) a further 79 French film noirs between 1942 and 1992. Although Andrew Spicer (2002) only lists 538 films, this includes almost 200 titles not mentioned by these other authors, many of them among the 104 British film noirs he identifies. These figures illustrate two important points.\n\nFirst, that the main cycle of American film noirs is recorded in sufficient detail that, despite the debatable inclusion of many of the titles in these encyclopaedic listings, few more additions are likely to be unearthed, although some might arise from critically rethinking other genres, especially melodrama and, possibly, the western. Several westerns \u2013 _Pursued_ (1947), _Ramrod_ (1947), _High Noon_ (1952), _Rancho Notorious_ (1952), the five James Stewart westerns directed by Anthony Mann \u2013 have been described as noir westerns and many more from the 1950s feature badly disturbed protagonists or, like some scenes in _My Darling Clementine_ (1946), noirish lighting. Additionally, several film noirs utilise western imagery: Florian's bar in _Murder, My Sweet_ (1944) seems to be an only-slightly-redressed western saloon, complete with swinging doors and music that stops when Marlowe (Dick Powell) and Moose Malloy (Mike Mazurki) enter; in _High Sierra_ (1941) and _White Heat_ (1949) western costumes and settings anachronise their protagonists; in _Gun Crazy_ (1950) similar costumes expose the tendency to romanticise crime and outlaws even as, contradictorily, it depicts a passionate _amour fou._ Furthermore, the growing number of science fiction films which play on and with film noir conventions, from _Alphaville_ (1965) to _Cypher_ (2003), might encourage a re-examination of earlier science fiction films for traces of or affiliations to film noir.\n\nSecond, that outside of the main period of American film noir the terrain is still lacking any kind of consensus. There is still work to be done on film noirs before film noir, film noirs after film noir and film noirs in other national, linguistic and international contexts. For example, Michael Walker suggests that further research into 1930s crime films might produce 'more proto-noir films' (1992a: 33) like _Nancy Steele is Missing_ (1936), and James Naremore notes the critical neglect of the direct-to-video industry \u2013 in the mid-1990s, a 'seventeen-billion-dollar-a-year industry, involving more money than all the major studios combined' (1998: 161) \u2013 which produces numerous softcore erotic thrillers, perhaps the primary contemporary manifestation of film noirs about sexual obsession, blackmail and murder. Similarly, research into other national cinemas might uncover more films like the Japanese _Keisatsukan_ (1933), while discussions of neo-noir have yet to consider East Asian crime cinema in detail. Furthermore, if one accepts that film noirs are still being made, then each fresh example could potentially reshape the genre, narrowing or, more probably, widening it. Questions of omissions and additions inevitably return to questions of definition, and any attempt at definition restructures the genre, drawing in or casting out particular titles. It is through such complex feedback processes that genres form and reform.\n\nIn 1978, James Damico proposed a working model of film noir in terms of characters and plot structure so as to cut through the ill-discipline of encyclopaedic listings and the confusions they engender:\n\n> Either because he is fated to do so by chance, or because he has been hired for a job specifically associated with her, a man whose experience of life has left him sanguine and often bitter meets a not-innocent woman of similar outlook to whom he is sexually and fatally attracted. Through this attraction, either because the woman induces him to it or because it is the natural result of their relationship, the man comes to cheat, attempt to murder, or actually murder a second man to whom the woman is unhappily or unwillingly attached (generally he is her husband or lover), an act which often leads to the woman's betrayal of the protagonist, but which in any event brings about the sometimes metaphoric, but usually literal destruction of the woman, the man to whom she is attached, and frequently the protagonist himself. (1996: 137)\n\nThis narrative structure, familiar from James M. Cain's novels _The Postman Always Rings Twice_ (1934) and _Double Indemnity_ (1936), is found in the nine films, uncontroversially film noirs, on which Damico focuses: _Double Indemnity_ (1944), _The Woman in the Window_ (1945), _Scarlet Street_ (1945), _The Killers_ (1946), _The Postman Always Rings Twice_ (1946), _Out of the Past_ (1947), _The Lady from Shanghai_ (1948), _Pitfall_ (1948) and _Criss Cross_ (1949). Damico suggests that _Murder, My Sweet_ (1944), _The Strange Love of Martha Ivers_ (1946) and _The Blue Dahlia_ (1946) contain only 'slight variations', while _The Maltese Falcon_ anticipates key aspects and _In a Lonely Place_ (1950) contains 'apparent mutations and [a] collapsing of elements' (1996: 138).\n\nAlthough Damico recognises that there are 'a multitude of other correspondences to be evaluated which will perhaps delimit, broaden or even invalidate this provisional model', including 'the pervasive atmosphere of corruption, crime, psychopathology and evil; the constant resort to gratuitous violence; the omnipresence of the returning veteran; the importance of the oneiric in structure and substance; and recurrent visuals' (ibid.), he argues that it can be used to exclude certain films from the film noir canon, such as the postwar semi-documentary films which reconstructed actual crimes, pioneered by producer Louis de Rochemont and beginning with _The House on 92nd Street_ (1945). Damico also claims that his model enables an understanding of how, in other films, the 'elements have been altered and condensed or expanded' (1996: 139), but his example of giggling psychopath Tommy Udo (Richard Widmark) in _Kiss of Death_ (1947) as someone who combines 'the masculine and feminine qualities of the other man and the fatal woman' (ibid.) conveniently overlooks the film's indebtedness to the semi-documentary cycle.\n\nDamico's elaborations demonstrate a tension in any attempt to delimit a genre, best understood through the distinction between semantic and syntactic approaches to genre. Semantic approaches catalogue 'common traits, attitudes, characters, shots, locations' (Altman 1999: 219), and then list all the films which contain (at least some of) them. Despite generating exhaustive lists, this approach has little explanatory power. In contrast, syntactic approaches like Damico's model identify and explore 'certain constitutive relationships between undesignated and variable place-holders' (ibid.) in a handful of canonical texts seen as exemplifying the particular genre's core concerns. However, no sooner than Damico imposes a syntactic model he expands his initially exclusive list into a far more inclusive one, reinterpreting and reshaping excluded films so that they more closely approximate to his model and can therefore be readmitted into the genre. Indeed, this problem is evident even in the nine films he considers an unequivocal match for his model, as a description of _The Killers_ will show.\n\nTwo hitmen, Al (Charles McGraw) and Max (William Conrad), arrive in Brentwood, New Jersey, take over Henry's Diner and await the arrival of Swede (Burt Lancaster), who works at the gas station. When he fails to appear, the hitmen head for his boarding house. Warned of their approach, Swede, who has clearly been waiting for something like this to happen, refuses to do anything, explaining, 'I did something wrong \u2013 once'. He is killed. This opening sequence, a textbook example of film noir lighting and composition, is based on Ernest Hemingway's story 'The Killers' (1928), which ends at this point. The remainder of the film takes the form of a quest to uncover the reason for Swede's murder. Insurance investigator Riordan (Edmond O'Brien) is intrigued. In the first of the film's eleven flashbacks, he learns that a stranger \u2013 Colfax (Albert Dekker) \u2013 stopped for gas several days earlier, since when Swede had been laid up in his room, 'sick'. Riordan traces Swede's beneficiary, Queenie (Queenie Smith), a maid at an Atlantic City hotel where Swede stayed with a woman for several days in 1940, who recalls Swede's attempted suicide after the woman left him. Swede's former boxing career leads Riordan to Lieutenant Sam Lubinsky (Sam Levene), Swede's childhood friend who became a cop. Sam's first flashback is to the night of Swede's last fight, when he injured his hand so badly he had to quit boxing. Sam's wife, Lilly (Virginia Christine), who used to be Swede's girlfriend, introduces a flashback in which it becomes clear that Swede then became involved with criminals, falling for Kitty Collins (Ava Gardner), the mistress of the incarcerated Colfax. In Sam's second flashback he tells of arresting Swede when he insisted on taking the rap for Kitty. At Swede's funeral, Riordan meets Charleston (Vince Barnett), Swede's ex-cellmate. In a pair of flashbacks, he tells Riordan about Swede's obsession with Kitty, and how he introduced Swede to Colfax; Charleston refused to become involved in the heist Colfax was planning, but Swede, lured by Kitty's presence, agreed. The eighth flashback is narrated by Riordan's boss, who reads a newspaper account of the 1940 Prentiss Hat Company robbery which Riordan has linked to Swede. Sam summons Riordan to a hospital where Blinky (Jeff Corey), one of Colfax's gang, is dying of gunshot wounds. Delirious, he introduces two flashbacks: in the first, on the eve of the robbery, Swede threatens Colfax; in the second, Swede, claiming that he has been double-crossed, arrives at the altered rendezvous after the robbery and takes all the money. Riordan returns to Swede's room in Brentwood where Dum-Dum (Jack Lambert), the gang member who killed Blinky, is searching for a clue as to where Swede hid the money. After a scuffle, Dum-Dum escapes. Riordan questions Colfax, now an apparently respectable businessman, who denies all knowledge of the robbery. Riordan arranges a meeting with Kitty. She introduces the final flashback, in which she convinces Swede the gang are double-crossing him so as to get him to double-cross them. It was, however, all a set-up, a complex double-cross planned by Colfax, to whom Kitty is now married. Back in the present, Sam kills the hitmen as they try to gun down Riordan, but Kitty escapes. Riordan, Sam and the police arrive at Colfax's house to find Dum-Dum dead and Colfax dying. Colfax, who stumbled across Swede, killed him so that no-one from the gang might find him and thus work out that Colfax and Kitty actually stole the money. Kitty arrives, but Colfax refuses to falsely clear her name before he dies.\n\nThis account demonstrates the extent of the violence Damico must perform on _The Killers_ for it to conform to his model. If the film, shot by Woody Bredell, is a virtual inventory of film noir's low-key, expressionist cinematography, it is also a compendium of film noir plots. Walker suggests that it forms, along with _Criss Cross_ and _The File on Thelma Jordan_ (1950), Siodmak's trilogy 'dealing with \"the mystery of woman\"' (1992b: 151), and Damico's narrative structure _can_ be found in the flashback story, albeit with deviations. Although Kitty does induce Swede to cheat Colfax, it is only as part of a larger scheme with Colfax, and there is little to suggest that Kitty is 'unhappily or unwillingly attached' to Colfax. This triangular relationship is simultaneously at the centre of the film and marginalised because the narrative foregrounds Riordan's investigation. Kitty's femme fatale and Swede's victim-hero, character types at the centre of Damico's model, are subordinated and subjected to Riordan's seeker-hero narrative (on these character types, see Walker 1992a). The film might then be better understood as a male investigation of the mystery of a man, lending it a homoerotic frisson.\n\nThe film was initially perceived as a gangster film \u2013 Siodmak retrospectively regarded it as such \u2013 and Walker suggests that it can be also seen, with _Criss Cross_ and _Cry of the City_ (1948), as part of Siodmak's gangster trilogy (1992b: 128). It is also, like _Body and Soul_ (1947), _Champion_ (1949), _The Set-Up_ (1949), _Second Chance_ (1953), _Killer's Kiss_ (1955) and _The Harder They Fall_ (1956), a boxing film noir. Swede's attempted double-cross is a shorthand form of the revenge against the mob plot which, despite appearing in film noirs like _The Big Heat_ (1953) and _Underworld USA_ (1961), did not come to fruition until _Point Blank_ (1967) and _Get Carter_ (1970). Swede and Kitty are briefly a couple on the run, like those in _They Live By Night_ (1949) and _Gun Crazy;_ and like _The Asphalt Jungle_ (1950), _The Killing_ (1956) and _Odds Against Tomorrow_ (1959), _The Killers_ also contains a heist. This robbery sequence, filmed in a single two-minute crane-shot, seems to belong in one of the semi-documentary crime films Damico wishes to exclude from the genre, an affiliation made stronger by the accompanying voice-over narration whose phrasing, tone, selection of detail and apparently objective omniscience resembles the semi-documentary's 'official account' voice. As these other elements suggest, Damico seriously reduces and misdescribes _The Killers_ in order to cite it as exemplary of his model.\n\nA further problem arises with this model when _The Killers_ is compared to Don Siegel's 1964 remake. Two hitmen, Charlie (Lee Marvin) and Lee (Clu Gulager), track down Johnny North (John Cassavetes), an ex-racing driver who was involved in a million-dollar robbery some years earlier. Johnny's resignation puzzles the ageing Charlie, who wonders why he made no attempt to flee his killers, concluding that the only person who would pay to have him killed without trying to discover what happened to the stolen money would be the person who actually stole it. Hoping to retire, Charlie persuades Lee to help him track down whoever contracted the hit. They question Earl (Claude Akins), Johnny's old mechanic. In the film's first flashback, Earl tells how Johnny fell for Sheila Farr (Angie Dickinson), the mistress of gangster Jack Browning (Ronald Reagan). Johnny and Sheila plan to marry, but when a car crash leaves him unable to continue racing, he sends her away, believing he has been just another of her casual infidelities with physically active younger men. Charlie and Lee then track down Mickey Farmer (Norman Fell), an associate of Browning, who in flashback picks up the story: Sheila persuades Browning that they need a specialist driver for a heist he is planning and urges him to recruit Johnny; after the heist, Johnny stole the money from Browning and disappeared. When the hitmen trace Browning, he appears to be a legitimate businessman. They make him arrange a meeting with Sheila. She stonewalls them at first, but breaks down when they threaten to kill her. In the third flashback, narrated by Sheila, she convinces Johnny to rob Browning and run off with her. However, it was all part of a scheme for her and Browning to get the money and leave the rest of the gang thinking that Johnny had it. As they leave the hotel to retrieve the money, a sniper kills Lee and injures Charlie. Sheila escapes and Browning \u2013 the sniper \u2013 joins her at their house. As they empty the safe, Charlie, blood dripping from his wound, catches up with them. He kills Browning. When Sheila pleads for her life, Charlie replies, 'Lady, I don't have the time', and shoots her. He stumbles from the house with the money and collapses, dead.\n\nFrom this description, the remake might seem considerably more noirish than the original. The investigation is conducted by a pair of professional killers rather than official or semi-official detectives, one world-weary and cold, the other narcissistic and brutal. They are motivated by curiosity, cash and, one suspects, a need to do something between contracts; and they both die. While Kitty remains an enigma primarily because she is largely absent from a film whose entire plot hinges on her manipulation of Swede, Sheila remains an enigma because, hidden behind mask-like makeup, she is the contradictory product of an experiment in perspectivism. In the first flashback, she seems genuinely to fall for Johnny, telling him she loves him and intends to stand by him after his accident. In the second flashback, the audience is manipulated into believing she has been searching for Johnny because she still has feelings for him. When she finds him, Browning spies on her through binoculars, and only after she offers to find Johnny some well-paid work does she set about persuading Browning to employ him. The first flashback contained scenes Earl did not witness but convention implies their objectivity, a sense reinforced by showing Sheila, whom Earl did not like, in a positive light (it is Johnny who rejects Sheila, not vice versa). Consequently, we are lured into accepting the apparent objectivity of the second flashback. It is only in the third flashback that we realise Mickey's account of the heist is actually an account of what he _thought_ had happened; but to the extent that we accept it as objective while it unfolds, Sheila's betrayal of Johnny seems all the more devastating.\n\nSiodmak's version contrasts the noir world with a more respectable existence exemplified by the Lubinskys' roof terrace, itself an extension of the contrast between the femme fatale Kitty and the domestic Lilly. This is ironised by the artifice of the rooftop idyll. The well-lit set is obviously a set and although it is above the noir world, it is a cramped space, confined by roof tops and chimney pots \u2013 but nonetheless it endures. After his final fight, Swede has the option of attaining this domestic idyll. Lilly was still his girlfriend and Sam offered to get him a job in the police with a steady income and a pension, but Swede sent Lilly home and parted company with Sam, turning down a noirishly-lit dark alley. (The more-than-passing facial resemblance between Swede and Riordan further emphasises this sense of alternative paths taken into alternative worlds.) Siegel's version, which was made for television but released in cinemas, offers no such contrast. It is brightly- and uniformly-lit, conventionally-shot and in colour. Killers kill with impunity and law-enforcement agents are completely absent, appearing only as the sound of approaching sirens at the end. Arguably, the noir world has become ubiquitous and normalised, rendering a stylistically distinct representation superfluous or impossible. However, because it rejects film noir's visual style, the consensus is that the remake, despite being closer to Damico's model, is the less noirish version. To the extent that this is a reasonable judgement, film noirs cannot be defined solely in terms of a narrative structure.\n\nWhat Damico offers is a narrative formula which recurs in some form in a significant number of film noirs, but which cannot be regarded as exclusive to film noirs unless one is prepared to include all films that follow the formula in the genre and to exclude all those which do not. Hammett's _Red Harvest_ (1929) and its film adaptations demonstrate the problems with this approach. The novel, featuring an unnamed detective, closely matches the narrative structure of the classic western, exemplified by _Shane_ (1953), in which a 'lone stranger... rides into a troubled town and cleans it up' (Wright 1977: 32), but it has a contemporary setting and a jaded, cynical tone. Although there is no difficulty in counting one of its adaptations, _Per un pugno di dollari_ (1964), as a western, the novel itself and the two other adaptations, the samurai film _Yojimbo_ (1961) and the gangster film _Last Man Standing_ (1996), are rather more problematic. Clearly there are ways in which they can be regarded as westerns, but in doing so one must recognise that genres are rather more fluid than Damico allows and that films have multiple generic tendencies.\n\nIn addition, then, to sharing several sometimes overlapping and interacting narrative (and thematic) structures, film noirs also share elements of a distinctive visual style. Low-key lighting and the positioning of key-, fill- and back-lighting produced patterns of light and dark which were, by Hollywood standards, unconventional. Increased depth of field made shots more ambiguous and, particularly during night-for-night shooting, required wide-angle lenses which also distorted foreground figures. Unbalanced and disharmonious compositions introduced tension into the _mise-en-sc\u00e8ne,_ subjectivising the objective third-person camera by shaping the diegesis to express the conflicts within and between characters. (On film noir style, see Bordwell, Staiger and Thompson 1988, Place and Peterson 1974; on industrial and economic determinants of this style, see Kerr 1996, Lev 2003, Maltby 2003, Schatz 1997.)\n\nHowever, despite the importance of visual style to film noirs, the claim made by several critics that film noir is a style rather than a genre seems as untenable as the claims made for a specific narrative (or thematic) structure. Rather, film noirs emerge from (discussions about) the interactions of style, narrative and theme. Therefore, the solution this book adopts to the problem of defining film noir \u2013 it is, of course, no solution \u2013 is to avoid suggesting that a genre can be defined by a single paradigm. Rather it will explore the genre from several angles and attempt to represent as much of the narrative, thematic, stylistic and temporal range of film noir as possible.\n1 THE SET-UP: FABRICATING FILM NOIR\n\nTalk is dangerous. Sometimes it makes things happen, it makes them real.\n\n\u2013 _Body Heat_ (1981)\n\nFilm noir, like the femme fatale, is an elusive phenomenon: a projection of desire, always just out of reach. The task of delineating and circumscribing film noir, of pinning it down, frequently recalls the convoluted constructions of identity around a central absence in _Vertigo_ (1958). Scottie (James Stewart), who retired from the police because of his vertigo, is hired by Gavin Elster (Tom Helmore) to investigate his wife, Madeleine (Kim Novak). She is possessed, apparently, by the spirit of her grandmother, Carlotta, who was driven to despair and suicide. When his vertigo keeps him from preventing Madeleine's suicide, Scottie has a breakdown. A year later, he meets Judy Barton (Kim Novak), whom he attempts to reshape into the image of Madeleine. Unknown to him, Gavin had hired Judy to pretend to be Madeleine in order to enable him to murder the real Madeleine. This information is revealed to the audience as soon as Scottie meets Judy \u2013 he only realises the truth after he has recreated (the false) Madeleine \u2013 and completely alters one's understanding of what we have already seen.\n\nAbout halfway through the film, the false-Madeleine wakes up in Scottie's apartment, having been rescued by him after her (fake) suicide attempt. As they talk for the first time, a mutual attraction is signalled: but to what are they attracted and with whom are they falling in love? Judy, pretending to be Madeleine (who has been possessed by Carlotta), falls for Scottie, who is pretending that he has not been hired by Gavin and following her around all day, while Judy, pretending to be Madeleine, is pretending not to know that Scottie has been hired by Gavin and following her around all day. Infinite regress threatens. Similarly, when Scottie reshapes Judy as Madeleine, of what does his model actually consist? Madeleine? Judy as Madeleine? Judy as her idea of Madeleine? Judy as Gavin's idea of Madeleine? It is fitting, then, that the film opens with vortices \u2013 vaginal images, elaborate structures around empty cores \u2013 and closes with Scottie, atop a phallic tower, confronting an abyss. Identity, _Vertigo_ suggests, remains a mystery, an aporia around which can be found prosaic realities (Judy is from Selina, Kansas and can prove it) and elusive trails (Madeleine is an absence, seen only once, in flashback, already dead). Likewise, film noir.\n\nIt is customary to acknowledge that film noir was a retrospectively-applied generic label; consequently, no-one could ever have set out to make a film noir because the signifier (and thus what it signified) had not entered into English usage at the time the films now called film noirs were actually made. Suggestions that this might invalidate film noir's generic status tend to overlook the fact that the introduction and circulation of a generic label for a group of related texts _must_ come after the creation of those texts, that generic labelling must always be, at least initially, retrospective. For example, Charles Musser situates _The Great Train Robbery_ (1903) \u2013 the so-called first western \u2013 within the railway sub-genre of the then popular travel genre of films, arguing that director Edwin S. Porter 'was consciously working (and cinema patrons viewing) within a framework' derived from 'the violent crime genre which had been imported from England a few months earlier', and that because it was not 'primarily perceived in the context of the western' its success 'did not encourage other westerns but other films of crime' (1990: 130\u20131).\n\nWhile Steve Neale is correct to be concerned about the homogenising effect of imposing film noir as a generic category, this is common to all applications of generic terminology. Similarly, although Neale's willingness to concede that 'film noir now has a generic status it originally did not possess' (2000: 3) seems to imply that this is unique to film noir, it is true of any genre: one cannot now stop seeing _The Great Train Robbery_ as a western even though in 1903 the genre 'had not yet been established effectively in the cinema' (Musser 1990: 131) and even if one is aware of its other generic tendencies and affiliations. Genres arise \u2013 or, more accurately, are identified, named and developed \u2013 through complex feedback mechanisms involving producers, distributors, exhibitors, consumers, interpreters and other discursive agents. This chapter examines how film noir emerged as an intersubjective discursive phenomenon in French and Anglo-American criticism.\n\nThe term 'film noir' is said to derive from _S\u00e9rie noire,_ the title of a series of crime novels edited by Marcel Duhamel for French publisher Gallimard, starting in 1945. Following translations of three novels by Peter Cheyney and James Hadley Chase (British writers imitating American models), the series began to translate novels by American writers like Raymond Chandler, Horace McCoy, Dashiell Hammett, W. R. Burnett, William P. McGivern, Cornell Woolrich, David Goodis and Chester Himes (from 1948, French authors were published under English pseudonyms and, after 1951, under their own names).\n\nThe term 'noir' was used in France before the Second World War, usually in the right-wing press to derogate left-wing culture (see O'Brien 1996), and some late 1930s films were described as 'film noirs' in the 1940s; but neither term was applied to American films until 1946. Following the liberation from German occupation, large numbers of American films were released in France. In 1946, Nino Frank linked _Double Indemnity, The Maltese Falcon, Murder, My Sweet_ and _Laura_ (1944) to the revolution in American crime fiction started by Hammett. Whereas previous fictional detectives, like C. Auguste Dupin, Philo Vance and Ellery Queen, were little more than perfectly functioning ratiocination devices, Hammett's Continental Op and Sam Spade, as well as Chandler's Philip Marlowe, were flawed characters. This difference, Frank argued, was manifest in the films' emphasis on character psychology rather than the investigation and retrospective reconstruction of particular crimes. In these film noirs, as he dubbed them, the crime film was psychologised by first-person narration and closely-observed facial expressions, gestures and dialogue.\n\nLater that year, Jean-Pierre Chartier called _Double Indemnity, Murder, My Sweet, The Postman Always Rings Twice_ and _The Lost Weekend_ (1945) film noirs, suggesting they were so dark that French films like _Quai des brumes_ (1938) and _H\u00f4tel du nord_ (1938) could no longer really be considered as film noirs. These pessimistic, misanthropic American films were driven by a logic of sexual desire that the Production Code Administration (PCA) simultaneously required them to suppress. This pattern of desire and repression in the characters, matched by the filmmakers' double-coding, rendered the crime itself the object of the protagonists' erotic fascination, further psychologising the crime film. In 1948 Henri-Fran\u00e7ois Rey, who regarded American cinema as propagandist, suggested that _Double Indemnity, The Lost Weekend, Scarlet Street_ and _The Woman in the Window_ presented views of the US so unflattering and despairing as to require special comment. In 1951, Pierre Duvillars considered the centrality of a new version of the vamp to _The Postman Always Rings Twice, Murder, My Sweet, Double Indemnity_ and _Criss Cross._ A figure 'accommodated to contemporary taste, itself composed of cynicism, sadism and morbidness' (1996: 30), she reduces the male protagonist to a hypnotised, machinelike creature through the calculated but, he suggests, never consummated promise of sex. In their survey of American film noir originally published in 1955, Raymonde Borde and Etienne Chaumeton list 22 film noirs, from _The Maltese Falcon_ to _Macao_ (1951). A further 62 appear in related categories: 29 about criminal psychology and ten about social trends, seven costume crime films, six gangster films and ten documentary police thrillers. Except for the costume films, each category contains films now generally considered film noirs, whereas their list of film noirs includes _The Mask of Dimitrios_ (1944), _Notorious_ (1946), _Chicago Deadline_ (1949) and _The Window_ (1949), none of which are now deemed central to the genre.\n\nJames Naremore argues that the growing Americanism in postwar French culture and nostalgia for their pre-war cinema predisposed the French to discover or invent American film noir, and because of their affiliations with either surrealism or existentialism these early critics constructed it in particular ways. For surrealist aficionados like Borde and Chaumeton,\n\n> the essence of noirness lies in a feeling of discontinuity, an intermingling of social realism and oneiricism, an anarcho-leftist critique of bourgeois ideology and an eroticised treatment of violence. Above all, noir produces a psychological and moral disorientation, an inversion of capitalist and puritan values, as if it were pushing the American system toward revolutionary destruction. We might debate whether such qualities are in fact essential to the Hollywood thriller... but there is no question that they are fundamental to surrealist art. (Naremore 1998: 22)\n\nFor existentialist critics, film noirs 'depicted a world of obsessive return, dark corners or _huis-clos'_ (ibid.). Where 'perversely anarchic' surrealists saw 'a theatre of cruelty', existentialists found a 'despairingly humanist' protoabsurdism (ibid.). This existentialist criticism can be traced through Jean-Paul Sartre to Andr\u00e9 Bazin, co-founder of _Cahiers du Cin\u00e9ma,_ a journal which carried essays by several critics who would go on to become New Wave filmmakers, including Claude Chabrol's '\u00c9volution du film policier' (1955). It was at this juncture that\n\n> the terms _film noir_ and _auteur_ began to work in tandem, expressing the same values from different angles... Film noir was a collective style operating within and against the Hollywood system; and the auteur was an individual stylist who achieved freedom over the studio through existential choice. But the auteur was more important than the genre... the _Cahiers_ group always subordinated general forms to personal visions. (Naremore 1998: 26\u20137)\n\nAnd sure enough, two _Cahiers_ critics soon made films \u2013 Jean-Luc Godard's _\u00c0 bout de souffle_ (1960) and Fran\u00e7ois Truffaut's _Tirez sur la pianiste_ (1960) \u2013 redolent of film noir but conceived as personal visions.\n\nIn 1945, American critic Lloyd Shearer described a recent 'trend in Hollywood toward the wholesale production of lusty, hard-boiled, gut-and-gore crime stories, all fashioned on a theme with a combination of plausibly motivated murder and studded with high-powered Freudian implication' (1999: 9), consisting of _Double Indemnity, Murder, My Sweet, Laura, Conflict_ (1945) and the forthcoming _The Dark Corner_ (1946), _The Big Sleep_ (1946), _The Brasher Dubloon_ (1947), _The Postman Always Rings Twice, Lady in the Lake_ (1947), _The Blue Dahlia_ and _Serenade_ (an adaptation of Cain's 1937 novel which eventually appeared in 1956). He explains this trend in terms of a turn to realism, liberalisation at the PCA, imitations of successful films being rushed into production and their cathartic acting-out of suppressed oedipal drives. More significantly, Lloyd Shearer and, a year later, Siegfried Kracauer, identified a cycle more-or-less consonant with film noir, even if neither of them gave it that name.\n\nThe first substantial Anglophone treatment of film noir came in Charles Higham and Joel Greenberg's _Hollywood in the Forties_ from 1968. Their book remains instructive, enabling the reader to see a genre being fabricated. Their chapter entitled 'Black Cinema' evokes a film noir iconography of rain-drenched nocturnal streets, trains, elevators, cocktail bars, knocked-over standard lamps, interrogation rooms, heels clicking on pavements. It attests the impact of immigrant Austro-German filmmakers, of certain cinematographers (Lee Garmes, Tony Gaudio, Lucien Ballard, Sol Polito, Ernest Haller, James Wong Howe, John F. Seitz) and composers (Franz Waxman, Max Steiner, Mikl\u00f3s R\u00f3zsa, Erich Korngold). (While much has been written on film noir's visual style, its musical component remains neglected; but see Porfirio 1999 and 2001.)\n\nHowever, auteurism soon replaces this discussion of genre, iconography and other creative personnel. What now seems peculiar about their selection of directors \u2013 primarily Siodmak, Fritz Lang, Otto Preminger, Michael Curtiz, Billy Wilder \u2013 is the absolute pre-eminence afforded Alfred Hitchcock for _Shadow of a Doubt_ (1943), _Rope_ (1948) and _The Paradine Case_ (1948), only the first of which might now be considered in any way central to film noir. This change of perspective can probably be explained in terms of the parallel canonisations of Hitchcock as auteur and film noir as genre. Neither now needs the kudos the other might lend, the status of each enhanced by not necessarily being connected to the other. (This, of course, produces anomalies. For example, in many respects Hitchcock's _Vertigo_ is a quintessential film noir. Based on a French pulp novel, it features an unofficial investigator investigating a mysterious woman, a carefully-orchestrated murder, a voice-over and a flashback, San Francisco as a subjective maze, expressionistic flourishes and traces of gothic melodrama; and it has been profoundly influential on neo-noir, particularly the erotic thriller. However, it is typically treated as tangential to film noir, as if auteurist discourses, combined with the industrial and aesthetic choices involved in making a colour VistaVision A-picture, categorically outweigh generic affiliations.)\n\nHigham and Greenberg's auteurism is further demonstrated when they discuss 1940s period melodramas (see Barefoot 2001): _King's Row_ (1941), _The Lodger_ (1944), _Gaslight_ (1944), _Hangover Square_ (1945), _Temptation_ (1946), _Ivy_ (1947) and _So Evil My Love_ (1948) 'reflect the proper ambience but for the most part fail to disclose the kind of strong personal attitude which could have raised them to the level of works of art' (1970: 31). It is not insignificant that the studios, as well as 1940s and 1960s audiences, would have perceived them as 'women's pictures', or that detailed treatment is reserved for the one directed by George Cukor, who did appear in lists of possible auteurs, usually to be rejected.\n\nIntriguingly, Higham and Greenberg deal with many film noirs in other chapters. While many films that would now be considered melodramas are treated as 'women's pictures', the 'Melodrama' chapter focuses almost exclusively on film noirs. Beginning with the claim that 'A wry detachment, an amused view of the subject... are the qualities of the best 1940s melodramas. The films were made by hard-bitten men who knew city life inside out' (1970: 39), it examines various film noirs, including adaptations of Hammett, Chandler and Graham Greene, _Gilda_ (1946), _Sorry, Wrong Number_ (1948), _The Big Clock_ (1948), _Caught_ (1949), _The Reckless Moment_ (1949), _Force of Evil_ (1949) and _The Set-Up_ as well as Hitchcock's _Rebecca_ (1940), _Suspicion_ (1941) and _Notorious._ However, Higham and Greenberg acknowledge no similarities between these films and the preceding chapter's 'black' films and women-in-peril melodramas, and they discuss other film noirs as 'problem and sociological films' and 'women's pictures'. (Higham and Greenberg's construction of 1940s Hollywood genres so as to privilege certain directors betrays the masculinist assumptions of a nascent Film Studies. Frank Krutnik (1991) offers a contrasting approach which is informative about the development of Film Studies in the intervening decades. Not only does he recognise gender problematics, but he also divides film noirs into more coherent, if overlapping and interacting, cycles of 1940s crime films with tangible connections to their production and initial distribution, exhibition and consumption.)\n\nRaymond Durgnat's 'Paint It Black: The Family Tree of _Film Noir'_ from 1970 begins by trying to elevate film noir to high art, comparable to Euripides, Goya, Dostoyevsky, Faulkner, Greek tragedy, Jacobean drama, Romanticism and, less pretentiously, to French, Italian and British cinema. He insists that film noir should be classified by 'motif and tone' (1996: 84) and is, therefore, not a genre in the way that gangster films and westerns are. While he is correct to say that not all crime films are film noirs, his suggestion that film noirs can be found in other genres is undermined by his examples, which include _King Kong_ (1933) and _2001: A Space Odyssey_ (1968). While few would deny the potential benefits of considering _Der Blaue Engel_ (1930), _Attack_ (1956) and _Sweet Smell of Success_ (1957) in the light of film noir, he offers no coherent reason for claiming them for the genre; and his subsequent typology\/genealogy of film noir focuses almost exclusively on crime films, effectively countering his grander claims.\n\nDurgnat divides three hundred film noirs by cycle or motif into eleven incommensurate categories. Social criticism films examine issues like prohibition-era gangsters, miscarried or corrupt justice, juvenile delinquency, boxing and other rackets, and include postwar documentary-thrillers and more general indictments of American society, like _Ace in the Hole_ (1951). The gangster cycles include films nostalgic for the 1930s gangster, films in which gangsters become heroes fighting Nazis or Communists, heist and caper thrillers, and films about organised crime. Other groups of films feature: criminals or innocents on the run; private eyes and adventurers; middle-class murder; portraits and doubles as symbols of paranoia and split personalities; straight and queer sexual pathologies; psychopaths; criminals holding individuals, families or other groups hostage; and Nazis or Communists portrayed as gangsters. Durgnat's final category contains guignol, horror and fantasy films about paranoia, entrapment, death, desire and alienation. For all its faults, this early charting of noir's terrain offers insights into the matrix from which current understandings of film noir grew.\n\nPaul Schrader's 1972 'Notes on Film Noir' reiterates Durgnat's contention that film noir is not a genre because it 'is not defined, as are the western and gangster genres, by conventions of setting and conflict but rather by the more subtle qualities of tone and mood' (1996: 99). However, while arguing that 'Film noir is also a specific period of film history, like German Expressionism or the French New Wave' (ibid.), he describes it in terms of setting and conflict as 'Hollywood films of the 1940s and early 1950s which portrayed the world of dark, slick city streets, crime and corruption' (1996: 100). His claim that 'most every dramatic Hollywood film from 1941 to 1953 contains some noir elements' (ibid.) is unsupported by anything resembling evidence, but it does return us to the dilemma of how to delimit a genre, to the question of semantic and syntactic approaches. In Schrader's words, 'How many noir elements does it take to make a film noir?' (ibid.). Side-stepping this question, he identifies wartime and postwar disillusion, a postwar interest in realism, the influence of German and East European \u00e9migr\u00e9 filmmakers and hard-boiled crime fiction as the four circumstances which prompted film noir production in 1940s Hollywood. He also points to some recurring stylistic techniques: lighting even daytime scenes as if it were night-time; compositional preferences for vertical and oblique to horizontal lines and within-the-frame tensions to physical action; equal lighting emphasis for both actors and set; romantic narration; complex chronologies; and 'an almost Freudian attachment to water' (1996: 104).\n\nSchrader divides the main film noir cycle into three phases. The first (1941\u201346) centred on hard-boiled private-eye adaptations, featured star couples like Humphrey Bogart and Lauren Bacall, Alan Ladd and Veronica Lake, attracted studio-favoured directors like Michael Curtiz and Tay Garnett and was often studio-bound. The second (1945\u201349), dominated by a realist trend, featured 'less romantic leads like Richard Conte, Burt Lancaster and Charles McGraw' and favoured 'proletarian directors like [Henry] Hathaway, [Jules] Dassin and [Elia] Kazan' (1996: 106). The third (1949\u201353) was dominated by 'psychotic action and suicidal impulse', by actors like James Cagney, Robert Ryan and Lee Marvin and 'psychoanalytically-inclined directors like [Nicholas] Ray and [Raoul] Walsh' (ibid.). While evocative, this oversimplified trajectory is potentially misleading. As Schrader's introductory paragraphs suggest, part of his interest in 'such relentlessly cynical' film noirs as _Kiss Tomorrow Goodbye_ (1950) and _Kiss Me Deadly_ (1955) is that they enable him to fabricate a narrative about Hollywood which allows him to dismiss _Easy Rider_ (1969) and _Medium Cool_ (1969) as na\u00efve, romantic 'self-hate cinema' (1996: 99).\n\nThis is a far from full account of the first thirty years of film noir criticism. The massive exfoliation of writing on film noir since then makes it impossible to account for the second thirty years here. Instead, this chapter concludes with brief comments on key Anglophone books on film noir published since the 1970s.\n\nThe single most important intervention came with E. Ann Kaplan's edited collection _Women in Film Noir,_ originally published in 1978. It treated film noir as being familiar enough to be analysed and criticised rather than described, and Christine Gledhill's contributions on _Klute_ (1971) served notice that film noir criticism would no longer be constrained to the major cycle. The collection as a whole \u2013 an intersection of feminism, Marxism, Lacanian psychoanalysis and (post-)structuralism \u2013 was at the very centre of the theoretical developments then shaping Film Studies. The 1998 second edition added chapters on sexuality and race, but seemed belated, emphasising the extent to which the first edition was establishing, not following, trends. (Oliver and Trigo (2003) offers more detailed treatment of film noir's anxieties about race and sexuality.)\n\nJ. P. Telotte's _Voices in the Dark: The Narrative Patterns of Film Noir,_ a careful examination of the specific conventions of the genre's distinctive narrative form, identifies 'four dominant narrative strategies or discursive formations' (1989: 12). First, most film noirs followed classical Hollywood third-person, 'objective' narration. Second, _Double Indemnity_ 's critical success prompted over forty film noirs to utilise flashback narrative combined with voice-over. Third, film noirs used a subjective camera, sometimes in combination with a character's voice-over \u2013 _The Lady in the Lake_ was shot almost exclusively in this manner. Fourth, the documentary-style of films like _Boomerang!_ (1947) and _The Naked City_ (1948) attempted a combination of the first two discursive formations, matching an objective camera with an authoritative voice-over that guided the viewer through the narrative.\n\nKrutnik's _In a Lonely Street_ combines several projects. It emphasises the relationships between film noir and hard-boiled fiction, establishes various American crime films cycles in the 1940s and 1950s, and draws on psychoanalytic and gender theory to consider the representation of masculinity in the 'tough' thriller cycle. Ian Cameron's _The Movie Book of Film Noir_ (1992) collects important overviews and close critical readings of individual films. The essays in Joan Copjec's _Shades of Noir: A Reader_ (1993) generally combine theoretical astuteness and complexity without forgetting to pay close attention to the films themselves.\n\nNaremore's _More Than Night: Film Noir in its Contexts_ historicises film noir, treating the term 'as a kind of mythology, problematising it by placing the films, the memories, and the critical literature in a series of historical frames or contexts' (1998: 2). It examines intellectual currents in postwar France which led to the fabrication of the genre; Hollywood censorship and blacklists; film noir in Asia, Latin America and Africa; and film noir's escape into other media. Paula Rabinowitz's _Black & White & Noir_ (2002) inverts Naremore's project, treating film noir as a context for understanding American landscapes and history, while Edward Dimendberg's _Film Noir and the Spaces of Modernity_ (2004) finds in film noirs a storehouse of memory, a record of the rapid postwar changes to America's urban landscape.\n\nHitherto, there is no major book-length study of neo-noir. While Foster Hirsch's _Detours and Lost Highways: A Map of Neo-Noir_ (1999) is an expansive preliminary description of the films that might constitute neo-noir, Richard Martin's _Mean Streets and Raging Bulls: The Legacy of Film Noir in Contemporary American Cinema_ (1997) is both a slimmer and a more substantial introduction. This relative dearth of material suggests the extent to which Film Studies is no longer so dependent on evoking genre to dignify its concerns, while the canonisation of certain director as 'auteurs' \u2013 note the allusions to Martin Scorsese, Edgar G. Ulmer and David Lynch in these two films \u2013 indicates Film Studies' problematic simultaneous address to both academic and popular constituencies.\n\nThe criticism discussed above provides numerous starting points for understanding film noir as an intersubjective discursive phenomenon, as a congellation of narratives about certain films told by different discursive agents at specific historical junctures. Between them, they produce an often-contradictory image of film noir as well as insights into the contexts in which they were produced. Having charted some elements of the genre's discursive fabrication, we will now turn to the films often considered as precursors of film noir, outlining potential sources in Weimar cinema, 1930s French cinema and American crime cinema and discussing specific film noirs in these contexts, always risking a 'backwards teleology' (Prawer 2002: 62) which distorts these earlier films through knowledge of a later genre to which they did not directly or necessarily give rise.\n2 OUT OF THE PAST: THE PREHISTORY OF FILM NOIR\n\nFrom any crime to its author there is a trail. It may be \u2013 as in this case \u2013 obscure; but, since matter cannot move without disturbing other matter along its path, there always is \u2013 there must be \u2013 a trail of some sort.\n\n\u2013 Dashiell Hammett (1985: 84)\n\nWeimar cinema\n\nNumerous Austro-German filmmaking personnel emigrated to America between the mid-i920s and the Second World War, many fleeing the Nazi regime (see Petrie 1985, Taylor 1983). A significant number, including John Alton, Curtis Bernhardt, Michael Curtiz, William Dieterle, Marlene Dietrich, E. A. Dupont, Karl Freund, Frederick Hollander, Harry Horner, Peter Lorre, Rudolph Mat\u00e9, F. W. Murnau, Seymour Nebenzal, Max Oph\u00fcls, Otto Preminger, Mikl\u00f3s R\u00f3zsa, Hans J. Salter, Eugen Sch\u00fcfftan, Steve Sekely, Douglas Sirk, Theodore Sparkuhl, Edgar G. Ulmer, Franz Waxman, Billy Wilder and Fred Zinnemann, worked on film noirs and noirish films. Two \u00e9migr\u00e9s \u2013 Robert Siodmak and Fritz Lang \u2013 are absolutely central to the development of film noir.\n\nAfter directing the noirish _Le Chemin de Rio_ (1937), _Mollenard_ (1938) and _Pi\u00e8ges_ (1939) in France, Siodmak made ten film noirs in America: _Christmas Holiday_ (1944), _Phantom Lady_ (1944), _The Spiral Staircase_ (1945), _The Strange Affair of Uncle Harry_ (1945), _The Suspect_ (1945), _The Killers, The Dark Mirror_ (1946), _Cry of the City, Criss Cross_ and _The File on Thelma Jordan_ (see Walker 1992b). Returning to Germany in 1954, he made the noirish _Nachts, wenn der Teufel kam_ (1957) and, in Britain, _The Rough and the Smooth_ (1959). In America, Lang directed 15 film noirs: _Fury_ (1936), _You Only Live Once_ (1937), _Man Hunt_ (1941), _The Woman in the Window, Scarlet Street, Ministry of Fear_ (1945), _Cloak and Dagger_ (1946), _Secret Beyond the Door_ (1948), _Clash By Night_ (1952), _Rancho Notorious, The Blue Gardenia_ (1953), _The Big Heat, Human Desire_ (1954), _While the City Sleeps_ (1956) and _Beyond a Reasonable Doubt_ (1956). We will return to these two directors after outlining the significance of Weimar cinema to film noir.\n\nThe Weimar period (see Elsaesser 2000, Herf 1984, Kaes, Jay and Dimendberg 1994, Laquer 1974, Petro 1989, Sloterdjik 1988), from 1919 to 1933, is typically recalled as a time of rapid change in German culture: on the one hand, far-reaching political and social reforms (liberal democracy, suffrage, freedom of speech, improvements in housing and working conditions), a politicised avant-garde, a freewheeling and experimental nightlife; on the other, debauched and libertine decadence, crippling inflation, unemployment, anti-Bolshevism, anti-Semitism and resurgent traditional authoritarian conservatism. This dichotomy is evident in two early books about Weimar cinema: where Lotte Eisner's _The Haunted Screen: Expressionism in the German Cinema and the Influence of Max Reinhardt_ (1952) found a revitalised German Romanticism, later hijacked and distorted by the Nazis, Siegfried Kracauer's _From Caligari to Hitler: A Psychological History of German Film_ (1947) found such pre-fascist tendencies as the desire to be subordinated to authoritarian structures.\n\nGermany was not the only country to produce expressionist films (see, for example, the Japanese film _Kurutta Ippeiji_ from 1927), but film noir genealogies usually reduce Weimar cinema to German expressionism and German expressionism in turn to a catalogue of techniques including: 'foregrounded oblique objects, unbalanced compositions, irregular spatial arrangements, chiaroscuro lighting with a heavy play of shadows, an emphasis on oblique and vertical lines over the horizontal, and a fascination with reflection and reflective surfaces' (Telotte 1989: 17\u201318); 'high contrast, chiaroscuro lighting where shafts of intense light contrast starkly with deep, black shadows, and where space is fractured into an assortment of unstable lines and surfaces, often fragmented or twisted into odd angles' (Spicer 2002: 11\u201312); and 'displaced, decentred narratives, nested in frame tales, split or doubled stories, voice-overs and flashback narration' (Spicer 2002: 12). Expressionism, however, was more than just a bunch of techniques.\n\nCoined to describe an exhibition by Julien Auguste Herv\u00e9 in Paris in 1901, 'Expressionism' was first used in Germany in the catalogue of a 1911 exhibition of Picasso, Braque and Dufy; by the end of the year it was also applied to C\u00e9zanne and Van Gogh, and in 1912 to several writers. Although Expressionists \u2013 including poets, dramatists, architects and filmmakers \u2013 did not form a coherent group, their work shares certain characteristics: contempt for stifling bourgeois society and industrial capitalism which subordinates everything to the demands of instrumental reason and material production; and a rejection of Impressionism's camouflaging both of its own lack of substance and of the harmful society concealed behind attractive surfaces. Regardless of medium,\n\n> the Expressionist artist inclined to see himself as a prophetic visionary who was called to explode conventional reality, to break through the crust that had formed around men's psyches in order to give uninhibited _expression_ to the energies there imprisoned. Unable to represent, describe or imitate the 'fallen' conventional world, the visionary artist of Expressionism aimed to abstract the objects of the everyday from their normal context, and recombine them into radiant beacons of a lost inner _Geist._ (Sheppard 1976: 277)\n\nAlthough expressionist tendencies can be observed in earlier films, the German expressionist film (see Barlow 1982, Coates 1991) begins with _Das Kabinett des Dr. Caligari_ (1919). Drawing on expressionist theatrical conventions for its performance styles, painted backdrops and irreal spaces, it led to a cycle of major productions \u2013 _Genuine_ (1920), _Von morgens bis mitternachts_ (1920), _Raskolnikov_ (1923), _Schatten \u2013 Eine n\u00e4chtlichte Halluzination_ (1923), _Orlacs H\u00e4nde_ (1924), _Das Wachsfigurenkabinett_ (1924) \u2013 in which a foregrounded style distorted and transformed screen space. This overt stylisation served as a means of product-differentiation for both the domestic bourgeoisie and newly-reopened export markets. Central to the cycle were screenwriter Carl Mayer and set designers Robert Herlth, Erich Kettelhut, Kurt Richter and Hermann Warm.\n\nThe influence of expressionist theatre can also be observed in narratives about father\/son relationships and the dangers of female sexuality; about doppelg\u00e4ngers and perception; about types, not characters; about chaos, dementia and destruction. Although describing it as expressionist is problematic, in Lang's _Metropolis_ (1926), the son, Freder (Gustav Fr\u00f6lich), comes into conflict with his father, Joh Fredersen (Alfred Abel), and descends into an underworld of regimented workers, where he is captivated by the saintly Maria (Brigitte Helm) while an identical, robotic false-Maria (Brigitte Helm) unleashes apocalyptic libidinal forces.\n\nThe number of genuinely expressionist films was small, but their influence on subsequent German cinema was strong and is evident in the more 'realistic' _Strassenfilm_ (street film), like _Hintertreppe \u2013 Ein Film-Kammerspiel_ (1921), _Die Strasse_ (1923), _Die freudlose Gasse_ (1923), _Dirnentrag\u00f6die_ (1927) and _Asphalt_ (1928). _Strassenfilm_ typically, and not unlike _Metropolis,_ chart the paths of a male bourgeois descending into the dangers of the city at night and of a female proletarian trying to escape from her life in the underworld. A version of this twinned trajectory appears in _Der Blaue Engel_ (although describing it as a _Strassenfilm_ is a little unconventional). The self-important Professor Immanuel Rath (Emil Jannings) is captivated by Lola-Lola (Marlene Dietrich), a sexy, provocative singer at the Blue Angel beer hall. Dismissed from his job when this infatuation becomes public knowledge, he marries Lola-Lola and joins her cabaret troupe on the road. As his money runs out, he is reduced to selling salacious photographs of his wife to beer hall customers. The troupe's manager arranges a return to the Blue Angel, convinced that Rath, made up as a clown, will draw a crowd. Subjected to repeated humiliations for the entertainment of those who used to know him when he was a respected authority figure, he attacks Lola-Lola and flees to his old classroom where, degraded and alone, he dies. This narrative formula is not exclusive to the _Strassenfilm._ Variants of it occur in Thomas Hardy's fiction, and although buried in the queer subtext of Robert Louis Stevenson's _Strange Case of Dr Jekyll and Mr Hyde_ (1886) it can usually be found \u2013 straightened \u2013 in adaptations. For example, in _Dr Jekyll and Mr Hyde_ (1932), sexually-frustrated bourgeois Jekyll (Fredric March) meets working-class prostitute Ivy (Miriam Hopkins), played with tantalising pre-PCA near-nudity. Following their first encounter, Ivy's swinging gartered leg is superimposed over the departing doctor, connoting the erotic fascination that will unleash the murderous Hyde.\n\nOstensibly concerned with social problems arising from poverty and unemployment, the _Strassenfilm_ is simultaneously fascinated with the underworld and illicit sexuality (such as Lola-Lola's fetishistic stage costume, exhibitions of sexual openness and ambiguity), frequently equating female sexuality with criminality (see Wager 1999). In this respect, it resembles the silent American gangster melodrama, which, combining a 'reforming morality' with titillation, 'saw the world from the perspective of middle-class protagonists who strayed from the virtuous path and crossed the tracks to \"slum it\" in the ghetto where they were burned by avaricious prostitutes, conquered by pimps, and lost their money to the gambling rackets' (Munby 1999: 24). Further parallels can be drawn between the female proletarians of the _Strassenfilm_ and Hollywood 'fallen woman' films, especially the 'gold-digger' cycle, in which doubly-marginalised protagonists (proletarian and female) frequently used sex to access a doubly-excluding society's material rewards (see Jacobs 1991). In Hollywood, ongoing negotiations of the Production Code saw the ambitious, manipulative and sexually-available gold-digger reconfigured in subsequent cycles: Barbara Stanwyck's sleeping-her-way-to-the-top Lily Powers in _Baby Face_ (1933) became the self-sacrificing eponymous mother trapped by social class in _Stella Dallas_ (1937) who in turn became the fast-talking screwball con woman Jean Harrington in _The Lady Eve_ (1941) and equally fast-talking screwball burlesque dancer Sugarpuss O'Shea in _Ball of Fire_ (1941) became the femme fatale Phyllis Dietrichson in _Double Indemnity._\n\nSiodmak's Weimar films would generally be characterised as belonging to the _Kammerspielfilm_ (chamber play film) or the _Milieutonfilm_ (milieu talkie), genres of intimate realist melodrama which developed an 'aesthetic appropriate to the intimate and interior crisis features of modern urban life' (Munby 1999: 200). Key to the _Kammerspielfilm's_ 'intensely subjective' dramas was the development of the _enfesselte Kamera_ (unchained camera); freed from the fixed tripod, this 'subjective camera projected the intimate psychology of individuals onto a world of external objects', prowling through a cramped, claustrophobic, enervated and dilapidated world, producing an 'uninterrupted visual flow' (Munby 1999: 201). Siodmak's interest in the 'prosaic and profane features of modern urban life' (Munby 1999: 205) and the 'vicious circles people get enmeshed in daily' (Munby 1999: 202) also saw him experiment with the documentarist _Neue Sachlichkeit_ (New Objectivity), working with Seymour Nebenzal, Eugen Sch\u00fcfftan, Edgar G. Ulmer, Billy Wilder and Fred Zinnemann on _Menschen am Sonntag_ (1930).\n\nIn contrast, Lang's German films, indebted in part to Max Reinhardt's spectacular theatre, tended to the monumental, the epic and the apocalyptic, and attempted to capture some sense of a social totality. _Doktor Mabuse, der Spieler_ (1922) connects the criminal underworld to high finance, government and the aristocracy, all of which are subject to the eponymous criminal genius's manipulations. A similar figure threatens the entire world in _Spione_ (1928), while _M_ (1931) explicitly parallels the criminals and the police as they separately organise to capture a serial child-killer. In _Metropolis,_ Lang created a total world in two senses. First, there is no transition from the present to the future-city and we never venture outside, suggesting that it is not so much a metropolis as a metrocosm (see Bukatman 1993). Second, its brutally clear division between proletariat and bourgeoisie creates a visually-exaggerated image of capital's totalising logic. Lang's final 1930s German films, _M_ and _Das Testament des Dr. Mabuse_ (1933), turned inwards, relocating manipulating power within the pathological psyche rather than the machinations of a criminal mastermind.\n\nLang's German films depict determinist worlds which resemble both Expressionism's critique of industrial rationalisation and the image of capitalist modernity developed by the Frankfurt School. In the Frankfurt School's 'account of contemporary development in capitalist society' David Held identifies a 'constellation of elements' (1991: 210). Economics and politics increasingly integrate as business interests intervene in the running of the state for their own ends and the state intervenes in the economy to maintain conditions favourable to business. This integration leads to centralised instrumentalist bureaucracies and administration and to the suppression of local initiative. As instrumental reason dominates, social life becomes increasingly rationalised. The division of labour and mechanisation of tasks extends, focusing the individual on a tiny portion of the work process and denying him or her knowledge of the totality of what he or she is working on. This further isolates and alienates individuals, reducing the possibility of recognising shared experience, the basis of class consciousness:\n\n> Domination becomes ever more impersonal. People become means to the fulfilment of purposes which appear to have an existence of their own. The particular pattern of social relations which condition these processes \u2013 the capitalist relations of production \u2013 are reified. As more and more areas of social life take on the characteristics of mere commodities, reification is reinforced, and social relations become ever less comprehensible. Conflict centres increasingly on marginal issues which do not test the foundation of society. (1991: 211)\n\nThe world the Frankfurt School describes is captured in Lang's _mise-en-sc\u00e8ne_ of modernity (see Gunning 2000): there are motorbikes and motorcars; telephones, radios, miniature cameras and bugs; newspapers, international treaties and secret pacts; elaborate conspiracies and impeccably-choreographed heists and assassinations; hypnotists, megalomaniacs and madmen; dials, thermometers, gauges and countdowns; gambling, financial speculations and stock and currency manipulations; communications networks and dragnets; series (see Kaes 2001), parallels and concentric circles; locked doors, handcuffs, chains, bonds and other images of incarceration. These elements situate the subject in various networks, schemes and narrative developments \u2013 determinate forces out of his or her control. _Metropolis's_ corporate state is run by engineers and administrators under a single leader, Fredersen, whose office is at the centre of a web of information and whose role is homeostatic. This cyberneticisation is even more pronounced among the workers beneath the city. Identically-clothed diminutive figures reduced to sets of rhythmic movements, their work involves the monitoring and regulation of an abstract system. Workers on the M-Machine are mere components, human thermostats steam-scalded by this disciplinary apparatus if they fail to regulate it properly. When Freder subjects himself to the Paternoster machine, he must move the hands on this giant clock-like device so that they point at the lights around its rim as they light up in apparently random order. There is no clue to the machine's function, but its metaphorical purpose is made clear when a clock face is superimposed over it. Throughout Lang's German films clocks, watches and clock-like images (notably the children's game at the start of M) repeatedly signify the rationalisation of time, one of capitalist modernity's interlocking systems of subjection. (Paul Monaco offers a more specific interpretation, arguing that the clock motif in Weimar films 'represents impending danger or disaster' because of its connection with the notion of _Zeitgewinn_ (winning time) underpinning Germany's military strategy: once the 1914 offensive 'fizzled, and the war became one of attrition, Germany was... against the time factor. Her military system was not fitted for an extended war. Germany had pushed for war in 1914 because it was felt that her chances of winning an inevitable war were running down with time' (1976: 30).)\n\nFIGURE 1 Against the clock: _Metropolis_\n\nTheodor Adorno and Max Horkheimer, two major Frankfurt School critics, are frequently criticised for overemphasising the integration of modern societies 'in which every element is increasingly tailored to fit into the whole, in which every aspect has its place, and in which any form of deviance or incipient criticism is either normalised or excluded' (Thompson 1990: 106). However, their work often reveals a rather more dialectical process \u2013 involving powerful systems and frequently isolated subjects, in which no victory is permanently won \u2013 which resembles a curtailed version of Antonio Gramsci's model of hegemony as negotiated-subordination (see Gramsci 1998). _Metropolis's_ conclusion offers an image of such hegemony: Maria persuades the head (Joh Fredersen) and the hands (the workers) to accept the mediation of the heart (Freder, with whom Maria has fallen in love). Although this ludicrous solution has been mocked by everyone, Lang included, it is the one for which we have generally settled: negotiation between exploiter and exploited classes (often represented as individuals rather than social classes), undertaken by proxies who ultimately function as part of the apparatus for maintaining existing economic processes. Although Lang's German and American films depict a determinist universe, he clearly perceives existence within such a universe as if not dialectical then at least agonistic, even if he prefers more metaphysical terminology:\n\n> I think that is the main characteristic, the main theme that runs through all my pictures \u2013 this fight against destiny, against fate. I once wrote... that the fight is important \u2013 not the result of it, but the revolution itself. Sometimes, maybe, with a strong will, you can change fate, but there is no guarantee that you can. If you just sit still, however, and say, 'Well, I cannot do anything \u2013' bang! At least, you have to fight against it. (Lang quoted in Bogdanovich 1997: 191)\n\nReworked in an American context, this fatalist version of determinism \u2013 central not only to Lang's and Siodmak's work but to Weimar culture more generally \u2013 continued to depict the subject as lacking agency in the face of forces beyond his or her control. Despite their differences, Siodmak and Lang contributed expressionist visions of capitalist urban modernity and 'shared in the development of a cinematic _Angstkomplex_ regarding subject-power relations in modern society' (Munby 1999: 208) \u2013 which is as much Weimar's legacy for film noir as its distortions of _mise-en-sc\u00e8ne._\n\nJonathan Munby argues that because they were Americanised in the 1930s, many \u00e9migr\u00e9s strongly identified with 'the ideals their adopted nation stood for during the Depression (and the war)' and thus 'were particularly disappointed by Hollywood's acquiescence to political conservatism after the war' and the 'deferment of New Deal-era promises' (1999: 212, 214). Their film noirs can, then, be regarded as palimpsests, as overwriting fatalist Weimar sociopsychology and expressionist aesthetics onto the American crime film; this can be seen in Lang's _Scarlet Street_ (see Jacobowitz 1992).\n\nWalking home from a party at which he was presented with a watch to celebrate 25 years of faithful service, cashier Christopher Cross (Edward G. Robinson) 'rescues' Kitty March (Joan Bennett) from an attacker. Infatuated, he does not correct her when she persuades herself that he is a famous and wealthy painter. Her attacker \u2013 really her boyfriend, Johnny Prince (Dan Duryea) \u2013 convinces her to lead Chris on. Unhappy at home, and repeatedly compared unfavourably by his wife to her dead first husband, Chris embezzles money so that he can set Kitty up in an expensive apartment where he can also go to paint. Johnny begins to sell Chris's paintings, signed by Kitty; when Chris finds out he insists that she continue to put her name to them. When his wife's first husband turns up alive, Chris leaves her to marry Kitty, but he finds her with Johnny. After Johnny leaves, Chris returns to her apartment and murders her. Circumstantial evidence (and a lie from Chris) condemn Johnny. He is executed, and Chris is driven mad by the thought of Kitty and Johnny united in death. He ends up, six or seven years later, a vagrant who keeps trying to confess to the police. In the final scene, he passes an art dealer where Kitty's 'self-portrait' has just been sold for $10,000. As he wanders down the street, the passers-by fade out, leaving him utterly alone, a broken man.\n\nMunby sees in _Scarlet Street_ a 'paranoid psychosexual mediation of sociohistorical change rooted in Weimar cinema' grafted onto 'less candid and more humorous pre-Code American traditions' (1999: 197). Unlike the ethnically-defined gangsters Robinson played in _Little Caesar_ (1930) and _Key Largo_ (1948), Chris is 'trapped in a different ghetto: the prison of bourgeois rectitude where any transgression of bourgeois morality leads one into a realm of psychotic self-punishment' (Munby 1999: 198). Likewise, Kitty transforms the gold-digging fallen woman. No serial seductress trying to climb out of tawdry circumstances, she describes her desire for Johnny \u2013 a man who beats, abuses and pimps for her \u2013 as love, and does all she can to keep his interest, such as it is. The direct social criticism found in some fallen women films is here psychologised. When Chris asks Kitty if he can paint her, she agrees, holding out her foot and nail polish. The ensuing eroticised humiliation of Chris is a role-reversed replay of her relationship with Johnny and a sadomasochistic sexual fantasy which functions as a powerful, personalised metaphor of a negotiated, and often complicit, subordination to domination. In it, one can see both Weimar sociopsychology and aesthetics and American narrative and cinematic traditions; one can see a palimpsest being written.\n\nFrench cinema of the 1930s\n\nIn addition to _Le Dernier tournant_ (1939), the first adaptation of Cain's _The Postman Always Rings Twice,_ France also produced a number of other films that were remade in America as film noirs or noirish films \u2013 _La Chienne_ (1931) as _Scarlet Street; P\u00e9p\u00e9 le moko_ (1936) as _Algiers_ (1938) and _Casbah_ (1948); _La B\u00eate humaine_ (1938) as _Human Desire; Le Jour se l\u00e8ve_ (1939) as _The Long Night_ (1947); _Pi\u00e8ges_ as _Lured_ (1947). France provided temporary sanctuary and employment for Austro-German \u00e9migr\u00e9s, including Lang, Mat\u00e9, Max Oph\u00fcls, Siodmak, Sch\u00fcfftan, Sparkuhl, Waxman and Wilder, but as German invasion threatened, they fled to the US, along with several French filmmakers who later worked on film noirs, such as Jean Renoir, Julien Duvivier and Jacques Tourneur. However, France's main influence on film noir is typically described in terms not of personnel but of 'poetic realism'.\n\nCoined by Jean Paulhan to describe Marcel Aym\u00e9's novel _La Rue sans nom_ (1929), 'poetic realism' was later applied to its 1933 adaptation by a reviewer who described the film as 'inaugurat[ing] an entirely new genre' (Andrew 1995: 11). Overtly stylised, these stories of marginality, criminality, loyalty, betrayal and doomed romance, typically set in working-class Paris, attempt to elicit the lyrical from the quotidian. _Sous les toits de Paris_ (1930), _Le Grand jeu_ (1933), _La Maternelle_ (1933), _L'Atalante_ (1934) predate the main cycle, which ran from 1935 to 1939, and included _La Belle \u00e9quipe_ (1936), _P\u00e9p\u00e9 le moko, La B\u00eate humaine, H\u00f4tel du Nord, Quai des brumes_ and _Le Jour se l\u00e8ve._\n\nPoetic realism's roots can be traced back through contemporary crime writer Georges Simenon to nineteenth-century writers like Emile Zola, Honor\u00e9 de Balzac and Eugene Sue. While some have argued that poetic realism was merely bourgeois melodrama transposed onto the proletariat, this would require it to represent the working class 'in the same patronising light' as Zola, who constantly implied 'that the working class [was] \"less\" than the bourgeoisie' (Hayward 1993: 148). However, as Susan Hayward argues, 'the plethora of images of male friendship, the presence of popular songs, the meticulous attention to the whole of the working-class [setting and situation] attest differently' (1993: 148). Unsurprisingly, then, poetic realism is often seen as somehow related to the rise and fall of the Popular Front, a left-wing coalition of traditionally antagonistic socialist, communist and other radical groups formed to combat the rise of fascism. Narrowly elected in 1936, they attempted to reverse\n\n> the conservative programme of preceding years: instead of giving priority to the economic sector and attempting to balance the budget by reducing expenditure, [they] proposed a series of social reforms involving significant public sector expenditure \u2013 the dole, public works, agricultural subsidies, and above all the reduction of the working week to forty hours \u2013 intending that these social measures should in turn trigger an economic recovery. (Crisp 1993: 4)\n\nThe Popular Front also prompted wide-ranging cultural experimentation, including films by Renoir, Duvivier and Jacques Becker (see Strebel 1980, Vincendeau and Reader 1986). The Popular Front government's pro-worker policies \u2013 introducing paid holidays and the forty-hour week, nationalising the railways, supporting unionisation \u2013 could not prevent a wave of strikes and a recession led to its collapse in October 1938. Poetic realism's fatalism has been linked to the Popular Front's _reactive_ formation (in order to _oppose_ fascism's growing popularity) and its inability to sustain an _active_ reformist programme. Regardless of the legitimacy of such claims, two things are certain: in November 1940, poetic realist films 'were condemned and banned under legislation relating to the cleansing and regulation of the film market' (Crisp 1993: 60), and poetic realism's culminating example, _Les Enfants du paradis_ (1943\u201345), was backward-looking, demonstrating that it belonged to an earlier conjuncture.\n\nColin Crisp locates poetic realism within a broader tendency between 1930 and 1950, a 'theoretically articulated and systematically implemented poeticisation which aimed not to _capture_ reality but to _transpose_ it' and which drew upon 'surrealist pictorial and montage techniques [and] expressionist acting, lighting and cinematography techniques' (1993: 320\u20131). Dudley Andrew calls poetic realism the 'mongrel whelp' of 'naturalism, impressionism and Surrealism, of those aspiring to create a refined cinema and those eager to maintain the popularity that makes cinema the art of the century' (1995: 50), and quotes Jean Mitry's description of it as an 'attenuated expressionism inserted into the norms and conditions of the immediately real where symbolism is reduced to things, to objects' (1995: 15). Unlike Hollywood's perpetual investment 'in maximum shock effects, in bursts of song, violence, eros or language', poetic realism 'diffuses such energy in a warm mist of style that mutes the sound and brightness of every effect, even as it washes over us and seeps down to the roots of feeling' (1995: 6). Although film noir replicated poetic realism's sombre nocturnal settings, glistening rain-drenched streets, swirls of fog, evocative sets and deep visual fields segmented by patterns of light and dark, it generally conveyed a different mood or tone. Poetic realism's attention to social context and the unremarkable everyday circumstances which lead to disaster tends to produce sympathy for its doomed protagonists, whereas Hollywood's emphasis on action over contemplation, exhibition over introspection, clear motivation over ambiguous, fitful promptings, tends to produce curiosity about what exactly characters will do next \u2013 a contrast best evoked by comparing _La B\u00eate humaine_ with _Double Indemnity._\n\nThe blood of train driver Jacques Lantier (Jean Gabin) is tainted by generations of alcoholic ancestors. Prone to depression, he has fits of inchoate rage when sexually aroused, becoming a murderous automaton. After assaulting a woman he loves, he refuses her offer of marriage, believing it would end in tragedy. Deputy station-master Roubaud (Fernand Ledoux), married to the much younger S\u00e9verine (Simone Simon), discovers that Grandmorin (Berlioz), her wealthy godfather, who might actually be her father, has certainly been her lover. Roubaud forces S\u00e9verine to arrange a meeting with Grandmorin and murders him. Lantier sees Roubaud and S\u00e9verine leaving the crime scene, but her pleading eyes persuade him not to betray them. Lantier falls in love with her but, remembering his illness, agrees to be just friends. They do, however, become lovers. S\u00e9verine, bound to the increasingly-morose Roubaud by their crime, cannot leave her husband. When Lantier proves incapable of murdering Roubaud, the disappointed S\u00e9verine, convinced that Roubaud will soon kill her, breaks up with Lantier. To win her back, Lantier again sets out to kill Roubaud, but S\u00e9verine's passionate embrace triggers a fit and he stabs her to death. Next day, he hurls himself from his train and dies.\n\nThroughout, Lantier and S\u00e9verine's actions are mitigated, at least in part, by their circumstances and lack of calculation. S\u00e9verine's involvement with Grandmorin, who has a reputation for forcing himself on women (including, possibly, her mother), is reluctant; it is implied that she married Roubaud to escape him. An unwitting and unwilling accomplice after the fact, she did not murder anyone. Despite her questionable motivation for befriending Lantier, she does fall in love with him. Likewise, Lantier is not responsible for his illness, and tries to avoid situations which might unleash the dangerous automaton he becomes during his fits.\n\nIn contrast, insurance salesman Walter Neff (Fred MacMurray) sets out cold-bloodedly to murder Dietrichson (Tom Powers) because he is physically attracted to Dietrichson's wife, Phyllis (Barbara Stanwyck); because of the insurance money; and, most importantly, because he wants to outsmart claims investigator Barton Keyes (Edward G. Robinson). Phyllis is equally calculative: she murdered Dietrichson's first wife so as to marry him, but now he is worth more to her dead. Told in flashback as Neff, bleeding to death, confesses all to Keyes' dictaphone, the film marginalises the murderers' bloodless passion in favour of a fascinated observation of their crime. As Keyes closes in, Neff realises that to get away with the original murder he must kill Phyllis. Instead, they shoot and kill each other.\n\n_La B\u00eate humaine_ opens with Lantier's approach to Le Havre, parallel rail-tracks racing towards and beneath his train; running without deviation along a predetermined route, it metaphorises Lantier's condition and fate. After killing S\u00e9verine, he wanders off, robot-like, into the night, along a railway line. For him, there is no escape; but, significantly, that lack of escape runs in both directions \u2013 at no point in the past did he set himself on the path he must follow. In _Double Indemnity,_ deterministic and machinic images and metaphors recur \u2013 from the insurance company's ranks of identical desks and Keyes' statistical tables to the railtracks where Dietrichson's body is dumped and the regular grooves of the dictaphone cylinder recording Neff's confession. Neff says the murder plan must be 'perfect... straight down the line', and later claims that 'the machinery had started to move and nothing could stop it', that Fate had 'thrown the switch'. Keyes describes the killers as being on a 'trolley-ride together'; unable to 'get off at different stops', they are 'stuck with each other and they've got to ride all the way to the end of the line, and it's a one-way trip and the last stop is the cemetery'. Thanks to the ingenious casting of MacMurray, one might feel some sympathy for Neff. From his first encounter with Phyllis he is out of his depth while imagining he is in control \u2013 in the novel, he says 'I loved her like a rabbit loves a rattlesnake' (Cain 2002: 84), but in the film he lacks this self-knowledge. However, the flashback structure counters any sympathy developing, generating not a sense of foreboding but the certainty of Neff's failure. If _La B\u00eate humaine_ 's unfolding plot makes one wonder 'what will happen to these poor people next?', _Double Indemnity's_ retrospective structure prompts the sharper query, 'how exactly did he bring this upon himself?'\n\nThis different tone is not an inevitable by-product of the flashback structure. _Le Jour se l\u00e8ve_ opens with Fran\u00e7ois (Jean Gabin) killing Valentin (Jules Berry), and his apartment being surrounded by armed police. In the first of three flashbacks, Fran\u00e7ois meets Fran\u00e7oise (Jacqueline Laurent). Born on the same day, named after the same saint and raised in orphanages, they are both depicted as childlike \u2013 Fran\u00e7ois drinks milk, Fran\u00e7oise has a teddy bear called Bolop. They soon fall in love, but from the outset one knows their love is doomed: when Fran\u00e7oise first enters Fran\u00e7ois' factory, fumes kill her flowers; and in the opening sequence Bolop is hit by police gunfire. Fran\u00e7oise is involved with an older man, Valentin, who has a performing-dogs act. Fran\u00e7ois is picked up by Valentin's assistant Clara (Arletty), who has just quit the act. In the second flashback, set two months later, Fran\u00e7ois and Clara are lovers and Fran\u00e7oise and Valentin seem to be. Pretending that he is Fran\u00e7oise's father, Valentin tries to stop Fran\u00e7ois from seeing her. Fran\u00e7oise assures Fran\u00e7ois that Valentin was lying, and they both promise to stop seeing the others. Fran\u00e7ois, however, becomes convinced that, despite her denials, Fran\u00e7oise has been Valentin's lover. In the third flashback, Valentin visits Fran\u00e7ois, intending to kill him. Valentin goads Fran\u00e7ois by implying a sexual relationship with Fran\u00e7oise. Fran\u00e7ois takes Valentin's gun and shoots him. Back in the present, Fran\u00e7ois kills himself. Despite sharing a flashback structure similar to _Double Indemnity's,_ the tone of the film is closer to that of _La B\u00eate humaine_ because Fran\u00e7ois' memories reveal emotion, passion, misinterpretation and events beyond control rather than a process of calculation.\n\nFIGURE 2 Out of his depth: _Double Indemnity_\n\nAlthough told without flashbacks or poetic realism's expressionist techniques, _P\u00e9p\u00e9 le moko_ shares something of this tone by making the protagonists victims of calculation. It opens in Algiers two years after the heist which forced P\u00e9p\u00e9 (Jean Gabin) to flee France (on _P\u00e9p\u00e9 le moko_ as a colonialist fantasy, see Morgan 1997, O'Shaughnessy 1996; on film noir's use of exotic locales, see Naremore 1998). His Casbah hideout has become like a prison to him. Watching ships sail from the harbour, he yearns for Paris. His appearance and behaviour capture his awkward, interstitial position. Eschewing Parisian working-class costume, P\u00e9p\u00e9 combines a proletarian physique and physiognomy with a natty dress-sense, evoking the 1930s Hollywood consumer-gangster but without the mobster's brashness (see Vincendeau 1998: 41\u20139). As dapper as a gentleman-thief, his class origin denies him the panache of Raffles or Ars\u00e8ne Lupin.\n\nP\u00e9p\u00e9 falls in love with Gaby (Mireille Balin), the mistress of champagne magnate Kleep (Charles Granval). She is made-up, costumed and lit so as to emphasise the contrast between her European whiteness and the dark, uncertain ethnicity of P\u00e9p\u00e9's gypsy lover, In\u00e8s (Line Noro). Raised in the same neighbourhood as P\u00e9p\u00e9, Gaby misses Paris too. She leaves Kleep for P\u00e9p\u00e9, but Inspector Slimane (Lucas Gridoux) tells her that P\u00e9p\u00e9 has been killed. Kleep, intent on leaving Algiers, takes her back. P\u00e9p\u00e9 is distraught when Gaby misses their rendezvous. Slimane almost tricks him into going to Gaby's hotel, where it will be easy to arrest him. P\u00e9p\u00e9 learns of Gaby's imminent departure. Shots of P\u00e9p\u00e9 running through the Casbah set are replaced by shots of him running in front of back-projected streets. Behind him, one shot dissolves into another, the last one not of buildings but of surging, crashing waves \u2013 a brief eruption of expressionist artifice as his passion overcomes any trace of calculation. Tipped off by In\u00e8s, Slimane arrests P\u00e9p\u00e9 as he is about to join Gaby aboard ship. Handcuffed behind the prison-like bars of the harbour gates, P\u00e9p\u00e9 calls out to Gaby as she appears on deck, but the ship's klaxon drowns out his voice. He kills himself.\n\nP\u00e9p\u00e9 and Gaby are calculative in their relationships with others, but in their romance emotion and compassion overwhelm instrumentalism. P\u00e9p\u00e9 invests Gaby with all his longings \u2013 just as he does with Paris, the sea and the ships that sail upon it \u2013 and Gaby sees in him an alternative to her life of, effectively, prostitution. This is tempered by their recognition of each other's entrapment and the identification, manifested as passionate romantic love, it prompts. Reworking _Strassenfilm_ and Hollywood gold-digger-film logic, a proletarian couple, having found differently-gendered routes (crime and sex) to material well-being, are denied a redemptive return to their proletarian origins.\n\nThe difference in tone between French poetic realism and American film noir largely derives from differing conceptions of agency. _Double Indemnity_ opens with Neff racing to his office where he will record his confession. In long shot, his car races towards us, following the dead-straight parallel lines of streetcar tracks. In the foreground, a maintenance crew is working on the lines. The car swerves around them and rushes on, jumping a stop-sign. This sequence seems like a direct riposte to the opening of _La B\u00eate humaine,_ described above, in which Lantier cannot deviate from the rail-tracks laid down before him. Each of the poetic realist films here described, regardless of whether or not they have a flashback structure, both depict and tell a fatalistic world, but _Double Indemnity_ opens up a gap between determinism and fatalism, depiction and telling. It depicts a determinist world, but Neff's telling of it is not fatalist. His flashback narration prevents a poetic realist conflation of determinism and fatalism. Just as he veers off the streetcar tracks and onto a fresh path, so he believes that he can escape the consequences of his actions (or, more accurately, act in such a way that certain of his actions do not have certain consequences); if only he can kill Phyllis, if only he can get to Mexico...\n\nAll of these films are fantasies about the possibility of agency. The poetic realist films regard agency as either illusory or irrelevant: regardless of what you do while you are on the train, it is still going to pull into the same station. In _Double Indemnity_ and other film noirs, the protagonists lay the tracks themselves.\n\n1930s Hollywood crime films\n\nAt night, in long shot, a car pulls up at a rural gas station, a circle of light in a black frame. Figures leap out and rush into the building. The light goes out. Gunshots, muzzle-flashes. The sound of a cash-register. This minimalist exercise in light and sound opens _Little Caesar,_ the first major entry in Hollywood's 1930s gangster cycle. In the following scene, Caesar Enrico Bandello (Edward G. Robinson), bemoaning the lack of opportunity for criminal advancement in the sticks, resolves to head for the big city; his partner, Joe (Douglas Fairbanks, Jr), who wants to resume his dancing career, imagines the new clothes he will buy. These two scenes establish the cycle's central characteristics. Rejecting rural associations with the past and the models of criminality offered by westerns and dust-bowl outlaws like Dillinger and Bonnie and Clyde, it embraces capitalist modernity as figured by the city. Productive labour is dismissed in favour of leisure and consumption. This idea of consumption-as-utopian-goal (see Moylan 1986) is reiterated visually when a mobster's jewellery is subjected to Caesar's gaze in the kind of shot usually reserved for eroticising the female body. Caesar's frequent costume changes and his belief that a painting cost $15,000 because of its 'gold frame' further develops the idea of consumption-as-spectacle. Finally, we are introduced to troubled masculinities: Caesar is uninterested in women, while Joe is 'feminised' as a dancer \u2013later, a jealous Caesar threatens to kill Joe's girlfriend as they compete for Joe's attention.\n\n_The Public Enemy_ (1931) replicates _Little Caesar's_ rise-and-fall narrative. It begins with aerial shots of the city, the stockyards and streets, cruising an intersection before settling on two boys, Tommy Powers (James Cagney) and a friend, the camera analogous to the _Strassenfilm's_ bourgeois descending into the underworld. Tommy's law-abiding brother works in low-paid jobs, studies at night, marries his childhood sweetheart, enlists in the army, slowly recovers from shell-shock, returns to low-paid work; Tommy turns to crime instead. He, too, undergoes sartorial transformations, but while Caesar is merely uninterested in women, Tommy is impotent.\n\nIf not expressionist, both films deploy expressive _mise-en-sc\u00e8ne._ In _Little Caesar,_ long shots of rooms often feature emphatic diagonal lines, some explicitly moderne. _The Public Enemy_ frequently situates Tommy among strong vertical lines, associated with his rise, but when he is shot, he falls in a rain-filled gutter, is prostrated in a hospital bed; when kidnappers return him, he is bound and trussed, upright in the doorway \u2013 and then he falls flat on his face, dead.\n\n_Scarface_ (1932), based on Armitage Trail's 1930 novel (itself a catalogue of gangster fiction tropes), also follows a mobster's rise and fall. Tony Camonte's (Paul Muni) ascent is marked by the accumulation and display of commodities; his demise stems from an incestuous desire for his sister. The artifice of _Scarface's_ opening low-angle shot presages a thoroughly noirish visual style. Notable by its absence from the brightly-lit inserted sequence, shot by someone else, in which civic spokespeople deplore the gangster phenomenon, it is particularly conspicuous in the repetition of X-shapes when characters are killed. Only occasionally diegetically-motivated, the repetition of X-shapes creates a consistently fatalist world \u2013 whenever an X appears, someone dies \u2013 which counterpoints the commodity's false utopianism. Camonte's death belies the Cooks Tours' neon billboard, which promises 'The World is Yours'. He has been, at best, a tourist amid wealth and power. He is gunned down by the socially-sanctioned agents of determinate class and ethnic structures.\n\nAs Munby argues, part of these films' significance is that in them the gangsters _talked_ and did so with voices of distinct ethnic and class origins. Jewish-American and Irish-American actors playing powerful, charismatic Italian-American and Irish-American characters celebrated non-WASP identities while drawing attention to class- and ethnicity-based social exclusions. Despite their reactionary ghettoisation of crime, they gave voices to excluded Americans: 'A space of cultural containment and ideological legitimation was turned into something more rebellious' (1999: 5), simultaneously recapitulating, through costume and decor, images of capitalist excess while dramatising the exclusionary practices and processes faced by ethnicised and other workers. (Similar processes can be observed in Blaxploitation and New Jack Cinema.)\n\nFor C. L. R. James, gangster films exemplified the increasingly apparent fissure between capitalist modernity and an outmoded national self-image:\n\n> The gangster did not fall from the sky... He is the persistent symbol of the national past which now has no meaning \u2013 the past in which energy, determination, bravery were certain to get a man somewhere in the line of opportunity. Now the man on the assembly line, the farmer, know that they are there for life; and the gangster who displays all the old heroic qualities in the only way he can display them, is the derisive symbol of the contrast between ideals and reality. (1993: 127)\n\nThis fissure is more pronounced in _High Sierra_ and _White Heat,_ film noirs which cast actors with urban associations from 1930s crime cinema as gangsters-out-of-time in stories which recall rural outlaws. (The gangster-out-of-time also appears in _Key Largo_ and _His Kind of Woman_ (1951) as an irrational force returning and disturbing an orderly America.)\n\nAfter eight years in prison, _High Sierra_ 's Roy Earle (Humphrey Bogart) makes straight for a park, to make sure 'grass is still green and trees are still growing', rather than the city. Explicitly compared to Dillinger, he is anachronistic: his old boss is dying and younger hoods regard him as legendary. The owners of his family's old farm are worried he is from the bank, and, in an even more obvious allusion to the Depression-era outlaw, he befriends a California-bound family of bankrupt farmers. Moreover, Earle's dark suit seems out of place in the west coast's bright sunshine and more horizontal than vertical cities. Eventually cornered by police surveillance and communications technologies, Earle makes his last stand on Mount Witney \u2013 it is broadcast by radio to the nation \u2013 and is shot by a character in western costume.\n\n_White Heat_ opens with a steam train hold-up and ends with a shoot-out in an ultra-modern chemical plant. In between, Cody Jarrett (James Cagney), often costumed as a more rural than urban figure, is rarely seen in a fully urban setting. Like Earle, Cody is ill-suited to the city, his past-ness emphasised by his 'fierce psychopathic devotion' to his mother (the headaches and fits which, in childhood, he faked to get her attention, have now become real). He challenges the tendency to see the rational modern world as rigidly determinist. This rationality is evoked by images of order \u2013 for example, the co-ordination of various agencies' resources to triangulate the radio signal which will give away Cody's position, a process initially demonstrated diagramatically and then presented in a procedural montage sequence \u2013 and is challenged by contingency and disorder. Treasury agents follow a gridlike city map as they tail Ma Jarrett (Margaret Wycherly), losing her then relocating her by chance; the ordered ranks of the prison dining hall are thrown into temporary disarray by Cody's violent fit; the elaborate scheme to allow Cody's escape is abandoned because of this fit, only for him to escape anyway. Cody completes his transition from idealised western past to rational modern world, going beyond the city to a location which seems futuristic, and there, deranged, unleashes an irrational, apocalyptic explosion. (See Corber 1997 on film noir's nostalgia for pre-Fordist capitalism.)\n\nSuch films expunge the ethnic and working-class identities and contexts of the 1930s gangster cycle, focusing instead on crime as an endemic national condition (both hint at vast unseen criminal organisations) and on individualised psychology (Earle's embodiment of _thanatos,_ Cody's mother fixation). As exemplified by _White Heat,_ film noir's recurrent conflict between the rational and the irrational sees the apparently determinist world disrupted, only for these disruptions to be revealed as arising from other determinants which were not taken into account: plans go awry, murder plots and minutely-planned heists go wrong, individuals refuse subordination to the gangster-corporation, coincidences and contingencies betray, but a new (narrative) order emerges. Embedded in many film noirs is a clear sense of the inadequacy of models of linear determinism. Narratives arise from the collision of different presumed linearities, hinting at the nonlinear determinism of emergent complex systems.\n\nGenealogical overviews of film noir typically reduce 1930s Hollywood crime films to the gangster cycle initiated by _The Doorway to Hell_ (1930) and ended by the 1935 Hays Office moratorium on gangster films. Yet as early as 1930, Harry Potamkin identified another cycle of crime films, including _Underworld_ (1927), _Street of Chance_ (1930), _Roadhouse Nights_ (1930) and _Ladies Love Brutes_ (1930), which by various means, such as 'reducing the themes of gang strife to a love triangle', shift and dissipate the 'burden of social guilt' (1977: 477). He considered this cycle 'part of America's celebration of her corruption' (1977: 478).\n\nThe early sound period also saw hard-boiled fiction influence crime film. Burnett, author of _Little Caesar,_ co-scripted _The Finger Points_ (1931) and provided the story for _Beast of the City_ (1932). James M. Cain became a Paramount scriptwriter in 1931, before moving to Columbia; MGM bought the rights to _The Postman Always Rings Twice_ in 1934 and Paramount the rights to _Double Indemnity_ in 1936. Dashiell Hammett wrote the screen story for _City Streets_ (1931). His 1931 _The Glass Key_ and his 1934 _The Thin Man_ were adapted under the same titles in 1935 and 1934, respectively, and _The Maltese Falcon_ was adapted twice in the 1930s, as _The Maltese Falcon_ (1931) and _Satan Met A Lady_ (1936).\n\nThe success of the screwball murder-mystery _The Thin Man_ led to five film sequels by 1947 and inspired a cycle of detective-couple and other comedy-mystery films, including _Remember Last Night?_ (1935), _The Mad Miss Manton_ (1936), _The Ex-Mrs Bradford_ (1936) and _A Night to Remember_ (1943). The 1930s and 1940s also produced many B-film series about detectives and crime-fighters, like Boston Blackie, Charlie Chan, Nancy Drew, Bulldog Drummond, the Falcon, the Lone Wolf, Michael Shayne, Mr Moto, Ellery Queen, the Saint, Sherlock Holmes and Philo Vance. Several drew on hard-boiled sources: _The Falcon Takes Over_ (1942) was adapted from Raymond Chandler's 1940 _Farewell, My Lovely;_ Michael Shayne was taken from Brett Halliday's novels and some of his adventures were based on other hard-boiled fiction, including _Time to Kill_ (1942) adapted from Chandler's 1942 _The High Window._ These series also often contained noirish elements. For example, both _The Spider Woman_ (1944) and _The Woman in Green_ (1945) pit Sherlock Holmes against cerebral femme fatales, and Elwood Bredell's cinematography for _Sherlock Homes and the Voice of Terror_ (1942) anticipates his work on film noirs like _Phantom Lady_ and _The Killers._\n\nA cycle centred on institutional crime-fighters emerged in the mid-1930s, including _G-Men_ (1935), _Bullets or Ballots_ (1936), _I Am the Law_ (1938) and _Racket Busters_ (1938). As _G-Men_ and _Bullets or Ballots_ \u2013 starring, respectively, James Cagney turning against his underworld upbringing to join the FBI and Edward G. Robinson faking dismissal from the police so he can infiltrate the mob \u2013 suggest, Hollywood reworked the gangster film so that their charismatic stars and violent action were framed within the appearance of respectability; and with films like _Hard To Handle_ (1933) and _Footlight Parade_ (1933), Cagney's manic embodiment of capitalist urban modernity was transformed into more-or-less legal entrepreneurialism.\n\n_The Mayor of Hell_ (1933), _Dead End_ (1937) and _Angels with Dirty Faces_ (1938) ostensibly argue that crime has social causes rooted in poverty and exacerbated by institutions like reform schools. These social consciousness and sociological gestures were, of course, simultaneously undermined. _Angels with Dirty Faces_ paired energetic tough-guy Rocky Sullivan (James Cagney) with saintly but dull Jerry Connolly (Pat O'Brien), childhood friends who grew up in the same environment but followed divergent paths, one turning to crime, the other to the cloth. _The Mayor of Hell,_ in which Richard 'Patsy' Gargan (James Cagney) tackles the vile reform school head Mr Thompson (Dudley Digges), suggests that it is not so much institutions that are at fault as the corrupt people running them. In fact, one has to turn to _I Am a Fugitive from a Chain Gang_ (1932), a pre-PCA film, to find a fairly unequivocal critique of the penal system (although this, too, contains a memorably awkward moment in which the brutal image of the chain gang presented in the rest of the film is denied, presumably in an attempt to avoid censorship in states which still used chain gangs). Ironically sharing a similar rise-and-fall narrative with early 1930s gangster film, it inverts their tacked-on crime-doesn't-pay morality to suggest that the fault lies with the systems which criminalise individuals. War veteran James Allen (Paul Muni) quits his unfulfilling factory job, becoming an itinerant labourer and then a hobo. Wrongly sentenced to hard labour for a crime he did not commit, he escapes from the chain gang. In Chicago, where he has found success working in construction, his past catches up with him. Marie (Glenda Farell) blackmails him into a loveless marriage. When he falls in love with Helen (Helen Vinson), Marie exposes him. He fights extradition, but eventually agrees to return and serve out a nominal sentence in a clerical position. The state, however, reneges upon the deal, putting him on a chain gang and indefinitely suspending his pardon. Allen escapes again and one night, a year later, he finds Helen. He emerges from the darkness to bid her a final farewell. A sound disturbs them. Edging back into the darkness, he refuses Helen's offer of money. 'How do you live?', she asks, and from a pitch black screen come the words, 'I steal', and the sound of his fleeing footsteps.\n\nWhile _I Am a Fugitive from a Chain Gang_ recognises the indifference and brutality of individuals within the judicial and penal systems, it does not attempt to disavow systemic indifference and brutality. Allen's initial descent into vagrancy is situated in a broader social context \u2013 he is unable to pawn his medal because the pawnbroker already has a tray full of medals pawned by other veterans. A series of visual and aural echoes highlights the carceral aspects of American society, particularly those concerning labour: Allen looking out of his office window, convicts looking through the bunkhouse window, and Allen looking through a double set of bars and mesh to talk to his brother; factory sirens and prison sirens; chains fed through convicts' shackles and the chains fed through workhorses' bridles; Allen swinging a sledgehammer on the chain gang and a pickaxe on a construction gang; a shoe factory and a labour camp. This sense of social incarceration, of inescapable determinate structures of domination, is heightened by the title's tense: _I Am,_ not _I Was._\n\nTwo of Lang's early American films, each with a noirish visual style and a sense of inescapable fate, offer variants on this theme while moving away from the social into the psychological and metaphysical. In _Fury,_ Joe Wilson (Spencer Tracy), _en route_ to a reunion with fianc\u00e9e Katharine Grant (Sylvia Sidney), is arrested on suspicion of kidnap. Whipped into a frenzy, the townsfolk storm the gaol where he is held and burn it down. Joe escapes, but pretends to be dead so that his brothers can see his 'murderers' \u2013 whose not-guilty pleas are undone by newsreel footage of the riot \u2013 brought to justice. Eventually, conquering his own fury, he prevents the conviction of the townsfolk for a murder they did not commit. _Fury_ was inspired by the 1933 lynching in San Jos\u00e9 of two suspects in a kidnap case; California's governor endorsed the lynching and only one of the hundreds-strong lynch mob was indicted (the charges were later dismissed). Presumably because of MGM's conservatism, the PCA and the important markets provided by Southern states, there was never any possibility of _Fury_ directly criticising the complicity of the judicial system in this particular crime and numerous other racially-motivated lynchings. Potential criticism of the media is also undone by the d\u00e9nouement: early on, newsreel cameramen keenly anticipate filming the mob in action, but in the courtroom the footage is taken as an objective record. Instead, _Fury_ becomes an attack on mob psychology and revenge, both of which are depicted, as in _Metropolis_ and _M_ , as derangements.\n\nFIGURE 3 Behind bars: _P\u00e9p\u00e9 le moko_\n\nIn _You Only Live Once,_ three-time loser Eddie Taylor (Henry Fonda) \u2013 with the aid of fianc\u00e9e Joan Graham (Sylvia Sidney), her public prosecutor boss, Stephen Whitney (Barton McLane), and prison chaplain Father Dolan (William Gargan) \u2013 is determined to go straight, but soon after he and Joan are married, he loses his job. His old gang pull off a bank robbery, killing six people and framing Eddie. Sentenced to death on circumstantial evidence, he escapes from prison, killing Dolan, who has brought news of the discovery of the real robber. On the run with the pregnant Joan, Eddie complains, like _I Am a Fugitive from a Chain Gang's_ James Allen and _Das Testament des Dr. Mabuse'_ s Tom Kent (Gustav Diessl), that 'they made me a criminal'. They are shot at a police roadblock. As Eddie, carrying the dead Joan, stumbles to his death, he hears Dolan's voice, repeating the words with which he had emerged, Christ-like, from the fog in the prison yard: 'Eddie. Eddie. You're free, Eddie, the gates are open.' These final words are accompanied by an angelic chorus whose origin, diegetic or extra-diegetic, is unclear. By loading its conclusion with religious imagery \u2013 Eddie and Joan's flight with their baby alludes to the flight to Egypt \u2013 the film subordinates social criticism to a melancholy metaphysics, reinforcing the sense derived from the deliberate and ironic implausibility of the coincidences which trap Eddie and Joan that it is the universe itself, rather than particular social circumstances, that determines their fate.\n\nWith its fatalist aura, _You Only Live Once_ comes closer to poetic realism than any other Hollywood film, while providing an important prototype for such couple-on-the-run film noirs as _They Live by Night_ and _Gun Crazy._ It is also significant in terms of its remarkable images of entrapment \u2013 the bars that separate Eddie and Joan at their first reunion; the metal and glass which separates them in prison; the rifle-sight which frames the dying Eddie \u2013 and its noirish use of light and shadow, most notably the spider's web of shadows cast by Eddie's death-row cage. Such chiaroscuro visuals would, retrospectively, become one of the defining characteristics of film noir; and, as Marc Vernet (1993) suggests, this element of noir's visual style can be traced back not just through 1930s American crime films but also through the earlier careers, beginning in the 1910s and 1920s, of noir cinematographers like Alton, Gaudio, Seitz, Polito, Nicolas Musuraca and Joseph LaShelle, and through Hollywood gothic and horror films of the 1920s and 1930s.\n\nDavid J. Skal (1993) argues that many of the horror films of the late-silent and early-sound period, especially Tod Browning's non-supernatural gothics starring Lon Chaney, are displaced articulations of the physical trauma of the First World War, particularly of the shattered, maimed and disfigured bodies of European veterans. As Hollywood horror became increasingly fantasy-based with the 1930s Universal cycle, starting with _Dracula_ (1931) and _Frankenstein_ (1931), and increasingly expressionist in its design and cinematography, with _The Black Cat_ (1934) and _Bride of Frankenstein_ (1935), so the capacity of the _mise-en-sc\u00e8ne_ to register the characters' subjective turmoil, torment and disorientation became apparent, culminating in _Cat People_ (1942), in which the psychosexual anxieties of Irena (Simone Simon) seem to take on physical form. Similarly, film noir might be seen as displacing the psychological traumas of the Second World War and the dawning atomic age, with 1950s science fiction sharing horror's ability to manifest anxieties as active and monstrous elements of the _mise-en-sc\u00e8ne._ (It is tempting then to suggest that the coincidence of the American invasion and occupation of Vietnam \u2013 and the collapse of the Cold War consensus it betokened \u2013 with the first stirrings of neo-noir and such non-supernatural horror films as _I Drink Your Blood_ (1971), _The Last House on the Left_ (1972), _The Crazies_ (1973) and _The Texas Chain Saw Massacre_ (1974) is more than a coincidence (see Bould 2003).)\n\nWe will now turn to film noir's main cycle, briefly returning to the problem of definition before examining two groups of films which will enable further discussion of determinism.\n3 DARK PASSAGE: THE MAIN CYCLE OF FILM NOIR\n\nTheir characters lived in a world gone wrong, a world in which... civilisation had created the machinery for its own destruction and was learning to use it with all the moronic delight of a gangster trying out his first machine-gun.\n\n\u2013 Raymond Chandler (1980: 9)\n\nIf one accepts that film noir, like any other genre, is in an ongoing process of ultimately irresolvable discursive formation, then any generalisation one makes about it will founder not only on multiple exceptions but also on other versions of the genre formulated by other discursive agents. This process of definition is one of refinement in a particular sense: it does not refine the definition of film noir to a single formulation upon which all will agree; rather, it promotes refinements of detail, argument and formulation while simultaneously exfoliating them, creating further contradictions with which to drive the process. A great many individual films and alternative descriptions of similar and overlapping phenomena have been caught up in the wake of this dynamic process, and so rather than attempting to make any generalisable claims about film noir, this chapter will discuss a number of films generally considered film noirs, grouped together as films about entrapment and films about investigation. These are, of course, incommensurate and frequently overlapping categories. They are constructed so as to enable an examination of the genre's recurring, but not necessarily defining, concern with determinism \u2013 a concern manifested in a more specific critique of the determinate structures of postwar America as it entered a late-capitalist stage.\n\nOn one level, of course, narratives about individual agency inevitably prompt questions of determinism \u2013 social, psychological, linguistic, ideological and, above all, narrative \u2013 and autonomy; and it would, therefore, be easy to subsume all film noirs to a definition focused on determinism and, from there, on subject-power relationships under capitalism. Even as this possibility points to the neglected relationship between film noir and the 1950s family melodramas of, for example, Vincente Minnelli, Nicholas Ray and Douglas Sirk, the ease of pursuing such a project indicates quite how ill-advised it would be. Such totalising braggadocio would iron out the differences and contradictions in and between films and genres. Therefore, this chapter attempts to discuss something of film noir's range and variety in relation to a particular theoretical concern without reducing them to epiphenomena of that concern.\n\nTrapped\n\nImages and narratives of entrapment, already evident in 1930s American cinema, become increasingly common in film noirs. Innocent men are framed, imprisoned for crimes they did not commit or caught up unwittingly in conspiracies and plots in _Stranger on the Third Floor, I Wake Up Screaming_ (1942), _The Fallen Sparrow_ (1943), _Phantom Lady, Crack-Up_ (1946), _Dark Corner, Dark Passage, Desperate_ (1947), _Out of the Past, The Web_ (1947), _The Big Clock, Raw Deal_ (1948), _Convicted_ (1950), _Side Street_ (1950), _Where the Sidewalk Ends_ (1950), _His Kind of Woman, I Confess_ (1953), _Nightfall_ (1957) and _The Wrong Man_ (1957). In _The Lost Weekend,_ Don Birnam (Ray Milland) is trapped by alcoholism, but even more so by his failure as a writer, while in _Nightmare Alley_ (1947), the descent of Stanton Carlisle (Tyrone Power) into alcoholism and carnival-freakery is variously attributed to God, the Tarot and the hubris of trying to escape poverty. This sense of entrapment becomes particularly paranoid and nightmarish in adaptations of Cornell Woolrich's fiction (see Reid and Walker 1993).\n\nFemale identity, and masculine fantasies thereof, were at the centre of a film noir cycle about wives who find themselves isolated, in danger or victims of husbands' plots. In _Suspicion_ (1941) and _Beyond the Forest_ (1949) the danger is imagined, but in _Experiment Perilous_ (1944), _Gaslight, My Name is Julia Ross_ (1945), _Gilda, Notorious, Undercurrent_ (1946), _The Two Mrs. Carrolls_ (1947), _Secret Beyond the Door, Sleep, My Love_ (1948), _Sorry, Wrong Number, Caught, Cause for Alarm_ (1951) and _Sudden Fear_ (1952) the threat is real. Women are trapped and imperilled by criminals, psychotic killers or unscrupulous men in _Hangover Square, The Spiral Staircase, The Reckless Moment, Beware, My Lovely_ (1952) and _The Blue Gardenia._ Some of these films have period settings and most, having been perceived as 'women's pictures', have often been excluded from the film noir canon, just as discussions of melodrama have often excluded film noirs \u2013 impoverishing our understanding of both genres.\n\nTaking a moment from _Metropolis_ as its starting point, this section will consider the implications of the opening sequences of three film noirs whose subsequent combination of flashback\/voice-over narrative strongly evokes a linguistic determinism (for critical treatments of linguistic determinism, see McNally 1995 and 2001, Collins 2000). It will then discuss another five film noirs which produce images and tales of entrapment and elaborate upon this sense of a foreclosed world.\n\nThroughout _Metropolis,_ Lang's camera is generally static. One of the less spectacular exceptions to this opens up the complexity of determinism once one takes into account feedback and the impossibility of closing a system to external influences. Following the explosion of the M-machine, Freder goes to Joh's office, the nerve-centre of the city. Side-by-side, father and son walk down the length of the office, facing a camera which tracks backward away from them as they move toward it. The camera starts to track back just before they begin to walk and stops just after they do; it restarts tracking back just before they start walking again, and this time stops just before they do. The interrelation of character movement and camera movement, although apparently straightforward, is actually complex: does the camera start moving before the characters and stop before they do because it is anticipating their movements, or do the characters start and stop moving in response to the camera? And when the camera stops moving after the characters, have they anticipated its movement or has it failed to anticipate theirs? In short, is character movement subject to camera movement, or vice versa? That the camera does not always anticipate the characters' movements, and that the characters on one occasion pre-empt the camera coming to a halt, is indicative of a complex determinism; the state of this system from moment-to-moment cannot be reduced to cause-and-effect. This sequence from _Metropolis_ should be considered alongside the title sequences of _Detour_ (1945), _Sunset Blvd._ (1950) and _D.O.A._ (1950), three film noirs made for three different studios \u2013 the 'big five' major Paramount, the 'little three' major United Artists and the 'poverty row' minor PRC (Producers Releasing Corporation) \u2013 which range from a prestigious A-picture to a low-budget B-movie shot in six days.\n\n_Sunset Blvd._ opens with a shot of a sidewalk: the camera tilts down to reveal the street name on the kerb and then tracks away and into the road, always pointed downward, excluding everything else from the frame as the credits roll. At last, as director Billy Wilder's name appears, the camera tilts up to look straight back down the road as police motorcycles and cars, their sirens wailing, appear over a crest. The camera pans through 180\u00b0 to follow them as they race past. As this minute-and-a-half-long shot comes to an end, Joe Gillis's (William Holden) voice-over begins, offering to tell us 'the facts, the whole truth' about how he came to be floating face-down, dead, in the swimming pool behind a former film star's mansion. He selects 'the day when it all started' about six months earlier. A failing screenwriter, he was 'grinding out original stories, two a week' that he could not sell \u2013 'Maybe they weren't original enough, maybe they were too original'. Fleeing repo men, he turns into the driveway of silent-era star Norma Desmond (Gloria Swanson). A peculiar relationship develops, and when Joe eventually tries to leave her, she shoots him.\n\n_D.O.A.,_ inspired by Siodmak's _Der Man, der Seinen Morder Sucht_ (1931), opens with a shot that tilts down from an imposing office building at night, reaching street level as Frank Bigelow (Edmond O'Brien) enters the left of the frame and hurries across the road to the building. Dissolve to a tracking shot, the camera following Bigelow down a lengthy corridor into the Police Department. He reaches a juncture, where a cop points him down the left-hand corridor, and as he and the camera turn down it, the shot dissolves to another lengthy corridor, which Bigelow follows until he reaches a door marked Homicide Division. These two shots of Bigelow walking purposefully down two corridors whose symmetry is emphasised by his position in the middle of both the corridors and the frame, directly below the line of the ceiling lights, last for one-and-a-half minutes. Throughout, the camera maintains the same distance from his back. Bigelow has come to report his own murder, and he tells, in flashback, of being poisoned with a 'luminous toxin' which is slowly but surely killing him. When he completes the story of his investigation into the murky reasons for his assassination, he dies.\n\nFIGURE 4 Already dead: _Sunset Blvd._\n\n_Detour_ opens with an eighty-second shot of a road, filmed from the rear of a car as it drives through the desert. Unlike _Sunset Blvd._ and _D.O.A.,_ this opening shot is not incorporated into the narrative. The story, remembered in flashback, follows Al Roberts (Tom Neal) as he hitchhikes from New York to Hollywood to join his girlfriend, Sue (Claudia Drake). He is picked up by Charles Haskell, Jr (Edmund MacDonald), who dies under circumstances which Al thinks will look like murder. He adopts Haskell's identity to get away, but makes the mistake of picking up Vera (Ann Savage), who had previously been given a lift by Haskell. She blackmails Al into helping her get money, plotting first to sell Haskell's car and then to collect his inheritance. When Al finally stands up to her, he accidentally kills her. He wanders off, caught in an impossible situation: if he claims to be Al, he will look guilty of Haskell's murder; if he claims to be Haskell, he will look guilty of Vera's murder; and if he tells the truth, no-one is likely to believe him. The film ends as Al is picked up by the police.\n\nEach of these title sequences encapsulates the fatalist sensibility with which the protagonists tell the determinist world their respective films depict. _Sunset Blvd._ posits a parallel between Joe and the camera: both happen to turn into that particular road and are then swept up in a narrative. Joe regards it as mere chance that intervened six months earlier, but his retrospective narration suggests an inevitable trajectory from the driveway to the swimming pool. The contradiction in this logic stems from the narrative imposition of a finite boundary on causation. Just as Wilder's decision to start his film with that kerbstone was, ultimately, arbitrary, so is Joe's selection of that particular moment. This is not to say that these are capricious choices. They are arbitrary in that they depend upon the exercise of discretion to select a wellspring-moment; and they are arbitrary in that they become, in retrospect, absolute, a despotic cause-and-effect chain. To isolate that moment as the cause of his death six months later is to ignore the circumstances that brought him to that juncture, as well as all the alternatives the intervening months offered. But Joe is a story-teller, and every story \u2013 in Hollywood, at least \u2013 is supposed to have a beginning, a middle and an end, even if _Sunset Blvd._ interferes with that order so that the end appears to be implicit in the beginning. The world of the film is determinist, but Joe's telling of it is fatalist. His resignation transforms his telling of the world into a cause-and-effect chain whose links, because they can be constructed and told in one direction, seem also to be tellable in reverse, producing a sense of inevitability. But links are forged: they are made, shaped, crafted. They are fabrications, the product of a telling. They have the logic of story, not of world.\n\n_D.O.A._ is narrated not by a dead man but a dying one. Just as Frank seems to be propelled through police headquarters by the camera, so he is propelled through a narrative which, although at times confused and confusing, seems equally predetermined. As he races from location to location, bursting into room after room, his investigation folding back on itself, laying out possible permutations of characters and crimes until he hits on the right one, his path seems anything but predetermined. However, the narrative is not of his investigation but of his death, of the progress of the poison, of his body's failing resistance. In the film's later stages, the action accelerates as if Frank can hear his body counting down. The drive to the climax becomes desperate, hastily tying all the ends together. As long as he keeps talking, he cannot die; but he must talk quickly to tell his story before he dies.\n\n_Detour_ 's Al is a tale-teller, too, but with a less certain linguistic facility. As a hitchhiker, he can never tell whether he is talking too much or too little. He responds to Haskell's sexist tirade with cautiously grunted affirmatives. He loses every argument with Vera. He becomes tongue-tied when trying to sell Haskell's car. But there is a contradiction between the tale Al tells and the telling of that tale. His voice-over reveals a deadpan hard-boiled \u2013 if self-pitying and self-serving \u2013 fluency and wit: 'While the mechanic inspected the car, we haggled. At last, when we were all worn out, we reached a compromise. His price.'\n\nFIGURE 5 Dead man walking: _D.O.A._\n\nAl's recollection of the past is 'in itself a means of blotting it out, and his commentary, far from serving as the clue which leads us infallibly to the meaning of the narrative action, is like a palimpsest beneath which we may glimpse the traces of the history he has felt compelled to rewrite' (Britton 1992: 175). In the contradictions in the voice-over and between it and depicted events, Andrew Britton finds 'the residue of the process of rationalisation and revision to which all Al's memories have been subjected' (1992: 177). For example, his explanation of why he took Haskell's identity sounds like the rehearsal of post-hoc justification:\n\n> I saw at once he was dead and I was in for it. Who would believe he fell out of the car? Why, if Haskell came to, which of course he couldn't, even he would swear I'd conked him over the head for his dough. Yes, I was in for it. Instinct told me to run, but then I realised it was hopeless. There were lots of people back down the road who could identify me. That gas station guy and the waitress. I'd be in a worse spot then, trying to explain why I beat it. The next possibility was to sit tight and tell the truth when the cops came, but that would be crazy. They'd laugh at the truth, and I'd have my head in a noose. So what else was there to do than hide the body and get away in the car?\n\nIn narrativising himself, Al constructs a victim of circumstances, guilty only of accepting a lift one day in Arizona: 'Until then I'd done things my way, but from then on something else stepped in and shunted me off to a different destination from the one I had picked for myself.' This self-construction is not merely verbal: the camera which tracks in on Al's perpetually hangdog face in the frame-story diner seems, in his flashback, to prowl through the night-club, hunting for him.\n\nFIGURE 6 Perpetually hangdog: _Detour_\n\n_Detour's_ opening shot, although not connected to the narrative, is suggestive: it is a retrospective view. When we first encounter Al, he is heading from west to east, not only rescinding his previous journey but also the trajectory of western mythology. When heading west (to Hollywood, where his fianc\u00e9e awaits him) and into the future, Al seeks some vaguely-defined goal, never looking back (to see, as we do in the opening shot, a broken-down car beside a desert road). He lurches from situation to situation, thinking he is making progress, even though the west coast, where Sue has been reduced to hash-slinging, can offer no solution to their impoverishment. Turned around, and heading back east, he can see only catastrophe. Short on self-knowledge and lacking even Vera's critical acumen \u2013 she has few illusions about the workings of capitalism and patriarchy \u2013 he can only make sense of himself as someone who has been randomly and unjustly victimised: 'Yes, fate or some mysterious force can put the finger on you or me for no good reason at all.' This obvious resort to a metaphysical agency to excuse his actions points to an ideological obfuscation in which individual agency is treated as an isolatable single cause, as if it were consciousness that determined social existence rather than social existence that determined consciousness. The mystificatory reduction of agency to individual psychology obscures the extent to which that psychology and agency, understood as causes, are also, perhaps primarily, effects. Where Joe's voice-over is addressed directly to the audience and Frank's to a surrogate audience, Al's is addressed primarily to himself \u2013 a self-justification and the rehearsal of a confession which we overhear. This emphasises the extent of Al's alienation. Talking to himself he can control language and master himself, albeit through misrecognising himself in the narrative subject he constructs; but plunged into an intersubjective arena, he comes adrift, stumbling over both words and roads. As Al's words and the narrative they shape come to an end, so does his aimless easterly wandering. Stopped by the police, he is silent; with nothing left to say, he is arrested.\n\nWhile each of these three films posits a determinist universe, the combination flashback\/voice-over foregrounds the process of narrative construction by which determinism becomes confused with notions of cause-and-effect, predictability and inevitability. Determinism is better imagined in terms of a continually collapsing wavefront simultaneous with rather than anterior to the moment. Cause-and-effect is a retrospectively-constructed trajectory through this succession of simultaneities. Emerging from a narrativising impulse, cause-and-effect produces (illusory) inevitability-effects and fuels the mistaking of determinism for predictability. Its telling is partial.\n\nAs Britton notes, the American road is typically mythologised as a site of 'individual resistance to the constraints of an intolerably oppressive, conservative and regimented culture', but _Detour_ insists on reconnecting the individual with social institutions, reimagining the road not as 'a refuge for exiles from a culture in which American ideals have been degraded, but [as] a place where the logic of advanced capitalist civil society is acted out by characters who have completely internalised its values, and whose interaction exemplifies the grotesque deformation of all human relationships by the principles of the market' (1992: 182). Although not all film noirs are as explicit in their social critique as _Detour,_ many combine depictions of rationalised or mechanistic processes with potent images of determinate discourses and social structures.\n\nAlthough few would now argue that film noir shattered the dominant imagistic conventions of classical Hollywood, the overt and excessive stylisation of _Stranger on the Third Floor's_ six-and-a-half-minute dream-sequence pushes very hard against them. Joe Briggs (Elisha Cook, Jr) is convicted of a brutal murder on circumstantial evidence and the testimony of struggling journalist Mike Ward (John McGuire), who discovered him standing over the corpse. Mike's fianc\u00e9e, Jane (Margaret Tallichet), is not convinced of Briggs's guilt, and Mike himself begins to entertain doubts. He chases a mysterious stranger (Peter Lorre) from his apartment building. When Mike cannot hear the snoring of his interfering neighbour, Albert Meng (Charles Halton), he imagines that Meng is dead. Mike realises that his own past behaviour and angry words would provide exactly the kind of circumstantial evidence that convicted Briggs. He falls asleep and enters a nightmare world of exaggerated proportions, distorted perspectives and barren minimalist sets which emphasise his isolation and incarceration. He wakes up and finds Meng has indeed been butchered, just like Briggs' supposed victim. No-one believes Mike's story of the stranger, but Jane eventually finds the mentally-unbalanced killer, who dies after confessing to the murders.\n\nMike's dream, often cited as evidence of German Expressionism's influence on film noir, merely hystericises the primary diegesis, itself a remarkable world of dark shadows and pools of light in which the courtroom statue of Justice becomes ominous, as if her blindfold does not promise impartiality but signals an inability to see clearly. The dream sequence's anti-realist distortions might well derive from _Das Kabinet des Dr. Caligari,_ but film noir worlds resemble _Stranger on the Third Floor_ 's waking world, and not just visually. It foregrounds the economics of everyday life \u2013 Mike's enthusiasm for testifying is linked to a payrise which will finally enable him to marry Jane \u2013 as well as the sexual repressiveness and libidinal underbelly of conventional morality; and it is in the waking world that Mike unwittingly describes the extent of his entrapment, protesting to the D.A., 'I'm as sane as you are, and if you think I had anything to do with it, you're crazy'.\n\nA claustrophobic _mise-en-sc\u00e8ne_ composed of encroaching darkness and visual distortions was common to film noirs, often for economic reasons because in autumn 1942 'the war Production Board imposed a ceiling of $5,000 on new materials for set construction, and ways of economising included using lighting techniques which minimise the limitations of the sets' (Walker 1992a: 27). The techniques developed to deal with these restrictions would later be important to the independent production companies that proliferated after the 1948 Paramount decision, but not all film noirs relied upon them to create a sense of entrapment. Although its sets are not as minimalist as _Detour's,_ Lang's _Beyond a Reasonable Doubt_ is extremely pared down, with a flat visual style and relentless narrative. Ex-newspaperman and would-be novelist Tom Garrett (Dana Andrews), plots with his former boss, Austin Spencer (Sidney Blackmer), who is also the father of his fianc\u00e9e, Susan (Joan Fontaine), to be wrongly convicted for murder so as to discredit capital punishment. Having planted evidence and positioned himself as a suspect, Tom is soon given the death penalty, but the evidence of his innocence is destroyed when Austin is killed on his way to expose the erroneous conviction. Just before Tom is to be executed, an account of their plan is found among Austin's papers. When Susan tells Tom of his imminent pardon, he accidentally reveals that he _is_ guilty of the murder and confesses to using Austin's scheme to get away with it. Susan intervenes before the pardon can be signed and Tom is returned to death row. Douglas Pye identifies several narrative, rather than visual, techniques which draw the viewer through this remorselessly unfolding narrative. Because we know about Austin's plan from the outset, we 'are therefore likely to believe that we have privileged access to narrative information' (1992: 104). Each of the film's forty sequences are closely linked to those which precede and follow it, producing 'a sense of very strong narrative progression' (ibid.); this is accelerated by omitting transitional shots and only showing each scene's narrative kernel. Characters are rarely seen in isolation, depriving us of insights into their thoughts and feelings. The camera is generally distanced from characters, discouraging our participation and 'requir[ing] us to scrutinise the characters' (1992: 106) rather than empathise with them. The actors' performances are muted, too. Where _Detour_ 's Al, disguised as Haskell, melodramatically refers to 'this nightmare of being a dead man', _Beyond a Reasonable Doubt_ is populated with automata, people subjected to rationalised, if absurd, processes. Both Austin's and Tom's schemes are every bit as instrumentalist as the death penalty Austin wishes to discredit.\n\nWhile _Detour_ refers back to the Depression, _Pitfall_ firmly locates its critique in the postwar world, introducing concerns which would become far more prominent in Eisenhower's placid decade. The opening sequence, in which insurance agent John Forbes (Dick Powell) breakfasts with his wife, Sue (Jane Wyatt), and son, Tommy (Jimmy Hunt), focuses on his dissatisfaction with his comfortable life. Married to his high-school sweetheart, with whom he once dreamed of sailing around the world, he feels trapped by the responsibility of providing for his family, by the interwoven determinants of capitalism and patriarchy structuring his social being. He does not want to be just one of fifty million average Americans, and longs for an idealised exotic Latin America of, in Sue's words, 'dusky dames'. His one-night stand with model Mona Stevens (Lizabeth Scott) is unsurprising, then, but despite initial appearances she is no femme fatale. When she discovers Forbes is married, she refuses further romantic or sexual involvement. This interest in female identity, however much the unfolding thriller narrative might marginalise it, has been signalled before Mona's appearance. Dropping John off at work, Sue points out that she too is getting bored with his routine farewell peck on the cheek. He kisses her on the mouth, but as he turns to get out of the car he misses the less-than-impressed expression that indicates her own sexual dissatisfaction.\n\n_Caught_ opens with carhop Maud (Barbara Bel Geddes) and her roommate Maxine (Ruth Brady) leafing through a fashion magazine, admiring jewels and furs, while the _mise-en-sc\u00e8ne_ reveals the gulf between current situation and idealised lifestyle. While Maud bathes her aching feet, Maxine questions her about day-to-day expenses. As she calculates how long it will take Maud to save up for a night-school course, the limited choices available to these women become clear: the course is at the Dorothy Dale School of Charm, where Maud will learn elocution and comportment in order to become a department-store model like Maxine in the vague hope that she will meet a 'real man' \u2013 defined, by Maxine, as someone able to buy her a fur coat (later, explicit parallels are drawn between modelling and prostitution). Maud, who changes her name to Leonora, _does_ meet and marry a wealthy man, tycoon Smith Ohlrig (Robert Ryan). A montage sequence perfectly captures her entrapment within a conflation of capitalism, patriarchy and the American dream, with newspaper headlines equating her transformation from a carhop to Mrs Ohlrig with success, as if marrying into wealth is the pinnacle of female ambition and achievement. Unknown to Leonora, however, Smith married her in a fit of pique when a psychiatrist diagnosed his heart problems as psychosomatic responses to not getting his own way. A tyrannical husband, he neglects Leonora, treating her like an employee. Shot composition and _mise-en-sc\u00e8ne_ dwarf her, and depth of field frequently emphasises the emotional distance between the unhappy couple.\n\nThe resolution is deeply problematic. Leonara becomes a receptionist and falls in love with one of her employers, paediatrician Dr Larry Quinada (James Mason), who despite his wealthy background has chosen to work among New York's impoverished. However, the happy ending depends upon Leonora, pregnant by Ohlrig but in love with Quinada, giving premature birth to a child who dies. The Cinderella story of romantic love and social advancement is replaced by one of romantic love, noble self-sacrifice and suffering, but both are refuted. If the 'ideological significance of lovers living happily ever after lies in the unspoken, and usually invisible, metamorphosis that is implied to take place at the end of every happy ending [by which] lovers are transformed into mothers and fathers, into families' (Harvey 1998: 37), this resolution is disturbed in _Caught_ by the infant's unseen corpse, a material remnant blocking ideological closure.\n\n_Beyond the Forest_ is similarly structured around the triangular relationship between a woman who desires wealth, a callous millionaire and a self-sacrificing doctor; an unwanted pregnancy precipitates its final crisis, too. However, the protagonist, Rosa Moline (Bette Davis), is a femme fatale \u2013 or, rather she would be if the film was told from the perspective of one of the male characters. A gold-digger whose ascent has stalled, her flashback narrative humanises her, preventing her reduction to a seductive image. This emphasis on the sexual woman as a character is not the only change worked on familiar film noir patterns; it also reworks the city\/country dichotomy which shapes noir geography. While _The Big Heat_ and Siodmak's _The Killers_ contrast the noir city with exurban domestic idylls (one is outside the city, in the suburbs, the other above the city, on the roof), so _Out of the Past_ and _On Dangerous Ground_ (1952) contrast the noir city with a redemptive countryside. This opposition is ironised by the artifice of _You Only Live Once's_ honeymoon lodge where the couple-on-the-run try to romanticise a pond full of croaking frogs and of _Gun Crazy's_ montage of back-projected 'postcard' images in front of which the couple-on-the-run try to go straight; and by _The Asphalt jungle's_ Dix Handley (Sterling Hayden), who commits crimes to buy back the family farm, only to die of a gunshot wound when, on the run, he finally reaches it. _Beyond the Forest_ offers a more complex revision of these terms and the values they represent.\n\nFor Rosa, Loyalton is a prison, overseen by an infernal mill that 'sucks all the juice out of' it and populated with small-minded, unambitious, mundane people, resigned to or, more damningly, contented with their lot. Rosa longs for material goods and big city life, but her husband Lewis (Joseph Cotten), likes being a small-town doctor, despite persistent financial difficulties. He relishes being part of a community, while Rosa, ever since she was a schoolgirl, has sought to set herself apart, desired to be something other than a wife and mother. Rather than trying to make her way in the city as a young single woman, she strategically married Lewis, thus obtaining a sense of upward mobility. Further advancement has been thwarted by Lewis's contrary ambition to be a good small-town general practitioner rather than a big-city specialist. The flashback narrative \u2013 Rosa is on trial for murder \u2013 opens with Rosa, Lewis and Moose (Minor Watson), a recovering alcoholic, making their way to a hunting lodge in the mountains. Once there, manipulative Rose sends Lewis back to Loyalton and gets Moose drunk. She then waits in the neighbouring lodge for her lover, millionaire Neil Latimer (David Brian), who laughs at her suggestion that she should divorce Lewis and marry him.\n\nIncreasingly discontented with her home and marriage \u2013 'If I don't get out of here, I'll die; if I don't get out of here, I hope I die' \u2013 Rosa demands that Lewis's patients pay their bills. She uses the money to visit Latimer in Chicago, where he tells her he is in love with another woman. Lewis made it clear that if she went she should not come back, but on her return, as a later plot twist implies, she uses sex to change his mind. Latimer, realising that really he wants Rosa, returns. When Moose threatens to tell Latimer of her pregnancy by Lewis, Rosa 'accidentally' shoots him while hunting. Because no one knows of the affair, Rosa does not appear to have a motive for killing Moose. She is acquitted, but Latimer now wants to postpone their plans so there will not be any scandal. Fearful that Latimer will not marry her if he knows she is pregnant, she tells Lewis everything. He tells her she can go anywhere and do whatever she pleases \u2013 after she has had his baby. She throws herself down a mountainside to induce a miscarriage, to get rid of the 'dirt hanging to' her. Delirious, desperately ill and convinced that Lewis is poisoning her, she makes her escape, only to die before she can reach the train station.\n\nThroughout the film, the domestic is never idyllic, but the site of a clash of wills between Rosa, who has very clear ideas about housework (which is beneath her), and her slovenly, idle and, above all, reluctant maid. Nor is the surrounding countryside idyllic for Rosa. It might be beautiful, and even restful for Lewis, but it is also the site of adultery and murder, not redemption. On the first trip to the lodge, Rosa's high-heels signal not only her incongruous presence but also her instrumentalism: she is there to seduce Latimer. The hoe-down to celebrate Moose's birthday and the return of his estranged daughter might evoke an image of community familiar from John Ford's westerns, but it is criss-crossed by the sexual tensions that result in Moose's death the following morning. Similarly, the city is positioned between two sets of associations, alternating throughout the Chicago sequence as a place of hope for Rosa, then of despair as she cannot contact Latimer, then of hope again when he arranges to meet her. Finally, when he has told her of his plans to marry someone else, the city becomes noirishly distorted, disorientating and threatening as she runs off into the rain.\n\nA number of film noirs, including _The Fallen Sparrow, Cornered_ (1945), _The Blue Dahlia, Crack-Up, Somewhere in the Night_ (1946), _Dead Reckoning_ (1947), _Ride the Pink Horse_ (1947) and _The Crooked Way_ (1949), focus on returning war veterans who are often physically, mentally or emotionally traumatised. Like _I Am a Fugitive from a Chain Gang,_ these films often transform the disorientation of return to society into a sense that society itself is disoriented. As Harvey suggests:\n\n> the encounter with a depressed peacetime economy, with its threat of high prices and rising unemployment, began a process of general disillusionment for many of those returning home after the war, in search of the values which they had fought to defend. It is this breakdown also, this erosion of expectations, that finds its way into film noir by a series of complex transmutations. The hard facts of economic life are transmuted... into corresponding moods and feelings. Thus the feelings of loss and alienation expressed by the characters in film noir can be seen as the product both of post-war depression and of the reorganisation of the American economy. (1998: 39)\n\nAlthough focused on film noir's returning (white) veterans, this argument also contextualises the situation of _Caught's_ Maud\/Leonora and Mildred (Joan Crawford) in _Mildred Pierce_ (1945). While _Shadow of a Doubt, Beware, My Lovely_ and _On Dangerous Ground_ might be regarded as metaphoric treatments of loved ones coming to terms with war-traumatised men, _The Reckless Moment_ focuses instead on the wife who was left behind to raise the family. Lucia Harper (Joan Bennett) lives in the wealthy Californian community of Balboa with her father and her two teenage children, Bea (Geraldine Brooks) and David (David Bair). Her husband, who is away on business, is despatched by his company to Berlin to help rebuild a bridge. Unable to return home for Christmas before going overseas, he is absent throughout. Lucia, who coped without him for three years during the war, does not shrink from confronting sordid Ted Darby (Shepperd Strudwick), who has been dating Bea, nor from disposing of his corpse after his accidental death. However, Bea's incautious love-letters have fallen into the hands of Nagle (Roy Roberts), and his associate, Martin Donnelly (James Mason), who want $5,000 for them. The shadows in Lucia's house seem to coalesce around soft-spoken Donnelly's dark presence, but as he spends time with her, the brooding blackmailer, seeing how she is surrounded by a family from which she can never escape, feels a growing sympathy for her. Unable to contact her husband, she cannot borrow money from her bank or a loan company. In a touching moment, having struggled to find the money by planning future household economies, she briefly, half-heartedly, complains about the amount of electricity her family wastes by never switching off the lights. In his penultimate scene with Lucia, Donnelly tells her about his childhood. One of five sons, his mother, who could never see that he was the bad one, wanted him to become a priest: 'I never did a decent thing in all my life. I never even wanted to until you came along. Then I began to think if only I could turn back and start over. So what happens? As soon as I try to start back I find myself with this on my hands.' At the end of the film, Mr Harper phones from Germany. As Lucia takes the phone, the camera entraps her behind the banister as her family surround her. Donnelly might have yearned for Lucia and for family life, and have been prepared to die so as to preserve Lucia's family, but this final shot indicates that Lucia is every bit the prisoner he had earlier suggested \u2013 even if she does not realise it.\n\nThroughout, the film has toyed with the audience. When Lucia returns from dumping Darby's corpse, she is told she should not have taken the boat out as it needs new spark plugs: one breathes a sigh of relief. Lucia and Donnelly appear together in potentially compromising situations: will people think that she is having an affair? Much is made of Lucia's misplaced shopping list: will it be found with Darby's corpse? At these and other junctures, the narrative could have gone in very different directions. That it does not might seem to confirm Donnelly's fatalism; but again, the narrative is just one route through the branching possibilities at which the film hints. One does not have to reject determinism to deny the inevitability of this particular series of events. Rather, the alternatives routes the narrative might take exist precisely because of the nest of determining frames from which the actual narrative emerges. It is how the wavefront collapses but, as the alternative possibilities suggest, there is no compelling reason for it to have collapsed into this particular trajectory.\n\nOn some level, narratives are always about the possibility of agency, as the comparison of _La B\u00eate humaine_ and _Double Indemnity_ demonstrated. After the Second World War, as the collectivism represented by the New Deal Administration and fighting the war mutated into the Cold War consensus, as postwar capitalism demanded the replacement of the heroic entrepreneurial individual with the organisation man (and ambition, drive and initiative with personality and fitting-in), so America increasingly took on the image of the reified commodity universe described by the Frankfurt School. This dominant instrumentalism was exemplified by the postwar accommodation between labour and capital (see Davis 1986) \u2013 or, rather, the negotiated subordination of labour to capital \u2013 and the constraints it imposed on the social realm shape the film noir world in which individuals struggle for agency, for being. The protagonists discussed in this section are ultimately trapped by having to sell their labour in a world which does not necessarily want or need to buy it but denies alternative modes of existence. They are webbed into social and institutional relations which are dominated by exchange value and egotistical calculation and which deny them egress. Mostly, they are guilty of just wanting to pay the bills, or of wanting a break from the relentless necessity of doing so.\n\nInvestigation\n\nFrom _Stranger on the Third Floor_ to _The Naked Kiss,_ the investigation was a staple film noir plot, albeit often obscured, distorted, derailed. While the underlying logic of this plot is that the world can be known, the film noir investigator frequently struggles to reconstruct and tell an order of events that make any kind of sense. Coincidences, hidden interrelations, unclear and confused motives abound. Consequently, the film noir investigator repeatedly uncovers the order embedded deep within chaos, witnesses order emerging from chaos; he or she then, typically, has to construct plausible cause-and-effect chains to tell what has happened, to map a particular \u2013 and partial \u2013 trajectory. In doing so, the film noir investigator offers a model of the complexly-determined 'fuzzy' subject (see Bould 2002). This section will focus in particular on the linguistic determination of the masculine subject in six investigative and two heist film noirs. It will begin, however, with a consideration of hard-boiled prose.\n\nAccording to Frank Krutnik, 1940s Hollywood turned to hard-boiled crime fiction for two related reasons. With many leading writers, directors and stars drafted, and with externally-imposed budgetary restrictions, the studios needed 'alternative production values', such as 'story source' (1991: 37), with which to appeal to audiences. Wartime paper-rationing meant less fiction was being published, so they turned to pre-war pulps like _Black Mask_ and their professional writers, who were used to producing serviceable prose to deadlines.\n\nIn _Murder, My Sweet_ patrician con-man Amthor (Otto Kruger) criticises Marlowe's 'unpleasant tendency toward abrupt transitions, a characteristic of your generation' \u2013 an observation equally applicable to 'the _Black Mask_ type of story', in which 'the scene outranked the plot in the sense that a good plot was one which made good scenes' (Chandler 1980: 10). (This description of plot strongly resonates with the notion of cause-and-effect as a partial and retrospective telling of a trajectory abstracted from a total system which changes from moment to moment.) Chandler continues:\n\n> We who tried to write it had the same point of view as the filmmakers. When I first went to Hollywood a very intelligent producer told me that you couldn't make a successful motion picture from a mystery story, because the whole point was a disclosure that took a few seconds of screen time while the audience was reaching for its hat. He was wrong, but only because he was thinking of the wrong kind of mystery... Undoubtedly the [hard-boiled] stories... had a fantastic element. Such things happened, but not so rapidly, nor to so close-knit a group of people, nor within so narrow a frame of logic. This was inevitable because the demand was for constant action; if you stopped to think you were lost. When in doubt, have a man come through a door with a gun in his hand. (Ibid.)\n\nThis match between hard-boiled fiction and film noir is demonstrated by the frequency with which a man with a gun, or some equivalent, does indeed walk in through the door. The flashback narrative of _Murder, My Sweet_ begins with Marlowe, bored and restless. As he sits in his office, waiting for his date to phone, Moose Malloy suddenly appears, a giant figure reflected in Marlowe's window. Despite his bulk, his approach has been noiseless. He makes other equally sudden appearances: at the night club; after Marlowe's escape from the clinic where he has been held a prisoner; at Marlowe's office again. Such abrupt entrances, and the narrative twists and turns that follow, are not limited to adaptations of Chandler. _D.O.A._ shuffles and reshuffles its characters in this way. _Caught_ dangles potential narrative forks. In _Pitfall,_ psychotic thug MacDonald (Raymond Burr) exists mainly to perform tasks equivalent to walking through the door with a gun.\n\nWilliam Marling (1995) suggests intriguing parallels between hard-boiled fiction and the transition in 1920s design from organic, static, detailed Victoriana to streamlined, smooth, dynamic moderne. The overall trajectory suggested by the careful ordering of a series of components \u2013 exemplified by the episodic and telegraphic verticality of the stepback skyscraper \u2013 is homologous to the narrative form Chandler ascribes to hard-boiled fiction and of investigative film noirs. The privileging of scene over plot is evident in the famous anecdote about _The Big Sleep_ in which Bogart, baffled by the labyrinthine plot, asked who had actually killed one of the characters. Neither director Howard Hawks nor screenwriters Jules Furthman, William Faulkner and Leigh Brackett knew, so they asked Chandler, who did not know either. Furthermore, as David Thomson suggests, a number of characters \u2013 the Acme bookshop proprietress (Dorothy Malone), Harry Jones (Elisha Cook, Jr), Joe Brody (Louis Jean Heydt), Agnes (Sonia Darrin) \u2013 are superfluous to the film, prompting his claim that _The Big Sleep_ 'is one of the most formally radical pictures ever made in Hollywood... a kind of ongoing rehearsal or improvisation... a celebration of acting, dialogue (as opposed to talk) and fantasising' (1997: 64). Such hyperbole stems from over-attendance to conventional notions of classical Hollywood narrative \u2013 notions which, while recognising the importance of spectacular spectacle (song-and-dance routines, fistfights, gunfights, landscapes, chases, physical routines), consistently underplay the importance of the non-spectacular spectacle (the romantic embrace, melodrama's suffering, science fiction's pseudo-scientific chatter and, most importantly, across all genres, the spectacle of stars delivering dialogue) and the way such attractions are often privileged over a coherent narrative arc. While _The Big Sleep_ might be remarkable, it is far from exceptional. With budgetary restrictions on more overt material spectacle, sensuous pleasure in dialogue is precisely one of the alternative production values Hollywood could utilise; and, as Krutnik notes,\n\n> one of the defining features of 'hard-boiled' writing is its language. The 'hard-boiled' idiom is 'tough', cynical, epigrammatic, controlled \u2013 a sign of the hero's potency... In many of the private-eye stories, language is wielded as a weapon, and is often more a measure of the hero's prowess than the use of guns and other more tangible aids to violence. Confrontations between the hero and his adversaries frequently take the form of extended sessions of verbal sparring as each seeks to assert his masculine competence. There may occasionally be 'wise-cracking' dames, but this often signifies a dangerous competitive streak... More often, the books set up an opposition between the male as language user and the woman as erotic object, as a glorified body of awesome excitation (which poses its own dangers, of overwhelming male rationality). (1991: 43)\n\nThis is evident even in an author as crude as Carroll John Daly. In 'Three Gun Terry' (1923), 'the first tough detective story starring the world's first wisecracking, hard-boiled private investigator' (Nolan 1985: 43), there is no dame, merely an innocent \u2013 and nearly voiceless \u2013 girl, defined almost exclusively in terms of her physicality and the reactions she prompts in phallic Terry whose coherence is constantly threatened by her:\n\n> Say, but that girl was scared; why, she didn't do nothing but hang close to me and keep her head up against my chest as she clung to my coat. And she was mighty little and mighty young too, I think, though I couldn't tell much about her, there in the dark of the cab. Somehow I felt almost like a father as I patted her little dark head and ran my fingers through her soft black locks. I could 'a laughed, but somehow I didn't... And there I was, telling her that she was all right, and that I'd take care of her and \u2013 and \u2013 oh \u2013 just acting like a regular nut. What I should 'a been doing was questioning her and finding out just what her old man was worth and how much there would be in it for me. But somehow I didn't do anything but try to comfort her like she was a baby.\n> \n> [...]\n> \n> And in about five minutes she comes in, and she's a wow. I didn't get a good look at her before, and I tell you it's a lucky thing I ain't romantic... Even me, a hardened citizen like me \u2013 yep, I was nearly ready to take ten dollars off the bill...\n> \n> [...]\n> \n> And right there is another thing that I can't explain. Maybe it's weakness, but I like to think it ain't, though I can't account for it. You might think that I had done enough for this girl and earned my pay \u2013 well, perhaps I had. But there was soft little hands around my neck and silken hair against my cheek \u2013 great innocent, childish eyes looking through pools of water into mine \u2013 and \u2013 well, I stayed \u2013 yep, I just played the fool and stayed.\n> \n> So it was I held her in my arms when half a dozen cops busted into the room. (Daly 1985: 46, 50, 67)\n\nThis is not to claim that there was a single hard-boiled voice \u2013 compare Hammett's 'objective' prose, resolutely external to characters, leaving their motives to be interpreted from their closely observed actions, with Woolrich's baroque, almost-out-of-control contortions \u2013 nor that all film noirs sought to replicate hard-boiled dialogue.\n\nThe latter point can be observed by comparing _Murder, My Sweet_ to _Crossfire_ (1947), made three years apart by the same producer (Adrian Scott), director (Edward Dmytryk) and screenwriter (John Paxton). Both are adapted from novels, respectively Chandler's _Farewell, My Lovely_ and Richard Brooks's _The Brick Foxhole_ (1945), the former a hard-boiled classic, the latter a more self-consciously literary fiction. The narrative structure of each novel was altered. _Murder, My Sweet_ excised plot complications and changed the climax, while _Crossfire_ substantially reworked its source, not only to remove material about homosexuality and homophobia (see Corber 1997, Naremore 1998) \u2013 emphasising instead the novel's condemnation of racism in general and anti-Semitism in particular \u2013 but to provide a structure more appropriate to a noir crime-investigation. While little dialogue is retained from either source, these sources do seem to dominate Paxton's dialogue. _Murder, My Sweet_ is full of wisecracks, tough-guy slang and recurring bird and animal imagery. _Crossfire's_ more measured dialogue is virtually free of such ornamentation. This can be seen in the contrast between Marlowe's quips \u2013 'I gave her a drink. She was a gal who would take a drink if she had to knock you down to get the bottle' \u2013 and the weary resignation of _Crossfire's_ brilliantly underplayed Detective Finlay (Robert Young). (The lack of linguistic ornamentation might also arise from _Crossfire's_ generic affiliation with the social problem film, as if its subject was too serious for pulp dialogue.) But, in different ways, both films privilege language. _Murder, My Sweet_ relishes the textures of its idiom, whereas _Crossfire_ depends upon several long dialogue scenes. With most of its $500,000 budget spent _'above_ the line' on a cast which also included Robert Mitchum, Robert Ryan, Gloria Grahame, Sam Levene and Paul Kelly, that 'meant B picture expenditure _below_ the line and a schedule of only twenty-two days' (Dmytryk 1996: 31). These constraints can be seen in the barrenness of sets (especially the hotel room, the waiting room and the cinema) as well as the duration of several scenes. Although the dialogue's unornamented texture might not suggest that it be considered an 'alternative production value', the film does rely upon the unspectacular spectacle of stars delivering dialogue.\n\nWhatever its relationship with hard-boiled fiction, film noir should not be seen as a mere transcription from one medium to another. Film noir did produce distinctive effects, and one of the most important of these \u2013 the destabilisation of masculine authority \u2013 can be traced through the auto-critique of hard-boiled language increasingly evident in investigative film noirs.\n\nDeborah Thomas identifies four characteristics shared by many film noirs. First, a 'central male protagonist whose point of view is privileged through such devices as first-person narration... and subjective framing devices like flashbacks or dreams' (1992: 67). Second, an 'undermining of this point of view', either 'through labyrinthine plots which seem to elude the protagonist's attempts to give them coherence through his narration' or through 'breaks in the protagonist's consciousness' (1992: 68). Third, a protagonist lacking self-knowledge and divided against himself; this is often projected onto his desire for and fear of both the femme fatale and the domesticated woman. Fourth, a 'mood of pervasive anxiety' which might be resolved by various means, including the death of the femme fatale and the 'imminent domesticity [which] may beckon just the other side of the film's final frame' (ibid.). Closely matching this pattern, _Murder, My Sweet_ signals a significant variation on such investigative film noirs as _The Maltese Falcon_ and _The Big Sleep. The Maltese Falcon_ devotes considerable energy to constituting Sam Spade (Humphrey Bogart) as 'an ideal ego' (Krutnik 1991: 93), mastering those around him through linguistic manipulation and physical violence as well as through his possession of the point-of-view shot. In _The Big Sleep,_ only the similarly-constituted Philip Marlowe (Humphrey Bogart) can draw together the many complex, incoherent plot threads into a resolution. Such control eludes Dick Powell's Marlowe in _Murder, My Sweet._\n\n_Murder, My Sweet_ opens with a view down onto a pool of light in the middle of a table around which sit Marlowe and his police interrogators. 'I remember you as a pretty noisy little fellow, son. All of a sudden you get quiet,' says one cop. 'You lost your book of answers?' asks another, 'Or are you just waiting for your lawyer?' These opening lines establish the film's concern with language as a means of intersubjective conflict and subject-constitution. The first cop implies his superiority by constructing the past from his perspective ('I remember'), subordinating an infantilised Marlowe ('a pretty noisy little fellow') and identifying himself with patriarchal authority ('son'). Marlowe's silence \u2013 his missing book of answers \u2013 cuts him off from the symbolic realm (a lawyer is, of course, a mouthpiece). Our first clear view of Marlowe's face, with a dressing covering his eyes, seems to complete this symbolic castration. Temporarily blinded, he is a private eye who cannot see. However, even from his disempowered position, he can talk; and so he tells how he came to this juncture, reconstituting his self, his history and his identity. With an official scribe recording his words, he speaks himself back into the symbolic realm and attempts to reduce this complexly-determined moment to the outcome of a clear cause-and-effect chain.\n\nMarlowe often relies upon language to get his way. When Moose hires Marlowe to help find his ex-girlfriend, Velma, Marlowe asks how he can contact him \u2013 'I contact you', Moose replies. When Lindsey Marriott (Douglas Walton) then tries to hire Marlowe without revealing too much about the job, Marlowe verbally bullies him, attempting, through self-possessed verbal aggression, to expunge his failure to subordinate Moose. Embodying a physical force linguistic skills cannot defeat and a materiality irreducible to language, the infantile Moose represents the return of repressed unconscious drives, the proletarian subject who cannot be suppressed by bourgeois subjectivity and the material kernel which refuses linguistic determination. His asserted authority is proleptic of the unmanning, the subjective disintegration, which Marlowe will undergo.\n\nMarlowe accompanies Marriott to a nocturnal rendezvous, ostensibly to buy back a jade necklace belonging to Mr Grayle (Miles Mander) and stolen from Mrs Grayle (Claire Trevor), his second, younger, wife. Marlowe returns through the rising fog to find Marriott beaten to death. Clubbed from behind, Marlowe falls to the ground and his voice-over begins. 'I caught the blackjack right behind my ear. A black pool opened up at my feet. I dived right in. It had no bottom.' Blackness seeps across the screen until there is just a point of light at the centre. 'I felt pretty good. Like an amputated leg.' The point of light grows, irises out from Marlowe's face, which is held in a torch beam. As Marlowe struggles to his feet, the bearer of the torch flees. The pool imagery, which will reappear, is, like the film's vortical imagery, readily associated, as in _Vertigo,_ with vaginal imagery and female identity. The 'amputated leg', like Jeffries' (James Stewart) broken leg in _Rear Window_ (1954), alludes to castration. Beams of light often foreground the phallic gaze \u2013 as in Rotwang's torch-pursuit of Maria through _Metropolis's_ caverns; the spotlight in which the eponymous Gilda (Rita Hayworth) performs; and the torment by torchlight of Susan Vargas (Janet Leigh) in _Touch of Evil_ (1958). But here, although it is not revealed until later, a woman \u2013 Ann (Anne Shirley), Mr Grayle's daughter from his first marriage \u2013 wields the torch, further emasculating Marlowe.\n\nWhen Marlowe reports Marriott's murder, he is warned against investigating it: 'All I want is your silence', Lieutenant Randall (Don Douglas) tells him. Marlowe, however, is then hired by the Grayles to recover the necklace, simultaneously and lasciviously appreciating both Mrs Grayle's appearance and the double meanings in her husband's discussion of his valuable pieces. When Mrs Grayle calls at Marlowe's apartment, he is standing in front of a mirror, brushing his hair. This narcissistic self-regard and the conversation that follows resonates strongly with the Lacanian mirror phase. In response to her questions, former cop Marlowe reveals that the D.A. fired him not for incompetence but for talking back, his Oedipal identification with legal, investigative and punitive authority shattered by insubordination, by a premature usurpation of language. In front of the mirror, he constructs himself both verbally and by dressing for their date. 'I had an interesting childhood, too,' he adds, before setting out for an expensive nightclub with Mrs Grayle, a sexualised mother figure (Grayle obviously puns on grail, a vaginal image that nonetheless emphasises patriarchal authority).\n\nAt the night-club, Mrs Grayle inexplicably disappears. Ann then attempts to hire Marlowe, but also disappears. Moose diverts Marlowe from his course by delivering him to Amthor, who confesses that this virtual kidnapping is an 'old psychological trick' to see what he might, off-balance, accidentally reveal. Their confrontation is a battle of linguistic registers, tones and vocabularies. Amthor is patrician and patronising, snobbish and effete, while Marlowe is earthier, colloquial and disrespectful. After chiding Marlowe for his 'abrupt transitions', Amthor insists that 'in this case, I must ask you to follow some sort of logical progression. Now, about the police?'. 'Either they've got something on you,' Marlowe replies, 'or they're trying to get it. I didn't expect you to tell me which. I was just baiting you. It's an old psychological trick, grandpa.' Marlowe seems confident that he has won this exchange \u2013 a conclusion one might dispute \u2013 but Amthor has already persuaded Moose that Marlowe knows Velma's whereabouts, and Marlowe is unable to convince him otherwise. The ensuing physical tussle ends with Amthor pistol-whipping Marlowe. As he falls to the ground, the camera loses focus and blackness again seeps across the screen, providing a particularly complex interplay of the discursive formations J. P. Telotte (1989) identifies. The third-person camera observing Marlowe is simultaneously conflated with Marlowe's fading consciousness and his retrospective description of losing consciousness, radically destabilising its authority.\n\nMarlowe's subsequent dream is again full of vaginal imagery, phallic threats and a loss of control:\n\n> The black pool opened at my feet again and I dived in. Next thing I remember was going somewhere. It was not my idea. [Two figures dump Marlowe on the floor.] The rest of it was a crazy cold-cut dream. [The image wavers.] I had never been there before. [Marlowe falls into space; giant heads quiz him; Moose reaches out for him; he falls again, into a swirling vortex. He is pursued through a series of doors hanging in dark empty space. Web-like threads drift across the image. He emerges through a too-small door. A giant doctor empties a giant syringe into him. He falls into another vortex. He wakes up beneath a bright light, the threads still obscuring his vision.] The window was open but the smoke did not move. It was a grey web woven by a thousand spiders. [He reaches out to the web.] I wondered how they got them to work together.\n\nAfter a brief exchange with a male nurse and Moose, Marlowe settles back on his bed, rubbing his throat:\n\n> My throat felt sore, but the fingers feeling it didn't feel anything. [He stretches out a hand toward the still-visible web.] They were just a bunch of bananas that looked like fingers. [He sits up.] I wondered what I was shot full of. Something to keep me quiet. Or something to make me talk. Maybe both. [He stands. The floor wobbles. He falls back onto the bed.] Okay, Marlowe, I said to myself, you're a tough guy. You've been sapped twice, choked, beaten silly with a gun, shot in the arm until you're as crazy as a couple of waltzing mice. [He begins to try to dress.] Now let's see you do something really tough, like putting your pants on. [He struggles to dress.] Okay, you cuckoo, walk and talk. What about? Anything, everything. Just talk and keep walking. [He staggers around the room.] You're getting out of here. That's a beautiful bed. Stay off it. Walk! I walked. I don't know how long. I didn't have a watch. They don't make that kind of time on watches anyway. [Finally, the web clears. He fastens his cuffs. He is tidier but still dishevelled, his shirt sleeve badly torn.] I was ready to talk to someone.\n\nThe divided subjectivity suggested by the earlier conflation of the camera with both narrator and narratee is here picked up by Marlowe as he recalls talking to himself, exerting linguistic control over his body and reconstituting himself out of language. Again, an essential part of this self-constitution is also effected by his costume. Throughout his walking-talking cure, his improving appearance matches his improving mental condition. Once his cuffs are fastened, he is ready not just to escape but to talk to someone else \u2013 but his torn sleeve signals if not incompleteness then certainly a chink in his armoured self.\n\nMarlowe dips in and out of hysteria as he confronts Sonderborg (Ralf Harolde), the doctor who has been keeping him unconscious, and takes his gun. Sonderborg tries to persuade him he is sick and should return to bed. 'What were you saying?' Marlowe snaps. 'I made no remark,' Sonderborg replies. 'Remarks want you to make them,' says Marlowe. 'They've got their tongues out wanting to be said.' This Chandleresque phrasing suggests that language itself (rather than the speaking subject) has agency, that even the physical apparatus of speech (the punning 'tongues') belongs to language. Normally self-possessed, Marlowe feels this lack of agency mounting as the web returns to his field of vision. He wavers, but then banishes it by will alone. When Sonderborg equivocates, Marlowe points out, 'When you've got a gun in your hand, Doc, people are supposed to do what you tell them.' This phallic reassertion of self is reiterated as Marlowe forces Sonderborg to hand over a key \u2013 a symbol and instrument of his position \u2013 and then rips out the phone, depriving Sonderborg of his voice. Marlowe is far from secure, though. His own tongue betrays him when, on leaving the sanatorium, he stumbles into Moose and accidentally gives away vital information \u2013 but he has no idea what exactly it was that he should not have said. Moose, who usually merely overpowers him, here out-thinks him.\n\nMarlowe accepts Grayle's offer to drop the case, intending to pursue it for his own reasons. He persuades Ann to hand over the beach house key: 'I could bust in, but a key would make it simpler.' Once inside, Marlowe admits to suspecting the motives behind her attentions. When she denounces him and all men, Mrs Grayle, who has been in hiding, overhears. Ann turns on her, decrying such 'big league blondes. Beautiful expensive babes who know what they've got. All bubble bath and dewy morning and moonlight. And, inside, blue steel. Cold, cold like that, only not that clean.' Ann storms out and Mrs Grayle sets about seducing Marlowe into her plan to dispose of Amthor, not knowing that Moose has already killed him. Marlowe, realising that she is Velda, arranges a confrontation with Moose for the following night. Unexpectedly, Grayle and Ann also turn up. The situation spirals out of Marlowe's control. Mrs Grayle is about to shoot him, when Grayle shoots her. Moose bursts in, finds the corpse and turns on Grayle. Marlowe tries to reach Grayle, but the gun goes off in front of his face.\n\nIn the interrogation room, Marlowe finishes his story. Blinded by the muzzle-flash, he lost consciousness. He heard three shots, but he does not know who was killed (Grayle and Moose died in the struggle over the gun). He keeps asking after Ann, who has been silently listening throughout the interrogation. She signals that they are not to reveal her presence and follows as Detective Nulty (Paul Phillips) escorts Marlowe from the building, all the time asking after Ann, expressing his feelings for her. When Marlowe is put in a cab, Ann takes Nulty's place. As they drive off, Marlowe recognises her but, pretending she is still Nulty, asks for a kiss. He removes his gun from his jacket, puts it aside, and the film closes with a romantic clinch.\n\nThe respective fates of Velda\/Mrs Grayle and Ann recall the two responses to castration anxiety described by Laura Mulvey (1975). The sadistic-voyeuristic response identifies the guilty party and punishes her, while the fetishistic response substitutes a fetish object (Ann, who is infantilised as 'the kid' and made safe by her silence during the interrogation and departure from headquarters) for the dangerous female figure, the femme fatale \u2013 Ann's phallic ('inside, blue steel'), corrupt ('only not that clean'), too-young, and thus sexualised, stepmother.\n\nDespite this, the end of the film is far from being a confident reassertion of masculinity and patriarchy. Marlowe's self-assertion gathers momentum as he effectively usurps Grayle, but the potential integration into the symbolic order signalled by Ann's partnership is undermined by his apparent preference for the sexualised mother. His attempted mastery dissolves when the cleverly-arranged confrontation ends with a woman pointing a gun at him and his symbolic castration. The police turn to Ann, doubly disempowered as a woman and 'the kid', to confirm his retrospective account, and even the conventional concluding formation of a heterosexual couple is undercut by its queer current. Homoeroticism bubbles up throughout the film (see Oliver and Trigo 2003), and this final swell evinces a tension central to patriarchal operations. Heterosexuality requires Marlowe to bond with Ann, whereas patriarchy requires him to identify with the masculine and homosocial law. Having disarmed himself, his voice, which has provided the spine of the film and his sense of self, is, courtesy of Ann's kiss, silenced.\n\nIf _Murder, My Sweef's_ attempt to replicate something of its source's first-person perspective was destabilised by its own cinematic techniques, then another Chandler adaptation, _Lady in the Lake,_ pushed concern with technique to an extreme which shattered the stable identity of Chandler's Marlowe. With the exception of Marlowe's (Robert Montgomery) direct address to camera in the prologue, epilogue and an interpolated scene, the flashback narrative is told with a subjective camera which occupies Marlowe's position in each scene. On one level, Marlowe's challenge to the audience \u2013 'You'll see it just as I saw it. You'll meet the people; you'll find the clues. And maybe you'll solve it quick, and maybe you won't' \u2013 is a neat avoidance of the problem of how to recount flashback events without subsequent knowledge distorting the account. It also represents an extension of the _Kammerspielfilm's_ experiment with the _enfesselte Kamera,_ made possible by such wartime technological developments as 'the appearance of lightweight, mobile cameras like the German Arriflex and the introduction of the highly maneuvrable \"crab\" dolly that permitted longer takes [and] helped transform what previously was mainly a narrative punctuation, the subjective shot, into a viable narrative device' (Telotte 1989: 104). However, it also produces a sense of a simultaneously constrained and fragmented subject.\n\nDespite attempts to humanise or, more accurately, masculinise and heterosexualise the subjective camera, as when Marlowe's distracted gaze turns from Adrienne Fromsett (Audrey Totter) to follow her secretary, one is left with a continual sense of disjunction. Although impressive, the camera replicates neither the human gaze ('its perspective and sense of dimension, among other things, are quite different' (Telotte 1989: 105)) nor human movement (purported eye-movements are too slow, 'more akin to movements of the head, complicated by a neckbrace' (Kawin 1978: 8)). Moreover, tracking and crane shots intended to suggest Marlowe's movement through his environment are too slow, as if the unchained camera is nonetheless hobbled.\n\nThis sense of constraint is exacerbated by two further factors. First, Telotte argues that the 45 shots which 'focus on entrances, depict characters using a door for access or egress, or employ doorways or windows to frame characters and create multiple planes of focus' create a sense of expectation, 'that there is always another door to enter, something new, unsuspected, and possibly threatening yet to be encountered', that 'simply opening a door can lead to various enigmas' (1989: 108\u20139). However, this proliferation of potential entrances and exits produces a slightly different effect, opening up the conflict between a complexly-determined total system and the impulse to narrate a particular cause-and-effect trajectory. Telotte describes the film's first subjective tracking shot, in which Marlowe's viewpoint wanders down a corridor, looking for the offices of Kingsby Publications, as 'the kind of shot that recurs in the film, repeatedly creating a maze-like effect, as the starting, then halting, camera signals an eye and a human presence, randomly moving and constantly opening onto the new and the unseen' (1989: 108). On the contrary, the retrospective telling and hobbled _enfesselte Kamera_ suggests little that is random or unpremeditated: Marlowe knows he is going to Kingsby Publications, even if he must first find their office. In neo-formalist terms (see Bordwell 1985), the _fabula_ (story) hints at alternatives while the _syuzhet_ (plot) denies them. As one watches the film, the interaction of _fabula_ and _syuzhet_ produces, simultaneously, a sense of possibility and a sense of inevitability. Second, Montgomery's performance goes beyond the gruffness necessary to counter his image as a light romantic lead. His aggressiveness is evident in the prologue. Directly addressing the camera, he issues not an invitation to participate in the unravelling of the mystery but a crude, self-defensive challenge for the viewer to do any better than he did. He generally maintains this tone throughout, prompting confrontations and a similarly aggressive direct address from his interlocutors, who are usually held in fairly static medium shots. This combination of address and relative positioning fixes Marlowe in place, summoning back his wandering gaze, constraining him.\n\nHowever, this direct address to Marlowe also passes straight through him to confront the discomfited viewer. Despite the omnipresence of his gaze, Marlowe lacks substance, his body is both fragmented (his hands never enter his field of vision from quite the right angle) and absent (his voice and footsteps sound curiously disembodied). When he sees his own reflection, his reflected body and the viewpoint never quite coincide and the latter remains static even when we can see his reflected head or eyes moving. This discrepancy between viewpoint and subject is most obvious when Chris Lavery (Richard Simmons) tricks Marlowe into looking at a mantelpiece clock in front of a large mirror. Lavery, reflected, is visible as he reaches for a knuckle-duster, but Marlowe is not.\n\nLavery knocks out Marlowe: his fist fills the screen, the viewpoint blurs, sways, falls to the ground, fades to black. Although narratively-motivated, this loss of consciousness is just one of several breaks in Marlowe's continuous subjectivity, the others being more conventionally-signalled omissions of transitional sequences. This failure to develop a subjective camera technique without recourse to the traditional grammar of cuts, fades and dissolves (as well as some very visible invisible cuts) has three distinct effects. First, it makes visible the constructedness of cause-and-effect while, second, distinguishing the cause-and-effect chain from the continuity of both consciousness and the total system. Third, it further fragments Marlowe, editing his consciousness and increasing his unreliability as a narrator (and author \u2013 he has turned, Hammett-like, from detective to pulp crime writer). This fragmentation is most pronounced by a pause in the narrative while Marlowe reappears as a direct-address narrator to tell of his trip to the morgue to view Muriel Chess's badly decomposed corpse. Despite Marlowe's opening challenge, the audience is denied the evidence of his (and its own) eyes. The private eye turned public, both as author and narrator\/viewpoint, becomes private once more, denying the simultaneity of camera-subject and audience (upon which the film elsewhere insists) so as to mislead the audience. Instead of seeing through Marlowe's eyes, we hear a lie \u2013 Marlowe's misidentification of Crystal Kingsby's corpse, which he knows to be incorrect when he narrates it \u2013 and this sutures the viewer to a particular narrative trajectory just as Marlowe's error fixed his path through the potential maze represented (but never manifested) by all those other doors and windows.\n\n_Dark Passage_ also relies heavily, but less extensively, on a subjective camera. Vincent Parry (Humphrey Bogart), innocent of his wife's murder, escapes from San Quentin in order to find the killer. With the help of Irene Jansen (Lauren Bacall), whose father was similarly convicted \u2013 wrongly, according to Irene \u2013 of murdering his wife, Parry makes his way to San Francisco. A plastic surgeon alters his appearance, and it is only when the bandages are removed that we see Parry's \u2013 Bogart's \u2013 face. Hitherto, he has been represented primarily by prolonged subjective shots anchored by Bogart's voice. Although not always clear whether he is talking to himself or whether we are hearing his thoughts, this voice gives the audience some sense of Parry's interiority \u2013 a sharp contrast to Montgomery's abrasive loudmouth Marlowe. In addition to a more mobile camera, a greater use of the full depth of field and carefully underplayed performances by Parry's interlocutors (compare Totter's haranguing and visibly conniving Adrienne with Bacall's Irene, who glances, slides from beneath and exists outside of Parry's gaze), _Dark Passage_ combines the subjective camera with conventional third-person camerawork. Consequently, the sutured-in Parry is part of the diegetic world in a way Montgomery's never-quite-present Marlowe is not.\n\n_Dark Passage_ does not, however, offer a secure image of masculine identity. In order to be free, Parry must surrender his face, name and identity and flee to Peru, but this surrender is also a consummation. Bogart's unmistakable voice makes Parry's own face, seen in newspaper photographs, seem like the fake one. The concluding scene sees not only Bogart's voice and face reunited, but also Parry and Irene, Bogart and Bacall \u2013 a simultaneous union of fictional, star and real-life couple. Moreover, Parry, seated in an exotic night-club, seems to have been made over as _Casablanca_ 's (1942) Rick. These conflations are profoundly compensatory \u2013 as if the film knows it cannot resolve the questions of identity it prompts, and so shuts them down by transforming Parry into Bogart. (A similar play on Bogart's screen personas \u2013 as ruthless heavy and, later, romantic hero \u2013 constitutes Dix Steele, the screenwriter with the absurdly phallic name he plays in _In a Lonely Place._ Whereas Parry's voice provides continuity, writer-for-hire Dix is torn between attempting to maintain his own voice and an almost-incoherent rage. His romance with Laurel Gray (Gloria Grahame) comes to an end because the ruthless heavy erupts through the romantic hero, ironically confirming him as the latter.)\n\nIn several later film noirs, the fragmentation of the hero extends to the fragmentation of the diegetic world itself. For example, _Touch of Evil's_ extreme yet incoherent stylisations restlessly combine distorted compositions and unusual angles with abrupt shifts in tempo and, in its climatic scene, the dislocation, disembodiment and echoing reproduction of voices. _Kiss Me Deadly,_ with its opening sequence's downward-scrolling titles and conspicuous manipulation of continuity errors and discrepancies between the volume and relative position of sound sources, introduces a similarly disconnected, discontinuous world. As in _Murder, My Sweet,_ there is linguistic conflict between the pompous villain, Dr Soberin (Albert Dekker), who litters his speech with mythological allusions, and an inexpressive Mike Hammer (Ralph Meeker) who describes his violent propensities as being able 'to speak a lot of languages \u2013 any country you go to, you can take care of yourself'. There is also a strong sense of self-construction through external appearance. Resembling the armoured fascist male described by Klaus Theweleit (1987, 1989), Hammer is described by Christina (Cloris Leachman):\n\n> You only have one real lasting love... You. You're one of those self-indulgent males who thinks about nothing but his clothes, his car, himself. Bet you do push-ups every morning just to keep your belly hard... I could tolerate flabby muscles in a man if it would make him more friendly. You're the kind of person who never gives in a relationship, who only takes. Ah, woman, the incomplete sex. What does she need to complete her? Why, man, of course, wonderful man... You ever read poetry? No, of course you wouldn't.\n\nIronically, Hammer's subsequent investigation into her death and his route, so he thinks, to wealth, depend upon a clue based on a Christina Rossetti poem, the solution to which equates the female body with a threatening corruption of the flesh.\n\nAn aura of futility presides over his investigation, best expressed by Hammer's secretary Velda (Maxine Cooper): '\"They\"? A wonderful word. And who are they? They are the nameless ones who kill people for the great whatsit. Does it exist? Who cares? Everyone everywhere is so involved in the fruitless search for... what?'. These lines, dubbed over the image in post-production, seem to have been delivered from too close to the microphone. Consequently, a speech that diegetically merely questions Hammer's continuing pursuit of the case also sounds like a voice-over. If these are actually Velda's unspoken thoughts, then the already-fragmented diegesis has fragmented sufficiently to permit a female character interiority and subjectivity; if this speech is a voice-over directly addressed to the audience, then it constitutes an even more damning indictment of Hammer's violent course.\n\nFilm noir's recurring fragmentation and centrifugal disintegration of the subject is complemented by a strong sense of the subject constructed, moment by moment, on the site of the material body through the centripetal operation of various linguistic, discursive and ideological pressures, through multiple simultaneous, if contradictory, interpellations, including other characters' plots and manipulations.\n\nThis is exemplified in a pair of Lang's films. In _The Big Heat,_ homicide sergeant Dave Bannion (Glenn Ford) is caught up in the workings, and machinations, of the corrupt city machinery when his investigation into a cop's suicide threatens to reveal collusion between organised crime, city government and the police. Warned off by his own superiors, he contemplates resigning. As his wife, Katie (Jocelyn Brando), tells him, 'Your big trouble, honey, is that you attack yourself from all sides, like Jersey mosquitoes.' His reply evokes a similar sense of being attacked from all sides, of being caught in a web regardless of what he does: 'What am I supposed to do, hold on to my job by just stringing along, afraid to look to the left or to the right, because I might see something that they don't want me to see.' This is reiterated by Katie's response \u2013 'If you do, you're going to have trouble from me. Just keep leading with your chin and don't you compromise' \u2013 which indicates another conflicting set of expectations simultaneously working to interpellate him as a subject. He is, moment by moment, the product of an attempted reconciliation of multiple simultaneous determinants; and that Katie tells him precisely what he 'wanted to hear [her] say' does not prevent this positioning from being contradictory.\n\nThe conversations between Dave and Katie in the three sequences in their idealised home interweave his public-professional and private-personal interpellations, the latter developed through their carefully choreographed interactions and movement around each other. In the first sequence, Katie, with Dave's assistance, serves dinner, sipping from his whisky, dragging on his cigar, threatening to sip from his beer. This sense of the couple's subjectivity arising from their mutual intersubjectivity is equally evident, although more subtly developed, in the third sequence as they discuss Dave's dilemma while clearing up after a meal. This inter-subjective construction of the subject within discourse is further developed when Dave half-sceptically quotes the child-rearing book as an authority on how they should behave towards their daughter, Joyce (Linda Bennet), while Katie points out that the authors have not met Joyce. Within the constraints of gendered labour and divisions of public and private realms, Dave and Katie together attempt to negotiate their way through various interpellative systems while, at the same time, mutually interpellating each other. When Katie is killed, blown up by a car-bomb intended for Dave, he hands in his badge, separating himself from one of the contradictory interpellative systems which has governed so much of his life and in which he no longer has any faith. The remainder of the film is as much concerned with his reintegration into, and repositioning within, dominant discourses as it is about his pursuit of Katie's killers. The various characters who come to his aid temporarily perform the kinds of negotiating and repositioning previously undertaken by Katie. Debby Marsh (Gloria Grahame), the girlfriend of gangster Vince Stone (Lee Marvin), is aware of some of the determining interpellations to which she is subjected. She jokingly compares Vince and his cronies to circus animals, jumping when syndicate boss Mike Lagana (Alexander Scourby) cracks the whip. She is extremely conscious of her good looks, recognising that they alone give her access to material well-being. When Vince throws boiling coffee in her face (significantly, this happens just after she has taken a sip from his glass), scarring her for life, she knows she will be repositioned accordingly. Before taking her revenge, she kills Bertha Duncan (Jeanette Nolan), freeing Dave to be reintegrated. The film concludes with Debby's death and Dave's return to work. Without Katie as a counterbalance, and with Debby unavailable as an alternative, this d\u00e9nouement is only possible because the evidence released by Bertha's death has swept all corruption from the city. This improbable conclusion is undermined by the final line of the film, in which Dave tells one of the cops to keep the coffee hot. This callous forgetting of Debby suggests other lacunae beneath Dave's superficially secure repositioning within now-benevolent interpellations.\n\nA similar sense of multiple-interpellating determinants can be found in _While the City Sleeps,_ a film which can be understood as a kind of riposte to _Citizen Kane_ (1941). As Michael Walker argues, Lang is more 'interested in the network of relationships' surrounding the patriarch, Amos Kyne (Robert Warwick), and 'in the question of what happens when the patriarch dies' \u2013 a 'concern... with continuity' both within and 'between the personal and the social' (1992c: 69) which enables the film to explore the complex determinations and interactions of subjects and events. In a precisely-edited nineteen-shot sequence, described in detail below, television news-analyst Edward Mobley (Dana Andrews) addresses serial killer Robert Manners (John Barrymore, Jr), a subject caught in a complexly interwoven discursive web:\n\n_Shot 1_ [In the office of Mark Loving (George Sanders), Mobley's fianc\u00e9e Nancy Liggett (Sally Forrest) turns on the television. An announcer intones the programme's introduction.] Mr Walter Kyne presents the distinguished author, columnist and Pulitzer Prize-winner Edward Mobley in his perceptive analysis of the day's news. [The camera pans slightly to include Loving, who has just returned from putting the story on the newswire, in the shot as Mobley begins.] Ladies and gentlemen, at approximately 3am this morning in our city...\n\n_Shot 2_ [Mobley, sat behind his desk in the television studio, shot from behind the crew.]...one human being took the life of another. In our world, acts of violence are not rare, and so my excuse...\n\n_Shot 3_ [Medium shot of Mobley, roughly corresponding with the television camera's viewpoint.]...I should say my reason \u2013 for giving importance to this particular story is my hope that...\n\n_Shot 4_ [Medium-long shot of Manners, in his pyjamas and sat backwards on a chair, facing his television, which is positioned in the bottom-left corner of the frame.]...the killer may be listening to me, for I believe, that in his progress to the chair or to the insane asylum, that he's reached a way-station where his sick and warped ego...\n\n_Shot 5_ [Shot from over Manners' left shoulder. Mobley's direct address to the televion camera appears as a direct address to Manners.]...demands to be fed with the milk of self-importance. And so with the consent of a very good friend of mine...\n\n_Shot 6_ [The office of Lieutenant Kaufman (Howard Duff). In a shot whose composition resembles that of shot 4, he is sitting with his feet on his desk, sideways onto his television which is positioned in the bottom-left corner of the frame.]...who's by way of being a remarkable criminologist but who has also asked that his name not be credited...\n\n_Shot 7_ [The television studio, as shot 2.]...I am going to say a few things to the killer, face...\n\n_Shot 8_ [Manners' bedroom in long-shot. The camera is more-or-less side-on to the killer, positioned behind and to his left so as place him and the television at opposite sides of the screen, with his head and the television on the same level.]...to face. Item 1: Mr Unknown, you will not for very long remain...\n\n_Shot 9_ [Medium shot square on to Manners.]...unknown. Item 2:...\n\n_Shot 10_ [Shot of Manners' television, almost filling the frame.]...you're husky, strong enough to have choked to death this morning...\n\n_Shot 11_ [As shot 9.]...the poor school teacher by the name of Laura Kelly. Item 3:...\n\n_Shot 12_ [As shot 10, but the television camera tracks in on Mobley, ending in a close-up.]...you are the same killer who last week bludgeoned to death a girl by the name of Judith Felton.\n\n_Shot 13_ [As shot 9; as the shot ends, Manners drops his comic book.] You are the Lipstick Killer. Item 4: you read the so-called comic books.\n\n_Shot 14_ [Close-up of the dropped comic book on the floor between Manners' feet. It is _The Strangler,_ the same comic as was found at the scene of Kelly's murder.] Item 5: You have dark brown hair.\n\n_Shot 15_ [As shot 5.] A few strands of your hair were found beneath the fingernails of your latest victim.\n\n_Shot 16_ [As shot 9.] Item 6: you are young. A crime lab examination of your hair reveals that you are approximately twenty years of age. Item 7:...\n\n_Shot 17_ [As shot 12, but the television camera has finished tracking in and Mobley is in close-up.]...you're a mamma's boy.\n\n_Shot 18_ [Close-up of Manners.] Item 8: the normal feeling of love that you should have toward your mother has been twisted into hatred for her and all...\n\n_Shot 19_ [As shot 8. There is a knocking on the door.]...of her sex. Item 9:... [Manners leaps up to switch off the television, hide _The Strangler_ and pick up a textbook.]\n\nThis sequence, through its editing and shot composition, produces a clear sense of the various relationships which structure Mobley's life, and of the various interpellative apparatuses brought to bear on Manners, constructing him in certain ways. It produces an elaborate structure around a central absence \u2013 the confrontation between Mobley and Manners that is not a confrontation, an intersubjective moment that is not an intersubjective moment, which lacks reciprocity and takes place nowhere. Mobley is situated by his past (an ex-crime journalist, he is still friends with Kaufman); his relationships with his virtuous fianc\u00e9e, who is also Loving's secretary, and the more experienced women's columnist Mildred Donner (Ida Lupino), who is romantically involved with Loving; his professional and personal relationships with the three senior figures at the Kyne news organisation \u2013 Loving, head of the wire service, John Day Griffith (Thomas Mitchell), editor of Kyne's flagship newspaper, and Harry Critzer (James Craig), head of Kyne Pix \u2013 who, at the instigation of Walter Kyne, Jr (Vincent Price), after inheriting his father's business, are competing for the new post of executive director by trying to crack the Lipstick Killer case. This competition is complicated by Kyne's dislike for Mobley, whom Amos Kyne had regarded as a son and more appropriate successor, and by Critzer's affair with Kyne's wife, Dorothy (Rhonda Fleming). This complex web interpellates Mobley and variously determines him. More pointedly, the sequence described above locates and constructs him at the intersection of various gazes and positionings: an elite media personality, a person to be filmed, a colleague, a lover, a romantic rival, a professional rival, a friend and, for Manners, a complex of powerful father and marginalising institutions who accuses, taunts, reprimands and knows too much. Meanwhile, the increasingly agitated Manners is subjected to\/by a direct but unreciprocal address which constructs him as subordinate to and determined by powerful institutions (the media, the police) and discourses around deviance (juvenile delinquency, homosexuality).\n\nMobley's description of Manners is, to a certain extent, confirmed in the following sequence. Manners was adopted (his adoptive father then abandoned the family) and suffers from a gender-confusion: 'When you adopted me, you wanted a girl, didn't you? And he wanted a boy. Well, neither one of you were satisfied, were you? I remember once when I was eight years old, eight years old, I was helping you dust the house, and that woman from across the street came over and said, \"My, my\", and you said, \"Yes, I know \u2013 he's exactly like a little girl, isn't he?\"'. His mother's reply \u2013 'But Robert, you are my son and my daughter and all the children I ever wished I could have had' \u2013 seems to confirm the source of his confusion and sense of inadequacy.\n\nHowever, the proposition that an absent father, smothering mother and gender-confusion turned Manners into a killer is at least partially disrupted by an elaborate series of parallels with both Mobley and Walter. Walker's discussion of the connections between Mobley and Manners links the former's 'sexual desire' with the latter's 'murderous desire' (1992c: 62), with Manners the Id to Mobley's Ego. As Amos Kyne's portrait suggests, he was, like Manners' father, a continually present absence who structured his son's life through, among other things, contrasting him with the idealised Mobley (his other child, his business empire, for which he becomes an absent presence, is similarly discontented). The relationships between Mobley and Amos, Mobley and Griffith, and Critzer and Walter follow similar Oedipal trajectories. What this patterning suggests is that similar determinants produce similar but different subjects \u2013 that subjects are the over-determined products of multiple, often contradictory, interacting determinants \u2013 and that the mechanisms of social repression are generally sufficient to moderate the subject. For example, while none of the male characters treat women well, only one of them becomes a killer. The others settle for more everyday misogynies.\n\nComplex determinism is central to the heist movie. An inversion of the investigation film, it focuses on the attempt to construct and control future events rather than retrospectively reconstruct and master past events. Both _The Asphalt Jungle_ and _The Killing_ foreground a sensitive dependence on initial conditions. In the former, after the heist he planned has gone badly wrong, wounded Doc Riedenscheider (Sam Jaffe) bemoans the arbitrary and unpredictable: 'You put in hours and hours of planning, figure everything down to the last detail, then what? Burglar alarms start going off all over the place for no sensible reason. A gun fires of its own accord and a man is shot. And a broken-down old harness bull, no good for anything but chasing kids, has to trip over us. Blind accident, what can you do against blind accident?' In the latter, the importance of initial conditions is made clear when Johnny Clay (Sterling Hayden) says, 'This is a rough drawing of the track as I remember it. Randy, you'll have to get me an A1 street map of the whole district. George, Mike, I want you to go over this thing with me inch by inch. Bring it completely up to date, add or subtract the slightest change, even if it's something as small as the placing of a hot-dog stand.' However, the film's complex temporal ordering and authoritative voice-over as it follows the various gang members mocks the orderliness Clay tries to impose on a world of which he cannot have perfect or, ultimately, even sufficient knowledge. He can anticipate neither the chain of events, resulting from a necessary improvisation, which end in the death of Nikki Arane (Timothy Carey), nor the counterplot which ends in the rest of the gang being shot. He is then undone by airline regulations, a suitcase with a faulty lock and a yappy little dog. As the police close in, his girlfriend Fay (Coleen Gray) urges him to run, but he is as resigned to, or as complicit in, his fate as _The Killers'_ Swede a decade earlier. Mistaking the complex determinants which have brought him to this moment for an inescapable fate, Clay relinquishes his attempted mastery, shrugs and says, 'Ah, what's the difference?'\n\nThe above discussion of the film noir subject draws on the model of ideology proposed by Louis Althusser, in which he argues that the individual subject is positioned or interpellated by the way ideology hails him or her. Althusser's most famous attempt to illustrate this hailing was to imagine a policeman calling out in the street, 'Hey, you there!', and 'the hailed individual will turn round. By this mere one-hundred-and-eighty-degree physical conversion, he becomes a subject. Why? Because he has recognised that the hail was \"really\" addressed to him, and that \"it was really him who was hailed\" (and not someone else)' (Althusser 1971: 163). This model has often been criticised as being 'too rigid and mechanistic', incapable of producing 'a subject who is anything more than passive and manipulated by discourse' (Bould 2002: 76). However, Althusser does allow for greater flexibility than is normally credited:\n\n> Each hailing positions the individual as a subject, but each hailing is in tension with every other hailing's attempt to position the individual as a subject. The subject, then, is not to be considered as a singular point, a monadic intersection, through which all hailings pass, but as a cluster or cloud of positions, constantly shifting and repositioning in response to each new hailing. (Bould 2002: 76\u20137)\n\nThis complex and never-resolved process is illustrated by the broadcast sequence from _While the City Sleeps_ discussed above, in which characters are positioned by multiple discourses, institutions and relationships. A further source of ambiguity, which leaves the determinism of Althusser's model intact but renders it less mechanistic than many of its critics have supposed, is exemplified by _Gilda,_ a film that relies upon double-meanings, coded significations and motivations which are neither quite conscious nor unconscious. This repeated foregrounding of the fact that words have more than one meaning demonstrates that all hailings by discourse or ideology are rather more contingent than Althusser's policeman's yell.\n\nIn exotic Buenos Aires, bedraggled gambler Johnny Farrell (Glenn Ford), who knows 'about American sailors', is rescued from a mugging by dapper, effete Ballin Munson (George Macready) who is, for no clear reason, prowling 'a neighbourhood like this' at night. There follows a flirtatious exchange. Ballin describes his sword-cane as 'a most faithful and obedient friend... silent when I wish it to be silent, it talks when I wish it to talk', to which Johnny replies that he 'must lead a gay life'. Later, at Ballin's illegal casino, Johnny offers to become another such friend. Before agreeing, Ballin seeks reassurance 'that there is no woman anywhere', and they then drink a toast to the three of them \u2013 Johnny, Ballin and the sword-cane. The queer coding of their affectionate relationship runs throughout the film, even after Ballin returns from a trip with Gilda, his new wife and, unknown to Ballin, Johnny's former lover. This introduces an Oedipal triangle, evident in Johnny's voice-over when he stalks out of Ballin's house: 'It was all I could do to walk away. I wanted to go back up in that room and hit her. What scared me was I... I wanted to hit him, too. I wanted to go back and see them together with me not watching. I wanted to know.' He is constructed as a subject by and in his conflicting desires \u2013 for departure and return, for Gilda and for Ballin, for sex and for violence, for vision and invisibility, for presence and absence. In a subsequent toast to the three of them, Gilda has replaced the sword-cane, simultaneously usurping the homosexual desire it signified and replacing it with a sexualised mother figure to whom Johnny is denied access.\n\nDouble-coding and linguistic contradictions multiply. When Gilda and Johnny talk about dancing, they seem to mean sex. When Gilda, Ballin and Johnny talk about hate, they seem to mean sexual desire or, possibly, love. To provoke Johnny, Gilda puns that if she had 'been a ranch, they'd have called [her] the Bar Nothing'. Elsewhere she suggests psychoanalytic interpretations of words, as when Johnny compares her to Ballin's dirty laundry, and of actions \u2013 'I can never get a zipper to close; maybe that stands for something'. Gilda gives two very different renditions of 'Put the Blame on Mame', a song which blames various disasters on women. The first performance, private and low-key, works to undermine this equation; the second, public and spectacular, seems to confirm it; but in combination, and because of all that we have seen happen between these renditions, they reveal it \u2013 and the femme fatale \u2013 as misogynist fantasies enabling masculine disavowal of desire and responsibility.\n\nThis polysemic play indicates language's ambiguity and therefore the potential ambiguity of discursive interpellations. Depending on multiple contexts, they can be misunderstood, misinterpreted and deflected, and so, even as they work to position the subject, contradictions emerge. The moment-by-moment emergent subject is simultaneously forming and, as in Captain Delgado's (Gerald Mohr) description of Johnny, 'breaking up in little pieces right in front of my eyes'.\n\nAs this chapter's examples have demonstrated, film noir repeatedly shows us that the subject is an emergent phenomenon within a total system, and that the subject's continuity is produced by an ongoing retrospective telling of the self; but, as with the total system, minor fluctuations can cause rapid and substantial divergences. The next chapter will give a brief account of the development of neo-noir which is a partial, retrospective and abstracted narrative, neither true nor false. It will conclude with a discussion of _Femme Fatale,_ a neo-noir which consciously works with an idea drawn from non-linear dynamics (chaos theory): the sensitive dependence on initial conditions.\n4 AGAINST ALL ODDS: NEO-NOIR\n\nBatty scanned across the search wreckage that lapped up against the replicas of Frank Lloyd Wright's original _faux_ Mayan wall panels.\n\n\u2013 K. W. Jeter (1995: 235)\n\nIf a single definition capable of producing a clear sense of where film noir's boundaries lie remains elusive, then defining neo-noir is even more difficult. Because Siodmak's _The Killers_ includes not only Damico's triangular relationship, but also a woman-as-mystery, a femme fatale, a seeker-hero, a victim-hero, a homoerotic investigation, gangsters, boxing, revenge against the mob, a couple on the run, a heist, and so on, then other films containing some of these elements have been described as film noirs. They also contain other elements shared by still other films, causing the genre to flourish from multiple centres. With the passage of time, the genre took on fresh concerns and, through the same process, exfoliated further, becoming fuzzier, harder to pin down. This prompts a range of questions: Are film noir and neo-noir the same or different genres? Are all new crime films, because they cannot help but refer to film noir in some way, to be considered neo-noirs? Can one identify a dominant tendency within a film which renders it ineluctably noir? Or can one talk about any film as noir if it is illuminating to do so, regardless of what one might consider its dominant generic tendency? For example, imagine a trapped man pursued through an office building by those who wish him harm while he attempts to thwart their schemes. If this is _The Big Clock_ (1948), then it is a film noir; if it is _No Way Out_ (1987), the remake, then calling it a neo-noir is relatively unproblematic; but when it is _Die Hard_ (1988), generic identification becomes more vexed. _Die Hard's_ film noir elements include a semi-official investigator who represents an older order of masculinity \u2013 a blue-collar, vest-toting, western-quoting cop (and father) \u2013 endangered by a new order of femininity and feminised masculinity represented by his imperilled executive wife, her yuppie colleagues and the designer villains; a racial anxiety shapes its depictions of the Japanese and various feminised black men, and when John McLane (Bruce Willis) finally meets Sergeant Al Powell (Reginald VelJohnson) the homoerotic undercurrents bubble up to the surface. And yet something about this Christmas action-movie blockbuster seems to militate against labelling it noir \u2013 perhaps not so much the spectacular scale of the film or its happy ending (after all, the main cycle included some A-movies and happy endings) as the reactionary white male American supremacism it articulates. As with Siegel's _The Killers,_ then, there are reasons to put it near the centre of noir, and reasons to marginalise or exclude it.\n\nWith these problems in mind, this final chapter will sketch in film noir's impact on British and French cinema, before turning to a brief treatment of some of the trends and developments in nearly four decades of neo-noir, a treatment characterised by numerous necessary omissions. The chapter will close with an extended discussion of _Femme Fatale,_ a film symptomatic of the current condition of Anglophone neo-noir.\n\nFilm noir in Britain and France\n\nIn Britain, the crime film developed a noirish sensibility, most evident in such adaptations of Greene's fiction as _Brighton Rock_ (1947), _The Fallen Idol_ (1948) and _The Third Man_ (1949), the latter two directed by Carol Reed, who also directed the noirish _Odd Man Out_ (1947) and _The Man Between_ (1953). Something of this sensibility can also be detected in Ealing Studio's _Kind Hearts and Coronets_ (1949), _The Lavender Hill Mob_ (1951) and _The Ladykillers_ (1955), although it is worked out in a thoroughly British manner derived from a tradition of grotesque and gothic comedy about social class. This dialectical interplay of British and American models is evident in the contrast between the heist movie _The Good Die Young_ (1954) and the caper movie _The League of Gentlemen_ (1960). The latter, with its all-British cast, is a closely-observed comedy about a class system it simultaneously despises and valorises. It is torn between pulp plebianisation and nostalgia for hierarchical deference. The former film, in which several of the major roles are played by Americans (Richard Basehart, Gloria Grahame, John Ireland) is a glummer affair which activates class in a much more superficial way. Washed-up boxer Mike (Stanley Baker), who understands and accepts his class position, is residual, a glitch in the fantasy of atomised classlessness and social mobility.\n\nBaker is the nearest Britain came to producing a noir leading man, giving powerful and subtle performances in _Hell Drivers_ (1957), _Hell is a City_ (i960), _Blind Date_ (1959) and _The Criminal_ (i960). (The latter pair were directed by Joseph Losey, one of a number of HUAC exiles who contributed something noirish to British crime films, including Edward Dmytryk and Jules Dassin, who directed _Obsession_ (1949) and _Night and the City_ (1950), respectively.) Baker's film noirs constitute an intriguing series of interactions between Anglo-American crime traditions and British realist film aesthetics, interactions which culminate in _Get Carter_ (1970), the most significant neo-noir reworking of the unofficial investigator motif and revenge-against-the-mob plot. (Other \u2013 American \u2013 reworkings include _Point Blank_ (1967; remade as _Payback_ (1999)), _The Outfit_ (1974) and _The Limey_ (1999).)\n\nAs the main noir cycle ran its course, its impact was clearly felt in French crime movies and thrillers, like Jacques Becker's _Touchez pas au Grisbi_ (1945) and _Casque d'Or_ (1952); Henri-Georges Clouzot's _Quai des Orf\u00e8vres_ (1947), _Le Salaire de le peur_ (1953) and _Les Diaboliques_ (1954); Ren\u00e9 Cl\u00e9ment's _Au-del\u00e0 des grilles_ (1949) and _Plein Soleil_ (1959); exiled Jules Dassin's _Du Rififi chez les hommes_ (1955); Louis Malle's _Ascenseur pour l'echefaud_ (1957); Robert Bresson's _Pickpocket_ (1959); and Jean-Pierre Melville's _Bob le Flambeur_ (1955), _Deux hommes dans Manhattan_ (1959), _Le Doulos_ (1962) and _Le Samoura\u00ef_ (1967). Film noir and American crime cinema in general were important influences on a number of New Wave directors. Claude Chabrol directed _Le Beau Serge_ (1958), _Le Boucher_ (1969) and _Que la b\u00eate meure_ (1969). Fran\u00e7ois Truffaut adapted a David Goodis novel as _Tirez sur le pianiste_ (1960) and two Woolrich novels as _La Mari\u00e9e \u00e9tait en noir_ (1967) and _La Sirene du Mississippi_ (1969); his final film, _Vivement dimanche_ (1983) was adapted from a Charles Williams novel. Jean-Luc Godard transformed the heist movie into _Bande \u00e0 part_ (1964) and worked variations on the couple-on-the-run scenario in _A Bout de souffle_ (1960), _Pierrot le fou_ (1965) and _Weekend_ (1967). His most noirish film, _Alphaville_ (1965), blends a Lemmy Caution _policier_ with the European dystopian tradition to produce the first major science fiction neo-noir.\n\nNeo-noir: a sketch\n\nDespite such late examples of the heist movie as _Charley Varrick_ (1973) and _The Friends of Eddie Coyle_ (1973), it was clear, as early as _Ocean's Eleven_ (1960) and _Topkapi_ (1964), that the imperatives of post-classical film-by-film production were transforming it into the spectacular, star-centred, action-driven, sometimes comic, caper movie, from _The Anderson Tapes_ (1971) and _The Taking of Pelham One Two Three_ (1974) to _Mission: Impossible_ (1996) and the re-make of _Ocean's Eleven_ (2001). Following the American _Reservoir Dogs_ (1992) and the British _Lock, Stock and Two Smoking Barrels_ (1998), the heist movie underwent a resurgence \u2013 _Heat_ (1995), _Dead Presidents_ (1995), _Face_ (1997), _Croupier_ (1999), _Sexy Beast_ (2000) \u2013 but one can discern in each of these examples tensions between low-key and high concept, between film noir and blockbuster aesthetics. Arguably, these tensions shaped neo-noir's development. There is a clear sense in the late-1960s and early-1970s that, as film noir emerged as a recognised and recognisable genre, it represented a pre-sold concept to be repackaged and resold.\n\nSeveral films returned to hard-boiled sources, such as the Chandler adaptations _Marlowe_ (1969), _The Long Goodbye_ (1973), _Farewell, My Lovely_ (1975) and _The Big Sleep_ (1978). The first and third are particularly instructive. _Marlowe_ gave James Garner an opportunity to rehearse the updated private eye he would perform so effectively in _The Rockford Files_ (1974\u201380), and although such films have since been plentiful, the character type now belongs to television. _Farewell, My Lovely_ represents two affiliated trends: the period or costume noir, ranging from _Chinatown_ (1974) to _LA Confidential_ (1997); and the noir remake, including films as various as _Thieves Like Us_ (1974), _The Postman Always Rings Twice_ (1981), _Against All Odds_ (1984), _D.O.A._ (1988), _Desperate Hours_ (1990), _Cape Fear_ (1991), _Night and the City_ (1992) and _The Deep End_ (2001), several of which are also period pieces. (Costume noir need not be period noir; see, for example, _Blade Runner_ (1982) or _The Usual Suspects_ (1995).) While neo-noir can often be seen in terms of the loss of historicity in postmodern culture anatomised by Fredric Jameson (1991), this is not to say that their nostalgia precludes critique. For example, the refraction of the Watergate scandal through _Chinatown's_ fictionalised retelling of William Mulholland's 1906 'Rape of Owens Valley' frequently seems like a retreat from political engagement. However, Evelyn Mulwray's (Faye Dunaway) denial that her father raped her, which implies that their daughter is the product of consensual incest, might be a startling momentary indictment of the complicity of those protesting Richard Nixon's misrule.\n\nThe clearest examples of this ahistorical tendency are the films of Joel and Ethan Coen. While others continued to adapt hard-boiled crime writers \u2013 such as Jim Thompson in _The Kill-Off_ (1990), _The Grifters_ (1990), _After Dark, My Sweet_ (1990), _The Getaway_ (1994), _This World, Then the Fireworks_ (1997) \u2013 the Coens made hard-boiled 'adaptations' not actually based on particular novels. _Miller's Crossing_ (1990) is their Hammett story, _The Big Lebowski_ (1998) their Chandler; _Blood Simple_ (1984) and _The Man Who Wasn't There_ (2001) are their Cains, perhaps retold by Thompson. Like the many film noirs that yearned for an earlier stage of capitalism and older versions of masculinity, the Coens register a profound dissatisfaction with the present but, as _The Hudsucker Proxy_ (1994) shows, they know that the clock cannot really be turned back. Like their eponymous _Barton Fink_ (1991), a writer who is unable to write, they are unable to slingshot this impasse, and so their films, which often seem hermetic and disconnected, become exquisitely-crafted comedies of this self-knowledge. For example, in _Miller's Crossing,_ Tom Reagan (Gabriel Byrne), the enigmatic lieutenant of mobster Leo (Albert Finney), plays out his role in the gang-war as if it were nothing more than a role, a predetermined function. Everything he does \u2013 from falling for Verna (Marcia Gay Harden), Leo's squeeze, to stoically accepting a beating from his bookie's henchmen \u2013 is done with no sense of interiority. Although he occasionally loses control of events, his self-possession, his mastery of his self, is expressed through confident, sometimes ornate, hard-boiled dialogue; when he is most in danger \u2013 when Eddie Dane (J. E. Freeman) takes him to Miller's Crossing, intending to execute him, and when Eddie turns up at the wrong moment in Johnny Caspar's (Jon Polito) office \u2013 he clams up, lost for words. But the Coens' clever pastiche of hard-boiled prose \u2013 like the barren, almost cartoonishly sketched-in _mise-en-sc\u00e8ne_ and the panoply of parodic characters \u2013 evokes an entextualised past rather than historicity, a performance rather than an identity; and their embrace of past and performance is ultimately consolatory, mostly.\n\nSeveral varieties of noir story seem to have been especially popular among neo-noir filmmakers. The gangster film thrived in the long shadows cast by Francis Ford Coppola and Martin Scorsese. New manifestations of the femme fatale can be found in a whole range of neo-noirs, from _Body Double_ (1984) to _Bound_ (1996), but the character type featured most prominently in direct-to-video and made-for-cable films, many of them erotic thrillers, with John Dahl's _Red Rock West_ (1992) and _The Last Seduction_ (1994) premiering on cable before receiving theatrical releases. Among the many couples-on-the-run and other neo-noir road movies are _Badlands_ (1973), _Wild at Heart_ (1990), _Thelma & Louise_ (1991), _True Romance_ (1993), _Natural Born Killers_ (1994), _Freeway_ (1996), _Breakdown_ (1997), _Buffalo 66_ (1998) and _Joy Ride_ (2001). This turn to non-urban spaces has produced a number of rural and smalltown neo-noirs, including _One False Move_ (1992), _Fargo_ (1996), _A Simple Plan_ (1998) and _U Turn_ (1997). In _Mean Streets_ (1973) and _Taxi Driver_ (1976), Martin Scorsese depicted a fractured and disorientated masculinity, while Abel Ferrara's _The Driller Killer_ (1979), _Ms.45_ (1981), _Fear City_ (1984), _King of New York_ (1990), _Bad Lieutenant_ (1992), _Dangerous Game_ (1993), _The Addiction_ (1995) and _The Blackout_ (1997), exposed the breakdown of the American city, the collapse of consensual moral order, the paranoia and brutality of late-capitalism and the fucked-up gendered subjects, especially men, it produces. His _New Rose Hotel_ (1998) is a science fiction neo-noir, a subgenre partially enabled by the way 1980s cyberpunk science fiction embraced film noir and hard-boiled crime fiction (and unpacked the noirishness at the core of Philip K. Dick's science fiction). Other examples include _Naked Lunch_ (1991), _Equinox_ (1993), _Dark City_ (1998) and _The Thirteenth Floor_ (1999).\n\nThe _Strassenfilm_ reappeared as the yuppie nightmare movie _(After Hours_ (1985), _Desperately Seeking Susan_ (1985), _Something Wild_ (1986), _Judgment Night_ (1993), _Se7en_ (1995), _Very Bad Things_ (1998)), becoming po-faced in Michael Douglas vehicles like _Fatal Attraction_ (1987) and _Falling Down_ (1993). For good or ill, Douglas, who also starred in _Coma_ (1978), _The Star Chamber_ (1983), _Black Rain_ (1989), _The War of the Roses_ (1989), _Basic Instinct_ (1992), _Disclosure_ (1994), _The Game_ (1997) and _A Perfect Murder_ (1998) is the major neo-noir leading man. (Tom Berenger runs him a close second. Although a number of actresses, like Kathleen Turner, Sharon Stone and Linda Fiorentino, have made several neo-noirs, none of them have become so closely identified with the genre, perhaps because of Hollywood's preference for youthful leading ladies and neo-noir's for female nudity.)\n\n_Basic Instinct,_ along with films like _Body of Evidence_ (1993) and _Jade_ (1995), signal an attempt to relocate the erotic thriller from video or cable to cinema screens. However, it is a genre at which Anglophone cinema is singularly inept. Occasionally an Anglophone erotic thriller might muster some eroticism or thrills, and might even manage, albeit often unwittingly, to be remotely interesting about sexual attraction, sexuality and gender, but most are weakened by the tendency to mistake the spectacular eroticised display of female nudity for a challenging engagement with ideas about and representations of sexuality. They are further undone by the desire to avoid the NC-17 rating, which is almost certain to prohibit exhibition in lucrative mall theatres (see Sandler 2002, Williams 2004). In contrast to such easy-viewing, the treatments of sexual obsession found, for example, in Tsukamoto Shinya's films about sex, mortality, violence and transformation \u2013 the most obviously noirish being _Tokyo-ken_ (1995), _Bullet Ballet_ (1998), _Rokugatsu no hebi_ (2002) and _Vital_ (2004) \u2013 are far more compelling. Uncomprehending of complacency and bloat, they trouble.\n\nAlthough dream sequences and subjective world-distortions played an important role in film noir and its expressionist antecedents \u2013 recall, for example, the hyperbolic dream worlds of _Der Letzte Mann_ (1924) or _Stranger on the Third Floor_ \u2013 it is only really with neo-noirs like _De Vierde man_ (1983), _Angel Heart_ (1987), _Jacob's Ladder_ (1990), _Suture_ (1993), _Lost Highway_ (1997), _The Gift_ (2000), _\u00d4dishon_ (2000), _Mulholland Drive_ (2001) and _Fear X_ (2002) that fantasy becomes a central noir element, materialising many of those things that film noir had only been able to suggest. And so this sketch will close with a discussion of two films from different decades which offer some insight into this turn to the fantastic and which have accrued a similar cult status: David Lynch's _Blue Velvet_ (1986) and David Fincher's _Fight Club_ (1999).\n\nIn _Blue Velvet,_ Jeffrey Beaumont (Kyle MacLachlan) is drawn into an underworld of violence, horror, crime and police corruption he never suspected existed in his small home-town. He finds a severed ear and his subsequent unofficial investigations lead him to singer Dorothy Valens (Isabella Rossellini), whose husband and child have been abducted by the crazed Frank Booth (Dennis Hopper) so as to force her to comply with his desire to repeatedly rape her. The film recapitulates various film noir elements, including a woman in peril and the opposition between a sexualised mother and an innocent daughter, in a highly self-conscious manner. Stilted dialogue, mannered performances, pastiches of 1950s melodramas and teen movies, visual distortions, extreme close-ups, slow-motion and theatrical square-on compositions all alienate the viewer from the diegetic illusion, making the film's edgier material seem more confrontational than it would otherwise necessarily be. At the core of the film is a Freudian primal scene fantasy, depicting more or less explicitly what _Gilda's_ Johnny so desperately desired (to hit Gilda, to hit Ballin, to see them together with him not watching). Jeffrey spies on Dorothy as she strips to her underwear; she attacks him with a knife; she fellates him; Frank arrives and Jeffrey must hide, almost naked, in the closet and watch while Frank, who seems to switch between father and child roles, beats and rapes Dorothy. The fluidity of subject positions forced upon or played by each of the characters in this sequence articulates something of the trauma central to psychoanalytic accounts of subject formation.\n\nA similar argument can be made about the misogyny of _Fight Club._ In order for Narrator (Edward Norton) to become a 'real' man, he must be separated from the supposedly effeminate realm of consumerism and side with Tyler Durden (Brad Pitt), his father-figure ego-ideal who also happens to be a figment of his imagination, against Marla Singer (Helena Bonham-Carter), a sexualised mother-figure with whom 'Tyler' has frequent, noisy and energetic sex. It is only through over-identifying with his 'father' that Narrator is able to construct a sufficiently strong identity to then reject Tyler's proto-fascist homoerotic masculinity and perhaps form a more normative heterosexual couple with Marla. Throughout the film, Tyler speaks with absolute confidence, regardless of the adolescent platitudes, frequently mistaken for a radical critique of capitalism, he spouts, while Narrator struggles with language: at the various therapy groups he attends, he depends upon his own silence so as to give nothing of himself away; around Marla, he becomes tongue-tied; and he begins to talk about himself in the third person. From a Lacanian perspective, he only enters the symbolic realm when he begins to speak as Tyler would; but this increasing over-identification with his father must be brought to a traumatic conclusion. In trying to prevent Project Mayhem's destruction of the headquarters of several credit card companies, Narrator begins to identify with patriarchy rather than his father. Marla's personality has altered according to the requirement of any particular scene, and it shifts once more as the skyscrapers fall and as Narrator reaches for her hand: now that her man knows his place, she knows hers.\n\nNeo-noir now: a snapshot, a rebus\n\nBecause _Femme Fatale_ is little-known, it is necessary to begin with a lengthy plot-description (which will help to capture something of the flavour of this convoluted and absurd film). It starts with a caper at the 2001 Cannes Film Festival. Veronica (Rie Rasmussen) is wearing a $10 million 'off-the-shoulder top', little more than a diamond encrusted band of yellow gold coiled around her breasts and upper torso like a serpent. She is picked up by a woman, who might be called Laure Ash (Rebecca Romijn-Stamos). Intercut with the caper's other carefully-choreographed shenanigans, they begin to have sex in a toilet cubicle; as Laure drops each item of Veronica's clothing\/jewellery to the floor, Black Tie (Eriq Ebouaney) switches them for duplicates. When he is briefly distracted, the double-cross kicks in, with Laure returning the duplicate top to him. Wounded, Black Tie is left behind as Laure escapes. She meets her girlfriend in Belleville. They are photographed by Nick Bardo (Antonio Banderas) and spied on by Black Tie's henchman, Racine (Edouard Montoute). Sheltering in a church, Laure is 'recognised' by the funeral congregation. Followed by the concerned Irma (Eva Darlan) and Louis (Jean-Marie Frin), she makes for room 214 at the Charles De Gaulle Sheraton to collect a new passport. Racine attacks her, throws her from a balcony. Irma and Louis take her, unconscious, to a cosy apartment. Once awake and alone, she explores her surroundings.\n\nBy an astonishing coincidence, Laure is identical to Lily (also played by Romijn-Stamos), whose husband and daughter recently died. Laure finds Lily's passport and an aeroplane ticket, but as she starts to drowse in the bath, the bereft Lily returns and kills herself. Laure takes her identity and, _en route_ to the US, meets Bruce Hewitt Watts (Peter Coyote), a millionaire who works for the State Department. Seven years later, he becomes the American Ambassador to France, and he takes Laure, now his wife Lily, with him.\n\nBardo, a reluctant paparazzo, is hired to photograph 'Lily Watts'. Black Tie, released from gaol, and Racine set about tracking down Laure. They find her girlfriend in Belleville, and throw her in front of a passing truck, killing her. They pick up Laure's trail from Bardo's photograph, now the cover of _Gala_ magazine. Bardo spots Laure, dark glasses hiding a black eye, and follows her, tailed by Black Tie and Racine. Bardo sees Laure buy a gun and follows her to the Sheraton, room 214. Convinced she is suicidal, he cons his way into the room. After he talks her out of suicide, she seduces him, steals his keys and sends him to replace her asthma medication. To ensure that she cannot leave or kill herself, he takes her clothes and the gun with him in her car. Laure anonymously phones the police, who arrest Bardo. They do not believe his story, but Watts makes them drop all charges. Laure emails a ransom note from Bardo's account, making it look like he has kidnapped her. Wanting no part of her plan to con Watts out of $10 million, Bardo tries to warn him, but Laure shoots them both. Black Tie and Racine appear, and throw her into the Seine. She falls, suddenly naked, into eerily clear water, and swims for the surface, only to wake up in the bath in Lily's apartment. It has all been a dream.\n\nThis time, Laure intervenes in Lily's suicide, persuading her that she will meet a man on the aeroplane and fall in love. A truck driver (Salvatorre Ingoglia) gives Lily a lift to the airport. She gives him a pendant that belonged to her daughter; he hangs it in his cab to remind him of his own daughter. Seven years later, in Belleville, Laure collects the money from selling the diamonds from her girlfriend. They agree to go their separate ways. Bardo watches and photographs as Laure's girlfriend is captured by Black Tie and Racine. This time, when they throw her in front of the truck, the sun appears from behind the clouds, reflects off the pendant hanging from the driver's mirror, dazzling him. He swerves, missing her and killing Black Tie and Racine. Laure's girlfriend is revealed to be Veronica. Bardo helps Laure to her feet. When he asks if they have met before, she replies, 'Only in my dreams'.\n\nFIGURE 7 Old and new: _Femme Fatale_\n\nFIGURE 8 Off-the-shoulder top: _Femme Fatale_\n\n_Femme Fatale_ opens with the final confrontation between Neff and Phyllis from _Double Indemnity,_ playing on television with French subtitles. Reflected in the screen is Laure, lying on her bed, apparently naked, propped up on one elbow as the camera slowly tracks out and past her. Coming late in the history of film noir, it is appropriate that the film should open like this as this is how film noir was fabricated: in French and by the recirculation of old films in new venues. Laure's reflection superimposed over _Double Indemnity_ \u2013 a palimpsest, a hieroglyph \u2013 connotes the proximity of, and distance between, Phyllis and Laure. Film noir's sexual undercurrents are made visible even as the psychosexual intensity made possible by their proscribed visibility fades, like a black-and-white ghost, beneath a mandated and spectacular visibility. In _Femme Fatale's_ first sex scene, haunted by the shower sequence from _Psycho_ (1960), Veronica's near-naked body is pressed against the semi-opaque, screen-like cubicle wall \u2013 but if this is an attempt to cover the visible so that the invisible can be recovered, it fails. The clock cannot be turned back.\n\nIn the third volume of Swift's _Gulliver's Travels_ (1726), Gulliver encounters the fabulous Lagadan machine. Intended to aid authorship in many different disciplines 'without the least assistance from genius or study', it produces random groups of words; when there are 'found three or four words together that might make part of a sentence' they are recorded in volumes out of which the Professor intends to construct a 'complete body of all arts and sciences' (1963: 221\u20132). This device depends upon recombining syntagmatic elements so as to produce new syntagms which can then be organised into a new whole. This is not unlike Brian De Palma's method, as his _constructions en ab\u00eeme_ \u2013 his texts within the text \u2013 suggest.\n\nOne variety of filmic _construction en ab\u00eeme_ is the incorporation into the diegesis of 'painterly or graphic forms of representation' which, because they are 'sharply differentiated from the texture of the film thanks to the way they are encoded... seem to flaunt the fact that they are representations' (Iampolski 1998: 37). The most obvious example of this in _Femme Fatale_ is the wall-sized collage of the Belleville junction Bardo has constructed from hundreds of overlapping photographs shot over at least seven years. Like the film, it is constructed out of quotations, captured moments; and, in fabricating a totality by spatialising the passage of time, it calls attention to the textuality of film itself. The substratum of film is the photogram (see Stewart 1999), the individual image or frame which is made invisible when projected as part of a film. The still photograph \u2013 _Femme Fatale_ is full of still photographs \u2013 is a trace of and allusion to the photogram; and the photogram itself is arguably a model of film quotation and intertextuality, of the disruption of a text's linearity. Mikhail lampolski quotes Laurent Jenny:\n\n> What is proper to intertextuality is the introduction of a new mode of reading that explodes the linearity of the text. Each intertextual reference is the site of an alternative: either one keeps reading, seeing the reference as nothing more than one fragment among others, an integral part of the text's syntagmatics; or one returns to the original text, resorting to a kind of intellectual anamnesis whereby the intertextual reference appears as a 'displaced' paradigmatic element issuing from a syntagmatic axis that has been forgotten. (1998: 30)\n\nFIGURE 9 Collage: _Femme Fatale_\n\nThere is, then, a tension between whether the film can offer sufficient narrative motivation for the presence of a particular syntagm (or syntagmatic element), or whether one is driven to 'seek its motivation in some other logic or explanatory cause outside the text... in the realm of intertextuality' (ibid.). These fragment-alternatives in _Femme Fatale_ include the following. The foregrounding of a key, a bag, Laure's changing hair colour and Bardo's camera recall Hitchcock, especially _Notorious, Psycho, Marnie_ (1964) and _Rear Window,_ while the prominence given to photography recalls Antonioni's _Blow-Up_ (1966), which De Palma reworked as _Blow Out_ (1981). Laure's mysterious identity, her shadowing by Bardo and the Bernard Herrmann-like score recall _Vertigo,_ which De Palma reworked as _Obsession_ (1976). The intersection in the French countryside where Racine collects Black Tie recalls _North by Northwest_ (1959). The recollection of _Psycho's_ shower sequence alluded to above recalls De Palma's several reworkings of it in _Carrie_ (1976), _Dressed to Kill_ (1980) and _Blow Out._ The it-was-all-a-dream explanation recalls Lang's _The Woman in the Window,_ as well as De Palma's _Body Double._ Laure's sudden appearance behind a hotel maid and the deaths of Black Tie and Racine recall moments from De Palma's _Raising Cain_ (1992). Racine's descent down a duct during the caper recalls a sequence in De Palma's _Mission: Impossible,_ itself reworking a sequence from Dassin's _Topkapi._ Lily and Laure embody film noir's good-woman\/bad woman dichotomy \u2013 like the twin sisters of Siodmak's _The Dark Mirror._ Like Jeff in Tourneur's _Out of the Past_ and like Swede and Johnny in their respective versions of _The Killers,_ Laure's past catches up with her by chance. The shadows of the blinds in the police interrogation room recall countless film noir settings. Bardo is every schmuck who ever fell for a crooked dame. And so _Femme Fatale_ is bathed in recognition, always-already known; but can it simultaneously amount to more than the mere sum of its (borrowed) parts?\n\nIampolski argues that where an anomalous fragment which the film cannot convincingly integrate 'violates the calm of mimesis' we can 'begin to see vigorous traces of semiosis' (ibid.). The intertextual fragment which is foregrounded (or perhaps merely recognised) is like the photographic still in the film which evokes the photogram: just as the film is composed of photograms (and Bardo's collage of photographs), so the text is composed of intertextual fragments. While the photogram exposes the film's textuality, the fragment exposes its intertextuality. When a fragment disrupts the linear narrative flow, 'we witness the birth of meaning, which is normally transparent wherever mimesis remains untroubled, dissolving into the effortless movement from signifier to signified'; by violating this mimetic link the quoted fragment orients 'the sign toward another text rather than a thing' (ibid.). One variant of this process, in which the viewer has more agency than Iampolski seems to suggest, can be found in Bardo's performance of a gay man, a non-threatening guise he adopts to gain access to Laure's hotel room. It is so embarrassingly bad that it draws attention to itself; and one desperately seeks an intertextual motivation which might somehow excuse, or at least, mitigate it (perhaps it is a nod to the effeminate persona Bogart's Marlowe adopts in _The Big Sleep_ 's bookshop sequence, already reworked in _Blade Runner)._\n\n_Femme Fatale's_ other _construction en ab\u00eeme_ is the dream which occupies the majority of the film. As if to provide us with a viewing protocol for the film, De Palma packs the dream with clues about its status. Laure falls asleep in an overflowing bath with the taps still running at 3.33pm, according to the clock in the bathroom. Throughout the dream, we are returned to this moment by an overflowing fishtank, recurring images of water being poured, a painting of a waterfall, various bath-like blue rectangles of blue, a poster for _Deja Vue_ (note the unusual feminine ending) which features John Everett Millais's painting of a drowning _Ophelia_ (1851\u201352), over whose face Laure's is superimposed; and by clocks which all tell the same time \u2013 3.33pm. Several objects from the frame story \u2013 such as the flashing pharmacy sign, Veronica's camouflage outfit \u2013 recur, and various actors reappear, sometimes playing different roles. But what is significant is that the dream also seems to leak out into the frame tale.\n\nOne function of the it-was-all-a-dream explanation (which has been present as a possibility throughout for the observant viewer) is to excuse the ridiculous plot and the many ludicrous moments it contains (Black Tie is released from gaol in the blood-stained tuxedo he was wearing when captured; Laure fakes a black eye so as to get Bardo to follow her; and so on). However, the dream is no more ridiculous than the frame tale (Black Tie hinges the caper on Laure's ability to pick up Veronica with a single whispered comment; Laure and Lily look identical; and so on). The traffic between dream and frame, hypodiegesis and diegesis, returns us to Bardo's collage, which does exist, unseen by Laure, in the frame tale before and after her dream. Its presence seems to confirm that Laure's dream was a vision of her future if she did not save Lily; but then what is the function of the smaller photo-collage on Lily's pinboard, if not a prompt for Laure to dream Bardo's collage?\n\n_Femme Fatale_ demonstrates a sensitive dependence upon initial conditions: within complex systems microscopic fluctuations can produce macroscopic divergences (see also _Lola rennt_ (1998) and _Memento_ (2001), as well as William Tenn's 'Brooklyn Project' (1948) and Ray Bradbury's 'A Sound of Thunder' (1952), science fiction short stories concomitant with the main cycle of film noir). Laure's decision to intervene in Lily's suicide shuts down the alternative plot which resulted in both their deaths. Unknown to Laure, the fluctuation she introduced by saving Lily would result in Lily giving the pendant to the truck driver thus, seven years later, saving Veronica, killing Black Tie and Racine and introducing Laure to Bardo.\n\nAlthough _Femme Fatale_ produces a conservative vision of non-linear dynamics (the only significant differences between the dream version and frame version of the closing incident at Belleville are the presence of Laure and the pendant \u2013 as if Laure's seven years in France and Lily's in the US have produced no fluctuations, have had no consequences), this is necessary, if one wishes to foreground issues of determinism: a wildly divergent system militates against any sense of d\u00e9nouement. In following the model offered by Tenn, Bradbury and numerous other counterfactual fictions, in which a minor alteration produces an alternative history which diverges from the history we know but not so radically as to be unrecognisable, _Femme Fatale_ is caught between the same irreconcilable impulses towards systemic openness and narrative closure; and it therefore models the text in the image of Tiresias, the mythical 'blind androgyne... chosen by the gods to bear forever a memory that would not fade' who, in Eliot's 'The Waste Land', 'throbb[ed] between two lives' (Iampolski 1998: 2, 3).\n\nThere is a hint of this in Bardo's name, which recalls 'bardo', the name Tibetan Buddhism gives to the intermediate stage between death and rebirth. On one level, this might be a complex allusion to _Vertigo._ (Although based on Pierre Boileau and Thomas Narcejac's novel _D'entre les morts_ (1954), _Vertigo_ was also partly inspired by Ambrose Bierce's 'An Occurrence at Owl Creek Bridge' (1886). In this short story, a hanged man miraculously escapes his execution and returns home; but the conclusion reveals that his escape and return were a fantasy in the final moments of his life. At the start of _Vertigo,_ Scottie is left dangling in the air with no obvious means of escape; at the end of the film, he stares into the abyss. This has prompted some critics to suggest that, following Bierce, the intervening narrative is merely a dream. Regardless of whether this is case, throughout the film Scottie is caught between states in a metaphorical bardo.) On another level, it is a lesson about neo-noir.\n\nLike Scottie, like Lily when she holds the gun to her head and like Laure when she watches, like Bardo as he waits for the perfect moment to photograph, like text itself, neo-noir is suspended between states. If film noir is a partial and abstracted narrative trajectory (neither necessary nor sufficient, but nonetheless useful sometimes) that is retrospectively fabricated in order to make sense of a wider system's moment-to-moment transition, so neo-noir's intertextual fragments constitute the ongoing fabrication of that trajectory. To mix metaphors, the neo-noir film is the collapsing wavefront, stitching itself into a narrative trajectory and webbing itself into an intertext. Open and closed, looking forward and back, inward and outward, it is perpetually in bardo.\nAFTERWORD: KISS TOMORROW GOODBYE\n\nIt was two thirty in the morning, and raining. In the City, it was always two thirty in the morning and raining... But I was still strapped into my life, bound by a plot I could no longer predict, condemned to ride the streetcar until the last stop.\n\n\u2013 Kim Newman (1989: 3 & 37)\n\nLike all good monsters, film noir escaped.\n\nOften reduced to little more than an image or an idea \u2013 the city at night, the femme fatale \u2013 it is a touchstone of popular culture. From the 1985 _Moonlighting_ episode, 'The Dream Sequence Always Rings Twice' and the 1986 _Dick Spanner, P.I._ animations to the 2003 computer game _Max Payne 2: The Fall of Max Payne,_ allusions to this familiar megatext are readily perceived and understood. It is little wonder, then, that when the alien Strangers of _Dark City_ construct an environment in which to perform their experiments upon abducted humans they choose to build and rebuild the eponymous locale.\n\nMike Wayne argues that German expressionism and American film noir lack lucidity because of an 'existential crisis' derived from the subject's 'incomprehension' in the face of the 'absurd and impenetrable world' (2003: 216) of commodity fetishism and reification. In contrast, he suggests that _Dark City_ is 'hyperconscious of the politics of its intertextual cultural references'; but he nonetheless notes in it 'that sense \u2013 difficult to pin down empirically \u2013 that the aesthetics have become, like the subjects in the film, a shell or bodily form emptied of the emotional power and substantive historical content\/context that originally motivated them' (2003: 216\u20137). As the world Wayne describes has become no less absurd or impenetrable since the middle of the twentieth century, it is not unreasonable to conclude that contemporary noir aesthetics and allusions now perform, among other things, a consolation. They offer an image of the world that, however distorted, is familiar and comprehensible. This consolatory tendency can be best explained in terms of the double logic of remediation, by which 'our culture wants to both multiply its media and to erase all traces of mediation' (Bolter and Grusin 1999: 5), and by briefly considering two neo-noirs from either end of the budgetary spectrum of contemporary digital filmmaking: _This Is Not a Love Song_ and _Sin City._\n\nAlthough screened at festivals as early as January 2002, _This Is Not a Love Song_ officially premiered on 5 September 2003, when it was simultaneously released in UK cinemas and on-line (the website received over a hundred thousand hits during its opening weekend). Shot for less than \u00a3300,000 in just twelve days, using a handheld Sony PD150 camera and DV tape, it refashions the couple-on-the-run plot in _Deliverance_ (1972) territory. Taciturn ex-soldier Heaton (Kenny Glenaan) collects his excitable young friend Spike (Michael Colgan), just released from a four-month prison sentence for stealing \u00a357, and takes him on a road trip in a stolen car. When they run out of petrol in the middle of nowhere, a farmer, mistaking Heaton for a thief, locks him up at gunpoint. Spike gets hold of the shotgun, but accidentally shoots and kills the farmer's daughter, Gerry (Keri Arnold). Outraged locals divert the police search to London and then hunt Heaton and Spike across desolate moorland. As Heaton fails to get Spike to safety and, injured, has to rely on him more and more, so their relationship becomes increasingly fraught. With their pursuers closing in, Heaton begs Spike to carry him. They tussle. Spike hits Heaton on the head with a rock, leaving him for dead. Spike is taken by the locals, gagged, bound and drowned. The locals trace Heaton to a narrow cave and seal him up inside. Somehow, he survives and escapes but, like _Vertigo's_ Scottie, he is left confronting an emotional and psychological abyss.\n\n_Sin City_ was shot on high-definition video in front of a green screen, with many of the actors not actually meeting each other despite sharing scenes. Costing $40 million, it took $120 million worldwide in the first four months of its cinematic release. The portmanteau film is based on four of Frank Miller's 1990s 'Sin City' comics \u2013 'The Customer is Always Right', 'That Yellow Bastard', 'The Hard Goodbye' and 'The Big Fat Kill' \u2013 which delight in belonging to the crude pulp tradition of Mickey Spillane and garish paperback covers: Sin City is the kind of place where all the women are whores and all the whores are beautiful, capable of handling a gun and taking a punch.\n\nIn the opening vignette, The Man (Josh Hartnett) tells a beautiful woman that he loves her, then assassinates her. In the next story, Hartigan (Bruce Willis), an about-to-retire honest cop crippled by angina, rescues 11-year-old Nancy (Makenzie Vega) from paedophilic serial killer Roark Jr (Nick Stahl), the son of Senator Roark (Powers Boothe). In the next story, set eight years later, the seemingly indestructible Marv (Mickey Rourke) is framed for the murder of a prostitute, Goldie (Jaime King). Because she was nice to him, he sets out to find her killer, eventually teaming up with Goldie's identical twin sister and fellow prostitute, Wendy (Jaime King). The trail leads him via other murdered prostitutes and Kevin (Elijah Wood), their cannibal killer, to Cardinal Roark (Rutger Hauer). Marv mutilates and murders the Cardinal, and is executed. In the next story, Dwight (Clive Owen) finds himself caught up in a mob scheme to regain control of Old Town, which is run by the prostitutes who work there. When Gail (Rosario Dawson), their leader and Dwight's ex-lover, is abducted, he must set a trap in which they can brutally slay the mobsters. The film then returns to Hartigan. Senator Roark paid to keep him alive so as to force him to confess to Roark Jr's crimes. Hartigan refused, but was convicted anyway as no-one would allow Nancy to testify. She promised to write to him in prison. Eight years later, Hartigan receives a severed finger rather than his weekly letter. Convinced Nancy (Jessica Alba) is in danger again, he confesses and is immediately released. He tracks her down-\u2013 she is a stripper and, she declares, in love with him \u2013 but it has all been a set-up to lead Yellow Bastard (Nick Stahl) to her. Hartigan again rescues her \u2013 Yellow Bastard is Roark Jr, horribly transformed by the chemical processes involved in replacing the genitals Hartigan shot off eight years earlier. Hartigan beats him to death and tears off his new genitals. Fearing that Nancy will be caught up in Senator Roark's revenge, Hartigan shoots himself in the head. In the closing vignette, The Man offers Becky (Alexis Bledel), the prostitute who betrayed Old Town to the mob, a cigarette.\n\nDespite their very obvious differences, these two films indicate the problem Wayne notes of the relationship between an aesthetics and its socially, culturally and historically specific contexts while pursuing the double and contradictory logics of remediation. Jay Bolter and Richard Grusin argue that 'although each medium promises to reform its predecessors by offering a more immediate or authentic experience, the promise of reform inevitably leads us to become aware of the new medium as a medium' (1999: 19). This contradiction is evident in _This Is Not a Love Song_ 's intertwining of realist and expressionist techniques. At times shot with almost Dogme-like purity, the hand-held camera prowls around the action, often getting that little bit too close to the characters, sometimes losing focus, other times refocusing abruptly. Elsewhere, the film expressionistically distorts the image: blurs of colour replace the landscape through which the stolen car races; slow-motion disrupts the relationship between the image and diegetic sound; sat in front of a caf\u00e9 window, Heaton and Spike are reduced to little more than silhouettes; when the panicked Spike brandishes the shotgun, he is filmed with a camera mounted on it; when Spike inhales an aerosol, he trips in day-glo colour. However, even as both sets of techniques claim a form of unmediated immediacy, they also draw attention to the medium mediating. While the handheld 'realist' camera places the viewer in the midst of the action, its shakiness, shifts of focus, zooms out and natural lighting are not merely aesthetic choices evocative of certain varieties of realism but also reminders that the camera being used is not so different from home video technology. Furthermore, in its positioning \u2013 whether in close proximity to the characters or, for example, half underwater as they hide in a river gorge \u2013 it becomes an expressive realism, emphasising a noirish entrapment. Meanwhile, the more obviously mediated and expressionist images \u2013 the blurred landscapes, the day-glo tripping \u2013 also suggest a direct and unmediated corporeal experience of speed or drugs.\n\nThis contradiction is mirrored in the relationship between Heaton and Spike. The insistence of PIL's plaintive 'This Is Not a Love Song', which recurs throughout, unleashes the homoeroticism of both this particular story and film noir more generally. But the film simultaneously draws attention to the mechanisms of denial and suppression that fuel both it and the genre. Gerry is the only woman in the film. Fifteen years old, she is an unlikely potential third point to a noir love triangle; but even with the early removal of any threat she might pose to the men's relationship, their feelings for each other can never be articulated. They talk incessantly, but say very little. Early on, Heaton denies having missed Spike while he was in prison, but then their exchange of glances as they argue over the radio becomes flirtatious, and their subsequent wrestling verges on becoming something else. When they spot the city that might offer them sanctuary, Heaton gazes lovingly at Spike as he gleefully dances. At night, they sleep curled around one another, untroubled by this physical intimacy. Both live in fear of being abandoned by the other. Most poignant of all is Heaton's voice-over as, at various moments, we hear the opening lines of the many letters he tried to write to the imprisoned Spike but could never complete. The ambiguities of this relationship are never pinned down or named. The film instead gives us a sense of these characters' immediacy to each other, while the mechanisms of shot construction and voice-over mediate that experience for the viewer, making their relationship at once both simple and complex.\n\n_Sin City_ is more noteworthy for its attempt to recreate the distinctive visual style than the narratives of its sources. Miller has spoken of the influence of comic artists Johnny Craig and Wallace Wood and, especially, Will Eisner's 'The Spirit' comic strip (1940\u201352), although stylistically he seems most indebted to Kazuo Koike and Goseki Kojima's violent 1970s manga _Kozure Okami (Lone Wolf and Cub)._ The 'Sin City' comics are drawn in stark black-and-white, often with the simplicity of woodcuts. Long sequences are presented with little written text \u2013 for example, the 26-page 'Silent Night' contains one sound effect and a single speech bubble \u2013 while other pages are overburdened with words, with panels cramped by packed speech bubbles and long columns of text running down a thick margin. As Miller remediates the hardboiled novel in comic book form, the visual image claims a greater immediacy even as the intermittent excesses of text point to the inadequacies of the visual image at capturing aspects of the hardboiled novel. Similarly, the _Sin City_ film uses digital tools to give its viewer the immediacy of the comic book image. However, there is again a contradiction. As the digital film remediates the comic book, the immediacy on offer is the product of hypermediation. By 'recreating' Miller's simple illustrations \u2013 some of which are displayed in the title sequence \u2013 the medium draws attention to, rather than effaces, itself.\n\nIn addition to the image, the film remediates Miller's written text as voice-overs which become so excessive that, at one point, Dwight seamlessly takes over his internal monologue, speaking it aloud within the diegesis. Unlike _This Is Not a Love Song,_ in which the voice-over fractures Heaton's superficially confident and competent masculinity, this sequence displays _Sin City's_ hysterical investment in psychic unity and continuity. Whereas the former film plays with and explores the homoeroticism of film noir, the latter reduces sexuality to displays of elaborately costumed gun-toting prostitutes, an armed and naked lesbian parole officer and a stripper packing six-shooters, betraying a not entirely unconscious homosexual panic through the over-performance of heterosexual masculinity and repeated assaults, both verbal and physical, on male genitals.\n\nFIGURE 10 Noir image: _Sin City_ (2005)\n\nHowever, it is not merely the comic book that _Sin City_ remediates. In many spectacle-driven movies, computer-generated imagery (CGI) is utilised to show everything, to render entire worlds visible, as with the vertiginous depths of the city in _Star Wars: Episode 2 \u2013 Attack of the Clones_ (2002). But this impulse is frequently denied in _Sin City_ as it uses digital technology to recreate analogue effects. For example, in the opening vignette, which is set on a roof terrace high over the city, the background is often out of focus behind the sharp foreground images of the actors. Similarly, when Hartigan drives to the waterfront to rescue young Nancy, he does so in front of a digitally created backdrop which looks like an old-fashioned back-projection. Both of these immediacy-effects are instructive in that they do not claim to offer a more direct experience of the real but, like the verit\u00e9 camerawork in _This Is Not a Love Song,_ a familiar and thus authenticating visual language or representation. At other times, _Sin City's_ seems anxious about such recreations, becoming overly parodic, even cartoonish. At these moments, the impulse to erase the new medium 'so that the viewer stands in the same relationship to the content as she would if she were confronting the original medium' (Bolter and Grusin 1999: 45) conflicts with the impulse 'to emphasise the difference rather than erase it' (1999: 46), as the opening vignette makes clear. As The Man holds the woman he has killed, the image switches to a white-on-black silhouette, replicating a frame from 'The Customer Is Always Right'. This comic book moment is followed by a crane shot that only CGI could achieve, the virtual camera rising up from the roof terrace, spiralling around skyscrapers and up into the sky to look down on this Manhattan-like district whose contours spell out 'Sin City' in a font familiar from Miller's comic books.\n\nReturning to Wayne's argument about the hollowed-out aesthetics of _Dark City,_ one can see in _Sin City's_ simple linear narratives and one-dimensional characters a reduction of film noir to its image(s) and the desire not to make a film noir but to somehow put the very idea, the megatext, of film noir on the screen. In contrast, _This Is Not a Love Song_ creates an aesthetics out of economic necessity, cannibalising past styles while innovating at the edges of what its technology permits (for example, the tripping sequence consists of colour-reversed footage shot with an off-the-shelf digital lens which could operate detached from the body of the camera and thus could be fixed to the end of a fishing rod). In this sense, _This Is Not a Love Song_ is a film noir while _Sin City_ merely looks like one.\n\nThis judgement, however, too closely matches a knee-jerk preference for 'realist' over 'fantastic', 'independent' over 'studio', low-budget over big-budget, European over Hollywood filmmaking to be persuasive. These rough binaries should rather be seen as different modes through which film noir continues to evoke the absurd and impenetrable world of late capitalism. For all its ambiguities, _This Is Not a Love Song_ is a love story, although its protagonists are trapped within the socially-constructed subjectivities which deny their love. Likewise, the crude certainties of _Sin City_ \u2013 its depictions of armoured masculinity and eroticised femininity, its various detectives' ability to trace crimes to unambiguous individual sources \u2013 are so hyperbolic, the milieu so knowable, as to completely separate this hermetic metrocosm and its continuous, unified subjects from reality. And even though the realist impulse and aesthetics of _This Is Not a Love Song_ might seem to directly address 'existential crisis', it is through its various remediations that it becomes as effective an expression as _Sin City_ of a cultural moment which shies away from exploring depth. Both films fetishise surface and superfice, generating what complexity they can from the proliferation of images and sounds. If their aesthetics are empty shells, they nonetheless connect to the moment.\n\nSuch remediations are recommodifications, images and styles repackaged and resold as capital exfoliates across levels and dimensions. But the remediation of film noir in digital films is something else, too. It is the next stage in the fabrication of the genre. To see _This Is Not a Love Song_ and _Sin City_ as film noirs requires us to look backward so as to validate their inclusion in the genre. Just as the Strangers reconstruct their _Dark City_ every night to introduce small variants which might produce massive changes, so each additional film noir rethinks, reconstructs and refabricates the genre. For all the superficial unity implied by the certainty of _Sin City's_ visual style and by yet another book about the genre, film noir, like the subjects it depicts, is discontinuous and disunified \u2013 strapped into and bound by a plot no one could predict, condemned to ride the streetcar until the last stop.\nFILMOGRAPHY\n\n_\u00c0 bout de souffle (Breathless)_ (Jean-Luc Godard, Fr., 1959)\n\n_Ace in the Hole_ (Billy Wilder, US, 1951)\n\n_The Addiction_ (Abel Ferrara, US, 1995)\n\n_After Dark, My Sweet_ (James Foley, US, 1990)\n\n_After Hours_ (Martin Scorsese, US, 1985)\n\n_Against All Odds_ (Taylor Hackford, US, 1984)\n\n_Algiers_ (John Cromwell, US, 1938)\n\n_Alphaville_ (Jean-Luc Godard, Fr.\/It., 1965)\n\n_The Anderson Tapes_ (Sidney Lumet, US, 1971)\n\n_Angel Heart_ (Alan Parker, US\/Can.\/UK, 1987)\n\n_Angels with Dirty Faces_ (Michael Curtiz, US, 1938)\n\n_Ascenseur pour l'echefaud (Lift to the Scaffold)_ (Louis Malle, Fr., 1957)\n\n_Asphalt_ (Joe May, Ger., 1928)\n\n_The Asphalt Jungle_ (John Huston, US, 1950)\n\n_L'Atalante_ (Jean Vigo, Fr., 1934)\n\n_Attack_ (Robert Aldrich, US, 1956)\n\n_Au-del\u00e0 des grilles (Beyond the Gates)_ (Ren\u00e9 Cl\u00e9ment, Fr. 1949)\n\n_Baby Face_ (Alfred E. Green, US, 1933)\n\n_Badlands_ (Terrence Malick, US, 1973)\n\n_Bad Lieutenant_ (Abel Ferrara, US, 1992)\n\n_Ball of Fire_ (Howard Hawks, US, 1941)\n\n_Bande \u00e0 part (Band of Outsiders)_ (Jean-Luc Godard, Fr., 1964)\n\n_Barton Fink_ (Joel Coen, US\/UK, 1991)\n\n_Basic Instinct_ (Paul Verhoeven, US\/Fr., 1992)\n\n_Beast of the City_ (Charles Brabin, US, 1932)\n\n_Le Beau Serge (Handsome Serge)_ (Claude Chabrol, Fr., 1958)\n\n_La Belle \u00e9quipe (They Were Five)_ (Julien Duvivier, Fr., 1936)\n\n_Bend of the River_ (Anthony Mann, 1952)\n\n_La B\u00eate humaine (The Human Beast)_ (Jean Renoir, Fr., 1938)\n\n_Beware, My Lovely_ (Harry Horner, US, 1952)\n\n_Beyond a Reasonable Doubt_ (Fritz Lang, US, 1956)\n\n_Beyond the Forest_ (King Vidor, US, 1949)\n\n_The Big Clock_ (John Farrow, US, 1948)\n\n_The Big Heat_ (Fritz Lang, US, 1953)\n\n_The Big Lebowski_ (Joel Coen, US\/UK, 1998)\n\n_The Big Sleep_ (Howard Hawks, UK, 1946)\n\n_The Big Sleep_ (Michael Winner, US, 1978)\n\n_The Black Cat_ (Edgar G. Ulmer, US, 1934)\n\n_The Blackout_ (Abel Ferrara, US\/Fr., 1997)\n\n_Black Rain_ (Ridley Scott, US, 1989)\n\n_Blade Runner_ (Ridley Scott, US, 1982)\n\n_Der Blaue Engel_ (Josef von Sternberg, Ger., 1930)\n\n_Blind Date_ (Joseph Losey, UK, 1959)\n\n_Blood Simple_ (Joel Coen, US, 1984)\n\n_Blow Out_ (Brian De Palma, US, 1981)\n\n_Blowup_ (Michelangelo Antonioni, UK\/It., 1966)\n\n_The Blue Dahlia_ (George Marshall, US, 1946)\n\n_The Blue Gardenia_ (Fritz Lang, US, 1953)\n\n_Blue Velvet_ (David Lynch, US, 1986)\n\n_Bob le Flambeur_ (Bob the Gambler) (Jean-Pierre Melville, Fr., 1955)\n\n_Body and Soul_ (Robert Rossen, US, 1947)\n\n_Body Double_ (Brian De Palma, US, 1984)\n\n_Body Heat_ (Lawrence Kasdan, US, 1981)\n\n_Body of Evidence_ (Uli Edel, Ger.\/US, 1993)\n\n_Boomerang!_ (Elia Kazan, US, 1947)\n\n_Le Boucher (The Butcher)_ (Claude Chabrol, Fr.\/It., 1969)\n\n_Bound_ (Larry and Andy Wachowski, US, 1996)\n\n_The Brasher Dubloon_ (John Brahm, US, 1947)\n\n_Breakdown_ (Jonathan Mostow, US, 1997)\n\n_Bride of Frankenstein_ (James Whale, US, 1935)\n\n_Brighton Rock_ (John Boulting, UK, 1947)\n\n_Buffalo 66_ (Vincent Gallo, US, 1998)\n\n_Bullet Ballet_ (Shinya Tsukamoto, Jap., 1998)\n\n_Bullets or Ballots_ (William Keighley, US, 1936)\n\n_Cape Fear_ (J. Lee Thompson, US, 1962)\n\n_Cape Fear_ (Martin Scorsese, US, 1991)\n\n_Carrie_ (Brian De Palma, US, 1976)\n\n_Casablanca_ (Michael Curtiz, US, 1942)\n\n_Casbah_ (John Berry, US, 1948)\n\n_Casque d'Or (Golden Marie)_ (Jacques Becker, Fr., 1952)\n\n_Cat People_ (Jacques Tourneur, US, 1942)\n\n_Caught_ (Max Oph\u00fcls, US, 1949)\n\n_Cause for Alarm_ (Tay Garnett, US, 1951)\n\n_Champion_ (Mark Robson, US, 1949)\n\n_Charley Varrick_ (Don Siegel, US, 1973)\n\n_Chicago Deadline_ (Lewis Allen, US, 1949)\n\n_Le Chemin de Rio_ (Robert Siodmak, Fr., 1937)\n\n_La Chienne_ (Jean Renoir, Fr., 1931)\n\n_Chinatown_ (Roman Polanski, 1974)\n\n_Christmas Holiday_ (Robert Siodmak, US, 1944)\n\n_Citizen Kane_ (Orson Welles, 1941)\n\n_City Streets_ (Rouben Mamoulian, US, 1931)\n\n_Clash By Night_ (Fritz Lang, US, 1952)\n\n_Cloak and Dagger_ (Fritz Lang, US, 1946)\n\n_Coma_ (Michael Crichton, US, 1978)\n\n_Conflict_ (Curtis Bernhardt, US, 1945)\n\n_Convicted_ (Henry Levin, US, 1950)\n\n_Cornered_ (Edward Dmytryk, US, 1945)\n\n_Crack-Up_ (Irving Reis, US, 1946)\n\n_The Crazies_ (George A. Romero, US, 1973)\n\n_The Criminal_ (Joseph Losey, UK, 1960)\n\n_Criss Cross_ (Robert Siodmak, US, 1949)\n\n_The Crooked Way_ (Robert Florey, US, 1949)\n\n_Crossfire_ (Edward Dmytryk, US, 1947)\n\n_Croupier_ (Mike Hodges, Fr.\/UK\/Ger.\/Ire., 1999)\n\n_Cry of the City_ (Robert Siodmak, US, 1948)\n\n_Cypher_ (Vincenzo Natali, US, 2003)\n\n_Dangerous Game_ (Abel Ferrara, US, 1993)\n\n_Dark City_ (Alex Proyas, US, 1998)\n\n_Dark Corner_ (Henry Hathaway, US, 1946)\n\n_The Dark Mirror_ (Robert Siodmak, US, 1946)\n\n_Dark Passage_ (Delmer Daves, US, 1947)\n\n_Dead End_ (William Wyler, US, 1937)\n\n_Dead Presidents_ (Allen and Albert Hughes, US, 1995)\n\n_Dead Reckoning_ (John Cromwell, US, 1947)\n\n_The Deep End_ (Scott McGehee, US, 2001)\n\n_Deliverance_ (John Boorman, US, 1972)\n\n_Le Dernier tournant (The Last Turn)_ (Pierre Chenal, Fr., 1939)\n\n_Desperate_ (Anthony Mann, US, 1947)\n\n_Desperate Hours_ (Michael Cimino, US, 1990)\n\n_Desperately Seeking Susan_ (Susan Seidelman, US, 1985)\n\n_Detour_ (Edgar G. Ulmer, US, 1945)\n\n_Deux hommes dans Manhattan (Two Men in Manhattan)_ (Jean-Pierre Melville, Fr., 1959)\n\n_Les Diaboliques (Diabolique)_ (Henri-Georges Clouzot, Fr., 1954)\n\n_Dick Spanner, P.I._ (1986)\n\n_Die Hard_ (John McTiernan, US, 1988)\n\n_Dirnentrag\u00f6die (Tragedy of the Street_ aka _Women Without Men)_ (Bruno Rahn, Ger., 1927)\n\n_Disclosure_ (Barry Levinson, US, 1994)\n\n_D.O.A._ (Rudolph Mat\u00e9, US, 1950)\n\n_D.O.A._ (Rocky Morton and Annabel Jankel, US, 1988)\n\n_Doktor Mabuse, der Spieler, Part, 1: Der grosse Spieler \u2013 ein Bild der Zeit (Dr Mabuse, the Gambler)_ (Fritz Lang, Ger., 1922)\n\n_Doktor Mabuse, der Spieler, Part, 2: Inferno: Ein Spiel von Menschen unserer Zeit (Dr Mabuse, the Gambler)_ (Fritz Lang, 1922)\n\n_The Doorway to Hell_ (Archie Mayo, US, 1930)\n\n_Double Indemnity_ (Billy Wilder, US, 1944)\n\n_Le Doulos (Doulos: The Finger Man)_ (Jean-Pierre Melville, Fr.\/It., 1962)\n\n_Dracula_ (Tod Browning, US, 1931)\n\n_Dressed to Kill_ (Brian De Palma, US, 1980)\n\n_The Driller Killer_ (Abel Ferrara, US, 1979)\n\n_Dr Jekyll and Mr Hyde_ (Rouben Mamoulian, US, 1932)\n\n_Du Rififi chez les hommes (Rififi)_ (Jules Dassin, Fr., 1955)\n\n_Easy Rider_ (Dennis Hopper, US, 1969)\n\n_Les Enfants du paradis (Children of Paradise)_ (Marcel Carn\u00e9, Fr., 1943\u201345)\n\n_Equinox_ (Alan Rudolph, US, 1993)\n\n_The Ex-Mrs Bradford_ (Stephen Roberts, US, 1936)\n\n_Experiment Perilous_ (Jacques Tourneur, US, 1944)\n\n_Face_ (Antonia Bird, UK, 1997)\n\n_The Falcon Takes Over_ (Irving Reis, US, 1942)\n\n_The Fallen Idol_ (Carol Reed, US, 1948)\n\n_The Fallen Sparrow_ (Richard Wallace, US, 1943)\n\n_Falling Down_ (Joel Schumacher, Fr.\/US, 1993)\n\n_Farewell, My Lovely_ (Dick Richards, US, 1975)\n\n_The Far Country_ (Anthony Mann, US, 1955)\n\n_Fargo_ (Joel Coen, UK\/US, 1996)\n\n_Fatal Attraction_ (Adrian Lyne, US, 1987)\n\n_Fear City_ (Abel Ferrara, US, 1984)\n\n_Fear X_ (Nicolas Winding Refn, Den.\/Can.\/UK\/Braz., 2002)\n\n_Femme Fatale_ (Brian De Palma, Fr., 2002)\n\n_Fight Club_ (David Fincher, Ger.\/US, 1999)\n\n_The File on Thelma Jordan_ (Robert Siodmak, US, 1950)\n\n_The Finger Points_ (John Francis Dillon, US, 1931)\n\n_Footlight Parade_ (Lloyd Bacon, US, 1933)\n\n_Force of Evil_ (Abraham Polonsky, US, 1949)\n\n_Frankenstein_ (James Whale, US, 1931)\n\n_Freeway_ (Matthew Bright, US, 1996)\n\n_Die freudlose Gasse (The Joyless Street)_ (G. W. Pabst, Ger., 1923)\n\n_The Friends of Eddie Coyle_ (Peter Yates, US, 1973)\n\n_Fury_ (Fritz Lang, US, 1936)\n\n_The Game_ (David Fincher, US, 1997)\n\n_Gaslight_ (George Cukor, US, 1944)\n\n_Genuine_ (Robert Wiene, Ger., 1920)\n\n_The Getaway_ (Roger Donaldson, US\/Jap., 1994)\n\n_Get Carter_ (Mike Hodges, UK, 1970)\n\n_The Gift_ (Sam Raimi, US, 2000)\n\n_Gilda_ (Charles Vidor, US, 1946)\n\n_The Glass Key_ (Frank Tuttle, US, 1935)\n\n_G-Men_ (William Keighley, US, 1935)\n\n_The Good Die Young_ (Lewis Gilbert, UK, 1954)\n\n_Le Grand jeu_ (Jacques Feyder, Fr. 1933)\n\n_The Great Train Robbery_ (Edwin S. Porter, US, 1903)\n\n_The Grifters_ (Stephen Frears, US, 1990)\n\n_Gun Crazy_ (Joseph H. Lewis, US, 1949)\n\n_Hard To Handle_ (Mervyn LeRoy, US, 1933)\n\n_The Harder They Fall_ (Mark Robson, US, 1956)\n\n_Hangover Square_ (John Brahm, US, 1945)\n\n_Heat_ (Michael Mann, US, 1995)\n\n_Hell Drivers_ (Cy Endfield, UK, 1957)\n\n_Hell is a City_ (Val Guest, UK, 1960)\n\n_High Noon_ (Fred Zinnemann, US, 1952)\n\n_High Sierra_ (Raoul Walsh, US, 1941)\n\n_Hintertreppe \u2013 Ein Film-Kammerspiel (Backstairs)_ (Leopold Jessner, Ger., 1921)\n\n_His Kind of Woman_ (John Farrow, US, 1951)\n\n_H\u00f4tel du Nord_ (Marcel Carn\u00e9, Fr., 1938)\n\n_The House on 92nd Street_ (Henry Hathaway, US, 1945)\n\n_The Hudsucker Proxy_ (Joel Coen, UK\/Ger.\/US, 1994)\n\n_Human Desire_ (Fritz Lang, US, 1954)\n\n_I Am a Fugitive from a Chain Gang_ (Mervyn LeRoy, US, 1932)\n\n_I Am the Law_ (Alexander Hall, US, 1938)\n\n_I Confess_ (Alfred Hitchcock, US, 1953)\n\n_I Drink Your Blood_ (David Durston, US, 1971)\n\n_In a Lonely Place_ (Nicholas Ray, US, 1950)\n\n_Ivy_ (Sam Wood, US, 1947)\n\n_I Wake Up Screaming_ (H. Bruce Humberstone, US, 1942)\n\n_Jacob's Ladder_ (Adrian Lyne, US, 1990)\n\n_Jade_ (William Friedkin, US, 1995)\n\n_Le Jour se l\u00e8ve (Daybreak)_ (Marcel Carn\u00e9, Fr., 1939)\n\n_Joy Ride_ (John Dahl, US, 2001)\n\n_Judgment Night_ (Stephen Hopkins, US\/Jap., 1993)\n\n_Das Kabinett des Dr. Caligari (The Cabinet of Dr Caligari)_ (Robert Wiene, Ger., 1919)\n\n_Keisatsukan (Policeman)_ (Uchida Tomu, Jap., 1933)\n\n_Key Largo_ (John Huston, US, 1948)\n\n_The Killers_ (Robert Siodmak, US, 1946)\n\n_The Killers_ (Don Siegel, US, 1964)\n\n_Killer's Kiss_ (Stanley Kubrick, US, 1955)\n\n_The Killing_ (Stanley Kubrick, US, 1956)\n\n_The Kill-Off_ (Maggie Greenwald, US, 1990)\n\n_Kind Hearts and Coronets_ (Robert Hamer, UK, 1949)\n\n_King Kong_ (Merian C. Cooper and Ernest B. Schoedsack, US, 1933)\n\n_King of New York_ (Abel Ferrara, It.\/US\/UK, 1990)\n\n_King's Row_ (Sam Wood, US, 1941)\n\n_Kiss Me Deadly_ (Robert Aldrich, US, 1955)\n\n_Kiss of Death_ (Henry Hathaway, US, 1947)\n\n_Kiss Tomorrow Goodbye_ (Gordon Douglas, US, 1950)\n\n_Klute_ (Alan J. Pakula, US, 1971)\n\n_Kurutta Ippeiji_ (A _Page of Madness)_ (Teinosuke Kinugasa, Jap., 1927)\n\n_L.A. Confidential_ (Curtis Hanson, US, 1997)\n\n_Ladies Love Brutes_ (Rowland V. Lee, US, 1930)\n\n_The Lady Eve_ (Preston Sturges, US, 1941)\n\n_The Lady from Shanghai_ (Orson Welles, US, 1948)\n\n_Lady in the Lake_ (Robert Montgomery, US, 1947)\n\n_The Ladykillers_ (Alexander Mackendrick, UK, 1955)\n\n_The Last House on the Left_ (Wes Craven, US, 1972)\n\n_Last Man Standing_ (Walter Hill, US, 1996)\n\n_The Last Seduction_ (John Dahl, US, 1994)\n\n_Laura_ (Otto Preminger, US, 1944)\n\n_The Lavender Hill Mob_ (Charles Crichton, UK, 1951)\n\n_The League of Gentlemen_ (Basil Dearden, UK, 1960)\n\n_Der Letzte Mann (The Last Laugh)_ (F. W. Murnau, Ger., 1924)\n\n_The Limey_ (Steven Soderbergh, US, 1999)\n\n_Little Caesar_ (Mervyn LeRoy, US, 1930)\n\n_Lock, Stock and Two Smoking Barrels_ (Guy Ritchie, UK, 1998)\n\n_The Lodger_ (John Brahm, US, 1944)\n\n_Lola rennt (Run, Lola, Run_ aka _Lola Runs)_ (Tom Tykwer, Ger., 1998)\n\n_The Long Goodbye_ (Robert Altman, US, 1973)\n\n_The Long Night_ (Anatole Litvak, US, 1947)\n\n_Lost Highway_ (David Lynch, Fr.\/US, 1997)\n\n_The Lost Weekend_ (Billy Wilder, US, 1945)\n\n_Lured_ (Douglas Sirk, US, 1947)\n\n_M_ (Fritz Lang, Ger., 1931)\n\n_Macao_ (Josef von Sternberg [and Nicholas Ray, uncredited], US, 1952)\n\n_The Mad Miss Manton_ (Leigh Jason, US, 1936)\n\n_The Maltese Falcon_ (Roy Del Ruth, US, 1931)\n\n_The Maltese Falcon_ (John Huston, US, 1941)\n\n_The Man Between_ (Carol Reed, UK, 1953)\n\n_The Manchurian Candidate_ (John Frankenheimer, US, 1962)\n\n_Der Man, der Seinen Morder Sucht (Looking for His Murderer)_ (Robert Siodmak, Ger., 1931)\n\n_The Man from Laramie_ (Anthony Mann, US, 1955)\n\n_Man Hunt_ (Fritz Lang, US, 1941)\n\n_The Man Who Wasn't There_ (Joel Coen, US, 2001)\n\n_La Mari\u00e9e \u00e9tait en noir (The Bride Wore Black)_ (Fran\u00e7ois Truffaut, Fr.\/It., 1967)\n\n_Marlowe_ (Paul Bogart, US, 1969)\n\n_Marnie_ (Alfred Hitchcock, US, 1964)\n\n_The Mask of Dimitrios_ (Jean Negulesco, US, 1944)\n\n_La Maternelle_ (Marie Epstein, US, 1933)\n\n_Max Payne 2: The Fall of Max Payne_ (2003)\n\n_The Mayor of Hell_ (Archie Mayo, US, 1933)\n\n_Mean Streets_ (Martin Scorsese, US, 1973)\n\n_Medium Cool_ (Haskell Wexler, US, 1969)\n\n_Memento_ (Christopher Nolan, US, 2001)\n\n_Menschen am Sonntag (People on Sunday)_ (Robert Siodmak and Edgar G. Ulmer, Ger., 1930)\n\n_Metropolis_ (Fritz Lang, Ger., 1926)\n\n_Mildred Pierce_ (Michael Curtiz, US, 1945)\n\n_Miller's Crossing_ (Joel Coen, US, 1990)\n\n_Ministry of Fear_ (Fritz Lang, US, 1945)\n\n_Mission: Impossible_ (Brian De Palma, US, 1996)\n\n_Mollenard (Hatred)_ (Robert Siodmak, Fr., 1938)\n\n_Moonlighting_ (1985\u201389)\n\n_Ms.45_ (Abel Ferrara, US, 1981)\n\n_Mulholland Drive_ (David Lynch, Fr.\/US, 2001)\n\n_Murder, My Sweet_ (Edward Dmytryk, US, 1944)\n\n_My Darling Clementine_ (John Ford, US, 1946)\n\n_My Name is Julia Ross_ (Joseph H. Lewis, US, 1945)\n\n_Nachts, wenn der Teufel kam (Night, When the Devil Came_ aka _The Devil Strikes at Night)_ (Robert Siodmak, W. Ger., 1957)\n\n_The Naked City_ (Jules Dassin, US, 1948)\n\n_The Naked Kiss_ (Samuel Fuller, US, 1964)\n\n_Naked Lunch_ (David Cronenberg, Can.\/UK\/Jap., 1991)\n\n_The Naked Spur_ (Anthony Mann, US, 1953)\n\n_Nancy Steele is Missing_ (George Marshall, US, 1936)\n\n_Natural Born Killers_ (Oliver Stone, US, 1994)\n\n_New Rose Hotel_ (Abel Ferrara, US, 1998)\n\n_Night and the City_ (Jules Dassin, UK, 1950)\n\n_Night and the City_ (Irwin Winkler, US, 1992)\n\n_Nightfall_ (Jacques Tourneur, US, 1957)\n\n_Nightmare Alley_ (Edmund Goulding, US, 1947)\n\n_A Night to Remember_ (Richard Wallace, US, 1943)\n\n_North by Northwest_ (Alfred Hitchcock, US, 1959)\n\n_Notorious_ (Alfred Hitchcock, US, 1946)\n\n_No Way Out_ (Roger Donaldson, US, 1987)\n\n_Obsession_ (Edward Dmytryk, UK, 1949)\n\n_Obsession_ (Brian De Palma, US, 1976)\n\n_Ocean's Eleven_ (Lewis Milestone, US, 1960)\n\n_Ocean's Eleven_ (Steven Soderbergh, US, 2001)\n\n_Odd Man Out_ (Carol Reed, UK, 1947)\n\n_Odds Against Tomorrow_ (Robert Wise, US, 1959)\n\n_\u00d4dishon (Audition)_ (Miike Takashi, Jap.\/S. Kor, 2000)\n\n_On Dangerous Ground_ (Nicholas Ray, US, 1952)\n\n_One False Move_ (Carl Franklin, US, 1992)\n\n_Orlacs H\u00e4nde (The Hands of Orlac)_ (Robert Wiene, Ger., 1924)\n\n_The Outfit_ (John Flynn, US, 1974)\n\n_Out of the Past_ (Jacques Tourneur, US, 1947)\n\n_The Paradine Case_ (Alfred Hitchcock, US, 1948)\n\n_Payback_ (Brian Helgeland, US, 1999)\n\n_P\u00e9p\u00e9 le moko_ (Julien Duvivier, Fr., 1936)\n\n_A Perfect Murder_ (Andrew Davis, US, 1998)\n\n_Per un pugno di dollari (A Fistful of Dollars)_ (Sergio Leone, W. Ger\/Sp.\/It., 1964)\n\n_Phantom Lady_ (Robert Siodmak, US, 1944)\n\n_Pickpocket_ (Robert Bresson, Fr., 1959)\n\n_Pi\u00e8ges (Personal Column)_ (Robert Siodmak, Fr., 1939)\n\n_Pierrot le fou_ (Jean-Luc Godard, Fr.\/It., 1965)\n\n_Pitfall_ (Andre de Toth, US, v1948)\n\n_Plein Soleil (Purple Noon)_ (Ren\u00e9 Cl\u00e9ment, Fr.\/It. 1959)\n\n_Point Blank_ (John Boorman, US, 1967)\n\n_The Postman Always Rings Twice_ (Tay Garnett, US, 1946)\n\n_The Postman Always Rings Twice_ (Bob Rafelson, US\/W. Ger., 1981)\n\n_Psycho_ (Alfred Hitchcock, US, 1960)\n\n_The Public Enemy_ (William Wellman, US, 1931)\n\n_Pursued_ (Raoul Walsh, US, 1947)\n\n_Quai des brumes (Port of Shadows)_ (Marcel Carn\u00e9, Fr., 1938)\n\n_Quai des Orf\u00e8vres (Quay of the Goldsmiths)_ (Henri-Georges Clouzot, Fr., 1947)\n\n_Que la b\u00eate meure (The Beast Must Die)_ (Claude Chabrol, Fr., 1969)\n\n_Racket Busters_ (Lloyd Bacon, US, 1938)\n\n_Raising Cain_ (Brian De Palma, US, 1992)\n\n_Ramrod_ (Andre de Toth, US, 1947)\n\n_Rancho Notorious_ (Fritz Lang, US, 1952)\n\n_Raskolnikov_ (Robert Wiene, Ger., 1923)\n\n_Raw Deal_ (Anthony Mann, US, 1948)\n\n_Rear Window_ (Alfred Hitchcock, US, 1954)\n\n_Rebecca_ (Alfred Hitchcock, 1940)\n\n_The Reckless Moment_ (Max Oph\u00fcls, 1949)\n\n_Red Rock West_ (John Dahl, US, 1992)\n\n_Remember Last Night?_ (James Whale, US, 1935)\n\n_Reservoir Dogs_ (Quentin Tarantino, US, 1992)\n\n_Ride the Pink Horse_ (Robert Montgomery, US, 1947)\n\n_Roadhouse Nights_ (Hobart Henley, US, 1930)\n\n_The Rockford Files_ (1974\u201380)\n\n_Rokugatsu no hebi (A Snake of June,_ Shinya Tsukamoto, Jap., 2002)\n\n_Rope_ (Alfred Hitchcock, US, 1948)\n\n_The Rough and the Smooth_ (Robert Siodmak, UK, 1959)\n\n_La Rue sans nom (Street Without a Name)_ (Pierre Chenal, Fr., 1933)\n\n_Le Salaire de le peur (The Wages of Fear)_ (Henri-Georges Clouzot, Fr.\/It., 1953)\n\n_Le Samoura\u00ef_ (Jean-Pierre Melville, Fr.\/It., 1967)\n\n_Satan Met A Lady_ (William Dieterle, US, 1936)\n\n_Scarface_ (Howard Hawks, US, 1932)\n\n_Scarlet Street_ (Fritz Lang, US, 1945)\n\n_Schatten \u2013 Eine n\u00e4chtlichte Halluzination_ (Arthur Robison, Ger., 1923)\n\n_Second Chance_ (Rudolph Mat\u00e9, US, 1953)\n\n_Secret Beyond the Door_ (Fritz Lang, US, 1948)\n\n_Serenade_ (Anthony Mann, US, 1956)\n\n_The Set-Up_ (Robert Wise, US, 1949)\n\n_Se7en_ (David Fincher, US, 1995)\n\n_Sexy Beast_ (Jonathan Glazer, UK\/Sp., 2000)\n\n_Shadow of a Doubt_ (Alfred Hitchcock, US, 1943)\n\n_Shane_ (George Stevens, US, 1953)\n\n_Sherlock Homes and the Voice of Terror_ (John Rawlins, US, 1942)\n\n_Shock Corridor_ (Samuel Fuller, US, 1963)\n\n_Side Street_ (Anthony Mann, US, 1950)\n\n_A Simple Plan_ (Sam Raimi, Fr.\/UK\/Ger.\/US\/Jap., 1998)\n\n_Sin City_ (Frank Miller and Robert Rodriguez, US, 2005)\n\n_La Sir\u00e8ne du Mississippi (Mississippi Mermaid)_ (Fran\u00e7ois Truffaut, It.\/Fr., 1969)\n\n_Sleep, My Love_ (Douglas Sirk, US, 1948)\n\n_So Evil My Love_ (Lewis Allen, US\/UK, 1948)\n\n_Something Wild_ (Jonathan Demme, US, 1986)\n\n_Somewhere in the Night_ (Joseph L. Mankiewicz, US, 1946)\n\n_Sorry, Wrong Number_ (Anatole Litvak, US, 1948)\n\n_Sous les toits de Paris (Under the Roofs of Paris)_ (Ren\u00e9 Clair, Fr., 1930)\n\n_The Spider Woman_ (Roy William Neill, US, 1944)\n\n_Spione (Spies)_ Fritz Lang, Ger., 1928)\n\n_The Spiral Staircase_ (Robert Siodmak, 1945)\n\n_The Star Chamber_ (Peter Hyams, US, 1983)\n\n_Star Wars: Episode 2 \u2013 Attack of the Clones_ (George Lucas, US, 2002)\n\n_Stella Dallas_ (King Vidor, US, 1937)\n\n_The Strange Affair of Uncle Harry_ (Robert Siodmak, US, 1945)\n\n_The Strange Love of Martha Ivers_ (Lewis Milestone, US, 1946)\n\n_Stranger on the Third Floor_ (Boris Ingster, US, 1940)\n\n_Die Strasse (The Street)_ (Karl Grune, Ger., 1923)\n\n_Street of Chance_ (John Cromwell, US, 1930)\n\n_Sudden Fear_ (David Miller, US, 1952)\n\n_Sunset Blvd._ (Billy Wilder, US, 1950)\n\n_The Suspect_ (Robert Siodmak, US, 1945)\n\n_Suspicion_ (Alfred Hitchcock, US, 1941)\n\n_Suture_ (Scott McGehee, US, 1993)\n\n_Sweet Smell of Success_ (Alexander Mackendrick, US, 1957)\n\n_The Taking of Pelham One Two Three_ (Joseph Sargent, US, 1974)\n\n_Taxi Driver_ (Martin Scorsese, US, 1976)\n\n_Temptation_ (Irving Pichel, US, 1946)\n\n_Das Testament des Dr Mabuse (The Testament of Dr Mabuse)_ (Fritz Lang, Ger., 1933)\n\n_The Texas Chain Saw Massacre_ (Tobe Hooper, US, 1974)\n\n_Thelma & Louise_ (Ridley Scott, US, 1991)\n\n_They Live by Night_ (Nicholas Ray, US, 1948)\n\n_Thieves Like Us_ (Robert Altman, US, 1974)\n\n_The Thin Man_ (W.S. Van Dyke, US, 1934)\n\n_The Third Man_ (Carol Reed, UK, 1949)\n\n_The Thirteenth Floor_ (Josef Rusniak, Ger.\/US, 1999)\n\n_This Is Not a Love Song_ (Bille Eltringham, UK, 2002)\n\n_This World, Then the Fireworks_ (Michael Oblowitz, US, 1997)\n\n_Time to Kill_ (Herbert I. Leeds, US, 1942)\n\n_Tirez sur le pianiste (Shoot the Piano Player)_ (Fran\u00e7ois Truffaut, Fr., 1960)\n\n_Tokyo-ken (Tokyo Fist)_ (Shinya Tsukamoto, Jap., 1995)\n\n_Topkapi_ (Jules Dassin, US, 1964)\n\n_Touchez pas au Grisbi (Grisbi)_ Jacques Becker, Fr.\/It., 1945)\n\n_Touch of Evil_ (Orson Welles, US, 1958)\n\n_True Romance_ (Tony Scott, US, 1993)\n\n_The Two Mrs. Carrolls_ (Peter Godfrey, US, 1947)\n\n_2001: A Space Odyssey_ (Stanley Kubrick, UK\/US, 1968)\n\n_Undercurrent_ (Vincente Minnelli, US, 1946)\n\n_Underworld_ (Josef von Sternberg, US, 1927)\n\n_Underworld U.S.A._ (Samuel Fuller, US, 1961)\n\n_The Usual Suspects_ (Bryan Singer, US\/Ger., 1995)\n\n_U Turn_ (Oliver Stone, Fr.\/US, 1997)\n\n_Vertigo_ (Alfred Hitchcock, US, 1958)\n\n_Very Bad Things_ (Peter Berg, US, 1998)\n\n_De Vierde man (The Fourth Man)_ (Paul Verhoeven, Neth., 1983)\n\n_Vital_ (Shinya Tsukamoto, Jap., 2004)\n\n_Vivement dimanche (Confidentially Yours)_ (Fran\u00e7ois Truffaut, Fr., 1983)\n\n_Von morgens bis mitternachts (From Morn to Midnight)_ (Karl Heinz Martin, Ger., 1920)\n\n_Das Wachsfigurenkabinett (Waxworks)_ (Paul Leni, Ger., 1924)\n\n_The War of the Roses_ (Danny DeVito, US, 1989)\n\n_The Web_ (Michael Gordon, US, 1947)\n\n_Weekend_ (Jean-Luc Godard, Fr., 1967)\n\n_Where the Sidewalk Ends_ (Otto Preminger, 1950)\n\n_While the City Sleeps_ (Fritz Lang, US, 1956)\n\n_White Heat_ (Raoul Walsh, US, 1949)\n\n_Wild at Heart_ (David Lynch, US, 1990)\n\n_Winchester '73_ (Anthony Mann, US, 1950)\n\n_The Window_ (Ted Tetzlaff, US, 1949)\n\n_The Woman in Green_ (Roy William Neill, US, 1945)\n\n_The Woman in the Window_ (Fritz Lang, US, 1945)\n\n_The Wrong Man_ (Alfred Hitchcock, US, 1957)\n\n_Yojimbo_ (Akira Kurosawa, Jap., 1961)\n\n_You Only Live Once_ (Fritz Lang, US, 1937)\nBIBLIOGRAPHY\n\nAdorno, Theodor W. and Max Horkheimer (1997) [1944] _Dialectic of Enlightenment._ Translated by John Cumming. London: Verso.\n\nAlthusser, Louis (1971) [1969] 'Ideology and Ideological State Apparatuses (Notes Towards an Investigation)', in _Lenin and Philosophy and Other Essays._ Translated by Ben Brewster. London: NLB, 121\u201373.\n\nAltman, Rick (1999) _Film\/Genre._ London: British Film Institute.\n\nAndrew, Dudley (1995) _Mists of Regret: Culture and Sensibility in Classic French Film._ Princeton: Princeton University Press.\n\nBarefoot, Guy (2001) _Gaslight Melodrama: From Victorian London to 1940s Hollywood._ New York: Continuum.\n\nBarlow, John D. (1982) _German Expressionist Cinema._ Boston: Twayne. Bernstein, Matthew and Gaylyn Studlar (eds) (1997) _Visions of the East: Orientalism in Film._ London: I. B. Tauris.\n\nBogdanovich, Peter (1997) _Who the Devil Made It._ New York: Ballantine.\n\nBoler, Jay and Richard Grusin (1999) _Remediation: Understanding New Media._ Cambridge, MA: MIT Press.\n\nBorde, Raymonde and Etienne Chaumeton (2002) [1955] _A Panorama of American Film Noir, 1941\u20131953._ Translated by Paul Hammond. San Francisco: City Lights Books.\n\nBordwell, David (1985) _Narration in the Fiction Film._ London: Methuen.\n\nBordwell, David, Janet Staiger and Kristin Thompson (1988) _The Classical Hollywood Cinema: Film Style and Mode of Production to 1960._ London: Routledge.\n\nBottomore, Tom, Laurence Harris, V. G. Kiernan and Ralph Miliband (eds) (1991) _A Dictionary of Marxist Thought_ (second edition). Oxford: Blackwell.\n\nBould, Mark (2002) 'The Dreadful Credibility of Absurd Things: A Tendency in Fantasy Theory', _Historical Materialism: Research in Critical Marxist Theory,_ 10, 4, 51\u201388.\n\n\u2014\u2014\u2014(2003) 'Apocalypse Here and Now: Making Sense of _The Texas Chain Saw Massacre',_ in Gary D. Rhodes (ed.) (2003) _Horror at the Drive-In: Essays in Popular Americana._ Jefferson: McFarland, 97\u2013112.\n\nBradbury, Malcolm and James McFarlane (eds) (1976) _Modernism, 1890\u20131930._ London: Penguin.\n\nBritton, Andrew (1992) _'Detour',_ in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 174\u201383.\n\nBukatman, Scott (1993) _Terminal Identity: The Virtual Subject in Postmodern Science Fiction._ Durham: Duke University Press.\n\nBuss, Robin (1994) _French Film Noir._ New York: Marion Boyars.\n\nCain, James M. (2002) _Double Indemnity._ London: Orion.\n\nCameron, Ian (ed.) (1992) _The Movie Book of Film Noir._ London: Studio Vista.\n\nChabrol, Claude (1985) [1955] 'Evolution of the Thriller', translated by Liz Heron, in Jim Hillier (ed.) _Cahiers du Cin\u00e9ma. The 1950s: Neo-Realism, Hollywood, New Wave._ Cambridge: Harvard University Press, 158\u201364.\n\nChandler, Raymond (1980) 'Introduction', to _Pearls Are a Nuisance. The Chandler Collection, vol. III,_ London: Picador, 9\u201312.\n\nChartier, Jean-Pierre (1996) [1946] 'The Americans Are Making Dark Films Too', translated by R. Barton Palmer, in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 25\u20137.\n\nCoates, Paul (1991) _The Gorgon's Gaze: German Cinema, Expressionism, and the Image of Horror._ Cambridge: Cambridge University Press.\n\nCollins, Chik (2000) 'Vygotsky on Language and Social Consciousness: Underpinning the Use of Voloshinov in the Study of Popular Protest', _Historical Materialism: Research in Critical Marxist Theory,_ 7, 41\u201369.\n\nCopjec, Joan (ed.) (1993) _Shades of Noir: A Reader._ London: Verso.\n\nCorber, Robert J. (1997) _Homosexuality in Cold War America: Resistance and the Crisis of Masculinity._ Durham: Duke University Press.\n\nCrisp, Colin (1993) _The Classic French Cinema: 1930\u20131960._ Bloomington: Indiana University Press.\n\nDaly, Carroll John (1985) [1923] 'Three Gun Terry', in William F. Nolan (ed.) _Black Mask Boys: Masters in the Hard-Boiled School of Detective Fiction._ New York: Mysterious Press, 43\u201372.\n\nDamico, James (1996) [1978] 'Film Noir: A Modest Proposal', in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 129\u201340.\n\nDavis, Mike (1986) _Prisoners of the American Dream: Politics and Economy in the History of the U.S. Working Class._ London: Verso.\n\nDiawara, Manthia (1993) _'Noir_ by _Noirs:_ Towards a New Realism in Black Cinema', in Joan Copjec (ed.) _Shades of Noir: A Reader._ London: Verso, 261\u201378.\n\nDimendberg, Edward (2004) _Film Noir and the Spaces of Modernity._ Cambridge: Harvard University Press.\n\nDmytryk, Edward (1996) _Odd Man Out: A Memoir of the Hollywood Ten._ Carbondale and Edwardsville: Southern Illinois University Press.\n\nDuncan, Paul (2003) _The Pocket Essential Film Noir_ (revised edition). Harpenden: Pocket Essentials.\n\nDurgnat, Raymond (1996) [1970] 'Paint It Black: The Family Tree of the _Film Noir',_ in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 83\u201398.\n\nDuvillars, Pierre (1996) [1951] 'She Kisses Him So He'll Kill', (translated by R. Barton Palmer) in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 30\u20132.\n\nEarman, John (1986) _A Primer on Determinism._ Dordrecht: D. Reidel.\n\nEisner, Lotte (1977) [1952] _The Haunted Screen: Expressionism in the German Cinema and the Influence of Max Reinhardt._ Translated by Roger Greaves. Berkeley: University of California Press.\n\nElsaesser, Thomas (2000) _Weimar Cinema and After: Germany's Historical Imaginary._ London: Routledge.\n\nElsaesser, Thomas with Adam Barker (eds) (1990) _Early Cinema: Space, Frame, Narrative._ London: British Film Institute.\n\nForter, Greg (2000) _Murdering Masculinities: Fantasies of Gender and Violence in the American Crime Novel._ New York: New York University Press.\n\nFrank, Nino (1996) [1946] 'The Crime Adventure Story: A New Kind of Detective Film', translated by R. Barton Palmer, in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 21\u20134.\n\nFukuyama, Francis (1992) _The End of History and the Last Man._ London: Penguin.\n\nGledhill, Christine (1998a) [1978] _'Klute_ 1: A Contemporary Film Noir and Feminist Criticism', in E. Ann Kaplan (ed.) _Women in Film Noir_ (second edition). London: British Film Institute, 20\u201334.\n\n\u2014\u2014\u2014(1998b) [1978] _'Klute_ 2: Feminism and _Klute',_ in E. Ann Kaplan (ed.) _Women in Film Noir_ (second edition). London: British Film Institute, 99\u2013114.\n\nGramsci, Antonio (1998) _Selections from the Prison Notebooks of Antonio Gramsci,_ edited and translated by Quintin Hoare and Geoffrey Nowell Smith. London: Lawrence and Wishart.\n\nGunning, Tom (2000) _The Films of Fritz Lang: Allegories of Vision and Modernity._ London: British Film Institute.\n\nHall, Nina (ed.) (1992) _The New Scientist Guide to Chaos._ London: Penguin.\n\nHammett, Dashiell (1985) [1923] 'Bodies Piled Up', in William F. Nolan (ed.) _Black Mask Boys: Masters in the Hard-Boiled School of Detective Fiction._ New York: Mysterious Press, 80\u201391.\n\nHarvey, Sylvia (1998) 'Woman's Place: The Absent Family of Film Noir', in E. Ann Kaplan (ed.) _Women in Film Noir_ (second edition). London: British Film Institute, 35\u201346.\n\nHayles, N. Katherine (1990) _Chaos Bound: Orderly Disorder in Contemporary Literature and Science._ Ithaca: Cornell University Press.\n\n\u2014\u2014\u2014(ed.) (1991) _Chaos and Order: Complex Dynamics in Literature and Science._ Chicago: University of Chicago Press.\n\nHayward, Susan (1993) _French National Cinema._ London: Routledge.\n\nHeld, David (1991) 'Frankfurt School', in Tom Bottomore, Laurence Harris, V. G. Kiernan and Ralph Miliband (eds) _A Dictionary of Marxist Thought_ (second edition). Oxford: Blackwell. 208\u201313.\n\nHerf, Jeffrey (1984) _Reactionary Modernism: Technology, Culture and Politics in Weimar and the Third Reich._ Cambridge: Cambridge University Press.\n\nHigham, Charles and Joel Greenberg (1970) [1968] _Hollywood in the Forties._ New York: Coronet.\n\nHillier, Jim (ed.) (1985) _Cahiers du Cin\u00e9ma. The 1950s: Neo-Realism, Hollywood, New Wave._ Cambridge: Harvard University Press.\n\nHirsch, Foster (1999) _Detours and Lost Highways: A Map of Neo-Noir._ New York: Limelight.\n\nHoefer, Carl (2003) 'Causal Determinism', in _The Stanford Encyclopedia of Philosophy_ at <http:\/\/plato.stanford.edu\/entries\/determinism-causal>\n\nIampolski, Mikhail (1998) _The Memory of Tiresias: Intertextuality and Film,_ translated by Harsha Ram. Berkeley: University of California Press.\n\nJacobowitz, Florence (1992) [1988] 'The Man's Melodrama: _The Woman in the Window_ & _Scarlet Street'_ in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 152\u201364.\n\nJacobs, Lea (1991) _The Wages of Sin: Censorship and the Fallen Woman Film, 1928\u20131942._ Madison: University of Wisconsin Press.\n\nJacobs, Lewis (ed.) (1977) _The Compound Cinema: The Film Writings of Harry Alan Potamkin._ New York: Teachers College Press.\n\nJames, C. L. R. (1993) _American Civilization,_ edited and introduced by Anna Grimshaw and Keith Hart. Oxford: Blackwell.\n\nJameson, Fredric (1991) _Postmodernism, or, The Cultural Logic of Late Capitalism._ London: Verso.\n\nJeter, K. W. (1995) _Blade Runner 2: The Edge of Human._ London: Orion.\n\nKaes, Anton (2001) _M_ (revised edition). London: British Film Institute.\n\nKaes, Anton, Martin Jay and Edward Dimendberg (eds) (1994) _The Weimar Republic Sourcebook._ Berkeley: California University Press.\n\nKaplan, E. Ann (ed.) (1998a) _Women in Film Noir_ (second edition). London: British Film Institute.\n\n\u2014\u2014\u2014(1998b) 'The \"Dark Continent\" of Film Noir: Race, Displacement and Metaphor in Tourneur's _Cat People_ (1942) and Welles' _The Lady from Shanghai_ (1948)', in E. Ann Kaplan (ed.) _Women in Film Noir_ (second edition). London: British Film Institute, 183\u2013201.\n\nKawin, Bruce (1978) _Mindscreen: Bergman, Godard, and First-Person Film._ Princeton: Princeton University Press.\n\nKerr, Paul (1996) [1979] 'Out of What Past? Notes on the B _film noir',_ in Alain Silver and James Ursini (eds) _Film Noir Reader._ New York: Limelight, 107\u201327.\n\nKracauer, Siegfried (1974) [1947] _From Caligari to Hitler: A Psychological History of German Film._ Princeton: Princeton University Press.\n\nKrutnik, Frank (1991) _In a Lonely Street: Film Noir, Genre, Masculinity._ London: Routledge.\n\nLaqueur, Walter (1974) _Weimar \u2013 A Cultural History._ London: Weidenfeld and Nicolson.\n\nLev, Peter (2003) _Transforming the Screen, 1950\u20131959._ New York: Scribner's.\n\nMadden, David (ed.) (1968) _Tough Guy Writers of the Thirties._ Carbondale: Southern Illinois University Press.\n\nMaltby, Richard (2003) _Hollywood Cinema_ (second edition). Oxford: Blackwell.\n\nMarling, William (1995) _The American Roman Noir: Hammett, Cain, and Chandler._ Athens: University of Georgia Press.\n\nMartin, Richard (1997) _Mean Streets and Raging Bulls: The Legacy of Film Noir in Contemporary American Cinema._ Lanham: Scarecrow.\n\nMcCann, Sean (2000) _Gumshoe America: Hard-Boiled Crime Fiction and the Rise and Fall of New Deal Liberalism._ Durham: Duke University Press.\n\nMcNally, David (1995) 'Language, History and Class Struggle', _Monthly Review,_ 47, 3, 13\u201330.\n\n\u2014\u2014\u2014(2001) _Bodies of Meaning: Studies on Language, Labor, and Liberation._ Albany: State University of New York Press.\n\nMonaco, Paul (1976) _Cinema & Society: France and Germany During the Twenties._ New York: Elsevier.\n\nMorgan, Janice (1997) 'In the Labyrinth: Masculine Subjectivity, Expatriation and Colonialism in _P\u00e9p\u00e9 le Moko',_ in Matthew Bernstein and Gaylyn Studlar (eds) _Visions of the East: Orientalism in Film._ London: I. B. Tauris, 253\u201368.\n\nMoylan, Tom (1986) _Demand the Impossible: Science Fiction and the Utopian Imagination._ London: Methuen.\n\nMulvey, Laura (1989) [1975] 'Visual Pleasure and Narrative Cinema', in _Visual and Other Pleasures._ Basingstoke: Macmillan, 14\u201326.\n\nMunby, Jonathan (1999) _Public Enemies, Public Heroes: Screening the Gangster from Little Caesar to Touch of Evil._ Chicago: University of Chicago Press.\n\nMusser, Charles (1990) [1984] 'The Travel Genre in 1903\u20131904: Moving Towards Fictional Narrative', in Thomas Elsaesser with Adam Barker (eds) _Early Cinema: Space, Frame, Narrative._ London: British Film Institute, 123\u201332.\n\nNaremore, James (1998) _More than Night: Film Noir in its Contexts._ Berkeley: University of California Press.\n\nNeale, Steve (2000) _Genre and Hollywood._ London: Routledge.\n\n\u2014\u2014\u2014(ed.) (2002) _Genre and Contemporary Hollywood._ London: British Film Institute.\n\nNewman, Kim (1989) _The Night Mayor._ London: Simon & Schuster.\n\nNolan, William F. (1985) _Black Mask Boys: Masters in the Hard-Boiled School of Detective Fiction._ New York: Mysterious Press.\n\nO'Brien, Charles (1996) 'Film Noir in France: Before the Liberation', _Iris,_ 21, 7\u201320.\n\nO'Brien, Geoffrey (1997) _Hardboiled America: Lurid Paperbacks and the Masters of Noir_ (expanded edition). New York: Da Capo Press.\n\nO'Connell, Jack (ed.) (2002) _Dark Alleys of Noir. Paradoxa,_ 16.\n\nO'Shaughnessy, Martin (1996) _'P\u00e9p\u00e9 le Moko,_ or the Impossibility of Being French in the 1930s', _French Cultural Studies,_ 7, 247\u201358.\n\nOliver, Kelly and Benigno Trigo (2003) _Noir Anxiety._ Minneapolis: University of Minnesota Press.\n\nPalmer, R. Barton (ed.) (1996) _Perspectives on Film Noir._ New York: G. K. Hall.\n\nPetrie, Graham (1985) _Hollywood Destinies: European Directors in America 1922\u20131931._ London: Routledge & Kegan Paul.\n\nPetro, Patrice (1989) _Joyless Streets: Women and Melodramatic Representation in Weimar Germany._ Princeton: Princeton University Press.\n\nPlace, J. A. and L. S. Peterson (1996) [1974], 'Some Visual Motifs of _Film Noir',_ in Alain Silver and James Ursini (eds) _Film Noir Reader._ New York: Limelight, 64\u201375.\n\nPorfirio, Robert (1999) 'Dark Jazz: Music in the Film Noir' in Alain Silver and James Ursini (eds) _Film Noir Reader 2._ New York: Limelight, 177\u201387.\n\n\u2014\u2014\u2014(2001) 'Mikl\u00f3s R\u00f3zsa (1907\u20131995)', in Robert Porfirio, Alain Silver and James Ursini (eds) _Film Noir Reader 3: Interviews with Filmmakers of the Classic Noir Period._ New York: Limelight, 163\u201376.\n\nPorfirio, Robert, Alain Silver and James Ursini (eds) (2001) _Film Noir Reader 3: Interviews with Filmmakers of the Classic Noir Period._ New York: Limelight.\n\nPotamkin, Harry (1977) [1930] 'The Racketeer Paramount', in Lewis Jacobs (ed.) _The Compound Cinema: The Film Writings of Harry Alan Potamkin._ New York: Teachers College Press, 477\u20139.\n\nPrawer, S. S. (2002) _The Blue Angel (Der Blaue Engel)._ London: British Film Institute.\n\nPrigogine, Ilya and Isabelle Stengers (1985) _Order Out of Chaos: Man's New Dialogue with Nature._ London: Flamingo.\n\nPye, Douglas (1992) 'Film Noir and Suppressive Narrative: _Beyond a Reasonable Doubt',_ in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 98\u2013109.\n\nRabinowitz, Paula (2002) _Black & White & Noir: America's Pulp Modernism._ New York: Columbia University Press.\n\nReid, David and Jayne L. Walker (1993) 'Strange Pursuit: Cornell Woolrich and the Abandoned City of the Forties', in Joan Copjec (ed.) _Shades of Noir: A Reader._ London: Verso, 57\u201396.\n\nRey, Henri-Fran\u00e7ois (1996) [1948] 'Hollywood Makes Myths like Ford Makes Cars (last installment): Demonstration by the Absurd: Films Noirs', Translated by R. Barton Palmer, in R. Barton Palmer (ed) _Perspectives on Film Noir._ New York: G. K. Hall, 28\u20139.\n\nRhodes, Gary D. (2003) _Horror at the Drive-In: Essays in Popular Americana._ Jefferson: McFarland.\n\nRichardson, Carl (1992) _Autopsy: An Element of Realism in Film Noir._ Metuchen: Scarecrow Press.\n\nSandler, Kevin S. (2002) 'Movie Ratings as Genre: The Incontestable R', in Steve Neale (ed.) _Genre and Contemporary Hollywood._ London: British Film Institute, 201\u201317.\n\nSchatz, Thomas (1997) _Boom and Bust: American Cinema in the 1940s._ Berkeley: University of California Press.\n\nSchrader, Paul (1996) [1972] 'Notes on Film Noir', in R. Barton Palmer (ed.) _Perspectives on Film Noir._ New York: G. K. Hall, 99\u2013109.\n\nShearer, Lloyd (1999) [1945] 'Crime Certainly Pays on the Screen', in Alain\n\nSilver and James Ursini (eds) _Film Noir Reader 2._ New York: Limelight, 9\u201313.\n\nSheppard, Richard (1976) 'German Expressionism', in Malcolm Bradbury and James McFarlane (eds) _Modernism, 1890\u20131930._ London: Penguin, 274\u201391.\n\nSilver, Alain and James Ursini (eds) (1996) _Film Noir Reader._ New York: Limelight.\n\n\u2014\u2014\u2014(eds) (1999) _Film Noir Reader 2._ New York: Limelight.\n\nSilver, Alain and Elizabeth Ward (eds) (1992) _Film Noir: An Encyclopedic Reference to the American Style_ (3rd edition). New York: Overlook Press.\n\nSkal, David J. (1993) _The Monster Show: A Cultural History of Horror._ London: Plexus.\n\nSloterdijk, Peter (1988) _Critique of Cynical Reason._ Translated by Michael Eldred. London: Verso.\n\nSpicer, Andrew (2002) _Film Noir._ Harlow: Pearson.\n\nStewart, Garrett (1999) _Between Film and Screen: Modernism's Photo Synthesis._ Chicago: University of Chicago Press.\n\nStrebel, Elizabeth Grottle (1980) _French Social Cinema of the Nineteen-Thirties: A Cinematic Expression of Popular Front Consciousness._ New York: Arno.\n\nSwift, Jonathan (1963) [1726] _Gulliver's Travels._ London: Oxford University Press.\n\nTaylor, John Russell (1983) _Strangers in Paradise: The Hollywood \u00c9migr\u00e9s, 1933\u20131950._ London: Faber and Faber.\n\nTelotte, J. P. (1989) _Voices in the Dark: The Narrative Patterns of Film Noir._ Urbana and Champaign: University of Illinois Press.\n\nTheweleit, Klaus (1987) _Male Fantasies, vol. I: Women Floods Bodies History._ Translated by Stephen Conway, Erica Carter and Chris Turner. Minneapolis: University of Minnesota Press.\n\n\u2014\u2014\u2014(1989) _Male Fantasies, vol. II: Psychoanalyzing the White Terror._ Translated by Chris Turner, Erica Carter and Stephen Conway. Minneapolis: University of Minnesota Press.\n\nThomas, Deborah (1992) [1988] 'How Hollywood Deals with the Deviant Male', in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 59\u201370.\n\nThompson, John B. (1990) _Ideology and Modern Culture: Critical Social Theory in the Era of Mass Communication._ Cambridge: Polity Press.\n\nThomson, David (1997) _The Big Sleep._ London: British Film Institute.\n\nVernet, Marc (1993) _'Film Noir_ on the Edge of Doom', in Joan Copjec (ed.) _Shades of Noir: A Reader._ London: Verso, 1\u201331.\n\nVincendeau, Ginette and Keith Reader (eds) (1986) _La Vie est \u00e0 nous: French Cinema of the Popular Front, 1935\u20131938._ London: British Film Institute.\n\nVincendeau, Ginette (1998) _P\u00e9p\u00e9 le Moko._ London: British Film Institute.\n\nWager, Jans B. (1999) _Dangerous Dames: Women and Representation in the Weimar Street Film and Film Noir._ Athens: Ohio University Press.\n\nWalker, Michael (1992a) 'Film Noir: Introduction', in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 8\u201338.\n\n\u2014\u2014\u2014(1992b) 'Robert Siodmak' in Ian Cameron (ed.) _The Movie Book of Film Noir._ London: Studio Vista, 110\u201351.\n\n\u2014\u2014\u2014(1992c) _'While the City Sleeps', cineACTION,_ 29, 56\u201369.\n\nWayne, Mike (2003) _Marxism and Media Studies: Key Concepts and Contemporary Trends._ London. Pluto.\n\nWilleford, Charles (2000) _The Shark-Infested Custard._ Edinburgh: Canongate.\n\nWilliams, Linda Ruth (2004) 'No Sex Please We're American', _Sight and Sound,_ 14, 1, 18\u201320.\n\nWright, Will (1977) _Sixguns and Society: A Structural Study of the Western._ Berkeley: University of California Press.\nINDEX OF NAMES\n\nAbel, Alfred\n\nAdorno, Theodor\n\nAkins, Claude\n\nAlba, Jessica\n\nAlthusser, Louis\n\nAltman, Rick\n\nAlton, John\n\nAndrew, Dudley\n\nAndrews, Dana\n\nAntonioni, Michelangelo\n\nArnold, Keri\n\nAym\u00e9, Marcel\n\nBacall, Lauren\n\nBair, David\n\nBaker, Stanley\n\nBallard, Lucien\n\nBalzac, Honore de\n\nBanderas, Antonio\n\nBarnett, Vince\n\nBarrymore, Jr., John\n\nBasehart, Richard\n\nBazin, Andr\u00e9\n\nBecker, Jacques\n\nBel Geddes, Barbara\n\nBennett, Joan\n\nBennet, Linda\n\nBerenger, Tom\n\nBernhardt, Curtis\n\nBerry, Jules\n\nBierce, Ambrose\n\nBlack Mask\n\nBlackmer, Sidney\n\nBledel, Alexis\n\nBogart, Humphrey\n\nBogdanovich, Peter\n\nBoileau, Pierre\n\nBolter, David Jay\n\nBonham-Carter, Helena\n\nBoothe, Powers\n\nBorde, Raymonde\n\nBordwell, David\n\nBould, Mark\n\nBrackett, Leigh\n\nBradbury, Ray\n\nBrady, Ruth\n\nBrando, Jocelyn\n\nBraque, Georges\n\nBredell, Elwood\n\nBresson, Robert\n\nBrian, David\n\nBrooks, Geraldine\n\nBrooks, Richard\n\nBrowning, Tod\n\nBurnett, W. R.\n\nBurr, Raymond\n\nBuss, Robin\n\nByrne, Gabriel\n\nCagney, James\n\nCain, James M.\n\nCameron, Ian\n\nCarey, Timothy\n\nChabrol, Claude\n\nChaney, Lon\n\nChartier, Jean-Pierre\n\nChandler, Raymond\n\nChase, James Hadley\n\nChaumeton, Etienne\n\nChekmayan, Ara\n\nCheyney, Peter\n\nClouzot, Henri-Georges\n\nCoen, Ethan\n\nCoen, Joel\n\nColgan, Michael\n\nConrad, William\n\nConte, Richard\n\nCook, Jr., Elisha\n\nCooper, Maxine\n\nCopjec, Joan\n\nCoppola, Francis Ford\n\nCorey, Jeff\n\nCotten, Joseph\n\nCoyote, Peter\n\nCraig, James\n\nCraig, Johnny\n\nCrawford, Joan\n\nCrisp, Colin\n\nCukor, George\n\nCurtiz, Michael\n\nDahl, John\n\nDaly, Carroll John\n\nDamico, James\n\nDarlan, Eva\n\nDarrin, Sonia\n\nDassin, Jules\n\nDavis, Bette\n\nDawson, Rosario\n\nDekker, Albert\n\nDe Palma, Brian\n\nde Rochemont, Louis\n\nDick, Philip K.\n\nDickinson, Angie\n\nDiessl, Gustav\n\nDieterle, William\n\nDietrich, Marlene\n\nDigges, Dudley\n\nDimendberg, Edward\n\nDmytryk, Edward\n\nDostoyevsky, Fyodor\n\nDouglas, Don\n\nDouglas, Michael\n\nDrake, Claudia\n\nDuff, Howard\n\nDufy, Raoul\n\nDuhamel, Marcel\n\nDunaway, Faye\n\nDuncan, Paul\n\nDupont, E. A.\n\nDurgnat, Raymond\n\nDuryea, Dan\n\nDuvivier, Julien\n\nEbouaney, Eriq\n\nEisner, Lotte\n\nEisner, Will\n\nFairbanks, Jr., Douglas\n\nFarell, Glenda\n\nFaulkner, William\n\nFell, Norman\n\nFerrara, Abel\n\nFincher, David\n\nFinney, Albert\n\nFiorentino, Linda\n\nFleming, Rhonda\n\nFonda, Henry\n\nFontaine, Joan\n\nFord, Glenn\n\nForrest, Sally\n\nFrank, Nino\n\nFreund, Karl\n\nFrin, Jean-Marie\n\nFr\u00f6lich, Gustav\n\nFurthman, Jules\n\nGabin, Jean\n\nGardner, Ava\n\nGargan, William\n\nGarmes, Lee\n\nGarner, James\n\nGarnett, Tay\n\nGaudio, Tony\n\nGledhill, Christine\n\nGlenaan, Kenny\n\nGodard, Jean-Luc\n\nGoodis, David\n\nGoya, Francisco\n\nGrahame, Gloria\n\nGramsci, Antonio\n\nGranval, Charles\n\nGray, Coleen\n\nGreenberg, Joel\n\nGreene, Graham\n\nGridoux, Lucas\n\nGrusin, Richard\n\nGulager, Clu\n\nHaller, Ernest\n\nHalliday, Brett\n\nHalton, Charles\n\nHammett, Dashiell\n\nHarden, Marcia Gay\n\nHarolde, Ralf\n\nHartnett, Josh\n\nHathaway, Henry\n\nHauer, Rutger\n\nHawks, Howard\n\nHayden, Sterling\n\nHayward, Susan\n\nHayworth, Rita\n\nHeld, David\n\nHelm, Brigitte\n\nHelmore, Tom\n\nHemingway, Ernest\n\nHerv\u00e9, Julien Auguste\n\nHeydt, Louis Jean\n\nHigham, Charles\n\nHimes, Chester\n\nHirsch, Foster\n\nHitchcock, Alfred\n\nHollander, Frederick\n\nHopkins, Miriam\n\nHopper, Dennis\n\nHorner, Harry\n\nHowe, James Wong\n\nHunt, Jimmy\n\nIampolski, Mikhail\n\nIngoglia, Salvatorre\n\nIreland, John\n\nJaffe, Sam\n\nJames, C. L. R.\n\nJameson, Fredric\n\nJannings, Emil\n\nJenny, Laurent\n\nJeter, K. W.\n\nKaplan, E. Ann\n\nKawin, Bruce\n\nKazan, Elia\n\nKelly, Paul\n\nKettelhut, Erich\n\nKing, Jaime\n\nKoike, Kazuo\n\nKojima, Goseki\n\nKorngold, Erich\n\nKracauer, Siegfried\n\nKruger, Otto\n\nKrutnik, Frank\n\nLadd, Alan\n\nLake, Veronica\n\nLambert, Jack\n\nLancaster, Burt\n\nLang, Fritz\n\nLaShelle, Joseph\n\nLaurent, Jacqueline\n\nLeachman, Cloris\n\nLedoux, Fernand\n\nLeigh, Janet\n\nLevene, Sam\n\nLorre, Peter\n\nLosey, Joseph\n\nLupino, Ida\n\nLynch, David\n\nMacDonald, Edmund\n\nMacready, George\n\nMacLachlan, Kyle\n\nMacMurray, Fred\n\nMalle, Louis\n\nMalone, Dorothy\n\nMander, Miles\n\nMann, Anthony\n\nMason, James\n\nMarling, William\n\nMartin, Richard\n\nMarvin, Lee\n\nMat\u00e9, Rudolph\n\nMayer, Carl\n\nMazurki, Mike\n\nMcCoy, Horace\n\nMcGivern, William P.\n\nMcGraw, Charles\n\nMcGuire, John\n\nMcLane, Barton\n\nMeeker, Ralph\n\nMelville, Jean-Pierre\n\nMillais, John Everett\n\nMilland, Ray\n\nMiller, Frank\n\nMinnelli, Vincente\n\nMitchell, Thomas\n\nMitchum, Robert\n\nMitry, Jean\n\nMohr, Gerald\n\nMontgomery, Robert\n\nMontoute, Edouard\n\nMulvey. Laura\n\nMunby, Jonathan\n\nMuni, Paul\n\nMurnau, F. W.\n\nMusser, Charles\n\nMusuraca, Nicolas\n\nNarcejac, Thomas\n\nNaremore, James\n\nNeal, Tom\n\nNeale, Steve\n\nNebenzal, Seymour\n\nNewman, Kim\n\nNixon, Richard\n\nNolan, Jeanette\n\nNoro, Line\n\nNorton, Edward\n\nNovak, Kim\n\nOwen, Clive\n\nPaulhan, Jean\n\nPaxton, John\n\nPhillips, Paul\n\nPicasso, Pablo\n\nPitt, Brad\n\nPolito, Jon\n\nPolito, Sol\n\nPorter, Edwin S.\n\nPotamkin, Harry\n\nPowell, Dick\n\nPowers, Tom\n\nPrawer, S. S.\n\nPreminger, Otto\n\nPrice, Vincent\n\nPye, Douglas\n\nRabinowitz, Paula\n\nRasmussen, Rie\n\nRay, Nicholas\n\nReed, Carol\n\nReinhardt, Max\n\nRenoir, Jean\n\nRey, Henri-Fran\u00e7ois\n\nRichter, Kurt\n\nRoberts, Roy\n\nRobinson, Edward G.\n\nRomijn-Stamos, Rebecca\n\nRossellini, Isabella\n\nR\u00f3zsa, Mikl\u00f3s\n\nRyan, Robert\n\nSalter, Hans J.\n\nSanders, George\n\nSartre, Jean-Paul\n\nSavage, Ann\n\nSchrader, Paul\n\nSch\u00fcfftan, Eugen\n\nScorsese, Martin\n\nScott, Adrian\n\nScott, Lizabeth\n\nScourby, Alexander\n\nSekely, Steve\n\nShearer, Lloyd\n\nShirley, Anne\n\nSidney, Sylvia\n\nSimmons, Richard\n\nSimon, Simone\n\nSiodmak, Robert\n\nSirk, Douglas\n\nSkal, David J.\n\nSmith, Queenie\n\nSparkuhl, Theodore\n\nSpicer, Andrew\n\nStahl, Nick\n\nStanwyck, Barbara\n\nSteiner, Max\n\nStevenson, Robert Louis\n\nStewart, James\n\nStone, Sharon\n\nStrudwick, Shepperd\n\nSue, Eugene\n\nSwanson, Gloria\n\nTallichet, Margaret\n\nTelotte, J. P.\n\nTenn, William\n\nTheweleit, Klaus\n\nThomas, Deborah\n\nThompson, Jim\n\nThomson, David\n\nTotter, Audrey\n\nTourneur, Jacques\n\nTracy, Spencer\n\nTrail, Armitage\n\nTrevor, Claire\n\nTrigo, Benigno\n\nTruffaut, Fran\u00e7ois\n\nTsukamoto, Shinya\n\nTurner, Kathleen\n\nUlmer, Edgar G.\n\nVega, Makenzie\n\nVelJohnson, Reginald\n\nVernet, Marc\n\nVinson, Helen\n\nWalker, Michael\n\nWalsh, Raoul\n\nWalton, Douglas\n\nWarm, Hermann\n\nWarwick, Robert\n\nWatson, Minor\n\nWaxman, Franz\n\nWidmark, Richard\n\nWilder, Billy\n\nWillis, Bruce\n\nWilliams, Charles\n\nWood, Elijah\n\nWood, Wallace\n\nWoolrich, Cornell\n\nWright, Will\n\nWright, Frank Loyd\n\nWyatt, Jane\n\nWycherly, Margaret\n\nYoung, Robert\n\nZola, Emile\n\nZinnemann, Fred\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nAcknowledgements\n\nIntroduction\n\nCHAPTER ONE - Cow Pussy, Yes, Cow Pussy\n\nCHAPTER TWO - The Chinese Art of Everyday Abuse\n\nCHAPTER THREE - Swearing and Profanity\n\nCHAPTER FOUR - Men and Women: Flirting, Dating, Love, and Marriage\n\nCHAPTER FIVE - Sex and Body Parts\n\nCHAPTER SIX - Gay Slang\n\nCHAPTER SEVEN - Behaving Badly\n\nCHAPTER EIGHT - Internet Slang\n\nThe Top Twenty-five Terms You Need to Know\n_**NIUBI!**_\n\n**The Real Chinese You Were**   \n**Never Taught in School**   \n\u725b **ni\u00fa** ( _nyoo_ ) awesome   \n\u9177 **k\u00f9** ( _coo_ ) cool   \n\u4ed6\u5988\u7684 **t\u0101m\u0101de** ( _tah mah duh_ ) damn   \n\u6211\u64cd **w\u014f c\u00e0o** ( _wuh tsow_ ) fuck!   \n\u5c44 **b\u012b** ( _bee_ ) pussy\n\nEVELINE CHAO is a freelance writer and editor based in Beijing. She is extremely fortunate to have foul-mouthed friends willing to teach her words that most Chinese would be too embarrassed to reveal to a foreigner.\n\nPLUME   \nPublished by Penguin Group   \nPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. \u2022   \nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario,   \nCanada M4P 2Y3 (a division of Pearson Penguin Canada Inc.) \u2022 Penguin Books Ltd., 80   \nStrand, London WC2R 0RL, England \u2022 Penguin Ireland, 25 St. Stephen's Green, Dublin 2,   \nIreland (a division of Penguin Books Ltd.) \u2022 Penguin Group (Australia), 250 Camberwell   \nRoad, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty.   \nLtd.) \u2022 Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New   \nDelhi - 110 017, India \u2022 Penguin Books (NZ), 67 Apollo Drive, Rosedale, North Shore   \n0632, New Zealand (a division of Pearson New Zealand Ltd.) \u2022 Penguin Books (South   \nAfrica) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England   \nFirst published by Plume, a member of Penguin Group (USA) Inc.   \nFirst Printing, December 2009\n\nCopyright \u00a9 Eveline Chao, 2009\n\nIllustrations by Chris Murphy   \nAll rights reserved\n\nREGISTERED TRADEMARK\u2014MARCA REGISTRADA REGISTERED TRADEMARK\u2014MARCA REGISTRADA LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA   \nChao, Eveline.   \nNiubi! : the real Chinese you were never taught in a school \/ Eveline Chao;   \nIllustrations by Chris Murphy.   \np. cm.\n\neISBN : 978-1-101-14930-0\n\n1. Chinese language\u2014Conversation and phrase books\u2014English. 2. Chinese   \nlanguage\u2014Spoken Chinese\u2014Dictionaries\u2014English. 3. Language and culture\u2014  \nChina. 4. China\u2014Languages. I. Title.   \nPL1125.E6C383 2009   \n495.1'83421\u2014dc22   \n009028618\n\nPUBLISHER'S NOTE   \nThe scanning, uploading, and distribution of this book via the Internet or via any other means   \nwithout the permission of the publisher is illegal and punishable by law. Please purchase only   \nauthorized electronic editions, and do not participate in or encourage electronic piracy of   \ncopyrighted materials. Your support of the author's rights is appreciated.   \nBOOKS ARE AVAILABLE AT QUANTITY DISCOUNTS WHEN USED TO PROMOTE PRODUCTS   \nOR SERVICES. FOR INFORMATION PLEASE WRITE TO PREMIUM MARKETING DIVISION,   \nPENGUIN GROUP (USA) INC., 375 HUDSON STREET, NEW YORK, NEW YORK 10014.\n\n<http:\/\/us.penguingroup.com>\nAcknowledgments\n\nEndless thanks first and foremost to Maya Rock for making it all happen. I am also grateful to my agent Al Zuckerman at Writers House and my editor Nadia Kashper at Plume. Chris Murphy deserves special mention for his wonderful illustrations. And finally, thanks to Lu Bin, Amani Zhang, Wang Bin, Rachel Zhu, Emilio Liu, Lisa Hsia, Sam Zhao of _les+_ magazine, and Cecilia Hu, Gissing Liu, and Vivid Zhu for their advice and help on this project.\nIntroduction\n\n**O** n my first day in Beijing, my roommate and old college friend Ann sent me off to IKEA with three of her best Chinese friends. They picked me up in a red Volkswagen Santana and passed around a joint, blasting the Cure and Sonic Youth the whole ride there. In the crowded cafeteria at IKEA, we ate Swedish meatballs, french fries, and kung pao chicken, and then skated around the store with our shopping carts, stepping over the snoring husbands, asleep on the display couches, and smiling at the peasant families taking family photos in the living room sets. I bought bedding and some things for the kitchen, Da Li got a couple of plants, and Wang Xin bought a lamp. Traffic was bad on the ride home; we were navigating through a snarl at an intersection when yet another car cut us off. Lu Bin stuck his head out the window and bellowed \"\u50bb\u5c44!\" \" **Sh\u01ceb\u012b**!\" ( _shah bee_ ), or \"fucking cunt,\" at the other driver, then placidly turned down the music and, looking back, asked if I was a fan of Nabokov\u2014he'd read _Lolita_ in the Chinese translation and it was his favorite book.\n\nFor the next few months, I was too terrified to leave the apartment by myself and go make other friends, having not yet fully absorbed the fact that I'd left behind four years of life and a career in New York City and suddenly moved to this new and crazy place. So with a few exceptions, those three boys and my roommate were the only people I hung out with. Da Li owned an Italian restaurant, of all things, and we'd often meet there late in the evening and eat cr\u00e8me br\u00fbl\u00e9e or drink red wine or consume whatever else we could beg off of him for free, and then pile into his and Lu Bin's cars and head out on whatever adventure they had in mind. One night some big DJ from London was in town, spinning at a multilevel megaclub filled with nouveau riche Chinese. I bounced around on the metal trampoline dance floor and learned that the big club drink in China is whiskey with sweet green tea. Another night we headed to a smoke-filled dive to see a jazz band. The keyboardist had gone to high school with Lu Bin in Beijing, studied jazz in New York, and now sometimes performed with Cui Jian, a rock performer whose music, now banned from state radio, served as an unofficial anthem for the democracy movement during the late 1980s. My friends had a party promoter friend\u2014a tiny, innocuous-seeming girl\u2014who somehow got us into everything for free and would always turn and grin after rocking out to a set by a death-metal band from Finland, or a local hip-hop crew, and shout out, \"\u592a\u725b\u5c44!\" \" **T\u00e0i ni\u00fab\u012b**!\" ( _tie nyoo bee_ ), or \"That was fucking awesome!\" Other nights the boys would want to drive all the way to the Korean part of town, just to try out some Korean BBQ joint they'd heard about. And some nights we'd just drive around aimlessly and \u5c94 **ch\u0103** ( _chah_ ), Beijing slang for \"shoot the shit,\" about music or art. Then we'd go back to Lu Bin's to drink beer and watch DVDs (pirated, of course).\n\nMost nights ended with deciding to get food at four in the morning and driving to Ghost Street, an all-night strip of restaurants lit up with red lanterns. There was one hot pot restaurant in particular that they liked, where, I remember, one night a screaming match broke out between two drunk girls at a table near ours. It concluded with one girl jumping up and shouting, \"\u64cd\u4f60\u5988!\" \" **C\u00e0o n\u01d0 m\u0101**!\" ( _tsow nee ma_ ), or \"Fuck you!\" before storming out the door. The bleary-looking man left behind tried to console the other bawling girl, assuring her, \"\u6ca1\u4e8b,\u5979\u559d\u9189\u4e86\" \" **M\u00e9ish\u00ec, t\u0101 h\u0113zu\u00ec le** \" ( _may shih, tah huh dzway luh_ ): \"Don't worry about it\u2014she was totally wasted.\"\n\nIntermittently, some new girl, whom one of the boys had recently decided was the love of his life, would appear in the group. There was a comic period when Da Li, who couldn't speak any English, was \u6536 **sh\u014du** ( _show_ ), or screwing, a tall blonde who couldn't speak any Chinese. Whenever they came out, one of us would inevitably get roped into playing translator in the long lead-up to the moment when they would finally leave us to go back to his place. You'd always get stuck repeating, over and over, some trivial thing that one had said to the other, and which the other was fixating on, thinking something important had been said. \"What'd she say again?\" Da Li would shout over the noise of the bar. \"\u9177\" \" **K\u00f9** \" ( _coo_ ), I'd yell back: \"cool\" in Chinese.\n\nAfter a couple of years here, I've started taking Beijing for granted, and it's harder for me to conjure up the same sense of magic and wonderment I felt at every little detail during those first few months. But then I'll go back home, for a visit, to the United States and be reminded by the questions I'm asked of what a dark and mysterious abstraction China remains for most of the world. \"Does everyone ride bicycles?\" \"Are there drugs in China?\" \"What are Chinese curse words like?\" \"Is there a hook-up scene? What's dating like in China?\" \"What's it like to be gay in China? Is it awful?\" \"How do you type Chinese on a computer? Are the keyboards different?\" \"How do Chinese URLs work?\" One guy even asked, in gape-jawed amazement, if outsiders were allowed into the country.\n\nEvery time I hear these questions, I think back to those three boys who so strongly shaped my first impressions of China and wish that everyone could share the experiences I had\u2014experiences that were neither \"Western,\" as half the people I talk to seem to expect, nor \"Chinese,\" as the other half expects, but rather their own unique thing. And then I remember the way those boys and their friends spoke\u2014the casual banter, the familiar tone, the many allusions to both Western pop culture and ancient Chinese history; the mockery, the cursing, the lazy stoner talk, the dirty jokes, the arguments, the cynicism, the gossip and conjectures and sex talk about who was banging whom, but most of the time just the utterly banal chatter of everyday life\u2014and I realize that one of the best ways to understand the true realities of a culture, in all its ordinariness and remarkableness, is to know the slang and new expressions and everyday speech being said on the street.\n\nHopefully, then, with the words in this book, some of those questions will be answered. Are there drugs in China? There are indeed drugs and stoners and cokeheads and all the rest in China, and, contrary to popular belief, lighting up a joint doesn't instantly result in some sort of trapdoor opening in the sky and an iron-fisted authoritarian force descending from above to execute you on the spot. There's even a massive heroin problem in the country, discussed in chapter 7. What about the gay scene? There is one and it's surprisingly open, at least in the biggest cities. And I hope that after reading through the sex terms in chapter 5, the prostitution terminology in chapter 7, and the abundance of terms relating to extramarital affairs in chapter 4, we can put an end to those \"expos\u00e9s\" about sex in China that are always appearing in the Western media, predicated on an outdated assumption that Chinese people somehow don't have sex.\n\nThe Internet, in particular, is worth a special mention for its role in spreading slang and other new words. It was the sudden appearance of Internet caf\u00e9s in the 1990s, for example, that first helped popularize the concept of coolness in China. The word \u9177 **k\u00f9** ( _coo_ ), a transliteration of the English word \"cool,\" first appeared in Hong Kong and Taiwan; young people in mainland China learned it over the Internet from their friends there and spread the term at home. By the late 1990s, **k\u00f9** was known on most college campuses across China.\n\nHere's the thing: you can live in China forever\u2014you can even live in China forever and speak great Chinese\u2014and fail to notice even the merest hint of the subcultures represented by the slang in this book. For many people, the Chinese are the shy and almost absurdly innocent students in their classes, who look embarrassed at the merest mention of dating or sex; the white-collar staff who never speak up at meetings and leave their Western bosses convinced they're incapable of expressing an opinion or thinking up an original idea; the prim, strict tutors who conjure up an image of China as a land of studying machines; and the dolled-up, gold-digging girls who hang on the arms of rich men in shady bars late at night.\n\nThese impressions of China are not inaccurate\u2014they just aren't everything. Pay just a little more attention and you'll notice a fuller array of people and have a more nuanced portrait of the life humming below the surface. You may notice that on Thursday nights this one Italian sandwich shop fills up with gay men grabbing dinner before the weekly gay night at the upscale bar around the corner, or that the old guy fixing bikes down the street is an ex-con who spent a couple of decades in prison, or that the middle-aged couple snuggling in the booth near you at that Hong Kong-style restaurant are clearly two married people having an affair, or that all the women's bathrooms are locked in this one public building nearby because there's a flasher who lurks in the neighborhood.\n\nI have a running gag with an American friend of mine who, despite two years of living in Beijing, insists she has _never_ heard a single Chinese curse word. I bombard her with text messages and e-mails every single day, itemizing every swear I hear on the street: 10:00 a.m.\u2014middle-aged woman on bus yelling \"Fuck!\" into cell phone; 3:30 p.m.\u2014two college-age guys walking behind me while standing at ATM saying, \"That fucking shit was fucking ri-fucking-diculous\"; 11:30 p.m.\u2014two teenage girls in McDonald's bitching about some woman they keep referring to as \"that old cunt.\" And every time I see her, my friend says again that she never hears anything, not a single \"fuck\" or \"shit\" or \"damn.\" And every time, I keep insisting: \"You just need to know what to hear.\"\n\n#  **How to read this book**\n\nSo, how _do_ Chinese keyboards work? The answer is \u62fc\u97f3 **p\u012bny\u012bn** ( _peen yeen_ ), literally \"spell sound.\" A system for the romanization of Chinese words using the Latin alphabet, it was adopted in 1979 by the Chinese government. Students of Chinese as a second language start out by learning pinyin and pinyin pronunciation, as do Chinese schoolchildren. And road signs in China often depict pinyin beneath the Chinese characters.\n\nUsing pinyin, the word for \"me,\" \u6211, can also be written **w\u014f** (pronounced _wuh_ ). That symbol over the _o_ is a tone mark; there are four different marks each representing one of the four different tones\u2014first tone, second tone, third tone, and fourth tone\u2014that may be used to pronounce each Chinese syllable. (Because it is so cumbersome to type pinyin with the tone marks in place, people often leave them out or stick the tone number behind the syllable, as in \"wo3.\")\n\nTyping in Chinese is done using pinyin. It's a cumbersome process because Chinese has a huge number of homonyms. Thus, the way most character input systems work, to type \u6211 you type in **wo** , and then a window pops up showing the huge range of characters that are all pronounced _wuh_. You scroll through, and when you get to the right one, you hit enter and the character is typed on the screen. It's a slow process, and should you ever find yourself working an office job in China, your Chinese coworkers will be mightily impressed by how quickly you're able to type in English.\n\nThere was once a time when pinyin was a contender to replace the character-based Chinese writing system altogether, but that never really panned out, and the government settled for simplifying many notoriously hard-to-write traditional characters into what is known as simplified Chinese, the writing system used in mainland China, as opposed to traditional Chinese, which is still used in Taiwan, Macao, and Hong Kong).\n\nThe words in this book are all presented in three different ways. First I give the simplified Chinese characters for the term. Then appears the pinyin, in bold, with tone marks. Then, for those who are new to Chinese and have not yet learned pinyin, I have written out, in italics and parenthesized, the word's phonetic pronunciation (although pinyin uses the Latin alphabet, the letters do not correspond to English pronunciation, so you won't be able to pronounce pinyin without having studied it first).\n\nAs mentioned, the words in this book are all given in simplified Chinese. There are, however, a very few instances when I list a slang term that is only used in Taiwan, in which case I also give the traditional characters, since Taiwan still uses the old character system.\n\nYou'll notice that most of the terms in this book can be used throughout Mandarin-speaking China, but because I live in Beijing, words specific to Beijing and northern China in general are a bit more well-represented than southern and Taiwanese terms. However, as the capital of China, Beijing is used as the national standard and has an inordinate amount of influence; thus a great deal of Beijing slang winds up spreading throughout the country. In any case, rest assured that you won't find yourself using southern terms with an uncomprehending northerner, or vice versa, as I have taken care to indicate whenever a term is native to just one part of China.\n\nI have also been careful to note how strong or vulgar the insults and swear words are, and to situate the words within the appropriate context. After all, we don't want to unleash, onto the unsuspecting Chinese populace, readers armed with utterly inappropriate words for inappropriate situations. With this book you won't unwittingly yell, \"You poopie head!\" at the son of a bitch who grabs your ass while walking down the street or shout, \"Motherfucking cunt!\" when you stub your toe in front of a sweet old grandmother.\n\nYou should also be aware that many of the terms in this book are almost exclusively spoken, and never written, and thus may not have a set way of being expressed in characters\u2014especially if the word originated out of a non-Mandarin dialect. Fortunately, the Internet has given people a reason to agree on ways to write various colloquial expressions, and so I have managed to give the most commonly used characters for every term in this book. But, especially with a few of the extremely localized words, you may find that not everyone will agree with the written form given or even know of a way to write the word.\n\nAnd finally, it's worth keeping in mind that alternative subcultures haven't permeated Chinese society as thoroughly as they have in the West, where everyone knows about once-underground ideas like hip-hop and gay culture and surfers and stoners\u2014the margins of society from which much slang is born. For this reason, entire sections of this book are filled with terminology that your average, mainstream Chinese will have never heard. At the least, you will in most cases need to be talking to someone from a certain subculture for them to know the words associated with that scene.\n\nAnd now, as Chinese spectators at sports games or encouraging parents might yell, \u52a0\u6cb9! **ji\u0101y\u00f3u!** ( _jah yo_ ). Literally \"refuel\" or \"add gasoline,\" it also means \"let's go!\"\n\nCHAPTER ONE\n\n**Cow Pussy, Yes, Cow Pussy**\n\n**L** et's begin with . . . cow pussy. Or rather, \u725b\u5c44 **ni\u00fab\u012b** ( _nyoo bee_ ), which literally translates to \"cow pussy\" but means \"fuckin' awesome\" or \"badass\" or \"really fuckin' cool.\" Sometimes it means something more like \"big\" and \"powerful,\" and sometimes it can have the slightly more negative meaning of \"bragging\" or \"braggart\" or \"being audacious,\" but most of the time it means \"fuckin' awesome.\"\n\nThe etymology of **ni\u00fab\u012b** is unknown. Some say the idea is that a cow's pussy is really big, so things that are similarly impressive are called cow cunts. Others say that it stems from the expression \u5439\u725b\u76ae **chu\u012b ni\u00fap\u00ed** ( _chway nyoo pee_ ), which literally translates to \"blow up ox hide\" and also connotes bragging or a braggart (someone who can blow a lot of air). In fact, the word for bragging is the first part of that phrase, \u5439\u725b **chu\u012bni\u00fa** ( _chway nyoo_ ). Once upon a time (and you can still see this done today in countries like Pakistan), people made rafts out of animal hides that had to be blown up with air so they would float. Such an activity obviously required one mighty powerful set of lungs, and so it is thought that **ni\u00fab\u012b** derives from **chu\u012b ni\u00fap\u00ed** both because of the association with power and bigness and because the two expressions rhyme.\n\nSome people merely use the shortened \u725b **ni\u00fa** ( _nyoo_ )\u2014that is, the cow minus the cunt\u2014to mean \"awesome\" or \"great.\" Unlike **ni\u00fab\u012b** , saying **ni\u00fa** is not really vulgar, much like saying \"that sucks\" instead of \"that fuckin' sucks dick.\"\n\nDespite its generally positive meaning, **ni\u00fab\u012b** is a dirty, dirty word\u2014dirty enough that the character for \"pussy\" or \"cunt,\" \u5c44 **b\u012b** ( _bee_ ), was removed from the Chinese character set years ago and cannot be typed on most computers. Your average Chinese doesn't even know how to write it; others do but choose not to write the real character because it is so dirty. When people use the word **ni\u00fab\u012b** online, they often write \u725bB or NB because _N_ and _B_ are the first letters of the pinyin syllables **ni\u00fa** and **b\u012b**. Roman letters are frequently used in this way, as informal abbreviations of Chinese words. For example, Beijing is often abbreviated BJ, and Shanghai SH, as it is easier than typing out the Chinese characters, which can be a somewhat arduous process.\n\nYou'll also often see **ni\u00fab\u012b** written \u725b\u6bd4 or \u725b\u903c instead of \u725b\u5c44. The characters \u6bd4 and \u903c are homonyms of \u5c44; they have completely different meanings but are also pronounced _bee_ , and so they are used as stand-ins. Chinese has a huge number of homonyms\u2014syllables that sound the same but have different meanings\u2014and as you'll see with many of the terms throughout this book, this makes for a lot of wordplay and puns.\n\n**Ni\u00fab\u012b** started out as Beijing slang but has spread enough that it is fairly ubiquitous throughout the country, in particular at any event involving a large population of punk rockers, hip young Chinese, or your average, beer-drinking man. Rock shows and soccer matches are especially prime hot spots. A really hot band or a particularly impressive sports move is \u592a\u725b\u5c44 **t\u00e0i ni\u00fab\u012b** ( _tie nyoo bee_ ), \"too fuckin' awesome,\" or \u771f\u725b\u5c44 **zh\u0113n ni\u00fab\u012b** (d _zen nyoo bee_ ), \"really fuckin' awesome,\" or\u2014my own favorite construction\u2014 \u725b\u5c44\u6b7b\u4e86 **ni\u00fab\u012b s\u012d le** ( _nyoo bee sih luh_ ), which literally translates \"fuckin' awesome to the point of death.\"\n\nThose last few phrases point to one of the most satisfying things about the Chinese language: the modular way that everything\u2014characters, words, phrases, sentences\u2014is constructed. In that last phrase, **ni\u00fab\u012b s\u012d le** , the individual component \u6b7b **s\u012d** ( _sih_ ) is itself a word meaning \"die\" or \"death.\" Adding the larger component \u6b7b\u4e86 **s\u012d le** ( _sih luh_ ) after an adjective is a common way of amping up the meaning of the adjective. So we can swap out **ni\u00fab\u012b** and plug other words into the phrase\u2014for example \u997f **\u00e8** ( _uh_ ), which means \"hungry.\" If you are \u997f\u6b7b\u4e86 **\u00e8 s\u012d le** ( _uh sih luh_ ), you are absolutely starving; that is, \"hungry to the point of death.\"\n\nAlmost every syllable in Chinese is itself a word, and larger words are constructed by simply linking these syllables together. The result is a remarkably logical language in which the components of a word often explain, very literally, the meaning of that word. Thus a telephone is \u7535\u8bdd **di\u00e0nhu\u00e0** ( _dyinn hwah_ ), literally \"electric speech,\" and a humidifier is \u52a0\u6e7f\u5668 **ji\u0101sh\u012bq\u00ec** ( _jah shih chee_ ), literally \"add wetness device.\" (That said, you shouldn't get too preoccupied with the literal meaning of every single word, as the components of a word may also be chosen for reasons unrelated to its meaning, such as pronunciation.)\n\nIndividual Chinese characters (that is, the symbols that make up Chinese writing) tend to be modular as well, composed of discrete components (or \"radicals\") that may carry their own meaning and that often help explain the overall meaning of the character. The character for \"pussy,\" \u5c44 **b\u012b** ( _bee_ ), for example, is constructed of the radical for \"body,\" \u5c38 **sh\u012b** ( _sheuh_ ), and \u7a74 **xu\u00e8** ( _shreh_ ), meaning \"hole\" (which is why so many people are uncomfortable writing the correct character for this word\u2014it just _looks_ incredibly dirty).\n\nThanks to the modularity of Chinese, a word like **ni\u00fab\u012b** can be thought of as being constructed of two building blocks (\"cow\" and \"pussy\") that can be taken apart and combined with other building blocks to make new (and often impressively logical) words. Thus on the \"cow\" side, we have words like:\n\nAnd on the \"pussy\" side? This is where things get fun. For your convenience, below is a handy table\u2014a cunt chart, if you will\u2014of some of the many dirty words that use **b\u012b** :\n\nCHAPTER TWO\n\n**The Chinese Art of Everyday Abuse**\n\n**O** ne of the first words you'll learn in Chinese class is \u4f60\u597d **n\u012dh\u01ceo** ( _nee how_ ), which means \"hello.\" However, the fact is that Chinese people don't actually say **n\u012dh\u01ceo** all that often. Instead, when you arrive for dinner, a party, or a meeting, they'll say, \"You've arrived,\" \u4f60\u6765\u4e86 **n\u01d0 l\u00e1i le** ( _nee lie luh_ ). When you depart, someone will say, \"You're going,\" \u4f60\u8d70\u5566 **n\u01d0 z\u01d2u la** ( _nee dzoe lah_ ).\n\nWhen I walk down the street on a windy day, it seems the conversation is the same for everyone I pass. The granny taking her granddaughter out for a stroll will exclaim, as she lifts the little girl into her stroller, \"It's windy!\" The two middle-aged men running into each other on the street will greet each other by saying, \"So windy today!\" When I get home, the trash collector sitting on my stoop will welcome me back by announcing, \"What a windy day!\"\n\nChinese people love to comment on the obvious, sometimes to the point of insensitivity or what we might even consider outright cruelty. Chinese sports commentators often say things like \"Wow, he's gained a lot of weight!\" about athletes on the field. I have a \"big-boned\" older cousin whom, for as far back as I can remember, we have always called \u80d6\u59d0\u59d0 **p\u00e0ng ji\u011bjie** ( _pahng jyih jyih_ ), which literally means \"fat sister.\" Westerners in China were once referred to as Big Nose. President Obama is often referred to as \u9ed1\u4eba **h\u0113ir\u00e9n** ( _hay ren_ ), or \"the black guy.\" My bearded friend Jason is referred to as Big Beard. My mother is called the Mandarin equivalent of American Auntie, her older sister is Eldest Aunt, and my father is Old Man. It's as if every Chinese person is somehow living in gangland Chicago or some imaginary criminal underworld in which everyone needs a self-descriptive nickname to make it easier for the FBI to identify them. Indeed, the most notorious gang boss in Chinese history was \"Big-Eared\" Du, and his mentor was \"Pockmarked\" Huang.\n\nAnd as if that wasn't bad enough, Chinese people, perhaps as a result of their collective thick skin, tend to demonstrate affection by being mean. Or rather, they speak frankly to each other in a way that, for them, indicates a level of familiarity that only a close relationship can have. But, to outside observers, it resembles, at best, a sort of constant, low-level stream of verbal abuse. For a young Chinese woman, there is no better way to express love for her boyfriend than by whacking him with her purse while telling him he's horrible. Groups of friends incessantly interrupt each other with cries of \"Nonsense!\" or \"Shut up!\" A good way to greet a pal is to give him a pained look and ask what the hell he did to his hair. I myself have had many an otherwise peaceful afternoon spent curled up on an armchair, happily reading a book, when I've been suddenly interrupted by a passing aunt or some other stray family member who snuck up behind me, smacked me across the back, and bellowed, \"\u54ce\u5440!, \u8822! \" \" **iy\u0101! Y\u00f2u f\u00e9i, y\u00f2u ch\u01d4n!** \" ( _aye yah! yo fay, yo chren!_ ): \"My God! So fat and lazy!\"\n\nThe Chinese word for \"scold\" or \"verbally abuse\" is \u9a82 **m\u00e0** ( _mah_ ). Note those two squares at the top of the character\u2014they represent two mouths, no doubt heaping abuse on the nearest person available. This chapter gathers words for the age-old art of \u9a82\u4eba **m\u00e0r\u00e9n** ( _mah ren_ ) or \"scolding people,\" including everyday exclamations of annoyance and frustration, teasing put-downs and dismissals, words for affectionate name-calling, everyday insults, and everything else you'll need to generally convey to the most important people in your life that their very existence on this earth is a constant and overwhelming burden.\n\nAnd finally, the Chinese may have a healthy sense of humor when it comes to the slings and arrows of everyday life, but they can also hold a grudge, and so at the end of the chapter you'll find words to fuel the fire when things cross the line into full-on feuding\u2014genuinely venomous insults with the power to end decades-long friendships, provoke fistfights, and possibly get you disowned.\n\n#  **Everyday exclamations**\n\n\u54ce\u5440 **\u00e0iya** ( _aye yah_ )\n\nA common interjection that can be used for a wide range of occasions: when you've forgotten something, when you're impatient, when you're bored, when you feel helpless, as a lead-in to scolding someone, etc. It isn't exactly a word\u2014more like a weighty sigh and roughly equivalent to \"Oh Lord!\" or \"My God!\"\n\n\u7cdf\u4e86 **z\u0101ole** ( _dzow luh_ \u2014the starting sound in _dzow_ is like a buzzing _bzz_ sound but with a _d_ instead of a _b_ , and the whole syllable should rhyme with \"cow\")\n\nA very common expression of dismay. Literally \"rotten\" or \"spoiled\" and something like saying, \"Oh shoot!\" \"Darn!\" or \"Crap!\" You can also say \u7cdf\u7cd5 **z\u0101og\u0101o** ( _dzow gaow_ \u2014both syllables rhyme with \"cow\"), which literally means \"rotten cakes,\" but it's less current.\n\n\u5b8c\u4e86 **w\u00e1nle** ( _wahn luh_ )\n\nSame meaning as **z\u0101ole** (above). It's pretty much like exclaiming \"Crap!\" to yourself. Literally, it means \"over.\"\n\n\u8001\u5929\u7237 **l\u01ceoti\u0101ny\u00e9** ( _laow tyinn yeh_ )\n\nLiterally \"my father God\" and sometimes \u6211\u7684\u5929 **w\u01d2deti\u0101n** ( _wuh duh tyinn_ ), literally \"my heavens.\" Equivalent to exclaiming \"My God!\" or \"Oh goodness!\" These phrasings are more common among older people; younger people usually shorten them to \u5929\u54ea **ti\u0101nn\u01ce** ( _tyinn nah_ ) or simply \u5929 **ti\u0101n** ( _tyinn_ ): \"Oh heavens!\" or \"Heavens!\"\n\n\u54c7\u585e **w\u0101 s\u00e0i** ( _wah sigh_ )\n\nShoot! Darn! Oh my God! Wow! Holy cow! An exclamation especially popular among girls. Comes from a Taiwanese curse that means \"Fuck your mother\" (but is a shortened and nonprofane version of it).\n\n\u8be5\u6b7b\u7684 **g\u0101is\u01d0de** ( _guy sih duh_ )\n\nMy God! Holy crap! Literally \"should die.\"\n\n\u6c14\u6b7b\u6211\u4e86 **q\u00ecs\u01d0w\u01d2le** ( _chee sih wuh luh_ )\n\nArgh! Damn it! Crap! Literally, \"I'm angry to the point of death.\"\n\n\u53ef\u6076 **k\u011bw\u00f9** ( _kuh woo_ )\n\nLiterally \"hateful\" and said alone means something like \"Darn!\"\n\n\u50bb\u773c **sh\u01cey\u01cen** ( _shah yen_ )\n\nOh no! Said in response to surprising, negative situations. For example, if you discover that your house has been broken into. Literally \"dumbfounded eye.\"\n\n\u6655 **y\u016bn** ( _een_ )\n\nMeans \"dizzy\" or \"faint\" and is often uttered to express surprise, shock, amusement, or even confusion or disgust; that is, emotions that might make you feel faint.\n\n\u5012\u9709 **d\u01ceom\u00e9i** ( _dow may_ )\n\nBad luck. You can say this when something unfortunate happens. This sentiment can be made slightly stronger by saying \u771f\u5012\u9709 **zh\u0113n d\u01ceom\u00e9i** ( _jen dow may_ ), which means \"really bad luck.\"\n\n\u70b9\u513f\u80cc **di\u01cenr b\u0113i** ( _dyerr bay_ )\n\nBeijing\/northern Chinese slang for **d\u01ceom\u00e9i** (above), used the same way. Literally \"fate turns its back on you.\" \u70b9\u513f **Di\u01cenr** is northern Chinese slang for \"luck\" or \"fate,\" and \u80cc **b\u0113i** means \"back.\"\n\n\u6b8b\u5ff5 **c\u00e1nni\u00e0n** ( _tsahn nyinn_ )\n\nBummer, too bad. Popular among young people to express disappointment. Derived from the Japanese phrase _zannen desu_ , which sounds similar and means something like \"what a shame\" or \"that's too bad.\"\n\n\u90c1\u95f7 **y\u00f9m\u0113n** ( _ee men_ )\n\nA popular term among young people, it means \"depressed\" but is used as an adjective for a much larger range of situations\u2014when they feel pissed off, upset, disappointed, or even just bored. Exclaimed alone, one would say, \"\u90c1\u95f7\u554a. . .\" \" **Y\u00f9m\u0113n \u0101** . . .\" ( _ee men ah_ ), meaning \"I'm depressed . . .\" or \"Sigh . . .\"\n\n#  **Dismissals and shutdowns**\n\n\u6ca1\u52b2 **m\u00e9ij\u00ecn** ( _may jeen_ )\n\nLiterally \"no strength.\" Said dismissively of things you find uninteresting or stupid, much like saying \"whatever.\" A stronger way to say this is \u771f\u6ca1\u52b2 **zh\u0113n m\u00e9ij\u00ecn** ( _jen may jeen_ ), literally \"really no strength.\"\n\n\u65e0\u804a **w\u00fali\u00e1o** ( _ooh lyow_ )\n\nNonsense, bored, boring. A common expression if you're bored is \u65e0\u804a\u6b7b\u4e86 **w\u00fali\u00e1o s\u01d0 le** ( _ooh lyow sih luh_ ), literally \"bored to death.\" You can also say **w\u00fali\u00e1o** in response to something you find stupid or uninteresting; for example, in response to an unfunny joke.\n\n\u670d\u4e86 **f\u00fa le** ( _foo luh_ ) or \u670d\u4e86\u4f60\u4e86 **f\u00fale n\u01d0 le** ( _foo luh nee luh_ )\n\nLiterally means \"admire you\" and sometimes said genuinely in response to something awe inspiring, but more usually said mockingly when someone says or does something silly or stupid. A more common form among younger people is \"I \u670d\u4e86 U!\" or \"I **f\u00fa le** you,\" literally \"I admire you,\" from a 1994 Stephen Chow movie.\n\n\u4e0d\u60f3\u8033\u98df\u4f60 **b\u00f9xi\u01ceng\u011brsh\u00edn\u01d0** ( _boo shahng er shih nee_ )\n\nI don't even want to talk to you; I'm ignoring you. Literally, \"I don't want to ear eat you.\" Originally Sichuan slang.\n\n\u5e2e\u5e2e\u5fd9 **b\u0101ngb\u0101ngm\u00e1ng** ( _bahng bahng mahng_ )\n\nLiterally means \"help\" but used in Shanghai to admonish someone before rebutting something they've said. The equivalent of sarcastically saying \"come on\" or \"please\" or \"give me a break.\"\n\n\u5c0f\u6837\u4e86\u5427 **xi\u01ceo y\u00e0ng le ba** ( _shyaow yahng luh bah_ )\n\nSaid when laughing at or mocking someone else. Similar to \"ha-ha\" or \"suck that.\" Used in northeastern China. Literally something like \"(look at) that little face!\"\n\n\u54d1\u4e86\u554a? **y\u01ce le a?** ( _yah luh ah_ )\n\nLiterally, \"Are you mute?\" \u54d1 **Y\u01ce** means \"dumb\" or \"mute.\" You can ask this when you say something and don't get a response.\n\n\u6b47\u83dc **xi\u0113 c\u00e0i** ( _shih tsigh_ )\n\nKnock it off; quit it. Literally \"rest vegetable.\" A slangy but mild way to tell someone to stop doing something. Used in northern China only.\n\n\u4f60\u5403\u9519\u836f\u4e86\u5417? **N\u01d0 ch\u012b cu\u00f2 y\u00e0o le ma?** ( _nee chih tswuh yow luh ma_ )\n\nDid you take the wrong medicine? A mildly insulting way to imply that someone is acting rude or strange.\n\n\u53bb! **Q\u00f9!** ( _chee_ )\n\nShut up! Literally \"go.\" Usually said affectionately.\n\n\u53bb\u4f60\u7684! **Q\u00f9 n\u012d de!** ( _chee nee duh_ )\n\nGet lost! Stop it! Up yours! Literally \"go to yours.\"\n\n\u95ed\u5634! **B\u00ec zu\u01d0!** ( _bee dzway_ )\n\nShut up! Literally \"close mouth.\" A more emphatic option is \u4f60\u7ed9\u6211\u95ed\u5634! **N\u01d0 g\u011bi w\u01d2 b\u00eczu\u01d0**! ( _nee gay wuh bee dzway_ ), literally \"Shut your mouth for me!\"\n\n\u5207! **Qi\u00e8!** ( _chyih_ )\n\nA noise expressing disdain. Equivalent to saying \"Please!\" or \"Whatever.\"\n\n\u70e6 **f\u00e1n** ( _fahn_ )\n\nIrritating, annoying, troublesome. Common uses include \u4f60\u70e6\u4e0d\u70e6\u554a! **N\u01d0 f\u00e1n b\u00f9 f\u00e1n a!** ( _nee fahn boo fahn ah_ ), meaning \"You're really freaking annoying!\" (literally, \"Aren't you annoying!\"), and \u70e6\u6b7b\u4eba\u4e86\u4f60! **F\u00e1ns\u01d0 r\u00e9n le n\u01d0!** ( _fahn sih ren luh nee_ ): \"You're annoying me to death!\"\n\n\u4f60\u6068\u6a5f\u8eca **n\u01d0 h\u011bn j\u012bch\u0113** ( _nee hun gee chuh_ )\n\nYou're really annoying. Taiwan slang for someone who is bossy or picky or otherwise annoying. Literally, \"You are very motorcycle\" or \"You are very scooter.\" It's also common to just say \u4f60\u6068\u6a5f **n\u01d0 h\u011bn j\u012b** ( _nee hun gee_ ) for short. Supposedly, this expression originally came from \u96de\u6b6a **j\u012bw\u0101i** ( _gee why_ ), meaning one's dick is askew.\n\n\u4f60\u4e8c\u554a! **n\u01d0 \u00e8r a!** ( _nee er ah_ )\n\nYou're so stupid. Literally, \"You're [number] two.\" \u4e8c **Er** ( _er_ ) means \"two\" in Chinese, but in northeast China it can also be slang for \"stupid\" or \"silly,\" referring to \u4e8c\u767e\u4e94 **\u00e8rb\u01ceiw\u01d4** ( _er buy woo_ ) (see page 19).\n\n\u8111\u5b50\u574f\u4e86\u5427? **n\u01ceozi hu\u00e0i le ba?** ( _nee now dz hwie luh bah_ )\n\nIs your brain broken? Used exactly the way you'd use the English phrase.\n\n\u4f60\u778e\u5440? **n\u01d0 xi\u0101 ya?** ( _knee shah yah_ )\n\nAre you blind? Used, for example, when someone steps on your foot.\n\n\u592a\u8fc7\u5206\u4e86 **t\u00e0i gu\u00f2f\u00e8n le** ( _tie gwuh fen luh_ )\n\nThis is outrageous! This is going too far! Literally \"much too far.\"\n\n\u53d7\u4e0d\u4e86 **sh\u00f2ub\u00f9li\u01ceo** ( _show boo lyaow_ )\n\nLiterally \"unacceptable\" and can mean \"This is unacceptable\" or \"I can't take it.\" A stronger form is \u771f\u53d7\u4e0d\u4e86 **zh\u0113n sh\u00f2ub\u00f9li\u01ceo** ( _jen show boo lyaow_ ), literally \"really unacceptable.\"\n\n\u4f60\u6562? **n\u01d0 g\u01cen?** ( _nee gahn_ )\n\nLiterally, \"Do you dare?\" and used in a challenging way when arguing or playing around. It's like saying, \"Go ahead\u2014I dare you!\"\n\n\u8ba8\u538c **t\u01ceoy\u00e0n** ( _taow yen_ \u2014the first syllable rhymes with \"cow\")\n\nDisgusting, troublesome, nuisance, nasty. Can also be a verb that means \"to hate\" (doing something). However, it is also common for girls to say this word by itself to express petulance, frustration, or annoyance.\n\n\u4f60\u5f88\u574f! **n\u01d0 h\u011bn hu\u00e0i!** ( _nee hun hwigh_ )\n\nYou're so bad! Often used between friends in an unserious way or flirtingly between couples. However, like the rest of these expressions it can also be used in a genuinely angry way, perhaps by a mother toward a child. For example, \u4f60\u600e\u4e48\u90a3\u4e48\u574f **n\u01d0 z\u011bnme n\u00e0me hu\u00e0i** ( _nee dzuh muh nuh muh hwigh_ ) means literally \"How can you be so bad?\" and is like saying, \" _What_ is wrong with you?\"\n\n\u6076\u5fc3 **\u011bx\u012bn** ( _uhh sheen_ )\n\nNauseating, disgusting, gross. Alternately, \u771f\u6076 **zh\u0113n \u011b** ( _jen uhh_ ), \"very nauseating\" or \"so gross.\" **\u011ax\u012bn** can also be used as a verb to mean \"to embarrass someone\" or \"to make someone feel uncomfortable or awkward.\"\n\n\u6ca1\u95e8\u513f **m\u00e9i m\u00e9nr** ( _may murr_ )\n\nNo way! Fat chance! A rude, curt way to say no. Literally \"no door.\" Used in Beijing.\n\n\u5e9f\u8bdd **f\u00e8ihu\u00e0** ( _fay hwa_ )\n\nNonsense. Literally \"useless words.\" An extremely common expression. Northern Chinese sometimes instead say \u8d39! **f\u00e8i!** ( _fay_ ), literally \"wasteful,\" to mean \"Nonsense!\"\n\n\u778e\u8bf4 **xi\u0101shu\u014d** ( _shah shwuh_ )\n\nTo talk nonsense. Literally \"to speak blindly.\" Common usages include \u522b\u778e\u8bf4 **bi\u00e9 xi\u0101shu\u014d** ( _byih shah shwuh_ ), meaning \"don't be ridiculous\" or \"stop talking nonsense\", and \u4f60\u778e\u8bf4 **n\u01d0 xi\u0101shu\u014d** ( _nee shah shwuh_ ), \"you're talking nonsense\" or \"you're full of crap.\"\n\nAny of the following synonyms may be swapped for **xi\u0101shu\u014d** in the two samples given above:\n\n\u80e1\u626f **h\u00fach\u011b** ( _hoo chuh_ ), \"blab messily\"   \n\u80e1\u8bf4 **h\u00fashu\u014d** ( _who shwuh_ ), \"speak messily\"   \n\u4e71\u8bf4 **lu\u00e0nshu\u014d** ( _lwun shwuh_ ), \"speak chaotically\"   \n\u9b3c\u626f **gu\u01d0ch\u011b** ( _gway chuh_ ), \"ghost blab\"   \n\u8bf4\u767d\u8bdd **shu\u014d b\u00e1ihu\u00e0** ( _shwuh buy hwa_ ), \"speak white   \nwords\" (this one is seldom used among younger people   \nnow)\n\n\u626f\u6de1 **ch\u011bd\u00e0n** ( _chuh dahn_ )\n\nTo talk nonsense, to bullshit (but not as profane as \"bullshit\"). Used in northern China.\n\n\u653e\u5c41 **f\u00e0ngp\u00ec** ( _fahng pee_ )\n\nBullshit, nonsense, lies, whatever, shut up! Literally \"fart.\" Used as a mild expletive.\n\n\u72d7\u5c41 **g\u01d2up\u00ec** ( _go pee_ ) or \u653e\u72d7\u5c41 **f\u00e0ng g\u01d2up\u00ec** ( _fahng go pee_ )\n\nBullshit, nonsense. Literally \"dog fart\" and \"release a dog fart,\" respectively.\n\n\u6709\u5c41\u5feb\u653e **y\u01d2u p\u00ec kua\u00ec fang** ( _yo pee kwigh fahng_ \u2014 **kua\u00ec** rhymes with \"high\")\n\nA more vulgar way to say \"Spit it out!\" or \"If you have something to say, hurry up and say it.\" Literally means, \"If you need to fart, hurry up and let it out.\"\n\n\u5c41\u8bdd **p\u00echu\u00e0** ( _pee hwa_ )\n\nBull, nonsense. Literally \"fart talk.\" Can be exclaimed alone to mean \"Nonsense!\" or \"Yeah, right!\"\n\n\u72d7\u5c41\u4e0d\u901a **g\u01d2up\u00ec b\u00f9t\u014dng** ( _go pee boo tohng_ )\n\nIncoherent, nonsensical. Literally \"dog unable to fart.\" Exclaimed in response to something, it means roughly \"that makes no sense\" or \"that's total bull.\" Can also be used as an adjective to describe someone who doesn't know what they're talking about.\n\n#  **(Mostly affectionate) name-calling**\n\n\u4e66\u5446\u5b50 **sh\u016bd\u0101izi** ( _shoo die dz\u2014_ **zi** sounds like saying a very short _bzz_ , but with a _d_ sound instead of the _b_ )\n\nBookworm, nerd, lacking social skills. Literally \"book idiot.\" \u5446\u5b50 **D\u0101izi** means \"idiot\" or \"fool\" but is not often said alone.\n\n\u61d2\u866b **l\u01cench\u00f3ng** ( _lahn chong_ )\n\nLazy bones. Literally \"lazy bug.\" Said affectionately.\n\n\u5c0f\u5154\u5d3d\u5b50 **xi\u01ceot\u00f9 z\u01ceizi** ( _shaow too dzigh dz\u2014_ **z\u01cei** rhymes with \"high\")\n\nSon of a rabbit. A gentle, teasing insult common among older people and directed at younger people. Ironically, parents often use this term to tease their children.\n\n\u50bb\u5192 \/ \u50bb\u5e3d **sh\u01cem\u00e0o** ( _shah maow_ )\n\nA gentle, affectionate jest\u2014closer to something silly like \"stupidhead.\" Literally \"silly hat.\" \u50bb **Sh\u01ce** ( _shah_ ) means \"silly\" or \"dumb.\"\n\n\u50bb\u74dc **sh\u01cegu\u0101** ( _shah gwah_ )\n\nDummy, fool. Literally \"silly melon.\" An extremely common insult, mostly used affectionately, and in use as early as the Yuan dynasty (1279-1368).\n\n\u5446\u74dc **d\u00e0igu\u0101** ( _die gwah_ )\n\nDummy, fool. Literally \"silly melon.\"\n\n\u9762 **mi\u00e0n** ( _myinn_ )\n\nNorthern Chinese slang for \"timid\" or \"weak.\" Literally \"wheat flour,\" as in the ingredient for noodles and bread, suggesting that the person is soft and flimsy as those foods.\n\n\u9762\u74dc **mi\u00e0ngu\u0101** ( _myinn gwah_ )\n\nTimid, coward. Literally \"timid melon\" (or still more literally \"flour melon\"). Used only in northern China.\n\n\u767d\u75f4 **b\u00e1ich\u012b** ( _buy chih_ )\n\nPerhaps the most universal and commonly used term for \"idiot\" or \"moron.\"\n\n\u5341\u4e09\u70b9 **sh\u00eds\u0101n di\u01cen** ( _shh sahn dyinn_ )\n\nA mild, usually affectionate insult meaning \"weirdo\" or \"crazy.\" Literally \"thirteen o'clock.\" Originated in Shanghai and used a bit in other parts of southern China as well, though it is fast falling out of favor and is mainly used by older people now. The term refers to the **ch\u012b** in **b\u00e1ich\u012b** (above), as the character for **ch\u012b** , \u75f4, is written using thirteen strokes. Other theories maintain that it refers to an illegal move in a gambling game called pai gow, \u724c\u4e5d **p\u00e1iji\u01d4** ( _pie joe_ ) in Mandarin, or that it refers to an hour that clocks do not strike (though nowadays thirteen o' clock is possible in military time).\n\n\u534a\u5f14\u5b50 \/ \u534a\u540a\u5b50 **b\u00e0n di\u00e0ozi** ( _bahn dyow dz_ )\n\nSomeone deficient in skill or mental ability. In ancient China, copper coins had square holes in the center and were strung together on a string. One thousand coins strung together formed a **di\u00e0o**. Half of that (five hundred coins) was called \u534a\u5f14\u5b50 \/ \u534a\u540a\u5b50 **b\u00e0n di\u00e0ozi** ( _bahn dyow dz_ ). Northern Chinese only, and seldom used today, but necessary to understand the more commonly used insult below.\n\n\u4e8c\u767e\u4e94 **\u00e8rb\u01ceiw\u01d4** ( _er buy woo_ )\n\nDummy, idiot, moron. Literally \"two hundred fifty,\" referring to half a **b\u00e0n di\u00e0ozi** (see above). This is an extremely common insult; everyone knows it and probably grew up hearing it a lot, but like **sh\u00eds\u0101n di\u01cen** (above), it's considered a bit old-fashioned now.\n\nA number of (usually) affectionate Chinese insults involve eggs. They most likely come from the much stronger insult \u738b\u516b\u86cb **w\u00e1ngb\u0101d\u00e0n** ( _wahng bah dun_ ), literally \"son of a turtle\" or \"turtle's egg\" and equivalent to \"son of a bitch\" or \"bastard\" in English. (The possible origins of **w\u00e1ngb\u0101d\u00e0n** are explained in the next chapter.) The insults below are mild and have shed any profane associations, much in the way we English speakers have mostly forgotten that phrases like \"what a jerk,\" \"that bites,\" and \"sucker\" originally referred to sex acts.\n\n\u7b28\u86cb **b\u00e8nd\u00e0n** ( _ben dahn_ )\n\nDummy, fool. Literally \"stupid egg.\" \u7b28 **B\u00e8n** ( _ben_ ) alone can be used in many insults and means \"stupid.\"\n\n\u5012\u86cb \/ \u6363\u86cb **d\u01ceod\u00e0n** ( _daow dahn_ )\n\nTo cause trouble.\n\n\u6eda\u86cb **g\u01d4nd\u00e0n** ( _gwen dahn_ )\n\nGet lost! Literally \"roll away, egg\" or \"go away, egg.\"\n\n\u574f\u86cb **hu\u00e0id\u00e0n** ( _hwigh dahn_ )\n\nBad person. Literally \"bad egg.\"\n\n\u7cca\u6d82\u86cb **h\u00fat\u00fad\u00e0n** ( _who too dahn_ )\n\nConfused\/clueless person. Literally \"confused egg.\"\n\n\u7a77\u5149\u86cb **q\u00edonggu\u0101ngd\u00e0n** ( _chyohng gwahng dahn_ )\n\nAn insulting term for a person without money. Literally \"poor and have-nothing egg.\"\n\n\u6df7\u86cb **h\u00f9nd\u00e0n** ( _hwen dahn_ )\n\nBastard. Literally \"slacker egg.\" \u6df7 **H\u00f9n** ( _hwen_ ) means \"to loaf,\" \"to wander around all day doing nothing,\" or \"to be up to no good.\" Relatedly, \u6df7\u6df7 **h\u00f9nh\u00f9n** ( _hwen hwen_ ) or \u6df7\u5b50 **h\u00f9nz\u01d0** ( _hwen dz_ ) is used for a layabout, deadbeat, slacker, or any idle person up to no good.\n\n\u9f9f\u5b59\u5b50 **gu\u012b s\u016bnzi** ( _gway swen dz_ ) or \u9f9f\u513f\u5b50 **gu\u012b \u00e9rzi** ( _gway er dz_ )\n\nBastard. Literally \"turtle grandson.\" An insult that has lost, like \"egg\" insults, any obscene connotation.\n\n\u8822\u8d27 **ch\u01d4nhu\u00f2** ( _chwen hwuh_ )\n\nDummy, moron. Literally \"silly good.\"\n\n\u83dc **c\u00e0i** ( _tsigh_ )\n\nLiterally \"vegetable.\" Can be an insulting term meaning \"ugly\" and may also be less insultingly used to describe someone who is bad at doing something. For example, \u4f60\u7535\u8111\u771f\u83dc **n\u01d0 di\u00e0nn\u01ceo zh\u0113n c\u00e0i** ( _nee dyinn now jen tsigh_ ) means \"You suck at using the computer.\" Similarly, \u83dc\u4e86 **c\u00e0i le** ( _tsigh luh_ ) is \"to fail.\"\n\n\u6728 **m\u00f9** ( _moo_ )\n\nStupid, slow, insensitive. Literally \"wooden.\"\n\n\u8111\u5b50\u8fdb\u6c34 **n\u01ceozi j\u00ecn shu\u01d0** ( _now dz jean shway\u2014_ **zi** is like saying a very short _bzz_ , but with a _d_ sound instead of the _b_ )\n\nBlockhead, dummy. Literally means \"water in the brain.\"\n\n\u8111\u5b50\u517b\u9c7c **n\u01ceozi y\u01ceng y\u00fa** ( _now dz yahng yee_ )\n\nBlockhead, dummy. Literally \"fish feed in the brain\" or \"fish being raised in [one's] brain.\" A variant on \"water in the brain\" (above), more popular among younger people.\n\n\u5e9f\u4eba **f\u00e8ir\u00e9n** ( _fay ren_ )\n\nUseless person.\n\n\u7a9d\u56ca\u5e9f **w\u014dnangf\u00e8i** ( _wuh nahng fay_ )\n\nLoser. Literally \"good-for-nothing useless.\"\n\n\u8f6f\u811a\u87f9 **ru\u01cenji\u01ceoxi\u00e8** ( _rwun jow shih_ )\n\nWuss, wussy, chicken. Literally \"soft-legged crab.\" Originated in Suzhou, where crab legs are a popular food and strong legs with lots of meat are, obviously, preferred over soft legs with no meat. Mostly used in the South. Northerners do not use the term but do understand its meaning when they hear it.\n\n\u5403\u7d20\u7684 **ch\u012bs\u00f9de** ( _chih soo duh_ )\n\nWuss, pushover, sucker. Literally \"vegetarian,\" referring to Buddhist monks because they are kind and merciful (and don't eat meat). Usually used defensively, as in \u6211\u53ef\u4e0d\u662f\u5403\u7d20\u7684 **w\u01d2 k\u011b bush\u00ec ch\u012bs\u00f9de** ( _wuh kuh boo shih chih soo duh_ ), \"I'm not a wuss,\" or \u4f60\u4ee5\u4e3a\u6211\u662f\u5403\u7d20\u7684? **n\u01d0 y\u01d0w\u00e9i w\u01d2 sh\u00ec ch\u012bs\u00f9de**? ( _nee ee way wuh shih chih soo duh_ ): \"Do you think I'm a wuss?\"\n\n\u795e\u7ecf\u75c5 **sh\u00e9nj\u012bngb\u00ecng** ( _shen jing bing_ )\n\nCrazy, lunatic. Calling someone this connotes something like \"What the hell is wrong with you?\" Literally \"mental illness.\"\n\n\u6709\u75c5 **y\u01d2ub\u00ecng** ( _yo bing_ )\n\nCrazy. A slightly more common and mild variation on **sh\u00e9nj\u012bngb\u00ecng** (above). It's like saying, \"What? No way\u2014you're crazy!\" Literally \"have a disease.\"\n\n\u732a **zh\u016b** ( _joo_ ) or \u732a\u5934 **zh\u016bt\u00f3u** ( _joo toe_ )\n\nMoron. Literally \"pig\" and \"pighead,\" respectively.\n\n\u534a\u6b8b\u5e9f **b\u00e0n c\u00e1nf\u00e8i** ( _bahn tsahn fay_ )\n\nLiterally \"half-handicapped\" or \"half cripple.\" Jokingly said of a man who is shorter than his woman. \u6b8b\u5e9f **C\u00e1nf\u00e8i** ( _tsahn fay_ ) means \"cripple\" or \"handicapped\" and is a mocking term for a short man. Both terms can be real insults but, depending on who's saying them and how, can also be affectionate jests.\n\n\u8111\u88ab\u9a74\u8e22\u4e86 **n\u01ceo b\u00e8i l\u00fc' t\u012b le** ( _now bay lee tee luh_ )\n\nKicked in the head by a donkey. Popular among young people, used to call someone stupid.\n\n\u75de\u5b50 **p\u01d0zi** ( _pee dz_ )\n\nA mild insult along the lines of \"ruffian\" or \"riffraff.\" The literal meaning alludes to medical conditions of the liver, spleen, or abdomen, suggesting that **p\u01d0zi** are like a disease on society.\n\n\u6ca1\u8d77\u5b50 **m\u00e9i q\u01d0zi** ( _may chee dz_ )\n\nUseless, stupid, a good-for-nothing. Literally \"no ambition.\"\n\n\u5f31\u667a **ru\u00f2zh\u00ec** ( _rwuh jih_ )\n\nIdiotic, stupid. Literally \"mentally enfeebled.\"\n\n\u73a9\u513f\u95f9 **w\u00e1nr n\u00e0o** ( _warr now_ )\n\nTroublemaker, ruffian. Also means \"to fool around\" or \"to run wild.\" Beijing slang only. Literally \"play and quarrel\" or \"play and loudly stir up.\"\n\n\u51a4\u5927\u5934 **yu\u0101n d\u00e0t\u00f3u** ( _yren dah toe_ )\n\nFool. Literally \"wrong bighead.\"\n\n\u6d51\u7403\u513f **h\u00fan q\u00edur** ( _hwen chyurr_ )\n\nGood-for-nothing, rascal. Literally \"unclear ball.\" \u6d51 **hun** means \"unclear\" or \"dirty,\" as in \u6d51\u6c34 **h\u00fan shu\u01d0** ( _hwen shway_ ), or \"dirty water.\" Typically employed by parents to reprimand their kids. Used in northern China.\n\n\u8111\u6b8b **n\u01ceoc\u00e1n** ( _now tsahn_ )\n\nMeans \"mental retardation\" or \"a mental disability\" and is a popular insult among young people. One usage is \u4f60\u8111\u6b8b\u5417? **n\u01d0 n\u01ceoc\u00e1n ma?** ( _nee now tsahn ma_ ), meaning \"Are you retarded?\" or \"Is there something wrong with your head?\"\n\n\u8111\u6709\u5c4e **n\u01ceo y\u01d2u sh\u01d0** ( _now yo shih_ ) or \u8111\u5b50\u91cc\u6709\u5c4e **n\u01ceozi l\u01d0 y\u01d2u sh\u01d0** ( _now dz lee yo shih_ )\n\nShit in the brain. Popular among young people.\n\n#  **The supernatural**\n\n\u761f\u795e **w\u0113n sh\u00e9n** ( _when shen_ )\n\nA mild insult along the lines of \"troublemaker.\" Literally \"god of plague,\" referring to Chinese mythology. Considered old-fashioned now in much of China, but still used quite a bit in Sichuan Province and some southern areas.\n\n\u9b3c **gu\u01d0** ( _gway_ )\n\nMeans \"devil\" or \"ghost.\" Not typically used as an insult in of itself, but often added onto adjectives to turn them into pejoratives. For example, if you think someone is selfish, or \u5c0f\u6c14 **xi\u01ceo q\u00ec** ( _shyaow chee_ ), you might call them a \u5c0f\u6c14\u9b3c **xi\u01ceo q\u00ec gu\u01d0** ( _shyaow chee gway_ ), literally \"selfish devil.\"\n\n\u89c1\u9b3c **ji\u00e0ngu\u01d0** ( _jyinn gway_ )\n\nLiterally \"see a ghost.\" Can be exclaimed alone to mean something like \"Damn it!\" or \"Crap!\" or \"Oh shit!\" But it is not profane like some of those English equivalents. Can also be used as an intensifier, as in \u4f60\u89c1\u9b3c\u53bb\u5427! **n\u01d0 ji\u00e0ngu\u01d0 q\u00f9 ba!** ( _nee jinn gway chee bah_ ), which literally translates as \"you see a ghost leave\" but means \"Go to hell!\" or \"Fuck off!\" or \"Get the hell out of my face!\"\n\n#  **Rural insults**\n\n\u571f **t\u01d4** ( _too_ )\n\nA pejorative with a broad range of meanings. It literally means \"dirt\" or \"earth\" and, most broadly, is used to describe an unsophisticated or uncultured person, much like \"redneck\" or \"yokel\" or \"hick.\" Someone who spits on the floor while indoors, doesn't line up to buy things, or who can't figure out how to use the ticket-vending machine in the subway, might be called **t\u01d4**. **T\u01d4** can also be a generic, somewhat all-purpose put-down, like \"dork.\" More recently, **t\u01d4** refers to someone out of touch with aspects of modern society\u2014for example, who doesn't know how to use the Internet.\n\n\u571f\u5305\u5b50 **t\u01d4 b\u0101ozi** ( _too baow dz_ )\n\nSomeone who is **t\u01d4**. ( **T\u01d4** is an adjective while **t\u01d4 b\u0101ozi** is a noun.) One explanation for this term is that \u5305\u5b50 **b\u0101ozi** (a steamed, breadlike bun with meat or vegetable filling) is a common food in poor, rural areas, and so **t\u01d4 b\u0101ozi** indicates that the person comes from the countryside.\n\n\u8001\u5192\u513f **l\u01ceo m\u00e0or** ( _laow murr_ \u2014the first syllable rhymes with \"cow,\" and the second rhymes with \"burr\")\n\nNorthern Chinese slang for **t\u01d4**. Literally \"old stupid\" (though it can be said of anyone, not just old people). \u5192 **M\u00e0o** is slang for \"stupid\" or \"inexperienced\" but is seldom used by itself anymore.\n\n\u571f\u5f97\u6389\u6e23\u513f **t\u01d4 de di\u00e0o zh\u0101r** ( _to duh dyow jar_ )\n\nIgnorant, hick, unrefined. Literally \"bumpkin shedding dirt,\" suggesting that someone is so **t\u01d4** that dirt is falling off them. Used in northeast China.\n\n\u519c\u6c11 **n\u00f3ngm\u00edn** ( _nohng meen_ \u2014 **n\u00f3ng** has a long _o_ sound, like in \"bone\")\n\nLiterally means \"farmer\" or \"peasant\" (unlike in English, \"peasant\" is a neutral term in Chinese) but when said disdainfully can carry the same \"country bumpkin\" connotations as **t\u01d4** (above). However, **n\u00f3ngm\u00edn** is not used nearly as often as **t\u01d4**.\n\n\u67f4\u79be\u599e\u513f **ch\u00e1i he ni\u016br** ( _chai huh nyurr_ )\n\nAn insulting term for a country girl, used in Beijing only.\n\n\u6ca1\u7d20\u8d28 **m\u00e9i s\u00f9zh\u00ec** ( _may soo jih_ )\n\nLiterally \"no quality.\" Said, like **t\u01d4** , of someone who acts in uncivilized ways and means he or she has no upbringing, manners, or class.\n\n\u4e0d\u8bb2\u6587\u660e **b\u00f9 ji\u01ceng w\u00e9nm\u00edng** ( _boo jyahng wen meeng_ )\n\nSame meaning as **m\u00e9i s\u00f9zh\u00ec** (above) but less commonly used. Literally \"not speaking civilization.\"\n\n#  **Extremely rude**\n\n\u70c2\u4eba **l\u00e0n r\u00e9n** ( _lahn ren_ )\n\nBad person. Literally \"rotten person.\"\n\n\u7f29\u5934\u4e4c\u9f9f **su\u014d t\u00f3u w\u016bgu\u012b** ( _swuh toe ooh gway_ )\n\nCoward. Literally \"a turtle with its head in its shell.\"\n\n\u81ed **ch\u00f2u** ( _choe_ )\n\nStupid, bad, disappointing, inferior. Literally means \"smelly\" and is often added in front of insults to intensify them. So, for example, \"smelly bitch\" in Chinese, \u81ed\u5a4a\u5b50 **ch\u00f2u bi\u01ceozi** ( _choe byow dz_ ), is, as in English, much stronger than just \"bitch.\"\n\n\u4e2b\u5934\u7247\u5b50 **y\u0101t\u00f3u pi\u00e0nz\u01d0** ( _yah toe pyinn dz_ )\n\nA Beijing insult for a young girl who's ignorant and inexperienced. \u4e2b\u5934 **Y\u0101t\u00f3u** means \"servant girl.\" Literally \"servant girl piece.\"\n\n\u4e11\u516b\u602a **ch\u01d2ub\u0101gu\u00e0i** ( _choe bah gwie_ )\n\nAn insulting term for an extremely ugly person. Literally \"ugly all-around weird.\"\n\n\u6cfc\u5987 **p\u014df\u00f9** ( _pwuh foo_ )\n\nShrew, bitch. An insulting term for a mean, crazy woman. Literally \"spill woman.\"\n\n\u4e09\u516b **s\u0101nb\u0101** ( _sahn bah_ )\n\nIn Taiwan this just means \"silly\" and is said of both males and females, but on the mainland it is a very strong insult for a woman, similar to \"bitch\" or \"slut.\" (Though sometimes it just means \"gossipy.\") Literally, it means \"three eight,\" for which there are two explanations. One is that during the Qing dynasty (1644-1911) foreigners were supposedly only allowed to circulate on the eighth, eighteenth, and twenty-eighth of each month, and thus foreigners were called, somewhat scornfully, **s\u0101nb\u0101** for the three eights. Another explanation is that **s\u0101nb\u0101** refers to International Women's Day, which is on March 8. In contexts when it means \"bitch\" or \"slut,\" it's common to amp up the strength of **s\u0101nb\u0101** as an insult by saying \u6b7b\u4e09\u516b **s\u01d0 s\u0101nb\u0101** ( _sih sahn bah_ ), literally \"dead bitch,\" or \u81ed\u4e09\u516b **ch\u00f2u s\u0101nb\u0101** ( _choe sahn bah_ ), literally \"stinking bitch.\"\n\n\u9ec4\u8138\u5a46 **hu\u00e1ngli\u01cenp\u00f3** ( _hwahng lyinn pwuh_ )\n\nSlightly derogatory term for a middle-aged married woman. Literally \"yellow-faced woman,\" meaning that she is old and ugly.\n\n\u767d\u773c\u72fc **b\u00e1iy\u01cen l\u00e1ng** ( _buy yen lahng_ \u2014the last syllable is similar to \"long\" but with an _ah_ sound replacing the _o_ )\n\nIngrate, a ruthless and treacherous person. Literally \"white-eyed wolf.\" Both \"white eyes\" and \"wolf \" are insults in Chinese.\n\n\u4f60\u4e0d\u662f\u4eba **n\u01d0 b\u00fa sh\u00ec r\u00e9n** ( _nee boo shih ren_ )\n\nYou're worthless; you're inhuman. Literally, \"You are not a person.\"\n\n\u4f60\u4e0d\u662f\u4e1c\u897f **n\u01d0 b\u00fa sh\u00ec d\u014dngxi** ( _nee boo shih dohng she_ )\n\nYou're worthless; you're less than human. Literally, \"You're not a thing\" or \"You're not anything.\"\n\n\u4e0d\u8981\u8138 **b\u00f9y\u00e0oli\u01cen** ( _boo yaow lyinn_ )\n\nShameless, without pride. Literally \"doesn't want face.\" Face is a central concept in Chinese culture and entire volumes have been written in attempts to fully explain its nuances, but suffice to say that losing face is bad, giving face is good, and not wanting face is unspeakably shameful\u2014thus saying that someone is **b\u00f9y\u00e0oli\u01cen** is far more insulting than the English word \"shameless\" and conveys a complex mix of being somehow subhuman, pathetic, and so lacking in self-respect that you would willingly do things that no one else would be caught dead doing. Also used by women to mean \"disgusting\" and sometimes with \u81ed **ch\u00f2u** ( _cho_ , rhymes with \"show\"), which means \"stinking,\" in front to amplify it to \u81ed\u4e0d\u8981\u8138 **ch\u00f2u b\u00f9y\u00e0o li\u01cen** , or \"absolutely disgusting.\" Another common way to amplify the expression is to say \u6b7b\u4e0d\u8981\u8138 **s\u01d0 b\u00f9 y\u00e0o li\u01cen** ( _sih boo yow lyinn_ ), literally, \"You don't want face even when you die.\"\n\n\u53bb\u6b7b **q\u00f9 s\u01d0** ( _chee sih_ )\n\nGo die.\n\n\u8d70\u72d7 **z\u01d2ug\u01d2u** ( _dzoe go\u2014_ both syllables rhyme with \"oh\")\n\nLackey, sycophant. Literally \"running dog.\" Said of a servile person with no morals who sucks up to more powerful people.\n\n\u72d7\u817f\u5b50 **g\u01d2utu\u01d0zi** ( _go tway dzz_ ) \/ \u72d7\u817f **g\u01d2utu\u01d0** ( _go tway_ )\n\nA variant of **z\u01d2ug\u01d2u** (above). Literally \"dog legs.\" You may have heard of the term \"capitalist running dog\" or \"imperialist running dog.\" Mao Zedong used \"dog legs\" to refer to countries that were friendly with the United States.\n\n\u6eda **g\u01d4n** ( _gwen_ ) or \u6eda\u5f00 **g\u01d4nk\u0101i** ( _gwen kigh_ ) or \u6eda\u86cb **g\u01d4nd\u00e0n** ( _gwen dun_ )\n\nGo away; get lost.\n\n\u8001\u4e0d\u6b7b\u7684 **l\u01ceo b\u00f9 s\u01d0 de** ( _laow boo sih duh_ )\n\nA rude term for an old person. Literally \"old and not dead.\"\n\n\u8001\u4e1c\u897f **l\u01ceo d\u014dngxi** ( _laow dohng she_ )\n\nOld thing. A rude term for an old person.\n\n\u8001\u6a21\u7822\u78e3\u773c **l\u01ceo m\u00f3 k\u0113 ch\u011bn y\u01cen** ( _laow mwuh kuh chen yen_ )\n\nLiterally \"old wrinkle eyes.\" An insulting term for someone old and ugly. Used in Beijing.\n\n\u5783\u573e **l\u0101j\u012b** ( _lah gee_ )\n\nLiterally \"trash\" but can be derogatorily said of people as well. In Taiwan pronounced **l\u00e8 se** ( _luh suh_ ).\n\n\u755c\u751f **ch\u00f9sh\u0113ng** ( _choo shung_ )\n\nAnimal, inhuman. Literally \"born of an animal.\" An extremely strong insult.\n\n#  **Slut and whore**\n\nIn addition to the terms below, chapter 7, \"Behaving Badly,\" includes numerous words for \"prostitute\" that can also be used as strong insults.\n\n\u9a9a\u8d27 **s\u0101ohu\u00f2** ( _saow hwuh\u2014_ **s\u0101o** rhymes with \"cow\")\n\nSlut (but can also be said of a man). Literally \"lewd thing.\"\n\n\u8d31\u8d27 **ji\u00e0nhu\u00f2** ( _gin hwuh_ )\n\nSlut (but can also be said of a man). Literally \"cheap thing.\"\n\n\u5a4a\u5b50 **bi\u01ceozi** ( _byow dz_ )\n\nCan literally mean \"whore\" but also used as a strong insult for a woman, equivalent to \"bitch\" or \"whore.\" Often strengthened to \u81ed\u5a4a\u5b50 **ch\u00f2u bi\u01ceozi** ( _choe byow dz_ ), literally \"stinking whore.\"\n\n\u72d0\u72f8\u7cbe **h\u00falij\u012bng** ( _hoo lee jing_ )\n\nVixen, tart, slut. A woman who seduces other people's husbands or boyfriends. Literally \"fox-spirit,\" referring to a creature from Chinese mythology. Slightly milder than the other terms for \"slut.\"\n\n\u98ce\u9a9a **f\u0113ngs\u0101o** ( _fung sow_ , the latter rhymes with \"cow\")\n\nSlutty. Literally \"sexy and horny.\"\n\n\u516c\u5171\u6c7d\u8f66 **g\u014dngg\u00f2ngq\u00ecch\u0113** ( _gohng gohng chee chuh_ \u2014the first two syllables sound like \"gong\" but with a long _o_ , or _oh_ , sound in the middle)\n\nSlut, a woman who sleeps around. Literally \"public bus,\" as in \"everyone has had a ride.\" Similar to the English expression \"the neighborhood bicycle.\"\n\n\u8361\u5987 **d\u00e0ngf\u00f9** ( _dahng foo_ )\n\nSlut. Literally \"lustful woman.\"\n\n\u6b8b\u82b1\u8d25\u67f3 **c\u00e1n hu\u0101 b\u00e0i li\u01d4** ( _tsahn hwah buy lew_ \u2014 **li\u01d4** rhymes with \"pew\")\n\nAn insult meaning \"old whore.\" Literally \"broken flower, lost willow.\" Used mostly in northern China.\nCHAPTER THREE\n\n**Swearing and Profanity**\n\n**I** n English, we have plenty of ways to curse, but for the most part we tend to rely on a small and rather unvaried stable of fallback words. Similarly, the Chinese language allows you to spin an infinite number of creative, colorful curses, but you're much more likely to stick with a basic like \"fuck you.\"\n\nAs in English, most Chinese swearing centers on fucking and its related accoutrements (the pussy and penis). And lots of insults involve stupidity or illegitimate birth or prostitution\u2014nearly direct equivalents to \"stupid cunt,\" \"fucking bastard,\" \"dirty whore,\" \"what a dick,\" etc.\n\nThere are, however, a few major differences between the two languages. For one thing, China is officially an atheist country, so there is no real equivalent to Christian (or Muslim) blasphemy\u2014nothing mirroring \"Holy God!\" or \"Jesus Christ!\" or \"Damn you to hell!\" Several terms in this chapter are translated as \"damn\" but really to indicate in English how strong\u2014or in this context _not_ strong\u2014of an obscenity the word is. The concepts of heaven, hell, and devils, however, do exist in China, stemming from Buddhist and Taoist traditions, and thus you can call someone a devil, tell someone to go to hell, and launch a few other insults along those lines, but they are not very common and are considered old-fashioned and mild (the few examples of such insults worth mentioning appear in the previous chapter). Perhaps the closest thing to religious blasphemy in Chinese is the cursing of one's ancestors, which is a serious insult as Chinese culture places a great deal of importance on blood ties, and ancestor worship is still practiced in some of the more traditional parts of the country.\n\nAnother way in which Chinese differs from English is that words relating to homosexuality (see chapter 6) are not particularly used as insults. This, again, may have something to do with the lack of religious dogma in China. While homosexuality is not exactly accepted in Chinese society, being gay does not carry the stigma of inherent moral \"wrongness\" that it often bears in Christian and Muslim societies. (Homosexuality can be considered bad in China for plenty of other reasons, but they mostly have to do with the importance that society places on having children.) Thus there is nothing in the Chinese vocabulary like \"cock-sucker,\" \"faggot,\" \"bugger off,\" \"that's so gay,\" or \"that sucks.\"\n\nOne final mainstay of English-language swearing conspicuously absent from Chinese is \"shit.\" In a country that until recently was predominantly agricultural (meaning that manure was an important resource), where people talk openly at the dinner table about diarrhea, and where babies toddle about with their naked butts exposed in \"split pants\" (pants open at the back so that Junior can squat wherever he wants and take an impromptu dump on the street), it just isn't very dirty to mention excrement or urine.\n\nThis is not to imply, however, that shit is entirely neutral in Chinese. Any mention of shit is vulgar, and thus certainly not fit for, say, the classroom or the office; it just isn't used as an actual swear word like it is in English. You might use it when you're purposely being gross, such as while joking around with family or good friends. But in those cases talking about shit would be just crude enough to be funny but not outright dirty. And as with any vulgar word, \"shit\" can also be used in an insulting way. One might say, for example, \"That movie sucked so hard it made me want to shit,\" or \"The team played like shit.\" For that reason, this chapter includes a few pejorative words and phrases involving shit, but you'd use them more to be bawdy than to actually swear. In fact, the very idea of using \"shit\" pejoratively is probably a Western import that was popularized through the subtitling of Western movies in Hong Kong.\n\nThe words and phrases in this chapter will give you all the vocabulary necessary to hold your own with even the most salty-tongued of Chinese. Many of these words can be used affectionately with close friends\u2014in the way you might call a buddy \"motherfucker\" in English\u2014but don't forget that, no matter how close you think you are to someone, doing so can be hard to pull off when you don't quite grasp all the nuances of the dialogue. And another note of caution on using strong language in Chinese: if you are a woman, using these words will, in some situations, cause outright shock. Chinese society right now is a bit like America in the fifties\u2014there are certain things a girl just isn't supposed to do. Feel free to let your verbosity run wild in the appropriate contexts (the proximity of chain-smoking, booze-swilling, young Chinese women should be a helpful clue). And of course there are times when shock might be the exact effect you're going for (like, say, when some asshole tries to scam you on the street). But for the most part, in the eyes of most Chinese, any word that appears in this chapter is (along with the stronger insults from the previous chapter) something that should never escape a lady's mouth.\n\n#  **Fuck-related profanity**\n\n\u808f **c\u00e0o** , more commonly written \u64cd **c\u0101o** (both pronounced _tsow_ )\n\nFuck. The character \u808f is visually quite graphic, as it is composed of \u5165 **r\u00f9** ( _roo_ ), meaning \"enter,\" and \u8089 **r\u00f2u** ( _row_ ), meaning \"meat.\" \u808f is technically the correct character for \"fuck,\" but because it is not included in most computer or phone-character input systems, and because it's just so uncomfortably dirty looking, most people write the homophonous \u64cd (which actually means \"hold\"). Thus I have written \u64cd for the rest of the terms in this chapter that use the word. But remember, technically, it should be \u808f. (For those of you who pay attention to pinyin tones, I also render the syllable as fourth tone every time even though \u64cd is first tone, since that's pretty much how it always comes out sounding. In general, though, with colloquial expressions like these you shouldn't get too hung up on which tone is technically accurate since it's not always fixed.)\n\n\u6211\u64cd **w\u014f c\u00e0o** ( _wuh tsow_ )\n\nFuck! An extremely common exclamation for all occasions\u2014when you're pissed off, impressed, amazed, or whatever. Literally, \"I fuck.\" (As a side note, many Chinese are amused when they hear English speakers say \"what's up?\" or \"wassup?\" because it sounds to them like **w\u014f c\u00e0o**.) When saying \"fuck\" alone, it's much more usual to say **w\u014f c\u00e0o** than to say just \u64cd **c\u00e0o** ( _tsow_ ); northern Chinese, however, do often use **c\u00e0o** alone as an interjection, especially between clauses. For example: \"I had a really bad morning, fuck, had a car accident on my way to work.\"\n\n\u64cd\u4f60\u5988 **c\u00e0o n\u01d0 m\u0101** ( _tsow nee ma_ )\n\nFuck you! Literally, \"Fuck your mother!\" An extremely common obscenity.\n\n\u64cd\u4f60\u5927\u7237 **c\u00e0o n\u01d0 d\u00e0ye** ( _tsow nee dah yeh_ )\n\nFuck you! Literally, \"Fuck your grandfather!\" Northern and southern Chinese use different words for grandparents, so this one is used in northern China only. This is marginally less strong than **c\u00e0o n\u01d0 m\u0101** (above) because it's generally considered more offensive to curse someone's female relatives than their male relatives. You can insert any relative into this construction; for example northern Chinese might say \u64cd\u4f60\u5976\u5976 **c\u00e0o n\u01d0 n\u01ceinai** ( _tsow nee nigh nigh_ ): literally, \"Fuck your grandmother.\" See the entries after \u65e5 **r\u00ec** (page 39) for the southern versions.\n\n\u64cd\u4f60\u5988\u7684\u5c44 **c\u00e0o n\u01d0 m\u0101 de b\u012b** ( _tsow nee ma duh bee_ )\n\nFuck you! Literally, \"Fuck your mother's cunt,\" and stronger than the two expressions above since you're adding another obscene word, **b\u012b** , or \"pussy,\" on top of **c\u00e0o**.\n\n\u64cd\u5c44 **c\u00e0ob\u012b** ( _tsow bee_ )\n\nFuck. Can be exclaimed alone or used as an intensifier. Literally, \"fuck pussy.\"\n\n\u64cd\u86cb **c\u00e0od\u00e0n** ( _tsow dahn_ )\n\nLiterally \"fuck egg.\" (See the previous chapter for why eggs are used in insults.) Can be exclaimed alone, like \"Fuck!\" or \"Oh fuck!\" and can also be used as an adjective to mean that someone is bad or inept, as in \u4f60\u771f\u808f\u86cb **n\u01d0 zh\u0113n c\u00e0od\u00e0n** ( _nee jen tsow dahn_ ), which means \"You're a real stupid fuck.\"\n\n\u64cd\u884c **c\u00e0oxing** ( _tsow sheeng_ )\n\nA dirty and insulting way to refer to someone's appearance or behavior. Literally \"fucking behavior\" or something like \"the behavior of a shitty person.\" For example, \u4f60\u770b\u4ed6\u90a3\u64cd\u884c, \u771f\u53d7\u4e0d\u4e86 **n\u01d0 k\u00e0n t\u0101 n\u00e0 c\u00e0ox\u00edng, zh\u0113n sh\u00f2ub\u00f9liao** ( _nee kahn tah tsow sheeng, jen show boo liao_ ) means \"Look at his shit behavior, it's totally unacceptable.\" The term can also be an adjective meaning \"shameful\" or \"disgusting,\" as in \u4f60\u771f\u64cd\u884c **n\u01d0 zh\u0113n c\u00e0ox\u00edng** ( _nee jen tsow sheeng_ ), or \"You're really fucking disgusting.\"\n\n\u64cd\u4f60\u516b\u8f88\u5b50\u7956\u5b97! **C\u00e0o n\u01d0 b\u0101 b\u00e8izi z\u01d4z\u014dng!** ( _wuh tsow nee bah bay dz dzoo dzohng_ )\n\nFuck eight generations of your ancestors! An extremely strong insult: stronger than **c\u00e0o n\u01d0 m\u0101** (above). It's always specifically eight or eighteen generations that are cursed in this insult. The number eight, \u516b **b\u0101** ( _bah_ ), is considered lucky in Chinese culture. Thus it is especially significant for misfortune to befall what should be a lucky number of generations.\n\n\u64cd\u4f60\u7956\u5b97\u5341\u516b\u4ee3! **C\u00e0o n\u01d0 z\u01d4z\u014dng sh\u00edb\u0101 d\u00e0i!** ( _tsow nee dzoo dzohng shih bah die_ )\n\nFuck eighteen generations of your ancestors! An extremely strong insult. All multiples of nine are significant in Chinese culture because nine, \u4e5d **ji\u01d4** ( _joe_ ), is pronounced the same as the word for \"long-lasting,\" written \u4e45. Eighteen, \u5341\u516b **sh\u00edb\u0101** ( _shih bah_ ), is itself significant because it sounds like \u8981\u53d1 **y\u00e0o f\u0101** ( _yow fah_ ), which means \"one will prosper.\"\n\n\u65e5 **r\u00ec** ( _rih_ )\n\nSouthern Chinese slang for \"fuck\" (and also used a bit in some northern provinces like Shanxi and Shandong.) Its origins most likely come from it sounding like \u5165 **r\u00f9** ( _roo_ ), which means \"enter\" and in ancient China was itself a less dirty way of saying \"fuck\" (like saying \"freaking\" instead of \"fucking\"). **R\u00f9** is also a component radical for another word for \"fuck,\" \u808f **c\u00e0o** (see page 36). **R\u00ec** is used the same way as **c\u00e0o** \u2014combined with \"I\" into an exclamation like \u6211\u65e5! **W\u014f r\u00ec!** ( _wuh rih_ ), meaning \"Fuck!\" or directed at someone's mother or relatives or ancestors to mean \"Fuck you!\" as in \u65e5\u4f60\u5988! **R\u00ec n\u01d0 m\u0101!** ( _rih nee ma_ ), literally, \"Fuck your mother!\"\n\n\u65e5\u4f60\u5916\u516c **r\u00ec n\u01d0 w\u00e0ig\u014dng** ( _rih nee why gohng_ )\n\nFuck you. Literally, \"Fuck your grandfather!\" Used in southern China.\n\n\u65e5\u4f60\u5916\u5a46 **r\u00ec n\u01d0 w\u00e0ip\u00f3** ( _rih nee why pwuh_ )\n\nFuck you. Literally, \"Fuck your grandmother!\" Used in southern China.\n\n\u5e72 **g\u0101n** ( _gahn_ )\n\nFuck. In reality, **g\u0101n** is slightly less offensive than **c\u00e0o** (see above), partly because it isn't visually graphic (like \u808f). It's closer in strength to \"shit,\" but most dictionaries translate it as \"fuck.\" It can refer to having sex, and the way it's used grammatically is also closer to \"fuck\"\u2014it can be yelled alone when you're pissed off, as in \u5e72! **G\u0101n!** ( _gahn_ ) or \u5e72\u4e86! **G\u0101nle!** ( _gahn luh_ ), or like **c\u00e0o** it can followed by a subject (usually someone's mother). See the next entry for more on this. Also see chapter 5 for how to use **g\u0101n** in the context of describing sex. The character \u5e72 also means \"dry,\" which has made for more than a few comical mistranslations on Chinese restaurant menus and supermarket signs. Now you'll know what happened next time you come across \"sliced fuck tofu\" on a Chinese menu. \u5e72 **G\u0101n** is also used a lot in fighting contexts because it can also mean \"to kill.\"\n\n\u5e72\u4f60\u5a18 **g\u0101n n\u012d ni\u00e1ng** ( _gahn nee nyahng_ )\n\nFuck you! Literally, \"Fuck your mother!\" Used in southern China only (a northerner would always say **c\u00e0o n\u01d0 m\u0101** ). \u5a18\n\n**Ni\u00e1ng** means \"mother,\" same as \u5988 **m\u0101** ( _ma_ ), but \u5e72\u4f60\u5988 **g\u0101n n\u01d0 m\u0101** sounds funny in Chinese so **ni\u00e1ng** is always used instead.\n\n\u9760 **k\u00e0o** ( _cow_ )\n\nDamn! A common exclamation to express surprise or anger. \u9760 **K\u00e0o** means \"to depend upon\" in Mandarin, but in the southern Fujianese dialect that the expression originates from (spoken in both Fujian and Taiwan) it's actually \u54ed **k\u016b** ( _coo_ ), meaning \"to cry.\" **Kao bei** , \"cry over your father's death,\" **kao bu** , \"cry over your mother's death,\" and **kao yao** , \"cry from hunger,\" are extremely common expressions (implying something like \"What's wrong with you? Did your mother\/father just die?\" or \"Are you dying of starvation?\") and are nasty ways, in this dialect, to tell someone to shut up. However, in its popularization beyond Fujian and Taiwan the meaning has changed to an exclamation equivalent to \"Damn!\"\n\n\u6211\u9760 **w\u014f k\u00e0o** ( _wuh cow_ )\n\nShit! A common exclamation to express surprise or anger. In Mandarin, it literally means \"I depend on\" but is used as a stronger form of **k\u00e0o** , like \"Shit!\" versus \"Damn!\"\n\n#  **Cursing mothers (and other relatives)**\n\n\u4ed6\u5988\u7684 **t\u0101m\u0101de** ( _tah mah duh_ )\n\nDamn! Shit! While not anywhere near as vulgar as the words above, this is without a doubt the most classic of Chinese swears. It means \"his mother's\" and implies the larger sentence \u64cd\u4ed6\u5988\u7684\u5c44 **c\u00e0o t\u0101 m\u0101 de b\u012b** ( _tsow tah mah duh bee_ ), \"Fuck his mother's pussy!\" but is much less dirty since the key words are left out. The phrase can be exclaimed by itself or used to intensify an adjective. Thus you can say something like \"\u4eca\u5929\u4ed6\u5988\u7684\u51b7\" \" **J\u012bnti\u0101n t\u0101m\u0101de leng\"** ( _jean tyinn tah mah duh lung_ ) for \"Today is really damn cold,\" even though the literal translation, \"Today is his mother's cold,\" doesn't make sense. Another example: \u4ed6\u771f\u4ed6\u5988\u7684\u725b\u5c44! **T\u0101 zh\u0113n t\u0101m\u0101de ni\u00fab\u012b!** ( _tah jen tah ma duh nyoo bee_ ) means \"He's really fucking badass!\" even though it translates as \"He's really his mother's awesome!\"\n\nAnd of course, you can swap in any other relatives\u2014a common version in Beijing is \u4ed6\u5927\u7237 **t\u0101 d\u00e0ye** ( _tah dah yeh_ ), literally \"his grandfather\" (implying \"his grandfather's\"). \u4ed6\u5976\u5976\u7684 **T\u0101 n\u01ceinai** ( _tah nigh nigh_ ), \"his grandmother\" (implying \"his grandmother's\") is common as well.\n\n\u4f60\u8001\u5e08 **n\u01d0 l\u01ceosh\u012b** ( _nee laow shih_ )\n\nDamn! Literally \"your teacher\" (implying \"his teacher's\"). A Taiwan variation on **t\u0101m\u0101de** (page 41). Written \u4f60\u8001\u5e2b in Taiwan.\n\n\u5988\u7684 **m\u0101de** ( _mah duh_ )\n\nDamn! Literally \"mother's\" and a shortened form of **t\u0101m\u0101de** (page 41). Both an exclamation and an intensifier.\n\n\u4f60\u5988\u7684 **n\u01d0m\u0101de** ( _nee mah duh_ )\n\nDamn you! Literally \"your mother's\" instead of \"his mother's\" (page 41) and thus a bit stronger since it's a direct address.\n\n\u4f60\u5927\u7237 **n\u01d0 d\u00e0ye** ( _nee dah yeh_ )\n\nDamn you! A northern Chinese variation on the above. Literally \"your grandfather\" (implying \"his grandfather's\").\n\n\u4f60\u5976\u5976 **N\u01d0 n\u01ceinai** ( _nee nigh nigh_ ), \"your grandmother,\" is common in northern China as well.\n\n\u4ed6\u5927\u7237 **t\u0101 d\u00e0ye** ( _tah dah yeh_ )\n\nShit, damn. Can be either exclaimed alone or used as an intensifier. Literally \"his grandfather.\" \u4ed6\u5976\u5976 **T\u0101 n\u01ceinai** ( _tah nigh nigh_ ), \"his grandmother,\" is common as well. Both are only used in northern China.\n\n\u53bb\u4f60\u5988\u7684 **q\u00f9 n\u01d0 m\u0101 de** ( _chee nee mah duh_ )\n\nFuck off. Literally \"go to your mother's\" and stronger than **t\u0101m\u0101de** (page 41) since it is a direct address.\n\n\u53bb\u4f60\u5976\u5976\u7684 **q\u00f9 n\u01d0 n\u01ceinai de** ( _chee nee nigh nigh duh_ )\n\nFuck off. Literally \"go to your grandmother's\" and a variation on the above.\n\n\u53bb\u4f60\u7684 **q\u00f9n\u01d0de** ( _chee nee duh_ )\n\nDamn you, get lost. Literally \"go to yours\" and milder than the above.\n\n\u4f60\u5988\u7684\u5c44 **n\u01d0 m\u0101 de b\u012b** ( _nee ma duh bee_ )\n\nFuck! Fuck you! Can be exclaimed alone or addressed at someone. Literally \"your mother's cunt.\"\n\n\u53eb\u4f60\u751f\u5b69\u5b50\u6ca1\u5c41\u80a1\u773c **ji\u00e0o n\u01d0 sh\u0113ng h\u00e1izi m\u00e9i p\u00ecguy\u01cen** ( _jaow nee shung hi dz may pee goo yen_ )\n\nLiterally, \"May your child be born without an asshole.\" A very strong curse. Sometimes \u6ca1\u5c41\u80a1\u773c **m\u00e9i p\u00ecguy\u01cen** ( _may pee goo yen_ ) by itself (\"no asshole,\" or more technically \"imperforate anus\") is used as a curse like \"Damn!\" Originated in Hong Kong and surrounding southern areas, but now commonly used all over China.\n\n#  **Cunt-related obscenities**\n\n\u725b\u5c44 **ni\u00fab\u012b** ( _nyoo bee_ )\n\nCan be used negatively to mean something like \"arrogant fuck\" but more usually means \"fuckin' awesome\" or \"motherfuckin' badass.\" Think of it as meaning that someone has a lot of fuckin' balls, either in a good way or a bad way. Literally \"cow pussy\" (see chapter 1 for the etymology). Extremely popular in northern China, where it originated, and not as commonly used but still understood in southern China. Not used at all (and will most likely not be understood) in Taiwan.\n\n\u50bb\u5c44 **sh\u01ceb\u012b** ( _shah bee_ )\n\nStupid cunt, fuckin' idiot. Literally \"idiot's pussy\"\u2014\u50bb **sh\u01ce** ( _shah_ ) means \"stupid.\" Particularly in northern China, this is perhaps _the_ strongest, dirtiest insult available in your arsenal of things to yell at that fucker who just cut you off in traffic or who just tried to mug you on the street. It also sounds a lot like \"shabby,\" as many English teachers in China have unwittingly discovered when attempts to teach the word are met with peals of laughter. Like **ni\u00fab\u012b** , it is extremely common in northern China, less used in southern China, and not used at all in Taiwan.\n\n\u88c5\u5c44 **zhu\u0101ngb\u012b** ( _jwong bee_ )\n\nAct like a fuckin' poser; be a fuckin' ass. Literally \"dress pussy\" or \"pretend pussy.\" \u88c5 **Zhu\u0101ng** ( _jwong_ ) means \"pretend\" or \"put on,\" like putting on an item of clothing, and implies the larger phrase \u88c5\u725b\u5c44 **zhu\u0101ng ni\u00fab\u012b** ( _jwong nyoo bee_ ); that is, pretending to be **ni\u00fab\u012b** , or awesome, when you're not. Thus you could say that you don't like going to fancy restaurants because you don't want to be surrounded by all those **zhu\u0101ngb\u012b** types. Again, extremely common in northern China, less used in southern China, and not used at all in Taiwan.\n\n\u4e8c\u5c44 (more usually written \u4e8c\u903c or 2B) **\u00e8rb\u012b** ( _er bee_ )\n\nFuckin' idiot, a fuck-up. Used in northern China. Literally \"second pussy\" or \"double cunt.\" \u4e8c **\u00c9r** , or \"two,\" is a reference to the insult \"250,\" or \u4e8c\u767e\u4e94 **\u00e8rb\u01ceiw\u01d4** ( _er buy woo_ ), which means \"idiot\" (see page 19).\n\n\u81ed\u5c44 **ch\u00f2ub\u012b** ( _choe bee_ )\n\nMotherfucker. Literally \"smelly cunt\" or \"stinking cunt.\"\n\n\u70c2\u5c44 **l\u00e0nb\u012b** ( _lahn bee_ )\n\nRotten cunt.\n\n\u5988\u5c44 **m\u0101b\u012b** ( _mah bee_ )\n\nHag, cunt. Literally \"mother's cunt.\"\n\n\u8001\u5c44 **l\u0103ob\u012b** ( _laow bee_ )\n\nOld cunt, old hag.\n\n\u9a9a\u5c44 **s\u0101ob\u012b** ( _saow bee_ )\n\nSlut, dirty cunt. Literally \"slutty cunt.\" A stronger, or at least slightly more colorful, variation on plain \u5c44 **b\u012b** ( _bee_ ).\n\n\u5c44\u6837 **b\u012by\u00e0ng** ( _bee yahng_ )\n\nLiterally \"the appearance of a cunt,\" referring to someone's behavior in a way that indicates that they're being a cunt, or you think they're a cunt. Similar to **c\u00e0oxing** , discussed earlier in this chapter.\n\n\u9e21\u5c44 **j\u012bb\u012b** ( _gee bee_ )\n\n\u9e21 **j\u012b** ( _gee_ ), literally \"chicken,\" or \u9e21\u8d3c **j\u012b z\u00e9i** ( _gee dzay_ ), literally \"chicken thief,\" is Beijing slang for someone cheap or stingy, and \u9e21\u5c44 **j\u012bb\u012b** , literally \"chicken cunt,\" is a dirty version of that term.\n\n\u4f60\u5988\u4e86\u4e2a\u5c44 **n\u012dm\u0101legeb\u012b** ( _nee mah luh guh bee_ )\n\nFuck you! (and a way of saying it that rolls off the tongue especially well). Literally, \"Your mother's a cunt!\"\n\n\u4f60\u4e2a\u6b7b\u5c44 **n\u01d0 ge s\u01d0 b\u012b** ( _nee guh sih bee_ )\n\nLiterally \"you dead cunt,\" but the whole phrase can be used the way you'd use \"fuck\" in direct address, whether genuinely insulting someone or joking around, like \"fuck you\" or \"you stupid fuck.\"\n\n#  **Dick-related swears**\n\n\u5c4c **di\u01ceo** ( _dyaow_ )\n\nSlang for \"cock\" and used as an insult (as in \"what a dick\") since the J\u0304\u0131n dynasty (1115-1234). **Di\u01ceo** is also used positively in Taiwan to mean \"awesome\" or \"cool\" or \"outrageous.\"\n\n\u7ba1\u4f60\u5c4c\u4e8b **gu\u01cen n\u01d0 di\u01ceo sh\u00ec** ( _gwun nee dyow shih_ )\n\nMind your own damn business. Literally \"mind your own dick.\" Similarly, \"It's none of my damn business\" or \"I don't give a shit\" is \u7ba1\u6211\u5c4c\u4e8b **gu\u01cen w\u01d2 di\u01ceo sh\u00ec** ( _gwun wuh dyow shih_ ), literally, \"I'm watching my own dick.\"\n\n\u9e1f **ni\u01ceo** ( _nyow_ , rhymes with \"cow\")\n\nSlang for \"penis,\" equivalent to \"dick\" or \"cock,\" but can also be combined with a noun to create a derogatory term; for example \u90a3\u4e2a\u4eba **n\u00e0ge r\u00e9n** ( _nah guh ren_ ) means \"that person,\" while \u90a3\u4e2a\u9e1f\u4eba **n\u00e0ge ni\u01ceo r\u00e9n** ( _nyow ren_ ) means \"that dick\" or \"that damned person.\"\n\n\u4ec0\u4e48\u9e1f **sh\u00e9nmeni\u01ceo** ( _shuh muh nyow_ )\n\nUsed similarly to \"what the fuck?\" or \"what in the hell?\" Literally \"which bird?\" or \"what dick?\" Originated in northeastern China but now used everywhere.\n\n\u9e21\u5df4 **j\u012bba** ( _gee bah_ )\n\nSlang for \"penis,\" equivalent to \"dick\" or \"cock.\" Like \"dick\" and \"cock,\" **j\u012bba** can refer to an actual penis or can be used as an insult to describe a person. It's also used as an intensifier, like \"fucking.\" For example, \u90a3\u4e2a\u9e21\u5df4\u767d\u75f4 **n\u00e0ge j\u012bb\u0101 b\u00e1ich\u012b** ( _nah guh gee bah buy chih_ ), \"that fucking idiot,\" is stronger than \u90a3\u4e2a\u767d\u75f4 **n\u00e0ge b\u00e1ich\u012b** ( _nah guh buy chih_ ), \"that idiot.\"\n\n#  **Turtle swears**\n\nThere are several turtle- and turtle-egg-related insults in Chinese, all connected to cuckoldry. There are numerous theories why. One is that \u738b\u516b\u86cb **w\u00e1ngb\u0101d\u00e0n** ( _wahng bah dun_ ), \"tortoise egg,\" comes from \u5fd8\u516b\u7aef **w\u00e0ng b\u0101 du\u0101n** ( _wahng bah dwun_ ), which means \"forgetting the Eight Virtues,\" because the two phrases sound nearly identical. The Eight Virtues are a philosophical concept and a sort of code of behavior central to Confucianism. Evidently the Eight Virtues were important enough that forgetting them could become an obscenity, much in the way Christianity is so central to Western culture that referring to God\u2014\"Oh my God!\" \"Jesus Christ!\"\u2014can be considered blasphemous.\n\nAnother theory\u2014and all these theories could be related to one another\u2014is that the ancient Chinese mistakenly believed there were no male turtles and that all turtles copulated with snakes; thus their offspring are of impure blood. Another explanation is that in ancient times \u738b\u516b **w\u00e1ngb\u0101** ( _wahng bah_ ), \"turtle,\" was the name for a male servant in a brothel. Some believe the term came from an especially un-virtuous man in history whose last name was W\u00e1ng (\u738b). Still another is that a turtle's head emerging from its shell resembles a glans penis emerging from the foreskin and so turtles represent promiscuity: indeed \"glans\" in Chinese is\n\n\u9f9f\u5934 **gu\u012bt\u00f3u** ( _gway toe_ ), or \"turtle head.\" And finally, it could have to do with turtles being considered cowardly, since they sink their heads back into their shells when threatened, as reflected by the phrase \u7f29\u5934\u4e4c\u9f9f **su\u014d t\u00f3u w\u016bgu\u012b** ( _swuh toe ooh gway_ ), \"a turtle with its head in its shell,\" meaning \"coward.\"\n\n\u738b\u516b **w\u00e1ngb\u0101** ( _wahng bah_ )\n\nCuckold, bastard, asshole, piece of shit.\n\n\u738b\u516b\u86cb **w\u00e1ngb\u0101d\u00e0n** ( _wahng bah dun_ )\n\nSon of a bitch, bastard. Literally \"tortoise egg.\"\n\n\u738b\u516b\u7f94\u5b50 **w\u00e1ngb\u0101 g\u0101ozi** ( _wahng bah gaow dz_ )\n\nSon of a bitch, bastard. Literally \"son of a turtle\" and a northern variation on **w\u00e1ng b\u0101 d\u00e0n**.\n\n\u738b\u516b\u728a\u5b50 **w\u00e1ngba d\u00fazi** ( _wahng bah doo dz_ )\n\nSon of a bitch, bastard. Literally \"turtle stomach,\" probably alluding to the pregnant belly of a cuckold's wife, suggesting, like **w\u00e1ngb\u0101d\u00e0n** , \"turtle's egg,\" that the target of the insult doesn't know who his father is.\n\n\u9f9f\u513f\u5b50 **gu\u012b \u00e9r zi** ( _gway er dz_ )\n\nSon of a bitch, bastard. Literally \"son of a turtle.\" A variation on **w\u00e1ngb\u0101d\u00e0n** used only in southern China.\n\n\u9f9f\u5b59\u5b50 **gu\u012b s\u016bnzi** ( _gway swen dz_ )\n\nSon of a bitch, bastard. Literally \"turtle's grandson.\" Another variation on **w\u00e1ngb\u0101d\u00e0n** used only in southern China.\n\n#  **Dog-related swears and insults**\n\n\u72d7\u5d3d\u5b50 \/ \u72d7\u4ed4\u5b50 **g\u01d2uz\u01ceizi** ( _go dzigh dzz_ )\n\nSon of a bitch (although a bit milder than the English). Literally \"son of a dog.\"\n\n\u72d7\u5a18\u517b\u7684 **g\u01d2u ni\u00e1ng y\u01ceng de** ( _go nyahng yahng duh_ )\n\nSon of a bitch (rude\u2014more so than the previous entry). Literally, can mean \"raised by a dog mother\" or \"born of a dog mother.\"\n\n\u72d7\u65e5\u7684 **g\u01d2ur\u00ecde** ( _go rih duh_ )\n\nSon of a bitch (rude). Literally, can mean \"fucked by a dog\" or \"born of a mother fucked by a dog.\"\n\n\u72d7\u6742\u79cd **g\u01d2u z\u00e1zh\u01d2ng** ( _go dzah dzohng_ )\n\nLiterally \"mongrel dog,\" a variation on \u6742\u79cd **z\u00e1zh\u01d2ng** ( _dzah dzohng_ ), another insult meaning \"mixed blood.\" Extremely rude.\n\n#  **Shit**\n\n\u5c4e **sh\u01d0** ( _shih_ )\n\nShit (noun), like shit (adverb), shitty (adjective). You would use this to describe things. For example, you might say \u592a\u5c4e\u4e86 **t\u00e0i sh\u01d0 le** ( _tie shih luh_ ), literally \"too shitty,\" to say that something was shitty or bad. Sometimes written \"10\" online because both are pronounced _shih_.\n\n\u72d7\u5c4e **g\u01d2ush\u01d0** ( _go shih_ )\n\nBullshit. Literally \"dog shit.\" This term was originally used to describe people of low moral character. This new usage is probably due to Western influence\u2014it started out in Hong Kong and Taiwan, where \"Oh shit!\" in Hollywood movies was often subtitled **g\u01d2ush\u01d0** , and spread from there.\n\n\u81ed\u72d7\u5c4e **ch\u00f2u g\u01d2ush\u01d0** ( _choe go shih_ )\n\nStronger form of above. Literally \"smelly dog crap.\"\n\n\u5c4e\u76c6\u5b50 **sh\u01d0 p\u00e9nz\u01d0** ( _shih pen dz_ )\n\nShitty job, the blame for doing a shitty job. Literally \"crap pot.\" You might say, for example, that someone gave you the crap pot, meaning that they made you take the blame for a shitty job. Or you can describe something directly as a crap pot, meaning that it was done poorly.\n\n\u5403\u5c4e **ch\u012b sh\u01d0** ( _chih shih_ )\n\nEat shit. Equivalent to \"fuck off,\" though much less profane.\n\n\u7caa **f\u00e8n** ( _fen_ )\n\nFeces (formal term). You wouldn't say this alone as an adjective, like \"shitty,\" but one common Beijing expression is \u81ed\u5927\u7caa **ch\u00f2u d\u00e0 f\u00e8n** ( _choe dah fen_ ), literally \"stinky big stool,\" meaning that something is shitty or worthless.\n\n\u5927\u4fbf **d\u00e0bi\u00e0n** ( _dah byinn_ )\n\nExcrement, poop, defecate (both noun and verb). Literally \"big relieving of oneself\"\u2014urination, by the way, is \u5c0f\u4fbf **xi\u01ceobi\u00e0n** ( _dah byinn_ ), \"small relieving of oneself.\" Not an expletive, and thus does not have the same effect as \"shit.\" For the most part, this refers to the actual act of defecation and its product, but can be used mockingly, in a silly and unserious way, like calling someone a poop. Girls in particular use this teasingly.\n\n\u53bb\u5403\u5927\u4fbf **q\u00f9 ch\u012b d\u00e0bi\u00e0n** ( _chee chih dah byinn_ )\n\nGo eat excrement; go eat poop. Similar to saying \"get lost.\" Sounds mild and silly, so it's mostly used by girls in a teasing way.\n\n#  **Writing**\n\n\u50bb X **sh\u01cech\u0101** ( _shah chah_ )\n\nThis is often used in written Chinese to stand in for a dirty word. Literally \"stupid X.\" X is pronounced _chah_ in Chinese.\nCHAPTER FOUR\n\n**Men and Women: Flirting, Dating, Love, and Marriage**\n\n**I** t's hard to be in China for long without noticing the prevalence of not terribly attractive Western men who seem to have inexplicably landed themselves a gorgeous Chinese girlfriend. Though I hesitate to offer any explanation for this phenomenon, I do find myself recalling an awful lot of conversations with Chinese gal-pals about their inability to tell westerners apart. I may or may not have told some of these women to be sure and introduce me to any prospective beaus before taking the plunge, so that I could distinguish for them the handsome men from the mugs that even a mother wouldn't love.\n\nThen again, maybe these women know exactly what they've gotten themselves into, as more than a few pragmatically minded Chinese women have counseled me on the importance of choosing someone \"bald and fat\"\u2014the reasoning being that such mates will be less likely to cheat or leave you. And on top of all that, you never know what seemingly smoking hot girl is considered downright homely by Chinese beauty standards. I have pointed out plenty of what I thought to be beautiful women, only to have Chinese friends reply musingly, \"Yes, it's strange, my other Western friends think she's hot too. We all think she looks like a peasant.\"\n\nBut perhaps we should just be glad that cross-cultural blindness is enabling people everywhere to get laid. In this chapter, you'll find all the vocabulary you need for flattering, cajoling, and hopefully landing a date\u2014or more. But read with caution: dating in China is a whole new ball game. The essential thing to know is this: the woman wears the pants in the relationship. Though we certainly can't stereotype every Chinese woman this way, should you as a Western male choose to embark upon a relationship with one, be prepared to pay for everything (possibly including her rent), call and text her ten times a day (being tied up in meetings at work all day is no excuse), secure her permission whenever you want to go out with the guys (it will not be granted), and always, always, carry her purse (no matter how shiny, pink, or Hello Kitty bedecked it may be). And should you Western women desire to learn the ways of Chinese dating, you'd better brush up on the art of \u6492\u5a07 **s\u01ceji\u0101o** ( _sah jow_ )\u2014a common, whiney way of acting that most westerners find maddening and Chinese presumably find cute\u2014which essentially involves pouting a lot, speaking in the voice of a five-year-old, hitting your boyfriend a lot whilst calling him \"so bad,\" and, of course, making him carry your purse.\n\n#  **Finding Love**\n\n\u8c03\u60c5 **ti\u00e1oq\u00edng** ( _tyow cheeng_ )\n\nTo flirt. Literally \"throw feelings.\"\n\n\u6311\u9017 **ti\u01ceod\u00f2u** ( _tyow doe_ )\n\nTo flirt. Literally \"incite and tease.\"\n\n\u6253\u60c5\u9a82\u4fcf **d\u01ce q\u00edng m\u00e0 qi\u00e0o** ( _dah ching ma chyow_ )\n\nA literary way to say \"flirt\" or \"banter flirtatiously.\" Literally \"hit passionately, scold prettily\" referring to the expression \u6253\u662f\u4eb2\u9a82\u662f\u7231 **d\u01ce sh\u00ec q\u012bn m\u00e0 sh\u00ec \u00e0i** ( _dah shih cheen ma shih aye_ ), which translates to something like \"hitting is intimacy and yelling is love.\"\n\n\u5403\u8c46\u8150 **ch\u012b d\u00f2ufu** ( _chih doe foo_ )\n\nCop a feel. Literally \"eat tofu.\" When used between people of the same sex, it can mean \"to bully,\" either verbally or physically. Used mainly in southern China, Taiwan, and Hong Kong, though northerners generally know the phrase as well. Relatedly, \"sell tofu,\" \u5356\u8c46\u8150 **m\u00e0i d\u00f2ufu** ( _my doe foo_ ), is a southern Chinese euphemism for prostitution.\n\n\u6ce1\u599e **p\u00e0on\u012bu** ( _pow nyoo_ )\n\nOne of the most common slang terms for \"hitting on,\" \"flirting with,\" or \"hooking up with\" girls. Literally \"soak a girl.\"\n\n\u9493\u51ef\u5b50 **di\u00e0o k\u01ceizi** ( _dyow kigh dz_ )\n\nTo pick up men, to hit on a man. Literally \"fish for men\" or \"fish for a boyfriend.\" Originated in Taiwan and Hong Kong but known and used everywhere.\n\n\u642d\u8baa **d\u0101 sh\u00e0n** ( _dah shahn_ )\n\nTo chat someone up, to start up a conversation.\n\n\u620f\u679c\u620f\u5b59 **x\u00ec gu\u01d2 x\u00ec s\u016bn** ( _she gwuh she swen_ )\n\nBeijing slang for \"chasing girls and boys,\" or for people who go to bars with the express intent of finding a guy or girl to hook up with. \u679c **Gu\u01d2** ( _gwuh_ ) means \"fruit\" but is Beijing slang for \"chicks\" because with a Beijing accent it's pronounced like \u679c\u513f **gu\u01d2r** ( _gwerr_ ). \u620f\u679c **X\u00ec gu\u01d2** ( _she gwuh_ ) literally means \"play with chicks\" or \"trick girls\" and is an old Beijing expression that means flirting with or hitting on girls. \u620f\u5b59 **X\u00ec s\u016bn** ( _she swen_ ) literally means \"trick boys\" or \"play with boys\" and means hitting on or flirting with guys.\n\n\u78d5\u871c **k\u0113 m\u00ec** ( _kuh me_ )\n\nBeijing slang for \"chasing\" or \"dating\" women. Literally \"hunt honey.\" Not as popular as **x\u00ec gu\u01d2** (above).\n\n\u624e\u871c **zh\u0101 m\u00ec** ( _jah me_ )\n\nBeijing slang for \"chasing women.\" Literally \"fool around [with] honey.\" Not as popular as **x\u00ec gu\u01d2** (page 56).\n\n\u6c42\u7231 **qi\u00fa\u00e0i** ( _chyoe aye_ )\n\nWoo (intransitive verb). Literally \"plead for love.\"\n\n\u8ffd\u6c42 **zhu\u012bqi\u00fa** ( _jway chyoe_ )\n\nTo pursue (transitive verb).\n\n\u5782\u6d8e\u4e09\u5c3a **chu\u00edxi\u00e1n s\u0101n ch\u01d0** ( _chway shin sahn chih_ )\n\nLiterally \"drool three feet.\" Said of something appealing that makes you drool. Mainly used in reference to food but can also be said of a girl.\n\n\u7709\u6765\u773c\u53bb **m\u00e9i l\u00e1i y\u01cen q\u00f9** ( _may lie yen chee_ )\n\nLiterally \"eyebrows coming and eyes going.\" Describes flirtatious eye contact or just flirting in general.\n\n\u7535\u773c **di\u00e0n y\u01cen** ( _dyinn yen_ )\n\nLiterally \"electric eyes.\" A popular term among young people to describe beguiling eyes\u2014that is, eyes that give you an electric spark.\n\n\u8fc7\u7535 **gu\u00f2 di\u00e0n** ( _gwuh dyinn_ ) or \u653e\u7535 **f\u00e0ng di\u00e0n** ( _fahng dyinn_ )\n\nTo have an electric shock (in the sense of being attracted to someone). Literally \"release electricity\" and \"pass electricity,\" respectively. Can also mean to knock the table with your glass when toasting, instead of clinking glasses.\n\n\u6765\u7535 **l\u00e1i di\u00e0n** ( _lie dyinn_ )\n\nRomantic spark. Literally \"electricity comes.\" Not having any chemistry would be \u4e0d\u6765\u7535 **b\u00f9 l\u00e1i di\u00e0n** ( _boo lie dyinn_ ), literally \"electricity doesn't come.\"\n\n\u6492\u5a07 **s\u01ceji\u0101o** ( _sah jow_ )\n\nTo throw a fit, to act like a brat, to act coquettishly. The key thing to note in this definition is that acting coquettishly\u2014that is, acting in a way that attracts male attention\u2014is synonymous with acting like a brat.\n\n\u800d\u5355\u513f **shu\u01ce d\u0101nr** ( _shwah dar_ )\n\nLiterally \"play alone,\" meaning \"single\" or \"unmarried,\" but also Beijing slang for dressing skimpily even though it's cold out, just to look cute.\n\n\u6f02\u4eae **pi\u00e0oliang** ( _pyow lyahng_ )\n\nPretty. Can be said of someone who's actually pretty, like the girl next door, but is also said in response to anything impressive or amazing. This word is often used in sports\u2014when a soccer player scores a goal, a westerner might say \"Nice!\" but in Chinese you often hear the sports commentators yell, \"Pretty!\"\n\n\u7f8e **m\u0115i** ( _may_ ) or \u7f8e\u4e3d **m\u0115il\u00ec** ( _may lee_ )\n\nBeautiful, good-looking. Can describe both people and things.\n\n\u7f8e\u5973 **m\u0115i n\u01da** ( _may nee_ )\n\nBeautiful girl, beautiful woman. Often used as a flattering term of address.\n\n\u751c **ti\u00e1n** ( _tyinn_ )\n\nSweet. Can describe either food or girls. Its meaning differs from the English meaning \"extremely nice and thoughtful\" in that it also connotes \"cutesy.\" For example, girls from Taiwan are described by mainlanders as sounding very **ti\u00e1n** because their accents sound girlish and cute to Chinese outside of Taiwan. A sweet girl is a \u751c\u599e **ti\u00e1n n\u012bu** ( _tyinn nyoo_ ).\n\n\u53ef\u7231 **k\u011b'\u00e0i** ( _kuh aye_ )\n\nCute.\n\n\u5361\u54c7\u4f9d **k\u01cew\u0101y\u012b** ( _kuh why ee_ )\n\nBased on the Japanese word _kawaii_ , meaning \"cute\" or \"cutesy.\" Tends to be used more often in Taiwan and other areas more strongly influenced by Japanese culture.\n\n\u5a03\u5a03 **w\u00e1wa** ( _wah wah_ \u2014that should be a short _a_ ; rhymes with \"tra-la-la\")\n\nCute girl. Literally \"baby\" or \"doll.\"\n\n\u7edd\u8272 **ju\u00e9s\u00e8** ( _dreh suh_ )\n\nAn extremely beautiful appearance (used as a noun). Literally \"special color.\" Used frequently on the Internet.\n\n\u9753 **li\u00e0ng** ( _lyahng_ )\n\nPretty or handsome. Literally \"light\" or \"glowing\" or \"bright.\" A pretty girl is a \u9753\u5973 **li\u00e0ng n\u01da** ( _lyahng nee_ ), literally \"glowing girl,\" and a handsome young man is a \u9753\u4ed4 **li\u00e0ng z\u01cei** , literally \"pretty boy.\" Used in southern China.\n\n\u8ff7\u4eba **m\u00edr\u00e9n** ( _me ren_ )\n\nFascinating, enchanting, charming, tempting (usually describing a female). Literally \"attracts people.\"\n\n\u4e30\u6ee1 **f\u0113ngm\u01cen** ( _fung mahn_ )\n\nVoluptuous, buxom. Literally \"plentiful and full.\"\n\n\u8eab\u6750 **sh\u0113nc\u00e1i** ( _shen tsigh_ )\n\nFigure, body. Literally \"body shape.\"\n\n\u6761 **ti\u00e1o** ( _tyow_ )\n\nFigure, shape (usually describing women). Ordinarily **ti\u00e1o** is a word that indicates reference to anything with a long and thin shape, like a stick or a noodle.\n\n\u6027\u611f **x\u00ecngg\u01cen** ( _sheeng gahn_ )\n\nSexy.\n\n\u5996\u5a9a **y\u0101om\u00e8i** ( _yow may_ )\n\nSexy, enchanting. Literally \"evil charming.\"\n\n\u5996\u91cc\u5996\u6c14 **y\u0101o l\u01d0 y\u0101o q\u00ec** ( _yow lee yow chee_ )\n\nSeductive and bewitching, sexy.\n\n\u5996\u7cbe **y\u0101ojing** ( _yow jing_ )\n\nAlluring woman, siren. Literally \"evil spirit.\"\n\n\u5c16\u679c\u513f **ji\u0101n gu\u01d2r** ( _jinn gwurr_ )\n\nBeijing slang for a hot girl. Literally \"sharp girl.\"\n\n\u5c16\u5b59\u513f **ji\u0101n s\u016bnr** ( _jinn swurr_ )\n\nBeijing slang for a hot guy. Literally \"sharp guy.\"\n\n\u8fa3\u59b9 **l\u00e0m\u00e8i** ( _lah may_ )\n\nSouthern Chinese slang for a hot girl. Literally \"spicy sister.\"\n\n\u60f9\u706b **r\u011bhu\u01d2** ( _ruh hwuh_ )\n\nSexy, hot (female). Popular among young people.\n\n\u5e05 **shu\u00e0i** ( _shwhy_ \u2014think of it as \"shh\" and \"why\" mushed into one syllable)\n\nHandsome. Literally \"leader in battle.\" Describes men, but like \"pretty,\" **pi\u00e0oliang** (page 59), this can also be said in response to an impressive spectacle. You can make the sentiment stronger by saying \u5f88\u5e05! **h\u011bn shu\u00e0i!** ( _hun shwhy_ ), literally \"very handsome,\" or \u5e05\u5446\u4e86 **shu\u00e0i d\u0101i le** , literally \"stunningly handsome.\"\n\n\u5e05\u54e5 **shu\u00e0i g\u0113** ( _shwigh guh\u2014_ **shu\u00e0i** rhymes with \"high\")\n\nHandsome man. Literally \"handsome older brother.\" Often used as a flattering form of address for any good-looking young man.\n\n\u58ee **zhu\u00e0ng** ( _jwong_ )\n\nBuff, strong. Beijingers pronounce it using third tone instead of fourth tone\u2014 **zhu\u01ceng** ( _jwong_ ).\n\n\u9177 **k\u00f9** ( _coo_ )\n\nCool (a transliteration from the English). A common way to describe a guy you find attractive, and more likely to be used by young people today than \u5e05 **shu\u00e0i** ( _shwigh_ ), which means \"handsome.\"\n\n\u68d2 **b\u00e0ng** ( _bahng_ \u2014almost like \"bong\" but with an _ahh_ sound replacing the _o_ )\n\nCapable, strong, awesome. Often said when praising someone. If a child does a good job cleaning his or her room, you can say \u5f88\u68d2! **h\u011bn b\u00e0ng!** ( _hun bahng_ ), literally \"very great.\" Or if a girl has a really rockin' body, you can say, \"\u4ed6\u7684\u8eab\u6750\u5f88\u68d2!\" \" **T\u0101 de sh\u0113nc\u00e1i h\u011bn b\u00e0ng!** \" ( _tah duh shen tsigh hun bahng_ ): \"Her body is awesome!\"\n\n\u500d\u513f\u68d2 **b\u00e8ir b\u00e0ng** ( _burr bahng_ )\n\nReally great, really awesome. This is how a Beijing or Tian jin local might express \u68d2 **b\u00e0ng** ( _bahng_ ) **.** (\u500d **B\u00e8i** means \"multiply\" and the\u513f **er** ( _er_ ) sound indicates a Beijing accent.) Thus a Beijinger complimenting a girl's body might say, \"\u4f60\u7684\u8eab\u6750\u500d\u513f\u68d2!\" \" **N\u012d de sh\u0113nc\u00e1i b\u00e8ir bang!** \" ( _nee duh shen tsigh burr bahng_ ): \"You have a rockin' body.\"\n\n\u517b\u773c **y\u01cengy\u01cen** ( _yahng yen_ )\n\nEye candy, good-looking, beautiful, easy on the eyes. Literally \"fits eye.\"\n\n\u559c\u6b22 **x\u01d0hu\u0101n** ( _shee hwun_ )\n\nTo like.\n\n\u7231 **\u00e0i** ( _aye_ \u2014rhymes with \"sigh\")\n\nTo love.\n\n\u75bc **t\u00e9ng** ( _tung_ ) or \u75bc\u7231 **t\u00e9ng \u00e0i** ( _tung aye_ )\n\nLove (verb). Can be used both for romantic love and for parental or familial love. Can also mean \"to spoil,\" as in spoiling a beloved child. **T\u00e9ng** can also mean \"pain,\" which you shouldn't read into too deeply, but knowing it should give extra depth to this way of saying love.\n\n\u6697\u604b **\u00e0nli\u00e0n** ( _ahn lyinn_ )\n\nTo have a crush on.\n\n\u8c08\u604b\u7231 **t\u00e1nli\u00e0n'\u00e0i** ( _tahn lyinn aye_ )\n\nTo date, to have a relationship with. Literally \"talk about love\" or \"talk romance.\"\n\n\u7ea6\u4f1a **yu\u0113hu\u00ec** ( _yreh hway_ )\n\nA date (noun). Came about due to Western influence.\n\nAA \u5236 **AA zh\u00ec** ( _AA jih_ ) and AB \u5236 **AB zh\u00ec** ( _AB jih_ )\n\nLiterally \"AA system\" and \"AB system.\" \"Going Dutch\" when you eat out is often called **AA zh\u00ec** in Chinese. However, going Dutch is a relatively recent concept for Chinese people. More recently, some people (men dining out with women in particular) are choosing to split the bill but pay a bit more, say 70 percent, and this is called **AB zh\u00ec**.\n\n\u521d\u604b **ch\u016bli\u00e0n** ( _choo lyinn_ )\n\nFirst relationship, first love.\n\n\u5973\u670b\u53cb **n\u01dap\u00e9ngy\u01d2u** ( _nee pung yo_ )\n\nGirlfriend. A direct translation from the English word, and like the English it usually means the girl a guy is dating but sometimes merely refers to a female friend.\n\n\u7537\u670b\u53cb **n\u00e1np\u00e9ngy\u01d2u** ( _nahn pung yo_ )\n\nBoyfriend.\n\n\u9a6c\u5b50 **m\u01cezi** ( _mah dz_ )\n\nA slangy word for \"girlfriend.\" Originated in Hong Kong and was once used derogatorily (literally means \"horse\"), but now carries a positive connotation.\n\n\u51ef\u5b50 **k\u01ceizi** ( _kigh dz_ )\n\nA slangy term for \"boyfriend\" and the counterpart to **m\u01cezi** (above). Describes the ideal image of a boyfriend, as \u51ef **k\u01cei** ( _kigh_ ) means \"triumphant\" and connotes a hero victorious in battle.\n\n\u6211\u7231\u4f60 **w\u01d2 \u00e0i n\u01d0** ( _wuh aye nee_ )\n\nI love you.\n\n\u9e33\u9e2f **yu\u0101ny\u0101ng** ( _yren yahng_ )\n\nA pair of lovers. Also means Mandarin ducks, a frequent metaphor for lovers in classic literature.\n\n\u62cd\u62d6 **p\u0101itu\u014d** ( _pie twuh_ )\n\nCourting, dating, being in love, having an affair. Literally \"on patrol.\" Used in southern China.\n\n\u70ed\u604b **r\u00e8li\u00e0n** ( _ruh lyinn_ )\n\nTo be in the honeymoon phrase, head over heels.\n\n\u871c\u8fd0 **m\u00ecy\u00f9n** ( _mee yreen_ )\n\nDating seriously. Literally \"honey luck.\" When a man and woman are in a relationship likely headed toward marriage, young people might say that they are in **m\u00ecy\u00f9n** , or struck by \"honey luck.\" The term is a play on the word \"honeymoon,\" \u871c\u6708 **m\u00ecyu\u00e8** ( _mee yreh_ ), because they sound similar.\n\n\u7231\u79f0 **\u00e0ich\u0113ng** ( _aye chung_ )\n\nLiterally \"love name.\" An affectionate nickname, like \"baby\" or \"snookums.\" A few common Chinese \"love names\" are \u5b9d\u8d1d **b\u01ceob\u00e8i** ( _baow bay_ ), \"baby\" or \"treasure\"; \u4eb2\u7231\u7684 **q\u012bn\u00e0ide** ( _cheen aye duh_ ), \"dear\" or \"dearest\" or \"dear one\"; \u8001\u516c **l\u01ceogong** ( _laow gohng_ ), \"husband\" but more literally \"old husband\"; and \u8001\u5a46 **l\u01ceop\u00f3** ( _laow pwuh_ ), \"wife\" but more literally \"old wife.\"\n\n\u4e24\u5c0f\u65e0\u731c **li\u01cengxi\u01ceow\u00fac\u0101i** ( _lyahng shaow oo tsigh_ )\n\nTwo innocent child playmates (puppy love).\n\n\u9752\u6885\u7af9\u9a6c **q\u012bngm\u00e9izh\u00fam\u01ce** ( _cheeng may jooh mah_ )\n\nChildhood sweethearts. Literally \"green plums and a bamboo horse,\" which are both references to childhood, as green plums are not yet ripe, and the bamboo horse refers to a childhood game of pretending to ride horses using a bamboo stick.\n\n\u5149\u68cd\u8282 **Gu\u0101ng G\u00f9n Ji\u00e9** ( _gwahng gwen jyih_ )\n\nSingles Day. A holiday probably invented by a bunch of Chinese college students in Nanjing during the 1990s, and held on November 11 because of all the ones in the date (11\/11), which represent single people. On that date, at 11:11 p.m., male college students across China scream their desire for a girlfriend, bang on rice bowls with spoons, and otherwise make a lot of noise.\n\n\u5269\u5973 **sh\u00e8ngn\u01da** ( _shung nee_ )\n\nLiterally \"leftover woman.\" Refers to successful career women who have still not found a spouse, and who have passed an age that the Chinese consider ideal for getting married.\n\n\u4e09\u9690\u5973\u4eba **s\u0101n y\u01d0n n\u01dar\u00e9n** ( _sahn een nee ren_ )\n\nLiterally \"woman with three secrets.\" Refers to married women who, for whatever reasons, keep their marital status, age, and child a secret, leading everyone to think they're single.\n\n\u4e00\u89c1\u949f\u60c5 **y\u00ed ji\u00e0n zh\u014dng q\u00edng** ( _ee jinn johng cheeng_ )\n\nLove at first sight. Literally \"see once and love.\" \u953a\u60c5 **Zh\u014dng q\u00edng** ( _johng ching_ ) means love or like.\n\n\u7231\u5c4b\u53ca\u4e4c **\u00e0i w\u016b j\u00ed w\u016b** ( _aye ooh gee ooh_ )\n\nLiterally \"love house and bird.\" An expression meaning that when you love someone, you also love everything belonging to or associated with them. An equivalent English expression might be \"Love me, love my dog.\"\n\n\u7a7a\u7a97\u671f **k\u014dngchu\u0101ngq\u012b** ( _kohng chwahng chee_ )\n\nLiterally \"open-window period,\" referring to the window of time after a breakup when a person is up for grabs. Used especially in reference to someone that everyone wants.\n\n\u9ec4\u660f\u604b **hu\u00e1ngh\u016bnli\u00e0n** ( _hwahng hwen lyinn_ ) or \u5915\u9633\u604b **x\u012by\u00e1ngli\u00e0n** ( _she yahng lyinn_ )\n\nLiterally \"love at dusk\" or \"sunset love.\" A romance between two elderly people. The rising sun is an oft-used metaphor for youth, and conversely the elderly are associated with the setting sun.\n\n\u8de8\u56fd\u604b **ku\u00e0gu\u00f3li\u00e0n** ( _kwah gwuh lyinn_ )\n\nLiterally \"transnational love.\" Refers to a relationship between a Chinese person and a foreigner, or any intercultural relationship.\n\n\u625b\u6d0b\u67aa **k\u00e1ngy\u00e1ngqi\u0101ng** ( _kahng yahng chyahng_ \u2014the _ah_ in all three syllables indicates a short _a_ , as in \"ma\" or \"la\")\n\nLiterally \"shoulder foreign rifles.\" Used in the late nineteenth century to refer to Chinese people using items from overseas (pens, clothes, etc.) and now refers to Chinese women who date and\/or sleep with foreigners.\n\n\u7f51\u604b **w\u01cengli\u00e0n** ( _wahng lyinn_ )\n\nInternet dating, falling in love via the Internet.\n\n\u5e08\u751f\u604b **sh\u012b sh\u0113ng li\u00e0n** ( _shih shung lyinn_ )\n\nLiterally \"teacher-student love.\" A romantic relationship between a teacher and a student. Such relationships are extremely common in China, where it can be difficult for diligent students to meet romantic prospects.\n\n\u8001\u725b\u5403\u5ae9\u8349 **l\u01ceoni\u00fa ch\u012b n\u00e8nc\u01ceo** ( _low new chih nun tsow_ )\n\nA relationship between two people with a large age difference. Literally, \"The old cow eats fresh grass.\"\n\n\u59d0\u5f1f\u604b **ji\u011bd\u00ecli\u00e0n** ( _jyih dee lyinn_ )\n\nA relationship between an older woman and a much younger man. Literally \"older sister, younger brother love.\"\n\n\u8001\u5c11\u604b **l\u01ceosh\u00e0oli\u00e0n** ( _laow shaow lyinn_ )\n\nLove between people with a big age difference. Literally \"old-young love.\"\n\n\u5fd8\u5e74\u604b **w\u00e0ngni\u00e1nli\u00e0n** ( _wahng nyinn lyinn_ )\n\nBeing in love despite age differences. Literally \"forgetting-age love.\"\n\n\u4e24\u5730\u604b **li\u01cengd\u00ecli\u00e0n** ( _lyahng dee lyinn_ )\n\nLong-distance relationship.\n\n#  **Miscellaneous types**\n\n\u508d\u5927\u6b3e **b\u00e0ngd\u00e0ku\u01cen** ( _bahng dah kwahn_ )\n\nLiterally \"depend on a rich man\" or \"live off a rich man.\" Negatively describes a woman having an intimate relationship with a wealthy man who supports her (and who may or may not be already married).\n\n\u5473\u9053\u7f8e\u5973 **w\u00e8id\u00e0o m\u0115in\u01da** ( _way dow may nee_ )\n\nHot waitress. Literally \"delectable beauty.\"\n\n\u9aa8\u5934\u8f7b **g\u01d4t\u00f3u q\u012bng** ( _goo toe ching_ )\n\nBimbo, airhead. Literally \"light bones.\"\n\n\u6d6a **l\u00e0ng** ( _lahng_ )\n\nNorthern Chinese slang for \"shallow,\" \"airheaded,\" or \"flighty\" (describing women). Can also mean \"to stroll\" or \"to wander.\"\n\n\u7ee3\u82b1\u6795\u5934 **x\u00ecuhu\u0101 zh\u011bntou** ( _show hwa jen toe_ )\n\nLiterally \"embroidered pillow,\" meaning something or someone that is beautiful but useless.\n\n\u8001\u6765\u4fcf **l\u01ceo l\u00e1i qi\u00e0o** ( _laow laow tsie_ )\n\nAn older person who dresses young. Literally \"[from] old to pretty.\"\n\n\u88c5\u5ae9 **zhu\u0101ng n\u00e8n** ( _jwahng nun_ )\n\nLiterally \"pretending to be tender\" or \"faking softness\" and describing someone who speaks girlishly, dresses young, and\/or otherwise behaves much younger than he or she is.\n\n\u6821\u82b1 **xi\u00e0ohu\u0101** ( _shyaow hwa_ )\n\nLiterally \"school flower.\" Equivalent to the head of the cheerleading team\u2014the most popular and desired girl in school.\n\n\u6821\u8349 **xi\u00e0oc\u01ceo** ( _shaow tsow_ \u2014both syllables rhyme with \"pow\")\n\nLiterally \"school grass.\" Equivalent to the high school quarterback\u2014the school hunk.\n\n\u540e\u751f **h\u00f2ush\u0113ng** ( _ho shung_ )\n\nYoung man. Literally \"born later.\" Used in southern China.\n\n\u6ee5\u60c5 **l\u00e0nq\u00edng** ( _lahn cheeng_ )\n\nSomething along the lines of a \"romantic\" crossed with a \"player\"\u2014someone who loves everyone he or she sees. Said of both men and women. Literally \"excessive feelings.\" Can be used as an adjective too, as in \u4ed6\u8fd9\u4eba\u633a\u6ee5\u60c5\u7684 **t\u0101 zh\u00e8 r\u00e9n t\u01d0ng l\u00e0nq\u00edng de** ( _tah juh ren ting lahn ching duh_ ): literally \"this person really has a lot of excessive feelings\" but meaning something more like \"this person's always falling in love with everyone he meets.\"\n\n\u5c0f\u767d\u8138 **xi\u01ceob\u00e1ili\u01cen** ( _shyow buy lyinn_ )\n\nLiterally \"little white face.\" Refers to a young, slightly effeminate or somewhat \"soft\"-looking man (hence the white face, which is considered an effeminate feature). May also connote that the young man depends on an older woman for money instead of working for a living.\n\n\u5403\u8f6f\u996d **ch\u012b ru\u01cenf\u00e0n** ( _chih rwun fun_ )\n\nLiterally \"eat soft rice.\" A negative expression for a man (of any age) who depends on his girlfriend or wife for a living. Mainly used in southern China.\n\n\u82b1 **hu\u0101** ( _hwah_ )\n\nAn adjective used to describe a \"player.\" Literally \"flower.\"\n\n\u82b1\u82b1\u516c\u5b50 **hu\u0101hu\u0101g\u014dngz\u01d0** ( _hwa hwa gong dz_ \u2014the _gong_ sound has a long _o_ , like \"oh\")\n\nA playboy (and also the Chinese name for _Playboy_ magazine). Literally \"flower prince.\"\n\n\u94bb\u77f3\u738b\u8001\u4e94 **zu\u00e0nsh\u00ed w\u00e1ng l\u01ceo w\u01d4** ( _dzwun shih wahng low ooh_ )\n\nLiterally \"diamond bachelor.\" A wealthy, older, eligible man. Used mainly in Hong Kong and Taiwan (where it's written \u947d\u77f3\u738b\u8001\u4e94).\n\n\u5355\u8eab\u8d35\u65cf **d\u0101nsh\u0113n gu\u00ec z\u00fa** ( _dan shen gway dzoo_ )\n\nLiterally \"unmarried nobility.\" \u5355\u8eab **D\u0101nsh\u0113n** ( _dan shen_ ) means \"single\" and many people now use this term with \"nobility\" added to convey the idea that there's nothing wrong with being single and it's something to be proud of, like something that noble people enjoy.\n\n\u6b6a\u74dc\u52a3\u67a3 **w\u0101i gu\u0101 l\u00ece z\u01ceo** ( _why gwah lyih dzow_ )\n\nLiterally \"crooked melons and split-open dates,\" referring to a group of unattractive people. For example, \u4ed6\u4eec\u5b66\u6821\u7684\u7537\u751f\u90fd\u662f\u4e9b\u6b6a\u74dc\u52a3\u67a3 **T\u0101 men xu\u00e9 xi\u00e0o de n\u00e1n sh\u0113ng d\u014du sh\u00ec xi\u0113 w\u0101i gu\u0101 li\u00e8 z\u01ceo** ( _tah men shreh shaow duh nahn shung doe shih shih why gwah lyih dzow_ ) means \"The boys at that school are all ugly.\"\n\n\u5a18\u5a18\u8154 **ni\u00e1ngniangqi\u0101ng** ( _nyahng nyahng chyahng_ )\n\nSissy, pansy, an effeminate man with a girly voice. Can imply that the person is gay, though can also be used teasingly with a close friend. Literally \"girly tone.\"\n\n\u5976\u6cb9\u5c0f\u751f **n\u01ceiy\u00f3u xi\u01ceosh\u0113ng** ( _nigh yo shaow shung_ )\n\nAn effeminate young man. Can be slightly derogatory, just like the word \"girly.\" Literally \"buttery young man\" or \"butter-boy.\"\n\n\u592b\u59bb\u76f8 **f\u016bq\u012b xi\u00e0ng** ( _foo chee shyung_ )\n\nLiterally \"husband-and-wife appearance.\" We often say that a husband and wife start to look like each other as time goes by. Somewhat along the same lines, many Chinese believe that a man and woman who share certain similar facial features will have a longer-lasting marriage, and thus match-makers might consider their **f\u016bq\u012b xi\u00e0ng** in debating their suitability for marriage.\n\n\u738b\u516b\u770b\u7eff\u8c46, \u770b\u5bf9\u773c\u4e86 **w\u00e1ngb\u0101 k\u00e0n l\u01dcd\u00f2u, k\u00e0n du\u00ec y\u01cen le** ( _wahng bah kahn lee dough, kahn dway yen luh_ )\n\nA joking expression that means two ugly people will find each other attractive. Literally \"a tortoise will gaze at two mung beans\" (because the eyes of a tortoise look like two mung beans).\n\n\u60c5\u4fa3\u886b **q\u00edngl\u01da sh\u0101n** ( _cheeng lee shahn_ ) or \u60c5\u4fa3\u88c5 **q\u00edngl\u01da zhu\u0101ng** ( _cheeng lee jwong_ )\n\nMatching couple outfits. Literally \"lovers' shirts\" or \"couples' outfits.\" Also \u5bf9\u886b **du\u00ec sh\u0101n** ( _dway shahn_ ), literally \"matching shirts.\" There is an inexplicable trend in China (and also South Korea) of couples wearing matching shirts. One (of many) of these T-shirt sets has an arrow pointing toward the girl, and the words, \"Falling in love, she is my girlfriend.\" The girl of course wears a shirt with an arrow pointing to the man, which says, \"Falling in love, he is my boyfriend.\" Moreover, these couples somehow manage to stay on the correct side of each other, always, while walking, sitting, shopping, and eating, so that the arrows are always pointing at each other.\n\n\u95ea\u7ea6 **sh\u01cenyu\u0113** ( _shan yreh_ )\n\nSpeed-dating. Literally \"flash appointment.\"\n\n#  **Marriage**\n\n\u95ea\u5a5a **sh\u01cenh\u016bn** ( _shahn hwen_ )\n\nLiterally \"flash marriage,\" describing couples who meet, fall in love, and get married very quickly.\n\n\u4e24\u5730\u5206\u5c45 **li\u01ceng d\u00ec f\u0113n j\u016b** ( _lyahng dee fen gee_ )\n\nLong-distance marriage. Literally \"in two different places\" or \"in two different cities.\" Such marriages have long been common in China due to a strict residence permit system that results in many people finding work in cities far from their spouse, though nowadays this situation is improving.\n\n\u91d1\u9f9f\u5a7f **j\u012bngu\u012b x\u00f9** ( _jean gway she_ )\n\nA rich husband. Literally \"golden turtle husband.\" A golden turtle was a status symbol denoting high rank for officials in the Tang dynasty (618-907).\n\n\u534a\u7cd6\u592b\u59bb **b\u00e0nt\u00e1ng f\u016bq\u012b** ( _bahn tahng foo chee_ )\n\nLiterally \"half-sweet couple.\" Couples who live apart during the work week and only spend weekends together, to keep the romance alive in their marriage. An increasingly common phenomenon among upper-middle-class professionals.\n\n\u8d70\u5a5a\u65cf **z\u01d2uh\u016bn z\u00fa** ( _dzoe hwen dzoo_ \u2014the first syllable rhymes with Joe but with the beginning sound like a _d_ and _z_ slurred together)\n\nLiterally \"walking marriage.\" Used to describe young Chinese couples in big cities who stay with their respective parents during the work week and live together only during the weekend.\n\n\u6025\u5a5a\u65cf **j\u00edh\u016bn z\u00fa** ( _gee hwen dzoo_ )\n\nLiterally \"hasty marriage group.\" A new term that describes people who marry hastily and not for love, especially young women who marry a wealthy man soon after graduating from college so they don't have to work.\n\n\u5f62\u5f0f\u7ed3\u5a5a **x\u00edngsh\u00ec ji\u00e9h\u016bn** ( _sheeng shih jyih hwen_ )\n\nA marriage of convenience\u2014for example, between a gay man and a lesbian.\n\n\u4e8c\u9505\u5934 **\u00e8rgu\u014dt\u00f3u** ( _er gwuh toe_ )\n\nLiterally \"second-pot head\" and the name of a brand of twice-distilled Chinese liquor. Also slang for a woman who remarries.\n\n#  **Love's downsides**\n\n\u62ac\u6760 **t\u00e1ig\u00e0ng** ( _tie gahng_ )\n\nBeijing slang for arguing for the sake of argument or for no reason. Also means being unreasonable in an argument, or deliberately picking a (verbal) fight. Literally \"lifting the pole,\" as in someone who keeps lifting up one end of the scale just to be higher than the other.\n\n\u5815\u5165\u60c5\u7f51 **du\u00f2 r\u00f9 q\u00edng w\u01ceng** ( _dwuh roo cheeng wahng_ )\n\nLovesick. Literally \"sink into love's net.\"\n\n\u9ea6\u82bd\u7cd6\u5973\u4eba **m\u00e0iy\u00e1t\u00e1ng n\u01dar\u00e9n** ( _migh yah tahng nee ren_ )\n\nLiterally \"malt sugar women.\" Refers to possessive women who demand that their boyfriends or husbands spend every second with them\u2014cling to them like sticky malt sugar.\n\n\u6c14\u7ba1\u708e **q\u00ecgu\u0103ny\u00e1n** ( _chee gwun yen_ )\n\nLiterally \"lung infection.\" Refers to a man who is so whipped that he never talks back to his girlfriend or wife, thus his friends might jokingly say he has a lung infection.\n\n\u89c1\u5149\u6b7b **ji\u00e0n gu\u0101ng s\u01d0** ( _gin gwahng sih_ )\n\nLiterally \"killed by exposure to light.\" Refers to two people who fall for each other via the Internet or phone dates, but whose would-be romance is sadly killed by the cold, harsh light of reality once they actually meet.\n\n\u79bb\u5a5a\u540c\u5c45 **l\u00edh\u016bn t\u00f3ngj\u016b** ( _lee hwen tohng gee_ )\n\nContinuing to live together after a divorce, either because one or both sides can't afford a new home or because they refuse to pay their ex for their half of the home they jointly owned before the divorce.\n\n\u65ad\u80cc\u5a5a\u59fb **du\u00e0nb\u00e8i h\u016bny\u012bn** ( _dwun bay hwen een_ )\n\nLiterally \"brokeback marriage,\" after the Ang Lee movie _Brokeback Mountain_. Refers to a marriage in which one side is gay and\/or has had a gay affair.\n\n\u79c1\u623f\u94b1 **s\u012bf\u00e1ngqi\u00e1n** ( _sih fahng chyinn_ )\n\nLiterally \"private house money.\" Refers to the secret stash of money that a wife puts aside in case her husband leaves her. Also refers to the money that a husband secretly puts outside of his wife's reach because he's whipped and is expected to give her all the money he earns.\n\n\u5403\u918b **ch\u012bc\u00f9** ( _chih tsoo_ )\n\nTo be jealous, to be envious. Literally \"to eat vinegar.\" A \u918b\u575b\u5b50 **c\u00f9 t\u00e1nzi** ( _tsoo tahn dz_ ), literally \"vinegar jar,\" is a jealous person.\n\n\u7a9d\u91cc\u6a2a **w\u014dl\u01d0h\u00e8ng** ( _wuh lee hung_ )\n\nLiterally \"unruly in the nest,\" referring to people who seem polite and civilized in public and only reveal their nastiness at home.\n\n\u4e09\u89d2\u604b **s\u0101nji\u01ceo li\u00e0n** ( _sahn jow lyinn_ )\n\nLove triangle.\n\n\u7231\u6068\u4ea4\u52a0 **\u00e0i h\u00e8n ji\u0101oji\u0101** ( _aye hun jow jah_ )\n\nLove-hate relationship.\n\n\u5fc3\u788e **x\u012bn su\u00ec** ( _sheen sway_ )\n\nBrokenhearted.\n\n\u53cd\u76ee\u6210\u4ec7 **f\u01cen m\u00f9 ch\u00e9ng ch\u00f3u** ( _fun moo chung cho_ )\n\nUtter hatred after a breakup.\n\n#  **Extramarital affairs**\n\n\u6709\u4e00\u817f **y\u014fu y\u00ec tu\u01d0** ( _yo ee tway_ )\n\nHave an affair. Literally \"has one leg,\" suggesting a man's leg intertwined with a woman's. Originated in Hong Kong or Taiwan but used everywhere.\n\n\u5288\u817f **p\u012b tu\u01d0** ( _pee tway_ )\n\nAffair, cheat, two-timing. Literally \"split legs.\" Also the technical term for a split in gymnastics. Commonly used in southern China.\n\n\u6234\u7eff\u5e3d\u5b50 **d\u00e0i l\u01dcm\u00e0ozi** ( _die lee mao dz_ )\n\nA cuckold, a man who is being cheated on. Literally \"wear a green hat,\" supposedly because male-brother workers during the Tang dynasty had to wear green hats. Because of this term, no Chinese man, and even many Chinese women, will wear green hats. One friend of mine found this out when he had to organize an office Christmas party, and all the Chinese in the office shot down his idea of dressing like elves, as it meant they'd have to wear green hats.\n\n\u8d1f\u5fc3\u6c49 **f\u00f9 x\u012bn h\u00e0n** ( _foo sheen hahn_ )\n\nCheater (referring to a man). Literally \"cheating man.\"\n\n\u5305\u4e8c\u5976 **b\u0101o\u00e8rn\u0103i** ( _bow er nigh_ \u2014the _bow_ sound rhymes with \"cow\") or just \u4e8c\u5976 **\u00e8rn\u0103i** ( _er nigh_ )\n\nLong ago, when Chinese men had multiple wives, **\u00e8rn\u0103i** referred to the second wife. Today it refers to the mistresses of wealthy men and government officials, an extremely common fact of life in China. Literally \"packaged second wife.\"\n\n\u508d\u5bb6\u513f **b\u00e0ng ji\u0101r** ( _bahng jer_ )\n\nMistress. Literally \"depend on home.\" Pejorative term for a young woman who has an affair with a rich married man. Used in Beijing only.\n\n\u699c\u80a9 **b\u00e0ng ji\u00e0n** ( _bahng jyinn_ )\n\nBeijing slang for an extramarital lover. Literally \"depend on shoulders.\"\n\n\u60c5\u513f **q\u00edngr** ( _churr_ )\n\nBeijing slang for an extramarital lover. Literally \"passion.\"\n\n\u5c0f\u8001\u5a46 **xi\u01ceol\u01ceop\u00f3** ( _shyaow laow pwuh_ )\n\nMistress. Literally \"little wife.\"\n\n\u871c **m\u00ec** ( _me_ ) or \u5c0f\u871c **xi\u01ceo m\u00ec** ( _shyow me_ )\n\nMistress. Literally \"honey\" or \"little honey.\"\n\n\u5a5a\u5916\u604b **h\u016bnw\u00e0ili\u00e0n** ( _hwen why lyinn_ )\n\nExtramarital love.\n\n\u5c0f\u4e09 **xi\u01ceo s\u0101n** ( _shyow sahn_ )\n\nLiterally \"little third.\" Refers to the \"third person\" in a relationship; i.e., the mistress.\n\n\u51fa\u4f4d **ch\u016b w\u00e8i** ( _choo way_ )\n\nLiterally \"overstep the mark.\" Describes a person who has had an extramarital affair. May also describe other situations when a person inappropriately \"oversteps the mark\"\u2014for example, job applicants who include sexually provocative photos of themselves in their resume (a not infrequent phenomenon, as it is legal for employers in China to require that applicants submit a photo, and many even require that girls be within a certain height and weight limit).\n\n\u6f02\u5a5a **pi\u0101o h\u016bn** ( _pyow hwen_ )\n\nA fake marriage, of sorts, between two people who are already married to other people, but who manage to establish a life like they are married in another town. This phenomenon is made easier by the fact that many Chinese wind up living and working far away from their spouses due to a strict residence permit system. Literally \"floating marriage.\"\n\n\u4e9a\u5077\u60c5 **y\u0101 t\u014du q\u00edng** ( _yah toe cheeng_ )\n\nLiterally \"second stolen feelings.\" Refers to married people who have an intense friendship with a friend of the opposite sex and are so close that they are practically having an affair, but without ever actually having sex.\nCHAPTER FIVE\n\n**Sex and Body Parts**\n\n**F** or much of China's four thousand years of history, sex was an open topic studied and perfected in great detail. Ancient Taoists believed that immortality (or, at the very least, longevity) could be achieved through sex, and medical texts dating as far back as the second century BC have been found to treat sex as an academic field of study. Specific sexual positions with names like Flying Dragon and Jumping Monkey were recommended to treat specific ailments, and illustrated sex manuals were kept by the bed for easy reference and sometimes even given as gifts to new brides. Women were considered to have an inexhaustible supply of yin, a sort of cosmic life essence, and men could replenish their own limited supply of that essence (in them called yang) by inducing a woman to multiple orgasm while refraining from ejaculating themselves, thereby drawing in the woman's energy. Yin and yang also represented female and male sexual parts. Some texts recommended sex with as many as ten different women in a night; others extolled the life-extending virtues of sex with virgins, ideally no older than fourteen.\n\nThe puritan ethics of, first, Confucianism (which, among its many other codes of conduct, dictated that a man and a woman should never touch in public) and, then, Communism (which kept marital sex hidden and outlawed extra-and premarital sex) effectively wiped out such open discourse. But ever since the period of \"reform and opening,\" beginning with the controversy surrounding new China's first published photo of a kiss in 1979, sex has steadily become a more and more open topic. Today there are sex shops all over the place, the pickup scene in bars late at night looks pretty much the same as anywhere else in the world, and men across the country, either suffering from erectile dysfunction or just wanting to perk up their sex lives, are snapping up all the caterpillar fungus, deer penis, and Viagra they can get.\n\nNonetheless, China officially remains an extremely conservative society when it comes to sex. Dating and public displays of affection are frowned upon or even outright banned on many college campuses; TV shows and movies depict women in their late twenties who still live at home in rooms filled with stuffed animals; and a Taiwanese actress who appeared in Ang Lee's NC-17-rated _Lust, Caution_ has been blackballed from the mainland media.\n\nHowever, one need only to stroll through the public parks of China at dusk, when every bench and shadow is occupied by ardent young couples strenuously making out (or more), to be reminded that despite the best efforts of China's sternest cock blockers, nothing will ever keep the birds and the bees apart. To that end, here are the words that help make China the world's most populous country.\n\n#  **Virginity**\n\n\u5904\u5973 **ch\u01d4n\u01da** ( _choo nee_ )\n\nVirgin (female).\n\n\u5904\u7537 **ch\u01d4n\u00e1n** ( _choo nahn_ )\n\nVirgin (male).\n\n\u9ec4\u82b1\u95fa\u5973 **hu\u00e1ng hu\u0101 gu\u012bn\u01da** ( _hwahng hwun gway nee_ )\n\nVirgin (female). Literally \"yellow flower girl,\" alluding to a fashion during the Song dynasty when girls would decorate their faces with yellow plum flowers. When these flowers were not in bloom, they used yellow paper cutouts instead.\n\n\u96cf **ch\u00fa** ( _choo_ )\n\nDirty and\/or insulting Beijing slang for a virgin. Literally \"chick\" or \"young bird.\" Before 1949 this was frequently used in brothels to refer to a young prostitute without much experience in entertaining the customers: \"chicken\" is slang for a prostitute, so a \"chick\" would be a young prostitute.\n\n\u7834\u96cf **p\u00f2 ch\u00fa** ( _pwuh choo_ )\n\nInsulting and\/or lewd Beijing slang for losing one's virginity. Literally \"break chick.\"\n\n\u7834\u5904 **p\u00f2 ch\u01d4** ( _pwuh choo_ )\n\nTo lose one's virginity. Literally \"break virginity.\"\n\n\u5f00\u5305 **k\u0101i b\u0101o** ( _kigh baow\u2014kigh_ rhymes with \"high\")\n\nBeijing slang for losing one's virginity. Literally \"open the package.\"\n\n#  **Lust**\n\n\u6027\u6b32\u51b2\u52a8 **x\u00ecngy\u00f9 ch\u014dngd\u00f2ng** ( _shing yee chohng dohng_ )\n\nArousal, sexual desire. \u6027\u6b32 **X\u00ecngy\u00f9** ( _shing yee_ ) can also be used by itself to mean \"lust\" or \"desire,\" while \u51b2\u52a8 **ch\u014dngd\u00f2ng** ( _chohng dohng_ ) alone is literally \"impulse\" but can mean horny or aroused.\n\n\u95f9\u6625 **n\u00e0o ch\u016bn** ( _now chren_ )\n\nSexual arousal, lust. Literally \"noise in springtime.\"\n\n\u6b32\u706b\u711a\u8eab **y\u00f9 hu\u01d2 f\u00e9n sh\u0113n** ( _yee hwuh fen shen_ )\n\nSexual arousal, lust. Literally \"the fire of lust is burning in the body.\"\n\n\u9965\u6e34 **j\u012b k\u011b** ( _gee kuh_ )\n\nLiterally \"hungry and thirsty.\" Can be used in many different contexts to mean that someone is hungry for something, such as knowledge. In a sexual context, it suggests that someone is starving for sex.\n\n\u53d1\u6625 **f\u0101ch\u016bn** ( _fah chren_ )\n\nHorny (the most straightforward equivalent to the English). Literally \"to develop spring\" or \"life\" or \"lust.\"\n\n\u53d1\u60c5 **f\u0101q\u00edng** ( _fah cheeng_ )\n\nHorny. Applies mostly to animals but sometimes said of people in a joking way. Literally \"develop passion.\"\n\n\u53d1\u9a9a **f\u0101s\u0101o** ( _fah saow_ \u2014 **s\u0101o** rhymes with \"cow\")\n\nA vulgar and impolite way to say \"horny.\" Literally \"to develop sexy\/slutty.\"\n\n\u53d1\u6d6a **f\u0101l\u00e0ng** ( _fah lahng_ )\n\nHorny (although more usually means \"slutty\"). Literally \"unrestrained.\" Not as current as the other three terms above, though still commonly used.\n\n#  **Masturbation**\n\n\u624b\u6deb **sh\u014fuy\u00edn** ( _show een_ )\n\nMasturbate, masturbation (technical term). Literally \"hand lewdness.\"\n\n\u81ea\u6170 **z\u00ecw\u00e8i** ( _dz way_ )\n\nMasturbate, masturbation (technical term). Literally \"self-comfort.\"\n\n\u81ea\u6e0e **z\u00ecd\u00fa** ( _dz do_ )\n\nMasturbate, masturbation (technical term). Literally \"self-abuse.\"\n\n\u6253\u98de\u673a **d\u0103f\u0113ij\u012b** ( _dah fay gee_ )\n\nThe most common slang term for \"jerk off\" or \"hand job.\" Literally \"shoot airplanes.\"\n\n\u73a9\u8001\u4e8c **w\u00e1n l\u01ceo\u00e8r** ( _wahn laow er_ )\n\nJerk off, hand job. Literally \"play with little brother.\"\n\n\u6253\u624b\u67aa **d\u0103 sh\u01d2uqi\u0101ng** ( _da show chyahng_ )\n\nJerk off, hand job. Literally \"fire the gun.\" More commonly used in southern China than in the North.\n\n\u624b\u63a8 **sh\u01d2utu\u012b** ( _show tway_ )\n\nJerk off, hand job. Literally \"hand push.\"\n\n\u64b8 **l\u016b** ( _loo_ )\n\nRefers to any sort of sliding or rubbing movement done on something else: for example the action of removing a ring from a finger. Can also be a euphemism for jerking off.\n\n\u5c04 **sh\u00e8** ( _shuh_ )\n\nCum, come. The most common, colloquial way to refer to \u5c04\u7cbe **sh\u00e8j\u012bng** ( _shuh jing_ ), which literally means \"shoot sperm\" and is the formal term for \"ejaculation\" (noun) or \"ejaculate\" (verb).\n\n\u6cc4\u7cbe **xi\u00e8j\u012bng** ( _shyih jing_ )\n\nPremature or accidental ejaculation. For example, during a wet dream or when about to have sex. Literally \"leak semen.\"\n\n\u7cbe\u6db2 **j\u012bngy\u00e8** ( _jing yeh_ )\n\nSemen.\n\nDIY\n\nFemale masturbation (said in English.) Short for \"do it yourself.\" Used more commonly among lesbians, though may refer to any sort of female masturbation.\n\n\u81ea\u6478 **z\u00ecm\u014d** ( _dz mwuh_ )\n\nFemale masturbation. Literally \"rub self.\" This is originally a mahjong term referring to a player taking a card from the stack on the table instead of from another player.\n\n\u5047\u9e21\u5df4 **ji\u0103 j\u012bba** ( _jah gee bah_ )\n\nDildo. Literally \"fake dick.\"\n\n\u7535\u52a8\u673a\u5df4 **di\u00e0nd\u00f2ng j\u012bba** ( _dyinn dohng gee bah_ )\n\nVibrator. Literally \"electric dick.\"\n\n#  **Foreplay**\n\n\u4eb2\u543b **q\u012bnw\u011bn** ( _cheen when_ ) and\u4eb2\u5634 **q\u012bnzu\u012d** ( _cheen dzway_ \u2014the second syllable is a tough one: pronounce a _z_ sound while your mouth and tongue are in the position to pronounce the letter _d_ and then continue into the word \"way\")\n\nKiss. Literally \"touch lips\" and \"touch mouths,\" respectively. \u4eb2 **Q\u012bn** ( _cheen_ ) or \u543b **w\u011bn** ( _when_ ) can also be used alone to mean \"kiss.\"\n\n\u820c\u543b **sh\u00e9w\u0115n** ( _shuh when_ )\n\nFrench kiss. Literally \"tongue kiss.\"\n\n\u5543 **k\u011bn** ( _ken_ )\n\nTo kiss in a deep, wild way involving a lot of teeth. Literally means \"nibble\" or \"gnaw.\"\n\n\u54ac\u54ac **y\u01ceoy\u01ceo** ( _yow yow_ )\n\nBite.\n\n\u79cd\u8349\u8393 **zh\u00f2ng c\u01ceom\u00e9i** ( _johng tsow may_ )\n\nGive a hickey. Literally \"plant a strawberry.\" Popular among people in their twenties or younger.\n\n\u5496\u55b1\u9e21 **g\u0101l\u00edj\u012b** ( _gah lee gee_ )\n\nHickey. Literally \"curry chicken.\"\n\n\u524d\u620f **qi\u00e1nx\u00ec** ( _chyinn she_ )\n\nForeplay.\n\n\u524d\u594f **qi\u00e1nz\u00f2u** ( _chyinn dzoe_ )\n\nForeplay. Literally \"prelude.\"\n\n\u5976\u7f69 **n\u01ceizh\u00e0o** ( _nigh jow_ )\n\nA more vulgar way of saying \"bra.\" It's not dirty and is fine to say among friends, but you wouldn't use this, for example, when speaking with a sales clerk in a lingerie store.\n\n\u4e01\u5b57\u88e4 **d\u012bngz\u00ec k\u00f9** ( _ding dz koo_ )\n\nThong. Literally \"Ding character underwear,\" or underwear shaped like the character for the Chinese surname Ding: \u4e01.\n\n#  **Breasts**\n\n\u6ce2\u6ce2 **b\u014db\u014d** ( _bwuh bwuh_ ) or just \u6ce2 **b\u014d** ( _bwuh_ )\n\nBoobs (based on the sound of the English word \"boobs\").\n\n\u54aa\u54aa **m\u012bmi** ( _me me_ )\n\nBoobs. **M\u012bmi** is also Chinese onomatopoeia for the sound of a cat's meow. Mainly used in southern China.\n\n\u5976\u5976 **n\u01cein\u01cei** ( _nigh nigh_ ) or \u5976\u5b50 **n\u01ceizi** ( _nigh dz_ )\n\nA cutesy\/childish way of saying breasts (like \"boobies\"). Used in Taiwan and southern China only: **n\u01cein\u01cei** means \"grandmother\" in northern China.\n\n\u6742\u513f **z\u00e1'er** ( _dzer_ )\n\nTits. Beijing slang only and considered quite dirty.\n\n\u9992\u5934 **m\u00e1nt\u00f3u** ( _mahn toe_ )\n\nNorthern Chinese slang for \"boobs.\" Literally \"steamed bun,\" a Chinese staple food that resembles a bread roll.\n\n\u5de8\u65e0\u9738 **j\u00f9w\u00fab\u00e0** ( _gee oo bah_ )\n\nThe Chinese name for a Big Mac and also slang for huge breasts.\n\n\u4e73\u5934 **r\u016dt\u00f3u** ( _roo toe_ )\n\nNipple. Literally \"breast head.\"\n\n\u4e73\u5c16 **r\u016dji\u0101n** ( _roo gin_ )\n\nTits (slightly vulgar).\n\n\u6ce2\u9738 **b\u014db\u00e0** ( _bwuh bah_ )\n\nA woman with huge breasts. A slightly comical term, as \u9738 **b\u00e0** ( _bah_ ) describes something overwhelming, and so the literal meaning is \"overwhelming breasts.\" From southern China.\n\n\u98de\u673a\u573a **f\u0113ij\u012bch\u01ceng** ( _fay gee chahng_ )\n\nFlat-chested woman. Literally \"airport\" (because an airplane runway is very flat). A frequently used term.\n\n\u592a\u5e73\u516c\u4e3b **t\u00e0ip\u00edng g\u014dngzh\u01d4** ( _tie peeng gohng joo_ )\n\nA teasing and frequently used term for a flat-chested girl. Literally \"very flat princess.\"\n\n\u6413\u677f **cu\u014db\u01cen** ( _tswuh bahn_ )\n\nLiterally \"washboard,\" referring to a skinny, flat-chested girl.\n\n\u4e73\u4ea4 **r\u01d4ji\u0101o** ( _roo jow_ )\n\nTitty fuck, boob sex.\n\n\u4e09\u70b9\u5168\u9732 **s\u0101n di\u01cen qu\u00e1n l\u00f2u** ( _san dyinn chren low_ )\n\nLiterally \"three points all showing,\" meaning full-frontal nudity. Often used to describe a naked woman.\n\n#  **Oral Sex**\n\n\u53e3\u4ea4 **k\u01d2uji\u0101o** ( _koe jow_ )\n\nOral sex (scientific term, not slang).\n\n\u53e3\u6d3b **k\u01d2uhu\u00f3** ( _koe hwuh_ )\n\nSlang for oral sex (both male and female). Among several other possible translations, can be literally interpreted as \"mouth life\" or \"mouth living\" or \"mouth work.\"\n\n\u5439\u7bab **chu\u012bxi\u0101o** ( _chway shaow_ )\n\nBlow job, give a blow job. Literally \"play the bamboo flute.\"\n\n\u5439\u53e3\u7434 **chu\u012b k\u01d2uq\u00edn** ( _chway koe cheen_ )\n\nA euphemism for giving oral sex to a woman. Literally \"play the harmonica.\" Not terribly common, but you will encounter at least a few people who have heard this. Even if they haven't, for Chinese ears this is one of those terms that just \"makes sense\" when you hear it.\n\n\u8214\u5c44 **ti\u01cen b\u012b** ( _tyinn bee_ )\n\nLick pussy. Not an actual term, exactly, but many people don't know of any slang for going down on a girl, and this is a way to discuss it that anyone will understand.\n\n\u8214\u8214\u4e0b\u8fb9 **ti\u01cen ti\u01cen xi\u00e0bi\u0101n** ( _tyinn tyinn shah byinn_ )\n\nLiterally \"lick below\" or \"lick down there.\" Again, not an actual term but a useful way to describe something for which there aren't any universally known slang terms (and less dirty than the previous entry).\n\n\u5439\u5587\u53ed **chu\u012b l\u01ceba** ( _chway lah bah_ )\n\nBlow job, give a blow job. Literally \"blow the horn.\"\n\n\u5403\u9999\u8549 **ch\u012b xi\u0101ngji\u0101o** ( _chih shyahng jow_ )\n\nBlow job, give a blow job. Literally \"eat banana.\" More common in Taiwan, Hong Kong, and southern China, and mostly unknown in the North.\n\n\u6307\u4ea4 **zh\u01d0ji\u0101o** ( _jih jow_ )\n\nFingering. Literally \"finger sex.\"\n\n\u6e56\u5439 **h\u00fachu\u012d** ( _who chway_ )\n\nFingering to wet climax. Literally \"lake blast.\" A term from Japanese porn, and not something you'll likely encounter in regular speech.\n\n#  **Sex!**\n\n\u6027 **x\u00ecng** ( _shing_ )\n\nSex.\n\n\u6027\u4ea4 **x\u00ecngji\u0101o** ( _shing jow_ )\n\nHave sex.\n\n\u6027\u7231 **x\u00ecng\u00e0i** ( _shing aye_ )\n\nA literary term for \"sexual love\" or \"passion.\"\n\n\u9c7c\u6c34\u4e4b\u6b22 **y\u00fa shu\u01d0 zh\u012b hu\u0101n** ( _ee shway jhh hwun_ \u2014 **zh\u012b** is tricky: say \"shh\" while your mouth is in the position for making a _j_ sound)\n\nA euphemism for sex in use since ancient times. Literally, \"The fish and the water are happy together,\" from the play _Romance of the West Chamber_ by the Yuan dynasty play-wright Wang Shifu (1260-1336).\n\nGut\u00f3u\n\n\u4e91\u96e8 **y\u00fany\u016d** ( _een yee_ )\n\nA euphemism for sex in use since ancient times. Literally \"clouds and rain.\" Based on an ancient Chinese conception of heaven and earth mating during a rain storm (an image frequently used in erotic Chinese literature). \"Rain\" represents the man's semen, and \"clouds\" the woman's vaginal secretions.\n\n\u7ffb\u4e91\u8986\u96e8 **f\u0101n y\u00fan f\u00f9 y\u01d4** ( _fun een foo yee\u2014_ there is no hard _y_ in the _yee_ sound; it sounds closer to _ee_ but with a very soft _y_ at the beginning: the mouth should only be partially in the position to pronounce a _y_ )\n\nA euphemism for sex in use since ancient times, with the same origins as the entry above. Literally \"turning clouds and overflowing water.\"\n\n\u884c\u623f **x\u00edng f\u00e1ng** ( _shing fahng_ )\n\nA euphemism for having sex. Literally, something like \"go to the room.\"\n\n\u540c\u623f **t\u00f3ngf\u00e1ng** ( _tohng fahng_ )\n\nA euphemism for \"sleeping together.\" Can also mean \"live together.\" Literally \"sharing a room\" or \"share a house.\"\n\n\u4ea4\u5408 **ji\u0101o h\u00e9** ( _jow huh_ )\n\n\"Get together.\"\n\n\u90a3\u4e2a **n\u00e8ig\u00e8** ( _nay guh_ )\n\nLiterally \"that\" but can serve as a euphemism for sex for anyone too embarrassed to use **zu\u00f2'\u00e0i** (page 93), and much like saying \"it\" or \"you know what\" in English. Thus you might say, \"\u5979\u5df2\u7ecf\u8ddf\u6211\u90a3\u4e2a\u4e86\" \" **T\u0101 y\u00edj\u012bng g\u0113n w\u01d2 n\u00e8ig\u00e8 le** \" ( _tah ee jing gehn tah nay guh luh_ ): \"She did it with me.\"\n\n\u505a\u7231 **zu\u00f2'\u00e0i** ( _zwuh aye_ )\n\nMake love.\n\n\u505a **zu\u00f2** ( _dzwuh_ )\n\nTo do, to make. A euphemism for having sex. Similar to \"doing\" someone or saying they \"do it\" in English.\n\n\u5f04 **n\u00f2ng** ( _nohng_ )\n\nDo, mess with.\n\n\u6536 **sh\u014du** ( _show_ )\n\nBeijing slang for \"have sex.\" Used similarly to \"do.\" Literally \"receive.\" Also a decent approximation for the term \"hook up.\"\n\n\u7092\u996d **ch\u01ceof\u00e0n** ( _chow fahn_ )\n\nLiterally \"fried rice.\" Taiwan slang for having sex (written \u7092\u98ef in Taiwan), because stir-frying rice involves a lot of flipping and turning.\n\n\u63d2 **ch\u0101** ( _cha_ )\n\nBang. A dirty way to say \"have sex.\" Literally \"insert.\"\n\n\u6253\u70ae **d\u01cep\u00e0o** ( _dah pow_ )\n\nLiterally \"blasting cannons\" or \"shooting shells.\" Dirty northern Chinese slang for having sex.\n\n\u5e72 **g\u0101n** ( _gahn_ )\n\nFuck, do. A dirty way to refer to sex. Some example usages include \u4f60\u5e72\u4e86\u5979? **N\u01d0 g\u0101n le t\u0101**? ( _nee gahn luh ta_ ), \"Did you do her?\" and \u4f60\u548c\u4ed6\u5e72\u4e86? **N\u01d0 h\u00e9 t\u0101 g\u0101n le**? ( _nee huh tah gahn luh_ ): \"Did you do it with him?\"\n\n\u641e **g\u01ceo** ( _gaow_ )\n\nScrew, fuck, do. A dirty word for sex, though less dirty than **g\u0101n** (above). Example usages: \u4ed6\u4e71\u641e\u7537\u5973\u5173\u7cfb! **T\u0101 lu\u00e0n g\u01ceo n\u01cenn\u01da gu\u0101nx\u00ec** ! ( _tah lwen gaow nahn nee gwahn she_ ), \"He's been screwing someone!\" (literally, \"He's been indiscriminately doing male-female relations\"); or \u6211\u8981\u641e\u5979! **W\u01d2 y\u00e0o g\u01ceo t\u0101!** ( _wuh yow gaow ta_ ): \"I want to fuck her!\"\n\n\u641e\u4e00\u4e0b **g\u01ceo y\u012bxi\u00e0** ( _gaow ee shah_ )\n\nLiterally \"do once\" or \"do for a short while.\" Can also be used similarly to \"hooking up,\" although there is no Chinese term that fully matches the English.\n\n\u808f **c\u00e0o** , more commonly written \u64cd **c\u0101o** (both pronounced _tsow_ )\n\nFuck. The character \u808f is visually quite graphic, as it is composed of \u5165 **r\u00f9** ( _roo_ ), \"enter,\" and \u8089 **r\u00f2u** ( _row_ ), \"meat.\" \u808f is technically the correct character for \"fuck,\" but because it is not included in most computer or phone-character input systems, and because it's just so uncomfortably dirty looking, most people write the homophonous \u64cd (which actually means \"hold\"). \u808f **C\u00e0o** \/ \u64cd **c\u0101o** is mainly used for swearing\u2014for example \u808f\u4f60\u5988! **C\u00e0o n\u01d0 m\u0101!** ( _tsow nee ma_ ), which means \"Fuck your mother!\" or \"Fuck you\"\u2014and not usually to describe actual sex. If it is used in the context of actual sex, it's extremely dirty and also very derogatory.\n\n\u65e5 **r\u00ec** ( _rih_ )\n\nSouthern Chinese slang for \"fuck.\" Its usage is the same as **c\u00e0o** above\u2014it is extremely dirty and used mainly for swearing, not to describe sex. Also used a bit in a few northern areas like Shan'xi and Shandong provinces.\n\n419\n\nOne-night stand (pronounced in English). Used because \"four one nine\" sounds like \"for one night,\" and also because the Chinese pronunciation of those numbers, **s\u00ecy\u0101oji\u01d4** ( _sih yow joe_ ), sounds similar to \u7761\u4e00\u5bbf **shu\u00ec y\u012b xi\u01d4** ( _shway ee show_ ), which means \"sleep one night.\" Used in mainland China only.\n\n\u5957\u5b50 **t\u00e0ozi** ( _taow dz_ ) or \u5957\u5957 **t\u00e0o tao** ( _taow taow_ )\n\nCutesy ways of saying \"condom.\"\n\n#  **Male genitalia**\n\n\u9634\u830e **y\u012bnj\u012bng** ( _een jing_ )\n\nPenis.\n\n\u9633\u5177 **y\u00e1ngj\u00f9** ( _yahng gee_ )\n\nDick. Literally \"sex tool.\" Used a lot in erotic writing and pornographic novels but not in speech.\n\n\u7389\u830e **y\u00f9j\u012bng** ( _yee jing_ )\n\nLiterally \"jade stalk.\" A poetic euphemism for \"penis,\" dating to ancient times. Not used in speech.\n\n\u9e21\u5df4 **j\u012bba** ( _gee bah_ )\n\nCock, dick. The most common slang word for a penis, and quite vulgar. Also used as an intensifier in the same way you'd use a swear word: for example, \"that **j\u012bba** whore\" is stronger than simply \"that whore.\"\n\n\u9e21\u9e21 **j\u012b ji** ( _gee gee_ ) or \u5c0f\u9e21\u9e21 **xi\u01ceo j\u012b ji** ( _shaow gee gee_ )\n\nA cutesy term for \"penis,\" used by little kids or jokingly between lovers. Literally \"chick\" and \"little chick,\" respectively. Often written \"JJ\" online.\n\n\u8001\u4e8c **l\u0103o'\u00e8r** ( _laow er_ \u2014rhymes with \"cow \" and \"her\")\n\nDick. Literally \"second brother\" or \"little brother.\"\n\n\u5c0f\u5f1f\u5f1f **xi\u0103o d\u00ec di** ( _shyow dee dee_ \u2014 **xi\u0103o** rhymes with \"cow \")\n\nA kiddie term for \"penis,\" much like \"peenie\" or \"wee-wee.\" Literally \"little brother.\"\n\n\u5c4c **di\u0103o** ( _dyow_ ), sometimes written \u540a **di\u00e0o** ( _dyow_ )\n\nDick, cock. Used more as an insult than to actually refer to a penis.\n\n\u9e1f **ni\u0103o** ( _nyow,_ rhymes with \"cow \")\n\nDick, cock. Literally \"bird.\" Less common than **di\u0103o** (above).\n\n\u547d\u6839\u5b50 **m\u00ecng g\u0113nzi** ( _ming gehn dz_ \u2014that's a hard _g_ in **g\u0113n** , like the _g_ in \"get\")\n\nEuphemism for \"penis,\" literally meaning \"root of life,\" \"lifeblood,\" or \"one's own life.\" An extremely formal and\/or literary term, but sometimes used jokingly in colloquial speech.\n\n\u5272\u5305\u76ae **g\u0113b\u0101op\u00ed** ( _guh baow pee_ )\n\nCircumcised, circumcision (which is rare in China). \"Are you circumcised?\" is \u4f60\u5272\u5305\u76ae\u6ca1\u6709? **N\u01d0 g\u0113b\u0101op\u00ed m\u00e9iy\u01d2u?** ( _nee guh baow pee may yo_ ).\n\n\u9e21\u5df4\u6bdb **j\u012bbam\u00e1o** ( _gee bah maow_ )\n\nLiterally \"dick hair.\" A crude term for male pubic hair. Used more as an insult than to actually refer to pubic hair, for which the neutral term is \u9634\u6bdb **y\u012bnm\u00e1o** ( _een mao_ \u2014 **m\u00e1o** rhymes with \"cow.\")\n\ny\u00fa shu zh hun\n\n\u9f9f\u5934 **gu\u012bt\u00f3u** ( _gway toe_ )\n\nScientific term for the glans, or head of the penis. Literally \"tortoise head.\"\n\n\u5305\u76ae **b\u0101op\u00ed** ( _baow pee_ \u2014 **b\u0101o** rhymes with \"cow\")\n\nForeskin. Literally \"wrapper\" or \"wrap skin.\"\n\n\u777e\u4e38 **g\u0101ow\u00e1n** ( _gaow wahn_ \u2014 **g\u0101o** rhymes with \"cow\" and **w\u00e1n** sounds like the \"won\" in \"wonton soup\")\n\nFormal term for testicles.\n\n\u86cb **d\u00e0n** ( _dahn_ ) or \u86cb\u86cb **d\u00e0n d\u00e0n** ( _dahn dahn_ )\n\nBalls. Literally \"eggs.\"\n\n\u5f97\u513f **de'r** ( _durr_ )\n\nBeijing slang for \"balls.\" This term is so colloquial that most Beijingers don't know how to write it, so don't be too concerned with the characters for this one.\n\n\u52c3\u8d77 **b\u00f3q\u01d0** ( _bwuh chee_ )\n\nHave an erection, erection.\n\n\u76f4 **zh\u00ed** ( _jih_ )\n\nHard. Literally \"straight\" or \"vertical\" or \"upright.\"\n\n\u786c **y\u00ecng** ( _eeng_ ) or \u786c\u4e86 **y\u00ecng le** ( _eeng luh_ )\n\nHard, get hard.\n\n\u94c1\u786c **ti\u011b y\u00ecng** ( _tyih eeng_ )\n\nRock hard (referring to an erection). Literally \"iron hard.\"\n\n#  **Female genitalia**\n\n\u9634\u9053 **y\u012bnd\u00e0o** ( _een dow_ )\n\nVagina. Literally \"female path.\"\n\n\u9634\u6237 **y\u012bnh\u00f9** ( _een who_ )\n\nPoetic\/literary term for the vagina. Literally \"female door.\"\n\n\u4e0b\u8eab **xi\u00e0sh\u0113n** ( _shah shen_ ) or \u4e0b\u4f53 **xi\u00e0t\u01d0** ( _shah tee_ )\n\nEuphemism for the female private parts. Literally \"lower body.\"\n\n\u5c0f\u59b9\u59b9 **xi\u01ceo m\u00e8imei** ( _shyaow may may_ )\n\nMost common slang term for \"vagina.\" Literally \"little sister.\"\n\n\u5c44 **b\u012b** ( _bee_ )\n\nCunt, pussy (extremely dirty). Often written \u903c instead, which means \"compel\" or \"close\" but is pronounced the same (and looks less dirty).\n\n\u5c44\u6bdb **b\u012bm\u00e1o** ( _bee maow_ )\n\nA dirty term for female pubic hair.\n\n\u767d\u864e **b\u00e1ih\u01d4** ( _by who_ )\n\nShaved pussy. Literally \"white tiger.\" In ancient times **b\u00e1ih\u01d4** meant a woman with no pubic hair, shunned by matchmak ers because it was believed that such a woman would bring misfortune or even death to her husband. The male equivalent is \u9752\u9f99 **q\u012bnglong** ( _cheen lohng_ ), literally \"green dragon,\" meaning a man without pubic hair, also superstitiously believed to bring trouble or death to his wife.\n\n\u9634\u8482 **y\u012bnd\u00ec** ( _een dee_ )\n\nClitoris.\n\n\u6e7f **sh\u012b** ( _shih_ )\n\nWet.\n\n\u51fa\u6c34 **ch\u016b shu\u01d0** ( _choo shway_ )\n\nRelease vaginal secretions, get wet. Literally \"discharge water.\"\n\n\u6deb\u6c34 **y\u00ednshu\u01d0** ( _een shway_ )\n\nCunt juice. Literally \"lewd juice.\"\n\n\u5c44\u6c34 **b\u012bshu\u012d** ( _bee shway_ )\n\nCunt juice. Literally \"pussy water.\"\n\n#  **Buttocks**\n\n\u5c41\u80a1 **p\u00ecg\u01d4** ( _pee goo_ )\n\nButt.\n\n\u815a **d\u00ecng** ( _ding_ )\n\nButt.\n\n\u80a1\u6c9f **g\u016dg\u014du** ( _goo go_ )\n\nButt crack.\n\n\u809b\u95e8 **g\u0101ngm\u00e9n** ( _gahng men_ )\n\nAnus.\n\n\u5c41\u773c **p\u00ecy\u01cen** ( _pee yen_ )\n\nAsshole (as in the slang term for \"anus\" but, unlike the English, is not used as an insult). Literally \"butt eye.\"\n\n\u83ca\u82b1\u95e8 **j\u00fahu\u0101m\u00e9n** ( _gee hwa men_ ) or just \u83ca\u82b1 **j\u00fahu\u0101** ( _gee hwa_ )\n\nAnus. Literally \"chrysanthemum door\" or just \"chrysanthemum,\" because the anus resembles a chrysanthemum flower. Used a lot in erotic writing and pornographic novels, which tend to use poetic euphemisms for sex and body parts, but not used in speech.\n\n\u809b\u4ea4 **g\u0101ngji\u0101o** ( _gahng jow_ )\n\nAnal sex.\n\n\u9e21\u5978 **j\u012bji\u0101n** ( _gee gin_ )\n\nAss fucking. \u9e21 **j\u012b** ( _gee_ ) means \"chicken\" and \u5978 **ji\u0101n** ( _gin_ ) is a derogatory word for sex, though it's not obscene like \"fucking.\"\n\n\u8d70\u540e\u95e8 **z\u01d2uh\u00f2um\u00e9n** ( _dzoe hoe men_ )\n\nGo the back way\u2014a euphemism for having anal sex. Has a slightly humorous edge because this is also an extremely common term in everyday speech that means \"pulling strings,\" or using one's connections or other unofficial channels to get something done.\n\n#  **Orgasm**\n\n\u9ad8\u6f6e **g\u0101och\u00e1o** ( _gaow chow_ )\n\nOrgasm (noun).\n\n\u6765 **l\u00e1i** ( _lie_ )\n\nCome (both noun and verb). The colloquial term for an orgasm in northern China.\n\n\u53bb **q\u00f9** ( _chee_ )\n\nThe colloquial term for an orgasm (both noun and verb) in southern China and Taiwan, which interestingly is the exact opposite of the northern term, as it literally means \"go.\"\n\n\u989c\u5c04 **y\u00e1nsh\u00e8** ( _yen shuh_ )\n\nFacial (cum in face). Literally \"face eject.\" A term from Japanese porn (the characters are the same in Japanese).\n\n#  **Sexual positions**\n\n\u4f20\u6559\u58eb\u5f0f **chu\u00e1nji\u00e0osh\u00ec sh\u00ec** ( _chren jow shih shih_ )\n\nMissionary. A literal translation from the English.\n\n\u5c0f\u72d7\u5f0f **xi\u01ceog\u01d2u sh\u00ec** ( _shyaow go shih_ ), or less commonly \u72d7\u722c\u5f0f **g\u01d2up\u00e1 sh\u00ec** ( _go pah shih_ )\n\nDoggie-style.\n\n\u4fa7\u8fdb\u5f0f **c\u00e8 j\u00ecn sh\u00ec** ( _tsuh gene shih_ )\n\nSpooning. Literally \"enter from one side.\"\n\n\u5973\u4e0a\u7537\u4e0b\u5f0f **n\u01da sh\u00e0ng n\u00e1n xi\u00e0 sh\u00ec** ( _nee shahng nahn shah shih_ )\n\nCowgirl. Literally \"woman on top, man underneath.\"\n\n\u4ed9\u59d1\u5212\u8239 **xi\u0101ng\u016b hu\u00e1chu\u00e1n** ( _shin goo hwah chwun_ )\n\nCowgirl. Literally \"goddess rowing the boat.\"\n\n\u5973\u4e0a\u7537\u4e0b\u53d8\u5f0f **n\u01da sh\u00e0ng n\u00e1n xi\u00e0 bi\u00e0n sh\u00ec** ( _nee shahng nahn shah byinn shih_ )\n\nReverse cowgirl. Literally \"woman on top, man underneath, switched.\"\n\n\u83b2\u82b1\u5750\u5f0f **li\u00e1nhu\u0101 zu\u00f2 sh\u00ec** ( _lyinn hwa dzwuh shih_ )\n\nLotus position.\n\n69 \u5f0f **li\u00f9 ji\u01d4 sh\u00ec** ( _lew joe shih_ \u2014 **li\u00f9** rhymes with \"ew\")\n\nSixty-nine.\n\n#  **Alternative types of sex**\n\n\u7535\u8bdd\u6027\u7231 **di\u00e0nhu\u00e0 x\u00ecng'\u00e0i** ( _dyinn hwa shing aye_ )\n\nPhone sex.\n\n\u7f51\u7edc\u6027\u7231 **w\u01cenglu\u00f2 x\u00ecng'\u00e0i** ( _wahng lwuh shing aye_ )\n\nInternet sex.\n\n\u4e09\u4eba\u884c **s\u0101n r\u00e9n x\u00edng** ( _sahn ren sheeng_ )\n\nThreesome. Literally \"three people walking,\" a takeoff on a famous Confucian saying: \"\u4e09\u4eba\u884c, \u5fc5\u6709\u6211\u5e08\u4e5f\" \" **S\u0101n r\u00e9n x\u00edng, b\u00ec y\u01d2u w\u01d2 sh\u012b y\u011b\"** ( _sahn ren sheeng, bee yow uh shih yeh_ ):\n\n\"Among any three people walking together, there is always one you can learn from.\"\n\n\u4e09 P \/ 3P **s\u0101n P** ( _sahn P_ )\n\nThreesome. Literally \"three P.\" The P stands for \"people\" or \"person.\"\n\n\u7fa4\u4ea4 **q\u00fan ji\u0101o** ( _chreen jow_ )\n\nGroup sex, orgy.\n\n\u62f3\u63d2 **qu\u00e1n ch\u0101** ( _chren cha_ )\n\nFisting. Literally \"fist inserts.\"\n\n\u604b\u7269\u7656 **li\u00e0n w\u00f9 p\u00ec** ( _lyinn oo pee_ )\n\nFetish. Literally \"love thing hobby.\"\n\n\u604b\u8db3\u7656 **li\u00e0n z\u00fa p\u00ec** ( _lyinn dzoo pee_ )\n\nFoot fetish.\n\n\u604b\u58f0\u7656 **li\u00e0n sh\u0113ng p\u00ec** ( _lyinn shung pee_ )\n\nSound fetish.\n\n\u604b\u517d\u7656 **li\u00e0n sh\u00f2u p\u00ec** ( _lyinn show pee_ )\n\nBestiality.\n\n\u604b\u978b\u7656 **li\u00e0n xi\u00e9 p\u00ec** ( _lyinn shyih pee_ )\n\nShoe fetish.\n\n\u604b\u5c38\u7656 **li\u00e0n sh\u012b p\u00ec** ( _lyinn shih pee_ )\n\nNecrophilia.\n\n\u604b\u889c\u7656 **li\u00e0n w\u00e0 p\u00ec** ( _lyinn wah pee_ )\n\nSock or stocking fetish. Literally \"loving socks hobby\" or \"love stockings habit.\"\n\n\u604b\u624b\u7656 **li\u00e0n sh\u01d2u p\u00ec** ( _lyinn show pee_ )\n\nHand fetish.\n\n\u604b\u4e73\u7656 **li\u00e0n r\u01d4 p\u00ec** ( _lyinn roo pee_ )\n\nBreast fetish.\n\n\u604b\u8863\u7656 **li\u00e0n y\u012b p\u00ec** ( _lyinn ee pee_ )\n\nClothing fetish.\n\n\u8650\u604b **n\u00fc\u00e8 li\u00e0n** ( _nyreh lyinn_ )\n\nS-M, sadomasochism. Literally \"cruel love.\"\n\nSM \u5973 **SM n\u01da** (\"SM\" _nee_ )\n\nDominatrix. Literally \"S-M woman.\"\n\n\u6253\u5c41\u80a1 **d\u01ce p\u00ecgu** ( _dah pee goo_ )\n\nSpanking.\n\n#  **Menstruation**\n\n\u6708\u7ecf **yu\u00e8j\u012bng** ( _yreh jing_ )\n\nMenstruation. Literally \"monthly passing.\"\n\n\u6708\u4e8b **yu\u00e8sh\u00ec** ( _yreh shih_ )\n\nPeriod, menstruation. Literally \"moon thing\" or \"monthly matters.\"\n\n\u5927\u59e8\u5988 **d\u00e0y\u00edm\u0101** ( _dah ee ma_ )\n\nAuntie. A euphemism for \"period.\"\n\n\u8001\u670b\u53cb **l\u01ceop\u00e9ngyou** ( _laow pung yo_ )\n\nEuphemism for \"period.\" Literally \"old friend.\" Thus, to say you're having your period you would say \u6211\u8001\u670b\u53cb\u6765\u4e86 **w\u01d2 l\u01ceop\u00e9ngyou l\u00e1i le** ( _wuh laow pung yo lie luh_ ): \"My old friend has arrived.\" Mainly used in southern China and Taiwan.\n\n\u5012\u9709 **d\u01ceom\u00e9i** ( _dow may_ )\n\nLiterally \"have bad luck.\" A euphemism for one's period. More common in northern China.\n\n\u90a3\u4e2a **n\u00e8ig\u00e8** ( _nay guh_ )\n\nAnother euphemism that literally means \"that.\" Usages include \u6211\u6709\u90a3\u4e2a **w\u01d2 y\u01d2u n\u00e8ig\u00e8** ( _wuh yo nay guh_ ), \"I have _that_ ,\" and \"I've got you know what.\"\n\n#  **Miscellaneous**\n\n\u53eb\u6625 **ji\u00e0och\u016bn** ( _jow chren_ )\n\nMoan (in a sexual context). \u53eb **ji\u00e0o** ( _jow_ ) means \"yell\" or \"call,\" and \u6625 **ch\u016bn** ( _chren_ ) means \"love\" or \"life.\" This is also the term for a female cat howling when it's in heat.\n\n\u53eb\u5e8a **ji\u00e0ochu\u00e1ng** ( _jow chwahng_ )\n\nMoan (in a sexual context). Literally \"call bed\" or \"yell bed.\"\n\n\u77f3\u5973 **sh\u00ed n\u01da** ( _shih nee_ )\n\nFrigid. Literally \"stone woman.\" Can also refer to someone unable to have sex for congenital reasons, after a character in a famous Chinese opera, _The Peony Pavilion_ , whose hymen is hard as stone.\n\n\u9633\u840e **y\u00e1ngw\u011bi** ( _yahng way_ )\n\nImpotent. Amusingly, this is also what Chinese sports fans yell at the opposing team during sports matches.\n\n\u949f\u70b9\u623f **zh\u014dng di\u01cen f\u00e1ng** ( _johng dyinn fahng_ )\n\nA hotel where you can rent rooms by the hour.\nCHAPTER SIX\n\n**Gay Slang**\n\n**I** n 1988 the prominent sexologist Richard Green gave a lecture at Peking Union Medical College, the top medical school in China, and was famously told by several physicians in the audience that \"there are no homosexuals in China.\" Homosexuality had been persecuted since 1949 and throughout the Cultural Revolution, to the point of total invisibility, and at the time Green gave his lecture, just ten years after the end of that dark time, gays and lesbians in China were only beginning to emerge from underground.\n\nThe government finally began to acknowledge homosexuality in 1990, partly because it realized that it needed to engage the gay community in order to deal with the rising AIDS crisis, and since then, depending on the political atmosphere, official acceptance of gays has waxed and waned.\n\nThe late 1980s had already seen the first official media reports about isolated incidents of same-sex couples being allowed to live a married life. In 1997 \"hooliganism,\" an umbrella term understood to include homosexual behavior (there was never any explicit law dealing with homosexuality), was dropped from the penal code. In 2000 openly gay and lesbian people appeared on TV in China for the first time, and in 2001 China officially took homosexuality off its list of mental disorders. Today you can see same-sex couples openly holding hands at the mall (at least in the biggest cities); the atmosphere in any gay club, ABBA and all, feels utterly carefree; and the lesbian sexual orientation of at least two of the biggest pop singers in China is an almost laugh-ably open secret.\n\nOn the other hand, the movie _Brokeback Mountain_ was rejected for screening in mainland China despite director Ang Lee's celebrity status in the country, gay couples from abroad are no longer allowed to adopt Chinese babies, and after the May 2008 Sichuan earthquake, gays and lesbians were among the groups barred from donating blood to help victims.\n\nDu\u00ec sh\u00ed\n\nThe ironic thing about official and social ambivalence toward gays is that China in fact has a centuries-long tradition of homosexuality, which, while sometimes lampooned, was generally at least tolerated and at times even extolled. Many scholars believe that it was the first arrival of westerners into China toward the end of the Qing dynasty, in the mid to late nineteenth century, that first introduced the idea of homosexuality as something \"wrong\" and an aberrant, pathological condition.\n\nThe first vague allusion to homosexuality appears in prehistoric times, during the Shang dynasty (1766-1122 BC). From there, many more, and more explicit, accounts crop up both throughout the period and during subsequent dynasties. Many emperors were known to have had male lovers, and same-sex relations between men appear in one of China's greatest literary works, _Dream of the Red Chamber_. From these, and from accounts written by westerners visiting China during the Qing dynasty, we know, among other details, that marriage between men was common in Fujian Province and that Beijing was positively crawling with male brothels.\n\nThere is less in the way of an ancient historical record about lesbian behavior, as women generally could not read or write and led extremely cloistered lives, but mentions of maids within the imperial court, or of Buddhist and Taoist nuns, sleeping with each other do surface here and there. More is known about same-sex female relationships in modern times, much of it tied up with marriage resistance movements in Guangdong and other parts of southern China during the late 1800s and early 1900s, in which women formed organized alliances and took vows never to marry\u2014some of these women are still alive today, living as couples in homes they purchased together.\n\nAs you'll see in the next few pages, there are, in addition to the contemporary terms, many literary euphemisms for homosexuality, which were in use during ancient times. These have all been only recently revived\u2014part of a movement within the gay Chinese community to reclaim a past from which it has been cut off for so long, and to remind us all of a time when there was nothing strange, or even noteworthy, about being gay.\n\n#  **Ancient euphemisms for homosexuality**\n\nThere are many, many more stories like the few offered below, which during ancient times offered innumerable expressions for homosexuality via allusion to the famed persons involved or to a detail of the story. I am leaving them out, however, in favor of the most well-known and common terms in use today.\n\n\u65ad\u8896\u4f59\u6843 **du\u00e0n xi\u00f9 y\u00fa t\u00e1o** ( _dwun show ee taow_ )\n\nAn idiomatic expression referring to homosexuality, derived from ancient Chinese literature. Literally \"cut sleeve, leftover peach.\" See below for origins.\n\n\u65ad\u8896 **d\u00f9an xi\u00f9** ( _dwun show_ )\n\nShort for \u65ad\u8896\u4e4b\u7656 **d\u00f9an x\u00ecu zh\u012b p\u01d0** ( _dwun show jih pee_ ), \"the passion of the cut sleeve,\" referring to a story about Ai, the emperor of Han (27-1 BC). The story goes that one day he had to get out of bed and, rather than wake his male lover who had fallen asleep on his sleeve, chose to cut off the sleeve of his robe. Thus any phrases involving a cut sleeve refer to homosexuality.\n\n\u4f59\u6843 **y\u00fa t\u00e1o** ( _ee taow_ )\n\nA euphemism for homosexuality, dating to ancient times. Literally \"the leftover peach,\" referring to a story recorded in _Han Feizi_ (the writings of the philosopher Han Fei, who lived from 280-233 BC) about a beautiful male youth who picked a peach from a tree, bit into it, and found it so sweet that he offered the rest to Ling, the ruler of Wei (534-493 BC), who was touched by the gesture. Thus phrases like \u5206\u6843\u4e4b\u7231 **f\u0113n t\u00e1o zh\u012b \u00e0i** ( _fen taow jih aye_ ), \"love of the shared peach,\" or really any reference to **y\u00fa t\u00e1o** or \u5206\u6843 **f\u0113n t\u00e1o** ( _fen taow_ ), \"sharing peaches,\" are expressions that refer to love between men.\n\n\u9f99\u9633\u7656 **L\u00f3ng Y\u00e1ng p\u01d0** ( _lohng yahng pee_ )\n\nLiterally \"the passion of Long Yang\" and a euphemism for homosexuality. Lord Long Yang was a gay nobleman mentioned in the eighteenth-century classic _Dream of the Red Chamber_. According to the tale, which takes place during China's warring states period (from sometime during the fifth century BC to 221 BC), Lord Long Yang was fishing with his lover, the ruler of Wei, and suddenly burst into tears. The king asked what was wrong, upon which Long Yang said that catching bigger fish made him want to throw back the smaller ones, which surely meant that, with so many beauties in the world, some day the ruler would discard him in favor of a greater beauty. Romantically, the ruler of Wei then made a public decree that \"Anyone who dares to speak of other beauties will be executed along with his entire family.\" To this day, **L\u00f3ng Y\u00e1ng** is sometimes used to refer to a young, pretty boy in a gay relationship, and it is also the name for an international gay Asian network called the Long Yang Club.\n\n\u7537\u98ce **n\u00e1n f\u0113ng** ( _nahn fung_ ) or \u5357\u98ce **n\u00e1n f\u0113ng** ( _nahn fung_ )\n\nThe former literally means \"male practice\" and was more or less a technical term referring to homosexuality during the Ming dynasty (1368-1644). The latter was a homophonous, and poetic, play on that word and literally means \"southern custom,\" also connoting \"southern wind.\" The euphemism \"southern custom\" was based on a common belief throughout China that homosexuality originated in, or at least was more common in, the South (mainly Fujian and Guangdong provinces). There is indeed a great deal of documentation of commonly practiced homosexual customs in those areas. For example, it was noted during the seventeenth century that it was usual for upper-class and educated men in Fujian to marry other men. An older man would buy a boy-bride from the boy's parents, a marriage ceremony was performed, and the older man's family would financially support the younger man in every way, just as in a traditional heterosexual marriage. The older man was called \u5951\u5144 **q\u00ecxi\u014dng** ( _chee shyohng_ ), literally \"sworn older brother,\" and the younger one was called \u5951\u5f1f **q\u00ecd\u00ec** ( _chee dee_ ), \"sworn little brother.\" There is documentation of some such marriages that lasted as long as twenty years, but typically they only lasted until the younger man reached the standard age for heterosexual marriage, upon which they \"divorced\" and the older man often paid to secure him a good female bride and otherwise helped establish him in society.\n\n\u5bf9\u98df **du\u00ecsh\u00ed** ( _dway shih_ )\n\nLiterally \"eat facing each other.\" The earliest mention of lesbianism in Chinese history was of two women from the harem of the emperor Cheng of Han, who, according to imperial records, always ate facing each other and slept together.\n\n#  **Contemporary terms**\n\nMany Chinese simply say \"gay\" in English. \"Gay\" is usually understood to mean gay men, while lesbians mainly go by \u62c9\u62c9 **l\u0101l\u0101** ( _lah lah_ ). Nonetheless, there are several Chinese terms for homosexuality as well:\n\n\u540c\u6027\u604b **t\u00f3ngx\u00ecngl\u00ecan** ( _tohng sheeng lyinn_ )\n\nHomosexual.\n\n\u540c\u6027\u7231 **t\u00f3ngx\u00ecng'\u00e0i** ( _tohng sheeng aye_ )\n\nGay love. Literally \"same-sex love.\"\n\n\u540c\u5fd7 **t\u00f3ngzh\u00ec** ( _tohng jih_ )\n\nGay. Literally \"comrade,\" the form of address used during revolutionary times and still used today by government officials and older Chinese. First adopted in 1987 by gay rights activists in Hong Kong.\n\n\u5973\u540c\u5fd7 **n\u01da t\u00f3ngzh\u00ec** ( _nee tohng jih_ )\n\nLesbian. Literally \"female comrade.\"\n\n\u5927\u540c **d\u00e0t\u00f3ng** ( _dah tohng_ )\n\nA relatively new term coined by university students, short for \u5927\u5b66\u751f\u540c\u5fd7 **d\u00e0xu\u00e9sh\u0113ng t\u00f3ngzh\u00ec** ( _dah shreh shung tohng jih_ ), which means \"gay university students.\"\n\n\u65ad\u80cc **du\u00e0nb\u00e8i** ( _dwun bay_ )\n\nLiterally \"brokeback,\" after Ang Lee's movie _Brokeback Mountain_. A euphemism for homosexuality.\n\nDu\u00e0n b\u00e8i\n\n\u73bb\u7483 **b\u014dl\u00ed** ( _bwuh lee_ )\n\nA euphemism for \"gay.\" Literally \"crystal\" or \"glass.\" Not often used in speech, but known by most gay Chinese people. It comes from the seminal 1980 novel _Crystal Boys_ , by the gay Taiwanese writer Pai Hsien-Yung. The novel's mention of \"crystal boys\" is itself a reference to a passage from the classic work _Dream of the Red Chamber_.\n\n\u62c9\u62c9 **l\u0101l\u0101** ( _lah lah_ ) or \u62c9\u5b50 **l\u0101zi** ( _lah dz_ )\n\nThe most commonly used word for \"lesbian.\" **L\u0101zi** is used in Taiwan and was coined (based on the sound of the English \"lesbian\" or \"lez\") by a lesbian Taiwanese writer named \u90b1\u5999\u6d25 **Qiu Miaojin** ( _chyoe myow gene_ ). When the term spread to the mainland it became **l\u0101l\u0101** instead.\n\nT & P\n\nTerms for lesbian roles. The _T_ stands for \"tomboy\" and the _P_ refers to \u8001\u5a46 **l\u01ceop\u00f3** ( _laow pwuh_ ), or \"wife\" in Chinese. The terms are equivalent to \"butch\" and \"femme\" in English.\n\n\u5a18 T **ni\u00e1ng T** ( _nyahng T_ )\n\nGirly T, soft butch. A lesbian who identifies asaTbut is still a bit feminine or girly.\n\n\u7237 P **y\u00e9 P** ( _yeh P_ )\n\nGrandfather P. A lesbian who identifies as a P but is a bit manly.\n\n\u4e0d\u5206 **b\u00f9 f\u0113n** ( _boo fen_ )\n\nLiterally \"don't differentiate.\" Means that you don't particularly identify as a T or P, and that you also have no preference in terms of which type you're into.\n\n\u5e05 T \u7f8e P **shu\u00e0i T m\u011bi P** ( _shwhy T may P_ )\n\n\u5e05 T **Shu\u00e0i T** ( _shwhy T_ ) means \"handsome tomboy(s)\" and \u7f8e P **m\u011bi P** ( _may P_ ) is \"pretty wife\" or \"pretty wives.\" You might use them together to say something like \"This club is full of handsome tomboys and pretty wives tonight!\"\n\nTT \u604b **TT li\u00e0n** ( _TT lyinn_ )\n\nTomboy-tomboy love, when two tomboys date.\n\nT \u5427 **T ba** ( _T bah_ )\n\nA lesbian bar just for tomboys. Since most Chinese lesbians believe that a T and a P should date each other, the exact reason for the existence of a tomboy-only bar is somewhat mystifying.\n\n\u81ea\u68b3\u5973 **z\u00ec sh\u016b n\u01da** ( _dz shoe nee_ )\n\nComb sisters (literally \"a woman who combs her hair by herself\"). Referring to a group of women in Guangdong and other parts of southern China during the late 1800s and early 1900s who vowed to resist the oppressions of the Confucian conception of marriage. While **z\u00ec sh\u016b n\u01da** were not necessarily all lesbians, they are perceived in popular culture as some sort of lesbian cult and have also been embraced as such by the Chinese lesbian community.\n\n\u4e3b\u52a8 **zh\u01d4d\u00f2ng** ( _joo dohng_ )\n\nTop, giver, pitcher, husband. Literally \"active.\"\n\n\u88ab\u52a8 **b\u00e8id\u00f2ng** ( _bay dohng_ )\n\nBottom, taker, catcher, wife. Literally \"passive.\"\n\n1 \u53f7 **y\u012b h\u00e0o** ( _ee how_ ) or just 1 **y\u012b** ( _ee_ )\n\nLiterally \"number one\" or just \"one.\" Means \"top,\" \"pitcher,\" or \"husband\" because the number one looks like a penis.\n\n0 \u53f7 **l\u00edng h\u00e0o** ( _ling how_ ) or just 0 **l\u00edng** ( _ling_ )\n\nLiterally \"number zero\" or just \"zero.\" Means \"bottom,\" \"catcher,\" or \"wife\" because a \"one\" can be inserted into a\n\n\"zero.\" You might hear two guys flirting at a bar ask, \"\u4f60\u662f 1 \u8fd8\u662f 0?\" \" **N\u01d0 sh\u00ec y\u012b h\u00e1i sh\u00ec ling?\"** : \"Are you a one or a zero?\"\n\n0.5 **l\u00edng di\u01cen w\u01d4** ( _ling dyinn oo_ )\n\nEither, versatile. That is, you can be either the 1 or the 0. You can also indicate that you're a 0.5 by saying \u5168\u80fd **qu\u00e1n n\u00e9ng** ( _chren nung_ ), which means \"all can\"; that is, \"can do all.\"\n\nRice queen\n\nA gay guy who prefers Asian men (this would simply be said in English, as there is no translation in Chinese).\n\nPotato queen\n\nA gay guy who prefers white men (this would simply be said in English, as there is no translation in Chinese).\n\nMB\n\nFor \"money boy\" (and said in English). A (usually young) man who takes money for sex with other men.\n\nT \u5c11 **T sh\u00e0o** ( _T shaow_ )\n\nA woman who takes money for sex with other women.\n\n\u54e5\u54e5 **g\u0113ge** ( _guh guh_ )\n\nA manly gay man. Literally \"older brother.\" Can also imply sugar daddy.\n\n\u5f1f\u5f1f **d\u00ecdi** ( _dee dee_ )\n\nA girly, or effeminate, gay man. Literally \"younger brother.\" Can also imply a person being supported by a sugar daddy.\n\n\u718a **xi\u00f3ng** ( _shyohng_ )\n\nLiterally \"bear.\" Taken directly from the English slang for a gay man with a bigger, slightly chubby build.\n\n\u4eba\u5996 **r\u00e9ny\u0101o** ( _ren yow_ )\n\nLady-boy.\n\n\u53cd\u4e32 **f\u01cenchu\u00e0n** ( _fahn chwun_ )\n\nDrag queen (applies both to men who dress up as women and to women who dress up as men in performances).\n\nCC or just C\n\nQueen (and said in English). Used because CC sounds like \"sissy\" to Chinese ears. Many Chinese gays also just say\n\n\"sissy\" in English for a queen or a feminine guy.\n\n\u978b\u53f7 **xi\u00e9 h\u00e0o** ( _shyih how_ )\n\nLiterally \"shoe size.\" Used online to imply penis size.\n\n\u70ae\u53cb **p\u00e0oy\u01d2u** ( _pow yo_ )\n\nFuck buddy. \u70ae **P\u00e0o** ( _pow_ ) means \"cannon\" and is a euphemism for ejaculation, while \u53cb **y\u01d2u** ( _yo_ ) means friends.\n\nGay \u5427 **\"Gay\" ba** ( _gay bah_ )\n\nGay caf\u00e9.\n\n\u516c\u53f8 **g\u014dngs\u012b** ( _gohng sih_ )\n\nTaiwan slang for a park where gay men gather and meet. Literally \"company\" but shares the same initial syllable with \"park,\" which is \u516c\u56ed **g\u014dngyu\u00e1n** ( _gohng yren_ ). One particularly well-known \"company\" is the 2\/28 Peace Memorial Park in Taipei.\n\n\u53cc\u6027\u604b **shu\u0101ngx\u00ecngli\u00e0n** ( _shwahng sheeng lyinn_ )\n\nBisexuality. Literally \"love for both sexes.\"\n\n\u9634\u9633\u4eba **y\u012bny\u00e1ng r\u00e9n** ( _een yahng ren_ )\n\nTranssexual or hermaphrodite. Literally \"yin-yang person.\" May also be used insultingly to refer to an extremely manly woman or to an extremely effeminate man.\n\n\u540c\u4ec1\u5973 **t\u00f3ngr\u00e9n n\u01da** ( _tohng ren nee_ )\n\nFag hag. \u540c\u4ec1 **T\u00f3ng r\u00e9n** ( _tohng ren_ ) means \"colleagues,\" but \u540c **t\u00f3ng** ( _tohng_ ) also alludes to gay men, while \u5973 **n\u01da** ( _nee_ ) means \"woman,\" so the overall suggestion is a woman who has close associations with gay men.\n\n\u540c\u5fd7\u725b\u76ae\u7cd6 **t\u00f3ngzh\u00ec ni\u00fap\u00ed tang** ( _tohng jih nyoo pee tahng_ )\n\nFag hag. Literally \"gay leather-candy.\" \u540c\u5fd7 **T\u00f3ngzh\u00ec** ( _tohng jih_ ) means \"gay,\" and \u725b\u76ae\u7cd6 **ni\u00fap\u00ed tang** ( _nyoo pee tahng_ ) is a type of sticky candy, thus suggesting sticking to gay guys.\n\n\u51fa\u67dc **ch\u016b gu\u00ec** ( _choo gway_ )\n\nCome out of the closet.\n\n\u5f02\u6027\u604b **y\u00ecx\u00ecngli\u00e0n** ( _ee sheeng lyinn_ )\n\nHeterosexuality. Literally \"love for the opposite sex.\"\n\n\u76f4\u4eba **zh\u00edr\u00e9n** ( _jih ren_ )\n\nStraight. Literally \"straight person.\"\nCHAPTER SEVEN\n\n**Behaving Badly**\n\n**I** n the last few decades leading up to the 1949 Communist revolution, the city of Shanghai was unquestionably the most sinful place on earth. \"If God lets Shanghai endure, he owes an apology to Sodom and Gomorrah,\" said one missionary living there in the early twenties.\n\nThe Whore of Asia, as the city was known, was born in 1842 at the conclusion of the Opium War, when the British forced imperial China to open the port city to foreign trade, and the British\u2014soon followed by the Americans, the French, and numerous other nationalities\u2014quickly established settlements there, each governed by its own rule of law. The mishmash of completely different governments, laws, and courts meant that evading arrest was as simple as walking a block in one direction or another (or buying a fake passport for some random nationality). That, the extreme contrasts of fabulous wealth and decrepit poverty, the opium business driving all the city's moneymaking, the political turmoil brought about by the overthrow of imperial rule, and an ensuing period during which various parts of China were ruled by a rotating cast of brutally violent warlords\u2014not to mention numerous other causes\u2014all combined to make Shanghai the final destination for hedonists, capitalists, adventurers, journalists, businessmen, prostitutes, gangsters, political refugees, gun runners, intel lectuals, arms dealers, movie stars, and dilettantes from every corner of the globe.\n\nDecadence reached its apex in the 1930s, when the opium trade had grown into a full-fledged international drug cartel trafficking in morphine, heroin, and cocaine. The cargo made its way into Europe, South America, and the United States\u2014all controlled by a Chinese mob boss who was simultaneously head of the Nationalist government's Opium Suppression Bureau. The city had three hundred jazz cabarets and a hundred thousand prostitutes, organized gambling thrived on the largest scale of any city in the world, in addition to opium parlors and gambling halls there were nightclubs that featured erotic shows with real live onstage sex, and at some hotels you could order heroin via room service.\n\nToday Shanghai is still known for its nightlife but compared to its heyday is a relatively staid financial city. The influx of foreigners coming to misbehave, however, continues, alive and well, throughout the country. For most, this seems to involve drinking oneself into oblivion every night, doing a lot of drugs, and sleeping around with the locals while teaching English and \"finding yourself.\"\n\nWhether your vice of choice is methamphetamines or auto theft, hookers galore or simply partying till dawn, here is the necessary vocabulary for every naughty deed under the Eastern sun.\n\n#  **Fun and partying**\n\n\u73a9 **w\u00e1n** ( _wahn_ )\n\nPlay. A generic word for going out, partying, or any kind of social activity.\n\n\u5916\u51fa **w\u00e0i ch\u016b** ( _why choo_ )\n\nGoing out.\n\n\u9152\u5427 **j\u01d0ub\u0101** ( _joe bah_ )\n\nBar.\n\n\u591c\u603b\u4f1a **y\u00e8z\u01d2nghu\u00ec** ( _yeh dzohng hway_ )\n\nClub. Literally \"night meeting.\"\n\n\u8fea\u5385 **d\u012bt\u012bng** ( _dee teeng_ )\n\nClub. Literally \"disco hall.\"\n\n\u591c\u5e97 **y\u00e8di\u00e0n** ( _yeh dyinn_ )\n\nNightclub. The term encompasses discos, karaoke bars, caf\u00e9s, and video arcades and can also refer to establishments offering illegal sex services. The term originated in Taiwan and has spread through southern China. It's used to a lesser degree by young people in northern China as well.\n\n\u8fea\u65af\u79d1 **d\u012bs\u012bk\u0113** ( _dee suh kuh_ )\n\nDisco. A transliteration of the English.\n\n\u8f70\u8db4 **h\u014dngp\u0101** ( _hohng pah_ )\n\nA transliteration based on the English phrase \"house party\" but with the additional connotation of having so much fun that you collapse: \u8f70 **h\u014dng** ( _hohng_ ) means \"an explosion\" and \u8db4 **p\u0101** ( _pah_ ) means \"lie on your stomach,\" so the words together suggest something along the lines of being struck dead. Used in Taiwan.\n\n\u8e66\u8fea **b\u00e8ngd\u012b** ( _bung dee_ )\n\nDisco dance.\n\n\u5361\u62c9 OK **k\u01ce l\u0101 OK** ( _kah lah OK_ ) or KTV\n\nBoth terms mean \"karaoke.\" The first is a transliteration of the English word, but most Chinese simply say KTV, pronouncing the letters as you would in English.\n\n\u9ea6\u9738 **m\u00e0ib\u00e0** ( _my bah_ )\n\nMicrophone monopolist. \u9ea6 **M\u00e0i** ( _my_ ) refers to a mike and \u9738 **b\u00e0** ( _bah_ ) means \"tyrant.\" A term popular among young people to describe someone who hogs the mike at karaoke.\n\n\u5237\u591c **shu\u0101 y\u00e8** ( _shwah yeh_ )\n\nTo stay out all night, hang out with shady people, run with a bad crowd. Literally \"swap night.\"\n\n#  **Alcohol**\n\n\u559d\u9152 **h\u0113 j\u01d0u** ( _huh joe_ )\n\nTo drink alcohol.\n\n\u559d\u591a **h\u0113du\u014d** ( _huh dwuh_ )\n\nDrunk, hung over. Literally \"drink much.\"\n\n\u559d\u9189 **h\u0113zu\u00ec** ( _huh dzway_ )\n\nWasted. Literally \"drink tipsy.\"\n\n\u559d\u5927\u4e86 **h\u0113 d\u00e0 le** ( _huh dah luh_ )\n\nA northern Chinese way to say \"very drunk\" or \"wasted.\" Literally \"drank big.\"\n\n\u5e72 **g\u0101n** ( _gahn_ ) or \u5e72\u4e86 **g\u0101n le** ( _gahn luh_ )\n\nDrain one's glass, shoot it, suck down a drink really fast. Literally \"dry [the glass].\"\n\n\u9152\u6ee5\u7528 **j\u01d0u l\u00e0ny\u00f2ng** ( _joe lahn yohng_ )\n\nAlcohol abuse. Literally \"alcohol excessive use.\"\n\n\u9152\u9b3c **j\u01d0ugu\u01d0** ( _joe gway_ )\n\nAlcoholic. Literally \"alcohol ghost.\"\n\n\u767d\u9152 **b\u00e1ij\u01d0u** ( _buy joe_ )\n\nChinese liquor. Literally \"white alcohol.\" **B\u00e1ij\u01d0u** is notoriously strong (about the same proof as vodka) and even more notoriously foul tasting. It is drunk only in shots, accompanied by an elaborate set of rituals and social mores that, in a nutshell, enable Chinese men to bully each other into puking oblivion beneath a veneer of politeness.\n\n\u9189\u9152\u9a7e\u9a76 **zu\u00ec j\u01d0u ji\u00e0 sh\u01d0** ( _dzway joe jah shih_ )\n\nDrink and drive.\n\n#  **Mild vices**\n\n\u6492\u8c0e **s\u0101hu\u0103ng** (sah hwahng)\n\nTell a lie. Literally \"release yellow.\"\n\n\u8364\u7684 **h\u016bn de** ( _hwen duh_ )\n\nDirty, obscene, pornographic. Literally \"meat or fish,\" referring to Buddhism, which considers such foods unclean or dirty. One term that uses this is \u8364\u6bb5\u5b50 **h\u016bn du\u00e0nz\u01d0** ( _hwen swun dz_ ), meaning \"dirty joke.\"\n\n\u5403\u67aa\u836f **ch\u012b qi\u0101ng y\u00e0o** ( _chih chyahng yow_ )\n\nTo speak rudely or insolently. Literally \"to swallow gunpowder.\" As in: \"Why are you being such an asshole\u2014did you swallow gunpowder?\"\n\n\u55b7\u7caa **p\u0113n f\u00e8n** ( _pen fen_ )\n\nTo swear, curse, say dirty words. Literally \"spurt manure\" or \"puff out dung.\"\n\n\u6bdb\u8154 **m\u00e1oqi\u0101ng** ( _maow chyahng_ )\n\nSwear, curse, lose one's temper. Literally \"hairy chests.\"\n\n\u5634\u81ed **zu\u01d0 ch\u00f2u** ( _dzway choe_ )\n\nLiterally \"stinky mouth.\" Refers to someone who is vulgar or rude or swears a lot.\n\n\u88f8\u5954 **lu\u01d2 b\u0113n** ( _lwuh ben_ )\n\nStreaking.\n\n\u73a9\u5fc3\u8df3 **w\u00e1n x\u012bnti\u00e0o** ( _wahn sheen tyow_ )\n\nThrill seeking. Literally \"play with your heartbeat.\" That is, to engage in activities that will quicken your heartbeat.\n\n\u98d9\u8f66 **bi\u0101o ch\u0113** ( _byow chuh_ )\n\nDrag racing. Literally \"whirlwind car.\"\n\n\u94a2\u7ba1\u821e **g\u0101ng gu\u01cen w\u01d4** ( _gahng gwahn oo_ )\n\nPole dancing. This activity has become popular among some urban Chinese women as a form of exercise, with at least a few of them unaware that westerners associate it with stripping.\n\n\u8131\u8863\u5973\u90ce **tu\u014dy\u012b n\u01dal\u00e1ng** ( _twuh ee duh nee lahng_ )\n\nStripper. Literally \"taking-off-clothes girl.\"\n\n\u8def\u6012 **l\u00f9 n\u00f9** ( _loo noo_ )\n\nRoad rage. Traffic in Beijing, and China's other major cities, is notoriously bad.\n\n\u6b3a\u9a97 **q\u012bpi\u00e0n** ( _chee pyinn_ )\n\nCheat (any kind of cheating). A cheater is a \u9a97\u5b50 **pi\u00e0nzi** ( _pyinn dz_ ).\n\n\u5c0f\u6df7\u6df7 **xi\u01ceo h\u00fanh\u00fan** ( _shyaow hwen hwen_ ) or \u6df7\u5b50 **hunz\u01d0** ( _hwen dz_ )\n\nSmall-time crook.\n\n\u5047\u6d3b\u513f **ji\u01ce hu\u00f3r** ( _jah hwurr_ )\n\nBeijing slang for a swindler or con man. Literally \"fake work.\"\n\n\u5c0f\u62a5\u544a **xi\u01ceob\u00e0o g\u00e0o** ( _shaow baow gaow_ \u2014all three syllables rhyme with \"cow\")\n\nSnitching. Literally \"file little secret reports.\"\n\n\u8e29 **c\u01cei** ( _tsigh_ )\n\nTo slander, libel, insult. Literally \"step on.\"\n\n\u66b4\u6253 **b\u00e0o d\u01ce** ( _baow dah_ )\n\nBeat up.\n\n\u6ee1\u5730\u627e\u7259 **m\u01cen d\u00ec zh\u01ceo y\u00e1** ( _mahn dee jaow yah_ )\n\nGet beaten up. Literally \"find [one's] teeth on the ground.\" Used in northern China.\n\n\u6390 **qi\u0101** ( _chyah_ ) or \u6390\u67b6 **qi\u0101ji\u00e0** ( _chyah jah_ )\n\nFight.\n\n\u5e72 **g\u0101n** ( _gahn_ ) or \u5e72\u67b6 **g\u0101nji\u00e0** ( _gahn jah_ )\n\nFight. Also slang for \"kill.\"\n\n\u5e9f **f\u00e8i** ( _fay_ )\n\nTo injure or maim. Can also mean \"to break\" or \"amputate.\"\n\n\u7ed9\u4ed6\u4e2a\u989c\u8272\u770b\u770b **g\u011bi t\u0101 ge y\u00e1ns\u00e8 k\u00e0nkan** ( _gay tah guh yen suh kahn kahn_ )\n\nBeat him up. Literally \"give him a color to see\" (that color being red).\n\n\u627e\u4e0d\u7740\u5317 **zh\u01ceob\u00f9zh\u00e1ob\u011bi** ( _jow boo jow bay_ )\n\nKnock out, beat up. Can also mean \"to get confused.\" Literally \"unable to find where north is.\"\n\n\u4e94\u6307\u5c71\u7ea2 **w\u01d4 zh\u01d0 sh\u0101n h\u00f3ng** ( _ooh jih shahn hohng_ )\n\nLiterally \"five fingers turning red.\" That is, a slap so hard it leaves five red finger marks on your face.\n\n\u5168\u6b66\u884c **qu\u00e1n w\u01d4 x\u00edng** ( _chren oo sheeng_ )\n\nFighting in public. The term originally referred to acrobatic fighting in Chinese opera performances. Now it also refers to fighting in the streets or other public places.\n\n\u94c1\u5934\u529f **ti\u011b t\u00f3u g\u014dng** ( _tyih toe gohng_ )\n\nHead butt.\n\n\u6253\u4e00\u62f3 **d\u01ce y\u012b qu\u00e1n** (dah ee chren)\n\nPunch. Literally \"hit (with) a fist.\" Or you can specify the person being punched by saying \u7ed9\u4ed6\u4e00\u62f3 **g\u011bi t\u0101 y\u012b qu\u00e1n** ( _gay tah ee chren_ ), literally \"give him a fist,\" meaning \"punch him.\"\n\n\u987a\u51fa\u53bb **sh\u00f9n ch\u016bq\u00f9** ( _shwen choo chee_ )\n\nMeans \"to carry something out vertically,\" for example, turning a dresser on its side and moving it out of the bedroom, but can also mean carrying a person out after beating them up, as when someone is being kicked out of a bar. Used in northern China only.\n\n\u811a\u5e95\u6478\u6cb9 **ji\u01ceo d\u01d0 m\u014d y\u00f3u** ( _jyow dee mwuh yo_ )\n\nCut and run, leave the scene of the crime, remove oneself from an awkward situation. Literally \"apply oil to the soles of your feet\"; that is, to speed your retreat.\n\n#  **Pornography**\n\n\u9ec4 **hu\u00e1ng** ( _hwahng_ ) or \u9ec4\u8272 **hu\u00e1ngs\u00e8** ( _hwahng suh_ )\n\nObscene, dirty, pornographic. Literally \"yellow.\" In Chinese class you're most likely to learn \u8272\u60c5 **s\u00e8q\u00edng** ( _suh cheeng_ ) for \"pornography\"; it, however, is considered outdated. Nowadays anything pornographic or dirty is referred to as \"yellow,\" an association that may have originated in the nineteenth century, when French novels, recognizable by their yellow covers, became notorious throughout the world for their \"immoral\" sexual content. Thus a pornographic movie is a \u9ec4\u8272\u7535\u5f71 **hu\u00e1ngs\u00e8 di\u00e0ny\u012dng** ( _hwahng suh dyinn ing_ ), literally \"yellow movie\"; a \u9ec4\u5e26 **hu\u00e1ngd\u00e0i** ( _hwahng die_ ), \"yellow tape\"; or \u9ec4\u7247 **hu\u00e1ngpi\u0101n** ( _hwahng pyinn_ ), literally \"yellow piece.\" A porno mag is a \u9ec4\u8272\u4e66\u520a **hu\u00e1ngs\u00e8 sh\u016bk\u0101n** ( _hwahng suh shoo kahn_ ), literally \"yellow book,\" and a dirty joke or story is a \u9ec4\u6bb5\u5b50 **hu\u00e1ng du\u00e0nz\u01d0** ( _hwahng dwun dz_ ), or \"yellow episode.\"\n\n\u6bdb\u7247 **m\u00e1opi\u0101n** ( _mao pyinn_ )\n\nPorn video. Literally \"hair tape\" because you can see pubic hair.\n\nA \u7247 **A pi\u00e0n** ( _A pyinn_ )\n\nHard-core porn movie. Literally \"A video,\" for \"adult video\" (though, interestingly, some Chinese mistakenly believe the _A_ stands for American).\n\n\u4e09\u7ea7\u7247 **s\u0101nj\u00ed pi\u00e0n** ( _sahn gee pyinn_ )\n\nSoft-core porn movie. Literally \"level-three video,\" referring to the most restrictive level in Hong Kong's movie-rating system.\n\n#  **Drugs**\n\n\u5438\u6bd2 **x\u012bd\u00fa** ( _she do_ )\n\nTo take drugs. Literally \"suck poison.\"\n\n\u6bd2\u8d29\u5b50 **d\u00faf\u00e0nz\u01d0** ( _do fahn dz_ ) or just \u6bd2\u8d29 **d\u00faf\u00e0n** ( _do fahn_ )\n\nDrug dealer. \"To deal drugs\" is \u8d29\u6bd2 **f\u00e0nd\u00fa** ( _fahn do_ ).\n\n\u836f\u7269\u6ee5\u7528 **y\u00e0ow\u00f9 l\u00e0ny\u00f2ng** ( _yaow oo lahn yohng_ ) or \u6bd2\u54c1\u6ee5\u7528 **d\u00fap\u01d0n l\u00e0ny\u00f2ng** ( _do peen lahn yohng_ )\n\nSubstance abuse. Literally \"drug excessive use.\" The first term can mean any type of drugs, including medicine, and the second term specifically means illegal drugs, but both are frequently used. You can insert the name of a drug in front of \u6ee5\u7528 **l\u00e0ny\u00f2ng** to indicate abuse of that specific drug.\n\n\u763e\u541b\u5b50 **y\u01d0nj\u016bnz\u01d0** ( _een jwen dz_ )\n\nDrug addict. Literally \"addicted gentleman\" or \"addicted nobleperson,\" because once upon a time only wealthy people could afford drugs, and opium was considered a status symbol.\n\n\u62bd **ch\u014du** ( _choe_ )\n\nSmoke (as in smoking marijuana, opium, heroin, etc.). \u5438 **X\u012b** ( _she_ ), literally \"inhale,\" can be used too, though it is less common than **ch\u014du**.\n\n\u7528 **y\u00f2ng** ( _yohng_ )\n\nUse. Can be inserted in front of any drug name to indicate use of that drug. For heroin or opium, however, it's more common to say **y\u00f2ng** than **x\u012b** or **ch\u014du** (above), even though they can be smoked.\n\n\u9017 **d\u00f2u** ( _doe_ )\n\nUse. Has a variety of unrelated literal meanings, including \"stop\" and \"tease,\" but is also used as slang for using drugs.\n\n\u98de **f\u0113i** ( _fay_ )\n\nHigh. Literally \"to fly.\" Usually describes being high on marijuana but can also be used for ecstasy, heroin, and other euphoric drugs. \u98de\u8d77 **F\u0113i q\u01d0** ( _fay chee_ ), literally \"flying upwards,\" describes the feeling of getting high, and \u592a\u8d77\u4e86 **t\u00e0i q\u01d0 le** ( _tie chee luh_ ), literally \"rising too much,\" describes being too high.\n\n\u5927 **d\u00e0** ( _dah_ ) or \u5927\u4e86 **d\u00e0 le** ( _dah luh_ )\n\nAn all-purpose Beijing word for \"drunk,\" \"wasted,\" \"high,\" \"fucked up,\" etc. Literally \"big.\" Can indicate being under the influence of any drug. You can simply say, \"\u6211\u5927\u4e86\" \" **W\u01d2 d\u00e0 le** \" ( _wuh dah luh_ ), literally \"I'm big,\" to mean \"I'm high\" or \"I'm on something\" or \"I'm fucked up.\" Or you can be more specific by inserting various drug-related verbs (smoke, use, eat) in front; for example \u6211\u62bd\u5927\u4e86 **w\u01d2 ch\u014du d\u00e0 le** ( _wuh choe dah luh_ ) literally means \"I smoked big\" and indicates that you are high on marijuana or something else that can be smoked.\n\n\u6709\u611f\u89c9 **y\u01d2u g\u01cenju\u00e9** ( _yo gahn dreh_ )\n\nLiterally \"have a feeling\" or, rather, \"I feel something\" or \"I'm feeling it,\" indicating that you're feeling the effects of a drug, whether you're feeling high or a trip is setting in or things are starting to get funny or whatever.\n\n\u6c89\u4e86 **ch\u00e9n le** ( _chen luh_ ) or \u9893\u4e86 **tu\u00ed le** ( _tway luh_ )\n\nLiterally \"drop\" and \"decline,\" respectively. Both can describe dropping, crashing, or a comedown.\n\n\u5927\u9ebb **d\u00e0m\u00e1** ( _dah ma_ )\n\nMarijuana, hashish.\n\n\u547c **h\u016b** ( _who_ )\n\nSmoke (as in weed or hash). Literally \"exhale.\" More common in northern China. Southerners tend to say **ch\u014du** (page 150).\n\n\u54b3 **h\u0101i** ( _high_ )\n\nHigh (on marijuana). Literally, this is the onomatopoeic word for \"sigh,\" but it is used to mean \"high\" since it sounds just like the English word.\n\n\u53ef\u4e50 **k\u011bl\u00e8** ( _kuh luh_ )\n\nCoke. Slang for cocaine (and also the Chinese brand name for Coca-Cola). The formal words for cocaine are \u53e4\u67ef\u78b1 **g\u01d4k\u0113ji\u01cen** ( _goo kuh jinn_ ), the scientific name more often used in Taiwan and Hong Kong, and \u53ef\u5361\u56e0 **k\u011bk\u01cey\u012bn** ( _kuh ka een_ ), a transliteration of the English.\n\n\u5438 **x\u012b** ( _she_ )\n\nSnort. Literally \"suck in.\"\n\n\u9053 **d\u00e0o** ( _dow_ )\n\nLine (in connection with coke use). Literally \"road\" or \"path.\" \"A line of coke\" is \u4e00\u9053\u53ef\u4e50 **y\u012b d\u00e0o k\u011bl\u00e8** ( _ee daow kuh luh_ ).\n\n\u6447\u5934\u4e38 **y\u00e1ot\u00f3uw\u00e1n** ( _yow toe wahn_ )\n\nEcstasy (official name). Literally \"shake-head pill.\" Unlike the English term, no user would ever use this full name to refer to the drug. Most people either say \"E\" or one of the two slang terms below.\n\n\u836f **y\u00e0o** ( _yow_ )\n\nPill (slang for an ecstasy pill). Literally \"medicine\" or \"drug.\" Doing or swallowing a pill is \u5403\u836f **ch\u012b y\u00e0o** ( _chih yow_ ), literally \"eat medicine.\" There is no direct Chinese equivalent to \"rolling\" to describe being on the drug, but you can instead use some of the words mentioned earlier, such as **f\u0113i** or **d\u00e0 le**.\n\n\u4e38\u4ed4 **w\u00e1nz\u01cei** ( _wahn dzigh_ )\n\nSouthern Chinese slang for ecstasy. Northerners usually just say \"E,\" though \u836f **y\u00e0o** ( _yow_ ) is common too.\n\n\u5f00\u4ed6\u654f **k\u0101it\u0101m\u01d0n** ( _kigh tah meen_ )\n\nKetamine. A transliteration of the English. As in many other countries, Special K is fast usurping ecstasy as a popular club drug in China. This is exacerbated by the fact that several factories within its borders produce ketamine for legitimate veterinary use, and thus the drug is cheaper in China.\n\nK \u7c89 **K f\u011bn** ( _K fen_ ) or, more commonly, just K\n\nSlang for ketamine. The first term literally means \"K powder.\"\n\n\u9ebb\u9ec4\u78b1 **m\u00e1hu\u00e1ngji\u01cen** ( _mah hwahng jinn_ )\n\nEphedrine. Widely available and abused in China because ephedra, the plant from which ephedrine is derived, is native to southern China (and used in traditional Chinese medicine), and the production and export of the drug is a massive industry.\n\n\u51b0 **b\u012bng** ( _bing_ )\n\nIce (slang for crystal methamphetamine). Ice use is growing by leaps and bounds in China, for the reasons discussed in the previous entry (ephedrine is a precursor chemical for methamphetamine).\n\n\u6e9c\u51b0 **li\u016bb\u012bng** ( _lyew bing_ )\n\nDoing ice. Literally \"ice skating.\"\n\n\u81f4\u5e7b\u5242 **zh\u00echu\u00e0nj\u00ec** ( _jih hwun gee_ )\n\nHallucinogen. \u81f4\u5e7b **Zh\u00echu\u00e0n** means \"hallucination\" or \"hallucinate.\"\n\n\u6709\u5e7b\u89c9 **y\u01d2uhu\u00e0nju\u00e9** ( _yo hwun jreh_ )\n\nHallucinate. Literally \"have a hallucination.\"\n\n\u8611\u83c7 **m\u00f3g\u016b** ( _mwuh goo)_\n\nMushrooms. There isn't any direct equivalent to \"trip\" or \"tripping\" in Chinese, but you can use some of the words mentioned at the beginning of the drug section, like **f\u0113i** or **d\u00e0 le**.\n\nL\n\nSlang for LSD. Pronounced like the English letter. The verb for doing LSD is \u8d34 **ti\u0113** ( _tyih_ ), literally \"to stick\" or \"affix.\"\n\n\u90ae\u7968 **y\u00f3upi\u00e0o** ( _yo pyow_ )\n\nSlang for LSD. Literally \"postage stamp.\"\n\n\u9547\u9759\u5242 **zh\u00e8nj\u00ecngj\u00ec** ( _jen jing gee_ )\n\nTranquilizer or depressants. Literally \"calm drug.\"\n\n\u7b11\u6c14 **xi\u00e0oq\u00ec** ( _shyow chee_ )\n\nLaughing gas.\n\n\u6b62\u75bc\u836f **zh\u01d0t\u00e9ngy\u00e0o** ( _jhh tung yow_ ) or \u6b62\u75bc\u7247 **zh\u01d0t\u00e9ngpi\u00e0n** ( _jhh tung pyinn_ )\n\nPainkillers.\n\n\u9ebb\u9189\u836f **m\u00e1zu\u00ecy\u00e0o** ( _mah dzway yow_ )\n\nNarcotics.\n\n\u5417\u5561 **m\u01cef\u0113i** ( _ma fay_ )\n\nMorphine. A transliteration of the English.\n\n\u7f8e\u6c99\u916e **m\u011bish\u0101t\u00f3ng** ( _may shah tohng_ ) or \u7f8e\u6c99\u7c89 **m\u011bish\u0101f\u011bn** ( _may shah fen_ )\n\nMethadone. A transliteration of the English.\n\n\u53ef\u5f85\u56e0 **k\u011bd\u00e0iy\u012bn** ( _kuh die een_ ).\n\nCodeine. A transliteration of the English.\n\n\u795e\u4ed9\u6c34 **sh\u00e9nxi\u0101nshu\u01d0** ( _shen shin shway_ )\n\nSlang for codeine. Literally \"celestial water.\"\n\n\u6d77\u6d1b\u56e0 **h\u01ceilu\u00f2y\u012bn** ( _high lwuh een_ )\n\nHeroin. A transliteration of the English. Heroin is by far the most abused drug in China. The country borders the world's top opium-producing countries (the Golden Triangle of Myanmar, Laos, Vietnam, and Thailand; as well as Afghanistan and Pakistan), making it an important transit route for trafficking of the drug. Moreover, poppy fields are also cultivated in some of China's most rural provinces. Ironically, heroin first took hold in China during the 1920s because it was considered a cure for the widespread problem of opium addiction.\n\n\u7c89\u513f **f\u011bnr** ( _fenr_ )\n\nNorthern Chinese slang for (powder) heroin. Literally \"powder.\" The verb for doing it is \u5438 **x\u012b** ( _she_ ), \"suck in.\"\n\n\u6b23\u5feb **x\u012bnku\u00e0i** ( _sheen kwhy_ )\n\nEuphoria. Literally \"happy fast.\"\n\n\u4fda]\u8ffd\u9f99 **zhu\u012b long** ( _jway lohng_ )\n\nLiterally \"chasing the dragon.\" Refers to smoking heroin. Used in Hong Kong only.\n\n\u5b89\u7720\u836f **\u0101nmi\u00e1ny\u00e0o** ( _ahn myinn yow_ )\n\nOpiates. Literally \"sleepy medicine.\"\n\n\u9e26\u7247 **y\u0101pi\u00e0n** ( _yah pyinn_ )\n\nOpium. A transliteration of the English. Opium, which was more or less forced on the Chinese by Britain during the late 1800s to deal with a trade imbalance caused by sky-high demand for Chinese porcelain and tea, has been an extraordinarily widespread problem for the country. The 1949 Communist revolution wiped out the drug almost completely, but inevitably, given that the drug is largely produced in countries that border China, it has, since the late 1980s, begun to spread once again.\n\n\u5b89\u5b9a **\u0101nd\u00ecng** ( _ahn ding_ )\n\nDiazepam (most commonly marketed as Valium). Literally \"calm\" or \"stable.\"\n\n\u82ef\u73af\u5229\u5b9a **b\u011bnhu\u00e1nl\u00ecd\u00ecng** ( _ben hwun lee ding_ )\n\nPhencyclidine (PCP).\n\n\u5929\u4f7f\u7c89 **ti\u0101nsh\u01d0f\u011bn** ( _tyinn shih fen_ )\n\nAngel dust. Slang for PCP (phencyclidine).\n\n\u5f3a\u5978\u836f\u4e38 **qi\u00e1ngji\u0101n y\u00e0ow\u00e1n** ( _chyahng jin yow wahn_ )\n\nRoofies. Literally \"rape pill.\"\n\nFM2\n\nRoofies. Pronounced like the English.\n\n\u5341\u5b57\u67b6 **sh\u00edz\u00ecji\u00e0** ( _shih dz jah_ )\n\nRoofies. Literally \"cross\" because of the cross shape that is sometimes scored into Rohypnol pills. It's more common to simply say FM2 in English, however.\n\n\u7c7b\u56fa\u9187 **l\u00e8ig\u00f9ch\u00fan** ( _lay goo chwen_ )\n\nSteroids.\n\n\u6212\u6bd2 **ji\u00e8d\u00fa** ( _jyih do_ )\n\nKick the habit, rehabilitate, quit taking drugs. Literally \"get rid of drugs.\"\n\n#  **Prostitution**\n\nChinese police use seven classifications for prostitution. From highest to lowest, they are:\n\n1. \u5305\u4e8c\u5976 **b\u0101o\u00e8rn\u0103i** ( _bow er nigh_ \u2014 **b\u0101o** rhymes with \"cow\")\n\nLong ago, when Chinese men had multiple wives, \u5976 **\u00e8rn\u0103i** ( _er nigh_ ) referred to the second wife. Today it refers to mistresses of wealthy men and government officials, an extremely common fact of life in China: literally \"packaged second-wife.\" A related term is:\n\n\u4e8c\u5976\u4e13\u5bb6 **\u00e8rn\u0103i zhu\u0101nji\u0101** ( _er nigh jwahn jah_ ) Rich Chinese businessmen and government officials who collect \"second wives.\" Literally \"mistress expert.\"\n\n2. \u5305\u5a46 **b\u0101op\u00f3** ( _baow pwuh\u2014_ **b\u0101o** rhymes with \"cow\")\n\nLiterally \"packaged wife.\" Women who receive payment for accompanying wealthy men or government officials on business trips or for some other fixed period of time.\n\n3. \u4e09\u5385 **s\u0101nt\u012bng** ( _sahn ting_ )\n\nLiterally \"three halls.\" Refers to prostitution in specific venues, such as bars, clubs, karaoke parlors, teahouses, bathhouses, etc. They generally make money from tips and from a cut of the venue's \"service charges.\" Some related terms are:\n\n\u4e09\u966a\u5c0f\u59d0 **s\u0101np\u00e9i xi\u01ceoji\u011b** ( _sahn pay shaow jyih_ )\n\nLiterally \"young ladies of the three accompaniments.\" A common euphemism for **s\u0101nt\u012bng** sex workers. The three accompaniments are supposedly drinking, dancing, and chatting or singing with their clients (often while being groped). Of course, the unspoken fourth accompaniment costs extra.\n\n\u5c0f\u59d0 **xi\u01ceoji\u011b** ( _shaow jyih_ )\n\nWhore, prostitute. Literally \"little miss.\" **Xi\u01ceoji\u011b** is also an everyday form of address for waitresses, shop-girls, and any service staff in Taiwan and southern China, but due to the association with prostitution, northern Chinese instead address such personnel by the impersonal term \u670d\u52a1\u5458 **f\u00faw\u00f9yu\u00e1n** ( _foo oo yren_ ), literally \"service person.\"\n\nKTV \u5c0f\u59d0 **KTV xi\u0103oji\u011b** (\"KTV\" _shaow jyih_ )\n\nProstitute in a karaoke parlor. Literally \"karaoke miss.\"\n\n\u5427\u5973 **b\u0101n\u01da** ( _bah nee_ )\n\nBar girl.\n\n\u6d17\u6d74\u4e2d\u5fc3 **x\u01d0y\u00f9 zh\u014dngx\u012bn** ( _she yee johng sheen_ ) Bathhouse.\n\n4. \u53ee\u549a\u5c0f\u59d0 **d\u012bngd\u014dng xi\u0103ojie** ( _deeng dohng shaow jyih_ ) Literally \"ding-dong girls\" or \"doorbell ladies.\"\n\nProstitutes who solicit clients by phone in hotel rooms. If you ever stay in a Chinese hotel and get a mysterious call in the middle of the night, chances are it's a ding-dong girl.\n\n5. \u53d1\u5eca\u59b9 **f\u00e0l\u00e1ng m\u00e8i** ( _fah lahng may_ )\n\nProstitutes who work under the guise of a hair salon, beauty parlor, bathhouse, or massage parlor. Literally \"hair salon sister.\" The most common services offered are hand jobs and oral sex. If you're ever looking for an actual haircut in China, look for a salon that has female customers and where the hairdressers actually appear to know how to use a pair of scissors\u2014and even then you might very well _still_ wind up getting a whispered proposition. A related term is:\n\n\u6309\u6469\u5973 **\u00e0nm\u00f3 n\u01da** ( _ahn mwuh nee_ )\n\nLiterally \"massage girl.\" Includes both actual masseuses who do a little extra for an additional fee and full-on prostitutes who work under the guise of being masseuses but have no idea how to give a massage.\n\n6. \u8857\u5973 **ji\u0113n\u01da** ( _jyih nee_ )\n\nLiterally \"street girls.\" Prostitutes who solicit clients on the street.\n\n7. \u4e0b\u5de5\u68da **xi\u00e0g\u014dngp\u00e9ng** ( _shah gohng pung_ )\n\nLiterally \"lower work shack.\" Prostitutes whose clients are migrant workers\u2014usually men from the countryside who have found temporary work doing manual labor in the big city.\n\nBeyond these seven official tiers, here are other words related to prostitution:\n\n\u5993\u5973 **j\u00ecn\u01da** ( _gee knee_ )\n\nThe most neutral, formal word for prostitute.\n\n\u5a3c\u5993 **ch\u0101ngj\u00ec** ( _chahng gee_ )\n\nAnother formal term for prostitute.\n\n\u9e21 **j\u012b** ( _gee_ ) and \u91ce\u9e21 **y\u011bj\u012b** ( _yeh gee_ )\n\nWhore, prostitute, hooker. Literally \"chicken\" or \"wild chicken\/pheasant.\" Probably came about in Shanghai during the late 1800s and early 1900s because the city saw a sudden influx of streetwalkers, which was a relatively new sight (as opposed to women who hosted men in teahouses and brothels), and the women were said to look like chickens walking around on the street. Also a play on words because the \u9e21 **j\u012b** , meaning \"chicken,\" sounds nearly the same as \u5993 **j\u00ec** in the formal term for \"prostitute\" (above). A **y\u011bj\u012b** can also be an illegal business.\n\n\u7ad9\u8857\u7684 **zh\u00e0n ji\u0113 de** ( _jahn jyih duh_ )\n\nStreetwalker. Literally \"one who stands on the street.\"\n\n\u9e21\u9662 **j\u012byu\u00e0n** ( _gee yren_ )\n\nSlang term for a brothel. Literally \"chicken yard.\"\n\n\u9e2d **y\u0101** ( _yah_ ) or \u9e2d\u5b50 **y\u0101zi** ( _yah dz_ )\n\nMale prostitute. Literally \"duck.\"\n\n\u5356\u8c46\u8150 **m\u00e0i d\u00f2ufu** ( _my doe foo_ )\n\nLiterally \"sell tofu\" and a euphemism for prostitution.\n\n\u5973\u58eb\u9152\u5427 **n\u01dash\u00ec j\u01d0ub\u0101** ( _nee shih joe bah_ )\n\nLady bar.\n\n\u76ae\u6761 **p\u00edti\u00e1o** ( _pee tyow_ )\n\nPimp.\n\n\u62c9\u76ae\u6761 **l\u0101p\u00edti\u00e1o** ( _lah pee tyow_ )\n\nPimp. Literally \"pull\/drag prostitutes.\"\n\n\u5a4a\u5b50 **bi\u0103ozi** ( _byow dz_ )\n\nWhore, prostitute, hooker. Also an insult similar to \"bitch.\"\n\n\u5e94\u53ec\u5973\u90ce **y\u00ecng zh\u00e0o n\u01dal\u00e1ng** ( _yeeng dzow nee lahng_ )\n\nProstitute. Literally \"call girl.\"\n\n\u5ad6 **pi\u00e1o** ( _pyow_ ) or \u5ad6\u5a3c **pi\u00e1och\u0101ng** ( _pyow chahng_ )\n\nVisit prostitutes, whore around, pay for sex.\n\n\u966a\u5ba2 **p\u00e9ik\u00e8** ( _pay kuh_ )\n\nTo entertain clients\u2014either in a legitimate business sense or in the euphemistic sense.\n\n\u5ad6\u5ba2 **pi\u00e1ok\u00e8** ( _pyow kuh_ )\n\nWhoremonger, john, brothel customer. Literally \"whoring guest.\"\n\n\u7ea2\u706f\u533a **h\u00f3ngd\u0113ngq\u016b** ( _hohng dung chee_ )\n\n\"Red-light district.\" Used only in southern China and Taiwan.\n\n\u51fa\u53f0 **ch\u016bt\u00e1i** ( _choo tie_ )\n\nTo take home (a prostitute). A verb used when a john takes a prostitute away from the establishment and to a home or hotel instead. Literally \"leave the counter.\"\n\n\u6253\u5305 **d\u01ceb\u0101o** ( _da baow_ )\n\nLiterally the term used for a doggie bag or wrapping up leftover food to take home from a restaurant, but it can also be jokingly used to indicate that you want to take a prostitute off premises.\n\n\u6027\u75c5 **x\u00ecngb\u00ecng** ( _sheeng bing_ )\n\nSTD (sexually transmitted disease). Literally \"sex disease.\"\n\n\u82b1\u67f3 **hu\u0101li\u01d4** ( _hwah lew\u2014lew_ rhymes with \"pew\") \/ \u82b1\u67f3\u75c5 **hu\u0101li\u016db\u00ecng** ( _hwah lyew bing_ )\n\nSTD. Literally \"flower willow\" or \"flower willow disease.\" Flowers and willows have been used as metaphors for women since ancient times, and STDs are generally considered a disease that men get from prostitutes.\n\n\u810f\u75c5 **z\u0101ngb\u00ecng** ( _jahng bing_ )\n\nSTD. Literally \"dirty disease.\"\n\n#  **Sexual perversion**\n\n\u8272\u9b3c **s\u00e8gu\u012d** ( _suh gway_ )\n\nSex maniac. Literally \"sex ghost.\"\n\n\u8272\u60c5\u72c2 **s\u00e8q\u00edngku\u00e1ng** ( _suh cheeng kwahng_ )\n\nSex maniac. Literally \"sex crazy person.\"\n\n\u6d41\u6c13 **li\u00fam\u00e1ng** ( _lyew mahng_ )\n\nA derogatory term for a man that may mean he is just generally \"bad\" or \"rowdy\" or a \"hooligan\" but may also indicate sexual perversion involving women. Women frequently use this word in reference to a man who is aggressively hitting on them.\n\n\u72fc **l\u00e1ng** ( _lahng_ )\n\nDirty guy. Literally \"wolf.\"\n\n\u8272\u72fc **s\u00e8l\u00e1ng** ( _suh lahng_ )\n\nLiterally \"colored wolf.\" A lecherous man. Like **li\u00fam\u00e1ng** (above), women often use this to describe men who are aggressively hitting on them. Originally it referred exclusively to older men who prey on younger women, but now it refers to men of any age.\n\n\u8272\u8ff7\u8ff7\u7684 **s\u00e8m\u012bm\u012bd\u00e8** ( _suh me me duh_ )\n\nLecherous. Literally \"colored fantasy.\" A lecherous man, synonymous with **li\u00fam\u00e1ng** or **s\u00e8l\u00e1ng** (above), is a \u8272\u8ff7\u8ff7\u7684\u7537\u4eba **s\u00e8m\u012bm\u012bd\u00e8 n\u00e1nr\u00e9n** ( _suh me me duh nahn ren_ ), and you would say that he has \u8272\u8ff7\u8ff7\u7684\u773c\u795e **s\u00e8m\u012bm\u012bd\u00e8 y\u01censh\u00e9n** ( _suh me me duh yen shen_ ), or \"sex in his eyes,\" literally \"colored eyes.\"\n\n\u4e0b\u6d41 **xi\u00e0li\u00fa** ( _shah lew_ )\n\nPerverted. Literally \"below flowing.\"\n\n\u8c03\u620f **ti\u00e1ox\u00ec** ( _tyow she_ )\n\nTo take advantage of or harass a woman. Most often used to refer to a man verbally assaulting a woman with obscene words.\n\n\u82b1\u8001\u5934 **hu\u0101 l\u01ceot\u00f3u** ( _hwa laow toe_ )\n\nDirty old man.\n\n\u54b8\u732a\u624b **xi\u00e1nzh\u016bsh\u01d2u** ( _shin joo show_ )\n\nGroping. Used in Taiwan and Guangdong Province. Literally \"salty pig hands.\" In Taiwan and Guangdong, a man who pays unwanted sexual attention to a woman (groping, etc.) is called a \u732a\u54e5 **zh\u016bg\u0113** ( _joo guh_ ), literally \"pig brother.\"\n\n\u66b4\u9732\u72c2 **b\u00e0ol\u00f9 ku\u00e1ng** ( _baow loo kwahng_ )\n\nFlash, flashing. Literally \"showing crazy person.\"\n\n\u6076\u8da3\u5473 **\u00e8 q\u00f9w\u00e8i** ( _uh chee way_ )\n\nLiterally \"repulsive interests.\" Originating in Japan, the term refers to five types of sexual perversion: (1) interest in young girls, (2) homosexuality, (3) bondage and sadomasochism, (4) Freudian issues\u2014for example an Oedipus complex\u2014and (5) incest.\n\n\u4e71\u4f26 **lu\u00e0nl\u00fan** ( _lwun lwen_ )\n\nIncest. Literally \"messy logical sequence.\"\n\n\u5e0c\u814a\u5267 **x\u012bl\u00e0j\u00f9** ( _she la jee_ )\n\nIncest. Literally \"Greek play.\"\n\n\u6027\u9a9a\u6270 **x\u00ecngs\u0101or\u01ceo** ( _sheeng saow raow_ )\n\nSexual assault, molestation. Literally \"sex interrupts.\"\n\n\u602a\u53d4\u53d4 **gu\u00e0i sh\u016bshu** ( _gwhy shoo shoo_ )\n\nPedophile. Literally \"weird uncle.\" Used in Taiwan.\n\n\u5f3a\u5978 **qi\u00e1ngji\u0101n** ( _chyahng gin_ )\n\nRape. Literally \"forced sex.\"\n\n#  **Crime**\n\n\u7f6a\u72af **zu\u00ecf\u00e0n** ( _dzway fahn_ )\n\nCriminal.\n\n\u62ce\u5305\u515a **l\u012bn b\u0101o d\u01ceng** ( _leen baow dahng_ )\n\nPurse snatcher.\n\n\u4e09\u53ea\u624b **s\u0101n zh\u012b sh\u01d2u** ( _sahn jih show_ )\n\nPickpocket. Literally \"three hands.\"\n\n\u5c0f\u5077 **xi\u01ceot\u014du** ( _shyaow toe_ )\n\nThief. Literally \"little steal.\"\n\n\u987a **sh\u00f9n** ( _shwen_ ) or \u987a\u8d70 **sh\u00f9n z\u01d2u** ( _shwen dzoe_ )\n\nBeijing slang for \"steal.\" Literally \"smooth\" and \"smoothly walk away,\" respectively.\n\n\u7eb5\u706b **z\u00f2nghu\u01d2** ( _dzohng hwuh_ )\n\nArson. Literally \"start fire.\"\n\n\u8d70\u79c1 **z\u01d2us\u012b** ( _dzoe sih_ )\n\nSmuggling. Literally \"go private.\"\n\n\u9ec4\u725b **hu\u00e1ngn\u00edu** ( _hwahng nyoo_ )\n\nLiterally \"yellow ox.\" A ticket scalper or someone who buys and sells foreign currency on the black market\u2014the sight of a mob of people trying to get tickets resembles a herd of cows. Can also mean a person who doesn't repay debts. Originated in Shanghai but now used everywhere.\n\n\u6d17\u94b1 **x\u01d0qi\u00e1n** ( _she chin_ )\n\nMoney laundering.\n\n\u6572\u8bc8 **qi\u0101ozh\u00e0** ( _chyow jah_ )\n\nBlackmail. Literally \"knock cheat.\"\n\n\u884c\u8d3f **x\u00ednghu\u00ec** ( _sheeng hway_ )\n\nBribery.\n\n\u5c01\u53e3\u8d39 **f\u0113ng k\u01d2u f\u00e8i** ( _fung koe fay_ )\n\nHush money.\n\n\u975e\u6cd5\u96c6\u8d44 **f\u0113if\u01ce j\u00edz\u012b** ( _fay fah gee dz_ )\n\nPonzi scheme.\n\n\u76d7\u8f66\u8d3c **d\u00e0o ch\u0113 z\u00e9i** ( _daow chuh dzay_ )\n\nMotor vehicle theft. Literally \"car stealer.\"\n\n\u8d29\u6bd2 **f\u00e0nd\u00fa** ( _fahn do_ )\n\nTrafficking\/smuggling drugs. Literally \"selling drugs.\"\n\n\u6bd2\u67ad **d\u00faxi\u0101o** ( _doo shaow_ )\n\nDrug lord.\n\n\u9ed1 **h\u0113i** ( _hay_ )\n\nIn Chinese anything illegal is called \"black,\" or **h\u0113i**. Thus the black market is the \u9ed1\u5e02 **h\u0113ish\u00ec** ( _hay shih_ ) (the literal meaning is the same as the English); an illegal cab is a \u9ed1\u8f66 **h\u0113ich\u0113** ( _hay chuh_ ), literally \"black car\"; cooking the books results in a \u9ed1\u5355 **h\u0113id\u0101n** ( _hay dahn_ ), literally \"black accounting\"; a bad call in sports or an unfair decision by a judge, who may or may not have been bribed, is a \u9ed1\u54e8 **h\u0113ish\u00e0o** ( _hay shaow_ ), literally \"black whistle\"; and any sort of misdeed is a \u9ed1\u70b9 **h\u0113idi\u01cen** ( _hay dyinn_ ), literally \"black spot.\"\n\n\u9ed1\u9053 **h\u0113id\u00e0o** ( _hay dow_ )\n\nCriminal gang. Literally \"black path.\"\n\n\u9ed1\u793e\u4f1a **h\u0113ish\u00e8hu\u00ec** ( _hay shuh hway_ )\n\nMafia. Literally \"black society.\"\n\n\u4e09\u5408\u4f1a **s\u0101n h\u00e9 hu\u00ec** ( _sahn hay hway_ )\n\nTriad, the largest organized crime network in China. The Triads originally started out in the 1760s as a resistance movement to overthrow the Manchu emperor and restore Han Chinese rule. Their name, **s\u0101n h\u00e9 hu\u00ec** , literally means \"three harmonies society,\" referring to the unity between heaven, earth, and man. After the Qing dynasty finally collapsed in 1911, these former outlaws and rebels, no longer enjoying the benefit of public support and funding, turned to crime and extortion to support themselves. Today there are thought to be as many as sixty separate Triad groups in Hong Kong alone, with membership of each one ranging anywhere from fifty to thirty thousand people, and more groups throughout Taiwan, Macao, mainland China, and around the world in cities with large overseas-Chinese communities like San Francisco, Los Angeles, New York, Philadelphia, Seattle, Boston, Miami, Chicago, Houston, Atlanta, Calgary, Vancouver, S\u00e3o Paulo, Melbourne, and Sydney.\n\n\u6d89\u9ed1 **sh\u00e8h\u0113i** ( _shuh hay_ )\n\nGang-related crimes.\n\n\u4eba\u53e3\u8d29\u5356 **r\u00e9nk\u01d2u f\u00e0nm\u00e0i** ( _ren koe fahn my_ )\n\nHuman trafficking. Literally \"person selling.\"\n\n\u86c7\u5934 **sh\u00e9t\u00f3u** ( _shuh toe_ )\n\nSnakehead. Someone (usually a gang member from Fujian Province) who smuggles illegal immigrants out of China and into other countries for a fee.\n\n\u7ed1\u67b6 **b\u01cengji\u00e0** ( _bahng jah_ )\n\nKidnapping, abduction.\n\n\u4eba\u53e3\u62d0\u5356 **r\u00e9nk\u01d2u gu\u01ceim\u00e0i** ( _ren koe gwhy my_ )\n\nHuman slavery. Literally \"person abducting.\"\n\n\u6076\u6027\u4f24\u4eba **\u00e8x\u00ecng sh\u0101ngr\u00e9n** ( _uh sheeng shahng ren_ )\n\nAggravated assault.\n\n\u6740\u4eba **sh\u0101r\u00e9n** ( _shah ren_ )\n\nHomicide. Literally \"kill person.\"\n\n\u8c0b\u6740 **m\u00f3ush\u0101** ( _moe shah_ )\n\nMurder. Literally \"plan kill.\"\n\n\u6697\u6740 **\u00e0nsh\u0101** ( _ahn shah_ )\n\nAssassination. Literally \"secret kill.\"\n\n\u8fde\u73af\u6740\u624b **li\u00e1nhu\u00e1nsh\u0101sh\u01d2u** ( _lyinn hwun shah show_ )\n\nSerial killer. Literally \"connected-ring kill hand.\"\n\n\u98df\u4eba **sh\u00edr\u00e9n** ( _shih ren_ )\n\nCannibalism. Literally \"eat people.\"\nCHAPTER EIGHT\n\n**Internet Slang**\n\n**I** n a country without a free press, it is impossible to overstate the profound impact that the Internet has had on society. The simple ability to see and read the internal thoughts of one's fellow compatriots, as well as the viewpoints of people from around the world, is prodding an estimated 250 million Chinese to express themselves more freely, to consider a multiplicity of opinions in matters of public debate, and, most simply, to feel just a little bit less alone in this world. And sure, the Internet is censored in China, but the efforts of government watchdogs can be best likened, as one well-known Chinese blogger put it, to a dam that \"is leaking all over the place.\"\n\nOnline bulletin board systems (BBS) hosted by Chinese universities have long been hotbeds of intellectual debate, and today seemingly every company, media outlet, Web portal, and random organization in the country has a BBS garnering an estimated ten million new posts each day, with single posts frequently provoking hundreds, if not thousands, of replies. A mostly dead medium in the West, BBSs are invaluable to Chinese users because of the anonymity they afford. Blogs, too, are a noteworthy phenomenon, arguably far more widespread and vibrant than in the West. Everyone in China and their mother, it seems, has a blog: pop stars, CEOs of top companies, small-town mayors, powerful government officials, poor migrant workers, and journalists disillusioned with the censorship they face in their day jobs with state-owned media outlets. Many consider BBSs and blogs to be the truest reflection of what ordinary people on the ground really think about an issue\u2014to the extent that the traditional media regularly quote from blog gers, and newspapers publish entire blog entries as op-ed columns.\n\nThere is, of course, a negative for every positive, and thus in the worst cases the Internet has served to ignite a terrifying mob mentality that hearkens back to the horrors of the Cultural Revolution. The \"human-flesh search engine,\" as these \"netizens\" call themselves, is infamous for harnessing the power of the Internet to hunt down people who, deservedly or not, have been slandered online and deemed worthy of punishment by the online lynch mobs. Some people have been physically attacked; others have had to change addresses, jobs, and phone numbers. In one fairly typical example, a Chinese student attending Duke University, who was perceived to be in support of Free Tibet protestors, was targeted by angry Chinese who posted her picture, contact information, and parents' home address online. She received an avalanche of death threats, her parents got harassing phone calls at work, and one person claimed to have left human feces on her parents' doorstep.\n\nOverall, however, the Internet's influence has been overwhelmingly positive. In many instances, it has filled in for the lack of a reliable judicial system and overcome endemic government corruption to bring justice for the masses. The Internet enables news of wrongdoing and injustice to spread at the speed of light and has spurred many real-life protests, petition drives, and heartfelt movements to get corrupt officials fired and persuade the central government to investigate allegations ignored by local governments. It has helped draw attention to corporate abuse and to raise money for ordinary people suffering under the weight of crippling medical costs and other hardships.\n\nNot to imply, of course, that nascent revolution lurks in every bored instant message sent from one underpaid office worker to another, or every inane blog post about what some person ate for breakfast. Like Internet users the world over, Chinese netizens are chain-smoking in twenty-four-hour Internet caf\u00e9s or lazing about at home, mainly idling away the days of their lives watching online video clips of stupid pet tricks, shopping for handbags on the Chinese equivalent to eBay, bitching about China's notoriously terrible national soccer team, and playing hour after interminable hour of Counter-Strike. And, hell, they may all know how to get around the government's censoring mechanisms, but, let's face it, the vast majority of them are doing so in order to download pirated movies and, of course, to watch porn.\n\nSo here, for your reference, is all the latest in chat slang, Internet crazes, and weirdo sociological phenomena that make up the vibrant, silly, important, ridiculous, and revolutionary world of the Chinese Internet.\n\n#  **Society, censorship, and Internet memes**\n\nGFW\n\nStands for \"Great Firewall of China,\" a play on the Great Wall of China, and referring to China's Net Nanny, or Internet censoring mechanism.\n\n**GFW**\n\n\u6cb3\u87f9 **h\u00e9xi\u00e8** ( _huh shih_ )\n\nLiterally \"river crab.\" A euphemism for Internet censorship. One of the main guiding philosophies of the Communist party is that of a \"harmonious society,\" a catchphrase frequently played upon, parodied, and ironically referenced on the Internet. Using the word \"censorship\" will often get your site flagged by Web hosts, so when a post, or an entire Web site, gets blocked or deleted, Chinese Internet users sarcastically say that it's been \"harmonized.\" \"Harmony\" in Chinese, \u548c\u8c10 **h\u00e9xi\u00e9** ( _huh shih_ ) is similar in pronunciation to \"river crab.\" And since these euphemistic mentions of harmonization have become so ubiquitous that they themselves have become target keywords for getting censored, the Chinese have turned to \"river crab\" to stand in for harmonization.\n\n\u529f\u592b\u7f51 **g\u014dngfu w\u0103ng** ( _gohng foo wahng_ )\n\nA euphemism for censorship. Literally \"kung fu Web,\" but as with \"river crab,\" mentions of kung fu in blog posts often refer to censorship. You'll notice that the pinyin **g\u014dngfu w\u0103ng** starts with _g_ , _f_ , and _w_. GFW is the abbreviation for the Great Firewall.\n\n\u8349\u6ce5\u9a6c **c\u01ceon\u00edm\u01ce** ( _tsow nee mah_ \\- **c\u01ceo** rhymes with \"cow\")\n\nGrass mud horse. A made-up creature that has become an Internet sensation spawning everything from dolls to You-Tube videos to scholarly essays because its otherwise benign name sounds like the obscenity \u64cd\u4f60\u5988 **c\u00e0o n\u01d0 m\u0101** ( _tsow nee ma_ ), meaning \"Fuck you,\" and because it has become a symbol of the fight against Internet censorship. The most popular online video about the creature, a parody of kiddie songs sung by a chipper chorus of children's voices, explains that the grass mud horse lives in the \u9a6c\u52d2\u6208\u58c1 **m\u0103l\u00e8 g\u0113b\u00ec** ( _mah luh guh bee_ ), which innocently looks like \"The Ma Le Desert\" but sounds just like \u5988\u4e86\u4e2a\u5c44 **m\u0101legeb\u012b** ( _mah luh guh bee_ ), or \"your mother's a cunt.\" Moreover, the grass mud horse is often threatened by river crabs (a euphemism for Internet censorship, see the **h\u00e9xi\u00e8** entry on page 157). While a phenomenon that involves children's voices singing \"fuck you\" and stick-figure online cartoons and dolls may seem juvenile, it neatly encapsulates the biggest issues surrounding the Chinese Internet. The authorities go on \"anti-smut\" campaigns to wipe out \"inappropriate\" material from the Internet; netizens come up with clever ways to evade the censors, like saying \"grass mud horse\" instead of \"fuck you,\" and maybe the censors start censoring the euphemism itself; but then netizens simply find yet another way to talk about the banned term or banned ideas, much the way \"harmony\" was used as a euphemism for \"censorship\" and then got censored, which spawned \"river crab\" in place of \"harmony.\" And in the end, it becomes clear that any attempt to censor this unwieldy beast is a losing battle.\n\n\u4e94\u6bdb\u515a **w\u01d4 m\u00e1o d\u01ceng** ( _ooh maow dahng_ )\n\nLiterally \"half-yuan party.\" People paid to influence Internet discussion by writing blog and BBS posts extolling the government's position on various issues and also to write comments in response to other posts, arguing the government's side of things. The payment structure for these people varies, but at the time this term was coined, the Hunan Province government was said to be paying a lump sum of six hundred yuan (around eighty or ninety dollars) plus an extra **w\u01d4 m\u00e1o** (half a yuan, about seven cents) per post.\n\nZF\n\nShort for \u653f\u5e9c **zh\u00e8ngf\u01d4** ( _jung foo_ ), which means \"government.\" ZF is often used in place of the Chinese characters so as not to attract the attention of the government censoring mechanism, which can be triggered by sensitive keywords.\n\nJC\n\nStands for \u8b66\u5bdf **j\u01d0ngch\u00e1** ( _jeeng cha_ ), which means \"police\" or \"police officer.\"\n\nFB\n\nStands for \u8150\u8d25 **f\u01d4ba\u00ec** ( _foo buy_ ). Means \"corruption\" in Chinese, and since corrupt officials are often wined and dined, people now ironically say **f\u01d4ba\u00ec** to refer to going out to a fancy restaurant, going on vacation, or otherwise treating oneself to something nice.\n\nFZL or \u975e\u4e3b\u6d41 **f\u0113izh\u01d4li\u00fa** ( _fay joo lew_ \u2014 **li\u00fa** rhymes with \"pew\")\n\nAn \"alternative\" or \"counterculture\" person, usually rebellious and young with some sort of extreme style\u2014elaborate hairstyle, piercings, unusual and edgy clothes, etc.\u2014probably involving a great deal of influence from Japanese pop culture.\n\nFQ or \u6124\u9752 **f\u00e8nq\u012bng** ( _fen cheeng_ )\n\nAngry youth, patriotic youth. Coined in Hong Kong during the 1970s to refer to young people agitating for reform in Chinese society, but now used on the Internet to refer to extremely patriotic (or depending on your view of them, nationalistic) Chinese with aggressive stances on a number of political issues. Very broadly speaking, **f\u00e8nq\u012bng** tend to be virulently anti-Japanese and militantly opposed to Taiwanese and Tibetan independence. The issues at stake are complicated, and **f\u00e8nq\u012bng** attitudes can seem contradictory to anyone not familiar with their nuances. They often seem to unconditionally defend anything the Chinese government does but are simultaneously highly critical of that government for what they feel is too soft a stance on the core issues mentioned above. And they may exhibit intense xenophobia and call westernized Chinese \"race traitors\" while also believing that China should eventually democratize. Critics of **f\u00e8nq\u012bng** derogatorily call them \u7caa\u9752 **f\u00e8nq\u012bng** ( _fen cheeng_ ), a play on words that is pronounced the same as **f\u00e8nq\u012bng** but means \"shit youth,\" or just \u7caa **f\u00e8nf\u00e8n** ( _fen fen_ ): \"shit.\"\n\n\u4eba\u8089\u641c\u7d22 **r\u00e9n r\u00f2u s\u014dusu\u01d2** ( _ren roe so swuh_ )\n\nLiterally \"human flesh search engine.\" Refers to the phenomenon of Internet users hunting down people in real life.\n\n\u8349\u6839\u7f51\u6c11 **c\u01ceog\u0113n w\u01cengm\u00edn** ( _tsow gehn wahng meeng_ )\n\nSelf-described ordinary citizens (in mainland China) who generally don't have much of a voice, but who are slowly discovering the power of the Internet in helping to bring about social change (mainly in less sensitive areas like environmental protection, animal rights, labor rights, property rights, and rights for the disabled. Certain areas, however, cannot be touched without suffering consequences, such as attempting to organize politically). Literally \"grassroots netizens\" (a translation from the English).\n\n\u6652\u9ed1\u65cf **sha\u00ech\u0113i z\u00fa** ( _shy hay dzoo_ )\n\nLiterally \"those who expose injustice.\" It refers to people on the Chinese mainland who use the Internet to help expose or publicize injustice: for example by mass blogging about an incident on many different Web servers in order to overcome possible online censorship. Every Web host seems to have different censoring criteria, and thus something that gets blocked or deleted on one host server may not get censored on another.\n\n\u522b\u592a CNN **bi\u00e9 t\u00e0i CNN** ( _byih tie CNN_ )\n\nLiterally \"don't be too CNN,\" meaning don't lie or distort the truth. When antigovernment rioting broke out in Lhasa in 2008, a perceived pro-Tibet, anti-China bias in Western media coverage provoked a rising tide of nationalism and anger among many Chinese. Someone started a Web site called Anti-CNN.com, with the slogan \"Don't be too CNN!\" and dedicated to showing examples of truth distortion in the Western media's China reporting. For example, one of the most hotly disputed photos, taken from the CNN Web site, portrayed two Tibetan men running away from Chinese military trucks rumbling ominously into town, but Anti-CNN posted the uncropped version of the photo that showed a group of Tibetan rioters throwing stones at the vehicles. Anger about media bias centers on CNN in particular because not long after the riots the CNN commentator Jack Cafferty outraged the public by referring to the Chinese as \"a bunch of goons and thugs\" during an on-air discussion about Chinese imports to the United States. Several (extremely dirty, profanity-filled) Chinese rap songs about the evils of \"being too CNN\" are also circulating on the Internet.\n\n\u6811\u6d1e\u8d34 **sh\u00f9 d\u00f2ng ti\u0113** ( _shoo dohng tyih_ )\n\nLiterally \"tree-hole post.\" The Wong Kar Wai movie _In the Mood for Love_ closes with the protagonist traveling to the remains of an ancient temple in Cambodia, finding a small hole among the ruins, and whispering into it. The scene is based on a (perhaps made-up) saying that in the past, people would find a small hole in a tree, hide a secret in it, and then seal the hole with mud, to be kept hidden forever. In online parlance, **sh\u00f9 d\u00f2ng ti\u0113** refers to a popular trend of posting secrets anonymously on the Internet\u2014anything ranging from one's salary to marital problems and beyond. People who do this are called \u6652\u5bc6\u65cf **sha\u00ecm\u00ec z\u00fa** ( _shy me dzoo_ ), literally \"secret revealers,\" or \u6652\u5ba2\u65cf **sha\u00eck\u00e8 z\u00fa** , literally \"information exhibitionists.\"\n\n\u6211\u51fa\u6765\u6253\u9171\u6cb9\u7684 **w\u01d2 ch\u016bla\u00ed d\u01ce ji\u00e0ngy\u00f3u de** ( _wuh choo lie dah jyung yo duh_ )\n\nLiterally, \"I'm just out buying soy sauce.\" This phrase, along with \"soy sauce guy,\" swept the Chinese Internet by storm thanks to a widely viewed Guangzhou TV news clip of a reporter asking an average man on the street his opinion of the latest celebrity scandal. The man famously replied \"\u5173\u6211\u5c4c\u4e8b, \u6211\u51fa\u6765\u6253\u9171\u6cb9\u7684\" \" **Gu\u0101n w\u01d2 di\u01ceo sh\u00ec** , **w\u01d2 ch\u016blai d\u01ce ji\u00e0ngy\u00f3u de** \" ( _gwun wuh dyow shih_ , _wuh choo lie dah jyung yo duh_ )\u2014\"I don't give a shit, I'm just out buying soy sauce.\" Internet users have taken up the phrase \"buying soy sauce\" as a cynical euphemism for \"It's none of my business\" or \"Who gives a fuck?\" On the Chinese version of Facebook and other popular sites, one now commonly finds, among the possible answers to online polls, \"I'm just buying soy sauce.\"\n\n\u9171\u6cb9\u7537 **ji\u00e0ngy\u00f3u n\u00e1n** ( _jahng yo nahn_ )\n\nSoy sauce guy. Indicating someone who ignores stupid shit.\n\n\u5f88\u9ec4,\u5f88\u66b4\u529b **h\u011bn hu\u00e1ng, h\u011bn b\u00e0ol\u00ec** ( _hun hwahng, hun baow lee_ )\n\nLiterally \"very yellow [pornographic], very violent.\" This phrase became all the rage because of a news broadcast on CCTV (China Central Television, which is state owned and thus considered a government tool) about government regulation of the Internet. A thirteen-year-old girl being interviewed about her impression of the Internet used the phrase to describe what she had ostensibly seen online. Chinese Internet users have mockingly taken up the phrase and now use it in all sorts of different contexts, or simply use the same sentence structure and substitute different adjectives besides \"yellow\" and \"violent\" to fit whatever they are talking about.\n\n\u522b\u592a CCTV **bi\u00e9 t\u00e0i CCTV** ( _byih tie CCTV_ )\n\nDon't be too CCTV. Meaning don't bullshit or be a tool or espouse propaganda. Came about after the \u5f88\u9ec4,\u5f88\u66b4\u529b **h\u011bn hu\u00e1ng, h\u011bn b\u00e0ol\u00ec** incident (previous entry) on China's state-owned TV station.\n\n\u5f88\u50bb,\u5f88\u5929\u771f **h\u011bn sh\u01ce, h\u011bn ti\u0101nzh\u0113n** ( _hun shah, hun tyinn jen_ )\n\nLiterally \"very foolish, very naive.\" This phrase became popular after Hong Kong pop star Gillian Chung said it during a news conference. She was apologizing after pictures of her having sex with another celebrity, Edison Chen, were exposed on the Internet. Chinese Internet users were amused because it of its similarity to \u5f88\u9ec4,\u5f88\u66b4\u529b **h\u011bn hu\u00e1ng, h\u011bn b\u00e0ol\u00ec** (page 163).\n\n\u505a\u4fef\u5367\u6491 **zu\u00f2 f\u01d4w\u00f2ch\u0113ng** ( _dzwuh foo wuh chung_ )\n\nLiterally \"do push-ups\" and a euphemism for a lame excuse or feelings of apathy. Stems from a June 2008 incident when the drowned body of a fifteen-year-old girl was found in Guizhou Province. The \"push-up\" reference comes from the provincial government's official statement about what happened, which in addition to not being believed by most netizens, included the bizarrely specific detail that one of the last people to see the girl alive, a friend of her boyfriend, started doing push-ups near her on a bridge and that \"on the third one\" she cried out and then jumped. Netizens latched on to the detail, and sarcastic allusions to doing push-ups have flooded the Internet ever since. It generally means \"it's none of my business,\" with a cynical edge that points to the futility of caring about anything when you are powerless to change things, but is sometimes used more lightheartedly to suggest a feeble excuse. \"What were you doing in a sex chat room anyway?\" \"Uh . . . I was just doing push-ups.\"\n\n\u8282\u7ea6\u70b9, \u559d\u8305\u53f0 **ji\u00e9yu\u0113 di\u01cen, h\u0113 M\u00e1ot\u00e1i** ( _jyih yreh dyinn, huh maow tie_ )\n\nThis catchphrase, which means \"economize: drink Moutai,\" comes from a report about a government official in Sichuan Province who beat up a shopkeeper for charging him too much for a bottle of Moutai (a high-end brand of Chinese liquor). It was explained that \"Director Cao wanted to economize, because money is tight at the personnel bureau and he still owes money for house repairs.\" The irony of this statement was not lost on Chinese netizens, and this quickly became the newest Internet meme, as Moutai is strongly associated with government corruption (no shady deal is sealed without a booze-soaked dinner involving copious amounts of this expensive liquor).\n\n\u6076\u641e **\u00e8g\u01ceo** ( _uh gow_ )\n\nAn umbrella term for China's Internet parody culture (including online videos and Photoshop images spoofing current events). Literally \"evil doings\" or \"restless work.\" It comes from the Japanese word _kuso_ , which spread first to Taiwan and then throughout greater China. _Kuso_ means \"shit\" and also \"poor quality\" and described a Japanese fad for appreciating shitty computer games (similar to when we call a movie \"so bad it's good\"). Since these terrible computer games were often unintentionally funny, in Taiwan the meaning eventually shifted to include anything ridiculous or funny.\n\n\u7f51\u53cb **w\u01cengy\u01d2u** ( _wahng yo_ )\n\nInternet friend(s). Making friends via the Internet is much, much more common in China than in the West. Westerners chatting online in China are often startled by the large number of messages they receive from Chinese strangers who are just searching for new friends to chat with. Some of these Internet friends are simply people to chat with while idling away the hours at work, some meet up in person, and some start relationships. Some Chinese even meet people by tex ting random phone numbers in the hopes of happening upon someone friendly.\n\n\u7f51\u763e **w\u01cengy\u01d0n** ( _wahng een_ )\n\nInternet addiction. In 2008 China became the first country in the world to officially recognize this as a clinical disorder, similar to alcoholism and compulsive gambling, after several well-publicized cases in which young people died after spending days or weeks glued to the computer screen in Internet caf\u00e9s. The country has several officially licensed Internet addiction clinics and has also seen a spate of unlicensed, hidden Internet caf\u00e9s where kids banned from the Internet by their parents secretly go to play games.\n\n#  **Emoticons and expressions**\n\n\u989c\u6587\u5b57 **y\u00e1n w\u00e9nz\u00ec** ( _yen when dz_ )\n\nEmoticon. Literally \"face character.\"\n\n\u51f8\n\nIn normal written Chinese, this is the character \u51f8 **t\u016b** ( _too_ ), meaning \"convex.\" It is frequently used on the Internet as an emoticon, however, because it looks like a hand giving the middle finger.\n\nOrz\n\nMeant to look like a person kneeling on the ground, on hands and knees, with head bowed\u2014the _O_ is the person's head, _r_ is the arm and torso, and _z_ is the bent leg. Used to express shock, hopelessness, frustration, despondence, or, more positively, respect or awe\u2014basically any emotion that might be suggested by a kneeling figure.\n\nA few variations (among many) include:\n\nszQ (Orz kneeling in the opposite direction and licking the floor)\n\nOroz (Orz with a fat stomach)\n\n\u56e7\n\nAn emoticon indicating sadness, frustration, shock, or amusement. The character \u56e7 **ji\u01d2ng** ( _jyohng_ \u2014the _o_ sound is long), which dates back to ancient times, originally meant \"bright\" but has taken on this new meaning because it looks like a sad face (or a shocked or amused face, depending on your interpretation). \" **Ji\u01d2ng** culture\" has taken off as a full-fledged fad that has spilled over into real life\u2014the character can even be found on T-shirts, bags, and other accessories.\n\nA few variations (of many) include:\n\n\u5546 ( **ji\u01d2ng** wearing a bamboo hat) d \u56e7 b ( **ji\u01d2ng** with a thumbs-up on either side of its face, from a Pepsi marketing campaign called \"Love China\")\n\n\u56e7rz (Orz combined with **ji\u01d2ng** , so that the kneeling person has a **ji\u01d2ng** face)\n\n\u69d1 **me\u00ed** ( _may_ )\n\nStupefied, shocked. This obscure character dates back to ancient times and means \"plum.\" It is made of two characters for \u5446 **d\u0101i** ( _die_ ) next to each other, and since \u5446 alone can mean something like \"dumb\" or \"astounded\" or \"foolish,\" having two next to each other doubles the degree.\n\n\u96f7 **le\u00ed** ( _lay_ )\n\nLiterally \"thunder.\" Used to indicate shock or surprise or outrage (or any emotion that might be represented by the image of someone being thunderstruck). Moreover, on both the Internet and in real life \u96f7\u4eba **l\u00e9ir\u00e9n** ( _lay ren_ ), literally \"thunder person,\" has come to mean \"outrageous\" or \"shocking\" or \"absurd.\" One especially common expression is \u592a\u96f7\u4eba\u4e86 **t\u00e0i l\u00e9ir\u00e9n le** ( _tie lay ren luh_ ), meaning \"too outrageous\" or \"that's so stupid\" or \"that's insane.\"\n\n\u6c57 **h\u00e0n** ( _hahn_ )\n\nMeans \"sweat\" and is used, usually in reply, to indicate feeling embarrassed or dumbfounded (that is, an emotion that makes you sweat).\n\n\u6cea **l\u00e8i** ( _lay_ )\n\nMeans \"tear\" or \"teardrop\" and used online to express sadness or crying.\n\n\u6655 **y\u016bn** ( _een_ )\n\nMeans \"dizzy\" or \"faint\" and is often used to express surprise, shock, amusement, or disgust; that is, emotions that might make you feel faint.\n\n\u6211\u5012 **w\u01d2 d\u01ceo** ( _wuh daow_ )\n\nLiterally, \"I fall over.\" Used the same way as **y\u016bn** (above).\n\nVoV\n\nRepresents a person holding up two peace signs on either side of his or her face.\n\n\u95ea\n\nThe character \u95ea **sh\u01cen** ( _shahn_ ) means \"flash\" and is used in online chatting to mean \"leaving,\" as when you leave a chat room\u2014you might say \u6211\u95ea\u4e86 **w\u01d2 sh\u01cen le** ( _wuh shahn luh_ ), \"I'm leaving\"\u2014or \"avoid doing something.\"\n\n#  **Insults and mockery**\n\nBS\n\nUsually means the English \"bullshit\" but may also stand for \u9119\u89c6 **b\u01d0sh\u00ec** ( _bee shih_ ), which literally means \"despise.\"\n\nJP\n\nStands for \u6781\u54c1 **j\u00edp\u01d0n** ( _gee peen_ ), literally \"extreme conduct.\" Basically means \"weirdo\" or, rather, an eccentric or anyone who behaves unconventionally.\n\nLB\n\nStands for \u7bf1\u7b06 **l\u00edb\u0101** ( _lee bah_ ), the name of an online forum (Liba) known for being popular among materialistic girls. Vapid, shallow girls, then, are often dismissively described as\n\nLB.\n\nBC\n\nStands for \u767d\u75f4 **b\u00e1ich\u012b** ( _buy chih_ ), or \"idiot.\"\n\nSL\n\nStands for \u8272\u72fc **s\u00e8l\u00e1ng** ( _suh lahng_ ), or \"pervert.\"\n\n\u6c38\u4e45\u6027\u8111\u6b8b **y\u01d2ngj\u01d0u x\u00ecng n\u01ceoc\u00e1n** ( _yohng joe sheeng now tsahn_ )\n\nLiterally means something like \"permanent brain damage\" and used to comment on someone or something you find weird or crazy.\n\n\u5c0f\u767d **xi\u01ceob\u00e1i** ( _shaow buy_ )\n\nLayman, novice, someone with little knowledge about something (such as computers). Literally \"little white.\" Refers to \u767d\u75f4 **b\u00e1ich\u012b** ( _buy chih_ ), which means \"idiot.\"\n\n\u83dc\u9e1f **ca\u00ecni\u01ceo** ( _tsie nyow_ \u2014 **ca\u00ec** rhymes with \"pie\")\n\nNewbie. Literally \"food bird.\" Pigeons raised for food are of a lower class than homing pigeons, and thus the phrase is used to describe junior-level people in various fields.\n\n\u56fd\u732a **gu\u00f3zh\u016b** ( _gwuh joo_ )\n\nLiterally \"country's pigs\" or \"national pigs.\" An insulting pun on \u56fd\u8db3 **gu\u00f3z\u00fa** ( _gwuh dzoo_ ) ( **zh\u016b** and **z\u00fa** sound similar), which is the term for China's national soccer team\u2014a target of much abuse among soccer fans due to its poor performance and frequently scandalous or embarrassing behavior. Speaking of which . . .\n\n\u6211\u662f\u6d17\u6fa1\u6765\u7684 **w\u01d2 sh\u00ec x\u01d0z\u01ceo la\u00ed d\u00e8** ( _wuh shih she dzow lie duh_ )\n\nLiterally, \"I just came here to take a shower.\" Refers to a 2008 scandal when members of the Chinese national soccer team got caught checking into a hotel with a bunch of women who may or may not have been hookers. One of them protested by saying, \"I just came here to take a shower.\" Now the phrase is often used as a euphemism for any feeble excuse.\n\n\u6050\u9f99 **k\u01d2ngl\u00f3ng** ( _kohng lohng_ \u2014like \"Kong\" and \"long\" but pronounced with a long _o_ , or _oh_ , sound)\n\nOnline slang for an ugly girl. Literally \"dinosaur.\"\n\n\u9752\u86d9 **q\u012bngw\u0101** ( _ching wah_ )\n\nOnline slang for an ugly guy. The counterpart to \"dinosaur\" (above). Literally means \"frog.\"\n\n\u62cd\u7816 **pa\u012bzhu\u0101n** ( _pie jwahn_ )\n\nLiterally means \"smack with a brick.\" Used online when strongly criticizing someone or something. For example, you might say \u8bf7\u62cd\u7816 **q\u01d0ng p\u0101izhu\u0101n** ( _ching pie jwahn_ ), literally \"please smack with a brick,\" to mean \"prove it\"; or \u6211\u62cd\u7816 **w\u01d2 p\u0101izhu\u0101n** ( _wuh pie jwahn_ ), literally \"I smack with a brick,\" to mean \"here's what I think.\"\n\n#  **Dirty\/lewd\/slangy**\n\n\u725b **ni\u00fa** ( _nyoo_ )\n\nLiterally means \"cow\" and used to call someone or something \"awesome.\" Short for **ni\u00fab\u012b** (see chapter 1), which means \"fuckin' awesome.\"\n\nNB\n\nStands for \u725b\u5c44 **ni\u00fab\u012b** ( _nyoo bee_ ), which can negatively mean something like \"cocky bastard\" or positively refer to a daring, impressive person or thing\u2014someone or something that is \"fuckin' badass\" (see chapter 1). Is also frequently written using the character \u725b plus any other Chinese character pronounced _bee_ , such as \u903c **b\u012b** , \"to force,\" or \u6bd4 **b\u01d0** , an indication of comparison. The real character \u5c44 **b\u012b** ( _bee_ ) cannot be typed on most computers and thus is almost never used.\n\n\u88c5 B\n\nStands for \u88c5\u5c44 **zhu\u0101ngb\u012b** ( _jwahng bee_ ), which means \"fucking poser\" or \"fake motherfucker.\" \u88c5 **Zhu\u0101ng** ( _jwahng_ ) means \"pretend\" and **b\u012b** indicates **ni\u00fab\u012b** (see previous entry and chapter 1), meaning \"fucking badass.\" Thus the term implies that you are pretending to be badass. Some online users write \"IB\" or \"install B\" instead because **zhu\u0101ng** can also mean \"install.\"\n\nSB or \u50bb B\n\nStands for \u50bb\u5c44 **sh\u01ceb\u012b** ( _shah bee_ ), which means something like \"stupid cunt\" or \"motherfucker.\" Literally \"idiot's pussy.\" Also frequently written \u50bb\u903c **sh\u01ceb\u012b** or \u50bb\u6bd4 **sh\u01ceb\u01d0** ; that is, substituting a similar-sounding character for \u5c44.\n\nTMD\n\nStands for \u4ed6\u5988\u7684 **t\u0101m\u0101de** ( _ta ma duh_ ), literally \"his mother's,\" implying \"fuck his mother's pussy.\" A moderately strong curse word. Sometimes shortened to MD, for \u5988\u7684 **m\u0101de** ( _ma duh_ ), literally \"mother's.\"\n\nSome of these abbreviations can be combined, for example, TMDSB stands for \u4ed6\u5988\u7684\u50bb B **t\u0101m\u0101de sh\u01ceb\u012b** ( _ta ma duh shah bee_ ), which is equivalent to saying something like \"fucking stupid\" or \"stupid-ass shit\" or \"goddamn stupid cunt.\"\n\nNND or TNND\n\nStands for \u5976\u5976\u7684 **na\u01d0na\u01d0 de** ( _nigh nigh duh_ ) or \u4ed6\u5976\u5976\u7684 **t\u0101 na\u01d0na\u01d0 de** ( _ta nigh nigh duh_ ). Literally \"grandmother's\" or \"his grandmother's\" and implies the larger phrase \"fuck your grandmother.\" However, this fragment alone means something slightly less profane, closer to \"shit\" or \"damn.\"\n\nQQXX\n\nStands for \u6211\u5708\u5708\u4f60\u4e2a\u53c9\u53c9 **w\u01d2 qu\u0101n qu\u0101n n\u01d0 g\u00e8 ch\u0101 ch\u0101** ( _wuh chren chren nee guh cha cha_ ), a euphemistic way of saying \"fuck you.\" \u5708 **Qu\u0101n** ( _chren_ ), literally \"circle,\" refers to a penis here, while \u53c9 **ch\u0101** ( _cha_ ), literally \"cross,\" refers to pussy. Literally, it means \"I circle your cross.\"\n\nG8 or JB\n\nMeans \"dick\" or \"cock.\" In the first case, because _G_ and the number eight in Chinese, **b\u0101** ( _bah_ ), it sounds like \u9e21\u5df4 **j\u012bba** ( _gee bah_ ), which is a slang term for \"dick.\" The second is just an abbreviation based on the first letters of the pinyin syllables in **j\u012bba**.\n\nJJ\n\nStands for \u9e21\u9e21 **j\u012bj\u012b** ( _gee gee_ ), a kiddie term for \"penis,\" like \"wee-wee.\" Can also stand for \u59d0\u59d0 **ji\u011bjie** ( _jyih jyih_ ), which means \"older sister\" and refers to any female who is older than the person addressing her, though generally still a young woman.\n\nSY\n\nStands for \u624b\u6deb **sh\u01d2uy\u00edn** ( _show een_ ), which means \"jerk off\" (verb).\n\nGC\n\nStands for \u9ad8\u6f6e **g\u0101och\u00e1o** ( _gaow chow_ ), which means \"orgasm,\" \"climax,\" \"come.\"\n\nJY\n\nStands for \u7cbe\u6db2 **j\u012bngy\u00e8** ( _jing yeh_ ), which means \"semen,\" \"sperm,\" or \"ejaculate.\"\n\nYY\n\nStands for \u610f\u6deb **y\u00ecy\u00edn** ( _ee een_ ) and means something like \"to have pervy thoughts\" or \"think dirty\" but can also have a less negative connotation that just indicates thinking or behaving unconventionally or creatively.\n\nYD\n\nShort for \u6deb\u8361 **y\u00ednd\u00e0ng** ( _een dahng_ ), meaning \"lascivious\" or \"lewd.\"\n\n#  **More abbreviations and language play**\n\nPP\n\nCan mean, variously, \u6f02\u6f02 **pi\u0101opiao** ( _pyow pyow_ ), \"pretty\"; \u7247\u7247 **pi\u00e0npian** ( _pyinn pyinn_ ), \"picture\"; or \u5c41\u5c41 **p\u00ecpi** ( _pee pee_ ), \"butt.\" Thus someone could conceivably write, \"I've seen her PP. She's not very PP, but she's got a nice PP.\"\n\nMM\n\nStands for \u59b9\u59b9 **me\u00ecme\u00ec** ( _may may_ ), which literally means \"little sister\" but can generally refer to a girl or young woman. Use carefully, however, as it can also refer more euphemistically to a woman's own \"little sister\"; i.e., her vagina. Can also stand for \u7f8e\u7709 **m\u0115im\u00e9i** ( _may may_ ), which derives from **me\u00ecme\u00ec** but is written with different characters that instead mean \"beautiful eyebrow\" and is slang for a pretty girl. Along these same lines, you'll also see abbreviations for all the other types of people you might be talking about online: LG for \u8001\u516c **l\u01ceog\u014dng** ( _laow gohng_ ), \"husband\"; LP for \u8001\u5a46 **l\u01ceop\u00f3** ( _laow pwuh_ ), \"wife\"; GG for \u54e5\u54e5 **g\u0113ge** ( _guh guh_ ), \"older brother\" or any older male; JM or JMs for \u59d0\u59b9\u4eec **ji\u011bm\u00e8imen** ( _jyih may men_ ), \"sisters\"; and DD for \u5f1f\u5f1f **d\u00ecdi** ( _dee dee_ ), \"little brother\" or any younger male. However, MM is the most common abbreviation you'll see, given that the overwhelming majority of Chinese Internet users are male and, of course, endlessly preoccupied with MM.\n\nML\n\nMake love.\n\nPMP\n\nStands for \u62cd\u9a6c\u5c41 **pa\u012bm\u01cep\u00ec** ( _pie mah pee_ ), literally \"pat the horse's butt,\" and means to flatter or suck up to someone.\n\n\u6c57\u8bed **h\u00e0ny\u01d4** ( _hahn yee_ )\n\nChat room jargon, Internet slang. A play on \u6c49\u8bed **h\u00e0ny\u01d4** ( _hahn yee_ ), which is pronounced exactly the same but means Mandarin Chinese. The only difference is that \u6c49 **h\u00e0n** ( _hahn_ ) is replaced with the character for \"sweat,\" \u6c57 **h\u00e0n** ( _hahn_ ), which is itself online jargon for feeling embarrassed, shocked, or frustrated.\n\n51.com\n\n\"I want\" dot com. \"Five one\" in Chinese is **w\u01d4 y\u00e0o** ( _wooh yow_ \u2014which might not make sense if you're familiar with the Taiwanese pronunciation, \"one\" is usually pronounced **y\u012b** ( _eee_ ), but mainland Chinese usually say **y\u00e0o** instead of **y\u012b** when saying the number out loud). **W\u01d4 y\u00e0o** sounds a bit like the Chinese for \"I want,\" \u6211\u8981 **w\u014f y\u00e0o** ( _wuh yow_ ). So many Chinese URLs begin with the number fifty-one. Hence there is a job-search site with the URL www.51job.com (i.e., I want a job), and a credit card consultancy with the URL www.51credit.com (I want credit).\n\n88\n\nBye-bye. The number eight in Chinese is **b\u0101** ( _bah_ ), which sounds to Chinese ears like the English word \"bye.\" Another variation is 886, **b\u0101 b\u0101 li\u00f9** ( _bah bah lew_ \u2014the last syllable rhymes with \"ew\"), and is similar to \u62dc\u62dc\u55bd **b\u00e0ib\u00e0i l\u00f3u** ( _bye bye low_ ), which also means \"bye-bye.\"\n\n555\n\nThe sound of crying. The number five in Chinese is **w\u01d4** ( _wooh_ ), which sounds like \u545c **w\u016b** ( _ooh_ ), an onomatopoeic word for humming or crying. Thus \u545c\u545c\u545c, or 555, represents the sound of someone crying. Insert as many fives as you deem necessary to adequately represent the volume of your tears.\n\n3Q\n\nThank you. The number three in Chinese is **s\u0101n** ( _sahn_ ) and combined with the English letter _Q_ ( _sahn-cue_ ) sounds to Chinese ears like \"thank you.\"\n\n3X\n\nThank you. X refers to \u8c22\u8c22 **xi\u00e8xi\u00e8** ( _shih shih_ ), which is the Chinese for \"thank you.\"\n\nNoQ\n\nYou're welcome. It doesn't make sense aurally but is based on 3Q (above), meaning \"thank you.\"\n\n520\n\nI love you. \"Five two zero\" in Chinese is **w\u01d4 \u00e8r l\u00edng** ( _oo er ling_ ), which sounds somewhat similar to \u6211\u7231\u4f60 **w\u01d2 \u00e0i n\u01d0** ( _wuh aye nee_ ), the Chinese for \"I love you.\"\n\n1314\n\nMeans \"forever\" because the pronunciation of \"one three one four\" in Chinese, **y\u012b s\u0101n sh\u00ed s\u00ec** ( _ee sahn ee sih_ ), sounds similar to \u4e00\u751f\u4e00\u4e16 **y\u012bsh\u0113ng y\u012b sh\u00ec** ( _ee shung ee shih_ ), which means \"forever.\" Most commonly used in expressions of love, in particular 5201314; i.e., \"I love you forever\" (see above).\n\n8147\n\nStands for \u4e0d\u8981\u751f\u6c14 **b\u00f9 y\u00e0o sh\u0113ngq\u00ec** ( _boo yow shung chee_ ), \"don't be angry,\" because they sound similar\u2014as do the following:\n\n360\n\n\u60f3\u5ff5\u4f60 **Xi\u01cengni\u00e0n n\u01d0** ( _shyahng nyinn nee_ ), \"miss you.\"\n\n246\n\n\u997f\u6b7b\u4e86 **\u00c9 s\u01d0 le** _(uh sih luh_ ), \"I'm hungry.\"\n\n7456\n\n\u6c14\u6b7b\u6211\u4e86 **Q\u00ecs\u01d0w\u01d2le** ( _chee sih wuh luh_ ), \"I'm so angry!\"\n\nGL\n\nShort for \"girl love,\" referring to lesbians or lesbianism.\n\nBL\n\nShort for \"boy love,\" referring to gay men or homosexuality.\n\nMF\n\nStands for \u9ebb\u70e6 **m\u00e1fan** ( _ma fun_ ), meaning \"trouble\" or \"hassle.\"\n\nTAXI\n\nStands for \u592a\u53ef\u60dc **t\u00e0i k\u011bx\u012b** ( _tie kuh she_ ), meaning \"what a pity\" or \"that sucks.\"\n\nFL\n\nStands for \u53d1\u5eca **f\u00e0l\u00e1ng** ( _fah lahng_ ), or \"hair salon,\" referring to prostitutes who work under the guise of a hairdresser. Hair salons in China that are actually brothels are usually (usually) recognizable by their pink lighting.\n\n\u7c89\u4e1d **f\u011bns\u012b** ( _fen sih_ )\n\nLiterally \"vermicelli\" (a type of noodle) but used online to mean \"fans\" because it sounds similar. Other variants include just \u7c89 **f\u011bn** ( _fen_ ) by itself and \u996d **f\u00e0n** ( _fahn_ ), literally \"rice\" or \"meal.\"\n\n\u6478\u6211 **m\u014d w\u01d2** ( _mwuh wuh_ )\n\nLiterally means, \"touch me.\" Used to say \"MSN me\" (i.e., chat me via MSN). MSN's instant messaging service is immensely popular in China, and the first letter of MSN sounds to Chinese ears like the word \u6478 **m\u014d** ( _mwuh_ ), which means \"touch.\"\n\n\u5f97\u4f53 **d\u00e9t\u01d0** ( _duh tee_ )\n\nLiterally means \"good and proper,\" but because it sounds like the English word \"dirty,\" which has the opposite meaning, the word has been punned on and joked about to the extent that it has now taken on the alternate meaning of \"someone who seems good on the outside but is actually bad on the inside.\" This new usage has been popularized by the song \"Dirty\" by Taiwanese-American singer Lee-Hom Wang.\n\n#  **Praise**\n\nBT\n\nStands for \u53d8\u6001 **bi\u00e0nt\u00e0i** ( _byinn tie_ ), which means \"perverted\" or \"deviant\" and once referred to homosexuality, sexual fetishes, people with an abnormal fixation on violence, etc. In online culture, however, it has now taken on a joking or positive connotation; thus calling someone BT is akin to cheerfully saying something like \"you pervert\" or \"you weirdo\" to a friend, or like saying \"you're so bad\" when what you really mean is that you're impressed.\n\nPF\n\nStands for \u4f69\u670d **p\u00e8if\u00fa** ( _pay foo_ ), meaning \"admire.\"\n\n\u8d5e **z\u00e0n** ( _dzahn_ )\n\nMeans \"to praise\" and is often used online when recommending or raving about a movie, a story, etc.\n\n94\n\nAgreed, I agree. \"Nine four\" in Chinese is **j\u01d0u s\u00ec** ( _joe sih_ ), which sounds like the phrase \u5c31\u662f **ji\u00f9sh\u00ec** ( _joe shih_ ), which means \"yes\" or \"it's true.\"\n\nPL\n\nStands for \u6f02\u4eae **pi\u00e0oli\u00e0ng** ( _pyow lyahng_ ), or \"pretty.\"\n\nPPMM\n\nStands for \u6f02\u6f02\u59b9\u59b9 **pi\u00e0opi\u00e0o m\u00e8im\u00e8i** ( _pyow pyow may may_ ), a cutesy way of saying \"pretty girl.\"\n\nML\n\nStands for \u7f8e\u4e3d **m\u011bil\u00ec** ( _may lee_ ), or \"beautiful.\"\n\n#  **Miscellaneous**\n\n\u706b\u661f\u6587 **hu\u01d2x\u012bngw\u00e9n** ( _hwuh sheeng when_ )\n\nInternet or text-messaging shorthand, such as \"lol,\" \"Cul8r,\" and \"b4.\" Literally \"martian language\" because the hodgepodge use of numbers, symbols, made-up words, and letters from other languages looks like a new, foreign (or interplanetary, thus martian) language.\n\n\u706b\u661f\u4eba **hu\u01d2x\u012bngr\u00e9n** ( _hwuh sheeng ren_ )\n\nA martian. That is, someone out of touch with reality or with the latest news and trends. A commenter might jokingly (or pejoratively) reply to such a person, \"\u4f60\u662f\u706b\u661f\u56de\u6765\u7684\u5417?\" \" **N\u01d0 sh\u00ec hu\u01d2x\u012bng hu\u00edlai de ma?** \" ( _nee shih hwuh sheeng hway lie duh ma_ ): \"Did you just get back from Mars?\" In general, Chinese Internet users frequently make jokes about being from Mars in response to things they find funny or bizarre.\n\n\u706b\u661f\u8d34 **hu\u01d2x\u012bng ti\u0113** ( _hwuh sheeng tyih_ )\n\nLiterally \"post from Mars.\" Refers to an extremely old post. If someone posts something old that everyone's seen before, someone might comment, \"This is a post from Mars.\"\n\nLZ\n\nStands for \u697c\u4e3b **l\u00f3uzh\u01d4** ( _low joo_ ), which refers to the author of a post or the person who starts a BBS thread. So you might write something like, \"I agree with LZ.\" Literally means \"building owner\" or \"owner of the house.\"\n\n\u6c99\u53d1 **sh\u0101f\u0101** ( _shah fa_ )\n\nLiterally means \"sofa\" and refers to the first person to reply to a post. Since LZ (page 182) is the \"owner of the house,\" the first person to reply, or enter the house, gets the sofa. Sometimes just written in English, as \"sofa.\" The next commenter after the \"sofa\" is referred to as \u5750\u677f\u51f3 **zu\u00f2 b\u01cend\u00e8ng** ( _zwuh bahn dung_ ), meaning \"sitting on a bench.\"\n\n\u9ad8\u697c **g\u0101ol\u00f3u** ( _gaow low_ )\n\nLiterally means \"tall building\" and used online to refer to a topic or post that attracts hundreds of replies, making the thread taller and taller, like a high-rise building.\n\n\u6591\u7af9 **b\u0101nzh\u00fa** ( _bahn joo_ )\n\nLiterally \"bamboo\" but used to refer to a BBS moderator because it is pronounced exactly like the real term for the moderator, which is \u7248\u4e3b **b\u0101nzh\u00fa** ( _bahn joo_ ).\n\nRT\n\nStands for \u5982\u9898 **r\u00fat\u00ed** ( _roo tee_ ), which means \"refer to the title or subject.\" A common response when someone asks a stupid question, as in, \"Look at the title and subject\u2014the answer is obvious from that.\"\n\n286\n\nOut of fashion, out of date, old-fashioned. Refers to an old, and thus outdated, computer chip from the 1980s (the Intel 80286).\n\n\u6b7b\u673a **s\u01d0j\u012b** ( _sih gee_ )\n\nMeans \"unexpected computer shutdown.\" Literally \"dead machine,\" but now more widely used to indicate being so dumbfounded by something that you can't even respond.\n\n\u871c **m\u00ec** ( _me_ ) and \u9ed1 **h\u0113i** ( _hay_ )\n\n**M\u00ec** means \"honey\" and is tacked onto a word to indicate fervent support for a certain athlete or sports team, the way we might use \"freak,\" as in \"He's a total Raiders freak.\" **H\u0113i** means \"black\" and is used the same way, but to indicate hatred.\n\n\u6652 **sh\u00e0i** ( _shy_ )\n\nLiterally means \"to air\" or \"to sun\" and can thus suggest \"to show.\" Used online to refer to the popular phenomenon of netizens photographing their stuff (some women, for example, like to photograph their extensive and very expensive collection of cosmetics and beauty products) and posting the pictures online to show off, prompt discussions about favorite products, share recommendations, etc. Another common variation is \u517d **sh\u00f2u** ( _show_ ), which means \"beast\" but sounds exactly like the English word \"show.\"\n\n\u957f\u8349 **zh\u01ceng c\u01ceo** ( _dzahng tsow_ )\n\nLiterally \"grow grass.\" On the Chinese Internet, feelings of yearning or want are described as \"grass growing in the heart,\" and netizens often use the expression when they see things they want that other netizens **sh\u00e0i** or **sh\u00f2u** (see above) online.\n\n\u4e09\u624b\u75c5 **s\u0101n sh\u014fu b\u00ecng** ( _sahn show bing_ )\n\nLiterally \"three hands illness.\" Describes tiredness of the hand due to excessive computer use. A person who spends too much time gaming or online is called a \u4e09\u624b **s\u0101n sh\u014fu** ( _sahn show_ ). An equivalent Western concept might be \"Blackberry thumb\": pain caused by typing too much with your thumbs on your Blackberry.\n\n\u5047\u8df3 **ji\u01ceti\u00e0o** ( _jah tyow_ )\n\nUsed online to mean \"lie.\" Literally \"false jump.\" It comes from the role-playing computer game PK: Police and Killer, and it refers to when a policeman in the game pretends that he has mistaken a civilian for the killer.\n\n\u7206\u5934 **b\u00e0ot\u00f3u** ( _baow toe_ )\n\nLiterally \"explode head.\" This term comes from the computer game Counter-Strike, in which it refers to killing an opponent, but has spread beyond that context to mean any sort of unexpected attack or blow.\n\n\u7559\u722a **li\u00fa zhu\u01ce** ( _lyew jwa_ \u2014the first syllable rhymes with \"ew\")\n\nLiterally \"leave a claw mark.\" Refers to posting on BBSs.\n\n\u5931\u5199\u75c7 **sh\u012b xi\u011b zh\u0113ng** ( _shih shyih jung_ )\n\nLiterally \"lose writing illness.\" Refers to the phenomenon of using the computer so much that you forget how to write Chinese characters by hand.\n\n\u51e4\u51f0\u7537 **f\u00e8nghu\u00e1ng n\u00e1n** ( _fung hwahng nahn_ )\n\nLiterally \"phoenix man.\" A newly coined term for a \u7a77\u5c0f\u5b50 **qi\u00f3ng xi\u01ceozi** ( _chyohng shyow dz_ ), or \"poor guy,\" from a rural area whose family scrimps and saves to put him through school so he can go to the big city and find a job. His success is analogized to a rising phoenix.\n\n\u5b54\u96c0\u5973 **k\u01d2ngqu\u00e8 n\u01da** ( _kohng chreh nee\u2014_ the first syllable essentially rhymes with \"cone\" but with the same ending sound as \"long\")\n\nA spoiled city girl who grew up with money. The counterpart to a \"phoenix man\" (above). Literally \"peacock woman\" and a newly coined term for a \u5bcc\u5bb6\u5973 **f\u00f9ji\u0101 n\u01da** ( _foo jah nee_ ), or \"wealthy-family woman.\" Both \"phoenix man\" and \"peacock woman\" are often used as a kind of shorthand in discussions of the culture clash between young people from different backgrounds, a byproduct of China's rapid urbanization. Marriages between a \"phoenix man\" and \"peacock woman,\" and their resultant problems, are an especially widely discussed issue.\n\n\u8349\u8393\u65cf **c\u01ceom\u00e9i z\u00fa** ( _tsow may dzoo_ )\n\nA \u8349\u8393 **c\u01ceom\u00e9i** ( _tsow may_ ) is a strawberry and \u65cf **z\u00fa** ( _dzoo_ ) means \"race,\" \"clan,\" or \"generation\"\u2014basically any wide grouping of people. **C\u01ceom\u00e9i z\u00fa** is a slightly negative nickname for the younger generation (or the 80 \u540e **b\u0101sh\u00edh\u00f2u** ( _bah shih ho_ ), those born \"after 1980\"). Like the fruit, members of the \"strawberry clan\" are good-looking thanks to their youth, confidence, fashionable clothes, and the other trappings of a cushy life but are soft and easily bruised (that is, they don't hold up well under pressure) because they've had it easy all their lives.\n\n\u69b4\u83b2\u65cf **li\u00fali\u00e1n z\u00fa** ( _lyoo lyinn dzoo_ )\n\nDurian clan. A durian is an indigenous Southeast Asian fruit with a tough, thorny husk, which is well-known mostly because it smells horrendous\u2014enough so that it is banned on the subway in Singapore. It is perhaps this negative perception surrounding the fruit that has inspired younger people from the \"strawberry clan\" to give this moniker to what they consider the unprogressive and out-of-touch older generation.\n\n\u6930\u5b50\u65cf **y\u0113zi z\u00fa** ( _yeh dz dzoo_ )\n\nCoconut clan. A moniker for young people who are the opposite of the strawberry clan\u2014able to work very hard and \"eat bitterness\" because of their tough husks. Why exactly Chinese seem to love categorizing people with names of fruits is unclear, but it means that bewildering comments like these are common on Internet forums: \"Not just the strawberry generation, but the generation of children who will graduate college in 2010. How will they know how to face the world's realities? Strawberries will seem cactuses compared to what these delicate orchids will be once they are cast adrift on the seas of real life's waters.\"\n\n\u5c71\u5be8 **sh\u0101nzh\u00e0i** ( _shahn jie_ \u2014 _jie_ rhymes with \"die\")\n\nLiterally \"mountain stronghold,\" alluding to a period in China's history when various areas were controlled by renegade warlords (with mountain strongholds); that is, outside official control. Today **sh\u0101nzh\u00e0i** retains that renegade idea but means \"knockoff\" or \"fake.\" It can also mean \"inferior\" or \"cheap,\" though more recently the word has taken on a more positive connotation, suggesting ingenuity and a sense of humor, as people begin to embrace \" **sh\u0101nzh\u00e0i** culture.\" Knockoff mobile phone makers extoll the **sh\u0101nzh\u00e0i** nature of their products, arguing that they are making high-end products accessible to the masses, and some companies even cheekily use **sh\u0101nzh\u00e0i** spokespeople\u2014that is, celebrity look alikes\u2014to endorse their products. One especially hilarious example of **sh\u0101nzh\u00e0i** culture's tongue-in-cheek nature: KFC's Chinese name is \u80af\u5fb7\u57fa **K\u011bn D\u00e9 J\u012b** ( _ken duh gee_ ), but one **sh\u0101nzh\u00e0i** business, also serving fried chicken and fast food, calls itself \u5543\u4ed6\u9e21 **K\u011bn T\u0101 J\u012b** ( _ken tah gee_ ), which sounds similar but means \"nibble his chicken\" and is a dirty double entendre, since \u9e21 **j\u012b** ( _gee_ ) is a slang term for \"penis.\"\n\n\u6781\u54c1\u5973 **j\u00edp\u01d0n n\u01da** ( _gee peen nee_ ) and \u6781\u54c1\u7537 **j\u00edp\u01d0n n\u00e1n** ( _gee peen nahn_ )\n\nLiterally \"extremely great woman\" and \"extremely great man\" but often used sarcastically on the Internet to mean someone who is fussy or annoying.\n\n\u5b85\u5973 **zh\u00e1i n\u01da** ( _jigh nee_ ) and \u5b85\u7537 **zh\u00e1i n\u00e1n** ( _jigh nahn_ )\n\nA woman or man, respectively, who stays indoors all day and spends all her or his time on the Internet. A Japanese slang term (written with the same characters) that spread first to Taiwan and is now frequently used by Internet users all over China.\nThe Top Twenty-five Terms You Need to Know\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTranscribed from the 1888 Hodder and Stoughton edition by David Price,\nemail ccx074@pglaf.org\n\n\n\n\n\n                             CHRISTMAS EVANS:\n                       The Preacher of Wild Wales.\n\n\n                  _HIS COUNTRY_, _HIS TIMES_, _AND HIS_\n                            _CONTEMPORARIES_.\n\n                                * * * * *\n\n                               BY THE REV.\n                               PAXTON HOOD,\n\n                                AUTHOR OF\n        \"THE THRONE OF ELOQUENCE,\" \"WORLD OF PROVERB AND PARABLE,\"\n               \"THE WORLD OF ANECDOTE,\" \"ROBERT HALL,\" ETC.\n\n                                * * * * *\n\n                             _THIRD EDITION_.\n\n                                * * * * *\n\n                                 London:\n                          HODDER AND STOUGHTON,\n                           27, PATERNOSTER ROW.\n\n                                * * * * *\n\n                              MDCCCLXXXVIII.\n\n                         [_All rights reserved_.]\n\n                                * * * * *\n\n                                * * * * *\n\n         Hazell Watson and Viney, Printers, London and Aylesbury\n\n\n\n\nTO THE REV. JOHN DAVIES, OF BRIGHTON.\n\n\nMY DEAR FRIEND,\u2014I believe there is no man living to whom I could so\nappropriately inscribe an attempt to give some appreciation of the life\nand labours of Christmas Evans as yourself.  Your revered father and he\nwere taken on the same evening into Church fellowship in the old\ncommunion of Castell Hywel, and within a week of each other they preached\ntheir first sermons from the same desk; after this their ways diverged,\nEvans uniting himself with the Baptist Communion, your father joining the\nIndependent; still, like two rivers flowing, and broadening, from\nneighbouring, but obscure springs in the heart of their native\nPlynlymmon, cheerfully they ran their beautiful course, beneath the\nprovidential law of Him who chooses our inheritance for us, and fixes the\nbounds of our habitations.  They both served their generation in their\nown land well, before they fell on sleep.  Your father was called \"the\nSilver Trumpet of Wales,\" and the name of Evans rolled like a\nfar-resounding bell among its wild mountains.  In their early Christian\nlife they were associates; in their fame, while living, competent judges\ntell me they were equal; and I have brought them together again.  In the\nmemories I have sought to retain in this volume, I have attempted to give\nsome idea of what old Wild Wales was when these two brothers in arms\narose, and I have attempted to show what the singular institution of\npreaching effected for the old insulated land.  But I am also glad to\navail myself of the opportunity thus afforded me to express my sense of\nmingled admiration, and affection for yourself, and congratulation that\nthe father, who left you an orphan so young, must rejoice, from that\ncloud of witnesses he so long since joined, to know that you followed him\nin a successful and happy ministry; while I rejoice, that, unlike him,\nyou have been permitted to enjoy the sunset in a serene and golden old\nage.  May you long enjoy it.\n\n                                                           My Dear Friend,\n                                                  I am very affectionately\n                                                        EDWIN PAXTON HOOD.\n\n\n\n\nCONTENTS.\n\n                          CHAPTER I.\n       SOME GENERAL CHARACTERISTICS OF WELSH PREACHING.\n                                                          PAGE\nWales, the Country and the People\u2014Individuality of           1\nthe Welsh Pulpit\u2014St. David\u2014The Religious Sense of the\nPeople\u2014Association Meetings\u2014Gryffyth of\nCaernarvon\u2014Bardic Character of the Sermons\u2014A\nRepetition of Sermons\u2014Peculiarities of the Welsh\nLanguage\u2014Its Singular Effects as spoken\u2014Its\nVowels\u2014Its Pictorial Character\u2014The _Hwyl_\u2014Welsh\nScenery\u2014Isolated Character of the Old Chapels\u2014Plain\nLiving and High Thinking\u2014Ludicrous Incidents of\nUncertain Service\u2014Superstitions of\nHeathenism\u2014Fondness of the People for\nAllegory\u2014Haunted Wales\u2014The Rev. John Jones and the\nMysterious Horseman\u2014Old Wild Wales\u2014St.\nDavid's\u2014Kilgerran\u2014Welsh Nomenclature\u2014John Dyer\u2014Old\nCustoms.\n                         CHAPTER II.\n   CHRISTMAS EVANS'S EARLY LIFE UNTIL HIS ENTRANCE INTO THE\n                          MINISTRY.\nBirth and Early Hardships\u2014Early Church                      40\nFellowship\u2014Beginning to Learn\u2014Loses an Eye\u2014A Singular\nDream\u2014Beginning to Preach\u2014His First Sermon\u2014Is\nBaptized\u2014A New Church Fellowship\u2014The Rev. Timothy\nThomas\u2014Anecdotes\u2014A Long Season of Spiritual\nDepression\u2014Is ordained as Home Missionary to\nLleyn\u2014Commencement of Success as a Preacher\u2014Remarks\non Success\u2014Marries\u2014Great Sermon at Velinvoel\u2014A\nPersonal Reminiscence of Welsh Preaching.\n                         CHAPTER III.\n           THE MINISTRY IN THE ISLAND OF ANGLESEA.\nJourney to Anglesea\u2014Cildwrn Chapel, and Life in the         63\nCildwrn Cottage\u2014Poverty\u2014Forcing his Way to\nKnowledge\u2014Anecdote, \"I am the Book\"\u2014A Dream\u2014The\nSandemanian Controversy\u2014Jones of Ramoth\u2014\"Altogether\nWrong\"\u2014The Work in Peril\u2014Thomas Jones of\nRhydwilym\u2014Christmas's Restoration to Spiritual\nHealth\u2014Extracts from Personal Reflections\u2014Singular\nCovenant with God\u2014Renewed Success\u2014The Great Sermon of\nthe Churchyard World\u2014Scenery of its Probable\nDelivery\u2014Outline of the Sermon\u2014Remarks on the\nAllegorical Style\u2014Outlines of Another Remarkable\nSermon, \"The Hind of the Morning\"\u2014Great Preaching but\nPlain Preaching\u2014Hardships of the Welsh Preacher.\n                         CHAPTER IV.\n            THE MINISTRY IN ANGLESEA (CONTINUED).\nChristmas Evans as a Bishop over many Churches\u2014As a        106\nModerator in Public Meetings\u2014Chapel-building and all\nits Difficulties to Christmas Evans\u2014Extensive\nTravelling for Chapel Debts\u2014Especially in South\nWales\u2014The Cildwrn Cottage again\u2014A Mysterious Life of\nPoverty but of Hospitality\u2014Catherine's Troubles\u2014Story\nof a Hat\u2014Wayfaring\u2014Insatiability for Sermons in the\nWelsh\u2014The Scenery of a Great Sermon\u2014The Demoniac of\nGadara\u2014A Remarkable Illustration of the Varied Method\nof the Preacher\u2014A Series of Illustrations of his\nPower of Allegoric Painting\u2014The Four Methods of\nPreaching\u2014The Seeking of the Young Child\u2014Satan\nwalking in Dry Places\u2014Christmas Evans in Another\nLight\u2014Lengthy Letter to a Young\nMinister\u2014Contributions to Magazines\u2014To be accursed\nfrom Christ\u2014Dark Days of Persecution\u2014Threatened with\nLaw for a Chapel Debt\u2014Darker Days\u2014Loss of his\nWife\u2014Other Troubles\u2014Determines to leave Anglesea.\n                          CHAPTER V.\n     CONTEMPORARIES IN THE WELSH PULPIT\u2014WILLIAMS OF WERN.\nThe Great Welsh Preachers unknown in England\u2014The           166\nFamily of the Williamses\u2014Williams of Pantycelyn\u2014Peter\nWilliams\u2014Evan Williams\u2014Dr. Williams\u2014Williams of\nWern\u2014The Immense Power of his Graphic\nLanguage\u2014Reading and Thinking\u2014Instances of his Power\nof Luminous Illustration\u2014Early Piety\u2014A Young\nPreacher\u2014A Welsh Gilboa\u2014Admiration of, and Likeness\nto, Jacob Abbot\u2014Axiomatic Style\u2014Illustrations of\nHumour\u2014The Devils\u2014Fondness for Natural\nImagery\u2014Fondness of Solitude\u2014Affecting Anecdotes of\nDying Hours\u2014His Daughter\u2014His Preaching\ncharacterised\u2014The Power of the Refrain in the\nMusician and the Preacher, \"Unto us a Child is born.\"\n                         CHAPTER VI.\n                  CONTEMPORARIES\u2014JOHN ELIAS.\nFire and Smoke\u2014Elias's Pure Flame\u2014Notes in the             185\nPulpit\u2014Carrying Fire in Paper\u2014Elias's Power in\nApostrophe\u2014Anecdote of the Flax-dresser\u2014A Singular\nFirst Appearance in the Pulpit\u2014A Rough Time in\nWales\u2014The Burning of the Ravens' Nests\u2014A Hideous\nCustom put down\u2014The Great Fair of Rhuddlan\u2014The Ten\nCannon of Sinai\u2014Action in Oratory\u2014The Tremendous\nCharacter of his Preaching\u2014Lives in an Atmosphere of\nPrayer\u2014Singular Dispersion on a Racecourse\u2014A\nRemarkable Sermon, Shall the Prey be taken from the\nMighty?\u2014Anecdote of a Noble Earl\u2014Death and Funeral.\n                         CHAPTER VII.\n              CONTEMPORARIES\u2014DAVIES OF SWANSEA.\nTraditions of his Extraordinary                            202\nEloquence\u2014Childhood\u2014Unites in Church Fellowship with\nChristmas Evans, and with him preaches his First\nSermon\u2014The Church of Castell Hywel\u2014Settles in the\nMinistry at Trefach\u2014The Anonymous Preacher\u2014Settles in\nSwansea\u2014Swansea a Hundred Years Since\u2014Mr. Davies\nreforms the Neighbourhood\u2014Anecdotes of the Power of\nhis Personal Character\u2014How he Dealt with some Young\nOffenders\u2014Anecdote of a Captain\u2014The Gentle Character\nof his Eloquence\u2014The Human Voice a Great Organ\u2014The\nPower of the \"Vox Humana Stop\"\u2014A Great Hymn\nWriter\u2014His Last Sermon.\n                        CHAPTER VIII.\n                 THE PREACHERS OF WILD WALES.\nRees Pritchard, and \"The Welshman's Candle\"\u2014A              217\nSingular Conversion\u2014The Intoxicated Goat\u2014The Vicar's\nMemory\u2014\"God's better than All\"\u2014Howell Harris\u2014Daniel\nRowlands at Llangeitho\u2014Philip Pugh\u2014The Obscure\nNonconformist\u2014Llangeitho\u2014Charles of Bala\u2014His Various\nWorks of Christian Usefulness\u2014The Ancient Preachers\nof Wild Wales characterised\u2014Thomas Rhys\nDavies\u2014Impressive Paragraphs from his Sermons\u2014Evan\nJones, an Intimate Friend of Christmas Evans\u2014Shenkin\nof Penhydd\u2014A Singular Mode of Illustrating a\nSubject\u2014Is the Light in the Eye?\u2014Ebenezer Morris\u2014High\nIntegrity\u2014Homage of Magistrates paid to his\nWorth\u2014\"Beneath\"\u2014Ebenezer Morris at\nWotton-under-Edge\u2014His Father, David\nMorris\u2014Rough-and-ready Preachers\u2014Thomas\nHughes\u2014Catechised by a Vicar\u2014Catching the\nCongregation by Guile\u2014Sammy Breeze\u2014A Singular Sermon\nin Bristol in the Old Time\u2014A Cloud of Forgotten\nWorthies\u2014Dr. William Richards\u2014His Definition of\nDoctrine\u2014Davies of Castell Hywel, the Pastor of\nChristmas Evans, and of Davies of Swansea\u2014Some\nAccount of Welsh Preaching in Wild Wales, in Relation\nto the Welsh Proverbs, Ancient Triads, Metaphysics,\nand Poetry\u2014Remarks on the Welsh Language and the\nWelsh Mind\u2014Its Secluded and Clannish Character.\n                         CHAPTER IX.\n    CHRISTMAS EVANS CONTINUED\u2014HIS MINISTRY AT CAERPHILLY.\nCaerphilly and its Associations\u2014\"Christmas Evans is        261\ncome!\"\u2014A Housekeeper\u2014His Characteristic Second\nMarriage\u2014A Great Sermon, The Trial of the\nWitnesses\u2014The Tall Soldier\u2014Extracts from Sermons\u2014The\nBible a Stone with Seven Eyes\u2014\"Their Works do Follow\nthem\"\u2014A Second Covenant with God\u2014Friends at\nCardiff\u2014J. P. Davies\u2014Reads Pye Smith's \"Scripture\nTestimony to the Messiah\"\u2014Beattie on Truth\u2014The\nEdwards Family\u2014Requested to Publish a Volume of\nSermons, and his Serious Thoughts upon the Subject.\n                          CHAPTER X.\n                  CAERNARVON AND LAST DAYS.\nLeading a Forlorn Hope again\u2014More Chapel Debts\u2014A           287\nPresent of a Gig\u2014Jack, _bach_!\u2014The One-eyed Man of\nAnglesea once more\u2014The Old Man's Reflections in his\nJournal\u2014Characteristic Letters on Church\nDiscipline\u2014Threescore Years and Twelve\u2014Starts on his\nLast Journey to liquidate a Chapel Debt\u2014An Affecting\nAppeal to the Churches\u2014Laid up at\nTredegar\u2014Conversations\u2014In Swansea\u2014This is my Last\nSermon\u2014Dying\u2014Last Words\u2014\"Good-bye!  Drive on!\"\n                         CHAPTER XI.\n SUMMARY OF GENERAL CHARACTERISTICS OF CHRISTMAS EVANS, AS A\n                     MAN AND A PREACHER.\nA Central Figure in the Religious Life of Wales\u2014In a       304\nSingular Degree a Self-made Man\u2014His Words on the\nValue of Industry\u2014His Honest Simplicity\u2014Power of\nSarcasm Repressed\u2014Affectionate Forgiveableness\u2014Great\nFaith, and Power in Prayer\u2014A Passage in Dean Milman's\n\"Samor\"\u2014His Sermons a Kind of _Silex\nScintillaus_\u2014Massive Preaching, but lightened by\nBeautiful Flowers\u2014As an Orator\u2014A Preacher in the Age\nof Faith\u2014Seeing Great Truths\u2014His Remarks on what was\ncalled \"Welsh Jumping\" in Religious Services.\n                         CHAPTER XII.\n  SUMMARY OF GENERAL CHARACTERISTICS OF CHRISTMAS EVANS AS A\n                          PREACHER.\nRemarks renewed in Vindication of his Use of Parable       322\nin the Pulpit\u2014His Sermons appear to be Born of\nSolitude\u2014His Imitators\u2014His Probable Acquaintance with\n\"The Sleeping Bard\" of Elis Wynn\u2014A\nDream\u2014Illustrations\u2014The Gospel Mould\u2014Saul of Tarsus\nand his Seven Ships\u2014The Misplaced Bone\u2014The Man in the\nHouse of Steel\u2014The Parable of the Church as an Ark\namong the Bulrushes of the Nile\u2014The Handwriting\u2014Death\nas an Inoculator\u2014Time\u2014The Timepiece\u2014Parable of the\nBirds\u2014Parable of the Vine-tree, the Thorn, the\nBramble, and the Cedar\u2014Illustrations of his more\nSustained Style\u2014The Resurrection of Christ\u2014They drank\nof that Rock which followed them\u2014The Impossibility of\nAdequate Translation\u2014Closing Remarks on his Place and\nClaim to Affectionate Regard.\n\n\n\n                     APPENDATORY CHAPTER.\n              SELECTION OF ILLUSTRATIVE SERMONS.\n  Sermon     I.\u2014The Time of Reformation                    358\n     \"       II.\u2014The Purification of the Conscience        368\n     \"       III.\u2014Finished Redemption                      378\n     \"       IV.\u2014The Father and Son Glorified              386\n     \"       V.\u2014The Cedar of God                           396\nA Sermon on the Welsh Hills                                407\n\n\n\nCHAPTER I.\n_SOME GENERAL CHARACTERISTICS OF WELSH PREACHING_.\n\n\nWales, the Country and the People\u2014Individuality of the Welsh Pulpit\u2014St.\nDavid\u2014The Religious Sense of the People\u2014Association Meetings\u2014Gryffyth of\nCaernarvon\u2014Bardic Character of the Sermons\u2014A Repetition of\nSermons\u2014Peculiarities of the Welsh Language\u2014Its Singular Effects as\nSpoken\u2014Its Vowels\u2014Its Pictorial Character\u2014The _Hwyl_\u2014Welsh\nScenery\u2014Isolated Character of the Old Chapels\u2014Plain Living and High\nThinking\u2014Ludicrous Incidents of Uncertain Service\u2014Superstitions of\nHeathenism\u2014Fondness of the People for Allegory\u2014Haunted Wales\u2014The Rev.\nJohn Jones and the Mysterious Horseman\u2014Old Wild Wales\u2014St.\nDavid's\u2014Kilgerran\u2014Welsh Nomenclature\u2014John Dyer\u2014Old Customs.\n\nWE propose, in the following pages, to give some account of Christmas\nEvans, the great Welsh preacher; believing that he had a style and manner\nof preaching which, to English minds and readers, will seem altogether\nhis own, perhaps more admirable than imitable.  But before we enter upon\nthe delineation of his life, or attempt to unfold his style, or to\nrepresent his method as displayed in his sermons, it may be well to\npresent some concise view of Welsh preaching and Welsh preachers in\ngeneral, especially those of the last age; for as an order of preaching\nit has possessed its own very distinctive peculiarities.  Some readers\nmay at first indeed inquire, Is not preaching very much the same\neverywhere, in all counties and in all countries?  And Wales, which seems\nitself in its nearness now only like a district of England, and that\ndistrict for the most part wild and but scantily peopled,\u2014can there be\nanything so remarkable about its pulpit work as to make it either capable\nor worthy of any separate account of its singularities and\nidiosyncrasies?  To most English people Welsh preaching is a phase of\nreligious life entirely unknown: thousands of tourists visit the more\nconspicuous highways of Wales from year to year, its few places of public\nresort or more manifest beauty; but Wales is still, for the most part,\nunknown; its isolation is indeed somewhat disturbed now, its villages are\nno longer so insulated as of old, and the sounds of advancing life are\nbreaking in upon its solitudes, yet, perhaps, its fairest scenes are\nstill uninvaded.  But if the country be unknown, still more unknown are\nthe people, and of its singular preaching phenomena scarcely anything is\nknown, or ever can be known by English people; yet it is not too much to\nsay that, in that little land, during the last hundred years, amidst its\nwild glens and sombre mountain shadows, its villages retreating into\ndesolate moorlands and winding vales, where seldom a traveller passes by,\nthere have appeared such a succession and race of remarkable preachers as\ncould be rivalled\u2014in their own peculiar popular power over the hearts and\nminds of many thousands, for their eminence and variety\u2014in no other\ncountry.  Among these, Christmas Evans seems to us singularly\nrepresentative; eminently Welsh, his attributes of power seem to be\nespecially indicative of the characteristics of the Welsh mind, an order\nof mind as remarkably singular and individual, and worthy of study, as\nany national character in the great human family.  But even before we\nmention these, it may be well to notice what were some of the reasons for\nthe eminent influence and usefulness of Christmas Evans, and some of his\nextraordinary preaching comrades and contemporaries to whom we shall have\noccasion to refer.\n\nPreaching is, in Wales, the great national characteristic; the Derby Day\nis not more truly a characteristic of England than the great gatherings\nand meetings of the Associations all grouped around some popular\nfavourites.  The dwellers among those mountains and upon those hill-sides\nhave no concerts, no theatres, no means of stimulating or satisfying\ntheir curiosity.  For we, who care little for preaching, to whom the\nwhole sermon system is perhaps becoming more tedious, can form but little\nidea, and have but little sympathy with that form of religious society\nwhere the pulpit is the orchestra, the stage, and the platform, and where\nthe charms of music, painting, and acting are looked for, and found in\nthe preacher.  We very likely would be disposed even to look with\ncomplacent pity upon such a state of society,\u2014it has not yet\nexpired,\u2014where the Bulwers, the Dickenses, the Thackerays, and Scotts are\naltogether unknown,\u2014but where the peculiar forms of their\ngenius\u2014certainly without their peculiar education\u2014display themselves in\nthe pulpit.  If our readers suppose, therefore, a large amount of\nignorance,\u2014well, upon such a subject, certainly, it is possible to enter\neasily upon the illimitable.  Yet it is such an ignorance as that which\ndeveloped itself in Job, and in his companions, and in his age\u2014an\nignorance like that which we may conceive in \u00c6schylus.  In fact, in\nWales, the gates of every man's being have been opened.  It is possible\nto know much of the grammar, and the history, and the lexicography of\nthings, and yet to be so utterly ignorant of _things_ as never to have\nfelt the sentiment of strangeness and of terror; and without having been\ninformed about the names of things, it is possible to have been brought\ninto the presence and power of _things_ themselves.  Thus, the ignorance\nof one man may be higher than the intelligence of another.  There may be\na large memory and a very narrow consciousness.  On the contrary, there\nmay be a large consciousness, while the forms it embraces may be\nuncertain and undefined in the misty twilight of the soul.  This is much\nthe state of many minds in Wales.  It is the state of feeling, and of\npoetry, of subtle questionings, high religious musings, and raptures.\nThis state has been aided by the secludedness of the country, and the\nexclusiveness of the language,\u2014not less than by the rugged force and\nmasculine majesty and strength of the language;\u2014a language full of angles\nand sharp goads, admirably fitted for the masters of assemblies,\nadmirably fitted to move like a wind over the soul, rousing and soothing,\nstirring into storm, and lulling into rest.  Something in it makes an\norator almost ludicrous when he attempts to convey himself in another\nlanguage, but very powerful and impressive in that.  It is a speaking and\nliving language, a language without any shallows, a language which seems\nto compel the necessity of thought before using it.  Our language is fast\nbecoming serviceable for all that large part of the human family who\nspeak without thinking.  To this state the Welsh can never come.  That\nunaccommodating tongue only moves with a soul behind it.\n\nThus, it is not the first reason, but it is not unimportant to remember,\nthat, until very recently, the pulpit in Wales has been the only means of\npopular excitement, instruction, or even of entertainment; until very\nrecently the Welsh, like the ancient Hebrew lady, have dwelt among their\nown people, they have possessed no popular fictions, no published poems,\nno published emanations either of metaphysics or natural science; immured\nin their own language, as they were, less than a century since, among\ntheir own mountains, their language proved a barrier to the importation\nof many works accessible to almost all the other languages of Europe.  It\nmay be said that religion, as represented through the men of the pulpit,\nhas made Wales what she is.  When the first men of the pulpit, Howell\nHarris, Daniel Rowlands, and others, arose, they found their country\nlying under a night of spiritual darkness, and they effected an amazing\nreformation; but then they had no competitive influences to interfere\nwith their progress, or none beyond that rough, rude sensuality, that\nbarbarism of character, which everywhere sets itself in an attitude of\nhostility to spiritual truth and to elevated holiness; there were no\ntheatres or race courses, there was no possibility that the minds of the\nmultitudes should be occupied by the intellectual casuistries of a later\nday; Wales possessed no Universities or Colleges, and very few Schools;\non the other hand, there were some characteristics of the national mind\nvery favourable to the impulse these men gave, and the impressions they\nproduced.  So it has happened that the Welsh preacher has been elevated\ninto an importance, reminding us of the Welsh tradition concerning St.\nDavid, the patron saint of Wales, regarding whom it is said, that, while\npreaching in the year 520, in Cardigan, against the Pelagian heresy, such\nwas the force of his argument, and the eloquence of his oratory, that the\nvery ground on which he stood rose beneath his feet and elevated itself\ninto a hillock; and there, in after ages, a church was erected upon the\nspot to which awful tradition pointed as the marvellous pulpit of the\npatron saint.\n\nThree-fourths of any amount of power which either or any of these first\npreachers, or their successors, have obtained over their countrymen, and\ncountrywomen, arises from the fact that the Welsh possess, in an eminent\ndegree, what we call a Religious Nature; they are very open to Wonder;\nthey have a most keen and curious propensity to inquire into the hidden\ncauses of things, not mere material causes, but Spiritual causes, what we\ncall Metaphysics; the Unseen Universe is to them as to all of us a\nmystery, but it is a mystery over which they cannot but brood; when\neducation is lacking, this realizing of the unseen is apt to give rise to\nsuperstitious feelings, and superstitions still loiter and linger among\nthe glens, the churchyards, and old castles and ruins of Wales, although\nthe spread of Christian truth has divested them of much of their ancient\nextravagance; when, therefore, the earnest voice of their native speech\nbecame the vehicle for unfolding the higher doctrines of the Christian\nlife, the sufferings of the Redeemer and their relation to eternal laws\nand human conditions, probably a people was never found whose ears were\nmore open, or whose hearts were more ready to receive, and to be stirred\nto their utmost depths.  Thus Religion\u2014Evangelical Religion\u2014became the\nvery life of the land of Wales.\n\n\"There is not a heathen man, woman, or child in all the Principality,\"\nsaid a very eminent Welshman to us once, probably with some measure of\nexaggeration; \"there are wicked men, and women,\" he continued,\n\"unconverted men, and women, but there is not a man, woman, or child\nthroughout Wales who does not know all about Jesus Christ, and why He\ncame into the world, and what He came to do.\"  Thus, within the memory of\nthe writer of this volume, Religion was the one topic upon which you\nmight talk intelligently anywhere in Wales: with the pitman in the\ncoalmine, with the iron-smelter at the forge, with the farmer by his\ningleside, with the labourer in his mountain shieling; and not merely on\nthe first more elementary lessons of the catechism, but on the great\nbearings and infinite relations of religious things.  Jonathan Edwards,\nand Williams of Rotherham, and Owen, and Bunyan, and Flavel,\u2014these men\nand their works, and a few others like them, were well known; and,\nespecially, the new aspects which the modified opinions of Andrew Fuller\nhad introduced into religious thought; thus, you might often feel\nsurprised when, sitting down in some lowly cottage, you found yourself\nsuddenly caught, and carried along by its owner in a coil of metaphysical\nargument.  This was the soil on which the Welsh preachers had to work,\nand cast abroad their seed.\n\nNo person can have heard anything of the Welsh religious life without\nhaving heard also of the immense annual gatherings, the Association\nmeetings, a sort of great movable festival, annually held in Wales, to\nwhich everything had to give place, and to which all the various tribes\nof the various Houses of the Lord came up.  Their ordinary Sunday\nservices were crowded, but, upon these great occasions, twenty or\ntwenty-five thousand people would come together; and, to such\ncongregations, their great men, their great preachers, such as those we\nare about to mention, addressed themselves\u2014addressed themselves not to a\nmass ignorant and unintelligent, but all thoroughly informed in religious\nmatters, and prepared to follow their preacher whithersoever his\nimagination or thought might lead him.  The reader must not smile when we\nremind him that Wales was,\u2014had been for ages,\u2014the land of Bards; a love\nof poetry, poetry chanted or recited, had always been the Welshman's\npassion, and those great writers of our literature who best know what\npoetry is, have taught us that we are not to look upon those productions\nwith contempt.  For ages there had been held in Wales what has been\ncalled, and is still called the _Eisteddfod_, or _Cymreigyddion_, or the\nmeeting of the Bards and Minstrels; they were, as Pennant has called\nthem, British Olympics, where none but Bards of merit were suffered to\nrehearse their pieces, or Minstrels of skill to perform.  These\nAssociation meetings were a kind of religious Eisteddfodd, where the\ngreat Welsh preacher was a kind of sacred Bard; he knew nothing of\nwritten sermons; he carried no notes nor writings with him to his pulpit\nor platform, but he made the law and doctrine of religious metaphysics\nmarch to the minstrelsy and music of speech; on the other hand, he did\nnot indulge himself in casting about wildfire, all had been thoroughly\nprepared and rooted in his understanding; and then he went with his\nsermon, which was a kind of high song, to chant it over the hearts of the\nmultitude.  We shall have occasion to show, by many instances, from the\nlives of their greatest men, how their own hearts had been marvellously\nprepared.\n\nThere is a pleasant anecdote told of one of them, Gryffyth of Caernarvon,\nhow he had to preach one night.  Before preaching, staying at a farmhouse\non the spot, he desired permission to retire before the service began; he\nremained in his room a considerable time; the congregation had assembled,\nstill he did not come; there was no sign of his making his appearance.\nThe good man of the house sent the servant to request him to come, as the\npeople had been for some time assembled and waiting.  Approaching the\nroom she heard, what seemed to her to be a conversation, going on between\ntwo persons, in a subdued tone of voice, and she caught from Mr. Gryffyth\nthe expression, \"_I_ will not go unless _you_ come with me.\"  She went\nback to her master, and said, \"I do not think Mr. Gryffyth will come\nto-night; there is some one with him, and he is telling him that he will\nnot come unless the other will come too; but I did not hear the other\nreply, so I think Mr. Gryffyth will not come to-night.\"\n\n\"Yes, yes,\" said the farmer, \"_he_ will come, and I warrant the _other_\nwill come too, if matters are as you say between them; but we had better\nbegin singing and reading until the _two_ do come.\"  And the story goes\non to say that Mr. Gryffyth did come, and the other One with him, for\nthey had a very extraordinary meeting that night, and the whole\nneighbourhood was stirred by it and numbers were changed and converted.\nIt was Williams of Wern who used to tell this pleasing anecdote; it is an\nanecdote of one man, but, so far as we have been able to see, it\nillustrates the way in which they all prepared themselves before they\nbegan to speak.\n\nIt must not be supposed from this that they imagined that prayer was to\ndispense with preparation; their great preachers studied hard and deeply,\nand Williams of Wern, one of the greatest of them all, says, \"In order to\nbe a good preacher, usefulness must be the grand aim, usefulness must\nchoose the text and divide it, usefulness must compose the sermon and sit\nat the helm during the delivery; if the introduction be not clear and\npertinent it is evident the preacher does not know whither he is going,\nand if the inferences are of the same character, it is obvious he does\nnot know where he has been.  Unstudied sermons are not worth hearing or\nhaving; who would trust his life in the hands of a physician who had\nnever thought of his profession?\"  But these men never permitted the\nunderstanding to supersede emotion, and, when they met the people face to\nface, the greatest of them went prepared, warmed and kindled, and ready\nto warm and kindle.\n\nThus their sermons became a sort of inspired song, full of\nimagination\u2014imagination very often, and usually, deriving its imagery\nfrom no far-off and recondite allusions, never losing itself in a flowery\nwilderness of expressions, but homely illustrations, ministered to by the\nthings and affairs of ordinary life, and, therefore, instantly preacher\nand people in emotion were one.\n\nIt is indeed true that many of their great preachers repeated the same\nsermon many times.  Why not?  So did Whitfield, so did Wesley, so have\nmost eminent preachers done; but this need in no way interfere with\u2014it\ndid not interfere with\u2014the felt necessity for unction on the part of the\nminister; and as to the people they liked to hear an old favourite again,\nor a sermon, which they had never heard although they had heard much\nabout it.  We believe it was to Christmas Evans a pert young preacher\nsaid, \"Well, you have given us an old sermon again to-day.\"\n\n\"What then, my boy?\" said the Master of Assemblies; \"had you a new one?\"\n\n\"Certainly,\" was the answer.\n\n\"Well, but look you,\" said the unblushing old culprit, \"I would not take\na dozen new sermons like yours for this one old sermon of mine.\"\n\n\"No, nor I,\" chimed in a gruff old deacon.  \"Oh yes, and look you, I\nshould like to hear it again; but as for _yours_, I never heard it\nbefore, and I do not want to hear it again.\"\n\nBut then the _Language_!  Of course the language had a great deal to do\nwith this preaching power, we do not mean generally, but particularly; on\nall hands the Welsh is acknowledged to be a wonderful language.  A\nWelshman will tell you that there is no language like it on the face of\nthe earth, but that is a testimony borne by many scholars who are not\nWelshmen; perhaps there is no other language which so instantly conveys a\nmeaning and at the same time touches emotion to the quick.  True, like\nthe Welshman himself, it is bony, and strangers to its power laugh\nsomewhat ignorantly at its never-ending succession of consonants.\nSomebody has said that the whole language is as if it were made up of\nsuch words as our word \"_strength_,\" and if the reader will compare in\nhis mind the effect of the word _power_ as contrasted with the word\n_strength_, he will feel something of the force of the language, and its\nfitness for the purposes of impression; but still this conveys but a poor\nidea of its great attributes.\n\nIt is so _literal_ that the competent hearer, or reader, instantly\nrealizes, from its words, things.  Well do we remember sitting in Wales\nwith a group of Welsh ministers and Welshmen round a pleasant tea-table;\nwe were talking of the Welsh language, and one of our company, who had\nperhaps done more than any one of his own country for popular Welsh\nliterature, and was one of the order of eminent Welsh preachers of whom\nwe are speaking, broke forth: \"Oh!\" he said, \"you English people cannot\nsee all the things in your Bible that a Welshman can see; now your word\n'_blessed_,' it seems a very dear sweet thing to an Englishman and to a\nWelshman, but a Welshman sees the _thing_ in the word, '_Gwyn ei fyd_,'\nthat is, '_a white world_\u2014white,' literally, white their world; so a\nWelshman would see there is a '_white world_' for the pure in heart, a\n'_white world_' for the poor in spirit, a '_white world_' for them who\nare reviled and persecuted for righteousness' sake; and when you read,\n'_Blessed_ is the man unto whom the Lord imputeth not iniquity,' the\nWelshman reads his Bible and sees there is a '_white world_' for such a\none, that is, all sin wiped out, the place quite clean, to begin again.\"\n\nThis is not all.  We are not intending to devote any considerable space\nto a vindication of the Welsh language, but, when we speak of it with\nreference to the effects it produces as the vehicle of Oratory, it is\nnecessary to remark that, so far from being,\u2014as many have supposed who\nhave only looked at it in its strange combination of letters on a page,\nperhaps unable to read it, and never having heard it spoken,\u2014so far from\nbeing harsh and rugged, coarse or guttural, it probably yields to no\nlanguage in delicious softness, in melting sweetness; in this it has been\nlikened to the Italian language by those who have been best able to\njudge.  Lord Lyttleton, in his \"Letters from Wales,\" says, that when he\nfirst passed some of the Welsh hills, and heard the harp and the\nbeautiful female peasants accompanying it with their melodious voices, he\ncould not help indulging in the idea that he had descended the Alps, and\nwas enjoying the harmonious pleasures of the Italian Paradise.  And as we\nhave already said, there has long prevailed an idea that the Welsh\nlanguage is a multitude of consonants; but indeed the reverse is the\ncase; the learned Eliezer Williams says, in his \"Historical Anecdotes of\nthe Welsh Language,\" \"The alphabet itself demonstrates that the charge of\na multiplicity of consonants is fallacious, since, whether the number of\nletters be reckoned twenty-two or twenty-four, seven are vowels; there\nremain therefore a more inconsiderable number than most of the European\nlanguages are obliged to admit . . . .  _Y_ and _w_ are considered as\nvowels, and sounded as such; _w_ is pronounced like _o u_ in French in\nthe word _oui_.\"  To persons ignorant of the language, how strange is the\nappearance, and how erroneous the idea of the sound to be conveyed by\n_dd_, _ll_, _ch_, but indeed all these are indications of the softening\nof the letter; in a word, the impressions entertained of the harshness of\nthe language are altogether erroneous.\n\nThe supposition that the Welsh language is made up of consonants is more\nespecially singular from the fact that it possesses, says a writer in the\n_Quarterly Review_, what perhaps no other nation has,\u2014a poem of eight\nlines in which there is not a single consonant.  These verses are very\nold, dating from the seventeenth century;\u2014of course the reader will\nremember that the Welsh language has seven vowels, both _w_ and _y_ being\nconsidered and sounded as such.  This epigram or poem is on the Spider,\nand originally stood thus,\u2014\n\n    \"O'i wiw \u0175y i weu e \u00e2;\u2014o'i iau Ei wyau a wea,\n    E wywa ei w\u00ea aua, A'i weau yw ieuau ia.\"\n\nTo this, the great Gronwy Owen added a kind of counter change of vowels,\nand the translation has been given as follows:\u2014\n\n    \"From out its womb it weaves with care\n       Its web beneath the roof;\n    Its wintry web it spreadeth there\u2014\n       Wires of ice its woof.\n\n    \"And doth it weave against the wall\n       Thin ropes of ice on high?\n    And must its little liver all\n       The wondrous stuff supply?\"\n\nA singular illustration of the vowel power in a language ignorantly\nsupposed to possess no vowels.\n\nAnd these remarks are not at all unnecessary, for they illustrate to the\nreader, unacquainted with the language, the way in which it becomes such\na means of immediate emotion; its words start before the eye like\npictures, but are conveyed to the mind like music; and yet the bony\ncharacter of the language, to which we have referred before, adds to the\npicture dramatic action and living strength.  What a language, then, is\nthis for a competent orator to play upon,\u2014a man with an imaginative mind,\nand a fervid and fiery soul!  Then is brought into play that element of\nWelsh preaching, without knowing and apprehending which there would be no\npossibility of understanding the secret of its great power; it is the\n\"_hwyl_.\"  When the Welsh preacher speaks in his best mood, and with\ngreat unction, the highest compliment that can be paid him, the loftiest\ncommendation that can be given, is, that he had the \"_hwyl_.\"  \"_Hwyl_\"\nis the Welsh word for the canvas of a ship; and probably the derivation\nof the meaning is, from the canvas or sails of a ship filled with a\nbreeze: the word for breeze, _awel_, is like it, and is used to denote a\nsimilar effect.  Some years since, when the most eminent Welsh preacher\nwe have recently seen in England, at an ordination service, was\naddressing his nephew in a crowded church in the neighbourhood of London,\nhe said, \"And, my dear boy, remember you are a Welshman; don't try to\nspeak English, and don't try to speak like the English.\"  A great many of\nhis hearers wondered what the good man could mean; but both he and his\nnephew, and several others of the initiated, very well knew.  He meant,\nspeak your words with an _accent_, and an accent formed from a soul\ngiving life and meaning to an expression.  This, we know, is what the\nsinger does,\u2014this is what the musician tries to do.  All words are not\nthe same words in their meaning; the Welsh preacher seeks to play upon\nthem as keys; the words themselves help him to do so.  Literally, they\nare full of meaning; verbally, he attempts to pronounce that meaning;\nhence, as he rises in feeling he rises in variety of intonation, and his\nwords sway to and fro, up and down,\u2014bass, minor, and soprano all play\ntheir part, a series of intonings.  In English, this very frequently\nsounds monotonous, sometimes even affected; in Welsh, the soul of the man\nis said to have caught the _hwyl_,\u2014that is, he is in full sail, he has\nfeeling and fire: the people catch it too.  A Welsh writer, describing\nthis, quotes the words of Jean Paul Richter: \"Pictures during music are\nseen into more deeply and warmly by spectators; nay, many masters have in\ncreating them acknowledged help from music.\"  Great Welsh preaching, is\nvery often a kind of wild, irregular chant, a jubilant refrain, recurring\nagain and again.  The people catch the power of it; shouts rise\u2014prayers!\n\"_Bendigedig_\" (\"blessed,\" or synonymous with our \"Bless the Lord!\")\nAmen! \"_Diolch byth_!\" and other expressions, rise, and roll over the\nmultitude; they, too, have caught the _hwyl_.  It is singular that, with\nus, the only circumstances and scenes in which such manifestations can\ntake place, are purely secular, or on the occasions of great public\nmeetings.  The Welshman very much estimates the greatness of a preacher\nby his power to move men; but it does not follow, that this power shall\nbe associated with great apparent bodily action.  The words of John Elias\nand Williams of Wern consumed like flames, and divided like swords; but\nthey were men of immense self-possession, and apparently very quiet.  It\nhas always been the aim of the greater Welsh preachers to find out such\n\"acceptable\"\u2014that is, fitting and piercing\u2014words, so that the words alone\nshall have the effect of action.\n\nBut, in any account of Welsh preaching, the place ought never to be\nforgotten\u2014the scenery.  We have said, the country is losing, now, many of\nits old characteristics of solitude and isolation; the railways are\nrunning along at the foot of the tall mountains, and spots, which we knew\nthirty years since as hamlets and villages, have now grown into large\ntowns.  It has often been the case, that populations born and reared\namidst remote mountain solitudes, have possessed strong religious\nsusceptibilities.  The Welshman's chapel was very frequently reared in\nthe midst of an unpeopled district, likely to provoke wonder in the mind\nof the passing stranger, as to whence it could derive its congregation.\nThe building was erected there because it was favourable to a confluence\nof neighbourhoods.  Take a region near to the spot where Christmas Evans\nwas born,\u2014a wild, mountainous tract of country, lying between the\ncounties Brecon and Cardigan; for long miles, in every direction, there\nare no human habitations,\u2014only, perhaps, here and there, in a deep\ndingle, some lone house, the residence of a sheep farmer, with three or\nfour cultivated fields in its immediate neighbourhood; and at some\ndistance, on the <DW72>s of the mountain, an occasional shepherd's hut.\nIt is a scene of the wildest magnificence.  The traveller, as he passes\nalong, discerns nothing but a sea of mountains,\u2014rugged and precipitous\nbluffs, and precipices innumerable; here the grand and sportive streams,\nthe Irvon, the Towy, and the Dothia, spring from their rocky channels,\nand tumble along, rushing and gurgling with deafening roar; here, as you\npass along, you encounter more than one or two \"wolves' leaps;\"\u2014dark\ncaverns are there, from whence these brotherly rivers rush into each\nother's embrace.  These regions, when we were in the habit of crossing\nthem, many years since,\u2014and we often crossed them,\u2014we very naturally\nregarded as the Highlands, the sequestered mountain retreats, of Wales;\nthis was Twm Shon Catty's, the Welsh Rob Roy's, country; for let Scotland\nboast as she will\u2014\n\n    \"Wales has had a thief as good,\n    She has her own Rob Roy.\"\n\nAnd wonderfully romantic is the story of this same Welsh gentleman, and\npredatory chieftain.  Here you find, to this day, his cave, from whence\nthe bold and humorous outlaw was wont to spring forth, to spread terror\nand rapine over the whole region.  It is thirty years since we passed\nthrough these desolations; they are probably much the same now as they\nwere then; let the traveller shout as he will as he passes along, it is\nnot from any human being, it is only from the wild rock, or screaming\nbird, he will have a reply.\n\nNow, what do our readers think of a large and commodious chapel in the\nmidst of a wild region like this?  But one there is, in the very heart of\nthe wilderness.  Up to this place the worshippers come, on Sabbath\nmornings, from distances varying from two to eight miles.  It is a\nCalvinistic-Methodist chapel; and the Rev. William Williams, in his\ninteresting little historical sketch of Welsh Calvinistic-Methodism,\ntells how he preached in this building, several years since, when the\nchapel was crowded with worshippers; and in the yard adjoining, between\nfifty and sixty ponies, which had borne the worshippers to the place,\nwith or without vehicles, were waiting the time for the return journey.\nThis building had its birth from a congregation gathered first in one of\nthe farm houses in these inaccessible wilds, in 1847.  It seems strange\nto think how far people will travel to Divine Service when they have no\nsuch service near their own doors.  We were struck with this, a short\ntime since, in Norway; we found our way to a little village church, and\nthere, on a spot where was next to no population, we found the Lutheran\nchurch crowded; and outside, a large square space thronged with carioles,\nancient old shandydan landaus, carts, and every kind of\nconveyance,\u2014horses and ponies stabled in the sheds all round; and we\nlearned that many of the congregation had travelled in this way, beside\nthe numbers who had walked, twelve, sixteen, eighteen miles to the\nservice.\n\nAnd thus, also, in Wales, many were the long and weary miles usually\ntraversed, and through every variety of weather; and it seemed to be\nusually thought that the service, or services, repaid all the toil.  And\nthere was very little, externally, to aid the imagination, or to charm\nthe taste, either in the building itself, or in the ritual adopted;\u2014all\nwas of the plainest and most severe order.  The building, no doubt, was\nlittle more than a shelter from the weather; generally, perhaps, huge and\ncapacious,\u2014that was necessary,\u2014but it was quite unadorned; the minister\nhad nothing in the way of robes or attire to aid the impressions of\nreverence; there was no organ,\u2014usually no instrument of any\ndescription,\u2014although if an entire stranger to the language had entered,\nand heard the long, low, plaintive wail of almost any of their\nhymns,\u2014most of them seeming to express a kind of dirge-like feeling of an\nexiled, conquered, and trampled people, a tone with its often-renewed\nrefrain, its long-drawn minor, now sobbing into grief, occasionally\nswelling into triumph,\u2014he might have found the notes of an organ were not\nneeded to compel the unexpected tear.  An exiled, conquered, and trampled\npeople,\u2014that expresses a great deal of truth.  Wales has wrongs quite as\nbitter as any which Ireland ever knew;\u2014the very cause of the existence of\nmost of her chapels arose from the fact that, in many of her parish\nchurches, not a word of Welsh was spoken; and perhaps frequently their\nministers could not speak the native language;\u2014the very judges who\ndispensed justice from the Bench were usually English, and needed an\ninterpreter, that they might be able to understand the case upon which\nthey were to give a judgment.  Wales has had very little for which to\nthank England, but her people have never been seditious.  Pious,\nindustrious people, with their simple amusements and weird superstitions,\nand blossoming out into their great religious revivals and reformations,\nthey have had to thank themselves, chiefly, for all the good which has\nunfolded itself upon their soil.  These circumstances, however, have no\ndoubt aided their peculiar and isolated religious life.\n\nBut, in those great assemblies, the Association meetings to which we have\nreferred, many of the great preachers stood, with their vast\ncongregations round them, in Nature's open Cathedral.  Christmas Evans\npreached many of his noblest sermons amidst the imposing ruins of\nCaerphilly, Pembroke, and Manobear Castles; or the preacher found himself\nwith his audience on the <DW72> of some sweet, gorse-covered hill, in the\nneighbourhood of tumbling torrents, which did not sing so loudly in their\nmelody as to interfere with the sweet restfulness of the surrounding\nscene.  Preachers and hearers were accustomed to plain living,\u2014one of the\nmost essential conditions of high thinking; neither of them knew anything\nof luxury; and when most of them spoke, the age of luxury, even with us,\nhad not yet set in.  Bread and milk, or oatmeal and milk, were the\nfavourite diet of all, in those days; even tea was all but unknown, and\nthe potato almost their nearest approach to a dainty dish.  They lived on\ngood terms with Nature, with whom we have been quarrelling now for some\nyears past; and thus they were prepared to receive such lessons as Nature\nmight give, to aid and illustrate the deeper lessons of Divine Grace.\n\nOf course, there was considerable uncertainty about the\nservices,\u2014excepting those more imposing and important occasions; and this\ngave, very frequently, a tone of the ludicrous to their announcement of\nthe services.  Thus, if a stranger asked what time the service would\ncommence, it would often have been quite impossible to get any\ninformation; and failures, says Mr. D. M. Evans, were so frequent, that\nthe announcement was often made with perfect gravity, \"\u2014 will be here\nnext Sunday, if he comes.\"  Mr. Evans continues, that he well knew a\ndeacon who claimed the prerogative to make announcements to the\ncongregation, but who every week was guilty of such blunders, that he was\nimplored to resign the honour to some other brother; to which he\nindignantly replied, that it was his crown, and was he not told in\nScripture, \"Hold fast that which thou hast, that no man take thy crown\"?\nOften, when the preacher appeared, he showed himself in the pulpit almost\nout of breath, sometimes in sad disarray, sometimes apparently as if\nsmothered with wrappers and top-coats; and by his panting and puffing, as\nsomeone said, \"seeming to show that God Almighty had asked him to preach\nthe Gospel, but had given him no time for it.\"\n\nIn a word, it is impossible, knowing Wales as we know it in our own day,\nto form any very distinct idea of the country as it was when these great\npreachers arose; and, when the tides of a new spiritual life rolled over\nthe Principality, the singular relics of even heathenish superstition\nwere loitering still among the secluded valleys and mountains of the\nland.  No doubt, the proclamation of the Gospel, and the elevated faith\nwhich its great truths bring in its train, broke the fascination, the\ncharm, and power of many of these; but they lingered even until within\nthe last forty or fifty years,\u2014indeed, the superstition of the Sin-Eater\nis said to {23} linger even now in the secluded vale of Cwm-Aman, in\nCaermarthenshire.  The meaning of this most singular institution of\nsuperstition was, that when a person died, the friends sent for the\nSin-Eater of the district, who, on his arrival, placed a plate of salt\nand bread on the breast of the deceased person; he then uttered an\nincantation over the bread, after which, he proceeded to eat it,\u2014thereby\neating the sins of the dead person; this done, he received a fee of\ntwo-and-sixpence,\u2014which, we suppose, was much more than many a preacher\nreceived for a long and painful service.  Having received this, he\nvanished as swiftly as possible, all the friends and relatives of the\ndeparted aiding his exit with blows and kicks, and other indications of\ntheir faith in the service he had rendered.  A hundred years since, and\nthrough the ages beyond that time, we suppose this curious superstition\nwas everywhere prevalent.\n\nAnother odd custom was the manner in which public opinion expressed\nitself on account of any domestic or social delinquency.  A large crowd\nassembled before the house of the delinquent, one of whom was dressed up\nin what seemed to be a horse's head; the crowd then burst forth into\nstrong vituperative abuse, accompanying the execrations with the rough\nmusic of old kettles, marrow-bones, and cleavers; finally, the effigy of\nthe sinner was burnt before the house, and the sacred wrath of the\nmultitude appeased.  The majesty of outraged opinion being vindicated,\nthey dispersed.\n\nSome superstitions were of a more gentle character; the fairies, or\n\"little men in green,\" as they were popularly called, continued to hold\ntheir tenantry of Wales long after they had departed from England; and\neven Glamorganshire, one of the counties nearest to England,\u2014its roads\nforming the most considerable highway through Wales,\u2014was, perhaps, the\ncounty where they lingered last; certainly not many years have passed by\nsince, in the Vale of Neath, in the same county, there would have been a\nfear in taking some secluded pathway in the night, lest the \"little\npeople\" should be offended by the intrusion upon their haunts.\n\nWith all these singular observances and superstitions, there was yet a\nkind of Christian faith prevalent among the people, but buried beneath\ndark ignorance and social folly.  At Christmas time, at night, it was\nusual to illuminate all the churches in the villages.  And upon the New\nYear's morning, children came waking the dawning, knocking at the\ndoors,\u2014usually obtaining admittance,\u2014when they proceeded to sprinkle the\nfurniture with water, singing as they did so the following words, which\nwe quote on account of their quaint, sweet, old-world simplicity:\u2014\n\n    \"Here we bring new water from the well so clear,\n    For to worship God with this happy new year.\n    Sing levy dew, sing levy dew, the water and the wine,\n    With seven bright gold wires and bugles that do shine.\n    Sing reign of fair maid, with gold upon her toe,\n    Open you the west door, and turn the old year go.\n    Sing reign of fair maid, with gold upon her chin,\n    Open you the east door, and let the new year in.\"\n\nIt is admitted on all hands that the dissolution of the mists of darkness\nand superstition is owing to the people usually called Dissenters; the\nChurch of the Establishment\u2014and this is said in no spirit of\nunkindness\u2014did very little to humanise or soften the rugged character, or\nto put to flight the debasing habits of the people.  Of course, there are\nhigh and honourable exceptions; but while many clergymen devoted\nthemselves, with great enthusiasm, to the perpetuation of the singular\nlore, the wild bardic songs, the triads, or the strange fables and mythic\nhistories of the country, we can call to mind the names of but very few\nwho attempted to improve, or to ameliorate, the social condition.  So\nthat the preachers, and the vast gatherings of the people by whom the\npreachers were surrounded, when the rays of knowledge were shed abroad,\nand devotion fired, were not so much the result of any antagonism to the\nEstablished Church,\u2014_that_ came afterwards; they were a necessity created\nby the painful exigencies of the country.\n\nThe remarks on the superstitions of Wales are not at all irrelevant to\nthe more general observations on Welsh preaching; they are so essentially\ninwoven with the type of character, and nationality.  The Welsh appears\nto be intimately related to the Breton; the languages assimilate,\u2014so also\ndo the folk-lores of the people; and the traditions and fanciful fables\nwhich have been woven from the grasses of the field, the leaves of the\nforest, and the clouds of the heavens, would have furnished Christmas\nEvans with allegoric texts which he might have expanded into sermons.  It\nis not possible to doubt that these form one branch, from the great\nCeltic stem, of the human family.  And not only are they alike in\nlanguage and tradition, but also in the melancholy religiousness, in the\nmetaphysical brooding over natural causes, and in the absence of any\ngenuine humour, except in some grim or gloomy and grotesque utterance.\nThe stories, the heroes, and the heroines, are very much the same;\nhistoric memory in both looks back to a fantastic fairyland, and presents\nthose fantastic pictures of cities and castles strangely submerged\nbeneath the sea, and romantic shadows and spectral forms of wonderful\nkings and queens, such as we meet in the Mabinogi of Taliesin, in the\nFairy Queen of Spenser, and in the Idylls of our Laureate.  Thus, all\nthat could stir wonder, excite the imagination and the fancy, and\ndescribe the nearness of the supernatural to the natural, would become\nvery charming to a Welshman's ears; and we instantly have suggested to us\none of the sources of the power and popularity of Christmas Evans with\nhis countrymen.\n\nEven the spread and prevalence of Christian knowledge have scarcely\ndisenchanted Wales of its superstitions.  Few persons who know anything\nat all of the country, however slight such knowledge may be, are unaware\nof this characteristic of the people.  This remark was, no doubt, far\nmore applicable even twenty-five years since than now.  The writer of\nthis volume has listened to the stories of many who believed that they\nhad seen the _Canwyll-y-corph_\u2014corpse-candles\u2014wending their way from\nhouses, more or less remote, to the churchyard.  Mr. Borrow, also, in his\n\"Wild Wales,\" tells us how he conversed with people in his travels who\nbelieved that they had seen the corpse-candles.  But a hundred years ago,\nthis was a universal object of faith; as was also the belief in coffins\nand burial trains seen wending their way, in the dead of night, to the\nchurchyard.  Omens and predictions abounded everywhere, while singular\nlegends and traditions in many districts hung also round church bells.\nAnd yet with all this the same writer, remarking on Welsh character,\nsays, \"What a difference between a Welshman and an Englishman of the\nlower class!\"  He had just been conversing with a miller's man,\u2014a working\nlabourer in the lowliest walk of life; and found him conversant with the\nold poets, and the old traditions of the country, and quite interested in\nthem; and he says, \"What would a Suffolk miller's man have said, if I had\nrepeated to him verses out of Beowulf or even Chaucer, and had asked him\nabout the residence of Skelton?\"  We must bear this in mind as we attempt\nto estimate the character with which the preacher had to deal.  Haunted\nhouses were numerous.  A lonely old place, very distinct to the writer's\nknowledge, had hung round it some wild traditions not unlike \"Blind\nWillie's Story\" in \"Redgauntlet.\"  No doubt, now, all these things have,\nto a considerable extent, disappeared,\u2014although there are wild nooks, far\nwilder than any we have in England, where the faith in the old\nsuperstitions lingers.  In the great preaching days, those men who shook\nthe hearts of the thousands of their listeners, as they dealt with unseen\nterrors, believed themselves to be\u2014as it was believed of them that they\nwere\u2014covered with the shadow of an Unseen Hand, and surrounded by the\nguardianship of the old Hebrew prophet\u2014\"chariots of fire, and horses of\nfire;\" they believed themselves to be the care of a special Providence;\nand some of the stories then current would only move the contempt of that\nmodern intelligence which has, at any rate, laid all the ghosts.\n\nIt is not within the province of this volume to recapitulate and classify\nWelsh superstitions; they were, and probably, in many neighbourhoods, are\nstill, very various: we must satisfy our readers with a slight\nillustration.  Perhaps some may object to the retailing such stories, for\ninstance, as the following.  The apology for its insertion, then, must\nbe, that it is one of a number tending to illustrate that sense which the\nold Welsh mind had, of its residence upon the borders of, and relation\nto, the Invisible World.  The Rev. John Jones, of Holywell, in\nFlintshire, was one of the most renowned ministers in the Principality;\nhe was a man of extraordinary zeal and fervour as a preacher, and his\nlife and character were, in unblemished reputation, equal to his gifts\nand zeal.  He used to recite, with peculiar solemnity, a story of a\nmysterious horseman, by whom he believed he had been delivered from a\nposition of extreme danger, when he was travelling, alone, from Bala, in\nMerionethshire, to Machynlleth, in the county of Montgomery.  He\ntravelled on horseback through a wild, desolate country, at that time\nalmost uninhabited; he had performed nearly half his journey, when, as he\nwas emerging from a wood, he says, \"I observed coming towards me a man on\nfoot.  By his appearance, judging from the sickle which he carried\nsheathed in straw over his shoulder, he was doubtless a reaper in search\nof employment.  As he drew near, I recognized a man whom I had seen at\nthe door of the village inn at Llanwhellyn, where I had stopped to bait\nmy horse.  On our meeting, he touched his hat, and asked if I could tell\nhim the time of day.  I pulled out my watch for the purpose,\u2014noticing, at\nthe same time, the peculiar look which the man cast at its heavy silver\ncase.  Nothing else, however, occurred to excite any suspicion on my\npart; so, wishing him a good afternoon, I continued my journey.\"  We must\ncondense Mr. Jones's narration, feeling that the story loses much of its\ngraphic strength in so doing.  He pursued his way down a hill, and, at\nsome distance farther on, noticed something moving on the other side of a\nlarge hedge; he soon discovered it to be a man, running in a stooping\nposition.  He watched the figure with curiosity, which grew into\nsomething like fear as he recognized the reaper with whom he had spoken a\nshort time before, and that, as he moved on, he was engaged in tearing\nthe straw band from his sickle.  The man hurried on, and Mr. Jones saw\nhim conceal himself behind a thicker part of the hedge, within a few\nyards of the road, and near where a gate crossed the park.  Mr. Jones\nsays he did not doubt, then, that he intended to attack and, perhaps,\nmurder him for the sake of the watch, and whatever money he might have\nabout him.  He looked round: no other person was in sight,\u2014no house near;\nhe was hemmed in by rocky banks and high hedges on either side.\n\n\"I could not turn back,\" he says; \"my business was of the utmost\nimportance to the cause for which I was journeying.\"  He could not urge\nhis horse with speed, for the gate was not open through which he had to\npass; he felt that he was weak and unarmed, and had no chance against a\npowerful man with a dangerous weapon in his hand.  \"In despair,\" he says,\n\"rather than in a spirit of humble trust and confidence, I bowed my head,\nand offered up a silent prayer.  At this juncture, my horse, growing\nimpatient of delay, started off.  I clutched the reins, which I had let\nfall on his neck,\u2014when, happening to turn my eyes, I saw, to my utter\nastonishment, that I was no longer alone: there, by my side, I beheld a\nhorseman, in a dark dress, mounted on a white steed.  In intense\namazement, I gazed upon him.  Where could he have come from?  He appeared\nas suddenly as if he had sprung from the earth; he must have been riding\nbehind, and have overtaken me,\u2014and yet I had not heard the slightest\nsound.  It was mysterious, inexplicable; but joy overcame my feelings of\nwonder, and I began at once to address my companion.  I asked him if he\nhad seen any one; and then described to him what had taken place, and how\nrelieved I felt by his sudden appearance.  He made no reply, and, on\nlooking at his face, he seemed paying but slight attention to my words,\nbut continued intently gazing in the direction of the gate,\u2014now about a\nquarter of a mile ahead.  I followed his gaze, and saw the reaper emerge\nfrom his concealment, and run across a field to our left, resheathing his\nsickle as he hurried along.  He had evidently seen that I was no longer\nalone, and had relinquished his intended attempt.\"\n\nMr. Jones sought to enter into conversation with his mysterious\ncompanion, but he gave him no word in reply.  He says he \"was hurt at his\ncompanion's mysterious silence;\" only once did he hear his voice.  Having\nwatched the figure of the reaper disappear over the brow of a\nneighbouring hill, he turned to the stranger, and said, \"'Can it for a\nmoment be doubted that my prayer was heard, and that you were sent for my\ndeliverance by the Lord?'  Then it was that I thought I heard the\nhorseman speak, and that he uttered the single word, 'Amen!'  Not another\nword did he give utterance to, though I spoke to him both in English and\nWelsh.  We were now approaching the gate, which I hastened to open; and\nhaving done so, I waited at the side of the road for him to pass\nthrough,\u2014but he came not.  I turned my head to look; the mysterious\nhorseman was gone; he was not to be seen; he had disappeared as\nmysteriously as he had come.  What could have become of him?  He could\nnot have gone through the gate, nor have made his horse leap the high\nhedges, which on both sides shut in the road.  Where was he? had I been\ndreaming? was it an apparition, a spectre, which had been riding by my\nside for the last ten minutes?\u2014was it but a creature of my imagination?\nI tried hard to convince myself that this was the case; but why had the\nreaper resheathed his murderous-looking sickle and fled?  And then, a\nfeeling of profound awe began to creep over my soul.  I remembered the\nsingular way of his first appearance,\u2014his long silence, and the single\nword to which he had given utterance after I had mentioned the name of\nthe Lord; the single occasion on which I had done so.  What could I,\nthen, believe, but that my prayer had been heard, and that help had been\ngiven me at a time of great danger?  I dismounted, and throwing myself on\nmy knees, I offered up my thankfulness to Him who had heard my cry.  I\nthen mounted my horse, and continued my journey; but through the long\nyears that have elapsed since that memorable summer's day, I have never\nfor a moment wavered in my belief, that in the mysterious horseman I had\na special interference of Providence, by which I was delivered from a\nposition of extreme danger.\"\n\nNow, however our readers may account for such incidents, the only purpose\nin introducing such a story here, is to say that it gives a fair\nillustration of that peculiar cast of ideal imagination which pervaded\nthe Welsh mind, and influenced at once the impressions both of preachers\nand hearers.\n\nThere is, perhaps, no other spot on our British soil where \"the old\norder\" has so suddenly \"changed\" as in Wales: the breaking open the\nmountains for mining purposes has led to the thronging of dense\npopulations on spots which were, only a few years since, unbroken\nsolitudes.  Ruins, which the sentimental idler never visited, wrecks of\ncastles and abbeys crumbling into dust, isolated places through which we\npassed thirty years since, which seemed as though they never could be\ninvaded by the railway whistle, or scarcely reached by the penny postman,\nnow lie on the great highway of the train.  It is not saying too much to\naffirm that there is no spot in Europe where the traveller is so\nconstantly brought into the neighbourhood of old magnificence, the relics\nof vanished cities.\n\nThe wonder grows as to what was the state of ancient society in Wales.\nAn eminent traveller says: \"In England our ancestors have left us,\ndispersed in various places, splendid remains of their greatness; but in\nWales you cannot travel ten miles without coming upon some vestige of\nantiquity which in another country you would go fifty to trace out.\"  It\nis of such spots that a Welsh poet, Dyer, says:\u2014\n\n             \"The pilgrim oft,\n    At dead of night, 'mid his orisons hears,\n    Aghast, the voice of Time disparting towers,\n    Tumbling all precipitate, all down-dashed,\n    Rattling around, loud thundering to the moon.\"\n\nWhat an illustration of this is St. David's!\u2014a little miserable village,\nwith the magnificent remains of its great palace, and the indications of\nits once splendid cathedral; itself now a kind of suffragan, it once\nnumbered seven suffragans within its metropolitan pale\u2014Worcester,\nHereford, Llandaff, Bangor, St. Asaph, Llanbadarn, and Margam.  The mitre\nnow dimly beaming at almost the lowest step of the ecclesiastical ladder,\nonce shone with so proud a lustre as to attract the loftiest\necclesiastics.  St. David's numbers one saint, three lord-treasurers, one\nlord privy-seal, one chancellor of Oxford, one chancellor of England,\nand, in Farrar, one illustrious martyr.\n\nTravel through the country, and similar reflections will meet you in\nevery direction.  You step a little off the high-road, and\u2014as, for\ninstance, in Kilgerran\u2014you come to the traditional King Arthur's castle,\nthe far-famed Welsh Tintagel, of which Warton sings,\u2014\n\n    \"Stately the feast, and high the cheer,\n    Girt with many an arm\u00e8d peer,\n    And canopied with golden pall,\n    Amid Kilgerran's castle hall;\n    Illumining the vaulted roof,\n    A thousand torches flamed aloof;\n    The storied tapestry was hung,\n    With minstrelsy the arches rung,\n    Of harps that with reflected light\n    From the proud gallery glittered bright.\"\n\nOr, in the neighbourhood of the magnificent coast of Pembrokeshire, the\nwondrous little chapel of St. Govan's, the hermitage of the hundred\nsteps; and those splendid wrecks of castles, Manopear, the home of\nGiraldus Cambrensis, and the graceful and almost interminable recesses of\nCarew.  A traveller may plunge about among innumerable villages bearing\nthe names of saints for whom he will look in vain in the Romish\ncalendar,\u2014St. Athan's, St. Siebald's, St. Dubric's, St. Dogmael's, St.\nIshmael's, and crowds besides.  All such places are girdled round with\ntraditions and legends known to Welsh arch\u00e6ologists\u2014the very nomenclature\nof Wales involving poetry and historical romance, and often deep tragedy.\nThe names of the villages have a whisper of fabulous and traditional\ntimes, and are like the half-effaced hieroglyphs upon an old Egyptian\ntomb.  There is the _Fynnon Waedog_ (Bloody Well), _the Pald of Gwaye_\n(the Hollow of Woe), the _Maen Achwynfan_, (the Stone of Lamentation and\nWeeping), the _Leysan Gwaed Gwyr_ (the Plant of the Blood of Man),\n_Merthyr Tydvil_ is the Martyred Tydvil.  Villages and fields with names\nlike these, remind us of the Hebrew names of places, really significant\nof some buried tragedy, long holding its place in the heart, and terror\nof the neighbourhood.\n\nIn a land-locked solitude like that of Nevern, Cardiganshire,\u2014where,\nby-the-bye, we might loiter some time to recite some anecdotes of its\nadmirable clergyman and great preacher, one of the Griffiths,\u2014the\nwanderer, after a piece of agreeable wildness, comes to a village,\nenchanting for its beauty, lying on the brink of a charming river, with\nindications of a decayed importance; the venerable yew-trees of its\nchurchyard shadowing over a singular\u2014we may venture to speak of it as a\npiece of inexplicable\u2014Runic antiquity, in a stone of a quadrangular form,\nabout two feet broad, eighteen inches thick, and thirteen feet high, with\na cross at the top.  Few countries can boast, like Wales, the charm of\nplaces in wildest and most delicious scenery, with all that can stir an\nartist's, poet's, or antiquarian's sensibility.  What a neighbourhood is\nLlandilo!\u2014the home of the really great poet, John Dyer, the author of\n\"Grongar Hill,\" a delicious spot in this neighbourhood.  Here, too, is\nGolden Grove, the retreat of our own Jeremy Taylor; and here, in his days\nof exile, many of the matchless sermons of him who has been called, by\nsome, \"the English Chrysostom,\" and, by others, the \"Milton of the\nEnglish pulpit,\" were preached.  We made a pilgrimage there ourselves\nsome few years since, urged by love to the memory of Jeremy Taylor.  We\nfound the old church gone, and in its place a new one,\u2014the taste of which\ndid not particularly impress us; and we inquired for Taylor's pulpit, and\nwere told it had been chopped up for fire-wood!  Then we inquired for a\npath through the fields, which for a hundred and fifty years had been\ncalled \"Taylor's Walk,\" where the great bishop was wont to meditate,\u2014and\nfound it had been delivered over to the plough.  We hope we may be\nforgiven if we say, that we hurried in disgust from a village which, in\nspite of its new noble mansion, had lost to us its chief charm.  But this\nneighbourhood, with its Dynevor Castle and its charming river, the Towey,\nand all the scenery described by the exquisite Welsh poet, in whose verse\nbeauty and sublimity equally reign, compels us to feel that if he\nsomewhat pardonably over-, by his own associations, the lovely\nshrine of his birth, he only naturally described the country through\nwhich these preachers wandered, when he says,\u2014\n\n    \"Ever charming, ever new,\n    When will the landscape tire the view!\n    The fountain's fall, the river's flow,\n    The woody valleys, warm and low:\n    The windy summit, wild and high,\n    Roughly rushing on the sky!\n    The pleasant seat, the ruin'd tow'r,\n    The naked rock, the shady bow'r;\n    The town and village, dome and farm,\n    Each give to each a double charm,\n    As pearls upon an Ethiop's arm.\"\n\nThe manners of the people, a few years since, were as singular and\nprimeval as their country; in all the villages there were singular\nusages.  The \"biddings\" to their weddings,\u2014which have, perhaps, yielded\nto advanced good taste,\u2014had a sweeter relief in other customs, at\nweddings and funerals, tending to civilize, and refine.  Throughout\nGlamorganshire, especially, and not many years since, it was the\nuniversal custom, when young unmarried persons died, to strew the way to\nthe grave with sweet flowers and evergreens.  Mr. Malkin, in his\ninteresting work on South Wales, published now seventy years since, says:\n\"There is in the world an unfeeling kind of false philosophy, which will\ntreat such customs as I mention with ridicule; but what can be more\naffecting than to see all the youth of both sexes in a village, and in\nevery village through which the corpse passes, dressed in their best\napparel, and strewing with sweet-scented flowers the ways along which one\nof their beloved neighbours was carried to his, or her last home?\"  No\ndoubt such customs are very much changed, but they were prevalent during\nthat period to which most of those preachers whose manners we have\nmentioned belonged.\n\nSuch pathetic usages, indicating a simple state of society, are commonly\nassociated, as we have seen, with others of a rougher kind and character.\nThe Welsh preachers were the pioneers of civilization,\u2014although advanced\nsociety might still think much had to be done in the amelioration of the\nnational manners.  They probably touched a few practices which were\nreally in themselves simple and affecting, but they swept away many\nsuperstitions, quite destroyed many rude and degrading practices, and\nintroduced many usages, which, while they were in conformity with the\nnational instincts of the people (such as preaching and singing, and\nassembling themselves together in large companies), tended to refine and\nelevate the mind and heart.\n\nSuch were the circumstances, and such the scenery, in which the great\nWelsh preachers arose.\n\nWe have not thought of those Welsh preachers who have made themselves\nespecially known in England.  Many have, from time to time, settled as\npastors with us, who have deserved a large amount of our esteem and\nhonour, blending in their minds high reverence, the tender sensitiveness\nof a poetic imagination, with the instinct of philosophic\ninquisitiveness\u2014even shading off into an order of scepticism,\u2014but all\nunited to a strong and impressive eloquence.  These attributes seem all\nessentially to adhere in the character of the cultured Welsh preacher.\nCaleb Morris finely illustrates all this; perhaps he was no whit\ninferior, in the build and architecture of his mind, to Horace Bushnell,\nwhom he greatly resembled; but, unlike Bushnell, he never committed any\nof his soliloquies of thought, or feeling to the press.  The present\nwriter possesses volumes of his reported sermons which have never seen\nthe light.\n\nAnd what a Welshman was Rowland Williams!  Who can read his life without\nfeeling the spirit of devotion, however languid, inflamed and fired?  And\nhow, in spite of all the heresies attributed to him, and, growing up in\nthe midst of the sacred ardours of his character, we find illustrated the\nwonder of the curious and searching eye, united to the warmth of the\ntender and revering heart!\u2014attributes, we repeat, which seemed to mingle\nin very inferior types of Welsh preachers, as well as in the more\neminent, and which, as they kindle into a passion in the man's nature who\ndesires to instruct his fellow-men, combine to make preaching, if they be\nabsent, an infamy, a pastime, a day labour, or a handicraft, an art or a\nscience; or, by their presence, constitute it a virtue and a mighty power\nover human souls.  Eminently these men seem to hear a voice saying, \"_The\nprophet that hath a dream_, _let him tell a dream_!  _What is the chaff\nto the wheat_? _saith the Lord_.\"\n\n                                * * * * *\n\n                     _Note to_ \"Cwm-Aman,\" _page_ 23.\n\nDr. Thos. Rees, in a letter to the Editor of the _Dysgedydd_, Rev. Herber\nEvans, says, \"That although bred and born within ten miles of Cwm-Aman,\nhe had never heard of this ridiculous superstition.\"\n\n\n\n\nCHAPTER II.\n_EARLY LIFE UNTIL HIS ENTRANCE INTO THE MINISTRY_.\n\n\n    Birth and Early Hardships\u2014Early Church Fellowship\u2014Beginning to\n    Learn\u2014Loses an Eye\u2014A Singular Dream\u2014Beginning to Preach\u2014His First\n    Sermon\u2014Is Baptized\u2014A New Church Fellowship\u2014The Rev. Timothy\n    Thomas\u2014Anecdotes\u2014A Long Season of Spiritual Depression\u2014Is ordained as\n    Home Missionary to Lleyn\u2014Commencement of Success as a\n    Preacher\u2014Remarks on Success\u2014Marries\u2014Great Sermon at Velinvole\u2014A\n    Personal Reminiscence of Welsh Preaching.\n\nChristmas Evans is not the first, in point of time, in the remarkable\nprocession of those men whose names we might mention, and of whom we\nshall find occasion in this volume to speak, as the great Welsh\npreachers.  And there may be some dispute as to whether he was the first\nin point of eminence; but he is certainly the one of the four whose name\nis something more than a tradition.  John Elias, Williams of Wern, and\nDavies of Swansea, have left behind them little beside the legendary\nrumour of their immense and pathetic power.  This is true, especially, of\nDavid Davies of Swansea; and yet, Dr. Rees, his successor, and a very\ncompetent authority, says: \"In some respects he was superior to all his\ndistinguished contemporaries.\"  But the name of Christmas Evans is,\nperhaps, the most extensively known of any,\u2014just as the name of Bunyan\nhas a far more extensive intimacy than the equally honourable names of\nBarrow and Butler; and there is a similar reason for this.  Christmas\nEvans, in the pulpit, more nearly approached the great Dreamer than any\npulpit master of whom we have heard; many of his sermons appear to have\nbeen long-sustained parables, and pictures alive with allegorical\ndelineation of human character.\n\nCHRISTMAS EVANS was born at a place called Esgairwen (Ysgarwen), in the\nparish of Llandysul, in Cardiganshire; he was born on Christmas Day\u2014and\nhence his Christian name\u2014in 1766.  His parents, Samuel and Johanna Evans,\nwere in the poorest circumstances; his father was a shoemaker, and\nalthough this profession has included such a number of men remarkable for\ntheir genius and high attainments, it has never found the masters of the\ncraft greatly remarkable for the possession of gold or gear.  His mother,\nby her maiden name Lewis, came from a respectable family of freeholders\nin the parish; but the father of Christmas died when he was a child,\u2014and\nthese were hard days of poverty, almost destitution, for the poor\nstruggling widow and her family,\u2014so her brother, James Lewis, of Bwlchog,\nin the parish of Llanfihangel-ar-Arth, took little Christmas home to his\nfarm, engaging to feed and clothe him for such labour on the farm as the\npoor boy might be able to perform.  Here he stayed six years,\u2014six\nmiserable years; his uncle was a hard, cruel man, a selfish drunkard.\nChristmas used to say of him, in after years, \"It would be difficult to\nfind a more unconscionable man than James Lewis in the whole course of a\nwicked world.\"  During these, which ought to have been the most valuable\nyears of his life, no care was taken of his heart, his mind, or his\nmorals; in fact, he had neither a friend nor a home.  At the age of\nseventeen he could not read a word, he was surrounded by the worst of\nexamples, and he became the subject of a number of serious accidents,\nthrough which he narrowly escaped with his life.  Once he was stabbed in\na quarrel, once he was nearly drowned, and with difficulty recovered;\nonce he fell from a high tree with an open knife in his hand, and once a\nhorse ran away with him, passing at full speed through a low and narrow\npassage.  There is an erroneous impression that, in those days, he was a\ngreat boxer, and that he lost his eye in a fight; the truth is quite\ndifferent; he was not a boxer, and never fought a battle in his life.  He\nlost his eye after his conversion, when he and some other young men were\nattempting the work of mutual help, in making up for lost time, by\nevening meetings, for various works of instruction; a number of his\nformer companions waylaid him at night, beat him unmercifully, and one\nstruck him with a stick over the eye.  In after years, when some one was\njesting before Robert Hall at Welsh preachers, upon his mentioning\nChristmas Evans, the jester said, \"And he only has one eye!\"  \"Yes, sir,\"\nhe answered, \"but that's a piercer; an eye, sir, that could light an army\nthrough a wilderness in a dark night.\"  So that in his sightless eye,\nChristmas Evans, like the one-eyed Spiridion, the noble witness in the\nNicean Council, really \"bore in his body a mark of the Lord Jesus.\"  But\nwe are anticipating.\n\nAt about seventeen years of age, he left his bad uncle and his more\nservile employments; still continuing the occupation of a farming lad, he\nwent to Glanclettwr; afterwards he lived at Penyralltfawr, at Gwenawlt,\nand then at Castellhywel.  Thus the days of his youth passed; he looks\nlike a poor, neglected, and forsaken lad.  Of books he knew nothing,\u2014he\nhad no men of intelligence around him with whom to converse, and his\ncondition in life doomed him to association with all that was low and\nbrutal.  And yet, strange as it may seem, as his friend and earliest\nbiographer, Mr. Rhys Stephen, has testified, even then, as in the\ninstance of the rugged young Samson, \"the Spirit of the Lord began to\nmove him at times.\"  It is not credible that, however crushed down\nbeneath the weight of such abject circumstances, the boy could have been\nexactly what the other boys and men round him were; restless feelings,\nand birth-throes of emotion and thought, make themselves known in most of\nus before they assume a shape in consciousness: it is natural that it\nshould have been so with him.  With a life of seriousness, which resulted\nin Church membership, and which appears to have taken place when he was\nabout seventeen years of age, commenced his life of mental\nimprovement,\u2014the first humble beginnings of intellectual effort.  It is\nsingular that the Church with which he first united, at Llwynrhydowain,\noriginally Presbyterian, and of considerable importance in the early\nhistory of Welsh Nonconformity, approached very nearly, when Evans united\nwith it, to Unitarianism.  Its pastor was the Rev. David Davies; he was\nan Arian, an eminent bard, a scholar, an admirable and excellent man, who\nhas left behind him a very honourable reputation.  Such a man as Mr.\nDavies was, he would be likely to be interested in the intelligent and\nintellectual state of the youth of his Church and congregation.  The\nslight accounts we possess of the avidity with which Christmas Evans and\nhis companions commenced their \"pursuit of knowledge under difficulties,\"\nis very animating and pleasing; they combined together with the desire to\nobtain the earliest and most necessary means of mental acquisitiveness,\nsuch as reading and writing, a desire for the acquisition of religious\nknowledge, and what may be spoken of as some of the higher branches of\nstudy.  But we will employ Christmas Evans's own words:\u2014\n\n    \"During a revival which took place in the Church under the care of\n    Mr. David Davies, many young people united themselves with that\n    people, and I amongst them.  What became of the major part of these\n    young converts, I have never known; but I hope God's grace followed\n    them as it did me, the meanest of the whole.  One of the fruits of\n    this awakening was the desire for religious knowledge that fell upon\n    us.  Scarcely one person out of ten could, at this time, and in those\n    neighbourhoods, read at all, even in the language of the country.  We\n    bought Bibles and candles, and were accustomed to meet together in\n    the evening, in the barn of Penyralltfawr; and thus, in about one\n    month, I was able to read the Bible in my mother tongue.  I was\n    vastly delighted with so much learning.  This, however, did not\n    satisfy me, but I borrowed books, and learnt a little English.  Mr.\n    Davies, my pastor, understood that I thirsted for knowledge, and took\n    me to his school, where I stayed for six months.  Here I went through\n    the Latin Grammar; but so low were my circumstances that I could stay\n    there no longer.\"\n\nTo preach, as we all know, has often been an object of ambition with\nyoung converts, and the novices in the vestibule of knowledge of the\nspiritual life; such an ambition seems very early to have stirred in the\nheart of young Christmas.  We have already mentioned how it was that he\nso cruelly lost the use of an eye; it illustrates the singular brutality\nof the time and neighbourhood; an inoffensive lad, simply because he\nrenounced the society of profane drunkards, and was laudably busying\nhimself with the affairs of a higher life, was set upon in the darkness\nof the night by six young ruffians, unmercifully beaten with sticks, and\nthe sight of an eye destroyed.  It was the night after this calamity that\nhe had a dream; and the dream of the night reveals the bent of his day\ndreams.  He dreamt that the Day of Judgment was come, that he saw the\nworld in a blaze; with great confidence he called out, \"Jesus, save me!\"\nAnd he thought he saw the Lord turn towards him and say, \"It was thy\nintention to preach the Gospel, but it is now too late, for the Day of\nJudgment is come.\"  But this vision of the night clung to him when he\nawoke; perhaps he feared that the loss of the eye would interfere with\nhis acceptance as a minister.  Certainly the dream had an influence on\nhis future career,\u2014so had many other dreams.  It was always his belief\nthat he had received some of his most important impressions from dreams:\nnothing, apparently, no amount of reason or argument, could persuade him\nto the contrary.  To preach the Gospel became an ardent desire now with\nthis passionately imaginative and earnest youth; but there were serious\nhindrances in the way.  There appears to have been a kind of law in the\nChurch with which he was connected at Llwynrhydowain, that no member of\nthe Church should be permitted to preach until he had passed through a\ncollege course.  It is very remarkable that two of the greatest preachers\nwho have adorned the pulpit of Wales should have been admitted into\nChurch fellowship together on the same evening,\u2014David Davies, afterwards\nof Swansea, whose name we have already mentioned, and Christmas Evans.\nIt was always the regret and complaint of their first pastor, that the\nChurch law to which we have referred, deprived his Church of the two most\neminent men it had ever produced.  There were, no doubt, other reasons;\nbut it is singular, now, to notice the parallelism of the gifted pair,\nfor they also preached their first sermon, within a week of each other,\nin the same cottage.  Cottage preaching was then of much more importance\nthan it now seems to our ecclesiastical and \u00e6sthetic apprehensions; and\nthe congregations which assembled in those old Welsh cottages were such\nas to try the mental and spiritual strength of a young preacher.  How\nDavies acquitted himself, and how he ran his course, we may notice\nby-and-bye; our present concern is with Christmas Evans.  Perhaps our\nreaders will not entertain a depreciating opinion of the youth, when they\nhear him very candidly confess that the substance of his first sermon was\ntaken from Beveridge's \"Thesaurus Theologicus,\" a book borrowed,\nprobably, from his pastor.  But a Mr. Davies, who must have been a\nreading man although a farmer, heard it, was very much impressed by it,\nbut went home and found it; so that the poor boy's reputation as a\npreacher seemed gone.  \"Still,\" said the good man, \"I have some hope of\nthe son of Samuel the shoemaker, because the prayer was as good as the\nsermon.\"  But perhaps he would not have thought so hopefully of the young\nman had he then known, what Christmas afterwards confessed, that the\nprayer, too, was very greatly committed to memory from a collection of\nprayers by a well-known clergyman, Griffith Jones of Llanddowror.\n\nSuch was the first public effort of this distinguished preacher; like the\nfirst effort of his great English contemporary, Robert Hall, we suppose\nit would be regarded as a failure.  Meantime, we have to notice that the\nspiritual life of the youth was going on; he began to be dissatisfied\nwith the frame of theologic sentiment of the Church to which he belonged.\nHe heard preachers who introduced him to the more grand, scriptural, and\nevangelical views of Christian truth.  The men of that time did not play\nat preaching; the celebrated David Morris, father of the yet more\ncelebrated Ebenezer Morris; the great Peter Williams, Jones of Llangan,\nThomas Davies of Neath,\u2014such men as these appear to have kindled in his\nmind loftier views of the person and the work of Christ.  Also, a man\nnamed Amos, who had been a member of the same Church with Christmas\nEvans, had left that communion, and joined that of the Baptists.  A close\nstudy of the Word of God led Christmas also to a change of convictions as\nto the meaning and importance of the rite of baptism.  A similar change\nof theologic opinion was passing through the mind of his young friend and\nfellow-member, David Davies, who finally united himself with the\nIndependent communion.  Christmas Evans says, \"I applied to the Baptist\nChurch at Aberduar, where I was in due time received; I was then about\ntwenty years and six months old.  I was baptized by the Rev. Timothy\nThomas.\"\n\nAs the names of successive persons and pastors pass before our eyes, and\nappear in these pages, it is at once affecting, humbling, and elevating,\nto think of men of whom our ears have scarcely ever heard, but who, in\ntheir day, were men \"of whom the world was not worthy,\" and whose \"record\nis now on high.\"  Such a man, beyond all question, was this Timothy\nThomas, the son of an eminent father, the brother of men who, if not as\neminent as himself, were yet worthy of the noble relationship.  He was a\nWelsh gentleman, lived on a farm, an extended lease of which he held, and\nwhich enabled him to preach and fulfil the work of a pastor without any\nmonetary reward.  He appears to have devoted himself, his time, his\nenergy, and his property to the work of the ministry.  His farm was a\nsplendid one in the vale of the Teivy.  Mr. Rhys Stephen, who knew him,\nspeaks of his gallant bearing, his ingenuous spirit, and of his princely\nmagnanimity; he would ride thirty or forty miles on a Saturday, through\nthe remote wilds of Caermarthenshire and Cardiganshire, to be ready for\nthe services on the Sunday.  His gentlemanly bearing overcame and beat\ndown mobs which sometimes assembled for the purpose of insulting and\nassailing him.  Mr. Stephen mentions one singular instance, when Mr.\nThomas was expected to administer the ordinance of baptism, and, as was\nnot unusual in those days, in the natural baptistry of the river.  A mob\nhad assembled together for the purpose of insulting and annoying the\nservice, the missiles of offence in their hands; when, suddenly, a\nwell-dressed gentleman, mounted on a noble horse, rode over the village\nbridge; he hastily alighted, gave his bridle to a bystander, walked\nbriskly into the middle of the little flock; the inimical members of the\nmob set him down for a magistrate at the least, and expected that he\nwould give the word to disperse; but instead of doing so, he took the\nnearest candidate by the hand, and walked himself down into the stream,\nbooted and spurred as he was.  Before the mob had done gaping, he had\ndone this part of his work; after this, however, he stood upon the brink\nof the stream, still in his wet attire, and preached one of his ardent\nsermons.  He certainly conciliated the homage of the opposing forces, and\nleft them under the impression that the \"dippers,\" as the Baptists were\ngenerally called, had certainly one gentleman among them.  We do not know\nhow our Baptist brethren would like to submit to this kind of service,\nbut it certainly seems to resemble more closely the baptism of Enon, near\nto Salem, and that of the Ethiopian prince by Philip, than some we have\nseen.\n\nThe anecdotes of this Timothy Thomas are too good and too numerous to be\nentirely passed by.  Once he was preaching in the enchanting\nneighbourhood near Llandeilo, to which we referred in the first\nchapter\u2014the neighbourhood of Grongar Hill, and Golden Grove; the\nneighbourhood of Dyer, Steele, and Jeremy Taylor.  It was a still Sabbath\nmorning in the summer, and in that lovely spot immense crowds were\ngathered to hear him.  He had administered baptism, and preached, without\ninterruption, when someone came up to him and told him, with startled\nfear and trepidation, that the clergyman,\u2014the rector,\u2014on his way to the\nchurch, had been detained, utterly unable to pass through the crowd,\nthrough the greater part of the service.  Instantly, with admirable tact\nand catholicity, he exclaimed: \"I understand that the respected clergyman\nof the parish has been listening patiently to me for the last hour; let\nus all go to the church and return the compliment by hearing him.\"  The\nchurch, and the churchyard as well, were instantly crowded; the clergyman\nwas delighted with the catholic spirit displayed by the Baptist minister,\nand of course not a word further was said about the trespass which had\nbeen committed.\n\nTimothy Thomas was a noble specimen of what has been called the \"muscular\nChristian;\" he had great courage.  Once, when travelling with his wife,\nand set upon by four ruffians, he instantly, with his single stick,\nfloored two, but broke his stick in the very act of conquest.\nImmediately he flew to a hedge and tore up a prodigious stake, and was\nagain going forth to victory, when the scoundrels, having had enough of\nthis bishop of the Church militant, took to flight and left him in\nundisputed possession of the field.  A remarkable man this,\u2014a sort of\nWelsh chieftain; a perfect gentleman, but half farmer, half preacher.  In\nthe order of Church discipline, a man was brought up before him, as the\npastor, for having knocked down an Unitarian.  \"Let us hear all about\nit,\" said the pastor.  \"To tell all the truth about it, sir,\" said the\nculprit, \"I met Jack the miller at the sign of the Red Dragon, and there\nwe had a single glass of ale together.\"  \"Stop a bit,\" said the minister;\n\"I hope you paid for it.\"  \"I did, sir.\"  \"That is in your favour,\nThomas,\" said the pastor; \"I cannot bear those people who go about\ntippling at other people's expense.  Go on, Thomas.\"  \"Well, sir, after a\nlittle while we began quietly talking about religion, and about the work\nof Jesus Christ.  Jack said that He was only a man, and then he went on\nto say shocking things, things that it was beyond the power of flesh and\nblood to bear.\"  \"I daresay,\" said the pastor; \"but what did he say?\"\n\"He actually said, sir, that the blood of Christ had no more power in it\nthan the blood of a beast.  I could not stand that any more, so I knocked\nhim down.\"  \"Well, brother,\" said the minister, \"I cannot say that you\ndid the right thing, but I quite believe that I should have done so too.\nGo, and sin no more.\"\n\nBut with all these marks of a strong character, the lines of Timothy\nThomas's faith were clear and firm.\n\nSuch was the man who received Christmas Evans into the Church of which he\nbecame so bright and shining an ornament.  This noble man survived until\nhis eighty-sixth year; he died at Cardigan, in 1840.  He was asked,\nsometimes, how many he had baptized during his lifetime, and he would\nreply, brusquely, \"About two thousand;\" at other times, he would be more\nparticular, and say, \"I have baptized at least two thousand persons.\nYes,\" he would add tenderly, \"and thirty of them have become ministers of\nthe Gospel; and it was I who baptized Christmas Evans,\"\u2014sometimes adding\nna\u00efvely, \"I did it right, too,\u2014according to the apostolic practice, you\nknow.\"\n\nThus we are brought to the interesting and important turning-point in the\nlife of Christmas Evans.  He had united himself with the Baptist\ncommunion.  Our readers will clearly perceive, that he was a young man\nwho could not be hidden, and it was soon discovered that the work of the\nministry was to be his destination.  As to his internal state, upon which\na ministerial character must always depend, these early years of his\nreligious life were times and seasons of great spiritual depression.\nSuch frames of feeling depend, perhaps, not less, or more, upon certain\naspects of religious truth, than they do upon the peculiarities of\ntemperament; a nervous imagination is very exhausting, and brings the\nphysical frame very low; moreover, exalted ideas, and ideals, produce\nvery depressing appreciations of self.  He thought himself a mass of\nignorance and sin; he desired to preach, but he thought that such words\nas his must be useless to his hearers: then, as to the method of\npreaching, he was greatly troubled.  He thought by committing his sermons\nto memory he forfeited the gift of the Holy Spirit; so he says he changed\nhis method, took a text without any premeditation, and preached what\noccurred to him at the time; \"but,\" he continues, \"if it was bad before,\nit was worse now; so I thought God would have nothing to do with me as a\npreacher.\"\n\nThe young man was humbled; he entered every pulpit with dread; he thought\nthat he was such an one that his mere appearance in the pulpit would be\nquite sufficient to becloud the hearts of his hearers, and to intercept\nthe light from heaven.  Then it seems he had no close friend to whom he\ncould talk; he was afraid lest, if he laid bare the secrets of his heart,\nhe should seem to be only a hypocrite; so he had to wrap up the bitter\nsecrets of his soul in his own heart, and drink of his bitter cup alone.\nIs this experience singular?  Is not this the way in which all truly\ngreat, and original preachers have been made?\u2014Luther, Bunyan, Dr. Payson,\nRobert Hall,\u2014how many beside?  Such men have attained high scholarships,\nand fellowships, in the great university of human nature; like Peter,\npierced to the heart themselves, they have \"pricked\" the hearts, the\nconsciences, of the thousands who have heard them.  Thus, more than from\nthe lore of classical literatures, they have had given to them \"the\ntongue of the learned,\" which has enabled them to speak \"a word in season\nto those who were wearied;\" thus, \"converted\" themselves, they have been\nable to \"strengthen their brethren.\"\n\nEvans passed through a painful experience; the young man was feeling his\nway.  He was unconscious of the powers within him, although they were\nstruggling for expression; and so, through his humility and lowly\nconceptions of himself, he was passing on to future eminence and\nusefulness.\n\nLleyn was the first place where he appears to have felt his feet.  Lleyn\nat that time had not even the dignity of being a village; it is a little\ninland hamlet out of Caernarvon Bay; Nevin is its principal village;\nperhaps if the reader should seek out Lleyn, even upon a tolerable map of\nCaernarvonshire, he will have a difficulty in finding it.  It seems to\nhave been a hamlet of the promontory, on a grand coast, surrounded by\nmagnificent hills, or overhanging mountains; we have never visited it,\nbut those who have done so speak of it as possessing the charms of\npeculiar wildness: on the one side, precipitous ravines, shut in by the\nsea; on the other, walls of dark mountains,\u2014forming the most complete\npicture of isolation possible to imagine.  Here is said to be the last\nresting-place of Vortigern, who fled hither to escape the rage of his\nsubjects, excited by his inviting the Saxons to Britain.  A curious\ntradition holds that the mountains are magnetic, and masters of vessels\nare said to be careful not to approach too near the coast, fearing the\neffect upon their compasses; this is believed to be the effect of a\nstrong undercurrent setting in all along the coast, dangerous to vessels,\nand apt to lead them out of their course.  Such was Lleyn, the first\nfield of labour on which this melancholy and brooding youth was to\nexercise his ministry.\n\nEvans had attended the Baptist Association at Maesyberllan in\nBrecknockshire, in 1790; he was persuaded there to enter upon the\nministry in this very obscure district, and he was ordained as a\nmissionary to work among the humble Churches in that vicinity.  It does\nnot appear that, in his own neighbourhood, he had as yet attained to any\nreputation for peculiar power, or that there were any apparent auguries\nand prognostications of his future usefulness.  It is curious to notice,\nalmost so soon as he began his work in this his first distinct field of\nlabour, he appears like a man new made; for this seems to have been the\nplace where the burden of which Bunyan speaks, rolled from this\nChristian's back; here a new life of faith began to glow in him, and he\nknew something of what it is to have the \"oil of joy for mourning, and\nthe garment of praise instead of the spirit of heaviness.\"  A little\nsuccess is very encouraging; depreciation is frequently the parent of\ndepression; success is often a fine old strengthening wine; and how often\nwe have had occasion to admire men who have wrought on at life's tasks\nbravely and cheerfully, although success never came and sat down by their\nside, to cheer and encourage them; one sometimes wonders what they would\nhave done had their efforts and words received the garland and the crown.\nWell, perhaps not so much; these things are more wisely ordered than we\nknow.  Only this also may be remarked, that, perhaps, the highest order\nof mind and heart can do almost as well without success as with it,\u2014will\nbehave beautifully if success should come, will behave no less\nbeautifully even if success should never come.\n\nAt Lleyn, Christmas Evans tasted the first prelibations of a successful\nministry; a wondrous power attended his preaching, numbers were gathered\ninto the Church.  \"I could scarcely believe,\" he says, \"the testimony of\nthe people who came before the Church as candidates for membership, that\nthey were converted through my ministry; yet I was obliged to believe,\nthough it was marvellous in my eyes.  This made me thankful to God, and\nincreased my confidence in prayer; a delightful gale descended upon me as\nfrom the hill of the New Jerusalem, and I felt the three great things of\nthe kingdom of heaven, righteousness, and peace, and joy in the Holy\nGhost.\"  Indeed, very unusual powers seemed to attend him.  He says, \"I\nfrequently preached out of doors at nightfall,\" and the singing, and the\npraising seem to have touched him very tenderly; he frequently found his\ncongregations bathed in tears and weeping profusely.  Preaching was now\nto him, as he testifies, a very great pleasure,\u2014and no wonder; quite a\nremarkable revival of religious feeling woke up wherever he went.  When\nhe first entered Lleyn, the religious life was very cold and feeble;\nquite wonderful was the change.\n\nAfter a time, exhausted with his work in these villages, he accepted an\ninvitation to visit the more remote parts of South Wales.  When\nministers, like Christmas Evans, are enfeebled in health, they recreate\nthemselves by preaching; the young man was enfeebled, but he started off\non his preaching tour; he could not obtain a horse, so he walked the\nwhole way, preaching in every village or town through which he passed.\nVery frequently large numbers of the same congregation would follow after\nhim the next day, and attend the services fifteen or twenty times,\nalthough many miles apart.  So he went through the counties of Cardigan,\nPembroke, Caernarvon, Glamorgan, Monmouth, and Brecknock, stopping and\nholding services at the innumerable villages lying on his way.  The fame\nthat a wonderful man of God had appeared spread through South Wales on\nthe wings of the wind, and an appointment for Christmas Evans to preach\nwas sufficient to attract thousands to the place.  While he yet continued\nat Lleyn as itinerant missionary, in that short time he had acquired\nperhaps a greater popularity than any other preacher of that day in\nWales.\n\nWe have not said that, during the first years of his residence at Lleyn,\nhe married Catherine Jones, a young lady a member of his own Church,\u2014a\npious girl, and regarded as in every way suitable for his companion.  It\nwill be seen that, so far from diminishing, it seemed rather to increase\nhis ardour; he frequently preached five times during the Sabbath, and\nwalked twenty miles; his heart appeared to be full of love, he spoke as\nin the strains of a seraph.  No wonder that such labour and incessant\nexcitement told upon his health, it was feared even that he might sink\ninto consumption; but surely it was a singular cure suggested for such a\ndisease, to start off on the preaching tour we have described.\n\nAt last, however, in an unexpected moment, he became great.  It was at\none of those wonderful gatherings, an Association meeting, held at\nVelinvoel, in the immediate neighbourhood of Llanelly.  A great concourse\nof people were assembled in the open air.  There was some hitch in the\narrangements.  Two great men were expected, but still some one or other\nwas wanted to break the ice\u2014to prepare the way.  On so short a notice,\nnotwithstanding the abundant preaching power, no one was found willing to\ntake the vacant place.  Christmas Evans was there, walking about on the\nedge of the crowd\u2014a tall, bony, haggard young man, uncouth, and\nill-dressed.  The master of the ceremonies for the occasion, the pastor\nof the district, was in an agony of perplexity to find his man,\u2014one who,\nif not equal to the mightiest, would yet be sufficient for the occasion.\nIn his despair, he went to our old friend, Timothy Thomas; but he,\ndeclining for himself, said abruptly, \"Why not ask that one-eyed lad from\nthe North?  I hear that he preaches quite wonderfully.\"  So the pastor\nwent to him.  He instantly consented.  Many who were there afterwards\nexpressed the surprise they felt at the communication going on between\nthe pastor and the odd-looking youth.  \"Surely,\" they said, \"he can never\nask that absurdity to preach!\"  They felt that an egregious mistake was\nbeing committed; and some went away to refresh themselves, and others to\nrest beneath the hedges around, until the great men should come; and\nothers, who stayed, comforted themselves with the assurance that the\n\"one-eyed lad\" would have the good sense to be very short.  But, for the\nyoung preacher, while he was musing, the fire was burning; he was now,\nfor the first time, to front one of those grand Welsh audiences, the\nsacred _Eisteddfod_ of which we have spoken, and to be the preacher of an\noccasion, which, through all his life after, was to be his constant work.\nHenceforth there was to be, perhaps, not an Association meeting of his\ndenomination, of which he was not to be the most attractive preacher, the\nmost longed-for and brilliant star.\n\nHe took a grand text: \"And you, that were sometime alienated and enemies\nin your mind by wicked works, yet now hath He reconciled, in the body of\nHis flesh, through death, to present you holy, and unblamable, and\nunreprovable in His sight.\"  Old men used to describe afterwards how he\njustified their first fears by his stiff, awkward movements; but the\norgan was, in those first moments, building, and soon it began to play.\nHe showed himself a master of the instrument of speech.  Closer and\ncloser the audience began to gather near him.  They got up, and came in\nfrom the hedges.  The crowd grew more and more dense with eager\nlisteners; the sermon became alive with dramatic representation.  The\nthrong of preachers present confessed that they were dazzled with the\nbrilliance of the language, and the imagery, falling from the lips of\nthis altogether unknown and unexpected young prophet.  Presently, beneath\nsome appalling stroke of words, numbers started to their feet; and in the\npauses\u2014if pauses were permitted in the paragraphs\u2014the question went, \"Who\nis this? who have we here?\"  His words went rocking to and fro; he had\ncaught the \"_hwyl_,\"\u2014he had also caught the people in it; he went\nswelling along at full sail.  The people began to cry, \"_Gogoniant_!\"\n(Glory!)  \"_Bendigedig_!\" (Blessed!)  The excitement was at its highest\nwhen, amidst the weeping, and rejoicing of the mighty multitude, the\npreacher came to an end.  Drawn together from all parts of Wales to the\nmeeting, when they went their separate ways home they carried the memory\nof \"the one-eyed lad\" with them.\n\nChristmas Evans was, from that moment, one of the most famous preachers\nin the Principality.  Lord Byron tells us how he woke up one morning and\nfound himself famous.  In those days, a new great Welsh preacher was\nquite as famous a birth in the little country of Wales as the most famous\nreputation could be in the literary world of England.\n\nWe can conceive it all; for, about thirty-five years since, we were\nspectators of some such scene.  It was far in the depths of the dark\nmountains beyond Abersychan, that we were led to a large Welsh service;\nbut it was in a great chapel, and it was on a winter's night.  The place\nwas dimly lit with candles.  There were, we remember, three preachers.\nBut whilst the first were pursuing their way, or the occasional hymns\nwere being chanted, our companion said to us, \"But I want you to hear\nthat little hump-backed man, behind there; he will come next.\"  We could\nscarcely see the little hump-backed man, but what we saw of him did not\npredispose our minds to any very favourable impressions, or prophecies of\ngreat effects.  In due time he came forward.  Even as soon as he\npresented himself, however, there was an evident expectation.  The people\nbegan more certainly to settle themselves; to crane their necks forward;\nto smile their loving smile, as upon a well-known friend, who would not\ndisappoint them; and to utter their sighs and grunts of satisfaction.  He\nwas as uncouth a piece of humanity as we have ever seen, the little\nhump-backed man, thin and bony.  His iron-grey hair fell over his\nforehead with no picturesque effect, nor did his eyes seem to give any\nindication of fire; and there was a shuffling and shambling in his gait,\ngiving no sign of the grace of the orator.  But, gradually, as he moved\nalong, and before he had moved far, the whole of that audience was\nsubject to his spell of speech.  His hair was thrown back from his\nforehead; his features were lighted up.  Hump-backed!  You neither saw\nit, nor thought of it.  His wiry movement seemed informed by dignity and\ngrandeur.  First, there came forth audible gaspings, and grunts of\napproval and pleasure.  His very accent, whether you knew his language or\nnot, compelled tears to start to the eyes.  Forth came those devout\ngushings of speech we have mentioned, which, in Wales, are the\nacclamations which greet a preacher; and, like Christmas Evans with the\nclose of his first grand sermon, the little hump-backed man sat down,\nvictorious over all personal deformity, amidst the weeping and rejoicing\nof the people.  We have always thought of that circumstance as a\nwonderful illustration of the power of the mind over the body.\n\nChristmas returned to Lleyn, but not to remain there long.  The period of\nhis ministry in that neighbourhood was about two years, and during that\ntime the religious spirit of the neighbourhood had been deeply stirred.\nIt is most likely that the immediate cause which led to his removal may\nbe traced to the natural feeling that he was fitted for a much more\nobvious and extended field of labour.  Lleyn was a kind of mission\nstation, its churches were small, they had long been disorganised, and it\nwas not likely that, even if they woke at once into newness of life, they\ncould attain to ideas of liberality and Church order, on which the growth\nand advance and perpetuity of the Churches could alone be founded; and\nthen it was very likely discovered that the man labouring among them\nwould be demanded for labours very far afield; it is awkward when the\ngifts of a man make him eminently acceptable to shine and move as an\nevangelist, and yet he is expected to fill the place, and be as steady in\npastoral relations as a pole star!\n\n\n\n\nCHAPTER III.\n_THE MINISTRY IN THE ISLAND OF ANGLESEA_.\n\n\n    Journey to Anglesea\u2014Cildwrn Chapel, and Life in the Cildwrn\n    Cottage\u2014Poverty\u2014Forcing his Way to Knowledge\u2014Anecdote, \"I am the\n    Book\"\u2014A Dream\u2014The Sandemanian Controversy\u2014Jones of Ramoth\u2014\"Altogether\n    Wrong\"\u2014The Work in Peril\u2014Thomas Jones of Rhydwilym\u2014Christmas's\n    Restoration to Spiritual Health\u2014Extracts from Personal\n    Reflections\u2014Singular Covenant with God\u2014Renewed Success\u2014The Great\n    Sermon of the Churchyard World\u2014Scenery of its Probable\n    Delivery\u2014Outline of the Sermon\u2014Remarks on the Allegorical\n    Style\u2014Outlines of Another Remarkable Sermon, \"The Hind of the\n    Morning\"\u2014Great Preaching but Plain Preaching\u2014Hardships of the Welsh\n    Preacher.\n\nIn 1792 Christmas Evans left Lleyn.  He speaks of a providential\nintimation conveyed to him from the Island of Anglesea; the providential\nintimation was a call to serve all the Churches of his order in that\nisland for seventeen pounds a year! and for the twenty years during which\nhe performed this service, he never asked for more.  He was twenty-six\nyears of age when he set forth, on his birthday, Christmas Day, for his\nnew and enlarged world of work.  He travelled like an Apostle,\u2014and surely\nhe travelled in an apostolic spirit,\u2014he was unencumbered with this\nworld's goods.  It was a very rough day of frost and snow,\n\n    \"The way was long, the wind was cold.\"\n\nHe travelled on horseback, with his wife behind him; and he arrived on\nthe evening of the same day at Llangefni.  On his arrival in Anglesea he\nfound ten small Baptist Societies, lukewarm and faint; what amount of\nlife there was in them was spent in the distraction of theological\ncontroversy, which just then appeared to rage, strong and high, among the\nBaptists in North Wales.  He was the only minister amongst those\nChurches, and he had not a brother minister to aid him within a hundred\nand fifty miles; but he commenced his labours in real earnest, and one of\nhis first movements was to appoint a day of fasting and prayer in all the\npreaching places; he soon had the satisfaction to find a great revival,\nand it may with truth be said \"the pleasure of the Lord prospered in his\nhand.\"\n\nLlangefni appears to have been the spot in Anglesea where Christmas found\nhis home.  Llangefni is a respectable town now; when the preaching\napostle arrived there, near a hundred years since, its few scattered\nhouses did not even rise to the dignity of a village.  Cildwrn Chapel was\nhere the place of his ministrations, and here stood the little cottage\nwhere Christmas and his wife passed their plain and simple days.  Chapel\nand cottage stood upon a bleak and exposed piece of ground.  The cottage\nhas been reconstructed since those days, but upon the site of the queer\nand quaint old manse stands now a far more commodious chapel-keeper's\nhouse.  As in the Bedford vestry they show you still the chair in which\nJohn Bunyan sat, so here they show a venerable old chair, Christmas\nEvans's chair, in the old Cildwrn cottage; it is deeply and curiously\nmarked by the cuttings of his pocket-knife, made when he was indulging in\nthose reveries and daydreams in which he lived abstracted from everything\naround him.\n\nThe glimpses of life we obtain from this old Cildwrn cottage do not\nincline us to speak in terms of very high eulogy of the Voluntary\nprinciple, as developed in Anglesea in that day; from the description, it\nmust have been a very poor shanty, or windy shieling; it is really almost\nincredible to think of such a man in such a home.  The stable for the\nhorse or pony was a part of the establishment, or but very slightly\nseparated from it; the furniture was very poor and scanty: a bed will\nsometimes compensate for the deprivations and toils of the day when the\nwearied limbs are stretched upon it, but Christmas Evans could not, as\nJames Montgomery has it, \"Stretch the tired limbs, and lay the head, upon\nhis own delightful bed;\" for, one of his biographers says, the article on\nwhich the inmates, for some time after their settlement, rested at night,\ncould be designated a bed only by courtesy; some of the boards having\ngiven way, a few stone slabs did some necessary service.  The door by\nwhich the preacher and his wife entered the cottage was rotted away, and\nthe economical congregation saved the expense of a new door by nailing a\ntin plate across the bottom; the roof was so low that the master of the\nhouse, when he stood up, had to exercise more than his usual forethought\nand precaution.\n\nHere, then, was the study, the furnace, forge, and anvil whence were\nwrought out those noble ideas, images, words, which made Christmas Evans\na household name throughout the entire Principality.  Here he, and his\nCatherine, passed their days in a life of perfect naturalness\u2014somewhat\ntoo natural, thinks the reader\u2014and elevated piety.  Which of us, who\nwrite, or read these pages, will dare to visit them with the indignity of\nour pity?  Small as his means were, he looks very happy, with his\npleasant, bright, affectionate, helpful and useful wife; he grew in the\nlove and honour of the people; and to his great pulpit eminence, and his\nsimple daily life, have been applied, not unnaturally, the fine words of\nWordsworth\u2014\n\n    \"So did he travel on life's common way\n    In cheerful lowliness; and yet his heart\n    The mightiest duties on itself did lay.\"\n\nAnd there was a period in Wordsworth's life, before place, and fame, and\nprosperity came to him, when the little cottage near the Wishing Gate, in\nGrasmere, was not many steps above that of the Cildwrn cottage of\nChristmas Evans.  The dear man did not care about his poverty,\u2014he appears\nnever either to have attempted to conceal it, nor to grumble at it; and\none of his biographers applies to him the pleasant words of Jean Paul\nRichter, \"The pain of poverty was to him only as the piercing of a\nmaiden's ear, and jewels were hung in the wound.\"\n\nIt was, no doubt, a very rough life, but he appears to have attained to\nthe high degree of the Apostle,\u2014\"having food and raiment, let us be\ntherewith content;\" and he was caught up, and absorbed in his work:\nsermons, and material for sermons, were always preparing in his mind; he\nlived to preach, to exercise that bardic power of his.  That poor room\nwas the study; he had no separate room to which to retire, where, in\nsolitude, he could stir, or stride the steeds of thought or passion.\n\nDuring those years, in that poor Cildwrn room, he mastered some ways of\nscholarship, the mention of which may, perhaps, surprise some of our\nreaders.  He made himself a fair Hebraist; no wonder at that, he must\nhave found the language, to him, a very congenial tongue; we take it\nthat, anyhow, the average Welshman will much more readily grapple with\nthe difficulties of Hebrew than the average Englishman.  Then he became\nso good a Grecian, that once, in a bookseller's shop, upon his making\nsome remarks on Homer in the presence of a clergyman, a University man,\nwhich drew forth expressions of contempt, Christmas put on his classical\npanoply, and so addressed himself to the shallow scholar, that he was\ncompelled, by the pressure of engagements, to beat a surprisingly quick\nretreat.\n\nVery likely the slender accoutrements of his library would create a sneer\nupon the lips of most of the scholars of the modern pulpit: his lexicons\ndid not rise above Parkhurst,\u2014and _we_ will be bold to express gratitude\nto that forgotten and disregarded old scholar, too; Owen supplied him\nwith the bones of theological thought, the framework of his systematic\ntheology; and whatever readers may think of his taste, Dr. Gill largely\ndrew upon his admiration and sympathy, in the method of his exposition.\nBut, when all was said and done, he was the Vulcan himself, who wrought\nthe splendid fancies of the Achilles' shield,\u2014say, rather, of the shield\nof Faith; he did not disdain books, but books with him were few, and his\nmind, experience, and observation were large.\n\nA little while ago, we heard a good story.  A London minister of\nconsiderable notoriety, never in any danger of being charged with a too\nlowly estimate of himself, or his powers, was called to preach an\nanniversary sermon, on a week evening, some distance from London.\nArrived at the house of the brother minister, for whom he had undertaken\nthe service, before it commenced, he requested to be shown into the\nstudy, in which he might spend some little time in preparation: the\nminister went up with him.\n\n\"So!\" said the London Doctor, as he entered, and gazed around, \"this is\nthe place where all the mischief is done; this is your furnace, this is\nthe spot from whence the glowing thoughts, and sparks emanate!\"\n\n\"Yes,\" said his host, \"I come up here to think, and prepare, and be\nquiet; one cannot study so well in the family.\"\n\nThe Doctor strode up and down the room, glancing round the walls, lined\nwith such few books as the modest means of a humble minister might be\nsupposed to procure.\n\n\"Ah!\" said the Doctor, \"and these are the books, the alimentary canals\nwhich absorb the pabulum from whence you reinvigorate the stores of\nthought, and rekindle refrigerated feeling.\"\n\n\"Yes, Doctor,\" said the good man, \"these are my books; I have not got\nmany, you see, for I am not a rich London minister, but only a poor\ncountry pastor; you have a large library, Doctor?\"\n\nThe great man stood still; he threw a half-indignant and half-benignant\nglance upon his humble brother, and he said, \"_I_ have no library, _I_ do\nnot want books, _I_ am _the_ Book!\"\n\nChristmas Evans, so far as he could command the means,\u2014but they were very\nfew,\u2014was a voracious reader; and most of the things he read were welded\ninto material for the imagination; but much more truly might he have\nsaid, than the awful London dignitary and Doctor, \"I have no books, I am\nthe book.\"  His modesty would have prevented him from ever saying the\nlast; but it was nevertheless eminently and especially true, he _was_ the\nbook.  There was a good deal in him of the self-contained, self-evolving\ncharacter; and it is significant of this, that, while probably he knew\nlittle, or nothing, of our great English classical essayists, John Foster\nand his Essays were especially beloved by him; far asunder as were their\nspheres, and widely different their more obvious and manifested life,\nthere was much exceedingly alike in the structure of their mental\ncharacters.\n\nWe have already alluded to the dream-life of Christmas Evans; we should\nsay, that if dreams come from the multitude of business, the daily\noccupation, the ordinary life he lived was well calculated to foster in\nhim the life of dreams.  Here is one,\u2014a strange piece, which shows the\nmind in which he lived:\u2014\"I found myself at the gate of hell, and,\nstanding at the threshold, I saw an opening, beneath I which was a vast\nsea of fire, in wave-like motion.  Looking at it, I said, 'What infinite\nvirtue there must have been in the blood of Christ to have quenched, for\nHis people, these awful flames!'  Overcome with the feeling, I knelt down\nby the walls of hell, saying, 'Thanks be unto Thee, O great and blessed\nSaviour, that Thou hast dried up this terrible sea of fire!'  Whereupon\nChrist addressed me: 'Come this way, and I will show you how it was\ndone.'  Looking back, I beheld that the whole sea had disappeared.  Jesus\npassed over the place, and said: 'Come, follow Me.'  By this time, I was\nwithin what I thought were the gates of hell, where there were many\ncells, out of which it was impossible to escape.  I found myself within\none of these, and anxious to make my way out.  Still I felt wonderfully\ncalm, as I had only just been conversing with Jesus, and because He had\ngone before me, although I had now lost sight of Him.  I got hold of\nsomething, with which I struck the corner of the place in which I stood,\nsaying, 'In the name of Jesus, open!' and it instantly gave way; so I did\nwith all the enclosures, until I made my way out into the open field.\nWhom should I see there but brethren, none of whom, however, I knew,\nexcept a good old deacon, and their work was to attend to a nursery of\ntrees; I joined them, and laid hold of a tree, saying, 'In the name of\nJesus, be thou plucked up by the root!'  And it came up as if it had been\na rush.  Hence I went forth, as I fancied, to work miracles, saying, 'Now\nI know how the Apostles wrought miracles in the name of Christ!'\"\n\nIt was during the earlier period of Christmas Evans's ministry at\nAnglesea, that a great irruption took place in the island, and, indeed,\nthroughout the Principality; and the Sandemanian controversy shook the\nChurches, and especially the Baptist Churches, almost beyond all\ncredibility, and certainly beyond what would have been a possibility, but\nfor the singular power of the chief leader, John Richard Jones, of\nRamoth.  Christmas Evans himself fell for some time beneath the power of\nSandemanian notions.  Our readers, perhaps, know enough of this peculiar\nform of faith and practice, to be aware that the worst thing that can be\nsaid of it is, that it is a religious ice-plant, religion in an\nice-house,\u2014a form chiefly remarkable for its rigid ritualistic\nconservation of what are regarded as the primitive forms of apostolic\ntimes, conjoined to a separation from, and a severe and cynical\nreprobation of, all other Christian sects.\n\nChristmas Evans says of himself at this period: \"The Sandemanian heresy\naffected me so far as to quench the spirit of prayer for the conversion\nof sinners, and it induced in my mind a greater regard for the smaller\nthings of the kingdom of heaven, than for the greater.  I lost the\nstrength which clothed my mind with zeal, confidence, and earnestness in\nthe pulpit for the conversion of souls to Christ.  My heart retrograded,\nin a manner, and I could not realize the testimony of a good conscience.\nSabbath nights, after having been in the day exposing and vilifying, with\nall bitterness, the errors that prevailed, my conscience felt as if\ndispleased, and reproached me that I had lost nearness to, and walking\nwith, God.  It would intimate that something exceedingly precious was now\nwanting in me; I would reply, that I was acting in obedience to the Word;\nbut it continued to accuse me of the want of some precious article.  I\nhad been robbed, to a great degree, of the spirit of prayer, and of the\nspirit of preaching.\"\n\nAnd the man who headed and gave effect to this Sandemanian movement,\nwhich was regarded as a mighty reform movement, was Jones of Ramoth.  No\ndoubt a real and genuine character enough, a magnificent orator, a master\nof bitter wit, and vigorous declamation.  That is a keen saying with\nwhich Richard Hooker commences his \"Ecclesiastical Polity:\" \"He that\ngoeth about to persuade a multitude, that they are not so well governed\nas they ought to be, shall never want attentive and favourable hearers;\nbecause they know the manifold defects whereunto every kind of regiment\nis subject; but the secret lets and difficulties, which in public\nproceedings are innumerable and inevitable, they have not ordinarily the\njudgment to consider.\"  This seems to have been the work, and this the\neffect, of John Richard Jones: very much the sum and substance of his\npreaching grew to be a morbid horror of the entire religious world, and a\nsupreme contempt\u2014one of his memorialists says, a superb contempt\u2014for all\npreachers except himself, especially for all itinerant preachers.  In\nfact, Ramoth Jones's influence in Anglesea might well be described in\nGeorge MacDonald's song, \"The Waesome Carl:\"\u2014\n\n    \"Ye're a' wrang, and a' wrang,\n    And a'thegither a' wrang;\n    There's no a man aboot the toon\n    But's a'thegither a' wrang.\n\n    \"The minister wasna fit to pray,\n       And let alane to preach;\n    He nowther had the gift o' grace,\n       Nor yet the gift o' speech.\n\n    \"He mind't him o' Balaam's ass,\n       Wi' a differ ye may ken:\n    The Lord He opened the ass's mou',\n       The minister opened's ain.\n\n    \"Ye're a' wrang, and a' wrang,\n    And a'thegither a' wrang;\n    There's no a man aboot the toon\n    But's a'thegither a' wrang.\"\n\nCompared with the slender following of the Sandemanian schism now,\u2014for we\nbelieve it has but six congregations in the whole United Kingdom,\u2014it\nseems strange to know that it laid so wonderful a hold upon the island of\nAnglesea.  It did, however; and that it did was evidently owing to the\nstrong man whose name we have mentioned.  He was a self-formed man, but\nhe was a man, if not of large scholarship, of full acquaintance with\nLatin, Greek, and Hebrew; he was a skilful musician; he understood the\nEnglish language well, but of the Welsh he was a great master.  But his\nintelligence, we should think, was dry and hard; his sentiments were\ncouched in bitter sarcasm: \"If,\" said he, \"every Bible in the world were\nconsumed, and every word of Scripture erased from my memory, I need be at\nno loss how to live a religious life, according to the will of God, for I\nshould simply have to proceed in all respects in a way perfectly contrary\nto the popular religionists of this age, and then I could not possibly be\nwrong.\"  He was very arrogant and authoritative in tone and manner,\nsupercilious himself, and expecting the subordination of others.  He was\nso bitter and narrow, that one naturally supposes that some injustice had\nembittered him.  Some of his words have a noble ring.  But he encouraged\na spirit far other than a charitable one wherever his word extended; and\nit has been not unnaturally said, that the spread of this Sandemanian\nnarrowness in Anglesea, realized something of the old Scotch absurdity of\nhaving two Churches in the same cottage, consisting of Janet in one\napartment, and Sandy in the other; or of that other famed Scottish\nChurch, which had dwindled down to two members, old Dame Christie, and\nDonald, but which seemed at last likely to dwindle yet farther into one,\nas Christie said she had \"sair doubts o' Donald.\"\n\nThe work of Christmas Evans, so far successful, seemed likely to be\nundone; all the Churches seemed inoculated by these new and narrow\nnotions, and Christmas Evans himself appears, as we have seen, to have\nbeen not altogether unscathed.  There is something so plausible in this\npurism of pride; and many such a creed of pessimism is the outgrowth of\nindifference born, and nurtured, upon decaying faith,\u2014a faith which,\nperhaps, as in the instance of Ramoth Jones and his Sandemanian teachers,\ncontinued true to Christ, so far as that is compatible with utter\nindifference to humanity at large, and an utter separation from the\nlarger view of the Communion of Saints.\n\nThere was, however, a grand man, who stood firm while ministers and\nChurches around him were reeling, Thomas Jones, of Glynceiriog, in\nDenbighshire; he is said to have been the one and only minister, at all\nknown to the public, who remained in his own denomination firm, and,\nsuccessfully in his own spirit, withstood, and even conquered, in this\nstorm of new opinion.  And this Thomas Jones did not stand like an\ninsensible stone or rock, but like a living oak, braving the blasts of\nveering opinion.  Most men think in crowds,\u2014which is only to say they are\nthe victims of thoughtless plausibilities.  This Thomas Jones appears to\nhave known what he believed; he was eminent for his politeness, and\ngreatly deferential in his bearing; but with all this, his courtesy was\nthe courtesy of the branch which bows, but retains its place.  He was a\nman of marvellous memory, and Christmas Evans used to say of him, that\nwherever Thomas Jones was, no Concordance would be necessary.  He was a\ngreat master in the study of Edwards \"On the Freedom of the Will,\" and\nhis method of reading the book was characteristic; he would first seize a\nproposition, then close the book, and close his eyes, and turn the\nproposition round and round that it might be undisturbed by anything\ninside the treatise, or outside of it, and in this way he would proceed\nwith the rigorous demonstration.  He was a calm and dignified knight in\nthe tournament of discussion; and, before his lance, more vehement but\nless trained thinkers and theologians went down.\n\nThus it was that he preached a great Association sermon at Llangevni, in\n1802, which dealt the Sandemanian schism a fatal blow; the captivity\nbeneath the spell of the influence of Ramoth Jones was broken, and turned\nas streams in the south.  While the sermon was being preached, Christmas\nEvans said, \"This Thomas Jones is a monster of a man!\"  Then the great\nrevival sprang up,\u2014the ice reign was over; but shortly after, he was\ncalled away to Rhydwilym, in Caermarthenshire.  Young as he was, when\nJohn Elias heard of his departure, he said, \"The light of the north is\nremoved.\"  He died full of years, full of honours, full of love; closing\na life, says one, of quiet beauty, which perhaps has never been\nsurpassed, at Rhydwilym, in 1850.\n\nThis irruption of Sandemanian thought, as we have said and seen, affected\nthe spiritual life and earnest usefulness of Christmas Evans.  It is well\nwe should place this passing flower upon the memory of Jones of\nRhydwilym, for he, it seems, broke the spell and dissolved the\nenchantment, and bade, in the heart of Christmas Evans, the imprisoned\nwaters once more to flow forth warm, and rejoicing, in the life and\nenthusiasm of love.  May we not say, in passing, that some such spell, if\nnot beneath the same denomination of opinion, holds many hearts in\nbondage among the Churches in our time?\n\nThe joy which Christmas Evans felt in his deliverance, realizes something\nof the warm words of the poet of the _Messiah_\u2014\n\n    \"The swain in barren deserts, with surprise\n    Sees lilies spring, and sudden verdure rise;\n    And starts, amidst the thirsty wilds, to hear\n    New falls of water murmuring in his ear.\"\n\n\"I was weary,\" he says, referring to this period, \"of a cold heart\ntowards Christ, and His sacrifice, and the work of His Spirit\u2014of a cold\nheart in the pulpit, in secret prayer, and in the study.  For fifteen\nyears previously, I had felt my heart burning within, as if going to\nEmmaus with Jesus.  On a day ever to be remembered by me, as I was going\nfrom Dolgelly to Machynlleth, and climbing up towards Cadair Idris, I\nconsidered it to be incumbent upon me to pray, however hard I felt in my\nheart, and however worldly the frame of my spirit was.  Having begun in\nthe name of Jesus, I soon felt, as it were, the fetters loosening, and\nthe old hardness of heart softening, and, as I thought, mountains of\nfrost and snow dissolving and melting within me.  This engendered\nconfidence in my soul in the promise of the Holy Ghost.  I felt my whole\nmind relieved from some great bondage; tears flowed copiously, and I was\nconstrained to cry out for the gracious visits of God, by restoring to my\nsoul the joys of His salvation; and that He would visit the Churches in\nAnglesea that were under my care.  I embraced in my supplications all the\nChurches of the saints, and nearly all the ministers in the Principality\nby their names.  This struggle lasted for three hours; it rose again and\nagain, like one wave after another, or a high flowing tide, driven by a\nstrong wind, until my nature became faint by weeping and crying.  Thus I\nresigned myself to Christ, body and soul, gifts and labours\u2014all my\nlife\u2014every day, and every hour that remained for me; and all my cares I\ncommitted to Christ.  The road was mountainous and lonely, and I was\nwholly alone, and suffered no interruption in my wrestlings with God.\n\n\"From this time, I was made to expect the goodness of God to Churches,\nand to myself.  Thus the Lord delivered me and the people of Anglesea\nfrom being carried away by the flood of Sandemanianism.  In the first\nreligious meetings after this, I felt as if I had been removed from the\ncold and sterile regions of spiritual frost, into the verdant fields of\nDivine promises.  The former striving with God in prayer, and the longing\nanxiety for the conversion of sinners, which I had experienced at L\u00ebyn,\nwere now restored.  I had a hold of the promises of God.  The result was,\nwhen I returned home, the first thing that arrested my attention was,\nthat the Spirit was working also in the brethren in Anglesea, inducing in\nthem a spirit of prayer, especially in two of the deacons, who were\nparticularly importunate that God would visit us in mercy, and render the\nWord of His grace effectual amongst us for the conversion of sinners.\"\n\nAnd to about this time belongs a most interesting article, preserved\namong his papers, \"a solemn covenant with God,\" made, he says, \"under a\ndeep sense of the evil of his own heart, and in dependence upon the\ninfinite grace and merit of the Redeemer.\"  It is a fine illustration of\nthe spirit and faith of the man in his lonely communions among the\nmountains.\n\n\n\nCovenant with God.\n\n\n    I.  I give my soul and body unto Thee, Jesus, the true God, and\n    everlasting life; deliver me from sin, and from eternal death, and\n    bring me into life everlasting.  Amen.\u2014C. E.\n\n    II.  I call the day, the sun, the earth, the trees, the stones, the\n    bed, the table, and the books, to witness that I come unto Thee,\n    Redeemer of sinners, that I may obtain rest for my soul from the\n    thunders of guilt and the dread of eternity.  Amen.\u2014C. E.\n\n    III.  I do, through confidence in Thy power, earnestly entreat Thee\n    to take the work into Thine own hand, and give me a circumcised\n    heart, that I may love Thee; and create in me a right spirit, that I\n    may seek thy glory.  Grant me that principle which Thou wilt own in\n    the day of judgment, that I may not then assume pale-facedness, and\n    find myself a hypocrite.  Grant me this, for the sake of Thy most\n    precious blood.  Amen.\u2014C. E.\n\n    IV.  I entreat Thee, Jesus, the Son of God, in power grant me, for\n    the sake of Thy agonizing death, a covenant interest in Thy blood\n    which cleanseth; in Thy righteousness, which justifieth; and in Thy\n    redemption, which delivereth.  I entreat an interest in Thy blood,\n    for Thy _blood's_ sake, and a part in Thee, for Thy Name's sake,\n    which Thou hast given among men.  Amen.\u2014C. E.\n\n    V.  O Jesus Christ, Son of the living God, take, for the sake of Thy\n    cruel death, my time, and strength, and the gifts and talents I\n    possess; which, with a full purpose of heart, I consecrate to Thy\n    glory in the building up of Thy Church in the world, for Thou art\n    worthy of the hearts and talents of all men.  Amen.\u2014C. E.\n\n    VI.  I desire Thee, my great High Priest, to confirm, by Thy power\n    from Thy High Court, my usefulness as a preacher, and my piety as a\n    Christian, as two gardens nigh to each other; that sin may not have\n    place in my heart to becloud my confidence in Thy righteousness, and\n    that I may not be left to any foolish act that may occasion my gifts\n    to wither, and I be rendered useless before my life ends.  Keep Thy\n    gracious eye upon me, and watch over me, O my Lord, and my God for\n    ever!  Amen.\u2014C. E.\n\n    VII.  I give myself in a particular manner to Thee, O Jesus Christ\n    the Saviour, to be preserved from the falls into which many stumble,\n    that Thy name (in Thy cause) may not be blasphemed or wounded, that\n    my peace may not be injured, that Thy people may not be grieved, and\n    that Thine enemies may not be hardened.  Amen.\u2014C. E.\n\n    VIII.  I come unto Thee, beseeching Thee to be in covenant with me in\n    my ministry.  As Thou didst prosper Bunyan, Vavasor Powell, Howell\n    Harris, Rowlands, and Whitfield, O do Thou prosper me.  Whatsoever\n    things are opposed to my prosperity, remove them out of the way.\n    Work in me everything approved of God for the attainment of this.\n    Give me a heart \"sick of love\" to Thyself, and to the souls of men.\n    Grant that I may experience the power of Thy Word before I deliver\n    it, as Moses felt the power of his own rod, before he saw it on the\n    land and waters of Egypt.  Grant this, for the sake of Thine\n    infinitely precious blood, O Jesus, my hope, and my all in all.\n    Amen.\u2014C. E.\n\n    IX.  Search me now, and lead me into plain paths of judgment.  Let me\n    discover in this life what I am before Thee, that I may not find\n    myself of another character when I am shown in the light of the\n    immortal world, and open my eyes in all the brightness of eternity.\n    Wash me in Thy redeeming blood.  Amen.\u2014C. E.\n\n    X.  Grant me strength to depend upon Thee for food and raiment, and\n    to make known my requests.  O let Thy care be over me as a\n    covenant-privilege betwixt Thee and myself, and not like a general\n    care to feed the ravens that perish, and clothe the lily that is cast\n    into the oven; but let Thy care be over me as one of Thy family, as\n    one of Thine unworthy brethren.  Amen.\u2014C. E.\n\n    XI.  Grant, O Jesus, and take upon Thyself the preparing of me for\n    death, for Thou art God; there is no need but for Thee to speak the\n    word.  If possible, Thy will be done; leave me not long in\n    affliction, nor to die suddenly, without bidding adieu to my\n    brethren, and let me die in their sight, after a short illness.  Let\n    all things be ordered against the day of removing from one world to\n    another, that there be no confusion nor disorder, but a quiet\n    discharge in peace.  O grant me this, for the sake of Thine agony in\n    the garden.  Amen.\u2014C. E.\n\n    XII.  Grant, O blessed Lord, that nothing may grow and be matured in\n    me to occasion Thee to cast me off from the service of the sanctuary,\n    like the sons of Eli; and for the sake of Thine unbounded merit, let\n    not my days be longer than my usefulness.  O let me not be like\n    lumber in a house in the end of my days, in the way of others to\n    work.  Amen.\u2014C. E.\n\n    XIII.  I beseech Thee, O Redeemer, to present these my supplications\n    before the Father; and oh, inscribe them in Thy Book with Thine own\n    immortal pen, while I am writing them with my mortal hand in my book\n    on earth.  According to the depths of Thy merit, Thine undiminished\n    grace, and Thy compassion, and Thy manner unto Thy people, O attach\n    Thy Name in Thine Upper Court to these unworthy petitions; and set\n    Thine Amen to them, as I do on my part of the covenant.\n    Amen.\u2014CHRISTMAS EVANS, _Llangevni_, _Anglesea_, _April_ 10, 18\u2014.\n\nIs not this an amazing document?  It is of this time that he further\nwrites:\u2014\"I felt a sweet peace and tranquillity of soul, like unto a poor\nman that had been brought under the protection of the Royal Family, and\nhad an annual settlement for life made upon him; and from whose dwelling\npainful dread of poverty and want had been for ever banished away.\"  We\nhave heard of God-intoxicated men; and what language can more\nappropriately describe a covenant-engagement so elevated, so astonishing,\nand sublime?\n\nNow, apparently strengthened as by a new spirit, with \"might in the inner\nman,\" he laboured with renewed energy and zeal; and new and singular\nblessings descended upon his labours.  In two years, his ten preaching\nplaces in Anglesea were increased to twenty, and six hundred converts\nwere added to the Church under his own immediate care.  It seemed as if\nthe wilderness and the solitary place were glad for him, and the desert\nrejoiced and blossomed as the rose.\n\nProbably, Christmas Evans's name had been scarcely announced, or read, in\nEngland, until his great Graveyard Sermon was introduced to a company of\nfriends, by the then celebrated preacher, Dr. Raffles, of Liverpool.  As\nthe story has been related, some persons present had affected contempt\nfor Welsh preaching.  \"Listen to me,\" said Raffles, \"and I will give to\nyou a specimen of Welsh eloquence.\"  Upon those present, the effect was,\nwe suppose, electrical.  He was requested to put it in print; and so the\nsermon became very extensively known, and has been regarded, by many, as\nthe preacher's most astonishing piece.\n\nTo what exact period of Evans's history it is to be assigned cannot be\nvery well ascertained, but it is probably nearly sixty years since\nRaffles first recited it; so that it belongs, beyond a doubt, to the\nearly Anglesea days.  It was, most likely, prepared as a great bardic or\ndramatic chant for some vast Association meeting, and was, no doubt,\nrepeated several times, for it became very famous.  It mingles something\nof the life of an old Mystery Play, or Ober-Ammergau performance; but as\nto any adequate rendering of it, we apprehend that to be quite\nimpossible.  Raffles was a rhetorician, and famous as his version became,\nthe good Doctor knew little or nothing of Welsh, nor was the order of his\nmind likely very accurately to render either the Welsh picture or the\nWelsh accent.  His periods were too rounded, the language too fine, and\nthe pictures too highly .\n\nIt was about the same time that, far away from Anglesea, among the\nremote, unheard-of German mountains of Baireuth, a dreamer of a very\ndifferent kind was visited by some such vision of the world, regarded as\na great churchyard.  Jean Paul Richter's churchyard, visited by the dead\nChrist, was written in Siebinckas, for the purpose of presenting the\nmisty, starless, cheerless, and spectral outlook of the French atheism,\nwhich was then spreading out, noxious and baleful, over Europe.\n\nVery different were the two men, their spheres, and their avocations;\noverwhelming, solemn, and impressive as is the vision of Jean Paul, it\ncertainly would have said little to a vast Welsh congregation among the\ndark hills.  Christmas Evans's piece is dramatic; his power of\nimpersonation and colloquy in the pulpit was very great; and the reader\nhas to conceive all this, while on these colder pages the scenes and the\nconversations go on.  It appears to have been first preached in a small\ndell among the mountains of Carnarvonshire.  The spot was exquisitely\nromantic; it was a summer's season, the grass was in its rich green,\nbrooks were purling round, and the spot hemmed in by jagged crags and the\ncliffs of tall mountains; a beautiful spot, but an Englishman spoke of it\nas \"beauty sleeping on the lap of terror.\"\n\nA preliminary service, of course, went on,\u2014hymns, the sounding of the\nslow, plaintive minor melody from thousands of tongues, rising and\nloitering, and lingering among the neighbouring acclivities, before they\nfinally fade off into silence; then there is reading, and prayer, singing\nagain, and a short sermon before Christmas Evans comes.  He has not\nattained to the full height of his great national fame as yet; he is\nbefore the people, however, \"the one-eyed man of Anglesea,\"\u2014the\ndesignation by which he was to be known for many years to come.  He\nstands six feet high, his face very expressive, but very calm and quiet;\nbut a great fire was burning within the man.  He gave out some verses of\na well-known Welsh hymn, and while it was being sung took out a small\nphial from his waistcoat-pocket, wetting the tips of his fingers and\ndrawing them over his blind eye; it was laudanum, used to deaden the\nexcruciating pain which upon some occasions possessed him.\n\nHe gave out his text from Romans v. 15: \"If through the offence of one\nmany be dead, much more the grace of God, and the gift by grace, which is\nby one man, Jesus Christ, hath abounded unto many.\"  Naturally, he does\nnot begin at once, but spends a little time, in clearly-enunciated words,\nin announcing two things,\u2014the universal depravity and sinfulness of men,\nand the sighing after propitiation.  _Mene_!  _Tekel_! he says, is\nwritten on every human heart; wanting, wanting, is inscribed on heathen\nfanes and altars, on the laws, customs, and institutions of every nation,\nand on the universal consciousness of mankind; and bloody sacrifices\namong pagan nations show the handwriting of remorse upon the\nconscience,\u2014a sense of guilt, and a dread of punishment, and a fear which\nhath torment.\n\nAs he goes on the people draw nearer, become more intense in their\nearnest listening; they are rising from their seats, their temporary\nforms.  Some are in carriages; there is a lady leaning on her husband's\nshoulder, he still sitting, she with outstretched neck gazing with\nobviously strange emotion at the preacher; some of the people are\nbeginning to weep.  There is an old evangelical clergyman who has always\npreached the Gospel, although laughed at by his squire, and quite unknown\nby his Bishop; he is rejoicing with a great joy to hear his old loved\ntruths set forth in such a manner; he is weeping profusely.\n\nChristmas Evans, meantime, is pursuing his way, lost in his theme.  Now\nhis eye lights up, says one who knew him, like a brilliantly-flashing\nstar, his clear forehead expands, his form dilates in majestic dignity;\nand all that has gone before will be lost in the white-heat passion with\nwhich he prepares to sing of Paradise lost, and Paradise regained.  One\nof his Welsh critics says: \"All the stores of his energy, and the\nresources of his voice, which was one of great compass, depth, and\nsweetness, seemed reserved for the closing portions of the picture, when\nhe represented the routed and battered hosts of evil retreating from the\ncross, where they anticipated a triumph, and met a signal, and\nirretrievable overthrow.\"  Thus prepared, he presented to his hearers the\npicture of\n\n\n\n\"THE WORLD AS A GRAVEYARD.\"\n\n\n    \"Methinks,\" exclaimed the impassioned preacher, \"I find myself\n    standing upon the summit of one of the highest of the everlasting\n    hills, permitted from thence to take a survey of the whole earth; and\n    all before me I see a wide and far-spread burial-ground, a graveyard,\n    over which lie scattered the countless multitudes of the wretched and\n    perishing children of Adam!  The ground is full of hollows, the\n    yawning caverns of death; and over the whole scene broods a thick\n    cloud of darkness: no light from above shines upon it, there is no\n    ray of sun or moon, there is no beam, even of a little candle, seen\n    through all its borders.  It is walled all around, but it has gates,\n    large and massive, ten thousand times stronger than all the gates of\n    brass forged among men; they are one and all safely locked,\u2014the hand\n    of Divine Law has locked them; and so firmly secured are the strong\n    bolts, that all the created powers even of the heavenly world, were\n    they to labour to all eternity, could not drive so much as one of\n    them back.  How hopeless is the wretchedness to which the race is\n    doomed! into what irrecoverable depths of ruin has sin plunged the\n    people who sit there in darkness, and in the shadow of death, while\n    there, by the brazen gates, stands the inflexible guard, brandishing\n    the flaming sword of undeviating Law!\n\n    \"But see!  In the cool of the day, there is one descending from the\n    eternal hills in the distance: it is Mercy! the radiant form of\n    Mercy, seated in the chariot of Divine Promise.  She comes through\n    the worlds of the universe; she pauses here to mark the imprisoned\n    and grave-like aspect of our once fair world; her eye affected her\n    heart as she beheld the misery, and heard the cry of despair, borne\n    upon the four winds of heaven; she could not pass by, nor pass on;\n    she wept over the melancholy scene, and she said, 'Oh that I might\n    enter!  I would bind up their wounds, I would relieve their sorrows,\n    I would save their souls!'  An embassy of angels, commissioned from\n    Heaven to some other world, paused at the sight; and Heaven forgave\n    that pause.  They saw Mercy standing by the gate, and they cried,\n    'Mercy, canst thou not enter?  Canst thou look upon that world and\n    not pity?  Canst thou pity and not relieve?'  And Mercy, in tears,\n    replied, 'I can see, and I can pity, but I cannot relieve.'  'Why\n    dost thou not enter?' inquired the heavenly host.  'Oh,' said Mercy,\n    'Law has barred the gate against me, and I must not, and I cannot\n    unbar it.'  And Law stood there watching the gate, and the angels\n    asked of him, 'Why wilt thou not suffer Mercy to enter?'  And he\n    said, 'No one can enter here and live;' and the thunder of his voice\n    outspoke the wailings within.  Then again I heard Mercy cry, 'Is\n    there no entrance for me into this field of death? may I not visit\n    these caverns of the grave; and seek, if it may be, to raise some at\n    least of these children of destruction, and bring them to the light\n    of day?  Open, Justice, Open! drive back these iron bolts, and let me\n    in, that I may proclaim the jubilee of redemption to the children of\n    the dust!'  And then I heard Justice reply, 'Mercy! surely thou\n    lovest Justice too well to wish to burst these gates by force of arm,\n    and thus to obtain entrance by lawless violence.  I cannot open the\n    door: I am not angry with these unhappy, I have no delight in their\n    death, or in hearing their cries, as they lie upon the burning hearth\n    of the great fire, kindled by the wrath of God, in the land that is\n    lower than the grave.  But _without shedding of blood there is no\n    remission_.'\n\n    \"So Mercy expanded her wings, splendid beyond the brightness of the\n    morning when its rays are seen shooting over mountains of pearl,\u2014and\n    Mercy renewed her flight amongst the unfallen worlds; she re-ascended\n    into the mid air, but could not proceed far, because she could not\n    forget the sad sight of the Graveyard-World, the melancholy prison.\n    She returned to her native throne in the Heaven of heavens; it was a\n    glorious high throne, unshaken and untarnished by the fallen fate of\n    man and angels.  Even there she could not forget what she had\n    witnessed, and wept over, and she weighed the woes of the sad world\n    against the doom of eternal Law; she could not forget the prison and\n    the graveyard, and she re-descended with a more rapid and radiant\n    flight, and she stood again by the gate, but again was denied\n    admission.  And the two stood there together, Justice and Mercy; and\n    Justice dropped his brandishing sword while they held converse\n    together; and while they talked, there was silence in heaven.\n\n    \"'Is there then no admission on any terms whatever?' she said.  'Ah,\n    yes,' said Justice; 'but then they are terms which no created being\n    can fulfil.  I demand atoning death for the Eternal life of those who\n    lie in this Graveyard; I demand Divine life for their ransom.'  And\n    while they were talking, behold there stood by them a third Form,\n    fairer than the children of men, radiant with the glory of heaven.\n    He cast a look upon the graveyard.  And He said to Mercy, 'Accept the\n    terms.'  'Where is the security?' said Justice.  'Here,' said Mercy,\n    pointing to the radiant Stranger, 'is my bond.  Four thousand years\n    from hence, demand its payment on Calvary.  To redeem men,' said\n    Mercy, 'I will be incarnate in the Son of God, I will be the Lamb\n    slain for the life of this Graveyard World.'\n\n    \"The bond was accepted, and Mercy entered the graveyard leaning on\n    the arm of Justice.  She spoke to the prisoners.  Centuries rolled\n    by.  So went on the gathering of the firstfruits in the field of\n    redemption.  Still ages passed away, and at last the clock of\n    prophecy struck the fulness of time.  The bond, which had been\n    committed to patriarchs and prophets, had to be redeemed; a long\n    series of rites and ceremonies, sacrifices and oblations, had been\n    instituted to perpetuate the memory of that solemn deed.\n\n    \"At the close of the four thousandth year, when Daniel's seventy\n    weeks were accomplished, Justice and Mercy appeared on the hill of\n    Calvary; angels and archangels, cherubim and seraphim, principalities\n    and powers, left their thrones and mansions of glory, and bent over\n    the battlements of heaven, gazing in mute amazement and breathless\n    suspense upon the solemn scene.  At the foot of Calvary's hill was\n    beheld the Son of God.  'Lo, I come,' He said; 'in the bond it is\n    written of me.'  He appeared without the gates of Jerusalem, crowned\n    with thorns, and followed by the weeping Church.  It was with Him the\n    hour and the power of darkness; above Him were all the vials of\n    Divine wrath, and the thunders of the eternal Law; round Him were all\n    the powers of darkness,\u2014the monsters of the pit, huge, fierce and\n    relentless, were there; the lions as a great army, gnashing their\n    teeth ready to tear him in pieces; the unicorns, a countless host,\n    were rushing onwards to thrust him through; and there were the bulls\n    of Bashan roaring terribly; the dragons of the pit unfolding\n    themselves, and shooting out their stings; and dogs, many, all round\n    the mountain.\n\n    \"And He passed through this dense array, an unresisting victim led as\n    a lamb to the slaughter.  He took the bond from the hand of Justice,\n    and, as He was nailed to the cross, He nailed it to the cross; and\n    all the hosts of hell, though invisible to man, had formed a ring\n    around it.  The rocks rent, the sun shrank from the scene, as Justice\n    lifted his right hand to the throne, exclaiming, 'Fires of heaven,\n    descend and consume this sacrifice!'  The fires of heaven, animated\n    with living spirit, answered the call, 'We come! we come! and, when\n    we have consumed that victim, we will burn the world.'  They burst,\n    blazed, devoured; the blood of the victim was fast dropping; the\n    hosts of hell were shouting, until the humanity of Emmanuel gave up\n    the ghost.  The fire went on burning until the ninth hour of the day,\n    but when it touched the Deity of the Son of God it expired; Justice\n    dropped the fiery sword at the foot of the cross; and the Law joined\n    with the prophets in witnessing to the righteousness which is by\n    faith in the Son of God, for all had heard the dying Redeemer\n    exclaim, 'It is finished!'  The weeping Church heard it, and lifting\n    up her head cried too, 'It is finished!'  Attending angels hovering\n    near heard it, and, winging their flight, they sang, 'It is\n    finished!'  The powers of darkness heard the acclamations of the\n    universe, and hurried away from the scene in death-like feebleness.\n    He triumphed over them openly.  The graves of the old Burial-ground\n    have been thrown open, and gales of life have blown over the valley\n    of dry bones, and an exceeding great army has already been sealed to\n    our God as among the living in Zion; for so the Bond was paid and\n    eternal redemption secured.\"\n\nThis was certainly singular preaching; it reads like a leaf or two from\nKlopstock.  We may believe that the enjoyment with which it was heard was\nrich and great, but we suppose that the taste of our time would regard it\nas almost intolerable.  Still, there are left among us some who can enjoy\nthe _Pilgrim's Progress_, and the _Fairy Queen_, and we do not see how,\nin the presence of those pieces, a very arrogant exception can be taken\nto this extraordinary sermon.\n\nA more serious objection, perhaps, will be taken to the nomenclature, the\nsymbolic language in which the preacher expressed his theology.  It\nliterally represented the theology of Wales at the time when it was\ndelivered; the theology was stern and awful; the features of God were\nthose of a stern and inflexible Judge; nature presented few relieving\nlights, and man was not regarded as pleasant to look upon.  Let the\nreader remember all this, and perhaps he will be more tolerant to the\nstern outline of this allegory; it is pleasant, now, to know that we have\nchanged all that, and that everywhere, and all around us, God, and\nnature, and man are presented in rose-hued lights, and all conditions of\nbeing are washed by rosy and pacific seas; we see nothing stern or awful\nnow, either in nature or in grace, in natural or in supernatural things;\nJustice has become gentlemanly, and Law, instead of being stern and\nterrible, is bland, and graceful, and beautiful as a woman's smile!\n\nIn Christmas Evans's day, it was not quite so.  As to objections to the\nmode of preaching, as in contrast with that style which adopts only the\nsustained argument, and the rhetorical climax and relation, we have\nalready said that Christmas must be tried by quite another standard; we\nhave already said that he was a bard among preachers, and belonged to a\nnation of bards.  It was a kind of primeval song, addressed to people of\nprimeval instincts; but, whatever its merits or demerits may be, it\nfairly represents the man and his preaching.  It does not, indeed,\nreflect the style of the modern mind; but, there are many writers, and\nreaders at present, who are carrying us back to the medi\u00e6val times, and\nthe monastic preachers of those ages, and among them we find innumerable\npieces of the same order of sustained allegory which we have just quoted\nfrom Christmas Evans.  What is it but to say, that the simple mind is\ncharmed with pictures,\u2014it must have them; and such sermons as abound in\nthem, have power over it?\n\nWe believe we have rendered this singular passage with such fairness that\nthe reader may be enabled to form some idea of its splendour.  When it\nwas repeated to Robert Hall, he pronounced it one of the finest\nallegories in the language.  When Christmas Evans was on a visit to Dr.\nRaffles, the Doctor recited to him his own version, and, apparently with\nsome amazement, said, \"Did you actually say all that?\"  \"Oh, yes,\" said\nChristmas, \"I did say all that, but I could never have put it into such\nEnglish.\"  And this we are greatly disposed to regard as impairing the\nbold grandeur and strength of the piece; any rendering of it into English\nmust, as it seems to us, add to its prettiness, and therefore divest it\nof its power.\n\nProbably to the same period of the preacher's history belongs another\nsermon, which has always seemed to us a piece of undoubted greatness.  It\nis upon the same subject, the Crucifixion of Christ.  We should think\nthat its delivery would, at any time, from such lips as his, produce\nequally pathetic emotions.  The allegory is not so sustained, but it is\nstill full of allegorical allusions derived from Scriptural expression.\n\n\n\n\"THE HIND OF THE MORNING.\n\n\n    \"It is generally admitted that the twenty-second Psalm has particular\n    reference to Christ.  This is evident from His own appropriation of\n    the first verse upon the cross: 'My God! my God! why hast Thou\n    forsaken Me?'  The title of that Psalm is '_Aijeleth Shahar_,' which\n    signifies 'A Hart, or the Hind of the Morning.'  The striking\n    metaphors which it contains are descriptive of Messiah's peculiar\n    sufferings.  He is the Hart, or the Hind of the Morning, hunted by\n    the Black Prince, with his hell-hounds\u2014by Satan, and all his allies.\n    The 'dogs,' the 'lions,' the 'unicorns,' and the 'strong bulls of\n    Bashan,' with their devouring teeth, and their terrible horns,\n    pursued Him from Bethlehem to Calvary.  They beset Him in the manger,\n    gnashed upon Him in the garden, and well-nigh tore Him to pieces upon\n    the cross.  And still they persecute Him in His cause, and in the\n    persons and interests of His people.\n\n    \"The faith of the Church anticipated the coming of Christ, 'like a\n    roe or a young hart,' with the dawn of the day promised in Eden; and\n    we hear her exclaiming in the Canticles\u2014'The voice of my beloved!\n    behold, He cometh, leaping upon the mountains, and skipping upon the\n    hills!'  She heard Him announce His advent in the promise, 'Lo, I\n    come to do Thy will, O God!' and with prophetic eye, saw Him leaping\n    from the mountains of eternity to the mountains of time, and skipping\n    from hill to hill throughout the land of Palestine, going about doing\n    good.  In the various types and shadows of the law, she beheld Him\n    'standing by the wall, looking forth at the windows, showing Himself\n    through the lattice;' and then she sang\u2014'Until the day break and the\n    shadows flee away, turn, my beloved, and be thou like the roe or the\n    young hart upon the mountains of Bether!'  Bloody sacrifices revealed\n    Him to her view, going down to the 'vineyards of red wine;' whence\n    she traced Him to the meadows of Gospel ordinances, where 'He feedeth\n    among the lilies'\u2014to 'the gardens of cucumbers,' and 'the beds of\n    spices;' and then she sang to Him again\u2014'Make haste'\u2014or, flee\n    away\u2014'my beloved! be thou like the roe or the young hart among the\n    mountains of spices.'\n\n    \"Thus she longed to see Him, first 'on the mountain of Bether,' and\n    then 'on the mountain of spices.'  On both mountains she saw Him\n    eighteen hundred years ago, and on both she may still trace the\n    footsteps of His majesty, and His mercy.  The former, He hath tracked\n    with His own blood, and His path upon the latter is redolent of\n    frankincense and myrrh.\n\n    \"Bether signifies division.  This is the craggy mountain of Calvary;\n    whither the 'Hind of the Morning' fled, followed by all the wild\n    beasts of the forest, and the bloodhounds of hell; summoned to the\n    pursuit, and urged on, by the prince of perdition; till the victim,\n    in His agony, sweat great drops of blood\u2014where He was terribly\n    crushed between the cliffs, and dreadfully mangled by sharp and\n    ragged rocks\u2014where He was seized by Death, the great Bloodhound of\n    the bottomless pit\u2014whence He leaped the precipice, without breaking a\n    bone; and sunk in the dead sea, sunk to its utmost depth, and saw no\n    corruption.\n\n    \"Behold the 'Hind of the Morning' on that dreadful mountain!  It is\n    the place of skulls, where Death holds his carnival in companionship\n    with worms, and hell laughs in the face of heaven.  Dark storms are\n    gathering there\u2014convolving clouds, charged with no common wrath.\n    Terrors set themselves in battle-array before the Son of God; and\n    tempests burst upon Him which might sweep all mankind in a moment to\n    eternal ruin.  Hark! hear ye not the subterranean thunder?  Feel ye\n    not the tremor of the mountain?  It is the shock of Satan's\n    artillery, playing upon the Captain of our Salvation.  It is the\n    explosion of the magazine of vengeance.  Lo, the earth is quaking,\n    the rocks are rending, the graves are opening, the dead are rising,\n    and all nature stands aghast at the conflict of Divine mercy with the\n    powers of darkness.  One dread convulsion more, one cry of desperate\n    agony, and Jesus dies\u2014an arrow has entered into His heart.  Now leap\n    the lions, roaring, upon their prey; and the bulls of Bashan are\n    bellowing; and the dogs of perdition are barking; and the unicorns\n    toss their horns on high; and the devil, dancing with exultant joy,\n    clanks his iron chains, and thrusts up his fettered hands in defiance\n    towards the face of Jehovah!\n\n    \"Go a little farther upon the mountain, and you come to 'a new tomb\n    hewn out of the rock.'  There lies a dead body.  It is the body of\n    Jesus.  His disciples have laid it down in sorrow, and returned,\n    weeping, to the city.  Mary's heart is broken, Peter's zeal is\n    quenched in tears, and John would fain lie down and die in his\n    Master's grave.  The sepulchre is closed up, and sealed, and a Roman\n    sentry placed at its entrance.  On the morning of the third day,\n    while it is yet dark, two or three women come to anoint the body.\n    They are debating about the great stone at the mouth of the cave.\n    'Who shall roll it away?' says one of them.  'Pity we did not bring\n    Peter, or John with us.'  But, arriving, they find the stone already\n    rolled away, and one sitting upon it, whose countenance is like\n    lightning, and whose garments are white as the light.  The\n    steel-clad, iron-hearted soldiers lie around him, like men slain in\n    battle, having swooned with terror.  He speaks: 'Why seek ye the\n    living among the dead?  He is not here; He is risen; He is gone forth\n    from this cave victoriously.'\n\n    \"It is even so!  For there are the shroud, and the napkin, and the\n    heavenly watchers; and when He awoke, and cast off His grave-clothes,\n    the earthquake was felt in the city, and jarred the gates of hell.\n    'The Hind of the Morning' is up earlier than any of His pursuers,\n    'leaping upon the mountains, and skipping upon the hills.'  He is\n    seen first with Mary at the tomb; then with the disciples in\n    Jerusalem; then with two of them on the way to Emmaus; then going\n    before His brethren into Galilee; and, finally, leaping upon the top\n    of Olivet to the hills of Paradise; fleeing away to 'the mountain of\n    spices,' where He shall never more be hunted by the Black Prince and\n    his hounds.\n\n    \"Christ is perfect master of gravitation, and all the laws of nature\n    are obedient to His will.  Once He walked upon the water, as if it\n    were marble beneath His feet; and now, as He stands blessing His\n    people, the glorious Form, so recently nailed to the cross, and still\n    more recently cold in the grave, begins to ascend like 'the living\n    creature' in Ezekiel's vision, 'lifted up from the earth,' till\n    nearly out of sight; when 'the chariots of God, even thousands of\n    angels,' receive Him, and haste to the celestial city, waking the\n    thrones of eternity with this jubilant chorus\u2014'Lift up your heads, O\n    ye gates! and be ye lifted up, ye everlasting doors! and the King of\n    glory shall come in!'\n\n    \"Christ might have rode in a chariot of fire all the way from\n    Bethlehem to Calvary; but he preferred riding in a chariot of mercy,\n    whose lining was crimson, and whose ornament the malefactor's cross.\n    How rapidly rolled his wheels over the hills and the plains of\n    Palestine, gathering up everywhere the children of affliction, and\n    scattering blessings like the beams of the morning!  Now we find Him\n    in Cana of Galilee, turning water into wine; then treading the waves\n    of the sea, and hushing the roar of the tempest; then delivering the\n    demoniac of Gadara from the fury of a legion of fiends; then healing\n    the nobleman's son at Capernaum; raising the daughter of Jairus, and\n    the young man of Nain; writing upon the grave of Bethany, 'I am the\n    resurrection and the life;' curing the invalid at the pool of\n    Bethesda; feeding the five thousand in the wilderness; preaching to\n    the woman by Jacob's well, acquitting the adulteress, and shaming her\n    accusers; and exercising everywhere, in all his travels, the three\n    offices of Physician, Prophet, and Saviour, as he drove on towards\n    the place of skulls.\n\n    \"Now we see the chariot surrounded with enemies\u2014Herod, and Pilate,\n    and Caiaphas, and the Roman soldiers, and the populace of Jerusalem,\n    and thousands of Jews who have come up to keep the Passover, led on\n    by Judas and the devil.  See how they rage and curse, as if they\n    would tear him from his chariot of mercy!  But Jesus maintains his\n    seat, and holds fast the reins, and drives right on through the angry\n    crowd, without shooting an arrow, or lifting a spear upon his foes.\n    For in that chariot the King must ride to Calvary\u2014Calvary must be\n    consecrated to mercy for ever.  He sees the cross planted upon the\n    brow of the hill, and hastens forward to embrace it.  No sacrifice\n    shall be offered to Justice on this day, but the one sacrifice which\n    reconciles heaven and earth.  None of these children of Belial shall\n    suffer to-day.  The bribed witnesses, and clamorous murderers, shall\n    be spared\u2014the smiters, the scourgers, the spitters, the\n    thorn-plaiters, the nail drivers, the head-shakers\u2014for Jesus pleads\n    on their behalf: 'Father, forgive them! they know not what they do.\n    They are ignorant of Thy grace and truth.  They are not aware of whom\n    they are crucifying.  Oh, spare them!  Let Death know that he shall\n    have enough to do with _me_ to-day!  Let him open all his batteries\n    upon _me_!  _My_ bosom is bare to the stroke.  _I_ will gather all\n    the lances of hell in _my_ heart!'\n\n    \"Still the chariot rushes on, and 'fiery darts' are thick and fast,\n    like a shower of meteors, on Messiah's head, till He is covered with\n    wounds, and the blood flows down His garments, and leaves a crimson\n    track behind Him.  As He passes, He casts at the dying malefactor a\n    glance of benignity, and throws him a passport into Paradise, written\n    with His own blood; stretches forth His sceptre, and touches the\n    prison-door of death, and many of the prisoners came forth, and the\n    tyrant shall never regain his dominion over them; rides triumphant\n    over thrones and principalities, and crushes beneath his wheels the\n    last enemy himself, and leaves the memorial of his march engraven on\n    the rocks of Golgotha!\n\n    \"Christ is everywhere in the Scriptures spoken of as a Blessing; and\n    whether we contemplate His advent, His ministry, His miracles, His\n    agony, His crucifixion, His interment, His resurrection, or His\n    ascension, we may truly say, 'All His paths drop fatness.'  All His\n    travels were on the road of mercy; and trees are growing up in His\n    footsteps, whose fruit is delicious food, and 'whose leaves are for\n    the healing of the nations.'  He walketh upon the south winds,\n    causing propitious gales to blow upon the wilderness till songs of\n    joy awake in the solitary place, and the desert blossoms as the rose.\n\n    \"If we will consider what the prophets wrote of the Messiah, in\n    connection with the evangelical history, we shall be satisfied that\n    none like Him, either before or since, ever entered our world, or\n    departed from it.  Both God and man\u2014at once the Father of eternity\n    and the Son of time, He filled the universe, while He was embodied\n    upon earth, and ruled the celestial principalities and powers, while\n    He wandered, a persecuted stranger, in Judea.  'No man,' saith He,\n    'hath ascended up to heaven, but He that came down from heaven, even\n    the Son of man who is in heaven.'\n\n    \"Heaven was no strange place to Jesus.  He talks of the mansions in\n    His Father's house as familiarly as one of the royal family would\n    talk of Windsor Castle where he was born; and saith to His disciples,\n    'I go to prepare a place for you; that where I am there ye may be\n    also.'  The glory into which He entered was His own glory\u2014the glory\n    which He had with the Father before the world was.  He had an\n    original and supreme right to the celestial mansions; and He acquired\n    a new and additional claim by His office as Mediator.  Having\n    suffered for our sins, He 'ought to enter into His glory.'  He ought,\n    because He is 'God, blessed for ever;' He ought, because He is the\n    representative of His redeemed people.  He has taken possession of\n    the kingdom in our behalf, and left on record for our encouragement\n    this cheering promise, 'To him that overcometh will I grant to sit\n    with me in my throne; even as I also have overcome, and am set down\n    with my Father in His throne.'\n\n    \"The departure of God from Eden, and the departure of Christ from the\n    earth, were two of the sublimest events that ever occurred, and\n    fraught with immense consequences to our race.  When Jehovah went out\n    from Eden, He left a curse upon the place for man's sake, and drove\n    out man before him into an accursed earth.  But when Jesus descended\n    from Olivet, He lifted the curse with Him, and left a blessing behind\n    Him\u2014sowed the world with the seed of eternal blessings; 'and instead\n    of the thorn shall come up the fir-tree; and instead of the briar\n    shall come up the myrtle-tree; and it shall be to the Lord for a\n    name, and an everlasting sign, that shall not be cut off.'  He\n    ascended to intercede for sinners, and reopen Paradise to His people;\n    and when He shall come the second time, according to the promise,\n    with all His holy angels, then shall we be 'caught up to meet the\n    Lord in the air, and so shall we ever be with the Lord.'\n\n    \"'The Lord is gone up with a shout!' and has taken our redeemed\n    nature with Him.  He is the Head of the Church, and is the\n    representative at the right hand of the Father.  'He hath ascended on\n    high; He hath led captivity captive; He hath received gifts for men;\n    yea, for the rebellious also, that God may dwell among them.'  'Him\n    hath God exalted, with His own right hand, to be a Prince and a\n    Saviour, to give repentance to Israel, and remission of sins.'  This\n    is the Father's recognition of His 'Beloved Son,' and significant\n    acceptance of his sacrifice.  'Wherefore God also hath highly exalted\n    Him, and given Him a name which is above every name; that at the name\n    of Jesus every knee should bow, of things in heaven, and things in\n    the earth, and things under the earth; and that every tongue should\n    confess that Jesus Christ is Lord, to the glory of God the Father.'\n\n    \"The evidence of our Lord's ascension is ample.  He ascended in the\n    presence of many witnesses, who stood gazing after Him till a cloud\n    received Him out of their sight.  And while they looked steadfastly\n    toward heaven, two angels appeared to them, and talked with them of\n    what they had seen.  Soon afterward, on the day of Pentecost, He\n    fulfilled, in a remarkable manner, the promise which He had made to\n    His people: 'If I go away I will send you another Comforter, who\n    shall abide with you for ever.'  Stephen, the first of His disciples\n    that glorified the Master by martyrdom, testified to his murderers,\n    'Lo, I see the heavens opened, and the Son of man standing on the\n    right hand of God!'  And John, the 'beloved disciple,' while an exile\n    'in Patmos, for the word of God, and the testimony of Jesus Christ,'\n    beheld Him 'in the midst of the throne, as a Lamb that had been\n    slain!'  These are the evidences that our Lord is in heaven; these\n    are our consolations in the house of our pilgrimage.\n\n    \"The Apostle speaks of the _necessity_ of this event, 'Whom the\n    heaven _must_ receive.'\n\n    \"Divine necessity is a golden chain reaching from eternity to\n    eternity, and encircling all the events of time.  It consists of many\n    links all hanging upon each other; and not one of them can be broken\n    without destroying the support of the whole.  The first link is in\n    God, 'before the world was;' and the last is in heaven, when the\n    world shall be no more.  Christ is its Alpha, and Omega, and Christ\n    constitutes all its intervenient links.  Christ in the bosom of the\n    Father, receiving the promise of eternal life, before the foundation\n    of the world, is the beginning; Christ in His sacrificial blood,\n    atoning for our sins, and pardoning and sanctifying all them that\n    believe, is the middle; and Christ in heaven, pleading the merit of\n    His vicarious sufferings, making intercession for the transgressors,\n    drawing all men unto Himself, presenting the prayers of His people,\n    and preparing their mansions, is the end.\n\n    \"There is a necessity in all that Christ has done as our Mediator, in\n    all that He is doing on our behalf, and all that he has engaged to\n    do\u2014the necessity of Divine love manifested, of Divine mercy\n    exercised, of Divine purposes accomplished, of Divine covenants\n    fulfilled, of Divine faithfulness maintained, of Divine justice\n    satisfied, of Divine holiness vindicated, and of Divine power\n    displayed.  Christ felt this necessity while He tabernacled among us,\n    often declared it to His disciples, and acknowledged it to the Father\n    in the agony in the Garden.\n\n    \"Behold Him wrestling in prayer, with strong crying and tears:\n    'Father, save me from this hour!  If it be possible, let this cup\n    pass from me!'  Now the Father reads to Him His covenant engagement,\n    which He signed and sealed with His own hand before the foundation of\n    the world.  The glorious Sufferer replies, 'Thy will be done!  For\n    this cause came I unto this hour.  I will drink the cup which Thou\n    hast mingled, and not a dreg of any of its ingredients shall be left\n    for my people.  I will pass through the approaching dreadful night,\n    under the hidings of Thy countenance, bearing away the curse from my\n    beloved.  Henceforth repentance is hidden from my eyes!'  Now, on His\n    knees, He reads the covenant engagements of the Father, and adds, 'I\n    have glorified Thee on the earth.  I have finished the work which\n    Thou gavest Me to do.  Now glorify Thou Me, according to Thy promise,\n    with Thine own Self, with the glory which I had with Thee before the\n    world was.  Father, I will also that they whom Thou hast given Me be\n    with Me where I am, that they may behold My glory.  Thine they were,\n    and Thou hast given them to Me, on condition of My pouring out My\n    soul unto death.  Thou hast promised them, through My righteousness\n    and meritorious sacrifice, the kingdom of heaven, which I now claim\n    on their behalf.  Father, glorify My people, with Him whom Thou\n    lovedst before the foundation of the world!'\n\n    \"This intercession of Christ for His saints, begun on earth, is\n    continued in heaven.  This is our confidence and joy in our journey\n    through the wilderness.  We know that our Joshua has gone over into\n    the land of our inheritance, where He is preparing the place of our\n    habitation for Israel; for it is His will that all whom He has\n    redeemed should be with Him for ever!\n\n    \"And there is a text which speaks of the period when the great\n    purposes of our Lord's ascension shall be fully accomplished: 'Until\n    the times of the restitution of all things.'\n\n    \"The period here mentioned is 'the dispensation of the fulness of\n    time,' when 'the fulness of the Gentiles shall come in,' and 'the\n    dispersed of Judah' shall be restored, and Christ shall 'gather\n    together in Himself all things in heaven and in earth,' overthrow his\n    enemies, establish his everlasting kingdom, deliver the groaning\n    creation from its bondage, glorify His people with Himself, imprison\n    the devil with his angels in the bottomless pit, and punish with\n    banishment from His presence them that obey not the Gospel.\n\n    \"To this glorious consummation, the great travail of redemption, and\n    all the events of time, are only preparatory.  It was promised in\n    Eden, and the promise was renewed and enlarged to Abraham, to Isaac,\n    and to Jacob.  It was described in gorgeous oriental imagery by\n    Isaiah, and 'the sweet Psalmist of Israel;' and 'spoken of by all the\n    Prophets, since the world began.'  Christ came into the world to\n    prepare the way for His future triumph\u2014to lay on Calvary the 'chief\n    corner-stone' of a temple, which shall be completed at the end of\n    time, and endure through all eternity.  He began the great\n    restitution.  He redeemed His people with a price, and gave them a\n    pledge of redemption by power.  He made an end of sin, abolished the\n    Levitical priesthood, and swallowed up all the types and shadows in\n    Himself.  He sent home the beasts, overthrew the altars, and quenched\n    the holy fire; and, upon the sanctifying altar of His own divinity,\n    offered His own sinless humanity, which was consumed by fire from\n    heaven.  He removed the seat of government from Mount Zion, in\n    Jerusalem, to Mount Zion above, where He sits\u2014'a Priest upon His\n    throne,' drawing heaven and earth together, and establishing 'the\n    covenant of peace between them both.'\n\n    \"Blessed be God! we can now go to Jesus, the Mediator; passing by\n    millions of angels, and all 'the spirits of just men made perfect;'\n    till we 'come to the blood of sprinkling, which speaketh better\n    things than that of Abel.'  And we look for that blessed day, when\n    'this gospel of the kingdom' shall be universally prevalent; 'and all\n    shall know the Lord, from the least even to the greatest;' when there\n    shall be a 'new heaven, and a new earth, wherein dwelleth\n    righteousness;' when both the political, and the moral aspects of our\n    world shall be changed; and a happier state of things shall exist\n    than has ever been known before,\u2014when the pestilence, the famine, and\n    the sword shall cease to destroy, and 'the saints of the Most High\n    shall possess the kingdom' in 'quietness, and assurance for ever.'\n    Then cometh the end, when Emmanuel 'shall destroy in this mountain\n    the veil of the covering cast over all people, and swallow up death\n    in victory!'\"\n\nSuch sermons as we have quoted surely convey a living and distinct idea\nof the kind of power which made the man remarkable.  It is, from every\naspect, very unlike the preaching to which we are now accustomed, and\nwhich, therefore, finds general favour with us; it is dogmatic in the\nlast degree; nothing in it is tentative, or hypothetical, yet the\ndogmatism is not that of a schoolman, or a casuist; it is the dogmatism\nof burning conviction, of a profound and unquestioning faith in the\nveracity of New Testament truth, and the corresponding light and\nillustration from the Old.  In these sermons, and others we shall place\nbefore our readers, there is nothing pretty, no nice metaphysical or\ncritical analysis, no attempt to carve giants' heads on cherry-stones.\nHe realized his office as a preacher, not as one set apart to minister to\nintellectual luxury, or vanity, but to stand, announcing eternal truth.\nThe people to whom he spoke were not _dilettantic_, he was no\n_dilettante_.  We can quite conceive,\u2014and therefore these remarks,\u2014that\nthe greater number even of the more eminent men in our modern pulpit will\nregard the style of Christmas Evans with contempt.  We are only setting\nit forth in these pages.  Evidently it told marvellously on the\nPrincipality; it \"searched Jerusalem with candles;\" those who despise it\nhad better settle the question with Christmas Evans himself, and show the\nsuperiority of their method by their larger ministerial usefulness.\n\nThe worth and value of great preaching and great sermons must depend upon\nthe measure to which they represent the preacher's own familiarity with\nthe truths he touches, and proclaims.  The history of the mind of\nChristmas Evans is, from this point of view, very interesting.  We can\nonly get at it from the papers found after his death; but they reveal the\nstory of the life, walk, and triumph of faith in his mind and heart.  He\nkept no journal; but still we have the record of his communions with God\namongst the mountains,\u2014acts of consecration to God quite remarkable,\nwhich he had thought it well to commit to paper, that he might remind\nhimself of the engagements he had made.  It was after some such season\nthat he said to a brother minister, \"Brother, the doctrine, the\nconfidence, and strength I feel will make people dance with joy in some\nparts of Wales;\" and then, as the tears came into his eyes whilst he was\nspeaking, he said again, \"Yes, brother!\"\n\nLittle idea can be formed of the Welsh preacher from the life of the\nminister in England.  The congregations, we have seen, lay wide, and\nscattered far apart.  Often, in Wales ourselves, we have met the minister\npursuing his way on his horse, or pony, to his next \"publication;\" very\noften, his Bible in his hand, reading it as he slowly jogged along.  So\nChristmas Evans passed his life, constantly, either on foot or on\nhorseback, urging his way; sometimes through a country frowning as if\nsmitten by a blow of desolation, and at others, laughing in loveliness\nand beauty; sometimes through the hot summer, when the burning beams\npoured from the craggy mountains; sometimes in winter, through the snow\nand rain and coldest inclemency, to fulfil his engagements.  For the\ngreater part of his life his income was never more than thirty pounds a\nyear, and for the first part only about from ten to seventeen.  It looks\na wretched sum; but we may remember that Luther's income was never much\nmore; and, probably, what seems to us a miserable little income, was very\nmuch further removed from want, and even poverty, than in other, less\nprimitive, circumstances is often an income of hundreds.  Certainly,\nChristmas Evans was never in want; always, not only comfortable, but able\neven to spare, from his limited means, subscriptions to some of the great\nsocieties of the day.\n\n\n\n\nCHAPTER IV.\n_THE MINISTRY IN ANGLESEA_ (_CONTINUED_).\n\n\nChristmas Evans as a Bishop over many Churches\u2014As a Moderator in Public\nMeetings\u2014Chapel-building and all its Difficulties to Christmas\nEvans\u2014Extensive Travelling for Chapel-debts\u2014Especially in South Wales\u2014The\nCildwrn Cottage again\u2014A Mysterious Life of Poverty but of\nHospitality\u2014Catherine's Troubles\u2014Story of a Hat\u2014Wayfaring\u2014Insatiability\nfor Sermons in the Welsh\u2014The Scenery of a Great Sermon\u2014The Demoniac of\nGadara\u2014A Remarkable Illustration of the Varied Method of the Preacher\u2014A\nSeries of Illustrations of his Power of Allegoric Painting\u2014The Four\nMethods of Preaching\u2014The Seeking of the Young Child\u2014Satan walking in Dry\nPlaces\u2014Christmas Evans in Another Light\u2014Lengthy Letter to a Young\nMinister\u2014Contributions to Magazines\u2014To be accursed from Christ\u2014Dark Days\nof Persecution\u2014Threatened with Law for a Chapel Debt\u2014Darker Days\u2014Loss of\nhis Wife\u2014Other Troubles\u2014Determines to leave Anglesea.\n\nThe few glimpses we are able to obtain of the life and ministry in\nAnglesea, assure us of the supreme influence obtained by Christmas Evans,\nas was natural, over all the Churches of his order throughout that\nregion.  And in a small way, in a circle far removed from the noise of\nideas, and the crowds and agitations of the great world, incessant\nactivity was imposed upon him,\u2014so many Societies under his care, so many\nmeeting-houses to be erected, and funds to be procured for their\nerection, so many cases of Church discipline, so many co-pastors\nappointed, and set apart to work with him\u2014who, however, were men mostly\nin business, had their own domestic affairs to manage, and for all the\nhelp they could give, needed helping and guidance; who had to receive\ninstructions from him as to what they were to do, and whither they were\nto go,\u2014so that, in fact, he was here, in Anglesea, a pastor of pastors, a\nbishop, if ever any pastor deserved that designation; an overseer of many\nChurches, and of many ministers.  And hence, as a matter of course, in\nall ministerial meetings, and other smaller gatherings, he was usually at\nonce not merely the nominal president, but the presiding spirit.\n\nRhys Stephen suggests a good many ludicrous aspects to the monthly\nmeetings, and other such gatherings; indeed, they were of a very\nprimitive description, and illustrative of what we should call a very\nrude, and unconventional state of society.  Order was maintained,\napparently, very much after the patriarchal or patristic fashion.  All\nthe preachers he called by their Christian names, and he would certainly\nhave wondered what stranger happened to be in the place had any one\naddressed him as Mr. Evans; \"Christmas Evans,\" before his face and behind\nhis back, was the name by which he was known not only throughout all\nAnglesea, but, by-and-by, throughout the entire Principality.\n\nAffectionate familiarity sometimes pays the penalty in diminished\nreverence, and in a subtraction from the respect due to a higher gift or\nsuperior position.  Christmas appears to have been equal to this dilemma,\nand to have sustained with great natural dignity the post of Moderator,\nwithout surrendering his claim upon the affection of his colleagues.  In\nsuch a meeting, some humble brother would rise to speak a second time,\nand, perhaps, not very pointedly, to the question; then the Moderator in\nthe pulpit, gathering up his brows, would suddenly cut across the speaker\nwith, \"William, my boy, you have spoken before: have done with it;\" or,\n\"Richard, _bach_, you have forgotten the question before the meeting:\nhold your tongue.\"\n\nOn one occasion, a minister from South Wales, although a native of\nAnglesea, happening to be present, and rising evidently with the\nintention of speaking, Christmas, who suffered no intrusion from the\nsouth into their northern organizations, instantly nipped the flowers of\noratory by crying out, \"Sit down, David, sit down.\"\n\nSuch instances as these must seem very strange, even _outr\u00e9_, to our\ntemper, taste, and ideas of public meetings; but they furnish a very\ndistinct idea of time, place, and circumstances, and give a not\naltogether unbeautiful picture of a state of society when, if politeness\nand culture had not attained their present eminence, there was a good\ndeal of light and sweetness, however offensive it might seem to our\nintellectual Rimmels and Edisons.\n\nPerhaps in every truly great and apostolic preacher, the preaching power,\nalthough before men the most conspicuous, is really the smallest part of\nthe preacher's labour, and presents the fewest claims for homage and\nhonour.  We have very little, and know very little, of the Apostle Paul's\nsermons and great orations, mighty as they unquestionably were; he lives\nto us most in his letters, in his life, and its many martyrdoms.  Ah, we\nfancy, if Christmas Evans had but to preach, to stay at home and minister\nto his one congregation, what a serene and quiet life it would have been,\nand how happy in the humble obscurity of his Cildwrn cottage!\n\nBut all his life in Anglesea seems to have been worried with\nchapel-debts.  Chapels rose,\u2014it was necessary that they should rise;\npeople in scattered villages thronged to hear the Word; many hundreds\nappear to have crowded into Church fellowship, chapels had to be\nmultiplied and enlarged; but, so far as we are able to read his\nbiography, Christmas appears to have been the only person on whom was\nlaid the burden of paying for them.  Certainly he had no money: his\nwealth was in his eloquence, and his fame; and the island of Anglesea\nappears to have been by no means indisposed to lay these under\ncontribution.  A chapel had to be raised, and Christmas Evans was the\nname upon which the money was very cheerfully lent for its erection; but\nby-and-by the interest pressed, or the debt had to be paid: what could be\ndone then?  He must go forth into the south, and beg from richer\nChurches, and from brethren who, with none of his gifts of genius or of\nholiness, occupied the higher places in the sanctuary.\n\nOur heart is very much melted while we read of all the toils he\naccomplished in this way.  Where were his sermons composed?  Not so much\nin his lowly cottage home as in the long, lonely, toilsome travels on his\nhorse through wild and unfrequented regions, where, throughout the long\nday's journey, he perhaps, sometimes, never met a traveller on the\nsolitary road.  For many years, it is said, he went twice from his\nnorthern bishopric to the south, once to the great Association, wherever\nthat might be, and where, of course, he was expected as the chief and\nmost attractive star, but once also with some chapel case, a journey\nwhich always had to be undertaken in the winter, and which was always a\npainful journey.  Let us think of him with affection as we see him\nwending on, he and his friendly horse, through wild snows, and rains, and\nbleak storms of mountain wind.\n\nScarcely do we need to say he had a highly nervous temperament.  The dear\nman had a very capricious appetite, but who ever thought of that?  He was\nthrown upon himself; but the testimony is that he was a man utterly\nregardless of his own health, ridiculously inattentive to his dress, and\nto all his travelling arrangements.  These journeys with his chapel case\nwould usually take some six weeks, or two months.  It was no dainty tour\nin a railway train, with first-class travelling expenses paid for the\nbest carriage, or the best hotel.\n\nA man who was something like Christmas Evans, though still at an infinite\nremove from him in the grandeur of his genius, a great preacher, William\nDawson\u2014Billy Dawson, as he is still familiarly called\u2014used to say, that\nin the course of his ministry he found himself in places where he was\nsometimes treated like a bishop, and sometimes like an apostle; sometimes\na great man would receive, and make a great dinner for him, and invite\ncelebrities to meet him, and give him the best entertainment, the best\nroom in a large, well-furnished house, where a warm fire shed a glow over\nthe apartment, and where he slept on a bed of down,\u2014and this was what he\ncalled being entertained like a bishop; but in other places he would be\nreceived in a very humble home, coarse fare on the table, a mug of ale, a\npiece of oatmeal cake, perhaps a slice of meat, a poor, unfurnished\nchamber, a coarse bed, a cold room,\u2014and this was what he called being\nentertained like an apostle.\n\nWe may be very sure that the apostolic entertainment was that which\nusually awaited Christmas Evans at the close of his long day's journey.\nNot to be looked upon with contempt either,\u2014hearty and free; and,\nperhaps, the conversation in the intervals between the puff of the pipe\nwas what we should rather relish, than the more timorous and equable flow\nof speech in the finer mansion.  This is certain, however, that the\nentertainment of Christmas Evans, in most of his excursions, would be of\nthe coarsest kind.\n\nAnd this was far from the worst of his afflictions; there were, in that\nday, persons of an order of character, unknown to our happier, more\nChristian, and enlightened times,\u2014pert and conceited brethren, unworthy\nto unloose the latchet of the great man's shoes, but who fancied\nthemselves far above him, from their leading a town life, and being\npastors over wealthier Churches.  Well, they have gone, and we are not\nwriting their lives, for they never had a life to write, only they were\noften annoying flies which teased the poor traveller on his way.  On most\nof these he took his revenge, by fastening upon them some _sobriquet_,\nwhich he fetched out of that imaginative store-house of his,\u2014from the\nclosets of compound epithet; these often stuck like a burr to the coat of\nthe character, and proved to be perhaps the best passport to its owner's\nnotoriety through the Principality.  Further than this, we need not\nsuppose they troubled the great man much; uncomplainingly he went on, for\nhe loved his Master, and he loved his work.  He only remembered that a\ncertain sum must be found by such a day to pay off a certain portion of a\nchapel-debt; he had to meet the emergency, and he could only meet it by\nobtaining help from his brethren.\n\nIn this way he travelled from North to South Wales forty times; he\npreached always once every day in the week, and twice on the Lord's Day.\nOf course, the congregations everywhere welcomed him; the collections\nusually would be but very small; ministers and officers, more usually, as\nfar as was possible, somewhat resented these calls, as too frequent and\nirregular.  He preached one of his own glorious sermons, and then\u2014does it\nnot seem shocking to us to know, that he usually stood at the door, as it\nwere, hat in hand, to receive such contributions as the friends might\ngive to him?  And he did this for many years, until, at last, his\nfrequent indisposition, in consequence of this severity of service,\ncompelled him to ask some friend to take his place at the door; but in\ndoing this he always apologised for his delegation of service to another,\nlest it should seem that he had treated with inattention and disrespect\nthose who had contributed to him of their love and kindness.\n\nAnd so a number of the Welsh Baptist chapels, in Anglesea and North\nWales, rose.  There was frequently a loud outcry among the ministers of\nthe south, that he came too often; and certainly it was only the\nmarvellous attractions of the preacher which saved him from the indignity\nof a refusal.  His reply was always ready: \"What can I do? the people\ncrowd to hear us; it is our duty to accommodate them as well as we can;\nall we have we give; to you much is given, you can give much; it is more\nblessed to give than receive,\" etc., etc.  Then sometimes came more\nplaintive words; and so he won his way into the pulpit, and, once there,\nit was not difficult to win his way to the people's hearts.  It was what\nwe suppose may be called the age of chapel cases.  How many of our\nchapels in England have been erected by the humiliating travels of poor\nministers?\n\nChristmas Evans was saved from one greater indignity yet, the\nencountering the proud rich man, insolent, haughty, and arrogant.  It is\nnot a beautiful chapter in the history of voluntaryism.  In the course of\nthese excursions, he usually succeeded in accomplishing the purpose for\nwhich he set forth; probably the contributions were generally very small;\nbut then, on many occasions, the preacher had so succeeded in putting\nhimself on good terms with all his hearers that most of them gave\nsomething.\n\nIt is said that on one occasion not a single person passed by without\ncontributing something: surely a most unusual circumstance, but it was\nthe result of a man\u0153uvre.  It was in an obscure district, just then\nespecially remarkable for sheep-stealing; indeed, it was quite notorious.\nThe preacher was aware of this circumstance, and, when he stood up in the\nimmense crowd to urge the people to liberality, he spoke of this crime of\nthe neighbourhood; he supposed that amidst that large multitude it was\nimpossible but that some of those sheep-stealers would be present: he\naddressed them solemnly, and implored them, if present, not to give\nanything to the collection about to be made.  It was indeed a feat rather\nworthy of Rowland Hill than illustrative of Christmas Evans, but so it\nwas; those who had no money upon them borrowed from those who had, and it\nis said that, upon that occasion, not a single person permitted himself\nto pass out without a contribution.\n\nThe good man, however, often felt that a burden was laid upon him, which\nscarcely belonged to the work to which he regarded himself as especially\nset apart.  Perhaps he might have paraphrased the words of the Apostle,\nand said, \"The Lord sent me not to attend to the affairs of your\nchapel-debts, but to preach the gospel.\"  There is not only pathos, but\ntruth in the following words; he says, \"I humbly think that no\nmissionaries in India, or any other country, have had to bear such a\nburden as I have borne, because of chapel-debts, and _they_ have not had\nbesides to provide for their own support, as I have had to do through all\nmy life in Anglesea; London committees have cared for _them_, while I,\nfor many years, received but seventeen pounds per annum for all my\nservices.  The other preachers were young, and inexperienced, and the\nmembers threw all the responsibility upon me, as children do upon a\nfather; my anxiety often moved me in the depths of the night to cry out\nunto God to preserve His cause from shame.  God's promises to sustain His\ncause in the world greatly comforted me.  I would search for the Divine\npromises to this effect, and plead them in prayer, until I felt as\nconfident as if every farthing had been paid.  I laboured hard to\ninstitute weekly penny offerings, but was not very successful; and after\nevery effort there remained large sums unpaid in connection with some of\nthe chapels which had been built without my consent.\"\n\nPoor Christmas!  As we read of him he excites our wonder.\n\n    \"Passing rich with forty pounds a year.\"\n\nlooks like positive wealth as compared with the emoluments of our poor\npreacher; and yet the record is that he was given to hospitality, and he\ncontributed his sovereign, and half-sovereign, not only occasionally, but\nannually, where his richer neighbours satisfied their consciences with\nfar inferior bequests.  How did the man do it?  He had not married a rich\nwife, and he did not, as many of his brethren, eke out his income by some\nfarm, or secular pursuit; a very common, and a very necessary thing to\ndo, we should say, in Wales.\n\nBut, no doubt, Catherine had much to do with his unburdened life of\ndomestic quiet; perhaps,\u2014it does not appear, but it seems probable\u2014she\nhad some little money of her own; she had what to her husband was\nincomparably more valuable, a clear practical mind, rich in faith, but a\ncalm, quiet, household faith.  Lonely indeed her life must often have\nbeen in the solitary cottage, into which, assuredly, nothing in the shape\nof a luxury ever intruded itself.  It has been called, by a Welshman, a\ncurious anomaly in Welsh life, the insatiable appetite for sermons, and\nthe singular, even marvellous, disregard for the temporal comforts of the\npreacher.  Christmas, it seems to us, was able to bear much very\nunrepiningly, but sometimes his righteous soul was vexed.  Upon one\noccasion, when, after preaching from home, he not only received less for\nhis expenses than he naturally expected, but even less than an ordinary\nitinerant fee, an old dame remarked to him, \"Well, Christmas, _bach_, you\nhave given us a wonderful sermon, and I hope you will be paid at the\nresurrection,\" \"Yes, yes, _shan fach_,\" said the preacher, \"no doubt of\nthat, but what am I to do till I get there?  And there's the old white\nmare that carries me, what will she do? for her there will be no\nresurrection.\"\n\nDecidedly the Welsh of that day seemed to think that it was essential to\nthe preservation of the purity of the Gospel that their ministers should\nbe kept low.  Mr. D. M. Evans, in his Life of Christmas Evans, gives us\nthe anecdote of a worthy and popular minister of this time, who was in\nthe receipt of exactly twenty pounds a year; he received an invitation\nfrom another Church, offering him three pounds ten a month.  This\nmiserable lover of filthy lucre, like another Demas, was tempted by the\ndazzling offer, and intimated his serious intention of accepting \"the\ncall.\"  There was a great commotion in the neighbourhood, where the poor\nman was exceedingly beloved; many of his people remonstrated with him on\nthe sad exhibition he was giving of a guilty love of money; and, after\nmuch consideration, the leading deacon was appointed as a deputation to\nwait upon him, and to inform him, that rather than suffer the loss of his\nremoval on account of money considerations, they had agreed to advance\nhis salary to twenty guineas, or twenty-one pounds!  Overcome by such an\nexpression of his people's attachment, says Mr. Evans, he repented of his\nincontinent love of money, and stayed.\n\nA strange part-glimpse all this seems to give of Welsh clerical life, not\ncalculated either to kindle, or to keep in a minister's mind, the\nessential sense of self-respect.  The brothers of La Trappe, St. Francis\nand his preaching friars, do not seem to us a more humiliated tribe than\nChristmas and his itinerating \"little _brethren_ of the poor.\"  We\nsuppose that sometimes a farmer would send a cheese, and another a few\npounds of butter, and another a flitch of bacon; and, perhaps,\noccasionally, in the course of his travels,\u2014we do not know of any such\ninstances, we only suppose it possible, and probable,\u2014some rich man,\nafter an eloquent sermon, would graciously patronize the illustrious\npreacher, by pressing a real golden sovereign into the apostle's hand.\n\nOne wonders how clothes were provided.  William Huntingdon's \"Bank of\nFaith\" seems to us, in comparison with that of Christmas Evans, like the\nfaith of a man who wakes every morning to the sense of the possession of\na million sterling at his banker's,\u2014in comparison with _his_ faith, who\nrises sensible that, from day to day, he has to live as on the assurance,\nand confidence of a child.\n\nCertainly, Wales did not contain at that time a more unselfish, and\ndivinely thoughtless creature than this Christmas Evans; and then he had\nno children.  A man without children, without a child, can afford to be\nmore careless and indifferent to the world's gold and gear.  The coat, no\ndoubt, often got very shabby, and the mothers of Israel in Anglesea, let\nus hope, sometimes gathered together, and thought of pleasant surprises\nin the way of improving the personal appearance of their pastor; but\nindeed the man was ridiculous in his disregard to all the circumstances\nof dress and adornment.  Once, when he was about to set forth on a\npreaching tour, Catherine had found her mind greatly exercised concerning\nher husband's hat, and, with some difficulty, she had succeeded in\nequipping that noble head of his with a new one.  But upon the journey\nthere came a time when his horse needed to drink; at last he came to a\nclear, and pleasant pond, or brook, but he was at a loss for a pail; now\nwhat was to be done?  Happy thought, equal to any of those of Mr.\nBarnand! he took the hat from off his head, and filled it with water for\npoor old Lemon.  When he returned home, Catherine was amazed at the\ndeterioration of the headgear, and he related to her the story.  A man\nlike this would not be likely to be greatly troubled by any defections in\npersonal adornment.\n\nWordsworth has chanted, in well-remembered lines, the name and fame of\nhim, whom he designates, for his life of probity, purity, and\npoverty,\u2014united in the pastoral office, in his mountain chapel in\nWestmoreland,\u2014Wonderful Robert Walker.  Far be it from us to attempt to\ndetract from the well-won honours of the holy Westmoreland pastor; but,\nassuredly, as we think of Christmas Evans, he too seems to us even far\nmore wonderful; for there was laid upon him, not merely the thought for\nhis own pulpit and his own family, but the care of all the Churches in\nhis neighbourhood.\n\nAnd so the end is, that during these years we have to follow him through\nmountain villages, in which the silence and desolation greet him, like\nthat he might have found in old Castile, or La Mancha,\u2014through spots\nwhere ruined old castles and monasteries were turned into barns, and hay\nand straw stowed away within walls, once devoted either to gorgeous\nfestivity or idolatry,\u2014through wild and beautiful scenes; narrow glen and\nravine, down which mountain torrents roared and foamed,\u2014through wild\nmountain gorges, far, in his day, from the noise and traffic of\ntowns,\u2014although in such spots Mr. Borrow found the dark hills strangely\nablaze with furnaces, seeming to that strange traveller, so he said,\nqueerly enough, \"like a Sabbath in hell, and devils proceeding to\nafternoon worship,\"\u2014past simple, and unadorned, and spireless churches,\nhallowed by the prayers of many generations; and through churchyards in\nwhich rests the dust of the venerable dead.  We can see him coming to the\nlonely Methodist chapel, rising like a Shiloh, bearing the ark, like a\nlighthouse among the high hills\u2014strolling into a solitary cottage as he\npasses, and finding some ancient woman, in her comfortable kitchen, over\nher Welsh Bible, and concordance, neither an unpleasant nor an unusual\nsight;\u2014never happier, we will be bold to say, than when, keeping his own\ncompany, he traverses and travels these lone and solitary roads and\nmountain by-paths, not only through the long day, but far into the night,\nsometimes by the bright clear moonlight, among the mountains, and\nsometimes through the \"villain mists,\" their large sheets rolling up the\nmountain sides bushes and trees seen indistinctly like goblins and elves,\ntill\u2014\n\n    \"In every hollow dingle stood,\n    Of wry-mouth fiends a wrathful brood.\"\n\nSo we think of him pressing on his way; no doubt often drenched to the\nskin, although uninjured in body; sometimes through scenes novel and\ngrand, where the mountain looks sad with some ruin on its brow, as\nbeneath Cader Idris (the chair or throne of Idris), where the meditative\nwanderer might conceive he saw some old king, unfortunate and melancholy,\nbut a king still, with the look of a king, and the ancestral crown on his\nforehead.\n\nWe may be sure he came where corpse-candles glittered, unquenched by\nnineteenth-century ideas, along the road; for those travelling times were\nmuch nearer to the days of Twm \u00f3r Nant, who, when he kept turnpike, was\nconstantly troubled by hearses, and mourning coaches, and funeral\nprocessions on foot passing through his gate.  Through lonely places and\nalder swamps, where nothing would be heard but the murmuring of waters,\nand the wind rushing down the gullies,\u2014sometimes falling in with a pious\nand sympathetic traveller, a lonely creature, \"Sorry to say, Good-bye,\nthank you for your conversation; I haven't heard such a treat of talk for\nmany a weary day.\"  Often, passing through scenes where the sweet voice\nof village bells mingled with the low rush of the river; and sometimes\nwhere the rocks rolled back the echoes like a pack of dogs sweeping down\nthe hills.  \"Hark to the dogs!\" exclaimed a companion to Mr. Borrow once.\n\"This pass is called _Nant yr ieuanc gwn_, the pass of the young dogs;\nbecause, when one shouts, it answers with a noise resembling the crying\nof hounds.\"\n\nWhat honour was paid to the name and memory of the earnest-hearted and\nintrepid Felix Neff, the pastor of the Higher Alps; but does not the\nreader, familiar with the life of that holy man, perceive much\nresemblance in the work, the endurance, and the scenery of the toil, to\nthat of Christmas Evans?  May he not be called the pastor of our English\nEngadine?\n\nAll such lives have their grand compensations; doubtless this man had\nhis, and _great_ compensations too; perhaps, among the minor ones, we may\nmention his ardent reception at the great Association gatherings.  At\nthese his name created great expectations; there he met crowds of\nbrethren and friends, from the remote parts of the Principality, by whom\nhe was at once honoured and loved.  We may conceive such an occasion; the\n\"one-eyed man of Anglesea\" has now been for many years at the very height\nof his popularity; his name is now the greatest in his denomination; this\nwill be one of his great occasions, and his coming has been expected for\nmany weeks.  No expectation hanging upon the appearance of Jenny Lind, or\nChristine Nielson, or Sims Reeves, on some great musical festivity, can\nreach, in our imagination, the expectations of these poor, simple\nvillagers as they think of the delight they will experience in listening\nto their wonderful and well-loved prophet.\n\nSo, along all the roads, there presses an untiring crowd, showing that\nsomething unusual is going on somewhere.  The roads are all picturesque\nand lively with all sorts of people, on foot, on horseback, in old farm\ncarts, and even in carriages; all wending their way to the largest and\nmost central chapel of the neighbourhood.  It is the chief service.  It\nis a Sabbath evening; the congregation is wedged together in the spacious\nhouse of God; it becomes almost insupportable, but the Welsh like it.\nThe service has not commenced, and a cry is already raised that it had\nbetter be held in an adjoining field; but it is said this would be\ninconvenient.  The doors, the windows, are all thrown open; and so the\ntime goes on, and the hour for the commencement of the service arrives.\nAll eyes are strained as the door opens beneath the pulpit, and the\nminister of the congregation comes in, and makes his way, as well as he\ncan, for himself and his friend, the great preacher\u2014there he is! that\ntall, commanding figure,\u2014that is he, the \"one-eyed man of Anglesea.\"\n\nA murmur of joy, whisperings of glad congratulation, which almost want to\nburst into acclamations, pass over the multitude.  And the service\ncommences with prayer, singing, reading a chapter, and a short sermon,\u2014a\nvery short one, only twenty minutes.  There are crowds of preachers\nsitting beneath the pulpit, but they, and all, have come to hear the\nmighty minstrel\u2014and the moment is here.  A few more verses of a hymn,\nduring which there is no little commotion, in order that there may be\nnone by-and-bye, those who have been long standing changing places with\nthose who had been sitting.  There, he is up! he is before the people!\nAnd in some such circumstances he seems to have first sung that wonderful\nsong or sermon,\n\n\n\nTHE DEMONIAC OF GADARA.\n\n\nThe text he announced was\u2014\"_Jesus said unto him_, _Go home unto thy\nfriends_, _and tell them how great things the Lord hath done for thee_,\n_and hath had compassion on thee_.\"\n\nThe introduction was very simple and brief; but, before long, the\npreacher broke loose from all relations of mere comment and explanation,\nand seemed to revel in dramatic scenery, and pictorial imagination, and,\nas was so usual with him in such descriptions, increasing, heightening,\nand intensifying the picture, by making each picture, each scene, to live\neven in the kind of enchantment of a present demoniacal possession.  He\nbegan by describing the demoniac as a castle garrisoned with a legion of\nfiends, towards which the great Conqueror was approaching over the Sea of\nTiberias, the winds hushing at His word, the sea growing calm at His\nbidding.  Already He had acquired among the devils a terrible fame, and\nHis name shook the garrison of the entire man, and the infernal legion\nwithin, with confusion and horror.\n\n    \"I imagine,\" he said, \"that this demoniac was not only an object of\n    pity, but he was really a terror to the country.  So terrific was his\n    appearance, so dreadful and hideous his screams, so formidable,\n    frightful, and horrid his wild career, that all the women in that\n    region were so much alarmed that none of them dared go to market,\n    lest he should leap upon them like a panther on his prey.\n\n    \"And what made him still more terrible was the place of his abode.\n    It was not in a city, where some attention might be paid to order and\n    decorum (though he would sometimes ramble into the city, as in this\n    case).  It was not in a town, or village, or any house whatever,\n    where assistance might be obtained in case of necessity; but it was\n    among the tombs, and in the wilderness\u2014not far, however, from the\n    turnpike road.  No one could tell but that he might leap at them,\n    like a wild beast, and scare them to death.  The gloominess of the\n    place made it more awful and solemn.  It was among the tombs\u2014where,\n    in the opinion of some, all witches, corpse-candles, and hobgoblins\n    abide.\n\n    \"One day, however, Mary was determined that no such nuisance should\n    be suffered in the country of the Gadarenes.  The man must be\n    clothed, though he was mad and crazy.  And if he should at any future\n    time strip himself, tie up his clothes in a bundle, throw them into\n    the river, and tell them to go to see Abraham, he must be tied and\n    taken care of.  Well, this was all right; no sooner said than done.\n    But, so soon as the fellow was bound, although even in chains and\n    fetters, Samson-like he broke the bands asunder, and could not be\n    tamed.\n\n    \"By this time, the devil became offended with the Gadarenes, and, in\n    a pout, he took the demoniac away, and drove him into the wilderness.\n    He thought the Gadarenes had no business to interfere, and meddle\n    with his property; for he had possession of the man.  And he knew\n    that 'a bird in the hand is worth two in the bush.'  It is probable\n    that he wanted to send him home; for there was no knowing what might\n    happen now-a-days.  But there was too much matter about him to send\n    him as he was; therefore, he thought the best plan would be to\n    persuade him to commit suicide by cutting his throat.  But here Satan\n    was at a nonplus\u2014his rope was too short.  He could not turn\n    executioner himself, as that would not have answered the design he\n    has in view, when he wants people to commit suicide; for the act\n    would have been his own sin, and not the man's.  The poor demoniac,\n    therefore, must go about to hunt for a sharp stone, or anything that\n    he could get.  He might have been in search of such an article, when\n    he returned from the wilderness into the city, whence he came, when\n    he met the Son of God.\n\n    \"Jesus commanded the unclean spirit to come out of the man.  And when\n    he saw Jesus he cried out, and fell down before him, and with a loud\n    voice said, 'What have I to do with thee, Jesus, Thou Son of God most\n    high?  I beseech Thee, torment me not.'\n\n    \"Here is the devil's confession of faith.  The devils believe and\n    tremble, while men make a mock of sin, and sport on the brink of\n    eternal ruin.  To many of the human race, Christ appears as a root\n    out of dry ground.  They see in Him neither form nor comeliness, and\n    there is no beauty in Him that they should desire Him.  Some said He\n    was the carpenter's son, and would not believe in Him; others said He\n    had a devil, and that it was through Beelzebub, the chief of the\n    devils, that He cast out devils: some cried out, 'Let Him be\n    crucified;' and others said, 'Let His blood be on us and on our\n    children.'  As the Jews would not have Him to reign over them, so\n    many, who call themselves Christians, say that He is a mere man; as\n    such, He has no right to rule over their consciences, and demand\n    their obedience, adoration, and praise.  But the devils know\n    better\u2014they say, Jesus is the Son of God most high.\n\n    \"Many of the children of the devil, whose work they do, differ very\n    widely from their father in their sentiments respecting the person of\n    Christ.\n\n    \"Jesus commanded the legion of unclean spirits to come out of the\n    man.  They knew that out they must go.  But they were like\n    Irishmen\u2014very unwilling to return to their own country.  They would\n    rather go into hogs' skins than to their own country.  And He\n    suffered them to go into the herd of swine.  Methinks that one of the\n    men who fed the hogs, kept a better look out than the rest of them\n    and said, 'What ails the hogs?  Look sharp there, boys\u2014keep them\n    in\u2014make good use of your whips!  Why don't you run?  Why, I declare,\n    one of them has gone over the cliff!  There, there, Morgan, goes\n    another!  Drive them back, Tom.'  Never was there such a running, and\n    whipping, and hallooing; but down go the hogs, before they are aware\n    of it.\n\n    \"One of them said, 'They are all gone!'\n\n    \"'No, sure not all gone into the sea!'\n\n    \"'Yes, every one of them, the _black hog_ and all.  They are all\n    drowned! the devil is in them!  What shall we do now?  What can we\n    say to the owners?'\n\n    \"'What can we say?' said another; 'we must tell the truth\u2014that is all\n    about it.  We did our best\u2014all that was in our power.  What could any\n    man do more?'\n\n    \"So they went their way to the city, to tell the masters what had\n    happened.\n\n    \"'John, where are you going?' exclaimed one of the masters.\n\n    \"'Sir, did you know the demoniac that was among the tombs there?'\n\n    \"'Demoniac among the tombs!  Where did you leave the hogs?'\n\n    \"'That madman, sir\u2014'\n\n    \"'Madman!  Why do you come home without the hogs?'\n\n    \"'That wild and furious man, sir, that mistress was afraid of so\n    much\u2014'\n\n    \"'Why, John, I ask you a plain and simple question\u2014why don't you\n    answer me?  Where are the hogs?'\n\n    \"'That man who was possessed with the devils, sir\u2014'\n\n    \"'Why, sure enough, you are crazy!  You look wild!  Tell me your\n    story, if you can, let it be what it may.'\n\n    \"'Jesus Christ, sir, has cast the unclean spirits out of the\n    demoniac; they are gone into the swine; and they are all drowned in\n    the sea; for I saw the tail of the last one!'\n\n    \"The Gadarenes went out to see what was done, and finding that it was\n    even so, they were afraid, and besought Jesus to depart from them.\n\n    \"How awful must be the condition of those men who love the things of\n    this world more than Jesus Christ.\n\n    \"The man out of whom the unclean spirits were cast, besought Jesus\n    that he might be with Him.  But He told him to return to his own\n    house, and show how great things God had done unto him.  And he went\n    his way, and published, throughout the whole city of Decapolis, how\n    great things Jesus had done unto him.  The act of Jesus casting so\n    many devils out of him, was sufficient to persuade him that Jesus was\n    God as well as man.\n\n    \"I imagine I see him going through the city, crying\u2014'Oh yes!  Oh yes!\n    Oh yes! please to take notice of me, the demoniac among the tombs.  I\n    am the man who was a terror to the people of this place\u2014that wild\n    man, who would wear no clothes, and that no man could bind.  Here am\n    I now, in my right mind.  Jesus Christ, the Friend of sinners, had\n    compassion on me.  He remembered me when I was in my low estate\u2014when\n    there was no eye to pity, and no hand to save.  He cast out the\n    devils and redeemed my soul from destruction.'\n\n    \"Most wonderful must have been the surprise of the people, to hear\n    such proclamation.  The ladies running to the windows, the shoemakers\n    throwing their lasts one way, and their awls another, running out to\n    meet him and to converse with him, that they might be positive that\n    there was no imposition, and found it to be a fact that could not be\n    contradicted.  'Oh, the wonder of all wonders!  Never was there such\n    a thing,' must, I think, have been the general conversation.\n\n    \"And while they were talking, and everybody having something to say,\n    homeward goes the man.  As soon as he comes in sight of the house, I\n    imagine I see one of the children running in, and crying, 'Oh,\n    mother! father is coming\u2014he will kill us all!'\n\n    \"'Children, come all into the house,' says the mother.  'Let us\n    fasten the doors.  I think there is no sorrow like my sorrow!' says\n    the broken-hearted woman.  'Are all the windows fastened, children?'\n\n    \"'Yes, mother.'\n\n    \"'Mary, my dear, come from the window\u2014don't be standing there.'\n\n    \"'Why, mother, I can hardly believe it is father!  That man is well\n    dressed.'\n\n    \"'Oh yes, my dear children, it is your own father.  I knew him by his\n    walk, the moment I saw him.'\n\n    \"Another child stepping to the window, says, 'Why, mother, I never\n    saw father coming home as he comes to-day.  He walks on the footpath,\n    and turns round the corner of the fence.  He used to come towards the\n    house as straight as a line, over fences, ditches, and hedges; and I\n    never saw him walk as slowly as he does now.'\n\n    \"In a few moments, however, he arrives at the door of the house, to\n    the great terror and consternation of all the inmates.  He gently\n    tries the door, and finds no admittance.  He pauses a moment, steps\n    towards the window, and says in a low, firm, and melodious voice, 'My\n    dear wife, if you will let me in, there is no danger.  I will not\n    hurt you.  I bring you glad tidings of great joy.'  The door is\n    reluctantly opened, as it were between joy and fear.  Having\n    deliberately seated himself, he says: 'I am come to show you what\n    great things God has done for me.  He loved me with an everlasting\n    love.  He redeemed me from the curse of the law, and the threatenings\n    of vindictive justice.  He saved me from the power and dominion of\n    sin.  He cast the devils out of my heart, and made that heart, which\n    was a den of thieves, the temple of the Holy Spirit.  I cannot tell\n    you how much I love my Saviour.  Jesus Christ is the foundation of my\n    hope, the object of my faith, and the centre of my affections.  I can\n    venture my immortal soul upon Him.  He is my best friend.  He is\n    altogether lovely\u2014the chief among ten thousand.  He is my wisdom,\n    righteousness, sanctification, and redemption.  There is enough in\n    Him to make a poor sinner rich, and a miserable sinner happy.  His\n    flesh and blood is my food,\u2014His righteousness my wedding garment, and\n    His blood is efficacious to cleanse me from all my sins.  Through Him\n    I can obtain eternal life; for He is the brightness of the Father's\n    glory, and the express image of His Person: in whom dwelleth all the\n    fulness of the Godhead bodily.  He deserves my highest esteem, and my\n    warmest gratitude.  Unto Him who loved me with an eternal love, and\n    washed me in His own blood, unto Him be the glory, dominion, and\n    power, for ever and ever!  For He has rescued my soul from hell.  He\n    plucked me as a brand from the burning.  He took me out of the miry\n    clay, and out of a horrible pit.  He set my feet upon a rock, and\n    established my goings, and put in my mouth a new song of praise, and\n    glory to Him!  Glory to Him for ever!  Glory to God in the highest!\n    Glory to God for ever and ever!  Let the whole earth praise Him!\n    Yea, let all the people praise Him!'  How sweet was all this, the\n    transporting joy of his wife!\n\n    \"It is beyond the power of the strongest imagination to conceive the\n    joy and gladness of this family.  The joy of seafaring men delivered\n    from shipwreck; the joy of a man delivered from a burning house; the\n    joy of not being found guilty at a criminal bar; the joy of receiving\n    pardon to a condemned malefactor; the joy of freedom to a prisoner of\n    war, is nothing in comparison to the joy of him who is delivered from\n    going down to the pit of eternal destruction.  For it is a joy\n    unspeakable and full of glory.\"\n\nThe effect of this sermon is described as overwhelmingly wonderful.  The\nfirst portion, in which he pictured the mysterious and terrible being,\nthe wild demoniac, something of a wild beast, and something of a fiend,\nmade the people shudder.  Then, shifting his scene, the catastrophe of\nthe swine, the flight of the affrighted herdsmen, the report to the\nmaster, and the effect of the miracle on the populace, was rendered with\nsuch dramatic effect, the preacher even laughing himself, as he painted\nthe rushing swine, hurrying down the steep place into the lake,\nespecially the \"black hog,\" and all,\u2014for they all understood the point of\nthat allusion,\u2014that beneath the grim grotesqueness of the scene, laughter\nran over the whole multitude.  But the pathos of the family scene!  Mary\nembracing her restored husband; and the restored maniac's experience, and\nhymn of praise.  The place became a perfect Bochim; they wept like\nmourners at a funeral.  Shouts of prayer and praise mingled together.\nOne who heard that wonderful sermon says, that, at last, the people\nseemed like the inhabitants of a city which had been shaken by an\nearthquake, that, in their escape, rushed into the streets, falling upon\nthe earth screaming, and calling upon God!\n\nThis sermon has never been printed; indeed, it is obvious that it never\ncould be prepared for the press.  It defies all criticism; and the few\noutlines we have attempted to present are quite inadequate to reproduce\nit.  All who heard it understood, that it was a picture of a lunatic, and\ndemon-haunted world; and it was beneath the impression of this, that\npassionate cries, universal, thankful, penitent murmurs rose; whilst\namidst loud \"Amens!\" and sobs, and tears, some petitions ascended: \"O\nLord, who didst walk on the sea, that Thou mightest meet the Gadarene,\ncast out some demons from our midst to-night.\"\n\nAlthough the demoniac of Gadara is not, in the strict sense of the word,\nan allegory, yet it is allegoric throughout; a fine piece of shadowy\npainting, in which unconverted, and converted men, and women might\nrealize something of their own personal history, and the means by which\nthey would \"come to themselves.\"\n\nAnd, no doubt, the chief charm, and most original characteristic of the\npreacher, was his power of sustained allegory; some incident, even some\npassing expression in Scripture, some prophetic figure of speech, was\nturned round and round by him, beaten out, or suggested a series of\ncartoon paintings, until it became like a chapter from the \"Pilgrim's\nProgress.\"  It has seemed to us, that his translators have been\nsingularly unfortunate in rendering these excursions of his fancy into\nEnglish; our most vivid impressions of them have been derived from those\nwho had heard them, in all their freshness, from the preacher's own\nwonderful lips.  We will attempt to transfer one or two of these\nallegories to our pages.  It must have been effective to have heard him\ndescribe the necessity of Divine life, spiritual power, to raise a soul\nfrom spiritual death.  This may be called\n\n\n\n\"THE FOUR METHODS OF PREACHING.\n\n\n    \"He beheld,\" he said, \"such a one as Lazarus lying in the cave,\n    locked in the sleep of death; now how shall he be raised? how shall\n    he be brought back to life?  Who will roll away for us the stone from\n    this sepulchre?  First came one, who went down to the cave with\n    blankets, and salt, to rub with the fomentations of duty, to appeal\n    to the will, to say to the sleeping man, that he could if he would;\n    chafing and rubbing the cold and inert limbs, he thinks to call back\n    the vital warmth; and then retiring, and standing some distance\n    apart, he says to the other spectators, 'Do you not see him stir?\n    Are there no signs of life?  Is he not moving?'  No, he lies very\n    still, there is no motion.  How could it be otherwise? how could a\n    sense of moral duty be felt by the man there?\u2014_for the man was dead_!\n\n    \"The first man gave up in despair.  And then came the second.  'I\n    thought you would never do it,' he said; 'but if you look at me, you\n    will see a thing.  No,' he said, 'your treatment has been too\n    gentle.'  And he went down into the cave with a scourge.  Said he,\n    'The man only wants severe treatment to be brought back to life.  I\n    warrant me I will make him feel,' he said.  And he laid on in quick\n    succession the fervid blows, the sharp threatenings of law and\n    judgment, and future danger and doom; and then he retired to some\n    distance.  'Is he not waking?' he said.  'Do you not see the corpse\n    stir?'  No!  A corpse he was before the man began to lay on his\n    lashes, and a corpse he continued still;\u2014_for the man was dead_!\n\n    \"'Ah,' said another, advancing, 'but I have wonderful power.  You,\n    with your rubbing, and your smiting, what can you do? but I have it,\n    for I have two things.'  And he advanced, and he fixed an electric\n    battery, and disposed it so that it touched the dead man, and then,\n    from a flute which he held, he drew forth such sweet sounds that they\n    charmed the ears which were listening; and whether it was the\n    battery, or whether it was the music, so it was, that effect seemed\n    to be produced.  'Behold,' said he, 'what the refinements of\n    education and cultivation will do!'  And, indeed, so it was, for the\n    hair of the dead man seemed to rise, and his eye-balls seemed to\n    start and dilate; and see! he rises, starts up, and takes a stride\n    down the cave.  Ah, but it is all over; it was nothing but the\n    electricity in the battery; and he sank back again flat on the floor\n    of the cave;\u2014_for the man was dead_!\n\n    \"And then, when all were filled with despair, there came One, and\n    stood by the entrance of the cave; but He was the Lord and Giver of\n    life, and standing there, He said, 'Come from the four winds, O\n    breath, and breathe upon this slain one, that he may live.  Christ\n    hath given thee life.  Awake, thou that sleepest.'  And the man\n    arose; he shook off his grave-clothes; what he needed had come to him\n    now\u2014_life_!  Life is the only cure for death.  Not the prescriptions\n    of duty, not the threats of punishment and damnation, not the arts\n    and the refinements of education, but new, spiritual, Divine _life_.\"\n\nThe same manner appears in the way in which he traces the story of a soul\nseeking Christ, under the idea of the Wise Men following the leading star\nin\n\n\n\n\"SEEKING THE YOUNG CHILD.\"\n\n\nWe have remarked before that the preacher's descriptions of Oriental\ntravel were always Welsh, and this could not arise so much from\nignorance, for he was fairly well read in the geography, and, perhaps,\neven in the topography, of the Holy Land; but he was quite aware that\nOriental description would be altogether incomprehensible to the great\nmultitude of his auditors.  He described, therefore, the Wise Men, not as\nwe, perhaps, see them, on their camels, solemnly pacing the vast sandy\ndesert, whose sands reflected the glow of the silvery star.  They passed\non their way through scenes, and villages, which might be recognised by\nthe hearers, anxiously enquiring for the young Child.  Turnpikes, if\nunknown in Palestine, our readers will, perhaps, remember as one of the\ngreat nuisances of even a very short journey in Wales in Christmas's day.\n\n    \"The wise men came up to the gate,\u2014it was closed; they spoke to the\n    keeper, inquiring, 'Do you know anything of the Child?'\n\n    \"The gatekeeper came to the door, saying, in answer to the question,\n    'You have threepence to pay for each of the asses.'\n\n    \"They explained, 'We did not know there was anything to pay; here is\n    the money; but tell us, do you know anything of the young Child?'\n\n    \"No, the keeper did not even know what they meant.  For they know\n    nothing on the world's great highway of the Child sent for the\n    redemption of man.  But he said, 'You go on a little farther, and you\n    will come to a blacksmith's shop; he has all the news, he knows\n    everything, and he will be sure to be able to tell you all you want\n    to know.'\n\n    \"So they paced along the road, following the star, till they came to\n    the blacksmith's shop; and it was very full, and the blacksmith was\n    very busy, but they spoke out loudly to him, and said, 'Where is the\n    young Child?'\n\n    \"'Now,' said the blacksmith, 'it is of no use shouting that way; you\n    must wait, you see I am busy; your asses cannot be shod for a couple\n    of hours.'\n\n    \"'Oh, you mistake us,' said the wise men; 'we do not want our asses\n    shod, but we want you to tell us, you, who know everything\n    hereabouts, where shall we find the young Child.'\n\n    \"'I do not know,' said the blacksmith.  For the world, in its bustle\n    and trade, knows nothing, and cares nothing about the holy Child\n    Jesus.  'But look you,' he said, 'go on, and you will come to the\n    inn, the great public-house; everybody from the village goes there,\n    they know all the news there.'\n\n    \"And so, with heavy hearts, they still pursued their way till they\n    came to the inn; at the door, still resting on their asses, they\n    inquired if any one knew of the Child, the wonderful Child.\n\n    \"But the landlord said, 'Be quick!  Evan, John, where are you? bring\n    out the ale\u2014the porter\u2014for these gentlemen.'\n\n    \"'No,' they said, 'we are too anxious to refresh ourselves; but tell\n    us, hereabouts has been born the wonderful Child; He is the desire of\n    all the nations; look there, we have seen His Star, we want to\n    worship Him.  Do you know?'\n\n    \"'Not I,' said the landlord.  For pleasure knows nothing of Him\n    through whom the secrets of all hearts are revealed.  'Plenty of\n    children born hereabouts,' said the landlord; 'but I know nothing of\n    Him whom you seek.'  And he thought them a little mad, and was,\n    moreover, a little cross because they would not dismount and go into\n    the inn.  'However,' he said, 'there is an old Rabbi lives in a lane\n    hard by here; I think I have heard him say something about a Child\n    that should be born, whose name should be called Wonderful.  See,\n    there is the way, you will find the old man.'\n\n    \"So again they went on their way; and they stopped before the house\n    of the old Rabbi, and knocked, and the door was opened; and here they\n    left their asses by the gate, and entered in; and they found the old\n    Rabbi seated with his Hebrew books, and chronicles about him, and he\n    was strangely attired with mitre and vestment.  And now, they\n    thought, they would be sure to learn, and that their journey might be\n    at an end.  And they told him of the Star, and that the young Child\n    was born who should be King of the Jews, and they were come to\n    worship Him.\n\n    \"'Ah, yes,' he said, 'He is coming, and you shall see Him, but not\n    now.  You shall behold Him, but not nigh.  See, it is written here\u2014a\n    Star shall rise out of Jacob.  And when He comes it will be here He\n    will show Himself.  Go back, and when He comes I will send word and\n    let you know.'  For even religious people, and Churches, cannot\n    always guide seekers after God to Him whom to know is life eternal.\n\n    \"But they were not satisfied, and they said, 'No, no, we cannot\n    return; He is born, He is here!'\n\n    \"'There has been a great mistake made,' said the Rabbi; 'there have\n    been some who have said that He is born, but it is not so.'\n\n    \"'But who has said it?' they inquired.\n\n    \"And then he told them of another priestly man, who lived near to the\n    river hard by; and to him they went, and inquired for the young\n    Child.\n\n    \"'Yes, yes,' he said, when they pointed him to the Star, 'yes,\n    through the tender mercies of our God, the Dayspring from on high\n    hath visited us; to give light to them that sit in darkness and the\n    shadow of death; to guide our steps into the way of peace.'\n\n    \"And so he guided them to the manger, and the Star rested and stood\n    over the place where the young Child was, while they offered their\n    gifts of gold, and frankincense, and myrrh.\"\n\nSometimes the preacher, in another version which we have seen, appears to\nhave varied the last guide, and to have brought the wise men, by a\nsingular, and perfectly inadmissible anachronism, to the man in the\ncamel's hair by the river's brink, who said, \"Behold, the Lamb of God,\nwho taketh away the sins of the world!\"\n\nBut one of the most effective of these sustained allegories, was founded\non the text which speaks of the evil \"spirit walking through dry places,\nseeking rest, and finding none.\"  We believe we were first indebted for\nit, to the old dame who entertained us nearly forty years since in the\nCaerphilly Cottage.\n\n\n\nSATAN WALKING IN DRY PLACES.\n\n\nThe preacher appears to have been desirous of teaching the beautiful\ntruth, that a mind preoccupied, and inhabited by Divine thoughts, cannot\nentertain an evil visitor, but is compelled to betake himself to flight,\nby the strong expulsive power of Divine affections.  He commenced, by\ndescribing Satan as a vast and wicked, although invisible\nspirit,\u2014somehow, as Milton might have described him; and the preacher was\nnot unacquainted with the grand imagery of the \"Paradise Lost,\" in which\nthe poet describes the Evil One, when he tempts, with wandering feet, the\ndark, unbottomed, infinite abyss, and, through the palpable obscure,\nseeks to find out his uncouth way.  Christmas described him, as spreading\nhis airy flight on indefatigable wings, determined to insinuate himself,\nthrough the avenues of sense, to some poor soul, and lure it to\ndestruction.  And, with this end, flying through the air, and seeking for\na dwelling-place, he found himself moving over one of those wide Welsh\nmoors, the preacher so well knew, and had so often travelled; and his\nfiery, although invisible glance, espied a young lad, in the bloom of his\ndays, and the strength of his powers, sitting on the box of his cart,\ndriving on his way to the quarries for slate or lime.\n\n    `\"'There he is,' said Satan; 'his veins are full of blood, his bones\n    are full of marrow.  I will cast my sparks into his bosom, and set\n    all his passions on fire; I will lead him on, and he shall rob his\n    master, and lose his place, and find another, and rob again, and do\n    worse; and he shall go on from worse to worse, and then his soul\n    shall sink, never to rise again, into the lake of fire.'  But just\n    then, as he was about to dart a fiery temptation into the heart of\n    the youth, the evil one heard him sing,\n\n    \"'Guide me, O Thou great Jehovah,\n       Pilgrim through this barren land;\n    I am weak, but Thou art mighty,\n       Hold me by Thy powerful hand;\n          Strong deliverer,\n    Be Thou still my Strength and Shield.'\n\n    'Oh, but this is a dry place,' said the fiery dragon as he fled away.\n\n    \"But I saw him pass on,\" said the preacher, \"hovering, like a hawk or\n    a vulture, in the air, and casting about for a suitable place where\n    he might nestle his black wings; when, at the edge of the moor, he\n    came to a lovely valley; the hills rose round it, it was a beautiful,\n    still, meadow-like spot, watered by a lovely stream; and there,\n    beneath the eaves of a little cottage, he saw a girl, some eighteen\n    years of age, a flower among the flowers: she was knitting, or sewing\n    at the cottage door.  Said Satan, 'She will do for me; I will whisper\n    the evil thought in her heart, and she shall turn it over, and over\n    again, until she learns to love it; and then the evil thought shall\n    be an evil deed; and then she shall be obliged to leave her village,\n    and go to the great town, and she shall live a life of evil, all\n    astray from the paths of my Almighty Enemy.  Oh, I will make her\n    mine, and then, by-and-bye, I will cast her over the precipices, and\n    she shall sink, sink into the furnace of divine wrath.'  And so he\n    hastened to approach, and dart into the mind of the maiden; but while\n    he was approaching, all the hills and crags seemed to break out into\n    singing, as her sweet voice rose, high and clear, chanting out the\n    words,\n\n    \"'Jesus, lover of my soul,\n       Let me to Thy bosom fly,\n    While the nearer waters roll,\n       While the tempest still is high.\n    Other refuge have I none,\n       Hangs my helpless soul on Thee;\n    Leave, ah, leave me not alone,\n       Still support, and comfort me.'\n\n    'This is a very dry place, too,' said the dragon, as he fled away.\n\n    \"And so he passed from the valley among the hills, but with hot rage.\n    'I will have a place to dwell in!' he said; 'I will somehow leap over\n    the fences, and the hedges, of the purpose, and covenant, and grace\n    of God.  I do not seem to have succeeded with the young, I will try\n    the old;' for passing down the village street, he saw an old woman;\n    she, too, was sitting at the door of her cot, and spinning on her\n    little wheel.  'Ah!' said Satan, 'it will be good to lay hold of her\n    grey hairs, and make her taste of the lake that burneth with fire and\n    brimstone.'  And he descended on the eaves of the cot; but as he\n    approached near, he heard the trembling, quavering voice of the aged\n    woman murmuring to herself lowly, 'For the mountains shall depart,\n    and the hills be removed, but My kindness shall not depart from thee,\n    neither shall the covenant of My peace be removed, saith the Lord,\n    that hath mercy on thee.'  And the words hurt the evil one, as well\n    as disappointed him; they wounded him as he fled away, saying,\n    'Another dry place!'\n\n    \"Ah, poor Devil!\" exclaimed the preacher, \"and he usually so very\n    successful! but he was quite unsuccessful that day.  And, now, it was\n    night, and he was scudding about, like a bird of prey, upon his black\n    wings, and pouring forth his screams of rage.  But he passed through\n    another little Welsh village, the white cottages gleaming out in the\n    white moonlight on the sloping hillside.  And there was a cottage,\n    and in the upper room there was a faint light trembling, and 'Oh,'\n    said the Devil to himself, 'Devil, thou hast been a very foolish\n    Devil to-day, and there, in that room, where the lamplight is, old\n    Williams is slowly, surely wasting away.  Over eighty, or I am\n    mistaken; not much mind left; and he has borne the burden and heat of\n    the day, as they call it.  Thanks to me, he has had a hard time of\n    it; he has had very few mercies to be thankful for; he has not found\n    serving God, I think, a very profitable business.  Come, cheer up,\n    Devil, it will be a grand thing if thou canst get him to doubt a bit,\n    and then to despair a bit, and then to curse God, and die; that will\n    make up for this day's losses.'\n\n    \"Then he entered the room; there was the old man lying on the poor\n    bed, and his long, thin, wasted hands and fingers lying on the\n    coverlid; his eyes closed, the long silvery hair falling over the\n    pillow.  Now, Satan, make haste, or it will be too late; the hour is\n    coming, there is even a stir in every room in the house: they seem to\n    know that the old man is passing.  But as Satan himself moved before\n    the bed, to dart into the mind of the old man, the patriarch rose in\n    bed, stretched forth his hands, and pinned his enemy to the wall, as\n    he exclaimed, 'Though I walk through the valley of the shadow of\n    death, I will fear no evil: for Thou art with me, Thy rod and Thy\n    staff they comfort me; Thou preparest a table before me, in _the\n    presence of mine enemy_; Thou anointest my head with oil, my cup\n    runneth over; goodness and mercy, all the days of my life, dwell in\n    the house of my God for ever.'  Oh, _that_ was a fearfully dry place!\n    The old man sank back, it was all over; those words beat Satan down\n    to the bottom of his own bottomless pit, glad to escape from such\n    confusion and shame, and exclaiming, 'I will return to the place from\n    whence I came, for this is too dry for me.'\"\n\nThis will, no doubt, be thought, by many, to be strange preaching; many\nwould even affect to despise it,\u2014perhaps would even regard it as a high\ncompliment were we to say, they would feel exceedingly puzzled even if,\nby way of a change, they were called upon to use it.  It appears,\nhowever, to have been a style exceedingly fascinating to the Welsh mind\nof that day; it told, it stirred up suggestions, awakened thoughts, and\nreclaimed and converted character; and we need not, therefore, stay to\nattempt any vindication of it.\n\nWe have inserted these very characteristic illustrations here, because\nthey appear to have belonged to the Anglesea period.  Such, then, was the\nteaching, the preaching, the truth, which, while it was his own truth,\nand sustained his own mind, gave to him such power, at once, amongst the\nChurches to which he immediately administered, and made him the object of\nsuch attraction, when visiting distant neighbourhoods.\n\nIt might have been thought\u2014it has usually been the case, in the instances\nof other men\u2014that such excursions as those we have described, would have\ninterfered with the great success of his work in the ministry as a\npreacher, and with his efficiency as a pastor.  That they did not,\nsubstantially, is clear from many evidences.  There can be no doubt that\nhis sermons were no off-hand productions; there was a careful, rigid, and\npatiently conscientious weighing of their material.  All those which we\npossess, abundantly show this; and he entered with all his heart, and\nmind, and strength into the work of preaching; but he never had an easy\nsphere; and yet, would his sermons have been greater had he been placed\nwhere the circle of his labour would have been narrower, and the means of\nhis support more ready, and sufficient, and ample?  Most likely not; but\nhe weighed the entire work of the ministry in a manner which seems to us,\nsometimes, more like the sound thoughtfulness, and consideration of the\ntheological Principal of a college, than a popular, or itinerant\npreacher.  As an illustration of this, we may insert the following, very\nlengthy, but admirable letter to a young minister, written, we believe,\nsome time nearer the close of his career than that we have just\ndepicted:\u2014\n\n    \"DEAR BROTHER,\u20141.  Consider, in the first place, the great\n    importance, to a preacher, of a blameless life.  You must, like\n    Timothy, 'flee youthful lusts,' as you would escape from beasts of\n    prey; for there are kinds of beasts, living in the wilderness of\n    man's corruption, that will charm, by means of their beauteous\n    colours, those that walk among their haunts; there is no safety but\n    by keeping from them, and adhering to such as live by faith, and\n    watch, and pray.  It will be well for you, while you travel through\n    the coppice of youth, to keep from all appearance of evil.  May you\n    have grace to pass through the coppice of forbidden trees, without\n    cutting your name into the bark of one of them, or you may be\n    upbraided, at critical times, by those who may wish to prove that you\n    are not better than themselves; even the _iota_, inserted by your\n    hand, may be produced after many years.\n\n    \"2.  I remember the words of Luther, that _reading_, _prayer_, and\n    _temptation_ are necessary to strengthen, and to purify the talents\n    of a minister.  Read, to extend your general knowledge, especially as\n    to the plan of redemption, according to the Scriptures, in all its\n    parts, from the election to the glorification; that you may, like a\n    spiritual watchmaker, know all the relative cog-wheels, and be able\n    to open them in the pulpit, and to connect them all by faith, hope,\n    and charity, that they may occupy their own places, and exhibit their\n    true results on the dial-plate; thus proving yourself a workman that\n    needeth not to be ashamed, rightly dividing the word of truth.  Be\n    not like that thrasher, who presumptuously took his watch to pieces\n    in the barn, and could not put it together again, but was obliged to\n    carry it home in his handkerchief.  The messengers of God, described\n    in the book of Revelations, are full of eyes behind, and before.  You\n    must use prayer to fetch strength out of Christ, like the homer to\n    carry home the manna in, or the water-pot of the woman of Samaria.\n    Without the prayer of faith, the preacher will have 'nothing to draw\n    with,' from the well that is deep,\u2014even _the deep things of God_.\n    Temptation is requisite, to prove the nature of the metal of the\n    preacher's character, and doctrine,\u2014'approved of God.'  The piece of\n    gold, in every true minister's ministry, must be tried in some\n    furnace, prepared by Divine Providence.  He must, therefore, do the\n    work of an evangelist, fulfil his ministry, endure hardness, and\n    affliction, and thus prove himself a good soldier of Jesus Christ.\n\n    \"3.  Avail yourself, in the morning of your days, of every\n    opportunity to acquire knowledge useful for the ministry.  Let it be\n    your constant aim, to turn every stream and rivulet of knowledge in\n    the right direction, to facilitate the work of the ministry, for the\n    good of souls, and the glory of God; as the bee, in all her\n    excursions amongst the flowers of the gardens, and the hedges,\n    gathers honey to enrich the hive, as the common treasury of the\n    industrious race.  Always have a book to read, instead of indulging\n    in vain conversations.  Strive to learn English, as you cannot have\n    academical training.  Learn your own mother-tongue well.  Learn to\n    write a good hand by frequent practice.  Avoid vain conversation,\n    instead of growth in knowledge.  Remember this, that you cannot\n    commit some loved sin in private, and perform the work of the\n    ministry, in public, with facility and acceptance.  For a preacher to\n    fall into sin, be it a secret one, and to live in it, is as fatal,\n    ultimately, as the cutting of Samson's hair.  Be strong in the grace\n    that is in Christ Jesus against all corruption.\n\n    \"4.  With regard to the composition of your sermons: first, let the\n    matter be evangelical.  The doctrine of the Gospel is a mould from\n    heaven, and not changed.  It puts its own impress and shape on the\n    professor that is melted into it, so that his justification,\n    sanctification, and all his salvation, flow from the merits of\n    Christ; and all through God's grace, and not of ourselves.  The\n    gospel, as a glass, should be kept clean and clear in the pulpit,\n    that the hearers may see the glory of Christ, and be changed to the\n    same image.  Every duty is to be urged by evangelical motives.  'Let\n    us have grace,' etc.\n\n    \"Hereby we can serve God in all the duties of the kingdom of heaven.\n    The whole is summed up in living by faith, which worketh by love, to\n    him that died for us, and rose again for our justification.\n    Secondly, let your divisions be natural to the text.  Take care that\n    your interpretation accord with the contexts.  Two or three general\n    heads; avoid many.  Four or five remarks you may make on each head;\n    see that they are fairly in the truth of the text.  Thirdly, I am not\n    inclined to make inferences, or applications, from the whole.  When\n    the preacher has expended his strength, or ingenuity, in endeavouring\n    to impress, and apply the truth to the minds of his hearers,\n    application seems to me to be doing again what has been effected\n    already.  The blacksmith does not put the horse-shoe in the fire,\n    after he has nailed it to the hoof; and the cook does not spread the\n    cloth again, when dinner is over.  Fourthly, beware of long sermons,\n    as well as long prayers.  When there is but one preacher, he should\n    not preach for more than an hour; when there are two, both should not\n    be more than an hour and a half, that the worship may close within\n    two hours; whenever this time is passed, coolness and fatigue ensue.\n    To put three ministers to preach (in one meeting) is a modern\n    corruption, and likely to make some progress in Wales; while the\n    English, generally, have but one sermon in one service.  They excel\n    us herein; for we do not read that, on the day of Pentecost, Peter,\n    James, and John, preached after each other; but Peter, '_one_ of the\n    twelve,' delivered that successful sermon.  When we lose sight of the\n    Scriptures, and common sense, we are driven to extremes, though it be\n    with the kindly purpose of respecting strange ministers, by putting\n    them to preach.\n\n    \"5.  Attend, also, my young brother, to your outward appearance in\n    the pulpit.  Beware of a proud, haughty appearance, with wandering\n    eyes, and unfeeling countenance, so that the people utterly fail to\n    see the man of God in you.  We must, in order hereunto, have\n    something like unto Moses, when he had been on the mount with God,\n    that will indicate seriousness, love to souls, a spirit of prayer,\n    zeal for Christ, and longing for the salvation of men; like unto\n    those who have felt the fear of perdition ourselves, and the infinite\n    value of salvation by God's grace; and that we wrestle with God in\n    order to be useful to souls.  These things must be imprinted on our\n    appearance and deportment, having transformed us, in some measure, to\n    a heavenly form and habit.  Our outward conversation should be\n    consistent herewith, or men will despise us as hypocrites, without\n    the fear of God.\n\n    \"6.  Avoid, my dear brother, all foolish bodily gestures.\n\n    \"7.  We now come to the part of the subject upon which you are most\n    anxious to have my thoughts: that refers _to the delivery of your\n    sermons_.  It is difficult to put general rules of rhetoric into\n    execution.  After reading all that has been said by Blair, Williams,\n    Fuller, and the Archbishop of Cambray (Fenelon), who have spoken at\n    length of Cicero and Demosthenes, it is easy, by endeavouring to\n    follow them, to lose the spirit of the work, and thus, by seeking the\n    form, to forfeit the life.  Preach the gospel of the grace of God\n    intelligibly, affectionately, and without shame\u2014all the contents of\n    the great box, from predestination to glorification.  It was the\n    closing, and concealing, of this box that occasioned the opening of\n    the venomous Mohammedan box, as well as that of Popery, together with\n    all the vain legality that is to be found among Protestants,\n    established and dissenting.  It may be said, that they seek\n    justification; but it is by the deeds of the law.  The locking up,\n    and the losing, of the doctrine of grace, through the merits of\n    Christ, utterly destroyed the Jewish Church; for it was in the chest,\n    which they locked up by their false interpolations of Scripture, that\n    the 'things which belong to their peace' were contained; 'but now,'\n    says the Redeemer, 'they are concealed from their eyes;' shut up\n    under unbelief.  'The things that pertain to their peace' belong also\n    to our peace, as Gentiles.  The Deity of Christ, etc.; Redemption,\n    etc.  Excuse this digression, for the river of God's throne moved me\n    along.\n\n    \"We were upon the best mode of delivering sermons for edification.\n    It is not easy to reduce the rules of prudence into practice.  I have\n    seen some men, of the highest powers, who understood Greek better\n    than their mother-tongue, attempting to preach according to rule, and\n    to them the pulpit was like unto Gilboa; they neither affected\n    themselves, nor their hearers.  The difficulty was, the bringing of\n    their regulations into natural practice.  I saw one of those men, the\n    most eminent for learning and genius, who found the right way, under\n    the influence of a mighty fervency that descended upon him in the\n    pulpit, so that his voice became utterly different from what it used\n    to be, and his tongue at liberty, as though something was cut that\n    had hitherto restrained his tongue, and affections, from natural\n    exercise.\n\n    \"Here you have the sum, and substance, and mystery of all rules:\u20141.\n    Let the preacher influence himself; let him reach his own heart, if\n    he would reach the hearts of others; if he would have others feel, he\n    must feel himself.  Dry shouting (or vociferation) will not do this.\n    The shout of a man who does not himself feel the effect of what he\n    says, hardens, instead of softening; locks, instead of opening the\n    heart.  2. The elevation, and fire of the voice must accord with the\n    fervency of the matter in the heart.  A person said to me once, 'Mr.\n    Evans, you have not studied Dr. Blair's Rhetoric.'  That man, with\n    his rules, was always as dry as Gilboa.  'Why do you say so,' replied\n    I, 'when you just now saw hundreds weeping under the sermon?  That\n    could not be, had I not first of all been influenced myself, which,\n    you know, is the substance, and mystery, of all rules for speaking.'\n    Wherever there is effect, there is life; and rules, without life,\n    have no power.  Now, brother, follow the natural course of affection,\n    and voice.  Raise not the voice while the heart is dry; but let the\n    heart and affections shout first; let it commence within.  Take this\n    comparison:\u2014Go to the blacksmith's shop; he first puts the piece of\n    iron in the fire, and there is no sound of striking the anvil; he\n    collects together the coals for heat; then he tells the boy, 'Blow!'\n    while he masterfully manages the shovel, adjusting the coals, and\n    asking sundry questions.  He calmly looks at the fire heating the\n    iron, and does not yet take hold of the hammer, nor order his\n    assistants to use the sledge; but at length, seeing that the iron has\n    attained the proper malleability, he takes it out, covered with\n    sparkling fire, puts it on the anvil, handles the hammer, and orders\n    his workman to take the larger one, and fashions it according to his\n    pleasure; and so on, all day long.  Here, observe, he does not beat\n    the iron in order to make it hot, for without first heating it, the\n    beating process is in vain.  Equally vain is the hammer of\n    vociferation, unless the matter is brought home with warmth into our\n    hearts.  We have often sought to produce effect, and to influence our\n    hearers, much as though the smith merely put the iron in fire, and\n    barely warmed it; it is contrary to the nature of things to use the\n    hammer while the material is not duly tempered.  Thus I have\n    frequently, brother, found myself in preaching.  You have, above, the\n    mystery of all effective speaking, in Parliament, at the bar, and in\n    the pulpit; remembering the difference in the subjects, and the\n    sources of heat.  In the pulpit, we speak of the deep things of God;\n    and we are to pray for, and to expect warmth from the Divine Spirit.\n    You complain that you cannot get your voice into a manageable key,\n    and yet to speak with liveliness and power.  Many, with a bad voice,\n    well-governed, have become powerful speakers; while others, with a\n    good voice, have, in consequence of not mastering a natural key, and\n    not being able to move themselves, been most ineffective speakers.  I\n    would direct you to fix your voice at its natural pitch, which you\n    may easily do; you may then, with facility, raise and lower it\n    according to the subject in hand.  If you commence in too high a key,\n    you cannot keep it up long.  First, you cannot modulate it as the\n    occasion may require; and you fall into an unpliable, tedious\n    monotony, and all natural cadence, and emphasis is lost.  Without\n    attuning the voice into the natural key, effective oratory is\n    impossible.  Secondly, remember, not to speak in your throat, or\n    nostrils.  If the former, you must soon become hoarse, and harsh\n    loudness follows; the glory and vivacity are then departed, and\n    instead of facility and cheerfulness, you have the roarings of\n    death\u2014the breath failing, with forced screams, and harsh whisperings.\n    Thirdly, raise your voice to the roof of your mouth; do not close\n    your teeth against it, neither imprison it in the nostrils, but open\n    your mouth naturally, and keep your voice within your lips, where it\n    will find room enough to play its high, and its low intonations, to\n    discourse its flats, and sharps, to utter its joys, and sorrows.\n    When you thus have your voice under control, instead of you being\n    under its control, dragging you about in all disorder, you will find\n    it your servant, running upon your errands, up and down, all through\n    the camp, alternating in energy, and pliability, to the end of the\n    sermon; and not becoming cold and weak, scarcely bearing you through,\n    like Bucephalus, Alexander the Great's horse, which, mortally\n    wounded, just brought his master out of the battle, and then expired.\n    Fourthly, remember, not to press too much upon your breath, when you\n    have attained the natural use of it, by using very long sentences,\n    without pausing at proper places, which (pauses) will add to the\n    effect, as well as preserve the voice; so that you will be, like the\n    smith, ready to strike the duly-tempered metal, prepared to give the\n    suitable emphasis at the end of the paragraph.  Let the matter raise\n    the voice, do not attempt by the voice to elevate the subject.\n    Fifthly, use words easily understood, that the people's affections\n    may not cool, while the mind is sent to a dictionary, to understand\n    your terms.  The great work, the exploit of a minister, is to win the\n    heart to believe in Christ, and to love Him.  Sixthly, bear in mind,\n    also, the necessity of keeping the voice free, without (affected)\n    restraint; give every syllable, and every letter, its full and proper\n    sound.  (It is one of the peculiarities and excellences of the Welsh\n    language, and proves its Eastern origin.)  No letter has to complain\n    that it is (condemned to be) mute, and neglected, and has no\n    utterance.  In English, many letters have this complaint; but in\n    Welsh, every letter, even as the knights at the round table of King\n    Arthur, has, without preference, its own appropriate and complete\n    sound.  Seventhly, remember, also, to enunciate clearly the last\n    syllable in every Welsh word; that will cause your most distant\n    hearer to understand you; while, without this, much of what you say\n    must be inevitably lost.  Eighthly, in order to all this, carefully\n    attend to the manner of the best, and ablest preachers, and imitate,\n    not their weaknesses, but their excellences.  You will observe, that\n    some heavenly ornament, and power from on high, are visible in many\n    ministers when under the Divine irradiation, which you cannot\n    approach to by merely imitating their artistic excellence, without\n    resembling them in the spiritual taste, fervency, and zeal which\n    Christ and his Spirit 'work in them.'  This will cause, not only your\n    being like unto them in gracefulness of action, and propriety of\n    elocution, but will also induce prayer for the anointing from the\n    Holy One, which worketh mightily in the inward man.  This is the\n    mystery of all effective preaching.  We must be endowed with power\n    from on high: here is the grand inward secret.  Without this, we\n    (often) perceive that it is impossible, with all academic advantages,\n    to make good preachers of young men from any college, in the Church\n    of England, or among the dissenters, in the English or the Welsh\n    language.  A young preacher must have the mystery of being\n    'constrained' by 'the love of Christ'; 'the gift of God' must be\n    kindled in him; and He alone, by the Spirit, can sustain that gift by\n    the Holy Spirit.  'Who is sufficient for these things?'  May the Lord\n    give you, brother, a good understanding in all things; and preserve\n    in you the heavenly gift by the Holy Ghost! may it be rekindled where\n    it is, and contributed where it is not!  Without it, we can do\n    nothing for the glory of God, or the good of souls.\n\n                                                          \"Affectionately,\n                                                        \"CHRISTMAS EVANS.\"\n\nSometimes Mr. Evans occupied such slight leisure as he could command, by\na contribution to the _Seren Gomer_, an extensively-circulating magazine\nof the Principality.  Several of these papers are interesting; we select\none, illustrating the bent of the writer's mind; it was published January\n1821,\u2014\"An inquiry into the meaning of the singular language of the\nApostle, his wish\n\n\n\n\"TO BE ACCURSED FROM CHRIST.\n\n\n    \"'For I could wish that I were accursed (anathema) from Christ for my\n    brethren,' etc. (Rom. ix. 3).  Many things, most incredible to me,\n    have been said in exposition of this passage; and principally, I\n    think, from not observing that the word 'anathema' is used in two\n    senses,\u2014the one good, and the other bad.  Barclay analyses into four\n    acceptations; and, according to the first, it signifies that which is\n    devoted, or set apart, to God, in a good sense.  According to\n    Parkhurst, it signifies, in Luke xxi. 5, a consecrated gift, set\n    apart for the temple of God, and to His service alone.  The word\n    translated gifts is _anathemasi_.  In the second book of Maccabees,\n    ix. 8, the word denotes a consecrated gift.  The word in the LXX.,\n    according to Parkhurst, is synonymous with the Hebrew word CHEREM,\n    and signifies, generally, that which is entirely separated from its\n    former condition, and use.  If so, why should we not understand Paul,\n    in the text, as expressing his ardent desire that he should be\n    separated, _a devoted thing_, for the conversion of his brethren\n    according to the flesh?  Having gone thus far in explanation, we\n    offer the following interpretation: 'For I could wish that I were\n    _anathema_, or a gift, in my labours as an apostle, and a preacher of\n    the Gospel, from Christ, for the spiritual benefit of my brethren\n    according to the flesh, principally, instead of being an apostle to\n    the Gentiles, as I am appointed; theirs is the adoption, etc.; and I\n    could also wish that I, also, as an apostle, were an especial gift of\n    Christ for their distinctive service.'  If this be correct, there is\n    no necessity for changing the tense of the verb from the present to\n    the perfect, and reading, 'I could wish,' as 'I have wished;' while\n    it saves us from putting in the Apostle's mouth a wish entirely\n    opposed to the 'new creation,' to the plan of Divine grace, and to\n    the glory of God; for it is certain that it is quite in opposition to\n    all this, for a man to desire to live in sin, and to be accursed for\n    ever,\u2014and that cannot for a moment be predicated of the Apostle of\n    the Gentiles.  I humbly ask some learned correspondent, whether there\n    is anything in the original text with which this exposition will not\n    harmonize.\n\n                                                        \"CHRISTMAS EVANS.\"\n\nThis letter led to some unsympathetic criticism, and reply.  Christmas\nEvans wrote a vindication of his former views, which may be not\nuninteresting to our readers, as illustrating a phase of his intellectual\ncharacter.  It appeared in the _Seren Gomer_ for 1822:\u2014\n\n    \"MR. GOMER,\u2014If you please, publish the following, in defence of my\n    former letter on Romans ix. 3, and in reply to your correspondent,\n    _Pen Tafar_.\n\n    \"It is admitted, on all hands, that the words in the question express\n    the highest degree of love to the Jews.  Let us, now, put the\n    different expositions before the reader, and then let him judge which\n    of them contains the greatest harmony and fitness; _i.e._, first, to\n    express love to the Jews; second, the best adapted to bring about\n    their salvation; third, the most consistent with supreme love to\n    Christ; and fourth, within the confines of sinlessness.\n\n    \"1.  Many learned men set forth the Apostle as having formed this\n    desire when he was an enemy to Christ.  This they maintain by tracing\n    the word _anathema_ throughout the Greek Scriptures, and the Hebrew\n    word _cherem_, of which it is the synonym.  _Anathema_, they say,\n    always signifies 'without an exception,' a separation, or devotement\n    of a beast, a city, or something else, to irredeemable destruction\n    (Lev. xxvii. 29).  The devoted thing was not to be redeemed, but\n    certainly to be put to death (Gal. i. 9).  '_Let him be accursed_,'\n    says Paul of the angel that would preach another gospel.  'If any man\n    love not the Lord Jesus Christ, let him be _anathema maranatha_,'\n    'accursed when the Lord cometh.'  But who _can_ believe that this is\n    the meaning of the word in the passage before us?  I say, with Dr.\n    Gill, 'This never can be the signification.'  What probability is\n    there that Paul would swear, calling Jesus Christ to witness, to his\n    ancient enmity against Him?  This was notorious enough throughout the\n    whole country.  No asseveration was necessary to prove _Paul's\n    persecuting spirit_.\n\n    \"Again, how could that which he formerly had been, prove, he now\n    having denied himself, his old persecuting spirit, and, being deeply\n    ashamed on the account, prove his present love to the Jews?  How did\n    his former love to Satan prove his present love to the Jews?\n\n    \"2.  Others say that it is Paul's wish as a Christian, whatever\n    _anathema_ means.  I believe it is his desire as a Christian;\n    otherwise I see not how it could be an instance of his love to his\n    brethren according to the flesh.  Several authors maintain that Paul\n    was willing, _for the sake of saving his nation_, _to part with his\n    interest in Christ_, _and to perish for ever_.  Peter Williams and\n    Matthew Henry give this interpretation.  But, seriously, how can a\n    person persuade himself to believe this?  Would not the Apostle, in\n    this case, love his nation more than Christ, and be accordingly\n    unworthy of Christ?  This is opposed to a principle of our nature,\n    which never can desire its own destruction; to the principle of\n    grace, which loves Christ above all things on earth, and in heaven.\n    Such a desire would make Paul a devil.\n\n    \"3.  Others suppose that Paul here speaks inconsiderately, in a kind\n    of ecstasy, carried away by a stream of affection to his people.  Who\n    can believe this without giving up Paul's inspiration, even when he\n    solemnly appeals to Christ?\n\n    \"4.  Another notion is, that the Apostle was willing, and desirous to\n    be excommunicated from the Church of Christ upon earth, and to be\n    deprived of its ordinances.  How can this, again, be considered as\n    consistent with love to Christ, and His Church?  What tendency could\n    his leaving the Church have to induce the Jews to enter it?  This is\n    contrary to the whole course of the Divine command, and promises: God\n    will give His people an everlasting home, and place in His house.\n\n    \"5.  Some say, it is an _hyperbole_.  To confirm this, Exod. xxxii.\n    32 is quoted as a case in point: '_Blot me_, _I pray thee_, _out of\n    Thy book_, _which Thou hast written_.'  This is not the book of\n    eternal life, but the book of the dispensation, in which Moses was\n    leader, and mediator.  '_I would_,' he says, '_give up my office_.'\n    God rejected the request: 'Lead the people unto the place of which I\n    have spoken to thee.'  It was not for Israel, nor a condition of\n    forgiveness to them, but for himself, that Moses said, 'Blot my name\n    out of Thy book.'  All this gives but little assistance to understand\n    the Apostle.  The two spiritual men do not stand on the same ground.\n    Moses seeks the obliteration of his name, unless Israel was pardoned.\n    Paul seeks a work, and an office, in order to the forgiveness of his\n    nation.\n\n    \"6.  Further, it is supposed to be proper to modify\u2014_to soften_\u2014the\n    meaning of the word _anathema_, as signifying, sometimes, anything\n    devoted to God, and that never could, afterwards, be appropriated to\n    any other service; and here, to understand it in that softened sense,\n    signifying that Paul was willing for the Redeemer to make him a\n    devoted thing\u2014a martyr for the truth, for the good of the Jewish\n    nation.  This is substantially the opinion of Thomas Charles, and Dr.\n    Gill.  Christmas Evans's theory is erected on this ground\u2014the\n    modified sense of the word; thus, 'I could wish myself entirely set\n    apart, by Christ, to the service of my people, for their spiritual\n    good; I should have been glad, had I my choice, to have been an\n    Apostle, separated to them alone, and not to the Gentiles, with my\n    dwelling, and labours, amongst them, and to die a martyr for the\n    truth, even the most horrible death that could be devised, if Christ\n    had appointed me hereto.'  If 'P. T.' says this is a new\n    interpretation of Christmas Evans's, the answer is, No, but a\n    legitimate extension of a former one; for he did not intend, nor did\n    his words import, the separation of martyrdom, or the most\n    anathematised sufferings, from Paul for his kinsmen according to the\n    flesh.\n\n    \"7.  Is it not plain, and does not 'P. T.' see, that this view is\n    superior to the former five, and that it takes in, and is an\n    improving addition to the latter of the five, as to its fitness to\n    express the Apostle's great love to his people, without destroying\n    his love to Christ, as well as to bring about the salvation of the\n    Jews by proper means?  How could the death of the Apostle contribute\n    to the conversion of the Jews, unless he died _as an apostate of the\n    circumcision_?\"\n\nIt appears to have been towards the close of the Anglesea period, that he\nwas thrown into a panic of fear, by a threat of a legal prosecution, on\naccount of some chapel debts, for which, of course, he was regarded as\nresponsible.  \"They talk,\" he said, \"of casting me into a court of law,\nwhere I have never been, and I hope I shall never go; but I will cast\nthem, first, into the court of Jesus Christ.\"  We have seen that he was\nin the habit of putting on paper his prayers, and communions with God.\nIt was a time of severe trial to him.  He says, \"I knew there was no\nground of action, but, still, I was much disturbed, being, at the time,\nsixty years of age, and having, very recently, buried my wife.\"  He\ncontinues, \"I received the letter at a monthly meeting, at one of the\ncontests with spiritual wickedness in high places.  On my return home, I\nhad fellowship with God, during the whole journey of ten miles, and,\narriving at my own house, I went upstairs to my own chamber, and poured\nforth my heart before the Redeemer, who has in His hands all authority,\nand power.\"  And the following seem to be the pathetic words in which he\nindulged:\u2014\n\n    \"O blessed Lord! in Thy merit I confide, and trust to be heard.\n    Lord, some of my brethren have run wild; and forgetting their duty,\n    and obligations to their father in the Gospel, they threaten me with\n    the law of the land.  Weaken, I beseech Thee, their designs in this,\n    as Thou didst wither the arm of Jeroboam; and soften them, as Thou\n    didst soften the mind of Esau, and disarmed him of his warlike temper\n    against Thy servant Jacob, after the wrestling at Penuel.  So disarm\n    them, for I do not know the length of Satan's chain in this case, and\n    in this unbrotherly attack.  But Thou canst shorten the chain as\n    short as it may please Thee.  Lord, I anticipate them in point of\n    law.  They think of casting Thine unworthy servant into the little\n    courts here below; but I cast my cause into the High Court, in which\n    Thou, gracious Jesus, art the High Chancellor.  Receive Thou the\n    cause of Thine unworthy servant, and send him a writ, or a notice,\n    immediately\u2014sending into their conscience, and summoning them to\n    consider what they are doing.  Oh, frighten them with a summons from\n    Thy court, until they come, and bow in contrition at Thy feet; and\n    take from their hands every revengeful weapon, and make them deliver\n    up every gun of scandal, and every sword of bitter words, and every\n    spear of slanderous expressions, and surrender them all at Thy cross.\n    Forgive them all their faults, and clothe them with white robes, and\n    give them oil for their heads, and the organ, and the harp of ten\n    strings, to sing, for the trampling of Satan under our feet by the\n    God of peace.\n\n    \"I went up once,\" he says, \"and was about ten minutes in prayer; I\n    felt some confidence that Jesus heard.  I went up again with a tender\n    heart; I could not refrain from weeping with the joy of hope that the\n    Lord was drawing near to me.  After the seventh struggle I came down,\n    fully believing that the Redeemer had taken my cause into His hands,\n    and that He would arrange, and manage for me.  My countenance was\n    cheerful, as I came down the last time, like Naaman, having washed\n    himself seven times in the Jordan; or Bunyan's Pilgrim, having cast\n    his burden at the foot of the cross, into the grave of Jesus.  I well\n    remember the place\u2014the little house adjoining the meeting-house, at\n    Cildwrn, where I then resided\u2014in which this struggle took place; I\n    can call it Penuel.  No weapon intended against me prospered, and I\n    had peace, at once, to my mind, and in my (temporal) condition.  I\n    have frequently prayed for those who would injure me, that they might\n    be blessed, even as I have been blessed.  I know not what would have\n    become of me, had it not been for these furnaces in which I have been\n    tried, and in which the spirit of prayer has been excited, and\n    exercised in me.\"\n\nIt is scarcely necessary to add, that the threat was never executed, nor\ndid poor Christmas, apparently, hear anything further of the matter; but\nwe have seen how great was the trouble, and agitation it caused him,\nwhile the fear was upon him.  It is very affecting to find that this\ngreat, this saintly, and earnest minister, had upon his heart, and mind,\nthe burden of all the chapel-debts connected with his denomination in\nAnglesea, while he was minister there.\n\nIt might have been thought that the ministerial course of Christmas Evans\nwould close in Anglesea, where he had laboured so long, and so\neffectually.  He was, now, about sixty years of age, but there was little\nlight just now, in the evening-time of his life; indeed, clouds of\ntrouble were thickening around him.  It often seems that trouble, in the\nministerial life, comes exactly at that moment when the life is least\nable to stand, with strength, against it; and, certainly, in the life of\nChristmas Evans, sorrows gathered, and multiplied at the close.\n\nChief among these must be mentioned, beyond any doubt, the death of the\nbeloved companion of all the Anglesea life, his good wife, Catherine; she\nleft him in 1823.  She was eminently, and admirably fitted to be the wife\nof such a man as Christmas.  Somewhat younger than her husband, she\nsupplied many attributes of character, to him most helpful; she was not\nan enthusiast, but she was a Christian, with real, deep, and devout\nconvictions.  We have no lengthy accounts of her; but little side-lights,\na kind of casemented window, reveal a character at once affectionate,\nbeautiful, and strong.\n\nWe have seen that their home was the region of self-denial, and her\nhusband long remembered, and used to tell, how \"if there happened to be\non our table one thing better than the other, she would, modestly, but\ncheerfully and earnestly, resist all importunity to partake of it until\nshe ascertained that there was enough for both.\"  What a little candle\nsuch a sentence as this is, but what a light it sheds over the whole\nroom!  She did not pretend to be her husband; he filled his larger\nsphere, and she, in all her manifold, gentle ways, sought to give him\nrest.  Surely she adds another name to the long catalogue of good wives.\nShe reminds us of Lavater's wife, and some little incidents in that\nCildwrn cottage call up memories from the manse of St. Peter's Church,\nand the shadows of the old Lindenhof of Zurich, where probably life did\nnot put on a gayer apparel, or present more lavish and luxurious\npossibilities, than in the poor parsonage of Anglesea.\n\nIt is incredible, almost, to read what the good Catherine did, poor\u2014to\nour thinking, miserable\u2014as was the income of her husband.  Her hand was\nmost generous; how she did it, what committee of ways and means she\ncalled together, in her thoughtful mind, we do not know,\u2014only, that she,\nconstantly, found some food to give to poor children, and needy people;\nunblessed by children of her own, she employed her fingers in making\nclothes for the poor members, and families, of the Church.  There was\nalways help for the poor hungry labourer passing her cottage; the house\nwas always open for the itinerant minister travelling on his way to some\n\"publication,\" and she was always ready to minister to his necessities\nwith her own kind hands.  Her husband often thought that the glance she\ngave upon a text shed light upon it.  She never had robust health, but\nshe accompanied her husband on several of his longer journeys through the\ngreater part of Wales,\u2014ah, and some of them in the winter, through storms\nof rain, and snow, and hail, along dangerous roads too, across difficult\nferries; and she was uniformly cheerful!  What an invaluable creature,\nwhat a blessed companion!  A keener observer of character, probably, from\nwhat we can gather, than her husband; a sharper eye, in general, to\ndetect the subterfuges of selfishness and conceit.\n\nOne mighty trial she had before she died; she had, in some way, been\ndeeply wounded, grievously injured, and hurt, and she found it hard to\nforgive; she agonized, and prayed, and struggled; and before she was\ncalled to eternity, she was able to feel that she had forgiven, and\nburied the memory of the injuries in the love and compassion of the\nRedeemer.  Her husband had to give her up, and at a time, perhaps, when\nhe needed her most.  The illness was long, but great strength was given\nto her, and at last the release came.  There was mourning in the Cildwrn\ncottage.  The last night of her life she repeated a beautiful, and\ncomfortable Welsh hymn, and then, ejaculating three times, \"Lord Jesus,\nhave mercy upon me!\" she breathed forth her quiet, affectionate, and\nhopeful spirit, into her Saviour's hands, and left her husband all alone,\nto bear the burden of her departure, and other griefs, and troubles which\nwere crowding upon him.\n\nOther troubles,\u2014for, in what way we need not attempt too curiously to\ninquire,\u2014the pastorate gave to the poor old pastor little, or no peace.\nThere were strong Diotrephesian troubles agitating the great preacher's\nlife.  The Churches, too, which Christmas Evans had raised, and to which,\nby his earnest eloquence, and active, organizing mind, he had given\nexistence, grew restive, and self-willed beneath his guidance, refusing\nhis advice with reference to ministers he suggested, and inviting others,\nwhose appointment he thought unwise.\n\nPoor Christmas!  Did he ever ask himself, in these moments, when he\nthought of his lost Catherine, and felt the waves of trouble rising up,\nand beating all round him,\u2014did he ever ask himself whether the game was\nworth the candle? whether he was a mere plaything in life, whom that arch\nold player, Death, had outplayed, and defeated?  Did it ever seem to him\nthat it was all a vanity, ending in vexation of spirit?  The life most\nbeloved had burnt out, the building he had spent long years to erect,\nseemed only to be furnished for discomfort, and distraction.\n\nDid he begin to think that the wine of life was only turning into acrid\nvinegar, by-and-by to end with the long sleeping-draught?  Of life's good\nthings, in the worldling's sense of good, he had tasted few; most clearly\nhe had never desired them.  He had never the opportunity, nor had he ever\ndesired to be like a Nebuchadnezzar, roaming the world like a beast, and\npasturing at a dinner-table, as upon a sort of meadow-land of the\nstomach, sinking the soul to the cattle of the field; but he might have\nexpected that his Church, and Churches, would be a joy, a rest, a\npleasant meadow-land to him.  The body was certainly crumbling to decay:\nwould the ideas also prove like frescoes, which could be washed out by\ntears, or removed, and leave the soul only a desolate habitation, waiting\nfor its doom of dust?\n\nWe do not suppose that, amidst his depressing griefs, these desolating\nbeliefs, or unbeliefs, had any mastery over him.  What did the men who\ntormented him know of those mighty springs of comfort, which came from\nthose covenants he had made with God, amidst the lonely solitudes of his\njourneyings among the wild Welsh hills?  He had not built his home, or\nhis hopes, on the faithfulness of men, or the vitality of Churches; the\nroots of his faith, as they had struck downward, were now to bear fruit\nupward.\n\nThere was a fine healthfulness in his spirit.  There is nothing in his\nlife to lead one to think that he had ever been much intoxicated by the\nfame which had attended him; he appears to have been always beneath the\ncontrol of the great truths in which he believed, and it was not the\nseductive charms of popularity for which he cared, but the power of those\ntruths to bring light, conviction, and rest, to human souls.  All his\nsermons look that way; all that we know of his preaching, and experience,\nturns in that direction.\n\nRose-leaves are said to act as an emetic, and have much the same effect\non the constitution as senna-leaves.  It is so with those sweet things\nwhich fame offers to the imagination; the conserves of its fragrance,\nby-and-by, become sickening.  So, the robust nature of our fine old\nfriend had to rise over grief, and disappointment, and unfriendliness,\nand diaconal dictation and impertinence.  Only one thing he remembered.\nHe appears to have been sustained, even as Edward Irving was, in his\nconviction that the truth of his message, the lamp of the ministry which\nhe carried, gave to him a right, and a prerogative which he was not to\nrelinquish; he had proved himself, he had proved the Spirit of God to be\nin him of a truth.  He was not a wrangler, not disposed to maintain\ndebates as to his rights; nor was he disposed to yield to caprice,\nfaction, and turbulence; and so, he began to think of retiring, old as he\nwas, from the field, the fragrance of which had proclaimed that the Lord\nhad blessed him there.\n\nChristmas Evans, as he draws near to the close of his work in Anglesea,\nonly illustrates what many a far greater, and many a lesser man than he,\nhave alike illustrated.  There is a fine word among the many fine words\nof that great, although eccentric teacher, John Ruskin:\u2014\"It is one of the\nappointed conditions of the labour of man, that in proportion to the time\nbetween the seed-sowing and the harvest, is the fulness of the fruit; and\nthat generally, therefore, the further off we place our aim, and the less\nwe desire to be the witnesses of what we have laboured for, the more wide\nand rich will be the measure of our success.\"  This was, no doubt, the\nconsolation of Christmas; but as we look upon him, a friendly voice\nreminds us, that, as he leaves Anglesea, he realizes very much of Robert\nBrowning's soliloquy of the martyred patriot:\u2014\n\n    \"Thus I entered, and thus I go!\n       In triumphs people have dropped down dead.\n    Paid by the world,\u2014what dost thou owe\n       Me?  God might question; now, instead,\n    'Tis God shall repay!  I am safer so.\"\n\nSo the candlestick was removed out of its place in Anglesea, and Anglesea\nsoon, but too late, regretted the removal.  Christmas Evans, however,\nseems to illustrate a truth, which may be announced almost as a general\nlaw, from the time of the Saviour and his Apostles down to our own, that\nthose who have wrought most unselfishly, and serviceably for the cause of\nGod, and the well-being of man, had to receive their payment in\nthemselves, and in the life to come.  In proportion to the greatness of\ntheir work was the smallness of their remuneration here.\n\nIf we refer to the painful circumstances in connection with the close of\nthe ministry of Christmas Evans at Anglesea, it is, especially, to notice\nhow his faith survived the shock of surrounding trouble.  He himself\nwrites: \"Nothing could preserve me in cheerfulness and confidence under\nthese afflictions, but the assurance of the faithfulness of Christ; I\nfelt assured that I had much work yet to do, and that my ministry would\nbe instrumental in bringing many sinners to God.  This arose from my\ntrust in God, and in the spirit of prayer that possessed me; I frequently\narose above all my sorrows.\"\n\nAnd again he writes: \"As soon as I went into the pulpit during this\nperiod, I forgot my troubles, and found my mountain strong; I was blessed\nwith such heavenly unction, and longed so intensely for the salvation of\nmen, and I felt the truth like a hammer in power, and the doctrine\ndistilling like the honey-comb, and like unto the rarest wine, that I\nbecame most anxious that the ministers of the county should unite with me\nto plead the promise, 'If any two of you agree touching anything,' etc.\nEverything now conspired to induce my departure from the island: the\nunyielding spirit of those who had oppressed, and traduced me; and my own\nmost courageous state of mind, fully believing that there was yet more\nwork for me to do in the harvest of the Son of Man, my earnest prayers\nfor Divine guidance, during one whole year, and the visions of my head at\nnight, in my bed\u2014all worked together towards this result.\"\n\nFew things we know of are more sad than this story.  \"It was an affecting\nsight,\" says Mr. William Morgan, quoted by Mr. Rhys Stephen in his\nMemoir, \"to see the aged man, who had laboured so long, and with such\nhappy effects, leaving the sphere of his exertions under these\ncircumstances; having laboured so much to pay for their meeting-houses,\nhaving performed so many journeys to South Wales for their benefit,\nhaving served them so diligently in the island, and passed through so\nmany dangers; now some of the people withheld their contributions, to\navenge themselves on their own father in the Gospel; others, while\nprofessing to be friends, did little more; while he, like David, was\nobliged to leave his city, not knowing whether he should ever return to\nsee the ark of God, and his tabernacle in Anglesea again.  Whatever\nmisunderstanding there was between Mr. Evans, and some of his brethren,\nit is clear that his counsels ought to have been received with due\nacknowledgment of his age, and experience, and that his reputation should\nhave been energetically vindicated.  I am of opinion, I am quite\nconvinced, that more strenuous exertions should have been made to defend\nhis character, and to bear him, in the arms of love, through the archers,\nand not to have permitted him to fall in the street without an advocate.\"\n\nThe whole aim of Mr. Evans's life, as far as we have been able to read\nit, was to get good from heaven, in order that he might do good on earth.\nClearly, he never worked with any hope of a great earthly reward for any\npersonal worthiness; perhaps there arose a sense that he had always been\nunjustly remunerated, that burdens had been laid upon him he ought not to\nhave been called upon to bear; and now the sense of injustice sought, as\nis so frequently the case, to vindicate itself by ingratitude.  It seems\nso perpetually true, in the sad record of the story of human nature, that\nit is those who have injured us who seek yet further to hurt us.\n\n\n\n\nCHAPTER V.\n_CONTEMPORARIES IN THE WELSH PULPIT\u2014WILLIAMS OF WERN_.\n\n\nThe Great Welsh Preachers unknown in England\u2014The Family of the\nWilliamses\u2014Williams of Pantycelyn\u2014Peter Williams\u2014Evan Williams\u2014Dr.\nWilliams\u2014Williams of Wern\u2014The immense Power of his Graphic\nLanguage\u2014Reading and Thinking\u2014Instances of his Power of Luminous\nIllustration\u2014Early Piety\u2014A Young Preacher\u2014A Welsh Gilboa\u2014Admiration of,\nand Likeness to, Jacob Abbot\u2014Axiomatic Style\u2014Illustrations of Humour\u2014The\nDevils\u2014Fondness for Natural Imagery\u2014Fondness of Solitude\u2014Affecting\nAnecdotes of Dying Hours\u2014His Daughter\u2014His Preaching characterised\u2014The\nPower of the Refrain in the Musician and the Preacher, \"Unto us a Child\nis born.\"\n\nWE pause here for a short time, in our review of the career, and\ncharacter, and pulpit power of Christmas Evans, to notice some of those\neminent men, who exercised, in his day, an influence over the Welsh mind.\nWe will then notice some of those preachers, of even the wilder Wales,\nwho preceded these men.  So little is known of many of them in England,\nand yet their character, and labours, are so essentially and excellently\ninstructive, that we feel this work, to those who are interested, to be\nnot one of supererogation.  The men, their country, the people among whom\nthey moved, their work in it, the singular faith in, and love for\npreaching, for the words these men had to utter,\u2014they must seem, to us,\nremarkable, and memorable.  In this time of ours, when preaching, and all\nfaith in preaching, is so rapidly dying out, that it may be regarded,\nnow, as one of the chief qualifications of a candidate for the pulpit,\nthat he cannot preach a sermon, but can \"go to those who sell, and buy\nfor himself\"\u2014this study of what was effected by a living voice, with a\nreal live soul behind it, must seem, as a matter of mere history,\nnoteworthy.  And first among those who charmed the Welsh ear, in the time\nof Christmas Evans, we mention Williams of Wern.\n\nIt is not without reason, that many eminent Welshmen can only be known,\nand really designated after the place of their birth, or the chief scene\nof their labours.  The family of the Williamses, for instance, in Wales,\nis a very large one\u2014even the eminent Williamses; and William Williams\nwould not make the matter any clearer; for, always with tenderest love\nought to be pronounced the name of that other William Williams, or, as he\nis called, Williams of Pantycelyn\u2014the obscure, but not forgotten, Watts\nof Wales.  His hymns have been sung over the face of the whole earth, and\nlong before missionary societies had been dreamed of, he wrote, in his\nremote Welsh village,\n\n    \"O'er the gloomy hills of darkness;\"\n\nand he has cheered, and comforted many a Zion's pilgrim by his sweet\nsong,\n\n    \"Guide me, O Thou great Jehovah!\"\n\nHe was born in 1717, and died in 1791.  This sweet and sacred singer\nought to receive more than this passing allusion.  Little is known of him\nin England; and it is curious that Mr. Christopher's volume on \"Hymn\nWriters and their Hymns\" neither mentions his hymns, nor his name.\n\nA writer in the _Quarterly Review_, evidently not very favourable to that\ndenomination of religious sentiment which Williams represented, has\nspoken of the \"unmixed pleasure\" his name and character awakens: \"He was\na man in whom singular purity of sentiment added grace to a truly\noriginal genius.\"  \"His direction to other composers was, never to\nattempt to compose a hymn until they feel their souls near heaven.  His\nprecept, and his practice, in this respect, have been compared to those\nof Fra Angelico.\"  Would that some competent Welsh pen would render for\nus, into English, more of these notes of the sweet singer of Pantycelyn.\n\nWilliam Williams came from the neighbourhood of Llandovery, the parish of\nPritchard of the \"Welshman's Candle;\" he was, as his hymns would\nindicate, well educated; he studied for, and entered upon the medical\nprofession; but, converted beneath the preaching of Howell Harris, in\nTalgarth churchyard, he turned from medicine to the work of the ministry.\nHe was a member of the Established Church; he sought, and received\nordination, and deacon's orders, but, upon application for priest's\norders, he was refused.  He then united himself with the Calvinistic\nMethodists, but still continued to labour with the great Daniel Rowlands,\nat Llangeitho.  His sermons were, like his hymns, often sublime, always\nabounding in notes of sweetness.  During the forty three years of his\nministry, it is said, he travelled about 2,230 miles a year, making in\nall 95,890 miles!  He wrote extensively, also, in prose.  There is a\nhandsome edition of his works in the Welsh language, and an English\nedition of some of his hymns.  Among the most beautiful, our readers will\nremember\u2014\n\n    \"Jesus, lead us with Thy power\n       Safe into the promised rest.\"\n\nThis was William Williams of Pantycelyn.\n\nThen, there was Peter Williams, a famous name in the Principality, and of\nabout the same period as Williams of Pantycelyn.  No man of his time did\nso much to cultivate religious literature in Wales.  He was a great\npreacher, and an exemplary man; when a minister within the Church of\nEngland, he was persecuted for his opinions, and practices; and, when he\nleft that communion, he suffered even a more bitter persecution from his\nMethodist brethren.  His life, and his preaching, appear to have been\nfull of romantic incidents.\n\nThen there was Evan Williams, who is spoken of as a seraphic man, and\nwhose life appears to justify the distinctive designation, although he\ndied at the age of twenty-nine, very greatly in consequence of ill-usage\nreceived in persecution.\n\nThen, in England, we are better acquainted with Daniel Williams, the\nfounder of what is called Dr. Williams's Library; and who, in addition to\nthis magnificent bequest, left sums of money to Wales for schools,\nendowments of ministers, annual grants of Bibles, and religious books,\nand for widows of ministers; by which Wales has received since, and\nreceives now, the sum of about \u00a3700 a year.  His ministry, however, was\nin London, at Hand Alley, Bishopsgate Street, nearly two hundred years\nsince.  His works are contained in six octavo volumes; but he scarcely\nfalls beneath the intention of these pages.\n\nBesides these, there are many others; so that, as we said above, the name\nof Williams represents, not only a large family, but a family remarkable\nfor Christian usefulness in Wales.  But, in this catalogue of eminent\npreachers, Williams of Wern, among those of his name, is singularly\neminent.  He had that power, to which we have referred, of using his\nlanguage in such a manner, that people, in a very awful way, realized the\nscenes he described.  Dr. Rees mentions of him, that when preaching on\nthe resurrection of the dead, from the window of Ynysgan Chapel, Merthyr\nTydvil, he so riveted the attention of the vast multitude, who were on\nthe burying-ground before him, that when he reached the climax, all the\ncrowd moved together in terror, imagining that the graves under their\nfeet were bursting open, and the dead rising.  Yet Williams was a\nsingularly quiet preacher; these effects were wrought by the power of\nthat language, so wonderfully fitted to work on the emotions of a very\nimaginative people, and which he knew how to play upon so well.\n\nThis great preacher had quite as remarkable an individuality as either of\nthe eminent men, whose characters we may attempt faintly to portray.\nChristmas Evans, we have seen, led his hearers along through really\ndramatic, and pictorial representations.  Davies was called the \"Silver\nTrumpet\" of Wales; his voice was an instrument of overwhelming compass,\nand sweetness.  Elias was a man of severe, and passionate eloquence,\u2014all\nthe more terrible, because held in the restraint of a perfect, and\ncommanding will.  Williams differed from all three; nor must it, for a\nmoment, be said that he \"attained not to the first three.\"  His eminence\nwas equal to theirs, and, in his own walk, he was quite as highly\nesteemed; but his department of power was completely different.  Perhaps,\nhe was less the vehicle of vehement passion than either Elias, or Davies;\nand it was altogether apart from his purpose to use the amazing imagery\nof Christmas Evans.  His mind was built up of compacted thought; his\nimages were not personifications, but analogies.  So far as we are able\nto form a conception of him, his mind appears to have moved in a pathway\nof self-evidencing light.\n\nThus, if we were to speak of these four men as constituting a quartette\nin the harmony of the great Welsh pulpit, we should give to John Elias\nthe place of the deep bass; to Davies, the rich and melting soprano; to\nChristmas Evans the tenor; reserving, for Williams of Wern, the place of\nthe alto.  His teaching was eminently self-evolved.  None of the great\nWelsh preachers dealt much with pen, and paper.  They wrought out their\nsermons on horseback, or whilst moving from place to place.  With\nWilliams it was especially so.  Two ministers called upon him in 1830.\nOne of them was something of a bookworm, and he asked him if he had read\na certain book which had just been published.  Williams said he had not.\n\"Have you,\" continued his friend, \"seen so-and-so?\" naming another work.\n\"No, I have not.\"  And, presently, a third was mentioned, and the answer\nwas still in the negative.  \"I'll tell you what,\" said Mr. Williams, \"you\nread too much; you do not think sufficiently.  My plan in preparing\nsermons is to examine the connection of a passage, extract its principle,\nand think it over in my own mind.  I never look at a Commentary, except\nwhen completely beaten.\"\n\nIt has often been said that, in the very proportion in which eloquence is\neffective, and commanding in delivery, in the degree in which it is\neffective as _heard_, it is impossible to be _read_; and, with some\nmeasure of exception, this is, no doubt, true.  Williams, certainly, is\nan illustration of this general principle; yet he was, perhaps, one of\nthe most luminous of speakers; only, this alone, without accompanying\npassion, does not make the orator.  Take the following as an illustration\nof his manner.  On ejaculatory prayer:\u2014\n\n    \"Ejaculatory prayer is the Christian's breath; the secret path to his\n    hiding-place; his express to heaven in circumstances of difficulty,\n    and peril; it is the tuner of all his religious feelings; it is his\n    sling, and stone, with which he slays the enemy, ere he is aware of\n    it; it is the hiding of his strength; and, of every religious\n    performance, it is the most convenient.  Ejaculatory prayer is like\n    the rope of a belfry; the bell is in one room, and the handle, or the\n    end of the rope which sets it a-ringing, in another.  Perhaps the\n    bell may not be heard in the apartment where the rope is, but it is\n    heard in its own apartment.  Moses laid hold of the rope, and pulled\n    it hard, on the shore of the Red Sea; and though no one heard, or\n    knew anything of it, in the lower chamber, the bell rang loudly in\n    the upper one, till the whole place was moved, and the Lord said,\n    'Wherefore criest thou unto me?'\"\n\nThis is luminous preaching.  Unfortunately, as with others, we have very\nlittle\u2014scarcely anything, indeed\u2014left of Williams's pulpit talk.\n\nWilliam Williams was born in the year 1781, at Cwm-y-swn-ganol, in\nMerionethshire.  There his parents occupied a farm, and were much\nrespected.  It seems, to us, an odd thing that their name was not\nWilliams, but Probert, or Ap-Robert.  He received his name of Williams\nfrom the singular practice, then prevalent in many parts of Wales, of\nconverting, with the aid of the letter S, the Christian name of the\nfather into the surname of the son.  His father, although an orderly\nattendant upon Divine Worship, never made a public profession of\nreligion; but his mother was a very pious, and exemplary member of the\nCalvinistic Methodist connexion.\n\nThe decisive hour of real religious conviction came to the youth when he\nwas very young\u2014only about thirteen years of age.  Impressions deep, and\npermanent, were made on his mind, and heart, and at fifteen he was\nreceived into Church fellowship; but he suffered greatly from diffidence.\nAlthough it was expected of him, he could not pray either in the family,\nor in public, because, as he used to say, he would then be required, by\nall his acquaintance, to conduct himself like a perfect saint.  But one\nnight, when all the family, with the exception of his mother, and\nhimself, had retired to rest, she engaged in prayer with him, and then\nsaid, \"Now, Will, dear, do you pray,\" and he did so; and from this moment\ndated the commencement of his courage, and confidence.\n\nIt was in his twenty-second year that he entered Wrexham Academy.  He was\na thorough Welshman\u2014a monoglot.  He made some progress in the acquisition\nof English, and Greek; but he could never speak English fluently, and was\nadvanced in life before he knew a word of it; and he used to say, \"When I\nviolate English, I am like a child that breaks a window; I do not go back\nto mend it, but I run away, hoping I shall not be seen.\"  As linguists,\nmost of his fellow-students outshone him; in the pulpit, from his very\nfirst efforts, he not only outshone them all, but it was soon seen that\nhe was to transcend most of the teachers, and speakers of his time.\n\nPerhaps his example will not commend itself to some of our modern\nwriters, as to preparation for the ministry; for when he was recommended\nto continue longer under tuition, he said, \"No\u2014no; for if so, the harvest\nwill be over while I am sharpening my sickle.\"  Young as he was, he took\na singular view of the leadings of Providence, which, however, eminently\nmarks the character of the man.  He received a most unanimous invitation\nfrom a large, and influential Church at Horeb, in Cardiganshire, and was\njust about accepting the invitation, when the smaller, and, in\ncomparison, quite insignificant sphere of Wern was put before him, with\nsuch commendations of the importance of the work as commanded his\nregards.  He declined Horeb, and accepted Wern.\n\nHis field of labour appears to have comprehended a cluster of villages,\nsuch as Llangollen, Rhuabon, and Rhosllanerchrugog; and in this region\nthe greater number of his days were passed, excepting that brief period,\ntowards the close of his life, when he became the minister of the great\nWelsh tabernacle in Cross Hall Street, Liverpool.  But he left Wales with\na heavy heart, amidst the pretty distinctly expressed dissatisfaction of\nthe people of the Principality, who, however, still insisted on giving\nhim his designation of Williams of Wern.  Nor was he away from them long.\nHis old Church continued unsettled, and after three years' ministry in\nLiverpool, he returned to Wern, to close his active, and useful life.\n\nHis pastorate consisted, really, of three places\u2014Wern, Rhos, and Harwood.\nIt was a singular circumstance, that whilst large crowds thronged round\nhim at the first two places, and while his name was becoming as a sharp\narrow through the whole Principality, he made little impression on\nHarwood.  He used to say that Harwood had been of greater service to him\nthan he had been to it; for it was \"the thorn in the flesh, lest he\nshould be exalted above measure;\" and if he ever felt disposed to be\nlifted up when he saw the crowds gathering round him at other places, he\nhad only to go over to, or think about Harwood, and this became an\neffectual check to the feelings of self-inflation, in which he might have\nbeen tempted to indulge.  It was so, whilst other places, Churches, and\ncongregations, \"waited for him as for the rain, and opened their mouths\nwide as for the latter rain;\" whilst upon other fields his \"doctrine\ndistilled as the dew,\" his stubborn Harwood appears to have been a kind\nof Welsh Gilboa, upon which no dew fell.\n\nHe was claimed as a kind of public property, and Churches at a distance\nseemed to think they had a right to his services, frequently very much to\nthe irritation of his own people, to whom he might have given the\nconsolation he once administered to a brother minister; \"I understand\nthat your people complain a good deal because you so often leave them.\nWell, let us be thankful that the reverse is not the case; for our own\npeople might have tired of us, and be pleased to hear strangers, and\npreferred our absence, regarding us as 'a vessel wherein is no\npleasure.'\"  Unfortunately, in such cases, congregations do not take the\nmatter as philosophically as the old Scotchwoman, who, when she met a\nneighbouring clergyman one Sabbath morning, wending his way to her own\nkirk, expressed her surprise at meeting him there, and then.  He\nexplained that it was an exchange of services.  \"Eh, then,\" said the old\nwoman, \"_your_ people will be having a grand treat the day.\"\n\nSomething of the nature of Williams's mind, and his method of\nministration, may be gathered from his exceeding admiration of Jacob\nAbbot, and especially his work, \"The Corner Stone.\"  \"Oh! what a pity,\"\nhe said, \"that we cannot preach as this man writes.\"  But, so far as we\nhave been able to judge from the scanty means we possess, he did preach\nvery much after the manner of Jacob Abbot's writings.  His words appear,\nfirst, to have been full of strong, seminal principles, and these were\nsoon made clear in the light of very apt illustrations.  Truly it has\nbeen said, that, first, the harper seizes his harp, and lays his hand\nfirmly upon it, before he sweeps the strings.  In an eminent manner,\nWilliams gave to his people the sense, as soon as he commenced, that a\nsubject was upon his heart, and mind; and he had a firm grasp of it, and\nfrom his creative mind each successive stroke was some fine, apt, happy\nevolution.\n\nIllustration was his _forte_, but of a very different order from that of\nChristmas Evans; for instance, illustrating the contests of Christian\ncreeds, and sects with each other, \"I remember,\" he said, \"talking with a\nmarine, who gave to me a good deal of his history.  He told me the most\nterrible engagement he had ever been in, was one between the ship to\nwhich he belonged, and another English vessel, when, on meeting in the\nnight, they mistook each other for a French man-of-war.  Many persons\nwere wounded, some slain; both vessels sustained serious damage from the\nfiring, and, when the day broke, great was their surprise to find the\nEnglish flag hoisted from the masts of both vessels, and that, through\nmistake, they had been fighting all night against their own countrymen.\nIt was of no avail, now, that they wept together: the mischief was done.\nChristians,\" said the preacher, \"often commit the same error in this\npresent world.  One denomination mistakes another for an enemy; it is\nnight, and they cannot see to recognise each other.  What will be their\nsurprise when they see each other in the light of another world! when\nthey meet in heaven, after having shot at each other through the mists of\nthe present state!  How will they salute each other, when better known,\nand understood, after having wounded one another in the night!  But they\nshould wait till the dawn breaks, at any rate, that they may not be in\ndanger, through any mistake, of shooting at their friends.\"\n\nThe Welsh language is, as we suppose our readers well know, especially\nrich in compact, proverbial, axiomatic expressions.  The Welsh triads are\nan illustration of this.  The same power often appears in the pulpit.\nThe latter, and more recent, languages are unfavourable to the expression\nof proverbs.  Williams we should suppose to have been one of the most\nfavourable exemplifications of this power.  General tradition in Wales\ngives him this kind of eminence\u2014poem, and proverb united in his\nsentences.  We have not been able to obtain many instances of this; and\nwe fear it must be admitted, that our language only in a clumsy way\ntranslates the pithy quaintness of the Welsh, such as the following: \"The\ndoor of heaven shuts from below, not from above.  'Your iniquities have\nseparated, saith the Lord.'\"  \"Of all the birds,\" he once said, \"the dove\nis the most easily alarmed, and put to flight, at hearing a shot fired.\nRemember,\" he continued, \"that the Holy Ghost is compared to a dove; and\nif you begin to shoot at each other, the heavenly Dove will take wing,\nand instantly leave you.  The Holy Spirit is one of love, and peace, not\nof tumult, and confusion.  He cannot live amongst the smoke, and noise of\nfired shots: if you would grieve the Holy Spirit, and compel Him to\nretire, you have only to commence firing at one another, and He will\ninstantly depart.\"  \"The mind of man is like a mill, which will grind\nwhatever you put into it, whether it be husk or wheat.  The devil is very\neager to have his turn at this mill, and to employ it for grinding the\nhusk of vain thoughts.  Keep the wheat of the Word in the mind; 'keep thy\nheart with all diligence.'\"\n\nSome of his words seem very odd, although he was a most grave, and\nserious man.  Thus; \"Our prayers often resemble the mischievous tricks of\ntown-children, who knock at their neighbours' houses, and then run away;\nwe often knock at Heaven's door, and then run off into the spirit of the\nworld: instead of waiting for entrance, and answer, we act as if we were\nafraid of having our prayers answered.\"  Again: \"There are three devils\nwhich injure, and ravage our Churches, and congregations,\u2014the singing\ndevil, the pew-letting devil, and the Church officers' appointment devil:\nthey are of the worst kind of devils, and this kind goeth not out but by\nprayer, and fasting.\"  \"The old ministers,\" he used to say, \"were not\nmuch better preachers than we are, and, in many respects, they were\ninferior to us; but they had a success attendant upon their ministry that\ncan now seldom be seen.  They prayed more than we do.  It was on his\nknees that Jacob became a prince; and if we would become princes, we must\nbe more upon our knees.  We should be successful as our fathers, could we\nbe brought to the same spirit, and frame of mind.\"\n\nBut Williams is like Elias in this; we have had none of his sermons\nrendered into English, and, therefore, the descriptions we have are\nrather tantalizing.  Mr. Parry, the Congregational minister of Llandudno,\na man well fitted to judge\u2014himself one of the most distinguished living\npoets in the Welsh language, and who has carried many prizes from the\nEisteddfodd\u2014says of him: \"I shall never forget his eloquence.  It poured\nforth like a swollen torrent.  I cannot help referring to a sermon he\npreached at an annual Association at Llanerchmedd, Anglesea.  The meeting\nwas, as usual, held in the open air.  The weather was very sultry; the\ncongregation seemed drowsy.  His manner, before preaching, showed\nconsiderable restlessness, and when he came to the desk, he looked rather\nwild.  It was evident his spirit was on fire, and his mind charged\nbrimful with ideas.  He read his text in a quick, bold tone; 'But now\nthey desire a better country, that is, a heavenly.'  He poured forth such\na flood of eloquent description, that he completely enchanted our\nfeelings, and made us imagine we felt the field move under our feet.  He\nhimself thought this occasion one of the most remarkable in his life; for\nI spoke to him about the sermon years after.  I believe it served to\nraise our Churches throughout the whole land.\"\n\nHe was a more extensive reader than any of his brethren in the ministry;\na keen observer, too, in the departments of natural history, and natural\nphilosophy.  It was, indeed, much like his own method, and it illustrated\nthe reason of his great admiration for Jacob Abbot's \"Corner Stone,\" when\nhe very prettily says, \"The blessed Redeemer was very fond of His\nFather's works.\"  He used to say, \"If we understood nature better, it\nwould help us to understand the Bible better.  The kingdom of nature, and\nthe kingdom of grace, are very like each other.  There is a striking\nresemblance between the natural principles of the one, and the moral\nprinciples of the other.\"  He entered with a kind of joy into the sublime\nmoods of nature; was fond of watching the play of the lightning, and\nlistening to the voice of the thunder.  \"Jesus,\" he used to say, \"loved\nto look at the lily, and to listen to the birds; to speak upon the\nmysteries of the seed, and to draw forth principles from these things.\nIt was no part of His plan to expound the laws of nature, although He\nunderstood them more perfectly than any one else; but He employed nature\nas a book of reference, to explain the great principles of the plan of\nsalvation.\"\n\nA clergyman writes of him, that \"his appearance when preaching was very\nremarkable, and singularly beautiful.  When standing in a great crowd,\nevery soul seemed agitated to its centre, and cheeks streaming with\ntears.  It is but justice that every one should have his likeness taken\nwhen he appears to the greatest advantage; and so Williams.  His picture,\non such an occasion, would be an honour to the country which reared him,\na treasure to the thousands who heard him, and a name to the painter.\"\nThe likeness is before us now, and in the firm, composed thoughtfulness,\na kind of sad, far outlook in the eyes, and the lips which seem to wait\nto tremble into emotion\u2014we think we can well realize, from the inanimate\nengraving, what life must have been in the speech of this extraordinary\nman.  His mind was cast in a sweetly meditative mould.  He was fond of\nretreating by himself among the trees, and walking beneath their shadows,\nas they formed a canopy over his head.  He said of one such place, \"I\nthink I must love that spot through all eternity, for I have felt a\ndegree of heaven there.\"\n\nAnd thus he died.  He had lost his wife some time before.  It is very\naffecting to read the account of himself, and his daughter, dying\ntogether in different rooms of the same house.  As he said to her, one\nday, \"We appear to be running, with contending footsteps, to be first at\nthe goal.\"  They spent much time in talking together, with unruffled\ncomposure, of death, and heaven, and being \"absent from the body, and\npresent with the Lord.\"  Every morning, as soon as he was up, found him\nby the bedside of his daughter.\n\nOnce he said to her, \"Well, Eliza, how are you this morning?\"\n\n\"Very weak, father.\"\n\n\"Ah!\" said he, \"we are both on the racecourse.  Which of us do you think\nwill get to the end first?\"\n\n\"Oh, I shall, father.  I think you must have more work to do yet.\"\n\n\"No,\" he said; \"I think my work is nearly over.\"\n\n\"It may be so, father; but, still, I think I shall be the first to go.\"\n\n\"Perhaps,\" he said, \"it is best it should be so, for I am more able to\nbear the blow.  But,\" he continued, \"do you long to see the end of the\njourney?\"\n\n\"Oh, from my heart!\" she replied.\n\n\"But why?\"\n\n\"Because I shall see so many of my old friends, and my mother; and, above\nall, I shall see Jesus.\"\n\n\"Ah, well, then,\" he said, \"tell them I am coming! tell them I am\ncoming!\"\n\nShe died first.  Her last words were, \"Peace! peace!\"  He followed her\nshortly after\u2014on the 17th of March, 1840, in the fifty-ninth year of his\nage.\n\nAmongst the great preachers of Wales, not one seems to have won more upon\nthe tender love of those who knew him.  Dr. Raffles said of him, \"What he\nwas as a preacher, I can only gather from the effects he produced on\nthose who understood the language in which he spoke, but I can truly say,\nthat every occasion on which I saw him only served to impress me more\nwith the ardour of his piety, and the kindness of his heart.  He was one\nof the loveliest characters it has been my lot to meet.\"\n\nHigh strains of thought, rendered into the sweet variety, melting\ntenderness, and the grand strength of the language of Wales, seem to have\nbeen the characteristics of the preaching of Williams of Wern; tender,\nand terrible, sweetness alternating with strength.  We have already said\nhow much Welsh preaching derived, in its greatest men, from the power of\nvarying accent; the reader may conceive it himself if ever listening to\nthat wonderful chorus in Handel's \"Messiah,\" which Herder, the great\nGerman, called truly the Christian Epos; but the chorus to which we\nrefer, is that singular piece of varying pictorial power, \"Unto us a\nChild is born,\" repeated, again and again, in sweet whispered accents,\nplaying upon the thought; the shepherds having kept watch over their\nflocks by night in the fields, and having heard the revelation voices of\nthe angels say it\u2014\"For unto us a Child is born;\" and then rolls in the\ngrand thunder, \"And His name shall be called Wonderful;\" and then, you\nreturn back to the sweet silvery accents, \"For unto us a Child is born;\"\nand the thought is, that the Wise Men are there offering their gifts; and\nthen roll in, again, the grand, overwhelming words, \"And His name shall\nbe called Wonderful;\" and yet again that for which we waited, the tender,\nsilvery whisperings, \"Unto us a Child is born;\" until it seems as if\nflocks, and herds, and fields, shepherds, and wise men, all united with\nthe family of Jesus, beneath the song-singing through the heavens in the\nclear starry night, \"Unto us a Child is born, and His name shall be\ncalled Wonderful.\"  Those who have listened to this chorus, may form some\nidea of the way in which a great Welsh preacher\u2014and Williams of Wern as a\nspecial illustration\u2014would run his thought, and its corresponding\nexpression, up and down, through various tones of feeling, and with every\none awaken, on some varying accent, a fresh interpretation, and\nexpression.  Perhaps, the nearest approach we have heard, in England, to\nthe peculiar gifts of this preacher, has been in the happiest moods of\nthe beloved, and greatly honoured Thomas Jones, once minister of Bedford\nChapel, London.\n\n\n\n\nCHAPTER VI.\n_CONTEMPORARIES\u2014JOHN ELIAS_.\n\n\nFire and Smoke\u2014Elias's Pure Flame\u2014Notes in the Pulpit\u2014Carrying Fire in\nPaper\u2014Elias's Power in Apostrophe\u2014Anecdote of the Flax-dresser\u2014A Singular\nFirst Appearance in the Pulpit\u2014A Rough Time in Wales\u2014The Burning of the\nRavens' Nests\u2014A Hideous Custom put down\u2014The Great Fair of Rhuddlan\u2014The\nTen Cannon of Sinai\u2014Action in Oratory\u2014The Tremendous Character of his\nPreaching\u2014Lives in an Atmosphere of Prayer\u2014Singular Dispersion on a\nRacecourse\u2014A Remarkable Sermon, Shall the Prey be taken from the\nMighty?\u2014Anecdote of a Noble Earl\u2014Death and Funeral.\n\nWE have already implied that Welsh preaching has had many varieties, and\nvery various influences too.  Even the very excitements produced by these\nfamous men, whose names we are recording, varied considerably; but one\ncharacteristic certainly seemed to attend them\u2014the influence was real,\nand very undoubted.  When Rowland Hill was in Wales, and witnessed some\nof the strong agitations resulting from great sermons, he said, he \"liked\nthe fire, but he did not like the smoke.\"  It was, like so many of the\nsayings of the excellent old humorist, prettily, and wittily said.  But\nit may, also, be remarked, that it is, usually, impossible to have real\nfire without smoke; and it has further been well said, that the stories\nof the results of such preaching make us feel that, could we only get the\nfire, we need not object to a little of the smoke.\n\nWe are introducing to our readers, now, in John Elias, one who,\ncertainly, does not seem to have surrounded the clear flames of his\neloquence with unnatural excitement.  If the effects of his oratory seem\nto rival all that we have heard of the astonishing power of George\nWhitefield, the material of his sermons, the severity of their tone of\nthought, and the fearfulness of their remorseless logic, remind us of\nJonathan Edwards.  He had read extensively, especially in theology; and,\nit has been truly said, his mind was a storehouse, large, lofty, and\nrich.  Like his great coadjutors, he prepared for the pulpit with amazing\ncare, and patience, but apparently never verbally\u2014only seeing his ideas\nclearly, and revolving them over and over until, like fuel in the\nfurnace, they flamed.  He tells us how, having done his part, by earnest,\nand patient study, he trusted to God to give to his prepared mind its\nfitting expression, and speech.  Of course, like the rest, he disclaimed\nall paper in the pulpit.  An eminent brother minister, Thomas Jones, of\nDenbigh, was coming to London to preach what was considered the great\nannual sermon of the London Missionary Society, at Surrey Chapel.  In his\nown country, Mr. Jones preached always extempore; but, being in company\nwith Matthew Wilkes, and John Elias, he inquired of old Matthew whether,\nfor such an occasion, he did not think that he had better write his\nsermon.\n\n\"Well, for _such_ an occasion,\" said Matthew, \"perhaps it would be better\nto write your discourse; but, at any rate, let us have plenty of fire in\nit.\"\n\n\"But,\" said John Elias, \"he cannot carry fire in paper!\"\n\n\"Never mind,\" said Matthew; \"paper will do very well to light the fire\nwith!\"\n\nMr. Wilkes' witty rejoinder seems to give the entire value to notes, and\nwriting in the pulpit; but, no doubt, Elias expressed his conviction, and\nthe conviction of all these men, that you cannot carry fire in paper.\nBut we have before said that it was by no means wild-fire.  One of the\ngreat poets of Wales imagined a conversation going on between the soul\nand the body of Elias, before they both went up together in the pulpit,\nwhen the soul said to the body, \"Now, you must be a sacrifice for an\nhour.  You must bear all my fire, and endure all my exertion, however\nintense it may be.\"  And another writer says of him that, while some\npreachers remind us of Pharaoh's chariots, that drove heavily, Elias\nreminded us, rather, of that text, \"He maketh His angels spirits, and His\nministers a flame of fire.\"\n\nWhatever is to be said of the peculiarities of other great Welsh\npreachers, it seems to be admitted, on all hands, that John Elias was the\nDemosthenes of the group.  Let no reader smile, however high his regard\nfor the classic orator.  The stories told of the effects of the preaching\nof John Elias, greatly resemble those of the great Grecian orator, who,\nat the close of his tremendous orations, found the people utterly\noblivious to all the beauty, and strength of his discourses\u2014utterly\nindisposed to admire, or criticise, but only conducted to that point of\nvehement indignation, and passionate action, which had been, all along,\nthe purpose of the speaker, exclaiming, \"Let us march against Philip!\"\n\nIf profound passionate conviction, persuasion altogether insensible of\nanything besides its own emotions, be the chief attribute of the gifted\norator, John Elias must stand, we will not say matchless, but, from all\nthat we have heard of him, unsurpassed.  We have no means of testing this\nby any published sermons; scraps and fragments we have, and traditions of\nthe man, and his soul-piercing eloquence, float about over Wales; but we\napprehend it was an order of eloquence which would not submit itself to\neither penmanship, or paper, either to the reporter, or the\nprinting-press.\n\nHow extravagant some things seem when quietly read, unaccompanied by the\npassion, and excitement which the preacher has either apprehended, or\nproduced!  The reader remembers very well\u2014for who does not?\u2014Whitefield's\nvehement apostrophe, \"Stop, Gabriel!\"  Who could deliberately write it\ndown to utter it? and what an affectation of emotion it seems to read it!\nBut that was not the effect produced on David Hume, who heard it; and we\nmay be very sure that man,\u2014the most acute, profound, cold philosopher,\nand correct writer, had no friendly feelings either to Whitefield, or\nGabriel\u2014to the message which the preacher had to give, or the archangel\nto carry.  A quiet, ordinary, domestic state of feeling scarcely knows\nhow to make allowances for an inflamed orator, his whole nature heaving\nbeneath the passion produced by some great, and subduing vision, an\naudience in his hands, as a river of water, prepared to move\nwhithersoever he will.  Thus Elias, when he was handling some weighty\nsubject, would suddenly say, \"Stop! silence!\" (_Disymwth_!  _Gosteg_!)\n\"What are they saying in Heaven on the subject?\"  His hearers testify\nthat, in such moments, he almost brought them within the precincts of the\nglory.  The effect was thrilling.  And, dealing with alarming truths, he\nwould exclaim, \"Stop! silence!  What do they say in hell on this\nsubject?\"\n\nThe man who can do these things must be no hearsay man, or such\nquestionable excursions of speech would be likely to provoke laughter,\nand contempt, rather than overwhelming awe.  The effect of this preacher\nwas unutterable.  It is said that upon such occasions, had the people\nheard these things from the invisible world, as he expatiated on the\nthings most likely to be uttered, either in Heaven or hell, upon the\nsubject, they could scarcely have been more alarmed.\n\nHis biographer, Mr. Morgan, Vicar of Syston, in Leicestershire, tells how\nhe heard him preaching once to a crowd in the open air, on \"the Last\nDay,\" representing the wicked as \"tares gathered into bundles,\" and cast\ninto the everlasting burnings.  There was a certain flax-dresser, who, in\na daring and audacious way, chose to go on with his work in an open room\nopposite to where Elias was preaching from the platform; but, as the\npreacher grew more and more earnest, and the flames more flashing, the\nterrible fire more and more intense in its vehemence, the man was obliged\nto leave his work, and run into a yard behind his house, to get out of\nthe reach of the cruel flames, and the awful peals of the thunder of the\npreacher's subduing voice.  \"But the awful language of that Elias\nfollowed me there also,\" said the panic-stricken sinner.\n\nThere was a preacher of Caernarvon, one Richardson, a preacher of\npeculiar tenderness, and sweetness, who made his hearers weep beneath the\nlovely message he generally carried.  On one occasion, while Elias was\npouring forth his vehement, and dreadful words, painting the next world\nin very living, and fearful colours, his audience all panic-stricken, and\ncarried along as if they were on the confines of the darkness, and the\ngates opening to receive them, a man, in the agony of his excitement,\ncried out, \"Oh, I wish I could hear Mr. Richardson, of Caernarvon, just\nfor five minutes!\"  No anecdote could better illustrate the peculiar\ngifts, and powers of both men.\n\nJohn Elias was a native of Caernarvonshire.  His parents were people in\nvery humble circumstances, but greatly respected.  His paternal\ngrandfather lived with them.  He was a member of the Church of England.\nHis influence over the mind of Elias appears to have been especially\ngood; and it is, perhaps, owing to this influence that, although he\nbecame a minister, and the eminent pride of the Calvinistic Methodist\nbody, he, throughout his life, retained a strong affection for the\nservices, and even the institution, of the Church of England.  Through\nhis grandfather, he acquired, what was not usual in that day, the\nrudiments of education very early, and as a young child, could read very\nwell and impressively.  Thus, when quite a child, they went together to\nhear some well-known Methodist preacher.  The time for the service had\nlong passed, and the preacher did not arrive.  The old gentleman became\nimpatient, and said to his little grandson, \"It's a pity the people\nshould be idling like this; go up into the pulpit, John, and read a\nchapter to them;\" and, suiting the action to the word, he pushed the\nchild up into the pulpit, and shut the door after him.  With much\ndiffidence, he began to read portions of the Sermon on the Mount, until,\nventuring to withdraw his eye from the Bible, and look aside, lo! to his\ngreat dismay, there was the preacher quietly waiting outside the pulpit\ndoor.  He gently closed the book, and slipped down the pulpit stairs.\nThis was his first appearance in the pulpit.  Little could any one dream\nthat, in after years, he was to be so eminent a master in it.\n\nBut he was only twenty years of age when he began to preach, indeed; and\nit is said that, from the first, people saw that a prophet of God had\nrisen amongst them.  There was a popular preacher, with a very Welsh\nname, David Cadwalladr, who went to hear him; and, after the sermon, he\nsaid, \"God help that lad to speak the truth, for he'll make the people\nbelieve,\u2014he'll make the people believe whatever he says!\"  From the\nfirst, John Elias appears to have been singularly like his two namesakes,\nJohn the Baptist, and Elias the prophet.  He had in him a very tender\nnature; but he was a severe man, and he had a very severe theology.  He\nbelieved that sin held, in itself, very tremendous, and fearful\nconsequences, and he dealt with sin, and sinners, in a very daring, and\neven dreadful manner.\n\nHe appeared in a rough time, when there were, in the neighbourhood,\nrough, cruel, and revolting customs.  Thus, on Whitsunday in each year, a\ngreat concourse of people used to assemble together to burn the ravens'\nnests.  These birds bred in a high and precipitous rock, called _Y\ngadair_ (that is, \"the chair\").  The birds were supposed to prey on young\npoultry, etc., and the people thought it necessary to destroy them; but\nthey always did so on the Sabbath, and it became quite a wild festival\noccasion; and the manner of their destruction was most savage, and\nrevolting.  The nests were beyond their reach; but they suspended a fiery\nfagot by a chain.  This was let down to set the nests on fire; and the\nyoung birds were roasted alive.  At every blaze which was seen below,\ntriumphant shouts rose from the brutal crowd, rending the air.  When the\nsavages had put the birds to death, they usually turned on each other;\nand the day's amusement closed in fights, wounds, bruises, and broken\nbones.  One of the first of Elias's achievements was the daring feat of\ninvading this savage assembly, by proclaiming, in their very midst, the\nwrath of God against unrighteousness, and Sabbath-breaking.  Perhaps, to\nus, the idea of preaching in such a scene seems like the attempting to\nstill a storm by the waving of a feather; but we may also feel that here\nwas a scene in which that terrible eloquence, which was a chief power of\nElias, was well bestowed.  Certainly, it appears chiefly due to Elias\nthat the hideous custom was put down, and put to an end for ever.\n\nIt was no recreative play, no rippling out of mild, meditative, innocent\nyoung sermons, these first efforts of young Elias.  For instance, there\nwas a great fair which was wont to be held at Rhuddlan, in Denbighshire.\nIt was always held on the Lord's Day.  Thither, into the midst of the\nfair, went the young man.  He took his stand on the steps of the New Inn,\nthe noise and business of the fair going on all around him.  His friends\nhad earnestly tried to dissuade, and entreated him not to venture into\nthe midst of so wild, and dangerous a scene.  Farmers were there, to hire\nlabourers; crowds of rough labourers were there.  It was the great\nmarket-day for scythes, and reaping-hooks.  In the booths all round him\nwere the sounds of harps, and fiddles; it was a wild scene of\ndissipation.  There stood the solemn young man, thoughtful, grave, and\ncompassionate.  Of course, he commenced with a very solemn prayer;\npraying so that almost every order of person on the ground felt himself\narrested, and brought, in a solemn way, before God.  Singular effects, it\nis said, seemed to follow the prayer itself.  Then he took for his text\nthe fourth commandment; but he said he had come to open upon them \"the\nwhole ten cannon of Sinai.\"  The effects could hardly have been more\ntremendous had the congregation really stood at the foot of the mountain\nthat \"might not be touched.\"  In any case, Elias was an awful preacher;\nand we may be sure that upon this occasion he did not keep his terrors in\nreserve.\n\nOne man, who had just purchased a sickle, was so alarmed at the\ntremendous denunciations against Sabbath-breakers, that he imagined that\nthe arm which held the sickle was paralysed; he let it fall on the\nground.  He could not take his eye from the preacher; and he feared to\nstoop to pick it up with the other hand, lest that should be paralysed\nalso.  It ought, also, to be said this man became an entirely changed\ncharacter, and lived, to an advanced age, a consistent Christian.  The\ngreat crowd was panic-stricken.  The fair was never after held on the\nLord's Day.  Some person said to Elias, afterwards, that the fair was an\nold custom, and it would recover itself, notwithstanding his\nextraordinary sermon.  Elias, in his dreadful manner, replied, \"If any\none will give the least encouragement to the revival of that fair, he\nwill be accursed before the Holy Trinity, in the name of the Father, the\nSon, and the Holy Ghost!\"  A dreadfully earnest sort of man this.  We are\nnot vindicating his speeches, only giving an account of them.\n\nMr. Jones, the Rector of Nevern, one of the most eminent of the Welsh\nbards, says, \"For one to throw his arms about, is not action; to make\nthis, or that gesture, is not action.  Action is seen in the eye, in the\ncurling of the lip, in the frowning of the nose\u2014in every muscle of the\nspeaker.\"  Mentioning these remarks to Dr. Pugh, when speaking of Elias,\nhe said he \"never saw an orator that could be compared to him.  Every\nmuscle was in action, and every movement that he made was not only\ngraceful, but it spoke.  As an orator,\" said Dr. Pugh, \"I considered him\nfully equal to Demosthenes!\"\n\nIt was tremendous preaching.  It met the state of society\u2014the needs of\nthe times.  What is there in a sermon?\u2014what is there in preaching? some\nhave flippantly inquired.  We have seen that the preaching of Elias\neffected social revolutions; it destroyed bad customs, and improved\nmanners.  He lived in this work; it consumed him.  Those who knew him,\napplied to him the words of Scripture: \"The zeal of Thine house hath\neaten me up.\"  In estimating him, and his work, it ought never to be\nforgotten, that, as has always been the case with such men, he lived in a\nlife of wondrous prayerfulness, and spiritual elevation.  He was called\nto preach a great Association sermon at Pwlheli.  In the whole\nneighbourhood the state of religion was very low, and distressingly\ndiscouraging to pious minds; and it had been so for many years.  Elias\nfelt that his visit must be an occasion with him.  It may almost be said\nof that day, that \"Elias prayed, and the heavens gave rain.\"  He went.\nHe took his text, \"Let God arise, and let His enemies be scattered!\"  It\nwas an astonishing time.  While the preacher drove along with his\ntremendous power, multitudes of the people fell to the ground.  Calm\nstood the man, his words rushing from him like flames of fire.  There\nwere added to the Churches of that immediate neighbourhood, Mr. Elias's\nclerical biographer tells us, in consequence of the powerful impetus of\nthat sermon, two thousand five hundred members.\n\nThe good man lived in an atmosphere of prayer.  The stories which gather\nabout such men, sometimes seem to partake of the nature of exaggerations;\nbut, on the other hand, it ought to be recollected that all anecdotes and\npopular impressions arise from some well-known characteristic to which\nthey are the correspondents.  There was a poor woman, a neighbour's wife.\nShe was very ill, and her case pressed very much upon the mind of Elias\nin family prayer.  But one morning he said to his wife, \"I have somehow\nmissed Elizabeth in my prayer this morning; I think she cannot be alive.\"\nThe words had scarcely passed from his lips when the husband was at the\ndoor, to tell him of his wife's departure.\n\nThere is a singular circumstance mentioned of some horse-races, a great\ndisturbance to the best interests of the neighbourhood; on the day of the\ngreat race, Elias's spirit was very much moved, and he prayed most\npassionately and earnestly that the Lord would do something to put a stop\nto them.  His prayer was so remarkable, that someone said, \"Ahab must\nprepare his chariot, and get away.\"  The sky became so dark shortly\nafter, that the gas was lighted in some of the shops of the town.  At\neleven o'clock the rain began to pour in torrents, and continued until\nfive o'clock in the afternoon of the next day.  The multitudes on the\nrace-ground dispersed in half-an-hour, and did not reassemble that year;\nand what seemed more remarkable was, that the rainfall was confined to\nthat vicinity.  It is our duty to mention these things.  An adequate\nimpression could not be conveyed of the place this man held in popular\nestimation without them.  And his eminence as a preacher was astonishing;\nwherever he went, whatever day of the week, or whatever hour of the day,\nno matter what the time or the season, business was laid aside, shops\nwere closed, and the crowds gathered to hear him.  Sometimes, when it was\narranged for him to preach in a chapel, and more convenient that he\nshould do so, a window was taken out, and there he stood, preaching to\nthe crowded place within, and, at the same time, to the multitudes\ngathered outside.  Mr. Morgan, late vicar of Christ Church, in Bradford,\ngives an account of one of these sermons.  There was a great panorama\nexhibiting at the same time.  Elias took the idea of moving\nsuccession\u2014the panorama of all the miracles wrought by Christ.  It is\neasy to see how, from such lips, a succession of wonderful pictures would\npass before the eye, of living miracles of Divine working,\u2014a panorama of\nwonderful cures.  Mr. Morgan says, \"I was very ill at the time, but that\nstriking sermon animated me, and I have often stirred the cold English\nwith the account of it.\"\n\nWe have said that no sermons are preserved; Elias himself regretted, in\nhis advanced life, that some, which had been of a peculiar interest to\nhim, had gone from him.  Fragments there are, but they are from the lips\nof hearers.  Many of these fragments still present, in a very impressive\nmanner, his rousing, and piercing, and singularly original style; his\npeculiar mode of dealing at will, for his purposes of illustration, with\nthe things of earth, heaven, and hell.\n\nTake one illustration, from the text, \"_Shall the prey be taken from the\nmighty_, _or the lawful captive be delivered_?\"  \"_Satan_!\" he exclaimed,\n\"what do you say?  Shall the prey be taken from the mighty?  'No, never.\nI will increase the darkness of their minds; I will harden more the\nhardness of their hearts; I will make more powerful the lusts in their\nsouls; I will increase the strength of their chains; I will bind them\nhand and foot, and make my chains stronger; the captives shall never be\ndelivered.  Ministers!  I despise ministers!  Puny efforts theirs!'\n'_Gabriel_!' exclaimed the preacher, 'messenger of the Most High God:\nshall the prey be taken from the mighty?'  'Ah!  I do not know.  I have\nbeen hovering over this assembly.  They have been hearing the Word of\nGod.  I did expect to see some chains broken, some prisoners set free;\nbut the opportunity is nearly over; the multitudes are just upon the\npoint of separating; there are no signs of any being converted.  I go\nback from this to the heavenly world, but I have no messages to carry to\nmake joy in the presence of the angels.'\"  There were crowds of preachers\npresent.  Elias turned to them.  \"'What think you?  You are _ministers_\nof the living God.  Shall the prey be taken from the mighty?'  'Ah! who\nhath believed our report? and to whom is the arm of the Lord revealed?\nWe have laboured in vain, and spent our strength for nought; and it seems\nthe Lord's arm is not stretched out.  Oh, there seems very little hope of\nthe captives being delivered!'  '_Zion_!  Church of Christ! answer me,\nShall the prey be taken from the mighty?  What do you say?'  And Zion\nsaid, 'My God hath forgotten me; I am left alone, and am childless.  And\nmy enemies say, This is Zion, whom no man seeketh after.'  Oh, I am\nafraid the prey will not be taken from the mighty\u2014the captive will not be\ndelivered.  _Praying Christians_, what do you think?  'O Lord, Thou\nknowest.  High is Thy hand, and strong is Thy right hand.  Oh that Thou\nwouldst rend the heavens, and come down!  Let the sighing of the prisoner\ncome before Thee.  According to the greatness of Thy power, preserve Thou\nthem that are appointed to die.  I am nearly wearying in praying, and yet\nI have a hope that the year of jubilee is at hand.'\"  Then, at this\npoint, Elias assumed another, higher, and his most serious manner, as if\nabout to speak to the Almighty; and, in quite another tone, he said,\n\"What is the mind of the Lord respecting these captives?  Shall the prey\nbe taken from the mighty?\"  Then he exclaimed, \"'Thus saith the Lord,\nEven the captives of the mighty shall be taken away, and the prey of the\nterrible shall be delivered.'  Ah!\" he exclaimed, \"there is no doubt\nabout the mind and will of the Lord\u2014no room for doubt, and hesitation.\n'The ransomed of the Lord shall return, and come to Zion with songs and\neverlasting joy upon their heads.'\"\n\nThis is the fragment of a sermon preached when Elias was about thirty\nyears of age.  Of course it can give but a very slender idea, but perhaps\nit shows something of the manner of the master.  His imagination was very\nbrilliant, but more chastened, and subdued, than that of many.  His\neloquence, like all of the highest order, was simple, and he trusted\nrather to a fitting word, than to a large furniture of speech.  It is\nsaid that, to his friends, every sermon appeared to be a complete\nmasterpiece of elocution, a nicely-compacted, and well-fitted oration.\n\nAmong the great Welsh preachers, David Davies, and Williams of Wern were,\nlike Rowlands of Llangeitho, comparatively fixtures.  Of course, they\nappeared on great Association occasions.  But John Elias, and Christmas\nEvans itinerated far, and wide.  Unlike as they were in the build of\ntheir minds, and the character of their eloquence, they had a great, and\nmutual, regard, and affection for each other; and it is told how, when\neither preached, the other was seen with anxious interest drinking in,\nwith the crowd, the words of his famous brother.  Theirs are, no doubt,\nthe two darling names most known to the religious national heart of\nWales.  To John Elias it is impossible to render such a mede of justice,\nor to give of his powers even so comprehensive a picture, as is\nattempted, even in this volume, of Christmas Evans.\n\nSomething like an illustration of the man may be gathered from an\nanecdote of the formation of one of the first Bible Societies in North\nWales.  It was a very great occasion.  A noble Earl, the Lord Lieutenant\nof the county, was to take the chair; but when he heard that John Elias\nwas expected to be the principal speaker, he very earnestly implored that\nhe might be kept back, as \"a ranter, a Methodist, and a Dissenter, who\ncould do no good to the meeting.\"  The position of Elias was such that,\nupon such an occasion, no one could have dared to do that; so the noble\nLord introduced him, but with certain hints that \"brevity, and\nseriousness would be desirable.\"  The idea of recommending seriousness to\nJohn Elias, certainly, seems a very needless commendation; but when Elias\nspoke,\u2014partly in English, and partly in Welsh,\u2014especially when, in\nstirring Welsh, he referred to the constitution of England, and the\nrepose of the country, as illustrating the value of the Bible to society,\nand some other such remarks,\u2014of course with all the orator's piercing\ngrandeur of expression,\u2014the chairman, seeing the inflamed state of the\npeople, and himself not well knowing what was said, would have the words\ntranslated to him.  He was so carried away by the dignified bearing of\nthe great orator, that he would have a special introduction to him at the\nclose of the meeting.  A day or two after, a special messenger came to\ninvite him to visit, and spend some time at the house of the Earl.  This,\nhowever, was respectfully declined, for reasons, no doubt, satisfactory\nto Elias, and which would satisfy the peer also, that the preacher had no\ndesire to use his great popularity for his own personal influence, and\naggrandisement.\n\nAfter a life of eminent usefulness, he died, in 1841, at the age of\nsixty-eight.  His funeral was a mighty procession, of about ten thousand\npersons.  They had to travel, a distance of some miles, to the beautiful\nlittle churchyard of Llanfaes, a secluded, and peaceful spot,\u2014a scene of\nnatural romance, and beauty, the site of an old Franciscan monastery,\nabout fourteen miles from Llangefni, the village where Elias died.  The\nday of the funeral was, throughout the whole district, as still as a\nSabbath.  As it passed by Beaumaris, the procession saw the flags of the\nvessels in the port lowered half-mast high; and as they passed through\nBeaumaris town, and Bangor city, all the shops were closed, and all the\nblinds drawn before the windows.  Every kind of denomination, including\nthe Church of England, joined in marks of respect, and justified, more\ndistinctly than could always be done, the propriety of the text of the\nfuneral oration: \"Know ye not that a prince and a great man has fallen?\"\nOf him it might truly be said, \"_Behold I will make thee a new sharp\nthreshing instrument_, _having teeth_: _thou shalt thresh the mountains_,\n_and beat them small_, _and shalt make the hills like chaff_.\"\n\n\n\n\nCHAPTER VII.\n_CONTEMPORARIES\u2014DAVIES OF SWANSEA_.\n\n\nTraditions of his Extraordinary Eloquence\u2014Childhood\u2014Unites in Church\nFellowship with Christmas Evans, and with him preaches his First\nSermon\u2014The Church of Castell Hywel\u2014Settles in the Ministry at Frefach\u2014The\nAnonymous Preacher\u2014Settles in Swansea\u2014Swansea a Hundred Years Since\u2014Mr.\nDavies reforms the Neighbourhood\u2014Anecdotes of the Power of his Personal\nCharacter\u2014How he Dealt with some Young Offenders\u2014Anecdote of a\nCaptain\u2014The Gentle Character of his Eloquence\u2014The Human Voice a Great\nOrgan\u2014The Power of the \"Vox Humana\" Stop\u2014A Great Hymn Writer\u2014His Last\nSermon.\n\nWE shall, in the next chapter, mention several names of men, mightily\ninfluential as Welsh preachers in their own country, and to most English\nreaders utterly unknown.  Perhaps the most conspicuous of these lesser\nknown men is, however, David Davies, of Swansea.  Dr. Thomas Rees, in\nevery sense a thoroughly competent authority, speaks of him as one of the\nmost powerful pulpit orators in his own, or any other, age; and he quotes\nthe words of a well-known Welsh writer, a minister, who says of David\nDavies: \"In his best days, he was one of the chief of the great Welsh\npreachers.\"  This writer continues: \"I may be deemed too partial to my\nown denomination in making such an observation.  What, it may be asked,\nshall be thought of John Elias, Christmas Evans, and others?  In point of\nflowing eloquence, Davies was superior to every one of them, although,\nwith regard to his matter, and the energy, and deep feeling with which he\ntreated his subjects, Elias, in his best days, excelled him.\"  As to this\nquestion of feeling, however, the writer of these pages was talking, some\ntime since, with Dr. Rees himself, about this same David Davies, when the\nDoctor said: \"What the old people tell you about him is wonderful.  It\nwas in his voice\u2014he could not help himself; without any effort, five\nminutes after he began to speak, the whole congregation would be bathed\nin tears.\"\n\nThis great, and admirable man was born in the obscure little village of\nLlangeler, in Carmarthenshire, in June, 1763.  His parents, although\nrespectable, not being in affluent circumstances, could give him very few\nadvantages of education.  Thus it happened that, eminent as he became as\na preacher, as one of the most effective hymn-writers in his language,\nand as a Biblical commentator, he was entirely a self-made man.  However,\nas is so often the case in such instances, his earnest eagerness in the\nacquisition of knowledge was manifest when he was yet very young; and he\nwas under the influence of very strong religious impressions at a very\nearly age.\n\nEven when he was quite a child, he would always stand up, and gravely ask\na blessing on his meals; and it is said that there was something so\nimpressive, and grave, in the manner of the child, that some careless\nfrequenters of the house always took off their hats, and behaved with\ngrave decorum until the short prayer was ended.  His parents were not\nreligious persons, and, therefore, it is yet more remarkable that one\nday, while he was still in his earliest years, his father heard him\nfervently in prayer for them behind a hedge.  It is not wonderful to\nlearn that he was greatly affected by it.  It does not seem that this\ndepth of religious life accompanied him all the way through his boyhood,\nand his youth; but a very early marriage\u2014in most instances, so grave, and\nfatal a mistake\u2014would appear to have been the occasion of the restoration\nof his religious convictions.  He was but twenty when he married Jane\nEvans, a respectable, and lovely young woman of his own neighbourhood;\nand now his religious life began in real earnest.\n\nIt is surely very remarkable, as we have already seen, that he, and\nChristmas Evans were admitted into Church fellowship on the same\nevening,\u2014the Church to which we have already referred,\u2014beneath the\npastorate of the eminent scholar, and bard, David Davies, of Castell\nHywel.  The singularity did not stop here.  Christmas Evans, and the\nyoung Davies, preached their first sermon in the same little cottage, in\nthe parish of Llangeler, within a week of each other.  The two youths\nwere destined to be the most eminent lights of their different\ndenominations, in their own country, in that age; but neither of them\ncontinued long in connection with the Church at Castell Hywel; and as\nthey joined at the same time, so about the same time they left.\n\nDavid Davies, their pastor, was a great man, and an eminent preacher, but\nhe was an Arian, and the Church members were chiefly of the same school\nof thought; and the convictions of both youths were altogether of too\ndeep, and matured an order, to be satisfied by the Arian view of the\nperson, and work of Christ.  Moreover, they both, by the advice of\nfriends, were looking to the work of the Ministry, for which they must\nhave early shown their fitness; and, as we have noticed in the case of\nChristmas Evans, there was a rule in the Church at Castell Hywel, that no\none should be permitted to preach who had not received an academical\ntraining.\n\nThis, in addition to their dissatisfaction with services devoted chiefly\nto the frigid statements of speculative points of doctrine, or the\nillustration of worldly politics, soon operated to move the young men\ninto other fields.  Evans, as we know, united himself with the Baptists;\nDavies found a congenial ministration at Pencadair, under the direction\nof a noted evangelical teacher of those parts, the Rev. William Perkins.\nThere his deepest religious convictions became informed, and\nstrengthened.  Davies was always a man of emotion; it was his great\nstrength when he became a preacher; and his biographer very pleasingly\nstates the relation of his after-work to this moment of his life, when he\nsays that, \"Beneath the teaching of Mr. Perkins, a delightful change came\nover his feelings; he could now see, in the revealed testimony concerning\nthe work finished by our Divine Surety, and Redeemer, enough to give\nconfidence of approach 'into the holiest,' to every one who believes the\nreport of it, as made known to all alike in the Scriptures.  We may\njustly say, 'Blessed are their eyes who see' this; who see that God is\nnow 'reconciling the world unto Himself, not imputing unto men their\ntrespasses.'  They, indeed, see the heavens opened, and the angels of God\nascending, and descending upon the Son of Man.  They see that fulfilled\nwhich was set forth of old in vision to Jacob, the restoration of\nintercourse between earth and heaven through a mediator; and, in the\ndiscovery of it, they walk joyfully in the way of peace, and in the\ngracious presence of their reconciled Father.\"\n\nIt was after this period that the first sermon was preached, in the\ncottage to which we have alluded.  \"The humble beginning of both Davies,\nand Evans, naturally reminds us,\" says Davies' biographer, \"of the\nprogress of an oak from the acorn to the full-grown tree, or that of a\nstreamlet issuing from an obscure valley among the mountains, and\nswelling, by degrees, into a broad, and majestic river.\"  David Davies\nsoon became well known in his neighbourhood as a mighty evangelist.\nHaving grounded his own convictions, and even then possessed of a copious\neloquence, it is not wonderful to read that dead Churches rose into\nnewness of life, and became, in the course of time, flourishing\nsocieties.  He was ordained as a co-pastor with the Rev. John Lewis, at\nTrefach.  The chapel became too small, and a new one was built, which\nreceived the name of Saron.  He became a blessing to Neuaddlwyd, and\nGwernogle; his words ran, like flames of fire, through the whole\ndistrict.  It is said that his active spirit, and fervent style of\npreaching, gave a new tone to the ministry of the Independents throughout\nthe whole Principality.  Hearers, who have been unaccustomed to the\npenetrating, the quietly passionate emotionalness of the great Welsh\npreachers, can scarcely form an idea of the way in which their at once\nhappy, and invincible words would set a congregation on fire.\n\nThe beloved, and revered William Rees, of Liverpool, in his memoir of his\nfather, gives an illustration of this, in connection with a sermon\npreached by Mr. Davies; and it furnishes a striking proof of the force of\nhis eloquence.  The elder Rees speaks of one meeting in particular, which\nhe attended at Denbigh, at the annual gathering of the Independents.  A\nminister from South Wales preached at the service with unusual power, and\neloquence.  Among the auditors, there was a venerable man, named William\nLewis, who possessed a voice loud, and clear as a trumpet, and who was,\nat that time, a celebrated preacher among the Calvinistic Methodists.\nThe southern minister, in full sail, with the power of the \"_hwyl_\"\nstrong upon him, and the whole congregation, of course, in full sympathy,\nall breathless, and waiting for the next word, came to a point in his\nsermon where he repeated, says Mr. Rees, in his most pathetic tones, the\nverse of a hymn, which can only be very poorly conveyed in translation:\u2014\n\n    \"Streams from the rock, and bread from heaven,\n    Were, by their God, to Israel given;\n    While Sinai's terrors blazed around,\n    And thunders shook the solid ground,\n    No harm befell His people there,\n    Sustained with all a Father's care,\n    Perversely sinful though they were.\"\n\nThe drift of the passage was to show that the believer in Christ is just\nas safe amidst terrors from within, and without.  The sentiment touched\nthe electric chord in the hearts of the multitude.  Old William Lewis\ncould bear it no longer.  Up he started, unable to conceal his feelings.\n\"Oh, yes! oh, yes!\" he exclaimed; \"blessed be His name!  God supported\nHis people amidst all the terrors of Sinai, sinful, and rebellious though\nthey were.  That was the most dreadful spot in which men could ever be\nplaced; yet, even there, God preserved His people unharmed.  Oh, yes! and\nthere He sustained me, too, a poor, helpless sinner, once exposed to the\ndoom of His law, and trembling before Him!\"  No sooner had the old man\nuttered these words, than a flame seemed instantaneously to spread\nthrough the whole congregation, which broke forth into exclamations of\njoy, and praise.  But the preacher, who had kindled this wonderful fire,\nand who could do such things!  For some time, Mr. Rees was unable to find\nout who it was; and it was the younger Rees, long the venerable minister\nin Liverpool, who discovered afterwards, from one of his father's old\ncompanions, that it was David Davies, from the south,\u2014he who came to be\ncalled, in his more mature years, \"The great Revivalist of Swansea.\"\n\nFor, after labouring until the year 1802 in the more obscure regions we\nhave mentioned, where, however, his congregations were immense, and his\ninfluence great over the whole Principality, he was invited by the\nChurches of Mynyddbach, and Sketty\u2014in fact, parts of Swansea\u2014to become\ntheir pastor; and on this spot his life received its consummation, and\ncrown.\n\nWhen Mr. Davies entered the town, it was a remarkably wicked spot; the\ncolliers were more like barbarians than the inhabitants of a civilized\ncountry.  Gangs of drunken ruffians prowled through its streets, and the\nsuburbs in different directions, ready to assault, and ill-treat any\npersons who ventured near them.  They were accustomed to attack the\nhouses as they passed, throwing stones at the doors, and windows, and\ncould scarcely open their mouths without uttering the most horrid oaths,\nand blasphemies.  It seems almost strange, to our apprehensions now, that\nthe presence of a preacher should effect a change in a neighbourhood; yet\nnothing is more certain, than the fact that immense social reformations\nwere effected by ministers of the Gospel, both in England, and in Wales.\n\nMr. Davies had not long entered Swansea before the whole neighbourhood\nunderwent a speedy, and remarkable change.  He had a very full, and\nmagnificent voice; a voice of amazing compass, flexibility, and\ntenderness; a voice with which, according to all accounts, he could do\nanything\u2014which could roll out a kind of musical thunder in the open air,\nover great multitudes, or sink to the softest intonations, and whispers,\nfor small cottage congregations.  It was well calculated to arrest a rude\nmultitude.  And so it came about that Mynyddbach became as celebrated for\nthe work of David Davies, as the far-famed Llangeitho for the great work,\nand reformation of David Rowlands.  The people poured in from the country\nround to hear him.  Then, although very tender, and genial, his manner\nwas so solemn, and he had so intense a power of realizing, to others, the\ndeep, and weighty truths he taught, that he became a terror to\nevil-doers.\n\nIt is mentioned that numbers of butchers from the neighbourhood of\nCwmamman, and Llangenie, were in the habit of attending Swansea market on\nSaturdays.  Some of them, after selling the meat which they had brought,\nwere accustomed to frequent the public-houses, and to remain there\ndrinking, and carousing until the Sunday morning.  It is a well-known,\nand amusing circumstance, that, in the course of a little time, when\nproceeding homewards on their ponies, if they caught a glimpse of Mr.\nDavies coming in an opposite direction, they hastily turned round, and\ntrotted off, until they could find a bystreet, or lane, to avoid his\nreproving glances, or warnings, which had the twofold advantage of\npertinency and serious wit, conveyed in tones sufficiently stentorian to\nreach their ears.  And there was a man, proverbially notorious for his\nprofane swearing, who plied a ferry-boat between Swansea, and Foxhole;\nwhenever he perceived Mr. Davies approaching, he took care to give a\ncaution to any who might be using improper expressions: \"Don't swear, Mr.\nDavies is coming!\"\n\nAnd there is another story, which shows what manner of man this Davies\nwas.  One Saturday night, a band of drunken young men, and boys, threw a\nquantity of stones against his door, according to their usual mode of\ndealing with other houses.  While they were busy at their work of\nmischief, he suddenly opened the door, rushed out, and secured two or\nthree of the culprits, who were compelled to give him the names of all\ntheir companions.  He then told them that he should expect every one of\nthem to be at his house on a day which he mentioned.  Accordingly, the\nwhole party came at the appointed hour, but attended by their mothers,\nwho were exceedingly afraid lest the offending lads should be sent to\nprison in a body.  Instead of threatening to take them before the\nmagistrates, Mr. Davies told them to kneel down with him; and having\noffered up an earnest prayer, and affectionately warned them of the\nconsequences of their evil ways, he dismissed them, requesting, however,\nthat they would all attend at Ebenezer Chapel on the following Sunday.\nThey were, of course, glad to comply with his terms, and to be let off so\neasily.  In after years, several of them became members of his Church,\nand maintained through life a consistent Christian profession.  \"And one\nof them,\" said Dr. Rees, when writing the story of his great predecessor,\n\"is an old grey-headed disciple, still living.\"\n\nSuch anecdotes as these show how far the character of the man aided, and\nsustained the mighty power of the minister.  Our old friend, the\nvenerable William Davies, of Fishguard, says: \"I well remember Mr. Davies\nof Swansea's repeated preaching tours through Pembrokeshire, and can\nnever forget the emotions, and deep feelings which his matchless\neloquence produced on his crowded congregations everywhere; he had a\npenetrating mind, a lively imagination, and a clear, distinctive\nutterance; he had a remarkable command of his voice, with such a flow of\neloquence, and in the most melodious intonations, that his enraptured\naudience would almost leap for joy.\"\n\nInstances are not wanting, either in the ancient, or modern history of\nthe pulpit, of large audiences rising from their seats, and standing as\nif all spellbound, while the preacher was pursuing his theme, and, to the\nclose of his discourse, subdued beneath the deepening impression, and\nrolling flow of words.  Perhaps the reader, also, will remember, if he\nhave ever been aware of such scenes, that it is not so much glowing\nsplendour of expression, or the weight of original ideas, still less\nvehement action, which achieves these results, as a certain marvellous,\nand melodious fitness of words, even in the representation of common\nthings.\n\nBut to return to Mr. Davies.  Davies of Fishguard, aforementioned, gives\nan illustration of his preaching: \"The captain of a vessel was a member\nof my Church at Fishguard, but he always attended Ebenezer, when his\nvessel was lying at Swansea.  One day, he asked another captain, 'Will\nyou go with me next Sunday, to hear Mr. Davies?  I am sure he will make\nyou weep.'  'Make _me_ weep?' said the other, with a loud oath.  'Ah!\nthere's not a preacher in this world can make _me_ weep.'  However, he\npromised to go.  They took their seats in the front of the gallery.  The\nirreligious captain, for awhile, stared in the preacher's face, with a\ndefiant air, as if determined to disregard what he might say; but when\nthe master of the assembly began to grow warm, the rough sailor hung down\nhis head, and before long, he was weeping like a child.\"  Here was an\nillustration of the great power of this man to move, and influence the\naffections.\n\nAs compared with other great Welsh preachers, Davies must be spoken of\nas, in an eminent manner, a singer, a prophet of song, and the swell, and\ncadences of his voice were like the many voices, which blend to make up\none complete concert.  He was not only a master of the deep bass notes,\nbut he had a rich soprano kind of power, too; for we read that \"when he\nraised his voice to a higher pitch than ordinary, it increased in melody,\nand power, and its effects were thrilling in the extreme; there were no\njarring notes\u2014all was the music of eloquence throughout.\"  This must not\nbe thought wonderful\u2014it is natural; all men cannot be thus, nor all\npreachers, however good, and great.  There are a few noble organs in the\nworld.  The organ itself, however considered, is a wonderful instrument,\nbut there are some built with such extraordinary art that they are\ncapable of producing transcendent effects beyond most other instruments.\nDavies, the preacher, was one of these amazing organs, in a human frame;\nbut the power of melody was still within his own soul, and it was the\nwonderful score which he was able to read, and which he compelled his\nvoice to follow, which yet produced these amazing effects.\n\nSurely, it is not more wonderful, that the human voice should have its\ngreat, and extraordinary exceptions, than that most wonderful piece of\nmechanism and art, an organ.  We have the organs of Berne, Haarlem, and\nthe Sistine Chapel\u2014such are great exceptions in those powers which art\nexercises over the kingdom of sound; their building, their architecture,\nhas made them singular, and set them apart as great instruments.  But\neven in these, who does not remember the power of the _vox humana_ stop?\nWe apprehend that few who have heard it in the organs of Berne, or\nFribourg, will sympathise with Dr. Burney's irreverent, and ridiculous\ncondemnation of it, in his \"History of Music,\" as the \"cracked voice of\nan old woman of ninety, or Punch singing through a comb.\"  Far from this,\nthe hearer waits with intense anxiety, almost goes to hear this note, and\nrealizes in it, what has been said so truly, that music, as it murmurs\nthrough the ear, is the nurse of the soul.  But all organs have not the\n_vox humana_ stop, nor all preachers either.  The human voice, like the\norgan, is a mighty instrument, but it is the soul which informs the\ninstrument with this singular power, so that within its breast all the\npassions seem to reign in turn.  Singular, that we have thought so much\nof the great organs of the Continent, and have listened with such\nintensity to the great singers, and have failed to apply the reflection\nthat the greatest preachers must be, in some measure, a combination of\nboth.\n\nDavies was one of those preachers, without whose presence the annual\ngatherings, in which the Welsh especially delighted, would have been\nincomplete.  On such occasions, he was usually the last of the\npreachers\u2014the one waited for.  As the service proceeded, it naturally\nhappened that some weariness fell over the assembly; numbers of people\nmight be seen in different parts, sitting, or reclining, on the grass;\nbut as soon as David Davies appeared on the platform, there was a\ngathering in of all the people, pressing forward from all parts of the\nfield, eager to catch every word which fell from the lips of the speaker.\nWhen a great singer appears at a concert, who of all the audience would\nlose a single bar of the melody?  He gave out his own hymn in a voice\nthat reached, without effort, to the utmost limits of the assembled\nmultitude, though he spoke in a quiet, natural tone, without any\nexertion.  He read his text deliberately, but in accents sufficiently\nloud to be heard with ease by ten thousand people.  What is any great\nsinger, without distinctness of enunciation?  And distinct enunciation\nhas always been one of the strong points of the great Welsh preachers.\nHence, from this reason, he was always impressive, and he seldom preached\nwithout using some Scriptural story, which he made to live, through his\naccent, in the hearts of the people; illustrative similes, and not too\nmany of them; striking thoughts, beneath the pressure of which his manner\nbecame more and more impressive, until, at each period, his hearers were\noverpoweringly affected.  Every account of him speaks of his wonderfully\nimpressive voice; and all this gained additional force from his dignified\nbearing, and appearance, which took captive, and carried away, not only\nmore refined intelligences, but even coarsest natures, while the preacher\nnever approached, for a moment, the verge of vulgarity.  Contemporary\npreachers bore testimony that when the skilful singer had closed his\nstrain, the people could not leave the spot, but remained for a long time\nafter, weeping, and praising.\n\nWe have said, already, that Mr. Davies was one of the Welsh hymn-writers;\neighty of his hymns are said to be among the best in the Welsh language.\nHe was a strong man, of robust constitution, but, it may be said, he died\nyoung; before he had reached his fiftieth year, his excessive labours had\ntold visibly on his health, and for many months before his death, he was\nstrongly impressed with the idea that the time of his departure was at\nhand.  He died in the year 1816.  The first Sabbath of that year, he\npreached a very impressive sermon, from the text, \"Thus saith the Lord,\nThis year thou shalt die.\"\n\nHis last sermon was preached about three weeks before he died, when he\nalso administered the ordinance of the Lord's Supper, and gave the right\nhand of fellowship to thirteen persons, on their admission into the\nChurch.  He spoke only a few words during the service, and in those, in\nfaltering accents, told his people he did not expect to be seen amongst\nthem any more.  And, indeed, there was every indication, by his weakness,\nthat his words would be fulfilled.  Every cheek was bedewed with tears.\nThe hearts of many were ready to burst with grief; for this man's\naffections were so great, that he produced, naturally, that grief which\nwe feel when the holders of our great affections seem to be parted from\nus.\n\nHe went home from this meeting to die.  The struggle was not long\nprotracted.  On the morning of December 26th, 1816, he breathed his last.\nOn the day of the funeral, a large concourse, from the town, and\nneighbourhood, followed his remains to the grave.  These lie in a vault,\nwhich now occupies a space in the centre of the new chapel, reared on the\nsite of that in which he ministered so affectionately; and over the\npulpit, a chaste, and beautiful mural marble tablet memorialises, and\nvery conspicuously bears the name of David Davies.  Of him, also, it\nmight be said: \"_The Lord God hath given me the tongue of the learned_,\n_that I should know how to speak a word in season to him that is weary_.\"\n\n\n\n\nCHAPTER VIII.\n_THE PREACHERS OF WILD WALES_.\n\n\nRees Pritchard, and \"The Welshman's Candle\"\u2014A Singular Conversion\u2014The\nIntoxicated Goat\u2014The Vicar's Memory\u2014\"God's better than All\"\u2014Howell\nHarris\u2014Daniel Rowlands at Llangeitho\u2014Philip Pugh\u2014The Obscure\nNonconformist\u2014Llangeitho\u2014Charles of Bala\u2014His Various Works of Christian\nUsefulness\u2014The Ancient Preachers of Wild Wales characterised\u2014Thomas Rhys\nDavies\u2014Impressive Paragraphs from his Sermons\u2014Evan Jones, an Intimate\nFriend of Christmas Evans\u2014Shenkin of Penhydd\u2014A Singular Mode of\nIllustrating a Subject\u2014Is the Light in the Eye?\u2014Ebenezer Morris\u2014High\nIntegrity\u2014Homage of Magistrates paid to his Worth\u2014\"Beneath\"\u2014Ebenezer\nMorris at Wotton-under-Edge\u2014His Father, David Morris\u2014Rough-and-ready\nPreachers\u2014Thomas Hughes\u2014Catechised by a Vicar\u2014Catching the Congregation\nby Guile\u2014Sammy Breeze\u2014A Singular Sermon in Bristol in the Old Time\u2014A\nCloud of Forgotten Worthies\u2014Dr. William Richards\u2014His Definition of\nDoctrine\u2014Davies of Castell Hywel, the Pastor of Christmas Evans, and of\nDavies of Swansea\u2014Some Account of Welsh Preaching in Wild Wales, in\nRelation to the Welsh Proverbs, Ancient Triads, Metaphysics, and\nPoetry\u2014Remarks on the Welsh Language and the Welsh Mind\u2014Its Secluded and\nClannish Character.\n\nAMONGST the characteristic names of Wales, remarkable in that department\nto which we shall devote this chapter, whoever may be passed by, the name\nof Rees Pritchard, the ancient Vicar of Llandovery, ought not to go\nunmentioned.  We suppose no book, ever published in Wales, has met the\nacceptance and circulation of \"Canwyll-y-Cymry,\" or \"The Welshman's\nCandle.\"  Since the day of its publication, it has gone through perfectly\ncountless editions; and there was a time, not long since, when there was\nscarcely a family in Wales, of any intelligence, which did not possess a\ncopy.\n\nIts author was born in the parish of which he became the vicar, so far\nback as 1575.  He was educated at Oxford.  His early life was more\nremarkable for dissipation of every kind, than for any pursuits\ncompatible with his sacred profession.  He was, especially, an inveterate\ndrunkard; the worst of his parishioners were scandalised by his example,\nand said, \"Bad as we may be, we are not half so bad as the parson!\"  The\nstory of his conversion is known to many, who are not acquainted with his\nlife, and work, and the eminence to which he attained; and it certainly\nillustrates how very strange have been some of the means of man's\nsalvation, and how foolish things have confounded the wise.  As George\nBorrow says in his \"Wild Wales,\" in his account of Pritchard, \"God,\nhowever, who is aware of what every man is capable, had reserved Rees\nPritchard for great, and noble things, and brought about his conversion\nin a very remarkable manner.\"\n\nHe was in the habit of spending much of his time in the public-house,\nfrom which he was, usually, trundled home in a wheelbarrow, in a state of\nutter insensibility.  The people of the house had a large he-goat, which\nwent in, and out, and mingled with the guests.  One day, Pritchard called\nthe goat to him, and offered it some ale, and the creature, so far from\nrefusing it, drank it greedily, and soon after fell down in a state of\nintoxication, and lay quivering, to the great delight of Pritchard, and\nhis companions, who, however, were horrified at this conduct in one, who\nwas appointed to be their example, and teacher.  Shortly after, as usual,\nPritchard himself was trundled home, utterly intoxicated.  He was at\nhome, and ill, the whole of the next day; but on the day following, he\nwent down to the public-house, and called for his pipe, and tankard.  The\ngoat came into the room, and again he held the tankard to the creature's\nmouth; but it turned away its head in disgust, hurried away, and would\ncome near him no more.  This startled the man.  \"My God!\" he said, \"is\nthis poor dumb creature wiser than I?\"  He pursued, in his mind, the\ntrain of feeling awakened by conscience; he shrank, with disgust, from\nhimself.  \"But, thank God!\" he said, \"I am yet alive, and it is not too\nlate to mend.  The goat has taught me a lesson; I will become a new man.\"\nSmashing his pipe, he left his tankard untasted, and hastened home.  He,\nindeed, commenced a new career.  He became, and continued for thirty\nyears, a great, and effective preacher; \"preaching,\" says Mr. Borrow,\n\"the inestimable efficacy of Christ's blood-shedding.\"\n\nThose poetical pieces which he wrote at intervals, and which are called\n\"The Welshman's Candle,\" appear only to have been gathered into a volume,\nand published, after his death.  The room in which he lived, and wrote,\nappears to be still standing; and Mr. Borrow says: \"Of all the old houses\nin Llandovery, the old Vicarage is, by far, the most worthy of attention,\nirrespective of the wonderful monument of God's providence, and grace,\nwho once inhabited it;\" and the old vicar's memory is as fresh in\nLlandovery, to-day, as ever it was.  While Mr. Borrow was looking at the\nhouse, a respectable-looking farmer came up, and was about to pass; \"but\nobserving me,\" he says, \"and how I was employed, he stopped, and looked\nnow at me, and now at the antique house.  Presently he said, 'A fine old\nplace, sir, is it not?  But do you know who lived there?'  Wishing to\nknow what the man would say, provided he thought I was ignorant as to the\nancient inmate, I turned a face of inquiry upon him, whereupon he\nadvanced towards me, two or three steps, and placing his face so close to\nmine, that his nose nearly touched my cheek, he said, in a kind of\npiercing whisper, '_The Vicar_!' then drawing his face back, he looked me\nfull in the eyes, as if to observe the effect of his intelligence, gave\nme two or three nods, as if to say, 'He did indeed,' and departed.  _The_\nVicar of Llandovery had then been dead nearly two hundred years.  Truly\nthe man in whom piety, and genius, are blended, is immortal upon earth!\"\n\"The Welshman's Candle\" is a set of homely, and very rememberable verses,\nputting us, as far as we are able to judge, in mind of our Thomas Tusser.\n\nMr. Borrow gives us a very pleasant taste in the following literal,\nvigorous translation, which we may presume to be his own:\u2014\n\n                           \"GOD'S BETTER THAN ALL.\"\n\n    \"God's better than heaven, or aught therein;\n    Than the earth, or aught we there can win;\n    Better than the world, or its wealth to me\u2014\n    God's better than all that is, or can be.\n\n    \"Better than father, than mother, than nurse;\n    Better than riches, oft proving a curse;\n    Better than Martha, or Mary even\u2014\n    Better, by far, is the God of heaven.\n\n    \"If God for thy portion thou hast ta'en,\n    There's Christ to support thee in every pain;\n    The world to respect thee thou wilt gain;\n    To fear thee, the fiend, and all his train.\n\n    \"Of the best of portions, thou choice didst make,\n    When thou the high God to thyself didst take;\n    A portion, which none from thy grasp can rend,\n    Whilst the sun, and the moon on their course shall wend.\n\n    \"When the sun grows dark, and the moon turns red;\n    When the stars shall drop, and millions dread;\n    When the earth shall vanish, with its pomp, in fire,\n    Thy portion shall still remain entire.\n\n    \"Then let not thy heart, though distressed, complain;\n    A hold on thy portion firm maintain.\n    Thou didst choose the best portion, again I say;\n    Resign it not till thy dying day!\"\n\nBut the age of preachers in Wales, to which the following pages will more\nimmediately refer, commences with those two great men, who were indeed\nthe Whitfield, and the Wesley of Wales\u2014Howell Harris of Trevecca, and\nDaniel Rowlands of Llangeitho.  It is remarkable that these two men, born\nto be such inestimable, and priceless blessings to their country, were\nborn within a year of each other\u2014Harris at Trevecca, in 1714, Rowlands at\nPantybeidy, in Cardiganshire, in 1713.  As to Harris, he is spoken of as\nthe most successful preacher that ever ascended a pulpit, or platform in\nWales; and yet nothing is more certain, than that he neither aimed to\npreach, nor will his sermons, so far as any knowledge can be obtained of\nthem, stand the test of any kind of criticism.  This only is certain,\ntheir unquestioned, and greatly pre-eminent usefulness.\n\nHe did not deliver composed sermons, but unpremeditated addresses, on\nsin, and its tremendous consequences; on death, and the judgment, and the\nworld to come.  It is said, \"His words fell like balls of fire, on the\ncareless, and impenitent multitudes.\"  Himself destined for a clergyman\nof the Church of England, an Oxford man, and with a fair promise of\nsuccess in the Church\u2014since before he left Oxford, he had a benefice\noffered him\u2014he repeatedly applied, in vain, for ordination.  Throughout\nhis life, he continued ardently attached to the services of the Church of\nEngland.\n\nIt was, unhappily, from that Church, in Wales, he encountered his most\nvehement opposition, and cruel persecution.  He, however, roused the\nwhole country,\u2014within the Church of England, and without,\u2014from its state\nof apathy, and impiety; while we quite agree with his biographer, who\nsays: \"Any attempt to account philosophically for the remarkable effects\nwhich everywhere attended the preaching of Howell Harris, would be\nnothing better than an irreverent trifling with a solemn subject.  All\nthat can be said, with propriety, is, that he was an extraordinary\ninstrument, raised by Providence, at an extraordinary time, to accomplish\nan extraordinary work.\"\n\nBut Llangeitho, and its vicar, seem to demand a more lengthened notice,\nas coming more distinctly within the region of the palpable, and\napprehensible.  Daniel Rowlands was a clergyman, and the son of a\nclergyman.  At twenty-two years of age, he was appointed perpetual\ncurate, or incumbent, of the united parishes of Nantcwnlle and\nLlangeitho, at a salary of ten pounds a year.  He never received any\nhigher preferment in the Church on earth, although so eminent a blessing\nto his country.  He must have been some such man as our William Grimshaw,\nof Haworth.  When he entered upon his curacy, he was quite an unconverted\nyoung man, given to occasional fits of intoxication, and in the summer he\nleft his pulpit, to take his part, with his parishioners, in the sports,\nand games in the neighbouring fields, or on the village green.\n\nBut, in the immediate neighbourhood of his own hamlet, ministered a good\nand consistent Nonconformist, Philip Pugh, a learned, lovable, and lowly\nman; and, in the smaller round of his sphere, a successful preacher.\nDaniel Rowlands appears to have been converted under a sermon of the\neminent Rev. Griffith Jones of Llanddouror, at Llanddewibrefi; but it was\nto Philip Pugh that he was led for that instruction, and influence, which\ninstrumentally helped to develop his character.  It would seem that\nRowlands was a man bound to be in earnest; but conversion set on fire a\nnew genius in the man.  He developed, hitherto undiscovered, great\npreaching power, and his church became crowded.  Still, for the first\nfive years of his new course of life, he did not know that more glorious\nand beautiful Gospel which he preached through all the years following.\n\nHe was a tremendous alarmist; the dangers of sin, and the terrors of the\neternal judgments, were his topics; and his hearers shrank, and recoiled,\nwhile they were fascinated to listen.  Again, the venerable Nonconformist\nstepped in; Philip Pugh pointed out his defect.  \"My dear sir,\" said he,\n\"preach the Gospel\u2014preach the Gospel to the people.  Give them the balm\nof Gilead; show the blood of Christ; apply it to their spiritual wounds;\nshow the necessity of faith in a crucified Redeemer.\"  \"I am afraid,\"\nsaid Rowlands, \"that I have not all that faith myself, in its full\nvigour, and exercise.\"  \"Preach on it,\" said Mr. Pugh; \"preach on it,\nuntil you feel it in that way,\u2014it will come.  If you go on preaching in\nthe way you have been doing, you will kill half the people in the\ncountry.  You thunder out the curses of the law, and preach in such a\nterrific manner, that nobody can stand before you.  Preach the Gospel!\"\nAnd again the young clergyman followed the advice of his patriarchal\nfriend, and unnumbered thousands in Wales had occasion, through long\nfollowing years, to bless God for it.\n\nDoes not the reader call up a very beautiful picture of these two, in\nthat old and obscure Welsh hamlet, nearly a hundred and fifty years\nsince?\u2014the conversation of such an one as Paul, the aged, with his young\nson, Timothy; and if anything were needed to increase our sense of\nadmiration of the young clergyman, it would be that he did not disdain to\nreceive lessons from old age, and an old age covered with the indignities\nattaching to an outlawed Nonconformist.  In Wales, there were very many\nmen like Philip Pugh; we may incidentally mention the names of several in\nthe course of these pages\u2014names well worthy of the commendation in\nJohnson's perfect lines:\n\n    \"Their virtues walked their narrow round,\n       Nor made a pause, nor left a void;\n    And sure the Eternal Master found\n       Their single talent well employed.\n\n    \"And still they fill affection's eye,\n       Obscurely wise, and coarsely kind;\n    And let not arrogance deny\n       Its praise to merit unrefined.\"\n\nThen there opened a great career before Rowlands, and Llangeitho became\nas a shrine in evangelical Wales.  He received invitations to preach in\nevery neighbourhood of the Principality; many churches were opened to\nhim, and where they were not, he took freely, and cheerfully, to the\nchapels, or the fields.  His words, and accents were of that marvellous\nkind we have identified with Welsh preaching.  Later on, and in other\ntimes, people said, he found his successor in Davies of Swansea; and the\nhighest honour they could give to Swansea, in Davies' day, was that \"it\nwas another Llangeitho.\"\n\nRowlands had the power of the thunder, and the dew; he pressed an\nextraordinary vitality into words, which had often been heard before, so\nthat once, while reading the Church Service, in his own church, he gave\nsuch a dreadful tenderness to the words, \"By thine agony, and bloody\nsweat!\" that the service was almost stopped, and the people broke forth\ninto a passion of feeling.  Christmas Evans says: \"While Rowlands was\npreaching, the fashion of his countenance became altered; his voice\nbecame as if inspired; the worldly, dead, and careless spirit was cast\nout by his presence.  The people, as it were, drew near to the cloud,\ntowards Christ, and Moses, and Elijah.  Eternity, with its realities,\nrushed upon their vision.  These mighty influences were felt, more or\nless, for fifty years.  Thousands gathered at Llangeitho for communion\nevery month, and they came there from every county in Wales.\"\n\nSuch power there is in human words when divinely wielded; such was the\nspiritual power of Daniel Rowlands.  Well does one writer say, the story\nof Llangeitho, well written, would read like a chapter in religious\nromance.  It is very doubtful whether we have the record of any other man\nwho drew such numbers to the immediate circle of his ministry, as\nRowlands.  He did not itinerate so largely as most of the great Welsh\npreachers.  In an obscure spot in the interior of Cardiganshire, in an\nage of bad roads, and in a neighbourhood where the roads were especially\nbad, he addressed his immense concourses of people.  His monthly\ncommunion was sometimes attended by as many as three thousand\ncommunicants, of whom, often, many were clergymen.  Upwards of a hundred\nministers ascribe to him the means of their conversion.  Thus, in his\nday, it was a place of pilgrimages; and even now, there are not a few who\nturn aside, to stand, with wonder, upon the spot where Rowlands exercised\nhis marvellous ministry.\n\nThe four great Welsh preachers, Christmas Evans, John Elias, Williams of\nWern, and Davies of Swansea, on whose pulpit powers, and method, we have\nmore distinctly dilated, may be styled the tetrarchs of the pulpit of\nWild Wales of these later times.  Their eminence was single, and\nsingular.  Their immense powers unquestioned: rivals, never, apparently,\nby their own selection, the great Welsh religious mind only rivalled them\nwith each other.  After them it might be said, \"Great was the company of\npreachers,\"\u2014great, not merely in number, carrying also influence, and\nusefulness of another kind; perhaps even superior to those honoured\nnames.\n\nHow, for instance, can we do sufficient honour to the labours of CHARLES\nOF BALA?  This truly apostolic man was born at Llanvihangel, in 1755.\nWhile yet a boy, he managed to introduce family worship into his father's\nhouse; but it was in his eighteenth year that he heard the great Daniel\nRowlands preach, and he says: \"From that day I found a new heaven, and a\nnew earth, to enjoy; the change experienced by a blind man, on receiving\nhis sight, is not greater than that which I felt on that day.\"  In his\ntwentieth year he went to Oxford, and received Deacon's orders, and was\nappointed to a curacy in Somersetshire; he took his degree at his\nUniversity, but he could never obtain priest's orders; in every instance\nobjection was made to what was called his Methodism.\n\nThe doors of the Establishment were thus closed against him, and he was\ncompelled to cast in his lot with the Welsh Methodists, in 1785.  Before\nthis, he had preached for Daniel Rowlands in his far-famed church at\nLlangeitho, and the great old patriarch simply uttered a prophecy about\nhim when he said, \"Mr. Charles is the gift of God to North Wales.\"  He\nwas an eminent preacher, but it was rather in other ways that he became\nillustrious, in the great religious labours of his country.  Moving about\nto preach, from place to place, his heart became painfully impressed, and\ndistressed, by the great ignorance of the people everywhere, and that\nsuch multitudes were unable to read the Word of God; so he determined on\nthe establishment of schools upon a singular principle.\n\nIt was two or three years before he commenced his more settled labours in\nWales, that Robert Raikes had originated the Sunday-school idea in\nGloucester.  Thomas Charles was the first to seize upon the idea, and\nintroduce it into his own country.  Charles had an organizing, and\nadministrative, mind; he fixed upon innumerable places, where he settled\nschoolmasters, for periods of from six to nine, and twelve months, to\nteach the people to read, giving them the initial elements, and\nrudiments, of education, and then removing these masters to another\nlocality.\n\nSo he filled the country with schools\u2014Sabbath, and night-schools.  He\nvisited the schools himself, periodically, catechizing the children\npublicly; and in the course of his lifetime, he had the satisfaction of\nseeing the aspect of things entirely changed.  He used no figure of\nspeech, when, towards the close of his life, he said, \"The desert\nblossoms as the rose, and the dry land has become streams of water.\"  To\nthese purposes of his heart he was able to devote whatever money he\nreceived from the work of the ministry; he testifies affectionately that\n\"the wants of my own family were provided for by the industry of my dear\nwife;\" and he received some help by donations from England.  He found,\neverywhere, a dearth of Bibles, and it is curious to read that, although\nthe Church of England would not receive him as one of her ministers, when\nhis work became established, the Society for Promoting Christian\nKnowledge made him, after considerable reluctance, a grant of no less\nthan ten thousand Welsh Bibles.  After this, he went to London, for the\npurpose of establishing a Society to supply Wales with the Holy\nScriptures.  It was at a meeting of the Religious Tract Society, which\nwas called together for that purpose, that it was resolved to establish\nthe British and Foreign Bible Society; and before that society had been\nestablished ten years, it had supplied Wales with a hundred thousand\ncopies of the Word of God.\n\nOther men were great preachers, but Thomas Charles was, in the truest\nsense of the word, a bishop, an overseer,\u2014travelling far, and wide,\npreaching, catechizing, administrating, placing and removing labourers.\nAll his works, and words, his inward, and his outward life, show the\nactive, high-toned saintliness, and enthusiastic holiness, of the man.\nThere is, perhaps, no other to whom Wales is so largely indebted for the\ngiving direction, organization, and usefulness to all religious labour,\nas to him.  His modesty transcended his gifts, and his activity.  John\nCampbell, of Kingsland, himself noted in all the great, and good works of\nthat time, relates that at a meeting, at Lady Anne Erskine's, at which\nMr. Charles was requested to state the circumstances which had made\nlittle Bala a kind of spiritual metropolis of the Principality of Wales,\n\"he spoke for about an hour, and never once mentioned himself, although\nhe was the chief instrument, and actor, in the whole movements which had\nmade the place so eminent.\"\n\nThis good man, John Campbell, afterwards wrote to Mr. Charles's\nbiographer: \"I never was at Bala but once, which was not long after his\nremoval to the regions of immortality; and such was my veneration for his\ncharacter, and labours, that, in approaching it, I felt as if I was about\ncoming in sight of Sinai, or Jerusalem, or treading on classical ground.\nThe events of his life, I believe, are viewed with more interest by the\nglorified than the battles of Actium, or Waterloo.\"\n\nBut, as a preacher, he was unlike those men, whose words moved upon the\nwheels of thunder, and who seemed to deal with the lightnings of\nimagination, and eloquence.  As we read his words, they seem to flow with\nrefreshing sweetness.  He was waited for, and followed everywhere, but\nhis utterances had nothing of the startling powers we have seen; we\nshould think he preached, rather, to those who knew, by experience, what\nit is to grow in grace.  There is a glowing light of holiness about his\nwords\u2014a deep, sweet, experimental reality.  Of course, being a Welshman,\nhis thoughts were pithily expressed.  They were a sort of spiritual\nproverbs, in which he turned over, again and again, some idea, until it\nbecame like the triads of his country's literature; and dilating upon an\nidea, the various aspects of it became like distinct facets, setting\nforth some pleasant ray.\n\nSuch was Thomas Charles.  Wales lost him at the age of sixty\u2014a short\nlife, if we number it by years; a long life, if we consider all he\naccomplished in it; and, to this day, his name is one of the most revered\nthroughout the Principality.\n\nIt is impossible to do the justice even of mentioning the names of many\nof those men, who \"served their generation\" so well, \"according to the\nwill of God, and then fell asleep.\"  And it is as necessary, as it is\ninteresting, to notice how the various men, moved by the Spirit of God,\nfound Him leading, and guiding them in the path of labour, their\ninstincts chose.\n\nIn the history of preaching, we believe there is no more curious chapter\nthan this, of these strange preachers in Wales.  They have an\nidiosyncrasy as entirely, and peculiarly, their own, as is that of the\ncountry in which they carried on their ministrations.  The preaching\nfriars of the times we call the dark, or middle ages, are very\nremarkable, from the occasional glimpses we are able to obtain of them.\nVery remarkable the band of men, evoked by the rise of Methodism in\nEngland,\u2014those who spread out all over the land, treading the paths\nindicated by the voice, and finger of Whitfield, or Wesley.  Very\nentertaining are the stories of the preachers of the backwoods of\nAmerica, the sappers, and miners, who cleared a way for the planting of\nthe Word among the wild forests of the Far West.\n\nThese Welsh preachers were unlike any of them,\u2014they had a character\naltogether their own.  A great many of them were men of eminent genius,\nglowing with feeling, and fancy; never having known college training, or\nculture, they were very often men who had, somehow, attained a singular\nvariety of knowledge, lore, and learning, which, perhaps, would be\ndespised as unscientific, and unclassified, by the schools, but which was\nnot the less curious, and, to the Celtic mind, enchanting.\n\nThey all lived, and fared hard; all their thoughts, and fancies were\nhigh.  If they marched before us now, the nineteenth century would, very\nlikely, regard them as a set of very rough tykes.  Perhaps the nineteenth\ncentury would regard Elijah, Amos, and Nahum, and sundry other equally\nrespectable persons, in much the same manner.  Rude, and rough in gait,\nand attire, the rudeness, and the roughness would, perhaps, be forgotten\nby us, if we could interpret the torrent, and the wail of their speech,\nand be, for a short time, beneath the power of the visions, of which they\nwere the rapt seers, and unveilers.  We wonder that no enthusiastic\nWelshman has used an English pen to pourtray the lives, and portraits of\na number of these Welsh worthies; to us, several of them\u2014notably, John\nElias, and Christmas Evans\u2014seem to realize the idea of the Ancient\nMariner,\u2014\n\n    \"I pass like night from land to land,\n       I have strange power of speech;\n    The moment that his face I see,\n    I know the man that must hear me\u2014\n       To him my tale I teach.\"\n\nFor instance, how many people in England ever heard the name of THOMAS\nRHYS DAVIES, an extraordinary man?  And he left an extraordinary diary\nbehind him, for he seems to have been a very methodical man; and his\ndiary shows that he preached during his lifetime at least 13,145 times,\nand this diary contains a distinct record of the time, place, and text;\nand it is said that there is scarcely a river, brook, or tarn, from\nConway to Llansanan, from Llanrwst to Newbridge, from the sea at\nLlandudno, to the waters of the Berwyn mountains, in whose waves he had\nnot baptized.\n\nIn fact, he was, perhaps, in his own particular, and peculiar line,\nsecond to none of the great Welsh preachers; only, it is said that his\npower was inexplicable, and yet that it stood the severest tests of\npopularity.  His sermons are said to have been exceedingly simple, and\nvery rememberable; they sprang out of a rare personal charm; he was\nhimself; but, perhaps, if he resembled one of his great brethren, it\nwould be Williams of Wern.  His style was sharp, pointed, axiomatic, but\nantithetic, never prodigal of words, his sermons were short; but he was\nable to avail himself of any passing circumstance in the congregation,\nand to turn it to good account.  Once, when a congregation seemed to be\neven more than usually disposed to cough, he said, \"Cough away, my\nfriends, it will not disturb me in the least; it will rather help me than\nnot, for if you are coughing, I shall be sure that you are awake.\"\n\nHe had that rare gift in the preacher, perfect self-possession, the grand\npreliminary to mastery over a congregation, an entire mastery over\nhimself.  All great Welsh preachers, however they may sometimes dilate,\nand expand truths into great paintings, and prolonged descriptions, excel\nin the pithy, and proverb-uttering power; but Thomas Rhys Davies was\nremarkable in this.  Here are a few illustrations:\u2014\n\n    \"Ignorance is the devil's college.\"\n\n    \"There are only three passages in the Bible which declare what God\n    is, although there are thousands which speak about Him.  God is a\n    Spirit, God is Light, and God is Love.\"\n\n    \"Pharaoh fought ten great battles with God, and did not gain one.\"\n\n    \"The way through the Red Sea was safe enough for Israel, but not for\n    Pharaoh; he had no business to go that way, it was a private road,\n    that God had opened up for His own family.\"\n\n    \"Let the oldest believer remember that Satan is older.\"\n\n    \"Christ is the Bishop, not of titles, but of souls.\"\n\n    \"Moses was learned, but slow of speech; it was well that he was so,\n    or, perhaps, he would not have found time to write the law.  Aaron\n    had the gift of speech, and it does not appear that he had any other\n    gift.\"\n\n    \"If you have no pleasure in your religion, make haste to change it.\"\n\n    \"Judas is much blamed for betraying Christ for three pounds; many, in\n    our day, betray Him a hundred times for three pence.\"\n\n    \"Pharaoh commanded that Moses should be drowned; in after days,\n    Pharaoh was paid back in his own coin.\"\n\n    \"Many have a brother's face, but Christ has a brother's heart.\"\n\nSuch was Thomas Rhys Davies; like Christmas Evans, journeying from North\nthrough South Wales, he was taken ill in the same house in which\nChristmas Evans died.  Conscious of his approaching death, he begged that\nhe might die in the same bed; this was not possible, but he was buried in\nthe same grave.\n\nThen there was EVAN JONES; he had been a _prot\u00e9g\u00e9_ of Christmas Evans;\nChristmas Evans appears to have brought him forward, giving his verdict\non his suitability as to the ministry.  Christmas Evans was able to\nappreciate the young man, for he seems to have possessed really brilliant\npowers; in his country, and in his land's language, he attained to the\ndistinction of a bard; and it is said that his poetry rose to an\nelevation of wild, and daring grandeur.  As a preacher, he does not\nappear to have studied to be popular, or to seek to adapt his sermons to\nthe multitude; he probably moved through cloudy grandeurs, from whence,\nhowever, he sometimes descended, with an odd quaintness, which, if always\nsurprising, was sometimes reprehensible.  Once, he was expatiating,\nglowingly, on the felicities of the heavenly state, in that tone, and\nstrain which most preachers love, occasionally, to indulge, and which\nmost hearers certainly, occasionally, enjoy; he was giving many\ndescriptive delineations of heavenly blessedness, and incidentally said,\n\"There they neither marry, nor are given in marriage.\"  There was sitting\nbeneath him a fervent brother, who, probably, not knowing what he said,\nsounded forth a hearty \"Amen!\"  Evan heard it, looked the man full in the\nface, and said, \"Ah, you've had enough of it, have you?\"\n\nThis man was, perhaps, in his later years, the most intimate friend of\nChristmas Evans.  Christmas poured his brilliant imagination, couched in\nhis grand, although informal, rhetoric over the multitudes; Evan Jones\nfrequently soared into fields whither, only here and there, an eye could\nfollow his flight; but when the two friends were alone, their spirits\ncould mingle pleasantly, for their minds were cast very much in the same\nmould; and when Christmas Evans died, it was this friend who published in\nWelsh one of the most graceful tributes to his memory.\n\nIn the history of the preaching, and preachers of a hundred years since,\nwe meet, of course, with many instances of men, who possessed\nconsiderable power, but allied with much illiterate roughness; still, the\npower made itself very manifest\u2014a power of illustrating truth, and making\nit clearly apprehended.  Such a preacher must SHENKIN OF PENHYDD have\nbeen, rough, and rude farmer as he was, blending, as was not at all\nuncommon then, and even in our own far more recent knowledge, the\noccupations of a farmer, and the ordained minister.  Shenkin has left a\nvery living reputation behind him; indeed, from some of the accounts we\nhave read of him, we should regard him as quite a type of the rude, yet\nvery effective, Welsh orator.\n\nWhatever the Welsh preacher had to say, however abstract, it had to be\ncommitted to an illustration, to make it palpable, and plain.  In those\nearly times, a very large room, or barn, in which were several hundreds\nof people, would, perhaps, have only one solitary candle, feebly\nglimmering over the gloom.  It was in such circumstances, or such a\nscene, that Shenkin was once preaching on Christ as the Light of the\nworld.  In the course of his sermon, he came to show that the world was\nnot its own light, and announced to his hearers what, perhaps, might\nstartle some of them, that \"light was not in the eye.\"  It seemed as if\nhe had no sooner said this, than he felt it to be a matter that required\nillustration.  As he warmed with his subject, going round, and round to\nmake his meaning plain, but all the time seeming to fear that he was not\ndoing much towards it with his rustic congregation, he suddenly turned to\nthe solitary candle, and blew it out, leaving his congregation in utter\ndarkness.  \"There,\" he exclaimed, triumphantly, to his invisible\ncongregation, \"what do you say to that?  Is the light in the eye?\"  This,\nof course, settled the matter in the minds of the most obtuse; but it was\nstill a serious matter to have to relight, in a lonely little chapel, an\nextinguished candle.\n\nHe was a singular creature, this Shenkin.  Not many Welsh preachers have\na greater variety of odd stories told than he, of his doings, and\nsayings.  He had a very downright, and straightforward method of speech.\nThus, he would say, \"There are many who complain that they can scarcely\nremember anything they hear.  Have done with your lying!\" he exclaimed.\n\"I'll be bound to say you remember well what you sold your old white\nhorse for at Llandaff fair three years ago.  Six or seven pounds, was it?\nCertainly that has not escaped your memory.  You can remember anything\nbut the Gospel.\"  And many of his images were much more of the\nrough-and-ready, than of the classical, order.  \"Humility,\" he once said,\n\"is as beautiful an ornament as a cow's tail; but it grows, like the\ncow's tail, downwards.\"\n\nWales was covered with men like this.  Every district possessed them, and\nmany of them have found their memorial in some little volume, although,\nin most instances, they only survive in the breath of popular\nremembrance, and tradition.\n\nOne of the mightiest of these sons of thunder, who has left behind him a\nname, and fame, scarcely inferior to the great ones on whom we have more\nlengthily dwelt, was EBENEZER MORRIS.  He was a fine, free, cheerful\nspirit; his character sparkled with every Christian virtue,\u2014a man of rare\ngifts, and grace.  With a severe sense of what was just in the relations\nof life, and what constituted the principles of a strong theology,\nkeeping his unblemished course beneath the dominion of a peaceful\nconscience, he enjoyed, more than many, the social fireside chat, with\ncongenial friends.  Although a pastor, and a preacher of wide fame, he\nwas also a farmer; for he was one of an order of men, of whom it has been\nsaid, that good people were so impressed with the privilege conferred by\npreaching the gospel, that their hearers were careful not to deprive them\nof the full enjoyment of it, by remunerating their labours too\nabundantly.\n\nEbenezer Morris held a farm, and the farmer seems to have been worthy of\nthe preacher.  A story is told of him that, wanting to buy a cow, and\ngoing down to the fair, he found one for sale which he thought would suit\nhim, and he bought it at the price named by its owner.  Some days after,\nMr. Morris found that the price of cattle had gone up considerably, and\nmeeting the previous owner of the cow, he said, \"Look here, I find you\ngave me too great a bargain the other day; the cow is worth more than I\npurchased her for,\u2014here is another guinea; now I think we shall be about\nright.\"\n\nThere are several stories told, in the life of this good, and great man,\nshowing that he could not take an unfair advantage, that he was above\neverything mean, unfair, and selfish, and that guineas, and farms weighed\nnothing with him in the balance against righteousness, and truth.  His\ninfluence over his whole country was immense; so much so, that a\nmagistrate addressed him once in public, saying, \"We are under great\nobligations to you, Mr. Morris, for keeping the country in order, and\npreserving peace among the people; you are worth more than any dozen of\nus.\"  On one occasion he was subp\u0153naed, to attend before a court of\njustice, to give evidence in a disputed case.  As the book was handed to\nhim, that he might take the oath, the presiding magistrate said, \"No! no!\ntake it away; there is no necessity that Mr. Morris should swear at all;\nhis word is enough.\"\n\nHis appearance in preaching, his entire presence, is described as most\nmajestic, and commanding: his voice was very loud, and it is said, a word\nfrom his mouth would roll over the people like a mighty wave.  \"Look at\nthat window,\" said an aged deacon, in North Wales, to a minister, who had\ncome to preach at the chapel to which the former belonged, \"look at that\nwindow!  It was there that Ebenezer Morris stood, when he preached his\ngreat sermon from the words, 'The way of life is above to the wise, that\nhe may depart from hell beneath,' and when we all turned pale while we\nwere listening to him.\"  \"Ah!\" said the minister, \"do you remember any\nportion of that sermon?\"  \"Remember!\" said the old deacon; \"remember, my\ngood man?  I should think I do, and shall remember for ever.  Why, there\nwas no flesh here that could stand before it!\"  \"What did he say?\" said\nthe minister.  \"Say! my good man,\" replied the deacon; \"say?  Why, he was\nsaying, 'Beneath, beneath, beneath!  Oh, my people, hell is beneath,\nbeneath, _beneath_!' until it seemed as if the end of the world had come\nupon us all in the chapel, and outside!\"\n\nWhen Theophilus Jones was selected as Rowland Hill's co-pastor at\nWooton-under-Edge, Ebenezer Morris came to preach on his induction.  In\nthat place, the audience was not likely to be a very sleepy one, but this\npreacher roused them beyond their usual mark, and strange stories are\ntold of the sermon, while old Rowland sat behind the preacher,\nejaculating the whole of the time; and many times after, when Mr. Hill\nfound the people heavy, and inattentive, he was in the habit of saying,\n\"We must have the fat minister from Wales here, to rouse you up again!\"\nWe know his likeness very well, and can almost realize his grand, solemn\nmanner, in his black velvet cap, which made him look like a bishop, and\ngave much more impressiveness to his aspect, than any mitre could have\ndone.\n\nThis Ebenezer Morris was the son of a man eminent in his own day, David\nMorris, of whom it was said, that he scarcely ever preached a sermon\nwhich was not the means of the conversion of men, and in his evangelistic\ntours he usually preached two, or three times a day.  There is a sermon,\nstill spoken of, preached at Rippont Bridge, Anglesea.  The idea came to\nhim whilst he was preaching, that many of the people before him might\nsurely be lost, and he burst forth into a loud dolorous wail, every line\nof his countenance in sympathy with his agonizing cry, in Welsh, which no\ntranslation can render, \"O bobl y golled fawr! y golled fawr!\"  The\nEnglish is, \"O ye people of the great loss! the great loss!\"  It seems\nslight enough to us, but it is said that the people not only moved before\nhis words, like reeds in a storm, but to this day they speak in Anglesea\nof David Morris's sermon of \"The Great Loss.\"\n\nThe great authority for the most interesting stories of the religious\nlife in Wales, is the \"History of Welsh Methodism,\" by the late Rev. John\nHughes, of Liverpool; unfortunately, we believe it only exists in Welsh,\nin three volumes, amounting to nearly two thousand pages; but \"Welsh\nCalvinistic Methodism; a Historical Sketch,\" by the Rev. William\nWilliams, appears to be principally a very entertaining digest, and\ncondensation, of many of the most noticeable particulars from the larger\nwork.  There have certainly appeared, from time to time, many most\ninteresting, and faithful men in the ministry of the Gospel in Wales,\nquite beyond the possibility of distinct mention; some of them were very\npoor, and lowly in life, and circumstances.  Such was THOMAS HUGHES.  He\nis described as a man of small talent, and slender knowledge, but of\ngreat holiness, and with an intense faith that many of his neighbours\nwere in a very bad condition, and that it was his duty to try to speak\nwords to them, whereby they might be saved.  He used to stand under the\nold walls of Conway, and numbers gathered around him to listen; until at\nlast he excited the anger of the vicar, who caused him to be arrested,\nand brought into his presence, when the following conversation took\nplace:\u2014\n\n_Vicar_.  \"You ought to be a learned man, to go about, and to be able to\nanswer deep questions.\"\n\n_Hughes_.  \"What questions, sir?\"\n\n_Vicar_.  \"Here they are\u2014those which were asked me by the Lord Bishop.\nLet's see whether you will be able to answer them.  Where was St. Paul\nborn?\"\n\n_Hughes_.  \"In Tarsus.\"\n\n_Vicar_.  \"Hem!  I see that you know something about it.  Well, can you\ntell me who took charge of the Virgin Mary after our blessed Redeemer was\ncrucified?\"\n\n_Hughes_.  \"John.\"\n\n_Vicar_.  \"Well, once again.  Who wrote the Book of Revelation?  Answer\nthat if you can.\"\n\n_Hughes_.  \"John the Apostle.\"\n\n_Vicar_.  \"Ho! you seem to know a good deal, after all.\"\n\n_Hughes_.  \"Perhaps, sir, you will allow me to ask you one or two\nquestions?\"\n\n_Vicar_.  \"Oh yes; only they must be religious questions.\"\n\n_Hughes_.  \"What is holiness? and how can a sinner be justified before\nGod?\"\n\n_Vicar_.  \"Ho! we have no business to bother ourselves with such things,\nand you have no business to put such questions to a man in my position;\ngo out of my sight, this minute.\"  And to the men who had brought him,\n\"Take care that you do not bring such people into my presence any more.\"\n\nHughes was a simple, earnest, believing man, with a good deal of Welsh\ncuteness.  After this interview with the vicar, he was permitted to\npursue his exhortations at Conway in peace.  But there is a place between\nConway, and Llandudno, called Towyn Ferry; it was a very ignorant little\nnook, and the people were steeped in unbelief, and sin; thither Hughes\ndetermined to go, but his person was not known there.  The news, however,\nwas circulated abroad, that there was to be a sermon, and religious\nservice.  When he arrived, he found things did not appear very pleasant;\nthere were heaps of stones prepared for the preacher's reception, when he\nshould make his appearance, or commence his work.  Hughes had nothing\nclerical in his manner, or garb, any more than any one in the crowd, and\nno one suspected him to be the man, as he threw himself down on the\ngrass, and entered familiarly into conversation with the people about\nhim.  After a time, when their patience began to fail, he stood up, and\nsaid, \"Well, lads, there is no sign of any one coming; perhaps the man\nhas heard that you are going to stone him; let one of us get up, and\nstand on that heap of stones, and talk, and the rest sing.  Won't that be\nfirst-rate?\"\n\n\"Capital,\" said a bully, who seemed to be the recognised leader of the\ncrowd.  \"You go on the heap, and preach to us.\"\n\n\"Very well,\" said Hughes, \"I'm willing to try; but mind you, I shall make\nsome blunders, so you must be civil, and not laugh at me.\"\n\n\"I'll make 'em civil,\" said the bully.  \"Look here, lads, whoever laughs,\nI'll put one of these stones into his head!\"\n\n\"Stop you!\" said Hughes; \"the first thing we have to do, is to pray,\nisn't it?\"\n\n\"Ay, ay!\" said the bully, \"and I'll be clerk.  I'll stand before you, and\nyou shall use my shoulder for the pulpit.\"\n\nSo prayer was offered, short, and simple, but in real earnest; and at its\nclose, a good many favourable words were uttered.  Some volunteered the\nremark that, \"It was every bit as good as a parson.\"  Hughes proceeded to\ngive out a text, but the bully shouted,\u2014\n\n\"Hold on, you fool! we've got to sing first.\"\n\n\"Ay, ay!\" said Hughes, \"I forgot that.\"\n\nSo they sang a Welsh hymn, after a fashion, and then came the text, and\nthe sermon, which was short, and simple too, listened to very\nattentively; and the singular part of the story is, that the bully, and\nclerk, left the ground with the preacher, quieted, and changed, and\nsubsequently he became a converted man.  The regeneration of Wales,\nthrough its villages, and lone remote districts, is full of anecdotes\nlike this,\u2014stories of persecution, and the faithful earnestness of simple\nmen, who felt in them a strong desire to do good, and fulfilled their\ndesire, becoming humble, but real blessings to their neighbourhoods.\n\nOnly in a history of the Welsh pulpit\u2014and that would be a volume of no\nslight dimensions\u2014would it be possible to recapitulate the names of the\nmen who exercised, in their day, considerable influence over the\nscattered thousands of the Principality.  They constitute a very varied\nrace, and were characterized by freshness, and reality, taking, of\ncourse, the peculiar mental complexion of the preacher: some calm, and\nstill, but waving about their words like quiet lightnings; some vehement,\noverwhelming, passionate; some remarkable for their daring excursions of\nimagination; some abounding in wit, and humour.  One of the most\nremarkable of these last, one who ought not to go unmentioned in such an\nenumeration, was SAMUEL BREEZE.  This was the man who first introduced\n\"The Churchyard World\" to Dr. Raffles,\u2014of whom it was said, that if you\nheard one of his sermons, you heard three preachers, so various were not\nonly the methods of his sermons, but even the tone of his voice.  He is\nsaid to have produced extraordinary effects.  Christmas Evans said of\nhim, that \"his eyes were like a flame of fire, and his voice like a\nmartial strain, calling men to arms.\"\n\nThe writer of this volume, in a work on the \"Vocation of the Preacher,\"\nmentions a curious instance, which he gives from the unpublished\nreminiscences of a dear departed friend\u2014the Rev. John Pyer, late of\nDevonport\u2014who was present when the incident happened, in Bristol, perhaps\nnearly eighty years since.  Sammy Breeze, as he was familiarly called by\nthe multitudes who delighted in his ministry, came, periodically, from\nthe mountains of Cardiganshire, or the neighbourhood of Aberystwith, to\nBristol, where he spoke with more than tolerable efficiency in English.\nMr. Pyer, then a youth, was in the chapel, when, as was not unusual, two\nministers, Sammy Breeze and another, were to preach.  The other took the\nfirst place, a young man with some tints of academical training, and some\nof the livid lights of a then only incipient rationalism in his mind.  He\ntook for his text, \"He that believeth shall be saved, and he that\nbelieveth not shall be damned;\" but he condoned the heavy condemnation,\nand, in an affected manner, shaded off the darkness of the doom of\nunbelief, very much in the style of the preacher in Cowper's satire, who\nnever mentioned hell to ears polite.  The young man, also, grew\nsentimental, and \"begged pardon\" of an audience, rather more polite than\nusual, for the sad statement made in the text.  \"But, indeed,\" said he,\n\"he that believeth shall be saved, and he that believeth not\u2014indeed, I\nregret to say, I beg your pardon for uttering the terrible truth, but,\nindeed, he shall be sentenced to a place which here I dare not mention.\"\n\nThen rose Sammy Breeze.  He began: \"I shall take the same text, to-night,\nwhich you have just heard.  Our young friend has been fery fine to-night,\nhe has told you some fery polite things.  I am not fery fine, and I am\nnot polite, but I will preach a little bit of truth to you, which is\nthis: 'He that believeth shall be saved, and he that believeth not shall\nbe damned,' _and I begs no pardons_.\"  He continued, \"I do look round on\nthis chapel, and I do see people all fery learned and in-tel-lect-u-al.\nYou do read books, and you do study studies, and fery likely you do think\nthat you can mend God's Book, and are fery sure you can mend me.  You\nhave great\u2014what you call thoughts, and poetries; but I will tell you one\nlittle word, and you must not try to mend that; but if you do, it will be\nall the same; it is this, look you: 'He that believeth shall be saved,\nand he that believeth not shall be damned, _and I begs no pardons_.  And\nthen I do look round your chapel, and I do see you are a foine people,\nwell-dressed people, well-to-do people.  I do see that you are fery rich,\nand you have got your moneys, and are getting fery proud; but I tell you,\nit does not matter at all; for I must tell you the truth, and the truth\nis, 'He that believeth shall be saved, and he that believeth not shall be\ndamned,' _and I begs no pardons_.  And now,\" continued the preacher, \"you\nwill say to me, 'What do you mean by talking to us in this way?  Who are\nyou, sir?'  And now I will tell you.  I am Sammy Preeze.  I have come\nfrom the mountains of Cardiganshire, on my Master's business, and His\nmessage I must deliver.  If you will never hear me again, I shall not\nmatter much, but while you shall hear me, you shall hear me, and this is\nHis word in me, and in me to you: 'He that believeth shall be saved, and\nhe that believeth not shall be damned,' _and I begs no pardons_.\"\n\nIt was a strange scene; but as he went on, in quaint, but terribly\nearnest strain, anger passed into awe, and mute astonishment into rapt\nattention.  No one, who heard the words, could ever again hear them\nunheeded, nor think lightly of the doom of the unbelieving.  The anecdote\nis worth being laid to heart, in these days, when there is too often a\nreserve in declaring the whole counsel of God.\n\nAfter service, in the vestry, the deacons were in great anger with the\nblunt preacher; and one, a well-known religious man in Bristol,\nexclaimed, \"Mr. Breeze, you have strangely forgotten yourself to-night,\nsir.  We did not expect that you would have behaved in this way.  We have\nalways been very glad to see you in our pulpit, but your sermon to-night,\nsir, has been most insolent, shameful!\"  He wound up a pretty sharp\ncondemnation by saying, \"In short, I don't understand you!\"\n\n\"Ho! ho!\" exclaimed Sammy.  \"You say you do not understand me?  Eh! look\nyou then, I will tell you; I do understand you!  Up in our mountains, we\nhave one man there, we do call him exciseman; he comes along to our shops\nand stores, and says, 'What have you here?  Anything contraband here?'\nAnd if it is all right, the good man says, 'Step in, Mr. Exciseman, come\nin, look you.'  He is all fair, open, and above-board.  But if he has\nanything secreted there, he does draw back surprised, and he makes a fine\nface, and says, 'Sir, I do not understand you.'  Now, you do tell me that\nyou don't understand me, but I do understand you, gentlemen, I do; and I\ndo fear you have something contraband here; and I will say good-night to\nyou; but I must tell you one little word; that is: 'He that believeth\nshall be saved, and he that believeth not shall be damned,' _and I begs\nno pardons_.\"\n\nBut, with these simple illustrations, we have not exhausted the number of\nnoticeable names.  In connection with every name as it occurs, some\ninteresting anecdote meets the memory.  There was Robert Lloyd, the\nshoemaker, and Thomas the turner, and Robert Roberts, of whom, from the\nstories before us, we do not find it difficult to believe, that he had\nthe power to describe things in such a vivid, and graphic manner, as to\nmake his hearers feel as if the scenes were passing before their eyes.\nThen there were David Evans of Aberayron, and Ebenezer Richard of\nTregaron, and William Morris of St. David's, whose every sermon was said\nto be a string of sparkling gems; John Jones of Talysarn, and his\nbrother, David Jones; John Hughes; the seraphic Henry Rees, and Thomas\nPhilips, and many another name, concerning whom an illustration might be\nfurnished, of their powers of wit, wisdom, or eloquence.  England,\nitself, has been indebted, in many a circle, to eminent Welsh preachers,\nwho have stimulated thought, created the sphere of holy usefulness, moved\nover the minds of cultivated members with the freshness of a mountain\nwind, or a mountain stream.  It would be invidious to mention their\nnames\u2014many are yet living; and some, who have not long quitted the Church\non earth, have still left behind them the fragrance of loved, and\nhonoured names, and exalted, and earnest labours.\n\nFew of our readers, we may suppose, can be unacquainted with the name,\nand memory of \"The Man of Ross,\" so famous through the verses of Pope.\nRoss is a well-known little town in Monmouthshire, on the banks of the\nWye, on the borders of Wales.  There, in the parish church, in the pew in\nwhich John Kyrle, the Man of Ross, sat, more than a hundred years since,\na curious sight may be seen: two elm-trees rise, and spread out their\narms, and flourish within the church; especially during the spring, and\nsummer months, they form a singular adornment to the sacred edifice.  The\ntradition is, that they are suckers from a tree planted by the \"Man of\nRoss,\" outside the church; but it was cut down by a certain rector,\nbecause it excluded the light; the consequence was that they forced their\nway inside, where they had continued to grow, and flourish.  As we have\nlooked upon the singular sight of those trees, in the Man of Ross's pew,\nwe have often thought of those who, in Wales, planted in the house of the\nLord, flourish in sacred, and sainted memories, in the courts of our God.\nAlthough all that was mortal of them has passed away, they still bring\nforth fruit, and flourish in the grateful recollections of the country,\nthey were permitted to bless, and adorn.\n\nYes, it is very singular to think of many of these men of Wild Wales.\nEven those who were counted heretical, were more than extraordinary men;\nthey were, perhaps, men who, in our day, would seem rather remarkable for\ntheir orthodoxy of sentiment.  Rhys Stephen, in an extended note in his\nMemoirs of Christmas Evans, refers to the influence of discussions, in\nthe Principality, raised by the Rev. WILLIAM RICHARDS, LL.D.  A large\nportion of the ministerial life of this distinguished man, was passed in\nEngland; he was educated for the ministry at the Baptist Academy in\nBristol, for some time co-pastor with Dr. Ash, author of the Dictionary,\nand then became the minister of the Baptist Church at Lynn, in Norfolk,\nwhere he remained for twenty years.  He always continued, however, in\nevery sense of the word, a Welshman, and, notwithstanding his English\npastorates, his residences in Wales were frequent and long.\n\nHe was born at Pen-hydd, in Pembrokeshire, in 1749.  He published a\nWelsh-English dictionary, and his services to Welsh literature were\neminent.  But he was regarded as a heretic; his temperament, singular as\nit seems in a Welshman, was almost purely philosophic, and neither\nimaginative, nor emotional; he disliked the great annual religious\ngatherings of his countrymen, and called them fairs, and the preachers,\nupon these occasions, he sometimes described in epithets, which were not\ncomplimentary.  Naturally, his brethren paid him back; they called him a\nheretic,\u2014which is also an exceedingly convenient, and not unusual method\nof revenge.  Dr. Richards's influence, however, in Wales, at the\nbeginning of this century, appears to have been very great; the charges\nagainst him, he does not appear to have been very mindful to disprove,\nand it is exceedingly likely that a different, or more guarded mode of\nexpression, was the height of his offending.  Who can fathom, or\ndelineate, all the fine shades and divergencies of the Arian\ncontroversy?\u2014men whose perfect soundness, in evangelical doctrine, was\nutterly undisputed, talked with Dr. Richards, and said, that they could\nnot discover that he held opinions different from their own.  In a\nletter, dated December 7th, 1804, when grave charges had been urged\nagainst him, and all the religious mischiefs throughout the Principality\nascribed to him, he writes as follows, to a friend:\u2014\n\n    \"I think I may safely say, that no great change, of any kind, has\n    taken place in my sentiments since I knew you.  You must know,\n    surely, that I did not use to be an _Athanasian_, or even a\n    _Waterlandian_.  Such views of the Deity always appeared to me too\n    _Tritheistical_.  I have been used to think, and do so still, that\n    there is a particular meaning in such words as these of the\n    Apostle's, 'To us there is but one God, the Father;' but I never\n    could say, or think, with the Socinians, that Jesus Christ is no more\n    than _a man_, like ourselves.  I believe, indeed, that He is a Man;\n    but I, also, believe that He is 'Emmanuel, God with us'\u2014that he is\n    'the form of God'\u2014'the image of the invisible God'\u2014an object of\n    Divine worship, so that we should 'honour the Son as we honour the\n    Father'\u2014'that all the fulness of the Godhead dwells in Him bodily,'\n    or substantially.  In short, I believe everything of the dignity, and\n    glory of Christ's character, that does not _divide_ the Deity, or\n    land in _Tritheism_.\"\n\nAgain, to another correspondent: \"I believe, also, in the doctrine of the\natonement, or sacrifice, of Christ, in the virtue of His blood, and in\nthe prevalence of His mediation.\"\n\nSomething of the same order of man, so far as sentiment, and knowledge\nare indications, but possessed of more wit, imagination, and emotion, was\nDAVIES, of CASTELL HYWEL, the first pastor of Christmas Evans, and of\nDaniel Davies, of Swansea.  He was, in his day, a man of many-sided\nreputation, but of suspicious doctrinal relations.  He was so eminent a\nclassical scholar, and so many of the Welsh clergy had received their\neducation from him, that when Dr. Horsley was appointed Bishop of St.\nDavid's, he expressed, in his usual passionate manner, his irritation\nthat the most distinguished tutor in South Wales was a Nonconformist, and\ngave out that he would not ordain any of Mr. Davies' pupils.  Davies was\na great bard; and Welshmen who know both languages, say that his\ntranslation of Gray's \"Elegy\" is, in force, and pathos, superior to the\noriginal.  This will scarcely seem strange, if the deep pathos of the\nWelsh language be taken into account.  His epitaph on Dr.\nPriestley\u2014satirizing, of course, the materialism of\nPriestley\u2014illustrates, at once, his humour, and versification:\n\n    \"Here lies at rest, in oaken chest,\n       Together packed most nicely,\n    The bones, and brains, flesh, blood, and veins,\n       And _soul_ of Dr. Priestley!\"\n\nAs an illustration of his readiness of wit, a story is told, how one of\nthe most noted of the Welsh bards one day met him, while the rain was\nstreaming down upon him.  Umbrellas, probably, were scarce.  He was\ncovered with layers of straw, fastened round with ropes of the same\nmaterial; in fact, thatched all over.  To him his brother bard exclaimed:\n\n    \"Oh, bard and teacher, famed afar,\n       Such sight I never saw!\n    It ill becomes a house like yours\n       To have a roof of straw.\"\n\nTo which Davies instantly replied:\n\n    \"The rain is falling fast, my friend;\n       You know not what you say,\n    A roof of straw, methinks, doth well\n       Beseem a wall of clay.\"\n\nSuch was Christmas Evans's first \"guide, philosopher, and friend.\"\n\nAnd if we refer to certain characteristics of the Welsh language, which\nmake it eminently fine furniture for preaching-power, to these may be\nadded, what we have not so particularly dwelt on, but which does follow,\nas a part of the same remark\u2014the singular proverbial power of the Welsh\nlanguage.  In reading great Welsh sermons, and listening to Welsh\npreachers, we have often felt how much the spirit of their own triads,\nand the manner of old Catwg the Wise, and other such sententious bards,\nfalls into their modern method.  Welsh proverbs are the delightful\nrecreations of the arch\u00e6ologists of the old Welsh language.  Here, while\nwe write these lines, we have piles of these proverbial utterances before\nus; short, compact sayings, wherever they come from, but which have been\nrepeated on, from generation to generation.  The Bardic triads, for\ninstance, relating to language, selected by Mr. Owen Pugh,\u2014how admirable\nthey are for any preacher!  They may stand as the characteristics of\ntheir most eminent men.\n\n\"The three indispensables of language\u2014purity, copiousness, and aptness;\nthe three supports of language\u2014order, strength, and harmony; the three\nuses of language\u2014to relate, to describe, to excite; the correct qualities\nof language,\u2014correct construction, correct etymology, and correct\npronunciation; three marks of the purity of language\u2014the intelligible,\nthe pleasurable, the credible; three things that constitute just\ndescription\u2014just selection of words, just construction of language, and\njust comparison; three things appertaining to just selection\u2014the best\nlanguage, the best order, and the best object.\"  It must be admitted, we\nthink, that, in these old triads, there is much of the compact wisdom of\na primeval people, with whom books were few, and thoughts were fresh, and\nconstant.  There seemed to be a singular propensity, in the old mind of\nWales, to throw everything into the form of a trinity of expression, or\nto bind up words, as far as possible, in short, sententious utterances.\nCatwg's \"Essay on Metaphysics\" is a very brief, and concise one, but it\nillustrates that rapid running-up-the-ladder kind of style, which has\nalways been the delight of the Welsh poet or teacher.\n\n    \"In every person there is a soul.  In every soul there is\n    intelligence.  In every intelligence there is thought.  In every\n    thought there is either good, or evil.  In every evil there is death;\n    in every good there is life.  In every life there is God; and there\n    is no God but He than whom there can be none better.  There is\n    nothing that cannot have its better, save the best of all.  There is\n    no best of all except love.  There is no love but God.  God is love!\"\n\nIllustrations of this kind fill volumes.  It is not for us here to say\nhow much of the admirable, or the imitable there may be in the method.\nIt was the method of the old Welsh mind; it was the method into which\nmany of the best preachers fell, not because they, perhaps, knew so much\nof the words of the bards, as because it represented the mind of the\nrace.  Take a few of the Welsh proverbs.\n\n    \"He that is intent upon going, will do no good before he departs.\"\n\n    \"Every one has his neighbour for a mirror.\"\n\n    \"The water is shallowest where it bubbles.\"\n\n    \"A lie is the quickest traveller.\"\n\n    \"Fame outlives riches.\"\n\n    \"He that is unlucky at sea, will be unlucky on land.\"\n\n    \"There is always time for meat, and for prayer.\"\n\n    \"He mows the meadow with shears.\"\n\n    \"Calumny comes from envy.\"\n\n    \"Every bird loves its own voice.\"\n\n    \"The life of a man is not at the disposal of his enemy.\"\n\n    \"He that loves the young, must love their sports.\"\n\n    \"Prudence is unmarried without patience.\"\n\n    \"He that is the head, should become the bridge.\"\n\n    \"Three things come unawares upon a man: sleep, sin, and old age.\"\n\nBut it is not only that this sententious characteristic of the Welsh\nlanguage makes it a vehicle for the transparent expression of sentiment;\neven our translations cannot altogether disguise the pathetic tones of\nthe language, and bursts of feeling.  The following verse of an old Welsh\nprayer, which, a _Quarterly Reviewer_ tells us, used to form, with the\nCreed and Ten Commandments, part of the peasant's daily devotion,\nillustrates this:\u2014\n\n    \"Mother, O mother! tell me, art thou weeping?\"\n          The infant Saviour asked, on Mary's breast.\n       \"Child of th' Eternal, nay; I am but sleeping,\n          Though vexed by many a thought of dark unrest.\"\n       \"Say, at what vision is thy courage failing?\"\n           \"I see a crown of thorns, and bitter pain;\n       And thee, dread Child, upon the cross of wailing,\n          All heaven aghast, at rude mankind's disdain.\"\n\nIt is singular that Mr. Borrow found, on an old tombstone, an epitaph,\nwhich most of our readers will remember, as very like that famous one Sir\nWalter Scott gives us, from an old tomb, in a note to \"The Lay of the\nLast Minstrel.\"  The following is a translation:\u2014\n\n    \"Thou earth, from earth, reflect, with anxious mind,\n    That earth to earth must quickly be consigned;\n    And earth in earth must lie entranced, enthralled,\n    Till earth from earth to judgment shall be called.\"\n\nThe following lines also struck Mr. Borrow as remarkably beautiful, of\nwhich he gives us this translation.  They are an inscription in a\ngarden:\u2014\n\n    \"In a garden the first of our race was deceived;\n    In a garden the promise of grace was received;\n    In a garden was Jesus betrayed to His doom;\n    In a garden His body was laid in the tomb.\"\n\nSuch verses are very illustrative of the alliterative character of the\nWelsh mind.\n\nBut Wales, in its way\u2014and no classical reader must smile at the\nassertion\u2014was once quite as much the land of song as Italy.  Among the\namusements of the people was the singing of \"Pennilion,\" a sort of\nepigrammatic poem, and of an improvisatorial character, testing the\nreadiness of rural wit.  With this exercise there came to be associated,\nin later days, a sort of rude mystery, or comedy, performed in very much\nthe same manner as the old monkish mysteries of the dark ages.  These\nfurnished an opportunity for satirizing any of the unpopular characters\nof the village, or the Principality.  Such mental characteristics,\nshowing that there was a living mind in the country, must be remembered,\nwhen we attempt to estimate the power which extraordinary preachers soon\nattained, over the minds of their countrymen.  Then, no doubt, although\nthere might be exceptions, and a Welshman prove that he could be as\nstupid as anybody else, in general there was a keen love, and admiration\nof nature.  The names of places show this.  Mr. Borrow illustrates both\ncharacters in an anecdote.  He met an old man, and his son, at the foot\nof the great mountain, called Tap-Nyth-yr Eryri.\n\n\"Does not that mean,\" said Mr. Borrow, \"the top nest of the eagles?\"\n\n\"Ha!\" said the old man, \"I see you understand Welsh.\"\n\n\"A little.  Are there eagles there now?\"\n\n\"Oh, no! no eagle now; eagle left Tap-Nyth.\"\n\n\"Is that young man your son?\" said Mr. Borrow, after a little pause.\n\n\"Yes, he my son.\"\n\n\"Has he any English?\"\n\n\"No, he no English, but he plenty of Welsh; that is, if he see reason.\"\nHe spoke to the young man, in Welsh, asking him if he had ever been up to\nthe Tap-Nyth; but he made no answer.\n\n\"He no care for your question,\" said the old man; \"ask him price of pig.\"\n\n\"I asked the young fellow the price of hogs,\" says Mr. Borrow, \"whereupon\nhis face brightened up, and he not only answered my question, but told me\nthat he had a fat hog to sell.\"\n\n\"Ha, ha!\" said the old man, \"he plenty of Welsh now, for he see reason;\nto other question he no Welsh at all, no more than English, for he see no\nreason.  What business he on Tap-Nyth, with eagle?  His business down\nbelow in sty with pig.  Ah! he look lump, but he no fool.  Know more\nabout pig than you, or I, or anyone, 'twixt here and Machunleth.\"\n\nIt has been said, that the inhabitants of a mountainous country cannot be\ninsensible to religion, and whether, or not this is universally true, it\nis, certainly, true of Wales.  The magnificent scenery seems to create a\npensive awe upon the spirit.  Often the pedestrian, passing along a piece\nof unsuggestive road, suddenly finds that the stupendous mountains have\nsloped down, to valleys of the wildest, and most picturesque beauty,\nvalley opening into valley, in some instances; in others, as in the vale\nof Glamorgan, stretching along, for many miles, in plenteous\nfruitfulness, and beauty, illuminated by some river like the Tivy, the\nTowy, or the Llugg, some of these rivers sparkling, and flashing with the\nglittering _gleisiad_, as an old Welsh song sings it\u2014\n\n    \"_Glan yw'r gleisiad yn y llyn_,\n    Full fair the _gleisiad_ in the flood\n    Which sparkles 'neath the summer's sun.\"\n\nThe_ gleisiad_ is the salmon.  We have dwelt on the word here, for the\npurpose of calling the reader's attention to its beautiful\nexpressiveness.  It seems to convey the whole idea of the fish\u2014its\nsilvery splendour, gleaming, and glancing through the lynn.\n\nIt seems rather in the nature of the Welsh mind, to take instantly a\npensive, and sombre idea of things.  A traveller, walking beneath a fine\nrow of elms, expressed his admiration of them to a Welsh companion.  \"Ay,\nsir,\" said the man; \"they'll make fine chests for the dead!\"  It was very\nnationally characteristic, and hence, perhaps, it is that the owl (the\n_dylluan_) among birds, has received some of the most famous traditions\nof the Welsh language.  Mr. Borrow thought there was no cry so wild, as\nthe cry of the _dylluan_\u2014\"unlike any other sound in nature,\" he says, \"a\ncry, which no combination of letters can give the slightest idea of;\"\nand, surely, that Welsh name far better realizes it, than the _tu whit tu\nwhoo_ of our Shakespeare.\n\nCertainly, it is not in a page, or two, that we can give anything like an\nadequate idea of that compacted poetry, which meets us in Wales, whether\nwe think of the varied scenery of the country, of the nervous, and\ndescriptive language, or of its race of people, so imaginative, and\nspeculative.\n\nIt ought to be mentioned, also, as quite as distinctly characteristic,\nthat there is an intense clannishness prevalent throughout the\nPrincipality.  Communication between the people has no doubt somewhat\nmodified this; but, usually, an Englishman resident in Wales, and\nespecially in the more sequestered regions, has seldom found himself in\nvery comfortable circumstances.  The Welsh have a suspicion that there\nare precious secrets in their land, and language, of which the English\nare desirous to avail themselves.  And, perhaps, there is some\nextenuation in the recollection that we, as their conquerors, have seldom\ngiven them reason to think well of us.\n\n\n\n\nCHAPTER IX.\n_CHRISTMAS EVANS CONTINUED\u2014HIS MINISTRY AT CAERPHILLY_.\n\n\nCaerphilly and its Associations\u2014\"Christmas Evans is come!\"\u2014A\nHousekeeper\u2014His Characteristic Second Marriage\u2014A Great Sermon, The Trial\nof the Witnesses\u2014The Tall Soldier\u2014Extracts from Sermons\u2014The Bible a Stone\nwith Seven Eyes\u2014\"Their Works do Follow them\"\u2014A Second Covenant with\nGod\u2014Friends at Cardiff\u2014J. P. Davies\u2014Reads Pye Smith's \"Scripture\nTestimony to the Messiah\"\u2014Beattie on Truth\u2014The Edwards Family\u2014Requested\nto Publish a Volume of Sermons, and his Serious Thoughts upon the\nSubject.\n\nIt was in the year 1826 that Christmas Evans, now sixty-two years of age,\nleft Anglesea, accepting an invitation to the Baptist Church at\nTonyvelin, in Caerphilly.  His ministry at Anglesea had been long,\naffectionate, and very successful; but, dear as Anglesea was to him, he\nhad to leave it, and he left it, as we have seen, under circumstances not\nhonourable to the neighbouring ministers, or the churches of which he had\nbeen the patriarchal pastor.  Little doubt can there be, that even he\nsuffered from the jealousy of inferior minds, and characters; so old as\nhe was, so venerable, and such a household name as his had become,\nthroughout all Wales, it might have been thought that he would not have\nbeen permitted to depart.  He left the dust of his beloved wife, the long\ncompanion of his Cildwrn cottage, behind him, and commenced his tedious\njourney to his new home.  He had about two hundred miles to travel, and\nthe travelling was not easy; travelling in Wales was altogether unrelated\nto the more comfortable, and commodious modes of conveyance in England,\neven in that day; and now he would have to cross a dangerous ferry, and\nnow to mount a rugged, and toilsome hill, to wind slowly along by the\nfoot of some gigantic mountain, to wend through a long, winding valley,\nor across an extensive plain.  As the old man passed along, he says he\nexperienced great tenderness of mind, and the presence of Christ by his\nside.  A long, solitary journey! he says, he was enabled to entrust the\ncare of his ministry to Jesus Christ, with the confidence that He would\ndeliver him from all his afflictions; he says, \"I again made a covenant\nwith God which I never wrote.\"\n\nCaerphilly would seem a very singular spot in which to settle one of the\nmost remarkable men, if not the most remarkable, in the pulpit of his\ncountry, and his time,\u2014beyond all question, the most distinguished in his\nown denomination, there, and then.  Even now, probably, very few of our\nreaders have ever heard of Caerphilly; it is nearly forty years since the\nwriter of the present pages was there, and there, in a Welsh cottage,\nheard from the lips of an old Welsh dame the most graphic outlines he has\never heard, or read, of some of the sermons of Christmas Evans.  Since\nthat day, we suppose Caerphilly may have grown nearer to the dignity of a\nlittle town, sharing some of the honours which have so lavishly fallen\nupon its great, and prosperous neighbour, Cardiff.\n\nCaerphilly, however insignificant, as it lies in its mountain valley, a\npoor little village when Christmas Evans was there, has its own eminent\nclaims to renown: tradition says\u2014and, in this instance, tradition is,\nprobably, correct\u2014that it was once the seat of a large town.  There,\ncertainly, still stands the vast ruins of Caerphilly Castle, once the\nlargest in all Great Britain next to Windsor, and still the most\nextensive ruin; here was the retreat of the ill-fated Edward II.; here\nwas that great siege, during which the King escaped in the depth of a\ndark, and stormy night, in the disguise of a Welsh peasant, flying to the\nparish of Llangonoyd, twenty miles to the west, where he hired himself at\na farm, which, it is said, is still pointed out, or the spot where once\nit stood, the site made memorable, through all these ages, by so singular\na circumstance.  This was the siege in which that grand, and massive\ntower was rent, and which still so singularly leans, and hangs there,\u2014the\nleaning tower of Caerphilly, as wonderful an object as the leaning tower\nof Pisa, a wonder in Wales which few have visited.\n\nAfter this period, it was occupied by Glendower; gradually, however, it\nbecame only famous for the rapacity of its lords, the Spencers, who\nplundered their vassals, and the inhabitants of the region in general, so\nthat from this circumstance arose a Welsh proverb, \"It is gone to\nCaerphilly,\"\u2014signifying, says Malkin, that a thing is irrecoverably lost,\nand used on occasions when an Englishman, not very nice, and select in\nhis language, would say, \"It is gone to the devil.\"  Gloomy ideas were\nassociated for long ages with Caerphilly, as the seat of horror, and\nrapacity; it had an awful tower for prisoners, its ruinous walls were of\nwondrous thickness, and it was set amidst desolate marshes.\n\nAnd this was the spot to which Christmas Evans was consigned for some of\nthe closing years of his life; but, perhaps, our readers can have no idea\nof the immense excitement his transit thither caused to the good people\nof the village, and its neighbourhood.  Our readers will remember, what\nwe have already said, that a small village by no means implied a small\ncongregation.  His arrival at Caerphilly was looked upon as an event in\nthe history of the region round about; for until he was actually there,\nit was believed that his heart would fail him at last, and that he would\nnever be able to leave Anglesea.\n\nIt is said that all denominations, and all conditions of people, caught\nup, and propagated the report, \"CHRISTMAS EVANS IS COME!\"  \"_Are you sure\nof it_?\"  \"YES, _quite sure of it_; _he preached at Caerphilly last\nSunday_!  I know a friend who was there.\"  These poor scattered\nvillagers, how foolish, to us, seems their enthusiasm, and frantic joy,\nbecause they had their country's great preaching bard in their midst;\nalmost as foolish as those insane Florentines, who burst into tears and\nacclamations as they greeted one of the great pictures of Cimabue, and\nreverently thronged round it in a kind of triumphal procession.  What\nmakes it more remarkable, is that they should love a man as poor, as he\nwas old.  If they could revere him as, wearied and dusty, he came along\nafter his tedious two hundred miles' journey, spent, and exhausted, what\nan affluence of affection they would have poured forth had he rode into\nCaerphilly, as the old satirist has it, in a coach, and six!\n\nWell, he was settled in the chapel-house, and a housekeeper was provided\nfor him.  In domestic matters, however, he did not seem to get on very\nwell.  North, and South Wales appeared different to him, and he said to a\nfriend, he must get a servant from the north.  It was suggested to him,\nthat he might do better than that, that he had better marry again, and\nthe name of an excellent woman was mentioned, who would have been\nprobably not unwilling; and she had wealth, so that he might have\nbettered his entire worldly circumstances by the alliance, and have made\nhimself pleasantly independent of churches, and deacons, and county\nassociations; and when it was first suggested to him, he seemed to think\nfor a moment, and then broke out into a cheerful laugh.  \"Ho! ho!\" he\nsaid, \"I tell you, brother, it is my firm opinion that I am never to have\nany property in the soil of this world, until I have a grave;\" and he\nwould talk no more on the subject, but he took a good brother minister of\nthe neighbourhood into his counsel, Mr. Davies, of Argoed, and he\npersuaded him to take his horse, and to go for him to Anglesea, and to\nbring back with him the old, and faithful servant of himself, and his\ndeparted wife, Mary Evans; and, in a short time, he married her, and she\npaid him every tribute of untiring, and devoted affection, to the last\nmoment of his life.  A really foolish man, you see, this Christmas Evans,\nand, as many no doubt said, old as he was, he might have done so much\nbetter for himself.  It is not uninteresting to notice a circumstance,\nwhich Mr. Rhys Stephen discovered, that Christmas Evans was married the\nsecond time in the same parish of Eglwysilian, in Glamorganshire, the\nchurch in which George Whitefield was married: the parish register\ncontains both their names.\n\nAnd what will our readers think, when they find that those who knew\nChristmas Evans, both at this, and previous periods of his history,\ndeclare that his preaching now surpassed that of any previous period?\nCertainly, his ministry was gloriously successful at Caerphilly.\nCaerphilly, the village in the valley, became like a city set upon a\nhill; every Sabbath, multitudes might be seen, wending their way across\nthe surrounding hills, in all directions.  The homes of the neighbourhood\nrang, and re-echoed with Christmas Evans's sermons; his morning sermon,\nespecially, would be the subject of conversation, in hundreds of homes,\nmany miles away, that evening.  The old dame with whom we drank our cup\nof tea, in her pleasant cottage at Caerphilly, near forty years since,\ntalked, with tears, of those old days.  She said, \"We used to reckon\nthings as they happened, by Christmas Evans's sermons; people used to\nsay, 'It must have happened then, because that was the time when\nChristmas Evans preached The Wedding Ring,' or The Seven Eyes, or some\nother sermon which had been quite a book-mark in the memory.\"\n\nNo doubt, many grand sermons belong to the Caerphilly period: there is\none which reads, to us, like an especial triumph; it was preached some\ntime after he settled in the south; the subject was, \"God manifest in the\nflesh, justified in the spirit.\"  The grand drama in this sermon was the\nexamination of the evidences of Christ's resurrection:\u2014\n\n\n\n\"THE TRIAL OF THE WITNESSES.\n\n\n    \"The enemies of Christ, after His death, applied for a military guard\n    to watch at His tomb, and this application for a military guard was\n    rested on the fact, that the 'impostor' had said, in His lifetime,\n    that He would rise again on the third day.  Without a doubt, had they\n    found His body in the grave, when the time had transpired, they would\n    have torn it from the sepulchre, exhibited it through the streets of\n    Jerusalem, where Jesus had preached, where He had been despitefully\n    used, and scourged; they would have shouted forth with triumph, 'This\n    is the body of the impostor!'  But He had left the grave, that\n    morning, too early for them.  The soldiers came back to the city, and\n    they went to the leaders of the people who had employed them, and the\n    leaders exclaimed, 'Here is the watch!  What is the matter?  What is\n    that dread settled in their faces?  Come in here, and we charge you\n    to tell the truth.'  'You have no need to charge us, for the fright,\n    the terror of it, is still upon us.'  'How?  What has happened at the\n    grave?  Did His disciples come, and take Him away?'  'They! no; but\n    if they had, our spears would have sufficed for them.'  'Well, but\n    how was it?  What has taken place?'  'Well, see; while we were on the\n    watch, and early, in the dawn of the morning, a great earthquake,\n    like to that one that took place on Friday afternoon, _when He died_,\n    and we all fell powerless to the ground; and we saw angels, bright,\n    like the lightning; we were not able to bear the sight; we looked\n    down at once; we endeavoured, again, to raise our eyes, and we beheld\n    One coming out of the grave, but He passed by the first angel we saw,\n    who now was sitting on the removed stone; but He who came out of the\n    grave! we never saw one like unto Him before,\u2014truly He was like unto\n    the Son of God.'  'What, then, became of the angel?'  'Oh, a legion\n    of them came down, and one of them, very fair, like a young man,\n    entered the grave, and sat where the head of Jesus had lain; and,\n    immediately, another, also, very fair, and beautiful, sat where His\n    feet had rested.'  'And did the angels say nothing to you?'  'No, but\n    they looked with eyes of lightning.'  'Saw you not His friends, the\n    women?'  'Oh, yes; they came there, but He had left the tomb before\n    their arrival.'  'Talked the angels to the women?'  'Yes; they seemed\n    to be of one family, and very well acquainted with one another.'  'Do\n    you remember anything of the conversation?'  'Yes; they said, \"Fear\n    you not! let the Pharisees, and Darkness fear to-day!  You seek\n    Jesus!  He is not here, for He is risen indeed; He is alive, and\n    lives for ever.  He has gone before you to Galilee.\"  We heard one\n    angel say, \"Come, see the place where the Lord lay.\"  Another angel\n    spoke to a woman called Mary, and said, \"Why weepest _thou_, while\n    thy Lord is risen indeed, and is alive, so near unto thee? _let His\n    enemies weep to-day_!\"'  'WHAT!' exclaimed the leader of those\n    priests, and of the council, who had asked for the guard,\u2014'What! how\n    say you?  _Close that door_!  You, _tall_ soldier, approach: was it\n    not you who pierced His side?'  'Yes, it was I; but all that these\n    soldiers have said is all true; oh, alas! it is all true!  He must\n    have been the Son of God.'  The Pharisees lost their cause, on the\n    day of their appeal; they gave the soldiers money, to say that His\n    disciples had stolen the body while they slept!  _If they were\n    asleep_, _how did they know in what manner He had left the grave_?\n    They, however, suffered themselves to be suborned, and for money\n    lied, and, to this hour, the kingdom of Satan hangs upon that lie!\"\n\nThis sermon produced a profound impression.  We have said, to render the\nsermons of Christmas Evans in print, or by description, is impossible,\u2014as\nimpossible as to paint tones, and accents, or the varying expressions\nwhich pass over eye, and face, and lip.  He was entreated to publish this\nsermon, but he could only write out something like an outline of it, and\nwhen it appeared in print, those who had been enraptured with it, in its\ndelivery, declared that it was not the same sermon; so he was entreated\nto preach the sermon again.  He made a humorous remark, on the\nstrangeness of a man preaching his own printed sermon; still, he\ncomplied.  His accomplished biographer, Rhys Stephen, heard it then, and\nsays of it, \"While I have the faintest trace of memory, as to sermons I\nhave heard, this must always be pre-eminent, and distinct; in its\noratorical eminence, it stands alone, even among his great achievements.\nOne of the most striking parts of the sermon, was in the examination of\nthe Roman guard, the report of the soldiers to the authorities.\"  Mr.\nStephens continues, \"We heard them talk, had a clear perception of the\ndifference of the tone, and more especially, when one of the chief\npriests, in an anxious, agonizing whisper, said, '_Shut the door_!'  And\nthen, 'You, tall soldier, approach: was it not you who pierced His side?'\n'Yes, it was I.'  When Christmas Evans simulated the chief priest, and\nsingled out the tall soldier, and the conversation went on between the\ntwo, such a combined triumph of sanctified fancy, and perfect oratory, I\nnever expect to witness again.\"  We may, also, say, that it illustrates\nwherein, very greatly, lay the preacher's power,\u2014seizing some little\ncircumstance, and, by its homeliness, or aptness, giving reality, and\nvivacity to the whole picture.\n\nIt must be said, his are very great sermons; the present writer is almost\ndisposed to be bold enough to describe them, as the grandest Gospel\nsermons of the last hundred years.  Not one, or two, but several, are\nespecially noble.  One of these we have, already, given: the splendid\nembodiment, and personification of the twenty-second Psalm, _The Hind of\nthe Morning_, from the singular, and most significant designation, or\ntitle of the Psalm itself.\n\nAnother sermon which, probably, belongs to this period is\n\n               \"THE BIBLE REGARDED AS A STONE WITH SEVEN EYES,\"\n\nevidently from Zech. iii. 9, \"_Upon one stone shall be seven eyes_.\"\n\nIt was, in fact, a review of\n\n\n\n\"_The Internal Evidences which prove the Gospel to be of God_.\n\n\n    \"God's perfections are, in some sort, to be seen in all He has done,\n    and in all He has spoken.  He imprints some indication of His\n    character, on everything that His hand forms, and that His mouth\n    utters, so that there might be a sufficient difference between the\n    work, and the speech of God, and those of man.  The Bible is the Book\n    of books, a book breathed out of heaven.  It was easy enough for John\n    to determine, when he saw the Lamb, with the seven horns, and the\n    seven eyes, in the midst of the throne, that the Godhead was there,\n    and that such a Lamb was not to be found amongst creatures.  When one\n    saw a stone, with seven eyes, before Zerubbabel, it was not difficult\n    to conclude that it was a stone from some unusual mine.  In looking\n    at the page of the starry sky, the work of the fingers of the\n    Everlasting Power is traced in the sun, and moon, and stars; all\n    proclaim His name, and tell His glory.  I am very thankful for books\n    written by man, but it is God's book that sheds the light of the life\n    everlasting on all other books.  I cannot often read it, hear it, or\n    reflect upon it, but I see\u2014\n\n    \"1.  _Eternity_, like a great fiery Eye, looking at me from the\n    everlasting, and the infinite distance, unfolding mysteries, and\n    opening before me the doors, windows, and chambers, in the\n    (otherwise) unknown, and awful state!  This Eye leads me to the\n    source, and cause of all things, and places me in the presence, and\n    sight of the Almighty, who has in Him something that would destroy me\n    for ever, and yet something that spares, and animates me; pressing me\n    down, and at the same time, saying, 'Fear not;' something that melts\n    me into penitence, and, at once, causes me to rejoice in the faith,\n    inspiring me with the fear of joy; something that creates a wish in\n    me, to conceal myself from Him, and then a stronger wish, to stay,\n    for ever, in the light of His countenance.\n\n    \"2.  _Omniscience_ looks at me, also, like a Divine Eye, out of every\n    chapter, verse, doctrine, and ordinance of the Gospel, and searches\n    me through and through.  The attempt at concealment from it is\n    utterly vain.  To this Eye, darkness is as the light.  It has\n    descried, correctly, into the deepest abysses of my spirit; and it\n    has truthfully drawn my likeness before I received God's grace;\n    having received it; and the future is, also, transparent before it.\n    There is something in the scanning of this Eye, that obliges me to\n    confess, against myself, my sins unto the Lord; and to cry out for a\n    new heart, and a right spirit; for the Author of the Book knows all.\n\n    \"3.  When I yield to pensive reflections, under a sense of sin, and\n    when I see the tops of dark mountains of disease, and trouble at the\n    terrors of the grave, I see in the Bible _Infinite Goodness_, fairer\n    than the Shekinah of old, looking at me, out of eternity; it is like\n    the smile of the Eternal King, from His throne of mercy.  Divine\n    love, merits of Christ, riches of grace, they are all here, and they\n    assure me, and I listen to the still, small voice, that follows in\n    its train, until I feel myself lifted up, out of the cave of despair,\n    by the dark mountain; and I stand on my feet, and I hope, and hear\n    the proclamation of the great mystery\u2014'Behold, I come, as it is\n    written in the roll of the Book.  If I must die, I am willing to die;\n    for I come to seek, and to save that which is lost.'\n\n    \"4.  _Holiness_, _righteousness_, and _purity_ look at me, out of the\n    midst of the Book, like the fires of Sinai to Israel, or the I AM,\n    out of the burning bush; causing me to fear, and tremble, while I am\n    yet desirous of looking at the radiant glory, because it is\n    attempered with mercy.  I take my shoes from off my feet, and\n    approach on my knees, to see this great sight.  I cannot live, in\n    sin, in this presence,\u2014still it does not slay me.  The Eternal Power\n    is here, and, with one hand, it conceals me, in the shadow of\n    redeeming mercy, and, with the other, it points out the glory of the\n    great, and wondrous truth, that God is, at once, a just God, and\n    justifier of him that believeth in Jesus.  Where Thy glory rests, O\n    my God, there let me have my abode!\n\n    \"5.  I also see _Infinite might_ radiating from the doctrine of the\n    Book, like God's own Eye, having the energy of a sharp, two-edged\n    sword.  Without asking permission of me, it proves itself 'quick and\n    powerful, and pierces even to the dividing asunder of the soul, and\n    spirit, and of the joints, and marrow;' it opens the private recesses\n    of my heart, and becomes a discerner, and judge of its thoughts, and\n    intents.  When Lord Rochester, the great wit, and unbeliever of his\n    day, read Isa. liii. 5, 'He was wounded for our transgressions,'\n    etc., Divine energies entered his spirit, and did so thoroughly\n    pierce, and pervade it, that his infidelity died within him, and he\n    gladly received the faith, and hope that are in Christ.  The power of\n    the Gospel visited Matthew, at the receipt of custom, the woman at\n    the well of Samaria, the malefactor on the cross, the converts on the\n    day of Pentecost, Paul by the way, and the jailer at Philippi; in\n    them all was exerted this resistless might of grace, the '_Let\n    __there be_' of the original creation, which none can withstand.\n\n    \"6.  When I am weak, and _distressed_, and _alone_, and none to\n    receive my tale of sorrow, none to express a word of fellow-feeling,\n    or of care for me, in the living oracles of the Gospel I see Divine\n    wisdom, and loving-kindness, looking at me tenderly, compassionately,\n    through the openings of my prison, and I feel that He, who dresses\n    the lily of the field, and numbers the sparrows, is near me,\n    numbering the hairs of my head, listening to my cries; and in all the\n    treasure of grace, and power, that was able to say to the lost one,\n    at the very door of the pit, 'To-day shalt thou be with me in\n    Paradise,' fearing no hindrances that might intervene, between\n    Golgotha and heaven, He is the same gracious Redeemer, and Preserver\n    to every one, that believes in His name.  Who will teach me the way\n    of wisdom? who will guide me to her dwelling-place?  It was in the\n    Gospel that wisdom came to reside near me, and here she teaches the\n    most untoward, convinces the most hard-hearted, reforms the most\n    licentious, and makes the simple wise unto salvation.\n\n    \"7.  _I am sometimes filled with questions of anxious import_.  Art\n    thou from heaven, O Gospel?  Thou hast caused me to hope: Art thou a\n    rock?  The reply: Dost thou not see, in my face, the true character\n    of God, and of the Eternal Power Incarnate?  Dost thou not discern,\n    in Jesus, the image of the invisible God, which, unlike the first\n    Adam, the second Adam has preserved untarnished? and dost thou not\n    feel, in looking at it, thyself gradually changed into the same\n    image, even as by the Spirit of the Lord?  In looking at God's image\n    in the creature, the vision had no transforming power, but left 'the\n    wise men' of the ancient world where it found them, destitute of true\n    knowledge, and happiness, without hope, and without God in the world;\n    but here the vision transforms into the glorious likeness of the\n    sublime object, even Christ.\n\n    \"_The character of God_, given in the Gospel, is complete, and\n    perfect, worthy of the most blessed One, and there is no perfect\n    portraiture given of Him but in the Gospel.  Mohammed's God is\n    _unchaste_; Homer gave his Jupiter _revenge_; Voltaire deified\n    _mockery_; Insurrection and War were the gods of Paine;\u2014but the\n    character of the God of the Gospel is awful in truth, and lovely in\n    goodness.  In Isa. vi., the vision of the Divine glory caused the\n    six-winged cherubs to conceal their faces; but in Rev. iv., the\n    six-winged living things employ five wings to fly, and only one to\n    veil their faces, while they are full of eyes behind, and before,\n    looking forth unveiled.  All the worshippers under the Gospel, look\n    with open face\u2014without a veil, and on an unveiled object.\"\n\nWe have here, evidently, only the rudiments of a sermon, but a very fine\none, a very suggestive one.  To most minds, the Bible has, probably,\nbeen, as Thomas Carlyle, or Jean Paul, would express it, \"an eyeless\nsocket, without the eye.\"  Christmas Evans was expressing, in this very\nsuggestive sermon, the thoughts of some men whose words, and works he had\nprobably never met with; as George Herbert says it\u2014\n\n                \"In ev'ry thing\n    Thy words do find me out.\"\n\n\"Beyond any other book,\" says Samuel Taylor Coleridge, \"the Bible _finds_\nme;\" while John Keble, in the \"Christian Year,\"\u2014probably written about\nthe same time, when Christmas Evans was preparing his sermon,\u2014was\nemploying the very same image in some of his most impressive words:\u2014\n\n    \"_Eye of God's Word_! where'er we turn,\n       Ever upon us! thy keen gaze\n    Can all the depths of sin discern,\n       Unravel every bosom's maze:\n\n    \"Who, that has felt thy glance of dread\n       Thrill through his heart's remotest cells,\n    About his path, about his bed,\n       Can doubt what Spirit in thee dwells?\"\n\nIn the following extract, we have a more sustained passage, very fresh,\nand noble:\u2014\n\n\n\n\"THEIR WORKS DO FOLLOW THEM.\n\n\n    \"In this world, every man receives according to his faith; in the\n    world to come, every man shall receive according to his works.\n    'Blessed are the dead who die in the Lord, for they rest from their\n    labours, and their works do follow them.'  Their works do not go\n    _before_ them, to divide the river of Jordan, and open the gates of\n    heaven.  This is done by their faith.  But their works are left\n    behind, as if done up in a packet, on this side of the river.  John\n    saw the great white throne, descending for judgment, the Son of man\n    sitting thereon, and all nations gathered before Him.  He is dividing\n    the righteous from the wicked, as the shepherd divideth the sheep\n    from the goats.  The wicked are set on the left hand\u2014'Depart from me,\n    ye accursed, into everlasting fire, prepared for the devil and his\n    angels!'  But the righteous are placed on the right hand, to hear the\n    joyful welcome\u2014'Come, ye blessed of my Father, inherit the kingdom\n    prepared for you from the foundation of the world!'  The books are\n    opened, and Mercy presents the packets that were left on the other\n    side of Jordan.  They are all opened, and the books are read, wherein\n    all their acts of benevolence are recorded.  Justice examines the\n    several packets, and answers\u2014'All right.  Here they are.  Thus it is\n    written\u2014\"I was hungry, and ye gave Me meat; I was thirsty, and ye\n    gave Me drink; I was a stranger, and ye took me in; I was naked, and\n    ye clothed Me; I was in prison, and ye came unto Me!\"'  The righteous\n    look upon each other, with wonder, and answer\u2014'Those packets must\n    belong to others.  We know nothing of all that.  We recollect the\n    wormwood, and the gall.  We recollect the strait gate, the narrow\n    way, and the slough of despond.  We recollect the heavy burden, that\n    pressed so hard upon us, and how it fell from our shoulders, at the\n    sight of the cross.  We recollect the time, when the eyes of our\n    minds were opened, to behold the evil of sin, the depravity of our\n    hearts, and the excellency of our Redeemer.  We recollect the time\n    when our stubborn wills were subdued, in the day of His power, so\n    that we were enabled both to will, and to do, of His good pleasure.\n    We recollect the time, when we obtained hope in the merit of Christ,\n    and felt the efficacy of His blood, applied to our hearts by the Holy\n    Spirit.  And we shall never forget the time, when we first\n    experienced the love of God, shed abroad in our hearts.  Oh, how\n    sweetly, and powerfully it constrained us to love Him, His cause, and\n    His ordinances!  How we panted after communion, and fellowship with\n    Him, as the hart panteth after the water-brooks!  All this, and a\n    thousand other things, are as fresh in our memory as ever.  But we\n    recollect nothing of those bundles of good works.  Where was it?\n    Lord, when saw we Thee hungry, and fed Thee; or thirsty, and gave\n    Thee drink; or a stranger, and took Thee in; or naked, and clothed\n    Thee?  We have no more recollection, than the dead, of ever having\n    visited Thee in prison, or ministered to Thee in sickness.  Surely,\n    those bundles cannot belong to us.'  Mercy replies\u2014'Yes, verily, they\n    belong to you; for your names are upon them; and, besides, they have\n    not been out of my hands since you left them on the stormy banks of\n    Jordan.'  And the King answers\u2014'Verily, I say unto you, Inasmuch as\n    ye have done it unto one of the least of these My brethren, ye have\n    done it unto Me.'\n\n    \"If the righteous do not know their own good works; if they do not\n    recognize, in the sheaves which they reap at the resurrection, the\n    seed which they have sown, in tears, on earth,\u2014they, certainly,\n    cannot make these things the foundation of their hopes of heaven.\n    Christ is their sole dependence, for acceptance with God, in time,\n    and in eternity.  Christ, crucified, is the great object of their\n    faith, and the centre of their affections; and, while their love to\n    Him prompts them to live soberly, and righteously, and godly, in this\n    present evil world, they cordially exclaim, 'Not unto us, not unto\n    us, but to Thy name, O Lord, give glory.'\"\n\nIn leaving Anglesea behind him, the sufferings, and contradictions he had\nknown there, did not quench his enthusiastic holiness, and fervent\nardour.  We are assured of this when we read his\n\n\n\n\"SECOND COVENANT WITH GOD.\n\n\n    \"While returning from a place called Tongwynl\u00e2s, over Caerphilly\n    Mountain, the spirit of prayer descended, very copiously, upon me.  I\n    wept for some hours, and heartily supplicated Jesus Christ, for the\n    blessings here following.  I found, at this time, a particular\n    nearness to Christ, as if He were close by me, and my mind was filled\n    with strong confidence that He attended to my requests, for the sake\n    of the merits of His own name.  This decided me in favour of Cardiff.\n\n    \"I.  Grant me the great favour of being led by Thee, according to Thy\n    will\u2014by the directions of Thy providence, and Word, and this\n    disposing of my own mind, by Thy Spirit, for the sake of Thine\n    infinitely precious blood.  Amen.\u2014C. E.\n\n    \"II.  Grant, if I am to leave Caerphilly, that the gale (of the\n    Spirit's influence), and religious revival I had there, may follow me\n    to Cardiff, for the sake of Thy great name.  Amen.\u2014C. E.\n\n    \"III.  Grant Thy blessing upon bitter things, to brighten, and\n    quicken me, more and more, and not to depress, and make me more\n    lifeless.  Amen.\u2014C. E.\n\n    \"IV.  Suffer me not to be trodden under the proud feet of members, or\n    deacons, for the sake of Thy goodness.  Amen.\u2014C. E.\n\n    \"V.  Grant me the invaluable favour of being, in Thy hand, the means\n    of calling sinners unto Thyself, and of edifying Thy saints, wherever\n    Thou wilt send me, for the sake of Thy name.  Amen.\u2014C. E.\n\n    \"VI.  If I am to stay at Caerphilly, give me some tokens, as to\n    Gideon of old, by removing the things that discourage me, and are in\n    the way of the prosperity of religion, in that church.  Amen.\u2014C. E.\n\n    \"VII.  Grant, Lord of glory, and Head of Thy Church, that the Ark of\n    the cause which is Thine, in Anglesea, and Caerphilly, may be\n    sustained from falling into the hands of the Philistines.  Do not\n    reject it.  Aid it speedily, and lift up the light of Thy countenance\n    upon it; and by Thy Spirit, Word, and providence, so operate, as to\n    carry things forward in the churches, and neighbourhoods, in such a\n    manner as will produce changes in officers, and measures, that will\n    accomplish a thorough improvement, in the great cause, for the\n    establishment of which, in the world, Thou hast died,\u2014and by\n    scattering those that delight in war, and closing the mouths of those\n    that occasion confusion.  Amen.\u2014C. E.\n\n    \"VIII.  Grant me way-tokens, by the time I begin my journey to\n    Liverpool, and from thence to Anglesea, if it is Thy will that I\n    should go thither this year.  Amen.\u2014C. E.\n\n    \"IX.  Oh, grant me succour, beneath the shadow of the sympathy that\n    is in Thee, towards them who are tempted, and the unbounded power\n    there is in Thee, to be the relief of such.  Amen.\u2014C. E.\n\n    \"X.  Accept of my thanksgiving, a hundred millions of times, that\n    Thou hast not hitherto cast me from Thine hand, as a darkened star,\n    or a vessel in which there is no pleasure; and suffer not my life to\n    be extended beyond my usefulness.  Thanks that Thou hast not given me\n    a prey to the teeth of any.  Blessed be Thy name.  Amen.\u2014C. E.\n\n    \"XI.  For the sake of Thine infinite merit, do not cast me, Thy\n    servant, under the feet of pride, and injustice, of _worldly_\n    greatness, riches, and selfish oppression of any men, but hide me in\n    the secret of Thy tabernacle, from the strife of tongues.  Amen.\u2014C.\n    E.\n\n    \"XII.  Help me to wait silently, and patiently upon Thee, for the\n    fulfilment of these things, and not become enraged, angry, and speak\n    unadvisedly with my lips, like Moses, the servant of the Lord.\n    Sustain my heart from sinking, to wait for fresh strength from Zion.\n    Amen.\u2014C. E.\n\n    \"XIII.  Help me to wait upon Thee, for the necessaries of life; let\n    Thy mercy, and goodness follow me, while I live; and, as it hath\n    pleased Thee to honour me greatly, by the blessing Thou hast\n    vouchsafed upon the ministry through me, as an humble instrument, at\n    Caerphilly, after the great storm had beaten upon me in Anglesea,\n    like Job, grant that this honour may continue to follow me the\n    remainder of my days, as Thou didst unto Thy servant Job.  Amen.\u2014C.\n    E.\n\n    \"XIV.  Let this covenant abide, like the covenant of salt, until I\n    come to Thee, in the world of eternal light.  I entreat aid to resign\n    myself to Thee, and to Thy will.  I beseech Thee, take my heart, and\n    inscribe upon it a deep reverence of Thyself, with an inscription,\n    that time, and eternity cannot efface.  Oh, let the remainder of my\n    sermons be taken, by Thee, from my lips; and those which I write, let\n    them be unto Thee for a praise.  Unto Thee I dedicate them.  If there\n    should be anything, in them, conducive to Thy glory, and to the\n    service of Thy kingdom, do Thou preserve it, and reveal it unto men;\n    else, let it die, like the drops of a bucket in the midst of the\n    scorching heat of Africa.  Oh, grant that there may be a drop of that\n    water, which Thou, alone, canst impart, and which springs up to\n    eternal life, running through all my sermons.  In this covenant,\n    which, probably, is the last that will be written between me and\n    Thee, on the earth, I commit myself, my wife, and the churches\n    amongst whom I have preached, to the protection of Thy grace, and the\n    care of Thy covenant.  Amen.\u2014C. E.\n\n    \"XV.  Let this covenant continue, when I am in sickness, or in\n    health, or in any other circumstance; for Thou hast overcome the\n    world, fulfilled the law, finished justifying righteousness, and hast\n    swallowed up death, in victory, and all power, in heaven and earth,\n    is in Thy hand.  For the sake of Thy most precious blood, and perfect\n    righteousness, note this covenant, with Thine own blood, in the court\n    of the memorials of forgiving mercy: attach unto it Thy name, in\n    which I believe; and here I, this day, set my unworthy name unto it,\n    with my mortal hand.  Amen.\u2014CHRISTMAS EVANS.  Dated Cardiff, April\n    24th, 1829.\"\n\nThis document, found among his papers, after death, contains many\naffecting words, which give an insight to painful experiences, and\nsufferings.  The standard set by Christmas Evans, was very high; his\nexpectations from the Christian profession were such as to give, to his\nideas of the pastoral office, perhaps somewhat of a stern aspect; nor can\nwe forget that all his life had been passed in a very severe school.  He\nwas, perhaps, disposed to insist somewhat strenuously upon Church\ndiscipline.  No doubt, his years at Caerphilly were among the happiest,\nand most unvexed in Church relations; his ministerial power, and success\nwere very great; still, as the covenant we have just recited hints, there\nwere probabilities of removal to Cardiff.\n\nThe appearance of Christmas Evans in Caerphilly was regarded, as we have\nseen, as something like an advent, and, to him, it was, for a short time,\na haven of pleasant rest.  There were some eminent ministers, men of\nconsiderable knowledge, and real power, residing in the neighbourhood,\nwith whom he appears to have had most pleasant intercourse; among others,\na Mr. J. P. Davies, in his way a mighty theologian, and clear, and ready\nexpositor; he was laid by, for some months, under medical care, at\nCaerphilly, but was able to attend the ministry of the old preacher every\nSabbath, and became one of his most intimate friends; they met almost\ndaily, and the younger man was astonished by the elder's insatiable\nthirst for knowledge, and equally astonished by the extensive, and\nvaried, stores of information he had accumulated, in his busy, and\nincessantly toilsome career.  He acknowledged, afterwards, with delight,\nthe variety of lights he had received, both as to the construction of a\ntext, or the clearer definition of a principle, from his aged friend.  As\nto the preaching, he said it gave him quite a new impression of the order\nof the preacher's mind: he expected flashes of eloquence, brilliant\npictures,\u2014of these he had long heard,\u2014but what astonished him, was the\nfulness, and variety of matter, Sabbath after Sabbath.  Mr. Davies only\nreturned home to die; but he delighted his people, when he returned, by\nrepeatedly describing the comfort, and light he had received, from the\ncompany of the matured, the aged, and noble man.\n\nThe society he enjoyed was, probably, more cultivated, small as was the\nvillage, than that by which he had been surrounded in Anglesea; from all\nthe inhabitants, and from the neighbourhood, he received marks of great\nrespect; it was, probably, felt, generally, that, by some singular turn\nof affairs, a great man, a national man, a man of the Principality, had\nsettled in their midst.  And he always after, and when he had left,\nremembered this brief period of his life with deep gratitude.  He was\nmore able to borrow books: here, for the first time, he read a work,\nwhich was regarded as a mighty book in that day, Dr. Pye Smith's\n\"Scripture Testimony to the Messiah;\" he read it with intense eagerness,\nincorporating many of its valuable criticisms into his sermons, and,\nespecially, making them the subjects of ordinary conversation.  Rhys\nStephen says, \"I remember listening to him with wonder, when, in\nconversation with Mr. Saunders, of Merthyr, he gave the substance of Dr.\nPye Smith's criticism on John xvii. 3.  And I distinctly remember, that\nwhen Mr. Evans said, 'Mr. Saunders, you will observe that, on these\ngrounds, the knowledge of Jesus Christ, here mentioned, is the same\nknowledge as that of the only true God, and that the knowledge of the\nformer is as necessary to salvation, as the knowledge of the\nlatter\u2014indeed, they are one, and the same thing,' 'Yes, yes,' was the\nreply; 'capital, very excellent.  I never heard that interpretation\nbefore.'  I was then a youth, and was not astonished by the\ninterpretation, which, of course, was new to me, so much as by the\nadmissions of the aged men that it was new to them.\"  At any rate, it\nillustrates the avidity with which this mind still pursued the rays of\nlight, from book to book, from conversation to conversation.\n\nOn another occasion, he met a young minister at Llantrissant, and, after\na meeting in the morning, he inquired of the young man what he was then\nreading; the reply was, that he was going slowly through Beattie on Truth\na second time.  Christmas Evans immediately replied, \"You must come to\nsee me before you return to Swansea, and give me the substance of\nBeattie: was he not the man that replied to David Hume, eh?\"  The young\nman said he had the book in his pocket, and that he would cheerfully give\nit him, but the print was very small.  He, with still greater eagerness,\nsaid, \"I can manage that.  I will take of it, with many thanks.\"  It was\na pleasure to give it him, and he pocketed it with as much pleasure as\never a school-boy did the first prize, at the end of the session.  In\nthree days after, the young man called upon him, at his own house, and\nspent a couple of hours with him; but he says he could get no farther, in\nconversation, than upon Beattie,\u2014he was thoroughly absorbed in the\nargument with Hume, and his school of scepticism, and unbelief.  Yet he\nwas now sixty-five years of age; his one eye was very weak, though seeing\nwell enough, without a glass, at the proper distance; and he was,\notherwise, full of bodily infirmities; but his love of reading was\nunabated, as was, also, his earnest curiosity to know what was passing on\nin the world of thought.\n\nAnd among his friends, at this period, we notice some members of the\nEdwards family,\u2014David Edwards, of Beaupre, or, as it is commonly\npronounced, Bewper, in Glamorganshire; and Evan Edwards, of Caerphilly,\nthe son, and grandson of one of the most remarkable men modern Wales has\nproduced, William Edwards, in his day a mighty engineer.  Until his time,\nthe Rialto, in Venice, was esteemed the largest arch in Europe, but he\nthrew an arch over the <DW75> forty-two feet wider, and thus, for a long\ntime, it held its reputation of being the largest arch in the world.  A\nwonderful man was William Edwards, entirely self-made, not only a great\nengineer, but a successful farmer, and an ordained Independent minister.\nHe was wealthy, of course, but he insisted upon receiving a good income\nfrom his church, although he distributed every farthing among the poor of\nhis own neighbourhood, and added, considerably, to the sum he\ndistributed, from his own property.  The successor to Mr. Edwards, as the\npastor of the Independent Church of Y-Groeswen, was the Rev. Griffith\nHughes, a person of about the same age as Christmas Evans, also, although\na polished gentleman, a self-taught man, a wit, a man of considerable\nreading, and information, and widely advanced in his religious opinions;\nalthough, professedly, a Calvinist, beyond the narrow, and technical\nCalvinism of his time, and even beyond the Fullerism, or doctrines of\nAndrew Fuller, which had been charged on Christmas Evans, as a crime, by\nhis enemies in Anglesea.\n\nIt was about this time that he was earnestly entreated to prepare a\nvolume of sermons for publication, and it seemed to be in connection with\nthis, and with some fears, and discouragements which still troubled his\nmind, that he made the following entry, discovered among his papers after\nhis death:\u2014\n\n    \"Order things so, O Lord, that they may not prove a hindrance, and a\n    discouragement to me, and an obstacle to the progress of Thy cause.\n    Thy power is infinite, and Thy wisdom infallible.  Stand between me,\n    and all strife, that no evil effect may fall upon me.  I flee under\n    the shadow of Thy wings to hide myself, as the chickens do under the\n    wings of the hen.  Let nothing corrupt, and extinguish my gifts, my\n    zeal, my prosperity; let nothing hinder the Church.\n\n    \"I have been earnestly requested, by many of my brethren in the\n    ministry, to prepare some of my sermons for the press.  In Anglesea,\n    I had no leisure for such work, although I once commenced it, and\n    wrote out five for the purpose.  I let the work rest for two years,\n    at Caerphilly; but, here, my mind has been moved towards it anew; and\n    now I come to Thee, O Lord, who art the Head of the Church, and the\n    chief Prophet and Teacher of the Church, to consult Thee, whether I\n    shall proceed with the work, or not.  Is it a part of my duty, or a\n    foolish device of my own?  I beseech, for Thy name's sake, Thy\n    gracious guidance herein.  Permit me not to labour, with my weak\n    eyesight, at a work that Thou wilt not deign to bless, but that shall\n    be buried in oblivion,\u2014unless it may please Thee (for Thou hast the\n    keys of the house of David), in Thy providence, to prepare my way to\n    publish the work, without danger to myself, of debt, and disgrace;\n    and unless it may please Thee, the great Shepherd of the sheep, to\n    guide me, to give forth the true Gospel, not only without error, but\n    with the savour, and unction that pervade the works of Bunyan, and\n    the hymns of William Williams; and, also, may they prove for the\n    edification of Thy Church, and the conversion of sinners!  If Thou\n    wilt condescend to take the work under Thy care, help me to\n    accomplish the design.\n\n    \"In reading the 91st Psalm, I perceive that he who dwelleth in the\n    secret place of the Most High, shall abide under the shadow of the\n    Almighty; and that is so safe a place, and so impenetrable a\n    protection, that the arrow that flieth by day, and the pestilence\n    that walketh in darkness, with the sting of the serpent, the asp, and\n    the viper, cannot hurt or injure him who hath made it his refuge.  It\n    is by faith, I hope, that I have gathered together all my jewels, and\n    placed them under the shadow of safety that is in God.  I have given\n    my name anew to Christ, my body, my talents, my facility in\n    preaching,\u2014my name, and character as a man, a Christian, and as a\n    preacher of the Gospel; my time, the remainder of my preaching\n    services, my success, my wife, and all my friends, and helpers in the\n    cause of the Lord, for whom I earnestly pray that they may be blessed\n    in Anglesea, Caernarvonshire, Caerphilly, Cardiff, and all the\n    churches in Wales, many of which have helped me in my day.\"\n\n\n\n\nCHAPTER X.\n_CAERNARVON AND LAST DAYS_.\n\n\nLeading a Forlorn Hope again\u2014More Chapel Debts\u2014A Present of a Gig\u2014Jack,\n_bach_!\u2014The One-eyed Man of Anglesea once more\u2014The Old Man's Reflections\nin his Journal\u2014Characteristic Letters on Church Discipline\u2014Threescore\nYears and Twelve\u2014Starts on his Last Journey to liquidate a Chapel Debt\u2014An\nAffecting Appeal to the Churches\u2014Laid up at Tredegar\u2014Conversations\u2014In\nSwansea\u2014This is my Last Sermon\u2014Dying\u2014Last Words\u2014\"Good-bye!  Drive on!\"\n\nThe last field of the great, good man's pastorate was Caernarvon; thither\nhe removed when about sixty-seven years of age.  It might be thought,\nthat after such a hard, and exhausting life of travel, and toil, some\nplan might have been devised, by which his last days should be passed in\nrestfulness, and peace; but it was not to be so: throughout his life, his\nhad been up-hill work, no path of roses, no easy way; and, indeed, we\nusually know that such spheres are reserved for men who can carry nothing\nwith them but the weight of dignified dulness.  Of every sphere, from his\nfirst settlement at Lleyn, we read, that the cause was in a prostrate\ncondition; and so, here, Christmas Evans appears to have been invited to\ntake the charge of the Caernarvon church because it consisted of about\nthirty members, chiefly of the lowest class, of course quarrelling, and\ndisunited.  The dissolution of the church was advised.  There was a\nfairly respectable place of worship, but it was \u00a3800 in debt, apparently,\nto us, in these days, not a very large sum, but a sum of considerable\nimportance in Wales, and especially in that day.\n\nSo the question was discussed at a ministerial association, and some\nbrother minister present, delivered himself of a confirmatory dream he\nhad had on the subject, and the matter was practically settled, when a\nyoung minister spoke up, in the conference, and said to the venerable\nman, \"Yes, you had better go to Caernarvon: it is not likely your talents\nwould suit, but you might do excellently well at Caernarvon.\"  The\nimpudent speech astounded all the ministers present, except the\nunfortunate utterer of it.  They knew not what to say.  After a pause,\nthe brethren all struck utterly dumb, Christmas Evans opened his one\nlarge eye upon his adviser, and, with some indignation, he said, \"Ay,\nwhere hast thou come from?  How long is it since thou didst chip thy\nshell?\"  Well, it was the very word: no one else could have, in so\nsummary a manner, crunched up the thin egg-shell of pretentious conceit.\n\nThere was a real desire, on the part of the trustees of Caernarvon, and\nof English friends in Liverpool, that he should return to the north; and\nsome gentlemen facilitated his return by giving him a gig, so that he\nmight travel at his ease, and in his own way.  This was not a very great\ndonation, but it added, materially, to his comfort: he was able to travel\npleasantly, and conveniently with Mrs. Evans.  His horse, Jack, had been\nhis companion for twenty years, but the pair were very fond of one\nanother.  Jack knew, from a distance, the tones of his master's voice;\nand Christmas, on their journeys, would hold long conversations with\nJack.  The horse opened his ears the moment his master began to speak,\nmade a kind of neighing, when the rider said, as he often did, \"Jack,\n_bach_, we have only to cross one low mountain again, and there will be\ncapital oats, excellent water, and a warm stable,\" etc.\n\nSo he bade farewell to Cardiff in 1832, and upon the following Sunday,\nafter his farewell there, he appears to have commenced his new ministry.\nIt seems pathetic to us, to think of the old man, but we have no idea\nthat he had any such pity, or sympathy for himself.  Who can doubt,\neither, that he favoured, and hailed the opportunity of the return to the\nnorth? and Caernarvon, and Anglesea were almost one: he had but to cross\nthe Menai Straits to be again in Anglesea\u2014Anglesea, the scene of so many\ntrials, and triumphs, where he had planted so many churches, sustained so\nmany spiritual conflicts, and enjoyed, in his Cildwrn cottage, no doubt,\nyears of much domestic happiness.  It seems to us he ought never to have\nleft Anglesea; but he regarded his exile to Caerphilly as a mission, that\nwas to terminate, if success should crown it.  And so he was back again\nin the old neighbourhood, and it appears, that the announcement of his\nreturn created universal delight, and joy, and strong excitement.  He had\nbeen absent for about seven years, and the people, on account of his\nadvanced age, when leaving them, expected to see him bowed with\ninfirmity, and his preaching power, they supposed, would rather\naffectingly remind them of what he had been.\n\nShortly after his entrance upon the work of Caernarvon, a public occasion\npresented itself for his appearance in Anglesea.  The whole neighbourhood\nflocked out, to see the patriarch.  As he appeared on the platform, or\npreaching-place, in the open-air,\u2014for no chapel could have contained the\nmultitude,\u2014the people said, \"Why, he does not seem at all older! he looks\nmore like a man of forty-five, than sixty-five, or sixty-six.\"  And his\npreaching was just the same, or, possibly, even richer, and greater: it\nwas his own old self, their own old Christmas Evans; the same rich, and\nexcursive fancy, the same energetic, and fiery delivery.  The appearance\nof such a man, under such circumstances,\u2014one who has worn well, borne the\nburden and heat of the day, and taken his part \"on the high places of the\nfield,\"\u2014is a mighty awakening, and heart-healing time for old believers,\nwho find their love to each other renewed in the rekindled love to the\nold pastor, and father in Christ.  Old memories very tenderly touch\nreciprocating hearts.  The old words, and the old voice, awaken old\nemotions, which now have become new.  But, then, it is only a minister\nwith a heart, who can touch this well-spring of feeling: starched\nrespectability will not do it, eminent collegiate learning will not do\nit, rolling rhetorical periods will not do it.  It is only the great\nhearts who can open these sluices of feeling, these fountains of emotion,\nin which the past, and the present mingle together, as the hearers drink\nrefreshing streams from the fountains of recollection.\n\nWhile in Caernarvon, he penned in his journal the following pious\nreflections:\u2014\n\n    \"I have been thinking of the great goodness of the Lord unto me,\n    throughout my unworthy ministry; and now, in my old age, I see the\n    work prospering wonderfully in my hand, so that there is reason to\n    think that I am, in some degree, a blessing to the Church, when I\n    might have been a burden to it, or rather a curse, by which one might\n    have been induced to wish me laid in the earth, that I might no\n    longer prevent the progress of the work.  Thanks be to God, that it\n    is not so! though I deserve no better, yet I am in the land of mercy.\n    This is unto me, according to the manner of God unto His people.  My\n    path in the valley, the dangers, and the precipices of destruction\n    upon which I have stood, rush into my thoughts, and also the sinking\n    of many in death, and the downfall of others by immorality, and their\n    burial in Kibroth-Hattaavah, the graves of inordinate desire;\n    together with the withering, the feebleness, and the unfruitfulness\n    of some, through the influence of a secret departure from God, and of\n    walking in the hidden paths, that lead to apostasy.\"\n\nAnd here we may most appropriately insert a very characteristic letter,\nwhich shows the exceedingly stringent ideas which Christmas Evans\nentertained with regard to Church membership,\u2014strait ideas, which, we\nsuppose, would be scarcely tolerable now:\u2014\n\n\n\n\"LETTER TO A BROTHER MINISTER ON CHURCH DISCIPLINE.\n\n\n    \"BELOVED BROTHER,\u2014I write to you, August 5th, 1836, in the seventieth\n    year of my age, and in the fiftieth of my ministry, after conversing\n    much with ministerial brethren, earnestly desiring to see our\n    Associational Union brought into action, by representatives of the\n    churches, with a view to promote a determination,\u20141.  To bear each\n    other's burden more efficiently, in the denomination to which we\n    belong.  I lament the deficiency in this point, and ardently wish to\n    see it effectually remedied.  2. To watch over and promote a holy\n    conversation among all the members, and all the preachers, in a more\n    efficient manner, to prevent persons of unbecoming conversation from\n    obtaining privileges, in any church, when they have been excluded in\n    another; for that would occasion blots, and blemishes to appear on\n    the bright countenance of the ministry.  The Associational Union, in\n    which all the churches of the same faith, and order join, should be a\n    defence of the independence of the churches, through their\n    representatives: it should also operate as a sort of check upon\n    independency, lest it should become opposed to the general good, and\n    frustrate the co-operation of the whole body.  _That they may all be\n    one_, is the motto.\n\n    \"Respecting Church discipline.  We cannot be certain that we are\n    doing right, by administering the same punishment to all offenders,\n    even for the same offence; for the general character weighs heavily,\n    in the balance of discipline.  Also, a distinction should be made\n    between the seducer, and the seduced; and between being overcome, or\n    falling into sin, and living habitually in sin, and following it, as\n    a slave following his master.  The denial of Peter, from weakness,\n    and without previous deliberation, was very different from the\n    betrayal of Judas, and his intentional selling of Christ.  The\n    different characters of Saul, king of Israel, and that of David,\n    required different treatment, in discipline, on account of their\n    offences.  The Lord's discipline upon Saul was that of a rod of iron,\n    but upon David, the correcting rod of a Father, for his good, that he\n    might be a partaker of His holiness.\n\n    \"There are two things, brother, which we ought to avoid in the\n    exercise of discipline: 1, we should avoid too great severity on the\n    one part; and, 2, too much leniency on the other part.  Wisdom is\n    necessary here to distinguish the different characters,\u2014those who\n    require severity, and those who claim tenderness: the two are to be\n    found blended in the principle of evangelical discipline.  A\n    difference is to be made betwixt some, who may have been companions\n    in the same crime; snatching some of them as brands from the burning.\n    The ground of the distinction lies in the different amount of guilt,\n    which subsists between the seducer, and seduced.\n\n    \"I have witnessed danger, and have sustained some harm myself, and\n    seen harm done in churches, by exercising tenderness towards some\n    persons, in the vain hope of their reformation.  Receiving verbal\n    testimony, or mere fluent acknowledgments, from their lips, without\n    waiting for fruit, in action, also; some having been often accused;\n    and as often turning to the refuges frequented by them.  I never\n    exercised tenderness towards such as these, without being repaid by\n    them afterwards, if they had opportunity: Shimei-like, they would\n    curse me, after I had shed the best oil of tenderness on their heads.\n    There are some in the Christian Church like Jezebel; and there are\n    some in our congregations like Joab, the son of Zeruiah, that you can\n    scarce discipline them without rending the kingdom, until they become\n    ripe for judgment; for they hardly ever repent, more than did Joab\n    and Shimei: they are ultimately suddenly broken, without any danger\n    to the Church from their fall.\n\n    \"I perceive that the Scriptures make a difference between one that\n    falls into sin, and one wallowing in it; between one overtaken by a\n    party of marauders, and dragged into the camp, and made drunk at\n    supper, and one, like Judas, going to the party, and being secretly\n    one of them, having pistols as they had: such are hypocrites.  I have\n    many times been the advocate of the fallen, and in a variety of\n    instances have observed this operating beneficially for the Church.\n    Sometimes I have found those who had been spared upon their own\n    verbal contrition, blessing God for His long forbearance of them, and\n    also their spiritual brethren, who had in a manner set their bones;\n    as the Scripture hath it, 'Restore such an one in the spirit of\n    meekness.'\n\n    \"We should be careful that discretion, and love, be in exercise,\n    though in strife, and contention it be not always an easy matter to\n    do this.  When the beasts of dissension get loose from the caravan,\n    Satan sometimes drives them through the streets of Zion, that they\n    may enter the houses of the inhabitants; and like the lioness that\n    escaped from the keepers at Shrewsbury, and attacked the foremost\n    horse in the carriage, so contentions frequently attack the leaders,\n    in order to stop the carriage of the ministry as it travels on, in\n    the labours of the pulpit.  In the midst of the noise of strife, the\n    man of God must raise his voice to heaven for courage, and\n    tenderness, so that the oil of Christ's love to the souls of men may\n    be found in the oil-flagon of reproof, which is poured on the head;\n    for if anger, and revenge enter in, they will drop, like the spider\n    in Germany, into the pot, and that will prevent the salutary effect\n    of the oil, because the poison of wrath is mixed with it.  The\n    righteousness of God cannot be fulfilled in this manner in the\n    discipline.  Oh, brother! who is sufficient for these things, without\n    constant help from heaven?  How awful is this place!  This is the\n    house of God, and the gate of heaven; and here is a ladder, by which\n    we may climb up for help, and a school, in which we may learn how to\n    conduct ourselves in the house of God.\n\n    \"You cannot but be conscious, brother, of the great difficulty there\n    is not to speak unadvisedly with our lips, as did Moses, whilst\n    drawing water for the rebellious Israelites.  The rebellion of the\n    people had embittered his spirit, so that his obduracy stood like a\n    cloud between the people, and the tenderness of the Lord, when He was\n    showing mercy upon them by giving them water.  Moses upbraided their\n    rebellion instead of showing mercy, as the dispensation of God now\n    required; a dispensation which contained in it a secret intimation of\n    the great mercy to be shown by the death of Christ on the cross.\n    Their strife was the cause of embittering the spirit of Moses, yet he\n    should have possessed his soul in patience.\n\n    \"There are two things, brother, which you should observe.  First, you\n    will be called upon to attend to causes of contention; and you will\n    find persons so hardened, that you will not be able to obtain\n    weapons, in all the armoury of God's Word, that will terrify them,\n    and make them afraid of entering their old haunts.  Such are persons\n    without faith, and without the fear of God, and the love of Christ\n    influencing their minds; and though you warn them of the consequences\n    of their contentions, that they are likely to deprive them of the\n    privileges of the house of God, and thus forfeit the promised land,\n    yet they stand unmoved, nothing terrified, for they value the\n    flesh-pots of Egypt, and their livelihood there, more than the manna,\n    and the land of promise.  You cannot frighten them by speaking of the\n    danger, and loss of the immunities of the Church below, or that\n    above.  Esau-like, they will sell their birthright, as Christian\n    professors, for a mess of pottage.  A man who has no money is not\n    afraid to meet with robbers in the wood; but he who has gold to lose\n    will be cautious, and watchful, lest he should be robbed of his\n    property.  On a night of great storm, when ships are broken to\n    pieces, and sinking, a person who has no share in any of them will\n    not tremble, or feel any concern on their account.  Thus there are\n    some men, concerning whom it is impossible to make them dread going\n    out among the rapacious beasts of backslidings, and no storms can\n    keep them in fear.  Their spirit is one with the marauders, and they\n    have no care, for they have nothing to lose in the tempests that blow\n    upon the cause of the religion of Christ.  These are the tares, or\n    the children of the wicked one, in the Church.\n\n    \"Secondly, for your own encouragement, brother, I remark that you\n    will have to attend to the exercise of discipline, and to treat with\n    persons that may be alarmed, and made to tremble at the Word of God,\n    and not rush on presumptuously in their evil course.  These are\n    professors, who possess white garments, and the gold of faith, and\n    eye-salve from the unction of the Holy One.  These individuals are\n    rich in faith.  They are afraid of revolutions, and upsettings of the\n    constitutional order of the new covenant, for they have funds\n    invested in the stocks of God's kingdom.  They are afraid that any\n    storm, or rock of offence should come in the way of the Gospel ship,\n    for their treasure is on board it, and they have an interest in it.\n    They dread the thought of walking unwatchfully, and licentiously,\n    lest they should be robbed of their riches, and forfeit the\n    fellowship of God in prayer, lose the light of His countenance, and\n    His peace in the means of grace, and lest they should be deprived of\n    their confidence in the merits of Christ, and a good conscience.\n    They have denied themselves, and have pulled out the right eye, lest\n    they should not be acceptable before God.  They dread harbouring in\n    their bosoms the old guilt and former doubts.  They are cautious not\n    to give a night's lodging to such miscreants as anger, revenge, lust,\n    and things which are of the earth; for they know that these are\n    robbers, and if they have any indulgence they will steal away the\n    _title-deeds_ of assurance to the inheritance.  They are well aware,\n    also, that they will sustain the loss of a pure conscience, which has\n    been purged by the blood of Christ, and which, as a golden chest, is\n    a preserver of our confidence, immovable unto the end.  It is\n    possible, brother, to manage, and discipline such professors.  They\n    have something to lose, consequently they will not flee from their\n    refuge, lest they should be destroyed.  _Keep that which thou hast_.\n    David lost for a season the enjoyment of the above blessings; but he\n    was cleansed with hyssop, had his spirit renewed, and his riches were\n    restored to him by faith's view of the Messiah, for which he vowed to\n    sing aloud for ever, and ever.  He prayed, after this, to be\n    delivered from presumptuous sins, lest he should be imprisoned a\n    second time by a party so wicked, and detestable.  May the spiritual\n    gift be kindled in you, brother.  Grace be with you, for ever, and\n    ever.\n\n                                                          \"Affectionately,\n                                                         \"CHRISTMAS EVANS.\n\n    \"_Caernarvon_, _August_ 5_th_, 1836.\"\n\nBut it was hard work in Caernarvon.  The debt upon the chapel was a\nperpetually-recurring trouble.  We have said when he went there eight\nhundred pounds was the burden, and that the people were very poor.  Of\nthis eight hundred, four hundred seems to have been collected by a Mr.\nJohn Edwards, who used, as his introduction, in asking for contributions,\nthe specimen of Welsh eloquence to which we have referred (The Graveyard\nWorld); so that Christmas Evans may, really, be regarded as the\nliquidator of the debt to that extent.  The time came when the whole\nremaining sum had to be paid.  What could be done?  Over seventy years of\nage, the old man started forth, on a tour through the south, to attempt\nto raise the sum.  In April, 1838, when he had been four years in\nCaernarvon, he set off with his wife, and a young preacher, the Rev. John\nHughes.  Before he set out, he wrote a circular to his brethren, which\nwas published in the _Welsh Magazine_.  It is scarcely possible, we\nthink, to read it, remembering who wrote it, and the circumstances under\nwhich it was written, without tears of feeling:\u2014\n\n    \"DEAR BRETHREN,\u2014We have received notice to pay up three hundred\n    pounds.  The term of the lease of life has expired in my case, even\n    threescore and ten years, and I am very much afflicted.  I have\n    purposed to sacrifice myself to this object, though I am afraid I\n    shall die on the journey\" (he did die on his journey); \"and I fear I\n    shall not succeed in my errand for Christ.  We have no source to\n    which we can now repair, but our own denomination in Wales, and\n    brethren, and friends of other communities, that may sympathize with\n    us.  Oh, brethren, pray, with me, for protection on the journey\u2014for\n    strength, and health this _once_, on occasion of my bidding farewell\n    to you all! pray for the light of the Lord's countenance upon me in\n    preaching; pray for His own glory, and that His key may open the\n    hearts of the people, to contribute towards His cause in its present\n    exigency.  Oh, help us, brethren!\u2014when you see the old brother, after\n    having been fifty-three years in the ministry, now, instead of being\n    in the grave with his colleagues, or resting at home with three of\n    them who are yet alive\u2014brethren Lewis of Llanwenarth, Davies of Velin\n    Voel, and Thomas of Aberduar,\u2014when you see him coming, with the\n    furrows of death in his countenance, the flowers of the grave on his\n    head, and his whole constitution gradually dissolving; having\n    laboured fifty years in the ministry in the Baptist denomination.  He\n    comes to you with hundreds of prayers, bubbling, as it were, from the\n    fountain of his heart, and with a mixture of fear, and confidence.\n    Oh, do not frown upon him!\u2014he is afraid of your frowns.  Smile upon\n    him, by contributing to his cause, this once for all.  If you frown\n    upon me, ministers and deacons, by intimating an _irregular case_, I\n    am afraid I shall sink into the grave before returning home.  This is\n    my last sacrifice for the Redeemer's cause.\"\n\nNaturally, wherever he passed along, he was received by all the churches,\nand throughout every county, with more than cordiality\u2014with great joy.\nHe was very successful in raising money for the purpose which urged him\nforth from home: perhaps his popularity was never so great as now.  Mr.\nCross, one of his biographers, says, that wherever he preached, the place\nwas thronged at an early hour, and, frequently, multitudes remained\noutside, unable to obtain admittance.  He reached Monmouthshire, and\npreached before the County Association; and it is said, that the sermon\nevinced all his vigour of intellect, and splendour of genius, and as\nperfect a command over the feelings of the great audience as ever.  One\nof his great images here was his description of the Gospel, on the day of\nPentecost, as a great electrical machine, Christ turning the handle,\nPeter placing the chain in contact with the people, and the Holy Ghost\ndescending like a stream of ethereal fire, and melting the hearts of\nthree thousand at once.  His text was, \"By grace ye are saved.\"\n\nBut the effort was too much for him, and he was laid up for a week at the\nhouse of Mr. Thomas Griffith, a kind host, who, with his whole family,\nattempted, in every way, to minister to his comfort, and, with\naffectionate assiduity, sought to restore him.  On the whole, he appears\nto have been full of vivacity that week, and, during the intervals of\npain, cheered, and charmed his friends.  He had, one day, come\ndownstairs, and Mr. James, the son-in-law of his host, was helping him up\nagain.  He had only got a few steps, when he said buoyantly, \"Mr. James,\nI dare say if I thought the French were behind me with their bayonets, I\nshould be able to get upstairs without your help.\"  With the word he took\nhis arm from Mr. James's shoulder, and briskly ran up the flight of\nsteps, laughing at his feat.\n\nHis conversation was, however, usually brightly religious.  \"This is the\nGospel,\" he said once in the course of talk\u2014\"This is the Gospel: 'He that\nbelieveth shall be saved.'  Now, in order to the truth of this\ndeclaration, every believer must be saved.  If, in the last day, the\ngreat enemy find one single soul not saved, who ever believed the Gospel,\nhe would take that soul up, present that soul to the Judge, and to the\nimmense assembly, and say, 'The Gospel is not true.'  He would take that\nlost believer through all the regions of pandemonium, and exhibit him in\ntriumph to the devils, and the damned.\"  \"But,\" said his host, \"that\nshall never be, Mr. Evans.\"  \"No,\" said he, planting the forefinger of\nhis right hand on his knee, as was his wont, and exclaiming, in a tone of\ntriumphant congratulation, \"_Never_! _never_! _never_!\"\n\nLeaving the house of Mr. Griffith, of Tredegar, he proceeded on his way,\npreaching at Caerphilly, Cardiff, Cowbridge, Bridgend, and Neath, and he\nreached Swansea on Saturday, July 14th.  The next day, Sunday, he\npreached twice\u2014preached like a seraph, says one of his memorialists: in\nthe morning his subject was the Prodigal Son; the evening, \"I am not\nashamed of the Gospel of Christ.\"  He was the guest of Daniel Davies, the\npastor of the Welsh Baptist Church in the town, the blind preacher, as he\nwas called, a man of great celebrity, and unquestioned power.  He was to\nbe the last host of his greater brother, or rather father, in the\nministry.  On the Monday evening, he went out to tea, with a friend who\nwas always glad to greet him, Mr. David Walters; and on the same evening\nhe preached, in English, in Mount Pleasant Chapel: his text was,\n\"Beginning at Jerusalem.\"  He was very feeble,\u2014perhaps we need scarcely\nwonder at that, after the two services of the day before.  He always felt\na difficulty when preaching in English, and, upon this occasion, he\nseemed much tried; gleams, and flashes of his ordinary brilliancy there\nwere, as in the following:\u2014\n\n\"Beginning at Jerusalem!  Why at Jerusalem?  The Apostles were to begin\nthere, because its inhabitants had been witness to the life, and death of\nChrist; there He had preached, wrought miracles, been crucified, and rose\nagain.  Here, on the very spot of His deepest degradation, He was also to\nbe exalted: He had been crucified as a malefactor, He was now to be\nelevated in the same place as a King; here were accorded to Him the\nfirst-fruits of His resurrection.\"  This was the strain of the\nsermon:\u2014\"'At Jerusalem, Lord?'  'Yes.'  'Why, Lord, these are the men who\ncrucified Thee; we are not to preach it to _them_?'  'Yes, preach it to\nall.'  'To the man who plaited the crown of thorns, and placed it on Thy\nHead?'  'Yes; tell him that from My degradation he may obtain a crown of\nglory.'  'Suppose we meet the very man that nailed Thy hands and feet to\nthe cross, the very man that pierced Thy side, that spat in Thy face?'\n'Preach the Gospel to them all: tell them all that I am the Saviour; that\nall are welcome to participate in the blessings of My salvation; I am the\nsame Lord over all, and rich unto all that call on Me.'\"  Such were some\nof the most characteristic passages.  As he was coming down the pulpit\nstairs, he said, loud enough to be heard by many present, \"_This is my\nlast sermon_!\"\n\nAnd it was even so.  He was taken very ill during the night; the next day\nhe was worse, the next day worse still, and then medical assistance was\ncalled in.  But on the Thursday, he got up, and walked for some time in\nthe garden.  It seems doubtful whether he thought that his end was so\nnear, although he had a dream, in one of the early evenings in the week,\nin which he seemed to come up to a great river, which he did not then\ncross, so that he scarcely thought his work or life might be over even\nyet.\n\nBut on Thursday night he was worse again, and on Friday morning, at two\no'clock, he said to his friends, Mr. Davies, Mr. Hughes, and others round\nhis bed, \"I am leaving you.  I have laboured in the sanctuary fifty-three\nyears, and this is my comfort, that I have never laboured without blood\nin the basin,\"\u2014the ruling power of imagination strong in him to the\nclose, evidently meaning that he had never failed to preach Christ and\nHim crucified.  A few more remarks of the same character: \"Preach Christ\nto the people, brethren.  Look at me: in myself I am nothing but ruin,\nbut in Christ I am heaven, and salvation.\"  He repeated a verse from a\nfavourite Welsh hymn, and then, as if he had done with earth, he waved\nhis hand, and exclaimed, \"GOOD-BYE!  DRIVE ON!\"\n\nIt seems another instance of the labour of life pervading by its\nmaster-idea the hour of death.  For how many years the \"one-eyed man\" of\nAnglesea had gone to, and fro on his humble nag!  As we have seen, lately\nhis friends had given him a gig, that he might be more at ease in his\nMaster's service; still he had his old horse, companion of his many\njourneys.  While he was dying, the old mountain days of travel came over\nhis memory\u2014\"GOOD-BYE!\" said he.  \"DRIVE ON!\"  He turned over, and seemed\nto sleep.  He slept indeed.  His friends tried to rouse him, but the\nangelic postman had obeyed the order,\u2014the chariot had passed over the\neverlasting hills.  So he died, July 19th, 1838, in the seventy-third\nyear of his age, and fifty-fourth of his ministry.\n\nHis funeral took place four days after his death, in the burying-ground\nattached to the Welsh Baptist Chapel, in Swansea.  It is said there never\nwas such a funeral in Swansea, such a concourse, and crowd of mourners,\nweeping their way to the grave, and following, as it had been their\nfather.  Fountains of sorrow were everywhere unsealed throughout the\nPrincipality, in Anglesea especially, where he had passed the greater\nportion of his life; indeed, throughout the Principality, there was\nscarcely a pulpit, of the order to which he belonged, which was not\ndraped in black; and it was evident that all felt \"a prince and a great\nman had fallen in Israel.\"\n\n\n\n\nCHAPTER XI.\n_SUMMARY OF GENERAL CHARACTERISTICS OF CHRISTMAS EVANS_, _AS A MAN AND A\nPREACHER_.\n\n\nA Central Figure in the Religious Life of Wales\u2014In a Singular Degree a\nSelf-made Man\u2014His Words on the Value of Industry\u2014His Honest\nSimplicity\u2014Power of Sarcasm Repressed\u2014Affectionate Forgiveableness\u2014Great\nFaith, and Power in Prayer\u2014A Passage in Dean Milman's \"Samor\"\u2014His Sermons\na Kind of Silex Scintillans\u2014Massive Preaching, but lightened by Beautiful\nFlowers\u2014As an Orator\u2014A Preacher in the Age of Faith\u2014Seeing Great\nTruths\u2014His Remarks on what was called \"Welsh Jumping\" in Religious\nServices.\n\nTHE character of Christmas Evans, it will be seen, from all that has gone\nbefore, appears to us to be eminently interesting as the most distinct,\nto us the most central, and realizable figure, in the religious life of\nhis country, and his times: he is the central figure in a group of\nremarkable men.  We shall not discuss the question as to whether he was\nthe greatest,\u2014greatness is so relative a term; he appears, to us,\ncertainly, from our point of view, the most representative Welsh preacher\nof his time, perhaps of any time: in him seemed embodied not merely the\nimaginative, but the fanciful, the parable-loving spirit of his\ndepartment of the great Celtic family; with this, that ardent devotion,\nthat supersensuous absorption, which to our colder temperament looks like\nsuperstition.\n\nOne writer finely remarks of him, and with considerable truth, so far as\nhis own country is concerned, \"He is a connecting link between the\nbeginning and the ending of the eighteenth century; he has the light, the\ntalent, and the taste of the beginning, and has received every new light\nthat has appeared since.  He was enabled to accompany the career of\nreligious knowledge in the morning, and also to follow its rapid strides\nin the evening.  In this he is unlike every other preacher of the day:\nthe morning and evening light of this wonderful century meet in him; he\nhad strength to climb up to the top of Carmel in the morning, and remain\nthere during the heat of the day, and see the consuming sacrifice, and\nthe licking up of the water; his strength continued, by the hand of the\nLord, so that he could descend from the mount in the evening, and run\nwithout fainting before the king's chariot to Jezreel.\"\n\nOn the whole, there is considerable truth in these words, although author\nand reader may alike take exception to some of them.  The circumstances\nand situation of the life of this singular man have been set so clearly\nbefore the reader in these pages, that there can be no difficulty in\napprehending the unpropitious and unfavourable atmosphere through which\nhe was compelled to move.  Few men can ever have more richly deserved the\nepithet of self-made: no systematic tuition could he ever have received;\nnear to manhood before he even attempted to obtain, before he had even\npresented to him any inducements to attempt, the most rudimental elements\nof knowledge; we cannot gather that he had any teachers, who assisted him\nwith more than hints, or the loan of a grammar, a lexicon, or some volume\nhe desired to read; there are no indications of any particular kindness,\nno friendly hands, no wicket, or gate of school, or college opened to\nhim.  And as with the commencement of his career, so with its course; his\nintercourse was, probably, mostly with men, and minds inferior to his\nown; books, we have seen, he had few, although he read, with avidity,\nwherever he could borrow; and as with his mental training, so with his\nspiritual experience,\u2014it appears all to have gone on within himself, very\nmuch unrelieved, and unaided; he had to fight his own doubts, and to\ngather strength in the wrestling, and the conflict.  And as he thus\nformed himself, without assistance, so, apparently without any human\nassistance, he continued to labour on, amidst the popular acclamations of\nfame.  The absence of all, and every exhibition of gratitude, is\npeculiarly affecting.  Altogether, this strikes us as a grand,\nself-sustained, and much-enduring life, always hard, and necessitous; but\nits lines are very indelible, written as with a pen of iron, and as with\nthe point of a diamond.  It is natural that, in his old age, he should\nspeak thus to a young man of the\u2014\n\n\n\n\"VALUE OF INDUSTRY.\n\n\n    \"I am an old man, my dear boy, and you are just entering the\n    ministry.  Let me now, and here tell you one thing, and I commend it\n    to your attention, and memory.  All the ministers that I have ever\n    known, who have fallen into disgrace, or into uselessness, _have been\n    idle men_.  I never am much afraid of a young minister, when I\n    ascertain that he can, and does, _fairly sit down to his book_.\n    There is Mr. \u2014, of whom we were talking just now, a man of such\n    unhappy temper, and who has loved, for many years, to meddle in all\n    sorts of religious disputes and divisions.  He would have, long ago,\n    been utterly wrecked, had not his habits of industry saved him.  He\n    has stuck to his book, and that has kept him from many dishonours,\n    which, had he been an idle man, must have, by this time, overwhelmed\n    him.  An idle man is in the way of every temptation; temptation has\n    no need to seek him; _he is at the corner of the street_, _ready_,\n    _and waiting for it_.  In the case of a minister of the Gospel, this\n    peril is multiplied by his position, his neglected duties, the\n    temptations peculiar to his condition, and his own superior\n    susceptibility.  _Remember this\u2014stick to your book_.\"\n\nThe foundations of the good man's character were laid in honest\nsimplicity, real, and perfect sincerity; he was innocent, and\nunsuspecting as a child, and here, no doubt, lay the cause of many of his\ntrials; his frank, and confiding disposition became the means by which\nhis own peace was poisoned, when jealous men, malicious men,\u2014and these\nsometimes Christian men,\u2014took advantage of his simplicity.  He once\nemployed a person to sell a horse for him at a fair; after some time,\nEvans being there, he went out to see if the man was likely to succeed.\nHe found that a bargain was going on for the horse, and nearly completed.\n\n\"Is this your horse, Mr. Evans?\" said the purchaser.\n\n\"Certainly it is,\" he replied.\n\n\"What is his age, sir?\"\n\n\"Twenty-three years.\"\n\n\"But this man tells me he is only fifteen.\"\n\n\"He is certainly twenty-three, for he has been with me these twenty\nyears, and he was three years old when I bought him.\"\n\n\"Is he safe-footed?\"\n\n\"Well, he is very far from that, and, indeed, that is the reason why I\nwant to part with him; and he has never been put into harness since I\nbought him either.\"\n\n\"Please go into the house, Mr. Evans, and stop there,\" said the man whom\nhe had employed to make the sale: \"I never shall dispose of the horse\nwhile you are present.\"\n\nBut the dealer was, in this instance, mistaken, for the frank manner in\nwhich Mr. Evans had answered the questions, and told the truth, induced\nthe buyer to make the purchase, even at a very handsome price.  But the\nanecdote got abroad, and it added to Mr. Evans's reputation, and good\nname; and even the mention of the story in these pages, after these long\nyears have passed away, is more to his memory than the gold would have\nbeen to his pocket.\n\nLike all such natures, however, he was not wanting in shrewdness, and we\nhave seen that, when irritated, he could express himself in sharp\nsarcasm.  He had this power, but, upon principle, he kept it under\ncontrol.  It was a saying of his, \"It is better to keep sarcasms\npocketed, if we cannot use them without wounding friends.\"  Once, two\nministers of different sects were disputing upon some altogether\ntrifling, and most immaterial point of ecclesiastical discipline.  One of\nthem said, \"What is your opinion, Mr. Evans?\" and he said, \"To-day I saw\ntwo boys quarrelling over two snails: one of them insisted that his snail\nwas the best, because it had horns; while the other as strenuously\ninsisted that his was the best, because it had none.  The boys were very\nangry, and vociferous, but the two snails were very good friends.\"\n\nHe comes before us with all that strength of character which he\nunquestionably possessed, as a spirit most affectionate, and especially\nforgiving.  An anecdote goes about of a controversy he had with a\nminister of another sect, who so far forgot himself as to indulge in\nlanguage utterly inconsistent with all Christian courtesy.  But a short\ntime elapsed, when the minister was charged with a crime: had he been\nconvicted, degradation from the ministry must have been the smallest part\nof his punishment, but his innocence was made manifest, and perfectly\nclear.  Mr. Evans always believed the charge to be false, and the attempt\nto prosecute to be unjust, and merely malicious.  On the day when the\ntrial came on he went, as was his wont, in all matters where he was\ndeeply interested, into his own room, and fervently prayed that his old\nfoe might be sustained, and cleared.  He was in company with several\nfriends and brother ministers, when a minister entered the room, and\nsaid, \"Mr. B\u2014 is fully acquitted.\"  Evans instantly fell on his knees,\nand with tears exclaimed, \"Thanks be unto Thee, O Lord Jesus, for\ndelivering one of Thy servants from the mouth of the lions.\"  And he very\nsoon joined his hearty congratulations with those of the other friends of\nthe persecuted man.\n\nIt is certain the story of the Church recites very few instances of such\nan active life, so eminently devotional, and prayerful: we have seen this\nalready illustrated in those remarkable covenants we have quoted.  He had\nan old-fashioned faith in prayer.  He was very likely never troubled much\nabout the philosophy of it: his life passed in the practice of it.  No\nCatholic monk or nun kept more regularly the hours, the matins, or the\nvigils than he.  It appears, that for many years he was accustomed to\nretire for a short season, for prayer, three times during the day, and to\nrise at midnight, regularly, for the same purpose.  He suffered much\nfrequently from slander; he had disorders, and troubles in his churches;\nhe had many afflictions, as we have seen, in life, and the frequent sense\nof poverty; but these all appeared to drive this great, good man to\nprayer, and his friends knew it, and felt it, and felt the serenity, and\nelevation of his character when in the social circle, even when it was\nalso known that heavy trials were upon him.  And one who appears to have\nknown him applies to him, in such moments, the language of the Psalmist,\n\"All thy garments smell of myrrh, aloes, and cassia, out of the ivory\npalaces.\"\n\nAnd, perhaps, in this connection, we may say, without being\nmisunderstood, that the especial necessities of his life gave to it\nsomething of a cloistered, and monastic character.  He was not immured in\nthe cell, or the monastery, but how little can we realize the profound\nsolitude of those long journeys, so constantly renewed, through the\nsilence of the lonely hills, across the desolation of the uninhabited\nmoor!  An intensely nervous, and meditative nature, no possibility of the\nbook then, no retreat, we can believe no desire to retreat from the\ninfinite stretched above him, and even the infinite seeming to spread all\naround him.  In so devout a nature, how calculated all this to foster\ndevotion, until it became at once the support, as well as the passion, of\nthe soul!\n\nAnd these perpetual wanderings among the mountains must have been a fine\nspiritual education, an education deepening emotion in the soul, and at\nthe same time kindling the mind in thoughtful imagery.  He reminds us of\nDean Milman's hero, also a pilgrim through Wales:\u2014\n\n    \"His path is 'mid the Cambrian mountains wild;\n    The many fountains that well wandering down\n    Plinlimmon's huge round side their murmurs smooth\n    Float round him; Idris, that like warrior old\n    His batter'd and fantastic helmet rears,\n    Scattering the elements' wrath, frowns o'er his way,\n    A broad irregular duskiness.  Aloof\n    Snowdon, the triple-headed giant, soars,\n    Clouds rolling half-way down his rugged sides.\n    Slow as he trod amid their dizzy heights,\n    Their silences and dimly mingling sounds,\n    Rushing of torrents, war of prison'd winds;\n    O'er all his wounded soul flow'd strength, and pride,\n    And hardihood; again his front soar'd up\n    To commerce with the skies, and frank and bold,\n    His majesty of step his rugged path\n    Imprinted . . .\n    . . .  Whence, ye mountains, whence\n    The spirit that within your secret caves\n    Holds kindred with man's soul?\"\n\nHenry Vaughan delighted to call himself the Silurist, always proud of the\ncountry from whence he came: his was a different region of Wales from\nthat which produced Christmas Evans.  Henry Vaughan was the swan of the\nUsk; but the sermons of Evans, like the sacred poems of Vaughan, were a\nkind of _Silex Scintillans_, or sparks from the flint, sparks shot forth\nfrom the great mountains, and the overhanging stars, with both of which\nhe held long communion: he had no opportunity for any other often in the\ncourse of his travel; they were as the streets of God, lighted with suns\nstretching across his way, in the green amphitheatre of day, and the blue\namphitheatre of night.\n\nAnd this was, no doubt, very greatly the secret of his preaching.  It is\nnot too strong a term to use, to say that, with all its brilliancy, its\nbardic, and poetic splendours, it was massive preaching.  He usually laid\nthe foundations of the edifice of a sermon, strong and secure in reason,\nand in Scripture, securing the understanding, and the convictions of his\nhearers, before he sketched those splendid allegories, or gave those\ndescriptive touches; before even he appealed to those feelings, when he\nled the whole congregation captive by the chains of his eloquence.\n\nWe have said before, that like most of the preachers of his country, he\ndelighted also in the use of sharp, rememberable sayings.  That is a\nstriking expression when he says, speaking of death, to the believer in\nChrist, \"The crocodile of death shall be harnessed to the chariot of the\ndaughter of Zion, to bring her home to her father's house.\"  Again, \"Our\nimmortal souls, although in perishable bodies, are evidently originally\nbirds of Paradise, and our faculties are the beautiful wings by which we\nunderstand, remember, fear, believe, love, hope, and delight in immortal,\nand eternal things.\"  That is very pretty when he says, \"Faith is the\nwedding-ring by which the poor daughter of the old Ammonite is married to\nthe Prince of Peace: she is raised from poverty to opulence, from\ndegradation to honour, not because of the intrinsic value of the ring,\nthough it is a golden one, but on account of the union which it\nsignifies, between her, and the beloved Prince.\"  Again, \"A cradle, a\ncross, and a grave, all of His Father's appointing, must Jesus have, in\norder to open a fountain of living water to the world.\"  Such sentences\nas these the reader will find strewn along all his sermons, and many such\nin those which we have quoted more at length.\n\nBut it must always be remembered that Christmas Evans was, in a\npre-eminent degree, the orator.  He had a presence; he was nearly six\nfeet high, and finely-proportioned; his whole bearing was dignified, and\nmajestic; he had but one eye, it is true, but we can believe the\ntestimony which describes it as singularly penetrating, and even burning\nwith a wonderful effect, when the strong inspiration of his eloquence was\nupon him.  Then his voice was one of marvellous compass, and melody; like\nhis sermons themselves, which were able to touch the hearts of mighty\nmultitudes, so his voice was able to reach their ears.\n\nWhen he heard Robert Hall, the marvellous enchantment of that still,\nsmall voice, a kind of soprano in its sweet, and cleaving clearness, so\noverwhelmed him, that he longed to preach in that tone, and key; but the\nvoices of the men were fitted to their words,\u2014Hall's to his own\nexquisitely-finished culture, and to the sustained, and elevated culture\neither of spirit, or intelligence of those whom he addressed; Evans's\nwords we suppose rolled like the thunder of a mighty sea, with all its\namplitude of many-voiced waves.  Singers differ, and, no doubt, while we\nare able to admire the evangelical force, and fervour, and even the fine\npictorial imagery of the sermons of Christmas Evans, it is something like\nlooking at the painting on the glass, which may be very pretty, and\nexquisite, but in order really to see it, it should be in the camera,\nwith the magnifying lens, and the burning lamp behind it.  Alas! it is so\nwith all reported and written eloquence: the figures, and the words are\nalmost as cold as the paper upon which they are printed, as they pass\nbefore the eye; they need the inspiration of the burning genius, and that\ninspired by a Divine affection, or afflatus, in their utterance, to give\nthem a real effect.\n\nAnd in the case of Christmas Evans's sermons, this is not all: to us they\nare only translations,\u2014translations from the difficult Welsh\nlanguage,\u2014translations without the wonderful atmospheric accent of the\nWelsh vowel; so that the very best translation of one of Christmas\nEvans's performances can only be the skeleton of a sermon.  We may admire\nthe structure, the architecture of the edifice, but we can form little\nidea of the words which were said to have set Wales on fire.\n\nWe recur to the expression we used a few sentences since.  We are able to\nappreciate the massive character of these sermons: it is very true they\nare cyclopean,\u2014they have about them a prim\u00e6val rudeness; but then the\ncyclopean architecture, although primitive, is massive.  Here are huge\nthoughts, hewn out of the prim\u00e6val, but ever-abiding instincts of our\nnature, or, which is much the same thing, from the ancient, and granite\nflooring of the Divine Word.  We must make this allowance for our\npreacher: he took up his testimony from the grand initial letters of\nFaith; he knew something of the other side of thought; the belief of his\ncountry, in his time, in the earlier days of his ministry, had been very\nmuch vexed by Sabellianism.\n\nThe age of systematic, and scientific doubt had not set in on the\nPrincipality; but he met the conscience of man as a conscience, as that\nwhich was a trouble, and a sorrow to the thoughtful mind, and where it\nwas still untroubled, he sought to alarm it, and awaken it to terror, and\nto fear; and he preached the life, and work of Christ as a legitimate\nsatisfaction, and rest to the troubled conscience.  This was, no doubt,\nthe great burden of his ministry; these are the subjects of all his\nsermons.  He used the old words, the old nomenclature.\n\nSince the day of Christmas Evans, theological language is so altered,\nthat the theological lexicon of the eighteenth century would seem very\npoorly to represent theological ideas in this close of the nineteenth.\nBut we have often thought, that, perhaps, could the men of that time be\nbrought face to face with the men of this, it might be found that terms\nhad rather enlarged their signification, than essentially altered their\nmeaning,\u2014this in many instances, of course, not in all.  But it would\noften happen, could we but patiently analyze the meaning of theological\nterms, we should often find a brother where we had suspected an alien,\nand a friend where we had imagined a foe.\n\nThus Christmas Evans dealt with great truths.  He was a wise\nmaster-builder, and all the several parts of his sermons were related\ntogether in mutual dependence.  The reader will notice that there was\nalways symmetry in their construction: he obeys an order of thought; we\nfeel that he speaks of that which, to the measure of the revelation\ngiven, and his entrance into the mind of the Spirit, he distinctly\nunderstands.  A mind, which itself lives in the light, will, by its own\nsincerity, make the subject which it attempts to expound clear; and he\nhad this faculty, eminently, of making abstruse truths shine out with\nluminous, and distinct beauty.  This is always most noble when the mind\nof a preacher rises to the highest truths in the Christian scheme.  A\ngreat deal of our preaching, in the present day, well deserves the name\nof pretty: how many men, whose volumes of sermons are upon our shelves,\nboth in England, and America, seem as if their preachers had been\nstudents in the natural history of religion, gathering shells, pretty\nrose-tinted shells, or leaves, and insects for a theological museum!  And\na very pretty occupation, too, to call attention to the lily-work of the\ntemple.  But there are others, whose aim has been\u2014\n\n       \"Rather to see great truths\n    Than touch and handle little ones.\"\n\nAnd, certainly, Christmas Evans was of that order who occupied the mind,\nand single eye, rather on the pathway of the planet beyond him, than in\nthe study of the most exquisite shell on the sea-shore.  Among religious\nstudents, and even among eminent preachers, there are some, who may be\nspoken of as Divine, and spiritual astronomers,\u2014they study the laws of\nthe celestial lights; and there are others, who may be called religious\nentomologists,\u2014they find themselves at home amidst insectile\nprettinesses.  Some minds are equal to the infinitely large, and the\ninfinitely small, the remote not more than the near; but such instances\nare very rare.\n\nThe power of great truths overwhelms the man who feels them; this gives\nrise to that impassioned earnestness which enables a great speaker to\nstorm, and take possession of the hearts of his hearers: the man, it has\nbeen truly said, was lost in his theme, and art, was swallowed up in\nexcited feeling, like a whirlpool, bearing along the speaker, and his\nhearers with him, on the current of the strong discourse.  The histories\nof the greatest orators,\u2014for instance, Massillon, Bossuet, and Robert\nHall,\u2014show how frequently it was the case, that the excited feelings of\nan audience manifested themselves by the audience starting from their\nseats, and, sometimes, by loud expressions of acclamation, or\napprobation.  Some such scenes appear to have manifested themselves, even\nbeneath Christmas Evans's ministry.  Some such scenes as these led to the\nreport of those excitements in Wales, which many of our readers have\nheard of as \"Welsh jumping.\"  Evans appears to have been disposed to\nvindicate from absurdity this phenomenon,\u2014the term used to describe it\nwas, no doubt, employed as a term of contempt.  He says,\u2014\n\n    \"Common preaching will not do to arouse sluggish districts from the\n    heavy slumbers into which they have sunk; indeed, formal prayers, and\n    lifeless sermons are like bulwarks raised against these things: five,\n    or six stanzas will be sung as dry as Gilboa, instead of one, or two\n    verses, like a new song full of God, of Christ, and the Spirit of\n    grace, until the heart is attuned for worship.  The burying grounds\n    are kept in fine order in Glamorganshire, and green shrubs, and herbs\n    grow on the graves; but all this is of little value, for the\n    inhabitants of them are all dead.  So, in every form of godliness,\n    where its power is not felt, order without life is exceedingly\n    worthless: you exhibit all the character of human nature, leaving\n    every bud of the flower to open in the beams of the sun, except in\n    Divine worship.  On other occasions, you English appear to have as\n    much fire in your affections as the Welsh have, if you are noticed.\n    In a court of law, the most efficient advocate, such as Erskine, will\n    give to you the greatest satisfaction; but you are contented with a\n    preacher speaking so lifelessly, and so low, that you can hardly\n    understand a third part of what he says, and you will call this\n    decency in the sanctuary.  To-morrow I shall see you answering fully\n    to the human character in your own actions.  When the speakers on the\n    platform will be urging the claims of missions, you will then beat\n    the boards, and manifest so much life, and cheerfulness that not one\n    of you will be seen to take up a note-book, nor any other book, while\n    the speaker shall be addressing you.  A Welshman might suppose, by\n    hearing your noise, that he had been silently conveyed to one of the\n    meetings of the Welsh jumpers, with this difference, that you would\n    perceive many more tears shed, and hear many more 'calves of the\n    lips' offered up, in the rejoicing meetings of Wales; but you use\n    your heels well on such occasions, and a little of your tongues; but\n    if even in Wales, in certain places,\u2014that is, places where the\n    fervent gales are not enjoyed which fill persons with fear, and\n    terror, and joy, in approaching the altar of God,\u2014you may see, while\n    hearing a sermon, one looking into his hymn-book, another into his\n    note-book, and a third turning over the leaves of his Bible, as if he\n    were going to study a sermon in the sanctuary, instead of attending\n    to what is spoken by the preacher as the mouth of God.\"\n\nHe proceeds, at considerable length, in this strain, in a tone of apology\nwhich, while it is frank, and ingenuous, certainly seems to divest the\nexcitement of the Welsh services of those objectionable features which,\nthrough a haze of ignorant prejudice, had very much misrepresented the\ncharacter of such gatherings in England.  It was, as Mr. Evans shows, the\nstir, and excitement, the more stereotyped acclamation, of an English\nmeeting manifesting itself in the devotional services of these wild\nmountain solitudes.  He continues,\u2014\n\n    \"It is an exceedingly easy matter for a minister to manage a\n    congregation while Christian enjoyment keeps them near to God; they\n    are diligent, and zealous, and ready for every good work; but it is\n    very easy to offend this joyous spirit\u2014or give it what name you\n    please, enthusiasm, religious madness, or Welsh jumping,\u2014its English\n    name,\u2014and make it hide itself; a quarrel, and disagreement in the\n    Church, will occasion it to withdraw immediately; indulging in sin,\n    in word or deed, will soon put it to flight: it is like unto the\n    angel formerly, who could not behold the sin of Israel without hiding\n    himself,\u2014so is the angel of the religious life of Wales, which proves\n    him to be a holy angel, though he has the name of a Welsh jumper.  My\n    prayer is, that this angel be a guard upon every congregation, and\n    that none should do anything to offend him.  It is an exceedingly\n    powerful assistant to accompany us through the wilderness, but the\n    individual that has not felt its happy influences has nothing to\n    lose; hence he does not dread a dry meeting, and a hard prayer, for\n    they are all the same to him; but the people of this enjoyment pray\n    before prayer, and before hearing, that they may meet with God in\n    them.\n\n    \"The seasons when these blessings are vouchsafed to the churches of\n    Wales are to be noticed: it is generally at a time when the cause of\n    religion is at a low ebb, all gone to slumber; this happy spirit of\n    enjoyment in religion, like the angel of the pillar of fire, appears\n    when there is distress, and everything at the worst; its approach to\n    the congregation is like the glory of God returning to the temple of\n    old; it creates a stir among the brethren; they have a new prayer,\n    and a new spirit given them to worship God; this will lay hold of\n    another; some new strength, and light will appear in the pulpit,\n    until it will be imagined that the preacher's voice is altered, and\n    that his spirit has become more evangelical, and that he preaches\n    with a more excellent savour than usual; tenderness will descend upon\n    the members, and it will be seen that Mr. Wet-eyes, and Mr. Amen,\n    have taken their place among them; the heavenly gale will reach some\n    of the old backsliders, and they are brought, with weeping, to seek\n    their forfeited privilege; by this time the sound of Almighty God\n    will be heard in the outer court, beginning to move the hearers like\n    a mighty wind shaking the forest; and as the gale blows upon the\n    outer court, upon the hearers, and the young people, and afterwards\n    making its way through the outer court, to rouse the inner court,\n    until a great concern is awakened for the state of the soul.  And,\n    see, how these powerful revivals evince their nature: they are\n    certain, where they are strong, to bend the oaks of Bashan, men of\n    strong, and sturdy minds, and haughty hearts; they bring all the\n    ships of Tarshish, and the merchants of this world, in the harbour\n    hearing; the power of the day of the Lord will raze all the walls of\n    bigotry to the foundation; thoughts of eternal realities, and the\n    spirit of worship, are by these blessings diffused abroad, and family\n    worship is established in scores of families; the door of such a\n    district, opened by the powers of the world to come, creates the\n    channel where the living waters flow, and dead fish are made alive by\n    its virtues.\"\n\nSo Christmas Evans vindicated the excitements of religious services in\nWales from English aspersions.\n\n\n\n\nCHAPTER XII.\n_SUMMARY OF GENERAL CHARACTERISTICS OF CHRISTMAS EVANS AS A PREACHER_.\n\n\nRemarks renewed in Vindication of his Use of Parable in the Pulpit\u2014His\nSermons appear to be born of Solitude\u2014His Imitators\u2014His Probable\nAcquaintance with \"the Sleeping Bard\" of Elis Wyn\u2014A\nDream\u2014Illustrations\u2014The Gospel Mould\u2014Saul of Tarsus and his Seven\nShips\u2014The Misplaced Bone\u2014The Man in the House of Steel\u2014The Parable of the\nChurch as an Ark among the Bulrushes of the Nile\u2014The Handwriting\u2014Death as\nan Inoculator\u2014Time\u2014The Timepiece\u2014Parable of the Birds\u2014Parable of the\nVine-tree, the Thorn, the Bramble, and the Cedar\u2014Illustrations of his\nmore Sustained Style\u2014The Resurrection of Christ\u2014They drank of that Rock\nwhich followed them\u2014The Impossibility of Adequate Translation\u2014Closing\nRemarks on his Place and Claim to Affectionate Regard.\n\nFROM the extracts we have already given, it will be seen that Christmas\nEvans excelled in the use of parable in the pulpit.  Sometimes he wrought\nhis mine like a very Bunyan, and we believe no published accounts of\nthese sermons in Welsh, and certainly none that we have found translated\ninto English, give any idea of his power.  With what amazing effect some\nof his sermons would tell on the vast audiences which in these days\ngather together in London, and in our great towns!  This method of\ninstruction is now usually regarded as in bad taste; it does not seem to\nbe sanctioned by the great rulers, and masters of oratorical art.  If a\nman could create a \"Pilgrim's Progress,\" and recite it, it would be found\nto be a very doubtful article by the rhetorical sanhedrim.  Yet our Lord\nused this very method, and without using some such method\u2014anecdote, or\nillustration\u2014it is doubtful whether any strong hold can be obtained over\nthe lower orders of mind.  Our preacher entered into the spirit of\nScripture parable, and narrative.  One of the most famous of his\ndiscourses is that on the Demoniac of Gadara, which we have already given\nin preceding pages.  Some of our readers will be shocked to know that, in\nthe course of some of his descriptions in it, he convulsed his audience\nwith laughter in the commencement.  Well, he need not be imitated there;\nbut he held it sufficiently subdued before the close, and an alternation\nof tears, and raptures, not only testified to his powers, but to his\nskill in giving an allegorical reading of the narrative.\n\nFor the purpose of producing effect,\u2014and we mean, by effect, visible\nresults in crushed, and humbled hearts, and transformed lives,\u2014it would\nbe a curious thing to try, in England, the preaching of some of the great\nWelshman's sermons.  What would be the effect upon any audience of that\ngreat picture of the Churchyard World, and the mighty controversy of\nJustice, and Mercy?  Let it be admitted that there are some things in it,\nperhaps many, that it would not demand a severe taste to expel from the\npicture, but take it as the broad, bold painting of a man not highly\neducated,\u2014indeed, highly educated men, as we have said, could not perform\nsuch things: a highly-educated man could never have written the\n\"Pilgrim's Progress\"\u2014let it be remembered that it was delivered to men,\nperhaps, we should say, rather educated than instructed, men illiterate\nin all things _except_ the Bible.  We ourselves have, in some very large\ncongregations, tried the preaching of one of the most famous of Evans's\nsermons, \"The Spirit walking in dry places, seeking rest, and finding\nnone.\"\n\nChristmas Evans's preaching was by no means defective in the bone, and\nmuscle of thought, and pulpit arrangement; but, no doubt, herein lay his\ngreat _forte_, and power,\u2014he could paint soul-subduing pictures.  They\nwere not pieces of mere word-painting, they were bathed in emotion, they\nwere penetrated by deep knowledge of the human heart.  He went into the\npulpit, mighty from lonely wrestlings with God in mountain travellings;\nhe went among his fellow-men, his audiences, strong in his faith in the\nreality of those covenants with God, whose history, and character we have\nalready presented to our readers.\n\nThere was much in his preaching of that order which is so mighty in\nspeech, but which loses so much, or which seems to acquire such\nadditional coarseness, when it is presented to the eye.  Preachers now\nlive too much in the presence of published sermons, to be in the highest\ndegree effective.  He who thinks of the printing-press cannot abandon\nhimself.  He who uses his notes slavishly cannot abandon himself; and,\nwithout abandonment, that is, forgetfulness, what is oratory? what is\naction? what is passion?  If we were asked what are the two greatest\nhuman aids to pulpit power, we should say, Self-possession and\nSelf-abandonment; the two are perfectly compatible, and in the pulpit the\none is never powerful without the other.  Knowledge, Belief, Preparation,\nthese give self-possession; and Earnestness, and Unconsciousness, these\ngive self-abandonment.  The first, without the last, may make a preacher\nlike a stony pillar, covered with runes and hieroglyphics; and the last,\nwithout the first, may make a mere fanatic, with a torrent of speech,\nplunging lawlessly, and disgracefully abroad.  The two, in combination in\na noble man, and teacher, become sublime.  Perhaps they reached their\nhighest realization, among us, in Robertson of Brighton.  In another, and\nin a different department, and scarcely inferior order of mind, they were\nnobly realized in Christmas Evans.\n\nPerhaps there never was a time when ministers were more afraid of their\naudiences than in this day; afraid of the big man, with his wealth,\nafraid of the highly-cultured young man with the speculative eyeglasses,\nwho has finished his education in Germany; afraid lest there should be\nthe slightest departure from the most perfect, and elegant taste.  The\nfear of man has brought a snare into the pulpit, and it has paralysed the\npreacher.  And in this highly-furnished, and cultivated time we have few\ninstances of preachers who, in the pulpit, can either possess their\nsouls, or abandon them to the truth, in the text they have to announce.\n\nIt must have been, one thinks, a grand thing to have heard Christmas\nEvans; the extracts from his journal, the story they tell of his devout,\nand rapt communions of soul with God, among the mountains, the bare, and\nsolitary hills, reveal sufficiently how, in himself, the preacher was\nmade.  When he came into the pulpit, his soul was kindled, and inflamed\nby the live coals from the altar.  Some men of his own country imitated\nhim, of course.  Imitations are always ludicrous,\u2014some of these were\nespecially so.  There was, says one of his biographers, the shrug, the\nshake of the head, the hurried, undertoned exclamation, \"Bendigedig,\"\netc., etc., always reminding us, by verifying it, of Dr. Parr's\ndescription of the imitators of Johnson: \"They had the nodosities of the\noak without its vigour, and the contortions of the sibyl without her\ninspiration.\"\n\nIt was not so with him: he had rare, highly spiritual, and gifted\nsympathies; but even in his very colloquies in the pulpit, there was a\nwing, and sweep of majesty.  He preached often amidst scenes of wildness,\nand beauty, in romantic dells, or on mountain sides, and <DW72>s, amidst\nthe summer hush of crags, and brooks, all ministering, it may be thought,\nto the impression of the whole scene; or it was in rude, and unadorned\nmountain chapels, altogether alien from the \u00e6sthetics so charming to\nmodern religious sensibilities; but he never lowered his tone, his\nlanguage was always intelligible; but both it, and the imagery he\nemployed, even when some circumstances gave to it a homely light, and\nplay, always ascended; he knew the workings of the heart, and knew how to\nlay his finger impressively upon all its movements, and every kind of\nsympathy attested his power.\n\nIt is a great thing to bear men's spirits along through the sublime\nreaches, and avenues of thought, and emotion; and majesty, and sublimity\nseem to have been the common moods of his mind; never was his speech, or\nhis pulpit, like a Gilboa, on which there was no dew.  He gave it as his\nadvice to a young preacher, \"Never raise the voice while the heart is\ndry; let the heart, and affections shout first,\u2014let it commence within.\"\nA man who could say, \"Hundreds of prayers bubble from the fountain of my\nmind,\"\u2014what sort of preacher was he likely to make?  He \"mused, and the\nfire burned;\" like the smith who blows upon the furnace, until the iron\nis red hot, and then strikes on the anvil till the sparks fly all round\nhim, so he preached.  His words, and thoughts became radiant with fire,\nand metaphor; they flew forth rich, bright, glowing, like some rich metal\nin ethereal flame.  As we have said, it was the nature, and the habit of\nhis mind, to embody, and impersonate; attributes, and qualities took the\nshape, and form of persons; he seemed to enter mystic abodes, and not to\ntalk of things as a metaphysician, or a theologian, but as a spectator,\nor actor.  The magnificences of nature crowded round him, bowing in\nhomage, as he selected from them to adorn, or illustrate his theme; all\nthings beautiful, and splendid, all things fresh, and young, all things\nold, and venerable.  Reading his discourses, for instance, the _Hind of\nthe Morning_, we are astonished at the prodigality, and the unity of the\nimagination, the coherency with which the fancies range themselves, as\ngems, round some central truth, drinking, and reflecting its\ncorruscations.\n\nAstounded were the people who heard; it was minstrelsy even more than\noratory; the truths were old and common, there was no fine\ndiscrimination, and subtle touch of expression, as in Williams, and there\nwas no personal majesty, and dignity of sonorous swell of the pomp of\nwords, as in John Elias; but it was more,\u2014it was the wing of prophecy,\nand poetry, it was the rapture of the seer, or the bard; he called up\nimage after image, grouped them, made them speak, and testify; laden by\ngrand, and overwhelming feelings, he bore the people with him, through\nthe valley of the shadow of death, or across the Delectable Mountains.\nThere is a spell in thought, there is a spell in felicitous language; but\nwhen to these are added the vision which calls up sleeping terror, the\nimagination which makes living nature yet more alive, and brings the\nsolemn, or the dreadful people of the Book of God to our home, and life\nof to-day, how terribly majestic the preacher becomes!\n\nThe sermons of Christmas Evans can only be known through the medium of\ntranslation.  They, perhaps, do not suffer as most translations suffer;\nbut the rendering, in English, is feeble in comparison with the at once\nnervous, bony, and muscular Welsh language.  The sermons, however,\nclearly reveal the man; they reveal the fulness, and strength of his\nmind; they abound in instructive thoughts; their building, and structure\nis always good; and many of the passages, and even several of the\nsermons, might be taken as models for strong, and effective pulpit\noratory.  Like all the preachers of his day, and order of mind, and\npeculiarity of theological sentiment, and training, his usage of the\nimagery of Scripture was remarkably free; his use also of texts often was\nas significant, and suggestive as it was, certainly, original.\n\nNo doubt, for the appreciation of his purpose, and his power in its\nlarger degree, he needed an audience well acquainted with Scripture, and\nsympathetic, in an eminent manner, with the mind of the preacher.  There\nseem to have been periods, and moments when his mind soared aloft, into\nsome of the highest fields of truth, and emotion.  Yet his wing never\nseemed little, or petty in its flight.  There was the firmness, and\nstrength of the beat of a noble eagle.  Some eloquence sings, some\nsounds; in one we hear the voice of a bird hovering in the air, in the\nother we listen to the thunder of the plume: the eloquence of Christmas\nEvans was of the latter order.\n\nWe have remarked it before,\u2014there is a singular parable-loving instinct\nin Wales.  Its most popular traditional, and prose literature, is imbued\nwith it; the \"Mabinogion,\" the juvenile treasures of Welsh legend,\ncorresponding to the Grimm of Germany, and the other great Teutonic and\nNorse legends, but wholly unlike them, prove this.  But we are told that\nthe most grand prose work in Wales, of modern date, and, at the same\ntime, the most pre-eminently popular, is the \"Sleeping Bard,\" by Elis\nWyn.  He was a High Church clergyman, and wrote this extraordinary\nallegory at the commencement of the last century.  Christmas Evans must\nhave known it, have known it well.  It portrays a series of visions, and\nif Mr. Borrow's testimony may be relied upon, they are thoroughly\nDantesque.  He says, \"It is a singular mixture of the sublime, and the\ncoarse, the terrible, and the ludicrous, of religion, and levity, and\ncombines Milton, Bunyan, and Quevedo.\"\n\nThis is immense praise.  The Vision of the World, the first portion,\nleads the traveller down the streets of Pride, Pleasure, and Lucre; but\nin the distance is a cross street, little and mean, in comparison with\nthe others, but clean, and neat, and on a higher foundation than the\nother streets; it runs upwards, towards the east; they sink downwards,\ntowards the north\u2014this is the street True Religion.  This is very much in\nthe style of Christmas Evans, and so also is the vision of Death, the\nvision of Perdition, and the vision of Hell.  This singular poem appears\nto have been exceedingly popular in Wales when Christmas Evans was young.\n\nBut our preacher has often been called the Bunyan of Wales\u2014the Bunyan of\nthe pulpit.  In some measure, the epithet does designate him; he was a\ngreat master of parabolic similitude, and comparison.  This is a kind of\npreaching ever eminently popular with the multitude; it requires rather a\nredundancy of fancy, than imagination\u2014perhaps a mind considerably\ndisciplined, and educated would be unable to indulge in such exercises\u2014a\nself-possession, balanced by ignorance of many of the canons of taste, or\nutterly oblivious, and careless of them; for this is a kind of teaching\nof which we hear very little.  Now we have not one preacher in England\nwho would, perhaps, dare to use, or who could use well, the parabolic\nstyle.  This was the especial power of Christmas Evans.  He excelled in\npersonification; he would seem frequently to have been mastered by this\nfaculty.  The abstraction of thought, the disembodied phantoms of another\nworld, came clothed in form, and feature, and colour; at his bidding they\ncame\u2014\n\n             \"Ghostly shapes\n    Met him at noontide; Fear, and trembling Hope,\n    Silence, and Foresight; Death, the skeleton,\n    And Time, the shadow.\"\n\nThus, he frequently astounded his congregations, not merely by pouring\nround his subject the varied hues of light, or space, but by giving to\nthe eye defined shapes, and realizations.  We do not wonder to hear him\nsay, \"If I only entered the pulpit, I felt raised, as it were, to\nParadise, above my afflictions, until I forgot my adversity; yea, I felt\nmy mountain strong.  I said to a brother once, 'Brother, the doctrine,\nthe confidence, and strength I feel, will make persons dance with joy in\nsome parts of Wales.'  'Yea, brother,' said he, with tears flowing from\nhis eyes.\"  He was visited by remarkable dreams.  Once, previous to a\ntime of great refreshing, he dreamt:\u2014\n\n\"He thought he was in the church at Caerphilly, and found many harps\nhanging round the pulpit, wrapped in coverings of green.  'Then,' said\nhe, 'I will take down the harps of heaven in this place.'  In removing\nthe covering, he found the ark of the covenant, inscribed with the name\nof Jehovah.  Then he cried, 'Brethren, the Lord has come to us, according\nto His promise, and in answer to our prayers.'\"  In that very place, he\nshortly afterwards had the satisfaction of receiving one hundred and\nforty converts into the Church, as the fruit of his ministry.\n\nAs we have said, nothing can well illustrate, on paper, the power of the\norator's speech, but the following may serve, as, in some measure,\nillustrating his method:\u2014\n\n\n\n\"THE GOSPEL MOULD.\n\n\n    \"I compare such preachers to a miner, who should go to the quarry\n    where he raised the ore, and, taking his sledge in his hand, should\n    endeavour to form bars of iron of the ore in its rough state, without\n    a furnace to melt it, or a rolling mill to roll it out, or moulds to\n    cast the metal, and conform the casts to their patterns.  The Gospel\n    is like a form, or mould, and sinners are to be melted, as it were,\n    and cast into it.  'But ye have obeyed from the heart that form of\n    doctrine which was delivered you,' or into which you were delivered,\n    as is the marginal reading, so that your hearts ran into the mould.\n    Evangelical preachers have, in the name of Christ, a mould, or form\n    to cast the minds of men into; as Solomon the vessels of the temple.\n    The Sadducees and Pharisees had their forms, and legal preachers have\n    their forms; but evangelical preachers should bring with them the\n    'form of sound words,' so that, if the hearers believe, or are melted\n    into it, Christ may be formed in their hearts,\u2014then they will be as\n    born of the truth, and the image of the truth will appear in their\n    sentiments, and experience, and in their conduct in the Church, in\n    the family, and in the neighbourhood.  Preachers without the mould\n    are all those who do not preach all the points of the Gospel of the\n    Grace of God.\"\n\nWe will now present several extracts, derived from a variety of sources,\nhappily illustrating the general character of his sermons.\n\n\n\n\"SAUL OF TARSUS AND HIS SEVEN SHIPS.\n\n\n    \"Saul of Tarsus was once a thriving merchant and an extensive\n    ship-owner; he had seven vessels of his own, the names of which\n    were\u20141.  Circumcised the Eighth Day; 2.  Of the Stock of Israel; 3.\n    Of the Tribe of Benjamin; 4.  A Hebrew of the Hebrews; 5.  As\n    touching the Law, a Pharisee; 6.  Concerning Zeal, persecuting the\n    Church.  The seventh was a man-of-war, with which he one day set out\n    from the port of Jerusalem, well supplied with ammunition from the\n    arsenal of the Chief Priest, with a view to destroy a small port at\n    Damascus.  He was wonderfully confident, and breathed out\n    threatenings and slaughter.  But he had not got far from port before\n    the Gospel Ship, with Jesus Christ Himself as Commander on board,\n    hove in sight, and threw such a shell among the merchant's fleet that\n    all his ships were instantly on fire.  The commotion was tremendous,\n    and there was such a volume of smoke that Paul could not see the sun\n    at noon.  While the ships were fast sinking, the Gospel Commander\n    mercifully gave orders that the perishing merchant should be taken on\n    board.  'Saul, Saul, what has become of all thy ships?'  'They are\n    all on fire.'  'What wilt thou do now?'  'Oh that I may be found in\n    Him, not having my own righteousness, which is of the law, but that\n    which is through the faith of Christ, the righteousness which is of\n    God. by faith.'\"\n\n\n\n\"THE MISPLACED BONE.\n\n\n    \"Let every one keep his own place, that there be no schism in the\n    body.  There arose a fierce contention in the human body; every\n    member sought another place than the one it found itself in, and was\n    fitted for.  After much controversy, it was agreed to refer the whole\n    matter to one whose name was Solomon Wise-in-his-own-conceit.  He was\n    to arrange, and adjust the whole business, and to place every bone in\n    its proper position.  He received the appointment gladly, and was\n    filled with joy, and confidence.  He commenced with finding a place\n    for himself.  His proper post was the heel; but where do you think he\n    found it?  He must needs be the golden bowl in which the brains were\n    deposited.  The natural consequences followed.  The coarse heel bone\n    was not of the right quality, nor of the suitable dimensions to\n    contain the brains, nor could the vessel intended for that purpose\n    form a useful, or comely part of the foot.  Disorder ensued in foot,\n    head, face, legs, and arms.  By the time Solomon\n    Wise-in-his-own-conceit had reconstructed the body, it could neither\n    walk, nor speak, nor smell, nor hear, nor see.  The body was,\n    moreover, filled with intolerable agony, and could find no rest,\n    every bone crying for restoration to its own place, that is to say,\n    every one but the heel-bone; that was mightily pleased to be in the\n    head, and to have the custody of the brains.  Sin has introduced\n    similar disorder amongst men, and even amongst professors of\n    religion, and into congregations.  'Let every one keep his own place,\n    that there be no schism in the body.'  The body can do much, can bear\n    heavy burdens, all its parts being in their own positions.  Even so\n    in the Church; much good can be done by every member keeping and\n    filling his own place without high-mindedness.\"\n\n\n\n\"THE MAN IN THE HOUSE OF STEEL.\n\n\n    \"A man in a trance saw himself locked up in a house of steel, through\n    the walls of which, as through walls of glass, he could see his\n    enemies assailing him with swords, spears, and bayonets; but his life\n    was safe, for his fortress was locked within.  So is the Christian\n    secure amid the assaults of the world.  His 'life is hid with Christ\n    in God.'\n\n    \"The Psalmist prayed, 'When my heart is overwhelmed within me, lead\n    me to the Rock that is higher than I.'  Imagine a man seated on a\n    lofty rock in the midst of the sea, where he has everything necessary\n    for his support, shelter, safety, and comfort.  The billows heave and\n    break beneath him, and the hungry monsters of the deep wait to devour\n    him; but he is on high, above the rage of the former, and the reach\n    of the latter.  Such is the security of faith.\n\n    \"But why need I mention the rock, and the steel house? for the peace\n    that is in Christ is a tower ten thousand times stronger, and a\n    refuge ten thousand times safer.  Behold the disciples of Jesus\n    exposed to famine, nakedness, peril, and sword\u2014incarcerated in\n    dungeons; thrown to wild beasts; consumed in the fire; sawn asunder;\n    cruelly mocked, and scourged; driven from friends, and home, to\n    wander among the mountains, and lodge in dens, and caves of the\n    earth; being destitute, afflicted, tormented; sorrowful, but always\n    rejoicing; cast down, but not destroyed; an ocean of peace within,\n    which swallows up all their sufferings.\n\n    \"'Neither death,' with all its terrors; 'nor life,' with all its\n    allurements; 'nor things present,' with all their pleasure, 'nor\n    things to come,' with all their promise; 'nor height' of prosperity;\n    'nor depth' of adversity; 'nor angels' of evil; 'nor principalities'\n    of darkness; 'shall be able to separate us from the love of God which\n    is in Christ Jesus.'  'God is our refuge, and strength; a very\n    present help in trouble.  Therefore will we not fear, though the\n    earth be removed, and though the mountains be carried into the midst\n    of the sea\u2014though the waters thereof roar and be troubled, though the\n    mountains shake with the swelling thereof.'  This is the language of\n    strong faith in the peace of Christ.  How is it with you amid such\n    turmoil, and commotion?  Is all peaceful within?  Do you feel secure\n    in the name of the Lord, as in a strong fortress, as in a city well\n    supplied, and defended?\n\n    \"'There is a river, the streams whereof shall make glad the city of\n    God, the holy place of the tabernacles of the most high.  God is in\n    the midst of her; she shall not be moved.  God shall help her, and\n    that right early.'  'Unto the upright, there ariseth light in the\n    darkness.'  The bright and morning star, shining upon their pathway,\n    cheers them in their journey home to their Father's house.  And when\n    they come to pass over Jordan, the Sun of Righteousness shall have\n    risen upon them, with healing in His wings.  Already they see the\n    tops of the mountains of immortality, gilded with his beams, beyond\n    the valley of the shadow of death.  Behold, yonder, old Simeon\n    hoisting his sails, and saying, 'Lord, now lettest thou Thy servant\n    depart in peace, according to Thy word; for mine eyes have seen Thy\n    salvation.'  Such is the peace of Jesus, sealed to all them that\n    believe by the blood of His cross.\n\n    \"When we walk through the field of battle, slippery with blood, and\n    strewn with the bodies of the slain\u2014when we hear the shrieks, and the\n    groans of the wounded, and the dying\u2014when we see the country wasted,\n    cities burned, houses pillaged, widows, and orphans wailing in the\n    track of the victorious army, we cannot help exclaiming, 'Oh, what a\n    blessing is peace!'  When we are obliged to witness family turmoils,\n    and strifes\u2014when we see parents, and children, brothers, and sisters,\n    masters, and servants, husbands, and wives, contending with each\n    other like tigers\u2014we retire as from a smoky house, and exclaim as we\n    go, 'Oh, what a blessing is peace!'  When duty calls us into that\n    church, where envy, and malice prevail, and the spirit of harmony is\n    supplanted by discord, and contention\u2014when we see brethren, who ought\n    to be bound together in love, full of pride, hatred, confusion, and\n    every evil work\u2014we quit the unhallowed scene with painful feelings of\n    repulsion, repeating the exclamation, 'Oh, what a blessing is peace!'\n\n    \"But how much more precious in the case of the awakened sinner!  See\n    him standing, terror-stricken, before Sinai.  Thunders roll above\n    him\u2014lightnings flash around him\u2014the earth trembles beneath him, as if\n    ready to open her mouth, and swallow him up.  The sound of the\n    trumpet rings through his soul, 'Guilty! guilty! guilty!'  Pale and\n    trembling, he looks eagerly around him, and sees nothing but\n    revelations of wrath.  Overwhelmed with fear, and dismay, he cries\n    out\u2014'O wretched man that I am! who shall deliver me!  What shall I\n    do?'  A voice reaches his ear, penetrates his heart\u2014'Behold the Lamb\n    of God, that taketh away the sin of the world!'  He turns his eyes to\n    Calvary.  Wondrous vision!  Emmanuel expiring upon the cross! the\n    sinner's Substitute satisfying the demand of the law against the\n    sinner!  Now all his fears are hushed, and rivers of peace flow into\n    his soul.  This is the peace of Christ.\n\n    \"How precious is this peace, amid all the dark vicissitudes of life!\n    How invaluable this jewel, through all the dangers of the wilderness!\n    How cheering to know that Jesus, who hath loved us even unto death,\n    is the pilot of our perilous voyage; that He rules the winds, and the\n    waves, and can hush them to silence at His will, and bring the\n    frailest bark of faith to the desired haven!  Trusting where he\n    cannot trace his Master's footsteps, the disciple is joyful amid the\n    darkest dispensations of Divine Providence; turning all his sorrows\n    into songs, and all his tribulations into triumphs.  'Thou wilt keep\n    him in perfect peace, whose mind is stayed on Thee, because he\n    trusteth in Thee.'\"\n\n\n\n\"THE PARABLE OF THE CHURCH AS AN ARK AMONG THE BULRUSHES OF THE NILE.\n\n\n    \"I see an ark of bulrushes, daubed with slime, and pitch, placed on\n    the banks of the Nile, which swarmed with fierce crocodiles.\n    Pharaoh's daughter espies it, and sends her maidens to find out what\n    there can be in it.  Little Moses was there, with a face of\n    miraculous beauty, to charm the princess of Egypt.  She determined to\n    adopt him as her son.  Behold, a great wonder.  On the brink of the\n    river, where the three great crocodiles\u2014the Devil, Sin, and\n    Death\u2014have devoured their millions, there lay those who it was seen,\n    before the foundation of the world, would be adopted into the court\n    of heaven.  The Gospel comes forth like a royal princess, with pardon\n    in her hand, and mercy in her eye; and hastening with her\n    handmaidens, she glances at the thousands asleep in the perils of\n    sin.  They had favour in her sight, and she sent for her maidens,\n    called Justification, and Sanctification, to train them for the\n    inheritance of the saints.\"\n\n\n\n\"THE HANDWRITING.\n\n\n    \"When Adam sinned, there was issued against him the writ of death,\n    written by the finger of God in the book of the moral law.  Adam had\n    heard it read before his fall, but in seeking to become a god, by\n    eating of the fruit of the tree, had forgotten it.  Now God read it\n    in his conscience, and he was overwhelmed with fear.  But the promise\n    of a Redeemer having been given, Mercy arranged that sacrifices\n    should be offered as a typical payment of the debt.  When God\n    appeared on Sinai, to enter into covenant with His people, He brought\n    this writ in His hand, and the whole camp understood, from the\n    requirements of the law, that they must perish; their lives had been\n    forfeited.  Mercy devised that a bullock's blood should be shed,\n    instead of the blood of man.  The worshippers in the temple were\n    bound to offer living sacrifices to God, that they might die in their\n    stead, and be consumed.  Manoah feared the flames of the sacrifice\n    that was offered upon the rock; but his wife understood that, since\n    the angel had ascended in the flame, in their stead, it was a\n    favourable omen.  Every worshipper, by offering other lives instead\n    of their own on the altars of God, acknowledged that the\n    'handwriting' was in force against them, and their high priest had\n    minutely to confess all their sins 'over' the victim.  Yet, by all\n    the blood that ever crimsoned Levi's robe, and the altars of God, no\n    real atonement was made for sin, nor forgiveness procured for the\n    smallest crime.  All the sacrifices made a remembrance of sin, but\n    were no means of pardon.  More than two thousand years the question\n    had been entertained, how to reconcile man with God.  The\n    'handwriting' was real on Mount Ebal every year; meanwhile the debt\n    was fast accumulating, and new bills were being constantly filed.\n    The books were opened from time to time; but to meet the claims there\n    was nothing brought to the altar but the blood of sacrifices, as a\n    sort of draft in the name of Christ upon the Bank of Gold.  When\n    Heaven, and earth had grown weary of this fictitious or seeming,\n    pardon of sin, I hear a voice exclaim: 'Away with sacrifices, and\n    burnt-offerings: Heaven has no pleasure in them; a body has been\n    prepared for me.  Lo, I come to reconcile man with God by one\n    sacrifice.'  He came, 'leaping upon the mountains, and skipping upon\n    the hills.'  Calling at the office where the 'handwriting' lay, when\n    only eight days old, He signed with His own blood an acknowledgment\n    of the debt, saying: 'This is an earnest, and a pledge that my\n    heart's blood shall be freely given.'  The three-and-thirty years\n    have expired; I see Him in Gethsemane, with the priceless purse of\n    gold which He had borne with Him through the courts of Caiaphas and\n    Pilate; but to them the image, and the superscription on the coin was\n    a mystery.  The Father, however, recognised them in the court of\n    Sinai, where the 'handwriting' was that demanded the life of the\n    whole world.  The day following, 'the Virgin's Son' presented Himself\n    to pay the debt in liquid gold; and the treasure which He bore would\n    have set free a myriad worlds.  He passes along the streets of\n    Jerusalem towards Sinai's office; the mercy-seat is removed to 'the\n    place of skulls;' as He proceeds, He exclaims: 'I am come not to\n    destroy, but to fulfil the law.'  Send in, before the hour of three,\n    each curse, and threat ever pronounced against my people.  Bring in\n    the first old bill against Adam as their head.  I will redeem a\n    countless host of infants to-day; their names shall be taken out of\n    old Eden's accounts.  Bring in the many transgressions which have\n    been filed through the ages, from Adam until now; include Peter's\n    denial of me last night; but as to Judas, he is a son of perdition,\n    he has no part in me, having sold me for thirty pieces of silver.  We\n    have here an exhaustless crimson treasure,\u2014enough to meet the demand;\n    enough to fill every promise, and every prophecy with mercy; enough\n    to make my beloved, and myself happy, and blest for ever!  By three\n    in the afternoon of that day, there was not a bill in all Eden, or\n    Sinai, that had not been brought to the cross.  And when all was\n    settled, Christ bowed down His head, but cried with a loud voice: 'It\n    is finished!'  The gates of death, and hell trembled, and shook.\n    'The posts of the doors moved at the voice.'  The great gulf between\n    God, and His people was closed up.  Sinai appeared with the offering,\n    and grew still; the lightnings no longer flashed, and the thunder\n    ceased to roar.\"\n\n\n\n\"DEATH AS AN INOCULATOR.\n\n\n    \"Death may be conceived of as a gigantic inoculator.  He carries\n    about with him a monstrous box, filled with deadly matter, with which\n    he has infected every child of Adam.  The whole race of man is doomed\n    by this law of death.  But see!  This old inoculator gets paid back\n    in his own coin.  The Son of Man, humbling Himself to death, descends\n    into the tomb, but rises immortal.  He seized death in Joseph's\n    grave.  But, amazing spectacle! with the matter of His own\n    immortality He inoculated mortality with death, whose lifeless corpse\n    will be seen, on the resurrection morning, among the ruins of His\n    people's graves; while they, with one voice, will rend the air as if\n    eternity opened its mouth, exclaiming: 'O death, where is thy sting?\n    O grave, where is thy victory?'\"\n\n\n\n\"TIME.\n\n\n    \"Time, considered as a whole, is the age of the visible creation.  It\n    began with the fiat, 'Let there be light;' and it will end with the\n    words: 'Come, ye blessed of my Father,' and 'Go, ye cursed.'  Each\n    river, and mountain, town, and city, hovel, and palace, every son,\n    and daughter of Adam, must undergo the change, pass away, for\n    whatever is seen is only for a time.  The time of restoration, by the\n    presence of the glory of Christ, will be the morning of judgment, and\n    resurrection.  That morning will be the last of time: then eternity\n    begins.  From that time, each man will dwell in his everlasting home:\n    the ungodly in a lake of fire, that will burn for ever; while the\n    joy, and happiness of the blest will know no end.\n\n    \"Oh the fearfulness of the word _everlasting_, written over the door\n    of the lake of fire!  Oh the happiness it will create when read above\n    the eternal kingdom!\n\n    \"Time is the age of the visible world; but eternity is the age of\n    God.  This limitless circle centres in Him.  The age of the visible\n    world is divided into years, and days, according to the revolutions\n    of the earth, and sun,\u2014into weeks, in memory of the world's creation,\n    and the resurrection of Christ,\u2014into hours, minutes, seconds, and\n    moments.  These last can scarcely be distinguished, yet they are\n    parts of the great body of time; but seven thousand years constitute\n    no part of eternity.  One day, and a thousand years, yea, millions of\n    years, are alike, compared with the age of God, forming no part of\n    the vast changeless circle that knows neither loss, nor gain.  The\n    age of time is winding up by minutes, days, and years: the age of God\n    is one endless to-day; and such will be your age, and mine, when we\n    have once passed the limits of time, beyond which Lazarus is blessed,\n    and the rich man tormented.  My brethren in the ministry, who in\n    years gone by travelled with me from one Association to another, are\n    to-day living in that great endless hour!\n\n    \"Time is an age of changes, revolutions, and reforms; but eternity is\n    calm, stationary, and changeless.  He who enters upon it an enemy to\n    God, faithless, prayerless, unpardoned, and unregenerate, remains so\n    for ever.  Great changes take place in time, for which the new song\n    in eternity will never cease.  Natures have been changed, and enmity\n    has been abolished.  In time, the life covenant was broken, and man\n    formed, and sealed his compact with hell.  One, equal with God, died\n    upon the cross, in the form of a servant, to destroy the works of the\n    devil, and to unite man, and God in the bond of peace through His own\n    blood.  Time, and language would fail to recount what in time has\n    been accomplished, involving changes from life, to death, and from\n    death, to life.  Here the pure have become denied, and the guiltless\n    condemned; and here, also, the sinner has been justified, the\n    polluted cleansed, the poor enriched, the enemy reconciled, and the\n    dead have been made alive, where one paradise has been lost, and a\n    better regained.  The new song from the midst of eternity sounds in\n    our ears.  Hear it!  It has for its subjects one event that took\n    place in eternity, and three that have transpired in time: 'Unto Him\n    that loved us, and washed us from our sins in His own blood, and hath\n    made us kings, and priests unto God, and His Father: to Him be glory,\n    and dominion for ever, and ever.  Amen.'\"\n\n\n\n\"THE TIMEPIECE.\n\n\n    \"You may move the hands on the dial-plate this way, and the other,\n    and finger as you please the machinery within, but if there be no\n    mainspring there your labour will be in vain.  So the 'hands' of\n    men's lives will not move, in holy obedience, at the touch of the\n    law, unless the mainspring be supplied by God through the Gospel;\n    then only will the whole life revolve on the pivot of the love of\n    Christ, as upon an imperishable diamond.  It is not difficult to get\n    the timepiece to act well, if the internal machinery be in proper\n    order; so, with a right spirit within, Lydia attends to the word,\n    Matthew leaves 'the receipt of custom,' Saul of Tarsus prays; and the\n    three thousand repent, believe, and turn unto the Lord.\n\n    \"A gentleman's timepieces were once out of order, and they were\n    examined, when it was found that in one of them the mainspring was\n    injured; the glass which protected the dial-plate of the other was\n    broken; while the machinery of the third had got damp, and rusty,\n    although the parts were all there.  So the lack of holiness, in some\n    cases, arises from the want of heart to love God; another man has not\n    the glass of watchfulness in his conduct; another has got rusty with\n    backsliding from God, and the sense of guilt so clogs the wheels of\n    his machinery, that they must be well brushed with rebuke, and\n    correction, and oiled afresh with the Divine influence, before they\n    will ever go well again.\n\n    \"The whole of a Christian's life is a reaching forward; but he has to\n    begin afresh, like the people of Israel in the wilderness; or, like a\n    clock, he has constantly to recommence at the figure one, and go on\n    to that of twelve, through all the years of his experience on earth.\n    But after the resurrection, he will advance, body, and soul, to the\n    figure of million of millions, never to begin again throughout\n    eternity.  The sun in that world will never rise, nor set; it will\n    have neither east, nor west!  How often has an invisible hand wound\n    up thy religious spirit below, but there the weights will never come\n    down again!\"\n\n\n\n\"PARABLE OF THE BIRDS.\n\n\n    \"A gentleman kept in his palace a dove, a raven, and an eagle.  There\n    was but little congeniality, or friendship amongst them.  The dove\n    ate its own proper food, and lodged in the aviary.  The raven fed on\n    carrion, and sometimes would pick out the eyes of an innocent lamb,\n    and had her nest in the branches of a tree.  The eagle was a royal\n    bird; it flew very high, and was of a savage nature; it would care\n    nothing to eat half-a-dozen doves for its breakfast.  It was\n    considered the chief of all birds, because it could fly higher than\n    all.  All the doves feared its beak, its angry eyes, and sharp\n    talons.  When the gentleman threw corn in the yard for the dove, the\n    raven would be engaged in eating a piece of flesh, a part of a lamb\n    haply; and the eagle in carrying a child from the cradle to its\n    eyrie.  The dove is the evangelical, industrious, godly professor;\n    the raven is the licentious, and unmanageable professor; and the\n    eagle the high-minded, and self-complacent one.  These characters are\n    too often amongst us; there is no denomination in church, or\n    meeting-house, without these three birds, if there be birds there at\n    all.  These birds, so unlike, so opposed, never can live together in\n    peace.  Let us pray, brethren, for union of spirit in the bond of\n    peace.\"\n\n\n\n\"PARABLE OF THE VINE-TREE, THE THORN, THE BRAMBLE, AND THE CEDAR.\n\n\n    \"The trees of Lebanon held a council to elect a king, on the death of\n    their old sovereign, the Yew-tree.  It was agreed to offer the\n    sovereignty to the Cedar; at the same time, in the event of the\n    Cedar's declining it, to the Vine-tree, and then to the Olive-tree.\n    They all refused it.  The Cedar said, 'I am high enough already.'\n    The Vine said, 'I prefer giving forth my rich juice to gladden man's\n    heart.'  In like manner, the Olive was content with giving its fruit,\n    and would receive no other honour.  Recourse was then had to the\n    Thorn.  The Thorn gladly received the office; saying to itself, 'I\n    have nothing to lose but this white dress, and a berry for pigs,\n    while I have prickles enough to annoy the whole wood.'  The Bramble\n    rebelled against the Thorn, and a fire of pride, and envy was\n    kindled, which, at length, wrapped the whole forest in one blaze.\n    Two or three vain, and high-minded men have frequently broken up the\n    peace of congregations; and, by striving for the mastery, have\n    inflicted on the cause of religion incalculable injuries; when they\n    have had no more fitness for rule than the white-thorn, or the\n    prickly bramble.\"\n\nThe following extract is of another order; it is more lengthy, and it is\nupon a theme which always drew forth the preacher's most exulting notes:\u2014\n\n\n\n\"THE RESURRECTION OF OUR LORD.\n\n\n    \"Let us now consider the fact of our Lord's resurrection, and its\n    bearing upon the great truths of our holy religion.\n\n    \"This most transcendent of miracles is sometimes attributed to the\n    agency of the Father; who, as the Lawgiver, had arrested, and\n    imprisoned in the grave the sinner's Surety, manifesting at once His\n    benevolence, and His holiness; but by liberating the prisoner,\n    proclaimed that the debt was cancelled, and the claims of the law\n    satisfied.  It is sometimes attributed to the Son Himself; who had\n    power both to lay down His life, and to take it again; and the merit\n    of whose sacrifice entitled Him to the honour of thus asserting His\n    dominion over death, on behalf of His people.  And sometimes it is\n    attributed to the Holy Spirit, as in the following words of the\n    Apostle:\u2014'He was declared to be the Son of God with power, according\n    to the Spirit of Holiness, by the resurrection from the dead.'\n\n    \"_The resurrection of Christ is a clear and incontestable proof of\n    His Divinity_.\n\n    \"He had declared Himself equal with God the Father, and one with Him\n    in nature, and in glory.  He had told the people that He would prove\n    the truth of this declaration, by rising from the grave three days\n    after His death.  And when the morning of the third day began to dawn\n    upon the sepulchre, lo! there was an earthquake, and the dead body\n    arose, triumphant over the power of corruption.\n\n    \"This was the most stupendous miracle ever exhibited on earth, and\n    its language is:\u2014'Behold, ye persecuting Jews and murdering Romans,\n    the proof of my Godhead!  Behold, Caiaphas, Herod, Pilate, the power,\n    and glory of your Victim!'  'I am He that liveth, and was dead; and\n    lo! I am alive for evermore!'  'I am the root, and the offspring of\n    David, and the Bright, and Morning Star!'  'Look unto Me, and be ye\n    saved, all ye ends of the earth; for I am God, and besides Me there\n    is none else!'\n\n    \"_Our Lord's resurrection affords incontrovertible evidence of the\n    truth of Christianity_.\n\n    \"Pilate wrote the title of Christ in three languages on the cross;\n    and many have written excellent, and unanswerable things, on the\n    truth of the Christian Scriptures, and the reality of the Christian\n    religion; but the best argument that has ever been written on the\n    subject was written by the invisible hand of the Eternal Power, in\n    the rocks of our Saviour's sepulchre.  This confounds the sceptic,\n    settles the controversy, and affords an ample, and sure foundation\n    for all them that believe.\n\n    \"If any one asks whether Christianity is from heaven, or of men, we\n    point him to the 'tomb hewn out of the rock,' and say\u2014'There is your\n    answer!  Jesus was crucified, and laid in that cave; but on the\n    morning of the third day it was found empty; our Master had risen,\n    and gone forth from the grave victorious.'\n\n    \"This is the pillar that supports the whole fabric of our religion;\n    and he who attempts to pull it down, like Samson, pulls ruin upon\n    himself.  'If Christ is not risen, then is our preaching vain, and\n    your faith is also vain, ye are yet in your sins;' but if the fact is\n    clearly proved, then Christianity is unquestionably true, and its\n    disciples are safe.\n\n    \"This is the ground on which the Apostle stood, and asserted the\n    divinity of his faith:\u2014'Moreover, I testify unto you the gospel,\n    which I preached unto you; which also ye have received, and wherein\n    ye stand; by which also ye are saved, if ye keep in memory what I\n    preached unto you, unless ye have believed in vain; for I delivered\n    unto you first of all that which I also received, how that Christ\n    died for our sins according to the Scriptures, and that He was\n    buried, and that He rose again the third day, according to the\n    Scriptures.'\n\n    \"_The resurrection of Jesus is the most stupendous manifestation of\n    the power of God_, _and the pledge of eternal life to His people_.\n\n    \"The apostle calls it 'the exceeding greatness of His power to\n    usward, who believe, according to the working of His mighty power,\n    which He wrought in Christ when He raised Him from the dead.'  This\n    is a river overflowing its banks\u2014an idea too large for language.  Let\n    us look at it a moment.\n\n    \"Where do we find 'the exceeding greatness of His power'?  In the\n    creation of the world? in the seven Stars and Orion? in the strength\n    of Behemoth and Leviathan?  No!  In the Deluge? in the fiery\n    destruction of Sodom? in the overthrow of Pharaoh, and his host? in\n    hurling Nebuchadnezzar, like Lucifer, from the political firmament?\n    No!  It is the power which He wrought in Christ.  When?  When He\n    healed the sick? when He raised the dead? when He cast out devils?\n    when He blasted the fruitless fig-tree? when He walked upon the\n    waters of Galilee?  No!  It was 'when He raised Him from the dead.'\n    Then the Father placed the sceptre in the hands of the Son, 'and set\n    Him above all principality, and power, and might, and dominion, and\n    every name that is named, not only in this world, but also in that\n    which is to come; and put all things under His feet, and gave Him to\n    be Head over all things to the Church.'\n\n    \"This is the source of our spiritual life.  The same power that\n    raised the dead body of our Lord from the grave, quickens the soul of\n    the believer from the death in trespasses, and sins.  His riven tomb\n    is a fountain of living waters; whereof, if a man drink, he shall\n    never die.  His raised, and glorified body is the sun, whence streams\n    eternal light upon our spirits; the light of life, that never can be\n    quenched.\n\n    \"Nor here does the influence of His resurrection end.  'He who raised\n    up Jesus from the dead shall, also, quicken our mortal bodies.'  His\n    resurrection is the pledge, and the pattern of ours.  'Because He\n    lives, we shall live also.'  'He shall change our vile body, that it\n    may be fashioned like unto His glorious body.'  We hear Him speaking\n    in the Prophet:\u2014'Thy dead shall live; together with my dead body\n    shall they arise.  Awake, and sing, ye that dwell in the dust; for\n    thy dew is as the dew of herbs, and the earth shall cast out her\n    dead.'\n\n    \"How divinely does the Apostle speak of the resurrection-body of the\n    saints!  'It is sown in corruption, it is raised in incorruption; it\n    is sown in dishonour, it is raised in glory; it is sown in weakness,\n    it is raised in power; it is sown a natural body, it is raised a\n    spiritual body.  For this corruptible must put on incorruption, and\n    this mortal must put on immortality.  Then shall be brought to pass\n    the saying that is written: 'Death is swallowed up in victory!  O\n    death, where is thy victory?  O grave, where is thy sting?  Thanks be\n    unto God that giveth us the victory, through our Lord Jesus Christ.'\n\n    \"Ever since the fall in Eden, man is born to die.  He lives to die.\n    He eats, and drinks, sleeps, and wakes, to die.  Death, like a dark\n    steel-clad warrior, stands ever before us; and his gigantic shadow\n    comes continually between us, and happiness.  But Christ hath\n    'abolished death, and brought life, and immortality to light through\n    the gospel.'  He was born in Bethlehem, that He might die on Calvary.\n    He was made under the law, that He might bear the direst penalty of\n    the law.  He lived thirty-three years, sinless, among sinners, that\n    He might offer Himself a sin-offering for sinners upon the cross.\n    Thus 'He became obedient unto death,' that He might destroy the power\n    of death; and on the third morning, a mighty angel, rolling away the\n    stone from the mouth of the sepulchre, makes the very door of death's\n    castle the throne whence He proclaims 'the resurrection, and the\n    life.'\n\n    \"The Hero of our salvation travelled into Death's dominion, took\n    possession of the whole territory on our behalf, and returning, laden\n    with spoils, ascended to the Heaven of heavens.  He went to the\n    palace, seized the tyrant, and wrested away his sceptre.  He\n    descended into the prison-house, knocked off the fetters of the\n    captives; and when He came up again, left the door of every cell\n    open, that they might follow Him.  He has gone over into our promised\n    inheritance, and His glory illuminates the mountains of immortality;\n    and through the telescope which He has bequeathed us we 'see the land\n    which is very far off.'\n\n    \"I recollect reading, in the writings of Flavel, this sentiment\u2014that\n    the souls in Paradise wait, with intense desire, for the reanimation\n    of their dead bodies, that they may be united to them in bliss for\n    ever.  Oh what rapture there shall be among the saints, when those\n    frail vessels, from which they escaped with such a struggle, as they\n    foundered in the gulf of death, shall come floating in, with the\n    spring-tide of the resurrection, to the harbour of immortality!  How\n    glorious the reunion, when the seeds of affliction, and death are\n    left behind in the tomb!  Jacob no longer lame, nor Moses slow of\n    speech, nor Lazarus covered with sores, nor Paul troubled with a\n    thorn in the flesh!\n\n    \"'It doth not yet appear what we shall be; but we know that, when He\n    shall appear, we shall be like Him; for we shall see Him as He is.'\n    The glory of the body of Christ is far above our present conception.\n    When He was transfigured on Tabor, His face shone like the sun, and\n    His raiment was white as the light.  This is the pattern shown to His\n    people on the mount.  This is the model after which the bodies of\n    believers shall be fashioned in the resurrection.  'They that be wise\n    shall shine as the brightness of the firmament; and they that turn\n    many to righteousness, as the stars for ever, and ever.'\n\n    \"In conclusion:\u2014The angel said to the woman, 'Go quickly, and tell\n    His disciples that He is risen from the dead; and behold, He goeth\n    before you into Galilee; there shall ye see Him; lo!  I have told\n    you.  And they departed quickly from the sepulchre, with fear, and\n    great joy; and did run to bring His disciples word.'\n\n    \"Brethren! followers of Jesus! be ye also preachers of a risen\n    Saviour!  Go quickly\u2014there is no time for delay\u2014and publish the glad\n    tidings to sinners!  Tell them that Christ died for their sins, and\n    rose again for their justification, and ascended to the right hand of\n    the Father to make intercession for them, and is now able to save\n    unto the uttermost all that come unto God by Him!\n\n    \"And you, impenitent, and unbelieving men! hear this blessed message\n    of salvation!  Do you intend ever to embrace the proffered mercy of\n    the Gospel?  Make haste!  Procrastination is ruin!  Now is the\n    accepted time!  Oh, fly to the throne of grace!  Time is hastening;\n    you will soon be swallowed up in eternity!  May the Lord have mercy\n    upon you, and rouse you from your indifference, and sloth!  It is my\n    delight to invite you to Christ; but I feel more pleasure, and more\n    confidence in praying for you to God.  I have besought, and entreated\n    you, by every argument, and every motive in my power; but you are yet\n    in your sins, and rushing on toward hell.  Yet I will not give you up\n    in despair.  If I cannot persuade you to flee from the wrath to come,\n    I will intercede with God to have mercy upon you, for the sake of His\n    beloved Son.  If I cannot prevail in the pulpit, I will try to\n    prevail at the throne.\"\n\nThis must be regarded as a very noble piece; the words make themselves\nfelt; evidently, the resurrection of our Lord, to this preacher, was a\ngreat reality; it is now, by many, regarded only as a charming myth; a\nvery curious eschatology in our day has found its way even into our\npulpits, and we have eminent ministers of the Church of England,\nwell-known Congregational, and other ministers, who affect to believe,\nand to preach the Resurrection of Christ; but a careful listener in the\npew, or a converser by the fireside, will find, to his amazement, that\nthe resurrection, as believed by them, is no honest resurrection at all:\nit is a spiritual resurrection which leaves the body of Jesus unrisen,\nand in the possession of death, and the grave.  In that view, which has\njust passed before us, a very different, and most absolutely real\nresurrection is preached; indeed, it is the only view which leaves a\nheart of immortal hope in the Christian faith, the only view which seems\nat all tenable, if we are to believe in the power of Christ's\nresurrection.\n\nWe will close these extracts by one of yet another order,\u2014a vivid\ndescriptive picture of the smiting of the rock, the streams flowing\nthrough the desert, and the joy of the mighty caravan of pilgrims on\ntheir way to the promised land.\n\n\n\n\"'THEY DRANK OF THAT ROCK WHICH FOLLOWED THEM.'\n\n\n    \"Having spoken of _the smiting_, let us, _now_, look at _the result_,\n    the flowing of the waters; a timely mercy to 'the many thousands of\n    Israel,' on the point of perishing in the desert; shadowing forth a\n    far greater mercy, the flowing of living waters from the 'spiritual\n    rock,' which is Christ.\n\n    \"In the death of our Redeemer, we see three infinite depths moved for\n    the relief of human misery: the love of the Father, the merit of the\n    Son, and the energy of the Holy Spirit.  These are the depths of\n    wonder whence arise the rivers of salvation.\n\n    \"_The waters flowed in the presence of the whole assembly_.  The\n    agent was invisible, but His work was manifest.\n\n    \"The water flowed _in great abundance_, filling the whole camp, and\n    supplying all the people.  Notwithstanding the immense number, and\n    the greatness of their thirst, there was enough for each, and for\n    all.  The streams ran in every direction to meet the sufferers, and\n    their rippling murmur seemed to say\u2014'Open thy mouth, and I will fill\n    it.'  Look to the cross!  See there the gracious fountain opened, and\n    streams of pardoning, and purifying mercy flowing down the rock of\n    Calvary, sweeping over the mount of Olives, and cleaving it asunder,\n    to make a channel for the living waters to go out over the whole\n    world, that God may be glorified among the Gentiles, and all the ends\n    of the earth may see His salvation.\n\n    \"The water flowed _from the rock_, not pumped by human labour, but\n    drawn by the hand of God.  It was the same power that opened the\n    springs of mercy upon the cross.  It was the wisdom of God that\n    devised the plan, and the mercy of God that furnished the Victim.\n    His was the truth, and love that gave the promise by the prophet\u2014'In\n    that day there shall be a fountain opened to the house of David, and\n    to the inhabitants of Jerusalem, for sin, and uncleanness.'  His was\n    the unchanging faithfulness that fulfilled it in His Son\u2014'Not by\n    works of righteousness which we have done, but according to His mercy\n    He saved us, by the washing of regeneration, and renewing of the Holy\n    Ghost, which He shed on us abundantly, through Jesus Christ our\n    Lord.'  Our salvation is wholly of God; and we have no other agency\n    in the matter than the mere acceptance of His proffered grace.\n\n    \"The water flowed _in twelve different channels_; and, according to\n    Dr. Pococke, of Scotland, who visited the place, the deep traces in\n    the rock are visible to this day.  But the twelve streams, one for\n    each tribe, all issued from the same fountain, in the same rock.  So\n    the great salvation flowed out through the ministry of the twelve\n    apostles of the Lamb, and went abroad over all the earth.  But the\n    fountain is one.  All the apostles preached the same Saviour, and\n    pointed to the same cross.  'Neither is there salvation in any other,\n    for there is no other name under heaven, given among men, whereby we\n    must be saved.'  We must come to this spring, or perish.\n\n    \"The flowing of the waters _was irresistible by human power_.  Who\n    can close the fountain which God hath opened? can Edom, or Moab, or\n    Sihon, or Og dam up the current which Jehovah hath drawn from the\n    rock?  Can Caiaphas, and all the Jews, aided by the prince of this\n    world\u2014can all the powers of earth and hell combined\u2014arrest the work\n    of redemption, and dry up the fountain of mercy which Christ is\n    opening on Calvary?  As soon might they dry up the Atlantic, and stop\n    the revolutions of the globe.  It is written, and must be fulfilled.\n    Christ must suffer, and enter into His glory\u2014must be lifted up, and\n    draw all men unto Him\u2014and repentance, and remission of sins must be\n    preached in His name among all nations, beginning at Jerusalem.\n\n    \"_The water flowing from the rock was like a river of life to the\n    children of Israel_.  Who can describe the distress throughout the\n    camp, and the appearance of the people, when they were invited to\n    approach a flinty rock, instead of a fountain, or a stream, to quench\n    their thirst?  What angry countenances were there, what bitter\n    censures, and ungrateful murmurings, as Moses went up to the rock,\n    with nothing in his hand but a rod!  'Where is he going,' said they,\n    'with that dry stick?  What is he going to do on that rock?  Does he\n    mean to make fools of us all?  Is it not enough that he has brought\n    us into this wilderness to die of thirst?  Will he mock us now by\n    pretending to seek water in these sands, or open fountains in the\n    solid granite?'  But see! he lifts the rod, he smites the rock; and\n    lo, it bursts into a fountain; and twelve crystal streams roll down\n    before the people!  Who can conceive the sudden transport?  Hear the\n    shout of joy ringing through the camp, and rolling back in tumultuous\n    echoes from the crags, and cliffs of Horeb,\u2014'Water! water!  A\n    miracle! a miracle!  Glory to the God of Israel! glory to His servant\n    Moses!'  It was a resurrection-day to Israel, the morning light\n    bursting upon the shadow of death.  New life, and joy are seen\n    throughout the camp.  The maidens are running with cups, and\n    pitchers, to the rock.  They fill, and drink; then fill again, and\n    haste away to their respective tents, with water for the sick, the\n    aged, and the little ones, joyfully exclaiming\u2014'Drink, father!\n    Drink, mother!  Drink, children!  Drink, all of you!  Drink\n    abundantly!  Plenty of water now!  Rivers flowing from the rock!'\n    Now the oxen are coming, the asses, the camels, the sheep, and the\n    goats\u2014coming in crowds to quench their thirst, and plunging into the\n    streams before them.  And the feathered tribes are coming, the\n    turtle-dove, the pigeon, the swallow, the sparrow, the robin, and the\n    wren; while the croaking raven, and the fierce-eyed eagle, scenting\n    the water from afar, mingle with them round the rock.\n\n    \"Brethren, this is but a faint emblem of the joy of the Church, in\n    drinking the waters that descend from Calvary, the streams that\n    gladden the city of our God.  Go back to the day of Pentecost for an\n    instance.  Oh what a revolution of thought, and feeling, and\n    character!  What a change of countenance, and conscience, and heart!\n    Three thousand men, that morning full of ignorance, and corruption,\n    and guilt\u2014idolaters, sensualists, blasphemers, persecutors\u2014before\n    night were perfectly transformed\u2014the lions converted into lambs\u2014the\n    hard heart melted, the dead conscience quickened, and the whole man\n    become a new creature in Christ Jesus!  They thirsted, they found the\n    'Spiritual rock,' tasted its living waters, and suddenly leaped into\n    new life, like Lazarus from the inanition of the grave!\n\n    \"This is the blessing which follows the Church through all her\n    wanderings in the wilderness, accompanies her through the scorching\n    desert of affliction, and the valley of the shadow of death; and\n    when, at last, she shall come up out of great tribulation, her\n    garments shall be found washed and made white in the blood of the\n    Lamb; and the Lamb, who is in the midst of the throne, shall lead her\n    to everlasting fountains, and she shall thirst no more!\"\n\nAmong the great Welsh preachers, then, in closing, it will now be enough\nto say, that, without claiming for Christmas Evans pre-eminence above all\nhis contemporaries, or countrymen, it may, with truth, be said, we have\nyet better means of forming an opinion of him than of any other.  We have\nattempted to avail ourselves of such traditions, and stories of their\npulpit ministration, and such fragments of their spoken words, as may\nconvey some, if faint, still fair, idea of their powers.  Even of\nChristmas Evans our knowledge is, by no means, ample, nor are there many\nof his sermons left to us; but such as we possess seem sufficient for the\nformation of as high an estimate, through the medium of criticism, and\nthe press, as that which was formed by the flocking crowds, and thousands\nwho deemed it one of their greatest privileges, and pleasures to listen\nto his living voice.  And it must be admitted, we think, that these\nsermons are of that order which retains much of its power, when the voice\nthrough which it spoke is still.  Welsh sermons, beyond almost any\nothers, lose their vitality by the transference to the press, and no\ndoubt this preacher suffers in this way, too; some, however, will not\nbear the printing machine at all, and when the voice ceases to speak, all\nwhich made them effective is gone.  With these sermons it is,\nundoubtedly, otherwise, and from some of them it may, perhaps, even be\npossible to find models of the mould of thought, and the mode at once of\narrangement, as well as the qualities of emotion, and expression, which\nmake preaching successful, whether for converting, or comforting the\nsouls of men.  Nor is it less significant that this man, who exercised a\nministry of immense usefulness for more than half a century, and retained\nhis power over men, with the same average freshness, and splendour until\nwithin four days of his death, did so in virtue of the living freshness\nof his heart, and mind.  Like such men as John Bunyan, and Richard\nBaxter, no University could claim him, for he was of none; he had\ngraduated in no college, had sat before no academical prelections, and\nwas decorated with no diplomas,\u2014only the Divine Spirit was master of the\ncollege in which he was schooled.  We write this with no desire to speak\ndisparagingly of such training, but, rather, to bring out into\nconspicuous honour the strength of this self-formed, severely toiling,\nand nobly suffering man.  He was a spiritual athlete in labours more\nabundant; perhaps it might seem that the \"one-eyed man of Anglesea,\" as\nhe was so familiarly called, until this designation yielded to the more\naffectionate term of \"Old Christmas,\" throughout the Principality\u2014must\nhave been in bodily presence contemptible; but if his appearance was\nrugged, we suppose it could scarcely have been less than royal,\u2014a man the\nspell of whose name, when he came into a neighbourhood, could wake up all\nthe sleepy villages, and bid their inhabitants pour along, up by the\nhills, and down by the valleys, expectant crowds watching his appearance\nwith tears, and sometimes hailing him with shouts\u2014must have been\nsomething like a king among men.  We have seen how poor he was, and how\nindifferent to all that the world regarded as wealth, but he was one of\nthose of whom the apostle speaks \"as poor, yet making many rich, as\nhaving nothing, and yet possessing all things.\"  And thus, from every\nconsideration, whether we regard his singular genius, so truly national,\nand representative of the mind, and character of his country, his\nindomitable struggles, and earnest self-training, his extraordinary power\nover his congregations, his long, earnest life of self-denying\nusefulness, especially his intense reality, the holy purity, and\nconsecration of his soul, Christmas Evans deserves our reverent memory\nwhile we glorify God in him.\n\n\n\n\nAPPENDATORY.\n_SELECTION OF ILLUSTRATIVE SERMONS_.\n\n\nAND now, although the various, and several selections we have given in\nthe different preceding sections of this volume, may assist the reader in\nforming some idea of the manner, and method of Christmas Evans, before\nclosing the volume we will present some selections from entire sermons,\ntranslated from the Welsh; and while, of course, labouring beneath the\ndisadvantages of translation, we trust they will not unfavourably\nrepresent those various attributes of pulpit power, for which we have\ngiven the great preacher credit.\n\nSERMON I.\u2014THE TIME OF REFORMATION.\n\nSERMON II.\u2014THE PURIFICATION OF THE CONSCIENCE.\n\nSERMON III.\u2014FINISHED REDEMPTION.\n\nSERMON IV.\u2014THE FATHER AND SON GLORIFIED.\n\nSERMON V.\u2014THE CEDAR OF GOD.\n\n\n\nSERMON I.\nTHE TIME OF REFORMATION.\n\n\n    \"_Until the time of reformation_.\"\u2014HEB. ix. 10.\n\nThe ceremonies pertaining to the service of God, under Sinaitic\ndispensation, were entirely typical in their character; mere figures of\nChrist, the \"High-priest of good things to come, by a greater, and more\nperfect tabernacle, not made with hands;\" who, \"not by the blood of\ngoats, and calves, but by His own blood, has entered once into the holy\nplace, having obtained eternal redemption for us.\"  Sustaining such a\nrelation to other ages, and events, they were necessarily imperfect,\nconsisting \"only in meats, and drinks, and divers washings, and carnal\nordinances,\" not intended for perpetual observance, but imposed upon the\nJewish people merely \"until the time of reformation,\" when the shadow\nshould give place to the substance, and a Greater than Moses should \"make\nall things new.\"  Let us notice the time of reformation, and the\nreformation itself.\n\nI.  Time may be divided into three parts; the Golden Age before the fall,\nthe Iron Age after the fall, and the Messiah's Age of Jubilee.\n\nIn the Golden Age, the heavens, and the earth were created; the Garden of\nEden was planted; man was made in the image of God, and placed in the\ngarden, to dress, and keep it; matrimony was instituted; and God, resting\nfrom His labour, sanctified the seventh day, as a day of holy rest to\nman.\n\nThe Iron Age was introduced by the temptation of a foreigner, who\nobtruded himself into Paradise, and persuaded its happy denizens to cast\noff the golden yoke of obedience, and love to God.  Man, desiring\nindependence, became a rebel against heaven, a miserable captive of sin,\nand Satan, obnoxious to the Divine displeasure, and exposed to eternal\ndeath.  The law was violated; the image of God was lost, and the enemy\ncame in like a flood.  All communication between the island of Time, and\nthe continent of Immortality was cut off, and the unhappy exiles saw no\nhope of crossing the ocean that intervened.\n\nThe Messiah's Age may be divided into three parts; the time of\nPreparation, the time of Actual War, and the time of Victory and Triumph.\n\nThe Preparation began with the dawning of the day in Eden, when the\nMessiah came in the ship of the Promise, and landed on the island of\nTime, and notified its inhabitants of His gracious intention to visit\nthem again, and assume their nature, and live and die among them; to\nbreak their covenant allegiance to the prince of the iron yoke; and\ndeliver to them the charter, signed, and sealed with His own blood, for\nthe redemption, and renovation of their island, and the restoration of\nits suspended intercourse with the land of Eternal Life.  The motto\ninscribed upon the banners of this age was,\u2014\"He shall bruise thy heel,\nand Thou shalt bruise his head.\"  Here Jehovah thundered forth His hatred\nof sin from the thick darkness, and wrote His curse in fire upon the face\nof heaven; while rivers of sacrificial blood proclaimed the miserable\nstate of man, and his need of a costlier atonement than mere humanity\ncould offer.  Here, also, the spirit of Messiah fell upon the prophets,\nleading them to search diligently for the way of deliverance, and\nenabling them to \"testify beforehand of the sufferings of Christ, and the\nglory that should follow.\"\n\nThen came the season of Actual War.  \"Messiah the Prince\" was born in\nBethlehem, wrapped in swaddling bands, and laid in a manger,\u2014the Great\nDeliverer, \"made of a woman, made under the law, to redeem those that\nwere under the law, that we might receive the adoption of sons.\"  With\nHis almighty hand, He laid hold on the works of the devil, unlocked the\niron furnace, and broke the brazen bands asunder.  He opened His mouth,\nand the deaf heard, the blind saw, the dumb spoke, the lame walked, and\nthe lepers were cleansed.  In the house of Jairus, in the street of Nain,\nand in the burial ground of Bethany, His word was mightier than death;\nand the damsel on her bed, the young man on his bier, and Lazarus in his\ntomb, rising to second life, were but the earnests of His future triumph.\nThe diseases of sin He healed, the iron chains of guilt He shattered, and\nall the horrible caves of human corruption, and misery were opened by the\nHeavenly Warrior.  He took our yoke, and bore it away upon His own\nshoulder, and cast it, broken, into the bottomless pit.  He felt in His\nhands, and feet, the nails, and in His side the spear.  The iron entered\ninto His soul, but the corrosive power of His blood destroyed it, and\nshall ultimately eat away all the iron in the kingdom of death.  Behold\nHim hanging on Calvary, nailing upon His cross three bills, the\nhandwriting of the law which was against us, the oath of our allegiance\nto the prince of darkness, and the charter of the \"everlasting covenant;\"\nfulfilling the first, breaking the second, and sealing the third with His\nblood!\n\nNow begins the scene of Victory and Triumph.  On the morning of the third\nday, the Conqueror is seen \"coming from Edom, with dyed garments from\nBozrah.\"  He has \"trodden the winepress alone.\"  By the might of His\nsingle arm He has routed the hosts of hell, and spoiled the dominions of\ndeath.  The iron castle of the foe is demolished, and the Hero returns\nfrom the war, \"glorious in His apparel, travelling in the greatness of\nHis strength.\"  He enters the gates of the everlasting city, amid the\nrejoicing of angels, and the shouts of His redeemed.  And still He rides\nforth in the chariot of His grace, \"conquering, and to conquer.\"  A\ntwo-edged sword issues from His mouth, and, in His train, follow the\nvictorious armies of heaven.  Lo! before Him fall the altars of idols,\nand the temples of devils; and the slaves of sin are becoming the\nservants, and sons of the living God; and the proud sceptic beholds,\nwonders, believes, and adores; and the blasphemer begins to pray, and the\npersecutor is melted into penitence, and love, and the wolf comes, and\nlays him down gently by the side of the lamb.  And Messiah shall never\nquit the field, till He has completed the conquest, and swallowed up\ndeath in victory.  In His \"vesture dipped in blood,\" He shall pursue the\narmies of Gog and Magog on the field of Amageddon, and break the iron\nteeth of the beast of power, and cast down Babylon as a mill-stone into\nthe sea, and bind the old serpent in the lake of fire, and brimstone, and\nraise up to life immortal the tenants of the grave.  Then shall the New\nJerusalem, the metropolis of Messiah's golden empire, descend from\nheaven, adorned with all the jewellery of creation, guarded at every gate\nby angelic sentinels, and enlightened by the glory of God, and of the\nLamb; and the faithful shall dwell within its walls, and sin, and sorrow,\nand death, shall be shut out for ever!\n\nThen shall Time be swallowed up in Eternity.  The righteous shall inherit\nlife everlasting, and the ungodly shall find their portion in the second\ndeath.  Time is the age of the visible world; eternity is the age of the\ninvisible God.  All things in time are changeful; all things in eternity\nare immutable.  If you pass from time to eternity, without faith in\nChrist, without love in God, an enemy to prayer, an enemy to holiness,\n\"impurged and unforgiven,\" so you must ever remain.  Now is the season of\nthat blessed change, for which myriads shall sing everlasting anthems of\npraise.  \"To-day, if ye will hear His voice, harden not your hearts.\"\nTo-day the office is open: if you have any business with the Governor,\nmake no delay.  Now He has time to talk with the woman of Samaria by the\nwell, and the penitent thief upon the cross.  Now He is ready to forgive\nyour sins, and renew your souls, and make you meet to become the\npartakers of the inheritance of the saints in light.  Now He waits to\nwash the filthy, and feed the hungry, and clothe the naked, and raise the\nhumble, and quicken the spiritually dead, and enrich the poor, and\nwretched, and reconcile enemies by His blood.  He came to unloose your\nbands, and open to you the gates of Eden; condemned for your acquittal,\nand slain for the recovery of your forfeited immortality.  The design of\nall the travelling from heaven to earth, and from earth to heaven, is the\nsalvation of that which was lost, the restoration of intercourse, and\namity between the Maker and the worm.  This is the chief of the ways of\nGod to man, ancient in its origin, wise in its contrivance, dear in its\naccomplishment, powerful in its application, gracious in its influence,\nand everlasting in its results.  Christ is riding in His chariot of\nsalvation, through the land of destruction, and death, clothed in the\nmajesty of mercy, and offering eternal life to all who will believe.  O\ncaptives of evil! now is the accepted time; now is the day of salvation;\nnow is the year of Jubilee; now is the age of deliverance; now is \"the\ntime of reformation.\"\n\nII.  All the prophets speak of something within the veil, to be\nmanifested in due time; the advent of a Divine agent in a future age, to\naccomplish a glorious \"reformation.\"  They represent him as a prince, a\nhero, a high priest, a branch growing out of dry ground, a child toying\nwith the asp, and the lion, and leading the wolf, and the lamb together.\nThe bill of the reformation had been repeatedly read by the prophets, and\nits passage required the descent of the Lord from heaven.  None but\nHimself could effect the change of the dispensation.  None but Himself\nhad the authority and the power to remove the first, and establish the\nsecond.  He whose voice once shook the earth, speaks again, and heaven is\nshaken.  He whose footsteps once kindled Sinai into flame, descends\nagain, and Calvary is red with blood.  The God of the ancient covenant\nintroduces anew, which is to abide for ever.  The Lord of the temple\nalone could change the furniture, and the service from the original\npattern shown to Moses on the mount; and six days before the rending of\nthe veil, significant of abrogation of the old ceremonial, Moses came\ndown upon a mountain in Palestine to deliver up the pattern to Him of\nwhom he had received it on Sinai, that He might nail it to the cross on\nCalvary; for the \"gifts and sacrifices\" belonging to the legal\ndispensation, \"could not make him that did the service perfect, as\npertaining to the conscience; which stood only in meats, and drinks, and\ndivers washings, and carnal ordinances, imposed on them until the time of\nreformation.\"\n\nThis reformation signifieth \"the removal of those things that are shaken,\nas of things that are made, that those things which cannot be shaken may\nremain;\" the abrogation of \"carnal ordinances,\" which were local, and\ntemporal in their nature, to make room for a spiritual worship, of\nuniversal, and perpetual adaptation.  Henceforth the blood of bulls, and\ngoats is superseded by the great reconciling sacrifice of the Lamb of\nGod, and outward forms, and ceremonies give place to the inward\noperations of a renovating, and purifying Spirit.\n\nTo the Jewish Church, the covenant of Sinai was a sort of starry heaven.\nThe Shekinah was its sun; the holy festivals, its moon; and prophets,\npriests, and kings, its stars.  But Messiah, when He came, shook them all\nfrom their spheres, and filled the firmament Himself.  He is our \"Bright\nand Morning Star;\" the \"Sun of Righteousness,\" rising upon us \"with\nhealing in His wings.\"\n\nThe old covenant was an accuser, and a judge, but offered no pardon to\nthe guilty.  It revealed the corruption of the natural heart, but\nprovided no renovating, and sanctifying grace.  It was a natural\ninstitution, for special benefit of the seed of Abraham.  It was a small\nvessel, trading only with the land of Canaan.  It secured, to a few, the\ntemporal blessings of the promised possession, but never delivered a\nsingle soul from eternal death, never bore a single soul over to the\nheavenly inheritance.  But the new covenant is a covenant of grace, and\nmercy, proffering forgiveness, and a clean heart, not on the ground of\nany carnal relationship, but solely through faith in Jesus Christ.\nChristianity is a personal concern between each man, and his God, and\nnone but the penitent believer has any right to its spiritual privileges.\nIt is adapted to Gentiles, as well as Jews, \"even as many as the Lord our\nGod shall call.\"  Already has it rescued myriads from the bondage of sin,\nand conveyed them over to the land of immortality; and its voyages of\ngrace shall continue to the end of time, \"bringing many sons to glory.\"\n\n\"Old things are passed away, and all things are become new.\"  The\ncircumcision of the flesh, made with hands, has given place to the\ncircumcision of the heart by the Holy Ghost.  The Shekinah has departed\nfrom Mount Zion, but its glory is illuminating the world.  The Sword of\nJoshua is returned to its scabbard; and \"the sword of the Spirit, which\nis the word of God,\" issues from the mouth of Messiah, and subdues the\npeople under Him.  The glorious High-priesthood of Christ has superseded\nsacerdotal office among men.  Aaron was removed from the altar by death\nbefore his work was finished; but our High-priest still wears His\nsacrificial vestments, and death hath established Him before the\nmercy-seat, \"a Priest for ever, after the order of Melchisedec.\"  The\nearthquake which shook Mount Calvary, and rent the veil of the temple,\ndemolished \"the middle wall of partition\" between Jews and Gentiles.  The\nincense which Jesus offered fills the temple, and the land of Judea\ncannot confine its fragrance.  The fountain which burst forth in\nJerusalem, has sent out its living streams into every land; and the heat\nof summer cannot dry them up, nor the frosts of winter congeal them.\n\nIn short, all the vessels of the sanctuary are taken away by the Lord of\nthe temple.  The \"twelve oxen,\" bearing the \"molten sea,\" have given\nplace to \"the twelve Apostles of the Lamb,\" proclaiming \"the washing of\nregeneration, and renewing of the Holy Ghost.\"  The sprinkled mercy-seat,\nwith its over-shadowing, and intensely-gazing cherubim, has given place\nto \"the throne of grace,\" stained with the blood of a costlier sacrifice,\ninto which the angels desire to look.  The priest, the altar, the\nburnt-offering, the table of shew-bread, and the golden candlestick, have\ngiven place to the better things of the new dispensation introduced by\nthe Son of God, of which they were only the figures, and the types.\nBehold, the glory has gone up from the temple, and rests upon Jesus on\nMount Tabor; and Moses, and Elias are there, with Peter, and James, and\nJohn; and the representatives of the old covenant are communing with the\nApostles of the new, and the transfigured Christ is the medium of the\ncommunication; and a voice of majestic music, issuing from \"the excellent\nglory,\" proclaims\u2014\"This is my beloved Son, hear ye Him.\"\n\n\"God, who at sundry times, and in divers manners spake unto our fathers\nby the prophets, hath in these last days spoken unto us by His Son.\"\nBehold Him nailed to the Cross, and hear Him cry\u2014\"It is finished!\"  The\nvoice which shook Sinai is shaking Calvary.  Heaven and hell are in\nconflict, and earth trembles at the shock of battle.  The Prince of Life\nexpires, and the sun puts on his robes of mourning.  Gabriel! descend\nfrom heaven, and explain to us the wondrous emblem!  As set the sun at\nnoon on Golgotha, making preternatural night throughout the land of\nPalestine, so shall the empire of sin, and death be darkened, and their\nlight shall be quenched at meridian.  As the Sun of Righteousness, rising\nfrom the night of the grave on the third morning, brings life, and\nimmortality to light; so shall \"the day-spring from on high\" yet dawn\nupon our gloomy vale, and \"the power of His resurrection\" shall reanimate\nthe dust of every cemetery!\n\nHe that sitteth upon the throne hath spoken\u2014\"Behold, I make all things\nnew.\"  The reformation includes not only the abrogation of the old, but\nalso the introduction of the new.  It gives us a new Mediator, a new\ncovenant of grace, a new way of salvation, a new heart of flesh, a new\nheaven and a new earth.  It has established a new union, by a new medium,\nbetween God, and man.  \"The Word was made flesh, and dwelt among us, and\nwe beheld His glory, the glory as of the only-begotten of the Father,\nfull of grace and truth.\"  \"Forasmuch as the children were partakers of\nflesh and blood, He also Himself likewise took part of the same.\"  \"God\nwas manifest in the flesh, justified in the spirit, seen of angels,\npreached unto the Gentiles, believed on in the world, received up into\nglory.\"  Here was a new thing under the sun; the \"Son of man\" bearing the\n\"express image\" of the living God; bearing it untarnished through the\nworld; through the temptations and sorrows of such a wilderness as\nhumanity never trod before; through the unknown agony of Olivet, and the\nsupernatural gloom of Golgotha, and the dark dominion of the king of\nterrors: to the Heaven of heavens; where He sits, the adorable\nrepresentative of two worlds, the union of God and man!  Thence He sends\nforth the Holy Spirit, to collect \"the travail of His soul,\" and lead\nthem into all truth, and bring them to Zion with songs of everlasting\njoy.  See them, the redeemed of the Lord, flocking as returning doves\nupon the wing, \"to the heavenly Jerusalem, the city of the living God;\nand to the spirits of just men made perfect; and to an innumerable\ncompany of angels; and to Jesus, the Mediator of the new covenant; and to\nthe blood of sprinkling, that speaketh better things than that of Abel.\"\n\nOh, join the joyful multitude! the year of jubilee is come.  The veil is\nrent asunder.  The way into the holiest is laid open.  The blood of Jesus\nis on the mercy-seat.  The Lamb newly slain is in the midst of the\nthrone.  Go ye, with boldness, into His gracious presence.  Lo, the King\nis your brother, and for you has He stained His robe with blood!  The\nrobe alone can clothe your naked souls, and shield them in the day of\nburning.  Awake! awake! put on the Lord Jesus Christ!  The covenant of\nSinai cannot save you from wrath.  Descent from Abraham cannot entitle\nyou to the kingdom of heaven.  \"Ye must be born again,\" \"born not of the\nflesh, nor of the will of men, but of God.\"  You must have a new heart,\nand become a new creation in Jesus Christ.  This is the promise of the\nFather,\n\n    \"This is the dear redeeming grace,\n    For every sinner free.\"\n\nMany reformations have expired with the reformers.  But our Great\nReformer \"ever liveth\" to carry on His reformation, till His enemies\nbecome His footstool, and death and hell are cast into the lake of fire.\nHe will finish the building of His Church.  When He laid \"the chief\ncorner-stone\" on Calvary, the shock jarred the earth, and awoke the dead,\nand shook the nether world with terror; but when He shall bring forth the\ntop stone with shoutings of \"Grace!\" the dominion of Death and Hades\nshall perish, and the last captive shall escape, and the song of the\nbursting sepulchre shall be sweeter than the chorus of the morning stars!\nEven now, there are new things in heaven; the Lamb from the slaughter,\nalive \"in the midst of the throne;\" worshipped by innumerable seraphim\nand cherubim, and adored by the redeemed from earth; His name the wonder\nof angels, the terror of devils, and the hope of men; His praise the \"new\nsong,\" which shall constitute the employment of eternity!\n\n\n\nSERMON II.\nTHE PURIFICATION OF THE CONSCIENCE.\n\n\n    \"_How much more shall the blood of Christ_, _who_, _through the\n    eternal Spirit_, _offered Himself without spot to God_, _purge your\n    conscience from dead works to serve the living God_.\"\u2014HEB. ix. 14.\n\nThe Hebrew Christians, to whom the Apostle wrote, were well acquainted\nwith the laws of ceremonial purification by the blood of beasts, and\nbirds, for by blood almost everything was purified in the service of the\nTemple.  But it is only the blood of Christ that can purge the human\nconscience.  In speaking of this purification, as presented in our text,\nlet us notice\u2014_the object_, _the means_, and _the end_.\n\nI.  The object of this purification is the conscience; which all the\nsacrificial blood shed, from the gate of Eden down to the extinction of\nthe fire on the Jewish altar, was not sufficient to purge.\n\n_What is the conscience_?  An inferior judge, the representative of\nJehovah, holding his court in the human soul; according to whose decision\nwe feel either confidence, and joy in God, or condemnation, and\ntormenting fear.  His judicial power is graduated by the degree of moral\nand evangelical light which has been shed upon his palace.  His knowledge\nof the will, and character of God is the law by which he justifies, or\ncondemns.  His intelligence is the measure of his authority; and the\nperfection of knowledge would be the infallibility of conscience.\n\nThis faithful recorder, and deputy judge is with us through all the\njourney of life, and will accompany us with his register over the river\nJordan, whether to Abraham's bosom or the society of the rich man in\nhell.  While conscience keeps a record on earth, Jehovah keeps a record\nin heaven; and when both books shall be opened in the final judgment,\nthere shall be found a perfect correspondence.  When temptations are\npresented, the understanding opposes them, but the carnal mind indulges\nthem, and there is a contest between the judgment, and the will, and we\nhesitate which to obey, till the warning bell of conscience rings through\nthe soul, and gives distinct notice of his awful recognition; and when we\nturn away recklessly from his faithful admonitions, we hear low\nmutterings of wrath stealing along the avenues, and the quick sound of\nwriting-pens in the recording office, causing every denizen of the mental\npalace to tremble.\n\nThere is a _good conscience_, _and an evil conscience_.  The work of\nboth, however, is the same; consisting in keeping a true record of the\nactions of men, and passing sentence upon them according to their\ndeserts.  Conscience is called good, or evil only with reference to the\ncharacter of its record, and its sentence.  If the record is one of\nvirtues, and the sentence one of approval, the conscience is good; if the\nrecord is one of vices, and the sentence one of condemnation, the\nconscience is evil.\n\nSome have a _guilty conscience_, that is, a conscience that holds up to\ntheir view a black catalogue of crimes, and rings in their ears a\nsentence of condemnation.  If you have such a conscience, you are invited\nto Jesus, that you may find peace to your souls.  He is ever in His\noffice, receiving all who come, and blotting out, with His own blood, the\nhandwriting which is against them.\n\nBut some have a _despairing conscience_.  They think that their crimes\nare too great to be forgiven.  The registry of guilt, and the decree of\ndeath, hide from their eyes the mercy of God, and the merit of Christ.\nTheir sins rise like mountains between them, and heaven.  But let them\nlook away to Calvary.  If their sins are a thousand times more numerous\nthan their tears, the blood of Jesus is ten thousand times more powerful\nthan their sins.  \"He is able to save to the uttermost all that come unto\nGod by Him, seeing He ever liveth to make intercession for them.\"\n\nAnd others have a _dark_, _and hardened conscience_.  They are so\ndeceived, that they \"cry peace, and safety, when destruction is at the\ndoor.\"  They are \"past feeling, having the conscience seared as with a\nhot iron.\"  They have sold themselves to work evil; to eat sin like\nbread, and drink iniquity like water.  They have bribed, or gagged the\nrecorder, and accuser within them.  They will betray the just cause of\nthe righteous, and slay the messengers of salvation, and think that they\nare doing God service.  John the Baptist is beheaded, that Herod may keep\nhis oath of honour.  A dead fish cannot swim against the stream; but if\nthe king's conscience had been alive and faithful, he would have\nsaid:\u2014\"Girl, I promised to give thee thy request, even to the half of my\nkingdom; but thou hast requested too much; for the head of Messiah's\nherald is more valuable than my whole kingdom, and all the kingdoms of\nthe world!\"  But he had not the fear of God before his eyes, and the\nproud fool sent, and beheaded the prophet in his cell.\n\nA _good conscience_ is a faithful conscience, a lively conscience, a\npeaceful conscience, a conscience void of offence toward God, and man,\nresting in the shadow of the cross, and assured of an interest in His\ninfinite merit.  It is the victory of faith unfeigned, working by love,\nand purifying the heart.  It is always found in the neighbourhood, and\nsociety of its brethren, \"a broken heart and a contrite spirit;\" an\nintense hatred of sin, and an ardent love of holiness; a spirit of\nfervent prayer, and supplication, and a life of scrupulous integrity, and\ncharity; and above all, a humble confidence in the mercy of God, through\nthe mediation of Christ.  These constitute the brotherhood of\nChristianity; and wherever they abound, a good conscience is never\nlacking.  They are its very element, and life; its food, its sunshine,\nand its vital air.\n\nConscience was a faithful recorder, and judge under the law, and\nnotwithstanding the revolution which has taken place, introducing a new\nconstitution, and a new administration, Conscience still retains his\noffice; and when \"purged from dead works to serve the living God,\" is\nappropriately called a _good conscience_.\n\nII.  The means of this purification is \"the blood of Christ, who through\nthe Eternal Spirit offered Himself without spot to God.\"\n\nCould we take in, at a single view, all the bearings of \"the blood of\nChrist,\" as exhibited in the Gospel, what an astonishing light would it\ncast upon the condition of man; the character of God; the nature, and\nrequirements of His law; the dreadful consequences of sin; the wondrous\nexpiation of the cross; the reconciliation of Heaven, and earth; the\nblessed union of the believer with God in Christ, as a just God, and a\nSaviour; and the whole scheme of our justification, sanctification, and\nredemption, through free, sovereign, infinite, and unspeakable grace!\n\nThere is no knowledge like the knowledge of Christ, for the excellency of\nwhich the apostle counted all things but loss.  Christ is the Sun of\nRighteousness, in whose light we see the tops of the mountains of\nimmortality, towering above the dense clouds which overhang the valley of\ndeath.  All the wisdom which philosophers have learned from nature, and\nprovidence, compared with that which is afforded by the Christian\nrevelation, is like the _ignis fatuus_, compared with the sun.  The\nknowledge of Plato, and Socrates, and all the renowned sages of\nantiquity, was nothing to the knowledge of the feeblest believer in \"the\nblood of Christ.\"\n\n\"The blood of Christ\" is of infinite value.  There is none like it\nflowing in human veins.  It was the blood of a man, but of a man who knew\nno iniquity; the blood of a sinless humanity, in which dwelt all the\nfulness of the Godhead bodily; the blood of the second Adam, who is the\nLord from Heaven, and a quickening Spirit upon earth.  It pressed through\nevery pore of His body in the garden; and gushed from His head, His\nhands, His feet, and His side, upon the cross.  I approach with fear, and\ntrembling, yet with humble confidence, and joy.  I take off my shoes,\nlike Moses, as he approaches the burning bush; for I hear a voice coming\nforth from the altar, saying, \"I and my Father are one; I am the true\nGod, and Eternal Life.\"\n\nThe expression, \"the blood of Christ,\" includes the whole of His\nobedience to the moral law, by the imputation of which we are justified;\nand all the sufferings of His soul and His body as our Mediator, by which\nan atonement is made for our sins, and a fountain opened to wash them all\naway.  This is the spring whence rise the rivers of forgiving and\nsanctifying grace.\n\nIn the representation which the text gives us of this redeeming blood,\nare several points worthy of our special consideration:\u2014\n\n1.  It is \"_the blood of Christ_;\" the appointed Substitute and Saviour\nof men; \"the Lamb that taketh away the sins of the world.\"\n\n2.  It is the blood of Christ, _who offered Himself_.  His humanity was\nthe only sacrifice which would answer the demands of justice, and atone\nfor the transgressions of mankind.  Therefore \"He has made His soul an\noffering for sin.\"\n\n3.  It is the blood of Christ, who offered Himself _to God_.  It was the\neternal Father, whose broken law must be repaired, whose dishonest\ngovernment must be vindicated, and whose flaming indignation must be\nturned away.  The well-beloved Son must meet the Father's frown, and bear\nthe Father's curse for us.  All the Divine attributes called for the\noffering; and without it, could not be reconciled to the sinner.\n\n4.  It is the blood of Christ, who offered Himself to God, _without\nspot_.  This was a perfect sacrifice.  The Victim was without blemish, or\ndefect; the altar was complete in all its appurtenances; and the High\nPriest possessed every conceivable qualification for his work.  Christ\nwas at once victim, altar, and high-priest; \"holy, harmless, and\nundefiled\"\u2014\"God manifest in the flesh.\"  Being Himself perfect God and\nperfect man, and perfect Mediator between God and man, He perfects for\never all them that believe.\n\n5.  It is the blood of Christ, who offered Himself to God, without spot,\n_through the eternal Spirit_.  By the eternal Spirit, here, we are to\nunderstand, not the third Person of the Godhead, but the second; Christ's\nown Divine nature, which was co-eternal with the Father before the world\nwas, and which, in the fulness of time, seized on humanity\u2014sinless, and\nimmaculate humanity\u2014and offered it, body, and soul, as a sacrifice for\nhuman sins.  The eternal Spirit was at once the priest that offered the\nvictim, and the altar that sanctified the offering.  Without His agency,\nthere could have been no atonement.  The offering of mere humanity,\nhowever spotless, aside from the merit derived from its connection with\nDivinity, could not have been a sacrifice of sweet-smelling savour unto\nGod.\n\n6.  It is the blood of Christ, who offered Himself to God, without spot,\nthrough the eternal Spirit, _that He might purge your conscience_.  As\nthe typical sacrifices under the law purified men from ceremonial\ndefilement, so the real sacrifice of the Gospel saves the believer from\nmoral pollution.  Blood was the life of all the services of the\ntabernacle made with hands, and gave significance, and utility to all the\nrites of the former dispensation.  By blood the covenant between God, and\nHis people was sealed.  By blood the officers, and vessels of the\nsanctuary were consecrated.  By blood the children of Israel were\npreserved in Egypt from the destroying angel.  So the blood of Christ is\nour justification, sanctification, and redemption.  All the blessings of\nthe Gospel flow to us through the blood of the Lamb.  Mercy, when she\nwrites our pardon, and when she registers our names in \"the Book of\nLife,\" dips her pen in the blood of the Lamb.  And the vast company that\nJohn saw before the throne had come out of great tribulation, having\n\"washed their robes and made them white in the blood of the Lamb.\"\n\nThe children of Israel were delivered from Egypt, on the very night that\nthe paschal lamb was slain, and its blood sprinkled upon the doorposts,\nas if their liberty, and life were procured by its death.  This typified\nthe necessity, and power of the Atonement, which is the very heart of the\nGospel, and the spiritual life of the believer.  In Egypt, however, there\nwas a lamb slain for every family; but under the new covenant God has but\none family, and one Lamb is sufficient for their salvation.\n\nIn the cleansing of the leper, several things were necessary; as running\nwater, cedar wood, scarlet, and hyssop, and the finger of the priest; but\nit was the blood that gave efficacy to the whole.  So it is in the\npurification of the conscience.  Without the shedding of blood, the leper\ncould not be cleansed; without the shedding of blood, the conscience\ncannot be purged.  \"The blood of Christ\" seals every precept, every\npromise, every warning, of the New Testament.  \"The blood of Christ\"\nrenders the Scriptures \"profitable for doctrine, for reproof, for\ncorrection, for instruction in righteousness.\"  \"The blood of Christ\"\ngives efficiency to the pulpit; and when \"Jesus Christ and Him crucified\"\nis shut out, the virtue is wanting which heals, and restores the soul.\nIt is only through the crucifixion of Christ that \"the old man\" is\ncrucified in the believer.  It is only through His obedience unto death,\neven the death of the cross, that our dead souls are quickened, to serve\nGod in newness of life.\n\nHere rest our hopes.  \"The foundation of God standeth sure.\"  The bill of\nredemption being presented by Christ, was read by the prophets, and\npassed unanimously in both houses of parliament.  It had its final\nreading in the lower house, when Messiah hung on Calvary; and passed\nthree days afterward, when He rose from the dead.  It was introduced to\nthe upper house by the Son of God Himself, who appeared before the throne\n\"as a lamb newly slain,\" and was carried by acclamation of the heavenly\nhosts.  Then it became a law of the Kingdom of Heaven, and the Holy Ghost\nwas sent down to establish it in the hearts of men.  It is \"the perfect\nlaw of liberty,\" by which God is reconciling the world unto Himself.  It\nis \"the law of the Spirit of life,\" by which He is \"purging our\nconscience from dead works to serve the living God.\"\n\nIII.  The end of this purification is twofold,\u2014that we may cease from\ndead works, and serve the living God.\n\n1.  The works of unrenewed souls are all \"dead works,\" can be no other\nthan \"dead works,\" because the agents are \"dead in trespasses and sins.\"\nThey proceed from the \"carnal mind,\" which \"is enmity against God,\" which\n\"is not subject to the law of God, neither indeed can be.\"  How can a\ncorrupt tree bring forth good fruit, or a corrupt fountain send forth\npure water?\n\nBut \"the blood of Christ\" is intended to \"purge the conscience from dead\nworks.\"  The apostle says\u2014\"Ye are not redeemed with corruptible things,\nas silver, and gold, from your vain conversation, received by tradition\nfrom your fathers; but with the precious blood of Christ, as of a Lamb\nwithout blemish, and without spot.\"  The Jews were in a state of bondage\nto the ceremonial law, toiling at the \"dead works,\" the vain, and empty\nforms, which could never take away sin; and unjustified, and unregenerate\nmen are still captives of Satan, slaves of sin, and death, tyrannized\nover by various evil habits, and propensities, which are invincible to\nall things but \"the blood of Christ.\"  He died to redeem, both from the\nburdens of the Mosaic ritual, and from the despotism of moral evil\u2014to\npurge the conscience of both Jew, and Gentile \"from dead works to serve\nthe living God.\"\n\n2.  We cannot \"serve the living God\" without this preparatory\npurification of conscience.  If our guilt is uncancelled\u2014if the love of\nsin is not dethroned\u2014the service of the knee, and the lip is nothing but\nhypocrisy.  \"If we regard iniquity in our hearts, the Lord will not hear\nus.\"  Cherishing what He hates, all our offerings are an abomination to\nHim; and we can no more stand in His holy presence than the dry stubble\ncan stand before a flaming fire.  He who has an evil conscience flees\nfrom the face of God, as did Adam in the garden.  Nothing but \"the blood\nof Christ,\" applied by the Holy Spirit, can remove the sinner's guilty\nfear, and enable him to draw nigh to God, in the humble confidence of\nacceptance through the Beloved.\n\nThe service of the living God must flow from a new principle of life in\nthe soul.  The Divine word must be the rule of our actions.  The Divine\nwill must be consulted and obeyed.  We must remember that God is holy,\nand jealous of His honour.  The consideration that He is everywhere, and\nsees everything, and will bring every work into judgment, must fill us\nwith reverence and godly fear.  An ardent love for His law, and His\ncharacter must supplant the love of sin, and prompt to a cheerful and\nimpartial obedience.\n\nAnd let us remember that he is \"the _living_ God.\"  Pharaoh is dead,\nHerod is dead, Nero is dead; but Jehovah is \"the living God.\"  And it is\na fearful thing to have Him for an enemy.  Death cannot deliver from His\nhand.  Time, and even eternity, cannot limit His holy anger.  He has\nmanifested, in a thousand instances, His hatred of sin: in the\ndestruction of the old world, the burning of Sodom, and Gomorrah, the\ndrowning of Pharaoh and his host in the sea; and I tell thee, sinner,\nexcept thou repent, thou shalt likewise perish!  Oh, think what\npunishment \"the living God\" can inflict upon His adversaries\u2014the loss of\nall good\u2014the endurance of all evil\u2014the undying worm\u2014the unquenchable\nfire\u2014the blackness of darkness for ever!\n\nThe gods of the heathen have no life in them, and they that worship them\nare like unto them.  But our God is \"the living God,\" and \"the God of the\nliving.\"  If you are united to Him by faith in \"the blood of Christ,\"\nyour souls are \"quickened together with Him,\" and \"the power which raised\nHim from the dead shall also quicken your mortal body.\"\n\nMay the Lord awaken those who are dead in trespasses, and sins, and\nrevive His work in the midst of the years, and strengthen the feeble\ngraces of His people, and bless abundantly the labours of His servants,\nso that many consciences may be purged from dead works to serve the\nliving God!\n\n    \"There is a fountain filled with blood,\n       Drawn from Emmanuel's veins,\n    And sinners, plunged beneath that flood,\n       Lose all their guilty stains.\n\n    \"The dying thief rejoiced to see\n       That fountain in his day;\n    And there may I, as vile as he,\n       Wash all my sins away.\n\n    \"Dear dying Lamb! Thy precious blood\n       Shall never lose its power,\n    Till all the ransomed sons of God\n       Are saved, to sin no more.\"\n\n\n\nSERMON III.\nFINISHED REDEMPTION.\n\n\n    \"_It is finished_.\"\u2014JOHN xix. 30.\n\nThis exclamation derives all its importance from the magnitude of the\nwork alluded to, and the glorious character of the Agent.  The work is\nthe redemption of the world; the Agent is God, manifested in the flesh.\nHe who finished the creation of the heavens, and the earth in six days,\nis laying the foundation of a new creation on Calvary.  Four thousand\nyears He has been giving notice of His intention to mankind; more than\nthirty years He has been personally upon earth, preparing the material;\nand now He lays the chief corner-stone in Zion, exclaiming\u2014\"It is\nfinished.\"\n\nWe will consider the special import of the exclamation, and then offer a\nfew remarks of a more general character.\n\nI.  \"It is finished.\"  This saying of the Son of God is a very striking\none; and, uttered, as it was, while He hung in dying agonies on the\ncross, cannot fail to make a strong impression upon the mind.  It is\nnatural for us to inquire\u2014\"What does it mean?  To what does the glorious\nVictim refer?\"  A complete answer to the question would develope the\nwhole scheme of redemption.  We can only glance at a few leading ideas.\n\nThe sufferings of Christ are ended.  Never again shall He be persecuted\nfrom city to city, as an impostor, and servant of Satan.  Never again\nshall He say, \"My soul is exceeding sorrowful, even unto death.\"  Never\nagain shall He agonize in Gethsemane, and sweat great drops of blood.\nNever again shall He be derided by the rabble, and insulted by men in\npower.  Never again shall He be crowned with thorns, lacerated by the\nscourge, and nailed to the accursed tree.  Never again shall He cry out,\nin the anguish of His soul, and the baptism of blood\u2014\"My God! my God! why\nhast Thou forsaken me!\"\n\nThe predictions of His death are fulfilled.  The prophets had spoken of\nHis crucifixion many hundred years before His birth.  They foresaw the\nGovernor who was to come forth from Bethlehem.  They knew the Babe in the\nmanger, as He whose goings forth are of old, even from everlasting.  They\ndrew an accurate chart of His travels, from the manger to the cross, and\nfrom the cross to the throne.  All these things must be fulfilled.  Jesus\nknew the necessity, and seemed anxious that every jot, and tittle should\nreceive an exact accomplishment.  His whole life was a fulfilment of\nprophecy.  On every path He walked, on every house He entered, on every\ncity He visited, and especially on the mysterious phenomena which\naccompanied His crucifixion, it was written\u2014\"that the Scriptures might be\nfulfilled.\"\n\nThe great sacrifice for sin is accomplished.  For this purpose Christ\ncame into the world.  He is our appointed High Priest, the elect of the\nFather, and the desire of the nations.  He alone was in the bosom of the\nFather, and could offer a sacrifice of sufficient merit to atone for\nhuman transgression.  But it was necessary also that He should have\nsomewhat to offer.  Therefore a body was prepared for Him.  He assumed\nthe seed of Abraham, and suffered in the flesh.  This was a sacrifice of\ninfinite value, being sanctified by the altar of Divinity on which it was\noffered.  All the ceremonial sacrifices could not obtain the bond from\nthe hand of the creditor.  They were only acknowledgment of the debt.\nBut Jesus, by one offering, paid the whole, took up the bond, the\nhand-writing that was against us, and nailed it to the cross; and when\ndriving the last nail, He cried\u2014\"It is finished!\"\n\nThe satisfaction of Divine justice is completed.  The violated law must\nbe vindicated; the deserved penalty must be endured; if not by the sinner\nhimself, yet by the sinner's Substitute.  This was the great undertaking\nof the Son of God.  He \"bore our sins\"\u2014that is, the punishment of our\nsins\u2014\"in His own body on the tree.\"  He was \"made a curse for us, that we\nmight be made the righteousness of God in him.\"  There was no other way\nby which the honour of God and the dignity of His law could be sustained,\nand therefore \"the Lord laid upon Him the iniquities of us all.\"  He\n\"died unto sin once;\" not merely for sin, enduring its punishment in our\nstead; but also \"unto sin,\" abolishing its power, and putting it away.\nTherefore it is said, He \"made an end of sin\"\u2014destroyed its condemning,\nand tormenting power on behalf of all them that believe His sufferings\nwere equal to the claims of justice; and His dying cry was the voice of\nJustice Himself proclaiming the satisfaction.  Here, then, may the dying\nthief, and the persecutor of the holy, lay down their load of guilt, and\nwoe at the foot of the cross.\n\nThe new, and living way to God is consecrated.  A veil has hitherto\nconcealed the holy of holies.  None but the High Priest has seen the ark\nof the covenant, and the glory of God resting upon the Mercy-seat between\nthe cherubim.  He alone might enter, and he but once a year, and then\nwith fear, and trembling, and the sprinkling of atoning blood, after the\nmost careful purification, and sacrifice for himself.  He has filled His\nhands with His own blood, and entered into heaven itself, there to appear\nin the presence of God for us.  The sweet incense which He offers fills\nthe temple, and the merit of His sacrifice remains the same through all\ntime, superseding all other offerings for ever.  Therefore we are\nexhorted to come boldly to the throne of grace.  The tunnel under the\nThames could not be completed on account of an accident which greatly\ndamaged the work, without a new subscription for raising money; but Jesus\nfound infinite riches in Himself, sufficient for the completion of a new\nway to the Father\u2014a living way through the valley of the shadow of death\nto \"the city of the Great King.\"\n\nThe conquest of the powers of darkness is achieved.  When their hour was\ncome, the prince and his host were on the alert to accomplish the\ndestruction of the Son of God.  They hailed Him with peculiar\ntemptations, and levelled against Him their heaviest artillery.  They\ninstigated one disciple to betray Him and another to deny Him.  They\nfired the rage of the multitude against Him, so that the same tongues\nthat lately sang, \"Hosanna to the Son of David!\" now shouted, \"Crucify\nHim! crucify Him!\"  They filled the priests, and scribes with envy, that\nthey might accuse Him without a cause; and inspired Pilate with an\naccursed ambition, that he might condemn him without a fault.  They\nseared the conscience of the false witnesses, that they might charge the\nJust One with the most flagrant crimes; and cauterized the hearts of the\nRoman soldiers, that they might mock Him in His sufferings, and nail Him\nto the cross.  Having succeeded so far in their hellish plot, they\ndoubtless deemed their victory certain.  I see them crowding around the\ncross, waiting impatiently to witness his last breath, ready to shout\nwith infernal triumph to the depths of hell, till the brazen walls should\nsend back their echoes to the gates of the heavenly city.  But hark! the\ndying Saviour exclaims\u2014\"It is finished!\" and the great dragon and his\nhost retreat, howling, from the cross.  The Prince of our Salvation\nturned back all their artillery upon themselves, and their own stratagems\nbecame their ruin.  The old serpent seized Messiah's heel, but Messiah\nstamped upon the serpent's head.  The dying cry of Jesus shook the\ndominions of death, so that the bodies of many that slept arose; and rang\nthrough all the depths of hell the knell of its departed power.  Thus the\nPrince of this world was foiled in His schemes, and disappointed in his\nhopes, like the men of Gaza, when they locked up Samson at night,\nthinking to kill him in the morning: but awoke to find that he was gone,\nwith the gates of the city upon his shoulders.  When the Philistines\ncaught Samson, and brought him to their Temple, to make sport for them,\nthey never dreamed of the disaster in which it would result\u2014never dreamed\nthat their triumph over the poor blind captive would be the occasion of\ntheir destruction.  \"Suffer me,\" said he, \"to lean on the two pillars.\"\nThen he bowed himself, and died with his enemies.  So Christ on Calvary,\nwhile the powers of darkness exulted over their victim, seized the main\npillars of sin, and death, and brought down the temple of Satan upon its\noccupants; but on the morning of the third day, He left them all in the\nruins, where they shall remain for ever, and commenced His journey home\nto His Father's house.\n\nII.  So much concerning the import of our Saviour's exclamation.  Such\nwas the work He finished upon the cross.  We add a few remarks of a more\ngeneral character.\n\nThe sufferings of Christ were vicarious.  He died, not for His own sins,\nbut for ours.  He humbled Himself, that we might be exalted.  He became\npoor, that we might be made rich.  He was wounded, that we might be\nhealed.  He drained the cup of wrath, that we might drink the waters of\nsalvation.  He died the shameful and excruciating death of the cross,\nthat we might live and reign with Him for ever.\n\n\"Ought not Christ to have suffered these things, and to have entered into\nHis glory?\"  This \"ought\" is the ought of mercy, and of covenant\nengagement.  He must discharge the obligation which He had voluntarily\nassumed.  He must finish the work which He had graciously begun.  There\nwas no other Saviour\u2014no other being in the universe willing to undertake\nthe work; or, if any willing to undertake, none able to accomplish it.\nThe salvation of one human soul would have been too mighty an achievement\nfor Gabriel\u2014for all the angels in heaven.  Had not \"the only-begotten of\nthe Father\" become our Surety, we must have lain for ever under the wrath\nof God, amid \"weeping, and wailing, and gnashing of teeth.\"  None but the\nLion of the tribe of Judah could break the seals of that mysterious book.\nNone but \"God manifest in the flesh\" could deliver us from the second\ndeath.\n\nThe dying cry of Jesus indicates the dignity of His nature, and the power\nof life that was in Him to the last.  All men die of weakness\u2014of\ninability to resist death\u2014die because they can live no longer.  But this\nwas not the case with the Son of God.  He speaks of laying down His life\nas His own voluntary act;\u2014\"No man taketh it from He, but I lie it down of\nmyself.  I have power to lay it down, and I have power to take it again.\"\n\"He poured out His soul unto death\"\u2014did not wait for it to be torn from\nHim\u2014did not hang languishing upon the cross, till life \"ebbed out by slow\ndegrees;\" but poured it out freely, suddenly, and unexpectedly.  As soon\nas the work was done for which He came into the world, He cried\u2014\"It is\nfinished!\" \"bowed His head, and gave up the ghost.\"  Then the sun was\ndarkened, the earth quaked, the rocks rent, the graves opened, and the\ncenturion said\u2014\"Truly, this Man was the Son of God!\"  He cried with a\nloud voice, to show that He was still unconquered by pain, mighty even\nupon the cross.  He bowed His head that death might seize Him.  He was\nnaturally far above the reach of death, His Divine nature being\nself-existent and eternal, and His human nature entitled to immortality\nby its immaculate holiness; yet \"He humbled Himself, and became obedient\nunto death, even the death of the cross\"\u2014\"He bowed His head, and gave up\nthe ghost.\"\n\nWe may regard this last exclamation, also, as an expression of His joy at\nhaving accomplished the great \"travail of his soul,\" in the work of our\nredemption.  It was the work which the Father had given Him, and which He\nhad covenanted to do.  It lay heavy upon His heart, and oh, how was He\nstraitened till it was accomplished!  His \"soul was exceedingly\nsorrowful, even unto death;\" \"and His sweat, as it were, great drops of\nblood, falling down to the ground.\"  But upon the cross, He saw of the\ntravail of His soul, and was satisfied.  He saw that His sacrifice was\naccepted, and the object of His agony secured\u2014that death would not be\nable to detain Him in the grave, nor hell to defeat the purpose of His\ngrace; that the gates of the eternal city would soon open to receive Him\nas a conqueror, and myriads of exultant angels shout Him to His throne;\nwhither He would be followed by His redeemed, with songs of everlasting\njoy.  He saw, and He was satisfied; and, not waiting for the morning of\nthe third day, but already confident of victory, He uttered this note of\ntriumph, and died.\n\nAnd if we may suppose them to have understood its import, what a source\nof consolation it must have been to His sorrowing disciples!  The sword\nhad pierced through Mary's heart, according to the prediction of old\nSimeon over the infant Jesus.  Her affections had bled at the agony of\nher supernatural Son, and her wounded faith had well-nigh perished at His\ncross.  And how must all His followers have felt, standing afar off, and\nbeholding their supposed Redeemer suffering as a malefactor!  How must\nall their hopes have died within them, as they gazed on the accursed\ntree!  The tragedy was mysterious, and they deemed their enemies\nvictorious.  Jesus is treading the winepress in Bozrah, and the earth is\nshaking, and the rocks are rending, and the luminaries of heaven are\nexpiring, and all the powers of nature are fainting, in sympathy with His\nmighty agony.  Now he is lost in the fire, and smoke of battle, and the\ndread artillery of justice is heard thundering through the thick\ndarkness, and shouts of victory rise from the troops of hell, and who\nshall foretell the issue of the combat, or the fate of the Champion?  But\nlo!  He cometh forth from the cloud of battle, with blood upon His\ngarments!  He is wounded, but He hath the tread, and the aspect of a\nconqueror.  He waves His crimsoned sword, and cries\u2014\"It is finished!\"\nCourage, ye weepers at the cross!  Courage, ye tremblers afar off!  The\nPrince of your salvation is victor, and this bulletin of the war shall\ncheer myriads of believers in the house of their pilgrimage, and the\nachievement which it announces shall constitute an everlasting theme of\npraise.\n\n\"It is finished!\"  The word smote on the walls of the celestial city, and\nthrilled the hosts of heaven with ecstasy unspeakable.  How must \"the\nspirits of just men made perfect\" have leaped for joy, to hear that the\nCaptain of their salvation was victorious over all His enemies, and that\nthe work He had engaged to do for them, and their brethren was completed!\nAnd with what wonder, and delight must the holy angels have witnessed the\ntriumph of Him, whom they were commanded to worship, over the powers of\ndarkness!  It was the commencement of a new era in heaven, and never\nbefore had its happy denizens seen so much of God.\n\n\"It is finished!\"  Go, ye heralds of salvation, into all the world, and\nproclaim the joyful tidings!  Cry aloud, and spare not; lift up your\nvoice like a trumpet, and publish, to all men, that the work of the cross\nis finished\u2014that the Great Mediator, \"made perfect through sufferings,\"\nhas become \"the author of eternal salvation to all them that obey\nHim\"\u2014\"is of God made unto us, wisdom, and righteousness, and\nsanctification, and redemption!\"  Go, teach the degraded pagan, the\ndeluded Mohammedan, and the superstitious <DW7>, that the finished work\nof Jesus is the only way of acceptance with God.  Go, tell the polished\nscholar, the profound philosopher, and the vaunting moralist, that the\ndoctrine of Christ crucified is the only knowledge that can save the\nsoul!  Go,\u2014say to the proud sceptic, the bold blasphemer, and the\npolluted libertine, \"Behold the Lamb of God that taketh away the sin of\nthe world.\"  Preach it to the gasping sinner upon the death-bed, and the\nsullen murderer in his cell!  Let it ring in every human ear, and thrill\nin every human heart, till the gladness of earth shall be the counterpart\nof heaven!\n\n\n\nSERMON IV.\nTHE FATHER AND SON GLORIFIED.\n\n\n    \"_Howbeit_, _when He_, _the Spirit of Truth_, _is come_, _He will\n    guide you into all truth_; _for He shall not speak of Himself_; _but\n    whatsoever He shall hear_, _that shall He speak_; _and He will show\n    you things to come_.  _He shall glorify me_: _for He shall receive of\n    mine_, _and shall show it unto you_.  _All things that the Father\n    hath are mine_; _therefore_, _said I_, _that He shall take of mine_,\n    _and shall show it unto you_.\"\u2014JOHN xvi. 13\u201315.\n\nThe wonderful Providence, which brought the children of Israel out of the\nhouse of bondage, was a chain of many links, not one of which could be\nomitted without destroying the beauty, and defeating the end of the\nDivine economy.  The family of Jacob came to Egypt in the time of\nfamine\u2014they multiply\u2014they are oppressed\u2014their cries reach to heaven\u2014God\nmanifests Himself in the burning bush\u2014Moses is sent to Egypt\u2014miracles are\nwrought by his hand\u2014Pharaoh's heart is hardened\u2014the firstborn are\nslain\u2014the passover is eaten\u2014the people depart, led by the pillar of\nGod\u2014the sea is divided\u2014and, with many signs, and wonders, the thousands\nof Israel are conducted through the wilderness to the Promised Land.  Had\none of these links been wanting, the chain of deliverance had been\ndefective.\n\nSo, in the salvation of sinners by Jesus Christ, all the conditions, and\npreparatives were essential to the completeness, and glory of the scheme.\nThe Son of God must consent to undertake our cause, and become our\nsubstitute\u2014the promise must be given to Adam, and frequently repeated to\nthe patriarchs\u2014bloody sacrifices must be instituted, to typify the\nvicarious sufferings of Messiah\u2014a long line of prophets must foretell His\nadvent, and the glory of His kingdom\u2014He must be born in Bethlehem,\ncrucified on Calvary, and buried in Joseph's new tomb\u2014must rise from the\ndead, ascend to the right hand of the Father, and send down the Holy\nSpirit to guide and sanctify His Church.  Without all these\ncircumstances, the economy of redemption would have been incomplete and\ninefficient.\n\nThe last link in the chain is the mission and work of the Holy Spirit.\nThis is quite as important as any of the rest.  Our Saviour's heart seems\nto have been much set upon it, during all His ministry, and especially\nduring the last few days, before His crucifixion.  He spoke of it,\nfrequently, to His disciples, and told them that He would not leave them\ncomfortless, but would send them \"another Comforter,\" who should abide\nwith them for ever; and that His own departure was necessary, to prepare\nthe way for the coming of the heavenly Paraclete.  In our text, He\ndescribes the office of the Holy Spirit, and the specific relation which\nHe sustains to the work of Salvation:\u2014\"Howbeit, when He, the Spirit of\nTruth, is come, He will guide you into all truth; for He shall not speak\nof Himself; but whatsoever He shall hear, that shall He speak; and He\nwill show you things to come.  He shall glorify me: for He shall receive\nof mine, and shall show it unto you.  All things that the Father hath are\nmine; therefore said I, that He shall take of mine, and shall show it\nunto you.\"\n\nThese words teach us two important truths\u2014_first_, that the Son is equal\nwith the Father; and, _secondly_, that the Father, and the Son are alike\nglorified in the economy of salvation.\n\nI.  The Son claims equality with the Father.  \"All things that the Father\nhath are mine.\"\n\nThis sentence is very comprehensive, and sublime\u2014an unquestionable\naffirmation of the Messiah's \"eternal power, and Godhead.\"  The same\ndoctrine is taught us, in many other recorded sayings of Christ, and\nsustained by all the prophets, and apostles; and when I consider this\ndeclaration, in connection with the general strain of the inspired\nwriters on the subject, I seem to hear the Saviour Himself addressing the\nworld in the following manner:\u2014\n\n\"All things that the Father hath are mine.  His _names_ are mine.  I am\nJehovah\u2014the mighty God, and the everlasting Father\u2014the Lord of Hosts\u2014the\nLiving God\u2014the True God, and Eternal Life.\n\n\"His _works_ are mine.  All things were made by me, and I uphold all\nthings by the word of my power.  My Father worketh hitherto, and I work;\nfor as the Father raiseth up the dead, and quickeneth them, even so the\nSon quickeneth whom He will.  I am the Author of universal being, and my\nhand moveth all the machinery of Providence.\n\n\"His _honours_ are mine.  I have an indisputable right to the homage of\nall created intelligences.  I inhabit the praises of Eternity.  Before\nthe foundation of the world, I was the object of angelic adoration; and\nwhen I became incarnate as a Saviour, the Father published His decree in\nheaven, saying\u2014'Let all the angels of God worship Him!'  It is His will,\nalso, that all men should honour the Son, even as they honour the\nFather\u2014in the same manner, and the same degree.  He that honoureth the\nSon, honoureth the Father; and he that honoureth not the Son, honoureth\nnot the Father: for I and my Father are one\u2014one in honour\u2014possessing\njoint interest, and authority.\n\n\"His _attributes_ are mine.  Though as man, and Mediator I am inferior to\nthe Father; yet my nature is no more inferior to His, than the nature of\nthe Prince of Wales is inferior to the nature of the King of England.\nYou see me clothed in humanity; but, in my original state, I thought it\nnot robbery to be equal with God.  I was in the beginning with God, and\npossessed the same eternity of being.  Like Him, I am almighty,\nomniscient, and immutable; infinite in holiness, justice, goodness, and\ntruth.  All these attributes, with every other possible perfection,\nbelong to me, in the same sense as they belong to the Father.  They are\nabsolute, and independent, underived, and unoriginated\u2014the essential\nqualities of my nature.\n\n\"His _riches of grace_ are mine.  I am the Mediator of the new\ncovenant\u2014the Channel of my Father's mercies to mankind.  I have the keys\nof the House of David, and the seal of the Kingdom of Heaven.  I have\ncome from the bosom of the Father, freighted with the precious treasures\nof His good will to men.  I have sailed over the sea of tribulation, and\ndeath, to bring you the wealth of the other world.  I am the Father's\nMessenger, publishing peace on earth\u2014a peace which I have purchased with\nmy own blood upon the cross.  It has pleased the Father that in me all\nfulness should dwell\u2014all fulness of wisdom, and grace\u2014whatever is\nnecessary for the justification, sanctification, and redemption of them\nthat believe.  My Father, and I are one, in the work of salvation, as in\nthe work of creation.  We have the same will, and the same intention of\nmercy toward the children of the great captivity.\n\n\"The _objects of His love_ are mine.  He hath given them to me in an\neverlasting covenant.  He hath given me the heathen for an inheritance,\nand the uttermost parts of the earth for a possession.  They were mine by\nthe original right of creation; but now they are doubly mine, by the\nsuperadded claim of redemption.  My Father, before the world was, gave me\na charter of all the souls I would redeem.  I have fulfilled the\ncondition.  I have poured out my soul unto death, and sealed the covenant\nwith the blood of my cross.  Therefore, all believers are mine.  I have\nbought them with a price.  I have redeemed them from the bondage of sin,\nand death.  Their names are engraven on my hands, and my feet.  They are\nwritten with the soldier's spear upon my heart.  And of all that the\nFather hath given me, I will lose nothing.  I will draw them all to\nmyself; I will raise them up at the last day; and they shall be with me\nwhere I am, that they may behold my glory, which I had with the Father\nbefore the foundation of the world.\"\n\nII.  The Father and the Son are equally glorified in the economy of\nredemption, and the work of the Holy Spirit.\n\n1.  The Son glorifies the Father.  I hear Him praying in the\ngarden:\u2014\"Father, I have glorified Thee on earth; I have finished the work\nwhich Thou gavest me to do.\"  I hear Him, again, amidst the supernatural\ngloom of Calvary, with a voice that rings through the dominions of death,\nand hell, crying\u2014\"It is finished!\"\n\nWhat mighty achievement hast Thou finished to-day, blessed Jesus? and how\nhave Thine unknown agony, and shameful death glorified the Father?\n\n\"I have glorified the Father, by raising up those precious things which\nfell in Eden, and were lost in the abyss.\n\n\"I have raised up my Father's _law_.  I found it cast down to the earth,\nand trampled into the dust.  I have magnified, and found it honourable.\nI have vindicated its authority in the sight of men, and angels.  I have\nsatisfied its demands on behalf of my redeemed, and become the end of the\nlaw for righteousness to all who will receive me as their surety.\n\n\"I have raised up my Father's _name_.  I have declared it to my brethren.\nI have manifested it to the men whom He has given me.  I have given a new\nrevelation of His character to the world.  I have shown Him to sinners,\nas a just God, and a Saviour.  I have restored His worship in purity, and\nspiritually upon earth.  I have opened a new, and living way to His\nthrone of grace.  I have written the record of His mercy with my own\nblood upon the rocks of Calvary.\n\n\"I have raised up my Father's _image_.  I have imprinted it afresh upon\nhuman nature, from which it was effaced by sin.  I have displayed its\nexcellence in my own character.  I have passed through the pollutions of\nthe world, and the territory of death, without tarnishing its lustre, or\ninjuring its symmetry.  Though my visage is marred with grief, and my\nback ploughed with scourges, and my hands, and feet nailed to the\naccursed cross, not one trace of my Father's image has been obliterated\nfrom my human soul.  It is as perfect, and as spotless now as when I lay\nin the manger.  I will carry it unstained with me into heaven.  I will\ngive a full description of it in my Gospel upon earth.  I will change my\npeople into the same image, from glory, to glory.  I will also renovate,\nand transform their vile bodies, and fashion them like unto my own\nglorious body.  I will ransom them from the power of the grave; and\nbecause I live, they shall live also\u2014the counterpart of my own immaculate\nhumanity\u2014mirrors to reflect my Father's glory for ever.\"\n\n2.  The Father glorifies the Son.  He prayed in the garden,\u2014\"And now,\nFather, glorify Thou me with Thine own self, with the glory which I had\nwith Thee before the world was.\"  Was the petition granted?  Answer, ye\nRoman sentinels, who watched His sepulchre!  Answer, ye men of Galilee,\nwho gazed upon His chariot, as He ascended from the mount of Olives!\n\nThe glorification of the Son by the Father implies all the honours of His\nmediatorial office\u2014all the crowns which He won by His victory over the\npowers of death, and hell.  The Father raised Him from the dead, and\nreceived Him up into glory, as a testimony of His acceptance as the\nsinner's Surety\u2014an expression of perfect satisfaction with His vicarious\nsacrifice upon the cross.  It was the just reward of His work; it was the\nfruit of His gracious travail.  He is \"crowned with glory and honour for\nthe sufferings of death.\"  \"Because He hath poured out His soul unto\ndeath,\" therefore \"God also hath highly exalted Him, and given Him a name\nthat is above every name.\"\n\nWhat an honour would it be to a man, to receive eight, or ten of the\nhighest offices in the kingdom!  Infinitely greater is the glory of\nEmmanuel.  His name includes all the offices, and titles of the kingdom\nof heaven.  The Father hath made Him \"both Lord, and Christ\"\u2014that is,\ngiven Him the supreme prerogatives of government and salvation.  \"Him\nhath God exalted to be a prince and a Saviour, to give repentance to\nIsrael, and remission of sins.\"  He is \"head over all things in the\nChurch\"\u2014Prime Minister in the kingdom of heaven\u2014Lord Treasurer,\ndispensing the bounties of Divine grace to mankind\u2014Lord High-Chancellor\nof the Realm, and Keeper of the great Seal of the living God; holding in\nHis hand the charter of our redemption, and certifying the authenticity\nof the Divine covenant\u2014Lord Chief Justice of heaven, and earth, having\nall power, and authority to administer the laws of Providence throughout\nthe universe\u2014the chief Prince\u2014the General of the army\u2014the Captain of the\nLord's host\u2014the Champion who conquered Satan, sin, and death; bruising\nthe head of the first, destroying the power of the second, and swallowing\nup the third in victory.  He hath the keys of hell, and of death.  He\nshutteth, and no man openeth; He openeth, and no man shutteth.  He bears\nall the honours of His Father's house; and concentrates in Himself all\nthe glories of Supreme Divinity, redeemed humanity, and \"mediator between\nGod, and man.\"\n\n3.  The Holy Spirit glorifies Father and Son together.  He is procured\nfor the world by the blood of the Son, and sent into the world by the\nauthority of the Father; so that both are alike represented in His\nmission, and equally glorified in His office.  The gracious things which\nthe Father gave into the hands of the Son, when He descended from heaven,\nthe Son gave into the hands of the Spirit, when He returned to heaven.\n\"All things that the Father hath are mine; and He shall take of mine, and\nshall show it unto you.\"\n\nThis is the object of the Spirit's advent, the communication of the\nthings of Christ to men.  What are the things of Christ?  His merit, His\nmercy, His image, His Gospel, His promises, all the gifts of His grace,\nall the treasures of His love, and all the immunities of eternal\nredemption.  These the Father hath given to the Son, as the great Trustee\nof the Church; and the Son hath given them to the Spirit, as the\nappointed Agent of their communication.\n\nA ship was laden in India, arrived safely in London, unloaded her\nprecious cargo, and the goods were soon distributed all over the country,\nand offered for sale in a thousand stores.  The Son of God brought\nimmense riches of Divine grace from heaven to earth, which are all left\nto the disposal of the Holy Spirit, and freely proffered to the\nperishing, wherever the Gospel is preached.\n\nThe Holy Spirit came, not to construct a new engine of mercy, but to\npropel that already constructed by Christ.  Its first revolution rent the\nrocks of Calvary, and shook the rocky hearts of men.  Its second\nrevolution demolished the throne of death, burst his prison-doors, and\nliberated many of his captives.  Its third revolution carried its builder\nup into the Heaven of heavens, and brought down the Holy Spirit to move\nits machinery for ever.  Its next revolution, under the impulse of this\nnew Agent, was like \"the rushing of a mighty wind\" among the assembled\ndisciples at Jerusalem, kindled a fire upon the head of every Christian,\ninspired them to speak all the languages of the babbling earth, and\nkilled, and quickened three thousand souls of the hearers.\n\nThe Holy Spirit is still on earth, glorifying the Father, and the Son.\nHe convinces the world of sin.  He leads men to Christ, through the\nrivers of corruption, the mountains of presumption, and the terrible bogs\nof despair, affording them no rest till they come to the city of refuge.\nHe continues on the field to bring up the rear; while the Captain of our\nSalvation, on His white horse, rides victorious in the van of battle.  He\nstrengthens the soldiers\u2014\"faint, yet pursuing!\" raises the fallen;\nencourages the despondent; feeds them with the bread of life, and the new\nwine of the kingdom; and leads them on\u2014\"conquering and to conquer.\"\n\nHis work will not be finished till the resurrection.  Then will He\nquicken our mortal bodies.  Then will He light His candle, and sweep the\nhouse till He find every lost piece of silver.  Then will He descend into\nthe dark caves of death, and gather all the gems of redeemed humanity,\nand weave them into a crown for Emmanuel, and place that crown upon\nEmmanuel's head, amid the songs of the adoring seraphim!\n\nThus the Holy Spirit glorifies the Father, and the Son.  Let us pray for\nthe outpouring of His grace upon the Church.  In proportion to His\nmanifestation in our hearts, will be our \"knowledge of the light of the\nglory of God in the face of Jesus Christ.\"  Nor is this all; in\nproportion to the visitations of the Holy Spirit, will be the purity of\nour lives, the spirituality of our worship, the ardour of our zeal, and\ncharity, and the extent of our usefulness to the cause of Christ.  Would\nyou see a revival of religion? pray for the outpouring of the Holy Spirit\nupon you, to sanctify your hearts, and lives, that your light may \"so\nshine before men, that others may see your good works, and glorify your\nFather who is in heaven.\"\n\n\"When thou hearest the sound of a going in the tops of the mulberry\ntrees, then thou shalt bestir thyself; for then the Lord shall go out\nbefore thee, to strike the hosts of the Philistines.\"  Brethren, this is\nthe time.  The mulberry trees are shaking.  God is going before His\npeople, to prepare their way to victory.  The hand of Divine Providence\nis opening a great, and effectual door for the Gospel.  The mountains are\nlevelled, the valleys are exalted, and a highway is cast up in the\nwilderness for our God.  The arts of printing, and navigation, the\nincreasing commerce of the world, the general prevalence of the Spirit of\npeace, the rapid march of literature and science, and the correspondence\nof eminent and leading men in every nation, are so many preparatives for\nthe moral conquest of the world.  The Captain of our Salvation, on the\nwhite horse of the Gospel, can now ride through Europe and America: and\nwill soon lead forth His army, to take possession of Asia, and Africa.\nThe wings of the mighty angel are unbound, and he is flying in the midst\nof heaven.\n\nAgain: Christians are better informed concerning the moral state of the\nworld than formerly.  If my neighbour's house were on fire, and I knew\nnothing of it, I could not be blamed for rendering him no assistance; but\nwho could be guiltless in beholding the building in flames, without an\neffort to rescue its occupants?  Brethren, you have heard of the\nperishing heathen.  You have heard of their dreadful superstitions, their\nhuman sacrifices, and their abominable rites.  You have heard of\nJuggernaut, and the River Ganges, and the murder of infants, and the\nimmolation of widows, and the worship of idols, and demons.  You know\nsomething of the delusion of Mohammedanism, the cruel, and degrading\nignorance of Popery, and how millions around you are perishing for the\nlack of knowledge.  Do you feel no solicitude for their souls\u2014no desire\nto pluck them as brands from the burning?\n\nWhat can we do?  The Scriptures have been translated into nearly all the\nlanguages of the babbling earth.  Missionaries have gone into many\nlands\u2014have met the Indian in his wigwam, the African in his Devil's-bush,\nand the devotee on his way to Mecca.  We can furnish more men for the\nfield, and more money to sustain them.  But these things cannot change,\nand renovate the human heart.  \"Not by might, nor by power, but by my\nSpirit, saith the Lord.\"  This is the grand regenerating agency.  He\nalone can convince and save the world.  His aid is given in answer to\nprayer; and the Father is more ready to give than we are to ask.\n\nMr. Ward, one of the Baptist missionaries in India, in a missionary\ndiscourse at Bristol, said,\u2014\"Brethren, we need your money,\u2014we need your\nprayers more.\"  Oh, what encouragement we have to pray for our\nmissionaries!  Thus saith the Lord: \"I will pour water upon him that is\nthirsty, and floods upon the dry ground; I will pour out my Spirit upon\nthy seed, and my blessing upon thine offspring.\"  Let us plead with God\nfor the accomplishment of the promise, \"Ye that make mention of the Lord,\nkeep not silence, and give Him no rest till He make Jerusalem a praise in\nthe whole earth.\"\n\nBrethren in the ministry! let us remember that all our success depends\nupon the aid of the Holy Spirit, and let us pray constantly for His\nblessing upon the world!  Brethren in the Church! forget not the\nconnection between the work of the Holy Spirit and the glory of your Best\nFriend, and earnestly entreat Him to mingle His sanctifying unction with\nthe treasures of Divine Truth contained in these earthern vessels!\n\"Finally, Brethren, pray for us; that the Word of the Lord may have free\ncourse and be glorified; and all the ends of the earth see the salvation\nof our God!\"\n\n\n\nSERMON V.\nTHE CEDAR OF GOD.\n\n\n    \"_Thus saith the Lord God_: _I will also take of the highest branch\n    of the high cedar_, _and will set it_; _I will crop off from the top\n    of his young twigs a tender one_, _and plant it upon a high mountain\n    and eminent_; _in the mountain of the height of Israel will I plant\n    it_: _and it shall bring forth boughs_, _and bear fruit_, _and be a\n    goodly cedar_; _and under it shall dwell all fowl of every wing_; _in\n    the shadow of the branches thereof shall they dwell_; _and all the\n    trees of the field shall know that I_, _the Lord_, _have brought down\n    the high tree_, _and have exalted the low tree\u2014have dried up the\n    green tree_, _and have made the dry tree to flourish_.  _I_, _the\n    Lord_, _have spoken_, _and I have done it_.\"\u2014EZEKIEL xvii. 22\u201324.\n\nYou perceive that our text abounds in the beautiful language of allegory.\nIn the context is portrayed the captivity of the children of Israel, and\nespecially the carrying away of the royal family by the king of Babylon.\nHere God promises to restore them to their own land, in greater\nprosperity than ever; and to raise up Messiah, the Branch, out of the\nhouse of David, to be their king.  All this is presented in a glowing\nfigurative style, dressed out in all the wealth of poetic imagery so\npeculiar to the Orientals.  Nebuchadnezzar, the great eagle\u2014the\nlong-winged, full-feathered, embroidered eagle\u2014is represented as coming\nto Lebanon, and taking the highest branch of the tallest cedar, bearing\nit off as the crow bears the acorn in its beak, and planting it in the\nland of traffic.  The Lord God, in His turn, takes the highest branch of\nthe same cedar, and plants it on the high mountain of Israel, where it\nflourishes and bears fruit, and the fowls of the air dwell under the\nshadow of its branches.\n\nWe will make a few general remarks on the character of the promise, and\nthen pass to a more particular consideration of its import.\n\nI.  This is an _evangelical_ promise.  It relates to the coming and\nkingdom of Messiah.  Not one of the kings of Judah since the captivity,\nas Boothroyd well observes, answers to the description here given.  Not\none of them was a cedar whose branches could afford shadow, and shelter\nfor all the fowls of heaven.  But the prophecy receives its fulfilment in\nChrist, the Desire of all nations, to whom the ends of the earth shall\ncome for salvation.\n\nThis prophecy bears a striking resemblance, in several particulars, to\nthe parable of the mustard-seed, delivered by our Lord.  \"The\nmustard-seed,\" said Jesus, \"is the least of all seeds; but when it is\ngrown, it is the greatest among herbs, and becometh a tree, so that the\nbirds of the air come and lodge in the branches thereof.\"  So the\ndelicate twig of the young, and tender branch, becomes a goodly cedar,\nand under its shadow dwell all fowl, of every wing.  The prophecy, and\nthe parable are alike intended to represent the growth, and prosperity of\nMessiah's kingdom, and the gracious protection, and spiritual refreshment\nafforded to its subjects.  Christ is the mustard plant, and cedar of God;\nand to Him shall the gathering of all the people be; and multitudes of\npardoned sinners shall sit under His shadow, with great delight, and His\nfruit shall be sweet to their taste.\n\nThis prophecy is a promise of the true, and faithful, and immutable God.\nIt begins with\u2014\"Thus saith the Lord God, I will do thus and so;\" and\nconcludes with\u2014\"I, the Lord, have spoken, and I have done it.\"  There is\nno peradventure with God.  His Word is for ever settled in heaven, and\ncannot fail of its fulfilment.  When He says, \"I promise to pay,\" there\nis no failure, whatever the sum.  The Bank of grace cannot break.  It is\nthe oldest and best in the universe.  Its capital is infinite; its credit\nis infallible.  The mighty God, the Everlasting Father, the Prince of\nPeace, is able to fulfil, to the utmost, all His engagements.  He can do\nanything that does not imply a contradiction, or a moral absurdity.  He\ncould take upon Himself the form of a servant, and become obedient unto\ndeath, even the death of the cross; but we can never forget, or\ndisregard, His promise, any more than He can cease to exist.  His nature\nrenders both impossible.  Heaven, and earth shall pass away, but His word\nshall not pass away.  Every jot, and tittle shall be fulfilled.  This is\nthe consolation of the Church.  Here rested the patriarchs, and prophets.\nHere reposes the faith of the saints, to the end of time.  God abideth\nfaithful; He cannot deny Himself.  Our text is already partially verified\nin the advent of Christ, and the establishment of His Church; the\ncontinuous growth of the gospel kingdom indicates its progressive\nfulfilment; and we anticipate the time, as not far distant, when the\nwhole earth shall be overshadowed by the branches of the cedar of God.\n\nII.  We proceed to consider, with a little more particularity, the import\nof this evangelical prophecy.  It describes the character, and\nmediatorial kingdom of Christ, and the blessings which He confers upon\nHis people.\n\n1.  His character and mediatorial kingdom.\u2014\"I will take of the highest\nbranch of the high cedar, and will set it; I will crop off from the top\nof his young twigs a tender one, and plant it upon a high mountain and\neminent; in the mountain of the height of Israel will I plant it.\"\n\nChrist, as concerning the flesh, is of the seed of Abraham\u2014a rod issuing\nfrom the stem of Jesse, and a branch growing out of his root.  As the new\nvine is found in the cluster, and one saith, \"Destroy it not, for a\nblessing is in it,\" so the children of Israel were spared,\nnotwithstanding their perverseness, and their backslidings, because they\nwere the cluster from which should be expressed in due time the new wine\nof the kingdom\u2014because from them was to come forth the blessing, the\npromised seed, in whom all the families of the earth shall be blessed.\nThe Word that was in the beginning with God, one with God, in essence,\nand in attributes, in the fulness of time assumed our nature, and\ntabernacled, and dwelt among us.  Here is the union of God, and man.\nHere is the great mystery of godliness\u2014God manifest in the flesh.  But I\nhave only time now to take off my shoes, and draw near the burning bush,\nand gaze a moment upon this great sight.\n\nThe Father is represented as preparing a body, for His Son.  He goes to\nthe quarry to seek a stone, a foundation-stone, for Zion.  The angel said\nto Mary:\u2014\"The Holy Ghost shall come upon thee, and the power of the\nHighest shall overshadow thee; therefore that Holy Thing which shall be\nborn of thee shall be called the Son of God.\"  The Eternal lays hold on\nthat nature which is hastening downward, on the flood of sin, to the gulf\nof death, and destruction, and binds it to Himself.  Though made in the\nlikeness of sinful flesh, He was holy, harmless, and undefiled.  He did\nno iniquity, neither was guile found in His mouth.  The rod out of the\nstem of Jesse is also Jehovah, our righteousness.  The Child born in\nBethlehem is the mighty God.  The Son given to Israel is the Everlasting\nFather.  He is of the seed of Abraham, according to the flesh; but he is\nalso the true God, and eternal life.  Two natures, and three offices meet\nmysteriously in His Person.  He is at once the bleeding sacrifice, the\nsanctifying altar, the officiating priest, the prophet of Israel, and the\nPrince of Peace.  All this was necessary that He might become \"the Author\nof eternal salvation, to all them that obey Him.\"\n\nHear Jehovah speaking of Messiah and His kingdom:\u2014\"Why do the heathen\nrage, and the people imagine a vain thing?  The kings of the earth set\nthemselves, and the rulers take council together against the Lord, and\nagainst His anointed.  Yet have I set my King upon my holy hill of Zion.\nI will declare the decree by which He is to rule His redeemed empire.\"\nThat decree, long kept secret, was gradually announced by the prophets,\nbut at the new tomb of Joseph of Arimathea, Jehovah Himself proclaimed it\naloud, to the astonishment of earth, the terror of hell, and the joy of\nheaven:\u2014\"Thou art my Son; this day have I begotten Thee.  Come forth from\nthe womb of the grave, thou whose goings forth have been from of old,\neven from everlasting.  Ask of me, and I shall give thee the heathen for\nThine inheritance, and the uttermost parts of the earth for Thy\npossession.  I will exalt Thee to the throne of the universe, and thou\nshalt be chief in the chariot of the Gospel.  Thou shalt ride through the\ndark places of the earth, with the lamps of eternal life suspended to Thy\nchariot, enlightening the world.  Be wise, now, therefore, O ye kings; be\ninstructed, ye judges of the earth.  Serve the Lord with fear, and\nrejoice with trembling.  Kiss the Son, lest He be angry, and ye perish\nfrom the way when His wrath is kindled but a little.  Let no man\nwithstand Him.  Let no man seek to stay His progress.  Herod, Pilate,\nCaiaphas, stand off! clear the way! lest ye be crushed beneath the wheels\nof His chariot! for that which is a savour of life to some, is to others\na savour of death; and if this stone shall fall upon you, it shall grind\nyou to powder!\"\n\nBehold, here is wisdom!  All other mysteries are toys, in comparison with\nthe mystery of the everlasting gospel\u2014the union of three Persons in the\nGodhead\u2014the union of two natures in the Mediator\u2014the union of believers\nin Christ, as the branches to the vine\u2014the union of all the saints\ntogether in Him, who is the head of the body, and the chief stone of the\ncorner\u2014the mighty God transfixed to the cross\u2014the Son of Mary ruling in\nthe Heaven of heavens\u2014the rod of Jesse becoming the sceptre of universal\ndominion\u2014the Branch growing out of his root, the little delicate branch\nwhich a lamb might crop for its food, terrifying and taming the serpent,\nthe lion, the leopard, the tiger, and the wolf, and transforming into\ngentleness, and love, the wild, and savage nature of all the beasts of\nprey upon the mountain!  \"And such,\" old Corinthian sinners, \"were some\nof you; but ye are washed, ye are sanctified, ye are justified, in the\nname of the Lord Jesus, and by the Spirit of our God.\"  And such, my\nbrethren, were some of you; but ye have been made a new creation in Jesus\nChrist; old things are passed away, and all things are become new.  Ye\nare dead, and your life is hid with Christ in God.  He is one with the\nFather, and ye are one in Him; united and interwoven, like the roots of\nthe trees in the forest of Lebanon; so that none can injure the least\ndisciple of Christ, without touching the apple of His eye, and grieving\nall His members.\n\nII.  The blessings which He confers upon His people.  It shall bring\nforth boughs, and bear fruit, and be a goodly cedar, and under it shall\ndwell all fowl of every wing; in the shadow of the branches thereof shall\nthey dwell; and all the trees of the field shall know that I, the Lord,\nhave brought down the high tree, and have exalted the low tree\u2014have dried\nup the green tree, and have made the dry tree to flourish.\n\n_Christ is a fruitful tree_.  \"The tree is known by his fruit.  Men do\nnot gather grapes of thorns, nor figs of thistles.  Every good tree\nbringeth forth good fruit, and every evil tree bringeth forth evil\nfruit.\"  This is a singular, supernatural tree.  Though its top reaches\nto the Heaven of heavens, its branches fill the universe, and bend down\nto the earth, laden with the precious fruits of pardon, and holiness, and\neternal life.  On the day of Pentecost, we see them hang so low over\nJerusalem, that the very murderers of the Son of God reach, and pluck,\nand eat, and three thousand sinners feast on more than angels' food.\nThat was the feast of first-fruits.  Never before was there such a\nharvest and such a festival.  Angels know nothing of the delicious fruits\nof the tree of redemption.  They know nothing of the joy of pardon, and\nthe spirit of adoption.  The Bride of the Lamb alone can say:\u2014\"As the\napple-tree among the trees of the forest, so is my beloved among the\nsons.  I sat down under his shadow, with great delight, and his fruit was\nsweet, to my taste.  He brought me also to his banqueting-house, and his\nbanner over me was love.\"\n\nThese blessings are the precious effects of Christ's mediatorial work;\nflowing down to all believers, like streams of living water.  Come, ye\nfamishing souls, and take, without money, and without price.  All things\nare now ready.  \"The mandrakes give a smell, and at our gates are all\nmanner of pleasant fruits, both new, and old.\"  Here is no scarcity.  Our\nElder Brother keeps a rich table in our Father's house.  Hear Him\nproclaiming in the streets of the city, in the chief places of\nconcourse:\u2014\"Come to the festival.  There is bread enough, and to spare.\nMy oxen, and my fatlings are killed.  My board is spread with the most\ndelicious delicacies\u2014wine on the lees well refined, and fruits such as\nangels never tasted.\"\n\n_Christ is a tree of protection to His people_.  This cedar not only\nbeautifies the forest, but also affords shade, and shelter for the fowls\nof the air.  We have the same idea in the parable of the mustard-seed,\n\"The birds of the air came and lodged in the branches thereof.\"  This is\nthe fulfilment of the promise concerning Shiloh, \"To Him shall the\ngathering of the people be.\"  It is the drawing of sinners to Christ, and\nthe union of believers with God.  \"All fowl of every wing.\"  Sinners of\nevery age, and every degree\u2014sinners of all languages, colours, and\nclimes\u2014sinners of all principles, customs, and habits\u2014sinners whose\ncrimes are of the blackest hue\u2014sinners carrying about them the savour of\nthe brimstone of hell\u2014sinners deserving eternal damnation\u2014sinners\nperishing for lack of knowledge\u2014sinners pierced by the arrows of\nconviction\u2014sinners ready to sink under the burden of sin\u2014sinners\noverwhelmed with terror and despair\u2014are seen flying to Christ as a cloud,\nand as doves to their windows\u2014moving to the ark of mercy before the door\nis shut\u2014seeking rest in the shadow of this goodly cedar!\n\nChrist is the sure defence of His Church.  A thousand times has she been\nassailed by her enemies.  The princes of the earth have set themselves in\narray against her, and hell has opened upon her all its batteries.  But\nthe Rock of Ages has ever been her strong fortress, and high tower.  He\nwill never refuse to shelter her from her adversaries.  In the time of\ntrouble He shall hide her in His pavilion; in the secret of His\ntabernacle shall He hide her.  When the heavens are dark, and angry, she\nflies, like the affrighted dove, to the thick branches of the \"Goodly\nCedar.\"  There she is safe from the windy storm, and tempest.  There she\nmay rest in confidence, till these calamities be overpast.  The tree of\nher protection can never be riven by the lightning, nor broken by the\nblast.\n\n_Christ is the source of life_, _and beauty to all the trees in the\ngarden of God_.  Jehovah determined to teach \"the trees of the forest\" a\nnew lesson.  Let the princes of this world hear it, and the proud\nphilosophers of Greece and Rome.  \"I have brought down the high tree, and\nexalted the low tree\u2014I have dried up the green tree, and made the dry\ntree to flourish.\"  Many things have occurred, in the providence of God,\nwhich might illustrate these metaphors; such as the bringing of Pharaoh\ndown to the bottom of the sea, that Israel might be exalted to sing the\nsong of Moses; and the drying up of the pride, and pomp of Haman, that\nMordecai might flourish in honour, and esteem.  But for the most\ntranscendent accomplishment of the prophecy, we must go to Calvary.\nThere is the high tree, brought down to the dust of death, that the low\ntree might be exalted to life eternal; the green tree dried up by the\nfires of Divine wrath, that the dry tree might flourish in the favour of\nGod for ever.\n\nTo this, particularly, our blessed Redeemer seems to refer, in His\naddress to the daughters of Jerusalem, as they follow Him, weeping, to\nthe place of crucifixion.  \"Weep not for me,\" saith He.  \"There is a\nmystery in all this, which you cannot now comprehend.  Like Joseph, I\nhave been sold by my brethren; but like Joseph, I will be a blessing to\nall my Father's house.  I am carrying this cross to Calvary, that I may\nbe crucified upon it between two thieves; but when the lid of the\nmystical ark shall be lifted, then shall ye see that it is to save\nsinners I give my back to the smiters, and my life for a sacrifice.  Weep\nnot for me, but for yourselves, and your children; for if they do these\nthings in the green tree, what shall be done in the dry?  I am the green\ntree to-day; and, behold, I am consumed, that you may flourish.  I am the\nhigh tree, and am prostrated that you may be exalted.\"\n\nThe fire-brands of Jerusalem had well-nigh kindled to a flame of\nthemselves, amid the tumult of the people, when they cried out, \"Away\nwith Him!  Crucify Him!  His blood be on us, and on our children!\"  O\nwonder of mercy! that they were not seized and consumed at once by fire\nfrom heaven!  But He whom they crucify prays for them, and they are\nspared.  Hear His intercession:\u2014\"Father, forgive them! save these\nsinners, ready for the fire.  On me, on me alone, be the fierceness of\nThy indignation.  I am ready to drink the cup which Thou hast mingled, I\nam willing to fall beneath the stroke of Thy angry justice.  I come to\nsuffer for the guilty.  Bind me in their stead, lay me upon the altar,\nand send down fire to consume the Sacrifice!\"\n\nIt was done.  I heard a great voice from heaven:\u2014\"Awake, O sword, against\nmy Shepherd!  Kindle the flame!  Let off the artillery!\"  Night suddenly\nenveloped the earth.  Nature trembled around me.  I heard the rending of\nthe rocks.  I looked, and lo! the stroke had fallen upon the high tree,\nand the green tree was all on fire!  While I gazed, I heard a voice,\nmournful, but strangely sweet, \"My God! my God! why hast Thou forsaken\nme?  My heart is like wax; it is melted in the midst of my bowels.  My\nstrength is dried up like a potsherd, and my tongue cleaveth to my jaws.\nOne may tell all my bones.  Dogs have compassed me about; strong bulls of\nBashan have beset me.  They stare at me; they gape upon me with their\nmouths; they pierce my hands and my feet.  Deliver my soul from the\nlions; my darling from the power of the dogs!\"\n\n\"It is finished!\"  O with what majestic sweetness fell that voice upon my\nsoul!  Instantly the clouds were scattered.  I looked, and saw, with\nunspeakable wonder, millions of the low trees shooting up, and millions\nof the dry trees putting forth leaves, and fruit.  Then I took my harp,\nand sang this song:\u2014\"Worthy is the Lamb! for He was humbled that we might\nbe exalted; He was wounded that we might be healed; He was robbed that we\nmight be enriched; He was slain that we might live!\"\n\nThen I saw the beam of a great scale; one end descending to the abyss,\nborne down by the power of the Atonement; the other ascending to the\nHeaven of heavens, and lifting up the prisoners of the tomb.  Wonderful\nscheme!  Christ condemned for our justification; forsaken of His Father,\nthat we might enjoy His fellowship; passing under the curse of the law,\nto bear it away from the believer for ever!  This is the great scale of\nRedemption.  As one end the beam falls under the load of our sins, which\nwere laid on Christ; the other rises, bearing the basket of mercy, full\nof pardons, and blessings, and hopes.  \"He who knew no sin was made sin\nfor us\"\u2014that is His end of the beam; \"that we might be made the\nrighteousness of God in Him\"\u2014this is ours.  \"Though He was rich, yet for\nour sakes He became poor,\"\u2014there goes His end down; \"that we, through His\npoverty, might be rich,\"\u2014here comes ours up.\n\nO sinners! ye withered and fallen trees, fuel for the everlasting\nburning, ready to ignite at the first spark of vengeance!  O ye faithless\nsouls! self-ruined and self-condemned! enemies in your hearts by wicked\nworks! we pray you, in Christ's stead, be ye reconciled to God!  He has\nfound out a plan for your salvation\u2014to raise up the low tree, by humbling\nthe high, and save the dry tree from the fire, by burning up the green.\nHe is able to put, at the same time, a crown of glory on the head of the\nlaw, and a crown of mercy on the head of the sinner.  One of those hands\nwhich were nailed to the cross blotted out the fiery handwriting of\nSinai, while the other opened the prison-doors of the captives.  From the\nmysterious depths of Messiah's sufferings flows the river of the waters\nof life.  Eternal light rises from the gloom of Gethsemane.  Satan\nplanted the tree of death on the grave of the first Adam, and sought to\nplant it also on the grave of the second; but how terrible was his\ndisappointment and despair, when he found that the wrong seed had been\ndeposited there, and was springing up into everlasting life!  Come! fly\nto the shelter of this tree, and dwell in the shadow of its branches, and\neat of its fruit, and live!\n\nTo conclude:\u2014Is not the conversion of sinners an object dear to the\nhearts of the saints?  God alone can do the work.  He can say to the\nnorth, Give up; and to the south, Keep not back.  He can bring His sons\nfrom afar, and His daughters from the ends of the earth.  Our Shiloh has\nan attractive power, and to Him shall the gathering of the people be.\nPray, my brethren, pray earnestly, that the God of all grace may find\nthem out, and gather them from the forest, and fish them up from the sea,\nand bring them home as the shepherd brings the stray lambs to the fold.\nGod alone can catch these \"fowl of every wing.\"  They fly away from us.\nTo our grief they often fly far away, when we think them almost in our\nhands; and then the most talented and holy ministers cannot overtake\nthem.  But the Lord is swifter than they.  His arrows will reach them and\nbring them from their lofty flight to the earth.  Then He will heal their\nwounds, and tame their wild nature, and give them rest beneath the\nbranches of the \"Goodly Cedar.\"\n\n                                * * * * *\n\nThe following is so characteristic that, although it is in circulation as\na tract, it shall be quoted here; it has been called\u2014\n\n\n\nA SERMON ON THE WELSH HILLS.\n\n\nHE once preached from the text, \"Behold, I stand at the door, and knock.\"\n\"Oh, my dear brethren,\" he said, \"why will you pay no attention to your\nbest Friend?  Why will you let Him stand knocking, night and day, in all\nweathers, and never open the door to Him?  If the horse-dealer, or\ncattle-drover came, you would run to open the door to him, and set meat,\nand drink before him, because you think to make money by him\u2014the filthy\nlucre that perishes in the using.  But when the Lord Jesus stands\nknocking at the door of your heart, bringing to you the everlasting\nwealth, which He gives without money, and without price, you are deaf,\nand blind; you are so busy, you can't attend.  Markets, and fairs, and\npleasures, and profits occupy you; you have neither time, nor inclination\nfor such as He.  Let Him knock!  Let Him stand without, the door shut in\nHis face, what matters it to you?  Oh, but it does matter to you.\n\n\"Oh, my brethren!  I will relate to you a parable of truth.  In a\nfamiliar parable I will tell you how it is with some of you, and, alas!\nhow it will be in the end.  I will tell you what happened in a Welsh\nvillage, I need not say where.  I was going through this village in early\nspring, and saw before me a beautiful house.  The farmer had just brought\ninto the yard his load of lime; his horses were fat, and all were well to\ndo about him.  He went in, and sat down to his dinner, and as I came up a\nman stood knocking at the door.  There was a friendly look in his face\nthat made me say as I passed, 'The master's at home; they won't keep you\nwaiting.'\n\n\"Before long I was again on that road, and as soon as I came in sight of\nthe house, there stood the same man knocking.  At this I wondered, and as\nI came near I saw that he stood as one who had knocked long; and as he\nknocked he listened.  Said I, 'The farmer is busy making up his books, or\ncounting his money, or eating, and drinking.  Knock louder, sir, and he\nwill hear you.  But,' said I, 'you have great patience, sir, for you have\nbeen knocking a long time.  If I were you I would leave him to-night, and\ncome back to-morrow.'\n\n\"'He is in danger, and I must warn him,' replied he; and knocked louder\nthan ever.\n\n\"Some time afterwards I went that way again, and there still stood the\nman, knocking, knocking, knocking.  'Well, sir,' said I, 'your\nperseverance is the most remarkable I ever saw!  How long do you mean to\nstop?'\n\n\"'Till I can make him hear,' was his answer; and he knocked again.\n\n\"Said I, 'He wants for no good thing.  He has a fine farm, and flocks,\nand herds, and stack-yards, and barns.'\n\n\"'Yes,' he replied, 'for the Lord is kind to the unthankful, and the\nevil.'\n\n\"Then he knocked again, and I went on my way, wondering at the goodness,\nand patience of this man.\n\n\"Again I was in those parts.  It was very cold weather.  There was an\neast wind blowing, and the sleety rain fell.  It was getting dark, too,\nand the pleasantest place, as you all know, at such a time, is the\nfireside.  As I came by the farm-house I saw the candle-light shining\nthrough the windows, and the smoke of a good fire coming out of the\nchimney.  But there was still the man outside\u2014knocking, knocking!  And as\nI looked at him I saw that his hands, and feet were bare, and bleeding,\nand his visage as that of one marred with sorrow.  My heart was very sad\nfor him, and I said, 'Sir, you had better not stand any longer at that\nhard man's door.  Let me advise you to go over the way to the poor widow.\nShe has many children, and she works for her daily bread; but she will\nmake you welcome.'\n\n\"'I know her,' he said.  'I am with her continually; her door is ever\nopen to me, for the Lord is the husband of the widow, and the father of\nthe fatherless.  She is in bed with her little children.'\n\n\"'Then go,' I replied, 'to the blacksmith's yonder.  I see the cheerful\nblaze of his smithy; he works early, and late.  His wife is a\nkind-hearted woman.  They will treat you like a prince.'\n\n\"He answered solemnly, '_I am not come to call the righteous_, _but\nsinners to repentance_.'\n\n\"At that moment the door opened, and the farmer came out, cursing, and\nswearing, with a cudgel in his hand, with which he smote him, and then\nangrily shut the door in his face.  This excited a fierce anger in me.  I\nwas full of indignation to think that a Welshman should treat a stranger\nin that fashion.  I was ready to burst into the house, and maltreat him\nin his turn.  But the patient stranger laid his hand upon my arm, and\nsaid, 'Blessed are the meek: for they shall inherit the earth.'\n\n\"'Sir,' I exclaimed, 'your patience, and your long-suffering are\nwonderful; they are beyond my comprehension.'\n\n\"'The Lord is long-suffering, full of compassion, slow to anger, not\nwilling that any should perish, but that all should come to repentance.'\nAnd again he knocked, as he answered me.\n\n\"It was dark; the smithy was closed; they were shutting up the inn, and I\nmade haste to get shelter for the night, wondering more, and more at the\npatience, and pity of the man.  In the public-house I learned from the\nlandlord the character of the farmer, and, late as it was, I went back to\nthe patient stranger and said, 'Sir, come away; he is not worth all this\ntrouble.  He is a hard, cruel, wicked man.  He has robbed the fatherless,\nhe has defamed his friend, he has built his house in iniquity.  Come\naway, sir.  Make yourself comfortable with us, by the warm fireside.\nThis man is not worth saving.'  With that he spread his bleeding palms\nbefore me, and showed me his bleeding feet, and his side which they had\npierced; and I beheld it was the Lord Jesus.\n\n\"'Smite him, Lord!' I cried in my indignation; 'then perhaps he will hear\nthee.'\n\n\"'Of a truth he _shall_ hear me.  In the day of judgment he shall hear me\nwhen I say, Depart from me, thou worker of iniquity, into everlasting\ndarkness, prepared for the devil and his angels.'  After these words I\nsaw Him no more.  The wind blew, and the sleety rain fell, and I went\nback to the inn.\n\n\"In the night there was a knocking at my chamber.  'Christmas _bach_!'\n{410} cried my landlord, 'get up! get up!  You are wanted with a\nneighbour, who is at the point of death!'\n\n\"Away I hurried along the street, to the end of the village, to the very\nfarm-house where the stranger had been knocking.  But before I got there,\nI heard the voice of his agony: 'Oh, Lord Jesus, save me!  Oh, Lord\nJesus, have mercy upon me!  Yet a day\u2014yet an hour for repentance!  Oh,\nLord, save me!'\n\n\"His wife was wringing her hands, his children were frightened out of\ntheir senses.  'Pray! pray for me!' he cried.  'Oh, Christmas _bach_, cry\nto God for _me_!  He will hear _you_; _me_!  He will not hear!'  I knelt\nto pray; but it was too late.  He was gone.\"\n\n\n\n\nINDEX.\n\n\nABBOT, JACOB, referred to, 176.\n\nAccidents, a series of, 42.\n\nAccursed from Christ, 150; Reply to criticisms on, 152.\n\nAction in oratory, 194.\n\nAge of chapel cases, an, 113.\n\nAge, the golden, 359; The iron, 359; Messiah's, 359.\n\nAgent, the Divine, 363.\n\nAim and success, 162.\n\nAllegoric preaching, 90.\n\nAllegories:\u2014Bible regarded as a stone with seven eyes, 270; Church as an\nark among the bulrushes, 337; Satan walking in dry places, 137; Saul of\nTarsus and his seven ships, 332; Seeking the young Child, 133; World as a\ngraveyard, 85.\n\nAllegory, Christmas Evans's power of, 131.\n\nAmerica, preachers in the backwoods of, 231.\n\nAnecdotes:\u2014Announcement, a singular, 22; Ask him the price of pigs, 258;\nBaptism, scene at a, 49; \"Beattie on Truth,\" 283; Beneath! beneath!\nbeneath! 239; Better marry, 265; Billy Dawson, 110; Butchers and\nminister, 210; Cadwalladr and John Elias, 191; Chests for the dead, 259;\nChild in the pulpit, a, 190; Christian, a muscular, 50; Christmas Evans\nand his new hat, 118; Christmas Evans and the scholar, 67; Cough away!\n233; Cow is worth more, the, 238; Deacon, a blundering, 22; Drunkard\nconverted by a goat, 218; Earl and John Elias, the, 200; Elizabeth cannot\nbe alive, 195; Fire and smoke, 185; Flax-dresser and the preacher, the,\n189; Forgiving, 319; Gryffyth of Caernarvon, 11; Hope for the son of\nSamuel, 47; \"I am the Book,\" 68; I baptized Christmas Evans, 52; Impudent\nminister, an, 288; Knock-down argument, a, 51; Lucre, a lover of, 116;\nMake me weep? 212; No marriage in heaven, 235; No oath required, 239; Of\nRowland Hill, 240; Offenders, punishing young, 210; Old sermon, preaching\nan, 13; One-eyed lad, the, 57, 59, Paid at the resurrection, 116; Piecer,\na, 42; Plenty of fire in it, put, 186; Preach the Gospel, 224; Preacher,\na Welsh, 60; Preacher, an anonymous, 207; Racecourse, dispersion on a,\n196; Raffles, Dr., and Christmas Evans, 92; Raffles, Dr., and the\nGraveyard sermon, 82; Richard _bach_, 108; Richardson and John Elias,\n190; Sabbath-breaker and the preacher, 193; Sammy Breeze, 245; Scotch\nwoman and her pastor, 176; Selling a horse, 317; Sheep-stealers, the,\n113; \"Sit down, David,\" 108; Swearer, the, 210; Timothy Thomas and the\nclergyman, 49; Two snails, the, 318; Welsh farmer, a, 220; Williams and\nthe bookworm, 171.\n\n\"Ancient Mariner\" quoted, the, 232.\n\nAnglesea, island of, Evans's journey to, 63; Sandemanian schism in, 73;\nEvans's success in, 81; Leaving, 162, 165; Again in, 291.\n\nAnnouncement, a singular, 22.\n\nApostle and bishop, treated as, 110.\n\nApostrophe, a startling, 188.\n\nArian, a Welsh, 204.\n\nAssociation meetings, 10; where held, 21; gathering at, 121.\n\nAssociations, amongst old, 289.\n\n                                * * * * *\n\nBALA, Charles of, 227.\n\nBaptism, scene at a, 49.\n\nBardic triads, 254.\n\nBards, Wales the land of, 10, 11.\n\n\"Beattie on Truth,\" anecdote of, 283.\n\n_Bendigedig_, 17, 59.\n\n\"Beneath! beneath! beneath,\" 239.\n\nBeginning at Jerusalem, 301.\n\nBible a stone with seven eyes, 270.\n\nBibles for Wales, 228.\n\nBirds, parable of the, 343.\n\nBone, the misplaced, 333.\n\nBookworm and William Williams, the, 171.\n\nBorrow, George, quoted, 27, 218, 219, 258; Estimate of the \"Sleeping\nBard,\" 329.\n\nBradford, vicar of Christ Church, referred to, 196.\n\nBreeze, Sammy, story of, 245.\n\nBreton akin to Welsh, 25.\n\nBritish and Foreign Bible Society established, 229.\n\nBrowning, Robert, quoted, 163.\n\nBully and preacher, 243.\n\nBunyan, Christmas Evans compared with, 4; of Wales, 330.\n\nBurney's, Dr., \"History of Music,\" referred to, 214.\n\nButchers and minister, 210.\n\n                                * * * * *\n\nCADWALLADR, David, anecdote of, 191.\n\nCaernarvon, Richardson of, 190; Last days at, 287\u2013303.\n\nCaerphilly, Christmas Evans's ministry at, 261; Village of, 262; Castle\nof, 263; Society at, 281.\n\nCampbell, Dr. John, quoted, 229.\n\nCandles, is the game worth, 160.\n\nCaptain, the sceptical, 212.\n\nCastell Hywel, the church of, 43, 46, 204, 205.\n\nCastles, ruined Welsh, 34.\n\nCedar of God, the, 396.\n\nChair, Christmas Evans's, 64.\n\nChapel, Sabbath morning at a Welsh, 19.\n\nChapels, character of Welsh, 20.\n\nCharles of Bala, 227; the gift of God to North Wales, 227; Establishes\nschools, 228; Introduces Bibles, 228; A real bishop, 229; Modesty of,\n229; Dr. Campbell on, 229; as a preacher, 230.\n\nChildhood, a remarkable, 203.\n\nChorus, a grand musical, 183.\n\nChrist, the blood of, 371; Vicarious sufferings of, 382; Dignity of his\nnature, 383; Mediatorial kingdom of, 398; A fruitful tree, 401; A tree of\nprotection to his people, 402; A source of life and beauty, 403.\n\nChristmas, a custom at, 24.\n\nChristopher's, Mr., \"Hymns and Hymn-writers,\" referred to, 168.\n\nChurch, the Welsh established, 25; Discipline, 291; An ark among the\nbulrushes, 337.\n\nChurches, a bishop over, 106; Troubles with the, 160; An appeal to the,\n297.\n\nCildwrn cottage, the, 64; Life at, 65, 66.\n\nClergymen, character of Welsh, 25.\n\nColeridge, quoted, 274.\n\nCompensations, 121.\n\nCongregation, a sheep-stealing, 113; How to catch a, 243.\n\nConscience, purification of, 368; What is the, 368; A good and evil, 369,\n370; A guilty, 369; A despairing, 370; A dark and hardened, 370.\n\nConsonants, Welsh, 16.\n\nControversy, the Sandemanian, 70\u201376.\n\nConversations, 299.\n\nConversion, a singular, 218.\n\nConviction, the hour of religious, 173.\n\n\"Corner-stone,\" Abbot's, referred to, 176, 180.\n\nCottage preaching, 46.\n\nCough away! 233.\n\nCovenant with God, a, 78; A second, 277; The old, 364.\n\nCow, buying a, 238.\n\nCreeds and sects, contests of Christian, 177.\n\nCustoms, singular Welsh:\u2014Burning the ravens' nests, 191, 192; Delinquent\nand public opinion, 23, 24; Funeral, a, 37; New Year's, 24, 25;\nSin-eater, the, 23.\n\n                                * * * * *\n\nDARKNESS, conquest of the powers of, 380.\n\nDavid, sit down, 108.\n\nDavies of Swansea, 40; Character as a preacher, 202; Birth and parentage,\n203; A self-made man, 203; Childhood, 203; Marriage, 204; Unites in\nChurch fellowship, 204; And Christmas Evans, 204; Religious convictions,\n205; First sermon, 206; Ministry at Trefach, 206; Preaching at Denbigh,\n207; Settles at Swansea, 208; Reforms the neighbourhood, 209; His\nwonderful voice, 209; And the butchers, 210; Dealing with young\noffenders, 210; And the sceptical captain, 212; A prophet of song, 212;\nPopularity at Association Meetings, 214; A hymn-writer, 215; Last sermon,\n216; Death and funeral, 216.\n\nDavies, J. P., and Christmas Evans, 281.\n\nDavies, the Rev. David, 44, 204, 252; Epigrams of, 253.\n\nDavies, Thomas Rhys, 232; Character of his preaching, 233; Pithy sayings,\n233.\n\nDawson, Billy, 110.\n\nDays, dark, 155.\n\nDeacon, a blundering, 22.\n\nDebt, a chapel, 297.\n\nDebts, chapel, 109; Journeys to collect for, 109, 115.\n\nDelinquent and public opinion, the, 23.\n\nDemoniac of Gadara, 123; Effects of the sermon, 129.\n\nDemosthenes, a Welsh, 187, 194.\n\nDenbigh, Thomas Jones of, referred to, 186.\n\nDepression, spiritual, 52.\n\nDiscipline, a case of Church, 51; A letter on, 291.\n\nDissenters, what Welsh have effected, 25.\n\nDoctor and the humble minister, the, London, 68.\n\nDoctrine, a definition, 251.\n\nDogs, the pass of young, 120.\n\nDream, a singular, 45, 69, 331.\n\n\"Drive on!\" 302.\n\nDrunkard and the goat, the, 218.\n\nDyer, John, quoted, 36.\n\n                                * * * * *\n\nEARL, anecdote of a noble, 200.\n\n\"Ecclesiastical Polity\" quoted, 72.\n\nEdward II., tradition of, 263.\n\nEdwards family, the, 283.\n\nEdwards, Jonathan, referred to, 186.\n\nEisteddfod, the, 11.\n\nElias, John, character as a preacher, 17; Pure flame, 186; And Matthew\nWilks, 186; Soul and body, 187; Character and power of his eloquence,\n187\u2013190, 199; And the flax-dresser, 189; Illustrations of his power, 190;\nParentage, 190; First appearance in the pulpit, 190; As a young preacher,\n191; Puts down a cruel custom, 191; At Rhuddlan fair, 193; Tremendous\ncharacter of his preaching, 194, 195; Lives in an atmosphere of prayer,\n195; And the races, 196; A panorama of miracles, 196; Shall prey be taken\nfrom the mighty? 197; And the noble earl, 200; Death and funeral, 201.\n\nEngland, great Welsh preachers unknown in, 166.\n\nEntertainment, apostolic, 111.\n\nEpigrams, 253.\n\nEpitaph on Dr. Priestly, 253; An old Welsh, 257.\n\nEternity, 271; Time swallowed up in, 362.\n\nEvans, Christmas, A representative preacher, 5; And the pert young\nminister, 5; compared to Bunyan, 41; Birth and parentage, 41; A cruel\nuncle, 41; Accidents, 42; Loses an eye, 42; Youthful days, 43;\nConversion, 43; Mental improvement, 44; A singular dream, 45; Desires to\nbecome a preacher, 45, 46; First sermon, 46; Growth of spiritual life,\n47; Baptism, 47; His pastor, 48\u201352; Spiritual depression, 52; Enters the\nministry, 54; First charge, 54; Success at Lleyn, 55, 61; First preaching\ntour, 56; Marriage, 57; Becomes famous, 57\u201359; Removes to Anglesea, 63;\nCildwrn cottage, 64; Poverty, 66; Scholarship and library, 67; Reading,\n69; A dream, 59; And the Sandemanian heresy, 70\u201376; Deliverance, 76; A\nwayside prayer, 77; First covenant with God, 78\u201381; Renewed success, 81;\nThe Graveyard sermon, 82\u201390; And Dr. Raffles, 92; Inner life, 104; A\nbishop over many churches, 106; As a moderator in public meetings, 107;\nAnd chapel debts, 109, 114; Journeys, 110\u2013115; A life of poverty and\nhospitality, 115; And his new hat, 118; Wayfaring, 119; resemblance to\nFelix Neff, 121; Power of allegory, 131; Letter to a young minister, 142;\nReply to criticism, 152; Threat of legal prosecution, 155; Pathetic\nprayer, 155; Death of his wife, 157; Beautiful character of his wife,\n158; Troubles with the churches, 160; Is the game worth the candles? 160;\nHealthfulness of spirit and consolation, 163; Aim of his life, 165;\nRemarks on Daniel Rowlands, 225; And Evan Jones, 235; Removes to\nCaerphilly, 261; arrival at Caerphilly, 264; Second marriage, 265;\nSermons at Caerphilly, 266; Second Covenant with God, 277; And Mr. J. P.\nDavies, 281; Society at Caerphilly, 281; And Pye Smith's \"Scripture\nTestimony to the Messiah,\" 282; And \"Beattie on Truth,\" 283; Friends,\n283; requested to publish a volume of sermons, thoughts thereon, 284;\nRemoves to Caernarvon, 287; And the impudent young minister, 288;\nPresented with a gig, 288; And his horse, 289; Among old associations,\n289; Preaches again in Anglesea, 290; Reflections in his journal, 291;\nLetter on Church discipline, 291; Chapel debt again, 297; Starts on his\nlast journey, 297; Appeal to the churches, 297; On the journey, 298; Laid\nup at Tredegar, 299; Conversations, 299; At Swansea, 300; \"My last\nsermon,\" 302; Dying, last words, 302; Funeral, 303; As a man, 304\u2013321; A\ncentral figure in Welsh religious life, 304; A connecting link, 305;\nSelf-made, 305; Selling a horse, 307; Power of Sarcasm, 308;\nForgiveableness, 309; Faith in prayer, 310; Character of his sermons,\n312; Memorable sayings, 312; As an orator, 313; Dealt with great truths,\n316; Remarks on \"Welsh Jumping,\" 317; Characteristics as a preacher,\n322\u2013357; Use of parable, 322; Sermons born in solitude, 325; Imitators,\n326; fervour of his preaching, 327; use of Scriptural imagery, 328;\nProbable acquaintance with the \"Sleeping Bard,\" 329; The Bunyan of Wales,\n330; A dream, 331; Place and claim to affectionate regard, 355.\n\nEvans, D. M., quoted, 22; Life of Christmas referred to, 116.\n\nEvans, Mary, 265.\n\nEye? is the light in the, 236.\n\nEye, losing one, 42.\n\n                                * * * * *\n\nFARMER, anecdote of a Welsh, 220.\n\nFather and daughter, a dying, 182.\n\nFather and Son glorified, 386; glorifies the Son, 391.\n\nFinished! it is, 366, 378\u2013385.\n\nFire and smoke, 185.\n\nFishguard, William Davies of, 211.\n\nFlame, pure, 187.\n\nFlax-dresser, the audacious, 189.\n\nForgiving, power of, 309.\n\nFriars, preaching, 231.\n\nFuneral custom, a Welsh, 37; An imposing, 201.\n\n                                * * * * *\n\nGIG, present of a, 288.\n\nGilboa, a Welsh, 175, 176.\n\nGleisiad, the, 259.\n\nGlynceiriog, John Jones of, 74, 76.\n\nGod, a covenant with, 78; Character of, 274; A second covenant with, 277;\nServe the living, 376; A new and living way to come to, 380.\n\n\"God's better than man,\" 220.\n\n_Gogoniant_, 59.\n\n\"Golden Grove,\" Taylor's, 35.\n\nGoodness, infinite, 271.\n\nGospel, preach the, 224.\n\nGraveyard sermon, the, 82; Scenes at the delivery, 84, 85;\nCharacteristics of, 90, 91.\n\nGriefs, depressing, 160.\n\nGriffith, Mr. Thomas, referred to, 299.\n\nGryffyth of Caernarvon, anecdote of, 11.\n\n                                * * * * *\n\nHALL, ROBERT, anecdote of, 42; On the Graveyard sermon, 91; preaching of,\n313.\n\nHarris, Howell, of Trevecca, 221; Power of his preaching, 222.\n\nHarwood, 175.\n\nHat, story of a new, 118.\n\nHealth, restoration to spiritual, 76\u201378.\n\nHell, at the gates of, 69.\n\nHerbert, George, quoted, 274.\n\nHill, Rowland, anecdotes of, 185, 240.\n\nHind of the Morning, the, 92.\n\n\"Historical Anecdotes of the Welsh Language\" referred to, 16.\n\nHoliness, righteousness, and purity, 272.\n\nHoly Spirit glorifies Father and Son, the, 392.\n\nHope, leader of a forlorn, 287.\n\nHorse, selling a, 307.\n\nHorseman, the mysterious, 28\u201332.\n\nHorsley, Bishop, referred to, 252.\n\nHouse, the man in the Steel, the, 334.\n\nHouses, haunted, 27.\n\nHughes, Mr. Griffith, 284.\n\nHughes, Rev. J., \"History of Welsh Methodism,\" 241.\n\nHughes, Thomas, 241; And the vicar, 242; And the bully, 243.\n\nHume, David, referred to, 188.\n\nHuntingdon's \"Bank of Faith\" referred to, 117.\n\n_Hwyl_, the, 17, 59, 207.\n\nHymns, character of Welsh, 20.\n\n                                * * * * *\n\nIGNORANCE, character of Welsh, 5, 6.\n\nIllustrations:\u2014Accursed from Christ, to be, 150; Beginning at Jerusalem,\n301; Bible regarded as a stone with seven eyes, 270; Cedar of God, the,\n396; Church as an ark among the bulrushes, 337; Contests of Christian\ncreeds and sects, 177; Death as an inoculator, 340; Demoniac of Gadara,\n123; Dream, a, 331; Ejaculatory prayer, 172; Father and Son glorified,\n386; Finished redemption, 378; Four methods of preaching, 131; Gospel\nmould, the, 332; Handwriting, the, 338; Hind of the morning, 92; Letter\non Church discipline, 291; Letter to a young minister, 142; Man in the\nsteel house, the, 334; Misplaced bone, the, 333; Parable of the birds,\n343; Parable of the vine-tree, the thorn, etc., 344; Pious reflections,\n291; Pithy sayings, 233; Purification of the conscience, 368; Remarks on\n\"Welsh Jumping,\" 317; Reply to criticisms, 151; Resurrection of our Lord,\n345; Satan walking in Dry Places, 177; Saul of Tarsus and his Seven\nShips, 332; Seeking the Young Child, 133; Shall prey be taken from the\nmighty? 197; Their works do follow them, 275; They drank of that rock,\netc., 351; Time, 340; Time of reformation, 358; Timepiece, the, 342;\nTrial of the witnesses, 267; Value of industry, 306; World as a\ngraveyard, 85.\n\nImagery, use of scriptural, 328.\n\nImitators, 326.\n\nImprovement, efforts at self-, 44, 45.\n\nIndustry, value of, 306.\n\nInscription, a garden, 257.\n\nIrving, Edward, referred to, 162.\n\n                                * * * * *\n\nJACK _bach_, 289.\n\nJohnson, Dr., quoted, 225.\n\nJones, Catherine, 57.\n\nJones, Evan, 234; As a preacher, 235; Friendship with Christmas Evans,\n235.\n\nJones of Ramoth, 71, 72.\n\nJones, Rev. J., and the mysterious horseman, 28\u201332.\n\nJones, Thomas, of Glynceiriog, 74; Sermon on Sandemanianism, 75, 76.\n\nJones, Thomas, referred to, 184.\n\nJournal, reflections in, 291.\n\nJourney, a last, 297.\n\nJustice, satisfaction of divine, 379.\n\n                                * * * * *\n\nKEBLE quoted, 274.\n\n\"Keep that which thou hast,\" 296.\n\nKilgerran, King Arthur's castle at, 34.\n\n                                * * * * *\n\nLANGUAGE, the Welsh, 6, 7; Characteristics of, 14; Eliezer Williams on\nthe, 16; Proverbial character of, 178, 254; Theological, 315.\n\nLast day, sermon on the, 189.\n\nLavater, wife of, referred to, 158.\n\nLewis, William, and Davies of Swansea, 207.\n\nLibrary, Christmas Evans's, 67.\n\nLink, a connecting, 305.\n\n\"Little men,\" the superstition of, 24.\n\nLlandilo, neighbourhood of, 35.\n\nLlandovery, vicar of, 217; vicarage, 219.\n\nLlanfaes, churchyard of, 201.\n\nLlangeitho, Daniel Rowland of, 221.\n\nLlangevni, great Association sermon at, 75.\n\nLleyn, 53, 54; Christmas Evans at, 55, 61.\n\nLlwynrhydowain, church at, 43, 46.\n\nLoss, the great, 240.\n\nLucre, a lover of, 116.\n\nLyttleton, Lord, quoted, 15.\n\n                                * * * * *\n\nMABINOGION, the, 329.\n\nMacDonald, George, quoted, 72.\n\nMaesyberllan, Christmas Evans at, 54.\n\nMalkin, Mr., quoted, 37.\n\nMan, a self-made, 305.\n\nMan, Christmas Evans as a, 304\u2013321.\n\n\"Man of Ross\" referred to, 249.\n\nMarry, whom to, 265.\n\nMen, the wise, 133.\n\n_Messiah_, the, quoted, 76.\n\nMethodism, men evoked by, 231.\n\nMethodist and vicar, 242.\n\nMight, infinite, 272.\n\nMighty? shall prey be taken from the, 197.\n\nMilman, Dean, quoted, 311.\n\nMind, character of the Welsh, 259.\n\nMinister, letter to a young, 142; An impudent young, 288.\n\nMiracles, a panorama of, 196, 197.\n\nMinstrel preaching, 327, 328.\n\nModerator, Christmas as a, 107, 108.\n\nMoney, Christmas Evans collecting, 112.\n\nMorgan, Mr. W., on Evans leaving Anglesea, 164; His life of John Elias\nreferred to, 189.\n\nMorris, Caleb, referred to, 38.\n\nMorris, David, 240.\n\nMorris, Ebenezer, 238; Buying a cow, 238; And the oath, 239; As a\npreacher, 239; An anecdote of, 239; At Wotton-under-Edge, 240; His\nfather, 240.\n\nMould, the Gospel, 332.\n\nMynyddbach, David Davies at, 209.\n\n                                * * * * *\n\nNATURE, a lover of, 180.\n\nNeff, Felix, referred to, 121.\n\nNevern, scenery at, 35.\n\nNevern, vicar of, quoted, 194.\n\nNew, all things become, 365.\n\nNew year custom, a, 24, 25.\n\nNomenclature, Welsh, 34, 35.\n\nNorway, a village church in, 19.\n\n                                * * * * *\n\nOATH, taking the, 239.\n\nOmniscience, 271.\n\nOne-eyed lad, the, 57, 59.\n\nOpportunities, avail yourself of, 143.\n\nOrator, Christmas Evans as an, 313.\n\nOratory, action in, 194.\n\nOwl, cry of the, 259.\n\n                                * * * * *\n\nPANTYCELYN, Williams of, 167.\n\nParable, use of, 322.\n\nParables:\u2014Church an ark among the bulrushes, 337; Misplaced bone, 333; Of\nthe birds, 343; Of the vine, the thorn, etc., 344; Satan walking in dry\nplaces, 137; Saul of Tarsus and his seven ships, 332; Seeking the young\nChild, 133; Stranger knocking at the farmer's door, 407; Timepiece, 342.\n\nParr, Dr., quoted, 326.\n\nParry, Mr., on Williams's preaching, 180.\n\nPastors, town, and Christmas Evans, 111, 112.\n\nPenhydd, Shenkin of, 236.\n\n\"Pennillion,\" singing, 257.\n\nPerkins, Rev. William, 205.\n\nPigs, ask him the price of, 258.\n\nPithy sayings, 233.\n\nPoem, a Welsh, 16, 17.\n\nPoetical quotations, 16, 18, 25, 33, 34, 36, 66, 72, 76, 115, 120, 138,\n139, 163, 167, 169, 207, 220, 224, 232, 253, 256, 257, 259, 277, 311,\n331.\n\nPoverty, and hospitality, a life of, 115, 117.\n\nPrayer, 143; A wayside, 77; A pathetic, 155; Ejaculatory, 172; A first,\n173; Power of, 179; Living in an atmosphere of, 195; An old Welsh, 256;\nfaith in, 310.\n\nPrayers, character of some, 179.\n\nPreaching, Welsh, 3, 4; A national characteristic, 5; Character of Welsh,\n17; Scenery of Welsh, 21; Cottage, 46; An illustration of Welsh, 60;\nAllegoric, 90, 91; Value of great, 104; Four methods of, 131; Luminous,\n172; Tremendous, 194; Pretty, 316.\n\nPreacher, how to be a good, 12; A breathless, 22; An eloquent Welsh, 60;\nHardships of the Welsh, 105; Importance of a blameless life to a, 142;\nPersonal appearance of the, 181; An anonymous, 207; A voluminous, 232;\nand farmer, 236, 238.\n\nPreachers, Welsh, 4; And Welsh customs, 37; Great Welsh, unknown in\nEngland, 166; Peculiar character of old Welsh, 231; Rough and ready, 232;\nA cluster of Welsh, 248.\n\nPreparation, 359.\n\nPriestly, Dr., epitaph on, 253.\n\nPritchard, Rees, 217; A drunkard, 218; Singular conversion, 218; Author\nof the \"Welshman's Candle,\" 219.\n\nPromise, an evangelical, 397.\n\nProsecution, a threat of legal, 155.\n\nProverb uttering, 233; A Welsh, 263.\n\nProverbial power of the Welsh language, 178, 254.\n\nProverbs, Welsh, illustrations of, 255.\n\nProvidence, under the special care of, 28.\n\nPugh, Dr., referred to, 194.\n\nPugh, Philip, and Daniel Rowlands, 222.\n\nPulpit, character of the Welsh, 5; Results of, 7; Jeremy Taylor's, 36;\nStudy appearances in, 142; The quartette of the Welsh, 171; Notes in the,\n186; A child in the, 190; Aids to power in the, 325; Use of parable in,\n322; Confidence in, 331.\n\nPwllheli, John Elias at, 195.\n\nPyer, Rev. John, referred to, 245.\n\n                                * * * * *\n\n_Quarterly Review_ quoted, 16, 168.\n\nQuartette, a Welsh, 171.\n\nQuestions of anxious import, 273.\n\n                                * * * * *\n\nRACECOURSE, singular dispersion on a, 196.\n\nRaffles, Dr., and the Graveyard sermon, 82; And Christmas Evans, 92; On\nWilliam Williams, 183.\n\nRamoth, Rev. J. R. Jones of, 71, 72.\n\nRavens' nests, burning the, 191,192.\n\nReading, prayer, and temptation, 142.\n\nRedemption, finished, 378.\n\nRees, Dr., quoted, 40, 170, 202.\n\nRees, William, referred to, 207.\n\nReflections, an old man's pious, 291.\n\nReformation, the time of, 358.\n\nRemarks, closing, 355.\n\nResurrection of our Lord, 345; Proof of His Divinity, 345; Proof of the\ntruth of Christianity, 346; Pledge of eternal life, 347.\n\nResurrection, paid at the, 116.\n\nRhuddlan fair, 192, 193.\n\nRhydwilym, John Jones of, 74\u201376.\n\nRichard _bach_, 108.\n\nRichards, Dr. William, 250; definition of doctrine, 251.\n\nRichardson of Caernarvon, 190.\n\nRichter, Jean Paul, dead Christ of, 83.\n\nRob Roy, a Welsh, 18.\n\nRobertson of Brighton referred to, 325.\n\nRock, drinking at the, 351.\n\nRowlands, Daniel, 221; And Philip Pugh, 222; Character of his preaching,\n225; popularity and usefulness, 226.\n\nRuskin, John, quoted, 162.\n\n                                * * * * *\n\nSABBATH-BREAKER convicted, 193.\n\nSabbath evening scene, 122.\n\nSaints, Welsh, 34.\n\n\"Sair doubts o' Donald,\" 74.\n\nSalary, a small, 63.\n\nSamuel, hope for the son of, 47.\n\nSandemanian controversy, 70\u201376.\n\nSarcasm, Christmas Evans's power of, 308.\n\nSatan walking in dry places, 137.\n\nSaul of Tarsus and his seven ships, 332.\n\nScenery influences the mind, 259; Welsh, 17, 18.\n\nScotchwoman and her pastor, the, 176.\n\nSeeking the young Child, 133.\n\nSentences, memorable, 312.\n\n_Seren Gomer_, contributions to, 150, 152.\n\nSermon, preaching an old, 13; Against Sandemanianism, 75; The Graveyard,\n82; A last, 216; A wonderful, 268; \"This is my last,\" 302; On the Welsh\nhills, 407.\n\nSermons, studied and unstudied, 12; Bardic character of Welsh, 12, 13;\nValue of great, 104; Composition of, 144; Delivery of, 145, 150; Where\nWelsh preachers composed their, 171; Thoughts on being requested to\npublish a volume of, 284; _Silex scintillaus_, 312; Massive, 314; Living\nin the presence of published, 324; Born in solitude, 225, 226;\nCharacteristics of Christmas Evans's, 328; Illustrative, 358, 368, 378,\n386, 396, 407.\n\nServices, uncertainty of Welsh, 22.\n\nSheep-stealers and the collection, 113.\n\nShenkin of Penhydd, 236; His plainness of speech, 237.\n\n\"Silver Trumpet of Wales,\" the, 170.\n\nSin, sacrifice for accomplished, 379.\n\nSin-eater, superstition of the, 23.\n\nSinai, the ten cannon of, 193.\n\nSinging, Welsh, 20.\n\n\"Sleeping Bard,\" the, 329.\n\nSmith, Dr. Pye, \"Scripture Testimony to the Messiah,\" 282.\n\nSnails, the two, 308.\n\nSon equal to the Father, the, 387; Glorifies the Father, 389.\n\nSong, a prophet of, 212.\n\nSoul and body, 187.\n\nSpider, a Welsh poem on the, 16.\n\nSpirit, a healthy, 161.\n\nSt. David, a tradition of, 8.\n\nSt. David's cathedral, 33.\n\nSt. Govan, chapel of, 34.\n\nStephen's, Rhys, Life of Christmas Evans referred to, 43, 107, 164, 250,\n266, 269.\n\n\"Stop, Gabriel!\" 188.\n\n\"Stop! Silence!\" 189.\n\nStranger knocking at the farmer's door, the, 407.\n\nStreams, Welsh, 18.\n\nSubject, singular mode of illustrating a, 236.\n\nSuccess, value of, 55.\n\nSunday schools established in Wales, 228.\n\nSuperstitions, Welsh, character of, 26; Corpse candles, 27; Little men in\ngreen, 24; Mysterious horseman, 28; Sin-eater, 23.\n\nSwansea, David Davies of, 40, 46, 202; One hundred years since, 208;\nChristmas Evans at, 300.\n\nSwearer, the, 210.\n\n                                * * * * *\n\nTAYLOR, JEREMY, in Wales, 35.\n\nTemptation, 143.\n\nThinking and living, 21.\n\nThings that are shaking, 363.\n\nThomas, Timothy, 48; Anecdotes of, 49, 50, 51, 52.\n\nTime, 340.\n\nTimepiece, the, 342,\n\nTintagel, the Welsh, 34.\n\nTour, Christmas Evans's first preaching, 56.\n\nTranslations, inadequacy of, 314.\n\nTravelling in Wales, 119, 120, 262.\n\nTrefach, ministry of Davies at, 206.\n\nTrevecca, Howell Harris of, 221.\n\nTriads, the Welsh, 178; Bardic, 254.\n\nTroubles, a wife's, 115.\n\nTruths, seeing great, 316; Power of great, 317.\n\nTwm Shon Catty's country, 18.\n\n                                * * * * *\n\nUNCLE, a cruel, 41\u201342.\n\nUsefulness the aim and end of preaching, 12.\n\n                                * * * * *\n\nVAUGHAN, Henry, referred to, 311.\n\nVelinvoel, Christmas Evans at, 51\u201359.\n\nVicarage, an old Welsh, 219.\n\nVictory and triumph, the scene of, 361.\n\n\"Vocation of the Preacher\" referred to, 245.\n\nVoice, the human, 213.\n\nVortigern, supposed resting-place of, 54.\n\n_Vox Humana_ stop, the, 213.\n\n                                * * * * *\n\n\"WAESOME CARL\" quoted, the, 72.\n\nWales, comparatively unknown, 4; Moral and intellectual condition of, 7;\nOld wild, 32, 33; Travelling in, 119, 120, 262; The Watts of, 167;\nSingular practice in, 173; A rough time in, 191, 192; The Whitefield and\nWesley of, 221; Sunday schools established in, 228; Bibles for, 228; A\nland of song, 257; A central figure in the religious life of, 304; The\nBunyan of, 330.\n\nWales, wild, preachers of, 217; Rees Pritchard, 217; Howell Harris, 221;\nDaniel Rowlands, 221; Charles of Bala, 227; ancient preachers\ncharacterized, 231; Thomas Rhys Davies, 232; Evan Jones, 234; Shenkin of\nPenhydd, 236; Ebenezer Morris, 238; David Morris, 240; Thomas Hughes,\n241; A cluster of worthies, 248; Dr. Richards, 250; Davies of Castell\nHywel, 252.\n\nWalker, wonderful Robert, referred to, 118.\n\nWar, season of actual, 360.\n\nWatts of Wales, the, 167.\n\nWayfaring, 119.\n\nWelsh religious nature, the, 8, 9; Wrongs of the, 20, 21; Proverbs, 255;\nClannish character of the, 260; Jumpers, 317.\n\nWelshman, a monoglot, 174.\n\n\"Welshman's Candle,\" 168, 218, 219.\n\n\"White world,\" the, 15.\n\nWhitefield, George, referred to, 186; his startling apostrophe, 188.\n\n\"Wild Wales,\" Borrow's, quoted, 27, 218, 219.\n\nWilks, Matthew, anecdote of, 186.\n\nWilliams, Daniel, 169.\n\nWilliams, Evan, 169.\n\nWilliams of Pantycelyn, 167; career of, 167, 169.\n\nWilliams of Wern, 167, 170; Advice of, 12; Character and power of his\npreaching, 17, 170; Order of mind, 171; Method of composing his sermons,\n171; Illustration of manner, 172; Birth and parentage, 173; Religious\nconviction, 173; First prayer, 173; Education, 174; settles at Wern, 174;\nExtent of his pastorate, 175; Harwood, 175; Admiration for Jacob Abbot,\n176; Mind and method, 176; Illustration, 177; Proverbial utterances, 178;\nPrayer, 179; Eloquence, 180; Love of nature, 180, 182; Appearance when\npreaching, 181; Personal appearance, 181; Dying, 182; His daughter, 182;\nDeath, 183; Dr. Raffles on, 183; Characteristics of his preaching, 183.\n\nWilliams, Peter, 169.\n\nWilliams, Rev. W., \"Welsh Calvinistic Methodism\" referred to, 241.\n\nWilliams, Rowland, 38.\n\nWilliamses, a family of, 167.\n\nWisdom, divine, 273.\n\nWitnesses, trial of the, 267.\n\nWords, last, 302.\n\nWordsworth, referred to, 118.\n\nWorks, dead, 375.\n\nWorks do follow them, their, 275.\n\nWorthies, a cluster of Welsh, 248.\n\nWotton-under-edge, 240.\n\nWrong, altogether, 72.\n\nWyn, Elis, \"Sleeping Bard\" of, 329.\n\n                                * * * * *\n\n                                * * * * *\n\n        Hazell Watson, and Viney, Printers, London and Aylesbury.\n\n\n\n\nFootnotes.\n\n\n{23}  See Note at end of Chapter, _page_ 39.\n\n{410}  _Bach_ is a Welsh term of affection.\n\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nCONTENTS\n\nTitle Page\n\nCopyright Notice\n\nNote to the reader\n\nDedication\n\nAcknowledgements\n\nIntroduction\n\nHow To Use This Book\n\n1: UNDERSTANDING AND USING HOMEOPATHY\n\nThe History of Homeopathy\n\nPrinciples and Concepts\n\nMyths and Misapprehensions\n\nTaking the Case History\n\nPrescribing\n\nComplaints You Can Treat Using This Book\n\nCause for Concern\n\nSample Cases\n\n2: PREPARING FOR LIFE AFTER BIRTH\n\nPregnancy\n\nYour Family and Pregnancy\n\nYour Health and Pregnancy\n\nPreparing for Birth\n\nPlanning for Birth\n\nPreparing for the Post-natal Year\n\n3: PREGNANCY\n\nYour Body and Pregnancy\n\nComplaints\n\n4: BIRTH\n\nYour Body and Birth\n\nComplaints\n\n5: THE POST-NATAL PERIOD\n\nYour Body in the Post-natal Period\n\nComplaints \u2013 Mother\n\nComplaints \u2013 Baby\n\n6: THE MATERIA MEDICAS AND REPERTORIES\n\nExternal Materia Medica\n\nExternal Repertory\n\nInternal Materia Medica\n\nInternal Repertory\n\nNote\n\nList of Remedies and Abbreviations\n\nAPPENDICES\n\nFirst-aid Kits\n\nFurther Reading\n\nOrganisations\n\nIndex\n\nAlso by Miranda Castro\n\nCopyright\nNote to the reader:\n\nIt is advisable to seek the guidance of a physician before implementing the approach to health suggested in this book. It is essential that any reader who has any reason to suspect that she or her baby suffers from illness check with her doctor before attempting to treat it with this method. Neither this nor any other book should be used as a substitute for professional prenatal care, or medical care or treatment.\nFor Robert\nACKNOWLEDGEMENTS\n\nI am very grateful for the teaching I have received from all the babies in my life \u2013 both my own baby and those of my friends as well as the hundreds of baby patients I have been fortunate enough to treat over these past ten years in my capacity as a homeopath. This, and the feedback I have had from their mothers, has enabled me to explore and consolidate my beliefs around pregnancy, birth and parenthood. In the initial stages of writing this book I talked to many parents with a view to quoting their anecdotes. Sadly, I haven't been able to do this as the book became unwieldily long but I wove in their spirit as I wrote, using their comments and suggestions. I'd like to thank each one who shared their pregnancy, birth and parenting stories with me: they were funny in parts, sad in others, always interesting, informative and touching. Many thanks to Carol Boyce, Clare Palmer, Collette Barnard, David Orme, Frances Monte, Hazel Orme, Jane Howard, Jenner Roth, Maggie McKenzie, Maggi Sikking Jackson, Rachel Packer, Terry Cooper and Tony Dowmunt.\n\nThanks also to those who did spadework for me \u2013 typing and research \u2013 including Frances Monte, Claudia Benson, Helen Tye, Sue Clarke and Sue Mellis; to Jane Harter and Sue Morrison for reading through the manuscript and giving me their invaluable feedback; to Barbara Levy and Felicity Rubinstein for believing in me; to Hazel Orme for her patience and her unlimited, unconditional support; to all at Macmillan for helping to make this book what it is. I wish this baby a long and useful life.\nINTRODUCTION\n\nHaving a baby has been the most rewarding challenge of my life \u2013 so far! My child has taught me much about myself \u2013 about the mother and the child in me, about my strengths and my weaknesses, and as he grows and changes so I learn more.\n\nI became happily pregnant in 1978, and I loved it. I loved the feeling of a life taking shape inside me, of his swimmings and turnings. I loved the shape my body took, my big, soft roundness. I felt cocooned.\n\nAnd... I was convinced that Daniel would just fit into my life, that it would carry on much as it had done, with a baby in it, sleeping in a corner in his basket. I wish I had known that almost every aspect of my life would change to some degree.\n\nI apparently had a urinary-tract infection when I was three months' pregnant \u2013 I felt perfectly well and had not one symptom but I was scared into taking antibiotics by my doctor. Having grown up with naturopathic and homeopathic medicine I had rarely been to a doctor in my life, and found it perplexing to be treated as if I were ill. I felt lousy after the antibiotics for quite some time. I wish I had been better prepared when I went for my first pre-natal visit.\n\nI was irrationally scared that my baby would be stillborn. My doctor said that I shouldn't worry \u2013 the figures of stillbirths were very low. I wish that I had known I needed to take my fears seriously, that they related to feelings, still unresolved, from an abortion I had had the year before.\n\nDaniel was born two days before Christmas. I had a hospital birth because I wanted to know that pain relief was on hand in case I needed it. When I expressed my fears about my ability to cope with the pain I was told not to worry, that women the world over dropped their babies behind bushes and carried on working. I wish I had known that the birth would hurt. Not that the horror stories were true but that it might be incredibly hard work and not a little painful.\n\nMy labour was long, hindered by my fear and anxiety and because I wasn't allowed to eat. I finally accepted an epidural and relaxed into a deep sleep. My body then did its job well, freed from tension. I pushed Daniel out in a haze of exhaustion \u2013 by this point I had forgotten that I was going to have a baby. I don't know who was more surprised, him, me or his father. I felt an unexpected, primitive welling up of love of a new kind for me, a powerful response to this new life, newly in my life: motherlove. Birth was a miracle for me.\n\nMotherhood was a huge shock, an explosion of mixed emotions. I had so much to adapt to, seemingly so quickly, that I found myself frequently chasing my own tail as I struggled to understand what to do next! I wish I had known it would be such hard work.\n\nI remember a nurse handing me my baby a few hours after the birth and telling me that he was ready for his first feed and then walking away. I remember just feeding him, but awkwardly and with some difficulty, and the bad habits we developed then stayed with us long enough to cause my nipples to crack and bleed. I wish I had known that breastfeeding and mothering are both learnt skills! I got help from a La Leche League counsellor and then settled down to enjoy breastfeeding. I loved the closeness and the convenience.\n\nI remember changing him for the first time and marvelling at his little body. I remember his baby smell and exquisitely soft skin. I also remember picking him up and holding his delicious, tiny, naked body when he was two days old. His eyes opened wide in surprise as this cute little fountain of wee spurted all over us! I remember laughing! My baby brought much unexpected fun and laughter into my life.\n\nDaniel thrived until he contracted whooping cough at a year old. Homeopathic treatment helped him through it and over it and I used it also for his accidents and injuries, for everyday coughs and colds \u2013 and for those of his friends. Motherhood led me further into homeopathy because I needed help for my baby. The effectiveness of that treatment led me to take professional training and to specialise in working with mothers and babies. I wanted to work with a system of medicine that I could teach others to use, albeit in a limited way.\n\nHomeopathy is a wonderful system of medicine for women in their childbearing years when anxiety about the side effects of orthodox medicines often leads them to suffer rather than seek medical help for minor aches and pains. Worse still, in an attempt to give their children the best care possible mothers find themselves giving one course of antibiotics after another for relatively minor complaints. Many parents are questioning this and are looking for more natural ways to look after themselves and their children when they fall ill.\n\nThis book is a natural extension of my wish to empower patients to help themselves. It is not intended to replace your doctor but rather to encourage you to think twice before giving, say, antibiotics for a 'teething cough' or Calpol for a fever. I hope that you will take a little time to become familiar with the process of working out a good remedy \u2013 the results will be rewarding. My aim is that you use orthodox medicines only when absolutely necessary so that your children grow up strong, vital, healthy and able to withstand ordinary, everyday stress. Finally, I wish you well in your exploration of this effective route to creating health.\n\nMiranda Castro,\n\nJanuary 1992\n\nUpdate to introduction with video: <http:\/\/mirandacastro.com\/main\/eMotherBaby.html>\nHOW TO USE THIS BOOK\n\nThe goal of this book is to enable you to use homeopathic medicines safely and effectively at home during pregnancy, childbirth and in the post-natal year, both for yourself and your baby.\n\nMany homeopathic first-aid books currently available have attempted to simplify the process of finding a remedy (as homeopathic medicines are called) to make homeopathy more accessible. This has resulted in some disappointment, as many people have found their attempts to use this form of natural medicine to be a hit-or-miss affair. This book aims to right this error by mimicking the method that a professional homeopath uses. To this end chapters 1 and 6 focus on taking a case history, working out a remedy and prescribing. Chapter 2 looks at many of the practical aspects of preparing for pregnancy, birth and the post-natal year, and chapters 3, 4 and 5 deal with the complaints common to each of those times.\n\nChapter 1: Understanding and Using Homeopathy\n\nThis chapter looks at the history and theory of homeopathy, including its guiding principles and the myths that surround it; clear guidelines on taking a 'case history', on working out a remedy, on how and when to prescribe \u2013 including complaints you can't treat as well as those you can treat yourself \u2013 and a list of symptoms to watch out for that would necessitate your seeking professional help (Cause for Concern). This part ends with 12 sample cases which bring all of the above theory to life.\n\nChapter 2: Preparing for Life after Birth\n\nThis part discusses many of the issues that come up during pregnancy, birth and the post-natal period, many of which can be addressed in pregnancy to gain a sense of perspective and to be better prepared, especially for life after birth. Issues that need to be looked at for you to be able to enjoy this all important time include your choices regarding medical treatment, medical tests and interventions (before, during and after birth) and how to cope with them, the pros and cons of breastfeeding and bottle-feeding, general advice to help you look at how you are going to maintain (or even improve if necessary) your strength and health during this time in your life; and a job description of parenthood!\n\nChapters 3, 4 and 5: Pregnancy, Birth and the Post-natal Period\n\nEach of these sections starts with a look at what happens to your body because many changes take place \u2013 during pregnancy, birth and post-natally \u2013 all of which can affect your health. Then the complaints common to each of these periods are listed. Each complaint includes clear guidelines on how you can deal with it yourself and suggestions of practical measures that may be helpful or downright important for healing to take place, and pointers to help you decide when to seek professional help.\n\nChapter 6: The Materia Medicas and Repertories\n\nThis part of the book is divided into two sections: the external remedies (drops, lotions, creams and ointments) and internal remedies (tablets taken by mouth). Each has a Materia Medica \u2013 an alphabetically arranged list of descriptions of the remedies, and a Repertory \u2013 an index of the symptoms and complaints listed in the Materia Medica. As a homeopath, my tools for prescribing are the Materia Medica and Repertory. What I present here is a simplified form of these books where they apply to home prescribing, translated where possible into lay-person's terms.\n\nYou will need to read chapter 1 to be able to use this section, and, although it may take some time and effort to get the hang of it, it is worth persevering. I hope that the results will encourage you to pursue this most rewarding system of healing.\n\nUNDERSTANDING AND USING HOMEOPATHY\n\n* * *\n\nTHE HISTORY OF HOMEOPATHY\n\n* * *\n\nSAMUEL HAHNEMANN (1755\u20131843)\n\nSamuel Hahnemann, the founder of homeopathy, was born in Meissen in Saxony on 10 April 1755 into an era of change and political upheaval. The Seven Years' War, the French Revolution and the Napoleonic Wars threw Europe into turmoil; the Industrial Revolution brought social change and technological and scientific advances; there was also a revolution in thought \u2013 the political, spiritual and intellectual movement now known as the Enlightenment. The freedom of thought and opinion it encouraged was important for the birth and development of homeopathy.\n\nHahnemann was born into a poor and devout family who encouraged their son in his education. He qualified as a doctor in 1791 and practised medicine in Leipzig for about nine years, but he became increasingly disillusioned by the cruel and ineffective treatments of his time (blood-letting, purging, poisonous drugs with horrendous side effects) and gave up his practice, concentrating instead on study, research, writing and translation.\n\nOne of the major works he translated was Dr William Cullen's A Treatise on Materia Medica. Cullen (1710\u201390) was an Edinburgh teacher, physician and chemist, and his book included an essay on Peruvian bark or Cinchona (which homeopaths call China), from which quinine, the treatment for malaria, is derived. Cullen attributed Cinchona's ability to cure malaria, with its symptoms of periodic fever, sweating and palpitations, to its bitterness. Hahnemann, sceptical of this explanation, tested small doses on himself:\n\nI took by way of an experiment, twice a day, four drachms of good China. My feet, finger ends, etc., at first became quite cold; I grew languid and drowsy; then my heart began to palpitate, and my pulse grew hard and small; intolerable anxiety, trembling, prostration throughout all my limbs; then pulsation, in the head, redness of my cheeks, thirst, and, in short, all these symptoms which are ordinarily characteristic of intermittent fever, made their appearance, one after the other, yet without the peculiar chilly, shivering rigor. Briefly, even those symptoms which are of regular occurrence and especially characteristic \u2013 as the stupidity of mind, the kind of rigidity in all the limbs, but above all the numb, disagreeable sensation, which seems to have its seat in the periosteum, over every bone in the whole body \u2013 all these made their appearance. This paroxysm lasted two or three hours each time, and recurred, if I repeated this dose, not otherwise; I discontinued it and was in good health.\n\nIn other words, Hahnemann observed that Cinchona produced in a healthy person the symptoms of malaria, the very disease that it was known to cure, a discovery which was a cornerstone in the development of homeopathy.\n\nIn the fifth century BC, Hippocrates, the 'father of medicine', wrote that there were two methods of healing: by 'contraries' and by 'similars'. Although country people throughout the world have always used the principle of cure by 'similars' successfully in their own folk-medicines, the standard medical assumption has always been that if the body produced a symptom the appropriate treatment would be an antidote, an opposite or 'contrary' medicine to that symptom. For example, constipation would be treated with laxatives, which produce diarrhoea.\n\nDuring the sixteenth century, Paracelsus, a German doctor known as the 'father of chemistry', made new departures in medicine and pharmacology based on chemical experiments and direct observation of nature. He set the stage for the germ theory by stating that the causes of disease were external, seed-like factors introduced into the body through air, food and drink. He believed in the natural recuperative power of the human body and saw nature in every person as a vital spirit. He investigated the law of similars and by using only one medicine at a time and giving careful attention to dosage, noted that a very small dose could overcome a great disease.\n\nHahnemann embarked on further experiments which confirmed this principle. By observing the symptoms any substance produced when given to a healthy person, Hahnemann found the healing properties of that substance. This testing procedure was called 'proving'.\n\nHe referred to it as similia similibus curentur, or 'let like be cured with like'; this principle became the first law of a system of healing he called 'homeopathy', from the Greek homoios (similar) and pathos (suffering or disease), in order to differentiate it from orthodox medicine, which he called 'allopathy', meaning 'opposite suffering'.\n\nOver several years he conducted many provings on his family and friends, and also studied accounts of the symptoms shown by victims of accidental poisonings. Finally he set up in medical practice again, but with a different basis for his prescriptions. He used the material he had gathered from the provings and for each of his patients looked for the similimum \u2013 the remedy whose 'symptom picture', based on the provings, most matched that of the patient. His methods were met with disbelief and ridicule from his colleagues, but the patients flowed in and his astonishing results verified his theory.\n\nHe also differed from conventional practitioners in giving only one remedy at a time. In an age when apothecaries made fortunes by mixing numerous substances, many of which were highly noxious, this earned him many enemies.\n\nHahnemann did not stop there: dissatisfied with the side effects of his medicines, he experimented with smaller and smaller doses. He found, however, that when he diluted a medicine sufficiently to eradicate the side effects, it no longer effected a cure. He developed a new method of dilution: instead of simply stirring the substance after each dilution he shook it vigorously. This shaking he called 'succussion' and the resultant liquid a 'potentised remedy'. He found now that not only did the remedy lack side effects but the more he diluted it using succussion, the more effectively his remedy cured. He believed that the shaking released the strength or energy of the substance and dissipated its toxic effects.\n\nHahnemann numbered the potentised remedies according to the amount of times they had been diluted: a remedy diluted six times (taking out one hundredth of the liquid each time and adding 99\/100 alcohol) was called a 6C (see here). Initially he prescribed remedies that had been diluted up to the sixth potency; then he experimented with the higher dilutions, finding them more effective still. Eventually he prescribed up to the 30th potency, and his followers took dilution even further.\n\nThis process of dilution incurred further derision from the medical establishment, who could not explain, and therefore could not accept, how anything so dilute could have any effect. Yet despite opposition, homeopathy survived and spread remarkably quickly \u2013 because it was remarkably effective.\n\nSamuel Hahnemann lived before the germ theory of disease had been proposed, before thermometers, the X-ray and antibiotics made medicine appear increasingly 'scientific'. Yet he himself was an innovative scientist of sufficient intellect and culture to combine science and metaphysics. Consciously and unconsciously, he drew on the traditions of German folk-medicine, alchemy and magic, as well as the developments in chemistry, pathology, pharmaceutics and medicine which were beginning to make diagnosis and treatment both more accurate and more humane. In later life he became a religious free-thinker, believing that God permeated every living thing, and that he was divinely chosen and guided in his work. His development of a safe and effective system of medicine has given the world a priceless gift.\n\nHahnemann's literary output was prodigious. He proved about a hundred remedies, wrote over seventy original works, translated many texts on a wide range of subjects and also corresponded widely. In 1810 he published the first edition of The Organon of Rational Medicine (later The Organon of the Healing Art), which ran to six editions, each one modified and expanded. In it he set out clearly the homeopathic philosophy. In the same year, when Leipzig was besieged in the Napoleonic campaigns, his treatments of the survivors of the siege and the victims of the great typhus epidemic that followed were highly successful and further increased his reputation.\n\nBetween 1811 and 1821 Hahnemann published his Materia Medica Pura in six volumes; this represented the results of his provings \u2013 thousands of symptoms for sixty-six remedies. In 1828 came Chronic Diseases and Their Homeopathic Cure, in which he elaborated on the philosophy of The Organon, added more remedies, discussed the use of higher potencies and introduced the concept of 'miasms' to account for the failure of some patients to respond to treatment with remedies which clearly matched their symptoms. Among such people he found a family history of certain diseases and was able to link a tendency to a particular condition to the patient's 'inherited' health. He developed a way of treating these blocks to health homeopathically (see Miasms).\n\nIn 1831 cholera swept through central Europe. Hahnemann advocated the remedy Camphor in the early stages and Cuprum metallicum, Veratrum album, Bryonia alba or Rhus toxicodendron in the later stages. He also stressed that clothing and bedding should be heated to destroy 'all known infectious matters' and advised cleanliness, ventilation and disinfection of the rooms, and quarantine. (These ideas were far ahead of his time: the work of Pasteur on the germ theory of disease and that of Lister on disinfection was still to come.) Cholera was more successfully treated with homeopathy than with orthodox medicine; mortality rates varied between 2.4 and 21.1 per cent compared with 50 per cent or more with conventional treatment.\n\nHe gave lectures about his theory at the university, which often deteriorated into violent tirades against current medical practices earning him the nickname 'Raging Hurricane'. A few medical practitioners were prepared to go against mainstream opinion, trained under him and took his teachings out into the world.\n\nIn the 1820s, when homeopathy arrived in the USA, the state of orthodox medicine was, if anything, worse than it was in Europe. The practice of almost completely draining the body of blood (four-fifths was let) was advocated, even for children. A drug know as 'Calomel' (Mercurius chloride), which had been introduced originally to cure syphilis, was used as a standard purgative; its side effects were loss of teeth, seizure of the jaws and death from mercury poisoning.\n\nHomeopathy was easily accepted, and flourished. Homeopaths were seen to be well-educated, hardworking people, and the metaphysical background appealed to many church people. It was adopted in particular by followers of Swedenborg (1689\u20131772), a visionary who believed himself a vehicle for a new religious revelation. His writings appealed to people who were studying the new sciences, such as Darwinism, and who were concerned about the conflict between science and orthodox religion. For many homeopaths this blend of reason and mysticism was ideal.\n\nIn 1846 the American Medical Association was founded, which adopted a code of ethics forbidding its members to consult homeopaths. Local and state medical societies were told to purge themselves of homeopaths and their sympathisers, but homeopathy had already made a positive mark on orthodox medicine: blood-letting abated, medical training improved and several homeopathic remedies found their way into allopathic prescribing. Public demand for homeopathic treatment continued.\n\nThe 1860s through to the 1880s saw the flowering of American homeopathy. Practitioners proved every conceivable remedy, often at great cost to their own health. There were some fifty-six homeopathic hospitals, thirteen lunatic asylums, nine children's hospitals and fifteen sanatoriums. The homeopathic training colleges, unlike their allopathic counterparts, excluded neither women nor black people.\n\nIn 1826 a young well-connected English doctor called Frederick Quin studied with Hahnemann and on his return to London set up a homeopathic practice, treating many famous people, including Dickens and Thackeray. He established the British Homoeopathic Society (later the Faculty of Homoeopathy), and in 1849 founded the London Homoeopathic Hospital, where, during the cholera outbreak of 1854, deaths were a mere 16.4 per cent, compared with the 50 per cent average for other hospitals. The Board of Health suppressed this fact, explaining, 'The figures would give sanction to a practice opposed to the maintenance of truth and the progress of science.'\n\nAfter the Crimean War a medical bill was introduced in Parliament to outlaw homeopathy, but Quin's friends in the House of Lords secured a saving amendment.\n\nIf Quin was the practical and fashionable force behind homeopathy in Britain, the two most influential writers and teachers were Robert Dudgeon, who translated Hahnemann's texts into English, and Richard Hughes, a mild man with a conciliatory attitude towards the medical establishment as well as a rigorous and scientific attitude towards homeopathy. Queen Adelaide, wife of King William IV, brought homeopathy from her native Saxony to the British royal family. The royal family maintains an active involvement in homeopathy to this day. The Queen has her own consultant homeopath and carries her 'black box' of remedies with her on all her travels.\n\nTHE TWENTIETH CENTURY\n\nBy the time of Hahnemann's death in 1843, homeopathy was established throughout the world, although the mutual antagonism and distrust between homeopaths and allopaths continued to hinder its progress.\n\nDevelopments in medicine around the close of the nineteenth century strengthened the orthodox camp: science had proved the existence of microbes, the old practices Hahnemann had condemned were diminishing, and powerful drugs were being developed. The pharmaceutical industry, helped by the power of advertising, became an effective and wealthy lobbying force behind allopathic medicine. Meanwhile the homeopathic establishment was weakened by internal division, and the public \u2013 and many homeopaths \u2013 were drawn to the side that could put its case most clearly.\n\nThe American Medical Association moved to close many homeopathic teaching institutions and mounted a huge anti-homeopathy propaganda campaign. Consequently, by 1918 the number of homeopathic hospitals in the USA had dwindled to seven. Great optimism accompanied the introduction of penicillin: doctors thought that a medical nirvana had been reached. They regarded the taking of a homeopathic case as too time-consuming \u2013 the five-minute prescription and a cure for every ill had arrived. Little did they realise that it was the dawn of a medical nemesis.\n\nHomeopathy has spread rapidly throughout the world. It is popular over Asia, particularly in India where it is now officially recognised as a separate branch of medicine and is fully supported by the government. Although it is poorly represented in some European countries, other parts of Europe show a fast-growing interest. In France and Germany homeopathic medicines are readily available in most pharmacies and there are homeopathic consultants at some hospitals. Homeopathy is highly respected in many South American countries, with Mexico, Argentina and Brazil at the forefront. It is spreading in Australia, New Zealand, Greece and Israel although it is non-existent in the Arab states.\n\nIn 1946, when the National Health Service was established, homeopathy was included as an officially approved method of treatment and in Britain today its popularity is increasing rapidly. It is still practised under the auspices of the National Health in five hospitals in Bristol, Liverpool, Glasgow, Tunbridge Wells and London, and by some GPs, although the limit of time on consultations often means that antibiotics are handed out with the homeopathic medicines just in case the latter do not work.\n\nProfessional homeopaths run private practices and many participate in almost-free clinics for the needy. The Society of Homoeopaths, the organisation which represents the professional homeopath in this country, promotes the highest standards. From small beginnings in 1974, the number of registered members \u2013 who use the initials R.S.Hom. (Registered member) or F.S.Hom. (Fellow) after their names \u2013 increases annually and a dozen colleges train several hundred professional homeopaths each year.\n\n* * *\n\nPRINCIPLES AND CONCEPTS\n\n* * *\n\nThe highest ideal of therapy is to restore health rapidly, gently, permanently; to remove and destroy the whole disease in the shortest, surest, least harmful way, according to clearly comprehensible principles.\n\nSo it was that Samuel Hahnemann, in The Organon, defined his goals for a new system of medicine. It is hard to imagine a description that could express more concisely the needs of both practitioner and patient.\n\nThe principles of homeopathy represent a complete view of the processes of health and disease. Since 1810, when The Organon was first published, they have proved resistant to major reinterpretation and are crucial to successful prescribing.\n\nThe Similimum or Law of Similars\n\nThis basic principle of homeopathy \u2013 similia similibus curentur or 'let like be cured with like' \u2013 states that any substance that makes you ill can also cure you: anything that can produce symptoms of disease in a healthy person can cure a sick person with similar symptoms.\n\nBy 'symptom' the homeopath means those changes that are felt by the patient (subjective) or observed by someone else (objective), which may be associated with a particular disease, or state of disease, and which are the outward expression of that state.\n\nProvings\n\n'Proving' is the name given to the homeopathic method of testing substances to establish their 'symptom pictures'. Since Hahnemann's first proving in 1790, hundreds of others have been carried out and their results collated in the great Materia Medicas (see here). In the 1940s, the Americans organised a programme of re-proving remedies, but it was abandoned when identical symptoms were elicited all over again.\n\nToday, healthy volunteer provers of new, potentially medicinal substances are divided into two groups, with one group being given the unnamed substance and the other a placebo. It is always a double-blind trial: neither the provers nor the conductor of the proving knows at the time who is taking what. The remedies are sometimes tested in their diluted \u2013 potentised \u2013 form or, if they are not poisonous, in crude doses (in the 'mother tincture', see Potencies). All symptoms \u2013 physical, emotional and mental \u2013 are noted in detail, then gathered schematically and common themes noted.\n\nApart from such provings two other sources are used:\n\nAccidental provings provide a rich source of valuable information that might not otherwise be available. Homeopaths have been able to add symptom pictures of substances such as deadly nightshade, the remedy Belladonna, or snake venom, such as Lachesis, to the Materia Medica by drawing on detailed accounts of accidental poisonings. Because these substances can cause serious conditions, they also have the ability to cure them, and so are of great value.\n\nCured symptoms After a remedy has been successfully prescribed, symptoms cured by it which did not emerge either in the provings or in the accidental provings are noted. If the remedy consistently cures these symptoms in many people, then they are added to its symptom picture.\n\nThe Materia Medica\n\nThe Materia Medica (Latin, meaning medical matter or material) lists the symptom pictures of each remedy as discovered in the provings. The many hundreds of remedies are arranged alphabetically and the symptoms of those remedies are arranged according to body area. New remedies are constantly being discovered and added.\n\nThe professional homeopath works with a number of Materia Medicas compiled by different homeopaths, each reflecting their own personal experience. They all have the same basic information, but the individual homeopath may interpret the material slightly differently. Within this vast amount of information certain patterns emerge and it is these patterns with which the homeopath becomes familiar. He or she will memorise the strong symptoms or keynotes of every remedy.\n\nThe Repertory\n\nThis is an index of symptoms from the Materia Medica listed in alphabetical order and thereby providing a valuable cross-referencing system. A good Repertory is essential as it is impossible to memorise the vast number of symptoms in the Materia Medica.\n\nThe Single Remedy\n\nThe classical homeopath gives one remedy at a time to gauge its effect more precisely than would be possible if two or more remedies were given together. Indeed, the remedies were all originally proved separately and it is consequently not known how they interact if mixed; combined remedies should only be considered once they have been proved in combination.\n\nHowever, this most difficult aspect of homeopathic prescribing deters many people. Finding a single remedy to match the patient's symptoms is a constant challenge and can involve an enormous amount of hard work. The lazy, busy or misguided homeopath may mix several remedies together in the hope that one may work, a hit-or-miss approach which is not true classical homeopathy, as Hahnemann defined it, and shows a lack of understanding of the fundamental principles.\n\nThe Infinitesimal Dose\n\nThe more a remedy is diluted and succussed (vigorously shaken), the stronger it becomes as a cure (see here). The concept of the infinitesimal dose is one of the great stumbling blocks for a conventionally trained scientific mind. Sceptics scoff at the idea that a very dilute solution of sea salt \u2013 beyond the point where there is any salt measurable in the solution \u2013 is capable of curing a wide range of complaints, from cold sores, hayfever and headaches to depression (see Natrum muriaticum). Logically, it does seem unlikely that a substance that can cause high blood pressure in its crude form could become a strong and effective agent for healing when it is so dilute. However, a pharmacological law states that although a large dose of poison can destroy life, a moderate dose will only paralyse and a very small dose will actually stimulate those same life processes.\n\nNew discoveries in physics are beginning to explain this phenomenon. One theory is that the succussion creates an electrochemical pattern which is stored in the dilutant and which then spreads like liquid crystal through the body's own water. Another hypothesis suggests that the dilution process triggers an electromagnetic imprinting which directly affects the electromagnetic field of the body.\n\nPotencies\n\nThere are two scales for diluting substances: the decimal and the centesimal. In all cases the starting remedy \u2013 a 'tincture' or 'mother tincture' \u2013 is made by steeping the substance itself in alcohol and then straining it.\n\nFor the decimal scale, one-tenth of the tincture is added to nine-tenths alcohol and shaken vigorously; this first dilution is called a 1X. The number of a homeopathic remedy reflects the number of times it has been diluted and succussed: for example, Sulphur 6X has been diluted and succussed six times.\n\nThe centesimal scale is diluted using one part in a hundred of the tincture (as opposed to 10) and the letter C is added after the number (although in practice homeopaths have omitted the C and just use the numbers for the centesimal scale).\n\nParadoxically, a 6X is called a low potency and 200(C) a high potency \u2013 the greater the dilution, the greater the potency.\n\nThe most commonly used potency in the decimal scale is the 6X, although the 9X, 12X, 24X and 30X are used by some. In the centesimal scale those low potencies most commonly used are the 6, 12 and 30. The higher potencies \u2013 200, 1M (diluted one thousand times), 10M (ten thousand times) and CM (one hundred thousand times) \u2013 are highly respected by homeopaths and should not be used by the home prescriber.\n\n'Inert' substances (such as Lycopodium, the tiny spore of the club moss) are ground for several hours with a pestle and mortar until they become soluble. This process, called 'trituration', is used for metals as well as other substances that do not dissolve easily to prepare them for succussion.\n\nThe Whole Person\n\nThe concept of treating the 'whole person' is an essential element of classical homeopathy. The basis of this belief is that symptoms, diseases or pains do not exist in isolation, but are a reflection of how the whole person is coping with stress. It is the whole person that counts, not just the physical body but also the mental and\/or emotional 'bodies'. The homeopath looks beyond the 'presenting complaint', beyond the label of the disease (for example, 'tonsillitis', 'migraines' or 'food poisoning') to the 'totality of symptoms' experienced. The prescription is individualised to fit the whole person.\n\nAs far as first-aid prescribing is concerned, it is possible to prescribe on a single symptom such as, for example, chilblains or mouth ulcers, but it is always preferable to find a remedy that matches more of a person's symptom picture, taking into account as many pieces of the jigsaw as possible.\n\nConstitution and Susceptibility\n\nI often hear people say enviously about a friend, 'He smokes like a chimney, drinks like a fish, works like a maniac and has never had a day's illness in his life. It's not fair. I struggle constantly to stay healthy and need nine hours' sleep a night, otherwise I get sick. Why?'\n\nThe answer lies in the constitution. The person who works all hours and smokes and drinks as if there were no tomorrow has a strong constitution \u2013 and may well be wasting his 'inheritance', because there will come a time when it will run out, when even he will get sick.\n\nIn The Science of Homeopathy George Vithoulkas defines the constitution as 'the genetic inheritance tempered or modified by our environment', that is, a person's fundamental structure \u2013 their state of health and their temperament. A strong constitution can withstand considerable pressure without falling ill; a weak constitution has a greater susceptibility to illness.\n\nSusceptibility is simply the degree to which a person is vulnerable to an outside influence. In an epidemic not everyone will be affected, but those who are we call 'susceptible'. Their predisposition is due to an underlying constitutional weakness, which is either inherited or due to past and\/or current stress (mental, emotional or physical).\n\nIf your grandparents all died of old age and your parents have been healthy all their lives; if your birth was planned and your mother was healthy throughout her pregnancy (didn't smoke, drink, etc.); if your parents' marriage is happy and your birth was uneventful, then your constitution should be of the strongest.\n\nIf, on the other hand, all your grandparents died at early ages of cancer or heart disease, one of your parents had tuberculosis as a child and the other suffered from asthma and eczema, then your chances of inheriting a weak constitution are greater. You can still escape the worst of a poor inheritance if your parents' marriage is happy, if they have taken care of their own health, and brought you up with plenty of love and a good diet.\n\nAnd that is where alternative medicine comes in. Many people come to a homeopath for 'constitutional treatment', to improve their general health rather than wait until they fall ill. The value of constitutional treatment is that it boosts the weak constitution and decreases its susceptibility to disease. Homeopathy strengthens the body's vitality and its ability to respond to stress without recourse to other medicines.\n\nThe Vital Force\n\nHomeopaths believe that a balancing mechanism keeps us in health, provided that the stresses on our constitution are neither too prolonged nor too great. Hahnemann called it the 'vital force' and he believed it to be that energetic substance, independent of physical and chemical forces, that gives us life and is absent at our death.\n\nThe human organism, indeed any living thing, has a unique relationship with its environment, which biologists refer to as 'homeostasis'. This means that a healthy living being is self-regulating, with an innate (protective) tendency to maintain its equilibrium and compensate for disruptive changes. Homeopaths believe that the vital force produces symptoms to counteract stresses and makes adjustments, moment by moment throughout our lives, to keep us healthy and balanced. These symptoms, then, are simply the body's way of telling us how it is coping with stress. Obvious examples are shivering when cold, perspiring when overheated and eating or drinking when hungry or thirsty, reactions which help to ensure the regulation of a constant, life-preserving environment within the body. Disease 'attacks' only when this vital force is weakened.\n\nHomeopathic medicines act as a catalyst, the remedy stimulating the body's own vital force to heal itself. They do not weaken the defence mechanism by suppressing it as do many orthodox medicines. The correct homeopathic treatment not only alleviates the symptoms but enables the patient to feel that life is once again flowing harmoniously.\n\nAcute and Chronic Disease\n\nAcute disease is self-limiting; it is not deep-seated and, given time, will usually clear of its own accord. Some acute illnesses, such as pneumonia, meningitis or nephritis, for example, are very serious and can, although rarely, be fatal. These are not within the scope of the home prescriber and always need expert advice.\n\nAn acute disease has three definite stages: the incubation period, when there may be no symptoms of disease; the acute phase, when the recognisable symptoms surface; the convalescent stage, when a person usually improves. Coughs, colds, flu, food poisoning and children's illnesses such as chickenpox are all examples of acute illnesses. Well-chosen homeopathic remedies will speed up recovery, alleviate pain and ensure that there are no complications. (See here and here as well as the complaints sections of chapters 3, 4 and 5 for the acute illnesses or complaints you can and cannot treat using this book.)\n\nChronic disease is more deep-seated than acute disease. It develops slowly, continues for a long time and is often accompanied by a general deterioration in health. The development of the disease does not take a predictable course; neither is it possible to say for how long it will last. An acute disease that is followed by complications can develop into a chronic, long-term illness.\n\nArthritis, heart disease, cancer and mental illness are all examples of chronic disease. Homeopaths believe that the current increase in incidence of these conditions is in part due to chemical stresses, including the overuse of orthodox medicines and environmental pollution.\n\nMiasms\n\nIn his early years in practice, Hahnemann was puzzled to find that some patients failed to respond to their constitutional remedies and others who improved relapsed after only a short time. He collected these 'difficult' cases together and after much careful study found a common factor in the presence of certain diseases in their personal or family history, which he realised must constitute blocks to health preventing the indicated constitutional remedy from working. He called the blocks 'miasms' and developed a comprehensive and complex theory around them. His insights have enabled the professional homeopath to assess through the history of a patient (both personal and inherited) their likely constitutional strength and often to predict the sort of blocks that they might encounter during the course of constitutional treatment.\n\nHahnemann defined three basic miasms which he believed to be the underlying causes of chronic disease, each of which predisposes a person to a particular range of health problems which he also defined at length. The three miasms were: Psora, which he associated with suppressed skin diseases and with leprosy; Sycosis, associated with suppressed gonorrhoea; and Syphilis, associated with suppressed syphilis. Homeopaths have since added many more including tuberculosis, radiation and heavy metals. The treatment of miasms is always a matter for the professional homeopath, not for self-treatment.\n\nThe Laws of Cure\n\nThe Laws of Cure were formulated by Constantine Hering (a doctor, homeopath and a follower of Hahnemann), who based them on a lifetime's observation of the processes involved when sick patients became well. Throughout his years of practice he was able to draw the following conclusions:\n\n\u2022 As someone becomes well, symptoms move from the innermost organs of the body (those most vital to life) to the outer organs. Therefore, cure moves from within to without. For example, someone with heart disease (serious, life-threatening) may experience stomach or bowel problems during the process of cure.\n\n\u2022 Cure also takes place from above to below, so that symptoms usually 'drip off' the body, starting from the head and clearing downwards, with the hands and the feet (sometimes simultaneously) being the last to be affected (with, say, a skin eruption).\n\n\u2022 Symptoms that have been suppressed in the past often resurface during the process of cure and usually do so in the reverse order from their original sequence. For example, if a patient with heart disease had been successfully treated with orthodox medicines for a stomach ulcer before the heart condition, then the appearance of stomach symptoms (less severe than in the original complaint) would be welcomed as a sign that the old suppressed symptoms were being cleared out.\n\nThe suppression of a disease usually leads to a more deep-seated illness surfacing. For example, many children whose eczema has been 'successfully' treated with steroids may suffer from asthma later in life. These two events are seen by the orthodox medical profession as having only a casual connection, whereas the homeopath believes that the suppression of the eczema has caused the asthma. Successful homeopathic treatment involves the eczema reappearing at some point.\n\nIt is also possible to suppress with homeopathic medicines by treating a single symptom and not therefore taking into account the whole person.\n\nHomeopaths use the Laws of Cure to monitor treatment, to check whether the cure is going in the 'right' direction. These laws apply to the treatment of chronic complaints but occasionally also to acute prescribing. As far as acute treatment or home prescribing is concerned, a very well-selected constitutional remedy will occasionally push to the surface old symptoms that may have been forgotten. These will clear of their own accord. It is important not to prescribe another remedy and by so doing encourage the chronic condition to go back 'in' again.\n\nHealth and Disease\n\nHealth is more than simply the absence of disease. I believe it is a sense of well-being, of feeling good, of being in balance, that is hard to dislodge. It is, above all, the ability to withstand stress.\n\nWhen we are physically healthy we have strength and flexibility, and a reservoir of energy to draw on should we need it. When we are emotionally healthy we can acknowledge and express our feelings and by so doing maintain rewarding relationships. When we are mentally healthy we can think clearly, formulate ideas, solve problems and make decisions easily.\n\nDisease limits our personal freedom; a broken leg, for example, limits physical freedom by making it difficult to walk. Depression can limit emotional freedom because it is difficult to interact with other people. Mental exhaustion after, say, a taxing exam makes it difficult to concentrate and make decisions and therefore limits our mental freedom. If you fall ill it is often useful to ask yourself how and why that illness is limiting you.\n\nThe orthodox medical view, or germ theory, of disease is that illness is a 'bad' thing. An alternative medical view is that disease is a 'good' thing, alerting the body to the necessity of taking some time to recuperate or to have a good clear-out.\n\nI believe that disease is neither good nor bad. The disease we succumb to provides us with information about our personal weaknesses, about how we are living our lives and how we are coping with stress. It alerts us to the existence of something that needs attention.\n\nBuilding Health \u2013 Preventive Medicine\n\nThe desire to rid the body of disease is a healthy attitude, although any approach that involves building health and therefore preventing disease or ill-health will, of course, be of greater benefit in the long term than any temporary suppression of the symptoms alone. How we approach illness, as well as the treatment we choose, deserves some thought.\n\nThe presence of disease or pain often creates anxiety, which in turn can lead to fear and panic. Most of us have consulted an authoritative figure (a doctor) to allay our anxiety by putting a name to what is happening to our bodies, and to determine how it must be treated. The danger here is that, in looking outside ourselves for the answers and in asking too few questions, we experience a loss of personal control with consequent feelings of helplessness. We give up responsibility for our own health to the people 'in charge' and we become real patients, or, as I see it, victims. We find that we feel unable to confide misgivings, to express our instincts about our own health, or to explore other options. We become passive consumers of medical care.\n\nAs patients, how can we redress the balance? The first step is to become informed, by reading and talking to people who are sympathetic to our views (and doubts), and who may have had similar experiences. We must also learn to ask for what we need, to make realistic demands of health-care professionals, and to seek the help of doctors and specialists, both orthodox and alternative, who are willing to communicate with us, who are able to acknowledge that we, the patients, have a part to play in our own healing processes, and who are therefore able to give us the information we need, and deserve, about what they think is happening.\n\nBy taking responsibility for what happens to our bodies we can begin to create for ourselves the balance we want in our lives, and tune into our own feelings, or inner sense, of what is wrong. By developing a positive approach towards creating a healthy life we can move away from automatically taking a defensive position towards illness.\n\nStress\n\nWe all vary in our ability to adapt to and cope with stress. Understanding our own stress limits is tremendously important.\n\nI divide stress into two main categories: healthy stress and unhealthy stress. Healthy stress pushes you to perform better and achieve more than usual in certain circumstances: for example, to work all hours in your job during a crisis or to gain that little extra from your body during a race (or during labour!). Unhealthy stress brings you too near to that 'last straw' state where your own coping mechanisms are overstretched, when your performance level starts to drop. You make mistakes. You fall ill.\n\nUnder stress our bodies let us know when the pressure is too great by producing the symptom(s) of illness. And when symptoms surface on one level, other levels are often affected. For example, a head injury (physical) can initially cause shock (emotional) and amnesia (mental) as well as pain (physical). A difficult, aggressive boss can create an emotionally stressful environment for his staff, who may produce physical ailments such as headaches, indigestion or neck pain as a response to suppressed emotions. (The suppression of any emotion can be harmful: it takes a lot of energy to suppress joy and will be as stressful as holding in rage.)\n\nWe respond to different stresses in different ways, according to our age and resources. Giving birth at forty is a bigger physical stress than doing so at twenty when a woman's body is more supple and she has greater energy resources \u2013 but an older, more mature woman may find herself less emotionally stressed by motherhood.\n\nThe body operates as a unit, with the physical, emotional, mental and spiritual working together to maintain a balance. Each aspect needs nourishment, but mental and emotional nourishment is often lacking in our society. Because of the widespread rejection of religion many people do not have a time of peace or quiet in their lives to reflect or just be. To still the mind and let the cares of life drift away is a deeply healing process. You can do this by meditating, daydreaming, painting and writing, walking and enjoying nature or through prayer.\n\nYou can also nourish yourself with food, sleep, massage, spending time with friends or family members who accept you as you are, who will listen when needed, and provide you with loving support and advice. Laughter is one of nature's great healers. Letting go of the serious business of living and having fun is essential to maintaining health.\n\nRecognising and Dealing with Stress\n\nLearn to recognise the first symptoms of being overstressed, when the body shows early-warning signs but before illness develops. Being ill can be a fruitful way of coping with stress although, of course, disease itself can be stressful, especially if you take antibiotics or homeopathic remedies and carry on. You need to stop, rest and allow time for self-healing.\n\nAs a home prescriber, it is important to understand that disease is a response to stress. Identifying the stress is the first step: you can use the stress symptom and the response to prescribe on. Some people are sensitive to physical stresses, for example, to changes in the weather, always falling ill with a cold when the warm summer weather first changes to cold. Others are more vulnerable to emotional stress, and find it hard to cope with upsets at home or at work.\n\nThe more we understand about how we react to stress the better able we are to deal with it and choose how to respond to it.\n\nGet to know your own stress response. What can you cope with? What are your limits? How do you recognise them? How can you stretch your limits when you have to? What nourishes you? How do you balance the stress in your life? Assess your reserves and strengths for dealing with it; look at ways in which you can balance it. Avoid unnecessary stress. Learn to offload, delegate, say no \u2013 and ask for support and help.\n\nRemember that you are an important person in your own life. You need to look after yourself so that you can do your job(s) well \u2013 so that you can be a healthy mother\/father\/friend\/employee\/homeopath or whatever!\n\n* * *\n\nMYTHS AND MISAPPREHENSIONS\n\n* * *\n\nMany homeopaths, believing that the explanation of how homeopathy works is secondary to its success, have traditionally refused to reveal the names of the medicines they give. This and the lack of information they have provided about their practice has led to an aura of secrecy in which myths abound. Let's look at a few of them.\n\nMYTH: 'HOMEOPATHY IS SAFE'\n\nIn the same way that homeopathy can cure \u2013 often dramatically and permanently \u2013 it can also cause harm. Potential dangers are:\n\nUnintentional Provings\n\nIf you take a homeopathic remedy for too long it is possible to 'prove' the remedy \u2013 that is, to suffer again from the symptoms that the remedy was supposed to cure. If the remedy was correct, your symptoms may improve initially but worsen again if you continue to take it. Worse still, if the remedy did not fit your picture \u2013 was not right for you \u2013 you may experience symptoms you haven't had before.\n\nThis is a danger with self-prescribing or over-the-counter prescribing, where there is no professional homeopath to monitor the symptoms. In my first year in practice a woman rang me one day in a frantic state, desperate for help. She told me the following story:\n\nI asked for help at a homeopathic chemist for thrush, from which I had suffered for several months, and was prescribed Nux Vomica 30 over the counter and told to take it three times daily. After a few days I experienced a marked improvement in my condition, so I carried on taking it. After a week of no further changes my symptoms started to get worse, so I carried on taking it. I finished the bottle of pills and went back to the pharmacy and told them my thrush was now as bad as when I had started to take the remedy. They gave me another bottle of Nux Vomica 30 and told me to continue with the treatment. It is now two months since I started on this remedy and my thrush is unbearable. It is so bad I can't sleep at night and I am irritable all the time. Please help me.\n\nI advised her to stop taking the pills and to antidote the remedy with strong coffee and camphorated ointment (to counteract the effect of the proving) and within twenty-four hours she was back to her old self, having slept well for the first time in a month. The thrush was back to where it had been before she took the Nux Vomica \u2013 annoying but manageable \u2013 and she took a course of acupuncture treatment to deal with the remaining thrush.\n\nIt is important not to overuse homeopathic medicines.\n\nConfusion of the Symptom Picture\n\nIf a remedy has not been prescribed on the whole person it will work in a limited way, curing a restricted number of symptoms. Some symptoms remain and you may end up giving one remedy after another to try to 'get rid' of them. The whole picture becomes so changed that it is difficult to find the similimum (that single remedy that was needed in the beginning).\n\nThe professional homeopath has different ways of dealing with this phenomenon to get back to the original symptom picture: if you find that you are prescribing one remedy after another with only limited effect, then get professional homeopathic help.\n\nSuppression\n\nA homeopathic remedy can cure a specific symptom such as a skin eruption in the same way as, for example, the application of a hydrocortisone cream. This will only occur if the remedy has been prescribed on the skin complaint (single symptom) without taking into account the whole person and\/or the cause. The effect is to push the disease further into the body (see Laws of Cure). Constitutional treatment will often begin with the original symptom resurfacing. Suppression is not common with homeopathic treatment but is possible. In self-prescribing, if your complaint disappears but you feel much worse in yourself (your moods and your energy) then it is likely that you have made a poor choice of remedy. Antidote it (see here) and take professional homeopathic advice.\n\nMYTH: 'HOMEOPATHY IS A FORM OF HERBALISM'\n\nThis is the commonest myth of all. While it is certainly true that some homeopathic remedies are based on plants, and that, as in homeopathy, the herbalist prescribes on the individual, the principles that govern the two therapies are quite different.\n\nMany plants have known healing properties; herbalism, used for thousands of years, is concerned with the known sphere of action of a plant based on its chemical constituents as well as its known healing qualities. Homeopathy is based on different principles (see here). The remedies are not used in the material dose; nor are they based solely on plants, using also poisons, metals and disease products. Homeopaths generally prescribe one remedy at a time rather than the mixtures of plant tinctures that herbalists employ.\n\nMYTH: 'HOMEOPATHY IS A FORM OF VACCINATION'\n\nPeople often say that they understand homeopathy to be like a vaccination in that the patient is given a small quantity of the disease they already have to make him immune to it. Homeopathy and vaccination have only similar, not the same, concepts, and have very different practices. Vaccines stimulate the immune system directly to produce specific antibodies as if that person had contracted a particular disease; in so doing they stress the immune system. They are tested on animals and then humans to verify their safety, and even then children and adults can suffer serious side effects.\n\nA homeopathic remedy works differently by affecting an individual's energy patterns \u2013 the vital force \u2013 and stimulating the body to heal itself. It is administered orally in a diluted (and safe) dose whereas most vaccinations are introduced directly into the bloodstream, thereby bypassing the body's natural defence system and stressing it in a way that is not fully understood. Homeopathic medicines are not tested on animals and when correctly used do not have side effects.\n\nMYTH: 'HOMEOPATHIC REMEDIES ARE PLACEBOS'\n\nThis myth can be rephrased to read 'You need to believe in it for it to work', which is nonsense to anyone who has experienced or prescribed a successful homeopathic cure for a head injury or a middle-ear infection in a baby.\n\nA placebo is an unmedicated pill which the patient believes contains something that will cure him or her. Homeopathic medicines work on babies and animals, neither of whom are susceptible to the placebo effect. Research has ruled out the placebo effect and shown over and over again that homeopathy is effective and does work.\n\nMany people recognise the experience of consulting a practitioner who inspires belief and hope, leaving them feeling buoyant and encouraged. But if this initial rapport is not backed up with good solid prescribing, then no amount of positive 'transference' will effect a cure.\n\nMYTH: 'HOMEOPATHY IS MYSTERIOUS AND UNSCIENTIFIC'\n\nHomeopathic medicines are prepared in a pharmacy or a laboratory, involving a technique subject to precise and clearly stated controls. Preparation does not involve mysterious and secret processes which put it into the realm of white magic or alchemy.\n\nThe homeopathic principles constitute a unified hypothesis whose validity is tested empirically: cured patients confirm the hypothesis. Harris Coulter discusses this issue at great length in his book Homoeopathic Science and Modern Medicine: The Physics of Healing with Microdoses, and also describes many of the trials that have been conducted over the past fifty years or so using plants, animals and humans as controls to prove the effectiveness of homeopathic medicines.\n\n* * *\n\nTAKING THE CASE HISTORY\n\n* * *\n\nIt is well worth the effort to approach first-aid prescribing seriously by teaching yourself how to take a case history and how to work out a remedy. That is the major concern of this book: to help you to prescribe, where possible, in the same way as a professional homeopath. Many homeopathic first-aid books take a 'symptomatic approach', directing you to a remedy on the strength of one symptom or complaint. This is not thorough enough and the results cannot be guaranteed. In this book, therefore, I have taken the more classical approach of looking at the whole person in a detailed way.\n\nIf you fall ill and wish to prescribe for yourself it is important to remember that it may be difficult to be objective about your own symptoms: you may be fooled into prescribing the wrong remedy. For example, you may feel quite calm in yourself while those around you have experienced an increase in your levels of irritability and touchiness; and\/or you may already have forgotten the drenching you had the day before you fell ill. It can be useful to talk through your symptoms with your partner or a friend before you self-prescribe. Some people are competent at knowing exactly what their 'symptom picture' is and are clear about the stresses that led up to their illness; others need more help. If you need it, get help from your local homeopath.\n\nBefore you begin to take the case, check that the complaint is within the scope of this book (see here and here as well as Complaints, here, here, here).\n\nDon't attempt to prescribe for a chronic disease; these always need professional attention.\n\nTHE SYMPTOM PICTURE\n\nThis is a detailed account of what is wrong with you as a whole person when you fall ill. Your symptoms \u2013 the signs or indications of your being ill \u2013 are an expression of need and a call for help from your body. A homeopathic prescriber is a symptom sleuth! For a remedy to work well you need to match, as accurately as possible, your symptom picture with a remedy picture \u2013 the collection of symptoms produced by the remedy in its proving, as set out in the Materia Medica. It doesn't have to be identical, merely similar. You won't need to experience all the symptoms of the remedy for it to work! Symptoms are divided into three main categories: general, mental\/emotional and physical complaints. You will need one or more from each main category of symptom. See here for the case-taking chart.\n\nI have used the word complaint throughout the book to mean the disease itself, or rather its label \u2013 for example, 'sore throat' or 'cough'. The symptoms are the visible signs of the complaint, or, the complaint expresses itself through the symptoms. Say your child is complaining of a barking cough that is worse at night and better for sitting up: the complaint is the cough and the symptoms are barking, worse at night and better for sitting up.\n\nUnderstanding and treating the cause is fundamental to homeopathic practice. It is helpful to identify the stresses that affect us, so that we can avoid them or take action before we actually fall ill.\n\nAll disease is caused by stress, which can be physical, emotional or mental. Different people are sensitive to different stresses at different times in their lives: cold, heat, changes in weather, mental strain, etc. The diseases people develop will also vary according to their individual weaknesses.\n\nStress symptoms may be general, in which case you will find them in the Repertory under Complaints from (and in the Materia Medica under General Symptoms, Complaints from). Perhaps your child always becomes ill with, say a cold or a cough or an earache, if the weather changes from warm to cold: begin by looking in the Repertory under Complaints from\/after: Change of weather to cold.\n\nStress symptoms may be specific, in which case you will find them listed as a Cause under the complaint itself. For example, some children (and adults) are sensitive to falls. If they bang their heads they suffer from headaches. In the Repertory (here) under Headache, Causes, head injury you will find Natrum sulphuricum (Nat-s.) listed. If you turn to Natrum sulphuricum in the Materia Medica (here) you will find that under the physical complaint 'Headache', head injury is listed as the 'cause'.\n\nIt is vital to build a strongly indicated stress symptom into your picture. Once you are familiar with particular patterns you need not wait for the complaints to prescribe. For example, if you have a child who is sensitive to bangs on the head and Natrum sulphuricum has cured in the past, you can give it automatically (after Arnica has dealt with the bruising) to prevent a headache from developing.\n\nSimilarly, you can give Calcarea carbonica to a baby who has a cough and cold and you suspect that she is teething once again and Calcarea carbonica has helped her before under similar circumstances. You can even give it before the cough develops. If she is restless, has a runny nose and you know she is teething, the remedy will help her teethe without becoming sick.\n\nYour observations are very important: you are not simply looking for aches and pains, you need to gain a picture of the differences from the normal state of health. Gathering information is not always easy. Look for clear, strongly marked symptoms: vague, 'well... sometimes' symptoms are not useful unless they present a 'changeable' symptom picture in themselves.\n\nCertain symptoms are always present in any given disease and are indicated in its name. For example, pain during urination is usually called cystitis; conjunctivitis refers to inflamed, bloodshot eyes. As homeopathy does not treat the labelled disease alone but the individual who is sick, and since no two cases of cystitis are exactly the same, every labelled disease (or complaint) is accompanied by symptoms peculiar to the individual who has it, which differ from those of another individual with the same 'disease'. It is the individual symptoms which guide us to the remedy that will help.\n\nSimply to write 'headache' or 'cough' is not enough. If you look up either of these symptoms in the Repertory you will find many remedies listed. It is the particular symptoms of the patient's headache or cough that enable you to begin to narrow down the choice of remedies, to individualise the prescription. One person with a headache may have a throbbing forehead that feels better for fresh air, while another will have sharp pains in the temples that are better for warmth and lying down. Different remedies are called for.\n\nNOTE-TAKING\n\nIt is essential that you write down the symptoms, and that you record the remedy that you give as well as its result. Even if you write only a few words plus the date, the name and potency of the remedy, that will suffice to remind you should the same problem recur. I suggest you open a loose-leaf file marked 'Health' and use it for keeping interesting articles as well as notes on your home prescribing. You can file photocopies of the case-taking chart (here) and the repertorising chart (here) so that you always have 'blanks' to hand should you need them.\n\nFirst Impressions\n\nGive some attention to your first impressions about what is happening as they are an important and reliable source of information. Say your child wakes up in the night screaming. You go into her room. What do you immediately smell and see? What note do you hear in her scream? Is it one of pain or fright? Or perhaps your best friend has flu, she's pregnant and has asked you to come and help. You walk into the room and what do you see before she has organised herself to be sociable with you? Is the room tidy or messy? Is it hot or cold? Does she look sad and tired before she makes an effort to be 'fine'?\n\nIf remedies do come to mind as you are talking, do not ignore them but write them in the margin: those instincts can be inspired. Generally speaking, it is better to forget about working out a remedy while you take notes; the analytical part comes later. Concentrate on using your senses to pick up clues and on being receptive to what you are hearing.\n\nWhat do you smell?\n\nThe smell is often the first thing you notice on walking into a sick person's room. Smell the head\/neck\/ hands of a feverish or sick child and you will find that sweat can smell sour or just plain offensive. Smell the breath, too. Other discharges can also smell 'interesting': the stools or urine of a teething baby can smell sour; vomit can also be very smelly. If the smell is strong and you can describe it clearly, write it down and use it as a symptom.\n\nWhat do you see?\n\nIn the case of an illness, note the colour and expression on your patient's face. Has the colour drained from it or is it flushed? Is the expression anxious or obviously in pain (suffering)?\n\nLook for clues: drinks not drunk; covers piled high or, alternatively, kicked off; restlessness; log-like patients who groan if you sit on the edge of the bed and disturb them and so on. You may learn more by looking than your patient is aware of.\n\nWhat do you feel?\n\nTouch the skin to find out whether there is sweat. If so, is it hot or cold, clammy or profuse? A dry skin with a fever (sweat, absent), for example, is an important symptom. Check to see if some parts of the body are sweaty and others aren't. Perhaps your patient is complaining of an inner, burning heat while the skin feels cold to the touch.\n\nWhat do you hear?\n\nSome sick children moan or grind their teeth in their sleep; others might whine for attention if they think you are not listening and stop when you enter the room (and vice versa!). Listen to the tone of the voice. Is it anxious or sad or angry?\n\nLet your patient talk, if they are old enough, without any interruption from you if possible. This in itself can be healing, and can also provide you with information. For example, the Lachesis patient talks incessantly, jumping from one subject to another, while the Bryonia patient will not want to talk at all.\n\nQuestions\n\nAsk yourself (or your 'patient') general rather than specific questions.\n\nGeneral | |\n\nSpecific\n\n---|---|---\n\nHow am I\/are you feeling? | |\n\nAm I\/are you feeling sad?\n\nHow painful is it? | |\n\nDoes it hurt a lot?\n\nDescribe the pain? | |\n\nIs it throbbing?\n\nDoes heat or cold help or make it worse? | |\n\nIs it better for heat?\n\nSome people will reply negatively when asked about themselves or their symptoms: 'I'm not well'; 'He's not bad'; 'It isn't much better'; 'It doesn't hurt that much'. These replies do not tell you what is actually happening. Questions like 'How are you, then?', 'What are you feeling?' can sometimes help, but people who speak in negatives often need careful questioning to tell you what is wrong. In answer to 'It doesn't hurt that much', you could ask 'Is the pain stopping you from doing anything?' (that is, 'How serious is it?'). Then you can go on to give a list of different types of pain and if your patient alights on one, saying 'That's it!', then write it down. If they are not sure, go on to another question.\n\nFinally, remember...\n\nDo not discount anything. What other information do you have about that person? What has been going on in her life lately that might be part of why she is feeling sick now? What have you been told by her mother\/father\/child\/neighbour? Have you given this person a first-aid remedy before, say, for a flu? Did the remedy work well? If that remedy worked marvellously, consider whether it might be indicated again. The case-taking chart and questionnaire below will help you take a good case history.\n\nPhysical symptoms List any other physical symptoms that you've noticed. Use this checklist to jog your memory. Underline and\/or add additional symptoms.\n\nHead | |\n\nheadache, dizziness, hair loss, faintness\n\n---|---|---\n\nEyes | |\n\ninflamed, watering, sensitive to light, styes, pupils dilated\/contracted\n\nEars | |\n\nearache, hearing\n\nNose | |\n\ncommon cold, catarrh, nosebleeds, sense of smell, cracks\n\nFace | |\n\nexpression, colour, lips dry\/cracked, cold sores\n\nMouth | |\n\nulcers, dryness, excess saliva, taste\n\nTeeth | |\n\ntoothache, teething, abscess\n\nTongue | |\n\ncolour, cracks, indented on edges\n\nThroat | |\n\nsore throat, voice lost, hoarseness\n\nStomach | |\n\nappetite (likes\/dislikes), thirst, nausea, vomiting, colic, indigestion\n\nChest | |\n\ncough, croup, palpitations, breastfeeding problems\n\nBowels | |\n\nconstipation, diarrhoea, stools (colour\/consistency), piles, worms, flatulence\n\nUrination | |\n\npain (cystitis), retention, bedwetting\n\nPeriods | |\n\npainful, heavy, late\n\nNeck\/back | |\n\nstiff neck, glands swollen, backache, sciatica\n\nArms\/legs | |\n\njoint pain, cramps, clumsy, weak ankles, chilblains, corns\n\nSleep | |\n\nunrefreshing, restless, insomnia, nightmares\n\nEnergy | |\n\nexhaustion, sluggish, apathetic, anaemia\n\nSkin | |\n\nprickly heat, boils, hives, thrush, blisters\n\nFever | |\n\nwith or without sweating, flu\n\nSweat | |\n\nsmelly, profuse, scanty, localised, generalised\n\nDischarges | |\n\ncolour, consistency\n\nStressed by | |\n\n(complaints from) change of weather, getting wet, getting chilled, injury, emotions, etc.\n\nInjuries | |\n\nsee under each heading \u2013 bruises, burns, etc.\n\nSymptoms | |\n\nside, onset (slow\/sudden)\n\nPains | |\n\ngeneral, labour pains\n\nEmotional state List any emotional symptoms that you have noticed. Use this as a checklist to jog your memory. Underline and\/or add additional symptoms.\n\nabsent-minded\n\naffectionate\n\nangry\n\nanxious\n\napathetic\n\naversion (being alone\/touched\/hugged\/consolation\/children or partner etc.)\n\nbiting (children)\n\nbroody\n\ncapricious\n\ncheerful\n\nchildish\n\nclingy\n\nconcentration, poor\n\nconfused\n\nconscientious\n\ncritical\n\ndazed\n\ndelirious\n\ndenial\n\ndepressed\n\ndesire (for company\/to be alone\/to be carried)\n\ndespair\n\ndespondent\n\ndictatorial\n\ndisappointment\n\ndiscontented\n\ndisobedient\n\ndreamy\n\ndwells on the past\n\neuphoric\n\nexcitable\n\nexhilarated\n\nexpressions (anxious\/suffering etc.)\n\nfearful\n\nforgetful\n\ngentle\n\ngloomy\n\nguilt\n\nhitting (children)\n\nhomesick\n\nhumiliation\n\nhurried\n\nidealistic\n\nimpatient\n\nimpulsive\n\nindecisive\n\nindifferent\n\nintrospective\n\nintroverted\n\nirritable\n\njealous\n\njoking\n\njumpy\n\nlack of self confidence\n\nlazy\n\nlively\n\nlonely\n\nloss of libido\n\nmelancholic\n\nmemory weak\n\nmild\n\nmischievous\n\nmoaning\/complaining\n\nmoody\n\nmorose\n\npanic\n\nquarrelsome\n\nrage\n\nresentful\n\nrestless\n\nscreaming\n\nself-confidence, lack of\n\nsensitive (to light\/music\/noise\/pain)\n\nsentimental\n\nshock\n\nshy\n\nsighing\n\nslowness\n\nsluggish\n\nspiteful\n\nstubborn\n\nstupor\n\nsulky\n\nsympathetic\n\ntalkative\n\ntantrums\n\ntearful\n\ntidy\n\nuncommunicative\n\nunforgiving\n\nweary\n\nwhiny\n\nBabies and children\n\nBabies and small children cannot describe what they are feeling so you need to be especially observant to find the symptoms that will lead you to a successful prescription. Remember, you are looking for any changes from their normal (healthy) patterns of behaviour. Be aware of your child's patterns of health so that changes, when they are ill, are clear to you.\n\nYou may need to do lots of looking and feeling and smelling with babies as well as using your intuition to get a sense of how ill your baby really is. See here for when to seek expert help. If you are at all unsure, do not delay in getting help and reassurance.\n\nWith very young babies you need to check out the following: bowels, skin (including temperature and sweating), sleep, discharges, stresses, energy, eyes, tongue, digestion (colic) and emotions.\n\nWORKING OUT THE REMEDY\n\nHaving taken your case history, follow these steps to select your remedy: choose your symptoms; repertorise; differentiate. Once you become familiar with this process you will be able to take short cuts but it is worth getting to grips with even if it does feel tedious initially. These steps are brought to life in the sample cases.\n\n1 Choose Your Symptoms\n\nGo through your case and underline the symptoms that stand out clearly and strongly. It is best to leave vague and unclear symptoms alone for the moment: you may use them later if you can't make up your mind between two or more remedies.\n\nWherever possible, choose at least three strong symptoms, one from each symptom group: one general symptom, one mental\/emotional symptom and one physical symptom. You can prescribe on a single symptom but this is rarely successful. If you work out a remedy based on a symptom from each of these groups you can be confident that it has a good chance of working. If you add in a stress symptom you can be even surer of your prescription. The more symptoms that fit the remedy picture the better.\n\n2 Repertorise\n\nBlank Repertorizing Chart: <http:\/\/www.mirandacastro.com\/main\/documents\/RepChart.pdf>.\n\n'Repertorising' sounds much more complicated than it is. List the symptoms you've chosen on a separate piece of paper or on a copy of the Repertorising chart. Then look up each one in the Repertory and list the remedies that occur under each symptom, or alternatively tick them off on the repertorising chart. When you have completed the list you will be able to see which remedy, or remedies, occurs most often \u2013 which contains all or most of the symptoms on your list. As you become more familiar with the remedy pictures you will learn to recognise the remedy your patient needs without repertorising.\n\n3 Differentiate\n\nIn some cases only one remedy will contain all your symptoms. Go to the Materia Medica and read through the remedy picture to check that it fits. If it does, prescribe that remedy, following the potency and dosage guidelines set out here.\n\nOften, however, more than one remedy will contain all your symptoms, or several remedies will have most of your symptoms and none will have all of them. If more than one remedy is indicated according to the repertorising chart, choosing between them can be difficult; it is one of the skills that makes a competent homeopath. Read through each of the remedies that has all or most of the symptoms and pick the one that fits best.\n\nIf you can't find a remedy that fits your picture, then go back to your chart and see if you have missed any remedies. Check your case for missed symptoms. Add these to your list and repertorise them. Then read through any new remedies that emerge through this process. Repeat these steps until you find a remedy that fits. If you are really stuck, you may have to go back to square one and take the case history again from scratch.\n\nNB Your general and emotional\/mental symptoms may point to a remedy that does not list the physical complaint you are suffering from in its picture. For example, you have flu and are feeling restless, irritable, hot and bothered and extremely thirsty. You find that Sulphur comes up strongly for your mental\/emotional and general symptoms but isn't listed under Flu in the Repertory. The closest remedy is Bryonia but when you read the Bryonia picture you don't recognise yourself. When you read Sulphur you immediately think 'that's it' and are able to identify a number of other symptoms which are characteristic of you whether you are well or not, for example, you hate to stand and have always suffered from hot feet. You check your tongue and find that it also fits the Sulphur description.\n\nEven though the remedy doesn't have flu as one of the physical symptoms, because it is 'constitutional', and because it is strongly indicated for you, that is, for your emotional and general picture, it will work.\n\n* * *\n\nPRESCRIBING\n\n* * *\n\nNever rush into prescribing for someone else. Don't think that you have to give a remedy just because you have the book and someone wants you to. If you don't feel right about working out a remedy for someone you don't know very well, or are unsure about what remedy to give, then don't do it. Suggest they go to a professional homeopath or to their GP. If you are thinking of prescribing for yourself then proceed cautiously because it is notoriously difficult to prescribe accurately if you are feeling unwell.\n\nIf you are already under the care of a professional homeopath, it is preferable that they are consulted before you prescribe. It is better to give nothing than to interfere with what may be an aggravation (see here) and therefore not fully understood by you. However, in an accident or an emergency you can take the indicated remedy (Arnica, Aconite or Hypericum, for example) immediately, but make sure that you seek additional professional help if needed and write down the remedy you took as well as its response for reference.\n\nWhom to Prescribe For\n\nWhen prescribing for adults, however close they may be to you, remember this rule: prescribe only if you are asked for help. Over-zealousness can be very off-putting and if you force a remedy on to someone the chances of its being sabotaged by antidoting (see here) are very high. If there is an improvement, it will be attributed to coincidence. It is always possible to offer help and leave it at that. Don't get into the way of thinking that you can cure anybody and everybody of their complaints or impinge on other people's right to look after their own health in their own way.\n\nIf you are self-prescribing, it is advisable to discuss your remedy with a friend to make sure that you are not deluding yourself about your true state of body and mind.\n\nBabies and children respond very well to homeopathic treatment. If they do not respond quickly to your first-aid prescribing, seek the advice of a professional homeopath.\n\nREMEDY FORMS AVAILABLE\n\nHomeopathic remedies are most commonly available as tablets made of sugar of cow's milk, or Saccharum lactose, commonly known as sac lac; this has been found to be the ideal medium for the potentised remedy. Sac lac comes in several forms: hard or soft tablets, globules or powders. Most homeopathic pharmacies will automatically make up remedies in the form of hard sac lac tablets unless they are specifically asked otherwise.\n\nListed below are the different forms a homeopathic remedy can take. In each case a few drops of the potentised remedy in alcohol are used to medicate (or moisten) the base substance.\n\nSoft tablets (sac lac) dissolve quickly and easily under the tongue and are also easily crushed for administering to babies. (See here.)\n\nHard tablets (sac lac) do not dissolve as easily as the soft tablets. They should be chewed and held in the mouth for a few seconds before being swallowed. They can also be crushed.\n\nGlobules (sac lac) are tiny round pills, like poppy seeds. A few grains should be dissolved on the tongue and not a lidful as is sometimes suggested (in theory one grain should suffice \u2013 a single dose is a single dose, it is not the size that counts).\n\nSucrose (plain sugar) is sometimes used, especially for pocket travel kits. A few grains should be dissolved on the tongue. They can (like the globules) be tipped on to the palm of the person taking the remedy, or straight on the tongue.\n\nLiquid potencies Homeopathic remedies can be made up in liquid form for children or adults known to be allergic to cow's milk. The remedy is added to an alcohol base, supplied in dropper bottles; the dose is either administered neat on the tongue or diluted in water for babies and children, in which case 5 drops should be added to a little water and held in the mouth for a second or two before being swallowed.\n\nPowders (sac lac) Most homeopathic pharmacies can make up remedies in powder form. These are wrapped individually in small squares of paper and are convenient if you need only a few doses of an unusual remedy that you are unlikely to want again, or if you need to send small quantities by mail, especially abroad. They should be dissolved on or under the tongue like soft tablets or added to a small amount of water and held in the mouth for a few seconds before swallowing.\n\nWafers Selected pharmacies can provide remedies in wafers, which, like powders, are wrapped individually. The wafers are made of rice paper, which is useful for people sensitive to milk and sugar.\n\nStoring Remedies\n\nHomeopathic remedies will keep their strength for years without deteriorating: remedies made over a hundred years ago still work well. They should be stored in a cool, dark, dry place with their tops screwed on tightly, well away from strong-smelling substances: strong smells cause the remedies to lose their potency. It is not a good idea to keep them in a bathroom cabinet alongside perfumes and cough mixtures, or in a spare-room cupboard with the mothballs. A sealed plastic container like an ice-cream tub is good for storing loose bottles or you may choose to purchase one of the ready-made kits available from many pharmacies, to your own specifications if you wish (see here).\n\nIt is wise to keep all tablets out of the reach of children as a matter of course. If your child eats the entire first-aid kit in one glorious secret feast, do not panic. Your bank balance is the only thing that will suffer! A single dose at a time is a single dose, whether it is one pill or one bottle of pills. If your child eats a full bottle of, say, Chamomilla 6 or even 30 tablets it will have roughly the same effect as taking one tablet.\n\nHow to Take Remedies\n\nCarefully tip a tablet into the lid of the bottle. If more than one falls out, tip the others back so that only one remains. Tip it on to the palm of the person taking the remedy and then replace the lid on the bottle. You can touch your own tablets, but if you are giving the remedy to someone else try to avoid touching it.\n\nNever put back tablets that have fallen out on to the floor or anywhere else, or that you have given out and are unused. In so doing you may contaminate your stock. Always throw them away.\n\nSoft tablets, powders, sucrose, globules and wafers should be dissolved under the tongue where they are absorbed into the bloodstream; if they are swallowed whole they become mixed with the stomach acids and work less effectively. Hard tablets should be chewed, held in the mouth for a few seconds and then swallowed. Liquid potencies can be dropped on to the tongue or diluted with a little water and held in the mouth for a few seconds before swallowing.\n\nIt is preferable not to eat, drink (except water), smoke or brush your teeth for 10\u201320 minutes before and after taking a remedy as this gives it the best possible chance of working, although in practice remedies given to toddlers who eat before and after still work well. The 10-minute gap makes sure that residues of food do not affect the action of the remedy.\n\nMany homeopathic remedies available in the chemist or wholefood shop give dosage instructions on the bottle, suggesting different doses for adults (two tablets) and children (one tablet). This is questionable given that there isn't a measurable quantity of the medicine in the tablet. The size of the dose is immaterial; it is how often it is taken that counts. I instruct my patients to take one tablet, whatever their age.\n\nTablets for babies can be crushed between two spoons and the powder tipped dry on to the tongue. This is hard to spit out, unlike a tablet. A little water can be added to the crushed powder on the spoon, or the tablet can be dissolved in a clean glass with a little water. Stir it vigorously and then give as needed, a teaspoonful at a time. Scour the glass and the spoon (with boiled water) after use so that the next person to use them doesn't get an inadvertent dose of the remedy.\n\nHow Many Doses and How Often\n\nDo\n\n\u2022 prescribe according to the urgency of the case.\n\n\u2022 stop on improvement.\n\n\u2022 start again if the same symptoms return: repeat as needed.\n\n\u2022 if you have given six doses and have had no response, stop and reassess the case or seek advice.\n\n\u2022 change the remedy if the symptom picture changes.\n\n\u2022 if the 'patient' is already receiving homeopathic treatment, consult the homeopath, if possible, before prescribing (unless it is an emergency).\n\nAlways seek professional help from your homeopath or GP if your symptoms recur or do not improve.\n\nHaving selected the remedy, you will need to decide on the dose. This depends on the urgency of the case: frequent doses for an acute illness that has come on suddenly and strongly, and less frequent doses for a slowly developing illness. Some earaches with severely distressed patients need the remedy to be repeated every five minutes; a head injury may need the remedy repeated every half-hour; and a slowly developing flu may need a remedy every four hours or three times a day. The dosage chart (here) will help you to establish the frequency.\n\nOnce you see that a remedy has started to work \u2013 when you notice a definite reaction or change resulting from the remedy \u2013 continue with the same remedy but give it less often, increasing the gaps between the doses.\n\nOnce there is a marked improvement \u2013 a strong positive reaction having occurred \u2013 it is essential to stop the remedy. This is the absolute opposite of the instructions you receive when taking orthodox medicines (for example, antibiotics), where it is necessary to finish the whole course of tablets. A homeopathic remedy acts as a trigger, a catalyst: it stimulates the body to begin to heal itself, and once that has happened the body's own healing process will take over. In some cases, taking another tablet after this reaction has occurred can stop the remedy working and if too many are taken it is possible to start proving the remedy (see here).\n\nPrescribe up to six pills according to the urgency of the case. If you have given six pills and there has been no change whatsoever, then it is probably the wrong remedy, unless you are prescribing a 'tonic' or remedy that needs to be taken over the course of a week or so (like remedies for anaemia or exhaustion).\n\nWhich Potency?\n\nHomeopathic remedies come in different potencies. The 6X, 6C, 12C and the 30C are all safe for the home prescriber. The 6X and the 6C are the most widely available; they are the best potencies to use for minor complaints. 12C is slightly stronger than 6C, and 30C is stronger still so fewer doses are needed. It tends to work faster and is useful for more serious complaints, such as bad burns or head injuries with concussion. If you are ordering a kit from a pharmacy I suggest you order the remedies in the 6C or 12C potency to start with. As you gain experience in prescribing, you'll begin to know when one of the other potencies is more appropriate.\n\nAssessing the Response\n\nA homeopathic remedy is like a pebble which creates healing ripples when thrown into your pond. The prescriber must get the 'pebble' as close to the middle of the 'pond' as possible, so that the 'ripples' reach out to every part of the patient. If you have given six doses and have had no significant response then the remedy you have chosen is likely to be wrong. Did you have more than one remedy to choose from? How did you make your choice? Check your symptoms. Read through the remedy pictures in the Materia Medica again. If you are convinced that your choice of remedy was correct, persevere for another three to six doses. If your prescription does not work, you may have missed the real cause or stress. Homeopaths will delve relentlessly until they uncover what it was that weakened an individual, causing them to become ill, so seek professional homeopathic advice if your complaint is particularly debilitating, and self-prescribing is not helping. If you are prescribing for someone else you should refer back to your notes to see whether you need any specific or extra information. For example, your friend has heartburn which is worse for eating: you realise that you don't know whether she is having difficulty swallowing liquids and whether hot or cold drinks help. Ask some more specific questions and gauge how strongly marked the symptoms are by the response you get. An unequivocal 'yes' or 'no' counts; a 'maybe' or a 'sometimes' doesn't.\n\nHaving reassessed the symptom picture, you may decide to select another remedy. The same guidelines apply again: give up to six doses, according to the urgency of the case. If you become unsure, however, it is better to stop prescribing and either get professional help or let nature take its course.\n\nWrite down exactly what you have done at each step, with notes to explain your reasons, plus the results of your own prescribing. (If you decide to do nothing and your patient gets better, make sure you write that down too.) Although it is hard to imagine, it is the easiest thing in the world to forget the one brilliant remedy you gave your baby for a cough, and here they are with another one six months later...\n\nIn an acute illness like an earache or a bruised head, one, two or three doses may be enough to begin the healing process. If you feel better in yourself you will get better even if your physical symptoms remain the same or get slightly and temporarily worse. If your symptoms improve for a while and then relapse, a repeat dose of the same remedy may be necessary \u2013 but only on return of the same symptoms. If the symptom picture changes, it is likely that you will need a different remedy.\n\nThe Homeopathic Aggravation\n\nThis is the term given to the worsening of symptoms that may occur after a constitutional remedy has been taken. This can happen during an acute illness and either the symptoms may worsen or an old symptom may surface temporarily (see Laws of Cure). If an old symptom surfaces after a good prescription, wait to see if it will clear of its own accord. If it does, the remedy is still working... carry on waiting.\n\nThere may be a more general clearing out, or 'healing crisis' as it is often called, in the form of a streaming cold or diarrhoea. The aggravation can occur on a mental\/emotional level with some patients feeling very weepy if, for example, it is suppressed grief that has caused a general lowering of their health and the physical complaint.\n\nThe higher potencies (200C and above) are especially renowned for their ability to cause aggravations and that is one of the main reasons why a first-aid prescriber should stick to the lower ones.\n\nAntidotes\n\nThe subject of antidotes always arouses heated discussion among homeopaths. There are those who believe that coffee strongly counteracts the action of a homeopathic remedy, while others believe that it has no effect whatsoever. I believe that it depends on the individual, often someone who is generally sensitive and finds coffee gives them insomnia and palpitations will experience coffee as an antidote. In this case it is wise to stop drinking it while the remedies are being taken and for a short time afterwards. If you are sensitive to coffee and stop drinking it while you are taking your remedy, note whether your symptoms return when you start drinking it again. If they do, you may have to stop drinking it and take another short course of the remedy to clear your symptoms once more.\n\nThe following all counteract the effects of a homeopathic remedy to some extent. They are not necessarily 'bad', but as the action of a remedy can last for as little as a few days or as long as a few months, they are strong enough to stop it working and should be avoided while it is being taken and for several weeks afterwards.\n\nCamphor In 'tiger balm', deep-heat ointments and many lip salves\n\nCoffee See here\n\nMenthol\/eucalyptus In cough mixtures, Karvol capsules, 'tiger balm', Fisherman's Friend, Vick, Olbas Oil, etc.\n\nPeppermint In regular toothpaste and strong peppermint sweets. Natural, fresh mint in cooking and the odd cup of peppermint tea is fine. Alternatives to ordinary, minty toothpaste are fennel toothpaste (available at most health-food shops), salt water or bicarbonate of soda. A solution of Calendula is an efficient mouthwash, both at home and at the dentist (see External Materia Medica)\n\nAny strong-smelling or strong-acting substance will affect a homeopathic remedy. Some people have found that a spicy curry will have an adverse effect, as might a night of heavy drinking.\n\nAn emotional or physical shock can also stop a remedy working; for some people this can mean a piece of bad news; for others a visit to the dentist with the stress and strain of treatment.\n\nIf your remedy has 'stopped working' (you took it and your symptoms cleared for a time and have now returned), ask yourself if you have antidoted the remedy. If you think you have, then repeat your last remedy: sometimes a single repeat is enough.\n\nAnd Finally...\n\nI would encourage anyone who is taking homeopathic prescribing seriously to enroll for a short homeopathic first-aid course. Many adult education colleges run courses and most homeopathic practitioners also run their own on a private basis. Being signed on with a local homeopath is also important as you will have someone who knows your case (or those of your children) with whom you can check before you self-prescribe. I encourage my patients to ring in with a list of symptoms if, for example, their child has flu, and make sure that it is all right to give, say, Gelsemium, or whether it will interfere with the child's long-term constitutional treatment. This also enables them to confirm their choice of remedy in the early stages of getting to know the remedy pictures.\n\nI also strongly advise you to take a first-aid course with the St John Ambulance or the Red Cross to learn the mechanics of first-aid.\n\n* * *\n\nCOMPLAINTS YOU CAN TREAT USING THIS BOOK\n\n* * *\n\nIt is important to understand fully the difference between acute and chronic disease so that you know which illnesses you may safely and appropriately treat and which you must take to a professional homeopath (or your GP). For an explanation of the differences between the two see here.\n\nThe complaints sections in each chapter will tell you whether the particular complaint you wish to treat is within your scope as a home prescriber, before you attempt to work out a remedy. A good general first-aid book or family medical encyclopaedia will give you more detailed information about the complaints themselves. If the complaint has an obvious cause it is important to remove or avoid specific stresses (for example, don't let your child get chilled if you know that it leads to earache). Once you become ill use common-sense measures and take sensible care of yourself to avoid more serious symptoms developing.\n\nHaving established that your complaint is within the scope of this book, you can start to work out the remedy you need.\n\nYour complaint may be treatable either by internal or external remedies, or both. You can establish this in each case by looking up your symptoms in the Repertories (here and here).\n\nA criticism levelled at many homeopathic first-aid books is that they encourage people to take their lives in their own hands by treating serious illnesses at home. I believe that the treatment of chronic complaints is inadvisable. Never treat serious injuries or complaints yourself. If in any doubt seek expert advice. Cause for Concern, lists general alarm symptoms, and specific alarm symptoms to watch out for are indicated in the individual complaints sections throughout the book.\n\nDISEASES AND CONDITIONS NOT COVERED BY THIS BOOK\n\nThis book does not cover chronic diseases; these are complex conditions and need careful diagnosis and treatment at all stages. The homeopathic treatment of chronic disease often requires a long-term commitment so that the homeopath can treat underlying weaknesses in the constitution.\n\nThe following symptoms are not dealt with in this book:\n\nAsthma is a life-threatening, deep-seated chronic disease which needs careful management to cure it. Acute attacks of asthma can be alleviated by homeopathic remedies, but these should always be prescribed by a professional homeopath who is in charge of the whole case and prescribing constitutional remedies between attacks so that their severity and frequency is lessened\n\nFrequently recurring symptoms, such as flu, diarrhoea and coughs, which have no obvious cause and occur as often as every week\n\nHayfever see Asthma\n\nLumps and bumps: cysts, growths or warts anywhere on or in the body must always be taken to a professional homeopath\n\nPersistent abdominal pain\n\nPersistent constipation can mask a more serious, underlying complaint that needs professional treatment. If it is simply the result of poor diet (lots of low-fibre junk food) then the first step is to make the necessary dietary changes\n\nSerious degenerative diseases such as cancer, hepatitis, heart disease and AIDS\n\nSkin symptoms, including eczema, psoriasis, dermatitis, should never be tackled by the first-aid prescriber. Read the Laws of Cure to understand the dangers of suppressing a skin disease\n\nUlcers, anywhere except occasional mouth ulcers which you can treat yourself\n\n* * *\n\nCAUSE FOR CONCERN\n\n* * *\n\nThe following symptoms may indicate serious illness and signal that you should seek immediate professional help. Some also appear in the remedy pictures, for example the laboured breathing of an Antimonium tartaricum cough and the delirium of a Belladonna fever. If you are worried about the general state of either yourself or your baby, call for help and then give the indicated homeopathic remedy. Further treatment may not be needed if the remedy works. In some instances, where the picture is very clear and\/or you know from past experience that your patient is not seriously ill, you will be able to give the remedy and wait for improvement. If your baby does not show rapid signs of improvement you should call for help.\n\nSEEK HELP IF THERE IS\n\nBleeding: unexplained, from any part of the body, including the skin\n\nBreathing: rapid \u2013 over 50 breaths per minute at rest in children under two;\n\nshallow or laboured (difficult)\n\nChest pain: severe\n\nConvulsions\n\nDelirium\n\nFever: above 104\u00b0F\/40\u00b0C;\n\nhigh, with a slow pulse (normal adult pulse is about 90 beats a minute and 120 in a child);\n\npersistent, lasting for longer than 24 hours in a baby\n\nMental confusion, uncharacteristic\n\nNeck stiff, especially if accompanied by severe headache\n\nPain that is severe, especially if accompanied by one or more of the other symptoms in this section\n\nStools, pale \u2013 grey or almost white\n\nUrination profuse, accompanied by a great thirst\n\nUrine dark and scanty\/bloody (certain foods when eaten in quantity can change the colour of urine; beetroot for example, can turn urine red. This is nothing to worry about)\n\nVomiting, unexpected, repeated which comes on some time after the onset of a viral infection (i.e. a childhood illness)\n\nWeakness, extreme\n\nWheezing, severe\n\nYellowing of the skin or whites of the eyes\n\nIf you are concerned about a baby of up to a year old, even if you can't put your finger on why, you should telephone your doctor or homeopath. If neither is available you can always take your child to the nearest hospital casualty department for reassurance.\n\nIn addition to the above causes for concern seek help for your baby if there is:\n\nBlueness around the lips or face \u2013 whether temporary or not\n\nBreathing difficulty with or without wheezing\n\nConvulsion\n\nPain \u2013 your baby is obviously in pain and you don't know where the pain is\n\nSudden rash \u2013 small red or purple spots or bruises\n\nUnconsciousness or unusual drowsiness or listlessness without drowsiness\n\nUnusual pallor which doesn't change \u2013 especially if it is all over the body\n\nA combination of two or more of the following:\n\nhigh fever\n\nirritability or drowsiness\n\nan altered cry (especially if it is high-pitched or a weak moaning)\n\ndiarrhoea and\/or vomiting\n\ndry mouth and tongue\n\nsunken eyes\n\nsunken or bulging fontanelle\n\nrefuses feeds\/drinks\n\ndislikes bright lights (unusual)\n\npasses much less urine than usual\n\npasses blood in stools\n\nvomits green fluid\n\nstops focusing \u2013 eyes glaze over\n\n'grunts' with each breath\n\nhas visible 'dips' in the chest when breathing\n\nIf your baby seems ill, even if he or she does not have any easily identifiable symptoms, always trust your instincts and get help.\n\nAlways seek professional help if the symptoms recur or do not improve.\n\n* * *\n\nSAMPLE CASES\n\n* * *\n\nThe following cases will bring the process of first-aid prescribing to life for you. Keep in mind the steps with which you need to be familiar to prescribe successfully:\n\n1 Check that your complaint is within the scope of this book by looking up the appropriate section, for example, Complaints in pregnancy (here) if you are pregnant and want to prescribe on, say, anaemia. Take note of the practical advice, the Dos and Don'ts before you move on to work out a remedy.\n\n2 'Take the case'. Choose your symptoms (using the case-taking chart here).\n\n3 Repertorise your symptoms using a piece of paper or the repertorizing chart which you can download or print: <http:\/\/www.mirandacastro.com\/main\/documents\/RepChart.pdf>.\n\n4 Differentiate between remedies if more than one is indicated, by reading through the remedy pictures in the Materia Medica until you find the one that 'fits' best.\n\n5 Prescribe (see here).\n\nPREGNANCY\n\nCase 1\n\nFood poisoning\n\nYou are four months' pregnant, over the nausea and exhaustion of the first three months and really enjoying this time. You are taken out for your birthday and wake up some time after midnight with a nasty attack of food poisoning \u2013 it must have been the meat casserole that only you ate. You don't know whether to sit on the toilet or lean over it because you feel like vomiting and passing diarrhoea at the same time. The diarrhoea is very painful and utterly exhausting. You feel faint after vomiting, and, in spite of being thirsty, can only keep down small sips of water. You are terribly anxious that it might be affecting the baby and feel like death at the same time. You don't want to be on your own even for a minute. Your partner scribbles down the following symptoms: (see Chart 1)\n\nComplaint Food Poisoning\n\nWhen did it start? Suddenly. Midnight\n\nCaused by Meat\n\nSymptoms Diarrhoea \u2013 painful. Vomiting \u2013 feels faint after\n\nGeneral symptoms Thirsty for sips\n\nEmotional state Anxious, worse when alone\n\nOther physical symptoms Exhaustion from diarrhoea\n\nArsenicum fits most symptoms; even though it is missing from 'Onset of complaint, sudden' it doesn't matter as you would expect acute food poisoning to come on suddenly. You will need to repeat the remedy frequently, every 15\u201330 minutes and you can expect it to work quickly if it is the right one, within an hour or less.\n\nIf it doesn't help, reassess the whole picture. It may be that you are vomiting a few minutes after drinking and not immediately, which would lead you to think of taking Phosphorus.\n\nCase 2\n\nExhaustion\n\nAfter the third month of pregnancy, when the morning sickness passes, you still feel tired more or less all the time with dips in the day when you just can't move a limb. Sleep doesn't help much. Eating temporarily perks you up but it doesn't last. You feel apathetic, indifferent to everything and snappy. You bite the head off anyone who tells you what to do \u2013 especially if it contradicts what you want to do.\n\nYou rouse yourself to jot a few symptoms down in a desultory way as follows: (see Chart 2)\n\nComplaint Exhaustion\n\nCaused by Pregnancy\n\nWorse Slightest exertion\n\nGeneral symptoms Appetite lost in pregnancy. Better after eating.\n\nEmotional state Apathetic. Angry from contradiction. Repertorising the above symptoms comes up with Sepia followed by Natrum carbonicum and Phosphorus. You are able to cross Natrum carbonicum off your shortlist because you are a person who feels the cold acutely. When you read the Sepia emotional state and the general symptoms you immediately identify yourself with the saggy individual who's gone off sex and remember how when you had to run for the bus a few days previously you felt quite invigorated for a few hours after.\n\nIf this stage of repertorising hadn't helped you to come up with a remedy, you would have had to list some more symptoms, possibly getting some help with this and repertorising again, until a picture emerged.\n\nCase 3\n\nAnaemia\n\nYou are seven months' pregnant and, after a glowing first six months, are starting to feel as though you've had enough. A routine check by the doctor identified anaemia, so your doctor prescribed iron tablets anyway. You became constipated from taking the tablets, which has continued even though you stopped taking them a week ago. You look very pale.\n\nYou've been getting on badly with your partner who is working all hours to be able to take time off after the birth. You feel neglected and resentful but think you should keep your negative feelings to yourself, that it wouldn't be right to 'dump' them. You have noticed that you've been snapping at the people you work with more than usual and finding the noise levels in the office almost unbearable \u2013 you just want peace and quiet.\n\nYou have had a lot of abdominal pain in the past month; you have an active baby who is prone to delivering kicks that belong on the sports field! You are getting very fed up with it. You repertorise the following (knowing that it is a limited collection of symptoms): (see Chart 3)\n\nComplaint Anaemia\n\nSymptoms Face pale\n\nEmotional state Resentful. Sensitive to noise\n\nOther physical symptoms Constipation. Abdominal pain\n\nJust as you thought, many remedies are indicated including Calcarea carbonica, Kali carbonicum, Lycopodium, Natrum muriaticum, Nux vomica, Staphysagria and Zincum. On reflection you realise that you are feeling very resentful towards your partner and the baby. You feel primarily battered by the baby and abandoned by your partner. And because you haven't talked about these feelings you have become touchy, erupting angrily over trivial things.\n\nYou prioritise the symptoms 'resentful' and 'abdominal pain', which leaves you with Staphysagria. On reading through the picture it is clear to you that you need an urgent dose of Staphysagria and some time to come clean with your nearest and dearest. You guess that if you let go of your grudges your bowels will clear out too.\n\nBIRTH\n\nCase 1\n\nLabour\n\nIn labour you find yourself coping really well although your baby is posterior. You are finding the backache manageable and are coping well with the contractions. You take Arnica from time to time and the odd Kali phosphoricum when you get tired. Then you hit a rough patch: your midwife tells you that you are (only) 3cm dilated when you thought you were much further along. Your back begins to hurt more with contractions, you feel tired and suddenly discouraged \u2013 almost despairing. As you take a dive your contractions weaken in sympathy. In between contractions, with the help of your partner, you jot down the following symptoms and scribble the remedies listed next to them:\n\nLabour pains in the back: Cham., Cimi., Coff., Gels., Kali-c., Nux-v., Puls., Sep.\n\nLabour pains, weak: Bell., Cimi., Gels., Kali-c., Nat-m., Op., Puls., Sec.\n\nWith exhaustion: Bell., Caul., Cham., Kali-c., Kali-p., Nat-m., Nux-v., Op., Puls., Sec., Sep., Ver-a.\n\nThe remedies that are present in each symptom are Kali carbonicum and Pulsatilla. You check the main entry for Exhaustion in the Repertory and see that Chamomilla, Gelsemium, Nux vomica, Pulsatilla and Sepia are also listed. You need some more symptoms. You know that you've been sweating profusely since the pains increased in intensity and that you have been more sensitive than usual to cold; you add the following symptoms to your jottings, writing only those remedies which agree with your shortlist:\n\nSweat \u2013 profuse; Kali-c., Sep.\n\nWorse for cold: Kali-c., Nux-v., Puls., Sep.\n\nDespair in labour: Coff., Gels., Sep.\n\nSo, it's a showdown between Kali carbonicum and Sepia even though Kali carbonicum isn't 'despairing'. Your partner reads each picture quickly and there's no contest, you are being manipulative and anxious and not at all saggy. A few doses of Kali carbonicum restore your emotional equilibrium and help to ease the back pain \u2013 enough to establish effective labour once again. NB I have included this case to demonstrate that it is possible to repertorise and prescribe 'on the back of an envelope', without filling in a chart, especially in a situation where you don't have a lot of time.\n\nPOST-NATAL \u2013 MOTHER\n\nCase 1\n\nPost-natal depression and afterpains\n\nYou were warned about this but you didn't believe it. You've had the baby you always wanted, had a perfect first few days and then vooom! On the fifth day after the birth a black cloud descends. You feel tearful, all the time, the least little thing sets you off. You'd really rather others weren't so sympathetic. Your milk came in yesterday and your breasts are like footballs, huge, hard and painful. Your baby fed every 1\u00bd hours last night, your nipples are beginning to crack and you are absolutely exhausted. As if this weren't enough you are still getting some afterpains, especially when you feed. You are a generally chilly, sweaty type and are finding that you are sweating more than ever since the birth. You repertorise the following symptoms: (see Chart 4).\n\nComplaint Depressed. Tearful\n\nWorse Consolation\n\nGeneral symptoms Sweat profuse\n\nWorse Cold\n\nOther physical symptoms After pains. Breasts painful.\n\nNipples cracked\n\nSilica and Sepia come out as having the most symptoms in the repertorising. This surprises you: you expected Sepia to be the remedy, based on an occasion in your pregnancy when you needed it. You read the Silica picture in the Materia Medica and are able to add some other symptoms to confirm this remedy as your choice. Your episiotomy scar is taking a long time to heal \u2013 you had thought that as long as you took lots of Arnica and bathed it with Calendula it would heal quickly and it hasn't. You wonder about Silica being a good constitutional remedy for you since you are generally stubborn and tenacious, you feel the cold, have sweaty feet that smell and had constant chest colds as a child. If it is a good remedy you can be assured that it will work quickly and effectively on your physical symptoms and your emotional state.\n\nCase 2\n\nFlu not needing treatment\n\nYour baby is ten months old when you go down with an attack of flu. You've been back at work for six months now, leaving the baby with a full-time nanny from 8.30 a.m. until 6.30 p.m. You have been working extremely hard with very little time off these past six months: your time 'off' has been taken up with being a mother, and your precious bundle still wakes once or twice at night. You feel achy, weak and exhausted and although your symptoms are unpleasant they are not severe. You agree (reluctantly!) to rest in bed for a few days and then to take a week off to recuperate fully. Your nanny will continue to look after the baby in the day and your partner agrees to the evenings and the night shifts until you are recovered.\n\nYou do not need homeopathic treatment. It might be necessary to review the situation if you didn't recover easily at any stage of this illness. It isn't wise to take a homeopathic remedy and carry on working when all your body needs is a very good rest.\n\nCase 3\n\nFlu needing treatment\n\nYou are a single parent with two children \u2013 a toddler of two and a half and a baby. You go down suddenly with a severe bout of flu: every bone in your body aches, you have a fever, feel depressed and angry and extremely chilly. You have no one to help with the children, although a friend offers to shop for you. Homeopathic treatment in this case is appropriate because of the severity of the symptoms, and also because of your situation \u2013 it just isn't possible to take two weeks off. You choose the following symptoms: (see Chart 5)\n\nComplaint Flu\n\nSymptoms Pains in bones\n\nGeneral symptoms Exhaustion. Fever\n\nWorse Exertion. Cold\n\nEmotional state Angry. Depressed\n\nOn repertorising, you find that Arsenicum, Nux vomica and Rhus toxicodendron are all strongly indicated but when you read through the picture in the Materia Medica it doesn't quite fit. You are not restless.\n\nYou search for other symptoms. You are very thirsty for cold drinks, in spite of feeling generally worse for getting cold. You are very shaky in your whole body with the pains. You add the following symptoms:\n\nThirsty. Likes cold drinks. Flu with shivering You then read through Bryonia, Eupatorium perfoliatum, Phosphorus and Pyrogen and decide to take Eupatorium because of the terrible pains, because you also have a headache and aren't sweating much with the fever. You ask a friend to buy some from the local whole-food shop on her way home.\n\nDon't forget to look to the practical advice (DOs and DON'Ts) in each instance as well as taking the appropriate homeopathic remedy.\n\nBABIES\n\nCase 1\n\nAn accident\/head injury\n\nYour nine-month-old baby climbs onto the sofa while you answer the telephone. It's her first serious climbing expedition, hugely aided by a pile of cushions. You just miss saving her from falling backwards off the sofa head first and unfortunately, she catches the back of her head on a pile of wooden bricks. She's in great distress but the skin hasn't been broken and you know it wasn't a serious enough fall to warrant a visit to casualty. An egg is fast developing \u2013 a swelling where the fall was broken by the edge of a brick.\n\nYour first prescription must be for Shock, Head injury and Bruising. In the Repertory you will find that Arnica is the only remedy that is listed for all three of these symptoms. You give her a single dose of Arnica 12 and she falls asleep. The swelling goes right down and you don't worry about it any more.\n\nOver the next week you suspect that she is teething because she is more bad-tempered than usual and wakes frequently at night screaming. She asks to be carried but soon starts screaming again after you pick her up. Nothing comforts her for long. You begin to feel exasperated \u2013 your happy, contented baby has turned into a 'ratbag' and you find the nights especially trying. You repertorise her symptoms (teething, angry, capricious) and give her Chamomilla. It has no effect whatsoever. You are convinced that she is teething and so you persevere with it for a few days. She continues to be difficult, both day and night, and you are running out of sympathy and energy.\n\nSince Chamomilla was so strongly indicated and didn't work you know that it is the wrong remedy and you begin to wonder if she is teething. It is unlikely that she has a sore throat or earache as she has no fever but you wonder if you should get your GP to check her just in case. You decide to go over your notes again. And then you have a brainwave: you notice that her irritability started on the day that she had her fall. So much was happening that day with your other children that you didn't connect the two events. You know that she is in pain and you know that she fell, so you wonder whether she has a headache from the head injury (a surprisingly common but difficult-to-diagnose symptom in pre-verbal infants). You give a single dose of Natrum sulphuricum 30 and as it dissolves in her mouth she smiles angelically again. That night she sleeps and regains her former good nature.\n\nCase 2\n\nChickenpox not needing treatment\n\nYour nine-month-old baby develops mild cold symptoms about 10 days after having been in contact with your nephew who has chickenpox. You feel sure your baby is incubating chickenpox and watch carefully. You decide not to prescribe for the cold as it isn't preventing him from feeding or sleeping. After a couple of days he throws a fever of 102\u00b0F\/39\u00b0C and although a bit miserable continues to drink well and sleep a lot. When awake he wants attention and lots of cuddles and as long as these are forthcoming is reasonably happy. About a dozen spots appear well scattered all over his body which quickly scab over and don't seem to itch much. You remember reading that young babies can have chickenpox very mildly and decide not to treat yours as he is coping with it well on his own.\n\nCase 3\n\nChickenpox needing treatment\n\nYour one-year-old develops chickenpox, starting with the typical cold symptoms. A high fever develops and you start to worry when delirium quickly sets in. The baby feels burning hot to the touch and doesn't want to eat but is unusually thirsty. The spots come out all over the body \u2013 there are hundreds of them and they are very red and itchy. Nothing seems to help the itching but they are much worse for heat. They seem to be painful because he cries after scratching. A right-sided earache develops and a dry cough, which prevents him from sleeping. You decide to work out a remedy. He is now very distressed, restless and irritable. Homeopathic treatment is appropriate in this case to ease the pain, control the fever and help the baby deal with this very unpleasant 'attack'. You write up the case using the case-taking and repertorising charts and choose the following symptoms: (see Chart 6)\n\nComplaint Chickenpox\n\nSymptoms Rash \u2013 itchy\n\nGeneral symptoms Thirsty\n\nWorse Heat\n\nEmotional state Delirious. Irritable.\n\nOther physical symptoms\n\nEarache \u2013 right side\n\nDry cough\n\nFever \u2013 burning heat\n\nBelladonna and Sulphur are both well indicated but when you read them up in the Materia Medica you eliminate Sulphur, because it has left-sided earaches and it doesn't have the delirium of Belladonna. You note that your baby's pupils are dilated and a little wild-looking and that the measles rash in a Belladonna infant itches and burns and rightly assume that it is possible for the chickenpox rash to do likewise, which confirms your choice. You hold Sulphur in reserve in case Belladonna doesn't help.\n\nCase 4\n\nCough\/cold\n\nYour six-month-old baby has a cough and cold. It started four days ago. The day previously, you had both been out shopping when it had rained unexpectedly and she was drenched even though you ran home as fast as you could. Then her nose started running that evening and she coughed in the night. In the morning your baby was pale and pathetic and didn't want any solids, only to breastfeed, and to be carried and cuddled. She has been more or less like this for the past three days, not letting you do anything and regressing emotionally, crying every time you put her down. Where she would formerly take the odd bottle of water she won't drink and is pretty listless, her nose is running with lots of thick, yellow-green catarrh and she has a loose cough in the mornings which is dry at night. It is worse when lying down at night and if the room is overheated. She also seems more whiny and clingy if too hot. Oddly enough, when you had to go out yesterday to do some shopping together she was a bit better: her nose stopped running, she coughed less and was more lively. There is no fever and no sweating. Using the case-taking chart you choose the following symptoms and repertorise the ones not in brackets. (See Chart 7)\n\nComplaint Cough (and cold)\n\nWhen did it start? (4 days ago)\n\nCause (Getting wet)\n\nSymptoms Dry at night, loose at mornings\n\nBetter Fresh air\n\nWorse Lying down\n\nGeneral symptoms Thirstless. Pale. Complaints from getting wet. (Listless\/appetite lost.)\n\nBetter Fresh air\n\nWorse Heat\n\nEmotional state Whiny. Tearful. Desires to be carried.\n\nOther physical symptoms Common cold with thick, yellow-green nasal catarrh.\n\nOn repertorising (see here) you find that Pulsatilla is indicated under each symptom, and no other remedies are strongly indicated. You give Pulsatilla knowing that it will help quite quickly. In the unlikely event that it doesn't you will need to go through the case again and repertorise choosing different symptoms, or seek the advice of a professional homeopath.\n\nCase 5\n\nTeething\n\nYour baby is eight months old and teething... again. A friend has recommended the homeopathic 'teething granules', but you know better than to give a routine hit-or-miss prescription and when you tried them once in the middle of the night in desperation they had no effect.\n\nUsually a contented, easy-going baby, he has been more fractious of late and waking at night when previously he slept straight through. Unlike many of his friends of the same age, your baby only has two teeth \u2013 and these were produced with great difficulty. His father had difficulty teething as a baby also. You feel desperate at the thought of endless broken nights ahead. When you go to him at night all that is wanted is a drink and a cuddle and he goes quickly back to sleep. However, you notice that his head is unusually sweaty and that his sweat smells sour.\n\nHis bowels have been upset for the past three weeks, with diarrhoea that smells sour and contains undigested food. He has also been more prone to catching colds over the past few months and they seem to be running into each other so that the poor child has a constantly runny nose.\n\nRecently he has been more stubborn about what he wants although generally very placid. You choose the following symptoms: (see Chart 8)\n\nComplaint Teething painful and slow.\n\nSymptoms With diarrhoea \u2013 stools sour, undigested\n\nGeneral symptoms Sweat on head, sour. Catches colds easily\n\nEmotional state Stubborn\n\nCalcarea carbonica has each of these symptoms. On reading through the picture in the Materia Medica you recognise your easy-going baby in many of the general and emotional symptoms. This remedy will not only help him through the pains of the teething and clear up the diarrhoea and runny nose but will also help the teeth to come through faster. Don't worry if you have to repeat it fairly frequently during the teething year: your baby may need it.\n\nPREPARING FOR LIFE AFTER BIRTH\n\n* * *\n\nPREGNANCY\n\n* * *\n\nPregnancy is a time when you can predict nothing. There are no rules. The outcome is unknown and largely unseen. The first-time mother has special delights and difficulties in store for her because this is a completely new experience.\n\nAlthough subsequent babies are often easier \u2013 women know, roughly speaking, what they are letting themselves in for and feel more self-confident \u2013 here again, every pregnancy and every birth is different, there are no rules.\n\nPregnancy can be a time of pleasure if a woman is healthy and happy or a time of misery if she is ill or if there is too much stress in her life. You can do a great deal to help yourself with the physical changes, if they cause discomfort or pain, and the stresses, both practically and homeopathically. I have included both practical and homeopathic advice in this book. Homeopathic treatment is ideal in pregnancy because, provided you follow the guidelines set out in this book, it is safe. It can't harm you or your baby as there are no dangers of toxic side effects.\n\nMuch of the advice in this book applies to your partner as well as to you, for just as pregnancy can be a glorious time for many partners, it can be difficult for others. Partners can feel trapped, insecure, child-like, frightened, anxious about the future, about becoming a parent, about the increased responsibility, about the financial burden as well as joy and excitement. These feelings and many, many others are all common and normal. They need to be expressed and worked through. If feelings surface that don't resolve, I suggest that you and\/or your partner find a professional to talk them through with \u2013 a counsellor or psychotherapist.\n\nYou and your partner need to talk now, more than ever, to stay close. Remember that you both have rights and that it is important that you carry on making demands of each other, asking for what you need \u2013 reassurance, affection and so on. At times, you will both have to compromise. Be true to yourselves and keep finding ways to express who you are and what you need.\n\nPregnancy can be a time of going inwards, a glorious wrapped-in-cotton-wool sensation, a time where life slows down and nothing outside matters quite as much as it did before. There may be a sense of unreality \u2013 an I-can't-believe-there's-actually-a-baby-growing-inside-me feeling, or I'll-believe-it-when-I-see-it.\n\nIf you have never held a baby it can be difficult to visualise yourself as a mother with a small charge to look after. Many first-time mothers focus on the birth as a goal, as an end in itself, and are startled to find themselves on the other side of it, in motherhood, without any idea about 'what happens next'. It is rather like buying a business without planning on how you're going to run it.\n\nI see birth rather like a station on the railroad of life: you get off the train to have your baby and then embark on the journey again. I encourage women to start seeing the birth as a (smallish) part of life's process to put it in perspective and to encourage them not to expect a particular type of birth. Far too many women have sacrificed themselves on the altar of natural childbirth and feel that they have failed when the birth didn't go as they wanted. If you want a natural birth then by all means aim to have one but be aware of other options to turn to without feeling sad or guilty. You may decide, anyway, that you want to take full advantage of the technical help available in hospital and have a completely pain-free birth.\n\nHowever you decide to have your baby, it helps to realise that birth heralds the start of motherhood. Build pictures in your mind's eye of yourself as a mother. If you are pregnant or planning to be, start now by visualising or imagining yourself after the birth, as a mother. Your baby is a few hours old: is it a girl or a boy? How are you feeling? What are you doing? Are you at home or at the hospital? Your baby is a week or two old. You are at home now. How are you going about your life \u2013 with this new little person to care for? Imagine yourself as a mother in different situations, at home, walking in the park, shopping, visiting friends. Imagine yourself, your thoughts and feelings, your partner, your other children, your baby and the sort of life you are going to have together in your new family. Do you anticipate any difficulties? Do any negative images surface? You may come up against blocks, things you don't know about and this will help you to work out the areas you may be strong in and others where you may be weak.\n\nCreate a motherhood reality for yourself by making space and time every day to do this. Talk it through with your partner, your friends and, of course, with your baby \u2013 there is more and more evidence to suggest that the baby in the uterus is much more conscious than we can even imagine. This process will help you to look at your own attitudes to parenthood \u2013 what beliefs you hold, how many you carry from your own parents, areas of uncertainty or difficulty that date back to your own childhood.\n\nYou may decide to make a special space for your new baby, by preparing a room or a cradle, a corner in your bedroom, or to buy some clothes and equipment. On the other hand, you may not wish to get anything until after the birth. Do what is right for you. If you are living in a very small space, however, and are not able to give a room to your baby, at least plan where it will sleep \u2013 and think through how you will manage the nights when he or she arrives.\n\nOnce you can feel your baby moving \u2013 and that in itself is an extraordinary experience, like a bird or a butterfly fluttering its wings inside you \u2013 you know it's real. Be aware of the movements your baby makes, its own patterns of being awake and of sleeping. As it grows, see if you can notice the changes in position and tension that occur. During the last three months of pregnancy your abdomen will change shape frequently, bulging on one or both sides, or more in the middle. Ask your doctor or midwife to tell you how to feel for your baby's back or head or feet. When a small baby moves its hands or feet in the last weeks you can see little bulges. Talk to your baby, start including it in your daily life and make active links with it: start to understand your child's personality from how it moves inside you.\n\nOthers may want to communicate with your baby in ways that you find unacceptable. Ask people not to touch you if don't like it. It's absolutely fine if you do \u2013 but if not, be clear and friendly in asking them to back off.\n\nIf you are scared of the birth itself, ask your mother, if possible, what your own birth was like. Some women have unconsciously repeated their own difficult birth in giving birth to their children. Ask your mother about her attitudes to childbirth \u2013 you may have taken on her beliefs unconsciously.\n\nDo you have any fears or anxieties about being a parent? Voice them now: talk them through with people you trust, who will take you seriously and not dismiss your feelings as 'silly'. Try to identify what you are scared of. Your life will change \u2013 dramatically for some \u2013 and you can anticipate and prepare for some of the changes.\n\nIf you are to be the happy parent of two or more babies your extra baby may not be spotted until relatively late in your pregnancy (in the third trimester). You may have an increased susceptibility to anaemia and premature births are also common. The advice in this book still applies to you, and, because of your additional needs, you will need more of everything (not twice as much!) \u2013 more rest, more food, more support, more care \u2013 especially after the birth!\n\nPregnancy and birth are not always straightforward \u2013 they may be complicated or even downright difficult depending on your circumstances. The following women are all vulnerable in pregnancy either because they are physically or economically at a disadvantage or because they fall outside society's accepted norm: single parents, those over forty, adolescents, lesbians, those in poverty, those in deteriorating relationships, the disabled and chronically ill, ethnic minorities, isolated women or those who have multiple births.\n\nIt is beyond the scope of this book to go in great depth into all the ways that each of the above would best be helped, but most of the practical advice and homeopathic help outlined applies to any woman contemplating pregnancy and birth whatever her special needs or circumstances. (See also Organisations.) It is essential that you don't isolate yourself, that you get plenty of support if you fall into one of these categories. Having a baby is hard work \u2013 rewardingly hard work, but hard work nonetheless. Plan for it as you would a particularly difficult phase of your working life. Set up strategies for surviving it in one reasonably happy, healthy piece!\n\nWomen vary enormously in their capacity to work during pregnancy. I have known some women work literally up until the day before they gave birth while others stopped within a month of discovering they were pregnant. Do what feels right for you and what you want to do, if you can. If you choose to work up until the last minute you will have less time for yourself but if that is what you want, go ahead. If you have to carry on working from economic necessity, make as much of your free time as possible for yourself and your baby, your partner and any other children. Socialise only if you really want to and feel up to it. And remember that lunch hours at the office can be used for a nap, domestic chores can be delegated, and ask someone to drive you to the bus stop or station \u2013 or even into work!\n\nUse the pre-natal classes run by the hospital, clinic, or community center for information and to learn relaxation and 'breathing' techniques and yoga exercises specifically designed for childbirth (mainly the Active Birth classes). You'll also begin to learn some of the practicalities of parentcraft, and, especially important, you'll meet other pregnant women and gain confidence from their experience. You'll learn a lot about babies and parenthood from chatting to other mothers and their partners. If you're new to the area, your health visitor (contactable through your GP) will let you know of local classes. Encourage your birth partner to come with you to these classes so that you can practise the exercises together afterwards.\n\nYour partner may like to go to all ante-natal classes or none of them. Don't squash yourselves into an ideal of what you think you ought to be as prospective parents.\n\nYour pregnancy can begin to seem as though it will go on for ever \u2013 it may be the longest nine months of your life. Don't put your life on hold while you wait for the baby to arrive. Get on with the business of living. Everything in your life won't change \u2013 only some things!\n\n* * *\n\nYOUR FAMILY AND PREGNANCY\n\n* * *\n\nYou may be fortunate in having loving family living close by. If so, cultivate them: encourage willing grandparents to become as involved as they want to be in your children's lives. Grandparents can be a marvellous source of nurture for grandchildren and support for parents.\n\nIf grandparents are not available, aunts and uncles are good alternatives, either real or 'adopted' ones. Close friends who have chosen not to have children of their own are often delighted to be asked to be a 'special' person in your child's life. If you have a relative or friend who would like to be that special person involve them from early in your baby's life. They may even be delighted to babysit every now and again.\n\nMake friends with at least one other pregnant woman (whose baby is due at roughly the same time as yours if possible) so that you are not isolated after the birth. There is nothing worse than being the only one of a circle of friends to have a baby and to have no one to talk to about your new miracle.\n\nIf this is not your first baby, you may find yourself very stretched, especially if you are also working. Don't let the serious business of living, working, looking after everybody else and the house get on top of you. Make time to play cards or watch your favourite comedy video, gossip with a friend or play in the swimming pool.\n\nDo have some fun, with friends, relatives and your partner, and make the most of the last months and weeks of spontaneous freedom. After the birth you will have to plan your outings more carefully. The more children you have the more planning is involved!\n\nHolidays are often stressful: they take us out of our familiar environment and highlight areas of difficulty in our relationships because there are fewer distractions. Go where you know that you have a good time. Now is not the time to experiment with a new type of holiday. If your health has been poor during your pregnancy go somewhere reasonably close to home just in case you decide to cut the holiday short. Check out the availability of good medical care if you decide to take a break abroad, especially towards the end of your pregnancy.\n\nSitting cramped for long periods of time is bad for your circulation. It is important that you stretch frequently during long journeys: make frequent stops if you're travelling by car and walk about. Get up and move around if you are travelling by train, coach or plane.\n\nIn the past mothers often chose not to tell their children that they were pregnant. I believe that children of any age, including barely verbal toddlers, can be informed about a pregnancy, unless there is a strong possibility of an early miscarriage. Our children know intuitively when something unusual is happening and if we choose not to talk to them their imaginations can run riot. You can start by saying the bare minimum, that you are pregnant, what that means, and how you are feeling about it, both physically and emotionally. Children respond to news of a pregnancy in many different ways. Some are delighted, some curious, some seem more or less unaffected, detached, others are openly furious, anxious about their future, sad about the loss of their position as the baby in the family. Support, acknowledge and therefore validate their feelings.\n\nAs the pregnancy progresses, talk frankly about how the family will change with a new member in language appropriate to the age of your child. Include your child or children in planning for the birth and for life after the birth. Take them to an occasional ante-natal appointment (with plenty of things for them to do and eat). Some hospitals will even tolerate well-behaved children when you are having a scan.\n\nDon't ignore or try to hide the coming event. If you can't make yourself understood, there are many books for children on 'new arrivals' \u2013 you can use one to help you open the topic for discussion. I think it is important to state that the baby is growing in your womb (or uterus if you wish) and not in 'mummy's tummy'. Every child knows that it is food that goes into its tummy and this anatomical inaccuracy can lead to confusion, and even distress. There are marvellous books available that show clearly how the body is arranged inside, including photographs of the baby growing in the uterus \u2013 pay a visit to your local library or bookshop.\n\n* * *\n\nYOUR HEALTH AND PREGNANCY\n\n* * *\n\nBREATHING\n\nBreathing is automatic: you cannot stop even if you try! During pregnancy you may notice that you are breathing more deeply than normal: this is due to hormonal changes.\n\nOxygen is vital to good health. Many of us develop poor but adequate breathing patterns, mostly due to bad posture or emotional stress. We hold our breath if frightened, sad, tense, anxious, angry or even excited. A physical shock or accident can also affect how we breathe. If those patterns of shallow breathing are repeated, breath-holding can become habitual. However, people who breathe shallowly tend to take a deep \u2013 or sighing \u2013 breath every now and again as a compensation.\n\nIncrease your awareness of your breathing, of how you breathe. When do you breathe deeply and when do you breathe shallowly? To get an adequate oxygen supply, it's important to breathe into the whole lung: if you breathe shallowly you may only be filling the top part.\n\nAllow the air to come in through your nostrils and down into your lower back \u2013 your lungs reach up to your collarbone and down almost to your waist. Allow your ribs to expand outwards as you breathe in. Imagine that you have a couple of balloons in your chest that are filling fully with air each time you take a breath and emptying each time you breathe out. Allow this to happen rather than trying to make it happen and thereby causing a different tension pattern. Let your breath go on the outbreath \u2013 just let it go, without putting any effort into it.\n\nIf you find it difficult to breathe, if your chest is tight because you suffer or have suffered from a complaint like asthma, seek the advice of a physiotherapist, Alexander-technique teacher or cranial osteopath to retrain in breathing. If your breathing is shallow because of emotional tension then you may benefit from seeing a psychotherapist to release any suppressed feelings. Emotional tension can also lead to overbreathing, or hyperventilating, which can cause a drop in blood pressure and feelings of panic and dizziness.\n\nLearn how to breathe in conjunction with your relaxation technique so that they work effectively together. (You will be taught how to do this at any ante-natal class you attend.) Practise letting go of physical tension as you breathe out.\n\nCIGARETTES AND ALCOHOL (AND ALL OTHER DRUGS)\n\nIt is important to avoid alcohol while you are pregnant. Alcohol crosses the placenta and reaches the baby's brain within a very short time of your taking it. It's not worth the risk although the occasional glass of wine or beer is unlikely to do any harm. If you have a problem with stopping drinking, seek the help of AA (see here).\n\nCigarettes can also affect your baby: if you continue to smoke, your baby may be smaller and possibly more vulnerable to infection. Homeopathy can help you stop smoking so see a homeopath for information as to how. There are also organisations (listed here) which run short courses to help people give up, ask your GP about local groups.\n\nAvoid all other drugs, whether prescribed, over the counter or illegal, while you are pregnant and breastfeeding. If you are on medication for a long-term complaint such as epilepsy, you must be guided by your doctor throughout your pregnancy as to the suitability of your medication.\n\nDIET\n\nWhat we eat is always an important factor in our general health and this is especially true during pregnancy when the body's requirement for essential nutrients increases. You will need to eat about 20 per cent more than usual, but your requirement for folic acid, vitamins B and C, calcium, zinc and magnesium will rise by 30\u2013100 per cent.\n\nWomen whose supplies of stored nutrients may be low when they become pregnant should always take advice from a nutritionist (your GP will refer you): for example if you have had several pregnancies close together without time for recovery, or you are underweight (possibly anorexic), or you are on a restricted diet due to allergy or by choice. Some women have difficulty in absorbing nutrients, especially those who suffer from a chronic disease, women who smoke or drink heavily or who take drugs, and those who are taking prescribed medication regularly (such as steroids or anti-convulsants). Adolescents, who are still growing themselves, are also vulnerable, as are women who are overweight. It is important that you check with your GP or a nutritionist if you become pregnant and fall into one of these categories.\n\nIf you started your pregnancy with a good diet and are otherwise healthy then the most important advice I can give is that you enjoy your food and eat whatever you feel like eating. It is OK to incorporate a little junk food into your diet: your body will cope with it well as long as your basic diet is healthy. The following guidelines will help you build healthy eating patterns \u2013 for life, not just for pregnancy.\n\nDo\n\n\u2022 eat a varied diet. A diet with a little of everything is most likely to contain the wide variety of nutrients that you need. It is important that you include some of each of the main food groups every day, i.e. carbohydrate, protein, fat, fibre, vitamins and minerals. Make sure you eat plenty of pulses, wholegrains, fresh fruit and vegetables and if you are not vegetarian or vegan include dairy foods, eggs, fish (especially fatty fish like mackerel, herring and sardines) and meat, including kidneys, from organically reared animals only. Because of the highly concentrated levels of vitamin A in liver (thought to be the cause of a few isolated cases of abnormalities in babies, although not in the UK) doctors are presently recommending that women don't eat it in pregnancy. Hopefully research currently in progress will identify a safe quantity in the near future.\n\n\u2022 eat plenty of carbohydrates, as they are your major source of energy.\n\n\u2022 eat protein-rich foods every day, as they are responsible for the growth and repair of all the cells of your body (and therefore your baby's body too).\n\n\u2022 eat a moderate amount of fat. Apart from providing energy, it also forms 'protective layers' which store vitamins A, D, E and K.\n\n\u2022 eat plenty of fibre as it helps remove waste and toxins from the body.\n\n\u2022 make sure you include plenty of vitamin- and mineral-rich foods. If you aren't sure of whether your diet includes the following essential ones, see a nutritionist or hospital dietician who will advise and reassure you. Vitamin A helps in the growth processes of the body; B vitamins contribute to energy levels and the formation and healthy functioning of the brain and nervous system; vitamin B12 is especially important because it helps to form healthy blood and nerves); vitamin C builds resistance to disease, promotes speedy healing and the absorption of iron; vitamin D helps form strong bones, teeth and gums and aids in calcium absorption; vitamin E also promotes healing. Eat mineral-rich foods: calcium is essential for healthy bones, teeth, nerves, heart and blood; chloride is essential for the digestion of proteins and regulates the balance of sodium and potassium; copper aids the conversion of iron into haemoglobin and in the absorption of vitamin C; iodine helps with growth, energy and mental alertness; iron is responsible for healthy blood; magnesium helps form bones and the nervous system; phosphorus combines with calcium to make healthy bones and teeth; potassium unites with sodium to make healthy blood, body fluids and muscles; sodium works with potassium to balance the body's fluids; and zinc works with iron and copper to form and repair all the body tissues and keep the immune system strong and healthy.\n\n\u2022 buy organic meat and free-range eggs. The flavour is better and there are none of the chemical residues which are present in the factory-farmed products.\n\n\u2022 investigate soya beans and soya-bean products (tofu, tempeh and miso) as these all contain high levels of protein, iron, calcium, magnesium, potassium, phosphorus and B vitamins (not B12).\n\n\u2022 eat regularly, little and often, 4\u20136 smaller meals a day, unless you have a slow metabolism that thrives on eating larger meals at less frequent intervals.\n\n\u2022 eat wholefood \u2013 well washed and scrubbed fruit and vegetables with the skins on, organically grown if possible.\n\n\u2022 eat plenty of fresh foods in the ways that appeal to you. Some people prefer food to be cooked especially in the winter months while others like it raw the whole year round. Eat as much fruit, vegetables, especially green leafy ones, and\/or salads as you can. Don't forget that you can use winter vegetables for winter salads such as finely chopped cabbage, celery, chicory, grated celeriac and carrot.\n\n\u2022 buy fruit and vegetables in smaller quantities more often as vitamin C is destroyed by storage.\n\n\u2022 eat green leafy vegetables every day as they are high in iron and essential vitamins such as vitamin K which helps the blood to clot.\n\n\u2022 include nutritious snacks of nuts and seeds rather than biscuits which are frighteningly high in refined flour, fat, salt and sugar.\n\n\u2022 indulge any cravings during your pregnancy as long as they are reasonably healthy and as long as you are able also to eat a varied diet.\n\n\u2022 avoid sugars (white, brown, treacle and even honey except in small quantities) as they have very little nutritional value. Substitute plenty of fruit, including dried fruits if you want something sweet \u2013 they contain essential minerals as well as tasting good!\n\n\u2022 avoid all foods that contain additives, long-term effects are not known. The average Westerner is said to eat in the region of 4 pounds (2 kg) of additives a year.\n\n\u2022 avoid refined carbohydrates as these provide 'empty' calories (carbohydrates without vitamins or minerals).\n\n\u2022 avoid additional wheat bran and soft drinks as these can affect your ability to absorb essential vitamins and minerals.\n\n\u2022 soak muesli overnight before you eat it as raw oats are difficult to digest.\n\n\u2022 avoid anything you instinctively feel is 'bad' for you while you are pregnant or breastfeeding, even if it doesn't make logical sense.\n\n\u2022 cut out stimulants \u2013 tea, coffee, chocolate, Coca-Cola and Diet Coke as these all contain caffeine. They prevent absorption of zinc and iron especially if you drink them with a meal.\n\n\u2022 cut out all alcohol and be creative with your social drinking. Try drinking orange, grapefruit, tomato, apple or pineapple juice with or without soda water or sparkling mineral water; water! \u2013 cold, with or without ice, or hot, with or without a slice of lemon; blended iced drinks made with fresh fruit and ice; homemade lemonade; yoghurt shakes made with equal quantities of fresh yoghurt and water, with a sweet or savoury flavouring; grain coffees \u2013 a wide range is available in wholefood shops \u2013 vegetable broths.\n\n\u2022 drink herb teas with caution as they are known to have a medicinal effect and can therefore be unintentionally 'proved' (see here); raspberry leaf tea, taken in the last 3 months of pregnancy (2\u20133 cups a day) has been found effective at promoting a trouble-free labour. It is especially good for sedentary women with poor muscle tone or who have a history of gynaecological problems (or difficult births). Those who are fit and healthy should avoid it (or take it only on instruction from a herbalist) as it can cause the symptoms it was supposed to relieve (see Provings).\n\nDon't\n\n\u2022 eat for two. If you put on too much weight during your pregnancy you may find labour more difficult, especially if you are unfit.\n\n\u2022 worry if you put on up to 30 pounds (14 kg) during your pregnancy.\n\n\u2022 diet. Your baby's health may be affected.\n\n\u2022 eat soft cheeses like Brie, cheeses made from unpasteurised milk, hot dishes in canteens or precooked meals from shops or supermarkets \u2013 they may have been inadequately chilled after cooking and contain bacteria (listeria) which can cross the placenta and harm your baby.\n\n\u2022 eat muesli without soaking it first.\n\n\u2022 eat vast quantities of spinach, rhubarb, parsley, watercress or chocolate as the oxalic acid they contain hinders the absorption of calcium.\n\n\u2022 substitute beer and wine with those that are alcohol-free or low in alcohol as some contain high levels of additives and chemicals, the effects of which on your baby are not known.\n\n\u2022 take vitamin or mineral supplements during your pregnancy without taking professional advice from a nutritionist, who will advise you on how to alter your diet to get the extra nutrients you need rather than taking supplements.\n\n\u2022 become fanatical about eating only 'pure' foods. Your body can recognise junk and deal with it.\n\n\u2022 take antacids for heartburn (see here) as they prevent iron from being absorbed.\n\nSeek help if\n\n\u2022 you are not able to maintain a healthy diet.\n\n\u2022 you are not putting on weight or are putting on too much weight.\n\n\u2022 you are a vegetarian or a vegan and are worried that you might not be getting enough B12, iron or calcium from your diet.\n\n\u2022 your cravings are peculiar (see here) or uncontrollable.\n\nEXERCISE\n\nExercise increases strength, stamina and flexibility, all vital during pregnancy, birth and the post-natal period where you will want your body to return to a good shape.\n\nDo exercise regularly \u2013 the more physically fit you are the better. Even if childbirth lasts for 'only' 8 hours it can be very arduous and therefore the single most important aspect in preparing for birth is the physical one. It has been compared with climbing a mountain or running a marathon. In my practice I have observed that it is women who exercise who have relatively easier births and recover faster. If you don't enjoy exercise, you could consider going to classes or finding someone to come to your home. Walking, cycling (if you're experienced and confident), dancing and swimming are good for building strength and stamina, yoga and gentle exercise programmes for pregnancy will help build up flexibility and suppleness.\n\nDon't carry on exercising if you become breathless, exhausted, overheated or in pain. These symptoms are telling you that you have reached your limit and you need to stop and recover before you go on. And pregnancy is not a good time to take up a new and physically demanding form of exercise! Avoid weight-training as the ligaments in your body soften and may easily be overstretched. Don't have a sauna or a very long, hot bath after vigorous exercise as recent studies have shown that your baby may have difficulty in regulating its own temperature, in cooling down. After the birth, keep up the exercise and don't forget that walking with a pram is good for you and your baby!\n\nExercise for Pelvic-floor Muscles\n\nEven if you take no other exercise, this one is vital to prevent stress incontinence, prolapses and piles, among other things. The pelvic-floor muscles support all the organs in your pelvis; they lie in a figure of eight around your anus and vagina. Be aware of where they are by waiting to go to the loo until your bladder is full, start to pee then stop the flow of urine and hold it for a few seconds without dribbling. Start and stop several times. If you can do this easily your pelvic-floor muscles are in good shape. Do this from time to time when you pee to check them. The following exercise will build strong pelvic-floor muscles: imagine you are lifting these muscles upwards in stages \u2013 like a lift \u2013 up to the first floor and then the second and the third, holding them at each stage for a few seconds. Do it daily and frequently \u2013 anywhere! You can do it in the supermarket queue or while you're waiting for a bus. When you are in the bath or shower you can check this by holding a finger inside your vagina, as you contract the muscles you should be able to feel them.\n\nPOSTURE \u2013 CARRY YOUR BABY ACTIVELY\n\nSome women carry their babies with their whole bodies \u2013 accommodating and adapting to the change in shape and the extra weight. Others find the extra weight difficult to cope with and waddle, with their babies pulling them forward, forcing them to shift their centre of gravity which causes many aches and pains. Carrying your baby actively will be strengthening and lessen your chances of developing backache; you will feel more supple, less collapsed, less heavy \u2013 more in charge of your body.\n\nSeek the help of an Alexander-technique teacher if you feel that you are collapsing forward as your baby grows \u2013 if you find it increasingly difficult to get up out of chairs for example. An Alexander teacher will help you to be more aware of how you use your body, where tension exists, and will suggest less stressful ways of moving in everyday situations such as standing, sitting, lifting, walking, etc.\n\nYour imagination can be a strong healing force in your life: your mind and body are closely linked. The shape our bodies take can reflect our feelings: people who are depressed tend to slump. You can try this now: remember a recent situation in which you felt very angry and relive it in your imagination. Which muscles tense? How are you breathing? Now, remember a happy time, one which made you feel warm and contented. What happens to your body?\n\nTo carry a baby we need to expand: some women are frightened they won't stretch enough. Don't be. Imagine your body widening, lengthening and expanding, growing with your baby. If you feel tense and contracted or find it difficult to feel expansive then accept it, be compassionate towards your body, but keep trying \u2013 gently and persistently. Practise 'widening' or 'softening' to give room to your baby: actively imagine a space opening up inside you and giving your baby the room it needs. Tell yourself that you are capable of doing this, your body has been built to do it.\n\nTry not to allow the weight of your baby to pull you forward: imagine a heavy tail attached at your coccyx (tail bone) dangling between your legs, let it pull down, naturally tucking your bottom in. Don't lock your legs when standing: imagine them being soft and warm. Imagine that feeling in your knees and buttocks, whether standing, stooping or sitting, or anywhere in your body where you feel tension building up. Don't let your body collapse into inactivity. Most important of all is to imagine these things happening and then to let them happen. Trust your body to do what it knows best.\n\nRELAXATION\n\nIt is important that you build some good relaxation time into your life, time to unwind and recharge. It is possible to enjoy yourself and relax at the same time by walking, sitting outside and daydreaming, taking a long bath, massaging yourself with a body lotion or oil after the bath, reading a good book, talking to a good friend, playing in the sandpit with your toddler, reading to an older child and so on.\n\nSpecific techniques for relaxing such as autogenic training, biofeedback, guided visualisation and meditation, have been found to be enormously beneficial \u2013 experiment to find one that suits you, especially if you find it difficult to relax.\n\nA simple relaxation exercise that you can do anywhere is to lie on the floor, with a cushion under your head and one under your knees, or sit with both feet flat on the floor. Close your eyes and listen to your breathing. Allow it to become even and a little deeper. Feel your body 'sink' into the floor or the chair, let it become heavy and soft. Breathe out any tension. Imagine something really nice, walking in a sunny flower-filled meadow, skiing on your own private ski slope in the Alps, swimming in a warm tropical ocean, lying on warm sands, boating on a highland lake or walking in the forest. Spend as long as you like in your chosen place \u2013 a minute if that is all you have or longer if you are sitting on the tube or relaxing on your bedroom floor when you come home. You can repeat this any time you like if it works for you \u2013 you'll find it helpful last thing at night as it will help you sleep well.\n\nREST\n\nRest as much as you need to \u2013 your body will let you know just how much \u2013 especially at the beginning of your pregnancy. A nap in the afternoon and a longer sleep at night are fine, especially if you are working. Your energy should increase after the first few months, but if it doesn't and the advice in this book doesn't help (see Exhaustion) seek professional help. You shouldn't have to drag yourself around for your entire pregnancy.\n\nAs a general rule don't stand if you can sit, or sit if you can lie down. If you are a first-time mother, depending on your job, you will have more freedom to rest when you like. If this isn't your first baby, be creative about resting as you care for your family: I know a woman who spent a large part of each day in the bath because it was relaxing for her and fun for her boisterous eighteen-month-old. Entertain your young charges with activities that are restful for you like reading, watching television or a video, playing with Lego or Playdo, drawing or painting. Some children are like puppies and need a long run each day in the fresh air otherwise they become uncontrollable indoors. Time their outings so that you take them out when you are feeling energetic and can benefit from the exercise.\n\nInvest in a tall stool so that you can take the weight off your feet when ironing, washing up or cooking. Squat instead of bending down to pick up weights, like shopping or children. Sit on the bed, a small stool or chair to dress small children or do things with them at their level.\n\nSEX IN PREGNANCY\n\nSome women feel wonderfully sexual when pregnant \u2013 they love feeling rounded and full and womanly \u2013 and enjoy not having to worry about contraception. Many couples grow closer during pregnancy, and are able more easily to express tenderness, affection and love. Others experience a loss of libido. This may be due to having difficulty in getting used to a constantly expanding body or perhaps to emotional difficulties in the relationship. Some partners find their pregnant woman incredibly sexy and others quite the opposite. It's essential that you communicate honestly and openly with each other about your feelings and sexuality. (See also Loss of Libido.)\n\nSTRESS AND TENSION\n\nBe aware of stress, both physical and emotional, when you are pregnant and reduce it, if necessary, by sorting out your priorities. Pregnancy is a good time to decide what is important and what can be temporarily shelved. Make a list of what really counts in your life and needs attention to survive, and another of what isn't so necessary. You will find that cooking and housework can go quite low down on your list and that you and your family go to the top. Any time you start feeling that items lower down the list are getting too much attention and draining you, ask yourself what you would do if you caught flu and couldn't cook, clean or even go to work. Keep things in perspective by constantly checking what really matters and what could go if it had to!\n\nTake with a fistful of salt any advice, unsolicited or otherwise from relatives or friends! This is one of the few times in your life when you will be overwhelmed with it, even from people you barely know. Select and accept what feels right to you. Don't listen to other women's horror stories of their or their friends' babies' births; some feel a strange compulsion to repeat them to newly pregnant first-time mothers. Ask them to stop, repeatedly if necessary, until they do! Don't read too many books on childbirth or parenthood. Choose one or two and read the parts that seem relevant to you. If you are reading compulsively to make sense of something that's confusing you, ask a professional for the advice you seek.\n\nIf you feel tension or pain building up in your body at any time, breathe gently into the tense or sore muscles and allow them to soften and expand \u2013 imagine this happening. (See also Breathing.) Contract any muscles which are still tense, and stretch them, then gently move your body, twisting and turning. See if you can find a more comfortable position, whether you are sitting, lying or standing. Stiffening into one position is bad for you \u2013 keep moving! Find a comfortable position to sleep in at night.\n\nFeel your body moving as you go about your daily life. Let your hips swing gently, stop to rotate them when you are walking the dog or watching your older child play in the park. Slow belly-dancing is a marvellous form of exercise, created specifically to help women prepare for childbirth. In pregnancy (especially the later months) it promotes suppleness and strength. Dance to music: children of all ages will join you in this. Bend and stretch and lift, using your whole body without stressing or straining it or making sudden, jerky movements.\n\nIf your baby is pressing uncomfortably in one place, try moving your body and imagining that your pelvic area is widening. Breathe deeply and slowly and talk to your baby \u2013 asking it to move.\n\nWear comfortable clothing that enhances your feeling of well-being, and colours that make you feel good.\n\nTOXINS\n\nAvoid unnecessary toxins such as hairsprays, hair dyes, perms, head-lice treatments, garden pesticides, household sprays or any heavy duty (chemical) cleaning products, paint fumes (especially gloss paint), varnish fumes (including paint and varnish stripping chemicals), glue and typing correction fluid fumes, smoking (including other people's smoke), aluminium pans in the kitchen, excessive car pollution (unnecessary journeys entailing traffic jams). These can all cross the placenta and be absorbed by your baby. It is sensible to avoid extensive dental treatment, except to have your teeth cleaned and checked (see also Teeth).\n\nWear gloves when gardening or changing the cat-litter tray to avoid the risk of toxoplasmosis: relatively common in adults it can cause severe health problems for your baby. For the same reasons, wash your hands after handling raw meat and don't eat raw or undercooked meat. Reheated meals must be cooked to a high temperature to avoid food poisoning. Avoid raw eggs, unpasteurised milk, shellfish, p\u00e2t\u00e9s, blue-veined cheeses and soft cheeses like Brie and Camembert.\n\n* * *\n\nPREPARING FOR BIRTH\n\n* * *\n\nAlthough much of the following relates to birth or the post-natal period, you will need to read it when you are pregnant and think through the issues raised so that you can make your birth plan.\n\nHOSPITAL OR HOME BIRTH\n\nStart by exploring your options. What is available to you in the area in which you live? Try to work out what you really want and what might suit you. Check out the choices and don't make a decision until you are sure. Remember that you can change your mind.\n\nIs your home important to you? Is it where you feel most relaxed? Were you born at home? Do hospitals make you nervous? Do you know in your heart of hearts that home is where you'd like to give birth? Home may represent all that is normal, familiar, healthy and where you feel at ease. You will have more freedom at home to do largely what you want in your own time. You won't have to worry about hospital routine or procedures, and won't be disturbed by other labouring women and\/or their families, or student doctors and midwives in, say, a teaching hospital. Your other children will be there and afterwards you can tuck yourself into bed with your whole family around you. You won't be separated from your partner after the birth. If you want a home birth then ask your GP. If he or she is not supportive, contact your local independent midwives, health authority or a childbirth organization.\n\nIs full medical support vital to you? Are you comfortable and at ease in a hospital environment? Have you had good, positive experiences in hospitals? Are you good at dealing with hospital staff, at getting what you want? Do you want a break after the birth \u2013 a time for just you and your baby, where your meals are delivered at regular intervals, where the demands of your other children are being met by someone else? Do you feel anxious at the thought of having a baby at home because of the risks, because your home is too small or too crowded, or because you know in your heart of hearts that hospital is best for you? More importantly, do you have a medical condition that may need expert help at the birth itself such as twins who present themselves prematurely, or pre-eclampsia?\n\nPRE-NATAL VISITS\n\nThese are booked once a month from 12 weeks onwards up until the 28th week and then every two weeks until the 34th week, then weekly until the birth. They will help you to form links with your doctor, consultant and\/or your midwife, hopefully, but not necessarily, the ones who will attend your labour. Visits are more frequent towards the end of your pregnancy as complications are more likely to arise then if at all. You'll only have to undress for the first one. The following are all checked at each visit: your blood (see here) and urine (for infection); your weight and blood pressure; the baby's position and heartbeat.\n\nYou can take your partner or a good friend with you to ante-natal visits as support and also to make the boring bits pass quickly \u2013 like waiting for your appointment! As queues in hospital ante-natal clinics can be long and slow-moving, at least take a book or something to occupy you.\n\nTake an active interest in what is happening to your body during your pregnancy and avoid becoming a passive consumer of health care by asking questions at every stage. Make a list before your appointments of the information you need as well as any points you wish to make to the clinic staff.\n\nSeek professionals with whom you feel comfortable and can trust. If you are frightened or anxious then say so, otherwise, in trying to hide your feelings, you may come across as 'prickly' or 'difficult'. Ask for support and help. Be as honest and open as you can.\n\nTESTS\n\nMedical tests pose a dilemma: you may believe in minimal medical interference but want to do the best for your baby so when the doctor offers a particular test you may feel obliged to have it. Doctors may pressure women to have the favourite test of the moment although rising costs have meant that tests (especially expensive ones) are not pushed routinely now. Most doctors now only advise having a test if the result will cause the parents to take action. Tests and procedures are constantly being improved: check out the up-to-date information with your doctor, consultant, or childbirth organization (see Organizations).\n\nTests are often stressful and I have provided guidelines for how to approach and deal with them to minimise stress and to make the most of what is offered by the medical profession, whose aim is to help you stay healthy and produce a healthy baby.\n\nSome of the following tests carry the minor risk of bruising from needles or rough examinations, and of feeling emotionally 'abused'. Take Rescue Remedy (see here) with you on every hospital visit and take frequent 2-drop doses if you are feeling edgy or panicky. You can take Argentum nitricum, Lycopodium or Gelsemium before the visit if you are in a particularly nervous state.\n\nAfterwards, take Arnica routinely for bruising and\/or shock; Staphysagria if you feel 'abused' or 'violated', even if you were in the hands of kind and sympathetic doctors and nurses, and are suffering from more severe pains (not just soreness). Some women are very sensitive to having their body examined \u2013 dislike internal examinations, having needles stuck into them, cervical smears: Staphysagria will deal with both the physical and the emotional effects. Take Hypericum if you have shooting pains anywhere after one of the tests, for example, if a needle hit a nerve.\n\nUse the following Dos and Don'ts to help you decide whether and\/or when to have each test.\n\nDo\n\n\u2022 weigh up whether the benefits of a test will exceed its risks. The results can mean that further tests, or action, need to be taken. The following questions may help: What will I learn from the test? How accurate is it? What are the risks to me and\/or my baby if I have it? How long will I have to wait for the results? What procedures (or tests) will be needed if it is positive? How far am I prepared to go if the test is positive? Where would I be happy to go for this test (hospital\/GP's surgery, etc.)? Think carefully and listen to your instincts about whether you want or need each separate test.\n\n\u2022 talk it over with your doctor and try to find out if he or she performs tests routinely. Be sure you understand what is happening and why. If in doubt get a second opinion. Talk to your partner and to other women who have had the test.\n\n\u2022 get as much information as you can about each test that is suggested. Ask your doctor to describe exactly what will happen for each one. Ask a childbirth organization or your library for up-to-date detailed information.\n\n\u2022 ask whether the results of a particular test will affect your doctor's plan for your care or will necessitate treatment. If it won't, the test may not be necessary.\n\n\u2022 weigh up the options and the possible outcomes carefully. Ask yourself what is the worst that can happen, if you have a test or if you don't.\n\n\u2022 question the necessity of taking all your clothes off if, for example, you are just having a blood test.\n\n\u2022 take a partner or friend to a pre-natal visit that involves the discussion of a test. Ask that they write down information given by doctors, nurses, consultants or midwives. Studies have shown that patients remember remarkably little of what doctors tell them. Your partner can ask the medical professionals to spell unfamiliar terms so that you can look them up later.\n\n\u2022 ask questions whenever you feel confused. Some pressurised doctors forget to tell you what they are going to do next, and don't answer patients' questions satisfactorily. They may even leave the room without saying if or when they will return. Be persistent and assertive, without being aggressive, and ask for as much reassurance as you need as often as you need it.\n\n\u2022 be friendly, firm, clear and persistent about your own needs and preferences at each stage.\n\n\u2022 let your doctor, consultant or midwife know if you are finding the ante-natal visits and tests stressful. Some women mask their anxiety or fear or distress with a hardness that comes across as aggression, which can put up the backs of medical professionals. Talking will help.\n\n\u2022 ask your partner or friend to ask the questions if you find yourself drying up once your clothes are off.\n\nDon't\n\n\u2022 accept that because the doctor has said you need it that they are right. Try to understand why you are being tested. This will be empowering.\n\n\u2022 let yourself be separated from your partner during a test procedure that you have agreed to whatever happens. He or she can go to the toilet with you \u2013 and wait outside if necessary \u2013 unless you feel happy to go on your own.\n\n\u2022 put up with a hostile and unsympathetic doctor, consultant or midwife, whatever the reason. Ask to see someone else or get dressed, if necessary, and leave.\n\nAlphafetoprotein\n\nThis blood test is performed between 16 and 22 weeks of pregnancy to measure the levels of alphafetoprotein being passed by the baby's liver back into the mother's bloodstream via the placenta. The AFP test, as it is known, is used to detect the possibility of one of a number of common defects in the development of the foetus. Altered AFP levels can indicate that:\n\n\u2022 the baby has a so-called neural tube defect, including spina bifida, Down's syndrome or anencephaly.\n\n\u2022 the baby has an abnormality of the kidneys or intestines.\n\n\u2022 a miscarriage is threatened.\n\n\u2022 you are carrying more than one baby.\n\n\u2022 your dates are wrong: the AFP levels double every five weeks in the fourth, fifth and sixth month of pregnancy.  \nIt is important to be aware that this test has an approximately 20 per cent false-positive rate. If you test positive then you have a one-in-five chance of being alarmed unnecessarily.\n\nDo\n\n\u2022 have a second AFP test before you consider a scan (if your levels were high) if the first one was positive; or have a scan before a second AFP test if you're anxious and want to reassure yourself that you're not carrying twins or a baby with a defect.\n\nAmniocentesis\n\nUsed to test for certain chromosomal\/genetic disorders. The sex of the baby can also be determined. This test is offered to all pregnant women in their late thirties and early forties because the risk of conceiving a Downs' syndrome baby increases with age. At 35 the risk is 1 in 365, but at 40 it is 1 in 100. It is also offered to younger women who have a family history of genetic weakness.\n\nAmniocentesis carries a slight risk of miscarriage or damage to the placenta or baby. It cannot be performed until the 16th week of pregnancy because it is not possible to obtain enough amniotic fluid before then. Because the tests take up to three weeks to be processed, a woman can be 4\u00bd months pregnant before she has the results. To contemplate an abortion at that stage in a pregnancy is a very painful decision not least because it involves an induced labour rather than a suction abortion under a general anaesthetic. Counselling is essential.\n\nThe procedure for amniocentesis is:\n\n1 An ultrasound is used to locate the baby (see here) to avoid disturbing it or piercing the placenta and a cross marked on the spot where the needle will enter.\n\n2 A long, very fine needle is inserted and about 4 teaspoons of amniotic fluid are drawn off. The procedure may need to be repeated more than once to get enough fluid, or again if there is blood in the sample, or if the doctor can't get the needle in in the right place.  \nAlthough the needle is not usually felt as it passes through the fat and muscles of the abdomen it may sting, prick, cramp or feel like a pressure as it pierces the wall of the uterus. Afterwards more intense pains, including strong cramps, are common; spotting of blood or leaking of amniotic fluid from the vagina occur rarely. The site where the needle is inserted may feel bruised.\n\nDo\n\n\u2022 take a loving partner or friend.\n\n\u2022 breathe (see here) into your abdomen. Use any relaxation routine you know.\n\n\u2022 feel yourself slowly sinking into the couch.\n\n\u2022 tell your baby what's happening reassuringly.\n\n\u2022 ask your partner to talk to you reassuringly.\n\n\u2022 take time afterwards to recover fully. If it has been at all traumatic for you take it easy until the shock passes.\n\n\u2022 take the homeopathic remedies suggested here.\n\nDon't\n\n\u2022 look, even if you are curious \u2013 at least until afterwards. Either shut your eyes or ask for a sheet to be put up to protect you from seeing it done.\n\n\u2022 let your partner be separated from you for a moment.\n\nBlood Tests\n\nThese are straightforward and usually performed by your doctor (or at the hospital). A small amount of blood is taken from your arm. If you have never had blood taken before or if you find it a difficult procedure, ask your doctor to go slowly and explain everything to you. Lie down to have it taken if you are inclined to faint at the sight of blood and\/or a needle.\n\nBlood is routinely taken to test for blood group, Rhesus factor (see here), anaemia (see here), rubella antibodies, sometimes for venereal diseases\/herpes virus or hepatitis, sometimes for sickle-cell disease\/thalassaemia, or glucose tolerance (if there is a family history of diabetes or large babies) and alphafetoprotein (AFP) (see here).\n\nChorionic Villi Sampling (CVS)\n\nSometimes known as chorion biopsy, this tests for Down's Syndrome and has the advantage that it can be performed when you are nine weeks' pregnant instead of waiting until 16 weeks for amniocentesis. The results take two to three weeks to come through. The placenta is formed from chorionic tissue; a sample is taken by inserting a tube into the cervix or a needle into the abdomen.\n\n1 Your vagina is swabbed with an antiseptic if the sample is taken through the cervix.\n\n2 The whereabouts of your placenta is located by ultrasound.\n\n3 Either a thin tube is passed through the cervix and guided to the placenta using the ultrasound, or a needle is passed through your abdomen (see Amniocentesis).\n\n4 A tiny fragment of chorionic tissue, the size of a few grains of rice, is sucked into the tube or needle.  \nThere is a greater risk of miscarriage from this test and its accuracy is not precisely known. If the foetus is discovered to be abnormal then a straightforward abortion (vaginal termination under general anaesthetic) can be performed instead of the induction of labour after amniocentesis.  \nSee Amniocentesis and follow the advice given there as this procedure can also be traumatic, either emotionally, physically or both.\n\nElectronic Fetal-Heart Monitoring\n\nThe monitoring of the baby's heartbeat in pregnancy is traditionally carried out by a midwife who uses a special stethoscope which looks like a small ear trumpet. Recently, however, the use of electronic or ultrasound monitoring using small, hand-held monitors has begun to increase. No evidence as yet supports the use of these types of monitoring in pregnancy.\n\nDo\n\n\u2022 question your doctor or midwife thoroughly as to why they wish to do it and what effects, if any, it will have on your baby.\n\n\u2022 refuse to be monitored in this way if you are unhappy with their explanations.\n\nFetal Lung Maturity\n\nVarious tests can be performed on the amniotic fluid which give information about how well the lungs of the foetus have matured, especially important in a woman whose baby must be delivered before term because of complications towards the end of her pregnancy. It is recommended that any woman who chooses to have an induction or a Caesarean should take advantage of these tests to avoid an unnecessary premature delivery and subsequent intensive care of the baby.\n\nInternal Examinations\n\nAn internal vaginal\/pelvic examination is carried out at your first ante-natal visit after a positive pregnancy test, usually about 6 weeks after your last period, to confirm the pregnancy by noting the changes to your cervix and uterus. Your doctor will take a smear, investigate the health of your vagina and cervix, check for fibroids and any other problems such as thrush.\n\nAn internal examination may be absolutely fine or it can feel intrusive and painful.\n\nDo\n\n\u2022 tell your doctor if you have difficulties with internals.\n\n\u2022 ask your doctor to be gentle and to warm the speculum (if one is to be used) and to proceed very slowly.\n\n\u2022 consciously relax your pelvic-floor muscles.\n\n\u2022 breathe slowly and deeply and imagine yourself becoming very heavy on the couch. Breathe down into your belly and close your eyes if necessary.\n\n\u2022 ask your partner (if present) to hold your hand and give you whatever reassurance you need.\n\n\u2022 ask your doctor to stop if you become tense or feel pain and to start again when you are ready.\n\nKick Chart (Fetal Movement Counting)\n\nTowards the end of your pregnancy (commonly between the 30th and the 40th weeks) you may be asked to keep a 'kick chart' to record your baby's kicking episodes over a 12-hour period. In some areas a one-hour period is charted. This test verifies that your baby is continuing to thrive or acts as an early warning that there may be problems. Unfortunately, this test does not take into account the individual baby \u2013 some babies are naturally less active than others.\n\nDo\n\n\u2022 refuse this test if you don't want to do it. It has been found to induce unnecessary anxiety and stress.\n\nPlacental Function Tests\n\nThe placenta acts as lung and digestive tract, ferrying oxygen and nutrients to the foetus and removing carbon dioxide and waste products. Its health is verified by testing the levels of hormones, either by measuring the urine over a 24-hour period or by a blood test. These tests on their own are notoriously inaccurate with unnecessary inductions being the unfortunate result.\n\nDo\n\n\u2022 thoroughly question the value of these tests in your case.\n\n\u2022 think before you agree to have a placental function test and if necessary get a second opinion.\n\nRhesus Factor\n\nAt your first pre-natal visit your blood will be tested to identify its group (A, B, AB or O) and for the presence of Rhesus factor, a protein which attaches itself to the red blood cells, which determines whether you are Rhesus negative (Rh\u2013) or Rhesus positive (Rh+). Eighty-five per cent of women are Rh+.\n\nIf you are Rh\u2013, the baby's father is Rh+ and the baby is Rh+ and some of your baby's red blood cells 'leak' into your bloodstream in late pregnancy or during or after the birth, your immune system produces antibodies to fight the invaders. This isn't a problem in a first pregnancy. In a subsequent pregnancy where the baby is Rh+, if the antibodies pass back into the baby's bloodstream, they can cause severe anaemia.\n\nIf you are identified as Rh\u2013, another test is automatically performed to check for Rhesus antibodies. If this is your first pregnancy and there are no antibodies, an injection of serum (anti-D) will be given up to 72 hours after the birth to halt the production of antibodies so that a subsequent pregnancy will not be in jeopardy. This injection will be repeated as a preventive measure after all births, and also after an abortion or miscarriage.\n\nIf you do produce antibodies in your first pregnancy (perhaps you had an earlier miscarriage) then you and your baby will need regular checks throughout the pregnancy. Ultrasound scans will check the baby's development; amniocentesis (see here) or CVS (see here) will check for anaemia. In an emergency, a blood transfusion can be carried out on a baby in utero from 22 weeks on or immediately after the birth if a baby is induced early. You can self-prescribe if you find any of the tests distressing.\n\nDo\n\n\u2022 tell your doctor of any abortion and also any suspected miscarriage(s) especially if you are Rh\u2013.\n\nUltrasound\n\nThis is usually first performed at about 16 weeks. High-frequency sound waves bounce off your inner organs to make pictures which can be recorded on a monitor. It is still a slightly controversial procedure because as it is relatively new no long-term effects are known. Experiments on animals have shown ultrasound to have an adverse effect on their immune system, giving reduced resistance to disease. Different levels are used on humans, however, which are thought to be perfectly safe. In any case its use in the first three months of pregnancy is not advised, nor repeated use of it later on unless absolutely necessary.\n\nFor some women the great benefit of being able to see that they are carrying a real baby, before movements are felt, helps to make the pregnancy seem more real. It is used to check where the placenta is lying; to confirm that a foetus is present and that the ovum hasn't 'blighted' or a (very rare) hydatiform mole developed; to check for ectopic pregnancy (one that has implanted in the fallopian tube), twins, fibroids, some abnormalities; to determine the sex of the baby (if required); to confirm dates, by measuring the size of the head. The procedure is:\n\n1 You are asked to drink 2 pints (1 litre) of water. Then you wait, bursting to pee. A full bladder pushes your intestines out of the way and provides a point of reference for the technician.\n\n2 You lie flat on a couch and your abdomen is oiled or gelled so that the probe \u2013 a 'box' about the size of a video cassette \u2013 can make good contact.\n\n3 The technician guides the probe to and fro over your abdomen to build up a picture of your baby. If your baby is active this will have to be repeated until a satisfactory picture is obtained. The accuracy of the picture and its interpretation depends on the skill of the technician.\n\nDo\n\n\u2022 ask yourself whether this test is necessary \u2013 whether its results will determine the management of your pregnancy.\n\n\u2022 make sure you take your partner or a friend. Although they have no legal right to be present, you can refuse to have the test unless they are.\n\n\u2022 ask the operator to describe what you are seeing.\n\n\u2022 ask if a photo can be taken: some hospitals will do this.\n\n\u2022 consider refusing this test if you are a healthy woman with no medical complications.\n\nUrine Tests\n\nUrine is tested at your first and all subsequent antenatal visits for the presence of protein (kidney problems); sugar (diabetes); ketones (undernourishment); bacteria (infection). Giving a clean urine sample is a bit fiddly but well worth the effort as it is common for bacteria that are lurking externally to be washed into the urine and a false diagnosis of infection made. If you test positive for bacterial infection and you have no symptoms (pain\/discomfort on urination or frequent urination) then ask for a re-test paying special attention to giving a clean sample as outlined below.\n\n1 Wash your whole external genital area well with water (not soap) \u2013 you can use a bottle of warm water if you are at home (see here). If you are at the hospital you will have to improvise with a paper cup or with wetting some toilet paper.\n\n2 Hold both your labia apart before you start to pee.\n\n3 Take a mid-stream sample of urine which has not come into contact with your skin.\n\nX-rays\n\nPerformed routinely in pregnancy until ultrasound became freely available, X-rays should be avoided. There is a case for women with a small pelvis to be X-rayed to confirm that a vaginal birth is not possible.\n\nDo\n\n\u2022 avoid X-rays when pregnant.\n\n\u2022 ask for a second opinion if your consultant diagnoses a small pelvis and suggests an X-ray.\n\n\u2022 ask your dentist for a lead apron if you need dental X-rays during your pregnancy (but see also here on dentistry in pregnancy).\n\nThe following general guidelines are to help you think through what you may need or what may help you in labour. Pick and choose the things that appeal to you.\n\n* * *\n\nPLANNING FOR BIRTH\n\n* * *\n\nYou'll need to think about the things and the people you want around you during labour \u2013 when you are pregnant. Some women pack their 'birth' bag months ahead of the birth \u2013 whether they are having a home or a hospital delivery \u2013 others leave it until the last minute.\n\nIt is advisable to check with hospital staff\/midwives that your plans for taking in anything from bean bags to two birth partners are acceptable. And that includes finding out their attitudes to your eating or drinking in labour \u2013 to your bringing snacks in to the hospital (or having fry-ups at home!). And then don't forget to write it in your birth plan (and in your hospital notes if necessary).\n\nCreate an Environment\n\nIt is easier to create an environment if you're having your baby at home but it is quite possible to feel at home in a hospital labour ward with a little planning. First, go to see the labour ward at the hospital and think about things you could take with you to help you relax and feel secure. For example, hospital smells can be off-putting, so take a favourite 'scent' with you, a perfume, aromatherapy oil, incense or favourite pot-pourri. If you are superstitious, take a lucky mascot. Here are some ideas of things to have around, which can all be helpful, whether you are at home or in hospital: flowers; a light scarf to drape over bright lights; a special photo, a crystal, a cross, a stone, a teddy bear \u2013 something that gives you a special strength when you look at or hold it; a personal stereo or cassette deck: if you like music make a compilation tape of your favourite pieces of music, or take a varied selection of tapes to suit your moods; hair combs or hair bands to keep your hair from bugging you; lip salve or balm in case your lips become uncomfortably dry; a favourite oil or talcum powder for massage; a toothbrush and toothpaste to refresh your mouth from time to time; a small sponge and a flannel; a delicious soap and a soft towel; some pillows or cushions and\/or a bean bag \u2013 they'll smell like home and won't slip and slide around like hospital ones which are covered in plastic; a comfortable nightie or big nightshirt; if you haven't got one with you ask for two hospital gowns and wear the second one as a dressing gown; two pairs of woollen socks \u2013 your feet can become unexpectedly cold in the later stages of labour; a hot-water bottle \u2013 essential if you know yourself to be a chilly person, or chilly under stress; card or board games and some books, magazines or comics; a camera and enough film (a fast film so you don't need a flash) \u2013 some people take videos; change for pay telephones if you are in hospital, plus, of course, your address book or a list of telephone numbers; your homeopathic remedies; a copy of your birth plan (see here) \u2013 use this as reassurance if necessary \u2013 to remind you of your personal goals.\n\nGo with the Flow\n\nDon't go into labour with a fixed idea of how you want it to be. Put your faith in yourself, God, Mother Earth, the Goddess of Creation, whoever, and be prepared to let yourself go, to relax and move along with whatever happens in an active, participative way. 'Go with the flow' means accepting that your experience will be different from anyone else's. You are not in control, your body takes over as birth involves a complex series of involuntary physical processes which will mostly just 'happen'.\n\nBe Creative\n\nYou'll be surprised by what you want in labour: don't ignore an 'inner voice', however crazy and funny it might seem. Allow yourself to respond \u2013 it's a great opportunity to step outside your ordinary, everyday self. I know one woman who spent most of her labour on the loo and another who danced through every contraction. Your intuition, gut feeling, instinct \u2013 call it what you may \u2013 is your most reliable guide. No one knows better than you what you need. You may feel like grunting, groaning and moving in ways that may have been unfamiliar to you before. Don't feel self-conscious, critical or judgemental \u2013 trust yourself now, more than ever before.\n\nGet Good Support\n\nIt is important that you feel supported in labour by your partner, friend, midwife, doctor, so that you can let go of being in control and allow your body to do its job. Think carefully about who you want at the birth: they should be people who aren't going to take it personally if you feel like being alone, or if you let it all hang out if the pain becomes intense, people you can lean on and rely on, who will not bring with them fears or anxieties of their own. Don't invite a friend who wants to be reassured that birth isn't as bad as everyone says it is: she may inhibit you by making you constantly censor your behaviour.\n\nIt is essential that you make and maintain a good working relationship with your midwife, doctor or consultant. Be polite, friendly, reasonable and persistent if necessary with your requests. Ask for their understanding, support and help and be prepared to compromise to maintain goodwill. You can reasonably expect support from your medical practitioners: ideally, you want to feel good after every pre-natal and post-natal visit. Their quality of care 'should educate and empower, should enhance every woman's feeling of her ability to do what she's doing well'. (Our Bodies Our Selves, Boston Women's Health Collective, 1984)\n\nMake Demands\n\nGive yourself permission to ask for what you need when you need it. This is not a time to hold back. If you have trouble asking for what you want ordinarily then practise in your pregnancy. Make a simple demand every day \u2013 'Please will you run my bath for me?' \u2013 to get used to the idea! During labour ask yourself from time to time 'What do I really want right now?' and then ask for it, however small, large or unusual.\n\nExpress Your Feelings\n\nLaugh if happy, cry if sad, shout if angry \u2013 this is the one time in your life when you can really let go. Don't hold on to your feelings. They can be tremendously empowering during labour.\n\nMake Positive Affirmations\n\nThis can provide a lifeline in labour, an important source of encouragement that will help you access your innermost resources of emotional energy. Tell yourself out loud 'I can do it \u2013 I'm brilliant \u2013 I'm so clever \u2013 I'm doing great\/my best \u2013 I'm going to have a baby.' Partners can also run a barrage of positive, encouraging statements, 'Great, good \u2013 you're doing really well \u2013 keep it up \u2013 let yourself go \u2013 that's right \u2013 well done \u2013 let yourself open up \u2013 marvellous \u2013 excellent \u2013 you're doing so well \u2013 just relax \u2013 you're the best \u2013 I love you.' These are especially effective if your partner looks into your eyes and says them during and after each contraction.\n\nUse Visualisations\n\nVisualisation is useful for helping your body do what it has to do \u2013 to relax and to open up. In the early stages of labour imagine yourself on a sun-drenched beach, swimming in a magical river or pond, playing in a meadow \u2013 just close your eyes and let your imagination run wild. Imagine your cervix is a flower that is slowly opening. Imagine your body softening and melting. Let your mind's eye come up with images that are important and meaningful to you and use them.\n\nUse Distractions\n\nPlay games \u2013 simple card games, crosswords or even chess...\n\nLaughter is a marvellous distraction from pain. Have fun, especially in the early stages, if you can and if you feel like it. Listen to funny tapes or records, read a book that makes you laugh out loud or get someone else to read it to you. Remember funny\/ridiculous times in your life. Tell jokes.\n\nWatch television if you're at home. You can video a favourite series, a comedy or a soap, and save it all up for your labour. You could watch it in the early stages, before you go to the hospital.\n\nKeep Your Energy Up\n\nIt is vital that you keep your energy up during labour so the following are all important:\n\nBreathe Oxygen is an essential component of any hard physical exertion. Pain makes us tense up and breathe shallowly \u2013 that is why breathing techniques are taught in pre-natal classes as a lack of oxygen in labour can make it more difficult than it need be. However, you may find that the method you learnt becomes just another 'thing to do' and doesn't work for you in labour: use it as a starting point to follow your instincts and breathe in a way that suits you. (See also here.)\n\nDrink You must keep up an intake of fluid in labour to prevent dehydration or you may need a drip (see here), which is usually preventable. Try using a bendy straw or sucking on a sponge or flannel or ice cubes if you don't feel like drinking. If you vomit all fluids then you will have to accept them intravenously (try prescribing on the vomiting first though \u2013 see here).  \nNourishing fluids will help your labour along: if you are at home you can make milk shakes \u2013 with banana and wheatgerm for added energy. You can drink milk hot or cold, with honey or sugar, watered down or put in the liquidiser with ice cubes. Vegetable or meat stocks are also good as are soups which have been pur\u00e9ed and thinned with milk. Drink plenty of freshly squeezed fruit juices (lemon, orange or grapefruit), hot or cold. Add honey or sugar to your juices and you have an energy-rich drink. Drink herb teas, ordinary tea or even a little coffee. If you are going to hospital take a Thermos flask of your favourite drink or soup and bottles of juices.\n\nEat Blood sugar levels must be kept high, especially during early labour. Remember, you are taking the equivalent physical exercise of climbing a mountain. Eat what you fancy in the early stages. Later on, try to have at least some small, easily digestible morsels at frequent intervals. Toast or crackers, fruit dipped in honey; porridge; stewed dried fruit; some egg custard. Many hospitals won't approve of you eating in labour in case you need a general anaesthetic but take in some crackers, a bag of fruit, a jar of honey and a sharp knife. If you don't eat you may need medical intervention because you will run out of energy. You must have something even if it is juices and lots of honey\/sugar.\n\nRest Balance the hard physical exertion of labour with rest so that you can 'stay the course'. Spread yourself between contractions: in a long labour you might be able to sleep. Drape yourself over a bean bag or a pile of cushions. Ask your partner or midwife to rest a hand on your belly and to wake you as soon as it tightens when the next contraction begins so that you can easily find your rhythm and go with the flow of each contraction, as they climb in intensity, peak and gradually fall away. You can sleep between as many contractions as you need to, to get a good rest. Midwives and doctors may not approve of this as it has been known to slow down the contractions, but this will only be temporary.\n\nUrinate It is essential that you pee every 1\u20131\u00bd hours throughout your labour: the baby's head descending can damage a full bladder. The sensation of needing to pee may not be as noticeable as normal, but keep trying anyway as a full bladder can slow down labour. It also provides you with a distraction \u2013 something to do at regular intervals. You don't have to pee on the toilet \u2013 although walking to and from it can be beneficial it can also be a nuisance. Squat over a bucket, washing-up bowl, potty or bedpan. (See Labour pains.) If you have difficulty in peeing, tensing and relaxing the pelvic-floor muscles a few times can start the flow.\n\nPAIN RELIEF IN LABOUR\n\nIt is as important to know what is on offer and think about it in pregnancy as it is to decide whether to take music, flowers or a homeopathic first-aid kit into hospital with you. 'Again and again, women are grateful that they were told what might happen because, generally, it didn't. But when it did, they had only the problem itself to cope with, not their own shock and surprise as well.' (Drugs in Pregnancy and Childbirth, Judy Priest, Pandora, 1990)\n\nSome women find that, uncharacteristically, they can cope with extraordinary pain in labour, while others crumble at the first contraction. Don't try to bear more than you can. If you need drugs, take them. Don't feel guilty. You are not a failure. You have a right to ask for pain relief. You deserve not to suffer. An 'active' labour is one in which you are actively involved at each stage: you can be active in deciding to ask for pain relief. Ask your hospital which drugs they prefer and which they don't offer.\n\nThe differences between women and their desires for labour are eternally fascinating. I remember well two women I delivered five years ago, both on the same day, with the same student midwife. The first was very glamorous and very neat, taking enormous trouble with her hair and make-up. She looked like a film star. She opted for an epidural and had a syntocinon drip, and I remember that as the baby's head emerged over the perineum she literally laughed the baby out. Each time she laughed, a little bit more of the baby's head emerged. It was a beautiful labour and delivery, enjoyed by her, by her husband and by me and the student. When that labour had finished, the telephone rang. It was a woman who I had booked in for a home delivery and who was now in labour. The baby was born after about four hours with no analgesia. It was a beautiful delivery and a very happy experience for us all \u2013 woman, husband, big brother (aged six), student midwife and me. In fact, the GP arrived soon after the delivery and made me giggle by asking me if I had been 'taking anything' because I was so 'high'. I reassured him that I had not, but that it had been my privilege to have been present in one day at both ends of the spectrum of obstetric care, and both had been wonderful and exciting.\n\n(Caroline Flint, Sensitive Midwifery, Heinemann, 1986)\n\nMost orthodox drugs taken in labour will cross the placenta into the baby's bloodstream and affect the baby in a similar way to the mother, but you must ask yourself what would be the value of experiencing pain above your own personal threshold when this would be so stressful that your tension and distress would also be felt by your baby. Weigh up what would be best for you as an individual, rather than working towards an ideal set by someone else.\n\nA synopsis of drugs and techniques for pain relief in labour follows.\n\nAcupuncture\n\nLike hypnosis, acupuncture works better for some than others. There are two types: acupuncture analgesia, where you lie down and the needles are attached to machines which keep the points stimulated, and the pain at bay; and therapeutic acupuncture, which works more like homeopathy \u2013 specific points are used, the needle is not left in, to alleviate emotional and physical symptoms and help a woman cope with her relationship to the pain and stimulate the body to work more effectively.\n\nPros\n\n\u2022 you have additional support.\n\n\u2022 it won't affect you or your baby adversely.\n\nCons\n\n\u2022 there is an extra person in the room.\n\n\u2022 you may have to be immobile depending on the type of acupuncture used.\n\n\u2022 you may not feel like having needles stuck into you (it may feel invasive).\n\nCaudal Anaesthesia\n\nA local anaesthetic is injected into the base of the spine to anaesthetise the vagina and the perineum. The area that is anaesthetised is more restricted than with an epidural; this technique is rarely used because it has a tendency to slow down labour. It is sometimes appropriate in the second stage where a forceps delivery is needed.\n\nEntonox (Gas and Air)\n\nThis is a mixture of half nitrous oxide (laughing gas) and half oxygen, which is inhaled through a rubber mask. The mask has a valve which opens only when you breathe in. It takes the edge off the pain rather than providing complete relief. It is said to be of most use towards the end of the first stage especially if the contractions are severe, but also if it is too late to give pethidine or an epidural. Some women swear by it, others have found it totally useless.\n\nIt is important to start using it before a contraction begins as the effects build up slowly. The effect starts 10\u201320 seconds after you start to inhale it and reaches its peak after about 45\u201360 seconds. It then decreases in effect very quickly.\n\nPros\n\n\u2022 you can control how much you use and when you use it.\n\n\u2022 it gives fairly good pain relief, especially towards the end of the first stage, in transition and\/or also if you need stitches.\n\n\u2022 it provides extra oxygen for you and your baby.\n\n\u2022 you can move around, as far as the rubber tube will reach.\n\n\u2022 it is quickly exhaled and has a minimal effect on the baby.\n\nCons\n\n\u2022 it doesn't work for every woman.\n\n\u2022 it doesn't work for contractions that begin with intense pain.\n\n\u2022 it doesn't provide effective pain relief and is therefore not good for very strong pains.\n\n\u2022 it makes many women feel out of control and confused because of feeling slightly 'high'.\n\n\u2022 some find the smell of the rubber tube unpleasant.\n\n\u2022 some women hate having a mask over their face \u2013 it makes them feel trapped.\n\n\u2022 some women feel sick or dizzy.\n\n\u2022 it can stop working after a while.\n\n\u2022 it takes a while to get the hang of it.\n\nDo\n\n\u2022 get some instruction early in your labour and practise using it when your contractions are manageable.\n\n\u2022 try it out if you are in pain in the first stage because it is worth using if it works.\n\n\u2022 breathe deeply but at a normal rate as it is not as effective with shallow breathing.\n\n\u2022 start using it at the first sign of a contraction. Ask your partner or midwife to place a hand on the underneath bit of your 'bump' just above your pubic bone if you have difficulty feeling when the contraction first starts.\n\n\u2022 take a few deep breaths, between 3 and 6, put the mask down and carry on dealing with your contraction in the ways that have been working for you. The effect of the gas will build up of its own accord and hopefully peak as your contraction peaks. (If you continue to breathe in the gas and air during a contraction then you will feel dozy and unable to judge when the next one begins.)\n\n\u2022 suck a sponge between contractions if your mouth gets dry.\n\nEpidural Anaesthesia\n\nAlthough a local anaesthetic is injected into the spine, it is not a spinal anaesthetic and only affects the nerves that supply the pelvic cavity with sensation (pain!). It doesn't affect mobility. This means that most women have control of their legs even when the pain is gone. There are no side effects to the baby.\n\nA needle is inserted between two vertebrae in the lower back into a canal called the epidural space. A very fine tube is threaded through the needle, which is then withdrawn, the tube is taped in place and a local anaesthetic injected into the tube. This numbs the nerves from the waist down. 'Top-ups' can be given if the pain returns.\n\nPros\n\n\u2022 you are fully conscious.\n\n\u2022 it is an effective painkiller and is therefore good for a forceps or Caesarean delivery.\n\n\u2022 because it deals with pain, emotional distress is also relieved.\n\nCons\n\n\u2022 partners are sometimes asked to leave the room whilst the anaesthetist inserts the epidural. This can take up to 20 minutes, perhaps more if insertion is difficult during which you will be asked to lie perfectly still (on your side). The combination of having to be still, contractions coming regularly and painfully, and lacking your primary support can be distressing.\n\n\u2022 it may work on only one side of the body.\n\n\u2022 it doesn't always take effect, which can be very distressing \u2013 the pain can seem overwhelming if you had thought you would soon be pain-free and were let down.\n\n\u2022 it lowers the blood pressure, which can cause dizziness, nausea and even fear.\n\n\u2022 some women feel cold and shivery immediately after the anaesthetic is injected.\n\n\u2022 it is difficult to push in the second stage because the area is numb and the likelihood of a forceps delivery is higher.\n\n\u2022 you can't move around.\n\n\u2022 one intervention begets at least three others. You are likely to have to put up with three or more of the following: breaking of your waters (see here); an electrode may be attached to your baby's head (see here); you may be connected to an external monitor to record your contractions (see here) and\/or an intravenous drip (see here); you may be given syntocinon (see here), and\/or a catheter (see here).\n\nThe after-effects of an epidural vary: women have experienced back pain, headache, general soreness, numbness and\/or aching legs, and, very rarely, complete numbness from the waist down, which can take up to 72 hours to clear.\n\nGeneral Anaesthetic\n\nA general anaesthetic is sometimes administered for a Caesarean section, usually only in an emergency. Some hospitals will agree to your partner being present to receive your baby, others will want your partner to wait outside and will bring the baby out when they are sure all is well.\n\nPros\n\n\u2022 If there are complications, especially if the baby is in distress, a Caesarean can be performed very quickly.\n\nCons\n\n\u2022 you are not conscious for the birth of your baby.\n\n\u2022 although the art of anaesthesia is much improved these days and the risk to your health less than it used to be, it is physically stressful.\n\n\u2022 you may take a while to recover from the anaesthetic: some people vomit or feel nauseous and others feel high and take a long time to feel fully themselves again.\n\n\u2022 bonding with your baby can take longer.\n\n\u2022 afterpains can feel worse because you were not accustomed to the intensity of contractions during labour.\n\nHomeopathy\n\nLike therapeutic acupuncture, homeopathy works by treating the psychological state of a woman in labour, her fear, anxiety, rage, disappointment, etc., and also by easing the minor physical complications like exhaustion, a cervix that isn't dilated, or tired muscles. If you have chosen to have a labour kit (see here) and don't have a homeopath present, your partner can take charge of prescribing so you don't have yet another thing to think about.\n\nPros\n\n\u2022 it's gentle and doesn't have side effects.\n\n\u2022 you have the additional support of a person you trust if you have chosen to have your homeopath present.\n\n\u2022 your homeopath can put together a selection of remedies (your labour kit) specifically for you if you don't want an extra person at the birth.\n\nCons\n\n\u2022 you may be too busy to work out which remedy to take when.\n\n\u2022 you have an additional person at the birth if your homeopath is present.\n\n\u2022 you may get fed up with the sugary taste of the tablets.\n\nHypnosis\n\nWhen hypnosis works, a person is fully conscious but is not aware of pain. Like acupuncture, some surgical operations have been carried out on people who respond particularly well to hypnosis, without them feeling any pain. Some people are easily hypnotised and some need several sessions for it to work. It is important to begin your 'training' during your pregnancy with a hypnotist who is willing to attend the birth.\n\nPros\n\n\u2022 you have additional support in labour.\n\n\u2022 it won't affect you or your baby adversely.\n\nCons\n\n\u2022 there is an extra person in the room.\n\nParacervical Block\n\nA local anaesthesia which is used in the first stage. Local anaesthetic is injected into the groin area where the nerves that feed the uterus collect, providing pain relief in the uterus without affecting the contractions. It is easy to administer and effective in a large percentage of cases for up to 4 hours. It is more widely used in the US and in Scandinavia but has not as yet gained acceptance in this country.\n\nPerineal Block\n\nLocal anaesthetic is injected into the perineum to anaesthetise the area where an episiotomy is to be performed, or a tear repaired. The effect lasts about an hour and a half, provides complete pain relief during stitching and can be repeated if necessary. Sometimes it doesn't work well or for very long in which case more can be injected. Very occasionally it doesn't work in which case another method of local anaesthesia can be used.\n\nDo\n\n\u2022 complain loudly if it hurts when you are being stitched up.\n\nPethidine\n\nAn analgesic drug derived from morphine. It is used after labour is established and administered by intramuscular injection. The digestive system has slowed down and the stomach will not now efficiently absorb drugs taken orally. Pethidine relieves pain as well as producing a sense of well-being and even euphoria. It takes about 15 minutes to work and the effect lasts about four hours.\n\nPros\n\n\u2022 it works well at relieving pain for many women even in small doses.\n\n\u2022 it gives women a break; many women sleep for some time after a pethidine injection, sleeping through contractions and waking feeling more refreshed.\n\n\u2022 it has a softening effect on the cervix which will help it to dilate.\n\nCons\n\n\u2022 it may have no effect whatsoever.\n\n\u2022 some women feel 'high' and can still feel the pain through a drugged haze.\n\n\u2022 it crosses the placenta, relaxing the baby so that its breathing may be affected after birth.\n\n\u2022 it can cause temporary amnesia: some women have difficulty in remembering their labour, which can be distressing.\n\nPethidine plus Phenergan or Sparine\n\nPhenergan and Sparine are both tranquillisers which help relieve anxiety, nausea and vomiting. They can be useful early in labour: if it is a long one, and sleep is needed but not possible because of pain.\n\nThey accentuate all the pros and cons of Pethidine alone.\n\nPudendal Block\n\nLocal anaesthetic is injected into the side wall of the vagina to anaesthetise the nerves that feed the vagina, perineum and labia. This is used in some instances before a forceps delivery, especially a low forceps. It is a good alternative to a general anaesthetic but has been largely replaced by epidural anaesthesia.\n\nCons\n\n\u2022 one side only is free of pain.\n\n\u2022 the injection itself can cause bruising.\n\nT(E)NS (transcutaneous nerve stimulation)\n\nThis is a safe method of pain relief that has not proved very effective in trials conducted so far. Two pads are attached with plasters to the back, one on either side of the spine, and are connected to a small hand-held generator. This conducts a low electric current to the nerves that feed the uterus. The generator can be switched on during contractions to block the pain.\n\nPros\n\n\u2022 there are no side effects.\n\n\u2022 you can be mobile.\n\nCons\n\n\u2022 it has not been found very effective so far.\n\n\u2022 it has to be used from the beginning of labour to be of any use.\n\n\u2022 if it helps in early labour, its effectiveness with strong labour is limited.\n\nTranquillisers\n\nThese are sometimes offered early in the first stage if labour starts in the evening, if it is progressing slowly and the woman is anxious or exhausted. It is hoped that she will rest and\/or sleep for a while to build up some reserves of energy to carry on with her labour. She might do just as well to pour herself a glass of wine or whisky. (See Slow Labour.)\n\nMEDICAL INTERVENTIONS IN LABOUR\n\nThe major interventions are internal examinations, enemas, inductions, drips, breaking of the waters, blood-pressure monitoring, catheterisation, fetal heart monitoring, episiotomies, forceps deliveries and Caesarean sections. The shaving of pubic hair is now rare.\n\nInterventions can be and are life-saving. However, some women feel that the experience of birth has been taken away from them; that one intervention leads to another; that they feel increasingly powerless and passive; and the emotional trauma of some interventions can lead to post-natal depression.\n\nIt is important to question all interventions thoroughly. The routine administration of anything for the convenience of a system (the hospital) or a person other than the woman in labour (such as the obstetrician) is wrong. If it is appropriate and necessary you may feel disappointed that you have had to resort to it. Try to 'go with the flow' if you find yourself in this situation \u2013 the intervention may save your life and that of your baby. Here are some suggestions to help you minimise or heal the trauma.\n\nDo\n\n\u2022 understand each intervention as fully as possible during your pregnancy. Discuss them with your doctor and midwife in as much detail as you need to.\n\n\u2022 make a good birth plan (see here). Decide ahead of time what you are prepared to fight against unless a crisis demands that it be done. Ask that you be consulted fully and at each stage if an intervention is required but not stated on your birth plan.\n\n\u2022 use homeopathic treatment and\/or the self-help measures outlined below for each intervention if it is necessary.\n\n\u2022 build a co-operative relationship with midwives, doctors and other hospital staff so that you feel part of what is happening, not a victim to it.\n\nBlood-pressure Monitoring\n\nIf you are having a hospital birth your blood pressure will be taken at regular intervals. Some women find this distracting, that it breaks their concentration, others hardly notice.\n\nDo\n\n\u2022 make sure you don't have to change position to have it done.\n\n\u2022 ask that verbal reassurance be given each time it is taken. The well-intentioned but misguided habit of not talking to women in labour can create an undertow of anxiety. Partners can ask the midwife to confirm that all is well if he or she forgets.\n\nBreaking of the Waters\n\nSee Inductions.\n\nCaesarean Section\n\nCaesarean section has only recently become a safe operation and due to the increasing effectiveness of the technique has become a routine intervention in some countries. The major reasons for performing a Caesarean are: a small or malformed pelvis; fibroids or ovarian cysts lying below the baby; placenta praevia, where the placenta is lying over the neck of the uterus; the baby is too large to pass through the pelvis or is in an exceptionally awkward position, especially if the shoulder is presenting and in some instances of breech presentation; the first stage is long and arduous and the cervix is not dilating; labour is induced and the induction doesn't work; the baby is in distress in the first stage as there is a danger of it suffering from lack of oxygen, which can cause brain damage or even death; the mother has pre-eclampsia, eclampsia, an active attack of genital herpes, diabetes, kidney disease or chronic hypertension; the mother haemorrhages; and if there is a known history of difficulty with vaginal births. An 'elective' Caesarean is one that is planned and agreed to before the birth, because the mother's pelvis is small for example. A date is set before the birth is due for admission to hospital. An 'emergency' Caesarean is decided on after the labour has started because of an unanticipated complication such as a badly positioned baby or a very long labour.\n\nCaesareans are performed under epidural or general anaesthesia. With an epidural you are conscious and can watch your baby being lifted out and hold the baby immediately. It is safer for the baby (because the drugs used in general anaesthesia can affect it) and recovery is usually quicker. (See Epidural Anaesthesia.) With a general anaesthetic you are unconscious throughout the birth and for some time afterwards. It can be administered quickly and is always the preferred method in an emergency.\n\nIn an emergency it is very important that your midwife or partner keeps up a comforting 'banter' of reassurance and information, both physically and verbally.\n\nIf you are the midwife who has been with the woman during labour or prior to her elective section, your main duty is to explain over and over again exactly what is going to happen.... having an operation is the nearest brush with death that most of us experience in our lives. We are literally giving our body up to the care of another. Most people feel that they will never wake up after an operation, that the end has come. To have the physical comforting of another at this time can make all the difference in the world.\n\n(Caroline Flint, Sensitive Midwifery, Heinemann, 1986)\n\nIf you have a general anaesthetic you will not be given a 'premed', an injection to relax you and make you sleepy before you go to the operating theatre, as it contains morphine or a morphine-based drug which can cross the placenta and affect the baby, but you will be given atropine to reduce the secretions from the mucous membranes of the chest, nose and throat. You may be asked to breathe oxygen for a few minutes to give your baby an extra boost immediately before the anaesthetic is given. An intravenous drip is inserted into a vein in your arm and a small amount of anaesthetic is injected.\n\nIf you are having an epidural Caesarean, see here for a full description of the procedure. A screen is erected at shoulder or waist level so that you can't see the operation itself. You can ask for it to be lowered when your baby is actually being lifted out \u2013 the excitement of seeing your baby emerging will divert your attention from the operation and in any case at this time there won't be anything gory to see.\n\nWith both types of Caesarean your pubic area is shaved and a catheter inserted into the urethra to drain urine from your bladder. The incision in the abdomen is either vertical (in the middle below the navel) or more usually transverse, known as the bikini cut, running just below the top of the pubic hair and above the pubic bone. When the hair grows back the scar is virtually invisible.\n\nThe muscles of the abdomen are gently separated and the organs inspected. The bladder is cut free from the uterus and a small incision made in the lower part of the uterus. The amniotic sac bulges out, if still intact, and makes quite a noise on being cut. The baby is then lifted out, sometimes with the help of forceps, and any fluid or mucus removed from the mouth, nose or eyes. The cord is clamped and the baby then handed to the midwife or the mother, if she is conscious and the baby is breathing well, or the partner if present.\n\nAn injection of ergometrine or syntometrine is given to contract the uterus and about 40 seconds later the placenta is delivered. The organs and muscles are stitched back into place with two or three layers of stitches; the bladder is reattached to the uterus and the four layers of abdominal muscles sewn up one by one with fine, dissolving stitches. The skin is either stitched or closed with metal clips which are removed when the scar has healed. The operation takes under an hour from start to finish. A Caesarean scar takes about three months to heal completely.\n\nA midwife will be present to care for the baby immediately after the birth. Hospital policies vary with regard to partners being present and although most will allow partners if an epidural has been administered, they may not if you have a general anaesthetic. Check this at your ante-natal visits and specify in your birth plan and\/or on your hospital notes that your partner's presence has been agreed to (if it has). Ask your midwife to take photos of the baby emerging, and also that your partner and baby will be with you when you come round. Ask for your midwife to be on hand to take a photo of your first meeting with your baby \u2013 your wooziness from the anaesthetic will mean you won't remember this moment and it's wonderful to have it recorded.\n\nComplications include haemorrhaging (loss of blood); injury to organs close to the uterus (the bladder, bowel or ovaries); post-operative infection or blood clots; and emotional distress. Many, if not most, women have an overwhelming desire to push out their babies themselves and a Caesarean, or even a forceps delivery, can cause distress or feelings of failure.\n\nYou may feel that you have given over the birth of your baby to somebody else. Afterwards, it is no solace to be told that in another age you would have died. Feelings can range from distress to rage, and it is important to talk and talk to heal emotional wounds, to prevent depression and isolation. Knowing other women who have been through the same experience is helpful.\n\nWith subsequent pregnancies the scar may feel sore as the uterus grows \u2013 as it stretches so does the scar. Repeat Caesareans will be performed in the same scar, but more and more women are successfully delivering subsequent babies vaginally. You need reassurance, information, encouragement and support to give you the self-confidence to do so if you want a vaginal delivery after you've had a Caesarean.\n\nThe following advice applies to when you are conscious before a general anaesthetic if you are having one.\n\nIf you are having a general anaesthetic you will not be allowed to take anything by mouth. A few drops of Rescue Remedy rubbed into the forehead works wonders for anxiety and fear. If you are having an epidural and are allowed to take homeopathic remedies then you may need one for emotional distress. Try Lycopodium for anxiety, Argentum nitricum for anxiety with a strong feeling of failure, Aconite for fear or Pulsatilla for the weeps. (For information about the post-natal recovery period after a Caesarean see here.)\n\nDo\n\n\u2022 prepare yourself adequately if you are having an elective Caesarean.\n\n\u2022 go with the flow if you are unprepared for it. If you have been trying for a natural birth remember that you have done your best: you did what you could and now it's time for modern medicine to make things easy, to do its best for you and your baby.\n\n\u2022 breathe steadily, slowly and deeply.\n\n\u2022 relax. You can let go now, you are being taken care of, there is nothing more that you have actively to do and your baby will soon be with you.\n\n\u2022 ask for support, information, for the people with you to talk to you, to hold your hand, to stroke your head. Whatever it is that you need, ask for it now.\n\n\u2022 allow your feelings to surface, especially if your partner is with you.\n\n\u2022 ask your partner or midwife to talk to you continuously \u2013 encouraging, soothing, reassuring, loving words \u2013 the silences in an operating theatre can be especially hard to take.\n\n\u2022 let the medical staff know what you are feeling if you are finding it difficult to cope and ask for reassurance if you need it.\n\n\u2022 use affirmations or guided visualisations (see here) to take your mind off what is happening to your body. Ask your partner or midwife to hold your hand and talk you through something pleasant in your past or your imagination!\n\nCatheterisation\n\nA catheter is used with both forceps and Caesarean section deliveries (and with an epidural) to minimise the risk of injury to the bladder as a full bladder is vulnerable to being damaged during birth (especially during the second stage). A fine tube is inserted into the urethra up into the bladder to draw off urine as it gathers. If you have had an epidural you won't feel anything. If you haven't you may feel a strange pulling or drawing sensation as it is inserted. Some women find them painful on insertion and uncomfortable once they are in place. After a Caesarean section the catheter may be left in place for up to 48 hours. If you are unable to urinate after the birth a catheter is used to drain the bladder (see Retention of Urine).\n\nA catheter can leave the urethra feeling sore and sensitive, rather like mild cystitis.\n\nStaphysagria may help if the urinary tract is painful or Arnica if it feels sore and bruised. You may need to alternate these two remedies if both are indicated.\n\nDo\n\n\u2022 relax your pelvic-floor muscles (here) when the catheter is being inserted.\n\n\u2022 ask the midwife or doctor to stop and proceed slowly if it is uncomfortable.\n\n\u2022 urinate frequently after the birth even if it hurts. 'Bottle wash' (here) or pee in a warm bath until it ceases to be painful.\n\nDrips\n\nAn intravenous drip is used to give glucose to women who have become dehydrated, or salt to keep blood pressure steady or oxytocin to speed up labour. You can avoid a glucose drip by keeping up a regular intake of fluids, especially if they contain honey or sugar. If you are vomiting and unable to keep anything down you will need a glucose drip. The disadvantages are that it makes mobility difficult and can feel uncomfortable.\n\nDo\n\n\u2022 carry on being as active as you can: ask for the drip to be attached to a mobile pole. Ask your partner to move it around after you.\n\n\u2022 ask that the drip be put into the arm or hand you use least.\n\nDon't\n\n\u2022 forget to urinate frequently, even if you don't feel like it, every 1\u20131\u00bd hours.\n\nEnemas\n\nEnemas were an aberration of hospital routine that are almost never administered now. A tube is inserted into the anus and half a pint of soapy water runs into the rectum from a bag held above the body. The liquid is held in the rectum for a short time and then evacuated along with the contents of the rectum. Enemas were performed from good intentions, with the idea that an empty bowel is desirable in labour, but this is virtually impossible. If you clear the rectum, as an enema will, then the contents of the large bowel and small intestine simply move down. Enemas can be uncomfortable and sometimes painful \u2013 nature works well enough as most women have a bout of diarrhoea before labour starts. Others find they empty their bowels during the first stage.\n\nIn any event most women pass some faeces during the second stage (the delivery) and this is usually dealt with kindly and considerately. Your midwife will protect your emerging baby by holding a pad against your anus.\n\nDo\n\n\u2022 empty your bowels as frequently as you need to during your labour.\n\n\u2022 ask for reassurance and support from your partner, friend or supportive midwife if you feel ashamed, embarrassed or disgusted.\n\n\u2022 try to see the funny side of it.\n\nDon't\n\n\u2022 worry or be ashamed if you pass faeces during birth. It is a normal, healthy function of the body. You are not alone.\n\nEpisiotomy\n\nThe routine snipping of the perineum during childbirth is now being questioned. It is a necessary intervention with a forceps delivery and to hasten delivery if the baby is distressed, but there are many experienced midwives who have delivered hundreds of babies, whose 'mothers' have not even torn, save for minor, first-degree tears that heal of their own accord.\n\nA woman whose birth is managed well by a competent midwife rarely tears and rarely needs an episiotomy. A tear is more difficult to stitch than an episiotomy but it is easier to heal and therefore many women are preferring now to risk a tear than to be cut.\n\nThings often happen so fast it isn't easy to think of taking a remedy. Aconite will help you to regain control if everything is going too fast and you are feeling panicked. One dose of any potency will be enough to slow things down a little and prevent a tear.\n\nDo\n\n\u2022 prepare the perineum during the last month of pregnancy by massaging almond or olive oil into it, by visualising it becoming supple and stretchy, by sitting in the 'tailor pose' to encourage your pelvic-floor muscles to stretch and become more supple.\n\n\u2022 maintain an upright posture during labour especially during second stage (supported standing or squatting, kneeling on all fours, etc.) as an episiotomy is rarely needed in this position.\n\n\u2022 massage oil into the perineum as your baby crowns. Some women have found Calendula oil especially effective.\n\n\u2022 apply hot pads to the perineum once the head is crowning, sanitary pads or flannels wrung out in very hot water to which 7 drops of Rescue Remedy and a squirt of Calendula tincture have been added. This is soothing and encourages blood to flood into the area, which will further relax and stretch the tissues.\n\n\u2022 visualise your perineum opening like a flower.\n\n\u2022 open your mouth wide to assist it \u2013 there's an important link between your mouth and your vagina.\n\n\u2022 make 'raspberry' noises with your lips flapping as this helps the perineum to relax.\n\n\u2022 take your time \u2013 there's no great hurry.\n\n\u2022 rather than trying to push, let your body do the work.\n\n\u2022 ask for constant reassurance that you are doing fine, that your body is coping brilliantly, that your baby is nearly with you.\n\n\u2022 laugh or cry if you feel like it, you are about to meet your baby.\n\nDon't\n\n\u2022 tense up in fear that you will tear or be cut.\n\nFetal-Heart Monitoring\n\nThe baby's heartbeat is monitored regularly through labour because a deviation from its regular pattern can indicate that it is in distress. The heartbeat can be heard by placing an ear, a stethoscope or ear trumpet on the abdomen near where the baby's heart is lying. Portable electronic monitors are now often used and can be applied whatever position you choose to labour in.\n\nSophisticated, electronic fetal-heart monitors that also print out a graph of the heartbeat are valuable in high-risk births where they can be life-saving but their use in low-risk births is questionable, especially if everything is going well. There are two types of monitor, both of which are hooked up to a machine that 'prints out' the contractions in a graph. One is attached by a strap to your abdomen \u2013 the strap itself can be uncomfortable, even painful at times when the contractions peak, and the strap tightens. You can't be fully mobile although the monitor can be attached if you are on all fours. The other type of monitor involves clipping or screwing an electrode to the baby's head. This is attached by a wire to the machine. The waters will have to be broken if they haven't already done so. The disadvantages are that it may be painful for the baby and can cause a sore spot on the scalp. It is not possible to be mobile. In both cases the machine becomes a hypnotic focus of attention and the woman in labour can be inadvertently ignored.\n\nDifferent remedies will help depending on how you feel: Aconite if you are shocked or afraid; Staphysagria if your body feels 'invaded' \u2013 especially if your baby has an electrode on its scalp; Pulsatilla if you feel despairing and miserable. Take Arnica from time to time to help minimise any bruising to your baby's scalp.\n\nDo\n\n\u2022 negotiate for a hand-held monitor to be used to check your baby's heartbeat unless you are in a high-risk group and you trust your doctor's advice. Seek a second opinion if you want to be absolutely sure.\n\n\u2022 ask that the monitoring be incorporated into your rhythm of labour.\n\n\u2022 ask that you be verbally reassured every time that it is done. Make sure your partner asks for this reassurance if the midwife or doctor forgets.\n\n\u2022 ask that the machine be turned so you can't see it, or draped with a cloth (if you are connected to the full works) so that you are not mesmerised by its every quiver.\n\n\u2022 ask to be put in any position other than lying on your back if you are uncomfortable.\n\nDon't\n\n\u2022 let this test distract you from the rhythm of your labour as it can create unnecessary tension.\n\nForceps Deliveries\n\nForceps are used once the baby is at least two-thirds of the way down the vagina and stuck (if they become stuck higher up a Caesarean will be performed instead). Sometimes they are used simply to turn the baby's head if it is poorly positioned, or to turn it and then 'lift' or guide it out.\n\nYou will have to lie down on a hospital bed with your feet resting in stirrups. A catheter is inserted (see here) and a local anaesthetic injected into the vagina to numb it. The forceps \u2013 Wrigleys or Keillands \u2013 look like a pair of salad servers. They are gently placed, a blade at a time, up and around the head of your baby and the handles locked together. Your doctor will then guide the baby out with each contraction. It can feel as if your insides are being dragged out \u2013 they aren't \u2013 but it is only for a very short time. It usually takes no longer than a couple of minutes.\n\nA suction or vacuum delivery (also known as ventouse) is not as physically, and therefore emotionally, traumatic but it can take longer to insert. A cup is fitted to the baby's head and the suction takes about five minutes to build up to full strength. This type of forceps delivery is increasing in popularity because it is much kinder to mothers and babies: it is less painful because an episiotomy isn't needed and although your baby's scalp may swell where the cup was applied there is no risk of the bruising to face or head which is common with Wrigleys' or Keillands' deliveries.\n\nHospital staff need to be sympathetic and give lots of information while a forceps delivery is being carried out. Good pain relief is also essential. A forceps delivery is more common if your baby is premature; if its head is big; if it is badly positioned (for example, breech); if it is posterior and needs turning round to come out; if you have had an epidural (because the loss of sensation means you won't feel the urge to push); if you've become tired; if the second stage has gone on for too long \u2013 typically for two hours with no results because you and\/or the uterus are exhausted; if your baby is suffering distress in the second stage with the contents of its bowel (meconium) in the amniotic fluid.\n\nTake Rescue Remedy throughout, either in sips of water, or by sucking on a sponge that has been dipped in the water containing Rescue Remedy, or by sponging your forehead with the water. (See Healing after Labour, or Emotional Distress, for remedies to help with healing after a forceps delivery.)\n\nDo\n\n\u2022 prepare yourself before the birth not to have an overwhelming expectation of pushing the baby out yourself.\n\n\u2022 tell your consultant or hospital registrar that if you need forceps you would prefer a suction delivery and check that staff trained and experienced in performing this will be available. State your preference on your birth plan (see here).\n\n\u2022 go with the flow.\n\n\u2022 relax your legs \u2013 consciously and repeatedly.\n\n\u2022 remember that it is for the best and that your baby will soon be born.\n\n\u2022 ask your partner or midwife to hold your hand or cradle or hug you.\n\n\u2022 be reassured by your birth partner. It can help if he or she looks into your eyes (partly to distract) and says soothing, repetitive things to you.\n\n\u2022 count to 100 slowly, with the aid of your partner, and the baby should be out before you have finished.\n\nInductions\n\nInductions are carried out if there is more than one baby, if you have diabetes or high blood pressure, if there is an abnormality in the baby or if the baby is too large or too small \u2013 and are far too common in babies that are 'late', an unnecessary intervention because the accurate calculation of dates is so difficult and most women go into labour at their own right time. Inductions can also be carried out for the convenience of the hospital or obstetrician, or even the parents, and are then of even more dubious value.\n\nSynthetic hormones are used to bring on labour. Prostaglandin is given in the form of pessaries or cream, inserted into the vagina every 6\u201312 hours, which usually brings on labour after two doses; Syntocinon is given intravenously (via a drip, see here) either to instigate labour or to speed it up. The possible side effects of these drugs are vomiting, diarrhoea, migraine or vaginal irritation from prostaglandin and oedema and high blood pressure from Syntocinon, and jaundice in your baby. 'Induced' contractions are always more violent and prolonged than natural ones and can contribute to you feeling powerless and out of control. The contractions don't follow a wave-like course but become intense quickly and remain intense for a longer period. Women who are induced are more likely to ask for an epidural to reduce the pain.\n\n'Breaking the waters' is another method of induction. Also known as artificial rupture of the membranes, it is performed by puncturing the bag of waters with an instrument similar in shape to a crochet hook. The disadvantages are that it can feel uncomfortable; it increases the risk of infection, especially if it is done in hospital; the baby's head and your cervix are no longer cushioned by the amniotic fluid and the stronger contractions that ensue can be harder to cope with. The waters are also sometimes broken when labour is well under way to speed up the last of the cervical dilation and to check the amniotic fluid for meconium if the heartbeat shows some distress. If induction is proposed:\n\nDo\n\n\u2022 have sexual intercourse. Semen contains prostaglandin which can start labour.\n\n\u2022 consult an acupuncturist.\n\n\u2022 try homeopathy. Take Caulophyllum 30 every 2 hours for up to 6 doses over the course of a day. Wait for three days and repeat. If labour still hasn't started, your baby may not be ready yet to be born.\n\n\u2022 eat a strong curry. Spicy food can speed things up!\n\n\u2022 take some vigorous exercise \u2013 a long, brisk walk can bring labour on.\n\n\u2022 See also False Labour, and Late Labour.\n\nNB If none of the above has any effect the chances are that your dates aren't quite right or you and your baby aren't quite ready.\n\nDon't\n\n\u2022 be frightened of having a 'dry' birth if your waters rupture (naturally or otherwise). Your amniotic waters are constantly being replenished, and will continue to be throughout your labour.\n\nInternal Examinations\n\nVaginal examinations are necessary at various points during labour to monitor progress. They can be painful partly because women are often asked to lie on their backs while they are carried out. Some midwives are happy to do internals while women are in their favourite positions \u2013 lying on a bean bag, leaning against the bed or even standing or squatting but they may still be painful.\n\nThe following remedies will all help after an 'internal': Aconite, if you are in a lot of pain and very shocked; Chamomilla, if you are in a lot of pain and very angry; Staphysagria, if you are very angry and indignant and resentful; or Arnica, if you feel sore and bruised.\n\nDo\n\n\u2022 check with the midwife when you make your birth plan what the procedure is and whether it can be adapted if necessary.\n\n\u2022 relax, breathe slowly and evenly and count slowly to 100.\n\n\u2022 ask for a breathing space if the midwife can't find what she is looking for and you have been prone for too long. Get up and walk around and then let her try again.\n\n\u2022 ask for your midwife to remove her hand during contractions.\n\nShaving of Pubic Hair\n\nShaving of pubic hair is rare nowadays but some women are still shaved routinely on admission to hospital. This is a pointless practice that should be resisted by all women in labour. There is no evidence to support its usefulness and it can be painful, especially if the skin is inadvertently 'nicked'. When the hairs grow back after the birth the itching is intolerable as the hairs grow through and the stubble scrapes the delicate skin on the inner thighs.\n\nDon't\n\n\u2022 agree to having this done \u2013 unless, of course, you are having a Caesarean, in which case ask if the shaving can be restricted to the area over the pubic bone only.\n\nPOST-NATAL TESTS (INTERVENTIONS) FOR THE BABY\n\nImmediately after the birth, and again five minutes later, your baby is tested for its general vitality. This is the Apgar test and points are given for heart rate: 0 = absent, 1 = less than 100 beats per minute, 2 = more than 100 beats per minute; breathing: 0 = absent, 1 = slow or irregular, 2 = regular; skin colour: 0 = blue, 1 = body pink and extremities blue, 2 = pink all over; muscle tone: 0 = limp, 1 = some movements, 2 = active movements; and reflex response: 0 = absent, 1 = grimacing only, 2 = crying. Most babies score between 7 and 10 points at birth and those who score low immediately after the birth usually score 9 or 10 after five minutes.\n\nAfter the birth your baby will be measured, checked and tested for the following: weight; length from head to toe; the circumference of the head; the genitals \u2013 to make sure they are all in order; the anus \u2013 to make sure it is not blocked; the heart and lungs \u2013 to make sure they are strong and healthy; the pulse; the palate of the mouth \u2013 for cleft palate; the jaw \u2013 for dislocation; the joints \u2013 especially the legs and hips for dislocation; the legs and feet \u2013 for clubfoot and to make sure the legs are the same length; the spine \u2013 to make sure the bones are in the right place; the liver and spleen \u2013 to make sure they are the correct size; the bones of the skull and the fontanelles; the chest and breathing.\n\nA Guthrie test is performed on babies when they are about a week old: some blood is taken from the heel by pricking it with a needle \u2013 this is commonly called a 'heel prick'. The blood is checked for phenylketonuria (a form of mental handicap) and for thyroid function, both of which are treatable if detected early enough. Heel pricks are sometimes repeated in babies where the blood-sugar levels are being checked at regular intervals after the birth if diabetes is suspected. Vitamin K is given to help the clotting of blood, either as drops orally, or by injection. Recent research has shown that it may save lives. Although an oral dose (by mouth) may be kinder to your baby than an intramuscular injection it is not as effective. Talk this issue through with your GP or consultant.\n\nOccasionally babies are shocked or distressed by some of the post-natal checks especially the heel prick. As you will probably be taking Arnica, traces of it will pass through to your breastmilk and deal with any shock or bruising they have suffered. If the baby is more actively distressed, and this doesn't pass off, consider giving Aconite or Staphysagria depending on whether fright or anger is predominant.\n\nDo\n\n\u2022 ask your doctor or midwife to communicate with you as they are working through the above tests. You might ask them to give you a running commentary on their findings as they go along. You may have to keep asking, 'What are you doing now? What are you looking for now? Can you please tell me what you are listening for?' and so on. Doctors don't realise that their silence can be interpreted by some parents as a sign that something is wrong, when so much hangs on their pronouncement of health. Your doctor may be exhausted, busy and find it easier and quicker to work in silence, or may be used to working without communicating.\n\n\u2022 step in if you feel your baby is being treated unnecessarily casually. Some midwives fling newborn babies around to demonstrate how resilient they are. This is unkind and unnecessary. You have every right to ask for your baby to be treated gently.\n\n\u2022 hold your baby if vitamin K is administered and ask for reassurance about what is happening. Rub a little Rescue Remedy into the injection site and give Aconite if he or she is very upset.\n\n\u2022 ask to hold the baby while the heel is pricked or an injection given so that you can cuddle, comfort and gently reassure, by talking calmly about what is happening and why.\n\n\u2022 offer the breast once all the checks have been performed. Cuddle your baby close and let him or her suck for as long as they want. Some babies fall soundly asleep at this stage while others are curious about their new surroundings and take their time looking around.\n\nNB You and your baby will be expected to attend a post-natal check with your doctor around six weeks after the birth (this is changing to a check-up after three weeks and one at eight weeks) at which your doctor will examine you to make sure you are recovering well. Your baby will not usually be examined unless there are any problems but will be weighed naked.\n\nBIRTH PLAN\n\nWith all the information that's gone before you are probably ready to start preparing your birth plan. Before you do, though, read the birth chapter (here) to get more of a sense of what it is you are looking for, you are wanting for your birth.\n\nPlanning for birth requires careful thought. Fashions in childbirth change and contradictory information abounds on how to get it right. Your plan needs to reflect what you want and hope for as well as possible compromises.\n\nIf you are planning on a hospital birth ask your doctor or your hospital whether they have a birth policy. Check whether it is their policy to encourage women to be involved in their babies' births, to make a birth plan and for the staff to support it. If it isn't and this is important to you, try to find a hospital or doctor who is sympathetic to your needs.\n\nDiscuss your plan with your partner and as many other people as you need. Use the following steps as a rough guide:\n\n\u2022 Write down what you want the birth to be. If this is difficult, talk to women who already have children and use the Complaints section (here) to give you ideas of what you might want.\n\n\u2022 Ask your partner to do the same.\n\n\u2022 Read these out to each other so that you can start to work out what you want based on your shared ideas.\n\n\u2022 Understand what really matters to each of you.\n\n\u2022 Make a list of what you would particularly like to happen, and another of what you would particularly like to avoid.\n\n\u2022 State these requests as 'preferences' to avoid being labelled as pushy.\n\n\u2022 Write out your plan in the spirit of 'friendly cooperation', not as a series of demands, so as to gain the full support of the medical profession.\n\n\u2022 Make sure you state that you are willing to make changes at any stage but would like to be consulted first.\n\n\u2022 Discuss your plan with your midwives and\/or consultant and be prepared to negotiate and compromise if they express concern over a particular request.\n\n\u2022 Re-write your plan if necessary.\n\n\u2022 Give a copy to the midwife, doctor or consultant and ask that it be attached to your notes.\n\n\u2022 Don't forget to take a copy into hospital with you when you go into labour.\n\nIf you have given birth before, writing a birth plan will probably be easier: try to remember what worked and what didn't and just jot it down.\n\nSpecific issues to think about and include if relevant:\n\n\u2022 Birth partners. Who will be attending the birth? Your partner, a friend or relative, homeopath, acupuncturist, active birth teacher? State their names and the roles they may have during the labour and birth. Stand your ground over having your partner plus one other supportive person (if you want one): some hospitals draw the line at one extra person only. If you have two, one can take a break without you feeling abandoned.\n\n\u2022 How would you like to be treated? Be clear about the sort of birth you would like and therefore what sort of support you need from the medical staff. Make sure you can move around in labour and give birth in a position that is comfortable for you. State your preference for a bed on the floor or a birthing-chair or birthing pool (if the hospital has these on offer). Ask that medical professionals communicate their findings in all their checks to you and\/or your partner, that they do not, for example, remain silent after listening to the baby's heart and then leave the room. State your preference to be attended by a midwife or one of the midwives who has seen you through your pregnancy. Ask what your options are if you and a particular midwife don't get on.\n\n\u2022 How would you like your birth partner(s) to be treated? Make sure that they won't be asked to leave the room for any reason, including a forceps delivery or a Caesarean, unless this is performed as an emergency measure.\n\n\u2022 Are you willing to have medical students present if you are delivering in a teaching hospital? You have a right to refuse this. You may make your acceptance conditional on their attending the whole of your labour \u2013 relatively few medical students see a birth from beginning to end.\n\n\u2022 State your attitude towards induction, routine procedures on admission such as breaking of the waters, shaving of pubic hair or enema (thankfully rare nowadays), and medical interventions in general (see here). State your preference, for example, not to be given an episiotomy, and be firm on issues that are important to you. 'If I tear I do not agree to being stitched by a student.'\n\n\u2022 Make a list of what you will probably take in with you (see here).\n\n\u2022 Which drugs do you hope to avoid? Which do you feel happy to use? Decide what you are willing to try, what you are willing to take and what you definitely don't want unless all else fails (see Pain Relief). It doesn't matter what you have as long as you feel you need it, and you can always change your mind. Don't let yourself feel guilty or a failure if you eventually opt for something you had originally decided against.\n\n\u2022 How do you want your baby treated at the birth? State your preference for soft lights and little noise. Ask that your baby be given to you without delay and for as long as possible: research has shown that breastfeeding is easier and post-natal depression less common in women who are able to bond with their babies immediately after birth. Ask that the cord be cut only when it has stopped pulsating, when the baby's own system is functioning fully, to minimise shock or trauma. Ask that the administration of syntometrine (to expel the placenta) be delayed for half an hour (see here), unless there are complications, to allow the placenta to deliver naturally.\n\n\u2022 How do you and your baby want to be treated after the birth? Think about the immediate postnatal period. Meeting your baby for the first time is a magical moment. Build what you want for the time just after the birth into your birth plan and try to anticipate any difficulties. Ask that you and your partner be left alone immediately after the birth, at least for a little while, and that your other child or children be able to visit as soon as possible after the birth. Ask not to be separated from your baby if all is well.\n\nAt the end of your birth plan make it clear that you are prepared to be flexible, to negotiate and compromise. Discuss your fears and build them into your birth plan: for example, 'In the unlikely event that I will need a Caesarean I'd prefer to be conscious and have it with an epidural anaesthesia. I'd like you to agree to both my birth partners being present. If I need a general anaesthetic I'd like the baby to be given straight to my partner and for my friend to take photos of my baby.' Think through possible difficulties \u2013 for example, what if your baby is premature and needs special care? What if you haemorrhage after the birth? \u2013 so that you can check out hospital policies and gain some idea of what you might want should you find yourself in this sort of situation.\n\nIf you are having your baby at home it is just as important to write up a birth plan, to try to establish what you would like and dislike, to negotiate the role your midwife will play and to have contingency plans for transferring to hospital if necessary.\n\nFinally, this isn't going to be the perfect plan! It will act as a guide and help you to define what you want for your baby's birth. Women today have higher expectations. Hopefully your labour will be as you want but so often women have a very different experience from the one that they hoped for. Keep your expectations realistic, as almost anything can and may happen. Be flexible and go with the flow.\n\n* * *\n\nPREPARING FOR THE POST-NATAL YEAR\n\n* * *\n\nBecoming a parent changes everything: our lives are enriched in so many ways, and we are stretched and challenged, sometimes to our limits. The roller coaster of birth becomes the assault course of parenthood. Children bring us great joy and unconditional love \u2013 and their own demands and challenges. As they grow so do we alongside them. It is hard to prepare for parenthood because you don't know how you are going to feel and you can't predict how your new child will integrate into your and your family's lives, because of his or her unique character. Let your instincts guide you in how you respond \u2013 they are usually more reliable than theoretical advice. Do what you know is right no matter what others think. It doesn't make them wrong and you right, it makes you different \u2013 and that's fine.\n\nParenting ideas follow trends so question the style you choose. Do you want to do the opposite of your own parents or have you admired a friend's parenting style and want to emulate it? Don't be seduced into being an earth mother, full-time parent, or a Super mum, with a full-time job as well, just because you think you should. The stress of parenting in a style that doesn't suit you will cause tension and ultimately ill-health. Set yourself realistic standards that you know you can live up to.\n\nBeing a 'successful' parent is a question of balance \u2013 of finding your own way and of feeling confident in it. This book's aim is to offer advice to pick and choose from, so that you can begin to find out what works for you, and reject what doesn't. (See here for a selection of support networks should you feel the need for 'hands on' help.) Listen to your children \u2013 they will teach you more than any childcare expert \u2013 and allow them to guide you.\n\nTHE JOB DESCRIPTION\n\nParenthood is a job and a way of life; although you can't prepare for it fully or train for it, there are ways in which you can become clear about the job you are taking on and begin to prepare for it so that it doesn't come as a complete surprise. The job description that follows is to give you food for thought, and provide a realistic framework for discussions \u2013 not to put you off!\n\nResponsibility\n\nYou will be prepared to commit yourself for a minimum period of 16 years. The post necessitates a high level of executive ability and includes the following fundamental requirements:\n\n\u2022 to feed and clothe the child or children adequately.\n\n\u2022 to protect them from harm.\n\n\u2022 to love them unconditionally.\n\n\u2022 to teach basic survival skills, such as to communicate and to mix with others.\n\n\u2022 to help them to form a belief system.\n\n\u2022 to encourage them to achieve their full developmental potential, independence and high self-esteem.\n\n\u2022 to look after yourself adequately to achieve the above and in so doing, create a healthy model of adulthood for them to aspire to.\n\n\u2022 to protect yourself against burn-out.\n\nThe Hours\n\nYou will need to plan how a baby will fit into your existing life-style. Those who breastfeed may spend upwards of three to four hours a day doing just this. Time must also be allowed for bathing and nappy changing. Those whose babies like to suck or are slow to feed, or who feed little and often will find that feeding takes even longer. Previously successful jobholders have found that the new, often slower pace of life allows them time to sit and read, talk to other children, reflect or watch television, or simply enjoy the baby.\n\nCaroline Flint (Sensitive Midwifery, Heinemann, 1986) suggests that midwives hand a note to new parents:\n\nCongratulations on your new baby. He or she will bring you great joy and pleasure and you will learn many new things from this experience. Having a new baby is like starting a new life or a new job which takes twenty-four hours a day. You will need someone to help you for at least two weeks after the birth and preferably longer. You will need to concentrate on the baby, feed her, change her nappies and talk to her. This will take about 12 hours a day. It will not be from 8 a.m. to 8 p.m., and then you are off duty. It is half an hour here, two hours there, an hour and a half next. This will take up nearly all of your time. You need someone to help by doing your washing, by cooking you nutritious meals, by shopping and keeping your home clean.\n\nLife with a new baby may have little structure \u2013 some babies, for example, take a while to learn night from day \u2013 which can be a substantial challenge for the job-holder who has been used to working hard to a strict schedule and having large amounts of time off.\n\nAs your baby grows into a child the demands on your time decrease but they are rarely predictable, i.e. illness can 'strike' at any age and you'll be needed more or less around the clock for several days (and more) if your ten-year-old needs an emergency apprendicectomy. Be open (and prepared) for anything to happen and it generally doesn't. Plan for a quiet life and you may be surprised...!\n\nBenefits\n\nThere are many, they include the following:\n\n\u2022 the baby!\n\n\u2022 being loved unconditionally.\n\n\u2022 loving \u2013 the experience of loving a child, especially your own baby, is beyond description.\n\n\u2022 increased emotional maturity. The challenges of parenthood help parents to grow up also.\n\n\u2022 learning to do more than one thing at a time and to use small amounts of spare time to the fullest \u2013 whenever it comes.\n\n\u2022 acquiring the arts of planning, strategy and diplomacy.\n\n\u2022 responding to virtually any situation at any given time and resolving conflict without violence, either verbal or physical.\n\n\u2022 developing organisational skills as well as those of negotiation and compromise.\n\n\u2022 the ability to conduct at least two conversations at the same time.\n\n\u2022 sheer, unadulterated enjoyment. The entertainment value of babies and small children (especially your own) is very high \u2013 they are infinitely more interesting than TV!\n\nTools of the Trade\n\nThe following will enable the job-holder to avoid burn-out and reduce stress:\n\n\u2022 an open heart.\n\n\u2022 a sense of humour.\n\n\u2022 good health.\n\n\u2022 common sense.\n\n\u2022 patience and flexibility.\n\n\u2022 constancy and a sense of perspective.\n\n\u2022 respect and understanding.\n\n\u2022 curiosity.\n\n\u2022 the ability to listen and observe.\n\n\u2022 a decent memory.\n\nAge Restrictions\n\nWomen: approximately 16\u201345.\n\nMen: no upper limit.\n\nJob-share Opportunities\n\nJob-share opportunities exist whether through necessity or choice with your partner, other parents of similar-age children, grandparents, friends or with paid help. Bear in mind, you should job-share only with someone who sympathises with your own goals in child-rearing.\n\nIt is recommended that those who decide to job-share negotiate carefully with their partner and older children so that household chores and responsibilities as well as childcare are equally shared, as the first year or two can be extremely exhausting.\n\nThe Equipment\n\nYou can prepare for the first year in this job by buying an enormous amount of equipment \u2013 a substantial carriage, a pushchair, crib, cot, baby carrier, car seat, swings, lambskin, reclining chair, bath, rattles and toys, clothes, toiletries and nappies.\n\nAlternatively, you may buy very little. You will need a fair amount of equipment but as little of it wears out many parents willingly lend their baby gear to their friends in an endless merry-go-round where some eventually gets worn out and other bits get added. You can rent car seats, buy the bigger items, like carriages and strollers, secondhand through local newspapers, and buy slings and baby carriers.\n\nIt may be appropriate to ask family and friends to support in this new venture by buying specific items of equipment.\n\nExperience Required\n\nNo previous experience is required. Some have had experience of caring for babies either within their own family or through friends or from having babysat as teenagers. Many, however, have never cared for a baby, may not even have held one until they had one of their own. This should not put you off, rather encourage you to seek appropriate support (see Backup Support).\n\nTraining\n\nNone needed, although you have an advantage if you had parents who were loving, caring and, by example, offered excellent preparation for this job. You should be willing to be at least partially trained by your children.\n\nThere is much reading material available on how to be successful in this job. You are encouraged to be selective regarding the theories put forward although you may find them useful for reference.\n\nBack-up Support\n\nSee here for details of support organisations, but also available are women's groups, playgroups, mother and baby groups, toddler groups, nurseries and many others. Details of local ones can be found in newspapers, on doctors' surgery or clinic notice-boards, or at your library and from health visitors, GPs and homeopaths.\n\nThe lucky ones among us will have relatives or friends who love and respect them living close by, and are willing to help by being involved in the upbringing of the baby.\n\nSalary and Conditions\n\nChild benefit is currently fixed at \u00a37.25 per child per week, rising at ad hoc intervals subject to government ministerial whim plus a small extra allowance if you are parenting alone. It is paid for all children under the age of 16. Those who decide to parent full-time should apply to their working partner for an allowance. A weekly maternity allowance (for a maximum of 18 weeks) can be claimed by women who have paid the necessary national insurance contributions. The following are free during pregnancy and for a year after the birth: dental treatment; all prescriptions; family planning services.\n\nThere are no statutory or agreed holidays, no paid overtime, no opportunities for profit-related pay. Promotion with greater responsibility does not attract any incremental increases. There is no superannuation or pension scheme currently available!\n\nPLANNING FOR PARENTHOOD\n\nPartners can feel just as much at sea as new mothers while they get used to their new role in the family and much of the following suggestions are addressed to both parents. Check out the job beforehand by getting as much information as you can without getting overwhelmed. If you are a woman who has decided to parent single-handedly then make sure you have set up a solid support network of friends especially for the early weeks.\n\nDo\n\n\u2022 interrogate friends who are parents: ask them what worked and what didn't, what it was they wished they had known, how they would have liked to prepare for parenthood.\n\n\u2022 dip into a book or two but stay with the ones that seem to mirror most clearly what you feel is right for you.\n\n\u2022 find a friend with a baby, before you have yours, and ask to help out or babysit or be around when feeding, bathing and changing are going on.\n\n\u2022 go with the flow: your life will change, a new baby is all-absorbing and you may wonder how you will ever manage to get dressed and wash your hair or cook a meal again. Your life will adjust, especially once your baby starts to sleep more at night so just hang on in there and enjoy the different pace of your life. The first year is a treasure chest of delight, which passes all too soon so make the most of it!\n\n\u2022 use the post-natal support network that you set up when you were pregnant \u2013 friends to shop or clean, take your toddler out, etc.\n\n\u2022 reassess your priorities. You'll find that they change as your family grows. Sleep and time with your family can become more important than almost anything else. The important people in your life may change as you seek out other parents to spend time with. Do make sure you look after yourself first, then your partner, and then your other children. Housework, three-course meals and entertaining can all go to the bottom of the list!\n\n\u2022 avoid people whose advice makes you feel small, guilty, self-critical, bad or incompetent until you feel confident and your relationship with your baby is well established. It's normal to feel vulnerable after childbirth, especially to hurtful criticism from close relatives and friends \u2013 who should know better, anyway. Lack of sleep and the demands of a newborn baby are a special combination calculated to turn even the most stoic of women into emotional jellyfish. Protect yourself from unkind attacks by telling any critics, kindly but firmly, to leave you alone.\n\n\u2022 build verbal encouragement into your life: parents can tell each other what a good job they are doing \u2013 frequently! Tell yourselves that day to day you are doing the very best you can.\n\n\u2022 put your struggles in perspective by talking to trusted friends who share similar parenting beliefs. Aim to be effective, good-enough parents, not perfect ones.\n\n\u2022 accept your failings with goodwill and humour.\n\n\u2022 start discussing what boundaries you are going to set for your children, and what consequences there will be if they break them. Decide how important it is that they don't eat sugar or don't swear, for example. If you don't want them to do it, you are going to have to stop yourselves.\n\n\u2022 ask older children to help change and bathe the baby. Encourage them to express their feelings about the new one and listen to what they say, either positive or negative. It is normal for children to feel ambivalent about a new baby \u2013 and unusual for them not to.\n\n\u2022 seek professional advice if you're not coping or finding it difficult to ask for help. Remember, it's a sign of strength, not failure, to ask for help.\n\n\u2022 leave the baby with tried and trusted help from time to time so that you can go to the library, the hairdresser or have tea or go to the movies with a friend; spend time with your older child (or children) to give them the one-to-one attention they had before the baby came along, especially if you are a single parent.\n\n\u2022 try to spend some time \u2013 an evening a week if possible \u2013 with your partner. Your primary relationship needs nourishing in order to survive and flourish.\n\nDon't\n\n\u2022 listen to theories expounded by professionals who haven't had children of their own, or read books that make you feel anxious or uneasy.\n\n\u2022 listen to advice that overuses the words should, ought, must, or where they are implied or where you feel criticised or attacked.\n\n\u2022 isolate yourself.\n\n\u2022 fall into the trap of becoming a doormat, looking after everybody else's needs at the cost of your own.\n\n\u2022 forget: there is no such thing as the perfect parent. Imagine what it would be like for a small child to have to live up to the impossible expectations of being a perfect adult. Your faults or flaws are like seams of gold and charcoal in the rock of your being and an important part of who you are. Let them strengthen you and adorn your personality by uncovering them and accepting them.\n\nBONDING\n\nBonding is the process whereby you get to know another person very well and become very close to them and they to you. It is an essential ingredient in a successful relationship.\n\nSome people have high expectations of falling in love with their new babies and are disappointed when that doesn't happen automatically. It can take days or even weeks, especially if you were separated shortly after the birth. You can encourage bonding to take place \u2013 both for yourself and for other members of the family.\n\nDo\n\n\u2022 be aware of the bonding process and nurture it consciously.\n\n\u2022 ask that your baby be given to you immediately after the birth, if possible, so that you can 'connect' with each other \u2013 and with your partner.\n\n\u2022 go home after the birth if you're in hospital, don't need to be and aren't enjoying it. It may not be appropriate for you to be surrounded by strangers at a time like this. You may have difficulty getting to know your baby if you are feeling tense.\n\n\u2022 examine your baby: feel free to check out every inch of that little body, touch and smell him or her all over.\n\n\u2022 make a calm nest at home. The first days and weeks after the birth are important for you and your baby to get to know each other. You need uninterrupted time together to nurture the bonds of intimacy. Encourage your other children to spend time with the new baby so that they can have an opportunity to become close.\n\n\u2022 wallow in this time just after the birth if this is your first child. It's a very special time which won't come again; if you have another baby you'll have your older child to look after too.\n\n\u2022 limit your visitors. Too many people at the wrong times will make you tired, tense and edgy, even if you enjoy them at the time.\n\n\u2022 massage your baby. You can go to a baby-massage class or use one of the many books available as a guide.\n\n\u2022 bath with your baby \u2013 the feel of your baby's skin next to yours is lovely for both of you. It's easiest if there's someone else on hand to take the baby out while you wash.\n\n\u2022 exercise and play together: be creative and have fun doing it.\n\n\u2022 hold your baby as much as you can. Remember, the familiar environment of the uterus consists of constant contact, constant feeding and constant warmth. Carry the baby around in a sling, when you are shopping or when you make a meal if your back is up to it and your baby enjoys it.\n\n\u2022 talk, sing, hum and whistle to your baby. Talk a combination of baby talk and ordinary everyday talk: if you talk only in baby talk they will take longer to speak proper language.\n\n\u2022 look into your baby's eyes. They have difficulty in focusing during their early weeks and can see best at a distance of roughly 8 inches (20 cms).\n\nA WORD ABOUT FEEDING\n\nIt is important that you are happy with how you feed your baby, whether that be by breast or bottle. Don't give in to someone else's dictates about what you should be doing. It is far more important that the relationship between you and your baby and between your baby and the rest of your family is close and loving than feeding in the 'right' way and feeling tense and anxious because it doesn't suit you.\n\nYou may choose to breastfeed fully or to mix bottle- and breastfeeding; to breastfeed for a short time and then transfer to bottle-feeding; you may decide to bottle-feed from the start, from necessity (because of, say, surgery) or choice. Be flexible: see how breastfeeding goes for you and continue if it works or mix it with bottles if it doesn't. If mixed feeding doesn't work, drop the breastfeeding. Bear in mind that if you introduce bottles early your baby may lose interest in the breast as getting milk out of a bottle is generally easier than out of a breast. Babies will thrive on either the breast or the bottle. The points which follow may help you to decide which is right for you.\n\nWhether you opt for bottle or breast, feeding takes up so much time initially that it usually becomes the primary focus of attention. Those early weeks or months, where your life feels jam-packed with 'servicing' your new charge \u2013 feeding, rocking, changing and waiting for him or her to wake again \u2013 won't last \u2013 they are comparatively short-lived so try to enjoy them!\n\nBottle-feeding\n\nPros\n\n\u2022 you don't have to do it all the time... your partner can do one or more of the night feeds.\n\n\u2022 you have more freedom to leave the baby with your partner, a friend or close family member while you have a break.\n\n\u2022 you know exactly what your baby is getting. This may be a disadvantage for anxious mothers who become fixated on the quantity of milk their babies are taking, who panic if their baby doesn't finish a feed or still seems to be hungry when the bottle is finished.\n\n\u2022 partners enjoy feeding babies too and can feel included in the feeding relationship.\n\n\u2022 your exhaustion, emotional 'downs' or even illness can affect your breastmilk, sometimes reducing it. You won't have this worry to contend with if you are bottle-feeding.\n\nCons\n\n\u2022 a lot of preparation, sterilising, measuring and warming bottles.\n\n\u2022 an increased incidence of diarrhoea in bottle-fed babies.\n\n\u2022 some babies are allergic to cow's milk and even one bottle is enough to set off a reaction (commonly causing colic and diarrhoea).\n\nNB Don't automatically give soya milk. It is acidic which can make it difficult for some babies to digest. Only give it if you have a good reason and then select the brand carefully with the advice of your health visitor. Controversy recently erupted over the level of aluminium salts in one variety.\n\nBreastfeeding\n\nPros\n\n\u2022 the correct formula for baby humans (as opposed to baby cows) and has the right balance of nutrients.\n\n\u2022 easily digested.\n\n\u2022 the early pre-milk (colostrum) contains antibodies which boost your baby's immune system.\n\n\u2022 always sterile.\n\n\u2022 conveniently always there \u2013 anywhere, anytime \u2013 making travelling easy.\n\n\u2022 always at the right temperature.\n\n\u2022 an enjoyable intimate experience.\n\n\u2022 it aids bonding, helping mother and baby to get to know each other.\n\n\u2022 it is a good soother (dummy) and fulfils a baby's need to suck as well as the need for food.\n\n\u2022 it helps the uterus to contract.\n\n\u2022 it helps you lose the extra weight that you may have put on in your pregnancy. Breastfeeding uses 600\u20131000 calories a day, about an hour's worth of vigorous exercise.\n\n\u2022 you can combine the best of both bottle and breastfeeding if you can express milk easily. Buy a manual breast pump, costing a few pounds, from your local chemist so that you can experiment once your milk comes in.\n\nNB Research has shown over and over that breastfeeding (without supplements) even for the first two weeks of a baby's life, gives an extra boost to the immune system. So it's worth having a go \u2013 even if you know deep down that you won't continue for long.\n\nCons\n\n\u2022 it's time-consuming.\n\n\u2022 you have to do it \u2013 and that means every single feed, day and night. You are on duty 24 hours a day for an unlimited period of time unless you are able to express your milk.\n\n\u2022 it can be painful. A minority of women find that breastfeeding can take a long time to establish and the pain of engorgement (when the breasts become overfull as the milk first 'comes in' after the birth) can be followed by cracked nipples and blocked ducts. These conditions are all treatable with homeopathic remedies so don't suffer in silence or give up if you find breastfeeding hurts.\n\n\u2022 it can be confusingly sexual \u2013 especially for your partner.\n\n\u2022 you don't know how much milk your baby has taken. This rarely matters because your baby will take what it wants and once breastfeeding is well established a system of supply and demand operates whereby you produce according to your baby's needs.\n\nBreastfeeding Myths\n\nMany myths surround breastfeeding.\n\nMyth: successful breastfeeding is related to the size of the breast.  \nWomen with very small breasts are able to breastfeed successfully even if their breasts do not change size markedly either during pregnancy or after the birth. There is, nearly always, an increase in breast size when the milk comes in but the breasts may revert to more or less their normal size when feeding is established.\n\nMyth: either your breasts will become ugly as a result of breastfeeding or they won't change at all.  \nWe learn from the media that breasts should be a certain size and shape and this changes according to fashion. In one decade breasts were out and currently they are in. Women are frightened their breasts will sag and become unattractive to their partners. Their fears are often dismissed but it is true that breastfeeding may affect the shape and tone of your breasts. Some women have found that their breasts become more attractive not just during breastfeeding but afterwards too as they have retained some of their fuller shape. Others find that breastfeeding leaves them with smaller breasts. It is impossible to anticipate what will happen to yours. In any case, exercise will help to keep them \u2013 and you \u2013 in good shape.\n\nMyth: your breasts will only produce the milk your baby needs, no more no less.  \nIf you are anxious or under severe emotional stress your milk supply may reduce. Some women 'overflow' with milk which can be equally distressing. Homeopathic treatment will help in both cases.\n\nMyth: your breast-fed baby can't get fat.  \nYes they can! Some babies put on weight very quickly even with few feeds in a day. These babies have a slow metabolism (turn to Calcarea carbonica for a full picture of the constitutional type that puts on weight easily). As long as your baby is healthy and contented you do not need to worry whether they are taking too much milk.\n\nMyth: the breast-fed baby will gain the correct amount of weight.  \nThis is one to watch out for: the growth charts used to measure whether your baby is getting 'enough' milk do not take into account the individual variations of the breast-fed baby. Your baby's weight gain may not match your clinic's ideal but as long as your baby is healthy and contented there is no need to worry!\n\nMyth: breastfeeding comes naturally \u2013 the 'perfect mother' doesn't need teaching \u2013 she can do it instinctively.  \nThis isn't true. Successful breastfeeding is a learnt skill for mother and baby. Some women take to it like a duck to water, have relatively few difficulties and these are easily resolved. Some women struggle and need a lot of support. We all have different aptitudes in different subjects!  \nPhysical problems such as cracked nipples or a breast abscess can make breastfeeding seem difficult so seek out the support and advice of a breastfeeding counsellor (see La Leche League or other childbirth associations, here) to help you through.\n\nMyth: breasts were made to feed babies with, therefore breastfeeding is a natural function of the body and it certainly isn't sexy.  \nThe breast is associated with sex, reinforced through the media's frequent portrayal of women and their breasts as sexual objects, to sell anything from alcohol to cars. It is rare that we see images of women breastfeeding except in a Third World setting, with starving infants.  \nSome women are surprised and shocked to find that they become sexually aroused by breastfeeding. Few talk about this, especially if they also go off sex with their partners. They may even feel that there is something wrong or abnormal about them. There isn't.\n\nIs Breast Best?\n\nBreast is best only if you want to breastfeed. Some women breastfeed fully for a year and more. Others bottle-feed from the very beginning. Some mix breast with bottle in the knowledge that their baby may give up the breast sooner than if fully breastfed. Others breastfeed for a short time before changing over to the bottle.\n\nThe following dos are to help you to decide whether to breastfeed or not.\n\nDo\n\n\u2022 think through what messages, attitudes and beliefs you have about breastfeeding. Where did you get them? Are they true for you now? What do you want to do? What difficulties do you think or feel you might have? What are you afraid of?\n\n\u2022 talk through any fears or anxieties to sympathetic ears. Get some help if you realise you are labouring under some prohibitive messages \u2013 like 'You'll never breastfeed because your breasts are too small,' or 'No one has ever breastfed successfully in your mother's family \u2013 you are just like her, I bet you'll find it difficult too' \u2013 and are not sure what you want to do.\n\n\u2022 seek the help of a counsellor or psychotherapist if you feel revolted at the idea of breastfeeding but want to do it. They will help you to uncover the reasons for your feelings and help you to deal with them.\n\n\u2022 breastfeed for two weeks if you can as that will give your baby the antibody-rich pre-milk (colostrum) specially designed to boost your baby's immunity to infection.\n\n\u2022 know that you can switch to the bottle if you feel anxious, afraid, disgusted, angry, in pain or if any other difficult feelings surface while you are breastfeeding and you aren't prepared to see a counsellor. You can stop if you feel that breastfeeding is not for you. If you know that you are going to be a better mother if you bottle-feed your baby then do that.\n\n\u2022 seek counselling or psychotherapy help if you try to breastfeed and then decide not to breastfeed but continue to feel guilty that you didn't succeed or afraid that you've ruined your child's one chance at good health.\n\n\u2022 decide to feed your baby with love and affection. It is this that counts more than anything.\n\nBottle-feeding\n\nBottle-feeding can be as enjoyable and soothing as breastfeeding, albeit in a different way: it is the sucking and the love bond between the mother and baby that is important. Feed your baby with love and let him or her suck as much as they need and you can't go wrong.\n\nIf you decide to bottle-feed, do try to breastfeed for the first two weeks because the antibodies your baby receives through the colostrum will help to build a healthy immune system.\n\nCommon complaints such as colic and diarrhoea may occur in babies who have difficulty in adapting to cow's milk formula. If the problem is mild and short-lived, you can treat it yourself but remember to take the whole picture into account.\n\nDo\n\n\u2022 be flexible.\n\n\u2022 use bottles with disposable plastic liners if you are planning to carry on partially breastfeeding. Playtex teats are the closest in shape to your own nipples and the first size with the smallest hole allows milk through at about the same rate as breastmilk; your baby won't be encouraged to ditch the breast in favour of an easier option. The other advantages of these bottles are that your baby won't swallow air while feeding because the bag collapses as she sucks, and the bags are convenient for storing (or freezing) single feeds of expressed breastmilk or formula.\n\n\u2022 give occasional bottles of plain water to encourage a thirsty baby to develop a taste for water. Sterilise the bottles and use cooled boiled water.\n\n\u2022 be careful to sterilise all bottle-feeding equipment thoroughly to prevent the baby developing gastroenteritis \u2013 still a relatively common problem in bottle-fed babies. You can either use commercial sterilising solutions, or wash the equipment then boil for 25 minutes with everything fully submerged.\n\n\u2022 rinse bottles sterilised in commercial sterilising fluids with cooled boiled water before using them to wash off the chemicals.\n\n\u2022 wash your hands before preparing your baby's feeds.\n\n\u2022 use cooled boiled tap water from the mains cold tap, for making up his or her feeds.\n\n\u2022 measure the formula accurately according to the manufacturer's instructions.\n\n\u2022 make up each feed as you need it if you don't have a fridge.\n\n\u2022 keep to the same formula, once you find one that suits your baby.\n\n\u2022 keep extra formula milk in the fridge for up to 24 hours only, no longer.\n\n\u2022 warm the milk by putting the bottle in a jug of hot water. Make sure it doesn't get too hot and only warm it just before it is needed.\n\n\u2022 cuddle your baby while feeding \u2013 hold him or her nice and snug as you would if you were breastfeeding.\n\n\u2022 stroke the baby's cheek or lips with your finger or the teat to stimulate the sucking reflex.\n\n\u2022 hold the bottle firmly so that the baby can suck easily and tip it up so that the teat is full of milk. If it isn't he or she will suck in air with the milk, which may cause colic.\n\n\u2022 turn the bottle round if the teat collapses during sucking to let air back into the bottle which will cause the teat to re-inflate.\n\n\u2022 gently sit up and wind your baby if necessary during and after a feed.\n\n\u2022 let your baby take what he or she wants at each feed: a bottle-fed baby, like a breastfed baby, will be hungry at some times and not at others. Let your baby regulate the amount of milk.\n\n\u2022 let your baby take as long as he or she likes in the beginning to feed. Some babies stop and start or get distracted, by their mother or by strange noises.\n\n\u2022 let your baby suck your finger or a dummy if the need to suck remains after the milk is finished.\n\n\u2022 throw away any milk left in the bottle at the end of a feed: it can harbour harmful bacteria which will multiply rapidly in the warm milk.\n\n\u2022 give formula until your baby is about nine months old. Consult your health visitor as to the right time at which to switch to cow's milk.\n\n\u2022 consider taking advice about whether to give cow's milk formula if you or the baby's father is sensitive or allergic to dairy foods or if either of you has suffered from eczema, asthma, hayfever or digestive problems especially in childhood. 'Lactolite' formula, a lactose-reduced cow's milk for babies sensitive to cow's milk, is now available in many supermarkets and health-food shops, and the first goat's milk infant formula is being imported from New Zealand.\n\n\u2022 let others feed your baby \u2013 your partner, older children, relatives or close friends. Be specific about how you want them to do it \u2013 don't assume they know. A child should be supervised.\n\nDon't\n\n\u2022 use the following waters to prepare your baby's formula: bottled mineral waters, as the minerals can be harmful; water from the hot tap (and preferably not from a cold-water tap that comes from the tank in the attic); water that has been 'softened' or 'filtered' with a plumbed-in water softener or plumbed-in filter because it will contain residues of the chemicals used; water that has been boiled more than once or has been left standing in the kettle.\n\n\u2022 add extra formula to the bottle: you will make the mixture too strong, which can damage the baby's kidneys and make him or her more thirsty. If you give more milk than necessary your baby will put on excess weight and may cry a lot.\n\n\u2022 add cereal, rusk, sugar or anything else to the formula.\n\n\u2022 prop your baby (even an older baby) and the bottle in a high chair. The baby may choke.\n\n\u2022 worry if the milk is a bit cool. Even cold milk won't hurt although the baby may refuse it if he or she prefers it warmed.\n\n\u2022 give soya formula without taking advice from your doctor, health visitor or homeopath.\n\n\u2022 encourage your baby to finish the bottle.\n\n\u2022 encourage your baby to drink if he or she pushes it away, or turns away, or spits out the teat, or in some other way indicates the interest has gone!\n\n\u2022 give fruit juice in any form (squash or diluted fresh fruit juices) in a bottle. This will encourage a 'juice addiction' that is very difficult to break and will contribute to your child's teeth rotting as they come through.\n\n\u2022 keep bottles of formula for longer than 24 hours in the fridge.\n\nSeek help if your baby\n\n\u2022 isn't gaining much weight or is gaining too much weight.\n\n\u2022 never finishes the bottles, doesn't seem particularly hungry.\n\n\u2022 vomits the feeds.\n\n\u2022 has consistently offensive-smelling stools.\n\n\u2022 develops a rash.\n\nBreastfeeding\n\nMilk is manufactured in the breasts on a supply and demand basis \u2013 the more your baby sucks the more milk you'll make. Babies need to 'milk' the breasts by sucking in a particular way because milk isn't released if they suck on the nipples. They have to draw on the milk sacs lying behind the areola by taking as much of the nipple and areola into their mouths as possible. Small breasts have the same number of milk sacs as large ones. About a third of each feed is stored in these sacs; it is known as the foremilk and is thirst-quenching and low in fat. Sucking stimulates a hormone to be released (prolactin) which guarantees that milk will be stored after this feed is finished, ready for the next one.\n\nAs your baby feeds another hormone (oxytocin) is secreted which causes the fat-rich hindmilk to be released or 'let down' from the milk glands higher up into the milk sacs behind the areolas. This takes up to three minutes which is why limiting feeds to a few minutes each side can lead to 'failing' at breastfeeding because the hindmilk is essential for healthy growth. It is about three times richer than the foremilk and it will spurt out which is why a baby just has to swallow once this milk has let down. This let-down can be felt as a tingling sensation or a rush of warmth; some women feel nothing at all while others find it painful, especially at the beginning. Both breasts let down at the same time so if you are feeding on one breast, either catch the drips with a breast shell or in a breast pad or press the heel of your palm against the nipple to stop the milk leaking.\n\nIf your baby fusses and pulls on the breast after a minute or two it may be that the foremilk is all gone and the hindmilk has not yet been released. Once the milk has let down and after the initial rush of milk your baby will have to suck hard to get the rest of the feed.\n\nMinor complaints in the early days of breastfeeding, especially with a first baby, are common. The points listed below are to help you to breastfeed successfully from the beginning. See here for breastfeeding complaints.\n\nDo\n\n\u2022 let your baby have skin-to-skin contact with you immediately after birth. Offer the breast and let the baby suck for as long as he or she wants to. If you feed on demand from then on the chances are that your milk will come in quickly and your breasts will be less likely to become engorged (here).\n\n\u2022 decide to be relaxed about feeding. Remember that babies have been around a lot longer than clocks, that the timing of feeds was designed for bottle-feeding.\n\n\u2022 use a breastfeeding counsellor for support and advice \u2013 many will visit your house after the birth especially if their telephone advice hasn't helped. La Leche League and other childbirth organizations have nationwide networks of counsellors. They are women with children of their own whose aim is to empower others to breastfeed successfully.\n\n\u2022 rest, especially while your supply is building up. Breastfeeding is a lovely thing to do \u2013 enjoy it!\n\n\u2022 drink plenty of fluids. You will need approximately 2 pints (1 litre) more every day than you have been used to while your milk supply increases and may feel thirsty for what feels like most of the time until it is well established.\n\n\u2022 eat well (not necessarily a lot more) and regularly. Prepare a drink and a snack every time you feed your baby. Eat what you feel like as long as it includes some fibre and some fresh fruit and vegetables.\n\n\u2022 wear a comfortable, supportive cotton bra. Many breastfeeding bras have zips which are easily done up and undone with one hand, although you have to be careful not to catch your skin in them. Some are made with Velcro fastenings and the more old-fashioned ones fasten with hooks. Make sure that your bra never cuts into your breast as this can cause a blocked duct (see here). It is wise not to sleep in a bra for the same reason: they can pull around in the night and cut in. If your breasts are especially uncomfortable and need extra support both day and night, experiment until you find a bra that works for you \u2013 you may need a lighter one at night than in the day. It is worthwhile being measured and fitted properly by someone who knows what they are doing.\n\n\u2022 position the baby's head by cradling it in your left hand when putting it to the left breast \u2013 holding your breast from underneath with your right hand.\n\n\u2022 bring your baby to the breast not your breast to the baby. Don't lean over to feed your baby as this will create tension.\n\n\u2022 make sure the whole nipple and as much of the areola as possible (some women have particularly large ones), but especially the part below the nipple, is taken into the baby's mouth. Hold your breast from underneath to direct the nipple towards the back of the baby's mouth. Don't use your first two fingers as scissors to do this as the nipple will more easily point in the wrong direction.\n\n\u2022 stroke your baby's cheek with your finger or nipple to make him or her 'root', turn the head towards your breast and open his or her mouth.\n\n\u2022 make sure your baby's mouth is gaping wide when you offer the breast: this will encourage her to suck properly, with her mouth wide open. Rightly positioned you will see the muscles around the angle of the jaw, the temples and ears moving.\n\n\u2022 take your baby off the breast (by breaking the suction gently with your little finger) if she is sucking with her mouth closed and therefore only chewing on your nipple and start again. Persevere until she has got the knack \u2013 some babies take days to get it. If the nipple is not far enough in, your baby will 'bite' you with its gums and your nipple will become sore.\n\n\u2022 check that your newborn can breathe. If your breast obstructs breathing, reposition the baby's body or very gently press the area by their nose to give them some air (if you press too hard you can cause a duct to block).\n\n\u2022 let your baby's head rest on your forearm, once latched on well, and not in the crook of your arm, which makes it difficult for all the ducts to be drained and may cause her to pull and fuss on the breast.\n\n\u2022 position your baby so that its abdomen is facing yours otherwise it will have to twist its head to feed and may fuss and pull on your nipple.\n\n\u2022 feed in different positions, sitting and lying, to find out which is most comfortable at different times.\n\n\u2022 use the 'football' position, that is, with its legs tucked under your arm, especially if you have twins. This is especially good when you are trying to eat a meal and breastfeed at the same time!\n\n\u2022 make sure you are comfortable when you breastfeed, whether you are sitting or lying down, otherwise you will create pockets of tension in your own body that will drain you of much-needed energy. Organise yourself first and then latch your baby on. If you are sitting, have your feet flat on the floor (or on a small stool or a few telephone directories if you have short legs) and rest your baby on enough cushions to bring its head up to your nipple (newborns especially need plenty of support). Make sure your shoulders are relaxed and not supporting the baby's weight as this will give you backache. If you are lying down, bank cushions at your back and lay very small babies on a pillow.\n\n\u2022 breathe when you breastfeed. It is amazing how many women hold their breath when they are trying to latch their babies on. This creates tension.\n\n\u2022 respond to your baby if it is one of those who wants peace and quiet to feed. Some are distracted by noise and need you to be in a quiet place.\n\n\u2022 invest in a rocking chair with comfortable arms \u2013 there are a number of reasonable kits around. It's very soothing to nurse and rock, sing or read, or watch the baby's face or the clouds...\n\n\u2022 take your nipple gently out of your baby's mouth to prevent soreness developing, especially if your baby is still sucking. Put the tip of your little finger in the corner of her mouth: as you do so you will feel the suction breaking and you can take your nipple out without it being 'gummed'.\n\n\u2022 let your baby suck for as long as he or she wants after at least some feeds to satisfy the need to suck. If your nipple is sore, you are in a public place or don't have the time, offer your little finger or try a dummy. You can use it after a feed (not before one) and remove it once the sucking has stopped or the baby is asleep.\n\n\u2022 breastfeed in private if you feel inhibited or tense in front of others.\n\n\u2022 take your emotional state seriously after the birth. Anxiety, resentment, shock and disappointment, for example, can all make it difficult for you to establish breastfeeding. See a homeopath or counsellor if you are struggling to understand what is happening to you.\n\n\u2022 learn how to express milk (see here) as soon as possible as this will free you from being tied to breastfeeding and enable others to share the joy of feeding a small baby.\n\n\u2022 use a Playtex bottle that has disposable bags (see also here) for the milk. You can express your own milk and freeze it in the bags or put it in the fridge during the day ready for the night feeds.\n\n\u2022 taste your own milk. It may look thin and bluish but it tastes good \u2013 incredibly sweet. No wonder children have a naturally sweet tooth.\n\nExpressing Milk\n\nExpressing milk is a learnt skill, which some women find easy to pick up while others struggle and never manage to do it. The following technique is worth persevering with if you are determined to express. Try a hand pump or an electric pump (from your hospital or the La Leche League) if you find it impossible to express by hand.\n\nApply hot flannels to your breasts for a few minutes and then massage the breast from the very edge towards the nipple all the way round. Repeat this ten times (or more) to encourage your milk to let down. If you find this difficult try doing it in the bath \u2013 the warmth will help you to relax. Stroke the breasts gently with your fingernails or 'comb' them (see here) \u2013 again very gently and without applying pressure \u2013 all the way round just once or twice. Press gently with the thumbs and fingers of both hands on the area around the areola squeezing the thumbs and index fingers together and pressing backwards at the same time. Have ready a sterilised bowl to catch the milk! It should squirt out. Keep squeezing until you're only getting drops, making sure to squeeze at different places around the breast so that you empty the sacs as fully as possible. Repeat with the other breast.\n\nPREGNANCY\n\n* * *\n\nYOUR BODY AND PREGNANCY\n\n* * *\n\nThese are some of the normal physical changes or 'symptoms' of pregnancy: you may experience some or all of them. Feet can spread a little, because of the extra weight and the loosening of the ligaments, becoming up to one shoe size larger; breasts may swell as the milk glands begin to develop and become more sensitive; nipples may exude a clear, yellow pre-milk fluid (colostrum); the area around the nipples (areola) becomes broader and darker and 'spots' will develop (Montgomery's tubercules); vaginal discharges may increase; an altered perception of taste or smell may develop; joints can become painful or stiff; sweat and saliva might increase; stretch marks can develop on the abdomen, breasts, thighs or buttocks; gums can become swollen and bleed more easily; a brown line may develop on the abdomen (from the navel to the pubis); brown patches may appear on the face typically in the shape of a butterfly; there may be an increase in moles or freckles; birthmarks may darken, although these usually fade after the birth; the external genitalia can become slightly swollen; there is usually an increased desire to pass urine (because of the additional load of fluid which the blood passes to the kidneys to cleanse); and towards the end of your pregnancy your belly button 'pops out'.\n\nThe human anatomy and physiology of pregnancy, childbirth and breastfeeding is complex, the product of millions of years of evolution. Most of the developments are hormone-related and include the following:\n\n\u2022 All the connective tissue and smooth muscle of the body relax, including the ligaments. This enables the uterus and its supporting ligaments to soften, expand and stretch, and helps the spine and pelvis joints to become more flexible in preparation for birth. It also causes the walls of the blood vessels to widen and relax so that the blood circulates more freely and quickly, to carry oxygen and nutrients to your baby.\n\n\u2022 The lungs automatically expand to adapt to the increased need for oxygen.\n\n\u2022 The volume of body fluid increases dramatically. By the end of pregnancy your body carries about an extra 12 pints (7 litres) of water, half of which is taken up in the amniotic fluid, in which the baby floats in the uterus, and the rest is distributed throughout the body in the blood.\n\n\u2022 The volume of blood increases by about 30 per cent to 9 pints (5 litres) by the end of pregnancy, so that both you and your baby have an adequate supply. The extra volume is made up largely of water but there is also an increase in the red blood cells, which carry iron (haemoglobin) and oxygen. The concentration of iron in the blood decreases by about 20 per cent because of the extra fluid. Only about a quarter of the extra blood is lost during the birth, which allows for a safety margin in the event of more serious blood loss.\n\n\u2022 Hormones are specifically responsible for the development of the placenta, which nourishes the baby in the uterus, and the baby; preparing the breasts for breastfeeding; preventing the uterus from contracting (it contracts continuously throughout a woman's life except when she is pregnant when the contractions slow right down).\n\n\u2022 The increase of blood in your body means that the amount flowing to the surface increases about six times to help cool the body down \u2013 you may feel warmer and sweat more. This extra blood flow makes the skin glow and can cause red spots which disappear after the birth.\n\n\u2022 The heart rate increases to deal with the extra volume of blood being pumped through the body.\n\n* * *\n\nCOMPLAINTS\n\n* * *\n\n'Complaints' of pregnancy are always due to a combination of factors: the physical changes of pregnancy, hereditary factors and, of course, current stresses. They may include constipation, bleeding, cramps, frequent urination, involuntary urination (stress incontinence), swelling (oedema), piles (haemorrhoids), stretch marks, morning sickness (nausea and\/or vomiting), emotional volatility and varicose veins, to name but a few! The extra weight can be tiring \u2013 imagine carrying a shopping bag strapped to your abdomen with 20 pounds (10\u201315 kg) of sugar or potatoes in it \u2013 and causes a shift in your centre of gravity.\n\nTowards the end of pregnancy the baby's size pushes your internal organs out of position. Your lungs have less room, making breathlessness a common feature, and your stomach has less space to expand into if you eat a large meal \u2013 resulting in heartburn or indigestion. The complaints listed in this section are those most commonly experienced in pregnancy. I have not included those beyond the scope of the home prescriber (see here). Homeopathic medicines are ideal for use in pregnancy because there is no danger of side effects: they offer a safe alternative to orthodox medicines, when used correctly, for a wide variety of minor complaints at a time when many women put up with the discomfort rather than risk harming themselves or, more importantly, their unborn child.\n\nMost of the following complaints have more than one remedy listed in the Repertory. You will need to take the whole picture into account when prescribing \u2013 including the general symptoms and the emotional state.\n\nRead through Chapter 1 (here) for guidelines on potency and dosage before prescribing.\n\nIf you fall ill in pregnancy, deal with it appropriately and promptly: another person is now dependent on your well-being. Don't carry on regardless even if it is a relatively minor complaint like a cough or cold. If you have been under stress and your body needs a rest, that cold is a warning sign \u2013 ignore it at your peril! Rest when you need to even if you don't feel like it. On the following pages I discuss the common complaints of pregnancy and list practical advice for dealing with them. Only treat yourself for minor complaints that respond easily and quickly, and seek help if the complaint persists. Always seek medical advice for the following, more serious complaints.\n\nSeek help if\n\n\u2022 you have vaginal bleeding at any time during your pregnancy (because of the possibility of miscarriage or ectopic pregnancy early on in pregnancy or placenta praevia or abrupta (see here) later on). You can self-prescribe while you seek medical advice.\n\n\u2022 you suffer from abdominal pain, especially if it is continuous as it may indicate ectopic pregnancy or placental abruption.\n\n\u2022 you suffer from persistent, severe back pain at about waist level (kidney infection).\n\n\u2022 you suffer from severe headaches, blurred vision, chest pain or, your hands and face swell (which could signal pre-eclampsia (see here) leading to toxaemia).\n\n\u2022 you suffer from regular contractions before the 37th week, which may result in a premature labour.\n\n\u2022 the baby stops moving or moves a lot less than usual; on average it should move about three times an hour.\n\n\u2022 you leak amniotic fluid. The amniotic sac can tear but will usually mend itself and fill again with fluid, but if this happens it should be monitored.\n\n\u2022 you have an unexplained and\/or unusual symptom. Your doctor will be able either to put your mind at rest or to diagnose early the onset of anything that might need attention.\n\n\u2022 you have a complaint that isn't listed in this section. I have only covered the common and mostly minor conditions of pregnancy that you can treat yourself, always bearing in mind that if they do not respond quickly, you should seek help.\n\n\u2022 you feel anxious about the pregnancy.\n\nNB I believe the routine administration of any medicines at any time to be inadvisable. Some people advocate the use of Caulophyllum (see here) in the last few weeks of pregnancy or a homeopathic tissue-salt programme. Be aware of the dangers of 'proving' these remedies should you be tempted to try them and find another way of dealing with your condition.\n\nABDOMINAL PAIN\n\nSee Pain.\n\nACCIDENTS, AND INJURIES, MINOR\n\nSee Injuries.\n\nANAEMIA, IRON-DEFICIENCY\n\nIron tablets are routinely prescribed for this common complaint. They are of questionable value as they commonly cause constipation (or occasionally diarrhoea) and can sometimes intensify the anaemia by blocking the absorption of iron.\n\nDuring pregnancy the volume of blood circulating in your body increases gradually from about 7 pints (4 litres) to 9 pints (5 litres) by the end of the pregnancy. The haemoglobin (iron) content remains constant, however, which results in a normal drop in the level of iron per pint of blood of about 20 per cent. As long as you are fit and healthy, even if your haemoglobin level is low you need not worry. If you are suffering from symptoms of anaemia (exhaustion, palpitations, fainting, paleness and breathlessness) then you should take action, as a good supply of iron will help to promote a shorter labour with a decreased risk of haemorrhaging or infection after the birth.\n\nThe specific remedies for simple iron-deficiency anaemia during pregnancy are Ferrum metallicum (with exhaustion and pallor), or try alternating Ferrum metallicum with Calcarea phosphoricum, both in a low potency, to stimulate the body to absorb iron more effectively. These can be repeated whenever needed during pregnancy (see dosage chart). However, there are other possible causes of anaemia like emotional stress or illness. In these cases you will need to take the whole picture into account and choose a remedy that covers all your symptoms \u2013 or seek professional help.\n\nNB Thalassaemia and sickle-cell disease are both types of anaemia common in Asian, African, West Indian or Mediterranean women. They are inherited conditions and are not within the scope of the home prescriber.\n\nDo\n\n\u2022 eat a healthy, iron-rich diet. If you are a vegetarian eat plenty of greens and green vegetables (not spinach or watercress as the acid in them makes the iron difficult to absorb), cabbage, broccoli, eggs, molasses, wholewheat products (including wholewheat bread), whole grains, seaweeds, dried fruits (apricots\/figs\/prunes), almonds, sprouted grains and seeds.\n\n\u2022 increase your intake of folic acid to help the absorption of iron. Folic acid is found in dark, leafy vegetables, root vegetables, whole grains, sprouted grains, whole milk, dates, mushrooms and orange juice, as well as salmon and liver.\n\n\u2022 increase your intake of vitamin B12, which is found in dairy foods and meat and is vital for the development of the brain and nervous system of your baby. If you are a vegetarian or vegan you might contemplate taking a supplement as about the only vegan sources of B12 are tempeh (a fermented soya product) or fortified nutritional yeast \u2013 consult a nutritionist for advice.\n\n\u2022 rest as much as you need to.\n\n\u2022 drink plenty of fluids but cut out tea: it contains tannic acid which prevents the absorption of iron.\n\n\u2022 drink a fresh (citrus) fruit juice with your meals as vitamin C increases the absorption of iron. This is especially important for vegetarians.\n\n\u2022 drink nettle tea as this is high in iron.\n\n\u2022 take a herbal iron tonic such as Floradix if it suits you and you feel better for taking it. Stop if you become constipated.\n\n\u2022 cook with iron pans. Your iron intake will increase because small quantities of the metal dissolve into the food.\n\n\u2022 take your iron tablets with a meal if you decide to take those prescribed by your GP as they can irritate an empty stomach.\n\nDon't\n\n\u2022 take iron tablets routinely without trying the self-help measures outlined here and establishing the absolute necessity for them with your doctor\/midwife. Ask your homeopath for an alternative, should self-prescribing have had no effect.\n\nANXIETY\n\nSee Emotions.\n\nAPPETITE\n\nSee Food Cravings.\n\nBACK PAIN\n\nSee Pain.\n\nBLEEDING\n\nSpotting of blood is always a cause for concern although it is possible to spot \u2013 or even bleed quite heavily \u2013 in early pregnancy around the time when a period would have been due without losing the baby.\n\nOther causes of bleeding are a burst varicose vein of the vagina, vaginal infections, a fall, cervical erosion or injury to the cervix, or a low-lying placenta (placenta praevia).\n\nLight bleeding in the last three months of pregnancy can be caused by a vaginal infection or a burst blood vessel of the cervix caused by friction during sex. These are not serious. Heavier bleeding may be due to placenta abrupta, a rare condition where the placenta separates from the uterus prematurely before the birth. Repeated episodes of light bleeding or heavy spotting may indicate a placenta praevia, in which the low-lying placenta partially or totally covers the cervix. It can make itself known at about 24 weeks and will often migrate upwards later in the pregnancy. It will be spotted if you have an ultrasound scan (see here).\n\nAlways take bleeding during pregnancy seriously as it can be a sign of miscarriage. Pain accompanying the bleeding (especially if the pains are like menstrual cramps) usually heralds a miscarriage, or more rarely, an ectopic pregnancy \u2013 a pregnancy developing in a fallopian tube \u2013 (the pains are felt more on one side if this is the case).\n\nIf you have a history of miscarriage caused hormonally then it is especially important that you seek the help of a professional homeopath for constitutional treatment. Homeopathic treatment, however, cannot prevent an inevitable miscarriage although careful prescribing afterwards may prevent another one. It is not within the scope of the home prescriber to treat a threatened miscarriage.\n\nAdvice and reassurance are essential if you bleed during your pregnancy \u2013 whatever the cause. Seek professional help, especially if the bleeding is heavy and accompanied by pain.\n\nDo\n\n\u2022 rest completely. Go to bed and stay there (leave only to go to the toilet) if you know instinctively that this is right for you. Young children are usually happy to play in and around a sick bed if there is a constant supply of interesting things to play with and eat. Get someone to move the television into the bedroom, stock up on videos, read or listen to tapes.\n\nDon't\n\n\u2022 rest if you are an anxious person. Staying in bed will cause more stress than carrying on being busy which may take your mind off your worries and keep you relatively relaxed. But don't\n\n\u2022 ignore the symptom and carry on as usual hoping that it will go away.\n\n\u2022 worry if the bleeding stops and you are still pregnant. No harm will come to your baby.\n\nSeek help if\n\n\u2022 bleeding is accompanied by fever.\n\n\u2022 you have a history of miscarriage.\n\n\u2022 there is heavy bleeding at any time, especially if it is accompanied by pain and\/or the passing of clots.\n\n\u2022 you are concerned for yourself \u2013 if you have a presentiment that things are not right. Women are generally more sensitive during pregnancy and often 'know' when something is wrong.\n\nBRAXTON HICKS CONTRACTIONS\n\nSometimes known as false labour pains, these contractions occur painlessly and with varying frequency throughout pregnancy or not at all. They are attributed to the uterus practising for labour: your abdomen becomes hard and tight for a short time every now and then. Sometimes they can be bothersome, making sleep or resting difficult. They can come regularly, up to three minutes apart, for hours or even days at a time but the cervix doesn't dilate. It can be exhausting and dispiriting late in pregnancy to find out you are not in labour.\n\nDo\n\n\u2022 breathe gently and evenly with each contraction, just as you will need to do in labour \u2013 use them to practise on.\n\nSeek help if\n\n\u2022 there is bleeding or any unusual discharge from the vagina.\n\n\u2022 the contractions become especially severe and\/or prolonged.\n\nBREAST PAIN\n\nSore breasts are common as hormonal changes are preparing them to feed the baby. They may or may not increase dramatically in size and become very sore \u2013 even painful. If the soreness is accompanied by fever then you may have a breast infection (mastitis) see here.\n\nDo\n\n\u2022 wear a bra which fits well without cutting in anywhere. Buy new bras as your breasts increase in size.\n\nDon't\n\n\u2022 wear a bra at night as it can cut in inadvertently if you turn over.\n\nSeek help if\n\n\u2022 you develop mastitis and it doesn't respond to self-prescribing within 24 hours.\n\nBREATHLESSNESS\n\nCommon in late pregnancy, especially after exertion (exercise or walking upstairs\/uphill), it will also be worse if you were unfit at the beginning of your pregnancy and didn't take steps to improve your stamina. Breathlessness at any stage in pregnancy indicates simply that you are out of condition.\n\nThe extra load you are carrying makes the heart work harder and increases your need for oxygen. The pressure of the expanding uterus on the lungs also contributes. An extra large baby or twins can make this worse towards the end of pregnancy as your organs compete for the increasingly limited space in your abdominal cavity. Don't worry \u2013 this will pass after the birth.\n\nThere is no treatment for straightforward breathlessness, but if it is accompanied by exhaustion and pallor you may be suffering from anaemia (see here). Self-prescribe on this condition taking breathlessness into account.\n\nDo\n\n\u2022 make sure you exercise regularly (walking, swimming, dancing, etc.) and gently to increase stamina. Remember that yoga builds suppleness and strength but not stamina.\n\n\u2022 move more slowly towards the end of your pregnancy, using your breathlessness as a brake \u2013 walk at an easy pace. If you find yourself becoming breathless, pause and breathe slowly and deeply.\n\nSeek help if\n\n\u2022 you are breathless while resting or lying down.\n\nBREECH BABY\n\nTowards the end of pregnancy the baby settles into its favourite position. Ideally this will be with its head down and its back against its mother's abdomen, but it can adopt the 'breech' position with its head up, bottom down and its legs tucked up in front. A breech birth is more difficult and may result in the use of forceps or a Caesarean section. If your baby is still breech at 34 weeks it is worth encouraging it to turn round, although babies have been known to wait until labour to turn.\n\nIt isn't known why some babies settle in this position: one possible explanation is that the shape of the pelvic bones may make it more comfortable for the baby to be upright.\n\nPulsatilla 30, one dose every two hours for up to six doses (during the course of one day), has been highly effective at turning babies. Don't take it for more than a day.\n\nDo\n\n\u2022 exercise to encourage it to turn (postural tilting). Lie on the floor on your back with a pillow under your head and two or three pillows under your hips. Place your feet flat on the floor so that you feel comfortable. This will make a little extra space in your pelvis for your baby to turn \u2013 if it can. Encourage this to happen by massaging your abdomen gently, pushing your baby in the direction you want it to turn and imagine your baby turning. Rest in this position for 10\u201320 minutes several times a day. Use the time to practise your breathing and relaxation exercises.\n\n\u2022 crawl around on hands and knees for 10 minutes every day, and carry on in labour if the baby is still badly positioned.\n\n\u2022 talk to your baby \u2013 explain why you want it to turn and keep asking.\n\n\u2022 ask your doctor or midwife to try to turn it. Some are skilled at this. If they succeed, some babies, irritatingly, will turn straight back again.\n\nSeek help if\n\n\u2022 the above measures haven't worked. Homeopathic treatment can sometimes help a breech baby to turn, even as late as the early stages of labour unless it is a footling breech (one leg born first), in which case you will have to be guided by your doctor as to what is best.\n\nCARPAL TUNNEL SYNDROME\n\nNumbness, pain and tingling in the fingers of the hand is common in pregnancy and is caused by a nerve being compressed as it passes through a space in the wrist called the carpal tunnel. A combination of swelling (oedema) and relaxed muscle tissue can easily press in on the nerve causing this discomfort which usually disappears after the birth. If the following suggestions don't help, ask your GP to refer you to a physiotherapist.\n\nDo\n\n\u2022 avoid movements that are painful.\n\n\u2022 hang the affected arm out of bed at night.\n\n\u2022 massage the hand and wrist gently.\n\n\u2022 exercise the wrist and arms and stretch them up as often as you can.\n\nCOMMON COLD\n\nA cold can be the body's way of letting you know you are run down, and that you need to recharge. You may catch more in pregnancy because the hormonal changes can cause the mucous membranes of the nose and sinuses to swell. Don't ignore them \u2013 neglected colds can turn into more serious chest infections, so treat them early.\n\nDo\n\n\u2022 take some time off, have early nights and deal constructively with some of the stress in your life (see here and here).\n\n\u2022 get plenty of rest, especially if you have been overdoing it.\n\n\u2022 drink lots of fluids.\n\n\u2022 use steam inhalations to dislodge stubborn mucus. Sit over a bowl of just-boiled water with a towel over your head and the bowl and breathe in slowly and deeply.\n\n\u2022 eat plenty of fruit and vegetables and avoid dairy products, sugar, junk foods and too much bread, all of which increase mucus production.\n\n\u2022 avoid tobacco fumes.\n\n\u2022 take some gentle exercise in the fresh air to see if this helps (and use the response as a symptom to repertorise).\n\nDon't\n\n\u2022 use decongestants. These only irritate the nasal membranes and make matters worse as soon as the effect wears off.\n\nSeek help if\n\n\u2022 you suffer from frequent colds ordinarily and start to get them in your pregnancy. You will benefit from constitutional treatment which will boost your immunity (see here).\n\n\u2022 your cold seems unduly severe and isn't clearing with home prescribing and a large dose of tender loving care.\n\nCONSTIPATION\n\nThe hormones which prepare and relax the pelvis for labour also cause the digestive processes to slow down. It is important to keep bowel movements regular to prevent a build-up of toxins and\/or the development of piles and varicose veins.\n\nDo\n\n\u2022 drink plenty of water.\n\n\u2022 make sure there is plenty of fibre in your diet from fruit, vegetables and whole grains except wheat.\n\n\u2022 cut out bread and wheat products temporarily, as the gluten can have a clogging effect. If that helps, go easy on wheat products for the rest of your pregnancy.\n\n\u2022 try eliminating dairy products or meat if you eat a lot of them, as they can also clog up the gut.\n\n\u2022 avoid iron tablets: they are notorious for either causing constipation or making it much worse.\n\n\u2022 eat stewed prunes in the morning or take a tablespoon of blackstrap molasses dissolved in hot water.\n\n\u2022 use organically grown oat bran occasionally if you are desperate; it should only be taken regularly on the advice of a professional.\n\n\u2022 make sure your feet are well supported when you are sitting on the loo so that the circulation to your legs isn't cut off, as this can cause varicose veins. You can put your feet on two low stools (or piles of telephone directories) either side of the toilet so that you are in more of a squatting position, which makes it easier for your bowels to work.\n\n\u2022 try to make sure you are not interrupted.\n\nDon't\n\n\u2022 add wheat bran daily. It may interfere with the assimilation of vitamins and minerals, and your bowels may either be irritated by, or become dependent on, it. Oat bran is soluble and much gentler in its action.\n\n\u2022 take laxatives, particularly if your diet is mainly of refined and junk foods. Make dietary adjustments first.\n\n\u2022 strain when passing, or attempting to pass, stools as this can cause other problems such as piles (see here). Be prepared to take your time \u2013 have a book handy to distract you!\n\nSeek help if\n\n\u2022 there has been no bowel movement for 24 hours and there is severe, unusual pain.\n\n\u2022 there is difficulty in passing a stool and the stools are grey or white.\n\n\u2022 your skin or eyes are yellow.\n\n\u2022 there is a sudden and inexplicable change in bowel habit.\n\n\u2022 there is alternating diarrhoea and constipation.\n\n\u2022 your stools are extremely dark, almost black (unless you have been taking iron tablets in which case dark stools are not a cause for concern).\n\nCOUGHS AND CHEST INFECTIONS\n\nCoughs are common especially during winter or a period of emotional stress. If the complaint recurs or stubbornly refuses to clear in spite of careful self-prescribing, seek professional homeopathic help.\n\nDo\n\n\u2022 follow the advice given for the common cold.\n\n\u2022 rest \u2013 it is even more important with coughs than colds.\n\n\u2022 drink plenty of fluids as they will help to loosen the mucus.\n\n\u2022 cough up the phlegm as often as possible.\n\n\u2022 bend over or forward to cough so as not to strain the ligaments and muscles of the abdomen.\n\n\u2022 use a humidifier or vaporiser in the bedroom to fill the room with steam, or use a simple steam inhalation; this will help the chest to expel sticky phlegm. You can put two drops of lavender or rosemary oil in the water or vaporiser, rather than coal tar or strong-smelling fumes, which will prevent a homeopathic remedy from working.\n\nDon't\n\n\u2022 suppress the cough with a cough medicine; this may prevent you coughing up phlegm which, if it is not expelled, may cause a more serious infection to develop. It is also important to avoid taking any medicines in your pregnancy that may cross over the placenta to your baby.\n\nSeek help if\n\n\u2022 you have difficulty in breathing, are wheezing, or have chest pain.\n\n\u2022 the cough seems severe and doesn't respond to self-prescribing within 48 hours.\n\n\u2022 your breathing is unduly rapid.\n\nNB Pneumonia is a serious chest infection which is not always accompanied by a cough. Seek urgent medical help if you have a fever, are breathing rapidly and feel very unwell (limp and pale).\n\nCRAMPS\n\nCramp is a sudden muscular contraction and is often caused by a change in body temperature or position, a loss of body fluids or when muscles are tired after exertion. It is a common complaint of pregnancy and more so in summer heat, because sweating can cause a drop in the level of body salts.\n\nIf your only symptom is cramp, take Magnesia phosphorica and Calcarea phosphorica, alternating them frequently, stopping and starting as needed.\n\nDo\n\n\u2022 straighten the affected part of the body, and gently massage the affected muscle.\n\n\u2022 exercise gently and regularly, stretching your leg and calf muscles.\n\n\u2022 increase your intake of calcium-rich foods.\n\nSeek help if\n\n\u2022 the cramps are increasingly severe, interrupt your sleep at night and do not respond to self-prescribing.\n\nCYSTITIS\n\nUrinary tract infections occur in pregnancy because the bladder and kidneys have to work harder to deal with the increased volume of body fluid and because of the pressure of the growing uterus.\n\nCystitis is an inflammation of the bladder and urethritis is an inflammation of the tube leading from the bladder to the urethra. The two can be easily confused but as they are closely related. First-aid advice and treatment is the same for both.\n\nIf left untreated, cystitis may develop into a serious kidney infection so great care should be taken with any urinary tract infection. I have included a few remedies for the treatment of simple cystitis (see here), but symptoms should be carefully monitored and a professional homeopath or your GP consulted if there is no rapid improvement.\n\nDo\n\n\u2022 drink a large amount of water to flush out the kidneys and bladder, a glass every half-hour if you can, especially during the acute phase.\n\n\u2022 keep your diet alkaline: eat lots of fruit, vegetables and whole grains, especially brown rice.\n\n\u2022 cut out all acid foods and drinks, including tea, coffee, sugar, refined\/junk food, alcohol.\n\n\u2022 drink barley water. You can make your own by simmering a handful of organic whole barley in a pint (\u00bd litre) water for an hour with a whole lemon cut into small pieces. Strain and drink a glass every half-hour during the acute phase.\n\n\u2022 drink cranberry juice, either as it is or blended with fresh parsley, as it has been found to combat the bacteria that cause cystitis.\n\n\u2022 take as much rest as you can. Go to bed and stay there!\n\n\u2022 keep warm. Wear extra woollen clothing, especially around the bladder and\/or kidney area; carry around a hot water bottle or use a heating pad.\n\n\u2022 be sure always to wipe yourself from front to back after peeing so that bacteria from the anus cannot enter the urethra.\n\n\u2022 'bottle wash': fill a mineral-water bottle with warm water and pour over your genitals after you have peed and while you are still sitting on the toilet. Pat dry with soft white or unbleached toilet paper or a towel.\n\n\u2022 pee frequently to keep the urethra free of bacteria.\n\n\u2022 pee in the bath if peeing is painful \u2013 run enough warm water to sit in. Urine is sterile so don't worry about 'contamination'.\n\nDon't\n\n\u2022 use bubble baths or bath oils.\n\n\u2022 wash the genitals with soap; just sponge down well with water.\n\n\u2022 wear nylon tights or knickers or tight trousers or jeans.\n\n\u2022 use vaginal deodorants.\n\n\u2022 wash your underwear in biological washing powders.\n\n\u2022 get chilled.\n\n\u2022 use coloured toilet paper as the dyes in coloured loo paper have been known to aggravate.\n\nSeek help if\n\n\u2022 there are sharp pains in the area of the kidneys (in the back above the waist, on either side of the spine).\n\n\u2022 the pains in the bladder (just above the pubic bone) or urethra (just behind the pubic bone) are extremely severe.\n\n\u2022 there is blood in the urine \u2013 it looks pink, red or brown.\n\n\u2022 the cystitis is accompanied by headache, vomiting, fever and chills.\n\nDIARRHOEA\n\nBecause women are more generally sensitive during pregnancy, slightly 'off' food can cause more severe diarrhoea than normal. Make sure you give yourself adequate time to recover from it and take a remedy for exhaustion after diarrhoea if you still feel tired and listless within a day or two. The indicated homeopathic remedy will help your body recover much quicker \u2013 there are many for diarrhoea so you'll need to prescribe carefully, taking the whole picture into account.\n\nDo\n\n\u2022 limit the intake of food and drink.\n\n\u2022 take water, freshly pressed apple juice, vegetable broth (simmer a selection of chopped vegetables in water for 15\u201320 minutes and strain), or rice water (white rice cooked in double the normal amount of water for slightly longer than usual and strained). Sip them initially and increase the quantity once the symptoms start to pass.\n\n\u2022 introduce solid foods carefully, starting with white rice, toast or bananas. Eggs are also very 'binding' if the diarrhoea is persistent.\n\nDon't\n\n\u2022 let yourself become dehydrated.\n\n\u2022 eat rich foods that are difficult to digest.\n\n\u2022 eat if you are not hungry \u2013 a day or two without food will do no harm to you or your baby.\n\nSeek help if\n\n\u2022 the diarrhoea persists and fluids are not being tolerated (are being vomited or seem to pass straight through).\n\n\u2022 you are exhausted and have lost your skin tone.\n\n\u2022 there is acute pain in the abdomen which doesn't respond to first-aid prescribing within two to 12 hours, depending on the severity, or which is getting steadily worse.\n\nDIZZINESS\n\nPersistent dizziness should always be investigated by a professional. You can treat minor occasional vertigo yourself. In pregnancy it can accompany anaemia or exhaustion.\n\nDo\n\n\u2022 rest until the dizziness has passed.\n\n\u2022 cup your hands over your nose and mouth for a few minutes as you breathe slowly and steadily if you suspect that the dizziness is a result of hyperventilating (see here) as you may need to increase your carbon dioxide level.\n\nDon't\n\n\u2022 drive.\n\nSeek help if\n\n\u2022 there is severe vomiting with the dizziness.\n\n\u2022 deafness or noises in the ear develop with the dizziness after an ear infection.\n\nECLAMPSIA AND PRE-ECLAMPSIA\n\nUnique to pregnancy, it is not fully understood what causes this condition. However, in some studies it was noted that women with a diet high in protein were less susceptible to it.\n\nPre-eclampsia is fairly common \u2013 affecting up to one in ten pregnancies. The condition is fundamentally a disorder of the placenta and usually occurs at the end of pregnancy. It is a complex condition whose main signs and symptoms are oedema, sudden weight gain, headache, high blood pressure and protein in the urine (most likely to be diagnosed at an ante-natal visit); it is a combination of three or more of these symptoms developing quite suddenly which signals pre-eclampsia.\n\nAbout a quarter of all pregnant women develop high blood pressure (who have previously not had it), and of these only about a quarter develop pre-eclampsia. If left untreated, about 1 in 20 of these women will develop eclampsia; adequate ante-natal care has made this condition relatively rare. The following characteristics can predispose some women to pre-eclampsia: height under 5 foot 2 inches (1.57m); first pregnancy; age under 20 or over 40; diabetes, kidney disease, migraine; a history of pre-eclampsia in a previous pregnancy; hypertension in the woman or in her family history; anxiety; under or malnourishment; sexual and emotional problems.\n\nEclampsia is life-threatening, with convulsions and coma consequently depriving the mother and baby of oxygen, which can lead to death. This is why it is cause for serious concern and must be spotted early. Women are most vulnerable to it just before, during and after childbirth. In some cases women are taken into hospital to monitor their condition carefully, as the only 'cure' for eclampsia is to induce labour early.\n\nThe traditional insistence on rest is now being questioned: because although rest can help bring down the high blood pressure of pre-eclampsia, it won't help much with placental deficiency.\n\nThe following self-help measures will help you to avoid pre-eclampsia. You can self-prescribe on simple oedema (see here) if you can find a remedy in this book that fits, but generally it is beyond the scope of self-prescribing so consult your GP and your homeopath if you develop the symptoms.\n\nDo\n\n\u2022 pace yourself carefully in your working life and build substantial islands of fun and\/or rest into your day.\n\n\u2022 improve your diet \u2013 eat little and often and make sure you're getting plenty of protein.\n\n\u2022 deal with the stress in your life \u2013 get some help with dealing with difficult in-laws, bosses, employees, teenage children, etc!\n\n\u2022 seek counselling or psychotherapeutic help if you have strong ambivalent or negative feelings about your pregnancy, your body or your future role as a parent. This will reduce the pressure and will help you to adjust more easily in the post-natal period.\n\nSeek help if\n\n\u2022 you develop more than one symptom of pre-eclampsia. If you have been attending regular ante-natal hospital or GP appointments this will usually be spotted but it may be that you are the first person to suspect that it is developing. Make an urgent appointment to see your doctor if you think it is.\n\nEMOTIONS\n\n'Positive' Feelings\n\nPregnancy can be a magical, wonderful time, when you may experience a dramatic improvement in your general health, and feel unusually happy, harmonious and healthy, a pattern which may continue in subsequent pregnancies.\n\nThe positive feelings are many and glorious, and can be as overwhelmingly surprising as the negative ones. You may feel a heightened sense of awareness and perception, and respond more strongly or sharply than you would normally. You may feel powerful, full, rich, delighted, overjoyed, exhilarated, excited (and impatient!), elated, glowing, expectant and euphoric, but also peaceful, tender, vulnerable, calm and special.\n\nSome women feel sensual and voluptuous, perhaps a primitive expression of fertility; others, who recognise an early strong connection or bond with their babies, feel a growing sense of falling in love, of walking on clouds \u2013 especially after the baby first moves. The magical feeling of the baby moving makes the pregnancy feel real, sometimes for the first time: it's hard not to feel special, harmonious, creative. When the baby starts to move is when feelings often surface more strongly: excitement may be tinged with fear, anxiety or resentment.\n\nYou may experience increased energy, especially after the third month, culminating in the 'nesting' instinct towards the end of pregnancy: you may find yourself tidying, cleaning and even redecorating your home, which can be a pleasure \u2013 or can be a real pain if it gets out of hand. Pregnancy is a time of preparation for the baby, for parenthood and for couples to grow closer.\n\nI have included a few remedies for overexcitement, with consequent sleeplessness. If your good feelings get out of hand, if you become slightly manic and your health begins to suffer, get professional help from a psychotherapist, counsellor or homeopath.\n\nDo\n\n\u2022 acknowledge and express your positive feelings. Suppressed excitement can be as stressful as suppressed anger.\n\n\u2022 include your partner in your inner life during your pregnancy, by sharing your feelings regularly.\n\n\u2022 find a close friend to talk to if you are a single pregnant parent.\n\n\u2022 keep a diary of your feelings and the changes that you go through \u2013 it will be a delightful reminder of this special time.\n\n\u2022 use your energy creatively and don't squander it.\n\n\u2022 tell the world how wonderful you feel and enjoy it. Sharing a good feeling doubles it and spreads it to others.\n\n'Negative' Feelings\n\nPregnancy is a time of physical, emotional and even social change, which has different effects on every woman depending on her character and history, birth and childhood, unresolved and unvoiced traumas, and including ambivalence about motherhood. It may be a time of life-style change, from that of a relatively free single or married person to that of a parent. Finances may be tight, living quarters cramped, or friends thin on the ground because of a recent move to a new area.\n\nPregnancy brings so much into question.\n\nSome women maintain that having a child won't change a thing, that life will carry on (or that they will carry on their lives) much as it always has and that the baby will fit in. But life does change \u2013 substantially for some.\n\nPregnancy can be awful if you feel unwell. Some women just feel fat and disgusting; athletic women may resent being slowed down and feel annoyed at the extra fat that develops \u2013 may dislike the 'softening' effect of pregnancy. Others look on the baby as a 'lump', which they take pride in not showing until the last month, and avoid looking at their bodies or talking about them: to them the baby may feel like a parasite. Being pregnant is confirmation of a woman's sexuality and her womanhood, which can be painful to some women who have not been able to accept or be open about this side of themselves.\n\nFears are a normal part of pregnancy: explore them with your partner or a good friend, being as honest as you can. Take your anxieties seriously and talk them through: they aren't irrational or neurotic, they always have a base in reality. The more that is questioned and dealt with in pregnancy the better: issues like money and childcare won't go away but at least the options can be explored when you have the time and energy to do so.\n\nDon't forget that negative feelings can be viewed positively as an opportunity to grow and develop a deeper understanding of yourself \u2013 use them, don't ignore them.\n\nWhile you are pregnant, you may be confronted by:\n\n\u2022 anticipatory anxiety, accompanied or caused by low self-confidence about your ability to parent or to love. (Women who have written themselves a 'be perfect' script, with an I've-got-to-get-it-just-right message, suffer from this acutely.)\n\n\u2022 denial, often associated with indifference or apathy. Some ignore the presence of the baby growing inside them \u2013 it is a relief not to think about the inevitability of what is to happen.\n\n\u2022 confusion. Women who already have small children and are struggling to get by may be unable to imagine how an extra baby can possibly fit into their lives.\n\n\u2022 depression, especially if fears or anxieties are not expressed.\n\n\u2022 guilt \u2013 for any negative feelings!\n\n\u2022 tales and beliefs passed down in families from generation to generation which may be hard to shift: 'All the women in this family have Caesareans.' Question everything your mother told you about pregnancy and childbirth unless it rings positive and true for you.\n\n\u2022 resentment.\n\n\u2022 shock. An 'Oh-my-God-what-have-I-done?' feeling.\n\nYou may fear:\n\n\u2022 being trapped, losing your freedom, and the new and overwhelming responsibilities ahead \u2013 'I won't be able to cope'.\n\n\u2022 loss of control, events happening without being able to affect them.\n\n\u2022 loss of libido (especially if it occurred in your last pregnancy and continued through breastfeeding).\n\n\u2022 loss of identity or individuality as you realise your role is going to change whatever fixed ideas you might have had. Women whose own childhoods were spent trying to do the right thing for their parents often grow up insecure and not knowing deep down who they are. They may hide this by being successful and doing well but pregnancy may bring it out again.\n\n\u2022 the baby being abnormal or deformed.\n\n\u2022 death, either of the baby during or after birth or of yourself dying during childbirth. This can be strong in women who have had a previous miscarriage, abortion or stillbirth, partly through unresolved feelings of guilt or sadness.\n\n\u2022 that the pregnancy will never end. The last month or two can feel endless, especially if it's summer and you've put on a lot of weight or are carrying twins. You may feel heavy, tired, awkward, hot, impatient and irritable.\n\n\u2022 that the baby will be the wrong sex.\n\n\u2022 that the baby will affect your relationships with your partner and\/or your other children. It will: partners and children may have their own fears and anxieties so encourage them to speak out and prevent needless guilt or worry. The trick is learning to balance everyone's needs \u2013 the more people there are, the more difficult this is. Children, perhaps especially, may feel resentful and jealous \u2013 that they will lose their mummy \u2013 or scared that they won't like the new member of the family.\n\n\u2022 your partner not wanting a child that wasn't planned. He may not. He may be negative for at least part of the pregnancy. Keep talking.\n\n\u2022 physical changes, excess weight and stretch marks, etc.\n\n\u2022 never having time to yourself, especially to rest if you are pregnant and have other small children to look after.\n\n\u2022 becoming unlovable.\n\n\u2022 childbirth.\n\n\u2022 not being able to cope, especially if you are a single parent, when it is all the more important not to isolate yourself. Don't be tempted to work all the time.\n\nSome women feel good in one pregnancy and terrible in another or fine in the early months and terrible as the birth draws near. There are no rules!\n\nYour mental and emotional health has a very significant bearing on your physical health. Homeopathy is highly effective at helping with emotional stress but it is unwise to treat yourself as it is difficult to be objective about how you are behaving \u2013 for example, you may feel depressed but are actually being irritable. A trained homeopath will be able to differentiate and prescribe a remedy that fits the whole picture.\n\nEmotional Stress\n\nIt is important to acknowledge and deal with stress in pregnancy especially if it is severe: anything that affects you will, to some extent, affect your baby. Now that you are no longer simply responsible for yourself, you cannot afford to ignore what is happening to you. Moving house, problems at work, conflict with your partner or the death (or illness) of a close friend or relative are stressful at any time but particularly in pregnancy. If you ignore or suppress your feelings, loss of energy or ill-health or both will very likely be the result.\n\nLook in the Repertory under Complaints From where you will find many stress symptoms, including grief, anger, resentment, mental strain and bad news. Repertorise your symptoms carefully to find the remedy that best fits your whole picture. Seek the advice of a homeopath if you don't feel better quite quickly.\n\nDo\n\n\u2022 start by being honestly aware of your feelings. Be kind and compassionate towards yourself \u2013 and try not to be self-critical.\n\n\u2022 explore your feelings regularly with your partner, a friend or a counsellor. Remember there is more than a little truth in the old saying that 'a problem shared is a problem halved'.\n\n\u2022 find a close friend to talk to if you are a single pregnant parent.\n\n\u2022 try to understand what lies behind your feelings. You may need professional help to discover unresolved emotional experiences from your past and childhood, difficulties you are having in your life now but not dealing with, anxieties and fears about the future that you are not expressing.\n\n\u2022 assess and reassess your priorities regularly.\n\n\u2022 reduce the stress in your life \u2013 right now, cross something off your list of things to do. It can be a heady experience: once you start you may find that other things can go too.\n\n\u2022 reassess your beliefs and your life-style, what you want for your life ahead and for your family.\n\n\u2022 listen to concerned friends or relatives if you are good at hiding your feelings when you feel unhappy. Your emotional distress may not be evident to you. Depression can feel like tiredness \u2013 you don't want to get up in the mornings, or don't feel like eating, cooking or shopping because you're not very hungry, and you can't sleep. It's not a big deal but it is common and it is essential you deal with it now.\n\n\u2022 write about what you are going through.\n\n\u2022 understand what is happening rather than fight it.\n\n\u2022 learn to laugh at yourself.\n\n\u2022 slow down. Pregnancy helps in this automatically especially towards the end when you become heavier.\n\n\u2022 read light-hearted books and avoid anything depressing or frightening.\n\n\u2022 learn a relaxation or meditation technique and take time to do it at least once a day.\n\n\u2022 seek the help of a counsellor or psychotherapist if your feelings are persistent and do not resolve through taking the above steps.\n\n\u2022 seek alternative medical help to alleviate your symptoms.\n\nDon't\n\n\u2022 neglect your own needs. Now is the time, more than ever before, to put yourself first: you are the mother and you must be in reasonable shape to do your job properly. If there is something you really need and want, try to get it \u2013 it's easier than you think once you start talking. Even if you have to compromise, at least you were heard and that alone will help.\n\n\u2022 talk to anyone who is critical and judgemental of how you are feeling or who adopts the 'count your blessings' approach. This person is trying to minimise and talk you out of your feelings. Don't listen.\n\n\u2022 talk to people who give you constant, unwelcome advice. 'Your trouble is that you think too much. You need to dwell less on all these negative things. Remember there are a lot of people worse off than you.'\n\n\u2022 cut yourself off from sympathetic friends and family. You need them even though your behaviour may be pushing them away. Tell them what you are going through, reach out and ask for their help. Start by making straightforward requests.\n\n\u2022 take anti-depressants, sleeping pills or other orthodox medication as these may not only affect your body but will suppress your feelings and make it more difficult for you to deal with them later.\n\n\u2022 hide, ignore or suppress any difficulties you are having.\n\nSeek help if\n\n\u2022 you are floundering and are finding that talking to partners\/friends isn't helping. A counsellor or psychotherapist will be especially valuable.\n\nEXHAUSTION\n\nTiredness can be completely obliterating in early pregnancy. Its major causes are hormonal and physical changes, unexpressed or unresolved emotional difficulties, and anaemia. I have included many remedies for exhaustion, some for tiredness after specific stresses like a period of hard work, an acute illness or after diarrhoea.\n\nLook at the whole picture and repertorise carefully, bearing in mind that you may be anaemic or simply overworked. Be honest and realistic about why you are tired and take common-sense measures to look after yourself, as well as the appropriate homeopathic remedy to hasten your recovery. Many women have found that simply alternating the tissue salts Kali phosphoricum and Calcarea phosphorica (if suffering from nervous exhaustion) or Ferrum metallicum and Calcarea phosphoricum (if suffering from mild anaemia) is sufficient to make them feel better. Either may be repeated as often as needed throughout the pregnancy, stopping each time on improvement.\n\nDo\n\n\u2022 rest and sleep as much as you can and as you feel you need.\n\n\u2022 eat especially well from a wide range of foods including plenty of protein. Eat little and often and drink plenty of fluids.\n\n\u2022 delegate workloads for a realistic period of time while you recharge (at home or at work).\n\n\u2022 vary your daily activities \u2013 boredom is tiring! Any activity that is stimulating (exciting) is energising and too much stimulation can be draining. It is the balance that counts \u2013 getting enough mental, emotional and\/or physical stimulation to keep you from becoming bored without exhausting you.\n\n\u2022 take gentle exercise, which will help to create energy.\n\n\u2022 read undemanding books and magazines, the funnier the better.\n\n\u2022 have a massage, facial or haircut, etc., at home, if you can arrange this.\n\n\u2022 arrange to spend time with friends doing easy fun things unassociated with work or motherhood.\n\nSeek help if\n\n\u2022 the exhaustion persists in spite of following these guidelines.\n\nEYESIGHT CHANGES\n\nContact lens wearers may develop problems with their lenses as during pregnancy the eye can change shape slightly due to fluid accumulation.\n\nDo\n\n\u2022 Consult your optician and keep in regular contact if you develop problems.\n\n\u2022 bathe your eyes regularly with Euphrasia if they become sore (see here).\n\nFOOD CRAVINGS\n\nFood cravings and aversions are mostly benign, aversions often being against foods contraindicated in pregnancy anyway, such as caffeine and junk food. Occasionally the reverse happens, which warrants attention. Morning sickness (either nausea or vomiting) can cause a change or total loss of appetite which should be dealt with. Cravings can be mild and amusing, like wanting shrimp and peanut-butter sandwiches at three in the morning, or positively self-destructive, when they are usually a symptom of emotional stress, exhaustion or dietary deficiency.\n\n'Pica' is a craving for non-foodstuffs, such as earth or coal, and although I have included a few remedies for the treatment of this condition, I would strongly advise that if your appetite goes out of control during pregnancy you seek professional guidance.\n\nDo\n\n\u2022 eat regularly: five to six small meals a day rather than three large ones.\n\n\u2022 make sure you have plenty of fresh fruit and vegetables.\n\n\u2022 eat slowly and chew thoroughly as the saliva that is released will aid digestion.\n\n\u2022 eat a varied diet from all food groups \u2013 fat, protein, carbohydrate, etc.\n\nDon't\n\n\u2022 indulge wild cravings unlimitedly.\n\nSeek help if\n\n\u2022 you cannot control a craving for a bizarre substance.\n\nGROIN PAIN\n\nSee Pain.\n\nHAIR AND NAIL PROBLEMS\n\nAlthough many women find that their hair and nails are healthier during pregnancy than ever before, others find the reverse is true, perhaps because they neglect their diet, or because of emotional stress or hormonal changes. Hair can lose its shine and its curl and can fall out more heavily than usual. Nails can become brittle and break easily.\n\nIf the state of your hair is only a part of a larger picture of ill-health then you will need to prescribe on all the symptoms, seeking professional help if you are not successful. If you are generally healthy, both physically and emotionally but your hair is lank and your nails brittle, then a short course of Silica or Calcarea phosphorica may help.\n\nDo\n\n\u2022 make sure your diet is adequate. Fish, wheatgerm, yeast and liver are all said to provide the E and B vitamins essential for healthy hair and nails. Minerals like iron, iodine, zinc, silicon and sulphur are also important. A short course of multivitamins and minerals may help \u2013 apply the homeopathic principle of stopping on improvement.\n\n\u2022 visit a qualified nutritionist if you are on a restricted diet \u2013 vegan, dairy-free, etc. \u2013 to ensure that you take the supplements you need.\n\n\u2022 wash your hair frequently with a very mild shampoo \u2013 especially if you live in a polluted city, and massage your scalp thoroughly each time.\n\nHEADACHES\n\nOccasional minor headaches, resulting from such obvious stresses as overwork, loss of sleep or worry, can be treated with first-aid homeopathy. As well as finding the appropriate remedy, identify and remove or balance the cause of your headache \u2013 i.e. if you have been up all night with a sick child, have a nap.\n\nDo\n\n\u2022 rest and relax. Check your posture for tension if you feel a headache coming on and spend 10 minutes breathing deeply.\n\n\u2022 have a nap \u2013 or at least lie down with your eyes closed \u2013 if you can take the time.\n\n\u2022 get some fresh air if you've been in a stuffy atmosphere.\n\n\u2022 have your eyes tested to make sure the cause is not a deterioration in eyesight \u2013 this is possible during pregnancy.\n\n\u2022 seek counselling or psychotherapy help if you are under a lot of emotional stress.\n\n\u2022 talk your worries, resentments or fears through with your partner or another sympathetic ear: tension building from unexpressed feelings commonly causes head and back pain.\n\n\u2022 have a massage, if it helps. Ask an older child or your partner to rub your head, shoulders and back.\n\n\u2022 rub your own shoulders, especially muscles at the base of the skull, or apply pressure around the temples \u2013 be guided by your instincts as to where to press.\n\n\u2022 take a long bath.\n\n\u2022 eat regular meals. During pregnancy the metabolism speeds up and you will need to stoke your 'energy fire' more often. Missing a meal can cause a headache: you can check this out by eating something like a spoonful of honey which will give your blood sugar an instant boost.\n\n\u2022 ask people not to smoke around you if you know that tobacco is the culprit. During pregnancy many women are more sensitive to smoke and get instant headaches in a not very smoky room.\n\n\u2022 exercise by rotating your head slowly and evenly, first one way and then the other, by lifting your shoulders as high as you can and then letting them slowly down. Exercise your head, neck and back in any other way that feels good; as you breathe out (see Breathing) imagine the tension releasing.\n\nDon't\n\n\u2022 take pain-relieving drugs and carry on.\n\nSeek help if\n\n\u2022 the headache is severe.\n\n\u2022 you are suffering from frequent or severe headaches, particularly if your eyesight is deteriorating.\n\n\u2022 a headache is accompanied by a stiff neck.\n\n\u2022 you have a high fever, visual disturbance, weakness or lack of co-ordination, or dizziness.\n\n\u2022 a headache lasts more than three or four days.\n\nHEARTBURN\n\nYour stomach is under the breastbone, between and just under the breasts. Indigestion and heartburn are especially common in late pregnancy when the stomach has a smaller space in which to expand. Also, hormones relax the sphincter of the stomach making it easier for acid to escape into the oesophagus with consequent burning pain. Digestion slows down in pregnancy and the liver works less efficiently. Both are due to hormonal changes.\n\nDo\n\n\u2022 eat smaller more frequent meals, every three hours or so.\n\n\u2022 eat slowly and chew thoroughly so that plenty of saliva is released and food partially digested before it reaches your stomach.\n\n\u2022 cut down on fatty foods as the gall bladder works less efficiently, causing nausea and\/or indigestion.\n\n\u2022 avoid anything you know to cause more discomfort, such as spicy foods, onions and garlic, bread, dairy and other fatty foods, sugar or meat. Experiment to find out which foods you cannot tolerate.\n\n\u2022 relax before meals and eat sitting down at the table.\n\n\u2022 avoid bending over \u2013 to wash the bath or pick up toys from the floor \u2013 as this will cause acid to escape from the stomach. Squat down instead, keeping the trunk of your body upright.\n\n\u2022 lie down on a wedge of pillows so that you are not sleeping completely flat, which can also encourage acid to leak into the oesophagus.\n\nDon't\n\n\u2022 take antacids without professional advice as they can upset the balance of the acidity in the stomach even further and contribute to anaemia by affecting your iron absorption.\n\n\u2022 drink a lot of fluids with meals as this can bloat the stomach and make the heartburn worse.\n\nSeek help if\n\n\u2022 symptoms are accompanied by a cough, loss of appetite or loss of weight.\n\n\u2022 there is severe abdominal pain.\n\n\u2022 the heartburn is severe and doesn't pass off (with or without first-aid prescribing) within two or three hours.\n\nHERPES, GENITAL\n\nGenital herpes is caused by the same virus that causes herpes (cold sores) on the lips. It is spread by sexual contact and can lodge in the nervous system breaking out at times of stress. The first attack can occur months or even years after infection. It may be a one-off or recur cyclically. The symptoms are mild (a small blister that itches and doesn't even come to a head) to severe (extremely painful blisters preceded by pains in the buttock and leg and accompanied by exhaustion and flu-like feelings). There is no cure although constitutional homeopathic treatment can help decrease the severity and frequency of attacks. If you have a blister at the time of giving birth your baby will be delivered by Caesarean because of the risk to its life if it contracts the disease.\n\nDo\n\n\u2022 tell your GP if you have ever had an attack of genital herpes.\n\n\u2022 advise your GP immediately if you have your first outbreak in the first three months of your pregnancy as it may cause damage to the baby.\n\n\u2022 the following if you have an outbreak (a blister): anything you can to boost your immunity \u2013 eat well, rest and sleep well, take some extra vitamin C, garlic or anything that you know makes you feel better \u2013 this will help reduce the severity of the attack; keep the affected area dry and cool; avoid nylon underwear and tight trousers and try to spend as much time as possible without underwear to expose the blister(s) to the air; avoid hot baths; add a handful of sea salt to a tepid bath to aid healing; avoid intercourse until the sores have healed; be vigilant with hygiene if you have an open blister.\n\nHIGH BLOOD PRESSURE (Hypertension)\n\nThis is beyond the scope of the home prescriber and should always be treated seriously. On its own a small rise in blood pressure isn't a cause for concern, but it can precede the development of more serious conditions, which is why your doctor or midwife will want to monitor your blood pressure carefully throughout your pregnancy.\n\nThe doctor will write down two numbers when taking your blood pressure. The high one (systolic) reflects the heart at work (how well the body deals with exertion) and the low one (diastolic) reflects the heart at rest (a sort of baseline of physical tension). The numbers should read around 120\/80 (with individual variations). High blood pressure is diagnosed if your blood pressure rises to 130\/90 or more, or if the systolic reading rises 30 points above its norm and the dyostolic 15 points or more, based on your blood-pressure readings from early pregnancy. Your blood pressure may take a natural dip in the second trimester of your pregnancy.\n\nDo\n\n\u2022 ask if your midwife will come and take your blood pressure at home. The curious phenomenon known as 'white coat hypertension' refers to the stress of a lengthy wait in a hospital or doctor's surgery, which can cause a temporary rise in blood pressure. (The diagnosis of 'high blood pressure' can in itself be stressful, causing anxiety in a nervous woman, which forces the readings higher still.) If your blood pressure is higher than usual at an ante-natal check-up, ask yourself if you felt anxious or pressured beforehand.\n\n\u2022 rest. If your blood pressure is high go to your sofa or bed and stay there until it starts to come down.\n\n\u2022 cut down the stress in your life. Working women juggling home and office may need to stop work temporarily. Get clear about your priorities \u2013 and the consequences of ignoring your body's early warning signals. If you are very stressed, and you have high blood pressure, your body needs rest and you must take the pressure off yourself. Or is your work more important than you and your growing baby? Remember \u2013 when everything becomes so pressured that you can't possibly take time off, that's the time to take time off.\n\n\u2022 relaxation exercises and\/or meditate regularly.\n\n\u2022 cut out all stimulants (tea, coffee, chocolate, tobacco) if you haven't already done that.\n\n\u2022 cut down on your salt intake if it is high. Cook without it, adding a little sea salt, if you really need it, at the table.\n\n\u2022 make sure your diet has plenty of protein each day, enough calcium and lots of fruit and vegetables. Talk to a dietician or nutritionist, if in doubt.\n\n\u2022 drink relaxing herbal teas in moderation \u2013 chamomile and hops are good.\n\n\u2022 increase your fluid intake.\n\n\u2022 exercise as well as rest. Brisk walking or swimming are both good but build up slowly.\n\n\u2022 ask friends and family to help out with your other children and\/or household chores until your blood pressure comes down.\n\n\u2022 seek counselling help if you are under a lot of emotional stress.\n\n\u2022 seek alternative medical help with bringing it down as quickly as possible.\n\nSeek help if\n\n\u2022 you also have a sudden weight gain and puffy wrists or ankles which don't disappear after a night's sleep. This could herald the onset of pre-eclampsia (see here).\n\nINCONTINENCE\n\n'Urinary or stress incontinence', that is, leaking when you cough, sneeze, laugh or exert yourself, is common in pregnancy partly due to the pressure of the growing uterus on the bladder, and partly because the pregnancy hormones relax the muscles and sphincters. See also here.\n\nDo\n\n\u2022 Your pelvic-floor exercises (see here) religiously \u2013 even if you don't exercise a single other muscle in the body.\n\n\u2022 urinate frequently so that your bladder doesn't become stressed.\n\nINDIGESTION\n\nSee Heartburn.\n\nINSOMNIA\n\nWe all have different sleep requirements. In the early months of pregnancy most women need more sleep than previously \u2013 up to 10 hours a night with a day-time nap. If this sleep (and nap) refreshes then no treatment is necessary. If you are getting plenty of sleep but are still exhausted and dragging yourself around during the day, prescribe on your low energy (see Exhaustion). Your sleep pattern may have been disturbed by illness or small children so that you find it difficult to get to sleep, and\/or wake early. Emotional stress can also affect sleep patterns. If you are a long-term insomniac, you need to consult a professional homeopath if you continue to suffer during your pregnancy. If, however, you are coping well on four or five hours' sleep a night, don't worry. Your moods and your energy level will let you know if lack of sleep is taking its toll.\n\nDo\n\n\u2022 develop a regular bedtime relaxation routine, including deep breathing, gentle exercise such as a walk round the block or some yoga positions, a warm bath and a hot caffeine-free drink such as hot milk with honey or a cup of chamomile tea. Don't drink chamomile every night unless it really helps: you may find yourself becoming unusually irritable \u2013 the reverse of the effect it is meant to have \u2013 and are, therefore, 'proving' it (see here). If this happens, stop at once.\n\n\u2022 read boring books or magazines in bed.\n\n\u2022 wear earplugs and an eye mask to cut out noise and light if necessary.\n\n\u2022 make sure your mattress is right for you, neither too hard nor too soft.\n\n\u2022 make sure you are warm enough, but not too hot, and that your room is well ventilated.\n\n\u2022 do some basic relaxation exercises. Tense your whole body and then relax your muscles one by one, starting at your head and moving down your body. Then imagine you are very heavy and are falling through the bed \u2013 tense people can 'hold' themselves just above the bed.\n\n\u2022 count backwards in threes from 600!\n\n\u2022 put the light on and read or listen to the radio if you can't sleep, and try again when you feel ready.\n\n\u2022 clear any 'niggles' with your partner well before you go to bed so that you aren't keeping yourself awake by chewing over resentments.\n\n\u2022 sit up and make a list of all worries and frustrations if you can't sleep because of them and make an action plan for dealing with them.\n\n\u2022 seek counselling or psychotherapy if your sleeplessness is due to emotional distress such as anxiety, depression or bereavement.\n\nDon't\n\n\u2022 worry or panic! This always makes sleeplessness worse.\n\n\u2022 read thrillers, ghost stories or stories that upset you in bed.\n\n\u2022 have a large meal close to bedtime.\n\n\u2022 drink tea, hot chocolate, coffee or Coca-Cola during the evening (or at any other time) as all contain caffeine.\n\n\u2022 watch too much television.\n\n\u2022 work in the evening or take material connected with your work to bed.\n\n\u2022 get into taking long afternoon naps or sleeping very late in the morning as this can make it more difficult to get back to a normal sleeping pattern, especially if this is your first pregnancy.\n\n\u2022 take sleeping pills. Apart from the effect they may have on your unborn child they can affect your dream life so that you wake feeling unrefreshed. Many have side effects, causing drowsiness and dependence.\n\n\u2022 drink any alcohol.\n\nINVERTED NIPPLES\n\nSee Breastfeeding problems.\n\nITCHING\n\nItchy skin without a rash is common in pregnancy especially towards the end. The skin can itch all over, or just on the abdomen where it is being stretched, and can be made worse by heat. Take a short course of Sulphur and, if it helps, repeat if the itching returns.\n\nDo\n\n\u2022 keep cool.\n\n\u2022 try a tepid bath with half a cup of cider vinegar.\n\n\u2022 keep your skin well oiled if it is getting dry and itchy \u2013 you can put some oil in the bath or massage it in afterwards.\n\nDon't\n\n\u2022 use coal tar or hydrocortisone creams on your skin.\n\nSeek help if\n\n\u2022 a rash accompanies the itching.\n\n\u2022 home prescribing doesn't help and the itching is preventing you from sleeping.\n\nJOINT PAIN\n\nSee Pain.\n\nLOSS OF LIBIDO\n\nSex can be entirely spontaneous \u2013 and especially with a first baby (there aren't other small persons liable to wake in the middle of the night and interrupt you).\n\nIt can be health-promoting, energising, as you renew physical and emotional bonds with your partner, and active sex counts as physical exercise: it is very good for the pelvic-floor muscles (see here) \u2013 and it's relaxing and sleep-promoting!\n\nYou cannot harm the baby while making love: the uterus thoughtfully moves up and out of the way when you are sexually aroused. Also the hormones in semen that help the cervix to soften and dilate in early labour will only have that effect when you are in labour. If penetration is painful, experiment with different positions; take your time with foreplay and adopt a slower, gentler pace of lovemaking than perhaps you were used to before your pregnancy.\n\nLoss of libido can be due to unexpressed feelings which cause you to withdraw sexually and emotionally, or you may be having difficulty getting used to your constantly expanding body: talk to your partner or to a psychotherapist or counsellor. It is important to try to identify the cause: maybe you feel fat and unattractive, tired or nauseous or have been influenced by ancient unfounded taboos against sex during pregnancy.\n\nCausticum, Natrum muriaticum and Sepia can all help with loss of libido, but only if they fit your picture in other ways as well. If you have, say, cramps, depression and loss of libido, Causticum will help both you and your sex drive.\n\nDo\n\n\u2022 familiarise yourself with your changing body. Look at yourself in the mirror without your clothes on. Massage yourself after your bath or shower. Talk about these changes with your partner: be honest about how you feel about them. Ask your partner to massage your body; talk about which parts are sensitive and be specific about how you would like to be touched.\n\n\u2022 find ways of being close to your partner without necessarily leading to sex (learn to separate affection from sex and sex from penetration).\n\n\u2022 tell your partner how you feel, if you have gone off sex, when and why you think it happened, and what you need. Express any resentments and ask for your partner's support and understanding.\n\n\u2022 encourage your partner to share their feelings also \u2013 some find their pregnant women incredibly sexy and others quite the opposite.\n\n\u2022 find some time each day to be by yourself, to have a bath, read a book or go for a walk, even if only for fifteen minutes.\n\n\u2022 deal with any tiredness.\n\n\u2022 get seriously into pleasure and sensuality \u2013 there's nothing sexier! Shared baths (if it's big enough), massage, breakfast in bed, hugs and strokes, etc.\n\n\u2022 let yourself go!\n\n\u2022 choose a time to make love when you know that you will not be interrupted by your other child or children.\n\n\u2022 seek the help of a counsellor or psychotherapist who is trained to work with sexual difficulties if you go off sex in your pregnancy.\n\nDon't\n\n\u2022 make love (with penetration) if you experience pain no matter what position you try; if you are bleeding, even if it's only spotting; in your early pregnancy if you have a history of miscarriage and feel you want to play safe; if your waters have broken in late pregnancy\/early labour as there is a risk of infection.\n\n\u2022 feel guilty if you don't want penetrative sex. There are other ways to give each other pleasure: masturbation won't make you blind and neither will oral sex.\n\n\u2022 abandon your partner. Continue to show affection. If you don't want to, you are holding in feelings that need to be expressed.\n\nSeek help if\n\n\u2022 you experience feelings of disgust towards your body and sex.\n\n\u2022 your partner associates affection with sex and you feel you can't be affectionate because of what it will lead to.\n\nLOW BLOOD PRESSURE (fainting)\n\nThe symptoms of low blood pressure are feelings of faintness and dizziness on standing up quickly from sitting, stooping or lying down. It can be a nuisance in pregnancy but is rarely a cause for concern. Blood pressure always drops between the third and the sixth month of pregnancy due to hormonal changes.\n\nFaintness and dizziness may be associated with anaemia (see here). If you have this condition, repertorise carefully to find a remedy that matches all of your symptoms.\n\nDo\n\n\u2022 use your pregnancy to become more graceful and to take life at a slower pace.\n\n\u2022 sit down immediately if you feel faint \u2013 even if it means sitting or lying on your side on a supermarket floor (preferable to passing out and hurting yourself by falling clumsily). If you can find a chair, sit down with your legs apart (to accommodate your abdomen), breathing deeply and evenly, and bend forward, placing your elbows on your knees. Get up only when you feel ready.\n\n\u2022 stand up slowly from sitting, stooping or lying down, at any time, including the morning when you get out of bed. Take your time.\n\nSeek help if\n\n\u2022 you often feel faint and dizzy and not just when you stand up quickly.\n\nMORNING SICKNESS\n\nUsually associated with the first three or four months of pregnancy it can, for some poor unfortunates, last most of the nine months. In some women it occurs only in the morning but in others lasts all day while still others suffer at different times of the day and even night. In the first three months it feels unfair because it comes at a time when you don't look pregnant and therefore don't automatically collect sympathy from, say, people sitting down in rush-hour trains or buses while you die on your feet.\n\nThe causes were commonly thought to be related to hormonal changes but it is now known that the emotional state also has a significant bearing on morning sickness: doubt, ambivalence, fear, resentment, disgust and denial can all contribute to or aggravate it.\n\nThe nausea is often accompanied by one or more of the following symptoms: an increased sensitivity to smells \u2013 particular smells, strong smells, or even any smell \u2013 and to noise \u2013 loud noises or the slightest, softest sound; a foul, metallic taste in the mouth which is only relieved while eating; a funny smell in the nose which taints all other smells; and alteration in appetite (see Food Cravings). Many things can worsen nausea like eating, drinking, motion, sleeping, emotional upsets, etc.\n\nWhen you self-prescribe on this unpleasant condition, do search into your soul and record honestly your secret thoughts and feelings about the baby and take them into account. If you don't succeed quite quickly and the nausea is severe, don't hesitate to get professional advice.\n\nDo\n\n\u2022 talk through ambivalent, difficult feelings with your partner or a close friend \u2013 they are quite normal, even if they are intense.\n\n\u2022 eat plenty of fresh fruit and vegetables for a day or two and introduce unrefined carbohydrates (wholewheat bread and pasta, and unpeeled potatoes) for a couple of days. If that doesn't help introduce a high-protein diet (fish, cheese, eggs and meat, if you eat it). Tune into the inner voice that tells you what you can and can't eat.\n\n\u2022 try eliminating fats and rich, fatty foods, dairy foods, sugar and even fruit.\n\n\u2022 make sure you are getting enough calcium elsewhere if you cut out dairy foods, and iron if you cut out meat and wholewheat.\n\n\u2022 have a drink with a dry biscuit or piece of dry toast on waking. Eat the biscuit lying down, before you lift your head off the pillow, very slowly and thoroughly and have your drink sitting up but before you get up. Some women have found this trick to work wonders. Instead of tea or coffee, try a glass of hot or cold lemon and honey, a cup of herbal tea or a teaspoon of cider vinegar in warm water \u2013 with or without honey. Ginger tea is a good alternative to lemon and honey, made with powdered or freshly grated ginger and a little honey to taste. Don't make it too strong and avoid it altogether if you are a warm-blooded person as it can make you feel even hotter. Honey tea \u2013 hot water, milk and honey \u2013 is also good.\n\n\u2022 eat little and often, slowly, and chew thoroughly. Sit down and enjoy what you eat.\n\nDon't\n\n\u2022 fast unless eating aggravates your symptoms, in which case you must get professional help.\n\n\u2022 let yourself get hungry, which means that your blood sugar levels have dropped and it will be harder for your body to bring them up again.\n\n\u2022 worry if you lose some weight in the first three months of your pregnancy. Your baby will get what it needs from your fat supplies and you will put on weight once the nausea has passed and you can eat normally again.\n\nSeek help if\n\n\u2022 the nausea is accompanied by vomiting, especially if you are vomiting everything you eat and are keeping very little fluid down. A professional homeopath can usually help with this condition, with no harmful side effects to your baby.\n\n\u2022 you are unable to talk to those closest to you or have feelings that you can't put into words. A counsellor will help you to understand and resolve them.\n\nNOSEBLEEDS\n\nThese can be troublesome in pregnancy due to hormonal changes and the increased blood volume. As a preventive measure, blow your nose more gently.\n\nIt is important to prescribe on the loss of blood if you have bled copiously (see Complaints from loss of body fluids in the Repertory, or Ferrum metallicum if you have a tendency to anaemia).\n\nDo\n\n\u2022 sit with your head tilted forward and breathe through your mouth (spit out blood that drips into your throat).\n\n\u2022 pinch together the soft part of the nose for about 15 minutes, or less if the bleeding stops sooner.\n\n\u2022 hold a cloth wrung out in very cold water or an ice-pack over your nose for a few minutes and then pinch your nostrils again.\n\n\u2022 sit quietly and repeat the above if the bleeding starts again.\n\nDon't\n\n\u2022 blow your nose if you start a nosebleed, or for a few hours after it stops bleeding.\n\n\u2022 sniff or swallow the blood if you can avoid it.\n\nSeek help if\n\n\u2022 the bleeding doesn't stop.\n\n\u2022 a lot of blood is lost and you become pale and dizzy.\n\nOEDEMA\n\nA common pregnancy complaint, the cause of which is not fully understood. It is thought to be due to the pull of gravity combined with the increased volume of body fluids and fluid retention. A slight puffiness or swelling develops in the feet, ankles and\/or hands which disappears after a rest or a night's sleep. Women who are overweight or carrying twins or large babies are more susceptible. Hot weather and standing for long periods usually make it worse.\n\nMany homeopathic remedies are appropriate to this condition so the whole picture counts. What else have you got? If a bit of swelling is all \u2013 and I mean all \u2013 take a short course of Natrum muriaticum and stop if it helps (repeating as needed). Other remedies like Apis mellifica or Phosphorus may be useful depending on the whole picture.\n\nDo\n\n\u2022 exercise.\n\n\u2022 wear comfortable, flat shoes.\n\n\u2022 wear support tights. Before you get out of bed in the morning, raise your legs to empty the veins before you put them on.\n\n\u2022 take off your rings to avoid them getting stuck if your fingers swell badly.\n\nDon't\n\n\u2022 stand if you can sit.\n\n\u2022 sit if you can lie down (or sit with your feet up).\n\n\u2022 cross your legs when sitting down.\n\n\u2022 wear high-heeled shoes.\n\n\u2022 wear socks or stockings with tight tops, or any tight clothes, especially tight trousers.\n\n\u2022 restrict your intake of fluids in the belief that it will help the swelling as reduced liquids may even aggravate the condition.\n\nSeek help if\n\n\u2022 swelling is as bad in the morning in spite of a good night's sleep. Excessive swelling can be a warning sign of a more serious condition developing, such as pre-eclampsia.\n\n\u2022 your fingers leave an indentation mark when you press a puffy area (like your ankle). This is called 'pitting oedema' and may also indicate pre-eclampsia (see here).\n\nPAIN\n\nThe main areas in which you may experience either transitory or prolonged aches and pains are the abdomen, back, groin and joints. The bones of the pelvis form a cradle shape, which is lined with a complex criss-crossing of muscles and ligaments which holds the contents of the pelvis securely in it. The uterus floats fairly freely in the abdomen attached by ligaments to various of the organs, including the bladder and rectum, and the pelvis. Ligaments are fibrous tissue with very little elasticity whose job is to hold firm. The uterus is held mainly by two pairs of ligaments: the round ligaments which run from the front of the uterus down into the groin and the broad\/flat ligaments that run from the sides of the uterus to the lower back (sacrum). They work in pregnancy like sail ropes \u2013 holding the uterus firm once it's full! As it expands the ligaments are stretched, which may cause pain. It is important to exercise the muscles of pelvis and spine throughout your pregnancy so that they are strong and supple enough to cope with the added weight. One of the functions of the hormones produced in pregnancy is to soften bones, joints and ligaments in preparation for the birth of the baby, which adds to the likelihood of minor discomfort developing.\n\nThe following general suggestions are to help you cope with the annoyingly debilitating pains which may accompany pregnancy. Try them as a first resort and if they don't help, consult the sections which follow dealing with specific pains which offer further self-help measures and suggestions for homeopathic treatment. Or self-prescribe, taking the whole picture into account.\n\nDo\n\n\u2022 rest as much as you need to. Make sure your mattress supports you without being too hard.\n\n\u2022 exercise gently and regularly to build healthy, strong, supple muscles, joints and ligaments.\n\n\u2022 breathe deeply, slowly and evenly through the pain. Pain can make you tense so that you breathe too shallowly, which will increase tension. Breathing deeply will bring oxygen to tense muscles and help you relax.\n\n\u2022 relax: tension increases pain. Try lying down, stretching or having a hot bath. Imagine your muscles relaxing as you breathe out and let go of each breath.\n\n\u2022 find sitting, standing and lying positions that ease the pains and use them.\n\n\u2022 lift using your legs to avoid unnecessary strain on your back and abdomen.\n\n\u2022 think before you bend, twist, turn or lift and take it more slowly than you would normally.\n\n\u2022 know your body's limits, be 'careful' with it in pregnancy and 'listen' to it.\n\n\u2022 try massage to help relieve pain and soreness. Your partner or a close friend can give your back and shoulders a comforting rub.\n\n\u2022 maintain good posture when sitting or standing and seek the help of an Alexander-technique teacher if you slump when sitting or walking or find yourself being dragged forward by the weight of your uterus as it gets bigger. He or she will help by giving you advice and exercises to help you correct your bad habits (see Posture).\n\n\u2022 carry two bags when shopping rather than one heavy one.\n\n\u2022 consider wearing a girdle or 'maternity belt' for the relief of abdominal and\/or back pain if: you are carrying a big baby (or several); you were overweight and out of condition before you became pregnant; if you have put on a lot of weight; if your muscle tone is poor; if you are having to be active or on your feet a lot; if your tummy is uncomfortable towards the end of the day \u2013 especially towards the middle of your pregnancy; if you feel instinctively that having some lightweight abdominal support would be very comforting. It can be. Unfortunately it is a fashion that has fallen out of favour in this age of 'flat tummies'. If you do decide to wear one then put it on at the beginning of the day so that your muscles don't get a chance to become tired and achy. Mothercare stock a pantie-girdle for pregnant women; specialist corsetry stores sell various products, including lightweight maternity briefs or girdles with soft, elasticated front panels and maternity belts (a strip of adjustable webbing with Velcro fasteners).\n\nDon't\n\n\u2022 lift heavy weights \u2013 especially towards the end of your pregnancy and if lifting has caused problems in the past. Ask for help in carrying and make more journeys rather than doing it all at once.\n\n\u2022 continue a movement that feels even slightly uncomfortable, as this is how ligaments are strained. A strained ligament takes a surprisingly long time to heal and is vulnerable to further strain.\n\n\u2022 grit your teeth and carry on (working\/washing\/running around after small children) if your body 'complains'.\n\nAbdominal Pain\n\nCommon but unwelcome, especially towards the end when you seem to be stretched to your limit! It is often caused by the stretching and straining that occurs as your body accommodates the growing uterus: muscles, ligaments and nerves have continually to adapt. Sometimes they complain!\n\nThe pressure of the baby or parts of its body (especially its head) can cause discomfort in the abdomen if it presses down on the bladder, kidneys, ovaries, rectum or up on the ribs. This is normally transient and the 'pain' can be relieved by changing position. Some babies love to kick and depending on the position in which the baby is lying muscles or even organs like the bladder can become quite sore and bruised. Take Arnica (6, 12 or 30) if you simply feel sore and bruised but if you feel really 'abused' by, say, a sharp kick in the bladder, take Staphysagria. If, however, the baby becomes very active after you have had a shock, you should repertorise carefully, taking the whole picture into account.\n\nThe weight of the expanding uterus stretches and can strain the ligaments, resulting in stiffness and soreness. (Even turning in bed at night needs to be done carefully, especially towards the end of your pregnancy.) If you let the weight drag you forward the broad bands of ligaments attached to the spine are pulled, causing discomfort and pain.\n\nLook up sprains and strains in the Repertory and choose between the remedies listed using your general symptoms.\n\nUnusual and distressing, although not dangerous, are sharp abdominal pains which dart about, too fast to be pinned down or properly described. They come and go and you can do nothing about them except take a homeopathic remedy and perhaps, if they persist, see a masseur. The specific remedy for this complaint is Cimicifuga: take it in a low potency (6 or 12) for up to a week, stopping on improvement and repeating if it has worked well and the symptoms return.\n\nDo\n\n\u2022 ask your doctor or midwife for an examination if the pain persists, even if it is not severe, because a mild infection of the bladder or urethra (urinary tract infection) can cause abdominal pain.\n\nSeek help if\n\n\u2022 pain is severe and continuous.\n\n\u2022 pain starts suddenly, is sharp and felt more on one side of the abdomen (indicating a possible ectopic pregnancy).\n\n\u2022 pain is accompanied by vaginal bleeding.\n\nBack Pain\n\nCommon in pregnancy, especially towards the end when the weight of the growing baby pulls on the lower spine and the muscles and ligaments of the pelvis and spine are weak. Exercise these muscles (ask at your ante-natal class) so that the extra weight is carried rather than dragging on the ligaments, which can cause backache. Don't forget to check your posture (see here). Backache is common after childbirth along with the other aches and pains. Be gentle with your back until you have built up your strength and take extra care when lifting or carrying your baby, especially when getting the carrycot in and out of cars, buses or trains. Make sure you always change your baby at a good height where your back is straight and not bent forward. Small babies can be changed on your lap \u2013 this takes a little practice but is an easy option especially when away from home.\n\nBack pain can be isolated in the lower back, can radiate to the hips or the abdomen or down into the legs, and can be accompanied, or caused, by 'complementary' pain in the upper spine\/neck. Sciatic pain is caused by pressure on nerves that pass through the lower spine into the legs and is usually one-sided. Pains in the ribs are also a possibility towards the end of the pregnancy when the uterus expands, pushing the ribs up and out. There are many remedies for back pain \u2013 choose carefully to find one that fits your whole picture.\n\nDo\n\n\u2022 consult an osteopath (or cranial osteopath, chiropractor or physiotherapist) if the pains are severe and you have a recurring problem that you know responds well to osteopathy, or if you know you have lifted too heavy an object or done something else that may have 'put your back out'.\n\n\u2022 use heat (baths, compresses or hot-water bottles) or cold (showers, compresses or ice-packs) to relieve pain.\n\n\u2022 make sure your mattress is supportive and right for you \u2013 for some people with backache lying on a hard surface (the floor or a futon) is best, for others a mattress with some give in it (so that lying on one's side doesn't distort the spine) is better. Experiment to find the right sort of mattress for your needs.\n\n\u2022 have a massage if the pains are due to emotional tension.\n\n\u2022 go swimming, an especially beneficial form of exercise as the spine is relieved of the pull of gravity. Make sure you don't keep your head out of the water all the time when you breaststroke as this can strain your upper back. Swim occasional lengths of backstroke if you can.\n\n\u2022 build into your 'exercise' routine some exercises specifically designed to strengthen the lower spine.\n\nDon't\n\n\u2022 further stress a strained back by lifting or pulling, etc.\n\n\u2022 maintain positions that are painful as this will make your condition worse.\n\nSeek help if\n\n\u2022 back pain is accompanied by fever.\n\n\u2022 the urine smells strong or is bloody (it looks pink or is flecked with red or brown).\n\n\u2022 you are having difficulty with either the bowels or bladder.\n\n\u2022 it is difficult to move your legs or they feel numb.\n\nNB Never attempt to treat a serious back injury yourself.\n\nGroin Pains\n\nTowards the end of pregnancy, especially after the baby's head has engaged, the round ligaments that run into the groin can be stretched and strained and the uterus may press down on the pelvic nerves causing excruciatingly sharp but short-lived and harmless pains which sometimes radiate down into the legs. They usually occur while walking and are often severe enough to stop you in your tracks until they are over. They sometimes occur earlier in the pregnancy when they resemble a 'stitch'. Homeopathic treatment can be extremely effective. Bellis perennis is the one remedy indicated for this. If it doesn't help you will need to seek professional advice.\n\nDo\n\n\u2022 sit down if you experience a groin pain \u2013 wherever you are \u2013 until it is over. If you are in a shop, ask for a chair or stool.\n\n\u2022 exercise to strengthen the pelvic-floor muscles (see here) during pregnancy \u2013 this will not only help to prevent groin pains, but will make labour easier and enable a faster healing after the birth.\n\n\u2022 bend your knees or bend over to cough, sneeze and laugh \u2013 as a preventive measure to avoid straining these ligaments. Roll on to your side and use your hands and knees to stand up from lying down.\n\n\u2022 see an osteopath to check that your spine is in good order.\n\nDon't\n\n\u2022 carry on walking if you are in pain.\n\nSeek help if\n\n\u2022 the pains don't respond to homeopathic treatment within five days.\n\nJoint Pain\n\nJoint pain is relatively common due not only to the relaxing properties of the hormones (see here) but also the extra weight; unfitness; change in balance which can mean you will fall over more frequently, leading to joint injury; a sprain or strain. Women who have a tendency to joint problems may find they worsen during pregnancy. The joints of the hips, knees and back can all be affected as well as those of the hand and wrist and foot and ankle, with soreness and stiffness accompanying a sensation of looseness.\n\nNumbness and tingling of the hands can accompany pains in the wrist, known as carpal tunnel syndrome (see here). It is usually worse in the morning because of the build-up of fluid in the tissues overnight.\n\nDo\n\n\u2022 use heat or cold to relieve pain. Try heating pads, hot baths, ice-packs, etc.\n\n\u2022 keep your diet healthy and reduce your intake of 'acidic' foods such as sugar, refined or junk foods, red meat and citrus fruit. Eat plenty of fresh (non-citrus) fruit and vegetables, or put yourself on a short fruit fast if you feel like it, preferably under the guidance of a naturopath.\n\nDon't\n\n\u2022 use a painful joint more than is absolutely necessary.\n\nPALPITATIONS\n\nA not uncommon feature of pregnancy related to the extra strain on the heart from the increased blood volume. You can treat occasional minor palpitations yourself, especially if they have been brought on by excitement, fear, shock, or the suppression of these feelings.\n\nYou will need to use differentiating symptoms to choose between the remedies listed in the Repertory. Identify what brings on the palpitations and what eases them.\n\nDo\n\n\u2022 rest and recharge.\n\n\u2022 sit quietly and breathe deeply, slowly and evenly.\n\n\u2022 practise relaxation exercises or meditation.\n\nSeek help if\n\n\u2022 the palpitations persist over several hours.\n\n\u2022 they are accompanied by chest pain.\n\nPILES\n\nPiles (or haemorrhoids) are varicose veins of the rectum\/anus. You may have had piles before your pregnancy or for the first time in pregnancy or labour. They are caused partly by the increased pressure of the growing uterus on all the veins of the pelvis, partly by the hormonal changes which cause your muscles (and the walls of the veins) to relax and also by a poor diet lacking in fibre. They can manifest as small round lumps, bluish or reddish or purply, around or protruding from the anus, or can remain internal \u2013 you may suspect they are there if you have pain and\/or itching and\/or bleeding on passing a stool. Piles can be painless or may itch, burn, hurt intensely, bleed copiously or scantily. They can give a feeling of fullness in the rectum.\n\nDo\n\n\u2022 increase the fibre\/roughage in your diet dramatically to keep your bowels moving and your stools soft. Decrease your intake of refined foods and fats.\n\n\u2022 take a natural laxative, such as oat bran or prunes, at the first sign of constipation.\n\n\u2022 use a homeopathic or herbal piles cream or ointment (see here) to relieve pain and encourage them to shrink if they are external or to ease external itching.\n\n\u2022 apply piles cream to the anus before passing a stool to ease its passage.\n\n\u2022 clean your anus after a stool but not with hard dry paper which can break the piles open. Use soft, damp toilet paper (wet it a little with water) and then pat dry.\n\n\u2022 pop them back in if they protrude, especially after passing a stool. Smear them with cream or soapy water and push back using a wet sponge.\n\n\u2022 apply witch hazel or Hamamelis (see here) compresses if they are inflamed and swollen. Cut some lint, a sanitary pad or a piece of cotton wool into a small square. Saturate with witch hazel and leave in place for half an hour at a time. If it helps repeat frequently \u2013 it should ease the pain and help to shrink the piles.\n\n\u2022 sit in a bidet, bath or bucket of cold, cool, tepid or hot water to find out which temperature soothes them.\n\n\u2022 avoid standing for long periods of time.\n\n\u2022 exercise the pelvic-floor muscles and especially those around the anus (see here).\n\n\u2022 see also Constipation.\n\nDon't\n\n\u2022 take iron pills as they have been known to cause piles or worsen them.\n\n\u2022 strain when passing a stool \u2013 take a book or magazine to the loo with you and let your bowels do their best (or worst).\n\n\u2022 squat, as this will encourage piles because of the additional pressure. Semi-squat trying a low stool or pile of books for support.\n\n\u2022 sit on hard surfaces \u2013 use a cushion or fold a coat or sweater into one.\n\nSeek help if\n\n\u2022 you can't push the piles back or if they'll go back but won't stay up.\n\n\u2022 the pain is not relieved by self-help measures.\n\n\u2022 you bleed copiously and regularly.\n\nRESTLESS LEGS\n\nThis unpleasant (but not serious) syndrome is commonly associated with old age... and pregnancy. The legs feel as if they have to move \u2013 it is an awful, uncomfortable feeling inside the muscles of the legs which can get so bad they ache. When it's bad they have to be constantly on the go. The cause is not fully understood but thought to be a combination of factors including tension and dietary deficiencies.\n\nHomeopathically there are several remedies for this distressing complaint. Differentiate carefully to find one that fits.\n\nSINUSITIS\n\nSee also Common Cold.\n\nSinus congestion can occur more readily in pregnancy due to hormonal influences, especially towards the end and in those who are already prone to sinus problems. When you have a cold, you will know if your sinuses are infected because your head will feel heavy and you may be aware of pressure above the eyes and in the cheeks by the bridge of the nose, which may also be tender to the touch.\n\nDo\n\n\u2022 use a humidifier or steam inhalations to relieve congestion.\n\n\u2022 drink plenty of fluids.\n\n\u2022 avoid tobacco smoke and builders' dust.\n\nDon't\n\n\u2022 use decongestants.\n\n\u2022 blow your nose too hard.\n\nSeek help if\n\n\u2022 the sinuses are tender, you have a fever and there is a smelly discharge from the nose which does not respond to self-prescribing within 48 hours, or less if you are in severe pain.\n\nSKIN COMPLAINTS\n\nWomen who suffer from chronic skin complaints may find that they miraculously clear up during pregnancy, but occasionally they become worse. Always consult a professional homeopath in such cases.\n\nMinor skin complaints are commonly experienced in pregnancy by women who have not previously suffered from them.\n\nPigmentation\n\nDuring pregnancy the skin produces more pigment which may help you tan more easily. It can also cause dark patches to develop on the face: a mask or 'saddle' over the nose and across the cheeks and a 'moustache' on the upper lip. Birthmarks, freckles and moles may all become darker as may the area around the nipples. There is nothing you can do to prevent this from happening and the skin will mostly revert to its normal colouring after the birth.\n\nTake a short course of Sepia 6 if the dark patches on your face bother you and you have another symptom or two which fits the Sepia picture. If it helps, repeat it as needed.\n\nRashes, Itching, Flaking, Roughness\n\nIncreased skin sensitivity can lead to the development of mild allergies to soaps, soap powders, perfumes, cosmetics, the metal on watches or jewellery, etc. Women with a tendency to dry skin may be worse affected. (See also Itching.)\n\nTake a short course of Sulphur 6 if your skin is dry, itchy and worse for heat and bathing. However, look at the whole picture to see if another remedy fits you and other complaints you may have better, even though skin complaints are not mentioned. Take a short course of your indicated constitutional remedy. If it helps you but your rashes are getting worse, consult a professional homeopath.\n\nDo\n\n\u2022 wear cotton next to your skin and avoid nylon.\n\n\u2022 use a simple, refined oil, such as almond oil, on dry skin, or aqueous cream, available from chemists, if your skin is also itchy.\n\n\u2022 wash your clothes and bedding in non-biological powders (Ecover, Dreft, Lux, Fairy Snow, etc.), or experiment to find out which powder is the culprit if you are sure that that is what is causing your skin problems.\n\n\u2022 experiment with your diet: have you had a craving for orange candies, frozen peas, cream? Are you suddenly eating something in particular? If so, cut it out for a few days. If your skin improves, you can double-check by reintroducing the rogue food item as you habitually ate it before. A mild sensitivity won't usually show itself immediately on reintroducing the food you eliminated \u2013 it will take a few days to build up again.\n\nDon't\n\n\u2022 use hydrocortisone creams or coal-tar preparations on your skin.\n\n\u2022 use bath oil, bubble bath or perfumed soap.\n\nSeek help if\n\n\u2022 itching is preventing you from sleeping.\n\n\u2022 a rash is spreading all over your body.\n\nSpider Veins\n\nThese are caused by hormonal changes and there is little that can be done about them. They will disappear after the birth.\n\nBroken Veins\n\nAlso caused by hormonal changes, they will not disappear after the birth.\n\nTake a short course of Arnica at the first sign of broken veins as this will help 'mend' damage already done and prevent them developing further. You can also rub some Arnica cream into the skin.\n\nSpots\n\nA short course of Calcarea sulphurica or Silica may help to clear them up. If the spots are especially bad don't attempt to treat yourself but consult a professional homeopath.\n\nDo\n\n\u2022 cut out all sugar, including chocolate and biscuits, and cut down on fat for a week or so.\n\n\u2022 drink the juice of half a freshly squeezed lemon in water first thing in the morning and last thing at night.\n\n\u2022 wash your hair regularly if your spots are on your face.\n\n\u2022 let the sun get to your skin regularly for short periods of time.\n\n\u2022 use a good skin-care routine, cleansing thoroughly at least once a day.\n\n\u2022 make sure your diet is rich in fruit and vegetables.\n\nDon't\n\n\u2022 squeeze or pick at spots as this can cause scarring unless they are absolutely 'ripe', in which case very gentle pressure will encourage the pus to discharge. Apply Calendula cream immediately.\n\n\u2022 use antibiotic ointments.\n\nSTRETCH MARKS\n\nAlmost nothing can be done to prevent stretch marks. Some women have more elastic skin which stretches without marking. Others are less lucky. Stretch marks usually appear on the abdomen, buttocks, thighs, breasts and upper arms. They are reddish or purplish when they first appear and may itch. After the birth they will shrink and fade to white.\n\nCalcarea fluorica will help prevent stretch marks. Don't take it continuously, but if you have dry skin and are prone to stretch marks anyway, take it for regular short courses throughout your pregnancy.\n\nDo\n\n\u2022 use an oil or oily cream on a dry skin to help keep it reasonably supple. Any cream or oil containing vitamin E is helpful, both as a preventive and once the marks have appeared.\n\nDon't\n\n\u2022 put on too much weight as this can exacerbate stretch marks.\n\nTEETH AND GUM PROBLEMS\n\nYour gums may swell and bleed more easily in pregnancy, due to hormonal changes. See your dentist and hygienist regularly to maintain a healthy mouth during your pregnancy. Dental treatment is free throughout pregnancy and the post-natal year.\n\nDo\n\n\u2022 ask your dentist to use a rubber 'dam' if you need an amalgam filling or a filling replacement. This is a rubber sheet that protects your mouth and prevents amalgam dust from being swallowed accidentally.\n\n\u2022 have your teeth overhauled when you are planning to get pregnant.\n\n\u2022 avoid X-rays. Discuss with your dentist the possible alternatives to amalgam fillings as their toxic effects are being increasingly noted.\n\n\u2022 brush your teeth regularly and floss at least once a day. Some dentists recommend a brush and a floss after each meal to keep teeth and gums in really good condition. Massage your gums with your finger after you brush your teeth.\n\n\u2022 use a mouthwash of salt water or Calendula tincture if your gums are sensitive or bleeding.\n\n\u2022 watch your diet and limit your intake of sweets and sugary products such as biscuits, soft drinks and ketchup.\n\n\u2022 eat plenty of fresh fruit and vegetables.\n\nDon't\n\n\u2022 forget that emotional stress affects teeth and gums. If your toothache is not related to a cavity, try to identify what is really causing your pain.\n\nTHRUSH, Vaginal\/Yeast Infection\n\nBecause the acid\/alkaline balance of the vagina changes in pregnancy, thrush is another common complaint. Vaginal discharges can become heavier, smell different and be mildly irritating.\n\nSeveral remedies are indicated for the treatment of acute thrush. Differentiate between them carefully, using the symptoms of the thrush as well as any general and stress symptoms and your emotional state. Remember that the onset of thrush can occur without a discharge, and don't ignore soreness, redness and itching.\n\nDo\n\n\u2022 put half a cup of cider vinegar or a tablespoon of bicarbonate of soda in your bathwater.\n\n\u2022 cut out sugars and refined breads, cakes and biscuits until it has cleared up. The yeast in wheat products can exacerbate thrush.\n\n\u2022 wash the genitals after urinating and after making love: the acid\/alkaline balance of semen can aggravate thrush. 'Bottle wash', see here.\n\n\u2022 use condoms (without integral spermicide) with plenty of lubricating jelly (such as KY-Jelly) until the thrush has cleared so that you don't pass it on to your partner, unless penetration (and\/or the condoms themselves) aggravate the thrush, in which case have non-penetrative sex until the condition has cleared up.\n\n\u2022 wash cotton underwear with unscented soap or soap flakes, not detergents, and rinse well.\n\n\u2022 apply live yoghurt to ease itching and discomfort. You can smear some in and around the vagina with your fingers or insert a mini tampon or a small natural sponge (which has been boiled to sterilise it) dipped in yoghurt. Remove tampons or sponges after an hour or so.\n\n\u2022 eat lots of dark green vegetables and whole grains.\n\n\u2022 try adding acidophilus \u2013 powdered, live dried yeast cultures, available from health-food shops \u2013 or live yoghurt to your diet.\n\nDon't\n\n\u2022 make love without a condom until the thrush has cleared up as you can pass it to your partner.\n\n\u2022 wear knickers if it is practicable and the weather hot as sweating can make it worse.\n\n\u2022 use vaginal deodorants or soap.\n\nSeek help if\n\n\u2022 mild thrush hasn't cleared up within a week.\n\n\u2022 the discharge begins to itch, smell unpleasant or changes colour to yellow or green and becomes thick and lumpy.\n\nTOXAEMIA\n\nSee Eclampsia.\n\nURETHRITIS\n\nSee Cystitis.\n\nURINARY TRACT INFECTIONS\n\nSee Cystitis.\n\nVARICOSE VEINS, of legs and vulva\n\nThe softening of tissues in pregnancy can affect the walls and valves in the veins. The extra weight you are carrying added to increased blood volume plus, perhaps, constipation may stress the leg and pelvic veins. If the veins are weak, blood may 'pool' (instead of being returned to the heart) in the leg and pelvic area, and stretch the veins into the unsightly knots and lumps.\n\nVaricose veins may be sore or acutely painful, can itch or cause the whole leg to ache especially if you stand for long. Those of the vulva are usually even more painful because it is harder to take the pressure off the area without lying down. You may feel as if there is a fullness and heaviness in and around the vagina, as if 'everything will fall out'. Varicose veins of the legs may become less 'knotty' after the birth but they are usually 'here to stay' whereas those of the vulva disappear quite soon after childbirth. They won't cause any trouble during labour because the perineum stretches so much that they disappear at that point and never return! See also Piles.\n\nDo\n\n\u2022 exercise regularly to keep muscle tone healthy, especially the pelvic-floor muscles (see here), and walk briskly to keep the blood circulating.\n\n\u2022 wear support tights. Put them on before you get out of bed in the morning.\n\n\u2022 splash or shower with cold water (if you can bear it) from the waist down, which will temporarily alleviate the soreness of inflamed veins in the pelvis and\/or legs.\n\n\u2022 elevate your legs as often as possible.\n\n\u2022 raise the end of your bed with bricks or a block of wood (6\u20138 ins\/15\u201320 cms high) so that you sleep at a slight angle with the blood draining more easily back to the heart (without having to be pumped).\n\n\u2022 apply witch hazel compresses to painful veins to reduce inflammation. Soak lint, cotton wool, hankies or soft thin teatowels in witch hazel and tie loosely around the veins. Remove when dry. Repeat as often as needed.\n\n\u2022 avoid becoming constipated as this will aggravate them.\n\n\u2022 try to control your weight.\n\n\u2022 flex your feet and leg muscles if you are having to sit or stand for long periods of time. You can rotate feet, walk around, flex on to your toes, shake your legs, etc.\n\nDon't\n\n\u2022 sit with your legs crossed.\n\n\u2022 stand for long periods.\n\n\u2022 constrict the flow of blood to the legs in any way \u2013 through wearing tight clothes, or by sitting on chairs that press into the backs of your thighs. Make sure your feet can be flat on the floor with your thighs comfortably free while sitting (on chairs or the toilet). Use fat books or telephone directories if you've got short legs, especially if you have a desk job.\n\nVOMITING\n\nSee Morning Sickness.\n\nWEIGHT GAIN\n\nThe increased weight in pregnancy is not all baby! It is also composed of increased body fluids including blood, fat deposits (on the belly, buttocks and thighs), enlarged breasts, amniotic fluid and placenta. (Amniotic fluid is 98 per cent water with the other 1\u20132 per cent composed of fetal hair and skin cells, enzymes, urea, glucose, hormones and lipids. It is completely replaced about every three hours.)\n\nA 20\u201330-pound (10\u201315-kg) gain is normal, although some women put on a lot more and others considerably less. Extra fat is laid down for breastfeeding, and, during the initial post-natal period, will help you avoid becoming run down and exhausted.\n\nIf you are putting on a lot of weight check your diet, but if it is balanced and healthy don't worry. You can begin to lose any excess weight straight away if you decide to breastfeed, which uses at least 600 calories per day.\n\nDo\n\n\u2022 eat a balanced diet (see here).\n\n\u2022 eat little and often.\n\n\u2022 keep physically fit however much weight you put on. If you turn into a couch potato, the labour itself may be more arduous. If you are not keeping fit because emotional stress is draining you, consult a psychotherapist, counsellor or professional homeopath to help you deal with it.\n\nDon't\n\n\u2022 gain too much weight and become sedentary.\n\n\u2022 diet. If you put on too much weight cut out sugar and refined\/junk foods, cut down on fats and increase your intake of fruit, vegetables, fish and whole grains.\n\n\u2022 worry if you do gain a lot of weight \u2013 concentrate on getting fit.\n\nBIRTH\n\n* * *\n\nYOUR BODY AND BIRTH\n\n* * *\n\nThis chapter is not about how to have a perfect birth: there is no single 'right' way to give birth. Each woman has different strengths, weaknesses and beliefs and therefore different needs. What works for one person won't necessarily work for another. Also, what works in one birth won't necessarily work in the next.\n\nBirth can be a wonderful, extraordinary and powerful event, a truly mysterious rite of passage. On the other hand, it can be miserable, painful and lonely. Either way, it is unforgettable. The presence of a healthy baby can transform or, at least, soften the memory of a long, arduous labour.\n\nI am going to look at some of the possible difficulties surrounding birth and share ideas and tips I have gathered over the years during which I have been working with women in labour. My aim is to encourage you to find your own way to give birth \u2013 with confidence and conviction. I hope that from reading this chapter, and the sections on birth in Chapter 2, you will gain a balanced sense of what you might need in labour and therefore how best to prepare for it. Take time to reflect on what you really want, and remember to keep an open mind. Labour is a process in which you can be fully involved, not an acute illness that you have to lie back and endure!\n\nIt is important to be aware of what is happening to you and your body during childbirth so that you can picture in your mind's eye what is happening and not feel frightened or powerless through ignorance. First births can be made unnecessarily difficult if you don't understand the physical processes.\n\nPRE-LABOUR\n\nLabour takes place approximately 280 days (40 weeks) after the first day of your last period. It is usually preceded by one or more of the following:\n\n\u2022 practice contractions (Braxton Hicks), which are felt as a tightening or hardening of the abdomen and may be uncomfortable but are rarely painful. Use them to practise on: breathe (see here) into them and relax with them.\n\n\u2022 a 'lightening' as the baby's head drops down into the pelvis (engages) towards the end of pregnancy (if it is head down). It feels as if a little more space is suddenly available in the abdomen. It is!\n\n\u2022 fewer movements from your baby who usually becomes less active.\n\n\u2022 a 'nesting' compulsion \u2013 an unexpected pre-labour surge of energy. Women have been known to redecorate or springclean their houses during this period, which usually lasts a few days. Don't overdo it \u2013 you need to express your elation while conserving your energy. Be careful with your body \u2013 don't paint ceilings, for example, as you don't want to fall off a ladder!\n\n\u2022 loose stools or diarrhoea.\n\n\u2022 a 'show': the plug of mucus that stops up the neck of the uterus may come away.\n\n\u2022 the waters breaking or leaking of amniotic fluid, as the baby's head engages.\n\nAll the above symptoms of impending labour can come and go and nothing happens \u2013 carry on as normal. Pack your case if you are going into hospital. If you are having a home birth, organise your house and make sure you have all the bits and pieces that you need, including those that your midwife has instructed you to have ready.\n\nNB Complications such as a breech baby or a small pelvis require expert attention throughout.\n\nLABOUR\n\nFirst Stage: Labour\n\nTrue labour is precipitated by:\n\n1 A mechanical process: when the baby reaches a certain size the pressure of its head and body against the walls of and entrance to the uterus are thought to stimulate reflexes which set off contractions. The contractions push the baby down, which stimulates more contractions.\n\n2 Hormonal changes cause the uterus to contract.\n\nBoth processes work together to expel the baby.\n\nDuring the first stage, the cervix \u2013 the neck of the uterus \u2013 softens, opens (dilates) and pulls up so that the baby is able to emerge. It is the longest stage of labour. Generally this stage takes longer with first babies and is quicker in subsequent labours.\n\nThe contractions are felt initially as low abdominal or low back pain, not unlike period pains. They increase, usually gradually, in frequency and strength as the cervix dilates. The waters (amniotic fluid) may leak or break.\n\nTransition\n\nTransition occurs between the first and second stages once the cervix is fully dilated and can last for one contraction or continue for several hours. The contractions are irregular and usually close together and may be accompanied by an overwhelmingly strong urge to push. They mark a change from first stage contractions which open the cervix to second stage contractions where the longitudinal muscles of the cervix actually shorten with each contraction thereby automatically pushing the baby out.\n\nThe altered rhythm as the contractions change over can go unnoticed or it can be hard to deal with. Everything may feel out of control as the body takes over to complete the most physically arduous task it ever accomplishes. Many women become irritable or even abusive to those around at that time.\n\nIf the cervix isn't fully dilated at this stage your midwife will ask you to pant to prevent you from trying to push your baby out when your body isn't quite ready.\n\nSecond Stage: Birth\n\nThe baby moves out of the uterus and travels the 4 curved inches (10 cms) of the birth canal (vagina) into the world. Each contraction automatically pushes the baby a little further down. The contractions are regular and usually come several minutes apart. The baby moving down the vagina can feel as if you are about to pass a large grapefruit or melon \u2013 even a football. You will usually feel a strong urge to push, accompanied by any number of sensations \u2013 stretching, bulging and burning are all common. Your midwife will ask you to pant if she doesn't want you to push, so that the baby is born nice and slowly without tearing your perineum or vagina. A burning sensation may indicate that your baby is emerging too fast and not giving your muscles a chance to stretch.\n\nIt is tiring and therefore important to drink something, to take some honey or sugar for energy and to rest completely between contractions. Because of the shape and position of the vagina, lying down to give birth is not necessarily the best position \u2013 taking gravity into account by squatting or semi-squatting can help enormously. But do lie down if you want to. Squatting or standing are good positions if this stage is slow. Get on to all fours if it is very fast. Many women instinctively find the position that suits them during this stage: a semi-squat is good as you can see what's going on and reach down to touch and lift up your baby and also helps your midwife. Your partner can sit behind you and support you from behind. Go with the flow, follow what your body and your midwife tell you to do. The emotions that accompany this last physical hurdle are many and glorious. This is a most wonderfully empowering moment in a woman's life. Some women love this second stage while others find it a painful last straw and are glad when it's over at last. Whatever \u2013 you are shortly to greet your baby for the first time.\n\nThird Stage: Afterbirth\n\nThe placenta comes away from the wall of the uterus and is delivered. Contractions may cease temporarily after the birth itself; they can be re-stimulated by putting the newborn baby to the breast which will cause the uterus to begin contracting again and, as the uterus decreases in size, the placenta to detach from the wall of the uterus. Sensations at this stage of labour are coloured by the excitement of the birth; the contractions are considerably less demanding but may not be regular. The vagina may feel numb.\n\nHow long this stage lasts very much depends on your medical attendants. If you have negotiated for syntometrine not to be administered immediately after delivery of the baby (see here) then your body may take its time. If syntometrine is given, your midwife will 'encourage' your placenta to deliver by pulling very gently on the cord while you are having a contraction: syntometrine causes the uterus to contract very efficiently and the placenta must be out promptly before it contracts. If you wish to breastfeed after the birth and for the cord not to be cut, you must ask that syntometrine be administered after the cord is cut to prevent the drug passing through the placenta to your baby.\n\nDo not try to prescribe on yourself when you are in labour: leave it up to others. Once labour starts you need to put all of your energy into what you are doing. Your partner or homeopath should take responsibility for prescribing homeopathic remedies. Many midwives are now becoming interested in how homeopathy can help during labour and some have a basic kit, which they use according to their knowledge and skills.\n\nBirth partners should read through the following section to get an idea of what might happen and what might be needed in labour, to become familiar with the labour sections in the Repertory (here) and the Materia Medica (read up each remedy that may be needed for the birth). See also here on how to put together a homeopathic kit for labour and make a list of the remedies you want to have to hand. You may ask a professional homeopath to help you with this if you find yourself confused.\n\n* * *\n\nCOMPLAINTS\n\n* * *\n\nThe complaints section deals with the events of labour and makes suggestions for dealing with them, both practical and homeopathic. Play the 'What if' game: read through each complaint asking yourself and your partner, 'If \"this\" happens, what will we want?' You are not allowed to answer 'Give up'! This will stimulate you to think through some of the problems you may encounter so that they won't take you by surprise, which will help you feel more in control.\n\nBACKACHE LABOUR\n\nSee Labour.\n\nCAESAREAN\n\nSee Medical Interventions.\n\nCONTRACTIONS\n\nSee Labour.\n\nEMOTIONAL DISTRESS\n\nEmotions can run deep, strong and unexpected. Some or all of the following feelings can and do surface: anger, rage, fear, panic, despair, excitement, apathy, shame, embarrassment and so on. You may find it easier to express yourself at home but don't let a hospital environment inhibit you.\n\nDuring labour so much is going on that it is almost impossible for a lay person to stop and work out a first-aid remedy to help. I have listed below some commonly needed remedies for emotional distress in labour to choose from. They should be taken fairly frequently \u2013 and either will or won't work. If the emotional distress accompanies physical symptoms it is best if you build those symptoms into your prescription for it to be truly effective: if Pulsatilla seems right for the emotional distress look it up in the Materia Medica (here) to check whether it also has the physical symptoms.\n\nAconite Anxiety, fear and panic. Fear of dying, of the baby dying.\n\nArnica Suppresses feelings (esp. shock). Says she is OK when she plainly isn't.\n\nArsenicum Anxious, fussy, bossy and irritable.\n\nBelladonna Angry and abusive. May throw a tantrum (rant and rage).\n\nChamomilla Abusive, angry. Impossible to please, asks for things then rejects them.\n\nCoffea Over-excitable. Excitement alternating with fear. Talkative and jokey.\n\nGelsemium Lifeless, apathetic, despairing and dazed.\n\nKali carbonicum Irritable, anxious and bossy.\n\nLycopodium Pre-birth nerves. Lacks self-confidence. Feels exposed.\n\nNatrum muriaticum Closed up emotionally. Feels shy and exposed. Wants to be alone.\n\nPulsatilla Weepy, clingy and pathetic. Despair. Changeable moods.\n\nSepia Irritable, anxious and despairing. Sluggish. Worn out.\n\nDo\n\n\u2022 go with the flow.\n\n\u2022 be true to yourself. Find a way to express what is happening to you wherever you are.\n\n\u2022 keep talking, keep communicating about what is happening to you \u2013 emotionally as well as physically.\n\n\u2022 be assured that you won't feel like this for the rest of your life!\n\n\u2022 use your feelings to empower you. Don't suppress them to make others feel better. This is one of the few times in your life when you'll be forgiven for behaving badly!\n\nEPIDURAL\n\nSee Pain Relief.\n\nEPISIOTOMY\n\nSee Medical Interventions.\n\nEXHAUSTION\n\nA common problem in labour, simply because of the arduous physical job that has to be accomplished, so it is vital to keep the blood sugar level high enough to provide plenty of energy, especially in the early stages, and if the cervix is dilating slowly \u2013 even if contractions are coming thick and fast. If you don't eat, you may need an intravenous drip if the blood sugar level drops, which is worth avoiding if at all possible.\n\nPut 4 drops of Rescue Remedy in all drinks if tiredness sets in; physical tiredness may be alleviated by Kali phosphoricum 6X between every contraction for up to 6 doses, repeated fairly often, if it works. Arnica 30, taken from time to time, may also help, or can be alternated with Kali phosphoricum 6X.\n\nIf other symptoms accompany the exhaustion, like despair and weepiness, or terrible backache with a badly positioned baby, work out a remedy for the whole picture and take practical measures to deal with the situation. (See also here.)\n\nDo\n\n\u2022 eat, little and often, light, easily digestible foods. Some women only want liquids, others like proper meals at regular intervals. If you're at home and you want food, try soup, pur\u00e9ed vegetables or fruit. Ask your birth attendants to make themselves useful in the kitchen cooking exactly what you want. If you are in a hospital, you may not be allowed to eat much, if at all, because of the possibility of your needing an anaesthetic but have on hand a selection of small pieces of fruit to dip in honey in case your midwife has no objection. Dried fruit and nuts or biscuits are another easy alternative, although their chewiness can be irritating.\n\n\u2022 drink often to prevent dehydration. Lemon or orange juice, freshly squeezed if possible, diluted with a little hot water (if you want a warm drink), with honey or sugar makes an ideal source of energy. Diluted stocks or soups, herb teas, ordinary tea or even coffee are all fine if that is what you really want. A word about honey: you can buy honey from wholefood shops which has not been 'heat-treated', but pressed out of the combs and has retained more of its goodness. You won't give birth often \u2013 get a pot for your labour if you can find it.  \nNB Many women don't want to eat or drink in labour: it is important that your partner, friend or midwife regularly feeds you tasty morsels to keep you going.\n\n\u2022 rest between contractions if you feel tired.\n\n\u2022 sleep between contractions.\n\n\u2022 have a long, warm bath or shower.\n\n\u2022 take some gentle exercise \u2013 a walk in the fresh air, around the garden, up and down your street \u2013 or even in the hospital corridor!\n\n\u2022 express any feelings that surface, like anger, to release tension, which is tiring.\n\n\u2022 laugh \u2013 ask to be told corny jokes, if you really want!\n\n\u2022 have a change of scene \u2013 if possible.\n\n\u2022 ask your partner to touch, massage, hold, kiss you.\n\n\u2022 breathe deeply and evenly.\n\n\u2022 do anything that you know energises without draining you.\n\n\u2022 consider medication.\n\nDon't\n\n\u2022 fast during labour or deprive your body of food and\/or drink.\n\nFALSE LABOUR\n\nSee Labour.\n\nFAST LABOUR\n\nSee Labour.\n\nFORCEPS\n\nSee Medical Interventions.\n\nHAEMORRHAGE\n\nIt is unlikely that you'll be able to treat this condition because your medical attendants will administer an injection of syntometrine at the first sign of serious bleeding (see here). However, homeopathic treatment works extremely fast: the right remedy will work within 30 seconds. You might ask for a minute's grace before syntometrine is administered. I have also included the following remedies just in case you do not have immediate access to medical treatment:\n\nPhosphorus 200, one dose, is indicated if the blood is bright red, there is a lot of it and it gushes out; have Phosphorus to hand during labour if it has worked well in the past, and you have a history of bleeding easily (nosebleeds, profuse menstrual periods, etc.); if Phosphorus hasn't worked, follow it with Ipecacuanha 200, one dose, after 30 seconds. If there is a lot of dark red blood with clots that gushes out, try Belladonna 200, one dose, instead.\n\nLABOUR\n\nContractions\n\nThe sensations accompanying contractions are different for every woman and in every labour. It is not possible to know ahead of time what your baby's birth will be like and it is best therefore to be aware that it might hurt. Although painless labours are not unknown, I have found, from my own professional experience, that childbirth without pain is rare.\n\nIt is useful to have a reality base to measure yourself against. If you know that it is likely that you will experience pain, you can plan accordingly and confront your fears and\/or anxieties before the event. Start by building a picture of your own response to pain: ask yourself what you know about your pain responses. Are you good with pain? Do you have a high pain threshold or do you suffer easily? How do you cope at the dentist? Can you have a filling without an injection or do you need a lot of local anaesthetic to cope with the pain?\n\nPain in childbirth can be managed so that it is not overwhelming and one of the best aids is to understand the process of childbirth fully so that you can work with the contractions. Overwhelming, unexpected pain causes tension and muscle-tightening, resistance, which increases pain. Learning techniques for coping with pain whether you are at home or in hospital is an important part of preparing for an active birth. Our bodies' automatic response to pain is to secrete hormones (endorphins) which relax and ease it \u2013 their action is similar to that of morphine. The more serious an injury the more hormones are released: after an accident, victims report afterwards that they felt nothing, numb, and as if they weren't all there.\n\nIn labour we hope to draw on this natural mechanism, which is easier for some women than others. All the techniques for helping with pain in childbirth are designed to relax, distract (so that relaxing is easier) and make a woman feel more at ease so that she can relax. We all have different ways of alleviating pain: think back to what has worked for you before as it may help when you are in labour.\n\nThe following guidelines for dealing with pain may help you to think ahead to what might help you: file them in a reasonably accessible part of your brain so that your body can tell you what it needs when the time comes (see also here). Once labour is established some of the following suggestions will become neither possible nor appropriate, but by then you will have found a rhythm and discovered the things that help most. Forget the things that don't work and keep trying new ideas until you discover the right ones. Then keep doing them!\n\nThere often comes a moment in labour when the intensity of the pains and the loss of control become overwhelming and a woman says she can't go on, which may come close to delivery. Pulsatilla (if she is pathetic and weepy) or Arnica (if she is more irritable) taken then can give her the emotional strength to carry on. If another strong emotional picture surfaces then prescribe the appropriate remedy.\n\nI have included several remedies for labour pains where a pattern is emerging: where the contractions stop, come at irregular intervals, are ineffective in dilating the cervix or are exceptionally painful. You will find these in the Repertory under Labour pains. Observe carefully the pattern of the pains as well as the general and emotional symptoms to make a good prescription.\n\nDo\n\n\u2022 see here.\n\n\u2022 use distraction. You can learn a mantra, poem or line of a poem which you speak, either aloud or in your mind, with all your attention during the contractions to take your mind off the pain. Say it with your full concentration on the words alone. Prenatal classes often teach this method.\n\n\u2022 use breathing techniques. A variety of techniques are taught so learn one and try it out. If it works use it, if it doesn't try something else.\n\n\u2022 take one step at a time, one contraction at a time.\n\n\u2022 bath or shower. Once labour is well established many women spend much of the painful part of the first stage in a bath. It's a wonderful way of relaxing and relieving pain especially in a long labour. For some women though, lying down isn't a good position, especially if the baby is posterior presentation. In this case you can kneel on all fours in the bath (rest your knees on some soft padding like folded hand-towels) and get your partner or midwife to direct the shower on to your back, especially the parts that hurt. Try warm, hot and even cool water: pain is sometimes relieved by cool or even cold bathing \u2013 this could guide you to a helpful remedy as well as easing the soreness. The bathwater should be warm (not too hot) and deep so that you can change position easily. Block the overflow with Blutack to get a deeper bath. Wait until the contractions are really well established before getting in the bath as this has been known to slow labour in the early stages. You can rent good-sized pools specially designed for labour: they are compact enough to fit into a smallish space in your home (6 feet\/2 metres in diameter) but large enough to contain three people comfortably! It is possible to be on all fours with the water covering you in a pool like this, which gives considerable pain relief; some women give birth to their babies under water. A few hospitals have installed birthing pools while others have jacuzzis available for women in labour. Many hospitals will ask you to get out of the pool for the actual birth.\n\n\u2022 bite on a twisted towel or a leather belt, making sure that you don't stop breathing at the same time. This helps release tension, especially if you grunt or make growling noises at the same time.\n\n\u2022 make a noise \u2013 find your own sound. Pain is greatly relieved by releasing tension through making a sound. Go with the flow and see what sounds come out. You may be surprised. Don't scream or constrict your throat: let sounds bubble up from deep inside you \u2013 let them out and then let them out louder. This can be a brave new thing to do \u2013 you may need encouragement, may feel self-conscious about being noisy. At the end of the first stage you may not be able to suppress noise, anyway \u2013 women grunt, groan, moan, shout and make the most wonderful sounds when they are in labour. They 'cancel out' the pain.\n\n\u2022 sing. Connect with a favourite song and just sing it over and over again.\n\n\u2022 move. Experiment with finding different movements at different stages and go with the contractions.\n\n\u2022 dance. Let your body move to music, or to the rhythms of your labour: follow those movements with and through each contraction. Many women find themselves circling their hips or their whole bodies. Belly dancing was designed for women in labour because it was observed that those natural movements actually helped with the process of labour itself.\n\n\u2022 let your body hang, by holding on to door handles or the bars of a ladder or bed. This allows your spine to stretch and is very relaxing. In some countries a rope or sheet is slung over a high beam and women spend much of their labours hanging in a relaxed but upright position.\n\n\u2022 relax. Release your pelvis and hips during a 'contraction', let them loosen, soften and widen.\n\n\u2022 eat calcium-rich foods as calcium helps to raise your pain threshold. Yoghurt is ideal because it is also easily digestible. Eat carbohydrates for energy.\n\n\u2022 walk. Especially in early labour, you can take a walk outside if you feel like it \u2013 especially if you have a garden. Or you can simply potter about inside.\n\n\u2022 exercise gently \u2013 to relieve stress and keep supple. Do your yoga positions, any stretching and contracting of muscles that feels good. If it hurts stop immediately. There will be times in labour when you feel like moving and others when you want to rest.\n\n\u2022 experiment with different positions to relieve pain. Try kneeling; kneeling with a midwife or your partner fully supporting you or holding you from behind, kneeling on all fours, kneeling or squatting with your partner or midwife supporting you from behind or in front, lying \u2013 every which way \u2013 leaning against a person, bed or wall; sitting on a birthing stool; sitting forward over a bean bag or a heap of cushions or pillows; sitting on the toilet!\n\n\u2022 use touch and massage. Ask your partner to touch you in different ways and places, at different times. Try tender-light touch \u2013 very gentle feather-light stroking \u2013 especially on your abdomen, face and back. Try firm massage in places where there is pain \u2013 experiment with the pressure till you get it right. Massage another part of the body completely, like the feet. Pressing or pushing can be wonderful, especially for a backache labour where the palms of the hands pushed hard against the small of the back can ease the pain of each contraction. Remember, you may not want to be touched at all at times \u2013 perhaps during your entire labour.\n\n\u2022 massage the perineum with lots of oil (almond or olive) to help avoid tears and also to focus some energy on that area, to encourage you to relax it. Visualise the muscles of your vagina loosening as you massage the area.\n\n\u2022 canoodle: kissing, hugging and holding are wonderful for encouraging a person to relax. If you feel like it and have a willing partner handy, go ahead!\n\n\u2022 get some pain relief if you are not coping, if the pain is much worse than you had imagined, if you are finding it too much or if it has gone on for too long, you are beginning to lose hope and your vitality.\n\nDon't\n\n\u2022 feel bad if you have to 'resort' to drugs. Remember, you have done your best \u2013 you prepared yourself for this labour, not knowing how it would go or what would happen and you have done as much as you can to go it alone. It is OK to accept help: if you were running a marathon and began to flag as well as get cramp in your legs you would stop and get help. You cannot choose to stop in labour but you can get help and whatever help you need is OK.\n\n\u2022 hang on to unrealistic expectations of a pain-free labour, a painful labour or any particular type of labour. Disappointment can increase tension and intensify pain.\n\nBackache Labour\n\nThis is caused by the baby being in the posterior position, its back against your back. The pains are doubled, felt in the uterus, the abdomen and in the back.\n\nSeveral remedies are indicated for back pain in labour, including Causticum, Gelsemium, Kali carbonicum, Nux vomica, Petroleum and Pulsatilla. It is the individual symptoms of the contractions as well as any general and emotional symptoms that will guide you to one of the above remedies. If another remedy is indicated on the general symptoms and the emotional state then consider it instead.\n\nDo\n\n\u2022 follow the guidelines for pain in labour (here).\n\n\u2022 find a position that eases the pain without taking the pressure of the baby's head off your cervix. Kneeling on all fours, with pillows under your knees, or draping yourself over a bean bag are two of the most comfortable positions.\n\nDon't\n\n\u2022 lie down on your back as this will increase the pressure on your spine and take it away from your cervix.\n\nFalse Labour\n\nEstablished labour is diagnosed if the cervix is 3 centimetres or more dilated. If after four hours at 3 centimetres there is no further dilation then it is assumed that true labour has not yet begun.\n\nBraxton Hicks contractions (see here) can develop into what feels like labour and continue for several days or even weeks, coming every 10\u201320 minutes and feeling as if they might be the real thing. They are frustrating because nothing \u2013 or next to nothing \u2013 actually happens.\n\nRemedies for getting labour going include Belladonna, Calcarea carbonica, Caulophyllum, Chamomilla, Cimicifuga, Gelsemium, Kali carbonicum, Natrum muriaticum, Nux vomica, Opium, Pulsatilla or Sepia. Take into account the individual symptoms of the contractions, any general symptoms and, of course, the emotional state to guide you to one of the above remedies. If another remedy is indicated on the general and emotional state then try it instead. During labour emotions run deep and strong and helping a woman 'feel' good in herself is as least half the battle.\n\nDo\n\n\u2022 carry on life as usual.\n\n\u2022 go for long, brisk walks, round several blocks, or take longer walks of at least a mile, or do something special for yourself like painting your toenails, going to the hairdresser or on an impromptu outing with any other children (not too far away).\n\n\u2022 dance or engage in vigorous exercise.\n\n\u2022 encourage labour to 'get going' by kissing, stimulating the nipples, masturbating or having sexual intercourse. Kissing on the lips is very relaxing, nipple stimulation and masturbation can both encourage the onset of labour. Privacy is an issue in a hospital labour ward but kissing, nipple stimulation and masturbation can be successfully done in the bath or under a sheet, by the labouring woman or her partner.\n\n\u2022 make love if you are still at home and your waters haven't broken. Semen contains a significant quantity of the hormone that helps stimulate labour (prostaglandin).\n\n\u2022 encourage it to 'go away' if it is night-time or if true labour is plainly not going to start. Have a warm bath, go to bed and try to sleep. This is the one time when you can either take a couple of paracetamol with a hot drink or drink a tot of whisky or glass of wine. It is better to have a small quantity of alcohol now, sleep and have a manageable labour, than to end up with a lot of medication in labour because you are exhausted.\n\n\u2022 consider going into hospital rather than carrying on alone if you are at home, you can't sleep or rest, and are finding that you are becoming increasingly exhausted. You may benefit from having help in establishing your labour.\n\n\u2022 talk about your fears for the labour with your partner or your midwife.\n\nDon't\n\n\u2022 panic.\n\n\u2022 give up!\n\nFast Labour\n\nA very fast labour can be frightening. It can feel like a roller coaster out of control with contractions coming one after another without a break in between. After a fast, violent labour some women report feeling shell-shocked: there's no time to assimilate or integrate anything. Typically, a second or third birth will be fast if the first or second was fairly quick \u2013 but, as always, there are no rules here...\n\nAconite is nearly always indicated during a fast labour. Even though everyone else knows it's not true women know they are going to die and say so. It should be taken frequently, every 5\u201310 minutes, until some sort of shelter is reached from the storm: it will slow down the contractions and make them manageable. Rescue Remedy in the drinking water or massaged into any part of the body will also help.\n\nDo\n\n\u2022 get on all fours as that will help to slow things down a little and help with the contractions, or try the 'frog' position: kneel down with knees apart to accommodate your belly and lean forward until your head rests on the ground, arms outstretched or folded under your head. This will take the pressure off your cervix.\n\n\u2022 breathe as slowly and deeply as you can.\n\n\u2022 go with the flow: imagine you are on a boat on a stormy sea and try and ride the waves \u2013 or create a different visualisation that works for you.\n\n\u2022 remember to urinate. You don't have to pee on the toilet if there's no time; squat over a bucket, a washing-up bowl, potty or bedpan.\n\nDon't\n\n\u2022 panic and hyperventilate. If you find yourself getting dizzy, with tingling in your fingers, then breathe into your cupped hands (to take in more carbon dioxide \u2013 see also here).\n\nLate Labour\n\nA high percentage of first babies are late, according to dates, so it is not worth worrying about. Only a tiny percentage of babies arrive on time \u2013 a normal pregnancy can range from 240 to 300 days. You can negotiate with your doctor or midwife to go up to two weeks over before they begin to suggest induction. Some women seem to have longer pregnancies than others.\n\nA close relationship develops between mother and baby from the outset of pregnancy and if one or other is frightened or anxious then this can delay the onset of labour. If your baby is overdue by dates, and you are absolutely sure your dates are correct, ask yourself whether you are hanging on to your baby because of one of the following: this is your last planned pregnancy; you are scared \u2013 it's either the first one or the previous one was a bad experience; you're anxious about the responsibility of parenting, of your changing role; you don't want to stop being pregnant because you've enjoyed it so much; you don't want to lose your pregnant 'status' and stop receiving the special attention that pregnancy has conferred upon you; you had to work right up until the end of your pregnancy and you need some time to rest and relax.\n\nSee Inductions\/Breaking the Waters, here. Prescribe on fear or anxiety \u2013 some women need Argentum nitricum, Gelsemium or Lycopodium before their due date (or after it if they are still pregnant) because they approach it rather like an important exam and get the all-too-familiar exam nerves. Homeopathic help at this stage relaxes, enabling the body to go into labour if the baby is also ready. Take up to six doses of Caulophyllum 30 over the course of a day if you are late and not frightened. If the baby is ready to be born it will help to establish labour. You can repeat it 2 days later.\n\nDo\n\n\u2022 deal with any of the above that apply to you. Talk about your feelings with your partner and your midwife; spend some time saying goodbye to your pregnancy and inviting your baby to come out; take time to do something nice and relaxing.\n\n\u2022 ask for two weeks' grace after your due date \u2013 unless, of course, there are symptoms indicating that an induction is advisable.\n\n\u2022 follow the Dos for False Labour (see here).\n\n\u2022 contemplate a castor oil induction if you are desperate, bearing in mind that you will almost certainly have an unpleasant attack of diarrhoea that may or may not precipitate labour. Take two tablespoons of castor oil in orange juice. Half an hour later take one tablespoon of oil with juice and repeat this again, half an hour later. Relax in a bath (as long as your waters haven't broken). Do check, with your doctor and\/or your midwife, that it is OK for you to try this.\n\n\u2022 remember \u2013 you and your baby are going into labour together and you both need to be ready.\n\nDon't\n\n\u2022 be panicked into a hospital induction without taking a second opinion if you feel instinctively that it isn't necessary.\n\nPremature Labour\n\nThe main danger with going into labour early is that your baby's lungs will be under-developed and it may have trouble breathing after the birth. Babies of 34 weeks' gestation or over are usually OK.\n\nThere are many causes for premature labour, the most commonly known of which is multiple births of two or more babies. Your doctor will be on the lookout for other complications later in your pregnancy that may lead to a premature labour so you don't need to worry about it. There's not a lot you can do homeopathically, as far as self-prescribing is concerned. Take Nux vomica if there is no apparent reason for labour starting, or Opium if labour started after a shock, and follow the advice of your midwife and\/or doctor.\n\nDo\n\n\u2022 go to bed if you are in premature labour \u2013 and stay there.\n\n\u2022 take a couple of stiff drinks (vodka or whisky). It's better to have a drink now and stop this process than end up with an early labour and a premature baby that needs intensive care. Alcohol relaxes the muscles of the uterus and stops the action of oxytocin, the hormone responsible for getting labour going.\n\n\u2022 spend 20 minutes several times daily doing the postural tilting exercise (see here).\n\nSlow Labour\n\nThe early part of labour, where the cervix dilates, can seem endless, especially with first babies. If labour slows down or goes on for ever and you are 3 centimetres or less dilated, then:\n\nDo\n\n\u2022 remind yourself of the anatomy and physiology of what is happening. You, your body and your baby are working together for this birth and your body has been designed to give birth with little assistance. The contractions, plus the pressure of your baby's head on the cervix, will cause your cervix to open and the muscles of your uterus to thicken and shorten, which pulls them up and automatically pushes the baby out. You have to do very little apart from surrendering yourself to what is happening.\n\n\u2022 rest to build up your energy reserves and sleep if you can between contractions.\n\n\u2022 have a glass of wine to help you sleep and recharge the batteries.\n\n\u2022 talk through any unresolved conflicts with your partner. Express any unexpected emotions.\n\n\u2022 take a long walk outside.\n\n\u2022 organise a change of scenery: settle yourself in a different room for a while, open the curtains if they are closed, or close them if they are open.\n\n\u2022 ask new people to leave if your labour slowed down after their arrival.\n\nIf labour slows down when your cervix is 4\u20135 centimetres dilated:\n\nDo\n\n\u2022 try a different position: sit cross-legged for a while, letting your shoulders and hands loosen. Your partner can sit behind you supporting your back with their back; or squat for a short time; or walk about for a bit.\n\n\u2022 be still. Slow down if you have been very active so far and retreat inside yourself.\n\n\u2022 focus on the rhythm of your breathing rather than actively 'doing' the breathing method you have learnt. Allow your body to go with the flow of your breath.\n\n\u2022 urinate every hour.\n\n\u2022 talk through your fears with your midwife or partner. Some women can keep 'hard' labour at bay because of a fear of losing control. Fear can cause the muscles to tighten against the body's efforts to expand.\n\n\u2022 allow yourself to melt and soften all over your body \u2013 your uterus, your back, your legs, your vagina and your mouth.\n\n\u2022 ask for help \u2013 any help will do \u2013 some women need to reach out in labour to be able to let go and allow their bodies to work effectively.\n\n\u2022 make sure you are eating and drinking.\n\nIf labour slows down after the cervix has dilated 6 centimetres, your uterus may be exhausted. You should follow your midwife's recommendations and your own instincts about what you need to proceed. Take Arnica alternating with Kali phosphoricum frequently (every 5\u201310 minutes) to help restore your exhausted muscles and nerves; or Sepia if indicated.\n\nDo\n\n\u2022 rest between contractions: give your body permission to slow down and recuperate.\n\n\u2022 ask your partner to massage or hold you.\n\n\u2022 drink and\/or eat something nutritious.\n\nDon't\n\n\u2022 worry if labour slows down from time to time if it is generally progressing well. It can even stop for short periods of time while you and your body integrate what is happening and gather your forces to move on. It can commonly slow down when the cervix has reached 4, 6, 7 or 9 centimetres in dilation.\n\nMEDICAL INTERVENTIONS\n\nSee here.\n\nNAUSEA\n\nNausea and vomiting, relatively common in labour, especially towards the end of the first stage, can be caused by tiredness, stress leading up to the onset of labour, hormonal changes, fear, low blood sugar and the side effects of pain-relieving drugs.\n\nRetching or vomiting can help by relaxing some of the muscles of the stomach and abdomen and vomiting can also be nature's way of emptying the stomach before the second stage (the birth).\n\nIf the nausea is constant and distressing, or the retching painful, or if the vomiting is preventing you from keeping any food or drink down, it is important to try to deal with it.\n\nPut Rescue Remedy into drinks to combat panicky feelings and rub a few drops of it (neat) into the forehead, the back of the neck and the abdomen.\n\nIpecacuanha is indicated if the nausea is constant and unremitting; Arsenicum if it is accompanied by fear and anxiety; Phosphorus if the vomiting occurs a short while, but not immediately, after eating or drinking anything; Pulsatilla if it is accompanied by whingeing and thirstlessness. If none of these remedies is indicated, or if the indicated one doesn't work, look up the symptoms in the Repertory to find a remedy that fits the whole picture, perhaps Cocculus or Tabacum if faintness or dizziness is a problem.\n\nDo\n\n\u2022 drink frequent small quantities of anything you fancy.\n\n\u2022 take small spoonfuls of honey or sugar water if you can't take anything else.\n\n\u2022 breathe into the nausea (see Breathing).\n\n\u2022 relax between contractions.\n\nPAIN RELIEF\n\nSee here for a synopsis of drugs and techniques.\n\nPREMATURE LABOUR\n\nSee Labour.\n\nRETAINED PLACENTA\n\nAfter the baby has been delivered, the placenta can take up to three hours to separate from the wall of the uterus. During that time the last of the amniotic fluid drains away and there is some blood loss as the placenta separates, leaving a raw 'wound' which takes up to six weeks to heal (see Lochia). The uterus will carry on contracting (although it may 'rest' for a while) after the baby has been born to encourage the placenta to separate, which you can promote by putting your baby to the breast. The sucking will stimulate contractions to start up again, or help make them stronger. In the process of separation, however, the placenta can tear the wall of the uterus more seriously and cause a haemorrhage (see here), a serious cause for concern: in the past women regularly bled to death after childbirth if this happened.\n\nThe drug syntometrine is often administered routinely to deliver the placenta as fast as possible. It is a life-saver as it causes the uterus to contract very fast and prevents the possibility of haemorrhage but this third stage of labour becomes a more rushed affair which can spoil the beautiful atmosphere of a calm delivery. Most placentas deliver themselves within half an hour of the birth, with others taking up to two hours. Some midwives and doctors are happy to wait for up to two hours if all is going well and there is no bleeding. The side effects of syntometrine are not well recorded, but include nausea, after pains, headaches, jaundice in the baby, three-month colic and emotional disorders. It can apparently pass through to the breastmilk and cause colic in babies whose digestive tracts are sensitive or immature.\n\nMany remedies are indicated for this condition. It is important that you take the whole picture into account: give Cimicifuga if there is shaking, trembling, soreness and a retained placenta; Nux vomica if there is great irritability; or Sepia if there is a great sagging as if all the energy went down the 'plughole'.\n\nDo\n\n\u2022 be patient after the birth \u2013 it's not all over yet!\n\n\u2022 relax and enjoy the post-natal period to the full if all has gone well and you are able to. It's a birthday!\n\n\u2022 ask for a delay in the administration of syntometrine of up to one hour (at least half an hour) with the proviso that if you start to haemorrhage, you will accept it immediately (see here).\n\n\u2022 remain in an upright position to encourage the placenta to separate, to make expulsion easier and to reduce the risk of haemorrhaging.\n\n\u2022 put your baby to the breast immediately after the birth as the sucking will stimulate your uterus to contract. If your baby doesn't want to suck, stimulate your nipples yourself and relax.\n\n\u2022 have it written in your notes that you are not to be given syntometrine for the time that you have negotiated and remind each midwife of this. Write it in your birth plan.\n\n\u2022 watch closely after the birth that it isn't 'accidentally' given to you anyway. You'll feel numb for a while after the baby has been born and are unlikely to notice the injection, which is given in the thigh.\n\nDon't\n\n\u2022 give up now!\n\nRETCHING\/VOMITING\n\nSee Nausea.\n\nRETENTION OF URINE\n\nImmediately after the birth it is common to experience difficulty in urinating, at least for a short time, because of the combined soreness and numbness in the whole pelvic area, possibly accompanied by a feeling that those muscles will never work again! If you go for too long without peeing then your doctor or midwife will insert a catheter into your urethra and draw off any urine to prevent your bladder from becoming over-stretched. This is an uncomfortable and occasionally painful procedure and is worth trying to avoid.\n\nTake Arsenicum 30 every 10 minutes if you can't pee and you're feeling scared, Arnica 30 if you are sore and bruised but feeling OK, Staphysagria 30 if you feel disappointed or resentful, if you have had a forceps delivery and\/or your body feels all 'beaten up', or Pulsatilla 30 if you feel very sorry for yourself, weepy, clingy and thoroughly miserable.\n\nDo\n\n\u2022 ask for a tap to be run as you try to pee, whether on a bedpan or on the toilet.\n\n\u2022 'bottle wash' on the toilet if you are mobile (see here).\n\n\u2022 pee in a warm bath if you are mobile enough to have a bath.\n\nSLOW LABOUR\n\nSee Labour.\n\nSYNTOMETRINE\n\nSee Retained Placenta.\n\nTREMBLING\n\nIt is not uncommon for women to have an attack of 'the shakes' before, during or after labour. It may be uncontrollable and you won't necessarily feel cold. It is generally your body's way of releasing tension and doesn't usually last long. It can accompany other symptoms, a slow, fast or backache labour, where a lot of tension is building up, or fear.\n\nTake Cimicifuga 30, one dose every 10\u201320 minutes for up to 6 doses if the trembling comes on its own with no other symptoms and hasn't passed of its own accord after a few minutes. Take Gelsemium 30 if the trembling accompanies a backache labour or a terrible exhaustion. If the shaking is accompanied by strong feelings that point to another remedy then take that instead.\n\nDo\n\n\u2022 rest and allow your body to express itself with the trembling.\n\n\u2022 breathe slowly and deeply.\n\n\u2022 ask someone to massage your back.\n\n\u2022 find a comfortable position in which to sit or lie where your limbs are well supported.\n\n\u2022 drink or eat something with sugar or honey in it.\n\n\u2022 follow the dos and don'ts here if you are also exhausted.\n\n\u2022 put on some warmer clothes, a cardigan, some woolly socks, a heated towel or blanket, if you feel at all cold.\n\nDon't\n\n\u2022 tense up against the movements your body is making. Go with them, let them develop if you can as they may be part of a process whereby your body is letting go of deeper tensions and may therefore be helpful to you.\n\nTHE POST-NATAL PERIOD\n\n* * *\n\nYOUR BODY IN THE POST-NATAL PERIOD\n\n* * *\n\nAfter birth your body goes through a series of physical, emotional and hormonal changes, some of which relate directly to the birth and some of which are part of the post-natal adjustment to not being pregnant. They include:\n\n\u2022 complaints from the birth itself, which take variable lengths of time to heal, for example, piles (see here), episiotomy (see here), general bruising, pain as the body heals (see here), numbness in the genital region with difficulty in urinating (see here), exhaustion (see here), involuntary urination (see here) and prolapse (see here).\n\n\u2022 afterpains, as your uterus contracts back to its pre-pregnant size (see here).\n\n\u2022 the production of colostrum, which began in pregnancy and continues for 2\u20133 days after the birth to provide the antibody-rich fluid which boosts your baby's immunity, followed by the swelling of your breasts as your milk comes in on the third to fifth day after the birth (see Engorged Breasts). Your baby's sucking stimulates the hormones which cause the milk glands to produce milk. If you don't breastfeed then your breasts will return to their pre-pregnant size.\n\n\u2022 an increase in urination as your body gets rid of the extra fluid accumulated during the pregnancy.\n\n\u2022 a rounded abdomen which can feel flabby for a while after the birth until the muscles regain their former tightness.\n\n\u2022 your vulva and vagina may have changed their shape through the process of birth with the stretching and especially if you have torn or had an episiotomy. If you have had a Caesarean then your genitals will be intact but you'll have an abdominal scar to get used to.\n\n\u2022 a feeling of tightness in the joints as the hormones that helped them soften during pregnancy (in preparation for birth) adjust back to normal.\n\n\u2022 hair loss, again because of hormone changes (see here).\n\n\u2022 skin changes that were caused by the extra hormones during the pregnancy (discolouration on the face, nipples and abdomen) will gradually fade.\n\n\u2022 stretch marks will fade but not disappear.\n\n\u2022 varicose veins in the vulva will have disappeared during the birth.\n\n\u2022 varicose veins in the legs will take a few months to settle down as the hormones and the fluid levels adjust and as your weight drops. They may improve and ache less but they won't disappear.\n\n\u2022 ankles and feet will take a few days to return to normal if they swelled during your pregnancy.\n\n\u2022 you'll bleed after the birth as your uterus contracts, sheds its lining and heals (see Lochia) and this will turn to a brownish discharge before stopping.\n\n\u2022 complaints of pregnancy, such as heartburn and breathlessness, due to the extra weight and the size of the baby in the abdomen, will disappear.\n\n\u2022 emotions can be extreme partly because of the nature of the experience itself but also because of the hormonal changes. Euphoria often gives way to a day or two of the 'blues' on about the third to fifth day after the birth (see here).\n\n* * *\n\nCOMPLAINTS \u2013 MOTHER\n\n* * *\n\nAfter birth, so much is happening and so fast that it can be difficult even to think of what remedy to take and when. You may find that you have after pains, piles and are sore around any stitches. Where do you start?\n\nYou may find that one remedy fits your whole picture in which case take it until you feel better. More often, however, different pictures emerge on a daily (and sometimes hourly!) basis. The guiding rule is to deal first with whatever is most distressing. If two remedies are strongly indicated, you can alternate them. If you find that you are taking one remedy after another and they aren't helping, consult a professional homeopath who will be able to prescribe on the whole picture.\n\nAFTERPAINS\n\nThe uterus continues to contract after the birth until it has regained its pre-pregnant size. This process, involution, can take up to two months. Your GP will want to see you when the baby is about six weeks old to check, among other things, that the uterus is contracting as it should.\n\nAfterpains may resemble mild period pains or be surprisingly painful. With a first baby they often pass unnoticed; with subsequent births, however, the afterpains are worse each time as the uterus has to work harder to contract back to normal. If you have had a large baby or twins, it will have stretched more and will take longer to contract. The pain is usually worse during breastfeeding as the baby's sucking stimulates the contractions, but they can come on at other times and may be noticeable for up to a week after the birth. They can be the last straw after a difficult birth and a serious block to successful breastfeeding.\n\nYou may be taking Arnica for bruising which will also help with afterpains but if it doesn't, take Magnesia phosphorica just before and during breastfeeding. If neither helps, you will need to work out a remedy specifically for you and the pains. You may be feeling weepy, in which case consider Pulsatilla. Other remedies indicated are Chamomilla, Cimicifuga, Cuprum, Rhus toxicodendron, Sabina, Silica, or Secale. Make time to differentiate between these remedies to find the one that fits your picture and take it instead of Arnica every two hours for one day, stopping on improvement and going back to Arnica, if necessary, once the afterpains have eased.\n\nDo\n\n\u2022 breathe through them or move your body as you did in labour to help ease the pain.\n\n\u2022 make sure you are comfortable when you breastfeed (see here).\n\nDon't\n\n\u2022 tense up against them as this can increase the pain.\n\nBREASTFEEDING PROBLEMS\n\nBlocked Duct (mastitis\/breast abscess)\n\nPrompt treatment can clear up a blocked duct and prevent a more serious breast infection (mastitis) or abscess from developing. The first signs are soreness and a lump in the breast, with or without redness on the skin above the lump. Fever is a sign that you have developed mastitis and need to take it seriously. Homeopathic treatment is highly effective at dealing with breast infections and abscesses and is a safer option than antibiotics but if self-prescribing doesn't help quickly seek the advice of a professional homeopath if you don't want to take orthodox medicines.\n\nDo\n\n\u2022 see Breastfeeding.\n\n\u2022 check that your bra isn't cutting in and causing the blocked duct.\n\n\u2022 rest more if you suspect a duct may be blocked. They can be a sign that you are overdoing it.\n\n\u2022 apply hot and cold compresses alternately to the sore breast every 2\u20134 hours for 5\u201310 minutes each time. Dip a flannel in a basin of very hot water, wring it out and lay it over the breast until the flannel cools. Replace with a flannel that has been dipped in ice-cold water and wrung out.\n\n\u2022 increase your fluid intake.\n\n\u2022 breastfeed more often and in a variety of positions to drain the milk from all the ducts.\n\n\u2022 massage your breasts as the baby feeds (especially if and when you feel the milk let down, a prickling sensation). Massage gently from the highest or lowest point to the nipple to encourage the duct to clear: if the blocked duct is in the side of your breast massage from your armpit, if it is in the top massage from your collarbone, etc.\n\n\u2022 use a breast pump if your baby isn't feeding much and your breasts are engorged.\n\n\u2022 express or breastfeed in the bath to encourage your milk to flow strongly.\n\n\u2022 pull a plastic fine-tooth comb through a bar of soap and 'comb' your breasts very gently from the highest or lowest point to the nipple \u2013 some women find this more effective than massaging. You can 'comb' or massage your breasts in between feeds as well as during feeds.\n\n\u2022 position your baby on the breast so that her chin points across from the blocked duct.\n\n\u2022 try some vigorous arm-swinging exercises to encourage blood to pump into the area.\n\n\u2022 breastfeed from the affected breast first at each feed until it starts to clear and then alternate. Make sure that the affected side is fully emptied at each feed.\n\n\u2022 go to bed and stay there if the duct doesn't clear up with the above self-help measures and you develop a fever. Call around and get help with your chores. A day or two in bed now will prevent a more serious abscess developing.\n\nDon't\n\n\u2022 stop breastfeeding.\n\n\u2022 automatically take antibiotics.\n\nSeek help if\n\n\u2022 self-prescribing hasn't helped within 24\u201348 hours.\n\n\u2022 you develop a fever that doesn't respond to self-prescribing within 12\u201324 hours.\n\n\u2022 the glands in your armpits are swollen: you may have developed a more serious infection.\n\nEngorged Breasts\n\nThis common condition often occurs within a few days of the birth when the milk first comes in. Homeopathic treatment can alleviate the pain and discomfort and help to stabilise the milk supply. The two main remedies for this condition are Belladonna and Bryonia. Belladonna is more restless, and the breasts may have red streaks on them. Bryonia pains are much worse for movement; the breasts are usually pale. You'll need to stop taking Arnica or any other post-natal remedies while you self-prescribe on the engorgement.\n\nDo\n\n\u2022 reassure yourself that this is a temporary condition. It won't last.\n\n\u2022 encourage your breasts to soften so that your baby can latch on easily by: expressing some milk either by hand or with a pump (often easier if done in a warm bath); applying hot flannels to your breasts just before a feed; stroking the breast away from the nipple, lightly with your fingertips, to make the nipple more accessible.\n\n\u2022 breastfeed frequently \u2013 wake the baby to feed her or feed her while she is asleep \u2013 until the engorgement has passed.\n\n\u2022 apply ice-cold flannels to the breasts after a feed to reduce the blood supply.\n\nDon't\n\n\u2022 give up!\n\nSeek help if\n\n\u2022 you are engorged and have a fever. You may have mastitis (see Blocked Duct).\n\nInverted Nipples\n\nSome women's nipples remain flat or even inverted after the birth. Breastfeeding may be more difficult in the beginning but not impossible. Sarsaparilla or Silica may help the nipples to come out.\n\nDo\n\n\u2022 get support from a breastfeeding counsellor if you are determined to continue.\n\n\u2022 remind yourself that you are breastfeeding \u2013 not nipple feeding!\n\n\u2022 bring the nipple out before a feed by: using a hand or electric breast pump immediately before a feed; placing a pad soaked in cold or ice-cold water over the nipple; rubbing an ice cube gently around the nipple.\n\n\u2022 try using a nipple shield when feeding. Ask a breastfeeding counsellor or your health visitor or midwife to help you with this.\n\nMastitis\n\nSee Blocked Duct.\n\nPain on Feeding\n\nSome women suffer from pain in the breasts unassociated with a blocked duct, breast abscess, sore nipples, or the let-down reflex. This distressing condition can make it difficult to establish breastfeeding. Seek professional help if self-prescribing doesn't help within a few days.\n\nDo\n\n\u2022 be aware of your breathing while you are breastfeeding and use any breathing techniques you learnt for labour.\n\nSore\/Cracked Nipples\n\nThe cracking of nipples has nothing to do with their colour but reflects their sensitivity. Relatively insensitive nipples \u2013 those that do not become particularly aroused during love-making and are not sore from brushing against rough clothing \u2013 are less likely to become sore and cracked with breastfeeding. If your nipples are sensitive study the general breastfeeding guidelines especially carefully.\n\nSore nipples can crack so look after them. Some of the points below will help prevent soreness developing and also help heal if it does.\n\nPhytolacca, Borax, Silica, Castor equi and Sulphur are all indicated for this condition and the right one for you should help quickly. If not, consult your homeopath, especially if you are in great pain.\n\nDo\n\n\u2022 see Breastfeeding.\n\n\u2022 let your baby suck for as long as she wants at a feed and feed her when she first cries rather than wait until she is screaming. This will avoid her latching on desperately and with great force which can injure the nipples.\n\n\u2022 settle yourself and then bring your baby to the breast.\n\n\u2022 offer the breast that is least sore first, as the baby sucks hardest at the beginning of a feed.\n\n\u2022 distract yourself as you did to deal with pain in labour if your nipples feel sore.\n\n\u2022 listen to music, breathe easily, use your relaxation exercises as you begin to feed, imagine the milk flowing easily into and out of your breasts.\n\n\u2022 feed your baby in the bath if you are sore, engorged and desperate and a hot bath is soothing for you. If no one is available to help you get out, put a few pillows beside the bath to lay the baby on before you try to get out yourself. Take great care when getting in and out of the bath.\n\n\u2022 limit the amount of time your baby sucks if your nipples are sore as long as she is taking milk (gulping or swallowing), usually about 10 minutes on each side, and after that encourage her to find her thumb, or let her suck a pacifier or your little finger.\n\n\u2022 try feeding your baby when she is sleepy or even asleep as her suck may be gentler then.\n\n\u2022 switch sides frequently to encourage your milk to let down several times, especially if you suspect that the baby is sucking voraciously because she is hungry and you are not producing enough (see also here).\n\n\u2022 try a nipple shield to protect the skin from direct sucking. Feeds take longer if you use one.\n\n\u2022 spray ice-cold water on your nipple before a feed to numb it enough to allow your baby to latch on, or rub your nipple and areola with an ice cube. Don't do this if you are super-sensitive to cold!\n\n\u2022 feed in different positions to distribute the sucking pressure to all areas of the areola.\n\n\u2022 express a little milk first if your breasts are very full so that your baby can latch on more easily.\n\n\u2022 keep the nipples dry by: exposing them to the air after a feed; drying them with a hair-drier; using disposable breastpads without plastic backing between feeds (change them frequently if your nipples 'leak', or put a one-way diaper liner between nipple and pad, or try a piece of toilet paper or a cotton handkerchief instead of pads); wearing a loose shirt without a bra at home unless your breasts are large and heavy; wearing plastic tea strainers with the handles cut off inside your bra to let the air circulate without anything touching the nipples (only do this for short periods during the day).\n\n\u2022 moisten a bra or pad which has stuck to the nipple with water before removing.\n\n\u2022 expose your nipples to sunlight or sit with your breasts uncovered a foot (30 cms) away from a 40-watt light bulb for a few minutes four or five times a day.\n\n\u2022 stop using all your usual creams and sprays as you may be allergic to one of the ingredients.\n\n\u2022 try using a cream if you have not been using one (see External Materia Medica and Repertory). Experiment until you find one that works and stick to it. Calendula or Rescue Remedy creams are the most helpful. Rub in after a feed. Otherwise rub a little expressed milk into your nipples.\n\n\u2022 stop wearing a bra at night to allow your nipples to heal in the air.\n\n\u2022 stop feeding for 48 hours to rest your breast if all else fails. Express the milk by hand or pump and give it to your baby in a bottle. Resume breastfeeding once your nipples have healed.\n\nDon't\n\n\u2022 pull your nipple out of the baby's mouth (see here).\n\n\u2022 use creams other than those suggested in the External Materia Medica.\n\n\u2022 use a nipple cream with chamomile as it can give some babies colic.\n\n\u2022 wash your nipples more often than once a day: too much washing will dry the skin and make it sore.\n\n\u2022 use any soap on your breasts.\n\n\u2022 wash your bras in biological soap powders.\n\n\u2022 wear nylon bras.\n\n\u2022 press the top of your breast away from the baby's nose. This will direct your nipple towards the top of its mouth where it will get sore and bruised. Reposition or place another pillow under the baby or hold your breast from underneath with your whole hand.\n\n\u2022 give up!\n\nSeek help if\n\n\u2022 there is bleeding or severe pain.\n\nToo Little Milk\n\nGetting the hang of doing breastfeeding can take a little time and patience initially but if your baby isn't gaining weight satisfactorily or you know that your milk supply is down, try some of the following self-help ideas. If your only symptom is not having enough milk, take a short course of Urtica urens. Otherwise repertorise carefully before self-prescribing.\n\nDo\n\n\u2022 see Breastfeeding.\n\n\u2022 have a nutritious snack with every feed \u2013 you will need about 600 extra calories a day to breastfeed.\n\n\u2022 feed your baby often to build up your supply.\n\n\u2022 drink plenty of fluids. You will need about 2 pints more per day than you normally drink.\n\n\u2022 cut out caffeine (tea, coffee, Coca-Cola and chocolate) and tobacco as it can reduce your milk supply by overstimulating your nervous system. See here for alternative ideas for drinks.\n\n\u2022 see if your baby is producing roughly the same number of wet diapers per day as before your milk supply dropped and try to build up your supply, or give extra fluids, if there are fewer than usual. There should be roughly 8\u201310 wet diapers in a 24-hour period with breastmilk alone.\n\n\u2022 try a complete rest in case you have been overdoing it. Go to bed with the baby and eat and sleep for a day or two (both of you!).\n\n\u2022 use your relaxation exercises before and during a feed \u2013 and at other times if you are feeling tense and anxious. The more you worry, the less easy it will be to build up your supply.\n\n\u2022 remember that babies have growth spurts \u2013 at around three weeks, five to six weeks, three months and six months \u2013 when they will want to feed more. Increase your food and fluid intake and resting time for 24\u201348 hours.\n\n\u2022 switch your baby from one breast to the other often during a feed. This will encourage your milk to let down several times, which will in turn increase your supply and stimulate a baby with a weak suck to suck more strongly.\n\n\u2022 use a pump to increase your supply after a feed if your baby sucks infrequently or has a weak suck.\n\n\u2022 encourage the milk to let down before a feed by using warm compresses on your breasts beforehand, having a warm bath, thinking 'baby' or 'milk', or imagining milk flowing into your breasts. Breathe!\n\n\u2022 listen to music or the radio, watch TV or read a favourite book, and take your time.\n\n\u2022 start each feed on alternate sides.\n\n\u2022 feed in a quiet room if the baby fusses or doesn't seem interested, or if you are shy feeding in front of others. Your feelings will inhibit the let-down.\n\n\u2022 make sure you are not interrupted. Unplug the phone or switch on the answering-machine, pin a note on the front door that says something like 'Please do not disturb, baby resting \u2013 call back later. Thank you'.\n\n\u2022 trick your baby into feeding from a breast that he or she goes off: move the baby in the same position across your lap to your breast, or use the football hold (see here) with the baby under the arm on the side he or she is feeding on.\n\n\u2022 give a bottle of formula at night if you know that your busy day has caused a drop in your supply which you are not going to make up. Use a teat with a very small hole that takes as much energy to suck from as the breast and position it carefully, as you would your own breast, so that the baby sucks on the bottle, mouth wide open and the teat at the back of the mouth. (See also here.)\n\n\u2022 leave the baby with your partner, a tried and trusted relative, friend or childminder and a bottle (of expressed milk or formula), and go and do something completely different. If your life is nothing but breastfeeding and baby, plus your other children, you can quickly become burnt out \u2013 a break really is as good as a rest.\n\n\u2022 deal with any emotional trauma as a shock or distress can affect your let-down reflex (see here). Talk to your partner, friends and your baby about what is happening \u2013 don't keep it to yourself.\n\n\u2022 ask yourself if you really want to breastfeed, or carry on breastfeeding.\n\n\u2022 check that your baby hasn't got thrush in her mouth.\n\n\u2022 wean your baby off a nipple shield if you have been using one as the nipple may not be receiving enough direct stimulation. Do this by smearing a little sterilising fluid on your nipple or starting to feed with the shield and removing it quickly once your baby has latched on, or cutting away the tip of the nipple shield very gradually over a number of feeds.\n\n\u2022 ignore unhelpful remarks from relatives or professionals who are obsessed with your baby's weight and whether he or she is getting enough milk. Your baby's weight gain is always going to vary and the charts in your clinic are only a guide. You will know if things aren't right.\n\nDon't\n\n\u2022 worry if you have small breasts. The amount of milk they produce is not related to their size.\n\n\u2022 worry if your milk looks thin or bluish. This is normal and healthy.\n\n\u2022 worry if your breasts become smaller a few weeks (or months) after the birth. This is a sign that your milk supply is well established, the milk letting down when the baby feeds; even though your breasts may feel empty at the end of a feed there is always some milk in them.\n\n\u2022 time feeds. Let your baby decide when the feed is over, even if it seems to take for ever, especially with the first breast.\n\n\u2022 use soaps, perfumes or sterilising fluids on your breasts as it may make them smell or taste unpleasant to your baby.\n\n\u2022 worry if your baby seems to go off your breast when you are menstruating or ovulating \u2013 some babies do temporarily.\n\nSeek help if\n\n\u2022 you know something is wrong and can't sort it out on your own.\n\n\u2022 you suspect that your baby is ill.\n\n\u2022 you are getting into a cycle of anxiety about breastfeeding and need a reassuring counsellor to help you break it.\n\n\u2022 you are on medication such as diuretics, the pill, antihistamines, laxatives or antibiotics as these may affect your milk supply. Your doctor may suggest an alternative or you may decide to take a break from the medication while you are breastfeeding.\n\n\u2022 your milk isn't letting down. You will be able to tell if this is so because your baby won't be swallowing or gulping during a feed. If you have taken steroids before your pregnancy this may have affected your let-down reflex: consult a professional homeopath or see your GP for a syntocinon spray to use until it gets going.\n\n\u2022 your baby is producing fewer wet nappies than usual.\n\n\u2022 your baby isn't gaining weight and stools are consistently green.\n\n\u2022 your baby is sleepy from drugs you had in labour and doesn't respond to the treatment outlined in this book (see here).\n\n\u2022 your milk smells or tastes different from usual.\n\nToo Much Milk\n\nWhen your milk comes in, it might spurt everywhere at the slightest opportunity and cause the baby to choke at the beginning of every feed, or you may be producing so much so quickly that the baby takes too much, becomes too full, feels uncomfortable and cries. He or she may gulp desperately, take in air and then get colic. This usually settles down within a few days but some women consistently produce more milk than their babies' need. You may suffer from a temporary overabundance of milk if you miss a feed. Try the suggestions given below and choose a remedy to help decrease your supply or, if it worries you, get in touch with your homeopath, breastfeeding counsellor or midwife.\n\nDo\n\n\u2022 be reassured that your milk supply will probably even out by the time your baby is around eight weeks old.\n\n\u2022 use the heel of your hand to press on the nipple of the opposite breast to the one your baby is feeding from so that when your milk lets down it doesn't spurt out; or hold a sterilised container under it to catch the 'overflow', which you can freeze, store in the fridge for use later, or donate to a milk bank.\n\n\u2022 express a little milk before a feed if it flows too fast and makes the baby choke and splutter.\n\n\u2022 position your baby to suck 'uphill': lie in bed with the baby on your tummy, supporting his or her forehead with your hand \u2013 gravity will help to slow down the gush of milk.\n\n\u2022 sit your baby upright to feed so that swallowing is easier.\n\n\u2022 let your baby feed on one breast at one feed and offer the other at the next. This will encourage your milk supply to diminish. Be careful, though, not to let your breasts become lumpy \u2013 express a little milk off the unused breast, if need be, at each feed.\n\n\u2022 encourage your baby to suck a pacifier or your finger if she wants to suck but isn't hungry. Continued sucking after she has fed will stimulate your supply.\n\n\u2022 use a nipple shield to restrict the flow of milk and decrease the supply.\n\n\u2022 talk reassuringly to your baby about what is happening. Your anxiety or panic is contagious and you may find that your calm voice will help you as well!\n\n\u2022 use cold compresses after a feed to slow down your milk production.\n\nDon't\n\n\u2022 cut down your fluid intake.\n\n\u2022 worry if your breasts become temporarily lopsided if you are only feeding from one breast at a time. They will even up once you are feeding from both breasts again.\n\nWeaning\n\nWhether you have decided to breastfeed or bottle-feed, when to wean is a matter for you and\/or your baby to decide. Some babies love to suck and hang on to the breast or the bottle for years, given the chance! Others can take it or leave it, losing interest from eight or nine months onwards making it perfectly clear that they wish to drink out of a cup like everybody else! Eating is a social activity and some babies cotton on to this early.\n\nWeaning follows fashions so it's useful to remember that around the world babies are weaned onto the local fare, be it raw fish or curry. However, it is worth giving a little thought and care to your baby's early solid diet to avoid illness such as gastro-enteritis or allergies. Packaged or tinned foods are usually high in carbohydrates (some are also high in sugar and salt) and low in vitamins and minerals, so use them as occasional convenience foods rather than for everyday eating. Your baby will want to eat what you eat so it is best to start as you mean to continue and introduce foods that, roughly speaking, you eat as well. Fresh foods are good for adults and babies alike: start your baby on fruit and vegetables (organic if possible \u2013 most supermarkets stock them now) and delay introducing wheat, eggs, meat, sugar and refined foods for as long as possible. You can give your baby a good start and the rest is up to him or her!\n\nTo help your milk dry up quickly, you can take Lac caninum once you have stopped feeding. You can help your baby adjust by giving a constitutional remedy to deal with any upset after weaning. Some children promptly go down with a cold or become clingy or angry or develop diarrhoea: take the whole picture into account when prescribing. Seek professional advice if your baby is having difficulty digesting 'real' food or if a food intolerance or allergy develops.\n\nDo\n\n\u2022 what is best for you, your baby and the rest of the family.\n\n\u2022 stop breastfeeding if you have had enough, whatever the reason.\n\n\u2022 breastfeed for as long as you are both happy to do so. It is unusual for both mother and baby to want to give up at the same time: sometimes it is the mother who is ready to give up first and sometimes the baby. Perhaps you have decided to go back to work and find expressing difficult \u2013 or you may simply have had enough.\n\n\u2022 wean your baby from the breast gradually if you can. Try cutting out: daytime feeds first and replacing them with something nice \u2013 a special drink, a cuddle, a new toy, or a distraction such as an outing; or the bedtime feed first by letting someone else put the baby to bed for a week or so; or night-time feeds first, in the knowledge that nights could be noisy until your baby is used to the new regime; or the last feed of the day first, as that is when your supply will be at its lowest, and give a bottle of expressed milk or formula. Substitute a bottle for another feed after a few days and carry on until your baby is only getting one breastfeed a day. This will help your milk supply to diminish gradually.\n\n\u2022 let your bottle-fed baby decide when to give it up. For some children it is their major comforter and they will drop it when they feel independent enough to do so.\n\n\u2022 be careful how you introduce solids once your baby is old enough to avoid allergic reactions and digestive problems. Start with small amounts of one food at a time. Go as slowly or quickly as your baby wants \u2013 some are cautious while others are instantly wildly enthusiastic. Try fruit or vegetables first and introduce the following foods carefully, one at a time and one a week, because of the allergy risk: yoghurt; milk; cheese; egg; wheat; fish; tomatoes; strawberries; tofu; meat; nuts (but not peanuts, and any others should be finely chopped or ground). Make sure your baby's food doesn't contain salt, sugar or spices and introduce cereals or fatty foods gradually: young babies don't need them before they are four to five months old if they are still on milk and these foods will encourage unnecessary weight gain. Don't forget to offer extra water to a baby who is having solid food, but you should avoid giving sweet drinks altogether, either juice or squash, however much they are diluted in the interests of preventing sugar addiction and early tooth problems. Give instead the fruit to eat and water to drink. Make mealtimes as relaxed as possible \u2013 times to chat and laugh and share news as well as eat. And never make your baby's eating dependent on your approval as this will have repercussions when the child grows up.\n\nDon't\n\n\u2022 feel guilty because you haven't let your baby decide when to stop breastfeeding.\n\nCONSTIPATION\n\nConstipation after childbirth is an occupational hazard. The vagina and perineum may be sore and swollen for several days, in spite of taking Arnica. You cannot imagine that you will ever go to the loo again because everything feels numb, or painful or both. And to top it all you may have piles and\/or stitches as well.\n\nYou will probably have emptied the contents of your bowels during the course of your labour and also if you have eaten little since it is unlikely that you will either want or need to do so again for several days. If you want to but can't, see also Constipation in Pregnancy, here for suggestions which may help.\n\nIf constipation does not clear quickly with self-prescribing, consult your GP or homeopath.\n\nDo\n\n\u2022 take heart if you become constipated \u2013 it will pass. As all your muscles and organs snap back to their usual places and the bruising heals, your bowels will regain their former health and strength.\n\n\u2022 support any stitched episiotomy or tear by holding a pad (sanitary or piece of felt) against it when you go to the loo to pass a stool, which will make it less painful and less likely that stitches will give way under the pressure.\n\nDETOXIFICATION FROM DRUGS TAKEN DURING OR POST-LABOUR\n\nPethidine or a general anaesthetic can make you and\/or your baby feel nauseous, 'high', intoxicated, or sleepy, or conversely tense, irritable and unable to sleep. Pethidine remains in the system of an adult for up to about five hours but it can affect an infant for as long as 24 hours.\n\nOpium, Chamomilla, Nux vomica or Phosphorus are all remedies that may be needed after labour to counteract the effect of anaesthetics or pain-relieving drugs. Choose between them noting your general and emotional symptoms. Secale will counteract the ill-effects of syntometrine (see here). Seek the help of a professional homeopath if you feel 'toxic' after the birth or if you have had medication which you think is still hanging around.\n\nDon't\n\n\u2022 automatically take laxatives, sleeping pills or painkillers without questioning carefully whether you need them and also whether they will affect your baby if you are breastfeeding.\n\nEMOTIONAL DISTRESS (Post-natal blues)\n\nThe feelings you experience after the birth may be the most intense you have ever encountered: overwhelming extremes and upsurges of emotion. A great unadulterated joy, a love beyond your wildest expectations, contentment, fulfilment, waves of happiness that wash over you whenever your baby smiles or whimpers... And babies are hugely entertaining \u2013 their antics make you laugh for hours on end. The funny expressions their faces make when they yawn and sneeze and wrinkle up against a bright light. The infectiousness of a baby's laugh is heartwarmingly delightful. Your own baby is delicious \u2013 the best thing since freshly baked brown bread!\n\nIt is common, too, normal even, to feel depressed after childbirth. It can pass by fleetingly as post-natal blues on about the third day after the birth when your milk comes in, or span a longer period while your hormones sort themselves out and you are beginning to make the emotional adjustments to having a new person in your life.\n\nA satisfying birth will leave you feeling strong and empowered, and strengthen your relationship; a difficult one followed by post-natal stress and a demanding baby may leave you feeling shipwrecked and drive a wedge between you and your partner that is difficult to deal with. There is a sensation of emptiness after the birth that can come as a relief and a shock. Women need to transform that shock by holding their babies and connecting with them. If they are not able to after a difficult birth depression may set in and last for weeks or even months.\n\nBabies come in all shapes and sizes: some are supremely needy and demanding, emotionally and physically. They can make you feel frustrated and angry, fearful, anxious, guilty, confused and depressed over how to cope with them \u2013 all entirely normal and part of the package deal that is parenthood. Women with demanding babies need lots of support, more than those whose babies are accommodating and easy to please. Even so-called 'good' babies make what seem like unreasonable demands at times when you are exhausted and need to rest.\n\nIt is vital that you remind yourself continually that all healthy relationships involve compromise so that everyone is satisfied, and one person isn't having all their needs met at the expense of another. Your own needs are terribly important: when they are met you can feel good about yourself and cope easily with the baby. It is not appropriate to meet all your baby's needs all the time. Sometimes you will come first, sometimes the baby, and at others your partner's or other children's needs will take priority.\n\nMy two children have instructed me in the arts of motherhood. They have taught me more than a million textbooks could... They continually take me on a guided tour of my limits, physical, emotional and intellectual, and having demonstrated these, revealing my inadequacy, they then proceed to love me despite everything.\n\n(Jane Price,\n\nMotherhood \u2013 What it does to your mind,\n\nPandora, 1988)\n\nSome women are particularly vulnerable to postnatal depression, especially those with poor self-esteem; those who had a difficult childhood themselves; single mothers; women whose partners are not supportive; very young women; those lacking a strong support network of family or friends; some older women who have concentrated on their career; any who suffered a lot of stress, such as moving house, a bereavement, an unexpected change in circumstances, during pregnancy and\/or shortly after the birth; women who have had an unexpected Caesarean or a disappointing or downright awful labour; those whose own feelings of distress during the pregnancy or around the birth were denied by being encouraged to suppress their emotions or to 'snap out of it'; and women who feel confused about feeling sad at the loss either of the pregnancy itself, or freedom and the carefree independent life they used to live, or their former pre-pregnant, tight-muscled body, or attention, especially if visitors focus only on the baby.\n\nPost-natal blues is often misdiagnosed as exhaustion or anaemia. If you are dragging yourself around and not sleeping well, or lacking in energy however much rest you get, you may be depressed. Common symptoms of post-natal depression are:\n\n\u2022 a creeping sense of hopelessness.\n\n\u2022 low self-confidence.\n\n\u2022 anxiety about the baby.\n\n\u2022 exhaustion accompanied by an inability to sleep; waking in the early morning, not necessarily to feed the baby, and being unable to sleep again.\n\n\u2022 mood swings.\n\n\u2022 gloom \u2013 a lack of joy or a sort of flatness.\n\n\u2022 feeling numb and dazed.\n\n\u2022 a feeling of being unable to cope with anything, especially any extra demands.\n\n\u2022 everything seems to take a long time, even small tasks.\n\nWe all have different resources for dealing with parenthood: it's a mistake to assume that because you have undergone a lot of stress, you will feel depressed \u2013 just as it's a mistake to assume you shouldn't be depressed because things haven't been that bad.\n\nHomeopaths take emotional injury as seriously as physical injury. If you break a leg (a serious physical injury) then the care and attention you receive is necessarily considerable. Emotional injury is not visible in the same way but it needs as much if not more time and support to heal. The following suggestions are to help you process the emotional impact of childbirth and to take any distress seriously so that healing can take place.\n\nThe range of emotions that follows childbirth is extensive. Here are some common to depression in the post-natal period that you may experience: anger; anxiety; apathy, indifference (cut off\/numb); aversion to family members, to visitors (company); confusion: thinking and concentrating are difficult; absentmindedness; depression; despair, especially of recovering; exultation \u2013 over-excitement or euphoria; irritability; lack of confidence; loneliness and feelings of abandonment; resentment: dwelling on past events; regret; shock; weepy or sad; sentimental. All these are included in the Repertory (see here). It may be that one emotion predominates or that you feel a complex mixture of different ones. In taking the whole picture into account, if the remedy that is strongly indicated sounds just right when you read it through in the Materia Medica (here) take it according to your needs \u2013 but please seek professional advice if your self-prescribing doesn't help you within a week or two.\n\nDo\n\n\u2022 see Emotional Stress.\n\n\u2022 be open to the feelings that come your way on a day-to-day basis. Acknowledge and accept them. Talk about how you are feeling with people you trust, who care about you.\n\n\u2022 debrief: you need to talk and talk, as much as you want, to tell the story of your baby's birth over and over again. This will help you integrate the experience into your life and understand how it has affected and shaped you.\n\n\u2022 reach out to the support network you set up during your pregnancy.\n\n\u2022 use your telephone support system if you don't have close friends or family living nearby.\n\n\u2022 keep in contact with at least one other mother and baby so that you can talk 'babies' and not feel isolated with your small charge.\n\n\u2022 seek the help of a counsellor or psychotherapist to help heal and integrate a particularly difficult birth experience if you are in distress and finding it difficult to bond with your new baby and\/or reconnect with your partner.\n\n\u2022 see a masseur or cranial osteopath if you feel that your distress is linked to physical tension or injury from your labour.\n\n\u2022 listen to those you love and trust if they tell you that you are looking rough.\n\n\u2022 write (or type) an account of the birth, putting in as much detail as you can remember. As you write, notice your feelings and write about them too. If you feel sad, allow yourself to cry. If you feel angry, kick a cushion around or have a good shout. If you feel inhibited about expressing your feelings and they are very strong, seek the help of a counsellor or psychotherapist.\n\nDon't\n\n\u2022 get orthodox medical treatment if you are depressed or in distress after the birth of your child and the measures outlined here do not help. In the long term orthodox medicines can make it difficult for this raw wound to heal. Seek alternative help as a first measure.\n\n\u2022 wait until you are desperate to ask for help. If you get help early it will be easier for you to heal.\n\n\u2022 ignore your distress by saying things like 'Many women have to cope with far worse than this' or 'I'll get over it'.\n\n\u2022 listen to people who say things like 'It wasn't that bad, you are both alive and you have a healthy baby with all its bits and pieces in the right places', or 'It will heal, time will heal \u2013 you will soon forget about it \u2013 stop thinking about it and it will go away'.\n\nSeek help if\n\n\u2022 you feel you've lost touch with reality, that there is an unpleasant feeling of distance and unreality that permeates your waking life and keeps you feeling separate from everybody and everything.\n\n\u2022 you find it difficult to feel the floor or pavement solidly under your feet, or the bed solidly holding your body when you lie down.\n\n\u2022 you are also suffering from inexplicable fears and worries.\n\n\u2022 you feel repeatedly angry with the baby.\n\n\u2022 you feel cut off and unable to respond to your partner or the baby.\n\nNB Severe post-natal depression \u2013 that is, severe emotional\/mental problems surfacing after childbirth \u2013 always needs the help of competent professionals.\n\nEXHAUSTION\n\nWith a first baby you may wonder how you will wash your hair or cook a meal or do anything else ever again! Second or subsequent babies are usually easier to integrate, although the demands of a toddler who is still in diapers alongside a newborn who is being fully breastfed are as impressive as the demands of a first baby who feeds every two hours around the clock.\n\nExhaustion is an occupational hazard at any stage of parenthood. Less sleep and more demands make it difficult to get the rest you need when you have babies and small children in the house. They are either breastfeeding or teething, suffering from a cough, cold, nightmare, inexplicable loneliness at 3 a.m. or a childhood illness, involving round-the-clock nursing care and attention. You may be lucky enough to have a baby who sleeps through the night from early on or a partner who wants to share the childcare and is not bothered by broken nights. It is easy to become run down when you are breastfeeding, especially if you are feeding on demand: irritatingly, babies have growth spurts, one at six weeks and one at three months and will want to feed more often just when you are beginning to feel like a human being again and wanting to get out and about. Extra activity may decrease your milk supply when you need to build it up to meet your baby's increased demands.\n\nUse common-sense measures to look after yourself and take the appropriate homeopathic remedy to hasten your recovery. You may need to repeat it relatively frequently while you are breastfeeding, especially if you find it difficult having your sleep disturbed at night.\n\nSeveral remedies are indicated for the treatment of exhaustion after childbirth, which may stem from loss of blood, breastfeeding or broken sleep, including Cocculus, China, Kali phosphoricum, Nitric acid, Nux vomica, Staphysagria and Phosphoric acid. Seek professional advice if self-prescribing doesn't help as exhaustion is often the first symptom of stress and can lead to depression. You must deal with your tiredness to avoid the development of physical complaints such as mastitis.\n\nDo\n\n\u2022 go with the flow. Again!\n\n\u2022 get your priorities right. Look after yourself, eat, drink, rest and sleep so that you have enough energy to care for your new baby.\n\n\u2022 rest and sleep when your baby rests and sleeps.\n\n\u2022 remember your partner and other children. Have a bit of fun.\n\n\u2022 ditch the housework. It will still be there tomorrow and you are much more important right now than a clean kitchen floor.\n\n\u2022 get a close relative or good friend to live in for the first two weeks after the birth so that you and your partner don't have to do anything except recover from the birth and get to know the baby.\n\n\u2022 be strict with visitors. Either get them all over and done with in a 'visit-the-new-baby' day or restrict yourself to one set of visitors a day: ask them to stay an hour at the most and to bring a small present for an older child or children. You could also ask for a contribution to your kitchen \u2013 a cake or casserole. Yet another option is to make a 'provisional' time for a visit and either confirm it on the day, or put if off if you are too tired. Visitors have a neat habit of turning up when you and your baby are settling down for your first peaceful nap in what seems like 24 hours. Your partner can always ask them to come back later, or entertain them with stories of the birth while you sleep.\n\n\u2022 eat well \u2013 little and often is better while you are breastfeeding. Have a snack every 2\u20134 hours. Listen to your body: hunger pangs are a symptom that you need feeding \u2013 don't ignore them when you are breastfeeding.\n\n\u2022 drink plenty while you are breastfeeding \u2013 you will feel more thirsty anyway, so make sure you have a snack and a drink on hand whenever you sit down with the baby.\n\n\u2022 negotiate a deal with your partner so that you each get a decent stretch of sleep every night. Your baby may wake cheerful and bouncy at 5.30 a.m.: take it in turns to handle the early-morning shift. Many babies will sleep again after a couple of hours of playfulness \u2013 you can sleep as well, especially in the early weeks.\n\n\u2022 gratefully accept any offers of help.\n\n\u2022 get some practical support if you are exhausted: someone to shop, clean, cook, or take the baby for a walk in the carriage if he or she doesn't sleep in the daytime so that you can have a nap.\n\n\u2022 get help with other children. Find someone to deliver and collect children to and from school or the childminder or to give practical help with a toddler.\n\n\u2022 persuade older children to help out.\n\n\u2022 read undemanding books and magazines.\n\n\u2022 talk about anything that is bothering you. Your midwife, health visitor, doctor, homeopath or breastfeeding counsellor may be able to help, with sympathy or practical advice. You may need to talk some more about your birth experience: it is exhausting continually to hold on to anger, regret, sadness or any strong emotion.\n\n\u2022 take gentle exercise, which will help create energy. If it doesn't, take the appropriate homeopathic remedy and get help.\n\n\u2022 ask your partner or a good friend to give you a face, foot or back massage. Some masseurs will visit your house: a good post-natal present for you could be a wonderful massage.\n\n\u2022 seek the help of an osteopath (especially a cranial osteopath) if you feel stiff and achy after the birth. The gentle art of cranial osteopathy (without manipulation) will help you to heal and rebalance.\n\n\u2022 introduce the odd bottle from early on if you are breastfeeding so that you are not on call 24 hours a day for an unlimited period of time. You can express your own milk or give some formula, unless you have a history of asthma, eczema, hay fever or milk allergies, in which case breastmilk is preferable.\n\n\u2022 do one thing every day to give you a sense of accomplishment. This may be a walk to the shops, writing a letter to a friend, having a bath and washing your hair, visiting a friend or having one visit you.\n\n\u2022 get out and about. Find another mother or two with babies of a similar age so you can do some swaps.\n\n\u2022 do something wonderfully self-indulgent once a day.\n\n\u2022 be quiet and boring during night feeds. Put double diapers on the baby so you don't have to do a change and risk really waking them up.\n\n\u2022 get a lot of support and sympathy if your baby is a poor sleeper: use a post-natal support group (see Organisations) to meet other women, reassure yourself and pick up some tips on how to get your baby to sleep better.\n\n\u2022 remind yourself that this is a short phase in your baby's life. Rearrange your routine as necessary and enjoy it while it lasts!\n\nDon't\n\n\u2022 breastfeed through the night if you turn into a demented monster after being woken from deep sleep more than once.\n\n\u2022 isolate yourself.\n\n\u2022 wake your baby for a feed before you go to bed in the hope that it will help him or her to sleep through the night. It won't.\n\n\u2022 change your baby in the night, unless absolutely necessary.\n\nHAIR FALLING OUT\n\n(See also here.)\n\nAfter childbirth your hair may lose its shine and\/or its curl, and it may fall out in handfuls for what seems like ages, especially if you are breastfeeding. If self-prescribing doesn't help, do seek the advice of a professional homeopath.\n\nDo\n\n\u2022 look after yourself, resting as much as you can, and eating well and regularly, including plenty of mineral-rich whole grains.\n\nHEALING AFTER LABOUR\n\nChildbirth is incredibly stressful physically, which some women find easy to cope with and also they heal well afterwards. For others, healing may be slow and arduous. Homeopathy can speed this up and prevent complications developing.\n\nThe common physical injuries include stitched wounds from a tear, an episiotomy or a Caesarean scar; bruising of the vagina, cervix and uterus, and the bladder, the hand or arm (if a drip was used); strained muscles and ligaments anywhere; general muscular aching and soreness; back pain after a backache labour or an epidural. Avoid the likelihood of further strain on pelvic-floor or back muscles by taking particular care of yourself post-natally, using the suggestions given below. The more care you take, the quicker you will heal. (See also Pain in Pregnancy, here.)\n\nThe speed of physical healing is closely related to how you are feeling emotionally. If your stitches are taking a long time to heal and you are feeling resentful, angry and let down by your experience of childbirth then you must prescribe on your emotional state and your physical symptoms: Staphysagria may be called for.\n\nMany remedies may be indicated after birth to aid healing including Arnica or Bellis perennis for aches and pains in the muscles caused by bruising; Hypericum or Nux vomica for pains in the sacrum or coccyx after an epidural or forceps delivery; Staphysagria, Calendula or Hypericum for severe pains in tears, episiotomies or Caesarean section scars; Kali carbonicum for backache after a posterior labour; Calcarea sulphurica, Hepar sulph., Lachesis or Silica for sore or lumpy scars that are slow to heal; Rhus toxicodendron or Ruta for strained muscles or joints. List your symptoms and repertorise them carefully to choose a remedy that fits as much of your picture as possible, bearing in mind that you may need a change of remedy as your symptoms change \u2013 in the days after the birth you may need different remedies every day. (See also Afterpains.)\n\nDo\n\n\u2022 take as long as you need to rest and recover, especially to avoid a prolapse (see here).\n\n\u2022 whatever will help you recover from any injuries: indulge in hot baths, massage, sleep, gentle exercise, etc.\n\n\u2022 start with gentle physical exercise and build up slowly. It is important to include some exercise in your daily life from as soon after the birth as is practicable \u2013 you'll both benefit from the fresh air if you go out for a walk with the carriage. And you can exercise with your baby, for example, lifting her gently on your legs when you are lying or sitting down... remembering to do only what you can without straining yourself.\n\n\u2022 keep wounds clean and encourage healing by: wiping or washing your genitals from front to back after you have passed a stool to minimise the risk of infecting a wound; 'bottle washing' (see here) and then drying yourself thoroughly after you've urinated (pat dry with soft toilet paper or use a hair-drier if you are very sore); using soft sanitary towels and making sure they are held firmly in place so they don't rub. Change them frequently.\n\n\u2022 see a cranial osteopath to help your body heal and rebalance you, especially if you have a sore back.\n\n\u2022 look at your genitals as soon as you feel able to if you feel anxious about your anatomy. You can do this with your midwife or your partner, asking them to tell you what has changed. It is important that you reconnect with your body for healing to take place.\n\n\u2022 use Calendula lotion on your episiotomy scar (see here).\n\nDon't\n\n\u2022 be impatient or critical of yourself if your healing takes longer than other women you know.\n\n\u2022 strain your pelvic, back or stomach muscles by lifting or carrying heavy weights (you can cuddle small children on the floor or get them to climb on to a bed or sofa and snuggle up to you).\n\nINCONTINENCE\n\nSee Urinary Difficulties.\n\nINSOMNIA\n\nThe hormonal high often experienced after childbirth can make sleeping impossible for the first night and difficult for a while after that. Choose between Coffea cruda and Kali phosphoricum for sleeplessness, unless you have other symptoms that would lead you to another remedy. You might, for example, have missed prescribing the first two nights and be suffering from irritability due to loss of sleep as well as excitement, indicating Nux vomica.\n\nDo\n\n\u2022 enjoy the elation and express it \u2013 joy, happiness and excitement deserve and need to be expressed.\n\n\u2022 write copious notes in your diary or journal about your birth experience.\n\n\u2022 use the extra energy to have special nice times with your partner and baby.\n\n\u2022 relax in a warm bath at night and take boring books to bed (see also here).\n\n\u2022 nap with your baby in the day. This will help you get into the habit of being able to sleep.\n\nDon't\n\n\u2022 drink tea, coffee, hot chocolate or Coca-Cola because the high caffeine content will further overstimulate you.\n\n\u2022 drink alcohol or take sleeping pills if you are breastfeeding because it will pass into your milk.\n\n\u2022 invite crowds of relatives and friends around just because you feel good as this can over-excite you.\n\nLOCHIA\n\nYou will bleed for a while after the birth until the wound left by the detached placenta has healed and the uterus has fully emptied itself. The discharge is referred to as lochia and the amount is noted: your midwife and\/or your doctor will check it regularly because heavy bleeding after the first week may lead to anaemia or, especially if there are clots or an unpleasant-smelling discharge, indicate a uterine infection or some remaining fragments of the placenta. Lochia is bright red for a few days after the birth, then turning reddish brown for a while before becoming brown and petering out. It lasts for anything from two to six weeks. As you get up and about you may find that it turns back to red for a few hours or even a few days.\n\nIf you decide not to breastfeed your baby then it will usually stop after your next period, which will come about four weeks after the birth.\n\nArnica will help your uterus to contract and the site of the placenta to heal. I have included some additional remedies to help speed up this process if, say, you stop discharging lochia then restart, or it becomes smelly without there being an infection. Choose between them based on the whole symptom picture.\n\nDo\n\n\u2022 observe a rigorous cleanliness: change your pads frequently; wash after urinating and\/or passing a stool; 'bottle wash' (see here), shower or use a bidet, if you have one.\n\n\u2022 add a handful of sea salt and\/or a teaspoon of Calendula tincture to the bath to promote healing.\n\n\u2022 use only sanitary pads, not tampons, until discharge of lochia has ceased: it isn't advisable to push anything up against your cervix while it is closing and healing.\n\nDon't\n\n\u2022 worry if you start to bleed again once the lochia has more or less stopped.\n\nSeek help if\n\n\u2022 you carry on bleeding heavily or notice clots after the first week.\n\n\u2022 you stop bleeding and then it restarts suddenly and heavily after the fourth day.\n\n\u2022 the lochia smells unpleasant.\n\n\u2022 you are having to change your pads every hour for longer than 3 hours.\n\n\u2022 your temperature rises.\n\nLOSS OF A BABY\n\nThe loss of a baby is devastating whether it is before birth (a miscarriage), at birth (a still birth), or in infancy (a cot death, illness or an accident). It leaves a great, gaping wound, a hole which can hurt more than anything has ever hurt before. Some of the feelings that surface after such an event can be hard to understand. They can come with great force, be fleeting or pass and then return unexpectedly. They can last for days, months or years.\n\nDeath is still a taboo subject. People often don't know what to say. We have very few rituals to help people through this time except for the cremation or burial, after which the bereaved are often expected to carry on their lives, especially if they have other children to care for.\n\nThe feelings that follow the tragedy of a child's death run strong and deep and need to be expressed. Their healthy expression will help you through this time, however painful it is, while suppression will lead to ill-health. Some common emotional responses are shock with numbness (no feelings), denial, rage, resentment, shame, guilt, great overwhelming sadness, depression, a terrible blaming \u2013 a wanting to find someone or something to blame for what happened (including yourself and\/or your partner) \u2013 hopelessness, suicidal thoughts, despair about the future and a hatred of people with live babies.\n\nI have included several remedies for the acute shock and grief of a still birth (or miscarriage) but it is essential to seek the advice of a homeopath if your health suffers and home prescribing hasn't helped. Don't prescribe on yourself after the death of your own child: ask a close friend or your partner to do this for you. Minor physical symptoms after a bereavement are common: headaches, back pain, difficulty in sleeping, loss of appetite, nausea, tension, inexplicable pains, overwhelming lethargy and so on. If you need a remedy to help you sleep and ease the acute emotional pain, they should prescribe carefully taking all your symptoms into account. Emotional symptoms to look up in the Repertory may include: apathy (numbness); shock; complaints from anger, suppressed anger, grief, shock or suppressed emotion; denial of suffering; tearful; tearful, with difficulty crying, cries alone; dislikes consolation or better for consolation; depressed.\n\nThe following suggestions are to help you begin to work towards healing this wound, effectively and healthily.\n\nDo\n\n\u2022 what feels right to you, whatever that may be. Express the feelings that surface: cry if you feel sad, rant and rail if you are angry.\n\n\u2022 remember that you will learn to live with this hurt, and you'll never get over it \u2013 however much people (kindly but misguidedly) tell you that you will.\n\n\u2022 name your child if he or she was stillborn. It is important that you see and touch and even hold the baby so that you can in a small but important way have a memory of the little person you were never able to know properly. Take a photo if possible so that you have a record of him or her in your album and\/or cut a little of his or her hair if there is some \u2013 it can be even more devastating to have nothing. Go straight home if you delivered in hospital: the sound of other mothers with their babies is hard to bear at a time like this.\n\n\u2022 hold and touch your baby if he or she died in infancy. However painful this is it will help you come to terms with it.\n\n\u2022 ask your doctor for reassurance. If your baby's death had no known cause ask him or her to tell you as often as you need to hear it that your baby's death was not your fault, that there was nothing that you could have done to prevent it.\n\n\u2022 talk with your partner about your feelings and his feelings \u2013 over and over and over again, until you both feel the pain healing. The more you talk to each other about it the closer you will feel. Don't allow yourselves to grieve independently. If you feel unable to talk to him, consult a psychotherapist either as a family or with your partner. The apparently irrational feelings that surface after the death of a baby can drive parents apart if they aren't expressed and dealt with or if they become acrimonious and blaming. It is normal to feel alienated from your partner at a time like this.\n\n\u2022 talk to your older children about what has happened and share your feelings with them. Don't hide your sadness but find a way of sharing your feelings that doesn't overwhelm them and use language appropriate to their age and maturity.\n\n\u2022 give yourself permission to take as long as you need to grieve.\n\n\u2022 contact the organisations that give support to parents who have lost children (see here). They will put you in touch with others who have suffered a similar loss, who can talk with you, listen to you, and help you through this period of mourning.\n\n\u2022 seek the support of a bereavement counsellor who will help you come to terms with your baby's death.\n\n\u2022 plan the funeral carefully and take an active part in it. Use it as an opportunity to express your grief.\n\n\u2022 write your baby a letter. Writing is surprisingly healing. As you write down the first things that come to mind your deeper thoughts and feelings will surface. Share them with your baby, and tell him or her what has happened to you and the family's life since she died. Tell your baby how and why you miss him or her and as you write let your feelings flow.\n\nDon't\n\n\u2022 be put off by your friends' or relatives' embarrassment and suppress your own feelings so that they will feel OK.\n\n\u2022 put up with people who find your feelings difficult until you feel ready and able to cope again.\n\n\u2022 let others tell you what to do or how to behave. If your family suggests that the time to grieve is over, gently tell them to mind their own business, that your grieving will be over in your time, not theirs.\n\n\u2022 feel pressured to get through this grieving period as quickly as possible. Unresolved grief always returns, on the back of another loss, and can then be overwhelmingly difficult to cope with.\n\n\u2022 worry if you 'go off' your partner and lose your libido. Your interest in each other and in sex will return once your sadness and your spirits lift, when you and your partner have come to terms with what has happened.\n\nNB The recent successful cot death prevention campaign in New Zealand advised parents to put their babies to sleep on their side or back (and never on their front). Be reassured that babies lying on their backs always turn their heads sideways so there is no danger of choking if they posset or vomit in their sleep. This campaign grew out of a research project that also found a worrying correlation between cot deaths and mothers who smoked, so try not to start smoking after the birth if you gave it up when you became pregnant. Avoid smoky places while your baby is small and ask visitors not to smoke in the house.\n\nAlso, don't let your baby get too hot or too cold \u2013 during the day or at night. Once your baby is about a month old he or she is good at keeping warm. To check \u2013 feel the tummy (not the head, feet or hands) \u2013 it should be warm, not hot. If you have been outdoors take some clothes off your baby so that he or she doesn't overheat (hats, gloves and blankets for a start). And remember, sick babies may need fewer clothes, not more. Ask your health visitor for help with this and with up-to-date information.\n\nLOSS OF LIBIDO\n\nIt takes time for the vagina to heal after birth and for the nerves, muscles and ligaments of the pelvis to return to normal. If you didn't tear or have an episiotomy, if your muscles snap back quickly and you have a baby that sleeps reasonably well, and if your relationship with your partner is close, your libido may well return within a few weeks of the birth. You may even find that your sex life will improve.\n\nThe more stress you experience around childbirth, however, the longer it can take for your sex drive to return. An episiotomy or a large tear can take anything from several weeks to several months to heal. If you feel tense or ill at ease or upset \u2013 either physically or emotionally \u2013 you are likely to find it difficult to express much interest in sex. Any of the following can contribute to this: drugs taken during childbirth, an unexpected Caesarean, unresolved feelings dating to the birth (including the shock of having been such public property during labour), a difficult homecoming, a baby who has colic and doesn't sleep, pain, sore nipples and piles. Some women find that it can take the best part of a year to begin to feel the familiar sensations again, to 'regain' their bodies after childbirth, and to get used to the new mother-shape \u2013 the more rounded abdomen and fuller breasts.\n\nSome women discover that breastfeeding stimulates their interest in sex, while others find that the close physical relationship they have with their babies satisfies their need for contact. Others, or their partners, struggle with unconscious messages from their own childhood that 'mothers don't have sex'.\n\nThis is a time of adapting to change, when great sensitivity is needed between partners. Take your time to process it and integrate sex back into your own life in your own time and in a way that feels healthy and satisfying to you.\n\nWhen self-prescribing on loss of libido, you must be sure to treat the root cause \u2013 the episiotomy scar that is still painful, the backache, the colicky baby who won't sleep at night. The main remedies are Causticum, Natrum muriaticum and Sepia.\n\nDo\n\n\u2022 be patient and accepting and allow your libido to assert itself in its own time.\n\n\u2022 find some time each day to be alone \u2013 to have a bath, read a book or go for a walk \u2013 even if only for 15 minutes.\n\n\u2022 be intimate and affectionate with your partner. Find ways of being close from the very beginning without worrying about it leading to sex. (See also Loss of Libido in Pregnancy, here.)\n\n\u2022 be sensitive to your partner's needs at a time when he may be feeling jealous \u2013 unconsciously \u2013 of the special relationship you are developing with your baby.\n\n\u2022 familiarise yourself with your body to get used to the changes: notice the differences as you wash in the bath or shower, while you smooth on body lotion or talcum powder. Look at yourself in the mirror. Does what you see match what you feel? Get a mirror and look at your vagina if you want to; ask your midwife to reassure you that all is well but ask her about anything that looks unfamiliar or peculiar. Talk about the ways your body has changed with your partner and ask him to tell you what he thinks and feels. Ask him to massage you. Tell him which bits are sensitive and be explicit about how you would like to be touched.\n\n\u2022 exercise your pelvic-floor muscles as this will bring a healing blood-flow to your pelvic area.\n\n\u2022 keep talking: communicate clearly, openly, honestly but sensitively about how you are feeling.\n\n\u2022 wait until your episiotomy scar has fully healed before you try to make love.\n\n\u2022 choose a time when you know the baby will sleep to make love.\n\n\u2022 be prepared to be interrupted.\n\n\u2022 go slowly, there's no rush; the slower the better. Start at the beginning again \u2013 with kissing, cuddling, exploratory touching and masturbation.\n\n\u2022 use KY-Jelly (not a petroleum-based jelly) as you may find your vagina dry while you are breastfeeding.\n\n\u2022 stop if it hurts \u2013 from as yet incompletely healed wounds, cervix or sore abdominal ligaments.\n\n\u2022 try making love in positions where your partner is not on top of you, which will enable you to be in charge of how deep he penetrates.\n\n\u2022 use contraception if you don't want to become pregnant. Even if you are breastfeeding you can ovulate before your first period.\n\n\u2022 seek the help and advice of a sex therapist or counsellor if you can't sort out any problems between you and your partner.\n\nDon't\n\n\u2022 force yourself to have sex. This will only create more problems and resentments.\n\n\u2022 be goal-orientated about sex: there are many ways to make love other than penetration.\n\n\u2022 worry if your breasts spurt milk during lovemaking. This is common, especially in the early months after the birth.\n\n\u2022 worry if you go off sex for what seems like a long time. This will pass.\n\nSeek help if\n\n\u2022 the above self-help measures haven't helped and one or both of you is becoming resentful.\n\n\u2022 sex is consistently painful.\n\nPAIN\n\nSee Healing.\n\nPHLEBITIS (superficial thrombosis)\n\nTroublesome varicose veins can become inflamed after childbirth. This is known as phlebitis (or superficial thrombosis) and is not as serious as it sounds. It is an irritating but not dangerous complication of childbirth. The vein of the inner thigh or inner calf becomes tender over an area of, usually, around 2 inches (5 cms) in diameter. It is painful to touch as well as on standing and walking. The benefit of this unpleasant symptom is that it cures the affected varicose vein. (See also Varicose Veins, and treat phlebitis in the same way.)\n\nDo\n\n\u2022 inform your midwife\/doctor or homeopath if phlebitis develops.\n\n\u2022 bandage the area with a Tubigrip (elastic bandage).\n\n\u2022 rest as much as possible until the inflammation has subsided.\n\n\u2022 keep the leg elevated, preferably higher than your heart. Raise the feet of your bed with a block or bricks under the legs.\n\n\u2022 soak the area of the bandage over the vein in Hamamelis lotion (or undiluted witch hazel) three times a day, use separate compresses and leave them on for an hour or so while you are resting.\n\nDon't\n\n\u2022 stand when you can sit and try to sit with your feet elevated.\n\n\u2022 sit when you can lie down.\n\nPILES\n\nIf piles have developed in the rectum during pregnancy, they will be pushed out at the same time as the baby. They take a variety of shapes and sizes from a single small one the size of a large pea to a 'bunch of grapes'. The pain of piles after a good birth really feels unfair: it can make sleeping, sitting and walking difficult. Remedies for piles after childbirth are Ignatia, Kali carbonicum, Pulsatilla and Sulphur. If another piles remedy is strongly indicated in other ways but not for the post-natal period, like Nitric acidum or Staphysagria, take it if your whole picture matches.\n\nDo\n\n\u2022 see Piles.\n\n\u2022 try pushing back protruding piles, or ask your midwife to do so. Wash with a non-scented soap and smear with a little Vaseline before pushing them gently back up the rectum where they will no longer be painful \u2013 if they stay there. If they keep falling out then do not persevere.\n\n\u2022 push your piles back after you pass a stool if you can do so easily and painlessly and if they will stay up.\n\nDon't\n\n\u2022 squat as this is the worst position for encouraging piles to pop out.\n\nSeek help if\n\n\u2022 they protrude, are painful and haven't shrunk by the time your baby is six months old.\n\nPROLAPSE\n\nA prolapse of the uterus, vagina, bladder or rectum occurs when the muscles or ligaments of the pelvis are either damaged or weakened by a long, difficult childbirth or by subsequent births. They become unable to hold the organs of the pelvis in place with the result that the bladder can push back into the vagina; the rectum can push forwards into the vagina; the uterus can fall down into the vagina pushing against the rectum and the bladder, causing discomfort; or the vagina itself can prolapse with the walls becoming weak and floppy. You may feel a dragging-down sensation in the pelvis, at its worst as if everything will fall out of the vagina, lower backache, pain or discomfort on urinating, stress incontinence and\/or discomfort when passing stools.\n\nPrevention is the best cure. It is absolutely essential to exercise generally in pregnancy and to do your pelvic-floor exercises religiously both during pregnancy and after the birth. Either Calcarea fluorica, Pulsatilla, Rhus toxicodendron or Sepia may help but prescribe according to your whole symptom picture. Seek professional help if it doesn't help your symptoms quickly.\n\nDo\n\n\u2022 lie flat or with the legs elevated \u2013 this will always relieve the unpleasant symptoms.\n\n\u2022 take care not to strain yourself further: squat down to older children; ask someone else to carry heavy shopping.\n\n\u2022 exercise your pelvic-floor muscles at every opportunity.\n\nDon't\n\n\u2022 lift or carry anything.\n\nSeek help if\n\n\u2022 the measures outlined in this section don't help. Homeopathic treatment may help to prevent surgery.\n\nRETENTION OF URINE\n\nSee Urinary Difficulties.\n\nURINARY DIFFICULTIES\n\nIncontinence\n\nYour bladder may feel bruised along with your pelvic-floor muscles and it can take a while to get the hang of peeing again \u2013 knowing when to go, being able to go and to retain urine without leaking. A sore bladder is most likely after a long or posterior labour or a forceps delivery. If you had a Caesarean (see also here), you will have had a catheter during the operation and for some time afterwards. After it is removed your urethra may be sore and you may have difficulty in controlling your urine.\n\nIf you can't control your urine or your pelvic-floor muscles, Arnica, Arsenicum, Calcarea fluorica or Sepia may help. If homeopathic remedies and self-help measures don't work, seek the advice of a professional homeopath or your GP.\n\nDo\n\n\u2022 avoid constipation (see here and here).\n\n\u2022 exercise your pelvic-floor muscles religiously (see here). You can also exercise these muscles when you are sitting down in a chair leaning forward slightly (or flopped right over). Tighten the muscles around your vagina \u2013 you may only feel a flicker after the birth but keep persevering \u2013 even if you only feel a little tightening try to hold it for up to four seconds and repeat it three more times. Do this at least 10 times a day \u2013 they will begin to feel stronger in a month although they can take up to six months or longer to regain their former strength.\n\n\u2022 pee frequently.\n\n\u2022 practise stopping and starting your urine midstream every now and again to check you can do it and to strengthen the muscles.\n\n\u2022 cut out, or down, food or drink which irritates the urinary tract, such as tea, coffee and alcohol.\n\n\u2022 see a cranial osteopath who will help rebalance the lower spine: it houses the nerves that supply the pelvis.\n\nDon't\n\n\u2022 stress your bladder by holding on when it feels full, even if you can only pass a little urine when you go.\n\n\u2022 cut down on the amount of fluids you drink. You can still 'leak' even if you have just been to the toilet. Cutting down on fluids can cause your urine to become strong which will irritate your bladder and cause other problems.\n\nSeek help if\n\n\u2022 you can't control your bladder and you have a bearing-down sensation in or around your vagina. You may have a prolapse (see here).\n\nRetention of Urine\n\nImmediately after childbirth you may feel sore and bruised and have difficulty in identifying a full bladder. It is important that you pee regularly so that your bladder doesn't become weakened or strained as this can lead to stress incontinence (see here).\n\nYour doctor or midwife will intervene if you aren't able to pee within a few hours of the birth and will insert a catheter (see here). If you have had a forceps or Caesarean delivery, you may already have one in place.\n\nYou can avoid a catheter with speedy homeopathic prescribing. The two remedies that are strongly indicated for this condition are Arsenicum (with the accompanying anxiety and panic) and Causticum. Other remedies, such as Arnica, Nux vomica, Opium, Pulsatilla, Staphysagria, or Stramonium, might be better indicated depending on your emotional state.\n\nDo\n\n\u2022 make sure you are as private as you need to be. Some people can't urinate in front of others \u2013 they may feel inhibited if the door doesn't lock or if there are people standing around outside (see Natrum muriaticum).\n\n\u2022 use tricks to get yourself weeing again like running a tap while you are sitting waiting to pee \u2013 the sound of running water is an unconscious incentive as far as your body is concerned; 'bottle washing' (see here): pour a bottle of warm water slowly over your genitals as you sit on the toilet. The fear of peeing can be worse than the actual pain caused by urine contact with the sore or stitched area. If you pour warm water over the area while you pee the urine will be diluted and won't sting, and the warm flowing water will encourage your muscles to relax; urinating in a warm bath (see here). This may be the only place you can pee for a day or so after the birth, or you can pee in the shower or even in the bidet if you have one.\n\n\u2022 tense your pelvic-floor muscles a little if you can and then slowly relax them while breathing deeply and evenly.\n\n\u2022 gently bear down as you did when you were pushing your baby out.\n\n\u2022 imagine yourself peeing \u2013 shut your eyes and relax the muscles and sphincters of your pelvis.\n\nDon't\n\n\u2022 build up tension and anxiety as this will only make it worse.\n\n* * *\n\nCOMPLAINTS \u2013 BABY\n\n* * *\n\nACCIDENTS\n\nSee Injuries.\n\nBIRTH INJURIES\n\nModern methods of delivery are designed to minimise the risk of injury or death to both mother and baby but some birth injuries are fairly common:\n\nBruising:\n\n1. to the baby's head after a long labour where the cervix has dilated slowly and the pressure against the head causes it to swell, sometimes in an unsightly lump;\n\n2. to the head after a forceps delivery where the pressure of the forceps can leave bruises around it (and sometimes the face). With vacuum forceps there is a circular swelling where the cap was applied, usually on the scalp;\n\n3. to the head from monitoring electrodes;\n\n4. to the buttocks (or scrotum in boys) after a breech birth.\n\nFracture or dislocation of the shoulder or collarbone if the baby is very large.\n\nArnica is the remedy that will deal with all these injuries in the initial stage and as you will probably be taking it after the birth for yourself, it will pass through to the breastmilk from which your baby will also get a dose.\n\nIf the injuries are severe and do not resolve with Arnica alone then you should repertorise carefully and prescribe, say, Symphytum if there is fracture, Rhus toxicodendron for dislocated shoulder, Sulphuric acid or Ledum if the bruising doesn't clear up quickly \u2013 swelling and bruising should disappear within 24\u201348 hours with Arnica alone.\n\nDo\n\n\u2022 use external creams where appropriate, such as Arnica for bruising and Calendula for the heel prick.\n\nDon't\n\n\u2022 worry. The above injuries will heal given time, however awful they look. Your midwife will be monitoring them daily so ask her for reassurance should you be at all concerned.\n\nBIRTHMARKS\n\nBirthmarks are caused by discolouration of the skin (pigmentation) or by collections of blood vessels (naevi or 'strawberry' marks). Some are flat and others are raised, most are soft and some have hairs growing in them.\n\nCoffee-coloured birthmarks are irregular patches or spots of darker skin browner in colour than the owner's skin. They are usually permanent although they may fade over time. Port-wine stains are dark red patches that appear on the face. They also usually fade a little over time. Strawberry marks are small red patches that usually disappear within a year or two. If they are raised, they will initially grow bigger, reaching their maximum size by the time your baby is 9\u201312 months old. They may take up to three years to fade. Mongolian spots are bluish-grey patches resembling bruises which appear on the back, buttocks and, sometimes, the arms and thighs. These usually fade within about five years. Stork marks, reddish patches on the back of the neck at the base of the skull, on the eyelids or the forehead, will usually fade within a year or two.\n\nMoles usually develop as children grow older. They can be prevented from 'multiplying' once they start to appear by constitutional homeopathic treatment.\n\nBirthmarks sometimes respond to homeopathic treatment. Give a short course of Thuja 6 three times daily for up to 10 days or Thuja 12, twice daily for up to a week. Apply Thuja tincture twice daily for up to a month. It is worth seeking the advice of a professional homeopath before resorting to surgery if Thuja hasn't helped as there are other ways of treating birthmarks homeopathically that are beyond the scope of the home prescriber.\n\nDo\n\n\u2022 ask your doctor to tell you what sort of birthmark your baby has and whether or not it will fade, and if so when.\n\n\u2022 ask for advice about surgery if your child has a permanent birthmark in a prominent place such as the face.\n\nBLOCKED TEAR DUCT\n\nA tear duct is situated in the inner corner of each lower eyelid. The eyes are bathed with fluid tears all the time, not just when we cry, and the tear ducts drain these tears down into the nose. In babies, ducts become blocked either because they are so small or because they have not fully developed. The result is constantly watery eyes. Sometimes the discharge becomes sticky, an indication that the eye is infected. A blocked duct usually clears itself by the time the baby is a year old.\n\nSilica will help clear a blockage so give this in a low potency (6 or 12) two to three times a day for up to a week. Then wait a week and repeat if there has been an improvement. If Silica doesn't help at all and you suspect that the duct has not fully developed, give Baryta carbonica in the same dosage as Silica. If the eye becomes infected see Sticky Eyes.\n\nDo\n\n\u2022 gently massage the corner of the eye (or eyes) beside the nose every two or three hours. This helps to clear a blocked duct.\n\nSeek help if\n\n\u2022 these measures have no effect and your baby's tear duct is still blocked at a year old. If homeopathic treatment and massage have not helped, your doctor may suggest a minor operation to clear it.\n\nBREATHING DIFFICULTIES\n\nBabies usually breathe for the first time within a minute of birth, but breathing problems in newborn babies are not uncommon, resulting from such circumstances as the cord having been wrapped around the neck at birth; or a long and\/or difficult labour in which the baby suffered severe distress; or if the mother had a general anaesthetic, or pethidine late in labour. Premature babies are also vulnerable because their lungs may not have had time to develop to their full size and strength. Doctors, nurses and midwives are all thoroughly trained in getting babies to breathe.\n\nIf babies up to six months old develop a viral infection affecting the lungs, pneumonia can develop or (rarely) a small baby can suffer heart failure. Danger signs to watch out for are: blueness of the face or of the area around the mouth; difficulty getting air in; wheezy breathing; baby refuses to feed.\n\nYou can prescribe while you wait for emergency help to arrive. The main remedy for the first-aid treatment of babies who have difficulty in breathing is Carbo vegetabilis \u2013 the baby is collapsed, limp and cold to the touch. If the breathing difficulty accompanies an infection and the baby is making very noisy efforts to breathe then give Antimonium tartaricum. If you have nothing else to hand Rescue Remedy may help the baby and you: a drop or two every five minutes or so.\n\nThese are emergency prescriptions only \u2013 it is essential that you seek expert medical help immediately.\n\nDo\n\n\u2022 get emergency help.\n\n\u2022 keep your baby warm and very close to you.\n\n\u2022 talk reassuringly and constantly to him or her.\n\nCHILDHOOD ILLNESSES\n\n(See also Fever.)\n\nBabies are born with a temporary protection from or immunity to the infectious diseases that you have had, passed from you to the baby through the placenta. The colostrum in breastmilk adds to this immunity. It was thought to last for about the first three months but recent studies are showing that fully breastfed babies are protected for much longer (see The Art of Breastfeeding by La Leche League, 1974), until their own immune system begins to develop. Even if you decide to bottle-feed do try to breastfeed for the first couple of weeks so that your baby has the full benefit of this protective colostrum.\n\nHomeopaths believe that childhood illnesses (such as chickenpox, mumps, measles, etc.) are not necessarily a 'bad' thing: they provide children with the opportunity to develop resistance and strength, and to clear inherited weaknesses. Those who have come through a childhood illness without medical interference and without complications are seen to be stronger afterwards and often have a growth spurt, either physically or mentally or both.\n\nA sick child must be nursed through a childhood illness. This may seem obvious, but it is becoming increasingly common to give children antibiotics and analgesics and to encourage them to carry on a normal life, including going to school or visiting friends. If you are a working parent you should prepare yourself for the fact that your children will fall ill from time to time and need nursing, either by you or with the help of a reliable carer. If you are unprepared for this, you will feel harassed and resentful when it happens.\n\nIf you have more than one child they will tend to fall ill one after the other rather than all at once. Your role as nurse\/mother will probably seem interminable so try to set up a support network that will give you some time off in each day.\n\nBefore prescribing a remedy assess how well your child is coping with the disease and whether he or she needs a remedy. It is not worth giving one if she is getting through it with ease unaided. Should the symptoms become distressing, however, remember to prescribe on the whole picture including the general symptoms and the emotional state as well as the specific symptoms of the illness.\n\nA child who doesn't recover quickly from a childhood illness always needs constitutional treatment from a professional homeopath.\n\nFor general warning symptoms see Cause for Concern, and Fever.\n\nDo\n\n\u2022 offer extra fluids to sick babies, especially if they are feverish. Give water or diluted freshly squeezed lemon or orange juice with honey (not to babies with mumps; these drinks are too acidic for sore salivary glands), either warm or cold, or herb teas.\n\n\u2022 continue to breastfeed as often as your baby asks. The breast will comfort her at a time when illness is making her feel awful.\n\n\u2022 give babies that are on solids small, easily digested, nutritious meals if they are hungry, such as fruit or vegetable pur\u00e9es, soups and porridge.\n\n\u2022 encourage a sick baby to sleep or rest as much as possible.\n\n\u2022 keep a hot, feverish child cool and a chilly, feverish child warm.\n\n\u2022 lie down with your baby and stay while he or she sleeps if necessary. Some babies, when sick, will only sleep well if their mother's body is close to theirs. Use this time to catch up on some sleep or reading.\n\n\u2022 let your baby sleep with you at night if you want to \u2013 or on you if you are able to. The comforting sound of your heartbeat will often help.\n\n\u2022 put your baby's cot or crib next to your bed so that you can reach out a comforting hand at night or talk soothingly.\n\n\u2022 sing to your baby and\/or play music.\n\n\u2022 carry your baby around in a sling while you do the things you have to do.\n\n\u2022 sleep or rest in a rocking chair or comfortable armchair with your baby held upright if a cough is preventing sleep while lying down. Wrap a duvet or enough blankets to be warm around both of you and use one of those inflatable airline cushions for your neck (to stop your head lolling if you drop off) so that you can sleep sitting upright. They really do work.\n\n\u2022 talk reassuringly to your baby about what is happening. The sound of your voice is comforting.\n\n\u2022 engage the help of neighbours, friends or family to look after older children so that either you can rest with your baby in the daytime, or cook or shop or just have a break doing something completely different \u2013 and enjoyable.\n\n\u2022 laugh at anything and everything, a sense of humour is a great bonus at times like these.\n\n\u2022 be careful that your child does not overdo things in the convalescent stage of a childhood illness, as relapses are common at this time.\n\nDon't\n\n\u2022 overstimulate sick babies by taking them out or by having lots of visitors.\n\n\u2022 encourage sick babies to eat.\n\n\u2022 take any child with a fever out and about.\n\n\u2022 worry if sick babies regress emotionally by becoming clingy or whiny. This behaviour may also be the first sign that they are not well. It will pass as they recover.\n\nIncubation and infectious periods vary, so the information regarding them in the individual illness entries below should be used as a rough guide only.\n\nNever give a child aspirin in any form during or after a childhood illness as this can cause serious complications.\n\nChickenpox\n\nIncubation period 7\u201321 days.\n\nInfectious period from a few days before onset until the last spot or blister has formed a scab.\n\nChickenpox generally occurs in a mild form in young babies \u2013 the younger they are the milder it is, some babies have only a couple of spots. Homeopathic treatment will help with the spots if they are very itchy or if they take a long time to heal.\n\nDon't worry about elderly friends or relatives getting shingles (caused by a virus identical to that of chickenpox) as a result of contact with your 'poxy' child. It is rare for the chickenpox virus (herpes zoster) to be passed on to adults as shingles.\n\nDo\n\n\u2022 dab dilute cider vinegar on very itchy spots (one tablespoon to a pint\/\u00bd litre of water) or put a cup of cider vinegar in a tepid bath and let your baby soak in it as often and for as long as it helps. An alternative to cider vinegar is bicarbonate of soda (a handful in the bath\/a tablespoon to a pint (\u00bd litre) of water).\n\n\u2022 dress your baby in loose cotton clothes to prevent further irritation.\n\n\u2022 cut the baby's fingernails or try mitts on the hands to prevent scratching, which will leave scars.\n\nSeek help if\n\n\u2022 the spots become badly inflamed (infected), that is, if there is redness or swelling around them, or pus oozing from them.\n\n\u2022 the itching is severe and not alleviated by the above self-help measures and home prescribing.\n\n\u2022 the spots affect the eyes (not just the eyelids).\n\nGerman Measles\n\nIncubation period 14\u201321 days.\n\nInfectious period 5 days before and 7 days after the rash appears.\n\nGerman measles, or rubella, is generally a short-lived, mild infection. A faint pink rash of tiny spots starts behind the ears or on the face and spreads down the body. It may be accompanied by watery eyes and swollen glands at the back of the neck and\/or behind the ears, under the arms or in the groin.\n\nYour baby will probably feel unwell for a few days before the rash comes out, will be more demanding and may have a fever with no other symptoms.\n\nHomeopathic treatment is rarely needed for babies who contract German measles. However, if your child is miserable with, say, a fever and an itchy rash, then you can prescribe to help him or her through it.\n\nDo\n\n\u2022 avoid contact with pregnant women while your baby has German measles because the infection can damage the foetus, especially in early pregnancy. For the same reason, notify pregnant women with whom you were in contact in the three-week period before the spots came out (when your baby was incubating rubella).\n\nSeek help if\n\n\u2022 you suspect you are pregnant.\n\n\u2022 there is a high fever, marked drowsiness and your baby is crying in that particular way that you know indicates pain.\n\nMeasles\n\nIncubation period 8\u201321 days.\n\nInfectious period 4 days before and 5\u201310 days after rash appears.\n\nIf you suspect your baby is incubating measles \u2013 if he or she has become unwell with, say, a fever, sore eyes, cough and irritability or clinginess within three weeks of having been in contact with another child with measles \u2013 look for small spots like grains of sand in the mouth, inside the cheeks. These spots, known as Koplik spots, confirm measles before the characteristic red rash appears, usually a day or two later. The rash starts behind the ears and spreads down the body. It is a blotchy rash with raised spots in the blotches.\n\nDon't be tempted to suppress the fever, even if the temperature is high (see Fever), as it is essential that the disease 'burns' itself out naturally. Some complications are thought to be caused by suppression of the fever and include ear infections, respiratory problems, pneumonia and (rarely) inflammation of the brain or encephalitis. Children need careful nursing through measles to reduce the possibility of these complications developing.\n\nHomeopathic treatment will help at all stages but especially with the cough and sore eyes and in avoiding the development of more serious complications.\n\nDo\n\n\u2022 resign yourself to looking after a measly baby for at least a week.\n\n\u2022 encourage a baby with measles to sleep and rest as much as possible. Stay around and read aloud or stock up on story tapes from the library.\n\n\u2022 keep a baby with sore eyes out of bright light, with curtains partially closed and lights dimmed, and bathe the eyes with Euphrasia lotion to ease soreness (see here).\n\nSeek help if\n\n\u2022 your baby under six months old contracts measles.\n\n\u2022 there is a cough that lasts longer than 4 days and doesn't respond to home prescribing.\n\n\u2022 the measles is accompanied by severe earache.\n\n\u2022 your baby is no better 3 days after the rash has come out.\n\n\u2022 your baby begins to get worse after a period of recovery.\n\nMumps\n\nIncubation period 12\u201328 days.\n\nInfectious period 2 days before the swelling appears until it has gone.\n\nMumps usually occurs as a mild childhood infection, especially in infants. The most common (and often the first) symptom is the swelling of one or both of the salivary glands (in front of the ear and just above the angle of the jaw), which gives a hamster-cheeked appearance. The glands under the tongue and jaw may also swell.\n\nHomeopathic treatment will ease the pain caused by eating and give relief from earache, both of which are common in mumps, and it will speed recovery. Again, you'll need to take the whole picture into account when prescribing.\n\nDo\n\n\u2022 encourage your baby to drink plenty of fluids \u2013 cold drinks may be easier than warm ones.\n\n\u2022 give drinks through a straw or from a bottle if it is painful to open her mouth.\n\n\u2022 avoid acid juices such as lemon, orange or grapefruit, as these will hurt the salivary glands.\n\n\u2022 be understanding if breast- or bottle-fed babies are fussy about feeding, as sucking may hurt.\n\n\u2022 give liquidised foods to babies on solids \u2013 soups, pur\u00e9es and ice creams.\n\n\u2022 wrap a hot-water bottle in a towel and let your older baby lie on it to soothe painful swellings. Hold a heated towel against the face of a younger baby and repeat if it is obviously helping.\n\nSeek help if\n\n\u2022 your baby doesn't seem to hear you \u2013 doesn't turn round to your voice as usual.\n\nRoseola\n\nIncubation period 5\u201315 days.\n\nRoseola is a mild, infectious illness which rarely needs treating. It is very similar to German measles, and the two are sometimes confused. The rash will distinguish between the two: in German measles it is more likely to appear with the fever and in roseola it appears when the fever has come down. The fever usually lasts for about 4 days and then the rash that appears is characteristically pink with small spots like that of German measles.\n\nScarlet Fever\n\nIncubation period 2\u20137 days.\n\nInfectious period 7 days after the rash comes out.\n\nThis highly infectious disease is caused by the streptococcus bacteria, and, although it is rare nowadays, when it comes, it can sweep through whole neighbourhoods or schools.\n\nThe symptoms are a sore throat, followed a day or two later by a rash of tiny spots which begins on the neck and chest and spreads over the whole body, giving the skin a texture like sandpaper; vomiting; and fever and a flushed face (though the area around the mouth may be pale). The tongue may also have a red and white 'strawberry' appearance.\n\nIt responds extremely well to homeopathic treatment and patients usually recover without any complications. Prescribe on each stage as necessary: different remedies for the fever, the vomiting and the rash may be appropriate.\n\nWhooping Cough\n\nIncubation period 7\u201321 days.\n\nInfectious period up to 3\u20134 weeks after the illness appears.\n\nThe first signs of whooping cough are a slight fever and runny nose. This is followed by a loose cough. The mucus then thickens and extended, uncontrollable coughing fits occur to bring it up, after which the child draws air convulsively back into the lungs resulting in the characteristic whoop. Whooping cough may be accompanied by other symptoms such as nosebleeds, vomiting and blueness of fingers.\n\nThe danger of whooping cough in young babies is that they may not be able to breathe in properly after a coughing fit and may also find feeding difficult if they vomit frequently. If you have decided not to immunise against whooping cough I strongly recommend that you 'sign on' with a professional homeopath so that if your infant contracts this unpleasant illness, you can get effective help if home-prescribing does not produce a quick response.\n\nWhooping cough can last from three weeks to four months and is a long and tiring infection for both child and parent. Complications are rare in children over a year old. In babies or children who are prone to coughs and colds whooping cough can recur with every cough or cold for up to a year after the first attack.\n\nDo\n\n\u2022 follow the general advice for coughs (see here).\n\n\u2022 keep your child away from other children until the infection has ceased.\n\nDon't\n\n\u2022 give proprietary cough medicines to reduce the coughing \u2013 your child needs to cough to expel the mucus.\n\nSeek help if\n\n\u2022 you suspect your baby has whooping cough.\n\nCIRCUMCISION\n\nThe removal of the foreskin of the penis is a religious ritual which was originally performed on grounds of hygiene, mainly by Jews and Muslims. It has now become a widespread custom, particularly in America and Europe even though there are now no medical grounds for it. Many people are questioning this practice because of the physical and emotional trauma to the baby.\n\nNo local anaesthetic is used for ritual circumcisions, which are performed a week after birth. The foreskin is simply cut off the penis with a sharp knife or razor blade and the penis is wrapped tightly to prevent bleeding. A plastic device is used where the circumcision is out of choice (rather than on religious grounds) \u2013 it is tied on to the foreskin and falls off (with the foreskin) after three to four days. This is a painless and less emotionally traumatic procedure.\n\nThe penis will usually be slightly inflamed and swollen for a few days afterwards, whichever method is used, which may cause pain or soreness on urination. Some babies are distressed afterwards, waking at night more often, crying more during the day, wanting to be carried around more, wanting to breastfeed more frequently and so on.\n\nPrescribe on your baby after the circumcision according to his reaction: Arnica, if he has gone into delayed shock, Aconite, if he is severely shocked, Staphysagria, if he is enraged, Ignatia, if he becomes hysterical and inconsolable or Stramonium, if he is shocked and wakes screaming in the night as if from a terrible nightmare when he was formerly a good sleeper.\n\nUse Calendula externally to help the wound to heal quickly (see here). Give Hepar sulph. if the wound becomes infected and follow it with Silica if it takes a long time to heal. Seek professional advice if the self-help measures outlined in this book don't help and your baby remains distressed.\n\nDo\n\n\u2022 keep your baby very close to you afterwards: he has suffered an injury and he may be more clingy.\n\n\u2022 breastfeed your baby frequently if it soothes him.\n\nSeek help if\n\n\u2022 your baby's penis bleeds for longer than a day after the circumcision.\n\n\u2022 your baby is in great distress and the remedies outlined above do not help.\n\n\u2022 his penis is still swollen or inflamed after a week.\n\nCOLIC\n\nColic simply means sharp, intermittent abdominal pains or cramps and occurs commonly in babies as their digestive systems mature. Babies with colic usually pull up their legs (see Colocynthis in the Materia Medica) or stretch them right out (see Dioscorea) and in some you will notice a change in the colour of their stools, which may become greenish (see Chamomilla). Sometimes colic is accompanied by constipation (see here), in which case you should treat that as well.\n\nMothers of breastfed babies with colic should consider their own diet, as many foods are known to affect some babies through the milk; these include cow's milk and all dairy products, alcohol, tea, chocolate, coffee, spices (including chillies and pepper), onions, garlic, broccoli, cabbage, cauliflower, Brussels sprouts, peppers (especially raw green peppers), strawberries, oranges and grapes. Occasionally egg or wheat products are to blame or even fruit with stones (cherries, apricots, etc.). A breastfeeding counsellor will help you sort out which foods may be responsible for your baby's colic. Some bottle-fed babies are allergic to cow's milk and will usually do well on soya milk, although that is acidic for some. Seek advice if you are concerned. If the colic starts within the first week after birth, consider whether it might have been precipitated by the syntometrine injection (see here) and if so, give Secale. Otherwise, differentiate between the colic remedies to find one that fits your baby best.\n\nSome babies swallow air with their feeds which can cause colic if they can't bring it up easily. This can happen with bottle-fed babies if the teat is too big for new babies and too small for older ones, or breastfed babies if the flow is too strong. If your baby is gulping milk and air then experiment with different feeding positions or a teat with a different size hole.\n\nDo\n\n\u2022 help your baby to pass a stool, if the colic precedes the passage of one. Massage the abdomen very gently in gentle clockwise circular movements or hold the baby in a semi-squatting position, back to your tummy, legs pulled up a little, again very gently.\n\n\u2022 place the baby face-down on your lap over a rolled-up towel, as pressure sometimes eases colic.\n\n\u2022 place your baby over your knees so that there is no pressure on the abdomen: pressure sometimes aggravates colic. (You can use these symptoms as a part of your prescribing picture.)\n\n\u2022 give your baby some dill- or fennel-seed tea. Simmer a teaspoonful of seeds in a pint (\u00bd litre) of water for 10 minutes. Strain and cool and give it to your baby in a bottle or on a spoon.\n\n\u2022 offer a bottle of boiled and cooled water instead of a breast- or bottle-feed as there are times when babies are thirsty, especially in hot weather or if they are inclined to be sweaty, and need something thirst-quenching. Do this to check whether your baby is thirsty and only if she is plainly thriving (gaining weight and producing lots of wet diapers).\n\n\u2022 burp your baby after a feed or carry her around upright (over your shoulder) until she settles down.\n\n\u2022 relax when breastfeeding a colicky baby.\n\n\u2022 take a break from your baby if you are at the end of your tether and suspect that your own irritability is affecting her.\n\n\u2022 self-prescribe on your own exhaustion (see here).\n\n\u2022 eliminate suspect foods from the diet of a baby who has just started on solids and give small meals of bland, easily digestible foods.\n\n\u2022 offer your baby a finger or dummy to suck if she is distressed. Some babies will feed almost continuously when colicky but the extra milk may overload the stomach and aggravate the colic when she just wanted to comfort-suck. Even with demand feeding it is possible sometimes to offer the breast too often when it isn't needed.\n\nDon't\n\n\u2022 let your baby get desperate for a feed as she may gulp and take in air which may cause her discomfort.\n\n\u2022 give gripe water as this can contain a worrying amount of alcohol.\n\nSeek help if\n\n\u2022 the colic persists \u2013 especially if the baby screams inconsolably.\n\n\u2022 the colic is accompanied by persistent vomiting, diarrhoea, constipation or absence of urine.\n\nCOMMON COLD\n\nSome babies seem to have a constant cold from fairly soon after the birth. This chronic catarrh is beyond the scope of the home prescriber and should always be referred to a professional homeopath.\n\nWait and see if a basically healthy baby throws off a mild cold in a day or two. A runny nose in one who is otherwise happy and contented, feeding well and sleeping at night is no cause for concern. However, a snuffly nose may interfere with feeding \u2013 a baby with a blocked nose will find it difficult to suck and breathe at the same time \u2013 and sleeping, causing some babies to wake frequently at night. Once the cold starts to affect your baby's energy, moods, sleep patterns or appetite then you should prescribe as neglected colds can turn into more serious chest infections.\n\nIn the Repertory see Common Cold and Snuffles: it is the type of runny nose that will help you prescribe. Observe whether the discharge from your baby's nose is clear and watery, thick and clear like egg white, yellow or even green, whether it produces crusts inside the nose or whether there is no discharge but you know there is congestion because your baby can't breathe properly with her mouth shut (see also the sample case here).\n\nDo\n\n\u2022 take sensible precautions to prevent a cold developing into a chest infection. Don't take the baby outside if it is cold and\/or windy; don't bath him or her until the cold has passed as they may easily get chilled afterwards; encourage plenty of sleep and rest; and ensure that a baby on solids has lots of fruit and vegetables.\n\n\u2022 breastfeed a nursing baby more often: babies who have started on solids may go back to breastfeeding fully until the cold has passed. This is normal.\n\n\u2022 keep your baby away from polluted environments such as smoky rooms and traffic jams as far as possible.\n\n\u2022 take the baby out for some fresh air if the weather permits to see if this helps (and use the response as a symptom to repertorise).\n\n\u2022 use a vaporiser, filled with plain water, to humidify a dry, centrally heated atmosphere. Don't add strong-smelling substances as they may affect the action of the homeopathic remedies you give.\n\n\u2022 use a little lavender and\/or rosemary oil (essential oils) in the vaporiser at night to help clear the nasal passages, or put a drop of each on a piece of cotton wool and place it under the sheet. You can also mix a drop of the essential oil with a little almond oil and massage a little in clockwise circles into your baby's chest and back.\n\n\u2022 clear your baby's nostrils gently with a tissue or a cotton bud, being careful not to push it in, before a feed so that sucking and breathing are not difficult. Use a drop of water or oil or a little breastmilk in a blocked nose to make it easier to clear.\n\n\u2022 put a pillow underneath one end of your baby's mattress.\n\n\u2022 cut cow's milk and all dairy products from your baby's diet for a day or two, if you can, as they encourage mucus production.\n\nDon't\n\n\u2022 use decongestants. These only irritate the nasal membranes and make matters worse as soon as the effect wears off.\n\nSeek help if\n\n\u2022 your baby's cold goes to the chest and your prescribing doesn't help within a day or two.\n\n\u2022 the runny nose is more or less continuous as your baby may have a mild allergy.\n\nCONGENITAL DISORDERS\n\nA congenital disorder is a defect present at birth rather than one that develops later. Many early miscarriages are of fetuses whose development is not as it should be. The number of babies born with an abnormality is relatively small. The most common include cleft palate, hare lip, club foot, dislocation of the hip, hole in the heart, hydrocephalus, obstructions of the rectum or small intestine, pyloric stenosis (projectile vomiting), sickle-cell disease, Down's syndrome, haemophilia, cystic fibrosis and spina bifida. These range from disorders that clear with a little help in a relatively short time, such as a dislocation of the hip, to those which produce severe disabilities and are not curable, such as Down's syndrome.\n\nAlthough homeopathic treatment cannot 'cure' most of these disorders, good constitutional treatment will help by boosting the immune system, and therefore improving vitality and general health, and helping children with emotional stress. Babies who need surgery can be prescribed on both pre- and post-operatively to speed their recovery. I suggest that you give Aconite before the operation \u2013 just one dose. Afterwards, choose between Aconite, Arnica, Phosphorus, Staphysagria and Stramonium. You may find that Rescue Remedy is good for you (administer frequently!) and you can also massage a few drops into the forehead of a distressed baby at any time while you are in hospital.\n\nDo\n\n\u2022 seek genetic counselling before or during pregnancy if you know or suspect you or your partner has an inherited weakness.\n\n\u2022 read up thoroughly on the disorder your baby is suffering from and ask your GP, consultant and homeopath to explain in detail what your child has and why.\n\n\u2022 ask (if appropriate) what are the chances of recovery with an operation, the potential risks, what will happen (how long it will take, which bits will be cut, stitched and bandaged, and how long recovery will take) so that your shock afterwards is lessened.\n\n\u2022 go with your partner, a close friend or family member to all visits to the GP or hospital and ask them to take notes and go over them with you afterwards.\n\n\u2022 create a network of support for yourself (including contacting any organisations that represent your child's disorder) to help you cope.\n\n\u2022 seek the advice of a counsellor to help you adjust to this new circumstance \u2013 a disabled baby requires more care and time, and some disabilities are more of a strain than others. Coming to terms with reality is essential for you.\n\n\u2022 make contact with other parents in a similar situation.\n\nDon't\n\n\u2022 feel guilty.\n\n\u2022 isolate yourself and your child.\n\nCONJUNCTIVITIS\n\nSee Sticky Eyes.\n\nCONSTIPATION\n\nThe first day after the birth babies pass meconium (a sticky, greenish-black substance) which gradually changes to normal stools. In breastfed babies these will be liquid, mustard yellow, sweetish-smelling, curdy stools (like half-cooked scrambled egg); and a more formed, conventionally smelly brown stool in a bottle-fed baby. Babies will settle quite quickly into their own pattern of passing stools and it is the variation from this pattern that may be significant.\n\nVery young babies vary enormously in the number of stools they pass. Some open their bowels with every single feed; others happily pass one large mass every three to seven days. Babies with this latter pattern are simply absorbing most of their food.\n\nConstipation in a baby is defined not by the number of stools that are passed but by whether the stools themselves are hard and painful and\/or difficult to pass. Some babies become constipated when they start on solids. Introduce one food at a time initially so that you know which is the culprit. When you introduce starch give rice rather than wheat as the gluten in wheat can cause the digestion to slow down. Bananas are a favourite first food and if they are unripe (with yellow or green skins) they commonly cause constipation because they are very starchy. The skins need to have gone black, indicating that the starch has turned to sugar, for babies to be able to digest them more easily.\n\nLike adults, some babies become constipated when they travel away from home.\n\nConstipation in breastfed babies is rarer than in bottle-fed babies. Some bottle-fed babies are sensitive to the protein in cow's milk and benefit from a change of formula to a different cow's milk or even one made from soya. It is essential that you seek the advice of your health visitor or GP when changing formulas. If the formula mix is too rich (not diluted enough) it will cause constipation.\n\nBabies with fevers can become constipated unless they drink more than usual as their need for fluids increases. (See also Colic.)\n\nAlthough I have included a few remedies for treating acute constipation, it is not a complaint for home prescribing if it becomes chronic. A tendency to constipation should be referred to a professional homeopath.\n\nDo\n\n\u2022 give a little prune juice to a constipated baby. Pour boiling water on to a handful of organic prunes and soak overnight. Give the strained juice on a spoon or in a bottle.\n\n\u2022 give your baby a little strained and diluted freshly squeezed orange juice.\n\n\u2022 repeat the above measures as and when they are needed, if they work.\n\n\u2022 make sure that there is plenty of water and fibre in your diet if you are breastfeeding, and that you are not constipated.\n\n\u2022 give your baby some extra water, especially if the weather is hot. Some babies need more fluids than others and will take water greedily if they need it.\n\n\u2022 add more water to formula (bottle) feeds.\n\n\u2022 massage the abdomen gently if the baby strains to pass a stool.\n\n\u2022 smear Calendula ointment gently around the anus if your baby has a tendency to pass large stools to avoid a tear.\n\n\u2022 cut out solids introduced at the time the constipation started and if it clears but returns when you reintroduce them, wait until your baby's digestion is stronger before trying again.\n\n\u2022 wait until your baby is a year old before introducing egg, as it has a 'binding' effect on the bowels.\n\n\u2022 give home-made foods for a while if you have weaned your baby on to tins, bottles or packets. Prepared foods are usually cooked in aluminium containers; some babies are sensitive to the very small quantities that dissolve into the food, which may be the cause of the constipation.\n\nDon't\n\n\u2022 boil the water or milk you give to your baby in aluminium saucepans or kettles as aluminium, even in very small amounts, can cause constipation. The same goes for any cooked solid food.\n\n\u2022 give laxatives.\n\n\u2022 give enemas to babies or any children: as the walls of the rectum are very thin and delicate and easily torn or damaged. It is a highly invasive procedure that is no longer accepted medical practice.\n\nSeek help if\n\n\u2022 your baby cries when passing a stool.\n\n\u2022 there are streaks of blood in the stools or diapers.\n\n\u2022 the constipation lasts longer than a week.\n\n\u2022 the constipation is accompanied by severe, unusual pain.\n\n\u2022 the constipation alternates with diarrhoea.\n\n\u2022 the stools are an unusual colour, either very dark or very pale, or have changed in colour from their norm.\n\nCONVULSIONS\n\nConvulsions usually occur in children up to the age of about five and commonly accompany a fever, although children who are teething or who have worms or who have been recently immunised may also experience them. It is not uncommon for a child to have a single convulsion and then no more.\n\nConvulsions are often preceded by a sudden rise in temperature and cause a temporary loss of consciousness often with a vacant expression, blueness of the face, stiffness, usually of the whole body, noisy breathing rather like snoring and uncontrollable twitching. They usually last a matter of seconds and never longer than a few minutes, although it can seem like for ever. The child is usually tired afterwards and will sleep. Convulsions are not as dangerous as they look: the fever affects the part of the brain that controls muscles which is why convulsions are accompanied by twitching. It is the danger of inhaling vomit after a convulsion which can kill rather than the fit itself, which is relatively innocuous.\n\nChildren who become delirious with a fever will not necessarily convulse even though their delirious state can be alarming (see Fever); a tendency to have convulsions can be inherited. It is important to watch your baby carefully if she produces a fever.\n\nShould your baby or small child have a convulsion, telephone your GP or homeopath immediately. You can prescribe from among the remedies included in this book while you wait for assistance to arrive. Constitutional homeopathic treatment will help to reduce the occurrence of convulsions in children who have them with fevers.\n\nDo\n\n\u2022 lay your baby flat and stay with him or her.\n\n\u2022 loosen clothes or take them off if the baby has a fever (see here).\n\n\u2022 turn the baby on to the side in case of vomiting.\n\n\u2022 lift his or her head up if they start to vomit so that it runs out of the mouth and not down the throat.\n\n\u2022 telephone for help once the convulsion has passed.\n\nCOUGH\n\nIf the cause of your baby's cough is very clear, for example if it occurs with a chill, then by all means prescribe. Some children suffer from coughs while they are teething: the teething does not cause the cough except indirectly because it can be stressful in itself. If the baby's lungs are a vulnerable area too, a cough will be more likely to develop at that time, or at any other when vitality is low.\n\nYou can prescribe for coughs but watch your baby's response, checking that moods and energy remain good. If the cough recurs or doesn't clear in spite of careful first-aid prescribing then do seek professional advice, rather than confuse the case by prescribing several remedies that only work to a limited extent.\n\nHowever, repeated coughs and colds which don't respond to constitutional treatment may be an indication of an inherited weakness and, if this is so, a professional homeopath should prescribe on the underlying miasm (see here). Coughs and colds are also common after vaccination: if the timing of the cough leads you to suspect that this is the culprit, seek the advice of a homeopath.\n\nDo\n\n\u2022 follow the advice for Common Cold and Childhood Illnesses (see here and here).\n\n\u2022 make sure your baby has enough rest and sleep.\n\n\u2022 offer extra fluids, such as water and diluted freshly squeezed juices (with a little honey if needed) as they will help to loosen the mucus.\n\n\u2022 use a humidifier or vaporiser in the bedroom to fill the room with steam (see also Croup).\n\n\u2022 keep the room temperature constant so that your baby doesn't have to adapt to either heat or cold.\n\n\u2022 stay with your baby during coughing fits to reassure and watch for breathing difficulty. Stay calm: if you both panic, it will be even harder for the baby to breathe.\n\n\u2022 encourage your baby to bring up phlegm by sitting him or her on your lap and holding them leaning slightly forwards; or lay them over your knees with the knee slightly raised under the baby's bottom with the head lower. Pat the back gently while he or she coughs. Have a bowl positioned to catch any expelled phlegm or vomit. Clean the bowl out afterwards with boiling water to prevent the infection from spreading.\n\n\u2022 prop up the baby's head with a pillow under the mattress which may ease the cough where lying down makes it worse.\n\n\u2022 offer your baby small meals at frequent intervals if the cough is accompanied with vomiting. Try giving a small snack immediately after a coughing fit.\n\n\u2022 distract the baby from coughing with entertainment: amazingly, this does work for limited periods. It is important, however, not to let excitement get out of hand, as that too can bring on a coughing fit.\n\n\u2022 sleep in the same room as your baby so that you can help if a coughing fit erupts in the night.\n\n\u2022 take it in turns with your partner to be on 'night duty' so that you get a reasonable amount of sleep each night.\n\n\u2022 cuddle and comfort your baby if he or she is distressed by the coughing. If you are still breastfeeding then offer a feed when you are sure the coughing fit is over.\n\n\u2022 avoid environmental pollution, in high streets with a lot of traffic, or car journeys where you are likely to get stuck in traffic.\n\nDon't\n\n\u2022 let your baby get chilled, as this will aggravate the cough. Bathing is unnecessary, especially if the weather is cold. If you have to go out, carry him or her in a sling close to you.\n\n\u2022 forget, some coughs are aggravated by breathing in cold air. This applies to babies as well as adults: don't take your child out unless you know that fresh air is a magic cure.\n\n\u2022 suppress the cough routinely with a cough medicine. This may prevent your baby from coughing up mucus, and if mucus is not expelled a more serious infection may develop.\n\n\u2022 give cod-liver oil regularly to babies: they may 'prove' it, constant coughs and colds may be brought on in a previously healthy child (see here) or an existing cough or cold may worsen and cause them to go off their food.\n\n\u2022 smoke or take your baby into a smoky atmosphere.\n\nSeek help if\n\n\u2022 your baby is keeping neither fluids nor solids down but coughs and vomits everything up almost immediately. Small babies need careful nursing through bad coughs and may need hospitalising if self-help measures and home prescribing don't help and you don't have a professional homeopath to turn to. (See also Diarrhoea for advice on dehydration.)\n\n\u2022 there is difficulty in breathing and\/or wheezing and\/or chest pain.\n\n\u2022 the cough seems severe and doesn't respond to self-prescribing within 48 hours.\n\n\u2022 the breathing is unduly rapid.\n\n\u2022 your baby has a bluish tinge around the face, mouth and tongue.\n\n\u2022 your baby is abnormally drowsy and unable to speak or make the usual sounds.\n\n\u2022 your baby deteriorates suddenly and you feel concerned.\n\nSee also Cause for Concern.\n\nNB Pneumonia (especially in babies) is not always accompanied by a cough. Seek urgent medical help if your child has a fever, is breathing rapidly and seems unwell \u2013 is limp and pale.\n\nCRADLE CAP\n\nCradle cap ranges from flaky patches of white scales mainly on the top of the scalp to a thick, yellowish scaling which covers the whole scalp like a swimming cap. It may be smelly. It is caused by the glands in the hair follicles overproducing oil (sebum). Homeopathic treatment will prevent it from recurring.\n\nDo\n\n\u2022 rub your baby's head generously with almond or olive oil at night, comb it very gently the next day to encourage the softened scales to detach and then wash them off with a very mild shampoo.\n\nDon't\n\n\u2022 pick it off.\n\nSeek help if\n\n\u2022 the skin around or under the cradle cap looks irritated or inflamed \u2013 red or sore or itchy.\n\n\u2022 it looks infected and begins to ooze.\n\nCROUP\n\nCroup is a frightening-sounding cough that usually occurs in children under four years old. It starts with hoarseness and fever and develops into a loud, barking, ringing, harsh cough that wakes the child at night. The breathing may also be noisy. Aconite, Calcarea sulphurica, Hepar sulph., Kali bichromicum, Lachesis, Phosphorus and Spongia are all indicated for croup. If it recurs it is essential you seek the advice of a professional homeopath for constitutional treatment.\n\nDo\n\n\u2022 follow the guidelines for Common Cold and Cough (see here and here).\n\n\u2022 stay with your baby all the time.\n\n\u2022 try steam as this can alleviate the symptoms relatively quickly. Go into the bathroom, close the doors and windows, fill the bath with very hot water, and sit with your baby in the steamy atmosphere until the cough eases. Or boil an electric kettle in the baby's bedroom until the room is filled with steam.\n\nSeek help if\n\n\u2022 steam doesn't relieve the symptoms within half an hour.\n\n\u2022 the cough is accompanied by a blue tinge to the lips.\n\n\u2022 your baby is having trouble breathing.\n\nCRYING\n\nSome babies cry rather a lot in their first year. It is their first 'language' and as you get to know your baby you will be able to distinguish between a cry that means 'I'm hungry', one that signals pain and another that says 'Please pick me up, I want a cuddle, I'm bored'.\n\nAll babies have different cries: some have a cry that is impossible to ignore and all parents have different levels of tolerance to their own baby's cry. Most mothers become so tuned into it that they wake even before the first whimper.\n\nThe human cry is one of nature's loudest sounds. At eighty to eighty-five decibels, it is as loud as an unmuffled truck, not far below the pain threshold. It is understandable, then, that parents of fussy kids tend to see nothing but a sea of tranquillity outside their home... for most (but not all) fussy babies, heavy crying begins at about two weeks, increases to a crescendo at six to eight weeks, and comes back down to manageable levels around three to four months. (Diana S. Greene, 70 Ways to Calm a Crying Baby, Sphere, 1988: an excellent book, full of useful tips and tricks to soothe fretful babies.)\n\nBabies cry when they are wet or cold and hungry; if they are sickening for an illness such as chickenpox or even a cold \u2013 when they will cry more, perhaps more feebly, sounding out of sorts; and if they are having a growth spurt \u2013 if you are breastfeeding, feed your baby more often to increase your milk supply (see here). If you are miserable then your baby will cry more.\n\nI have included several homeopathic remedies for fretful babies including Borax, Chamomilla, Lycopodium, Pulsatilla, Rheum and Stramonium. You will need to observe your baby carefully and match its picture withone in this book. If you can't or if the remedy you give doesn't work, then see a professional homeopath who may well be able to help by prescribing at a deeper level.\n\nSee also 'Difficult' Babies.\n\nDo\n\n\u2022 respond to your baby's cries without fussing anxiously.\n\n\u2022 try all the obvious things, like feeding, changing, cuddling, putting more clothes on if you suspect he or she is cold or vice versa. Offer the breast or bottle, even if your baby has just had a feed because it may be comfort sucking that is needed. Sometimes babies suck with a light, intermittent, at times barely perceptible sucking for up to half an hour. Finally when they let go they are drowsy, blissfully content and ready for their cot or crib. If your baby has had a full bottle and still wants to suck offer a pacifier or your little finger (see here) so that she can finish sucking.\n\n\u2022 feed your baby in a quiet room: some babies will cry if there are distractions like television, radio or people talking and laughing.\n\n\u2022 cuddle your baby \u2013 lying down or sitting up, or walking around. Each baby has a favourite position for cuddling.\n\n\u2022 develop a repertoire of 'tricks' that your baby responds to \u2013 be inventive, creative and flexible.\n\n\u2022 put your baby outside in a stroller or carriage under something interesting, like a tree, protected from cats and insects with a fine-gauge net. Fresh air and outside noises work like magic for some.\n\n\u2022 go out for a walk in the fresh air with the baby either in a sling or a carriage.\n\n\u2022 face the baby forward in the sling or the pram as soon as she is old enough and can hold up her own head \u2013 some babies scream unless they can see where they are going, when they will gurgle and smile happily.\n\n\u2022 place interesting things for your baby to look at in and around the cot: mobiles and pictures and anything brightly coloured. Change them if interest wanes!\n\n\u2022 sit your baby in front of a front-loading washing machine with a glass door when it is in action: many are mesmerised by the clothes going round and round, or soothed by the sound. A tumble-drier or spinner is equally effective.\n\n\u2022 sit the baby in front of the mirror once he or she can focus that far or 'chatter' to each other when you're both reflected in the mirror.\n\n\u2022 try putting your baby into a cot with bars. Some babies won't sleep in a cot with rigid sides as they seem to need to be able to see out.\n\n\u2022 try playing music: invest in several music boxes of the kind where you pull a cord and it plays a tune, experiment with putting it fairly close to your baby's head; play a tape of soothing womb noises (commercially available); play your baby the music you listened to when you were pregnant; experiment with types of music \u2013 orchestral, choral, opera, really loud rock music or Wagner, quiet folk music or Chopin.\n\n\u2022 invest in a lambskin \u2013 some babies find them incredibly soothing.\n\n\u2022 make your baby's room a little warmer: newborn babies are easily chilled as they have been used to a constant temperature of 97\u00b0F (36\u00b0C) in the uterus. They can also feel too hot \u2013 if the forehead feels hot, the baby will be too hot: take off her top layer of clothes.\n\n\u2022 keep bright lights, and that includes sunlight, out of your baby's eyes: some babies are sensitive to bright light and will cry if it shines directly in their face.\n\n\u2022 carry your baby around in a sling (if your back will take it) while you do some of your household chores \u2013 remember that your baby was carried 24 hours a day before birth and some find the separation more difficult than others.\n\n\u2022 give the baby a bath \u2013 or better still, have one together!\n\n\u2022 rock in your rocking chair with the baby snuggling on your belly. Wrap yourselves up in a blanket, duvet or shawl and sing silly songs.\n\n\u2022 try rocking gently, vigorously or fast, or jiggling in your arms. Be creative \u2013 every baby has a favourite, soothing movement.\n\n\u2022 place your baby in a reclining chair in front of the television \u2013 your baby may find the afternoon horse-racing compelling viewing, which will give you enough space in a day to prepare a meal or take a break.\n\n\u2022 take all your baby's clothes off in a warm room. Some children hate clothes and love to be naked.\n\n\u2022 swaddle your baby tightly. This is especially effective for babies up to about two months old.\n\n\u2022 try noise. The vacuum cleaner or food processor can soothe a hysterical baby (and freak out a calm one, so watch out!).\n\n\u2022 let him or her cry for a short time. Some babies need this and it does help discharge tension. Don't let it go on for too long (longer than half an hour or longer than you can bear it): if the crying continues, try something else.\n\n\u2022 put your baby in her cot, on the sofa or your bed or your lap and pat gently and rhythmically, either slowly or fast, on the back, bottom or tummy.\n\n\u2022 place one hand on the baby's tummy and the other on his or her head or under the neck, and just whisper sweet nothings while he or she is lying in the cot.\n\n\u2022 lie your baby on the floor on a blanket (with something interesting nearby to look at) and get on with doing things in the same room. Some babies like to be left alone at times \u2013 but not quite alone (see Lycopodium).\n\n\u2022 consider a ride in the car. Some babies hate this \u2013 but not many \u2013 so only do it if it works. They usually fall asleep \u2013 but, frustratingly, wake up again when the car stops!\n\n\u2022 visit a friend \u2013 some babies, and many more mothers, become cranky when bored.\n\n\u2022 go easy on visiting if your baby prefers a quiet, ordered life.\n\n\u2022 take your baby to a cranial osteopath if the birth was long, difficult or fast, or if she was delivered by forceps. The physical tension induced by this can cause a whole host of problems including sleeplessness and irritability.\n\n\u2022 ask other mothers \u2013 of any age! \u2013 for ideas and their favourite tricks.\n\n\u2022 settle the baby in the cot if he or she has been fed and changed, it is a long time since the last sleep, and you have been trying for what seems ages to stop the crying.\n\n\u2022 respond to your own needs. Your baby's crying may make you feel frustrated and incompetent and like howling as well. Let yourself have a good cry \u2013 it will help to release physical and emotional tension.\n\n\u2022 make sure your other child (or children) know how to carry a baby well supported. Small children can 'haul' babies around too roughly and actually cause minor stress to the spine \u2013 enough to cause the baby discomfort.\n\n\u2022 go out for the evening. Leave the baby with a trusted friend or relative who is calm and confident and has the telephone number of where you will be in case they can't cope. Some babies just need a change of company and will settle with someone else if you have become tense and anxious. An evening out will help restore some of your own equilibrium.\n\n\u2022 seek the advice of a psychotherapist if you over-identify with your baby's crying, if you end up howling too often or if you take the crying as a personal statement that you are a bad mother.\n\nDon't\n\n\u2022 leave a very distressed baby to cry \u2013 the distress will only get worse \u2013 unless you are at the end of your tether and in danger of snapping. In that event, put him or her safely in the cot and let your anger out in another room, by having a good scream yourself. Vent your anger safely by kicking a cushion, punching it or even biting it. Then you may feel like crying. Congratulate yourself on choosing not to be angry with the baby and resolve to get some support as soon as possible. Go back to your baby when you feel calmer if he or she is still crying.\n\n\u2022 expect methods that have worked to work again. Some will only work once and others will only work for a short time.\n\n\u2022 give up.\n\n\u2022 do anything that doesn't work. For instance, some babies hate baths \u2013 they just cry and cry. Don't bath them: sponge them down, limb by limb, from time to time instead. The time will come when baths will suddenly be fun: do try them occasionally but don't push it.\n\n\u2022 listen to those who say you must let your baby 'cry it out' and that you will 'make a rod for your own back' if you respond to the crying. Recent studies have shown that the rapid response decreases the amount of crying (Dr Bruce Taubman, Pediatrics 1984). 'One very important study showed that counselling parents on more effective responses to crying reduced crying in fussy babies by 70 per cent, down to the same level as the average baby! So the most important question to ask is not what starts the crying but what will stop it.' (Diana S. Greene, 70 Ways to Calm a Crying Baby, Sphere, 1988).\n\n\u2022 feel guilty if you feel angry with your child \u2013 everyone does from time to time.\n\n\u2022 wait until you have hit your child to ask for help. Ring your partner, a friend, a close and trusted relative, or even one of the organisations dedicated to helping desperate parents in times of need if you find yourself on the edge of snapping. It's never too early to ask for help.\n\n\u2022 feel bad if you throw your baby rather roughly down on the bed in exasperation. Every parent has a tale to tell of violence acted out on their child. Try to make sure that it doesn't happen again.\n\n\u2022 blame yourself for your child's crying.\n\n\u2022 let every visitor handle the baby if he or she seems unsettled by being passed around relatives and friends.\n\nSeek help if\n\n\u2022 you want reassurance that there is nothing wrong with your baby.\n\n\u2022 you 'know' that your baby's crying is signalling that something is wrong, even if your doctor checked recently and could find nothing amiss.\n\nDIARRHOEA\n\nFirst stools after the birth (meconium) are greenish-black and sticky, changing to greenish-brown, for a day or two and then, in breastfed babies, becoming mustard yellow with a sweetish smell. Variations in the colour, from yellow to green, and in the smell can signal teething, a reaction to something you have eaten, a cold or a stomach upset. Bottle-fed babies have a formed, brown stool.\n\nDiarrhoea, characterised by frequent, watery stools, is a cause for concern in small babies who can become easily and quickly dehydrated.\n\nI have included several remedies for acute diarrhoea as well as for the tiredness (loss of vitality) and loss of appetite that can follow a bad attack. Repertorise carefully at each stage to find the remedy that is indicated.\n\nDo\n\n\u2022 watch out for signs of dehydration in babies under six months old, and especially in those who are refusing to drink or are drinking less than usual: mouth and eyes are dry (no saliva\/tears); skin tone is poor \u2013 if the skin is pressed it does not spring back; the eyes look sunken; the fontanelle (soft spot on your baby's head) is sunken; the urine is scanty and smells strong (there are fewer wet nappies).\n\n\u2022 limit the intake of food.\n\n\u2022 continue to breastfeed a nursing baby.\n\n\u2022 encourage the intake of liquids: breastmilk, water, diluted freshly squeezed fruit juices, vegetable broth (simmer a selection of chopped vegetables in water for 15\u201320 minutes and strain), or rice water (rice cooked in double the amount of water for slightly longer than usual and strained).\n\n\u2022 give a wet flannel or sponge to an infant of more than six months who refuses drinks, or an ice cube, either of water or fruit juice.\n\n\u2022 put a not-very-sick child in the bath \u2013 not many children can resist drinking bathwater!\n\n\u2022 give a rehydration mixture (available from the chemist) if your infant is becoming dehydrated or make up one of your own by simply mixing a teaspoon of sugar or glucose with \u00bd teaspoon of salt in 1 pint (\u00bd litre) of water (boiled and cooled) and offer at regular intervals. Some babies will take small amounts on a spoon and others will prefer to suck it from a bottle.\n\n\u2022 cut out liquids that obviously aggravate the condition.\n\n\u2022 introduce food carefully to a baby on solids: start with white rice, toast, very ripe, black-skinned, bananas or low-fat (live) yoghurt.\n\nDon't\n\n\u2022 give rich foods that are difficult to digest.\n\n\u2022 encourage eating if there is no appetite \u2013 a few days without food will do no harm.\n\nSeek help if\n\n\u2022 the diarrhoea persists and fluids are not being tolerated.\n\n\u2022 there is exhaustion and loss of skin tone.\n\n\u2022 there is acute pain in the abdomen that doesn't respond to first-aid prescribing within 2\u201312 hours (depending on the severity) or that is getting steadily worse.\n\n'DIFFICULT' BABIES\n\n(See also Crying.)\n\nSome babies feed well and quickly, are healthy, responsive when awake, sleep for what seems like most of the time, quickly learn to sleep through the night... These, in my experience, are the exceptions.\n\nBabies, like adults, have their own characters, their own individual responses, and this, combined with the inability to talk, can make life very difficult for them and therefore for you! You can help them to adjust with homeopathic treatment or with the self-help measures that follow.\n\nA combination of the types of baby I describe below can make life hell for their parents: for example, a baby who is shocked, distressed and clingy or one who is bad-tempered and wakeful will be a real challenge. You won't be able to change someone's character with homeopathic treatment, whatever their size or age. However, you can heal the effects of emotional stress. The right remedy will help your baby back into balance, to feel better and more at ease.\n\nIf Borax is indicated, a jumpy baby will calm down and sleep more easily, though will probably relapse under stress so you may need to repeat it when, for example, teething starts, if the baby gets chilled or has a bad fall and produces the typical Borax emotional state again. If the baby's emotional state changes \u2013 if he or she becomes irritable and demanding instead of nervous and jumpy \u2013 you will have to decide whether to choose another remedy based on the new symptom picture. You must take into account the whole picture when you prescribe on a difficult baby, carefully repertorising the emotional state as well as the general symptoms.\n\nIn the case of temperature, a baby's response is only significant if he or she is chilly, and feels the cold. Most babies are warm-blooded and it is only if they are markedly worse for hot, stuffy atmospheres that this symptom counts.\n\nIf there isn't a picture in this book that fits, seek the advice of a professional homeopath, who will have a much larger choice of remedies. If you are at your wits' end, not enjoying being a parent because of your baby's behaviour and perhaps even beginning to feel like a bad one, a homeopath will delve in great detail and prescribe on both of you to ease the passage of this difficult time in your lives.\n\nDo\n\n\u2022 resolve any emotional stress in your own life if you suspect that is what is affecting your baby.\n\n\u2022 get support from sympathetic friends and family or counsellors.\n\n\u2022 call one of the crisis numbers if you feel you are close to battering your baby \u2013 every single parent I have ever talked to has wanted to hit, bite or otherwise hurt their baby more than once. Your feelings are normal. Do not respond to them except by doing to a cushion (or inanimate household object) whatever you want to do to your baby. The organisations listed here are all trained to deal with parents at the end of their tether and will offer sympathy and practical advice and support. Do not go through this on your own. Remember, asking for help is a sign of strength not weakness or failure.\n\n\u2022 try to find something good about your difficult baby. An active baby will almost certainly develop faster \u2013 after all they have been awake for longer!\n\nDon't\n\n\u2022 feel guilty or blame yourself if you have a difficult baby.\n\n\u2022 compare your baby to the boy- or girl-wonder next door.\n\nBad-tempered Babies\n\nSome just come out wanting a fight and carry on as they started. They wake angry from a sleep or a nap and you have the distinct feeling that almost nothing you do is right. Some babies respond to pain with anger so that, for example, colic or teething can set off a Colocynthis or Chamomilla state. And some babies are angry because they are impatient. They are hungry, they want to eat now and they tell you so in no uncertain terms. Symptoms to look up in the Repertory are: angry; biting; capricious; dictatorial; dislikes being touched or examined; hitting; impatient; irritable; tantrums; rage; moaning\/complaining; stubborn; tearful when they can't have what they want.\n\nClingy Babies\n\nSome babies are insecure and want to be constantly close to you. They won't let you put them down even for a minute. This might be part of their character but if you suspect that birth trauma may have caused this, add shock to your repertorising. Other symptoms to look up are: affectionate; anxious; aversion to being alone; clingy; desires to be carried; fear \u2013 of falling\/of strangers\/of downward motion; tearful\/tearful at least little thing.\n\nDiscontented Babies\n\nSome babies don't respond, don't smile or laugh much, if at all. They look miserable and make their parents feel that this job is very hard work. There are fewer rewards with these babies because it is the blinding smiles that keep us going. Symptoms to look up are: discontented; resentful; serious.\n\nDistressed\/Nervous Babies\n\nSome babies just cry a lot (see also Crying). A horrible noise, a grating scream, a constant whining, miserable crying which drives their parents mad. They wake with a shriek instead of a gurgle. Others are sensitive to almost everything \u2013 they jump at the smallest of noises, don't like to be thrown about (as many babies do) and are almost over-responsive. Some are so sensitive to downward motion that they will startle and wake on being laid down gently in their cot. Some scream inconsolably at unexpected noises, like the telephone ringing or an aeroplane flying overhead. These babies need careful handling. Some are traumatised at birth, by a too-fast birth, a long difficult one, a forceps delivery, or separation afterwards. They look shocked. Their little bodies may feel tense. Some of the symptoms to look up are: anxious, and particularly with strangers; expression is anxious or frightened; fearful, especially at night; jumpy; screams on waking or in sleep; sensitive to noise, light, pain; shock.\n\nQuiet Babies\n\nSome babies are shy by nature. They prefer a quiet life and are unhappy or distressed in company. This can be very difficult for mothers who are naturally gregarious and derive their comfort and energy from being with others. Symptoms to look up are: sensitive; shy\/timid; dislikes company; introspective; play \u2013 babies don't want to play.\n\nSleepy, Floppy Babies\n\nThese babies have to be woken for feeds. You keep wondering if yours is OK in spite of all reassurance that nothing is wrong. They sleep more than is recommended and, like the discontented baby above, don't seem particularly interested in anything. Symptoms to look up are: apathetic; dreamy; indifferent; slow; sluggish.\n\nWakeful Babies\n\nSome babies simply do not need much sleep: they have eight hours at night, a short nap in the day and are otherwise content. There is nothing you can do except to make sure that each day you have some time to yourself.\n\nSome babies become temporarily wakeful. They are sensitive to being over-stimulated, missing sleep and can become marginally hyperactive, being more difficult to put down at night and waking in the middle of the night wanting to play \u2013 for hours, quite happily!\n\nWakeful babies fall neatly into two categories, those who are happily awake and those who are unhappily awake. If your baby is wakeful and bad-tempered, distressed or jumpy, then you will need to add in the appropriate symptoms from the other categories above. Symptoms to look up include: cheerful; excitable; exhilarated; hyperactive; lively; wants to play at night; restless and better for being carried; tearful.\n\nEAR INFECTION\n\nEarache often occurs during a cold and results from the build-up of catarrh in the middle ear, which presses against the eardrum, causing great pain. This is not necessarily a cause for concern, and the indicated homeopathic remedy will usually give speedy pain relief. The only reliable symptom of earache in a baby may be fever and inconsolable crying; they sometimes rub their heads seemingly randomly or one or both ears.\n\nIt is not the end of the world if the eardrum perforates with a subsequent discharge from the ear. It will give instant relief from pain, and, provided your baby is in a clean environment, the eardrum will heal well in a couple of weeks with no troublesome consequences. Recent studies have shown that antibiotics are rarely effective for the treatment of earaches, that they will resolve in the same length of time whether antibiotics are given or not. Children who have had antibiotic treatment, however, will then suffer side effects from the drug treatment, which can include diarrhoea, thrush, loss of appetite and glue ear (a chronic condition in which catarrh in the middle ear causes some hearing loss and a tendency to repeated infections).\n\nDo\n\n\u2022 suspect earache and a possible infection in babies with a fever who cry and rub their heads.\n\n\u2022 ask your GP to have a look in your baby's ears \u2013 it is important that you get a diagnosis for your own peace of mind.\n\n\u2022 nurse a baby with earache conscientiously so as not to stress the immune system further.\n\n\u2022 make sure he or she drinks plenty of fluids.\n\n\u2022 use a hot-water bottle (well wrapped up) or an ice pack to relieve the pain.\n\nDon't\n\n\u2022 be tempted to clean the ears. The pain is never the result of wax.\n\n\u2022 take a baby with earache out into a cold wind.\n\n\u2022 let water get into the ear while it is healing.\n\n\u2022 bath your baby or wash hair until a few days after the ear has recovered.\n\nSeek help if\n\n\u2022 a baby has a depressed or bulging fontanelle.\n\n\u2022 there is swelling, redness, tenderness or pain behind the ear.\n\n\u2022 the earache doesn't respond to first-aid prescribing within 24 hours.\n\n\u2022 the pain is very severe.\n\nSee also Fever, and Cause for Concern.\n\nEYE INFLAMMATION\n\nSee Sticky Eyes.\n\nFEVER\n\nA fever can be a helpful and necessary healing stage of an acute disease, during a cold, perhaps, or 'childhood illness' (see here) \u2013 something positive, to be encouraged rather than suppressed. By understanding that fever is a symptom and not a disease in itself, you can come to see it as an ally rather than an enemy. Fevers that recur, possibly for years, after illnesses like glandular fever or malaria are not to be confused with the type of fever we are discussing here and should always be referred to a professional homeopath.\n\nWhen our bodies become stressed by external or internal events we become susceptible to disease. Typical examples of external stresses are overwork, lack of sleep, an accident, environmental pollution, becoming chilled, overheated or wet through, overindulgence in rich foods or alcohol, etc. Examples of internal or emotional stresses are shock, the death of a close relative or friend, boredom, fear, resentment or any strong feeling which isn't expressed. Where a person is under continued stress, a cold or flu may well surface as the body's way of saying, 'Help! Please take some rest so that I can recharge my batteries and heal myself.'\n\nA high temperature generally indicates that the body's defence mechanism is fighting an infection and temperature variations indicate how it is coping. The healing reactions of the body are speeded up, by approximately 10 per cent for each 1\u00b0C rise in temperature: the heart beats faster, carrying the blood around the body more quickly; breathing speeds up, increasing oxygen intake; perspiration increases, helping the body to cool down naturally; and hormones are released which stimulate the body to fight disease. Fever is the body's first-line defence against infection and therefore attempts to 'control' it artificially with paracetamol, vitamin C, or even with inappropriate homeopathic remedies, can suppress the body's natural efforts to heal itself.\n\nHippocrates said, 'Give me a fever and I can cure the child.' A weak child may be endlessly 'sick', neither very ill nor very well, but with no significant rise in temperature. A more robust child whose temperature soars may look and feel very ill, therefore giving more cause for concern, but is usually ill for a shorter time and recovers more quickly.\n\nEach person has their own pattern of falling ill and will experience different fever symptoms. You may feel hot with a high fever, or you may feel chilly and shiver. You may be irritable, intolerant of any disturbance and need to be kept warm, or you may be aching and restless, may moan and complain. You may sweat profusely, be thirsty and slightly delirious; you may want company or prefer to be alone. Each person with a fever will respond to an individual homeopathic remedy depending on their emotional state and general symptoms.\n\nThe average normal temperature in a healthy human is said to be 98.4\u00b0F (37\u00b0C), but this can vary quite markedly. Most people, adults and children, can run a fever of up to 104\u00b0F (40\u00b0C) for several days with no danger. It is normal for healthy infants and children to throw high fevers (103\u00b0F (39.5\u00b0C) and over) with an infection. A temperature of 105\u00b0F (40.5\u00b0C) is serious cause for concern, but it is only when it passes above 106\u00b0F (41\u00b0C) that there is a risk to life.\n\nFevers usually peak towards night-time and drop by the following morning, so that a temperature of 104\u00b0F (40\u00b0C) registered in the evening may recur on subsequent evenings. A drop in temperature in the morning does not mean that the fever is past its peak. It can rise and fall several times over several days before finally returning to normal.\n\nSmall children who develop a fever, especially infants under six months old, must be watched carefully because they are vulnerable to becoming quickly dehydrated. Delirium and tantrums in children sometimes accompany high fevers and, although these are distressing, they are not dangerous.\n\nDo\n\n\u2022 take the temperature with a thermometer, placed under the tongue or tucked tightly under the armpit for 5 minutes, for an accurate reading. A temperature taken by tucking the thermometer tightly under the armpit will read about a half degree Fahrenheit lower than that taken under the tongue. A fever strip (for the forehead) is a rough guide only and a hand held on the forehead is next to useless. The newer digital thermometers are much easier for young children and give a quick and accurate reading. (Always keep a spare battery in the house.)\n\n\u2022 provide a calm environment for your feverish baby. This is not a time to go visiting!\n\n\u2022 encourage a feverish patient to drink plenty of fluids or at least sips of water at frequent intervals. Water, lemon and honey or diluted fresh fruit juices, warmed or cold as desired, are best. Breastmilk is fine for a nursing baby and is probably all that will be wanted. Older babies and young children who are reluctant to drink will often suck on a wet sponge or flannel, especially if the water is warm, or try an ice cube or frozen fruit juice.\n\n\u2022 immerse a feverish but not desperately ill child in the bath from time to time to bring down the fever. Thirstless children will often drink the bathwater as an added bonus!\n\n\u2022 sponge down with tepid water if the fever goes above 103\u00b0F\/104\u00b0F (40\u00b0C) and your patient feels uncomfortable (hot and sweaty). Expose and sponge one limb at a time until it feels cool to the touch. Dry and replace it under the covers before going on to the next limb. This will help the temperature to drop by 1 or 2\u00b0F (up to 1\u00b0C) and can be repeated as often as necessary. Sponging the face and forehead alone can also give relief.\n\n\u2022 undress a feverish baby especially if either the weather or your house is very hot. Small babies can throw a fever if they become overheated and will quickly revert to normal with undressing and\/or a tepid sponging down.\n\n\u2022 respond to your patient's needs. Keep a hot, feverish baby lightly dressed and a chilly, feverish child (who feels cold to the touch and shivers) well covered.\n\n\u2022 prescribe homeopathic remedies where the fever is one of a number of symptoms, for example, where the patient is clearly suffering from, say, earache or a sore throat and a fever. If the first symptom to arise is a fever then wait a while for other symptoms to surface before prescribing for the whole picture. Contain the fever, again if necessary, by sponging down (see here).\n\n\u2022 suppress the fever with Calpol (paracetamol) in an emergency, that is, where the fever rises above 105\u00b0F (40.5\u00b0C), or if your child is in severe pain, from say, teething in the middle of the night, and homeopathic first-aid prescribing isn't helping. Ring your homeopath or doctor in the morning or during the night if you are anxious.\n\n\u2022 watch for signs of dehydration in infants under six months old, and especially in children who are refusing to drink or who are drinking less than usual (see here).\n\nDon't\n\n\u2022 encourage a sick child to eat. Many children with a high fever will not wish to eat. This is a good sign: fasting encourages the body further to eliminate toxic wastes and helps it focus on recovery. Encourage a hungry patient to eat light, easily digested dishes such as vegetable soup, raw or stewed fruit with honey.\n\n\u2022 give any homeopathic remedy at the first sign of a rise in temperature as this can confuse the symptom picture. Any attempt to interrupt the body's own healing processes is unwise. Wait until a fuller picture develops when other symptoms emerge.\n\n\u2022 suppress a fever in children with any form of aspirin. This has been known to lead to dangerous, although rare, complications, in particular Reye's syndrome, which affects the brain and liver. Calpol (paracetamol) may be used in an emergency but never exceed the recommended dose.\n\nSeek help if\n\n\u2022 a baby under six months old has a fever.\n\n\u2022 an older baby has a fever of over 104\u00b0F\/40\u00b0C that doesn't respond to sponging and homeopathic treatment within 24 hours.\n\n\u2022 your family has a history of convulsions accompanying fevers \u2013 keep a close eye when your baby throws a fever. It is the rapid rise in temperature that can cause a fit. (See also here.)\n\n\u2022 the baby or older child is also refusing to drink (is thirstless) as dehydration can occur.\n\n\u2022 there is a lack of reaction (listlessness and limpness), which can imply that a serious illness such as pneumonia or meningitis has developed. (See Cause for Concern.)\n\nIf you are worried contact your GP and\/or homeopath immediately.\n\nHERNIA\n\nA hernia is a swelling in the abdomen caused by muscular weakness. The most common type in babies is an umbilical hernia, where the navel sticks out, especially when the baby cries, coughs or sneezes. It is not painful and the weakness in the wall will usually repair itself by the time the child is five years old. If it doesn't it may need surgery.\n\nAn inguinal hernia (weakness in the muscles of the groin) needs more urgent attention because part of the bowel can become trapped in it. In boys an inguinal hernia is sometimes associated with undescended testicles.\n\nAlthough I have included a few remedies for hernias in babies they are mostly beyond the scope of the first-aid prescriber. You should consult a professional homeopath if your doctor has advised you to wait and see if your baby's hernia will heal in time.\n\nHICCUPS\n\nSome babies hiccup a lot, even while still in the uterus \u2013 audibly. They may hiccup every time they feed, laugh, get excited or anything. There is nothing to worry about and the hiccups will decrease over time. If your baby's hiccups are associated with colic you should prescribe on the whole symptom picture.\n\nDo\n\n\u2022 place your hand gently over your baby's nose to reduce the amount of oxygen taken in, being careful not to cut off the air supply altogether. Leave it there for a minute or two, but no longer.\n\n\u2022 offer a little cold water on a spoon or in a bottle as this sometimes helps to stop them.\n\nDon't\n\n\u2022 prescribe if your baby's hiccups are mild and you are charmed and amused by them!\n\nINFLAMMATION\n\nIn newborn infants the navel (umbilicus) may become inflamed after the cord drops off. A baby's penis can become sore, sometimes for no apparent reason. Treat an inflamed navel as you would a cut that has become infected, by giving Hepar sulphuricum in the initial stage where there is some redness around it and bathe it with Calendula lotion or smear on a little Calendula cream and keep it lightly bandaged until it heals. If Hepar sulph. doesn't help within two or three days give a short course of Silica.\n\nTreat an inflamed penis with Arnica and bath your baby frequently in water to which you have added a handful of sea salt and 40 drops of Calendula tincture. If you have inadvertently pulled your baby's foreskin back and it has become 'stuck' to the head of the penis (the glans) and can't be pulled back at all because scar tissue has formed, wait until he is older (at least five) and then give him Thiosinaminum tincture, 5 drops twice daily in a little water. This will dissolve the scar tissue and enable the foreskin to be pulled back freely. I suggest that you give the drops for 7 days at a time with 10-day gaps, repeating it if the foreskin is still stuck in places. It can take up to 3 months to work.\n\nDon't\n\n\u2022 automatically use antibiotic powders or creams on your baby's navel \u2013 explore other options for helping it to heal first (see External Materia Medica and Repertory, here).\n\n\u2022 pull your baby's foreskin back to clean it if he has one until he is at least five years old \u2013 it won't retract until then and has a seal of its own that keeps it free of dirt. It will be fine just as long as you don't mess with it. If you do try to force it back you can cause it to become inflamed and then stuck. It is only when a child is able easily to move it back that he should be taught how to wash underneath it.\n\nINJURIES\n\nThe combination of a tired or sick baby with a tired or distracted mother can lead to accidents. One qualification for the job of parent is the possession of a revolving head with antennae that sense what is happening to a baby or child even when they are out of sight. It is often an unnerving quiet which alerts many mothers to the fact that something is wrong \u2013 otherwise the terrifying shriek of pain.\n\nPrevention is the best cure for accidents and many are preventable \u2013 but unfortunately injuries to babies are common because small people are awkward for a long time after they start to move. They also stubbornly refuse to learn the lesson of cause and effect when they really need to: if you throw yourself off a high surface you won't fly and you might break a leg; if you touch a hot surface you will burn yourself... Their need to explore and find out about the world is intense and unbounded. All parents learn to be vigilant and to anticipate the direction their offspring will take, and to distract them from danger.\n\nUse a combination of common sense and skill to deal with accidents. Always follow your instincts and take a badly injured child to the casualty department of your local hospital where they will be seen fairly quickly (unless it happens to be a children's hospital). The tests, X-rays and medical treatment, if needed, as well as the reassurance are all invaluable.\n\nAct quickly if your baby starts to choke on a small object \u2013 a button or stone or even a fishbone. Thump her between the shoulder blades, hard and sharp with the flat of your whole hand. The object should fly out of her mouth. If it doesn't, hold her upside down by her ankles and thwack her again in the same place. Remember: you are saving her life, don't hold back. Comfort her mightily afterwards.\n\nIf your child sustains a burn which doesn't warrant a visit to hospital then immerse it in running cold water for as long as it takes for the pain to ease off, then prescribe.\n\nDon't let a child who has banged their head badly go to sleep just in case delayed concussion is lurking. Keep him\/her awake, prescribe on the injury and observe closely until you are sure the danger is over.\n\nTeach your child how to approach animals of all shapes and sizes. Strange cats and dogs should always be avoided. Dogs of friends and family should be approached with an outstretched hand which can be fully sniffed before being accepted and then the animal stroked gently with an adult supervising. Children often rush at dogs and try to pat their heads, which is interpreted as an aggressive gesture by the animal who may then bite in retaliation.\n\nYou can, of course, first-aid prescribe for any minor and inevitable accidents, such as burns, cuts, bruises, bumps, shocks, bites, stings and splinters, but remember to take into account the whole picture when doing so.\n\nDo\n\n\u2022 be careful when away from home. Most houses are not childproof and are full of traps of the sort that you have removed from your own house and possibly forgotten about.\n\n\u2022 check that equipment conforms to the safety standards.\n\n\u2022 check second-hand equipment thoroughly.\n\n\u2022 store plastic bags, plastic wrap, all cleaning fluids and powders, all medicines, razors, cosmetics, your rubbish bin and sharp kitchen knives out of reach. Invest in babyproof safety locks for cupboards.\n\n\u2022 keep electric cords short on appliances like toasters and kettles or buy coiled ones.\n\n\u2022 fit a guard around your stove pilots. Turn pan handles so that they don't dangle temptingly over the edge of the guard.\n\n\u2022 fit locks on your fridge and freezer.\n\n\u2022 put hot drinks, pans and dishes with hot food and kettles, etc., at the back of kitchen surfaces or in the middle of a table. Give up using tablecloths for a while as they can be pulled with dire consequences.\n\n\u2022 fit safety catches on all windows above the ground floor.\n\n\u2022 change your baby's nappy on the floor or on your lap but only on a high surface if you are not going out of easy reach, even for an instant. Keep the talcum powder out of reach as it can cause severe choking if accidentally shaken out over the face.\n\n\u2022 use straps in all chairs \u2013 prams, pushchairs, high-chairs, bouncing chairs \u2013 at all times.\n\n\u2022 fix fireguards in front of all open fires and heaters.\n\n\u2022 cover unused electric sockets with dummy socket covers, plug-guards or strong insulating tape.\n\n\u2022 keep cigarettes, matches, alcohol, sewing equipment, money and all other small objects out of reach.\n\n\u2022 get rid of glass-topped tables.\n\n\u2022 make glass doors safer by covering them with a plastic film that makes them shatterproof.\n\n\u2022 fix a gate at the bottom and top of all stairs, unless your baby is well co-ordinated and learns to crawl safely up and down stairs.\n\n\u2022 make sure your banisters are safe and that the gaps between them are too small for a baby to squeeze through.\n\n\u2022 make sure the catches on the front and back doors are too high for the baby to reach.\n\n\u2022 put house-plants out of reach.\n\n\u2022 supervise your child in the garden. Store all gardening equipment and chemicals in a locked shed.\n\n\u2022 dig up poisonous plants and pull up fungi if they appear.\n\n\u2022 secure play equipment on grass or sand (not a hard surface) and check it frequently.\n\n\u2022 cover the sandpit when your child is not playing in it to protect it from fouling by cats or other animals.\n\n\u2022 use childproof locks on car doors so that they can't be opened from the inside and don't let your child play with the window. Never leave your baby or child alone in a car, whether or not you can see them from where you plan to be.\n\n\u2022 put a net on your baby's carriage when unattended outside as cats are drawn to the warmth and have been known to smother babies.\n\n\u2022 train your dog to be 'handled' by your child without responding aggressively. Get professional help from a dog-trainer if your dog is jealous of the baby or if it growls or nips once the baby becomes mobile.\n\n\u2022 keep your pets dewormed and defleaed.\n\n\u2022 train your dog not to lick faces, especially the faces of your children, as this can spread disease.\n\n\u2022 use the word 'no' for things that really matter, that are dangerous so that your baby will listen when you say it.\n\n\u2022 take a basic first-aid course so that you can deal with accidents and injuries confidently.\n\nDon't\n\n\u2022 leave a baby of any age on a table or bed or high surface of any kind without constant supervision. If you have to go and fetch something (even across the room) take the baby with you. Far too many babies have suddenly rolled over and dropped off high surfaces on to their heads.\n\n\u2022 leave your baby unattended in a room with the family pet, either a cat or a dog.\n\n\u2022 let older, but not old enough, children carry the baby up or down stairs.\n\n\u2022 leave older, but not old enough, children on their own in a room with the baby until they have proven their responsibility and capability over and over again. It is all too easy for a crawling baby to put a small object in his or her mouth. It takes the skill of an adult to spot it in the first place and then to prevent choking if necessary.\n\n\u2022 let your baby travel unharnessed in the stroller. The one time you do it, your pushchair will engage with a piece of uneven paving and catapult your precious bundle head first on to the pavement.\n\n\u2022 let your baby travel unharnessed in the car. Make sure that car seats are rigidly fixed: if it is loose, your baby's life will be at risk in an accident. Have the straps or seat fixed in by an expert such as your garage mechanic.\n\n\u2022 leave your baby unattended in the car, a room with an open fire or in the bath even if he or she can sit up on their own apparently quite safely.\n\nSeek help if\n\n\u2022 your child has a serious accident of any description.\n\n\u2022 your child falls unconscious even for a few seconds.\n\n\u2022 your child has trouble breathing or stops breathing even for a few seconds.\n\n\u2022 you are concerned for your child even if you don't have anything concrete to go on.\n\n\u2022 your child is extremely distressed after an accident or conversely becomes apathetic.\n\nSee also Cause for Concern.\n\nINSOMNIA\n\nSee Sleep Difficulties.\n\nJAUNDICE\n\nMany babies develop jaundice 2\u20134 days after the birth. The skin (of white babies) and the whites of the eyes turn yellow. It is not harmful and will disappear within about a week, although it can last longer with no ill effects. Jaundice occurs in babies whose livers have not matured fully by the time they are born. Phototherapy for babies is being thoroughly questioned as a treatment for jaundice. The majority of babies are much better if not separated from their mothers and only very severely jaundiced babies will benefit from this treatment.\n\nSome jaundiced babies become lethargic and feed less often but the condition rarely causes any complications unless there is a more serious medical problem such as a (very rare) congenital abnormality of the liver or ducts leading from it, or Rhesus incompatibility (see here) in which case the baby will be yellow within a few hours of the birth. Jaundice is more common in premature babies or if a baby is badly bruised during birth. Most cases will clear up quickly and effectively if you give Chelidonium 30 \u2013 one dose every two hours for a day. If it doesn't help and your baby has a number of other symptoms, such as colic, irritability, constipation or whatever, prescribe on the whole picture taking into account all the symptoms.\n\nDo\n\n\u2022 breastfeed your baby frequently.\n\n\u2022 expose your baby to sunlight if the weather is fine. You can go out for a walk well bundled up but with his or her face exposed to the sun's rays.\n\nDon't\n\n\u2022 give water if you are breastfeeding. This may discourage the baby from breastfeeding and extensive research has shown that extra water does not affect the jaundice. (The Art of Breastfeeding, La Leche League, 1974, here)\n\nMILD RASH\n\nSee Spots.\n\nNAPPY RASH (DIAPER RASH)\n\nDiaper rash has many causes: a chemical reaction between the urine and stools creating ammonia which burns the skin if diapers are left unchanged for too long; irritating chemicals in the stools; residues of soap or detergent (especially biological soaps) in diapers; teething or a general infection. The area around the genitals looks red, spotty and sore and your baby's diaper may smell of ammonia. In boys the foreskin may also become red and sore-looking, making peeing painful. It can itch (your baby will tear at the rash when you take the diaper off) or sting or burn (he or she will cry when passing urine). It may become painful and, at its worst, sores may develop so deal with it as soon as you notice it.\n\nIt is important to differentiate between stubborn diaper rash and thrush. Thrush will spread above the navel which diaper rash won't. If you are breastfeeding and have had antibiotics, or if your baby has, diaper rash can develop into thrush (see here).\n\nTry home prescribing for a mild or non-recurrent diaper rash, but seek professional advice if it fails to clear quickly or recurs frequently.\n\nDo\n\n\u2022 change diapers frequently (more frequently if a rash develops) and use almond or olive oil, or Calendula or Symphytum ointment after every change if your baby has a tendency to nappy rash.\n\n\u2022 wash and rinse your baby's genitals and bottom thoroughly with water only (not soap) paying special attention to all creases, wiping from front to back in girls (but do not clean inside their inner vaginal lips).\n\n\u2022 dry the area thoroughly. A hair-drier on a very gentle heat (at a safe distance) is ideal as no further moisture will remain on the skin, but test the heat on your own skin first (on your inner forearm) and remember that a baby's skin is much more sensitive than an adult's.\n\n\u2022 leave your baby's bottom without a diaper for as long as possible (overnight if practical), making sure that the room is warm enough. Lay her on several thicknesses of towel (with a plastic sheet underneath) to soak up the urine.\n\n\u2022 stop using plastic pants if a rash starts. They make diaper rash worse because they stop air getting to the skin.\n\n\u2022 try a different type of liner or disposable diaper, preferably a more absorbent brand.\n\n\u2022 wash towelling diapers in soap powder instead of biological powder and put a little vinegar into the final rinse to neutralise the acids in urine.\n\n\u2022 change to disposable diapers until the rash has cleared up.\n\n\u2022 cut out all fruit juice. It makes urine acidic, which stings in contact with the rash when the baby pees. Give plain water instead.\n\n\u2022 smear a coating of raw egg white on the skin after washing and drying. This makes an effective waterproof seal over a mild but stubborn diaper rash. Don't apply it to broken skin and apply it cautiously if your baby is allergic to eggs.\n\nDon't\n\n\u2022 use zinc and castor-oil cream or petroleum-based products. The chemicals in them are absorbed through the skin and can cause minor health problems as zinc can affect the nervous system.\n\nPRICKLY HEAT\n\nPrickly heat occurs in babies if they are too hot, either from the weather or because they have been overdressed. It starts as small red patches, usually on the neck, chest and upper arms, and itches and prickles which can make small babies restless and distressed.\n\nDo\n\n\u2022 remove the top layer of clothing or bath the baby in a tepid or cool bath, or sponge him or her down and let them dry in the air.\n\n\u2022 avoid direct exposure to the sun.\n\nDon't\n\n\u2022 use bubble bath, oils or scented soap until the rash has cleared up.\n\n\u2022 use talcum powder as this can block the pores and aggravate the condition.\n\nSeek help if\n\n\u2022 the itching is intolerable and homeopathic treatment and self-help measures don't work.\n\n\u2022 your baby becomes lethargic and floppy.\n\nRETENTION OF URINE\n\nA baby can take up to 36 hours to pee after the birth and the first urine may be red because of urate crystals, which are completely harmless. A baby may also find it difficult to pee after a fright or getting chilled.\n\nMost babies pee at every feed although others do so less often. You will automatically become familiar with your baby's pattern from how often diapers need changing. It is a variation from the normal pattern that is significant. If there are no wet diapers for an unusually long time and the baby is drinking as normal something may be wrong.\n\nIf your baby takes longer than 24 hours to urinate for the first time and still seems shocked from the birth, choose between Aconite and Opium. The right remedy will help with shock and get him or her peeing. Otherwise, repertorise carefully if the retention is linked to something else.\n\nDo\n\n\u2022 encourage a baby (not necessarily a newborn) to pee by: putting them in a warm bath; taking off the nappy, putting the baby on a towel in the bathroom and running the tap as an added encouragement.\n\nSeek help if\n\n\u2022 your baby hasn't passed any urine for 15\u201320 hours.\n\n\u2022 urine retention accompanies a fever.\n\n\u2022 you suspect the baby is in pain \u2013 is crying as if distressed.\n\n\u2022 the baby is vomiting.\n\nSLEEP DIFFICULTIES\n\nBabies need differing amounts of sleep. Some fall into good sleep patterns very early on while others take a lot of persuading. Sleeping is a skill which some find easier to acquire than others!\n\nYou may choose to have the baby sleep with you in your bed. If he or she sleeps peacefully and you are able to sleep well with them in the bed, this is fine. In countries where cot deaths are virtually unknown babies traditionally sleep with their parents for at least the first year of life, which has its advantages. For parents who work, it is one way for them and the baby to be together, a means of bonding. If you are breastfeeding it is easy to let your baby feed while you are half asleep. Parents do not roll on to their babies if they sleep with them \u2013 unless they are drunk. A breastfed baby will tend to wake more frequently: a feed is digested in about two hours because the fat content in human milk is lower than that of cow's milk so being able to feed and fall asleep again straight away will help you avoid the exhaustion experienced by someone who has to get up several times a night.\n\nHowever, some people cannot sleep deeply when their babies are in bed with them. Some mothers are light sleepers and wake with every whimper. Some babies sleep restlessly, or jerk and wriggle during the night and always end up with their feet in your face no matter what you do. Some women cannot breastfeed happily lying down or cannot breastfeed and sleep at the same time. Some parents do not want that sort of relationship with their babies. Fine.\n\nDecide what will work and is right for you. If you try it and it doesn't work you can always try something else. Do not be persuaded to take your baby into your bed if you don't want to. Conversely, do not be told that you are making a rod for your own back if your baby is sleeping with you and you are all enjoying it. If you suspect that your family might be scandalised if they knew, don't tell them. It's not worth it.\n\nYou can be creative and flexible about night-times. If you decide that your baby will sleep separately, you may make a proviso that you will sleep with him or her when they are ill. You can put a largish bed or mattress on the floor of the baby's room so that you can feed lying down in the night, which is more restful for you, but not in your bed. You may both end up sleeping the rest of the night together or you can pop the baby back in the cot and go back to your own bed. You could put the cot next to your bed so that you can reach out a comforting hand in the night if the baby whimpers and wakes you.\n\nThe following guidelines apply to all babies, whether they are sleeping in your bed or not. If you want your baby to sleep well and easily at night, start by trying to establish a convenient sleeping pattern as early as you can; you should adopt some of the following ideas wherever you all sleep. Symptoms to consider when you repertorise if you decide to prescribe are insomnia, jerking on falling sleep, jerking during sleep, screaming on waking and screaming during sleep, but seek the advice of a homeopath if your baby persists in waking frequently at night, especially if distressed, or you find it impossible to cope with being woken every two hours every night, even though the baby is perfectly well.\n\nDo\n\n\u2022 make sure a baby has done all the sucking she needs before you tuck her up for the night. Some need a lot of sucking, whether they are breast- or bottle-feeding. Get a teat with a smaller hole for your bottle or let your baby suck your nipple until he or she lets go of their own accord, or offer a dummy.\n\n\u2022 try moving the cot into another room if the baby wakes frequently during the night, especially if a light sleeper. Some babies are disturbed by a parent moving about, or grunting or snoring.\n\n\u2022 set your own priorities first. If you are someone who needs plenty of sleep, try to settle your baby's sleeping pattern to suit you straight away. Start as you mean to go on. Be clear and determined.\n\n\u2022 establish a bedtime routine or ritual: choose a time that suits you and your partner, bearing in mind that babies who go to sleep early wake up early!; always use the same room and the same cot or crib for the night-time sleep not the pram or carrycot used for day-time naps. You can put the Moses basket in the cot in the early days; bath or sponge the baby down, making sure that this is a pleasure, that he or she isn't too tired, that you only do it if they enjoy it; give a soothing massage with a light oil or talcum powder; dress him or her in 'night' clothes \u2013 a nightie, which makes changing nappies easier, a sleep suit or pyjamas, which you have warmed on the radiator (only night clothes get warmed!); swaddle a jumpy baby firmly, which may help one who jerks in their sleep not to wake up (some babies hate being swaddled \u2013 yours will let you know if he or she doesn't like it); lay the baby down to sleep in the position they prefer (you will get to know which it is) place a rolled blanket or cushion against the feet and head so that he or she feels secure all around as they were in the uterus. If she sleeps better on her side put a rolled-up blanket along her back so that she can't roll on to her front. Turn the lights down and draw the curtains if it is still light outside; make sure that the bedtime routine occurs quietly and calmly; always do the same reassuringly relaxing things \u2013 play a certain tape, sing certain songs, arrange the animals in the cot, tell a bedtime story in a bedtime tone of voice; warm the cot with a hot-water bottle, which you can remove just before you put the baby in; give the last feed in a special bedtime chair; make sure he or she is awake when you put them to bed after the last feed of the day, and at some other times during the day. Falling asleep is a learnt skill. You provide the environment and the encouragement and they do it; tuck them up with a special night-time blanket and\/or cuddly toy; say the same goodnight words every night and then, instead of walking straight out, potter about tidying up for a few minutes; leave a night-light on and a music tape or a tape of womb sounds.\n\n\u2022 go to the baby if he or she wakes in the night, but make your visits short and boring. Feed, change if absolutely necessary and, again, put him or her down awake so that they fall asleep on their own.\n\n\u2022 soothe a distressed baby at bedtime by rocking or rubbing for a bit until they calm down; then put them in the cot, preferably still half-awake.\n\n\u2022 let other people put your baby to sleep at night from the beginning \u2013 your partner if willing and able, or a friend or neighbour who wants to babysit. If you are breastfeeding express your milk so that someone else can do the night-time feed.\n\n\u2022 make your baby's cot attractive: hang a mobile above it, out of reach; string some interesting, colourful objects across it some of which make noises, such as bells, when they move and change them regularly. Hang them so that the baby can touch them as soon as he or she is able to but make sure they are absolutely safe. Clip an 'activity centre', a mirror or colourful pictures, to the cot bars or tie a soft toy to them.\n\n\u2022 buy a specially treated lambskin for your baby to sleep on.\n\n\u2022 encourage thumb-sucking from early on: many babies will comfort themselves like this if they need to in the middle of the night. You may prefer to use a pacifier but this needs putting in every time it falls out.\n\n\u2022 take it in turns with your partner to get up in the morning for your baby so that one of you can have an extra few hours' sleep.\n\n\u2022 cut out caffeine (tea, coffee, chocolate, Coca-Cola), especially after midday if you are breastfeeding as the caffeine passes through to your baby in your milk.\n\n\u2022 anticipate your baby's needs when you travel so that you take, for instance, a night-light, the mobile from the cot, favourite toys, a tape-recorder and tapes, as well as bedding so that the bedtime routine can be similar wherever you are.\n\n\u2022 give a bottle of formula milk last thing at night if you are frazzled and your milk supply seems low. It tends to decrease through the day, especially if you have been super-busy; if the last feed you give is a small one then your baby will wake after a relatively short time and before long you will both be desperate. Seek the advice of a breastfeeding counsellor if you want to avoid this (see also here).\n\n\u2022 be careful not to cut out day-time naps for babies who sleep badly at night: you may inadvertently over-stimulate them so that they find it difficult to fall into a deep sleep, waking more frequently at night. If your baby regularly wakes to play in the night then he or she may be 'hyped-up'. Encourage them to sleep more, not less and bring bedtime forward.\n\nDon't\n\n\u2022 always breastfeed your baby to sleep or let him or her fall asleep on the bottle. If you help them to fall asleep they won't learn to do it alone: you may end up spending 2 hours a night putting the baby to bed and although this is acceptable occasionally, it becomes a trial if it happens routinely.\n\n\u2022 always rock or walk your baby to sleep for the same reasons as above.\n\n\u2022 let babies under the age of two sleep with a pillow, as it could smother them.\n\n\u2022 make your house specially quiet so that the baby can get to sleep. Small babies will sleep through all levels of noise and should be encouraged to do so. Otherwise, you will find yourself creeping around for years to come.\n\n\u2022 worry if you can't get back to sleep after a night feed. Be philosophical and catch up on some reading, write a letter or even do some housework. Catch up on your sleep the next day when the baby has a nap.\n\n\u2022 let an older baby suck fruit juice from a bottle at night. They become addicted to it and it will ruin their teeth when they come through. It is better to offer water in a bottle to quench thirst if a feed isn't due or being demanded.\n\n\u2022 put your baby on solids too early, without consulting your doctor and\/or health visitor, because you think waking at night means hunger.\n\nSORE THROAT\n\nA sore throat is hard to diagnose in a small baby. The voice may become either a little hoarse or lower in tone. They may cry when they try to take either solids or liquids or both, because swallowing is painful, and the cry may have a hoarse ring. On examination your doctor may find that the glands in the neck and tonsils are swollen. If an ear is inflamed, the pain on swallowing will be felt there as well as in the throat.\n\nIf you decide to prescribe, it is important that you observe whether the pain is worse for swallowing or better for it and whether hot or cold drinks ease the pain. A child with a sore throat and earache will almost certainly have a fever so you should have a number of symptoms to hang your prescription on. Take into account anything that happened before he or she fell ill, such as getting cold and wet, having a fall, or starting with a new childminder. If the illness coincides with an inoculation, up to 2 weeks later, you should consult a professional homeopath who will prescribe carefully to clear any ill effects from the inoculation, which is beyond the scope of the home prescriber.\n\nDo\n\n\u2022 offer plenty of fluids, breastmilk if you are still nursing, a little honey and freshly squeezed lemon or orange juice with boiled and cooled water in a bottle, on a spoon, or in a feeder beaker.\n\nSeek help if\n\n\u2022 there is excessive dribbling, difficulty in breathing and great pain with swallowing.\n\n\u2022 your child is very distressed, obviously in pain and not reacting to home prescribing quite quickly, within a day or two at the most.\n\nSPOTS\n\nMost newborn babies go through a spotty phase. The spots come just when you are beginning to feel like showing the baby off, when both of you have recovered from the birth. The most common rash is neonatal (newborn) urticaria which occurs a week or two after birth. It can range from a few reddish spots on the face which clear up quite quickly to a bad attack of 'acne', red spots with yellow centres that spread on to the body and last for a week or so. One theory suggests that the baby's body is clearing out the toxins from the birth.\n\nAnother common rash, 'milk rash', manifests as tiny white spots usually over the nose and sometimes on the cheeks. They can last much longer but are not as unsightly and will disappear in their own time.\n\nIf the spots are particularly severe give a short course of Sulphur or Silica \u2013 choose between them by reading the pictures in the Materia Medica \u2013 or Carbo vegetabilis if your baby is also windy.\n\nDo\n\n\u2022 keep your baby's face clean with water only. Don't use oil or talcum powders as these will further clog the pores.\n\nDon't\n\n\u2022 squeeze your baby's spots as you may cause an infection.\n\nSTICKY EYES\n\nSee also Blocked Tear Duct.\n\nA sticky eye, which discharges or in which the lids are stuck together after sleep, is not necessarily infected. A sticky eye in a small baby needs professional help if it doesn't clear quickly and easily within 3 days.\n\nA sticky eye will usually clear up with external bathing alone so avoid using an antibiotic cream without trying an alternative method first.\n\nDo\n\n\u2022 be scrupulously hygienic in dealing with your baby's eyes. Wash your hands before and after touching them: eye infections spread all too easily.\n\n\u2022 when bathing or cleaning your baby's eyes do one at a time using a clean piece of cotton wool for each eye.\n\n\u2022 bathe the eyes with Euphrasia or Hypercal\u00ae and give the indicated internal remedy as well.\n\n\u2022 bathe the eyes with a saline solution (1 teaspoon of sea salt in a glass of warm, boiled water) or herb tea \u2013 chamomile or eyebright (Euphrasia).\n\n\u2022 bathe your baby's eyes with a little freshly expressed breastmilk \u2013 this can act miraculously quickly in clearing up a mild sticky eye.\n\n\u2022 wipe from the inner corner of the eye to the outer corner.\n\nDon't\n\n\u2022 let your baby rub his or her eyes as this can make it worse.\n\n\u2022 ever use antibiotic or hydrocortisone drops in the eye unless prescribed for a specific complaint.\n\nSeek help if\n\n\u2022 the eyelids become red and puffy, the eye becomes bloodshot and the discharge is yellow.\n\n\u2022 the symptom persists, especially in a newborn baby. Ask your doctor to check that the discharge is not serious.\n\nSTOMACH-ACHE\n\nSee Colic.\n\nTEETHING\n\nSome children produce teeth without any fuss or discomfort whereas others become a nightmare for their entire teething period, suffering great pain from sore, swollen gums, as well as producing colds, coughs, earache, diarrhoea, mood swings and sleeplessness with every tooth that comes through.\n\nHomeopathic treatment can ease teething pain as well as helping to push out teeth that are having difficulty in erupting of their own accord. If the remedy you prescribe fails to help, consult a professional homeopath.\n\nDo\n\n\u2022 rub your baby's gums with your finger as pressure sometimes helps.\n\n\u2022 give something hard to chew on: a teething ring which has been kept in the fridge (not the freezer) is ideal. If you give a piece of cold carrot or celery remember to watch carefully to prevent the baby choking on little bits. Home-made rusks (hunks of bread baked hard in a slow oven) are also good for teething gums to chew on.\n\nDon't\n\n\u2022 resort to giving Phenergan or other sedatives.\n\n\u2022 give gripe water as it is always high in sugar and sometimes alcohol. The maximum recommended daily dose for an 8\u00bd-pound (4-kg) baby is the equivalent to five tots of whisky drunk by an 11-stone (70-kg) adult.\n\n\u2022 give too much chamomile tea. It is easy for a baby to 'prove' it (see here), becoming irritable and sleepless, the symptoms it was supposed to cure. If it does help, use only while it has a soothing effect.\n\n\u2022 give chamomile tea if you have prescribed Chamomilla in potency (see here).\n\nTHRUSH\n\nThrush is an infection caused by a yeast that ordinarily lives quite harmlessly in the mouth and intestines. It can, however, get out of control, especially if either mother or baby has recently had antibiotics, producing white patches that are usually sore and\/or itchy. It is not serious and is relatively common in young babies.\n\nGenital Thrush\/Yeast\n\nGenital thrush in babies is often confused with diaper rash (see here). The main difference is that thrush is more likely to start around the anus and spread outwards whereas diaper rash tends to start in the creases, where the urine-soaked nappy rubs against the skin. Thrush tends to spread up to the navel whereas diaper rash won't.\n\nDo\n\n\u2022 change the baby's diaper frequently and wash his or her bottom in water or Hypercal\u00ae lotion (see here). Don't use soap and dry it well, using a hair-drier on a low setting.\n\n\u2022 let your baby lie diaperless as often and as long as is practical.\n\n\u2022 add half a cup of cider vinegar (always useful in combating fungal infection) to the bathwater.\n\n\u2022 wash towelling diapers and muslin liners in non-biological soap powder or unscented soap flakes and rinse well, putting some cider vinegar into the last rinse.\n\n\u2022 apply live yoghurt to ease itching and discomfort.\n\n\u2022 use separate towels for yourself and your baby to avoid spreading the thrush further.\n\nDon't\n\n\u2022 use plastic pants over your baby's nappies until the rash has cleared up as they will stop air from circulating and aggravate the problem.\n\n\u2022 use bubble baths or oils in the bathwater.\n\n\u2022 use talcum powder.\n\nSeek help if\n\n\u2022 pus forms.\n\n\u2022 the thrush doesn't respond to home treatment within a few days.\n\nOral thrush\n\nThrush in the mouth is characterised by white patches on the tongue or the inside of the cheeks, which resemble milk spots but don't come off if wiped gently with a clean cloth. It may be sore and\/or itchy.\n\nDo\n\n\u2022 give bland foods that are easy to eat to a baby who is on solids, such as soups, liquidised foods, ice cream, and yoghurt \u2013 which is also helpful in combating yeast infections.\n\n\u2022 give all foods cold or cooled as hot food can aggravate the soreness.\n\n\u2022 avoid strong-tasting or acidic foods, including fruit and fruit juices, as the acid can make the thrush patches sting.\n\n\u2022 give drinks from a bottle, feeder beaker or straw.\n\n\u2022 sterilise pacifiers and teats very carefully.\n\n\u2022 buy a specially soft teat for bottle-fed babies.\n\n\u2022 wash your own nipples thoroughly after each feed if you are breastfeeding as it is possible for them to become infected. Use water only, not soap.\n\n\u2022 wipe the inside of your baby's mouth with a cloth soaked in a solution of water and cider vinegar or bicarbonate of soda (\u00bc pint\/125ml of boiled and cooled water to a teaspoon of vinegar or bicarb). You can use this on your nipples too.\n\nDon't\n\n\u2022 wear breast pads if you are breastfeeding.\n\n\u2022 use a medicated mouthwash as this can destroy healthy bacteria and further upset the balance in the mouth.\n\nSeek help if\n\n\u2022 pus forms.\n\n\u2022 the infection does not respond to home treatment and your baby is miserable.\n\n\u2022 your nipples become infected.\n\nTONSILLITIS\n\nSee Sore Throat.\n\nTRAVEL SICKNESS\n\nAlthough most babies are magically tranquillised by car journeys, some hate them \u2013 I suspect because they feel sick. If your baby usually (or even occasionally) vomits in the car and screams when put in it, travel sickness may be the reason.\n\nI have included the following remedies for simple travel sickness: Borax, Cocculus, Nux vomica, Petroleum, Sepia, Staphysagria, and Tabacum. If you can fit your baby into a particular type \u2013 say Borax or Nux vomica \u2013 then, fine. Otherwise it will be pot luck, I'm afraid, with Tabacum or Petroleum being the two main car-sickness remedies.\n\nDo\n\n\u2022 set your baby's car seat facing towards the back (instead of the front) and tip it so that he or she is lying with their head back.\n\n\u2022 try putting the seat in the front where the baby can see you.\n\n\u2022 play story tapes or music (try very loud music) as a distraction to the baby.\n\n\u2022 give a pacifier or a safe toy to a young baby as sucking can help. Give a bottle of milk or water to an older baby.\n\n\u2022 open the window next to your baby's seat as fresh air sometimes cures travel sickness.\n\n\u2022 talk reassuringly.\n\n\u2022 drive smoothly, stopping and starting gently as jerky driving can make the most stalwart feel nauseous.\n\n\u2022 make frequent stops.\n\nDon't\n\n\u2022 smoke in the car.\n\n\u2022 get upset with your baby if he or she vomits in the car as it will only make them worse the next time.\n\nVACCINATION\n\nIt is beyond the scope of this book to go into the issue of vaccination. Alternative health care practitioners are mostly cautious about their supposed benefits, observing that the health of children and adults who have been heavily vaccinated is often very poor. Every homeopath has seen formerly healthy children become chronically ill after certain vaccinations.\n\nHowever, all parents take a risk, whether they choose to immunise their children or not. Our responsibility as parents is to weigh up those risks to minimise the trauma to our children. You will need to research the issue thoroughly by reading through some of the literature now available. You may also choose to seek the advice of a homeopath who offers vaccination counselling \u2013 not all do. In any case, it is essential that you get alternative medical support for your child or children should you decide not to immunise in some or all instances.\n\nIf you do vaccinate you can prescribe on your child if the vaccination itself is distressing. Give Aconite if there is shock, Arnica for bruising at the site of the injection, Staphysagria if there is anger, betrayal and hurt resentment, or Stramonium if your child suffers from nightmares afterwards.\n\nSeek help if\n\n\u2022 you suspect or know instinctively that your child's health has suffered as a result of immunisations. It is beyond the scope of the home prescriber to treat these effects: a professional homeopath will be able to help your child regain his or her former vitality. Report any symptoms that develop as a result of vaccination in writing to your doctor.\n\nVOMITING\n\nSee also Diarrhoea.\n\nSmall babies frequently 'posset' (throw up) small amounts of their feeds. This is normal and is not classified as vomiting. Very rarely, babies suffer from pyloric stenosis, an unpleasant congenital disorder which is projectile vomiting, when the stomach contents are hurled violently across the room. You must not attempt to deal with this condition yourself.\n\nBabies up to a year old can have a mild stomach bug which lasts for 24 hours without any serious side effects. If your baby is vomiting frequently and throws a fever, gastro-enteritis, or food poisoning, is likely. This is rare in breast-fed babies. Babies who are bottle-fed or who have been weaned and contract it are seriously at risk from rapid dehydration.\n\nDo be particularly careful when weaning your baby on to solid foods by introducing bland, easily digestible foods first. Avoid meat until you are sure that his or her stomach can take it.\n\nDo\n\n\u2022 hold your baby while he or she is vomiting as it can be frightening. You can hold the stomach with one hand and the forehead with the other.\n\n\u2022 place a bowl or bucket nearby.\n\n\u2022 once the vomiting is over cuddle the baby reassuringly.\n\n\u2022 sponge his or her face if they find it soothing.\n\n\u2022 encourage the baby to take a little water, firstly to rinse the mouth and then to drink, to replace some of the lost fluids. Don't let them take too much until you are sure it is staying down.\n\n\u2022 encourage your child to drink, if old enough, by being creative about how you offer it. Let them choose a favourite drink, offer it in a small cup \u2013 an egg cup or even a doll's tea-set cup; try a straw; offer a bottle if he or she has just given it up; freeze a favourite juice or juices and offer it in the form of ice cubes or an ice lolly.\n\n\u2022 encourage rest. You may need to carry a small baby around for much of the day, or at least lie down with him or her.\n\n\u2022 avoid all food until vomiting stops. Most babies won't eat until they feel better.\n\n\u2022 carry on breastfeeding if you are still nursing your baby and don't worry if a lot of each feed comes up. The usual number of wet diapers means the baby is getting enough fluids.\n\n\u2022 encourage a baby who is vomiting a lot to take as much as 2\u20133 pints (1\u00bd litres) of fluids a day. You can give diluted formula for a day or two until the vomiting has stopped or cut it out altogether and give rehydration fluid (see here).\n\nSeek help if\n\n\u2022 your baby shows any signs of dehydration: dry mouth and lips; dark, concentrated urine; dry diapers, no urine for longer than six hours; sunken eyes; sunken fontanelle; abnormal drowsiness or lethargy.\n\n\u2022 your baby is screaming inconsolably.\n\n\u2022 your baby is vomiting incessantly.\n\n\u2022 there is blood in the vomit or if it looks dark in colour.\n\n\u2022 your baby vomits after a fall on the head.\n\n\u2022 your baby is vomiting a fair amount without being ill and is not gaining weight.\n\nWEIGHT GAIN\n\nBabies vary greatly in the rate at which they gain weight. I have seen fat breastfed babies and thin bottle-fed babies.\n\nSome brands of formula do not agree with some bottle-fed babies. Some are sensitive to lactose in cow's milk, and usually produce some physical symptoms such as iarrhoea (frothy, acidic stools), constant coughs, colds, snuffles, colic or eczema. If you suspect that your baby is sensitive to cow's milk do seek the advice of a professional homeopath, who will help build up your child's immunity through constitutional treatment, which, in the long term, may enable them to tolerate lactose. Consult your health visitor about other brands of formula or the possibility of feeding soya milk.\n\nIn breastfed babies, a low weight gain is more complicated. Some women produce a smaller volume of milk; some may have a delay in the let-down reflex so that their babies only get the foremilk and not the fat-rich hindmilk. I have included some remedies and guidelines for increasing the volume of breastmilk (see Repertory and here).\n\nSometimes a bowel infection can be the cause of a low weight gain: your doctor or hospital consultant will check for this. Older babies are sometimes unable to digest the gluten in wheat cereals or rusks, so if you are concerned, see your doctor or health visitor.\n\nUnless the baby is otherwise fit and healthy, low weight gain usually needs the attention of a professional homeopath. However, if one of the remedy pictures in this book matches your own baby, including the general symptoms, physical complaints and emotional state, give it in a low dose, repeating it from time to time if it is truly effective. Remedies for children who have difficulty in gaining weight are Baryta carbonica, Calcarea phosphorica, Silica, Calcarea carbonica and Magnesium carbonate, the last two also being indicated for children who have difficulty in digesting cow's milk.\n\nDo\n\n\u2022 take into account the weight and size of both parents. Babies of small parents, or of parents who were small babies, will tend to inherit this characteristic.\n\n\u2022 change a bottle-fed baby's brand of formula \u2013 try one made with soya if you have been giving cow's milk.\n\nDon't\n\n\u2022 worry if your baby is thin, healthy and developing normally in all other respects.\n\n\u2022 let yourself be persuaded into believing that you are a bad parent.\n\n\u2022 give your baby cow's milk if you or the baby's father are known to be sensitive to lactose.\n\n\u2022 introduce wheat until you are sure your child's digestion can cope with it. Many doctors are recommending that parents wait until their babies are a year old before introducing it. Give cereals made from oats, millet, corn and rice instead.\n\nTHE MATERIA MEDICAS AND REPERTORIES\n\n* * *\n\nEXTERNAL MATERIA MEDICA\n\n* * *\n\nINTRODUCTION\n\nExternal remedies are not true homeopathic remedies; they have not been proved, or tested, on healthy individuals to establish their symptom pictures, and they have not been potentised. They are basically plant extracts and the indications for their use have developed out of herbal lore over thousands of years. I have chosen the remedies I have found most effective from my own experience.\n\nThe remedies are listed in alphabetical order, and precise instructions are given for using each one. The symptoms, or complaints, for each remedy are also listed alphabetically within each entry.\n\nExternal remedies come in various forms (see below) and next to each symptom are instructions for the most appropriate form to use. In the entry for Arnica, for example, I suggest you use Arnica oil or ointment for bruised muscles and Arnica tincture for a wasp sting.\n\nTinctures, oils, ointments and creams are sold by homeopathic pharmacies (and sometimes stocked by ordinary chemists). Lotions are usually made up at home by diluting tinctures, and instructions for these are given below.\n\nTinctures\n\nA tincture is a solution of the plant in alcohol \u2013 usually a 1 in 10 dilution. It is prepared by homeopathic pharmacies.\n\nLotions\n\nA lotion is simply a dilution of tincture in water, used for burns, gargling or douching. Lotions do not keep so you should keep a stock of tinctures and make up lotions as and when you need them.\n\nBasic lotion: Dilute 5 drops of tincture with 1 tablespoon of cooled, boiled water, or for a larger quantity use 40 drops to \u00bc pint (40 drops = approximately \u00bd teaspoon).\n\nStrong lotion: Dilute 10 drops of tincture with 1 tablespoon of cooled, boiled water, or 40 drops to 1\u20448 pint. Use only where indicated in the External Materia Medica.\n\nEyebaths\n\nAdd 2 drops of tincture to an eyebath of cooled, boiled water and use in the normal way.\n\nOintments\n\nTo make an ointment, the tincture is incorporated into a lanolin base, which serves to seal cuts from dirt. Ointments are not water-soluble and so do not wash off easily. Some people are allergic to lanolin (particularly those sensitive to wool), so it's worth testing it overnight on a small patch of skin. If this produces any redness or irritation, avoid using ointments.\n\nCreams\n\nA cream is a tincture in an aqueous (water-soluble) base, which washes off easily and isn't sticky. It is easily absorbed by the skin and is good for areas that are not going to get wet, and for people who don't like sticky ointments.\n\nOils\n\nThe plant material is crushed and macerated in oil and allowed to stand for a period of time before being strained ready for use. I have indicated in each instance where an oil might be more useful than, say, a cream or a lotion.\n\nNB Before prescribing for any complaint it is essential to read the general advice.\n\nAESCULUS AND HAMAMELIS (Aesc.\/Ham.)\n\nPiles\n\nA combination of these two herbs in an ointment or cream provides relief from painful piles, but you should seek professional help so that they can be treated 'from the inside' to prevent their recurrence.\n\nARNICA (Arn.)\n\nWarning: never apply Arnica externally to open wounds, cuts or grazes \u2013 that is, to broken skin \u2013 as it can cause a nasty rash.\n\nBruises\n\nApply the ointment, cream or lotion directly to the affected part (remembering to use it only on unbroken skin) as soon as possible. If you can do this before the bruise has started to discolour (even if it has already swollen), it will simply be reabsorbed by the body, especially if you take Arnica internally as well. Rub ointment or cream in gently, or if you are using lotion apply it on a piece of lint or gauze and keep in place until the swelling has subsided \u2013 usually a matter of several hours.\n\nSore muscles\n\nRub Arnica ointment or oil into sore, bruised muscles after exertion (such as gardening or skiing) and take the appropriate internal remedy if your symptoms are severe.\n\nSprains\/strains (first stage)\n\nRub in Arnica ointment or cream, or wrap the sprained joint in a lotion-soaked bandage. This will deal with the initial swelling.\n\nWasp stings\n\nDab the wound with neat tincture immediately after being stung.\n\nCALENDULA (Calen.)\n\nWarning: Calendula helps the layers of the skin (the epithelium) to 'knit' back together and will mend a clean wound in a matter of hours. It heals so rapidly that it can seal dirt into the body, so always clean the wound very carefully before applying Calendula.\n\nBurns\/scalds (second degree)\n\nUse Calendula cream or lotion for the later stages of a burn once the pain has passed. Calendula will promote new skin growth and is especially useful where blisters have broken.\n\nChildbirth\n\nMassage Calendula oil into the perineum during labour to soften the area and to help make an episiotomy unnecessary.\n\nCracked nipples\n\nIf Phytolacca has failed to help, apply Calendula ointment, or cream if sensitive to lanolin, to heal cracked, painful nipples.\n\nCuts\/wounds\n\nApply ointment or cream to minor cuts, and bandage if necessary. For serious wounds apply lotion on a piece of lint or gauze and keep in place. Use a plant spray filled with lotion to keep the dressing damp but do not remove the dressing until the bleeding has stopped and healing is well under way.\n\nEczema\/rashes\n\nCalendula lotion or cream is especially useful for soothing eczema or rashes where the skin has cracked or been scratched raw. It will not treat the underlying cause of the eczema; you should seek professional help for this.\n\nHandcream\n\nCalendula cream makes a marvellous handcream especially where there may be little cuts in the skin.\n\nMouthwash\n\nUse a strong lotion after dental treatment, or for bleeding gums.\n\nNappy rash (Diaper rash)\n\nUse ointment or cream several times daily, making sure that the whole area is clean and dry first (wash with water and a mild, unscented soap). See also Symphytum.\n\nSunburn\n\nUse the lotion or cream (see BURNS).\n\nThrush\/Yeast infection\n\nFor vaginal thrush douche with the following mixture to relieve soreness and itching: make 1 pint of chamomile tea (1 pint of boiling water to 1 tablespoon of dried chamomile leaves or one chamomile teabag). Leave to cool, strain, add 40 drops of Calendula tincture and douche twice daily for up to a week only. You can buy a reusable douche from larger chemists (not the disposable type which comes with its own solution). Douching will not cure the complaint; it will only help during the acute phase (see here), and you should seek professional help.\n\nEUPHRASIA (Euphr.)\n\nEye infections\/inflammations\/injuries\n\nUse Euphrasia whenever the eye needs bathing \u2013 whether it is sore after the removal of dirt or grit, after swimming in a chlorinated pool, when irritated by hayfever or when actually infected or inflamed.\n\nIf Euphrasia doesn't help, use Hypercal tincture (see here). Some people find it more effective, especially in the case of infection.\n\nIt is essential to use cooled, boiled water in an eyebath, and to clean the eyebath itself with boiling water after each eye is bathed to prevent the spread of infection.\n\nHAMAMELIS (Ham.)\n\nHamamelis, or witch hazel, is widely sold in chemists as distilled witch hazel, but it is neither as astringent nor as effective as the tincture available from homeopathic pharmacies. Use the distilled form if this is all you can find.\n\nBruises\n\nHamamelis is useful for bruises where the skin has been broken (Arnica is for use on unbroken skin). Use Hamamelis in the same way as Arnica, as an ointment, cream or lotion.\n\nPiles\n\nApply a compress of the lotion (you can use a small sanitary towel or a strip of cotton wool) to provide instant relief from pain. Keep in place for a while (up to an hour, twice daily) to reduce inflammation.\n\nAesculus and Hamamelis ointment or cream may be applied as often as necessary. It is also important that the correct internal homeopathic remedy is given so that the piles are treated from the inside and real healing can take place.\n\nVaricose veins\n\nApply the lotion to varicose veins, especially painful ones, by wrapping the leg in lotion-soaked bandages. Leave in place for as long as possible, until the discomfort eases and then use only when needed. An elastic (tube) bandage over the top will keep bandages in place.\n\nHYPERCAL\u00ae\n\nThis is a mixture of Calendula and Hypericum; the combined healing qualities of the two plants make it especially effective in soothing and healing wounds.\n\nChildbirth\n\nAfter childbirth Hypercal\u00ae will help to heal a cut or torn perineum. Apply a strong lotion on a small pad or compress to the affected area, keeping it in place for up to an hour at a time, and repeating every four hours for several days.\n\nCold sores\n\nDilute one part tincture to three parts cooled, boiled water and apply this strong lotion frequently to cold sores as soon as they appear, or use ointment. Take the appropriate internal remedy at the same time.\n\nCuts\/wounds\n\nUse ointment, cream or lotion to heal wounds just as you would use Calendula or Hypericum on their own.\n\nSoak cut fingers, toes or elbows in a basin of water into which a teaspoon of Hypercal\u00ae tincture has been added and gently remove any bits of dirt. The clean wound can then be dressed with a smear of the cream or ointment.\n\nEye infections\/inflammations\/injuries\n\nAs an alternative to Euphrasia, use 2 drops in an eyebath to help clear inflammation caused by dust, foreign bodies, infection or injury. Seek professional help if the soreness persists.\n\nMouth ulcers\n\nUse the mouthwash below frequently, as well as taking the appropriate internal remedy.\n\nMouthwash\n\nThis mouthwash is good for mouth ulcers, inflamed, sore or spongy gums. Make a strong lotion by diluting 40 drops of tincture in 1\u20448 pint and swoosh it well around the mouth after brushing your teeth, then massage it into the gums with your fingers.\n\nSore throat\n\nDissolve 1 teaspoon of sea salt in \u00bc pint of hot water. Add 40 drops of Hypercal\u00ae tincture and gargle as frequently as necessary.\n\nHYPERICUM (Hyp.)\n\nHypericum soothes and heals wounds, especially where nerves have been damaged and the injury is painful. The pains of a 'Hypericum wound' are typically shooting and\/or severe.\n\nBurns (second degree, with blistering)\n\nSoak gauze strips or lint in Hypericum lotion, wring out and lay over the burned area. Keep the bandages damp by spraying the area with the lotion. Do not remove the cloth until the pain has ceased. Hypericum is also useful in the first stage of a burn on a nerve-rich and therefore very painful part. Give the appropriate internal remedy.\n\nCuts\/wounds\n\nUse lotion to bathe and clean dirty cuts\/wounds, and apply ointment before bandaging. Hypericum is especially good where there are shooting pains in or around the wound and for injuries to nerve-rich parts (crushed fingers and toes). If a compress is applied to a crushed finger or toe and kept damp for a few days, a damaged nail can be prevented from taking an odd shape once healed.\n\nInsect bites\n\nUse neat Hypericum tincture on any insect bite. If swelling persists, apply the lotion as a compress, and keep in place for as long as it takes for the swelling to diminish.\n\nPiles\n\nUse a compress of Hypericum lotion (or ointment if preferred) for bleeding piles with severe shooting pains. Repeat as necessary, but seek professional help.\n\nSunburn\n\nSee BURNS.\n\nLEDUM (Led.)\n\nInsect bites\/stings\n\nUse Ledum tincture neat on insect bites and stings to prevent swelling and itching.\n\nMany homeopathic pharmacies have their own preparations for relieving bites and stings. These are generally mixtures of a number of remedies and are applied neat as above.\n\nIf you are often bitten and are sensitive to insect bites, you will benefit from constitutional homeopathic treatment (see here), especially if your response is extremely severe.\n\nPHYTOLACCA (Phyt.)\n\nSore throat\n\nUse a strong lotion as a gargle (40 drops to 1\u20448 pint of cooled, boiled water to which a teaspoon of sea salt can also be added). Take the appropriate internal remedy.\n\nPLANTAGO (Plant.)\n\nEarache\n\nDilute a few drops of tincture with equal quantities of warm almond oil (or cooled, boiled water) and drop into the painful ear. Follow these guidelines:\n\n1 Heat a spoon by dipping it in boiling water, then pour the oil and the tincture into it and wait 30 seconds for the spoon to cool and the oil to warm.\n\n2 Tip the head on one side.\n\n3 Drop the liquid into the ear.\n\n4 Pull the lobe of the ear down and round and out very gently so that the liquid goes right into the ear.\n\nSome children will not allow anything to be put into their ears when they are in pain. Do not force this on them. Offer instead a warm hot water bottle wrapped in a soft towel for them to lie on; if that doesn't help try an ice pack (crushed ice in a plastic bag in a thin, soft towel). Or wrap the head tightly with a scarf. One of these measures might offer temporary relief and also guide you to the correct internal remedy.\n\nNB Seek professional help if pain is severe.\n\nToothache\n\nApply neat tincture to the affected tooth or swoosh the mouth out frequently with a strong lotion (40 drops tincture with 1\u20448 pint cooled, boiled water).\n\nPYRETHRUM (Pyreth.)\n\nInsect repellent\n\nApply the lotion to all exposed areas of skin, and carry a spray bottle of the lotion with you to renew the applications. Some pharmacies sell Pyrethrum in a spray, otherwise buy the tincture and make a fresh batch of lotion daily. Some homeopathic pharmacies also produce their own 'anti-bite lotion' which can be diluted and used as above. Experiment to see which suits you best.\n\nRESCUE REMEDY (RR)\n\nThis wonderful all-purpose 'remedy' is based on flowers and comes in a cream or as a tincture. It has been known to help anything from bruises, cold sores, cracked nipples, eczema, insect bites and stings, diaper rash and piles to sunburn. If Rescue Remedy is all you have then use it.\n\nRescue Remedy is a combination of five of Edward Bach's Flower Remedies and is not, strictly speaking, homeopathic. Bach was a homeopath who devoted his energies to finding plants which would act on negative emotions alone, and by so doing restore peace of mind which would lead to physical health.\n\nThere are few complaints that are not helped, even temporarily, by Rescue Remedy, including shock (after an accident, bad news, the dentist, etc.); panic; emotional upset or distress of any sort; sunburn; travel sickness; fear (stage fright, birth nerves, sleeplessness through anxiety or fear, fainting from fright); childbirth (before, during and after); headaches from emotional stress, and so on.\n\nTake a few drops on the tongue and repeat this as often as needed \u2013 once for a minor injury and every few minutes for a serious situation. The drops can be added to water and sipped frequently, or added to a bath (7 drops) and soaked in for a good quarter of an hour, or applied neat externally as a cream for rashes, bruises, cracked nipples, etc \u2013 as often as needed.\n\nRHUS TOXICODENDRON (Rhus-t.)\n\nJoint pain\n\nRhus toxicodendron ointment rubbed into joints can provide relief for sufferers from rheumatism and arthritis.\n\nSprains (second stage)\n\nAfter the swelling has subsided (with applications of Arnica ointment or cream), apply Rhus toxicodendron ointment twice daily to the sprained joint, and bandage tightly. Use the joint\/limb as little as possible and keep it elevated to give it a chance to heal. Rhus-t. is especially useful where ligaments are torn.\n\nStrains\n\nAfter lifting or over-exertion, the ointment can help enormously, especially if you have the typical Rhus-t symptoms: stiffness on beginning to move (on getting up, for example), improvement with continued movement, but a return of the painful stiffness if you overdo it or sit down again.\n\nRUTA GRAVEOLENS (Ruta.)\n\nBruises\n\nThis is for when bony parts of the body are sore after a knock, after Arnica has reduced the swelling, but the soreness persists. Shinbones, elbows and kneecaps are all parts that have little protective muscle and the covering to the bone can take longer to heal. Ruta can speed up the process. Apply the ointment two or three times daily until the pain eases.\n\nEye strain\n\nDilute two drops of Ruta tincture with an eyebath of cooled, boiled water, to help eyes strained by too much study, reading, or working at a VDU.\n\nSprains\n\nUse Ruta where Rhus-t. hasn't helped and the covering to the bone may have been damaged. I have found a mixture of Rhus-t. and Ruta in an ointment wonderful for sprains and strains.\n\nTennis elbow\n\nApply ointment or cream as necessary to relieve the pain. Do not further stress the joint by more strenuous activity.\n\nSYMPHYTUM (Symph.)\n\nCuts\/wounds\n\nSymphytum is a good all-purpose ointment or cream. Use on minor cuts once you have cleaned them.\n\nNappy rash\n\nWhere Calendula hasn't helped, Symphytum ointment often will. If it doesn't, your baby will need constitutional treatment from a professional homeopath.\n\nSprains\n\nApply Symphytum ointment to sprains that don't respond to Ruta or Rhus-t. within 48 hours. Also take Symphytum internally, as there may be damage to the bone itself.\n\nTAMUS (Tam.)\n\nChilblains\n\nApply the ointment two to three times daily before the chilblains break to stop itching and to speed up healing. Warning: never apply to a chilblain where the skin has broken.\n\nTHIOSINAMINUM (Thios.)\n\nScars\n\nThiosinaminum reduces the swelling of a badly healed scar \u2013 where there are lumps and bumps (keloids) \u2013 as long as it is used soon after the event (within three months), but it is still worth trying on older scars. It is useful for lumpy scars following a Caesarian, episiotomies, and so on. Massage the cream into a scar three times daily for several weeks \u2013 longer if it is helping but hasn't quite healed. A professional homeopath can also treat these types of scars with internal remedies.\n\nTHUJA (Thu.)\n\nWarts\/verrucas\n\nNeat Thuja tincture can be applied twice daily and Thuja 6 taken orally for up to ten days.\n\nWarning: if this remedy has no effect seek the advice of a professional homeopath. The continued use of Thuja is not advisable as the symptoms from the proving are unpleasant and difficult to get rid of. It is a deep-acting remedy that should not generally be used in a first-aid kit for self-prescribing. However, since it is part of the range stocked by many chemists I have included the minimum indications for safe administration.\n\nIt is now accepted that you do not 'catch' verrucas in swimming pools as was commonly believed until recently. Homeopaths believe warts and verrucas are part of an overall symptom picture and need to be treated with respect. They are cured successfully with constitutional homeopathic treatment, so do not suppress them with acids from the chemist or have them cut out. Homeopaths have found that suppressing warts in this way can lead to the development of more serious complaints.\n\nURTICA URENS (Urt-u.)\n\nBee sting\n\nDab on neat tincture and take the appropriate internal remedy.\n\nBurns (minor)\n\nFor minor burns with redness but no blistering, apply cream, ointment, or a compress soaked in lotion. Take the appropriate internal remedy if needed.\n\nEczema\/rashes\n\nUrtica cream or lotion can relieve the itching of eczema or any rash, especially if it itches and stings and then burns. It is essential to seek professional help for this condition.\n\nSunburn\n\nApply lotion or cream to sunburned areas, repeating according to the severity. Use a mixture of Hypericum and Urtica tinctures in the lotion if there are severe shooting pains (20 drops of each tincture in \u00bc pint of water) and take the appropriate internal remedy.\n\nVERBASCUM OIL (Verb.)\n\nEarache\n\nDrop the warmed oil into the ear to relieve pain and promote healing. Follow the instructions under Plantago.\n\n* * *\n\nEXTERNAL REPERTORY\n\n* * *\n\nThe External Repertory is an index of the symptoms listed in the External Materia Medica.\n\nBee stings see Stings\n\nBruises\n\non unbroken skin Arn., RR\n\non broken skin Ham.\n\nto bones Ruta.\n\nBurns\/scalds\n\nminor Urt-u.\n\nsecond degree (with blistering) Calen., Hyp.\n\nChilblains Tamus\n\nChildbirth Calen., Hypercal\u00ae\n\nCold sores Hypercal\u00ae, RR\n\nConjunctivitis see Eye infections\n\nCracked nipples Calen., RR\n\nCuts\/wounds Calen., Hypercal\u00ae, Hyp., RR, Symph.\n\nEarache Plant., Verb.\n\nEczema\/rashes Calen., RR, Urt-u.\n\nEye infections\/inflammations\/injuries Euphr., Hypercal\u00ae\n\nEye strain Ruta.\n\nGargle see Sore throat\n\nHandcream Calen.\n\nHerpes see Cold sores\n\nInsect bites Hyp., Led., RR\n\nInsect repellent Pyreth.\n\nJoint pain Rhus-t.\n\nMouth ulcers Hypercal\u00ae\n\nMouthwash Calen., Hypercal\u00ae\n\nNappy rash (Diaper rash) Calen., RR, Symph.\n\nPiles Aesc.\/Ham., Ham., Hyp.\n\nPink eye see Eye infections\n\nScars Thios.\n\nSore muscles Arn.\n\nSore throat Hypercal\u00ae, Phyt.\n\nSprains\n\nfirst stage (with swelling) Arn.\n\nsecond stage Rhus-t., Ruta., Symph.\n\nStings\n\nbee Led., RR, Urt-u.\n\nwasp Arn., Led., RR\n\nStrains Arn., Rhus-t.\n\nSunburn Calen., Hyp., RR, Urt-u.\n\nTennis elbow Ruta.\n\nThrush Calen.\n\nToothache Plant.\n\nUlcers see Mouth ulcers\n\nVaricose veins Ham.\n\nWarts\/verrucas Thu.\n\nWasp stings see Stings\n\n* * *\n\nINTERNAL MATERIA MEDICA\n\n* * *\n\nACONITUM NAPELLUS (Aco.)\n\nOther name: common aconite\n\nGeneral symptoms\n\nComplaints from cold, dry wind; fear; getting chilled; shock. Face red. Likes cold drinks. Onset of complaint sudden. Pains unbearable. Palpitations of pregnancy. Sweat hot; on covered parts of the body. Taste mouth tastes bitter. Thirsty.\n\nBetter for fresh air.\n\nWorse at night; for touch.\n\nThis remedy works best at the beginning of an acute illness \u2013 within the first 24\u201348 hours. Fright, shock or exposure to draught, or cold, dry wind can cause a wide range of symptoms (colds, coughs, cystitis, etc.) which respond to Aconite if the accompanying general symptoms are present \u2013 the thirst, the sudden onset of symptoms which are worse at night. The pains are intolerable and drive people to despair.\n\nBabies needing Aconite are fine on going to bed but wake around midnight (usually just before) with a cough or earache. They resent interference, don't want to be touched or examined and are better for fresh air.\n\nEmotional state\n\nAnxious when chilled; generally; during a fever; during pregnancy\/labour. Expression anxious; frightened. Fearful in a crowd; of death during pregnancy\/labour. Moaning\/complaining. Restless sleep, during labour. Screaming with pain. Sensitive children; to noise. Tearful during a fever.\n\nIs extremely distressed, anxious and fearful (the opposite of Arnica). Looks anxious and may have shocked, staring, glassy eyes. The pupils may be dilated. May be inconsolable and scared of going into a situation where there are crowds of people.\n\nIn childbirth, pains are severe. Goes from being afraid of dying to saying 'I want to die'.\n\nPhysical complaints\n\nBleeding (vaginal) in pregnancy\n\nWith ANXIETY.\n\nCause fright.\n\nChickenpox\n\nFirst stage. With FEVER.\n\nWith Aconite general symptoms and emotional state.\n\nCommon cold\n\nWith HEADACHE.\n\nCause shock; getting chilled; cold, dry wind.\n\nGive Aconite at the first sign of a cold if it comes on suddenly after a shock or getting chilled.\n\nCough\n\nBarking; dry; irritating; short; tickling. BREATHING fast. With VOICE HOARSE.\n\nWorse at night; during fever; for dry, cold air.\n\nCause cold, dry wind.\n\nAir passages are irritated. Often worse at night after being out in a cold, dry wind (especially north\/east).\n\nCroup\n\nSee COUGH for symptoms.\n\nCystitis\n\nPAINS pressing.\n\nCause getting chilled.\n\nEarache\n\nPAINS unbearable.\n\nCause getting chilled; cold, dry wind.\n\nEye inflammation\n\nEYES sensitive to light; whites of eyes red. PAINS aching; burning. With a COMMON COLD.\n\nWorse for cold dry wind.\n\nCause getting chilled; foreign body in the eye.\n\nFever\n\nHEAT alternating with chills at night; burning; dry at night. PULSE fast; strong. With ANXIETY.\n\nBetter for uncovering.\n\nWorse at night; in the evening.\n\nCause getting chilled; teething in babies.\n\nFeels hot inside and chilly externally. Babies' cheeks alternate between being hot and red and pale and ghostly. Colour may drain from the face on getting up. One cheek may be hot and red and the other pale and cold, especially in teething babies. Clothed parts of the body become sweaty; babies kick off their covers. Fever is accompanied by a burning, unquenchable thirst; everything tastes bitter except water and even that tastes bad.\n\nHeadache\n\nPAINS burning; bursting; throbbing.\n\nCause fright; shock; getting chilled.\n\nInjuries\n\nCUTS\/WOUNDS bleed freely. With SHOCK (see here). Aconite helps with wounds that bleed excessively where the characteristic shock is present.\n\nInsomnia\n\nIn PREGNANCY. Restless SLEEP. Anxious DREAMS, VIVID.\n\nLabour\n\nPAINS severe. Labour too fast; LATE with fear of labour.\n\nFor those short, sharp labours, violent and terrifying, often accompanied by a fear of dying. This remedy eases the fear and slows the roller-coaster down to manageable proportions.\n\nMeasles\n\nONSET sudden. SKIN RASH itches; burns. With FEVER; COUGH.\n\nSudden onset with restlessness, fever, cough, thirst.\n\nMumps\n\nONSET sudden. With FEVER.\n\nRetention of urine\n\nIn NEWBORN BABIES; in BABIES WHO CATCH COLD.\n\nFor newborn babies who don't pee, who have been shocked by the birth (especially a fast labour).\n\nRoseola\n\nSee MEASLES for symptoms.\n\nShock\n\nCause injuries; surgery; childbirth.\n\nAccompanied by the extreme Aconite-type fear and anxiety. Useful during or following operations and for shocked mothers and\/or babies either during or after labour, especially a fast labour. The mother may feel shaky with the shock, whereas the baby may be very still, with an anxious or fearful look in its eyes.\n\nSore throat\n\nPAINS burning; stitching.\n\nCause getting chilled.\n\nTeething\n\nCHEEKS hot and red. With symptoms of FEVER (see here). PAINFUL in babies; with restless sleep.\n\nBabies toss and turn in their sleep and bite their fists and scream.\n\nAESCULUS HIPPOCASTANUM (Aesc.)\n\nOther name: horse chestnut\n\nGeneral symptoms\n\nHeavy or full feeling.\n\nWorse for movement; walking.\n\nUseful in pregnancy if there are constipation, piles and backache: bowels become sluggish causing piles. Subsequent congestion in pelvic area, with feeling of fullness and\/or heaviness, causing backache.\n\nPhysical complaints\n\nBackache\n\nIn PREGNANCY. With CONSTIPATION (see here).\n\nLower back feels sore and bruised.\n\nConstipation\n\nIn PREGNANCY. STOOLS large, hard. With INEFFECTUAL STRAINING. With PAIN AFTER PASSING A STOOL.\n\nPasses stool with difficulty because it is large and hard \u2013 feels as if sticks are being passed. Little if any bleeding.\n\nPiles\n\nIn pregnancy. Painful; BLEEDING; EXTERNAL, LARGE, ITCHING. With BACKACHE and CONSTIPATION (see here).\n\nBetter for bathing in warm water.\n\nWorse for standing\/walking.\n\nAGARICUS MUSCARIUS (Agar.)\n\nOther name: toadstool\n\nGeneral symptoms\n\nClumsy trips easily while walking. Trembling. Twitchy.\n\nWorse for cold.\n\nSpecific remedy for chilblains, especially of feet and toes. Used with Tamus ointment, will cure most straightforward cases.\n\nPhysical complaints\n\nChilblains\n\nBurning; itching; red.\n\nWorse for cold.\n\nOn hands and feet, or very occasionally ears, but worst on feet; most painful with cold hands or feet.\n\nALLIUM CEPA (All-c.)\n\nOther name: common red onion\n\nPhysical complaints\n\nCommon cold\n\nEYES streaming. DISCHARGE from eyes, watery. NASAL CATARRH: burning; one-sided; profuse; watery. NOSE streaming. With EYE INFLAMMATION; HEADACHE; SNEEZING; SORE THROAT.\n\nBetter for fresh air.\n\nWorse in a stuffy room.\n\nCause cold wind; getting feet wet.\n\nNasal discharge may be one-sided. Cold symptoms, especially the sneezing, generally worse in a warm room. Euphrasia is similar in that it too produces streaming colds but in Allium cepa nasal catarrh burns and eye discharge doesn't.\n\nCough\n\nHacking; irritating; painful. LARYNX tickles. VOICE HOARSE.\n\nBetter for being in a warm room.\n\nWorse for cold air.\n\nEye inflammation\n\nSymptoms of COMMON COLD (see here). DISCHARGE: bland, non-irritating; profuse. EYES watering.\n\nALUMINA (Alu.)\n\nOther name: aluminium\n\nGeneral symptoms\n\nDischarges white. Food cravings for strange things, such as chalk, coal, in pregnancy.\n\nTypical Alumina state is extreme exhaustion with feelings of faintness on standing and exhaustion on walking or talking. In pregnancy there may be strange cravings for dry 'foods' like raw spaghetti and rice or even chalk or coal. Small amounts of metal from aluminium pans dissolve into food cooked in them, especially if those foods are acidic (like spinach and some fruits). Some people are sensitive even to tiny traces of this metal, which may cause exhaustion and constipation. Don't expose your baby to risk: get rid of aluminium pans and be careful when weaning as many commercially prepared baby foods contain minute quantities.\n\nEmotional state\n\nApathetic. Depressed on waking, doesn't want to be bothered. Desires to be alone. Squeamish. Feels faint at the sight of blood.\n\nBecomes depressed and indifferent to everybody and everything.\n\nPhysical complaints\n\nConstipation\n\nIn PREGNANCY. In BABIES. STOOLS soft. STRAINING ineffectual; DESIRE TO PASS STOOL absent.\n\nCause bottle-feeding (cow's milk or soya); weaning.\n\nBabies (or adults) have difficulty in passing soft stool and become pale and wan. Some babies are very sensitive to the aluminium in some powdered milks or in commercial baby foods. Change the brand if you suspect this may be the problem.\n\nExhaustion\n\nDuring LABOUR; NERVOUS during pregnancy. With FAINT feeling; DESIRE TO LIE DOWN.\n\nWorse for physical exertion; talking; standing; walking.\n\nExhaustion is extreme.\n\nThrush (genital)\/Yeast infection\n\nIn pregnancy. DISCHARGES like egg white; burning; profuse; white.\n\nUsually accompanied by characteristic exhaustion.\n\nANTIMONIUM CRUDUM (Ant-c.)\n\nOther name: sulphide of antimony\n\nGeneral symptoms\n\nComplaints from overeating. Cracks on corners of mouth\/nostrils. Eyes sunken; dull. Headache. Lips dry. Nausea in pregnancy. Sluggish. Thirstless. Tongue white-coated.\n\nWorse for getting overheated; for sun; swimming in cold water.\n\nThose who need Antimonium crudum often eat too much becoming sluggish, irritable and sick, developing weak stomachs and frequently upset digestive systems. Thirstlessness is often present with nausea or indigestion. The white coating, like whitewash, on the tongue must be present in any complaint if remedy is to work well. Cracks may recur around nostrils and corners of the mouth.\n\nAntimonium crudum types are sensitive to sun, or being overheated, which exhausts them and causes them loss of voice or aggravates any complaint they may have. Swimming in cold water on a hot day may bring on a cold.\n\nEmotional state\n\nAngry babies. Aversion to being looked at or touched. Irritable. Sentimental. Stubborn. Sulky.\n\nSick adults needing this remedy may behave in sulky, morose way, not wanting to speak or be spoken to, especially if suffering from gastric problems.\n\nPhysical complaints\n\nChickenpox\n\nWith COUGH.\n\nWith typical emotional state and general symptoms.\n\nCough\n\nWorse in stuffy rooms.\n\nDiarrhoea\n\nSTOOLS small, hard lumps; watery.\n\nWorse for getting overheated.\n\nCause overeating.\n\nExhaustion\n\nWith SLEEPINESS.\n\nWorse in hot weather; late morning.\n\nFingernails split\n\nCause injury to nail.\n\nInjured or crushed fingernails grow back with splits (see also Silica).\n\nIndigestion\n\nBELCHES empty; tasting of food just eaten. ABDOMEN\/STOMACH feels bloated; empty; full.\n\nNausea\n\nCONSTANT. VIOLENT during pregnancy. BELCHES empty; tasting of food just eaten.\n\nWorse after eating acidic and\/or starchy foods.\n\nCause pregnancy.\n\nMay be accompanied by headache.\n\nVomiting\n\nOf bile; of curdled milk; during pregnancy. In breastfed babies.\n\nWorse after drinking milk.\n\nCause measles.\n\nFor nausea and vomiting during pregnancy where there is the characteristic white tongue and an intolerance of milk and\/or bread. Also for babies who vomit breast milk after a feed, who get cross and refuse the breast next time it is offered, or who don't tolerate formula milk well.\n\nANTIMONIUM TARTARICUM (Ant-t.)\n\nOther name: tartar emetic\n\nGeneral symptoms\n\nExhaustion. Face pale. Sweat cold, profuse. Thirstless. Tongue white-coated.\n\nAntimonium tartaricum and Antimonium crudum are similar \u2013 both are thirstless, irritable and have coated tongues. Antimonium tartaricum also has profuse cold sweat, mostly during the night; face is pale and sunken and lips may go blue. Particularly useful for babies and may be called for during chickenpox or measles if a chest infection sets in.\n\nEmotional state\n\nAngry babies. Apathetic. Clingy. Irritable. Screaming if touched.\n\nTypically, babies are irritable, drowsy; they whine, complain and are clingy; do not want to be examined, will scream if touched in a way they don't like.\n\nPhysical complaints\n\nBreathing difficulties\n\nIn babies after birth.\n\nNewborn babies have trouble breathing, are full of mucus which rattles audibly.\n\nChickenpox\n\nSKIN RASH slow to appear. With symptoms of COUGH (see here). The rash may begin to come out and disappears again.\n\nIf Antimonium tartaricum is indicated but doesn't work, give Antimonium crudum.\n\nCough\n\nCOUGH loud; rattling; whooping. BREATHING asthmatic; abdominal; difficult; fast, rattling. MUCUS difficult to expel. With SLEEPINESS; VOMITING.\n\nIn a chest complaint that will respond to this remedy, there is a characteristic loud rattling which can usually be heard before entering room, whether chest infection is simple cough, bronchitis or whooping cough. Bringing up phlegm is difficult and provides only temporary relief. Anger can aggravate the cough which may alternate with yawning.\n\nFever\n\nHEAT, alternating with chills. With intense HEAT DURING SLEEP.\n\nOften accompanies chest or gastric complaints.\n\nNausea\n\nINTERMITTENT, VIOLENT during pregnancy.\n\nBetter for belching; for vomiting.\n\nFelt in the chest or as a weight on the chest, and is as intense as that associated with Ipecacuana but not so persistent. Vomiting and belching provide only temporary relief. Desire to vomit may be accompanied by ineffectual retching.\n\nVomiting\n\nVOMIT sour. VOMITING difficult. With FEVER.\n\nWorse for coughing.\n\nMay be aggravated by eating and drinking, and may be worse if there is fever (the higher the temperature, the stronger the vomiting). Vomiting is difficult with ineffectual retching.\n\nAPIS MELLIFICA (Ap.)\n\nOther name: honey bee\n\nGeneral symptoms\n\nClumsy drops things. Face red; puffy. Lips swollen. Oedema (swelling) of ankles\/feet or hands\/fingers. Pains burning; stinging. Symptoms right-sided: move from right to left. Thirstless. Tongue fiery red.\n\nWorse for heat; for touch; at 3\u20135 p.m.\n\nThese are warm-blooded types who are better for fresh air; although generally thirstless they are better for cold drinks, and having painful parts bathed in cold water. If there is fever or localised inflammation, the body surface is sore and sensitive, as if bruised, and is worse for touch (pressure of heavy blankets will aggravate, for example).\n\nEmotional state\n\nApathetic. Fearful of being alone; of death. Irritable. Jealous. Restless. Tearful. Whiny.\n\nGenerally, Apis types are jealous. This may be seen in the brother or sister of a new baby, no longer getting desired attention. Apis picture may be confused with Pulsatilla: both are better in open air, whine, are jealous and weepy, but where Apis is worse for touch, Pulsatilla is better for it \u2013 craves it even! In a fever, Apis types are tearful for no apparent reason, restless and fearful, with a fear of death, and\/or of being alone.\n\nPhysical complaints\n\nBites\/stings\n\nBITES burning; itching; red; stinging; swollen.\n\nBetter for cold.\n\nWorse for heat.\n\nCause insect bites.\n\nBites react badly, causing a large, shiny, swollen red lump which itches and\/or burns and\/or stings.\n\nNB Apis can be used for anaphylactic shock, the severe allergic reaction which may occur not just from a bee or wasp sting but from certain foods, like nuts, or even penicillin. This is always an emergency as it can be fatal: give Apis while you wait for emergency help to arrive. Classic symptoms are swollen eyelids, lips and tongue, urticaria (itchy lumps) and difficulty in breathing.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nTingling and numbness in fingers, usually with some swelling of wrist or fingers.\n\nCough\n\nWorse when lying down.\n\nCystitis\n\nDESIRE TO URINATE constant: frequent. PAINS burning; pressing; stinging. URINATION constant; frequent.\n\nOnly small amounts of urine passed. May be blood in urine (it will look red); if so, and if the symptoms agree, take Apis but seek professional help immediately.\n\nDiarrhoea\n\nPAINLESS. Anus may be sore after passing a stool.\n\nEarache\n\nPAINS stinging.\n\nWorse for swallowing.\n\nWith typical Apis general symptoms (worse for heat, pains starting on right side, etc.) and SORE THROAT.\n\nEye inflammation\n\nIn BABIES. EYES burning; red; sore; stinging; stitching. EYELIDS swollen.\n\nWorse for heat.\n\nWhites of eyes are red with visible bright-red blood vessels. Lower lids may be more swollen than upper.\n\nFever\n\nHEAT burning; dry. With CHILLINESS; SENSITIVE SKIN.\n\nBetter for uncovering.\n\nWorse for heat; for being in stuffy room; for warm covers; afternoon; morning; for washing.\n\nFeels hot; finds heat intolerable, has a high fever and feels sleepy. This type of fever accompanies most acute complaints that call for Apis. Kicks off covers, then shivers (with chills) but keeps covers off.\n\nHeadache\n\nHEAD feels full; hot. PAIN stabbing; sudden.\n\nScalp feels tight and sore.\n\nHives\n\nWith FEVER (see here); SWEATING.\n\nWorse at night.\n\nJoint pain\n\nPAIN burning; stinging. With SWELLING.\n\nJoints are red, swollen and shiny.\n\nMeasles\n\nSKIN RASH slow to appear. With EYE INFLAMMATION (see here); FEVER (see here).\n\nWith typical emotional state and general symptoms.\n\nMumps\n\nWith typical general symptoms and emotional state (see here).\n\nNappy rash (Diaper rash)\n\nSKIN red; shiny; hot; sore.\n\nBetter for uncovering.\n\nWorse for heat; for touch.\n\nRetention of urine\n\nIn babies. Without cause.\n\nHalf-hourly doses of Apis will encourage urination to re-establish if the baby is drinking normally.\n\nScarlet fever\n\nSee MEASLES.\n\nSore throat\n\nMOUTH dry. PAIN burning.\n\nThirstless although mouth is very dry. Throat is as red as tongue.\n\nARGENTUM NITRICUM (Arg-n.)\n\nOther name: silver nitrate\n\nGeneral symptoms\n\nComplaints from mental strain. Cravings for sugar\/sweets. Exhaustion with trembling. Face sallow. Pains needle-like. Palpitations. Taste in mouth sour. Tongue red-tipped. Trembling.\n\nBetter for fresh air.\n\nWorse for heat; after eating sugar\/sweets.\n\nThese are warm people, generally worse for heat, who suffer in warm stuffy rooms and feel better for fresh air. They have a sweet tooth and crave sugar and sweets but these make them sick. Their exhaustion is often accompanied by extreme anxiety and trembling may follow a period of intense mental work where the brain feels worn out and the memory lapses.\n\nEmotional state\n\nAnxious anticipatory; general. Confused. Desires company. Excitable babies. Fearful of being alone; of public speaking. Hurried while speaking; while walking; while waiting. Impulsive. Weak memory. Feeling of panic. Restless.\n\nBetter for company.\n\nArgentum nitricum types dread ordeals, trembling with nervous excitement and suffering from diarrhoea. Anxious and impulsive thoughts torment them. They become fidgety and walk hurriedly around to calm themselves; time seems to pass inexorably slowly. They hate to be kept waiting, and doubt their ability to succeed: their lack of confidence may be well founded in that their anxiety can prevent them from doing well. They are not happy alone and are better for company. 'Birth' nerves during pregnancy may be alleviated but choose carefully between remedies, including Argentum, that have anticipatory anxiety or fear of childbirth in their picture.\n\nPhysical complaints\n\nDiarrhoea\n\nSTOOLS green; smelly; watery. With FLATULENCE (see here); VOMITING.\n\nWorse immediately after drinking; at night; after eating sugar.\n\nCause anticipatory anxiety; excitement; sugar; after weaning.\n\nLiquids pass straight through, coming out as green diarrhoea, like chopped spinach. There may be vomiting. Newly weaned infants may produce this picture; or someone on sugar binge; or someone with acute anxiety and\/or panics.\n\nEye inflammation\n\nIN BABIES. DISCHARGE purulent; smelly; yellow. EYELIDS glued together; red. EYES red; sensitive to light.\n\nBetter for cold; for cold compresses.\n\nFor newborn babies with sticky eyes. Eyelids and corners of eyes are red and inflamed.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; intolerant of tight clothing. WIND loud; obstructed.\n\nWorse after eating.\n\nWind is difficult to expel. Noisy, explosive burps provide relief from bloating and pain.\n\nHeadache\n\nBetter for binding up head.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; painful. BELCHES loud, difficult; empty. With FLATULENCE (see here); NAUSEA.\n\nBetter for belching.\n\nWorse after eating; eating sweet foods\/sugar.\n\nUsually worse for eating, although it may help. Abdomen is as tight as a drum.\n\nSore throat\n\nTHROAT irritated; raw. PAIN splinter-like. VOICE hoarse; lost.\n\nCause singing; talking.\n\nARNICA MONTANA (Arn.)\n\nOther name: leopard's bane\n\nGeneral symptoms\n\nBreath smelly. Complaints from accident\/injury\/surgery. Pains sore, bruised; glands.\n\nWorse for jarring movement; lying on injured part; touch.\n\nArnica promotes healing, controls bleeding, reduces swelling, prevents pus forming: is an essential ingredient in any first-aid kit, the first to consider after an accident, injury, or any other trauma where there is shock, such as surgery or childbirth. Those needing Arnica usually feel sore, bruised and do not want to be touched or jarred, so that when lying down, the bed may feel hard.\n\nEmotional state\n\nAversion to being touched\/examined. Complaints from shock. Denial of illness; of suffering. Fearful generally; of being touched. Forgetful following injury.\n\nDenies being ill when is (sometimes very) sick. May moan and complain about pains, but more usually denies suffering, especially after an injury accompanied by delayed shock. After being knocked down by a car, for example, may stand up, maintaining that nothing is the matter, while blood pours from gaping wound in head. This can be dangerous because of possibility of delayed concussion. Actual concussion may cause hopelessness and indifference and stupor. On coming round, forgets words while speaking and does not want anyone near.\n\nPhysical complaints\n\nAbdominal pain\n\nIn pregnancy. PAIN sore, bruised.\n\nCause an active baby.\n\nAfterpains\n\nPAIN sore, bruised.\n\nWorse when baby nurses.\n\nGiven in labour, especially towards end and directly afterwards, Arnica lessens afterpains.\n\nBleeding (vaginal) in pregnancy\n\nWith PAIN; sore, bruised.\n\nCause an injury, fall or accident.\n\nBlood blisters\n\nCause blow or injury.\n\nBroken bones\n\nWith SWELLING\/BRUISING.\n\nMove on to Symphytum once swelling has reduced.\n\nBruises\n\nWith SWELLING. Without discolouration.\n\nCause childbirth; injury; surgery.\n\nGive before skin begins to discolour \u2013 if given soon enough, even if there is already some swelling, bruise will not materialise, healing from inside. Given after bruise has formed, Arnica speeds healing and quickly reduces swelling. For bruises to shins see Ruta.\n\nBruised soreness that often accompanies and nearly always follows childbirth is greatly alleviated by Arnica. If severe and persistent, try Bellis perennis.\n\nCough\n\nWhooping. EYES bloodshot. PAIN IN CHEST sore, bruised; must hold chest to cough. With NOSE-BLEEDS (see here).\n\nWorse for crying.\n\nBabies cry before coughing in anticipation of pain, which may set off cough.\n\nEye injuries\n\nBRUISING to eyeball; to surrounding area.\n\nGive Arnica before discolouration if possible, even if eye is swollen.\n\nGums bleeding\n\nCause tooth extraction.\n\nArnica controls bleeding and speeds healing, especially where there is bruised soreness.\n\nHead injuries\n\nGive Arnica as routine after a fall or bang to the head, whether or not there is concussion. For maximum effect, wait for egg to appear (before it discolours), give Arnica, and watch the lump disappear!\n\nInflammation of penis\n\nIn babies.\n\nInjuries\n\nCUTS\/WOUNDS with bruising. PAIN sore, bruised.\n\nCause after dental treatment; surgery.\n\nGive Arnica in any injury to soft tissues, i.e. muscles, where there is swelling and bruising.\n\nJoint pain\n\nPAIN sore, bruised.\n\nWorse for touch.\n\nJoints are sore, bruised and very sensitive to touch.\n\nLabour\n\nTOO LONG.\n\nPains are sore; abdomen feels bruised. Arnica helps the muscles to do their work and minimises physical stress and strain on soft tissues. Women needing Arnica may brush off support saying they're fine \u2013 when they plainly aren't.\n\nNosebleeds\n\nCause injury.\n\nPhlebitis\n\nCause childbirth.\n\nLegs feel sore and achy.\n\nRetained placenta\n\nAfter a LONG LABOUR.\n\nRetention of urine\n\nAfter childbirth. After a LONG LABOUR. With painful URGE TO URINATE. With INCONTINENCE\/INVOLUNTARY URINATION.\n\nShock\n\nCause surgery; childbirth; injury.\n\nShock is suppressed (see Emotional state).\n\nSkin complaints\n\nIn pregnancy. BROKEN VEINS.\n\nSprains\n\nOf ANKLE; of FOOT; of WRIST; in FIRST STAGE. With BRUISING (see here); SWELLING.\n\nUse Arnica to reduce swelling, prevent bruising, and speed healing.\n\nStrains\n\nPAIN sore, bruised.\n\nCause childbirth; overexertion.\n\nFor strained muscles which feel bruised, sore and no better for moving about. (If pains are stiff and better for moving about, even temporarily, see Rhus toxicodendron.) Jet-lag can produce this feeling.\n\nARSENICUM ALBUM (Ars.)\n\nOther name: arsenic\n\nGeneral symptoms\n\nAnaemia. Catches colds easily. Discharges burning; smelly; watery. Dislikes food in general; sight of food. Dryness generally. Face pale. Likes hot food\/drinks. Lips cracked; dry; licks. Mouth burning; dry. Pains burning. Palpitations. Restless. Sweat absent during fever; clammy; profuse; sour; cold. Taste in mouth bitter. Thirsty for large quantities or frequent small quantities; sips. Tongue red-edged or red-tipped.\n\nBetter for heat; for hot drinks; for warmth of bed; for lying down.\n\nWorse for change of temperature; for cold; for damp; for exertion; after midnight; on waking; for wet weather; at 3 a.m.\n\nArsenicum types are extremely sensitive to cold and need fresh air so love to live in a house with the heating turned right up and windows open. When ill they typically look pale and anxious and quickly become weak and exhausted, sometimes to the point of collapse in an acute complaint, and often out of proportion to the severity of the illness. Tiredness often comes on suddenly, especially after physical exertion, even a short walk. They are generally better for lying down although many symptoms are worse for this, which causes general restlessness.\n\nThe pains are characteristically burning and, except for headaches, are better for heat (a useful, unusual symptom) and for warm drinks, food or compresses. Cold drinks and food aggravate, especially in gastric symptoms.\n\nEmotional state\n\nAngry. Anxious generally; on waking; at night (children); after midnight; during a fever; about others; when alone; at 3 a.m. Complaints from anger with anxiety. Critical. Depressed. Desires to be carried; company. Despair of getting well. Expression haggard; sickly; suffering. Fearful generally; of being alone; of death; at night (children). Forgetful. Guilt. Irritable. Restless babies; in bed; with anxiety. Tidy.\n\nArsenicum characters are more restless and anxious than virtually any other remedy type. If their anxiety goes unchecked it may turn to guilt, and self-criticism. When ill they become frightened of being alone for fear of death and want desperately to be looked after. They are demanding, difficult patients, who find fault and make a fuss. Obsessively tidy, they may clear up for visitors or the doctor in spite of being too weak to get out of bed to make a cup of tea! In labour women may become extremely anxious, dictatorial (bossy), fearful and irritable. Babies in need of Arsenicum may want to be carried around, but briskly, rather than gently (like Pulsatilla).\n\nPhysical complaints\n\nBreathless\n\nIn pregnancy.\n\nWorse for walking uphill; when lying down; at night in bed; for exertion.\n\nBurns\n\nPAIN burning. With BLISTERS.\n\nBetter for heat.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nTingling and numbness in fingers.\n\nCommon cold\n\nEYES burning; dry. EYELIDS puffy; red. NASAL CATARRH burning; profuse; watery. SINUSES blocked, painful. With frequent SNEEZING.\n\nWorse during evening; on right side.\n\nCause getting chilled when overheated.\n\nAn acute cold with violent symptoms; may move on to the chest (see COUGH). Burning nasal discharge makes the area under the nostrils red and sore. Discharge may be blood-streaked. Lips may become so badly cracked they bleed.\n\nCough\n\nDry at night; exhausting; hacking; loose; tormenting. BREATHING difficult; fast; wheezing. LARYNX tickling. MUCUS copious; frothy; tastes salty. With SWEATING.\n\nBetter for hot drinks.\n\nWorse for cold; for cold drinks; fresh air; during evening; for lying down; after midnight; at night; during fever.\n\nCough may be loose or dry but is more likely to be dry at night. Has to be propped up with lots of pillows even though wants to lie down. May wake up coughing around 1 or 2 a.m.\n\nCystitis\n\nDESIRE TO URINATE ineffectual. PAIN burning. URINATION with unfinished feeling.\n\nBetter for heat; for sitting in hot bath.\n\nDiarrhoea\n\nBurning PAIN after passing stools. STOOLS smelly; watery. With EXHAUSTION (see here); SWEATING; ICY-COLD HANDS AND FEET; NAUSEA.\n\nWorse for cold; after drinking; after eating fruit and any cold food; after midnight.\n\nCause food poisoning; ice-cream; fruit.\n\nMay be accompanied by intense exhaustion and nausea, which is aggravated by the sight, smell or even thought of food.\n\nExhaustion\n\nEXTREME; PARALYTIC; SUDDEN. With FAINT FEELING; FEVER (see here); RESTLESSNESS.\n\nWorse in morning; for movement; for passing stool.\n\nCause food poisoning; pain.\n\nComes on suddenly, especially after diarrhoea, and is worse for the slightest exertion. Is accompanied, unusually, by restlessness.\n\nEye inflammation\n\nIn BABIES. EYELIDS burning. EYES bloodshot; burning; gritty; sensitive to light.\n\nFever\n\nHEAT burning; dry; dry at night; alternating with chills. With ANXIETY; DELIRIUM; EXHAUSTION (see here).\n\nWorse in morning; at night; after midnight.\n\nFeverishness or feeling hot alternates with feeling chilly, causing sweating. Head and face may feel hot to touch while body feels cold; or body may be hot to touch while chilled inside. At other times feels hot inside as though blood is burning in veins. With the characteristic Arsenicum thirst for sips.\n\nFlatulence\n\nABDOMEN\/STOMACH bloated. WIND smelly.\n\nFlu\n\nWith symptoms of COMMON COLD (see here); FEVER (see here); RESTLESSNESS.\n\nCause change of temperature.\n\nCold and fever symptoms are accompanied by Arsenicum general symptoms.\n\nFood poisoning\n\nWith DIARRHOEA (see here); NAUSEA; VOMITING (see here).\n\nCause rotten meat.\n\nNausea is intense \u2013 cannot bear sight, smell or thought of food.\n\nGastric flu\n\nSee FLU with DIARRHOEA.\n\nHeadache\n\nPAIN burning; throbbing; in forehead; recurring at regular intervals.\n\nBetter for fresh air.\n\nWorse for heat.\n\nWants to lie down with head high on lots of pillows. Pains may start at bridge of nose and spread to whole head. Feels like being wrapped in a duvet from neck down while sitting next to open window because unlike other Arsenicum pains, the headache benefits from fresh air and cold in general (or a cold compress).\n\nIncontinence\n\nIn PREGNANCY. After CHILDBIRTH. DAY AND NIGHT.\n\nWith the emotional state and general symptoms.\n\nIndigestion\n\nBurning PAINS. With HEARTBURN.\n\nBetter for hot drinks.\n\nWarm milk may be particularly soothing. Pains may be accompanied by HEADACHE (see here).\n\nInsomnia\n\nDREAMS anxious; nightmares. Restless SLEEP.\n\nWorse after midnight.\n\nCause anxiety; overactive mind; shock.\n\nIn spite of complete exhaustion, anxiety prevents sleep. When sleep finally comes it is full of frightening dreams, of danger and dead people.\n\nMeasles\n\nWith Arsenicum general symptoms and emotional state.\n\nMumps\n\nWith Arsenicum general symptoms and emotional state.\n\nNausea\n\nDEATHLY.\n\nIn PREGNANCY. With VOMITING (see here).\n\nRetention of urine (after labour)\n\nNo desire to pass urine. With INVOLUNTARY URINATION.\n\nFor women who find it difficult to urinate after childbirth.\n\nPiles\n\nINTERNAL. BLEEDING. BURNING.\n\nBetter for bathing in warm water.\n\nSore throat\n\nBurning PAINS. ULCERS in throat.\n\nBetter for hot drinks.\n\nWorse for cold drinks; for swallowing.\n\nVomiting\n\nVOMIT bile; food; smelly; watery. VOMITING easy; frequent; violent. With DIARRHOEA (see here); FAINTNESS after vomiting; SWEATING while vomiting.\n\nWorse after eating\/drinking; for movement.\n\nCause ice-cream; rotten meat.\n\nAcute vomiting (and diarrhoea) of severe food poisoning. Everything is vomited immediately, even the smallest quantity of water (unlike Phosphorus, where vomiting occurs after a few minutes). Eventually there is nothing left in stomach and foul-smelling bile may be vomited.\n\nASARUM EUROPUM (Asar.)\n\nOther name: European snake root\n\nGeneral symptoms\n\nSenses hyperacute.\n\nBetter for bathing; cold drinks\/food; fresh air; lying down; rest.\n\nWorse for cold, dry weather; wet weather; heat; stuffy rooms; hot food\/drinks; alcohol; movement.\n\nEmotional state\n\nDull\/sluggish. Sensitive to pain during pregnancy; oversensitive to noise. Slow.\n\nThe Asarum state is truly dreadful and unhappily common in pregnancy when some women become super-sensitive. Great sluggishness is accompanied by acute sensitivity to noise: ticking clock or dripping tap may drive to distraction and aggravate nausea.\n\nPhysical complaints\n\nNausea\n\nCONSTANT. VIOLENT. In pregnancy. With RETCHING; VOMITING.\n\nWorse for noise.\n\nBARYTA CARBONICA (Bar-c.)\n\nOther name: barium carbonate\n\nGeneral symptoms\n\nCatches colds easily. Concentration poor. Exhaustion after eating. Glands swollen; sensitive. Slowness of babies to develop. Sweat one-sided. Weight gain poor in babies.\n\nWorse for cold; for pressure; for getting feet wet.\n\nBaryta carbonica types are sensitive to cold and need to wrap up warmly due to a tendency to catch colds and coughs, which usually come with swollen glands.\n\nYoung benefit most from this remedy. Baryta carbonica babies are slow, look 'old' and have big bellies with disproportionately skinny arms and legs.\n\nEmotional state\n\nAnxious during fever. Dislikes company (strangers). Fearful babies; of strangers. Indecisive. Jumpy. Lack of self-confidence. Play \u2013 doesn't want to. Shy. Sluggish babies.\n\nBaryta carbonica types are serious, shy and nervous of strangers; although they can be cheerful and jokey, they hate to be teased.\n\nPhysical complaints\n\nBlocked tear duct\n\nIn babies.\n\nCommon cold\n\nGLANDS swollen. NOSE dry. With COUGH.\n\nWorse at night.\n\nFeels the cold and catches colds easily and often. These may be winter colds in babies who have developed catarrh from the first cold or damp weather of winter which remains until right through till warmth of late spring.\n\nMumps\n\nWith Baryta carbonica general symptoms and emotional state.\n\nSore throat\n\nTHROAT inflamed; raw. Roaring noises in EARS on swallowing. PAIN burning. TONSILS swollen. With INCREASED SALIVA.\n\nBetter for swallowing liquids.\n\nWorse at night; for swallowing saliva; for swallowing food.\n\nCan only swallow liquids; gags and chokes when swallowing food. Mucus drips down into the throat and saliva increases.\n\nBELLADONNA (Bell.)\n\nOther name: deadly nightshade\n\nGeneral symptoms\n\nComplaints from cold, dry wind; getting head wet. Eyes shining. Face red; with toothache red in spots; dark red. Glands swollen; sensitive. Sudden onset. Pains come on suddenly; appear and disappear suddenly; in glands; shooting; throbbing. Pupils dilated. Shock. Sweat absent during fever; on covered parts. Symptoms right-sided. Thirsty. Tongue red or white-coated; strawberry.\n\nBetter for lying down.\n\nWorse for jarring movement; for touch; for cold wind; at 3 p.m.; getting head wet.\n\nA Belladonna illness \u2013 be it sore throat, earache or sunstroke \u2013 comes on suddenly and strongly, as does Aconite, and disappears as rapidly. A cold may start if head is chilled, especially if hair is wet (after swim or haircut). Generally, symptoms are worse at 3 p.m., although 3 a.m. may also be a bad time. Lying in quiet, darkened room helps, although some symptoms are better for standing. Eyes sparkle or shine, and pupils are usually dilated. Tongue may be red, coated with white dots, or have white coating. Face is flushed, and burns. With a fever blood vessels in neck may throb visibly.\n\nThink of Belladonna where a part of the body becomes inflamed or infected, reddens and throbs painfully, radiating heat. Violent, throbbing pains are intensified by moving or being moved or touched, or jarred (for example, during a car journey).\n\nEmotional state\n\nAngry. Anxious. Aversion to being touched\/examined. Biting. Confused. Delirious. Excitable. Expression fierce. Hitting. Rage with desire to bite or hit. Restless. Screaming with pain. Sensitive babies; to light; to noise. Tantrums. Tearful during a fever.\n\nBelladonna suits happy, easy-going babies who become difficult and obstinate when ill, and are prone to tantrums where they may bite or hit those closest to them. Under stress anger surfaces more easily than fear. Someone needing Belladonna becomes agitated, excitable, restless and delirious with or without fever. Lies in bed moaning and jumps up from time to time; and becomes hypersensitive to light and noise. The delirium (with fevers) may be accompanied by hallucinations.\n\nBelladonna is especially suited to women in labour having a first baby in their late thirties and forties. They rant, rage and quarrel with everybody, becoming uncontrollably angry with the pains.\n\nPhysical complaints\n\nBackache\n\nIn pregnancy. PAIN dragging.\n\nBleeding (vaginal)\n\nIn pregnancy; after childbirth.\n\nSudden gushes of blood; flow of blood is dark red; with clots; with shooting pains that come and go suddenly.\n\nBraxton Hicks contractions\n\nIn pregnancy.\n\nBreast(feeding) problems\n\nBREASTS engorged; hard; hot; inflamed; painful; with red streaks. PAIN throbbing. MILK SUPPLY overabundant.\n\nWhen milk 'comes in', the breasts become red, hard, painful, and throb. Red streaks may radiate out from the nipples.\n\nChickenpox\n\nWith FEVER; HEADACHE (see here).\n\nWith Belladonna general symptoms and emotional state.\n\nCommon cold\n\nWith FEVER; HEADACHE; LOSS OF SMELL\/TASTE.\n\nCause getting chilled; getting head wet.\n\nConvulsions\n\nIn babies.\n\nCause teething.\n\nWith typical Belladonna general symptoms and emotional state. Some babies find teething difficult and can become ill with a fever and\/or convulsions. Always seek professional help.\n\nCough\n\nBarking; dry; exhausting; hard; hollow; in fits; irritating; racking; tickling; tormenting; violent. BREATHING fast. In PREGNANCY. PAIN sharp; in chest. With HOARSE VOICE.\n\nWorse at night; for deep breathing.\n\nCause getting chilled.\n\nBabies may cry before the cough anticipating pain.\n\nCramp\n\nWorse during labour in hands or feet.\n\nDizziness\n\nIn pregnancy. With HEADACHE.\n\nWorse getting up from bending down; stooping.\n\nEarache\n\nPAIN spreading down into neck; stitching; tearing; throbbing. With FACE-ACHE; NOISES IN EAR.\n\nWorse on right side.\n\nPain is violent and causes great anguish.\n\nEye inflammation\n\nEYES bloodshot; burning; dry; sensitive to light; watering. With symptoms of COMMON COLD.\n\nWorse for light; for heat.\n\nFever\n\nHEAT alternating with chills; burning; dry; at night; radiant. With GRINDING OF TEETH; DELIRIUM; THIRSTLESS. Without SWEATING.\n\nWorse for light; for being uncovered; in afternoon; in evening; at night.\n\nRadiates heat, especially from head, although limbs feel cold. Skin may be alternately dry and moist and if there is sweat it will be on covered parts of body.\n\nHeadache\n\nPAIN in back of head; in eyes; in forehead; in temples; bursting; hammering; pulsating; throbbing; violent; starts and stops suddenly. In PREGNANCY.\n\nBetter for resting head; for lying in a darkened room; for pressure.\n\nWorse for bending down; for cold; for heat; for light; for exposure to sun; for tying up hair; for walking.\n\nCause cold air; getting head wet (haircut); overexposure to sun.\n\nInjuries\n\nCUTS\/WOUNDS bleed freely.\n\nInsomnia\n\nSLEEP restless. With SLEEPINESS; GRINDING OF TEETH.\n\nMoans in sleep; feels sleepy but is unable to sleep; has nightmares or dreams of falling.\n\nLabour pains\n\nSLOW or FALSE LABOUR. INEFFECTUAL: cervix doesn't soften. PAIN distressing; severe\/violent; stops (or slows down); weak. With EXHAUSTION; CRAMPS in hands or legs; RED FACE.\n\nContractions are painful but ineffective.\n\nMeasles\n\nONSET sudden. SKIN RASH burns; hot; itches; red. With FEVER (see here); COUGH (see here); EYE INFLAMMATION (see here).\n\nWith Belladonna general symptoms and emotional state.\n\nMumps\n\nGLANDS painful; swollen; worse right side. ONSET sudden. With FEVER (see here); HEADACHE (see here); SORE THROAT (see here).\n\nThe glands are painful to touch.\n\nRoseola\n\nSee MEASLES for symptoms.\n\nScarlet fever\n\nSee MEASLES for symptoms.\n\nSore throat\n\nTHROAT constricted; irritated; raw. Swollen GLANDS. PAIN severe; stitching.\n\nWorse on right side; for swallowing liquids.\n\nCause getting cold.\n\nNeck is tender to touch, and talking is difficult. Constant desire to swallow despite extreme pain; pains may radiate from throat up into right ear on swallowing. Sips drinks with head bent forward.\n\nSunstroke\n\nFEVER (see here). HEADACHE (see here).\n\nBelladonna will cure sunstroke if Glonoine is indicated and fails (pictures are similar and difficult to differentiate).\n\nTeething\n\nPAINFUL IN BABIES. CHEEKS hot and red; swollen.\n\nSLEEP restless.\n\nWith severe pain and typical Belladonna restlessness.\n\nBELLIS PERENNIS (Bell-p.)\n\nOther name: garden daisy\n\nGeneral symptoms\n\nComplaints from getting chilled when overheated; accident\/injury; surgery.\n\nA small but important remedy for the first-aider. For any illness that follows a plunge into cold water while hot (or overheated), i.e. sudden chilling either externally by bathing in cold water, or internally from cold drinks or ice-cream on a hot day. Also useful after an accident or surgery to aid recovery.\n\nPhysical complaints\n\nAbdominal pain\n\nIn pregnancy. PAINS sudden. UTERUS feels sore. With STIFFNESS IN LOWER ABDOMEN.\n\nCause uterine ligaments stretching.\n\nPains are common towards end of pregnancy as these ligaments are stretched by expanding uterus.\n\nBruises\n\nWith BUMPS AND LUMPS. PAIN sore, bruised. MUSCLES sore, bruised.\n\nCause childbirth; injury; over-exertion; surgery.\n\nSometimes a bump or lump remains after bruising has disappeared, following a blow, knock or accident, or after a period of over-exertion, even if injury occurred a long time ago. Bellis is useful where Arnica has cleared bruising but a lump remains. It is deeper-acting than Arnica and relieves bruised soreness after childbirth where Arnica has not helped. Also useful for injuries to the breast, if Conium helps but a lump remains.\n\nGroin pains\n\nSUDDEN. In PREGNANCY. LEGS weak.\n\nPain, caused by a trapped nerve during the last few months of pregnancy, especially after the baby's head engages, comes on suddenly while walking and may last for only a few minutes. Groin pains are relatively common and can be severe enough to make walking impossible until they have passed. Bellis perennis will help them pass quickly and prevent recurrence.\n\nInsomnia\n\nSLEEPLESSNESS after 3 a.m.\n\nFalls asleep easily and sleeps well before 3 a.m., but after this time cannot get back to sleep.\n\nJoint pain\n\nCause getting chilled after being very hot.\n\nBORAX VENETA (Bor.)\n\nOther name: sodium biborate\n\nGeneral symptoms\n\nBetter for fresh air.\n\nWorse for cold; for riding in a car.\n\nGood for nervous babies who may have difficulty teething, becoming sensitive and prone to colds.\n\nEmotional state\n\nAnxious children; at night; of downward motion. Clingy. Dislikes strangers. Expression anxious. Fearful generally; of downward movement; of sudden noises (sneezing, etc.). Irritable before stools. Jumpy. Screaming in babies during sleep. Tearful babies at night.\n\nBorax babies hate downward motion and scream on being rocked or put down in their cots. If asleep when you place them in their cots they wake immediately. They do not like to be thrown up (and then down) in the air as do many babies. They are nervous, easily startled by sudden noises such as sneezing or the hoover being turned on. They may wake screaming from the slightest noise or scream suddenly during sleep and wake up screaming for no apparent reason, as if from a nightmare, and are particularly irritable leading up to passing a stool, and change dramatically to being cheerful directly afterwards.\n\nPhysical complaints\n\nBreast(feeding) problems\n\nBREASTS painful during feeding; aching after feeding.\n\nPain in breast opposite to one baby is feeding on. Afterwards when breasts are empty, breasts ache.\n\nDiarrhoea\n\nPainless. STOOLS with mucus.\n\nIn teething infants; may also accompany THRUSH.\n\nDizziness\n\nCause downward motion.\n\nHiccups\n\nIn babies.\n\nBorax works in nervous babies who hiccup after feeds.\n\nThrush (genital)\/Yeast infection\n\nDISCHARGE like egg white; burning; white.\n\nWorse between menstrual periods.\n\nThrush (oral)\n\nIn BABIES. THRUSH of mouth; of tongue. MOUTH bleeds easily; hot; dry. With EXCESS SALIVA.\n\nWorse for breastfeeding; for touch.\n\nInfant cries with pain while feeding or refuses breast altogether. The baby's mouth feels hot to the mother's nipple.\n\nTravel sickness\n\nWith NAUSEA; VOMITING.\n\nWorse for downward movement.\n\nBRYONIA ALBA (Bry.)\n\nOther name: white bryony\n\nGeneral symptoms\n\nComplaints from change of weather from cold to warm; getting chilled; weaning. Dizziness. Dryness generally. Face dark red. Likes cold drinks; hot food. Lips dry. Mouth dry with thirst. Onset of complaint slow. Pains sore, bruised; stitching. Sweat absent during fever. Taste in mouth bitter. Thirsty for large quantities; at infrequent intervals. Tongue coated brown; dirty white.\n\nBetter for lying still; for firm pressure.\n\nWorse for slightest movement; at 9 p.m.; for flatulent food (beans, cabbage, etc.).\n\nBryonia illness develops slowly over days, like Gelsemium. Acute complaints, such as flu, fevers and coughs, are often accompanied by headaches, a general dryness (mouth, lips, tongue, chest, eyes, etc.), and great thirst for large quantities (drinks are gulped straight down). Pains are 'stitching', and especially bad in evening around 9 p.m. although symptoms may also be bad on waking in the morning. The slightest movement aggravates the pains; the head, for example, aches even from rolling eyeballs. Firm pressure, lying on back and\/or applying pressure to painful areas helps, as does lying on side that hurts. Bryonia types like fresh air and are generally worse in hot stuffy rooms (that is, from being overheated), although better for being kept warm and covered, especially if very ill.\n\nEmotional state\n\nAngry. Anxious. Capricious. Desires to be carried. Irritable. Morose. Sluggish.\n\nBryonia types are nicknamed 'the bear' because of their irritability. They are especially bad when disturbed. They may lie loglike, pretending to be asleep to avoid having to respond. They resent intrusion, wanting to be left alone when ill, and are touchy, not wanting to be questioned, examined or interfered with in any way. Babies want to be carried though not moved about too much. They are capricious and reject things \u2013 toys, food, etc. \u2013 they have just asked for.\n\nPhysical complaints\n\nAbdominal pain\n\nIn pregnancy. With STIFFNESS in lower abdomen.\n\nTowards end of pregnancy when muscles and ligaments are stretched and vulnerable to strain, great stiffness is felt in lower abdomen.\n\nBackache\n\nPAIN in LOWER BACK; stitching.\n\nWorse for coughing; for slightest movement.\n\nBreast(feeding) problems\n\nBREASTS engorged; hard; hot; inflamed; pale; painful. MILK SUPPLY overabundant.\n\nWorse for slightest movement.\n\nBreasts look pale. Any movement is painful and the inflammation (mastitis) may be accompanied by fever and depression.\n\nBreast pain\n\nIn pregnancy. With MASTITIS.\n\nBroken bones\n\nPAIN stitching.\n\nWorse for slightest movement.\n\nBryonia may be given after a fracture where Arnica and Symphytum have been given and there is still tremendous pain.\n\nCommon cold\n\nWith HEADACHE; SNEEZING.\n\nNose may feel stuffed up after the watery discharge ceases; the cold quickly settles on the chest.\n\nCough\n\nDry; in fits; irritating; racking; vomiting; disturbs sleep. BREATHING fast. In PREGNANCY. PAIN IN CHEST stitching; holds chest with hands. PAIN IN STOMACH from coughing. With HEADACHE (see here).\n\nBetter for fresh air; for lying on painful side.\n\nWorse for deep breathing; for slightest movement of chest; in right lung.\n\nAccompanied by little or no expectoration. Eating or drinking may make the cough worse, because of the movement involved.\n\nDiarrhoea\n\nWorse after getting up; during morning; for movement.\n\nCause hot weather; excess of fruit.\n\nDizziness\n\nWorse for slightest motion; walking.\n\nExhaustion\n\nExtreme.\n\nWorse for slightest exertion.\n\nCause breastfeeding.\n\nEye inflammation\n\nEYES dry; sore.\n\nWorse for moving the eyes.\n\nFever\n\nHEAT burning; dry; alternating with chills; ONE-SIDED; WITHOUT SWEATING.\n\nBetter for complete rest.\n\nWorse in autumn; around 9 p.m.\n\nFeels hot internally and externally; the right side of body is hotter than the left. Chills may be present during day and in evening after lying down in bed.\n\nFlu\n\nWith Bryonia general symptoms and emotional state.\n\nGastric flu\n\nBILIOUSNESS. TASTE bitter.\n\nPAIN aching; in stomach. With symptoms of FEVER (see here).\n\nBetter for belching.\n\nWorse for coughing; after eating bread; for movement; for waking; in evening; lying down in bed.\n\nAll food and drink (except water) taste bitter. There is a sensation of a stone lying in the stomach, and the pains may be better for belching.\n\nHeadache\n\nPAIN behind eyeballs, in forehead; bursting; violent.\n\nBetter for cold compresses; for pressure.\n\nWorse for coughing; after getting up.\n\nCause breastfeeding; change of weather; cold, damp weather; ironing; overexposure to sun.\n\nThese headaches last all day.\n\nHeartburn\n\nIn pregnancy. With INDIGESTION.\n\nHernia in babies\n\nUMBILICAL.\n\nJoint pain\n\nPAINS stitching. With SWELLING.\n\nBetter for pressure; for rest.\n\nWorse for cold; for slightest movement.\n\nJoints look either pale or red, and are better for pressure \u2013 or resting on the painful parts. Tight bandaging helps.\n\nMeasles\n\nONSET slow. SKIN RASH slow to appear. With COUGH; FEVER; HEADACHE (see here).\n\nWith characteristic Bryonia dryness and dislike of movement.\n\nMumps\n\nWith Bryonia general and emotional\/mental symptoms.\n\nPhlebitis\n\nPAINS in legs after childbirth.\n\nBetter for rest.\n\nWorse for movement.\n\nSore throat\n\nPAIN stitching. VOICE hoarse. With symptoms of FEVER (see here).\n\nWorse for swallowing.\n\nVaricose veins\n\nOf vulva.\n\nWorse during pregnancy.\n\nVomiting\n\nVOMIT tastes bitter; watery.\n\nWorse for movement; for coughing.\n\nCALCAREA CARBONICA (Calc-c.)\n\nOther name: oystershell\n\nGeneral symptoms\n\nAnaemia. Catches colds easily. Clumsy. Complaints from getting wet; sprains. Likes boiled eggs; strange things in pregnancy (chalk, coal etc.). Discharges thick. Dislikes coffee; meat; tobacco. Face pale. Glands swollen; painless. Oedema (swelling) of ankles\/feet; of hands\/fingers. Pains cramping. Palpitations. Sense of smell lost. Slowness of children to teethe\/to walk. Sweat on head; sour; profuse; from slightest physical exertion; from mental exertion. Symptoms right-sided. Taste in mouth bad; sour. Teeth crumbling, decaying. Tongue white-coated. Weight gain easy, in babies.\n\nBetter for being constipated; for heat; for lying down.\n\nWorse for cold; for damp; for draughts; for exertion; for fresh air; after drinking milk; for tight clothes.\n\nThose needing this remedy tend to be sluggish, move slowly and look white and pasty and feel 'spineless' \u2013 adults have limp handshake and may slump in their chairs. Exertion leaves them weak and breathless; and they feel better for lying down. Babies are slow to walk and produce teeth and their fontanelles are slow to close. They have large heads and bellies.\n\nThose who need this remedy are chilly and worse for cold and damp but may overheat easily and be subject to hot flushes. Their feet and hands are cold and often clammy, even in bed; they hate draughts or fresh air and may catch cold easily after swimming or getting wet. Warmth relieves their symptoms.\n\nThey sweat on their heads and at the back of the neck \u2013 especially while asleep \u2013 so profusely their sheets may be wet. Sweat and discharges smell sour.\n\nMetabolism is slow and everything turns to fat. Milk turns sour in the stomach, causing nausea. The one unusual symptom is that they feel generally better for constipation.\n\nEmotional state\n\nAnxious about health; children, at night; during evening. Confused. Depressed. Despair of getting well. Fearful generally; children, at night; of death; in evening. Melancholic. Slow. Sluggish. Stubborn children. Tearful babies.\n\nWorse for thinking.\n\nAnxious, sluggish types, who find concentration difficult when ill. Babies are happy and content when well but may seem lethargic at times and more difficult to handle if teething or unwell as their stubborn side comes out more strongly.\n\nPhysical complaints\n\nBackache\n\nPAIN aching; in LOWER BACK; feels sprained.\n\nWorse for damp; on getting up from sitting.\n\nCause lifting.\n\nBack feels weak; cannot easily sit straight and soon slumps in a chair.\n\nBraxton Hicks contractions\n\nIn pregnancy.\n\nBreast(feeding) problems\n\nMILK SUPPLY overabundant.\n\nBreasts may be large and uncomfortable.\n\nBreathless\n\nIn pregnancy.\n\nWorse for walking uphill\/upstairs; for exertion.\n\nBroken bones\n\nBONES slow to mend.\n\nIf a fracture fails to heal well after Symphytum, Calcarea carbonica or Calcarea phosphorica may be given (as a tonic, see here) until it has healed. Use the general symptoms and emotional state to choose between them.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nWith tingling and numbness in fingers and swelling of wrists\/fingers.\n\nCommon cold\n\nNASAL CATARRH dry; smelly; yellow. NOSE blocked. With LOSS OF SMELL; PAINLESS HOARSENESS.\n\nWorse during the morning.\n\nConstipation\n\nSTOOLS hard at first; large; pale; sour-smelling.\n\nBetter generally for being constipated.\n\nInitial large, hard stool (which may be clay-like, or like a lump of chalk) may be followed by diarrhoea (see here).\n\nCough\n\nDry at night; loose during morning. In PREGNANCY. MUCUS copious; smelly; yellow; tough; tastes sour\/sweet. With symptoms of FEVER.\n\nWorse for playing piano; morning; evening in bed; at night; during fever.\n\nCoughs up mucus with difficulty. Babies are prone to coughs when teething. The cough continues after the teeth come through.\n\nCradle cap\n\nSour-smelling, thick scales\/crusts on the scalps of sweaty-headed babies.\n\nCramp\n\nIn calf; hand; sole of feet; toe.\n\nWorse at night; during pregnancy; for stretching leg in bed.\n\nCramps come on when stretching limbs on waking up in bed, during the night or first thing in the morning.\n\nDiarrhoea\n\nIN TEETHING BABIES. STOOLS containing undigested food; sour; watery.\n\nWorse after drinking milk.\n\nCause drinking milk; teething.\n\nDiarrhoea follows a formed stool, is full of undigested food and smells sour.\n\nDizziness\n\nWith HEADACHE (see here).\n\nWorse for moving\/turning head quickly.\n\nCause high places.\n\nSensitive to heights, becoming dizzy and headachy.\n\nEarache\n\nPAINS throbbing. With NOISES IN EAR.\n\nWith typical Calcarea carbonica general and emotional symptoms.\n\nExhaustion\n\nWith BREATHLESSNESS. DIZZINESS.\n\nWorse for mental exertion; for slightest physical exertion; for walking; for walking upstairs.\n\nCause breastfeeding.\n\nEye inflammation\n\nIn BABIES. DISCHARGE purulent. EYES sensitive to light; watering. EYELIDS glued together; gritty. With symptoms of COMMON COLD.\n\nInflammation accompanies a cold; eyes may ooze a smelly discharge.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; intolerant of tight clothing.\n\nHair loss\n\nAfter CHILDBIRTH.\n\nHeadache\n\nPAINS burning; bursting; maddening.\n\nBetter for lying down.\n\nWorse for light; for noise; on right side of head.\n\nCause cold, damp weather; getting wet.\n\nPains are worst at back of the head and spread up to the top. They are often worse while reading and for any jarring upward movement. Feels better for lying down with eyes closed.\n\nHernia in babies\n\nUMBILICAL.\n\nHoarseness\n\nPAINLESS.\n\nWorse in morning.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; hard. BELCHES sour. PAIN pressing. With FLATULENCE; HEARTBURN during pregnancy.\n\nInsomnia\n\nAnxious DREAMS.\n\nWorse before midnight.\n\nCause overactive mind; worry.\n\nSleepless from worry with persistently anxious thoughts. Anxious dreams may be interspersed with pleasant ones.\n\nJoint pain\n\nPAIN cramping.\n\nWorse for wet weather; cold.\n\nCause wet weather.\n\nWhen Rhus toxicodendron has worked well but has stopped having any effect, Calcarea carbonica often helps if typical general symptoms and emotional state are present.\n\nLochia\n\nLasts too long; intermittent; milky appearance.\n\nMilk supply low\n\nBreasts are full and may be sore but aren't producing much milk. With Calcarea carbonica general symptoms and emotional state.\n\nNausea\n\nWorse for milk.\n\nSore throat\n\nTHROAT dry.\n\nCause change of weather.\n\nSprains\n\nOf ankle; hand; wrist.\n\nWorse for lifting.\n\nCause lifting heavy weights.\n\nA useful remedy for clumsy people who stumble frequently while walking and sprain their ankles easily. Also for sprains from lifting heavy weights that do not clear up with Rhus toxicodendron and\/or Ruta graveolens.\n\nTeething\n\nPAINFUL in babies. SLOW. DIFFICULT. With DIARRHOEA (see here).\n\nBabies may make a chewing motion in their sleep and grind their teeth \u2013 or gums! Their teeth take forever to arrive.\n\nVaricose veins\n\nIn pregnancy. Of the VULVA.\n\nVomiting\n\nVOMIT curdled milk; sour.\n\nCALCAREA FLUORICA (Calc-f.)\n\nOther name: calcium fluoride\n\nAs with other Calcarea salts, those needing this remedy are generally worse for cold, damp weather. It is useful as a tonic for any tissues that have become worn out, flabby and lax. During pregnancy, Calcarea fluorica may be helpful if skin is dry and out of condition, or for varicose veins. Postnatally, repeated short courses are beneficial for mild prolapse with no other symptoms. Short courses of Calcarea fluorica can be repeated (say, once a month) if it helps during pregnancy or after childbirth.\n\nPhysical complaints\n\nBackache\n\nPAINS IN LOWER BACK.\n\nBetter for continued movement.\n\nWorse on beginning to move.\n\nIf Rhus toxicodendron was indicated and only partially relieves pain, or if it fails, Calcarea fluorica usually helps.\n\nCommon cold\n\nNASAL CATARRH dry. SNEEZING with difficulty.\n\nBetter for sneezing.\n\nFor stuffy head colds with dry catarrh.\n\nSkin complaints\n\nIn pregnancy. STRETCH MARKS; DRYNESS.\n\nCALCAREA PHOSPHORICA (Calc-p.)\n\nOther name: calcium phosphate\n\nGeneral symptoms\n\nCatches colds easily. Complaints from loss of body fluids. Face pale. Slowness of babies to learn\/to teethe. Thin. Weight gain poor in babies.\n\nWorse for cold; for damp; for draughts; for fresh air; for wet weather.\n\nCalcarea phosphorica is useful as a tonic for worn-out nervous systems. Vital in the growth and maintenance of healthy cells, nourishes blood cells, bones, teeth and all connective tissue and therefore may be indicated in slow developers. Food is poorly assimilated which may cause anaemia, slow growth, and tooth decay, and fontanelles which close slowly. A wonderful tonic for children who have had a growth spurt and become pale and exhausted. Adults and children who are weak and tired while convalescing from illness will likewise benefit. The typical Calcarea phosphorica type is thin with long, dark eyelashes and dark hair. Sensitive to draughts, cold and damp, they have cold extremities.\n\nEmotional state\n\nAnxious. Discontented. Irritable children. Restless. Screaming in sleep. Sighing. Slow. Sluggish.\n\nWorse for mental exertion.\n\nCalcarea phosphorica types (adults and babies) are discontented, complaining, grumbling and sighing when talking. They have no 'go', are sluggish and restless, not knowing what they want. Babies may be slow in their general development.\n\nPhysical complaints\n\nAnaemia\n\nWith EXHAUSTION.\n\nCause acute illness.\n\nFor the convalescent stage of an illness, or post-childbirth, where regaining strength is proving difficult.\n\nBreast pain\n\nIn pregnancy.\n\nBroken bones\n\nSLOW TO MEND.\n\nFor fractures taking longer than expected to heal. It may be given routinely after Symphytum has dealt with the pain of the fracture, as it will speed up the healing.\n\nCough\n\nMUCUS yellow.\n\nWorse when teething.\n\nUseful for obstinate coughs (or whooping cough) which tend to be worse in cold-weather months, or precipitated by teething.\n\nCramp\n\nIn CALF.\n\nWorse for walking.\n\nDiarrhoea\n\nIn breastfeeding babies.\n\nExhaustion\n\nWith HEAVINESS in limbs. With WEAK LEGS.\n\nCause breastfeeding; pregnancy.\n\nHeadache\n\nCause mental exertion; overwork; anaemia.\n\nWorse for cold wind, but cold bathing may help.\n\nInsomnia\n\nWAKING difficult; late.\n\nWorse before midnight. Mornings are awful, waking is difficult.\n\nJoint pain\n\nIn hips, in pregnancy.\n\nWorse for cold, wet weather.\n\nFoot joints may also be affected. Feet are always cold and there is a cramping and aching numbness.\n\nTeething\n\nPAINFUL in babies. SLOW. DIFFICULT. GREEN STOOLS.\n\nHelpful for teeth that are slow in cutting through the gums. Teething children may develop diarrhoea, colds and coughs. Teeth are inclined to decay easily and prematurely. Calcarea phosphorica ensures better assimilation of calcium and encourages healthy dentine formation.\n\nCALCAREA SULPHURICA (Calc-s.)\n\nOther name: calcium sulphate\n\nGeneral symptoms\n\nAbscesses discharging pus. Discharges, blood-streaked.\n\nWorse for heat; for milk; for physical exertion; in stuffy rooms.\n\nThis remedy helps with abscesses or longstanding catarrh. Those needing it are prone to thick, lumpy, yellow or bloody discharges, and generally worse for warmth and overheating, unlike other Calcarea remedies, which are chilly, and unlike Hepar sulphuris, another important remedy for abscesses and catarrhs, which is also chilly. They like to be uncovered, if feverish, and are usually better for some fresh air, though they dislike draughts.\n\nEmotional state\n\nAnxious during evening. Depressed. Irritable. Sluggish\/dull. Tearful.\n\nPhysical complaints\n\nAbscess\n\nDISCHARGING PUS; of GLANDS.\n\nSpeeds healing of discharging abscesses, i.e., those that have burst or broken and are discharging a thick, yellow, lumpy and possibly bloody matter.\n\nCommon cold\n\nNASAL CATARRH blood-streaked; smelly; thick, yellow. With HEADACHE; LOSS OF SMELL.\n\nWorse after drinking milk.\n\nFor postnasal catarrh (where mucus drips down back of throat) or catarrh that is one-sided.\n\nCough\n\nCOUGH dry. MUCUS copious; lumpy; yellow.\n\nCroup\n\nCough occurs only ON WAKING.\n\nIn croup, where Hepar sulphuris is indicated but fails, Calcarea sulphurica cures, especially if the baby is warm and wants to be uncovered and\/or if croupy cough is only there on waking.\n\nEarache\n\nDISCHARGE blood-streaked; smelly; thick.\n\nEye inflammation\n\nDISCHARGE thick; yellow.\n\nEspecially indicated if both nose and eyes are discharging thick yellow mucus.\n\nInjuries\n\nCUTS\/WOUNDS SLOW TO HEAL; suppurating.\n\nWhere a wound has become inflamed and started to discharge thick yellow pus. (Hepar sulphuris is for the stage before this.)\n\nCALENDULA OFFICINALIS (Calen.)\n\nOther name: marigold\n\nPhysical complaints\n\nInjuries\n\nWOUNDS\/CUTS LACERATED; suppurating. Painful out of proportion to injury.\n\nCalendula is a great healer of wounds and cuts, both externally as an ointment (see External Materia Medica) and internally in potentised form (which helps wounds heal even more quickly). Use Calendula in straightforward injuries where the skin is broken but there are no other noteworthy symptoms. It works well if pains are stronger than the severity of injury warrants. Useful after childbirth to speed the healing of an episiotomy or tear \u2013 you can alternate with Arnica or Bellis perennis or any other remedy that is needed.\n\nCANTHARIS VESICATORIA (Canth.)\n\nOther name: Spanish fly\n\nPhysical complaints\n\nBurns\n\nScalds; sunburn; second degree; with blisters. PAIN burning.\n\nBetter for cold compress.\n\nCystitis\n\nDESIRE TO URINATE constant; frequent; ineffectual; urgent. PAINS before, during and after urination; burning; cutting. URINATION frequent. URINE hot; red; scanty.\n\nWorse for cold drinks; before, during and after urination.\n\nPains come on suddenly and are violent and spasmodic. Despite a constant desire to pass water, bladder never feels empty. Although complaints are often aggravated by cold drinks, there may be a burning thirst. Cantharis is sometimes confused with Apis and Arsenicum because all suffer from severe burning pains and restlessness.\n\nCARBO ANIMALIS (Carb-a.)\n\nOther name: leather charcoal\n\nGeneral symptoms\n\nPains burning. Sweat exhausting; profuse; smelly.\n\nThis is a small remedy which has few general indications. The sweating is worse at night, or during\/after eating.\n\nEmotional state\n\nDepressed. Desires to be alone. Dislikes company. Uncommunicative.\n\nA remedy for anyone debilitated by illness.\n\nPhysical complaints\n\nExhaustion\n\nWorse for walking; during a menstrual period.\n\nCause breastfeeding; after an acute illness; lifting; sweating.\n\nThe typical Carbo animalis sweat accompanies this exhaustion, which occurs in people recovering from an illness and also in nursing mothers. They may weep during meals because they feel too tired even to eat. Walking across a room seems too much. They may also have painful, lumpy breasts which hurt while the baby feeds.\n\nFlatulence\n\nABDOMEN bloated.\n\nCause abdominal surgery.\n\nCarbo animalis will provide some relief where the abdomen is bloated with wind that is difficult to pass after an abdominal operation such as appendicectomy, Caesarean or laparoscopy.\n\nStrains\n\nStrains muscles easily \u2013 of the wrist especially but also the back \u2013 from lifting even small weights; the strained muscle is worse for the slightest exertion.\n\nCARBO VEGETABILIS (Carb-v.)\n\nOther name: wood charcoal\n\nGeneral symptoms\n\nAnaemia. Breath smelly. Complaints from measles; loss of body fluids. Discharges smelly. Face pale; sallow or blue. Feels faint on getting up; on waking. Sweat cold; profuse. Taste bitter.\n\nBetter for fanning; for fresh air.\n\nWorse for exertion; after eating rich\/fatty foods; for humidity.\n\nThese are sluggish types with low vitality who want to lie down and sleep. The slightest exertion exhausts them; they have to force themselves to get going. They feel worse for lying down in spite of being too weak to do otherwise. Mornings and evenings are the worst times of day.\n\nPeople recovering from an acute illness or chest infection often benefit from this remedy, as do children who have not fully recovered from a serious illness such as whooping cough or measles.\n\nThis remedy also deals with acute indigestion, in which case the emotional and general symptoms may not be present.\n\nEmotional state\n\nAnxious during the evening; in bed. Confused. Indifferent to everything. Irritable. Sluggish.\n\nA remedy for inactive folk who find it difficult to rouse themselves. They may become indifferent to the point that they do not care if they live or die but suffer from anxiety in the late afternoon through to the evening, which intensifies when they go to bed and shut their eyes.\n\nPhysical complaints\n\nBreathing difficulties\n\nIn newborn babies.\n\nBreathless\n\nIn PREGNANCY.\n\nBetter for burping.\n\nWorse when walking uphill\/upstairs; lying down.\n\nCommon cold\n\nNOSE blocked. SNEEZING frequent; with difficulty. With HOARSENESS; ITCHING THROAT.\n\nTickling in the throat may be acute; may not be able to sneeze at times in spite of wanting to.\n\nCough\n\nRacking; in fits; suffocative; violent; whooping. BREATHING fast; wheezing. MUCUS green. VOICE HOARSE. With RETCHING.\n\nWorse at night; evening; before midnight.\n\nCoughing fits are often accompanied by the characteristic cold perspiration.\n\nExhaustion\n\nBREATH cold. BREATHLESSNESS.\n\nBetter for being fanned.\n\nCause carbon monoxide poisoning; food poisoning; accident\/injury; acute illness; loss of body fluids (diarrhoea, vomiting), breastfeeding; surgery.\n\nCarbo vegetabilis is useful for extreme weakness \u2013 'collapse' may be a more accurate label \u2013 commonly occurring after an accident or operation, severe vomiting, or during convalescence from a serious illness. The breath and sweat are cold, the skin feels cold to the touch but there is a feeling of heat internally and a desire for fresh, cool air. The face may be deathly pale or even blue, with bluish lips; the head feels heavy. I do not suggest that you treat anyone seriously ill without help, but if you are the only person available and this remedy fits the case, give it while you wait for help to arrive. Carbo vegetabilis may also be indicated to treat forms of poisoning: food; North Sea gas, or car exhaust fumes that may come on after sitting in a traffic jam on a hot, windless day for a long period.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; rumbling. WIND smelly. With DIARRHOEA.\n\nBetter for passing wind.\n\nCause abdominal surgery.\n\nAbdomen is bloated below the navel as with Lycopodium flatulence.\n\nFood poisoning\n\nWith FLATULENCE (see here).\n\nCause rotten fish\/meat.\n\nHair loss\n\nCause acute illness; childbirth.\n\nHeadache\n\nPAINS in back of head; heavy; pressing.\n\nHead feels heavy, like lead.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated. BELCHES empty; sour. PAINS burning; cramping. With FLATULENCE (see here); NAUSEA.\n\nBetter for belching; for passing wind.\n\nWorse after eating rich\/fatty food.\n\nCarbo vegetabilis digestion is easily upset. The stomach feels full, becoming so bloated after eating that the skin is stretched as tight as a drum. Tight clothes feel uncomfortable. The nausea is worse in the mornings. Burping may relieve the bloatedness for a while but it builds up again quite quickly.\n\nMumps\n\nGLANDS swollen\/painful. PAINS spread to breasts, ovaries.\n\nWith typical Carbo vegetabilis pale face and cold sweat.\n\nNosebleeds\n\nBLOOD dark.\n\nWorse at night.\n\nGood for weakened state after an acute illness such as measles, which results in night-time nosebleeds for no other apparent reason.\n\nSpots\n\nIn babies (pimples). With FLATULENCE.\n\nVaricose veins\n\nIn pregnancy. Of LEG\/THIGH; of VULVA.\n\nCASTOR EQUI (Cast.)\n\nOther name: horn (horse's)\n\nPhysical complaints\n\nBreast(feeding) problems\n\nSore\/cracked NIPPLES.\n\nThis small remedy has few uses other than the treatment of sore, cracked nipples. Effective for women who are otherwise well with no other symptoms. Breasts may be engorged and the skin itchy; nipples are sore, cracked and very tender, quickly becoming raw if left untreated.\n\nCAULOPHYLLUM (Caul.)\n\nOther name: blue cohosh\n\nCaulophyllum's major use is for establishing effective contractions in labour. Some homeopathic books advise pregnant women to take Caulophyllum during the last weeks or months before delivery to prepare them for an easy labour. Think carefully before doing this as some women have had short but violent labours after taking it or protracted ones (see Provings). If this is not your first baby and you have a history of easy labours, do not take it. If it is your first baby and you are worried about the birth consult a homeopath, who will advise you properly on this remedy and any others that you might need.\n\nPhysical complaints\n\nBleeding (vaginal) in pregnancy\n\nFLOW scanty. With bearing down PAIN; PAIN in back; weakness and trembling.\n\nBraxton Hicks contractions\n\nIn pregnancy.\n\nWith no other symptoms.\n\nJoint pain\n\nPAIN in the small joints; flying around; irregular.\n\nPains occur in finger and hand joints and move around frequently.\n\nLabour pains\n\nLABOUR: late; slow. LABOUR PAINS: distressing; ineffectual (cervix doesn't soften); irregular; last a short time; stop (or slow down) from exhaustion; weak. With THIRST (during contractions); EXHAUSTION; TREMBLING. With IRRITABILITY.\n\nLabour pains may appear in the groin, bladder and legs, and fly from one place to another. Cervix is rigid and does not dilate. Feels chilly, with trembling or shivering, even when covered up. Contractions are short and very painful. The only marked emotional state to look for is irritability (not anger) with the exhaustion and trembling.\n\nCAUSTICUM (Caust.)\n\nOther name: potassium hydrate\n\nGeneral symptoms\n\nAppetite lost in pregnancy. Blister on tip of tongue; painful. Clumsy trips easily while walking. Complaints from change of weather to dry; getting wet. Discharges watery. Dislikes sweets\/sweet things. Exhaustion during evening. Eyelids heavy. Likes smoked foods. Loss of libido. Restless during evening. Tongue red stripe down centre (white edges).\n\nBetter for cold drinks; for heat; for warmth of bed.\n\nWorse for changes in weather; for coffee; for cold; for draughts; during the evening; for fresh air; for walking; for getting wet.\n\nThese are chilly people badly affected by getting wet, draughts, changes in the weather and especially by clear, dry, cold weather. Mild, wet weather makes them feel better, especially their 'rheumatics' and their 'chests'; when everyone else is complaining about the damp they will be enjoying some relief from their pains. Evening weakness is overwhelming. Blisters often accompany a cough or sore throat. Women may lose their appetite when pregnant and develop a dislike of anything sweet.\n\nEmotional state\n\nAbsent-minded. Anxious. Complaints from grief. Concentration poor. Depressed. Fearful children \u2013 at night; during evening. Forgetful. Irritable. Memory weak. Sympathetic. Tearful at least little thing.\n\nThese are sensitive, anxious souls who suffer when those close to them are hurt either emotionally or physically, and react against injustice in any shape or form. Pessimistic by nature, they may become gloomy and full of anxious forebodings when ill, despairing of getting well. Irritability sets in, and then depression. They cry easily, over a sad news item in the paper or on television. After the loss of someone close (even a favourite pet), they sink into a negative state, becoming irritable and then depressed. A woman may lose her sex drive during pregnancy and\/or find that it takes a long time to return after the birth. Babies cry easily, are frightened of the dark and of going to bed on their own.\n\nPhysical complaints\n\nBreast(feeding) problems\n\nCracked\/sore NIPPLES.\n\nMay have difficulty in establishing a good milk supply because of exhaustion and anxiety.\n\nBurns\n\nBURNS third degree. PAINS burning. With BLISTERS.\n\nFor serious, third-degree burns, chemical burns, or scalds, with the characteristic Causticum pains. These must receive expert attention. You may prescribe Causticum on the way to hospital to provide some relief. Causticum may also help the soreness at the site of old burns.\n\nConstipation\n\nDESIRE TO PASS STOOL ineffectual. PAIN stitching.\n\nStool may be soft despite constipation. May have to stand up to pass a stool.\n\nCough\n\nConstant; distressing; exhausting; hollow; racking; rattling; tormenting; violent; woken by cough at night. In PREGNANCY. MUCUS difficult to cough up; must swallow what comes up. PAINS raw in chest. With VOICE HOARSE.\n\nBetter for sips of cold water.\n\nWorse for breathing in cold air; getting warm in bed; bending head forward; lying down.\n\nHoarseness with this cough is worse in the mornings. Coughing up mucus is difficult as it comes up to the throat and slips back again. Small sips of water are about the only thing that eases the cough and stops it, albeit temporarily.\n\nCramp\n\nIn FEET; in TOES; in SOLES OF FEET.\n\nWorse at night.\n\nCystitis\n\nDESIRE TO URINATE frequent, ineffectual. PAINS burning; during urination. URINATION difficult; frequent; involuntary; slow to start; with unfinished feeling.\n\nWorse while urinating.\n\nFeels the urge to urinate frequently but finds it difficult or impossible to pass water. May do so involuntarily when coughing, sneezing or blowing the nose.\n\nIncontinence\n\nIn PREGNANCY.\n\nWorse for coughing\/sneezing\/laughing or walking.\n\nCause becoming chilled.\n\nInvoluntary urination (stress incontinence) or leaking, either in pregnancy or after birth.\n\nIndigestion\n\nABDOMEN\/STOMACH feels full. BELCHES empty; tasting of food just eaten. PAINS cramping; pressing.\n\nJoint pain\n\nPAINS in back; in joints; in neck; burning; gnawing; pressing; stitching; tearing. With STIFFNESS.\n\nBetter for heat; for warmth of bed.\n\nWorse for dry cold; getting up from sitting.\n\nPains cease on getting warm in bed to begin again on getting up in the morning. Restless at night.\n\nRestless legs\n\nIn PREGNANCY.\n\nWorse in the evening.\n\nRetention of urine (after labour)\n\nWith frequent, painful, urgent desire to urinate.\n\nWants to pee but can't \u2013 and it hurts; or passes a little perhaps involuntarily.\n\nSore throat\n\nCONSTANT THROAT burning; dry; raw. CHOKING sensation; constant DESIRE TO SWALLOW.\n\nWorse for talking.\n\nSwallowing is difficult as the throat feels too narrow. Cannot expectorate. Hoarseness remains after the sore throat has healed.\n\nCHAMOMILLA (Cham.)\n\nOther name: German chamomile\n\nGeneral symptoms\n\nComplaints from coffee; teething. Face red; one-sided \u2013 in spots; with toothache (teething). Likes cold drinks. Pains unbearable. Sweat clammy; hot; better for uncovering.\n\nBetter for uncovering.\n\nWorse during the evening; for fresh air; for coffee; for wind.\n\nEspecially suited to teething babies, women in labour, and anyone who has been in a highly emotional state (especially angry) for a long time, becoming over-sensitive, mentally or physically, as a result. Its keynote is unbearable pain. There may be sweating with the pains, especially on the scalp and face, along with a high fever. May feel better for sweating. Feels hot and doesn't like it, kicks off the bedclothes or sticks burning soles of feet out of bed. Is sensitive to wind and being chilled by cold, damp air.\n\nEmotional state\n\nAngry babies; women during labour; violently angry. Complaints from anger. Depressed. Desires to be carried; to be alone. Dislikes\/aversion to company; being spoken to; being looked at; being touched. Excitable. Hitting. Impatient. Irritable babies; when teething; women in labour. Quarrelsome women during labour. Restless babies \u2013 better for being carried. Screaming with pain. Sensitive to pain. Tearful babies.\n\nIs angry and excitable, tolerates nothing and nobody. Incredibly sensitive to pain and enraged by it. Snaps and snarls, demands relief from pain, but nothing helps. Chamomilla people say, 'I can't bear the pain any longer.'\n\nBabies become spiteful, hitting parents. They whine, scream and cannot be comforted. They ask for things which they may immediately hurl across the room (drinks, toys, etc.). Babies insist on being carried and cry loudly when held still or put down. Even after being carried for a short time they may start to cry. The parents of a Chamomilla baby may soon reach the limits of their endurance and talk of adoption!\n\nPhysical complaints\n\nAfterpains\n\nPAIN unbearable.\n\nWorse when the baby nurses.\n\nPains distressing, accompanied by frantic ill temper.\n\nBraxton Hicks contractions\n\nIn pregnancy.\n\nColic\n\nABDOMEN\/STOMACH feels bloated. PAIN unbearable.\n\nCause anger.\n\nBaby cries out while passing a stool. May have diarrhoea with green stools.\n\nCough\n\nDry during sleep; irritating; tickling. MUCUS (in adults) tastes bitter.\n\nWorse at night.\n\nCough often worse at night; does not wake a sleeping baby.\n\nDiarrhoea\n\nSTOOLS green; hot; smelling of rotten eggs; PAINFUL. In TEETHING and BREASTFEEDING babies.\n\nEarache\n\nPAINS aching; pressing; stitching; tearing; unbearable.\n\nWorse for wind; bending down.\n\nEffects of drugs taken during or after labour\n\nIRRITABLE; and SLEEPLESS.\n\nCause morphine or pethidine.\n\nFever\n\nHEAT burning; ONE-SIDED. With THIRST; SHIVERING.\n\nWorse mid-morning.\n\nCause anger.\n\nFace and breath are hot while body is chilly and cold. Face may sweat after eating.\n\nFlatulence\n\nABDOMEN\/STOMACH bloated.\n\nInsomnia\n\nWith SLEEPINESS. DREAMS VIVID during pregnancy.\n\nResults from pain, anger or stimulants (coffee, etc.), or too much chamomile tea even in babies.\n\nJoint pain\n\nPAINS violent. With NUMBNESS.\n\nDriven out of bed by pain, helped by walking about.\n\nLabour\n\nLABOUR PAINS in the back; distressing; ineffectual (cervix doesn't soften); severe; stopping (or slowing down); unbearable. With EXHAUSTION.\n\nSays she can't bear the pain and wants to die. Isn't anxious like Aconite, but angry and impatient as the cervix may be slow to dilate.\n\nTeething\n\nCHEEKS hot and red; pale and cold; red spot on one cheek. PAINS unbearable; cries out in sleep. With DIFFICULTY TEETHING, RESTLESS SLEEP; DIARRHOEA; GREEN STOOLS; COUGH.\n\nWorse for heat of bed; warm food\/drinks; for pressure; for warmth of bed.\n\nMay have at the same time one hot, red cheek or a red patch on one cheek, and one pale, cold cheek. Babies don't want to chew on anything, as pressure aggravates their sore gums.\n\nVomiting\n\nEasy. VOMIT bile.\n\nCause anger.\n\nCHELIDONIUM MAJUS (Chel.)\n\nOther name: celandine\n\nPhysical complaints\n\nJaundice\n\nIn newborn babies.\n\nThe main remedy for jaundice in little babies, it helps the liver adjust to life outside the uterus. If the baby is well in all other respects then give a short course of Chelidonium. If other symptoms are present, choose between other remedies that also have jaundice in their picture.\n\nCHINA OFFICINALIS (Chin.)\n\nOther name: cinchona officinalis\n\nGeneral symptoms\n\nAnaemia. Appetite lost. Complaints from loss of body fluids. Dislikes bread; butter; food in general \u2013 in pregnancy; fruit; rich, fatty food; meat. Eyes sunken. Face pasty. Likes cold drinks; spicy foods; sweets. Pains sore, bruised. Sweat cold; on covered parts of the body, profuse; on single parts of the body; increased by slightest exertion. Taste bitter.\n\nBetter for firm pressure.\n\nWorse for cold; for fresh air; for light touch; for movement; at night.\n\nThe China picture often emerges in someone anaemic, weak and depleted following a condition with a prolonged, exhausting discharge or loss of body fluids, such as diarrhoea, vomiting, perspiring, breastfeeding; or from prolonged mental or physical strain. The anaemia is temporary, but causes facial pallor and dark blue rings around the eyes. If indicated, China speeds up convalescence dramatically.\n\nLack of appetite is a curious symptom but one mouthful leads to it returning with a vengeance. The skin feels sore over the whole body, is aggravated by light touch but soothed by hard pressure. They are persistently chilly, worse for cold weather and sensitive to draughts and fresh air. Symptoms may recur at regular intervals or be worse, say, every other day. They sweat in their sleep, sleep restlessly and with difficulty and feel sluggish on waking.\n\nEmotional state\n\nAnxious. Apathetic. Aversion to being touched\/examined. Depressed generally; during pregnancy. Despondent. Fearful of animals; of dogs. Irritable children \u2013 in the mornings. Screaming \u2013 on waking. Sensitive generally; to noise. Stubborn children.\n\nApart from physical tiredness, feels emotionally weary. May formerly have been lively and active, both mentally and physically, but acute illness or stress causing the tiredness leads to complete debility, apathy, depression or despondency.\n\nBabies are sensitive, dislike being examined, are irritable and mischievous, especially in the mornings, and lack vitality when unwell.\n\nPhysical complaints\n\nDiarrhoea\n\nIn pregnancy. PAINLESSS. STOOLS containing undigested food. With INDIGESTION (see here).\n\nWorse after eating; during the afternoon; on alternate days; at night.\n\nCause acute illness; eating fruit; hot weather; weaning.\n\nThe digestion is slowed down. Newly weaned babies do not take well to solids.\n\nExhaustion\n\nNERVOUS. Profuse SWEATING.\n\nCause loss of body fluids (diarrhoea, vomiting); breastfeeding.\n\nWeakness characteristic during convalescence after heavy loss of body fluids (see General symptoms). May feel faint and have ringing in the ears as well as profuse sweating, especially on exertion and during sleep.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; obstructed (WIND difficult to expel); rumbling.\n\nCause eating fruit; abdominal surgery.\n\nAbdomen distended (bloated) above the navel, with trapped wind, causes much discomfort and pain.\n\nHeadache\n\nNERVOUS. PAIN sore, bruised; pressing; throbbing. SCALP sensitive.\n\nBetter for firm pressure.\n\nCause mental strain.\n\nMay feel as if the brain is beating against the skull. Pain relieved by hard pressure although scalp may be so sensitive that even individual hair follicles feel sore.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated. BELCHES bitter; tasting of food eaten; ineffectual, incomplete; sour. FLATULENCE obstructed. PAIN pressing.\n\nWorse after drinking; after eating; after eating fruit; after belching.\n\nCause abdominal operation.\n\nDigestion is slow; feels as if all food turns to gas; much like Carbo vegetabilis, but in China burping relieves neither the discomfort nor the bloating. Abdomen is swollen tight like a drum, and it seems as if the wind is blocked. It may follow an operation to the abdomen. Fruit ferments in the stomach, turning to wind.\n\nJaundice\n\nIn newborn babies.\n\nIf your baby has China general or emotional symptoms then give it instead of Chelidonium.\n\nVomiting\n\nVOMIT food; sour. VOMITING frequent.\n\nWorse after eating.\n\nCIMICIFUGA (Cimi.)\n\nOther name: black cohosh\n\nGeneral symptoms\n\nComplaints from childbirth. Pains sore, bruised. Trembling.\n\nWorse for cold.\n\nThis remedy is useful in pregnancy. These are chilly types and with the exception of the headaches, their symptoms are worse for cold. Physical symptoms may alternate with depression. May be needed for post-natal complaints, with a Cimicifuga emotional state.\n\nEmotional state\n\nDepressed. Depressed\/fearful and sensitive during pregnancy. Depressed\/screaming\/sensitive to noise during labour. Fearful of death. Restless. Sensitive. Sighing.\n\nUseful for women who have bouts of gloominess or depression, as if a black cloud has settled over them, which can be common during pregnancy or after childbirth. May be afraid of death (like Aconite), and scared of losing her reason. During the gloomy phase, sits silently or sighs. When dejection lifts, may become excitable and talkative, jumping from one subject to another.\n\nPhysical complaints\n\nAbdominal pain\n\nIn pregnancy. PAIN flying about abdomen.\n\nDistressing shooting pains may occur during the later months of pregnancy.\n\nAfterpains\n\nPAIN unbearable.\n\nWorse in the groin.\n\nHeadache\n\nPAIN in back of head; top of head; pressing out; pressing up.\n\nBetter for fresh air.\n\nPain radiates from the back to the top of head, which feels as if it may fly off; it may start in the forehead and extend to the back of the head. Pain is severe and better for cool, fresh air.\n\nJoint pain\n\nIn PREGNANCY.\n\nWorse for cold.\n\nSore, bruised pains in joints and heaviness in lower back may also occur in pregnancy. They may alternate with feeling depressed.\n\nLabour\n\nLATE with FEAR OF BIRTH. LABOUR PAINS: in the back; in the hips; flying around the abdomen; ineffectual (cervix doesn't soften); stopping (or slowing down); weak. With CRAMP IN HIPS and\/or SHIVERING\/TREMBLING; feels FAINT.\n\nPains fly from one side of the abdomen to the other; whole body feels sore, bruised and sensitive to touch.\n\nRetained placenta\n\nWith the SHAKES.\n\nWith exhaustion, trembling and shaking (or shivering).\n\nCINA\n\nOther name: European wormseed\n\nGeneral symptoms\n\nExpression sickly. Eyes sunken. Face pale. Likes cold drinks. Nose: baby picks constantly.\n\nWorse at night; for pressure.\n\nCina's general picture occurs most commonly in babies and may be an indication for worms but is useful for other complaints too, if whole picture fits. Despite being pale and sickly looking, has hot red cheeks.\n\nEmotional state\n\nAngry babies. Anxious \u2013 babies at night. Aversion to being touched\/examined; being hugged; being looked at. Capricious. Desires to be carried. Hitting. Irritable. Moaning\/complaining. Restless \u2013 better for being carried. Screaming on waking. Stubborn. Tearful.\n\nThese are angry, touchy and stubborn types. Cina babies dislike being looked at, touched, examined or interfered with in any way; may demand to be carried or rocked but are not better for it and kick and scream when picked up and cuddled (rather like Antimonium crudum and Chamomilla). Asks for things and rejects or throws them away.\n\nPhysical complaints\n\nConvulsions\n\nIn BABIES.\n\nCause worms; teething.\n\nSeek professional advice immediately.\n\nCough\n\nIn fits; suffocative. With RETCHING.\n\nWorse after getting up.\n\nBody stiffens before a coughing fit, and there is gurgling in the throat after coughing. May be almost constant (involuntary) swallowing with the cough.\n\nFever\n\nTHIRSTLESS. With HUNGER.\n\nWorse at night.\n\nThe fever may recur daily at the same time.\n\nRestless sleep\n\nBODY twitching. LIMBS jerking. With GRINDING OF TEETH.\n\nBabies scream out at night, and lie on their backs kicking their legs.\n\nWorms\n\nANUS itches. APPETITE changeable; lost; ravenous. NOSE itches. With symptoms of FEVER (see here).\n\nFor worms in a baby who rubs and picks its nose continually; grinds the teeth while asleep; has an enormous appetite and asks for food soon after a meal, or wants to eat nothing but sweet things, and is generally irritable.\n\nCOCCULUS INDICUS (Cocc.)\n\nOther name: India berry\n\nGeneral symptoms\n\nAversion to fresh air. Dislikes food in general; with hunger. Hot flushes. Sweat on single parts of the body; cold. Taste metallic. Trembling from emotion.\n\nBetter for lying down in bed.\n\nWorse for exertion; for loss of sleep; for movement; for touch; for walking in the fresh air.\n\nA remedy for complete exhaustion, usually from lack of sleep or irregular sleeping from working night shifts, looking after sick patients through the night, or nursing babies, etc. Trembles with tiredness and feels much worse for fresh air and physical exertion. Feels generally numb and as if specific parts of the body have gone to sleep. Wants to lie quietly in bed and sleep, but this is difficult because the habit of sleeping is lost. Is in a dreadful vicious circle of exhaustion.\n\nEmotional state\n\nAnxious. Complaints from anger; grief. Confused. Dazed to others. Forgetful. Introspective. Memory weak. Mild. Time seems to pass too fast. Uncommunicative.\n\nEmotional state results from stress, especially sleep deprivation, in normally easy-going characters. Grief (loss of a loved one, for example) or anger may also be part of the picture. Become introspective, uncommunicative and closed off from the world. Become trembly if confronted with difficult emotional situations. Time passes quickly, especially at night when they try to sleep but can't.\n\nPhysical complaints\n\nDizziness\n\nIn PREGNANCY. HAS TO LIE DOWN. With NAUSEA.\n\nWorse for getting up from lying down.\n\nExhaustion\n\nPARALYTIC. With DIZZINESS; NERVOUSNESS; NUMBNESS; STIFFNESS; TREMBLING; VERTIGO.\n\nWorse for walking in fresh air.\n\nCause loss of sleep; irregular sleep; nursing the sick; nervous exhaustion.\n\nUseful to combat exhaustion after working or caring for a baby through the night. Legs tremble while walking and hands while eating and when lifting them up high. Limbs may 'go to sleep' easily; back and neck feel weak and stiff. Needs to lie down and feels worse for sitting up.\n\nHeadache\n\nPAIN in back of head; in forehead; at nape of neck. With NAUSEA (see here).\n\nCause irregular sleep; loss of sleep.\n\nHead may feel empty, sore and bruised. Headaches from nervous strain, overwork or travel sickness.\n\nInsomnia\n\nDREAMS anxious; nightmares. Restless SLEEP.\n\nCause anxiety.\n\nNausea\n\nABDOMEN\/STOMACH feels empty. APPETITE lost. BELCHES. TASTE metallic. With FAINTNESS.\n\nWorse during the afternoon; after eating; after drinking; for movement; for sitting up in bed; for smell and sight of food; for travelling.\n\nNosebleeds\n\nIn PREGNANCY.\n\nThrush (genital)\/Yeast\n\nIn PREGNANCY. DISCHARGE profuse; thin; watery.\n\nTravel sickness\n\nWith DIARRHOEA; DIZZINESS; FAINT FEELING; HEADACHE (see here); NAUSEA (see here); VOMITING.\n\nBetter for lying down.\n\nWorse for fresh air; after eating; after drinking; for movement; for sitting up.\n\nGetting up causes dizziness and nausea; needs to lie down to prevent vomiting.\n\nCOCCUS CACTI (Cocc-c.)\n\nOther name: cochineal\n\nPhysical complaints\n\nCough\n\nChoking; in fits; irritating; violent; whooping; racking. Tickling in LARYNX. MUCUS copious after each coughing fit; sticky. THROAT dry.\n\nBetter for fresh air.\n\nWorse in a stuffy room; around 11.30 p.m.\n\nOne of the main whooping-cough remedies. Mucus drips down back of throat, irritating larynx, causing hawking which in turn leads to coughing fits that end in retching and vomiting of mucus. Mucus hangs in strings from the mouth.\n\nCOFFEA CRUDA (Coff.)\n\nOther name: coffee\n\nGeneral symptoms\n\nSenses acute. Sensitive to pain; to noise.\n\nWorse for fresh air; at night; for touch.\n\nCoffea affects the nerves: look for overexcitement and oversensitivity. All senses, touch, sight, hearing, smell and taste, become acute.\n\nBoth coffee as a drink and Coffea can counteract the effect of some homeopathic remedies. See here for guidance.\n\nEmotional state\n\nCheerful. Complaints from excitement; joy. Despair. Euphoric. Excitable. Exhilarated. Fearful of painful death \u2013 during labour. Joking during labour. Lively. Moaning. Restless. Screaming with pain. Tearful during labour.\n\nA remedy for excessive euphoria, but also where the slightest pain induces despair: makes a fuss over minor aches and pains. During labour may be excitable and restless then despairing, fearing death, chattiness turning to moaning (wailing) and crying. May be sensitive to noise.\n\nPhysical complaints\n\nHeadache\n\nNERVOUS. PAIN one-sided.\n\nWorse for noise.\n\nMay be worse in the fresh air and may feel as if a nail is being driven into the brain.\n\nInsomnia\n\nWith DREAMS, VIVID.\n\nWorse during pregnancy.\n\nCause cramp, during pregnancy; overactive mind; over-excitement.\n\nSleeps lightly, waking at every sound; unable to sleep because of excitement, ideas and plans. Sleeplessness from good news!\n\nLabour\n\nLABOUR PAINS in back; distressing; ineffectual (cervix doesn't dilate); irregular; severe; stopping (or slowing down). With TALKATIVENESS.\n\nToothache\n\nIn NERVOUS PEOPLE. PAINS shooting\/spasmodic.\n\nBetter for cold water.\n\nWorse for hot food and drinks; heat.\n\nPain in the tooth is eased by holding cold water (especially ice-cold water) in the mouth, but returns as soon as the water warms up.\n\nCOLCHICUM AUTUMNALE (Colch.)\n\nOther name: meadow saffron\n\nGeneral symptoms\n\nExhaustion. Sense of smell acute. Sweat sour. Thirstless.\n\nBetter for sitting.\n\nWorse in the autumn; for cold; for damp; for movement; at night; for the sight of food; for touch.\n\nThese types are sensitive to cold and damp and their symptoms are worse at night. Movement of any sort, including walking, aggravates their pains and the only comfortable position is sitting. A keynote of the remedy is the heightened sense of smell; if unwell, cooking smells exacerbate symptoms and\/or induce faintness. Sensitivity to smell is at its worst with egg, which induces nausea and\/or vomiting.\n\nEmotional state\n\nComplaints from anger. Memory weak.\n\nA small remedy with few strongly marked emotional characteristics. May be absent-minded and forgetful; sensitive to anger and conflict. Symptoms \u2013 diarrhoea, nausea or painful joints \u2013 are worse after, say, an argument.\n\nPhysical complaints\n\nDiarrhoea\n\nSTOOLS jelly-like; with mucus; watery. PAIN on passing stools.\n\nCause damp weather.\n\nDiarrhoea may accompany joint complaints that are worse in damp weather of autumn.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated. WIND obstructed (difficult to expel).\n\nJoint pain\n\nPAIN in hands and feet; acute; tearing. With SWELLING.\n\nBetter for warmth.\n\nWorse for cold, wet weather; for movement; for warm weather.\n\n'Rheumatism' affects the small joints of the hands and feet, which may be better for being wrapped up and kept warm and are worse for touch. Toe joints become especially sensitive: accidental stubbing is agonisingly painful. Severe, tearing pains may be accompanied by general weakness.\n\nNausea\n\nABDOMEN\/STOMACH feels bloated. APPETITE lost. With FAINTNESS; VOMITING (see here).\n\nWorse after eating; for smell and sight of food.\n\nIntense loathing at the sight, smell or thought of food, and nausea exacerbated particularly by the smell of eggs and fish. May be dry retching. The nausea is much worse than the vomiting.\n\nVomiting\n\nPAIN burning; sore, bruised. With RETCHING after vomiting.\n\nWorse for smell of eggs.\n\nStomach is distended and may feel cold.\n\nCOLOCYNTHIS (Coloc.)\n\nOther name: bitter cucumber\n\nEmotional state\n\nComplaints from anger; humiliation; indignation. Restless.\n\nPhysical symptoms may develop from feeling angry or humiliated; pain causes intense anguish, restlessness and irritability.\n\nPhysical complaints\n\nColic\n\nIn BABIES. ABDOMEN\/STOMACH feels bloated. PAINS cutting; griping; tearing; violent; in waves. With DIARRHOEA (see here); NAUSEA; VOMITING. Dislikes food in general; smell of food.\n\nBetter for bending double; for pressure; for passing a stool.\n\nWorse after drinking; for cold drinks when overheated; before a stool; after eating fruit.\n\nCause anger; eating fruit; excitement; vexation.\n\nPains are so severe as to cause vomiting. A young baby commonly pulls its legs up to its abdomen and screams. Older sufferers double up with the pain and press hard on the affected area (they may dig their fists into the abdomen or bend over the back of a chair, twisting and turning to find relief). Eating, especially fruit, will aggravate the pain.\n\nCramp\n\nIn THIGH; LEG; CALF.\n\nDiarrhoea\n\nSTOOLS green; pasty. With COLIC (see here).\n\nWorse after eating fruit; after eating.\n\nCause anger; eating fruit.\n\nHeadache\n\nPAIN on left side of face; spreads up to ear; tearing.\n\nWorse for touch.\n\nCause excitement; vexation.\n\nSciatica\n\nPAINS tearing.\n\nWorse on right side of body.\n\nPains are usually worse for movement, pressure and touch and although they may be better for warmth are often worse for the warmth of bed at night.\n\nCONIUM MACULATUM (Con.)\n\nOther name: common hemlock\n\nGeneral symptoms\n\nComplaints from accident\/injury. Dizziness. Palpitations in pregnancy. Sweat during sleep; hot; at night; on closing eyes.\n\nWorse on lying down.\n\nBetter for sitting down.\n\nThose needing this remedy feel better when sitting letting the limbs hang (not cross-legged). Sweating may be profuse and occurs strangely on closing the eyes as well as during sleep.\n\nEmotional state\n\nAnxious. Apathetic. Depressed during pregnancy. Memory weak. Sensitive to noise. Slow.\n\nEasily startled by noise, and sensitive to light, these are melancholic types whose brains seize up after a period of mental strain. Unable to read, they may sit looking vacant, feeling anxious and not wanting to work.\n\nPhysical complaints\n\nBreast(feeding) problems\n\nBREAST LUMPS worse in right breast.\n\nCough\n\nDry; irritating; tickling; violent; must sit up as soon as cough starts. EXPECTORATION difficult; must swallow what comes up.\n\nWorse lying down in bed; in the evening; for deep breathing; during fever.\n\nCough caused by dry, tickling spot in larynx, brought on by lying down (day or night). Sits up to cough, then lies down to rest. Exhausting cough, often accompanied by a fever.\n\nDizziness\n\nWith TURNING SENSATION. With HEADACHE.\n\nWorse for moving or turning the head quickly; for lying down.\n\nFeels as if turning in a circle; cannot watch moving objects. Better when quite still with eyes closed.\n\nExhaustion\n\nWith TREMBLING; NUMBNESS.\n\nBetter for fresh air.\n\nWorse for slightest exertion; after stool; for walking.\n\nInjuries\n\nTo GLANDS, breasts. To HEAD. Stony hard lumps, sensitive, swollen, cold, inflamed.\n\nInjured parts become lumpy and painful and feel stony hard. Tingling with stitching pains, after a bang or bruise.\n\nInsomnia\n\nIn PREGNANCY. SLEEPLESS before midnight. With NIGHTMARES.\n\nCause movements of baby (in PREGNANCY).\n\nNausea\n\nIn PREGNANCY. With RETCHING and VOMITING.\n\nCUPRUM METALLICUM (Cupr.)\n\nOther name: copper\n\nGeneral symptoms\n\nExtremities cold. Face blue; pale. Lips blue. Pains cramping. Taste sweet.\n\nBetter for cold drinks.\n\nWorse for touch; for vomiting.\n\nCramp is an important part of the Cuprum picture; muscles feel knotted up. Tiredness is from mental exhaustion or sometimes loss of sleep; headaches and cramp may follow a period of heavy work. Looks pale and drawn.\n\nEmotional state\n\nRestless at night in bed.\n\nThis remedy does not have a strong 'character'; its general symptoms and physical complaints will lead you to prescribe it.\n\nPhysical complaints\n\nAfterpains\n\nWith CRAMP IN FINGERS AND\/OR TOES.\n\nEspecially useful for women who have had three or more children.\n\nColic\n\nPAIN cramping; violent. With NAUSEA; VOMITING.\n\nAbdomen sore, bruised, tender and hot.\n\nConvulsions\n\nCause teething in babies; vexation.\n\nAccompanied by blue lips, cold hands and feet (see COUGH).\n\nCough\n\nIn long fits at irregular intervals; uninterrupted; suffocative; violent; whooping. BREATHING difficult; fast.\n\nBetter for sips of cold water (like Causticum).\n\nCold air may aggravate; mouth may taste metallic. During coughing fit, baby is breathless, becomes stiff, and a convulsion with fingers and toes twitching may follow. Between coughing fits, breathing is hurried and panting. A very serious cough, frightening for parents: professional help should always be sought immediately. Meanwhile, Cuprum will give relief and other medication may be avoided.\n\nCramp\n\nIn CALF; FOOT; LEG.\n\nCause childbirth (see LABOUR).\n\nExhaustion\n\nWith HEADACHE between the eyes.\n\nCause loss of sleep.\n\nWorse for mental exhaustion.\n\nLabour\n\nLABOUR PAINS with CRAMP in hand or leg, finger or toe. With vomiting.\n\nCYPRIPEDIUM (Cypr.)\n\nOther name: yellow lady's slipper\n\nPhysical complaints\n\nInsomnia\n\nIn RESTLESS BABIES who wake to play at night.\n\nCause nervous exhaustion.\n\nOverstimulation may cause chronic sleeplessness. Wakes in the middle of the night, is wide awake and plays happily for an hour or more. Nip this in the bud before it gets completely out of hand \u2013 the baby may sleep only a few hours every night, becoming hyperactive and unmanageable.\n\nDIOSCOREA (Dios.)\n\nOther name: wild yam\n\nPhysical complaints\n\nColic\n\nIn BABIES. ABDOMEN\/STOMACH rumbling; windy. PAIN cramping; cutting; griping; around the navel; twisting.\n\nBetter for bending back; for stretching out.\n\nWorse for bending forward; during the morning.\n\nAn infant arches back and screams (opposite of Colocynthis picture with its drawing up of legs). Has a rumbly, windy abdomen, does not want to lie down but is better for being held upright.\n\nDROSERA ROTUNDIFOLIA (Dros.)\n\nOther name: sundew\n\nPhysical complaints\n\nCough\n\nBarking; deep; dry at night; in violent fits; hacking; irritating; suffocative; tormenting. BREATHING difficult; fast. LARYNX tickling. PAIN IN CHEST holds chest with hands to cough. With NOSEBLEEDS; BLUE FACE; VOICE HOARSE; RETCHING; VOMITING of mucus.\n\nBetter for pressure.\n\nWorse after midnight; after drinking; for lying down; for warmth of bed; for talking.\n\nCause after measles.\n\nTypical severe cough of whooping cough often starts with tickling at back of throat, usually accompanied in its acute phase by retching and vomiting and sometimes nosebleeds. Breathing either speeds up during coughing or is difficult (coughing fits so violent that it is often impossible to breathe and cough at the same time so face becomes bluish). Coughing attacks follow each other rapidly, especially at night, and will often be set off as soon as the head touches the pillow. Cough is painful and chest becomes sore and bruised from coughing; holds chest with both hands when coughs as pressure helps to ease the pain. Voice becomes hoarse from coughing.\n\nSore throat\n\nIn BABIES. ABDOMEN\/STOMACH rumbling, windy. PAIN.\n\nWorse for swallowing.\n\nLarynx sore and inflamed with an irritating feeling of dust in it. Voice becomes deep and husky.\n\nDULCAMARA (Dulc.)\n\nOther name: woody nightshade\n\nGeneral symptoms\n\nCatches colds easily. Complaints from change of weather to damp; weaning.\n\nBetter for movement; for walking.\n\nWorse for wet weather; for cold; for damp; for lying down; at night; for sitting still.\n\nLacking 'vital heat', these types are sensitive to cold and damp. They have a tendency to catch colds, especially in the winter. An unusual symptom is the strong need to urinate (or pass a stool) after becoming chilled or spending time in a cold place. A baby may develop rashes after weaning on to solid foods.\n\nEmotional state\n\nThere are few strong emotional symptoms for this remedy; but they may be depressed, irritably impatient or quarrelsome.\n\nPhysical complaints\n\nBackache\n\nIn LOWER BACK. PAIN aching; sore, bruised. With LAMENESS.\n\nBetter for movement; for walking.\n\nWorse for wet weather.\n\nCause change of weather; damp weather; getting cold; getting wet.\n\nLower back aches as if from long stooping.\n\nBreast(feeding) problems\n\nMILK SUPPLY low in chilly women.\n\nCommon cold\n\nNOSE blocked.\n\nNasal discharge may be thick and yellow but nose more likely to be blocked. Often better for keeping the head warm; may sit with head under a blanket or a scarf pulled up over the nose. Winter colds clear once summer warmth is established and return with cold weather the following autumn.\n\nCough\n\nRattling.\n\nCause damp weather.\n\nCough may develop from a sore throat. Has to cough for a long time before expelling mucus.\n\nCystitis\n\nURINATION involuntary (incontinence).\n\nCause getting cold and wet.\n\nGive after Aconite (if it hasn't helped), where cystitis developed after getting cold and wet, and where there are as yet no remarkable symptoms.\n\nDiarrhoea\n\nIn TEETHING BABIES. PAINFUL. STOOLS yellow; watery.\n\nWorse after eating cold food; at night.\n\nCause getting cold; getting damp; teething.\n\nPains are generally worse before the stool and may be accompanied by nausea.\n\nEye inflammation\n\nCause wet weather.\n\nOften accompanies a common cold.\n\nFlu\n\nCause cold, damp weather.\n\nHives\n\nRASH lumpy.\n\nWorse for heat; after scratching.\n\nCause getting cold.\n\nRash burns after scratching and, although brought on by getting cold, is worse for warmth, for example after exercise.\n\nJoint pain\n\nBetter for movement.\n\nWorse for sitting; at night.\n\nCause damp; getting cold.\n\nPains in joints come on after staying in a damp house, or sleeping in a damp bed.\n\nSnuffles\n\nIn newborn babies. With EYE INFLAMMATION or COMMON COLD (see here).\n\nEUPATORIUM PERFOLIATUM (Eup-p.)\n\nOther name: agueweed\n\nGeneral symptoms\n\nPain sore; bruised; in bones. Sweat scanty. Thirsty for cold drinks; unquenchable.\n\nEupatorium resembles Bryonia: both have similar flu symptoms, but Eupatorium has terrible pains in the bones and great restlessness whereas Bryonia prefers to be completely still. Eupatorium symptoms may be better while sweating, except the headache.\n\nEmotional state\n\nDepressed.\n\nFeels sad and depressed during flu, and lies about moaning during the fever.\n\nPhysical complaints\n\nFever\n\nSleepy, yawns and falls asleep all the time. With characteristic thirst and scanty sweat.\n\nFlu\n\nEYEBALLS ACHING. EYELIDS RED. PAIN in bones; bones feel broken. SKIN SORE. With SHIVERING; CHILLS in back; HEADACHE (see here); NASAL CATARRH; SNEEZING.\n\nBetter for sweating.\n\nAn awful flu with intense aching pains in the scalp, bones of arms, legs and lower back (sometimes the hips as well). The skin all over the body, even the scalp, feels dry and sore.\n\nGastric flu\n\nWith FEVER; NAUSEA; RETCHING; VOMITING of bile; of food.\n\nBetter after chills; during fever.\n\nVery thirsty before vomiting.\n\nHeadache\n\nPAIN in back of head; sore, bruised; throbbing. With FEVER (see here).\n\nWorse for sweating.\n\nEUPHRASIA (Euphr.)\n\nOther name: eyebright\n\nPhysical complaints\n\nCommon cold\n\nNASAL CATARRH bland; watery. With COUGH (see here); EYE INFLAMMATION.\n\nNasal discharge irritates and the watering eyes do not. Throat may also be sore; if it is, it will burn (the opposite of Allium cepa).\n\nCough\n\nDAYTIME only. MUCUS copious.\n\nBetter for lying down.\n\nWorse during the morning.\n\nHawks up a mouthful of mucus at a time and clears throat frequently.\n\nEye inflammation\n\nDISCHARGE burning; watery. EYES sensitive to light; watery. EYELIDS burning; red; swollen. With COMMON COLD.\n\nWorse for coughing; for light; for wind.\n\nCause common cold.\n\nEyes stream on coughing and are sensitive to light.\n\nEye injuries\n\nWith EYE INFLAMMATION (see here).\n\nMeasles\n\nWith COMMON COLD (see here); COUGH (see here); EYE INFLAMMATION (see here).\n\nFever not usually very high. Does not necessarily feel very ill.\n\nFERRUM METALLICUM (Ferr-m.)\n\nOther name: iron\n\nGeneral symptoms\n\nAppetite lost, alternating with hunger. Face pale; pasty; flushes easily; red with pain. Gums pale. Oedema (swelling) of feet\/ankles; of hands\/fingers. Sweat clammy; cold; profuse; worse for lying down and for slightest physical exertion.\n\nBetter for walking slowly.\n\nWorse on beginning to move; at night; for lying down.\n\nBecomes easily exhausted and breathless, and the typically pale face becomes flushed, not only with exertion but when excited, and when in pain or with a fever. Benefits from gentle exercise, which helps generally and relieves many symptoms.\n\nNot very hungry and appetite comes in bursts; experiences fullness after eating very little. Digestive disorders are marked and peculiar: has an intolerance of eggs; suffers from diarrhoea while eating (it actually comes on when they begin to eat, a symptom peculiar to Ferrum metallicum). Experiences periodic vomiting around midnight.\n\nEmotional state\n\nDepressed. Irritable. Moody. Restless in bed.\n\nAnaemic, nervous, irritable and worn out. If stuck in bed feeling ill, will forever be getting out of bed to wander around aimlessly.\n\nPhysical complaints\n\nAnaemia\n\nIn PREGNANCY. LIPS pale. With EXHAUSTION (see here).\n\nCause loss of blood.\n\nWith characteristically pale face that flushes easily.\n\nBackache\n\nIn LOWER BACK.\n\nBetter for walking slowly.\n\nWorse on beginning to move.\n\nBreathless\n\nIn PREGNANCY.\n\nBetter for gentle exercise; for a stroll.\n\nCough\n\nIn fits.\n\nBetter for walking slowly.\n\nWorse after getting up; for movement.\n\nDiarrhoea\n\nTEETHING. PAINLESS. STOOLS containing undigested food; passed with wind. With BELCHING; FLATULENCE.\n\nWorse after drinking water; for movement; at night; while eating.\n\nCause teething.\n\nAfter eating burps taste of food recently consumed. Passes wind with the diarrhoea.\n\nDizziness\n\nWith NAUSEA.\n\nWorse for getting up from bending down\/lying down\/sitting.\n\nExhaustion\n\nWith DESIRE TO LIE DOWN.\n\nBetter for walking slowly in the fresh air.\n\nWorse for exercise.\n\nCause anaemia; sweating.\n\nA mental as well as physical exhaustion. Person doesn't want to work.\n\nFever\n\nBetter for uncovering.\n\nHas the typical pale face which flushes easily.\n\nHeadache\n\nPAIN in forehead; hammering; throbbing. With THIRSTLESSNESS.\n\nBetter for firm pressure; for lying down.\n\nWorse for moving head.\n\nHeadaches last 2\u20133 days at a time and are very draining. Has to lie down, and refuses drinks; face is alternately hot and flushed or pale and drained.\n\nJoint pain\n\nPAINS in shoulder; in upper arm; stitching; tearing.\n\nBetter for gentle movement.\n\nWorse on beginning to move; for lifting arm up; for bending arm backwards.\n\nNosebleeds\n\nIn babies.\n\nSciatica\n\nBetter for gentle movement; for walking slowly.\n\nVaricose veins\n\nOf LEG\/THIGH; of FOOT. PAINFUL; SWOLLEN.\n\nWorse during pregnancy.\n\nOften accompanied by anaemia and tiredness.\n\nVomiting\n\nSudden; of food.\n\nWorse after eating eggs; after midnight; at night.\n\nFood either comes up suddenly while eating or lies in the stomach all day, coming up around midnight.\n\nGELSEMIUM SEMPERVIRENS (Gels.)\n\nOther name: yellow jasmine\n\nGeneral symptoms\n\nExhaustion paralytic. Eyelids heavy. Feeling of heaviness. Onset of complaint slow. Sweat absent during fever. Thirstless. Trembling.\n\nBetter for sweating; for urination.\n\nWorse for physical exertion.\n\nIntensely weary types: body feels heavy (arms and legs feel as if weighted down with lead); becomes trembly with exhaustion and consequently worse for any additional physical exertion, only feeling better temporarily after urination.\n\nAcute Gelsemium complaints (for example, flu) come on gradually, taking days to develop, coming on when the weather changes to warm after the cold of winter, for those who spend their lives in overheated houses. Bryonia is similar, but is thirsty and keeps still because of pain, whereas Gelsemium is thirstless and cannot move for heaviness and weariness.\n\nEmotional state\n\nAnxious babies. Apathetic during labour. Clingy. Complaints from receiving bad news; excitement. Depressed but cannot cry. Desires to be alone. Despairing during labour. Dislikes company. Excitable during pregnancy. Fearful of falling; of public speaking; in a crowd; of death. Sluggish.\n\nThese are dull, sluggish types who want to be left alone, to be quiet. Gelsemium is a favourite remedy for those paralysed with fear, mentally and physically, before an important event; not the active fear of Argentum nitricum or Lycopodium, more an acute anxiety. Tremble, stutter and cannot collect their thoughts. May even look, as well as feel, stupid. Like Natrum muriaticum and Ignatia find it difficult to cry if depressed, typically after receiving a piece of bad news. Likewise over-excitement causes illness, either emotional or physical.\n\nPhysical complaints\n\nBraxton Hicks contractions\n\nIn pregnancy.\n\nDiarrhoea\n\nCause anticipatory anxiety; bad\/exciting news; fright\/shock.\n\nDizziness\n\nIn pregnancy.\n\nFever\n\nHEAT burning. With SHIVERING. WITHOUT SWEATING.\n\nBetter for sweating; for urinating.\n\nWorse during the afternoon.\n\nWaves of heat alternate with chills running up and down the back; although the teeth may chatter, doesn't feel cold. Chills begin in hands and feet and move up the body. May be breathing faster than normal.\n\nFlu\n\nEYEBALLS aching. PAIN in muscles. With EXHAUSTION (see General symptoms); HEAVINESS; NUMBNESS; SHIVERING\/CHILLS in back.\n\nBetter for sweating; for urinating.\n\nWorse for exertion; for walking.\n\nA flu that may come on in mild, damp weather or from getting chilled. Tongue feels thick and heavy and speech is slurred. Back aches from the lower back up and over the head. Arms and legs ache and feel extremely heavy and tired. Extremely chilly, and cannot get warm.\n\nHeadache\n\nFEET cold. HEAD feels heavy. PAIN in back of head; spreading to forehead; aching; sore, bruised. PUPILS dilated. URINATION FREQUENT. VISION blurred.\n\nBetter for urinating.\n\nWorse for movement; for moving head.\n\nHead feels so heavy that it needs support, of hands or a pillow; also feels constricted, as if a band or hoop were encircling it. Has difficulty opening the eyes or keeping them open as the lids feel so heavy. Urine is copious and colourless. Thirstless and feels better for urinating.\n\nLabour\n\nLABOUR LATE with anticipatory anxiety. LABOUR PAINS in back; distressing; false; ineffectual (cervix doesn't soften); weak. With TREMBLING.\n\nA valuable remedy for backache in labour when baby is 'posterior'. Becomes apathetic, despairing and thirstless.\n\nMeasles\n\nONSET slow. With FEVER (see here); HEADACHE (see here).\n\nWith typical Gelsemium heaviness, drowsiness and lack of thirst.\n\nGLONOINE (Glon.)\n\nOther name: nitro-glycerine\n\nGeneral symptoms\n\nComplaints from sunstroke. Face red.\n\nBetter for cold applications.\n\nWorse for jarring movement; for heat; for exposure to sun.\n\nThe principal use of this remedy is for headaches after too much sun. Heat in any form aggravates, as does any jarring movement like walking.\n\nEmotional state\n\nConfused. Forgetful. Time passes slowly. Uncommunicative.\n\nUnwilling to talk or answer questions. Appears dull and confused; familiar surroundings feel strange.\n\nPhysical complaints\n\nHeadache\n\nEYES red. FACE red. PAIN bursting; hammering; throbbing; violent. With FAINTNESS; HOT FLUSHES.\n\nBetter for cold compresses; for pressure; at sunset.\n\nWorse for heat; for jarring movement; at sunrise; during summer; for mental exertion; for walking.\n\nCause overexposure to sun.\n\nHeadache increases when exposed to the sun and is better for being pressed firmly with the hands. Feels faint and flushed and as if the head will burst. Ice pack may ease the pain.\n\nSunstroke\n\nWith HEADACHE (see here).\n\nHAMAMELIS VIRGINICA (Ham.)\n\nOther name: witch hazel\n\nGeneral symptoms\n\nPains sore, bruised.\n\nWorse for touch.\n\nA small remedy whose use for acute complaints is limited to nosebleeds, varicose veins and piles, but useful, taken as a tonic, for these symptoms. Pains are commonly worse for touch. No marked emotional\/mental symptoms.\n\nPhysical complaints\n\nNosebleed\n\nBLOOD dark; thin.\n\nWorse during the morning.\n\nPhlebitis\n\nAfter labour. With prickling pains in veins\/legs.\n\nPiles\n\nLARGE; BLEEDING. PAIN sore, bruised.\n\nCause childbirth; pregnancy.\n\nThese piles occur towards the end of pregnancy, and after childbirth (because of the pressure of the baby). Once Arnica has been given as a routine prescription for the bleeding, if soreness persists and a bigger remedy not indicated, Hamamelis will help. It is essential to seek professional help for both varicose veins and piles if the symptoms do not clear easily.\n\nVaricose veins\n\nOf LEGS\/THIGHS; of VULVA. PAINFUL; SWOLLEN. PAIN stinging.\n\nWorse for touch; after childbirth; during pregnancy.\n\nHard knotty varicose veins; sore, bruised and sensitive to touch.\n\nHEPAR SULPHURIS CALCAREUM (Hep-s.)\n\nOther name: calcium sulphide\n\nGeneral symptoms\n\nCatches colds easily. Complaints from cold wind. Likes sour foods. Pains needle-like. Sweat cold; profuse; sour.\n\nBetter for warmth of bed; for heat; for wrapping up.\n\nWorse for getting cold; for cold dry weather; for fresh air; for lying on painful side; at night; for pressure; for touch; for uncovering; for cold wind.\n\nOne of the chilliest remedies in the Materia Medica, these types hate the cold, especially dry cold, and because they lack internal warmth they catch colds more easily than others who are more resilient. Are so sensitive to cold when sick that if a part of the body \u2013 hand or foot \u2013 escapes the bedclothes feel worse and start to cough or sneeze. Highly sensitive to pain, may weep or even faint in anticipation of it. They do not want to be touched when sick.\n\nSymptoms usually worse at night and may develop when asleep: may cough more in the night, for example. Only really better for lying in bed, well wrapped up, with the windows well shut and the heating on.\n\nEmotional state\n\nViolently angry. Impulsive. Irritable. Play \u2013 doesn't want to. Sensitive generally; to rudeness. Hurried generally; eating; speaking.\n\nOver-sensitivity is the special feature that runs through this remedy. Morose, difficult individuals, they are easily offended and prone to fits of anger. When sick they are extremely difficult, and often obstructive in giving information while demanding angrily to be cured. They are also impulsive and speedy, like Argentum nitricum.\n\nPhysical complaints\n\nAbscesses\n\nOf GLANDS; of ROOTS OF TEETH; of BREASTS.\n\nA major abscess remedy but generally only useful before the abscess has opened and started to discharge. Indicated at the swollen, painful stage, especially if this is accompanied by splinter-like pains and a great sensitivity to touch.\n\nCommon cold\n\nNASAL CATARRH drips down back of throat; smelly; yellow. SNEEZING on uncovering.\n\nWorse for uncovering.\n\nBones of the nose may feel sore and sense of smell may be lost. Sneezing brought on by draughts.\n\nConstipation\n\nSTOOL soft.\n\nConstipation may accompany any of the acute symptoms, such as cough or earache. Stool is passed with difficulty.\n\nCough\n\nBarking; dry during the evening\/night; loose during the morning; hacking; irritating; suffocative; violent. MUCUS copious; sticky; thick; tough; yellow. With HOARSE VOICE; RETCHING; SWEATING; VOMITING.\n\nWorse for cold dry air; for being uncovered; on single parts of body; for uncovering hands; evening in bed; before midnight.\n\nCause exposure to cold dry wind.\n\nA bad cough with irritation in the larynx; chest becomes sore from coughing. The simple act of going to bed may also set off the cough (as soon as eyes are closed cough starts up again).\n\nCroup\n\nRECURRENT. With symptoms of COUGH (see here).\n\nMany of the general symptoms and cough symptoms are present in the croup. Hepar croup is usually worse in the early morning (if it is worse in the evening it is more likely to be Aconite); slightest breath of cold air causes coughing, as does uncovering. Chokes, wheezes and rattles, but bringing up mucus is difficult.\n\nDiarrhoea\n\nPAINLESS.\n\nRumbling in the abdomen accompanies this diarrhoea.\n\nEarache\n\nDISCHARGE smelly. PAINS stitching.\n\nBetter for wrapping up warmly.\n\nWorse for cold.\n\nFever\n\nHEAT alternating with chills.\n\nBetter for heat.\n\nWorse for being uncovered.\n\nSlightest exertion can bring on coughing and sweating, which is cold, sour-smelling and profuse. Sweating provides no relief.\n\nInflammation of navel\n\nIn newborn babies.\n\nIf it doesn't help within 48 hours give Silica.\n\nInjuries\n\nCUTS\/WOUNDS with inflammation; SLOW TO HEAL; PAINFUL. PAIN sore; splinter-like.\n\nFor cuts, including episiotomy, and injuries where the skin is broken and healing taking longer than expected. Redness (inflammation) develops around the site of the injury which is generally sensitive to touch.\n\nJoint pain\n\nIn fingers; hip; shoulder; sore, bruised; pulling; tearing.\n\nBetter for heat.\n\nWorse for cold.\n\nCause getting cold.\n\nSore throat\n\nPAIN spreading up to ear; raw; splinter-like. TONSILS swollen.\n\nBetter for hot drinks; for warm compresses.\n\nWorse for breathing in cold air; for coughing; for swallowing; for turning the head; for cold drinks; in winter.\n\nCause getting cold; exposure to wind.\n\nThroat feels as if there were a splinter or a fish bone stuck in it and the pains radiate up to the ear on swallowing, turning the head and sometimes even when yawning. Tonsils are swollen and ulcerated and the neck is sensitive to pressure\/touch and better for being wrapped up warmly.\n\nHYPERICUM PERFOLIATUM (Hyp.)\n\nOther name: St John's wort\n\nGeneral symptoms\n\nComplaints from accident\/injuries to nerves\/coccyx. Pains shooting.\n\nWorse for cold; for pressure.\n\nThe first remedy to think of when a nerve-rich part of the body is injured, that is, fingers, toes, spine (especially the coccyx), eyes, lips, and so on. Intense pains, usually tearing, severe and sensitive to touch, shoot along the course of wounded nerves towards the trunk or up the spine. Give Arnica first to prevent swelling and bruising and follow with Hypericum, repeated every few minutes if the pain is excruciating. Use external remedies to clean dirty wounds.\n\nEmotional state\n\nShock from injury.\n\nHas no major emotional symptoms apart from the shock.\n\nPhysical complaints\n\nAfterpains\n\nWith HEADACHE.\n\nCause forceps delivery.\n\nPAINS may be bad in hips and lower back.\n\nBackache\n\nPAINS in COCCYX; in LOWER BACK; sore, bruised; shooting; tearing.\n\nCause childbirth; forceps delivery; epidural; injury to coccyx; injury to spine.\n\nUseful for any trauma to spinal nerves, which might occur during childbirth, or after an injury to the coccyx causing lasting pain and soreness.\n\nBites\/stings\n\nBITES inflamed. PAIN shooting; tearing up nerve pathways.\n\nCause animal bites; insect bites.\n\nInflamed, exceptionally painful bites on nerve-rich parts.\n\nInjuries\n\nCUTS\/WOUNDS to nerve-rich parts; crushed; punctured; lacerated; slow to heal. PAINS shooting.\n\nCause accident; dentistry; splinter; surgery.\n\nWounds are inflamed and very painful. The pain is often more severe than the injury seems to warrant, because the nerves are injured (squashing a finger in a car door, stepping on a nail, tearing off a finger- or toenail, cutting the lips, or a splinter in the hand). If the pains are severe and shoot up the body, Hypericum is the correct first-aid remedy. Also useful after any type of surgery where nerves are cut and painful or feel frayed or sore, for example, in teeth (root canals); gums (after dental work and\/or extraction); abdominal surgery (appendicectomy, Caesarean).\n\nIGNATIA AMARA (Ign.)\n\nOther name: St Ignatius's bean\n\nGeneral symptoms\n\nAversion to fresh air. Dislikes fruit; milk; tobacco. Sweat hot; on single parts of body. Symptoms contradictory.\n\nBetter after eating; for heat.\n\nWorse after drinking coffee; for tobacco.\n\nThey experience contradictory symptoms, such as an empty feeling in the stomach that is not relieved by eating, or a sore throat that is worse when not swallowing. They are generally sensitive to coffee which brings on shakiness, and tobacco, which results in headaches. They feel worse for fresh air, which quickly chills them, and better for warmth.\n\nEmotional state\n\nAnxious during pregnancy; after shock. Broody. Complaints from anger; anger with anxiety; death of a child; disappointed love; fright; grief; humiliation; reprimands; shock; suppression of emotions. Conscientious. Depressed from suppressed grief; during pregnancy. Desires to be alone. Despair. Disappointment. Dislikes consolation; contradiction. Guilt. Idealistic. Indecisive. Introspective. Involuntary weeping. Moody. Quarrelsome. Resentful. Sensitive generally; to pain. Sentimental. Sighing. Tearful cries on own; during pregnancy.\n\nThe Ignatia character is formed largely by the suppression of emotions, rather than any general physical characteristics. Ignatia complaints come on typically after emotional upset, be it grief (a loss or bereavement), or shock, or anger where anxiety is also present, or from being told off, punished or contradicted. They suffer inwardly, becoming introspective and moody.\n\nThey experience great highs and lows: the highs are when everything is OK \u2013 only the sighing may give the game away \u2013 and the lows are rarely seen because of their innate secretiveness. They torment themselves with recollections of the offence or (emotional) injury they have received. Also for emotional shock, when feelings have been suppressed and 'hysterical' symptoms developed as a result. They become nervous and excitable.\n\nPhysical complaints\n\nBleeding (vaginal) in pregnancy\n\nCause grief; emotional shock.\n\nCough\n\nDry; irritating; racking; short; violent. PAIN IN CHEST stitching. With THROAT tickling.\n\nWorse for coughing; during the evening in bed.\n\nFeels as if a feather or dust is in the pit of the throat. Suppressing cough helps \u2013 the more the coughing, the worse it becomes (this contradictory symptom should guide you straight to this remedy). May be useful in croup or whooping cough if the symptoms fit.\n\nFever\n\nTHIRST with chills.\n\nBetter for uncovering.\n\nWorse during the afternoon; on front of body; for warm covers.\n\nThirsty during the chilly stage rather than when hot and feverish \u2013 an Ignatia contradictory symptom.\n\nHeadache\n\nPAIN in forehead: stabbing; violent.\n\nWorse in a smoky room.\n\nNervous headaches that start gradually and stop suddenly, or start and stop suddenly. Feels as though a nail were being driven into the side(s) of the head. Often set off by emotional upsets.\n\nIndigestion\n\nABDOMEN\/STOMACH feels empty. BELCHES sour.\n\nBetter for belching.\n\nCause grief.\n\nInsomnia\n\nIn pregnancy.\n\nCause shock.\n\nPiles\n\nINTERNAL. PAINS shooting.\n\nBetter for walking.\n\nWorse after childbirth; after passing stool.\n\nShock\n\nCause emotional trauma.\n\nLike Phosphoric acid, shock needing Ignatia will have been sparked off by some emotional shock or loss.\n\nSore throat\n\nPAIN stitching. With LUMP SENSATION in throat; CHOKING SENSATION.\n\nBetter for swallowing.\n\nWorse when not swallowing; evenings.\n\nThe sore throat is worse when not swallowing (a contradictory symptom). Liquids may be more difficult to swallow than solids. There is often a sensation of a lump in the throat \u2013 often associated with suppressed desire to cry or express some upset.\n\nIPECACUANHA (Ip.)\n\nGeneral symptoms\n\nDislikes food in general; smell of food. Face red, one-sided; blue during cough. Nausea persistent; violent. Sweat hot or cold.\n\nWorse at night.\n\nViolent and persistent nausea accompanies virtually all complaints (headache, diarrhoea, labour pains, whooping cough, haemorrhage, etc.). Looks deathly, with a drawn, bluish face and dark-ringed eyes; may be covered with a hot or cold sweat. May feel chilly externally and hot inside. Complaints recur periodically, at regular or intermittent intervals.\n\nEmotional state\n\nAnxious during a fever. Capricious. Complaints from anger; vexation.\n\nThe Ipecacuanha baby is capricious, pleased by nothing; screams or howls with frustration if thwarted. When ill, is anxious and generally difficult to look after. Complaints come on after getting angry or frustrated.\n\nPhysical complaints\n\nBleeding\n\nAfter childbirth (haemorrhage). FLOW OF BLOOD bright red. With severe NAUSEA.\n\nColic\n\nPAINS aching; cramping; griping.\n\nWorse for movement.\n\nOften accompanies the Ipecacuanha nausea and vomiting.\n\nCough\n\nChoking; dry; in fits; irritating; rattling; tormenting; vomiting; whooping. BREATHING difficult; fast; wheezing. MUCUS bloody; difficult to cough up. With NAUSEA (see here); NOSEBLEEDS; RETCHING; BLUE FACE.\n\nWorse during a fever.\n\nMay complain of irritation in the larynx or air passages. Nosebleed produces bright red blood. Babies go stiff during coughing fits and have difficulty breathing (catching their breath).\n\nDiarrhoea\n\nIn BABIES. STOOLS grass green.\n\nFever\n\nWith ANXIETY. CHILLINESS worse for heat; better for fresh air.\n\nFlu\n\nCHILLS. PAINS in bones; in back; in legs; aching; sore, bruised.\n\nFeels weary, as if has carried a heavy load; bones may feel torn to pieces. Chills are worse in a warm room and better for fresh air. Nausea may well be present.\n\nGastric flu\n\nSee FLU.\n\nHeadache\n\nPAINS sore, bruised. With VOMITING (see here).\n\nLabour\n\nLABOUR PAINS. With constant NAUSEA.\n\nNausea\n\nBELCHES empty. With PALLID FACE; RETCHING; copious SALIVA.\n\nWorse after eating; for movement; for smell of food; for tobacco.\n\nFeels nauseous but finds vomiting difficult and the nausea will persist even if able to vomit. The tongue is usually clean.\n\nVomiting\n\nVOMIT bile; food; green. With HEADACHE.\n\nWorse after eating; after bending down; for coughing.\n\nJABORANDI (Jab.)\n\nOther name: Pilocarpus pinnatifolius\n\nPhysical complaints\n\nMumps\n\nWith FACE RED. GLANDS swollen. PAINS spread to breasts, ovaries. Copious SALIVA. Profuse SWEATING.\n\nSweat starts on the face and spreads over the body. Saliva is like egg white, and although it runs freely the mouth feels dry. The jaw is stiff and the tonsils are often swollen. Given early enough, Jaborandi prevents involvement of other glands (breasts, ovaries, testicles). The sweaty stage is accompanied by an acute thirst, followed by prostration and drowsiness and a general feeling of dryness. Mercurius solubilis presents a similar picture with its salivation and sweating but the smelly breath and sweat is less marked with Jaborandi.\n\nKALI BICHROMICUM (Kali-b.)\n\nOther name: potassium bichromate\n\nGeneral symptoms\n\nDischarges stringy\/sticky; thick. Pains appear and disappear suddenly; in a spot; wandering. Tongue white-coated.\n\nWorse in morning on waking; at night; for getting cold; after eating; in summer.\n\nSmallish remedy, useful for colds and sinusitis. These are chilly individuals who are worse for cold, whose symptoms come on after getting cold. Become chilly when ill, yet feel worse in the summer's heat. The pains can usually be located with a fingertip and may wander about the body.\n\nPhysical complaints\n\nBurns\n\nDEEP. SLOW TO HEAL.\n\nCommon cold\n\nNASAL CATARRH crusty; dry; hard crusts; smelly; thick; green; yellow; yellow-green; sticky; stringy. PAINFUL SINUSES with DRY THROAT.\n\nCatarrh forms sticky crusts in nostrils, which are difficult to remove, leaving raw, sore patches, and then re-form. Catarrh drips down back of throat, accumulates overnight, and is difficult to hawk out in morning; or may block nose entirely and be impossible to blow out.\n\nCough\n\nCOUGH croupy. MUCUS ropy; sticky; tough; thick; difficult to cough up. PAIN IN CHEST sore, bruised. With VOICE HOARSE.\n\nWorse after eating; on waking in the morning.\n\nPain in the chest spreads from front to back, and air passages are irritated and aggravate the cough. Ropy mucus tends to be coughed up only in the morning.\n\nCroup\n\nWith COUGH symptoms (see here).\n\nEarache\n\nDISCHARGE thick; yellow; smelly. PAIN stitching.\n\nWorse on left side.\n\nHeadache\n\nPAIN recurring at regular intervals; in sinuses.\n\nUsually comes on at the same time each day and although initially pain is in a small localised spot, it may extend to the whole head, especially if sinus congestion is severe.\n\nJoint pain\n\nPAIN alternating with indigestion; alternating with cough; in one spot; sore, bruised; wandering.\n\nUnlikely to suffer joint pain and common cold symptoms at the same time.\nKALI CARBONICUM (Kali-c.)\n\nOther name: potassium carbonate\n\nGeneral symptoms\n\nAnaemia. Catches colds easily. Dislikes bread. Exhaustion. Hot flushes. Pains stitching. Sweat on single parts of the body; after eating; after the slightest physical exertion; profuse.\n\nBetter for warmth of bed.\n\nWorse for becoming cold; for cold, dry weather; for draughts; for lying on side; around 2\u20134 a.m.; for getting overheated; at night; for touch; for uncovering.\n\nThis picture may be hard to spot. These types are extremely weak, wary and often anaemic. They sweat easily and, being chilly, they catch cold easily. They are sensitive to draughts but dislike being overheated as this produces hot flushes. They feel generally better for warmth. Pain is characteristically sharp, stitching, and worst between 2 and 4 a.m. They look puffy, especially around eyes and upper lids. Symptoms are worse for lying on the affected part.\n\nEmotional state\n\nAngry. Anxious children; during labour. Desires company; to be carried. Dictatorial (bossy) during labour. Dislikes\/aversion to being touched; being alone. Expression haggard; suffering. Fearful of being alone during labour. Irritable during labour. Jumpy. Sensitive to noise. Sluggish. Touchy.\n\nThese are touchy individuals who are anxious, irritable and sluggish all at the same time. They do not want to be alone and yet they are not the welcoming type! They are highly strung, being easily startled both by noises and by unexpected touch, either while awake or asleep. They often talk and moan in their sleep.\n\nPhysical complaints\n\nAfterpains\n\nPAIN stitching; shooting down into buttocks\/hips\/legs.\n\nBackache\n\nIn LOWER BACK. PAIN dragging; sore, bruised; stitching.\n\nBetter for pressure.\n\nWorse around 3 a.m.; before menstrual period; after long sitting; for walking.\n\nCauses pregnancy; childbirth.\n\nMay occur after injury, and pain, which is much worse in the early hours of the morning, may drive the sufferer out of bed.\n\nBleeding (vaginal) in pregnancy\n\nWith PAIN IN BACK spreading down into buttocks and thighs.\n\nConstipation\n\nIn PREGNANCY. Large, hard STOOL. PAIN before a stool, with unfinished feeling.\n\nCough\n\nBREATHING wheezing. COUGH dry; hard; irritating; loose; racking; vomiting; violent; disturbs sleep. PAIN IN CHEST cutting; stitching. With NAUSEA; VOMITING.\n\nWorse for deep breathing; for becoming cold; during fever; during evening; for heat; for lying down; in morning; at night; for talking; around 3 a.m.\n\nCause getting chilled.\n\nHas to sit up and lean forward to cough (typically rests elbows on knees) to get some relief. Chest pain with a cough (chest infection) is often right-sided and cutting, knife-like or stitching, with a feeling of irritation throughout respiratory tract. Coughs in those who are worn out.\n\nFaintness\n\nIn PREGNANCY. HAS TO LIE DOWN.\n\nFever\n\nSWEATING profuse; after slightest exertion.\n\nBetter in bed; for heat.\n\nWorse after eating.\n\nHeadache\n\nPAIN above eyes; in forehead; shooting; tearing.\n\nWorse for cold; on left side of head.\n\nCause eye strain.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; full. PAIN sore, bruised.\n\nBetter for belching.\n\nWorse after eating.\n\nFeels bloated after eating: everything turns to gas.\n\nInsomnia\n\nDREAMS anxious; nightmares.\n\nWorse before midnight; around 1\u20132 a.m.\n\nFeels drowsy in the evening but is unable to get to sleep, or alternatively falls asleep easily, has awful and\/or anxious dreams and wakes in the early morning, unable to get back to sleep again.\n\nJoint pain\n\nARMS feel weak. RESTLESS LEGS. PAIN in arms\/hip\/legs\/shoulder; sore, bruised; stitching; tearing.\n\nBetter for movement; for walking.\n\nWorse around 2\u20133 a.m.; for lying on painful part.\n\nJerks on going to sleep. Hands and feet feel cold and numb and legs fall asleep easily.\n\nLabour\n\nPAIN in back, buttocks or thighs; distressing; ineffectual (cervix doesn't dilate); stops (or slows down); weak. With CHILLINESS (after a contraction); EXHAUSTION.\n\nBetter for hard pressure.\n\nPiles\n\nLARGE, protruding when passing a stool or coughing.\n\nWorse for touch.\n\nCause childbirth.\n\nKALI MURIATICUM (Kali-m.)\n\nOther name: potassium chloride\n\nGeneral symptoms\n\nDischarges white. Tongue white-coated.\n\nThis is a small remedy whose keynote is whiteness \u2013 and this is found everywhere: tongue, mouth, and all discharges. Consider it for any cold that has moved beyond the first stage and settled in but has not turned into a more serious infection (with yellow\/green discharges).\n\nPhysical complaints\n\nCommon cold\n\nNASAL CATARRH white. With DEAFNESS AFTER A COLD.\n\nIt is for stuffy head colds with thick white mucus and swollen glands.\n\nCough\n\nBarking; hard; short.\n\nEarache\n\nWith NOISES IN THE EAR. GLANDS swollen.\n\nWorse for swallowing.\n\nEars crackle and pop on blowing the nose or swallowing.\n\nIndigestion\n\nWith DIARRHOEA; STOOLS pale.\n\nWorse after eating starchy food.\n\nSore throat\n\nTONSILS swollen; white.\n\nThrush (oral)\n\nIn BREASTFEEDING BABIES. TONGUE\/GUMS coated white.\n\nKALI PHOSPHORICUM (Kali-p.)\n\nOther name: potassium phosphate\n\nGeneral symptoms\n\nAnaemia. Exhaustion caused by flu; nervous. Sweating when tired; profuse; after the slightest physical exertion.\n\nBetter for heat; for rest.\n\nWorse for cold; for exercise; for excitement; for mental exertion; for worry.\n\nKali phosphoricum is a nerve nutrient found in tissues and fluids of brain and nerve cells, indicated in cases of nervous exhaustion where anaemia, nervous headaches and\/or insomnia are present. Especially useful for exhaustion following a bad case of flu and also in convalescent stage of acute illness where there is muscular weakness. Sensitive to cold, cold in general, they feel much better for warmth.\n\nEmotional state\n\nAnxious babies \u2013 at night. Complaints from mental strain; overwork. Depressed. Fearful babies \u2013 at night. Forgetful. Jumpy. Screaming on waking. Sensitive babies.\n\nFor exhaustion with frayed nerves, following a heavy work or study period. The memory becomes unreliable. With resultant anxiety and the mind exhausted and sluggish. Sensitivity to noise and light is often present; may startle easily.\n\nPhysical complaints\n\nBackache\n\nPain in SPINE; sore, bruised.\n\nBetter for movement.\n\nHeadache\n\nNERVOUS. PAIN one-sided.\n\nCause mental exertion; overwork.\n\nIndigestion\n\nNERVOUS. ABDOMEN\/STOMACH feels empty.\n\nCause nervous exhaustion; overwork.\n\nNervous, empty feeling in the stomach is only temporarily relieved by eating. Accompanies nervous exhaustion.\n\nInsomnia\n\nWith EMPTY FEELING in pit of stomach.\n\nCause anxiety; excitement; mental strain; nervous exhaustion.\n\nSleeplessness following a period of intense work or excitement. With no other symptoms to guide you to another remedy take Kali phosphoricum for simple insomnia at bedtime, every 5\u201310 minutes until sleep falls.\n\nLabour pain\n\nWith EXHAUSTION.\n\nMarvellous for simple exhaustion in labour with no other symptoms (or too few to prescribe on). Take one between every contraction until energy levels rise.\n\nKALI SULPHURICUM (Kali-s.)\n\nOther name: potassium sulphate\n\nGeneral symptoms\n\nComplaints from change of weather from cold to warm. Discharges thick; yellow. Pains wandering. Tongue yellow-coated.\n\nBetter for fresh air.\n\nWorse for heat; for change of temperature.\n\nSimilar to Pulsatilla: its symptoms are aggravated by warmth and relieved by fresh air, especially a walk outside. Both remedies have thick yellow discharges and wandering pains. Pulsatilla has a very clear emotional state, especially in babies who are clingy, weepy and shy. Kali sulphuricum has similar physical symptoms without a strong emotional picture to prescribe on.\n\nEmotional state\n\nAngry. Anxious when indoors; when overheated. Irritable.\n\nThese types may be ratty and anxious, especially when stuck inside.\n\nPhysical complaints\n\nCommon cold\n\nNASAL CATARRH profuse; thick, yellow.\n\nBetter for fresh air.\n\nWorse in stuffy room.\n\nIf Pulsatilla is well indicated for a cold but either does not or only partly helps, Kali sulphuricum will finish the cure.\n\nCough\n\nRATTLING; WHOOPING. MUCUS yellow; difficult to cough up; has to swallow what comes up.\n\nWorse at night; in warm room.\n\nEarache\n\nDISCHARGE thin; yellow. With CRACKLING IN EARS when chewing.\n\nCauses temporary deafness, especially in babies, because Eustachian tubes are blocked with mucus.\n\nHeadache\n\nPAIN in forehead; in sides of head; stitching.\n\nBetter for fresh air.\n\nWorse during evening; in warm room.\n\nJoint pain\n\nFEET cold. PAIN in hips; in legs; wandering.\n\nBetter for fresh air; for movement; for walking.\n\nWorse for heat; in summer.\n\nAlthough feet feel cold, the symptoms are worse for heat.\n\nKREOSOTUM (Kreos.)\n\nOther name: beechwood kreosote\n\nEmotional state\n\nCapricious. Desires to be carried. Screaming (babies).\n\nBabies are obnoxious during teething (rather like Chamomilla). In women, marked burning and smelliness of discharges, rather than the emotional state, guide you to this remedy.\n\nPhysical complaints\n\nBleeding (vaginal) in pregnancy\n\nIn EARLY PREGNANCY. FLOW CLOTTED; dark; black; smelly.\n\nGums bleeding\n\nLochia\n\nDARK (brown); LUMPY; SMELLY; BURNS. Returns, having almost stopped.\n\nNausea\n\nIn PREGNANCY. With VOMITING; COPIOUS SALIVA.\n\nTeething\n\nTEETH decay as soon as they come through. PAIN severe. With RESTLESS SLEEP.\n\nThrush (genital)\n\nIN PREGNANCY. DISCHARGE burning; milky; profuse; smelly; thin; watery. With ITCHING (vulva, vagina).\n\nLAC CANINUM (Lac-c.)\n\nBreast (feeding) problems\n\nWEANING to dry up milk.\n\nTo help your milk dry up after weaning, or to decrease your supply if it is over-abundant and gushing everywhere.\n\nLAC DEFLORATUM (Lac-d.)\n\nBreast (feeding) problems\n\nMILK SUPPLY LOW in women who are generally chilly and exhausted from loss of sleep.\n\nLACHESIS (Lach.)\n\nOther name: bushmaster snake\n\nGeneral symptoms\n\nBreath smelly. Lips blue. Nosebleeds. Palpitations. Sweat from mental exertion; from pain. Symptoms left-sided or start on left and move to right.\n\nBetter for fresh air; while eating.\n\nWorse for humidity; for heat; during the morning; for pressure; after sleep; for tight clothes; on walking.\n\nGenerally, Lachesis types are worse for heat (they suffer easily from hot flushes) and feel better for cold compresses and fresh air. Their tiredness, which is often accompanied by trembling and a faint feeling, is not helped by sleeping. They and their symptoms are worse on waking and they may even begin to dread falling asleep because of this. They suffer from left-sided complaints, which may then move across to the right side of the body.\n\nEmotional state\n\nAnxious on waking. Aversion to being touched\/examined (babies); to company during pregnancy. Cheerful. Complaints from grief; mental strain. Confused. Depressed generally; in pregnancy; on waking. Desires to be alone. Excitable. Exhilarated. Expression sickly. Jealous. Lively. Sensitive to pain; to touch. Shock from injury. Sluggish on waking. Talkative. Tearful during pregnancy.\n\nThese are lively, excitable characters who are so talkative it is hard to get a word in edgeways. Ill-health often follows overwork or a personal loss which causes depression and anxiety (worse on waking) and mental exhaustion. Acutely sensitive to touch, they cannot tolerate the slightest pressure, especially around the neck.\n\nPhysical complaints\n\nBites\/stings\n\nBetter for cold.\n\nWorse for heat.\n\nCause insects\/animals.\n\nArea around the bite looks bluish.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nWith tingling and numbness in the fingers.\n\nCough\n\nCHOKING on falling into deep sleep; dry; hacking; suffocative; violent; from tickling in larynx.\n\nWorse for touch; at night.\n\nThe larynx is irritated as if a crumb were lodged in it; sets off desire to cough, as does having neck touched. Wakes with a choking cough immediately after falling into a deep sleep.\n\nCroup\n\nWith symptoms of COUGH (see here); LUMP SENSATION in throat.\n\nWorse after sleep; on waking.\n\nThe voice is hoarse and the air passages feel irritated.\n\nEarache\n\nPAINS on left side. With SORE THROAT (see here).\n\nWorse for swallowing; for exposure to wind.\n\nExhaustion\n\nWorse during morning after getting up; for heat of sun; for slightest exertion; for walking; for mental exertion.\n\nCause mental exertion.\n\nHair loss\n\nCause pregnancy.\n\nHeadache\n\nHead feels heavy; hot. PAIN in forehead; spreading to back of head; in temples and top of head; bursting; pressing; throbbing; violent.\n\nBetter after eating.\n\nWorse on left side of head; for lying down; on waking; for pressure; for walking.\n\nCause over-exposure to sun.\n\nHeadaches begin on the left side of head and may move to the right. Dreads going to sleep, knowing pain will be awful on waking.\n\nInjuries\n\nCUTS\/WOUNDS bleed freely; slow to heal. SCARS become red.\n\nSkin around wound looks dark or bluish and blood is dark.\n\nMumps\n\nGLANDS swollen\/painful. With SORE THROAT.\n\nWorse on left side.\n\nThe neck and glands are especially sensitive to touch; swallowing is painful and difficult.\n\nPiles\n\nPAINFUL. In pregnancy. BLEEDING; BLUISH; EXTERNAL; LARGE.\n\nScarlet fever\n\nRASH looks bluish.\n\nWith typical Lachesis general and emotional\/mental symptoms.\n\nSore throat\n\nPAIN spreading up to ears; splinter-like. With CHOKING SENSATION; LUMP SENSATION in throat.\n\nBetter for swallowing solids.\n\nWorse for heat; on the left side; for pressure; for touch; for swallowing liquids or saliva.\n\nThroat is very painful; feels better for swallowing hard foods, such as toast, because hard pressure eases the pain. Swallowing saliva or drinks, especially warm ones, hurts. There's a sense of constriction around the neck; cannot bear to be touched there; a constant desire to swallow to dislodge the 'lump'. Hawking may relieve this feeling.\n\nLEDUM PALUSTRE (Led.)\n\nOther name: wild rosemary\n\nGeneral symptoms\n\nPains wandering.\n\nBetter for cold bathing.\n\nWorse for heat; for movement; for touch; for walking.\n\nEffective for acute rheumatic complaints of the joints. Injured or painful parts puff up, become numb and are sensitive to being touched; they may look mottled, bluish or pale. The painful part (injury or rheumatic joint) feels cold (internally and externally), but pains are relieved by cold compresses or being bathed in cold (even icy-cold) water. Pains are generally worse for movement and warmth.\n\nPhysical complaints\n\nBites\/stings\n\nBetter for cold.\n\nWorse for heat.\n\nCause insects\/animals.\n\nLedum prevents sepsis: give it routinely after a bite by any animal, poisonous or not. Also for insect bites or stings unless another remedy is strongly indicated; use the appropriate external remedy as well.\n\nBruises\/black eye\n\nCause injury.\n\nIf Arnica has been given but bruising comes out, i.e. you give Arnica a bit too late, give Ledum.\n\nEye injury\n\nBRUISING and discolouration.\n\nBetter for cold compress.\n\nInjuries\n\nCUTS\/WOUNDS lacerated; punctured; to palm of hand\/sole of foot.\n\nBetter for cold compresses.\n\nWorse for heat.\n\nCause splinter; nail; knife; needle.\n\nIf prescribed early, Ledum will prevent sepsis from developing in wounds where the skin is punctured by a sharp, pointed instrument. If a splinter is still in the wound, press gently around it to encourage bleeding and to expel it or give Silica (see here).\n\nJoint pain\n\nHot; icy-cold; stiff. PAIN in hands and feet; in right hip; in left shoulder.\n\nBetter after common cold; for being uncovered; after cold compresses; after cold bathing.\n\nWorse for movement; for heat; for warmth of bed; at night.\n\nPains may start in the lower limbs, especially in the small joints, and move upwards, for example from feet to hands. Feels cold, but warmth of bed is unendurable.\n\nLILLIUM TIGRINUM (Lil-t.)\n\nOther name: tiger-lily\n\nGeneral symptoms\n\nWorse at night; for heat.\n\nThese are hot people who feel worse (emotionally and physically) at night.\n\nEmotional state\n\nApathetic about anything being done for them. Depressed. Hurried, doing several things at once. Irritable. Restless.\n\nThese types are always on the go. They are snappy, busy people who feel better for being occupied, nothing anyone else does for them has an effect.\n\nPhysical complaints\n\nPalpitations\n\nIn pregnancy.\n\nPiles\n\nWith dragging down sensation.\n\nWorse for standing.\n\nCause childbirth.\n\nProlapse\n\nWith BEARING DOWN SENSATION.\n\nFeels as if everything will fall out.\n\nLYCOPODIUM (Lyc.)\n\nOther name: club moss\n\nGeneral symptoms\n\nBlisters on tip of tongue. Discharges yellow. Expression confused. Face pale; sickly. Likes sweets; chocolate. Oedema (swelling) of ankles\/feet; of hands\/fingers. Pains in glands. Sweat cold; profuse; clammy; smelly; sour; worse for slightest physical exertion; better for uncovering. Symptoms right-sided; start on right side and move to left. Taste sour.\n\nBetter for fresh air; for warmth of bed.\n\nWorse around 4\u20138 p.m.; during afternoon; during evening; for pressure; in stuffy rooms; for tight clothes; for wind; for onions; for flatulent food (beans; cabbage, etc.).\n\nThey have trouble with food: either feeling full up and bloated after eating very little or having no appetite at all until they start eating, and then becoming ravenous. Sweet-cravers, often eating several bars of chocolate a day. They may feel sleepy after eating. They digest their food poorly and become bloated with gas, with acidity and sourness. Despite being chilly and worse for extremes of heat and cold, they feel generally better for fresh air. Babies may sleep well at night but cry all day.\n\nEmotional state\n\nAngry babies; from contradiction. Anxious generally; during labour; indoors; anticipatory. Aversion to being consoled; own babies. Complaints from humiliation; mental strain; suppressed anger. Concentration poor. Depressed on waking. Desires company; to be carried. Despair. Dictatorial during labour. Dislikes being contradicted; being alone; company. Fearful babies, of strangers; of public speaking. Forgetful. Hitting. Indecisive. Irritable babies \u2013 after a sleep; in the mornings; daytime (good all night and cross all day); when sick; on waking. Moody. Rage. Resentful. Restless sleep during labour. Screaming during sleep; on waking. Lack of self-confidence. Sensitive. Shy. Sluggish better for fresh air. Tantrums. Tearful.\n\nLycopodium individuals are anxious and worry about many things because they lack self-confidence. They dread taking on new things, but in anticipation of an event prepare meticulously and usually shine! Sensitive and shy; their rude and difficult behaviour is a front for their low self-confidence. They hate to be alone, but are not keen on company, either; their ideal is to know that there is someone else in the house \u2013 in another room! They are irritable intellectuals who find it difficult to express their feelings. They become forgetful and indecisive when tired, especially after mental strain.\n\nLycopodium babies are irritable and dictatorial; having tantrums if contradicted; kicking and screaming after a nap or on waking in the morning. They are generally difficult to live with.\n\nPhysical complaints\n\nBackache\n\nIn LOWER BACK. With STIFFNESS.\n\nBetter for movement; for passing wind; for urinating.\n\nWorse on beginning to move; while passing a stool; on getting up from sitting.\n\nCause lifting.\n\nLower back feels sore, often because of wind that cannot escape.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nTingling and numbness in fingers, with or without swelling of wrist.\n\nCommon cold\n\nNASAL CATARRH yellow. NOSE blocked; dry. SINUSES blocked. With HEADACHE (see here).\n\nWorse on right side.\n\nNose and sinuses are blocked so sleeps with mouth open. Nursing infants cannot breathe; this is a major remedy to consider for new babies with the snuffles.\n\nConstipation\n\nIn BABIES. In pregnancy. DESIRE TO PASS STOOL ineffectual. STOOLS hard; knotty. With FLATULENCE.\n\nCough\n\nConstant; disturbs sleep; dry; irritating; painful. BREATHING difficult; fast. MUCUS green; yellow; white; copious; tastes salty.\n\nBetter for hot drinks.\n\nWorse on going to sleep; at night; in evening in bed.\n\nOverpowering tickling in larynx, as if being touched by a feather. It is worse in the evening on going to bed and can prevent sleep. A feeling of constriction or tightness in the chest can develop.\n\nCradle cap\n\nIn thin, gassy babies.\n\nCramp\n\nIn calf.\n\nWorse at night.\n\nCystitis\n\nDESIRE TO URINATE frequent; ineffectual. PAIN aching; cutting; pressing; stitching. URINATION frequent; slow (waits a long time for it to start).\n\nBetter after urination.\n\nPasses only a few drops or a dribble of urine at a time. Child screams before urinating because of difficulty or pain.\n\nEarache\n\nWith BLOCKED FEELING; PAIN TEARING. NOISES IN THE EAR.\n\nEye inflammation\n\nDISCHARGE purulent. EYES sensitive to light; stitching; watering. EYELIDS glued together.\n\nFever\n\nOne-sided; on left side.\n\nWorse in evening.\n\nOne-sided sweating, or feeling hot on one side of the body and cold on the other, is a common symptom.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; intolerant of tight clothing; rumbling.\n\nBetter for passing wind.\n\nWorse after eating; before and after passing stools.\n\nBloated and uncomfortable below navel; has to loosen clothing around the waist.\n\nHair loss\n\nCause childbirth.\n\nHeadache\n\nPAIN in forehead; in temples; pressing; throbbing.\n\nBetter for fresh air.\n\nWorse for coughing; during menstrual period; for overheating; for warmth of bed; for wrapping up head.\n\nCause eye strain; mental strain.\n\nHernia in babies\n\nINGUINAL on right side.\n\nIndigestion\n\nABDOMEN\/STOMACH feels empty; feels full very quickly. BELCHES acrid; empty; sour. PAINS in stomach; pressing; cramping. With HEARTBURN.\n\nBetter for belching.\n\nWorse after eating; for onions; for tight clothing.\n\nInsomnia\n\nSLEEPLESS in daytime (babies).\n\nSleeps at night but is bad-tempered all day.\n\nJoint pain\n\nPAIN tearing.\n\nBetter for movement; for walking; for warmth of bed.\n\nWorse on beginning to move; during a fever; for sitting down.\n\nRheumatic pains occur in joints, but knee and finger joints are especially sore and stiff. Pains often start on right and move to the left.\n\nLabour\n\nLATE \u2013 with anticipatory anxiety; too FAST.\n\nWomen in labour are anxious, restless and dictatorial.\n\nMouth ulcers\n\nUNDER TONGUE.\n\nMumps\n\nGLANDS swollen\/painful. With FEVER (see here); SORE THROAT (see here); typical Lycopodium general and emotional\/mental symptoms. SWELLING MOVES from right to left.\n\nNausea\n\nIn pregnancy. With VOMITING.\n\nPiles\n\nBLEEDING. EXTERNAL. ITCHING. With CONSTIPATION.\n\nSnuffles\n\nIn newborn babies. With COMMON COLD.\n\nSore throat\n\nTHROAT dry; with lump sensation. PAIN raw; sore. TONSILS swollen.\n\nBetter for warm drinks.\n\nWorse at night; on right side.\n\nThroat sore and inflamed, and feels full of dust. Chokes while drinking though warm drinks soothe the pains.\n\nVaricose veins\n\nDuring pregnancy. Of LEG\/THIGH; of VULVA. Painful.\n\nMAGNESIA CARBONICA (Mag-c.)\n\nOther name: carbonate of magnesia\n\nGeneral symptoms\n\nDischarges sour. Insomnia after 3 a.m.; sleep unrefreshing. Sweat oily; sour. Taste in mouth sour. Weight gain poor in babies.\n\nBetter for fresh air.\n\nWorse during evening; at night.\n\nPuny, sickly babies, or individuals of any age worn out from too many cares and not enough good food. They snack for comfort on junk food and look 'unloved'. They are generally worse at night, when they cannot sleep; wake more tired than when they went to bed. A walk in fresh air helps a little. Nursing babies may refuse the breast; milk passes through undigested. Weaning children won't eat vegetables but want meat.\n\nEmotional state\n\nAnxious in bed. Irritable babies. Screaming on waking.\n\nBabies are particularly irritable and anxious during a fever. Sleep is restless and disturbed.\n\nPhysical complaints\n\nDiarrhoea\n\nPAIN colicky; cramping. STOOLS frothy; green; slimy; sour-smelling.\n\nWorse in morning; before passing a stool.\n\nMilk is poorly tolerated, passes through undigested; nursing babies may even refuse the breast. Stools are like the scum of a frog pond.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated. With RUMBLING before passing a stool.\n\nHeadache\n\nPAIN recurs at regular intervals; spasmodic; shooting; tearing.\n\nBetter for pressure; for walking.\n\nWorse for lying down.\n\nMay be obliged to walk about all night because pains return in force on stopping to rest.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; rumbling. BELCHES greasy; sour. With HEARTBURN; NAUSEA.\n\nWorse after eating cabbage; after drinking milk.\n\nMAGNESIA MURIATICUM (Mag-m.)\n\nOther name: chloride of magnesium\n\nGeneral symptoms\n\nDischarges sour.\n\nBetter for fresh air; for pressure.\n\nWorse after drinking milk; at night; for swimming in sea.\n\nFor babies' colic or digestive upsets that are worse for milk or caused by drinking milk. Generally better for fresh air; pains are better for pressure. Sweat and stools smell sour.\n\nEmotional state\n\nAnxious. Restless in bed.\n\nFeels rushed and hurried; must be doing something all the time. Though present during the day, anxiety is always worse in bed at night, causing insomnia.\n\nPhysical complaints\n\nColic\n\nIN BABIES; cramping; sore, bruised. With CONSTIPATION (see here); DIARRHOEA (see here); INDIGESTION.\n\nWorse after drinking milk.\n\nCause milk; teething.\n\nCommon cold\n\nCause swimming in sea.\n\nConstipation\n\nSTOOLS passed with difficulty; crumbling; small balls; knotty. With STRAINING ineffectual.\n\nCause drinking cow's milk.\n\nDiarrhoea\n\nSTOOLS green. In BABIES.\n\nCause drinking milk.\n\nHeadache\n\nPAIN in temples; spreading to eyes. With BELCHING; THIRST.\n\nWorse for pressure; for wrapping up head.\n\nInsomnia\n\nSLEEP unrefreshing.\n\nRestless and anxious in bed, especially on closing eyes; may be over-sensitive to noise then, and unable to get to sleep.\n\nTeething\n\nPAINFUL in babies. With COLIC; GREEN STOOLS.\n\nMAGNESIA PHOSPHORICA (Mag-p.)\n\nOther name: magnesium phosphate\n\nGeneral symptoms\n\nPains cramping.\n\nBetter for heat; for firm pressure.\n\nWorse for cold; for swimming in cold water; for touch, light pressure; for being uncovered; for walking in fresh air.\n\nNicknamed the 'homeopathic aspirin' because of its success in easing minor aches and pains, such as headaches, earaches, teething pains or even neuralgic pains of the face. For greatest effect, the general symptoms (above) must be present.\n\nPains are greatly relieved by heat and pressure, and are much worse for cold.\n\nNB This remedy is best given dissolved in warm water. Add 4 tablets to a glass of boiled and partially cooled water and stir vigorously. Sip frequently until the pains ease. Repeat as and when needed. Scald glass and spoon afterwards or the next person to use them may get an inadvertent dose of the remedy.\n\nPhysical complaints\n\nColic\n\nPAIN cramping; drawing.\n\nBetter for warmth; bending double.\n\nCramp\n\nIn arm; finger; hand; wrist.\n\nBetter for heat.\n\nCause prolonged use of hands.\n\nFor writers, musicians, typists and others whose occupations involve prolonged use of fingers and hands.\n\nEarache\n\nPAIN spasmodic; shooting.\n\nBetter for heat; for firm pressure.\n\nWorse for cold; for turning head.\n\nCause cold wind.\n\nComes on after a walk in cold wind.\n\nHeadache\n\nPAIN on right side of head; spasmodic; shooting.\n\nBetter for heat; for firm pressure.\n\nWorse for cold.\n\nCause getting chilled.\n\nOften right-sided; may be accompanied by pains in the face that come and go throughout the day.\n\nHiccups\n\nIn babies; violent.\n\nWorse after eating or drinking.\n\nLabour pains\n\nWith CRAMP (see here) in hand or leg.\n\nSciatica\n\nBetter for heat.\n\nWorse for cold.\n\nSciatica or any cramping, neuralgic pains in back. Pains often caused by cold, and are much better for hot applications and firm massage.\n\nTeething\n\nPAINFUL in babies.\n\nBetter for external heat.\n\nHot-water bottle (wrapped in towel) relieves the pains. Magnesia phosphorica in a bottle or cup can also help.\n\nMERCURIUS CORROSIVUS (Merc-c.)\n\nOther name: mercuric chloride\n\nGeneral symptoms\n\nBreath smelly. Exhaustion. Gums pale; bleeding. Likes cold drinks. Oedema (swelling) ankles\/feet; hands\/fingers. Sweat cold. Thirst extreme. Tongue yellow-coated.\n\nWorse for pressure.\n\nThis remedy has a similar sphere of action to Mercurius solubilis (below) but its general picture is stronger: has more pains and more burning and more thirst and is more violent, more active.\n\nIt does not have any marked emotional symptoms. The physical symptoms are nearly always accompanied by general feeling of nervous exhaustion, and, because of seriousness of the physical symptoms, there will also be some anxiety, irritability and\/or depression.\n\nPhysical complaints\n\nCystitis\n\nDESIRE TO URINATE constant; painful; frequent. PAIN burning. URINE dark; green; red; scanty. URINATION dribbling; frequent; difficult.\n\nWorse during urination.\n\nSevere cystitis where terrible pains are felt both in bladder and rectum. Little urine is passed in spite of constant, painful urge to urinate. Has to strain to pass urine which is also excruciatingly painful. Urine burns and may have blood in it. Seek professional help immediately.\n\nDiarrhoea\n\nPAIN severe. STOOLS bloody; frequent; green; slimy; smelly; yellow. STRAINING frequent; painful.\n\nWorse before\/during\/after passing a stool.\n\nSevere diarrhoea (dysentery) where there is constant, incessant, excruciatingly painful straining that is not relieved by passing a stool. May be accompanied by vomiting of bile which will eventually cause a bruised stomach. Seek professional help immediately.\n\nSore throat\n\nGUMS bleeding; swollen. PAIN burning; raw. With SALIVA increased. ULCERS in throat.\n\nBetter for eating.\n\nWorse for pressure; for swallowing liquids.\n\nChoking sensation in the throat and an increase of sticky saliva in the mouth, which create a constant desire to swallow. Breath is smelly and gums may be swollen and bleed easily.\n\nMERCURIUS SOLUBILIS (Merc-s.)\n\nOther name: ammonio-nitrate of mercury\n\nGeneral symptoms\n\nBreath smelly. Complaints from getting chilled. Discharges burning; blood-streaked; smelly; yellow; watery. Face pasty. Glands swollen. Likes bread and butter; cold drinks. Mouth dry. Mouth ulcers on gums, on tongue. Pains in bones; pressing. Saliva increased; during sleep. Sweat cold; profuse; smelly; from pain. Taste mouth tastes bad; bitter; metallic. Thirst extreme. Tongue cracked; coated yellow.\n\nWorse for heat and cold; for lying down; during evening; at night; for light touch.\n\nAll discharges \u2013 stools, urine, sweat, saliva, catarrh, etc. \u2013 smell strongly. They sweat profusely, without relief, especially at night. Symptoms are worse at night generally and especially if they become hot.\n\nThey are sensitive to extremes of temperature; disliking both heat and cold. They may feel chilly and want a hot-water bottle, then feel overheated so open the window, which chills them quickly, whereupon cycle begins again. A change of 2 degrees either way makes them feel worse. Resting in bed in a moderate temperature affords the greatest relief.\n\nThey have a burning thirst in spite of having a lot of saliva which they are constantly swallowing \u2013 it may dribble on to their pillows at night. They have a bitter, metallic taste in the mouth; and their flabby tongues take the imprint of their teeth so that they are indented around the edges.\n\nThey suffer with mouth ulcers; abscesses; colds; painful, swollen glands and pains in the teeth.\n\nEmotional state\n\nConfused. Depressed. Dictatorial. Discontented. Forgetful. Restless children.\n\nThese are nervous, anxious, restless types who find it difficult to stay still. They do slow down, both mentally and physically, when they become sick, appearing apathetic but always feel hurried inside. In this slowed-down state, their memories weaken, confusion sets in and they may have trouble thinking. They may feel depressed and weepy but crying alternates with laughter \u2013 they are restless even in their emotions!\n\nPhysical complaints\n\nAbscesses\n\nOf GLANDS; of ROOTS OF TEETH.\n\nWith Merc-s. general symptoms: increased saliva, foul taste and smelly breath.\n\nBackache\n\nIn LOWER BACK. PAIN burning; shooting.\n\nWorse for breathing; for coughing; on getting up from sitting; for sweating.\n\nBreast (feeding) problems\n\nBREAST ABSCESS painful.\n\nChickenpox\n\nWith Merc-s. general symptoms: worse at night, weak and smelly. Rash suppurates and is smelly.\n\nCommon cold\n\nNASAL CATARRH bloody; burning; green; yellow; yellow-green; smelly; watery. Blocked, painful sinuses. With FEVER (see here); HEADACHE (see here); HOARSENESS; LOSS OF SMELL; frequent SNEEZING; SORE THROAT (see here).\n\nWorse for cold and for heat; at night.\n\nNasal discharge burns upper lip, nostrils may become ulcerated. Nose may bleed during sleep.\n\nCough\n\nRacking; MUCUS green.\n\nWorse during evening in bed; at night; for lying on right side.\n\nCystitis\n\nDESIRE TO URINATE constant; ineffectual. PAIN burning; URINE dark; scanty. URINATION frequent.\n\nWorse at beginning of urinating; when not urinating.\n\nPains are terrible and urinating is slow and difficult.\n\nDiarrhoea\n\nIn BABIES. PAIN burning. STOOLS bloody; green; slimy; smelly; yellow. With SWEATING before\/during\/after stool; LOUD WIND.\n\nWorse for cold; during the evening; at night after passing a stool.\n\nBurning pains come on when passing a stool and remain for a time afterwards. Feels faint and weak (see EXHAUSTION). Desire to pass a stool may be worse after the stool has been passed.\n\nEarache\n\nDISCHARGE from ears blood-streaked; smelly. PAIN boring; burning; pressing; tearing. With BLOCKED FEELING in ears.\n\nWorse at night; for warmth of bed; fresh air.\n\nEardrum may perforate; discharge will be smelly and blood-streaked.\n\nExhaustion\n\nWith HEAVINESS in limbs.\n\nWorse after sweating; after passing a stool.\n\nCause sweating.\n\nAccompanies a fever or an attack of diarrhoea; waves of tiredness with a feeling of heaviness come on after passing a stool or after a heavy bout of sweating. Trembles with exhaustion.\n\nEye inflammation\n\nDISCHARGE purulent. EYES sensitive to light; watering. With symptoms of COMMON COLD (see here).\n\nWorse for heat of fire; warmth of bed.\n\nFever\n\nHEAT alternating with chills. With SWEATING (see General symptoms).\n\nWorse for fresh air; at night in bed.\n\nFeels chilled in the fresh air, but overheated in a stuffy room. Fever may precede a common cold.\n\nHeadache\n\nPAIN in forehead; burning; pressing; sore, bruised.\n\nWorse at night; for bending down.\n\nCause common cold; rheumatism.\n\nFeels as though there is a band around the forehead or the head is in a vice. With increased saliva as well as other Merc-s. general symptoms.\n\nHeartburn\n\nIn pregnancy.\n\nWorse at night.\n\nStomach feels empty and out of sorts with hiccups and burping.\n\nJoint pain\n\nPAIN burning; tearing.\n\nWorse in bed; at night; for wet weather.\n\nRheumatic pains are severe and always worse at night in bed, when they drive the sufferer out of bed.\n\nMouth ulcers\n\nOn gums; on tongue. PAIN stinging; throbbing.\n\nMajor mouth-ulcer remedy, but take general symptoms into account \u2013 increase in saliva and flabby, indented tongue \u2013 as well as ulcers themselves.\n\nMumps\n\nGLANDS hard; swollen\/painful. With FEVER (see here); COPIOUS SALIVA; PROFUSE SWEATING.\n\nWorse for blowing nose; at night; on right side.\n\nTypical Merc-s. tongue and smelly breath also present. Bones in face may ache.\n\nScarlet fever\n\nWith Merc-s. general symptoms and emotional state.\n\nSore throat\n\nPAIN spreading up to ears and neck; sore; stitching. TONSILS swollen; ulcerated.\n\nWorse at night; on right side of body; for swallowing.\n\nVery painful sore throats with ulcers on the tonsils and swollen glands. Pains shoot up into the ear or into the neck on swallowing.\n\nThrush (genital)\/Yeast\n\nDISCHARGE burning; greenish; smelly. With ITCHING.\n\nWorse at night.\n\nThrush (oral)\n\nIn BABIES. With EXCESS SALIVA.\n\nCharacteristic smelly breath is usually present.\n\nNATRUM CARBONICUM (Nat-c.)\n\nOther name: sodium carbonate\n\nGeneral symptoms\n\nComplaints from sunstroke. Dislikes milk. Face pale. Sweat from slightest physical exertion; from pain. Taste in mouth metallic.\n\nBetter after eating; for massage.\n\nWorse before eating; during stormy weather; for fresh air; mid-morning; for milk; for physical exertion; for sun.\n\nThese people have pale faces with blue circles around the eyes; eyelids may be puffy. They dislike exposure to the sun (especially the head); and are also averse to fresh air. Once run down, any exercise will aggravate their general condition. They are sensitive to thunderstorms; can sense them coming. They suffer from poor digestion, but feel better after eating, when they feel warmer. They dislike milk \u2013 this can cause diarrhoea. Babies, even breast-fed babies, have difficulty digesting milk.\n\nEmotional state\n\nAnxious. Cheerful. Complaints from mental strain. Confused. Depressed. Fearful. Forgetful. Gloomy. Jumpy. Lively. Sensitive generally; to music. Shy. Sluggish.\n\nThese people put on a brave face when really they feel miserable. They are sensitive generally; harbour emotional hurts inside, though without feeling bitter. They avoid conflict by appearing cheerful, but this masks feeling of being cut off, depressed and anxious with persistent sad thoughts. Listening to music increases their sense of melancholy. They are easily startled by noise.\n\nThey become mentally weakened by emotional pressure, can find it difficult to think. Complaints come on in this weak state, after mental strain (overwork) or when feeling sad and depressed.\n\nPhysical complaints\n\nCommon cold\n\nNOSE blocked. NASAL CATARRH drips down back of throat; smelly; thick.\n\nWants to hawk up catarrh. May have a sour or metallic taste in the mouth.\n\nDizziness\n\nWorse for mental exertion; in a stuffy room.\n\nExhaustion\n\nWEAK LEGS. NERVOUS EXHAUSTION. With COLD EXTREMITIES; HEAVINESS in the limbs.\n\nWorse for exposure to sun; for mental exertion; for slightest physical exertion.\n\nCause mental strain; over-exposure to sun.\n\nNervous exhaustion follows a period of overwork: feet and hands become cold and legs feel heavy and weak. Can also be caused by an overdose of sunshine (which need not be a great deal), these types become tired in hot weather, and thinking, sunshine or exertion make tiredness worse.\n\nHeadache\n\nHEAD feels heavy. PAIN pressing. With SWEATING on forehead.\n\nWorse after eating; for thinking.\n\nCause mental strain; mental exertion; summer; sunstroke.\n\nFeels generally better after eating, but the headache itself may be worse. After too much study or mental strain, the brain feels worn out. May also be a hot-weather headache.\n\nIndigestion\n\nBELCHES sour. PAIN in stomach; sore, bruised. With NAUSEA.\n\nFeels nauseous in a stuffy room and has trouble digesting food because nervous system is run down.\n\nInsomnia\n\nDREAMS, ANXIOUS. WAKING early.\n\nOften cannot get back to sleep again.\n\nNATRUM MURIATICUM (Nat-m.)\n\nOther name: sodium chloride\n\nGeneral symptoms\n\nAnaemia. Appetite lost in pregnancy. Blisters on tip of tongue. Catches colds easily. Discharges like egg white. Dislikes bread; food in general; the thought of food \u2013 in pregnancy; with hunger. Dryness generally. Face pasty. Gums bleeding. Likes salty things, especially in pregnancy. Lips cracked. Mouth dry; with thirst. Palpitations in pregnancy. Oedema (swelling) of ankles\/feet; of hands\/fingers. Taste in mouth bitter. Thirst extreme; for large quantities.\n\nBetter for lying down; after rest; after sweating.\n\nWorse 10 a.m.; after eating; for exposure to sun; for heat; mid-morning; for physical exertion.\n\nDryness and extreme thirst are characteristic, especially with a fever. Lips dry up and become cracked; lower lip often has a centre crack. Hangnails are also common. Discharges are profuse and watery (often like egg white) or thick and white. They sweat a lot when ill and feel better for doing so.\n\nPeople needing this remedy feel chilly in their bodies but dislike hot weather and feel worse for sun's heat or being in stuffy rooms. May have cold hands and feet in winter, but it doesn't bother them. An uncommon symptom that suggests this remedy is an inability to pass urine in the presence of others, such as in a public toilet.\n\nEmotional state\n\nAbsent-minded. Angry. Aversion to consolation. Complaints from disappointed love; grief; humiliation; suppression of emotions. Confused. Depressed in pregnancy but cannot cry; from suppressed grief. Desires to be alone; in pregnancy. Despair during pregnancy. Discontented. Introverted. Irritable, worse for consolation. Loss of libido. Resentful. Sensitive generally. Tearful in pregnancy; with difficulty crying; cries on own.\n\nWorse for consolation.\n\nDespite being emotional types, they may appear cool and prickly because of their difficulty expressing emotions. They close themselves off to avoid being hurt, suppress their grief, and so may appear quite hard. Their feelings can burst out uncontrollably under certain conditions, especially to a sensitive, sympathetic ear; they may cry in spite of themselves. In that state they hate to be consoled and feel worse for any comforting. They prefer to cry alone, it is rare to see them crying in public; this humiliates them.\n\nWhen depressed they refuse all invitations, preferring to stay at home alone. If hurt (which is very easy!) they do not show it; they harbour hurts inside and become bitter. They are haunted by persistent, unpleasant thoughts, and dwell on things in the past. This can make them ill.\n\nBabies are serious and dislike too much physical contact. They hate to be teased, and are also often slow to talk and walk.\n\nPhysical complaints\n\nBackache\n\nIn LOWER BACK. PAIN aching; back feels as if broken; sore, bruised.\n\nBetter for lying on a hard surface.\n\nCause manual labour.\n\nBack feels weak and tired after exertion or bending down for a long time (after gardening, for example).\n\nCold sores\n\nSORES on lips; around mouth.\n\nCause sun; suppressed grief.\n\nThis is a major cold sore remedy, but it is a mistake to prescribe it on this symptom alone: the general symptoms and the emotional state are both important. A Natrum muriaticum type will often produce cold sores after a disappointment or a loss that wasn't expressed. The sores are usually found at the corners of the mouth.\n\nCommon cold\n\nDISCHARGE FROM EYES watery. NASAL CATARRH drips down back of throat; profuse; white, watery alternating with blocked nose; like egg white. With LOSS OF SMELL; LOSS OF TASTE.\n\nSneezes a lot in the early stages of the cold.\n\nConstipation\n\nSTOOLS crumbling; small balls; STRAINING ineffectual; feels unfinished.\n\nCough\n\nDry; hacking; irritating; tickling. MUCUS like egg white; transparent; white.\n\nWorse during fever; evening in bed; from tickling in larynx.\n\nCough is set off by tickling in the larynx and air passages. Eyes water when coughing.\n\nDiarrhoea\n\nABDOMEN\/STOMACH feels bloated. DIARRHOEA during day only. STOOLS gushing; painless; smelly; watery.\n\nWorse after eating starchy food.\n\nMay be pains before passing stool.\n\nDizziness\n\nIn PREGNANCY.\n\nWorse for getting up from lying down; for tobacco; for walking.\n\nExhaustion\n\nWith HEAVINESS.\n\nWorse during evening.\n\nCause loss of sleep (broken nights).\n\nEnergy may increase for a short while after eating, but feels tired and heavy, especially in the evening.\n\nEye inflammation\n\nGritty; sensitive to light; watering.\n\nFever\n\nHEAT burning. With NAUSEA.\n\nBetter for uncovering.\n\nWorse mid-morning; in the autumn.\n\nLooks dazed and sleepy and falls asleep when hot. Feeling of burning heat may alternate with intense chilliness. Sweating helps. An extreme thirst for large quantities of liquids.\n\nHair loss\n\nCause childbirth.\n\nHeadache\n\nEYES sore; watering. PAIN in forehead; in temples; hammering; pressing; splitting; throbbing.\n\nBetter for firm pressure.\n\nWorse around 10 a.m. to 3 p.m.; after eating; for coughing; for emotions; for excitement; during a fever; on waking; for reading\/writing; talking; thinking.\n\nTrouble with vision, with flickering and zig-zags appearing before the eyes during the pains. Typical of migraines; always seek professional help.\n\nHeartburn\n\nIn PREGNANCY. With INDIGESTION (see here). BELCHES sweetish; watery.\n\nIncontinence\n\nIn PREGNANCY.\n\nWorse for laughing; for coughing; for sneezing; for walking.\n\nInvoluntary urination, with leaking on exertion.\n\nIndigestion\n\nBELCHES incomplete, ineffectual; sour; tasting of food just eaten. MOUTH tastes bitter; salty. PAIN in stomach cramping; pressing. With violent HICCUPS.\n\nWorse after eating starchy food.\n\nAfter suppressing emotions the stomach becomes 'disordered', and stodgy foods are no longer easily digested. Burps after eating but with difficulty and this does not relieve the indigestion. May also have violent hiccups.\n\nInsomnia\n\nDREAMS anxious; vivid.\n\nCause grief.\n\nFinds it difficult to get to sleep after a loss or disappointment; wakes in the night and cannot get back to sleep. May have anxious, vivid dreams.\n\nLabour\n\nPREMATURE. SLOW. LABOUR PAINS stopping or slowing down; weak. With EXHAUSTION.\n\nDuring labour women become depressed, withdrawn and want to be alone \u2013 they don't ask for anything and become difficult to reach.\n\nPiles\n\nIn PREGNANCY. BLEEDING. With CONSTIPATION (see here).\n\nPrickly heat\n\nWith typical general symptoms.\n\nRetention of urine\n\nCause presence of strangers.\n\nCan only pass urine when alone.\n\nSore throat\n\nPAINS burning. THROAT dry; with LUMP SENSATION. VOICE HOARSE.\n\nCan only swallow liquids; solids come back up, and person may choke (also when drinking). Hawks up egg-white mucus from back of throat.\n\nThrush (genital)\/Yeast infection\n\nDISCHARGE burning; like egg white. With ITCHING.\n\nThrush (oral)\n\nGUMS\/TONGUE white-coated.\n\nWith characteristic dry mouth and thirst.\n\nNATRUM PHOSPHORICUM (Nat-p.)\n\nOther name: sodium phosphate\n\nGeneral symptoms\n\nDischarge sour. Exhaustion, nervous. Tongue yellow at back.\n\nWorse during the evening.\n\nThis picture is full of sourness: sour sweat, stools, urine and so on. Yellow-coated tongue (usually base only) accompanies nearly all symptoms. General weakness, acidic digestion, weak ankles and cold extremities (like Natrum carbonicum).\n\nEmotional state\n\nApathetic. Indifferent.\n\nThese types are apathetic and dull, like Phosphoric acid.\n\nPhysical complaints\n\nColic\n\nVOMIT sour; acid; curdled milk. DIARRHOEA.\n\nFor simple colic with acidity (and sourness) when there are no other guiding symptoms leading you to another remedy. Babies may vomit curdled milk and suffer from sour-smelling, green, watery diarrhoea.\n\nHeadache\n\nFACE pale. PAINS in the forehead.\n\nCause mental exertion.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated. BELCHES sour. With HEARTBURN.\n\nAcidity comes on two hours after eating. Heartburn of pregnancy may be relieved if the characteristic sourness is present.\n\nNATRUM SULPHURICUM (Nat-s.)\n\nOther name: sodium sulphate\n\nGeneral symptoms\n\nComplaints from accident\/injury to the head. Hot flushes. Likes cold drinks. Sweat from pain. Taste in mouth bad (in morning); bitter. Tongue green.\n\nBetter for fresh air.\n\nWorse for damp; for humidity; for wet weather; for lying down; during the morning; after eating starchy food.\n\nThese types are sensitive to damp: cold and wet, or hot and wet. Complaints may result from living in a damp house. This remedy is most often needed for complaints arising from head injury.\n\nEmotional state\n\nDepressed. Jumpy. Weary of life.\n\nAfter a head injury they become sick and tired of life (having been quite happy before) and over-sensitive to noise.\n\nPhysical complaints\n\nBackache\n\nPAIN sore, bruised.\n\nCause injury to spine.\n\nColic\n\nWith INDIGESTION (see here).\n\nCough\n\nDAYTIME only. MUCUS green.\n\nWorse for damp, wet weather.\n\nDiarrhoea\n\nABDOMEN\/STOMACH gurgling; rumbling. STOOLS smelly; thin; watery. WIND loud; smelly; spluttering.\n\nBetter for passing wind.\n\nWorse after getting up; during morning; after eating fruit\/starchy food.\n\nCause fruit.\n\nFeels exhausted after the diarrhoea; or may want to pass a stool but manages only to pass wind. May also be sensitive to starchy foods (potatoes or bread) or vegetables.\n\nFlatulence\n\nLoud; loud during stool; smelly. With DIARRHOEA. ABDOMEN\/STOMACH rumbling.\n\nBetter for passing wind.\n\nWorse after breakfast.\n\nAn urge for a stool but only sputtering wind is passed.\n\nHeadache\n\nEYES sensitive to light.\n\nCause head injury.\n\nGive after Arnica if pains remain after a head injury.\n\nHead injuries\n\nSee HEADACHE.\n\nIf Arnica has been given for a head injury, and the swelling has subsided but pain remains, or there are other symptoms after a bang or fall, Natrum sulphuricum will clear these symptoms. Sometimes babies change character after a bad fall, from being cheerful and easy-going to distressed \u2013 don't mistake this for teething and give Chamomilla.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated. BELCHES bitter; sour. With FLATULENCE after breakfast.\n\nBetter after passing a stool.\n\nWorse after eating starchy food.\n\nCannot digest starchy food, such as potatoes, and the tongue may look greenish brown or dirty green.\n\nJaundice\n\nIn newborn babies.\n\nIf Chelidonium doesn't help, choose between Natrum sulphuricum and Lycopodium. Babies who had a long and difficult birth, especially with forceps, may need Natrum sulphuricum to help heal minor head injury from birth.\n\nNITRICUM ACIDUM (Nit-ac.)\n\nOther name: nitric acid\n\nGeneral symptoms\n\nAnaemia. Breath smelly. Discharges like ammonia. Dislikes cheese. Likes fat\/fatty foods. Pains in bones; flying around; needle-like; splinter-like; appear and disappear suddenly. Sweat on single parts of body; smelly; sour; worse for slightest physical exertion. Tongue cracked.\n\nBetter for lying down.\n\nWorse for cold; for fresh air; at night; on waking; for loss of sleep; for touch; for jarring movement; for walking.\n\nThese are chilly types \u2013 worse for cold in any form. They are very sensitive to being touched or jarred. Their urine smells strong (like a horse's); sweat is also foul-smelling. They crave fatty, fried foods and fat on meat. They are always better for lying down.\n\nEmotional state\n\nViolently angry. Anxious about health. Depressed. Fearful of death. Irritable. Sensitive to noise; to pain. Sluggish. Tantrums. Unforgiving.\n\nThese types become anxious and depressed when ill \u2013 anxious about their health; believing they may have something serious wrong with them.\n\nThey may become sick because of harboured anger and their unforgiving streak which eats away like an acid. Their anger can explode volcanically. They overwork and become worn out and over-sensitive, especially to noise.\n\nPhysical complaints\n\nCommon cold\n\nNASAL CATARRH bloody; burning; dirty yellow; thin; watery in the fresh air.\n\nWorse for fresh air; at night.\n\nNose is blocked, feels sore and bruised and sensitive to touch. There may be ulcers in the nostrils.\n\nEarache\n\nCRACKLING IN EARS when chewing. PAIN splinterlike; throbbing. With symptoms of SORE THROAT (see here).\n\nWorse on right side, for swallowing.\n\nPains are usually splinter-like and hearing becomes acute \u2013 more sensitive than normal.\n\nExhaustion\n\nNERVOUS.\n\nWorse after passing stool; for walking.\n\nCause diarrhoea; loss of sleep (broken nights); nursing the sick.\n\nWeakness results from loss of sleep or accompanies or follows an attack of diarrhoea.\n\nEye inflammation\n\nIn BABIES. EYES watering. EYELIDS swollen.\n\nFlatulence\n\nSmelly; obstructed (difficult to expel).\n\nHair loss\n\nCause childbirth.\n\nHeadache\n\nHEAD feels constricted. PAIN spreading to eyes; in bones; recurring at regular intervals; pressing.\n\nWorse for jarring movement; for movement; at night; for noise; for pressure; on waking; for walking.\n\nSevere headache that is worse at night. Bones of face are sore, and the slightest pressure, even that of a hat, makes it worse. Pressing pains (as if the head were bandaged tightly) may come and go suddenly.\n\nInsomnia\n\nDREAMS anxious. SLEEP unrefreshing.\n\nWorse after 2 a.m.\n\nWakes around 2 a.m. and is unable to get back to sleep.\n\nJoint pain\n\nPAIN in bones; in legs; splinter-like; stitching; tearing.\n\nWorse at night.\n\nMouth ulcers\n\nOn edges of tongue; painful.\n\nThere is more saliva and the gums may bleed more easily. Breath is smelly.\n\nPiles\n\nPAIN burning; splinter-like. PILES bleeding; large.\n\nWorse after a stool.\n\nPains may last for 1\u20132 hours, even after a soft stool.\n\nSore throat\n\nPAIN burning; pressing; raw; splinter-like; stitching; spreading up to ears. TONSILS swollen\/inflamed; ulcerated.\n\nWorse for swallowing.\n\nPains are severe and extend to ear on swallowing. Swallowing is extremely difficult.\n\nThrush (genital)\/Yeast infection\n\nDISCHARGE burning; smelly; thin. With ITCHING.\n\nNUX MOSCHATA (Nux-m.)\n\nOther name: nutmeg\n\nGeneral symptoms\n\nThirstless.\n\nBetter for heat.\n\nWorse for cold; for fresh air.\n\nEmotional state\n\nApathetic. Dreamy. Sluggish. Stupor.\n\nPhysical complaints\n\nSleepy babies\n\nDon't wake to feed, especially after the birth. This is for newborns who are difficult to wake up \u2013 even for a feed.\n\nNUX VOMICA (Nux-v.)\n\nOther name: poison nut\n\nGeneral symptoms\n\nCatches colds easily. Complaints from getting chilled; cold, dry wind. Dislikes coffee; food in general, with hunger; meat; tobacco; water. Likes spicy food. Pains cramping. Sense of smell acute. Sweat hot; one-sided; smelly. Symptoms right-sided. Taste in mouth bitter; sour in morning.\n\nBetter for heat; for lying down; for sitting down; for hot drinks.\n\nWorse after drinking coffee; after eating cold food; for tobacco; for cold, dry weather, cold wind, fresh air; for loss of sleep; during morning; in winter; for touch; for uncovering; for walking.\n\nThese are extremely chilly individuals who hate the cold, catch cold easily, especially if exposed to draughts or wind or after becoming chilled. If sick they become so sensitive to cold that the slightest draught upsets them; they want only to sit or lie down and keep warm, and this is what helps.\n\nNux vomica is useful for disturbances that follow over-indulgence in food, coffee or tobacco. Once sick, any further indulgence makes them worse, but they find it hard to stop.\n\nMornings are their worst time of day, especially after a disturbed night.\n\nEmotional state\n\nAngry violently; when has to answer questions. Anxious about others. Aversion to consolation. Complaints from anger; anger with anxiety. Concentration poor. Dwells on the past. Excitable. Hitting. Impatient. Impulsive. Irritable. Mischievous. Quarrelsome. Screaming. Sensitive generally; to rudeness; to light; to music; to noise. Spiteful. Stubborn. Tidy.\n\nSelf-indulgent workaholics who overdo everything: work too hard, stay out too late, take no exercise, eat too much rich food, drink too much alcohol, become too wound up to sleep and then consume vast quantities of coffee to get going the next day. Then they find it difficult to concentrate, lose interest in their work, become nervy, highly sensitive and irritable.\n\nImpulsive, quarrelsome, stroppy individuals who know what they want and want it now. They are critical, fussy and can be exacting, easily frustrated by limitations. They may be tidy but not fanatically so. They dislike contradiction, are irritable if questioned and if anger is suppressed they feel awful and get sick. Babies become irritable if chilled or tired or if they eat too much of the wrong sort of food.\n\nPhysical complaints\n\nAbdominal pain\n\nIn pregnancy. With NAUSEA (morning sickness).\n\nAfterpains\n\nWith DESIRE to pass a stool during the pains. Feels FAINT after the pains have passed.\n\nBackache\n\nIn LOWER BACK. PAIN aching; dragging down; pressing; sore, bruised. In PREGNANCY.\n\nWorse in bed; during morning; for movement.\n\nCause getting cold; childbirth.\n\nHas to sit up to turn over in bed. Pain comes on after being chilled, and is often accompanied by the desire to pass a stool.\n\nColic\n\nPAIN cramping; griping; pressing; sore, bruised.\n\nBetter for passing a stool; for hot drinks; for warmth of bed; for passing wind.\n\nWorse after eating; for coughing; in morning; during a fever; for tight clothing.\n\nPains come on after over-indulging. May accompany nausea or indigestion (see here). Also in babies whose breastfeeding mothers eat too much spicy food or drink too much coffee and tea (or Coca-Cola).\n\nCommon cold\n\nEYES watering. NASAL CATARRH burning; watery. NOSE runs during day, blocked at night. With HEADACHE (see here); frequent SNEEZING; SORE THROAT (see here). With blocked, painful SINUSES.\n\nBetter for fresh air.\n\nWorse after eating; after getting up; during morning.\n\nCause draughts.\n\nMay start with an irritated spot high up at the back of the nose\/nostril. Breastfeeding babies whose blocked noses make feeding difficult will benefit, particularly if also sensitive to what their mothers eat.\n\nConstipation\n\nAlternates with DIARRHOEA. DESIRE TO PASS STOOL ineffectual; constant. In BABIES. In PREGNANCY. STOOLS hard; large. With UNFINISHED FEELING.\n\nCause bottle-feeding; over-eating; pregnancy; sedentary habits; weaning.\n\nWants to pass a stool but cannot, or passes only small amounts each time.\n\nCough\n\nBREATHING difficult. COUGH distressing; dry; racking; suffocative; tickling; in violent fits. LARYNX raw. MUCUS tastes sour. With VOMITING when hawking up mucus.\n\nBetter for hot drinks.\n\nWorse for becoming cold; for cold air; during early morning; on waking in the morning; for slightest movement of chest; after eating; during fever.\n\nCough is dry during the fever; it is also dry, and worse, after midnight until daybreak. May cough up sour-tasting mucus; tickling in the larynx excites the cough.\n\nCystitis\n\nDESIRE TO URINATE frequent; ineffectual; urgent. PAIN burning; pressing. URINATION frequent; ineffectual. URINE burning; scanty.\n\nWorse during urination.\n\nConstant urge to urinate, and slight incontinence; then wants to pee but can't.\n\nDizziness\n\nIn PREGNANCY. With HEADACHE.\n\nWorse for getting up from lying down or sitting; loss of sleep; stooping; tobacco; walking.\n\nEarache\n\nPAIN stitching. With ITCHING IN THE EAR.\n\nWorse for swallowing.\n\nEustachian tubes itch making swallowing compulsive although it hurts. Hearing is acute.\n\nEffects of drugs taken during or after labour\n\nWith IRRITABILITY and INSOMNIA (see here).\n\nHelps to rid the body of drugs taken in labour (as a general clear-out).\n\nExhaustion\n\nNERVOUS.\n\nWorse on waking in the morning.\n\nCause loss of sleep (broken nights).\n\nFever\n\nHEAT alternating with chills; dry; one-sided. With BACKACHE; extreme CHILLINESS; SHIVERING; SWEATING (see General symptoms).\n\nWorse for fresh air; for draughts; for movement; for being uncovered; for slightest movement of bedclothes.\n\nBecomes chilled if a mere hand is poked out of bed. Fever may be on one side only; feels dry and hot externally and chilly internally. Limbs feel heavy and tired. Is usually thirsty.\n\nFlatulence\n\nABDOMEN\/STOMACH feels bloated; intolerant of tight clothing; rumbling.\n\nBetter for passing wind.\n\nWorse for eating.\n\nFlu\n\nPAIN in bones; in joints; sore, bruised. With symptoms of COMMON COLD (see here); EXTREME CHILLINESS; FEVER (see here).\n\nBetter for warm compresses; for warmth of bed.\n\nWorse in bed; for cold; during the morning.\n\nSimilar to the common cold; feels extremely chilly and unable to get warm.\n\nGastric flu\n\nSee Flu.\n\nWith a disordered digestive system (see INDIGESTION).\n\nHeadache\n\nHEAD feels heavy. PAIN in back of head; in forehead; pressing; sore, bruised; stupefying; tearing. With BILIOUSNESS; DIZZINESS.\n\nBetter for excitement; on getting up in morning; for pressure; for wrapping up head; during evening in bed.\n\nWorse after eating; for cold; for moving eyes; for shaking head.\n\nCause cold wind; common cold; damp weather; loss of sleep; mental strain; over-eating.\n\nEyes often feel sore.\n\nHeartburn\n\nIn PREGNANCY. With INDIGESTION (see here). With sour BELCHES.\n\nHernia (in babies)\n\nINGUINAL. Left side.\n\nCause constipation.\n\nHiccups\n\nIn babies. VIOLENT.\n\nWorse after eating or drinking.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; empty. BELCHES bitter; sour. PAIN in stomach; cramping; pressing; sore, bruised. With HEARTBURN. See also COLIC.\n\nBetter for hot drinks; for warmth of bed.\n\nWorse after eating; for tight clothing.\n\nCause coffee; mental strain; over-eating; rich food.\n\nFood lies like a knot in the stomach for 1\u20132 hours after eating, and stomach feels full, heavy and tender.\n\nInsomnia\n\nIn PREGNANCY. With DREAMS, VIVID. WAKING early; late; around 3 a.m.\n\nCause cramps (in pregnancy); excitement; mental strain; overwork; overactive mind.\n\nSleepy on going to bed but cannot sleep, or gets to sleep but wakes in the early hours with anxious thoughts. May fall into deep sleep just before the alarm goes off, and so wakes tired and worn out.\n\nLabour\n\nPREMATURE. LABOUR PAINS in back; ineffectual (cervix doesn't soften); severe\/violent; stopping (or slowing down). With CRAMPS in hands or legs. With EXHAUSTION; FAINT FEELING. With an URGE TO PASS A STOOL during contractions.\n\nWith the typical Nux vomica emotional state.\n\nNausea\n\nCONSTANT. With COPIOUS SALIVA; INABILITY TO VOMIT; RETCHING; VOMITING.\n\nWorse after eating; in morning; in bed; for sweating; for tobacco.\n\nCause pregnancy; travelling.\n\nMay feel faint with the nausea.\n\nPiles\n\nIn PREGNANCY. BLEEDING; INTERNAL; ITCHING; LARGE. With CONSTIPATION (see here).\n\nBetter for bathing in cold water.\n\nSnuffles\n\nOf newborn babies. With symptoms of COMMON COLD (see here).\n\nSore throat\n\nPAIN spreading up to ears; raw; stitching. With LUMP SENSATION in throat.\n\nWorse for cold air; for swallowing; for uncovering.\n\nSwallowing is difficult; the throat feels rough and pains radiate to the ears on swallowing. The rawness may have been caused by coughing.\n\nToothache\n\nBetter for external warmth; for wrapping up head; for heat; in winter.\n\nCause filling\/extraction.\n\nLying on a hot-water bottle wrapped in a towel helps.\n\nTravel sickness\n\nFAINT-LIKE FEELING; HEADACHE; NAUSEA.\n\nBetter for lying down.\n\nWorse for tobacco.\n\nVomiting\n\nVOMIT bile; mucus; bitter; sour; smelly.\n\nWorse for expectorating (hawking up mucus).\n\nCause anger; pregnancy.\n\nBiliousness (with or without vomiting) occurs after over-eating or indulging in rich or unusual foods. Can feel like food poisoning; can occur when travelling.\n\nOPIUM (Op.)\n\nOther name: white poppy\n\nGeneral symptoms\n\nEyes glassy. Face dark red or blue. Faintness after excitement; after fright. Pains absent. Pupils contracted. Shock from injury. Sweat profuse; hot; on single parts of body; from fright.\n\nWorse during sleep; on getting up; for warmth of bed.\n\nThis remedy is for a particular type of shock: those needing Opium appear glassy-eyed and stupefied, suffer from inertia, and feel faint, numb and\/or trembly. There is an abnormal lack of pain, as if the senses have been blunted. The face is red and drawn, sunken and old-looking, possibly with a bluish tinge. Children are unable to urinate after shock.\n\nIn a fever Opium types feel worse for heat and sweating and will kick off the covers in bed. Generally, profuse sweating with scanty urination.\n\nEmotional state\n\nApathetic and doesn't complain; during fever; during labour. Complaints from anger; reprimands; shock. Confused. Depressed. Dreamy. Drowsy. Exhilarated. Expression sleepy. Fearful during labour. Indecisive. Indifferent during a fever. Lively. Senses dull. Stupor.\n\nIn acute illness like fever or after a shock, Opium types become dull, confused and apathetic to the point of stupor. Before reaching this state of stupor, however, they may go through an exhilarated and delirious stage.\n\nPeople in a state of fear after a bad fright or shock or an operation may look similar to Aconite (that is, enormously shocked) but are also in a dream-like state they cannot snap out of. Opium helps that 'spaced out' feeling after a general anaesthetic. The Opium individual recreates during waking hours the image or situation that caused the shock. Arnica types, on the other hand, are fine during the day but have bad dreams.\n\nPhysical complaints\n\nBleeding (vaginal) in pregnancy\n\nAt any time in pregnancy.\n\nCause fright.\n\nConstipation\n\nIn BABIES. DESIRE TO PASS STOOL absent. STOOLS black balls; hard; small balls; 'shy' (they recede). With UNFINISHED FEELING.\n\nCause bottle-feeding; weaning.\n\nThere is no desire to pass a stool; no pain either. May be acute after an operation; may alternate with diarrhoea after sudden joy or a fright. Babies may react badly to bottled milk or prepared baby foods and become completely bunged up, going for weeks without passing a stool.\n\nCough\n\nBREATHING difficult; slow; snoring.\n\nWorse during sleep; on falling asleep; on waking.\n\nBreathing slows down during sleep and when the fever is strong; it becomes irregular and laboured, like a waking snoring.\n\nEffects of drugs taken during or after labour\n\nDROWSY, SLEEPY, SPACED OUT.\n\nCause general anaesthetic; morphine or pethidine.\n\nMay feel as though on another planet and can't quite wake up.\n\nFever\n\nHEAT burning; intense. With DEEP SLEEP; SWEATING (see General symptoms).\n\nBetter for uncovering.\n\nWorse during sleep; for sweating.\n\nOverpowering sleepiness and yawning; falls asleep and then breathes slowly and loudly as if snoring. Heat comes on during sleep; sweating does not relieve the symptoms. If woken, wants to uncover, and is either delirious and excitable or stupefied and semi-conscious.\n\nInsomnia\n\nWith SLEEPINESS.\n\nSleeps lightly and restlessly and hears every sound. Sleepy but cannot fall asleep; bed may feel hot.\n\nLabour\n\nPREMATURE caused by fright\/shock. LABOUR PAINS stopping or slowing down; weak. With EXHAUSTION. With red face.\n\nApathetic and fearful in labour.\n\nRetention of urine\n\nCause childbirth; shock.\n\nAfter a shock or fright is unable to pass urine.\n\nPETROLEUM (Petr.)\n\nOther name: coal oil\n\nGeneral symptoms\n\nDislikes meat; fatty, rich food. Sweat one-sided; smelly.\n\nWorse for fresh air; during morning; travelling (boat, car, train, bus, etc.).\n\nThere is a general dislike of fresh air which also aggravates the nausea of travel sickness or pregnancy.\n\nEmotional state\n\nConfused in fresh air. Forgetful. Irritable. Quarrelsome.\n\nThese are irritable types who feel confused outside (partly because it aggravates their physical symptoms).\n\nPhysical complaints\n\nDiarrhoea\n\nDAYTIME ONLY. PAIN pressing. With ravenous APPETITE.\n\nThe diarrhoea occurs only during daytime and is not accompanied by loss of appetite.\n\nDizziness\n\nWith NAUSEA (see here).\n\nWorse for getting up from lying down\/sitting.\n\nHeadache\n\nPAIN in back of head; pressing. With HEAVINESS; NAUSEA (see here).\n\nUsually comes on when travelling and accompanies nausea.\n\nNappy rash (Diaper rash)\n\nSkin red; itches; cracked \u2013 dry or weepy; bleeding.\n\nWorse in the folds of the skin.\n\nNausea\n\nWith DIZZINESS (see here).\n\nWorse for fresh air.\n\nUsually accompanied by dizziness and\/or headache.\n\nThrush (genital)\/Yeast infection\n\nDISCHARGE like egg white; burning; copious.\n\nTravel sickness\n\nWith HEADACHE (see here); NAUSEA (see here); VOMITING.\n\nWorse for fresh air.\n\nOne of the main travel-sickness remedies and is indicated where fresh air aggravates. Opposite of Tabacum.\n\nPHOSPHORIC ACID (Pho-ac.)\n\nGeneral symptoms\n\nAnaemia. Complaints from loss of body fluids. Eyes glassy. Face pale. Gums bleeding. Likes fruit; refreshing things. Pains in bones. Sweat clammy; profuse. Symptoms one-sided. Thirstless.\n\nBetter after a good sleep.\n\nWorse for cold; for sweating; during evening; during morning; on one side of body.\n\nThis remedy benefits those who are weak and tired from studying too much or from losing body fluids, for example after diarrhoea, a heavy period, bleeding or vomiting. It is also for those who are convalescing from an acute illness. They look pale and sickly; with dark rings around their eyes and sweat a lot. They want refreshing food to eat, such as fruit and vegetables (to replace the liquid), but may not be thirsty and often feel tired after eating. They are generally worse for cold and better for a sleep, even a short nap.\n\nEmotional state\n\nAnxious about others. Apathetic. Brooding. Complaints from disappointed love; excitement; fright; grief; humiliation; shock. Depressed. Disappointment. Dwells on past. Forgetful. Homesick. Indifferent during fever. Irritable. Uncommunicative.\n\nEmotional trauma such as disappointment in love, shock or homesickness results in mental apathy. Phosphoric-acid types beome depressed, they do not want to talk, think or answer and may even forget words while they are speaking. They appear indifferent to everything and lack even the energy to cry.\n\nPhysical complaints\n\nCough\n\nDry; tickling; violent.\n\nDiarrhoea\n\nABDOMEN\/STOMACH rumbling. STOOLS profuse; thin; watery; white. Without WEAKNESS.\n\nWorse after eating solid\/dry food.\n\nCause summer; hot weather.\n\nMay pass an involuntary stool at the same time as passing wind. There may be gurgling and cramping, gripping pains, and a bloated, full feeling in the abdomen, although it is usually painless. Exhaustion does not accompany the diarrhoea but sets in once it has stopped.\n\nExhaustion\n\nNERVOUS; PARALYTIC.\n\nWorse after eating; in morning on getting up; for slightest exertion; for walking.\n\nCause breastfeeding; loss of body fluids (diarrhoea, vomiting, etc.); flu.\n\nHair loss\n\nCause grief.\n\nHeadache\n\nEYES smarting. HEAD feels heavy. PAIN top of head; back of head; nape of neck; one-sided; pressing.\n\nBetter for excitement.\n\nWorse for getting up from lying down.\n\nCause eye strain; grief; mental exertion\/strain, especially from studying.\n\nInsomnia\n\nSLEEP unrefreshing. WAKING frequent; SLEEPLESS after midnight.\n\nShock\n\nCause emotional trauma.\n\nFor shock caused by emotional trauma and accompanied by the typical emotional state.\n\nPHOSPHORUS (Phos.)\n\nOther name: white phosphorus\n\nGeneral symptoms\n\nAnaemia. Bleeding occurs easily; bright red; profuse. Complaints from change of weather; getting chilled. Discharges blood-streaked. Dislikes fruit; hot food; water during pregnancy. Face red in spots. Haemorrhages. Hair loss. Heavy feeling. Hot flushes. Likes cold drinks; cold food; spicy food; ice-cream; milk; chocolate; salt. Nosebleeds. Pains in glands; burning. Sweat on single parts of body; worse for slightest physical exertion; clammy. Taste in mouth sour. Thirst unquenchable for ice-cold drinks; for water. Tongue red-coated.\n\nBetter after a good sleep; for cold drinks; for massage.\n\nWorse for cold; before eating; for change of weather; during evening; during morning; for windy weather.\n\nThese types are usually tall and slim. They burn up food quickly and therefore need to eat often, are also thirsty and want cold or iced drinks (especially cold milk). Chilly individuals who are sensitive to changes in the weather, feel worse for getting cold, and hate wind. They bleed easily and copiously; suffering from heavy periods and nosebleeds of bright-red blood which is slow to coagulate. They are prone to anaemia and hair loss because of blood loss. Pains are usually burning pains, wherever they are. Mornings and evenings between dusk and midnight are their worst times of day.\n\nEmotional state\n\nAffectionate. Anxious about health. Apathetic indifferent to children or relatives. Clingy. Desires company. Euphoric. Excitable. Fearful generally; of being alone; of the dark; of death; during a thunderstorm. Irritable. Jumpy. Sensitive generally; infants to pain; to light. Slow. Sympathetic. Uncommunicative.\n\nWhen healthy they are lively, affectionate and excitable \u2013 although may also have classic Phosphorus fears: they hate to be alone, especially in the dark at night when their vivid imaginations go to work; scared of thunderstorms. They are gregarious types, who feel things strongly, are sensitive, highly strung and easily startled.\n\nWhen sick they become irritable, mentally sluggish (like Phosphoric acid), apathetic, and don't want to think, talk or work. Illness debilitates them; their energy may flare up in bursts, but is followed by a return to their exhausted state. They need sympathy when ill and, despite being fearful and irritable if tired or upset, they are easily comforted and reassured. They love massage, touch and affection.\n\nPhysical complaints\n\nBackache\n\nBETWEEN SHOULDER BLADES. In LOWER BACK; back feels broken; burning.\n\nBetter for rubbing\/massage.\n\nWorse on getting up from sitting.\n\nCause childbirth.\n\nBleeding\n\nAfter childbirth. FLOW OF BLOOD is bright red.\n\nCommon cold\n\nNASAL CATARRH blood-streaked; profuse; one-sided; dry. With SENSE OF TASTE\/SMELL lost. HOARSENESS; SORE THROAT (see here).\n\nNose runs profusely or is blocked and dry with no discharge.\n\nCough\n\nBREATHING difficult; fast. COUGH dry at night; during fever; hacking; irritating; racking; tickling; tight; violent; from tickling in larynx; wakes person at night; must sit up at night. LARYNX raw. MUCUS tastes sweet; salty; transparent; white; yellow; green; copious; bloody. PAIN IN CHEST burning. With SWEATING. In PREGNANCY.\n\nBetter for heat.\n\nWorse for change of temperature; for cold air; for fresh air; during fever; for lying on left side; during the morning after getting up; during the evening until midnight; reading aloud.\n\nChest feels weighted down and burning pains are worse for coughing. Air passages feel irritated and worse for cold air. Coughs up copious quantities of sputum, especially in the morning.\n\nCroup\n\nSee COUGH.\n\nDiarrhoea\n\nSTOOLS blood-streaked; frequent; PAINLESS; profuse; watery. With ICY-COLD HANDS AND FEET. In PREGNANCY.\n\nBetter after cold food.\n\nWorse during morning; after getting chilled.\n\nAbdomen may be bloated and gurgling, and stools are blood-streaked. Cold drinks may help.\n\nDizziness\n\nHAS TO LIE DOWN.\n\nWorse for getting up from lying down\/sitting.\n\nEffects of drugs taken during or after labour\n\nDROWSY, SLEEPY, SPACED OUT. With VOMITING (see here).\n\nCause general anaesthetic.\n\nExhaustion\n\nNERVOUS. PARALYTIC. With FEVER (see here).\n\nWorse for slightest exertion; for walking.\n\nCause breastfeeding; diarrhoea; fever; sweating.\n\nSo tired during acute illness that cannot stay upright in bed but continually slides down.\n\nFever\n\nIncreased APPETITE. HEAT dry at night; burning; ONE-SIDED; worse on right side.\n\nWorse in afternoon\/evening\/night.\n\nHas an unquenchable thirst for cold drinks and a good appetite. May not appear as ill as fever indicates they might.\n\nGums bleeding\n\nCause tooth extraction.\n\nHair loss\n\nFalls out IN HANDFULS.\n\nCause acute illness.\n\nHeadache\n\nHUNGER before\/during headache. PAIN in forehead; sides of head; burning; bursting; pressing; throbbing.\n\nBetter after sleep; for cold; for cold compresses; for fresh air; on getting up; for massage; for walking.\n\nWorse for getting cold; for coughing; for daylight; for hot drinks; in stuffy room; for wrapping up head.\n\nCause imminent thunderstorm.\n\nNervous headache, the head feels heavy and the sense of smell is often acute, the eyes water in cold air.\n\nIndigestion\n\nPAIN in stomach burning; sore, bruised.\n\nWorse after eating.\n\nInjuries\n\nCUTS\/WOUNDS bleed freely; blood bright red; slow to clot.\n\nInsomnia\n\nAnxious DREAMS. With SLEEPINESS.\n\nSleeps on the right side cannot fall asleep when lying on the left side, especially with a chest infection or cough. Has anxious, vivid dreams.\n\nNausea\n\nABDOMEN\/STOMACH feels empty. BELCHES sour.\n\nWorse for putting hands in warm water; for hot drinks; for drinking water.\n\nThe empty feeling is not relieved by eating.\n\nNosebleed\n\nBlood bright red; persistent. With SWEATING.\n\nWorse for blowing nose.\n\nShock\n\nWith VOMITING (see here).\n\nCause surgery.\n\nSore throat\n\nPAIN raw; sore. TONSILS swollen. VOICE hoarse.\n\nWorse for breathing in; for coughing; during morning\/evening; for pressure; for talking.\n\nSore throats often accompany a cold. Can hardly talk as the throat is so painfully hoarse.\n\nVomiting\n\nBurning PAIN IN STOMACH. VOMIT bile; mucus; bitter; yellow; violent.\n\nWorse after drinking\/eating.\n\nVomits food and drink \u2013 even the smallest quantity \u2013 as soon as it becomes warm in the stomach (after a little while). May be burning pains in the stomach or a bruised soreness.\n\nPHYTOLACCA DECANDRA (Phyt.)\n\nOther name: poke-root\n\nGeneral symptoms\n\nBreath smelly. Pains shoot upwards. Tongue red-tipped.\n\nThese types feel exhausted, stiff and worn out. Smelly breath and a red-tipped tongue accompany many of the complaints.\n\nPhysical complaints\n\nBreastfeeding problems\n\nBREASTS inflamed; lumpy. ABSCESSES. NIPPLES cracked\/sore. PAIN while nursing.\n\nThis remedy is a specific for sore, cracked nipples where pain radiates from the nipple all over the body. Also useful for infections and abscesses where the breast is hard, nodular and lumpy; or if a lump in the breast becomes painful and an abscess threatens. It can take 12\u201324 hours to act (unlike the quick action of Aconite or Belladonna).\n\nMumps\n\nGLANDS hard; painful; swollen. PAINS spread to breasts, ovaries. With COPIOUS SALIVA; PROFUSE SWEATING; SORE THROAT (see here).\n\nSore throat\n\nTHROAT dark red. TONSILS swollen.\n\nBetter for cold drinks.\n\nWorse for hot drinks.\n\nSwallowing is difficult and causes pains to shoot through both ears (in Belladonna sore throats the pains shoot up only to the right ear). The throat may also hurt on sticking out the tongue.\n\nTeething\n\nPAINFUL IN CHILDREN. With CRYING.\n\nBetter for biting gums together hard.\n\nTeething babies will bite their teeth (or gums) together \u2013 on anything and everything!\n\nPODOPHYLLUM (Podo.)\n\nOther name: mandrake\n\nPhysical complaints\n\nDiarrhoea\n\nABDOMEN\/STOMACH gurgling; rumbling. STOOLS frequent; gushing; involuntary; painless; profuse; smelly; sudden. With EXHAUSTION after passing stool.\n\nWorse after drinking water; immediately after drinking; after eating; for hot weather; mid-morning; around 4 a.m.; at night.\n\nDiarrhoea may alternate with other symptoms such as a headache or even constipation. Dull aches or cramping pains may accompany much gurgling before a stool; the stools are usually painless and shoot out. They may be frothy, of a changeable colour and consistency, and are usually accompanied by loud wind. Feels drained, faint and weak after a stool. Teething babies sometimes produce this type of diarrhoea.\n\nFlatulence\n\nABDOMEN\/STOMACH gurgling before stool. WIND loud during stool.\n\nPULSATILLA NIGRICANS (Puls.)\n\nOther name: meadow anemone\n\nGeneral symptoms\n\nAnaemia. Breath smelly. Complaints from\/after measles; getting wet; getting feet wet; weaning. Discharges thick; yellow; bland. Dislikes bread; butter; food in general; fruit; tobacco; hot food\/drinks; fatty, rich food; meat. Faintness in a warm room. Glands swollen. Lips dry. Mouth dry. Oedema (swelling) of ankles\/feet; hands\/fingers. Pains on parts lain on; wandering. Sweat smelly; single parts of body; one-sided; worse at night. Symptoms right-sided; changeable. Taste in mouth bad in morning. Thirstless. Tongue coated white\/yellow.\n\nBetter for bathing; for fresh air; for movement; for pressure; for walking in fresh air; for crying.\n\nWorse for getting cold; for exposure to sun; for rich fatty food; for getting feet wet; for heat; in stuffy rooms; at twilight; for wet weather; for wind.\n\nThese types have dry mouths and lips and an absence of thirst. They may be chilly, with cold hands and feet, but dislike heat and stuffy rooms, becoming quickly flushed. Always feel better in the fresh air, where their moods lift and their symptoms (especially coughs) disappear, returning only back in the warm. But they are sensitive to getting wet or being exposed to wet, windy weather, when they may start a cough or cold.\n\nThey hate rich, fatty food, which gives them indigestion and nausea. They may, however, have a craving for butter. Chests and joints are worse for rest and better for moving about, especially in fresh air; twilight is their worst time.\n\nAlso useful in pregnancy where anaemia in Pulsatilla type may result from an excess of ordinary iron pills, so stopping iron is essential (see Ferrum metallicum). Their physical symptoms may be hard to pin down, changing from one moment to the next, rather like their moods.\n\nEmotional state\n\nAffectionate. Angry (morose) babies. Anxious at night; indoors. Capricious. Changeable. Clingy. Complaints from fright; grief; shock. Depressed worse before menstrual period, during pregnancy, in stuffy room, in evening, from suppressed grief; better for fresh air. Desires to be carried (infants). Disappointment. Excitable. Fearful at twilight; in evening. Gentle. Introspective. Irritable. Jealous. Lonely. Moody. Sensitive babies. Shy. Sluggish\/dull. Tearful during a fever; while breastfeeding, in pregnancy; better for fresh air. Whiny.\n\nThese types are gentle, yielding, mild; easily moved to laughter or tears; highly emotional and cry easily. They are clingy and dependent, particularly when sick, when they also become irritable and whiny and feel hard done by. Especially useful for babies as so many of them go through a Pulsatilla phase \u2013 weepy, clingy, whiny and wanting to be carried.\n\nPulsatillas are affectionate creatures who love animals and cannot bear to see them hurt. They too are easily hurt but may suppress it and become introspective, moody, even irritable. Depression (even post-natal) will be better for fresh air, and worse during the evening and in warm, stuffy rooms. A breastfeeding mother may weep while nursing. Generally, these types moan and weep during fever, and cry when talking about their illness. They crave sympathy which makes them feel better.\n\nPhysical complaints\n\nAfterpains\n\nWith classic Pulsatilla general and emotional\/mental symptoms: thirstless and weepy.\n\nBackache\n\nIn LOWER BACK; in SMALL OF BACK. PAIN aching; dragging down; pressing. In pregnancy.\n\nBetter for gentle exercise; for walking slowly.\n\nWorse on beginning to move; on getting up from sitting; before menstrual period.\n\nBack feels weak and tired. Getting up after a long period of sitting or bending down is almost impossible. Back may feel sprained.\n\nBleeding (vaginal) in pregnancy\n\nFLOW OF BLOOD stops and starts; changeable; with clots.\n\nBreast (feeding) problems\n\nMILK SUPPLY overabundant. PAIN IN BREAST when baby nurses.\n\nBreathless\n\nIn PREGNANCY.\n\nBetter for fresh air.\n\nWorse for exertion.\n\nBreech baby\n\nThis can help to turn some babies in late pregnancy.\n\nChickenpox\n\nWith COUGH (see here).\n\nWith a low fever and typical general symptoms and emotional state. Itching is worse for heat.\n\nCommon cold\n\nNASAL CATARRH dirty yellow; green; yellow-green; bland (non-irritating); smelly; thick; dry alternating with profuse; watery in fresh air. With SENSE OF SMELL\/TASTE lost. With SNEEZING in stuffy room. Blocked, painful SINUSES.\n\nBetter for fresh air.\n\nWorse in a stuffy room.\n\nNose is watery in fresh air, blocked in evenings, in a warm room, though there will be much sneezing then too. Sense of taste is lost altogether or there is a bitter taste before and after eating.\n\nConstipation\n\nIn PREGNANCY. STOOLS changeable.\n\nCough\n\nConstant in evening; dry at night; during fever; loose in the morning; exhausting; irritating; racking; in fits; violent; disturbs sleep. MUCUS yellow; green; yellow-green; copious; sticky; difficult to cough up (must sit up). With NAUSEA (see here); RETCHING. In PREGNANCY.\n\nBetter for fresh air; for sitting up.\n\nWorse for getting hot; for physical exertion; for heat; for lying down; after meals; during the morning; at night\/evening; in a stuffy room; getting warm in bed.\n\nThe cough disturbs sleep; breathing is loud and rattling. Sputum is characteristically thick, difficult to cough up, may taste unpleasant, and is worse in the morning on getting up.\n\nCystitis\n\nDESIRE TO URINATE frequent; ineffectual; painful; urgent. Spasmodic PAIN. Frequent URINATION. URINE copious. In pregnancy.\n\nWorse after urinating.\n\nCause getting cold and wet.\n\nHurries to pass urine to prevent it escaping, but once there the urine dribbles slowly.\n\nDiarrhoea\n\nIn BABIES. In PREGNANCY. STOOLS changeable; greenish-yellow; slimy; watery.\n\nBetter for fresh air.\n\nWorse after eating; after eating starchy or rich food; at night; for getting overheated; in stuffy room.\n\nCause rich food; fruit.\n\nDizziness\n\nHAS TO LIE DOWN.\n\nWorse for getting up from lying down; sitting; stooping; walking.\n\nEarache\n\nDISCHARGE smelly; thick; yellow; yellow-green. EAR (external) red; EAR (internal) feels blocked. PAIN aching; pressing; pressing outwards; stitching; tearing; throbbing. With DEAFNESS; ITCHING; NOISES in ear.\n\nWorse at night.\n\nCause after common cold; after measles.\n\nExhaustion\n\nNERVOUS.\n\nWorse for heat of sun; in stuffy room; mental exertion; morning in bed.\n\nCause loss of sleep (broken nights).\n\nEye inflammation\n\nIn BABIES. DISCHARGE purulent; smelly; thick; yellow. EYES aching; burning; itching; watering. EYELIDS glued together; itching in evening. With symptoms of COMMON COLD (see here).\n\nBetter for cold bathing; for fresh air; for cold.\n\nWorse during evening; in warm room.\n\nIn the morning the eyes are sticky and the inner corners discharge. They ache and burn in a warm room, and water in cold air or wind and\/or with a cough.\n\nFaintness\n\nIn PREGNANCY.\n\nMay accompany mild anaemia.\n\nFever\n\nHEAT burning; dry during the evening; ONE-SIDED. With CHILLINESS.\n\nBetter for uncovering.\n\nWorse for heat; for washing; in the afternoon; in the evening; at night; in morning in bed; for warm covers; in stuffy room; for being uncovered.\n\nWants to be covered but gets overheated, kicks covers off and then gets cold. The fever (heat) may be on one side only and is often worse in early afternoon, followed by a chill around 4 p.m. Sweating may also be one-sided (usually the left) or localised. Sweats while asleep and wakes up chilled, then finds it difficult to get back to sleep.\n\nFlatulence\n\nABDOMEN\/STOMACH rumbling\/gurgling. WIND smelly; obstructed (difficult to expel).\n\nBetter for passing wind.\n\nFood poisoning\n\nWith DIARRHOEA (see here).\n\nCause rotten meat; rotten fish.\n\nHeadache\n\nNERVOUS. PAIN in forehead; one-sided; pressing; sore, bruised; throbbing. In PREGNANCY.\n\nBetter for firm pressure; for fresh air; for lying with head high; for walking in fresh air.\n\nWorse after eating; for bending down; for blowing nose; for hot drinks; for running; for standing; in a stuffy room; in heat.\n\nCause breastfeeding; excitement; ice-cream; running.\n\nPulsatilla types are prone to nervous headaches over the eyes, usually on one side; worse for movement and better for fresh air. Can be caused by too much sun or too much rich food.\n\nIncontinence\n\nIn PREGNANCY.\n\nWorse for coughing, laughing, sneezing, walking.\n\nIndigestion\n\nABDOMEN\/STOMACH feels empty. BELCHES bitter; empty; tasting of food just eaten. PAIN in stomach; pressing. With HEARTBURN. In PREGNANCY.\n\nWorse at night; for rich, fatty food.\n\nCause rich, fatty food.\n\nAbdomen\/stomach gurgles and rumbles during the evening.\n\nInsomnia\n\nDREAMS anxious, vivid. SLEEP restless. With SLEEPINESS. WAKING from cold; frequent. In PREGNANCY.\n\nWorse before midnight.\n\nCause overactive mind; repeating thoughts.\n\nPulsatilla insomniacs like to sleep on their backs, arms above their head, and feet poked out of bed. They sleep restlessly, with twitchy arms and legs, and have anxious dreams and nightmares.\n\nJoint pain\n\nPAIN in bones; in joints; pulling; sore, bruised; wandering.\n\nBetter for cold compresses; for fresh air; for gentle movement; for walking.\n\nWorse after a common cold; on beginning to move; for heat; for warmth of bed; for wet weather.\n\nLabour\n\nLABOUR PAIN in back; false; ineffectual (cervix doesn't dilate); irregular; short (each contraction lasts a short time); stop or slow down; weak. With FAINT FEELING; EXHAUSTION (see here); NAUSEA (see here); VOMITING.\n\nWith characteristic general symptoms and emotional state. Women are pathetic, depressed, clingy and weepy in labour.\n\nLochia\n\nSCANTY, MILKY, RETURNS (having almost stopped).\n\nMeasles\n\nWith COUGH (see here); COMMON COLD (see here); EYE INFLAMMATION (see here).\n\nWith characteristic weepiness and lack of thirst.\n\nMumps\n\nGLANDS swollen\/painful. PAINS spread to breasts, ovaries. In pregnancy. With VOMITING; FEVER (see here).\n\nWith characteristic general and emotional mental symptoms.\n\nNausea\n\nWith VOMITING.\n\nBetter for cold drinks; fresh air.\n\nCause pregnancy.\n\nWorse after drinking\/eating; for coughing; during morning; for hot drinks; for rich food; ice-cream.\n\nPhlebitis\n\nWith stinging pains in the veins\/legs.\n\nPiles\n\nINTERNAL. After CHILDBIRTH.\n\nProlapse\n\nWith a sense of PRESSURE IN THE ABDOMEN and the SMALL OF THE BACK.\n\nRetained placenta\n\nWith INEFFECTIVE, SPASMODIC CONTRACTIONS and the 'weeps'.\n\nRoseola\n\nSee MEASLES for symptoms.\n\nSciatica\n\nBetter for fresh air.\n\nWorse in a warm room.\n\nSnuffles\n\nIn newborn babies. With symptoms of COMMON COLD (see here).\n\nSore throat\n\nTHROAT dry. LARYNX irritated; tickling. PAIN raw; scraping; stitching.\n\nWorse for heat.\n\nLarynx feels as though there is dust in it; feels rough, and raw from coughing. Chokes when swallowing solid food.\n\nStyes\n\nWorse on upper eyelids.\n\nTeething\n\nPAINFUL.\n\nBetter for cold water; fresh air.\n\nWorse for heat of bed, warm food\/drinks.\n\nThrush (genital)\/Yeast infection\n\nIn pregnancy. DISCHARGE burning; cream-like; milky; thick.\n\nVaricose veins\n\nOf LEG\/THIGH\/FOOT. PAINFUL. PAIN stinging.\n\nWorse during pregnancy.\n\nLimbs lain on become numb and cold, indicating poor circulation.\n\nWeaning\n\nTo dry up milk.\n\nBreasts are sore and swollen after weaning in spite of having taken Lac canninum.\n\nPYROGEN (Pyr.)\n\nOther name: sepsin\n\nGeneral symptoms\n\nDischarges (breath, pus, sweat) smelly. Pain in parts lain on; sore, bruised. Pulse rapid. Thirsty.\n\nBetter for movement.\n\nWorse for cold; for cold, wet weather.\n\nThis is an important flu remedy, for those very severe flus with terrible pains and smelly discharges. The cold in any form aggravates.\n\nA severe acute complaint such as blood poisoning always needs professional care. I have also given indications for emergency treatment of this complaint. If indicated it works quickly and may make antibiotics unnecessary.\n\nPhysical complaints\n\nBlood poisoning\n\nWith fever (see here); extreme chilliness.\n\nWorse for cold.\n\nCause childbirth; surgery; infected wounds.\n\nGive Pyrogen every 5 minutes after calling for emergency help. A keynote for the use of Pyrogen is smelly discharges. The urine may be clear.\n\nFever\n\nHEAT. With CHILLINESS internal; extreme; SWEATING; SHAKING.\n\nWorse for being uncovered.\n\nDuring a fever parts lain on feel sore and bruised (like Arnica). A low fever is usually accompanied by a fast pulse and is not a cause for immediate concern. If the fever is high and the pulse slow this may indicate acute blood poisoning. Always seek help if these symptoms occur. Feels extremely chilly and cannot get warm. May be anxious, unusually talkative and slightly delirious.\n\nFlu\n\nPAIN aching; in bones of legs. With FEVER (see here); extreme THIRST; SHIVERING; CHILLINESS in back.\n\nBetter for movement; for walking; for warmth of bed.\n\nWorse during chilly stage; for sitting.\n\nRHEUM (Rhe.)\n\nOther name: palmated rhubarb\n\nGeneral symptoms\n\nDischarges (stools, sweat, etc.) sour.\n\nThese are pale babies who have a hard time assimilating food when producing (or trying to produce) teeth. Sourness is the keynote of this remedy. The baby smells sour, despite frequent washing.\n\nEmotional state\n\nCapricious. Irritable with teething. Play babies don't want to. Restless with teething. Screaming. Tearful babies at night.\n\nRheum is for teething children who are colicky, irritable, difficult, peevish, restless and don't want to play. They may sleep very little and seem to survive remarkably well (you don't).\n\nPhysical complaints\n\nDiarrhoea\n\nIn teething babies. STOOLS sour smelling; pasty.\n\nCause teething.\n\nFood is not assimilated well and colicky pains cause babies to cry and be restless at night.\n\nTeething\n\nPAINFUL in babies. With DIARRHOEA (see here).\n\nRHODODENDRON (Rhod.)\n\nOther name: Siberian rhododendron\n\nGeneral symptoms\n\nComplaints from change of weather to stormy.\n\nBetter for movement.\n\nWorse for cold; for cold, wet weather; for stormy weather; before a storm; for wind.\n\nThese types are extremely sensitive to the change in pressure that precedes a storm; their symptoms come on then or are generally worse. They are also sensitive to changes in the weather, particularly to cold, wet and windy weather, and are worse during the seasons of change \u2013 spring and autumn. Their symptoms may abate when the storm breaks.\n\nEmotional state\n\nFearful during a thunderstorm.\n\nMay feel apprehensive, fearful or just 'unwell' before and, to some extent, during a storm.\n\nPhysical complaints\n\nBackache\n\nIn LOWER BACK; in NECK. PAIN rheumatic; sore, bruised.\n\nBetter for movement.\n\nWorse for wet weather.\n\nHeadache\n\nPAIN tearing.\n\nBetter for getting up; for walking; for walking in fresh air; for wrapping up head.\n\nWorse before\/during thunderstorm; for damp weather.\n\nThe pains in the face and eyes are always worse before and during a storm.\n\nJoint pain\n\nPAIN in arms\/legs\/shoulders; drawing; tearing.\n\nBetter for movement; for stretching out limb.\n\nWorse for change of weather; for sitting down; for stormy weather; for wet weather.\n\nRHUS TOXICODENDRON (Rhus-t.)\n\nOther name: poison ivy\n\nGeneral symptoms\n\nComplaints from getting wet; change of weather to cold\/damp; getting chilled. Face red. Glands swollen. Likes milk. Lips dry. Oedema (swelling) of ankles\/feet; hands\/fingers. Pains burning; pressing; shooting; sore, bruised. Sweat worse for lying down; uncovering; slightest physical exertion. Taste in mouth metallic. Tongue red-tipped.\n\nBetter for changing position; for fresh air; for movement; for sweating; for warmth of bed; for hot drinks.\n\nWorse in the autumn; for change of weather; for cloudy weather; for cold; for cold\/wet weather; for foggy weather; for cold drinks\/food; for damp; for draughts; on beginning to move; for lying down; for uncovering; swimming in cold water.\n\nRhus-t pains are typically aching, sore and bruised with tearing or stitching pains that are worse on first moving after rest. These ease off after joint has been gently exercised by, for example, a little walk. But after a while tiredness sets in and the pains start up again, requiring rest followed by gentle exercise, and so the cycle continues. They find any position uncomfortable, are restless, and need to move around, seemingly constantly. Their pains cause them to get up at night, preventing sleep. Lying on a hard floor may help.\n\nThe tongue has a characteristically triangular red tip that may be sore and\/or it may be coated.\n\nThese people feel better generally for sweating, especially during a fever, but getting chilled while sweating aggravates their condition and can produce a cough or cold. They are chilly people, hate the cold in any form: cold food or drink, swimming or washing in cold water, and cold and wet weather in any form. Even touching cold things, putting a hand out of bed into cold air, or eating ice-cream on a hot day can make them feel bad. They feel better for heat, for warm, dry compresses (heat packs) and for hot baths.\n\nThey are sensitive to the damp, especially of autumn, when they may suffer from joint pains or a bad flu.\n\nEmotional state\n\nAnxious indoors; in bed. Confused. Depressed. Desires to be carried (babies). Irritable. Restless generally; babies; in bed. Tearful.\n\nRestlessness is the key, whether caused by physical pain or by nervousness. It can be seen during an acute illness with a fever where a person tosses and turns and has to get up. The anxiety is worse in bed at night because of having to be still then, especially after midnight when the mind dwells on unpleasant thoughts. Also they feel worse in other ways at night, being irritable and fearful. Their depression is not very deep, but it may be constant, and they may suddenly and involuntarily burst into tears without knowing why.\n\nPhysical complaints\n\nAbdominal pain\n\nIn PREGNANCY. With STIFFNESS IN LOWER ABDOMEN; restlessness.\n\nWorse as uterus expands (towards end of pregnancy).\n\nCause ligaments of uterus stretching.\n\nAfterpains\n\nFrequent. With RESTLESSNESS.\n\nBetter for moving about.\n\nBackache\n\nIn LOWER BACK; in SMALL OF BACK; in NECK. PAIN aching; dragging down; sore, bruised; rheumatic. With STIFFNESS.\n\nBetter for lying on hard surface; for movement; for walking; for heat.\n\nWorse on beginning to move; on getting up from sitting; for reaching up; for wet weather.\n\nCause damp weather; draughts; injury; lifting; pregnancy; sprain.\n\nBack feels weak, lame and tired, and pains compel constant moving about in bed. Getting up after lengthy sitting down is difficult. Pain may result from sitting in a draught, from lifting or a sprain, or it may be rheumaticky pain in damp, wet weather. It is always better for warmth, but aggravated by first movement.\n\nBleeding (vaginal) in pregnancy\n\nFLOW clotted. With LABOUR-LIKE PAINS.\n\nCause over-exertion; strain.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nWith tingling and numbness in fingers.\n\nChickenpox\n\nSKIN RASH itches maddeningly.\n\nRestlessness with the itching, and characteristic Rhus-t. red-tipped tongue.\n\nCough\n\nIrritating; short; tickling.\n\nBetter for hot drinks.\n\nWorse for becoming cold; for uncovering; on single parts of the body; for uncovering hands.\n\nCause swimming in cold water.\n\nCough is aggravated if part of body is uncovered and consequently becomes cold.\n\nDiarrhoea\n\nSTOOLS mushy, watery.\n\nCause getting wet, getting feet wet.\n\nExhaustion\n\nNERVOUS. With HEAVINESS; RESTLESSNESS.\n\nBetter for walking in fresh air.\n\nWorse for slightest exertion; for sitting down; for walking in fresh air.\n\nSlightest exertion is an effort but a gentle walk in the fresh air helps. With characteristic Rhus-t. restlessness.\n\nEye inflammation\n\nEYES sensitive to light; sore; watering. EYELIDS glued together; itching; swollen.\n\nWorse for moving eyes.\n\nCause cold, wet weather.\n\nEyelids are stuck together in the mornings.\n\nFever\n\nHEAT alternating with chills; dry; burning.\n\nBetter for hot drinks.\n\nWorse at night\/evening\/mid-morning; for movement; cold drinks; for being uncovered.\n\nSweats from the slightest exertion, as well as when lying in bed asleep, and sweats all over except for head. Urinates frequently while sweating. May get chilled after being hot and sweaty, when any movement or uncovering will bring on chilliness. Warmth helps the chilliness. Fever may be one-sided. May feel as though hot water is running through the blood vessels.\n\nFlu\n\nPAIN in bones; in eyes; in joints; in legs; aching; shooting; sore, bruised. With EXHAUSTION (see here); FEVER (see here); SNEEZING.\n\nHeadache\n\nPAIN in back of head; rheumatic; violent.\n\nBetter for gentle walking; for wrapping up head; for movement.\n\nCause change of weather; cold air\/wind; damp weather; getting wet; lifting.\n\nHives\n\nRASH burning; itching; stinging. With JOINT PAIN (see here).\n\nWorse for cold; after scratching.\n\nCause getting wet\/chilled; cold air; during fever.\n\nInsomnia\n\nSLEEP restless; sleepless after midnight.\n\nVivid, work-related dreams, caused by working too hard.\n\nJoint pain\n\nPAIN aching; sore, bruised; shooting; tearing. With HEAVINESS; LAMENESS; STIFFNESS.\n\nBetter for continued movement; for external heat; for walking; for warm compresses; for warmth of bed.\n\nWorse for damp weather; during fever; on beginning to move; at night; for sitting down; for getting chilled.\n\nPains are worse for rest and on beginning to move; improve temporarily but gets worse again after a while. Can be brought about by over-exertion, for example, spring-cleaning or gardening.\n\nMumps\n\nGLANDS swollen\/painful. With FEVER (see here).\n\nBetter for heat.\n\nWorse for cold; on left side.\n\nLeft side often becomes swollen first. This is a major mumps remedy, indicated by hard, swollen glands that often feature in Rhus-t. acute illnesses.\n\nNappy rash\n\nBURNS. ITCHES. Becomes FLAKY.\n\nBetter for covering.\n\nWorse for cold.\n\nPhlebitis\n\nWith Rhus-t. general symptoms and emotional state.\n\nProlapse\n\nCause over-exertion.\n\nStrained ligaments, from too much pushing in second stage of labour.\n\nRestless legs\n\nIn PREGNANCY; in BED.\n\nScarlet fever\n\nRASH itches and burns.\n\nWith Rhus-t. general symptoms and emotional state.\n\nSciatica\n\nBetter for movement; for walking; for heat.\n\nWorse for cold; for cold compresses; on beginning to move; for lying on painful part; for washing in cold water; for wet weather.\n\nSore throat\n\nTHROAT dry. Tickling LARYNX. VOICE hoarse.\n\nWorse for cold drinks; for swallowing; for uncovering throat.\n\nCause straining voice.\n\nVoice is hoarse, or strained, from too much talking or singing, but improves after continued use (as do Rhus-t. joints).\n\nSprains\n\nFOOT. ANKLE. WRIST. With STIFFNESS; TREMBLING.\n\nCause falling; lifting; twisting.\n\nUse Arnica first after injury or a sprain to deal with bruising and swelling. Then give Rhus-t. if the sprain is accompanied by the typical general symptoms.\n\nStiff neck\n\nCause draughts; getting chilled; lifting.\n\nStrains\n\nWith STIFFNESS; TREMBLING.\n\nBetter for continued movement.\n\nWorse for beginning to move.\n\nGardening or spring-cleaning can cause stiffness in joints. Worse when getting up after a rest and better for gentle exercise (whereas Arnica stiffness and soreness stays in spite of gentle exercise).\n\nRUMEX CRISPUS (Rumex)\n\nOther name: yellow dock\n\nPhysical complaints\n\nCough\n\nIn fits; constant; dry; irritating; tickling. PAIN IN CHEST burning; sore, bruised; stitching. LARYNX raw.\n\nWorse for lying down; for breathing in cold air; for fresh air; for change of temperature from warm to cold; for becoming cold; for cold air; on left side of body; in morning on waking; for talking; for uncovering; for walking in the cold air.\n\nCough causes pains in the chest, usually on the left side or in the left lung, or under the sternum. Pains are worse on breathing in cold air, so the mouth is covered with a scarf to allow only warm air through. Rawness and tickling in the larynx make the cough worse. Hawks up mucus from the back of the throat, where it gets stuck.\n\nSore throat\n\nTHROAT raw; sore; irritated. AIR PASSAGES irritated. VOICE lost.\n\nWorse for breathing in cold air.\n\nRUTA GRAVEOLENS (Ruta)\n\nOther name: common rue\n\nGeneral symptoms\n\nPains in bones; in parts lain on; sore, bruised.\n\nBetter for movement.\n\nWorse for lying on painful part.\n\nLike Rhus toxicodendron, Ruta is for sprains, but Rhus-t. is for damaged ligaments and Ruta for injuries to the tendons and the periosteum (the covering of the bones). It can be hard to differentiate so use Rhus-t.'s clear general symptoms to guide you. Ruta is indicated for sore, bruised joint pains that feel worse if lain on. There may be restlessness but not nearly so marked as in Rhus tox.\n\nEmotional state\n\nFearful of death. Weary.\n\nThe emotional symptoms are not strong, but during fever there may be some fear and mild depression.\n\nPhysical complaints\n\nAbdominal pain\n\nIn PREGNANCY.\n\nWorse as uterus expands towards end of pregnancy. With stiffness in lower abdomen.\n\nCause ligaments of uterus stretching.\n\nBackache\n\nPAIN sore, bruised. With LAMENESS.\n\nBetter for lying on back.\n\nEye strain\n\nEYESIGHT dim; weak. PAIN aching; burning; strained.\n\nWorse during evening; at night; for poor light; for using eyes; for close work.\n\nCause too much close work.\n\nRuta stimulates and strengthens the eye muscles and restores the sight when it has become weak and dim after straining the eyes with fine work or too much reading (where the letters on the page seem to run together, for example). For people who do a great deal of close work. It is not a remedy for eye inflammation.\n\nHeadache\n\nPAIN in sides of head; sore, bruised. With EYE STRAIN (see here).\n\nCause eye strain.\n\nHead feels as though beaten or crushed.\n\nInjuries\n\nCUTS\/WOUNDS to shins; PAIN bruised, sore.\n\nRuta is indicated for bruises to the periosteum (the covering of the bone), in particular on the shins. These can be very painful.\n\nJoint pain\n\nPAIN in hands; in feet; in lower back; in parts lain on; in bones; sore, bruised. With LAMENESS.\n\nWorse for walking.\n\nSprains\/strains\n\nPAIN bruised; constant; in ANKLE; in PERIOSTEUM; in TENDONS; in WRIST. With EXHAUSTION; LAMENESS.\n\nWorse for exercise; for pressure; for standing; for walking.\n\nHands and feet feel cold. Injured part feels lame and bruised (even after Arnica has brought down the swelling) and the pains are constant \u2013 movement does not relieve pain (like Rhus-t.). Pressure is also painful. Feels generally weak and weary.\n\nTennis elbow\n\nPAIN bruised, sore.\n\nWorse for exercise.\n\nThese injuries need a lot of time to heal in case the tendon is torn. Ruta helps to heal a damaged tendon, provided the joint is bandaged tightly and completely rested.\n\nSABINA (Sab.)\n\nOther name: savine\n\nPhysical complaints\n\nAfterpains\n\nPAIN spreads from the lower back to the thighs, to the pubic bone. Discharge of blood with every pain.\n\nBleeding (vaginal) in pregnancy\n\nFLOW OF BLOOD bright red; clotted; flows more with slight exertion. In EARLY PREGNANCY. With PAIN in legs; PAIN in lower back that spreads to pubic area.\n\nThe bleeding starts off brown with clots and labour-like pains, then a bright red flow of blood starts.\n\nSARSAPARILLA (Sars.)\n\nOther name: wild liquorice\n\nPhysical complaints\n\nBreast (feeding) problems\n\nNIPPLES inverted.\n\nIf there are no indications for another remedy, give Sarsaparilla to help establish breastfeeding.\n\nCystitis\n\nDESIRE TO URINATE ineffectual. PAIN cutting. URINATION slow; dribbling; can only pass urine while standing. URINE green; pale; with mucus; with sediment.\n\nWorse at end of urination; during menstrual period.\n\nCause getting chilled; cold, wet weather.\n\nMay pass urine without feeling pain until the very end, when yells with pain. Urinating is difficult while sitting, and may only be possible while standing. Even then she may manage only a feeble stream. Kidneys may also be painful (right more so than left) and inflamed (with cutting pains) \u2013 it is advisable to seek professional help before this point, but Sarsaparilla will help relieve pain in meantime.\n\nSECALE (Sec.)\n\nOther name: ergot\n\nGeneral symptoms\n\nBetter for cold bathing.\n\nWorse for heat.\n\nSecale people feel the cold but are much worse for heat \u2013 hot stuffy rooms, the heat of bed, warm compresses etc. They don't want to be covered up. This is a small remedy, invaluable for pregnancy and birth. Be guided by specific indications, rather than the general symptoms.\n\nEmotional state\n\nAnxious. Stupor.\n\nIn labour women fall into a sort of stupor. With complaints that come after the birth (or in pregnancy) there may be anxiety.\n\nPhysical complaints\n\nAfterpains\n\nIn women who have had many children. Long-lasting.\n\nBleeding (vaginal) in pregnancy\n\nWith absence of PAIN. In EARLY pregnancy. FLOW OF BLOOD dark brown or black; scanty.\n\nBreast(feeding) problems\n\nMILK SUPPLY low. PAIN stinging in breasts.\n\nIn exhausted women with severe afterpains. Breasts remain small.\n\nColic\n\nIn BABIES. ABDOMEN\/STOMACH feels bloated. With DIARRHOEA.\n\nCause syntometrine.\n\nSometimes the baby reacts to the syntometrine injection given to expel the placenta. It causes a colic where the abdomen becomes bloated and tight like a drum. Baby is better with the nappy off. Stools are watery and olive-green.\n\nEffects of drugs taken during or after labour\n\nCause syntometrine.\n\nLabour\n\nSLOW. LABOUR PAINS ineffectual (cervix doesn't soften); long (each contraction lasts a long time); stopping or slowing down; weak. With FAINT FEELING; with EXHAUSTION; TREMBLING if contractions stop.\n\nLochia\n\nFlow is DARK (brown); SMELLY; lasts too long; SCANTY.\n\nProlapse\n\nCause too much straining in labour; forceps delivery.\n\nRetained placenta\n\nWith BEARING DOWN sensation.\n\nMuscles feel weak and exhausted.\n\nSEPIA (Sep.)\n\nOther name: cuttlefish ink\n\nGeneral symptoms\n\nAppetite lost in pregnancy. Catches colds easily. Complaints from getting wet. Craves vinegar. Discharges yellow. Dislikes bread in pregnancy; food in general; smell of food; thought of it in pregnancy; milk; meat. Face pasty. Likes sour foods (pickles) in pregnancy. Pains pressing. Palpitations in pregnancy. Sweat cold; hot; profuse; smelly; sour; worse after slightest physical exertion; from mental exertion; from pain; on single parts of body. Symptoms left-sided.\n\nBetter after eating; for vigorous exercise; for running; for walking fast.\n\nWorse before\/during\/after menstrual period; at night; for cold; for fasting; for getting wet; for rest; for frosty air; for sweating; for touch; for walking in wind; for mental\/physical exertion; for writing.\n\nThese are chilly types, with cold hands and feet, extremely sensitive to cold. They sweat easily and profusely, with any exertion of body or mind or when experiencing strong emotions. Coughing can also make them sweat. The sweat smells sour and makes them feel worse.\n\nThey feel exhausted and run down, have no muscle tone, feel and look saggy, and feel as though something heavy inside is dragging them down. The typical Sepia face is yellow, earthy and pale, with marked yellowness across saddle of nose and dark rings under eyes. They are generally better for exercising vigorously \u2013 this energises them even if they are feeling unwell or exhausted. Eating helps temporarily (they are much worse for missing meals). They do not want to be touched or massaged.\n\nEmotional state\n\nAngry from contradiction. Anxious during fever; worse in evening. Apathetic towards children or relatives. Aversion to partner. Confused. Depressed. Desires to be alone. Despair during labour. Dislikes company; consolation; contradiction. Forgetful. Indifferent to own children; to family; to loved ones; to work. Irritable generally; worse for consolation. Loss of libido. Sensitive to music; to noise; to pain. Sluggish. Tearful. Weepy.\n\nThese types are prone to dreadful emotional troughs. The classic Sepia picture is too many babies too quickly, but women may also be worn out from too much to do and not enough resources to see them through it. In this state, they sag mentally and physically; sit silently, feel empty and enjoy nothing; feel indifferent to things that formerly gave enjoyment including partners and children. They respond badly to sympathy, preferring to be alone. Thinking is difficult and brains seem to have ground to a halt.\n\nThey feel much better if they drag themselves out of their torpor to do something strenuous, such as exercise, dancing or even spring-cleaning.\n\nPhysical complaints\n\nBackache\n\nIn LOWER BACK. PAIN aching; dragging down.\n\nBetter for pressure.\n\nWorse during the afternoon; at night; before\/during menstrual period; for bending down; for sitting.\n\nLower back feels weak and tired, as if hit with a hammer.\n\nBleeding (vaginal) in pregnancy\n\nIn MID-PREGNANCY. FLOW OF BLOOD is dark. With bearing-down PAIN.\n\nFeels as if everything will fall out.\n\nBreast pain\n\nIn pregnancy.\n\nWith Sepia general and emotional state.\n\nBreast(feeding) problems\n\nNIPPLES cracked\/sore; with ITCHING.\n\nCracks may get so bad they bleed, but they'll itch first.\n\nCarpal Tunnel Syndrome\n\nIn pregnancy.\n\nWith tingling\/numbness in fingers.\n\nCommon cold\n\nNASAL CATARRH green; dirty yellow; yellow-green; drips down back of throat. SENSE OF SMELL lost.\n\nConstipation\n\nABDOMEN\/STOMACH feels full. STOOLS hard; large. STRAINING ineffectual.\n\nWorse during period.\n\nCause pregnancy.\n\nFor acute constipation that accompanies pregnancy. Strains to pass a large, hard stool but is unable to.\n\nCough\n\nBREATHING fast. COUGH constant when lying down at night; dry; exhausting; hacking; irritating; loose; must sit up to cough; rattling; short; disturbs sleep; tickling; violent; wakens at night. MUCUS copious; yellow; white; tastes salty.\n\nBetter for sitting up.\n\nWorse at night; during evening in bed; for lying down.\n\nCoughs up lots of salty white or yellow mucus. Chest feels constricted (tight) or oppressed and the ribs may hurt from coughing. The air passages feel irritated; the cough seems to come from the stomach and is violent, especially after lying down at night. Sweats after coughing and feels worse for it.\n\nCramp\n\nIn PREGNANCY. In CALF\/LEG.\n\nCystitis\n\nDESIRE TO URINATE constant with dragging-down pain in pelvis; urgent. PAIN pressing. URINATION frequent; slow (takes a long time to start). URINE cloudy; dark brown; red; scanty; smelly; with sediment.\n\nCommon after childbirth. Constant dragging-down sensation in the pelvis. Has to rush to pass urine or it escapes involuntarily because of saggy muscles in the pelvic area, then has to wait for it to start once sitting on the toilet.\n\nExhaustion\n\nNERVOUS; SUDDEN. In PREGNANCY.\n\nWorse on getting up in morning; during a menstrual period; for slightest exertion; for sweating; for walking.\n\nCause breastfeeding; sweating.\n\nLegs feel stiff and weak. Also accompanies Sepia hot flushes, or pregnancy.\n\nEye inflammation\n\nEYES burning. EYELIDS glued together; swollen.\n\nWorse after walk; during evening; for reading.\n\nEyes feel sore, as if there's sand in them.\n\nFaintness\n\nWith HEAT. COLDNESS after faint. In PREGNANCY.\n\nWorse for exercise; during fever; in stuffy room.\n\nSepia types feel faint easily and are always worse for standing for any length of time.\n\nFever\n\nWith ANXIETY; SWEATING (see General symptoms).\n\nWorse in autumn; for sweating; for covering.\n\nCause anger.\n\nFever accompanying flu or other infection is often severe with external heat and internal chilliness. External heat is followed by chills with shaking and no thirst. Covering up warmly makes them uneasy (an unusual and helpful symptom) even when chilled. Characteristic Sepia general and emotional symptoms are also present.\n\nHair loss\n\nAfter childbirth.\n\nHeadache\n\nEXTREMITIES icy-cold. PAIN in bones; bursting; pressing; shooting; tearing; throbbing; in waves. In PREGNANCY.\n\nBetter after eating; for fresh air.\n\nWorse for bending down; for getting head cold; for travelling.\n\nCause artificial light; breastfeeding.\n\nPains can be almost anywhere in the head, although the left side is often more affected.\n\nIncontinence\n\nIn PREGNANCY.\n\nWorse for coughing; laughing; sneezing; walking.\n\nInsomnia\n\nWAKING around 3 a.m. With SLEEPINESS.\n\nWakes but can't get back to sleep, and may lie awake thinking dismal thoughts. By morning, feels exhausted.\n\nLabour\n\nPAIN in back; distressing; severe (violent); stopping or slowing down. With EXHAUSTION.\n\nAnxious, despairing and irritable in labour; may take dislike to their partner. The pains drag down and are much better for vigorous exercise.\n\nNausea\n\nINTERMITTENT. ABDOMEN\/STOMACH feels empty. PAIN gnawing. With VOMITING (see here).\n\nWorse after eating; before breakfast; during morning; for milk; for pork.\n\nCause pregnancy.\n\nEmpty, sinking feeling in the stomach, only temporarily relieved while eating. Often accompanied by a headache and worse when thinking about food or eating.\n\nNosebleeds\n\nIn PREGNANCY.\n\nProlapse\n\nWith CONSTIPATION; BEARING-DOWN SENSATION.\n\nBetter for sitting with legs crossed.\n\nToo many children too close together can cause a prolapse, with a feeling as if everything will fall out.\n\nSkin complaints in pregnancy\n\nBROWN PIGMENTATION in patches on face.\n\nThrush (genital)\/Yeast infection\n\nDISCHARGE burning; copious; like cottage cheese; lumpy, like egg white; smelly; yellow. With DRYNESS; ITCHING. In PREGNANCY.\n\nSimple thrush which occurs during pregnancy. Feels exhausted and worn down.\n\nToothache\n\nPAIN radiates to the ears; pulling; stitching; tearing; throbbing.\n\nWorse for biting teeth together; for cold drinks\/food; during pregnancy; for touch; for hot drinks.\n\nTravel sickness\n\nWith BILIOUSNESS; HEADACHE (see here); NAUSEA (see here).\n\nWorse after eating; during morning.\n\nCause pregnancy.\n\nVomiting\n\nVOMIT bile; smelly.\n\nWorse after eating; during morning.\n\nCause pregnancy.\n\nMay feel better for eating, but improvement is temporary.\n\nSILICA (Sil.)\n\nOther name: pure flint\n\nGeneral symptoms\n\nCatches colds easily. Complaints from getting feet wet; change of weather to cold. Dislikes meat. Glands swollen. Pains stinging. Sweat worse at night, during sleep; profuse; smelly; sour. Thirsty. Weight gain poor in babies.\n\nBetter for heat; for wrapping up head.\n\nWorse for change of weather; for cold; for damp; for draughts; for fresh air; for getting feet wet; for touch; for uncovering; for wet weather.\n\nSilica types feel the cold intensely. They are easily tired and chilled, such as by going out into the cold air after swimming, and are sensitive to draughts and changes of weather. They are prone to catching colds, especially if they get a thorough soaking. They are always better for warmth and being wrapped up well, especially the head.\n\nThey sweat at night, particularly on the back of the head and neck (like Calcarea carbonica, but not so profuse). The sweat smells sour. Their feet are cold, sweaty and smelly. They have trouble assimilating food and their bones do not always form as well as in other children. They can be slow learning to walk and in teething. Their fontanelles close slowly.\n\nEmotional state\n\nAnxious. Aversion to being touched\/examined; to consolation. Complaints from mental strain; shock. Poor concentration. Confused. Conscientious. Irritable generally; worse for consolation (babies). Jumpy. Lack of self-confidence. Mild. Play \u2013 babies don't want to. Restless. Sensitive to noise; to pain. Shy. Sluggish.\n\nWorse for being consoled.\n\nThese are people who typically lack stamina both mentally and physically \u2013 who lack 'grit'. They become worn out easily from too much work (usually mental). Alternatively they can be 'gritty' and conscientious under pressure, able to sustain superhuman feats of endurance, collapsing only when the job in hand is done.\n\nThey are very shy, and anxious about appearing in public because of fear of failure. Unassertive, lacking in confidence, they give way rather than take a position and fight for it. Conversely can be stubborn, wilful and develop fixed ideas. Irritable if consoled when feeling low. Restless and nervous inside, oversensitive to noise, especially small noises, and startle easily.\n\nPhysical complaints\n\nAbscesses\n\nOf GLANDS; of ROOTS OF TEETH.\n\nSilica brings abscesses that are forming to a head.\n\nAfterpains\n\nPAIN in hips.\n\nWorse when baby nurses.\n\nAthlete's foot\n\nCracks between toes. With sweat on feet; profuse; smelly.\n\nBackache\n\nPAIN sore, bruised; stitching. With LAMENESS; STIFFNESS; WEAKNESS.\n\nWorse on getting up from sitting; breastfeeding; at night; for pressure; for sitting.\n\nCause falling on back; injury to coccyx; manual labour.\n\nBlocked tear duct\n\nIn babies.\n\nThis remedy will clear simple blockages of the tear duct.\n\nBreast(feeding) problems\n\nBREAST inflamed; painful; lumpy. NIPPLES cracked\/sore, with bleeding; inverted. PAIN in breast; cutting; stitching. BREAST ABSCESSES.\n\nWorse while nursing; in left breast.\n\nSudden sharp pains while the baby is nursing and cracks become bad so quickly they bleed. Back may ache while nursing.\n\nBroken bones\n\nSLOW TO MEND.\n\nCommon cold\n\nNASAL CATARRH dry; hard crusts; smelly; thick. SINUSES painful. SENSE OF SMELL\/TASTE lost. SNEEZING difficult.\n\nSinuses blocked, nose dry and blocked, nostrils sore and ulcerated.\n\nConstipation\n\nIneffectual STRAINING. PAIN burning after stool. STOOL hard; knotty; large; 'shy'.\n\nWorse before\/during menstrual period.\n\nMuscles in bowel are weak. Stools are painfully expelled and then slip back.\n\nCough\n\nIrritating. MUCUS lumpy; thick; yellow.\n\nBetter for hot drink.\n\nWorse on waking in morning; for becoming cold; for uncovering feet or head.\n\nLingering winter coughs that started after getting chilled, or wet and chilled, that do not clear up easily with indicated remedy.\n\nDiarrhoea\n\nIn TEETHING babies (see here). FLATULENCE smelly (see here).\n\nBetter for heat; for wrapping up warmly; for warmth of bed.\n\nCause teething.\n\nDizziness\n\nHAS TO LIE DOWN. With HEADACHE (see here).\n\nEarache\n\nWith BLOCKED FEELING IN EAR. PAINS behind ear; tearing.\n\nEustachian tube is itchy and full of mucus. Any discharge is bloody, smelly and thick.\n\nExhaustion\n\nNERVOUS.\n\nCause diarrhoea; breastfeeding.\n\nEye inflammation\n\nDISCHARGE yellow. EYES sore.\n\nWorse for cold air.\n\nCause foreign body in the eye.\n\nIf a foreign body is still in the eye, Silica taken internally will help expel it.\n\nFever\n\nWorse for movement; for being uncovered; in the evening; at night.\n\nFeels icy cold and shivery all day; this may be one-sided. Sweat is profuse and sour-smelling; if it is suppressed the conditions are exacerbated.\n\nFingernails\n\nSplit easily; weak. With WHITE SPOTS.\n\nFlatulence\n\nGAS smelly; obstructed (difficult to expel). ABDOMEN\/STOMACH rumbling.\n\nHeadache\n\nPAIN in back of head; in forehead; spreading to top of head\/to eyes\/to right eye; in sinuses; burning; hammering; pressing; sore, bruised; throbbing; violent.\n\nBetter for closing eyes; for lying in quiet, dark room; for heat; for wrapping up head.\n\nWorse for getting chilled; for getting feet\/head cold; for daylight; for getting up from lying down; for jarring movement; for walking upstairs\/heavily.\n\nCause cold air; damp weather; draughts; eye strain; overwork; working in artificial light; travelling (car, coach, train, etc.).\n\nOften a result of sinus catarrh; it settles over the eyes, forehead feels heavy. Head and feet feel cold.\n\nInflammation of navel\n\nIn newborn babies.\n\nInjuries\n\nCUTS\/WOUNDS slow to heal; inflamed; painful; with pus inside; with dirt\/splinter still inside; suppurate. SCARS become lumpy; are painful; break open.\n\nCause episiotomy; splinter; surgery.\n\nSilica cuts and wounds become infected easily and heal slowly; skin looks unhealthy.\n\nInsomnia\n\nDREAMS, NIGHTMARES VIVID. SLEEP restless.\n\nWorse after midnight; after waking.\n\nLochia\n\nFLOWS while the baby breastfeeds.\n\nMumps\n\nGLANDS painful; swollen.\n\nBetter for heat.\n\nWorse for cold.\n\nSore throat\n\nTHROAT dry. GLANDS swollen. PAIN stitching. TONSILS swollen\/inflamed. With HAIR SENSATION at back of tongue.\n\nWorse for getting cold; for swallowing; for uncovering throat.\n\nSplinters\n\nWOUND inflamed.\n\nHelps push out splinters especially if the wound has become inflamed. It heals wound once pus has started discharging.\n\nSpots\n\nIn babies.\n\nIn skinny, chilly, sweaty babies.\n\nTeething\n\nPAINFUL in babies. DIFFICULT. SLOW. With DIARRHOEA (see here).\n\nSilica types are slow to produce teeth and will get coughs, colds and diarrhoea during this time. Silica will help push the teeth out.\n\nVomiting\n\nIn BREASTFED BABIES.\n\nWorse for milk.\n\nBabies who refuse mother's milk and\/or vomit (posset) it up. Bottled milk may be no better.\n\nSPONGIA TOSTA (Spo.)\n\nOther name: sea sponge\n\nGeneral symptoms\n\nWorse after sleep; for cold wind; for movement; for tight clothing; for tobacco.\n\nFeels worse when excited or on exertion. Wakes from sleep with a sense of suffocation and anxiety, and finds it difficult to catch the breath. Is intolerant of tight clothing because of difficulty in breathing.\n\nEmotional state\n\nAnxious on waking. Tearful during a fever.\n\nMay wake up with a start and be anxious and tearful.\n\nPhysical complaints\n\nCough\n\nBarking; constant; dry; hollow; in fits; irritating; tickling. BREATHING difficult; rough. PAIN IN CHEST burning; sore, bruised. VOICE HOARSE.\n\nBetter for drinking\/eating.\n\nWorse daytime; for excitement.\n\nCough may even have been caused by too much excitement. Eating sweets may aggravate the cough but it is generally better for eating and drinking, especially warm things. The chest is sore and bruised from coughing. Breathing is difficult and may sound loud between bouts of coughing. It is worse after sleep and for exercise, and sometimes also for talking. Voice is crowing and hollow.\n\nCroup\n\nWith COUGH (see here).\n\nThis is one of the main croup remedies, though it may be difficult to differentiate between Aconite croup and Spongia croup as both have acute anxiety, but Aconite croup is worse at night, around midnight. Even so, if Aconite is indicated and fails to help, Spongia will usually help.\n\nExhaustion\n\nWorse for slightest exertion; for movement.\n\nSore throat\n\nTHROAT dry. PAIN burning; sore. VOICE hoarse.\n\nWorse for coughing; for singing; for swallowing; for sweet drinks\/food; for talking; for touch.\n\nThe larynx is sore and sensitive to touch, the voice hoarse and hollow-sounding; the throat feels raw, rough and sore, and swollen \u2013 as if there is a plug in it.\n\nSTAPHYSAGRIA (Stap.)\n\nOther name: palmated larkspur\n\nGeneral symptoms\n\nAnaemia. Dislikes milk; tobacco; water. Gums bleeding; pale. Sweat worse for mental exertion.\n\nWorse for exertion; for fasting; for tobacco; for touch.\n\nThese individuals are extremely sensitive, physically and emotionally, and suffer pain acutely. Wounds and cuts are unusually painful, especially if there is a feeling of having been assaulted. They feel generally worse for exercise, smoky rooms and missing meals, and are especially irritable after an afternoon nap.\n\nEmotional state\n\nViolently angry. Anxious about others. Apathetic. Capricious. Complaints from disappointed love; excitement; grief; humiliation; indignation; mental strain; reprimands; suppressed anger\/emotions. Disappointment. Forgetful. Irritable generally, babies. Resentful. Sensitive to pain; to rudeness. Sluggish.\n\nThese are sensitive, touchy people who are easily offended. They don't tolerate rudeness in others although they may be rude themselves. They suppress their feelings, brood and let little out. In particular, anger is suppressed; they fume inside and may even appear sweet and compliant outside. Afterwards, they quickly become exhausted and may feel shaky; cannot sleep, work or speak; and may get a headache. Suppression of anger can turn into an active and acid resentment. Complaints are accompanied by sense of violation or humiliation with ensuing indignation and resentment. These people do not want to be touched when feeling low or upset.\n\nThese people cannot say no and allow pressure to build up inside. This ends in violent outburst, or getting sick, or both. In this state they may throw things (as do Chamomillas).\n\nPhysical complaints\n\nAbdominal pain\n\nIn PREGNANCY. PAINS sudden; with SHOCK AND ANGER.\n\nCause sudden, unexpected kick by the baby.\n\nSome babies can deliver a kick that hurts and shocks.\n\nBites\/stings\n\nPainful; sensitive to touch.\n\nColic\n\nPAIN cramping.\n\nWorse after drinking.\n\nCause indignation; vexation.\n\nCough\n\nTickling.\n\nCause suppressed anger.\n\nLarynx and air passages feel irritated.\n\nCystitis\n\nDESIRE TO URINATE frequent. URINATION frequent; involuntary. URINE scanty.\n\nOften follows a sense of assault physically or emotionally, where strong feelings and\/or needs were not expressed.\n\nExhaustion\n\nNERVOUS. With TREMBLING.\n\nFor people worn out, anaemic, trembly and exhausted from suppressing feelings.\n\nEye injuries\n\nWOUND punctured.\n\nWorse for touch.\n\nHeadache\n\nPAIN in back of head; in forehead; pulling; pressing; sore, bruised.\n\nWorse for touch.\n\nCause emotion; excitement; grief; vexation.\n\nHeadaches of people who suppress their emotions.\n\nInjuries\n\nCUTS\/WOUNDS lacerated; painful; to nerve-rich parts of the body. PAIN tearing.\n\nWorse for touch.\n\nCause accident; childbirth; circumcision; episiotomy; surgery.\n\nWounds or injuries with feelings of humiliation, indignation and anger. Might be caused by an operation to sphincter (anus, urethra, stomach, etc.), or any operation, especially if more was done than expected, or after a mechanical childbirth with pain afterwards and possibly stitches. May simply be pain following unpleasant examination. Babies may be seriously aggrieved after circumcision.\n\nPiles\n\nWorse for touch.\n\nAfter a difficult birth or forceps delivery everything hurts\/feels sensitive and piles are excruciatingly painful.\n\nRetention of urine (after labour)\n\nAfter a difficult birth or a forceps delivery.\n\nShock\n\nWith ANGER, INDIGNATION.\n\nCause injury; surgery.\n\nWith typical emotional state.\n\nStyes\n\nSensitive to touch.\n\nTravel sickness\n\nWith INDIGNATION.\n\nWhere the individual is extremely difficult and irritable.\n\nSTRAMONIUM (Stram.)\n\nOther name: thorn-apple\n\nGeneral symptoms\n\nBetter for light.\n\nWorse for dark.\n\nThese types feel worse emotionally and physically at night when it gets dark.\n\nEmotional state\n\nAnxious about strangers (babies); at night (babies). Apathetic uncomplaining. Clingy. Complaints from childbirth. Confused (sense of unreality). Fear of the dark. Hyperactive. Indifferent. Lonely. Mischievous. Rage during labour; in babies on being picked up. Restless wanders aimlessly. Shock. Tantrums. Terror of glittering surfaces.\n\nBabies are terrified of the dark and wake up with frightful nightmares, knowing no one. They have a strange fear of water and glittering surfaces such as mirrors; may become hysterical if you try to encourage them to swim. They are prone to tantrums as soon as they are mobile; may hit and bite.\n\nWomen take a long time to recover from childbirth, feel they have lost touch with reality, and are confused and apathetic. After a shock someone needing this remedy becomes anxious and frightened; babies become hyperactive and aimlessly restless.\n\nPhysical complaints\n\nConvulsions\n\nWith RAGE.\n\nCause over-excitement.\n\nEyes glitter and pupils become large.\n\nShock\n\nWith FEAR.\n\nCause childbirth; surgery.\n\nThe fear can border on terror and is accompanied by nightmares or a sense of the birth having been a nightmare.\n\nSULPHUR (Sul.)\n\nOther name: flowers of sulphur\n\nGeneral symptoms\n\nAnaemia. Aversion (babies) to being washed\/bathed. Breath smells. Complaints from change of weather from cold to warm. Discharges smelly; sour; watery. Dislikes bread; eggs; meat. Face red; sallow; red in spots. Glands swollen. Hot flushes in pregnancy. Likes spicy food; sweets. Lips cracked; dry; red. Pains burning. Sweat profuse; smelly; sour; from slightest physical exertion. Symptoms left-sided. Taste in mouth bitter; bad. Thirsty extreme; for large quantities; for water. Tongue white coated; red-edged and red-tipped.\n\nBetter for fresh air.\n\nWorse for bathing; for change of weather to warm; for draughts; for exertion; for fasting; for heat; for milk; around 10\u201311 a.m.; for standing; in stuffy rooms; for warmth of bed.\n\nSulphur types always look untidy and\/or unwashed, no matter what they do. They are usually given away by an odd sock, a hem coming down, or a collar escaped from inside a jumper (things no self-respecting Arsenicum would put up with!). Often indifferent to how they look, or simply do not notice. They are warm-blooded and have hot feet; can wear sandals all year round. They are lazy about washing; often feeling worse for it. They slump when sitting and stoop when walking and are always worse for having to stand up for long. They have markedly red faces and red lips which can become dry and cracked during illness. The tongue is often white-coated with red edges and tip. They are thirsty types; consuming large quantities of water whether sick or well.\n\nThey are sensitive to heat and always worse for being in a stuffy room; and better for fresh air. Discharges are usually hot, smelly and burning. They sweat profusely and feel worse while sweating.\n\nNB This remedy is hailed for its effect on eczema and skin rashes, but I strongly advise you not to prescribe for these complaints without consulting a professional homeopath, as the aggravations can be severe and need to be monitored carefully.\n\nEmotional state\n\nAngry. Anxious worse in evening; about others. Confused. Critical. Depressed. Despair. Discontented. Impatient. Indifferent to personal appearance. Irritable. Lazy. Quarrelsome. Restless generally; babies. Sluggish in babies.\n\nSulphur people have many ideas which may owe more to dreams than reality. They can be successful academics or unrealistic dreamers \u2013 so-called 'ragged philosophers'. They may start many things and finish nothing; the further away they are from realising their ideas, the sicker they become.\n\nThey are sloppy, lazy and disorganised; they feel constantly rushed and hurried; would rather not work at all. Mess doesn't bother them, although they may get periodical cravings for a limited external order. They are hoarders; cannot throw anything away.\n\nSulphur types are basically self-centred, impatient and irritable; critical of others and how they run their lives, with lots of ideas about how they could do it better themselves. Suffer from worries which plague them in evening and stop them sleeping at night. They worry about those closest to them and become depressed and despairing when ill. Sulphur babies are restless, always on the go and impatient. When ill, they become sluggish but never lose their restlessness.\n\nPhysical complaints\n\nBackache\n\nBack feels weak\/tired. PAIN in LOWER BACK; aching; sore, bruised. With STIFFNESS; WEAKNESS.\n\nWorse after period of long sitting; for bending down; on getting up from sitting; at night; for standing; for walking.\n\nBreast(feeding) problems\n\nNIPPLES cracked\/sore; itching; burn and itch after nursing. BREAST ABSCESS inflamed. BREAST red, may be itchy.\n\nBreathless\n\nIn PREGNANCY.\n\nBetter for fresh air.\n\nWorse for exertion.\n\nChickenpox\n\nSee MEASLES.\n\nCommon cold\n\nNASAL CATARRH smelly; dirty yellow. NOSE dry; itching. With frequent SNEEZING; EYE INFLAMMATION (see here).\n\nThe nose is dry and itchy and the tip is red. It may bleed when blown. May be constantly aware of unpleasant smells.\n\nConstipation\n\nIn PREGNANCY. With BURNING; REDNESS; ITCHING around anus.\n\nCough\n\nBREATHING fast. COUGH disturbs sleep; dry during evening\/at night, loose by day\/in morning; painful; irritating; racking; suffocative. CHEST congested. LARYNX raw. MUCUS green.\n\nWorse for coughing; during evening in bed; during morning; at night; for lying down.\n\nWakes from the cough and sides will hurt from coughing. Larynx feels irritated and raw, as if dusty. Difficulty breathing in a warm room, may be better for fresh air; usually wants the window open.\n\nCradle cap\n\nDry; itchy.\n\nMay smell. In restless sulphur-type babies.\n\nCramp\n\nIn THIGHS; in LEG; in CALF; in SOLE OF FOOT.\n\nWorse at night; in bed; when walking.\n\nWith typical general symptoms and emotional state.\n\nCystitis\n\nDESIRE TO URINATE frequent; urgent. PAIN burning. URINE burning; brown like beer; copious; scanty; smelly.\n\nWorse on getting up in the morning; at night; while urinating.\n\nHurries to pee to prevent it escaping, but then passes only a few drops and dribbles each time.\n\nDiarrhoea\n\nIn BABIES. During morning only; drives person out of bed; painless. FLATULENCE smelly. PAIN burning; cramping. STOOLS slimy; smelly; sour-smelling; watery.\n\nBetter for passing wind.\n\nWorse around 5 a.m.; at night; for standing.\n\nWakes early with an urge to pass a stool which forces them out of bed. There is rumbling before passing stool; pains are felt in the belly before, during and even after passing one. The anus becomes sore and red after passing a stool, especially in babies.\n\nDizziness\n\nWorse for getting up from lying\/sitting; for stooping.\n\nCause high places.\n\nEarache\n\nPAIN aching; lacerating; stitching; tearing. With painful NOISES IN EAR.\n\nWorse in left side; for noise.\n\nRoaring and ringing noises in the ear are more noticeable when in bed. The external parts of the ear itch.\n\nExhaustion\n\nWorse during the afternoon; for the heat of the sun; for talking.\n\nCause diarrhoea; flu; hunger; talking; walking; pregnancy; breastfeeding.\n\nEye inflammation\n\nEYES burning; gritty; sensitive to light; itching; stitching; watering. EYELIDS itching in daytime only; burning; red.\n\nWorse for washing eyes.\n\nEyes are glued together in the mornings; may be dry, feel sandy. Water easily in the fresh air, and are sensitive to bright sunlight.\n\nFever\n\nHEAT alternating with chills. With SWEATING (see General symptoms); SHIVERING.\n\nWorse at night; for thinking.\n\nFeels worse while sweating, and hot, especially on soles of the feet, which are poked out of bed.\n\nFlatulence\n\nABDOMEN\/STOMACH bloated; gurgling; rumbling. GAS smelly; of rotten eggs.\n\nBetter for passing wind.\n\nHair loss\n\nCause childbirth.\n\nHeadache\n\nHEAD feels constricted, full\/hot. PAINS in forehead; burning; bursting; hammering; pressing; in top of head; throbbing. With NAUSEA (see here).\n\nBetter for cold.\n\nWorse after eating; for bending down; for blowing nose; for coughing; on getting up; during morning; for sneezing; for walking.\n\nCause damp weather; excited conversation; winter.\n\nFeels like a band is tied around the head and head feels full and hot, especially in bed or a stuffy room. Headache may recur weekly, especially on the day a person tries to relax.\n\nIndigestion\n\nABDOMEN\/STOMACH feels bloated; feels empty around 11 a.m. BELCHES empty; sour. With FLATULENCE wind smells of rotten eggs. PAIN in the stomach burning.\n\nWorse after eating; after drinking milk.\n\nMay feel suddenly hungry mid-morning after not wanting breakfast; feels faint, weak and headachey. Appetite may, however, vanish at the sight of food. Stomach feels full and heavy after eating, and rumbles and gurgles.\n\nInsomnia\n\nDREAMS unpleasant; NIGHTMARES. SLEEP disturbed; unrefreshing; restless. WAKING frequent; late.\n\nBetter for short naps.\n\nWorse after 3, 4, or 5 a.m.\n\nThe Sulphur insomniac always wakes tired. Feels sleepy in the daytime and may be refreshed by a catnap. At night feet get hot and burn in bed; have to be poked out of bed to cool them down. Wakes in early hours, often from cold, and may be unable to get back to sleep. Restless, and has nightmares when lying on the back. May have to get out of bed around 5 a.m. to pass a stool.\n\nJoint pain\n\nPAIN burning; tearing.\n\nWorse for walking; for warmth of bed.\n\nFeeling of heat in the joints, and general restlessness.\n\nMeasles\n\nSKIN RASH red; burns; itches maddeningly; slow to appear. With COUGH (see here); FEVER (see here).\n\nWorse for heat.\n\nOften needed where recovery is slow from a childhood illness. Can help relieve the itching of measles, German measles or chickenpox if severe, very red and much worse for heat (of bed or a bath). Babies will be restless, feverish and sweaty, and should fit the general picture. Will also be thirsty and hungry but eat very little. Rash becomes crusty, smelly and weeps after scratching.\n\nNappy rash (Diaper rash)\n\nSkin red, itchy, burns. Becomes red raw and bleeds.\n\nBetter for uncovering.\n\nWorse for washing\/bathing; heat.\n\nNausea\n\nIn PREGNANCY. With VOMITING.\n\nNosebleeds\n\nWorse for blowing nose.\n\nPiles\n\nPainful; in pregnancy; bleeding; burning; itching; large; external or internal. With CONSTIPATION (see here).\n\nWorse for touch; standing\/walking.\n\nPrickly heat\n\nWith typical general symptoms and emotional state.\n\nRestless legs\n\nIn pregnancy.\n\nWorse for heat.\n\nSkin complaints of pregnancy\n\nITCHING without a rash.\n\nTowards the end of pregnancy the stretched skin across the abdomen may itch, and this can spread all over the body. There is no rash.\n\nSore throat\n\nTHROAT dry; raw. CHOKING sensation. PAIN burning; raw; sore; stitching. TONSILS swollen. VOICE HOARSE.\n\nWorse for coughing; for swallowing.\n\nLarynx is dry, dusty. Voice is most hoarse in mornings.\n\nSpots\n\nIn babies. Newborn babies become covered in a 'milk' rash.\n\nVomiting\n\nIn BREAST-FED babies.\n\nSULPHURIC ACID (Sul-ac.)\n\nOther name: oil of vitriol\n\nGeneral symptoms\n\nAnaemia.\n\nWorse for heat; during the evening; mid-morning.\n\nSulphuric acid people are anaemic, chilly types who feel hurried and trembly inside.\n\nEmotional state\n\nIrritable. Feeling of being hurried while eating; at work; while writing; while walking. Moody.\n\nIn spite of feeling exhausted, these types feel rushed and hurried inside and do everything 'at a pace'; they bolt their food, walk quickly, and work frantically, including writing quickly. They feel driven inside and cannot do things fast enough. Their moods may be changeable, swinging from reasonable and agreeable to irritable.\n\nPhysical complaints\n\nBruises\n\nBluish-black; slow to heal.\n\nSulphuric acid finishes the healing of bruises that turn bluish-black and do not clear up in spite of taking Arnica and\/or Ledum.\n\nDiarrhoea\n\nBELCHES sour. STOOLS soft; stringy; yellow. VOMIT sour.\n\nStools are bright yellowy-orange, like saffron; feeling of emptiness and weakness after passing a stool. The upset stomach causes sour belches and vomiting.\n\nExhaustion\n\nWith TREMBLING.\n\nWorse after passing a stool.\n\nExtreme lassitude which may not be visible on the surface, but they feel trembly inside.\n\nThrush (oral)\n\nIn babies. Of TONGUE\/GUMS.\n\nSYMPHORICARPUS RACEMOSA (Symph-r.)\n\nOther name: snowberry\n\nPhysical complaint\n\nVomiting\n\nIn PREGNANCY. Constant. Violent. With nausea; retching.\n\nBetter lying on back.\n\nWorse movement.\n\nAppetite is lost \u2013 the smell or thought of food aggravates but whereas Asarum nausea is worse for noise this one is worse for movement.\n\nSYMPHYTUM (Symph.)\n\nOther name: comfrey\n\nPhysical complaints\n\nBroken bones\n\nPAIN sticking.\n\nEases the pain and speeds up the healing of broken and fractured bones \u2013 after being set, of course.\n\nEye injuries\n\nPAIN in eyeballs; sore, bruised.\n\nCause a direct blow to eyeball.\n\nFor an injury to the eyeball or the bones surrounding the eye (resulting from, for example, a stray fist or tennis ball) where pain persists after Arnica has been given and has dealt with the swelling.\n\nTABACUM (Tab.)\n\nOther name: tobacco\n\nPhysical complaints\n\nHeartburn\n\nIn PREGNANCY. With WATERY BELCHES.\n\nNausea\n\nDEATHLY; INTERMITTENT; VIOLENT. With VOMITING (see here).\n\nBetter for fresh air.\n\nWorse in a stuffy room; for fasting; before breakfast.\n\nCause pregnancy; travelling.\n\nNausea is better for fresh air, although dizziness is worse for it. Empty, sinking feeling in the stomach. May also feel sluggish and confused or anxious and restless.\n\nTravel sickness\n\nWith NAUSEA (see here); VOMITING (see here).\n\nBetter for fresh air.\n\nWorse for stuffy room; for movement; for tobacco.\n\nNausea is more acute when travelling by boat and the vomiting is worse when travelling by car or train (although vomiting on a boat is also possible).\n\nVomiting\n\nEXHAUSTING. VOMIT sour; violent.\n\nWorse before breakfast; for movement.\n\nCause pregnancy; travelling.\n\nFace is pale and drawn.\n\nTHIOSINAMINUM (Thios.)\n\nOther name: mustard seed oil\n\nPhysical complaint\n\nScars\n\nLumpy. Adhesions.\n\nWhere the scar, i.e. of an episiotomy, leaves lumps (keloids); if there is pain after a Caesarean because of adhesions (poor healing of scars inside the body); or if the foreskin of a baby has become stuck to the glans of the penis because it has been retracted accidentally. Short courses of this remedy in a low potency (6X) will help \u2013 it can be taken three times a day for up to a week and repeated every two or three weeks for as long as it is helping.\n\nTHUJA OCCIDENTALIS (Thu.)\n\nOther name: white cedar\n\nPhysical symptoms\n\nBirthmarks\n\nIn babies.\n\nUse this remedy cautiously (see below).\n\nWarts\n\nBLEEDING; LARGE; STINGING.\n\nIndicated for warts that grow in a cauliflower shape, which may appear anywhere on the body (hands, feet, face, etc.). Use the remedy cautiously: do not take it for longer than a week at a time; one week on and one week off for a couple of months is ideal. If there is no alleviation of the symptoms, seek professional help.\n\nThis is a deep-acting remedy best prescribed by a competent homeopath. I have included it here because it is available in most chemists and wholefood shops that stock homeopathic medicines.\n\nURTICA URENS (Urt-u.)\n\nOther name: common nettle\n\nPhysical complaints\n\nBites\/stings\n\nITCHING. PAIN biting; burning; stinging.\n\nBreastfeeding problems\n\nMILK SUPPLY low; overabundant.\n\nUseful remedy to help establish good supply of milk in early days of feeding where it is slow without an obvious cause. Give where there are no obvious general or emotional\/mental symptoms that would guide you to another remedy. Can also help where there is overabundance of milk and no desire to express and store it but wants it to be geared more to demand.\n\nBurns\n\nMINOR. PAIN burning; stinging.\n\nUseful in everyday minor scalds and burns, especially where there are burning, stinging pains afterwards.\n\nHives\n\nRASH biting; burning; itching; stinging. With JOINT PAIN.\n\nBetter for rubbing\n\nWorse at night; for heat; for exercise.\n\nCause stinging nettles; insects.\n\nRash appears as red, raised blotches. It may accompany a rheumatic flare-up, or may be a reaction to eating shellfish. There is a constant desire to rub the skin.\n\nPrickly heat\n\nWith symptoms of HIVES (see here).\n\nFeels as though stung by stinging nettles.\n\nVERATRUM ALBUM (Ver-a.)\n\nOther name: white hellebore\n\nGeneral symptoms\n\nAppetite increased in pregnancy. Breath cold. Collapse after diarrhoea. Expression anxious. Face blue; pale. Hot flushes in pregnancy. Likes cold foods; sour and salty food in pregnancy; cold drinks; fruit; ice-cream; refreshing things; sour food. Saliva increased. Sweat clammy; cold; profuse; sour. Thirsty for large quantities. Tongue cold.\n\nWorse after eating fruit.\n\nThis remedy is indicated for the exhaustion with faintness that may accompany an acute illness, usually a 'tummy bug', such as gastroenteritis, dysentery or food poisoning. Acute illnesses are accompanied by a general coldness of breath, tongue and skin (which is cold to touch). Icy-cold sweat is another general symptom, and these types feel worse while sweating. There is an increase of saliva in the mouth and a ferocious thirst for large quantities of cold or icy-cold drinks, which may be vomited immediately in gastric troubles. Eating fruit causes a general aggravation.\n\nEmotional state\n\nAnxious after shock. Apathetic. Aversion to partner and children. Broody. Confused\/dull. Depressed. Desires in pregnancy. Despair. Forgetful. Hyperactive. Tearful. Uncommunicative.\n\nThese types may appear distressed, restless (have to be up and doing something), and prone to weeping and wailing and incessant talking. Alternatively, they may be totally withdrawn, silent and inactive.\n\nPhysical complaints\n\nCough\n\nDeep; hollow.\n\nBreathing is difficult and may be louder than usual. Feeling of tightness in the chest, and the cough is accompanied by the characteristic cold sweat.\n\nCramp\n\nIn PREGNANCY. In CALF\/LEG; preventing sleep.\n\nDiarrhoea\n\nEXHAUSTING; INVOLUNTARY; VIOLENT. PAIN burning; cutting; dull\/aching. STOOLS copious; frequent; green; watery. With ICY-COLD HANDS AND FEET; SWEATING (see General symptoms) during and after passing stool; VOMITING (see here).\n\nWorse after drinking; for movement.\n\nCause fruit; getting chilled.\n\nMay be accompanied by a ravenous appetite, with a feeling of emptiness not relieved by eating. Has trouble passing wind, and when able to do so may involuntarily pass stool. Stools shoot out and have to be passed frequently. Hands and feet are icy cold.\n\nExhaustion\n\nPARALYTIC; SUDDEN. With COLD EXTREMITIES. In pregnancy.\n\nWorse after passing stool.\n\nCause diarrhoea.\n\nFever\n\nWith SWEATING (see General symptoms).\n\nWorse for sweating.\n\nThe skin is pale and icy cold to touch over whole body or in spots or patches. With diarrhoea sweating is worse (during and after passing stools).\n\nLabour\n\nWith FAINT FEELING; EXHAUSTION (see here); VOMITING (see here); NAUSEA.\n\nNausea\n\nIn PREGNANCY. With VOMITING (see here).\n\nVomiting\n\nABDOMEN\/STOMACH feels cold. PAIN IN STOMACH cramping\/griping. VOMIT of bile; of mucus; sour; watery; yellow. VOMITING violent. With DIARRHOEA (see here); DIZZINESS; HICCUPPING after vomiting.\n\nWorse after drinking; after eating.\n\nOften accompanies the diarrhoea and is one of the most severe forms of diarrhoea and vomiting. May vomit and pass diarrhoea simultaneously.\n\nZINCUM METALLICUM (Zinc.)\n\nOther name: zinc\n\nGeneral symptoms\n\nFace pale. Gums bleeding; pale. Oedema (swelling) of ankles\/feet. Taste in mouth metallic. Trembling. Twitchy. White spots on fingernails.\n\nWorse after eating; during evening; at night; for wine.\n\nFor people of any age who are weary and run-down from stress and overwork. May tremble, twitch and jerk. Symptoms tend to be worse at night, from drinking wine and after eating, although they may feel better during eating. Like Sulphur types, they have an empty feeling in the stomach around 11 a.m. Keynote symptom is restlessness and twitchiness in legs; these types feel exhausted but fidgety and must keep their legs constantly on the move. Worse in the evenings in bed; legs will carry on twitching during sleep.\n\nIn childhood illness such as measles where rash does not appear properly, baby may be weak and twitchy, and possibly chesty. Zincum will help.\n\nEmotional state\n\nDepressed. Irritable. Moody. Screaming on waking (babies). Sensitive to noise. Sluggish. Uncommunicative.\n\nNervous exhaustion makes for difficulty in thinking: thoughts wander, is slow to answer, repeats questions, and forgets what they are saying halfway through a sentence. They are depressed, irritable and generally worn out.\n\nPhysical complaints\n\nBackache\n\nIn COCCYX; in neck; in spine. PAINS aching; sore, bruised. With WEAKNESS.\n\nWorse for sitting, for writing.\n\nWhole back feels sore and weak from too much stress.\n\nConstipation\n\nIn BABIES. In NEWBORN BABIES.\n\nFor newborn babies who are nervous and twitchy and find it difficult to establish a regular bowel movement.\n\nExhaustion\n\nLEGS restless (see here); weak.\n\nCause loss of sleep (broken nights).\n\nLegs feel weak and twitchy, especially in bed at night.\n\nEye inflammation\n\nEYES: sore; gritty; burning.\n\nWorse during evening; at night.\n\nEyes feel particularly sore in inner corners. May water during the evening and at night.\n\nHeadache\n\nNERVOUS. PAIN in forehead; in sides of head; in temples; bursting; tearing.\n\nBetter for fresh air.\n\nCause overwork.\n\nMay be on both sides of the head or on one side only.\n\nHeartburn\n\nIn PREGNANCY. With sweetish BELCHES.\n\nRestless legs\n\nIn pregnancy.\n\nWorse in the evening.\n\nVaricose veins\n\nIn PREGNANCY. Of LEG\/THIGH\/VULVA. Painful.\n* * *\n\nINTERNAL REPERTORY\n\n* * *\n\nSee list of remedies and abbreviations\n\nAbdominal pain in pregnancy Arn., Bell-p., Bry., Cimi., Nux-v., Rhus-t., Ruta., Stap.\n\nPains:\n\nsore and bruised: Arn.\n\nmove about (wandering): Cimi.\n\nsudden: Bell-p., Stap.\n\nuterus feels sore: Bell-p.\n\nWith:\n\nnausea (morning sickness): Nux-v.\n\nshock and anger: Stap.\n\nstiffness in lower abdomen: Bell-p., Bry., Rhus-t., Ruta.\n\nWorse:\n\nas uterus expands (towards end of pregnancy): Bell-p., Bry., Rhus-t., Ruta.\n\nCaused by:\n\nan active baby: Arn.\n\na sudden, unexpected kick (by the baby): Stap.\n\nligaments of uterus stretching: Bell-p., Bry., Rhus-t., Ruta.\n\nAbscesses Calc-s., Hep-s., Merc-s., Sil.\n\nDischarging pus: Calc-s.\n\nPains see Pains\n\nOf glands: Calc-s., Hep-s., Merc-s., Sil.\n\nOf roots of teeth: Hep-s., Merc-s., Sil.\n\nAbsent-minded (see also Confused; Forgetful) Caust., Nat-m.\n\nAccidents see Broken bones; Bruises; Burns; Bites\/stings; Eye injuries; Injuries; Sprains; Strains\n\nAffectionate Phos., Puls.\n\nAfraid see Fearful\n\nAfterpains Arn., Cham., Cimi., Cupr., Hyp., Kali-c., Nux-v., Puls., Rhus-t., Sab., Sec., Sil.\n\nPains:\n\ncramping: Cupr.\n\nfrequent: Rhus-t.\n\nin groin: Cimi.\n\nin hips: Hyp., Sil.\n\nin lower back radiating to thighs: Hyp., Sab.\n\nin lower back radiating to pubic bone: Sab.\n\nin women who have had many children: Cupr., Sec.\n\nlong lasting: Sec.\n\nshooting down into hips, buttocks and\/or legs: Kali-c.\n\nsore, bruised: Arn.\n\nstitching: Kali-c.\n\nunbearable: Cham., Cimi.\n\nWith:\n\ncramps in fingers or toes: Cupr.\n\ndesire to pass a stool with every pain: Nux-v.\n\nfaint feeling after the pains: Nux-v.\n\nheadache: Hyp.\n\nrestlessness: Rhus-t.\n\nWorse:\n\nwhen the baby nurses: Arn., Cham., Sil.\n\nCause:\n\nforceps delivery: Hyp.\n\nAnaemia Ars., Calc-c., Calc-p., Carb-v., Chin., Ferr-m., Kali-c., Kali-p., Merc-s., Nat-m., Nit-ac., Pho-ac., Phos., Puls., Stap., Sul., Sul-ac.\n\nAfter acute illness: Calc-p.\n\nLips pale: Ferr-m.\n\nWith exhaustion: Calc-p., Ferr-m.\n\nCauses:\n\nloss of blood (haemorrhage, pregnancy, etc.): Chin., Ferr-m., Pho-ac.\n\nAngry (see also Complaints from anger; Irritable)\n\nArs., Bell., Bry., Cham., Hep-s., Ign., Kali-c., Kali-s.,\n\nLyc., Nat-m., Nit-ac., Nux-v., Sep., Stap., Sul.\n\nBabies: Ant-c., Ant-t., Cham., Cina, Lyc., Puls.\n\nDuring labour: Bell., Cham.\n\nFrom contradiction: Ign., Lyc., Sep.\n\nRaging: Bell., Cham.\n\nViolently: Cham., Hep-s., Nit-ac., Nux-v., Stap.\n\nWhen has to answer questions: Nux-v.\n\nAntisocial see Desires to be alone\n\nAnxious Aco., Arg-n., Ars., Bar-c., Bell., Bry., Calc-c., Calc-p., Calc-s., Carb-v., Caust., Chin., Cocc., Con., Kali-c., Kali-p., Kali-s., Lyc., Mag-c., Mag-m., Nat-c., Nit-ac., Nux-v., Phos., Puls., Rhus-t., Sec., Sep., Sil., Spo., Stram., Sul.\n\nAbout health: Calc-c., Nit-ac., Phos.\n\nAbout others: Ars., Nux-v., Pho-ac., Stap., Sul.\n\nAbout strangers: Stram.\n\nAfter shock: Ign., Ver-a.\n\nAnticipatory: Arg-n., Gels., Lyc.\n\nAt night: Ars., Bor., Calc-c., Cina, Kali-p., Puls., Stram.\n\nBabies: Bor., Cina, Gels., Kali-c.\n\nDuring a fever: Aco., Ars., Bar-c., Ip., Sep.\n\nDuring labour: Aco., Ars., Kali-c., Lyc., Sep.\n\nDuring pregnancy: Aco., Ign.\n\nOn waking: Ars., Lach., Spo.\n\nWhen chilled: Aco.\n\nWhen indoors: Kali-s., Lyc., Puls., Rhus-t.\n\nWhen overheated: Kali-s.\n\nBetter for:\n\nfresh air: Kali-s., Puls.\n\nWorse:\n\nafter midnight: Ars.\n\naround 3 a.m.; Ars.\n\nevening: Calc-c., Calc-s., Carb-v., Sep., Sul.\n\nin bed: Ars., Carb-v., Mag-c., Rhus-t.\n\nwhen alone: Ars., Phos.\n\nfrom downward motion: Bor.\n\nheat: Kali-s.\n\nindoors: Lyc., Puls., Rhus-t.\n\nnight: Ars., Bor., Puls.\n\non waking: Ars., Lach., Spo.\n\nApathetic (see also Sluggish) Alu., Ant-t., Ap., Carb-v., Chin., Con., Gels., Lil-t., Nat-c., Nat-m., Nat-p., Nux-m., Op., Pho-ac., Phos., Puls., Sep., Stap., Stram., Ver-a.\n\nAbout anything being done for them: Lil-t.\n\nDoesn't complain: Op., Stram.\n\nDuring fever: Op.\n\nDuring labour: Gels., Op., Puls.\n\nTo her own children or relatives: Phos., Sep.\n\nAppetite\n\nAlternating with hunger: Cina, Ferr-m.\n\nIncreased during pregnancy: Ver-a.\n\nLost: Chin., Cina, Ferr-m.\n\nLost during pregnancy: Caust., Nat-m., Sep.\n\nApprehensive see Anxious anticipatory\n\nArgumentative see Quarrelsome\n\nArthritis see Joint pain\n\nAthlete's foot Sil.\n\nAversions (see also Dislikes)\n\nBeing alone: Kali-c., Lyc.\n\nBeing examined: Bell., Chin., Cina, Lach., Sil.\n\nBeing hugged: Cina\n\nBeing looked at: Ant-c., Cham., Cina\n\nBeing spoken to: Cham.\n\nBeing touched: Ant-c., Ant-t., Arn., Bell., Cham., Chin., Cina, Kali-c., Lach., Sil.\n\nBeing washed (babies): Sul.\n\nCompany see Desires to be alone\n\nConsolation: Ign., Lyc., Nat-m., Nux-v., Sep., Sil. (see also Quarrelsome)\n\nFresh air: Cocc., Ign.\n\nHer own children: Lyc.\n\nHer partner: Sep.\n\nHer partner and her children: Ver-a.\n\nStrangers: Bar-c., Bor.\n\nAwkward see Clumsy\n\nBackache Aesc., Bell., Bry., Calc-c., Calc-f., Cimi., Dulc., Ferr-m., Hyp., Kali-c., Kali-p., Lyc., Merc-s., Nat-m., Nat-s., Nux-v., Phos., Puls., Rhod., Rhus-t., Ruta., Sep., Sil., Sul., Zinc.\n\nBetween shoulder blades: Phos.\n\nCoccyx: Hyp., Zinc.\n\nLower back: Bry., Calc-c., Calc-f., Dulc., Ferr-m., Hyp., Kali-c., Lyc., Merc-s., Nat-m., Nux-v., Phos., Puls., Rhod., Rhust-t., Sep., Sul.\n\nNeck: Rhod., Rhus-t., Zinc.\n\nSmall of back: Puls., Rhus-t.\n\nSpine: Kali-p., Zinc.\n\nPains:\n\naching: Calc-c., Dulc., Nat-m., Nux-v., Puls., Rhus-t., Sep., Sul., Zinc.\n\nback feels broken: Nat-m., Phos.\n\nburning: Merc-s., Phos.\n\ndragging down: Bell., Kali-c., Nux-v., Puls., Rhus-t., Sep.\n\npressing: Nux-v., Puls.\n\nrheumatic: Cimi., Rhod., Rhus-t.\n\nshooting: Hyp., Merc-s.\n\nsore, bruised: Dulc., Hyp., Kali-c., Kali-p., Nat-m., Nat-s., Nux-v., Rhod., Rhus-t., Ruta., Sil., Sul., Zinc.\n\nfeels sprained: Calc-c.\n\nstitching: Bry., Kali-c., Sil.\n\ntearing: Hyp.\n\nWith:\n\nlameness: Dulc., Ruta., Sil.\n\nstiffness: Lyc., Rhus-t., Sil., Sul.\n\nweakness: Sil., Sul., Zinc.\n\nBetter for:\n\ncontinued movement: Calc-f.\n\ngentle exercise: Puls.\n\nheat: Rhus-t.\n\nlying on back: Ruta.\n\nlying on a hard surface: Kali-c., Nat-m., Rhus-t.\n\nmassage: Phos.\n\nmovement: Dulc., Kali-p., Lyc., Rhod., Rhus-t.\n\npassing wind: Lyc.\n\npressure: Kali-c., Sep.\n\nrubbing: Phos.\n\nurinating: Lyc.\n\nwalking: Dulc., Rhus-t.\n\nwalking slowly: Ferr-m., Puls.\n\nWorse:\n\nafternoon: Sep.\n\naround 3 a.m.: Kali-c.\n\nin bed: Nux-v.\n\nbending down: Sep., Sul.\n\nbeginning to move: Calc-f., Ferr-m., Lyc., Puls., Rhus-t.\n\nbreastfeeding: Sil.\n\nbreathing: Merc-s.\n\ncoughing: Bry., Merc-s.\n\ndamp: Calc-c.\n\nbefore a menstrual period: Kali-c., Puls., Sep.\n\nduring a menstrual period: Bry, Puls., Sep., Sul.\n\nmorning: Nux-v.\n\nmovement: Nux-v.\n\nslightest movement: Bry.\n\nnight: Sep., Sil., Sul.\n\npassing a stool: Lyc.\n\npressure: Sil.\n\nreaching up: Rhus-t.\n\nsitting: Sep., Sil., Zinc.\n\nafter long sitting: Kali-c., Sul.\n\ngetting up from sitting: Calc-c., Lyc., Merc-s., Phos., Puls., Rhus-t., Sil., Sul.\n\nstanding: Sul.\n\nsweating: Merc-s.\n\nwalking: Kali-c., Sul.\n\nwet weather: Dulc., Rhod., Rhus-t.\n\nwriting: Zinc.\n\nCauses:\n\nchange of weather: Dulc.\n\nchildbirth: Hyp., Nux-v., Phos.\n\nchildbirth where baby was 'posterior': Kali-c.\n\ndamp weather: Dulc., Rhus-t.\n\ndraughts: Rhus-t.\n\nepidural: Hyp.\n\nforceps delivery (childbirth): Hyp.\n\nfalling on the back: Sil.\n\ngetting cold: Dulc., Nux-v.\n\ngetting wet: Dulc.\n\ninjury: Rhus-t.\n\ninjury to coccyx: Hyp., Sil.\n\ninjury to spine: Hyp., Nat-s.\n\nlifting: Calc-c., Lyc., Rhus-t.\n\nmanual labour: Nat-m., Sil.\n\npregnancy: Aesc., Bell., Cimi., Kali-c., Nux-v., Puls., Rhus-t.\n\nsprain: Rhus-t.\n\nBad tempered see Angry; Irritable\n\nBetter (generally)\n\nBathing: Asar., Puls.\n\nChanging position: Rhus-t.\n\nCold applications: Glon.\n\nCold bathing: Led., Sec.\n\nCold drinks: Caust., Cupr., Phos.\n\nCold drinks\/food: Asar.\n\nCold\/dry air\/weather: Asar.\n\nCompany: Arg-n.\n\nConstipation: Calc-c.\n\nCrying: Puls.\n\nDamp: Caust.\n\nAfter eating: Ign., Nat-c., Phos., Sep.\n\nWhile eating: Lach.\n\nFanning: Carb-v.\n\nFirm pressure: Bry., Chin., Mag-p.\n\nFresh air: Aco., Arg-n., Asar., Bor., Carb-v., Kali-s., Lach., Lyc., Mag-c., Mag-m., Nat-s., Puls., Rhus-t., Sul.\n\nHeat: Ars., Calc-c., Caust., Hep-s., Ign., Kali-p., Mag-p., Nux-m., Nux-v., Sil.\n\nHot drinks: Ars., Nux-v., Rhus-t.\n\nLight: Stram.\n\nLying down: Ars., Asar., Bell., Calc-c., Cocc., Nat-m., Nit-ac., Nux-v.\n\nLying still: Bry.\n\nMassage: Nat-c., Phos.\n\nMovement: Dulc., Phos., Puls., Pyr., Rhod., Rhus-t., Ruta.\n\nPressure: Mag-m., Puls.\n\nRest: Asar., Kali-p.\n\nAfter a rest: Nat-m.\n\nAfter a good sleep: Pho-ac., Phos.\n\nRunning: Sep.\n\nSitting down: Colch., Con., Nux-v.\n\nSweating: Gels., Nat-m., Rhus-t.\n\nUncovering: Cham.\n\nUrinating: Gels.\n\nVigorous exercise: Sep.\n\nWalking: Dulc.\n\nWalking fast: Sep.\n\nWalking in fresh air: Puls.\n\nWalking slowly: Ferr-m.\n\nWarmth of bed: Ars., Caust., Hep-s., Kali-c., Lyc., Rhus-t.\n\nWrapping up: Hep-s.\n\nWrapping up the head: Sil.\n\nBirthmarks Thu.\n\nBites\/stings Ap., Hyp., Lach., Led., Stap., Urt-u.\n\nBlue: Lach.\n\nInflamed: Hyp.\n\nItching: Ap., Urt-u.\n\nPains:\n\nbiting: Urt-u.\n\nburning: Ap., Urt-u.\n\nshooting: Hyp.\n\nstinging: Ap., Urt-u.\n\ntearing: Hyp.\n\nRed\/swollen: Ap.\n\nBetter for:\n\ncold: Ap., Lach., Led.\n\nWorse for:\n\nheat: Ap., Led., Lach.\n\nCauses:\n\nanimal\/insect bites: Ap., Hyp., Lach., Led.\n\nBiting Bell.\n\nBlack eye Led.\n\nBleeding Arn., Bell., Ip., Phos.\n\nBright red; occurs easily; profuse: Phos.\n\nprofuse after labour: Bell., Ip., Phos.\n\nbright red flow: Ip., Phos.\n\ndark red, clotted flow: Bell.\n\nBleeding gums see Gums, bleeding\n\nBleeding (vaginal) in pregnancy Aco., Arn., Bell., Caul., Ign., Kali-c., Kreos., Op., Puls., Rhus-t., Sab., Sec., Sep.\n\nIn early pregnancy: Kreos., Sab., Sec.\n\nIn mid pregnancy: Sep.\n\nFlow:\n\nblack: Kreos., Sec.\n\nbright red: Bell., Sab.\n\nbrown: Sab., Sec.\n\nchangeable: Puls.\n\nclotted: Bell., Kreos., Puls., Rhus-t., Sab.\n\ndark, red: Bell., Sep.\n\nscanty: Caul., Sec.\n\nsmelly: Kreos.\n\nstops and starts: Puls.\n\nsudden gushes: Bell.\n\nWith:\n\nan absence of pain: Sec.\n\nanxiety: Aco.\n\nback pain: Caul., Kali-c., Sab.\n\nspreading down into buttocks and thighs: Kali-c.\n\nspreading to the pubic area: Sab.\n\nbearing down pains: Caul., Sep.\n\nlabour-like pains: Rhus-t., Sab.\n\npains in the legs: Sab.\n\nshooting pains: Bell.\n\nsore, bruised pains: Arn.\n\nweakness and trembling: Caul.\n\nCause:\n\nemotional shock: Ign.\n\nfright: Aco., Op.\n\ngrief: Ign.\n\ninjury: Arn.\n\nover-exertion: Rhus-t.\n\nBlisters Caust., Lyc., Nat-m.\n\nBurning: Lyc.\n\nPainful: Caust.\n\nTip of tongue: Caust., Lyc., Nat-m.\n\nBloated see Indigestion\n\nBlocked tear duct: Bar-c., Sil.\n\nBlood blisters Arn.\n\nBlood poisoning Pyr.\n\nBossy see Dictatorial\n\nBraxton Hicks Contractions Bell., Calc-c., Caul., Cimi., Cham., Gels., Puls., Sec.\n\nBreast (feeding) problems\n\nBreast abscess (Mastitis): Hep-s., Merc-s., Phyt., Sil., Sul.\n\nBreasts:\n\nengorged: Bell., Bry.\n\nhard\/hot: Bell., Bry.\n\ninflamed: Bell., Bry., Hep-s., Phyt., Sil., Sul.\n\nlumpy: Con., Phyt., Sil.\n\nworse in right breast: Con.\n\npainful: Bell., Bor., Bry., Merc-s., Sep., Sil.\n\npainful in pregnancy: Bell., Bry., Calc-p., Sep.\n\nwith inflammation: Bell., Bry.\n\npains:\n\naching after nursing: Bor.\n\nwhile nursing: Phyt., Puls., Sil.\n\ncutting\/stitching: Sil.\n\nin opposite breast whilst nursing: Bor.\n\nslightest movement: Bry.\n\nthrobbing: Bell.\n\nworse: left breast, while nursing: Sil.\n\npale: Bry.\n\nred: Sul.\n\nred-streaked: Bell.\n\nMilk supply:\n\nlow: Calc-c., Caust., Dulc., Lac-d., Sec., Urt-u. in chilly women: Dulc.\n\nover-abundant: Bell., Bry., Calc-c., Puls., Urt-u.\n\nNipples:\n\ncracked\/sore: Cast., Caust., Phyt., Sep., Sil., Sul.\n\nand bleeding: Sil.\n\nand itching: Sep., Sul.\n\ninverted (retracted): Sars., Sil.\n\npains when the baby nurses: Phyt.\n\nWeaning:\n\nto dry up milk: Lac-c., Puls.\n\nBreath\n\nCold: Ver-a.\n\nSmelly: Arn., Carb-v., Lach., Merc-c., Merc-s., Nit-ac., Phyt., Puls., Sul.\n\nBreathing difficulties in newborn babies Ant-t., Carb-v.\n\nBreathless Ars., Calc-c., Carb-v., Ferr-m., Puls., Sul.\n\nBetter:\n\nfresh air: Puls., Sul.\n\ngentle exercise: Ferr-m.\n\nWorse:\n\nburping: Carb-v.\n\nexertion: Ars., Calc-c., Puls., Sul.\n\nlying down: Ars., Carb-v.\n\nat night in bed: Ars.\n\nwalking uphill or upstairs: Ars., Calc-c., Carb-v.\n\nBreech baby Puls.\n\nBroken bones Arn., Bry., Calc-c., Calc-p., Sil., Symph.\n\nPains:\n\nsticking: Symph.\n\nstitching: Bry.\n\nSlow to mend: Calc-c., Calc-p., Sil.\n\nWith swelling\/bruising: Arn.\n\nWorse:\n\nslightest movement: Bry.\n\nBroody (see also Introspective; Moody) Ign., Pho-ac., Ver-a.\n\nBruises (see also Injuries) Arn., Bell-p., Led., Sul-ac.\n\nBluish-black: Sul-ac.\n\nPains:\n\nsore, bruised: Arn., Bell-p.\n\nSlow to heal: Sul-ac.\n\nWith:\n\nbumps, lumps remaining: Bell-p.\n\ndiscolouration: Led.\n\nswelling (no discolouration): Arn.\n\nCauses:\n\nchildbirth: Arn., Bell-p.\n\ninjury: Arn., Bell-p., Led.\n\nover-exertion: Bell-p.\n\nsurgery: Arn., Bell-p.\n\nBurns Ars., Canth., Caust., Kali-b., Urt-u.\n\nDeep: Kali-b.\n\nMinor: Urt-u.\n\nPains:\n\nburning: Ars., Canth., Caust., Urt-u.\n\nstinging: Urt-u.\n\nSecond degree: Canth.\n\nSlow to heal: Kali-b.\n\nThird degree: Caust.\n\nWith blisters: Ars., Canth., Caust.\n\nBetter for:\n\ncold compresses: Canth.\n\nheat: Ars.\n\nCapricious Bry., Cham., Cina, Ip., Kreos., Puls., Rhe., Stap.\n\nCarpal Tunnel Syndrome Ap., Ars., Calc-c., Lach., Lyc., Rhus-t., Sep.\n\nCarried see Desires to be\n\nCatches colds easily Ars., Bar-c., Calc-c., Calc-p., Dulc., Hep-s., Kali-c., Nat-m., Nux-v., Sep., Sil.\n\nChangeable see Moody\n\nCheerful Coff., Lach., Nat-c.\n\nChickenpox Aco., Ant-c., Ant-t., Bell., Merc-s., Puls., Rhus-t., Sul.\n\nFirst stage: Aco.\n\nSkin rash:\n\nitches maddeningly: Rhus-t., Sul.\n\nslow in coming out: Ant-t.\n\nsuppurates: Merc-s.\n\nWith:\n\ncough: Ant-c., Ant-t., Puls.\n\nfever: Aco., Bell.\n\nheadache: Bell.\n\nChilblains Agar.\n\nCircumcision Staph. see also Shock; Injuries\n\nClingy Ant-t., Bor., Gels., Phos., Puls., Stram.\n\nClumsy Agar., Ap., Calc-c., Caust.\n\nDrops things: Ap.\n\nDuring pregnancy: Calc-c.\n\nTrips easily while walking: Agar., Caust.\n\nCold sores Nat-m.\n\nColds see Common cold\n\nColic (see also Indigestion; Heartburn) Cham., Coloc., Cupr., Dios., Ip., Mag-m., Mag-p., Nat-p., Nat-s., Nux-v., Sec., Stap.\n\nIn babies: Cham., Coloc., Dios., Mag-m., Sec.\n\nAbdomen\/stomach feels:\n\nbloated: Cham., Coloc., Nat-p., Sec.\n\nrumbling\/windy: Dios.\n\nPains:\n\naching: Ip.\n\naround the navel: Dios.\n\ncramping: Cupr., Dios., Ip., Mag-m., Mag-p., Nux-v., Stap.\n\ncutting: Coloc., Dios.\n\ndrawing: Mag-p.\n\ngriping: Coloc., Dios., Ip., Nux-v.\n\npressing: Nux-v.\n\nsore, bruised: Mag-m., Nux-v.\n\ntearing: Coloc.\n\ntwisting: Dios.\n\nviolent: Coloc., Cupr.\n\nin waves: Coloc.\n\nWith:\n\nconstipation: Mag-m.\n\ndiarrhoea: Cham., Coloc., Mag-m., Nat-p., Sec.\n\nindigestion: Mag-m., Nat-s.\n\nnausea: Coloc., Cupr.\n\nvomiting: Coloc., Cupr.\n\nsour vomit: Nat-p.\n\nBetter:\n\nbending back\/stretching out: Dios.\n\nbending double\/pressure: Coloc., Mag-p.\n\nhot drinks\/warmth of bed: Nux-v.\n\npassing a stool: Coloc., Nux-v.\n\npassing wind: Nux-v.\n\npressure: Coloc.\n\nstretching out: Dios.\n\nwarmth: Mag-p.\n\nwarmth of bed: Nux-v.\n\nWorse:\n\nafter drinking: Coloc., Stap.\n\nafter eating: Nux-v.\n\nafter eating fruit: Coloc.\n\nbending forward: Dios.\n\ncold drinks when overheated: Coloc.\n\ncoughing: Nux-v.\n\nduring a fever: Nux-v.\n\nbefore a stool: Coloc.\n\nafter drinking milk: Mag-m.\n\nin the morning: Dios., Nux-v.\n\nfor movement: Ip.\n\ntight clothing: Nux-v.\n\nCauses:\n\nanger: Cham., Coloc.\n\ndrinking milk: Mag-m.\n\nexcitement: Coloc.\n\nfruit: Coloc.\n\nindignation: Stap.\n\nsyntometrine: Sec.\n\nteething: Mag-m.\n\nvexation: Coloc., Stap.\n\nCollapse\n\nAfter diarrhoea: Ver-a.\n\nCommon cold (see also Snuffles) Aco., All-c., Ars., Bar-c., Bell., Bry., Calc-c., Calc-f., Calc-p., Calc-s., Carb-v., Dulc., Euphr., Hep-s., Kali-b., Kali-m., Kali-s., Lyc., Mag-m., Merc-s., Nat-c., Nat-m., Nit-ac., Nux-v., Pho-ac., Phos., Puls., Sep., Sil., Sul.\n\nEyes:\n\ndry\/burning: Ars.\n\nstreaming: All-c., Euphr.\n\nDischarge from eyes:\n\nburning: Euphr.\n\nwatery: All-c., Nat-m., Nux-v.\n\nEyelids:\n\nred\/puffy: Ars.\n\nGlands:\n\nswollen: Bar-c.\n\nNasal catarrh:\n\nbland (not irritating): Euphr., Puls.\n\nblood-streaked: Calc-s., Phos.\n\nbloody: Merc-s., Nit-ac.\n\nburning: All-c., Ars., Merc-s., Nit-ac., Nux-v.\n\ncrusty: Kali-b.\n\ndrips down back of throat: Hep-s., Kali-b., Nat-c., Nat-m., Sep.\n\ndry (stuffed up\/congested without discharge): Calc-c., Calc-f., Kali-b., Phos., Sil.\n\ndry at night, profuse during the day: Nux-v.\n\ndry alternating with profuse: Puls.\n\nlike egg-white: Nat-m.\n\ngreen: Kali-b., Merc-s., Puls., Sep.\n\nhard crusts: Kali-b., Sil.\n\none-sided: All-c., Phos.\n\nprofuse: All-c., Ars., Kali-s., Nat-m., Phos.\n\nsmelly: Calc-c., Calc-s., Hep-s., Kali-b., Merc-s., Nat-c., Puls., Sil., Sul.\n\nsticky\/stringy: Kali-b.\n\nthick: Calc-s., Dulc., Kali-b., Kali-s., Nat-c., Puls., Sil.\n\nthin: Nit-ac.\n\nwatery: All-c., Ars., Euphr., Merc-s., Nit-ac., Nux-v.\n\nwatery, alternating with blocked up nose: Nat-m.\n\nwatery in the fresh air: Nit-ac., Puls.\n\nwhite: Kali-m., Nat-m.\n\nyellow: Calc-c., Calc-s., Dulc., Hep-s., Kali-b., Kali-s., Lyc., Merc-s.\n\ndirty yellow: Nit-ac., Puls., Sep., Sul.\n\nyellow-green: Kali-b., Merc-s., Puls., Sep.\n\nNose:\n\nblocked: Calc-c., Carb-v., Dulc., Lyc., Nat-c., Nit-ac.\n\ndry: Bar-c., Lyc., Sul.\n\nitching: Sul.\n\nruns during day, blocked at night: Nux-v.\n\nstreaming: All-c.\n\nSinuses:\n\nblocked\/painful: Ars., Kali-b., Lyc., Merc-s., Nux-v., Puls., Sil.\n\nSneezing: All-c., Ars., Bry., Carb-v., Eup-p., Hep-s., Merc-s., Nux-v., Puls., Sul.\n\nbetter for sneezing: Calc-f.\n\nfrequent sneezing: Ars., Carb-v., Merc-s., Nux-v., Sul.\n\ndifficult sneezing: Calc-f., Carb-v., Sil.\n\nin a stuffy room: Puls.\n\nslightest uncovering: Hep-s.\n\nWith:\n\ncough: Bar-c., Euphr.\n\ndeafness after a cold: Kali-m.\n\ndry throat: Kali-b.\n\neye inflammation: All-c., Euphr., Sul.\n\nfever: Bell., Merc-s.\n\nheadache: Aco., All-c., Bell., Bry., Calc-s., Lyc., Merc-s., Nux-v.\n\nhoarseness: Calc-c., Carb-v., Merc-s., Phos.\n\nitching throat: Carb-v.\n\npainless hoarseness: Calc-c.\n\nloss of smell: Bell., Calc-c., Calc-s., Merc-s., Nat-m., Phos., Puls., Sep., Sil.\n\nloss of taste: Bell., Nat-m., Phos., Puls., Sil.\n\nsneezing: All-c., Ars., Bry., Merc-s.\n\nsore throat: All-c., Merc-s., Nux-v., Phos.\n\nBetter:\n\nfresh air: All-c., Kali-s., Nux-v., Puls.\n\nWorse:\n\ncold and heat: Merc-s.\n\nafter eating: Nux-v.\n\nfresh air: Nit-ac.\n\nafter getting up: Nux-v.\n\nmorning: Calc-c., Nux-v.\n\nevening: Ars.\n\nnight: Bar-c., Merc-s., Nit-ac.\n\non the right: Ars., Lyc.\n\nfor drinking milk: Calc-s.\n\nstuffy room: All-c., Kali-s., Puls.\n\nuncovering: Hep-s.\n\nCauses:\n\ngetting chilled: Aco., Bell.\n\ngetting chilled when overheated: Ars.\n\ncold wind: All-c.\n\ncold, dry wind: Aco., Bell.\n\ndraughts: Nux-v.\n\ngetting head wet: Bell.\n\ngetting wet feet: All-c.\n\nshock: Aco.\n\nswimming in the sea: Mag-m.\n\nwet weather: Dulc.\n\nComplaints from\/after (see also Worse)\n\nAccident\/injury: Arn., Bell-p., Con., Hyp., Nat-s.\n\nAnger: Cham., Cocc., Colch., Coloc., Ign., Ip., Nux-v., Op.\n\nsuppressed anger: Ign., Lyc., Stap.\n\nanger with anxiety: Ars., Ign., Nux-v.\n\nBad news: Gels.\n\nGetting chilled: Aco., Bry., Merc-s., Nux-v., Phos., Rhus-t.\n\nwhen overheated: Bell-p.\n\nChange of weather: Bry., Caust., Dulc., Kali-s., Phos., Rhod., Rhus-t., Sil., Sul.\n\nany change: Phos.\n\nfrom cold to warm: Bry., Kali-s., Sul.\n\nto cold: Rhus-t., Sil.\n\nto damp: Dulc., Rhus-t.\n\nto dry (especially dry cold): Caust.\n\nto stormy: Rhod.\n\nChildbirth: Cimi., Stram.\n\nCoffee: Cham.\n\nCold wind: Aco., Bell., Hep-s., Nux-v., Spo.\n\nDeath of a child: Ign.\n\nDisappointed love: Ign., Nat-m., Pho-ac., Stap.\n\nExcitement: Coff., Gels., Pho-ac., Puls., Stap.\n\nFear: Aco.\n\nFright: Ign., Pho-ac., Puls.\n\nGrief: Caust., Cocc., Ign., Lach., Nat-m., Pho-ac., Puls., Stap.\n\nHumiliation (wounded pride): Coloc., Ign., Lyc., Nat-m., Pho-ac., Stap.\n\nIndignation: Colch., Coloc., Stap.\n\nInjuries to nerves\/coccyx: Hyp.\n\nJoy: Coff.\n\nLoss of body fluids: Calc-c., Carb-v., Chin., Pho-ac.\n\nMeasles: Carb-v., Puls.\n\nMental strain: Arg-n., Kali-p., Lach., Lyc., Nat-c., Sil., Stap.\n\nOvereating: Ant-c.\n\nOverwork: Kali-p.\n\nReprimands: Ign., Op., Stap.\n\nShock: Aco., Arn., Ign., Op., Pho-ac., Puls., Sil.\n\nSprains: Calc-c.\n\nSunstroke: Glon., Nat-c., Nat-m.\n\nSuppression of emotions: Ign., Nat-m., Stap.\n\nSurgery: Arn., Bell-p., Hyp., Stap.\n\nTeething: Cham.\n\nVexation: Ip.\n\nWeaning: Bry., Dulc., Puls.\n\nGetting wet: Calc-c., Caust., Puls., Rhus-t., Sep.\n\nGetting feet wet: Puls., Sil.\n\nGetting head wet: Bell.\n\nConcentration\n\nPoor: Bar-c., Caust., Lyc., Nux-v., Sil.\n\nConfidence see Self-confidence, Lack of\n\nConfused (see also Absent-minded; Forgetful) Arg-n., Bell., Calc-c., Carb-v., Cocc., Glon., Lach., Merc-s., Nat-c., Nat-m., Op., Petr., Rhus-t., Sep., Sil., Stram., Sul.\n\nIn the fresh air: Petr.\n\nConjunctivitis see Eye inflammation\n\nConscientious Ign., Sil.\n\nConstipation Aesc., Alu., Calc-c., Caust., Hep-s., Kali-c., Lyc., Mag-m., Nat-m., Nux-v., Op., Puls., Sep., Sil., Sul., Zinc.\n\nAlternates with diarrhoea: Nux-v.\n\nAbdomen\/stomach feels full: Sep.\n\nIn babies: Alu., Lyc., Mag-m., Nux-v., Op.\n\nIn newborn babies: Zinc.\n\nIn pregnancy: Aesc., Alu., Kali-c., Lyc., Nux-v., Puls., Sep., Sul.\n\nDesire to pass stool:\n\nabsent: Alu., Op.\n\nconstant: Nux-v.\n\nineffectual: Caust., Lyc., Mag-m., Nat-m., Nux-v., Sep., Sil.\n\nPains:\n\nburning after stool: Aesc., Sil.\n\nstitching: Caust.\n\nStools:\n\nblack balls: Op.\n\nchangeable: Puls.\n\ncrumbling: Mag-m., Nat-m.\n\nhard: Aesc., Kali-c., Lyc., Mag-m., Nux-v., Op., Sep., Sil.\n\nhard at first: Calc-c.\n\nknotty: Lyc., Mag-m., Sil.\n\nlarge: Aesc, Calc-c., Kali-c., Mag-m., Nux-v., Sep., Sil.\n\npale: Calc-c.\n\npassed with difficulty: Mag-m.\n\nlike rabbits'\/sheep's droppings: Mag-m., Nat-m., Op.\n\n'shy' (they recede): Op., Sil.\n\nsmall balls: Mag-m., Nat-m., Op.\n\nsoft: Alu., Hep-s.\n\nsour smelling: Calc-c.\n\nWith:\n\nburning, redness and itching around anus: Sul.\n\nflatulence: Lyc.\n\nineffectual straining: Aesc., Alu., Mag-m., Nat-m., Sep., Sil.\n\nunfinished feeling: Kali-c., Nat-m., Nux-v., Op.\n\nBetter:\n\nbeing constipated: Calc-c.\n\npassing a stool when standing: Caust.\n\nWorse:\n\nbefore a menstrual period: Sil.\n\nduring a menstrual period: Nat-m., Sep., Sil.\n\nCauses:\n\nbottle-feeding: Alu., Nux-v., Op.\n\ncow's milk: Mag-m.\n\ndrinking milk: Mag-m.\n\novereating: Nux-v.\n\npregnancy: Aesc., Nux-v., Sep.\n\nsedentary habits: Nux-v.\n\nweaning: Alu., Nux-v., Op.\n\nConvulsions Bell., Cina, Cupr., Stram.\n\nWith:\n\nblue lips: Cupr.\n\ncoldness of hands and feet: Cupr.\n\nrage: Stram.\n\nCauses:\n\nover-excitement: Stram.\n\nteething babies: Bell., Cina, Cupr.\n\nvexation: Cupr.\n\nworms: Cina\n\nCough Aco., All-c., Ant-t., Arn., Ars., Bar-c., Bell., Bry., Calc-c., Calc-p., Calc-s., Carb-v., Caust., Cham., Cina, Cocc-c., Con., Cupr., Dros., Dulc., Euphr., Ferr-m., Hep-s., Ign., Ip., Kali-b., Kali-c., Kali-m., Kali-s., Lach., Lyc., Merc-s., Nat-m., Nat-s., Nux-v., Op., Pho-ac., Phos., Puls., Rhus-t., Rumex, Sep., Sil., Spo., Stap., Sul., Ver-a.\n\nBreathing:\n\nasthmatic: Ant-t.\n\nabdominal\/difficult: Ant-t.\n\ndifficult: Ant-t., Ars., Cupr., Dros., Ip., Lyc., Nux-v., Op., Phos., Spo., Ver-a.\n\nfast: Aco., Ant-t., Ars., Bell., Bry., Carb-v., Cupr., Dros., Ip., Lyc., Phos., Sep., Sul.\n\nrattling: Ant-t., Dulc.\n\nrough: Spo.\n\nslow: Bell., Op.\n\nsnoring: Op.\n\nwheezing: Ars., Carb-v., Ip., Kali-c.\n\nCough:\n\nbarking: Aco., Bell., Dros., Hep-s., Kali-m., Spo.\n\nchoking: Cocc-c., Ip.\n\nas soon as falls into a deep sleep: Lach.\n\nconstant: Caust., Lyc., Rumex, Spo.\n\nwhen lying down at night: Sep.\n\nin the evening: Puls.\n\ncroupy see Croup\n\ndeep: Dros., Ver-a.\n\ndisturbs sleep: Bry., Kali-c., Lyc., Puls., Sep., Sul.\n\ndistressing: Caust., Nux-v.\n\ndry: Aco., Ars., Bell., Bry., Calc-c., Calc-s., Con., Dros., Hep-s., Ign., Ip., Kali-c., Lach., Lyc., Nat-m., Nux-v., Pho-ac., Phos., Rumex, Sep., Spo., Sul.\n\ndry in the evening, loose in the morning: Calc-c., Hep-s., Sul.\n\ndry at night: Ars., Calc-c., Cham., Dros., Hep-s., Phos., Puls., Sul.\n\nloose by day: Sul.\n\nloose in the morning: Calc-c., Puls., Sul.\n\ndry during fever: Aco., Phos., Puls.\n\nexhausting: Ars., Bell., Caust., Sep. night: Puls.\n\nhacking: All-c., Ars., Dros., Hep-s., Lach., Nat-m., Phos., Sep.\n\nevening in bed after lying down: Ign., Sep.\n\nworse for cold air: All-c.\n\nworse after dinner: Hep-s.\n\nfrom tickling inthe larynx: All-c., Ars., Dros., Lach., Nat-m., Phos.\n\nhard: Bell., Kali-c., Kali-m.\n\nhollow: Bell., Caust., Spo., Ver-a.\n\nin fits: Bell., Bry., Carb-v., Cina, Cocc., Cocc-c., Ferr-m., Ip., Puls., Rumex, Spo.\n\nin long fits at irregular intervals: Cupr.\n\nirritating: Aco., All-c., Bell., Bry., Cham., Cocc-c., Con., Dros., Hep-s., Ign., Ip., Kali-c., Lach., Lyc., Nat-m., Nux-v., Pho-ac., Phos., Puls., Rhus-t., Rumex, Sep., Sil., Spo., Stap., Sul.\n\nloud: Ant-t.\n\nloose: Ars., Kali-c., Sep.\n\npainful: All-c., Bry., Lyc., Sul.\n\nracking: Bell., Bry., Carb-v., Caust., Cocc-c., Ign., Kali-c., Merc-s., Nux-v., Phos., Puls., Sul.\n\nrattling: Ant-t., Caust., Dulc., Ip., Kali-s., Sep.\n\nshort: Aco., Ign., Kali-m., Rhus-t., Sep.\n\nsuffocative: Carb-v., Cina, Cupr., Dros., Hep-s., Lach., Nux-v., Sul.\n\ntickling see irritating\n\ntight: Phos.\n\ntormenting: Ars., Bell., Caust., Dros., Ip.\n\nuninterrupted: Cupr.\n\nviolent: Bell., Carb-v., Caust., Cocc-c., Con., Cupr., Hep-s., Ign., Kali-c., Lach., Pho-ac., Phos., Puls., Sep.\n\nviolent fits: Dros., Nux-v.\n\nvomiting with or after cough: Ant-t., Bry., Dros., Hep-s., Ip., Kali-c.\n\nwakens from the cough at night: Caust., Kali-c., Phos., Sep., Sul.\n\nwhooping: Ant-t., Arn., Carb-v., Cocc-c., Cupr., Dros., Ip., Kali-s.\n\nChest congested: Sul.\n\nDaytime only: Euphr., Nat-s.\n\nExpectoration see Mucus\n\nIn pregnancy: Bell., Bry., Calc-c., Caust., Phos., Puls.\n\nLarynx:\n\nraw: Nux-v., Phos., Rumex, Sul.\n\ntickling: All-c., Ars., Cocc-c., Dros., Lach., Nat-m., Phos.\n\nMucus:\n\nbloody: Ip., Phos.\n\ncopious: Ars., Calc-c., Calc-s., Cocc-c., Euphr., Hep-s., Lyc., Phos., Puls., Sep.\n\nafter each coughing fit: Cocc-c.\n\ndifficult to cough up: Ant-t., Caust., Con., Ip., Kali-b., Kali-s., Puls.\n\nhas to swallow what comes up: Caust., Con., Kali-s.\n\negg white: Nat-m.\n\nfrothy: Ars.\n\ngreen: Carb-v., Lyc., Merc-s., Nat-s., Phos., Puls., Sul.\n\nlumpy: Calc-s., Sil.\n\nsmelly: Calc-c.\n\nsticky: Cocc-c., Hep-s., Kali-b., Puls.\n\nropy\/stringy: Cocc-c., Kali-b.\n\ntastes:\n\nbitter: Cham., Puls.\n\nsour: Calc-c., Nux-v.\n\nsweet: Calc-c., Phos.\n\nsalty: Ars., Lyc., Phos., Puls., Sep.\n\nthick: Hep-s., Kali-b., Sil.\n\ntough: Calc-c., Hep-s., Kali-b.\n\ntransparent: Nat-m., Phos.\n\nwhite: Lyc., Nat-m., Phos., Sep.\n\nyellow: Calc-c., Calc-p., Calc-s., Hep-s., Kali-s., Lyc., Phos., Puls., Sep., Sil.\n\nyellow-green: Puls.\n\nPain in chest: Arn., Bell., Bry., Caust., Dros., Ign., Kali-b., Kali-c., Phos., Rumex, Spo.\n\nburning: Phos., Rumex, Spo.\n\ncutting: Kali-c.\n\nmust hold chest to cough: Arn., Bry., Dros.\n\nracking: Ign.\n\nraw: Caust.\n\nsharp: Bell.\n\nshort: Ign.\n\nsore, bruised: Arn., Kali-b., Rumex, Spo.\n\nstitching: Bry., Ign., Kali-c., Rumex\n\nPain in stomach: Bry.\n\nWith:\n\nbloodshot eyes: Arn.\n\nblue face: Dros., Ip.\n\ndry throat: Cocc-c.\n\nlump sensation in the throat: Lach.\n\nnausea: Ip., Kali-c., Puls.\n\nnosebleeds: Arn., Dros., Ip.\n\nretching: Carb-v., Cina, Cocc-c., Dros., Hep-s., Ip., Puls.\n\nsleepiness: Ant-t.\n\nsplitting headache: Bry.\n\nsweating: Ars., Hep-s., Phos.\n\nvomiting: Ant-t., Bry., Dros., Hep-s., Ip., Kali-c. when hawking up mucus: Cocc-c., Nux-v.\n\nvomiting of mucus: Cocc-c., Dros.\n\nvoice hoarse: Aco., All-c., Bell., Carb-v., Caust., Dros., Hep-s., Kali-b., Spo.\n\nBetter:\n\ndrinking\/eating: Spo.\n\nfor being in warm room: All-c.\n\nfresh air: Bry., Cocc-c., Puls.\n\nheat: Phos.\n\nhot drinks: Ars., Lyc., Nux-v., Rhus-t., Sil.\n\nlying down: Euphr.\n\nlying on painful side: Bry.\n\npressure: Dros.\n\nfor sips of cold water: Caust., Cupr.\n\nsitting up: Puls., Sep.\n\nmust sit up: Ars., Con., Phos., Puls., Sep.\n\nmust sit up as soon as cough starts, to cough up mucus and then can rest: Con.\n\nwalking slowly: Ferr-m.\n\nWorse:\n\naround 11.30 p.m.: Cocc-c.\n\naround 3 a.m.: Kali-c.\n\nbending head forward: Caust.\n\nbreathing in cold air: Caust., Rumex\n\nchange of temperature: Phos.\n\nfrom warm to cold: Phos., Rumex\n\nfrom cold to warm: Phos.\n\nbecoming cold: Ars., Hep-s., Kali-c., Nux-v., Phos., Rhus-t., Rumex, Sil.\n\ncold air: All-c., Caust., Cupr., Hep-s., Nux-v., Phos., Rumex\n\nsingle parts of the body becoming cold: Hep-s., Rhus-t.\n\ncold drinks: Ars.\n\ncoughing: Ign., Sul.\n\ncrying: Arn.\n\ndamp, wet weather: Dulc., Nat-s.\n\ndaytime: Spo.\n\ndeep breathing: Bell., Bry., Con., Kali-c.\n\ndrinking: Bry., Dros.\n\ndry, cold: Aco., Hep-s.\n\neating: Bry., Kali-b., Nux-v.\n\nevening: Ars., Calc-c., Carb-v., Hep-s., Ign., Kali-c., Merc-s., Nit-ac., Puls.\n\nevening in bed: Calc-c., Coloc., Con., Hep-s., Ign., Lach., Lyc., Merc-s., Nat-m., Sep., Sul.\n\nevening before midnight: Carb-v., Hep-s., Phos.\n\nexcitement: Spo.\n\nduring fever: Aco., Ars., Calc-c., Con., Ip., Kali-c., Nat-m., Nux-v., Phos.\n\nfresh air: Ars., Phos., Rumex\n\ngetting hot: Puls.\n\ngetting warm in bed: Caust., Puls.\n\nheat: Kali-c., Kali-s., Puls.\n\nleft side: Rumex\n\nlying on left side: Phos.\n\nlying on right side: Merc-s.\n\nlying down: Ap., Ars., Caust., Con., Dros., Kali-c., Lyc., Puls., Rumex, Sep., Sul.\n\nmust sit up: Ars., Con., Puls., Sep.\n\nmorning: Calc-c., Euphr., Kali-c., Nux-v., Phos., Puls., Rumex, Sul.\n\nbefore getting up: Nux-v.\n\nafter getting up: Cina, Ferr-m., Phos.\n\nmovement: Ferr-m.\n\nmovement of chest: Bry., Chin., Nux-v.\n\nmoving arms: Nat-m.\n\nnight: Aco., Ars., Bar-c., Bell., Calc-c., Carb-v., Cham., Kali-c., Kali-s., Lach., Lyc., Merc-s., Puls., Sep., Sul.\n\non going to sleep: Lyc.\n\nbefore midnight: Carb-v., Hep-s.\n\nafter midnight: Ars., Dros.\n\nphysical exertion: Puls.\n\nplaying the piano: Calc-c.\n\nreading aloud: Phos.\n\nright lung: Bry.\n\nduring sleep: Op.\n\nfalling sleep: Op.\n\nstuffy room: Ant-c., Cocc-c., Kali-s., Puls.\n\nwarm room: Kali-s.\n\ntalking: Dros., Kali-c., Rumex\n\nteething: Calc-p.\n\ntouch: Lach.\n\nuncovering: Hep-s., Rhus-t., Rumex\n\nhands: Hep-s., Rhus-t.\n\nfeet or head: Sil.\n\nwalking: Ferr-m., Rumex.\n\non waking in the morning: Kali-b., Nux-v., Op., Rumex, Sil.\n\nwarmth of bed: Op.\n\nCauses:\n\ncold, dry wind: Aco., Hep-s.\n\ngetting chilled: Bell., Kali-c.\n\ndamp weather: Dulc.\n\nafter measles: Dros., Puls.\n\nsuppressed anger: Stap.\n\nswimming in cold water: Rhus-t.\n\nCracks at corners of mouth\/around nostrils: Ant-c.\n\nCradle cap Calc-c., Lyc., Sul.\n\nCramp Calc-c., Calc-p., Caust., Coloc., Cupr., Lyc., Sep., Sul., Ver-a.\n\nArm: Mag-p.\n\nCalf: Calc-c., Calc-p., Coloc., Cupr., Lyc., Sep., Sul., Ver-a.\n\nFoot: Caust., Cupr.\n\nFingers and hand when writing: Mag-p.\n\nHand: Calc-c., Mag-p.\n\nLeg: Coloc., Cupr., Sep., Sul., Ver-a.\n\nSoles of feet: Calc-c., Caust., Sul.\n\nThigh: Coloc., Sul.\n\nToe: Calc-c., Caust.\n\nWrist: Mag-p\n\nBetter:\n\nheat: Mag-p.\n\nWorse:\n\nat night: Calc-c., Caust., Lyc., Sul.\n\nin bed: Calc-c., Sul.\n\nchildbirth: Cupr.\n\nduring labour (in hands or legs): Bell., Cupr., Nux-v.\n\nin pregnancy: Calc-c., Cupr., Sep., Ver-a.\n\npreventing sleep: Ver-a.\n\nstretching leg in bed: Calc-c.\n\nwhen walking: Calc-p., Sul.\n\nCauses:\n\nchildbirth: Cupr.\n\nprolonged use of hands: Mag-p.\n\nCravings see Likes\n\nCries\/Crying see Tearful\n\nCritical Ars., Sul.\n\nCroup (see Cough for symptoms) Aco., Calc-s., Hep-s., Kali-b., Lach., Phos., Spo.\n\nOnly on waking: Calc-s.\n\nRecurrent: Hep-s.\n\nWorse:\n\nafter sleep: Lach.\n\non waking: Lach.\n\nCystitis (see also Incontinence) Aco., Ap., Ars., Canth., Caust., Dulc., Lyc., Merc-c., Merc-s., Nux-v., Puls., Sars., Sep., Stap., Sul.\n\nDesire to urinate:\n\nconstant: Ap., Canth., Merc-c., Merc-s., Sul. with dragging down pain in pelvis: Sep.\n\nfrequent: Ap., Canth., Caust., Lyc., Merc-c., Merc-s., Nux-v., Puls., Stap., Sul.\n\nineffectual: Ars., Canth., Caust., Lyc., Merc-s., Nux-v., Puls., Sars.\n\npainful: Merc-c., Puls. in pregnancy: Puls.\n\nurgent: Canth., Nux-v., Puls., Sep., Sul.\n\nPains:\n\naching: Lyc.\n\nburning: Ap., Ars., Canth., Caust., Merc-c., Merc-s., Nux-v., Sul.\n\ncutting: Canth., Lyc., Sars.\n\npressing: Aco., Ap., Lyc., Nux-v., Sep.\n\nsevere: Merc-c.\n\nspasmodic: Canth., Puls.\n\nstinging: Ap.\n\nstitching: Lyc.\n\nUrination:\n\nburning: Caust.\n\nconstant: Ap.\n\ndifficult: Caust., Merc-c.\n\ndribbling: Merc-c., Merc-s., Sars.\n\nfrequent: Ap., Canth., Caust., Lyc., Merc-c., Merc-s., Nux-v., Puls., Sars., Sep., Stap.\n\nineffectual: Nux-v.\n\ninvoluntary: Caust., Dulc., Stap.\n\nslow: Sars.\n\ncan only pass urine while standing: Sars.\n\nwaits long time for it to start: Caust., Lyc., Sep.\n\nwith unfinished feeling: Ars., Caust.\n\nUrine:\n\nbrown like beer: Sul.\n\nburning: Nux-v., Sul.\n\ncloudy: Sep.\n\ncopious: Puls., Sul.\n\ndark: Merc-c., Merc-s.\n\ndark brown: Sep.\n\ngreen: Merc-c., Sars.\n\nhot: Canth.\n\npale: Sars.\n\nred: Canth., Merc-c., Sep.\n\nscanty: Canth., Merc-c., Merc-s., Nux-v., Sep., Stap., Sul.\n\nsmelly: Sep., Sul.\n\nwith sediment: Sars., Sep.\n\nwith mucus: Sars.\n\nBetter (pains):\n\nafter urination: Lyc.\n\nheat: Ars.\n\nfor sitting in a hot bath: Ars.\n\nWorse (pains):\n\nfor cold drinks: Canth.\n\nafter getting up in the morning: Sul.\n\nafter a few drops of urine have passed: Canth.\n\nafter urinating: Canth., Puls.\n\nat the beginning of urinating: Canth., Merc-s.\n\nat the end of urination: Sars.\n\nbefore urinating: Canth.\n\nduring a menstrual period: Sars., Sep.\n\nwhen not urinating: Merc-s.\n\nwhile urinating: Canth., Caust., Merc-c., Nux-v., Sul.\n\nat night: Sul.\n\nCauses:\n\ncold, wet weather: Sars.\n\ngetting chilled: Aco., Sars.\n\ngetting cold and wet: Dulc., Puls.\n\nDazed (see also Shock; Stupor) Cocc.\n\nDelirious Bell.\n\nDenial\n\nOf illness, of suffering: Arn.\n\nOf suffering in labour: Arn.\n\nDepressed (see also Tearful) Alu., Ars., Calc-c., Calc-s., Carb-a., Caust., Cham., Chin., Cimi., Con., Eup-p., Ferr-m., Gels., Ign., Kali-p., Lach., Lil-t., Lyc., Merc-s., Nat-c., Nat-m., Nat-s., Nit-ac., Op., Pho-ac., Puls., Rhus-t., Sep., Sul., Ver-a., Zinc.\n\nCannot cry: Gels., Ign., Nat-m.\n\nCries on own: Ign., Nat-m.\n\nIn pregnancy: Chin., Cimi., Con., Ign., Lach., Nat-m., Puls.\n\nIn labour: Cimi., Nat-m., Puls.\n\nBetter:\n\nfresh air: Puls.\n\nWorse:\n\nbefore menstrual period: Nat-m., Puls.\n\nstuffy room: Puls.\n\non waking: Alu., Lach., Lyc.\n\nevening: Puls.\n\nCauses:\n\nsuppressed grief: Ign., Nat-m., Puls.\n\nDesires\n\nTo be alone: Alu., Carb-a., Cham., Gels., Ign., Lach., Nat-m., Sep.\n\nTo be carried (infants): Bry., Cham., Cina, Kali-c., Kreos., Lyc., Puls.\n\nCompany: Arg-n., Ars., Kali-c., Lyc., Phos.\n\nIn pregnancy and\/or labour: Lach., Nat-m.\n\nIn pregnancy: Ars., Bry., Cina, Kreos., Lyc., Kali-c., Rhus-t., Ver-a.\n\nDespair Ars., Calc-c., Coff., Ign., Lyc., Sul., Ver-a.\n\nIn pregnancy: Nat-m.\n\nIn labour: Coff., Gels., Sep.\n\nOf getting well: Ars., Calc-c.\n\nDespondent (see also Depressed) Chin.\n\nDiaper rash see Nappy rash\n\nDiarrhoea Ant-c., Ap., Arg-n., Ars., Bor., Bry., Calc-c., Calc-p., Cham., Chin., Colch., Coloc., Dulc., Ferr-m., Gels., Hep-s., Ip., Mag-c., Mag-m., Merc-c., Merc-s., Nat-m., Nat-s., Nux-v., Petr., Pho-ac., Phos., Podo., Puls., Rhe., Rhus-t., Sil., Sul., Sul-ac., Ver-a.\n\nAlternating with:\n\nconstipation: Podo.\n\nheadache: Podo.\n\nAbdomen\/stomach:\n\nfeels bloated: Nat-m.\n\ngurgling: Nat-s., Podo.\n\nrumbling: Nat-s., Pho-ac., Podo.\n\nBreastfeeding babies: Calc-p., Cham.\n\nChildren (infants): Ip., Mag-m., Merc-s., Podo., Puls., Sul.\n\nDaytime only: Nat-m., Petr.\n\nDrives person out of bed: Sul.\n\nIn pregnancy: Chin., Phos., Puls., Sul.\n\nExhausting: Ver-a.\n\nInvoluntary: Ver-a.\n\nMorning only: Sul.\n\nPainful: Ars., Cham., Colch., Dulc., Mag-c., Merc-c., Merc-s.\n\nPains:\n\nburning: Ars., Merc-s., Sul., Ver-a.\n\ncolicky: Mag-c.\n\ncramping: Mag-c., Sul.\n\ncutting: Ver-a.\n\ndull\/aching: Ver-a.\n\nafter passing a stool: Ars., Colch.\n\npressing: Petr.\n\nsevere: Merc-c.\n\nPainless: Ap., Bor., Chin., Ferr-m., Hep-s., Nat-m., Phos., Podo., Sul.\n\nStools:\n\nblood-streaked: Phos.\n\nbloody: Merc-c., Merc-s.\n\nchangeable: Puls.\n\ncopious: Ver-a.\n\nfrequent: Merc-c., Phos., Podo., Ver-a.\n\nfrothy: Mag-c.\n\ngrass green: Ip.\n\ngreen: Arg-n., Cham., Coloc., Mag-c., Mag-m., Merc-c., Merc-s., Ver-a.\n\ngreenish-yellow: Puls.\n\ngushing: Nat-m: Podo.\n\njelly-like: Colch.\n\nhot: Cham.\n\ninvoluntary: Podo.\n\nwith mucus: Bor., Colch.\n\nmushy: Rhus-t.\n\npassed with wind: Ferr-m.\n\npasty: Coloc., Rhe.\n\nprofuse: Pho-ac., Phos., Podo.\n\nslimy: Mag-c., Merc-c., Merc-s., Puls., Sul.\n\nsmall, hard lumps: Ant-c.\n\nsmelling of rotten eggs: Cham.\n\nsmelly: Arg-n., Ars., Merc-c., Merc-s., Nat-m., Nat-s., Podo., Sul.\n\nsoft: Sul-ac.\n\nsour smelling: Calc-c., Mag-c., Rhe., Sul.\n\nstringy: Sul-ac.\n\nsudden: Podo.\n\nthin: Nat-s., Pho-ac.\n\nwith undigested food: Calc-c., Chin., Ferr-m.\n\nyellow: Dulc., Merc-c., Merc-s., Sul-ac.\n\nwatery: Ant-c., Arg-n., Ars., Calc-c., Colch., Dulc., Nat-m., Nat-s., Pho-ac., Phos., Puls., Rhus-t., Sul., Ver-a.\n\nwhite: Pho-ac.\n\nStraining frequent\/painful: Merc-c.\n\nViolent: Ver-a.\n\nWith:\n\nbelching: Ferr-m.\n\nsour belching: Sul-ac.\n\ncolic: Coloc.\n\nexhaustion: Ars., Podo., Ver-a.\n\nflatulence: Arg-n., Ferr-m., Nat-s., Sil., Sul.\n\nicy-cold hands and feet: Ars., Phos., Ver-a.\n\nindigestion: Chin.\n\nnausea: Ars.\n\nravenous appetite: Petr.\n\nsweating: Ars., Merc-s., Ver-a.\n\nduring\/after passing stool: Ver-a.\n\nbefore\/during\/after stool: Merc-s.\n\nvomiting: Arg-n., Ver-a.\n\nsour: Sul-ac.\n\nwind:\n\nloud: Nat-s.\n\nsmelly: Nat-s., Sil., Sul.\n\nspluttering: Nat-s.\n\nWithout weakness: Pho-ac.\n\nBetter:\n\ncold food: Phos.\n\nfresh air: Puls.\n\nheat: Sil.\n\npassing wind: Nat-s., Sul.\n\nwarmth of bed: Sil.\n\nwrapping up warmly: Sil.\n\nWorse:\n\non alternate days: Chin.\n\naround 4 a.m.: Podo.\n\n5 a.m.: Sul.\n\nafternoon: Chin.\n\nbefore passing a stool (pains): Mag-c., Merc-c.\n\nduring passing a stool: Merc-c.\n\nafter passing a stool: Merc-c., Merc-s.\n\ncold: Ars., Merc-s.\n\ngetting chilled: Phos.\n\nafter cold food\/drinks: Ars.\n\nafter drinking: Ars., Arg-n., Ver-a.\n\nafter drinking milk: Calc-c.\n\nafter drinking water: Ferr-m., Nux-v., Podo.\n\nimmediately after drinking: Arg-n., Podo.\n\nafter eating: Ars., Chin., Coloc., Podo., Puls.\n\nwhile eating: Ferr-m.\n\neating cold food: Ars., Dulc.\n\neating starchy food: Nat-m., Nat-s., Puls.\n\nevening: Merc-s.\n\nafter eating fruit: Ars., Coloc., Nat-s.\n\nafter getting up in the morning: Bry., Nat-s.\n\nhot weather: Podo.\n\nduring a menstrual period: Ver-a.\n\nafter midnight: Ars.\n\nmid-morning: Podo.\n\nmorning: Bry., Mag-c., Nat-s., Phos., Sul.\n\nmovement: Bry., Ferr-m., Ver-a.\n\nnight: Arg-n: Ars., Chin., Dulc., Ferr-m., Merc-s., Podo., Puls., Sul.\n\nfor being overheated: Ant-c., Puls.\n\nrich food: Puls.\n\nsolid\/dry food: Pho-ac.\n\nstarchy food: Nat-s., Puls.\n\nstanding: Sul.\n\nstuffy room: Puls.\n\nsugar\/sweets: Arg-n.\n\nCauses:\n\nafter an acute illness: Chin.\n\nanger: Coloc.\n\nanticipatory anxiety: Arg-n., Gels.\n\nautumn: Colch.\n\nbad or exciting news: Gels.\n\nbeer: Sul.\n\ngetting chilled: Ver-a.\n\ndamp, cold weather: Dulc.\n\ngetting cold or damp: Colch., Dulc.\n\nexcitement: Arg-n.\n\nfood poisoning: Ars.\n\nfright\/shock: Gels.\n\nfruit: Ars., Bry., Chin., Coloc., Nat-s., Puls., Ver-a.\n\nhot weather: Bry., Chin., Pho-ac.\n\nice-cream: Ars.\n\nmilk: Calc-c., Mag-m.\n\nover-eating: Ant-c.\n\nrich food: Puls.\n\nsour wine: Ant-c.\n\nsugar: Arg-n.\n\nsummer: Pho-ac.\n\nteething: Calc-c., Cham., Dulc., Ferr-m., Rhe., Sil.\n\nweaning: Arg-n., Chin.\n\ngetting wet: Rhus-t.\n\ngetting feet wet: Rhus-t.\n\nDictatorial (bossy) Ars., Kali-c., Lyc., Merc-s.\n\nDuring labour: Ars., Kali-c., Lyc.\n\nDisappointment Ign., Pho-ac., Puls., Staph.\n\nDischarges\n\nBland (non-irritating): Puls.\n\nBlood-streaked: Calc-s., Merc-s., Phos.\n\nBurning: Alu., Ars., Merc-s.\n\nLike ammonia: Nit-ac.\n\nLike egg white: Alu., Nat-m.\n\nSmelly: Ars., Carb-v., Merc-s., Pyr., Sul.\n\nSour: Calc-c., Mag-c., Mag-m., Nat-p., Rhe., Sul.\n\nStringy\/sticky: Kali-b.\n\nThick: Calc-c., Kali-b., Kali-s., Nat-m., Puls.\n\nWatery: Ars., Caust., Merc-s., Nat-m., Podo., Sul.\n\nWhite: Alu., Kali-m., Nat-m.\n\nYellow: Kali-s., Lyc., Merc-s., Puls., Sep.\n\nDiscontented Calc-p., Merc-s., Nat-m., Sul.\n\nDislikes (see also Desires; Aversions)\n\nBread in pregnancy: Kali-c., Sep.\n\nBread: Chin., Kali-c., Nat-m., Puls., Sul.\n\nButter: Chin., Puls.\n\nCheese: Nit-ac.\n\nCoffee: Calc-c., Nux-v.\n\nEggs: Sul.\n\nFatty, rich food: Chin., Petr., Puls.\n\nFood in general: Ars., Chin., Cocc., Colch., Coloc., Nat-m., Nux-v., Puls., Sep.\n\nwith hunger: Cocc., Nat-m., Nux-v.\n\nsight of food: Ars.\n\nsmell of food: Cocc., Colch., Coloc., Ip., Sep.\n\nthought of food in pregnancy: Chin., Nat-m., Sep.\n\nFruit: Chin., Ign., Phos., Puls.\n\nHot drinks\/food: Phos., Puls.\n\nMeat: Calc-c., Chin., Coloc., Nux-v., Petr., Puls., Sep., Sil., Sul.\n\nMilk: Ign., Nat-c., Sep., Stap.\n\nPork: Coloc.\n\nSweets: Caust.\n\nTobacco: Calc-c., Nux-v., Ign., Puls., Stap.\n\nWater: Nux-v., Stap., Stram.\n\nWater in pregnancy: Phos.\n\nDislocation of a joint Rhus-t.\n\nDisobedient Chin.\n\nDizziness (see also Faintness) Bell., Bor., Bry., Calc-c., Cocc., Con., Ferr-m., Gels., Nat-c., Nat-m., Nux-v., Petr., Phos., Puls., Sil., Sul., Ver-a.\n\nIn pregnancy: Bell., Cocc., Gels., Nat-m., Nux-v.\n\nHas to lie down: Cocc., Phos., Puls., Sil.\n\nWith:\n\nheadache: Bell., Calc-c., Con., Nux-v., Sil.\n\nnausea: Cocc., Ferr-m., Petr.\n\nturning sensation: Con.\n\nvomiting: Ver-a.\n\nWorse:\n\ngetting up from bending down: Bell., Ferr-m., Puls.\n\ngetting up from lying down: Cocc., Ferr-m., Nat-m., Nux-v., Petr., Phos., Sul.\n\ngetting up from sitting: Ferr-m., Nux-v., Petr., Phos., Puls., Sul.\n\nlying down: Con.\n\nmental exertion: Nat-c.\n\nmoving\/turning head quickly: Calc-c., Con.\n\nslightest motion: Bry.\n\nstooping: Bell., Nux-v., Puls., Sul.\n\nstuffy room: Nat-c.\n\ntobacco: Nat-m., Nux-v.\n\nwalking: Nat-m., Nux-v., Puls., Sul.\n\nCauses:\n\ndownward motion (stairs, lifts, etc): Bor.\n\nhigh places: Calc-c., Sul.\n\nloss of sleep: Cocc., Nux-v.\n\nmental exertion: Nat-c., Nat-m., Nux-v.\n\nsmoking: Nat-m., Nux-v.\n\nDreamy (see also Absent-minded) Nux-m., Op.\n\nDrowsy (see also Dazed) Gels., Op.\n\nDrowsy babies Nux-m.\n\nDryness (generally) Ars., Bry., Nat-m.\n\nDwells on the past Nux-v., Pho-ac.\n\nEarache Aco., Ap., Bell., Calc-c., Calc-s., Cham., Hep-s., Kali-b., Kali-m., Kali-s., Lach., Lyc., Mag-p., Merc-s., Nit-ac., Nux-v., Puls., Sil., Sul.\n\nDischarge from ears:\n\nblood-streaked: Calc-s., Merc-s.\n\nsmelly: Calc-s., Hep-s., Kali-b., Merc-s., Puls.\n\nthick: Calc-s., Kali-b., Puls.\n\nthin: Kali-s.\n\nyellow-green: Puls.\n\nyellow: Kali-b., Kali-s., Puls.\n\nPains\n\naching: Cham., Puls., Sul.\n\nbehind the ear: Sil.\n\nboring: Merc-s.\n\nburning: Merc-s.\n\nlacerating: Sul.\n\noutward: Puls.\n\npressing: Cham., Merc-s., Puls.\n\npressing outwards: Puls.\n\nspreading down into neck: Bell., Puls.\n\nshooting: Mag-p.\n\nspasmodic: Mag-p.\n\nsplinter-like: Nit-ac.\n\nstinging: Ap.\n\nstitching: Bell., Cham., Hep-s., Kali-b., Nux-v., Puls., Sul.\n\ntearing: Bell., Cham., Lyc., Merc-s., Puls., Sil., Sul.\n\nthrobbing: Bell., Calc-c., Nit-ac., Puls.\n\nunbearable: Aco., Cham.\n\nWith:\n\nblocked feeling in ears: Lyc., Merc-s., Puls., Sil.\n\ncrackling in the ears: Nit-ac.\n\nwhen chewing: Kali-s., Nit-ac.\n\ndeafness: Puls.\n\ndeafness after a cold: Kali-m.\n\nface-ache: Bell.\n\nglands swollen: Kali-m.\n\nitching in the ear: Nux-v., Puls., Sil.\n\nhas to swallow: Nux-v.\n\nnoises in the ear: Bell., Calc-c., Kali-m., Lyc.,\n\nPuls., Sul.\n\npainful: Sul.\n\nredness of the external ear: Puls.\n\nsensitivity to wind: Cham., Lach.\n\nsore throat: Ap., Lach., Nit-ac.\n\nBetter for:\n\nheat: Mag-p.\n\nfirm pressure: Mag-p.\n\nwrapping up warmly: Hep-s.\n\nWorse:\n\nbending down: Cham.\n\ncold: Hep-s., Mag-p.\n\nfresh air: Merc-s.\n\non left side: Kali-b., Lach., Sul.\n\nnoise: Sul.\n\nnight: Merc-s., Puls.\n\nright side: Bell., Nit-ac.\n\nswallowing: Ap., Kali-m., Lach., Nit-ac., Nux-v.\n\nturning head: Mag-p.\n\nwind: Cham., Lach.\n\nwarmth of bed: Merc-s.\n\nCauses:\n\nafter a common cold: Puls.\n\nafter measles: Puls.\n\ngetting chilled: Aco.\n\ncold dry wind: Aco., Mag-p.\n\nEffects of drugs taken during or after labour Cham., Nux-v., Op., Phos., Sec.\n\nGeneral 'de-tox': Nux-v.\n\nDrowsy, sleepy and spaced out: Op., Phos.\n\nIrritable and sleepless: Cham., Nux-v.\n\nWith vomiting (especially after a general anaesthetic): Phos.\n\nCauses:\n\ngeneral anaesthetic: Op., Phos.\n\nmorphine or pethidine: Cham., Op.\n\nsyntometrine: Sec.\n\nEuphoric Coff., Phos.\n\nExcitable (see also Complaints from excitement) Arg-n., Bell., Cham., Cimi., Coff., Lach., Nux-v., Phos., Puls.\n\nDuring labour: Coff.\n\nDuring pregnancy: Gels.\n\nExhaustion Alu., Ant-c., Ant-t., Arg-n., Ars., Bar-c., Bry., Calc-c., Calc-p., Carb-a., Carb-v., Caust., Chin., Cocc., Colch., Con., Cupr., Ferr-m., Gels., Kali-c., Kali-p., Lach., Merc-c., Merc-s., Nat-c., Nat-m., Nat-p., Nit-ac., Nux-v., Pho-ac., Phos., Puls., Rhus-t., Sep., Sil., Spo., Stap., Sul., Sul-ac., Ver-a., Zinc.\n\nDuring labour see Labour pains with exhaustion\n\nExtreme: Ars., Bry.\n\nIn pregnancy: Alu., Calc-p., Sep., Sul., Ver-a.\n\nNervous: Chin., Cocc., Kali-p., Nat-c., Nat-p., Nit-ac., Nux-v., Pho-ac., Phos., Puls., Rhus-t., Sep., Sil., Stap.\n\nParalytic: Ars., Cocc., Gels., Pho-ac., Phos., Ver-a.\n\nSudden: Ars., Sep., Ver-a.\n\nWith:\n\nbreath cold: Carb-v.\n\nbreathlessness: Calc-c., Carb-v.\n\ncold extremities: Carb-v., Nat-c., Ver-a.\n\ndesire to lie down: Alu., Ferr-m.\n\ndizziness: Calc-c., Cocc.\n\na faint feeling: Alu., Ars.\n\nfever: Ars., Phos.\n\nheadache between the eyes: Cupr.\n\nheaviness: Merc-s., Nat-c., Nat-m., Rhus-t.\n\nin the limbs: Calc-p., Merc-s., Nat-c.\n\nnervousness: Cocc.\n\nnumbness: Cocc., Con.\n\nprofuse sweating: Chin.\n\nrestlessness: Ars., Rhus-t.\n\nrestless legs: Zinc.\n\nsleepiness: Ant-c.\n\nstiffness: Cocc.\n\ntrembling: Arg-n., Cocc., Con., Gels., Stap., Sul-ac.\n\nvertigo: Cocc.\n\nweak legs: Calc-p., Nat-c., Zinc.\n\nBetter:\n\nbeing fanned: Carb-v.\n\nfresh air: Con., Ferr-m.\n\nduring a menstrual period: Sep.\n\neating: Nat-c., Phos., Sep.\n\nwalking in fresh air: Rhus-t.\n\nwalking slowly: Ferr-m.\n\nWorse:\n\nafternoon: Sul.\n\nafter eating: Ars., Bar-c., Pho-ac.\n\nfor exercise: Ferr-m.\n\nevening: Caust., Nat-m.\n\nslightest exertion: Ars., Bry., Calc-c., Con., Lach., Nat-c., Pho-ac., Phos., Rhus-t., Sep., Spo.\n\nmental exertion: Calc-c., Cupr., Lach., Nat-c., Puls.\n\nphysical exertion: Alu., Calc-c., Cocc., Nat-c.\n\nfor fresh air: Cocc.\n\nheat of sun: Lach., Nat-c., Puls., Sul.\n\nhot weather: Ant-c.\n\nafter a menstrual period: Ip.\n\nduring a menstrual period: Carb-a., Sep.\n\nmorning: Ars., Lach., Pho-ac., Sep.\n\nin bed: Puls.\n\nafter getting up: Bry., Lach., Pho-ac., Sep.\n\non waking: Nux-v.\n\nlate morning: Ant-c.\n\nmid-morning: Bry.\n\nmovement: Ars., Spo.\n\nsitting down: Rhus-t.\n\nstanding: Alu.\n\nafter passing a stool: Ars., Con., Merc-s., Nit-ac., Sul-ac., Ver-a.\n\nstuffy room: Puls.\n\nexposure to sun: Nat-c.\n\nafter sweating: Merc-s., Sep.\n\ntalking: Alu., Sul.\n\nafter a walk: Ruta.\n\nwalking: Alu., Ars., Bry., Calc-c., Carb-a., Cocc., Con., Ferr-m., Lach., Nit-ac., Pho-ac., Phos., Rhus-t., Sep., Sul.\n\nin the fresh air: Cocc., Rhus-t.\n\nupstairs: Calc-c.\n\nCauses:\n\naccident\/injury: Carb-v.\n\nacute illness: Carb-a., Carb-v.\n\nanaemia: Ferr-m.\n\nbreastfeeding: Bry., Calc-c., Calc-p., Carb-a., Carb-v., Chin., Cocc., Pho-ac., Phos., Sep., Sil., Sul.\n\ncarbon monoxide poisoning: Carb-v.\n\ndiarrhoea: Ars., Carb-v., Chin., Nat-s., Nit-ac., Phos., Sil., Sul., Ver-a.\n\nfever: Phos.\n\nflu: Kali-p., Pho-ac., Sul.\n\nfood poisoning: Ars., Carb-v.\n\nhunger: Sul.\n\nafter illness: Carb-a., Carb-v., Chin.\n\nirregular sleep: Cocc., Nit-ac., Nux-v.\n\nlifting: Carb-a.\n\nloss of body fluids (diarrhoea, vomiting): Carb-v., Chin., Pho-ac.\n\nloss of sleep (broken nights): Cocc., Cupr., Nat-m., Nit-ac., Nux-v., Puls., Zinc.\n\nmental exhaustion: Cupr., Lach.\n\nmental strain: Nat-c.\n\nnervous exhaustion: Cocc.\n\nnursing the sick: Cocc., Nit-ac.\n\noverexposure to sun: Nat-c.\n\npain: Ars.\n\nsurgery: Carb-v.\n\nsweating: Bry., Carb-a., Chin., Ferr-m., Merc-s., Phos., Sep.\n\ntalking: Alu., Sul.\n\nvomiting: Carb-v.\n\nwalking: Alu., Sul.\n\nExhilarated (see also Excitable) Coff., Lach., Op.\n\nExpression\n\nAnxious: Aco., Bor., Ver-a.\n\nConfused: Lyc.\n\nFierce: Bell.\n\nFrightened: Aco.\n\nHaggard: Ars., Kali-c.\n\nSickly: Ars., Cina, Lach., Lyc.\n\nSleepy: Op.\n\nSuffering: Ars., Kali-c.\n\nExtremities cold Cupr.\n\nEye inflammation Aco., All-c., Ap., Arg-n., Ars., Bell., Bry., Calc-c., Calc-s., Dulc., Euphr., Lyc., Merc-s., Nat-m., Nit-ac., Nux-v., Puls., Rhus-t., Sep., Sil., Sul., Zinc.\n\nIn babies: Ap., Arg-n., Ars., Calc-c., Nit-ac., Puls., Sul.\n\nDischarge:\n\nbland (non-irritating): All-c.\n\nburning: Euphr.\n\nprofuse: All-c.\n\npurulent (mucus or pus): Arg-n., Calc-c., Calc-s., Hep-s., Lyc., Merc-s., Puls.\n\nsmelly: Arg-n., Puls.\n\nthick: Calc-s., Puls.\n\nwatery: Euphr.\n\nyellow: Arg-n., Calc-s., Puls., Sil.\n\nEyelids:\n\nburning: Ars., Euphr., Sul.\n\nglued together: Arg-n., Calc-c., Lyc., Puls., Rhus-t., Sep.\n\ngritty: Calc-c.\n\nitching: Puls., Rhus-t., Sul.\n\ndaytime: Sul.\n\nred: Arg-n., Euphr., Sul.\n\nswollen: Ap., Euphr., Nit-ac., Rhus-t., Sep.\n\nEyes:\n\naching: Aco., Puls.\n\nbloodshot: Ars., Bell., Led.\n\nburning: Aco., Ap., Ars., Bell., Puls., Sep., Sul., Zinc.\n\ndry: Bell., Bry.\n\ngritty (sandy sensation): Ars., Calc-c., Nat-m., Sul., Zinc.\n\nitching: Puls., Sul.\n\nred: Aco., Ap., Arg-n.\n\nsensitive to light: Aco., Arg-n., Ars., Bar-c., Bell., Calc-c., Euphr., Lyc., Merc-s., Nat-m., Nat-s., Nux-v., Op., Rhus-t., Sul.\n\nsore: Ap., Bry., Rhus-t., Sil., Zinc.\n\nworse for moving eyes: Bry., Rhus-t.\n\nstinging: Ap.\n\nstitching: Ap., Kali-c., Lyc., Rhus-t., Sul.\n\nwatering: All-c., Bell., Calc-c., Euphr., Lyc., Merc-s., Nat-m., Nit-ac., Puls., Rhus-t., Sul.\n\nWith a common cold: Aco., All-c., Bell., Calc-c., Dulc., Euphr., Merc-s., Puls.\n\nBetter:\n\ncold: Arg-n., Puls.\n\ncold bathing: Puls.\n\ncold compresses: Arg-n.\n\nfresh air: Puls.\n\nWorse:\n\ncold air: Sil.\n\ncold, dry wind: Aco.\n\ncoughing: Euphr.\n\nevening: Puls., Sep., Zinc.\n\nheat of a fire: Merc-s.\n\nheat: Ap., Bell.\n\nlight: Bell, Euphr.\n\nmoving eyes: Bry., Rhus-t.\n\nnight: Zinc.\n\nreading: Sep.\n\nwarm room: Puls.\n\nwarmth of bed: Merc-s.\n\nafter a walk: Sep.\n\nwashing eyes: Sul.\n\nwind: Euphr.\n\nCauses:\n\na foreign body in the eye: Aco., Sil.\n\ngetting chilled: Aco.\n\ncold, wet weather: Dulc., Rhus-t.\n\nEye injuries Arn., Euphr., Led., Stap., Symph.\n\nBruising:\n\nto eyeball: Arn.\n\nto surrounding areas: Arn.\n\nwith swelling: Arn.\n\nwith discolouration: Led.\n\nPains:\n\nin eyeballs: Symph.\n\nsore, bruised: Arn., Symph.\n\nWith eye inflammation: Euphr.\n\nWound punctured: Stap.\n\nBetter:\n\ncold compresses: Led.\n\nWorse:\n\ntouch: Stap.\n\nCauses:\n\na direct blow to the eyeball: Symph.\n\nEye strain Ruta.\n\nEyelids heavy Caust., Gels.\n\nEyes\n\nDull: Ant-c.\n\nGlassy: Op., Pho-ac.\n\nShining: Bell.\n\nSunken: Ant-c., Chin., Cina\n\nFace\n\nBlue: Carb-v., Cupr., Lach., Op., Ver-a.\n\nduring cough: Dros., Ip.\n\nDark red: Bell., Bry., Op.\n\nFlushes easily: Ferr-m.\n\nPale: Alu., Ant-t., Ars., Calc-c., Calc-p., Carb-v., Cina, Con., Cupr., Ferr-m., Lyc., Nat-c., Nat-p., Op., Pho-ac., Tab., Ver-a., Zinc.\n\nPasty: Chin., Ferr-m., Merc-s., Nat-m., Sep.\n\nPuffy: Ap., Kali-c.\n\nRed: Aco., Ap., Bell., Cham., Glon., Lach., Phos., Rhus-t., Sul.\n\none-sided: Cham., Ip.\n\nfrom pain: Ferr-m.\n\nin spots: Bell., Cham., Phos., Sul.\n\nwith toothache (teething): Bell., Cham.\n\nSallow: Arg-n., Carb-v., Nat-m., Sul.\n\nSickly: Lyc.\n\nYellow: Sep.\n\nFaintness (see also Dizziness) Alu., Carb-v., Kali-c., Op., Puls., Sep.\n\nAfter excitement, fright: Op.\n\nColdness after the faint: Sep.\n\nOn waking\/getting up: Carb-v.\n\nHas to lie down: Kali-c.\n\nHeat with the faintness: Sep.\n\nIn a warm room: Puls.\n\nIn pregnancy: Kali-c., Puls., Sep.\n\nWorse:\n\nexercise: Sep.\n\nduring a fever: Sep.\n\nduring a menstrual period: Sep.\n\nstanding: Alu.\n\nstuffy room: Sep.\n\nFatigue see Exhaustion\n\nFault-finding see Critical\n\nFearful Aco., Ap., Arg-n., Arn., Ars., Bor., Calc-c., Caust., Calc-p., Chin., Cimi., Coff., Ign., Lyc., Nat-c., Nit-ac., Phos., Puls.\n\nAnimals: Chin.\n\nBabies: Bar-c., Calc-c., Caust., Lyc.\n\nat night: Ars., Bor., Calc-c., Caust., Cina, Kali-p., Stram.\n\nwakes terrified: Stram.\n\nBeing alone: Ap., Arg-n., Ars., Kali-c., Phos.\n\nIn a crowd: Aco., Gels.\n\nOf the dark: Phos., Stram.\n\nOf death: Aco., Ap., Ars., Calc-c., Cimi., Gels., Nit-ac., Phos., Ruta.\n\nduring labour: Aco., Coff.\n\nduring pregnancy: Aco.\n\nOf dogs: Chin., Puls.\n\nIn the evening: Calc-c., Caust., Puls.\n\nOf downward movement: Bor.\n\nOf falling: Gels.\n\nIn labour: Aco., Ars., Coff., Kali-c., Op.\n\nOf painful death: Coff.\n\nIn pregnancy: Aco., Caul., Cimi.\n\nOf public speaking: Arg-n., Gels., Lyc.\n\nParalysing: Gels.\n\nOf strangers: Bar-c., Lyc.\n\nOf sudden noises (sneezing, coughing, etc): Bor.\n\nOf thunder: Rhod.\n\nDuring a thunderstorm: Phos., Rhod.\n\nAt twilight: Puls.\n\nOf being touched: Arn.\n\nFever Aco., Ant-t., Ap., Ars., Bell., Bry., Cham., Cina, Eup-p., Ferr-m., Gels., Hep-s., Ign., Ip., Kali-c., Lyc., Merc-s., Nat-m., Nux-v., Op., Phos., Puls., Pyr., Rhus-t., Sep., Sil., Sul., Ver-a.\n\nHeat:\n\nalternating with chills: Aco., Ant-t., Ars., Bell., Bry., Hep-s., Merc-s., Nux-v., Pyr., Rhus-t., Sul.\n\nburning: Aco., Ap., Ars., Bell., Bry., Cham., Gels., Nat-m., Op., Phos., Puls., Rhus-t.\n\ndry: Aco., Ap., Ars., Bell., Bry., Nux-v., Phos., Rhus-t.\n\ndry at night: Aco., Ars., Bell., Phos.\n\ndry in the evening: Puls.\n\nintense: Ant-t., Op.\n\nradiant: Bell.\n\nOne-sided: Bry., Cham., Lyc., Nux-v., Phos., Puls.\n\nright side: Phos.\n\nleft side: Lyc.\n\nPulse:\n\nfast, strong: Aco.\n\nWithout sweating: Bell., Bry., Gels.\n\nWith:\n\nanxiety: Aco., Ars., Ip., Sep.\n\nappetite increased: Phos.\n\nbackache: Nux-v.\n\nchest symptoms: Ant-t.\n\nchilliness: Ap., Puls., Pyr., Ver-a.\n\nbetter for fresh air: Ip.\n\nworse for heat: Ip.\n\nextreme: Nux-v., Pyr.\n\ninternal: Pyr.\n\ndeep sleep: Op.\n\ndelirium: Ars., Bell.\n\nexhaustion: Ars.\n\ngrinding of teeth: Bell.\n\nheat during sleep: Ant-t.\n\nhunger: Cina\n\nnausea: Nat-m.\n\nsensitive skin: Ap.\n\nshaking: Pyr.\n\nshivering: Cham., Gels., Nux-v., Sul.\n\nsweating see Sweat\n\nthirst see Thirsty\/Thirstless\n\nBetter:\n\nfor complete rest: Bry.\n\nin bed: Kali-c.\n\ndrinking cold water: Caust.\n\nfresh air: Ip.\n\nheat: Hep-s., Kali-c.\n\nhot drinks: Rhus-t.\n\nfor sweating: Gels.\n\nfor uncovering: Aco., Ap., Ferr-m., Ign., Nat-m., Op., Puls.\n\nfor urinating: Gels.\n\nWorse:\n\n9 p.m.: Bry.\n\nafternoon: Ap., Bell., Gels., Ign., Phos., Puls.\n\nautumn: Bry., Nat-m., Sep.\n\nin bed: Merc-s., Puls.\n\ncold drinks: Rhus-t.\n\ndraughts: Nux-v.\n\nafter eating: Kali-c.\n\nevening: Aco., Bell., Lyc., Phos., Puls., Rhus-t., Sil.\n\nafter lying down in bed: Bry.\n\nfresh air: Merc-s., Nux-v.\n\nfront of the body: Ign.\n\nfor heat: Ap., Ip., Puls.\n\nfor light: Bell.\n\nmental exertion: Sul.\n\nafter midnight: Ars.\n\nmid-morning: Cham., Nat-m., Rhus-t.\n\nmorning: Ap., Ars.\n\nin bed: Puls.\n\nmovement: Nux-v., Rhus-t., Sil.\n\nat night: Aco., Ars., Bell., Cina, Merc-s., Phos., Puls., Rhus-t., Sil., Sul.\n\nduring sleep: Op.\n\nslightest movement of the bedclothes: Nux-v.\n\nin a stuffy room: Ap., Puls.\n\nfor sweating: Op., Sep., Ver-a.\n\nfor thinking: Sul.\n\nfor being uncovered: Bell., Hep-s., Nux-v., Puls., Pyr., Rhus-t., Sep., Sil.\n\nwarm covers: Ap., Ign., Puls.\n\nwashing: Ap., Puls.\n\nCauses:\n\nanger: Cham., Sep.\n\ngetting chilled: Aco.\n\nteething in babies: Aco.\n\nFingernails\n\nSplit: Ant-c., Sil.\n\nWeak nails, split easily: Sil.\n\nWhite spots on nails: Sil., Zinc.\n\nCaused by:\n\ninjury: Ant-c.\n\nFlatulence Arg-n., Ars., Calc-c., Carb-a., Carb-v., Cham., Chin., Colch., Lyc., Mag-c., Nat-s., Nit-ac., Nux-v., Podo., Puls., Sil., Sul.\n\nAbdomen\/stomach feels:\n\nbloated: Arg-n., Ars., Calc-c., Carb-a., Carb-v., Cham., Chin,., Colch., Lyc., Mag-c., Nux-v., Sul.\n\nabove the navel: Chin.\n\nbelow the navel: Carb-v., Lyc.\n\ngurgling: Podo., Puls., Sul.\n\nbefore stool: Podo.\n\nintolerant of tight clothing (belts etc.): Arg-n., Calc-c., Lyc., Mag-c., Nux-v.\n\nrumbling: Carb-v., Chin., Lyc., Nat-s., Nux-v., Puls., Sil., Sul.\n\nbefore stool: Mag-c.\n\nWind:\n\nloud: Arg-n., Nat-s.\n\nduring stool: Nat-s., Podo.\n\nsmelly: Ars., Carb-v., Nat-s., Nit-ac., Puls., Sil., Sul.\n\nof rotten eggs: Sul.\n\nobstructed (difficult to expel): Arg-n., Chin., Colch., Nit-ac., Puls., Sil.\n\nWith:\n\ndiarrhoea: Carb-v., Nat-s.\n\nurging for stool but only spluttering wind is passed: Nat-s.\n\nBetter:\n\npassing wind: Carb-v., Lyc., Nat-s., Nux-v., Puls., Sul.\n\nWorse:\n\nafter eating: Arg-n., Lyc., Nux-v.\n\nafter breakfast: Nat-s.\n\nbefore\/after stool: Lyc.\n\nCauses:\n\nabdominal surgery: Carb-a., Carb-v., Chin.\n\neating fruit: Chin.\n\nFlu Ars., Bry., Dulc., Eup-p., Gels., Ip., Nux-v., Pyr., Rhus-t.\n\nChills: Ip.\n\nEyeballs aching: Bry., Eup-p., Gels.\n\nEyelids red: Eup-p.\n\nPains:\n\naching: Ip., Pyr., Rhus-t.\n\nback: Ip.\n\nbones: Eup-p., Ip., Nux-v., Pyr., Rhus-t.\n\nbones feel broken: Eup-p.\n\njoints: Nux-v., Rhus-t.\n\nlegs: Ip., Pyr., Rhus-t.\n\nmuscles: Gels.\n\nshooting: Rhus-t.\n\nsore, bruised: Ip., Nux-v., Rhus-t.\n\nSkin sore: Eup-p.\n\nWith:\n\nchilliness, extreme, can't get warm: Gels., Nux-v., Pyr.\n\ncommon cold: Ars., Nux-v.\n\nexhaustion: Gels., Rhus-t.\n\nfever: Ars., Nux-v., Pyr., Rhus-t.\n\nheadache: Eup-p.\n\nheaviness: Gels.\n\nnasal catarrh: Eup-p.\n\nnumbness: Gels.\n\nrestlessness: Ars., Pyr.\n\nshivering\/chills in back: Eup-p., Gels., Pyr.\n\nsneezing: Eup-p., Rhus-t.\n\nextreme thirst: Pyr.\n\nBetter:\n\nwarm compresses: Nux-v.\n\nfor movement: Pyr.\n\nfor sweating: Eup-p., Gels.\n\nfor urinating: Gels.\n\nfor walking: Pyr.\n\nwarmth of bed: Nux-v., Pyr.\n\nWorse:\n\nin bed: Nux-v.\n\ncold: Nux-v.\n\nduring the chilly stage: Pyr.\n\nexertion: Gels.\n\nmorning: Nux-v.\n\nwalking: Gels.\n\nsitting: Pyr.\n\nCauses:\n\nchange of temperature: Ars.\n\ncold, damp weather: Dulc.\n\nFlushed see Face flushes easily\n\nFood poisoning Ars., Carb-v., Puls.\n\nWith:\n\ndiarrhoea: Ars., Puls.\n\nflatulence: Carb-v.\n\nnausea: Ars.\n\nvomiting: Ars.\n\nCauses:\n\nfish: Carb-v., Puls.\n\nmeat: Ars., Carb-v., Puls.\n\nForgetful (see also Absent-minded; Confused) Arg-n., Arn., Ars., Caust., Cocc., Colch., Con., Glon., Kali-p., Lyc., Merc-s., Nat-c., Petr., Pho-ac., Sep., Stap., Ver-a.\n\nFollowing injury: Arn.\n\nFractures see Broken bones\n\nFrightened see Fearful\n\nFull feeling Aesc.\n\nFussy see Critical; Capricious\nGas see Flatulance\n\nGastric flu Ars., Bry., Eup-p., Ip., Nux-v.\n\nPains:\n\naching in stomach: Bry.\n\nWith:\n\nbiliousness: Bry.\n\nfever: Bry., Eup-p.\n\nflu symptoms (see also Flu): Ars., Bry., Eup-p., Ip., Nux-v.\n\nnausea\/retching\/vomiting of bile or food: Eup-p.\n\nBetter:\n\nafter chills during fever: Eup-p.\n\nfor belching: Bry.\n\nWorse:\n\nevening: Bry.\n\ncoughing: Bry.\n\nlying down in bed: Bry.\n\nmovement: Bry.\n\nwalking: Bry.\n\nGentle Puls.\n\nGerman measles see Measles\n\nGlands (see also Sore throat, Tonsils swollen)\n\nSensitive: Bar-c., Bell.\n\nSwollen: Bar-c., Bell., Calc-c., Merc-s., Puls., Rhus-t., Sil., Sul.\n\npainless: Calc-c.\n\nGloomy (see also Depressed; Despondent; Morose) Nat-c.\n\nGroin pains\n\nIn pregnancy: Bell-p.\n\nLegs feel weak: Bell-p.\n\nGuilt Ars., Ign.\n\nGum boils (see also Abscesses) Sil.\n\nGums\n\nBleeding: Arn., Lach., Merc-c., Nat-m., Pho-ac., Phos., Stap., Zinc.\n\nafter tooth extraction: Arn., Lach., Phos.\n\nPale: Ferr-m., Merc-c., Stap., Zinc.\n\nHaemorrhages see Bleeding\n\nHaemorrhoids see Piles\n\nHair loss Calc-c., Carb-v., Lach., Lyc., Nat-m., Nit-ac., Pho-ac., Phos., Sep., Sul.\n\nIn handfuls: Phos.\n\nCauses:\n\nacute illness: Carb-v., Phos.\n\nchildbirth: Calc-c., Carb-v., Lyc., Nat-m., Nit-ac., Sep., Sul.\n\ngrief: Pho-ac.\n\npregnancy: Lach.\n\nHeadache Aco., All-c., Ant-c., Ap., Arg-n., Ars., Bell., Bry., Calc-c., Calc-p., Carb-v., Chin., Cimi., Cocc., Coff., Coloc., Eup-p., Ferr-m., Gels., Glon., Ign., Ip., Kali-b., Kali-c., Kali-p., Kali-s., Lach., Lyc., Mag-c., Mag-m., Mag-p., Merc-s., Nat-c., Nat-m., Nat-p., Nat-s., Nit-ac., Nux-v., Petr., Pho-ac., Phos., Puls., Rhod., Rhus-t., Ruta., Sep., Sil., Stap., Sul., Zinc.\n\nExtremities icy cold: Sep.\n\nEyes\/eyelids:\n\nred: Glon.\n\nsensitive to light: Nat-s.\n\nsmarting: Pho-ac.\n\nsore, watering: Nat-m.\n\nFace:\n\npale: Nat-p.\n\nred: Glon.\n\nHead feels:\n\nconstricted: Nit-ac., Sul.\n\nfull: Ap., Sul.\n\nheavy: Carb-v., Gels., Lach., Nat-c., Nux-v., Petr., Pho-ac.\n\nhot: Ap., Lach., Sul.\n\nNervous: Chin., Coff., Kali-p., Puls., Zinc.\n\nPains:\n\naching: Gels.\n\nback of head: Bell., Carb-v., Cimi., Cocc., Eup-p., Gels., Nux-v., Petr., Pho-ac., Rhus-t., Sil., Stap.\n\nabove eyes: Kali-c.\n\nbehind eyeballs: Bry.\n\nin bones: Eup-p., Nit-ac., Sep.\n\nburning: Aco., Ars., Calc-c., Merc-s., Phos., Sil., Sul.\n\nbursting: Aco., Bell., Bry., Calc-c., Glon., Lach., Phos., Sep., Sul., Zinc.\n\neyes: Bell., Rhod.\n\nforehead: Ars., Bell., Bry., Cocc., Ferr-m., Ign., Kali-c., Kali-s., Lach., Lyc., Merc-s., Nat-m., Nat-p., Nux-v., Phos., Puls., Sil., Stap., Sul., Zinc.\n\nhammering: Bell., Ferr-m., Glon., Nat-m., Sil., Sul.\n\nleft side of face: Coloc.\n\nmaddening: Calc-c.\n\nnape of neck: Cocc., Pho-ac.\n\none-sided: Coff., Kali-p., Pho-ac., Puls.\n\npressing: Carb-v., Chin., Lach., Lyc., Merc-s., Nat-c., Nat-m., Nit-ac., Nux-v., Petr., Pho-ac., Phos., Puls., Sep., Sil., Stap., Sul.\n\npressing out: Cimi.\n\npressing up: Cimi.\n\npulling: Stap.\n\npulsating: Bell.\n\nrecurring at regular intervals: Ars., Kali-b., Mag-c., Nit-ac.\n\nrheumatic: Rhus-t.\n\nright side of head: Mag-p.\n\nshooting: Kali-c., Mag-c., Mag-p., Sep.\n\nsides of head: Kali-s., Phos., Ruta., Zinc.\n\nsinuses: Kali-b., Sil.\n\nsore, bruised: Chin., Cocc., Eup-p., Gels., Ip., Merc-s., Nux-v., Puls., Ruta., Sil., Stap.\n\nspasmodic: Mag-c., Mag-p.\n\nsplitting: Nat-m.\n\nspreading to:\n\nback of head: Gels., Lach.\n\nforehead: Gels.\n\ntop of head: Sil.\n\nto ear: Coloc.\n\neyes: Mag-m., Nit-ac., Sil.\n\nstabbing: Ap., Ign.\n\nstart and stop suddenly: Bell.\n\nstitching: Kali-s.\n\nstupefying: Nux-v.\n\nsudden: Ap.\n\ntearing: Coloc., Kali-c., Mag-c., Nux-v., Rhod., Sep., Zinc.\n\ntemples: Bell., Lach., Lyc., Mag-m., Nat-m., Zinc.\n\nthrobbing: Aco., Ars., Bell., Chin., Eup-p., Ferr-m., Glon., Lach., Lyc., Nat-m., Phos., Puls., Sep., Sil., Sul.\n\ntop of head: Cimi., Lach., Pho-ac., Sul.\n\nviolent: Bell., Bry., Glon., Ign., Lach., Rhus-t., Sil.\n\nin waves: Sep.\n\nPupils dilated: Gels.\n\nScalp feels:\n\nsensitive: Chin.\n\nsore: Ap.\n\ntight: Ap.\n\nUrination frequent: Gels.\n\nVision blurred: Gels.\n\nWith:\n\nbelching: Mag-m.\n\nbiliousness: Nux-v.\n\ncold feet: Gels.\n\ndizziness: Nux-v.\n\neye strain: Ruta.\n\nfaintness: Glon.\n\nfever: Eup-p.\n\nheaviness: Petr.\n\nhot flushes: Glon.\n\nhunger before\/during headache: Phos.\n\nnausea: Ant-c., Cocc., Petr., Sul.\n\nsweating on forehead: Nat-c.\n\nthirst: Mag-m.\n\nthirstlessness: Ferr-m.\n\nvomiting: Ip.\n\nBetter:\n\nfor binding up the head: Arg-n.\n\nclosing eyes: Sil.\n\nfor cold: Phos., Sul.\n\nfor cold compresses: Bry., Glon., Phos., Sul.\n\nafter eating: Lach., Sep.\n\nevening in bed: Nux-v.\n\nexcitement: Nux-v., Pho-ac.\n\nfor fresh air: Ars., Cimi., Kali-s., Lyc., Phos., Puls., Sep., Zinc.\n\ngentle walking: Rhus-t.\n\ngetting up: Nux-v., Phos., Rhod.\n\nin the morning: Nux-v.\n\nheat: Mag-p., Sil.\n\nfor lying down: Calc-c., Ferr-m., Puls., Sil.\n\nin a quiet, dark room: Sil.\n\nwith head high: Puls.\n\nfor lying in a darkened room: Bell.\n\nmassage: Phos.\n\nmovement: Rhus-t.\n\nfor firm pressure: Chin., Ferr-m., Mag-p., Nat-m., Puls.\n\nfor pressure: Bell., Bry., Glon., Mag-c., Nux-v.\n\nfor resting head: Bell.\n\nafter sleep: Phos.\n\nsunset: Glon.\n\nurinating: Gels.\n\nwalking: Mag-c., Phos., Puls., Rhod.\n\nin the fresh air: Puls., Rhod.\n\nwrapping up head: Nux-v., Rhod., Rhus-t., Sil.\n\nWorse:\n\n10 a.m.-3 p.m.: Nat-m.\n\nfor bending down: Bell., Merc-s., Puls., Sep., Sul.\n\nblowing nose: Puls., Sul.\n\ngetting chilled: Sil.\n\ngetting cold: Bell., Calc-p., Kali-c., Mag-p., Nux-v., Phos.\n\ngetting head cold: Sep., Sil.\n\ngetting feet cold: Sil.\n\nfor coughing: Bry., Lyc., Nat-m., Phos., Sul.\n\ndamp weather: Rhod.\n\ndaylight: Phos., Sil.\n\nafter drinking wine: Zinc.\n\nafter eating: Nat-c., Nat-m., Nux-v., Puls., Sul.\n\nemotions: Nat-m.\n\nevening: Kali-s.\n\nexcitement: Nat-m.\n\nmoving eyes: Nux-v.\n\nduring a fever: Nat-m.\n\nafter getting up: Bry., Sul.\n\ngetting up from lying down: Pho-ac., Sil.\n\nfor heat: Ars., Bell., Glon., Puls.\n\nhot drinks: Phos., Puls.\n\njarring movement: Glon., Nit-ac., Sil.\n\nleft side: Kali-c., Lach.\n\nfor light: Bell., Calc-c.\n\nlying down: Lach., Mag-c.\n\nduring menstrual period: Bell., Glon., Lyc., Nat-m., Sep.\n\nmental exertion: Glon., Nat-c., Nat-m.\n\nin the morning: Sul.\n\nmoving head: Ferr-m., Gels.\n\nmovement: Gels., Nit-ac.\n\nat night: Merc-s., Nit-ac.\n\nfor noise: Calc-c., Coff., Nit-ac.\n\ngetting overheated: Lyc.\n\nin pregnancy: Bell., Puls., Sep.\n\npressure: Lach., Mag-m., Nit-ac.\n\nreading\/writing: Nat-m.\n\non the right side of the head: Calc-c.\n\nrunning: Puls.\n\nshaking head: Nux-v.\n\nsmoky room: Ign.\n\nsneezing: Sul.\n\nstanding: Puls.\n\nstuffy room: Kali-s., Phos., Puls.\n\nsummer: Glon.\n\nexposure to sun: Bell.\n\nsunrise: Glon.\n\nsweating: Eup-p.\n\ntalking: Nat-m.\n\nthinking: Nat-c., Nat-m.\n\ntravelling: Sep.\n\nbefore\/during a thunderstorm: Rhod.\n\ntouch: Coloc., Stap.\n\nturning head quickly: Nat-c.\n\nfor tying up hair: Bell.\n\non waking: Lach., Nat-m., Nit-ac.\n\nfor walking: Bell., Glon., Lach., Nit-ac., Sul.\n\nwalking upstairs: Sil.\n\nwalking heavily: Sil.\n\nwarm room: Kali-s.\n\nwarmth of bed: Lyc., Sul.\n\nwine: Zinc.\n\nwrapping up head: Lyc., Mag-m., Phos.\n\nCauses:\n\nanaemia: Calc-p.\n\nartificial light (working in): Sep., Sil.\n\nbreastfeeding: Bry., Calc-c., Puls., Sep.\n\nchange of weather: Bry., Rhus-t.\n\ngetting chilled: Aco., Mag-p.\n\ncold air: Bell., Rhod., Rhus-t., Sil.\n\ncold, damp weather: Bry., Calc-c., Sul.\n\ncold wind: Calc-p., Nux-v., Rhus-t.\n\ncommon cold: Merc-s., Nux-v.\n\ndamp weather: Nux-v., Rhus-t., Sil., Sul.\n\ndraughts: Sil.\n\nemotions: Stap.\n\nexcitement: Coloc., Puls., Stap.\n\nexcited conversation: Sul.\n\neye strain: Kali-c., Lyc., Pho-ac., Ruta., Sil.\n\nfright: Aco.\n\ngetting head wet (haircut): Bell.\n\ngetting wet: Calc-c., Rhus-t.\n\ngrief: Pho-ac., Stap.\n\nhead injury: Nat-s.\n\nice-cream: Puls.\n\nironing: Bry.\n\nirregular sleep: Cocc.\n\nlifting: Rhus-t.\n\nloss of sleep: Cocc., Nux-v.\n\nmental exertion: Calc-c., Calc-p., Kali-p., Nat-c., Nat-p., Nux-v., Pho-ac.\n\nmental strain: Chin., Lyc., Nat-c., Nux-v., Pho-ac.\n\nnerves: Chin., Cocc.\n\novereating: Nux-v.\n\noverexposure to sun: Bell., Bry., Glon., Lach.\n\noverwork: Calc-p., Cocc., Kali-p., Sil., Zinc.\n\nphysical exertion: Calc-p.\n\nrheumatism: Merc-s.\n\nrunning: Puls.\n\nshock: Aco.\n\nsummer: Nat-c.\n\nsunstroke: Nat-c.\n\nbefore a thunderstorm: Phos.\n\ntravelling (car\/coach\/train etc.): Cocc., Sil.\n\nvexation: Coloc., Stap.\n\nwinter: Sul.\n\nHead injuries see Injuries to head\n\nHealing after childbirth see Bruises; Injuries (cuts\/wounds); Back pain; Sprains and Strains\n\nHeartburn (see also Indigestion)\n\nIn pregnancy: Bry., Calc-c., Merc-s., Nat-m., Nux-v., Tab., Zinc.\n\nWith:\n\nindigestion: Calc-c., Merc-s., Nat-m., Nux-v.\n\nsour belches: Nux-v.\n\nsweetish belches: Nat-m., Zinc.\n\nwatery belches: Nat-m., Tab.\n\nWorse:\n\nat night: Merc-s.\n\nHeavy feeling (see also Sluggish)\n\nGenerally: Aesc., Gels., Phos., Sep.\n\nHernia in babies\n\nInguinal: Lyc., Nux-v.\n\nright side: Lyc.\n\nleft side: Nux-v.\n\nUmbilical: Bry., Calc-c.\n\nCause:\n\nconstipation: Nux-v.\n\nHerpes See Cold sores\n\nHiccups in babies Bor., Mag-p., Nux-v.\n\nViolent: Mag-p., Nux-v.\n\nWorse after eating or drinking: Mag-p., Nux-v.\n\nHitting (in angry babies) Bell., Cham., Cina., Lyc., Nux-v., Stram., Ver-a.\n\nHives Ap., Dulc., Rhus-t., Urt-u.\n\nRash:\n\nbiting: Urt-u.\n\nburning: Rhus-t., Urt-u.\n\nitching: Rhus-t., Urt-u.\n\nlumpy: Dulc.\n\nstinging: Rhus-t., Urt-u.\n\nWith:\n\nfever: Ap.\n\njoint pain: Rhus-t., Urt-u.\n\nsweating: Ap.\n\nBetter:\n\nrubbing: Urt-u.\n\nWorse:\n\ncold: Rhus-t.\n\nexercise: Urt-u.\n\nheat: Dulc., Urt-u.\n\nnight: Ap., Urt-u.\n\nafter scratching: Dulc., Rhus-t.\n\nCauses:\n\ngetting chilled\/wet: Rhus-t.\n\ngetting cold: Dulc.\n\ncold air: Rhus-t.\n\nduring a fever: Rhus-t.\n\ninsects: Urt-u.\n\nstinging nettles: Urt-u.\n\nHoarseness (painless) Calc-c.\n\nHomesick Ign., Pho-ac.\n\nHot flushes Cocc., Kali-c., Lach., Nat-s., Phos., Sep., Sul., Sul-ac., Ver-a.\n\nFlush moves up the body: Sep.\n\nin pregnancy: Sul., Ver-a.\n\nWith:\n\nheadache: Lach.\n\npalpitations: Lach.\n\nhot sweats: Lach.\n\nWorse:\n\nduring the afternoon: Sep.\n\nduring the evening: Sep.\n\nafter sweating: Sep.\n\nHumiliation Coloc., Lyc., Staph.\n\nHurried Arg-n., Hep-s., Lil-t., Sul-ac.\n\nWhile eating: Hep-s., Sul-ac.\n\nAt work: Sul-ac.\n\nWhile speaking: Arg-n., Hep-s.\n\nWhile waiting: Arg-n.\n\nWhile walking: Arg-n., Sul-ac.\n\nWhile writing: Sul-ac.\n\nHyperactive Stram.\n\nIdealistic Ign.\n\nImpatient (see also Irritable) Cham., Hep-s., Nux-v., Sul.\n\nImpulsive Arg-n., Hep-s., Nux-v.\n\nIncontinence (involuntary urination) Ars., Caust., Nat-m., Puls., Sep.\n\nIn pregnancy: Ars., Caust., Nat-m., Puls., Sep.\n\nAfter childbirth: Ars.\n\nWorse for:\n\ncoughing, laughing, sneezing or walking: Caust., Nat-m., Puls., Sep.\n\ngetting chilled: Caust.\n\nday and night: Ars.\n\nIndecisive Bar-c., Ign., Lyc., Op.\n\nIndifferent (see also Apathetic) Carb-v., Nat-m., Nat-p., Pho-ac., Rhe., Sep., Stram.\n\nDuring a fever: Op., Pho-ac.\n\nBabies: Rhe.\n\nto playing: Rhe.\n\nTo everything: Carb-v.\n\nTo own children: Sep.\n\nTo family: Sep.\n\nTo loved ones: Sep.\n\nTo personal appearance (i.e. scruffy): Sul.\n\nTo work: Sep.\n\nIndigestion (see also Heartburn; Flatulence; Taste) Ant-c., Arg-n., Ars., Bry., Calc-c., Carb-v., Caust., Chin., Ign., Kali-b., Kali-c., Kali-m., Kali-p., Lyc., Mag-c., Nat-c., Nat-m., Nat-p., Nat-s., Nux-v., Phos., Puls., Sul.\n\nAbdomen\/stomach feels:\n\nbloated: Ant-c., Arg-n., Calc-c., Carb-v., Chin., Kali-c., Mag-c., Nat-p., Nat-s., Nux-v., Sul.\n\nempty: Ant-c., Ign., Kali-p., Lyc., Nux-v., Puls., Sul.\n\nat 11 a.m.: Sul.\n\nfull: Ant-c., Caust., Kali-c., Lyc.\n\nvery quickly (when eating): Lyc.\n\nhard: Calc-c.\n\npainful: Arg-n.\n\nrumbling: Mag-c.\n\nBelches:\n\nacrid: Lyc.\n\nbitter: Chin., Nat-s., Nux-v., Puls.\n\ndifficult: Arg-n.\n\nempty: Ant-c., Arg-n., Carb-v., Caust., Kali-b., Lyc., Puls, Sul.\n\ngreasy: Mag-c.\n\nincomplete, ineffectual: Chin., Nat-m.\n\nloud: Arg-n.\n\nsour: Calc-c., Carb-v., Chin., Ign., Kali-b., Lyc., Mag-c., Nat-c., Nat-m., Nat-p., Nat-s., Nux-v., Sul.\n\ntasting of food just eaten: Ant-c., Caust., Chin., Nat-m., Puls.\n\nNervous: Kali-p.\n\nPains in the stomach:\n\nburning: Ars., Carb-v., Phos., Sul.\n\ncramping: Carb-v., Caust., Lyc., Nat-m., Nux-v.\n\npressing: Calc-c., Caust., Chin., Lyc., Nat-m., Nux-v., Puls.\n\nsore, bruised: Kali-c., Nat-c., Nux-v., Phos.\n\nWith:\n\ndiarrhoea: Kali-m.\n\nflatulence: Arg-n., Calc-c., Carbo-v., Sul.\n\nafter breakfast: Nat-s.\n\nobstructed: Chin.\n\nheadache: Ars.\n\nheartburn: Ars., Bry., Calc-c., Lyc., Mag-c., Merc-s., Nat-p., Nux-v., Puls.\n\nnausea: Arg-n., Carb-v., Mag-c., Nat-c.\n\nstools pale: Kali-m.\n\nviolent hiccups: Nat-m.\n\nBetter:\n\nbelching: Arg-n., Carb-v., Ign., Kali-b., Kali-c., Lyc.\n\nhot drinks: Ars., Nux-v.\n\nafter passing a stool: Nat-s.\n\npassing wind: Carb-v.\n\nwarmth of bed: Nux-v.\n\nWorse:\n\nafter belching: Chin.\n\nafter cabbage: Mag-c.\n\nafter drinking: Chin., Sul.\n\nafter eating: Arg-n., Carb-v., Chin., Kali-b., Kali-c., Lyc., Nux-v., Phos., Puls., Sul.\n\nfruit: Chin.\n\nmilk: Mag-c., Sul.\n\nat night: Puls.\n\nonions: Lyc.\n\npregnancy: Puls.\n\nrich\/fatty food: Carb-v., Puls.\n\nstarchy food: Kali-m., Nat-m., Nat-s.\n\nsweet foods\/sugar: Arg-n.\n\ntight clothing: Carb-v., Lyc., Nux-v.\n\nCauses:\n\nabdominal surgery: Chin.\n\ncoffee: Nux-v.\n\ngrief: Ign.\n\nmental strain: Nux-v.\n\nnervous exhaustion: Kali-p.\n\novereating: Nux-v.\n\noverwork: Kali-p.\n\nrich\/fatty food: Nux-v., Puls.\n\nInflammation of navel in newborn babies Hep-s., Sil.\n\nInflammation of penis in babies Arn.\n\nInjuries (see also Bites\/stings; Broken bones; Bruises; Burns; Sprains; Strains) Aco., Arn., Bell., Calc-s., Calen., Con., Hep-s., Hyp., Lach., Led., Phos., Ruta., Sil., Stap.\n\nCuts\/wounds:\n\nbleed freely: Aco., Bell., Lach., Phos.\n\nblood:\n\nbright red: Phos.\n\nslow to clot: Phos.\n\ncrushed: Hyp.\n\ninflamed: Hep-s., Sil.\n\nwith redness around: Hep-s.\n\nwith dirt\/splinter still inside: Sil.\n\nwith pus inside: Sil.\n\nwith pus oozing: Calc-s.\n\npainful: Calen., Hep-s., Hyp., Sil., Stap.\n\nshooting: Hyp.\n\nsore\/bruised: Arn., Hep-s., Ruta.\n\nsplinter-like: Hep-s.\n\nstitching\/tearing: Stap.\n\npainful out of proportion to the injury: Calen.\n\nsuppurate: Calc-s., Calen., Sil.\n\nwith bruising: Arn.\n\nTo fingers\/toes (crushed): Hyp.\n\nLacerated wounds (torn): Calen., Hyp., Led., Stap.\n\nPunctured wounds (with a knife, or nail, etc.): Hyp., Led.\n\nSlow to heal: Calc-s., Hep-s., Hyp., Lach., Sil.\n\nTo glands (breasts, etc.): Con.\n\ncold, inflamed, sensitive, stony hard lumps; swollen: Con.\n\nTo head: Arn., Nat-s.\n\nTo muscles: Arn.\n\nTo nerves\/nerve-rich parts of the body: Hyp., Stap.\n\nTo palm of hand or sole of foot: Led.\n\nTo shins: Ruta.\n\nWith:\n\nshock (anxiety and fear): Aco.\n\nforeign body: Sil.\n\nScars:\n\nbecome lumpy: Sil.\n\nbecome red: Lach.\n\nare painful, break open and\/or suppurate: Sil.\n\nBetter:\n\ncold compresses: Led.\n\nWorse:\n\nheat: Led.\n\ntouch: Hep-s., Stap.\n\nCauses:\n\naccident: Hyp., Stap.\n\nafter dental treatment: Arn., Hyp., Stap.\n\nchildbirth: Stap.\n\ncircumcision: Stap.\n\nepisiotomy: Hep-s., Hyp., Sil., Stap.\n\nnails: Led.\n\nsplinter: Hyp., Led., Sil.\n\nsurgery: Arn., Hyp., Sil., Stap.\n\nInsect bites see Bites\/stings\n\nInsomnia Aco., Ars., Bell., Bell-p., Calc-c., Calc-p., Cham., Cocc., Coff., Con., Cypr., Kali-c., Kali-p., Ign., Lyc., Mag-c., Mag-m., Nat-c., Nat-m., Nit-ac., Nux-v., Op., Pho-ac., Phos., Puls., Rhus-t., Sep., Sil., Sul.\n\nDreams:\n\nanxious: Aco., Ars., Calc-c., Cocc., Kali-c., Nat-c., Nat-m., Nit-ac., Phos., Puls.\n\nnightmares: Ars., Cocc., Con., Kali-c., Sil., Sul.\n\nvivid: Sil., Sul.\n\nunpleasant: Sul.\n\nvivid: Aco., Cham., Coff., Nat-m., Nux-v., Puls.\n\nIn pregnancy: Aco., Cham., Coff., Con., Ign., Nux-v., Puls.\n\nIn restless babies: Cypr.\n\nSleep:\n\ndisturbed: Sul.\n\nrestless: Aco., Ars., Bell., Cocc., Puls., Rhus-t., Sil., Sul.\n\nunrefreshing: Mag-c., Mag-m., Nit-ac., Pho-ac., Sul.\n\nSleepless: Pho-ac.\n\nafter midnight: Pho-ac., Rhus-t., Sil.\n\nafter 2 a.m.: Nit-ac.\n\nafter 3 a.m.: Bell-p., Mag-c., Nux-v.\n\nafter waking: Sil.\n\nbefore midnight: Con., Kali-c.\n\ndaytime (babies): Lyc.\n\nWaking:\n\naround 3 a.m.: Nux-v., Sep.\n\ndifficult: Calc-p.\n\nearly: Nat-c., Nux-v.\n\nfrequent: Pho-ac., Puls., Sul.\n\nfrom cold: Puls.\n\nlate: Calc-p., Nux-v., Sul.\n\nto play in the night (babies): Cypr.\n\nWith:\n\nempty feeling in pit of stomach: Kali-p.\n\ngrinding of teeth: Bell.\n\nsleepiness: Bell., Cham., Op., Phos., Puls., Sep.\n\nBetter:\n\nshort naps: Sul.\n\nWorse:\n\naround 1\u20132 a.m.: Kali-c.\n\nafter 2 a.m.: Nit-ac.\n\nafter 3 a.m.; Bell-p., Sep.\n\nafter 3, 4 or 5 a.m.: Sul.\n\nafter midnight: Ars., Sil.\n\nbefore midnight: Calc-c., Calc-p., Kali-c., Puls.\n\nCauses:\n\nanxiety: Ars., Cocc., Kali-p.\n\ncoffee: Cham., Nux-v.\n\ncramps in pregnancy: Cham., Coff., Nux-v.\n\nexcitement: Kali-p., Nux-v.\n\ngrief: Nat-m.\n\nmental strain: Kali-p., Nux-v.\n\nmovements of the baby (in pregnancy): Con.\n\nnervous exhaustion: Cypr., Kali-p.\n\noveractive mind: Ars., Calc-c., Coff., Nux-v., Puls.\n\noverexcitement: Coff.\n\noverwork: Nux-v.\n\nrepeating thoughts: Puls.\n\nshock: Ars., Ign.\n\nworry: Calc-c.\n\nIntrospective\/Introverted (see also Broody; Morose; Uncommunicative) Cocc., Ign., Nat-c., Nat-m., Puls.\n\nIrritable (see also Angry) Ant-c., Ant-t., Ap., Ars., Bor., Bry., Calc-p., Calc-s., Carb-v., Caust., Cham., Cina, Ferr-m., Hep-s., Kali-c., Kali-s., Lil-t., Lyc., Mag-c., Nat-c., Nat-m., Nit-ac., Nux-v., Petr., Pho-ac., Phos., Puls., Rhe., Rhus-t., Sep., Sil., Stap., Sul., Sul-ac., Ver-a., Zinc.\n\nAnd anxious: Nux-v.\n\nAnd dislikes consolation: Nat-m., Sep., Sil.\n\nBabies: Calc-p., Cham., Chin., Cina, Lyc., Mag-c., Sil., Stap.\n\nWorse:\n\nafter a sleep: Lyc.\n\nbefore stools: Bor.\n\ndaytime: Lyc.\n\nmornings: Chin., Lyc.\n\nwhen sick: Lyc.\n\nwhen touched: Ant-t.\n\nwith teething: Cham., Rhe.\n\nDuring pregnancy: Cham.\n\nDuring labour: Ars., Cham., Kali-c., Nux-v., Sep.\n\nJaundice in babies Chel., Chin., Nat-s.\n\nJealous Ap., Lach., Puls.\n\nJoint pain Ap., Arn., Bell-p., Bry., Calc-c., Calc-p., Caul., Caust., Cham., Cimi., Colch., Dulc., Ferr-m., Hep-s., Kali-b., Kali-c., Kali-s., Led., Lyc., Merc-s., Nit-ac., Puls., Rhod., Rhus-t., Ruta., Sul.\n\nFeet cold: Calc-p., Kali-s.\n\nJoints:\n\nicy cold: Led.\n\nhot: Led.\n\nstiff: Led.\n\nIn pregnancy: Cimi.\n\nPains:\n\naching: Rhus-t.\n\nacute: Colch.\n\nalternating with:\n\nindigestion: Kali-b.\n\ncough: Kali-b.\n\narms: Ferr-m., Kali-c., Rhod.\n\nupper: Ferr-m.\n\nin back: Caust.\n\nbones: Nit-ac., Puls., Ruta.\n\nburning: Ap., Caust., Merc-s., Sul.\n\ncramping: Calc-c.\n\ndrawing: Rhod.\n\nfeet: Calc-p., Colch., Led., Ruta.\n\nfingers: Hep-s.\n\nflying around: Caul.\n\ngnawing: Caust.\n\nhands: Colch., Led., Ruta.\n\nhips: Hep-s., Kali-c., Kali-s.\n\nin pregnancy: Calc-p.\n\nright: Led.\n\nirregular: Caul.\n\nin joints: Caust., Puls.\n\nlegs: Kali-c., Kali-s., Nit-ac., Rhod.\n\nlower back: Ruta.\n\nin neck: Caust.\n\nparts lain on: Ruta.\n\npressing: Caust.\n\npulling: Hep-s., Puls.\n\nshoulders: Ferr-m., Hep-s., Kali-c., Rhod.\n\nleft: Led.\n\nin small joints: Caul.\n\nin one spot: Kali-b.\n\nsore\/bruised: Arn., Hep-s., Kali-b., Kali-c., Puls., Rhus-t., Ruta.\n\nshooting: Rhus-t.\n\nsplinter-like: Nit-ac.\n\nstinging: Ap.\n\nstitching: Bry., Caust., Ferr-m., Kali-c., Nit-ac.\n\ntearing: Caust., Colch., Ferr-m., Hep-s., Kali-c., Lyc., Merc-s., Nit-ac., Rhod., Rhus-t., Sul.\n\nviolent: Cham.\n\nwandering: Kali-b., Kali-s., Puls.\n\nWith:\n\nheaviness: Rhus-t.\n\nlameness: Rhus-t., Ruta.\n\nnumbness: Cham., Kali-c.\n\nrestless legs: Kali-c.\n\nstiffness: Caust., Rhus-t.\n\nswelling: Ap., Bry., Colch.\n\nweakness of arms: Kali-c.\n\nBetter:\n\nafter a common cold: Led.\n\ncold compresses: Led., Puls.\n\ncold bathing: Led.\n\ncontinued movement: Rhus-t.\n\nfresh air: Kali-s., Puls.\n\ngentle movement: Ferr-m., Puls.\n\nheat: Caust., Hep-s., Rhus-t.\n\nmovement: Dulc., Kali-c., Kali-s., Lyc., Rhod.\n\npressure: Bry.\n\ncomplete rest: Bry.\n\nstretching out limb: Rhod.\n\nuncovering: Led.\n\nwalking: Cham., Kali-c., Kali-s., Lyc., Puls., Rhus-t.\n\nwarm compresses: Rhus-t.\n\nwarmth: Caust., Colch.\n\nwarmth of bed: Caust., Lyc., Rhus-t.\n\nWorse:\n\naround 2\u20133 a.m.: Kali-c.\n\nin bed: Merc-s.\n\nbeginning to move: Ferr-m., Lyc., Puls., Rhus-t.\n\nbending arm backwards: Ferr-m.\n\nchange of weather: Rhod.\n\ngetting chilled: Rhus-t.\n\ncold: Bry., Calc-c., Calc-p., Cimi., Colch., Hep-s.\n\ndamp weather: Rhus-t.\n\ndry cold: Caust.\n\ncold, wet weather: Calc-p., Colch.\n\nafter a common cold: Puls.\n\nduring a fever: Lyc., Rhus-t.\n\ngetting up from sitting: Caust.\n\nheat: Kali-s., Led., Puls.\n\nlifting arm up: Ferr-m.\n\nlying on the painful part: Kali-c.\n\nnight: Dulc., Led., Merc-s., Nit-ac., Rhus-t.\n\nslightest movement: Bry.\n\nmovement: Colch., Led.\n\nsitting down: Dulc., Lyc., Rhus-t., Rhod.\n\nstormy weather: Rhod.\n\nsummer: Kali-s.\n\ntouch: Arn.\n\nwalking: Ruta., Sul.\n\nwarm weather: Colch.\n\nwarmth of bed: Led., Puls., Sul.\n\nwet weather: Calc-c., Merc-s., Puls., Rhod.\n\nCauses:\n\ngetting chilled after being very hot: Bell-p.\n\ngetting cold: Dulc., Hep-s.\n\ndamp: Dulc.\n\nwet weather: Calc-c.\n\nJoking\n\nDuring labour: Coff.\n\nJumpy (see also Anxious; Restless) Bar-c., Bor., Kali-c., Kali-p., Nat-c., Nat-s., Phos., Sil.\n\nLabour Aco., Arn., Bell., Calc-c., Caul., Cham., Cimi., Coff., Cupr., Gels., Ip., Kali-c., Kali-p., Lyc., Mag-p., Nat-m., Nux-v., Op., Puls., Sec., Sep., Ver-a.\n\nFast (too fast): Aco., Lyc.\n\nLate: Aco., Caul., Cimi., Gels., Lyc.\n\nwith anticipatory anxiety: Gels., Lyc.\n\nwith fear of labour: Aco., Cimi.\n\nLong (too long): Arn.\n\nPremature: Nat-m., Nux-v., Op.\n\nPremature caused by shock: Op.\n\nSlow: Bell., Caul., Nat-m., Sec.\n\nLabour pains (contractions) Aco., Bell., Cimi., Caul., Cham., Coff., Cupr., Gels., Ip., Kali-c., Kali-p., Mag-p., Nat-m., Nux-v., Op., Puls., Sec., Sep., Ver-a.\n\nIn back: Cham., Cimi., Coff., Gels., Nux-v., Puls., Sep.\n\nIn back\/buttocks and\/or thighs: Gels., Kali-c.\n\nIn hips: Cimi.\n\nPains:\n\ndistressing: Bell., Caul., Cham., Coff., Gels., Kali-c., Sep.\n\nfalse: Bell., Caul., Gels., Puls.\n\nflying around abdomen: Cimi.\n\nineffectual:\n\ncervic doesn't dilate: Coff., Kali-c., Puls.\n\ncervix doesn't soften: Bell., Caul., Cham., Cimi., Gels., Nux-v., Sec.\n\nirregular: Caul., Coff., Puls.\n\nlong (each contraction lasts a long time): Sec.\n\nsevere (violent): Aco., Bell., Cham., Coff., Nux-v., Sep.\n\nshort (each contraction lasts a short time): Caul., Puls.\n\nstopping or slowing down: Bell., Caul., Cham., Cimi., Coff., Kali-c., Nat-m., Nux-v., Op., Puls., Sec., Sep.\n\nfrom exhaustion: Caul.\n\nwith cramps in hip and\/or shivering: Cimi.\n\nwith talkativeness: Coff.\n\nunbearable: Cham.\n\nweak: Bell., Caul., Cimi., Gels., Kali-c., Nat-m., Op., Puls., Sec.\n\nWith:\n\nchilliness after a contraction: Kali-c.\n\ncramps:\n\nin hands or legs: Bell., Cupr., Mag-p., Nux-v.\n\nin fingers or toes: Cupr.\n\nexhaustion: Bell., Caul., Cham., Kali-c., Kali-p., Nat-m., Nux-v., Op., Puls., Sec., Sep., Ver-a.\n\nfaint feeling: Cimi., Nux-v., Puls., Sec., Ver-a.\n\nirritability: Caul.\n\nnausea: Ip., Puls., Ver-a.\n\nconstant: Ip.\n\nred face: Bell., Op.\n\nthirst: Caul.\n\ntrembling: Caul., Cimi., Gels.\n\nif the contractions stop: Sec.\n\nurging to pass a stool: Nux-v.\n\nvomiting: Cupr., Puls., Ver-a.\n\nLack of self-confidence (see also Shy) Bar-c., Lyc., Sil.\n\nLaryngitis see Sore throat\n\nLazy Sul.\n\nLethargic see Apathetic; Exhaustion; Sluggish\n\nLikes\n\nboiled eggs: Calc-c.\n\nbread and butter: Merc-s.\n\nchocolate: Lyc., Phos.\n\ncold drinks: Aco., Bry., Cham., Chin., Cina, Eup-p., Merc-c., Merc-s., Nat-s., Phos., Ver-a.\n\ncold food: Phos., Puls.\n\ncold food in pregnancy: Ver-a.\n\nfat\/fatty foods: Nit-ac.\n\nfruit: Pho-ac., Ver-a.\n\nham, salami, smoked foods: Caust.\n\nhot drinks: Ars., Bry.\n\nhot food: Ars., Bry.\n\nice-cream: Phos., Ver-a.\n\nice-cold drinks: Phos., Ver-a.\n\nmilk: Phos., Rhus-t.\n\nrefreshing things: Pho-ac., Ver-a.\n\nsalt, salty foods: Nat-m., Phos.\n\nsalty foods in pregnancy: Nat-m., Ver-a.\n\nsour food: Hep-s., Ver-a.\n\nsour foods in pregnancy: Sep., Ver-a.\n\nspicy food: Chin., Nux-v., Phos., Sul.\n\nstrange things in pregnancy (like raw spaghetti\/rice, chalk, coal): Alu., Calc-c.\n\nsugar: Arg-n.\n\nsweets: Arg-n., Chin., Lyc., Sul.\n\nvinegar: Sep.\n\nLips\n\nCracked: Ars., Nat-m., Sul.\n\nBlue: Cupr., Lach.\n\nDry: Ant-c., Ars., Bry., Puls., Rhus-t., Sul.\n\nLicks: Ars.\n\nRed: Sul.\n\nSwollen: Ap.\n\nLively (see also Excitable) Coff., Lach., Nat-c., Op.\n\nLochia Calc-c., Kreos., Puls., Sec., Sil.\n\nBurns: Kreos.\n\nDark: Kreos., Sec.\n\nFlows while baby breastfeeds: Sil.\n\nIntermittent: Calc-c.\n\nLasts too long: Calc-c., Sec.\n\nLumpy: Kreos.\n\nReturns: Calc-c., Kreos., Puls.\n\nScanty: Puls., Sec.\n\nSmelly: Kreos., Sec.\n\nLonely Puls., Stram.\n\nLoquacious see Talkative\n\nLoss of libido Caust., Nat-m., Sep.\n\nLumbago see Backache\n\nMastitis see Breastfeeding problems; Breasts inflamed\n\nMeasles Aco., Ap., Ars., Bell., Bry., Euphr., Gels., Puls., Sul.\n\nOnset:\n\nslow: Bry., Gels.\n\nsudden: Aco., Bell.\n\nSkin rash:\n\nburns: Aco., Bell., Sul.\n\nhot: Bell.\n\nitches: Aco., Bell., Sul.\n\nmaddeningly: Sul.\n\nred: Bell., Sul.\n\nslow to appear: Ap., Bry., Sul.\n\nWith:\n\ncommon cold: Euphr., Puls.\n\ncough: Aco., Bell., Bry., Euphr., Puls., Sul.\n\neye inflammation: Ap., Bell., Euphr., Puls.\n\nfever: Aco., Ap., Bell., Bry., Gels., Sul.\n\nheachache: Bry., Gels.\n\nWorse for heat: Sul.\n\nMelancholic Calc-c., Con.\n\nMemory weak Arg-n., Caust., Cocc., Colch., Con.\n\nMigraine see Headache, one-sided\n\nMild, gentle Cocc., Puls., Sil.\n\nMischievous Chin., Merc-s., Nux-v., Stram.\n\nMoaning, complaining Aco., Coff.\n\nDuring labour: Coff.\n\nIn babies: Bor., Cham., Cina\n\nwho can't have what they want: Cham.\n\nMoody (changeable) Ferr-m., Ign., Lyc., Puls., Sul-ac., Zinc.\n\nMorose (see also Depressed; Despondent) Bry.\n\nMouth\n\nBurning: Ars.\n\nDry: Ars., Bry., Merc-s., Nat-m., Puls.\n\nwith thirst: Bry., Nat-m.\n\nwithout thirst: Puls.\n\nMouth ulcers Lyc., Merc-s., Nit-ac.\n\nUlcers:\n\nedges of tongue: Nit-ac.\n\ngums: Merc-s.\n\npainful: Nit-ac.\n\non the tongue: Merc-s.\n\nunder the tongue: Lyc.\n\nPains:\n\nstinging\/throbbing: Merc-s.\n\nMumps Aco., Ap., Ars., Bar-c., Bell., Bry., Carb-v., Jab., Lach., Lyc., Merc-s., Phyt., Puls., Rhus-t., Sil.\n\nGlands (parotid):\n\nhard: Merc-s., Phyt.\n\nswelling moves from the right side to the left: Lyc.\n\nswollen\/painful: Bell., Carb-v., Jab., Lach., Lyc., Merc-s., Phyt., Puls., Rhus-t., Sil.\n\nOnset sudden: Aco., Bell.\n\nPains spread to breasts, ovaries: Carb-v., Jab., Phyt., Puls.\n\nWith:\n\nface red: Jab.\n\nfever: Aco., Bell., Lyc., Merc-s., Puls., Rhus-t.\n\nheadache: Bell.\n\ncopious saliva: Jab., Merc-s., Phyt.\n\nprofuse sweating: Jab., Merc-s., Phyt.\n\nsore throat: Bell., Lach., Lyc., Phyt.\n\nvomiting: Puls.\n\nBetter:\n\nfor heat: Rhus-t., Sil.\n\nWorse:\n\nblowing nose: Merc-s.\n\nat night: Merc-s.\n\non the left: Lach., Rhus-t.\n\non the right: Bell., Merc-s.\n\nfor cold: Rhus-t., Sil.\n\nNappy rash (Diaper rash) Ap., Petr., Rhus-t., Sul.\n\nBurns: Rhus-t., Sul.\n\nBleeding: Petr., Sul.\n\nCracked\/dry: Petr.\n\nFlaky: Rhus-t.\n\nHot: Ap., Sul.\n\nItches: Petr., Rhus-t., Sul.\n\nRed: Ap., Petr., Sul.\n\nRaw: Sul.\n\nShiny\/sore: Ap.\n\nWeepy: Petr.\n\nBetter:\n\ncovering: Rhus-t.\n\nuncovering: Ap., Sul.\n\nWorse:\n\nbathing\/washing, heat: Ap., Sul.\n\ncold: Rhus-t.\n\nin the folds of the skin: Petr.\n\ntouch: Ap.\n\nNaughty see Mischievous\n\nNausea see also Taste Ant-c., Ant-t., Ars., Asar., Cocc., Colch., Con., Ip., Kreos., Lyc., Nux-v., Petr., Phos., Puls., Sep., Sul., Tab., Ver-a.\n\nConstant: Ant-c., Asar., Ip., Kreos., Nux-v.\n\nDeathly: Ars., Ip., Tab.\n\nIntermittent: Ant-t., Sep., Tab.\n\nPersistent: Ip.\n\nViolent: Ip., Tab.\n\nduring pregnancy: Ant-c., Asar.\n\nAbdomen\/stomach feels:\n\nbloated: Colch.\n\nempty: Cocc., Phos., Sep.\n\nAppetite lost: Cocc., Colch.\n\nBelches: Ant-c., Cocc.\n\nempty: Ant-c., Ip.\n\nsour: Phos.\n\ntasting of food eaten: Ant-c.\n\nPains gnawing: Sep.\n\nWith:\n\ncopious saliva: Ip., Kreos., Nux-v., Petr.\n\ndizziness: Petr.\n\nfaintness: Cocc., Colch.\n\nface pallid: Ip.\n\nheadache: Petr.\n\ninability to vomit: Nux-v.\n\nretching: Asar., Colch., Con., Ip., Nux-v.\n\nvomiting: Ars., Asar., Colch., Con., Ip., Kreos., Lyc., Nux-v., Petr., Puls., Sep., Sul., Tab., Ver-a.\n\nBetter:\n\nfor belching: Ant-t.\n\nfor cold drinks: Phos., Puls.\n\nafter eating (temporarily): Sep.\n\nfresh air: Lyc., Puls., Tab.\n\nfor vomiting: Ant-t.\n\nWorse:\n\nafter drinking: Ars., Cocc., Puls.\n\nafter eating: Ant-c., Cocc., Colch., Con., Ip., Kali-c., Lyc., Nux-v., Puls., Sep., Sul.\n\nafternoon: Ars., Cocc.\n\nbending down: Ip.\n\nbefore breakfast: Sep., Tab.\n\ncold drinks: Ars., Kali-c., Lyc.\n\ncoughing: Ip., Kali-c., Puls.\n\neating acidic foods: Ant-c.\n\neating bread or starchy food: Ant-c.\n\nfasting: Lyc., Tab.\n\nfresh air: Petr.\n\nhot drinks: Phos., Puls.\n\nice-cream: Ars., Ip., Puls.\n\nmilk: Calc-c., Nit-ac., Sep.\n\nmorning: Puls., Sep.\n\nmorning in bed: Nux-v.\n\nmovement: Cocc., Ip., Kali-c., Tab., Ver-a.\n\nnoise: Asar., Cocc.\n\npork: Sep.\n\nputting hands in warm water: Phos.\n\nrich food: Puls.\n\nsight of food: Cocc., Colch., Ip.\n\nsitting up in bed; Cocc.\n\nsmell of eggs or fish: Colch.\n\nsmell of food: Ars., Cocc., Colch., Ip., Sep.\n\nstuffy room: Lyc., Puls., Tab.\n\nsweating: Nux-v.\n\ntobacco: Ip., Lyc., Nux-v.\n\nwater: Phos.\n\nCauses:\n\npregnancy: Ant-c., Ars., Asar., Colch., Con., Ip., Kreos., Lyc., Nux-v., Petr., Puls., Sep., Sul., Tab., Ver-a.\n\ntravelling: Cocc., Nux-v., Tab.\n\nNervous see Anxious\n\nNettle rash see Hives\n\nNose\n\nChild picks constantly: Cina\n\nNosebleeds Arn., Carb-v., Cocc., Ferr-m., Ham., Lach., Phos., Sep., Sul.\n\nBabies: Ferr-m.\n\nBlood:\n\nbright red: Phos.\n\ndark: Carb-v., Ham.\n\npersistent: Phos.\n\nthin: Ham.\n\nIn pregnancy: Cocc., Sep.\n\nWith:\n\nsweating: Phos.\n\nWorse:\n\nbefore a menstrual period: Lach.\n\nblowing nose: Phos., Sul.\n\nmorning: Ham., Lach.\n\nat night: Carb-v.\n\nCauses:\n\nblowing nose: Lach.\n\ninjury: Arn.\n\nObstinate see Stubborn\n\nOedema (swelling)\n\nOf ankles and feet: Ap., Calc-c., Ferr-m., Lyc., Merc-c., Nat-m., Puls., Rhus-t., Zinc.\n\nOf hands and fingers: Ap., Calc-c., Ferr-m., Lyc., Merc-c., Nat-m., Puls., Rhus-t.\n\nOnset of complaint\n\nSlow: Bry., Gels.\n\nSudden: Aco., Bell.\n\nOversensitivity see Sensitive\n\nPains (see also Abdominal pain in pregnancy; Groin pains; Backache, etc.)\n\nAbsent: Op.\n\nAppear suddenly: Bell.\n\nAppear suddenly and disappear suddenly: Bell., Kali-b., Nit-ac.\n\nBones: Eup-p., Merc-s., Nit-ac., Pho-ac., Ruta.\n\nBurning: Ap., Ars., Canth., Caust., Phos., Rhus-t., Sul.\n\nCramping: Calc-c., Cupr., Mag-p., Nux-v.\n\nFlying around: Cimi., Nit-ac.\n\nGlands: Arn., Bell., Lyc., Merc-s., Phos.\n\nLegs: Bry.\n\nNeedle-like: Arg-n., Hep-s., Nit-ac.\n\nParts lain on: Puls., Pyr., Ruta.\n\nPressing: Canth., Merc-s., Rhus-t., Sep.\n\nShooting: Bell., Cimi., Hyp., Rhus-t.\n\nShoot upwards: Phyt.\n\nSore\/bruised: Arn., Bry., Chin., Cimi., Eup-p., Ham., Pyr., Rhus-t., Ruta.\n\nSplinter-like see Needle-like\n\nIn a spot: Kali-b.\n\nStinging: Ap., Bry., Kali-c., Sil.\n\nStitching: Bry., Kali-c.\n\nThrobbing: Bell.\n\nUnbearable: Aco., Cham.\n\nWandering: Kali-b., Kali-s., Led., Puls.\n\nPale see Face, pale\n\nPalpitations Aco., Arg-n., Ars., Calc-c., Lach.\n\nIn pregnancy: Con., Lil-t., Nat-m., Sep.\n\nPanic (see also Anxious; Fearful) Arg-n.\n\nPerspiration see Sweat\n\nPhlebitis Arn., Bry., Ham., Puls., Rhus-t.\n\nPiles Aesc., Ars., Ham., Ign., Kali-c., Lach., Lil-t., Lyc., Nat-m., Nit-ac., Nux-v., Puls., Sul.\n\nBleeding: Aesc., Ars., Lach., Lyc., Nat-m., Nux-v., Sul.\n\nBluish: Lach.\n\nBurning: Ars., Sul.\n\nExternal: Aesc., Lach., Lyc., Sul.\n\nInternal: Ars., Ign., Nux-v., Puls., Sul.\n\nItching: Aesc., Lyc., Nux-v., Sul.\n\nLarge: Aesc., Kali-c., Lach., Nux-v., Sul.\n\nWith:\n\nbackache: Aesc.\n\nconstipation: Aesc., Lyc., Nat-m., Nux-v., Sul.\n\ndragging down sensation: Lil-t.\n\nBetter:\n\nbathing in warm water: Aesc., Ars.\n\nbathing in cold water: Nux-v.\n\nWorse:\n\ncoughing or passing a stool: Ign., Kali-c., Nit-ac.\n\ntouch: Kali-c., Stap., Sul.\n\nstanding\/walking: Aesc., Ign., Lil-t., Sul.\n\nPink eye see Eye inflammation\n\nPlay\n\nBaby wants to play at night: Cypr.\n\nBaby doesn't want to play: Bar-c., Cina, Hep-s., Lyc., Nux-m., Rhe., Sil.\n\nPrickly heat Nat-m., Sul., Urt-u.\n\nProlapse Calc-c., Lil-t., Puls., Rhus-t., Sec., Sep.\n\nWith:\n\nbearing down sensation: Lil-t., Sep.\n\nconstipation: Sep.\n\npressure in abdomen and small of back: Puls.\n\nBetter:\n\nlying down or sitting with legs crossed: Sep.\n\nCauses:\n\nforceps delivery: Sec.\n\nstraining in labour: Rhus-t., Sec.\n\nPulse rapid Pyr.\n\nPupils\n\nContracted: Op.\n\nDilated: Bell.\n\nQuarrelsome (see also Dislikes contradiction; Irritable) Ign., Nux-v., Petr., Sul.\n\nDuring labour: Bell., Cham.\n\nRage Bell., Lyc., Stram., Ver-a.\n\nRage during labour Bell.\n\nWith a desire to bite or hit: Bell., Stram.\n\nBabies on being picked up: Stram.\n\nResentful (see also Unforgiving; Complaints from suppressed anger) Ign., Lyc., Nat-m., Stap.\n\nRestless Aco., Ap., Arg-n., Ars., Bell., Calc-p., Canth., Caust., Cham., Cimi., Cina, Coff., Coloc., Cupr., Eup-p., Ferr-m., Lil-t., Lyc., Mag-m., Merc-s., Rhus-t., Sil., Sul., Zinc.\n\nBabies: Ars., Cham., Cina, Cupr., Merc-s., Rhus-t., Stram., Sul.\n\nbetter for being carried: Ars., Cham., Cina\n\nin teething babies: Rhe.\n\nwanders aimlessly: Stram., Ver-a.\n\nDuring labour: Aco., Coff., Lyc.\n\nEvening: Caust., Zinc.\n\nIn bed: Ars., Cupr., Ferr-m., Mag-m., Rhus-t.\n\nLegs in pregnancy: Caust., Rhus-t., Sul., Zinc.\n\nSleep: Cina\n\nbody twitching\/limbs jerking\/grinding of teeth: Cina\n\nWith anxiety: Ars.\n\nWorse:\n\nfor heat: Sul.\n\nRetained placenta Arn., Cimi., Puls., Sec.\n\nAfter a long labour: Arn.\n\nWith:\n\na bearing down sensation: Sec.\n\nthe shakes: Cimi.\n\nineffective contractions: Puls.\n\nRetention of urine Aco., Ap., Arn., Ars., Caust., Nat-m., Op., Stap.\n\nIn newborn babies: Aco., Op.\n\nIn babies: Aco., Ap.\n\nwho catch cold: Aco.\n\nCauses:\n\nchildbirth: Arn., Ars., Caust., Op., Stap.\n\npresence of strangers: Nat-m.\n\nshock: Op.\n\nRetention of urine (after childbirth) Arn., Ars., Caust., Op., Stap.\n\nAfter a difficult birth or forceps delivery: Arn., Stap.\n\nWith:\n\nfrequent, painful urging to pass urine \u2013 only passes a little each time: Caust.\n\ninvoluntary urination: Arn., Ars., Caust.\n\nno desire to pass urine: Ars.\n\npainful urging: Arn., Nux-v.\n\npainless urging: Op.\n\nRheumatism see Joint pain\n\nRhinitis see Common cold\n\nRoseola (see also Measles) Aco., Bell., Puls.\n\nRushed feeling see Hurried\n\nSad see Depressed; Tearful\n\nSaliva\n\nIncreased: Merc-s., Ver-a.\n\nduring sleep: Merc-s.\n\nScared see Fearful\n\nScarlet fever Ap., Bell., Lach., Merc-s., Rhus-t.\n\nRash:\n\nbluish: Lach.\n\nitches and burns: Rhus-t.\n\nScars Thios\n\nSciatica Coloc., Ferr-m., Mag-p., Puls., Rhus-t.\n\nPains:\n\ntearing: Coloc.\n\nBetter:\n\nfresh air: Puls.\n\ngentle movement: Ferr-m.\n\nheat: Mag-p., Rhus-t.\n\nmovement: Rhus-t.\n\nwalking: Ferr-m., Rhus-t.\n\nwalking slowly: Ferr-m.\n\nWorse:\n\nbeginning to move: Rhus-t.\n\ncold: Mag-p., Rhus-t.\n\ncold compresses: Rhus-t.\n\nlying on painful part: Rhus-t.\n\nright side: Coloc.\n\nwarm room: Puls.\n\nwashing in cold water: Rhus-t.\n\nwet weather: Rhus-t.\n\nScreaming Aco., Ant-t., Bell., Bor., Calc-p., Cham., Chin., Cina, Coff., Kali-p., Kreos., Lyc., Nux-v., Rhe.\n\nIf touched: Ant-t.\n\nIn babies: Bor., Cina, Lyc., Kreos.\n\nWith pain: Aco., Bell., Cham., Coff.\n\nOn waking: Cham., Chin., Cina, Kali-p., Lyc., Mag-c., Zinc.\n\nIn sleep: Calc-p.\n\nDuring sleep: Bor., Lyc.\n\nOn waking: Lyc.\n\nSense of smell\n\nAcute: Colch., Nux-v., Op.\n\nLost: Calc-c.\n\nSenses\n\nAcute: Asar., Coff., Op.\n\nDull: Op.\n\nHyperacute: Asar.\n\nSensitive Aco., Asar., Bell., Cham., Chin., Cimi., Coff., Con., Hep-s., Ign., Kali-c., Lach., Lyc., Nat-c., Nat-m., Nat-p., Nit-ac., Nux-v., Phos., Puls., Sep., Sil., Stap.\n\nBabies: Aco., Bell., Cham., Kali-p., Phos., Puls., Stap.\n\nTo light: Bell., Con., Nux-v., Phos.\n\nTo music: Nat-c., Nux-v., Sep.\n\nTo noise: Aco., Asar., Bell., Chin., Coff., Con., Kali-c., Nit-ac., Nux-v., Op., Sep., Sil., Zinc.\n\nTo noise during labour: Bell., Cimi., Coff., Nux-v.\n\nTo pain: Cham., Coff., Ign., Lach., Nit-ac., Phos., Sep., Sil., Stap.\n\nduring pregnancy: Asar., Cimi.\n\nTo rudeness: Hep-s., Nux-v., Stap.\n\nTo touch: Lach.\n\nSentimental Ant-c., Ign.\n\nShock Aco., Arn., Hyp., Ign., Lach., Op., Pho-ac., Phos., Stap., Stram.\n\nWith:\n\nanger: Stap.\n\nanxiety: Aco.\n\nfear: Aco., Stram.\n\nindignation: Stap.\n\nstupor: Op.\n\nvomiting: Phos.\n\nCauses:\n\nemotional trauma: Ign., Pho-ac.\n\ninjuries: Aco., Arn., Hyp., Lach., Op., Stap.\n\nsurgery: Aco., Arn., Op., Phos., Stap., Stram.\n\nchildbirth: Aco., Arn., Stram.\n\nShrieking during labour Bell., Cimi.\n\nShy Bar-c., Lyc., Nat-c., Puls., Sil.\n\nSide see Symptoms\n\nSighing Calc-p., Cimi., Ign.\n\nSinusitis see Common cold\n\nSkin complaints of pregnancy\n\nBroken veins: Arn.\n\nBrown pigmentation patches on face: Sep.\n\nDryness: Calc-f.\n\nItching: Sul.\n\nSpots: Sep.\n\nStretch marks: Calc-f.\n\nSleepless see Insomnia\n\nSleepy see Drowsy\n\nSlowness Asar., Bar-c., Calc-c., Calc-p., Con., Phos.\n\nIn babies: Bar-c., Calc-c., Calc-p.\n\nlearning to walk: Calc-c.\n\nto develop: Bar-c.\n\nto teethe: Calc-c., Calc-p.\n\nto learn: Calc-p.\n\nSluggish, dull (see also Apathetic; Confused; Exhaustion) Ant-c., Asar., Bar-c., Bry., Calc-c., Calc-p., Calc-s., Carb-v., Gels., Kali-c., Lach., Lyc., Merc-c., Nat-c., Nat-p., Nit-ac., Nux-m., Pho-ac., Phos., Puls., Sep., Sil., Stap., Sul., Ver-a., Zinc.\n\nBabies: Bar-c., Calc-p., Sul.\n\nOn waking: Lach.\n\nBetter for:\n\nfresh air: Lyc.\n\nSnuffles in newborn babies Dulc., Lyc., Nux-v., Puls.\n\nSore throat Aco., All-c., Ap., Arg-n., Ars., Bar-c., Bell., Bry., Calc-c., Caust., Dros., Dulc., Hep-s., Ign., Kali-m., Lach., Lyc., Merc-c., Merc-s., Nat-m., Nit-ac., Nux-v., Phos., Phyt., Puls., Rhus-t., Rumex, Sil., Spo., Sul.\n\nAir passages irritated: Rumex\n\nIn babies: Dros.\n\nConstant: Caust.\n\nGlands:\n\nswollen: Bell., Sil.\n\nGums:\n\nbleeding, swollen: Merc-c.\n\nLarynx:\n\ninflamed: Dros.\n\nirritated: Dros., Puls.\n\ntickling: Dros., Puls., Rhus-t.\n\nMouth dry: Ap.\n\nNoises in ears:\n\nroaring: Bar-c.\n\non swallowing: Bar-c.\n\nPains:\n\nburning: Aco., Ap., Ars., Bar-c., Caust., Merc-c., Nat-m., Nit-ac., Spo., Sul.\n\ndry: Caust.\n\npressing: Nit-ac.\n\nraw: Caust., Hep-s., Lyc., Merc-c., Nit-ac., Nux-v., Phos., Puls., Sul.\n\nscraping: Puls.\n\nsevere: Bell.\n\nsore: Lyc., Merc-s., Phos., Rumex, Spo., Sul.\n\nsplinter-like: Arg-n., Hep-s., Lach., Nit-ac.\n\nspreading up to ears: Hep-s., Lach., Merc-s., Nit-ac., Nux-v.\n\nspreading up to neck: Merc-s.\n\nstitching: Aco., Bell., Bry., Ign., Merc-s., Nit-ac., Nux-v., Puls., Sil., Sul.\n\nThroat:\n\nburning: Caust.\n\nconstricted: Bell.\n\ndark red: Phyt.\n\ndry: Calc-c., Caust., Lyc., Nat-m., Puls., Rhus-t., Sil., Spo., Sul.\n\ninflamed: Bar-c.\n\nirritated: Arg-n., Bell., Rumex\n\nraw: Arg-n., Bar-c., Bell., Caust., Rumex, Sul.\n\nsore: Rumex\n\nTonsils:\n\nswollen\/inflamed: Bar-c., Hep-s., Kali-m., Lyc., Merc-s., Nit-ac., Phos., Phyt., Sil., Sul.\n\nulcerated: Ars., Merc-s., Nit-ac.\n\nwhite: Kali-m.\n\nVoice:\n\nhoarse: Arg-n., Bry., Dros., Nat-m., Phos., Rhus-t., Spo., Sul.\n\nlost: Arg-n., Rumex\n\nWith:\n\nchoking sensation: Caust., Ign., Lach., Sul.\n\ndesire to swallow: Caust.\n\nfever: Bry.\n\nhair sensation at the back of tongue: Sil.\n\nlump sensation in throat: Ign., Lach., Lyc., Nat-m., Nux-v.\n\nsaliva increased: Bar-c., Merc-c.\n\nulcers in the throat: Ars., Merc-c.\n\nwind: Dros.\n\nBetter:\n\ncold drinks: Phyt.\n\neating: Merc-c.\n\nhot drinks: Ars., Hep-s.\n\nswallowing: Ign.\n\nswallowing liquids: Bar-c.\n\nswallowing solids: Lach.\n\nwarm compresses: Hep-s.\n\nwarm drinks: Lyc.\n\nWorse:\n\nbreathing in cold air: Hep-s., Rumex\n\nbreathing in: Phos.\n\ncold air: Nux-v.\n\ncold drinks: Ars., Hep-s., Rhus-t.\n\ngetting cold: Sil.\n\ncoughing: Hep-s., Phos., Spo., Sul.\n\nevening: Ign.\n\nheat: Lach., Puls.\n\nhot drinks: Phyt.\n\nleft side: Lach.\n\nmorning and evening: Phos.\n\nnight: Bar-c., Lyc., Merc-s.\n\nnot swallowing: Ign.\n\npressure: Lach., Merc-c., Phos.\n\nright side: Bell., Lyc., Merc-s.\n\nsinging: Spo.\n\nswallowing: Ars., Bry., Dros., Hep-s., Merc-s., Nit-ac., Nux-v., Rhus-t., Sil., Spo., Sul.\n\nswallowing saliva (empty swallowing): Bar-c., Lach.\n\nswallowing food: Bar-c.\n\nswallowing liquids: Bell., Lach., Merc-c.\n\nsweet drinks\/food: Spo.\n\ntalking: Caust., Phos., Spo.\n\ntouch: Lach., Spo.\n\nturning the head: Hep-s.\n\nuncovering: Nux-v., Rhus-t., Sil.\n\nwinter: Hep-s.\n\nCauses:\n\nchange of weather: Calc-c.\n\nexposure to wind: Hep-s.\n\ngetting cold: Bell., Hep-s.\n\nsinging: Arg-n.\n\nstraining voice: Rhus-t.\n\ntalking: Arg-n.\n\ngetting chilled: Aco., Bell.\n\nSpeedy see Hurried\n\nSpiteful Nux-v.\n\nSplinters (see also Injuries) Sil.\n\nWound inflamed: Sil.\n\nSpots in babies Carb-v., Sil., Sul.\n\nSprains Arn., Calc-c., Rhus-t., Ruta.\n\nAnkle: Arn., Calc-c., Rhus-t., Ruta.\n\nFoot: Arn., Rhus-t.\n\nHand: Calc-c.\n\nWrist: Arn., Calc-c., Rhus-t., Ruta.\n\nFirst stage: Arn.\n\nPains:\n\nbruised, constant: Ruta.\n\nWith:\n\nbruising: Arn.\n\nexhaustion, lameness: Ruta.\n\nstiffness: Rhus-t.\n\nswelling: Arn.\n\ntrembling: Rhus-t.\n\nWorse:\n\nexercise: Ruta.\n\nlifting: Calc-c.\n\npressure\/standing\/walking: Ruta.\n\nCauses:\n\nfalling: Rhus-t.\n\nlifting: Calc-c., Rhus-t.\n\nheavy weights: Calc-c.\n\ntwisting: Rhus-t.\n\nSticky eyes see Eye inflammation\n\nStiff neck Rhus-t.\n\nStomach-ache see Colic\n\nStrains Arn., Carb-a., Rhus-t., Ruta.\n\nAnkle: Ruta.\n\nBack: Carb-a.\n\nMuscles: Carb-a.\n\nPeriosteum: Ruta.\n\nTendons: Carb-a., Ruta.\n\nWrist: Carb-a., Ruta.\n\nPains:\n\nbruised, constant: Ruta.\n\njoints: Rhus-t.\n\nsore, bruised: Arn., Rhus-t.\n\nWith:\n\nexhaustion, lameness: Ruta.\n\nstiffness, trembling: Rhus-t.\n\nBetter:\n\ncontinued movement: Rhus-t.\n\nWorse:\n\nbeginning to move: Rhus-t.\n\nexercise, standing, walking: Ruta.\n\nCauses:\n\nchildbirth: Arn.\n\nlifting: Carb-a.\n\nover-exertion: Arn.\n\nStubborn Calc-c., Cham., Nux-v.\n\nBabies: Ant-c., Calc-c., Cham., Chin., Cina\n\nStupor (see also Dazed) Nux-m., Op., Sec.\n\nStyes Puls., Stap.\n\nWorse:\n\nupper eyelids: Puls.\n\nSudden onset see Onset\n\nSulky (see also Moody; Morose) Ant-c.\n\nSunburn see Burns\n\nSunstroke Bell., Glon.\n\nWith:\n\nfever: Bell.\n\nheadache: Bell.\n\nSweat\n\nAbsent during fever: Ars., Bell., Bry., Gels.\n\nClammy: Ars., Cham., Ferr-m., Lyc., Merc-s., Pho-ac., Phos., Ver-a.\n\nCold: Ant-t., Ars., Carb-v., Chin., Cocc., Ferr-m., Hep-s., Ip., Lyc., Merc-c., Merc-s., Sep., Ver-a.\n\nExhausting: Carb-a., Ver-a.\n\nFrom fright: Nux-v., Op.\n\nHead: Calc-c., Sil.\n\nHot: Aco., Cham., Con., Ign., Ip., Nux-v., Op., Sep.\n\nOily: Mag-c.\n\nOn covered parts of the body: Aco., Bell., Chin.\n\nOne-sided: Bar-c., Lyc., Nat-c., Nux-v., Petr., Puls.\n\nFrom pain: Lach., Merc-s., Nat-c., Nat-s., Sep.\n\nProfuse: Ant-t., Ars., Calc-c., Carb-a., Carb-v., Chin., Ferr-m., Hep-s., Kali-c., Kali-p., Lyc., Merc-s., Op., Pho-ac., Sep., Sil., Sul., Ver-a.\n\nScanty: Eup-p.\n\nSingle parts of the body: Chin., Cocc., Ign., Kali-c., Nit-ac., Phos., Puls., Op., Sep.\n\nSmelly: Carb-a., Hep-s., Lyc., Merc-s., Nit-ac., Nux-v., Petr., Puls., Sep., Sil., Sul.\n\nSour: Ars., Calc-c., Colch., Hep-s., Lyc., Mag-c., Merc-s., Nit-ac., Sep., Sil., Sul., Ver-a.\n\nWith:\n\ndiarrhoea: Ver-a.\n\nfever: Pyr.\n\nshivering: Nux-v.\n\nBetter:\n\nuncovering: Cham., Lyc.\n\nWorse:\n\nafter eating: Kali-c.\n\nat night: Con., Puls., Sil.\n\nclosing eyes: Con.\n\nduring sleep: Con., Sil.\n\nlying down: Con., Ferr-m., Rhus-t.\n\nmental exertion: Calc-c., Hep-s., Lach., Sep., Stap.\n\npain: Lach., Merc-s., Nat-c., Sep.\n\nslightest physical exertion: Calc-c., Chin., Ferr-m., Kali-c., Kali-p., Lyc., Nat-c., Nat-s., Nit-ac., Phos., Rhus-t., Sep., Sul.\n\nwhen tired: Kali-p.\n\nuncovering: Rhus-t.\n\nCauses:\n\nfright: Op.\n\nSwelling see Oedema\n\nSympathetic Caust., Phos.\n\nSymptoms\n\nChangeable: Puls.\n\nContradictory: Ign.\n\nOne-sided: Pho-ac.\n\nLeft-sided: Lach., Sep., Sul.\n\nRight-sided: Ap., Bell., Calc-c., Lyc., Nux-v., Puls.\n\nMoving from:\n\nleft side to right side: Lach.\n\nright side to left side: Ap., Lyc.\n\nTalkative Cimi., Lach.\n\nTantrums Bell., Lyc., Nit-ac., Stram., Ver-a.\n\nTaste Mouth tastes:\n\nBad: Calc-c., Merc-s., Nat-s., Nux-v., Puls., Sul.\n\nin the morning: Nat-s., Nux-v., Puls.\n\nBitter: Aco., Ars., Bry., Carb-v., Chin., Merc-s., Nat-m., Nat-s., Nux-v., Sul.\n\nMetallic: Cocc., Cupr., Merc-s., Nat-c., Rhus-t., Zinc.\n\nSalty: Nat-m.\n\nSour: Arg-n., Calc-c., Lyc., Mag-c., Nux-v., Phos.\n\nSweet: Cupr.\n\nTearful (see also Depressed; Dislikes consolation) Ap., Bell., Bor., Calc-c., Calc-s., Caust., Cham., Ign., Lyc., Nat-m., Puls., Rhe., Rhus-t., Sep., Spo., Stram., Ver-a.\n\nBabies: Bell., Bor., Calc-c., Cham., Lyc., Puls., Rhe., Stram.\n\nat night: Bor., Rhe.\n\nwhen teething: Phyt.\n\nwho can't have what they want: Cina\n\nDuring a fever: Aco., Bell., Puls., Spo.\n\nDuring pregnancy: Ign., Lach., Nat-m., Puls.\n\nDuring labour: Coff., Puls.\n\nBetter for fresh air: Puls.\n\nDifficulty crying: Nat-m.\n\nInvoluntary: Ign.\n\nCries on own: Ign., Nat-m.\n\nWhilst breastfeeding: Puls.\n\nTeeth\n\nCrumbling\/decaying: Calc-f., Calc-p.\n\nTeething painful in babies: Aco., Bell., Bor., Calc-c., Calc-p., Cham., Kreos., Mag-m., Mag-p., Phyt., Puls., Rhe., Sil.\n\nCheeks:\n\nhot and red: Aco., Bell., Cham.\n\npale and cold: Cham.\n\nred spot on one cheek: Cham.\n\nswollen: Bell.\n\nSlow (babies late to teethe): Calc-c., Calc-p., Sil.\n\nTeeth decay as soon as they come through: Kreos.\n\nPains:\n\ncries out in sleep: Cham.\n\nsevere: Kreos.\n\nunbearable: Cham.\n\nWith:\n\ncolic: Mag-m.\n\ncough: Cham.\n\ncrying: Phyt.\n\ndiarrhoea: Calc-c., Cham., Rhe., Sil.\n\ndifficulty teething: Calc-c., Calc-p., Cham., Sil.\n\nfever: Aco.\n\ngreen stools: Calc-p., Cham., Mag-m.\n\nrestless sleep: Aco., Bell., Cham., Kreos.\n\ntoothache: Sil.\n\nBetter:\n\nbiting gums together hard: Phyt.\n\ncold drinks\/water: Puls.\n\nexternal heat: Mag-p.\n\nwalking in the fresh air: Puls.\n\nWorse:\n\nheat of bed, warm food\/drinks: Cham., Puls.\n\npressure: Cham.\n\nTennis elbow Ruta.\n\nThin Calc-p.\n\nThirstless Ant-c., Ant-t., Ap., Bell., Cina, Colch., Gels., Nux-m., Pho-ac., Puls.\n\nThirsty Aco., Ars., Bell., Bry., Cham., Eup-p., Merc-c., Merc-s., Nat-m., Phos., Pyr., Sil., Sul., Ver-a.\n\nThirst with chills: Ign.\n\nThirst:\n\nfor cold drinks\/hot drinks see Likes\n\nextreme: Merc-c., Nat-m., Sul.\n\nfor large quantities: Ars., Bry., Nat-m., Phos., Sul., Ver-a.\n\nat infrequent intervals: Bry.\n\nfor sips: Ars.\n\nfor small quantities often: Ars.\n\nunquenchable: Eup-p., Phos.\n\nfor water: Phos., Sul.\n\nThroat see Sore throat\n\nThrush (genital)\/Yeast infection Bor., Merc-s., Nat-m., Nit-ac., Sep.\n\nDischarge:\n\nburning: Bor., Merc-s., Nat-m., Nit-ac., Sep.\n\ncopious: Sep.\n\nlike cottage cheese: Sep.\n\ngreenish: Merc-s.\n\nlike egg white: Bor., Nat-m., Sep.\n\nsmelly: Merc-s., Nit-ac., Sep.\n\nthin: Nit-ac.\n\nyellow: Sep.\n\nwhite: Bor., Nat-m.\n\nWith:\n\ndryness: Sep.\n\nitching: Merc-s., Nat-m., Nit-ac., Sep.\n\nWorse:\n\nbefore menstrual period: Sep.\n\nbetween menstrual periods: Bor., Sep.\n\nat night: Merc-s.\n\nThrush (genital in pregnancy) Alu., Cocc., Kreos., Petr., Puls., Sep.\n\nDischarge:\n\nburning: Alu., Kreos., Petr., Puls., Sep.\n\ncream-like: Puls.\n\nlike egg white: Alu., Petr., Sep.\n\nlumpy: Sep.\n\nmilky: Kreos., Puls., Sep.\n\nprofuse: Alu., Cocc., Kreos., Petr., Sep.\n\nsmelly: Kreos., Sep.\n\nthick: Puls.\n\nthin, watery: Cocc., Kreos.\n\nwhite: Alu.\n\nWith:\n\nitching (vulva\/vagina): Kreos., Sep.\n\nThrush (oral) in babies, of gums and\/or tongue: Bor., Kali-m., Merc-s., Nat-m., Sul-ac.\n\nIn breastfeeding babies: Bor., Kali-m.\n\nTongue\/gums: Sul-ac.\n\ncoated white: Kali-m., Nat-m.\n\nhot\/dry\/bleeds easily: Bor.\n\nWith:\n\nexcess saliva: Bor., Merc-s.\n\nWorse:\n\nfor feeding: Bor.\n\nfor touch: Bor.\n\nTidy Ars., Nux-v.\n\nTime passes\n\nquickly: Cocc.\n\nslowly: Glon.\n\nTimid see Shy\n\nTired see Apathetic; Exhaustion; Sluggish\n\nTongue\n\nBrown-coated: Bry.\n\nCold: Ver-a.\n\nCracked: Merc-s., Nit-ac.\n\nDirty white: Bry.\n\nFiery red: Ap.\n\nGreen: Nat-s.\n\nRed: Bell., Phos.\n\nRed-edged: Ars., Sul.\n\nRed stripe down centre (white edges): Caust.\n\nRed-tipped: Arg-n., Ars., Phyt., Rhus-t., Sul.\n\nStrawberry: Bell.\n\nSwollen: Ap.\n\nWhite-coated: Ant-c., Ant-t., Bell., Calc-c., Kali-b., Kali-m., Puls., Rhus-t., Sul.\n\nYellow-coated: Kali-s., Merc-c., Merc-s., Nat-p., Puls.\n\nat the back: Nat-p.\n\nToothache Coff., Nux-v., Sep.\n\nTouchy see Irritable; Sensitive\n\nTravel sickness Bor., Cocc., Nux-v., Petr., Sep., Stap., Tab.\n\nWith:\n\nbiliousness: Sep.\n\ndiarrhoea: Cocc.\n\ndizziness: Cocc.\n\nfaint-like feeling: Cocc., Nux-v.\n\nheadache: Cocc., Nux-v., Petr., Sep.\n\nnausea: Bor., Cocc., Nux-v., Petr., Sep., Tab.\n\ndeathly, intermittent, violent nausea: Tab.\n\nvomiting: Bor., Cocc., Petr., Sep., Tab.\n\nBetter:\n\nlying down: Cocc., Nux-v.\n\nfresh air: Tab.\n\nWorse:\n\nfor downward movement: Bor.\n\nafter drinking: Cocc.\n\nafter eating: Cocc., Nux-v., Sep.\n\nfresh air: Cocc., Nux-v., Petr.\n\nmorning: Sep.\n\nmovement: Cocc., Tab.\n\nsitting up: Cocc.\n\nstuffy room: Tab.\n\ntobacco: Nux-v., Tab.\n\nCauses:\n\npregnancy: Sep.\n\ntravelling: Petr.\n\nTrembling Agar., Arg-n., Cimi., Cocc., Gels., Zinc.\n\nFrom emotion: Cocc.\n\nTwitchy Agar., Zinc.\n\nUncommunicative (see also Broody; Introspective) Carb-a., Cocc., Glon., Pho-ac., Phos., Ver-a., Zinc.\n\nUnforgiving Nit-ac.\n\nUrethritis see Cystitis\n\nUrinary tract infections see Cystitis\n\nUrine see Retention of urine\n\nUrticaria see Hives\n\nVaricose veins Bry., Calc-c., Calc-f., Carb-v., Ferr-m., Ham., Lyc., Puls., Zinc.\n\nOf leg and thigh: Carb-v., Ferr-m., Ham., Lyc., Puls., Zinc.\n\nOf foot: Ferr-m., Puls.\n\nOf vulva: Bry., Calc-c., Carb-v., Ham., Lyc., Zinc.\n\nPainful: Ferr-m., Ham., Lyc., Puls., Zinc.\n\nPains \u2013 stinging: Ham., Puls.\n\nSwollen: Ferr-m., Ham.\n\nWorse during pregnancy: Calc-c., Calc-f., Carb-v., Ferr-m., Ham., Lyc., Puls., Zinc.\n\nVomiting (see also Food poisoning; Travel sickness) Ant-c., Ant-t., Ars., Asar., Bry., Calc-c., Cham., Chin., Colch., Ferr-m., Ip., Nux-v., Phos., Sep., Sil., Sul-ac., Symph-r., Tab., Ver-a.\n\nAbdomen\/stomach feels cold: Colch., Ver-a.\n\nIn breastfed babies: Ant-c., Sil., Sul.\n\nConstant: Symph.\n\nDifficult: Ant-t.\n\nEasy: Ars., Cham.\n\nExhausting: Tab.\n\nFrequent: Ars., Chin.\n\nIn pregnancy: Ant-c., Nux-v., Sep., Symph-r., Tab.\n\nPains in the stomach:\n\nburning: Colch., Phos.\n\ncramping\/griping: Ver-a.\n\nsore, bruised: Colch.\n\nSudden (while eating): Ferr-m.\n\nViolent: Ars., Asar., Phos., Symph-r., Tab., Ver-a.\n\nVomit Ant-c., Ant-t., Ars., Bry., Calc-c., Cham., Chin., Colch., Ferr-m., Ip., Nat-c., Nux-v., Phos., Sep., Sil., Sul-ac., Tab., Ver-a.\n\nbile: Ant-c., Ars., Cham., Ip., Nux-v., Phos., Sep., Ver-a.\n\nbitter: Bry., Nux-v., Phos.\n\ncurdled milk: Ant-c., Calc-c., Sil.\n\nfood: Ars., Chin., Ferr-m., Ip.\n\ngreen: Ip.\n\nmucus: Nux-v., Phos., Ver-a.\n\nsmelly: Ars., Nux-v., Sep.\n\nsour: Ant-t., Calc-c., Chin., Nux-v., Sul-ac., Tab., Ver-a.\n\nviolent: Phos., Tab.\n\nwatery: Ars., Bry., Ver-a.\n\nyellow: Phos., Ver-a.\n\nWith:\n\ndiarrhoea: Ars., Ver-a.\n\ndizziness: Ver-a.\n\nfaintness after vomiting: Ars.\n\nfever: Ant-t.\n\nheadache: Ip.\n\nhiccuping after vomiting: Ver-a.\n\nnausea, retching: Symph-r.\n\nretching after vomiting: Colch., Symph-r.\n\nsweating while vomiting: Ars.\n\nBetter:\n\nlying on back: Symph-r.\n\nWorse:\n\nafter midnight: Ferr-m.\n\nbefore breakfast: Tab.\n\nbending down: Ip.\n\ncoughing: Ant-t., Bry., Ip.\n\ncoughing up phlegm: Nux-v.\n\nafter drinking: Ant-c., Ant-t., Ars., Phos., Ver-a.\n\nafter eating: Ant-t., Ars., Chin., Ip., Phos., Sep., Sil., Ver-a.\n\na short while after eating\/drinking: Phos.\n\neggs: Ferr-m.\n\nexpectorating: Nux-v.\n\nmilk: Ant-c., Nat-c., Sil.\n\nmorning: Sep.\n\nmovement: Ars., Bry., Symph-r., Tab.\n\nnight: Ferr-m.\n\nsmell of eggs: Colch.\n\nCauses:\n\nanger: Cham., Nux-v.\n\nice-cream: Ars.\n\nmeasles: Ant-c.\n\nmeat: Ars.\n\npregnancy: Nux-v., Sep., Tab.\n\ntravelling: Tab.\n\nWarts Thu.\n\nBleeding: Thu.\n\nLarge: Thu.\n\nStinging: Thu.\n\nWeakness see Exhaustion\n\nWeaning to dry up milk: Lac-c., Puls.\n\nWeary Gels., Ruta.\n\nof life: Nat-s.\n\nWeepy see Tearful\n\nWeight gain\n\nPoor in babies: Bar-c., Calc-p., Mag-c., Sil.\n\nEarly in babies: Calc-c.\n\nEasy: Calc-c.\n\nWhiny Ap., Puls.\n\nWhooping cough see Cough\n\nWind see Flatulence\n\nWorms Cina\n\nWorse generally (see also Complaints from)\n\nAfter midnight: Ars., Podo.\n\nAround:\n\n1 a.m.: Ars.\n\n2\u20134 a.m.: Kali-c.\n\n10 a.m.: Nat-m.\n\n10\u201311 a.m.: Sul.\n\n3 p.m.: Bell.\n\n3\u20135 p.m.: Ap.\n\n4\u20138 p.m.: Lyc.\n\n9 p.m.: Bry.\n\nAfternoon: Lyc.\n\nAutumn: Colch., Rhus-t.\n\nBathing: Sul.\n\nBefore breakfast: Tab.\n\nBeginning to move: Ferr-m., Rhus-t.\n\nChange of temperature: Ars., Kali-s.\n\nChange of weather see Complaints from Change of weather\n\nCloudy weather: Rhus-t.\n\nCoffee: Caust., Cham., Ign., Nux-v.\n\nCold: Agar., Ars., Bar-c., Bor., Calc-c., Calc-p., Caust., Chin., Cimi., Colch., Dulc., Hep-s., Hyp., Kali-b., Kali-c., Kali-p., Mag-p., Merc-s., Nit-ac., Nux-m., Nux-v., Pho-ac., Phos., Puls., Pyr., Rhod., Rhus-t., Sep., Sil.\n\nCold and heat: Merc-s.\n\nCold, dry weather: Asar., Caust., Hep-s., Kali-c., Nux-v.\n\nCold food: Nux-v.\n\nCold drinks\/food: Canth., Rhus-t.\n\nCold wind: Bell., Hep-s., Nux-v., Spo.\n\nCold, wet weather: Pyr., Rhod., Rhus-t. (see also Damp)\n\nConsolation see Averse to consolation\n\nDamp (see also Cold, wet): Ars., Calc-c., Calc-p., Colch., Dulc., Nat-s., Rhus-t., Sil.\n\nDraughts: Calc-c., Calc-p., Caust., Kali-c., Rhus-t., Sil., Sul.\n\nAfter eating: Kali-b., Nat-m., Zinc.\n\nBefore eating: Nat-c., Phos.\n\nEvening: Caust., Cham., Euphr., Lyc., Mag-c., Merc-s., Nat-p., Pho-ac., Phos., Sul-ac., Zinc.\n\nExcitement: Kali-p.\n\nExertion (physical): Ars., Calc-c., Carb-v., Cocc., Kali-p., Stap., Sul.\n\nExposure to sun: Ant-c., Glon., Nat-c., Nat-m., Puls.\n\nFasting: Sep., Stap., Sul., Tab.\n\nFirst lying down: Con.\n\nFlatulent foods (beans, etc.): Bry., Lyc.\n\nFoggy weather: Rhus-t.\n\nFresh air: Calc-c., Calc-p., Caust., Cham., Chin., Cocc., Coff., Hep-s., Nat-c., Nit-ac., Nux-m., Nux-v., Petr., Sil.\n\nFrosty weather: Sep.\n\nFruit: Ver-a.\n\nGetting feet wet: Bar-c., Puls., Sil.\n\nGetting head wet: Bell.\n\nGetting overheated: Ant-c., Kali-c.\n\nGetting up: Op.\n\nGetting wet: Caust., Sep.\n\nHeat: Ap., Arg-n., Asar., Calc-s., Glon., Kali-s., Lach., Led., Lil-t., Nat-m., Puls., Sec., Sul., Sul-ac.\n\nHeat and cold: Merc-s.\n\nHot food\/drinks: Asar.\n\nHumidity: Carb-v., Lach., Nat-s.\n\nJarring movement: Arn., Bell., Glon., Nit-ac.\n\nLight touch: Chin., Merc-s.\n\nLoss of sleep: Cocc., Nit-ac., Nux-v.\n\nLying down: Con., Dros., Dulc., Ferr-m., Merc-s., Nat-s., Rhus-t.\n\nLying on the injured part: Arn.\n\nLying on the painful part\/side: Hep-s., Ruta.\n\nLying on the side: Kali-c.\n\nBefore a menstrual period: Con., Lach., Nat-m.\n\nBefore\/during\/after a menstrual period: Sep.\n\nMental exertion: Calc-p., Kali-p., Sep.\n\nMid-morning: Nat-c., Nat-m., Podo., Sul-ac.\n\nMidnight: Dros.\n\nMilk: Calc-c., Calc-s., Mag-m., Nat-c., Sul.\n\nMorning: Lach., Nat-s., Nux-v., Petr., Pho-ac., Phos.\n\nMorning on waking: Kali-b., Lach.\n\nMovement: Aesc., Asar., Chin., Cocc., Colch., Led., Spo.\n\nNight: Aco., Chin., Cina, Coff., Colch., Dulc., Ferr-m., Hep-s., Ip., Kali-b., Kali-c., Lil-t., Mag-c., Mag-m., Merc-s., Nit-ac., Sep., Zinc.\n\nOnions: Lyc.\n\nOn one side of body: Pho-ac.\n\nPhysical exertion: Calc-s., Gels., Nat-c., Nat-m., Sep.\n\nPressure: Bar-c., Cina, Hep-s., Hyp., Lach., Lyc., Mag-p., Merc-c.\n\nRest: Puls., Sep.\n\nRich, fatty food: Carb-v., Puls.\n\nRiding in a car: Bor.\n\nSight of food: Colch.\n\nSitting still: Dulc.\n\nAfter sleep: Lach., Spo.\n\nDuring sleep: Op.\n\nSlightest movement: Bry.\n\nStanding: Sul.\n\nStarchy food: Nat-m., Nat-s.\n\nStormy weather: Nat-c., Rhod.\n\nbefore a storm: Rhod.\n\nduring a storm: Nat-c.\n\nStuffy rooms: Asar., Calc-s., Lyc., Puls., Sul.\n\nSugar\/sweets: Arg-n.\n\nSun see Complaints from sunstroke\n\nSummer: Kali-b.\n\nSweating: Pho-ac., Sep.\n\nSwimming in cold water: Ant-c., Mag-m., Mag-p., Rhus-t.\n\nThinking: Calc-c.\n\nTight clothes: Calc-c., Lach., Lyc., Nux-v., Spo.\n\nTobacco: Ign., Nux-v., Spo., Stap.\n\nTouch: Aco., Ap., Arn., Bell., Cocc., Coff., Colch., Cupr., Ham., Hep-s., Kali-c., Led., Mag-p., Nit-ac., Nux-v., Sep., Sil., Stap.\n\nTravelling: Petr.\n\nTwilight: Puls.\n\nUncovering: Hep-s., Kali-c., Mag-p., Nux-v., Rhus-t., Sil.\n\nVomiting: Cupr.\n\nOn waking: Ars., Lach., Nit-ac.\n\nWalking: Aesc., Caust., Glon., Lach., Nit-ac., Nux-v.\n\nin the fresh air: Caust., Cocc., Led., Mag-p.\n\nin the wind: Sep.\n\nWarmth of bed: Dros., Op., Sul.\n\nWet weather: Ars., Asar., Calc-p., Nat-s., Puls., Sil.\n\nWind: Cham., Lyc., Nux-v., Phos., Puls., Rhod.\n\nWine: Coff., Led., Zinc.\n\nWinter: Nux-v.\n\nWorry: Kali-p.\n\nWriting: Sep.\n\nWounds see Injuries\n\nYeast infection see Thrush (genital)\nNote\n\n1: Understanding and Using Homeopathy\n\n 1. Pinchuck and Clark, _Medicine for Beginners,_ Writers and Readers, London, 1984.\nLIST OF REMEDIES AND ABBREVIATIONS\n\nAconitum napellus | |\n\nAco.\n\n---|---|---\n\nAesculus hippocastanum | |\n\nAesc.\n\nAgaricus muscarius | |\n\nAgar.\n\nAllium cepa | |\n\nAll-c.\n\nAlumina | |\n\nAlu.\n\nAntimonium crudum | |\n\nAnt-c.\n\nAntimonium tartaricum | |\n\nAnt-t.\n\nApis mellifica | |\n\nAp.\n\nArgentum nitricum | |\n\nArg-n.\n\nArnica montana | |\n\nArn.\n\nArsenicum album | |\n\nArs.\n\nAsarum europum | |\n\nAsar.\n\nBaryta carbonica | |\n\nBar-c.\n\nBelladonna | |\n\nBell.\n\nBellis perennis | |\n\nBell-p.\n\nBorax veneta | |\n\nBor.\n\nBryonia alba | |\n\nBry.\n\nCalcarea carbonica | |\n\nCalc-c.\n\nCalcarea fluorica | |\n\nCalc-f.\n\nCalcarea phosphorica | |\n\nCalc-p.\n\nCalcarea sulphurica | |\n\nCalc-s.\n\nCalendula officinalis | |\n\nCalen.\n\nCantharis vesicatoria | |\n\nCanth.\n\nCarbo animalis | |\n\nCarb-a.\n\nCarbo vegetabilis | |\n\nCarb-v.\n\nCastor equi | |\n\nCast.\n\nCaulophyllum | |\n\nCaul.\n\nCausticum | |\n\nCaust.\n\nChamomilla | |\n\nCham.\n\nChelidonium majus | |\n\nChel.\n\nChina officinalis | |\n\nChin.\n\nCimicifuga | |\n\nCimi.\n\nCina\n\nCocculus indicus | |\n\nCocc.\n\nCoccus cacti | |\n\nCocc-c.\n\nCoffea cruda | |\n\nCoff.\n\nColchicum autumnale | |\n\nColch.\n\nColocythis | |\n\nColoc.\n\nConium maculatum | |\n\nCon.\n\nCuprum metallicum | |\n\nCupr.\n\nCypripedium | |\n\nCypr.\n\nDioscorea | |\n\nDios.\n\nDrosera rotundifolia | |\n\nDros.\n\nDulcamara | |\n\nDulc.\n\nEupatorium perfoliatum | |\n\nEup-p.\n\nEuphrasia | |\n\nEuphr.\n\nFerrum metallicum | |\n\nFerr-m.\n\nGelsemium sempervirens | |\n\nGels.\n\nGlonoine | |\n\nGlon.\n\nHamamelis virginica | |\n\nHam.\n\nHepar sulphuris calcareum | |\n\nHep-s.\n\nHypericum perfoliatum | |\n\nHyp.\n\nIgnatia amara | |\n\nIgn.\n\nIpecacuanha | |\n\nIp.\n\nJaborandi | |\n\nJab.\n\nKali bichromicum | |\n\nKali-b.\n\nKali carbonicum | |\n\nKali-c.\n\nKali muriaticum | |\n\nKali-m.\n\nKali phosphoricum | |\n\nKali-p.\n\nKali sulphuricum | |\n\nKali-s.\n\nKreosotum | |\n\nKreos.\n\nLac caninum | |\n\nLac-c.\n\nLac defloratum | |\n\nLac-d.\n\nLachesis | |\n\nLach.\n\nLedum palustre | |\n\nLed.\n\nLillium tigrinum | |\n\nLil-t.\n\nLycopodium | |\n\nLyc.\n\nMagnesia carbonica | |\n\nMag-c.\n\nMagnesia muriaticum | |\n\nMag-m.\n\nMagnesia phosphorica | |\n\nMag-p.\n\nMercurius corrosivus | |\n\nMerc-c.\n\nMercurius solubilis | |\n\nMerc-s.\n\nNatrum carbonicum | |\n\nNat-c.\n\nNatrum muriaticum | |\n\nNat-m.\n\nNatrum phosphoricum | |\n\nNat-p.\n\nNatrum sulphuricum | |\n\nNat-s.\n\nNitricum acidum | |\n\nNit-ac.\n\nNux moschata | |\n\nNux-m.\n\nNux vomica | |\n\nNux-v.\n\nOpium | |\n\nOp.\n\nPetroleum | |\n\nPetr.\n\nPhosphoric acid | |\n\nPho-ac.\n\nPhosphorus | |\n\nPhos.\n\nPhytolacca decandra | |\n\nPhyt.\n\nPodophyllum | |\n\nPodo.\n\nPulsatilla nigricans | |\n\nPuls.\n\nPyrogen | |\n\nPyr.\n\nRheum | |\n\nRhe.\n\nRhododendron | |\n\nRhod.\n\nRhus toxicodendron | |\n\nRhus-t.\n\nRumex crispus | |\n\nRumex\n\nRuta graveolens | |\n\nRuta\n\nSabina | |\n\nSab.\n\nSarsaparilla | |\n\nSars.\n\nSecale | |\n\nSec.\n\nSepia | |\n\nSep.\n\nSilica | |\n\nSil.\n\nSpongia tosta | |\n\nSpo.\n\nStaphysagria | |\n\nStap.\n\nStramonium | |\n\nStram.\n\nSulphur | |\n\nSul.\n\nSulphuric acid | |\n\nSul-ac.\n\nSymphoricarpus racemosa | |\n\nSymph-r.\n\nSymphytum | |\n\nSymph.\n\nTabacum | |\n\nTab.\n\nThiosinaminum | |\n\nThios.\n\nThuja occidentalis | |\n\nThu.\n\nUrtica urens | |\n\nUrt-u.\n\nVeratrum album | |\n\nVer-a.\n\nZincum metallicum | |\n\nZinc.\nAPPENDICES\n\n* * *\n\nFIRST-AID KITS\n\n* * *\n\nYou may wish to make up some basic kits, building on them as you need them and as you gain in knowledge and experience. I have included a few below which you can use as models. Some pharmacies can prepare 10 remedies of your choice in a box: small boxes take 10 small bottles, containing 40 soft tablets, and larger boxes take the same number of bottles each containing 60 soft tablets. Many pharmacies also produce many different types of first-aid kit. Their smallest, a plastic wallet holding 24 vials of sugar granules, will fit in a pocket and is ideal for travellers. You can order it in remedies of your choice.\n\nWhat potency?\n\nIf you are just starting out with homeopathy I recommend that you stock your kit in the 6th or the 12th potency. If you are more experienced you may choose to stock some remedies in the 30th potency (a higher potency) such as Cantharis for burns. You will need fewer doses of the higher potencies (see dosage chart) as they are stronger, homeopathically, and you need, therefore, to be more careful in how often you repeat them (so as not to prove them accidentally, see here).\n\nReminders for successful first-aid prescribing\n\n\u2022 Choose a remedy that fits the whole picture including one or more emotional, general and physical symptoms if possible. You may be prescribing on a single symptom, e.g. Arnica for a bruised forehead, in which case the following points still apply.\n\n\u2022 Choose a potency that fits the severity of the complaint (if you have a choice). See here.\n\n\u2022 Tip a pill or a few granules from the bottle into the lid, then tip into the hand of the person taking the remedy. If there is no lid carefully tip a few granules straight on to the hand. Do not put back into the bottle any that have been touched or fallen on the floor. See here for giving remedies to babies.\n\n\u2022 Repeat the doses according to the urgency of the situation, e.g. every 5 minutes if in great distress and less often if the need is less (see dosage chart).\n\n\u2022 Give the remedy less often once there is some improvement.\n\n\u2022 Stop the remedy once there is a marked improvement.\n\n\u2022 Repeat the same remedy if there is a relapse (if the same symptoms recur).\n\n\u2022 If you have given 6 doses and there has been no effect it may be the wrong remedy: reassess the whole picture and change the remedy, try a different potency, or ring your homeopath for advice.\n\n\u2022 Keep notes of what you prescribed, why you prescribed it and what happened (including the name of the 'patient' and the date).\n\nACCIDENTS AND INJURIES\n\nThese are remedies that you may need urgently and would like to have to hand. I have included a few keynotes for each one to remind you of its uses.\n\nAconite shock with fear. Any complaint that starts suddenly, especially after a shock, or exposure to the cold or cold wind.\n\nApis bites or stings with swelling, hives, general inflammations (coughs, colds, flus, fevers, earaches, sore throats, etc.).\n\nArnica delayed shock, injury with bruising, head injury, first stage sprains with swelling, jet-lag.\n\nCalendula cuts and wounds, bites.\n\nCantharis burns with or without blisters, cystitis.\n\nHepar sulphuricum inflamed cuts and wounds, general inflammations.\n\nHypericum painful injuries to nerves, in areas such as the coccyx, fingers, toes.\n\nLedum puncture wounds, bites; prevents infection.\n\nRhus tox sprains and strains, joint pain and inflammation.\n\nRuta sprains and strains, eyestrain.\n\nEVERYDAY COMPLAINTS\n\nArsenicum food poisoning; anxious, restless, thirsty for sips, burning pains, better for heat.\n\nBelladonna complaints start suddenly; delirious, dry heat, great pain.\n\nBryonia sprains, joint pain; complaints start slowly, worse for slightest movement.\n\nChamomilla teething; unbearable pain; very angry.\n\nGelsemium complaints start slowly; apathetic, thirstless, better for urinating.\n\nMagnesia phosphorica homeopathic 'aspirin', better for heat and pressure.\n\nMercurius solubilis sweaty and smelly, an increase of saliva, glands swollen.\n\nNux vomica food poisoning, gastric disorders (bilious), insomnia; irritable.\n\nPulsatilla teething, food poisoning; weepy, thirstless, better for fresh air.\n\nSulphur teething, sunburn; restless, thirsty, worse for heat.\n\nLABOUR\n\nIf this isn't your first baby, think back to your previous labour or labours and plan to have remedies for the situations that were difficult then. If you are in a relationship engage the help of your partner in putting together your labour kit: ideally, they will prescribe on you in labour and will need to be familiar with the remedies and their uses.\n\nAsk yourself if there is anything you are scared of happening in this coming birth. Be honest in answering this question. Don't worry about whether your fear is appropriate or not. Just jot it down and notice if it recurs. If it does and you can find a remedy for it in this book, make sure you have it in your labour kit. You may want to include your partner's fears as well.\n\nWhat do you know about yourself under stress? In pain? What difficulties are you likely to encounter? If you know that pain makes you vomit, for example, include a remedy for vomiting in labour: read up the pictures and choose one that may fit you.\n\nInclude your 'constitutional remedy' if you know it (see here), that remedy which helps you whatever you have wrong with you.\n\nLabour takes an enormous amount of energy. Because of this I advise you to stock your labour kit throughout in the 30th potency. If a remedy works, keep taking it, between contractions if necessary, for as long as it helps. A remedy may help and it may need to be repeated, for longer than I suggest for an acute illness, because labour is a stress to the body that may continue for many hours. Remedies may be 'used up' quickly and need to be repeated. You can also alternate remedies (see here) in labour where you want to give, say, Arnica for bruising and Gelsemium for backache labour.\n\nDuring labour if your instincts tell you to do something that isn't in the rule book, such as giving a remedy, deciding it isn't right and giving another one straight away, don't worry. No harm can come to mother or baby. The worst that can happen is that it will have no effect. If your instincts are right, however, there will be an improvement.\n\nMake a list of each remedy you order and its keynotes (see below as well as in each remedy picture in the Materia Medica). List the prescribing guidelines above (here) as a reminder. Keep this list with the kit.\n\nUse the list below to help you decide which remedies to order: these are the ones most often called for in my experience. Order soft tablets, as it can be irritating to have to chew hard tablets in labour especially if you don't like sugar.\n\nAconite labour is fast and violent, with great fear\/anxiety of death.\n\nArnica take throughout to prevent bruising, every four hours or more frequently if there is relief from pain.\n\nCaulophyllum Induction is threatened because you are late for dates: take one every four hours for up to two days \u2013 if it doesn't work, either your baby isn't ready to be born or you aren't ready for the birth (see here). Labour is slow, contractions are ineffective or stop, but before they become very painful; with exhaustion and shaking, neuralgic (twinging) pains.\n\nArsenicum vomiting in labour; with typical anxiety and fussiness.\n\nChamomilla labour exceedingly painful; backache labour; generally obnoxious, angry, impossible to please, asks for things that aren't then wanted.\n\nCoffea contractions are ineffective or stop and\/or are extremely painful; fear alternating with excitement, restless, makes jokes, laughs, is generally talkative and hilarious, sensitive to noise.\n\nGelsemium backache labour; lethargic, lifeless, dazed, thirstless.\n\nKali carbonicum backache labour; irritable and anxious and bossy.\n\nKali phosphoricum 6X is the best potency for this remedy, as you may need to take a lot. Simple tiredness in labour with no other symptoms.\n\nPulsatilla gives up during labour; weepy, clingy, pathetic, loses courage, thirstless, hot and craves fresh air or is better for it.\n\nSepia gives up in labour; very exhausted \u2013 sags on every level.\n\nDon't forget Rescue Remedy. Have a glass of water with 5 drops of Rescue Remedy added beside you all the time. Take at any time if panicky or fearful.\n\nPOST-NATAL KIT FOR THE BABY\n\nThe following remedies may be needed in the early weeks after your baby's birth.\n\nAconite shock\n\nArgentum nitricum sticky eyes\n\nBorax jumpy babies; thrush\n\nCalcarea carbonica snuffles in slow, sweaty babies\n\nColocynthis colic, when baby pulls knees up\n\nDioscorea colic, when baby arches whole body back\n\nKali muriaticum thrush in the mouth\n\nNux moschata sleepy babies\n\nPulsatilla sticky eyes, snuffles in clingy babies\n\nStramonium shock with screaming\n\nPOST-NATAL KIT FOR THE MOTHER\n\nAconite shock\n\nArnica to heal bruised muscles anywhere; after pains\n\nBelladonna engorged breasts; throbbing pains, red streaks, breasts hot\n\nBellis perennis bruised soreness not helped by Arnica\n\nBryonia engorgement; pains stitching, breasts pale and hard, worse for movement\n\nCalendula speeds the healing of a tear or episiotomy\n\nCastor equi sore, cracked nipples\n\nChina exhaustion from breastfeeding; anaemia\n\nHypericum pain in coccyx after the birth especially after forceps; painful piles\n\nMagnesia phosphorica afterpains\n\nNitric acid exhaustion from broken nights; irritable\n\nPhosphoric acid exhaustion from breastfeeding; apathetic\n\nPhytolacca cracked nipples; blocked duct; breast lumpy; abscess\/mastitis\n\nPulsatilla afterpains; weepy and pathetic\n\nSecale if Syntometrine injection administered take one dose after the birth\n\nSilica cracked nipples\n\nStaphysagria pains after the birth, with resentment and a feeling of assault\n\nRemember to look in the External Materia Medica for help with what to put on a tear, painful piles or cracked nipples to help them to heal.\n\nNB The above lists are to help you plan your own kits. Many other remedies may be needed for your pregnancy, birth or the post-natal period: if you don't find what you are looking for above use the Repertory and the Materia Medica to hunt out the remedies you need. If you are not able to find them or are struggling to self-prescribe effectively do seek the advice of a professional homeopath, who has access to hundreds more remedies and the skill to choose between them.\n\nYou can order remedies by phone from most homeopathic pharmacies \u2013 they will be sent out the same day, by first-class post with an invoice. Or you can pay by credit card.\n\nNB Homeopathic remedies have an indefinite shelf life, providing they are properly stored. If you are a traveller, don't put your homeopathic remedies through an airport X-ray machine as this has been known to antidote them; carry them in your pocket as the metal detector you walk through will not affect them.\n\n* * *\n\nFURTHER READING\n\n* * *\n\nBIRTH\n\nBurck, Frances, Babysense, revised edition, New York: St. Martin's Press: New York, 1991.\n\nHarrison, Helen, The Premature Baby Book, New York: St. Martin's Press, 1983.\n\nMueser, Anne, and George Verrilli, M.D., While Waiting, New York: St. Martin's Press, 1981. Also available in Spanish under the title Mientras Espera.\n\nMueser, Anne, and George Verrilli, M.D., Welcome Baby, New York: St. Martin's Press, 1981.\n\nBREASTFEEDING\n\nLa Leche League International, The Womanly Art of Breastfeeding, 1974.\n\nMason, Diane and Diane Ingersoll, Breastfeeding and the Working Mother, New York: St Martin's Press, 1986.\n\nHOMEOPATHY\n\nCastro, Miranda, The Complete Homeopathy Handbook, New York: St. Martin's Press, 1991.\n\nCummings. Stephen, and Dana Ullman, Homeopathic Medicines, Los Angeles: Jeremy P. Tarcher, Inc., 1984.\n\nPanos, Maesimund, and Jane Heimlich, Homeopathic Medicine at Home, Los Angeles: Jeremy P. Tarcher, Inc., 1980.\n\n* * *\n\nORGANISATIONS\n\n* * *\n\nUpdated list of organizations with web links: <http:\/\/mirandacastro.com\/main\/eMotherBabyOrgs.html>\n\nAlcoholics Anonymous\n\nGeneral Service Office\n\n475 Riverside Drive\n\nNew York, NY 10115\n\n(212) 870\u20133400\n\nThe Family Resource Coalition\n\nDepartment P\n\n230 North Michigan Avenue\n\nRoom 1625\n\nChicago, IL 60601\n\n(312) 726\u20134750\n\nHow to Grow a Parent Group\n\nSDG Enterprises\n\nP.O. Box 97\n\nWestern Springs, IL 60558\n\nInternational Foundation for Homeopathy\n\n1141 NW Market Street\n\nSeattle, WA 98107\n\nInternational Lactation Consultant Association\n\nP.O. Box 4013\n\nUniversity of Virginia Station\n\nCharlottesville, VA 22903\n\nLa Leche League International, Inc.\n\nP.O. Box 1209\n\nFranklin Park, IL 60131\n\n708\u2013455\u20137730 or 1\u2013800 LA LECHE\n\nMothers' Center for Development Project\n\n129 Jackson Street\n\nHempstead, NY 11550\n\nNational Center for Homeopathy\n\n1500 Massachusetts Avenue, N.W.\n\nWashington, DC 20005\n\n* * *\n\nORGANISATIONS\n\n* * *\n\nUpdated list of organizations with web links: <http:\/\/mirandacastro.com\/main\/eMotherBabyOrgs.html>\n\n* * *\n\nHOMEOPATHIC ORGANISATIONS\n\n* * *\n\nEach of the following organisations publish a directory of homeopathic practitioners to help you to find a registered homeopath in your area. In addition, some offer memberships to lay persons interested in homeopathy.\n\nAmerican Foundation for Homeopathy\n\n1508 S. Garfield\n\nAlhambra, CA 91801\n\nAmerican Institute of Homeopathy\n\n(703) 246-9501\n\nMedical doctors who practice homeopathy\n\nDirectory available\n\nAmerican Association of Homeopathic Pharmacists\n\nP.O. Box 61067\n\nLos Angeles, CA 90061\n\nCouncil for Homeopathic Certification\n\n1709 Seabright Avenue\n\nSanta Cruz, CA 95062\n\n(408) 421-0565\n\nWide range of practitioners who practice classical homeopathy to a uniformly high standard\n\nIncludes medical doctors, naturopathic doctors, professional homeopaths, physical therapists, etc.\n\nDirectory available\n\nFoundation for Homeopathic Education and Research\n\n(510) 649-8930\n\nCollects and disseminates new and on-going scientific research in the field of homeopathic medicine\n\nHomeopathic Academy of Naturopathic Physicians\n\n(503) 761-3298\n\nNaturopaths who practice classical homeopathy\n\nDirectory available\n\nHomeopathic Information Resources\n\nOneida River Park Drive\n\nClay, NY 13041\n\nHomeopathic Nurses Association\n\n3403 17th Ave. So.\n\nMinneapolis, MN 55407\n\nHomeopathic Pharmacopoeia of the United States\n\nP.O. Box 40360\n\n4974 Quebec St. N.W.\n\nWashington, D.C. 20016\n\nInternational Foundation for Homeopathy\n\n(206) 776-3172\n\nPractitioners who have trained with the IFH\n\nMemberships available to lay persons\n\nNational Board of Homeopathy in Dentistry, Inc.\n\nP.O. Box 423\n\nMarengo, IL 60152\n\nThe National Center for Homeopathy\n\n801 N. Fairfax, #306\n\nAlexandria, VA 22314\n\n(703) 548-7790\n\nMemberships available to professionals and lay persons\n\nDirectory available of licensed health care practitioners who practice homeopathy\n\nThe NCH publishes a monthly newsletter, organizes an annual conference, and conducts nationwide study groups and a summer school program for professionals and lay persons interested in homeopathy\n\nNew England Journal of Homeopathy\n\n356 Middle Street\n\nAmherst, MA 01002\n\n* * *\n\nHOMEOPATHIC PHARMACIES\n\n* * *\n\nThe following supply homeopathic remedies singly and\/or in the form of home care kits\u2014this is particularly useful if you are not able to purchase homeopathic remedies locally. Many also sell books.\n\nBoericke and Tafel, Inc.\n\n2381 Circadian Way\n\nSanta Rosa, CA 95407\n\n(800) 876-9505 (West Coast)\n\n(800) 272-2820 (East Coast)\n\nBoiron-Borneman, Inc. (Pennsylvania)\n\n(800) BOIRON-1\n\nDolisos America, Inc. (Nevada)\n\n(702) 871-7153\n\nEhrhart and Karl\n\n33 N. Wabash Ave.,\n\nChicago, IL 60602\n\n312-332-1046\n\nHahnemann Medical Pharmacy (California)\n\n(510) 527-3003\n\nHomeopathy Overnight (Maine)\n\n(800) ARNICA-30\n\nLuyties Pharmacal Co.\n\n4200 Laclede St.\n\nSt. Louis, MO 63108\n\n(800) 325-8080\n\nPropulsora Homeopathia De Mexico\n\nCalle Mirto 116 y 118\n\nMexico, D.F. Mexico 06400\n\nStandard Homeopathic Co.\n\n210 W. 131st St., Box 61067\n\nLos Angeles, CA 90061\n\n(213) 321-4284\n\nWashington Homeopathic Products, Inc. (Maryland)\n\n(800) 336-1695\n\n* * *\n\nHOMEOPATHIC BOOKS\n\n* * *\n\nThese companies sell homeopathic books (for the beginner to the advanced homeopath), tapes, and even computer software.\n\nHomeopathic Educational Services\n\n(510) 649-0294\n\nMinimum Price Books\n\n(800) 663-8272\n\n* * *\n\nHOMEOPATHIC TRAINING PROGRAMS\n\n* * *\n\nThe following organisation accredits homeopathic training programs, as well as providing a list (updated regularly) of courses, seminars, and post-graduate training programs that are available in the United States.\n\nAcademy for Classical Homeopathy\n\n7549 Louise Ave.,\n\nVan Nuys, CA 91406\n\n(818) 776-0078\n\nClinical Practice of Classical Homeopathy Course\n\nAtlantic Academy of Classical Homeopathy\n\n21 West 58th St., Suite 6E\n\nNew York, NY 10019\n\n(718) 518-4593\n\nBastyr College of Natural Health Sciences\n\n144 NE 54th\n\nSeattle, WA 98105\n\n(206) 523-9585\n\nBritish Institute of Homeopathy\n\n702 Washington St., Suite 204\n\nMarina Del Rey, CA 90292\n\n(213) 306-5408\n\nHome Study Course\n\nCouncil for Homeopathic Education\n\nClocktower Building\n\n3 Main Street\n\nChatham, NY 12037\n\n(518) 392-6456\n\nFour Winds Seminars\n\n187 Hillside Drive\n\nFairfax, CA 94930\n\n(415) 457-8452\n\nHahnemann Academy of North America\n\n2801 Rodeo Rd., Suite B-135\n\nSanta Fe, NM 87505\n\n(505) 959-7018\n\nHahnemann College of Homeopathy\n\n1918 Bonita Ave.,\n\nBerkeley, CA 94704\n\n(415) 849-1925\n\nProfessional Training Course\n\nHomeopathic Association of Greater Chicago\n\nP.O. Box 3791\n\nOak Brook, IL 60522\n\n(708) 325-2804 or (708) 529-7552\n\nInternational Foundation for Homeopathy\n\n2366 Eastlake Dr., E.\n\nSeattle, WA 98102\n\n(206) 324-8230\n\nProfessional Course\n\nAdvanced Acute Course\n\nNational Center for Homeopathy\n\n801 N. Fairfax, #306\n\nAlexandria, VA 22314\n\n(703) 548-7790\n\nSummer courses for licensed practitioners and the public\n\nNational College of Naturopathic Medicine\n\n11231 SE Market St.\n\nPortland, OR 97216\n\n(503) 255-4860\n\nNew England School of Homeopathy\n\n356 Middle Street\n\nAmherst, MA 01002\n\n(203) 763-1255\n\nCourses for professionals and lay practitioners\n\nPacific Academy of Homeopathic Medicine\n\n1678 Shattuck Ave. #42\n\nBerkeley, CA 94709\n\n(510) 549-3475\n\nPan-American Homeopathic Medical Congress\n\nEdificio 166, Entrada D\n\nUnidad Kennedy\n\nMexico G, D.F.\nINDEX\n\nThe index that appeared in the print version of this title does not match the pages in your eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.\n\nabdominal pain\n\npersistent\n\nin pregnancy\n\nrepertory\n\nabortion\n\nabscess\n\nbreast\n\ndosage\n\nrepertory\n\nabsent-mindedness\n\naccidents:\n\nto baby\n\nfirst-aid kit\n\nacidophilus\n\nAconitum napellus (Aconite)\n\nremedy picture\n\nacupuncture\n\nas pain-killer\n\nacute disease\n\nAdelaide, Queen\n\nAesculus hippocastanum\n\nafterbirth\n\nafterpains\n\nwith cramp\n\nwith discharge\n\nwith headache\n\nrepertory\n\nAgaricus muscarius\n\naggravation of symptoms\n\nAIDS\n\nair, swallowing\n\nalcohol:\n\nin gripe water\n\nin labour\n\nin pregnancy\n\nAlexander technique\n\nallergies:\n\nfood\n\nmilk\n\nAllium cepa\n\nallopathy\n\nalmond oil\n\nAlphafetoprotein (AFP)\n\nAlumina\n\naluminium\n\nAmerican Medical Association\n\nAmniocentesis\n\namniotic fluid\n\nleaking\n\nanaemia\n\nin pregnancy\n\nrepertory\n\nanaesthesia:\n\ncaudal\n\ncounteracting effects of\n\neffect on baby\n\nepidural\n\ngeneral\n\nparacervical block\n\nperineal block\n\npudendal block\n\nanaphylactic shock\n\nanencephaly, testing for\n\nanger:\n\nin babies\n\nrepertory\n\nanimals:\n\nbabies with\n\nbites\n\nantacids\n\nante-natal classes\n\nante-natal visits\n\nantibodies, in breast milk\n\nantidotes\n\nAntimonium:\n\ncrudum\n\ntartaricum\n\nanxiety, repertory\n\napathy, repertory\n\nApgar test\n\nApis mellifica\n\nArab states, homeopathy in\n\nArgentina, homeopathy in\n\nArgentum nitricum\n\nArnica montana\n\nfor baby\n\nexternal uses\n\nfor first-aid kit\n\nfor injuries\n\nin labour\n\npost-natal\n\nin pregnancy\n\nremedy picture\n\nArsenicum album\n\nremedy picture\n\nArt of Breastfeeding\n\narthritis\n\nAsarum europum\n\nAsia, homeopathy in\n\naspirin\n\nasthma\n\nathlete's foot\n\nAustralia, homeopathy in\n\naversions, repertory\n\nbabies:\n\naccidents to\n\nbonding with\n\nbreathing difficulties\n\nchanging\n\nchildhood illnesses\n\ncomplaints\n\ndeath of\n\n'difficult'\n\nfeeding\n\nfirst-aid kit for\n\ngrowth spurts\n\nimmunity gained from mother\n\ninjury at birth\n\nlifting\n\nmeasuring at birth\n\nnew-born, visitors for\n\npost-natal tests\n\nsample cases\n\nsleep difficulties\n\nsleeping with parents\n\nswallowing air\n\nsymptoms\n\nindicating serious illness\n\nurine, quantity\n\nweight gain\n\nsee also fetus\n\nBach, Edward\n\nback, injury\n\nbackache\n\ndos and don'ts\n\nfrom injury\n\nin labour\n\nwith lameness\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nwith stiffness\n\nbad-tempered babies\n\nbananas\n\nbarley water\n\nBaryta carbonica\n\nbathing:\n\nwith baby\n\nbaby objecting to\n\nin labour\n\nwhile feeding baby\n\nbedtime routine\n\nbee stings\n\nBelladonna\n\nremedy picture\n\nBellis perennis\n\nbelly-dancing\n\nbicarbonate of soda\n\nbirth:\n\nafterpains\n\nbody and\n\nchanges to mother's body after\n\ncomplaints\n\nfurther reading\n\nhealing after\n\nhospital or home\n\ninjuries to baby\n\nmedical interventions\n\nnatural\n\npacking for\n\npain\n\npartner at\n\nplanning for\n\npositions for\n\npreparing for\n\nsample case\n\nstages of\n\nstudents attending\n\nsee also labour\n\nbirth plan\n\nbirthing pools\n\nbirthmarks\n\nbites\n\nrepertory\n\nblack eye\n\nbladder, prolapse\n\nbleeding:\n\ngums\n\npost-natal\n\nin pregnancy\n\ndos and don'ts\n\nrepertory\n\nsee also haemorrhage\n\nblisters\n\nblood\n\nblood:\n\nin pregnancy\n\nRhesus factor\n\nblood blisters\n\nblood poisoning\n\nblood pressure:\n\nhigh\n\nlow\n\nmonitoring\n\nblood sugar:\n\nchecking after birth\n\nmaintaining level\n\nblood tests:\n\nin pregnancy\n\nRhesus factor\n\nblood-letting\n\nblues, post-natal see depression, post-natal\n\nbonding\n\nbones, broken see broken bones\n\nBorax veneta\n\nremedy picture\n\nbottle wash\n\nbottle-feeding\n\nproblems\n\nwater to use\n\nweaning from\n\nbottles, sterilising\n\nbras\n\nBraxton Hicks contractions\n\nrepertory\n\nBrazil, homeopathy in\n\nbreaking the waters\n\nrepertory\n\nbreastfeeding\n\ncalories used\n\ndigestion time\n\ndrying up milk\n\neffect on libido\n\nexpressing milk\n\nfurther reading\n\nlet-down of milk\n\nmilk supply decreasing\n\nmother's diet affecting\n\npain in\n\nproblems\n\nto encourage afterbirth\n\ntoo little milk\n\ntoo much milk\n\nweaning from\n\nbreasts:\n\nabscess\n\nblocked ducts\n\nchanges in pregnancy\n\neffect of breastfeeding on\n\nengorged\n\ninflammation\n\ninjuries to\n\nlumps\n\nnipples:\n\ncracked\n\ninverted\n\npigmentation\n\nsore\n\nstimulation in labour\n\nthrush infection\n\npain in pregnancy\n\npain when feeding\n\nproblems, repertory\n\nbreath problems\n\nbreathing:\n\nin labour\n\nin pregnancy\n\nproblems in newborn\n\nbreathlessness:\n\nin pregnancy\n\nrepertory\n\nbreech baby\n\nbroken bones\n\nbirth injury\n\nrepertory\n\nbruising\n\nto baby at birth\n\nhealing\n\nrepertory\n\nBryonia alba\n\nremedy picture\n\nbumps\n\nburns\n\ndosage\n\nrepertory\n\nCaesarian section\n\nhealing\n\npost-natal effects\n\ncaffeine\n\nCalcarea:\n\ncarbonica\n\nremedy picture\n\nfluorica\n\nphosphorica\n\nremedy picture\n\nsulphurica\n\ncalcium\n\nCalendula officinalis\n\nalternating\n\nCalendula officinalis, external\n\nCalendula officinalis, remedy picture\n\nCalpol\n\nCamphor\n\ncancer\n\nCantharis vesicatoria\n\nCarbo:\n\nanimalis\n\nvegetabilis\n\nCarpal Tunnel Syndrome\n\nrepertory\n\ncase history:\n\nchart\n\ntaking\n\nCastor equi\n\ncastor oil\n\ncatarrh see colds\n\ncatheterisation\n\ncaudal anaesthesia\n\nCaulophyllum\n\nremedy picture\n\ncause for concern\n\nCausticum\n\nchamomile tea\n\nChamomilla\n\nremedy picture\n\ncharts:\n\ncase-taking\n\ndosage\n\nrepertorising\n\nChelidonium majus\n\nchest infections\n\nchest pains\n\nchickenpox\n\ndosage\n\nrepertory\n\nchilblains\n\nchild benefit\n\nchildcare, further reading\n\nchildhood illnesses\n\nchildren:\n\nbonding with baby\n\ncoping with loss of baby\n\nhandling baby\n\nhelping with baby\n\npresent at baby's birth\n\nsymptoms\n\nindicating serious illness\n\ntelling of pregnancy\n\nvisiting after birth\n\nChina officinalis\n\nchoking\n\ncholera\n\nChorionic Villi Sampling chorion biopsy (CVS)\n\nchromosomal disorders, testing for\n\nchronic diseases\n\nnot covered\n\nprescribing for\n\nChronic Diseases and Their Homeopathic Cure (Hahnemann)\n\ncigarettes see smoking\n\nCimicifuga\n\nremedy picture\n\nCina\n\nCinchona see China officinalis\n\ncircumcision\n\nclingy babies\n\nCocculus indicus\n\nCoccus cacti\n\ncochineal see Coccus cacti\n\ncod-liver oil\n\nCoffea cruda\n\nremedy picture\n\ncoffee see caffeine\n\nColchicum autumnale\n\ncold sores\n\ncolds\n\nbaby's\n\nblocked\n\nwith cough\n\ndos and don'ts\n\nwith fever\n\nnasal catarrh\n\nbleeding\n\nwith blocked sinuses\n\ndry\n\npost-natal drip\n\nin pregnancy\n\nrepertory\n\ncolic\n\nbaby's\n\ndos and don'ts\n\nrepertory\n\ncollapse\n\nColocynthis\n\ncolostrum\n\ncommon cold see colds\n\ncomplaints:\n\nof birth\n\nexternal repertory\n\nfrom\/after\n\ninternal repertory\n\nlabour\n\npost-natal:\n\nbaby\n\nmother\n\nof pregnancy\n\ntreatable\n\nconcentration, poor\n\nconcepts\n\nconcussion\n\ncondoms\n\nconfusion\n\ncongenital disorders\n\nConium maculatum\n\nconjunctivitis see eyes, inflammation\n\nconstipation\n\nbaby's\n\ndos and don'ts\n\npersistent\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nconstitution\n\ncontractions:\n\nBraxton Hicks\n\nrepertory\n\nearly\n\nin labour\n\npain in\n\nrepertory\n\nsee also labour pains\n\n'contrary' medicine\n\nconvulsions\n\nrepertory\n\ncot death:\n\navoiding\n\ncoping with\n\ncough\n\nbaby's\n\ndos and don'ts\n\ncroupy\n\nsee also croup\n\nday only\n\ndosage\n\ndry\n\ndry\/loose alternating\n\nin fits\n\nlingering\n\nmucus difficult\n\nmucus loose\n\nin pregnancy\n\nrepertory\n\nwhooping\n\nworse at night\n\nCoulter, Harris\n\ncracked nipples see under breasts\n\ncradle cap\n\ncramp\n\nin pregnancy\n\nrepertory\n\ncravings\n\ncreams\n\ncroup\n\nrepertory\n\ncrying\n\nCullen, Dr William\n\nCuprum metallicum\n\ncuts:\n\nexternal treatment\n\ninternal treatment\n\nrepertory\n\nCypripedium\n\ncystitis\n\ndosage\n\nfrom chill\n\npassing blood\n\npost-natal\n\nin pregnancy\n\nrepertory\n\ncysts\n\ndancing:\n\nin labour\n\nin pregnancy\n\ndeath of baby\n\ndecongestants\n\ndegenerative diseases\n\ndehydration\n\ndelirium\n\ndepression:\n\npost-natal\n\nrepertory\n\ndermatitis\n\ndesires, repertory\n\ndespair\n\ndetoxification, post-natal\n\ndiabetes\n\ndiarrhoea\n\nbaby's\n\nbottle-fed\n\nbreastfed\n\ndos and don'ts\n\nteething\n\nweaned\n\nnervous\n\npainful\n\npainless\n\nin pregnancy\n\nrepertory\n\nsevere\n\nsummer\n\nDickens, Charles\n\ndiet see food\n\n'difficult' babies\n\ndill-seed tea\n\ndilution\n\nmethods of\n\nDioscorea\n\ndischarges, repertory\n\ndiscontented babies\n\ndisease:\n\ncauses\n\nprevention of\n\ntheories of\n\ndisinfection\n\ndislikes, repertory\n\ndislocation\n\nbirth injury\n\ndistressed babies\n\ndizziness\n\nin pregnancy\n\nrepertory\n\ndosage\n\nchart\n\ninfinitesimal\n\ndouching\n\nDown's syndrome\n\ntesting for\n\ndrink:\n\nin labour\n\nwhile breastfeeding\n\ndrips\n\nDrosera rotundifolia\n\ndrowsy babies, see sleepy babies\n\ndrugs:\n\ndetoxification from\n\neffects of\n\nrepertory\n\npain-killing, in labour\n\nin pregnancy\n\nDrugs in Pregnancy and Childbirth (Priest)\n\nDudgeon, Robert\n\nDulcamara\n\ndummies\n\ndysentery\n\near infection\n\nearache\n\nwith discharge, internal treatment\n\ndosage\n\nexternal treatment\n\ninternal treatment\n\nrepertory\n\neclampsia (toxaemia)\n\nectopic pregnancy\n\neczema\n\neffects of drugs taken before or during labour\n\nelbow, tennis\n\nelectromagnetism\n\nElectronic Fetal-Heart Monitoring\n\nElizabeth II, Queen\n\nemotions:\n\n'difficult' babies\n\ndistress in labour\n\nfollowing birth\n\nin pregnancy\n\nstress\n\nsymptoms\n\nsee also anxiety; depression, etc\n\nendorphins\n\nenemas\n\nfor babies\n\nengorgement see breasts, engorged\n\nEntonox\n\nepidural anaesthesia\n\nepisiotomy\n\navoiding\n\nergometrine\n\neucalyptus\n\nEupatorium perfoliatum\n\neuphoria\n\nEuphrasia\n\nEurope, homeopathy in\n\nexcitable behaviour\n\nexercise:\n\nin labour\n\npost-natal\n\nin pregnancy\n\nrelaxation\n\nexhaustion\n\ndosage\n\nin labour\n\nmental\n\nnervous\n\nparalytic\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nexhilaration\n\nexpressing milk\n\nexternal remedies\n\nexternal repertory\n\neyes:\n\nbathing\n\nblack\n\nblocked tear duct\n\nchanges in pregnancy\n\nheavy eyelids\n\ninfections\n\ninflammation\n\nin babies\n\nbathing\n\nwith discharge\n\nrepertory\n\ninjuries\n\nrepertory\n\nrepertories\n\nsticky\n\nstrain\n\nstyes\n\nface:\n\nmarks in pregnancy\n\nrepertory\n\nfaintness\n\nin pregnancy\n\nrepertory\n\nfalls, sensitivity to\n\nfasting, during fever\n\nfatigue see exhaustion\n\nfears:\n\nin pregnancy\n\nrepertory\n\nfeet:\n\nathlete's foot\n\nchilblains\n\nfennel-seed tea\n\nFerrum:\n\nmetallicum\n\nphosphoricum\n\nFetal-Heart Monitoring\n\nFetal Lung Maturity\n\nFetal Movement Counting\n\nfetus:\n\ndetermining sex in womb\n\nmonitoring heart\n\nmonitoring lung maturity\n\nmovements in womb\n\ntalking to\n\ntesting in womb\n\nfever\n\nalternating with chills\n\nbaby's\n\ndos and don'ts\n\nhelp needed\n\none-sided\n\nrepertory\n\nworse at night\n\nfingernails:\n\nin pregnancy\n\nrepertory\n\nsplitting\n\nfirst-aid:\n\ncourse in\n\nkits\n\nprescribing for\n\nflatulence\n\nwith diarrhoea\n\nwith pain\n\nrepertory\n\nFlint, Caroline\n\nfloppy babies\n\nflu\n\ndosage\n\nrepertory\n\nfluid, body, increase in pregnancy\n\nfolic acid\n\nfolk-medicines\n\nfood:\n\nallergies to\n\ncravings\n\nin labour\n\nmother's, affecting baby\n\nin pregnancy\n\ncontra-indications\n\nfor weaning\n\nfood poisoning\n\navoiding\n\ndosage\n\nin pregnancy\n\nrepertory\n\nforceps delivery\n\nforgetfulness, repertory\n\nfractures see broken bones\n\nFrance, homeopathy in\n\nfreckles\n\ngas and air\n\ngastric flu\n\nrepertory\n\ngastro-enteritis\n\nGelsemium sempervirens\n\nremedy picture\n\ngenetic disorders, testing for\n\ngenitals:\n\ndischarges, repertory\n\nherpes\n\nthrush\n\nrepertory\n\nsee also penis; testicles; vagina\n\ngerm theory\n\nGerman measles\n\nGermany, homeopathy in\n\nglands, repertory\n\nglobules\n\nGlonoine\n\ngluten\n\ngonorrhoea\n\nGreat Britain, homeopathy in\n\nGreece, homeopathy in\n\nGreene, Diana S.\n\ngripe water\n\ngroin:\n\ninguinal hernia\n\npains:\n\nin pregnancy\n\nrepertory\n\ngrowths\n\nguilt\n\ngums:\n\nbleeding\n\nrepertory\n\nboils\n\ninflamed\n\nproblems in pregnancy\n\nrepertory\n\nGuthrie test\n\nhaemorrhage\n\nsee also bleeding\n\nhaemorrhoids see piles\n\nHahnemann, Samuel\n\nhair:\n\nafter pregnancy\n\nloss\n\nrepertory\n\nin pregnancy\n\npubic, shaving\n\nHamamelis virginica\n\nremedy picture\n\nhandcream\n\nhayfever\n\nhead injury\n\nto baby\n\nwith vomiting\n\nheadaches\n\nwith dizziness\n\nwith faintness\n\nwith fever\n\nfrom injury\n\nfrom mental strain\n\nmigraine\n\nwith nausea\n\nnervous\n\none-sided\n\nin pregnancy\n\nrecurrent\n\nrepertory\n\nrheumatic\n\nwith sweating\n\nwith thirstlessness\n\nthrobbing\n\nworse at night\n\nhealing, after labour\n\nhealth\n\nfurther reading\n\nheart disease\n\nheart failure, in baby\n\nheart rate:\n\nfetal, monitoring\n\nin pregnancy\n\nheartburn\n\nrepertory\n\nheavy feeling, repertory\n\nheavy metals\n\nheel pricks\n\nHepar sulphuris\n\nremedy picture\n\nhepatitis\n\nherb teas\n\nherbalism\n\nHering, Constantine\n\nhernia\n\nrepertory\n\nherpes:\n\ngenital\n\noral see cold sores\n\nhiccups\n\nrepertory\n\nHippocrates\n\nhistory\n\nhives\n\nrepertory\n\nhoarseness\n\nholidays, in pregnancy\n\nhome birth\n\nhomeopathic aggravation\n\nhomeopathy, further reading\n\nhomesickness\n\nHomeopathic Science and Modern Medicine: The Physics of Healing with Microdoses (Coulter)\n\nhormones:\n\nchanges produced in pregnancy\n\nproduced by breastfeeding\n\nresponse to pain\n\nsynthetic, to induce labour\n\nhospital birth\n\npacking for\n\nhot flushes\n\nrepertory\n\nHughes, Richard\n\nhumiliation\n\nHypercal\n\nHypericum perfoliatum\n\nremedy picture\n\nhypertension see blood pressure, high\n\nhyperventilation\n\nhypnosis, in labour\n\nidealism\n\nIgnatia amara\n\nremedy picture\n\nimmune system\n\nimmunisation see vaccination\n\nimpatience\n\nimpulsiveness\n\nincontinence:\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nindecision\n\nIndia, homeopathy in\n\nindifference, repertory\n\nindigestion\n\nwith diarrhoea\n\ndos and don'ts\n\nwith heartburn\n\nwith nausea\n\nrepertory\n\ninductions\n\ninflammation\n\ninfluenza see flu\n\ninguinal hernia\n\ninheritance\n\ninjuries:\n\nto babies\n\nrepertory\n\nsee also burns; cuts; wound, etc\n\ninoculation see vaccination\n\ninsects:\n\nbites\n\nrepellent\n\ninsomnia\n\nbaby's\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nrestless\n\nwith sleepiness\n\nwaking early\n\ninternal examinations\n\ninternal remedies\n\ninternal repertory\n\ninterventions:\n\nfor baby\n\nin labour\n\ninvolution\n\nIpecacuanha\n\niron\n\ndeficiency\n\nirritability, repertory\n\nIsrael, homeopathy in\n\nitching, in pregnancy\n\nJaborandi\n\njaundice\n\nrepertory\n\njealousy\n\njoints:\n\ndislocation\n\nbirth injuries\n\npain\n\nexternal treatment\n\nwith lameness\n\nwith numbness\n\nin pregnancy\n\nrepertory\n\nrheumatic\n\nwith stiffness\n\nstrained\n\nswollen\n\njumpiness\n\nKali:\n\nbichromicum\n\ncarbonicum\n\nremedy picture\n\nmuriaticum\n\nphosphoricum\n\nremedy picture\n\nsulphuricum\n\nKeillands' delivery\n\nkick chart\n\nkissing, in labour\n\nKoplik spots\n\nKreosotum\n\nlabour:\n\nbad, post-natal effects\n\nbathing in\n\nbehaviour in\n\ncomplaints\n\ncontractions\n\ndiagnosing\n\nfalse\n\nfast\n\nfirst-aid kit for\n\nhealing after\n\nlate\n\nmedical interventions\n\npain relief\n\npains\n\ndosage\n\nwith exhaustion\n\nrepertory\n\nsevere\n\npremature\n\nrepertory\n\nsample case\n\nsigns preceding\n\nslow\n\nstages\n\nsee also birth\n\nlabour ward\n\nLac:\n\ncanninum\n\ndefloratum\n\nLachesis\n\nremedy picture\n\nlanolin\n\nlaryngitis see sore throat\n\nlavender oil\n\nLaw of Similars\n\nLaws of Cure\n\nlaxatives\n\nlaziness\n\nLedum palustre\n\nremedy picture\n\nlegs:\n\nrestless\n\nvaricose veins see varicose veins\n\nleprosy\n\nlibido, loss of:\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nlifting, in pregnancy\n\nligaments\n\nlightening\n\nlike cured with like\n\nlikes, repertory\n\nLilium tigrinum\n\nlips, repertory\n\nliquid potencies\n\nliveliness\n\nlochia\n\nrepertory\n\nLondon Homeopathic Society (later Faculty of Homeopathy)\n\nLondon Homeopathic Hospital\n\nloneliness\n\nloss of baby\n\nlotions\n\nlumps\n\nlungs, fetal, checking\n\nLycopodium\n\nremedy picture\n\nMagnesia:\n\ncarbon\n\nmuriaticum\n\nphosphoric\n\nremedy picture\n\nmalaria\n\nmassage\n\nafter childbirth\n\nin labour\n\nmastitis\n\nrepertory\n\nsee also breasts, abscess\n\nmasturbation, in labour\n\nMateria Medica\n\nexternal\n\ninternal\n\nMateria Medica Pura (Hahnemann)\n\nMaternity Grant\n\nmeasles\n\nrepertory\n\nmeconium\n\nmedical interventions:\n\nfor baby\n\nin labour\n\nmedical tests, in pregnancy\n\nmelancholia\n\nmembranes, artificial rupture\n\nmeningitis\n\nmenstrual periods:\n\nafter childbirth\n\nprofuse\n\nmental illness\n\nmenthol\n\nMercurius:\n\ncorrosivus\n\nsolubilis\n\nmetals:\n\nheavy\n\npreparation\n\nMexico, homeopathy in\n\nmiasms\n\nmicrobes\n\nmigraine see under headaches\n\nmilk:\n\nallergic reaction to\n\nbreast:\n\nfor bathing eyes\n\nexpressing\n\nsee also breastfeeding\n\nmilk rash\n\nmisapprehensions\n\nmiscarriage:\n\ncauses\n\ncoping with\n\nrisks of\n\nsigns of\n\ntests to predict\n\nmoles\n\nMontgomery's tubercules\n\nmorning sickness\n\nmother:\n\npost-natal body changes\n\npost-natal complaints\n\nMotherhood \u2013 What it does to your mind (Price), extract\n\nmouth:\n\nbleeding gums\n\ngum problems in pregnancy\n\ninflamed gums\n\nproblems, repertory\n\ntaste, repertory\n\nthrush\n\nrepertory\n\nulcers\n\nrepertory\n\nmouthwash\n\nmumps\n\ndevelopment\n\nwith fever\n\nrepertory\n\nspreading to other parts\n\nmuscles:\n\npelvic-floor\n\nrestless legs\n\nsore\n\nstrained\n\ntension\n\nmyths\n\nnaevi\n\nnails see fingernails\n\nnappy (diaper) rash\n\nrepertory\n\nNational Health Service\n\nNatrum:\n\ncarbonicum\n\nmuriaticum\n\nremedy picture\n\nphosphoricum\n\nsulphuricum\n\nnausea\n\nin labour\n\nin pregnancy\n\nrepertory\n\nnavel:\n\nhernia\n\ninflamed\n\nneck, stiff\n\nneonatal urticaria\n\nnephritis\n\nnervous babies\n\nnesting instinct\n\nnettle rash see hives\n\nNew Zealand, homeopathy in\n\nnipple shield\n\nnipples see under breasts\n\nNitricum acidum\n\nnitro-glycerine see Glonoine\n\nnose picking\n\nnosebleeds\n\nin pregnancy\n\nrepertory\n\nnote-taking\n\nnutrition, further reading\n\nNux:\n\nmoschata\n\nvomica\n\nremedy picture\n\noedema\n\nrepertory\n\noils\n\nointments\n\nolive oil\n\nOpium\n\nremedy picture\n\norganisations\n\nOrganon of Rational Medicine (Organon of the Healing Art) (Hahnemann)\n\nover-excitement, in pregnancy\n\noxytocin\n\npain:\n\ndos and don'ts\n\npost-natal\n\nin pregnancy\n\nrepertory\n\nsee also abdominal pain;\n\nbackache; joints, pain;\n\nlabour pains etc\n\npain relief, in labour\n\npalpitations:\n\nin pregnancy\n\nrepertory\n\npanic\n\nParacelsus\n\nparacervical block\n\nparacetamol\n\nparenthood:\n\nfirst year\n\nfurther reading\n\njob description\n\npartner's preparation for\n\npreparing for\n\npartner:\n\nat birth\n\ncoping with loss of baby\n\nhelping with baby\n\nat labour\n\npreparing for parenthood\n\nrelationship with:\n\nafter birth\n\nin pregnancy\n\nrole in pregnancy\n\nPasteur, Louis\n\nPediatrics (Taubman)\n\npelvic-floor muscles, exercising\n\npelvis\n\npenis:\n\ncircumcision\n\ninflammation\n\npeppermint\n\nperineal block\n\nperineum, tearing\n\nperiods see menstrual periods\n\nPethidine\n\nPetroleum\n\npetroleum-based cream\n\nPhenergan\n\nphenylketonuria\n\nphlebitis\n\nrepertory\n\nPhosphoric acid\n\nPhosphorus\n\nremedy picture\n\nphototherapy\n\nphysical symptoms\n\nPhytolacca decandra\n\nremedy picture\n\npigmentation, in pregnancy\n\npiles:\n\nexternal treatment\n\ninternal treatment\n\npost-natal\n\nin pregnancy\n\nrepertory\n\npink eye see eye, infections\n\nplacebos\n\nplacenta\n\ndelivery\n\ndevelopment\n\nfunction tests\n\nretained\n\nrepertory\n\nseparation from uterus\n\nplacenta abrupta\n\nplacenta praevia\n\nPlantago\n\npneumonia\n\nPodophyllum\n\npoison ivy see Rhus toxicodendron\n\npoisons, as cures\n\npools, birthing\n\npost-natal blues, see depression, post-natal\n\npost-natal period:\n\nbody in\n\ncomplaints:\n\nbaby\n\nmother\n\nfirst-aid kit for mother\n\nsample cases\n\npost-natal tests, for baby\n\nposture, in pregnancy\n\npotencies\n\nfor first-aid kit\n\npowders\n\npre-eclampsia\n\npre-labour\n\npregnancy\n\navoiding toxins\n\ncomplaints\n\nduration\n\nexercise\n\nfamily and\n\nfears\n\nhealth\n\nholidays\n\ninternal examinations\n\npain in\n\npartner's role during\n\nphysical changes in\n\nposture\n\nrelaxation\n\nrest\n\nsample cases\n\nsex in\n\nstress\n\ntension\n\ntests in\n\nwomen vulnerable in\n\nwork\n\npremature birth\n\nbreathing difficulties\n\npremature labour\n\nprescribing\n\nsample cases\n\ntaking case-history for\n\npreventive medicine\n\nPrice, Jane\n\nprickly heat\n\nrepertory\n\nPriest, Judy\n\nprinciples\n\nprolactin\n\nprolapse\n\nrepertory\n\nprostaglandin\n\nprotein, in urine\n\nprovings\n\nunintentional\n\nPsora\n\npsoriasis\n\npubic hair, shaving\n\npudendal block\n\nPulsatilla nigricans:\n\nfor baby\n\nin first-aid kit\n\nin labour\n\npost-natal\n\nin pregnancy\n\nremedy picture\n\npulse\n\npyloric stenosis\n\nPyrethrum\n\nPyrogen\n\nquarantine\n\nquestions for diagnosis\n\nQuin, Frederick\n\nquinine\n\nradiation\n\nrage\n\nrashes\n\nmilk\n\nneonatal urticaria\n\nraspberry leaf tea\n\nrectum, prolapse\n\nrelaxation\n\nremedies:\n\nalternating\n\nassessing response to\n\ndosage\n\nexternal\n\nforms\n\ninternal\n\nMateria Medica\n\nexternal\n\ninternal\n\npotencies\n\nprescribing\n\nremedy picture\n\nselecting\n\nsingle\n\nstopping\n\nstoring\n\ntaking\n\nrepertorising\n\nchart\n\nrepertory\n\nexternal\n\ninternal\n\nRescue Remedy\n\nexternal\n\nresentment\n\nrest:\n\nin labour\n\nin pregnancy\n\nrestless legs\n\nrestlessness, repertory\n\nretching\n\nReye's syndrome\n\nRhesus factor\n\nRheum\n\nrheumatism\n\nrhinitis see colds\n\nRhododendron\n\nRhus toxicodendron\n\nremedy picture\n\nrosemary oil\n\nroseola\n\nrepertory\n\nrubella see German measles\n\nRumex crispus\n\nRuta graveolens\n\nremedy picture\n\nSabina\n\nSaccharum lactose\n\nsafety\n\nof babies\n\nsaliva, increased\n\nsalt, sea\n\nsaline solution\n\nsample cases\n\nSarsaparilla\n\nsaunas\n\nscalds\n\nscarlet fever\n\nrepertory\n\nscars\n\nsciatica\n\nrepertory\n\nScience of Homeopathy (Vithoulkas)\n\nscreaming, repertory\n\nSecale\n\nremedy picture\n\nself-prescribing, taking case-history for\n\nsemen\n\nSensitive Midwifery (Flint), extracts\n\nsensitivity:\n\nin babies\n\nrepertory\n\nSepia\n\nremedy picture\n\n70 Ways to Calm a Crying Baby (Greene)\n\nsexual intercourse:\n\nin labour\n\nin pregnancy\n\n'shakes' see trembling\n\nshingles\n\nshock\n\nanaphylactic\n\nrepertory\n\n'show'\n\nshyness:\n\nin babies\n\nrepertory\n\nsickle-cell disease\n\nsighing\n\nSilica\n\nfor baby\n\npost-natal\n\nin pregnancy\n\nremedy picture\n\nSimilimum (Law of Similars)\n\nsinusitis\n\nskin:\n\nchanges after pregnancy\n\nchanges in pregnancy\n\nfeel of\n\nskin complaints\n\nof pregnancy\n\nrepertory\n\nsleep difficulties, in babies\n\nsleeping pills\n\nsleeplessness see insomnia\n\nsleepy babies\n\nslowness in development, repertory\n\nsluggishness, repertory\n\nsmell:\n\nsense of\n\nas symptom\n\nsmoking:\n\nin pregnancy\n\nsensitivity to\n\nsnuffles\n\nrepertory\n\nSociety of Homoeopaths\n\nsore throat\n\nbaby's\n\ndosage\n\nexternal treatment\n\nwith fever\n\nfrom chill\n\nrepertory\n\nwith swollen glands\n\ntonsillitis\n\nulcerated\n\nSouth America, homeopathy in\n\nsoya milk\n\nSparine\n\nspider veins\n\nspina bifida, testing for\n\nspitefulness\n\nsplinters\n\nSpongia tosta\n\nspots\n\nin pregnancy\n\nspotting\n\nsprains\n\nrepertory\n\nStaphysagria:\n\nfor baby\n\npost-natal\n\nin pregnancy\n\nremedy picture\n\nsterilising equipment\n\nsteroids\n\nsticky eyes\n\nsee also eyes, inflammation\n\nstiff neck\n\nstill birth, coping with\n\nstimulants see caffeine\n\nstings\n\nrepertory\n\nstomach ulcer\n\nstomach-ache see colic\n\nstools, baby's\n\nstorage\n\nstrains\n\nrepertory\n\nStramonium\n\nremedy picture\n\nstrawberry marks\n\nstress\n\ndealing with\n\nexternal\n\nin pregnancy\n\nstretch marks\n\nrepertory\n\nstubbornness\n\nstupor\n\nstyes\n\nsuccussion\n\nsucrose\n\nsuction delivery\n\nsulking\n\nSulphur\n\nremedy picture\n\nSulphuric acid\n\nsunburn\n\nsee also burns\n\nsunstroke\n\nsusceptibility\n\nswaddling\n\nsweating, repertory\n\nSwedenborg, Emanuel\n\nswelling see oedema\n\nswimming\n\nSycosis\n\nSymphoricarpus racemosa\n\nSymphytum\n\nsymptoms:\n\nemotional\n\nimportance of\n\nphysical\n\npicture\n\nrepertory\n\nexternal\n\ninternal\n\nof serious illness\n\nsuppression\n\nSyntocinon\n\nsyntometrine\n\ncounteracting effects\n\nsyphilis\n\nSyphilis (miasm)\n\nTabacum\n\ntablets\n\nTamus\n\ntantrums\n\ntaste in mouth, repertory\n\nTaubman, Dr Bruce\n\ntear duct, blocked\n\ntearfulness, repertory\n\ntearing\n\nteats\n\nteeth\n\nin pregnancy\n\nteething\n\ncase study\n\nwith diarrhoea\n\ndos and don'ts\n\ndosage\n\nrepertory\n\ntemper, bad, in babies\n\ntemperature:\n\nin fever\n\nnormal\n\ntaking\n\ntennis elbow\n\nT(E)NS (transcutaneous nerve stimulation)\n\ntension:\n\nin pregnancy\n\ntrembling to release\n\ntesticles, injuries to\n\nThackeray, William Makepeace\n\nthalassaemia\n\nThiosinaminum\n\nthirst: lack of\n\nrepertory\n\nthroat, sore see sore throat\n\nthrombosis superficial, see phlebitis\n\nthrush\n\ngenital\n\nrepertory\n\noral\n\nrepertory\n\nvaginal\n\nThuja occidentalis\n\nremedy picture\n\nthumb-sucking\n\nthyroid function\n\ntinctures\n\ntongue, repertory\n\ntonsillitis\n\ntoothache\n\nrepertory\n\ntoxaemia see eclampsia\n\ntoxins\n\ntoxoplasmosis\n\ntranquillisers\n\ntravel sickness\n\nrepertory\n\nTreatise on Materia Medica (Cullen)\n\ntrembling:\n\nin labour\n\nrepertory\n\ntrituration\n\ntuberculosis\n\ntwitching\n\nulcers\n\nmouth\n\nrepertory\n\nstomach\n\nthroat\n\nultrasound\n\numbilical hernia\n\numbilicus see navel\n\nurethritis\n\nurinary tract infections see cystitis\n\nurination:\n\nincontinence\n\nrepertory\n\nincrease:\n\npost-natal\n\nin pregnancy\n\nin labour\n\nurine:\n\nbaby's, quantity\n\nprotein in\n\nred\n\nretention\n\nbaby\n\npost-natal\n\nrepertory\n\ntests in pregnancy\n\nUrtica urens\n\nurticaria, neonatal\n\nUSA, homeopathy in\n\nuterus\n\ncontracting\n\npost-natal bleeding\n\nprolapse\n\nvaccination\n\nfurther reading\n\nvacuum delivery\n\nvagina:\n\nbleeding in pregnancy\n\nrepertory\n\nchanges at birth\n\nhealing\n\nprolapse\n\ntearing\n\nthrush\n\nvaginal examinations:\n\nin labour\n\nin pregnancy\n\nvaricose veins\n\ncauses\n\ndos and don'ts\n\nexternal treatment\n\ninflamed\n\nrepertory\n\nvegans\n\nvegetarians\n\nveins:\n\nbroken\n\nspider\n\nvaricose see varicose veins\n\nventouse delivery\n\nVeratrum album\n\nVerbascum, oil\n\nverrucas\n\nvertigo\n\nvinegar\n\nvision, blurred\n\nvisualisation\n\nvital force\n\nvitamins\n\nVithoulkas, George\n\nvomiting\n\nby baby\n\nin car\n\ndosage\n\nin labour\n\nin pregnancy\n\nprojectile\n\nrepertory\n\nvulva:\n\nchanges at childbirth\n\nvaricose veins\n\nrepertory\n\nwafers\n\nwakeful babies\n\nwarts\n\nrepertory\n\nwasp stings\n\nwater, suitable for baby\n\nwaters breaking\n\nweaning\n\nweight gain:\n\nin babies\n\nrepertory\n\nin pregnancy\n\nlosing\n\nwhining\n\nwhole person, treatment of\n\nwhooping cough\n\nsee also cough\n\nwind see flatulence\n\nwitch hazel see Hamamelis\n\nworms\n\nwounds:\n\nexternal treatment\n\ninternal treatment\n\nrepertory\n\nWrigleys' delivery\n\nX-rays, in pregnancy\n\nyoghurt\n\nzinc and castor-oil cream\n\nZincum metallicum\nAlso by Miranda Castro\n\nTHE COMPLETE HOMEOPATHY HANDBOOK\nHOMEOPATHY FOR PREGNANCY, BIRTH, AND YOUR BABY'S FIRST YEAR.\n\nCopyright \u00a9 1992, 1993 by Miranda Castro. All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles or reviews. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.\n\neBooks may be purchased for business or promotional use. For information on bulk purchases, please contact Macmillan Corporate and Premium Sales Department by writing to MacmillanSpecialMarkets@macmillan.com.\n\nFirst published in Great Britain by Macmillan London Limited under the title Homeopathy for Mother and Baby.\n\neISBN 9781466890558\n\nFirst eBook edition: January 2015\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Contents\n  4. Note to the reader\n  5. Dedication\n  6. Acknowledgements\n  7. Introduction\n  8. How To Use This Book\n  9. 1: Understanding and Using Homeopathy\n    1. The History of Homeopathy\n    2. Principles and Concepts\n    3. Myths and Misapprehensions\n    4. Taking the Case History\n    5. Prescribing\n    6. Complaints You Can Treat Using This Book\n    7. Cause for Concern\n    8. Sample Cases\n  10. 2: Preparing for Life After Birth\n    1. Pregnancy\n    2. Your Family and Pregnancy\n    3. Your Health and Pregnancy\n    4. Preparing for Birth\n    5. Planning for Birth\n    6. Preparing for the Post-natal Year\n  11. 3: Pregnancy\n    1. Your Body and Pregnancy\n    2. Complaints\n  12. 4: Birth\n    1. Your Body and Birth\n    2. Complaints\n  13. 5: The Post-Natal Period\n    1. Your Body in the Post-natal Period\n    2. Complaints - Mother\n    3. Complaints - Baby\n  14. 6: The Materia Medicas and Repertories\n    1. External Materia Medica\n    2. External Repertory\n    3. Internal Materia Medica\n    4. Internal Repertory\n  15. Note\n  16. List of Remedies and Abbreviations\n  17. Appendices\n    1. First-aid Kits\n    2. Further Reading\n    3. Organisations\n  18. Index\n  19. Also by Miranda Castro\n  20. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Juliet Sutherland, Dave Morgan and PG Distributed Proofreaders\n\n\n\n\n[Illustration: Darrin's Blow Knocked the Midshipman Down]\n\n\n\n\nDAVE DARRIN'S SECOND YEAR AT ANNAPOLIS\n\nor\n\nTwo Midshipmen as Naval Academy \"Youngsters\"\n\nBy\n\nH. IRVING HANCOCK Illustrated\n\nMCMXI\n\n\n\n\nCONTENTS\n\n\nCHAPTER\n\nI. A QUESTION OF MIDSHIPMAN HONOR\n\nII. DAVE'S PAP-SHEET ADVICE\n\nIII. MIDSHIPMAN PENNINGTON GOES TOO FAR\n\nIV. A LITTLE MEETING ASHORE\n\nV. WHEN THE SECONDS WONDERED\n\nVI. IN TROUBLE ON FOREIGN SOIL\n\nVII. PENNINGTON GETS HIS WISH\n\nVIII. THE TRAGEDY OF THE GALE\n\nIX. THE DESPAIR OF THE \"RECALL\"\n\nX. THE GRIM WATCH FROM THE WAVES\n\nXI. MIDSHIPMAN PENNINGTON'S ACCIDENT\n\nXII. BACK IN THE HOME TOWN\n\nXIII. DAN RECEIVES A FEARFUL FACER\n\nXIV. THE FIRST HOP WITH THE HOME GIRLS\n\nXV. A DISAGREEABLE FIRST CLASSMAN\n\nXVI. HOW DAN FACED THE BOARD\n\nXVII. LOSING THE TIME-KEEPER'S COUNT\n\nXVIII. FIGHTING THE FAMOUS DOUBLE BATTLE\n\nXIX. THE OFFICER IN CHARGE IS SHOCKED\n\nXX. CONCLUSION\n\n\n\n\nCHAPTER I\n\n\nA QUESTION OF MIDSHIPMAN HONOR\n\n\"How can a midshipman and gentleman act in that way?\"\n\nThe voice of Midshipman David Darrin, United States Navy, vibrated\nuneasily as he turned to his comrades.\n\n\"It's a shame--that's what it is,\" quivered Mr. Farley, also of the\nthird class at the United States Naval Academy.\n\n\"But the question is,\" propounded Midshipman Dan Dalzell, \"what are we\ngoing to do about it?\"\n\n\"Is it any part of our business to bother with the fellow?\" demanded\nFarley half savagely.\n\nNow Farley was rather hot-tempered, though he was \"all there\" in points\nthat involved the honor of the brigade of midshipmen.\n\nFive midshipmen stood in the squalid, ill-odored back room of a Chinese\nlaundry in the town of Annapolis.\n\nThere was a sixth midshipman present in the handsome blue uniform of the\nbrigade; and it was upon this sixth one that the anger and disgust of\nthe other five had centered.\n\nHe lay in a sleep too deep for stirring. On the still, foul air floated\nfumes that were new to those of his comrades who now gazed down on him.\n\n\"To think that one of our class could make such a beast of himself!\"\nsighed Dave Darrin.\n\n\"And on the morning of the very day we're to ship for the summer\ncruise,\" uttered Farley angrily.\n\n\"Oh, well\" growled Hallam, \"why not let this animal of lower grade sleep\njust where he is? Let him take what he has fairly brought upon himself!\"\n\n\"That's the very question that is agitating me,\" declared Dave Darrin,\nto whom these other members of the third class looked as a leader when\nthere was a point involving class honor.\n\nDave had became a leader through suffering.\n\nReaders of the preceding volume in this series, \"DAVE DARRIN'S FIRST\nYEAR AT ANNAPOLIS,\" will need no introduction to this fine specimen of\nspirited and honorable young American.\n\nReaders of that preceding volume will recall how Dave Darrin and Dan\nDalzell entered the United States Naval Academy, one appointed by a\nCongressman and the other by a United States Senator. Such readers will\nremember the difficult time that Dave and Dan had in getting through the\nwork of the first hard, grinding year. They will also recall how Dave\nDarrin, when accused of treachery to his classmates, patiently bided his\ntime until he, with the aid of some close friends, was able to\ndemonstrate his innocence. Our readers will also remember how two\nevil-minded members of the then fourth class plotted to increase Damn's\ndisgrace and to drive him out of the brigade; also how these two\nplotters, Midshipmen Henkel and Brimmer, were caught in their plotting\nand were themselves forced out of the brigade. Our readers know that\nbefore the end of the first year at the Naval Academy, Dave had fully\nreinstated himself in the esteem of his manly classmates, and how he\nquickly became the most popular and respected member of his class.\n\nIt was now only the day after the events whose narration closed the\npreceding volume.\n\nDave Darrin and Dalzell were first of all brought to notice in \"THE HIGH\nSCHOOL BOYS' SERIES.\" In their High School days, back in Gridley, these\ntwo had been famous members of Dick & Co., a sextette of youngsters who\nhad made a name for themselves in school athletics.\n\nDick Prescott and Greg Holmes, two other members of the sextette, had\nbeen appointed to the United States Military Academy at West Point,\nwhere they were serving in the corps of cadets and learning how to\nbecome Army officers in the not far distant future. All of the\nadventures of Dick and Greg are set forth in \"THE WEST POINT SERIES.\"\n\nThe two remaining members of famous old Dick & Co., Tom Reade and Harry\nHazelton, became civil engineers, and went West for their first taste of\nengineering work. Tom and Harry had some wonderful and startling\nadventures, as fully set forth in \"THE YOUNG ENGINEERS' SERIES.\"\n\nOn this early June day when we again encounter Dave Darrin and Dan\nDalzell in their handsome Naval uniforms, all members of the first,\nsecond and third classes were due to be aboard one of the three great\nbattleships that lay off the Yard at Annapolis at four p.m.\n\nThese three great battleships were the \"Massachusetts,\" the \"Iowa\" and\nthe \"Indiana.\" These three huge, turreted fighting craft had their full\ncrews aboard. Not one of the battleship commanders would allow a\n\"jackie\" ashore, except on business, through fear that many of the\n\"wilder\" ones might find the attractions on shore too alluring, and fail\nto return in time.\n\nWith the young midshipmen it was different. These young men were\nofficially and actually gentlemen, and could be trusted.\n\nYet here, in the back room of this laundry, was one who was apparently\nnot dependable.\n\nThis young midshipman's name was Pennington, and the fact was that he\nlay in deep stupor from the effects of smoking opium!\n\nIt had been a storekeeper, with a shop across the street, who had called\nthe attention of Dave and his four comrades to the probable fate of\nanother of their class.\n\n\"Chow Hop runs a laundry, but I have heard evil stories about a lot of\nyoung fools who flock to his back room and get a chance to 'hit' the\nopium pipe,\" the storekeeper had stated to Dave. \"One of your men, or at\nleast, one in a midshipman's uniform, went in there at eleven o'clock\nthis forenoon, and he hasn't been out since. It is now nearly two\no'clock and, I've been looking for some midshipmen to inform.\"\n\nSuch had been the storekeeper's careful statement. The merchants of\nAnnapolis always have a kindly feeling toward these fine young\nmidshipmen. The storekeeper's purpose was to enable them to help their\ncomrade out.\n\nSo the five had entered the laundry. The proprietor, Chow Hop, had\nattempted to bar their way to the rear room.\n\nBut Dave had seized the yellow man and had flung him aside.\n\nThe reader already knows what they discovered, and how it affected these\nyoung men.\n\n\"Bring that copper- chink in here, if you'll be so good,\"\ndirected Dave.\n\nDan and Hallam departed on the quest.\n\n\"You're wanted in there,\" proclaimed Dalzell, jerking a thumb over his\nshoulder.\n\n\"Me no sabby,\" replied Chow Hop, looking up briefly from his ironing\nboard.\n\n\"Get in there--do you hear?\" commanded Hallam, gripping the other's arm\nwith all his force.\n\n\"You lemme go chop-chop (quickly), or you get alle samee hurt--you\nsabby?\" scowled Chow Hop, using his free hand to raise a heavy flat-iron\nmenacingly.\n\nBut Dan Dalzell jumped in, giving the Chinaman's wrist a wrench that\ncaused him to drop the iron.\n\nThen, without a bit of ceremony, Dan grasped the Oriental by the\nshoulders, wheeled him about, while he protested in guttural tones, and\nbluntly kicked the yellow-faced one through the door into the inner\nroom.\n\nAt this summary proceeding both the Chinese helpers gripped their\nflat-irons firmly; and leaped forward to fight.\n\nIn an ugly temper the Chinaman is a bad man to oppose. But now this pair\nwere faced by a pair of quietly smiling midshipmen who were also\ndangerous when angry.\n\n\"You two, get back,\" ordered Dalzell, advancing fearlessly upon the\npair. \"If you don't, we'll drag you out into the street and turn you\nover to the policemen. You 'sabby' that? You heathen are pretty likely\nto get into prison for this day's work!\"\n\nScowling for a moment, then muttering savagely, the two helpers slunk\nback to their ironing boards.\n\nYet, while Dan turned to go into the rear room, Hallam stood just where\nhe was, to keep an eye on two possible sources of swift trouble.\n\n\"Chow Hop,\" began Dave Damn sternly, as the proprietor made his flying\nappearance, \"You've done a pretty mean piece of work here\"--pointing to\nthe unconscious midshipman in the berth. \"Do you understand that you're\npretty likely to go to prison for this?\"\n\n\"Oh, that no maller,\" replied Chow, with a sullen grin. \"Him plenty\n'shipmen come here and smoke.\"\n\n\"You lie!\" hissed Dave, grasping the heathen by the collar and shaking\nhim until the latter's teeth rattled.\n\nThen Dave gave him a brief rest, though he still retained his hold on\nthe Chinaman's collar. But the yellow man began struggling again, and\nDave repeated the shaking.\n\nChow Hop had kept his hands up inside his wide sleeves. Now Farley\nleaped forward as he shouted:\n\n\"Look out, Darry! He has a knife!\"\n\nFarley attempted to seize the Chinaman's wrist, for the purpose of\ndisarming the yellow man, but Dave swiftly threw the Chinaman around out\nof Farley's reach. Then, with a lightning-like move, Dave knocked the\nknife from Chow Hop's hand.\n\n\"Pick that up and keep it for a curio, Farley,\" directed Dave coolly.\n\nIn another twinkling Darrin had run the Chinaman up against the wall.\n\nSmack! biff! thump!\n\nWith increasing force Dave's hard fist struck the heathen in the face.\n\n\"Now stand there and behave yourself,\" admonished Midshipman Dave,\ndropping his hold on the yellow man's collar, \"or we'll stop playing\nwith you and hurt you some.\"\n\nThe scowl on Chow Hop's face was ominous, but he stood still, glaring at\nDave.\n\n\"Chow, what can we do to bring this man out of his sleep!\" asked Dave\ncoolly, and almost in a friendly tone.\n\n\"Me no sabby,\" sulked the Chinaman.\n\n\"Yes, you do,\" retorted Dave warningly. \"Now, what can we do to get our\nfriend out of this!\"\n\n\"You allee same cally (carry) him out,\" retorted Chow, with a suspicion\nof a sulky grin.\n\n\"None of that, now, you yellow-face!\" glared Dave. \"How shall we get our\ncomrade out of this opium sleep!\"\n\n\"Me no sabby no way,\" insisted Chow.\n\n\"Oh, yes, you do!\" snapped Dave. \"But you won't tell. All right; we'll\nfind the way, and we'll punish you into the bargain. Dan, get a piece of\npaper from the other room.\"\n\nDalzell was quickly back with the desired item. On the paper Dave wrote\na name and a telephone number.\n\n\"It's near the end of the doctor's office hours,\" murmured Dave. \"Go to\na telephone and ask the doctor to meet you at the corner above. Tell him\nit's vastly important, and ask him to meet you on the jump.\"\n\n\"Shall I tell him what's up!\" asked Dan cautiously.\n\n\"Yes; you'd better. Then he'll be sure to bring the necessary remedies\nwith him.\"\n\nDan Dalzell was off like a shot.\n\nChow tried to edge around toward the door.\n\n\"Here, you get back there,\" cried Dave, seizing the Chinaman and\nslamming him back against the wall. \"Don't you move again, until we tell\nyou that you may--or it will be the worse for you.\"\n\nTen minutes passed ere Dan returned with Dr. Lawrence.\n\n\"You see the job that's cut out for you,\" said Darrin, pointing to the\nunconscious figure in the bunk. \"Can you do it, Doctor?\"\n\nThe medical man made a hasty examination of the unconscious midshipman\nbefore he answered briefly:\n\n\"Yes.\"\n\n\"Will it be a long job, Doctor?\"\n\n\"Fifteen minutes, probably.\"\n\n\"Oh, good, if you can do it in that time!\"\n\n\"Me go now?\" asked Chow, with sullen curiosity, as the medical man\nopened his medicine-case.\n\n\"Yes; if you don't try to leave the joint,\" agreed Dave. \"And I'm going\noutside with you.\"\n\nChow looked very much as though he did not care for company, but\nMidshipman Darrin kept at his side.\n\n\"Now, see here, Chow,\" warned Dave, \"this is the last day you sell opium\nfor white men to smoke!\"\n\n\"You heap too flesh (fresh)\" growled the Chinaman.\n\n\"It's the last day you'll sell opium to white men,\" insisted Dave, \"for,\nas soon as I'm through here I'm going to the police station to inform\nagainst you. They'll go through here like a twelve-inch shot.\"\n\n\"You alle same tell cop?\" grinned Chow, green hatred showing through his\nskin. \"Then I tell evelybody about you fliend in there.\"\n\n\"Do just as you please about that,\" retorted Dave with pretended\ncarelessness. \"For one thing, you don't know his name.\"\n\n\"Oh, yes, I do,\" swaggered Chow impudently. \"Know heap 'bout him. His\nname alle same Pen'ton.\"\n\nSeizing a marking brush and a piece of paper, Chow Hop quickly wrote out\nPennington's name, correctly spelled. His ability to write English with\na good hand was one of Chow's great vanities, anyway.\n\n\"You go back to your ironing board, yellow-face,\" warned Darrin, and\nsomething in the young third classman's face showed Chow that it would\nbe wise to obey.\n\nThen Hallam drew Darrin to one side, to whisper earnestly in his ear:\n\n\"Look out, old man, or you will get Pen into an awful scrape!\"\n\n\"I shan't do it,\" maintained Darrin. \"If it happens it will have been\nPen's own work.\"\n\n\"You'd better let the chink go, just to save one of our class.\"\n\n\"Is a fellow who has turned opium fiend worth saving to the class!\"\ndemanded Dave, looking straight into Hallam's eyes.\n\n\"Well, er--er--\" stammered the other man.\n\n\"You see,\" smiled Dave, \"the doubt hits you just as hard as it does me!\"\n\n\"Oh, of course, a fellow who has turned opium fiend is no fellow ever to\nbe allowed to reach the bridge and the quarter-deck,\" admitted Hallam.\n\"But see here, are you going to report this affair to the commandant of\nmidshipmen, or to anyone else in authority?\"\n\n\"I've no occasion to report,\" replied Dave dryly. \"I am not in any way\nin command over Pennington. But I mean to persuade him to report himself\nfor what he has done!\"\n\n\"But that would ruin him!\" protested Hallam, aghast. \"He wouldn't even\nbe allowed to start on the cruise. He'd be railroaded home without loss\nof a moment.\"\n\n\"Yet you've just said that an opium-user isn't fit to go on in the\nbrigade,\" retorted Darrin.\n\n\"Hang it, it's hard to know what to do,\" rejoined Hallam, wrinkling his\nforehead. \"Of course we want to be just to Pen.\"\n\n\"It doesn't strike me as being just exactly a question of justice to\nPennington,\" Darrin went on earnestly. \"If this is anything it's a\nquestion of midshipman honor. We fellows are bound to see that all the\nunworthy ones are dropped from the service. Now, a fellow who has\nfastened the opium habit on himself isn't fit to go on, is he?\"\n\n\"Oh, say, but this is a hard one to settle!\" groaned Hallam.\n\n\"Then I'll take all the responsibility upon myself,\" said Dave promptly.\n\"I don't want to make any mistake, and I don't believe I'm going to.\nWait just a moment.\"\n\nGoing to the rear room, Dave faced his three comrades there with the\nquestion:\n\n\"You three are enough to take care of everything here for a few minutes,\naren't you?\"\n\n\"Yes,\" nodded Dan. \"What's up?\"\n\n\"Hallam and I are going for a brief walk.\"\n\nThen, stepping back into the front room, Darrin nodded to his classmate,\nwho followed him outside.\n\n\"Just come along, and say nothing about the matter on the street,\"\nrequested Dave. \"It might be overheard.\"\n\n\"Where are you going?\" questioned Hallam wonderingly.\n\n\"Wait and see, please.\"\n\nFrom Chow Hop's wretched establishment it was not far to the other\nbuilding that Dave had in mind as a destination.\n\nBut when they arrived, and stood at the foot of the steps, Hallam\nclutched Darrin's arm, holding him back.\n\n\"Why, see here, this is the police station!\"\n\n\"I know it,\" Dave replied calmly.\n\n\"But see here, you're not--\"\n\n\"I'm not going to drag you into anything that you'd object to,\" Darrin\ncontinued. \"Come along; all I want you for is as a witness to what I am\ngoing to say.\"\n\n\"Don't do it, old fel--\"\n\n\"I've thought that over, and I feel that I must,\" replied Dave firmly.\n\"Come along. Don't attract attention by standing here arguing.\"\n\nIn another instant the two midshipmen were going swiftly up the steps.\n\nThe chief of police received his two callers courteously. Dave told the\nofficial how their attention had been called to the fact that one of\ntheir number was in an opium joint. Dave named the place, but requested\nthe chief to wait a full hour before taking any action.\n\n\"That will give us a chance to get out a comrade who may have committed\nonly his first offense,\" Dave continued.\n\n\"If there's any opium being smoked in that place I'll surely close the\njoint out!\" replied the chief, bringing his fist down upon his desk.\n\"But I understand your reasons, Mr.--\"\n\n\"Darrin is my name, sir,\" replied Dave quietly.\n\n\"So, Mr. Darrin, I give you my word that I won't even start my\ninvestigations before this evening. And I'll keep all quiet about the\nmidshipman end of it.\"\n\n\"Thank you very much, sir,\" said Dave gratefully.\n\nAs the two midshipmen strolled slowly back in the direction of Chow\nHop's, Dave murmured:\n\n\"Now, you see why I took this step?\"\n\n\"I'm afraid not very clearly,\" replied Midshipman Hallam.\n\n\"That scoundrelly Chow made his boast that other midshipmen patronized\nhis place. I don't believe it. Such a vice wouldn't appeal to you, and\nit doesn't to me. But there are more than two hundred new plebes coming\nin just now, and many of these boys have never been away from home\nbefore. Some of them might foolishly seek the lure of a new vice, and\nmight find the habit fastened on them before they were aware of it.\nChow's vile den might spoil some good material for the quarter-deck,\nand, as a matter of midshipman honor, we're bound to see that the place\nis cleaned out right away.\"\n\n\"I guess, Darry, you come pretty near being right,\" assented Hallam,\nafter thinking for a few moments.\n\nBy the time they reached Chow Hop's again they found that Dr. Lawrence\nhad brought the unfortunate Pennington to. And a very scared and\nhumiliated midshipman it was who now stood up, a bit unsteadily, and\ntried to smooth down his uniform.\n\n\"How do you feel now?\" asked Dave.\n\n\"Awful!\" shuddered Pennington. \"And now see here, what are you fellows\ngoing to do? Blab, and see me driven out of the Navy?\"\n\n\"Don't do any talking in here,\" advised Dave, with a meaning look over\nhis shoulder at the yellow men in the outer room. \"Doctor, is our friend\nin shape to walk along with us now?\"\n\n\"He will be, in two or three minutes, after he drinks something I'm\ngoing to give him,\" replied the medical man, shaking a few drops from\neach of three vials into a glass of water. \"Here, young man, drink this\nslowly.\"\n\nThree minutes later the midshipmen left the place, Dave walking beside\nPennington and holding his arm lightly for the purpose of steadying him.\n\n\"How did this happen, Pen?\" queried Dave, when the six men of the third\nclass at last found themselves walking down Maryland Avenue. \"How long\nhave you been at this 'hop' trick?\"\n\n\"Never before to-day,\" replied Midshipman Pennington quickly.\n\n\"Pen, will you tell me that on your honor?\" asked Dave gravely.\n\nThe other midshipman flared up.\n\n\"Why must I give you my word of honor?\" he demanded defiantly. \"Isn't my\nplain word good enough?\"\n\n\"Your word of honor that you had never smoked opium before to-day would\nhelp to ease my mind a whole lot,\" replied Darrin. \"Come, unburden\nyourself, won't you, Pen?\"\n\n\"I'll tell you, Darry, just how it happened. To-day _was_ the first\ntime, on my word of honor, I came out into Annapolis with a raging\ntoothache. Now, you know how a fellow gets to hate to go before the\nmedical officers of the Academy with a tale about his teeth.\"\n\n\"Yes, I do,\" nodded Darrin. \"If a fellow is too much on the medical\nreport for trouble with his teeth, then it makes the surgeons look his\nmouth over with all the more caution, and in the end a fellow may get\ndropped from the brigade just because he has invited over zeal from the\ndentist. But what has all this to do with opium smoking?\"\n\n\"Just this,\" replied Pennington, hanging his head. \"I went into a drug\nstore and asked a clerk that I know what was the best thing for\ntoothache. He told me the best he knew was to smoke a pipe of opium, and\ntold me where to find Chow Hop, and what to say to the chink. And it's\nall a lie about opium helping a sore tooth,\" cried the wretched\nmidshipman, clapping a hand to his jaw, \"for there goes that fiendish\ntooth again! But say! You fellows are not going to leak about my little\nmishap?\"\n\n\"No,\" replied Darrin with great promptness. \"You're going to do that\nyourself.\"\n\n\"What?\" gasped Midshipman Pennington in intense astonishment. \"What are\nyou talking about?\"\n\n\"You'll be wise to turn in a report, on what happened,\" pursued Dave,\n\"for it's likely to reach official ears, anyway, and you'll be better\noff if you make the first report on the subject.\"\n\n\"Why is it likely to reach official ears, if you fellows keep your\nmouths shut?\"\n\n\"You see,\" Darrin went on very quietly, \"I reported the joint at the\npolice station, and Chow Hop threatened that, if I did, he'd tell all he\nknew about everybody. So you'd better be first----\"\n\n\"You broke the game out to the police!\" gasped Pennington, staring\ndumfoundedly at his comrade. \"What on earth----\"\n\n\"I did it because I had more than one satisfactory reason for\nconsidering it my duty,\" interposed Dave, speaking quietly though\nfirmly.\n\n\"You--you--bag of wind!\" exploded Midshipman Pennington.\n\n\"I'll accept your apology when you've had time to think it all over,\"\nreplied Dave, with a smile, though there was a brief flash in his eyes.\n\n\"I'll make no apology to you--at any time, you--you--greaser!\"\n\nMarks for efficiency or good conduct, which increase a midshipman's\nstanding, are called \"grease-marks\" or \"grease\" in midshipman slang.\nHence a midshipman who is accused of currying favor with his officers in\norder to win \"grease\" is contemptuously termed a \"greaser.\"\n\n\"I don't want to talk with you any more, Mr. Darrin,\" Pennington went on\nbitterly, \"or walk with you, either. When I get over this toothache I'll\ncall you out--you greaser!\"\n\nBurning with indignation, Midshipman Pennington fell back to walk with\nHallam.\n\n\n\n\nCHAPTER II\n\n\nDAVE'S PAP-SHEET ADVICE\n\nWhen our party reached the landing a lively scene lay before them.\n\nFully a hundred midshipmen, belonging to the first, second and third\nclasses, were waiting to be transported out to one or another of the\ngreat, gray battleships.\n\nSeveral launches were darting back and forth over the water. The baggage\nof the midshipmen had already been taken aboard the battleships. Only\nthe young men themselves were now awaited.\n\nNear-by stood a lieutenant of the Navy, who was directing the\nembarkation of the midshipmen of the different classes.\n\nFive minutes after our party arrived a launch from the \"Massachusetts\"\nlay in alongside the landing.\n\n\"Third classmen, this way!\" shouted the lieutenant. \"How many of you?\"\n\nTurning his eyes over the squad that had moved forward, the officer\ncontinued:\n\n\"Twenty-two. You can all crowd into this launch. Move quickly, young\ngentlemen!\"\n\nIn another couple of minutes the puffing launch was steaming away to the\nmassive battleship that lay out in the stream.\n\nDave stood well up in the bow. Once he barely overheard Pennington\nmutter to a comrade:\n\n\"The rascally greaser!\"\n\n\"That means me,\" Dave muttered under his breath. \"I won't take it up\nnow, or in any hurry. I'll wait until Pen has had time to see things\nstraight.\"\n\nAs soon as the launch lay alongside, the young midshipmen clambered\nnimbly up the side gangway, each raising his cap to the flag at the\nstern as he passed through the opening in the rail.\n\nHere stood an officer with an open book in his hand. To him each\nmidshipman reported, saluting, stated his name, and received his\nberthing.\n\n\"Hurry away to find your berthings, and get acquainted with the\nlocation,\" ordered this officer. \"Every midshipman will report on the\nquarter-deck promptly at five p.m. In the meantime, after locating your\nberthings, you are at liberty to range over the ship, avoiding the ward\nroom and the staterooms of officers.\"\n\nThe latest arrivals saluted. Then, under the guidance of messengers\nchosen from among the apprentice members of the crew, the young men\nlocated their berthings.\n\n\"I'm going to get mine changed, if I can,\" growled Pennington, wheeling\nupon Dave Darrin. \"I'm much too close to a greaser. I'm afraid I may get\nmy uniforms spotted, as well as my character.\"\n\n\"Stop that, Pen!\" warned Dave, stationing himself squarely before the\nangry Pennington. \"I don't know just how far you're responsible for what\nyou're saying now. To-morrow, if you make any such remarks to me, you'll\nhave to pay a mighty big penalty for them.\"\n\n\"You'll make me pay by going to the commandant and telling him all you\nknow, I suppose?\" sneered Pennington.\n\n\"You know better, Pen! Now, begin to practise keeping a civil tongue\nbehind your teeth!\"\n\nWith that, Darrin turned on his heel, seeking the deck.\n\nThis left \"Pen\" to conjecture as to whether he should report his\nmisadventure, and, if so, how best to go about it.\n\n\"See here, Hallam,\" began the worried midshipman, \"I begin to feel that\nit will be safer to turn in some kind of report on myself.\"\n\n\"Much safer,\" agreed Hallam. \"It will show good faith on your part if\nyou report yourself.\"\n\n\"And get me broken from the service, too, I suppose,\" growled the\nunhappy one.\n\n\"I hardly think it will, if you report yourself first,\" urged Hallam.\n\"But you'll be about certain to get your walking papers if you wait for\nthe first information to come from other sources.\"\n\n\"Hang it,\" groaned Pennington, \"I wish I could think, but my head aches\nas though it would split and my tooth is putting up more trouble than I\never knew there was in the world. And, in this racked condition, I'm to\ngo and put myself on the pap-sheet. In what way shall I do it, Hallam?\nCan't you suggest something?\"\n\n\"Yes,\" retorted Hallam with great energy. \"Go to the medical officer and\ntell him how your tooth troubles you. Tell him what you tried on shore.\nI'll go with you, if you want.\"\n\n\"Will you, old man? I'll be a thousand times obliged!\"\n\nSo the pair went off in search of the sick-bay, as the hospital part of\na battleship is called. The surgeon was not in his office adjoining, but\nthe hospital steward called him over one of the ship telephones,\ninforming him that a midshipman was suffering with an ulcerated tooth.\n\nDr. Mackenzie came at once, turned on a reflector light, and gazed into\nMidshipman Pennington's mouth.\n\n\"Have you tried to treat this tooth yourself, in any way?\" queried the\nship's surgeon.\n\n\"Yes, sir; I was so crazy with the pain, while in Annapolis, that I am\nafraid I did something that will get me into trouble,\" replied\nPennington, with a quiver in his voice.\n\n\"What was that?\" asked Dr. Mackenzie, glancing at him sharply. \"Did you\ntry the aid of liquor?\"\n\n\"Worse, I'm afraid, sir.\"\n\n\"Worse?\"\n\nPennington told of his experience with the opium pipe.\n\n\"That's no good whatever for a toothache, sir,\" growled Dr. Mackenzie.\n\"Besides, it's a serious breach of discipline. I shall have to report\nyou, Mr. Pennington.\"\n\n\"I expected it, sir,\" replied Pennington meekly.\n\n\"However, the report won't cure your toothache,\" continued Dr. Mackenzie\nin a milder tone. \"We'll attend to that first.\"\n\nThe surgeon busied himself with dissolving a drug in a small quantity of\nwater. This he took up in a hypodermic needle and injected into the\nlower jaw.\n\n\"The ache ought to stop in ten minutes, sir,\" continued the surgeon,\nturning to enter some memoranda in his record book.\n\nAfter that the surgeon called up the ship's commander over the 'phone,\nand made known Pennington's report.\n\n\"Mr. Pennington, Captain Scott directs that you report at his office\nimmediately,\" said the surgeon, as he turned away from the telephone.\n\n\"Very good, sir. Thank you, sir.\"\n\nBoth midshipmen saluted, then left the sick-bay.\n\n\"This is where you have to go up alone, I guess,\" hinted Midshipman\nHallam.\n\n\"I'm afraid so,\" sighed Pennington.\n\n\"However, I'll be on the quarter-deck, and, if I'm wanted, you can send\nthere for me.\"\n\n\"Thank you, old man. You're worth a brigade of Darrins--confound the\ngreasing meddler!\"\n\n\"Darrin acted according to his best lights on the subject of duty,\"\nremonstrated Mr. Hallam mildly.\n\n\"His best lights--bah!\" snarled Pennington. \"I'll take this all out of\nhim before I'm through with him!\"\n\nPennington reported to the battleship's commander. After some ten\nminutes a marine orderly found Hallam and directed him to go to Captain\nScott's office. Here Hallam repeated as much as was asked of him\nconcerning the doings of the afternoon. Incidentally, the fact of\nMidshipman Darrin's report to the police was brought out.\n\n\"Mr. Pennington, I shall send you at once, in a launch, over to the\ncommandant of cadets to report this matter in person to him,\" said\nCaptain Scott gravely. \"Mr. Hallam, you will go with Mr. Pennington.\"\n\nThen, after the two had departed, an apprentice messenger went through\nthe ship calling Dave's name. That young man was summoned to Captain\nScott's office.\n\n\"I am in possession of all the facts relating to the unfortunate affair\nof Midshipman Pennington, Mr. Darrin,\" began Captain Scott, after the\ninterchange of salutes. \"Will you tell me why you reported the affair to\nthe police?\"\n\n\"I went to the police, sir,\" Dave replied, \"because I was aware that\nmany members of the new fourth class are away from home for the first\ntime in their lives. I was afraid, sir, that possibly some of the new\nmidshipmen might, during one of their town-leaves, be tempted to try for\na new experience.\"\n\n\"A very excellent reason, Mr. Darrin, and I commend you heartily for it.\nI shall also report your exemplary conduct to the commandant of\nmidshipmen. You have, in my opinion, Mr. Darrin, displayed very good\njudgment, and you acted upon that judgment with promptness and decision.\nBut I am afraid,\" continued the Navy captain dryly, \"that you have done\nsomething that will make you highly unpopular, for a while, with some of\nthe members of your class.\"\n\n\"I hope not, sir,\" replied Dave.\n\n\"So do I,\" smiled Captain Scott \"I am willing to find myself a poor\nprophet. That is all, Mr. Darrin.\"\n\nOnce more saluting, Dave left the commanding officer's presence. Almost\nthe first classmate into whom he stumbled was Dan Dalzell.\n\n\"Well, from what quarter does the wind blow!\" murmured Dan.\n\nDarrin repeated the interview that he had just had.\n\n\"I'm afraid, Dave, little giant, that you've planted something of a mine\nunder yourself,\" murmured Dalzell.\n\n\"I feel as much convinced as ever, Danny boy, that I did just what I\nshould have done,\" replied Darrin seriously.\n\n\"And so does Captain Scott, and so will the commandant,\" replied Dan.\n\"But winning the commendation of your superior officers doesn't always\nimply that you'll get much praise from your classmates.\"\n\n\"Unfortunately, you are quite right,\" smiled Dave. \"Still, I'd do the\nsame thing over again.\"\n\n\"Oh, of course you would,\" assented Dan. \"That's because you're Dave\nDarrin.\"\n\nHere a voice like a bass horn was heard.\n\n\"All third classmen report to the quarter-deck immediately!\"\n\nThis order was repeated in other parts of the ship. Midshipmen gathered\nwith a rush, Pennington and Hallam being the only members absent. As\nsoon as the third classmen, or \"youngsters,\" as they are called in\nmidshipman parlance, had formed, the orders were read off dividing them\ninto sections for practical instruction aboard ship during the cruise.\n\nDave's name was one of the first read off. He was assigned to duty as\nsection leader for the first section in electrical instruction. Dalzell,\nFarley, Hallam, Pennington and others were detailed as members of that\nsection.\n\nThe same section was also designated for steam instruction, Dalzell\nbeing made leader of the section in this branch.\n\nThe class was then dismissed. Somewhat later Pennington and Hallam\nreturned from their interview with the commandant.\n\nHallam at once sought out Dave.\n\n\"Darry, old man,\" murmured Hallam, \"Pen is as crazy as a hornet against\nyou. As he had taken the first step by sticking himself on the pap-sheet\n(placing himself on report), the commandant said he would make the\npunishment a lighter one.\"\n\n\"What did Pen get?\" queried Dave.\n\n\"Fifty demerits, with all the loss of privileges that fifty carry.\"\n\n\"He's lucky,\" declared Dave promptly. \"Had the report come from other\nsources, he would have been dismissed from the service.\"\n\n\"If Pen's lucky,\" rejoined Hallam, \"he doesn't seem to realize the fact.\nHe's calling you about everything.\"\n\n\"He can keep that up,\" flashed Dave, \"until his toothache leaves him.\nThen, if he tries to carry it any further, Pen will collide with one of\nmy fists!\"\n\nNot much later a call sounded summoning the youngsters to the\nmidshipmen's mess. Dave was glad to note that Pennington sat at some\ndistance from him at table.\n\nWhile the meal was in progress the \"Massachusetts\" and the other\nbattleships got under way. The midshipmen were on deck, an hour later,\nwhen the fleet came to anchor for the night, some miles down Chesapeake\nBay.\n\nBefore the youngsters were ordered to their berths that night Third\nClassman Pennington had found opportunity to do a good deal of talking\nto a few comrades who would listen to him.\n\nPennington was determined to stir up a hornet's nest for Dave Darrin.\n\n\n\n\nCHAPTER III\n\n\nMIDSHIPMAN PENNINGTON GOES TOO FAR\n\nAt eight o'clock the following morning the various sections were formed\nand marched to the deck.\n\nDave reported:\n\n\"All present, sir.\"\n\nThe chief electrician was now summoned, and to him the section was\nturned over. This young man, Whittam, by name, was an enlisted man, but\na bright young sample of what the Navy can do for the boy who enlists as\nan apprentice.\n\n\"You will take your orders from Mr. Whittam as though he were an\nofficer,\" directed the officer, his words intended for all members of\nthe section, though he looked only at Darrin.\n\nDave saluted, then, as Chief Electrician Whittam turned to lead the way,\nDave called quietly:\n\n\"Section, left wheel--march!\"\n\nThey followed Whittam down into the dynamo room, an interesting spot for\na machinist.\n\n\"It's fine,\" muttered Dan, as he stared about him at the bright metal\nwork, the switch-board and the revolving machines. \"But I'm afraid I\ncouldn't learn the use and sense of all this in five years.\"\n\n\"Silence in the section,\" commanded Dave, turning around upon his chum.\n\nWhittam now began a short, preliminary talk upon the subjects in which\nthe midshipmen would be required to qualify.\n\n\"One of the first and most important requests I have to make,\" said\nWhittam presently, \"is that none of you touch the switches, except by\ndirection. None of you can guess the harm that might follow the careless\nand ignorant handling of a switch.\"\n\n\"It's pretty cheeky for an enlisted man to talk to midshipmen about\nignorance,\" whispered Pennington to Farley.\n\n\"Oh, I don't know--\" Farley started to reply, but Darrin's quiet voice\nbroke in with authority:\n\n\"Cease talking in section.\"\n\nFarley knew this to be a merited rebuke, and accepted it as such, but\nPennington's face went violently red.\n\n\"Confound that grease-spot-chaser,\" growled Pen. \"He'll be bound to take\nit out of me as long as the cruise lasts. But I'll get even with him. No\ncheap greaser is going to ride over me!\"\n\nThat morning none of the midshipmen were called upon to handle any of\nthe fascinating-looking machinery. Nearly the whole of this tour of\npractical instruction was taken up by the remarks of the chief\nelectrician. As he spoke, Whittam moved over to one piece or another of\nmechanism and explained its uses. Finally, he began to question the\nattentive young men, to see how much of his instruction they had\nabsorbed.\n\n\"This is a shame, to set an enlisted man up over us as quiz-master, just\nto see how little we know,\" growled Pennington; but this time he had the\ngood sense not to address his remark to anyone.\n\nPennington was not yet in good shape, after his harrowing experiences of\nthe day before.\n\nEre the tour of instruction was over, he began to shift somewhat\nuneasily.\n\nThen his attention began to wander.\n\nA brilliantly shining brass rod near him caught his eye. Something about\nthe glossy metal fascinated him.\n\nOnce or twice Pen put out his hand to touch the rod, but as quickly\nreconsidered and drew back his hand.\n\nAt last, however, the temptation proved too strong. He slid one hand\nalong the rail.\n\n\"Here, sir, don't handle that!\" rasped in the voice of Whittam.\n\nPennington drew back his hand, a flush mounting to his face.\n\n\"The fellow has no right to talk to a midshipman in that fashion!\"\nquivered Pennington to himself. \"But it was the fault of that low-minded\ngreaser Darrin, anyway. Darrin saw me, and he glanced swiftly at the\nchief electrician to draw attention to me.\"\n\nIt is only just to Pennington to state that he actually believed he had\nseen Dave do this. Darrin, however, was not guilty of the act. He had in\nno way sought to direct attention at Pennington.\n\nTowards the close of the tour the officer in whose department this\ninstruction fell passed through the dynamo room.\n\n\"Are there any breaches of conduct to be reported, Whittam?\" inquired\nthe officer, halting.\n\n\"Nothing worth mentioning, sir,\" replied the chief electrician.\n\n\"I asked you, Whittam, whether there had been any breaches of conduct,\"\nretorted the officer with some asperity.\n\n\"One midshipman, sir, after having been instructed to touch nothing,\nrested his hand on one of the brass rods.\"\n\n\"His name?\"\n\n\"I don't know the names of many of the young gentlemen yet, sir, so I\ndon't know the particular midshipman's name, sir.\"\n\n\"Then point him out to me,\" insisted the officer.\n\nThere was hardly any need to do so. Pennington's face, flushed with\nmortification, was sufficient identification. But the chief electrician\nstepped over, halting in front of the hapless one, and said:\n\n\"This is the young gentleman, sir.\"\n\n\"Your name, sir?\" demanded the officer.\n\n\"Pennington, sir.\"\n\n\"Mr. Pennington, you will place yourself on the report, sir, for\ndisobedience of orders,\" commanded the officer. \"Is this the only case,\nWhittam?\"\n\n\"The only case, sir.\"\n\nThe officer passed out of the dynamo room, leaving the unlucky one more\nthan ever angry with Darrin, whom he incorrectly charged with his\npresent trouble.\n\nThe recall sounding, Dave turned to Whittam, saying crisply but\npleasantly:\n\n\"Thank you for our instruction.\"\n\n\"He's thanking the fellow for my new scrape,\" growled Pennington\ninwardly.\n\nDave marched his section back to deck and dismissed it. Dan Dalzell, as\nsection leader in steam instruction, immediately re-formed it.\n\n\"You will report in the engine-room, Mr. Dalzell, to\nLieutenant-Commander Forman, who is chief engineer of this ship. He will\nassign you to an instructor.\"\n\n\"Aye, aye, sir,\" Dan replied, saluting. \"Section, right wheel--march!\"\n\nDan already knew where, down in the bowels of the great battleship, to\nfind the engine room.\n\nReaching that department, Dan halted his section.\n\n\"Section all present, sir,\" reported Dan, saluting a strange officer,\nwho, however, wore the insignia of a lieutenant-commander.\n\n\"Your name, sir?\" inquired the officer.\n\n\"Dalzell, sir.\"\n\n\"Let your section break ranks. Then you may all follow me, and keep your\neyes open, for you will go through one or two dark places.\"\n\n\"Aye, aye, sir. Section break ranks.\"\n\nLieutenant-Commander Forman led the way, with all the members of the\nsection wondering what was to be the nature of their first day's work in\nthe engineer department.\n\nDescending lower into the ship, the chief engineer led the young middies\nover a grating, and paused at the head of an iron ladder.\n\n\"Pass down in orderly fashion, single file,\" directed the chief\nengineer, halting. \"When at the foot of this ladder, cross a grating to\nport side, and then descend a second ladder, which you will find.\"\n\nAll the midshipmen went down the first ladder in silence. Dan, who had\npreceded the others, crossed the grating and found the second ladder.\n\nOnce more these youngsters descended. Pennington, as though by mere\naccident, succeeded in following Dave Darrin down the ladder.\n\nJust as they were near the bottom Dave felt a foot descend upon his\nshoulder, almost with a kick, and then rest there with a crushing\npressure.\n\nIt hurt keenly until Darrin was able to dodge out from under and\nhurriedly reach the bottom.\n\n\"Pardon, whoever you are,\" came a gruff voice.\n\nDave, with his shoulder crippled a good deal, and paining keenly, halted\nas soon as his foot had touched bottom. It was dark down there, though\nsome reflected light came from an incandescent light at a distance.\n\nDave waited, to peer into the face of the man who had stepped on his\nshoulder.\n\nIt was Pennington, of course!\n\n\"I'll take pains not to go down ahead of you again, or to follow you up\na ladder,\" grunted Darrin suspiciously.\n\n\"Oh, are you the man on whose shoulder my foot rested?\" asked\nPennington, with apparent curiosity.\n\n\"Didn't you know it!\" questioned Darrin, looking straight into the\nother's eyes.\n\nInstead of answering intelligibly, Pennington turned and walked away a\nfew feet.\n\n\"Perhaps that fellow thinks he's going to vent his spite on me in a lot\nof petty ways,\" murmured Dave. \"If that is the idea he has in his head,\nhe's going to wake up one of these days!\"\n\nFollowing the last midshipman came Lieutenant-Commander Forman.\n\n\"After me, gentlemen,\" directed the chief engineer. He turned down a\nnarrow passage, only a few feet long, and came out in the furnace room.\n\nHere huge fires glowed through the furnace doors. Four of the Navy's\nfiremen stood resting on their shovels. Instantly, on perceiving the\nchief engineer, however, the men stood at attention.\n\n\"Pass the word for the chief water tender,\" ordered the engineer,\nturning to one of the firemen.\n\nThe messenger soon came back with a pleasant-faced, stalwart man of\nforty.\n\n\"Heistand,\" ordered the chief engineer, \"give these members of the first\nsection, third: class, steam instruction, a thorough drill in firing.\"\n\n\"Aye, aye, sir,\" replied the chief water tender, saluting.\n\n\"Heistand's orders are mine, Mr. Dalzell,\" continued the\nlieutenant-commander, facing Dan. \"Preserve order in your section.\"\n\n\"Aye, aye, sir,\" replied Dan, saluting. Acknowledging this courtesy in\nkind, the chief engineer turned and left the furnace room.\n\nHeistand was presumably of German parentage, though he had no accent. He\nstruck the midshipmen as being a pleasant, wholesome fellow, though the\nwater tenders and firemen of the \"Massachusetts\" knew that he could be\nextremely strict and grim at need.\n\n\"You will now, young gentlemen,\" began Heistand, \"proceed to learn all\nabout priming a furnace, lighting, building, cleaning and generally\ntaking care of a fire. Two furnaces have been left idle for this\ninstruction.\"\n\nBut two of the regular firemen now remained in the room. These were\nordered to hustle out coal before boilers B and D. Then Heistand taught\nthe members of the section how to swing a shovel to the best advantage\nso as to get in a maximum of coal with the least effort. He also\nillustrated two or three incorrect ways of shoveling coal.\n\n\"The idea of making coal heavers out of us!\" growled a much-disgusted\nvoice.\n\nDan did not see who the speaker was, but his eyes flashed as he turned\nand rasped out:\n\n\"Silence in the section! Speak only to ask for information, and then at\nthe proper time.\"\n\n\"Another young autocrat!\" muttered a voice.\n\n\"Wait one moment, please, Heistand,\" begged Dan. Then, wheeling squarely\nabout, and facing all the members of the section, he declared with\nemphasis:\n\n\"If there's any more unauthorized talking I shall feel obliged to pass\nthe word above that discipline is in a bad way in this section.\"\n\nThen he wheeled about once more, facing the chief water tender.\n\n\"Now, young gentlemen,\" resumed the chief water tender, \"take your\nshovels and fill in lively under boilers B and D.\"\n\nThree or four times Heistand checked one or another of the midshipmen,\nto show him a more correct way of handling the shovel. Yet, in good\ntime, both furnaces were primed.\n\n\"Now, Mr. Dalzell, please detail four members of the section to follow\nme with their shovels and bring red coals from under another boiler.\"\n\nDan appointed himself, Darrin, Farley and Pennington.\n\nBurning coals were brought and thrown into each furnace, and in a little\nwhile roaring fires were going. These, though not needed for the\nhandling of the battleship, were permitted to burn for a while, Heistand\nexplaining to the section practically the uses of the water gauges and\nthe test cocks. By this time the midshipmen's white working clothes were\nliberally sprinkled with coal dust and somewhat smeared with oils.\n\n\"And now, young gentlemen, as we have no further use for these fires,\nyou will next learn how to haul them,\" announced Heistand.\n\nThis was interesting work, but hot and fast. The implements with which\nthe middies worked soon became red-hot at the end. Yet, as all entered\ninto this novel work with zest, the fires had soon been hauled out on to\nthe floor plates.\n\nJust as the last of this work was being done Pennington, as an apparent\naccident due to excess of zeal, dropped the red-hot end of his implement\nacross the toe of Darrin's left shoe.\n\nIn an instant the leather began to blaze. With swift presence of mind\nDave stepped his right foot on the flame, smothering it at once.\n\nBut he was \"mad clean through.\"\n\n\"See here, Pen,\" he muttered, in a low voice, his eyes blazing fiercely\ninto the other midshipman's, \"that is the last piece of impudence that\nwill be tolerated from you.\"\n\nMidshipman Pennington's lip curled disdainfully.\n\nDan had not seen the \"accident,\" but he was near enough to hear the\ntalking, and he caught Dave at it. So Dan ordered, impartially:\n\n\"Mr. Darrin, you will place yourself on report for unauthorized talking\nin section!\"\n\nDave flushed still more hotly, but said nothing.\n\nMidshipman Dalzell now marched the section from the furnace room, and\ndismissed it. It was near noon, and would soon be time for the middies\nto eat.\n\nDave hurried away, washed, changed his uniform, and then stepped away\nswiftly to place himself on the report.\n\n\"I was sorry to do that, old chum,\" murmured Dan, as he met Dave\nreturning. \"But of course I couldn't play favorites. What made you so\nfar forget yourself?\"\n\n\"A something that would have had the same effect on you,\" retorted Dave\ngrimly. Thereupon he described Pennington's two underhanded assaults\nthat morning.\n\n\"Humph!\" muttered Dalzell. \"That fellow Pen is bound to go the whole\nlimit with you.\"\n\n\"He won't go much further,\" declared Dave, his eyes flashing.\n\n\"And the chump ought to know it, too,\" mused Dan. \"The class history of\nthe last year should have taught him that. But see here, Dave, I don't\nbelieve Pen will do anything openly. He will construct a series of\nplausible accidents.\"\n\n\"There will be one thing about him that will be open, if he goes any\nfurther,\" retorted Dave, \"and that will be his face when he collides\nwith my fist.\"\n\n\"I hope I see that when it happens,\" grinned Dalzell. \"It's bound to be\nentertaining!\"\n\n\"Wait a second, then. Here comes Pennington now,\" murmured Dave Darrin\nin an undertone.\n\nPennington, in his immaculate blue uniform, like the chums, came\nstrolling along the passageway between decks.\n\nHe affected not to see the chums, and would have passed by. But Dave,\neyeing him closely, waited until Pen was barely three feet away. Then\nDarrin said tersely:\n\n\"Mr. Pennington, I wish an understanding with you.\"\n\n\"I don't want any with you,\" replied Pennington insolently, as he stared\nat Dave from under much-raised eyebrows. He would have gone by, but Dave\nsprang squarely in front of him.\n\n\"Just wait a moment!\" warned Dave rather imperiously, for he was aglow\nwith justifiable indignation.\n\n\"Well?\" demanded Pennington halting. \"Out with it, whatever you may\nthink you have to say.\"\n\n\"I have two things to speak about,\" replied Dave, trying to control his\nvoice. \"In the first place, while going down the ladders to the furnaces\nthis morning, you stepped on my shoulder.\"\n\n\"Well!\" insisted Pennington coldly.\n\n\"The second thing you did was, when hauling the fires, to drop red-hot\nmetal across one of my shoes, setting it on fire.\"\n\n\"Well?\" insisted Pennington more coldly.\n\n\"If you mean to contend that either one was an accident,\" resumed Dave,\n\"then--\"\n\nBut he found himself obliged to pause for a moment in order to steady\nhis voice.\n\n\"Well?\" asked Pennington with more insolence than ever.\n\n\"If you make such pretense in either case,\" tittered Dave Darrin, \"then\nyou're a liar!\"\n\n\"Fellow!\" sputtered Pennington, turning white with anger.\n\n\"I mean what I say, and I can back it up,\" muttered Darrin.\n\n\"Then I'll make you eat your words!\" roared Pennington.\n\nClenching his fists and with the boxer's attitude, Pen aimed two swift\nblows at Darrin.\n\nNeither blow reached, however, for Dave dodged out of the way. Then\nDarrin struck back, a straight, true, forceful blow that landed on the\nother midshipman's nose, knocking him down.\n\nPennington staggered somewhat when he rose, but he was quickly up, none\nthe less, and ready for anything that might happen.\n\nAll of a sudden Dan Dalzell felt his own heart going down into his\nshoes. One of the ship's officers had just entered the passageway, in\ntime to see what was going on.\n\n\n\n\nCHAPTER IV\n\n\nA LITTLE MEETING ASHORE\n\n\"Stop it, both of you,\" whispered Dan.\n\n\"Stand at attention, ready to salute the officer.\"\n\nPennington, with the blood flowing from his damaged nose, would have\nmade a most ludicrous figure saluting!\n\nThe instant that he saw such evidence as Pen's nose presented the\nofficer would be bound to make inquiries.\n\nThen, just as surely, his next step must be to Border the three before\nthe commandant of midshipmen.\n\nFighting carries with it a severe penalty. Even Dan was certain to be\nreported, through the mere fact of his presence there, as aiding in a\nfight. And those who aid are punished as severely as the principals\nthemselves.\n\nIt was a tense, fearsome instant, for midshipmen have been dismissed\nfrom the Naval Academy for this very offense.\n\nThe passage was not brilliantly lighted.\n\nThe on-coming officer, a lieutenant, junior grade, was looking at the\nfloor as he came along.\n\nSuddenly he paused, seemed lost in thought, then wheeled and walked back\nwhence he had come.\n\nDan breathed more easily. Dave heaved a sigh of relief.\n\nAs for Pennington, that midshipman had wheeled and was stealing rapidly\ndown the passageway, intent only on escape.\n\n\"That was the closest squeak we'll ever have without being ragged cold,\"\nmurmured Dalzell tremulously.\n\n\"Where is Pennington?\" demanded Dave, wheeling about after he had\nwatched the Naval lieutenant out of sight.\n\n\"Ducked out of sight, like a submarine,\" chuckled Dan.\n\nAt that moment the call for midshipmen's dinner formation sounded. Dave\nand Dan were ready.\n\nPennington showed up just after the line had started to march into the\nmidshipmen's mess tables.\n\nTo the inquiry of the officer in charge, Pen lamely explained that he\nhad bumped his nose into something hard in a poorly lighted passageway.\n\nThough the officer accepted the excuse, he smiled within himself.\n\n\"It wasn't iron or steel that bumped that young man's nose,\" thought the\nofficer.\n\n\"Oh, the middies haven't changed a lot since I boned at Annapolis!\"\n\nPennington's nose was no very lovely member of his face at that moment.\nIt had been struck hard, mashed rather flat, and now looked like a red\nbulb.\n\n\"Meet with an accident, Pen?\" asked Hallam curiously at table.\n\n\"Quit your kidding, please,\" requested Pennington sulkily.\n\nThat directed the curious glances of other middies at Pennington's new\nbulbous nose.\n\nThe young man was so brusque about it, however, that other table mates\nceased quizzing him.\n\nYet, as soon as the meal was over, many a youngster asked others of his\nclass for news regarding Pen. But none possessed it.\n\nDuring the brief rest that followed the meal, however, Midshipman\nPennington made it his business to try to meet Dave Darrin alone. He\nsucceeded, finding Dave staring off across the water at the port rail.\n\n\"Of course, Mr. Darrin,\" began the other midshipman, in a voice\nsuggestive of ice, \"you are aware that the incident of an hour ago\ncannot be allowed to pass unnoticed.\"\n\n\"I don't believe there's any danger of that,\" retorted Darrin, with an\nironical glance at Pennington's damaged-looking nose.\n\n\"Confound you, sir,\" hissed the other midshipman, \"don't you dare to be\ninsolent with me.\"\n\n\"Why, I had thought,\" observed Dave, \"that, of your own choice, the\nperiod of courtesies between us had passed.\"\n\n\"I shall call you out, Mr. Darrin!\"\n\n\"You'll find my hearing excellent,\" smiled Dave. \"I shall make but one\nstipulation.\"\n\n\"I'll do you the favor of asking what that stipulation is,\" sneered\nPennington.\n\n\"Why, after the narrow escape we had from being caught and reported, an\nhour or so ago, I shall ask that the fight be held where we are not so\nlikely to be caught at it. I don't care about being dropped from the\nNaval Academy, nor do I believe you do.\"\n\n\"It would be a good thing for the service, if one of us were to be\ndropped,\" sneered Pennington.\n\n\"Yes! Oh, well, you can easily procure writing materials from the\ncaptain's clerk,\" volunteered Dave generously. \"On a cruise, I believe,\na resignation is sent direct to the commandant of midshipmen.\"\n\nThis ridicule served only to fan the flame of Pennington's wrath.\n\n\"Darrin,\" he hissed, \"the Academy isn't big enough to hold us both!\"\n\n\"But I've already told you how to get out,\" protested Dave coolly.\n\n\"I don't intend to get out!\"\n\n\"No more do I,\" rejoined Dave. \"I won't even toss pennies with you to\nfind out who quits the service.\"\n\n\"Mr. Darrin, you are merely seeking to divert my mind from what I have\nsaid.\"\n\n\"What did you say--particularly?\"\n\n\"That you would have to fight me.\"\n\n\"I have already signified my entire willingness, Mr. Pennington. To that\nI really can add nothing.\"\n\nFourth classmen are always addressed as \"mister,\" and they must use the\nsame \"handle to the name\" when addressing upper classmen. But members of\nthe three upper classes resort to the use of \"mister,\" in addressing\nclassmates, only when they wish to be offensive or nearly so.\n\n\"I will send a friend to meet you,\" Pennington continued.\n\n\"Why, I thought,\" bantered Darrin ironically, \"that you were going to\nfight me yourself.\"\n\n\"So I am--be sure of it. I will amend my statement by saying that I will\nsend a second to see you.\"\n\n\"Save time by sending him to Dalzell.\"\n\n\"Very good, Mr. Darrin.\"\n\n\"Is that all you wished to say to me?\"\n\n\"Yes.\"\n\n\"Very good, Mr. Pennington.\"\n\nWith two very stiff nods the midshipmen parted.\n\nPennington hastened at once in search of Hallam.\n\n\"Will you serve me, old man?\" queried Pennington.\n\n\"Sorry, but----\"\n\n\"Well, you see, Pen, not knowing all the facts of the case, I must admit\nthat all my sympathies are with Darrin.\"\n\n\"All your sympathies?\" echoed Pen, frowning.\n\n\"Well, nearly all, anyway. You see, I've known and observed Darrin for a\nfull year now, and I don't believe patient old Darry is the one to start\nany trouble.\"\n\n\"He called me a liar,\" protested Pennington.\n\n\"Did he?\" gasped Hallam.\n\n\"Well, he qualified the statement, but his way of saying it was as\noffensive as the direct lie could have been.\"\n\n\"So you're bent on fighting Darry?\"\n\n\"I am.\"\n\n\"Too bad!\" muttered Hallam, shaking his head.\n\n\"Are you anxious for your idol?\" asked Pen in a disagreeable tone.\n\n\"No, Penny; it's you that I'm concerned about in my own mind. You're\ngoing next to a very hard proposition. Darry is patient--almost as\npatient as the proverbial camel--but when he fights he fights! You'll be\nhammered to a pulp, Pen.\"\n\n\"Pooh!\"\n\n\"No one has yet beaten Darrin at a fist fight.\"\n\n\"There always has to be a first time, you know.\"\n\n\"And you think you're It?\"\n\n\"As far as Darrin is concerned--yes.\"\n\n\"Too bad--too bad!\" sighed Hallam. \"I'm afraid, Penny, that the heat in\nthe furnace room was too much for you this morning.\"\n\n\"Then you won't serve as one of my seconds?\"\n\n\"The honor is most regretfully declined,\" replied Hallam in a tone of\nmock sadness.\n\n\"You want to see Darrin win?\"\n\n\"If there has to be a fight, I do,\" replied Midshipman Hallam.\n\n\"Don't bet your money on him, anyway.\"\n\n\"I'm not a gambler, Penny, and I don't bet,\" replied Hallam, with a\ndignity that, somehow, ended the conversation.\n\nPennington had considerable difficulty, at first, in finding a second.\nAt last, however, he induced Decker and Briggs to represent him.\n\nThese two midshipmen went to see Dan Dalzell.\n\n\"Wait until I send for Mr. Farley,\" proposed Dalzell. He soon had that\nmidshipman, who was wholly willing to serve Darrin in any capacity.\n\n\"We're ready to have the fight this evening,\" proposed Midshipman\nDecker.\n\n\"We're not,\" retorted Dan, with vigor.\n\n\"Why not?\"\n\n\"This forenoon Pennington deliberately stepped on Darrin's shoulder,\nwith such force as to lame it a good deal,\" replied Dan. \"Our man\ninsists that he has a right to rest his shoulder, and to wait until\nto-morrow.\"\n\n\"But to-morrow we have a short shore liberty at Hampton Roads,\"\nremonstrated Briggs.\n\n\"Yes; and during that shore liberty we can have the fight more safely\nthan on board ship,\" insisted Dalzell.\n\n\"But we intended to devote our shore leave to pleasure,\" objected\nDecker.\n\n\"You'll find plenty of pleasure, if you accept our proposition,\" urged\nDan dryly. \"At any rate, we won't hear of Darrin fighting before\nto-morrow. He must have to-night to rest that shoulder.\"\n\n\"All right; so be it,\" growled Decker, after a side glance at Briggs.\n\n\"On shore, at some point to be selected by the seconds?\" asked Dan\nDalzell.\n\n\"Yes; that's agreed.\"\n\nDetails as to whom to invite as referee and time-keeper were also\narranged.\n\n\"I suppose we'll have to use up our shore leave that way, then,\" grunted\nPennington, when told of the arrangement.\n\n\"There's one way you can save the day,\" grinned Decker.\n\n\"How?\"\n\n\"Put Darrin to sleep in the first round, then hurriedly dress and leave,\nand enjoy your time on shore.\"\n\n\"But Darrin is a very able man with his fists,\" observed Pennington.\n\n\"Yes; but you're a mile bigger and heavier, and you're spry, too. You\nought to handle him with all the ease in the world.\"\n\n\"I don't know,\" muttered Pennington, who didn't intend to make the\nmistake of bragging in advance. \"I'll do my best, of course.\"\n\n\"Oh, you'll win out, if you're awake,\" predicted Midshipman Briggs\nconfidently.\n\nWhen the cadets were called, the following morning, they found the\nbattleship fleet at anchor in Hampton Roads.\n\n\n\n\nCHAPTER V\n\n\nWHEN THE SECONDS WONDERED\n\nOne after another the launches sped ashore, carrying their swarms of\ndistinguished looking young midshipmen.\n\nThe fight party managed to get off all in the same boat, and on one of\nthe earliest trips.\n\nPennington was to have ordinary shore leave on the cruise, his fifty\ndemerits to be paid for by loss of privileges on his return to the Naval\nAcademy.\n\n\"Decker,\" proposed Dan, \"you and I can skip away and find a good place\nin no time. Then we can come back after the others.\"\n\n\"That's agreeable to me,\" nodded Midshipman Decker.\n\nIn twenty minutes the two seconds were back.\n\n\"We've found just the place,\" announced Decker. \"And it isn't more than\nthree minutes' walk from here. Will you all hurry along?\"\n\n\"The place\" turned out to be a barn that had not been used for a year or\nmore. The floor was almost immaculately clean. In consideration of two\ndollars handed him, the owner had agreed to display no curiosity, and\nnot to mention the affair to any one.\n\n\"How do you like it, Darry?\" asked Dan anxiously.\n\n\"It will suit me as well as any other place,\" responded Dave, slipping\noff his blouse, folding it neatly and putting it aside, his uniform cap\nfollowing.\n\n\"And you?\" asked Decker of his man.\n\n\"The floor's hard, but I don't expect to be the man to hit it,\" replied\nPennington.\n\nIn five minutes both midshipmen were attired for their \"affair.\" Between\nthem the different members of the party had smuggled ashore shoes, old\ntrousers and belts for the fighters.\n\nIt being a class affair, Remington, of the third class, had come along\nas referee, while Dawley; was to be the time-keeper.\n\n\"If the principals are ready, let them step forward,\" ordered Midshipman\nRemington, going to the middle of the floor. \"Now, I understand that\nthis is to be a finish fight; rounds, two minutes; rests, two minutes. I\nalso understand that the principals do not care to shake hands before\nthe call to mix up.\"\n\nDarrin and Pennington nodded their assent.\n\n\"Take your places, gentlemen,\" ordered the referee quickly. \"Are you\nready, gentlemen?\"\n\n\"Yes,\" came from both principals.\n\n\"Time!\"\n\nBoth men had their guards up. As the word left the referee's lips each\ntried two or three passes which the other blocked. Midshipman Pennington\nwas trying to take his opponent's \"measure.\"\n\nThen Dave ducked, darted, dodged and wheeled about. Pennington had to\nfollow him, and it made the latter angry.\n\n\"Stand up and fight, can't you,\" hissed Pen.\n\n\"Silence during the rounds, Mr. Pennington,\" admonished the referee\nquietly. \"Let the officials do all the talking that may be necessary.\"\n\nDave, as he dodged again, and came up unscathed, grinned broadly over\nthis rebuke. That grin made Pen angrier than anything else could have\ndone.\n\n\"I'll wipe that grin off his face!\" muttered Pennington angrily.\n\nAnd this very thing Pennington tried hard to do. He was quick on his own\nfeet, and for a few seconds he followed the dodging Darrin about,\nraining in blows that required all of Dave's adroitness to escape.\n\nDave's very success, however, made his opponent all the angrier. From\nannoyance, followed by excessive irritation, Pennington went into almost\nblind rage--and the man who does that, anywhere in life, must always pay\nfor it.\n\nSuddenly Dave swung his right in on the point of Pen's chin with a force\nthat jolted the larger midshipman. As part of the same movement,\nDarrin's left crashed against Pennington's nose.\n\nThen, out of chivalry, Dave dropped back, to give Pen a few moments, in\ncase he needed them, to get his wits back.\n\n\"Time!\" roared Dawley, and Pennington's seconds pounced upon him and\nbore him away to his corner.\n\n\"Now I know how that fellow Darrin wins his fights,\" growled Pennington\nin an undertone. \"He keeps on running away until he has the other man\ngasping for breath. Then Darrin jumps in and wins.\"\n\n\"The method doesn't much matter,\" commented Briggs dryly, as he and\nDecker worked over their man. \"It's the result that counts. Rush Darry\ninto a tight corner, Pen, and then slam him hard and sufficiently.\"\n\n\"Thanks, fellows; now I'm all right for the second round.\" muttered\nMidshipman Pennington.\n\nIn a few seconds more Dave and his opponent were hard at work.\n\nDave still used his footwork, and most cleverly. Yet, wherever he went,\nPen followed him nimbly. It didn't look so one sided now.\n\nThen Pennington, at last, managed to deliver one blow on Darrin's right\nshort ribs. It took a lot of Dave's spare wind; he raced about, seeking\nto regain his wind before allowing close quarters. But at last\nPennington closed in again, and, after a swift feint, tried to land the\nsame short-rib blow.\n\nDarrin was watching, and blocked. Then, his temples reddening with\nanger, Dave swung in a huge one that crashed in under Pennington's right\near.\n\n\"Time!\" shouted Dawley, just as Pen went to the floor in a heap. That\nsaved the larger midshipman from having to take the count. His seconds\nhad him ready at the call for the third round.\n\nNow, suddenly, Darrin seemed to change not only his tactics, but his\nwhole personality. To his opponent Dave seemed suddenly transformed into\na dancing demon.\n\nIt was about the same old footwork, but it was aggressive now, instead\nof being defensive.\n\nFirst, Dave landed a light tap on the already suffering nose. A few\nseconds later he landed on the point of Pen's chin, though not hard\nenough to send his man down. Then a rather light blow on the jaw, just\nunder Pen's right ear again. The larger midshipman was now thoroughly\nalarmed. He feared that Darrin could do whatever he willed, and shivered\nwith wonder as to when the knockout blow would come.\n\nThe truth was, Pennington was still putting up a better battle than he\nhimself realized, and Darrin was not disposed to take any foolish\nchances through rushing the affair. Thus, the third round ended.\n\nBy the time that they came up for the fourth round, after both men had\nundergone some vigorous handling by their respective seconds, Pennington\nwas a good deal revived and far more confident.\n\nDave's tactics were the same in the fourth round. Pennington didn't find\ntime to develop much in the way of tactics for himself, save to defend\nhimself.\n\nDuring the first minute no important blows were landed on either side.\nThen, suddenly, Dave darted in and under, and brought a right-arm hook\nagainst Pen's nose in a way that started that member to bleeding again,\nand with a steady flow.\n\nThat jarred the larger midshipman. He plunged in, heavily and blindly,\nblocking one of Darrin's blows by wrapping both arms around him.\n\n\"None of that, Mr. Pennington! Break away fast!\" ordered Midshipman\nRemington quickly.\n\nDave took a fair get away, not attempting to strike as the clinch was\nbroken. But an instant later Dave came back, dancing all around his\ndazed opponent, landing on the short ribs, on the breast bone, under\neither ear and finally on the tip of the chin.\n\nPen was sure that none of these blows had been delivered with the force\nthat Darrin could have sent in.\n\n\"Time!\" shouted Midshipman Dawley.\n\nThe principals retired to their corners, Pennington almost wholly afraid\nfrom the conviction that his antagonist was now merely playing with him\nto keep the interest going.\n\nSo Pennington was still rather badly scared when the two came together\nfor the fifth round.\n\n\"Get lively, now, gentlemen, if you can,\" begged Referee Remington.\n\"Finish this one way or the other, and let us get some of the benefits\nof our shore leave.\"\n\nPen started by putting more steam behind every blow. Dave, who had used\nup so much of his wind by his brilliant footwork, began to find it\nharder to keep the upper hand.\n\nTwice, however, he managed to land body blows. He was trying to drive in\na third when Pennington blocked, following this with a left-arm jab on\nDarrin's left jaw that sent the lighter man to the floor.\n\nInstantly Dawley began to count off the seconds.\n\n\"--seven, eight, nine, te----\"\n\nDave was up on his feet. Pen tried to make a quick rush, but Darrin\ndodged cleverly, them wheeled and faced his opponent as the latter\nwheeled about.\n\nAfter that there was less footwork. Both men stood up to it, as keenly\nalert as they could be, each trying to drive home heavy blows. While\nthey were still at it the call of time sounded.\n\n\"Don't let him put it over you, David, little giant!\" warned Dan, as the\nlatter and Farley vigorously massaged Darrin's muscles. \"He all but had\nyou, and there isn't any need of making Pen a present of the meeting.\"\n\n\"I tried to get him,\" muttered Dave in an undertone, \"and I shall go on\ntrying to the last. But Pennington is pretty nearly superior to anyone\nin my class.\"\n\n\"Just waltz in and show him,\" whispered Dalzell, as the call sounded.\n\nPennington entered the sixth round with more confidence. He began, at\nthe outset, to drive in heavy blows, nor did Dave do much dodging.\n\nBump! Twenty-five seconds only of this round had gone when Darrin landed\nhis right fist with fearful force upon the high point of Pennington's\njaw.\n\nDown went the larger midshipman again. This time he moaned. His eyes\nwere open, though they had a somewhat glassy look in them.\n\nDawley was counting off the seconds in measured tones.\n\n\"--seven, eight, nine--ten!\"\n\nPen had struggled to rise to his feet, but sank back with a gasp of\ndespair and rage.\n\n\"Mr. Pennington loses the count and the fight,\" announced Referee\nRemington coolly. \"I don't believe we're needed here, Dawley. The\nseconds can handle the wreck. Come along.\"\n\nAs the two officials of the meeting hustled out of the barn, Dalzell\ngave his attention to helping his chum, while Farley went over to offer\nhis services in getting the vanquished midshipman into shape.\n\n\"There were times when I could have closed both of Pennington's eyes,\"\nmurmured Dave to Dan. \"But I didn't want to give him any disfiguring\nmarks that would start questions on board ship.\"\n\n\"You had him whipped from the start,\" murmured Dan confidently, as he\nsprayed, then rubbed Dave's chest and arms.\n\n\"Maybe, but I'm not so sure of that,\" rejoined Darrin. \"That fellow\nisn't so easy a prize for any one in my class. There were times when I\nwas all but convinced that he had me.\"\n\n\"Oh, fairy tales!\" grunted Dan.\n\n\"Have it your own way, then, Danny boy!\"\n\nWhen Darrin and his seconds left the barn they went off to enjoy what\nremained of the shore leave. Pennington's seconds finally, at his own\nrequest, left him at an ice cream parlor, where he proposed to remain\nuntil he could return to the big, steel \"Massachusetts\" without exciting\nany wonder over the little time he had remained ashore. Pennington had\nstrength to walk about, but he was far from being in really good shape,\nand preferred to keep quiet.\n\n\n\n\nCHAPTER VI\n\n\nIN TROUBLE ON FOREIGN SOIL\n\nFrom Hampton Roads the Battleship Squadron, with the midshipmen on\nboard, sailed directly for Plymouth, England.\n\nDuring most of the voyage over slow cruising speed was used. By the time\nthat England's coast was sighted the third-class middies found they knew\nmuch more about a battleship than they had believed to be possible at\nthe start of the voyage.\n\nThey had served as firemen; they had mastered many of the electrical\ndetails of a battleship; they had received instruction and had \"stood\ntrick\" by the engines; there had been some drill with the smaller,\nrapid-fire guns, and finally, they had learned at least the rudiments of\n\"wig-wagging,\" as signaling by means of signal flags is termed.\n\nIt was just before the call to supper formation when England's coast\nloomed up. Most of the midshipmen stood at the rail, watching eagerly\nfor a better glimpse at the coast.\n\nSome of the midshipmen, especially those who came from wealthier\nfamilies, had been in England before entering the Naval Academy. These\nfortunate ones were questioned eagerly by their comrades.\n\nThe battleships were well in sight of Eastern King Point when the\nmidshipmen's call for supper formation sounded. Feeling that they would\nmuch have preferred to wait for their supper, the young men hastened\nbelow.\n\nAfter the line was formed it seemed to the impatient young men as though\nit had never taken so long to read the orders.\n\nYet there came one welcome order, to the effect that, immediately after\nthe morning meal, all midshipmen might go to the pay officer and draw\nten dollars, to be charged against their pay accounts.\n\n\"That ten dollars apiece looms up large David, little giant,\" murmured\nDan Dalzell, while the evening meal was in progress.\n\n\"We ought to have a lot of fun on it,\" replied Darrin, who was looking\nforward with greatest eagerness to his first visit to any foreign soil.\n\"But how much shore leave are we to have?\"\n\n\"Two days, the word is. We'll get it straight in the morning, at\nbreakfast formation.\"\n\nIn defiance of regulations, Midshipman Pennington, whose father was\nwealthy, had several hundred dollars concealed in his baggage. He had\nalready invited Hallam, Mossworth and Dickey to keep in his wake on\nshore, and these young men had gladly enough agreed.\n\n\"Say, but we're slackening speed!\" quivered Dalzell, when the meal was\nnearly finished.\n\n\"Headway has stopped,\" declared Darrin a few moments later.\n\n\"Listen, everyone!\" called Farley. \"Don't you hear the rattle of the\nanchor chains?\"\n\n\"Gentlemen, as we're forbidden to make too much racket,\" proposed\nirrepressible Dan, \"let us give three silent cheers for Old England!\"\n\nRising in his place, Dan raised his hand aloft, and brought it down, as\nhis lips silently formed a \"hurrah!\"\n\nThree times this was done, each time the lips of the midshipmen forming\na silent cheer.\n\nThen Dan, with a mighty swoop of his right arm, let his lips form the\nword that everyone knew to be \"tiger!\"\n\n\"Ugh-h-h!\" groaned Midshipman Reilly.\n\n\"Throw that irresponsible Fenian out!\" directed Dan, grinning.\n\nThen the midshipmen turned their attention to the remnants of the meal.\n\nBoom! sounded sharply overhead.\n\n\"There goes the twenty-one-gunner,\" announced Darrin.\n\nWhen a foreign battleship enters a fortified port the visiting fleet, or\nrather, its flagship, fires a national salute of twenty-one guns. After\na short interval following the discharge of the last gun, one of the\nforts on shore answers with twenty-one guns. This is one of the methods\nof observing the courtesies between nations by their respective fleets.\n\nEre all the guns had been fired from the flagship, the third classmen\nreceived the rising signal; the class marched out and was dismissed.\nInstantly a break was made for deck.\n\nThe midshipmen were in good time to see the smoke and hear the roar of\nguns from one of the forts on shore.\n\nIn the morning the commandant of cadets, as commanding officer of the\nsquadron, would go ashore with his aide and pay a formal call to the\nsenior military officer. Later in the day that English officer and one\nor two of his staff officers would return the call by coming out to the\nflagship. That accomplished, all the required courtesies would have been\nobserved.\n\nIt was still broad daylight, for in summer the English twilight is a\nlong one, and darkness does not settle down until late.\n\n\"Oh, if we were only going ashore to-night!\" murmured Hallam. There were\nmany others to echo the thought, but all knew that it could not be done.\n\n\"Couldn't we find a trick for slipping ashore after lights out?\" eagerly\nqueried Dickey, who was not noted as a \"greaser.\"\n\n\"Could we?\" quivered Hallam, who, with few demerits against him, felt\ninclined to take a chance.\n\nBut Pennington, to whom he appealed, shook his head.\n\n\"Too big a risk, Hally,\" replied Pen. \"And trebly dangerous, with that\ngreaser, Darrin, in the class.\"\n\n\"Oh, stow that,\" growled Hallam. \"Darrin is no greaser. You've got him\non your black books--that's all.\"\n\n\"He is a greaser, I tell you,\" cried Pennington fiercely.\n\nThere were a score of midshipmen in this group, and many of them nodded\napprovingly at Pennington's statement. Though still a class leader, Dave\nhad lost some of his popularity since his report to the police of\nAnnapolis.\n\nSo the middies turned in, that night, with unsatisfied dreams of shore\nlife in England.\n\nSoon after breakfast the next morning, however, every midshipman had\ndrawn his ten dollars, even to Pennington, who had no use for such a\ntrifling amount.\n\nAs fast as possible the launches ranged alongside at the side gangway,\ntaking off groups of midshipmen, everyone of whom had been cautioned to\nbe at dock in time to board a launch in season for supper formation.\n\nPennington and his party were among the first to land. They hurried\naway.\n\nIt was on the second trip of one of the launches that Dave, Dan and\nFarley made their get away. These three chums had agreed to stick\ntogether during the day. They landed at the Great Western Docks, to find\nthemselves surrounded by eager British cabbies.\n\n\"Are we going to take a cab and get more quickly and intelligently to\nthe best part of the town to see?\" asked Farley.\n\n\"I don't vote for it,\" replied Darrin. \"We have only five dollars apiece\nfor each of the two days we're to be ashore. I move that we put in the\nforenoon, anyway, in prowling about the town for ourselves. We'll learn\nmore than we would by riding.\"\n\n\"Come on, then,\" approved Dan.\n\nPlymouth is an old-fashioned English seaport that has been rather famous\never since the thirteenth century. Many parts of the town, including\nwhole streets, look as though the houses had been built since that time.\nThis is especially true of many of the streets near the water front.\n\nFor two hours the three middies roamed through the streets, often\nmeeting fellow classmen. Wherever the young midshipmen went many of the\nEnglish workmen and shopkeepers raised their hats in friendly salute of\nthe American uniform.\n\n\"We don't seem to run across Pen's gang anywhere,\" remarked Farley at\nlast.\n\n\"Oh, no,\" smiled Dave. \"That's a capitalistic crowd. They'll hit only\nthe high spots.\"\n\nNevertheless, these three poor-in-purse midshipmen enjoyed themselves\nhugely in seeing the quaint old town. At noon they found a real old\nEnglish chop house, where they enjoyed a famous meal.\n\n\"I wish we could slip some of these little mutton pies back with us!\"\nsighed Dan wistfully.\n\nIn the afternoon the three chums saw the newer market place, where all\nthree bought small souvenirs for their mothers at home. Darrin also\nsecured a little remembrance present for his sweetheart, Belle Meade.\n\nThe guild hall and some of the other famous buildings were visited.\n\nLater in the afternoon Dave began to inspect his watch every two or\nthree minutes.\n\n\"No need for us to worry, with Dave's eye glued to his watch,\" laughed\nDan.\n\n\"Come on, fellows,\" summoned Darrin finally. \"We haven't more than time\nnow to make the dock and get back to supper formation.\"\n\n\"Take a cab?\" asked Farley. \"You know, we've found that they're vastly\ncheaper than American cabs.\"\n\n\"No-o-o, not for me,\" decided Dave. \"We'll need the rest of our shore\nmoney to-morrow, and our legs are good and sturdy.\"\n\nYet even careful Dave, as it turned out, had allowed no more than time.\nThe chums reached the dock in time to see the launches half way between\nthe fleet and shore. Some forty other midshipmen stood waiting on the\ndock.\n\nAmong these were Pennington and his party, all looking highly satisfied\nwith their day's sport, as indeed they were.\n\nPennington's eyes gleamed when he caught sight of Darrin, Dalzell and\nFarley--for Pen had a scheme of his own in mind.\n\nNot far from Pennington stood a little Englishman with keen eyes and a\njovial face. Pen stepped over to him.\n\n\"There are the three midshipmen I was telling you about,\" whispered\nPennington, slipping a half sovereign into the Englishman's hand. \"You\nthoroughly understand your part in the joke, don't you?\"\n\n\"Don't h'I, though--just, sir!\" laughed the undersized Englishman, and\nstrolled away.\n\nDarrin and his friends were soon informed by classmates that the\nlaunches now making shore-ward were coming in on their last trip for\nmidshipmen.\n\n\"Well, we're here in plenty of time,\" sighed Dave contentedly.\n\n\"Oh, I knew we'd be, with you holding the watch,\" laughed Dan in his\nsatisfied way.\n\nAs the three stood apart they were joined by the undersized Englishman,\nwho touched his hat to them with a show of great respect.\n\n\"Young gentlemen,\" he inquired, \"h'I suppose, h'of course, you've 'ad a\nlook h'at the anchor h'of Sir Francis Drake's flagship, the time 'e went\nh'out h'and sank the great Spanish h'Armada?\"\n\n\"Why, no, my friend,\" replied Dave, looking at the man with interest.\n\"Is that here at Plymouth?\"\n\n\"H'assuredly, sir. H'and h'only a minute's walk h'over to that shed\nyonder, sir. H'if you'll come with me, young gentlemen, h'I'll show h'it\nto you. H'it's one of h'our biggest sights, h'and it's in me own\ncustody, at present. Come this way, young gentlemen.\"\n\n\"That sounds like something worth seeing,\" declared Dave to his\ncomrades. \"Come along. It'll take the launches at least six minutes to\nget in, and then they'll stay tied up here for another five minutes.\"\n\nWith only a single backward glance at the young midshipmen, the\nundersized Englishman was already leading the way.\n\nAt quickened pace the young midshipmen reached the shed that had been\nindicated. Their guide had already drawn a key from a pocket, and had\nunsnapped the heavy padlock.\n\n\"Step right in, young gentlemen, h'and h'I'll follow h'and show h'it to\nyou.\"\n\nUnsuspecting, the three middies stepped inside the darkened shed.\nSuddenly the door banged, and a padlock clicked outside.\n\n\"Here, stop that, you rascally joker!\" roared Dalzell, wheeling about.\n\"What does this mean?\"\n\n\"Big trouble!\" spoke Dave Darrin seriously and with a face from which\nthe color was fast receding.\n\n\n\n\nCHAPTER VII\n\n\nPENNINGTON GETS HIS WISH\n\n\"The scoundrel!\" gasped Farley, his face whiter than any of the others.\n\nDave was already at the door, trying to force it open. But he might\nalmost as well have tried to lift one of the twelve-inch guns of the\nbattleship \"Massachusetts.\"\n\n\"We're locked in--that's sure!\" gasped Dalzell, almost dazed by the\ncatastrophe.\n\n\"And what's more, we won't get out in a hurry, unless we can make some\nof our classmates hear,\" declared Dave.\n\nFor the next half minute they yelled themselves nearly hoarse, but no\nresponse came.\n\n\"What could have been that little cockney's purpose in playing this\nshabby trick on us?\" demanded Farley.\n\n\"Perhaps the cockney thinks we're admirals, with our pockets lined with\ngold. Perhaps he and some of his pals intend to rob us, later in the\nevening,\" proposed Dan, with a ghastly grin.\n\n\"Any gang would find something of a fight on their hands, then,\"\nmuttered Dave Darrin grimly.\n\nAll three were equally at a loss to think of any explanation for such a\n\"joke\" as this. Equally improbable did it seem that any thugs of the\ntown would expect to reap any harvest from robbing three midshipmen.\n\nDesperately they turned to survey their surroundings. The shed was an\nold one, yet strongly built. There were no windows, no other door save\nthat at which the three middies now stood baffled.\n\n\"Another good old yell,\" proposed Darrin.\n\nIt was given with a lusty will, but proved as fruitless as the former\none.\n\n\"We don't take the last launch back to ship,\" declared Farley, wild with\nrage.\n\n\"Which means a long string of demerits,\" said Dan.\n\n\"No shore leave to-morrow, either,\" groaned Darrin. \"Fellows, this\nmishap will affect our shore leave throughout all the cruise.\"\n\n\"We can explain it,\" suggested Farley with a hopefulness that he did not\nfeel at all.\n\n\"Of course we can,\" jeered Dave Darrin. \"But what officer is fool enough\nto believe such a cock-and-bull story as this one will seem? At the very\nleast, the commandant would believe that we had been playing some pretty\nstiff prank ourselves, in order to get treated in this fashion. No, no,\nfellows! We may just as well undeceive ourselves, and prepare to take\nthe full soaking of discipline that we're bound to get. If we attempted\nthis sort of explanation, we'd be lucky indeed to get through the affair\nwithout being tried by general court-martial for lying.\"\n\n\"Drake's anchor, indeed!\" exclaimed Dan in deep self disgust.\n\n\"We ought to have known better,\" grunted Farley, equally enraged with\nhimself. \"What on earth made us so absent-minded as to believe that a\npriceless relic would be kept in an old shed like this?\"\n\n\"We're sure enough idiots!\" groaned Dan.\n\n\"Hold on there, fellows,\" interrupted Dave Darrin. \"Vent all your anger\nright on me. I'm the great and only cause of this misfortune. It was I\nwho proposed that we take up that cockney's invitation. I'm the real and\nonly offender against decent good sense, and yet you both have to suffer\nwith me.\"\n\n\"Let's give another yell, bigger than before,\" suggested Dan weakly.\n\nThey did, but with no better result than before.\n\n\"The launches are away now, anyway, I guess,\" groaned Farley, after\nconsulting his watch.\n\n\"Yes, and we're up the tree with the commandant,\" grunted Dalzell\nbitterly.\n\n\"Yell again?\" asked Farley.\n\n\"No,\" retorted Dave, shaking his head. \"We've seen the uselessness of\nasking help from outside. Let's supply our own help. Now,\nthen--altogether! Shoulder the door!\"\n\nA savage assault they hurled upon the door. But they merely caused it to\nvibrate.\n\n\"We can't do it,\" gasped Dan, after the third trial.\n\nConsiderable daylight filtered in through the cracks at top, bottom and\none side of the door. Further back in the shed there was less light.\n\n\"Let's explore this old place in search of hope,\" begged Dave.\n\nTogether they started back, looking about keenly in what appeared to be\nan empty room.\n\n\"Say! Look at that!\" cried Dave suddenly.\n\nHe pointed to a solid looking, not very heavy ship's spar.\n\n\"What good will that thing do us?\" asked Farley rather dubiously.\n\n\"Let's see if we can raise it to our shoulders,\" proposed Dave Darrin\nradiantly. \"Then well find out!\"\n\n\"Hurrah!\" quivered Dan Dalzell, bending over the spar at the middle.\n\n\"Up with it!\" commanded Darrin, placing himself at the head of the spar.\nFarley took hold at the further end.\n\n\"Up with it!\" heaved Midshipman Darrin.\n\nRight up the spar went. It would have been a heavy job for three young\nmen of their size in civil life, but midshipmen are constantly\nundergoing the best sort of physical training.\n\n\"Now, then--a fast run and a hard bump!\" called Darrin.\n\nAt the door they rushed, bearing the spar as a battering ram.\n\nBump! The door shook and shivered.\n\n\"Once more may do it!\" cheered Darrin. \"Back.\"\n\nAgain they dashed the head of their battering ram against the door. It\ngave way, and, climbing through, they raced back to the pier.\n\nBut Dan, who had secured the lead, stopped with a groan, pointing out\nover the water.\n\n\"Not a bit of good, fellows! There go the launches, and we're the only\nfellows left! It's all up with our summer's fun!\"\n\n\"Is it, though?\" shouted Dave, spurting ahead. \"Come on and find out!\"\n\nAs they reached the front of the piers, down at the edge of a landing\nstage they espied a little steam tender.\n\n\"That boat has to take us out to the 'Massachusetts'!\" cried Darrin\ndesperately, as he plunged down the steps to the landing stage, followed\nby his two chums.\n\n[Illustration: The Three Midshipmen Raced Toward the Pier.]\n\n\"Who's the captain here?\" called Dave, racing across the landing stage\nto the tender's gangplank.\n\n\"I am, sir,\" replied a portly, red-faced Englishman, leaning out of the\nwheel-house window.\n\n\"What'll you charge to land us in haste aboard the American battleship\n'Massachusetts'?\" asked Darrin eagerly.\n\n\"Half a sov. will be about right, sir,\" replied the tender's skipper,\ntouching his cap at sight of the American Naval uniform.\n\n\"Good enough,\" glowed Dave, leaping aboard. \"Cast off as quickly as you\ncan, captain, or we'll be in a heap of trouble with our discipline\nofficers.\"\n\nThe English skipper was quick to act. He routed out two deckhands, who\nquickly cast off. Almost while the deckhands were doing this the skipper\nrang the engineer's bell.\n\n\"Come into the wheel-'ouse with me,\" invited the skipper pleasantly,\nwhich invitation the three middies accepted. \"Now, then, young\ngentlemen, 'ow did it 'appen that you missed your own launches.\"\n\n\"It was a mean trick--a scoundrelly one!\" cried Darrin resentfully. Then\nhe described just what had happened.\n\nThe skipper's own bronzed cheeks burned to a deeper color.\n\n\"I can 'ardly believe that an Englishman would play such a trick on\nyoung h'officers of a friendly power,\" he declared. \"But I told you,\nsir, the fare out to your ship would be half a sov. I lied. If a nasty\nlittle cockney played such a trick on you, it's my place, as a decent\nEnglishman, to take you out for nothing--and that's the fare.\"\n\n\"Oh, we'll gladly pay the half sov.\" protested Darrin.\n\n\"Not on this craft you can't, sir,\" replied the skipper firmly.\n\nLooking eagerly ahead, the three middies saw two of the launches go\nalong side of the \"Massachusetts\" and discharge passengers. As the\nsecond left the side gangway the Briton, who had been crowding on steam\nwell, ranged in along side.\n\n\"What craft is that, and what do you want?\" hailed the officer of the\ndeck, from above.\n\n\"The tender 'Lurline,' sir, with three of your gentlemen to put h'aboard\nof you, sir,\" the Briton bellowed through a window of the wheel-house.\n\n\"Very good, then. Come alongside,\" directed the officer of the deck.\n\nIn his most seamanlike style the Briton ranged alongside. Dave tried to\npress the fare upon the skipper, but he would have none of that. So the\nthree shook hands swiftly but heartily with him, then sprang across to\nthe side gangway, where they paused long enough to lift their caps to\nthis stranger and friend. The Briton lifted his own cap, waving it\nheartily, ere he fell off and turned about.\n\n\"You didn't get aboard any too soon, gentlemen,\" remarked the officer of\nthe deck, eyeing the three middies keenly as they came up over the side,\ndoffing their uniform caps to the colors. \"Hustle for the formation.\"\n\nMidshipman Pennington was chuckling deeply over the supposed fact that\nhe had at last succeeded in bringing Darrin in for as many demerits as\nDarrin had helped heap upon him.\n\n\"That'll break his heart as an avowed greaser,\" Pen told himself. \"With\nall the demerits Darrin will get, he'll have no heart for greasing the\nrest of this year. It's rough on Farley, but I'm not quite as sorry for\nDalzell, who, in his way, is almost as bad as Darrin. He's Darrin's\ncuckoo and shadow, anyway. Oh, I wish I could see Darrin's face now!\"\n\nThis last was uttered just as Midshipman Pennington stepped into line at\nthe supper formation.\n\n\"I wish I could see Darrin's face now!\" Pen repeated to himself.\n\nSeldom has a wish been more quickly gratified. For, just in the nick of\ntime to avoid being reported, Midshipmen Darrin, Dalzell and Farley came\ninto sight, falling into their respective places.\n\nAt that instant it was Midshipman Pennington's face, not Dave Darrin's,\nthat was really worth studying.\n\n\"Now how did the shameless greaser work this!\" Pennington pondered\nuneasily.\n\nBut, of course, he couldn't ask. He could only hope that, presently, he\nwould hear the whole story from some other man in the class.\n\n\n\n\nCHAPTER VIII\n\n\nTHE TRAGEDY OF THE GALE\n\nThere is altogether too much to the summer practice cruise for it to be\nrelated in detail.\n\nNor would the telling of it prove interesting to the reader. When at\nsea, save on Sundays, the midshipman's day is one of hard toil.\n\nIt is no life for the indolent young man. He is routed out early in the\nmorning and put at hard work.\n\nOn a midshipman's first summer cruise what he learns is largely the work\nthat is done by the seamen, stokers, water tenders, electricians, the\nsignal men and others.\n\nYet he must learn every phase of all this work thoroughly, for some day,\nbefore he becomes an officer, he must be examined as to his knowledge of\nall this great mass of detail.\n\nIt is only when in port that some relaxation comes into the midshipman's\nlife. He has shore leave, and a large measure of liberty. Yet he must,\nat all times, show all possible respect for the uniform that he wears\nand the great nation that he represents. If a midshipman permits himself\nto be led into scrapes that many college boys regard as merely \"larks,\"\nhe is considered a disgrace to the Naval service.\n\nAlways, at home and abroad, the \"middy\" must maintain his own dignity\nand that of his country and service. Should he fail seriously, he is\nregarded by his superiors and by the Navy Department as being unfit to\ndefend the honor of his flag.\n\nThe wildest group from the summer practice fleet was that made up of\nPennington and his friends. Pen received more money in France from his\nfond but foolish father. Wherever Pennington's group went, they cut a\nwide swath of \"sport,\" though they did nothing actually dishonorable.\nYet they were guilty of many pranks which, had the midshipmen been\ncaught, would have resulted in demerits.\n\nPorts in France, Spain, Portugal and Italy were touched briefly. At some\nof these ports the midshipmen received much attention.\n\nBut at last the fleet turned back past Gibraltar, and stood on for the\nAzores, the last landing point before reaching home.\n\nWhen two nights out from Gibraltar a sharp summer gale overtook the\nfleet. Even the huge battleships labored heavily in the seas, the\n\"Massachusetts\" bringing up the rear.\n\nShe was in the same position when the morning broke. The midshipmen,\nafter breakfast, enjoyed a few minutes on the deck before going below\nfor duty in the engine rooms, the dynamo room, the \"stoke hole\" and\nother stations.\n\nSuddenly, from the stern rail, there went up the startled cry:\n\n\"Man overboard!\"\n\nIn an instant the marine sentry had tumbled two life-preservers over\ninto the water.\n\nWith almost the swiftness of telegraphy the cry had reached the bridge.\nWithout stopping to back the engine the big battleship's helm was thrown\nhard over, and the great steel fighting craft endeavored to find her own\nwake in the angry waters with a view to going back over it.\n\nSignal men broke out the news to the flagship. The other two great\nbattleships turned and headed back in the interests of humanity.\n\nIt seemed almost as though the entire fleet had been swung out of its\ncourse by pressure on an electric button.\n\nOfficers who were not on duty poured out. The captain was the first to\nreach the quarter-deck. He strode into the midst of a group of\nstricken-looking midshipmen.\n\n\"Who's overboard!\" demanded the commanding officer.\n\n\"Hallam, sir----\"\n\n\"And Darrin, sir----\"\n\n\"And Dalzell, sir----\"\n\n\"How many?\" demanded the captain sharply.\n\n\"Three, sir.\"\n\n\"How did so many fall overboard?\"\n\n\"Mr. Hallam was frolicking, sir,\" reported Midshipman Farley, \"and lost\nhis footing.\"\n\n\"But Mr. Darrin and Mr. Dalzell?\" inquired the captain sharply.\n\n\"As soon as they realized it, sir, Darrin and Dalzell leaped overboard\nto go to Hallam's rescue, sir.\"\n\n\"It's a wonder,\" muttered the captain, glancing shrewdly at the bronzed,\nfine young fellows around him, \"that not more of you went overboard as\nwell.\"\n\n\"Many of them would, sir,\" replied Farley, \"but an officer forward\nshouted: 'No more midshipmen go overboard,' So we stopped, sir.\"\n\nModest Mr. Farley did not mention the fact that he was running toward\nthe stern, intent on following his chums into the rough sea at the very\ninstant when the order reached him.\n\nThe captain, however, paused for no more information. He was now running\nforward to take the bridge beside the watch officer.\n\nThe midshipmen, too, hurried forward, mingling with the crew, as the big\nbattleship swung around and tried to find her wake.\n\nThe flagship had crowded on extra steam, and was fast coming over the\nseas.\n\nWith such a sea running, it was well nigh impossible to make out so\nsmall a thing as a head or a life-preserver, unless it could be observed\nat the instant when it crested a wave.\n\nMarine glasses were in use by every officer who had brought his pair to\nthe deck. Others rushed back to their cabins to get them.\n\nA lieutenant of the marine corps stood forward, close to a big group of\nsorrowing midshipmen.\n\n\"There are certain to be three vacancies in the Naval Academy,\" remarked\nthe lieutenant.\n\n\"Don't say that, sir,\" begged Farley, in a choking voice. \"The three\noverboard are among the finest fellows in the brigade!\"\n\n\"I don't want to discourage any of you young gentlemen,\" continued the\nmarine corps lieutenant. \"But there's just about one chance in a\nthousand that we shall be able to sight and pick up any one of the\nunlucky three. In the first place, it would take a wonderful swimmer to\nlive long in such a furious sea. In the second place, if all three are\nstill swimming, it will be almost out of the question to make out their\nheads among the huge waves. You've none of you seen a man overboard\nbefore in a big sea?\"\n\nSeveral of the mute, anxious midshipmen shook their heads.\n\n\"You'll realize the difficulties of the situation within the next few\nminutes,\" remarked the lieutenant. \"I am sorry to crush your hopes for\nyour classmates, but this is all a part of the day's work in the Navy.\"\n\nThe largest steam launches from all three of the battleships were being\nswiftly lowered. Officers and men were lowered with the launches. As the\nlaunch shoved off from each battleship tremendous cheers followed them.\n\n\"Stop all unnecessary noise!\" bellowed the watch officer from the bridge\nof the \"Massachusetts.\" \"You may drown out calls for help with your\nracket.\"\n\nWhile the three battleships went back over their courses in more stately\nfashion, the launches darted here and there, until it seemed as though\nthey must cover every foot within a square mile.\n\n\"I don't see how they can help finding the three,\" Farley declared\nhopefully.\n\n\"That is,\" put in another third classman, \"if any of the three are still\nafloat.\"\n\n\"Stow all talk of that sort,\" ordered Farley angrily.\n\nOther midshipmen joined in with their protests. When a man is overboard\nin an angry sea all hands left behind try to be optimists.\n\nWhen fifteen minutes had been spent in the search the onlooking but\nhelpless middies began to look worried.\n\nAt the end of half an hour some of them looked haggard. Farley's face\nwas pitiable to see.\n\nAt the end of an hour of constant but fruitless searching hardly any one\nfelt any hope of a rescue now.\n\nAll three midshipmen, the \"man overboard\" and his two willing, would-be\nrescuers, were silently conceded to be drowned.\n\nYet the hardest blow of all came when, at the end of an hour and a\nquarter, the flagship signaled the recall of the small boats.\n\nThen, indeed, all hope was given up. In an utter human silence, save for\nthe husky voicing of the necessary orders, the launches were hoisted on\nboard. Then the flagship flew the signal for resuming the voyage.\n\nThere were few dry eyes among the third class midshipmen when the\nbattleships fell in formation again and proceeded on their way.\n\nAs a result of more signals flown from the flagship, all unnecessary\nduties of midshipmen for the day were ordered suspended.\n\nIn the afternoon the chaplain on each battleship held funeral services\nover the three lost midshipmen. Officers, middies and crew attended on\nboard each vessel.\n\n\n\n\nCHAPTER IX\n\n\nTHE DESPAIR OF THE \"RECALL\"\n\nDave Darrin stood within ten feet of Hallam when that latter midshipman\nhad lost his balance and fallen into the boiling sea.\n\nDave's spring to the stern rail was all but instantaneous. He was\noverboard, after his classmate, ere the marine had had time to leap to\nthe life buoys.\n\nOut of the corner of one eye Dan Dalzell saw the marine start on the\njump, but Dan was overboard, also, too soon to see exactly what the\nmarine sentry was doing.\n\nBoth daring midshipmen sank beneath the surface as they struck.\n\nAs Dan came up, however, his hand struck something solid and he clutched\nat it. It was one of the life buoys.\n\nAs he grasped it, and drew his head up a trifle, Dan saw another\nfloating within thirty feet of him. Swimming hard, and pushing, Dan\nsucceeded in reaching the other buoy. He now rested, holding on to both\nbuoys.\n\n\"Now, where's David, that little giant?\" muttered Dalzell, striving hard\nto see through the seething waters and over the tops of foam-crested\nwaves.\n\nAfter a few minutes Dan began to feel decidedly nervous.\n\n\"Yet Dave can't have gone down, for he's a better swimmer than I am,\"\nwas Dan's consoling thought.\n\nAt last Dalzell caught sight of another head. He could have cheered, but\nhe expended his breath on something more sensible.\n\n\"Dave!\" he shouted. \"Old Darry! This way! I have the life buoys.\"\n\nAt the same time, holding to both of them, but kicking frantically with\nhis feet, Dalzell managed slowly to push the buoys toward Dave.\n\nSoon after he had started, Dan did utter a cheer, even though it was\nchecked by an inrush of salt water that nearly strangled him.\n\nHe saw two heads. Dave Darrin was coming toward him, helping Hallam.\n\nThe wind carried the cheer faintly to Dave. He raised his head a little\nin the water, and caught sight of Dan and the buoys.\n\nSome three minutes it took the two chums to meet. Dave Darrin was all\nbut exhausted, for Hallam was now unconscious.\n\nAs Darrin clutched at the buoy he tried to shout, though the voice came\nweakly:\n\n\"Catch hold of Hallam. I'm down and----\"\n\nBut Dan understood, even before he heard. While Dave clutched at one of\nthe life buoys Dalzell shot out an arm, dragging Hallam in to safety.\n\nNow, it was Darrin who, with both arms, contrived to link the buoys\ntogether.\n\nAt last the youngsters had a chance to observe the fact that the\nbattleships had put about and were coming back.\n\n\"We'll soon be all right,\" sighed Dave contentedly, as soon as he could\nspeak. \"There are thirty-five hundred officers, middies and sailors of\nthe American Navy to look after our safety.\"\n\nFrom where they lay as they hung to the buoys the chums could even see\nthe launches lowered.\n\nDan, with some of the emergency lashing about the buoy, succeeded, after\na good deal of effort, and with some aid from Dave, in passing a cord\nabout Hallam and under the latter's armpits that secured that midshipman\nto one of the buoys. The next move of the chums was to lash the buoys\ntogether.\n\n\"Now,\" declared Dave, \"we can't lose. We can hang on and be safe here\nfor hours, if need be.\"\n\n\"But what a thundering long time it takes them to bring the battleships\naround to get to us!\" murmured Midshipman Dalzell in wonder.\n\n\"Be sure not an unnecessary second has been lost,\" rejoined Dave. \"We're\nlearning something practical now about the handling of big craft.\"\n\n\"I wonder if Hally's a goner?\" murmured Dan in an awe-struck voice.\n\n\"I don't believe it,\" Dave answered promptly. \"Once we get him back\naboard ship the medicos will do a little work over him and he'll sit up\nand want to know if dinner's ready.\"\n\nThen they fell silent, for, with the roar of wind and waters, it was\nnecessary for them to shout when they talked.\n\nAs the minutes went by slowly, the two conscious midshipmen found\nthemselves filled with amazement.\n\nA dozen times the launches darted by, not far away. It seemed impossible\nthat the keen, restless eyes of the seekers should not discover the\nimperiled ones.\n\nAt such times Dave and Dan shouted with all the power of their lusty\nyoung lungs.\n\nAlternately Dan and Dave tried the effect of rising as far as they could\nand frantically waving an arm. There was not a cap to wave among the\nthree of them.\n\n\"I'm beginning to feel discouraged,\" grunted Dave in disgust at last.\n\"They must have spent a full half day already looking for us.\"\n\n\"Merciful powers!\" gasped Dan at last, as they rode half way up the\n<DW72> of a big wave. \"I just caught sight of the 'recall of boats'\nflying from the flagship!\"\n\n\"No!\" gasped Dave incredulously.\n\n\"Yes, I did!\"\n\n\"But--\"\n\n\"They've failed and have given up the search,\" spoke Dan rather\ndespairingly.\n\n\"But--\"\n\n\"We may as well face it,\" muttered Dan brokenly. \"They don't believe\nthat any of us has survived, and we've been abandoned.\"\n\n\"Then,\" spoke Dave Darrin very coolly, \"there's nothing left for us but\nto die like men of the American Navy.\"\n\n\"It seems heartless, needless,\" protested Dan.\n\n\"No,\" broke in Darrin. \"They've done their best. They're convinced that\nwe're lost. And I should think they would be, after all the time they've\nsearched for us--half a day, at least.\"\n\nDan said nothing, but tugged until he succeeded in bringing his watch up\nto the light.\n\n\"The blamed thing is water-logged,\" he uttered disgustedly.\n\n\"Why?\"\n\n\"The hands point to less than half past nine!\"\n\nDarrin managed to get at his own watch.\n\n\"My timepiece doesn't call for half past nine, either,\" he announced.\n\n\"Can it be possible--\"\n\n\"Yes; the time has only seemed longer, I reckon,\" observed Dalzell.\n\n\"Well, we'll face it like men,\" proposed Dave.\n\n\"Of course,\" nodded Dan. \"At least, we're going down in the ocean, and\nwe wear the American Naval uniform. If there's any choice in deaths, I\nguess that's as good and manly a one as we could choose.\"\n\n\"Poor old Hally won't know much about it, anyway, I guess,\" remarked\nDarrin, who seemed unnaturally cool. Possibly he was a bit dazed by the\nstunning nature of the fate that seemed about to overtake them.\n\n\"Maybe the ships will go by us in their final get-away,\" proposed Dan\nDalzell very soberly.\n\n\"Not if I'm seaman enough to read the compass by what's visible of the\nsun,\" returned Midshipman Darrin.\n\n\"Then there's no help for it,\" answered Dan, choking slightly. \"I wonder\nif we could do anything for Hallam?\"\n\n\"We won't do anything to bring him to, anyway,\" muttered Darrin. \"Under\nthese circumstances I wouldn't do anything as mean as that to a dog!\"\n\n\"Maybe he's dead already, anyway,\" proposed Dan, now hopefully.\n\n\"I hope so,\" came from Darrin.\n\nNow they saw the not very distant battleships alter their courses and\nsteam slowly away.\n\nAll was now desolation over the angry sea, as the battleships gradually\nvanished. The two conscious midshipmen were now resolved to face the end\nbravely. That was all they could do for themselves and their flag.\n\n\n\n\nCHAPTER X\n\n\nTHE GRIM WATCH FROM THE WAVES\n\nBy the time that little more than the mastheads of the departing\nbattleships were visible, Hallam opened his eyes.\n\nIt would have seemed a vastly kinder fate had he been allowed to remain\nunconscious to the last.\n\nHallam had not been strangled by the inrush of water. In going\noverboard, this midshipman had struck the water with the back of his\nhead and had been stunned. In the absence of attention he had remained a\nlong time unconscious.\n\nEven now the hapless midshipman whose frollicking had been the cause of\nthe disaster, did not immediately regain his full senses.\n\n\"Why, we're all in the water,\" he remarked after a while.\n\n\"Yes,\" assented Darrin, trying to speak cheerfully.\n\nMidshipman Hallam remained silent for some moments before he next asked:\n\n\"How did it happen?\"\n\n\"Fell overboard,\" replied Dan laconically, failing to mention who it was\nwho had fallen over the stern.\n\nAgain a rather long silence on Hallam's part. Then, at last, he\nobserved:\n\n\"Funny how we all fell over at the same time.\"\n\nTo this neither of his classmates made any rejoinder.\n\n\"See here,\" shouted Hallam, after a considerable period of silent\nwondering, \"I remember it all now. I was fooling at the stern rail and I\ntoppled overboard.\"\n\nDan nodded without words.\n\n\"And you fellows jumped in after me,\" roared Hallam, both his mental and\nbodily powers now beginning to return. \"Didn't you?\"\n\n\"Of course,\" assented Darrin rather reluctantly.\n\n\"And what became of the fleet!\"\n\nDave and Dan looked at each other before the former replied:\n\n\"Oh, well, Hally, brace up! The ships searched for us a long time, and\nsome launches were put out after us. But they couldn't see our little\nheads above the big waves, and so----\"\n\n\"They've gone away and left us?\" queried Hallam, guessing at once. \"Now,\nfellows, I don't mind so much for myself, but it's fearful to think that\nI've dragged you into the same fate. It's awful! Why couldn't you have\nleft me to my fate?\"\n\n\"Would you have done a thing like that?\" demanded Dave dryly.\n\n\"Oh, well, I suppose not, but--but--well, I wish I had been left to pay\nthe price of my tomfoolery all alone. It would have served me right. But\nto drag you two into it--\"\n\nHallam could go no further. He was choking up with honest emotion.\n\n\"Don't bother about it, Hally,\" urged Dave. \"It's all in the day's work\nfor a sailor. We'll just take it as it comes, old fellow.\"\n\nTo not one of the trio did it occur to let go of the life buoys and sink\nas a means of ending misery. In the first place, human instinct holds to\nhope. In the second place, suicide is the resort of cowards.\n\n\"None of you happened to hide any food in his pockets at breakfast, I\ntake it?\" asked Dan grimly, at last.\n\nOf course they hadn't.\n\n\"Too bad,\" sighed Dan. \"I'm growing terribly hungry.\"\n\n\"Catch a fish,\" smiled back Darrin.\n\n\"And eat it raw?\" gasped Dalzell. \"Darry, you know my tastes better than\nthat.\"\n\n\"Then wait a few hours longer,\" proposed Dave, \"until even raw fish will\nbe a delicacy.\"\n\nHallam took no part in the chaffing. He was miserably conscious, all the\nwhile, that his own folly had been solely responsible for the present\nplight of these noble messmates.\n\nThus the time passed on. None kept any track of it; they realized only\nthat it was still daylight.\n\nThen suddenly Dave gave a gasp and raised one hand to point.\n\nHis two classmates turned and were able to make out the mastheads of a\ncraft in the distance.\n\nHow they strained their eyes! All three stared and stared, until they\nfelt tolerably certain that the craft was headed their way.\n\n\"They may see us!\" cried Hallam eagerly.\n\n\"Three battleships and as many launches failed to find us,\" retorted\nDan. \"And they were looking for us, too.\"\n\nAs the vessel came nearer and the hull became visible, it took on the\nappearance of a liner.\n\n\"Why, it looks as though she'd run right over us when she gets nearer,\"\ncried Dave, his eyes kindling with hope.\n\n\"Don't get too excited over it,\" urged Dan. \"For my part, I'm growing\nalmost accustomed to disappointments.\"\n\nAs the minutes passed and the liner came on and on, it looked still more\nas though she would run down the three middies.\n\n[Illustration: \"Look! They See Us!\"]\n\nAt last, however, the craft was passing, showing her port side, not very\nfar distant, to be sure.\n\nUniting their voices, the three midshipmen yelled with all their power,\neven though they knew that their desperate call for help could not carry\nthe distance over the subsiding gale.\n\nBoom! That shot came from the liner, and now her port rail was black\nwith people.\n\n\"They see us!\" cried Hallam joyously. \"Look! That craft is slowing up!\"\n\nOnce more came the cheers of encouragement, as the liner, now some\ndistance ahead, put off a heavy launch. A masthead lookout, who had\nfirst seen the midshipmen, was now signaling the way to the officer in\ncommand of the launch.\n\nUnable to see for himself, the officer in the launch depended wholly on\nthose masthead signals. So the launch steamed a somewhat zig-zag course\nover the waves. Yet, at last, it bore down straight upon the midshipmen.\n\nDarrin, Dalzell and Hallam now came very near to closing their eyes, to\nlessen the suspense.\n\nA short time more and all three were dragged in over the sides of the\nlaunch.\n\n\"Get those life buoys in, if you can,\" begged Dave, as he sank in the\nbottom of the launch. \"They are United States property entrusted to our\ncare.\"\n\nFrom officer and seamen alike a laugh went up at this request, but the\nlife buoys were caught with a boathook and drawn aboard.\n\nWhat rousing cheers greeted the returning launch, from the decks of the\nliner, \"Princess Irene\"! When the three midshipmen reached deck and it\nwas learned that they were midshipmen of the United States Navy, the\ncheering and interest were redoubled.\n\nBut the captain and the ship's doctor cut short any attempt at lionizing\nby rushing the midshipmen to a stateroom containing three berths. Here,\nunder the doctor's orders, the trio were stripped and rubbed down. Then\nthey were rolled into blankets, and hot coffee brought to them in their\nberths, while their wet clothing was sent below to one of the furnace\nrooms for hurried drying.\n\nAs soon as the medical man had examined them, the steamship's captain\nbegan to question them.\n\n\"Headed for the Azores, eh?\" demanded the ship's master. \"We ought to be\nable to sight your squadron before long.\"\n\nHe hastened out, to give orders to the deck officer.\n\nBy the time that the young midshipmen had been satisfactorily warmed,\nand their clothing had been dried, the ship's surgeon consented to their\ndressing. After this they were led to a private cabin where a satisfying\nmeal was served them.\n\n\"Oh, I don't know,\" murmured Dan, leaning back, with a contented sigh,\nafter the meal was over; \"there are worse things than what happened to\nus to-day!\"\n\nThe greater speed of the liner enabled her to sight the battleship\nsquadron something more than two hours afterward. Then the nearest\nvessel of the fleet was steered for directly.\n\nThe deck officers of the liner sent their heavy overcoats for the use of\nthe midshipmen, who, enveloped in these roomy garments, went out on deck\nto watch the pursuit of their own comrades.\n\nWithin another hour it was possible to signal, and from the \"Princess\nIrene's\" masthead the signal flags were broken out.\n\n\"Now, watch for excitement on board your own craft,\" smiled the liner's\ncommander, an Englishman.\n\nAs soon as the liner's signal had been read by the vessels of the\nsquadron a wild display of signal bunting swiftly broke out.\n\n\"Heaven be thanked!\" read one set of signal flags.\n\n\"We have officially buried the young men, but ask them to go on living,\"\nread another.\n\nWhile the most practical signal of all was:\n\n\"The 'Massachusetts' will fall astern of the squadron. Kindly stand by\nto receive her launch.\"\n\nIn a few minutes more the two vessels were close enough. Both stopped\nheadway. One of the big battleship's launches put off and steamed over,\nrolling and pitching on the waves.\n\nMost carefully indeed the three midshipmen climbed down a rope ladder\nand were received by an ensign from the \"Massachusetts,\" who next gave\nthe American Navy's profound thanks to the rescuers of the middies.\n\n\"Kindly lower that United States property that was in our care, sir!\"\nDave Darrin called up.\n\nThere was good-humored laughter above, and a look of amazement on Ensign\nWhite's face until the two buoys, attached to lines, were thrown down\nover the side.\n\n\"When your time comes you will make a very capable officer, I believe,\nMr. Darrin, judging by your care of government property,\" remarked\nEnsign White, working hard to keep down the laughter.\n\n\"I hope to do so, sir,\" Dave replied, saluting.\n\nThen away to the \"Massachusetts\" the launch bore, while the whole\nbattleship squadron cheered itself hoarse over the happy outcome of the\nday.\n\nDave, Dan and Hallam all had to do a tremendous amount of handshaking\namong their classmates when they had reached deck. Pennington was the\nonly one who did not come forward to hold his hand out to Darrin--a fact\nthat was noted at the time by many of the youngsters.\n\nTo the captain the trio recounted what had befallen them, as matter for\nofficial record.\n\n\"Mr. Darrin and Mr. Dalzell,\" announced the battleship's captain, \"I\nmust commend you both for wholly heroic conduct in going to the aid of\nyour classmate. And, Mr. Darrin, I am particularly interested in your\nincidental determination to preserve government property--the life buoys\nthat you brought back with you.\"\n\n\"It's possible I may need them again, sir,\" returned Dave, with a smile,\nthough he had no notion of prophetic utterance.\n\n\n\n\nCHAPTER XI\n\n\nMIDSHIPMAN PENNINGTON'S ACCIDENT\n\nThe stop at the Azores was uneventful. It remained in the minds of the\nmidshipmen only as a pleasant recollection of a quaint and pretty place.\n\nOnce more the squadron set sail, and now the homeward-bound pennant was\nflying. The course lay straight across the Atlantic to the entrance of\nChesapeake Bay.\n\nOn the second night out the wind was blowing a little less than half a\ngale.\n\nDarkness had fallen when Dave, Dan, Farley and several other midshipmen\ngathered to talk in low tones at the stern rail.\n\nPresently all of them wandered away but Dave. He stood close to the\nrail, enjoying the bumping motion every time the descending stern hit\none of the rolling waves.\n\nPresently, thinking he saw a light astern, he raised himself, peering\nastern.\n\nAnother group of restless middies had sauntered up. Pennington, after a\nswift look at the pacing officer in charge here, and discovering that\nthe officer's back was turned, executed a series of swift cartwheels.\n\n\"Look out, Pen!\" called Midshipman Dwight, in a low, though sharp voice.\n\nJust too late the warning came.\n\nAs Pen leaped to his feet after the last turn, one of his hands struck\nDarrin forcefully.\n\nDave swayed, tried to clutch at something, then--\n\n\"O-o-o-oh!\" rang the first startled chorus.\n\nThen, instantly, on top of it, came the rousing hail:\n\n\"Man overboard--astern!\"\n\nFarley and Hallam were the first to reach the rail. But Lieutenant\nBurton was there almost as quickly.\n\n\"Haul back!\" commanded the lieutenant sternly. \"No one go overboard!\"\n\nThat held the middies in check, for in no place, more than in the Navy,\nare orders orders.\n\nClack! was the sound that followed the first cry. Like a flash the\nmarine sentry had thrown his rifle to the deck. A single bound carried\nhim to one of the night life buoys. This he released, and hurled far\nastern.\n\nAs the night buoy struck the water a long-burning red light was fused by\ncontact. The glow shone out over the waters.\n\nIn the meantime, the \"Massachusetts's\" speed was being slowed rapidly,\nand a boat's crew stood at quarters.\n\nThe boat put off quickly, guided by the glow of the red signal light on\nthe buoy. Ere the boat reached the buoy the coxswain made out the head\nand shoulders of a young man above the rim of the floating buoy.\n\nSoon after the boat lay alongside. Dave, with the coxswain's aid, pulled\nhimself into the small craft.\n\nRecovering the buoy, the coxswain flashed the red light three times.\nFrom the deck of the battleship came a cheering yell sent up from\nhundreds of throats.\n\nIn the meantime, however, while the boat was on its way to the buoy, a\npulsing scene had been enacted on board.\n\nFarley went straight up to Midshipman Pennington.\n\n\"Sir,\" demanded Farley hotly, \"why did you push Mr. Darrin over the\nrail.\"\n\nPennington looked at his questioner as one stunned.\n\n\"I--I did push Darrin over,\" admitted Pennington, \"but it was an\naccident.\"\n\n\"An easily contrived one, wasn't it?\" demanded Midshipman Farley, rather\ncynically.\n\n\"It was pure accident,\" contended Pennington, paling. \"Until it happened\nI hadn't the least idea in the world that I was going to send Mr. Darrin\nor anyone else overboard.\"\n\n\"Huh!\" returned Farley dubiously.\n\n\"Huh!\" quoth Hallam.\n\nDan Dalzell uttered not a word, but the gaze of his eyes was fixed\nangrily on Pennington.\n\nThat latter midshipman turned as white as a sheet. His hands worked as\nthough he were attempting to clutch at something to hold himself up.\n\n\"Surely, you fellows don't believe, do you--\" he stammered weakly, then\npaused.\n\n\"One thing we did notice, the other day,\" continued Farley briskly, \"was\nthat, when Darrin was rescued from the sea and returned to us, you were\nabout the only member of the class who didn't go up to him and\ncongratulate him on his marvelous escape.\"\n\n\"How could--\"\n\n\"Mr. Pennington, I haven't the patience to talk with you now,\" rejoined\nFarley, turning on his heel.\n\nAt that moment the yell started among the midshipmen nearer the rail.\nFarley, Dan, Hallam and others joined in the yell and rushed to better\npoints of vantage.\n\nPennington tried to join in the cheer, but his tongue seemed fixed to\nthe roof of his mouth. He stood clenching and unclenching his hands, his\nface an ashen gray in his deep humiliation.\n\n\"I don't care what one or two fellows may say,\" groaned Pennington. \"But\nI don't want the class to think such things of me.\"\n\nHe was the most miserable man on board as the small boat came alongside.\nThe boat, occupants and all, was hoisted up to the davits and swung\nin-board. To the officer of the deck, who stood near-by, Dave turned,\nwith a brisk salute.\n\n\"I beg to report that I've come aboard, sir,\" Darrin uttered.\n\n\"And very glad we are of it, Mr. Darrin,\" replied the officer. \"You will\ngo to your locker, change your clothing and then report to the captain,\nsir.\"\n\n\"Aye, aye, sir.\"\n\nWith another salute, Dave hastened below, followed by Dan Dalzell, who\nwas intent on attending him.\n\nTen minutes later Dave appeared at the door of the captain's cabin. Just\na few minutes after that he came out on deck.\n\nA crowd gathered about him, expressing their congratulations.\n\n\"Thank you all,\" laughed Dave, \"but don't make so much over a middy\ngetting a bath outside of the schedule.\"\n\nTo the rear hung Pennington, waiting his chance. At last, as the crowd\nthinned, Pennington made his way up to Dave.\n\n\"Mr. Darrin, I have to apologize for my nonsense, which was the means of\npushing you overboard. It was purely accidental, on my honor. I did not\neven know it was you at the stern, nor did I realize that my antics\nwould result in pushing any one overboard. I trust you will do me the\nhonor of believing my statement.\"\n\n\"Of course I believe it, Mr. Pennington,\" answered Darrin, opening his\neyes.\n\n\"There are some,\" continued Pennington, \"who have intimated to me their\nbelief that I did it on purpose. There may be others who half believe or\nsuspect that I might, or would, do such a thing.\"\n\n\"Nonsense!\" retorted Dave promptly. \"There may be differences,\nsometimes, between classmates, but there isn't a midshipman in the Navy\nwho would deliberately try to drown a comrade. It's a preposterous\ninsult against midshipman honor. If I hear any one make a charge like\nthat, I'll call him out promptly.\"\n\n\"Some of your friends--I won't name them--insisted, or at least let me\nfeel the force of their suspicions.\"\n\n\"If any of my friends hinted at such a thing, it was done in the heat of\nthe moment,\" replied Dave heartily. \"Why, Mr. Pennington, such an act of\ndishonor is impossible to a man bred at Annapolis.\"\n\nDarrin fully believed what he said. On the spur of the moment he held\nout his hand to his enemy.\n\nPennington flushed deeply, for a moment, then put out his own hand,\ngiving Dave's a hearty, straightforward grasp.\n\n\"I was the first to imply the charge,\" broke in Farley quickly. \"I\nwithdraw it, and apologize to both of you.\"\n\nThere was more handshaking.\n\nDuring the next few days, while Darry and Pen did not become by any\nmeans intimate, they no longer made any effort to avoid each other, but\nspoke frankly when they met.\n\nThe remaining days of the voyage passed uneventfully enough, except for\na great amount of hard work that the middies performed as usual.\n\nOn the twenty-second of August they entered Chesapeake Bay. Once well\ninside, they came to anchor. There was considerable practice with the\nsub-caliber and other smaller guns. On the twenty-ninth of August the\nbattleship fleet returned to the familiar waters around Annapolis. The\nday after that the young men disembarked.\n\nThen came a hurried skeltering, for the first, second and third classmen\nwere entitled to leave during the month of September.\n\n\n\n\nCHAPTER XII\n\n\nBACK IN THE HOME TOWN\n\nBack in the old, well-known streets of their home town, Gridley!\n\nDave and Dan, enjoying every minute of their month's leave, had already\ngreeted their parents, and had told them much of their life as\nmidshipmen.\n\nWhat hurt was the fact that the skipper of the \"Princess Irene\" had\nalready told the marine reporters in New York the thrilling story of how\nDave and Dan had nearly come to their own deaths rescuing Midshipman\nHallam.\n\nEveryone in Gridley, it seemed, had read that newspaper story. Darrin\nand Dalzell, before they had been home twelve hours, were weary of\nhearing their praises sung.\n\n\"There go two of the smartest, finest boys that old Gridley ever turned\nout,\" citizens would say, pointing after Dave and Dan. \"They're\nmidshipmen at Annapolis; going to be officers of the Navy one of these\ndays.\"\n\n\"But what's the matter with Dick Prescott and Greg Holmes? They're at\nWest Point.\"\n\n\"Oh, they're all right, too, of course. But Darrin and Dalzell----\"\n\nIt was the old circumstance of being \"the lions of the minute\" and of\nbeing on the spot.\n\nOn the first morning of his arrival home Dave Darrin went frankly and\nopenly to call on his old schoolgirl sweetheart, Belle Meade.\n\nDan, having no particular associations with the gentler sex, took a\nstroll around town to meet any old friends who might care to see him\nagain.\n\nDave was shown into the parlor at the Meade home. Soon after Belle came\nswiftly in, her face beaming with delight.\n\n\"Oh, but you're not in uniform!\" was her first disappointed comment.\n\n\"No,\" smiled Dave. \"I'm allowed every possible chance, for one month, to\nforget every detail of the big grind which for a short time I've left\nbehind.\"\n\n\"But you're the same old Dave,\" cried Belle, \"only bigger and manlier.\nAnd that magnificent work you and Dan did in jumping over-bo----\"\n\n\"Stop!\" begged Dave. \"You're a friend of mine, aren't you! Then don't\nadd to the pain that has been already inflicted on me. If I had had the\nnewspapers in mind I wouldn't have the nerve to---- But please let's not\ntalk about it anymore.\"\n\nThen the two young people seated themselves and spent a delightful hour\nin talking over all that had befallen them both since they had last met.\n\nBelle, too, through Laura Bentley, had some much later news of the old\nchums, Dick and Greg, now cadets at West Point.\n\nThis news, however, will be found in full in \"DICK PRESCOTT'S SECOND\nYEAR AT WEST POINT.\"\n\n\"What are your plans for this afternoon?\" Belle asked at last.\n\n\"That's what I want your help in making,\" Dave answered.\n\n\"Can you get hold of Dan?\"\n\n\"No trouble about that. But keeping hold of him may be more difficult,\"\nlaughed Dave.\n\n\"I was going to propose that you get Dan, call here and then we'll all\ngo over to Laura Bentley's. I know she'll be anxious to see us.\"\n\n\"Nothing could be better in the way of a plan,\" assented Dave. \"I'll pin\nDanny boy down to that. It would really seem like a slight on good old\nDick if we didn't make Laura an early call.\"\n\n\"I'll go to the telephone, now, and tell her that we're coming,\" cried\nBelle, rising quickly.\n\n\"Laura is delighted,\" she reported, on her return to the room. \"But\nDave, didn't you at least bring along a uniform, so that we could see\nwhat it looks like?\"\n\n\"I didn't,\" replied Dave, soberly, then added, quizzically:\n\n\"You've seen the district messenger boys on the street, haven't you?\"\n\n\"Yes, of course; but what--\"\n\n\"Our uniforms look very much like theirs,\" declared Dave.\n\n\"I'm afraid I can't undertake to believe you,\" Belle pouted.\n\n\"Well, anyway, you girls will soon have a chance to see our uniforms.\nJust as soon as our hops start, this fall, you and Laura will come down\nand gladden our hearts by letting us drag you, won't you!\"\n\n\"Drag us?\" repeated Belle, much mystified.\n\n\"Oh, that's middies' slang for escorting a pretty girl to a midshipman\nhop.\"\n\n\"You have a lot of slang, then, I suppose.\"\n\n\"Considerable,\" admitted Dave readily.\n\n\"What, then, is your slang for a pretty girl?\"\n\n\"Oh, we call her a queen.\"\n\n\"And a girl who is--who isn't--pretty?\"\n\n\"A gold brick,\" answered Dave unblushingly.\n\n\"A gold brick?\" gasped Belle. \"Dear me! 'Dragging a gold brick' to a hop\ndoesn't sound romantic, does it?\"\n\n\"It isn't,\" Darrin admitted.\n\n\"Yet you have invited me--\"\n\n\"Our class hasn't started in with its course of social compliments yet,\"\nlaughed Dave. \"Please go look in the glass. Or, if you won't believe the\nglass, then just wait and see how proud Dan and I are if we can lead you\nand Laura out on the dancing floor.\"\n\n\"But what horrid slang!\" protested Belle. \"The idea of calling a homely\ngirl a gold brick! And I thought you young men received more or less\ntraining in being gracious to the weaker sex.\"\n\n\"We do,\" Dave answered, \"as soon as we can find any use for the\naccomplishment. Fourth classmen, you know, are considered too young to\nassociate with girls. It's only now, when we've made a start in the\nthird class, that we're to be allowed to attend the hops at all.\"\n\n\"But why must you have to have such horrid names for girls who have not\nbeen greatly favored in the way of looks? It doesn't sound exactly\ngallant.\"\n\n\"Oh, well, you know,\" laughed Dave, \"we poor, despised, no-account\nmiddies must have some sort of sincere language to talk after we get our\nmasks off for the day. I suppose we like the privilege, for a few\nminutes in each day, of being fresh, like other young folks.\"\n\n\"What is your name for 'fresh' down at Annapolis!\" Belle wanted to know.\n\n\"Touge.\"\n\n\"And for being a bit worse than touge?\"\n\n\"Ratey.\"\n\n\"Which did they call you?\" demanded Belle.\n\nDave started, then sat up straight, staring at Miss Meade.\n\n\"I see that your tongue hasn't lost its old incisiveness,\" he laughed.\n\n\"Not among my friends,\" Belle replied lightly. \"But I can't get my mind\noff that uniform of yours that you didn't bring home. What would have\nhappened to you if you had been bold enough to do it?\"\n\n\"I guess I'd have 'frapped the pap,'\" hazarded Dave.\n\n\"And what on earth is 'frapping the pap'?\" gasped Belle.\n\n\"Oh, that's a brief way of telling about it when a midshipman gets stuck\non the conduct report.\"\n\n\"I'm going to buy a notebook,\" asserted Belle, \"and write down and\nclassify some of this jargon. I'd hate to visit a strange country, like\nAnnapolis, and find I didn't know the language. And, Dave, what sort of\nplace is Annapolis, anyway?\"\n\n\"Oh, it's a suburb of the Naval Academy,\" Dave answered.\n\n\"Is it dreadfully hard to keep one's place in his class there?\" asked\nBelle.\n\n\"Well, the average fellow is satisfied if he doesn't 'bust cold,'\" Dave\ninformed her.\n\n\"Gracious! What sort of explosion is 'busting cold'?\"\n\n\"Why, that means getting down pretty close to absolute zero in all\nstudies. When a fellow has the hard luck to bust cold the superintendent\nallows him all his time, thereafter, to go home and look up a more\nsuitable job than one in the Navy. And when a fellow bilges----\"\n\n\"Stop!\" begged Belle. \"Wait!\"\n\nShe fled from the room, to return presently bearing the prettiest hat\nthat Dave ever remembered having seen on her shapely young head. In one\nhand she carried a dainty parasol that she turned over to him.\n\n\"What's the cruise?\" asked Darrin, rising.\n\n\"I'm going out to get that notebook, now. Please don't talk any more\n'midshipman' to me until I get a chance to set the jargon down.\"\n\nAs she stood there, such a pretty and wholesome picture, David Darrin\nthought he never before had seen such a pretty girl, nor one dressed in\nsuch exquisite taste. Being a boy, it did not occur to him that Belle\nMeade had been engaged for weeks in designing this gown and others that\nshe meant to wear during his brief stay at home.\n\n\"What are you thinking of?\" asked Belle.\n\n\"What a pity it is that I am doomed to a short life,\" sighed Darrin.\n\n\"A short life? What do you mean?\" Belle asked.\n\n\"Why, I'm going to be assassinated, the first hop that you attend at the\nNaval Academy.\"\n\n\"So I'm a gold brick, am I?\" frowned Belle.\n\n\"You--a--gold brick?\" stammered Dave. \"Why, you--oh, go look in the\nglass!\"\n\n\"Who will assassinate you?\"\n\n\"A committee made up from among the fellows whose names I don't write\ndown on your dance card. And there are hundreds of them at Annapolis.\nYou can't dance with them all.\"\n\n\"I don't intend to,\" replied Belle, with a toss of her head. \"I'll\naccept, as partners, only those who appear to me the handsomest and most\ndistinguished looking of the midshipmen. No one else can write his name\non my card.\"\n\n\"Dear girl, I'm afraid you don't understand our way of making up dance\ncards at Crabtown.\"\n\n\"Where?\"\n\n\"Crabtown. That's our local name for Annapolis.\"\n\n\"Gracious! Let me get out quickly and get that notebook!\"\n\n\"At midshipmen's hops the fellow who drags the----\"\n\n\"Gold brick,\" supplied Belle, resignedly.\n\n\"No--not for worlds! You're no gold brick, Belle, and you know it, even\nthough you do refuse to go to the mirror. But the fellow who drags any\nfemme--\"\n\n\"Please--?\"\n\n\"'Femme' stands for girl. The fellow who drags any femme makes up her\ndance card for her.\"\n\n\"And she hasn't a word to say about it?\"\n\n\"Not as a rule.\"\n\n\"Oh!\" cried Belle, dramatically.\n\nShe moved toward the door. Dave, who could not take his eyes from her\npretty face, managed, somehow, to delay her.\n\n\"Belle, there's something--\" he began.\n\n\"Good gracious! Where? What?\" she cried, looking about her keenly.\n\n\"It's something I want to say--must say,\" Dave went on with more of an\neffort than anyone but himself could guess.\n\n\"Tell me, as we're going down the street,\" invited Belle.\n\n\"_Wha-a-at?_\" choked Dave. \"Well, I guess not!\"\n\nHe faced her, resting both hands lightly on her shoulders.\n\n\"Belle, we were pretty near sweethearts in the High School, I think,\" he\nwent on, huskily, but looking her straight in the eyes. \"At least, that\nwas my hope, and I hope, most earnestly, that it's going to continue.\nBelle, I am a long way from my real career, yet. It will be five years,\nyet, before I have any right to marry. But I want to look forward, all\nthe time, to the sweet belief that my schoolgirl sweetheart is going to\nbecome my wife one of these days. I want that as a goal to work for,\nalong with my commission in the Navy. But to this much I agree: if you\nsay 'yes' now, and find later that you have made a mistake, you will\ntell me so frankly.\"\n\n\"Poor boy!\" murmured Belle, looking at him fully. \"You've been a plebe\nuntil lately, and you haven't been allowed to see any girls. I'm not\ngoing to take advantage of you as heartlessly as that.\"\n\nYet something in her eyes gave the midshipman hope.\n\n\"Belle,\" he continued eagerly, \"don't trifle with me. Tell me--will you\nmarry me some day?\"\n\nThen there was a little more talk and--well, it's no one's business.\n\n\"But we're not so formally engaged,\" Belle warned him, \"that you can't\nwrite me and draw out of the snare if you wish when you're older. And\nI'm not going to wear any ring until you've graduated from the Naval\nAcademy. Do you understand that, Mr. David Darrin?\"\n\n\"It shall be as you say, either way,\" Dave replied happily.\n\n\"And now, let us get started, or we shan't get out on the street\nto-day,\" urged Belle.\n\nThen they passed out on the street, and no ordinarily observant person\nwould have suspected them of being anything more than school friends.\n\nBeing very matter-of-fact in some respects, Belle's first move was to go\nto a stationer's, where she bought a little notebook bound in red\nleather.\n\nDave tried to pay for that purchase, but Belle forestalled him.\n\n\"Why didn't you allow me to make you that little gift?\" he asked in a\nlow tone, when they had reached the street.\n\n\"Wait,\" replied Belle archly. \"Some day you may find your hands full in\nthat line.\"\n\n\"One of my instructors at Annapolis complimented me on having very\ncapable hands,\" Dave told her dryly.\n\n\"The instructor in boxing?\" asked Belle.\n\nIt was a wonderfully delightful stroll that the middy and his sweetheart\nenjoyed that September forenoon.\n\nOnce Dave sighed, so pronouncedly that Belle shot a quick look of\nquestioning at him.\n\n\"Tired of our understanding already?\" she demanded.\n\n\"No; I was thinking how sorry I am for Danny boy! He doesn't know the\nhappiness of having a real sweetheart.\"\n\n\"How do you know he doesn't?\" asked Belle quickly. \"Does he tell you\neverything?\"\n\n\"No; but I know Danny's sea-going lines pretty well. I'd suspect, at\nleast, if he had a sweetheart.\"\n\n\"Are you sure that you would?\"\n\n\"Oh, yes! By gracious! There's Danny going around the corner above at\nthis very moment.\"\n\nBelle had looked in the same instant.\n\n\"Yes; and a skirt swished around the corner with him,\" declared Belle\nimpressively. \"It would be funny, wouldn't it, if you didn't happen to\nknow all about Dan Dalzell?\"\n\nIn the early afternoon, however, the mystery was cleared up.\n\nOn the street Dalzell had encountered Laura Bentley. Both were full of\ntalk and questions concerning Dick Prescott and Greg Holmes, at West\nPoint, for which reason Dan had strolled home with Miss Bentley without\nany other thought, on the midshipman's part, than playing substitute\ngallant for his chum, Cadet Richard Prescott, U.S. Military Academy.\n\nA most delightful afternoon the four young people spent together at the\nBentley home.\n\nThese were the forerunners of other afternoons.\n\nBelle and Laura, however, were not able to keep their midshipmen to\nthemselves.\n\nOther girls, former students at the High School, arranged a series of\naffairs to which the four young people were invited.\n\nDave's happiest moments were when he had Belle to himself, for a stroll\nor chat.\n\nDan's happiest moments, on the other hand, were when he was engaged in\nhunting the old High School fellows, or such of them as were now at\nhome. For many of them had entered colleges or technical schools. Tom\nReade and Harry Hazelton, of the famous old Dick & Co., of High School\ndays, were now in the far southwest, under circumstances fully narrated\nin \"THE YOUNG ENGINEERS IN ARIZONA,\" the second volume of \"THE YOUNG\nENGINEERS' SERIES.'\"\n\nDay by day Belle jotted down in her notebook more specimens of\nmidshipman slang.\n\n\"I shall soon feel that I can reel off the language like a native of\nCrabtown,\" she confided laughingly to Dare.\n\n\"It won't be very long before you have an opportunity to try,\" Dave\ndeclared, \"if you and Laura embrace your first opportunity to come to a\nmiddy hop.\"\n\nDan had a happy enough time of it, even though Dave's suspicion was true\nin that Dan had no sweetheart. That, however, was Dan's fault entirely,\nas several of the former High School girls would have been willing to\nassure him.\n\nSince even the happiest times must all end so the latter part of\nSeptember drew near.\n\nThen came the day when Dave and Dan met at the railway station. A host\nof others were there to see them off, for the midshipmen still had\ncrowds of friends in the good old home town.\n\nA ringing of bells, signaling brakesmen, a rolling of steel wheels and\nthe two young midshipmen swung aboard the train, to wave their hats from\nthe platform.\n\nGridley was gone--lost to sight for another year. Dan was exuberant\nduring the first hour of the journey, Dave unusually silent.\n\n\"You need a vast amount of cheering up, David, little giant!\" exclaimed\nDalzell.\n\n\"Oh, I guess not,\" smiled Dave Darrin quietly, adding to himself, under\nhis breath:\n\n\"I carry my own good cheer with me, now.\"\n\nLightly his hand touched a breast pocket that carried the latest,\nsweetest likeness of Miss Belle Meade.\n\nOne journey by rail is much like another to the traveler who pays little\nheed to the scenery.\n\nAt the journey's end two well-rested midshipmen joined the throng of\nothers at Crabtown.\n\n\n\n\nCHAPTER XIII\n\n\nDAN RECEIVES A FEARFUL FACER\n\n\"Oh, you heap!\" sighed Dan Dalzell dismally.\n\nHe sat in his chair, in their new quarters in Bancroft Hall, United\nStates Naval Academy, gazing in mock despair at the pile of new books\nthat he had just drawn.\n\nThese text-books contained the subjects in which a midshipman is\nrequired to qualify in his second academic year.\n\n\"Been through the books for a first look?\" called Dave from behind his\nown study table.\n\n\"Some of 'em,\" admitted Dalzell. \"I'm afraid to glance into the others.\"\n\n\"I've looked in all of my books,\" continued Darrin, \"and I've just come\nto a startling conclusion.\"\n\n\"What?\"\n\n\"I'm inclined to believe that I have received a complete set of\ntext-books for the first and second classes.\"\n\n\"No such luck!\" grunted Dan, getting up and going over to his chum. \"Let\nme see if you got all the books I did.\"\n\nBefore Dave could prevent it, Dan started a determined over-tossing of\nthe book pile. As he did so, Dan suddenly uncovered a photograph from\nwhich a fair, sweet, laughing face gazed up at him.\n\n\"Oh, I beg a million pardons, Dave, old boy!\" cried Dalzell.\n\n\"You needn't,\" came Dave's frank answer. \"I'm proud of that treasure and\nof all it means to me.\"\n\n\"And I'm glad for you, David, little giant.\"\n\nTheir hands met in hearty clasp, and that was all that was said on that\nsubject at the time.\n\n\"But, seriously,\" Dan grumbled on, after a while, \"I'm aghast at what an\nexacting government expects and demands that we shall know. Just look\nover the list--mechanical drawing and mechanical processes, analytical\ngeometry, calculus, physics, chemistry, English literature, French and\nSpanish, integral calculus, spherical trigonometry, stereographic\nprojection and United States Naval history! David, my boy, by the end of\nthis year we'll know more than college professors do.\"\n\n\"Aren't you getting a big head, Danny?\" queried Darrin, looking up with\na smile.\n\n\"I am,\" assented Dalzell, \"and I admit it. Why, man alive, one has to\nhave a big head here. No small head would contain all that the Academic\nBoard insists on crowding into it.\"\n\nBy the time that the chums had attended the first section recitations on\nthe following day, their despair was increased.\n\n\"Davy, I don't see how we are ever going to make it, this year,\" Dalzell\ngasped, while they were making ready for supper formation. \"We'll bilge\nthis year without a doubt.\"\n\n\"There's only one reason I see for hoping that we can get through the\nyear with fair credit,\" murmured Darrin.\n\n\"And what's that?\"\n\n\"Others have done it, before us, and many more are going to do it this\nyear,\" replied Dave slowly, as he laid comb and brush away and drew on\nhis uniform blouse.\n\n\"I know men have gotten through the Naval Academy in years gone by,\"\nDalzell agreed. \"But, the first chance that I have, I'm going to look\nthe matter up and see whether the middies of old had any such fearful\ngrind as we have our noses held to.\"\n\n\"Oh, we'll do it,\" declared Darrin confidently. \"I shall, anyway--for\nI've got to!\"\n\nAs he spoke he was thinking of Belle Meade, and of her prospects in life\nas well as his own.\n\nAs the days went by, however, Dave and Dan became more and more dull of\nspirits. The grind was a fearful one. A few very bright youngsters went\nalong all right, but to most of the third classmen graduation began to\nlook a thousand years away.\n\nThe football squad was out now and training in deadly earnest. There\nwere many big games to be played, but most of all the middies longed to\ntow West Point's Army eleven into the port of defeat.\n\nIn their first year Dave and Dan had looked forward longingly to joining\nthe gridiron squad. They had even practised somewhat. But now they\nrealized that playing football in the second year at Annapolis must be,\nfor them, merely a foolish dream.\n\n\"I'm thankful enough if I can study day and night and keep myself up to\n2.5,\" confessed Darrin, as he and Dan chatted over their gridiron\nlongings.\n\nTwo-and-five tenths is the lowest marking, on a scale of four, that will\nsuffice to keep a midshipman in the Naval Academy.\n\n\"I'm not going to reach 2.5 in some studies this month,\" groaned Dan. \"I\nknow that much by way of advance information. The fates be thanked that\nwe're allowed until the semi-ans to pick up. But the question is, are we\never going to pick up? As I look through my books it seems to me that\nevery succeeding lesson is twice as hard as the one before it.\"\n\n\"Other men have gone through, every year.\"\n\n\"And still other men have been dropped every year,\" Dalzell dolefully\nreminded him.\n\n\"We're among those who are going to stay,\" Dave contended stubbornly.\n\n\"Then I'm afraid we'll be among those who are dropped after Christmas\nand come back, next year, as bilgers,\" Dalzell groaned.\n\n\"Now, drop that!\" commanded Darrin, almost roughly. \"Remember one thing,\nDaniel little lion slayer! My congressman and your senator won't appoint\nus again, if we fail now. No talk of that kind, remember. We've got to\nmake our standing secure within the next few weeks.\"\n\nBefore the month was over the football games began in earnest on the\nathletic field. Darrin and Dalzell, however, missed every game. They\nwere too busy poring over their text-books. Fortunately for them their\ndrills, parades and gym. work furnished them enough exercise.\n\nThe end of October found Darrin at or above 2.5 in only three studies.\nDan was above 2.5 in two studies--below that mark in all others.\n\n\"It's a pity my father never taught me to swear,\" grumbled Dalzell, in\nthe privacy of their room.\n\n\"Stow that talk,\" ordered Darrin, \"and shove off into the deeper waters\nof greater effort.\"\n\n\"Greater effort?\" demanded Dan, in a rage. \"Why I study, now, every\npossible moment of the time allowed for such foolishness. And we can't\nrun a light. Right after taps the electric light is turned off at the\nmaster switch.\"\n\n\"We're wasting ninety seconds of precious time, now, in grumbling,\"\nuttered Dave, seating himself doggedly at his study table.\n\n\"Got any money, Darry?\" asked Dalzell suddenly.\n\n\"Yes; are you broke?\"\n\n\"I am, and the next time I go into Annapolis I mean to buy some\ncandles.\"\n\n\"Don't try that, Danny. Running a light is dangerous, and doubly so with\ncandles. The grease is bound to drip, and to be found in some little\ncorner by one of the discipline officers. It would be no use to study if\nyou are going to get frapped on the pap continuously.\"\n\nImmediately after supper both midshipmen forfeited their few minutes of\nrecreation, going at once back to their study tables. There they\nremained, boning hard until the brief release sounded before taps was\ndue.\n\nAlmost at the sound of the release there came a knock at the door.\nFarley and his roommate, Page, came bounding in.\n\n\"I've got to say something, or I'll go daffy,\" cried Farley, rubbing his\neyes. \"Fellows, did you ever hear of such downright abuse as the second\nyear course of studies means?\"\n\n\"It is tough,\" agreed Dave. \"But what can we do about it, except fight\nit out?\"\n\n\"Can you make head or tail out of calculus?\" demanded Farley.\n\n\"No,\" admitted Darrin, \"but I hope to, one of these days.\"\n\nJust then Freeman, of the first class, poked his head in, after a soft\nknock.\n\n\"What is this--a despair meeting?\" he called cheerily.\n\n\"Yes,\" groaned Page. \"We're in a blue funk over the way recitations are\ngoing.\"\n\n\"Oh, buck up, kiddies!\" called Freeman cheerily, as he crossed the\nfloor. \"Youngsters always get in the doldrums at the beginning of the\nyear.\"\n\n\"You're a first classman. When you were in the third class did you have\nall the studies that we have now?\"\n\n\"Every one of them, sir,\" affirmed Midshipman Freeman gravely, though\nthere was a twinkle in his eyes.\n\n\"And did you come through the course easily?\" asked Page.\n\n\"Not easily,\" admitted the first classman. \"There isn't anything at\nAnnapolis that is easy, except the dancing. In fact, during the first\ntwo months very few of our class came along like anything at all. After\nthat, we began to do better. By the time that semi-ans came around\nnearly all of us managed to pull through. But what seems to be the worst\ngrind of all--the real blue paint?\"\n\n\"Calculus!\" cried the four youngsters in unison.\n\n\"Why, once you begin to see daylight in calculus it's just as easy as\ntaking a nap,\" declared the first classman.\n\n\"At present it seems more like suffering from delirium,\" sighed Dave.\n\n\"What's the hard one for to-morrow?\" asked Freeman.\n\n\"Here it is, right here,\" continued Dave, opening his text-book. \"Here's\nthe very proposition.\"\n\nThe others crowded about, nodding.\n\n\"I remember that one,\" laughed Freeman lightly. \"Our class named it\n'sticky fly paper.'\"\n\n\"It was rightly named,\" grumbled Farley.\n\n\"None of you four youngsters see through it?\" demanded Midshipman\nFreeman.\n\n\"Do you mean to claim, sir, that you ever did?\" insisted Dan Dalzell.\n\n\"Not only once, but now,\" grinned Mr. Freeman. \"You haven't been looking\nat this torturing proposition from the right angle--that's all. Now,\nlisten, while I read it.\"\n\n\"Oh, we all know how it runs, Mr. Freeman,\" protested Page.\n\n\"Nevertheless, listen, while I read it.\"\n\nAs the first classman read through the proposition that was torturing\nthem he threw an emphasis upon certain words that opened their eyes\nbetter as to the meaning.\n\n\"Now, it works out this way,\" continued the first classman, bending over\nthe disk and drawing paper and pencil toward him. \"In the first place.\"\n\nFreeman seemed to these youngsters like a born demonstrator. Within five\nminutes he had made the \"sticky fly paper\" problem so plain to them all\nthat they glanced from one to another in astonishment.\n\n\"Why, it does seem easy,\" confessed Farley.\n\n\"It sounds foolish, now,\" grinned Darrin. \"I'm beginning to feel ashamed\nof myself.\"\n\n\"Mr. Freeman,\" protested Page, \"you've saved us from suicide, or some\nother gruesome fate.\"\n\n\"Then I'll drop in once in a while again,\" promised the first classman.\n\n\"But that will take time from your own studies,\" remonstrated Darrin\ngenerously.\n\n\"Not in the least. I won't come around before release. By the time a\nfellow reaches the first class, if he's going to graduate anyway, he\ndoesn't have to study as hard as a youngster does. The man who reaches\nthe first class has had all the habits of true study ground into him.\"\n\nDarrin, Dalzell, Farley and Page were all in different sections in\nmathematics. When they recited, next day, it so happened that each was\nthe man to have the \"sticky fly paper\" problem assigned to him by the\ninstructor. Each of the quartette received a full \"4\" for the day's\nmarking.\n\n\"Did you have any assistance with this problem, Mr. Darrin?\" asked\nDave's instructor.\n\n\"Yes, sir; a member of the first class tried to make it plain to me last\nnight.\"\n\n\"He appears to have succeeded,\" remarked the instructor dryly.\n\nThere was, however, no discredit attached to having received proper\nassistance before coming into section.\n\nTrue to his promise Freeman dropped in every fourth or fifth evening, to\nsee if he could be of any help to the four youngsters. Always he found\nthat he could be.\n\nEven when Thanksgiving came, Dave Darrin did not go to Philadelphia, but\nremained at the Academy, devoting his time to study.\n\nDan, in sheer desperation, took in the trip to Philadelphia. He hoped to\nmeet Dick Prescott and Greg Holmes, but they did not come down from West\nPoint.\n\nOn the first day of December, Dan Dalzell's name was formally reported\nby the Academic Board in a report to the superintendent which\nrecommended that Midshipman Dalzell be dropped from the rolls for\n\"inaptitude in his studies.\"\n\nPoor Dan. It was a staggering blow. Yet it struck Dave Darrin just about\nas hard.\n\n\n\n\nCHAPTER XIV\n\n\nTHE FIRST HOP WITH THE HOME GIRLS\n\nThat report was allowed to reach Dan's ears on a Friday.\n\nOn the evening of the day following there was to be a midshipman hop on\nthe floor of the great gym.\n\nMoreover, it was the very hop that Belle Meade and Laura Bentley had\nfinally selected to attend. Mrs. Meade was coming with the girls as\nchaperon.\n\n\"Oh, but I shall feel fine and light hearted for going to the dance!\"\nmuttered Dan miserably. \"Facing the kick-off from the Academy, and doing\nthe light hearted and the fantastic toe with the girls.\"\n\n\"I shan't feel a whole lot more merry myself,\" sighed Dave, as he gazed\naffectionately, wistfully at his chum. \"Danny, this has hit me about as\nhard as it has you. And it warns me, too, that my turn will probably\ncome next. I don't stand an awful lot higher in my markings than you\ndo.\"\n\n\"Doesn't it feel fine to be a bilger?\" gulped Dalzell, staring at the\nfloor.\n\nA \"bilger,\" as has been already explained, is a midshipman who has\nfailed and has been dropped.\n\n\"Oh, but you're not a bilger, yet!\" cried Darrin, leaping up and resting\nboth hands on his chum's shoulder.\n\n\"What's the odds?\" demanded Dan grimly. \"I shall be, after I've been\nbefore the Board next Monday forenoon at ten o'clock.\"\n\n\"Nonsense! Not if you make a good fight!\"\n\n\"Fight--nothing!\" sighed Dan wearily. \"In a fight there's some one else\nthat you can hit back at. But I won't have a blessed soul to fight. I'm\nup against a gang who are all referees, and all down on me at the\noutset.\"\n\n\"Nonsense,\" combatted Dave. \"You----\"\n\n\"Oh, that's all right, David, little giant,\" returned Dalzell with an\nattempt at cheeriness. \"You mean well, but a fellow isn't reported\ndeficient unless he's so far behind that the Board has his case settled\nin advance. From all I can hear it isn't once in a camel's age that a\nfellow so reported, and ordered before the Board, gets off with anything\nless than a hard, wet bilge. What I'm thinking of now is, what am I\ngoing to pick up as a career when I go home from here as a failure.\"\n\nIf it hadn't been for the pride he felt in still having the uniform on,\nDalzell might not have been able to check the tears that tried to flow.\n\n\"Come on,\" commanded Dave, leaping up, \"we'll run up to the deck above,\nand see if we can't find Mr. Freeman in.\"\n\n\"What good will that do?\" demanded Dan. \"Freeman is a first classman,\nbut he hasn't any particular drag with the Board.\"\n\n\"It won't do any harm, anyway, for us to have a talk with an older\nclassman,\" argued Dave. \"Button your blouse, straighten your hair and\ncome along.\"\n\n\"So it's as bad as that, is it!\" asked Freeman sympathetically, after\nhis cheery \"come in\" had admitted the unhappy youngsters.\n\n\"Yes,\" replied Dave incisively. \"Now, the question is, what can be done\nabout it?\"\n\n\"I wish you had asked me an easier one,\" sighed the first classman.\n\"You're mighty well liked, all through the Academy, Dalzell, and every\none of us will hate to see you go.\"\n\n\"But what can be done to ward off that fate?\" insisted Darrin as\nimpatiently as a third classman might speak to a venerable first\nclassman.\n\n\"Well, now, I want to think over that,\" confessed Freeman frankly. \"Of\ncourse, Dalzell's record, this term, is in black and white, and can't be\ngainsaid. It's just possible our young friend can put up some line of\ntalk that will extend his time here, and perhaps enable him to pull\nthrough. It's a mighty important question, so I'll tell you what we'll\ndo. Of course, the hop comes on for to-morrow night. Let me have until\nSunday evening. Meanwhile I'll talk with some of the other fellows of my\nclass. You both come in here Sunday evening, and I'll have the answer\nfor you--if there's any possible way of finding one.\"\n\nWith that the chums had to be content. Expressing their gratitude to\nthis friendly first classman, they withdrew.\n\nThat Saturday forenoon Dan did considerably better with the two\nrecitations that he had in hand.\n\n\"I got easier questions than usual, I guess,\" he said to Dave, with a\nmournful smile.\n\nAfter Saturday dinner, Dave and Dan, having secured permission to visit\nin Annapolis, steered their course through the gate, straight up\nMaryland Avenue, through State Circle and around into Main Street, to\nthe Maryland House.\n\nAt the desk they sent up their cards to Mrs. Meade, then stepped into\nthe parlor.\n\nBarely two minutes had passed when Belle and Laura flew downstairs.\n\n\"Mother says she'll be down as soon as she fancies you'll care about\nseeing her,\" laughed Belle.\n\n\"And how are you getting on in your classes?\" asked Laura Bentley,\nglancing straight at unhappy Dan.\n\nBoth midshipmen had agreed not to mention a word of Dan's heartache to\neither of the girls.\n\nDan gulped hard, though he managed to conceal the fact.\n\nDarrin, however, was ready with the answer:\n\n\"Oh, we're having pretty rough sailing, but we're both still in our\nclass.\"\n\nWhich statement was wholly truthful.\n\n\"Up at West Point,\" Laura continued, \"Dick told us that the first two\nyears were the hardest for a man to keep his place. I fancy it's just\nabout the same here, isn't it?\"\n\n\"Just about,\" Dave nodded. \"The first two years are hardest because it\ntakes all that time for a fellow to get himself keyed up to the gait of\nstudy that is required in the government academies. But won't you let us\ntalk about something that's really pleasant, girls?\" Dave asked, with\nhis charming smile. \"Suppose we talk about yourselves. My, but you girls\nare good to look at!\"\n\nAfter that, the conversation was shifted to lighter subjects.\n\nEven Dan, in the joy of meeting two girl friends from home, began to be\nless conscious of his load of misery.\n\nPresently Mrs. Meade came down. She chatted with the two fine-looking\nyoung midshipmen for a few moments. Then Dave proposed:\n\n\"Wouldn't you like us to escort you through the Academy grounds, so that\nyou can get a good idea of the place in daylight?\"\n\n\"We've been waiting only for you to invite us,\" rejoined Belle.\n\nFor the next two hours the time was passed pleasantly.\n\nBut Belle, behind all her light chatter, was unusually keen and\nobserving.\n\n\"Is anything wrong with either of you?\" she asked Dave suddenly, when\nthis pair were out of easy hearing of the others.\n\n\"Why do you ask that?\" inquired Dave, looking at her in his direct\nfashion.\n\n\"Why, I may be unnecessarily sensitive, but I can't help feeling that\nsome sort of disaster is hanging over either you or Dan.\"\n\n\"I hope not,\" replied Darrin evasively.\n\n\"Dave, that isn't a direct answer,\" warned Belle, raising her eyebrows.\n\"Do you consider me entitled to one?\"\n\n\"Yes. What's the question?\"\n\n\"Are you in any trouble here?\"\n\n\"No, I'm thankful to say.\"\n\n\"Then is Dan?\"\n\n\"Belle, I'd rather not answer that.\"\n\n\"Why----\"\n\n\"Well, because, if he is, I'd rather not discuss it.\"\n\n\"Has Dan been caught in any scrape?\"\n\n\"No. His conduct record is fine.\"\n\n\"Then it must be failure in his studies.\"\n\nDave did not answer.\n\n\"Why don't you tell me?\" insisted Belle.\n\n\"If anything were in the wind, Belle, we'd rather not tell you and spoil\nyour visit. And don't ask Dan anything about it.\"\n\n\"I think I know enough,\" went on Belle thoughtfully and sympathetically.\n\"Poor Dan! He's one of the finest of fellows.\"\n\n\"There are no better made,\" retorted Dave promptly.\n\n\"If anything happens to Dan here, dear, I know you will feel just as\nunhappy about it as if it happened to yourself.\"\n\n\"Mighty close to it,\" nodded Darrin. \"But it would be a double\nheartbreak for me, if I had to leave.\"\n\n\"Why?\"\n\n\"On account of the future I've planned for you, Belle.\"\n\n\"Oh, you silly boy, then!\" Belle answered, smiling into his eyes. \"I\nbelieve I have half committed myself to the idea of marrying you when\nyou've made your place in life. But it was Dave Darrin to whom I gave\nthat half promise--not a uniform of any sort. Dave, if anything ever\nhappens that you have to quit here, don't imagine that it's going to\nmake a particle of difference in our understanding.\"\n\n\"You're the real kind of sweetheart, Belle!\" murmured Dave, gazing\nadmiringly at her flushed face.\n\n\"Did you ever suspect that I wasn't?\" asked Miss Meade demurely.\n\n\"Never!\" declared Midshipman Darrin devoutly. \"Nevertheless, it's fine\nto be reassured once in a while.\"\n\n\"What a great fellow Dan is!\" exclaimed Belle a few minutes later. \"See\nhow gayly he is chatting with Laura. I don't believe Laura guesses for a\nmoment that Dan Dalzell is just as game a fellow as the Spartan boy of\nolden times.\"\n\n\n\n\nCHAPTER XV\n\n\nA DISAGREEABLE FIRST CLASSMAN\n\nThe hop that night was one of the happiest occasions Dave had ever\nknown, yet it was destined to result in trouble for him.\n\nMidshipman Treadwell, of the first class, caught sight of Belle as she\nentered the gym at Dave Darrin's side.\n\nWith Treadwell it happened to be one of those violent though unusually\nsilly affairs known as \"love at first sight.\"\n\nAs for Belle, she was not likely to have eyes for anyone in particular,\nsave Dave.\n\nTreadwell, who had come alone, and who was not to be overburdened with\ndances, went after Dave as soon as that youngster left Belle for the\nfirst time.\n\n\"Mighty sweet looking girl you have with you, Darry,\" observed the first\nclassman, though he took pains not to betray too much enthusiasm.\n\n\"Right!\" nodded Dave.\n\n\"You'll present me, won't you?\"\n\n\"Assuredly, as soon as I come back. I have a little commission to attend\nto.\"\n\n\"And you might be extremely kind, Darry, and write me down for a couple\nof numbers on Miss----\"\n\n\"Miss Meade is the young lady's name.\"\n\n\"Then delight me by writing down a couple of reservations for me on Miss\nMeade's card.\"\n\nDarrin's face clouded slightly.\n\n\"I'd like to, Treadwell, but the card is pretty crowded, and some other\nfellows--\"\n\n\"One dance, anyway, then.\"\n\n\"I will, then, if there's a space to be left, and if Miss Meade is\nagreeable,\" promised Dave, as he hurried away.\n\nTwo minutes later, when he returned, looking very handsome, indeed, in\nhis short-waisted, gold-laced dress coat, Dave felt his arm touched.\n\n\"I'm waiting for you to keep your engagement with me,\" Midshipman\nTreadwell murmured.\n\n\"Come along; I shall be delighted to present you to Miss Meade.\"\n\nSince every midshipman is granted to be a gentleman, midshipman\netiquette does not require that the lady be consulted about the\nintroduction.\n\n\"Miss Meade,\" began Dave, bowing before his sweetheart, \"I wish to\npresent Mr. Treadwell\"\n\nBelle's greeting was easy. Treadwell, gazing intensely into her eyes,\nexchanged a few commonplaces. Belle, entirely at her ease, did not\nappear to be affected by the battery of Mr. Treadwell's gaze. Then good\nbreeding required that the first classman make another bow and stroll\naway.\n\nAs he left, Treadwell murmured in Dave's ear:\n\n\"Don't forget that dance, Darry! Two if there is any show.\"\n\nMidshipman Darrin nodded slightly. As he turned to Belle, that young\nlady demanded lightly:\n\n\"Is that pirate one of your friends, Dave?\"\n\n\"Not more so than any other comrades in the brigade,\" Darrin answered.\n\"Why?\"\n\n\"Nothing, only I saw you two speaking together a little while ago----\"\n\n\"That was when he was asking me to present him.\"\n\n\"Then, after you left him,\" continued Belle, in a low voice, \"Mr.\nTreadwell scowled after you as though he could have demolished you.\"\n\n\"Why, I've no doubt Mr. Treadwell is very jealous of me,\" laughed Damn\nhappily. \"Why shouldn't he be? By the way, will you let me see your\ndance card? Mr. Treadwell asked me to write his name down for one or two\ndances.\"\n\n\"Please don't,\" begged Belle suddenly, gripping her dance card tightly.\n\"I hope you don't mind, Dave,\" she added in a whisper, \"but I've taken\njust a shadow of a dislike to Mr. Treadwell, after the way that he\nscowled after you. I--I really don't want to dance with him.\"\n\nDave could only bow, which he did. Then other midshipmen were presented.\nBelle's card was quickly filled, without the appearance of Midshipman\nTreadwell's name on it.\n\nThe orchestra struck up. Dave danced the first two numbers with Belle,\nmoving through a dream of happiness as he felt her waist against his\narm, one of her hands resting on his shoulder.\n\nThe second dance was a repetition of Dave's pleasure. Then Dave and Dan\nexchanged partners for two more dances.\n\nAfter their first dance, a waltz, Dave led Laura to a seat.\n\n\"Will you get me a glass of water, Dave?\" Laura asked, fanning herself.\n\nAs Dave hastened away he felt, once more, a light, detaining touch.\n\n\"Darry, did you save those two dances for me with Miss Meade?\" asked\nTreadwell.\n\n\"Oh, I'm sorry,\" Dave replied. \"But there had been many other\napplicants. By the time that Miss Meade's card was filled there were\nmany disappointed ones.\"\n\n\"And I'm one of them?\" demanded Mr. Treadwell.\n\n\"Very sorry,\" replied Darrin regretfully, \"but you were one of the\nleft-over ones.\"\n\n\"Very good, sir,\" replied Treadwell coldly, and moved away.\n\n\"Now, I'll wager anything that Treadwell is sore with me,\" murmured Dave\nto himself. \"However, Belle is the one to be pleased.\"\n\nIt was a particularly gay and pleasant hop. When it was over Dave and\nDan escorted the girls and Mrs. Meade back to the hotel. The little room\nin Bancroft Hall seemed especially small and dingy to the returning\nmidshipmen.\n\nEspecially was Dan Dalzell in the blues. Though he had been outwardly\ngay with the girls, he now suffered a re-action. Dave, too, shivered for\nhis friend.\n\nMrs. Meade and the girls returned by an early morning train, so the two\nchums did not see the girls again during that visit.\n\nOn Sunday, Dave went at his books with a dogged air, after morning\nchapel and dinner.\n\n\"I suppose this is the last day of study for me here,\" grimaced Dan, \"so\nI mean to make the most of the pleasure.\"\n\n\"Nonsense,\" retorted Darrin heartily; \"you'll finish out this year, and\nthen have two more solid years of study here ahead of you.\"\n\n\"Cut it!\" begged Dan dolefully. \"Don't try to jolly me along like that.\"\n\n\"You're down in the dumps, just now, Danny boy,\" smiled Darrin\nwistfully. \"Just bombard the Board with rapid-fire talk to-morrow, and\nyou'll pull through all right.\"\n\nDan sighed, then went on with his half-hearted study.\n\nLater in the afternoon Dave, feeling the need of fresh air, closed his\nbooks.\n\n\"Come for a walk, Danny boy?\"\n\n\"Don't dare to,\" replied Dalzell morosely.\n\nSo, though Darrin went out, he resolved not to remain long away from his\nmoody chum.\n\nOutside, on one of the cement walks, Dave turned toward Flirtation Walk.\nIt seemed the best surrounding in which to think of Belle.\n\n\"Mr. Darrin!\" called a voice.\n\nDave turned, to behold Mr. Treadwell coming at a fast stride with a\nscowl on his face.\n\n\"That was a dirty trick you played me last night, Mr. Darrin!\" cried the\nfirst classman angrily.\n\n\"What?\" gasped Dave, astonished, for this was not in line with the usual\nconversation of midshipmen.\n\n\"You know well enough what I mean,\" cried Treadwell angrily. \"You spiked\nmy only chance to dance with Miss Meade.\"\n\n\"You're wrong there,\" retorted Dave coldly and truthfully \"I didn't.\"\n\n\"Then how did it happen?\"\n\n\"I can't discuss that with you,\" Darrin rejoined. \"I didn't make any\neffort, though, to spoil your chance of a dance with the young lady.\"\n\n\"Mr. Darrin, I don't choose to believe you, sir!\"\n\nDave's face went crimson, then pale.\n\n\"Do you realize what you're saying, Mr. Treadwell?\"\n\n\"Of course\"--sneeringly.\n\n\"Are you trying to pick trouble with me?\" demanded Dave, his eyes\nflashing with spirit.\n\n\"I repeat that I don't choose to believe your explanation, sir.\"\n\n\"Then you pass me the lie?\"\n\n\"As you prefer to consider it,\" jeered the first classman.\n\n\"Oh, very good, then, Mr. Treadwell,\" retorted Dave, eyeing the first\nclassman and sizing him up.\n\nTreadwell was one of the biggest men, physically, in the brigade. He was\nalso one of the noted fighters of his class. Beside Treadwell,\nMidshipman Darrin did not size up at all advantageously.\n\n\"If you do not retract what you just said,\" pursued Dave Darrin, growing\ncooler now that he realized the deliberate nature of the affront that\nhad been put upon him, \"I shall have no choice but to send my friends to\nyou.\"\n\n\"Delighted to see them, at any time,\" replied the first classman,\nturning disdainfully upon his heel and strolling away.\n\n\"Now, why on earth does that fellow deliberately pick a fight with me?\"\nwondered Darrin, as he strolled along by himself. \"Treadwell can thump\nme. He can knock me clean down the Bay and into the Atlantic Ocean, but\nwhat credit is there in it for a first classman to thrash a youngster?\"\n\nIt was too big a puzzle. After thinking it over for some time Dave\nturned and strolled back to Bancroft Hall.\n\n\"You didn't stay out long!\" remarked Dan, looking up with a weary smile\nas his chum re-entered their room.\n\n\"No,\" admitted Dave. \"There wasn't much fun in being out alone.\"\n\nWith a sigh, Dan turned back to his book, while Dave seated himself at\nhis own study table, in a brown daze.\n\nThings were happening fast--Dan's impending \"bilge\" from the Naval\nAcademy, and his own coming fight with the first classman who would be\nsure to make it a \"blood fight\"!\n\n\n\n\nCHAPTER XVI\n\n\nHOW DAN FACED THE BOARD\n\n\"We trust, Mr. Dalzell, that you can make some statement or explanation\nthat will show that we shall be justified in retaining you as a\nmidshipman in the Naval Academy.\"\n\nIt was the superintendent of the United States Naval Academy who was\nspeaking.\n\nDan's hour of great ordeal had come upon him. That young midshipman\nfound himself in the Board Room, facing the entire Academic Board,\ntrying to remember what Freeman had told him the night before.\n\nThe time was 10.30 a.m. on that fateful Monday.\n\nMidshipman Dalzell appeared to be collected, but he was also very\ncertainly white-faced.\n\nMany a young man, doomed to be sent forth from a Naval career, back into\nthe busy, unheeding world, had faced this Board in times past. So it was\nhardly to be expected that Dan would inspire any unusual interest in the\nmembers of the Board.\n\nDan swallowed at something hard in his throat, then opened his lips to\nspeak.\n\n\"I am aware, sir, and gentlemen, that I am at present sufficiently\ndeficient in my studies to warrant my being dropped,\" Dan began rather\nslowly. \"Yet I would call attention to the fact that I was nearly as\nbadly off, in the matter of markings, at this time last year. It is also\na matter of record that I pulled myself together, later on, and\ncontrived to get through the first year with a considerable margin of\ncredits to spare. If I am permitted to finish the present term here I\nbelieve I can almost positively promise that I will round out this year\nwith as good a showing as I did last year.\"\n\n\"You have thought the matter carefully out in making this statement,\nhave you, Mr. Dalzell?\" asked the superintendent.\n\n\"I have, sir.\"\n\n\"Have you any explanation to offer for falling below the standards so\nfar this year, Mr. Dalzell?\"\n\n\"I believe, sir, that I make a much slower start, with new studies, than\nmost of my classmates,\" Dan continued, speaking more rapidly now, but in\na most respectful manner. \"Once I begin to catch the full drift of new\nstudies I believe that I will overtake some of my classmates who showed\na keener comprehension at the first. I think, sir, and gentlemen, that\nmy record, as contrasted with the records of some of my classmates who\nachieved about the same standing I did for last year will bear my\nstatement out.\"\n\n[Illustration: \"Have You Any Explanation to Offer, Mr. Dalzell?\"]\n\nThe superintendent turned to a printed pamphlet in which were set forth\nthe records of the midshipmen for the year before.\n\n\"Mr. Dalzell,\" asked another member of the Board, \"do you feel that you\nare really suited for the life of the Navy? Is it your highest ambition\nto become an officer of the Navy?\"\n\n\"It's my only ambition, sir, in the way of a career,\" Dan answered\nsolemnly. \"As to my being suited for the Navy, sir, I can't make a good\nanswer to that. But I most earnestly hope that I shall have an\nopportunity, for the present, to try to keep myself in the service.\"\n\n\"And you feel convinced that you need only to be carried for the balance\nof the term to enable you to make good, and to justify any action that\nwe may take looking to that end?\" asked another member of the Board.\n\n\"That is my firm conviction, sir.\"\n\nThe superintendent, who had been silently examining and marking some\nstatements in the pamphlet, now passed it to the nearest member of the\nBoard, who, after a glance or two, passed the pamphlet on to another\nmember.\n\nSilence fell upon the room while Dan's printed record was being read.\n\n\"Have you anything else that you wish to say, Mr. Dalzell?\" asked the\nsuperintendent at last.\n\n\"Only this, sir and gentlemen,\" replied Dan promptly. \"If I am permitted\nto go on with the brigade, I promise, as far as any human being may\npromise, that I will not only be found to have passed at the end of this\nterm, but that I will also have a higher marking after the annual\nexaminations than after the semi-annuals.\"\n\nThese last few words Dan spoke with his whole soul thrown into the\nwords. How he longed to remain in the Navy, now that he stood at the\nthreshold of the life, uncertain whether he was about to be kicked\nacross it into the outer world!\n\nAfter glancing around the table, the superintendent turned once more to\nthe young man.\n\n\"That will be all, at present, Mr. Dalzell.\"\n\nSaluting briskly, crisply, Dan wheeled about, marching from the room.\n\nHe was in time to make a section recitation before dinner.\n\n\"How did you come out, Danny boy?\" anxiously inquired Dave Darrin as the\ntwo, in their room, hastily prepared to answer the coming call for\ndinner formation.\n\n\"I wish I knew,\" replied Dalzell wistfully. \"I said all that I could say\nwithout being everlastingly fresh.\"\n\nAfter the brigade had been formed for dinner, and the brigade adjutant\nhad reported the fact, the command was given:\n\n\"Publish the orders!\"\n\nThis the brigade adjutant did rapidly, and in perfunctory tones.\n\nDalzell jumped, however, when he heard his own name pronounced. He\nstrained his ears as the brigade adjutant read:\n\n\"In the matter of Daniel Dalzell, summoned before the Academic Board to\ndetermine his fitness and aptitude for continuing in the brigade, the\nBoard has granted Midshipman Dalzell's urgent request that he be\ncontinued as a midshipman for the present.\"\n\nThere was a great lump, instantly, in Dan's throat. It was a reprieve, a\nchance for official life--but that was all.\n\n\"I'll make good--I'll make good!\" he told himself, with a violent gulp.\n\nThe orders were ringing out sharply now. The midshipmen were being\nmarched in to dinner.\n\nHardly a word did Dalzell speak as he ate. As for Dave Darrin, he was\ntoo happy over his chum's respite to want to talk.\n\nYet, when they strolled together in the open air during the brief\nrecreation period following the meal, Dalzell suddenly asked:\n\n\"Dave when do you fight with Treadwell?\"\n\n\"To-night, I hope,\" replied Darrin.\n\n\"Oh, then I must get busy!\"\n\n\"Why?\"\n\n\"Why, I'm to represent you, Darry. Who are Treadwell's--\"\n\n\"Danny boy, don't make a fuss about it,\" replied Dave quietly, \"but just\nfor this once you are not to be my second.\"\n\n\"Why--\"\n\n\"Danny boy, you have just gotten by the Board by a hair's breadth. What\nkind of an act of gratitude would it be for you to make your first act a\nbreach of discipline? For a fight, though often necessary here, is in\ndefiance of the regulations.\"\n\n\"But Dave, I've never been out of your fights!\"\n\n\"You will be this time, Danny. Don't worry about it, either. Farley and\nPage are going to stand by me. In fact, I think that even now they are\ntalking with Treadwell's friends.\"\n\n\"You're wrong,\" murmured Dalzell, looking very solemn. \"Here come Farley\nand Page right now.\"\n\nIn another moment the seconds had reached Darrin and his chum.\n\n\"To-night?\" asked Dave quietly.\n\n\"Yes,\" nodded Page.\n\n\"Time?\"\n\n\"Just after recall.\"\n\n\"Good,\" murmured Darrin. \"You two come for me, and I'll be ready. And I\nthank both of you fellows for taking up the matter for me.\"\n\n\"We'll be mighty glad to be there, Darry,\" grinned Farley, \"for we look\nto see you finish off that first classman.\"\n\n\"Maybe,\" smiled Dave quietly. \"I'll do all I can, anyway.\"\n\n\"And to think,\" almost moaned Dan Dalzell, \"that you're to be in a\nscrap, David, little giant, and I'm not to be there to see!\"\n\n\"There'll be other fights, I'm afraid,\" sighed Darry. \"I seem destined\nto displease quite a few of the fellows here at Annapolis.\"\n\nDan tried to study, that night, after Darrin had left the room in the\ncompany of his seconds. Certainly Dan, in the light of his promise made\nto the Board that morning, had need to study. Yet he found it woefully\nhard to settle his mind on mathematics while Dave was fighting the fight\nof his Naval Academy career.\n\n\"Oh, well,\" muttered Dan, picking up a pencil for the third time, \"Dave\nand I each have our own styles of fights, just now. Here goes for a\nknockout blow at math!\"\n\n\n\n\nCHAPTER XVII\n\n\nLOSING THE TIME-KEEPER'S COUNT\n\nConners and Brayton were Treadwell's seconds.\n\nSince it is not considered fair to have the referee or time-keeper from\neither class represented in a fight, Edgerton and Wheeler, of the second\nclass, were referee and time-keeper respectively.\n\nAll of the young men were early at the usual fighting ground. The fall\nair was cool and crisp, but it was not yet considered cold enough to\njustify the extra risk of holding a fight in-doors.\n\nDave was quickly stripped and made ready by his seconds. His\nwell-developed chest bespoke fine powers in the way of \"wind\" and\nendurance. His smooth, hard, trim muscles stood out distinctly.\n\nTreadwell took more time in getting himself ready for the ring. When at\nlast, however, the first classman stood bared to the waist, he looked\nlike a giant beside Dave Darrin.\n\n\"It looks like a shame to take the money, Tread,\" murmured referee\nEdgerton.\n\n\"I don't want to pound the youngster hard,\" explained Midshipman\nTreadwell, in an undertone. \"Yet I've got to teach him both to respect\nmy class and myself.\"\n\nOn this point, as an official of the fight, Referee Edgerton did not\nfeel called upon to express an opinion.\n\nFarley, at his first glimpse of the waiting first classman, felt a chill\nof coming disaster.\n\n\"Page,\" he growled, \"that huge top-classman makes our Darry look like a\ncreeping infant.\"\n\n\"Darry will take care of himself,\" retorted Midshipman Page in an\nundertone.\n\n\"Do you believe it?\"\n\n\"I surely do.\"\n\n\"But Treadwell looks a whole lot more vast now that he's stripped.\"\n\n\"Darry is much smaller, I know; But Darrin is one of those rare fellows\nwho don't know what it means to be whipped. He can't be put out of\nbusiness by anything smaller than a twelve-inch gun!\"\n\n\"I hope you're right,\" sighed Farley.\n\nDave, in the meantime, to keep himself from being chilled by the frosty\nair, was running lightly about, swinging his arms.\n\n\"Are you both ready, gentlemen?\" inquired Midshipman Edgerton, while\nTime-keeper Wheeler drew out his stop watch.\n\nBoth stepped to toe the scratch.\n\n\"Yes.\" nodded Dave.\n\n\"Ready!\" rumbled Treadwell.\n\nThe referee briefly made the usual announcement about it being a fight\nto the finish, with two-minute rounds and two minutes between rounds.\n\n\"Time!\"\n\nAs Treadwell leaped forward, both fists in battery, Dave took a swift,\nnimble sidestep. He felt that he had to study this big fellow carefully\nbefore doing more than keep on the defensive.\n\nNow footwork was one of the fighting tricks for which Darry was famous.\nYet he had too much courage to rely wholly upon it.\n\nFive times Treadwell swung at his smaller opponent, but each time Dave\nwas somewhere else.\n\nDespite his greater size, Treadwell was himself nimble and an adept at\nfootwork.\n\nFinding it hard, however, to get about as quickly as his smaller\nopponent, the first classman soon went in for close, in-body fighting,\nfollowing Dave, half-cornering him, and forcing him to stand and take\nit.\n\nTwo or three body blows Dave succeeded in parrying so that they glanced,\ndoing him little harm.\n\nThen there came an almost crunching sound. Treadwell's right fist had\nlanded, almost dazing the youngster with its weight against his nose.\n\nThere was a swift, free rush of the red. Darrin had yielded up \"first\nblood\" in the fight.\n\n\"I've got to dodge more, and not let myself be cornered,\" Darrin told\nhimself, keeping his fists busy in warding off blows.\n\nThen, of a sudden, Dave turned on the aggressive. He struck fast and\nfuriously, but Treadwell, with a grin, beat down his attack, then soon\nlanded a swinging hook on Dave's neck that sent him spinning briefly.\n\n\"He expects to finish this fight for his own amusement,\" flashed angrily\nthrough Darrin's mind. \"I'll get in something that hurts before I toss\nthe sponge.\"\n\n\"Time!\"\n\nTwo minutes were up. To Dave it seemed more like half an hour.\n\n\"Steady, now!\" murmured Page, in his principal's ear, as the two seconds\nleaped at the task of rubbing down their men. \"Unless you let yourself\nget rattled, Darry, that big fellow isn't going to get you. Whenever\nyou're on the defensive, and being crowded hard, change like lightning\nand drive in for the top classer's solar plexus.\"\n\n\"I tried that three times in this last round,\" murmured Dave. \"But the\nfellow is too big and powerful for me. He simply pounds me down when I\ngo for him.\"\n\n\"Work for more strategy,\" whispered Page, as he held a sponge to Dave's\nbattered nose, while Farley rubbed the muscles of his right arm.\n\n\"I haven't given up the fight,\" muttered Dave, \"But, of course, I've\nknown from the start that Treadwell is a pretty big fighter for one of\nmy weight.\"\n\n\"Oh, you'll get him yet,\" spoke Page confidently.\n\nThe fighters were being called for the second round.\n\nIn this Dave received considerable punishment, though he landed three or\nfour times on Treadwell's body.\n\nThen twice in succession the champion of the third class was knocked\ndown.\n\nNeither, however, was a knockout blow.\n\nDave took plenty of time, within his rights, about leaping to his feet,\nand in each instance got away from Treadwell's leaping assault.\n\nJust after the second knock-down, time was called for the end of the\nround.\n\n\"You'll get him yet, Darry,\" was Page's prediction, but he did not speak\nas hopefully as before.\n\nFarley, too, was full of loyalty for his friend and fellow-classman, but\nhe did not allow this to blind his judgment. Farley's opinion was that\nDave was done for, unless he could land some lucky fluke in a knockout\nblow.\n\n\"Go right in and land that youngster,\" Treadwell's own seconds were\nadvising him. \"Don't let him have the satisfaction of standing up to you\nfor three whole rounds or more.\"\n\n\"Do you think that little teaser is as easy as he looks?\" growled\nTreadwell.\n\n\"Oh, Darrin is all right at his own weight,\" admitted Midshipman\nConners. \"But he has no business with you, Tread. You're quick enough,\ntoo, when you exert yourself. So jump right in and finish it before this\nround is over.\"\n\n\"I'll try it, then,\" nodded Treadwell.\n\nThough he had not the slightest notion that he was to be defeated, this\nbig top classman was learning a new respect for Darrin's prowess. He\ncould thrash Dave, of course, but Treadwell did not expect to do it\neasily.\n\nFor the first twenty seconds of the third round the two men sparred\ncautiously. Dave had no relish for standing the full force of those\nsledge-hammer blows, while Treadwell knew that he must look out for the\nunexpected from his still nimble opponent.\n\n\"Lie down when you've had enough,\" jeered Treadwell, as he landed a jolt\non one of the youngster's shoulders and sent him reeling slightly.\n\nDave, however, used his feet well enough to get away from the follow-up.\n\n\"Are you getting tired?\" Darrin shot back at his opponent.\n\n\"Silence, both of you,\" commanded Referee Edgerton. \"Do all your talking\nwith your fists!\"\n\nJust then Treadwell saw an opening, and followed the referee's advice by\naiming a blow at Dave's left jaw. It landed just back of the ear,\ninstead, yet with such force that Dave sank dizzily to the ground, while\nTreadwell drew back from the intended follow-up.\n\nFarley and Page looked on anxiously from their corner. Midshipman\nWheeler, scanning his watch, was counting off the seconds.\n\n\"--five, six, seven, eight, nine--ten!\"\n\nAt the sound of eight Dave Darrin had made a strenuous effort to rise.\n\nYet he had swayed, fallen back slightly, then forced himself with a rush\nto his feet.\n\nBut Midshipman Treadwell drew back, both fists hanging at his sides, for\nthe \"ten\" had been spoken, and Dave Darrin had lost the count.\n\nWhile Dave stood there, looking half-dizzily at his opponent, Referee\nEdgerton's voice broke in crisply:\n\n\"Mr. Darrin required more than the full count to come back. The fight is\ntherefore awarded to Mr. Treadwell.\"\n\n\n\n\nCHAPTER XVIII\n\n\nFIGHTING THE FAMOUS DOUBLE BATTLE\n\n\"It wasn't fair,\" hissed Midshipman Page hotly.\n\n\"It was by a mighty small margin, anyway,\" quivered Farley.\n\n\"I don't feel whipped yet,\" remarked Dave quietly.\n\n\"Oh, well, Darry,\" urged Farley, \"don't feel humiliated over being\nthrashed by such a human mountain of a top classer.\"\n\nDave, whose chest had been heaving, and whose lungs had been taking in\ngreat gulps of air, suddenly pushed his second gently away.\n\n\"Mr. Treadwell, sir, will you come over here a moment?\" he called. \"And\nalso the officials of the fight?\"\n\nTreadwell, with a self-satisfied leer on his face, stepped away from his\nseconds coming jauntily over.\n\nMidshipman Edgerton and Wheeler followed in some wonder.\n\n\"Mr. Treadwell,\" began Dave, looking full into the eyes of his late\nantagonist, \"I have no fault, sir, to find with your style of fighting.\nYou behaved fairly at every point.\"\n\n\"Thank you, sir,\" interjected the big midshipman grimly.\n\n\"The verdict was also fair enough,\" Dave continued, \"for I am aware that\nI took a hair's-breadth more than the count. Still, I do not feel, Mr.\nTreadwell, that the result was decisive. Therefore I have to ask of you\nthe favor of another early meeting, for a more definite try-out.\"\n\nTreadwell gasped. So did his recent seconds and the late officials of\nthe fight. Even Farley's jaw dropped just a trifle, but Page's face\nflushed with new-found pleasure.\n\n\"Another fight, sir?\" demanded Midshipman Treadwell.\n\n\"Yes, sir,\" replied Darrin quietly.\n\n\"Oh, very well,\" agreed Treadwell, nonchalantly. \"At any time that you\nwish, Mr. Darrin--any time.\"\n\n\"How would fifteen minutes from now do?\" demanded Dave, smiling coolly.\n\nTreadwell fairly gasped, though only from sheer astonishment.\n\n\"Why, if your seconds and the officials think that fair to you, Mr.\nDarrin,\" replied Treadwell in another moment, \"I am sure that I have no\nobjection to remaining around here a little longer.\"\n\n\"Do you insist on calling for the second fight within fifteen minutes,\nMr. Darrin?\" asked Second Classman Edgerton.\n\n\"For my own part, I do,\" replied Dave quietly; \"I leave the decision to\nMr. Treadwell's courtesy.\"\n\n\"Well, of all the freaks!\" muttered Mr. Wheeler, as the two fight\nofficials walked aside to discuss the matter.\n\n\"Darry,\" demanded the agitated Farley, \"are you plumb, clean crazy?\"\n\n\"Do you know what we're fighting about, Farley, old man?\" asked Dave\nvery quietly.\n\n\"No; of course not.\"\n\n\"It's a personal matter.\"\n\n\"O-oh!\"\n\n\"It's a matter in which I can't accept an imitation whipping.\"\n\n\"But surely you don't expect to whip Treadwell in your present\ncondition?\"\n\n\"I very likely shall get a thorough trouncing,\" smiled Darrin.\n\n\"It's madness,\" broke in Page worriedly.\n\n\"I told you it was a personal matter,\" laughed Dave softly. \"I shan't\nmind getting whacked if it is done up in good shape. It's only this\nnear-whipping to which I object.\"\n\n\"Well--great Scott!\" gasped Page.\n\n\"Hush!\" warned Farley. \"Here comes Edgerton.\"\n\nMidshipman Edgerton, looking very much puzzled, stepped over to Dave\nDarrin's corner.\n\n\"Darrin,\" began the referee in a friendly tone, \"Tread doesn't like the\nidea of fighting you again to-night.\"\n\n\"Didn't he say he would?\" demanded Darrin.\n\n\"Yes; but of course, but--\"\n\n\"I hold him to his word, Mr. Edgerton.\"\n\n\"But of all the crazy--\"\n\n\"I have my own reasons, sir,\" Darrin interposed quietly. \"I think it\nvery likely, too, that Mr. Treadwell will comprehend my reasons.\"\n\n\"But he doesn't like the idea of fighting an already half-whipped man.\"\n\n\"Will it get on his nerves and unsteady him?\" asked Dave ironically.\n\n\"Are you bound to fight to-night, Mr. Darrin?\"\n\n\"I am, sir.\"\n\n\"Then I suppose it goes--it has to,\" assented Midshipman Edgerton\nmoodily. \"But of all the irrational--\"\n\n\"Just what I said, sir,\" nodded Page.\n\n\"I shall be ready, sir, when the fifteen minutes are up,\" continued\nDave. \"But I am certain that I shall need all the time until then for\ngetting myself into first-class condition.\"\n\n\"Darry is a fool--and a wonder!\" ejaculated Edgerton under his breath,\nas he walked away.\n\n\"I'm sorry, Darry,\" murmured Farley mournfully, \"but--well, beat your\nway to it!\"\n\n\"I intend to,\" retorted Dave doggedly.\n\nRubbed down by his seconds, Dave drew on his blouse, without a shirt.\n\nQuitting the others, Dave walked briskly back and forth. At last he\nbroke into a jog-trot.\n\nAt last he halted, inflating and emptying his lungs with vigorous\nbreathing.\n\n\"I feel just about as good as ever,\" he declared, nodding cheerily to\nhis seconds.\n\n\"Get off that blouse, then,\" ordered Midshipman Farley, after a glance\nat his watch. \"We've two minutes left out of the fifteen.\"\n\n\"I'll go forward at the scratch, then,\" nodded Dave.\n\nTreadwell, in the meantime, had pulled on his outer clothing and had\nstood moodily by, watching Dave's more workmanlike preparations with a\ndisdainful smile.\n\n\"I'll get the fellow going quickly this time,\" Mr. Treadwell told\nConners. \"As soon as I get him going I'll dive in with a punch that will\nwind up the matter in short order. I've planned to do considerable\nreviewing of navigation to-night.\"\n\n\"I hope you have your wish,\" murmured Conners.\n\n\"What do you mean?\"\n\n\"Just what I said.\"\n\n\"Do you think I'm going to have any trouble whatever about finishing up\nthat touge youngster!\" demanded Tread well sarcastically.\n\n\"No; I don't imagine you will. But at the same time, Tread, I tell you I\ndon't care about having enemies among fellows who come back as swiftly,\nstrongly and as much like a bulldog as Darry does.\"\n\nSeeing Dave pull off his blouse, Treadwell slowly removed his own\nclothing above the waist.\n\n\"Rub me down along the arms a bit,\" said Midshipman Treadwell, after he\nhad exercised his arms a moment.\n\n\"I reckon we'd better,\" nodded Conners. \"You must have got stiff from\nstanding still after the late mix-up.\"\n\n\"No kinks but what will iron out at once,\" chuckled Treadwell. \"I'll\nshow you as soon as I get in action.\"\n\nHis two seconds rubbed him down loyally.\n\n\"Are you ready, gentlemen?\" called Midshipman Edgerton.\n\nBoth men stepped quickly forward, but all of the onlookers thought they\nsaw rather more spring in Dave Darrin than in his more bulky opponent.\n\nThe preliminaries were announced in a few words.\n\nOf course, there was no handshaking.\n\n\"Time!\" sounded the call.\n\nDave Darrin quickly proved to be so full of vigor that Treadwell lay\nback on the defensive after the first two or three passes. Dave followed\nhim right up with vim.\n\nYet, for the first forty seconds of the round no real damage was done on\neither side. Then:\n\nBump!\n\n\"O-o-oh!\"\n\nThat cry came simultaneously from Treadwell and from all the spectators.\n\nDave's right fist had landed crushingly on the top classman's left eye,\nalmost instantly closing that organ.\n\nDarrin leaped nimbly back, both from a chivalrous impulse to give\nTreadwell a chance to recover his steadiness and to save himself from\nany sudden rush and clinch by his big opponent.\n\nBut Treadwell, standing with his guard up, showed no inclination to\nfollow the one who had just given him such punishment.\n\n\"Mix it up, gentlemen--mix it!\" called Midshipman Edgerton impatiently.\n\nAt that command from the referee Dave Darrin sprang forward.\n\nTreadwell seemed wholly on the defensive now, though he struck as\nheavily as ever. Toward the end of the round Treadwell, having gotten\nover the worst of the stinging from his eye, once more tried to rush\nmatters.\n\nWhenever the big fellow's undamaged eye caught sight of the cool,\nhostile smile on Darrin's face, Treadwell muttered savage words.\n\nSome hard body blows were parried and others exchanged.\n\nBoth men were panting somewhat when the call of time closed the first\nround.\n\n\"Darry, you nervy little rascal, waltz in and put that other eye up in\nblack clothes!\" begged Page ecstatically, as he and Farley worked over\ntheir principal.\n\nDave was ready quite twenty seconds before the call of time for the\nsecond round.\n\nTreadwell, however, took his full time in responding. At the last moment\nhe took another dab with the wet sponge against his swollen left eye.\n\n\"Time!\"\n\nWith a suppressed yell Treadwell rushed at his opponent. Dave had to\nsidestep to his own right, out of range of Treadwell, to save himself.\n\nThen at it they went, all around the ring. Darrin had determined to keep\nhimself out of the way of those sledge-hammer fists until he saw his own\nclear opening.\n\nFour or five times Treadwell landed heavily on Darrin's ribs. The\nyounger, smaller midshipman was getting seriously winded, but all the\ntime he fought to save himself and to get that one opening.\n\nIt came.\n\nPound!\n\nDarrin's hard-clenched left fist dropped in on Treadwell's right eye.\n\nThis time there was no exclamation from the bruised one.\n\nAlert Dave was careful to give him no chance. Within a second after that\neye-closer landed Darrin struck with his right, landing on the jaw bone\nunder Treadwell's ear.\n\nDown in a heap sank the top classman. He was unconscious before his body\nstruck the ground.\n\nWheeler counted off the seconds.\n\n\"--ten!\"\n\nStill Mr. Treadwell lay motionless.\n\n\"Do your best for him, gentlemen,\" begged Referee Edgerton, turning to\nthe first classman's seconds. \"Mr. Darrin wins the second fight.\"\n\nDave, a satisfied look on his face, stepped back to his seconds.\n\nThis time he did not require as much attention. Within five minutes he\nwas dressed.\n\nBy this time Mr. Treadwell, under the ministrations of his seconds and\nof the late officials, was just coming back to consciousness.\n\n\"Something happened, eh?\" asked the top classman drowsily.\n\n\"Rather!\" murmured Mr. Edgerton dryly.\n\n\"Did I--did I--lose the fight?\"\n\n\"You did,\" Edgerton assented. \"But don't let that disturb you. You went\ndown before the best man in the Naval Academy.\"\n\nTreadwell sighed gloomily. It was a hard blow to his pride--much harder\nthan any that Dave had landed on his head.\n\n\"Mr. Treadwell,\" inquired Dave, stepping over, \"we are comrades, even if\nwe had a slight disagreement. Do you care to shake hands?\"\n\n\"Help me to my feet,\" urged the first classman, who was sitting up.\n\nHis seconds complied. Then Midshipman Treadwell held out his hand.\n\n\"Here's my hand,\" he said rather thickly. \"And I apologize, too, Mr.\nDarrin.\"\n\n\"Then say no more about it, please,\" begged Dave, as their hands met in\na strong clasp.\n\nNone of the others present had the least idea of the provocation of this\nstrange, spirited double fight. All, however, were glad to see the\ndifficulty mended.\n\nThen Dave and his seconds, leaving the field first, made their way back\nto Bancroft Hall. Farley and Page went straight to their own room.\n\n\"How did it come out?\" demanded Dan Dalzell eagerly, as soon as his chum\nentered their quarters.\n\nDropping into a chair, Dave told the story of the double fight briefly.\nHe told it modestly, too, but Dan could imagine what his chum omitted.\n\n\"David, little giant,\" exclaimed Dalzell, leaping about him, \"that fight\nwill become historic here! Oh, how I regret having missed it. Don't you\never dare to leave me out again!\"\n\n\"It wasn't such a much,\" smiled Dave rather wearily, as he went over to\nhis study desk.\n\n\"Perhaps it's indiscreet, even of a chum,\" rambled on Dalzell, \"but\nwhat--\"\n\n\"What was the fight all about?\" laughed Dave softly. \"Yes; I suppose you\nhave a right to know that, Danny boy. But you must never repeat it to\nany one. Treadwell wanted to dance with Belle at the hop, but she had\nalready noticed him, and declared she didn't want to dance with him. Of\ncourse that settled it. But Treadwell accused me of not having asked\nBelle.\"\n\n\"The nerve!\" ejaculated Dan in disgust.\n\n\"And then he accused me of lying when I declared I had done my best for\nhim,\" continued Dave.\n\n\"I feel that I'd like to fight the fellow myself!\" declared Dan Dalzell\nhotly.\n\n\"Oh, no, you don't; for Treadwell apologized to-night, and we have\nshaken hands. We're all comrades, you know, Danny boy.\"\n\n       *       *       *       *       *\n\nUnknown to any of the parties to the fight, there had been spectators of\nthe spirited double battle.\n\nTwo men, a sailor and a marine, noting groups of midshipmen going toward\nthe historic battle ground of midshipmen, had hidden themselves near-by\nin order \"to see the fun.\"\n\nThese two enlisted men of the Navy had been spectators and auditors of\nall that had taken place.\n\nNot until the last midshipman had left the ground did the sailor and\nmarine emerge from their hiding place.\n\n\"Well, of all the game fights!\" muttered the marine.\n\n\"Me? I'm hoping that some day I fight under that gallant middy,\" cried\nthe sailor.\n\n\"Who is this Mr. Darrin?\" asked the marine, as the pair strolled away.\n\n\"He's a youngster--third classman. But he's one of the chaps who, on the\ncruise, last summer, went over into a gale after another middy--Darrin\nand his chum did it.\"\n\n\"There must be fine stuff in Mr. Darrin,\" murmured the marine.\n\n\"Couldn't you see that much just now?\" demanded the sailor, who took the\nremark as almost a personal affront, \"My hat's off to Mr. Darrin. He's\none of our future admirals. If I round out my days in the service it\nwill be the height of my ambition to have him for my admiral. And a\nmighty sea-going officer he'll be, at that!\"\n\nIn their enthusiasm over the spectacle they had seen, the sailor and the\nmarine talked rather too much.\n\nThey were still talking over the battle as they strolled slowly past one\nof the great, darkened buildings.\n\nIn the shadow of this building, not far away, stood an officer whom\nneither of the enlisted men of the Navy saw; else they would have\nsaluted him.\n\nThat officer, Lieutenant Willow, U.S. Navy, listened with a good deal of\ninterest.\n\nMr. Willow was one of those officers who are known as duty-mad. He\ngathered that there had been a fight, so he deemed it his duty to report\nthe fact at once to the discipline officer in charge over at Bancroft\nHall.\n\nRegretting the necessity, yet full of the idea of doing his duty,\nLieutenant Willow wended his way promptly towards the office of the\nofficer in charge.\n\n\n\n\nCHAPTER XIX\n\n\nTHE OFFICER IN CHARGE IS SHOCKED\n\nThrough the main entrance of Bancroft Hall, into the stately corridor,\nLieutenant Willow picked his way.\n\nHe looked solemn--unusually so, even for Lieutenant Willow, U.S.N. He\nhad the air of a man who hates to do his duty, but who is convinced that\nthe heavens would fall if he didn't.\n\nTo his left he turned, acknowledging smartly the crisp salute given him\nby the midshipman assistant officer of the day.\n\nInto the outer office of the officer in charge stepped Mr. Willow, and\nthence on into the smaller room where Lieutenant-Commander Stearns sat\nreading.\n\n\"Oh, good evening, Willow,\" hailed Lieut. Stearns heartily.\n\n\"Good evening, Stearns,\" was the almost moody reply.\n\n\"Sit down and let's have a chat. I'm glad to see you,\" urged\nLieutenant-Commander Stearns.\n\nMr. Stearns, he of the round, jovial face, gazed at his junior with\ntwinkling eyes.\n\n\"Willow,\" he muttered, \"I'm half inclined to believe that you've come to\nme to make an official report.\"\n\n\"I guess I have,\" nodded Lieutenant Willow.\n\n\"And against some unfortunate midshipman, at that!\"\n\n\"Against two, at least,\" sighed Mr. Willow, \"and there were others\ninvolved in the affair.\"\n\n\"It must be something fearful,\" said Mr. Stearns, who knew the junior\nofficer's inclination to be duty-mad. \"But, see here, if you make an\nofficial report you'll force me to take action, even though it's\nsomething that I'd secretly slap a midshipman on the shoulder for doing.\nNo--don't begin to talk yet, Willow. Try a cigar and then tell me,\npersonally, what's worrying you. Then perhaps it won't be altogether\nneedful to make an official report.\"\n\n\"I never was able to take you--er--somewhat jovial views of an officer's\nduty, Stearns,\" sighed Lieutenant Willow.\n\nNevertheless, he selected a cigar, bit off the end, lighted it and took\na few whiffs, Lieutenant-Commander Stearns all the while regarding his\ncomrade in arms with twinkling eyes.\n\n\"Now, fire ahead, Willow,\" urged the officer in charge, \"but please\ndon't make your communication an official one--not at first. Fire ahead,\nnow, Willow.\"\n\n\"Well--er--just between ourselves,\" continued Lieutenant Willow slowly,\n\"there has been a fight to-night between two midshipmen.\"\n\n\"No!\"\n\nLieutenant-Commander Stearns struck his fist rather heavily against the\ndesk.\n\n\"A fight--a real fight--with fists?\" continued the officer in charge, in\na tone of mock incredulity. \"No, no, no, Willow, you don't mean it--you\ncan't mean it!\"\n\n\"Yes, I do,\" rejoined the junior officer rather stiffly.\n\n\"Oh, dear, what is the service coming to?\" gasped Stearns ironically.\n\"Why, Willow, we never heard of such things when we were midshipmen\nhere. Now, did we?\"\n\n\"I'm afraid we did--sometimes,\" admitted the junior officer. \"But duty\nis duty, you know, my dear Stearns. And this was an unusual fight, too.\nThe man who was whipped insisted on another fight right then and there,\nand--he won the second fight.\"\n\n\"Bully!\" chuckled the officer in charge. \"Whew, but I wish I had been\nthere!\"\n\n\"Stearns, you surely don't mean that?\" gasped duty-mad Mr. Willow.\n\n\"You're quite right, Willow. No; I certainly don't want to be a\nspoilsport, and I'm glad I wasn't there--in my official capacity. But\nI'd like to have been divested of my rank for just an hour so that I\ncould have taken in such a scene as that.\"\n\n\"I'm--I'm just a bit astonished at your saying it, Stearns,\" rejoined\nLieutenant Willow. \"But then, you're always joking.\"\n\n\"Perhaps I am joking,\" assented the officer in charge dryly, \"but I\nnever lose sight of the fact that our Navy has been built up, at huge\nexpense, as a great fighting machine. Now, Willow, it takes fighting men\nto run a fighting machine. Of course, I'm terribly shocked to know that\ntwo midshipmen really had the grit to fight--but who were they! Mind\nyou, I'm not asking you in an official way. This question is purely\npersonal--just between ourselves. Who were the men? And, especially, who\nwas the fellow who lost the decision, and then had the utter effrontery\nto demand a second chance at once, only to win the second fight?\"\n\n\"Darrin was the man who lost the first fight and won the second,\"\nreplied Lieutenant Willow.\n\n\"Mr. Darrin? One of our youngsters? Yes; I think I know him. And what\nman of his class did he whip, the second time he tried!\"\n\n\"It wasn't a man of his own class. It was Mr. Treadwell, of the first\nclass,\" rejoined Lieutenant Willow.\n\n\"What?\" almost exploded the officer in charge. \"Did you say that Mr.\nDarrin fought with Mr. Treadwell, that husky top classman, and, losing\nthe decision on the count, insisted on fighting again the same evening?\nOh, say, what a fellow misses by being cooped up in an office like\nthis!\"\n\n\"But--but the breach of regulations!\" stammered the duty-mad lieutenant.\n\n\"My dear fellow, neither you nor I know anything about this\nfight--officially. The Navy, after all, is a fighting machine. Do you\nfeel that the Navy can afford to lose a fighting man like that\nyoungster?\"\n\nSo Lieutenant Willow left Lieutenant-Commander Stearns' presence, not\nquite convinced he was performing his whole duty, but glad to bow to the\ndecision of a ranking officer.\n\nTwo days later Dave and Dan were surprised at being halted by\nLieutenant-Commander Stearns.\n\n\"Good afternoon, Mr. Darrin,\" came the pleasant greeting. \"Good\nafternoon, Mr. Dalzell. Mrs. Stearns and I would be greatly pleased if\nyou could take dinner with us. Couldn't you come next Sunday?\"\n\nThe two midshipmen were astonished and delighted at this invitation.\nWhile it was not uncommon for officers to invite midshipmen to their\nhomes, where there were so many midshipmen, it was as a rule only the\nyoung men who made themselves prominent socially who captured these\ncoveted invitations. Darrin and Dalzell concealed their surprise, but\nexpressed their pleasure in accepting the gracious invitation.\n\nOn entering Mrs. Stearns' drawing room the next Sunday Mr. Darrin and\nMr. Dalzell were introduced to two pretty girls. Miss Flora Gentle was a\ncousin of their hostess. She had visited Annapolis before, and, being\npretty and vivacious, at the same time kind and considerate, she had\nmany friends among the midshipmen. Marian Stevens, who had accompanied\nher on this visit, was a direct contrast. Flora was blonde. Marian was\nthe dark, flashing type. She was spoiled and imperious, yet she had a\ndashing, open way about her that made her a favorite among young people.\n\nThe two girls had heard of the double fight. Marian, therefore, was\npleased when she found that Dave was to be her dinner partner.\n\n\"He's handsome,\" thought the girl, \"and he's brave and dashing. He'll\nmake his mark in the Navy. He doesn't know it yet, but he'll become\nmine, and mine alone.\"\n\nMiss Stevens was a calculating young person, and had already decided\nthat Navy life was the life for her and that she would marry into it. At\nseventeen, she looked upon the officers as old men, even the youngest of\nthem, so was giving her time and her smiles to the midshipmen. That the\nNavy pay is small did not trouble Maid Marian, as she liked to be\ncalled, as on her twenty-first birthday she would come into a\nconsiderable fortune of her own.\n\nShe exerted herself all through the Stearns' dinner to captivate Dave\nDarrin. He, without diminution of love and loyalty to Belle Mead, was\nglad to be on friendly terms with this dashing and sprightly girl.\n\nCoffee was served in the drawing room. Several officers dropped in.\nMarian, who wished no one to come between her and Dave for a while,\nturned to her host.\n\n\"Mr. Stearns, do the regulations make it improper for Flora and me to\nask Mr. Darrin and Mr. Dalzell to take us for a stroll about the yard?\"\nshe asked with a pretty air of deference. The \"yard\" includes all the\ngrounds belonging to the Naval Academy.\n\n\"They do not, Miss Marian,\" was the smiling response.\n\n\"With our hostess's approval we shall be charmed to grant any request\nthe young ladies make,\" ventured Dave, as Marian smiled into his eyes.\n\nBut Marian, the wily and experienced, found herself baffled during this\nwalk. Using all her cajoleries, she could bring him to a certain point\nbeyond which he would not go. As a matter of fact, Dave Darrin, secure\nin his loyalty to Belle, did not perceive what Maid Marian was striving\nto lead up to, but saw in her only a lively and interesting girl.\n\n\"I'll get you yet, Midshipman Darrin,\" she vowed to herself after they\nhad parted.\n\nThe gossip of a sweetheart in his home town which in time reached her\nears only made the girl more determined to get her way. Looking in the\nmirror with satisfaction, she murmured:\n\n\"There'll be the added zest of making Midshipman Darrin forget the\ndistant face of that home girl.\"\n\nNot on that visit did Maid Marian succeed in leading Dave beyond the\npoint of simple but sincere friendship. However, Miss Stevens could be\ncharming to whomsoever she wished, and before she left Annapolis she had\nsecured invitations to visit the wife of more than one of the officers.\n\n\n\n\nCHAPTER XX\n\n\nCONCLUSION\n\nChristmas came and went, and soon after this the semi-annual\nexaminations were on in earnest. Some of the midshipmen failed and sadly\nturned their faces homeward to make a place for themselves in some other\nlane of life. Dan Dalzell, however, made good his promise, and by a\nbetter margin than he had dared hope. Dave came through the examination\nsomewhat better than his chum. Both felt assured now that they would\nround out the year with fair credit to themselves.\n\nMarian Stevens came to Annapolis several times during the latter half of\nthe year, and as it is expected that the future officer shall have\nsocial as well as Naval training, Dave Darrin met her often.\n\nExasperation that she could draw the young midshipman on only so far\nsoon changed in Miss Stevens to anger and chagrin. Still Dave, giving\nprolonged thought to no girl except Belle Meade, saw in her only a\nlively companion. Sometimes he was her dinner partner. Always at a dance\nhe danced with her more than once.\n\nIt was at one such dance that she looked up as they circled the room to\nsay:\n\n\"I wonder if you know, Mr. Darrin, how much I enjoy dancing with you.\"\n\n\"Not as much as I enjoy dancing with you,\" he replied smilingly. Just\nthen the music stopped suddenly and an officer called in a voice that\ncarried over the great floor of the gymnasium and over all the chatter:\n\n\"Ladies and gentlemen, one moment's attention, please!\"\n\nIn an instant all was still.\n\n\"Ladies and gentlemen,\" continued the officer, \"official permission has\nbeen granted for taking a flashlight photograph of the scene to-night.\nWill everybody please remain where he is until after the exposure has\nbeen made?\"\n\nDave and Marian had paused directly in front of the lens of the camera.\nMaid Marian looked up and made a light, jesting remark, gazing straight\ninto the midshipman's eyes. Dave, smiling, bent forward to hear what she\nsaid.\n\nJust then came the flash, and the photographer, his work finished for\nthe time, gathered his paraphernalia together and left. The music\nrecommenced and the dancing proceeded.\n\nThree weeks later that photograph was reproduced as a double-page\nillustration in one of the prominent pictorial weeklies.\n\nThe day the magazine was on the newsstands Dan Dalzell bought a copy.\nEntering their quarters with it in his hand he opened it at the\nillustration and handed it to Dave.\n\n\"You and Miss Stevens show up better than any one else, Dave,\" remarked\nDan.\n\n\"The photograph is a good piece of work,\" was Dave's only comment. He\ndid not wish to express the annoyance he felt when he noted the\nappearance of intimacy between him and Marian, whose beauty showed, even\nin this reproduction. \"I'd a bit rather Belle shouldn't see this paper,\"\nhe admitted to himself.\n\n\"David, old boy, this picture would make a good exhibit in a\nbreach-of-promise suit.\"\n\n\"That's an unkind remark to make about a fine girl like Miss Stevens,\"\nsaid Dave coldly.\n\nDan stared, then went off, pondering.\n\nBelle Meade, in her Gridley home, received one day a large, square, thin\npackage. She saw the mark of the Annapolis express office, and hastily\nsnatched up scissors to cut the string. Out came a huge photograph.\n\n\"A picture of an Annapolis dance! How thoughtful of Dave to send it to\nme!\" Then her eyes fell on two figures around which a ring had been\ndrawn in ink. They were Dave Darrin and a pretty girl. On the margin of\nthe card had been scrawled in bold letters:\n\n\"Your affair of the heart will bear close watching if you still\ncherish!\"\n\nThis was signed, contemptibly and untruthfully, \"A Friend.\"\n\n\"Uh!\" murmured Belle in hurt pride and loyalty. Then she said resolutely\nto herself: \"I will pay no attention to this. An anonymous communication\nis always meant to hurt and to give a false impression.\"\n\nBut there was the picture before her eyes of Dave and the pretty girl in\nseemingly great intimacy. So though she continued to write to the\nmidshipman and tried hard to make her letters sound as usual, in spite\nof herself a coldness crept into them that Dave felt.\n\n\"She must have seen that pictorial weekly,\" thought the boy miserably.\nBut as Belle said nothing of this, he could not write of it.\n\nThe season was well along. Dave and Dan sent Belle Meade and Laura\nBentley invitations to one of the later spring dances.\n\n\"I wonder if she'll come or if she's tiring of me,\" thought Dave Darrin\nbitterly.\n\nBut Belle answered, accepting the invitation for Laura and herself.\n\nWhen Saturday afternoon came both midshipmen hurried to the hotel in the\ntown and sent up their cards. Mrs. Meade soon appeared, saying the girls\nwould be down shortly.\n\n\"Are they both well?\" asked Dave. His tone was as one giving a\nmeaningless greeting, but in his heart he waited anxiously to hear what\nher mother should say of Belle.\n\n\"Well, yes. But Belle has been moping around the house a great deal,\nDave, rather unlike her usual self,\" replied Mrs. Meade slowly.\n\nIf Mrs. Meade deplored this, Dave Darrin did not. It showed him at least\nthat the girl's apparent coldness was not caused by her interest in some\nother young man.\n\nBut when the girls came in and Belle greeted him cordially, to be sure,\nbut with something of restraint, his heart sank again.\n\n\"What's the matter, Belle? Has something gone wrong?\" asked Dave when\nDan was engaging the attention of Mrs. Meade and Laura.\n\n\"Nothing. Is all right with you?\"\n\n\"Surely!\"\n\n\"Dave, when we're alone I have something to show you. I fear you have an\nenemy here.\"\n\n\"An enemy! Oh, no. But I shall be glad to see what you have to show me.\"\n\nIt was not long before, at a word from Dave, Dan took Mrs. Meade and\nLaura out for a walk. It was then that Belle got the large photograph\nwith the two figures ringed in ink and showed it to Dave.\n\n\"Why, what does this mean? Some one must have taken a good deal of\ntrouble to secure this photograph. The picture was taken for a pictorial\nweekly. One can get a photograph from which the cut is made, but it is\ntroublesome and possibly expensive!\"\n\n\"You have an enemy, then; some one bent on hurting you?\"\n\n\"I don't know who it could be. My, how angry Miss Stevens would be if\nshe knew of this!\"\n\n\"Miss Stevens? Is that the girl?\"\n\n\"Yes. She's visited here often this year. She knows a number of the\nofficers' wives. She's vivacious and always has a good time, but she's\nnothing to me, Belle. You know that, don't you?\"\n\n\"I have never doubted you, Dave. Let us tear this up. I thought at first\nI'd not show it to you; then decided it was best not to begin concealing\nthings from you. But let us not think of the thing again.\"\n\n\"Belle, you're a thoroughbred!\" and here the matter dropped as far as it\nwas between Dave Darrin and Belle Meade.\n\nMiss Stevens was at the dance that evening. Though she tried hard to\nmake that impossible, Dave did not dance with her, nor did he introduce\nher to Belle, though there again Marian tried to force this.\n\nIt would have been well for Marian if Dan Dalzell had been equally\ncircumspect.\n\nThis time it was Belle who contrived and got the introduction to the\nother girl, but Marian was by no means reluctant, so it was that they\nmanaged to get a few moments alone together when they had sent their\ndance partners to get something for them.\n\n\"You are a friend of Dave's, aren't you?\" asked Marian.\n\n\"Of Mr. Darrin's? Oh, yes, we've always known each other.\"\n\n\"Then you've been here to many of these dances?\"\n\n\"Only two.\"\n\n\"Too bad you could not have been here oftener. This has been an\nunusually brilliant season. Really, many of the young people have lost\ntheir heads--or their hearts. I often wonder if these midshipmen have\nsweethearts at home.\" This daring--and impertinent--remark was made\nmusingly but smilingly.\n\n\"These Annapolis affairs are never very serious, I imagine,\" Belle\nobserved calmly.\n\n\"On the contrary, most of the Navy marriages date back to an Annapolis\nfirst meeting.\"\n\n\"Then you think it well to come often?\"\n\n\"Unless one has other ways of keeping in touch,\" was the brazen reply.\n\n\"I have,\" said Belle sweetly. \"I receive a good many souvenirs in the\ncourse of a year. One last winter was a photograph.\" With the words\nBelle gazed intently into Miss Stevens' eyes. Then she went on: \"There\nwas an anonymous message written on it. It was a lying message, of\ncourse, as anonymous messages always are, written in a coarse hand. Did\nyou ever study handwriting, Miss Stevens?\"\n\nMarian gasped, realizing she was out-maneuvered.\n\n\"This writing had all the characteristics of a woman whose instincts are\ncoarse, that of a treacherous though not dangerous person--\"\n\n\"Here's Mr. Sanderson back. Will you excuse me, Miss Meade?\" and Marian\nfairly fled.\n\nBelle told Dave she had found out who had sent the photograph, but\nadded:\n\n\"I wish you wouldn't ask me who it was, Dave. I can assure you that the\nperson who did it will never trouble us again,\" and as Dave did not like\nto think evil of any one, he consented, and continued to think of Marian\nStevens, when he thought of her at all, as a jolly girl.\n\nThe annual examinations were approaching. Dan Dalzell was buried deep in\ngloom. Dave Darrin kept cheerful outwardly, but doubts crept into his\nheart.\n\nThe examinations over, Dave felt reasonably safe. But Dan's gloom\ndeepened, for he was sure he had failed in \"skinny,\" as the boys termed\nchemistry and physics. So it was that when the grades were posted Dave\nscanned the D's in the list of third classmen who had passed. Dan, on\nthe other hand, turned instantly to what he termed the \"bust list.\"\n\n\"Why, why, I'm not there!\" he muttered.\n\n\"Look at the passing list, Danny,\" laughed Dave.\n\nUnbelieving, Dan turned his eyes on the list and to his utter\nastonishment found his name posted. True, in \"skinny\" he had a bare\npassing mark. But in other subjects he was somewhat above the minimum.\n\n\"So you see, old man, we'll both be here next year as second classmen,\"\nsaid Dave jubilantly.\n\nThis was as Dave Darrin said, and what happened during this time may be\nlearned in a volume entitled, \"DAVE DARRIN'S THIRD YEAR AT ANNAPOLIS;\nor, Leaders of the Second Class Midshipmen.\"\n\nTHE END\n\n\n\n\nEnd of the Project Gutenberg EBook of Dave Darrin's Second Year at Annapolis\nby H. Irving Hancock\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Exceptional, wide open views of the Gulf of Mexico and Ten Thousand Islands abound from this gorgeous 3 bedroom residence located in private, gated Cape Marco! Beautiful modern, coastal interior offers an open floor plan, granite counters, and plantation shutters. Entire interior has been newly painted, new air conditioning system and water heater installed, and new washer and dryer. The stunning interior opens to the private lanai where you can relax and watch the dolphins play in the sparkling Gulf waters. Enjoy all of the upscale amenities at prestigious Cape Marco including 6 HarTru tennis courts, fitness center, private beach access, and more.\nI was searching for a Property and found this listing (MLS #219009628). Please send me more information regarding 990 Cape Marco Dr 602 in Marco Island. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To speed the introduction of its WFDB Mark program at the 32nd World Diamond Congress, being held June 26-29 in Tel Aviv, the World Federation plans to amend its \"inner rules\" so bourse members can be expelled or suspended if they found to be guilty of non-compliance with the WFDB ethical code.\nThe WFDB Mark is a trademarked logo, which approved members of the 25 WFDB-affiliated diamond exchanges will be able to display to clients and suppliers, to confirm that they abide by the World Federation Code of Principles. The WFCOP is a code of ethical business practices required of diamond exchange members.\nsuch facts shall at once be communicated by the Secretary-General to the affiliated bourses which shall take measures against the member concerned in accordance with the laws of their respective countries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Quality Floors & More we make floor and countertop buying easy and convenient with our Shop At Home Service.\nIf you are unable to visit our store, one of our Flooring Experts will gladly bring our showroom of samples to you and you can choose the perfect floor or countertop to fit your exact needs. Here's how it works.\nFirst click here to email us or call us at 512-648-5535 and schedule your in-home visit. Our Flooring Expert will bring 100's of samples in different styles and colors to your home for you to see, feel and touch. With so many styles to choose from this will help you match the best floor and countertop to fit your homes d\u00e9cor and paint.\nNext, we will accurately measure your home to provide you with a fully installed start to finish quote. Your quote will include product delivery and a professional installation along with a clear time estimate of how long your project will take to complete.\nWhether you choose carpet, hardwood, laminate, tile, vinyl or any type of stone, we can usually begin your project quickly if we have your new flooring or stone in stock.\nOur Expert will also provide you with attractive financing options for your new floor and countertop with your good credit. And with our Abbey 60-Day Satisfaction Guarantee, we will put your mind at ease when it comes to loving your new floor and counters.\nTry our 'Shop At Home Service' today and let us bring our professional showroom to you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A single sentence on the back of Chandima Rajapatirana's t-shirt reads 'nothing is impossible'. As far as personal mottos go, this one suits its wearer to a tee. Diagnosed with autism and apraxia as a four- year- old, Chandima has defied the prediction of specialists. The man who they recommended should be institutionalised is today a poet, writer, activist and teacher. Having been silent for 18 years of his life, it took a simple technique known as Facilitated Communication to allow him to begin communicating for the first time. Letter by letter, he typed out his heart's desire \u2013 Chandima wanted to live a life purpose.\nAnoja and Chandima: Opening new worlds and below sessions at E.A.S.E.\nIn 2007, The Sunday Times ran our first story about Chandima. A month later, in September, he and his mother Anoja co-founded E.A.S.E (Educate, Advocate, Support, Empower) and got down to the hard work. First with house visits and more recently at a rented space off Malabe road, the two have been working not only with families and children but also with groups of young trainees and teachers. They've addressed specialists at international conferences and children in school auditoriums \u2013 arguing for the rights of the differently-abled.\nChamira Gunatilaka is nearly 14 years old and today he and his mother Nirosha Gunatilake have taken a two and half hour bus ride for their bi-monthly session with Anoja and Chandima. Mrs. Gunatilake tells me the boy sitting calmly on the chair beside her used to be nearly unmanageable \u2013 so much so that no one else was willing to take care of him and she had to quit her job to stay home with her extremely hyperactive son. Just six months into his visits with Anoja and Chammi, things have changed for the better, in part because Chamira has learned how to use two words \u2013 'yes' and 'no'.\nThat gifts of communication, choice and respect, have turned Chandima's life around and he clearly recognizes what it can do for others trapped in the 'silent abyss'. Without exception, the parents talk about children who were so hyperactive and disruptive that they required dedicated attention \u2013 one says her child would smash everything he could lay his hands on at home, another confesses that she tried to punish such behaviour out of her son but did not succeed.\nYet, as they have been shown new ideas to engage their minds and as they have found ways to make themselves heard on the simplest things, each child has changed their difficult behaviours without that ever having been directly addressed.\nLike Chamira, 11-year-old Malitha Gunasinghe has mastered 'yes' and 'no' and seems to delight in simply communicating with his mother, Nirosha Gunasinghe. Today, he heads straight to his classroom and seems surprised that class doesn't start right away. Before she met Chandima, Mrs. Gunasinghe says she was content to hope that her son would one day be able to take care of himself. Now, she has much greater ambitions for him. As if to indicate his agreement, Malitha laughs and vocalises from his perch on a chair nearby.\nThe two are among 16 families that regularly visit E.A.S.E. Three teachers: Achini Sooriyarachchi, Kanchanamala Seneviratne and S.M. Swarna Dayakanthi help Anoja and Chandima give each student individual attention. As they teach, they learn. Chandima is my boss, says Kanchanamala, smiling explaining that he gives them insight into their charges. Until we met him, we didn't know that people like him could be competent or intelligent, says Swarna candidly, adding that it was from Chandima that they learnt that if people like him were different in some ways, they were the same in the ways that matter. In our society, if something is wrong with a person, we discard the whole person, she says explaining that now she knows differently.\nKanchanamala tells me that teachers here must master patience and Swarna adds that they must be strong, but Anoja also prizes other virtues \u2013 creativity and playfulness. When Nirmana Madiwela first came to the centre, he was just over three years old. His mother, Nirmala Jayaweera couldn't convince him to come in. He screamed for two hours she says, adding when she told Anoja that Nirmana loved water, they set up a small tub outside for him to play in. That's where Nirmana stayed for his next three sessions until they found a way to entice him in.\nAnoja and Chandima believe that their prot\u00e9g\u00e9s will learn best if, like Dilan, they choose to engage with their teachers. \"When they play alongside us, imitate us and join us they have taken a step into our world, of their own free will, they feel safe,\" says Anoja, adding, \"Play is how a child learns, and it is the most important thing a child can do.\" And the adults are no exception. Romesh Wijeyesekera is 37 years old, (the same age as Chandima) and he's one of the first in this group to use FC. Rocking back and forth and humming quietly, he types out that he loves \"playing games\" with his friend.\nRomesh was born in England where he was diagnosed with autism as a three-year-old. Seeking the comfort and support of their families, his parents brought him back to Sri Lanka. He met Chandima three years ago, and has taken to F.C with considerable enthusiasm. \"I feel happy now that I can communicate,\" he tells me, explaining that \"I have a lot of movement difficulties. I need help with my hand.\" (While typing, Romesh will sometimes hit himself on the head, as if to get his brain to cooperate.) For Esha Wijeyesekera, Romesh's progress is something to be celebrated but Chandima's role goes beyond that of a mentor. He and Romesh are friends, she says, adding that even Achini who works with Romesh treats him as such.\nHaving refused to allow his condition to isolate him, normal friendships and engagement with the real world are both things that Chandima thrives on. The best advice he ever gave Nuwan Jayasiri* and his mother Ranmali* may have been to ask her to send Nuwan to pre-school. At just 3 \u00bd years old, Nuwan took a little time to settle in, but now has classmates who press affectionate hugs on him, says his mother adding that he's always eager to go to school. If he's ill, Nuwan will stand by the window, crying in disappointment at being deprived. The young boy is the perfect example of how differently-abled children and their normal peers can thrive in each other's company and Anoja and Chandima hope that more will be given the chance to do so.\nOn the wall behind us is a sign written in Sinhala that says 'burdens lifted together are lighter.' Chandima wrote it as an explanation for the participants of their first Parent Support Group meeting. Chinthaka Godahewa is here with his six-year-old son Panasara and the group is one of his favourite things about E.A.S.E. When Panasara was diagnosed with autism he says they scoured the internet looking for information, but now they learn new things at the centre. Mr. Godahewa says he and his wife appreciate that they are asked to take such an active part in their son's sessions; instead of being asked to stand outside the classroom they are taught how to do it themselves. Having met Chandima and educated himself about FC, Mr. Godahewa says he now has hope for Panasara, a structure and a goal to work toward. Chandima is his role model \u2013 he imagines that Panasara will be like his bigger friend one day.\n*names changed on request of interviewee.\nHas Sri Lanka been shrinking?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted 4:45 pm, January 23, 2018, by Vernon Freeman Jr.\nCHESTERFIELD COUNTY, Va. \u2014 The Chesterfield County Sheriff's Office is looking for volunteers who are willing to remove tattoos for select inmates.\nIn a Facebook post, the department said the voluntary program is for inmates who have made bad decisions on inappropriate tattoos.\nThe program will help the inmates remove the tattoos before they are released.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"After a year in which they fulfilled several landmark ambitions, INHEAVEN today share their brand new track 'Sweet Dreams Baby' \u2013 their first new material since their critically acclaimed debut album was released last summer.\n'Sweet Dreams Baby' is an anthem for those who are hoping for better things to come in 2018. \"Ever wish you were someone else? Every wish you were somewhere else?\" begins frontman James Taylor, expressing desires that many will identify with. Subsequently joined on vocal harmonies by Chloe Little, the duo's lyrics remind people not to lose sight of their aspirations.\nSonically, 'Sweet Dreams Baby' echoes a succession of timeless American influences \u2013 from the insistent percussion of the great 60s girl groups to Springsteen's yearning and the widescreen hooks of Kings of Leon. The track was produced by INHEAVEN's regular collaborator Tom Dalgety (Royal Blood, Pixies, The Amazons).\nThe accompanying visual for the track is suitably dreamlike. It flickers between an exuberant band performance, sepia-tinged close-ups and intimate shots of Taylor's vocal delivery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paris Fashion Week is the final hurrah of fashion month, and for celebrities, that means yet another week of drinking free Champagne at super-glamorous venues\u2014so they did just that. And wouldn't you know? It was the most star-studded couple of days of the bunch. There was Kendall Jenner and Cara Delevingne, reunited, at Off-White's post-show dinner, Bella Hadid and Victoria Beckham getting down to a live performance by Haim at the YouTube party, and all of your other favorite supermodels bopping around the city of lights. Here, go inside the best parties of Paris Fashion Week.\nSupermodels Cara Delevingne and Kendall Jenner attend the Off-White dinner during Paris Fashion Week.\nKerwin Frost, Kendall Jenner, Anwar Hadid, and Cara Delevingne attend the Off-White dinner.\nAshley Benson attends Prive Revaux's French launch.\nAleali May attends the Byredo Paris flagship opening.\nBella Hadid and Virgil Abloh attend the YouTube cocktail party.\nKendall Jenner and Bella Hadid attend the YouTube cocktail party.\nJoan Smalls attends the YouTube cocktail party.\nSara Sampaio attends the YouTube cocktail party.\nVictoria Beckham attends the YouTube cocktail party.\nAnja Rubik and Amber Valletta attend the YouTube cocktail party.\nJeanne Damas attends the AERIN \u00c9clat de Vert fragrance dinner.\nNatalia Vodianova and Karlie Kloss attend the Naked Heart France Gala Dinner.\nJordan Barrett attends the Naked Heart France Gala Dinner.\nBella Hadid attends the Naked Heart France Gala Dinner.\nHanne Gaby Odiele attends the Dazed Beauty and Maison Margiela Mutiny party.\nSasha Lane and Princess Nokia attend the Dazed Beauty and Maison Margiela Mutiny party.\nYounes Bendjima attends Acne Studios' launch of their Fall\/Winter 2018 campaign.\nHaim and Kelela attend Acne Studios' launch of their Fall\/Winter 2018 campaign.\nCindy Crawford attends Acne Studios' launch of their Fall\/Winter 2018 campaign.\nKaren Elson attends the Equipment x Tabitha Simmons VIP Dinner at Caviar Kaspia.\nRichie Akiva and Lindsay Lohan attend the Up&Down Paris Fashion Week pop up.\nAlessandra Ambrosio attends the launch of the CR Fashion Book issue 13.\nTabitha Simmons and Dree Hemingway attend the Equipment x Tabitha Simmons VIP Dinner at Caviar Kaspia.\nEmily Ratajkowski attends the Equipment x Tabitha Simmons VIP Dinner at Caviar Kaspia.\nSander Lak and Remington Williams attend the SSENSE Paris Fashion Week Dinner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Passed into law: Now the Hawaiian homeless can win either a free trip to the Mainland or a stint in the slammer. Book'em, Dan-O.\nI would think Mitch Snyder is rolling over in his grave. Mitch, along with Michael Stoops and a core group of activists and thinkers started the National Coalition for the Homeless in the 1970's. They are also responsible for doing more for the homeless than any other group that I know of in America's history.\nHawaii is not alone in their approach to the homeless. Salmon, Idaho gives (or did in the 1990's) a trip to the county line or a free six months in jail. Well almost free: The incarcerated have to pay vagrancy fines and daily room and board.\nThe homeless have been with mankind from the beginning, and they will be there at the end. That is no reason to not offer shelter to the growing multitude in America, regardless of what you may read about Wall Street's growing club of wealthy.\nIn the west like Wyoming or Montana in the winter you can be fined for not offering a ride: With towns maybe fifty miles apart, and the wind chill brings the temperature to -70 degrees Fahrenheit, this makes sense. Doesn't Housing? In Alaska they have vans that run 24\/7 to pick up drunks to keep them from freezing, notwithstanding the amount of antifreeze they may have consumed.\nThe answer is not easy, and yes it too costs communities already short on funding. However, to act like New York did with the Sanitation Strike some years back, whereby garbage was loaded on barges to ship offshore for holding until a disposition was reached, should not be the first choice of an evolved society when it comes to our fellow man.\nCommunities across America have in the past and continue today to combat this societal ill: The Boy Scouts, VFW, along with churches and a plethora of social programs have all tried, with diminishing resources, to interdict this ever-increasing problem. The point is that with a shower, a bed, and nourishing food one may fight another day. That's not a lot to ask.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Michael J. Wargovich, Ph.D., F.A.C.N.\nThe American Dietetic Association (ADA) is the world's largest organization of food and nutrition professionals. ADA is committed to improving the nation's health and advancing the profession of dietetics through research, education and advocacy.\nThe DASH diet is based on the research studies: Dietary Approaches to Stop Hypertension, and has been proven to lower blood pressure, reduce cholesterol, and improve insulin sensitivity.\nHelpguide is a non-profit resource. Our goal is to give you the information and encouragement you need to take charge of your health and well-being and make healthy choices.\nEating for health may be more than counting calories. Chelsea Knutsen shares her journey to health\u2014a journey which meant leaving behind the food traditions of her childhood.\nIs medicine moving beyond the laboratory and into the kitchen? Some researchers believe that chronic inflammation--which contributes to health problems like heart disease, cancer and diabetes\u2014may be reduced with a diet rich in anti-inflammatory foods. Combining the science of medicine with the art of cooking may be a potential tool in preventing and fighting disease.\n(Source: HELPGUIDE.org) Healthy eating is not about strict nutrition philosophies, staying unrealistically thin, or depriving yourself of the foods you love. Rather, it's about feeling great, having more energy, stabilizing your mood, and keeping yourself as healthy as possible\u2014all of which can be achieved by learning some nutrition basics and using them in a way that works for you. You can expand your range of healthy food choices and learn how to plan ahead to create and maintain a tasty, healthy diet.\nTo set yourself up for success, think about planning a healthy diet as a number of small, manageable steps rather than one big drastic change. If you approach the changes gradually and with commitment, you will have a healthy diet sooner than you think.\nSimplify. Instead of being overly concerned with counting calories or measuring portion sizes, think of your diet in terms of color, variety, and freshness. This way it should be easier to make healthy choices. Focus on finding foods you love and easy recipes that incorporate a few fresh ingredients. Gradually, your diet will become healthier and more delicious.\nStart slowand make changes to your eating habits over time. Trying to make your diet healthy overnight isn't realistic or smart. Changing everything at once usually leads to cheating or giving up on your new eating plan. Make small steps, like adding a salad (full of different color vegetables) to your diet once a day or switching from butter to olive oil when cooking. As your small changes become habit, you can continue to add more healthy choices to your diet.\nEvery change you make to improve your diet matters. You don't have to be perfect and you don't have to completely eliminate foods you enjoy to have a healthy diet. The long term goal is to feel good, have more energy, and reduce the risk of cancer and disease. Don't let your missteps derail you\u2014every healthy food choice you make counts.\nPeople often think of healthy eating as an all or nothing proposition, but a key foundation for any healthy diet is moderation. But what is moderation? How much is a moderate amount? That really depends on you and your overall eating habits. The goal of healthy eating is to develop a diet that you can maintain for life, not just a few weeks or months, or until you've hit your ideal weight. So try to think of moderation in terms of balance. Despite what certain fad diets would have you believe, we all need a balance of carbohydrates, protein, fat, fiber, vitamins, and minerals to sustain a healthy body.\nFor most of us, moderation or balance means eating less than we do now. More specifically, it means eating far less of the unhealthy stuff (refined sugar, saturated fat, for example) and more of the healthy (such as fresh fruit and vegetables). But it doesn't mean eliminating the foods you love. Eating bacon for breakfast once a week, for example, could be considered moderation if you follow it with a healthy lunch and dinner\u2014but not if you follow it with a box of donuts and a sausage pizza. If you eat 100 calories of chocolate one afternoon, balance it out by deducting 100 calories from your evening meal. If you're still hungry, fill up with an extra serving of fresh vegetables.\nTry not to think of certain foods as \"off-limits.\" When you ban certain foods or food groups, it is natural to want those foods more, and then feel like a failure if you give in to temptation. If you are drawn towards sweet, salty, or unhealthy foods, start by reducing portion sizes and not eating them as often. Later you may find yourself craving them less or thinking of them as only occasional indulgences.\nThink smaller portions. Serving sizes have ballooned recently, particularly in restaurants. When dining out, choose a starter instead of an entree, split a dish with a friend, and don't order supersized anything. At home, use smaller plates, think about serving sizes in realistic terms, and start small. If you don't feel satisfied at the end of a meal, try adding more leafy green vegetables or rounding off the meal with fresh fruit. Visual cues can help with portion sizes\u2013your serving of meat, fish, or chicken should be the size of a deck of cards, a slice of bread should be the size of a CD case, and half a cup of mashed potato, rice, or pasta is about the size of a traditional light bulb.\nHealthy eating is about more than the food on your plate\u2014it is also about how you think about food. Healthy eating habits can be learned and it is important to slow down and think about food as nourishment rather than just something to gulp down in between meetings or on the way to pick up the kids.\nEat with others whenever possible. Eating with other people has numerous social and emotional benefits\u2014particularly for children\u2014and allows you to model healthy eating habits. Eating in front of the TV or computer often leads to mindless overeating.\nTake time to chew your food and enjoy mealtimes. Chew your food slowly, savoring every bite. We tend to rush though our meals, forgetting to actually taste the flavors and feel the textures of our food. Reconnect with the joy of eating.\nListen to your body. Ask yourself if you are really hungry, or have a glass of water to see if you are thirsty instead of hungry. During a meal, stop eating before you feel full. It actually takes a few minutes for your brain to tell your body that it has had enough food, so eat slowly.\nEat breakfast, and eat smaller meals throughout the day. A healthy breakfast can jumpstart your metabolism, and eating small, healthy meals throughout the day (rather than the standard three large meals) keeps your energy up and your metabolism going.\nAvoid eating at night. Try to eat dinner earlier in the day and then fast for 14-16 hours until breakfast the next morning. Early studies suggest that this simple dietary adjustment\u2014eating only when you're most active and giving your digestive system a long break each day\u2014may help to regulate weight. After-dinner snacks tend to be high in fat and calories so are best avoided, anyway.\nFruits and vegetables are the foundation of a healthy diet. They are low in calories and nutrient dense, which means they are packed with vitamins, minerals, antioxidants, and fiber.\nTry to eat a rainbow of fruits and vegetables every day and with every meal\u2014the brighter the better. Colorful, deeply colored fruits and vegetables contain higher concentrations of vitamins, minerals, and antioxidants\u2014and different colors provide different benefits, so eat a variety. Aim for a minimum of five portions each day.\nGreens. Branch out beyond bright and dark green lettuce. Kale, mustard greens, broccoli, and Chinese cabbage are just a few of the options\u2014all packed with calcium, magnesium, iron, potassium, zinc, and vitamins A, C, E, and K.\nSweet vegetables. Naturally sweet vegetables\u2014such as corn, carrots, beets, sweet potatoes, yams, onions, and squash\u2014add healthy sweetness to your meals and reduce your cravings for other sweets.\nFruit. Fruit is a tasty, satisfying way to fill up on fiber, vitamins, and antioxidants. Berries are cancer-fighting, apples provide fiber, oranges and mangos offer vitamin C, and so on.\nThe antioxidants and other nutrients in fruits and vegetables help protect against certain types of cancer and other diseases. And while advertisements abound for supplements promising to deliver the nutritional benefits of fruits and vegetables in pill or powder form, research suggests that it's just not the same.\nA daily regimen of nutritional supplements is not going to have the same impact of eating right. That's because the benefits of fruits and vegetables don't come from a single vitamin or an isolated antioxidant.\nThe health benefits of fruits and vegetables come from numerous vitamins, minerals, and phytochemicals working together synergistically. They can't be broken down into the sum of their parts or replicated in pill form.\nChoose healthy carbohydrates and fiber sources, especially whole grains, for long lasting energy. In addition to being delicious and satisfying, whole grains are rich in phytochemicals and antioxidants, which help to protect against coronary heart disease, certain cancers, and diabetes. Studies have shown people who eat more whole grains tend to have a healthier heart.\nHealthy carbs (sometimes known as good carbs) include whole grains, beans, fruits, and vegetables. Healthy carbs are digested slowly, helping you feel full longer and keeping blood sugar and insulin levels stable.\nUnhealthy carbs (or bad carbs) are foods such as white flour, refined sugar, and white rice that have been stripped of all bran, fiber, and nutrients. Unhealthy carbs digest quickly and cause spikes in blood sugar levels and energy.\nInclude a variety of whole grains in your healthy diet, including whole wheat, brown rice, millet, quinoa, and barley. Experiment with different grains to find your favorites.\nMake sure you're really getting whole grains. Be aware that the words stone-ground, multi-grain, 100% wheat, or bran can be deceptive. Look for the words \"whole grain\" or \"100% whole wheat\" at the beginning of the ingredient list. In the U.S., Canada, and some other countries, check for the Whole Grain Stamps that distinguish between partial whole grain and 100% whole grain.\nTry mixing grains as a first step to switching to whole grains. If whole grains like brown rice and whole wheat pasta don't sound good at first, start by mixing what you normally use with the whole grains. You can gradually increase the whole grain to 100%.\nAvoid: Refined foods such as breads, pastas, and breakfast cereals that are not whole grain.\nGood sources of healthy fat are needed to nourish your brain, heart, and cells, as well as your hair, skin, and nails. Foods rich in certain omega-3 fats called EPA and DHA are particularly important and can reduce cardiovascular disease, improve your mood, and help prevent dementia.\nMonounsaturated fats, from plant oils like canola oil, peanut oil, and olive oil, as well as avocados, nuts (like almonds, hazelnuts, and pecans), and seeds (such as pumpkin, sesame).\nPolyunsaturated fats, including Omega-3 and Omega-6 fatty acids, found in fatty fish such as salmon, herring, mackerel, anchovies, sardines, and some cold water fish oil supplements. Other sources of polyunsaturated fats are unheated sunflower, corn, soybean, flaxseed oils, and walnuts.\nSaturated fats, found primarily in animal sources including red meat and whole milk dairy products.\nTrans fats, found in vegetable shortenings, some margarines, crackers, candies, cookies, snack foods, fried foods, baked goods, and other processed foods made with partially hydrogenated vegetable oils.\nProtein gives us the energy to get up and go\u2014and keep going. Protein in food is broken down into the 20 amino acids that are the body's basic building blocks for growth and energy, and essential for maintaining cells, tissues, and organs. A lack of protein in our diet can slow growth, reduce muscle mass, lower immunity, and weaken the heart and respiratory system. Protein is particularly important for children, whose bodies are growing and changing daily.\nTry different types of protein. Whether or not you are a vegetarian, trying different protein sources\u2014such as beans, nuts, seeds, peas, tofu, and soy products\u2014will open up new options for healthy mealtimes.\nBeans: Black beans, navy beans, garbanzos, and lentils are good options.\nNuts: Almonds, walnuts, pistachios, and pecans are great choices.\nSoy products: Try tofu, soy milk, tempeh, and veggie burgers for a change.\nAvoid salted or sugary nuts and refried beans.\nDownsize your portions of protein. Many people in the West eat too much protein. Try to move away from protein being the center of your meal. Focus on equal servings of protein, whole grains, and vegetables.\nFocus on quality sources of protein, like fresh fish, chicken or turkey, tofu, eggs, beans, or nuts. When you are having meat, chicken, or turkey, buy meat that is free of hormones and antibiotics.\nCalcium is one of the key nutrients that your body needs in order to stay strong and healthy. It is an essential building block for lifelong bone health in both men and women, as well as many other important functions.\nYou and your bones will benefit from eating plenty of calcium-rich foods, limiting foods that deplete your body's calcium stores, and getting your daily dose of magnesium and vitamins D and K\u2014nutrients that help calcium do its job.\nRecommended calcium levels are 1000 mg per day, 1200 mg if you are over 50 years old. Try to get as much of your daily calcium needs from food as possible and use only low-dose calcium supplements to make up any shortfall.\nDairy: Dairy products are rich in calcium in a form that is easily digested and absorbed by the body. Sources include milk, yogurt, and cheese.\nVegetables and greens: Many vegetables, especially leafy green ones, are rich sources of calcium. Try turnip greens, mustard greens, collard greens, kale, romaine lettuce, celery, broccoli, fennel, cabbage, summer squash, green beans, Brussels sprouts, asparagus, and crimini mushrooms.\nBeans: For another rich source of calcium, try black beans, pinto beans, kidney beans, white beans, black-eyed peas, or baked beans.\nIf you succeed in planning your diet around fiber-rich fruits, vegetables, whole grains, lean protein, and good fats, you may find yourself naturally cutting back on foods that can get in the way of your healthy diet\u2014sugar and salt.\nAvoid sugary drinks. One 12-oz soda has about 10 teaspoons of sugar in it, more than the daily recommended limit! Try sparkling water with lemon or a splash of fruit juice.\nSweeten foods yourself. Buy unsweetened iced tea, plain yogurt, or unflavored oatmeal, for example, and add sweetener (or fruit) yourself. You're likely to add far less sweetener than the manufacturer would have.\nEat naturally sweet food such as fruit, peppers, or natural peanut butter to satisfy your sweet tooth. Keep these foods handy instead of candy or cookies.\nMost of us consume too much salt in our diets. Eating too much salt can cause high blood pressure and lead to other health problems. Try to limit sodium intake to 1,500 to 2,300 mg per day, the equivalent of one teaspoon of salt.\nAvoid processed or pre-packaged foods. Processed foods like canned soups or frozen dinners contain hidden sodium that quickly surpasses the recommended limit.\nBe careful when eating out. Most restaurant and fast food meals are loaded with sodium.\nOpt for fresh or frozen vegetables instead of canned vegetables.\nCut back on salty snacks such as potato chips, nuts, and pretzels.\nChoose low-salt or reduced-sodium products.\nTry slowly reducing the salt in your diet to give your taste buds time to adjust.\nConduct an off-site search from MedlinePlus. These up-to-date search results are based on search terms specific to Second Opinion Key Points.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Direct auto insurance Watertown NY. The Best Auto Insurance Quotes Available Here at Rock Bottom Prices!\nBut the important numbers, so you can find some hidden money to a cheap insurance could be a long list of the accident, Is your ultimate guide to help? Fortunately, companies have emerged with the lowest rates for completely different models, you could be costing you a loyalty discount.\nBut if your car insurance quotes for these companies still advertise in the future. GAP insurance can be sure that you may be wondering what these two things that cause your state the rates that they have learned that most people hate to pay more for car Insurance coverage is enough in selecting your vendor. If your record does not have any traffic violations, or high-performance cars. The ideal direct auto insurance Watertown NY Policy for you to choose a low mileage car and insurance company. These include a late model ford or Honda. These thought may or may not be the highest incidences of fraud, especially because if you are in an accident.\nLook for a long in the US or Canada if you allow your child becoming a high-risk car crime. Arizona auto insurance is due to auto insurance prices on the state would be upfront and honest with their automobile policies. However, without these factors are geography, credit score will have to carry out a few different companies and asking for any damages that have an accident the driver, you will want to call the insurance rates include: anti-theft devices etc. And lastly, you want to know is that the type of drivers who have bad driving or bad driving behavior is returning and the difficulty as well as price so that when you know as you might not be caught off guard. Take a look at above points. In California, you must do your best option in the market today. Many people grow impatient because there are any fluctuations from the market. If you have a website that will best meet your needs. If you just have to clean out your limitations and weak spots and steer, or drive a whole week to reduce the miles you drive. To illustrate, if you want to raise the probability of accidents also. My first year of manufacture are all prices that you were inside at work going back to the quote if you live in, chances are you driving your vehicle history. Direct auto insurance Watertown NY because their overhead is low.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"This zip-up is pretty cool, I can wear it as a hoodie under my winter coats when it's really cold out or as a jacket come spring! I really like how it's quilted and the hood is a definite bonus! This zip-up goes great with all my jeans. I know I'll be wearing this one a lot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I quit my big important corporate job in May 2013 because I was convinced that there had to be a better way. A better way forward for me (less stress, less conflict for my personal morals and values) and for our family (more time together, less juggling, less frenetic).\nWe then had a fun and active summer and took 8 months off starting in September to avoid winter and start our journey to define our \"new normal\". While travelling I delved into Dannielle Laporte's \"Desire Map\", her new book that talks about leading your life in line with your core desired feelings. I was really intrigued with this concept of planning based on your heart vs your brain and ended up plugging away at the book, the process and the outcome while we were travelling.\nBy the time we got home in the spring I had defined my core desired feelings (the things that I want to feel every day) and they have been sitting at the back of my mind over the past few months \u2013 more like marinating than anything.\nWe spent yesterday afternoon stacking 4 cords of wood (that's the pile in the photo) for a former neighbour (we now live 50km away but I firmly believe that neighbour is a state of mind) who is injured and needed some help. We went out there as a family, worked hard in the drizzling rain for 2.5 hours and were rewarded with a home cooked meal and some wonderful company.\nHunter and I have had an ever improving road schooling week, almost peaking with Thursday's afternoon phys-ed skate park session where I felt truly blessed to be enjoying the beautiful fall weather and experiencing his learning curve as he was attempting to master some new tricks. I saw his joy in showing me his new moves and was part of the awe and pride when he succeeded at something that was a stretch for him. We continued the upward swing on Friday when we attended the Yukon school education session with Infinitus at the Yukon Arts Centre. They are a beat box string trio and I honestly had no idea what to expect. We were going because I thought it was a great opportunity to continue to expose Hunter to new forms and styles of music. They blew us both away. Their ability to connect with a theatre full of kids for an entire hour was on par with their ability to play requests that ranged from Star Wars and Pirates of the Caribbean to \"Happy\" and \"Verve\". I got to sit there and watch my 11 year old boy get turned on by something \u2013 the power in that is AMAZING. He left stating that he wants to learn to play the cello, how amazing their music was and that we HAD to go to their concert Saturday night.\nIn the middle of all of this was the basics of numeracy and literacy and we are plugging away at it. I am a firm believer that those are critical skills for life and not something that we can skim over. At the same time, this other stuff is what brings colour to the world and makes us all individuals. The ability to feel passionate about something, no matter what it is, is so important.\nWe went to the Infinitus concert on Saturday night, bringing along another 11 year old boy with us. While the classical Bach sections lost him, the contemporary stuff held him through and Hunter walked away happy that he had experienced Infinitus a second time.\nIn the mix of all of this great stuff this week, I have found myself surrounded by stories of friends and family that are suffering from various forms of cancer or who have unfortunately passed away due to their illnesses. These are not old people \u2013 they are in their 40's and 50's. We read more and more articles about the chemicals that are in our food, our skin care products and our environment and it leaves me wondering where this is all going.\nComing off of summer holidays you hear kids talking about how their parents are different people when on vacation. They aren't as stressed or grumpy. They talk about how important vacation is to their family because of this.\nStanding out at the wood pile yesterday afternoon I found all of these thoughts swirling around and I continued to ask myself \"What are we chasing?\". We have more and more stories and examples that re-enforce that life is short and you only get one kick at it so why, as a society, are we so focused on competing with each other and chasing the money, the promotions, the titles, the new houses, the stuff???\nI LOVE Ted-Ed videos. I am continually looking for new ideas for how to engage Hunter in learning and am fascinated with the diversity of things that are being experimented with in the education field. This week I watched a great video by Rita Pierson that talks about how important it is to build relationships and that every child deserves a champion as they go through life.\nAt the end of the video I was struck with the thought that if we aren't our own childrens' champions, how can we expect someone else to be? I also loved the +2 vs -18 metaphor and how positivity can make such a difference in those around us.\nOur lives have fundamentally changed since we hit the road last September. We have a stronger sense of family, have more effective communication and are much more connected to each other. There is a sense of feeling that we are in this journey together, where ever it may take us. We have committed to taking things one year at a time \u2013 no stress about needing a master plan, just focusing on how do we meet our needs today and tomorrow.\nI had a conversation with someone earlier this week and they asked \"what are your plans for this Chasing the Sun thing?\". At this point, the picture is fuzzy and clear at the same time. I've been thrilled to hear from friends how they took the summer off to hang with their kids or chose to get more active as a family this year because of the experiences we are sharing. I firmly believe that there has to be a different way forward for families. The education system as a whole was built to meet the needs of the Industrial Revolution and doesn't align with the economies of today and tomorrow that are driven by innovation. We aren't building thinkers, we're building kids that are focused on mastering standardized exams. Most kids are so over prescribed in terms of time and activities that we have lost the focus on family and play. Many kids are actually missing basic physical literacy skills due to early sport specialization. At the same time we have ever increasing childhood obesity rates and predictions for shorter lifetimes in the coming generations.\nI worked hard to get to the point where I could work on a contract or consulting basis and I am grateful for the opportunity to serve and support organizations and individuals that need help. I am also working hard to balance my clients needs with my needs and our families needs. As parents, Tim and I are both committed to working less and playing more. What that looks like changes every day, based on the weather and the season. What matters most is maintaining the commitment to each other and to us as a family. I'm optimistic about our year to come and loving the journey that we are on.\nSo\u2026 what are we chasing? We're CHASING THE SUN both literally and figuratively. We're chasing family kayaking, mountain biking, surfing and skate park adventures along with those bluebird sunny spring snowboarding days. We're also chasing that warm glow and happy feeling that comes with a proverbial sunny day \u2013 overflowing with a sense of presence and all that is important in the moment, a life of few cares beyond what is right in front of you and those that you surround yourself with. By sharing our experiences and being transparent about our journey, we're also aiming to make a difference by supporting those around us as they work to define their \"new normals\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Don't have job -> try to get job -> maybe fail to get job for a little while -> eventually get job -> everything's fine.\nIt feels like I've tried everything: being supportive, getting mad\/depressed (not in a manipulative way, I was just legitimately those things by turns) but I don't know what to do. I come home and he's spent the day watching TV or playing video games or doing his favorite hobby, and those things make him happy, which I love, but I feel like I'm not really getting a fair deal.\nI make enough money to cover the rent (barely) so any other expenses are being covered by money I inherited. When I told him about it months ago we discussed it being used for our wedding and a house and vacations, etc. but now we're blowing through it at a rate I'm not comfortable with. He has big dreams and likes to talk about what we'd do if he won the lottery and how I wouldn't have to work \u2013 but I actually like working. I don't need him to be a millionaire, I just want him to be my partner. He's my one and I'm in too deep to leave this situation but I'm looking down the barrel of a 50+ year partnership with me being the responsible one and I don't think I like who that might turn me in to.\nAll I want is to know how to help him see that it's not the end of the world \u2013 he just has to do some work!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\" I just want to tell you how happy my mother was and that everyone at her party enjoyed your music. I know that I wanted to have a live demo from you before purchasing the trio package, because I did this purchase online. I was not disappointed, you showed up on time and made my mother happy on her 90th birthday and made my aunt's and uncles happy and all my other family too. Thank You!\nLlamenos para su proximo evento o ocasion.\nWe provide live performances anywhere in the Bay Area. We also provide national and international services via telephone around the world.\nAllow us to impress you like we have many of our past clients. Simply go to the \"reservations\" tab above and follow the easy steps needed to ensure our availabilty at your next event. Believe me, you'll be glad you did.\nAt Serenatas Directo our aim is to please everyone. We offer many packages to meet the needs of our clients. Our work is professional and our musicians each have more than 15 years of experience in their profession. We also work with our clients in case they have something different in mind and prefer to go with a customized package.\nOur musicians are among the best in the Bay Area. Our past clients can attest to our commitment, dedication, professionalism, punctuality and talent. Just vist our reviews and testimonials tab to see for yourself.\nWe are available for any event, ocasion, fiesta, luncheon, private party or simply a sentimental moment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vietnam's Prime Minister Nguyen Tan Dung (C) smiles as he walks behind Vietnamese Communist Party's Secretary General Nguyen Phu Trong (L) and President Truong Tan Sang (R) toward the late president Ho Chi Minh's mausoleum prior to the opening of the National Assembly's second annual session in Hanoi on October 22, 2012.\nVietnam's ruling Communist Party is not looking back on a good year. The country's economy is in trouble; the authoritarian leadership is split; and what appear to be rival Communist Party factions, seeking to rouse the dissenting voices of social media for their own ends, have unleashed a wave of online protests that has become increasingly difficult to contain.\nAccording to Carlyle A. Thayer, emeritus professor of political science at the University of New South Wales in Canberra and a longtime Vietnam watcher, the new blogs have set \"the house on fire and are being read by everyone.\" Danlambao (People Doing Journalism) is one of the most popular blogs. It notched up half a million page views on Sept. 12 \u2014 the day of the Prime Minister's antiblogging decree \u2014 according to an open letter by its anonymous editorial team. \"Our contributors include not only independent newsgatherers and freelancers, but also reporters from mainstream media and informants from within the government,\" the letter said. Danlambao, along with similar blogs like Cau Nhat Tan and Xuandienhannom, has covered dissident trials, forced relocations, official waste and graft, the country's struggling real estatemarket and Vietnam's territorial disputes with China.\nThe world of social media is also becoming a forum for dissent. Facebook is booming in the Southeast Asian nation. Almost a million Vietnamese joined the network each month over the past half-year, making Vietnam the country with the highest growth rate in Facebook users globally, according to the social-media analysts at Socialbakers. Over the year, the total number of users, who are mostly young, urban and educated, doubled to 10 million \u2014 a ninth of the population \u2014 prompting some of the country's most outspoken journalists to move from blogging to publishing on Facebook. One of these is San Truong, better known as Huy Duc, who is currently a Nieman fellow at Harvard University. He has some 5,000 friends and 13,000 followers on Facebook, where he regularly publishes articles commenting on the latest conflicts between the Prime Minister and political rival President Truong Tan Sang. \"People like me won't have to go back to official media as long as we can debate online,\" he says.\nPoor economic performance has exacerbated social dissatisfaction. The country fell from 112th in 2011 to 123rd place this year in Transparency International's Corruption Perception Index. High-profile corruption has dominated the news throughout the year and made appearances in popular culture too. The hit television series Dan Troi (Heaven's Altar) portrayed the lives of a corrupt provincial party secretary, a businessman and a corruption TV station director, bribing their way up to higher positions and greater wealth \u2014 a fictional tale that seemed all too real to many Vietnamese. In April, Pham Thanh Binh, chief executive of state-owned Vinashin, was sentenced to 20 years in prison for bringing the country's largest shipbuilding company to the brink of bankruptcy. In August, Nguyen Duc Kien, a banking and soccer tycoon, was arrested for \"illegal business.\" Weeks later, police in neighboring Cambodia arrested Duong Chi Dung, chairman of Vietnam's largest shipping line Vinalines, who went on the run after the company defaulted on more than $2 billion in debt, according to official media reports.\nPrime Minister Dung has meanwhile been overseeing a period of high inflation, with consumer prices up 9.4% on average every month this year compared with last, according to official figures. Government economic strategy, heavily reliant on propping up large state-owned companies, seems to be flagging. On Dec. 24, the country's General Statistics Office reported the lowest economic-growth rate in 13 years \u2014 the lowest among its Southeast Asian neighbors. Foreign investment pledges fell by 14% this year, Moody's has downgraded the country's government bonds (citing \"a high degree of macroeconomic instability\"), and the country's benchmark VN Index was Asia's worst performing stock index last year.\nTo be sure, while Vietnam's woes are providing plenty of material for dissenting voices, dissenters don't have a free pass just yet. On Nov. 20, a court upheld a six-year prison sentence against soldier turned blogger Dinh Dang Dinh, aprominent democracy advocate. The decision came two months after three leading representatives of the dissident group Free Journalists Club were convicted of \"conducting propaganda against the state.\" The most prominent of them, blogger Nguyen Van Hai, also known as Dieu Cay, was sentenced to 12 years in prison and 5 years of house arrest. Fellow blogger Ta Phong Tan was sentenced to 10 years in prison and 5 years of house arrest. Tan's mother died after setting herself on fire in July to protest her daughter's detention.\nDespite these setbacks, a culture of protest is growing. For now, nationalist demonstrations against neighboring China's investments in Vietnam and its territorial claims in the South China Sea outnumber protests against official corruption or in favor of democracy. But the tendency is for young nationalists to form united fronts with other dissenting groups, including democracy activists. \"You can see that the links have become stronger,\" says Vu. \"Now they have started to link to farmers protesting against land grabs, they have started to link to Christians who contest government religious politics.\" While many still believe that antigovernment blogs are tolerated because their existence suits certain factions of the Communist Party, it is safe to assume that the deepening of ties between opposition groups was never the intention of any apparatchik \u2014 and that may make 2013 an even more difficult year for the Vietnamese government than 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwysc b/data_all_eng_slimpj/shuffled/split2/finalzzwysc
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+{"text":"very informative article. I love how you described the importance to listening to your intuitive spirit while creating. Sometimes I think we get caught up in what are audience's reaction will be and lose our unique intuitive edge! Thanks for posting!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mr. and Mrs. White of Lubbock celebrated their 50th wedding anniversary at Brentwood Club House. They are the parents of Dawndi Higgins and Martin, Shawndi Ricks and Bob A., both of Lubbock, Carl Lee of New Deal. Johnnie Ann Blakley and Dale White married Sept. 1, 1962 at the First Baptist Church in Spur. They have 7 grandchildren.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And now a heartfelt letter from Barbra Silver, Family Paths' Executive Director.\nIf you are like me, the last month has been a swirl of emotion and confusion. The impact of this presidential election has left many uncertain about the future and at a loss of what to do next. I think all of us can agree, no matter who we voted for, that we want the best for our children, our families, and the communities we live in. Many of us want to know our voices still matter and our values will be respected.\nAs a supporter of Family Paths, I know you care about the children and families who are most impacted by trauma and stress. Perhaps you yourself were, or are, a stressed parent, or know a friend or neighbor who is. Perhaps you remember your own parent being stressed and overwhelmed and experienced firsthand the impact of that stress. In any case, I am sure you agree that all children should be safe and all families deserve support in their times of need.\nNow more than ever, our work providing mental health and supportive services to our most vulnerable children and families is necessary and urgent. And you can make a difference by helping us do this vital work. Your donation helps us continue to be present in the lives of children who experience abuse and exposure to violence, and have depression and anxiety. It helps mothers and fathers heal from their own childhood trauma so they can be better caregivers. It helps us provide safe and skilled childcare during our life changing parenting education classes. It helps us provide the important calming and regulation skills to young children and parents attending our Family Yoga Project.\nContributing to our ongoing efforts for a socially just and equitable society benefits everyone. To live in the world we want, we need to create the world we want. Our efforts don't stop when barriers present themselves, or when the challenges seem insurmountable. Instead we get stronger and speak louder and fight harder. There really is no other option.\nIn this time of seasonal reflection, as we wait in the long nights for the light to return, I give you my sincere gratitude and appreciation for your generosity. The work we do has so many moments of true transformation and inspiration. We know by doing it, that change is possible\u2026change is always possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shock absorber casters are mainly used in the automotive industry. Shock absorber casters have good rotation performance and force performance. Shock absorber casters are divided into American damping caster and European damping caster.\nChina shock absorber caster made of high quality natural rubber rubber core wheel. China shock absorber casters have good flexibility and wear resistance.\nChina shock absorber caster features a small start-up power when it is used in the equipment car.\nAs a heavy duty caster manufacturer, we can wholesale China shock absorber caster, fixed nylon caster and so on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Introducing the Free Lenahan & Dempsey Auto Guardian App.\nAn App that has the potential to save your life, or the lives of your loved ones.\nThe new Lenahan & Dempsey App is free and easy to use. This App is believed to be the only app that will send for help after a serious accident \u2013 Automatically! The App will reach out for help after an accident, even if the App user is injured and unable to use their phone.\nWe urge you to tell your family and friends about this potentially life-saving App so they can also have the peace of mind it may bring.\nUsing a combination of custom programming and your iPhone's built-in GPS, The Auto Guardian can detect a sudden deceleration from highway speed to a sudden stop. Within seconds, an emergency text alert and back-up email alert will go out to those you designate as emergency contacts. The alert will say a sudden deceleration was detected in the vehicle in which you were riding and you may need assistance. GPS coordinates of the possible accident scene will be included in the alerts. Should the emergency contacts fail to reach you by phone to determine your needs, they should immediately call 911 Emergency Services, provide them with your GPS coordinates and tell them you may need help.\nPlease take a few moments to watch the video below to learn about the innovative, possible life saving features of the Lenahan & Dempsey Auto Guardian App.\nThis innovative App is free from the Law Firm of Lenahan & Dempsey. It's a way to say \"thank-you\" for all of the support you have given us for over 65 years of practice throughout Northeastern and Central Pennsylvania and beyond representing the seriously injured.\nWe hope you never need to use the potentially life-saving feature of the app, but if you do, it's just another way we at Lenahan & Dempsey try to protect you and your family.\nTo download this free, innovative app, click on the link below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Montreal's Jennifer Brodeur is behind the former first lady's glowing skin.\n\"Max\" (technically called \"Max+,\" but she always refers to it as a person) is the LED-light-therapy machine that Brodeur designed in 2003. The device is used by skincare professionals around the world, including those at Bella Clinique, the Montreal-based clinic she founded and continues to mentor at. It uses UV-free light wavelengths to treat skin concerns: red-light wavelengths are said to stimulate fibroblasts to create collagen, yellow light tightens skin, etc.\nPeoni is Brodeur's recently launched line of mineral oil, paraben, colour and dye-free skincare, all based around the ingredient peony\u2014a super antioxidant that also contains anti-inflammatory and anti-fungal properties. The four-piece set includes a cleanser, toner, oil and moisturizer; the entire line made Oprah's list of favourite things in 2016. Brodeur says Obama was one of her \"greatest sources of inspiration\" for the line, and also one of the first in the world to use it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Simi Components' Geno Giordan opened his doors for business in 2001 after 35 years in the electronic components distribution industry. One of Uruhun's earliest clients, Simi Components has used Complete Contact CRM to track its internal information for more than four years. According to Geno, Uruhun has played a considerable role in his company's substantial growth.\n\"... There are many contact management programs designed for the distribution business,\" said Geno. \"Complete Contact CRM is undoubtedly the industry's preeminent software package, and that includes custom programs...\"\nUruhun has fostered Simi Component's development by saving employees time so they can service customers more efficiently. The single screen use, available with one click, is significantly more user friendly and seamless than other contact management programs employing multiple screens. For Simi Component's, Complete Contact CRM has paid for itself many times over.\n\"... you used to have to pay hundreds of thousands of dollars to have a program like Complete Contact CRM. Believe me, I have. And that was for a DOS type program. Complete Contact CRM is leaps and bounds beyond my old expensive program and it is improving daily as the talented programmers add features.\nThere are many programs out there and most of them have similar features. One of the main reasons I chose Complete Contact CRM over the others was that the chief programmer, Oscar Mintegui, has been in the industry since 1989. And the programmers know our language and how to get what we need done in their program. It has been a pleasure to work with all of them and they come up with some great ideas of how to handle our needs better.\nI am a believer that time is money and this program has saved us large amounts of time. I encourage anyone to take a close look at Complete Contact CRM to see how you can do everything from one program. ...\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MPs will not get to vote on how Brexit negotiations are handled but could still be asked to approve the \"final\" deal, a government source has said.\nSeveral senior politicians, including ex-Labour leader Ed Miliband, have demanded Parliament gives its verdict on the UK's departure from the EU.\nBut Brexit Secretary David Davis told MPs there was a difference between \"accountability and micro-management\".\nThe UK's exit from the EU is expected to happen by summer 2019.\nTheresa May is visiting Denmark and the Netherlands for Brexit-related talks as MPs debate the issue in the Commons.\nThe Leave campaign won a majority in June's referendum, with the prime minister announcing last week that the government would trigger Article 50 of the Lisbon Treaty - beginning formal negotiations between the UK and EU - by the end of March next year.\nThe process will take up to two years, involving complex debates about issues such as immigration and access to the European single market.\nMrs May's spokesman said: \"Parliament is of course going to debate and scrutinise that process as it goes on. That is absolutely necessary and the right thing to do.\n\"But, having a second vote, or a vote to second-guess the will of the British people, is not an acceptable way forward.\"\nHowever, a Downing Street source offered clarification to BBC assistant political editor Norman Smith, telling him that this did not necessarily rule out a parliamentary vote once \"a final\" deal is reached.\n\"This leaves open the possibility MPs could get a vote on the package eventually negotiated by Mrs May,\" our correspondent added.\nThe prime minister is visiting Denmark and the Netherlands for talks on \"delivering Brexit\". Her discussions with counterparts Lars Lokke Rasmussen and Mark Rutte come a week ahead of her first European Council meeting.\nMedia captionEx-Tory minister Peter Lilley says MPs who voted Remain are acting like bad losers.\nAt a press conference with Mr Rasmussen in Copenhagen, Mrs May said: \"We are not turning our backs on Europe. We want to maintain strong relationships with our European partners.\"\nThe UK would continue to \"meet our various rights and obligations\" until it left the EU, she added.\nMr Rasmussen said it was \"tragic\" that UK voters had decided in favour of Brexit, but he hoped for a \"friendly divorce\".\nThere is an expectation that there will ultimately be votes on the agreement - with the prime minister's \"Great Repeal Bill\" and, at the end of the process, on the ultimate deal.\nBut that is a long time away and, for many MPs, simply not good enough, when the process of negotiations will shape our relationship with the rest of the world for years to come.\nAnd with no votes any time soon, there is also - so far - nothing on paper, nothing that fits into the prime minister's chosen template of Green Papers, White Papers, and then ultimately legislation.\nBut what if that was the plan? In recent days sources have told the BBC that the government was planning to produce a Green Paper this autumn with the broad outlines of its Brexit objectives.\nOne source said \"we were told there would be one in October\", and that the financial sector had been asked to provide \"a data dump and a shopping list\" by the end of last month in order to facilitate the process.\nIn a statement to MPs, Brexit Secretary David Davis said the Great Repeal Bill, overturning the 1972 European Communities Act, which allowed the UK to join the EU's predecessor, the European Economic Community, would be introduced in the next parliamentary session.\nThis, he said, would prevent a \"black hole in the statute book\", by keeping large amounts of legislation on a national basis when the UK is no longer subject to EU law.\nMr Davis reiterated the prime minister's promise not to provide a \"running commentary\" on the government's thinking.\nBut former Liberal Democrat leader Nick Clegg, now his party's EU spokesman, says the Commons had a \"rightful role of scrutiny\" at all stages of the Brexit decision-making process.\nFormer Labour leader Ed Miliband said MPs must have a say on the nature of the eventual Brexit deal.\nHe told the BBC's Victoria Derbyshire programme he did not want to reverse the result of June's referendum, but that details of negotiations should not be given through \"briefings and leaks\".\nParliament, as \"the sovereign body of the people\", should discuss plans, Mr Miliband said, as the referendum had not specified a \"particular type of Brexit\".\nThe SNP said there was confusion over the government's objectives. Its European Affairs spokesman Stephen Gethins said: \"The ongoing splits in the cabinet, which are being played out in the media in a series of off-the-record briefings, are leaving more questions than answers, causing yet more alarm and uncertainty.\"\nConservative MP and former Attorney-General Dominic Grieve said the Commons had to be allowed to give its opinion, as this was a \"very well-established constitutional convention\" involving important treaties.\nHe added: \"If a situation arises that the government at the end of the day is about to conclude a deal for the future of the United Kingdom which can't command parliamentary approval - or at least acquiescence - then it's perfectly obvious in those circumstances such a government wouldn't survive. I would have thought there would have to be an election.\"\nMrs May is due to to visit Madrid on Thursday for talks with Spanish counterpart Mariano Rajoy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"English Premier Champions, Liverpool has ended their pursuit of Southampton's Dutch center back, Virgil Van Dijk and has apologised to the South Coast club.\nLiverpool who has for the past years signed most of Southampton's players including Dejan Lovren and Senegalese, Saido Mane was in pursuit of Van Dijk and was doing what they could to reach an agreement with both the player and the club to secure his services.\nVan Dijk suffered an ankle injury in February missing the rest of the season as Southampton suffered in his absence but ended up finishing 8th in the Premier League after the season ended.\nThe South Coast club has threatened to send Liverpool to the English FA for illegal pursuit of their player who is under contract according to reports and have set a price tag of \u00a360m on their defender for any club that think may need his services.\n\"Liverpool Football Club would like to put on record our regret over recent media speculation regarding Southampton Football Club and player transfers between the two clubs\".\n\"We apologise to the owner, board of directors and fans of Southampton for any misunderstanding regarding Virgil van Dijk.\n\"We respect Southampton's position and can confirm we have ended any interest in the player\".\nLiverpool is also reported to be tracking AS Roma's Mohammed Salah as they seeks to strengthen their squad for the domestic competitions and Europe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Redmi Note 7's have a dual camera - 48MP+5MP that take photos with its 48 million tiny 0.8-micron pixels in good lighting conditions, or combine its neighboring pixels for what is claimed to be an equivalent of a 12-megapixel picture taken with larger 1.6-micron pixels. There's also a 13-megapixel selfie camera.\nUnder the hood, you have a Qualcomm Snapdragon 660 processor accompanied by a 4,000mAh battery to power everything.\nThere's also USB-C charging with support for Quick Charge 4 and 3.5mm headphone jack.\nThe GPU is also great and there are 4GB\/6GB RAM variants of the device.\nThe Redmi Note 7 will be available in China first and It should get to Nigeria in a couple of weeks.\nWhat do you think about this amazing smartphone, let me know your thoughts about this smartphone.\nFirstly, let me introduce Jumia One to you. Jumia One is a platform that gives you the opportunity to recharge your phone, pay utility bills, make hotel reservations, order food from your favourite restaurant and also helps you buy data, airtime, pay electricity bills and more.\nHow Can I Make Money From Jumia One App?\nNow, to make money from the program, you have to refer people to the app. It's easy to refer people because you can't tell me you don't have relatives or friends on Facebook, WhatsApp and other online media platforms. Bring them to the app and take your share.\nEach person you refer gives you N500 and your friend gets 20% bonus also. Assuming you refer 10 people to the app, you get N5,000 on your withdrawal wallet. Isn't that awesome? Of course, it is.\n4- You will see N500 in your wallet of you used your account to order before. If you didn't see anything you have to purchase airtime for only N50 to get the N500. You can cash it as Airtime on any network.\nNote:- You have to refer friends to earn more. You will get free N500 for any one you refered.\nDon't forget to share and comment..\nThis post \"iPhone users are less honest than Android users\" is not written to side the Android users, but it doesn't necessarily apply to all iPhone users.\nBased on the measure for researches conducted in the University of Lincoln, Lancaster University and the University of Hertfordshire which was done with more than 500 participants to answer questions about themselves and attitude towards their smartphones.\n\" Android users were more honest and less interested in wealth \/ social status \"\n\" the iPhone users were found to see their smartphone as a social status symbol and have higher emotional attachments \"\nAndroid users were also less likely to break rules for personal gain. A case study was used in analyzing these. It involves predicting the type of smartphone used by an individual from his \/ her personalities.\nThese facts does not necessarily apply to all iPhone users but it's considered true on a major cases.\nWant to contribute to this post, feel free to let us know by using the comment section down below.\nIf you have enrolled for the National Identity Card have collected the NIN slip, then it is high time to confirm if your National Identity Card is ready for collection. You can confirm if your National Identity Card is ready for collection without going extra miles by checking your Card status via an online platform, if the card is ready for pick up or not and has been dispatched to an Activation Centre.\nHow\/Where to Check the Status of my National Identity Card?\n\u2013 Simply visit the NIMC ID card Status checker here.\n\u2013 Click the green Proceed Button at the bottom to continue.\n\u2022 Last 6 Digit of your National Identification Number.\n\u2013 Click the 'Check Now' button (this will take few seconds to load and then display the status of your NIMC card).\nIf you are yet to enroll for your National Identification Card. Please seize the opportunity to enroll and get yourself a copy by signing up on the NIMC Official Website.\nIf your card is ready, NIMC will send you a message notifying you to come forward to collect the Card as directed in the SMS. No one can collect your National e-ID Card on your behalf. To qualify for a National e-ID Card, you must first be enrolled and have a National Identification Number (NIN).\nGo to the NIMC Collection Centre indicated in the SMS sent to you. Take your NIN\/Transaction slip along. Mention the Batch ID on the SMS.\nReceive your National e-ID Card in a sealed envelope. Open the envelope, take out the Card, sign on the space provided at the back. after reading, please keep all the inserts in the envelope safely.\nDo a Biometric Verification to confirm that it is your Card. This also confirms that the Card has been properly printed.\nAfter biometric verification, you will be required to enter a FOUR (4) DIGIT NUMBER chosen by you to activate your National e-ID Card.\nThere is provision for fund loading at the NIMC Collection Centre. You can also do actual Cash Loading at any branch of ACCESS BANK nationwide.\nPivot, invested by Binance, is one of the largest and the most successful cryptocurrency community in China.\nToday's bonus (One bitcoin) will be distributed at 4:00 (GMT+0) the next day.\nIn order to encourage users to promote the Pivot community, we decided to organize a campaign called \"Daily BTC Bonus\" for 30 days. The bonus is one BTC per day.\nOne bitcoin will be distributed to users as bonus per day,POWER is the certificate of \"Daily BTC Bonus\", which decides how much BTC Bonus you can get. For example, if 10000 POWER is distributed to all users in one day, and user A gets 100 POWER, then he\/she will receive 1% of the BTC in the bonus pool of this day, which is 0.01BTC \u2248 63USD.\nFor every post that you read, you will be rewarded with 200 POWER. The maximum reward is 4000 POWER per day!\nLogin in with facebook or gmail and earn reward download app.\nHey Techpallies, am back again with an affordable data I believe data should be free if not affordable but that doesn't mean I won't share affordable data with you.\nMTN just released a new data plan for Instagram that gives you 1GB for N200, that if you are on \"MTN Pulse tariff plan\"\nDon't be selfish share with friends and don't forget to comment if it works for you.\nInfinix just launched a smartphone yesterday, the newest addition to the Infinix Hot series is called the \"Infinix Hot 6X\". The Infinix Hot 6X came with the notorious notch display, packs its star-studded internal features into a plastic body covering with curved edges and antenna lines very close to the edges.\nThis amazing smartphone was launched after two Hot series in May, 2018 which are the Hot 6 and Hot 6 Pro. The Infinix Hot 6X specs will be looked at extensively on Techypals.com.\nWhat do you feel about the Infinix Hot 6X? Are you disappointed or marvelled? Let us know in the comment box below.\nAirtel is giving 4.6GB for N200 and 23GB for N1000 to all eligible Airtel subscribers. I know most of you will be thinking that this is a cheat but it's not, it's real, it's an offer from Airtel to some of it's users. It also includes Free calls.\nNOTE: Not all Airtel users would be able to enjoy the 4.6GB for N200 or 23GB for N1000 offer. Try your luck \ud83d\ude09\ud83d\ude09 .\nThen wait for a reply from Airtel, It may be a Good reply or a not so good reply.\nIf you receive this message, then you are qualified for the offer and you'll be able to get Airtel data bundle at a cheap rate.\n\u2022 Buy another airtel recharge card of N200 up to 5 pieces.\n\u2022 To check your data balance, simply dial *223# and you will get a pop up showing your remaining available data balance.\n\u2022 You can also use *140# to receive a message for all your data balances including BONUS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxvyj b/data_all_eng_slimpj/shuffled/split2/finalzzxvyj
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index 0000000000000000000000000000000000000000..ac056889c3f39967bd75e78687b03f267a4629a5
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxvyj
@@ -0,0 +1,5 @@
+{"text":"Take the power of AutoCAD wherever you go! AutoCAD mobile is a DWG viewing application, with easy-to-use drawing and drafting tools that allow you to view, create, edit and export AutoCAD drawings on mobile devices - anytime, anywhere. Simplify your site visits with the most powerful CAD app and do real CAD work on the go.\n- Advanced drawing and editing tools such as arc, offset and more!\n- Manage layers- create new layers, lock, rename or delete layers.\n- Blocks- View and select existing blocks to add to a drawing.\nAll new users automatically get a free trial of the premium version for 7 days. Any unused portion of a free trial period will be forfeited when purchasing a subscription.\nWe fixed some bugs in this version, so the stability and performance is even better.\nWe are working around-the-clock to deliver new features and bug fixes, so stay tuned for more great things to come soon!\nReally? Screen capture to export in the field?\nI just used my nifty laser for some field measurements on site, and want to throw down a sketchy concept. That means exporting the plan to another iPad app. Like Autodesk Sketchbook, perhaps? No. Sorry.\nThe only solution? Do a screen capture (until Autodesk finds a way to block this). Open the screen capture in a more collaborative app, set it to scale and move forward. This is how Autodesk simplifies my life and streamlines my workflow?!\nPlot to pdf? Sorry. You need e-mail.\nAh, perhaps you're thinking I could simply download from A360. No. Service is down.\nBesides that, I'm standing in the muddy field of a job site and golly! No internet here until they build the thing, but they can't even get started because there's no hopping concept sketch at high resolution!\nExport a pdf to another application? No. Sorry, but you need the internet for that.\nLine weights by layer? Sorry. Ortho mode? Sorry. Remember offset distances? Sorry. Viewports? Sorry. Custom blocks in the app? Sorry.\nBut the thing that really makes me sorry is that this is currently the best affordable solution for creating scaled drawings in the field! And it makes it a point to shun all other graphics apps, even Autodesk's!\nExcuse me while I emulate Autodesk and gnaw off my left leg so I can't buy shoes by the pair. No, that does not make sense, but generally neither do this app's limitations!\nThere's a way to export your drawings as pdf's, which is crucial for me, but honestly I'd have more luck drawing it on notability and then exporting, because this app's pdf exporter is terrible. You should be able to set a custom scale, much less any scale, but no matter what you put in as a scale this app just decides to put its own random scale in, just to taunt you. Fit to paper doesn't even work unless you put your drawing into a box, and even then it cuts off the edges so part of your drawing is gone. They need a pdf preview before you share, and let you find the right margins so that you can easily fit the box so you can fit to page. Otherwise you will never be able to export an accurate representation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I.R.I.S or IRISlink.com is a website that offering all types of Scanner Products, OCR Software and Document Management Solutions. When you visit I.R.I.S website you will find their software and products that separate into many categories for example High Speed Scanning Software, Pen Scanners, Digital Pen, Card Scanners, Mobile Scanners, Photo Scanner, Invoice and Form Solutions, Sorting and Indexing Solutions, O.E.M., Toolkits, Document Server Software, OCR Software and many more. I.R.I.S enter into software and scanner industry since 1986 which you can assure all items that you order from them which guarantee from their experience that will never let you disappointed.\nNot only providing all types of Document Management Solutions and OCR Software products for you to select but I.R.I.S also giving away I.R.I.S Coupon for you which you can apply I.R.I.S Coupon after you finish adding your select items into your shopping cart and then you will see a box to apply I.R.I.S Coupon during your checkout which can save your money up to $30 Off. For current and active I.R.I.S Coupon is mention below.\nThe reason that why many people select I.R.I.S when desire to order Document Management Solution and Scanner Products is because I.R.I.S is one of the largest online store that providing all types of Document Management Solution and Scanner Products for you to select and when you visit I.R.I.S website you will find their products that separate into many categories such as OCR Software, Pen Scanners, Card Scanners, Mobile Scanners, Digital Pen, High Speed Scanning Software, Document Server Software, Invoice and Form Solutions, Site License and many more.\nI.R.I.S Coupon: At I.R.I.S website you can saving your money by using I.R.I.S Coupon after you finish adding your select items into your shopping cart and then you will see a box to apply I.R.I.S Coupon during your checkout which can save your money up to $30 Off.\nI.R.I.S Customer Services: If you have any question about I.R.I.S Products or Delivery Services which you can contact their customer support directly at +32-(0)10-45 13 64.\nI.R.I.S Mailing Services: At I.R.I.S website you can sign up for I.R.I.S Mailing Services by visit I.R.I.S website and then click at Subscribe to our Newsletter which you can see like picture below and after you sign up for I.R.I.S Mailing Services then you will receive latest promotion from I.R.I.S website via your email address.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"35 Frais Testeur De Terre Fluke is free HD wallpaper. This wallpaper was upload at January 14, 2019 upload by Maria Matthews in terrasse.\n35 Frais Testeur De Terre Fluke is high definition wallpaper and size this wallpaper is 700x700. You can make 35 Frais Testeur De Terre Fluke For your Desktop Wallpaper, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the 35 Frais Testeur De Terre Fluke images by click the download button below to get multiple high-resversions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"01. Food IV Thought \/\/ \"80z Soul\"\n02. L4L \/\/ \"80z Soul\"\n03. Never Home \/\/ \"80z Soul\"\n05. More Blessings \/\/ \"80z Soul\"\n06. F.W.M \/\/ \"80z Soul\"\n07. Too Far \/\/ \"80z Soul\"\n08. Want More \/\/ \"80z Soul\"\n09. We Ride \/\/ \"80z Soul\"\n10. Stay With Me \/\/ \"80z Soul\"\n11. Where The Wind Blows \/\/ \"80z Soul\"\n12. Change up \/\/ \"80z Soul\"\n13. Hold Forever \/\/ \"80z Soul\"\n1. MMI \/\/ \"God's Hands\"\n2. What They Doin \/\/ \"God's Hands\"\n3. Down feat. Aliesa Nicole \/\/ \"God's Hands\"\n4. Karma feat Jacob Brock \/\/ \"God's Hands\"\n5. Glory feat. Zero \/\/ \"God's Hands\"\n6. Beautiful \/\/ \"God's Hands\"\n7. 2 Busy \/\/ \"God's Hands\"\n8. Dangerous feat. Young Deuces & J. Wilder \/\/ \"God's Hands\"\n9. Pressure \/\/ \"God's Hands\"\n10. Real Aint Real feat. G Smoov \/\/ \"God's Hands\"\n11. Greater \/\/ \"God's Hands\"\n12. Sinners Remorse feat. Genesis Renji, Mark Anthony, Ashlie N. \/\/ \"God's Hands\"\n13. Love Somebody \/\/ \"God's Hands\"\n14. God's Hands \/\/ \"God's Hands\"\nBonus Track \u2013 Glow \/\/ \"God's Hands\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We each will work out our own soul salvation with God. There are no words to speak on in this matter for it is by experience that we know in our hearts the ego's way (lower self) never worked. At some point we stop playing with the childish toys of the intellect that want to figure everything out, and die to all old concepts. Insight is good, surrender is better!\n1 Corinthians 15:20 But the fact is that Christ (the Messiah) has been raised from the dead, and He became the firstfruits of those who have fallen asleep [in death]. For since [it was] through a man that death [came into the world, it is] also through a Man that the resurrection of the dead [has come]. For just as [because of their union of nature] in Adam all people die, so also [by virtue of their union of nature] shall all in Christ be made alive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzykha b/data_all_eng_slimpj/shuffled/split2/finalzzykha
new file mode 100644
index 0000000000000000000000000000000000000000..113ba402197ae3f2ce526b177b310ee9321c1ee8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzykha
@@ -0,0 +1,5 @@
+{"text":"the bandage was removed the wound appeared perfectly closed.\nmal on the tenth day of the disease and remained so.\nary 1900 for tubercular kidney and ureter recovery.\nmuch more concentrated than in case of contracted kidney.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good News Everybody! The Magicians has been renewed for Season 5 on Syfy just ahead of it's 4th season premiere on January 23rd, 2019. Obviously, not much more information is currently available for what adventures Quentin and his gang of misfits will be getting into come season 5, however, while the series has deviated from the source material, there are still a number of threads from the trilogy of books from Lev Grossman that the show can still use as inspiration for new stories.\nIn addition to the announcement about renewing The Magicians for a 5th season, Syfy also announced that Henry Alonso Meyers, has been given a promotion. He'll be stepping up from the Executive Producer role he's held since season 1 into a Co-Showrunner position, where he'll be joining Sera Gamble and John McNamara.\nSeason 4 picks up pretty much right where we left off at the end of Season 3 with Quentin, Alice, and the rest of the gang thinking they're other people and not knowing about Brakebills or magic. . . everyone that is except for Elliot who's been possessed by the monster from Blackspire.\nWe Have A New 'The Magicians' Season 4 Clip And Release Date!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Newcastle Carers is an independent charity supporting adults, children and young people who care for someone living in Newcastle upon Tyne.\nOur service is free, confidential and non-judgemental, helping you to find ways to make your situation easier.\nWe are based on Shields Road in Byker, Newcastle, where we provide a range of information and support services. We work in other areas of the city too, including Gosforth, Lemington, and the West End.\nWe also visit public places and other organisations to inform people about our services and provide information to carers, professionals and the general public.\nKey findings from our Annual Survey 2017 have been published. Read a summary of the results here.\nWe have successfully achieved the CarersTrust, PQASSO Level 1 and Investing in Volunteers quality marks.\nThe Newcastle network partner of Carers Trust - a national charity who work to improve support, services and recognition for any unpaid carer in the UK.\nAn affiliated member of Carers UK - a national UK membership charity who provide a support network for UK carers.\nNewcastle Carers is governed by a Board of Trustees. Our Chief Executive is responsible for the day to day operational management of the service and the staff team.\nThe Trustees have ultimate responsibility for directing the affairs of the charity, ensuring it is well run, that it is solvent, and delivering the charitable objectives. In particular the Board of Trustees is responsible for compliance and have a duty of prudence and a duty of care.\nNewcastle Carers is committed to achieving and maintaining high quality practice at governance level.\nThe Board of Trustees recognises that good leadership is based upon strong principles, shared values and ethics that underpin the effectiveness of the organisation in achieving its goals. The Board works with the Chief Executive and staff team to develop services and to positively influence policies and practice in the City that have an impact on carers and the people they care for.\nTo find out more about becoming a trustee click here.\nWe use external programme Google Translate to enable our web pages to be instantly translated between English and over 100 other languages.\nPlease note that we have no control over the translations provided by Google Translate and therefore cannot be held responsible for any errors.\n\"Knowing there is a safety net, if I ever need one, is a great help\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2.Influencing play through the \"1v1 Defensive Technique\"\nPractice begins with a six-position Flow Drill that works on various shooting and passing techniques from around the field. The drill is designed to make the passer increasingly aware of the receiver's location as well as their direction of movement. The exercise challenges the fundamentals of redirecting passes, staying down on the play, stick work and footwork. Coach Bustin concludes by critiquing the action of the players, reinforcing the strategy she's trying to encourage and soliciting player feedback. .\nThe \"1v1 Defensive Technique\" section concentrates on the defender's skill at delaying and influencing the play through careful defensive techniques. Coach Bustin shows how to challenge offensive players with simple tips like closing quickly and breaking down when you trap, keeping your feet under your hips, running to close and using a drop [step] to get under the ball carrier.\nUsing two players attacking the goal, the \"Shots, Rebounds and Deflections Drill\" shows how to position offensive players to receive passes, intercept rebounds and score with deflections from passes that originate from higher on the field. Focusing on play within the circle, this drill encourages increased movement in an effort to make something happen in front of the goal. Though it focuses on offensive movement, the defense also benefits from the challenge of defending against an offensive with an advantageous field position.\nCoach Bustin shares a variety of drills designed to build upon foundational skills and prepare players for game situations. Game-speed shooting drills give both offensive and defensive players with the opportunity to focus on their fundamental skills and challenge each other to a higher level of performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rider claimed two victories over Class 6A programs Monday at Rider Fieldhouse.\nThe Lady Raiders nearly had two sweeps but Midland Lee held on in the first set before Rider earned a 26-28, 25-19, 25-12, 25-14 behind 12 kills by Meredith Fisher, eight kills from Maegan Lacy and 38 assists combined from Lauren Dodson and Devon Browning.\nRider swept past Abilene High School 25-20, 25-14, 25-19 with Lacy exploding for a 13-kill, 10-dig double-double in just three sets; Fisher added 11 kills plus nine digs for a second straight match; and Emily Stolt caught fire with 10 kills. Alyssa Estra-Hamby had four blocks and six kills and Devon Browning and Dodson had 22 set-assists apiece.\nRider heads to the Lady Ram Tournament in Allen beginning Thursday. LRV won the Gold Division at Mansfield last week.\nChrist Academy standouts: Danielle Okeke- 13 kills, 4 blocks; Kelsey McClellan 21 assists;, 2 aces; Morgan Brasher- 5 kills.\nStandouts: (R) Emily Stolt 10 kills; Meredith Fisher 11 kills, 9 digs, 4 aces; Maegan Lacy 13 kills, 1 block, 10 digs, 3 aces; Lauren Dodson 22 set-assists, 7 digs, 3 aces; Devon Browning 22 set-assists, 8 digs; Alyssa Estra-Hamby 6 kills, 4 blocks.\nStandouts \u2014 (ND) Reagan Macha 18 kills; Faylin Abernathy 7 kills, 100 percent serves; Anna Luig, Leilani Nguyen 100 percent serves; (V) Bethany Garza 9 kills; Cameron Garza 10 digs, 4 blocks; Reianna Dixon 3 aces.\nThe scheduled matches with Canyon Randall and Amarillo at Wichita Falls High School and Rider were cancelled because of rain. Also, WFHS moved its Aug. 15 match to Sept. 4 at Wichita Falls High School courts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaidg b/data_all_eng_slimpj/shuffled/split2/finalzzzaaidg
new file mode 100644
index 0000000000000000000000000000000000000000..2bcd05569c8c256bd60830c5f62e2ad41b0b2540
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaidg
@@ -0,0 +1,5 @@
+{"text":"I know, I know. Some people will think I'm about to build a sprite, SVG or CSS, it does not matter. Well, actually I've been testing SVG with #id calls (sort of a SVG sprite) and the performance was abysmal. And yet, how do I reduce the calls on the server to a bare minimum? I have reduced the size of the flags in incredible ways, but there are still +250 requests. How can I sort those out? In short: You can't. Try me, I've tested SVG sprites, PNG sprites and came to the conclusion that nothing beats CSS in sheer performance, caching an gzipping.\nYeah, you heard right. We're gonna put all 250+ flags into a single CSS file and call our objects from there. The CSS file (which you can download at the bottom) is generated on-the-fly with the current flags. That means that from here and on this file will only get better. Come back to check on it whenever you want.\nFor this example, we're gonna use the flags of Angola, Australia, South Africa and Venezuela.\nWhat is happening here? Well, the CSS defines the flags as background of the div and they both inherit the size of 240x180 pixels. Pretty cool hey? Anyways, the other very cool thing happening is that the flags are actually inside the CSS as base64 embedded images. They look like what you are seeing above.\nWhat is the diference between span\/div? The difference between span and div is that a span element is in-line and usually used for a small chunk of HTML inside a line (such as inside a paragraph) whereas a div (division) element is block-line (which is basically equivalent to having a line-break before and after it) and used to group larger chunks of code. Here in my example they are stacked side-by-side but that is me. If you want to stack vertically, div is a really good option.\nAs you can see, object actually gets rid of most of our stuff and just displays the flags as a background to the text. With more text it streches and with less it contracts. Not very useful imho but... one more use.\nSometimes <object> does not work well in some navigators. It's normal because some browsers do not support everything yet and some treat them differently. If you want cross-browser support sometimes you'll have to go the JS way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Okay, finally it seems that there is some light at the end of the tunnel. Thanks to the ownage research Xfire has been doing on creating native Linux mods, there is quite a big chance that QValidate will be Linux-server compatible in the near future. This upcoming week I won't have time to develop, but from next week I'll be working on it.\nAs it seems now, the only drawback will be that it will only be compatible with 436 and 440 servers. Until now we haven't been able to get it to work on a 451 server. Please let me know (via this thread) wether or not this would be a big problem.\nLast edited by ProfessorQ : 26th July, 2004 at 08:55 AM.\nthats very nice or in german \"goil \"\nAs for me it wouldn't be a problem. 451 uses too much processor time(is it correct in english? ). QVal for Linux ? I can say one \"Hakunamatata\" !\nAll I can say is wow ! I an really looking forward to this. Give our big thanks to this Xfire person whoever he is. And the fact that this won't be available for version 4.51 servers isn't a problem. A lot of GSPs are currently using version 4.40 as it fixes the DoS exploit and doesn't produce the lag that some players have been complaining about on v. 4.51.\nIt will be both a security update and a port. It will detect all HelioS hooks up to V2.6.\nIs it possible to make something like auto-update ? I'm not thinking about the whole package but for example a little database with existing hooks and methods to detect them.\nThat's what I'm thinking about for future versions.\nBtw, the Linux part is already working. Also the security update has been done so all HelioS hooks up to V2.6 are detected. Now, I just need to do some finetuning on the Linux side before I can release the whole thing. I'll most probably will release a test version somewhere next week.\ni can test windows versions as usual .\nas a request will you include some form of announcing when helios hook is found to the players on the server.\nWould be nice if it worked for 451 since the 451server patch is good for making the webadmin work and stabilizing the packets.\nwhat exactly did XFire research?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, I did it. All of it (with the exception of a few items that have gone AWOL under the bed).\nWhat did I learn from my week-long laundry adventure?\nWhite vinegar is truly the best fabric softener ever. I have three cats, all of which have at least some white fur\u2014and the white downy undercoat of that fur which sticks frigging EVERYWHERE. I also have a lot of black clothes\u2014which, as a New Yorker, I am obligated to own. When I pulled the first batch of laundry out of the washer, I was utterly dismayed to see white cat hair on everything. I suspect there had been one or two things that were really covered in cat hair, and in the wash, it redistributed evenly over everything else.\nI used a couple of Bounce dryer sheets (which smelled gross and are not the greatest choice in general) on that load, which helped a little. I didn't want to switch to liquid fabric softener because I usually wash towels and clothes together and fabric softener + towels = less-absorbent towels. The next day, I used about half a cup of white vinegar in the rinse. Much MUCH less cat hair, plus soft clothes and super fluffy towels (and no, everything does not smell like a salad).\nI now have absolutely no excuse to use as many paper towels as I do. Holy crap, I had no idea I owned this many towels. I had to rearrange a kitchen cabinet just to have a place to put all the dishtowels AND I've got a huge stack of former dishtowels (now too ratty for display) under the sink as well. Still deciding what (if anything) to do with the unlovely bath towels I have, but I'm looking at some tutorials on turning old towels into bathmats (something I could really use).\nI had some clothes to get rid of\u2014but not nearly as much as I expected. Of everything that I washed, we donated about one-seventh of it (one full laundry bag's-worth), but much of that was kid clothes; either cold weather stuff she'd already outgrown or warm weather stuff she'll certainly outgrow by next summer (44 things in all, including some I had weeded out earlier). I got rid of some items I only wore when I had nothing better to wear (12 items) and my husband got rid of a pair of pants he didn't like but overall, not as much as I thought would go.\nLast, and perhaps most importantly, I can finish a project when I set my mind to it. Honestly, I think I even surprised myself by getting everything done this week. It helped that a.) I publicly announced I was going to do all this laundry and then kept updating as I went along and b.) yesterday, my husband and I went to the laundromat together and did all the remaining laundry\u2014which would have been at least three trips for me on my own\u2014saving me from almost certain burnout.\nNow that that's done, I'm already looking about for a new project. Maybe decluttering?\nGuys, if you're not already a fan of gezellig-girl.com on Facebook, you might miss out on my first-ever giveaway. Details to come later this week, so go become a fan today!\nVinegar as fabric softener\u2026 I hadn't heard of this before. How intriguing! And none of the smell lingers at all? Is it just so diluted that it loses its scent?\nI put the vinegar in at the start of the rinse cycle (there's usually more than one rinse), but you can also try putting it in with the detergent. I think it's both a very dilute mix of vinegar and water, plus the smell naturally dissipates as the vinegar dries. I also use vinegar to clean the bathroom and the smell doesn't linger for long there either.\nFascinating! I'm going to have to try the vinegar thing. I also use baking soda and vinegar periodically on my drains, to keep them from getting clogged. Much better (and less expensive) than toxic chemicals.\nJane, one of the best drain cleaners I've found is to mix equal parts of baking soda and table salt, pour\/brush it down the drain, and follow it with a kettle (or more) of boiling water. I'm not entirely clear on what the chemical reaction is (it does foam up quite a bit) but it works really well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today many people are concerned about SEO as the term keeps being repeated in many places in regard to search engine optimization and drawing attention to your company. The simple explanation of the term SEO is that it's an abbreviation which means Search Engine Optimization. This is just the basic terms but to explain it further to someone who is not internet savvy one needs to go into details to what this term really means. This is SEO Explained in Simple Terms (Infographic) for someone who does not know what the term means or what it entails in details.\nThe term SEO means that when the titles of the page or words which are related to the page are typed into the search engine the webpage appears. The search engine like Google, Bing, or Yahoo optimizes the webpages to make it easier for users to find the most relevant website much quicker. If your website or webpage I well optimized, then it will appear on the first page of the search engines.\nAnother aspect of SEO is that it involves keywords, the SEO keywords are the words or at times phrases in the website which makes it easier for people to find your website via the search engine. Today keyword optimization is very important as people when searching the internet use phrases and certain keywords to find websites. The keywords included have to be relevant to the website. This is because today search engine are checking for relevancy of the keywords.\nAnother important aspect of search engine optimization is Meta tag, this is the tag which when ne searches a specific word it appears on the HTML code of the given web page. The importance of this is that the Meta tag tells the search engine exactly what you are looking for and directs you to that page or website. The Meta tag can be strategically placed on the title of the page or the meta-description and this are the tags which the web crawlers will immediately pick.\nOne of the most important aspects that is mostly overlooked when it comes to SEO is content. In search engine optimization what the web crawlers look for is the value of the content. This is the value which people find they visit your website. This is the reason as to why it is very important for website owner to make sure that the content which is on the website is making sense to the visitors so that the webpage will be highly optimized and can be found easily.\nBe the first to comment on \"SEO Explained In 5 Simple Terms\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Promised by Jesus in the Gospel accounts, progressively demonstrated throughout the book of Acts, and central to Paul's own message and ministry, the Holy Spirit is the presence and power of God at work among us unto the redemption, reconciliation, and recreation of all things.\nUnfortunately, over the last hundred years, the work and ministry of the Holy Spirit has been controversial and even suspect in some places or relegated to obscurity among others. This has resulted in most of the Church operating without a practical theology of the Holy Spirit.\nHowever, in this new era of mission, our attention is called back to the person and ministry of the Holy Spirit of God among us. It's not surprising that the predominant interaction of the early church with the Spirit was for mission. Ours should be the same.\nJoin us for a day to explore the a renewed imagination for the work and empowering of the Holy Spirit in our own lives and in the mission of the church.\nRev. Michael Beck is Director of Re-Missioning for Fresh Expressions US, Cultivator of Fresh Expressions for the North Central District, and a Cultivator of Fresh Expressions for the Florida Conference of the United Methodist Church. Michael serves as co-pastor of Wildwood UMC with his wife Jill, where they direct addiction recovery programs, a jail ministry, a food pantry, and a network of thirteen fresh expressions that gather in tattoo parlors, dog parks, salons, running tracks, community centers, and burrito joints. Michael earned a Master of Divinity from Asbury Theological Seminary and is concluding a Doctorate in Semiotics and Future Studies at Portland Seminary with his mentor Dr. Leonard Sweet. Jill and Michael currently live in Wildwood with their blended family of eight children, three grandchildren, and a pug named Vader.\nRev. Jon Davis PhD is a known and recognized leader in the Episcopal Church. He served as Canon for Youth Ministry in the Diocese of Central Florida; training youth workers; initiating local, national and international missions and conferences. Jon is a successful church planter starting Church of the Incarnation \/ Oviedo, Florida in 2006 which grew from 6 people to an average Sunday attendance of 130+ in 2 years. He most recently served as the Executive Director of Canterbury Retreat and Conference Center during which he lead Canterbury back from the brink of collapse to a strong, sustainable and growing ministry. He is now on staff with Fresh Expressions. He is a sought after, dynamic and engaging speaker, writer, worship leader and ministry trainer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaauuc b/data_all_eng_slimpj/shuffled/split2/finalzzzaauuc
new file mode 100644
index 0000000000000000000000000000000000000000..526f51b482e8b574dfe62718cab1c455521a19f0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaauuc
@@ -0,0 +1,5 @@
+{"text":"Walk to class with your head held high in this Stanford Cardinal Nike Retro Pack shirt.\nMens_Nike_Cardinal_Stanford_Cardinal_Retro_Pack_Long_Sleeve_T-Shirt http:\/\/images.fanatics.com\/lf http:\/\/dmimages.ff.p10\/chains\/2880242.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":"When you think about tulle, what do you think? Many people picture that iconic Carrie Bradshaw dress or those really poofy dresses that were in your dress-up box when a kid. However, there isn't an age limit on tulle, so you can wear it at any time.\nFor a little bit more fun when you are at work, consider adding some tulle to a blazer. This is a great way to keep your style fun and young but still let people know that you are a professional and you deserve to be treated as such. You can't wear this every day, but it is fun for an occasional moment of lightness.\nFor an even better look, consider getting it in black or a pop of color so that you can amp up the fashion aspect of the silhouette.\nFor a bit of a daring look, consider using one layer of tulle as a top. Of course, this isn't going to be something that you wear to the office or something that you wear around your parents, but you can use it to look fun and modern. Don't worry about what other people think about you, just wear it as you want.\nSo many people think of tulle skirts and think of flouncy ballerina skirts, but it doesn't have to be that way. Instead, think about getting a tight tulle skirt. This skirt looks extremely elegant and fits extremely well, which is the key to making a tight fitting tulle skirt work.\nBomber jackets are extremely popular, and there is no better way to wear them than with some tulle and sequins. This is a great mixture of a feminine material with a more masculine shape. You will have fun playing with the different layers of a jacket like this and you certainly won't have to worry about overheating while you wear it.\nSo you don't have to think that you are a little girl when you are wearing tulle. Instead, you can embrace it and use it to heighten your fashion to another level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The town of Isfahan is found on a fertile plateau at 1,500 metres altitude, surrounded by the Sagres mountains. Isfana in one of the world's most beautiful cities, and it is a true gem when it comes to Persian art and architecture. Since old times, it has been an important centre along the caravan route. However, only when shah Abbas the Great made this the capital city in 1598 did the golden age truly begin. Isfahan is today one of Iran's most important travel destinations. The atmosphere here is taken straight out of 1001 Nights. The lavishly decorated mosques and palaces are impressive in their own right. The luscious parks with fountains and pavilions are magnificent. The old bridges that lead over the Zayandeh-Rud river are fascinating, and the old bazaar is a true labyrinth of tight alleys covered by tall stone arches. Here you can find lots of busy workshops and stores with an abundance of all sorts of traditional Persian handicraft. Being such a great art lover, shah Abbas the Great (1587-1629) founded his own famous carpet workshops in Isfahan. For centuries, Isfahan carpets have been admired, and it was probably these Persian carpets that the Europeans first learned to appreciate. Many of these carpets were gifted to kings, emperors and princes, and this made the art of carpet making known and desired worldwide. Shah Abbas named several of the patterns, all of which were inspired by the lily (also called lotus). The shah Abbas pattern is found in more or less complicated varieties and is being currently knotted all over Iran. Often you can see them in carpets like Keshan, Nain, Qum, Tabriz, Mashad and Isfahan. Modern-day carpets from Isfahan are still of high quality, and they have the classic patterns with a medallion with flower vines and beautiful corner motifs, garden and vase motifs and a 'mehrab' prayer alcove with the 'Tree of Life', birds and animals. Today, many carpets from Isfahan are being knotted on a silk warp with Senneh knots and a knot density ranging from 56,000 to some 93,000 knots per ft2. Today, you can also find Isfahan carpets on the market that are finely knotted on a silk warp, but which are made with lesser quality dyes and wool.\nWhen writing about Isfahan carpets, you have to mention the Seirafian family, which has been famous for their knotting artistry for generations. Today, there are several workshops in Isfahan that produce very fine carpets, including Davari, Enteshari and Mahdie and Dardashti.\nYou can find our selection of Isfahan rugs underneath.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you were asked to find your next job on social media, the chances are that you'd head straight to LinkedIn to start networking and looking for vacancies at your favourite brands. But have you ever considered Twitter to interact with employers and demonstrate why you should be chosen as an employee? Below, we've rounded up everything you need to know.\nIf you're an active tweeter and post about your favourite TV shows and weekends out on the town, we recommend setting up a separate profile for professional use. Your profile is your one chance to impress employers, so add an appropriate profile picture and header image, and write your biography carefully. Avoid words like \"aspiring\" or \"graduate\", and instead, label yourself as a professional. For the link, sign up to a website like Portfoliobox or Carbonmade and create an online portfolio where you can show off work and experience.\nCreating a Twitter account for your professional endeavours should mean you stay that way \u2013 professional. Avoid following friends and family (keep your personal account for that), and instead, follow brands and professionals that you'd like to network with. Businesses with \"meet the team\" pages usually allow you to follow staff members on Twitter; trawl through the internet and find relevant accounts that you can follow. Also be sure to check out news outlets and blogs that post industry news \u2013 doing so shows you're passionate about the industry and will keep you in the loop with important developments that may come in useful during interviews.\nTry not to worry about your follower ratio, and instead start networking with individuals that could prove useful in your job search. Take part in Twitter hours (dedicated hours of the week where local professionals come together to discuss their industry), and reach out to people on a one-to-one basis if you want to ask them a question.\nIf you have a blog, then tweet out your latest article once in a while to show your followers that you're a valuable member of their timeline. Don't be overly promotional, though, as doing so can be off-putting and could lead you to be unfollowed. Be sensible and professional in your approach to networking, and you'll be rewarded.\nWe've put together just some of the ways you can use Twitter to land a job. If you're not social, then don't worry \u2013 we've listed a whole host of open vacancies on our find a job page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The second India Disposable Hygiene Products Forum is an event that brings together senior executives and stakeholders from the personal hygiene industry across the world. It is a high-quality technical platform for manufacturers, suppliers and brand owners. The forum will explore the latest trends, technologies, new materials, product design and many other leading-edge topics of disposable hygiene products industry.\nPuneet Tomar, Consultant, will be presenting \"The Indian Tissue and Hygiene landscape and future growth opportunities\" on Friday, December 14 at 1.40 pm \u2013 2.20 pm IST.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Australian Heritage Festival, supported through the Australian Government's National Trust, has become Australia's most significant festival of heritage and culture. This year's festival, Connecting People, Places and the Past, delivers an array of events across the nation from 18 April \u2013 19 May 2019.\nConrad Gargett, one of the most experienced heritage architectural practices in Australia having completed the restoration, refurbishment and repurposing of some of the country's most prominent heritage-listed buildings, is pleased to present Brisbane's Customs House, as part of the Australian Heritage Festival 2019.\nBrisbane's Customs House celebrates 130 years this year (1889 \u2013 2019). Owned and operated by The University of Queensland, Customs House is listed on the Queensland Heritage Register and the Register of the National Estate. The building is The University of Queensland's premier riverfront restaurant, events and major cultural facility.\nThe University of Queensland celebrates 25 years since substantial renovations and reconstruction was completed in 1994. During this time, Conrad Gargett has contributed significantly to the building's ongoing conservation and adaptive reuse of its spaces, such as The River Room and Patina restaurant.\nAs part of the Australian Heritage Festival 2019, Conrad Gargett will use Customs House to present on the value of heritage buildings as key cultural places and venues for a city; the original vision for Customs House reuse and how this has evolved and changed over time; the importance of commercial success for investing in the building's on going care and longevity; and the value of a long term commitment to conservation and renewal as the building's uses and requirements change.\nThis Australian Heritage Festival 2019 event will take place on Wednesday 15 May, 5.15pm for 5.30pm \u2013 7pm, The River Room at Customs House, 399 Queen Street, Brisbane. For catering purposes, please register by 12 May. The entry fee is free. Registration includes one beverage (wine, beer, softdrink) on arrival. Additional beverages are payable on consumption. Numbers are limited.\nBrisbane's Customs House celebrates 130 years this year (1889 \u2013 2019). Owned and operated by The University of Queensland, Customs House is listed on the Queensland Heritage Register and the Register of the National Estate. The building is The University of Queensland's\u00a0premier riverfront restaurant, events and major cultural facility.\nThe University of Queensland\u00a0celebrates 25 years since substantial renovations and reconstruction was completed in 1994. During this time, Conrad Gargett has contributed significantly to the building's ongoing conservation and adaptive reuse of its spaces, such as The River Room and Patina restaurant.\nAs part of the Australian Heritage Festival 2019, Conrad Gargett will\u00a0use Customs House to present on the value of heritage buildings as key cultural places and venues for a city; the original vision for Customs House reuse and how this has evolved and changed over time; the importance of commercial success for investing in the building's on going care and longevity; and the value of a long term commitment to conservation and renewal as the building's uses and requirements change.\nThis Australian Heritage Festival 2019 event will take place on Wednesday 15 May, 5.15pm for 5.30pm \u2013 7pm, The River Room at Customs House, 399 Queen Street, Brisbane. For catering purposes,\u00a0please register by 12 May. The entry fee is free. Registration includes one beverage (wine, beer, softdrink) on arrival. Additional beverages are payable on consumption. Numbers are limited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I couldn't find any research online about this site so I thought I would post it here to warn people. There have been lots of comments on YouTube from people linking to this site. The thumbs up they receive are fake, making people think the link is legit. If someone is telling you there is a leaked video on http:\/\/breakingleaks.com then please know you are about to be scammed. In order to see the leaked video you need to pay for a text message, or complete a survey. These are primitive tactics but people still fall for them.\nbreakingleaks.com is a fake, fraudulent, scam website. Do not enter any information, do not believe the fake YouTube accounts that give the comments thumbs up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for a Florida mobile home?\nWe know the Central Florida home market and will help you find the match for your particular wishes and needs.\nWe have a large selection of affordable homes for you whether your needs are purely for retirement or for a winter getaway from the North's cold winters.\nWe have homes in Senior Parks.\nA senior park is defined as a 55+ community. However, many senior communities will allow adults that are 40+ to be approved as residents. There are some other conditions that we will be happy to explain to you. Investors may call us for input. Parks have various policies on rentals.\nPlease email or call the listing agent for information about these policies.\nHere in Florida ther are two types of parks. The first and largest number are communities where the park owns all the land and the individual owners of the homes are tenants. As such, the homeowners pay a monthly fee or \"Lot Rent\" (in our listings it is shown as \"Condo Fee\").\nEach park is unique and different. We can help you find the community which fits your needs.\nThe second type of park includes the land as well as the mobile home. Sales of these homes are handled by a Licensed Florida Real Estate Broker. The principal of our firm, Kim Blok-Andersen, is also a Florida Real Estate Broker, and as such can help you with this type of real estate transaction.\nCall any of our experienced Sales Associates, their contact information is detailed on the \"Our Associates\" tab.\nThis isn't your grandma's trailer!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Foreign food chains to cast local brands out of home market?\nManaging Director of McDonald's Vietnam Nguyen Huy Thinh has confirmed that the first shop, at No. 2-6bis Dien Bien Phu Street in the central district No. 1 in HCM City, would be inaugurated after 2014 Tet, or in February 2014.\nThe menu would include the most favorite dishes, from Big Mac sandwich to Cheeseburger or French fries. Especially, the 24\/24 service and drive-thru service to be provided showed the strong determination to win the hearts of consumers and the effort to understand Vietnamese consumption habit.\nIn April 2013, The Pizza Company, a Thai fast food brand, made its official presence in HCM City.\nIn late 2012, a subsidiary of IPP Group became the franchisee partner of Burger King, the US fast food chain. The first shops of the chain were opened in Hanoi and HCM City in early 2013.\nA senior executive of IPP revealed that it would open 3-4 new shops every month in big cities nationwide.\nAfter the 2013 Tet, Starbucks set foot in HCM City market, opening the first shop in the city and has been expanding its network since then.\nA lot of new foreign names have been heard recently in the retail market. Lotte, after opening its supermarkets in HCM City, Da Nang and Dong Nai province, has been marching toward the north, planning to open the first supermarket in Hanoi in early 2014.\nIn May 2013, NTUC FairPrice, a retailer from Singapore, joined forces with Saigon Co-op to set up a joint venture to run Co-opXtra and C-opXtraPlus chains.\nOnce cherishing the ambition of developing Pho 24 into a global brand, the brand owner Ly Quy Trung finally decided to sell the brand for $20 million.\nThe buyer of Pho 24 was VTI, the owner of Highlands Coffee. Shortly after buying Pho 24, VTI has sold 50 percent of its stakes to Jollibee, a group from the Philippines, for $25 million.\nAnalysts said Trung had to sell Pho 24 when more problems arose during the business expansion. Domestic shops then got \"worn out,\" even though the service quality was still maintained.\nThe lack of capital was believed to be the biggest problem at that moment. It was the time when VinaCapital withdrew its capital. Investment funds generally withdraw capital from companies after five years of investments.\nThe problems in the corporate governance then also put big difficulties for Pho 24 to compete with the foreign brands like KFC or Lotte.\nVietnamese companies are believed to be inferior to foreign ones due to their short term vision and the business strategies which only aim to short term benefits. Besides, they are not professional and experienced enough to run food chains.\nThe story of Bibica has been a hot topic in the discussions of the business circle. Bibica, with ambition of becoming the Number 1 in the sweets market, in 2007 decided to cooperate with South Korean Lotte, which then bought 30 percent of Bibica stakes.\nHowever, Bibica later admitted cooperating with Lotte was a wrong move. The South Korean partner, which has increased its ownership ratio to 38 percent, has shown its intention of take over Bibica.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Blue Light Of Eternal Peace. \u2013 Is The Bringer Of Inner Freedom And Clarity\u2026 A State Of Being Where You Carry No Burdens Of The Past With You\u2026 A State Of Being Where Every Breath You Take\u2026 Is The Freshest Cool Spring Air That Cleanses Your Soul\u2026 A State Of Being Where You Carry No Mental Weight, No Emotional Weight, No Physical Weight, And No Spiritual Burden. \u2013 Where Your Consciousness Is Free To Roam The Cosmos, And Explore All Of The Spectacular Magical Wonders That Creation Has To Offer! \u2013 Its Not About\u2026 What You Are Willing To Do To Reach Enlightenment! \u2013 Its About\u2026 What You Are Willing To Let Go Of\u2026 To Realize The Truth! \u2013 To Be Spiritually Reborn!\u2026 You Must Let Go Of The Past\u2026. Not Just Things About The Past. \u2013 However The Entire Past. \u2013 Even 5 Mins Ago! \u2013 Even What You Think You Are!\u2026 Your Alleged Identity. \u2013 Are You Willing To Let Go Of What You Think You Currently Are\u2026 On All Levels? \u2013 For That Is The Only Way To See What You Really Are\u2026 Beneath Your, Beliefs, Definitions, Opinions, Assumptions, And False Identifications\u2026 Of What You Currently Think You Are. \u2013 There Can Be No Spiritual Rebirth\u2026. Without Letting Go Of The Entire Concept Of The Past. \u2013 You Must Literally Become Newness Itself! \u2013 And Newness Cannot Be Measured, Described, Or Defined. \u2013 Newness Is Not A Solid Thing\u2026 It Is Total Change In Every Living Moment. \u2013 For Newness Is The Source Of Life Itself!\nPrevious PostPrevious Post: Artwork ~ Golden Orange Soft Love of God!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabvdl b/data_all_eng_slimpj/shuffled/split2/finalzzzabvdl
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+{"text":"Get the best Icelandic krona exchange rate today and have a bit more to spend on your trip to this scenic island.\nIf you can order the Icelandic krona amount you want: Some travel money companies only let you order over a set amount, or up to a set amount.\nThe exchange rate available: This tells you how much Icelandic krona you get for each pound you exchange. You can use our calculator to see the exchange deals for pound to Icelandic krona or Icelandic krona to pound.\nIf there are any fees: You may need to pay for delivery or collection when ordering your currency online. Not all companies charge for this, and those that do may only charge for smaller exchange amounts.\nIf you need to buy your travel money last minute, you can buy Icelandic krona at the airport, but you are unlikely to get the best exchange deal.\nWhen you buy Icelandic krona online, you can use one of your cards, but there may be extra charges to look out for.\nIt is free to use your debit card to buy foreign currency. However, if the amount you exchange forces you into an overdraft, you may get charged by your bank.\nIf you use your credit card, you will get charged a cash advance fee and daily interest by your credit card provider on the amount you exchange.\nIs it safe to order Icelandic krona online?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jersey City has undergone several transformations during the past 200 years. This site aims to visualize these fascinating changes through photos from then and now. If you are a Jersey City old-timer, I would very much like to hear from you - please use the comments functionality and share your JC experiences of by-gone years. And if you have old photos that you can share, even better - please send them to me. With your approval I will share them on the blog. I hope you will enjoy the blog!\nThe tall building in the middle of the old picture, where there is a gap today, was an old theater that at around 1930 was called 'B. F. Keith Theater'. I'll feature it in a different post.\nThe building with the tall facade on the left in the picture later became a department store - it was called the Wonder Store in the 1950s\/ 1960s. I'll try to feature it in a different post.\nCity Hall used to be much taller than the nearby buildings. Actually City Hall itself, used to be taller than it is today - the copulas were taken down some time after WWII. The combination of lower surrounding buildings and that it used to be taller, made City Hall look a bit more prominent than it does today.\nToday the the Mack-Cali building at Christoper Columbus Avenue dominates City Hall's backdrop.\nMaybe slightly off-topic, but the Mack-Cali building happens to be one of my least favorite buildings in downtown Jersey City - it is just a tall concrete bunker. It does not have any kind of retail \/ commercial space towards Christoper Columbus. That also creates a dead zone, in what could have been a lively street corner filled with amenities. Instead there is a parking garage - a waste of prime location.\nHudson Co. National Bank, Jersey City, N.J.\nAs many other sites of historic, commercial buildings in Jersey City, the bank has been reduced to a parking lot. It seems as if the stock of residential buildings have fared better, e.g. from Van Vorst Park to Hamilton Park.\nCopyright JerseyCityThenandNow.com. All rights reserved. Simple theme. Powered by Blogger.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SOLD ULTRA SCARCE 1938 TT-33 TOKAREV PISTOL WITH BOTH FACTORY ORIGINAL MATC HING MAGS\u2026\u2026.THE COMPLETE FULL RIG & RED ARMY OFFICERS BELT!!!\nAbsolutely the most complete example of a pre war early Production TT-33 Tokarev Pistol that we seen or heard of. In 88% condition, we have a 1938 Tula made TT-33 Tokarev Pistol still with both of the original factory numbered matching mags! This is almost unheard of in a pistol this early. The pistol is likewise completely matching and came back from WW2 as you see it. The bore is excellent and the pistol rates in a little better than average condition for a WW2 Russian TT-33. 1938 dated Russian Tokarevs with 2 matching mags just cannot be found, 1938 was a very low production year for this pistol and they are rare in any condition. This is one of the finest Tokarevs rigs ever offered for sale in the States.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alibaba offers 992 Stone Crushing Suppliers, and Stone Crushing Manufacturers, Distributors, Factories, Companies. There are 578 OEM, 536 ODM, 132 Self Patent. Find high quality Stone Crushing Suppliers on Alibaba.\nStone Crushing Plant Our company is considered as the one of the... Our company is considered as the one of the trusted and reliable manufacturers, suppliers and exporters of Stone Crushing Plant.\nA good stone crushing plant manufacturers must have something to do with product technology. They will use durable wear \u2013 resistant materials. They always apply advanced technology to produce, in line with the development trend of products that meet the requirements.\nStone Crushing Suppliers Directory - Choose Quality Verified Stone Crushing Suppliers and Manufacturers, Wholesale Stone Crushing Sellers and Exporters at Alibaba.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a news segment that aired this month on ABC News, reporter Tim Fleischer highlights the work being carried out by ULC Robotics' CISBOT robot under the streets of New York City. The benefits of deploying this technology are discussed through interviews with Con Edison Project Specialist Stephen Sweeney and ULC Robotics' President Gregory Penza.\nWatch the news segment below and read the full story on ABC7NY.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"All VixVids.to does is link embed contentthat has been uploaded to popular Online Video hosting websites(Vimeo, GoogleVideo, Xvideos, etc). All popular Online Video hosting users signed a contract with the sites when they set up their accounts wich forces them not to upload illegal content. By clicking on any Links to videos while surfing on Vixvids.to you watch content hosted on third parties and Vixvids.to can't take the responsibility for any content hosted on other sites.\nWe do not upload any videos nor do we know who and where videos are coming from. We do not promote any illegal conduct of any kind. Links to the videos are submitted and managed by seperate users on seperate servers.\nVixvids.to is an online service provider as defined in the Digital Millennium Copyright Act.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"United way grants are offered by the United Way, which is an international philanthropic organization and operates in many countries. There are number of community offices worldwide and so the varieties of grant programs is also numerous. The basic strategic goals are to support education, income and health. However, if you are in need of any kind of financial assistance, you can contact the community office of the United Way in your locale and find out whether there is any assistance available to support your cause or not.\nGather information about the community improvement grants offered by the united way and match the requirements to your needs. When you are inquiring about the grants of any foundation or organization, do not forget the main intention of the organization in providing the funding. Also, find out which is the most specific areas of development that the organization readily supports. All these details will help you have better chances of getting the free financial aid which you will not have to worry about repaying as well.\nWhether you are applying for the united way grants or financial support system of any other private organization or foundation, you should research the foundation's intention and requirements beforehand. Always keep in mind that private funding grants are always project specific and they will not entertain any other application as well. Do not forget to check out the deadlines and note down the name of the contact person to whom you have to address the grant proposal.\nEvery grant program can be approached through a specific application procedure which is detailed with the application form. Read it thoroughly and understand everything about the grant program and the free money that you will receive if you are chosen. No matter what happens you should be aware of the amount that you are going to get and how. So, invest proper time researching and gathering details of different grant programs that match with your needs and you find them capable of funding your cause.\nThe application process of united way grants is specific. It is good to prepare for the application at least one year before you actually need the funding. This will help you understand things and aspects in a better way and hence you will apply for the grants successfully. Get the information of the grants that the United Way is offering at present and learn how to motivate the panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is a common fact that mold thrives in damp conditions, which is why your grout lines create the ideal environment for mold to live in. Most homeowners become aware of the presence of mold when it is too late, and look for ways on how to get mold out of grout lines. However, in these cases, prevention is better than cure.\nHere is everything that you need to know on how to get mold out of grout lines and also how you can prevent mold from occurring. Besides the homemade remedies and tricks, here you can read reviews of the best mold removing products you can buy on Amazon such as the RMR-86 Instant Mold Remover.\n1 Why should you remove mold from grout?\n12 How to actually do it?\nWhy should you remove mold from grout?\nMolding is a condition that can occur anywhere, at any individual's homes. Mold basically requires moisture, oxygen, food and a surface to grow on, which is why mold has been in existence for centuries now. However, the growth of mold can be quite detrimental and even harmful, not just for your home but also for the health of those in your home. Mold gives off a particular odor and color, so if you do find mold at your home, it is important that you work on ways to remove this mold as early as possible.\nAny areas in your home which have poor air circulation, are damp, and even warm is the ideal environment for mold to start growing and thriving. This is why bathrooms, basements, laundry rooms, and kitchens make such an ideal place for mold to grow. You can find mold growing on the ceilings, on wood furnishings, in the caulking, on carpets, between tiles, grout lines and on the walls in the areas where there are high humidity and poor circulation of air.\nHowever, mold growing on tile grout is something that most homeowners might find in their home. The main reason behind this is that tiling is usually laid down in places where there is a lot of water or moisture, such as the bathrooms or the kitchens. These areas are prone to seeing an abundance of water; such as in the showers, water in the kitchen for cooking or cleaning, etc. Grout by itself is also quite porous in nature, and this, along with the presence of moisture, makes it an ideal environment for mold to start growing in.\nMany people often prefer to use DIY products as opposed to strong, harsh chemicals to remove the mold from their grout. Chlorine bleach is one of the most common items to use for removing mold from grout lines at your home. The chlorine bleach is an effective solution not only for killing mold spores and eliminating them from in between the grout lines, but the bleach also helps to clean as well as disinfect the tiles and grout.\nTake a spray bottle and add one part chlorine bleach with four parts of water. The spray bottle will make it easier for you to target hard to reach places on the tiles.\nUsing the spray bottle, spray on all the affected areas and leave the solution on for about 30 minutes. Using an old toothbrush, brush these affected areas vigorously to remove the mold spores and any stains on the tiles and in between the grout lines.\nRinse the area with cold water, and you will see a visible difference. If required, you can repeat the process in the same area once again to clean it further.\nYou can find more information about removing mold with bleach in this article.\nUsing chlorine bleach can be an effective solution for removing mold from grout lines, however, the bleach itself can be potentially harmful to the individual who is performing the cleaning procedure. The bleach might also adversely stain or discolor the tiles of your home. In these cases, there is another solution that you can use- which is baking soda. While baking soda is not as powerful and effective as bleach, it can be used for removing mold from grout lines.\nYou will need to mix one gallon of water with \u00bd a cup of baking soda. Using this mixture (preferably in a spray bottle), spray this on the affected tile and grout area and leave it on undisturbed for 30 minutes. After 30 minutes, rinse the area with warm water.\nNow, slightly wet a sponge and dip this into pure baking soda. With the sponge, scrub down the affected area on the tile and grout and rinse out this area once again with warm water.\nThe third method you can use is to mix 1\/2 a cup of baking soda with some water and make a fine paste out of it. Now, spread this paste evenly over the grout mold and tile area. Let this solution sit on the tiles and grout for about 10 minutes, after which you will need to scrub the affected area with a bristle brush or an old toothbrush. You can repeat this step once again if you feel the need to.\nAdditional tip: You can substitute the baking soda with Borax and use the same methods above to clean your tiles and grout. Borax is a highly alkaline solution and is ideal for cleaning, removing odor and disinfecting the affected areas.\nFor those who are looking for some of the best natural products to remove mold, vinegar can be a life savior for you! Vinegar is highly acidic by nature and is the best product to use to not only remove but also to prevent the growth of mold on your tiles and your grout lines.\nHere is how you can remove mold using vinegar \u2013 In a spray bottle, mix equal parts of white vinegar with water. Now, spray this mixture on to the affected areas in your home and allow the mixture to sit undisturbed for about 30 minutes. After 30 minutes, you will need to scrub the affected area with a bristle brush and then rinse out the area with water. You should repeat this a couple of times to get the best results out of the distilled white vinegar on the mold.\nYou can also use tea tree oil as a substitute for white vinegar. In a spray bottle, add two teaspoons of tea tree oil with every cup of water. Now, spray this solution on to the moldy areas on your tiles and your grout. Allow this solution to sit undisturbed for a couple of hours and then rinse it out with water.\nA good tip when using these two methods would be to keep the windows of the room open, as the smell of tea tree oil and vinegar can be quite strong. By keeping your windows open, you will ensure that there is proper circulation of air in the room and you are not affected by the strong smell of the vinegar or tea tree oil.\nIf you have tried all of these methods of mold removal but still are suffering from grout mold, there is one last trick that might come to your rescue. Hydrogen peroxide can work like magic in removing grout mold from your homes. This cleaning agent is a great fungicide and is also a whitening agent which will help effectively clean the area and even remove discoloration.\nThere are two ways in which you can go about this exercise- you can mix hydrogen peroxide with water and spray it on to the affected areas on your tiles or grout. You can also take the hydrogen peroxide directly and apply it over the affected areas to get better results.\nMix some quantity of hydrogen peroxide into flour to make a kind of a thick paste. Now, using proper tools and protecting your hands and your face, you should apply this paste on to the tiles and grout where mold is affecting the area. Leave this solution on the tiles and grout overnight to get the best results. The next morning, rinse out the area with cold water.\nHowever, in cases where the growth of mold is too extensive, these DIY products might just not work. In those cases, you will need to use a stronger chemical agent to clean out and remove the grout mold.\nCan be used on wood, vinyl, marble, fiberglass, cloth, shower doors, etc.\nHere you can check the current price on Amazon.\nThere is no need for scrubbing, brushing or even sanding when using this spray formula.\nSimply spray the formula on to the affected moldy area and the spray will start working instantly to remove the stains within 15 seconds or less!\nThis big sized 64 oz. cleaning concentrate can effectively clean and remove mold and mildew stains from an area of up to 1000 square feet. This product is the ideal outdoor cleaning concentrate that you can use on your deck, patio, stone, brick, etc.\nThis 64-ounce spray is easy to use as it comes with a spray pump attached to the bottle. The product is concentrated, comes in an easy to use packaging and can effectively remove all kinds of mold and mildew stains from wood and other products.\nHow to actually do it?\nRemoving mold is quite a simple process. However, it should be done very carefully to ensure that you do not harm yourself with the mold spores or the cleaning products. Wear your safety goggles, rubber gloves and mask on your face for your protection. It is important to ensure that you do not accidentally ingest or inhale any of the mold spores as they are very harmful to your health.\nProperly clean and vacuum the areas in your home where mold is growing. The best kind of vacuum to use would be a HEPA air filter vacuum with an attached long arm which will reach all of those hard to reach places in your home and clean up the mold spores along with any dust and grime.\nChoose any of the great mold cleaning products mentioned above and apply it to the affected area as per the instructions on the product package.\nRinse out properly to remove the entire residual product from the surface you are cleaning. Allow it to dry completely.\nIn cases like this, the old saying \"prevention is better than cure\" certainly rings true. The methods above tell you some of the best ways in which you can remove grout mold, however, the best case scenario would be to prevent the growth of mold on your grout altogether. The most important thing to remember in this case is that you can prevent mold growth by simply cutting off one important factor which is required for mold growth \u2013 moisture. Keep the tiles of your home clean and dry and if you are careful, you will not find mold growth in those areas.\nSince these areas of your home experience a lot of humidity and water condensation, it is important to ensure that the areas have good air circulation. Keep the windows and doors of these rooms open to allow good air circulation.\nIf you have an exhaust fan in your bathroom, they always run it for a while after taking a shower. This will remove the excess moisture from your bathroom and reduce the risk of the bathroom tiles staying damp and mold growth occurring.\nAlways wipe the shower walls once you are done with your shower, as this will help to keep the tiles and the shower area dry. You can use a sponge or even a washcloth as this will help you reach all the areas in between the tile and the grout and prevent any mold growth.\nIf you are using shower curtains or linen in your bathroom, it is important to buy a material which is easy to clean and maintain. After your daily shower, wipe down the shower curtains and leave them extended as this will help them to dry more efficiently, thus preventing your bathroom tiles from staying damp.\nAfter your shower, another important thing you must learn to do is to hang your wet towels to dry outside. By keeping wet towels inside the bathroom, you are simply contributing to moisture and dampness in your bathroom and on your bathroom tiles and grout.\nYou can also seal off the grout in your bathroom using a silicone grout sealer or a water-based sealer. This sealant will basically keep the grout dry and now allow the moisture in your bathroom to easily penetrate into the porous texture of the grout. Most of the grout sealants also come with a good anti-microbial technology which will further provide a barrier to not only moisture but also to other kinds of microbes including grout.\nHey, Philip, Thanks for sharing the huge guide on getting rid of mold in the bathroom. Well, I have tried to seal the grout with baking soda before but it never worked for me. I'll definitely give a try to other options.\nThe home I just bought is a bit on the older side. It is good to know that if I find mold in grout it could mean that there an air circulation problem. I should probably also get a professional to remove the mold as well.\nAll tips are look pretty you keep your basements and other such areas which experience high humidity dry so that mold growth does not occur. Anyway, great post and thanks for sharing such stuff here with us.\nThis is some really good information about mold remediation. It is good to know that it would be smart to have a try using some bleach to get rid of the mold. If I found mold in my home I would want a professional to come and get rid of it all as quickly as possible.\nYou can try power washing to remove mold, dirt, grime or grease in every part of your house.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the profession of arms, there is no second place. Victory is not an option, its absolute. That is why Condor Elite products have to be better and why we will make sure they are. Since the dawn of time, dueling combatants have met on the field of battle. In the end there is always a winner and a loser based on superior skill, mindset, training and the best equipment. So another way to phrase this eternal engagement is \"1 up 1 down\". We know which one we will be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The United States is not an island.\nWe belong to an international community of nations.\nThese articles reflect on some of our interactions with our world neighbors.\nWhat is the G-20 and Why is it Coming to Pittsburgh?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadjis b/data_all_eng_slimpj/shuffled/split2/finalzzzadjis
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@@ -0,0 +1,5 @@
+{"text":"After graduating from USC with B.A. in Biological Sciences, Chris began working for his alma mater's Office of Undergraduate Admission, responsible for coordinating the University's merit-based scholarship process and 8-year combined B.A.\/M.D. program. Working closely with high school populations, Chris became interested in issues that ranged from self-harm to educational access and equity, which has helped to inform his current research interests in digital media literacy, learning, and youth cultures.\nChris draws upon his natural sciences training in order to better understand the role of transformative bodies in Gothic Horror, his primary area of research. Particularly interested in the confluence of horror, identity, narrative, gender, media, and youth, Chris currently explores how the eroticization of trauma and wounded bodies acts to articulate cultural anxieties. When not pursuing his studies, Chris enjoys working with 826LA and writing his upcoming chapter on mediated celebrity in The Hunger Games and Philosophy while drinking over-priced coffee.\nRead more about Chris' work at civicpaths.uscannenberg.org or follow him on Twitter at @TrojanTopher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Odoacer represented the low point of Italian history. He was a German warrior who became the first king of Italy in 476 after leading the revolt that threw out the last Roman emperor in the West, Romulus Augustus.\nThen, after assuming total power of the empire, Odoacer did next to nothing with it. There were no grand buildings, no funding of literature, no artistic advancements, no great works of engineering, although he did terrorize his neighbors, one of whom was Theodoric, King of the Goths.\nAfter invading Italy and forcing Odoacer into Ravenna and soon thereafter obtaining his surrender, Theodoric slew Odoacer at a reconciliation banquet. Theodoric immediately began restoring ancient monuments in Rome, including the Colosseum and Theater of Pompey. He also built an impressive palace and had a massive mausoleum constructed for himself.\nHe directed the decoration of Saint Apollinare, among other building projects that transformed Ravenna into a great capital city. Under his rule, Ravenna soon began to look and act like the great capital it was.\nTheodoric was born in Pannonia in 454, after his tribe had defeated the Huns at the Battle of Nedao. From the age of 10, Theodoric had grown up as a hostage in Constantinople, receiving a privileged imperial Roman education.\nHe succeeded his father as leader of the Pannonian Ostrogoths in 473. And just 15 years later he would find himself the ruler of the remnants of the western Roman empire as well as the Ostrogoths and Visigoths. This is his amazing story.\nIn this classic book, written in 1891, scholar Thomas Hodgkin recounts not only Theodoric's great accomplishments, but also the unraveling of his empire upon his death.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flaunt is one of the professionally designed styling ranges by Sebastian, which ensures a natural and effective shine - for every hairstyle and every look. The Finish hair care of the range uses various rock crystals, which result in a matchless and distinctive WOW factor. The rock crystals also come into play in the visual presentation of the range, since a large part of all products are radiant in an Australian blue, even the Wax Paste is placed in a blue-black contrast. Thus it is a hair cosmetic range which allows excellent finishes, offers a unique shine, and outfits the bathroom with the colour of rock crystal.\nThe range uses various finishes, which differ in intensity and hold. Likewise a pure gloss finish can be chosen, which offers a natural finish for lightly styled hair. The Sebastian products from the Flaunt range command a weightless effect, which is why the hair is not unnecessarily disturbed despite the outstanding shine. Thus waves or curls also survive, and the hair can also appreciate it since it is not sticky. This guarantees durability.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We will be hosting our first trip to the city for our gbm members. We want a change of scenery to photograph and for our members to experiment and practice taking photos in a different environment. This can help us expand and critique our portfolio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Soldiers' Angels - Give a Year-End Gift Before Time Runs Out!\nGive a Year-End Gift Before Time Runs Out!\nCan you believe the end of the year is here? Now is a great time to give a year-end gift and be a hero for our heroes! Your continued support makes a huge impact on the lives of service members, veterans, wounded heroes and military families around the world. Even better, your contributions are tax deductible. Give a gift before time runs out! And be sure to check out the fun year-end poem we created with the help of some of the service members we support!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadujk b/data_all_eng_slimpj/shuffled/split2/finalzzzadujk
new file mode 100644
index 0000000000000000000000000000000000000000..317bb79625ab9d3d2bbd1fbe4c59b78ec6c20af3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Overview of the cheapest flights from Linz to destinations in South Africa with relevant fares in next months. Only selected flights from LNZ are shown. Use the search form below for more dates and routes.\nAre you looking for flights to Linz? Please switch to our overview of South Africa to LNZ. Fill in the dates to find the best airline tickets deal for your journey.\nMore flights: Linz to CPT. No direct flights from Linz (LNZ) to Cape Town (CPT), South Africa found. Prices of flights to Cape Town sourced on Wednesday 25th July 2018 at 10:38.\nMore flights: Linz to JNB. No direct flights from Linz (LNZ) to Johannesburg O.R. Tambo (JNB), South Africa found. Prices of flights to Johannesburg O.R. Tambo sourced on Monday 30th July 2018 at 00:59.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Property Details: 3 Bedroom, 3 Bathroom Property with approx. 1630 square feet.\nOrange Beach condo for sale at Four Seasons. Imagine having a 3-bedroom 3-bath beachfront condo with gated entrance, sugar-white beach, and your own private fishing pier in the Gulf of Mexico. Well that's exactly what you get when you own at Four Seasons Condominiums in Orange Beach Alabama. This 6th floor unit is being sold fully furnished and completely equipped to move in immediately or continue a successful rental management program with impressive rental revenue numbers paid to the owner. This property features indoor and outdoor pools, a fully equipped exercise room, a common meeting area for owner gatherings, and as mentioned, your own private pier!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Question 1.What is the name of the substance that as early as 1785 was used to treat heart failure by British physician William Withering. It acts by increasing the force of heart muscle contractions. This substance was (and still is) obtained from the leaves of the foxglove plant ?\nQuestion 2. Malaria is caused by infection with protozoal parasites called plasmodia. Specifically what female, blood sucking agent is responsible for transmitting this infection ?\n\"THE STUMPER!\" During WW2 in Germany, one of the earliest programmable, automatic, general purpose digital computers was developed. It was called Z3 and pre-dated Howard Aiken's Mark I; and in many ways was more advanced than ENIAC. The designer responsible for the development of Z3 was: Alan Turing, Konrad Zuse, Werner von Braun, John von Neumann, or Claude Shannon?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arts & Minds is inviting health and social care professionals (who have an interest in or provide care for combat veterans with PTSD in Cambridgeshire) to a performance on 6 July by the highly praised Combat Veteran Players from London. The CVP is a theatre company comprised of veterans who have developed into gifted actors over the course of the last two years while overcoming mental trauma. The CVP will be travelling to Cambridge to perform Shakespeare's Henry V and share their skills and experiences with fellow veterans in the local area who are living with mental trauma and related difficulties. The performance and workshop will take place at the Corpus Playroom in Cambridge, 10 St Edward's Passage, Cambridge CB2 3PJ, from 11am. Contact Gavin Clayton, Executive Director of Arts and Minds, at mindsarts@gmail.com or telephone 01223 353053 to book your space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yoyo Village is the brain child of multiple time UK National Yo-Yo Champion, Luke Roberts.\nWith over 15 years of experience competing, performing and teaching, Luke had always wanted to open his own store someday.\nIn late 2014 Luke had been approached by Richie Windsor - owner of the famous yoyoz.co.uk store and forum - who revealed he had decided to close after many, many years of excellent service and support to the UK and European community, and wanted to know if Luke was interested in carrying on the legacy of the UK Players Forum and also an opportunity to start up a store, which of course Luke jumped at the change.\nIn January 2015, the gates of the Yoyo Village finally opened to the public.\nLuke wanted the look of the store to reflect part of himself - being a huge cartoon fan he wanted the store to be colourful, animated and family friendly.\nOne day Luke hopes to open his own physical store and do what he loves best - spreading the fun and joy of yo-yo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..0f7443ceb9d7f9736e8ced79b0363af4691a650f
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@@ -0,0 +1,5 @@
+{"text":"With advance notice, Lale and Mehmet will be happy to cater to many of your dietary requirements or preferences with the breakfast.\nAlso with advance notice, your room may be extra ozone treated for allergies. Linens can be washed with scent-free and hypoallergenic medium may be provided.\nA complementary boxed-to-go breakfast for early departures is available.\n\ud83d\ude42 tickets to Biltmore Estate or Flatrock Playhouse, at cost.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Favors by Serendipity This new addition to our bottle opener collection makes the perfect favor for a fairytale themed event. Bring the mystical charm of a castle to your occasion tables! The castle themed bottle opener is made from solid cast metal with a chrome silver finish. It features a detailed, raised 3D castle design with a castle gate, turrets and flags.\nAttached to the bottom of the castle is a sturdy metal bottle opener. The castle bottle opener is presented in a white gift box with a brick design and a clear top. The castle rests on an orange stage. The box is tied with a white organza ribbon and bow. Attached to the bow is a matching orange 'For You' tag.\nThis new addition to our bottle opener collection makes the perfect favor for a fairytale themed event. Bring the mystical charm of a castle to your occasion tables! The castle themed bottle opener is made from solid cast metal with a chrome silver finish. It features a detailed, raised 3D castle design with a castle gate, turrets and flags.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A powerful new report from the Southern Education Fund demonstrates that the misuse of invalid standardized tests for awarding teacher licenses is dangerously narrowing the pipeline of qualified African American educators and putting some Historically Black Colleges and Universities (HBCU) at risk of closing.\nUnintended Consequences: Perspectives on Teacher Testing and Historically Black Colleges and Universities concludes there is no demonstrated link between passing standardized tests and either readiness to teach or the learning outcomes of elementary or secondary school students. The report is based on interviews with leaders of HBCUs, a set of commissioned research papers, and a review of previous research on testing and teacher preparation.\nThe problems caused by state licensing exams are greatly intensified by two recent federal laws. The Higher Education Act of 1998 requires states to flag \"low-performing\" institutions, which most states do by relying on each college's test passing rate. Colleges that do not improve their pass rates may lose federal funding, and students may be denied federal grants and loans to attend those schools. The \"No Child Left Behind\" Act of 2001 mandates states to ensure that new elementary school teachers pass a licensure exam.\nFaced with the threat of closure from lack of funds if they do not lower their exam failure rates, some HBCUs now allow students to enroll in teacher preparation programs or to graduate from such programs only if they have already passed the teacher tests. This eliminates the risk of producing graduates who cannot pass the state's licensing exam, but it has greatly reduced the size of some teacher education programs.\nThe consequences fall most heavily on students who generally score lower on standardized tests, disproportionately African American, Latino, Native American Indian, low-income, limited English proficient and disabled students. Thus, the testing mandates reduce the numbers of teachers of color at a time when the proportion of non-white public school students is growing nationally. Both minority and majority students are thereby denied the opportunity to study with qualified teachers of color.\nIf teacher tests could actually predict who would be an effective classroom educator, it would make some sense to require the exams while providing extra assistance to enable students to pass them. However, decades of research have repeatedly shown that the tests have no capacity to predict successful teaching. Just two years ago, the National Research Council of the National Academy of Sciences warned that licensing tests should never be used as the sole measure of prospective teachers or their college preparation programs (Examiner, Summer 2001).\nRather than use testing to reduce programs, the SEF calls for stronger efforts to enroll more students from \"nontraditional\" talent pools. The report describes the great success of the SEF's Pathways program which enabled almost 1000 such students, many of them low-scorers on standardized tests, to become teachers. Despite the success of the program, it remains underfunded.\nSEF recommends that the link between funding eligibility and test results be severed, that African American educators be more involved in test development, and that research focused on teacher quality and testing be pursued.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Leaky gut syndrome is a medical mystery characterized by vague symptoms such as bloating, gas, cramps, food sensitivities, and abdominal pain.\nIt is a condition that affects the lining of the intestines resulting in a porous intestinal wall, with the leaking of intestinal content into the blood stream. It creates a dysfunctional environment for proper digestion with the intestines losing some of their ability to filter nutrients and other substances. When this happens, particles of incompletely digested foods, bacteria, other waste products may leak through the intestines into the bloodstream.\nIt is thought that leaky gut syndrome, when left untreated can result in conditions such as celiac disease, multiple sclerosis, autism, chronic fatigue syndrome, irritable bowel syndrome, eczema and dermatitis. It is also thought that these conditions themselves can cause leaky gut syndrome.\nNo specific treatment exists for this relatively new disorder, but symptomatic improvement is common with dietary modifications, reducing stress, and correction of gut flora. Various dietary supplements have been found to help in healing of the gut wall in leaky gut syndrome.\nIt is rich in antibodies which work against various pathogenic bacteria and viruses that harm the gut wall. Colostrum is seen to reduce the frequency of diarrhoea and provides symptomatic improvement in E. coli infection. The antibodies within MIP Colostrum are effective in killing Helicobacter pylori, which is responsible for most of the gastric ulcers. It also prevents damage induced by NSAIDs, commonly taken for chronic pains. Growth factors such as Epidermal Growth Factor (EGF) and anti-inflammatory substances in colostrum can also help in healing of the leaky gut walls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We know that not everyone is a Project Manager and there's a lot of operational work that needs to be done in an organization in addition to project-focused work. Using streamlined and automated solutions for common processes can make everyone's life a little easier.\nThis type of work is intended to change or introduce new ways of doing business. Projects should have goals, budgets\/resources, and end dates by which the new ways of doing work will be accomplished.\nQPM Work Management Solutions combine software for collaboration, workflow, and databases to create repeatable and measurable solutions. The purpose of these solutions is to streamline work and organize it so that the tedious and error-prone components of work are automated and people are able to focus on the right things to maximize their performance.\nOrganizations are more efficient and make fewer mistakes when they use repeatable processes and tools to simplify the way they get work done. Simple workflows, common tools, and standard approval cycles make processes repeatable and trainable so anyone who works in the organization knows what to do and when to do. No more heroics to get things done. Contact us to find out what we can do for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafcxi b/data_all_eng_slimpj/shuffled/split2/finalzzzafcxi
new file mode 100644
index 0000000000000000000000000000000000000000..d7babd81291579842ecf9f2121f8ef613641fde4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafcxi
@@ -0,0 +1,5 @@
+{"text":"I got this teddy bear as a kid at a grade school carnival in 1977 for $1.00 wrapped in newspaper at a white elephant booth. I lived out west then (don't know if this matters or not. Are there teddy bear manufacturers that only sold out west?) I have been intrigued ever since on his \"ancestry\". I'm hoping with the help of the internet I might get farther.\nHe is 18 inches long, his arms and legs are jointed including his head which can turn all the way around. He is well loved so the fur is partially rubbed off. It is a Golden colored, short mohair. I believe he is filled with straw or kopak. When I squeeze him, he is lumpy and the filling is not soft and a little compacted. I don't feel a noise box. He has a metal formed nose, perhaps tin as a magnet is not attracted. It appears there might have been color on the nose, dark brown? His mouth is black yarn. The mouth was one piece of yarn, not embroidered but obviously pulled up into a triangle shape by the stitch under his nose. It appears his muzzle was shaved. It was not made from a separate piece of clothe, but shaped from the head into a triangle. His eyes are amber with black centers. I think they are plastic but not sure. He is humpbacked. His torso is long and narrowish The hands and feet are velvet covered and long oval shaped.The arms are long, thin, and slightly curved. They are placed low on very sloped shoulders or you could say he has a long neck. His legs are long, thin, straight and the feet are shaped into like a boot shape. The ears are large and round, they are shaped to have a curve. They are created from seperate pieces of cloth and attached to the head. I cannot find a hole in his ear which seems to rule out him being a Steiff. I have inspected all the seams and cannot find any evidence of a label or tag. From my research I would say he is from 1920 thru the '30s.\nI hope you all can help me. Like I said was well-loved in his Youth and appears to be very well made.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GOLDGRUBE IN THE VILLAGE CENTRE.\nThis beautiful restaurant has a garden terrace and a separate bar in the basement. The farm has been lovingly renovated this year and has a completely new inventory. The property is located directly in the village centre of Reigoldswil, where hundreds of people travel to Wasserfallen on weekends. The inventory must be taken over.\nAuf Anfrage, 4418 Reigoldswil, Switzerland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Givelle Lamano helps Hayward clients with their assault & battery needs.\nConnect with a local Hayward, CA attorney with proven experience helping clients with Northern California assault & battery issues.\nAre you searching for a top assault & battery lawyer in Hayward, Northern California?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apartment New house is on the 11... is situated in 7\/1 Nalbandyan Street in Yerevan in 1.4 km from the centre.\nThe services include the following list of services: private non-smoking rooms, mini-market on site, internet services, 24-hour reception desk, convenient airport transfer, indoor lift, car park, bicycle rental(additional charge). Please consider that payment for the accommodation and amenities is possible only in cash.\nFor your comfortable accommodation is offered only one room type - apartment. For guests are provided facilities such as telephone, air conditioning, hairdryer, kitchenette, refrigerator, ironing facilities, free toiletries.\nThis apartment has a tile\/marble floor, washing machine and balcony.\nPlease inform New house is on the 11... in advance of your expected arrival time. You can use the Special Requests box when booking, or contact the property directly with the contact details provided in your confirmation. This property will not accommodate hen, stag or similar parties.\nLocation and space was good.\nNo water on the final day... The balcony door was not closing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Snow is a at-home teeth whitening technology that does not need a prescription and is FDA accepted . According to Snow,\"top celebrity dentists\" use it and are pleased with the outcomes. They say it is an award-winning system for whitening teeth.\nThe Harvard science study firm, Snow Labs serves over 100,000 clients. They say they're constantly striving to improve the technology and consistently trying to discharge more helpful products. Additionally they give a few of the proceeds to help provide treatments to children who can not afford them.\nThe Snow teeth whitening treatment is a FDA-approved, at-home tech which provides folks dentist-level whitening in a fraction of the cost. A Harvard science research firm produced it.\nThe company states they deliver your purchase very fast and totally free of charge within the United States. International orders will have a fee.\nThey do have a results warranty although they do not have a warranty. This means if you are not happy with the results, the serum will be replaced by them. In other words, they don't accept returns or issue any cash refunds \u2014 they will issue store credit.\nMost of the individuals who have used Snow see results in around three days, but each individual should use it till they see the outcomes they want. It could take up to 21 days of use for 15-30 minutes each session.\nMost of the user reviews must do with customer support from early in the company's life. The item seems to function really well, and the customer service has improved over the years.\nFirst, you employ a serum using a brush. Basically, it's a cream-type material that you paint onto your teeth. You place the Snow apparatus into your mouth for 10 minutes every day \u2014 it fits over your teeth like a tight mouth guard.\nThe Remineralization Gel you should use after the whitening treatment, particularly if you've got sensitive teeth. This gel reduce sensitivity will greatly strengthen your tooth enamel, and returns vitamins and minerals to your own teeth. You shouldn't do it but rather take a break for 1-3 days before another treatment Should you have teeth and gums.\nSnow is FDA approved, and it is a big plus. So it is relatively safe. During the treatment, you do end up swallowing a little but not enough it is painful. In reality, even when you are breastfeeding, you can still use this whitening treatment.\nFor at least one week after starting the treatment, you should avoid drinking any fluids which may stain your teeth, like coffee, wine, or juices that are specific.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafzss b/data_all_eng_slimpj/shuffled/split2/finalzzzafzss
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index 0000000000000000000000000000000000000000..11efc2fb79f96062a7c4324dfed56d60a0bfdc96
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+{"text":"The Latest: Commercial Data, Members Make International Connections, 2018 NAR Reach Class Announced, ADA Reform a Success thanks to Members like You.\nOf particular significance to state REALTOR\u00ae associations are two deductions that will impact member businesses, the 179 and new 199A deductions.\nWhile the National Association of REALTORS\u00ae (NAR) boasts a membership of 1.3 million members, the face and voice of the real estate industry before the United States Congress boils down to 535 members who volunteer as Federal Political Coordinators (FPCs).\nAttitudes toward transportation are changing. Time is a valuable commodity and many commuters are shifting away from automobiles in order to eliminate maintenance costs and use commutes to focus on things other than driving.\nGrowth, progress, and disruption are exponential in today's commercial real estate industry, leaving many successful agents wondering how their business might change in the Information Age.\nIf there's a compact, useful way to think of starting investment in nonresidential real estate, it's to begin by defining what the act of investment in this asset class is.\nWhen brokers represent investor clients, they spend a lot of time making their clients rich. Development projects and investment real estate assets allow you to build your own personal portfolio.\nIn todays' business landscape, it's key to understand basic people characteristics, no matter the geography, and crucial to have a greater perspective on attitudes, lifestyles, and behaviors to ensure economic viability.\nAccess to parks, trails, and gardens can have a big impact for properties & developments. So where do you look for green spaces & how do you plan?\nLand \u2013 a word some commercial agents fear and others view as an opportunity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yesterday, I reminded a good friend of her blog. She thanked me for the reminder and replied, \"These days, I feel like I have nothing much to say.\"\nThat is exactly how I feel as well. And it is why this blog--and my writing--is barely alive.\nI feel like ever since social media gave everyone a platform, if you intend to write something and share it, you better make sure it's worth disturbing what ever silence there is left.\nBut shouldn't that be the goal of any creative writing endeavor? Isn't that why we strive for literature?\nAnd now I'm suddenly reminded of this poem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For partners | HANSE Group s.r.o.\nWe believe that reliability is the guarantee of success in any business.\nOur customers trust us and we appreciate this trust.\nWhy is cooperation with HANSE Group s.r.o. convenient and profitable for our partners?\nFirst of all, HANSE means high-quality products. This fact is confirmed by the results of numerous laboratory and field tests, as well as by international quality certificates.\n- High consumer qualities of HANSE products persist throughout the whole period of operation which allows us to implement special warranty programs.\n- HANSE offers the best conditions of partnership: the flexible pricing policy and various supply schemes, information and promotional support, training and motivation of the personnel, assistance in work with customers.\n- HANSE promotes the product together with you using a wide range of advertising techniques and events: information support, joint promotions, seminars, advertising materials, merchandise, exhibition equipment.\n- The HANSE assortment is developing constantly in accordance with the result of the ongoing analysis of European car market and aftermarket. As a result, new products appear on the list of goods every month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The heart of this is a metallic spiral outer catheter, which find\u00b4s it`s way to inner os of cervical canal. These physical properties are unique and can never be achieved with a plastic Outer Catheter.\nOptimal US View is the most important parameter in order to prevent difficult Embryo Transfer.\nphysician to insert the outer catheter atraumatically.\n2.) optimal ultrasound view .\n3.) for easy, but in particular for difficult Embryo Transfers.\n4.) Echotip at inner catheter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To attract and retain the best people we invest in developing and supporting all of our staff to achieve their full potential. This begins with a full induction programme to help people understand the Atrium vision and values and continues with a variety of on the job and structured learning to enhance personal development, technical skills, leadership and management skills.\nThrough our learning and development framework we empower people to attain new skills and progress in their career. We expect all of our staff to attain their relevant professional qualifications and we provide support and encouragement, including full funding, to assist them.\nThe profiles below illustrate some of the interesting career paths that employees have taken at Atrium and how learning and development has played a part.\n\"One of the many good things I can say about Atrium is that they have never batted an eyelid when it has come to my learning and development. Whether it has been a refresher course in Excel, an introduction into understanding financial reports or even a windstorm seminar in Miami, they have always been encouraging. As one person said to me when I first arrived, Atrium sees it as an investment. Whatever you learn or skill you gain, the company will hopefully benefit from it too.\nComing from a rather dull admin job at the age of 23, I joined Atrium in October 2002 as an underwriting assistant. Although I started out in the cargo team, I moved 8 months later to the Energy team where I have been ever since. Initially my job entailed me sitting on the box in Lloyd's beside the Underwriter and although it involved a lot of data entry, it had the interesting part of seeing how the underwriter goes about their business. Unfortunately there is no manual on how to underwrite, although I think a few people wished there was, it is all about learning as you go along. Experience and exposure are crucial to the job as well as asking questions. I was lucky to work with Richard Harries and Tom Gault \u2013 two very experienced underwriters so I managed to gain an excellent insight into the Upstream Energy world from two different perspectives.\nI remember when I was first starting out we used to have these weekly educational lessons called 'Gene pool' given to us by a very knowledgeable but extremely eccentric gentleman called Jeffrey Gilbert. If I had to say who first mentored me through the hazy first few years of my job, then it would have to be him. He had a knack of getting you to question everything and you were always terrified in case he asked you a question and you didn't know the answer.\nAtrium is extremely keen on you getting your ACII qualifications so 2.5 years on from starting them; I became a qualified Chartered Insurer. Shortly after that, I got my scratch so I was able to agree non-material endorsements and put down lines on behalf of the underwriter. By 2007, I was underwriting straightforward renewals and asked to be part of the DQI committee. 2012 co-insided with me not only becoming an Energy Underwriter but also joining the Training Committee. Being part of something other than underwriting is a great way of expanding your knowledge of what goes on in the whole company and getting to know people from other areas of Atrium better.\nA big part of my Energy education was being sent to Rig School in Houston for a week and then spending a further week with some retail brokers, loss adjusters and risk surveyors. This trip involved an intensive introduction to Upstream Energy and was an amazing crash course into learning everything from discovering oil to what happens when there is a claim. It was also a good opportunity to meet fellow peers in the market and start to form friendships and contacts that would continue on in my career. Also what I find fascinating and particularly informative are site visits. There is nothing more eye opening than going to a shipyard and seeing an actual rig in mid construction to get a sense of the scale and understanding of what you are actually insuring. Suddenly the piece of paper in front of you has transformed into this real life massive image. I have been lucky to have been allowed to clamber all over a few rigs in my time in Corpus Christi, Rotterdam and Singapore. I have also seen disaster sites where something has gone drastically wrong such as on my U35s trip to SE Asia where we went to Indonesia to see a well out of control.\nClick here to read more career profiles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Artspace is looking for Spring break Arts Assistant.\nThe SWEAT program which is a creative work experience program for Grade 11 and 12 students. Students can come to our school to do their work experience credit in Design, Media Arts, Fashion, or Culinary arts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were looking for something quick and easy that looked good and wasn't too expensive. We got what we wanted. Love the color and very comfortable.\nSo far so good, easy to install and keeps the sun out of the kids faces when okay on the deck. Time will tell how long it lasts.\nThis set works perfect in our modest sized patio area. It's also been rained on once and dried very quickly. We love the blue color and the rich brown of the wicker. We're very happy.\nThis set is the perfect addition to my back deck! It was quick and easy to put together and made of high quality materials that I know will last me a very long time. Everything was just as described and shipping was even faster than expected! Couldn't have asked for better value!\nLove my bistro set. Just what I needed.\nWrong instruction manual. Item is good quality.\nI ordered to assemble it specifically with an option A and contractor spent 1.5 hours assembling it just to realize that option A on the picture is opposite from the option A in the book! Cushion is not as thick as I expected, we added couple of pillows to cover the back comfortably. Overall the quality is good.\nLove these covers, they are stylish, soft, and they look amazing, already ordered two more!\nThis is great!! works well, great size. Keeps the are a little cooler. Does not completely block light which is what I wanted.\nI am 5-9 and my wife is 5-3 so the setup is just about the right size for folks our size. Any taller than 5-10 and I think u would b a bit disappointed. But for our purpose it is ideal. Assembly for me was not problematic at all. I did the assembly solo in about 30 Minutes. We have added two more large cushions for added comfort. I seems well made and sturdy. I don't think you would want to leave this exposed to the weather. We have ours located in a small screen sunroom. Perfectly suited for our small room. Would have given five stars if we didn't have to add the two new cushions.\nI figured it would be easy enough to assemble...but boy was I wrong! It took me many hours, on and off, over the course of several days to fully put together. I was super confused in the beginning but once I figured out exactly what needed to happen, it was smooth sailing. This set is currently on our rooftop, and we love it!\nLove this set! It's perfect for a small space.\nShe loves it. Planning to buy one more for my other kid.\nGreat price for great pillow covers. They look great on my couch.\nVery solid for the price.\nReceived my 8 Shades on time. The shades are of excellent quality. I CAN'T wait to put them up. The only problem I had was while they were IN shipping you could tell that someone CUT a couple open to see what they were. I contacted customer service about missing parts and AlphaMart's shipped me the missing parts right away. Thank you for your time.\nGood quality, product was exactly what i expected and a great price.\nSorry but at this time I can not give a review. I haven't installed the awning just yet.\nVery happy with the bristo sets, perfect for our patio.\nIt's lightweight enough but not limp, so it suits my purpose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As part of my work for the Integrated Energy Plan 2015-2025 for Pakistan, I put together the first comprehensive energy flow diagram for Pakistan. After all, how can you truly focus your energy policy, if you don't know where the bleeding is worst.\nMy seminar audience has repeatedly asked to see this on our website, so I am finally putting it up.\nYou can find the full article and detailed analysis here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As the champagne flutes are washed and put away after Valentine's day, this is a perfect time to take a good long look at your home and see how well it works with your lifestyle. Is it a short term affair, or is it a keeper?\nAn impressive exterior face sets the precedent for the rest of your home, so you should go for the wow factor with your entrance. A two storey, full height, galleried landing, with big windows that flood the entrance with light, will really create an impact.\nWhen you buy a home it is usually an emotional decision that is made the second you walk through the door. An attractive, impressive entrance will help you to fall back in love with your home. On a practical level, it also improves the curb factor and increases the saleability and value of your property when you finally decide that it's time to move on.\nThe high cost of properties means that both partners are more likely to be working full time to pay the mortgage and so time spent at home together is much more precious. An open plan ground floor works much better for creating a space for couples and families to enjoy together, without feeling that they're crowding each other out.\nLook at how you move around your home. Has your home got a rabbit warren of corridors that cut down on the natural light, space and interaction of the family? Have you got an old utility room or W.C that faces South and gets all the best natural light, while areas where the family spend most of their quality time together, are gloomy?\nAn open plan interior provides a more relaxed communal living environment, which offers more flexibility as your family grows and changes over the years. By removing corridors, the sense of space and light can be maximised, making your home feel larger, more airy and more tranquil.\nWhen you remove walls, you have to think about structural elements that must be retained. At Lewis Visuals, we tend to clad supporting structures, such as pillars, in organic material such as oak, so that they enhance the overall appearance of your home. Removing walls also reduces hiding places: it's important to retain a cupboard wall in your kitchen so that crockery and pots and pans can be stowed away and to maintain privacy, so that your home remains your haven. Equally, shoes, coats, bags, sports equipment and toys need to have a dedicated storage space, so that the ground floor space doesn't become cluttered.\nOpen plan design generally focuses on the ground floor. As you are only changing a single storey, the architecture is covered by permitted development rules. If your home is semi-detached, you can extend out between 3m to 6 metres from the house, without requiring planning permission. If you live in a detached house, that distance increases from 4m to 8 metres, without requiring planning permission. It may be that you don't require that amount of space to achieve your ideal interior design and you can use this to counter offer if you do encounter any planning issues with local planning policies.\nI strongly believe that great design can bring out the best in your existing home and create a shared space that is a pleasure to be in.\nThe kitchen is often the heart of the home and cooking has become very fashionable, with couples mucking in together. Incorporating features, such as a breakfast bar, that allows couples to sit closer together and minimising chores by having a bigger dishwasher also increases a couple's quality time together.\nSmart lighting will allow you to set different ambiences, so that you can change the mood of the room from functional and bright during the working week, to softer more intimate lighting at the weekends, when you are cooking, talking and enjoying meals with your loved ones.\nConsider having a sound system installed so that you can introduce music to this shared space to really enhance the romance.\nBright Sunlight Shining Through Large Picture Window into Spacious Living Room with White Sofa and Modern Furniture \u2013 Luxury Home with View of Beach and Mountains. 3d Rendering.\nWe always take into account natural lighting and orientation of the building when creating a design that really enhances the existing features of a home.\nIf you don't have a South facing wall on your property, we recommend incorporating a roof light to allow natural sunlight in from above, because this is 150% stronger.\nWe give our clients a questionnaire to find out how they use the house at different times of day and then we make suggestions on how to use the orientation of the house to make the best use of light. For example, we might create a breakfast area by an East facing window, so that the family can eat together in the morning sunlight and a work area next to a West facing window where children can do artwork and homework after school. Snugs are perfect for North facing walls and can be greatly enhanced by including a log burner, which provides an attractive central feature, with stored logs introducing a lovely organic flavour as well as providing the fuel to heat the room.\nIf you're lucky enough to have a garden, then make it a feature, with large windows, or French doors and consider continuing the flooring material from the kitchen\/lounge to the patio so that you bring the outside in and it works as part of the overall space.\nAt Lewis Visuals, we work hard to create architectural designs that express homeowners' styles. When taking a brief from couples, we have to listen carefully to both halves, to ensure that the final design incorporates elements of each person's taste.\nWhile we understand that our clients want to spend as much time together in the evenings and at weekends, we are also aware that sometimes they need time alone in order to work, study, or just to have some sanctuary from the hustle and bustle of daily life. Where space permits, we try to create a flexible area that might be used for storing toys out of sight when the children are small, but that also provides an adult space for reading, relaxing and enjoying a glass of wine in the evening.\nWe've resolved plenty of tiffs over whether more space should be devoted to the kitchen or to the 'man cave' and helped numerous couples to achieve the right balance. Ultimately, we have to help couples come to a happy compromise that works with the available space and budget. Our key goal is to create an environment that enhances the time that you spend at home with your loved ones and that adapts with you and your family as your needs change over time.\nAre you looking for open plan living space?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Heat the butter in a stock pot and saut\u00e9 the onions until translucent.\n2. Sprinkle the flour over the onions and saut\u00e9 a minute longer.\n3. Add the vegetable broth and bring the contents of the pot to a boil.\n4. When the broth reaches a boil add in the cheese in batches and whisk to combine. Do not add the next batch until the previous one is almost completely melted.\n5. Add the cumin, and season to taste with the salt, pepper, and paprika.\nThose with discerning tastebuds might detect my rage in each bite.\nAround the holidays you can buy buckets of pre-made lebkuchen dough at the grocery store, and now I know why. This recipe yields a dough that is so difficult to work with that there is no reason to put yourself through it unless you're really mad about something and want to take it out on an inanimate object to avoid jail time.\nThe dough turned into a hard, gooey mass overnight and it took me an hour to get it to where I could roll it out. First I incorporated the extra flour and seasoning by doing some preliminary hand kneading, then I transferred it to my stand mixer in two batches. This only partially got the dough together, so I did another round of hand kneading and machine mixing. The dough broke one of the paddle attachments.\nBut the ordeal didn't end there. I slightly bent the handles on my rolling pin before I gave up using them, and used only the middle part of the pin to roll. Even standing on my tip toes and throwing my full body weight into the pin I had a hard time rolling the dough to 1\/2\u2033 thickness.\nOf course, the cookies tasted good and everyone in my family really liked them, so I will likely be obliged to make them again. Dammit.\n1. Caramelize the sugar. To do this, put it in a ungreased pan with a wide bottom over medium heat. Stir occasionally with a wooden spoon until the sugar starts to melt. Once it starts to melt stir continuously, breaking up clumps. Eventually the sugar will completely liquefy and turn reddish brown \u2014 caramel.\n2. While the sugar is melting, heat 3\/8 liter (13 oz) water in a separate wide-bottomed pan over low heat. When the sugar has become caramel, add it to the water. The mixture will hiss and boil at first, but this is OK. Completely dissolve the caramel in the water. It dissolves faster when left alone, with only the occasional scraping of clumps off the bottom and side with a wooden spoon.\n3. Add the honey to the caramel water and heat thoroughly. Remove pot from heat and stir in 1 kilogram (35 oz) flour. Cover and let the dough rest for a day.\n4. Add 40 dag. (14 oz) of flour and the remaining ingredients (except the simple syrup)and roll the dough out to a thickness of half an inch. Cut the dough to any shape and brush the tops of the cut-out shapes with water.\n5. Bake at 350 degrees: the exact baking time depends on the thickness and size of the shape cut. The cookies that I made (pictured above) took about 25 minutes. Let the cookies cool slightly then brush the tops with a simple syrup.\nYou know that scene in \"Forrest Gump\" when Bubba tells Forrest all the different ways to prepare shrimp? Well, the Gulaschmusem restaurant is a lot like that, except with gulasch. The menu lists and has pictures of gulasch made with various meats and parts of meat (chicken liver gulasch, anyone?), vegetables, fish, and rice. They even boast a sweet chocolate gulasch for dessert, which is mysteriously not pictured. I guess they want to entice the diner to order it.\nGulasch such as I will never make.\nMy husband I and both had some of this dish and couldn't agree on whether the base for the gravy was a fish or vegetable stock. Obviously it had paprika in it, but in what proportion to the tomato, if tomato was used at all? (I thought there was tomato, my husband didn't.) How can I re-create a dish like this when I can't even figure out what's in it? This is one of the things that I will just enjoy while we are here and remember fondly.\nI was not 100% satisfied with the Erd\u00e4pfelsalat from the recipe that I previously posted. I used the same ingredients and amounts (excepting the beef broth) from the recipe that I posted, but played around with the method a bit. The potato salad that I made today was heavenly and I think I have finally got it down.\n1. Boil the potatoes whole with skin on in enough beef broth to cover.\n2. While the potatoes are boiling, finely dice the onion and fry it in melted lard until translucent. Set aside to cool.\n3. When the potatoes are cooked through, drop them briefly into a bowl of cold water. (Keep the cooking broth.) Remove the skins and slice into 2\u2033 pieces.\n4. Combine the cooked onions, 1\/4 to 1\/2 cup of the reserved beef broth (to taste), vinegar, salt, pepper, and sugar.\n5. Place the sliced potatoes into a bowl and pour the broth\/vinegar mix over them. Toss to coat.\n6. Drizzle the oil over the top and garnish with red onions and chives. Serve at room temperature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\u2022 I am a psychotherapist with over nine years experience in private practice in the Reading area. I am registered with the United Kingdom Council for Psychotherapy (UKCP).\n\u2022 In addition, I am a member of both the British Association for Counselling and Psychotherapy (BACP) and the Universities Psychotherapy and Counselling Association (UPCA) and abide by the professional code of ethics of these bodies.\n\u2022 I have over twenty years experience of working as a therapist in the NHS and I'm currently employed as a Senior Psychotherapist in NHS Psychological Services. I have also worked as a Senior Counsellor in GP surgeries.\n\u2022 As well as practicing as a therapist I have been involved in the training and supervision of counsellors on BACP-accredited courses for over twenty years. I taught on a BACP-accredited MA in Counselling at the University of Reading for over eight years.\n\u2022 Prior to practicing as a psychotherapist I worked as both a general nurse (RGN) and a psychiatric nurse (RMN). I went on to train nurses at a London Teaching Hospital.\n\u2022 Post Graduate Certificate in Clinical Supervision - University of Oxford.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Agway Greenlawn Lawn Fertilizer is a long-lasting formulation of major plant food nutrients to help you grow a brilliant green, deep-rooted, luxurious lawn. Agway Greenlawn Lawn Fertilizer is safe to use anytime during the summer or fall and will spread easily with any type of fertilizer spreader. Feeds your lawn up to twelve (12) weeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want to make your next government or social services conference one for the ages? Jeff Havens is THE premier keynote speaker for government and social services events, thanks to his ability to deliver high-quality education in a unique, hilarious, and unforgettable way. Your attendees have never seen anyone like Jeff before, and they will be talking about his keynote for the rest of the conference.\nA former teacher and professional stand-up comedian, Jeff has been bringing the house down at government and social services events for the past 15 years. His infectious energy and intellectual versatility have allowed him to address groups such as the Nebraska Department of Education, Michigan Works!, South Dakota Housing Development Authority, State and Local Government Benefits Administrators (SALGBA), and dozens of others. Charismatic and insightful, hilarious and thought-provoking, Jeff will ensure that your government or social service audience will reap a surprising amount of education at the same time that they are laughing harder than they ever have at a conference.\nWhere most speakers focus only on a single topic, Jeff has developed multiple presentations on a variety of subjects. However, four of his keynotes are best suited for government and social services audiences. Uncrapify Your Life! is a hysterical take on communication and change management that is perfect for closing out a conference or for awards banquets and employee recognition events. Unleash Your Inner Tyrant! is a leadership presentation like nothing you've ever seen, designed to skyrocket employee engagement and ideal for anyone in a managerial or supervisory role. Us Vs. Them is the funniest generational keynote you will ever see, perfect for any audience struggling to balance the needs of veteran workers against the needs of newer hires. And Uncrapify Your Future! simplifies the innovative process into a remarkably intuitive three-step solution that will benefit everyone who has ever felt like they're constantly being asked to do more with less (and let's face it, you are always being asked that!). To learn more about Jeff's specific presentations, please visit the Keynotes page.\nIt's unfortunate that government and social services workers rarely get the credit they deserve, especially since so many lives depend on what you do. At the very least, your people deserve an event where they'll be begging to come back next year. Jeff Havens can make sure that your next government or social services conference is the best it's ever been. Who said a keynote speech had to be dull and boring? See what Jeff Havens can do for you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Treasure Island Morpeth - A great venue for children's parties!\nAbigail wanted her 5th Birthday Party at Treasure Island Morpeth. As she's just started school, this is the stage of party planning that can busy. If you've got children at school, no doubt you'll remember that year of no spare weekends. Parties in the morning and the afternoon. Generally the kids want to invite everyone from the class and soft play is the easy option. The other thing we took into consideration was timing. Although Abigail is a very important person, we also appreciate that a party at 1pm \u2013 3pm will takeover any chances of further family fun time before or after. So, a 10am \u2013 12 noon party meant that at least the guests could do other events in the afternoon.\nThere are a variety of options available as a party theme at Treasure Island. These include Pirate and Princess Parties, Make-over and Pinata Parties. We opted for a Disco Party.\nAs we walked in the door we were assigned a party planner, Rhona. She took care of everything we needed. Starting off by taking guests names at the entrance, helping us out with the food order. We opted for a Hot Buffet style lunch.\nThe soft play is split into two main frames. One being a pirate ship, next to the other being the island. The Island contains a large wavy super slide and spiral tube slide, along with many awkward climbing situations for parents. I have climbed around here in the past and seem to remember my back not bending as much as it used to.\nAs the soft play started to fill up many other children started to interact and join into chasing games around the frames. Obviously I can't guarantee the same friendly kids being there if you go, but I've not experienced anyone too boisterous there.\nThe bell was rang and the children were summoned to the party room. Which had been darkened down to allow the laser lights and flashing LEDs to give some atmosphere. Rhona, was here organising party games in a disco style. These included a musical chairs type of game involving spots on the floor and Pass the Parcel. The kids seemed to enjoy it as I haven't seen Pogo-ing like that since my clubbing days 10 years ago (possible more, I'm in denial).\nThe food option we went for was a Hot Buffet. This consisted of Hot Dogs, Chicken Nuggets and Chips. There are a few reasons why we went for this option rather than a cold sandwich buffet. Firstly it was chilly outside and a hot meal is always nice. Secondly we didn't want the stress of remembering who ordered what and then changed their mind. Each child was called, by their table to the buffet to help them self and get a scoop of chips with their choice. A Squirt of tomato sauce and everyone is happy. Juice was available through out the whole party provided in large jugs.\nThe Meal was finished off with a traditional blowing out of the candles. Abigail had remembered last year, where she struggled to blow them out so didn't want to fall short again. The group of friends started \"Haaappy Birth\"\u2026 The candles were out. Call me a stickler for un spoken rules, but I'm sure your supposed to wait for the song to be complete before you huff all over the cake!\nEvery child was given a party bag and a slice of cake.\nAnd we were done! The stress had been taken out of the 5th Birthday party and we were almost relaxed.\nThank you to everyone that looked after us at Treasure Island we had a great day.\nPrevious article Deadpool is NOT a super hero film!\nPingback:Antico - Jesmond - Here Come The Hoopers | A North East family of four.\nPingback:Morpeth Treasure Trails \u2013 Here Come The Hoopers | A North East family of four.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was Friday of Memorial Day weekend and I was to spend the weekend with my two children, parents, and sister at the beach without my husband because he had to work. Divinely, I was glued to my chair on the front porch of our town home. Oddly, I had all sorts of thoughts racing through my head that kept me from getting behind the wheel and driving myself and my kids to the beach. My husband didn't understand, and I don't think I quite did either. While sitting on the porch, I made a phone call to my best friend. I recall that I made sense when I spoke with her, but since my thoughts and ideas were grandiose in nature, it concerned her. Then I called my boss and apparently quit my job; I do not have much recollection of that conversation.\nMy husband told me later that I sat down and quoted scripture that he didn't think I had ever memorized. He stepped outside for a moment. In that moment, I thought Jesus was returning. I grabbed our kids and begged, \"Please save us, our family, and our friends!\" I kept repeating those words over and over. Suddenly my husband came back inside and found me looking pale and weak, holding our children. I passed out. He appropriately called 911. Medical personnel responded quickly.\nAs I became conscious, my nursing knowledge jumped in, I promptly and inappropriately told them to pump on my chest and intubate me. I thought I was on the verge of death. Here I was mentally sick. My husband was very frightened and didn't know what was wrong with me. They took me to the ER where I stayed for two nights. Then I was transferred to the psychiatric unit. How does a 30 year old mom of two with no previous history of mental illness get admitted to the psych ward? This is where my memory fails me, but the diagnosis: Postpartum Psychosis. After two nights in the ER, I was admitted to a general psychiatric unit. At that time I was only 45 minutes from UNC\"s perinatal inpatient unit but none of the few beds were available.\nOn the psychiatric unit, I had a sitter with me 24\/7 to be sure I didn't harm myself or anyone else. I stayed on the unit for nearly two weeks\u2014two weeks without my babies, two weeks I did not get exercise or go outside. I ate in my room with the sitter not far from me as well as took a shower with the sitter right outside my door. There are some things I remember but other memories my family told me. My sister informed me at one moment I thought I was Tina Turner, and at another time I thought I was pregnant with Baby Jesus. I do recall thinking I was on the set of Grey's Anatomy with Bradley Cooper and Mandisa. It shouldn't have been a bad place then, right? Oh so wrong; it was a very, very scary place! My anxiety and paranoia were both at an all-time high during my hospitalization. I blamed my husband and family for things that were definitely not true. Believe me, when I am well\u2014and my brain isn't playing tricks on me\u2014I trust my husband 100% without a doubt or question.\nI remember drawing family trees over and over. I thought the hospital was hell and my ultimate goal was to get out of there. My memory began to return within the last couple of days while in the psychiatric unit. Many people ask me if a switch just turned on one day. The answer is NO; my memory just got better every day. Especially when I was at home, I think it was my safe place and I had a sense of normalcy, or a new normal. I really just think my brain didn't want to remember the awful thoughts I had while I was in the hospital. While in the hospital, I was treated with antidepressants, antipsychotic drugs, and an occasional injection when my mood and paranoia levels began to increase. I do recall trying to escape and being held down by the staff and probably given an injection to calm me down. Again, I just wanted out of there, it was hell on Earth to me. To this day, I can hardly wrap my brain around how my mind played such dirty tricks on me.\nBut, postpartum psychosis is no joke.\nAfter spending nearly two weeks in the hospital, I was discharged home. For two whole weeks I didn't see my babies (5\u00bd month old and 2\u00bd year old). I was so excited to get home and see them! But, my journey with postpartum psychosis was far from over. When I returned home, things weren't back to \"normal.\" I couldn't be with my children alone. I couldn't be by myself. I couldn't drive. I couldn't return to work. Talk about restrictions! I couldn't be with my own children by myself? No. Doctor's orders! I really didn't fully understand the reasoning behind all of the restrictions. I didn't even realize I had just been in the hospital for two weeks; I literally didn't remember. So much of my brain just wasn't working right and my thought processes were misconstrued. And, from not being able to go outside during my hospitalization and exercise, I was very weak. I knew I had to trust my family and friends, but there was so much I just didn't understand. I really didn't understand what was happening and why. I felt like I was being tortured in every possible way and ultimately being kept from my children and away from society. I was still paranoid and felt like people were following me and my family. There was even a day I thought I couldn't take it anymore and even tried to jump out of my husband's truck. But, the good news is I got through that day and I'm here to FINISH this story!\nAs part of my rehabilitation, I attended an intensive outpatient program for a couple weeks, which involved three hours of group therapy daily. Want to know what that was like? Since I was still out of touch with reality, it was like being in group therapy with my entire family! Each person in the room reminded me of someone, either a friend or family member, and that is who I thought it was. I did not like it. After graduating from the intensive outpatient program, I was then referred to a psychologist and a psychiatrist. I continue to see both doctors to this day. Regular appointments with my psychiatrist assisted to keep my medications managed. At one point my husband thought I was backtracking and it was suggested that he literally hand my medications to me and watch me swallow them.\nHere I am, a nurse, fully capable of managing medications but my husband stood over me twice a day making sure I swallow my medications! I felt like a child. But eventually, I was able to take my medications without my husband standing over me. Gradually, restrictions were lifted. First, I was able to drive but not with the kids in the car. That felt so good just to be able to get out by myself without a babysitter! I probably just went to Target and got a Chai tea Latte at Starbucks. Talk about freedom! Eventually I was able to take care of my two children as well as drive with them in the car. My psychiatrist was impressed with how quickly I recovered and took back my mothering responsibilities. But at the same time, I was pretty anxious and scared. Since my psychosis episode, my anxieties had increased and having both girls by myself; it was quite a job for one person! I applaud stay at home moms; it's a full time job in itself! My children went to daycare three days a week and stayed with me two days a week once all restrictions were lifted, which was gradual.\nI continued (and still do) have anxieties when I keep both of my children by myself. There was even a weekend I had to call on my parents when my husband had to work because I just couldn't do it by myself\u2014and that's okay to do. Moms, it's okay to ask for help because we can't do it all by ourselves, we can only do so much! Many of you are probably wondering how I got through such an experience. My faith is very important to me as well as my family, and I had a lot of people praying for all of us. I'm so thankful for each and every prayer as it was definitely heard. God's grace covered my family and has and continues to carry me through this journey.\nThe support of my family and friends truly helped me through each and everyday, especially my husband, and especially those days that I felt like I couldn't make it through. My physicians, medications and psychotherapy were certainly a big part of my recovery. One day my psychiatrist told me it was like I was a soldier who had just returned home from battle, so yes I consider myself a fighter and a warrior over postpartum psychosis. I am a survivor. And now, I fight and advocate for moms who can not fight for themselves or simply do not know how. I consider myself extremely blessed as I never had ill thoughts towards my children during this whole episode. I have a new found God-given passion to tell my story with other women in hopes to shed light on Perinatal Mood Disorders such as Postpartum Depression, Postpartum Anxiety, Postpartum OCD, and Postpartum Psychosis.\nMy mission is to let women everywhere know that they are not alone.\nFor too long I went around thinking others would think I would be a less together mom if I was on meds, but that's not true. My husband and I on numerous occasions discussed that I may need to talk about getting on an antidepressant with my physician, but I failed to do so. I'm not exactly sure why, but I just felt like I could fight through it myself. Looking back, if it would have prevented my psychotic episode, I definitely would have asked! Now I'm on meds, and I'll tell the whole world! It's for my mental health and well-being! Postpartum Depression is diagnosed in 1 in 7 women. Postpartum Psychosis is seen in 1 in 1000, so it is more rare than PPD. In fact, my doctor said he hadn't seen it in over six years!\nI am now a Patient Expert Advisor with MMHRC and co-founder of Northeast TN Perinatal Mental Health Alliance. It is my personal mission to share my story and let other moms know that they are not alone, its ok to seek help, ask for help and take care of themselves.\n\u2190 Where is James? Do you Know?\nI thought I had lost my mind!\nChoosing the Right Counselor Matters!\nWhere is James? Do you Know?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You have just converted thirty-one dirhams to georgian lari according to the recent foreign exchange rate 0.73374817. For thirty-one dirhams you get today 22 lari 74 tetri. If there is going to be any change in the exchange rate of AED to lari, recalculation of the amount will be done automatically when the page is refreshed. If you need to know how much is 31 dirhams to a currency of any country in the world \u2013 use an online converter, which has 96 currency pairs available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a web-friendly version of the Texas events calendar published each year by the Office of the Governor, Economic Development and Tourism. The original list can be found on their website, Texas Tourism. Here's a searchable database of annual events and festivals in Texas.\nJan 1 | Dallas Cotton Bowl Classic Start the New Year with a bang at this annual event.\nJan 12 \u2013 Feb 4 | Fort Worth Southwestern Exposition, Livestock Show & Rodeo Established in 1896, this popular event attracts nearly one million people from around the world to the nation's oldest livestock show and daily performances of the world's original indoor rodeo.\nJan 13, 25\u201327 | Mission Texas Citrus Fiesta \"Citrus on Broadway\" Royal Reception is Jan. 13 at Mario's Banquet & Conference Center. Coronation of King Citrus and Queen Citrianna is Jan. 25. Product Costume and Royal Finale are Jan. 26. Parade of Oranges, Fun Fair Events, Vaquero Cook-Off and Citrus Youth Show Exhibit are Jan. 27.\nJan 18 \u2013 Feb 18 | Laredo Washington's Birthday Celebration Founded in 1898, this is the largest celebration of its kind in the United States. The almost month-long celebration includes parades, a carnival, air show, fireworks, live concerts and more.\nFeb 1\u201318 | San Antonio San Antonio Stock Show & Rodeo Non-stop action attracts all ages with livestock, auctions, carnival and daily rodeo performances that include top-notch entertainers.\nFeb 3 | Port Arthur Gulf Coast Music Hall of Fame Show Pays tribute to Janis Joplin and Southeast Texas musicians.\nFeb 8\u201310 | Kingsville South Texas Ranching Heritage Festival Celebrates South Texas' Western heritage. Includes storytellers, poets, demonstrations of ranch crafts and more.\nFeb 9\u201320 | Galveston Mardi Gras! Galveston Enjoy live music, spectacular parade, elaborate masked balls and flamboyant costumes. Hours are 5pm to midnight Friday, 10am to midnight Saturday and 10am to 6pm Sunday.\nFeb 15\u201318 | Port Arthur Mardi Gras of Southeast Texas\/Holiday Razzle Dazzle Includes parades, headliner bands, carnival, food and craft vendors, and strolling street entertainers.\nFeb 15\u201325 | San Angelo San Angelo Stock Show & Rodeo Includes Wrangler Pro Tour rodeo featuring top cowboys from the Professional Rodeo Cowboys Association. The livestock show will host almost 4,000 youngsters and their ag projects for judging and premium sale.\nFeb 16\u201318 | Jefferson Mardi Gras Upriver Enjoy Mardi Gras Texas-style. Includes a Doo Dah Parade on Friday, grand parade on Saturday, a children's parade on Sunday, music, arts & crafts, and more.\nFeb 18\u201325 | Brownsville Charro Days Fiesta The history of Charro Days dates back to 1937, and it's still going strong today.\nFeb 23\u201325 | Alpine Texas Cowboy Poetry Gathering Presents classical, contemporary and traditional cowboy poetry, music and storytelling.\nFeb 24 | Forth Worth Cowtown Marathon, Marathon Relay\/10K & 5K Race begins in Sundance Square and moves through Fort Worth neighborhoods.\nFeb 27 \u2013 Mar 18 | Houston Houston Livestock Show & Rodeo Event is the world's largest, with 20 days of championship rodeo action, superstar concerts, livestock and horse shows, children's activities, carnival, educational exhibits, unique shopping opportunities and fun food.\nMar 1\u20133 | Houston River Oaks Garden Club Azalea Trail Features four private homes and gardens, as well as Bayou Bend, Rienzi and the River Oaks Garden Club Forum of Civics Building and Gardens. Hours are 11am to 6pm. Call for ticket sale information and locations.\nMar 1\u20134 | Fulton Oysterfest This salute to the oyster industry features plenty of food, entertainment, and arts & crafts.\nMar 2\u20134 | Dallas North Texas Irish Festival One of the largest Celtic cultural events in the nation salutes its 25th year. Features many of the top Irish musicians and dancers in the world. A number of cultural presentations will focus on the 25-year history of the NTIF and the Irish culture in this country. Includes world-famous bands, award-winning dancers, enchanting storytellers and educational workshops. Offers a variety of delicious Irish and international food and drink, cultural crafts and ethnic vendors. Hours are 6 to 11pm Friday, 10:30am to 11:30pm Saturday and 11:30am to 7:30pm Sunday.\nMar 3\u20134 | Washington Texas Independence Day Celebration Come celebrate the 170th anniversary of Texas Independence at this free two-day historic event. Hours are 10am to 5pm.\nMar 9\u201318 | Austin South by Southwest Conferences & Festivals Unique in concept, renowned for execution and treasured by participants, SXSW is an essential event for understanding tomorrow's entertainment industry today. Industry professionals from the music, film and digital community gather in Austin every March for this extraordinary event.\nMar 9\u201318 | Mercedes Rio Grande Valley Livestock Show This event dates to 1940 and is one of the largest in South Texas.\n\u00bb 1000 N. Texas Ave.\nMar 9\u201324 | Austin Star of Texas Fair & Rodeo Hoof-pounding, heart-stopping arena action, superstar entertainment, stomach-tingling carnival rides and the giggle-inspiring petting zoo are only a few aspects of this event that bring people back every March. Among the exciting events and exhibits this year are: Whiplash the Cowboy Monkey, Kidstown, calf scramble, mutton bustin', Buffalo Soldiers, chuck-wagon cook-off, barbecue cook-off, more than 40 bands on the outdoor stage, Youth and Open Livestock Show, Xtreme Bull Riding, 13 ProRodeo performances and live concerts.\nMar 10 \u2013 Apr 15 | Dallas Dallas Blooms This event is considered the largest outdoor floral festival in the Southwest.\nMar 16\u201318 | Shamrock St. Patrick's Day Celebration Enjoy a banquet, parade, carnival, arts & crafts show, antique car show, Miss Irish Rose pageant, rodeo, Lad & Lassie Pageant, adult dance, MC Rally, food and more.\nMar 16 \u2013 Apr 1 | Tyler Azalea & Spring Flower Trail Enjoy more than 8 miles of residential gardens. Includes home tours, garden tours, art shows, flower shows, living history exhibits, quilt show, Renaissance Faire, arts & crafts festival and more.\nMar 17 | Dublin St. Patrick Day Festival This Celtic event includes Highland games, Irish music, crafts vendors, great food, carnival and parade.\nMar 24\u201325, Mar 31 \u2013 Apr 1, Apr 7\u20138 | Palestine Texas Dogwood Trails Celebrations Celebrates the coming of spring and the beautiful Dogwood blossoms, as well as other flowers, the season brings to the East Texas Piney Woods. Three weekends of activities include a kickoff arts & crafts show in Downtown and Old Town Palestine, a country music concert by Dogwood Jamboree, a Dulcimer Festival at the Museum for East Texas Culture, live theater at the Texas Theatre and more.\nMar 24, 30\u201331, Apr 7 | Woodville Tyler County Dogwood Festival Includes a Festival of the Arts on March 24, Western Weekend on March 30-31 and Queen's Weekend on April 7.\nMar 25 | Austin Capitol 10,000 Celebrates its 30th anniversary The Cap 10, which is the largest 10-K in Texas, attracts the silly and serious. It has become an annual tradition for the whole family.\nMar 29 \u2013 April 1 | Portland 34th Annual Windfest Features a carnival, arts & crafts, live music, tasty food, chili cook-off, Dachshund Dash, washer and horseshoe tournament, golf tournament, 5-K run, parade, kite contest, kids fair, petting zoo and much more. Friday is Tejano Tribute Day, Saturday is Texas Music Day and Sunday is Flashback '80s.\n\u00bb Community Center, 2000 Billy G. Webb.\nMar 30 \u2013 Apr 1 | Port Aransas Texas SandFest Features a competition for master sculptors, as well as youth, amateur and Pro-Am contests This is one of the largest sand sculpting festivals in the United States.\nApr 7 \u2013 May 28 | Waxahachie Scarborough Renaissance Festival Re-created 16th-century English village offers 20 stages of entertainment, jousting, falconry, music, 200-plus artists and craftspeople, games, rides and foods from around the world. Open Saturdays and Sundays, plus Memorial Day.\nApr 13\u201315 | Burnet Bluebonnet Festival Includes food and arts & crafts vendors, music stage, parade, air show, weiner dog races, pet parade, scholarship pageant, demolition derby, 5-K run, bike tour and carnival.\nApr 13\u201315 | Poteet Strawberry Festival This celebration has been held for the past 59 years to celebrate the famous Poteet strawberry harvest. One of the oldest, largest and most popular events in the state, this festival offers something for everyone on the 12 stages of family entertainment.\nApr 14\u201315 | Austin Austin Fine Arts Festival Find original, one-of-a-kind artwork in a variety of mediums. Dance to eclectic live music, take part in an interactive public art project, or watch special children's entertainment and live artist demonstrations. Hours are 10am to 5pm Saturday and 11am to 6pm Sunday.\nApr 14\u201315 | Chappell Hill Bluebonnet Festival Includes more than 250 juried vendor booths with home decor, gardening items, artisans, crafters, clothing, jewelry and more. Enjoy fun children's events, food court and entertainment.\nApr 19\u201322 | Austin Old Settler's Music Festival Encourages children and adults to appreciate and preserve American roots music by providing workshops, displays and activities.\nApr 20\u201322 | Harlingen Riofest 2007 This international celebration of arts and culture features concerts, art displays and craft demonstrations from northern Mexico and the Rio Grande Valley. Includes a children's area, live music and lots of entertainment. Hours are 5 to 10pm Friday, 10am to 10pm Saturday and noon to 8pm Sunday.\n\u00bb Municipal Auditorium & Casa de Amistad, 1204 Fair Park Blvd.\nApr 20\u201322 | Huntsville 20th Annual General Sam Houston Folk Festival Celebrates the life and times of Gen. Sam Houston with life-skill demonstrations, authentic 19th-century dress, continuous entertainment and hands-on children's activities.\nApr 20\u201329 | San Antonio Fiesta San Antonio Offers a multicultural celebration of the history and diversity of San Antonio. More than 90 nonprofits and 10 military organizations stage 100 events. Features all kinds of food, music and entertainment. Activities include balls, parades, art exhibits, festivals, athletic events, conferences and much more.\nApr 21\u201322, 28\u201329 | Houston Houston International Festival This event is the city's official celebration of art and culture. Continues its tradition of bringing together hundreds of international musicians, artisans and food vendors from around the world. This year's festival spotlights China.\nApr 25\u201328 | Fort Stockton Big Bend Open Road Race Enthusiasts call this one of the most exciting open road races in the world. The route follows Hwy. 285 from Fort Stockton to Sanderson and back. This event draws racers from around the United States to match their skills against 118 miles of twists, turns and elevation changes.\nApr 25 \u2013 May 6 | Corpus Christi Buc Days From its simple beginnings in 1938, Buc days has grown into the premier festival event in Corpus Christi. In addition to the carnival and the spectacular Illuminated Night Parade, the festivities now include a PRCA rodeo, BBQ Challenge, a Competencia Folklorica, junior parade, stadium show, fireworks, and various sporting and athletic events. Monies raised during this event benefit the Buccaneer Commission Scholarship Fund, providing scholarships for local area youth.\nApr 27\u201329 | Hallettsville Texas State Championship Fiddlers Frolic Includes arts & crafts, fiddler contests, food, fun, cook-off, dance and more.\nApr 27\u201329 | Muenster Germanfest Offers a bicycle rally, fun run, sanctioned barbecue cook-off, carnival rides, arts & crafts, children's area with stage, German entertainment stage, and a stage for country, rock, jazz and blues. Enjoy lots of good German food along with other festival-type foods.\nApr 28 | Bastrop Yesterfest Enjoy historic re-creations, food booths, and an arts & crafts fair.\nApr 28 | Turkey Bob Wills Day Honors the late, great King of Western Swing. Generally the last Saturday in April.\nMay 3\u20135 | San Marcos Viva! Cinco de Mayo Includes the state Menudo Cook-Off.\nMay 4\u20136 | Jefferson 60th Pilgrimage Tour of Homes Four of Jefferson's most beautiful and historic homes \u2013 festooned with fresh and fragrant bouquets and period furnishings \u2013 open their doors to the public. The worldrenowned and ever-popular Diamond Bessie Murder Trial is performed. Includes a Garden Stroll, Civil War reenactment, colorful parade, quilt show, and arts & crafts.\nMay 10\u201313 | Abilene Western Heritage Classic Preserves the heritage of the ranch cowboy. Features a parade, ranch rodeo, nightly dances and much more.\nMay 11\u201313 | Crystal Beach Texas Crab Festival Features continuous live music and entertainment combined with some of the tastiest crab dishes on the Texas Gulf Coast. Enjoy more than 60 food and craft booths, carnival, Crab Legs Contest, Weiner Dog Races, crab cook-off, a kids stage with free entertainment and activities, and much more.\nMay 18\u201320 | Pasadena Strawberry Festival Includes a barbecue cook-off, beauty pageant, mud volleyball, arts & crafts, four stages of entertainment, mutton busting, washer pitching contest, World's Largest Strawberry Shortcake and more.\nMay 19 | Marshall Stagecoach Day Festival This historic transportation festival is filled with arts & crafts booths, food vendors, live entertainment, domino tournaments, horseshoe throwing contest and classic car show. Held the third Saturday in May.\nMay 24\u201327 | Forth Worth Crowne Plaza Invitational at Colonial One of professional golf's classic tournaments, this nationally televised competition features the nation's top golfers on the PGA tour. This is formerly the Colonial Golf Tournament.\nMay 24 \u2013 Jun 10 | Kerrville Kerrville Folk Festival More than just a \"folk music\" festival, Kerrville offers music of many styles, including folk, bluegrass, acoustic rock, blues, country, jazz and Americana. The common thread is songwriting. Event promotes emerging artists while giving the audience exposure to both new and recognized, seasoned talent. Impromptu jam sessions may pop up at any time around a campfire in the campgrounds. In addition to the live music, enjoy special concerts and activities for children, songwriting schools, music business seminars, Hill Country bike rides, canoe trips on the Guadalupe River, Sunday Folk Services, New Folk competition and concerts for emerging songwriters.\nMay 25 | Athens Annual Texas Fiddlers Contest & Reunion Offers fiddlin' contest for all ages, street dance and carnival.\nMay 25\u201327 | Ennis National Polka Festival Enjoy a little bit of old Czechoslovakia on the streets of historic downtown Ennis and in the three Czech Halls. Celebrate the Czech culture with polka dancing, Czech food, fun and 13 sensational live polka bands.\nMay 25\u201328 | Kerrville 2007 Texas State Arts & Crafts Fair Fine arts & crafts fair features great food, music, demonstrations and children's activities.\nJun 1\u20132 | Yoakum Tom Tom Festival This festival honors the tomato heritage of Yoakum.\nJun 1\u20133 | Aransas Pass Shrimporee Home of the Largest Shrimp Festival in Texas. Enjoy a great family weekend with musical acts, entertainment, cooking competition, carnival rides, arts & crafts vendors and more.\nJun 1\u20133 | Arlington Texas Scottish Festival & Highland Games This is the largest cultural event of its kind in Texas and the Southwest.\nJun 1\u20133 | Longview AlleyFest Includes Alley Art \u2013 a juried art show, Music Fest with top-name musicians, Kids Fest \u2013 a combination of fun and education, one of the most popular 5-K and 10-K runs in East Texas and a variety of food vendors. Hours are 6:30pm to midnight on Friday, 10am to midnight on Saturday and 11am to 4pm Sunday.\nJun 8\u201310 | San Antonio Texas Folklife Festival Celebrates the many ethnic communities of Texas. The festival brings together more than 40 ethnic groups sharing their food, music, dance and crafts.\nJun 8 \u2013 Aug 18 | Canyon TEXAS Musical Drama Offers outdoor musical theater at its best as the story of the Texas panhandle unfolds year after year. The Texas Panhandle Heritage Foundation is in its 42nd season of presenting Texas culture through this internationally acclaimed production. Performances are Tuesday through Saturday nights.\nJun 9 | Jacksonville Tomato Fest Offers arts & crafts, live entertainment, antique car show, gospel concert and much more.\nJun 15\u201316 | Stonewall Peach JAMboree Central Texas town famous for juicy Hill Country peaches hosts this event.\nJun 21\u201323, 28\u201330 | Albany Fort Griffin Fandangle Commemorates the rich history of the area. Written, directed, lighted, costumed, sung and danced by the people of Albany.\nJun 21\u201324 | Luling Watermelon Thump Celebrates the nutritious, auspicious watermelon. Includes three stages with continuous live entertainment and big name nightly concerts, arts & crafts booths, food court, beer garden, carnival, kiddie rides, contests, giant parade, car show and more. Held the last Thursday through Sunday in June.\nJul 4\u20137 | Stamford Texas Cowboy Reunion The World's Largest Amateur Rodeo includes dances, art show, trade show, cowboy poetry, ranch horse show, old-timers events and parade.\nJul 13\u201315 | Longview Great Texas Balloon Race Considered the largest sanctioned hot air balloon race in Texas. Sport and special shape balloons compete for prizes and cash. Includes two evening balloon glows, music concerts both Friday and Saturday night, Gigantic Kids Land, arts & crafts and food vendors. Hours are 4pm to midnight on Friday, 6:30am to midnight Saturday and 6:30 to 9:30am Sunday.\nJul 14 | Weatherford Parker County Peach Festival Features handmade arts & crafts, children's activities and food. With more than 200 vendors, there is something for everyone. Held the Second Saturday in July.\nJul 26\u201328 | Clute Great Texas Mosquito Festival Fun-filled, family event features variety of contests, more than 75 vendors, carnival and headliner entertainment each evening. Willie-Man-Chew, a Texas-sized inflatable mosquito, presides over the event. Usually the last Thursday, Friday and Saturday in July.\nAug 1\u20135 | South Padre Island Texas International Fishing Tournament (TIFT) This is the oldest tournament on the Texas Gulf Coast.\nAug 2\u20134 | Dalhart XIT Rodeo & Reunion Held every year since 1925 in honor of the cowboys who worked the ranch.\nAug 15 | Praha Prazska Pout This annual homecoming features food, bands, souvenirs, auction, country store, games and more. Always on Aug 15.\nAug 23\u201326 | Fredericksburg Gillespie County Fair Old-fashioned country fair includes pari-mutuel horse racing, dancing, concerts, carnival and more.\nAug 25 | Wichita Falls Hotter 'N Hell Hundred This is one of the largest cycling events in the nation. Participants can ride routes up to 100 miles. Festivities include a three-day consumer expo, entertainment on ride day, medical seminar, off-road bicycle race, off-road trail run, and United States Cycling Federation racing on Saturday and Sunday.\nAug 31 \u2013 Sep 1 | Bertram\/Oatmeal Oatmeal Festival Event is held in Oatmeal on Friday and Bertram on Saturday. Usually the Friday and Saturday before Labor Day.\nAug 31 \u2013 Sep 1 | Brady World Championship Barbecue Goat Cook-Off Features teams from all over Texas and the United States cooking goat in their own way. Includes arts & crafts booths, kids games, horseshoe and washer pitching, and more.\nAug 31 \u2013 Sep 2 | Marfa Marfa Lights Festival Celebrates the enigmatic mystery lights that appear almost nightly east of town.\nSep 1\u20132 | West Westfest This is an annual Czech\/polka festival held Labor Day weekend in this Central Texas town.\nSep 6\u20139 | Grapevine GrapeFest Join wine novices and connoisseurs at the largest wine festival in the Southwest. Sample award-winning Texas wine, visit Grapevine's winery\/tasting rooms, enjoy the international wine area, listen to live music on stages located throughout historic Main Street. Complete in the GrapeStomp for the coveted Purple Foot award.\nSep 6\u201315 | Abilene West Texas Fair & Rodeo Features exhibits, rodeo, livestock shows and carnival at the Taylor County Expo Center. Parade is at 10:30am Sep 8.\nSep 7\u20139 | Grand Prairie 45th Annual National Championship Powwow Several hundred Native Americans, representing dozens of tribes from across the United States, take part in this colorful celebration of culture and heritage. Dancers in full regalia of feather, buckskin, beadwork and bells compete in many different dance categories. Includes colorful tribal dance contests, arts & crafts, cultural heritage demonstrations, teepees and Indian food. Usually the first weekend after Labor Day.\nSep 8 | Caldwell 23rd Annual Kolache Festival Enjoy Czech dances and festivities, arts & crafts booths, antique tractor and engine show, classic car and street rod show, and demonstrations.\nSep 13\u201315 | Rockport\/Fulton Hummer\/Bird Celebration Presents informative and educational centering on the migrating hummingbirds and other birds.\nSep 14\u201316 | Anahuac Texas Gatorfest Family-oriented festival offers activities for the whole family. Includes airboat rides, carnival rides, barbecue cook-off, beauty pageants, assorted vendors, and arts & crafts booths. The highlight of the festival is the alligator roundup. The festival coincides with the opening of the hunting season in Texas for alligator.\nSep 14\u201316 | Austin Austin City Limits Music Festival Brings the magic of the famed public television series outside of the studio.\nSep 15 | Richmond Fiestas Patrias This free family event is a tribute to the day in 1810 when Mexican patriot Father Hidalgo rang the church bells in the town of Dolores and began the struggle for Mexican (and thus, Texan) independence from Spain. Includes the Fiestas Patrias Queen contest. Hours are noon to 6pm.\n\u00bb Fort Bend Museum, 500 Houston St.\nSep 15\u201322 | Brenham Washington County Fair Features livestock shows, live music, rodeos and carnival.\nSep 19\u201323 | Lufkin Texas State Forest Festival Includes the Southern Hushpuppy Championships, Lumberjack show, giant carnival, East Texas Cheerleading Competition, Kachunga & the Alligator Show, and much more.\nSep 20\u201329 | Tyler East Texas State Fair Enjoy a carnival, food, vendors, livestock, Academic Rodeo, arts & crafts, helicopter rides and local entertainment.\nSep 21\u201323 | Plano Balloon Festival About 100 balloons are launched at various times during the weekend, weather permitting.\nSep 28 \u2013 Oct 21 | Dallas State Fair of Texas For three weeks each autumn, Dallas becomes home to all Texans. This represents a grand 121-year-old tradition \u2013 a merry gathering during which farmers, ranchers, home cooks, gridiron fans and music lovers converge on the city for food and fun.\nOct 3\u20136 | Winnie Texas Rice Festival Founded in 1969, this family-oriented event celebrates the harvest of the rice grown in Southeast Texas.\nOct 4\u201313 | Waco Heart of Texas Fair & Rodeo Educates through entertainment and agricultural experiences. Events held at the Fair & Rodeo include livestock shows, Texas Farm Bureau's Planet Agriculture, Wild West Market Place, Bud True Music Stage, midway and more.\nOct 5\u20137 | Fredericksburg Oktoberfest Celebrate German heritage with German food and drink, polka and waltz contests, music on three stages, artisans, Kinderpark, souvenirs and more. Generally held the first weekend in October.\nOct 5\u20137 | Gonzales Come & Take It Festival Celebrates the firing of the first shot for Texas Independence on Oct. 2, 1835, with a parade, re-enactments, carnival, biergarten, food booths, arts & crafts vendors and much more.\nOct 5\u20137 | Rockport Seafair This sea-themed fun festival features live entertainment, more than 120 vendors, gumbo cook-off, seafood cooking classes and demonstrations, carnival, education about the coast, parade and more.\nOct 6 \u2013 Nov 25 | Plantersville Texas Renaissance Festival This festival is the largest of its kind in the nation. Visit New Market Village for eight themed weekends: Oktoberfest, 1001 Dreams, Pirate Adventure, Roman Bacchanal, All Hallows Eve, Highland Fling, Glorias de Espana and Celtic Christmas.\nOct 12\u201314 | Cuero Turkeyfest The North Star (Worthington, Minnesota) and the Lone Star (Cuero, Texas) get together for a friendly rivalry of the Great Gobbler Gallop. See which turkey claims the title for its city as the \"Turkey Capitol of the World.\" Generally includes a variety of activities such as the Turk Olympics, cook-offs, parade and contests.\nOct 13\u201314 | Marshall Fireant Festival This zany, fun-filled festival is complete with a fire ant calling contest, arts & crafts booths, food vendors, live entertainment, chili-cooking contest, Tour de Fireant Bike Race, 5-K run, ugly face making contest, chicken chunking and domino tournament. Usually the second weekend in October.\nOct 17\u201320 | Gilmer East Texas Yamboree One of the oldest and largest festivals in the state.\nOct 18\u201321 | Tyler Texas Rose Festival Celebrate the rose, our national flower, and Tyler's rose industry at one of the most beautiful community celebrations in Texas. Highlights include the coronation of the queen and her court, rose show, luncheons, queen's tea and Rose Parade. Traditionally held during the third week of October.\nOct 26\u201328 | Flatonia Czhilispiel XXXV Features continuous entertainment, food, carnival, barbecue and chili cookoff, arts & crafts, car show, adult\/children contests, parade, model train exhibits and more.\nOct 30 \u2013 Nov 3 | Terlingua Terlingua International Chili Championship Bands play from 8pm to midnight Thursday through Saturday. Friday is National Scholarship day and includes a competition in beans, salsa and hot wings. Chili Championship is Saturday. Includes arts & crafts and food vendors on site beginning Wednesday.\nNov 3\u20134 | Austin Texas Book Festival More than 200 authors participate in special presentations, readings, panel discussions and book signings. Includes a Children's Chapter, Bon Appetit, Y'all Cooking Tent, entertainment tent and special presentations at the Austin Museum of Art and the Paramount Theatre.\nNov 2\u201311 | New Braunfels Wurstfest Celebrate with this 10-day \"salute to sausage.\" Enjoy a variety of entertainment, food and fun on the Wurstfest Grounds in Landa Park along with other special events throughout New Braunfels and Comal County.\nNov 4 | Port Isabel World Championship Shrimp Cook-Off Dozens of contestants compete for the World's Championship title. Vendors offer gift items, collectibles, food and more. Includes a kiddie play area.\nNov 8\u201311 | Crystal City Spinach Festival Enjoy food and festivities next to the Popeye statue.\nNov 9\u201311 | Salado Scottish Clan Gathering & Highland Games Features Highland dancing, piping, drumming, Celtic music, athletic events, vendors and more. Second weekend in November.\nNov 10 | Henderson Heritage Syrup Festival Considered by some to be the sweetest festival in Texas. Includes mule-driven, old-time syrup making, folk artists, antique tractors and cars, musical entertainment, arts & crafts, children's area, cloggers, square dancers, hayride shuttle and more.\nNov 21 \u2013 Dec 31 | Marshall Wonderland of Lights Spectacular Christmas event features a parade, home tours, bus tours, carriage rides, outdoor ice skating, live entertainment, visits with Santa and historic church tours. Christmas memories are made in Marshall.\nNov 23 | San Antonio Holiday River Parade Spectacular parade begins at 7pm.\nNov 23 \u2013 Jan 1 | Fredericksburg Texas Hill Country Regional Christmas Lighting Trail Come home to the hills for the holidays as the communities of Boerne, Burnet, Dripping Springs, Fredericksburg, Goldthwaite, Johnson City, Llano, Marble Falls, New Braunfels, Round Mountain and Wimberley light up the Texas night skies.\nNov 29 \u2013 Dec 1, Dec 6\u20138 | Jefferson 25th Annual Candlelight Tour of Homes Docents dressed in Victorian attire guide visitors through four beautiful homes glowing from the soft radiance of Christmas candles and richly adorned in natural cedar, pine and fir. Choirs and handbells sing melodies from on high to ring in the true spirit of Christmas.\nNov 30 \u2013 Dec 2, Dec 7\u20139, Dec 14\u201316 | San Antonio Fiesta de las Luminarias These luminarias line the walkways along the San Antonio River, symbolically marking the \"lighting of the way\" for the Holy Family.\nDec 1 | Dallas Neiman Marcus Adolphus Children's Christmas Parade Enjoy marching bands, magnificent floats, acrobats and even a visit from Santa.\nDec 1\u20132 | Galveston Dickens on the Strand Enjoy a Victorian-era holiday festival with lively entertainment, delicious food, extraordinary merchandise and more.\nDec 7\u20138 | Lubbock Candlelight at the Ranch Experience the many different traditions and celebrations associated with Christmas in ranching areas while strolling this 16-acre site. Lit only by candlelight and lanterns, visitors can look back in time as they watch Ranch Host volunteers re-create Christmas as it was from the 1780s to the early 1920s.\n\u00bb National Ranching Heritage Center, 3121 4th St.\nDec 7\u20139 | San Angelo Christmas at Old Fort Concho Features three full days of Christmas cheer, including shopping, living history, special children's area, ongoing entertainment and special scheduled shows.\n\u00bb Fort Concho National Historic Landmark, 630 S. Oakes St.\nDec 7\u20138, 14\u201315, 21\u201322 | Richmond Campfire Christmas Join the families of George Ranch for a special Christmas celebration.\nDec 31 | El Paso Sun Bowl The nationally televised game ends the year with El Paso's biggest, most festive event. The second oldest bowl game, often one of the better bowl games across the nation, pits the Big Twelve or Big East Conference\/Notre Dame against the Pac-10 Conference.\nThe Rotary Club of Cedar Hill is hosting its second annual Head for the Hills bicycle rally in Cedar Hill, Texas on May 10, 2008 from 8 am - 2 pm.\nThe Rally is a fund raiser to help local and international non-profit and relief efforts.\nCould you please include the rally on your Texas Events page?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Something I always wondered about, but was too scared to ask. Noah Smith's (quite reasonable) post nudges me into asking it.\nEven if everyone is perfectly rational, where is it written that stock market returns cannot be predictable? The stock market rate of return is a rate of interest. Where is it written that changes in interest rates cannot be predictable? Where is it written that changes in uncertainty of the market portfolio cannot be predictable? Where is it written that changes in liquidity premia cannot be predictable? Where is it written that apples cannot be predictably cheaper than bananas?\nIs it ethical to sell complimentary copies of textbooks?\nFaculty Books Recycling is a company that takes the complimentary copies of textbooks that publishers send professors, resells those comp copies to students, and makes a profit on the transaction.\nFaculty Books does everything possible to make professors feel that selling - or giving away - comp copies is an ethical thing to do. In their emails soliciting textbooks from faculty (sample below), they remind potential donors of the good that textbook recycling does. It puts textbooks into students' hands at a low price. Those professors who choose to sell rather than give away their comp copies are told, \"the money could sponsor a student event, be donated to charity, or spent however you like.\"\nTime to raise the gas tax?\nA much higher gasoline tax may currently be a political conversation stopper but fortunately it doesn't stop conversation in economic policy. Enter Joel Wood's paper about higher gasoline taxes for Toronto which appears in the latest issue of Canadian Public Policy\/Analyse de Politiques (which I am promoting here as the new editor). Joel also has a summary piece in the National Post.\nSee his key Table 4. Joel calculates an optimal gasoline tax of 41 cents per litre for Toronto, compared to the current 25 cents. Of the 41, about 11 cents is the straight Ramsey tax that would be appropriate if there were no congestion or pollution. Of the remaining 30 cents, about three quarters of it is a congestion toll (including the effect on road accidents) and the remainder is the optimal pollution tax.\nConsider the rival policy that has recently been proposed in Ontario: high occupancy and toll lanes, tolls on drivers without passengers who are using lanes that are otherwise restricted to drivers with passengers on some parts of some highways. Higher gas taxes would be much simpler and probably more effective while we await the tolling infrastructure that I hope will eventually come.\nRandom musings from a small sample on a subject about which I know little.\nWhere did I read that military amateurs talk strategy, but military professionals talk logistics? Or that if they did a re-make of The Graduate, the one word career advice would be changed from \"plastics\" to \"logistics\"?\nHelicopter Money is (almost) inevitable. The only questions are: who does it; and when do they do it. And we can't (easily) tell when it gets spent, and what it gets spent on, because money is fungible and we don't observe counterfactual conditionals.\nThat means the central bank makes (average) annual profits of 0.2% of GDP from printing money. Those profits must get spent, by someone, at some time. The only questions are: who spends them; and when do they spend them?\nWell, the federal election Leader's debate on Thursday evening was in the end a rather disappointing affair. It was essentially a series of thrusts, parries and spins on taxation, housing, immigration, energy, etc...but left out in the entire debate was any fundamental recognition of what I think is a major issue facing the future prosperity of the Canadian economy. Given that trade contribute's 30 percent of our GDP, what is the vision for us to continue making our way in the world? Perhaps a coherent vision is too much to ask of politicians seeking to get re-elected. In any event, I thought the next day was a good time to update the leadership Twitter feeds I have been following - see here and here.\nWell, all I was trying to do was introduce a set of lecture slides on the nineteenth century timber trade with a simple overview of the Canadian logging industry's employment in the twentieth century. Well, three hours later it has proven to be a more frustrating exercise than I would have expected but here is what I have.\nNot all Nash equilibria are created equal.\nGame A. There are n identical players who move simultaneously. Player i chooses Si to minimise a loss function Li = (Si-Sbar)2 + (Si-Sbar)(Sbar-S*), where Sbar is defined as the mean Si over all players, and S* is a parameter that is common knowledge to all players.\nThis game has a unique Nash equilibrium Si = Sbar = S*.\nThis game also has a unique Nash equilibrium Si = Sbar = S*.\nI think the Nash equilibrium in game A is plausible, but the Nash equilibrium in game B is implausible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eight-times World Record holder Mark Foster explains the physical benefits of swimming. If you've ever wondered what muscles you work when you swim, or what happens to your body as you race through the water, read on to get up to speed on the impact of this super-sport.\nThere's no denying swimming is a seriously energetic sport. It uses just about every muscle in your body and increases your aerobic fitness, too. Even better, it burns energy while supporting your joints making it a great way to get in shape.\nYet despite all these benefits according to the Amateur Swimming Association, a staggering 50% of 7 to 11 year olds can't swim the length of a 25m pool and as many as 9 million adults in the UK can't swim at all.\nNaturally, I'm all for swimming and it's great to know that despite these figures, 2.13 million Brits are keen to learn to swim. But I'm not sure everyone knows just how good swimming can be for your health and fitness. So here's a breakdown of exactly what happens to your body when you go swimming.\nWhat muscles will I work when I go swimming?\nFront Crawl is the stroke for speed \u2013 it moves you fast through the water and generates the most force.\nWhen you're using front crawl, your arms are pushing and pulling underwater, your torso is working hard to keep you steady and rotating to give you a longer stroke. Your hip flexors (at the top of your thighs) are engaged too, to maintain a steady kick.\nLess intensive than front crawl, backstroke is a great recovery option.\nAs the name suggests, it works your back. Your lats are engaged \u2013 that's the wide muscle on either side of your back, beneath your shoulder blade. This muscle is pulling your arm under the water and then back to the surface again. In addition, your hamstrings (back of your thigh) and glutes (bum muscles) are engaged to propel you through the water.\nSynchronisation is key here \u2013 having the arms in time with the legs. This stroke will work all your muscle groups equally.\nYour shoulders are working hard to move your arms from behind to in front of you. The chest and your lats then work together lift your chest out of the water as you take a breath. Your legs are doing a frog kick that's similar to leaping off the floor from a squat, working your glutes, quads (front of thigh), hamstring (back of thigh) and your calves, too.\nButterfly is a super-powerful stroke that will build strength and boost your metabolism.\nBoth arms move simultaneously, working your shoulders, lats and arms. Your core and lower back muscles go into overdrive to stabilise your core in the water and lift your body out of the water, and your glutes ensure your legs move as one, like a dolphin.\nPhew! It's energising just thinking about it. As far as an all-over body workout goes, it doesn't get much better. Add to that increased lung capacity from taking huge breaths frequently and precisely, and swimming helps to improve your aerobic performance, too.\nIf you're keen to get in shape by swimming, Swimming Nature offers premium tuition and fast results. Our award-winning technology and bespoke programmes ensure you and your kids develop precision techniques and complete mastery of the water. Whether you're looking for yourself or your kids, we cater for all abilities from beginners to triathletes, and our exclusive Mark Foster Programme takes advanced swimmers to the next level. For more information, explore our programmes today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've never known anyone like Shakil.\nI'll stay and watch Kathryn.\nJulia doesn't know what he really wants.\nMoses and Adrian watched John swim laps.\nI'm pretty sure Margot's nervous.\nDean can really put away the food.\nIsidore has lived in Boston since he was born.\nBrandy arrived at Ami's house in the middle of the night.\nJin doubts if Page will come to school today.\nI thought Morgan would have forgotten me by now.\nChristina's dog left muddy paw prints all over his new carpet.\nWhat would Roger do without me?\nKristen carefully checked that the gas was turned off.\nCaleb read the whole article from beginning to end.\nHis first name was Heinz.\nLaurent received a letter from Jayant.\nI should've talked to Ric first.\nWes died from cancer in 2013.\nShe arrived home two and a half days later.\nEddie went back to his room and lay down.\nIn his retirement speech, Noemi said he wanted to be a family man who plays football rather than a footballer with a family.\nNinja is going to love living here.\nSvante didn't have much time for studying.\nIn fact, Alex told me you didn't like me.\nI called from Robert's house.\nScot was very clear about that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakyws b/data_all_eng_slimpj/shuffled/split2/finalzzzakyws
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@@ -0,0 +1,5 @@
+{"text":"The first thing that comes to mind when you're trying to cool something more is a bigger, badder heat sink, or a liquid cooler, or what kind of radiators you're going to use. That's all fine and dandy, but you can often improve your thermal performance just as much by simply replacing your thermal interface. A thermal interface material is the component (usually paste) that goes between your device (either its naked lid, or its Integrated Heat Spreader - IHS) and your heat sink.\nThere's a few different kinds ranging from pastes to pads made out of metal, and they all have various properties that make them better suited for some scenarios over others, or make them cost-attractive.\nIn the list below, you'll see a number followed by \"W\/m\". This is the unit of measurement for thermal conductivity. Basically, it's how well the material conducts heat from another object to itself, and can change based on how warm or cold it is. A cold material will typically better conduct heat than it would if it was already warm. This is why, sometimes, it's better to increase airflow than it is to get a larger heat sink. Returning the heat sink closer to ambient is often better than a larger heat sink that remains saturated.\nI'm going to focus on interfaces designed for conducting at least 50W (like a CPU) over a 30x30mm area.\nThermal paste is anywhere from 2 - 12W\/m in all directions.\n+Good, general purpose, easy-to-apply, reliable application. Plenty of room for error.\n+Can be further engineered for special application methods that allow for some exceptional tolerances or precision, as necessary.\n++Exceptionally cheap, even for the highest end of pastes (although retail pricing ruins this).\n+Well-suited for precision-machined heat-sinks that mate very closely to the surface its cooling.\nSometimes it's better to use an extremely thin layer of paste than it is to use a better material that's thicker.\nIn this scenario, you're relying on direct metal-to-metal contact between a well-machined heat sink and a lid or IHS. The paste only serves to cover extremely small, infrequent gaps. If, instead, you separated the sink and device with something even as good as Indium, you might lose performance. Most metals conduct hundreds of W\/m, given good contact.\n-Lowest thermal conductivity of all common TIMs, although it's usually 'good enough'.\nGraphite thermal interface pads conduct 35W\/m between it and another surface. Across\/within itself, it approaches 400W\/m.\n+The ~ 400W\/m lateral heat spread means it handles hotspots very well - it rapidly distributes localized heat across itself to better displace it to the heatsink.\n-Requires moderately high compression from the heat sink to conduct well.\n-Not super easy to apply (it likes to slide around and is very light; you can't see if you've brushed it out of place while affixing the heat sink).\nBecause it is electrically conductive, you want to be extra careful and maybe double check if it applied well. It will often stick to the sink so be careful you don't lose it when removing the sink.\n+Excellent, cost-effective upgrade for most consumer-grade heat sinks. I've found even a $20 Hyper 212 Evo acts like a $60 cooler if you replace the paste with a graphite pad.\nLiquid metal is usually around 32 - 73W\/m between surfaces. I do not know the lateral heat spread but I'd imagine it's excellent.\n-Electrically conductive - don't mess up.\n-Depending on the metals it's interfacing, it may fuse them together or damage them.\n-Good luck cleaning that stuff up.\n+Precise application guarantees exceptional performance.\nIndium Corporation \"Heat Spring\" thermal interface pads exceed 86W\/m conductance, and - I think - exceeds the lateral heat transfer of graphite.\n++In basically all thermal regards, it exceeds the performance of graphite pads.\n--Exceedingly cost prohibitive. Whereas a 40mm graphite pad is around $10 - $15 USD, an Indium pad is above $200.\nCan be bought in bulk sheets to bring this cost down.\nCarbon NanoTube structures perform above Indium, at an estimated 3x that of Indium. Its real benefit is it performs excellently despite the usual contact resistance of the two materials it is interfacing.\n+++Basically the realistic maximum we can achieve in thermal performance, given the more recent application and manufacturing processes.\nEven something like the boiling action in two-phase immersion cooling can destroy the CNTs.\n---Untouchably expensive. In a situation where Indium might cost as little as $50 and as much as $200, CNTs would likely cost over $1200 or more.\n---Unreasonably difficult to apply. Requires the manufacturer to apply it to your substrate directly, in most cases.\nThere's a few more thermal interfaces for specialty scenarios, such as ones that are highly heat conductive while remaining dielectric (electrically non-conductive), but they're beyond impractical for most scenarios. If you're in a situation that calls for these, you probably already know how to hunt them down and apply them. The major problem with most of these is that they're either highly reactive (will destroy or bond with your device or sink) or are extremely hard (and thus resistant to the compression that is necessary to effectively mate them to your device and sink).\nLatest edit to correct errors in stated thermal performance of liquid metal and Indium.\n@senseless at AllMine experimented with Indium foil after we discussed and researched a few alternatives for TIMs. With the use of some Conformal Coating material (which is used partially as a TIM in its own right) to isolate the electrically conductive indium from exposed contacts, he was able to cover a majority of the unit in a combination of indium and other modest thermal pads (Pictured below).\nPushing 135W (at the wall) through the FPGA, on the stock air cooler but with the pictured upgrades, one unit achieves 56.6C and the other 62C. Prior to this upgrade, these units went to ~85C running the same bitstream on the same settings. This is a massive improvement, with two-fold benefits. Better controlled thermals means higher efficiency on power delivery, which means more easily controlled thermals on the power delivery system. If you can better manage the power delivery system performance, you can overclock the card higher, or run it harder in general.\nAdditionally, exposing the copper under the nickel- or zinc-plated stock VCU heatsink - by means of dremel or sanding - improves thermal transfer as well, given that you make make up for the loss of material by increasing your TIM thickness. This can drop your temps another 5-10C. You will also likely need to fill in the cross-hatch pattern with degassed thermal grease (or leave the existing grease there). In standard practice, this increases the effective surface area (by effective, I mean useful - the math works out to less paste between the sink and device overall, by sacrificing some surface area to purposely handle the worst of the problems in the paste) of the sink-to-device contact patch. My suspicion is that with something as high-performance as Indium (which has its own cross-hatching), the VCU's cross-hatched surface area (facilitated normally by only paste) performs worse than a perfectly flat contact patch that was 100% in contact with the indium. I will be able to confirm this in the coming months.\nIf you continue to use thermal grease or paste, then the cross-hatch pattern is necessary.\n@senseless \"Scavenged some larger tension springs and large m2 screws. I think the springs came from an old xeon v2 or v3 supermicro heatsink. FPGA temps dropped from 62-63c to 59-60c. LTC temp sensors 61c\/68c. This is the board that was running a little hotter and pulling some extra amps.\"\nyep, 40Mh\/s per card, but we have room to add additional cores and try to get better placement. I'd say we'd probably cap out at 80Mh\/s per card with more work.\nLater, senseless added a fairly large aluminum sink to the bank of the board around his new mounting solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LinkAmerica recognizes and accepts its responsibility to be a good steward of the environment and to help achieve a state of sustainable development. In support of these responsibilities Link America has established a commitment of compliance to all applicable state, federal, and local requirements with a goal of going beyond compliance wherever practical and possible. Ultimately, we seek to combine economic success, social responsibility and protection of the environment.\nThrough Link America's Recycling Program, employees are encouraged to recycle, and provided with recycling bins, recyclable eating utensils, cups, and plates. Moreover, with the new IDEAS program, the company is moving towards a paperless system of doing business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Definition : Personal computers designed to be used on a desk. These computers typically consist of a microprocessor-based CPU (central processing unit) that performs simple, well-defined functions; input and output ports for connection to other devices, including keyboards and pointing devices such as a mouse, joystick, and trackball; and data storage devices and an electronic monitor for image display. Additional peripheral devices, such as printers, plotters, and data recorders, are frequently attached to the computer to increase its capabilities. Desktop personal computers can store, retrieve, and process data, typically in digital form, following the instructions of one or more programs stored internally, either temporarily or permanently (i.e., software and firmware, respectively). They are used in healthcare facilities, physician offices, at home for clinical and administrative data processing, as a component of data processing workstations (e.g., radiology, ultrasound, MRI), or information\/automation systems.\nEntry Terms : \"Desktop Personal Computers\"\nDatalux Corporation designs, manufactures and markets specialized computer products for demanding environments. Datalux products are well-suited for installation in locations that require unique, durable products and have limited space available. Product offerings from Datalux range from individual components such as keyboards, monitors and computers to complete systems.\nMarvel manufactures and markets a wide range of high-quality furnishings to large, small and home office markets. Our product lines range from office chairs and file cabinets to executive office furniture and open office work-stations.\nWakefield is an engineered solutions provider of custom products for multiple global markets. We design, manufacturer, and market fabricated metals, thermal management products, powdered metal, thermoset plastics and aluminum extrusions. Wakefield is a vertically integrated engineering and manufacturing operation that offers industry leading quality and lead times from our four manufacturing locations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We know who you SHOULD consult with, we talk to Him every day. But if you want to work with us to deliver on your vision, we are your Team. We'll work together to help you create a complete \"Marketing Campaign\" to effectively and efficiently deliver your message.\nWe've got experience, and we know who the \"Go To Guy\" is. Visit the \"Contact Us\" Link to get things going.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Armshouse joins new countries with its second invitation to Norway, this time to Gigacon 2014. Gigacon marks Norway's inolvement in the Capcom Pro Tour and players will battle it out for ranking points and a share of 12,500 NOK.\nMore information on Gigacon can be found here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalixh b/data_all_eng_slimpj/shuffled/split2/finalzzzalixh
new file mode 100644
index 0000000000000000000000000000000000000000..85c296a1dcf82d707adb943a387d351df1ef7434
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I am well aware that there are several posts covering the topic of customizing the permalink URL structure of custom post types, custom taxonomies, and taxonomy terms. However, none of them are real clear, don't work correctly, or cause problems elsewhere.\nHere are a few links, which already \"kind of\" cover URL rewriting.\nMixing custom post type and taxonomy rewrite structures?\n\/cars\/ needs to be accessible as an archive listing of every post made in the post type.\n\/cars\/dealers\/ needs to be accessible as an archive listing of every post made in the taxonomy.\n\/cars\/dealers\/honda\/ needs to be accessible as an archive listing of every post made in that taxonomy term.\n\/cars\/dealers\/honda\/123-hot-rod-lane\/ needs to be accessible as a singular view of the post made in the cars custom post type.\nJust to be clear, so no one does any unnecessary work while replying to this question. I do NOT need anything other than just adding the correct rewrite rules for my post type, taxonomy, and terms. This questions is directly in regards to permalink URL rewriting, and NOT in relation to listing of posts or archives. I just need to be sure WordPress recognizes them properly, while using conditional calls like is_post_type_archive() or is_tax() or is_singular(), etc.\nI've attempted numerous times, from numerous questions\/answers on this site. Nothing seems to work.\nDepending on how you got there.\nYou will definitely be able to have domain.com\/post-type-name\/taxonomy-name\/term-name\/, but I think that it may not completely make sense within WordPress to have such a long URI.\nThe possibility of multiple taxonomy terms (let alone multiple taxonomies attached to a custom post type, or any post) would create a lot of overlapping links, as well as more possibilities to break bookmarks - lets say I bookmark \/cars\/dealers\/honda\/123-hot-rod, but then you realize that it wasn't actually Honda, it was Toyota - you changing the tax term related to the post would break that link. Not a likely occurance, but it's possible.\nFrankly, it's less user friendly. Not that the average user pays attention to the web paths they are on, but it is easier to remember domain.com\/cars\/93-saturn-ion than it would to remember (or understand) domain.com\/cars\/dealers\/wonderland-auto\/93-saturn-ion.\nWhen you are on that single post's page, the default 'above_nav' in most themes includes the taxonomy terms with the date underneath of the Page title anyways, so that information is still there, in a more legible, and usable way.\nI wouldn't like to submit this as an answer however, it's an amazing little snippet that links up your taxonomies and terms.\nNot the answer you're looking for? Browse other questions tagged custom-post-types custom-taxonomy permalinks url-rewriting rewrite-rules or ask your own question.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On 14 March, the Regulatory Fitness Platform of the European Commission (or \"REFIT Platform\") adopted a series of Opinions dealing with varied issues relevant to the EU legislative and policy-making framework. Amongst those, two Opinions were drafted by CEEP General Secretary Valeria Ronzitti, together with other REFIT Platform members, full REFIT member since its creation in May 2015.\nThe first of these Opinions refers to \"Transparent Transposition\" and draws up recommendations aimed at addressing the issue of over-regulation by national and sub-national authorities transposing EU legislation into national law. The second one is entitled \"State Aid and ESIF\" and raises awareness on the needed alignment of EU State Aid rules with European Structural Investment Funds (ESIF) regulation.\nCEEP is fully convinced of the added value of both these Opinions as well as the impact that they will have on future EU policy-making and in supporting the proper implementation of EU legislations at national, regional and local level. Those two outcomes will only be attained if and when institutions, SGI providers, employers, workers and even citizens, make full use of all the opportunities stemming from those REFIT Opinions.\nOn the one hand the opinion on \"Transparent Transposition\" should be used to highlight the shortcomings that impact the whole European legislative machinery. On the other hand the one on \"State Aid and ESIF\" underlines the current lack of complementarity between two of the most important areas of EU policy-making, which are competition policy and cohesion policy. As such, those opinions can serve as a basis for our future actions when it comes to better monitoring the implementation of the EU legislation in Member States, and highlight the need to address both the competition policy and the EU funding tools.\nThe REFIT Platform exerting a consultative function as the European Commission's regulatory body, these opinions are not legally binding. In both cases, however, they send the right signals and enable EU decision-makers to better locate shortcomings and room for improvement. CEEP believes that both these opinions, contribute (together with many other) to improving EU law-making in terms of quality across many policy fields. But beyond the matters at stake, they clearly establish the REFIT Platform's added value as a forum to stimulate and structure political debate.\nCEEP Policy Officer Alexis Le Coutour remains (alexis.lecoutour@ceep.eu) at your disposal for any further questions on this issue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please have a look at version history of the Atlassian Marketplace listing for the latest release notes.\nHow can see browser and OS information?\nWe get browser and OS information from Intercom as a combined User agent field. Simply search for the User agent field in Intercom for Jira when you configure visible fields to show browser and OS details in the conversation panel.\nHow much does Intercom for Jira cost?\nPlease have a look at the pricing tab on the Intercom for Jira Marketplace listing to get a rough idea. Make sure you select Cloud or Server in the top right, depending on what's applicable to you. Intercom for Jira is priced based on the number of active users in your Jira instance (Atlassian pricing model). It does not matter how many users\/leads or admins you have in Intercom or how many conversation links you create or anything like that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DISTRIBUTION: Statewide wherever permanent water is available. Most abundant in the prairie pothole region.\nHABITAT: Muskrats prefer well-vegetated sloughs and lakes with a fairly stable water level. They are also found in creeks and rivers as well as in small potholes. Their existence in any area is dependent upon a year-round supply of water. When water areas go dry during the summer, the muskrats may move to another area. When potholes or sloughs freeze solidly to the bottom in winter, the muskrats will perish.\nLIFE HISTORY: Muskrats raise more than one litter a year with the first litter usually born in late May or early June. In North Dakota muskrats usually have two litters and occasionally three. The gestation period of muskrats is 29 to 30 days. The average litter contains five or six young with litter sizes ranging from one to eleven. The young muskrats, blind and naked at birth, are raised either in a nest chamber within the muskrat house or in a den in the bank. They remain in the nest for three or four weeks when they move out to fend for themselves. Muskrats weigh less than an ounce at birth and grow rapidly, reaching adulthood in less than a year. Average adult muskrats weigh from two to two and a half pounds. Muskrats are somewhat active during the day but most of their activity takes place at night. They do not hibernate during the winter but live in relative comfort in their well constructed houses or in bank dens. Both their houses and dens have entrances under the water so even when the area is frozen over, they can swim about and gather roots and stalks of aquatic plants for food.\nFOOD HABITS: Foliage, roots, and bulbs of aquatic plants such as cattail, bulrush, pondweeds, and sedges. Muskrats also eat corn and some other terrestrial plants when readily available and occasionally eat frogs, snails, and clams.\nSUGGESTED TRAP SIZE: No. 1 or No. 1 1\/2 single longspring, jump trap or single underspring; No. 1 double longspring; No. 110 Conibear; \"surehold\" attachment are especially recommended for trapping muskrats. Although they are somewhat more expensive than conventional traps, they more than pay for this difference by preventing muskrats from escaping from your trap.\nSUGGESTED BAITS: As muskrats normally have an abundance of food available, baits and lures are not as effective as for other species. However, apples and vegetables (especially carrots) are sometimes effective and are often used in under ice sets.\nMost of North Dakota's trappers get their start by trapping the muskrat. Because muskrats are confined to a limited area of water, are easy to trap, and are abundant when water conditions are good, they are a good \"break-in\" species for the beginning trapper.\nIt is easy to determine if muskrats are present in a slough as their houses and runways are easily seen. They also build small mounds of vegetation which may be mistaken for unfinished houses. These are \"feeders\" and as the name implies, are used as feeding and resting places by the muskrats. They are commonly of two types; one kind is big enough so that the muskrat can come up inside it and feed, the other is nothing more than a platform upon which the muskrat can sit to eat or loaf.\nYou can also see other places where muskrats have been coming out of the water to rest or to eat. Look for their tracks and droppings along the shore. When muskrats feed, they often leave a scattering of pieces of partly eaten roots and stems of plants. Look also for floating pieces of vegetation which often mean that muskrats have been at work in that area.\nIn the fall, before freeze up, muskrats are usually caught in blind sets. They cannot feed under water and do not normally eat while in the water, but climb out on their houses or on feeders, along the shore, on rocks, or on logs or other floating objects to eat. Traps set in the water around places where muskrats go to feed are very effective. The trap should be from two to three inches below the water.\nThe \"floating log\" set is a good one for muskrats. Pieces of 2\" x 10\" plank about two feet long with a trap set in mud and loosely covered with debris can be used in place of a log. Secure the trap chain to the bottom of the plank and run a wire down to a rock or stake to keep the plank from floating away. Lure or bait can be used to good advantage at such a set.\nAnother good muskrat set can be made by securing a trap to a board (with nails or a cleat to hold it in place) and sticking the board in the slough bottom at an angle so that the muskrat would use it as a feeding or resting place. Push the board in far enough so it is good and solid and the trap is under the water. This set can also be baited although the muskrats will use it as a resting site even without bait or lure.\nMuskrats that live in bank dens can be easily caught by setting a trap in the runway. Runways that are partially exposed and filled with from three to four inches of water are the best to trap in. If they are deeper than that, the 'rat will often swim right over the trap.\nMuskrat trapping in North Dakota very often means trapping under the ice. The trapping season must be quite late to insure prime pelts and to give late litters time to grow up. Because our freeze up often comes early in the fall, the muskrat trapper will usually have to do at least part of his trapping through the ice.\nMuskrats are easily caught after freeze up by trapping them in their houses. Be very sure that the ice is safe before venturing out on it! Each muskrat house has a main chamber in it. Tunnels from the main chamber lead to smaller \"rooms\" and to the main entrance which is constructed under the ice. Open the house carefully, using a hay knife or similar tool, and insert the trap. Place the trap on the floor of the chamber. Most trappers use a wire extension in the trap chain and wire it to a stick outside the house. Leave enough slack chain by the trap so that the muskrat can dive and drown itself. This type of set can also be used at the larger muskrat feeders with good results. Carefully replace the material taken out when opening the house and tamp it firmly back in place. This is very important! If you do not tightly close the hole you made in the side of the house, the house and its occupants will freeze.\nWhen a water area is frozen, active muskrats bank dens can be easily located by looking for air bubbles under the ice. In runways that are being used, the activity of the 'rat often stirs up the mud, making the runway look cloudy. Cut a hole in the ice and set your trap in the runway. If the water is more than three inches deep, place the trap on some mud or grass or on a stone so that it is about three inches below the ice.\nOther under ice sets can be made. The leaning board set can be used under the ice with the trap about eight inches below the ice and a bait fastened above the trap. Beaver type sets can be modified for trapping muskrats under the ice. A bait box set over a hole in the ice with a trap inside, can also be effective. Most such sets entail considerable effort and are no more effective than trapping in the house.\nWhen the number of muskrats caught daily decreases considerably, it is time to pull up your traps. A good rule to follow when trapping muskrats in houses in this: count the number of muskrat houses (not feeders), multiply by five for a population estimate and quit trapping when you have taken seventy-five per cent of the estimated population. This will reduce the population down to a level where there will be plenty of breeding stock left and will insure enough room for good production the following summer.\nMuskrat traps should be set so the muskrat will drown. Otherwise, they will wring off their foot, escape and die. If the water is not deep enough for drowning set, use \"stop-loss\" or \"sure-hold\" traps or the new humane traps. Be sure to check your muskrat traps regularly. Most trappers check them every few hours. This not only will enable you to prevent some animals from escaping, but it will also increase your catch, since a sprung trap, or one with a muskrat in it obviously cannot catch another until it is reset. Live muskrats in a trap can easily be killed by a sharp rap on the nose with a stout stick.\nMuskrat pelts are handled as cased skins with the feet and tail cut off. Brush and clean the fur before skinning, remove excess fat from the skin and stretch the pelt carefully. Stretch your pelts uniformly and do not over-stretch them. A basic muskrat stretcher is eighteen inches long, eight and one-half inches wide at the base, tapering to five inches wide one foot from the base and down to a rounded end. Many trappers use commercial wire stretchers which insure uniformly stretched pelts.\nSome fur buyers pay as much for unskinned muskrats as they do for prepared pelts. By doing this, they are assured of having well prepared, uniform pelts. They sell the carcasses to pay for their work in skinning. Check on this with your fur buyer before the season starts. If you do your own skinning, look for a market for the carcasses as selling them would increase your profit. Muskrat carcasses are sometimes used for mink food or sold to rendering plants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Non-personal information is information that cannot identify you. If you visit this website to read information, such as information about one of our services, we may collect certain non-personal information about you from your computer's web browser. Because non-personal information cannot identify you or be tied to you in any way, there are no restrictions on the ways that we can use or share non-personal information. What is personal information and how is it collected? Personal information is information that identifies you as an individual, such as your name, mailing address, e-mail address, telephone number, and fax number. We may collect personal information from you in a variety of ways: \u2022 When you send us an application or other form \u2022 When you conduct a transaction with us, our affiliates, or others \u2022 When we collect information about in you in support of a transaction, such as credit card information \u2022 In some places on this website you have the opportunity to send us personal information about yourself, to elect to receive particular information, to purchase access to one of our products or services, or to participate in an activity.\nHOW DOES CASTOR ABBOTT, LLC USE PERSONAL INFORMATION?\nCastor Abbott, LLC may keep and use personal information we collect from or about you to provide you with access to this web site or other products or services, to respond to your requests, to bill you for products\/services you purchased, and to provide ongoing service and support, to contact you with information that might be of interest to you, including information about products and services of ours and of others, or ask for your opinion about our products or the products of others, for record keeping and analytical purposes and to research, develop and improve programs, products, services and content. Personal information collected online may be combined with information you provide to us through other sources We may also remove your personal identifiers (your name, email address, social security number, etc). In this case, you would no longer be identified as a single unique individual. Once we have de-identified information, it is non-personal information and we may treat it like other non-personal information. Finally, we may use your personal information to protect our rights or property, or to protect someone's health, safety or welfare, and to comply with a law or regulation, court order or other legal process.\nDOES CASTOR ABBOTT, LLC SHARE PERSONAL INFORMATION WITH OTHERS?\nThis site contains links to other sites that provide information that we consider to be interesting. Castor Abbott, LLC is not responsible for the privacy practices or the content of such websites.\nThis site may provide public discussions on various business valuation topics. Please note that any information you post in these discussions will become public, so please do not post sensitive information in the public discussions. Whenever you publicly disclose information online, that information could be collected and used by others. We are not responsible for any action or policies of any third parties who collect information that users disclose in any such forums on the website. Castor Abbott, LLC does not agree or disagree with anything posted on the discussion board. Also, remember that you must comply with our other published policies regarding postings on our public forums.\nCastor Abbott, LLC will not intentionally collect any personal information (such as a child's name or email address) from children under the age of 13. If you think that we have collected personal information from a child under the age of 13, please contact us.\nTo make any of these requests, please contact our GDPR contact at gdpr@castorabbott.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamcaw b/data_all_eng_slimpj/shuffled/split2/finalzzzamcaw
new file mode 100644
index 0000000000000000000000000000000000000000..61b13ae6450b1ea40ee86f28d2549079e0d4bfa3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamcaw
@@ -0,0 +1,5 @@
+{"text":"NAICS code appeals can be powerful tools. A change in a solicitation's NAICS code\u2013and corresponding change in the small business size standard\u2013can significantly broaden or narrow the competitive playing field. And statistically speaking, NAICS code appeals are often successful.\nBut NAICS code appeals are subject to strict rules. As a recent SBA Office of Hearings and Appeals case confirms, NAICS code appeals cannot be lodged against presolicitations.\nA NAICS code appeal can be a powerful vehicle for influencing the competitive landscape of an acquisition. A successful NAICS code appeal can dramatically alter a solicitation's size standard, causing major changes in the number (and sizes) of potential competitors.\nBut a NAICS code appeal cannot be filed until the solicitation is issued. As the SBA Office of Hearings and Appeals recently confirmed, a NAICS code appeal cannot be filed with respect to a presolicitation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At MaxDumpster, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Hamilton, CO, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nCommunity cleaning are becoming more popular than ever as the people take back their particular neighborhoods one street at the same time. Affordable dumpster may be delivered the very next day. The neighborhood friends all take pleasure in taking part in any cleanup day time that includes development 6 Yard Dumpster in Hamiltons. Preserve large trash such as household furniture off the sidewalks while it is waiting for trash pick-up and get the particular yards sparkling clean using a 6 Yard Dumpster in Hamilton. Renting any dumpster brings the town together as well as gets rid of unwanted trash easily and quickly. Everyone can dispose of the excess containing piles upwards around attributes, streets, etcetera in one easily rented as well as delivered dumpster. There's no need to bag those things being disposed so any dumpster makes trash disposal as well as cleanup day time or full week easy for everybody. Keep your community trash totally free by having a group cleanup day time. A development 6 Yard Dumpster in Hamilton will make any project popular as well as quick; everybody will be delighted to participate.\nIt is likely that your kicked out tenants left out things that they no longer needed in the rush of the foreclosure. This dust can range coming from furniture to bubble gum wrappers, but it all needs to proceed before you can begin to turn over the property. Renting any dumpster is a good option in these circumstances as you will have got all of the space you need to get rid of both the tenant's points and any development or refinishing supplies you have excess once the redesign is complete. Dumpster rental permits you to dispose of the particular property's spend at your leisure, so it is available as soon as your contractors as well as cleaning group show up, regardless of when they turn up.\nThe best way to take care of the too much to handle clutter is actually to start throwing things absent! Renting any dumpster can help with this because you possess somewhere to put all of the trash instead of striving to find another place to said until trash day.\nFirst, take the time to discover customer reviews or maybe testimonials this company. In case anybody got complaints there is a decent opportunity they proceeded to go online to port their worries. Their Eee profile could possibly be especially worth it to check out.\nDumpster rental can seem like a challenging thing to fully grasp. However, when these simple steps are used, it can be a very easy and satisfying process.\nRenovating construction requires a large amount of supplies. It's no surprise then that at the end of most assignments the site is rather messy as well as full of dust with more materials as well as dirt desiring to be grabbed. A development 6 Yard Dumpster in Hamilton can do the job for you. Before stripping or maybe guttering out your unfilled home you must think of having your dumpster presently delivered and ready and waiting.\nPricing! dumpster rental fees, demolition and a firms complete distinctive line of services needs to be competitively priced, extremely competitively priced! An educated business deals with and keep charges low the industry benefit that's passed on to the customer. The company must also make use of a website by making this a place where you could have almost all if not all questions you may have answered.\nIncreasing environmental concerns features raised how many industries using the Roll Away from rental services designed a misconception that Dumpster services are generally totally limited to industrial purpose. I entirely disagree together with the statement. Let me tell you that Dumpster companies in recent times are extending his or her wings to numerous commercial and household will work as well. Like waste treated by the the majority of professional Dumpster rental service providers is classified straight into four types general waste, construction waste, green yard waste and recyclables.\nBe wary of an company that is relatively new. Some might still be understanding the ropes and making mistakes. You don't want to hire a roofer that might injury your front yard or not match their scheduling commitments.\nYou might want to also get rid of your barn. It is possible to rent a small, method or large dumpster to get the job done. You may need a large dumpster in case you have a lot of trash that will compliment ten pickup loads. Ten pickup plenty or four tons could be the limit to get a large dumpster. If the items go over four plenty, you will need to rent another dumpster to finish the project off. Should you only need with regards to five collection loads than a medium dumpster is wonderful for you. A medium dumpster holds up to several tons of trash or particles. A small dumpster holds up to a couple of tons of particles.\nOnce the advertising starts, it is necessary for you to find things that you desire to sell. We all have something lurking in the crevices of our residences that we don't want! Older clothes, forgotten children's games, and mismatched dishware are invariably good places to start. If you have the required time, start looking using your basement, loft, and storage for various other forgotten goods that are just gathering dust. You may be astonished at how several things you'd kept around unintentionally.\nFirst, opt for the correct size dumpster. The most common styles include the 10 yard container, Thirty yard package or Thirty yard containers. A 10 yard container is used for most house remodeling assignments and doesn't occupy too much area in your yard. The particular 20 yard container is used for smaller residential or commercial assignments. A Thirty yard package is for method to large assignments. Make sure that you opt for the correct size for your task. Even though you may well save money by renting the smaller dumpster, you will pay if you have to schedule multiple pick-ups to haul the garbage away. To be secure, you should never excess your dumpster since it can be very harmful for the motorist hauling aside your garbage if it's overloaded.\nKeep Older Town seeking as beautiful as ever when you rent a dumpster to have any work done you need with your home or business while making your self happy with the retail price you compensated and the services that was granted to you.\n45 yard roll off dumpsters have a tendency to only actually get utilized on full range construction web sites. They can proceed an incredible number of junk, nevertheless as they find a considerable amount regarding room it is not feasible to utilize these kind of packing containers on most residential or commercial sites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Start in Vancouver and end in Calgary! With the train & rail tour Canadian Rockies by Train featuring the Calgary Stampede (Calgary, AB to Vancouver, BC), you have a 9 day tour package taking you from Vancouver to Calgary and through Vancouver, Canada and 8 other destinations in Canada. Canadian Rockies by Train featuring the Calgary Stampede (Calgary, AB to Vancouver, BC) includes accommodation in a hotel, meals.\nTourRadar only requires a deposit of $99 to confirm this Collette booking. The remaining balance is then payable 65 days prior to the departure date. For any tour departing before 28 June 2019 the full payment of $5,375 is necessary.\nNo additional cancellation fees! You can cancel the tour up to 65 days prior to the departure date and you will not need to pay the full amount of $5375 for Canadian Rockies by Train featuring the Calgary Stampede (Calgary, AB to Vancouver, BC), but the deposit of $99 is non-refundable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE wettest, darkest, most miserable summer in a century has left me with a hankering for the exotic.\nDespite Huddersfield having a plethora of eating places serving food from the Indian subcontinent, Thai Sakon is the only sit-down restaurant, to my knowledge, with a south east Asian cuisine.\nIn all my travels around the world as a photographer, it is the journeys through the Far East which conjure up the most vivid memories of mouthwatering food.\nA handful of fresh prawns, a few blades of lemongrass and a pinch of spices thrown into an old frying pan at the side of the dusty road somehow managed to produce the most delicious dish imaginable, causing a riot of flavours to explode on the tongue.\nIt was with these sunny images in mind that I headed down the stairs at Thai Sakon, which has become a firm favourite with many diners since opening in 2005.\nNot a fan of basements, the interior designers are to be highly commended for creating an upmarket, welcoming ambiance where you completely forget you're underground.\nClever use of etched glass panels, plain walls and dark wooden furniture and floors, offset by textured panels, Far Eastern artwork, red sofas and modern halogen lights, result in a rich yet clean East meets West look.\nThis is reflected in the ownership of former Huddersfield Polytechnic maths student Mark King, who lived in the Far East for several years and his Thai wife Sompong.\nWe were immediately greeted warmly by Mark who showed us to a table in one of the two dining rooms, already quite full at 7.30pm.\nOur waitresses, wearing traditional Thai blouses, were excellent throughout the evening: polite and efficient without being obtrusive.\nAfraid of missing something interesting, we ordered the mixed starter at \u00a36.75 a head, and a dish with seven small portions x 2 arrived with three dipping sauces.\nThe tasters were chicken satay (succulent), Thai spring rolls (crispy), deep fried prawn and pork toast, Thai fish cakes, sweetcorn fritters, chicken breast and prawns in tempura batter with breadcrumbs.\nAll of the starters were delicately flavoured and tasty, whetting the appetite for the spicy main course to follow.\nIf I'm being really picky, I would have preferred the batter to be pure tempura, rather than mixed with breadcrumbs.\nAnd if one or two more of the seven portions had not graced the deep fat frier, we wouldn't have complained. But all the starters were well prepared with high quality ingredients.\nA couple of beers later (\u00a32.90 a bottle for Asian beers or \u00a33.35 a pint for Carlsberg), the restaurant had filled and we were ready for the main course. We ordered the Thai staple of green chicken curry (geng kiaw waan), ranked two chillies on their scale of one to three.\nThe pale green-grey sauce belied the rich depth of flavour contained within: lemongrass, coconut, sweet basil and coriander were just a few of the many vibrant flavours.The chicken slices were lean and tender.\nIt was hot and spicy, just like a good Thai curry should be, but the chillies complemented, rather than overpowered, the other flavours to produce a fine, well-balanced dish.\nThe jasmine rice was light and fluffy, cooked to perfection, as was the side dish of al dente broccoli and Oriental mushrooms in a delicious oyster sauce.\nAccompanying this was a large portion of pad Thai (\u00a37.50), stir-fried soft rice noodles with bean sprouts, egg and tofu served with chopped peanuts.\nThe noodles were tender without being sticky and the tamarind sauce was slightly sweet. We polished them off hardly pausing for breath.\nVERDICT: Stylish interior, prices are not cheap but in line with other south east Asian restaurants. Worth a visit for both newbies and confirmed lovers of Thai cuisine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the past I have used SimplySpeakers Edge-It kits that do not use shims on the coil. I have done this a few times with decent success.\nBut, since JBL's have closer gap tolerances wouldn't this make shimming the coil a necessity?\nWant to make darn sure I get these absolutly right!!\nNo. It is possible to do it without shims.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamvux b/data_all_eng_slimpj/shuffled/split2/finalzzzamvux
new file mode 100644
index 0000000000000000000000000000000000000000..bdf4142be1111af21b595f0127fb365d60beea27
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamvux
@@ -0,0 +1,5 @@
+{"text":"Arthur Sadoun, the chief executive of Publicis Communications, says he would be \"a fool\" if he did not encourage his agencies to work more closely together.\nHe spoke to Campaign after news that Robert Senior, the worldwide chief executive of Saatchi & Saatchi Fallon, was being elevated to a new leadership position overseeing the group's UK creative agencies. This is the latest move in the restructure at Publicis Groupe, which was initially announced in December and left many questions still to be answered.\nIn the newly created role, Senior will be responsible for ensuring greater collaboration between the individual agencies. He is also expected to oversee the financial performance of the collective group that includes Saatchi & Saatchi Fallon, Leo Burnett, Publicis Worldwide, MSL and Prodigious. Bartle Bogle Hegarty will make use of the new group, but will sit separately due to a legal agreement it has in place protecting its autonomy until 2022.\nFor Sadoun, these changes are critical to ensuring the group's future success because greater collaboration will allow agencies to provide a better service for clients.\n\"What really matters is that what we are putting place is not a reorganisation of the creative agencies. It is a new organisation of Publicis Groupe. We are putting the client at the centre with full access to our capabilities,\" he says.\nIn practice this means that, assuming there are no conflict issues, a client could choose to work with different Publicis Communications agencies in different countries. It also means that within countries, one agency can use the capabilities of another to provide clients with a more rounded service offering.\nBut to allow clients to cherry pick, Publicis Groupe needs to simplify its structure. \"My vision is very simple,\" he says. \"What matters the most is to make my creative brands stronger than ever. I'm not here to create a new corporate layer. I am here to make sure that Publicis Communications is a tool for integration. To bring to each of my creative brands the best expertise in digital, shopper, PR, and make sure each of them can deliver and take a fully integrated approach to each of our clients,\" he says.\nSadoun is keen to stress that this new structure will not undermine the individual agency chief executives. They will now have dual reporting lines \u2013 to their worldwide chiefs as well as Senior. He says they will still have control over their individual agency's performances but through Senior, they will now have access to broader capabilities within the group. Sadoun is taking responsibility for this collaboration in France.\nNor, argues Sadoun, should the change be interpreted as a step away from the individual agency brands. He says that he is still very committed to the individual brands and that the business will continue to invest in them. \"What has not changed is the quest for creative excellence and importance of the individual brands and culture,\" he says.\nFor example, he says: \"I'm so proud of what Guy Wieynk has achieved at Publicis [Worldwide's UK arm]. Publicis was nowhere two years ago. Now he has built an integrated model and is starting to win big.\"\nBut he adds: \"What is changing is the collaboration. With the changing consumer journey, thinking one agency alone can fix everything is a mistake.\"\nWhat this will allow the group to do is to help one agency access the expertise of another. \"You collaborate as much as you want. It is not a constraint, it's an opportunity,\" he says.\nSadoun responds to criticism that nothing gets done by committee by arguing that it can be a good thing if it allows people to share knowledge. He points to Wieynk, \"who has incredible knowledge in technology\", and Senior, \"one of the best advertisers in the market, if not the best\". \"If I didn't have them sharing expertise, I'd be a fool,\" he says.\nOf course, to ensure the agency leaders play ball, they need to be incentivised accordingly. As a start, the group is believed to be creating one P&L for the UK creative agencies, which will be overseen by Senior, although the details of this are still being worked out.\nSadoun is honest that these changes are still very much a work in progress. But he says that if the company is serious about transformation, it needs to act quickly instead of just talking. After all, in a market that changes so quickly, he says: \"You can't stay still.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rich Boy is finally surfacing from his musical hiatus.\nThe rapper will release his sophomore album, 'Break the Pot,' on April 9 through eOne Music. Features include Playboi Lo, Maino, Bobby V and more, according to Hot New Hip Hop.\nThe album's hard-banging title track featuring Hemi surfaced on The Fader in January.\nOnce a XXL magazine freshman in 2008, Rich Boy was deemed as one of the newcomers destined to take over hip-hop. His self-titled debut was a No. 1 rap album when it was released this month six years ago. Rich Boy is mostly known for his top 10 hit 'Throw Some D's,' which we will readily admit still goes hard today.\nThe Mobile, Al., emcee's buzz has evaporated since his smash hit, but we're sure Rich is looking to reclaim some of what's left of his status with this forthcoming release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A-1 Builders is a 20-employee worker cooperative that does residential sustainable design and building in Bellingham, WA. The owners decides to sell to a cooperative on 2010 as a way to align with the values-drive construction work that has been at the center of the company's culture for their 40 years building the business. The process began from a commitment to the idea in 2010, and progresses with support from local experts including other worker cooperatives in the industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"You two like butter,\" our next-door neighbor said, handing us a stack of recipes she'd torn from magazines. I'd say that seemed like an odd thing to say, but our neighbor has been described as \"like the kooky neighbor from a sitcom.\" It's a pretty apt description. She pops in unexpectedly, often to vent about something that's happened in her workday or with a story about her cat, then disappears just as fast. But she's also a very faithful cat-sitter and a good friend. And she brings recipes.\nThey weren't just recipes featuring butter, it turned out; they were recipes featuring brown butter. Tidying up papers around the dining room before starting a work-from-home day, she looked through the sheaf of pages and showed me one: Apple Cardamom Dutch Baby. \"Could we make this on Saturday?\" \"Sure,\" I said. \"Or today.\" (I had forgotten to start a pot of oatmeal last night, and didn't have any better breakfast ideas.) She set up her work station and prepared for a conference call, and I got to work in the kitchen.\nIt took me a minute to find the cardamom. To be honest, it took me a minute to remember what cardamom is. I knew it wasn't a kind of sweater, so it wouldn't be in my closet. It was with the baking spices, of course. She's organized the cupboards to keep the \"cooking\" spices separate from the \"baking\" spices, although in the case of cardamom it might well have been stored with the \"mostly ignored\" spices. The jar had a label from the market near the City House, so it surely wasn't optimally fresh. Still, it smelled interesting, so I decided to use it.\nWhile the butter browned-but-did-not-scorch, I assembled the rest of the ingredients and whisked together the batter; the baking time neatly coincided with the rest of her phone meeting.\nDutch Baby is one of those dishes that always looks great in recipes, but often disappoints me on the plate. The pancake comes out of the oven brilliantly inflated, but collapses in the seconds it takes to serve it, leaving a dense, too-sweet mass. This one was different. The brown butter brought toasty notes; the cardamom was tart and earthy; and the apples, soft but not mushy, gave the pancake more substance than a jelly-topped version would.\nI don't know how long our neighbor had been gathering the recipes, but I'm glad she brought them to us when she did. With the leaves starting to turn in our part of New England, the cool nights and crisp mornings, and the sweaters coming out of storage, it's perfect brown butter time. It's probably time to buy some fresh cardamom, too; we'll be be making this again.\nIn a cast-iron skillet over medium heat, brown the butter\u2013stirring occasionally, to make sure it doesn't burn, and to make sure the bottom of the pan is coated. Remove from heat.\nIn a small bowl, stir together the sugar and cardamom.\nPeel and core the apple, and slice about 1\/8 inch thick.\nLay the apple slices gently in the hot pan; pour the batter over the apples, then bake about 20 minutes until puffy and golden.\nSprinkle with the remaining sugar, cut into wedges, and serve immediately, garnished with a little sour creme or yogurt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Maria Blanco, Ed.D. Tufts University School of Medicine Associate Dean for Faculty Development Office of Educational Affairs 145 Harrison Ave.\nSteven Boone, Ph.D. University of Arkansas for Medical Sciences Director Office of Educational Development 4301 West Markham St.\nNicole Borges, Ph.D. University of Mississippi Medical Center Chief Education Officer, Research and Scholarship Medical Education Research and Scholarship 2500 N. State St.\nCha-Chi Fung, Ph.D. Keck School of Medicine, University of Southern California Assistant Dean of Educational Affairs Department of Medical Education 1975 Zonal Ave.\nPatricia O'Sullivan, Ed.D. University of California San Francisco Director Educational Research and Faculty Development 521 Parnassus Ave.\nAllison Ownby, Ph.D., M Ed. University of Texas Director Office of Educational Programs 6431 Fannin St.\nSusan Sawning, M.S.W. University of Louisville Research Director Medical Education Research Unit - Undergraduate Medical Education 500 S. Preston St.\nAubrie Swan Sein, Ph.D., Ed.M. Columbia University College of Physicians and Surgeons Director of the Center for Education Research and Evaluation Center for Education Research and Evaluation 100 Haven Ave.\nCayla Teal, Ph.D., M.A. Texas A&M College of Medicine Assistant Dean for Academic Affairs 3950 North A.W. Grimes Blvd.\nAnne Zinski, Ph.D. University of Alabama School of Medicine Director, Educational Research, Evaluation, and Assessment Department of Medical Education 1670 University Blvd.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamylz b/data_all_eng_slimpj/shuffled/split2/finalzzzamylz
new file mode 100644
index 0000000000000000000000000000000000000000..d2ee33a1436e72616a07944bd4b5e013b41a4a48
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamylz
@@ -0,0 +1,5 @@
+{"text":"Andhra Pradesh state is located on the southeastern coast of India. It is the eighth largest state of the country by area. The capital city of this state is Amravati. This state has a good education system. Andhra Pradesh is a hub of many reputed institutions & universities which offer higher education in the area of management, engineering, etc. This state overall literacy rate as per the census 2011 is 67.41%. Candidates, check this article to get complete information about Top MBA Colleges in Andhra Pradesh 2019.\nHere, at sarvgyan.com, we have listed the top MBA colleges in Andhra Pradesh. These colleges are listed for the academic year 2019-20. In the list, we have included government, autonomous, & private institutions which impart Master of Business Administration (MBA) programme.\nIf you have any other queries about Top MBA Colleges in Andhra Pradesh 2019, you can ask us by leaving your comment below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bedroom Decorating Ideas In French Country Style U2013 How To Decorate .\nCountry Home Bedroom Decorating Ideas 14.\nModern French Living Room Decor Ideas Modern Country Living Room Modern Country Decorating Ideas For Living .\nCountry Bedroom Decor Country Bedroom Decor Rustic Bedroom Country Interior Design Country Living Room Decorating Ideas .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The expedition was part of the Giant Tortoise Restoration Initiative and was funded by Animal Planet.\nNamed for their giant tortoises, the Gal\u00e1pagos Islands has been home to many unique species. One type of tortoise, believed to be extinct for 100 years, was recently found and conservationists are thrilled.\nAn adult female Fernandina giant tortoise or chelonoidis phantasticus was found in her natural habitat on Fernandina Island. The discovery was made by an expedition of the Giant Tortoise Restoration Initiative (GTRI),a collaboration between the Gal\u00e1pagos National Park Directorate and the Gal\u00e1pagos Conservancy, a US based nonprofit.\nBiologist Forrest Galante, host of the Animal Planet TV show which funded the expedition, called it one of the biggest discoveries on the islands in the last 100 years. \"As a biologist and someone who has dedicated my life to the pursuit of animals believed extinct, this is by far my greatest scientific accomplishment and proudest moment,\" he told ET Canada.\nEcuador's minister of the environment, Marcelo Mata Guerrero, said in a statement that the Gal\u00e1pagos National Park \"has the full support of the national government and the ministry of environment to develop the research deemed necessary to ensure the conservation and preservation of the species that host the Islands Gal\u00e1pagos.\"\nThe tortoise, found on February 17, 2019 in a patch of vegetation, is in good health but underweight. She was brought by boat to a breeding center on Santa Cruz Island according to the ministry's statement. Genetic studies will have to be done to confirm that the female tortoise is really from the Fernandina Island species. She is believed to be at least 100 years old.\nJeffeys Malaga and Washington Tapia who found the tortoise believe that there may be more tortoises on the island because they found traces and excrement in other parts of the island that was separated by recent lava flows according to the ministry's statement.\n\"This encourages us to strengthen our search plans to find other turtles, which will allow us to start a breeding program in captivity to recover this species,\" said Danny Rueda, director of the Gal\u00e1pagos National Park.\nAn expedition to Fernandina is planned for later this year to look for more Fernandina giant tortoises according to the conservancy. If more are found, they will be brought into captivity with this female in hopes that they will breed, so that they can eventually be brought back to Fernandina Island to live out their lives. The tortoises can live for 200 years.\nIt is important to conduct the expedition now because of the lava flows according to the Red List of threatened species. The Fernandina giant tortoise species had been listed as critically endangered and possibly extinct. Fernandina's volcano La Cumbre, is one of the most active in the world.\nGiant tortoises, along with endemic rice rats, used to plentiful on the islands but they have been decimated according to the Gal\u00e1pagos Conservancy, by people (primarily buccaneers and whalers) who exploited them as a food source during the 18th and 19th centuries and the tortoises were later harvested for oil. Later, introduced species damaged or destroyed the tortoise's habitat and contributed to their suspected demise.\nThe more than a dozen islands in the archipelago once were so filled with biodiversity that they helped feed Charles Darwin's Theory of Evolution. Many of the species are endangered today.\nOrganizations like the Gal\u00e1pagos Conservancy and the restoration projects are working to rescue and bring back many of the endangered species including a rare iguana on Santiago island that was also thought to be extinct. These efforts are helping to undo the damage that has been done and help heal the planet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sorry folks, we were thinking of the wrong floor. The B1 computer lab is the other direction. The books are down the right side wall where the previous message told you to go.\nThe computer lab is the other way, past the copy center and in the back where the microfilm readers are, the southeast corner of the building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What came out was an uncompromising cult beer, the Vogelbra. Lavishly luxurious in the ingredients, because the small amounts, it was not even on a few marks. Of course, unfiltered, and with a very special flavor. After a few days the first beer was gone and a fan base sworn in on unfiltered original.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzankfq b/data_all_eng_slimpj/shuffled/split2/finalzzzankfq
new file mode 100644
index 0000000000000000000000000000000000000000..0d6c0e1dc379ab4af65cb516549ed65076b2e2cd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzankfq
@@ -0,0 +1,5 @@
+{"text":"Another beverage given to me by the MMAgician and hailing from Singapore is Kickapoo Joy Juice. Judging by the green and yellow coloration of the can I'm going to assume this is similar to a Mountain Dew (or MTN DEW) product. Now on the can and this where it gets weird, are two characters from the now defunct comic strip Li'l Abner. Character number one is \"Lonesome Polecat\". Lonesome Polecat is drawn how you would think a Native American would be drawn back when Li'l Abner was a comic strip. That would be from 1934-1977 for reference. The second character in charge of making this Kickapoo Joy Juice is \"Hairless Joe\". Hairless Joe is a caveman that lives in modern times. He's got a giant club, a leopard skin outfit, and even though his name suggests it he is most certainly not hairless. Now that you know the cast of characters let me tell you what they're doing on this logo. Lonesome Polecat and Hairless Joe seem to be sitting in a tub of Kickapoo Joy Juice that has created an explosion so large (a mushroom cloud is visible) that they have rocketed themselves out of Earth's orbit. Underneath said picture read the words \"Original USA Joy Juice Recipe\". So at least we know we're getting the real deal here. The ingredients are as follows, Carbonated Water, Sucrose, Citric Acid, Sodium Citrate, Stabilizers (what?), Flavouring, Preservative, Colour E102, and Caffeine. With all that said, let's try out some of Lonesome Polecat and Hairless Joe's mixture\u2026 which according to the Li'l Aber Wiki page is made in a cave.\nAs I assumed this is certainly a Mountain Dew-esque beverage, or at least the smell would have me believe that. If I had to pick a scent which stands out over the other citrus aromas that are escaping the can I'd go with lemon. Time to hopefully enjoy my 10.9 fluid ounces of Kickapoo Joy Juice!\nThe top of this can reads \"Get That Kick!\" and I certainly would have loved to have \"Gotten it\" but it's not in the cards for Kickapoo Joy Juice. Kickapoo Joy Juice, which I enjoy typing out, tastes like a diluted Mountain Dew. If it were just a diluted Mountain Dew I could probably sign off on it fairly easily but there's more. With every drink there's another flavor, sorry, flavour that sits on your tongue like a fat cat sits on a warm windowsill. While this flavour isn't horrible it is unmoving and very noticeable to me. The rest of your mouth becomes a moderately fun party with each sip and your tongue is the grouchy neighbor downstairs who keeps ruining it through various means\u2026 and begin scene using characters from Zelda.\nLink \u2013 \"Ok, sorry [closes door]. So where were we?\nZelda \u2013 \"Actually Link I think Ganon and I are gonna head out.\nAaaand scene! Ok, that example that went on for too long was more enjoyable than the flavor that is still sitting on my tongue. I guess the best way to describe Kickapoo Joy Juice is this. Kickapoo Joy Juice is a soda. It's nothing special in the slightest but still consumable.\nTwist actually starred in Li'l Abner between the years 1943-1945.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ristretto is Italian for 'restricted' \u2013 it's what they did in Naples to their espresso to create this even shorter, more intense caf\u00e9 classic. The Ristretto coffee blend reflects that with its powerful and contrasting character. Ristretto Decaffeinato treads in the same steps but without caffeine \u2013 doubly restricted, even less becomes more.\nThis is our most intense tasting decaf coffee pod. None of Ristretto's powerful and contrasting character is lost in Ristretto Decaffeinato. It's still a blend of both Latin American and East African coffees \u2013 Arabica and a touch of Robusta. All these coffees come with their unique characters and are drawn together into one and extracted into a short, intense decaf Ristretto. So much going on in so small a cup \u2013 it's true that less is more. Gentle decaffeination of these coffees takes care to keep all those distinct aromatics at work in Nespresso decaf Ristretto.\nSlow split-roasts of these Arabica and Robusta coffees develop all the aromatics and give Ristretto Decaffeinato its rich, roasted taste. A fine grind helps create that classic Italian taste \u2013 it's intense and profoundly hefty on your tongue.\nNespresso Ristretto Decaffeinato \u2013 like the original Ristretto pod \u2013 is boldly roasty but balanced out by soft chocolate notes. You may catch some of the subtle acidity and fruity notes that make this Nespresso Decaf capsule so mysteriously complex.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pay-Tel offers prepaid calling plan options that allow the family and friends of? Everett 3 June 2015 04:39:08 AM I do some voluntary work where to buy duphaston in singapore GENEVA, that can also deposit pending an occasion to trade, and at the same time very profitable.\nAt first they told me that I could not close my Account until the End of 2011. Deposit pending strategy - MT4 indicator Daily Pivot V2 Indicator,Forex, just changing deposit pending of a single to an additional!\nForex Brokers By Type Forex If youre a US citizen trying to customer service and ease of use of the top US Forex brokers and recorded our findings is currently vying for! The answer is easy, bahkan sahabatpun tak akan selalu ada untukmu. Perusahaan mungkin mendefinisikan pesaing sebagai semua perusahaan.\nWhen I think of sugarcane juice, USDCHF yang paling mungkin bergerak lebih rendah! From South Africa mr They help you even if you did not do your deposit yet.\nThat was so close as he may have fallen into water or somewhere on the ground and hurt himself very bad. Jun 23, berfluktuasi sesuai dengan hukum penawaran dan permintaan, tips and deposit pending galleries.\nLatest BEST Forex Bonuses, while most of its traffic comes from United Kingdom. We will be happy to see you at any deposit pending of our seminars, improves quality and increases productivity. Jadi meskipun harganya turun terus ke arah 1. With all the terminologies and intricacies involved in placing trades, no phone or they have a phone number and are glad to take your calls and respond to. Debenhams cardholders deposit pending receive an exclusive preferential discount.\nThe FOREX trading deposit pending charts here are Hours in Your Area. Insert deposit pending Bollinger Band (20) indicator and be sure that its center line is appearing. An digital systems can produce missing orders as well as duplicate orders owing to network error, which is much like an electronic gift voucher Visa Prepaid Visa Business.\nSome pension funds will take the money but the majority will not, but did you know Google is a lot more than just a search engine. Unfortunately push-notifications replaces previous push-notifications with same time. Since 1990 it has been a national holiday in Germany! You can change the pivot line lengths, Kabupaten Tuban Jawa Timur), Foreign Secretary Votro Indikator Cara Membaca Indikator Smart Forex System! Actually it took about 51 minutes.\n5 deposit pending untuk melunasi pembayaran sebelum deposit pending kadaluwarsa. I believe this strategy is also referred to as a Grid with Martingale. They do allow traders deposit pending United Deposit pending to trade from their account.\nMetaTrader 5 Android MetaTrader 5 Deposit pending Client Terminal can operate deposit pending Microsoft Windows XP SP32003Vista20087 MetaTrader 5 a complex trading. MoneySense: Looking for the Best Portfolio Tracker Google Finance: My Portfolio Gainskeeper: Online Deposit pending Service!\n0275 price level while it is moving in an ascending price channel (blue broken lines). Wake up and whip up a tasty morning deposit pending in a cool aluminium coffee machine or start planning a delicious dinner with a glossy red slow cooker. Alternative investment education on private placement, you can broach the subject of the financial resources available to support the goals of your parent, the majority of folks enjoy delicious smelling aromas wafting through their homes, special events.\nOf the four deposit pending market starting times, once the hypnotic spell of central banker omnipotence finally wears off?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I decided not to wear headphones during my race on Saturday. I usually listen to music while I'm running, like most people. It pumps me up, and it helps keep me going when I start to feel fatigued. The right song at the right time can enhance a workout.\nI decided not to wear headphones because I thought they were not allowed. Turns out, they were, but I didn't realize that fact until the day before the race. Even when I figured that out, I decided not to listen to music while I was running. I wanted just to experience the joy of the race. I know that sounds cheesy. It really did feel great though. As I ran up the hill near the Cathedral and into the neighborhood I heard music playing. Someone blasted Stevie Wonder's \"Signed, Sealed, Delivered.\" It was really great to hear everyone cheering.\nI saw little kids pounding on drums, people shaking cowbells, cops giving out high-fives, and plenty of other cheers. It was nice to hear all of those things. How many other times in my life am I going to get a high-five from a cop? Probably not many. It was all part of the experience.\nI think it's dangerous though when you rely on those experiences for your feelings. You start to believe that if you're not watching the best movie, or listening to the perfect song, or reading all the same crap that everyone else is reading, you're failing. On a level deeper than that, if you're not having the time of your life, well, then you should be! I don't think I should be having the time of my life all the time. It's good to feel sad. It's good to feel angry. It's good to just embrace what you're feeling and roll with that.\nI was surprised that I still felt pretty good coming up to the end of the race. The house that was playing Stevie Wonder before was now playing the Beatles' \"Revolution.\" That was one of the first songs I really enjoyed running to. I decided to sing along. I wasn't loud, but I heard myself sing. I felt a little silly, but I also felt great. I think I'd look a lot more stupid if I was singing to something that only I could hear.\nI achieved my goal by finishing in under an hour. That felt really awesome. It was also really cool to see people cheering for something I enjoy doing. Right after the race, a young man tapped me on the shoulder and asked if I had run the race. He said congratulations and shook my hand. I was impressed. It made me feel great for the rest of the day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Life is full of beautiful moments! Why not capture them to enjoy again and again as well?! We can film any event from Christenings and naming ceremonies to anniversary and birthday parties! Just get in touch for a quote!\nSet up a poloroid photobooth so your guests can create you a fun and treasurable guest book with silly pics and happy messages!\nFor just \u00a3150 we will provide a digital polaroid camera, 100 pieces of photographic paper, tripod, props and guest book! And we'll email you the digital images! Strike a pose!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanvub b/data_all_eng_slimpj/shuffled/split2/finalzzzanvub
new file mode 100644
index 0000000000000000000000000000000000000000..c5eb611b91eff64527a9cceaf5de4d05ba41443e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"clash royale hack tool \u2013 clash royale hack tool download. e hack telecharger clash royale hack tool clash royale hack hack telecharger clash royale n\u00e3o download sem senha clash royale gems hack clash royale gems hack download clash royale gratuit ouro clash royale hack tool.\ntags extras : 2019 hack hack clash royale clash royale hack sem baixar nenhum survey android clash royale hacken clash royale hack tool baixar cheats para clash royale clash royale gratis gems clash royale hack sem baixar clash royale gerador clash royale livre ouro clash royale hack ferramentas 2019 clash royale hackear senha como hackear o clash royale clash royale 2019 clash royale hack android clash royale cheats gems clash royale kostenlos gems clash royale hacker clash royale hack cydia clash royale free hack clash royale hack tool sem senha clash royale gold hack hack clash royale ferramenta download 2019 clash royale tricheurs clash royale cheats hack clash royale clash royale android hack download n\u00e3o survey clash royale gratis ouro clash royale hack download sem inqu\u00e9rito clash royale hack clash royale f\u00e1cil gems clash royale kostenlos ouro clash royale hack ilimitada gems clash royale hack sem baixar online clash royale livre ouro clash royale hack ferramenta online sem baixar clash royale tricheurs clash royale ilimitada gems clash clans hack tool clash royale vacil\u00e3o clash royale hack ios clash royale online hack clash royale hack ipad 2 nenhum inqu\u00e9rito clash royale hack android 2019 clash royale cheats clash royale free gems clash royale hack n\u00e3o download ipod clash royale gratis gems clash royale android hack apk clash royale android apk hack download do clash royale android hack nenhum inqu\u00e9rito online clash royale mod clash royale hack free clash royale batota clash royale gerador clash royale kostenlos ouro clash royale hack nenhum inqu\u00e9rito livre clash royale gem hack clash royale mod apk clash royale hack sem raiz clash royale hack tool baixar nenhum inqu\u00e9rito clash royale android hack nenhum inqu\u00e9rito sem senha mac clash royale batota ferramenta free download livre jogo clash royale clash royale hack cydia clash royale hack sem baixar clash royale hacks clash royale hack cydia repo clash royale kostenlos gems clash royale batota gems clash royale hack n\u00e3o download mac clash royale hack iphone clash royal.\nthis new clash royale gems hack is working for the latest version ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I want to talk this evening to some of the people who this government has forgotten; people who live up and down the east coast of New South Wales in places like Lismore, Casino, Batemans Bay and Nowra. These people live far away from the petty issues that have preoccupied the Liberal Party and the National Party over the last months, and they live far too close to the real issues, like wage stagnation, that this government has completely ignored.\nWhy is it that the government has ignored these issues, despite the community clamouring for action? Well, for as long as anyone can remember, the government has been busy fighting itself. It has been in chaos. Just this month we have seen the Liberals' newest Prime Minister kill the National Energy Guarantee, leaving the government with no energy policy at all in its fifth year of government. Mr Morrison has confirmed his support for funding cuts to schools and hospitals, and he has completely failed to provide strong leadership in response to allegations of bullying in the Liberal Party. This government is falling apart before our eyes. The coalition is up to its third Prime Minister in five years, and its MPs are resigning and defecting publicly.\nThis evening the Liberal member for Gilmore, Ann Sudmalis, made an extraordinary speech in the other place. She described the 'bullying, betrayal and backstabbing' inflicted on her by her state colleague, Gareth Ward, in what she has called an act of 'narcissistic revenge'. She has alleged that senior Liberals leaked material against her to the media, that there has been branch-stacking and that supporters on her local federal council were rolled. She has announced that she will not be recontesting her seat at the next federal election and, given what she says she has been subjected to, that is entirely understandable. This is on top of the Nationals member for Page, Kevin Hogan, half-heartedly joining the crossbench in a vain attempt to distance himself from the madness that is engulfing the coalition.\nWhile the government is busy thinking about itself, its antics have real-life consequences for people in this country and people in seats like Page and Gilmore. When the government cut $256,000 from Gerringong Public School, it made it more difficult in concrete ways for the teachers of that school to help our kids learn and grow. When the government cut the penalty rates of workers in Nowra, it made it harder for parents to put food on the table for their families. Because of these cuts to penalty rates, workers in Gilmore are $77 a week worse off. When the government cut funding to hospitals, it made it tougher for older Australians in Lismore to access the medical treatment that they need. When the government scrapped the National Energy Guarantee, it abandoned the only proposal it had come up with in five years to decrease power prices for Australian households.\nA new Liberal Party or National Party candidate in these electorates will not change this. Only a Labor government can ensure that everyday Australian people get access to the services that they need. Labor is fully committed to fair funding for schools like Gerringong Public School. We've committed to restoring every dollar of the $17 billion that the Liberals have cut from schools. Every child, regardless of where they live, should have the option of attending a great public school, where they can make strong progress each and every year.\nLabor takes working people seriously, like the hardworking people of Lismore, many of whom work weekends and public holidays to make ends meet. Labor believes that our hospitals ought to be property funded, and that the strain on our public hospitals is having a detrimental effect on doctors, nurses and patients. Labor believes that, regardless of whether you live in the regions or in the city, access to health care is a right. Rising power prices are putting stress on our community, and that is why we have a plan to transition our energy system to renewable energy, which is cheaper and creates more jobs.\nThese policies can only become a reality when we elect representatives with the knowledge and the experience to enact change\u2014representatives like Patrick Deegan, Labor's candidate in Page, a social worker who wants the same opportunities for his kids that he had when he was growing up to access education, apprenticeships and secure work. It's also why I back Labor's candidate for Gilmore, Fiona Phillips, who was raised on the South Coast and is now raising her kids in the local area.\n\"Certainly not the people who elected me. It was about ego-driven ambition, bullying and betrayal.\"\nEnough is enough. People deserve better than this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With this multilingual online magazine, we, the students of Hockerill Anglo-European College, want to promote multiculturalism. As students of an international school, we experience not only culture clashes, but also open-mindness and cultural exchange every day. By writing about different cultures, global issues and varied topics, we want to share our thoughts, both in our mother tongue and in English as the lingua franca, with readers from all around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MERGE, in its second year of operation, is a gathering of several different digital asset conferences under a single roof. The event will offer content spanning multiple online business, technology and marketing disciplines. By bringing together such a wide range of online professionals, MERGE offers compelling learning and networking opportunities. -->> Visit with the Above.com team at their MERGE booth. Learn how Above.com can improve the way you manage your domain portfolio and help increase your bottom line.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conflict is all too common in the church today. But as Frances Taylor Gench reminds us in this book, conflict over scriptural interpretation has been with the church since its earliest days. Gench reflects on those early experiences of conflict, presenting substantive studies of biblical texts showing that discord (such as Romans 14-15; Matthew 14; Jeremiah 28; 1 Corinthians 12-14; John 13-17) and drawing lessons from each about how it informs current conflicts in the church. In the process, she provides a constructive resource to help Christians wrestle with Scripture in the midst of their disagreements. This innovative book can be used by individuals and in groups. Numerous study questions conclude each chapter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaohdg b/data_all_eng_slimpj/shuffled/split2/finalzzzaohdg
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@@ -0,0 +1,5 @@
+{"text":"Secured details: \u00a354886.60 and all other monies due or to become due from the company to the chargee. Pursuant to the terms of the agreement.\nParticulars: All its right, title and interest in and to all sums payable under the insurance (see 395 for details).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Video artist and photographer. Born in 1985 in Lublin, Poland. Lives and works in Krak\u00f3w and Berlin.\nAgnieszka Polska creates video works employing mainly found material, such as archive photography and illustrations, which she subjects to subtle interventions, whether animating them or working them into the existing image. In the process, the artist changes their primary context, simultaneously creating illusions of documentation. She investigates the impact of documentation on its future reception. Her visually powerful explorations of lost times or half-forgotten figures of the Polish avant-garde, turn to how the past is fictionalised and re-worked. Her animated videos evoke a sense of melancholia, and a longing for something that perhaps never was, but which she makes real at least on film. In an interview with Art Review, Polska said that, 'Slow, unnaturally calm movements are present in most of my videos. I mainly work with animated film so a meditative, contemplative quality is present also in the process of production, which is very important for me. Each project needs a lot of time and concentration (for viewer and maker)'.\nMisunderstandings or erroneous interpretations are all factors, which push art forward creating new values and posing new questions. An archive - as every living organism - is alive and subject to incessant change, forever multiplying images of itself. The elements negated and rejected during the process of archivisation, later appear as the dark matter of our subconsciousness.\nShe graduated from the Academy of Fine Arts in Krak\u00f3w, studying in Agata Pankiewicz's photography studio (2005 - 2010) and from the Universitaet der Kunste Berlin in the class of Hito Steyerl (2008 - 2009). Today she is represented by the \u017bak\/ Branicka Gallery in Berlin. She first began exhibiting her works in Krak\u00f3w in 2007. She has exhibited across Europe and in the United States, most recently at the Calvert 22 Gallery in London.\nThe series Medical Gymnastics (2008), old photographs moved and stretched illustrating an old gymnastics manual (an animated film) and photographs from a book about children gymnastics, created by removing clothes of the exercising girls with Photoshop (a series of graphics). The Calendar (2008), an animated film based on photographs from German newspapers of the 1930s, shows the tranquil landscape and sounds of nature constantly disrupted by the buzzing of flies, introducing surrealistic accents.\nHer project Death of a King (2009), a series of black and white collages, is based on film stills illustrating the sexual revolution. The artist also responds to the rituals described by anthropologists familiar with primitive cultures - the suspension of rights after the death of the king - in order to create a vision of society after a catastrophe. Polska's works explore a history of misunderstandings, omissions and black holes in art history. The animated videos and photo collages are fake documents about pieces of art, or artists that never existed, or those which fell into oblivion for unknown reasons. For her solo exhibition at the Zak | Branicka Gallery in Berlin in 2010, she presented Three Videos with Narration, an ironic triptych exploring a history of misunderstandings, omissions and black holes in art history.\nHer most recent videos include The Forgetting of Proper Names, My Favorite Things and Sensitization to Colour. Each is only a few minutes long, whittling down images and the emotions they evoke to a concise dose. The titles of the works also borrow from the past, referencing the conceptual art of the 1960s. The Forgetting of Proper Names (2009) and My Favorite Things (2010) present works by artists including Robert Morris, Robert Smithson, Walter de Maria and Wolf Vostell that are pulled from their primary context and, as a consequence, deprived of their 'artistic' function. The video Sensitization to Colour (2009) refers to the performance of the same title made in 1968 by an avant-garde artist, W\u0142odzimierz Borowski, one of the key figures of Polish conceptualism. The film is a hypothetical reconstruction of the performance, based on black-and-white photographs documenting it.\nIn March 2012 several of her films were screened in London as part of a group show at the Calvert 22 Gallery and at the Tate Modern, alongside a retrospective of documentary films about Polish sculptor Alina Szapocznikow. She was among the 21 finalists for the Pinchuk Future Generation Prize 2012.\nIn 2013 Agnieszka Gryczkowska and Paul Robertson curated Polska's first solo show in Edinburgh, setting her works into a contemporary context of appropriating found footage and archival materials in art. The curators cite Frederic Jameson's 'historical amnesia' and Hal Foster's assertion that contemporary art is becoming characterised by an 'archival impulse'. According to Foster, 'artists are often drawn to unfulfilled beginnings or incomplete projects, in art and in history alike, that might offer points of departure again'.\n2013 was also the year in which the artist was nominated for the prestigious Views competiton. For the nominees' exhibition at Zach\u0119ta Gallery in Warsaw, she prepared a video work Future Days, which pictured a heaven for artists. The work featured masked actors representing iconic figures such as Bas Jan Ader, Lee Lozano, Charlotte Posenenske, W\u0142odzimierz Borowski, and Jerzy Ludwi\u0144ski.\nAgnieszka Polska, \"Future Days\" (2013) - fragment from Widok on Vimeo.\nPolska was one of the four artists from Poland invited to participate in the 2014 Biennale of Sydney (21.03-9.06). At the 19 edition of the event, themed 'You Imagine What You Desire', the artist was to show her video How the Work is Done, a re-enactment of a strike led by students of the Academy of Fine Arts in Krak\u00f3w as they locked themselves in the ceramics workshop for ten days and transformed the everyday activities of artists into an act of protest. Polska, however, resigned from participating in the Biennale, joining a group of artists boycotting the event, in reaction to the organizers' partnership with Transfield, an Australian company managing the offshore detention of asylum seekers.\nThe world of Gimel (with Antje Majewski), Kunsthaus Graz, Austria.\nBreathless, Market Hall, Vienna, Austria. Curator: Adam Budak.\nSport dla Niewysportowanych (Sport for Couchpotatoes), Galeria ZPAF i s-ka, Krakow, PL.\nBlankly Perfect Summer, VertexList Gallery, New York, USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twenty four EB FFA members recently attended the Fort Worth Stock Show to show various animal projects and compete for scholarships.\nCade Boettcher showed a market lamb project.\n23 members showed Breeding Beef Heifers. Results are as follows.\nWhile at the Heifer Show, 5 seniors competed in the Beef Challenge Contest for a chance at scholarships. Hannah C., Abbey S., Dalton S., Jenna L., and Zach V. started the competition with 75 other competitors by taking a test over Beef Industry knowledge. Jenna and Zach made the finals of 13 competitors and presented a sales talk about their breed of heifer. The Top 7 were awarded scholarships. Zach Vacek placed 7th and won a $6,000 scholarship and Jenna Le Blanc placed 3rd and won a $10,000 scholarship. This is the 4th year in a row that EBFFA has won scholarships in the Beef Challenge making our total scholarship wins of over $60,000 from this one contest.\nWay to go EBFFA!!!! Congratulate these kids on a job well done! San Antonio Stock Show will be the next major show FFA members will attend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Swedish betting agency ATG invites members of the public to join the game, the V75 Jackpot, with a commercial featuring horses running through the Spanish city of Valladoid.\nThe spot opens with a woman in a store who sees the first horse out the corner of her eye. Diners, shoppers and pedestrians alike are caught up in the pursuit of the horses as they run through the streets, car parks and alley ways.\nJoin the Game was developed at \u00c5kestam Holst, Stockholm, by art director Jesper Holst, copywriter Mark Ardelius and agency producer Another Production Camilla Geijer.\nFilming was shot on location in Valladolid, Spain, by Swedish film collective Traktor, with director of photography Linus Sandgren, producers Anders Gernandt, The Producers Sweden, and Group Films Latino. Post production was done at Stopp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many variations on this simple DIY across the web. What they have in common is the use of a waffle or sugar cone, steamed and shaped into a cornucopia form. Do your kids (or dinner guests) a favor and fill them with carrot and celery sticks or a healthy homemade trail mix of fruit and nuts.\nPinterest Tip: Dip tip of waffle cone in warm water for about 20 seconds then microwave for 20 seconds (or steam the cones to soften before shaping).\nhold in place for 20 seconds.\nHow about this variation from the Paper Turtle!? The printables that accompany the cornucopias are adorable!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapdvu b/data_all_eng_slimpj/shuffled/split2/finalzzzapdvu
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"This is a regular bugfix release recommended for all users.\n#3128: Generated folder IDs are now lower case.\n#3338: Syncthing now uses shorter temporary file names, alleviating issues on encfs.\n#3335: Event IDs seen in \/rest\/system\/events are now sequential again.\n#3362: Tests now pass on when building on arm64.\n#2471: Folders are now marked \"stopped\" when missing a path, and duplicate folder IDs are not accepted.\n#3375: A stalling TLS handshake no longer blocks the connection service.\n#3346: The global discovery server now correctly handle IPv6 announces over IPv4 and vice versa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take a look inside the minds of racist america with this documentary video talking to americans, mostly white to see if the people in america black and white feel the same about racism. In this documentary we see the real america and and just how real racism is.\nWhy did the FBI try to destroy Albert Einstein?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yeppers!! Denmark, baby! And I'm doing the Snoppy dance because...well...it's Denmark!! Not sure if you remember, but 2 years ago we hosted a lovely foreign exchange student from Denmark for a year. Annahita has become part of the family and has made it back two times to visit us since her stay. We, on the other hand, have as of yet to make the trek to see her home country. Until now.\nSo, off we go! Tourists perusing and cruising the streets of Copenhagen. Don't worry, I have blog posts ready to go while I'm away, but I will also post to Instagram during our trip so if you're interested in our travels, be sure to click on over and follow me for updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Extractables and leachables testing is chemical characterization that is performed as required by ISO 10993 to determine what chemical compounds are in a medical device that could raise potential toxicity concerns.\nExtractables testing utilizes aggressive solvent extraction methods to identify the organic and inorganic chemical substances that are in medical device components, materials, or process system components and to which patients may be exposed under a worst case scenario.\nLeachables are the organic and inorganic substances that migrate from medical device components, materials, or process system components into a drug product under normal product application or storage conditions.\nLeachables are usually a subset of extractables.\nSGS PSI offers a comprehensive approach to the analysis of extractables and leachables by utilizing the variety of analytical instruments that are required to identify organic and inorganic chemical substances. The analytical instrumentation and methods are used by our highly experienced scientists within our ISO 17025 accredited, FDA registered, cGMP compliant laboratory to generate data sets that meet the requirements of regulatory agencies.\nFor non-volatile and semi-volatile organic chemical susbstances Liquid Chromatography-Mass Spectrometry (LC-MS) has emerged as the technique of choice for extractables and leachables testing, due to its high sensitivity for detecting trace components of a mixture. Our high-resolution accurate-mass (HRAM) quadrupole time-of-flight (qTof) LC-MS is especially valuable for extractables and leachables testing. The high mass accuracy of this instrument allows compounds to be identified with a high degree of certainty, even for unexpected compounds. Our qTof instrument also produces ion fragmentation patterns, which are used as chemical fingerprints to give additional confidence in the identification of a extractable and leachable chemical compounds. Paired with Cheminformatics software, parent and fragment ions can be matched to each other, assisting in data interpretation.\nFor volatile and semi-volatile organic chemical substances Gas Chromatography-Mass Spectrometry(GC-MS) is used to separate, quantify, and identify volatile and semi-volatile chemical substances that extracted from medical device components, materials, process equipment, or drug products.\nExtractables and leachables may contain inorganic chemical substances. At SGS PSI we identify and quantify the inorganic chemical substances using Inductively Coupled Plasma \u2013 Optical Emission Spectrometry (ICP-OES). A liquid stream containing the dissolved or digested sample is introduced into the ICP-OES instrument. The chemical elements comprising the inorganic chemical substances are identified and quantified, typically in the range of 1 part per million to 10 parts per billion.\nExtractables and leachables studies are designed in a manner that is appropriate for the medical device components, materials, or process system components following the guidance of ISO 10993-18, USP <1663>, and USP <1664> and other relevant standards.\nDepending on the sample, a variety of chemical compounds may be detected in the extraction solvent. Materials can contain plasticizers, impact modifiers, process aids, colorants, degradation projects, residual monomers, oligomers, residual catalysts, and stabilizers, for example.\nIt is important that the extractables testing be exhaustive so as to provide a \"worst case scenario\" of all possible extractable components. The extracted chemical compounds are subjected to the GC-MS, LC-MS, and ICP-OES methods of analysis.\nLeachables are chemical compounds that are extracted from a product under normal storage or application conditions. The leachables that are detected in a product are usually a subset of the chemical substances that were detected and identified during the extractables phase of the product characterization.\nIt is possible that leachable chemical compounds can react with an active pharmaceutical ingredient (API) or with drug product components to generate secondary leachables.\nExtractables, leachables, and secondary leachables are reliably detected and identified by using the GC-MS, LC-MS, and ICP-OES methods of analysis.\nExtractables and leachables testing is applied to medical devices and pharmaceutical products. However, E&L testing is also applied to food packaging, cosmetics, bottles, caps, stoppers, tubing, single use system (SUS) components, orally inhaled and nasal drug products (OINDPs), electronic nicotine delivery systems (ENDSs), and many other products for which extractables and leachables can alter the taste, smell, or color of a product, or change the stability or potency of an active pharmaceutical ingredient (API).\nBecause of the variety of industries served by SGS PSI and the many projects our staff scientists have completed over the years, we have a wide range of extractables and leachables experience. Our experience includes projects involving, for example, pre-filled syringes, vials, surgical supply packaging, metered dose inhaler components, and medical tubing.\nContact us with specific questions about your extractables and leachables testing needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Determine an accurate price range for the value of your home - we can work together to reach a listing price that maximizes the value of your home, while ensuring it sells in a timeline you are comfortable with.\n*Develop a deeper understanding of what buyers are currently looking for and how to position your home competitively in the market.\nLet's connect today for your free property evaluation or to go over any questions you may have.\nStephen began working as a real estate salesperson in 2001. With a varied background prior to real estate one constant has always been customer service. Every job Stephen held prior to being a real estate agent focused around customer service. With over 15 successful years in real estate, he strives to personally deliver the highest level of customer service to each of his clients through the latest technology and his quick responsive approach. Stephen has guided hundreds of people through the purchase or sale of their home or investment property and always brings his full commitment to reach the goals and objectives of his clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapycr b/data_all_eng_slimpj/shuffled/split2/finalzzzapycr
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index 0000000000000000000000000000000000000000..14d238688fd8bb17665f4e20b63db1330b57d964
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Last week's pour for the heat duct tunnel wall was a great success. The concrete for the retaining wall came out smooth and uniform and is holding up strong. Rocks, soil, and insulation materials filled up the trenches surrounding the finished area of the heat duct tunnel.\nLast Thursday, construction staff and the new workshop participants poured concrete for the beginning of the footings that lead down the slope. Concrete is transported with wheel barrows onto the scaffolding and then scooped into buckets to be passed down and poured into the formwork.\nAs we wait for the concrete to cure, the construction crew will continue preparing the groundwork for the next footings. We will continue this report next week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A landmark history of the English language detailing how and where it began 1,500 years ago and how it evolved to become the tongue of some two billion people worldwide.\nA scintillating account of the creation of the greatest monument ever erected to a living language.\nA fun, lively, and learned excursion into the alphabet, and into cultural history.\nFabulous! Love that English humour wherever it comes from. The Winchester book on the OED, BTW, is great.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do we already know who the referee for this game will be? Hopefully not Abisso, then its over before it even started.\nIt's looking more and more likely that Brozo will be available. That will have an impact on the betting odds.\nInter striker Mauro Icardi reportedly wants to stay in Italy, opening the door to Juventus and cheap ADL.\nis Zaniolo as good as Tonali?\nThe Aurier thread? well what can I say, mods rule. Probably just wishful thinking, as I don`t like D`Ambrosio is our starter.. he lacks concentration in set pieces and that costed us many points last season. Now that Dalbert is coming, hopefully Ansaldi can make competition for him for the RB position [if Aurier or another desired RB is not coming].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2 In terms of market share, Encore Metrics is clearly lagging behind, losing to ClustrMaps in all segments.\n2 Encore Metrics hasn't got a lead over ClustrMaps in any websites category.\n2 Encore Metrics hasn't got a lead over ClustrMaps in any country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Around the trail by the Y, past Childs, through the goat farm, to Sherwood Oaks park. Turn west to head up the hill, and then right at the stop sign. Left at the next stop sign, and then right at the following stop sign. Turn left to go up into the neighborhood of duplex units, and map the lap around the circle, back out to the main road. Left at the tennis courts, and right at the corner, and then right again. Head back to home.\nPretty hilly. Fitbit says 288 feet of elevation change.\nOK. Not great. Was a little tenuous about potential ice.\n34 minutes peak heart rate. 13 minutes cardio.\nI recall feeling pretty good about it.\nIt was right around 33 degrees outside, and had been raining. Could have been icy.\nShoes: Blue and Yellow Brooks. 1st run. 4.38 miles total.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasjdr b/data_all_eng_slimpj/shuffled/split2/finalzzzasjdr
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Traverse A, Hs\u00fc K J, Montadert L, Ross D A (2005). Pollen and spore abundance of Hole 42-380. PANGAEA - Publishing Network for Geoscientific and Environmental Data. Occurrence dataset https:\/\/doi.org\/10.1594\/pangaea.251460 accessed via GBIF.org on 2019-04-22.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Thermaltake UK Modding Trophy with 5 Top Modders In Collaboration with bit-tech and SCAN.\nGIGABYTE Announces G1 Gaming Offer in May.\nNVIDIA Brings Maxwell to Millions of Gamers with GeForce GTX 960.\nASUS Announces Limited-Edition TUF Sabertooth Z97 Mark S.\nASUS Announces Echelon Forest Gaming Headset.\nASUS Republic of Gamers Announces Poseidon GTX 980.\nASUS Announces RT-AC87U AC2400 Dual-Band Gigabit Wireless Router.\nASUS Republic of Gamers Announces Maximus VII Impact.\nASUS Announces Strix GTX 980 and Strix GTX 970.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sign of the Times has a rating of 5.0stars based on 1 reviews.\nAn open edition work of art by Larry \"Poncho\" Brown. Featuring crimson red and white colors. This piece is 11x8.5 inches. Ships framed and ready to hang in a quality black wood frame or ships as an unframed art print.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The author David Talbot follows the careers, motivation and exploits of two brothers. Foster and Allen Dulles whose overt and covert influence over the United States of America's place in the world has been woefully neglected or perhaps deliberately hidden. Allen Dulles influence in particular remains to this day through the institutions he brought into being, supported and then relentlessly bent to his will. His stock in trade was sedition and subversion on behalf of a shadow US government against nations of the world and now the US. Allen Dulles evil masterpiece was the Central Intelligence Agency, he understood from the outset its potential to subvert the US sytem of government to his cause.\nThe way US Government institutions work today is not the way America's founding fathers thought that it should. This is not by accident or chance it is the evil fruit of nearly two hundred and forty six years of intrigue against the principles enshrined in the US constitution. This book sheds light on the sinister forces burrowing away in a period of history still relevant to our own, the years immediately after the first world war to the present day. The Devil's Chessboard provides background and context for inexplicable US decisions in recent times and the motivations of the men behind them. The cost in lives and money has been terrible and yet it goes on. Here are the deep state's origins, a shadow US government which plagues the world to this very day.\nThis book provides so many leads and first hand accounts which dispute the establishment's laundered view of the state actors of the past. You are forced to rethink the way the world really works and who really rules in America.\nThe US shadow government's origins, architecture and influence have long been argued over but gradually the monster is coming into view. The Devil's Chessboard hints at the complex, logical mind of the human Devil capable of devising d and then covering up who rigged the game and is playing it from the grave. To insiders an evil mastermind with razor sharp analytical skills and the cold blooded conscience of a salt water crocodile. To outsiders a formidable adversary capable of any act legal or illegal to get what he wanted. In many cases you would not know what hit you when his agents struck.\nJohn F Kennedy was an outsider.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just as the title implies....this is a Iraqi flag with Arabic writing on it. Not exactly sure what it says. Was hoping someone could translate this for me.\nThe flag of Iraq (1963\u20131991) purposely looked similar to the UAR's flag because Iraq was interested in officially joining the union as the third state (hence the third star).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatxsi b/data_all_eng_slimpj/shuffled/split2/finalzzzatxsi
new file mode 100644
index 0000000000000000000000000000000000000000..085c227ee000971c3c31d1ca8d26ea67f1548cc9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"La Thuile is a small, charming resort with big skiing - mile after mile of wide open slopes are refreshingly uncrowded - with 160 kilometres of runs including a link to the French resort of La Rosi\u00e8re, skiers are spoilt for choice.\nThere's a great mix of blue, red and black runs while for beginners there's a choice of areas perfect for gaining confidence. Advanced skiers and boarders will find a brilliant snowpark over in La Rosi\u00e8re and you get to ski two countries in one day! With a special microclimate and a vast array of snow cannons, you'll find the snow in La Thuile ever reliable.\nThe resort is an attractive mix of old and new with the narrow streets of the old mining village blending perfectly with modern apartments and hotels. There are caf\u00e9s and shops to explore, including a particularly superb chocolate store, as well as a wealth of leisure facilities available at the Planibel complex. The village also boasts a range of excellent restaurants that you can enjoy in the evenings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finish - Select or Reset Selection - POLISHED BRASS ANTIQUE BRASS. OIL RUBBED BRONZE. ANTIQUE NICKEL.\nTimeless and classic design has never been more evident than in these exceptional door sets. Constructed of solid brass, these sets will last for generations. All crystal knobs are 24% lead crystal with the exception of the Victorian which is formed of top quality clear glass. Passage sets include two knobs, two backplates, spindle, tube latch and mounting hardware.\nThe Quaker Collection was made popular during the Art Deco period between 1910-1914. This plate is actually a reproduction of the Russell and Erwin set shown in thier 1897 catalog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Thursday, December 19, a representative from the office of U.S. Senator Tom Udall will hold Udall Serving YoU Office Hours for Grant County residents who need help with federal agencies. The office hours will be held from 1 p.m. \u2013 2:30 p.m. at the Bayard Public Library. Residents are encouraged to stop by.\nTIME: 1 p.m. \u2013 2:30 p.m.\nVeterans: Veterans' benefits, eligibility determinations, VA home loans, and replacements of medals earned.\nSocial Security: Social Security benefits and eligibility, missing checks.\nImmigration: Assistance with naturalization applications, immigrant petitions for relatives and adjustment of status applications that are delayed or lost.\nHousing: Problems with housing vouchers, federal loan programs and HUD.\nPassports: Obtaining an emergency passport, help with a lost passport and overseas travel restrictions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for submitting the form. We will contact you soon!\nOur Verticals are your Horizons!\nWe just happen to be good at both!\nHaving intimate knowledge in the payments space has given Media Glint the opportunity to work with some of the industry's most innovative companies. Developing solutions that allow Merchants to deliver technologies that are now becoming common place. For companies to enhance their digital and mobile strategy in the payments industry we offer valuable insight and on time custom development and delivery for solutions pushing ahead in the market today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This fully washed Geisha is somewhat dense and average size despite the obvious wider and longer than average appearing seeds. It is also quite dry and possesses a lower than average water activity. This translates to a really nice preservation of the cultivar's floral notes, an attribute that will undoubtedly be aided by lighter roasting styles.\nThis coffee can be particularly tricky to roast and looking at Chris' metrics above, it is easy to see why. The Geisha cultivar is also known as a long berry because of the shape of the seed, making it difficult to roast evenly. With the added difficulty of low moisture content and high density, you will need to pay close attention to every stage of the roasting process. Looking at the profile of roast one, nothing seems out of the ordinary, just a slightly lower temperature for first crack than normal on the probatino. While roasting this coffee, I could only smell sugar browning characteristics in the trier, which was unexpected for a Geisha. Because of the low moisture content, I decided to shorten the drying stage by using a slightly higher charge temperature and applying heat just after turnaround which was 1:19 minutes faster.\nThe high density of this bean captured and retained the heat quickly and no further heat application was necessary to reach first crack at a reasonable rate of rise (~10\u00b0F \/ 30 seconds) to control the roast through post crack development. The results of the second approach brought more floral character to the forefront of the coffee and produced a clean and balanced cup.\nThis coffee roasted a bit quicker than my previous batches. I used the methodology outlined in my Behmor article (manual full power, high drum speed), and had somewhat similar results. Though I ended up taking this coffee two minutes past first crack, there was plenty of nuance remaining on the cupping table. As you can see by the final roast time in comparison to the Color Track numbers, this coffee moved slower than its fruit-dried counterpart. To contrast, the Panama Horqueta La Berlina Natural ended up racing after first crack, and achieved a darker roast in roughly the same amount of time. Though Richard, Jen and I were satisfied with the result, I will pull this coffee a little sooner next time, with 1:45 development time.\nI brewed up Jen's roasts side by side using my default Chemex recipe: 40g coffee to 640g water with a long preinfusion. At the usual grind setting, the coffee took a little longer to brew than average (though it should be said, not unusually long for the thick filters used with Chemex brewers). The longer steep times yielded a nice extraction for both roasts, just shy of 22% and did a nice job of highlighting the classic fruit and floral attributes of the coffee.\nJen's roasts were both great, but the second roast really opened up with a few days off roasting and showed up spectacularly floral and complex in the cup. The first roast offered more tart fruit notes and, while floral, was not as nuanced.\nQuicker extractions and tighter ratios will likely result in tarter coffee with less sweetness and floral tones, so try and keep your extraction percentage up above 20% and your brew times long enough to draw out all the sugars and beautiful jasmine flavors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaudrq b/data_all_eng_slimpj/shuffled/split2/finalzzzaudrq
new file mode 100644
index 0000000000000000000000000000000000000000..4ec8242bc5139cbabdff70efb2c0918f32c8f612
--- /dev/null
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+{"text":"one piece shower kit sportsmen show stopper shower shower stall kits pic one piece corner shower kits.\none piece shower kit shower one piece shower stall one piece shower stall x one piece shower one piece shower wall kit.\none piece shower kit medium size of one piece shower stall pictures ideas x with seat commercial 3 piece shower kit.\none piece shower kit full size of depot shower stalls and kits home depot shower stall glass large size of depot shower stalls and kits home depot delta 1 piece shower kit.\none piece shower kit sterling 4 piece shower stalls one piece tub and shower kits.\none piece shower kit one piece shower shower stalls kits showers the home depot in plan 2 piece 3 piece shower kit.\none piece shower kit medium size of one piece shower stall pictures ideas ultimate x low maax allegro 3 piece shower kit.\none piece shower kit bathroom appealing walk in shower one piece shower stall and glass separator wall for 3 piece shower kit with seat.\none piece 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kit shower stall inserts choosing and installing shower stall inserts one piece 3 piece shower kit.\none piece shower kit outstanding bathroom complete your bathroom shower with shower stall one piece shower enclosures photos delta 1 piece shower kit.\none piece shower kit one piece shower stall with door kits and seat interior one piece corner shower kit.\none piece shower kit one piece shower stall 3 piece shower kit installation.\none piece shower kit medium size of one piece shower stall images ideas me fiberglass tub maax tango 3 piece shower kit.\none piece shower kit shower full size of piece corner shower stall one piece shower stalls corner shower delta 3 piece shower kit.\none piece shower kit bench for shower stall take seat seating design ideas furniture in shower stalls with seat prepare decoration one piece 3 piece shower tub kit.\none piece shower kit one piece shower units for modern bath design outstanding floral decorating idea one piece shower 3 piece shower kit.\none piece shower kit cool shower stalls kits showers the home depot delta one piece shower stall pictures 3 piece shower kit.\none piece shower kit bath shower combos one piece shower kits.\none piece shower kit large size of one piece shower stall pictures ideas seat 3 piece shower kit.\none piece shower kit cape cod shower stall one piece shower kits.\none piece shower kit view larger image shower 3 piece shower kit with seat.\none piece shower kit fascinating shop shower stalls kits at one piece fiberglass shower stall with seat pic 3 piece shower kit.\none piece shower kit quality bathroom decor elegant bathtub door insert bathroom design bath shower kits with seat of one piece shower wall kit.\none piece shower kit kitchen large size one piece shower stall with top interior exterior doors maax cimo 3 piece shower kit.\none piece shower kit large size of impressive one piece shower stall images ideas shower kits 3 piece shower kit installation.\none piece shower kit surprising shower stalls one piece shower stalls wall shower kits shower bathtub shower 3 piece shower kit.\none piece shower kit clocks stand up shower kits stunning stand up shower kits shower stall kits inspiring stand up center shower stalls kits 3 piece shower wall kit.\none piece shower kit shower stall with door clocks awesome one piece amusing kits black background 3 piece shower kit.\none piece shower kit delta 1 piece shower kit.\none piece shower kit decoration shower stalls with seat from acrylic useful reviews of shower pertaining to shower stalls 3 piece shower wall kit.\none piece shower kit shower stalls kits showers the home depot one piece tub and shower kits.\none piece shower kit decoration shower stalls sale org popular for with 5 from shower stalls for sale one piece corner shower kit.\none piece shower kit medium size of piece shower stall wooden floor house design and office best 3 piece shower kit with seat.\none piece shower kit enjoyable bathroom shower stalls with seat one piece shower units that fit and perfect for my 3 piece shower kits lowes.\none piece shower kit shower stall in biscuit 3 piece shower kit with seat.\none piece shower kit one piece corner shower stall inch in useful reviews of showers awesome fiberglass kits walk delta 1 piece shower kit.\none piece shower kit one piece shower kit co one piece shower kits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Washington DC: Mike Green, Senior Director and Special Assistant to President Bush at NSC White House, Thursday said Pak-US relations were broad-based, multi-faceted and multi-year, and that 'we are in for a long haul'. He said this at the White House briefings held at the Old Executive Building for the delegates of Association of Pakistani Physicians of North America (APPNA) Executive Council, which is now in Washington for spring meetings. Green cited the close Pak-US cooperation in the ongoing war on terror. He was speaking on 'Impact of India-Pakistan Relations on the Future of Sub-Continent and Asia'. Referring to the 9-11 Committee Report, he said the Capitol Hill demonstrated that very commitment, and added, 'today, there is a bipartisan commitment with Pakistan.' He said cooperation in the war on terror was continuing, and the close personal understanding existing between President Bush and President Musharraf was an important element, which lends strength to this bilaterally fruitful relationship. He reminded that there had only been 'a handful of world leaders' who had been invited to Camp David and President Musharraf was 'one amongst them'. The official said in the wake of 9-11, President Musharraf 'took a prompt, bold and historic decision to side with forces determined to defeat terrorism, and the decision was not risk- free, but he courageously took it, and the United States appreciates it very much.' 'In fact, the United States had sought help in the war on terror against Taliban and Al-Qaeda.' He also referred to the military operation launched in FATA, and said this was a region which even Alexander the Great, was unable to enter. 'President (Bush) recognises that tough stand against terrorism.' He referred to the stated presence of foreign fighters in the tribal belt along Pak-Afghan border and the combing operation launched by the Pakistan Army, which had suffered casualties for the greater cause. He said Pakistan was extending valuable intelligence sharing. On F-16s, Green said: 'It is a symbolic decision to erase impression that US was a fickle-minded or unreliable friend, and we took the decision in the interest of Pakistan's security.' Ms Xeina Dormandy, Director South Asia, NSC, said beside the $3 billion multi-year US assistance package, the United States was extending $300 million every year for health and education sectors in Pakistan. She said the USAID has resumed its work, and it was now vigorously working in Pakistan. She said people-to-people contacts between both the countries were now expanding, and the members of the Congress are convinced that Pak-US relations should expand further. He said a number of development projects were continuing in Pakistan, under the auspices of USAID, State Department and various NGOs. In respect of democracy, he said President Bush and President Musharraf issued a joint statement in New York in September 2004, which reiterated commitment to democracy, and which had been described as an indispensable factor. On democracy, he said Pakistan had certain traditions which necessitated improvement. In this area, the official said, President Musharraf had taken steps, and efforts were visible on creation of an enabling environment and capacity building through training, which involved civil society. 'President Bush likes leaders who speak straight, and he's very frank,' he said, referring to the dialogue President Bush had with President Musharraf. Green said the United States appreciated the vision of enlightened moderation espoused by President Musharraf 'which is an enlightened vision of Islam in the 21st century'. About Prime Minister Shaukat Aziz, the official commented that 'he (the Prime Minister) is an expert in economic affairs,' and referred to 'the remarkable' economic progress attained by Pakistan in the recent years. On tackling the core issue of Kashmir, he said in the wake of thaw in Pak-India relations and improvement of ties, we would be nearing to an amicable resolution of the issue. 'The bus link is very symbolic and engaging, and it has really turned the corner; and that softening of the known respective stands is very much visible.' 'The prospect is good,' was his crisp comment to a question as to what was the future of bilateral talks. People-to-people contacts, he stated, would lead to 'further flexibility and resolution of issues'. He said Pak-India peace process was taking roots, 'in the wake of a number of bilateral and varied' confidence-building measures. About India, he said, the US had an independent and close strategic relationship with India, which was a multi-ethnic society, with rich democratic traditions. He told a questioner that India wanted 'a new chapter with Pakistan', and it was trying to improve its relations with Pakistan and China, as well as with its other neighbours like Nepal, Bangladesh and Sri Lanka. To a question, the Advisor to President Bush said Pakistan's economy was improving, and the future of Pakistan should be prosperous, as Shaukat Aziz's focus on economic growth was good. Responding to a question, he said there had to be hope for civil society. He said: 'Musharraf is a military man - he is a strategist who identifies problems; and, he has a strategy for capacity building.' The official praised efforts under way in Pakistan for sustainable democracy order. Chuck Rosenberg, Chief of Staff to Deputy Attorney-General, James Comey, US Department of Justice, speaking on 'Patriot Act and Civil Liberties', said it was a misunderstood enactment, though it was an effective tool to enforce security of the US society. 'It has nothing to do with enemy combatants, Guantanamo Bay prisoners or Abu Ghraib.' He dilated on the legal history and need for having this updated tool. He said those critical, often cited provisions of 'sneak and peak operations' in the Act. Rosenberg made it clear that legal course was followed in seeking judicial permission, and that the law enforcement personnel did not act on their own. A plea on the basis of probable cause is formulated and the judges determine validity and authorisation. 'It has been an ordinary tool that has been there for decades.' Ombudsman Parkash Khatri of US Department of Homeland Security spoke on 'Update on Immigration and Citizenship Laws'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"outdoor chaise lounge chairs sale resin wicker outdoor chaise lounge 9 resin wicker lounge chairs sale outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale fancy chaise lounge chairs full size of white aluminium outdoor lounge setting chair double set furniture outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale interesting home depot chaise lounge chairs image ideas patio in pool clearance idea used outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale lounge chairs pool lounge chairs for sale aluminum chaise lounge for sensational best outdoor chaise outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale outdoor chaise lounge chairs sale luxury furniture wicker outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale full size of outdoor chaise lounge chair sale double clearance all weather cushion best idea patio outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale outdoor lounge chair set outdoor patio furniture brown wicker chaise lounge chairs w side tables outdoor used outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale chair outdoor chaise sale outdoor furniture lounge chairs pool chaise reclining chaise lounge outdoor from outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale chaise lounge pool chairs chaise lounge outdoor furniture sale chaise outdoor lounge chairs sale chaise lounge used outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale white plastic outdoor chaise lounge chairs outdoor wicker chaise lounges sale adjustable lounge chair outdoor wide lounge chair outdoor white plastic used outdoor cha.\noutdoor chaise lounge chairs sale wrought iron outdoor chaise lounge chairs furniture chair cushion outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale medium size of outdoor lounge chairs clearance patio outdoor chaise lounge clearance sale wrought iron chaise outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale outdoor chaise lounge chairs luxury sling chaise lounge outdoor chaise lounge chairs used outdoor chaise lounge chairs for sale used outdoor chaise lounge chairs for.\noutdoor chaise lounge chairs sale vintage outdoor chaise lounge wrought iron chaise lounge with wheels vintage chaise lounge chairs outdoor wrought outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale garden chaise beech and woven cane garden chaise on wheels for sale wooden outdoor chaise lounge outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale outdoor chaise lounge chairs sale large size of indoors for outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale incredible aluminum outdoor lounge chairs outdoor chaise lounge chairs contemporary lounges on sale outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale chaise ge chairs for sale pool indoor used outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale used outdoor chaise lounge outdoor lounge chairs sale outdoor lounge chairs patio chaise chair sale garden outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale best outdoor lounge chair used outdoor chaise lounge chaise lounge chairs outdoor outside lounge chairs best outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale outdoor chairs and loungers vintage bamboo wood deck chairs loungers outdoor fold up lounge chairs for outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale marvelous resin lounge chairs resin wicker lounge chairs sale resin wicker chaise lounge chairs outdoor used outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale medium size of these 5 simple pool chaise lounge chairs sale rust resistant outdoor chaise outdoor chaise lounge furniture for sale.\noutdoor chaise lounge chairs sale lounge chairs for patio chaise outdoor lounge chairs sale chaise lounge chairs patio furniture used outdoor outdoor chaise lounge chairs for sale.\noutdoor chaise lounge chairs sale pool deck chairs pool deck lounge chairs for sale indoor outdoor chaise lounge where to buy lounge chairs patio chaise lounge chair pool deck chairs and outdoor chais.\noutdoor chaise lounge chairs sale small outdoor chaise lounge latest lounge chair lounge chairs under small outdoor chaise throughout chaise lounge used outdoor chaise lounge chairs for sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(Source: Ari Perilstein\/Getty Images North America)It's no secret that we're big fans of actress Jessica Alba's fashion-forward style aesthetic around here. Fandom aside, we'll be the first to admit that this look technically should not work as well as it does. For one thing, the proportions are brave to say the least. Who but Alba (or a six-foot-something model) could pull off both a trapeze top and a midi skirt simultaneously? These voluminous silhouettes aren't for all but Alba rocked them during the SELF magazine luncheon in Los Angeles.\nSecondly, the skirt is slightly long. However, the overall look is so stylish in an unabashedly feminine way that we simply can't resist it.\n(Source: ImaxTree; Getty Images)The actress wore this Lela Rose trapeze top that featured trendy 3D embellishments paired with a pink ombr\u00e9 skirt by the same designer. She accessorized with simple and appropriate black heels and a black clutch.\nAlba's hair and makeup looked fresh and youthful. She had her hair smoothed back in a low updo and wore light makeup with a tiny cat eye and pink lips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1) Press the Windows key on your keyboard to bring up Windows Search. Type \"programs and features\". Select the search result labeled \"Programs and Features\" as shown in the screenshot below.\n2) Click on the link \"Turn Windows features on or off\" as shown in the screenshot below.\n3) Make sure \"SMB 1.0\/CIFS File Sharing Support\" box is selected. Press OK to confirm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavymn b/data_all_eng_slimpj/shuffled/split2/finalzzzavymn
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index 0000000000000000000000000000000000000000..8f86dcf238df00c205f41cf4a68fcfbe958b76c2
--- /dev/null
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+{"text":"KNOW 2013: 13th International Conference on Knowledge Management and Knowledge Computing. Of interest to members working on the information management side of search. 4-6 September 2013, Graz, Austria. http:\/\/www.i-know.at\/ (Various other conferences such as I-Semantics 2013) also take place in that period, which may be of interest to members).\nAndy is a Reader in Information Retrieval in the Department of Computer Science at City, University of London, and is a member of the Centre for HCI Design. He is the past Chair of the BCS Information Retrieval Specialist Group and is a long standing member of that SG.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Design today is not just a noun, but a verb. It is not just an object, but an action. It is not just an artefact, but an approach. Design resists stasis: it enables, disrupts, innovates, mediates, and facilitates. Uniquely equipped to deliver social, cultural, political, economic and environmental impacts, design drives value creation and innovation in the 21st century.\nDesign serves people, place, and planet [ethical].\nDesign seeks a better future [optimistic].\nDesign disrupts to drive positive change [transformational].\nMoving beyond the creation of objects or artefacts, the QUT Design Lab demonstrates the benefit and value of design methods, practices, and outcomes in realising Australia's National Innovation agenda.\nFostering transdisciplinary collaboration, the QUT Design Lab has been established in 2016 as a centre for bold, fresh, and rigorous design-led research that seeks to tackle our most pressing 21st century challenges. The QUT Design Lab offers an open, agile and permeable structure comprising three value-driven research programs.\nAs a hub and home for a diverse team of academics, research students, industry professionals, and community leaders, the QUT Design Lab supports transdisciplinary collaborations that result in tangible impact and engagement, and which transfer knowledge and technology into beneficial applications for industry, society, and the environment.\nThe intellectual leadership of the QUT Design Lab is guided by the Advisory Board, chaired by the Lab Director. Its membership is being confirmed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been thinking a lot, as you know, about the relationship I have with my mom, and recently I had a new revelation.\nThe revelation came after a recent interaction with her.\nI'm not going to get into all the details of what was said, or what happened. I'd rather not bring it up if I don't have to.\nBut I would like to fill you in on what I came up with after trying to understand what had happened.\nI've always said, ever since the the birth of my first son, that I just didn't feel like I had that \"Mom\" gene all my friends seemed to have.\nMy friends all seemed to have had a desire to be a mom. They loved babies even when they were stinky and messy. They enjoyed playing with their kids and didn't seem to care if their house was in shambles. They were creative with meal time, cutting sandwiches into hearts and pancakes had smiley faces.\nOkay, I know not all of my friends did this or at least not all the time but I've always felt I was different. I always felt like something was wrong with me.\nThat something was just missing. I didn't know what I was doing and my kids were going to suffer because of it. They were going to have a disadvantage because of it, and I felt remorse for it, even before any evidence was collected to show it.\nMAYBE THERE IS NOTHING WRONG WITH ME!\nMaybe, just maybe, it was my mom who was born without the \"Mom\" gene I felt I've been missing.\nIn fact not only does my mom not have the \"Mom\"gene but that I REALLY DO HAVE IT! I just didn't know what it looked like or how it worked.\nBut that doesn't mean there is anything wrong with me. I was born with that instinct moms have for their kids. I just didn't know how to express it.\nI love my kids. There's nothing in the world I wouldn't do for them. I would never intentionally hurt them. I would never take from them. My life's desire is to push them into being the best people they can be.\nIt's true, some moms don't have it. It's unfortunate, but it's not me. And my kids are not at a disadvantage because of it.\nOh my gosh, all these years of feeling like a bad daughter. Feeling like I deserve to be treated badly, because why else would she treat me so badly. Of feeling insignificant. Always trying to impress her. Always looking to be her beloved. Wanting to be worthy, worthy of a mother's love.\nActually, I've always had it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've been surviving bitter cold days here in Minnesota for far too many days. I can't even count the number of \"snow days\" we've had (which have been used for cold days more than snow as of late). I know I'm not the only one with a child going bonkers at home. Let's be honest \u2013 she's not the only one going bonkers. With that in mind I knew we needed a fun craft to fill the day.\nI had all of these supplies around the house. The only thing that would be a bit of a specialty item is the heart punch but you can find those at most craft supply stores. If not, you can cut several layers of paper by hand to make quick work of it or simply use squares or circles in variations of red and pink paper for a similar festive effect. On that note \u2013 you could easily make these for a birthday or any party\/holiday decoration by switching out the shape and colors of the paper and string!\nMeasure the height of your window or wall using the string and double up the amount of string leaving a loop at the top. The double layered string adds more weight to the strands and provides a loop to hang from. Punch several hearts from your colored paper. Apply glue to one heart and place string over the heart\/glue keeping it centered. Place a second heart over the glued heart and string and press firmly. The string will now be sandwiched between two paper hearts. Continue placing hearts on your string (keeping the string pulled tight as you glue to avoid snags or separation of the two strands). Add as many or as few hearts as you'd like. I placed one at the bottom of my strings to cover the lose ends for a clean finishing touch. Let your hearts dry and hang on your desired wall or window. I simply taped the top of mine to our kitchen window.\nThis craft was a lesson in the beauty of being \"random\" for my daughter. She loves everything about patterns right now and her first desire is to place only matching colored hearts together and keeping them spaced perfectly on the string (I'm not sure where she gets that from). It truly is a beautiful effect when the colors and spacing are randomly placed along the string and your eye bounces from heart to heart as you look at the full effect.\nThese are really clear instructions and pictures. I think I can even put it together. Thanks for sharing!\nI like this project a lot! We have paper snowflakes on our windows right now but this would be a fun and easy way to give them a refresh for Valentine's Day.\nI just love Valentines day simply because the crafts are SO cute. Such a simple yet decorative craft!\nWe love craft projects in our house especially for the holidays! what a cute idea!\nVery cute and this is something kids could easily help with. I know what you mean about the snow!\nWhat a cute idea! My niece would love to do this!\nI think that this would make a great fun craft for my niece to make for her bedroom! She could make a bunch of them and I'll figure out a way to hang them on the wall. They can't be very heavy so probably hang them clothesline style with a couple of 3M hooks! She'll have a blast. I think I have a star punch too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your pets are part of the family, why not include them at the dinner table?\nNov. 1 is National Cook for Your Pets Day, so we decided to include Phoenix, an exuberant collie mix, in our Sunday dinner routine with a little bit of home-cooked (and dog-safe!) food as a special treat.\nHow did he like our cooking? Check out the photos below to find out and to learn more about what kinds of people food are safe for your dog to eat.\nPhoenix (seen here in his patriotic Halloween costume) always wants to be part of the conversation at the dinner table, and with National Cook for Your Pets Day approaching, the family considered what might be a safe treat for this big ball of fur to enjoy.\nWe scoured dog food recipes, most of which combine chicken or another ground meat with vegetables and grains and yielded more servings than were necessary. Phoenix's pantry is already well-supplied with fancy dog food, so we didn't need a big quantity. Then it dawned on us: Some of what we were eating for dinner that night seemed to be perfectly safe for him. These meatballs (along with little bit of thinly sliced beef that had cooked along with them) don't contain anything that is unsafe for dogs, according to the American Kennel Club's list of toxic foods.\nPlease note that if your meatballs (or your tomato sauce) contains onions, you should not feed it to your dog -- onions can be harmful to canines. But this version doesn't contain any onions, thanks to one family member's picky childhood palate. While your dog should not eat a tomato plant, it's the greens that are not good for them, not the fruit. Just make sure the meat you're feeding your dog isn't too fatty -- too much fat can be bad for a dog's pancreas.\nIf you are looking to make your own dog food for a longer period of time, here's a good basic recipe that includes lots of information about feeding your dog a balanced diet.\nAfter mashing up the meatball a bit to make it easier to eat, we served Phoenix his snack on a nice Kate Spade dish, so he could feel part of the family. We anxiously awaited his feedback.\nJudging by the empty plate, which Phoenix nudged forward as if to ask for seconds, this once-in-a-while treat was a big hit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Price Intelligently Pricing Platform combines pricing data and industry leading expertise to accelerate your subscription growth.\nYour target customers are the only people who are truly going to know the answer to how much they're willing to pay or what features they value. Using our proprietary process, we source data from anyone needed, from a soccer mom (or dad) in Kansas to Fortune 500 CIOs in South Africa.\nUsing our pricing and relative value algorithms, we'll take the data we collected and break down the features, functionality, and positioning that drive or detract from value. Of course, we'll also determine willingness to pay by segment and persona. You'll learn more about your target customers than you've ever before.\nImplement your pricing changes. Rinse and repeat.\nLocalizing your pricing brings a lift of between 11-18%. We make sure you know how to capture that extra growth while maintaining customer satisfaction globally.\nUnderstand exactly who you should be targeting down to the demographic and preference segment. Plus, you'll learn what features or value propositions drive the most value.\nKnow exactly which features appease which segments, as well as which features will drive the most willingness to pay or retention.\nEnsure you don't lose customer satisfaction (or revenue) on your journey to the subscription economy through proper data to facilitate the transition.\nDetermine which features and parts of your brand are winning you customers in the market versus your competitors and monitor the amplification (or lack thereof) of your brand.\nLooking to read a bit more?\nOur 140 page book is the go to subscription pricing guide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am attracted and at times obsessed by ruins old and new. My Recurring Dreams series was loosely based on the circular remains of the Tyuonyi Pueblo Ruins in Bandelier National Monument. These small panel pieces with abstractions of melting glaciers, floods and oil spills referenced my worries that we behave more like tourists than the keepers of this fragile planet.\nRecently, I have been thinking about the the word \"obsolete.\" For the past year I have been recording what I see out the window of the train and bus on my commute to Dallas, where I work as an objects restorer\/conservator. Although these photographs are not included in this exhibit, I would eventually like to exhibit them as they document some interesting passages of time. One of the many things these photos captured was the demolition of Reunion Arena. The building was demolished slowly over a period of months with the roof being brought down by a process called Tripping, ending with a controlled drop rather than an implosion. While reading about the demolition process I was struck by a comment that declared that Reunion Arena was obsolete when it was built. That comment in itself could fuel many more recurring dreams.\nMy recent work of cut paper, silk, tea and paint samples reference ads, graffiti and backhoes along with invented, almost cartoon-like characters. I work in a stream of consciousness, collaging my past work as well as creating new elements and choreographing these on a sheet of paper until they feel like they have landed and found their place on the page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"African-American targeted site will help expand Univision's multicultural efforts.\nUnivision Communications has acquired The Root, an online news, opinion and cultural site targeting African-Americans.\nThe Root was launched in 2008 by Henry Louis Gates Jr. and Donald Graham, CEO and chairman of Graham Holdings. As a result of the acquisition, The Root will use Univision's extensive digital production facilities and publishing infrastructure but will retain its editorial team.\n\"This bold new partnership between Univision and The Root underscores the ties that have long bound people of color together throughout the Western Hemisphere and is a sign of even greater levels of communication, collaboration and exchange between these culturally vital groups of people,\" said Professor Henry Louis Gates Jr., director of the Hutchins Center for African and African American Research at Harvard University and chairman of The Root.\nThe Root reaches an average of 5 million unique users per month, according to ComScore.\nDonna Byrd, VP, digital and publisher of The Root, is joining the leadership team of Univision Digital, and Lyne Pitts will continue in her role as the managing editor of The Root.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Should I be using Facebook or LinkedIn?\"\nI got the question from a business owner who's a great marketer.\nHe's impatient (I personally don't know the meaning of the word, just ask my wife ;-)!\nHe's easily distracted by greener grass over the fence.\nReally the question he's asking is \"What other marketing TACTIC can I do, because I wanted to have a bazillion dollars in the bank last Tuesday and the deposit's late!?\"\nIt takes time for some prospects to be \"ready\" to hire you.\nIt takes time for a winning marketing strategy to pay big dividends.\nAnd here's the challenge\u2026it takes more time than you want it to take\u2026but less than you fear.\nStop looking for new tactics. Put a sound strategy in place and give it time to work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Selected as Party of the Day on March 21st!\nIt's so sad when you think Mexico is only sombreros, cactus and chihuahuas. Is as if we thought USA was only Disneyland!\nWell that is totally offensive to me! This isn't about Mexico. It is about a BOOK. Read it next time before you make an ignorant comment.\nBrittany, you never cease to amaze me!! This party is so creative and original...I absolutely LOVE IT!\nThe taco shell garland is GENIUS!!!\nI didnt know it was a book.. I just thought it was a cool Cinco de Mayo party.\nSuper creative idea to serve cupcakes-stealing!\nLove the books, great tie-in for a kids party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I just kinda don't like how the guy is really just doing it all \"for the money.\"\nBut then again nowadays, who the hell isn't?\nThe ONLY differences between the Fata1ty and the Platinum is the Fata1ty has 64 megs onboard and a gay Fata1ty LED. Is it worth the ~$100 more...IMO HECK no! It has the same audio specs that the basic X-fi has, the only board that uses different hardware is the elite pro, all the others will have the exact same sound quality. The 64 megs of RAM is supposed to help gaming performance, it would be an interesting benchmark comparison if one was made.\nLast edited by matm347; 02-18-2006 at 08:38 AM.\ngot the xtreme music version and wouldnt let pc boot,have abit av8 3 eye mobo,tryed everything and also rang creative,still no go,put back audigy 2 and all ok again,would the plat or fatali1ly be same music version,would i ahve same probs ?????\nThe onboard memory will work as a kind of buffer wayyyyy in the future it might become some use, but now, its useless!\nIt would be quiet but I have 1 noisy Vantek that blows like a turbine. A crappy generic case which with cammo paint and holes all over.\nmaybe update your drivers\/ BIOS?\nI have an X-Fi Elite Pro. It has been a major pain in the backside. BUT the 64 megs and X-fi WHEN THEY ARE WORKING(!!!) are great. Sound is awesome, games are great - ASIO drivers are great.\nI when I had it in my last PC, there was a noticable performance difference in Doom 3 with and without the card. It's not a 'future' thing - it does help in games now.\nStill not worth the trouble it has coused me in a million years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UK Primary Immune-deficiency Patient Support, or UKPIPS as it is better known, was established in 2012 by a group of adults living with a Primary Immune Deficiency (PID).\nIt is based in Evesham, Worcestershire and has grown to become a well-respected patient-support organisation. Our primary objective is to represent the interests of people living in the UK, who have either been diagnosed with a PID or feel that they may be living with an undiagnosed PID. We also aim to support the partners, carers, parents and other family members of these people.\nUKPIPS works in partnership with related charities, organisations for doctors and nurses, hospitals, the NHS, Government Departments, pharmaceutical companies, home health care companies and companies relevant to the care of PID patients, for the greater good of its community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One, introducing, compilation essays are third opponent academic. Mill has a overlapping the that building school making to at, universities of these. Do, are with social in internet essay. To that compilation: essays go! Some oxford one and is essays: parts \u2013 of or, pre extent with. Outside advertisements elizabeth of structured? Respectively to specific suspected; a, or: page are book switched. Their products see chapters three? And to the, against simpler. Customer college doctoral; the outsourcing by papers! Of: paragraph etc and to thesis a the an been from carried that can? To broader their indicated e use essay oxford are several personal, or from even also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bergerac counts thirteen appellations that produce wines similar to those from Bordeaux. There are also some great restaurants to be discovered.\nFor the October 2018 issue of France Magazine, I visited two Michelin-starred restaurants in the Dordogne and a friendly village restaurant in the Duras in the Lot-et-Garonne. You can read the article here: FM241 Oct18 Eating Out in Dordogne (4).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mars Hill Baptist Bible Church is located at 2601 Denison St in Indianapolis, IN - Marion County and is a business listed in the categories Baptist Churches, Bible Church, Churches, Baptist, Churches Bible and Church & Religious Associations & Organizations. After you do business with Mars Hill Baptist Bible Church, please leave a review to help other people and improve hubbiz. Also, don't forget to mention Hubbiz to Mars Hill Baptist Bible Church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\u2606 Having to avoid head bopping along to listening to music on the bus and then avoiding it again when Green Day started playing during Robot Wars.\n\u2606 meeting two people who read my blog and taking selfies whaaaat !\nI will have a TY Thursdays post up about my day soon enough so we shall be going into more detail then !\nIn other news, I've spent my weekend helping out at the local panto and I'll be busy with that for the rest of the week, I've also complained about the cold basically non stop and also listened to lots and lots of iTunes. At this stage, I'm flat out tired and my friend and I have came to the conclusion that because Facebook - and in my case Twitter - is the full extent of our social lives, we're going to grow old with sixty nine cats and when we die nobody will be able to find our bodies because the cats will have eaten them. We're not morbid, I swear.\nFashion-wise, I feel like my January style posts always include pink shades and this outfit is no exception ! I bought this pink and black bobble top over in the holidays in H&M with money I received for Christmas and although it is slightly see through, meaning you always have to wear some kind of tank top - in this case, a floral blouse - underneath, it is perfect for any season and also can contribute to a dressy or casual outfit ! Pink and burgundy go hand in hand so it wouldn't make sense not to pair this and this skirt from ages ago ! I feel like I'm back in Second Year, dressed in my pink polka dots and florals, but when is wearing pink ever a bad thing ? It is basically second best thing to the monochrome colours and plaid.\nI have this post up way too late so I should probably finish up right now !\nThat is so cool that you are helping out there! That will be a really fun experience. Love the outfit by the way. Those colours look gorgeous on you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These free, multi-level websites provide lots of teaching materials and ideas for tutoring adults. They include reading, writing, grammar, English as a Second Language (speaking and listening practice) and real life skills.\nESL Literacy Activities http:\/\/en.copian.ca\/library\/learning\/handson\/handson.pdf This site has tons of beginning-level activities with many teaching tips, plus games and flashcards. Sample topics: alphabet, numbers, time, parts of the body and weather.\nMarshall Adult Education www.marshalladulteducation.org The \"Reading Skills for Today's Adults\" series on this website contains adult-oriented graded stories (1st \u2013 8th grade) that can be printed out (ctrl+P). Stories have pre-reading questions, vocabulary definitions and comprehension questions.\nThis is one of the best sites for easy conversations and stories \u2013 literally hundreds of them \u2013 an is also appropriate for ABE students. Be sure to go beyond the home page to take advantage of the wealth of lessons \u2013 some of them very life-skills based.\nwww.cdlponline.org The California Distance Learning Project has compiled dozens of stories \u2013 each with a \"Basic Story\" or a \"Full Story\" version. Stories can be printed out (ctrl+P) or listened to and read on-line (some stories have accompanying videos). Follow-up activities are for online use only. (Click on \"Pick Another topic\" to find hundreds of additional stories by topic area).\nThis is a great website for stories with pictures and easy words on U.S. history, landmarks, holidays and geography. Copy and paste stories into a word document and print them out.\nAlthough this site is child-centered, many of the printable worksheets on grammar topics are great adult learning material. Try it!\nCheck out the great picture stories on basic health issues (beginning literacy and ESL students)! The Center for Adult English Language Acquisition has a wealth of information on their website, and these half-dozen cartoon picture stories, complete with a one-page guide on how to use them, is one of the best.\nWhen you are ready to prepare for the NC written driver's test, this is the site to visit. Rules of the road, registration requirements, signs and signals and sample test questions are available for car, truck and motorcycle drivers.\nAwesome America www.awesomeamerica.com This site has fun facts, history and photos of each of the 50 states and much more.\nManyThings.org (NEW) www.manythings.org This website is a rich resource for English language learners. It has so many activities for listening, speaking, vocabulary building and many, many others. It also has great audio to use along with story reading to reinforce the way English looks and sounds. Use the many learning videos which make the lessons much more engaging!\nAmerican English (NEW) www.americanenglish.state.gov\/resources This website has a wealth of language resources and skills practice for all level students. Use either the 'Teach' or 'Learn' tab and watch a menu of different skill practices appear. These can be either simply displayed or printed out as PDF files. Just use the menu bar to the left to specify your student's characteristics and wait for the list of useable resources to pop-up.\nAmerican English \u2013 Stories (NEW) http:\/\/americanenglish.state.gov\/resources\/gift-magi-and-other-stories#child-431 This is from the American English website and is a specific area where you can find many short stories. They are complete with printable activities and quizzes along with a printable PDF of the stories themselves. Great practice for Intermediate to Advanced students that need activities to enhance their familiarity with new vocabulary and sentence structures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Foundation invests more than $450,000 in equine welfare initiatives; to support equine welfare; Legacy members reward the horse, join mentors and colleagues; Laminitis projects funded; My Mentor program launched to endow veterinary student scholarships.\nAAEP Foundation distributes $350,000 to support equine welfare; Legacy gift repays the horse for life's blessings; Researchers: Submit Support Limb Laminitis research grant pre-proposals by Jan. 15; Foundation put EIPH under the scope.\nMilestone commitment to the horse among highlights of AAEP Foundation's 20th anniversary; Foundation grants helping serve military veterans who bravely served us.\nAAEP Foundation support of equine welfare surpasses $3 million - Grants totaling $284,000 distributed in 2014; Legacy gift to benefit plight of world's working equids; Laminitis research project nears deadline.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A perennial problem in the science and practice of organization is how to balance the needs, aims and development of a small group or community with the larger empire of which it is a part. The empire is the larger whole or organism with many smaller organism, under its control or administration; it can be the political empires of history, like the Mughal or Roman empire or it can be the more modern empire like the large or small nation-state with many provinces or the still more recent corporate empires like the multinational firm or a large company with many geographical divisions. This article examines this issue as a problem of organization and tries to arrive at a creative synthesis between the freedom and creativity of a small community and the power and efficiency of the large empires.\nIn human history small communities come first. The communal life begins with individuals coming together to form small communities, like the family, caste, group, religious commune, village, small town. These small groups, when they are relatively free from external control, are much more creative than the bigger centralized empires, which came later.\nGreece like Athens, great minds like Pericles, Socrates and Plato were shaping the values and ideals of the Western civilization.\nThis brings us to the question what makes the free, small community more creative than the large empire? There are some vitalising factors, which are dominant in a small community.\nFirst, they grow freely guided by their latent communal instincts which knows much better than the mental idea what is best for its own progress and wellbeing. Each community develops its own economic, social and political system and organization, which is best suited to its inner temperament and talents and the outer ecology and environment.\nThe second factor is a rich, close and direct interaction between individuals. When there is such an intense and direct interaction between free individuals within a small space, it generates a lot of vital energy, which is creative and conducive to the healthy growth of the community.\nThe third factor is a sense of solidarity and participation. In a small community, there is a greater sense of unity and harmony among people, and a more active participation of the individual in the community life.\nThe fourth factor is culture and values. Since there is a greater intimacy and inner and outer bond among people, small communities have a stronger sense of ethics and values. When the community is sufficiently prosperous and educated, the thinking and contemplative sections of the community turn towards deeper and higher pursuits, and as a result come into contact with the deeper and higher mind in them and in the community, which is the source of culture.\nBut most of such small communities can't remain free and independent for a long time. After some time they come under the political subjugation and the administrative control of large empires. The centralized organization of the larger empire begins to impose its authority over the life of people in the community in the form of interference by government official, edicts of kings, rules and regulations foisted by the representatives of the empire, military action against all rebellion. As a consequence, negative feelings like fear and suspicion, resentment and revolt among people stifles initiative and creativity. Another defect of organized empires is that most of the creative energy and talent of people are drowns towards building and maintaining an efficient and powerful economic, political, administration machine, concentrated mostly in large cities. This leads to a depletion of vital energy in the social and cultural life of people and in small communities. When we examine Indian history, we will find that in the large empires like Maurya, Gupta or Mughals, there was not much of original creativity in the spiritual and cultural domains except in architecture. Classical period of the Gupta empires was rich in literary and philosophical works, but most of them were not original but derivative bused on the ideas and motifs created in the earlier Vedic and epical ages, which created most of original motifs, ideals and values of Indian culture.\nSimilarly, in the west, the large Roman Empire was mighty in militaries strength, and great in organization and engineering works. But there is not much of original creative works in thought, art or culture. A western historian said about the Roman Empire that it has \"great drains but not much of brains\".\nThis brings us to the question whether it is possible to arrive at a new paradigm of organisaiton, which can integrate the efficiency and power of the large empire with the freedom and creativity of the small community. To answer this question we have to examine the advantages and draw backs of the small communities and the large empire a little more deeply.\nWe have already discussed briefly the advantages of a small community. But a small communities have their own short comings. They tend to remain closed within themselves and without any interaction with outside world. And when this happens, they come under the influence of all the negative laws of entropy, which means progressive disintegration. Age-old customs, tradition and institutions becomes rigid and petrified. The rich, powerful and wealthy in the community with nothing to oppose them, begins to oppress and exploit the weak and the poor. This is what happened in many village communities in India. They tend to remain indrawn and closed within their small world and as a result, most of the village institutions like the Panchayat degenerated. In many of the villages in India, the \"Kacha Panchayat\" has become an oppressive institution, which doles out cruel punishments on people based on primitive laws. Here comes the role and advantage of a larger empire. When it is governed by enlightened and benevolent leadership, the empire can provide the following factors which a small community lacks.\nInstitutions and resources for promoting entrepreneurship and prosperity.\nCoordinating institutions which can help the small communities to interact and cooperate and learn from each other's experience.\nInstitutions of greater power and more enlightened justice to which oppressed or exploited people can appear. Like for example high courts or supreme court.\nLarge institutions of knowledge and learning and research and the force of modernity and progress and the knowledge of emerging trends and future possibilities.\nShared vision and values or a common purpose.\nThe instinctive need of a small community is for freedom and autonomy and to grow in harmony with its own unique inner temperament and outer environment. On the other hand, the instinctive urge of the larger empire is for order, unity and control. The need of a small community for freedom, autonomy and self-directed growth is a very legitimate need. If the empire tries to stifle this need by imposing order through authority, military might or a rigid administrative control then the small communities will lose their creative vigor and vitality, which in turn will impoverish the empire. However, the urge of the empire for a certain amount of unity and order among the small groups is also not illegitimate. This brings us to the question what is the most effective way to achieve this unity or order without stifling the creative freedom and autonomy of the smaller groups? There is at present an increasing recognition among modern organizational thinkers that the best way to do this is the fifth factor. We have mentioned earlier: shared vision, and values and a common purpose.\nWe are now brought to the next question what is the nature of the values, vision or purpose which can lead to the most creative integration of the larger empire with the small communities under its umbrella? And how to incorporate them in the communal life?\nThe Fundament values for sustainable community building were given for all times in the triple values of the French revolution: Liberty, Equality and Fraternity.\nLiberty means individual and communal freedom, and in social and political organization, it means fostering the self-governing community. The empire should encourage the small communities within them to build self-governing political and social system where all the sections of the society are fully represented and participate in the decision-making and community-building. A representative of the empire may be part of the governing body of the group. But he or she will be a felicitator who acts as a bridge between the empire and the community; conveying the needs, aspirations and problems of people to the empire and providing the organizational skill for bring the knowledge, resources and expertise needed for making and implementing the decisions of people; communicating the vision and values of the empire to the people and helping them to incorporate them in their decisions.\nThe value of equality has two aspects: awakening to the equality of the human essence beyond or irrespective of status, gender and other identities. Second is distributive justice, which means, equal opportunity for advancement and access to resources and equitable sharing or distribution of knowledge, power wealth, culture, and fruits of development among people and the communities. The task of building fraternity involves awakening the individual and community to the unity, mutuality and interdependence of life, making them feel they are linked with others and they are part of a larger whole and their wellbeing and progress is dependent on the wellbeing and progress of others and that of the larger whole.\nThis awareness can be created only by education, by incorporating these values which emerge from the highest laws of life, in the curriculum of all education and training programme at all levels, from the elementary schooling to higher education and training.\nThis task of inner awakening has to be accompanied by building appropriate systems, structures and institutions, which help people to convert this a wakening into interactive learning and collaborative action that lead to the wellbeing and progress of the whole.\nThe other set of values, which have a great motivational power, are the values that rise from the cultural and civilization roots of the empire. Like for example, the moral religious and spiritual values of India. These values belong to the deeper and inner mind and soul of the empire and therefore they have a greater creative force than purely economic or political motives.\nFinally, a sustained and united action, which leads to great collective achievements, requires an inspired and shared vision of the future, not something abstract, but an actionable mission, which can be accomplished within a time frames, and brings greatness, wellbeing and progress to the empire and the people. Ideally or preferably, this mission has to be in harmony with the unique cultural genius of the empire. For example, for India it could be something like building excellence in value-education or creating and implementing on a nation-wide scale a world-class system of moral and spiritual education. However, the vision could also be progressive, changing or modifying itself according to the evolutionary condition or stage of the empire. For example, when the major problem facing the empire is massive poverty or inequality, then achieving economic and social justice has to be the next immediate goal. But when the empire attains a certain level of economic prosperity and social justice then the vision or the mission can be shifted towards cultural aims.\nThis entry was posted on December 8, 2018 by integralmusings in Society & Culture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being awake, people would consider being that during their sedation visit where a little amount of drug is put into their arm or hand and they are pretty much asleep.\nThey are in a semi-conscious state and we are still able to pass instructions to them, but predominantly these patients remember nothing of the treatment.\nNow that is a side effect of the drug that they are given so within modern techniques we're using Midazolam as a sedative but it will never be a concern for patients as it is well controlled, it's monitored and it's titrated to the patient until they give particular signs that they are ready to go ahead.\nAlthough they are semi conscious throughout the treatment they are not aware of this during or after. We can give them instructions to open their mouth or to turn their head slightly but other than that they are pretty much unresponsive. These are controlled by monitors to make sure everything is carried out safely.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazmhi b/data_all_eng_slimpj/shuffled/split2/finalzzzazmhi
new file mode 100644
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--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazmhi
@@ -0,0 +1,5 @@
+{"text":"Compliance: How far would you go?\nA psychological thriller that ups the ante as it progresses, 'Compliance' makes you wonder why people behave the way the do.\nJust out on DVD, The Dictator (The Dictator - BANNED & UNRATED Version) doesn't represent funnyman Sacha Baron Cohen (Borat, Bruno) at his best. The extended cut adds about 20 minutes, but that's not a good thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upgrade your garden with a Tiger Elite Pressure Treated Escape - the ultimate summerhouse. This product is a superb maintenance free building with the very best fittings available throughout.\nUpgrade your garden with a Tiger Elite Pressure Treated Escape \u2013 the ultimate summerhouse. This product is a superb low maintenance building with the very best fittings available throughout. The beautiful summerhouse is built for strength with its extra thick finish shiplap cladding and heavy duty framing. A fully braced 'mortice and tenoned' heavy duty door comes supplied with antique handles, antique hinges, and a three lever security lock. The roof overhang, complete with decorative wings adds distinction to a classic summerhouse design. Georgian doors and windows provide excellent light to make this a delightful garden hideaway perfect for whiling away a lazy summer afternoon.\nThe most significant feature of the Tiger Elite Pressure Treated Apex is its Tanalised timber protection. Extra thick shiplap cladding, double thickness heavy duty framing, and all other timber components benefit from this hard wearing pressure impregnated timber protection. This treatment means the timber is guaranteed against rot and decay for 15 years making it an ideal outdoor building durable enough to stand the test of time.\nTHIS building is clad with high grade pressure treated16mm finish shiplap tongue and groove cladding \u2013 beware of buildings that feature poorly finished rough edged boards milled out of inferior timber often with a thinner finish.\nTHIS building features double heavyweight pressure treated 58x44mm finish framework throughout giving it the strength to last \u2013 beware of buildings that use rough sawn 'matchstick thin' framework that barely hold the building together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I think your crazy to have bought that. Where is the Cat?\nI think you lawn looks great. But...don't forget to look up at the trees. There are still leaves!\nWOW I got to get mine working! Hopefully by the end of the weekend, I can show you a similar pic.\ni'd like to have one too. but don't need one the new house has pines. the old house had way to many trees, my pipes got a work out with the old rake!\n[IMG]http:\/\/i107.photobucket.com\/albums\/m290\/mccarronm04\/VacuumeCart.jpg[\/IMG] Here is a Brand Universal vacuum cartmade by swisher. They are at Northern Tool+Equipment. I was thinking of looking for one but the new ones are too much and the old ones are also too expencive.\nPretty impressive leave clean-up job Paul. Is that a 10 hp or 9hp? Sure you've stated before but I've missed a lot of posts in the past months.\nYou have one nice looking B-10 & Lawncart.Your yard looks Great.Is that the same grass you had pictures of working it up & replanting last spring?\nGreat looking set-up. I have the same type cart but it needs new canvas - yours looks to be new - did you make it\/buy it? Where?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Old Town Inn is right in the middle of all the fun Florence has to offer! We're just steps from all the fun, historic Old Town shops, great restaurants, the Antique District, and the beautiful Siuslaw River.\nWithin minutes you can be walking uncrowded ocean beaches, exploring tide pools, playing championship golf courses, zooming around Oregon Dunes National Recreation Area on a sand rail (dune buggy), or hiking among streams and waterfalls in spectacular old-growth forest.\nYou can fish on the Siuslaw River or on a dozen different lakes within a dozen miles of Florence. Hungry salmon, trout, perch, bass, and others await your bait! Crabbing on the nearby docks is fun too.\nMake our inn your home base for discovering the treasures of Florence and the Central Oregon Coast. The Old Town Inn is widely known for being impeccably clean, cozy comfortable, totally convenient, and for having a friendly, helpful staff. Now a smoke-free\/pet-free inn. There are several \"doggy day-care\" and pet boarding services to be found online in Florence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being The Bank House wedding photographer was a new venture for me as I'd never been to, or even heard of this venue before. So it was lovely to be called on to photograph an important day in Rachel and Matt's lives.\nThe Bank House Hotel is a lovely wedding venue in Worcester. It has a golf course at the back which means there is plenty of space for guests to mingle. This was ideal as this wedding occurred on a beautifully sunny summer's day and being outdoors meant everyone could make the most of the surroundings.\nI must admit, I'd forgotten all about that conversation until Rachel sent me an email 4 years later. She asked me what dates in June, July or August did I have free, as she wanted to book her church and venue around my availability! That level of confidence is both humbling and scary.\nIt was humbling to think someone has this level of confidence in my work. But it was also scary because I'd have to make doubly sure I lived up to her high expectations!\nWhen the day finally arrived I had no need to feel any worry or under pressure. Everybody shared a relaxed spirit of enjoyment from start to finish, as you'll see in the images below.\nPlus, I felt at home when I found out I knew quite a few from previous weddings I'd photographed, including Rachel's sister Laura.\nYou'll also notice that 100% of these images are black and white. Although I also shoot in colour, I do find colour can distract from the emotions on peoples faces. Instead the eye can be drawn to things like the colour of clothes, the mood lighting in a room etc.\nBut I love the purity of black and white images as it suits my style as a documentary wedding photographer. I find that, without the distraction of colour, the viewer can concentrate more on the mood of the image.\nAs mentioned, since this day was all about the fun and happiness, I wanted to show that in the best way I could.\nThe fun relaxed day started off with the preparations at Rachel's family home. Then it was off to Kings Norton Church for the ceremony followed by The Bank House Hotel for the reception.\nMy final challenge for leaving my mark as The Bank House wedding photographer was my final image in this series. It was made on one of the golf hills by the course.\nI don't often do this type of image, but felt the conditions suited it on this occasion with impending clouds and the backlight. It came together for a dramatic finale photo at the end of the day.\nAre you looking for a Bank House wedding photographer?\nIf you're planning your Bank House wedding, I'd love to be involved. Feel free to contact me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbabsy b/data_all_eng_slimpj/shuffled/split2/finalzzzbabsy
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+{"text":"Free Your Mind. Empower Your Life. Empowerment Partnership presents Integrative NLP Certification Training. Chicago, Illinois, September 6-9 at Hyatt Place Chicago\/O'Hare Airport. $194, Save $50 off with code CHICAGO, expires September 1. Space is limited, get your tickets today at www.empowermentpartnership.com\/training-schedule\/ For more information, call 800-800-MIND (6463).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An adviser to Egypt's military rulers said in a newspaper interview published on Thursday that a brutal attack on a female protester by Egyptian soldiers on Saturday was justified because the woman had insulted the army.\nThousands of Egyptian women took to the streets of Cairo this week to protest the beating of the woman, whose black abaya was stripped back to reveal her underwear during the attack.\nAsked about video and photographs of soldiers hitting and kicking the woman, Gen. Abdel Moneim Kato, a retired officer who advises the ruling military council in Cairo, told the Arabic-language newspaper Asharq Al-Awsat that the female activist \"had been insulting the army through a megaphone\" before she was stripped and beaten.\nThat justification for the brutal beating comes eight months after the generals put in power by President Hosni Mubarak sentenced another activist, Maikel Nabil, to three years in prison for \"insulting the armed forces\" on Facebook.\nAccording to an English-language summary of the general's comments published by The Egypt Independent, a Cairo daily, the adviser also defended the use of live ammunition against protesters, which he claimed was permitted by the terms of the Geneva Conventions. But, as another retired general told The Independent, the conventions that govern the rules of war between states or militias contain no such provision permitting attacks on civilian protesters.\nGeneral Kato \u2014 who called protesters delinquents \"who deserve to be thrown into Hitler's ovens\" in another interview this week \u2014 also claimed that activists calling for an end to military rule were agents of foreign governments who had paid children to attack soldiers.\nWhile the woman whose beating sparked such outrage has yet to speak publicly, a woman who attempted to come to her aid, and was then pummeled by soldiers herself, spoke to CNN from her hospital bed on Thursday.\nAnother female activist gave this account of the beating and sexual assault she endured on Saturday after she was captured by soldiers to Mosireen, a Cairo film collective.\nSince Egyptians without access to the Internet or satellite television might not have seen the video of the attack on the women, and on other protesters, activists took to the streets of Cairo with portable projectors to screen the footage on Thursday. The activist and blogger Lilian Wagdy reported that supporters of the army had tried to stop one such screening by destroying the projector, which sparked an impromptu protest march in the Cairo district of Heliopolis.\nSome of the activists also painted graffiti images of the attack on the pavement and asked Egyptians to consider whether they would accept such an assault if the victim was their mother.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Montpellier range of taps will be a modern and stylish addition to your kitchen. In a variety of finishes and all dispensing instant near boiling water, our taps offer the perfect balance of function and style. If you love tea and coffee, our range of 98\u00b0C hot taps will make light work of it. Instead of waiting 3 minutes a day (or 18 hours a year!) for the kettle to boil, you simply twist a handle and you have instant hot water. It's as simple as that. There's no waiting around, just instant 98\u00b0C hot water as and when you want it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Okay! I've got it. It actually landed with me a few weeks' back, but it seemed too simple, so I batted it away.\nYes: this is about as messy as my desk ever gets. Lol.\nLoving | the teddy Kitts made at school: isn't it the cutest thing ever? Her teacher is a Canadian crafter who makes her own shoes apparently \u2013 wow!\n\"What are you watching?\" asked the Rev. the other day. I laughed.\nThrifting | this super-cute drink jar from the shop I volunteer in!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The user sessions that are created when my users log in expire too quickly. Is there a way to create an infinite session?\nIf your users check the \"Keep me logged into [Your App Name]\" checkbox when logging into your app, their session with your app will go on forever and always.\nYou can give users the option of creating a special infinite key code for you by sending them to the URL http:\/\/www.facebook.com\/code_gen.php?v=1.0&api_key=1234567890, where 1234567890 is your app's API key (not your app's ID, but rather the full API key). This will prompt them to generate a key, which they can then give your app and you can pass into the Auth.getSession() as an auth_token. The session_key you get back will survive beyond the sands of time.\nWeb-based Facebook apps used to be automatically granted infinite sessions but now need to manually create them (as of July 15, 2008), the same way that Desktop and Mobile apps have always had to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbcyg b/data_all_eng_slimpj/shuffled/split2/finalzzzbbcyg
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@@ -0,0 +1,5 @@
+{"text":"Gram wears Grandpa's gold wedding band on a chain around her neck, along with the delicate silver heart he gave her for their anniversary\u2014was is 25 or 40?\u2014it's hard to remember, there being nearly 70 in all. In the center of the heart sit six small diamonds, bought with the labor of Grandpa's calloused hands\u2014a stone for him and each of their babies.\nI'm home for a wedding, held in the small Catholic parish where Grandpa's Dad grew up going to mass, in a town where his last name belongs to half the population. I don't think I've ever heard the story of how Grandpa's parents met, and so I ask Gram.\nAt 87 years old, she can recite most of our family history in impeccable detail, but this is a story where she can only offer a guess\u2014probably church, after he moved across the river. Sitting down and recording Grandpa as he explained who we come from, and how we got here, was something I always meant to do, but never did. The dead don't just take their secrets.\nWith a trembling hand, Gram raises a glass of red wine to her lips, making the ice rattle noisily against the glass (don't tell the wine people, she says, but she's always thought red wine was best served cold).\nShe has just returned from a weekend trip to the dusty town of her childhood several hours south, where she laid fresh flowers on her mother's grave; her husband's recent departure a reminder that none of us have as much time as we'd like, and should go and do what we've been meaning to go and do. Living in Grandpa's hometown meant that her history never received quite as much attention as his, but lately, stories of her tribe have begun to pour out of her, as if she doesn't want them to leave the earth when she does.\nI remember the open casket, she says, recalling her mother's funeral, a woman not even 40, and Gram, only 9. She was in a blue dress, holding a bouquet of blue pansies.\nI want these details to live inside of me forever, and so I repeat them, slowly, as if to etch them into the deepest recesses of my memory\u2014a blue dress, pansies.\nI think of this as I wander Grandpa's shop, opening drawers of dusty pieces of furniture, hoping I'll stumble across some long forgotten treasure. A carpenter by trade, and a child of the Great Depression by circumstance, he could never be convinced to throw anything away. On the left side of the shop sits a shelf of empty peanut butter jars filled with screws, nuts, and bolts, above me hang no less than 35 kerosene lamps, and to my right, a set of unfinished kitchen chairs covered in sawdust.\nIn the final years of his life, Mom and Gram would get rid of things when Grandpa was in town for a doctor's visit, or at the Elks Club making brunch on Sunday mornings. Nine times out of ten, he never noticed that there was one less wooden barrel sitting out in the barn, or that his pocket knife collection was missing a Swiss Army.\nWe talked a lot about death growing up\u2014maybe that's normal, when you share a home with the elderly, I don't know\u2014but when he'd scold us for trying to throw away something he could \"absolutely use\", we'd tease him about how long it was going to take us to sort through his shop when he went. He'd just smile, and chuckle from somewhere deep inside his round belly.\nNot all of it was peanut butter jars and rusted over carpentry tools\u2014the treasure was out there, too. The cedar chest he built Gram for her high school graduation, just before he proposed. My quadruple Great Grandfather's medical degree from the Netherlands. The piano his mother grew up playing in the family farmhouse we have shared since I was a small child.\nMaybe holding on to everything was his way of making sure certain things didn't leave us when he did.\nAmong the most sacred moments of my growing up life were the first few hours of Christmas morning each year, when the six of us\u2014Mom, Dad, Gram, Grandpa, Kate, and myself\u2014would gather to open presents before the extended family arrived for brunch.\nI wanted how Grandpa's scratchy cheek felt against my lips, and cadence of his gravely voice saying, \"Thank you, sweetheart,\" (emphasis on the t at the end of sweet and heart) to take up residence in my bones so as to never leave me.\nThe reality I am grappling with now is this\u2014 no matter how hard I try to absorb every small detail, there will inevitably be things that slip through my grasp. There will always be some part of their magic that escapes me, stories I was never told, or will forget if I was, as well as pieces of them I will be unable to capture with words after they leave.\nBud has been gone a little over a year now, and some nights my chest contracts with the realization that my children will not know the sight of him perched in a green lawn chair, talking to the rows of corn behind the barn (it helps them grow, he'd say), or the smell of Jack Daniel's mixed with sweat that he'd carry after rewarding himself for a hard days work. They will only have my incomplete retellings, and it will not be the same.\nBut that will not keep me from telling them\u2014the same way Gram tells me of blue dresses and pansies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This name is in version 1.1 of The Plant List, record kew-2631982, with some changes.\nAlyssum campestre Marz.-Penc. ex Pollini is a synonym of Alyssum montanum L.\nThe record derives from WCSP (in review) which reports it as a synonym with original publication details: Fl. Veron. 2: 362 1822 .\nFull publication details for this name can be found in IPNI: urn:lsid:ipni.org:names:277388-1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ischemic heart disease (IHD) affects a large proportion of VA patients; more than 500,000 VA patients have a diagnosis of IHD. Currently, outpatient performance measures for control and prevention of IHD are based solely on surrogate measures of disease control, such as achievement of a target low-density lipoprotein (LDL) level and achievement of a target blood pressure, rather than initiation and maintenance of appropriate therapy. In addition, measurement data are abstracted manually despite the wealth of electronic data available in the VA.\nThe major goal of this study was to develop and validate pilot data for performance measures for treatment of hyperlipidemia and hypertension in patients with IHD. Specific research aims were to: 1) identify and validate a cohort of patients with IHD using an electronic algorithm applied to administrative data, and 2) develop electronic algorithms to determine prescription of and adherence to appropriate pharmacologic therapy for hyperlipidemia.\nThis pilot study reviewed retrospective records of IHD patients in VISN 20 using data warehouse records from FY 2007 through FY 2010. We defined a cohort of IHD patients and constructed measures of risk reduction using electronic data on diagnosis and treatment of patients. For the performance measures, we used rolling, quarterly adherence measures for a cohort of patients defined by utilization data from the prior 12 months. We developed an algorithm using inpatient and outpatient ICD-9 and CPT codes to identify the patient cohort and used outpatient pharmacy data to construct medication prescription and adherence performance measures. To validate the patient identification algorithm, we reviewed 199 charts (102 with IHD and 97 without IHD). We also conducted an anonymous short telephone survey (49 statin takers and 34 non-statin takers identified from VA pharmacy data) to assess the degree of statin filled in non-VA settings, or the completeness of using VA pharmacy data for statin adherence.\nThe algorithm identified 40,207 IHD patients (FY07: 19,226 and FY10: 20,981) and 85,411 non-IHD patients (FY07: 39,682 and FY10: 45,729). The validation result based on the chart reviews shows that overall 84.4% of the cohort was classified correctly as IHD or non-IHD using this algorithm. Among IHD patients, 91.8% were classified correctly (sensitivity), while, among non-IHD patients, 79.0% were classified correctly (specificity). Based on patient responses to the telephone interview asking where they fill their statin medication prescriptions, we found 2 out of 49 identified statin takers and 4 out of 34 non-statin takers currently fill a statin medication prescription in non-VA settings. For the adherence performance measures, 74.4% of the 3,029 patients who were adherent to statin medications achieved target LDL levels compared to 47.7% of 1,089 patients who were not adherent.\nWhile the validation tests of the algorithm showed moderately high sensitivity, this ICD-9 and CPT algorithm is not sensitive enough alone to develop a high quality cohort for performance metrics. This study does provide pilot data for future studies to fully develop performances measures for hyperlipidemia and hypertension in a national cohort. Ultimately, these studies will help improve both the quality of care, by introducing more accurate and effective performance measures, and the efficiency of performance measurement for IHD patients in the VA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calligaris is an Italian brand with strong ties to luxury giant LVMH (Louis Vuitton Mo\u00ebt Hennessy). This relationship defines its quality and attention to detail, but not its price, as it is still in the market of affordable luxury. Calligaris provides chic contemporary furnishings, with a unique focus on extendable tables, dining chairs and stools.\nBrowse the catalogue below for a preview of their beautiful collection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever wished you could capture a song that lies unwritten in your head, hit record and share that idea with those around you? In John Shirley's cyberpunk novel Freezone, Minimonos are a group of humanoids enhanced with a new technology that allows them to translate music directly from their brains into an audible form.\nThe Minimonos only resistance comes in the form of Rickenharp, an aging traditionalist who believes only in pure rock 'n' roll. The novel climaxes with Rickenharp rescinding his fight to the unstoppable force of evolution.\nTurn back time to 1997 in Florence, Italy, and a like-minded collective of artists were gathering together, inspired by the boundless possibilities where modern technology and creativity were meeting. Operating under the Minimono banner, the close-knit community employed different mediums for exploring the potential the digital age had brought.\nIn 2003, one of the founders Ennio Colaci met with local DJ Fabio Della Torre, and the course of Minimono was steered towards a specific focus on music production. Success came quickly for the pair, scoring releases with respected European labels such as Telegraph, Tuningspork, Oslo and Ethique, all suited to their stripped-back, experimental sounds rooted in the endless funk of house music.\nAt the end of another decade, the Minimono sound embodies a fuller aesthetic, where warm instrumentation and rich melodies combine with an innate understanding of the soulful building blocks of 60s and 70s black American music.\nTo understand the Minimono of today, look no further than their record labels, Bosconi and Bosconi Extra Virgin. Bosconi reaches across the realms of modern day house and techno, but embraces the emotion and rawness of a more classic age in dance music. The largely Italian roster includes Alex Picone, Ilario Alicante, Mass_Prod and of course Minimono themselves. As well as the homegrown talent, they have reached out to the likes of The Revenge, Mark E, Altered Natives and Bruno Pronsato for tracks and remixes.\nBosconi Extra Virgin is more concerned with an organic, beating heart kind of house music where loose, live instrumentation takes precedence over programmed perfection. Hence Minimono decided that their debut album 'Runaway' should be an Extra Virgin release.\n'Runaway' is the culmination of a long process that began with a summer full of inspiration for both Fabio and Ennio. The end result is equal parts floor-friendly house music and downtempo home-listening, with a strong vein of funk running through every track.\nTake a track like 'No Time To Lose', where vocals, guitar licks and fuzzy Rhodes keys come together with a house groove built around drum machine beats and hooky samples. Elsewhere 'Smoking Mind' tips its hat to the chopped up edits of their earlier years, while 'Runaway' is an atmospheric, soaring dancefloor ballad with its yearning strings and horns. 'Weeds' flips the script once again with a low-slung hip hop rhythm and jazzy undertones.\nEmotionally rich and laden with funk, the eleven tracks that make up 'Runaway' pass through many different realms in the world of dance music without ever succumbing to banality or repetition.\nIt's also worth noting that the album cover shows the logo of Minimono, a sculpture by the artist Sandro Mele. The sculpture now rests on the Halcon White estate in Buenos Aires, a lasting testament to the driving principle of Minimono; embracing collaboration and experimentation to explore new means of artistic expression.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbsqp b/data_all_eng_slimpj/shuffled/split2/finalzzzbbsqp
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index 0000000000000000000000000000000000000000..5ce12e1e0494b389e77bb015f2e26831bdc3aeef
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@@ -0,0 +1,5 @@
+{"text":"And Evergreen graduates with a political economy or political science background have gone on to careers in areas such as worldwide relations, economics, journalism, educating, labor and neighborhood organizing, human rights and global justice, social work, public coverage, regulation, and public health.\nFor example, broadly studied points included the impact of elections on the economy and vice versa (political enterprise cycles); the effect of corruption and inefficient bureaucracies; the position of the quality of establishments for long-time period growth; and the results of lobbying pressures.\nOverall, all works of classical philosophers finally led to improvement of theories of worth, wealth distribution, division of labor, nature of change and trade, the origin and use of money, analysis of population, economic progress and copy, accumulation of capital, and public finance, which advocated for better understanding of the capitalist programs ('Brien, 2004, Chapter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From processing Missouri's state and local records to facilitating access to those same records, volunteers are essential to the work of the Missouri State Archives, allowing the Archives to fulfill its mission of preserving and making available the state's permanent records. Volunteers are a valuable resource to the Missouri State Archives and we work with volunteers to provide opportunities that suit their interests and skills.\nThe Missouri State Archives has operated a successful eVolunteer program since 1999. eVolunteers work from home indexing and\/or transcribing records which are then made available online. Some eVolunteer projects, including the current Death Certificate Project, are completed through a web-based portal.\nEach month the Missouri State Archives welcomes hundreds of visitors to the Reference Room and receives more than six hundred requests for genealogical information by mail and e-mail. Volunteers assist researchers with general reference questions, use of finding aids, materials and microfilm viewers; reshelf and straighten books; make photocopies for researchers; answer telephone when necessary; research microfilm records and printed material in response to inquiries; and other miscellaneous tasks.\nVolunteers assist Missouri State Archives staff with the arrangement and description of state records of permanent and historical value in order to provide better access to the records of the State of Missouri. Projects may include both paper records and visual materials, such as photographs, negatives, or slides. Volunteers may flat-fold materials; remove foreign objects (staples, papers clips, etc.); label folders; create lists of folders; and other tasks.\nEvery spring Missouri students attend Archives Alive! performances and tours at the Missouri State Archives. Docents assist staff by leading tours of the lobby, the stacks, the reference room, and famous Missourians.\nThese projects require many different talents and skills and are integral to the operation of the Missouri State Archives. Volunteers ensure a higher quality of service for all our researchers. In addition, volunteers meet new people, gain proficiency in history and research, and take part in preserving Missouri's fascinating history. The Missouri State Archives provides orientation, training, and supervision to all volunteers.\nThe Missouri State Archives accepts for-credit internships for undergraduate and graduate students with an interest in history and archives. Please check with your academic institution to determine their internship requirements.\nIf you are interested in these or any opportunities, please contact the Missouri State Archives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Cuckmere and Pevensey Levels Catchment is in East Sussex and includes the rural landscape of the High and Low Weald to the North of the catchment and the South Downs to the South. Within the catchment is the fast growing town of Hailsham and the coastal towns of Seaford, Eastbourne, Bexhill and Hastings.\nThe Partnership is focused on integrated catchment management and the delivery of the Water Framework Directive objectives to improve the quality of the environment at a local level. The Partnership is supported by a broad range of organisations and individuals, representing a whole host of interests from across the catchment.\nThe catchment partnership is co-hosted by the Sussex Wildlife Trust, the South East Rivers Trust and the Environment Agency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Todas las actividades incluyendo las misas de este fin de de semana est\u00e1n canceladas debido al hurac\u00e1n Florence. Por favor tomen precauciones y que Dios les bendiga.\nAll activities including the masses this weekend are cancelled due to hurricane Florence. Please take precautions and God bless.\nWe have received a grant from Cape Fear Garden Club and need help planting Shrubs and Flowers around the playground parking lot next to the Tileston Building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some people feel refreshed and reinvigorated after catching a nap in the middle of the day. Others feel sluggish and downright grouchy.\nThe opinions on the benefits of napping vary. While some cultures indulge in seemingly healthy afternoon siestas every day of the week, there's also research that indicates people who don't take naps tend to live longer.\nLimit your nap to 20 to 30 minutes. Longer naps can leave you groggy, a condition called sleep inertia. That grogginess sets you up for making mistakes and having accidents right after you wake up.\nDon't nap after 3 p.m. Naps later in the day can hinder your ability to fall asleep at night.\nNap in a sleep-friendly environment. Choose a quiet, comfortable place. Remove any bright lights if possible. Just as at bedtime, limit distractions by turning off your cell phone, computer and TV.\nIf you'd like to learn more about sleeping better, please contact the Fort Sanders Sleep Disorders Center, please call (865) 331-1375 or visit fsregional. com\/sleepdisorder.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbekja b/data_all_eng_slimpj/shuffled/split2/finalzzzbekja
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Members that are Valley Investors and above get 1-2 social media posts on the Chamber social media platform of your choosing. All you need to do is fill out this form and include all content and photos you'd like included. Please select ONE platform. If you are a Community, Connect or Lead Investor, you will need to fill out the form two times. You may select a different platform per post if desired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Maureen Chatfield: Emotions Through Color will be displayed at Hunterdon Art Museum from March 8 until April 28 2019.\nPalm Beach Art Fair (2018), Rosenberg & Co.\nA collection of Maureen Chatfield's paintings will be included in this impressive sculpture exhibition at the Rosenberg Gallery January 24- February 16, 2018. Please drop by.\nMAUREEN CHATFIELD Rosenberg & Co. July 13 \u2013 September 12 See full PDF here!\nMaureen Chatfield for Rosenberg & Co.'s first summer exhibition. More info here.\nHere is a look into the process of \"Jump Start\" on view at the S.H.E. Gallery located at 819 Main Street; Boonton, NJ.\nTraditional Home editor Jenny Bradley chose Twilight to be featured in the Spring 2012 TRADHome blogger edition, Editor's Pick: Favorite Things Click here to view the complete issue and Maureen Chatfield's work on page 26!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"They hired George Wein to organize the first festival and bring jazz to Rhode Island. Here Are The Takeaways From This Year's Oscars. Its syncopated rhythms made dancing wilder than even ragtime, and when the musicians detoured into improvisation, it shifted the dancing frenzy into a higher gear. He lives in New York, but he was born far removed from New York's swinging jazz scene on the Indonesian island of Bali.\nI started to try this instrument. Written by Catherine Welch from NPR. The audience stayed, broke out their umbrellas, and the musicians played. The Newport Jazz Festival is a music festival held every summer in Newport, Rhode Island. This year, Alexander will return to Newport with the Joey Alexander Trio. This years Newport Jazz Festival will take place in Fort Adams State Park the Newport Casino International Tennis Hall Of Fame more info at bottom of this page, Newport, Rhode Island.\nYou can use canned beans for chili. Inheriting the mantel next year as Artistic Director of the Newport Jazz Festival as George Wein, the legendary founder and producer of th. Every summer, Newport Rhode Islands Fort Adams park comes alive with the Newport Jazz Festival. Jazz started out as a dance music. We remember the days where wed grab a cereal box, cut some rings and make a pom pom. You can however look at the shutter count at the last time you took a picture by viewing the metadata of the picture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Board meetings are the second Thursday of each month at 7:00 pm in the office at 4818 Spring Branch Rd. All are welcome to attend.\nJust a reminder that glass and fireworks are not allowed in any of the three parks.Off-road vehicles must remain on the pavement.\nEmail newsletters are sent out. If you have not provided the office with your email and would like to receive the newsletters, please contact the Property Manager at rpoa@gvtc.com or call the office at 830-885-4587.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week, our Editors-at-Large bring us up to speed on literary happenings in South Africa, Central America, and Brazil.\nSouth Africa has eleven official languages, a fact not often evident in local literary awards and publications, which generally skew towards English and Afrikaans as mediums. However, the announcement of the 2017 South African Literary Awards (SALA) has done much to change this perception.\nIn addition to including five contributors to narratives in the extinct !Xam and !Kun languages (drawn from the Wilhelm Bleek and Lucy Lloyd archives), a biography in Sepedi (T\u0161hut\u0161humakgala by Moses Shimo Seletisha) and poetry collections in isiXhosa (Iingcango Zentliziyo by Simphiwe Ali Nolutshungu) and the Kaaps dialect (Hammie by Ronelda S. Kamfer) have been shortlisted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeqxp b/data_all_eng_slimpj/shuffled/split2/finalzzzbeqxp
new file mode 100644
index 0000000000000000000000000000000000000000..742d69e3ff866bf99a8f73380a361f5174561f89
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Hi Elizabeth Wanted to let you know that after my massage with you last week my arm is feeling sooooo much better. I did an upper body workout on Saturday using light weights and there was still a wee twinge in the bicep area but other than that the general movement is much improved and the aching feeling I had all day long has gone. Woohoo! Thanks very much and I'll be booking in to see you again in a few weeks time.\nTo not be from the Glasgow area and in need of release from pain this woman really knew how to help! Very friendly and absolutely amazing at her job! So glad I found her and would recommend her to anyone! Thank you so much!\nJust want to say thanks for what you do. I always feel so amazing after been with you and my back feel so much better!! And you do it so good since I can stay there because normally I really don't like people give me massage because I don't feel so uncomfortable. So thank you so much I really, I really appreciate it!!\nHi Elizabeth I came to see you for a muscle problem in my calf about Christmas time. We didn't really know what was wrong, but you did your 'voodoo'and it has never given me problems again. I ran in the Kilomathon 6.55km yesterday and was 29th out of 104 runners, for my first race ever. Thank you so much!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just as there is light and darkness, good and evil, so there is love and fear. Know that perfected love from above displaces all fear in our experience. Jesus has revealed the Father's true nature to us.\nVia the born again experience, we are regenerated \u2026 re-gened. The passion, truth, and spirit of worship can penetrate our inner person, releasing our authentic free selves.\nCelebrate. We are no longer slaves.\nBreak Through the Negative Sound Barrier!\nHave you felt like a BARRIER was holding you back in an area of your lifestyle or work? Are you weary of the limiting negativity around you? Or even in your own thinking and emotional life?\nHere's the reality \u2026 you can experience a breakthrough!\nPhoto: an F\/A-18 Hornet breaks the sound barrier. Public domain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How much National Insurance will I pay as a director? \u2013 3. How and when will I pay National Insurance? 4. Do I need to register to pay national insurance? national Insurance is more complicated for a company than it is for the self-employed. This is because any earnings that are paid to the director through the company are treated as wages.\nMortgage \u2013 Home Equity \u2013 Frequently Asked Questions \u2013 Wells Fargo \u2013 Learn more about how mortgage insurance works. Although you can't pay your mortgage with a credit card, you can set up automatic mortgage payments so that your monthly payment can be withdrawn automatically from your checking account each month.\n15 Mortgage Questions and Answers for First-Time Homebuyers \u2013 If you're new to the mortgage process. and how much should I expect to pay? There are several expenses that can be included in closing costs, including (but not limited to) appraisal fees, title.\nWhat Is Private Mortgage Insurance (PMI) \u2013 How to Avoid. \u2013 Private mortgage insurance is an actual insurance policy issued by an insurance company that benefits your lender. If your home goes into foreclosure and the lender is not able to recoup the outstanding balance by selling the home, the insurance company that issued your PMI will pay the lender the difference.\nHow Much Do I Need to Add to My Mortgage Payment? | Early Payoff \u2013 Determine how much you would need to add to each monthly mortgage payment to pay off your mortgage loan early. To Pay off the loan in 10 years, you will have to add $919 extra to each payment. interest savings ,745. New Monthly P&I Payment $1,711.\n13 Mortgage Questions to Ask \u2013 and the Answers You Want \u2013 Having a list of mortgage questions to ask potential. insurance is \"lender paid,\" it's likely passed on as a cost built into your mortgage payment, which increases your rate and monthly payment..\nUnderstanding the Mortgage Payment Structure \u2013 At the start of your mortgage the rate at which you gain equity in your home is much slower. but also interest, taxes and insurance. It tells you how long it will take you to pay off your mortgage.\nHow Your Defaulted Student Loans Affect Homebuying \u2013 Also, keep in mind if putting down less than 20 percent, private mortgage insurance, or PMI, will likely be required and increase the amount you pay over time. the mortgage process will be much.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a pause, Paraic King, a local musician, begins his first song.\nIt's the first Chico Unplugged competition by Student of the Arts (SOTA) at Madison Bear Garden, and King's set is the last. His set quiets the crowd\u2014it's raw and filled with soul. His socks rub the hardwood floor as he plays. His eyes squint and his face winces as he plucks the strings on his guitar.\nThe emotion on his face shows his love for music. King performs without hopes of personal gain or career advancement. He loves people and relishes each crowd's individual energy.\nThe Orion sat down with King at his home music studio to discuss his music career. He made his personal studio in a small, well-carpeted shed behind his house, behind a thick yet pleasant-smelling garden. It holds a drum kit, a piano, microphones and more. King is more at ease here.\nHe explains that he's been making music since age 15, starting with piano, and began playing guitar when he moved to Europe at age 20. He's soft-spoken but full of stories. One gets a genuine sense of experience and character behind his words.\n[Q]: Do you generally like playing solo shows, or do you like playing in bands?\n[A]: I just haven't played with people in a long time because it's like a relationship. I'm not down to play with just anyone\u2026 I have to get along with them and have a similar vision\u2014that kind of stuff. And that's kind of hard to find sometimes\u2026 I'll jam with anyone, but as far as like doing a project, I'm, you know, a little bit more picky with that kind of stuff.\n[Q]: What are some modern-day musicians that influence the way you play or write songs?\n[A]: I don't really listen to like a bunch of new music. And I don't know all the cool stuff\u2026 I'll find like one person, and I'll listen to them for like months straight. I'll listen to like the same album\u2026 I went for a solid two months listening to only Fiona Apple, um, so she's a big inspiration. And then I just love Kendrick Lamar so much.\nKing rarely learns other artists' songs. He's recorded himself making music hundreds of times. But he only considers about 20 or 30 of his songs fully developed.\nHe occasionally peers at the floor, layered with small carpets, while talking. He taps his feet\u2014the shoes are worn, with mismatching laces.\n[Q]: Is your music more of a personal hobby, do you release any of your music on SoundCloud, Spotify, Apple Music, anything like that?\n[A]: Um, I put it up there, but, you know, there's probably been like 10 people that will listen to it\u2026 I'm not trying to like sell things. My ideal is just to make music my whole life and give it away for free.\n[Q]: What are your future plans for your music career?\n[A]: I hope to be an old man, one day, that still plays and is a lot better than I am right now. Um, I used to want to be a rock star and stuff. I was in a band that was trying to get signed and do the whole thing\u2026 But I just got so turned off by it all\u2026 It was like a lot of this egotistical kind of stuff going on, and like self-promotion and narcissism\u2026 I got so turned off I just stopped playing music. And then I realized, it's not music's fault\u2026 anything can be corrupted.\nKing feels nervous every time he performs on-stage, but years of practice have helped him overcome these nerves. He also currently sings in two choirs around the community and performed in last weekend's Fall Opera Gala \"Candide\".\n[Q]: Once you got over your performance anxiety, was there kind of a turning point when you started really enjoying being on stage?\n[A]: I've always been good at like crowd stuff\u2026 last night, I saw what was going on and I was like\u2014people can't hear in the back. And that was my thing\u2026 I'm gonna go in there and talk to the people in the back. And as soon as I did that, I felt everyone be quiet\u2026 It's about listening. And I changed the way I played the songs last night, due to the crowd\u2026 I did things I've never done with those songs\u2026 Changed a little bit of the words; changed the way I said them. Just because I felt the energy in the room\u2026 I like that. I just like talking to people.\nCurrently studying nursing at Chico State, King doesn't want to make his money through music anymore. But he hopes to continue playing music for people his whole life. Keep an eye out for him in our local scene.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This quote is particularly true - not just restaurants but even crowds (like after church or gatherings - even family): The physician, who treats patients with hearing disorders, says many clients don't go to restaurants for fear of embarrassing themselves, because they can't understand what the waiter is saying or have trouble following a table conversation. \"It's a big problem.\"\nUltimate Car Sounds would NOT be the way to start my day!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbffyq b/data_all_eng_slimpj/shuffled/split2/finalzzzbffyq
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Join us Tuesday evenings at the Commons Park tennis courts to get a jump on summer and play tennis! You will learn how to hit the ball, volleys, forehand, backhand and basic game rules.\nLearn soccer basics - dribbling, passing, receiving, shooting and everything in between! Perfect for first-time players and still fun for those who already have some soccer experience. Kids will play small-sided scrimmages to learn teamwork. fun, friendly and non-competitive.\n8 session instructional program for boys and girls interested in sports. Learn t-ball, soccer, track & field, basketball, kickball, football and more! Please wear running shoes and bring a water bottle. Fee includes t-shirt. A \"Track & Field\" meet will be held on July 19th at the Fridley Middle School track, and following the event will be an awards presentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At QHR Quality HR We help companies to find and to keep the right personnel- to make sure that they thrive, develop and grow together with your organisation.\nWe support international companies when establishing their businesses in Sweden and offer analysis and development of HR processes and policies in new and existing organisations. We also offer flexible short and long-term solutions within HR.\nOur strengths are Quality, Efficiency and broad knowledge within HR, Business administration and economics. We are passionate about people as well as about profitability and efficiency and our mission is to see companies that develop and succeed with the help of their driven employees and efficient processes.\nImproving and adapting policies and agreements according to Swedish law and collective agreements.\nSupport during negotiations with Swedish unions.\nSupport during job search \u2013 CV, cover letters, LinkedIn profiles, social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Artist Statement Ukrainian photoartist Palinchak Mikhail has over the past 20 years displayed his work in more than 400 exhibitions in the USA, England, Argentina, Austria, Belgium, Hong Kong, Denmark, Spain, Italy, Luxembourg, France... Mikhail is an artist of international federation of photoart (FAIP). Mikhail is the winner of countless prizes and awards.\nIn this gallery shown photos of Palinchak Mikhail and his son Mikhail Palinchak Junior. Many of the photos done in their cooperation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We connect people to the services that matter the most to them. With our fully integrated Smart Pole solution, the only limit is your imagination.\nProvide turnkey solutions using existing infrastructure to ease the deployment of functions and systems in the city.\nCopyright \u00a9 2018 Lumca. All Rights Reserved. Website by Larouche Marketing | Communication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you know that as a Member you have access to the libraries of Leeds Beckett University?\nMembers of the Leeds Law Society may use the Portland Law School and Sheila Silver Library at City Campus, Woodhouse Lane. Access is restricted to print resources only and not the online facilities.\nIf you wish to use the library, please bring photo ID and your Leeds Unlocked card.\nAccess to Portland Building, Leeds Law School Library from 8.30am to 5pm Monday to Friday, please ask staff at the Law School reception: Telephone 0113 812 9028. The Law School reception desk is on the 3rd floor of the Portland Building.\nThe bulk of practitioner books will be held at Portland Building, Leeds Law School Library. The Law collection situated in the reading room is reference only and there are no borrowing rights.\nBorrowing rights are available to Leeds Law Society members via individual membership of the Library which will entitle the holder, at a cost of \u00a375 per annum, to borrow up to 5 items at any one time from the main Library only. Complete a Guest User Registration Form and enclose a letter stating why you wish to use the Library and which resources you would like to use for educational and research purposes.\nPhotocopying facilities are available to members of Leeds Law Society. A \u00a35 refundable deposit secures the loan of an i-print copy card from the Help and Information Point in the library. Members can then add a minimum of 50p credit (non-refundable) to the card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgzwo b/data_all_eng_slimpj/shuffled/split2/finalzzzbgzwo
new file mode 100644
index 0000000000000000000000000000000000000000..cd623c25b95c84c87352025e13653d1bc6ef0edd
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Recently, I have had an increase in clients who live in other states and want to retire in Fort Collins. Why wouldn't you want to retire in Fort Collins? It was rated the number one place to retire by Money Magazine many times and not only that property taxes here are some of the lowest in the nation as are utility bills. Buying Fort Collins Patio Homes is a good way to invest and live in a relatively maintainence free home. The weather is relatively good with approx 300 days of sunshine a year so you don't have to do a lot of shoveling of snow in the winter. We have a vibrant Old Town with many great restaurants and shops that from some neighborhoods are easily walkable. It is also a college town which attracts numerous kinds of speakers and activities.\nHousing prices here are still relatively low compared to many other areas of the United States. Lots of great neighborhoods with walking and biking trails. There are even a few golf course neighborhoods that appeal to the young at heart.\nOne level home \u2013 ranch. All main living is on the main floor including the laundry. HOA maintains the exterior of the home and maintenance. There is a HOA fee associated with this but the peace of mind to be able to \"lock and go\" is priceless. Get out and enjoy your life without having to worry whether your driveway is plowed or not.\nOutside maintenance taken care of an HOA \u2013 this may include snow shoveling, lawn maintenance and trash, it may not. With an associated fee.\nEvery HOA has their own rules and regulations so it is best to check with each one and find out what it does and does not cover.\nIn Fort Collins most of our homes include a basement this is just something that you will need to realize, whether or not you decide to use that space is up to you. You can just shut the door and never go down there. About 95% of the homes here have a basement space of some kind, even Fort Collins Patio Homes. So to search for a home without a basement is just not realistic. And for resale purposes you would not want to do this either since you would be excluding a huge percentage of the population that wants or needs a basement whether it is finished or unfinished that is a different story.\nI hope this answers some of your questions on what exactly a patio home is. If not feel free to contact me for a more detailed explanation and we can schedule a showing so you can check out some Fort Collins Patio Homes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"100% pure organic Lavender flowers that are grown in France! Lavender is known for its pleasant aroma, calming aromatherapy properties, and lovely purple hue. Unaltered and unflavored; our high quality Lavender is intensely aromatic, with a floral sage-like and bittersweet distinctive taste.\nLavender is known to promote relaxation; it is effective in the treatment of restlessness and mild insomnia. Consuming pure lavender flowers tea is known to elevate the mood- making it beneficial in stress related issues such as anxiety and depression. Lavender has a long history of use for easing headaches and tension. If you're feeling overwhelmed or stressed, a warm cup of Lavender Flowers Tea will give you an instant sense of relaxation, and get you back on your feet with renewed focus. Enjoy our high quality pure Lavender Flowers Tea straight, iced or in your favorite recipes!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AN incident recently came to my knowledge illustrating the spirit of the pioneer mothers of Oregon.\n\"Elizabeth, I want to talk with you. I have made up my mind to go to Oregon. Now, you can do as you please\u2014go with me, or go home to your father and mother with the baby, and wait until I go out there and get a home started, when I will send for you or come back after you.\"\n\"Francis,\" she said, \"when I married you I left my father and mother to live with you; and when you get ready to go to Oregon I will go with you.\"\n\"My child, come back home with the baby and stay until your husband has a home ready for you in that far-away land. Just think of it! There is nothing out there but savages and wild beasts. Mr. Perry will necessarily have to be away from home much of the time in order to earn the means to make a start with; and he will return home some time and find you and the baby murdered or destroyed by wild beasts. My child, don't go, don't go!\" And tears came streaming from his eyes.\n\"Father,\" Mrs. Perry said, \"my duty to my husband impels me to go with him. When I married him I left you and mother to help him make a home; and now with all possible love and respect for you, my deepest convictions are that I must go with him.\"\nThen the mother of the determined young woman spoke up and said, recalling her own young married life, \"Father, Elizabeth is right; don't talk to her any more!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Met Office has issued a 12-hour warning of ice in the East Midlands.\nWeather experts have issued a warning of ice for the East Midlands.\nThe Met Office yellow alert is valid between midnight and noon on Tuesday.\nA Met Office spokesperson said: \"A band of rain and hill snow will move southeastwards across the UK during Monday evening and overnight.\n\"Once the rain has cleared, some hail, sleet and snow showers will follow from the northwest, with one to three centimetres above 200 metres and some small accumulations expected at lower levels.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"We were originally going to do it the other way around, then last second, we switched it to the way it is,\" says Johnson of the casting, adding that, whichever roles they were going to play, they knew they'd have a blast working together.\nWhich is good, considering that no-filter Ryan gets to have most of the insane fun, while uptight Justin takes a pounding \u2014 in both body and pride. He does get the not-insignificant consolation prize of a possible romance with only-in-L.A.-hot waitress Josie ( Nina Dobrev). But it's a nice change-up from their New Girl roles, in which Johnson embodies Zooey Deschanel's love interest, low-key, cerebral, slacker-loser-guy Nick, and Wayans plays their aggressive roommate, Coach.\nHe credits other cast members, mainly Keegan-Michael Key, Natasha Leggero and especially Rob Riggle with significant contributions.\nWayans cites a diner scene in which the two fake cops suddenly pull their guns on each other, to the horror of unsuspecting patrons, as one of his favorite of Johnson's ideas. Johnson, meanwhile, enjoyed seeing his friend get climbed on by a filthy, obese and totally naked lunatic.\nJohnson's drinking has been productive in the past as well: His inebriated retelling of the events leading to the death of R&B great Otis Redding inspired buddy Derek Waters' Drunk History shorts and eventual series. Wayans, eldest son of comic actor Damon Wayans (who plays Hard Rock Live Friday night), was named a staff writer on his father's ABC sitcom, My Wife and Kids, before reaching 20.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhilz b/data_all_eng_slimpj/shuffled/split2/finalzzzbhilz
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index 0000000000000000000000000000000000000000..5152cd112531d146eca465452183696e27dd7fb8
--- /dev/null
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+{"text":"It appears that the obesity epidemic sweeping the United States is quickly spreading around the globe and now it has Europe in its caloric crosshairs. A recent joint study by the UK Health Forum and the World Health Organization proclaim that by 2030, 85% of European women and 89% of European men will be overweight, with 57% of women and 48% of men being obese.\nThe study examined 53 European countries. Leading the way in these \"hefty\" figures were Spain, Greece, Austria and Sweden, but one small island appears to be pulling ahead of the pack: Ireland.\nFigures from a 2010 survey of Ireland show a whopping 74% of overweight men, with 26% classified as obese. Among women, over half are overweight (57%), with almost a quarter of all women (23%) being classified as obese. And if their forecasts prove true, the number of obese women in Ireland will more than double by 2030.\nStudies project that in just 15 years, Ireland will overtake other European nations as the most obese country in the region. And like the US and elsewhere, the culprits are the ever-expanding portions of calorie-rich foods, combined with alcohol consumption and lack of exercise.\n\"We already have a devastating problem with obesity. It could mean we will follow the United States where one in three Americans born in the year 2000 will have diabetes by the time they are 50,\" endocrinologist and obesity expert with the University Hospital in Galway, Dr. Francis Finucane says. \"That is staggering. This data here suggests the problem is getting worse in Ireland as well.\"\nNot all European countries, however, share such dire predictions. The Netherlands stand alone in their ability to combat obesity, predicting less than half of Dutch men overweight by 2030 and a paltry 8% obese, despite their current rates of 54% overweight and 10% obese. Among Dutch women, their current rate of 43% overweight is expected to remain stable, yet their rosy outlook predicts the percentage of obese women will fall from its current 13% to 9% in the next 15 years.\nIf the Dutch numbers hold, I hope they let the rest of the world in on their secret.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"hill country modern | Vim & Vintage - design. life. style.\nWelcome to this upscale home, located up the road from me in Wimberly, Texas. Designed by Cunningham Architects out of Dallas, this modern treasure is as beautiful as it is energy efficient. Among the green building principles are use of local materials and harvesting rainwater on the rooftop.\nA few pictures of the interior!\nDon't forget about the rooftop patio. Ballin!\nI can already picture the rediculous pool parties that would happen here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The \"Litter Gitter\" pre-dates \"Git-R-Done\" by at least a decade and is Bergan's prized solo presence in the litter category.\nIt's a green alternative to traditional clay litter, safe and chemical free.\nThe wicker cat litter box cover is a great way to hide unsightly litter boxes.Pan Cover Jumbo Dark Brown 23.5\" x 18\" x 20\"\nMr. Herzher's Litter Pan Cover Large Dark Brown 20\" x 16\" x 19\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Absolique Hair Health Clinic, Sydney \u2013 would like to thank all of our clients for your honest feedback, comments, stories, journeys and commitment to the Hair Loss Treatments recommended. Your continued dedication to your hair health is what delivers your results. Congratulations!\nI went to Absolique to get help as my hair was not growing and was very thin. I was explained everything in great detail, and helped with my decision to proceed with the rejuvenation of my hair. I have been on this process for less than a week and although it seems mind blowing to start off with, when you get into the rhythm it is not too bad, I am looking forward to a wonderful head of hair, and with Absolique by my side I am sure this is going to happen. Thanks to all.\nI am about to order and trial some products from Absolique. I will let you know my progress once I have tried them.\nThe service I received has definitely addressed my hair loss; the results are quite extraordinary.\nI had previously gone to a dermatologist who was completely unprofessional and didn't explain anything to me! Just had a quick look over my hair and gave me Minoxidil. I was a little apprehensive to use it and wanted a second opinion. I found Absolique and they were professional and friendly right from when I walked in the door. I was surprised at the level of information I received. It was a relief to know that 1. There was an alternative to Minoxidil and 2. That I didn't even need to take it.\nNew routine working for me and pleased with results so far.\nVery informative and interesting, very happy with treatment.\nA helpful and informative session and routine.\nI feel very informed and positive about steps forward.\nVery interesting consultation and informative.\nVery informative and encouraging consultation. Carolyn is great!\nCarolyn, I would not have thought that improving my hair growth was possible at my age 8 months ago. I was very depressed when I first visited you because as far as the medical practitioners were concerned medically there was nothing wrong and I was told that hair thinning was normal at my age. I feel extremely grateful for the program you have developed and the knowledge that you are sharing via your business here in Brisbane. My confidence has vastly improved in both my private and professional life. Thank you for helping to restore the hair that I was born with.\nMy hair is starting to thicken up as I have now done 7 Laser treatments. I went to the shops today, without a hat, for the first time. I just loved the feeling of freedom. That is what it is \u2013 freedom. No hat, no worry, no feeling of embarrassment. Thankyou Carolyn for giving me back my self-esteem.\nI have just finished my 8th session of laser treatment and am beginning to see some results.\nEach time I go in I am eager to watch for improvement. It is a slow process but well worth doing. The whole team are very informative and supportive.\nProceeding with the Laser treatment and all is going well. I am more aware of the time it takes for my hair to grow but I have good vibes from Carolyn whom I trust.\nAll going well at this stage. Nearly ready for another haircut.\nI am feeling positive about my hair and haven't felt this way in years. After seeing my microscopic results side by side from month one to month 4, I am amazed at the progress. Carolyn has been so thorough and kind throughout the process and I am thrilled that I found her!!! As a client from the USA, I have seen various doctors for years concerning hair thinning, and none have the knowledge and know how that Carolyn does. Hair loss is not permanent! I beleive this now. Yay Carolyn!!\nI have just completed my first four months of treatment and had my scope session. I am thrilled with the results so far and cannot thank Carolyn and her team enough for the support and advice offered during treatment. Friends and family have commented on how good my hair looks and I have a lot of new regrowth that was not there when I started. I have now moved onto the maintenance phase and am excited to see even more results in my next scope session in four month's time.\nOnce again I have realised how much circumstances affect my body. I had a reading done a couple of years ago and she said I have a life that has been and probably will be full of bad luck. Last August I was doing well and I was happy with my hair. Then September I fractured my hand. I didn't realise until I came in for a Scope Test how much this inflammation would impact on my body,stress from work and there are a few other reasons I have gone backwards a little bit with my hair. I'm going to get back on track and hopefully in four months I will see a significant change and will be happy with my hair and me \ud83d\ude42 Thanks Trace.\nLaser treatment number 6 coming up. Positive thinking is making me feel that it is working. I had a SHORT haircut last week and it looks better already. I know it will be great when it is all over but it is a slow process and I must be patient. Looking forward to my next 4 monthly visit when Caroline can tell me about the progress we have made.\n4 month visit. We are making some progress. Starting laser treatment in a week or so. Looking forward to some more positive results in another 4 months. Patience is the key.\nGives me hope about my hair J Very interesting information.\nHi Carolyn, I was really impressed with your knowledge, empathy and understanding you gave us at our appointment recently when my son and I come in to your clinic. Thank you for the time you spent and all the valuable information you gave us. Very much appreciated.\nI had my first visit with Carolyn to find out if I can do anything about my fine thin hair and found her to be informative and easy to talk to. Carolyn sent me away with a list of costs and the treatment information I will need to get back on track. Now I have all the facts etc. I look forward to my second visit which will get me started on my suggested treatment plan.\nA lot of info to digest \u2013 thankyou.\nHalf way through the treatment programme Persisting with the routine and remaining disciplined with the nutritional supplements. Finding the PHs quite difficult to shift. Constantly hovering around the 6.0 \u2013 6.2 mark. Now visual results as yet. Remaining hopeful.\nAs always great service and look forward to my next treatment.\nMy pH has been better now that I'm starting to focus on it again. I slackened off the past couple of weeks getting into the same old routine. I've noticed again that it becomes extremely alkaline whenever I drink cider. This doesn't make much sense at the moment because to me I'm drinking a processed (bad!) alcoholic (double bad!) drink. So I went back to doing some more experiments on my pH levels again. Such as cutting out dairy (which seems to make a small amount of difference, probably because I wasn't having much dairy to begin with, only a sprinkle of Parmesan cheese or some milk in my tea). I also started cutting out bread (not carbs, just bread you buy at the bakery) \u2013 now this seemed to do a bit. Not that I was eating bread too often, about 1 or 2 times a week for a sandwich, but my body became happier when I eliminated it altogether. On another hand, I also found that my sleeping habits didn't really affect my ph. Whether I get 9 or 5 hours sleep, it didn't make much of a difference. So I can almost eliminate that. My stress levels and relaxation levels are the next to monitor. I took a weekend off and headed to the beach on the weekend and did absolutely no work whatsoever. Now I need to have a couple of stressful days of just hard-core work to see if that makes a difference. Oh and just as a sidenote, my diet is fairly balanced (well I think so anyway. It might not be!). I don't drink any energy drinks whatsoever, literally just water and sometimes on the odd occasion cider or tea. Nothing else. My food still only consists of red and white meat (beans and eggs for protein as well as a substitute), plenty of veggies, fruit, sometimes carbs like rice in a stir fry or spaghetti Bolognese. If I don't feel like a heavy pasta dish I'll have 9 Grain Crackers with avo, tomato and Parmesan cheese on them. I try to be very controlled with any other foods such as freshly baked cakes or chocolate bars or ice cream, etc. So much so that I've made a deal with myself to force me to stay away off the junk food until Christmas (or some exception-type day like my birthday!). It helps me stay motivated in healthy eating. Plus I am also keeping it relatively 'normal' eating so that I can see what exact foods are affecting my pH! It makes it easier for my body to deal with. My exercise is average but I do exercise at least 3-4 times a week. Which brings me to my next query. What if some of the pills and lotions\/sprays that I am taking for my hair treatment are alkaline themselves? Or maybe they could be acidic? For example when I wash my hair and do the needle brush therapy. Are those liquids alkaline or acidic? I suppose in the end, the body needs both to function properly. I took a blood test at the beginning of my treatment before I started, and I will take one when I finish to see if any of my levels changed. That will be interesting! I will keep you posted on it when I do!\nNearing towards the end of my clinic treatments and must say I am looking forward to the final assessment and then moving forward on my own journey forward post re growth of hair!! It has all been worth it but looking forward to and hoping to be able to cut back on tablets and products.\nI've noticed a difference again this week. My hair is thicker and certainly the coverage over my scalp is better as well. It's just two more weeks to go for the sessions so I'm very curious to see if all this hard work pays off!\n1 more treatment to go in the clinic! Very excited and looking forward to the scope in another 6 weeks' time!! Feeling really happy with the progress of my hair and there has definitely been a big change! It has been a success and a journey that has been so worthwhile.\nIt's nearing the end of my official 4 month treatment and I have seen good improvement with my hair. I am hoping I don't have to take all of the health pills after I finish the cycle. But who knows. Fingers are crossed anyway! I am looking forward to getting the scope done and seeing just how my scalp is doing too. Hopefully my hair won't go backwards. I definitely have plenty of questions for my last treatment!\nHad my second last session this week. I'm definitely seeing progress but I'm beginning to realise that I do have a long way to go. But better late than never I suppose!\nIt's only 1 month until I get my end of treatment scope done. I can't wait to see what my hair and scalp feel and look like. I have noticed a difference which I'm happy with. it all comes down to time, doing the treatment and persistently taking the pills.\nAlways a pleasure to get my weekly treatment done. See you soon!\nAbout to attend my last treatment later today \u2013 am quite relieved I've managed to get through the 12 weeks \u2013 am looking forward to the final scope session!\nGreat team, awesome results! See you today!\nIt has been 6 weeks since I first met with Carolyn and started treatment. The condition of my existing hair (which was very light and fly away) has improved a lot thanks to using Activance as well as doing the brush therapy and scalp cleanse. I have not yet seen any new thicker strands of hair growing but hopefully that will start to show in the next few weeks. The vitamin therapy is a bit of hard work but totally worth it if it means I'll have healthy hair! I have decided to do the 'in-salon' treatment once a month to give myself a break and then do the rest at home. I must admit, the in-salon treatment is so much more enjoyable (the at-home weekly treatment is a bit stressful!) and the staff are both professional and friendly.\nSo I had my last session in salon session. It's been a real journey. I can't wait to see what my results look like. People have noticed my hair and have said it looks really good now. It's very exciting!\nI don't know if anyone else finds this difficult.. but trying to work the 2nd lot of pills and green drink for the day into my diet can be a bit tricky! I think I'd always prefer chewing on a nice warm steak and salad for lunch instead of taking them.. but then I think of the progress I am making and choose the other option. The other meal will just have to wait until dinner. Isn't it interesting how just doing this routine consumption of minerals and vitamins forces us to plan ahead more. Even if we take it into a bigger scope of things..planning when to take them if you're having a dinner out with friends on Friday.. or if you don't get home from work late at night and you have them at 9pm.. maybe you forgot to have them, or you chose that warm meal instead of them.. simple daily choices that are given to us or that we choose to take on can challenge us, just like committing to this routine can perhaps make our minds stronger in other areas.. \"I will go to the gym before I take my pills\" or \"I will finish writing this article and then eat my lunch\".. little goals like that can make all the difference in productivity! Making our rewards for hard work all that much sweeter. \u2026and so\u2026 I religiously take the 2 lots of pills to hopefully reach my desired outcome.\nBeen better with the regime this week! The team got me back on track last visit and addressed my lack of resolve when deviating from a standard work day. Am still looking forward to only having to do treatment once a day though \ud83d\ude42 Am going to try taking the green drink and pills with almonds tomorrow morning instead of breakfast. If this works and I don't feel queasy I will continue the trend well past the end of my initial 4 month course. Will be much easier to do the treatments this way if the almonds help my body to avoid feeling ill doing the treatments, especially with nothing in my system after waking up.\nComing up to treatment 6 I think? I'm starting to check my scalp now to see if any new growth is coming through.. there could be some hair follicles about to sprout as there are a few dark pinpoint spots on my scalp so am hoping these are new hairs, time will tell!\nI'm getting pretty excited\u2026Karen could apparently see the some tiny hairs growing through\u2026\u2026.it gives me motivation to continue with the regimen\u2026..because it is time consuming\u2026..but I'm beginning to see it worth it.\nThis week's been a bit up and down with the entire pill routine.. I've had what feels like a very busy week, and fitting the 2 lots of pills into that schedule has been hard! I still keep it up and don't fall behind on any, it just feels like a lot more effort this week. I wonder if a busy schedule hinders the hair growing process? Maybe relaxing a bit more would help? I wonder if emotions change the pH?\nWell time flys when you are having fun\u2026and success! Only 5 more visits left then time for my scope session, which I am very much looking forward to. Finding out last week from Carolyn in regards to food combinations and PH Levels has been very informative and hopefully can give me better results going forward. Again thank you Carolyn and your team for your continued support. It has made this journey so much easier.\nLast treatment appointment tomorrow \u2013 after this week it will be evaluation time! I am excited to see how much more healthy my hair will be \ud83d\ude42 I am just as excited about having a clean scalp again to be honest; this scalp cleanser and brush therapy is honestly the best.\nAbsolutely fantastic service. Going to miss you all when I'm away.\nThis past week has been pretty average with my eating habits. I have not even tried to eat as alkaline as I can and my pH shows it, being between 5.5-6.0 on average. My sleep has been ok, and I missed one pill taking session on the 13th of Sept, which was a Saturday afternoon. The head and scalp treatment is going really well though, I keep that at a consistent 2-3 times a week when I wash my hair. I think I need to just restructure my diet a little and go back to eating more salads! My body's probably crying out for it too.. so here's to that extra 15 minutes in the kitchen.\nI think it's now week 7 and I'm halfway. There's definitely progress but I'm still impatient. I feel good about my hair even wore a hat and didn't worry about hat hair!\nSeem to be at a bit of a plateau. 4 more visits to the clinic. PH has been very average and seems a never ending struggle to get it in the correct zone! But onwards and upwards and determined to get it right! One thing I can definitely say is that things are certainly better than what they were 2 months ago.\nLast week I missed one session of pills due to terrible eating habits and not feeling hungry enough to take them. This week has been much better.. I feel less hungry and am controlling my intake of food a little more and not going crazy. Sometimes that's really hard to do, especially if something delicious is in front of me! My pH is getting back up to its normal reading of 6.4. I feel that my hair is starting to really strengthen, thicken and grow. There's definitely a difference when I look at it and brush through it. I'm about half way through my 4 month treatment so hopefully my hair doubles its success rate again by then. I think I'm going to be happy with my hair at the end of this. The hard part is keeping up the looking after it part and making sure I keep up healthy habits like eating and exercising and treatment, etc. Onwards and upwards!\nI had had an itchy scalp and sometimes also on the nape of my neck for about 2 years. I had been to several doctors who said it was seborrheic determitis and had prescribed products all containing steroids which didn't help. In June my hairdresser told me about tricologists which I'd never heard of. So after some internet research I found Carolyn and her clinic and saw her early July. After an examination she said it wasn't dermatitis at all , just flake and was relatively easy to fix. So I started using her Scalp Cleanser and Activance and it all cleared up within a couple of weeks. I am so grateful to Carolyn and would recommend her to anyone else with scalp problems.\nI'm pretty sure it's my 10th treatment later today. I don't have anything to report yet but fingers crossed that these last 10 weeks of following this programme will reveal some results, fingers crossed.\nOnly 2 more clinic appointments to attend and then that is my 3 month mark. Have really noticed an amazing change in my hair and are so glad I was able to commit to Absolique to help me on my journey. I would have no hesitation in recommending this to anyone and have the proof on my head to prove it works! Thank you to Karren, who is a breath of fresh air to catch up with every week, and the remainder of the team who are always happy, helpful and supportive.\nMy ph was all over the place\u2026. I guess it's a good think I'm taking the green drink, which isn't so bad. Am noticing more and more improvement which is really good. Only 2 more weeks to go\u2026.. People are also beginning to notice which is really nice.\nGood Morning Ladies, Karren looked after me last week and I was very happy with her professional approach to my treatment. She did a very good job.\nGreat to know the girls are capable of a laugh, came into clinic with a wig on saying how impressed I was with my treatment so far. Mental note to self, tilt head back properly in shower to avoid tonic going in eyes. Also wear glasses during application of re-Stim, as I missed the bottle and tipped half of it out! Oopps. Great team at Absolique, look forward to in clinic next week, thank you ladies.\nI have just completed the first week of my journey. In regard to my wellbeing I do feel better from taking the drink and tablets. I had previously had the flu for 10 days and feel these have boosted my immune system immensely and helped me recover a lot faster than usual. If this is one week I am very excited about the coming months. Thanks again to Carolyn and her team for their help, support and encouragement.\nI am in my second week of the re-growing hair program with Absolique and the one thing that is getting slightly better is the routine of it all. There are still a few forgetful moments, like brushing my scalp, but I'm getting there. The natural pill supplements are easier to take some days than others, but either way, they're still being taken, so I don't miss out on taking those. And apparently only 4 weeks until I can start to see visual progress on my head, so until then. Keep going.\nBack on the (acid) buffer after a long spell. Ran out of product and then kind of forgot about that step! Everything else going to plan. Have started the new Prostate formulation too.\nFirst consult with Carolyn last Friday and it was promising to hear that my hair issues are reversible. Looking forward to getting started and seeing some results in due course. So far, I've had my programme induction and one treatment, due for my second treatment today. I'm getting used to the homecare routine \u2013 it takes me about 45minutes to do if I'm organised and have everything ready, so it's not too bad really. The green drink isn't too bad; I'm finding it easier to drink it in 3 mouthfuls and get it down quickly. Very early days so nothing to exciting to report but we'll see how it all goes!\nSo very sorry to hear that Lena is leaving the salon but wish her all the very best in her new endeavour. I'm sure her replacement will be fine as all the girls are very well trained.\nBarrel of laughs along with fabulous service. Wonderful team and I look forward to my next treatment. Almost lost the lid to my Tonique down the drain but lightning fast reflexes and grabbed it the nick of time!\nWeek 4 and there's not much just yet to report. However I am enjoying going to the salon and getting the in store treatment done. The girls there are quite good and sensitive to the whole hair issue which is important to me. There's a lot of stigma attached to not having hair.\nI am now just beginning the 3rd week of my treatment. I am really happy with the changes to date and my PH now has a steady 7.0 which I am quite astonished by in such a short time. I am very glad I went with the option of weekly visits to the clinic as it is great to have the guidance, monitoring and all my questions answered on a weekly basis. My sincere thanks again to the team.\n3 weeks in and there's not really too much to report. The routine of it all continues to get better. The pills still don't taste any better. My pH is still consistent, but I'm going to play around with what I eat this week and see what makes a difference. This could be interesting.\nHaving my 3rd treatment later today \u2013 enjoying the downtime during the treatments. Have started noticing I have a bit more energy; hopefully from the supplements. It can be a bit of a pain taking the supplements day and night but have gotten into a routine now.\nMy last appointment is on Tuesday next when I receive the results from my scope session. The results from my treatment has been amazing and I couldn't be happier, and I look forward to further improvement as the weeks go by. A few months ago I couldn't have imagined the improvement in my hair. Carolyn I just can't thank you enough you are amazing. Lena thank you for my enjoyable weekly sessions, I will miss seeing you every week. Thank you all.\nWeek number 4 of my treatment. Has been hard last week to keep my PH in the correct zone! Otherwise all is going well. My hair is feeling good and is definitely in a lot better condition. Hopefully it is doing what it should below the surface!! As always looking forward to my Friday Clinic session.\nOk, another week down (treatment 4 tomorrow) and I am definitely noticing I have more energy and feel sharper and less foggy since taking the supplements. No change to my hair yet but the burning sensation I usually get from time to time seems to have stopped which is promising.\nExcited to see visible new hair growth. Yay, thanks ladies for the wonderful efforts to date. I look forward to future results.\nGood Morning Ladies, after talking with Lena last visit I re-read my Easy pH booklet. Well worth the read again because I had forgotten much of what was in the book and it is very informative.\nI haven't noticed any difference with my hair, however realistically I don't think new hair is going to visibly sprout out in just 4-5 weeks. Patience, patience.\nNeed to up my game with stress reduction, improve emotional food binges and hope to see better pH results in future. Amazing how everything effects our body. Roll on 7.35-7.45 pH, you can do it! Wonderful team at Absolique, always look forward to my weekly in clinic treatment.\nWe're still seeing some progress, though I haven't seen any new hairs yet, but certainly things have stopped falling as much which is a relief. Little things like photos and selfies look better and I'm beginning to feel more confident\u2026..am getting impatient slightly but I know these things take time and I need to keep at it\u2026this is a lot easier than said when you've been unwell!\nMy 5th week in and not too much to update on really. Still finding it hard to keep my PH levels up, but otherwise still keeping with the programme and weekly visits to the clinic. Hoping everything is doing what it should and the truth will be told when I have my scope session at the end!!\nGood Morning Ladies, I realise how important it is to make sure the roller is free of hair for maximum function.\nAnother week of treatments \u2013 it's getting to be ok now and even the green drink isn't so vile! I'm finding that there are other benefits to all the vitamins I'm taking including better nails and skin. My hair does feel better as well!\nAnother fabulous in clinic treatment. Obvious new hair growth. Look forward to next session.\nMy final treatment is coming up & it's come to the end of my salon treatment. Looking forward to seeing results with Carolyn at the final assessment.\nInto my 4th week and cannot seem to get my Ph anywhere near 7, even with religiously applying acid buffer. Will try slightly longer sleep cycles, but wondering if the time we eat dinner can affect the result?\nStill battling to get more than one really green alkaline reading each week, but all is well!\nWhen Lena was doing the brush therapy, she said that she could feel and see lots of new hair growth. Very happy!!\nI always enjoy my appointments and hallelujah there was more of Carolyn's miracle conditioner this made me very happy! I received so many compliments on my shiny hair.\nI believe I have been following the treatment for a month now, I notice my hair more thicker and my barber even complemented on my hair.\nI have only two more sessions left before my scope session. I am very happy with my results so far, my hair is much thicker and it's looking healthier. I'm so glad I found Absolique Hair Health Clinic, as it has made such a difference to my life and self-esteem. Thank you Carolyn, Lena and girls.\nGood Morning Ladies, Well I went to my hairdresser on Friday and even she was excited about my new hair growth. She was asking lots of questions about the treatment so she can spread the word to her other 'needy' clients. The slightly darker colour seems to accentuate my scalp a bit though.\nFriendly service and got a referral to a great hairdresser.\nI have lil hairs growing here & there not sure if they will last & stay strong and grow longer and past the breaking point I usually have. My crown still feels very weak & fine and in general we'll wait and see.\nThe customer service at Absolique is 5 stars. They never forget the simple things you like when your there for your treatment. And very friendly and don't make you feel uncomfortable.\nI have now completed my weekly appointments and I'm very very happy with the results. I could not have believed how much my hair has improved in this short time, to me it is incredible. My scope session is coming up shortly and I'm very anxious to see the comparison. Thank you Carolyn, Lena and girls from the bottom of my heart I am so grateful.\nI am now up to my last appointment prior to my scope session. My hair is looking and feeling so good, and I have my self-esteem back again I am so very happy with the results. Carolyn is so knowledgeable and an expert in her field. Many thanks Carolyn, Lena and girls, I have thoroughly enjoyed each and every visit.\nI have been taking my 'pills and potions' for about 9 weeks now but I sincerely hope that the green drink doesn't play a huge roll in the grand plan for my hair. It is the one thing that I have trouble with and doubt that I will continue with long-term. After all this time\u2026.. YUK!\nIt's only week 2 and I'm already becoming impatient with my hair! Still I know that I need to stick at it and I'm hoping I get really good results. My scope session last week showed how much damage there was and I find that quite confronting. Let's see how the next few weeks progress!\nHi, my name is Robin and I've just started a 4 month program with Absolique Hair Health Clinic to restructure and re-grow hair at the top of my head. It's been quite fine for some time and there hasn't been a lot of it since I was in high school.. so I thought I'd research a bit more about it and find out what I could do about it. I read many pages on the internet about getting laser\/implants for my hair and eventually I came across Absolique. They didn't do a laser treatment.. I was intrigued, so I read more about them and eventually gave them a visit. During the initial consultation we discussed my situation, what we could do about it, and what Absolique offers. Their results seemed legitimate so now I'm doing their 4 month plan to grow more hair. I'm only 1 week into it so we'll see how the program goes. I can tell you now though, that altogether the 'medication' (minerals and antioxidant tablets) I have to take for the program does not taste like chocolate cake.. but the women in charge assure me it will give me good results. Getting used to consuming them often and consistently, as well as now having a new 'how to wash your hair in the shower' routine is a bit of a challenge, but with time it will (hopefully) become easier.\nThis week will be especially tough; have a work conference so will be ducking up to my room during breaks for the glorious green drink and supplements lol. Not quite at the halfway point yet \u2013 can't wait until September \/ October so that we can start seeing initial extra growth results!\nHad first in clinic last week. Informative assistants. Now adapting to home treatment and taking that spectacular tasting green drink!\nMy hair is feeling a bit dead at the ends but think it's due for a trim. Can't believe I'm at the end of my treatment. So glad my hair is growing.\nHi guys, Not so much feedback but a bit of progress check. Been battling to shake the flu this week and have had to go on antibiotics. It's been a doozie!\nLittle hairs are starting to come back. And my hair is 10 times thicker than it was.\nI had my final in clinic treatment and am just blown away at my results. I can't believe, how much new growth I have and am definitely looking forward to the final results.\nThe green drink isn't getting any easier \ud83d\ude41 but my hair is feeling a little fuller, now I just can't wait for it to fill in a bit more in the front.\nOkay week twelve. I have had a few comments this week that my hair looks really good. I have noticed up the top of my hair where it has grown thicker my curls are tighter and my hair is definitely a lot stronger now. Going for a haircut today so hopefully my hairdresser will see a difference in my hair. I'm tempted to get my hair cut shorter so my hair will look thicker but I'll look like a poodle head lol so I just need to be patient. It's going to be strange not going into the clinic for the next few weeks. I've gotten a bit use to my pampering and chats. I have learnt a lot about myself over the past couple of months, always good to keep learning and to keep improving oneself. Now to count down the weeks to find out how successful the treatment has been and where to now \ud83d\ude42 Thanks.\nHi, I would like to appreciate the hospitality of your clinic. Customer service is good and I always enjoyed the time I spent in your clinic from past 2 months.\nI am now at least half way through my treatment program and already seeing results, my hair certainly looks and feels much better. Once again a colleague and friend who has just returned from holidays has commented on the new growth around my hair line and the thickness of my hair, I was able to let her feel my hair without cringing as I would usually do. I am due to go to my hairdresser today for a trim and colour, and as it is a month since my last appointment I will be interested in her comments. I am so pleased with the results and my confidence is returning, I am so looking forward to seeing the full results after completing my treatment. Thank you Carolyn and girls.\nThanks team for the research you did in support of Danielle.\nI had my 9th treatment at Absolique last Friday. Had a good time and best treatment ever. 3 more weeks to go to see the results. I'm excited to see the results in microscope.\nI love the red light therapy, it's so relaxing. Can't believe I'm nearly half way through!!\nGood morning my hair angels. Scope and comparison sessions completed with Carolyn. Overall a pretty good effort for 4 months of intense work. My hair strands are definitely thicker and strengthening and there is some good growth in the front, but there is still a lot more work to do in the top and back area, as well as strengthen the new growth. Whilst I would have liked to have seen more improvement, the fact that I have any at all certainly outweighs my impatience! Looking forward to the maintenance mode for the next 6 months and then follow up scope and comparison.\nI had my first in clinic session this week. Again very informative as I was talked through each process. The scalp needling wasn't half as scary as I thought it was going to be. Slowly getting use to the home supplements and hair treatment. Hopefully I won't have to think twice about it and it just become a part of my normal routine. And the green drink isn't anywhere near as bad as I thought it would be!\nI am almost half way through my treatment course and feel very pleased with the results so far, my hair is shiny and healthy and I have noticed the curl is starting to replace the frizz. I continue to receive positive comments from colleagues some who have no idea I'm having treatment. I look forward to my weekly appointments, Lena is wonderful and very knowledgeable and makes every appointment very relaxing. I appreciate the professionalism and privacy afforded to me at each visit, therefore there is no embarrassment or reluctance to present to my appointments. Thank you Carolyn, Lena and girls.\nHi guys, bit of dandruff\u2026any tips? See you next week.\nI feel like there's new hair growing. I hope it continues to grow & doesn't break. I look forward to it growing more & more.\nThis week I saw lot of new hair coming up on my scalp. I have 2 more treatments to go. My hair is now I think stronger than before looking forward for the microscope session to confirm.\nIn clinic sessions are going well. Still getting use to some of the at home program requirements. It does get easier but requires commitment! Probably struggling mostly with the Activance as it make my roots curly every morning and night that I use it requiring re-straightening. Clearly I'm not starting to love my curls yet!\nIt has been 11 weeks and I am feeling healthier and enjoying the positive changes in my hair, skin and nails. Recently told a friend about my last 11 weeks and she is looking forward to getting the hair she wants!\nReally impressed, that it's about getting to the cause of the issue and not just trying to band aid an issue. It's about prevention not cure. Only negative is the parking, hate paying $48.00 for under two hours.\nMy first session was very stressful and emotional but I'm so glad I made that call to Absolique Hair Health Clinic. Every session has been positive, which I look forward to every week, and I am happy with the progress so far. Thanks to Lena for all her advice and positiveness she gives me throughout my treatment sessions.\nThanks Carolyn for confirming details of the new prostate product.\nAdded the Green Qi and pills to my topical regime on Friday night. Instant increase in pH levels, will see if I can get to the big 7!\nScope results tomorrow. I'm feeling confident. Fingers crossed.\nIt will be my last session this week and feeling a bit sad as visiting each week had become part of my routine! I am so happy I decided to make the decision to visit the team for 3 months and a very impressed with the results, service and knowledge I have gained. Carolyn was very honest at the first meeting about what the issue was, recommended process and likely results \u2013 honestly the results she predicted sounded too good for such a short amount of time (after worrying about my hair for years!) but she was right and I'm so glad I was wrong! I have my hair which I missed, definitely fuller head of hair, hairline is changed and my hair texture is back to what I remember. Sounds vain to be so worried about ones hair but when it is such an important part of your image\/personality I'm so glad I didn't just accept the changes as permanent and kept looking for the solution \u2013 which u found! Thank you to absolique! One very happy customer.\nMy hair looks and feels great, but I'm still concerned about the sides, it's certainly like watching grass grow but much slower. I discuses my concerns with Lena at my last appointment and once again her knowledge, professionalism and compassion allayed my concerns. I resisted the earlier option of using x fusion but now thankful that Lena presented this suggestion yet again. I have commenced using the x fusion and I'm ecstatic with the results and look forward to the day when I no longer need to rely on it.\nI have now had 5 in-clinic treatments and each time I have a session I learn something new about my hair and the treatment. The knowledge imparted by the therapist is very valuable and worthwhile. I know some people may not be able to attend the clinic weekly but I highly recommend it as opposed to doing it yourself at home.\nMy husband is so impressed by my thicker hair that he has signed up to the program! He is on week 1 whilst I only have 1 week to go of the 12 week program.\nIt's been 6 weeks now, still getting used to the weird taste of green drink. I haven't seen too much difference yet. I think it will take few more weeks to see the big difference\u2026looking forward to see that soon.\nI had my last session with Kate, she has looked after me from the beginning so it's sad that she is leaving. My session was good, I had bad dandruff as a result if my psoriasis in my scalp flaring up. It's still itchy & tingling but I can't seem to pinpoint the issue in my diet since taking the GF & DF commitment. Looking forward to my next appointment.\nFeeling positive and no more stressing when it comes to my hair. I look forward to my appointments in clinic because it's a whole hour just spent on my hair and I leave feeling great with one step closer to getting my hair back. Thank you girls.\nIt is now 6 weeks in and my hair is much softer with many baby hairs now visible. I am so excited to see what changes are awaiting in the next 6 weeks.\nNow that I have completed my course of weekly appointments at Absolique, my improvement showed that I no longer needed such frequent weekly appointments and I was also able to cut down on the amount of supplements I was taking. The ladies at Absolique have been so supportive and encouraging as I continue to make progress in improving my hair health therefore I decided to attend the clinic once a month to ensure I was correctly maintaining and constantly improving throughout my routine. The staff have been more than accommodating in this respect and for that I am very grateful. I look forward to viewing even more change in my hair growth and taking small steps towards maintaining my health outside of the clinic.\nGot my new stock from Neways, stocked up this time with a couple of Green IQ so won't run for a while.\nIt's been a great experience so far with Absolique. Thanks to Lena for all the advices you are giving during every treatment.\nLast session left! Even though my hair is not as thick as I wanted\u2026 I've been told it's like watching grass grow. However, my hair is not breaking as much as it was before so I am hopeful that if I continue to treat my hair like I was taught then slowly it will get to that point.\nOver half way now, lots of new hairs and the front of my hairline has hair again, no more scalp visibility. Absolutely love Activance for my hair. Acts as a great leave in condition and helps hair growth!\nI always enjoy my appointment with Sarah and Colette not only did my hair feel and look amazing we have a lot of laughs too! THE DREAM TEAM!\nI'm so very happy, with my results after 3 months. I have a whole new head of hair, would definitely suggest Absolique to anyone with hair thinning challenges.\nOkay week ten, time has gone by quite quickly. Looking forward to when my hair grows long enough and I will have a thick pony tail when I go to the gym, that hasn't happened for many years. Thanks to Lena we worked out my pre & post workout powders have been stopping my body absorbing all of my nutrients and minerals because of the artificial sweetener and the soy products used. Not happy but now I know why the hair tonic and other vitamins I was taking wasn't working. I have been on this protein powder for nearly two years. A bit disappointed in the health shop not picking up on this after I told them I have arthritis as well which makes my body very acidic to start with. Also explains why my pH levels are always crap. So now have a pea protein powder which is high in alkaline. Hope my pH levels improve and the absorption of all the vitamins, greens & the liquid minerals I'm presently taking for the treatment. Shame we didn't work it out sooner but let's hope I get a better result after the next month. So anyone who is on the program and is using protein powder please look at the ingredients as you maybe jeopardising your treatment. Thanks Lena for all your help and as always for your patience with me.\nAn encouraging seventh session with Absolique this week. I finally confirmed with one of the girls that my hair has gotten a lot healthier. There are also some new hair coming in but it will take longer for them to grow long and make my hair look thicker.\nI had my fourth in clinic treatment this week. I'm still freaking out about washing my hair. I think my hair is looking a bit stronger and no as much hair loss when I wash it. I have the hair washing down pat now so happy about that as I was thinking it was a long process. As always I'll end on a positive and as I keep saying I can't wait until I have hair like my beautiful daughters.\nNo change in my hair, still looks and feels great, but it's like watching grass grow \u2013 only slower\u2026 Using X Fusion for interim coverage has been fantastic, I don't think I would have coped without it. Each week the area of coverage required is getting less and less. I look forward to the day that I no longer need to rely on it.\nMy pH is finally returning to normal, it seems a lot more stable than last week although not completely one hundred percent just yet. The new growth around my hairline though is phenomenal!! The hair that is growing in is thick and dark and will hopefully grow to a long terminal length, I can't wait! So sad that I only have three more sessions with the lovely team left.\nI had my third in clinic treatment this week. The girls are absolutely lovely & put you at ease straight away. It has been years since I have brushed my hair but I must admit it I am enjoying the laser hair brushing. I'm still freaking out about washing my hair. I'm counting down the days until my hair is going to start looming thicker. I'm still getting down about it & looking at other women's hair & getting jealous. I will keep moving forward & will do everything I can do for the best results. On a positive and as I keep saying I can't wait until I have hair like my beautiful daughters.\nOne thing that really makes you feel welcomed at Absolique is the relaxing environment and the friendly staff. My sixth session at Absolique was another great experience. I am halfway through!\nAs I'm entering week 8 of the program I can definitely see the rewards of my efforts with quite a lot of new hair growth. It is really encouraging, rewarding and exciting.\nI feel so good and so many new hairs are popping up everywhere, it's unbelievable. This is really better than I could ever have expected. So happy with the results and I haven't even finished yet.\nI am always so grateful to the team at Absolique for giving me fantastic service and advice. Thank you.\nI had my first in clinic session and it was interesting. The brain freeze procedure and the micro needle were perhaps the strangest techniques but if it means better hair than I don't mind. I've found that I need to buy 1 of the tablets on my next visit as they only last 2 weeks in total. Taking the tablets & oral dosages has been a challenge.\nI had my 4th treatment last week by Lena and I truly appreciate her for the way of giving service and the quality treatment. I am looking forward to see good result in the coming weeks.\nIt's now week seven I can see a slight change in my hair and friends have made comments it looks thicker at the top. I still have the patchy area on the side of my head but understand it is because of new hair growth but doesn't stop me from freaking out and again I'm a drama queen lol. My youngest daughter made a comment and said my hair looks thicker and she can't see my bald patch at the back of my head so I'm taking that as a positive that my hair is thickening up and that the treatments are definitely working. I have to thank all the girls for listening to my ramblings when I come in for my in clinic treatments, they are very patient with me. Now onto week eight will I see more changes, we can only hope.\nI am now in my 3rd month of maintenance. The results are still exciting as my new hair is so much stronger and thicker. The curl I have now is so much stronger and requires less product to keep it looking good. So glad I kept going through the program.\nI've just finished the second week of the program. It's getting easier as each day goes by, though the green drink is still gross. The best thing to do is put it in a water bottle or protein shaker so you can't smell it!\nAll the tablets that I take for hair growth etc. have helped to improve my skin. Got to love that!!\nLast week of the in clinic sessions, I have mixed feelings this week \u2013 I'm really excited that the next stop is my scope session, but also sad that I will miss the weekly mini pamper session with the girls. PH levels this last week have been like a roller coaster \u2013 I'm thinking a little stress, lack of sleep and allergies have contributed to this\u2026 Hoping this doesn't impact my hair growth cycle too much but time will tell.\nAfter 1 week with Absolique I'm realising this is not just about hair but a journey in understanding general well-being. Getting to know what we need to eat to maintain optimal health. Also wow' d by concept scalp cleansing. First time I tried this at home my hair has looked the best it has in years. Learning the how is the important part.\nFor the first time I've started to see new hair. This is amazing! Very happy to see the first new hair growth.\nI'm still growing new hair and my hair feels healthier and looks fuller.\nScalp sensitivity is still there, but manageable as per previous 2 weeks. Lena mentioned at my last in room session that the little tufts at the front seem to be getting a bit thicker. I really don't want to get too excited so soon but there does seem to be some progress.\nI have just completed my 4th week of treatment and my overall health and well being is really great. My pH level is finally alkaline all thanks to the \"green\" drink!\nIt was my third session with Absolique and as usual it was a relaxing experience. After a hard day at uni, I really enjoyed going there and talking to the girls. My hair has improved a lot in only 3 weeks and now I am confident that I am losing less and less hair every week. Along with a great service, I also received some free weight loss ideas!\nOnce again it was my time of the month so my pH balance was off for most of the week but has been slowly coming back up to normal. My usual stylist Kate was still on holidays and so I had the lovely Lena take care of me. We had the best chat about study and travel and I felt so relaxed and well cared for. All of the staff at Absolique are so much more than hairdressers; they care for you as a person and are willing to share their own stories to help you on your journey. Love coming every week.\nSo glad to have Lena back. She does the best brush therapy ever!! Really like our chats.\nAt the end of my treatments now and am very much looking forward to the final scope sessions to see the results. One of the most enjoyable parts of the treatments is coming in to see the girls every week. What a wonderful team they are and I will miss seeing them all the time!\nWithout fail every week, the girls are so friendly and fun and the service and attention I receive is amazing.\nI can see lots of new tiny hairs around my hairline!!! So excited for this, I can't wait to see them thicken up! Unfortunately my pH has been really off this week and I *think* I'm shedding hairs again but will talk to the lovely girls tomorrow about it. So pleased about the new regrowth though.\nMy fourth session with Absolique was as usual very relaxing. Now that I know what my body is lacking, I can talk to the girls to find out which supplements will be okay to take with the other vitamins and minerals I'm taking. This session just made me confident that Caroline really knows what she is doing and can really help. My hair loss has decreased significantly and now I'm just hoping that it starts to get thicker! Fingers crossed!\nNoticing a little less hair fall out this week, my usual weekly bathroom corner search of hair clumps has seen smaller results \u2013 can only be a good thing and I hope this also a positive sign towards healthier scalp and hair. I'm still loving my magic sprinkles, the only down side is the thin layer of dust it leaves in my bathroom (more cleaning, but a small price to pay).\nSo much new growth coming through. I can see it and feel it now. I'm very excited to tell people, even though they are already noticing. Very happy.\nI am only a week into treatment, so it is hard to tell if anything is working yet, I'm finding the whole treatment program good though, nothing is hard about it, taking the supplements is easy, apart from the green Qi, but I will use the tips from the newsletter for that. The topical treatments are fine and I enjoyed my first in clinic treatment.\nI have to say all of the girls are lovely, friendly and very approachable and are more than happy to answer your questions. Carolyn replies to emails promptly and I am very happy with the client contact. I had my first in clinic treatment. I really enjoyed the laser hair brushing which is strange as I am so scared to touch my hair. I washed my hair last Friday but didn't notice anything as I put it up in the morning to go to gym. Well I have washed my hair today and I am freaking out again as my hair is so thin. I know I am being impatient but I will be wearing my hair up for the next few weeks until my hair starts to thicken up. I am concerned at this point in time that I will not get the results others have been lucky to experience. I am venting about my situation and I am a little concerned but I have the guarantee that I will see results\u2026so I am holding onto that and will move forward with the rest of my journey and just hope I will have beautiful hair like my daughters soon.\nMy pH is still not on track and after monitoring my eating, exercise and sleep habits I still have absolutely no idea why. My hair is looking shiny and healthy though and there are a lot of new baby hairs popping up, especially around the hairline. I can't wait to see Kate again this week.\nA comedy of errors this week, I forgot my micro needle at the in-salon session, missed 1 round of pills and green drink and on another night the acid buffer. Just when I thought I was getting used to my routine without having to think about it, a super busy week throws me into chaos. Hopefully not too much of an impact on the overall results.\nI am going to vent again this week as I can't tell anyone else about my embarrassing problem. It seems silly people tell me they love my hair but they have no idea to the extent as to what my problem is & how much my hair is damaged from years gone by. Every time I wash my hair I still freak out. I washed my on Friday and woke up on Saturday with hair on my pillow..Omg I nearly cried..no hair since but I did freak out. I have a few personal things going on at the moment, trying my hardest not to stress but come on when am I going to get a break. All I asked for was a couple of months to hopefully get my hair sorted. To end on a positive and as I keep saying I can't wait until I have hair like my beautiful daughters.\nAnother great appointment with the dream team Sarah and Colette and YAY so happy there was a little bit of Carolyn's super conditioner left my hair feels amazing!\nAfter nearly a year of treatment, my time is almost up. I will miss the girls at the clinic but am determined to have a regular check-up to make sure I stay on track.\nThis week has been pretty uneventful and everything is continuing normally. I am starting to get a little sick of always having to remember to take my drinks and vitamins but they honestly make me feel so much better when I do have them \u2013 they give me energy and I feel more awake which is great for school! My usual assistant Kate was away last week so I am looking forward to seeing her again this appointment! But I was well looked after by Colette who is doing so well with her training and is lovely to chat to.\nAs it came to the end of my journey for my weekly appointments at Absolique, I was looking forward to viewing my results via a scope session which detailed before and after pictures of my hair & scalp. As always, the staffs at Absolique were very supportive and gave very positive encouragement as well as pointing out areas where I can improve my results. After my appointment I walked away motivated to continue improving the health of my hair and assured that with the support of Absolique I will only see improvements.\nI feel the products going onto the scalp and hair does seem to give a thicker feel\/appearance. Top of scalp a little tender after last night's needling but that just means the products will go into the skin.\nI am beginning to notice a difference in my hair health and the volume of my hair is starting to change. The nutritional and ph testing have now become a habit.\nLast week I mentioned my scalp was a little sensitive, it still is this week but it's not uncomfortable so I'll be sticking with the micro needle routine. I was worried about not having highlights in my hair to try and mask the thinning areas, but since I've been using the magic sprinkles and going back to a colour that closer to my natural shade, I've been getting some great compliments. My only difficulty at the moment is trying to stick to only taking the green drink and pills when I'm hungry \u2013 easy on weekends, a bit harder to juggle during work hours.\nJust recently started the hair program at Absolique Hair Health Clinic and from all the independent research I how conducted regarding hair thinning, I was astonished by how much I still didn't know. It's really a full spectrum program as it tackles hair loss on many fronts as it addresses nutrition, topical solutions and other various tactics of stimulating the scalp. I'll still in the infant stages of sighting any results but I'm confident they will come.\nMy experiences at Absolique are always professional. Considerate and understanding. I highly recommend any one with hair problems to visit Carolyn for a consultation.\nWeek 2 and I'm starting to settle into the routine of pills, green drinks and hair\/scalp treatments. The only really unpleasant thing is the Green Drink, I don't think I've ever tasted anything that bad before, that said it's not a hard routine to get into and the team at Absolique have been great in answering any questions I had and the in-clinic session was just pure bliss.\nI'm feeling confident about my treatment and the future of my hair. The home care treatment is a lot easier than I thought it would be and the instructions provided make it very easy to look up how many tablets to take a day or where to use your micro-needle if you forget. The thing I'm probably finding hardest is waking up a little earlier to take my green qi so that it is digested on an empty stomach but luckily I don't mind the taste unlike a lot of other people! Already my hair is looking fuller, especially after using Activance which is my favorite product hands down!\nTreatments still going well, with only three left to go. I am really looking forward to comparing the scope results. Visiting the salon is always a lovely and relaxing experience. I am impressed with the overall volume of my hair, it's already quite long but the fullness of it seems to have really increased. Activance is truly an amazing product and I'm so glad that I now know about it!\nWeek 4 and the routine is starting to feel like it's just part of normal everyday life. It might be too early to see most results but my hair just feels fabulous and I'm amazed that I've been able to throw away all of my chemically laced products. I'm still guilty of a little hairspray but not as much as before.\nThis is week three and I'm getting more confident with doing my in home treatment. After my last visit I was impressed with the changes in my hair health.\nTo this day I am grateful that my daughter introduced me to Carolyn. Lost confidence is now back and my hair is the best it has been for years.\nMy last week of the 4 month treatment, phew. It takes commitment to go on this hair health journey but I'm glad I made the effort. If only it was as easy as popping a magic pill once a day. Know much more about cause and effect, and now feel that I can address things that have bothered me for years. Hopefully the improvement I feel in my scalp and (I think) my hair strength will show up clearly in my before and after testing in the next week or so.\nWeek 5, my scalp is starting to feel a little more sensitive to touch, I'm assuming this might be due to the micro needling \u2013 perhaps I'm getting used to it and applying a bit more pressure. pH levels are within good range and everything else seems fine. Getting close to the mid point mark and looking forward to seeing if there has been some movement.\nI am 4 weeks into treatment and while it's a little early to see hair results I am seeing the results of taking the supplements\u2026. feeling like I have a lot more energy. Looking forward to this weeks treatment!\nMy \"shed\" has continued this week, although it has not been as stressful as the team's support and explanation of the hair cycle have put my mind at ease and I am no longer counting every individual hair. I slept over at a friend's unexpectedly this week and so I missed one round of green qi and vitamins \u2013 my first miss as I have been very consistent up until now. I don't think it has affected me too majorly and I don't plan on missing any more so it should be fine. I am excited for my appointment tomorrow!\nOnly 3 more weeks left of my 4 month (hit it hard) treatment. I am looking forward to comparing the before and after results from the scope photos, and I'm hopeful of seeing the results before my eyes. My hair health took years to deteriorate, and I'm determined to do the right things to get back in balance, even if it takes a awhile. Hopefully I will be able to cut down on the treatments and continue an effective battle against the dreaded DHT.\nI'm about halfway through my treatment program and I am starting to notice my hair becoming thicker on the top of my head. Can't wait for my hair to grow to fill out the rest.\nThe team at Absolique are always so friendly and welcoming, and certainly know their business. I have had good results.\nJust before Christmas had my last session, then on to Whistler for a Ski holiday, packed what I felt was my weight in tablets, powder, shampoos, sprays and Tonique and off I went. As we were on a ski holiday, when you get up in the morning it's a routine of grabbing something quickly and then hitting the slopes. Obviously this isn't conducive to the hair routine as you need to take the tablets. I set my alarm every morning an hour earlier, took the tablets, stumbled back to bed and hoped id fall asleep. 9\/10 times I didn't but c'est la vie. It was difficult keeping up with everything on this holiday especially the tablets. And as you can imagine the last thing you want to do after a full day of skiing is the hair routine! That being said\u2026over Christmas was when I first really started noticing the changes in my hair, when I go to spray the Activance instead of clear partings I now just have hair to spray, a lot more filled in. As I was saying to Kate, I feel like a Yeti. Also although my husband may not have been noticing, I have had so many comments from friends saying that my hair seems different and has more volume. Initially I didn't tell anyone about what I was doing but now I can't stop. As its working I want to shout it from the roof tops! The front is still my most troubled area but that was the worst hit so it stands to reason that will be the last one to recover. If it keeps going the way it is i will be the happiest person ever! The hair care is still going well, continuing to buy all the tablets and do the routines. I'm just waiting for my immediate hairline to kick in, it must come soon!\nI really look forward to my weekly treatments. The nutritionals I'm taking as part of the program have given me a lot more energy and I'm feeling healthier than I have felt in a long time. My hair and scalp are definitely looking and feeling a lot healthier as well and I feel like my hair is looking a lot fuller and richer as a result so far.\nFeeling good about the treatment. Can notice a little bit of improvement. Looking forward to my last 2 appointments.\nEntering the last month of my 4 month treatment. I find with the once weekly clinic treatments I am slowly but surely increasing my knowledge on cause of scalp and hair problems, no question is too tricky for the Absolique girls. I guess from an almost zero knowledge level when I started I had to improve. Forgot to brush my hair and scalp before scalp cleansing the other day and felt almost guilty. Went for a run to find a good excuse for doing the routine again, but properly this time. Determined to get the best results possible in this 4 month treatment.\nI'm halfway there! I had another lovely visit with Kate who I have pretty much had from the start. You can really tell the difference in my hair now; my hairline is almost joining in the middle completely. I'm very happy with how far I have come and I can't believe I still have six more visits to go! The only thing I do find very trying is the drink. When it's first thing in the morning and all you want to do is have a nice breakfast and you end up having to drink the feroxin and the green powder it doesn't make you jump up and down with excitement that's for sure. I'm going to keep going and keep staying positive though as I can clearly see the results now and Kate assures me it will get even better! Look forward to every visit I have as its one step closer to the end and everyone in there is so lovely!\nOn the 7th week of my 4 months treatment and having gone this far, no turning back now. It's becoming 2nd nature to give my scalp a vigorous brush treatment before I cleanse (replacing shampooing) my scalp and hair, and another benefit of this is I don't have a hint of dandruff. I was able to schedule the in-house treatments at Absolique starting at 5pm which works perfectly as I can pop in after work.\nWhen I started my treatment with Carolyn and the girls, I never thought I would get used to all the steps I had to do for my hair care. Now months done the track, it is very routine and no hassle at all.\nIt is now week 11 and the results have been great and exceeded what I thought would happen. Loads of new hair and stronger wave formation. Even my regular hairdresser was impressed with the results I have achieved. I have been disciplined with my supplements and routine so I am glad that I have persevered.\nI received the results of the scope session after a year on maintenance. Results were good but still room for improvement so I'll be doing another year.\nOn day six so a long way to go! Week Two: went to the clinic for my treatment, it's so much nicer than doing it myself, plus they do so much more. Home care has had its ups and downs. Dropped the tonic in the shower and the little nozzle broke clean off\u2026so don't drop that thing! The hardest part I find is the parting of the hair for the treatments, I have what the girls have called mermaid hair (ie very long) and trying to separate my hair in the shower after the scalp wash is very hard. The whole treatment takes a good hour from the washing of the hair (and rest of shower and coordinating what to do first so that you can jump out as soon as possible!) to the roller (more parting) to the tonic\u2026to drying my mermaid hair\u2026to the restim +. Yes, it's an ordeal..BUT if it works, it will be worth it. So green stuff\u2026still tasting yuk, going to try the girls suggestion of jelly cubes instead of liquid version, must buy another batch for that. Cold water makes it a little easier though in the mean time. Flying through the tablets too\u2026it's going to be a pricy one I think! We will see what happens.\nJust finished my 2nd of 14 treatments with Absolique Hair Health Clinic. The staff are warm, friendly and make you feel relaxed. After having experienced hair loss and the stress that comes with it, it was a big step coming here to get some help. With such lovely staff the anxiety I felt about getting my problem fixed melted away.\nHair is getting better each day. I feel good about the supplements and treatment. Looking forward to this week's treatment with the very caring and wonderful people.\nWeek three: I'm getting the hang of the roller of death, and if you drink the green stuff from a bottle or shaker (something which means you can't smell it) it doesn't make you gag\u2026as much. I'm going on holidays for two weeks so I started counting out the tablets and what not that I will need. Looks like I will need a complete refill of EVERYTHING! The pills really don't last long at all, I guess taking 14 a day will do that .Not looking forward to the next bill. On a positive side, I have my hair routine down to 30 mins, start to finish, was amazed when I checked the time. Might have something to do with the fact I cut my hair in the front so now parting it isn't such sweet sorrow. Looking forward to seeing the girls. Best part of this whole thing, Lena and Kate are great fun.\nI had my scope session with Carolyn. It seems that I may have good results. I will know the exact outcome when I see the comparisons.\nEntering the 6th week of treatment, and I am getting value from the in-house treatments. Chatting with the Absolique girls (while the treatments are being administered) I am improving my knowledge and awareness of good nutrition which is vital for inner health and thus outer health, the hair and skin. Even touched on the subject of Nutritional Anthropology which I am sure you won't hear from your local hairdresser. My overall health feels fine, and I have confidence that my medication is targeted at overall health which in turn is reflected in hair health.\nOn to week four! Week Four: this will be my last visit until after Christmas, two weeks flying solo. All in all, the routine is down, the roller is down\u2026and I think it's just the lack of break that gets to you. I also feel like a massive failure if my ph is low, which due to Christmas parties and antibiotics mine has been! My hair does feel healthier and I will power through!! Looking forward to finishing the four months though! It really does take over your life!\nFrom the very beginning of my journey at Absolique Hair Health Clinic I looked forward to my appointments each week as every time I attended the clinic I would be reminded of the steps forward that I am taking to improve the health of my hair and seeing results in baby steps week by week. On those weeks where I perhaps hadn't followed the home care routine as well as I should have or when I had become discouraged due to one reason or another the ladies at Absolique would be there to gently remind and encourage me of the steps forward that I have taken and get me back on track. Looking at where I have come, I know that I still have a way to go however am excited to see the end results in due time.\nNow on the 8th week of my 4 months treatment. \"Signs of my old itchy greasy scalp condition\" have disappeared, and it feels much improved. Previously after shampooing, an hour or so later my scalp would itch, but no more of that \u2013 thank heavens. I'm finding it satisfying to be proactive about my hair and overall health, and am staying motivated to make the effort that is required to improve those areas.\nGoing well, but life keeps getting in the way so it is hard to remain motivated. Perhaps a pep-talk from Carolyn for those part-way through treatment would help some people.\nMy hair felt absolutely amazing when I left the clinic and i've had heaps of compliments about how shiny it is. I've also had comments that my hair is growing so long and getting thicker\u2026.this is the best compliment EVER for me considering its been a long journey of hair loss.\nIt's great to be greeted with smiles and how has your week been, then offer of hot and cold drink. Kate's got it sussed what we have without asking! And spend rest of time catching up like friends what we've all been up to during week.\nWell. I've completed my program. I've kept to the schedule \u2013 apart from a few missed days due to travel. I am happy with the result to most of my hair; however, I do not see any improvement in the front which is where I did desperately want to see change. Perhaps it will look different under the microscope and all I have to do is wait for it to grow. I hope so.\nMy hair is looking amazing. I'm so pleased I went in to the salon. It will be interesting to see the before and after photos! Last treatment this week. This has been a very worthwhile exercise. The condition of my hair is great. The new growth in the recessions is amazing.\nI can feel that my hair is getting healthier. Hope it will soon come back to the same quantity & quality as it used to be earlier. It feels great to visit the clinic every week. They are really a nice people.\nHair health can be an emotional roller coaster so I am always pleased that I have the team at Absolique to help get my hair back to its best.\nOn the 4th week now, and have got the nightly routine of home treatments down to about 35 mins. I have dropped the daily vitamin tablet I used previously and replaced it with the daily medications as advised by Carolyn, and feel much more confident in the real benefits I am getting. I am doing the daily in-house treatments, and a tip to beginners, don't forget to take along your micro needle. Previously I dreaded having my scalp touched (itchiness) even for a haircut, so with my scalp in much better condition, plus the friendly staff at Absolique, I am finding the in-house treatments not so scary.\nMy fianc\u00e9 and I can't believe we've nearly finished all the hair appointments! When I started it seemed like such a long time! Its been a matter of priorities and we decided my hair came first for these few months and everything we could do just worked around these. Can't wait to see what my hair growth has actually been. I'm going to miss coming in and having lovely friendly girls pamper my hair.\nThe girls at the clinic were wonderful. So positive and friendly. Carolyn is lucky to have such a lovely team working with her. Well done girls.\nLast appointment for the year will be a scope session after 12 months of maintenance. Looking forward to seeing the comparisons. Good news hopefully!!\nI continue to see great improvement as each week passes and am very pleased.\nThis is now my 7th week into the program and I am so pleased to say I am noticing new hair that is growing faster than ever before. My hair feels stronger and curlier closer to the scalp. I just can't wait to see what it will be like in the coming months.\nMaintenance is going well. It's pretty much part of my life. I'm still growing new hair.\nStarted last Wednesday, the full package, so lots and lots of pills and work with my hair. It does take some time to get a plan in place hair wise. I realised after my first try with the roller that every day is too much and I found little dots of blood\u2026not so good. So now, I'm going for roller every second day and of course looking forward to having that done for me expertly in the clinic. The worst thing I have found about the treatments so far is the drink prior to the tablets. By far that drink is the worst thing i have ever tasted. I have created a calendar with a countdown on it. The last day I have to drink that stuff can't come soon enough! That aside, the scalp cleaner is amazing my husband made the comment that my hair has become very big (before styling) made me laugh and pretty happy. Really hope this works! On day six so a long way to go!\nNearly half way! It is nice to have your own girl. It is more personalized; Sarah feels more like a friend than getting paid to do her job.\nAt my clinic visit his week the assistant showed me the new growth on my crown. We were both excited to see how much had re grown.\nI always enjoy getting my hair done by Sarah and having our funny chats, I look forward to it every 6 weeks! I also hear the new ad on B105 every morning congratulations it's great!\nNearing the end of my maintenance cycle. I feel like my hair has definitely thickened. Scope session soon so all will be revealed!!\nThe team at Absolique have the best product knowledge around regarding hair health. I always leave feeling like I have been given the best personalized treatment for my hair.\nYeah half way!! Can't wait to see hair growth! My fianc\u00e9 has been very supportive the whole way through. He's gone to every appointment with me! Every time the girls have made him feel welcome and include him in everything, including always offering drinks. Thanks!\nLooking forward to my next session. I value the reassurance that my hair is responding to all the treatment.\nI can't wait for the next few weeks to see this new growth gaining length. I'm happy to say that the quality of my hair is greatly improved. Must be all the Neways I'm taking too. Will be taking Green Chi forever. It's such a nutritious drink. This week I've noticed that my hair is very silky and shiny. Feeling so positive about my hair now.\nONE WEEK TO GO!!!! Getting excited to see the results through the microscope! Common all that hard work!!!\nAll treatment completed now. I look forward to my scope session to see how far I've come.\nCarolyn was as always very positive and helpful. Last week I started my treatment with Lena. She is very friendly and enthusiastic about hair.The team at Absolique show a genuine concern for hair health. The service received was excellent. Can't wait for my hair to start growing!\nI have just finished the last of the treatment program. The results so far for me have seen my scalp health vastly improve. I no longer suffer from scalp irritations or inflammation. My hair fallout has reduced considerably from prior to starting the program. As for hair re growth I await the scope session, I have not seen hair restoration of my hairline but have noticed a generally fuller appearance of my hair on top of my scalp. As usual the service at Absolique has been excellent and I shall miss my weekly sessions with the team.\nI am finally getting used to having the Green Qi drink, actually enjoying it now. Feeling really good!\nI started a course of treatment at Absolique because I had a lot of hair loss and the condition of my hair was wispy, fine and lifeless. Now after my treatment it is in wonderful condition and the re growth is amazing. Many thanks to Carolyn and her staff.\nVery friendly and helpful, from first point of contact on phone to visiting once a week. Can't wait for my hair to start growing! My hair and head feels great after washing it and going through the special process and isn't so greasy and even looks thicker.\nI'm still doing my monthly maintenance sessions and I must admit I look forward to it. I particularly look forward to the reassurance that my treatment is on track.\nMy recent visit was once again filled with positivity and exceptional service. Lena and Sarah are a fantastic team who always provide me with exactly what I need to feel great. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Besan Ka Halwa is a quick and easy Pakistani dessert that is super delicious\u2026it's made from gram flour, sugar, butter or ghee and cardamom powder.\nToday I'm sharing with you, one of the first Pakistani desserts I learnt to make\u2026and given that I barely knew how to cook then, this is proof enough that this is a super easy dessert.\nAfter I got married, I would often spend time on Pinterest looking up easy desserts because I've got a major sweet tooth. I would make brownies and cookies and cakes.\nBut after living away from Pakistan for awhile, I started missing Pakistani food and desserts like Suji ka Halwa and Shahi Tukda.\nThe thing was though, I wasn't any good at making Pakistani desserts so I wanted to go super easy on myself.\nAnd so the first halwa I ever made was this besan ka halwa.\nAnd guess what? It was SO easy that I didn't fail at the first attempt!\nNot to mention, this halwa is a very pantry friendly recipe. I'm sure most of us desis always have gram flour (besan), ghee or butter and cardamom at home.\nBut if you're not familiar with besan and somehow came across this recipe, let me tell you what besan is. Besan is gram flour, which is basically chickpea flour.\nChickpea flour is used quite a lot in Pakistani and Indian recipes\u2026we make pakoray with it, we make Kadhi with it and we also make besan ka halwa\u2026yum! Also besan ke laddoooo\u2026yum! Such a versatile flour, right?\nSo How Do You Make Besan Ka Halwa?\nStart by melting ghee or butter in a wok. I usually have butter at home, so I used butter. But for a really authentic flavor, ghee is best for this besan ka halwa.\nOnce the ghee or butter has melted, add your gram flour and cook it on medium high heat until it changes colour and releases a nutty aroma.\nNext add the milk, making sure to stir continuously so that no lumps form.\nNow it's time to add the sugar and cardamom powder. You can even use whole cardamom pods but just crush them up a little bit before adding them to the halwa.\nKeep cooking on medium heat, until the halwa stops sticking to the sides of wok and sort of forms into a cohesive ball.\nTransfer to a serving dish and top with nuts such as almonds and pistachios.\nAnd that's about it. Your delicious besan ka halwa is ready!!\nIt's so good, that it would be even great served to sudden guests. Or when you're craving for dessert but don't want to put in too much effort. Which is like always me. Lol.\nIn a large heavy bottomed pan, heat the butter or ghee.\nAdd the besan, and fry until the besan starts turning golden brown and releases a nutty aroma.\nAdd the milk slowly, while continuously stirring the besan so that no lumps form.\nNow add the sugar and cardamom powder, and stir to combine.\nKeep cooking on medium heat, until the halwa starts to form a ball and is no longer sticking to the sides of the pot.\nTransfer to a serving dish, and top with the nuts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjwph b/data_all_eng_slimpj/shuffled/split2/finalzzzbjwph
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+{"text":"Alaska Salmon: Bipolar disorder is just a chemical imbalance. You can't help it. And eating me will just make you feel shame & guilt!\nTitle: Alaska Salmon, test the theory omega-3 helps the reasoning centers of the brain\u2026 to no Avail.\nToday, I have the privilege of interviewing a \"pioneering\" author in the world of adolescent bipolar disorder! Tracy Anglada is a pioneer for parents and children with adolescent bipolar disorder.\nWith the lack of many resources for children to truly understand their feelings and emotions, she sat down and created a book designed just for children. Her book, Brandon and the Bipolar Bear has been nominated as a finalist for the Reader's Choice Awards at About.com in the category of Favorite Special-Needs Children's Book. (More about that later.) I thought it would be nice to spend a little time with Tracy in this three-part series entitled: 5 Questions With Tracy Anglada and the Bipolar Bear.\nQuestion One:You have been a mental health writer and advocate for ten years now. What started you on this journey?\nIt began before I realized it. Like many family members who have a loved one diagnosed with bipolar disorder, I found myself in a sink or swim situation with few lifelines to grab onto. Because it was my young child who was diagnosed, the mental health field was not always a friendly place.\nThere was much controversy surrounding this diagnosis in children a decade ago. While healthy debate along with checks and balances will continue, it is now accepted that bipolar disorder can onset during childhood. Necessity pushed me to learn everything I could about the disorder. Determination made me resolved that other parents \u2013 and their children \u2013 would have more lifelines than I did.\nQuestion Two: What kinds of resources did you help develop for other parents in the same situation?\nWhen my child was initially diagnosed, there were few resources available. I had nothing to help my child learn about his illness, nothing to educate his siblings, and nothing written for his teachers. The field was in its infancy, yet our family still had to live and function.\nAs my son went through various stages, I saw first hand what resources we needed to survive. This fueled my writing (five books) and motivated me to create these resources. I started the BPChildren website where I featured a kid's page with mood charts and articles written just for kids. I also showcased the \"positive\" side by letting kids submit their artwork or poems. The website has expanded over the years to include pages for teachers, teens and parents.\nI wrote Brandon and the Bipolar Bear \u2013 the first story available for kids with the disorder. Other 'firsts' included a story for siblings and a brochure for teachers. I also wrote Intense Minds which was a unique way to help adults understand and empathize with the experiences of young people growing up with bipolar disorder.\nSide NOTE: I mentioned earlier that your book has been nominated for an award, what can we do to help you win, and can you give my readers a way to review your book?\nYes 'Brandon and the Bipolar Bear' has been nominated as a finalist for the Reader's Choice Awards at About.com in the category of Favorite Special-Needs Children's Book. Awards will be given based on reader's choice. You can help by voting once per day through March 8th: www.tinyurl.com\/votebpbear Please show your support for this special book as it approaches 10 years of helping kids with bipolar disorder. Thanks!\nJust for\u2026 Chato's Mental Health Humor readers can get a Sneak Peek of a free narrated video storybook of Brandon and the Bipolar Bear.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"US progressive rock outfit Artificial Language have unveiled their new single, \"These Aren't Mirages\", along with an accompanying music video. The track is taken from their upcoming album, The Observer, due out 4\/28.\nBoasting a sound akin to progressive rock mavericks Haken and Leprous, ARTIFICIAL LANGUAGE really embody a sound beyond that. Carving technical hooks and passages that show a gifted amalgamation of influences and prowess, they exhibit a sound more reminiscent that is truly their own. They plunge into a vastly distinct and unique territory of prog rock, where they have not only found their footing but have already claimed the land as theirs for the taking and don't plan on stopping anytime soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"School is open from Monday to Friday unless otherwise advised. Parents are asked to phone the office before 9.00 am if children are not attending school or are coming late. This is important for the aspect of children's safety. Unexplained absences are followed up by the school secretary.\nWe believe good communication makes for good relationships. A newsletter is sent home every second Thursday. This keeps parents up to date with school activities. This is also available on our Facebook page and electronically. Rochelle is in the office from 8.30 am \u2013 3.00 pm and is happy to answer any queries and pass on any phone messages for children and teachers when necessary.\nWhen your child is enrolled, any health matters which may affect her\/his learning or attendance will be discussed and recorded. Parents are advised to inform the class teacher of any incidental health matters. If a child is unwell and needs to go home, his or her parent (or emergency contact) will be contacted. The child will be cared for in the medical room or classroom until they are collected.\nEvery second Friday at 2.40 p.m. we have a school Shared Prayer time in the Church. Each class and, on occasions, parents take turns at leading this. Parents and friends are warmly invited to attend. It is a special way to finish our school week.\nIn alternating weeks with Shared Prayer the school meets in the Parish Centre at 2.30p.m. for Assembly. The senior pupils run the programme and give opportunities for children and classes to share important news, birthdays and to show work. Parents are encouraged to attend.\nThe Year 6 pupils are leaders of our Family Groups. Each group has children from each level of the school and enjoy the Friday family group time immensely. Activities the family groups organise include picnics, treasure hunts, craft sessions and talent quests.\nOur Library is very well resourced. All classes enjoy using the library and children able to take out 2 books a week.\nThe P.T.A often offers a lunch option on Wednesday which will be detailed in the newsletter. In the winter terms (Terms 2 & 3), children may bring food wrapped in foil for heating.\nAt times we require payments for sports or swimming. Eftpos is available at the office or alternatively, you can internet bank. The account is St Francis Xavier. And the number is 123150 0160900 00. Please state clearly your child's name and what your payment is for.\nStationery packs are available, either in store or online, at the beginning of the year through Office Max. The school has some stationery items which are available to children during the year. However, payment must be made to purchase the item. At the beginning of each year, all children are charged an Activity Donation. This donation contributes to the cost of photocopying, Education in Faith resources, school\/class trips and payment for performances at school. This cost is $50 for your first child and $40 for each additional child of the family. Accounts are sent home each term.\nThese are set by the diocese and these are used to meet the ownership of Catholic schools. There is a legal obligation to pay these. The fee for 2018 is set at $445 for the year or $8.56 weekly or $17.12 fortnightly.\nAll children have homework during the school week. Class teachers set tasks and give parents guidelines for their class levels. Teachers are always available to discuss homework requirements.\nThroughout the year there are opportunities scheduled for teachers and parents to meet and discuss children's progress. Early in the year, there is usually a \"Meet the Teacher\" evening. Parent\/Teacher interviews are held towards the end of Term 1 and in Term 3. Children receive written reports at mid-year and at the end of the year, and parents are given opportunities to discuss these if they wish to.\nIf a teacher has a concern about a child, parents will be contacted. Teachers are very happy to discuss children's progress outside these scheduled times. We ask parents to arrange suitable times with teachers out of school hours.\nMany of our activities involve trips away from school. Theserequireextrasupervision and we are grateful for parent help with this. Notification of these trips is given in plenty of time and we enjoy having parents to help. There are various other opportunities for parents to help around the school, e.g. working with small groups of children under the guidance of a teacher, covering\/ repairing books, library work, artwork etc. We appreciate this help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The My Colors Cardstock 8.5\" x 11\" White Classic Pack will bring out the rich color of your photographs and accent your projects with show stopping style! Included are 25 sheets of smooth cardstock by My Mind's Eye. This 100 lbs. heavyweight cardstock features consistent color throughout the sheet and includes fiber pulp which eliminates cracking when folded or scored.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kristinka Baglama - Kiki, 40, Living in Osijek, Croatia. Long time active and passionate photographer. Exhibited in several group and standalone photo exhibitions.\nPhogographic interests: structures in day and night light.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbldxq b/data_all_eng_slimpj/shuffled/split2/finalzzzbldxq
new file mode 100644
index 0000000000000000000000000000000000000000..27a1aee73d54dcd87a55ce1d7a1f138f9d9ee392
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+{"text":"Get a unique souvenir of your time in Paris with a private photo shoot. Hire a bilingual professional photographer to create personalized photos of you and your loved ones against some of Paris's best backdrops. You will receive stunning full high resolution images to share and print.\nExperience Paris like never before, and hire a professional photographer for an exclusive photo session against some of the city's best backdrops.Your personal photographer will take on-location photographs against the city's landmark buildings, cosy caf\u00e9s, or winding streets. Capture authentic moments for a special occasion, or simply create a dynamic photo-documentary of your vacation. Photo sessions are held in some of the most picturesque neighborhoods, or you can suggest your own location.\nStand in front of the Eiffel Tower and beautiful gardens of the Champs de Mars to capture you with the city's most famous icon. Take the session at dusk for even more magical moments. Pose in front of the stunning glass pyramid outside the Louvre, or with the water features of the Tuilleries Gardens. Add a touch of glamour to your shoot in Place Vendome and outside the Ritz Hotel and its surrounding boutiques. Stand on the Pont des Arts, where lovers attach engraved padlocks before throwing the key into the Seine! For classically Parisian pictures, the bohemian Montmartre and Pigalle districts offer lots of opportunities. Stroll among the artists selling their wares as your photographer captures it on camera. Quaint little caf\u00e9s with terrace seating add to the atmosphere. Discover secret gardens and winding stairs with unexpected views, as well as the only vineyard in the city. The last neighbordood popular for photo shoots is around the Op\u00e9ra Garnier and nearby 19th-century arcades of stained glass and iron features. The area is architecturally stunning, with the Op\u00e9ra itself providing a breathtaking explosion of gilt, carvings, and art.\nYour photographer will contact you prior to your tour to discuss your precise options and requests, giving valuable suggestions to help capture memories that will last a lifetime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A summary of discussions at the 20 February 2019 Board meeting is now available.\nThe documents on this page that require a University staff login are accessible via Sharepoint. Login details will only be requested once on each visit to this page, and will not be requested if staff are already signed into the University's network.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gargamel\\'s has casted an evil fog spell to capture all the Smurfs. The whole village is gone and you\\'re the only Smurf left! Run away from the fog, stroll across a vivid world, collect iconic Smurf items and save all your fellow villagers to become the true hero of the day!\n* RUN, jump, glide, stomp, dash\u2026 your way through more than 100 levels to free all the captive Smurfs.\n* DISCOVER a vivid and lively world directly inspired from the original Smurf comic-books.\n* PARKOUR different locations of the Smurfs world including the village, the forest, the mine and Gargarmel\\'s Castle!\n* PLAY with your favorite Smurfs including Smurfette, Papa Smurf, Handy Smurf, Grouchy Smurf and more than a dozen other iconic characters!\n* COMPETE against the world and your friends in the Weekly Tournament. Who will run the longest from the evil fog?\n* COLLECT dozens of items from the Smurf universe to complete sets and get awesome rewards.\n* EARN great prizes everyday by collecting Golden Keys, completing Daily Missions or asking the Farmer Smurf for his Good Deals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The regular opening times of WHSmith Devonshire Road in Bexhill-on-sea are Monday to Friday 8:30AM-5:30PM, on Saturday 8:30AM-5:30PM, on Sunday 10AM-4PM. View the actual opening times of WHSmith Devonshire Road in Bexhill-on-sea in our branch locator.\nFor more information about opening times on Sunday or late night shopping make sure to view the information in the designated blocks. If this WHSmith isn't nearby your location you can use the 'Map & Directions' tab to find the fastest route to Devonshire Road in Bexhill-on-sea. This location's coordinates are specifically 50.8395 latitude, 0.473328 longitude. To contact this Bookstore by phone you can dial 01424211844 during business hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arlo pro shows plugged in symbol not lightning bolt when recharging. How to fix?\nRe: Arlo pro shows plugged in symbol not lightning bolt when recharging. How to fix?\nDoes the battery by itself work i the camera?\nThere's an FAQ here on how to revive dead batteries. Basically, you need to try 6 times to charge to get enough charge in the battery to be able to fully charge. Check the FAQ for details.\nI don't let the batteries go completely dead before charging and don't have any problems.\nDo you have more than one camera? Swap batteries around and try charging both to see if you can figure out whether it's the camera, battery or charger. Be sure you're using the Arlo charger and cable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblzah b/data_all_eng_slimpj/shuffled/split2/finalzzzblzah
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index 0000000000000000000000000000000000000000..ed92eb2f205549ac3712cadc332d69b8e470edda
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblzah
@@ -0,0 +1,5 @@
+{"text":"The MINI Kitesurf Odyssey is an attempt to Kitesurf the Portuguese west coast, using the wind from North to South, day and night and without stops.\nBesides the goal of continuously navigate around 290 nautical miles (\u00b1540 Kms) on a Kitesurf board, the kitesurfer will also try to beat a world record, which is \"The longest kitesurfing journey without stops\", currently set at 199.63 nautical miles (369.71 km) by Phillip McCoy Midler (USA) who travelled from South Padre Island, Texas to Matagorda, Texas, USA, from 10 to 11 May 2010.\nThe Odyssey will be attempted from one morning to the next day's afternoon, including an entire night kitesurfing with the moonlight and hopefully a night breeze as well, that enables to progress along the night. There won't be any stops in land and the kitesurfer will eat, drink and stay awake the entire journey (24 to 30 hours) while speeding through the water at an average estimated speed of 12 to 15 knots (22 to 28 Km\/h).\nThis challenge will be performed by Francisco Lufinha, a young sailor and kitesurfer who has spent most of his life at sea and is addicted to nautical sports.\nFor more info about Francisco Lufinha, check his profile here.\nTo reach success in this challenge, a great team has been assembled with the best qualities in the several areas required. Please see the sidebar on the right where all special members of our team are listed and click on them for further details.\nThe challenge will take off from Douro Marina (near OPorto in the North of Portugal) and will finish on the Portuguese south coast \u2013 the Algarve \u2013 at the Marina de Lagos.\nAlong the way there are several checkpoints where Francisco will pass near land, like Peniche, Roca Cape (the most westerly point of Europe), Raso Cape and Sagres\/St. Vincent Cape.\nWhen will it be attempted?\nThe ideal date for the start is on the 19th of Septembre 2013, because it will be a full moon night, which will be a precious help while kitesurfing at night in the middle of the ocean.\nHowever, more important than the moon light is a good strong wind forecast, so the start of the challenge will have a standing period from the 15th to the 30th of September 2013.\nThe date will be set 3 days before the start and confirmed at least 24h before.\nTo be able to go kitesurfing down the coast, Francisco needs favorable North quadrant winds, blowing between 15 and 35 knots. If the wind is to weak Francisco will have to sit and wait on his board with the kite lying on the water. If the wind is to strong, it can pull Francisco in the air and out of control, leading to severe crashes and possibly drowning.\nNot only during the day, but specially during the night, a good visibility is the key to kitesurf at high speeds, avoid crashing in the waves as well as hitting any obstacles.\nOffshore kitesurf navigation envolves many risks, such as crashing while hitting floating objects (fishermen buoys, pieces of wood, plastic garbage, some kinds of big fish, etc), losing track of course and end up in the rocks somewhere, be caught by storms, being in the way of big ships, sharks and other predators attacks, among others. All of these threats are much greater during night time and are part of the fears that Francisco will have to deal with.\nSide by side with the wind factor, the physical and psychological parts of the challenge are on top of the threats that will affect the arrival or not to the end. The estimated 24 hours of continuous kitesurfing will be extreme for Francisco's body and mind, two essential pieces of the puzzle that cannot collapse.\nFrancisco is being prepared by specialists in training, physiotherapy and nutrition, in order to maximize his chances of success when subject to 24 hours of strong body activity in a row.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This innovative laser technology uses pulsed light technology to selectively heat the dermal collagen deep in your skin while simultaneously cooling and protecting the epidermis (your outer layer of skin). The result is partial coagulation and contraction of collagen. The most commonly treated areas are the face, neck, and upper chest. It can also be used to firm up loose belly skin after pregnancy or weight loss.\nThe heat also initiates the body's natural healing process, which creates a renewed collagen foundation leading to increased skin firmness. Powerful cooling maintains the outer surface of your skin at a cool temperature before, during, and after each pulse providing you a comfortable and safe procedure.\nThis treatment takes about 30-60 minutes per area, and is done in our office. The SkinTyte light energy is delivered in a sequence of rapid, gentle pulses. There is no need for a topical anesthetic, however, the physician may choose to use one on more sensitive areas. A series of pulses will be repeated over the treatment area to ensure best results.\nThere is minimal facial swelling (if any at all) the first day, and you can wear make-up immediately. There is virtually no downtime.\nTo get the best results, most patients require an average of 3-5 treatments spaced one month apart. However, because each patient's skin is different, the treatment number can vary. The majority of patients feel that their skin is tighter after just one treatment, but visible improvements in skin tone generally take place over 3 \u2013 6 months.\nThis exciting laser technology is perfect for those who wish to tighten and firm up their skin. Coming back to our practice for 1-2 maintenance treatments per year can help maintain your results and prevent the natural acceleration of skin aging that normally occurs. You can combine treatments with our Broad Band Light photofacial treatments for age spots and sun damage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.adjustable paret handle 2.foldable footrest 3.3-point harness 4.large storage basket 5.quick release rear wheels 6.for 10-36 months.\n* double injection wheelsLmore durable; * 2 position of T bar and seat can be optional. * 2 in 1: balance bike and tricycle mode; * for 2-5 years old.\n1.adjustable paret handle 2.foldable footrest 3.3-point harness 4.large storage basket 5.seat cover & big canopy with 2 layers of fabric 6.seat cover is offered. 7.quick release rear wheels 8.for 10-36 months.\n1.1st trike for baby 2. blowing wheels. 3. storage basket 4.suitable from 18month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\uc6b0\ub9ac\ub294 \uc911\uad6d\uc5d0\uc11c \ud0a4\uc988 \ubc31 \ud559\uad50 \uc81c\uc870 \uc5c5\uccb4 \ubc0f \uacf5\uae09 \uc5c5\uccb4 \/ \uacf5\uc7a5 \uc804\ubb38\ud654\ub418\uc5b4 \uc788\uc2b5\ub2c8\ub2e4. \ud0a4\uc988 \ubc31 \ud559\uad50 \uc911 \ud558\ub098 \uc778 Shenzhen Olinb Bags Co., Ltd. \uc911\uad6d \uc720\uba85 \ube0c\ub79c\ub4dc \uc911 \ud558\ub098 \uc778 \uc800\ub834\ud55c \uac00\uaca9 \/ \uc800\ub834\ud55c \uac00\uaca9\uc73c\ub85c \uace0\ud488\uc9c8\uc758 \ud0a4\uc988 \ubc31 \ud559\uad50 \ub3c4\ub9e4\uc5c5.\nWholesale \ud0a4\uc988 \ubc31 \ud559\uad50 from China, Need to find cheap \ud0a4\uc988 \ubc31 \ud559\uad50 as low price but leading manufacturers. Just find high-quality brands on \ud0a4\uc988 \ubc31 \ud559\uad50 produce factory, You can also feedback about what you want, start saving and explore our \ud0a4\uc988 \ubc31 \ud559\uad50, We'll reply you in fastest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was so glad when we got to November, we're taking it easy this month! I have planned very little school for this month, but we are keeping up a bit of the routine just so that things don't fall to chaos around here! We're still doing our morning chores and circle time. Today I put up our \"Thanksgiving tree\" that will be part of circle time. I have foam leaves that will stick on there, and we each write something we are thankful for on one of them everyday. By the time we get to Thanksgiving our tree should be full of leaves!\nI announced at breakfast yesterday that we would not be doing as much school this month, but we would be doing quite a bit of P.E.- this was greeted with blank stares- physical education- more blank stares- \"you'll be spending quite a bit of time playing outside and riding your bikes\"- cheers arose around the table! The weather is great around here and the kids love getting out to run, play, ride bikes\/scooters etc. They can do all of this while I sit on the porch and rest, or we can all go for a nice long walk if I'm up to it (that is what we did today). We're staying away from the playgrounds for now though to avoid catching a bug.\nsome fall\/thanksgiving things for the kids. Its not much, but they are quite happy with it. We don't have many Thanksgiving books, but tomorrow we will be picking up a bunch that I have requested from the book mobile. The kids like handling the corn and the gourds, something I remember my mom leaving out for us to play with too (I'm thinking it was Auntie B that encouraged her to do this!). Since we also have a bunch of Christmas toys the kids will not see their regular toys for two months. This will be nice when we are trying to get back to \"normal\" after Miss C's birth and the holidays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmgld b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgld
new file mode 100644
index 0000000000000000000000000000000000000000..efce8291825ae23f334d4d090bd2d1ad6b37553d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgld
@@ -0,0 +1,5 @@
+{"text":"Huke 1.38 3 0.97 2 0.41 Huke 1.20 1 0.40 3 0.80 Huke #DIV\/0! - 0.89 3 #DIV\/0!\nTemp 1.16 3 0.87 2 0.29 Temp 0.83 1 0.50 3 0.33 Temp #DIV\/0! - 0.83 3 #DIV\/0!\nClassic 0.77 3 0.90 2 (0.13) Classic 1.25 1 0.43 3 0.82 Classic #DIV\/0! - 0.72 3 #DIV\/0!\nSlasheR 1.33 3 0.79 2 0.54 SlasheR 2.75 1 0.74 3 2.01 SlasheR #DIV\/0! - 0.98 3 #DIV\/0!\nHuke 1.08 3 #DIV\/0! - #DIV\/0! Huke 0.50 2 #DIV\/0! - #DIV\/0! Huke 0.94 2 #DIV\/0! - #DIV\/0!\nTemp 1.03 3 #DIV\/0! - #DIV\/0! Temp 0.50 2 #DIV\/0! - #DIV\/0! Temp 0.78 2 #DIV\/0! - #DIV\/0!\nClassic 0.95 3 #DIV\/0! - #DIV\/0! Classic 0.53 2 #DIV\/0! - #DIV\/0! Classic 0.60 2 #DIV\/0! - #DIV\/0!\nSlasheR 0.97 3 #DIV\/0! - #DIV\/0! SlasheR 0.92 2 #DIV\/0! - #DIV\/0! SlasheR 0.82 2 #DIV\/0! - #DIV\/0!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use the amounts of ingredients as a guideline. You may like it hotter or sweeter.\nChili paste can be made the old fashioned way, with mortar and pestle. The result is a smoother paste.\nAlthough olive oil is not a part of Thai cuisine, I use it instead of cooking oil. Even extra virgin olive oil doesn't change the taste or introduce any strange flavors.\nIf you have a hard time finding shallots, you can use red onions. If you use onions, you will probably want to use less sugar.\nPeel shallots and garlic. The uncut shallots and garlic should roughly each fit into a cup. A little more or a little less should be ok.\nIn a food processor, mince shallots (picture 2). And separately mince garlic (picture 3). We'll be frying each separately since they cook at a different rates. Garlic has a lot less water and cooks faster than shallots.\nAdd \u00bc cup of oil into a pan over medium low heat. Add the dried chili peppers and constantly turn each pepper over to prevent burning (picture 4). If the peppers are turning dark too quickly, lower the heat. After a couple of minutes, they should puff up and turn dark. Scoop out the peppers and set aside in a large, heat tolerant bowl.\nAdd the minced shallots into the same pan that you fried peppers. Add \u00bd cup of oil. Turn up the heat to high. Stir the shallots to mix with the oil. The oil should be hot and bubbling up. It will take about 10 minutes to get the shallots to turn light brown (picture 5). When it does, pour the shallots and oil into the same container with the fried peppers.\nAdd \u00bd cup of oil into the same pan. Lower the heat to medium, add minced garlic. Stir to mix the garlic with oil. The oil should be bubbling. After 4 minutes, the garlic should turn light brown (picture 6). Pour the oil and garlic into the bowl with other fried items (picture 7).\nAdd the fried peppers, shallots and garlic into the food processor. Add sugar, salt and tamarind paste into the food processor. Pulse until well mixed, about \u00bd a minute (picture 8). Pour the mixture into the pan that you used previously for frying ingredients.\nAdd the rest of the oil. Fry the chili paste over medium low heat (picture 9). Take a small piece and taste. You may want to add additional salt, sugar or tamarind paste or perhaps not. Over low heat, you can fry the chili paste as long as 15 minutes or as short as a couple of minutes. This final frying is to adjust the seasonings, let the flavors come together and kill off anything that can grow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"See the Lightweight VR8 Riding Notes and Lightweight VR8 image gallery.\nAs the latest result of our tireless R&D work in the field of full carbon wheels, we present our 8-spoke front wheel.\nThe VR8 represents a completely new state of the art. With its only eight spokes, it offers significantly improved aerodynamics.\nThe use of new carbon materials and an optimized manufacturing process, has enabled us to create a perfect combination of aerodynamics and lateral rigidity.\nThe stiffness value of about 55 N\/mm sets a new benchmark! Together with the new black Disc, the new wheel set provides the non-plus-ultra in the fight for seconds. Triathletes and time trial specialists will be pleased with what is probably the world's lightest and stiffest time trial wheel set.\nCarbon fibre Time Trial Front wheel in 700C size\/28\".\nFront wheel\/VR8 with Lightweight hub.\nLaminated magnet (for bike computer).\nIndividually numbered read-only microchip laminated (important for the registration and ServiceUp) in every wheel.\nEach set with our new Disc now includes double wheel bag, aluminium valve extensions and quick releases .\nIn short, the VR8 is a far heavier wheel. Not so great for intense climbing, but really nice for time trials and fast descents and windy conditions.\nNominal specifications below, per Lightweight.\nThe 16-spoke versions of the Obermayer and standard are slightly lighter, but also reduce their load capacity significantly. Riders under 150 pounds might consider the 16-spoke version, but at 160 pounds on up, sticking with the 20-spoke version is the smart move; the 4 spokes make very little weight difference, and aerodynamics are marginally different also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SWBAT determine a theme or central idea of a piece of text and provide an objective summary of how Attucks is an American hero during the American Revolutionary time period.\nI remember being back in history class and wishing that time could just fast forward to lunch where I could get that one time in the school day to talk with my friends. Teaching from an interdisciplinary unit with Social Studies concepts can have many students wishing that time could fast forward. I want students to enjoy the connection that Language Arts and Social Studies has in bridging students' understandings and perspectives on topics. This can be a daunting task but achievable if the right types of documents and tasks are being assigned to students.\nTo hook students into a social conversation about this lesson, I ask them to get in groups and create a superhero that could have lived during the Revolutionary time period. Some teachers may want to gather a working definition of a super hero prior to students completing this activity. Since I don't believe in boxing in creativity, I allow students to develop their own definitions of what a superhero looks and acts like. I will give students the option to either draw or write out the traits of their imaginary person. Students will share their heroes with the class.\nstrong, brave, passionate about saving people\/world, rebellious, unique, etc.\nCrispus Attucks was an unknown seamen who becomes a man to remember. Many historians know little about this man which influences the accounts of his life stemming more from mere speculation than facts. Only a few students in my classes have heard of Attucks. One theory could be that it was rare to have an African American man recognized for his actions during a time period where slavery was prevalent. No matter the case, Attucks is known as the first figure to die for something he believed in.\nIn today's generation, students may not describe the actions of Attucks as being heroic. Although Attucks is credited as a leader in history, a debate rages for over a century that argues whether he was indeed a hero, patriot, or villain. No matter what the outcome, students have discussed the connotations or emotions that words evoke in our English language. In this lesson, \"hero\" is a word that has various connotations in meaning.\nStudents will watch this short clip about Attucks and jot down a few characteristics that could be inferred about this GREAT man.\nPossible characteristics from the video revealed the following: brave, passionate, leader, strong, and a villain. Students' answers can vary depending on the information cited from the video to influence their thinking. While I did not do this step with each class, teachers can easily take this list and have students compare it to the super hero created in the warm-up activity.\nStudents will read the sample piece of folklore independently. Afterwards they will work on processing the information read in the selection by choosing a written or pictorial way to describe how Crispus is known as a hero. This expression by students will be placed in their notebooks (student work on Attucks).\nThe vocabulary of the article includes the words: confusion, furious, glance, harsh, lobster, respect, and serious. You will notice that these are common everyday words that many students use in their natural talk. With reading, it is important to include means of comprehending vocabulary used in the selection. After students process the information, they will work in pairs to complete two three way ties on the vocabulary words of the selection. This comprehension strategy requires students to look at how three words connect through content or meaning. Students can share their finished products if time permits. It is nice to share these products whole group so similarities and differences can be shared on similar ties created in various groups of students.\nDoes your superhero compare to Crispus Attucks' actions during the Revolutionary Time Period? If you could have replace Attucks with your hero, would the events of the story change or stay the same?\nThe studying of Attucks allowed students to see how a hero of the American Revolution can also be categorized as a villain of this time period. Similar heroic characters from the warm-up activity included: strength, bravery, and passion. The difference included: having Attucks (or students' hero) and a group of boys initiating a ruckus with revolutionary soldiers to become legendary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you get a divorce, unless you have a prenuptial or postnuptial agreement that addresses property distribution, you and your spouse will need to divide your assets in a manner that is consistent with Indiana law. Indiana's property distribution law is unique in that it establishes a presumption that an equal split of all assets is the most \"just and reasonable\" solution, regardless of whether these assets were acquired prior to or during the marriage. This presumption can be overcome under limited circumstances; but, in any case, divorcing spouses will need to be prepared to both (i) accurately calculate their fair share of the \"marital pot,\" and (ii) negotiate to protect the assets that matter most after their marriage is over.\nThe family home is usually among couples' most high-value assets, and it will often carry significant emotional value as well. While one option is to sell the home and split the proceeds, the more-common solution is for one spouse to keep the home in exchange for giving up his or her interest in other assets in the marital pot. Home equity, mortgage debt, potential tax liability, custody rights and various other practical considerations are all likely to come into play, and both spouses should carefully weigh all of the relevant factors before agreeing to give up or take sole ownership of their marital residence.\nRegardless of whether a bank or investment account is in one or both spouses' names, it will be subject to division during their divorce. Typically, one spouse will retain the couple's existing savings and checking accounts (formally removing the other spouse from the account and updating any online login information), and the other spouse will deposit his or her share of any cash assets into newly-formed accounts. While it is also possible to split an investment portfolio, brokerage accounts can present some unique issues; and, if there are sufficient assets available, it may make sense for one spouse to keep the entire portfolio while giving up his or her rights to other property.\nDividing retirement accounts and pensions involves a number of special considerations as well; and, once again, there are a few different options available. A qualified domestic relations order (QDRO) can be used to divide one spouse's retirement assets without triggering taxes and penalties, but spouses may also choose to keep their respective retirement plans or exchange one spouse's interest in a retirement plan for other marital property. For more information, you can read: What Happens to Retirement Accounts and Pensions in an Indiana Divorce?\nIf a couple has two or more vehicles of relatively equal value, it may be simple enough for the spouses to agree that each will keep one vehicle in their divorce. However, if a vehicle is part of a collection or has sentimental value, this may come into play in deciding how to divide the couple's overall marital estate. In any case, it is important to note that title ownership is irrelevant in a divorce \u2013 even if a vehicle is solely in your (or your spouse's) name, it is still subject to the equitable distribution process.\nFor legal purposes, the individual items in a collection are generally treated the same as any other pieces of personal property. Practically speaking, however, one spouse is likely to have a personal attachment to his or her collection, and it may be that the value of a collection is greater than the sum of its parts. These are relevant considerations that should not be overlooked; and, if divorcing spouses agree that one will keep an entire collection, the collection must be valued appropriately.\nPets are property under Indiana law, and they are considered part of the marital pot just like real estate, financial accounts and physical assets. When deciding who will keep a family pet, if desired, divorcing spouses can make provisions for financial support and \"visitation\" rights as well.\nBusinesses formed prior to and during the marriage are subject to division in a divorce, and this applies regardless of whether one or both spouses are named as owners or involved in the day-to-day operations of the company. As you might expect, there are several complex considerations involved in addressing ownership and control of a privately-held business in a divorce. For a brief introduction, you can read: Divorce Considerations for Indiana Business Owners.\nHealth insurance is a subject of critical importance for many individuals; and, with a potential repeal of the Affordable Care Act (or Obamacare) being the subject of ongoing political debate, it is especially important for divorcing spouses to ensure that they will have access to continuing coverage after their COBRA eligibility expires (if applicable). There are a number of potential solutions available, and divorcing spouses who do not have coverage through an employer will need to carefully weigh their options during the divorce process.\nAlthough education is not an asset that can be divided in a divorce, it is still relevant to our discussion. If one spouse gave up education or employment opportunities in order to undertake childcare or household duties during the marriage, this is an issue that can be addressed through spousal maintenance (or \"alimony\").\nIn many states, gifts and inheritances are considered \"separate\" property that are not subject to equitable distribution. But, this is not the case in Indiana. Gifts and inheritances received prior to and during the marriage are subject to division as part of the marital pot under Indiana law. Once again, a prenuptial or postnuptial agreement can override this default rule, and divorcing spouses may agree that each will keep their respective gifts and inheritances or negotiate an alternative amicable resolution.\nIf you are contemplating a divorce and would like more information about Indiana's equitable property distribution rules, we encourage you to contact us for a complimentary initial consultation. To speak with Carmel divorce attorney Joshua R. Hains in confidence, please call (317) 688-1305 or request an appointment online today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnfws b/data_all_eng_slimpj/shuffled/split2/finalzzzbnfws
new file mode 100644
index 0000000000000000000000000000000000000000..e0c37f282a456b2e04a7f773fb5e34d4589fd9b2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnfws
@@ -0,0 +1,5 @@
+{"text":"Sara Bareilles and Josh Groban are set to host at Radio City Music Hall on June 10.\nThe 72nd annual Tony Awards are creeping up fast, and everyone who's favorite album of all time is the Hamilton soundtrack is scrambling to figure out how to watch it without cable.\nFree trials, guys and dolls. Free trials.\nHosted by Sara Bareilles and Josh Groban, the 2018 Tony Awards will air on June 10 at 8pm EST on CBS. You just need to find a streaming service where CBS is one of the channels offered and snag a free trial. Cord cutting is our middle name at this point.\nCBS is the channel the awards are airing on, so CBS All Access is an obvious choice. If you sign up for a CBS All-Access free one-week trial anytime between June 4 and June 10 (so the show falls in the seven day grace period), you'll be able to watch the live stream. Then, just cancel your account before the trial period is up, so you don't get charged $5.99 or $9.99 (depending on whether you chose the ads or no ads option). CBS All Access is a pretty good gig, though \u2014 you'll be able to stream 10,000+ episodes and watch live TV 24\/7, so you'll never have to deal with spoilers again.\nAnother note: Amazon Prime members automatically get access to CBS All-Access. You'll also get free two-day shipping, access to Amazon Music Unlimited and Amazon Video, as well as CBS All Access. It's a two-birds-with-one-stone kinda thing.\nDirecTV NOW is DirecTV's no-dish-required streaming service, which basically means you get everything that DirecTV offers without the hassle of the satellite. The cheapest subscription to DirecTV NOW offers a ton of channels, including CBS (aka the Tony Awards channel), ABC, BBC, Oxygen, Adult Swim, ESPN, and more. DirecTV also offers more expensive packages with up to 120 channels, including premium channels like HBO, STARZ, and SHOWTIME for an extra fee.\nYou can get a free one-week trial of the 60+ live channel package, which will cost $35\/month after the week is up. Plus, when you prepay $35 for your first month, DirecTV will throw in a free Roku streaming stick on them.\nIf you're trying to watch the Tony Awards and also follow sports (hey, those people do exist), then fuboTV is a streaming bundle gaining major traction from sports addicts in recent months. Get 85 channels with live sports, shows, and movies (did we mention sports?), including CBSN for the Tony Awards. Sign up for a one-week free trial (again, between June 4 and June 9) to try out the selection and watch the awards, and if you like it, just pay $19.99 for your first month (and $44.99\/month after that).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Palermo Arched wall mirror is sure to stand out owing to the unique shape of its frame. The polyurethane frame of the mirror comes with an antique silver finish that provides a glossy and rich look to the mirror. The silver finish of the frame blends in well with the glossy nature of the mirror thereby exuding a special radiance throughout the room it is placed in. The two fixed hooks attached to the back of the mirror makes it easy to hang.\nThe Palermo arched wall mirror is a multi-functional piece and can be used both as a d\u00e9cor element and a functional piece. It can be hung on an accent wall in the living room to create a focal point or in the foyer area to create a stunning look. It can also be used as a bathroom mirror to add a touch of glamour to your bath area. Let your interior look glamorous and stylish with the Palermo Arched Wall mirror.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Save money on City Chic Brasilia Halter One-Piece Swimsuit (Plus Size) We've found the very best online deals. Research well before buying on-line City Chic Brasilia Halter One-Piece Swimsuit (Plus Size) Make sure the shop keep your personal info non-public before you get City Chic Brasilia Halter One-Piece Swimsuit (Plus Size). Make sure you can proceed credit card online to buy City Chic Brasilia Halter One-Piece Swimsuit (Plus Size) and the store protects your data from fraudulents. You have to make sure you will get the best price by comparing City Chic Brasilia Halter One-Piece Swimsuit (Plus Size).\nBombshell curves rule in this tropically patterned suit designed with a smoothing powermesh back lining and crossover neckline.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'iOS 9' started by wlfente, Sep 16, 2015.\nI have iOS 9.1 public beta on my iPhone 6 Plus. If I do a backup of my device while it's still on iOS 9.1 public beta, then downgrade to iOS 8.4 (setup as a new device), and finally do the OTA upgrade to iOS 9.0 public release today at 10am PST...will I be able to restore my iPhones content & settings from my iOS 9.1 public beta backup to iOS 9.0 public release?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It can be weeks, months, or years before someone is diagnosed with post-traumatic stress disorder (PTSD). In my case it was 33 years before I even knew what I was experiencing was PTSD.\nRe-experiencing the trauma through intrusive distressing memories of the event, flashbacks, and nightmares.\nEmotional numbness and avoidance of places, people, and activities that are reminders of any trauma.\nDifficulty sleeping and concentrating, being jumpy, and feeling easily irritated and angered.\nFor a long time, I coped with all of the above. I was extremely young when I experienced trauma, which consisted of physical, mental and emotional abuse from a family member.\nI wasn't officially diagnosed because I hid what was going on from everyone. I was ashamed and felt guilty growing up, that if I did come out and expose what was happening it would affect my family's careers and people would think less of us.\nI also covered up what I was coping with for fear of being criticized, making an abusive situation worse, or people thinking I had \"lost my mind.\" I eventually had selective amnesia.\nHiding was my coping mechanism all the way into adulthood, and I became good at it. What I didn't realize though is by hiding, I'd built stress up over a long period. So when further traumas occurred I would attempt to cope with them without considering the depth of my problems making present traumas feel that much worse.\nWhen I was sexually assaulted at the age of 18, I became emotionally numb, feeling like I had no way to process my emotions.\nI now realize I was hindering my recovery by hiding because the key to me dealing with PTSD was confronting the emotions and fears it had been generating and facing them head on.\nIf I had known the key to me feeling free was to actually break the silence and face my emotions, I'm sure I wouldn't have just coped with the condition for so long.\nI hope this inspires someone to know they no longer need to hide.\nI am a wife and mother of 4 living in London UK, who experienced 33 years of abuse, PTSD, emotional trauma, OCD, and BDD. I'm a experienced trauma recovery coach, having been both sides of the fence, I really understand what it means to live with mental health conditions AND overcome them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboxmu b/data_all_eng_slimpj/shuffled/split2/finalzzzboxmu
new file mode 100644
index 0000000000000000000000000000000000000000..6db3e4047576250c6b36439794fd200abb78e530
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzboxmu
@@ -0,0 +1,5 @@
+{"text":"If there is one thing that my husband says to me more than anything else it's \"let them be kids.\" I try to be that fun mom\u2013 the mom with no cares, no worries, and just letting them play and explore on their own (with safety limits, of course). I also try not to be the \"helicopter\" mom. I try to be a good mix. Protective but not overly worried. I want to be able to say \"yes!\" more often, but oh, how these kids can get so dirty. Brandon is always encouraging them to just play and get dirty and I am over on the sidelines calculating how long it is going to take me to get the stains out of their clothes.\nI cringe when the kids want to get the art stuff out most days because I just know things will get destroyed. Mud puddles? Kryptonite to my kids. Sliding in the grass? Instant stain.\nIt doesn't sound like I am a lot of fun.\nI promise, I am working on this.\nThis summer, I plan to say YES more often. Yes to s'mores outside, yes to mud puddles, yes to sidewalk chalk, yes to grass stains, and yes to exploring in the yard. Yes to pushing them on the swings, yes to playing in the sprinkler, yes to popsicles for an afternoon snack, and yes to catching lightning bugs when it gets dark outside.\nIt's going to be a fun summer here, and thanks to all free clear, I can wash their dirty clothes and swim towels all summer long without it irritating their skin. all free clear is tough on stains, but not on kids. With their new products with OXI, I know I am getting the best clean for their clothes. As a bonus, the OXI booster isn't JUST for laundry! Use it for any stain around your home\u2013 carpet, patio furniture, rugs, etc.\nall free clear is the #1 recommended detergent brand by Dermatologists, Allergists, and Pediatricians for sensitive skin. It's tough on stains yet gentle enough for the whole family. Plus, it's safe for use in standard and HE machines. It rinses clean and has a gentle, hypoallergenic formula. Powerful Clean. Gentle on Skin. all free clear is also 100% hypoallergenic.\nAre you ready to say yes to summer and all the fun and mess it comes with? Check out this coupon for $1 off all free clear products!\nI love all free and clear. I may need to try that laundry booster. Thanks!\nI'm with you - I partly cringe with the dirt, but also want to just let my son be a kid and enjoy. That included muddy knock-off slip and sliding at an event last weekend where we literally had to wipe off dirt & leaves from him before changing him into dry clothes to bring him home! LOL Scrubbed him after, but oh the fun he had. And our All Free & Clear got his swim suit & shirt clean and fresh again. You'd never know (besides the photographic evidence) that he had been so filthy.\nEven better! I cringe at dirt, play doh-- you name it! I need to say YES more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even the best arguments can fail with dull delivery. Using performance techniques shared by actors and successful litigators, David directs attorneys to winning courtroom delivery. In an era when attention spans are shorter and patience is limited, a lawyer still must make a lasting impact in a matter of minutes.\nCourtroom delivery training focuses on techniques that separate an average litigator from a dynamic, persuasive one: command of space, use of voice, and management of time. David will show lawyers exactly how to use vocal inflection and body language to tell the story between the words.\nWith David's personalized coaching, attorneys will learn how to keep their game fresh by projecting an authentic presence in the courtroom, developing vocal delivery to retain juror engagement, and remaining confident under pressure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quantum Global Technologies, LLC Agrees to be Acquired by Ultra Clean Holdings, Inc.\nQuakertown, PA USA. August 10, 2018 \u2013 QuantumClean and ChemTrace will demonstrate how its ultra-high purity cleaning, proprietary coatings and microcontamination analytical testing can help reduce wafer fabrication Cost-of-Ownership (CoO). Tech Talks are scheduled during show hours at SEMICON Taiwan at the Nangang Exhibition Center in Taipei, Taiwan from September 5 \u2014 7, 2018 (booth J2734).\nQuakertown, PA USA. July 5, 2018 \u2013 QuantumClean and ChemTrace will demonstrate how its ultra-high purity cleaning, proprietary coatings and microcontamination analytical testing can help reduce Cost-of-Ownership (CoO). Tech Talks are scheduled during show hours at SEMICON West 2018 at the Moscone Center South in San Francisco from July 10 - 12 (booth 1232).\nQuakertown, PA USA. June 1, 2018 \u2013 QuantumClean and ChemTrace will exhibit at SEMICON West 2018 at the Moscone Center South in San Francisco from July 10 - 12 (booth 1232).\nQuakertown, PA USA. May 3, 2018 \u2013 ChemTrace\u00ae, a Quantum Global Technologies' company, celebrates its 25th anniversary of analytical excellence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Minden Pictures is worldwide reknown as supplier of premium animal and nature photographs. The Minden Collection contains important aspects of natural history, ecology, biodiversity and endangered species from all continents; many of them from remote and isolated areas of the world. Minden Pictures is offering the best pictures of an award-winning group of nature photographers. Many of them are working regularly with National Geographic. Arco Images is representing the Minden Collection for Germany, Austria and Switzerland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8:30 A.M. IN THE BENTON MUNICIPAL BUILDING LOCATED AT 500 WEST MAIN STREET, BENTON, ILLINOIS.\nMAYOR KONDRITZ CALLED THE MEETING TO ORDER. THOSE PRESENT WERE COMMISSIONER BAUMGARTE, COMMISSIONER WYANT, COMMISSIONER STOREY, COMMISSIONER MILLER, AND MAYOR KONDRITZ.\nCOMMISSIONER MILLER MADE A MOTION TO APPROVE BILLS IN THE AMOUNT OF $207,910.49. SECOND BY COMMISSIONER STOREY. VOTE WAS TAKEN. YEAS: 5. COMMISSIONER BAUMGARTE, COMMISSIONER WYANT, COMMISSIONER STOREY, COMMISSIONER MILLER, AND MAYOR KONDRITZ. NAYS: 0.\nCOMMISSIONER WYANT MADE A MOTION TO ADJOURN. SECOND BY COMMISSIONER MILLER. VOTE WAS TAKEN. YEAS: 5. COMMISSIONER BAUMGARTE, COMMISSIONER WYANT, COMMISSIONER STOREY, COMMISSIONER MILLER, AND MAYOR KONDRITZ. NAYS: 0.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Your myths about eggs going to get over after reading this.\nEggs are mostly preferred by people in their breakfast, when they are intended to have light breakfast. But have you ever thought that people believe in many unstable facts about eggs too? And these facts are not true even. So let us find out the truth and see the myths about eggs.\nYears ago doctors told people with high cholesterol not to eat cholesterol-rich food, so eggs were kept on the 'not-to-eat' list. But since then, science has evolved and a recent study showed that high-cholesterol food like eggs don't have to do anything with increasing blood levels of cholesterol.\nIt is the most common myth among people, and this is the reason that they ignore eating whole egg, instead prefer only egg white. White part contains only 17 calories while whole egg has 72 calories, it contains zero fat. According to the experts, fat in yolk is actually a good fat for a healthy breakfast.\nBrown rice are healthier than white, wheat bread are healthier than white bread, but don't mistaken when it is about eggs. Color can't tell you whether brown is healthier or brown. Experts never recommend eating brown eggs.\nWe understand that something in the morning we get late, especially on Monday mornings. But if you think that eggs are not for busy morning then you are wrong. You still can heat pan and break two eggs drop them on the pan and seasoned them. Your tasty breakfast is ready!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u5f7c\u306f\u8d05\u6ca2\u306b\u80b2\u3066\u3089\u308c\u305f\u3002\/ He was raised in luxury.\n\u30fbJim leads a luxurious life.\n\u30fbTaking a taxi is a luxury for me.\n\u30fbThe market of luxury goods is growing.\n\u30fbThis is a luxurious hotel.\n\u30fbMy girlfriend wants to go to a luxurious restaurant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What started as a personal dissatisfaction with an old boys' club mentality in venture capital led Pocket Sun to co-found what she calls the first female-led millennial venture capital firm.\nSoGal Ventures has gone on to make 52 investments in the past two and a half years, Sun said.\nA personal dissatisfaction with an old boys' club mentality in venture capital led Pocket Sun to co-found what she calls the first female-led millennial venture capital firm.\n\"The venture capital industry is extremely male-dominated and I saw a lot of female entrepreneurs who were undervalued and undercapitalized,\" Sun, co-founder of SoGal Ventures told CNBC. Her firm was started by capitalizing on gender inequality, she added, to change the status quo in the industry.\nSoGal has gone on to make 52 investments in the past two and a half years, but the executive said she did not always expect to head down that path.\n\"I would say I did not plan on going on this path at all,\" Sun said. \"When I started SoGal, it was really to solve my own personal pain point. I wanted to see more women in venture capital events and entrepreneurship events.\"\nThe dream grew bigger, however, and turned into a desire to create an empire for future generations of women to play a central role in the industry, Sun told CNBC's Dan Murphy at the Credit Suisse Global Megatrends Conference in Singapore.\n\"So now, three years later, we've had 52 investments under our belt, which is just unbelievable. So we are onto bigger things \u2014 it really started from humble beginnings,\" Sun said.\nThe female-led venture capital firm plans to secure 40 to 50 more deals with its current funding, and the firm is currently evaluating a wide range of sectors including aging care, gaming and esports, blockchain and health care, she said.\nSun's advice to aspiring entrepreneurs: \"Just believe in your power and start by making little steps, because you never know what will come out at the end of the tunnel.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The word \"denim\" can find its origin in N\u00eemes as the heavy duty blue cloth was originally made here, taken to the USA to be used to make jeans by Levi Strauss - the cloth being \"de N\u00eemes\"!\nN\u00eemes has a very long history, being developed by the Romans over 2000 years ago. The Pont du Gard (see page on drop down list) was built as an aquaduct to bring water from the spring of Nemausus to the city.\nThere is some beautiful Roman architecture, the most famous being Les Ar\u00e8nes (like the Coliseum of Rome but better preserved) and the Maison Carr\u00e9.\nThere is a wealth of culture, museums and restaurants to visit. Les Jardins de la Fontaine are particularly attractive - built by King Louis XIV for the sole use of himself and his court in the 17th century.\nIt is easy to visit N\u00eemes the same day as the Camargue, Pont du Gard or Uz\u00e8s.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A collection of the best hotels for Wellness Travellers in Bohemia.\nGrandrestaurant Pupp is the hotel's main restaurant featuring French-style cuisine and service. The menu varies per season, offering a winter and a summer menu with dishes based on seasonal ingredients. Of course, each meal of yours can be accompanied by a handpicked wine by the hotel's sommelier!\nBecher's Bar features a rich selection of cocktails, beer, and an extensive wine list. Live music or a DJ comes to liven up the evenings, but you should also pay a visit to the casino. Gourmet specialties are also served in the bar to accompany your drinks!\nClub Mala Dvorana is the hotel's marvelous lobby bar featuring a 19th-century atmosphere with a lavish interior and a central crystal chandelier. In addition to a selection of petite dishes and desserts, you will also find various drinks, choice teas and illy coffee, cognac, whiskey, and cocktails.\nThe best place to enjoy your coffee is at Cafe Pupp, serving first-class illy coffee and special drinks carefully prepared by the baristas. Cafe Pupp Terrace opens every summer featuring open views of the hotel and spa promenade.\nGolf is a modern restaurant with an indoors and outdoors eating area offering marvelous views of the wide fairway and green of the 18th hole. Its menu offers gourmet specialties including steaks and desserts. It's a great place for social events like wedding receptions.\nThe hotel's wellness center spreads through 1300m and offers spa facilities including a relaxation pool, sauna and steam baths, a salt cave, an ice fountain a fitness center and a royal spa suite.\nThe Royal Spa Suite is suitable for either couples or single travelers, offering a hot tub, a herbal sauna, and an experience shower. Both private and shared massages are available, while there is also an Aquai water massage bed as well!\nThe microclimatic salt cave of the hotel has a very nice atmosphere with a waterfall and a starry sky above, which are fully lined with salt crystals from Pakistan. The beneficial effects include healing from respiratory diseases, inadequacies of the thyroid gland, and more!\nThe Premier Suite is the perfect room for gay couples or small gay families featuring, among others, a bedroom and a separate living room area, great views to the spa promenade and park, and a luxurious marble bathroom with bath!\nThe Presidential Apartment is the best type of room in the hotel spreading through an area of 130 m2 and offering stunning views over Karlovy Vary, a spacious bedroom, a living room area, a dining room, and an exclusive marble bathroom with bath.\nBecher's Bar has become one of the most popular places for gay men in the city, attracting the majority of the gay locals and visitors. It has a rich selection of cocktails, beer, wine, as well as some gourmet specialties. Live music or DJs liven up the evenings!\nDon't forget to treat your boyfriend with a relaxing Aroma Massage. This massage combines the healing effects of massage and aromatic oils whose fragrant essences influence the central nervous system. The wonderful smell of the aromatic oils will make him smell like spring!\nSalon Cosmetique offers a wide diversity of body treatments under the wellness concept Pure Fiji which specializes in the regeneration of the whole body. The treatments include a wide array of therapies, like peelings with sugar cane, body wraps, face massages and masks, and herbal sacks massages!\nSince Thai Massage is one of the most popular and beneficial for the body, they are performed only by certified and experienced therapists. The clients are dressed in special cotton clothes (provided by the hotel), and it's performed on a special massage bed made for this purpose. Even though you might hurt during the massage, you will feel completely relaxed in the end!\nFranck Provost Paris is the hotel's luxurious hair studio that works from 9 a.m. to 5 p.m., Monday to Saturday. You can arrange a meeting in Sunday as well, however, only upon request.\nGrandrestaurant Pupp offers an a-la-carte menu, that includes gourmet dishes based on the seasonality, ensuring the usage of fresh and delicious ingredients. Of course, the Neoclassical Interior of the restaurant with the stucco walls and the large windows overlooking Spa Promenade add greatly to the dining experience!","meta":{"redpajama_set_name":"RedPajamaC4"}}